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1997-01-04T02:06:08 | 9701 | alg-geom/9701002 | en | https://arxiv.org/abs/alg-geom/9701002 | [
"alg-geom",
"math.AG"
] | alg-geom/9701002 | null | M. Cook | A smooth surface in P4 not of general type has degree at most 46 | 16 pages, Latex | null | null | null | null | This is the continuation of papers by Braun and Floystad, Cook, Braun and
Cook. We use Generic Initial Ideal Theory in conjunction with Liaison Theory to
further restrict the possible generic initial ideals of hyperplane sections of
smooth surfaces not of general type in P4.
| [
{
"version": "v1",
"created": "Sat, 4 Jan 1997 01:04:56 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Cook",
"M.",
""
]
] | alg-geom | \section{Definitions and Introduction}
First let us recall some of the language of generic ideal theory.
(For a full treatment refer to [B] and [G].)
As we will mainly concern ourselves with a generic hyperplane
section of a surface in $\pfour$ we will restrict our definitions to
curves in $\pthree$.
\vspace{2mm}
Let ${\rm \bf C}[x_0, x_1, x_2, x_3]$ be the ring of polynomials
of $\pthree$ with the reverse lexicographical ordering $\succ$.
Let $C$ be a curve of degree $d$ in $\pthree$, with ideal $I_C$.
After a generic change of basis, the monomial ideal of initial
terms of elements of $I_C$ under $\succ$ is called the {\it generic
initial ideal} of $C$ and denoted ${\rm gin}(I_C)$.
\vspace{2mm}
Let $\Gamma$ be a generic hyperplane section of $C$. Then
the generic initial ideal of $\Gamma$,
${\rm gin}(I_{\Gamma})$, is an ideal in ${\rm \bf C}[x_0, x_1, x_2]$
and can be defined by
$$ {\rm gin}(I_{\Gamma}) = ({\rm gin}(I_C)|_{x_3 =0})^{{\rm sat}}$$
where the saturation is with respect to $x_2$.
As $I_C$ and $I_{\Gamma}$ are saturated ideals,
${\rm gin}(I_C)$ is generated
by monomials of the form $x_0^{i}x_1^{j}x_2^{k}$
and the generic initial ideal of $\Gamma$ is of the form
$$ {\rm gin}(I_{\Gamma}) = (x_0^s,\ x_0^{s-1}x_1^{\lambda_{s-1}},
\ \dots , \ x_1^{\lambda_0}), $$
where $\sum \lambda_i = d$ and $\lambda_i \geq \lambda_{i+1}+1$.
Furthermore, as the points of $\Gamma$ are in uniform
position, the work of Gruson and Peskine [GP] tells us that
$\lambda_{i+1}+2 \geq \lambda_i $.
The $\{ \lm_0, \ \lm_1, \ \dots , \ \lm_{s-1} \}$
are called the {\it connected invariants} of $\Gamma$ or $C$.
\vspace{3mm}
{\bf Definition}.
A monomial $x_0^ax_1^bx_2^c$ is a {\it sporadic zero} of $C$
if $x_0^ax_1^bx_2^c \notin {\rm gin}(I_C)$,
but $x_0^ax_1^b \in {\rm gin}(I_{\Gamma}$).
(i.e. there exists $c' > c$ such that
$x_0^ax_1^bx_2^{c'} \in {\rm gin}(I_C)$. )
\vspace{2mm}
Notice that every generator $x_0^ax_1^bx_2^{c'}$ of ${\rm gin}(I_C)$
with $c' > 0$ gives rise to sporadic zeros $x_0^ax_1^bx_2^c$ for all
$0 \leq c < c'$.
\vspace{2mm}
In [BF], it was shown that if $S$ is a smooth surface of degree $d$
in $\pfour$ not of general type, whose generic hyperplane section,
$C$, has $\alpha_t$ sporadic zeros in degree $t$ and invariants
$\{ \lm_0, \ \lm_1, \ \dots , \ \lm_{s-1} \}$, then the following
equation must be satisfied
\begin{eqnarray}
0 \geq & d^2-&5d-18-10\sum_{i=0}^{s-1}(\binom{\lm_i}{2}+
(i-1)\lm_i) + \nonumber \\
& & 12 \sum_{i=0}^{s-1} (\binom{\lm_i+i-1}{3} -
\binom{i-1}{3}) - \sum_{t } \alpha_t (12t-22).
\end{eqnarray}
\vspace{2mm}
For each degree $d$, there are only a few possibilities for
$\{ \lm_0, \ \lm_1, \ \dots , \ \lm_{s-1} \}$ and for each
$d$ and $\{ \lm_0, \ \lm_1, \ \dots , \ \lm_{s-1} \}$ there is
an upper bound on the number of sporadic zeros by the work of Ellingsrud
and Peskine [EP].
Thus, in order to bound $d$, we need to find the smallest
possible upper bound for $A = \sum_{t=0}^{m} \alpha_t t$, for each
$d$ and $\{ \lm_0, \ \lm_1, \ \dots , \ \lm_{s-1} \}$.
\vspace{2mm}
We will bound $A$ by bounding the
degree of the generators of ${\rm gin}(I_C)$. We will show that
if there are $a$ generators in a high degree $\geq r > \frac{d}{2}$,
and all the others are in degree $\leq r-2$, then $a \geq 6$.
Thus we can find an upper bound for $A$ by assuming that either all
generators of ${\rm gin}(I_C)$ are in degree $\leq \frac{d}{2}$
or if there exists a generator in degree
$r > \frac{d}{2}$ then either there is a generator in degree $r-1$
or there are six generators in degree $\geq r$. Having done this we
find that $d \leq 48$. We will then use the connectedness of the
invariants of $C$ (see [C2]) to lower the bound to $46$ by considering
the last few examples on a case by case basis. We will also give an example
of a connected Borel-fixed monomial ideal in degree $46$ where
all the conditions of this paper are satisfied. (However, whether or not
this example corresponds to a actual curve is another matter.)
\section{Restricting the generators in high degree}
Let $S$ be a smooth surface not of general type in $\pfour$ with
generic hyperplane section $C$. Let
$\{ \lm_0, \ \lm_1, \ \dots \ , \ \lm_{s-1} \}$ be the invariants
of $C$. Assume that $d > (s-1)^2 +1$. In [BC] we found that
$s \leq 7$ and $d \leq 66$. Furthermore, if $s \leq 3, d \leq 8$
and if $s=6$ or $7$, $d \leq 44$.
Thus we may restrict ourselves to the case $s = 4$ or $5$.
Let $\gamma $ be the number
of sporadic zeros of $C$. [EP] show that
\begin{eqnarray}
{\rm for} \ s=4 & \gamma \leq
1 + \sum_{i=0}^{s-1}(\binom{\lm_i}{2}+(i-1)\lm_i)
-\frac{d^2}{8} + \frac{9d}{8}, \\
{\rm for} \ s=5 & \gamma \leq 1 +
\sum_{i=0}^{s-1}(\binom{\lm_i}{2}+(i-1)\lm_i)
-\frac{d^2-5d+10}{10}.
\end{eqnarray}
\vspace{2mm}
For equation (1) to hold for large degree, $A$ will need to
be large. Every generator of ${\rm gin}(I_C)$ of the form
$x_0^ax_1^bx_2^c$ with $c > 0$ gives a sporadic zero in each
degree $n$ for $a + b \leq n \leq a+ b+ c -1$.
So, one would like generators of ${\rm gin}(I_C)$
in as high a degree as possible.
We saw in [BC], that if there were only one generator in degree
$r > \frac{d}{2}$ and all others are in degree $\leq r-2$, this
would imply that there were an secant line of $C$ of order $r$
which leads to a contradiction if $d > 50$. In Appendix A we give a
slight improvement on this Lemma, showing that one can obtain a
contradiction for $d > 42$. (Although 42 is not optimal, in that we
could lower this bound on $d$ if we were more careful, it is quite
sufficient for our needs.) Thus, if there is one generator in degree
$r > \frac{d}{2}$, there must be another in degree $\geq r-1$.
We will continue with this argument and show that if there
are $a$ generators in degree $\geq r > \frac{d}{2}$ and
all the others are in degree $\leq r-2$, then $a \geq 6$.
\vspace{2mm}
Consider the following situation. Let $C$ be a
non-degenerate curve in $\pthree$ of degree $d$.
Suppose ${\rm gin}(I_C)$ has $a$ generators
in degree $\geq r > \frac{d}{2}$, where $1 \leq a \leq 5$
and the rest in degree $\leq r-2$. (Assume from this point,
that a generic
change of coordinates has been made so that
${\rm gin}(I_C) = {\rm in}(I_C)$, the initial ideal of $C$.)
Let $J$ be the ideal generated by elements of $I_C$ in degree
$\leq r-1$. Then the generators of ${\rm gin}(J)$ are the
generators of ${\rm gin}(I_C)$ in degree $\leq r-2$.
By considering the Hilbert function associated to $J$,
one finds that degree$(V(J)) = $ degree$\ (C)+a$.
Hence, $V(J) = C \cup X$ and degree$(X) = a$.
Thus $X$ contains a pure one-dimensional scheme $Y$ of degree $a$.
We will show that $Y$ must either be non-reduced or reducible.
Then given that $a \leq 5$, $Y$ must contain (perhaps with a
multiple structure) a line or a conic. Furthermore this line
(respectively conic) must meet $C$ in $> \frac{d}{2}$
(respectively $> d$) points (up to multiplicity).
We will then proceed as in [BC] to show in Propositon 4 that
$Y$ cannot contain such a line or such a conic.
\vspace{3mm}
{\bf Proposition 2}
If $I_C$ has two generators of degree $m$ and $n$ so
that $m + n -2 \leq \frac{d}{2}$, then $Y$ is either non-reduced or
reducible.
\vspace{2mm}
(The condition of Proposition 2 is easily satisfied, if our curves
arise as generic hyperplane sections of surfaces not of general type.
Each of these curves lies on a surface of degree $s \leq 7$
(and in the cases we are interested in $s = 4, \ {\rm or} \ 5$).
By connectedness $\lambda_{s-1} \leq \frac{d}{s}-\frac{s-1}{2} < \frac{d}{s}$. Furthermore, as there are only a few {\it chains} of sporadic zeros, the curve will also lie on a surface of degree
$\lambda_{s-1} +(s-1) +\epsilon$ where $\epsilon$ is small and
hence $s + \lambda_{s-1} +(s-1) +\epsilon$ will be much less than
$\frac{d}{2}$. )
\vspace{2mm}
{\bf Proof.}
Suppose, for a contradiction, that $Y$ is reduced and irreducible.
\vspace{2mm}
{\bf Step 1}
$H^1({\cal I}_Y(d)) = 0$ for all $d \leq 0$.
\vspace{2mm}
Let $H$ be a generic hyperplane section of $\pthree$, then
$Y \cap H$ is a set of points, $\Gamma' $, and we have the following
short exact sequence
$$ O \rightarrow {\cal I}_Y(d-1) \rightarrow
{\cal I}_Y(d) \rightarrow {\cal I}_{\Gamma'}(d) \rightarrow O. $$
$H^0({\cal I}_{\Gamma'}(d)) = 0 $ for $d \leq 0$, therefore
$$ O \rightarrow H^1({\cal I}_Y(d-1)) \rightarrow
H^1({\cal I}_Y(d)) \rightarrow $$
for $d \leq 0$ and hence
$h^1({\cal I}_Y(d-1)) \leq h^1({\cal I}_Y(d)) $ for $d \leq 0$.
Now, the restriction sequence
$$ O \rightarrow {\cal I}_Y \rightarrow
{\cal O}_{\pthree} \rightarrow {\cal O}_{Y}
\rightarrow O $$
gives the exact sequence of cohomology
$$ O \rightarrow H^0({\cal I}_Y) \rightarrow
H^0({\cal O}_{\pthree}) \rightarrow H^0({\cal O}_{Y})
\rightarrow H^1({\cal I}_Y) \rightarrow O. $$
As $Y$ is reduced and irreducible
$h^0({\cal O}_Y) = 1$. $ H^0({\cal I}_Y) =0 $ and hence
$ H^0({\cal O}_{\pthree}) \cong H^0({\cal O}_{Y})$.
This implies that $H^1({\cal I}_Y) = 0$
and hence $H^1({\cal I}_Y(d)) = 0$ for all $d \leq 0$.
\vspace{2mm}
{\bf Step 2}
All the generators of
${\rm gin}(I_{C \cup X}) = {\rm gin}(J)$ are in degree
$\leq t_0 \leq r-2$. We will show that all the generators of
${\rm gin}(I_{C \cup Y})$ are also in degree $\leq t_0 \leq r-2$.
(Recall that the maximum degree of the minimal generators of
${\rm gin}(I_C)$ is the same as the regularity of $C$. (See [B]))
\vspace{2mm}
Let $H$ be a generic hyperplane. Then
$(C \cup X) \cap H = (C \cup Y) \cap H = \Gamma$
and we have the following commutative diagram
$$ \begin{array}{ccccccc}
& 0 && 0 && 0 & \\
& \downarrow && \downarrow &&
\downarrow & \\
O \rightarrow & {\cal I}_{C \cup X}(t-1) & \rightarrow &
{\cal I}_{C \cup X}(t) & \rightarrow & {\cal I}_{\Gamma}(t) &
\rightarrow O \\
& \downarrow \alpha_1&& \downarrow \alpha_2 &&
\downarrow \alpha_3 & \\
O \rightarrow & {\cal I}_{C \cup Y}(t-1) & \rightarrow &
{\cal I}_{C \cup Y}(t) & \rightarrow & {\cal I}_{\Gamma}(t) &
\rightarrow O,
\end{array}
$$
where $\alpha_3 $ is an isomorphism and $\alpha_1 $ and
$\alpha_2$ are injections. All generators of ${\rm gin}(I_{C \cup X})$
are in degree $\leq t_0$, hence ${\cal I }_{C \cup X}$ is $t_0$-regular
and $H^1({\cal I}_{C \cup X}(t)) = 0$ for $t \geq t_0-1$.
Taking cohomology of the diagram above we get for
$t \geq t_0-1$
$$ \begin{array}{ccccccc}
& 0 & & 0 & \\
& \downarrow & & \downarrow & & &\\
\rightarrow &
H^0({\cal I}_{C \cup X}(t+1)) & \stackrel{\alpha}{\rightarrow} &
H^0({\cal I}_{\Gamma}(t+1)) &
\rightarrow & O & \\
& \downarrow \gamma &&
\downarrow \delta & \\
\rightarrow &
H^0({\cal I}_{C \cup Y}(t+1)) & \stackrel{\beta}{\rightarrow} &
H^0({\cal I}_{\Gamma}(t+1)) &
\rightarrow & H^1({\cal I}_{C \cup Y}(t)) & \rightarrow
\end{array}
$$
$\alpha$ is onto and $\delta$ is an isomorphism, therefore
$\delta \circ \alpha$ is onto.
$\beta \circ \gamma = \delta \circ \alpha$, and
so $\beta$ is onto. Hence the coker($\beta) = 0$ for all
$t \geq t_0-1$. This means there are no sporadic zeros of
$C \cup Y$ in degree $t \geq t_0$ and hence all generators
of ${\rm gin}(I_{C \cup Y})$ are in degree $\leq t_0$.
\vspace{2mm}
{\bf Step 3}
$C \subset c.i.(n, m) $ a complete intersection of type $(m, n)$,
where $m$ and $n \leq r-2.$ Thus $C$ is linked via this complete
intersection to a curve containing the curve $Y$. Let's call
this curve $C' \cup Y$.
$C'$ is also a curve (perhaps non-reduced or reducible) and it
is linked via the same complete
intersection to $C \cup Y$.
Now
$$ O \rightarrow {\cal O}_{C' \cup Y} \rightarrow
{\cal O}_{C'} \oplus {\cal O}_{Y} \rightarrow {\cal O}_{C' \cap Y}
\rightarrow O
$$
and hence for all $t$
$$ H^0({\cal O}_{C' \cup Y}(t)) \hookrightarrow
H^0({\cal O}_{C'}(t)) \oplus H^0({\cal O}_{Y}(t)). $$
Also for an arbitrary scheme $S \subset \pthree$
$$ O \rightarrow {\cal I}_{S} \rightarrow
{\cal O}_{\pthree} \rightarrow {\cal O}_{S}
\rightarrow O
$$
and hence for $t < 0$, $H^0({\cal O}_S(t)) \cong H^1({\cal I}_S(t))$
By Step1, $H^1({\cal I}_Y(t)) = 0$ for $t\leq 0$ and hence
$$ H^0({\cal O}_{C' \cup Y}(t)) \hookrightarrow
H^0({\cal O}_{C'}(t)) \ { \rm for \ all } \ t < 0 $$
or equivalently
\begin{eqnarray}
H^1({\cal I}_{C' \cup Y}(t)) \hookrightarrow
H^1({\cal I}_{C'}(t)) \ { \rm for \ all } \ t < 0.
\end{eqnarray}
Now ${\rm gin}(I_C)$ has a generator in degree $r$ and hence
$C$ has a sporadic zero in degree $r-1$ which means that
$H^1({\cal I}_C(r-2))\neq 0$. $C$ is linked to $C' \cup Y$
via a complete intersection of type $(m,n)$
and so by a fundamental theorem from liaison theory ([PS])
$$H^1({\cal I}_C(t)) \cong H^1({\cal I}_{C' \cup Y}(m + n - 4 - t)){\check{}}. $$
Hence
$H^1({\cal I}_{C' \cup Y}(m + n - 4 - (r-2))) \neq 0$.
On the other hand $C'$ is linked to $C \cup Y$ via a
complete intersection of type $(m,n)$ and all the sporadic zeros of
$C \cup Y$ are in degree $\leq t_0 -1 \leq r-3$. Hence
$H^1({\cal I}_{C \cup Y}(t)) = 0 $ for all $t \geq t_0-1$.
Again by the relationship between the deficiency modules
of linked curves
$$H^1({\cal I}_{C \cup Y}(t)) \cong H^1({\cal I}_{C'}(m + n - 4 - t)){\check{}}$$
and so $H^1({\cal I}_{C'}(m + n - 4 - t))= 0$ for all $t \geq t_0-1$.
$r-2 \geq t_0-1$, therefore $H^1({\cal I}_{C'}(m + n - 4 - (r-2)))= 0$.
Furthermore, by the assumptions of the proposition $m+n-r-2 < 0$.
This contradicts the injection in equation (4).
\qed
Let $I_C = (J, f_1, \ \dots, \ f_b)$ where $ 1 \leq b \leq a \leq 5$
and $\frac{d}{2} < r = $ deg$(f_1) \leq \ \dots \ \leq$ deg$(f_b)$.
As $Y$ is either non-reduced or reducible and the degree of $Y$ is at most 5, $Y$ must contain
either a line, $l$, or a conic, $Q$ perhaps with a multiple structure.
\vspace{3mm}
{\bf Lemma 3}
There exists $i, 1 \leq i \leq b$ such that $l \cap F_i$
(respectively $Q \cap F_i$) in
${\rm deg}(f_i) > \frac{d}{2}$ (respectively $2{\rm deg}(f_i) > d$)
points (up to multiplicity).
(Here $F_i = \{x \in \pthree | f_i(x) = 0 \}$.)
\vspace{2mm}
{\bf Proof}
As the line $l$ or conic $Q$, with perhaps a multiple structure,
is contained in $V(J)$, then the
reduced scheme will certainly be contained in $V(J)$ as the
non-reduced structure is really information about the embedding.
Thus $l$ or $Q$ is contained in $V(J)$.
If $b=1$, $I_C = (J, f_1)$ and $l$ (respectively $Q$)
$\notin F_1$. By Bezout's theorem $l \cap F_1$ (respectively
$Q \cap F_1$) in ${\rm deg}(f_1) > \frac{d}{2}$ \
(respectively $2 {\rm deg}(f_1) > d$) points (up to multiplicity).
If $b >1$, consider the ideals
$I_j = (J, f_1, \dots, \ \widehat{f_j}, \dots , \ f_b )$,
$j = 1, \dots , b$.
As ${\rm in}(f_i)$ is of the form $x_0^{i_0}x_1^{i_1}x_2^{i_2}$
with $i_2 > 0$, deg$V(I_j) > $ deg$(C)$. Hence $V(I_j)$ contains
a pure one dimensional scheme $Y_j$.
$C \subset C \cup Y_j \subset C \cup Y$ and hence $Y_j$
contains some component of $Y$.
If $Y_j$ contains
a line or a conic (with possible multiple structure) we are done
as in the case of $b=1$, replacing $J$ with $I_j$.
The only other possibility is that $Y_j$ only
contains the unique cubic or quartic (possibly) contained in $Y$.
However this is only possible for at most one
choice of $j$, (otherwise the cubic or quartic is contained in $C$.)
Therefore there exists some $j$ such that $Y_j$ contains
a line or a conic.
\qed
\vspace{3mm}
{\bf Proposition 4}
Let $S$ be a smooth surface not of general type of degree $d$
in $\pfour$ with generic hyperplane section $C$.
Suppose ${\rm gin}(I_C)$ has at least four generators, at least
two in degree $\leq r-2$ and $a$ in degree $\geq r > \frac{d}{2}$.
Then $d \leq 42$ or $a \geq 6$.
\vspace{2mm}
{\bf Proof}
The proof is in two parts. The first uses the (slightly improved)
result of [BC], which can be found at the end of this paper,
which states
that if $S$ is a smooth surface not of general type of degree $d$
in $\pfour$ with generic hyperplane section $C$ and $d > 42$,
then $C$ cannot have a secant line of order $r > \frac{d}{2}.$
\vspace{2mm}
The second deals with the conic.
Let $Q$ be the conic as above. Let
deg$(f_i) = t \geq r > \frac{d}{2}$ and $F = \{ f_i=0 \}$.
We may assume $Q$ is reduced and irreducible
and hence (as we are in $\pthree$) a smooth conic.
By Bezout's Theorem $ F \cap Q $ in $2t >d$ points
(up to multiplicity) and all these points must lie on $C$. Let
$F \cap Q = \sum m_ip_i $ where $p_i$ are the points of $C$.
\vspace{2mm}
{\bf Claim}
$Q$ meets $C$ at $p_i$ with multiplicity $m_i$.
\vspace{2mm}
Given the claim, as $Q$ is a conic $Q \subset P$ a plane
in $\pthree$ and so $C \cap P$ in at least $2t > d$ points. But
this means that $C$ is contained in the plane which is not possible.
\vspace{2mm}
{\bf Proof of the Claim}
The definition of intersection multiplicity given in [H] pg 427, is
as follows.
Let $X$ and $Y$ be varieties (in $\pthree$ for simplicity)
meeting at a point $p$,
then the intersection multiplicity
$$i(X, Y, p) = \sum_{j} (-1)^j
{\rm length}({\rm Tor}_j^A(A/{\bf a}, A/{\bf b})). $$
where $A ={\cal O}_{p, \pthree}$ is the local ring of $p$ and
${\bf a}= I_X$ and ${\bf b} = I_Y$ are the ideals of $X$ and $Y$
in $A$.
As $C$ is smooth, it is a local complete intersection, locally cut out
at $p$ by coprime polynomials $g_1$ and $g_2$. We may assume
that these polynomials correspond to polynomials in $I_C$ of
degree $t$.
Suppose the conic $Q$ meets $V(g_1)$ and
$V(g_2)$ at $p$ with multiplicity $m$ and thus
$$ m = \sum_{j} (-1)^j {\rm length}({\rm Tor}_j^A(A/{\bf a}, A/{\bf b}_i))$$
Where ${\bf a} = I_Q$ and ${\bf b}_i = (g_i)$ for $ i = 1, 2$.
We have the following short exact sequence
$$ 0 \rightarrow \frac{{\bf b_2}}{{\bf b_1}} \rightarrow
\frac{A}{{\bf b_1}} \rightarrow \frac{A}{(g_1, g_2)}
\rightarrow 0$$
and hence via the long exact sequence of Tor,
$$\begin{array}{ll}
m & = \sum_j (-1)^j {\rm length}({\rm Tor}_j^A(\frac{A}{{\bf a}})),
\frac{A}{{\bf b_1}}) \\
& = \sum_j (-1)^j{\rm length}({\rm Tor}_j^A(\frac{A}{{\bf a}}, \frac{A}{(g_1,g_2)}))
+\sum_j (-1)^j {\rm length}({\rm Tor}_j^A(\frac{A}{{\bf a}},
\frac{{\bf b_2}}{{\bf b_1}})).
\end{array}$$
Thus we need to show that
$$\sum_j (-1)^j {\rm length}({\rm Tor}_j^A(\frac{A}{{\bf a}},
\frac{{\bf b_2}}{{\bf b_1}}) )= 0. $$
Now,
$$ O \rightarrow {\bf b_1} \cap {\bf b_2} \rightarrow
{\bf b_2} \rightarrow \frac{{\bf b_2}}{{\bf b_1}}
\rightarrow O $$
Thus we need to show that
$$\sum_j (-1)^j {\rm length}({\rm Tor}_j^A(\frac{A}{{\bf a}},
{\bf b_1} \cap {\bf b_2}) )= \sum_j (-1)^j {\rm length}({\rm Tor}_j^A(\frac{A}{{\bf a}}, {\bf b_2}) ). $$
${\bf b_1} \cap {\bf b_2} = (g_1g_2)$ and
${\bf b_1} = (g_1)$ are principal ideals and hence have
minimal resolutions
$$ O \rightarrow A \rightarrow {\bf b_1} \cap {\bf b_2}
\rightarrow O $$
$$ O \rightarrow A \rightarrow {\bf b_1}
\rightarrow O $$
Therefore for any $A$-module $B$,
${\rm Tor}^A_i(B, {\bf b_1} \cap {\bf b_2}) =
{ \rm Tor}^A_i(B, {\bf b_1} ) = 0$ if $i \geq 1$.
Moreover
${\rm Tor}^A_0(\frac{A}{{\bf a}}, {\bf b_1} \cap {\bf b_2})
= \frac{A}{{\bf a}} \otimes {\bf b_1} \cap {\bf b_2}
= \frac{{\bf b_1} \cap {\bf b_2}}{{\bf a} \cap {\bf b_1} \cap {\bf b_2}}$
and
${\rm Tor}^A_0(\frac{A}{{\bf a}}, {\bf b_1})
= \frac{A}{{\bf a}} \otimes {\bf b_1}
= \frac{{\bf b_1}}{{\bf a} \cap {\bf b_1}}.$
Let $\Phi : \frac{{\bf b_1}}{{\bf a} \cap {\bf b_1}}
\rightarrow \frac{{\bf b_1} \cap {\bf b_2}}{{\bf a} \cap {\bf b}_1 \cap {\bf b_2}}$ be defined by
$\Phi(g_1a + {\bf a} \cap {\bf b_1}) = g_1g_2a + {\bf a} \cap {\bf b_1}\cap
{\bf b_2}$. This is an isomorphism, hence we are done.
\vspace{2mm}
Thus if $a \leq 5$, $Y$ either contains a secant line of $C$ of order
$t > \frac{d}{2}$ which contradicts [BC] if $d > 42$ or $Y$ contains a
conic meeting $C$ in $>d$ points which again leads to a contradiction.
Therefore either $d \leq 42$ or $a \geq 6$.
\qed
\vspace{2mm}
Note that if $Y$ were of degree 6, there is the possibility
that $Y$ is a union of two cubics, or a double cubic, which cannot
be eliminated by the above methods. However, given the bounds
on the number of sporadic zeros it would be very difficult
to have more than $6$ generators in degree $> \frac{d}{2}$. And as we will see in the last example this is not the obstruction to lowering the bound further.
\section{Bounding A}
\vspace{3mm}
Thus if the degree of $S > 42$, a generic hyperplane section,
$C$, of $S$ cannot have an $t$-secant line with $t > \frac{d}{2}$
nor a conic meeting the curve in $2t > d$ points.
In terms of the generic initial ideal of $C$, this means that
either
(1) all generators of ${\rm gin}(I_C)$ are in degree $\leq \frac{d}{2}$
or
(2) if there exists a generator of ${\rm gin}(I_C)$ in degree
$r > \frac{d}{2}$ then either there exists a generator in degree
$r-1$ or there exist a five more generators in degree $\geq r$.
\vspace{2mm}
The idea now is to build Borel-fixed monomial ideals
with as many generators in degree
$\geq \lfloor \frac{d}{2} \rfloor$ as possible.
If there can be 6 generators in degree
$\geq \lfloor \frac{d}{2} \rfloor+1$, let the ideal have
5 generators in degree
$\lfloor \frac{d}{2} \rfloor +1 $ and the sixth generator be in
as high a degree as possible using all the remaining sporadic zeros.
(This configuration is unlikely to arise from a curve.
However, for the purposes of Bounding $A$ it will give a good
enough upper bound.) If $s=5$ the bound
on the number of sporadic zeros means that it is impossible to have
6 or more generators in degree $\lfloor \frac{d}{2} \rfloor +1$, and
it is $s=5$ which is giving the upper bound on $d$.
Otherwise let the ideal have one generator in degree
$\lfloor \frac{d}{2} \rfloor$, one in degree
$\lfloor \frac{d}{2} \rfloor +1 $ and so on until all the
sporadic zeros have been used up.
The rest of the calculations are done by computer.
For each $d \leq 66$ (the bound we obtained in [BC]) we find all
connected invariants $\{ \lambda_{0}, \dots , \lambda_{s-1} \}$
with $s = 4$ or $5$ using a program of Rich Liebling.
We then use equations
$(2)$ and $(3)$ to find the maximal number of sporadic zeros, $z$.
We then use Mathematica to find the maximal $A$ using the criteria
above and see which examples satisfy equation $(1)$.
The following is a list of the examples with the highest degree
the Mathematica program gives which satisfy the criteria
above and equation (1).
\vspace{3mm}
${\bf s = 4}$
For $s=4$ the only possibilities are in degree $\leq 46$. Those of
highest degree being
\vspace{2mm}
\begin{center}
\begin{tabular}{|| c | c | c | c | c | c || } \hline
\# °ree & $z$ & $\{ \lambda_i \} $ & A-bound & neg \\
\hline
1 & 46 & 50 & 14, \ 12, \ 11, \ 9 & 921 & -2 \\
2 & 45 & 49 & 14, \ 12, \ 10, \ 9 & 882 & -18 \\
3 & 45 & 48 & 13, \ 12, \ 11, \ 9 & 854 & -2 \\
\hline
\end{tabular}
\end{center}
\vspace{3mm}
${\bf s = 5}$
For $s=5$ the only possibilities are in degree $\leq 48$. Those of
highest degree being
\vspace{2mm}
\begin{center}
\begin{tabular}{|| c | c | c | c | c | c || } \hline
\# & degree & $z$ & $\{ \lambda_i \} $ & A-bound & neg \\
\hline
1 & 48 & 45 & 13, \ 11, \ 10, \ 8, \ 6 & 810 & -14 \\
2 & 47 & 44 & 13, \ 11, \ 9, \ 8, \ 6 & 770 & -20 \\
3 & 47 & 43 & 12, \ 11, \ 10, \ 8, \ 6 & 743 & -4 \\
4 & 46 & 44 & 13, \ 11, \ 9, \ 7, \ 6 & 770 & -56 \\
\hline
\end{tabular}
\end{center}
\vspace{3mm}
{\bf Notes:}
\vspace{1mm}
{\bf 1.} $z$ is the maximal number of sporadic zeros.
\vspace{1mm}
{\bf 2.} {\it A-bound} is the upper bound on $A$ found using the
criteria above.
\vspace{1mm}
{\bf 3.} The column {\it neg} gives the value of equation (1) divided
by 12 then rounded up. Hence for each of the examples above in
degree $\geq 47$ we
need to show that $A < $A- bound $-$ neg in order to eliminate that
particular case.
\vspace{1mm}
{\bf 4.} The input list was not actually the full list of
possible connected invariants for each degree.
Certain configurations give rise to Arithmetically Cohen-Macaulay
(ACM) curves, or equivalently, curves without sporadic zeros.
If a curve has no sporadic zeros then $A =0 $, then in order to
satisfy equation (1) we must have
for $s = 4$, $d \leq 10$ and for $s= 5$, $d \leq 17$.
Invariants which give ACM curves (for $s \geq 4$) are
$(i)$ those whose consecutive invariants differ by two in which
case $C$ is a complete intersection of type $(s, \lambda_{s-1}+s-1)$.
For example the invariants $ \{ 6, 8, 10, 12, 14 \}$ correspond to
a complete intersection of type $(5, 10)$.
\vspace{1mm}
$(ii)$ those whose consecutive invariants differ by two
except $\lambda_0 = \lambda_1 + 1$ in which case $C$ is linked to a
line by a complete intersection of type $(s, \lambda_{s-1} + s-1)$.
Then as a line is ACM, $C$ is ACM. (See Rao [R] or
Migliore [M]) For example the invariants $ \{ 6, 8, 10, 12, 13 \}$
correspond to a
curve linked by a complete intersection of type $(5, 10)$ to a line.
\vspace{1mm}
{\bf 5.} We will show that example 4 for $s=5$ gives rise to
a Borel-fixed, connected monomial ideal
which satisfies all the conditions of this paper.
\vspace{2mm}
{\bf Eliminating} ${\bf (13, \ 11, \ 10, \ 8, \ 6\ )}$
\vspace{1mm}
We will eliminate the degree 48 configuration using
connectedness. The two configurations in degree 47 can
also be eliminated in a similar way.
\vspace{2mm}
If $x_0^4x_1^6$ is a generator of ${\rm gin}(I_C)$
then $C$ is contained in a complete intersection of type $(5, 10)$ and
$$ {\rm gin}(I_C) \supseteq (x_0^5, x_0^4x_1^6, x_0^3x_1^8, x_0^2x_1^{10}, x_0x_1^{12}, x_1^{14}). $$
This means that the only monomials of the form
$x_0^ax_1^bx_2^c$ with $c >0$ which can be generators of
${\rm gin}(I_C)$ are those with $(a, b) = (1, 11)$ or $(0, 13)$.
Then the best one could hope for is that one is of degree 24,
and the other is of degree 25 and
$$ A \leq \sum_{t = 12}^{23} t + \sum_{t=13}^{24}t = 156 < 810-14.$$
So this is not a possibility.
\vspace{1mm}
Therefore $x_0^4x_1^6$ is a sporadic zero.
As $z = 45$, there are at most three
generators in degree $\geq 24$ and taking connectedness
into account (See [C2]) there are only a few possibilities. The one giving
the best bound on $A$ would be if $x_1^{13}x_2^{12}$,
$x_0^1x_1^{11}x_2^{12}$ and $x_0^4x_1^6x_2^{16}$ are generators.
this leaves 5 sporadic zeros which one can use up by making
$x_0^3x_1^8x_2^{5}$ a generator. This gives an upper bound on $A$
of $777 < 810-14$, therefore this particular set of invariants is not
possible.
\vspace{2mm}
{\bf Example} ${\bf (13,\ 11, \ 9, \ 7, \ 6\ )}$
\vspace{1mm}
The degree 46 configuration for $s=5$ is actually possible.
(i.e. one can create a Borel-fixed monomial ideal which is connected
and still get an upper bound on $A$ which is big enough.)
Let ${\rm gin}(I_C)$ be defined as follows
$$\begin{array}{ll}
{\rm gin}(I_C) = (x_0^5, \ x_0^4x_1^6x_2, &x_0^4x_1^7, \
x_0^3x_1^7x_2^{13}, \ x_0^3x_1^8, \ x_0^2x_1^9x_2^{13}, \\
&x_0^2x_1^{10}, \ x_0x_1^{11}x_2^{13}, \ x_0x_1^{12}, \
x_1^{13}x_2^4, \ x_1^{14})
\end{array}$$
(Notice that $x_0^4x_1^6$ cannot be a generator of ${\rm gin}(I_C)$
otherwise $C$ would be linked to a plane quartic
and hence Arithmetically Cohen-Macaulay.)
In this case we get $A = 731 > 770-56$.
\vspace{3mm}
{\it Acknowledgments}.
This work was completed for the Summer School in Commutative
algebra at the Centre de Recerca Matem\`{a}tica, Barcelona in the
Summer of 1996. I would like to thank the Centre for allowing me to
participate. I would also like to thank the Association for
Women in Mathematics for their generous contribution toward
my travel expenses to the meeting.
\section*{Appendix A}
We saw in [BC], that if $S$ is a smooth surface not of general type
in $\pfour$, whose hyperplane section $C_h$ has an secant line
of order $r$, with $r > \frac{d}{2}$, then $S$ contains a plane
curve of degree $r > \frac{d}{2}$. In [BC] we showed that this was
impossible if $d > 50$, here we will improve the lemma slightly to
show that
this is impossible if $d >42$.
\vspace{3mm}
{\bf Lemma}
{\em If $S$ is a smooth surface not of general type in $\pfour$ of
degree $ d > 42$, $S$ cannot contain a plane curve of degree
$r > \frac{d}{2}$. }
\vspace{2mm}
{\bf Proof}
We may assume that $d > (s-1)^2+1$ then using the bounds
\begin{eqnarray}
1+ \sum_{i=0}^{s-1}(\binom{\lm_i}{2}+(i-1)\lm_i) & \leq &
\frac{d^2}{2s}+(s-4)\frac{d}{2}+1
\end{eqnarray}
\begin{eqnarray}
\sum_{t=0}^{s-1} (\binom{\lm_t+t-1}{3} - \binom{t-1}{3}) & \geq &
s\binom{\frac{d}{s}+\frac{s-3}{2}}{3}+1-\binom{s-1}{4}
\end{eqnarray}
found in [GP] and [BF] (respectively), equation (1) can be approximated by
\begin{eqnarray}
0 & \geq & d^2-5d-18-10(\frac{d^2}{2s}+(s-4)\frac{d}{2})
+12 s\binom{\frac{d}{s}+\frac{s-3}{2}}{3} \nonumber \\
&& + 12(1-\binom{s-1}{4}) - \sum_{t =0}^{m} \alpha_t (12t-22).
\end{eqnarray}
Let us first find a lower bound for the number of sporadic
zeros of a generic hyperplane section of $S$.
For $s=4 $, suppose that the number of sporadic zeros is
$\leq \frac{3d}{4}$ then, naively,
$A \leq \sum_{\lambda_0}^{\lambda_0 + \frac{3d}{4} -1} t.$
By connectedness $\lambda_0 \leq \frac{d}{4}+3$ and hence
$A \leq \frac{5}{32}d^2 + \frac{13}{8}d-3$. Substituting back into
equation $(7)$ we get
$$ 0 \geq \frac{d^3}{8}-\frac{23}{8}d^2-\frac{17}{2}d+33 $$
and hence $d \leq 25$. Therefore we may assume that the
number of sporadic zeros is $ > \frac{3d}{4}$, then
$ g(C_h) < \frac{d^2}{8} +1 -\frac{3d}{4}$.
Similarly for $s=5 $, suppose that the number of sporadic zeros is
$\leq \frac{2d}{5}$ then
$A \leq \sum_{\lambda_0}^{\lambda_0 + \frac{2d}{5} -1} t.$
By connectedness $\lambda_0 \leq \frac{d}{5}+4$ and hence
$A \leq \frac{4}{25}d^2 + \frac{7}{5}d$. Substituting back into
equation $(7)$ we get
$$ 0 \geq \frac{d^3}{25}-\frac{24}{25}d^2-10d-9 $$
and hence $d \leq 35$. Therefore we may assume that the
number of sporadic zeros is $ > \frac{2d}{5}$, then
$ g(C_h) < \frac{d^2}{10} +\frac{d}{2} +1 -\frac{2d}{5}$.
\vspace{2mm}
Let $C \subset P$ be a plane curve of degree
$r > \frac{d}{2}$ contained in $S$. Let $H$ be a hyperplane containing $C$.
Then
$S \cap H = C_h = C \cup C_{res} $.
We have
$$ 0 \rightarrow {\cal O}_{C \cup C_{res}} \rightarrow
{\cal O}_{C } \oplus {\cal O}_{C_{res}} \rightarrow {\cal O}_{C \cap C_{res}}
\rightarrow 0 $$
therefore
$$h^1 ({\cal O}_{C_h}) \geq h^1({\cal O}_{C }) +
h^1({\cal O}_{C _{res}})$$
and hence
$$ g(C_h) \geq g(C) + g(C_{res}) \geq g(C). $$
$C$ is a plane curve of degree $d_C \geq \frac{d}{2}$ and so
$$g(C) = \frac{(d_C-1)(d_C-2)}{2} - \delta
\geq \frac{(\frac{d}{2}-1)(\frac{d}{2}-2)}{2} $$
Hence
$$ \begin{array}{ll}
{\rm for} \ s=4 & \frac{d^2}{8} +1 -\frac{3d}{4} >
\frac{(\frac{d}{2}-1)(\frac{d}{2}-2)}{2} \\
{\rm for} \ s = 5 & \frac{d^2}{10} +\frac{d}{2} +1 -\frac{2d}{5}
\geq \frac{(\frac{d}{2}-1)(\frac{d}{2}-2)}{2} \\
{\rm for} \ s=6, 7 & \frac{d^2}{2s} + (s-4)\frac{d}{2} +1 \geq
\frac{(\frac{d}{2}-1)(\frac{d}{2}-2)}{2}
\end{array}
$$
This means for $s=4$ we have a contradiction and hence $d \leq 25$,
for $s=5$, $d < 34$ and for $s =6$ and $7$, $d \leq 42$,
\qed
\section*{References}
{\bf [B]} D. Bayer {\it The division algorithm and the Hilbert scheme},
Ph.D. Thesis, Harvard University (1982).
{\bf [BC]} R. Braun, M. Cook {\it A smooth surface in $\pfour$
not of general type has degree at most 66}, to appear in Compositio
Mathematica.
{\bf [BF]} R. Braun, G. Fl{\o}ystad {\it A bound for the degree
of smooth surfaces in $\pfour$ not of general type}, Compositio
Mathematica, Vol. 93, No. 2, September(I) (1994) 211-229.
{\bf [BPV]} W. Barth, C. Peters, A. Van de Ven
{\it Compact Complex Surfaces}, Springer-Verlag (1984).
{\bf [C1]} M. Cook {\it An improved bound for the degree of smooth
surfaces in $\pfour$ not of general type}, Compositio
Mathematica, Vol. 102, No. 2, June 1996, 141-145.
{\bf [C2]} M. Cook {\it The connectedness of space curve invariants}
to appear in Compositio Mathematica.
{\bf [EP]} G. Ellingsrud, C. Peskine {\it Sur les surfaces lisses de
$\pfour$}, Invent. Math. 95 (1989) 1-11.
{\bf [G]} M. Green {\it Generic Initial Ideals} Notes from the Summer School on Commutative Algebra, Centre de Recerca Matam\`{a}tica, Spain, 1996.
{\bf [GP]} L. Gruson, C. Peskine {\it Genres des courbes de l'espace
projectif}, Lecture Notes in Mathematics, Algebraic Geometry,
Troms{\o} 1977, 687 (1977) 31-59.
{\bf [H]} R. Hartshorne {\it Algebraic Geometry}, Springer-Verlag (1977).
{\bf [K]} L. Koelblen {\it Surfaces de $\pfour$ trac{\'e}es sur une
hypersurfaces cubique} Journal f{\"u}r die Riene and Angewandte
Mathematik, 433 (1992) 113-141.
{\bf [M]} J. Migliore {\it An introduction to deficiency modules and
liaison theory for subspaces of projective space} Seoul National University, Lecture Notes Series \# 24.
{\bf [PS]} C. Peskine, L. Szpiro {\it Liaison des vari{\'e}t{\'e}s
alg{|'e}brique I} Invent. Math. 26 (1974) 271-302.
{\bf [R]} P. Rao {\it Liaison among curves in $\pthree$.} Invent.
Math. 50 (1979) 205-217.
\vspace{5mm}
Michele Cook
Department of Mathematics
Pomona College
610 N. College Avenue
Claremont, CA 91711-6348
e-mail mcook\verb+@+pomona.edu
\end{document}
|
1997-01-23T12:38:40 | 9701 | alg-geom/9701010 | en | https://arxiv.org/abs/alg-geom/9701010 | [
"alg-geom",
"math.AG"
] | alg-geom/9701010 | Bill Oxbury | W.M. Oxbury, C. Pauly, E. Previato | Subvarieties of SU_C(2) and 2\theta-divisors in the Jacobian | LaTeX 41 pages, 2 figures; postscript including the figures available
at http://fourier.dur.ac.uk:8000/~dma0wmo/ | null | null | null | null | We explore some of the interplay between Brill-Noether subvarieties of the
moduli space SU_C(2,K) of rank 2 bundles with canonical determinant on a smooth
projective curve and 2\theta divisors, via the inclusion of the moduli space
into |2\theta|, singular along the Kummer variety. In particular we show that
the moduli space contains all the trisecants of the Kummer and deduce that
there are quadrisecant lines only if the curve is hyperelliptic; we show that
for generic curves of genus <6, though no higher, bundles with >2 sections are
cut out by \Gamma_00; and that for genus 4 this locus is precisely the
Donagi-Izadi nodal cubic threefold associated to the curve.
| [
{
"version": "v1",
"created": "Thu, 23 Jan 1997 11:36:19 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Oxbury",
"W. M.",
""
],
[
"Pauly",
"C.",
""
],
[
"Previato",
"E.",
""
]
] | alg-geom | \section{\@startsection {section}{1}{{\bf Z}} \def\n{{\bf N}} \def\q{{\bf Q}} \def\c{{\bf C}@}{-3.5ex plus
-1ex minus
-.2ex}{1.5ex plus .2ex}{\large\bf}
\def\subsection{\@startsection{subsection}{2}{{\bf Z}} \def\n{{\bf N}} \def\q{{\bf Q}} \def\c{{\bf C}@}{-3.25ex plus -1ex
minus
-.2ex}{1.5ex plus .2ex}{\normalsize\it}}
\let\emppsubsection\subsection
\newcommand{\numberequationsassubsubsections}
\newtheorem{prop}{Proposition}[section] \newtheorem{lemm}[prop]{Lemma}
\newtheorem{theo}[prop]{Theorem} \newtheorem{cor}[prop]{Corollary}
\newtheorem{rem}[prop]{\it Remark} \newtheorem{rems}[prop]{\it Remarks}
\newtheorem{ex}[prop]{Example} \newtheorem{exs}[prop]{Examples}
\begin{document}
\title{Subvarieties of ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2)$ and $2\theta$-divisors in the Jacobian}
\author{W.M. Oxbury, C. Pauly and E. Previato}
\date{}
\maketitle
Let ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,L)$ denote the projective moduli variety of semistable rank
2 vector bundles with determinant $L\in \pic(C)$ on a smooth curve $C$
of genus $g>2$; and suppose that $\deg L$ is even. It is well-known
that, on the one hand, the singular locus of ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,L)$ is isomorphic
to the Kummer variety of the Jacobian; and on the other hand that when
$C$ is nonhyperelliptic ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,\oo)$ has an injective morphism into the
linear series $|2\Theta|$ on the Jacobian $J_C^{g-1}$ which restricts
to the Kummer embedding $a \mapsto \Theta_a + \Theta_{-a}$ on the
singular locus. Dually ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ injects into the linear series
$|\ll|$ on $J_C^0$, where $\ll = \oo(2\Theta_{\kappa})$ for any theta
characteristic $\kappa$, and again this map restricts to the Kummer map
$J_C^{g-1} \rightarrow |2\Theta|^{\vee}} \def\cctil{\ctil_{\eta} = |\ll|$ on the singular locus.
This map to projective space (the two cases are of course isomorphic)
comes from the complete series on the ample generator of the Picard
group, and (at least for a generic curve) is an embedding of the moduli
space. Moreover, its image contains much of the geometry studied in
connection with the Schottky problem; notably the configuration of
Prym-Kummer varieties.
In this paper we explore a little of the interplay, via this embedding,
between the geometry of vector bundles and the geometry of
$2\theta$-divisors. On the vector bundle side we are principally
concerned with the Brill-Noether loci ${\cal W}^r \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ defined
by the condition $h^0(E) >r$ on stable bundes $E$. These are analogous
to the very classical varieties $W_{g-1}^r \subset J_C^{g-1}$. Unlike
the line bundle theory, however, general results---connectedness,
dimension, smoothness and so on---are not known for the varieties
${\cal W}^r$ (see \cite{BF}).
On the $2\theta$ side we shall consider the Fay trisecants of the
$2\theta$-embedded Kummer variety, and the subseries $\g00 \subset
|\ll|$ consisting of divisors having multiplicity $\geq 4$ at the
origin. This subseries is known to be important in the study of
principally polarised abelian varieties \cite{vGvdG}: in the Jacobian
of a curve its base locus is the surface $C-C \subset J_C^0$ (plus a
pair of isolated points in the case $g=4$) \cite{W}, whereas for a ppav
which is not a Jacobian it is conjectured that the origin is the only
base point (but see \cite{BD}).
The organisation and main results of the paper are as follows. In the
first two sections we study two families of lines on ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K) \subset
|\ll|$ (or equivalently ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,\oo) \subset |2\Theta|$), each of
dimension $3g-2$. These are the Hecke lines, coming from vector bundles
of odd degree, on the one hand, and lines lying inside $g$-dimensional
linearly embedded extension spaces (generating the lowest stratum of
the Segre stratification), on the other. We prove ({\bf 1.3} and {\bf
1.4}) that a line in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ lies in both of these families if and
only if it intersects the Kummer variety. Moreover, we show ({\bf 2.1})
that every trisecant of the Kummer is such a line, and in particular
lies on the moduli space. (This fact is certainly well-known to the
experts, but we were not aware of a reference in the literature.) As a
corollary we show ({\bf 2.2}) that the Kummer variety has quadrisecant
lines if and only if the curve is hyperelliptic.
In section 3 we introduce the subschemes ${\cal W}^r \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$, and
as a first step in their study we examine the stratification by $h^0$
of spaces of extensions, which will then map rationally into
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$. The natural formulation of this stratification turns out to
involve the Clifford index, and as an easy by-product we obtain the
inequality ({\bf 3.7}) $$ h^0(E) \leq g+1 -\cliff(C) $$ for any
semistable rank 2 bundle $E$ with $\det E = K$.
In section 4 we prove, using a spectral curve construction, that ({\bf
4.1}) $$ {\cal W}^2 = \g00 \cap {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K) $$ provided $C$ is nonhyperelliptic
of genus 3 or 4, or nontrigonal of genus 5. On the other hand, we show
later on ({\bf 8.3}) that the equality fails for all curves of
genus~6.
The remaining four sections of the paper are devoted to examining some
of the geometry in detail for each of the cases $g= 3,4,5,6$. For
genus~3 the moduli space ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is embedded in ${\bf P}^7$ as the
unique Heisenberg-invariant quartic singular along the Kummer
variety---the so-called {\it Coble quartic}. We examine the
configuration ${\cal W}^1 \subset {\cal W} \subset {\bf P}^6$, where ${\bf P}^6$ is the
hyperplane spanned by the `generalised theta divisor' ${\cal W}={\cal W}^0$. It
is known that ${\cal W}$ has a unique triple point ${\cal W}^2=\g00$; we show
({\bf 5.3}) that ${\cal W}^1$ is a Veronese cone (already to be found in the
classical literature \cite{C}) with vertex ${\cal W}^2$, and whose
generators are trisecants of the Kummer variety corresponding to a
natural embedding of $|K|$ in the parameter space of all trisecants. In
addition, we identify the tangent cone of ${\cal W}$ at the triple point
({\bf 5.5}): this is nothing but the secant variety of the Veronese
surface, with equation $\det A =0$ where $A$ is a symmetric $3\times 3$
matrix.
To each nonhyperelliptic curve of genus 4 one can associate a nodal
cubic threefold $\tt\subset {\bf P}^4$, which can be described in various
ways. The view we adopt here is that it is the rational image of
${\bf P}^3$ via the linear system of cubics containing the canonical curve.
There is an identification ${\bf P}^4 \mathbin{\hbox{$\widetilde\rightarrow$}}} \def\ext{{\rm Ext} \g00$, due to Izadi \cite{I},
and we prove ({\bf 6.4}) that this restricts to an isomorphism $\tt
\mathbin{\hbox{$\widetilde\rightarrow$}}} \def\ext{{\rm Ext} {\cal W}^2$, with the node mapping to ${\cal W}^3 = \{g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus
\trigg\}$, the direct sum of the two trigonal line bundles on the
curve. For the proof of this we make use of Izadi's description of the
lines in $\tt$ as pencils of $2\theta$-divisors. Note also that by
passing to the tangent cone at the origin, $\g00$ may be viewed as a
linear system of quartics in canonical space ${\bf P}^3$. We observe ({\bf
6.11}) that projection of the cubic ${\cal W}^2$ away the node can be
naturally identified with the quotient of $\Gamma_{00}$ by $q^2$ where
$q$ is the unique quadric vanishing on the canonical curve.
For genus 5 we show ({\bf 7.2}) that ${\cal W}^3$ is a Veronese surface
cutting the Kummer variety in the image of a plane quintic. If the
curve is nontrigonal this quintic is the discriminant of the net of
quadrics containing the canonical curve, while in the trigonal case it
is isomorphic to the projection of the canonical curve from a
trisecant.
Finally we show ({\bf 8.1}) that for a nontrigonal curve of genus 6 the
locus ${\cal W}^4$ is a single point, stable if $C$ is not a plane quintic
(this case was also observed by Mukai in \cite{Muk}), while ${\cal W}^4$ is
a line not meeting the Kummer in the trigonal case. In the generic case
${\cal W}^4$ is the vertex of a configuration of five ${\bf P}^6$s which form
the intersection of ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with $\g00$ residual to ${\cal W}^2$.
\medskip
{\it Acknowledgements.} The authors are grateful to R. Donagi, E. Izadi, P.
Newstead, S. Ramanan, E. Sernesi, and especially B. van Geemen for
various helpful comments. We would also like to thank Miles Reid for
his organisation of the Warwick Algebraic Geometry Symposium 1995--96
where much of this work was carried out, and the MRC for its
hospitality during this symposium; also L. Brambila-Paz and the
hospitality of IMATE-Morelia, Mexico during the Vector Bundles workshop
in July 1996. Research of the third named author was partly supported
by NSA grant MDA904-95-H-1031.
\vfill\eject \bigskip\noindent {\large\bf I Lines}
\section{The $g$-plane ruling}
For a line bundle $L$ on the curve $C$ we denote by ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,L)$ the
projective moduli variety of semistable rank 2 vector bundles with
determinant $L$; and in particular we shall be concerned with
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$. The semistable boundary of this space is the image of
$J_C^{g-1} \rightarrow {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ mapping $L\mapsto L\oplus KL^{-1}$;
this is the singular locus when $g>2$. Throughout the paper we shall
view both the moduli space and the Kummer variety---when $C$ is
nonhyperelliptic---as lying in the projective space $|2\Theta|^{\vee}} \def\cctil{\ctil_{\eta} =
{\bf P}^{2^g-1}$ in the standard way: by the complete linear series $|\ll|$
on ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$, where $\ll$ is the ample generator of $\pic\ {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)
\cong {\bf Z}} \def\n{{\bf N}} \def\q{{\bf Q}} \def\c{{\bf C}$, restricting to $\oo(2\Theta)$ on the Jacobian. A {\it line}
on ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is then an embedded ${\bf P}^1$ on which the restriction of
$\ll$ has degree one.
We shall consider the following subvariety of ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ ruled by
$g$-planes. For $x\in \pic^{g-2}(C)$ let ${\bf P}(x) = {\bf P} H^1(C,K^{-1}x^2)
\cong {\bf P}^g$. This parametrises isomorphism classes of extensions $$
\ses{x}{E}{Kx^{-1}}, $$ and thus has a moduli map to ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$, which
is linear (with respect to $\ll$) and injective (see also remark
\ref{bertram} below). Globally we have a ruling: $$ \begin{array}{rcl}
{\bf P} U & \map{\eee}& {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K) \\ &&\\
\downarrow {\bf P}^g&&\\ &&\\ \pic^{g-2}(C) &&\\ \end{array} $$ where $U =
R^1\pi_* K^{-1}{\cal N}} \def\mm{{\cal M}} \def\bb{{\cal B}} \def\ll{{\cal L}^{2}$, with ${\cal N}} \def\mm{{\cal M}} \def\bb{{\cal B}} \def\ll{{\cal L} \rightarrow C\times \pic^{g-2} (C)$
a Poincar\'e bundle and $\pi: C\times \pic^{g-2}(C) \rightarrow
\pic^{g-2}(C)$ the natural projection.
We shall need to make repeated use, in what follows, of the following
result of Lange--Narasimhan \cite{LN}. Consider any extension $$
\ses{n_0}{F}{n_0^{-1}\otimes \det F} $$ where $n_0 \subset F$ is a line
subbundle of maximal degree. This is represented by a point $f$ of the
extension space ${\bf P} H^1(C, n_0^2 \otimes \det F^{\vee}} \def\cctil{\ctil_{\eta}) = {\bf P} H^0(C,
Kn_0^{-2}\otimes \det F)^{\vee}} \def\cctil{\ctil_{\eta}$, into which the curve $C$ maps via the
linear series $|Kn_0^{-2}\otimes \det F|$. For an effective divisor $D$
on $C$, we shall denote by $\overline D$ the linear span in ${\bf P} H^1(C,
n_0^2 \otimes \det F^{\vee}} \def\cctil{\ctil_{\eta})$ of the image of this divisor. Then the
following is proposition 2.4 of \cite{LN}:
\begin{lemm} \label{ln} With the above notation there is a bijection,
given by $\oo (D) = n^{-1}n_0^{-1}\otimes \det F$, between:
\begin{enumerate} \item line subbundles $n\subset F$, $n\not= n_0$, of
maximal degree; and \item line bundles $\oo (D)$ with degree $\deg D =
\deg F - 2 \deg n_0 $ and such that $f\in \overline D$.
\end{enumerate} \end{lemm}
Let us return now to the $g$-planes ${\bf P}(x)\hookrightarrow {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$,
where $x\in \pic^{g-2}(C)$, and the following well-known facts. The
curve $C$ maps into ${\bf P}(x)$ via $|K^2x^{-2}|$, as a special case of
the Lange--Narasimhan picture. Moreover, a point of ${\bf P}(x)$
represents a stable bundle (with $x$ as maximal line subbundle)
precisely away from the image of $C$; while a point $q\in C\subset {\bf P}
(x)$ represents the equivalence class of the semistable bundle
$x(q)\oplus Kx^{-1}(-q)$. In other words there is a commutative
diagram: \begin{equation} \label{5} \begin{array}{rcl} C & \map{t_x} &
J_C^{g-1} \\ &&\\ |K^2x^{-2}|\downarrow && \downarrow\\ &&\\ {\bf P}(x)
&\map{\eee}& {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)\\ \end{array} \end{equation} where $t_x:q\mapsto
x(q)$ and the second vertical arrow maps $L\mapsto L\oplus KL^{-1}$.
The incidence relations between $g$-planes of this ruling can be given
as follows.
\begin{prop} \label{40} Suppose that $C$ is nonhyperelliptic. For $x,y
\in \pic^{g-2}(C)$ the intersection ${\bf P}(x) \cap {\bf P}(y)$ is either
empty, or: \begin{enumerate} \item the secant line $\overline{pq}$ of
the curve (in either of ${\bf P}(x)$ or ${\bf P}(y)$) if $x\otimes y =
K(-p-q)$; \item the point $x(p) \oplus Kx^{-1}(-p) \in
{\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$, if $H^0(C, Kx^{-1} y^{-1}) =0$ and $x\otimes y^{-1} =
\oo(q-p)$. \end{enumerate} \end{prop}
{\it Proof.}\ First we note that at any point $E\in {\bf P} (x)$ away from the curve
the residual $g$-planes ${\bf P}(y)$ through $E$ can be identified, via
lemma \ref{ln}, with the set of effective divisors $p+q$ such that $E$
lies on the secant line $\overline{pq}$; and the line bundles $x,y$ are
then related by \begin{equation} \label{30} x\otimes y = K(-p-q).
\end{equation} Note that the point $p$ on the curve in ${\bf P}(x)$
represents the bundle $$ x(p)\oplus Kx^{-1}(-p) = y(q)\oplus
Ky^{-1}(-q), $$ i.e. it coincides with the image of $q$ on the curve in
${\bf P}(y)$; and similarly $q\in C\subset {\bf P}(x)$ coincides with $p\in
C\subset {\bf P}(y)$. This shows that condition (\ref{30}) is equivalent to
$\overline{pq} \subset {\bf P}(x) \cap {\bf P}(y)$.
On the other hand, when $C$ is nonhyperelliptic ${\bf P}(x)$ and ${\bf P}(y)$
cannot intersect in dimension greater than one: for then a generic
point $E$ of the intersection would lie on distinct secant lines
$\overline{pq}$ and $\overline{rs}$, both satisfying (\ref{30}), and
hence $\oo(p+q) = \oo(r+s)$, a contradiction.
The only other possibility for nonempty intersection ${\bf P}(x) \cap
{\bf P}(y)$ is that this intersection is a single point, in which case it
must be a point of the Kummer, and we easily find case 2. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
Next we recall the Hecke correspondence between ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ and
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K(p))$ where $p\in C$ is a point of the curve. For a stable
bundle $F\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K(p))$ we shall write $l_F \cong {\bf P}^1 \subset
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ for the image of $$ \begin{array}{rcl} {\bf P} {\rm Hom}(F,\c_p)
&\rightarrow& {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)\\
\phi &\mapsto& \ker \phi.\\ \end{array} $$ This is called the {\it
Hecke line} associated to the bundle $F$. Our aim in the remainder of
this section will be to compare Hecke lines in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with lines
contained in the $g$-planes ${\bf P}(x)$.
\begin{theo} \label{10} Let $x\in \pic^{g-2}(C)$. A projective line
$l\subset {\bf P}(x)$ is Hecke if and only if it meets the image of the
curve $C\subset {\bf P}(x)$. \end{theo}
{\it Proof.}\ Suppose that a Hecke line $ l_F$ is contained in ${\bf P}(x)$. This
means that for all extensions of sheaves of the form $$
\ses{E}{F}{\c_p}, $$ where $K(p)$ is the determinant of $F$, the kernel
$E$ contains $x$ as a line subbundle. This means we have a pencil of
homomorphisms $x \hookrightarrow F$, for which we have two
possibilities.
{\it Either} the image subsheaf is constant and is in the kernel of
every homomorphism $F\rightarrow \c_p$. Then there is an inclusion of
sheaves $x(p) \subset F$, which by stability of $F$ is a line
subbundle, i.e. $F$ is an extension $$ \ses{x(p)}{F}{Kx^{-1}}. $$ But
the space ${\bf P} H^1(C,K^{-1} x^{2}(p))$ of such extensions is the image
of the projection of ${\bf P} (x) = {\bf P} H^1(C, K^{-1} x^{2})$ from the
point $p\in C \subset {\bf P} (x)$, i.e. the set of lines in ${\bf P} (x)$
passing through the point $p$. It is not hard to see that the line $l$
corresponding to $F$ in this manner is precisely $l_F$---and we note
that {\it every} line meeting the curve arises in this way.
{\it Or}---the second possibility---the image sheaf is non-constant, in
which case we have a subsheaf $x\oplus x\subset F$, with torsion
quotient supported on some effective divisor $D$. But then $\det F =
K(p)$ implies that $\oo(D) = Kx^{-2} (p)$. So $\deg D = 3$ and we
observe that $$ h^0(C, K^2x^{-2}(-D)) = h^0(C,K(-p)) = g-1, $$ i.e.
that $\dim \overline{D} = 1$. In this case the Hecke line $l_F$ is just
$\overline{D}$ and is trisecant to the curve. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
It follows by a dimension count that for $g>2$ a generic Hecke line in
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is not contained in any $g$-plane. We shall show next that
those that are are precisely the Hecke lines that meet the Kummer
variety (i.e. the singular locus).
\begin{theo} \label{20} Let $l_F \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ be any Hecke line.
\begin{enumerate} \item There is a canonical surjection from line
subbundles $n\subset F$ with $\deg n = g-1$ to points of intersection
$l_F \cap {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$ with the Kummer variety, which is bijective
if $l_F$ is not a tangent line of the Kummer. \item The intersection
$l_F \cap {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$ is nonempty if and only if $l_F$ is
contained in a $g$-plane ${\bf P}(x)$ for some $x\in \pic^{g-2}(C)$. If $C$
is nonhyperelliptic and $l_F \cap {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$ has cardinality $k$
then the number of such $g$-planes is $1+{k\choose 2}$.
\end{enumerate} \end{theo}
\begin{rems}\rm \label{21}
{\it (i)} We have two irreducible
families of lines in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$: the Hecke lines and the lines
contained in $g$-planes of the ruling. These families have the same
dimension $3g-2$, and we expect that each is a component of the Hilbert
scheme of all lines. Theorems \ref{10} and \ref{20} would then say that
{\it the intersection of these two components consists of the members
of each family which meet the Kummer variety}.
{\it (ii)} The
cardinality of $l_F \cap {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$ satisfies $k\leq 4$ if $C$ is
hyperelliptic and $k\leq 3$ otherwise; this follows from part 1 and
proposition 5.1 of \cite{Lan}. See also corollary \ref{4-secants}
below. \end{rems}
{\it Proof of theorem \ref{20}:} For $n\subset F$ consider the
diagram: $$ \begin{array}{rrcl} &&0&\\ &&\downarrow&\\ &&Kn^{-1}&\\
&&\downarrow&\\ 0\rightarrow n \rightarrow F & \rightarrow&
Kn^{-1}(p)&\rightarrow 0\\ &\phi \searrow&\downarrow&\\ &&\c_p&\\
&&\downarrow&\\ &&0&\\ \end{array} $$ Then if $E=\ker \phi$ we have a
surjective sheaf map $E\rightarrow Kn^{-1}\rightarrow 0$, and hence
$n\subset E$. So $E$ is S-equivalent to $n\oplus Kn^{-1}$ and defines a
point of intersection $l_F \cap {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$. To see that this is
surjective let $n\oplus K n^{-1} \in {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$ be a point of
intersection with the Hecke line $l_F$. This means there is an exact
sequence $$ \ses{E}{F}{\c_p} $$ where $E$ is S-equivalent to $n\oplus
Kn^{-1}$; i.e. at least one of $n$ or $Kn^{-1}$ is a line subbundle of
$E$, and hence of $F$.
In a moment we shall verify that in this construction we have:
\begin{equation} \label{tangent} n\oplus Kn^{-1} \subset F \quad
\Longleftrightarrow \quad \hbox{$l_F$ is tangent to ${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$
at $n\oplus Kn^{-1}$.} \end{equation} This will show that the
correspondence is bijective when $l_F$ is not a tangent line.
For part 2, we note first that if $l_F$ is contained in a $g$-plane
${\bf P}(x)$ then by theorem \ref{10} it meets the curve $C\subset {\bf P}(x)$
and hence the Kummer. For the converse it suffices, by the preceding
construction, to suppose that there is a degree $g-1$ line subbundle
$n\subset F$. Then letting $x = n(-p)$, it follows that $F$ is
represented by a point of the extension space ${\bf P} H^1(C, K^{-1}n^2
(-p)) = {\bf P} H^1(C, K^{-1}x^{2} (p))$ and hence determines---as in the
proof of theorem \ref{10}---a line $l \subset {\bf P} (x)$ through $p\in
C$, which coincides with the Hecke line $l_F$.
Notice that by lemma \ref{ln} any residual degree $g-1$ line subbundles
$m\subset F$ correspond to points $q\in C$ by the relation $K(p-q) =
n\otimes m$; and in this case $l_F$ must be the secant $\overline{pq}
\subset {\bf P}(x)$. In particular, $Kn^{-1}$ is a subbundle of $F$ if and
only if $l_F$ is the tangent line to $C\subset {\bf P}(x)$ at $p$---this
proves (\ref{tangent}).
Finally, if we fix any $g$-plane containing $l_F$ then by proposition
\ref{40} the residual such $g$-planes correspond bijectively to the
effective divisors $p+q$ such that $l_F = \overline{pq}$, i.e. to pairs
of intersection points of $l_F$ with the Kummer. And so we obtain
$1+{k\choose 2}$ for the number of such $g$-planes. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\section{Trisecants of the Kummer variety} \label{trisecants}
Recall that the quotient ${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$ of $J_C^{g-1}$ by the Serre
involution is embedded in ${\bf P}^{2^g -1}$ by the linear system
$|2\Theta|$, and that this embedding extends to the moduli space
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ containing the Kummer as its singular locus (when $g>2$). In
this embedding the Kummer possesses a unique irreducible 4-dimensional
family of trisecant lines, which characterise Jacobians amongst
principally polarised abelian varieties. We shall briefly describe this
family of trisecants (see~\cite{M} or~\cite{Deb}).
The base $\ff$ of the family is the fibre product: $$
\begin{array}{rcl} \ff &\rightarrow & S^4 C\\ &&\\ \downarrow
&&\downarrow \hbox{Abel-Jacobi}\\ &&\\ \pic^{g-3}(C) & \rightarrow &
\pic^4(C)\\ \end{array} $$ where the bottom map sends $a\mapsto K
a^{-2}$. An element of $\ff$, in other words, is a pair $(a,D) \in
\pic^{g-3}(C)\times S^4 C$ such that $a^{2} = \oo(K-D)$. Writing $D=
p+q+r+s$, one shows that the following three points of ${\bf P}^{2^g -1}$
are collinear: \begin{equation} \label{3points} \begin{array}{lllll}
\phi(a(q+r)) && \phi(a(p+r)) && \phi(a(p+q))\\ =\phi(a(p+s)), &&
=\phi(a(q+s)), &&=\phi(a(r+s)).\\ \end{array} \end{equation}
We shall refer to the lines of ${\bf P}^{2^g -1}$ parametrised by $\ff$ in
this way as the {\it Fay trisecants}.
\begin{theo} \label{fay} The Fay trisecants are precisely the Hecke
lines which are trisecant to the Kummer variety. In particular they
all lie on ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$. \end{theo}
{\it Proof.}\ We ask for the condition on a Hecke line $l_F \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ for
it to be trisecant to the Kummer. Let $\det F = K(p)$. For $l_F$ to
meet the Kummer $F$ must have a line subbundle $n \subset F$ of degree
$g-1$. Then by theorem \ref{20}, $l_F$ is a trisecant if and only if
$F$ has two further degree $g-1$ line subbundles. By stability these
are maximal, and so by lemma \ref{ln} the residual subbundles
correspond bijectively to points of $C$ mapping to the extension class
of $F$ under $$ C\ \map{|K^2 n^{-2}(p)|}\ {\bf P} H^1(C,K^{-1} n^2(-p)).
$$ Thus trisecants $l_F$ correspond to {\it nodes} of the image curve
under the linear series $|K^2 n^{-2}(p)|$; and the condition for such a
node is that for points $q,r \in C$, $$ \begin{array}{rcl} h^0(C, K^2
n^{-2}(p-q-r) & \geq & h^0(C, K^2 n^{-2}(p))-1\\ &=& g-1;\\ \end{array}
$$ or equivalently $h^0(C,K^{-1} n^2(-p+q+r)) \geq 1$. This in turn is
equivalent to $K^{-1}n^2(-p+q+r) = \oo(s)$ for some $s\in C$. We
conclude that the necessary and sufficient condition for $l_F$ to be a
trisecant $\overline{pqr}$ of $C \subset {\bf P}(x)$, $x=n(-p)$, is:
\begin{equation} \label{34} n^2 = K(p-q-r+s) \quad \hbox{or
equivalently} \quad x^2 = K(-p-q-r +s). \end{equation} One can now
verify, using (\ref{5}), that the points of intersection of $l_F$ with
the Kummer---i.e. with the curve $C\subset {\bf P} (x)$---are the three
points (\ref{3points}) where $a= x(-s) $. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{cor} \label{4-secants} \begin{enumerate} \item If $C$ is
nonhyperelliptic then no Fay trisecant has more than three intersection
points with the Kummer variety. \item If $C$ is hyperelliptic then all
Fay lines are exactly quadrisecant. \end{enumerate} \end{cor}
{\it Proof.}\ By theorems \ref{20} and \ref{fay} every Fay line lies in ${\bf P}(x)$
for some $x\in \pic^{g-2}(C)$. Let $D$ be an effective divisor on $C$
with $\dim \overline{D} = 1$ in ${\bf P}(x)$. This is equivalent to
$h^0(C,K^2x^{-2}(-D)) = g-1$, or, by Riemann-Roch, $$
h^0(C,K^{-1}x^{2}(D)) = \deg D - 2. $$ If $\deg D = 5$ then this says
that $|K^{-1}x^{2}(D)|$ maps $C$ birationally to a plane cubic, which
is impossible; while if $\deg D = 4$ then it is equivalent to:
\begin{equation} \label{35} K x^{-2} = \oo(D-H) \end{equation} where
$H$ is a hyperelliptic divisor. This proves part 1; for part 2 let
$\overline{pqr}\subset {\bf P}(x)$ be the trisecant constructed in the
proof of theorem \ref{fay}, and consider $D = p+q+r+\tau(s)$ where
$\tau:C\leftrightarrow C$ is the hyperelliptic involution. Then
(\ref{35}) follows from (\ref{34}) and we see that $\overline{pqr} =
\overline{D}$ is a quadrisecant. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\vfill\eject \bigskip\noindent {\large\bf II Loci}
\section{Brill-Noether loci in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$}
Let ${\cal W} \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ be the closure of the locus of stable
bundles $E$ for which $H^0(C,E) \not= 0$, i.e. the `theta divisor' for
rank 2 bundles. In terms of the map $\phi: {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K) \rightarrow
|2\Theta|^{\vee}} \def\cctil{\ctil_{\eta}$, ${\cal W}$ is the unique hyperplane section tangent to the
Kummer variety ${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J^{g-1}_C)$ along the image of the theta divisor
${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(\Theta)$.
The Brill-Noether loci are the subschemes $$ {\cal W} \supset {\cal W}^1 \supset
\cdots \supset {\cal W}^{g-1} \supset {\cal W}^g\\ $$ where ${\cal W}^r$ is the
closure of the set of stable bundles $E$ for which $h^0(C,E) \geq
r+1$. (We shall see in a moment that ${\cal W}^{g+1} = \emptyset$---see
proposition \ref{bound}).
\begin{rem}\rm \label{petri} The local structure of ${\cal W}^r$ is governed
by a symmetric Petri map $$ S^2 H^0(C,E) \rightarrow H^0(C,K\otimes \ad
E), $$ where $\ad E$ is the bundle of trace-free endomorphisms; as a
consequence ${\cal W}^r$ has expected codimension $r+2 \choose 2$ in
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$. (See for example \cite{BF}.)
In addition, it is not hard to show that {\it ${\cal W}^1$ is the union of
all Hecke lines meeting ${\cal W}^2$}. We shall see this illustrated for
curves of genus 3 in theorem~\ref{coblecone}. \end{rem}
In order to study the Brill-Noether loci ${\cal W}^r$ we shall analyse them
first in spaces ${\bf P} \ext^1(K-D,D)$ of extensions \begin{equation}
\label{basicext} \ses{\oo(D)}{E}{\oo(K-D)} \end{equation}
\noindent for some line bundle $\oo(D)\in \pic^d(C)$. Usually, though
not always, we shall think of $D$ as an effective divisor; indeed $E$
has sections if and only if it can be expressed as such an extension
with $D$ effective.
\begin{rems}\rm \label{nag}
{\it (i)} Note that by semistability
$d \leq g-1$ with equality if and only if $E$ is S-equivalent to
$\oo(D) \oplus \oo(K-D)$. Moreover, every $E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is such an
extension for some $D\in \pic^d(C)$ with $$ d\geq \Bigl[ {g-1 \over 2}
\Bigr]. $$ This follows from a classical result of Segre and Nagata
(see \cite{Lan2}) which says that every ruled surface of genus $g$ has
a section with self-intersection at most $g$.
{\it (ii)} It will
be convenient below to introduce the Clifford index $\cliff(D) = \deg D
- 2r(D)$, where $r(D)=h^0(D) -1$, into our notation. Recall that the
Clifford index $\cliff(C)$ of the curve is defined to be the minimum
value of $\cliff(D)$ for which $h^i(D)\geq 2$ for $i=0,1$ (see
\cite{GL}). Recall also that $$ \cliff(C) \leq \Bigl[ {g-1 \over 2}
\Bigr] $$ with equality for generic $C$. \end{rems}
As in section 1, the curve $C$ maps into the space of such extensions;
and we shall denote the rational coarse moduli map of this space by
$\eee_D$: $$ C\ \map{|2K-2D|}\ {\bf P} \ext^1(K-D,D)\cong {\bf P}^{3g-4-2d}
\ \map{\eee_D}\ {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K). $$ We shall denote by $\iic$ the ideal sheaf
of the image curve in ${\bf P}^{3g-4-2d}$; and we shall write ${\rm Bl}} \def\cliff{{\rm Cliff}} \def\secant{{\rm Sec}_C$ for
the blow-up along this curve and $\secant^n C$ for the variety of its
$n$-secant $(n-1)$-planes; although of course the map $C \rightarrow
{\bf P} \ext^1(K-D,D)$ is not necessarily an embedding or even birational.
We shall write $$ {\cal W}_D = \eee_D ({\bf P} \ext^1(K-D,D)). $$ Note here
that by $\eee_D (\Omega)$, where $\Omega \subset {\bf P} \ext^1(K-D,D)$, we
shall always mean the proper transform of $\Omega$, i.e. the closure in
${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ of the image of the domain of definition of $\eee_D$.
\begin{rem}\rm \label{bertram} The rational map $\eee_D$ has been
studied in detail by Bertram and others (see \cite{B}), and resolves to
a morphism ${\widetilde \varepsilon}} \def\pitil{{\widetilde \pi}_D$ of the blow-up: $$ \begin{array}{r} {\bf P}
\ext^1(K-D,D) \leftarrow {\rm Bl}} \def\cliff{{\rm Cliff}} \def\secant{{\rm Sec}_C \leftarrow {\rm Bl}} \def\cliff{{\rm Cliff}} \def\secant{{\rm Sec}_{\sectil_2 C} \leftarrow
\cdots \leftarrow {\rm Bl}} \def\cliff{{\rm Cliff}} \def\secant{{\rm Sec}_{\sectil_{g-2-d}C} \\ \\ \downarrow {\widetilde \varepsilon}} \def\pitil{{\widetilde \pi}_D\\
\\ {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)\\ \end{array} $$ Moreover, hyperplanes of $|\ll| =
|2\Theta|^{\vee}} \def\cctil{\ctil_{\eta}$ pull back to divisors of $|\iic^{g-2-d}(g-1-d)|$ on
${\bf P} \ext^1(K-D,D)$. We shall need these facts only in the cases $d=
g-2$ (already discussed in section 1) and $d=g-3$; in both of these
cases $\eee_D$ comes from the complete series $|\iic^{g-2-d}(g-1-d)|$.
\end{rem}
It is easy to analyse the filtration of each ${\bf P} \ext^1(K-D,D)$ by the
dimension $h^0(E)$. For any such extension we have
\begin{equation} \label{h^0(E)} \begin{array}{rcl} h^0(E) &=& h^0(D) +
h^0(K-D) - {\rm rank \ } \delta(E)\\ &=& g+1 - \cliff(D) - {\rm rank \ } \delta(E)\\
\end{array} \end{equation} where $\delta (E): H^0(K-D) \rightarrow
H^1(D)$ is the coboundary homomorphism in the cohomology sequence of
(\ref{basicext}). (Note that (\ref{h^0(E)}) gives an upper bound
$g+1-\cliff(C)$ on $h^0(E)$; see proposition \ref{bound} below.)
By Serre duality $\delta(E)$ is an element of $\otimes^2 H^1(D)$, while
its transpose $\delta(E)^t$ is the coboundary map for the dual sequence
tensored with $K$: $ \ses{\oo(D)}{K\otimes E^{\vee}} \def\cctil{\ctil_{\eta}}{\oo(K-D)}. $ But
$K\otimes E^{\vee}} \def\cctil{\ctil_{\eta} = E$ and so $\delta (E) = \delta(E)^t$. We have
therefore constructed a linear homomorphism \begin{equation} \delta :
\ext^1(K-D,D) \rightarrow S^2 H^1(D), \end{equation} with respect to
which $h^0(E)$ satisfies (\ref{h^0(E)}). But the {\it rank}
stratification of $S^2 H^1(D)$ coincides with the {\it secant}
stratification of its embedded Veronese variety $$ \begin{array}{rcl}
\ver : {\bf P} H &\hookrightarrow& {\bf P} S^2 H\\
\xi &\mapsto& \xi\otimes \xi,\\ \end{array} $$ where $H =
H^1(D)$. In other words $$ \secant^{n} (\ver {\bf P} H) = \{\ a \in {\bf P} S^2
H\ |\ {\rm rank \ } a \leq n\ \} $$ for $n= 1,\ldots ,\dim H = r(D) -d+g$. On
the other hand, the homomorphism $\delta$ is dual to the multiplication
map $ S^2 H^0(K-D) \rightarrow H^0(2K-2D) $ and so the above Veronese
embedding fits into the following commutative diagram:
\begin{equation} \label{verdiagram} \begin{array}{rcl} C & \map{|K-D|}
& {\bf P} H^1(D) \\ &&\\ \scriptstyle|2K-2D| \displaystyle\downarrow
&&\downarrow \ver \\ &&\\ {\bf P} \ext^1(K-D,D) &\map{\delta} & {\bf P} S^2
H^1(D) \\ &&\\ \eee_D \downarrow &&\\ &&\\ {\cal W}_D &&\\ \end{array}
\end{equation} We now define: $$ \begin{array}{rcl} \Omega^0_D &=& {\bf P}
\ker \delta, \\ \Omega^n_D &=& \delta^{-1} (\secant^n(\ver {\bf P}
H^1(D))), \quad n= 1,\ldots , g-d+r(D).\\ \end{array} $$ (When it is
convenient we shall drop the subscript and write $\Omega^n
=\Omega^n_D$.) Thus if $\Omega^0$ is nonempty then $\Omega^0 \subset
\Omega^1 \subset \cdots \subset \Omega^{g-d+r(D)} \subset {\bf P}
\ext^1(K-D,D)$ is a sequence of {\it cones} with vertex $\Omega^0$.
We can therefore state the main conclusion of this section as follows:
\begin{equation} \label{conethm} h^0(E) = g+1-\cliff(D)-n \qquad
\hbox{for $E\in \Omega_D^n \backslash \Omega_D^{n-1}$}. \end{equation}
\begin{ex} \label{d=g-2} $\bf d=g-2.$ \rm If $D\in \pic^{g-2}(C)$ then
${\bf P} (D) = {\bf P} \ext^1(K-D,D)$ is a $g$-plane of the ruling of section
1. The map $\eee_D: {\bf P}(D) \hookrightarrow {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is a linear
embedding, and we shall not distinguish between ${\bf P} (D)$ and its
image. In this case (\ref{conethm}) says: $$ {\cal W}_D \cap {\cal W}^r =
\Omega^{2r(D) + 2 -r}\subset {\bf P}(D), \qquad r= r(D), \ldots , 2r(D) +
2. $$ The cones $\Omega^n$ are constructed using $\delta :
\ext^1(K-D,D) \rightarrow S^2 H^1(D)$ where $\dim \ext^1(K-D,D) = g+1 $
and $\dim H^1(D) = h^0(D) + 1$.
If $h^0(D) =0$ then $\dim S^2 H^1(D) = 1$ and $\Omega^0 \subset
\Omega^1 = {\bf P}(D)$ is a hyperplane, on which $h^0(E) = 1$. In other
words $\Omega ^0 = {\bf P}(D) \cap {\cal W}$.
If $h^0(D) = 1$ then $\dim S^2 H^1(D) = 3$. Since $|K-D|$ is a pencil,
$S^2 H^0(K-D) \rightarrow H^0(2K-2D)$ is necessarily injective and so
again $\delta$ is surjective. In this case, therefore: $$
\begin{array}{rclr} \Omega^0 &\cong& {\bf P}^{g-3} &\hbox{on which $h^0(E)
= 3$,}\\ \Omega^1 &=& \hbox{quadric of rank 3,} &h^0(E) = 2,\\ \Omega^2
&= & {\bf P}(D),& h^0(E) =1.\\ \end{array} $$
If $h^0(D) = 2$ then the series $|K-D|$ maps $f: C\rightarrow {\bf P}^2$
with degree $g$; and the homomorphism $\delta$ is no longer surjective
in general. In fact surjectivity fails precisely when $f$ maps $C$
onto a conic, which cannot happen if $g$ is odd, but can occur for a
trigonal curve of genus 6, for example: if $|L| = g^1_3$ then take
$D=K-2L$. In case $\delta$ {\it is} surjective we have: $$
\begin{array}{rclr} \Omega^0 &\cong& {\bf P}^{g-6} &\hbox{on which $h^0(E)
= 5$,}\\ \Omega^1 &=& \hbox{cone over a Veronese}&\\ &&\hbox{surface in
${\bf P}^5$,} &h^0(E) = 4,\\ \Omega^2 &=& \hbox{cone over cubic} &\\
&&\hbox{hypersurface $S^2 {\bf P}^2 \hookrightarrow {\bf P}^5$,} &h^0(E) = 3,\\
\Omega^3 &= & {\bf P}(D),& h^0(E) =2.\\ \end{array} $$ And so on. \end{ex}
The `universal' case of (\ref{conethm}) is the case $D=0$. This says
that ${\cal W}^r$ is composed of the image of $\Omega^{g-r} \subset {\bf P}
\ext^1(K,\oo)\cong {\bf P}^{3g-4}$ together with those of the corresponding
cones in the exceptional divisors of the blow-up of
remark~\ref{bertram}: $$ {\cal W}^r = \bigcup_{D\geq 0\atop \deg D \leq
g-2} \eee_{D} \Omega^{g-r-\cliff(D)}. $$ Diagram (\ref{verdiagram})
becomes in this case:
\begin{equation} \label{verdiagram2} \begin{array}{rcccl} &&C&
\map{|K|} & {\bf P}^{g-1}\\ &&&&\\ &&\scriptstyle|2K|
\displaystyle\downarrow && \downarrow \ver\\ &&&&\\
\Omega^{g-r}&\subset & {\bf P}^{3g-4}&\map{\delta}& {\bf P} S^2 H^1(\oo)\\
&&&&\\ \downarrow &&\eee_0\downarrow &&\\ &&&&\\ {\cal W}^r & \subset& {\cal W}
&&\\ \end{array} \end{equation}
\begin{rem}\rm Note that if $C$ is nonhyperelliptic then by Noether's
theorem $\delta$ in (\ref{verdiagram2}) is injective. Then $\Omega^n$
is the intersection of ${\bf P}^{3g-4}\subset {\bf P} S^2 H^1(\oo)$ with the
secant variety $\secant^n (\ver {\bf P}^{g-1})$, and in particular contains
$\secant^n C\subset {\bf P}^{3g-4}$. One can show, in fact, that Green's
conjecture on the syzygies of the canonical curve (see \cite{GL})
implies: $$ \Omega^n = \secant^nC \subset {\bf P}^{3g-4} \qquad \hbox{for
$n<\cliff(C)$.} $$ One consequence of this statement is that
$\Omega^{\cliff(C)}$ is the smallest cone in the sequence containing
semistable extensions. \end{rem}
We conclude this section with two inequalities. The first, which will
be useful later, is due to Mukai (\cite{Muk}, proposition 3.1):
\begin{lemm} \label{muk} If $|D|$ is base-point-free then for any rank
2 vector bundle $E$ with $\det E = K$ we have $ h^0(E(-D)) \geq h^0(E)
- \deg D. $ \end{lemm}
\begin{prop} \label{bound} For all semistable bundles $E$ in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$
we have $$ h^0(E) \leq g+1-\cliff(C). $$ \end{prop}
\begin{rem}\rm In particular, this bound becomes: $$ h^0(E)\leq
\cases{g & for nonhyperelliptic $C$,\cr
g-1 & for $C$ not trigonal or a plane quintic,\cr \cdots&\cr
[g/2] + 2& for generic $C$.\cr} $$ The bound $h^0(E) \leq g$ for
nonhyperelliptic curves was observed by Laszlo \cite{L}, proposition
IV.2. \end{rem}
{\it Proof of proposition \ref{bound}.} We may assume $E$ comes from an
extension in ${\bf P} \ext^1(K-D,D)$ where, by remark \ref{nag} (i) $$
\Bigl[ {g-1 \over 2} \Bigr] \leq \deg D \leq g-1. $$ The right-hand
inequality implies $h^0(D) \leq h^1(D)$ so that if $h^0(D)\geq 2$ then
$\cliff(D) \geq \cliff(C)$ by definition. If, on the other hand,
$h^0(D) \leq 1$ then $\cliff(D) \geq \deg D \geq \cliff(C)$ by the
left-hand inequality together with remark \ref{nag} (ii). In either
case, therefore, the proposition follows from (\ref{h^0(E)}). {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\section{$\g00$}
We shall as usual identify ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with its image in $|2\Theta|^{\vee}} \def\cctil{\ctil_{\eta}
= |\ll|$ where $\ll = \oo(2\Theta_{\kappa}) \in \pic(J_C)$ for any
theta characteristic $\kappa \in \vartheta(C)$. Namely, a stable bundle
$E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is mapped to the divisor $D_E \in |\ll|$ defined by $$
D_E = \{ \ L\in J_C\ |\ h^0(C,L\otimes E)>0\ \}. $$ On the other hand,
one can consider the linear system $\g00\subset |\ll|$ defined by:
\begin{equation} \begin{array}{rcl} \g00 &=& \{\ D\in |\ll|\ |\ {\rm mult}} \def\spin{{\rm Spin}} \def\pin{{\rm Pin}_0
D \geq 4\ \}\\
&=& \{\ D\in |\ll|\ |\ C-C \subset {\rm supp}\ D\ \}.\\
\end{array} \end{equation} For the equivalence of these two definitions
see \cite{vGvdG} or \cite{W}; one can show, in addition, that $\g00$
has codimension $1+{1\over 2}g(g+1)$.
It is easy to verify that the Brill-Noether locus ${\cal W}^2$ is always
contained in the subspace $\g00$. The main result of this section is a
partial converse:
\begin{theo} \label{w2g00} ${\cal W}^2 \subset \g00 \cap {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K) \subset
{\cal W}^1$. Moreover, if $C$ is nonhyperelliptic of genus 4 or nontrigonal
of genus 5 then ${\cal W}^2 = \g00 \cap {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$. \end{theo}
\begin{rem}\rm We shall show later that ${\cal W}^2 \not= \g00 \cap
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ for curves of genus 6 (see remark \ref{21dimspan}). For genus
4 the embedding ${\cal W}^2 \subset \g00$ will be described in theorems
\ref{w2cubic} and \ref{quartics}.
\end{rem}
\begin{lemm} \label{4.3} Suppose $F$ is a semistable vector bundle of
rank 2 and degree $2d$ where $0\leq d \leq g-1$; and $k\geq 0$ an
integer. Then $h^0(F) \geq k$ if and only if $h^0(F(D)) \geq k $ for
all $D\in S^{g-1-d} C$. \end{lemm}
Before proving this lemma let us show how it implies theorem
\ref{w2g00}. We suppose that $C-C \subset D_E$ for a stable bundle
$E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$, and we show first that $h^0(E)\geq 2$: by hypothesis
$h^0(E(p-q)) \geq 1$ for all $p,q\in C$, so by the lemma we deduce that
$h^0(E(-q)) \geq 1$ for all $q\in C$. From this it follows that $h^0(E)
\geq 2$, since $h^0(E) \geq 1$ and equality would imply that every
section vanishes at arbitrary $q\in C$, a contradiction.
Now suppose that $h^0(E) = 2$ and consider the evaluation map $e_q :
H^0(E) \rightarrow E_q$ for $q\in C$. Since $h^0(E(-q)) \geq 1$ we have
${\rm rank \ } e_q \leq 1$ for all $q\in C$, and hence the sections of $E$
generate a line subbundle $L\subset E$ with $h^0(L) =2$. But by
stability of $E$ this must satisfy $\deg L < g-1$ so $C$ admits a
$g^1_{g-2}$. So if $C$ is nonhyperelliptic of genus 4 or is
nontrigonal of genus 5 we obtain a contradiction, and we conclude that
$h^0(E) \geq 3$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
For the lemma, we shall prove the following equivalent statement. Let
$E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,\oo)$ and $\xi \in \pic^d(C)$, $0\leq d \leq g-1$. Then
\begin{equation} \label{killingD} H^0(C,\xi \otimes E)\geq k \quad
\Longleftrightarrow \quad {H^0(C,\xi(D) \otimes E)\geq k \atop
\forall\,\, D\in S^{g-1-d}C.} \end{equation}
We shall introduce a {\it spectral curve} $q: B= B_s\rightarrow C$ (see
\cite{BNR}). Namely, $B_s$ is the subscheme of the total space of the
canonical line bundle $K \map{q} C$ with equation $x^2 = s$, where
$s\in H^0(C,K^2)$ is a generic section. This is a smooth double cover
of $C$ of genus $g_B = 4g-3$, and there is a dominant rational map of
finite degree of the Prym variety on to ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2)$: $$
\begin{array}{rcl} Q_s = \nm_q ^{-1}(K) &\rightarrow& {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2)\\ \zeta
&\mapsto & q_*\zeta.\\ \end{array} $$ Moreover, the images of these
rational maps cover the moduli space as the section $s$ varies, and so
for any $E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2)$ we may assume that $E=q_* \zeta$ for some line
bundle $\zeta\in Q_s$, for suitable $s\in H^0(C,K^2)$.
By the projection formula the left-hand side of (\ref{killingD}) is $$
H^0(C,\xi \otimes E) = H^0(B_s, L) \qquad \hbox{where $L = \zeta
\otimes q^*\xi$.} $$ Notice that for $d<g-1$, $\deg L = 2g-2+2d \leq
4g-6 = g_B -3$, and in particular the Serre dual linear series $|K_B
L^{-1}|$ is base-point-free for generic $\zeta\in Q_s$. By choosing $s$
generically we may assume, for any given $E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2)$ and $\xi \in
J^d$, that this is the case.
We shall need:
\begin{lemm} Suppose, for $q:B\rightarrow C$ a double cover as above,
that $|N|$ is a base-point-free linear series on $B$. Then {\it either}
$N=q^*N'$ for some $N'\in \pic\ C$ {\it or} $$ h^0(B, N\otimes
q^*\oo(-x)) = h^0(B,N) - 2 $$ for generic $x\in C$. \end{lemm}
{\it Proof.}\ Write $q^{-1}(x) = x_1 + x_2$ with $x_1 \not= x_2$. Then $h^0(B,N)
- h^0(B,N(-x_1-x_2)) $ is the rank of the evaluation map $H^0(B,N)
\rightarrow \c_{x_1} \oplus \c_{x_2}$; and either this rank is 2 for
generic $x\in C$ or it is $\leq 1$ for all $x\in C$. In the latter
case, the base-point-free hypothesis ensures that the image of the
evaluation map is not contained in either summand; this implies that
every divisor in $|N|$ is symmetric, so $N= q^*N'$ as asserted. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
We shall want to apply the lemma to $N = K_B L^{-1}$; we begin by
observing that this line bundle cannot be symmetric, as follows. Since
$K_B = q^* K_C^2$ and $L= \zeta \otimes q^*\xi$, $N =q^*N'$ would imply
that $\zeta = q^* \tau$ for some $\tau \in \pic\ C$. But then $$ E= q_*
\zeta = \tau \otimes q_* \oo_B = \tau \oplus K_C^{-1} \tau, $$
violating semistability.
So finally, consider the right-hand side of (\ref{killingD}). By the
projection formula this space is $$ H^0(C,\xi(D) \otimes E) = H^0(B,
L\otimes q^*\oo(D)). $$ We note that $\deg L\otimes q^*\oo(D) = 4g-4 =
g_B -1$, so by Riemann-Roch $$ h^0(C,\xi(D) \otimes E) = h^0(B, K_B
L^{-1}\otimes q^*\oo(-D)). $$ We now apply the lemma $e=g-1-d$ times
to $N = K_B L^{-1}$: this gives, for $D\in S^e C$ generic, $
h^0(C,\xi(D) \otimes E) = h^0(B, K_B L^{-1}) - 2e$.
\medskip {\it Proof of (\ref{killingD}).} Assuming the right-hand side
we have, by the last remark and by choosing $D$ generically, $h^0(B,
K_B L^{-1}) \geq k + 2e$. Consequently $$ \begin{array}{rcl} h^0(C,\xi
\otimes E) &=& h^0(B,L)\\ &=& h^0(B, K_B L^{-1}) + \deg L - g_B + 1 \\
&\geq& k+ 2e +\deg L - g_B + 1 \\ &=& k.\\ \end{array} $$ The converse
is trivial. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\vfill\eject \bigskip\noindent {\large\bf III Low genera}
\section{Genus 3}
In this section we shall take $C$ to be a nonhyperelliptic curve of
genus 3. Then ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is the {\it Coble quartic} associated to the
Kummer variety in ${\bf P}^7$ (see~\cite{NR} and \cite{C}, \S33). It is
well-known that in this case the 3-plane ruling $\eee : {\bf P} U
\rightarrow {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ of section 1 is surjective and has degree~8.
This follows easily from remark \ref{nag} (i) and lemma \ref{ln}.
The behaviour of $h^0(E)$ in each 3-plane of the ruling is given by
example~\ref{d=g-2}. Namely, if $h^0(x)=0$ then ${\cal W}\subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$
cuts ${\bf P}(x)$ transversally in a 2-plane along which $h^0(E) = 1$,
while:
\begin{equation} \label{g=3,w1} {\cal W}^1 = \bigcup_{p\in C} \Omega^1_p,
\qquad \Omega^1_p = \hbox{quadric cone} \subset {\bf P}(p). \end{equation}
In a moment we shall show that the vertices $\Omega^0_p$ of these cones
all coincide at a single point of ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ (see \ref{w2point} and
\ref{coblecone}).
\begin{rem} \label{x=etap} \rm Note that for each $p\in C$ the image of
$C$ in ${\bf P}(p)$ lies on the cone $\Omega^1_p$; and projecting along the
generators is the trigonality $f:C\rightarrow {\bf P}^1$ given by the
series $|K(-p)|$. (Conversely, one may show that the image of $C$ in a
3-plane ${\bf P}(x)$ lies on a quadric cone only if $x= \eta(p)$ for some
$p\in C$ and some square root of the trivial line bundle, $\eta^2 =
\oo$.) \end{rem}
We consider now the birational map $\eee_0 : {\bf P} \ext^1(K, \oo) = {\bf P}^5
\rightarrow {\cal W} \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ and the diagram (\ref{verdiagram2}).
The map $\delta$ is an isomorphism: its dual $S^2 H^0(K) \rightarrow
H^0(2K)$ is surjective by Noether's theorem and injective since the
canonical curve $C\subset {\bf P}^2$ is not contained in any quadric. Thus
$\Omega^1\subset {\bf P}^5$ is the Veronese surface; and it is well-known
that its variety of secant lines $\Omega^2$ is a cubic hypersurface
isomorphic to $S^2 {\bf P}^2$. Thus we have a tower of rational maps, where
$\eee_0$ is given by the complete linear series $|\iic(2)|$ on ${\bf P}^5$
(see remark~\ref{bertram}):
$$ \begin{array}{rcl} {\rm Bl}} \def\cliff{{\rm Cliff}} \def\secant{{\rm Sec}_C({\bf P}^5)&&\\ \downarrow&\searrow&\\ {\bf P}^5 =
\Omega^3 &\map{\eee_0} & {\cal W}\\ |&&|\\ S^2 {\bf P}^2 = \Omega^2
&\longrightarrow& {\cal W}^1\\ |&&|\\ C\subset \ver({\bf P}^2)= \Omega^1
&\longrightarrow& {\cal W}^2\\ \end{array} $$ First of all, this allows us
to recover the following result of Laszlo \cite{L} and
Paranjape--Ramanan \cite{PR}. The bundle $V$ appearing here is simply
the normal bundle of $C$ canonically embedded in its Jacobian.
\begin{prop} \label{w2point} ${\cal W}^2$ consists of a single point, i.e.
there is a unique stable bundle $V\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with $h^0(V) =3$.
\end{prop}
{\it Proof.}\ Since the canonical curve $C\subset {\bf P}^2$ has degree 4 any quadric
of the series $|\iic(2)|$ either contains the Veronese surface
$\Omega^1$ or has no further points of intersection. Thus $\Omega^1$
contracts to a single point $V\in{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ under~$\eee_0$.
On the other hand, for each $p\in C$ the 3-plane ${\bf P}(p) \subset
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is the image of the fibre ${\bf P} N_{C/{\bf P}^5}$ of the
exceptional divisor in the blow-up, by remark \ref{bertram}. Making
this identification the point $\Omega^0_p\in {\bf P}(p)$ is the normal
direction of $\Omega^1=\ver({\bf P}^2) \supset C$, and is therefore
contained in the closure of the image of $\Omega^1$. In other words
$\Omega_p^0 = \{V\}$, and since we've seen that there are no further
points of ${\cal W}^2$, this completes the proof. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
We now wish to give a geometric description of ${\cal W}^1 \subset
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$, and to this end we consider again the space $\ff$ of Fay
trisecants of the Kummer. Notice that for genus 3 there is a natural
inclusion of the canonical series $$ |K| \hookrightarrow \ff
\ \map{J[2]}\ S^4 C $$ given by $D\mapsto (0,D)$ (see section
\ref{trisecants}). For $D\in |K|$ let us denote the corresponding
trisecant by $t_D \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$. If $D=p+q+r+s$ then by the proof
of theorem \ref{fay}, $t_D$ lies in the four 3-planes
${\bf P}(p),{\bf P}(q),{\bf P}(r),{\bf P}(s)$; in ${\bf P}(p)$, $t_D$ is the trisecant line
$\overline{qrs}$, and similarly in the other three spaces. By remark
\ref{x=etap}, on the other hand, this line is a generator of the cone
$\Omega^1_p \subset {\bf P}(p)$, and conversely every generator is such a
trisecant. By (\ref{g=3,w1}), therefore, we conclude that
\begin{equation} {\cal W}^1 = \bigcup_{D\in |K|}t_D \subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K).
\end{equation}
From this we obtain the following description of ${\cal W}^1$.
\begin{theo} \label{coblecone} The subvariety ${\cal W}^1\subset
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)\subset {\bf P}^7 $ has the following structure: \begin{enumerate}
\item ${\cal W}^1$ is a cone over the Veronese surface $|K| = {\bf P}^2
\map{|\oo(2)|} {\bf P}^5$; \item ${\cal W}^1$ has point vertex ${\cal W}^2\in
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$; \item ${\cal W}^1$ intersects the Kummer variety in the theta
divisor ${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(\Theta)$, and projection along the generators of the cone
coincides with the 3 to 1 Gauss map ${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(\Theta) \rightarrow |K|$.
\end{enumerate} \end{theo}
{\it Proof.}\ We have already seen that each trisecant $t_{p+q+r+s}$, where
$p+q+r+s \in |K|$, passes through the point $V\in {\cal W}^2$. Assigning to
the divisor $p+q+r+s$ the tangent direction of $t_{p+q+r+s}$ at $V \in
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ therefore defines an injective map $$ \pi : |K| \rightarrow
{\bf P} T {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)|_{ V} \cong {\bf P}^5; $$ for which $\pi^*\oo(1) = \oo(2)$
on the pencils $|K(-p)|\subset |K|$, and hence on the whole plane.
Parts 1 and 2 of the theorem follow straightaway.
From (\ref{5}) we see that the trisecant $t_{p+q+r+s}$ meets the Kummer
in the three points
$$ \begin{array}{c} \oo(p+s)\oplus \oo(q+r), \\ \oo(q+s)\oplus
\oo(p+r), \\ \oo(r+s)\oplus \oo(p+q); \\ \end{array} $$ which is
equivalent to part 3. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm The Veronese cone of theorem \ref{coblecone} appears in
the work of Coble (\cite{C}, \S48). In particular, Coble exhibits a
uniquely determined cubic hypersurface in ${\bf P}^6$ which cuts out
${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(\Theta)$ on the cone ${\cal W}^1$. It would be interesting to
interpret this cubic in terms of vector bundles. \end{rem}
Finally, we shall sketch two proofs of the following fact.
\begin{theo} At the triple point ${\cal W}^2= \{V\}$ the theta divisor ${\cal W}$
has projectivised tangent cone $ {\bf P} T_V {\cal W} \cong \Omega^2 = S^2 {\bf P}^2
\subset {\bf P}^5 $. \end{theo}
{\it First proof.} Since $V$ is stable we can identify $T_V {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$
with $H^1(C,\ad V)$. We have already remarked (\ref{petri}) that since
$\det V = K$ the Petri map factors through the symmetric product $S^2
H^0(C,V) \rightarrow H^0(C,K\otimes \ad V)$; and this is dual to a map
$$ \mu: H^1(C,\ad V) \rightarrow S^2 H^0(C,V)^{\vee}} \def\cctil{\ctil_{\eta} \subset {\rm
Hom}(H^0(V),H^1(V)). $$ By standard Brill-Noether type arguments the
tangent cone $T_V {\cal W}$ is the pull-back under $\mu$ of the
homomorphisms with nontrivial kernel (see for example \cite{L}). On the
other hand, one can show that $\mu$ is an isomorphism in the present
case. The tangent cone is therefore precisely the locus of singular
quadratic forms on $H^0(C,V)$, and hence isomorphic to $\Omega^2 = S^2
{\bf P}^2$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
{\it Second proof} (due to B. van Geemen). This exploits the fact that
the hypersurface ${\cal W} \subset {\bf P}^6$ has degree 4 (since it is a
hyperplane section of the Coble quartic), while $V$ is a triple point
of ${\cal W}$ (\cite{L} proposition IV.7). It follows that ${\bf P} T_V {\cal W} $ is
the complement in ${\bf P}^5$ of the (Zariski open) image of ${\cal W}$ under
the rational projection map away from the point $V$.
We consider, then, the following diagram:
\begin{equation} \begin{array}{crccl} &&{\cal W}&\subset & {\bf P}^6\\ &&&&\\
&\hidewidth\eee_0 = \lambda_{|\iic(2)|}\nearrow&&\searrow&\downarrow
\pi_{V}\\ &&&&\\ {\bf P}^5&&\map{\Delta}&&{\bf P}^5\\ \end{array}
\end{equation} We have seen that ${\cal W}$ is the (closed) image of ${\bf P}^5$
under the rational map $\eee_0$ given by the complete linear series of
quadrics through the bicanonical curve, contracting $\ver({\bf P}^2)$ down
to the point $V$. Thus the rational map $\Delta$ is given by the
complete linear series of quadrics containing $\ver({\bf P}^2)$. It is
well-known that this can be identified with the inversion map of
symmetric $3\times 3$ matrices (geometrically, it sends a plane conic
to its dual conic). $\Delta$ is a birational involution, blowing up the
locus $\Omega^1 = \ver({\bf P}^2)$ of rank 1 conics and contracting the
exceptional divisor down to the locus $\Omega^2 = S^2 {\bf P}^2$ of rank 2
(dual) conics.
The image of $\Delta$, and hence of $\pi_V|_W$, is therefore the
complement of $\Omega^2$ and we are done. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\section{Genus 4}
To any nonhyperelliptic curve of genus 4 one can associate in a
canonical way a nodal cubic threefold $\tt \subset {\bf P}^4$ as follows
(see \cite{D}). The canonical curve $C\subset {\bf P}^3$ lies on a unique
quadric surface $Q\subset {\bf P}^3$ and is base-locus of a 4-dimensional
linear system of cubics; we define $\tt \subset {\bf P}^4$ to be the image
of the rational map $$ \lambda_{|\iic(3)|} : {\bf P}^3 \rightarrow {\bf P}^4.
$$ Let us denote by $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing},\trigg \in \Theta \subset \pic^3(C)$ the two
trigonal line bundles on the curve. These satisfy $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}\otimes \trigg
= K$ and coincide precisely when the curve has a vanishing theta-null.
The quadric surface $Q$ is ruled by trisecants $\overline{D}\subset
{\bf P}^3$ of the curve, where $D$ belongs to $|g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}|$ or $|\trigg|$ (and
$Q$ is singular precisely when the two pencils coincide); it therefore
contracts to a point $t_0 \in \tt$ under $\lambda_{|\iic(3)|}$. Any
hyperplane through $t_0 \in {\bf P}^4$ then pulls back to $Q$ plus a
residual hyperplane, and it follows that projection away from the point
$t_0$ is the birational inverse of $ \lambda_{|\iic(3)|}$:
\begin{equation} \begin{array}{crccl} &&\tt&\subset & {\bf P}^4\\ &&&&\\
&\hidewidth\lambda_{|\iic(3)|}\nearrow&&\searrow&\downarrow \pi_{t_0}\\
&&&&\\ {\bf P}^3&&=&&{\bf P}^3\\ \end{array} \end{equation} This allows us to
see that $\tt$ is a cubic: a general hyperplane $H\subset {\bf P}^4 =
|\iic(3)|^{\vee}} \def\cctil{\ctil_{\eta}$ identifies with ${\bf P}^3$ under the projection
$\pi_{t_0}$, and under this identification its intersection with $\tt$
is the cubic surface corresponding to the point of $|\iic(3)|$
annihilated by $H$.
\begin{prop} ${\rm mult}} \def\spin{{\rm Spin}} \def\pin{{\rm Pin}_{t_0} \tt=2 $ and the projectivised tangent cone
at this point is ${\bf P} T_{t_0} \tt = Q \subset {\bf P}^3$. \end{prop}
{\it Proof.}\ That ${\rm mult}} \def\spin{{\rm Spin}} \def\pin{{\rm Pin}_{t_0} \tt=2 $ follows at once from the fact that the
projection $\pi_{t_0}: \tt \rightarrow {\bf P}^3$ is birational and $\deg
\tt =3$. On the other hand, let $H\subset {\bf P}^4$ be any hyperplane
passing through $t_0$ and $H' = \pi_{t_0}(H) \subset {\bf P}^3$ its
projection. Then $H\cap \tt$ is the cubic surface obtained by blowing
up the six points $C\cap H' \subset {\bf P}^2$; these six points lie on the
conic $Q' = Q\cap H'$ and it is well-known that the resulting cubic
surface is nodal with projectivised tangent cone $Q'\subset {\bf P}^2$ at
the node. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm
From this we can easily write down an equation for $\tt$: choosing a
simplex of reference with $t_0 \in {\bf P}^4$ as one vertex, and the
opposite face corresponding to a choice of cubic surface $F\in
|\iic(3)|$, the threefold $\tt$
has equation $$ z_0 Q(z_1,\ldots ,z_4) + F(z_1,\ldots ,z_4) = 0. $$
This is the description given by Donagi. \end{rem}
We shall need next the Fano surface $F(\tt)$ of lines on $\tt$, which
is easy to describe. First note that there is an inclusion $$
\begin{array}{rcl} i: C &\hookrightarrow& F(\tt) \\ p&\mapsto& l_p =
\hbox{line joining $t_0$ to $p\in C\subset {\bf P}^3$.}\\ \end{array} $$ In
other words, $l_p\subset {\bf P}^4$ is the line joining $t_0$ to the point
$p$ on the canonical curve via the projection $\pi_{t_0}$, and it is
easy to see that these are precisely the lines through $t_0 \in {\bf P}^4$
which lie on $\tt$. We now map $$ \begin{array}{rcl} f: S^2 C
&\rightarrow& F(\tt) \\ p+q&\mapsto& l_{pq} = \hbox{residual line in
$\tt\cap {\rm Span}\{l_p,l_q\}$.}\\ \end{array} $$ (Note that this
still makes sense on the diagonal of $S^2 C$: if $p=q$ then ${\rm
Span}\{l_p,l_q\}$ is interpreted to mean the 2-plane spanned by $t_0$
and the tangent line to the canonical curve at $p\in C$.)
If the secant line $\overline{pq}\subset {\bf P}^3$ is not on $Q$ then
$\lambda_{|\iic(3)|}(\overline{pq}) = l_{pq}$; whilst if $\overline{pq}
\subset Q$ then it contracts down to $t_0$, but $l_{pq} = l_r$ where
$r\in \overline{pq}\cap C$ is the third point of the trisecant. Thus
$f$ is a birational morphism and is injective away from the two curves
$C\hookrightarrow S^2 C$ defined by $r\mapsto g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}(-r)$ and $r\mapsto
\trigg(-r)$, each of which it identifies with $i(C)$:
\begin{equation} \label{pic} figure
\end{equation}
Izadi \cite{I} makes use of the lines on $\tt$ to identify $\tt \subset
{\bf P}^4 \mathbin{\hbox{$\widetilde\rightarrow$}}} \def\ext{{\rm Ext} \g00$ (theorem \ref{izadi} below). Namely, for $r\in C$
and for $p+q\in F(\tt)\backslash i(C)$ (which we identify with the
corresponding subset of $S^2 C$ as above) she constructs pencils which
we shall denote by $l'_r,l'_{pq} \subset \g00$ respectively. These are
characterised by their base locus: for any $p,q \in C$ let
\begin{equation} \Sigma_{pq} = C-C \cup W_2 -p-q \cup p+q-W_2 \subset
J_C. \end{equation} Then the pencil $l'_r\subset \g00$ has base locus
$\Sigma_{pq}\cup \Sigma_{p'q'}$ where $f^{-1}(i(r)) = \{p+q,p'+q'\}$,
i.e. $p+q+r\in |g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}|$ and $p'+q'+r \in |\trigg|$; and the pencil
$l'_{pq} \subset \g00$ has base locus $\Sigma_{pq} \cup \Sigma(X)$
where (in Izadi's notation---see \cite{I}, \S7)
\begin{equation} \label{Sigma(X)} \begin{array}{rccl} \Sigma(X) &=&
\{s+t-s'-t'\ |& s,t,s',t'\in C,\\ &&& h^0(K-p-q-s-t)>0,\\
&&&h^0(K-p-q-s'-t')>0\}.\\ \end{array} \end{equation} (In this notation
$X$ denotes a curve of genus 5 in the fibre of the Prym map over $J_C$;
though this will not concern us here.)
Izadi's result, in our (nonhyperelliptic Jacobian) situation, can then
be stated as follows.
\begin{theo} \label{izadi} Let $C\subset {\bf P}^3$ be a canonical curve of
genus 4, and ${\bf P}^4 = |\iic(3)|^{\vee}} \def\cctil{\ctil_{\eta}$ be the ambient space of its
associated cubic threefold $\tt$. Then there is a natural
identification ${\bf P}^4 \mathbin{\hbox{$\widetilde\rightarrow$}}} \def\ext{{\rm Ext} \g00$ under which $l_{r} \mathbin{\hbox{$\widetilde\rightarrow$}}} \def\ext{{\rm Ext} l'_{r}$,
$l_{pq} \mathbin{\hbox{$\widetilde\rightarrow$}}} \def\ext{{\rm Ext} l'_{pq}$ and the node $t_0\in \tt$ maps to $\Theta-g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}
\cup \Theta-\trigg$. \end{theo}
We now return to consider the Brill-Noether loci in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)\subset
{\bf P}^{15}$ and to state our main result. Recall that ${\cal W}^2 = \g00 \cap
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ (by theorem~\ref{w2g00}).
\begin{theo} \label{w2cubic} If $C$ is a nonhyperelliptic curve of
genus 4 then ${\cal W}^2\subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)\subset {\bf P}^{15}$ is precisely the
Donagi-Izadi cubic threefold $\tt \subset \g00 = {\bf P}^4$; with node at
$g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg \in {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^3)$. \end{theo}
\begin{rem}\rm \label{w3node} Note that (up to S-equivalence) $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}
\oplus \trigg$ is the unique semistable bundle in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with $h^0
= 4$, and so ${\cal W}^3$ is by definition empty. This is a consequence of
Mukai's lemma~\ref{muk}: since $|g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}|$ is base-point-free, $h^0(E)
\geq 4$ would imply that $h^0({g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}}^{-1}\otimes E) \geq 1$, and hence
by semistability that $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \subset E$. So $E$ is S-equivalent to
$g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus K{g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}}^{-1} = g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg$.
(In fact, if $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \not= \trigg$ then this is an isomorphism since
$\trigg \subset E$ by the same argument. If, on the other hand, $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}
= \trigg$ then one can check using the arguments of section 3 that as
well as $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}$ there is, up to isomorphism, a unique
nonsplit extension $E$ with $h^0(E) =4$: the space of all such
extensions is ${\bf P} (g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}) = {\bf P} H^0(K)^{\vee}} \def\cctil{\ctil_{\eta}$, in which the canonical
curve lies on a quadric cone. $E$ is then the extension corresponding
to the vertex of the cone.)
Thus it makes sense to view ${\cal W}^3 = \{g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg\}$.
\end{rem}
\begin{lemm} Suppose that $C$ is nonhyperelliptic of genus 4 or
nontrigonal of genus 5. Then for every stable $E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with
$h^0(E)=3$ the exterior multiplication map $\phi_E: \bigwedge^2 H^0(E)
\rightarrow H^0(K)$ is injective. \end{lemm}
{\it Proof.}\ Since every element of $\bigwedge^2 H^0(E)$ is decomposable, i.e.
of the form $s\wedge t$, a nontrivial element of $\ker \phi_E$ would
give two independent sections $s,t$ generating a line subbundle
$L\subset E$. Then $r(L) \geq 1$ while by stability $\deg L \leq g-2$,
contrary to the hypotheses on $C$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
In the genus 4 case the lemma determines a rational map (defined away
from the single point $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg$) $ \pi: {\cal W}^2 \rightarrow
{\bf P}^3 = |K|^{\vee}} \def\cctil{\ctil_{\eta} $ sending $E\mapsto {\rm im \ }} \def\sym{{\rm Sym}} \def\ker{{\rm ker \ } \phi_E \subset |K|$. In
proving theorem \ref{w2cubic} we shall in fact prove slightly more,
namely that the following diagram commutes (and we shall also extend
this diagram in theorem \ref{quartics} below):
\begin{equation} \label{nodalcubic} \begin{array}{rcl} \tt & = & {\cal W}^2
\subset \g00\\ & \hidewidth\pi_{t_0} \searrow \qquad\swarrow \pi
\hidewidth&\\ & {\bf P}^3 &\\ \end{array} \end{equation}
\begin{prop} \label{6.8} For each $p\in C \subset {\bf P}^3$ the closure of
the fibre $\pi^{-1}(p)\subset {\cal W}^2$ is a Hecke line $l_F$ with $\det F
=K(p)$. Moreover, $l_F$ is the unique such Hecke line contained
in~${\cal W}^2$ and passing through the point $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg$.
\end{prop}
{\it Proof.}\ Consider a stable bundle $E\in \pi^{-1}(p)$. By definition the
sections of $E$ fail to generate $E$ at the point $p$, and we denote by
$D_p\subset E_p$ the line in the fibre at $p$ which is generated by
global sections. This line uniquely determines an extension $$
\ses{E}{F}{\c_p}, $$ and by construction the coboundary map $\delta :
\c \rightarrow H^1(E)$ vanishes, so that $h^0(F) = 4$ and
$H^0(E)\subset H^0(F)$ coincides with the subspace $H^0(F(-p))$.
Finally, since $E$ is stable $F$ is stable. This proves the first part
of the proposition, with $l_F \subset {\cal W}^2$.
Now choose a section $s\in H^0(F)$ not lying in $H^0(E)$. Then $s(p)
\not= 0$ and spans a line in the fibre $F_p$; we consider a nonzero
homomorphism $u: F\rightarrow \c_p$ such that this lines coincides with
$\ker u_p$. Then by construction $\ker u \subset F$ is a semistable
bundle with $h^0(\ker u) =4$ and hence by remark~\ref{w3node} $\ker u =
g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg$ up to S-equivalence; and this point therefore lies
on $l_F$.
It remains to show that a Hecke line with these properties is unique.
If $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \not= \trigg$ then by remark \ref{w3node} $F$ has subsheaf
$g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \oplus \trigg$, and by \cite{B1} lemma 3.2 this determines the
Hecke line $l_F$ uniquely. Alternatively one can argue similarly to the
vanishing theta-null case, to which we shall now restrict.
So assume that $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} =\trigg$. By theorems \ref{10} and \ref{20}
$l_F$ lies in some 4-plane ${\bf P}(x)$, meeting the curve at the image of
a point $q\in C$: thus $x = g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}(-q)$. By the proof of \ref{10}, {\it
either} $p=q$ {\it or} $l_F$ is a trisecant $\overline D$ where $\oo(D)
=Kx^{-2}(p)$. On the other hand, the second case does not occur for the
following reason: by (\ref{verdiagram}) and (\ref{conethm}) in section
3, $h^0 \geq 3$ in ${\bf P} (x)$ precisely along a line $\Omega^0_x$,
projection away from which maps $C$ onto a plane conic via the linear
series $|g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}(q)|$ with the single base point $q$. Thus (since $h^0
\geq 3$ along $l_F$) $l_F = \Omega^0_x$ and meets the curve only at one
point (with multiplicity 2). Thus $l_F = \Omega^0_{g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}-p}$ and is
uniquely determined. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm In the case $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \not= \trigg$ one can show that $l_F$
is the intersection of the two 4-planes ${\bf P}(g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}(-p))$ and
${\bf P}(\trigg(-p))$ (in the notation of section 1), and in each space is
the tangent line to the curve at the image of $p\in C$. In the case
$g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} = \trigg$ just discussed in the above proof, the curve $C\subset
{\bf P} (g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}(-p))$ has a cusp at $p\in C$. The line $l_F$, passing
through $p$, is not the tangent line but is the vertex of a rank 3
quadric containing the curve. \end{rem}
\begin{prop} \label{6.9} For each $p\in C$ the Hecke line of the
previous proposition coincides with Izadi's pencil $l'_p = l_p =
\pi_{t_0}^{-1}(p)$. \end{prop}
{\it Proof.}\ Consider a stable bundle $E\in \pi^{-1}(p) = l_F$. We shall show
that $E\in l'_p$; since both sets are lines the result will follow.
So we have to show that the divisor $D_E = \{ L\in J_C | h^0(C,L\otimes
E)>0 \}$ contains the surfaces $\Sigma_{st}$ and $\Sigma_{s't'}$ where
$p+s+t \in |g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}|$ and $p+s'+t' \in |\trigg|$. (Note that for $E\in
{\cal W}^2$ we have $D_E\in \g00$ and hence $C-C \subset D_E$ a priori---see
section 4.) Since $D_E$ is symmetric it is enough to prove that $W_2
-s-t \subset D_E$, i.e. that $h^0(E(p+q-s-t)) >0$ for all $p,q\in C$.
We shall show that $h^0(E(-s-t))\geq 1$ (and note that by proposition
\ref{4.3} this is actually equivalent); and similarly that
$h^0(E(-s'-t'))\geq 1$.
By hypothesis ${\rm im \ }} \def\sym{{\rm Sym}} \def\ker{{\rm ker \ } \phi_E = H^0(K(-p))$; and we have a natural
2-dimensional subspace $V\subset H^0(K(-p))$, namely $$ V =
H^0(K(-p-s)) = H^0(K(-p-t)) = H^0(K(-p-s-t)). $$ So consider the
subspace $\phi_E^{-1}(V) \subset \bigwedge^2 H^0(E)$ and choose
sections $u,v,w \in H^0(E)$ such that $u\wedge v, u\wedge w$ form a
basis of $\phi_E^{-1}(V)$. Since $v\wedge w \not\in \phi_E^{-1}(V)$ the
effective divisor $(u\wedge w)\in |K|$ is not supported at $s$ or $t$;
this implies that the sections $v,w$ generate $E$ at the points $s,t\in
C$. However, by construction $s+t \leq (u\wedge v)$ and $s+t \leq
(u\wedge w)$; and we claim that this can only occur if $u(s) = u(t) =
0$. For if $u(s) \not= 0$, for example, then $\c u(s) = \c v(s)$ and
$\c u(s) = \c w(s)$ and hence $s \in {\rm supp}(v\wedge w)$, a
contradiction. Hence we obtain a nonzero section $u\in H^0(E(-s-t))$;
and similarly we can do the same for $H^0(E(-s'-t'))$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
Let us return to the proof of theorem \ref{w2cubic}. In seeking stable
bundles with three sections we may consider extensions $E\in {\bf P}
\ext^1(K-D,D)$ with $\deg D = 1$ or 2 (using remark \ref{nag}). If
$E\in \Omega^n \backslash \Omega^{n-1}$ then by (\ref{conethm}) $$
h^0(E) = 5 - \cliff(D) - n. $$ Thus either $D=p\in C$, and $E\in
\Omega^1$; or $D=p+q\in S^2 C$, and $E\in \Omega^0_{p+q}\cong {\bf P}^1$.
The second case is that of example \ref{d=g-2}; we shall show next that
this case exhausts all such bundles.
\begin{prop} ${\cal W}^2 = \bigcup_{p+q\in S^2 C} \Omega^0_{p+q}$.
Moreover, $\Omega^0_{p+q}$ maps under $\pi: {\cal W}^2 \rightarrow {\bf P}^3$
onto the secant line $\overline{pq}$ if $p+q \not\in i(C)$, while
$\Omega^0_{p+q} = \pi^{-1}(r)$ if $p+q+r \in |g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}|$ or $|\trigg|$.
\end{prop}
{\it Proof.}\ We first observe (by considering diagram (\ref{verdiagram})) that
the line $\Omega^0_{p+q} \subset {\bf P}(p+q)$ meets the image of the curve
if and only if $f(p+q) \in i(C)$ (see (\ref{pic})); and in this case
meets the curve at a point $r\in C$ representing the bundle $g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}
\oplus \trigg$. By theorem \ref{10} $\Omega^0_{p+q}$ is a Hecke line
$l_F$, where one easily checks that $\det F = K(r)$. So by the
uniqueness statement in \ref{6.8} $\Omega^0_{p+q} = \pi^{-1}(r)$.
We may now assume, then, that $E\in{\cal W}^2$ is a stable bundle for which
$\pi(E)$ does not lie on the canonical curve; $\pi(E)$ then lies on
some secant line $\overline{pq}\subset {\bf P}^3$. This means that ${\rm im \ }} \def\sym{{\rm Sym}} \def\ker{{\rm ker \ }
\phi_E\subset H^0(K)$ is a hyperplane, distinct from $H^0(K(-p))$ and
$H^0(K(-q))$ but containing the 2-dimensional subspace $H^0(K(-p-q))$.
As in the proof of the previous proposition we can find a basis $u,v,w
\in H^0(E)$ such that $u\wedge v, u\wedge w$ are a basis of
$\phi_E^{-1} H^0(K(-p-q))$ which $v\wedge w$ completes to a basis of
$\bigwedge^2 H^0(E)$. Then we have $$ p+q\leq (u\wedge v), \quad
p+q\leq (u\wedge w), $$ while $p+q \not\leq (v\wedge w)$. As before,
it follows from this that $u(p) = u(q) = 0$, i.e. $h^0(E(-p-q))> 0$ and
so $E\in \Omega^0_{p+q}$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
{\it Proof of theorem \ref{w2cubic}.} By propositions \ref{6.8},
\ref{6.9} and theorem \ref{izadi} it suffices to check that the line
$\Omega^0_{p+q}\subset {\cal W}^2 \subset \g00$ coincides with the pencil
$l'_{pq}$ if $p+q \in F(\tt)\backslash i(C)$, i.e. when
$h^0(g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}(-p-q)) = h^0(\trigg(-p-q)) = 0$.
Consider a stable bundle $E\in \Omega^0_{p+q}$. Since $h^0(E(-p-q))> 0$
the symmetric divisor $D_E$ trivially contains the surfaces $W_2 - p-q$
and $p+q - W_2$; while $C-C \subset D_E$ since $E\in {\cal W}^2$. We will
show that $D_E$ also contains the surface $\Sigma(X)$ (see
(\ref{Sigma(X)})). Let $\lambda = \oo(s+t-s'-t') \in \Sigma(X)$. By
definition we have an exact sequence $$ \ses{\lambda(p+q)}{\lambda
\otimes E}{K\lambda (-p-q)} $$ with, say, extension class $f\in
\ext^1(K-p-q,p+q) = H^0(C,K^2(-2p-2q))^{\vee}} \def\cctil{\ctil_{\eta}$.
We have to show that $h^0(\lambda \otimes E) >0$. We can suppose that
$h^0(\lambda(p+q)) = 0$ (otherwise there is nothing to prove); so by
Riemann-Roch $h^1(\lambda(p+q)) = 1$. If $h^0( K\lambda (-p-q)) >1$
then $h^0(\lambda \otimes E) >0$ and we are done; so we assume $h^0(
K\lambda (-p-q)) =1$. In this case $h^0(\lambda \otimes E) >0$ if and
only if the coboundary map $$ \delta: H^0(K\lambda (-p-q)) \rightarrow
H^1(\lambda(p+q)) $$ vanishes, which in turn is equivalent to $\ker f$
containing the image of the multiplication map: $$ H^0(K\lambda
(-p-q)) \otimes H^0(K\lambda^{-1} (-p-q)) \rightarrow \ker f \subset
H^0(K^2 (-2p-2q)). $$ In fact we shall check that the image is
contained in the subspace $S^2 H^0(K(-p-q)) \subset \ker f$ (see
example \ref{d=g-2}).
This last assertion results from the definition (\ref{Sigma(X)}): we
can write $$ K = \oo(p+q+s+t+u+v) = \oo(p+q+s'+t'+u'+v'), $$ for some
$u,v,u',v' \in C$, and hence $$ \begin{array}{rclcl} K\lambda(-p-q) &=&
K(-p-q-s'-t'+s+t) &=& \oo(u'+v'+s+t), \\ K\lambda^{-1}(-p-q) &=&
K(-p-q-s-t+s'+t') &=& \oo(u+v+s'+t'). \\ \end{array} $$ By hypothesis
these divisors are unique in their linear equivalence classes and we
can write their sum as $$ (u'+v'+s+t)+(u+v+s'+t') =
(s+t+u+v)+(s'+t'+u'+v') $$ where $s+t+u+v$ and $s'+t'+u'+v' \in
|K(-p-q)|$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\medskip
We shall conclude this section by giving another interpretation of
diagram (\ref{nodalcubic}), as follows. First, we shall view
$\Gamma_{00} \hookrightarrow S^4 H^0(C,K)$ by assigning to each element
the leading terms of its Taylor expansion at $0\in J_C$; or
equivalently by assigning to a divisor its tangent cone at the origin.
This map is injective by \cite{I} lemma 2.1.1.
Next we note that there is a distinguished element $q^2 \in S^4
H^0(C,K)$, where $q\in S^2 H^0(C,K)$ is the equation of the quadric
$Q\subset {\bf P}^3$ containing the canonical curve. Under the above
inclusion this comes from the split divisor $\Theta-g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing} \cup
\Theta-\trigg \in \g00$.
Third, we identify ${\bf P}^3 = {\bf P} T_0 J_C$ with the space of
translation-invariant vector fields on the Jacobian. One can then map
$$ \begin{array}{rcrcl} \alpha &:& {\bf P} T_0 J_C &\rightarrow& {\bf P}
\bigl(S^4 H^0(K) / \c q^2 \bigr)\\ &&D &\mapsto & qDf - fDq,\\
\end{array} $$ where $f\in H^0(\iic(3))$ is any cubic through the
canonical curve. It is easy to check that this construction is
independent of the choice of $f$; moreover $\alpha$ is an isomorphism
onto the subspace ${\bf P} (\Gamma_{00} / \c q^2)$, as observed by
Beauville--Debarre \cite{BD}, pages 32--33.
\begin{theo} \label{quartics} The following diagram commutes: $$
\begin{array}{ccccc} {\cal W}^2 & \subset & \g00 &\subset & {\bf P} S^4
H^0(K)\\ &&&&\\ \pi \downarrow &&\downarrow&&\downarrow\\ &&&&\\ {\bf P}^3
& \map{\alpha}& {\bf P} \bigl(\Gamma_{00} / \c q^2\bigr)&\subset&
{\bf P}\bigl(S^4 H^0(K) / \c q^2 \bigr)\\ \end{array} $$ \end{theo}
{\it Proof.}\ We have to check commutativity of the left-hand square, and since
both vertical arrows are linear projections it is sufficient to check
commutativity over points $p\in C$ of the canonical curve. For such a
point denote by $D_p\in {\bf P} T_0 J_C$ the associated constant vector
field. By propositions \ref{6.8} and \ref{6.9} the line $\pi^{-1}(p)$
corresponds to the pencil $l_p'$ with base locus $\Sigma_{st}\cup
\Sigma_{s't'}$, where the points $s,t,s',t' \in C$ are defined by
$p+s+t \in |g^1_3} \def\trigg{h^1_3} \def\thsing{\Theta_{\rm sing}|$ and $p+s'+t' \in |\trigg|$. By tangent cones at the
origin, the pencil $l'_p$ corresponds to a pencil of quartics spanned
by $qDf -fdq$ and $q^2$, for some $D\in T_0 J_C$ and $f\in
H^0(\iic(3))$. We have to show that $D=D_p$.
Since the pencil is uniquely determined by (the tangent cone of) its
base locus, it is enough to check that the two quartics $qD_pf -fD_pq$
and $q^2$ contain the tangent cones of $C-C$, $W_2 -s-t$ and $W_2
-s'-t'$. These tangent cones are the canonical curve $C\subset {\bf P}^3$
and the two trisecants $\overline{pst}, \overline{ps't'} \subset {\bf P}^3$
respectively. The result now follows easily: $q$ vanishes on all three
curves; while $f$ (and hence $fD_p q$) vanishes on $C$, and---since the
two trisecants span the tangent plane to $Q$ at $p$---the derivative
$D_p q$ vanishes on the two lines. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\section{Genus 5}
Let $C$ be a curve of genus 5. If $C$ is nontrigonal then the canonical
curve $C\hookrightarrow {\bf P}^4$ is the complete intersection of a net of
quadrics $|\iic(2)| = {\bf P}^2$, in which the locus $\Gamma \subset {\bf P}^2$
of singular quadrics is a plane quintic curve, smooth if $C$ has no
vanishing theta-nulls, otherwise having ordinary double points
corresponding to quadrics of rank 3 (see \cite{ACGH}, page 270).
Let $\thsing = W^1_4$ be the singular locus of the theta divisor. This
is a curve, and by assigning to each point $x\in \thsing$ its
projectivised tangent cone ${\bf P} T_x \Theta = Q_x$ we have a double
cover $$ \begin{array}{rrcl} f:&\thsing & \rightarrow & \Gamma \subset
{\bf P}^2 \\
& x & \mapsto & Q_x \\ &&& \displaystyle = \bigcup_{D\in |x|}
\overline{D} \subset {\bf P}^4.\\ \end{array} $$ The sheet interchange of
$\thsing$ with respect to this double cover is induced by the Serre
involution of $J_C^4$.
\begin{lemm} $f^* \oo_{\Gamma}(1) = \oo_{\thsing} (\Theta)$. Moreover,
the induced restriction map $H^0(J_C^{g-1}, 2\Theta) \rightarrow
H^0(\Gamma, \oo(2))$ is surjective. \end{lemm}
{\it Proof.}\ The first part follows from \cite{G}. To prove that the pull-back
of hyperplane sections is surjective, it is sufficient to show this on
the image of the Kummer map $J_C \rightarrow |2\Theta|$, $a\mapsto
\Theta_a + \Theta_{-a}$. In other words, we consider the rational map
$$ \alpha : J_C \rightarrow |\oo_{\Gamma}(2)| \cong {\bf P}^5 $$ sending
$a\in J_C$ to the divisor whose pull-back to $\thsing$ is $(\Theta_a +
\Theta_{-a})\cap \thsing$. (Note that $\alpha$ is defined away from
$C-C \subset J_C$: this follows from \cite{W}, theorem 2.4.) One can
show that the map $$ \begin{array}{rcl} \beta: J_C &\rightarrow &
S^{10}(\thsing) \\ a &\mapsto& \Theta_{a}\cap \thsing\\ \end{array} $$
is injective (see, for example, \cite{ACGH}, pages 265--268); this
implies that $\alpha$ is a finite map and so we are done. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
It follows from this that the image of $\thsing$ under the Kummer map
is a Veronese embedding of $\Gamma \subset {\bf P}^2$:
\begin{equation} \label{mapv} \begin{array}{ccc} \thsing & \map{{\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}} &
{\bf P}^{31} \\ f\downarrow && \uparrow v \\ \Gamma & \subset & {\bf P}^2 \\
\end{array} \end{equation} where $v({\bf P}^2)\subset {\bf P}^5 \subset
{\bf P}^{31}$ is a Veronese surface.
\begin{theo} \label{g5ver} For any curve $C$ of genus 5 the
Brill-Noether locus ${\cal W}^3\subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ is a Veronese surface
intersecting the Kummer variety in the Veronese image of a plane
quintic $\Gamma \subset {\bf P}^2$. In particular: \begin{enumerate} \item
If $C$ is nontrigonal then ${\cal W}^3 = v({\bf P}^2)$, where $\Gamma$ is as in
(\ref{mapv}) and $v(\Gamma) = {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(\thsing)$. \item If $C$ is trigonal
then $\Gamma \subset {\bf P} H^1(g^1_3)$, where $g^1_3$ is the (unique)
trigonal line bundle, is the projection of the canonical curve away
from a trisecant; and its Veronese image cuts the Kummer in the
component $C+g^1_3$ of~$\thsing$. \end{enumerate} \end{theo}
{\it Proof of part 2.} This is easily dispatched. We first remark that
it is well-known that on a curve of genus $\geq 5$ a $g^1_3$ is unique
if it exists; while for a curve of genus 5 the following argument will
give another proof of this fact.
Let $|D| = g^1_3$; by lemma \ref{muk} any stable bundle $E\in
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with $h^0(E) \geq 4$ has line subbundle $\oo(D) \subset E$,
so $E$ belongs to the 5-plane ${\bf P}(g^1_3)$ of section 1. By example
\ref{d=g-2} we have seen that $h^0(E) = 4$ precisely along a Veronese
surface in ${\bf P}(g^1_3)$. This intersects the Kummer precisely in the
image of the curve---that is, in the Kummer image of $C+g^1_3$---and
from diagram (\ref{verdiagram}) this is the projection of the canonical
curve as asserted. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
From now on we shall assume that the curve $C$ is nontrigonal.
Before proving part 1 of the theorem we shall need to make some further
observations about the curve $\Gamma$; we consider the map
\begin{equation} \label{mapl} \begin{array}{rcl} l: S^2 C &\rightarrow&
({\bf P}^2)^{\vee}} \def\cctil{\ctil_{\eta} \\ D & \mapsto & |{\cal I}_{C\cup \overline{D}}(2)|.\\
\end{array} \end{equation} In other words $l(D)$ is the pencil of
quadrics containing $C$ and the line $\overline{D}$. Note that the base
locus of such a pencil is a quartic del Pezzo surface containing
sixteen lines, and so $\deg l = 16$.
For each $D\in S^2 C$ we shall identify the five quadrics
\begin{equation} \label{lDmeetG} l(D) \cap \Gamma = \{ Q_1, \ldots ,
Q_5\}. \end{equation} Projection away from the line $\overline{D}
\subset {\bf P}^4$ maps the canonical curve $C$ to a 5-nodal plane sextic
$C'\subset {\bf P}^2$. (Note, again, that the del Pezzo base locus of the
pencil $l(D)$ is obtained by blowing up ${\bf P}^2$ in the five nodes of
$C'$.) Let us denote by $D^{(1)}, \ldots , D^{(5)} \in S^2 C$ the
divisors over the five nodes of $C'\subset {\bf P}^2$. Then by
Riemann-Roch each $|D+D^{(i)}|$ is a $g^1_4$, and hence each
\begin{equation} \label{lDmeetG'} Q_i = Q_{D+D^{(i)}} =\bigcup_{D' \in
|D+D^{(i)}|}\overline{D'} \qquad i=1,\ldots , 5, \end{equation} is a
quadric of rank $\leq 4$ containing the line $\overline{D}$. These are
therefore the points of intersection (\ref{lDmeetG}).
We now return to the proof of theorem \ref{g5ver}. We consider stable
extensions $E\in {\bf P} \ext^1(K-D,D)$ where (by remark \ref{nag} (i)) we
may take $\deg D = 2$ or 3. For such an extension, by (\ref{conethm}),
$$ h^0(E) = 6-\cliff(D) -n $$ where $E\in \Omega_D^n\backslash
\Omega^{n-1}_D$. So for $h^0(E) =4$ we must have $n+ \cliff(D) =2$; if
$\deg D =3$ then this forces $|D| = g^1_3$ contrary to the hypothesis
that $C$ is nontrigonal. So the only possibilities we need to consider
are $D\in S^2 C$, and then $h^0(E) =4$ for $E\in \Omega_D^0 \subset {\bf P}
\ext^1(K-D,D)$.
In this situation diagram (\ref{verdiagram}) becomes:
$$ \begin{array}{rcr} C & \map{|K-D|} & C'\subset {\bf P}^2 \\ &&\\
\scriptstyle|2K-2D| \downarrow &&\ver \downarrow \\ &&\\ \Omega_D^0 =
{\bf P}^1 \subset {\bf P}^7 & \map{\delta} & {\bf P}^5 \\ \end{array} $$ where
$C'\subset {\bf P}^2$ is the 5-nodal sextic as above; and in particular
$\delta$ is surjective.
It follows that ${\cal W}^3 = \bigcup_{D\in S^2 C} \eee_D\Omega_D^0 \subset
{\bf P}^{31}$, where we observe that {\it each $\eee_D \Omega_D^0$ is a
nonsingular conic.} This is because by lemma \ref{bertram} the rational
map $\eee_D$ comes from the (complete) linear series $|\iic (2)|$ on
${\bf P}^7$; while $\Omega_D^0$ has no intersection with the base locus $C$
since $|K-D|$ has no base points---i.e. the canonical curve has no
trisecant lines since $C$ is nontrigonal.
Finally, theorem \ref{g5ver} will follow directly once we observe that
\begin{equation} \label{Dconic} \eee_D \Omega_D^0 = v(l(D)) \qquad
\hbox{for all $D\in S^2 C$,} \end{equation} where $v$ and $l$ are as
defined in (\ref{mapv}) and (\ref{mapl}) respectively. To prove
(\ref{Dconic}) it is sufficient, since both sides are nonsingular
conics, to show that both contain the five points $v(Q_1),\ldots ,
v(Q_5)$ (see (\ref{lDmeetG}) and (\ref{lDmeetG'})); and so it remains
only to check this for $\eee_D \Omega_D^0$.
First note that an extension $E\in \Omega_D^0$ fails to be stable (i.e.
maps to the Kummer variety) if and only if it lies on $\secant^2 C$;
and there are precisely five such points, which are the intersections
of $\Omega_D^0$ with the secant lines
$\overline{D^{(1)}},\ldots,\overline {D^{(5)}}$, where as before the
$D^{(i)}\in S^2 C$ are the nodal divisors over the curve $C'\subset
{\bf P}^2$. The corresponding extensions then contain $\oo(K-D-D^{(i)})$,
respectively, as line subbundles---in other words they map under
$\eee_D$ to the points
$$ \oo(D+D^{(i)}) \oplus \oo(K-D-D^{(i)}) \in {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1}). $$ By
(\ref{lDmeetG'}) and diagram (\ref{mapv}) these are precisely the
images $v(Q_1),\ldots,v(Q_5)$, which completes the proof. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\section{Genus 6}
By proposition \ref{bound} we have $h^0(E) \leq 6$ for all semistable
bundles $E$ in ${\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ on a nonhyperelliptic curve $C$ of genus 6,
and $h^0(E) \leq 5$ if $C$ is not trigonal or a plane quintic.
\begin{theo} Let $C$ be a nonhyperelliptic curve of genus 6.
\begin{enumerate} \item If $C$ is not trigonal or a plane quintic then
there exists a unique stable bundle $E\in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with $h^0(E) = 5$;
i.e. ${\cal W}^4 = \{E\}$. \item If $C$ is trigonal then ${\cal W}^4 \cong
{\bf P}^1$ is a line, along which $h^0(E) = 5$, and does not meet
${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$. \item If $C$ is a plane quintic then $h^0(E)\geq 5$
if and only if $E$ is in the S-equivalence class of the point $g^2_5
\oplus g^2_5 \in {\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$. \end{enumerate} \end{theo}
{\it Proof.}\ First suppose that $C$ is not a plane quintic, and observe that if
$h^0(E)\geq 5$ and $E$ is semistable then it is necessarily stable:
otherwise $E$ fits in an extension (\ref{basicext}) with $\deg D = 5$
(and with $D$ not necessarily effective). Then $h^0(D) = h^0(K-D) \leq
2$, since $C$ is not a plane quintic, and so $h^0(E) \leq 4$.
On the other hand, for any tetragonal pencil $g^1_4 = |D|$ one can
(following Mukai) apply lemma \ref{muk} to observe that $h^0(E(-D))
\geq 1$ for any such bundle. By stability this means that $\oo(D)
\subset E$ is a line subbundle, i.e. $E \in {\bf P} (D)$, the corresponding
6-plane of the ruling of section 1. By example \ref{d=g-2}, $h^0(E) =5$
exactly for $E\in \Omega^0$; if $C$ is nontrigonal this is a single
point, and part 1 is proved.
If $C$ is trigonal then we may take $D=K-2L$ where $|L| = g_3^1$;
$|K-D|$ maps $C\rightarrow {\bf P}^2$ with degree 3 onto a conic, so in
this case $\ker \delta$ is 2-dimensional and $\Omega_D^0 \subset
{\bf P}(D)$ is a line. This line does not meet the image of $C$ in ${\bf P}(D)$
since $|K-D|$ is base-point-free; so we have proved part 2.
For part 3, first note that by the reasoning of remark \ref{w3node} the
only semistable bundle with $h^0(E) =6$ is $E=g^2_5 \oplus g^2_5$. On
the other hand the reasoning of part 1 above yields extensions with
$h^0(E) =5$ in $\Omega^0_D$ for any $|D| = g^1_4$. In this case the
tetragonal pencils are precisely the projections from points of the
plane quintic, i.e. $D=L-p$ for $|L| = g^2_5$ and some $p\in C$. The
map $C\rightarrow {\bf P}^2$ given by the series $|K-D|$ is projection of
the canonical curve (which lies on a Veronese surface) away from the
conic in ${\bf P}^5$ spanned by $D$, and hence has base-point $p$. The
image is thus the plane quintic model of $C$; in particular $\delta$ is
surjective and $\Omega^0_D$ is a single point. But because $p\in C$ is
a base-point of $|K-D|$, the curve passes through $\Omega^0_D$ at the
image of $p$, which is the equivalence class of $g^2_5 \oplus g^2_5$.
This proves part 3. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm Part 1 of the theorem was observed by Mukai in
\cite{Muk}. Recall that the canonical curve lies on the (del Pezzo)
transverse intersection with a ${\bf P}^5 \subset {\bf P}^9$ of the Pl\"ucker
embedded Grassmannian of lines in ${\bf P}^4$. The bundle $E$ in the
theorem is then dual to the restriction to the curve of the
tautological bundle on the Grassmannian. \end{rem}
It is well-known that a generic curve of genus 6 possesses five
tetragonal pencils, so the point $E$ of part 1 is common to the
corresponding five 6-planes of the ruling. It is amusing to see this
using the results of section~1.
Let us denote the five by $x_0,\ldots , x_4 \in \pic^4(C)$; and let us
recall how they are related to each other. By Riemann-Roch $|x_0| =
g^1_4$ implies that $|Kx_0^{-1}| = g^2_6$. Thus the image of $$
\lambda_{|Kx_0^{-1}|} : C \rightarrow {\bf P}^2 $$ is a sextic with four
nodes, which we shall denote by $p_1,\ldots,p_4 \in {\bf P}^2$:
\begin{center}
figure
\end{center}
Let $D_i\in S^2C$,
$i=1,\ldots,4$, be the nodal divisors, i.e. $p_i =
\lambda_{|Kx_0^{-1}|}(D_i)$. If we denote by $H= Kx_0^{-1}$ the
hyperplane class on $C$ then by adjunction in the blow-up at the four
nodes we have $K = 3H - D_1-\cdots -D_4$ and hence $$ x_0 = \oo(2H -
D_1 - \cdots -D_4) $$ i.e. {\it $|x_0|$ is cut out by the pencil of
conics through the four nodes $p_1, \ldots ,p_4$}. In this model it is
easy to see the remaining four $g^1_4$s: {\it for $i=1,\ldots , 4$ the
pencil $|x_i|$ is cut out by the lines through $p_i \in {\bf P}^2$}.
Formally $x_i = \oo(H-D_i)$, and in particular we deduce that
\begin{equation} \label{tets} x_0 \otimes x_i = \oo(K - D_i).
\end{equation}
Consider again the five 6-planes ${\bf P}(x_0),\ldots , {\bf P}(x_4)$: we have
just seen that the bundles $E$ for which $h^0(E)=5$ are the points
$\Omega^0$ in these five spaces. Let us denote these five bundles by
$E_i \in {\bf P}(x_i)$. We are claiming that they all coincide:
\begin{equation} \label{5=} E_0 = E_1 = E_2 = E_3 = E_4 \in {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K).
\end{equation} To see this, let us work in ${\bf P}(x_0)$. In example
\ref{d=g-2} we have seen that $\Omega^0 = \{E_0\}$ is the vertex of a
Veronese cone $\Omega^1$ containing the image of $C$ (i.e. the
intersection of ${\bf P}(x_0)\subset {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)$ with ${\rm Kum}} \def\ver{{\rm Ver}} \def\endo{{\rm End}(J_C^{g-1})$), and
that projection away from the vertex maps $C$ to ${\bf P}^2$ via the linear
system $|Kx_0^{-1}|$. The image of this map is the 4-nodal sextic just
noted, and it follows that the four secant lines $\overline{D_i}
\subset {\bf P}(x_0)$, $i=1,\ldots ,4$, where $D_i \in S^2 C$ is the $i$-th
nodal divisor as above, all pass through the vertex~$E_0$. By
proposition \ref{40} together with (\ref{tets}) it follows that $$
\overline{D_i} = {\bf P} (x_0) \cap {\bf P}(x_i), \qquad i=1,\ldots , 4. $$
Thus $E_0 \in \overline{D_i}\subset {\bf P}(x_i)$ for each $i$, and
therefore coincides with the unique bundle $E_i \in {\bf P}(x_i)$ having
five sections---so again we have proved (\ref{5=}).
\begin{rem}\rm \label{21dimspan} By the proof of theorem \ref{w2g00} we
have, for a curve of genus 6, $ {\cal W}^2 \subset \g00 \cap {\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K)
\subset {\cal W}^2 \cup \bigcup_{x\in W^1_4} {\bf P}(x). $ We have seen that in
each such ${\bf P}(x)$, $|x| = g^1_4$, we have $h^0(E)\geq 3$ along a cubic
cone, which in particular spans ${\bf P}(x)$. It follows that $$ \g00 \cap
{\cal SU}_C} \def\iic{{\cal I}_C} \def\eee{\varepsilon(2,K) = {\cal W}^2 \cup \bigcup_{x\in W^1_4} {\bf P}(x), $$ and that this
intersection properly contains ${\cal W}^2$ since $h^0(E) =2$ at the generic
point of each ${\bf P}(x)$. For $C$ generic $\bigcup_{x\in W^1_4} {\bf P}(x)$
consists of five 6-planes meeting pairwise in ten lines concurrent at
the point ${\cal W}^4$. \end{rem}
|
1997-01-26T16:50:21 | 9701 | alg-geom/9701014 | en | https://arxiv.org/abs/alg-geom/9701014 | [
"alg-geom",
"math.AG"
] | alg-geom/9701014 | Donu Arapura | Donu Arapura | Geometry of cohomology support loci II: integrability of Hitchin's map | 17 pages, AMS LaTeX | null | null | null | null | In very rough terms, the main theorem is that the set, which consists of
semistable vector bundles with trivial rational Chern classes and nontrivial
kth cohomology on a smooth complex projective variety, is a degeneration of a
union of abelian varieties. More precisely, we consider the subset of the
moduli space of Higgs bundles satisfying the analogous cohomological condition.
We show that this set is Zariski closed and that if Sigma is the normalization
of an irreducible component containing a stable point, then a connected
component of a general fiber of the restriction of the Hitchin map to Sigma is
an abelian variety. This should be interpreted as a nonabelian version of a
theorem of Green and Lazarsfeld. The Hitchin map in this setting is in fact
Simpson's generalization of it. The key point is to show that the general
fibers of this map are lagrangian (where the target of the map is taken to be
the image). This hinges on the fact that this property is essentially
hereditary for hyperkaehler submanifolds. We establish various other properties
of these sets including a codimension estimate which may be viewed as a generic
vanishing theorem.
| [
{
"version": "v1",
"created": "Sun, 26 Jan 1997 14:21:42 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Arapura",
"Donu",
""
]
] | alg-geom | \section{ Introduction}
Green and Lazarsfeld \cite{gl2} have proven the following remarkable theorem:
\begin {thm}\label{thm:gl} Let $X$ be a smooth
complex projective variety. Then the set $S^{pq}(X)$ of line bundles in $Pic^0(X)$
satisfying $H^q(X, \Omega_X^p\otimes L) \not= 0$ is a union of a finite
number of translates of abelian subvarieties.
\end{thm}
In this paper, we seek a generalization for higher rank bundles. The
analogue of $Pic^0(X)$ is the moduli space $M=M_V(X,n)$ of
semistable bundles of a given rank $n$ with trivial Chern classes,
and within it a subset
$\{E\,|\, H^q(X,\Om^p\otimes E)\not= 0\}$ can be defined as above.
However, $M$ is very
far from an abelian variety in general, so it is not immediately clear what
the analogous theorem should even say.
A clue is provided by the following theorem of Hitchin \cite{hit2}:
\begin{thm}\label{intro:thm:hit} Suppose that $X$ is a smooth
projective curve and
$M^s\subset M$ the
smooth open set of stable bundles.
Then there is morphism
from the cotangent bundle $T^*M^s$ to an affine space
such that the general
fibers are open subsets of abelian varieties.
\end{thm}
Our ultimate goal then is to give a common generalization of both theorems.
We will establish a result similar to \ref{intro:thm:hit}
for any component of
the above cohomology support locus in $M$ when
$X$ has arbitrary dimension.
In order to delve deeper into the story, it will be necessary to
explain how to compactify Hitchin's map. For this, we need to make the
transition from vector bundles to Higgs bundles, which can be motivated
as follows. Suppose that $E$ is a stable
bundle corresponding to a smooth point $[E]\in M$. Then a cotangent
vector to $[E]$ is just a section $\theta\in H^0(X,\Omega_X^1\otimes End(E))$.
As $[E]$ is a smooth point, there is no obstruction to extending a first order
deformation of $E$ to one of second order. The dual
condition is $\theta\wedge\theta = 0$. The pair $(E,\theta)$ is an
example of a Higgs bundle. Simpson \cite{s2} has shown that
the set of isomorphism classes of Higgs bundles of rank $n$ with
vanishing rational Chern classes (and subject to a suitable semistability
condition weaker than semistability of the underlying vector bundle),
can be parameterized by a quasiprojective moduli scheme $M_{Dol}(X,n)$.
Furthermore there is a proper morphism $h$, the analogue of Hitchin's
map, from $M_{Dol}(X,n) $
to an affine space (which assigns to $(E,\theta)$ the characteristic
polynomial of $\theta$). We can define
$\Sigma_{m,Dol}^k(X,n) \subseteq M_{Dol}(X,n)$ as the set of
those pairs $(E,\theta)$ such that the appropriate $k$th cohomology
group has dimension at least $m$. In section 4, we prove the main
result:
\begin {thm} If $X$ is a smooth projective variety, then
$\Sigma_{m,Dol}^k(X,n)$ is a Zariski closed subset
of $M_{Dol}(X,n)$. If $\tilde \Sigma$ is the normalization
of an irreducible component of $\Sigma_{m,Dol}^k(X,n)$ (with its reduced
subscheme structure) containing a stable point, then
the connected components of the general fibers of the pullback
of $h$ to $\tilde\Sigma$ are abelian varieties.
\end{thm}
The case $m=k=0$ gives an analogue of Hitchin's result.
The above sets can be further subdivided into $(p,q)$ parts,
some components of
which form partial compactifications of the ``cotangent bundles'' of the
sets $\{E\,|\, H^q(X,\Om^p\otimes E)\not= 0\}$ considered earlier.
Similar results will be proved for these sets.
The analytic space associated to $M_{Dol}(X,n)$
has a second complex structure
$M_B(X,n)^{an}$ which comes about via a correspondence between Higgs
bundles (of the above type) and semisimple local systems \cite{s1,s2}.
The key observation is that when taken together, these yield a
quaternionic structure on this space, and the set
$\Sigma_{m,Dol}^k(X,n)$ is compatible with this structure.
We use this fact to show that the
general fiber of the restriction of $h$ is lagrangian with respect to a
suitable symplectic structure, then the theorem follows easily.
An important precedent for the use of the
quaternionic structure in this context
is Deligne's and Simpson's \cite{s3} approach to
proving theorem \ref{thm:gl}. One complication, absent
in the rank one case, is the presence of singularities.
Recent work of Verbitsky, discussed in the next section,
allows us to handle these issues.
The cohomology group of a Higgs bundle has a number of different
incarnations. To begin with, it can be defined as the hypercohomology
of an explicit complex. It is also (isomorphic to) the cohomology of the
associated local system. Both interpretations are needed in order to
verify that the cohomology support loci are quaternionic.
The first description will also be used establish the invariance
of these loci under a natural $\C^*$-action. This will imply that any
irreducible component contains a complex variation of Hodge structure
The second
point of view will be useful for establishing certain homotopy
invariance properties for these sets.
Finally, we will reinterpret the cohomology group of a Higgs bundle
as an Ext group for certain sheaves on the cotangent bundle of $X$.
Then using the local to global spectral sequence, we prove a generic
vanishing theorem in the spirit of \cite{gl1}. This leads to estimates
on the codimension of the cohomology support loci.
The final section of this paper contains some explicit examples. So
readers may wish to skip to it from time to time.
For the most part, schemes over $\C$ will be treated as sets of
$\C$-valued points. As usual, the superscript ``an'' indicates the
analytic space associated to a scheme.
\section{ Quaternionic geometry}
This section is completely expository. It is intended to give a quick
introduction to quaternionic geometry, and to some of
Verbitsky's work in particular. A nice discussion of some related ideas and
examples
can be found in \cite{hit3}. See also \cite{hit1}, \cite{fu},
\cite{s4}, \cite{v1}, \cite{v2} and references contained therein.
A quaternionic (or hypercomplex) manifold is a $\Ci$-manifold
$X$ with two complex structures $\I$ and $\J$ which induce the same real
analytic structure on $X$ and satisfies $\I\J = -\J\I$. Setting $\K = \I\J$
gives an action of the quaternions $\HH$ on the tangent bundle. Any
quaternionic vector space is naturally a quaternionic manifold.
A morphism of quaternionic manifolds is $\Ci$ map which is holomorphic
with respect to $\I,\J$ and $\K$ (it suffices to check holomorphicity with
respect to any two).
\begin{lemma} Let $V_1$ and $V_2$ be two finite dimensional
quaternionic vector spaces, and let $U_i\subseteq V_i$ be open
neighbourhoods of the origin with their induced quaternionic
structures. Any morphism $f:U_1\to U_2$ satisfying $f(0) = 0$ is
the restriction of an $\HH$-linear map.
\end{lemma}
\begin{proof} (Deligne, see \cite{s3}). Choose a point $x\in U_1$.
After identifying the tangent space at $x$ and $f(x)$ with $V_1$ and
$V_2$, the differential gives an $\R$-linear map $df_x:V_1\to V_2$
By assumption
$$df_x\I = \I df_x,\quad df_x\J = \J df_x$$
so in fact $df_x$ is $\HH$-linear. Thus $df$ can be viewed as a $\Ci$
map from $U_1$ to $Hom_\HH(V_1,V_2)$. Differentiating again yields the
Hessian, which is a symmetric $\R$-bilinear
form $H_x:V_1\times V_1\to V_2$.
$H_x$ is $\HH$-linear in one variable and therefore in both.
Now for the punchline:
$$\K H_x(\alpha,\beta) = \I H_x(\J\alpha,\beta)= H_x(\J\alpha,\I\beta) =
\J H_x(\alpha,\I\beta) = -\K H_x(\alpha,\beta)$$
Therefore $H_x = 0$ and the lemma follows immediately.
\end{proof}
A quaternionic submanifold of a quaternionic manifold is a
$\Ci$ submanifold such that the inclusion is a morphism. More generally
a quaternionic subvariety $Y$ of a quaternionic manifold
$X$ is a reduced real analytic subvariety whose complexified ideal is
locally defined by both $\I$ and $\J$ holomorphic functions.
\begin{cor} \label{cor:quat}Any quaternionic submanifold of a quaternionic vector
space $V$ is a translate of a linear subspace. A quaternionic
subvariety is a union of submanifolds.
\end{cor}
A hyperk\"ahler manifold is a $\Ci$ manifold with a Riemannian metric
$g$ and two anticommuting complex structures $\I$ and $\J$, such that
$g$ is K\"ahler with respect to both of these structures.
\begin{prop} (\cite[6.5]{v1}) If $X$ is hyperk\"ahler then the underlying real
analytic structures associated to $\I$ and $\J$ coincide. Therefore
$X$ is a quaternionic manifold.
\end{prop}
\begin{thm} (Verbitsky \cite{v2}). Let $(X,x)$ be a germ of a
hyperk\"ahler manifold. The the germ of quaternionic subvariety $(Y,x)$
is the union of a finite number of germs of quaternionic submanifolds.
\end{thm}
Here is an outline of the proof:
Let $R$ be the local ring of real analytic functions of $(X,x)$.
Given a complex structure $L$ on $X$, let $O_L$ be the local ring
of $L$-holomorphic functions. There is a natural inclusion
$O_L\subset R\otimes \C$ which splits: the complement is
the ideal generated by $L$-antiholomorphic functions vanishing
at $x$. Let $\phi$ be the composition of local homomorphisms:
$$O_\I \hookrightarrow R\otimes \C \to
O_\J\hookrightarrow R\otimes \C \to O_\I$$
A direct calculation shows that the induced endomorphism on the
cotangent space $m/m^2$ of $O_{\I}$
is a homothety associated to a scalar $\lambda\in\C^*$
which is not a root of unity. After a change of variables, one can
arrange $\I$-holomorphic coordinates so that $\phi(x_i) = \lambda x_i$.
Let $I$ be the ideal of $Y$ in $O_\I$. Then $I(R\otimes \C)$ is
also generated by $\J$-holomorphic functions. Therefore $\phi(I) \subset I$
and this implies that $I$ is homogeneous, and thus $(Y,x)$ is
isomorphic to the germ of its tangent cone.
The tangent cone is a quaternionic subvariety of a quaternionic
vector space, and therefore by \ref{cor:quat} is a union of manifolds.
\begin{cor}\label{cor:norm} The normalization of a quaternionic subvariety of a
hyperk\"ahler manifold with respect to $\I$ coincides with the
normalization with respect to $\J$ and it is smooth. Furthermore it
inherits a hyperk\"ahler structure from the ambient manifold.
\end{cor}
Given a \hyp manifold $X$, set $\omega_L(\alpha,\beta) =
g(L\alpha,\beta)$ for $L=\I,\J,\K$. These are just the K\"ahler forms
associated to the complex structures.
In particular, they define (real) symplectic structures on $X$.
The form $ \omega_\J + \sqrt{-1}\omega_\K$ defines an $\I$-holomorphic
symplectic structure on $X$.
\section{Lagrangian maps}
Let $(X,\omega)$ be a real or holomorphic symplectic manifold. A closed
submanifold $Y\subset X$ is lagrangian if the tangent spaces of $Y$ are
maximal isotropic subspaces of the tangent spaces of $X$ with respect
to the symplectic pairing induced by $\omega$. A map $h:X\to B$
($\Ci$ or holomorphic according to the category) to manifold $B$ will
be called lagrangian if its differential has maximal rank, and all its
fibers are lagrangian submanifolds. Note that a lagrangian map is the
same thing as a completely integrable system. A complete discussion of
these notions would take us too far afield, see for example
\cite{gs} for further details.
If $\omega$ is only defined on a dense open subset of $U'\subseteq X$,
a map $h:X\to B$ will be called generically lagrangian if there is a dense
nonsingular open set $U\subset B$ such that $h^{-1}(U)\subseteq U'$ and
$h^{-1}(U)\to U$ is lagrangian, the
complement of the largest such $U$ will be called the discriminant.
\begin{lemma} Let $X$ be a K\"ahler manifold with a real $\Ci$ symplectic
structure $\omega$ (not necessarily equal to the K\"ahler form).
If $h:X \to B$ is a proper
lagrangian holomorphic map, then the connected components of the
fibers are complex tori.
\end{lemma}
\begin{proof} After Stein factorization, we can assume that the fibers
are connected. Standard results in symplectic geometry \cite[page 353]{gs} show that
any fiber $F$ is diffeomorphic to a torus. It follows easily that
the Albanese map
$$F\to H^1(F,\R)^*/H_1(F,\Z) \cong H^0(F,\Omega_F^1)^*/H_1(F,\Z)$$
is a diffeomorphism and therefore a biholomorphism. Note that the
isomorphism $H^1(F,\R)\cong H^0(F,\Omega_F^1)$ is
only place where K\"ahler condition on $X$ is used, so the lemma
holds under considerably weaker hypotheses.
\end{proof}
Let $(X,g,\I,\J)$ be a \hyp manifold. We will usually take $\I$ as
the preferred complex structure with holomorphic symplectic structure
given by $\omega_\J + \sqrt{-1}\omega_K$. In particular, a lagrangian
map $h:X\to B$ will be assumed to be $\I$-holomorphic and lagrangian with
respect to the indicated symplectic structure. Note that such a map is also
lagrangian with respect to the real symplectic structure $\omega_J$,
consequently the above lemma applies.
\begin{thm} \label{thm:lagr}
Let $(X,g_X,\I,\J)$ and $(Y, g_Y,\I,\J)$ be a \hyp manifolds
Suppose that $h:X\to B$ is
a lagrangian map. If $f: Y\to X$ is a
finite quaternionic morphism such that $g_Y =
f^*g_X$ (away from critical points). Then $h\circ f: Y \to B'$ is a
a generically lagrangian map, where the image $B' = h(f(Y))$ is endowed
with the reduced analytic structure.
\end{thm}
\begin{proof}
For any $x\in X$, $V_x = ker\, dh_x$ is a maximal isotropic
subspace of the tangent space $T_x$ with respect to $\omega_\J$.
Thus $\J V_x$ is
orthogonal to $V_x$ with respect to $g_X$.
Since $$dim\, \J V_x = dim\, V_x = dim T_x/2,$$
$T_x = V_x \oplus \J V_x$.
Let $y\in Y$ be a general point, then we can identify $T_y$ with a
quaternionic subspace of $T_{f(y)}$ and $ker\, d(h\circ f)_y$ with
$V_{f(y)}\cap T_y$, and so
$T_{y} = ker\, d(h\circ f)_y\oplus \J ker\, d(h\circ f)_y$.
Therefore $ker\, d(h\circ f)_y$ is a maximal
isotropic subspace. So the general fibers of $h\circ f$ are lagrangian.
\end{proof}
\begin{cor} If in the above notation $h\circ f$ is proper, then
the connected components of its general fibers are complex tori.
\end{cor}
\section{ Cohomology support loci for Higgs bundles.}
Let $X$ be a smooth complex projective variety with a fixed ample
line bundle $L$.
A Higgs bundle on $X$ consists of an algebraic vector bundle
$E$ together with a section
$\theta\in H^0(X,\Omega_X^1\otimes End(E))$ satisfying
$\theta\wedge \theta = 0$. A Higgs bundle $(E,\theta)$ , with
$c_1(E)=0$ (in rational cohomology) is called stable if
$c_1 (F).c_1(L)^{dimX -1} < 0$ for any coherent
subsheaf $F\subset E$ satisfying $rk F < rk E$ and $\theta(F)\subseteq
\Om^1\otimes F$. A Higgs bundle with $c_1(E)= 0$ is called polystable if
it is a direct sum of stable Higgs bundles with vanishing first Chern class.
It will be convenient to combine the main results of Simpson
\cite{s1, s2} into one
big theorem:
\begin {thm} There is an affine scheme (of finite type
over $Spec \Z$) $M_B(X,n)$ whose complex points
parameterize the isomorphism classes of
semisimple representation of $\pi_1(X)$ into $Gl_n(\C)$.
There is a quasiprojective scheme $M_{Dol}(X,n)$, over $Spec \C$,
whose complex points parameterize
polystable rank $n$ Higgs bundles with vanishing first and
second rational Chern classes.
There are open subsets $M_{Dol}^s(X,n)\subseteq M_{Dol}(X,n)$
and $M_{B}^{irr}(X,n)\subseteq M_{B}(X,n)$ which parameterize stable
bundles and irreducible representations respectively.
The spaces $M_{Dol}(X,n)^{an}$ and
$M_{B}(X,n)^{an}$ are
homeomorphic,
and $M_{Dol}^s(X,n)$ and $M_{B}^{irr}(X,n)$ correspond under this
homeomorphism.
\end{thm}
\begin{remark}
0) $X$ and $n$ will be omitted from the notation when it is safe
to do so.
1) The word ``parameterize'' is a bit vague. The
correct statement is that these are coarse moduli spaces for the
appropriate moduli functors.
2) A semistable Higgs bundle with $c_1=0$
is an iterated extension of stable Higgs bundles with $c_1=0$.
Two semistable bundles are equivalent if the their stable
factors coincide (up to isomorphism). Every equivalence
class of semistable bundles has a unique polystable representative.
Thus $M_{Dol}$ parameterizes equivalence classes of semistable
bundles. Similarly $M_B$ parameterizes equivalence classes of arbitrary
representations, where two representations are equivalent if they
have isomorphic semisimplifications.
3) These moduli spaces may be nonreduced. However we will usually suppress the
scheme structure and just treat them as sets of $\C$-valued points.
For every (poly, semi)stable Higgs bundle
or semisimple representation $V$, let $[V]$ denote the corresponding
point in the moduli space.
\end{remark}
A family of Higgs bundles on $X$ parameterized by $T$ is a
vector bundle ${\cal E}$ on $X\times T$,
with a section $\Theta$ of $p_X^*\Om^1 \otimes End({\cal E})$
satisfying $\Theta\wedge\Theta = 0$.
$M_{Dol}$ is only a coarse moduli space, so there may not be
a universal family of Higgs bundles. However Simpson's construction gives a bit
more. Namely $M_{Dol}(X,n)$ is a quotient, in the sense of geometric invariant
theory, of a
locally closed subscheme $Q_n$ of an appropriate Quot or Hilbert scheme.
$X\times Q_n$ will in fact carry a family of Higgs bundles
$({\cal E}, \Theta)$
such that its restriction $({\cal E},\Theta)|_q$
to a slice $X\times \{q\}$ is semistable and corresponds
to the image of $q$ in $M_{Dol}(X,n)$. Over $M_{Dol}^s(X,n)$ it
possible to find
cross sections to $Q_n\to M_{Dol}(X,n)$ locally in the etale topology.
The following is an almost immediate consequence:
\begin{prop}\label{prop:closed}
Let $S$ be a set of semistable Higgs bundles on $X$.
Suppose that for every $T$ and semistable family of
Higgs bundles $({\cal E},\Theta)$
parameterized by $T$, the set
$S_T = \{t\,|\, ({\cal E},\Theta)|_t \in S\}$ is
Zariski closed. Then the set $S_M$ of all $m\in M_{Dol}$ which possess a
semistable representative in $S$ is Zariski closed.
\end{prop}
\begin{proof} $S_M$ is constructible because it
is the image of $S_Q$ under the canonical map. Given a limit point
$s$ of $S_M$, there is an irreducible curve $C$ such that $s\in C$ and
$C-\{s\}\subset S_{M}$.
As $Q\to M_{Dol}$ is surjective, there is an irreducible curve $C'\subset Q$ and
finite map $C' \to C$. One obtains a family of Higgs bundles
on $C'$ by restriction. $S_{C'}= C'$ since it is closed and contains
the preimage of $C-\{s\}$. Therefore $s\in S_{M}$ and so $S_M$ is closed.
\end{proof}
\begin{prop}\label{prop:restrict}
Let $C\subset X$ be a curve
obtained as a complete intersection
of divisors associated to an $N$th power of $L$, with $N >> 0$.
Then the restriction maps
$M_B(X,n) \to M_B(C,n)$ and $M_{Dol}(X,n) \to M_{Dol}(C,n)$
are injective (on $\C$-valued points)
and compatible with the homeomorphisms of the associated analytic spaces.
\end{prop}
\begin{proof} This well known to the experts, so we
will merely indicate the main ideas.
The first part follows from the Lefschetz hyperplane
theorem \cite{mi}. For the second part, first note that the restriction
of a semistable Higgs bundle is semistable thanks to Simpson's
generalization of the Mehta-Ramanathan theorem \cite{s1},
so the map is well defined.
As for injectivity, choosing $N >>0$ guarantees that
$$H^1(E_1^*\otimes E_2\otimes I_C) = 0$$
for any pair of Higgs bundles on $X$. Thus any isomorphism of their
restrictions can be lifted to map of the Higgs bundles
which can be seen to be
an isomorphism (using, for example, the fact that polystable bundles
are direct sums of simple bundles). The last part is just a
restatement of the naturality of the correspondence.
\end{proof}
\begin{remark} The images of $M_{Dol}^s(X,n)$ and
$M_B^{irr}(X,n)$ under restriction lie in the corresponding subsets
for $C$.
\end{remark}
Given a Higgs bundle $(E,\theta)$, define $H^i(E,\theta)$ to be
the $i$th hypercohomology of the
complex:
$$
E\stackrel{\theta\wedge}{\longrightarrow}
\Omega_X^1\otimes E \stackrel{\theta\wedge}{\longrightarrow}
\Omega_X^2\otimes E \ldots
$$
We define the cohomology support loci as:
$$\Sigma^k_{m,Dol}(X,n) =
\{[(E,\theta)] \in M_{Dol}(X,n)\,|\,
(E,\theta)\>{ polystable\> and\>}dim\, H^k(E,\theta) \ge m\} $$
We can make an analogous definition for $M_B$. Given a representation
of $\rho:\pi(X)\to Gl_n(\C)$, there exists (up to isomorphism) a unique
rank $n$ locally constant sheaf, or local system, with $\rho$ as
its monodromy representation. So we can, and will, view $M_B(X,n)$ as a moduli
space of local systems on $X$. Let
$$\Sigma^k_{m,B}(X,n) =
\{ V \in M_B(X,n)\, | \,
V\> { semisimple\> and}\> dim\, H^k(X,V) \ge m\}$$
\begin{prop} $\Sigma^k_{m,Dol}$ is Zariski closed in
$M_{Dol}$. $\Sigma^k_{m,B}$ is is Zariski closed in
$M_B$. $\Sigma^k_{m,Dol}$ and $\Sigma^k_{m,B}$ coincide under
the correspondence between $M_{Dol}^{an}$ and $M_B^{an}$.
\end{prop}
\begin{proof}
Note that if $(E_i,\theta_i),\, i=1,2$ are equivalent semistable
bundles with $(E_1,\theta_1)$ polystable, then
$$dim H^k(E_1,\theta_1) \ge dim H^k(E_2,\theta_2)$$
by subadditivity of cohomology.
Therefore the first statement follows from proposition
\ref{prop:closed} and
the semicontinuity theorem for cohomology \cite[7.7.5, 7.7.12]{ega3}
The second statement is proved in \cite{a2}. The last part follows
from \cite[2.2]{s1}.
\end{proof}
\begin{thm}(Hitchin\cite{hit1}) Let $C$ be a smooth projective curve.
Then the stable locus $M_{Dol}^{s,an}(C,n) $ is smooth and
carries a natural hyperk\"ahler structure $(g,\I,\J)$. Where $\I$ is the usual
complex structure and $\J$ is the structure induced from the
identification $M_{Dol}^{s,an} \cong M_B^{irr,an}$.
\end{thm}
\begin{remark} Hitchin stated this only when $n=2$, although his
proof presumably works for higher rank bundles.
In any case, Fujiki \cite{fu} has established a much more general
result.
\end{remark}
Given a rank $n$ Higgs bundle $(E,\theta)$ on $X$, let
$\theta^{i}\in H^{0}(X, S^{i}\Om^{1})$ be the $ith$ (symmetric) power
Let
$$h_X(E,\theta)\in S_n(X) = \bigoplus_{i=1}^n\, H^0(X,S^i\Omega_X^1)$$
be the map $ (E,\theta) \mapsto trace(\theta^{i})$.
This differs from Simpson's and Hitchin's
definition in that they use the
characteristic polynomial. However there is
an algebraic automorphism $\sigma$ of the target space such that
$\sigma\circ h$ agrees with their map, so they are essentially the same.
Therefore:
\begin{thm} (Simpson\cite{s2}, Hitchin\cite{hit2}) $h_X$ gives a
proper morphism from $M_{Dol} $
to $S_n(X)$. When $dim X = 1$, $h_X$ is surjective and
generically lagrangian with respect to the
hyperk\"ahler structure on $M_{Dol}^{s,an}$.
\end{thm}
Suppose that a curve $C\subseteq X$ has been chosen as in
\ref{prop:restrict},
then we obtain a commutative diagram:
$$\begin{array}{ccc}
M_{Dol}(X,n) & \hookrightarrow & M_{Dol}(C,n)\\
h_X\downarrow & & h_C\downarrow \\
S_n(X) & \stackrel{i}{\to} & S_n(C) \\
\end{array} $$
While $i$ need not be injective, we do have:
\begin{lemma} The map $h_X(M_{Dol}(X,n))\to h_C(M_{Dol}(C,n))$ is
finite.
\end{lemma}
\begin{proof} This follows from the properness of all the other maps
in the diagram.
\end{proof}
Consequently, fibers of $h_X$ are components of fibers of $i\circ h_X$.
Putting the above results together yields the main theorem:
\begin {thm} Let $X$ be a smooth projective variety.
If $\tilde \Sigma $ is a connected
component of the normalization
of $\Sigma^k_{m,Dol}$ (with its reduced subscheme structure) which
meets $M_{Dol}^s$,
and $\tilde h:\tilde \Sigma \to h(\tilde \Sigma)$ the
natural map.
Then $\tilde h$ is generically lagrangian.
In particular the connected
components of its general fibers
are abelian varieties of dimension half of that of $dim\,
\tilde\Sigma$
\end{thm}
\begin{proof} Let $C\subseteq X$ be a complete intersection curve of high
degree. Then $\Sigma_{m,Dol}^k(X,n)$ is a quaternionic subvariety of
$M_{Dol}(C,n)$ by the previous results. Therefore the theorem follows
from \ref{thm:lagr}.
\end{proof}
\begin{cor} \label{cor:fibdim} Any irreducible component $F$ of every
fiber of $\tilde h$ satisfies $dim F = dim\, \tilde \Sigma/2$.
\end{cor}
\begin {proof} By upper semicontinuity of dimensions, it is enough to
prove $dim F \le dim\, \tilde \Sigma/2$. By \ref{cor:norm}, $\tilde
\Sigma$ is compatible with the complex structure coming from
$\Sigma_{m,B}^k(X,n)$. With this structure $\tilde \Sigma^{an}$ is Stein,
therefore $H_i(\tilde \Sigma^{an}, \Z) = 0$ for $i > dim \tilde S$ \cite{n}.
On the other hand, with the original complex structure
$\tilde \Sigma$ is quasiprojective, and $F$ is a proper subvariety.
Therefore the fundamental class of $F$ defines a nonzero element of
$H_{2dim(F)}(\tilde\Sigma^{an},\Z)$, and this forces the inequality.
\end{proof}
\begin{remark} The theorem can also be used to recover a result of Biswas
\cite[8.2]{b} about Poisson commutivity of higher dimensional Hitchin
maps.
\end{remark}
In order to get a better analogue of the original theorem of Green and
Lazarsfeld, we need to break $\Sigma_m^k$ up into $(p,q)$ parts.
For a Higgs bundle $(E,\theta)$, let
$H^{pq}(E,\theta)$ be the $E^{pq}_\infty$ term
of the spectral sequence:
$$E_1^{pq} = H^q(X,\Omega_X^p\otimes E) \Rightarrow H^{pq}(E,\theta)$$
Then set $S^{pq}_m(X,n)$ equal to the closure of
$$\{(E,\theta) \in M_{Dol}(X,n) | dim\, H^{pq}(E,\theta) \ge m\}$$
Clearly
$$\Sigma_{m,Dol}^k(X,n)
= \bigcup_{ m_0+m_1+\ldots = m}\bigcap_{p}\, S^{p,k-p}_{m_p}(X,n)$$
\begin{cor} Let $\tilde S$ be a connected component of the
normalization of $S_m^{pq}$ which meets $M_{Dol}^s$.
The connected components of
general fibers of the restriction of $h$ to $\tilde S$
are abelian varieties. All fibers have dimension equal to one half
of that of $\tilde S$.
\end{cor}
This can be deduced from the theorem and the previous corollary
using the next lemma:
\begin{lemma} Let $X$ be a noetherian topological space. Suppose that
there are nested closed sets
$$X = X^i_0 \supseteq X^i_1 \supseteq \ldots$$
$i = 0,\ldots n$. Then any irreducible component of $X^i_m$ is an
irreducible component of some set of the form
$$Y_m = \bigcup_{m_0+m_1+\ldots = m}\bigcap_{ i}\,X_{m_i}^i$$
\end{lemma}
\begin{proof} Suppose that $S$ is an irreducible component of $X_{m_0}^0$.
We can assume that $S$ is not contained in $X_{m_0+1}^0$. Similarly,
let $m_1,\ldots$ be the largest integers for which $S$ is contained
in $X_{m_1}^1,\ldots$. Then is easily seen to be an irreducible component of
$Y_m$, where $m = m_0+m_1+\ldots$.
\end{proof}
If $(E,\theta)$ is a Higgs bundle, its dual is $(E^*,-\theta)$ where
$\theta$ is viewed as section of $\Om^1\otimes End(E^*)\cong
\Om^1\otimes End(E)$. This is compatible with the duality of local
systems. Simpson has already observed a duality theorem holds on
cohomology. This can be refined slightly:
\begin{prop} If $d = dim\, X$ then
$$H^{pq}(E,\theta) \cong H^{d-p,d-q}(E^*,-\theta)^*$$
\end{prop}
\begin{proof} Set $\omega_X = \Omega_X^d$.
A special case of the Grothendieck-Serre duality theorem is that
$$\HH^{d-i}({\cal H}om(V^\dt,\omega_X)) \cong \HH^i(V^\dt)^*$$
for any finite complex of locally free sheaves $V^\dt$.
The natural pairing of $\Om^p\otimes \Om^{d-p}\to \omega_X$ induces an
isomorphism of complexes
$${\cal H}om((\Om^\dt\otimes E,\theta), \omega_X) \cong
(\Om^\dt\otimes E^*,-\theta)[d]$$
upto sign. This respects the the filtrations
$${\cal H}om((\Om^{\le p}\otimes E,\theta), \omega_X) \cong
(\Om^{\ge d-p}\otimes E^*,-\theta)[d]$$
and induces isomorphisms on the associated graded parts. Therefore
there is an isomorphism between the associated spectral sequences
converging to the hypercohomology groups on the left and right hand
sides. The $E_{\infty}$ terms can be identified with the groups of the
proposition.
\end{proof}
\begin{cor}\label{cor:duality} $S^{pq}_m(X,n) = S^{d-p,d-q}_m(X,n)$
\end{cor}
\section{ Cohomology support loci for vector bundles.}
We will use the same notation as in the previous section. Let $M_V(X,n)$
be the closed subscheme of $M_{Dol}(X,n)$ parameterizing polystable
Higgs bundles of the form $(E,0)$.
Let $M^s_V$ be the open subset of stable bundles.
Note that $(E,0)$ is (poly, semi)stable
if and only if $E$ is (poly, semi)stable in the usual
sense (with respect to ``slope'').
So $M_V$ is just the coarse moduli space of semistable
vector bundles of rank $n$ with trivial Chern classes.
Note that $(E,\theta)$ is (poly, semi)stable if $E$ is.
Let $T^*M_V$ (respectively $T^*M_V^s$) be the set of all
Higgs bundles $[(E,\theta)]$ such that
$E$ is polystable (respectively stable).
$M_V$ may have singularities, so the notation $T^*M_V$ is
merely suggestive; some justification for it will be given below.
There is a morphism $T^*M_V \to M_V$ given
by projection.
A vector bundle is determined by a $1$-cocycle $g_{ij}\in Z^1({\cal U}, Gl_{n}(O_{X}))$,
and a first order deformation to it is described by a cocycle of the form
$$g_{ij}+\epsilon\gamma_{ij}\in Z^1({\cal U},Gl_{n}(O_{X}[\epsilon]/(\epsilon^{2})))$$
$\gamma_{ij}$ defines a cocycle with values in $End(E)$.
So the Zariski tangent space to $[E] \in M_{V}^s$ can be identified
with $H^{1}(X,End(E))$.
The obstruction to lifting a first order deformation $v\in H^{1}(X,End(E))$ to
one of second order is given by
$[v, v]\in H^{2}(X,End(E))$.
There are in fact no higher obstructions,
so the tangent cone to $[E]\in M_{V}^s$ is just
$\{v\in H^{1}(X,End(E))\, |\, [v, v] = 0\}$ \cite[9.4]{gm}.
By a theorem of Donaldson \cite{d} and Uhlenbeck-Yau \cite{uy}, $E$
carries unitary flat connection $\del$. Consequently, $End(E)$ also
carries a unitary flat connection. Hodge theory with unitary flat
coefficents and the self duality of $End(E)$ shows that there is a
conjugate linear isomorphism
$$H^{i}(X,End(E)) \cong H^{0}(X,\Om^{i}\otimes End(E))$$
preserving the graded Lie brackets. The Higgs fields on $E$ are
precisely the conjugates of the vectors in the tangent cone.
This implies that they have the same dimension as real algebraic
varieties, and therefore as complex algebraic varieties.
It is sometimes better to view
$T^{*}_{[E]}=H^{0}(X,\Om^{1}\otimes End(E))$ as dual to the tangent space,
via the hard Lefschetz pairing
$$<\alpha,\beta> = \int_{X}trace(\alpha\cup \beta)\cup L^{dimX -1}$$
In an analogous fashion , the tangent space to any stable point
$[(E,\theta)]\in M_{Dol}$ is $H^{1}(End(E,\theta))$ where
$End(E,\theta)$ is the Higgs bundle $(End(E),ad(\theta))$ \cite[10.5]{s2}.
The tangent cone is defined by the quadratic form associated to the
cup product
$$H^{1}(End(E,\theta)) \times H^{1}(End(E,\theta)) \to
H^{2}(End(E,\theta)).$$
\begin{prop}\label{prop:cotang} $T^*M^s_V$ is an open subset of $M_{Dol}$.
If $M\subseteq M_{V}$ is an irreducible component, then $dim M =
dim (\pi^{-1}M)/2 $ and $M$ is an irreducible component of
$h^{-1}(0)\cap \pi^{-1}M$.
\end{prop}
\begin{proof}
As pointed out in the remarks preceding \ref{prop:closed},
etale local cross sections to $Q^s\to M_{Dol}^s$ exist. Therefore
there is an
etale neighbourhood $T \to M_{Dol}$ of any stable point $(E_{0},\theta_{0})$
and family of Higgs bundles
$(E_{t},\theta_{t})$ parameterized by $T$, such that
$[(E_{t},\theta_{t})]$ is precisely the image of $t$. Stability is an
open condition \cite[3.7]{s2},
thus if $E_{0}$ were stable, then this would hold in open
neighbourhood of $0\in T$.
Note that $M_{V} \subseteq h^{-1}(0)$, and the dimension of any component
of $h^{-1}(0)$ is half the dimension of an irreducible component of
$M_{Dol}$ containing it by \ref{cor:fibdim}. Thus it suffices to prove that
$dim M \ge 2 dim(\pi^{-1}M)$.
Choose a general point $[E]\in M$. Then consider the terms of low
degree for the spectral sequence converging to the cohomology of
$End(E,0)$:
$$0\to H^{0}(X,\Om^{1}\otimes End(E)) \to H^{1}(End(E,0)) \to
H^{1}(X, End(E)) \to 0$$
It follows that the tangent cone $C_{1}$ of $M_{Dol}$ at $[(E,0)]$ maps to the
tangent cone $C_{2}$ of $M_{V}$ at $[E]$, and the fiber over $0$ is the
cone $\Ts_{[E]}$. This implies that the dimension of $C_{1}$ is less than
or equal to $dim C_{2} + dim \Ts_{[E]} = 2dim C_{2}$.
\end{proof}
The intersections of $S^{pq}_m$ with $M_V$ and
$T^*M$ have a rather concrete description which is very close
to the spirit of \cite{gl1}. Let us write $T^*S^{pq}_{m}$ for
$S^{pq}_m \cap T^*M_V$.
\begin{prop} If $(E,\theta)$ is polystable, then $[(E,\theta)] \in T^*S^{pq}_{m}$
if and only if the $p$th cohomology of the complex
$$\ldots H^q(X,\Om^p\otimes E)\stackrel{\theta\wedge}{\longrightarrow}
H^q(X,\Om^{p+1}\otimes E)\stackrel{\theta\wedge}{\longrightarrow} \ldots$$
has dimension greater than or equal to $m$. In particular
$(E,0)\in S^{pq}_1$ if and only if $H^q(X,\Om^p\otimes E) \not=0$.
\end{prop}
The proposition is a consequence of the next two lemmas.
\begin{lemma} If $E$ is polystable, then $H^{pq}(E,\theta)$ is just
the $p$th cohomology of the complex
$$\ldots H^q(X,\Om^p\otimes E)\stackrel{\theta\wedge}{\longrightarrow}
H^q(X,\Om^{p+1}\otimes E)\stackrel{\theta\wedge}{\longrightarrow} \ldots$$
\end{lemma}
\begin{proof}
This is equivalent to the assertion that the spectral sequence
$$E^{pq}_1 = H^q(X,\Om ^p\otimes E) \Rightarrow H^{p+q}(E,\theta)$$
degenerates at $E_2$.
By the theorem of Donaldson and Uhlenbeck-Yau,
$E$ carries a unitary flat connection $\nabla$. The lemma now follows by
applying \cite[III 3.6]{a2} to the complex constructed in the proof
of [loc.cit, IV 2.1]. (In the notation of that paper, the spectral
sequence associated to ${\cal V}(0)$ coincides with the one above.)
An alternative argument can be given by modifying the proof of
\cite[3.7]{gl1} by replacing $\partial$ and $\bar\partial$ by the
$(1,0)$ and $(0,1)$ parts of $\nabla$.
\end{proof}
\begin{lemma} If $(E_t,\theta_t)$ is a family of Higgs bundles
parameterized by a smooth curve $T$ such that $E_0$ is
polystable for some $0\in T$ and $dim \,H^{pq}(E_t,\theta_t) \ge m$
for $t\not= 0$ then $dim\, H^{pq}(E_0,\theta_0) \ge m$.
\end{lemma}
\begin{proof}
We can assume that $T = Spec R$.
Consider the complex of $R$-modules
$$\ldots H^q(X,\Om^p\otimes E)\stackrel{\theta\wedge}{\longrightarrow}
H^q(X,\Om^{p+1}\otimes E)\stackrel{\theta\wedge}{\longrightarrow} \ldots$$
Our assumptions imply that if one tensors this by the residue field
of $t\not= 0$, the $p$th cohomology has dimension greater than or equal
to $m$ (This is true regardless of whether the spectral sequence
degenerates for $t\not= 0$, because at any rate $dim\, E_2\ge dim
E_\infty$). Therefore this property persists for $t=0$, and the lemma
follows from the previous one.
\end{proof}
The next result gives a useful dimension estimate on the cohomology
support loci. It can also be deduced using Green's and Lazarsfeld's
deformation theory \cite{gl1}.
\begin{prop}\label{prop:dimest} Let $S_V$ be an irreducible
component of $S^{pq}_{m}\cap M_V$, and let $S$ be the irreducible component of $S^{pq}_m$
containing $S_V$.
Then for a general point $[E] \in S_V$
$$dim S_V = dim (T^*_{[E]}\cap S) =
\frac {dim S}{2}$$
\end{prop}
\begin{proof} By \ref{cor:fibdim} and \ref{prop:cotang},
$dim S_V = \frac {dim S}{2}$. The remaining equality follows
from $dim S = dim S_V + dim (T^*_{[E]}\cap S) $.
\end{proof}
\section{ $\C^*$-invariance}
In the last section, we made an explicit study of the intersection of
the cohomology support loci with $T^*M_V^s$. There are some features
of this geometry which extend to the whole space. First, recall:
\begin{thm} (Simpson \cite{s2}) There is an algebraic $\C^*$-action
on $M_{Dol}$ given by $t:(E\theta) \mapsto (E,t\theta)$. For any
point $e\in M_{Dol}$ the limit of $ te$ as $t\to 0$ exists.
\end{thm}
Of particular interest are the fixed points. A Higgs bundle
$(E,\theta)$ is called a complex variation of
Hodge structure if $E$ admits a grading $\oplus E^p$ such that
``Griffiths transversality'' $\theta(E^p) \subset E^{p-1}$
holds. Given a complex variation of Hodge structure,
let $T$ be the automorphism which acts by $t^{-p}$ on $E^p$ where
$t\in \C^*$. Then
$T$ induces an isomorphism $(E,\theta)\cong (E,t\theta)$. Therefore
$[(E,\theta)]$ is a fixed point. Conversely, Simpson has shown that all
fixed points arise this way, and in fact if the underlying Higgs bundle
$(E,\theta)$ is stable then the grading is uniquely determined (up to
to a shift of indices).
Note that if $\theta = 0$, then we can take $E^0 = E$
The previous theorem implies that the $\C^*$-action extends to a
morphism of reduced schemes
${\Bbb A}^1\times M_{Dol,red} \to M_{Dol,red}$. The image of
$\{0\}\times M_{Dol}$ is the fixed point set $F$. Thus we
get a morphism $\pi:M_{Dol,red} \to F_{red}$ by composing
$$M_{Dol,red} \cong \{0\}\times M_{Dol,red} \to F_{red}.$$
$\pi$ extends the map $T^*M_V\to M_V$ constructed earlier,
and exhibits $M_{Dol}$ as a family of cones.
The cohomology support loci are compatible with this conical
structure:
\begin{lemma} $\Sigma^k_{m,Dol}$ is invariant under the $\C^*$-action.
\end{lemma}
\begin{proof}
There is an isomorphism $H^k(E,\theta)\cong H^k(E,t\theta)$
given on the level of complexes by
multiplication by $t^{p}$ on $\Om^p\otimes E$.
\end{proof}
As these sets are closed, we obtain:
\begin{cor} $\pi(\Sigma^k_{m,Dol}) \subset \Sigma^k_{m,Dol}$, so
$\Sigma_{m,Dol}^k$ is
a family of subcones.
\end{cor}
\begin{cor} Any irreducible component of $\Sigma^k_{m,Dol}$ contains a
complex variation of Hodge structure.
\end{cor}
Conjectures of Simpson \cite{s1} and Pantev \cite{p} suggest that much more
should be true, for example every component should contain an integral
variation of Hodge structure.
In the rank one case, the $\C^*$ invariance of the above sets is a
powerful constraint. In fact, it leads to a proof
of theorem \ref{thm:gl}, \cite{a1,s3}.
\section{ Generic vanishing}
In this section, we relax the condition on the Higgs bundles.
We no longer
insist that the Chern classes vanish, or even that they are locally
free.
Let $X$ be a smooth projective $d$-dimensional variety
A Higgs sheaf on $X$ is a torsion free coherent $O_{X}$-module $E$
together with morphism $\theta:E \to \Om^{1}\otimes E$ satisfying
$\theta\wedge\theta = 0$. There is a useful alternative viewpoint
which we now recall.
Let $\pi:
T^{*}X \to X$ be the cotangent
bundle. A cotangent vector $\eta\in T^{*}X$ can also be viewed as
an element of the fiber of $\pi^*\Om^{1}$.
This defines a canonical section $\Theta$ of $\pi^*\Om^{1}$. If
$\cal E$ is a torsion free $O_{T^{*}X}$-module, such that $supp(\E)\to
X$ is finite, then $E = \pi_{*}{\cal E}$ is a torsion free coherent
sheaf, and the map
$$ [E\stackrel{\theta}{\rightarrow} \Om^1 \otimes E ]
\cong \pi_{*}(\E \otimes [O_{T^{*}X}\to \pi^*\Om^{1}])$$
defines a Higgs structure on $E$. In fact, every Higgs sheaf arises
this way from a unique $\E$ \cite{s1,s2}. The
support of $\E$ is
precisely the set of eigenforms for $\theta$. In other words,
$supp(\E)\cap T^{*}_{x}$ is the set
of cotangent vectors $\eta$ satisfying $\theta_{x}(v)= \eta v$
for some nonzero $v\in E_{x}$. Define the degeneracy locus of
$(E,\theta)$ as $supp( \E)\cap X$ where
$X\subset T^{*}X$ is identified with the zero section. More concretely,
if $E$ is locally free then
$x$ lies in the degeneracy locus if and only if $\theta_{x}$ has a zero eigenform or,
equivalently, is not injective. Let $degen(\theta)$ be the dimension
of the degeneracy locus of $(E,\theta)$ if it is nonempty, or $-1$
otherwise. It will simplify matters to define $dim\emptyset = -1$.
Given a Higgs sheaf $(E,\theta)$, the cohomology $H^{i}(E,\theta)$ can
be defined as the hypercohomology of the complex
$$
E\stackrel{\theta\wedge}{\longrightarrow}
\Omega_X^1\otimes E \stackrel{\theta\wedge}{\longrightarrow}
\Omega_X^2\otimes E \ldots
$$
as before.
\begin{prop}\label{prop:ext} If $X\subset T^{*}X$ is identified with the zero
section, then for any Higgs sheaf $(E,\theta)$ and corresponding
sheaf $\E$ on $\Ts X$,
$$H^{i}(E,\theta) \cong Ext^{i}(O_{X},\E)$$
\end{prop}
\begin{proof}
The zero locus of $\Theta$ is precisely $X$. Therefore $O_{X}$ is
quasiisomorphic to the Koszul complex
$$K_{\bullet} = \ldots \wedge^{2}\pi^{*}T_{X}\to \pi^{*}T_{X} \to O_{T^{*}X} \to 0$$
Therefore
$$Ext^{i}(O_{X},\E) \cong H^{i}({\cal H}om(K_{\dt}, \E))
\cong H^{i}(\pi_{*}(K_{\bullet}^{*}\otimes \E))$$
By the projection formula,
$$\pi_{*}(K_{\bullet}^{*}\otimes \E) = \Omega_{X}^{\bullet}\otimes E$$
and the differentials are easily seen to be given by $\theta\wedge$.
\end{proof}
\begin{thm} If $(E,\theta)$ is a Higgs sheaf and $i > degen(\theta) +
d$, then $H^i(E,\theta) = 0$. If $E$ is locally free then
$H^i(E,\theta)=0$
for $i < d -degen(\theta)$ also.
\end{thm}
\begin{proof} The support of ${\cal E}xt^{\bullet}(O_X,\E)$ lies
in the degeneracy locus $supp(\E)\cap X$.
Therefore the first part of the theorem follows from \ref{prop:ext}
and the spectral sequence
$$ H^p({\cal E}xt^q(O_X,\E)) \Rightarrow Ext^{p+q}(O_X,\E).$$
The second statement follows by duality \cite[2.5]{s1}.
\end{proof}
As an immediate corollary: $\Sigma^{i}_{1} \not= M_{Dol}$ if there
exists a Higgs bundle $(E,\theta)$ with $ |i -d| > degen(\theta) $.
When $E$ is stable, this can be interpreted as a generic vanishing
theorem:
\begin{cor} If $[E]\in M^{s}_{V}$, and $\theta$ is a Higgs field on
$E$. Then in any neigbourhood of $[E]$ in $M^{s}_{V}$ there is an $[E']$
such that
$$H^{q}(X,\Om^{p}\otimes E') = 0$$
for $|p + q -d|> degen(\theta)$.
\end{cor}
One gets a very concrete statement by restricting to diagonal
Higgs fields.
\begin{cor} If $\phi \in H^{0}(X,\Om^{1})$, then for any irreducible
component $M$ of $M_{V}(X,n)$, there exists a semistable vector
bundle $E$, with $[E]\in M$ and
$$H^{q}(X,\Om^{p}\otimes E) = 0$$
for $|p + q-d| > dim \{x\,|\, \phi_{x} = 0\} $.
\end{cor}
\begin{proof} Apply the theorem to $(E,\theta)$ where $[E]\in M$ is a
general point and $\theta = \phi I_{E}$.
\end{proof}
The above arguments can be refined to yield a codimension estimate.
Given a Higgs field $\theta$, clearly $degen(\lambda\theta) =
degen(\theta)$ for any $\lambda \not= 0$. Thus $degen$ gives a map,
from the projectivization $\PP\Ts_{[E]}$ to the set of natural numbers,
which is easily seen to be upper semicontinuous.
\begin{cor} If $S \subseteq \Sigma^{i}_{1}\cap M_{V}^s$ is an irreducible
component, and $[E]\in S$ a general point. Assume that $M_{V}$ is
smooth at $[E]$, then the codimension of $S$,
in the irreducible component of $M_{V}$ containing it, is greater than
$dim\, \{[\theta] \in \PP\Ts_{[E]}\, | \, degen(\theta) \ge |i-d| \}.$
\end{cor}
\begin{proof} The smoothness assumption implies that $\Ts_{[E]}$ is a vector space.
By \ref{prop:dimest},
the codimension of the projectivized cone $V = \PP(\Ts_{[E]}\cap
\Sigma^{i}_{1})\subset \PP \Ts_{[E]}$ coincides with the above
codimension.
By the theorem, $V$ cannot meet $\{[\theta] \in \PP\Ts_{[E]}\, | \,
degen(\theta) \ge |i-d| \}$. Thus the corollary is a consequence of
Bezout's theorem.
\end{proof}
\section{Homotopy invariance}
The cohomology support loci are clearly isomorphism invariants.
But much more is true, namely the $\Sigma_{m,B}^k$
depend only on the homotopy type of the space. We will give some
variants of this which will be quite useful for the construction of
examples.
\begin{prop} Suppose that $f:X \to Y$ is a morphism of smooth projective
varieties, with connected fibers, such that the induced map on homotopy groups
$\pi_i(X^{an})\to \pi_i(Y^{an})$ is an isomorphism for $i \le k$
and a surjection for $i = k+1$ with $k >0$. Then for all $m$,
$f^*(\Sigma_{m,Dol}^i(Y))$ is contained in $ \Sigma_{m,Dol}^i(X)$
when $i = k+1$ and equality holds when $i \le k$.
\end{prop}
\begin{proof} It suffices to prove the corresponding statement for
$\Sigma_B$. First note that by standard arguments in topology
\cite[pages 99-100]{sp} $f^{an}$ is homotopy
equivalent to a fibration of topological spaces $f':X'\to Y'$.
The homotopy long exact sequence and the hypothesis
implies that
the first $k$ homotopy groups of the fiber $F$
vanish. Therefore by Hurwitz theorem $H^j(F,V) = 0$ for $j \le k$
and $V$ and arbitrary coefficient group. Thus the Leray-Serre spectral
sequence yields an injection
$$H^i(Y^{an},V) \to H^i(X^{an},f^*V)$$
when $i = k+1$ and an isomorphism when $i \le k$.
\end{proof}
\begin{cor}\label{cor:lef} If $X\subset Y$ is general hyperplane
section where $dim Y \ge 3$, then under restriction:
$\Sigma_{m,Dol}^k(Y) \subseteq \Sigma_{m,Dol}^k(X)$ for $k =
dim X$ and equality holds when $k < dim X$.
\end{cor}
\begin{proof} The hypothesis of the theorem is fulfilled by the Lefschetz hyperplane
theorem \cite{mi}.
\end{proof}
\begin{prop} If $f:X\to Y$ is a surjective map of smooth projective
varieties, then $f^*\Sigma_{m,Dol}^k(Y) \subseteq \Sigma_{m,Dol}^k(X)$ for
$k \le dim Y$.
\end{prop}
\begin{proof} Once again, we will work with $\Sigma_B$. We will also
omit the superscript ``an''.
We will show that the map $f^*:H^k(Y,V)\to H^k(X,V)$ is injective
for any local system $V$.
To begin with, assume that $f$ is generically finite.
Then one gets a transfer or Gysin homomorphism $f_!:H^k(X,V)\to
H^k(Y,V)$ as the Poincar\'e dual of $H^{2dimY - i}(Y,V^*)\to
H^{2dimY-i}(X,V^*)$. Arguing as in the case of constant coefficients,
$\frac{1}{deg\,f}f_!$ splits $f^*$.
In general, choose a complete intersection $Z\subset X$ of hyperplane
sections, such that $f|_Z$ is finite. The map
$H^k(Y,V)\to H^k(Z,f^*V)$ factors through $f^*$. Consequently $f^*$ is
again injective.
\end{proof}
We extend the notation $\Sigma_B^k(X)$ so as to
allow $X$ to be any topological space. Of course, there won't be an
analogue $\Sigma_{Dol}$ in general. Our next task is to prove
a K\"unneth decomposition. Given two spaces $X_1, X_2$, there is a
morphism
$$\tau:M_B(X_1,n_1)\times M_B(X_2,n_2) \to M_B(X_1\times X_2, n_1n_2)$$
given by $\tau(V_1,V_2) = p_1^*V_1\otimes p_2^*V_2$. Also let
$$\sigma:M_B(X,n_1)\times M_B(X,n_2)\to M_B(X, n_1+n_2)$$
be the morphism $\sigma(V_1,V_2)\to V_1\oplus V_2$.
When the $X_i$
are smooth projective, similar morphisms can be defined for $M_{Dol}$.
\begin{prop}\label{prop:kunneth}
$\Sigma_{1,B}^k(X_1\times X_2\times \ldots X_N,n)$ is
the union of images of products
$$\sigma(\tau(\Sigma_{1,B}^{k_1}(X_1,n_1)\times\ldots
\Sigma_{1,B}^{k_N}(X_N,n_N))\times M_B(X_1\times\ldots
X_N,n-n_1n_2\ldots n_N))
$$
as $k_i$ and $n_i$ range over partitions of $k$ and factorizations
of integers no greater than $n$ respectively.
\end{prop}
We will prove the proposition when $N=2$. Let
$G_i =\pi_1(X_i)$, then of course $G=G_1\times G_2$ is the
fundamental group of the produce. The classification of irreducible
$G$-modules is well known, at least for finite groups.
\begin{lemma} Any irreducible finite dimensional complex representation
of $G$ is the form $V_1\otimes_\C V_2$ where $V_i$ are irreducible
$G_i$ representations.
\end{lemma}
\begin{proof} Let $V$ be a nonzero irreducible representation of $G$. Then
it contains a nonzero irreducible $G_1$-module $V_1$. The vector
space $G_2V_1 = \sum_{g\in G_2} gV_1$ is a $G$-module, so it
must coincide with $V$. Therefore $V$ is isomorphic to direct sum of
copies of $V_1$, so by standard arguments \cite[section 27]{cr},
$R=End_{G_1}(V)$ is isomorphic to
the $\C$-endomorphism ring of an irreducible $R$-submodule $V_2$. $V_2$ is
necessarily a $G_2$-module, and the natural map $V_1\otimes V_2\to V$
is easily seen to be an isomorphism. $V_2$ must be irreducible, since
otherwise, we could find a smaller submodule $V_2'$, and the $G$-module
$V_1\otimes V_2'$ would violate irreducibility of $V$.
\end{proof}
The above proposition is now a consequence of the usual K\"unneth
formula for cohomology. If $X$ is a product of subvarieties then,
there is a similar decomposition for $\Sigma_{Dol}^k$.
\section{Examples}
In this final section, we will give some explicit examples.
To simplify notation, we will drop the decorations ``an, Dol, B'';
it should be clear from context where we are.
In addition to the
previous results, we will need the following well known fact:
\begin{lemma}
The Euler characteristic of a rank $n$ local system $V$ on a variety $X$
is $n$ times the Euler characteristic $e(X)$ of $X$.
\end{lemma}
\begin{proof}
There are a number of ways to
prove this; perhaps the easiest is choose a finite triangulation of
$X$, then observe that each term $S^\dt(V)$ in the simplicial
cochain complex with coefficients in $V$ is (noncanonically) isomorphic to
$S^\dt(\C)^{\oplus n}$.
\end{proof}
\begin{ex} Let $C$ be a smooth projective curve of genus $g > 1$.
If $V$ is a nontrivial irreducible rank $n$ local system then
$H^0(C,V) = 0$ and $H^2(C,V) \cong H^0(C,V^*)^*=0$.
Therefore $dim\,H^1(C,V) = 2n(1-g)$. So that
$\Sigma_{2n(1-g),}^1(C,n) = M(C,n)$ because irreducible
local systems are dense in $M(C,n)$. If $m > m_0=n(1-g)$, then
$\Sigma_{2m}^1(C,n)$ is locus of semisimple local systems with a trivial
summand of rank $m-m_0$. If $X= C_1\times C_2$ is a product of two
curves, then
$$M(X,n) =
\bigcup_{a+b=n}M(C_1,a)\times M(C_2,b) $$ and
$V\in \Sigma_1^1(X,n)$ if and only if it has a direct summand of
the form $p_i^*V_i$
by \ref{prop:kunneth}. Hitchin's map respects this
decomposition. More precisely, the product
$$h_{C_1}\times h_{C_2}:M(C_1,a)\times M(C_2,b) \to S_a(C_1)\times
S_b(C_2)$$
factors through the restriction of $h_X$ to the above component, and
the map
$$h_X( M(C_1,a)\times M(C_2,b))\to S_a(C_1)\times S_b(C_2)$$
is a bijection. To see this, let
$$(E,\theta)=(p_1^*E_1\otimes p_2^*E_2,
p_1^*\theta_1\otimes I_{E_2} + I_{E_1}\otimes p_2^*\theta_2)$$
be a point
on this component.
Then the projection of $h_X(E,\theta)$ to $S_a(C_1)\times S_b(C_2)$ is
$(h_{C_1}(E_1,\theta_1), h_{C_2}(E_2,\theta_2))$.
Moreover, the value of $h_X(E,\theta)$ at any point
$x=(x_1,x_2)\in X$ is determined by the eigenvalues of
$\theta_i(x_i)$ which is, in turn, determined by
$h(E_i,\theta_i)(x_i)$.
\end{ex}
The generic vanishing theorem, shows that $\Sigma_1^k(X,n) $ is
proper whenever the $X$ has $1$-form without zeros. When $X$
is an abelian variety, we can say much more.
\begin{ex} Let $A$ be an abelian variety. Then for each $k$,
$\Sigma_{1}^k(A,n)$ is locus of semisimple local systems with
a trivial direct summand. To prove this, notice that
$A$ is homeomorphic to a product of circles, and by \ref{prop:kunneth}, it
suffices to treat the case of a single circle. In this case its
clear, as $H^0(S^1,V)$ and $H^1(S^1,V)$ are respectively
just the invariants and coinvariants of $V$.
\end{ex}
\begin{ex}
Let $X\subset A$ be an ample
divisors. Then $\Sigma_{1}^k(X,n)$ has the same description as in the
previous example when $k < dim X$ by \ref{cor:lef}.
The same holds for $k > dim X$ by duality \ref{cor:duality}.
An easy Chern class argument shows that the Euler
characteristic of $X$ is nonzero. Consequently $\Sigma_{1}^{dim X}(X,n)$
is all of $M(X,n)$.
\end{ex}
The last example has an abelian fundamental group.
Now we will show that any group $\Gamma$ which
can be realized as the fundamental group of a smooth projective
variety, is realizable as the fundamental group of smooth
projective variety with $\Sigma_m^k(x,n)$ as large as possible, for
any $m,n$ and $k\ge 2$. Note that $\Sigma^1$ cannot be changed without
altering the fundamental group.
\begin{ex}
Let $Y$ be smooth projective variety with fundamental group
$\Gamma$. We can arrange that $dim Y = k+1$ as follows. First after
replacing $Y$ with a product with a projective space, we can assume
that $dim Y \ge k+1$. Now replace $Y$ by a $k+1$ dimensional
generic complete intersection.
Choose a very ample line bundle $O_Y(1)$ on $Y$, and let $X_d$ be the
general member of the linear system associated to $O_Y(d)$. Then
$\Sigma_m^i(Y) = \Sigma_m^i(X_d)= \Sigma_m^{2k-i}(X_d)$
for $i < k$ by \ref{cor:lef} and
duality. Thus these sets are independent of $d$.
The Euler characteristic is just the Chern number $c_k(X_d)$,
and this can be computed to obtain a polynomial of degree $k$ in $d$. Thus
the Euler characteristic approaches $\pm\infty$ as $d\to \infty$.
Therefore after choosing $m$, we can find $d$
large enough so that $\Sigma_m^k(X_d,n) = M(X_d,n)$
\end{ex}
|
1997-01-18T12:12:27 | 9701 | alg-geom/9701007 | en | https://arxiv.org/abs/alg-geom/9701007 | [
"alg-geom",
"math.AG"
] | alg-geom/9701007 | Elisabetta Colombo | Bert van Geemen and Aise Johan de Jong | On Hitchin's Connection | 49 pages, LaTeX | null | null | null | null | The aim of this paper is to give an explicit expression for Hitchin's
connection in the case of rank 2 bundles with trivial determinant over curves
of genus 2. We recall the definition of this connection (which arose in Quantum
Field Theory) and characterize it for families of genus two curves. We then
consider a particular family (over a configuration space), the corresponding
vector bundles are (almost) trivial. The Heat equations which give the
connection these bundles are related to the Lie algebra so(6). We compute some
local monodromy representations. As a byproduct we obtain, for any
representation space V of so(2g+2), a flat connection on the trivial bundle
with fiber V over a configuration space and thus a monodromy representation of
a pure braid group.
| [
{
"version": "v1",
"created": "Sat, 18 Jan 1997 11:11:29 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"van Geemen",
"Bert",
""
],
[
"de Jong",
"Aise Johan",
""
]
] | alg-geom | \section{Introduction}
\subsection{}
The aim of this paper is to give an explicit expression for Hitchin's
connection in the case of rank 2 bundles with trivial determinant over
curves of genus 2.
We sketch the general situation.
Let $\pi:{\cal C}\rightarrow S$ be a family of projective smooth curves.
Let $p:{\cal M}\rightarrow S$ be the family of moduli spaces of
(S-equivalence classes of) rank $r$ bundles with trivial determinant
on ${\cal C}$ over $S$.
On ${\cal M}$ we have a naturally defined determinant linebundle ${{\cal L}}$.
The vector bundles $p_{\ast}({\cal L}^{\otimes k})$ on $S$ have a natural
(projective) flat connection (such a connection also exists for other
structure groups besides $SL(r)$).
This connection first appeared in Quantum Field Theory (conformal blocks)
and was subsequently studied by various mathematicians (see \cite{Hi} and
\cite{BM}
and the references given there). We follow the approach of Hitchin
\cite{Hi} who showed that this connection can be constructed along the
lines of Welters' theory of deformations \cite{W}.
In case the curves have genus 2 and the rank $r=2$,
the family of moduli spaces $p:{\cal M}\rightarrow S$
is isomorphic (over $S$) to a ${\bf P}^3$-bundle
$$
p:{\cal M}\cong{\bf P}\longrightarrow S.
$$
Under this isomorphism ${\cal L}$ corresponds to ${\cal O}_{{\bf P}}(1)\otimes p^*{\cal
N}$ for some line bundle ${\cal N}$ on $S$. Giving a
projective connection on $p_{\ast}{\cal L}^{\otimes k}$ is then the same
as giving a projective connection on $p_*{\cal O}_{{\bf P}}(k)$
$$
\nabla^{(k)}:p_\ast{\cal O}_{{\bf P}}(k)
\longrightarrow p_\ast{\cal O}_{{\bf P}}(k)\otimes\Omega^1_S.
$$
\subsection{}
In the first part of the paper (Section \ref{johan})
we give a characterization of
Hitchin's connection (strictly speaking, we define a new connection which
is almost characterized by Hitchin's requirements).
We first develop a bit of general theory
concerning heat operators. A heat operator is a certain type of map
$D:p^*({\cal T}_S)\rightarrow {\rm Diff}^{(2)}_{{\cal M}}({\cal L}^{\otimes k})$.
Thus a vector field $\theta$
on $S$ gives a second order differential operator $D(\theta)$ on
${\cal L}^{\otimes k}$, which determines a map
$D(\theta) : p_{\ast}{\cal L}^{\otimes k} \to p_{\ast}{\cal L}^{\otimes k}$.
These then determine a connection on $p_{\ast}{\cal L}^{\otimes k}$.
The symbol of $D$ is a map $\rho_D:p^*({\cal T}_S)\rightarrow S^2{\cal T}_{{\cal M}/S}$.
Hitchin showed that if a heat operator $D$ determines the desired
connection on $p_{\ast}{\cal L}^{\otimes k}$, then its symbol map:
$$
\rho_D^s:p^*({\cal T}_S)\longrightarrow S^2{\cal T}_{{\cal M}^s/S},
$$
where ${\cal M}^s$ is the locus
of stable bundles in ${\cal M}$, must be equal to a map
$1/(k+2)\rho^s_{Hitchin}$.
The map $\rho^s_{Hitchin}$ is determined by the deformation theory of
a typical fibre $C$ and of the stable bundles on $C$.
He also proved that
$\rho^s_{Hitchin}$ extends to a
$\rho_{Hitchin}:p^*({\cal T}_S)\rightarrow S^2{\cal T}_{{\cal M}/S}$
for bundles of rank $r$ on curves of genus $g$,
except maybe for the case $r=2,\;g=2$ (!) (in that case the non-stable
bundles have codimension 1 in ${\cal M}$, whereas in all other cases they have
codimension at least two). We show, using simple bundles,
that $\rho^s_{Hitchin}$ also extends to a $\rho_{Hitchin}$
in that case (Theorem \ref{symext}).
{}{}From a study of heat operators and their symbols we conclude that
we can find a (projective)
heat operator $D$ with the right symbol over all of ${\cal M}$
(Corollary \ref{exisD}). Such a $D$ need not be unique however.
The two-torsion group scheme ${\cal H}:=Pic^0_{{\cal C}/S}[2]$
acts on ${\cal M}$ (tensoring
a rank two bundle by a line bundle of order two doesn't change the
determinant). Then ${\cal H}$ acts (projectively)
on ${\cal L}^{\otimes k}$, this action can be linearized to the action of
a central extension ${\cal G}$ of ${\cal H}$ on ${\cal L}^{\otimes k}$
$$
1\longrightarrow {\bf G}_m\longrightarrow {\cal G}
\longrightarrow {\cal H}\longrightarrow 0.
$$
We show that there is a unique (projective) heat operator
$$
{\bf D}_k:p^*({\cal T}_S)\longrightarrow {\rm Diff}^{(2)}_{\cal M}({\cal L}^{\otimes k})\qquad
{\rm with}\quad \rho_{{\bf D}_k}=\mbox{${1\over {k+2}}$}\rho_{Hitchin}
$$
and which commutes with the action of ${\cal G}$
(Proposition \ref{existence+heat}).
The (projective) connection determined by ${\bf D}_k$ is called
Hitchin's connection (cf.\ \ref{H22}).
\subsection{}
To be able to compute ${\bf D}_k$ (and to establish
some of the results mentioned
above), we use a map
$f:{\rm Pic}^0_{{\cal C}/S}\rightarrow {\cal M}$ over $S$ whose image is
the family of Kummer surfaces ${\cal K}\rightarrow S$.
We show that the pull-back of ${\bf D}_k$ along $f$ determines
the connection on $q_*f^*{\cal L}^{\otimes k}$ given by the heat
equations on (abelian) theta functions. (The connection defined by
${\bf D}_k$ on $p_*{\cal L}^{\otimes k}$ does not pull-back to the `abelian'
connection however.)
We show that ${\bf D}_k$ is characterized by this property. Moreover, we
express this property in terms
of the (local) equation defining ${\cal K}\subset {\cal M}$ (\ref{crit} $(Eq.)$).
\subsection{}\label{defP}
To write down a completely explicit connection (on $p_*{\cal O}_{{\bf P}}(k)$)
we must now choose a
certain $S$ and a family of curves ${\cal C}$ over $S$. We take $S={\cal P}$,
the configuration space of $2g+2$ points in ${\bf C}^{2g+2}$ and take the
obvious family of hyperelliptic curves over it:
$$
\begin{array}{cll}
{\cal C}& & \supset {\cal C}_z:\quad y^2=\prod_i(x-z_i)\\
\Big\downarrow&&\phantom{\subset}\Big\downarrow \\
{\cal P}&:={\bf C}^{2g+2}-\{(z_1,\dots,z_{2g+2}):
\; z_i\neq z_j\;{\rm if}\;i\neq j\;\},\quad&\ni z
\end{array}
$$
We are in the case $g=2$, but the connections we define might
be interesting for any $g$.
The pull-back of the bundle $p:{\bf P}\rightarrow {\cal P}$ along an unramified
$2^4:1$-Galois cover $\tilde{{\cal P}}\rightarrow {\cal P}$ with group $H=({\bf Z}/2{\bf Z})^4$
is trivial. The bundle ${\bf P}$ can thus obtained as a quotient of
$\tilde{{\cal P}}\times {\bf P}^3$ by $H$ (\ref{PPdescent}).
Therefore, locally
(in the complex or the etale topology),
the vector bundles $p_*{\cal O}_{{\bf P}}(k)$ and the trivial
bundle $S_k\otimes {\cal O}_{\cal P}$ are canonically
isomorphic, with $S_k=H^0({\bf P}^3,{\cal O}(k))$,
the vector space of homogenous polynomials of degree $k$ in 4 variables.
Thus it suffices to write down the connection on the trivial bundle which
corresponds to Hitchin's connection on $p_*{\cal O}_{{\bf P}}(k)$.
This connection on $S_k\otimes {\cal O}_{\cal P}$ is given by
$$
\nabla:S_k\otimes{\cal O}_{\cal P}\longrightarrow S_k\otimes\Omega_{\cal P},\qquad
\nabla(w\otimes f)=w\otimes{\rm d} f-\mbox{$1\over {k+2}$}\sum_{i\neq
j}M_{ij}(w)\otimes{f{{\rm d} z_i}\over{z_i-z_j}},
$$
for certain endomorphism $M_{ij}\in {\rm End}(S_k)$.
We give two descriptions of the endomorphisms $M_{ij}$.
One is based on the Lie algebra $so(2g+2)$ and its (half)-spin
representations. We will in fact identify $S_k=S^kV(\omega_{g+1})$
where $V(\omega_{g+1})$ is a half spin representation of $so(6)$.
The other description uses the Heisenberg group action and we sketch
that description here.
The family ${\cal C}$ has $2g+2$ sections which are the Weierstrass points
on each fiber:
$$
P_i:{\cal P}\longrightarrow {\cal C},\qquad z\longmapsto (x,y)=(z_i,0).
$$
Each $P_i-P_j$ is a section in ${\rm Pic}^0_{{\cal C}/S}[2](S)\cong({\bf Z}/2{\bf Z})^{2g}$.
There is a central extension $G$ of this group which acts on the $S_k$'s
(the groupscheme ${\cal G}$ is a twist of the constant group scheme $G$, these
group schemes are isomorphic over $\tilde{P}$).
Let $\pm U_{ij}\in\mbox{End}(S_1)$ be the endomorphisms induced by $P_i-P_j$ with
the property that $U_{ij}^2$ is the identity. Since
$\mbox{End}(S_1)=S_1\otimes S_1^\ast$ and $S_1^\ast$
may be identified with the
derivations on $S:=\oplus S_k$ and the composition $U_{ij}U_{ij}$ may
be considered as a second order differential operator
which acts on any $S_k$. The symbol (that is, the degree two part) of
this operator is, up to a factor $-1/16$,
the desired $M_{ij}$ (which, in this sense, does not
depend on $k$).
The idea for trying these $\Omega_{ij}$'s comes from \cite{vGP}.
There the Hitchin map, a Hamiltonian system on the cotangent bundle
$T^*{\cal M}^s$, is studied.
The Hitchin map is basically the symbol of the heat operator ${\cal D}_k$
(cf.\ \cite{Hi} $\S$4).
Among the results of \cite{vGP} is an explicit description of
the Hitchin map for $g=2$ curves (obtained from line geometry in ${\bf P}^3$)
and implicitly in that description is
the one for the $M_{ij}$ given here.
\subsection{}
Now that the connection is determined it is interesting to consider its
monondromy representation. In the case of rank two bundes, this has been
considered by Kohno \cite{K2}. We verify that our connection gives has the
same local monodromy as Kohno's representation in Subsection
\ref{localmon}.
Kohno (and the physicists) use (locally on the base it
seems) the Knizhnik-Zamolodzhikov
equations (which define a flat connection over ${\cal P}$
using the Lie algebra $sl(2)$ \cite{Ka}, Chapter XIX).
It is not clear to us how the KZ-equations are related to our connection
(which is based on $so(2g+2)$).
We do not know which representations of the braid group one
obtains from our connections.
\section{Characterizing Hitchin's connection}\label{johan}
\subsection{Introduction}
We recall some deformation theory as described by Welters \cite{W}
and relate it to connections.
Then we introduce heat operators and show how they define
(projective) connections on certain bundles.
Next we give a criterion for the existence of a heat equation with a
given symbol and in Section \ref{modbun} we recall the definition of the
symbol of Hitchin's connection. In Section \ref{g=2} we combine these
results and give criteria (in \ref{crit} and \ref{H22})
which determine Hitchin's connection.
\subsection{Deformation theory}\label{def}
\subsubsection{Deformations}
Let $X$ be a smooth projective variety over ${\bf C}$ with tangent bundle
$T_X$.
Let $L$ be an invertible ${\cal O}_X$-module on $X$ and let $s\in H^0(X, L)$.
We are going to classify first order deformations of $X$, of the
pair $(X,L)$ and of the triple $(X,L,s)$. See \cite{W} for proofs.
Isomorphism classes of first order deformations $X_\epsilon$ of $X$
(over $\mathop{\rm Spec} {\bf C}[\epsilon]/(\epsilon^2)$) are classified by $H^1(X,T_{X})$.
We write $[X_\epsilon]$ for the class in $H^1(X,T_{X})$ of the deformation
$X_\epsilon$; similar notation will be used throughout.
The first order deformations $(X_\epsilon,L_\epsilon)$ of the pair $(X,L)$
are classified by $H^1(X,{\rm Diff}^{(1)}_X(L))$. Here ${\rm Diff}^{(1)}_X(L)$ is the
sheaf of first order differential operators on $L$. This sheaf sits in an
exact sequence:
$$
0\longrightarrow {\cal O}_{X}\longrightarrow
{\rm Diff}^{(1)}_X(L) \stackrel{\sigma}{\longrightarrow} T_{X}\longrightarrow 0
$$
where the map $\sigma:{\rm Diff}^{(1)}_X(L)\rightarrow T_{X}$ gives the symbol of the
operator. This sequence gives a map $\alpha:H^1(X,{\rm Diff}^{(1)}_X(L))\to H^1(X,T_X)$,
which maps $[(X_\epsilon,L_\epsilon)]$ to $[X_\epsilon]$.
Finally, we consider first order deformations of the triple $(X,L,s)$.
The evaluation map $d^1s:{\rm Diff}^{(1)}_X(L)\rightarrow L,\;D\mapsto Ds$ gives
a complex:
$$
0\longrightarrow {\rm Diff}^{(1)}_X(L)\stackrel{d^1s}{\longrightarrow} L
\longrightarrow 0.
$$
Let ${\bf H}^1(d^1s)$ be the first hypercohomology group of this
complex. This is the space classifying the isomorphism classes
of deformations
$(X_\epsilon,L_\epsilon,s_\epsilon)$ of $(X,L,s)$ (\cite{W}, Prop.\ 1.2).
The spectral sequence connecting hypercohomology to cohomology gives a
map $\beta: {\bf H}^1(d^1s) \to H^1({\rm Diff}^{(1)}_X(L))$, which maps
$[(X_\epsilon,L_\epsilon,s_\epsilon)]$ to $[(X_\epsilon,L_\epsilon)]$.
Any element of ${\bf H}^1(d^1s)$ may be represented by a Cech cocycle.
Let us choose an affine open covering ${\cal U} : X=\bigcup U_i$ and
write $U_{ij}=U_i\cap U_j$. A Cech cocycle is given by a
pair $(\{t_i\},\,\{D_{ij}\})$ satisfying the relations:
$$
t_i-t_j=D_{ij}(s),\quad
D_{jk}-D_{ik}+D_{ij}=0\qquad \qquad
(t_i\in L(U_i),\; D_{ij}\in {\rm Diff}^{(1)}_X(L)(U_{ij})).
$$
The maps $\beta$ and $\alpha\circ\beta$ correspond to forgetting
first $s_\epsilon$ and then $L_\epsilon$. They can be described
as follows. The cocycle $(\{t_i\},\,\{D_{ij}\})$ is mapped to
the 1-cocycle $\{D_{ij}\}$ in the sheaf ${\rm Diff}^{(1)}_X(L)$ under $\beta$.
This is then mapped to the 1-cocycle $\{{\rm Diff}^{(1)}(D_{ij})\}$ in
the sheaf $T_X$.
Let us make explicit a deformation $(X_\epsilon, L_\epsilon, s_\epsilon)$
associated to $(\{t_i\},\,\{D_{ij}\})$.
Write $U_i=\mathop{\rm Spec}(A_i)$ and $U_{ij}=\mathop{\rm Spec}(A_{ij})$. Let $M_i=L(U_i)$
and $M_{ij}=L(U_{ij})$. The section $s$ gives elements $s_i\in M_i$;
note that $t_i\in M_i$ as well.
Write $\theta_{ij}=\sigma(D_{ij})\in T_X(U_{ij})$.
The 1-cocycle $\{\theta_{ij}\}$ defines
a scheme $X_\epsilon$ over $\mathop{\rm Spec}({\bf C}[\epsilon])$ by glueing the rings
$A_i[\epsilon]$ via the isomorphisms
$$
A_{ij}[\epsilon]\longrightarrow
A_{ij}[\epsilon],\qquad
f+g\epsilon\longmapsto f+(\theta_{ij}(f)+g)\epsilon.
$$
Here we view $\theta_{ij}$ as a derivation. The 1-cocycle $\{D_{ij}\}$
in ${\rm Diff}^{(1)}_X(L)$
gives a deformation $L_\epsilon$ of $L$ by glueing the
$A_i[\epsilon]$-modules $M_{i}[\epsilon]$ via the
$A_{ij}[\epsilon]$-module isomorphisms:
$$
\phi_{ij}:M_{ij}[\epsilon]\longrightarrow M_{ij}[\epsilon],\qquad
m_{ij}+m'_{ij}\epsilon\longmapsto
m_{ij}+(D_{ij}(m_{ij})+m'_{ij})\epsilon.$$
Note one has $D_{ij}(fs)=fD_{ij}(s)+\theta_{ij}(f)s$ by
definition of the symbol $\sigma$ and $\sigma(D_{ij})=\theta_{ij}$.
The cocycle $(\{t_i\},\,\{D_{ij}\})$ in ${\bf H}^1(d^1s)$ defines a
deformation $s_\epsilon\in H^0(X_\epsilon,L_\epsilon)$. The section
$s_\epsilon$ is given by the family of elements:
$$
s_\epsilon:=\{s_{\epsilon,i}\}\quad(\in M_i[\epsilon])
\qquad{\rm with}\quad
s_{\epsilon,i}:=s_i+t_i\epsilon.
$$
The
cocycle relation $t_i-t_j=D_{ij}(s)$ shows that the
$s_{\epsilon,i}$'s glue: $\phi_{ij}(s_{\epsilon,j})=s_{\epsilon,i}$.
\subsubsection{Second order operators}\label{secondorder}
The following construction is useful to obtain elements in
${\bf H}^1(d^1s)$ (\cite{W}, (1.9)).
Let ${\rm Diff}^{(2)}_X(L)$ be the sheaf of second order differential operators
on $L$. The symbol gives an exact sequence which is the first row
of the complex:
$$
\begin{array}{ccccccccc}
0&\longrightarrow&{\rm Diff}^{(1)}_X(L)&\longrightarrow&{\rm Diff}^{(2)}_X(L)
&\longrightarrow&S^2T_X&\longrightarrow&0\\
&&\downarrow&&\downarrow&&\downarrow&&\\
0&\longrightarrow&L&\longrightarrow&L&\longrightarrow&0&\longrightarrow&0,
\end{array}
$$
the vertical maps are evaluation on $s$ ($D\mapsto Ds$), thus the
first column is the complex $d^1s$ considered before.
{}From the exact sequence of hypercohomology one obtains a map
$$
\delta_s:H^0(S^2T_X)\longrightarrow {\bf H}^1(d^1s).
$$
Let us give a description in terms of
cocycles of this map. Let ${\cal U}=\{U_i\}$ be an affine
open cover of $X$ as before. Let $w\in H^0(X, S^2T_X)$. We can find
operators $D^{(2)}_i\in {\rm Diff}^{(2)}_X(L)(U_i)$ which map to
$w|_{U_i}$ in $S^2T_X(U_i)$. Then
$$
\delta_s(w)=(\{D^{(2)}_i(s)\},\{D^{(2)}_i-D^{(2)}_j\})
\qquad(\in {\bf H}^1(d^1s)).
$$
It is easy to verify that $\delta_s(w)$ is indeed a cocycle for the
complex defined by $d^1s$.
Note $w=\sigma(D^{(2)}_i)=\sigma(D^{(2)}_j)$ on $U_{ij}$,
thus $D^{(2)}_i-D^{(2)}_j$ is a
first order operator.
Note that $\beta(\delta_s(w))=\{D^{(2)}_i-D^{(2)}_j\}\;(\in {\rm Diff}^{(1)}_X(L))$
is {\it independent}
of the choice of the section $s$. Therefore, given $w\;(\in H^0(X,S^2T_X))$
we get a deformation
$(X_\epsilon, L_\epsilon)$ such that {\it any} section $s\in H^0(X, L)$
is deformed to a section $s_\epsilon\in H^0(X_\epsilon, L_\epsilon)$.
\subsubsection{Infinitesimal connections}\label{infcon}
Let $(X_\epsilon,L_\epsilon)$ be a deformation of $(X,L)$.
We consider a differential operator $D$ of order at most two on $L_\epsilon$
which locally can be written as (cf.\ \ref{defho} below)
$$
D=\{D_i\}\quad(\in H^0(X_\epsilon,{\rm Diff}^{(2)}_{X_\epsilon}(L_\epsilon))),
\qquad
D_i=\partial_\epsilon+R_i:M_i[\epsilon]\longrightarrow M_i[\epsilon]
$$
with $\partial_\epsilon(n+\epsilon m)=m$ for $n,\,m\in M_i$, and
$R_i$ a
${\bf C}[\epsilon]$-linear map of
order at most
$2$ on $M_i$ (so $s_i\mapsto R_i(fs_i)-fR_i(s_i)$ has order at most
$k-1$ and
a map of order zero is by definition an $A_i[\epsilon]$-linear map).
Thus $R_i$ is a second order differential operator involving only
deriviatives in the fiber directions (and not in the base direction
$\epsilon$).
The symbol of such a $D$ is the second order part of the $D_i$ (and thus
of the $R_i$). The restriction of the symbol to
$X\;(\subset X_\epsilon)$ is:
$$
\bar{\rho}(D):=\{\sigma(\bar{R}_i)\}\quad{(\in H^0(X,S^2T_{X}))},
\qquad{\rm where}\quad R_i=\bar{R}_i+\epsilon \bar{R}'_i
$$
with $\bar{R}_i,\,\bar{R}_i':M_i\rightarrow M_i[\epsilon]$ (recall $R_i$ is
$\epsilon$-linear).
For any $s\in H^0(X,L)$ we then have a hyper cohomology class
$$
\delta_s(w):=(\{\bar{R}_i(s_i)\},\{\bar{R}_i-\bar{R}_j\})\qquad
(\in {\bf H}^1(d^1s)).
$$
First of all we show that the deformation $(X_\epsilon,L_\epsilon)$ is the deformation
determined by $-\beta(\delta_s(\bar{\rho}(D)))$:
$$
[(X_\epsilon,L_\epsilon)]+\beta(\delta_s(\bar{\rho}(D)))=0
\qquad (\in H^1(X,{\rm Diff}^{(1)}_X(L))).
$$
This can be verified as follows. Let $s_{\epsilon,i}\in M_i[\epsilon]$ such
that $s_{\epsilon,i}=\phi_{ij}(s_{\epsilon,j})$. Then we have
$\phi_{ij}(D_j(s_{\epsilon,j})=D_i(\phi_{ij}(s_{\epsilon,j})$.
With $s_{\epsilon,j}:=s_j+t_j\epsilon,\;(s_j\,t_j\in M_j)$ this gives
(for the `constant' term):
$$
D_{ij}(s_j)+t_j+\bar{R}_j(s_j)=
t_j+\bar{R}_i(s_j),
$$
which is zero for all such local sections iff
$$
D_{ij}+(\bar{R}_j-\bar{R}_i)=0\qquad \in
{\rm Diff}_X^{(1)}(L)(U_{ji}).
$$
Since the $\bar{R}_j$ are local lifts of $\bar{\rho}(D)$, we get that
$\beta\delta_s(\bar{\rho}(D))=\{\bar{R}_i-\bar{R}_j\}\;(\in
{\rm Diff}_X^{(1)}(L)(U_{ij})$.
Next we consider any deformation $(X_\epsilon,L_\epsilon,s_\epsilon)$
of $(X,L,\epsilon)$.
We say that a section $s_\epsilon=\{s_i+\epsilon t_i\}
\in H^0(X_\epsilon,L_\epsilon)$
is flat for $D$ if
$$
D_i(s_i+\epsilon t_i)\equiv 0\;{\rm mod}\;(\epsilon),\qquad
{\rm equivalently}\quad
t_i=-\bar{R}_i(s_i),
$$
in that case we write $\nabla(D)(s_\epsilon)=0$.
Since $[(X_\epsilon,L_\epsilon,s_\epsilon)]=(\{t_i\},\{D_{ij}\})\;(\in{\bf
H}(d^1s))$
we get:
$$
\delta_s(\bar{\rho}(D))=-[(X_\epsilon,L_\epsilon,s_\epsilon)]\quad
{\rm iff}\quad \nabla(D)(s_\epsilon)=0.
$$
\subsubsection{A remark on symbols}\label{normalization}
We use the conventions of \cite{W} concerning
differential operators and symbols. In particular
when we write $S^2 T_X$ we mean symmetric tensors
in the sheaf of vector fields on $X$. Note that if
$X={\bf A}^1_{\bf C}$ with coordinate $t$, then the symbol
of the operator $\partial^2/\partial^2t\;(\in {\rm Diff}^{(1)}_{\cal O})$ is the
symmetric section $2\partial/\partial t \otimes \partial/\partial t
\;(\in S^2T_X)$. By abuse of notation we will sometimes write
$\sigma(\partial^2/\partial^2t)=2\partial^2/\partial^2t$.
\subsection{Heat operators}
\subsubsection{Notation}
Let $p: {\cal X}\to S$ be a smooth surjective morphism of smooth
varieties over ${\bf C}$. Let ${\cal L}$ be an invertible ${\cal O}_{\cal X}$-module
over ${\cal X}$.
We write ${\cal T}_{\cal X}$ (resp.\ ${\cal T}_S$, resp.\ ${\cal T}_{{\cal X}/S}$) for the
sheaf of vector fields on ${\cal X}$ (resp.\ $S$, resp.\ ${\cal X}$ over $S$).
We denote ${\rm Diff}^{(k)}_{\cal X}({\cal L})$ the sheaf of differential operators
of order at most $k$ on ${\cal L}$ over ${\cal X}$. Further,
${\rm Diff}^{(k)}_{{\cal X}/S}({\cal L})\subset {\rm Diff}^{(k)}_{\cal X}({\cal L})$ denotes
the subsheaf of $p^{-1}({\cal O}_S)$-linear operators.
We remark that ${\rm Diff}^{(k)}_{\cal X}({\cal L})$ is a coherent (left)
${\cal O}_{\cal X}$-module and that ${\rm Diff}^{(k)}_{{\cal X}/S}({\cal L})$ is a
coherent submodule. For any $k$ there is a symbol map
$$ \sigma^{(k)} : {\rm Diff}^{(k)}_{\cal X}({\cal L}) \longrightarrow
S^k{\cal T}_{\cal X}={\rm Sym}^k_{{\cal O}_{\cal X}}({\cal T}_{\cal X}).$$
It maps ${\rm Diff}^{(k)}_{{\cal X}/S}({\cal L})$ into $S^k{\cal T}_{{\cal X}/S}$.
Consider a point $x\in {\cal X}({\bf C})$, an (\'etale or analytic)
neighbourhood $U$ of $p(x)$ in $S$, and an (\'etale or analytic)
neighbourhood $V$ of $x$ in ${\cal X}$ such that $p(V)\subset U$.
Assume we have coordinate functions $t_1,\ldots, t_r\in {\cal O}_S(U)$
and functions $x_1,\ldots, x_n\in {\cal O}_{\cal X}(V)$ such that
$t_1=t_1\circ p,\ldots, t_r=t_r\circ p, x_1,\ldots, x_n$
are coordinates on $V$. (In the \'etale case this means that
$\Omega^1_{U}=\bigoplus {\cal O}_U{\rm d}t_i$
and $\Omega^1_{V/U}=\bigoplus {\cal O}_V{\rm d}x_i$.)
Then the elements $\partial/\partial x_i$ form a basis for
${\cal T}_{{\cal X}/S}|_V$ over ${\cal O}_V$. The system $(V, t_1, \ldots,
t_r, x_1,\ldots,x_n)$ will be called a {\it coordinate patch}.
We can find coordinate patches around any point $x\in {\cal X}({\bf C})$.
\subsubsection{Definition}\label{defho}
A {\it heat operator $D$ on ${\cal L}$ over $S$} is a (left) ${\cal O}_{\cal X}$-module
homomorphism
$$D : p^\ast({\cal T}_S)\longrightarrow {\rm Diff}^{(2)}_{\cal X}({\cal L})$$
which satisfies the following property: for any coordinate patch
$(V, t_1, \ldots,t_r, x_1,\ldots,x_n)$ as above, and any
trivialization ${\cal L}|_V\cong {\cal O}_V$, the operator
$D(\partial/\partial t_1)$ has the following shape:
$$
D({\partial\over\partial t_1})= f + {\partial\over\partial t_1}
+ \sum_{i=1}^n f_i {\partial\over\partial x_i} +
\sum_{1\leq i\leq j\leq n} f_{ij} {\partial^2\over\partial x_i\partial x_j}
$$
for certain $f, f_i, f_{ij}\in {\cal O}_{\cal X}(V)$.
More precisely, let $${\cal W}_{{\cal X}/S}({\cal L})={\rm Diff}^{(1)}_{\cal X}({\cal L})+
{\rm Diff}^{(2)}_{{\cal X}/S}({\cal L})\subset {\rm Diff}^{(2)}_{\cal X}({\cal L})$$ be
the subsheaf of second order differential
operators on ${\cal L}$ whose symbol lies in
$S^2({\cal T}_{{\cal X}/S})\subset S^2({\cal T}_{\cal X})$. Note that ${\rm Diff}^{(1)}_{{\cal X}/S}({\cal L})
\subset {\cal W}_{{\cal X}/S}({\cal L})$ and that there is a canonical exact sequence
$$
0\longrightarrow
{\rm Diff}^{(1)}_{{\cal X}/S}({\cal L})\longrightarrow
{\cal W}_{{\cal X}/S}({\cal L}) \longrightarrow
p^\ast({\cal T}_S)\oplus S^2{\cal T}_{{\cal X}/S}\longrightarrow 0.
$$
Our condition on $D$ above means that $D$ is a map $D: p^\ast({\cal T}_S)
\to {\cal W}_{{\cal X}/S}({\cal L})$ whose composition with the map
${\cal W}_{{\cal X}/S}({\cal L})\to p^\ast({\cal T}_S)$ is the identity.
(So $D(\partial/\partial t_1)$ has no terms involving
$\partial/\partial t_i$ except for the
term $\partial/\partial t_1$ with coefficient 1.)
Such a map $D$ is determined by the ${\cal O}_S$-linear map
$p_\ast D: {\cal T}_S \to p_\ast({\rm Diff}_{\cal X}^{(2)}({\cal L}))$. Note that
${\cal O}_S$ is a subsheaf of $p_\ast({\rm Diff}_{\cal X}^{(2)}({\cal L}))$.
Any ${\cal O}_S$-linear map
$\bar D : {\cal T}_S \to p_\ast({\rm Diff}_{\cal X}^{(2)}({\cal L}))/ {\cal O}_S$
can locally be lifted to a map into $p_\ast({\rm Diff}_{\cal X}^{(2)}({\cal L}))$.
This means that any point $s\in S({\bf C})$ has a neighbourhood $U$
such that there is an ${\cal O}_U$-linear map $D_U : {\cal T}_S|_U\to
p_\ast({\rm Diff}_{\cal X}^{(2)}({\cal L}))|_U$ which reduces to $\bar D|_U$.
A {\it projective heat operator $\bar D$ on ${\cal L}$ over $S$}
is an ${\cal O}_S$-linear map
$$
\bar D : {\cal T}_S \longrightarrow p_\ast\big({\rm Diff}_{\cal X}^{(2)}({\cal L})\big)\Big/ {\cal O}_S
$$
such that any local lifting $D_U$ as above gives rise to a
heat operator on ${\cal L}$ over $U$.
The {\it symbol} of a heat operator $D$ is the ${\cal O}_{\cal X}$-linear
map
$$ \rho_D: p^\ast({\cal T}_S)\longrightarrow S^2{\cal T}_{{\cal X}/S}$$
given by composing $D$ with the symbol $\sigma^{(2)}$. We note
that the symbol of a projective heat operator is well-defined.
\subsubsection{Heat operators and connections}
We claim that a heat operator $D$ gives rise to a connection
$$\nabla(D) : p_\ast({\cal L})\longrightarrow
p_\ast({\cal L})\otimes_{{\cal O}_S}\Omega^1_S.$$
Indeed, suppose that $s\in p_\ast({\cal L})(U)={\cal L}(p^{-1}(U))$
and that $\theta\in {\cal T}_S(U)$. Then $D(p^{-1}\theta)$ is
a second order differential operator on ${\cal L}$ over $p^{-1}(U)$.
Hence $D(p^{-1}\theta)(s)\in {\cal L}(p^{-1}(U))=p_\ast({\cal L})(U)$. By our local
description of $D$ above we see that
$D(p^{-1}\theta)(fs)=fD(p^{-1}\theta)(s)+\theta(f)s$ for every
$f\in {\cal O}_S(U)$.
Hence the operation $(\theta, s)\mapsto D(p^{-1}\theta)(s)$ defines
a connection $\nabla(D)$ on $p_\ast {\cal L}$ over $S$. In the sequel we
will often write $D(\theta)$ in stead of $D(p^{-1}\theta)$.
If we have a projective heat operator $\bar D$, the connection
$\nabla(\bar D)=\nabla(D_U)$ is only defined locally, by choosing lifts
$D_U$.
The difference of two local lifts is given by a local section
$\eta\in \Omega^1_S(U)$, in which case the difference of the two
connections $\nabla(D_U)$ is
multiplication by $\eta$. Thus $\bar D$ defines unambiguously
a projective connection $\bar \nabla(\bar D)$.
\subsubsection{Heat operators and deformations}\label{heat+defo}
Let $D$ be a heat operator on ${\cal L}$ over $S$.
Let $0\in S({\bf C})$ be a point of the base. Put $X={\cal X}_0$ and $L={\cal L}_0$.
An element $\theta\in T_0S$ will also be considered as a morphism
$\theta:\mathop{\rm Spec}({\bf C}[\epsilon])\to S$. By base change we get a pair
$(X_\epsilon, L_\epsilon)$ over $\mathop{\rm Spec}({\bf C}[\epsilon])$. Clearly,
$w_\theta:=\rho_D(\theta)$ is an element of $H^0(X, S^2T_X)$. It can be
seen by working through the definitions that
$[(X_\epsilon, L_\epsilon)]+\beta(\delta_s(w_\theta))=0$ for any section
$s\in H^0(X, L)$. (Note that in Subsection \ref{secondorder}
we showed that $\beta(\delta_s(w_\theta))$ was independent of $s$.)
Let $s_\epsilon\in H^0(X_\epsilon, L_\epsilon)$ lift $s$. One can show
that $[(X_\epsilon, L_\epsilon, s_\epsilon)]=-\delta_s(w_\theta)$
if and only if the section $s_\epsilon$ is horizontal for the
connection $\nabla(D)$.
\subsubsection{Projective heat operators and change of line
bundle}\label{heat+change}
Let $D$ be a heat operator on ${\cal L}$ over $S$.
Let $g\in H^0(S, {\cal O}_S^\ast)$ be an invertible function on $S$.
Multiplication by $p^{-1}(g)$ defines an invertible operator on ${\cal L}$,
also denoted by $g$.
For any local section $\theta$ of ${\cal T}_S$, we have the obvious relation
$$
g^{-1}\circ D(\theta) \circ g = g^{-1}\theta(g) + D(\theta).
$$
We conclude that $g^{-1}\circ D \circ g$ is a heat operator and that
the projective heat operators $\bar D$ and
$\overline{g^{-1}\circ D \circ g}$ are equal. (The difference
is given by the section $g^{-1}{\rm d}g={\rm d}\log g$ of $\Omega^1_S$.)
Suppose that ${\cal M}$ is an invertible ${\cal O}_S$-module on $S$.
The above implies that the set of projective heat
operators on ${\cal L}$ over $S$ can be identified canonically with the
set of projective heat operators on ${\cal L}'={\cal L}\otimes_{{\cal O}_{\cal X}}p^\ast({\cal M})$.
Indeed, choose a covering of $S$ on whose members $U_i$ the line bundle
${\cal M}$ becomes trivial. This identifies the heat operators on
${\cal L}$ and ${\cal L}'$ over $U$. The difference in the local identifications
is given by the 1-forms ${\rm d}\log g_{ij}\in \Omega^1_{U_{ij}}$.
\subsubsection{Heat operators and flatness}\label{heat+flat}
We say that a heat operator $D$ on ${\cal L}$ over $S$ is {\it flat} if
$D([\theta,\theta'])=[D(\theta),D(\theta')]$ for any two local
sections $\theta, \theta'$ of ${\cal T}_S$ on $U\subset S$. It suffices
to consider local vector fields $\theta, \theta'$ on $S$ with
$[\theta,\theta']=0$ and to check that $[D(\theta),D(\theta')]=0$.
We remark that in this case the operator $[D(\theta),D(\theta')]$
is a section of ${\rm Diff}^{(3)}_{{\cal X}/S}({\cal L})$ over $p^{-1}(U)$.
We say that $D$ is {\it projectively flat} if we have
$D([\theta,\theta']) = h + [D(\theta),D(\theta')]$ for some function
$h=h_{\theta,\theta'}\in {\cal O}_S(U)$ for any $\theta,\theta'\in{\cal T}_S(U)$.
A projective heat operator $\bar D$ is called {\it projectively flat}
if any of the local lifts $D_U$ are projectively flat.
We remark that if the heat operator $D$ is flat, then the
associated connection $\nabla(D)$ on $p_\ast{\cal L}$ is flat as well.
In particular, if $p_\ast {\cal L}$ is a coherent ${\cal O}_S$-module,
then this implies that $p_\ast {\cal L}$ is locally free. In the
same vein we have that a projectively flat projective
heat operator $\bar D$ defines a projectively flat projective connection
$\bar\nabla(\bar D)$ on $p_\ast{\cal L}$.
\subsubsection{Heat operators with given symbol}\label{heat+symbol}
Let ${\cal X}\to S$ and ${\cal L}$ be as above, and assume that we are given
a ${\cal O}_{\cal X}$-linear map
$
\rho: p^\ast{\cal T}_S\rightarrow S^2{\cal T}_{{\cal X}/S}.
$
The question we are going to study is the following:
When can we find a (projective) heat operator $D$ on ${\cal L}$
over $S$,
$
D:p^\ast {\cal T}_S\rightarrow {\cal W}_{{\cal X}/S}\quad(\subset
{\rm Diff}^{(2)}_{{\cal X}}({\cal L}))$
with $\rho_D=\rho$?
So the
symbol $\rho_D$ (which is the compsition of $D$ with the map
$\sigma^{(2)}:{\rm Diff}^{(2)}_{{\cal X}}({\cal L}))\rightarrow S^2{\cal T}_{{\cal X}/S}$)
should be equal to the given $\rho$.
Let us define a number of canonical maps associated to the
situation. First we have the standard exact sequence
$$
0 \longrightarrow {\cal T}_{{\cal X}/S} \longrightarrow
{\cal T}_{\cal X} \longrightarrow p^\ast {\cal T}_S \longrightarrow 0.
$$
This gives us a ${\cal O}_S$-linear map (the {\it Kodaira-Spencer map})
$$
\kappa_{{\cal X}/S} \ \ :\ \ {\cal T}_S \longrightarrow R^1p_\ast {\cal T}_{{\cal X}/S}.
$$
Next we have the exact sequence
$$
0 \longrightarrow {\cal T}_{{\cal X}/S}\longrightarrow
{\rm Diff}^{(2)}_{{\cal X}/S}({\cal L})\big/ {\cal O}_{\cal X} \longrightarrow
S^2{\cal T}_{{\cal X}/S} \longrightarrow 0.
$$
This gives an ${\cal O}_S$-linear map
$$
\mu_{\cal L}\ \ :\ \ p_\ast S^2{\cal T}_{{\cal X}/S} \longrightarrow
R^1p_\ast {\cal T}_{{\cal X}/S}.
$$
The invertible ${\cal O}_{\cal X}$-module ${\cal L}$ is given by an element
of $H^1({\cal X},{\cal O}_{\cal X}^\ast)$. Using the map ${\rm d}\log : {\cal O}^\ast_{\cal X}
\to \Omega^1_{{\cal X}/S}$ we get an element of $H^1({\cal X}, \Omega^1_{{\cal X}/S})$,
and hence a section $[{\cal L}]$ in $H^0(S, R^1p_\ast(\Omega^1_{{\cal X}/S}))$.
This will be called the cohomology class of ${\cal L}$.
By cupproduct with
$[{\cal L}]$ we get another map $p_\ast S^2{\cal T}_{{\cal X}/S} \to R^1p_\ast
{\cal T}_{{\cal X}/S}$,
$w\mapsto w\cup [{\cal L}]$. (There is a natural pairing
$S^2{\cal T}_{{\cal X}/S}\otimes \Omega^1_{{\cal X}/S}\to {\cal T}_{{\cal X}/S}$.)
It is shown in \cite[1.16]{W} that we
have $\mu_{\cal L}(w) = - w\cup [{\cal L}] + \mu_{{\cal O}}(w) $.
Now let $\theta$ be a local vector field on $S$, i.e.,
$\theta\in {\cal T}_S(U)$, with $U$ affine.
We want to find a $D(\theta)\in {\cal W}_{{\cal X}/S}({\cal L})$ with symbol
$\rho(\theta)$.
Recall that ${\cal W}_{{\cal X}/S}({\cal L})$ is an extension:
$$
0\longrightarrow
{\rm Diff}^{(1)}_{{\cal X}/S}({\cal L})\longrightarrow
{\cal W}_{{\cal X}/S}({\cal L}) \longrightarrow
p^\ast({\cal T}_S)\oplus S^2{\cal T}_{{\cal X}/S}\longrightarrow 0.
$$
The obstruction against finding a lift
$D(\theta)\;(\in {\cal W}_{{\cal X}/S}({\cal L}))$
of the section $p^{-1}(\theta)\oplus\rho(p^{-1}\theta)$
of $p^\ast {\cal T}_S\oplus S^2{\cal T}_{{\cal X}/S}$ over
$p^{-1}(U)$
is an element $o(\theta, {\cal L})$
in $H^1(p^{-1}(U), {\rm Diff}^{(1)}_{{\cal X}/S}({\cal L}))=
H^0(U, R^1p_\ast({\rm Diff}^{(1)}_{{\cal X}/S}({\cal L})))$
(recall that $U$ was assumed affine).
There is an exact sequence
$$
0 \longrightarrow {\cal O}_{{\cal X}} \longrightarrow
{\rm Diff}^{(1)}_{{\cal X}/S}({\cal L})\longrightarrow {\cal T}_{{\cal X}/S}
\longrightarrow 0.
$$
We remark that the image of the class $o(\theta, {\cal L})$
in $R^1p_\ast{\cal T}_{{\cal X}/S}(U)$ is the section
$\kappa_{{\cal X}/S}(\theta) + \mu_{\cal L}(\rho(\theta))$.
The first condition that has to be satisfied for $\bar D$
to exist is therefore:
$$
\kappa_{{\cal X}/S} + \mu_{\cal L} \circ \rho = 0.
\leqno{(*)}
$$
If this condition is satisfied, then we can find a lift of
$p^{-1}(\theta)\oplus\rho(p^{-1}\theta)$ to an element
$\overline{D}_\theta$ in the sheaf ${\cal W}_{{\cal X}/S}({\cal L})/{\cal O}_{\cal X}$.
This element is well defined up to addition of a section
of ${\cal T}_{{\cal X}/S}$. Hence we get a second obstruction in
$$
{\rm Coker}\Big(p_\ast {\cal T}_{{\cal X}/S}
{\stackrel{[{\cal L}]\cup}{\longrightarrow}}
R^1p_\ast {\cal O}_X\Big) \leqno{(**)}
$$
Let us assume for the moment that both obstructions vanish for any
$\theta$ as above. Then we can locally find a $D(\theta)$ lifting the
element $p^{-1}(\theta)\oplus\rho(p^{-1}\theta)$. Thus, if we
have a basis $\theta_1,\ldots,\theta_r$ of ${\cal T}_S$ over $U$,
then we can define $D_U$ by the formula
$D_U(\sum a_i \theta_i)=\sum a_i D(\theta_i)$
for certain choices of the elements $D(\theta_i)$.
If there is some way of choosing the elements $D(\theta)$ uniquely
up to an element of $p^{-1}{\cal O}_S(U)$, e.g.\ if the map in $(**)$
is injective and $p_\ast({\cal O}_{\cal X})={\cal O}_S$, then $D_U$ determines
a unique projective heat operator over $U$. In this case the
map $\rho$ determines a
unique projective heat operator on ${\cal L}$ over $S$.
Here is another set of hypotheses which imply the existence of heat
operators with given symbol.
Let $0\in S({\bf C})$ be a point. Put $X={\cal X}_0$ and $L={\cal L}_0$
as in \ref{heat+defo}. Let $\theta\in T_0S$ and consider
$(X_\epsilon, L_\epsilon)$ over $\mathop{\rm Spec}({\bf C}[\epsilon])$
as in \ref{heat+defo}. We know from \ref{heat+defo}
that $[(X_\epsilon, L_\epsilon)]+\beta(\delta_s(\rho(\theta)))=0$
if a heat operator exists. Suppose that
\begin{enumerate}
\item $R^1p_\ast {\cal O}_{\cal X}$ and $R^1p_\ast{\cal T}_{{\cal X}/S}$
are locally free on $S$, and have fibres $H^1(X, {\cal O}_X)$
and $H^1(X, T_X)$ at any point $0$ in $S({\bf C})$.
\item for any $0$ and $\theta$ as above, we have
$[(X_\epsilon, L_\epsilon)]+\beta(\delta_s(\rho(\theta)))=0$.
\end{enumerate}
If these conditions are satisfied, then we can at least find
local liftings $D_U$ of $\rho$ to a heat operator. Again, we will
get a ``canonical'' projective heat operator if there is a
``preferred'' way of choosing the local lifts $D(\theta_i)$.
\subsubsection{Heat operators on abelian schemes}\label{heat+abelian}
In this subsection, we let ${\cal X}\to S$ be an abelian scheme
with zero section $o: S\to {\cal X}$. We assume the line bundle
${\cal L}$ is relatively ample on ${\cal X}$ over $S$, i.e., ${\cal L}$
defines a polarization of ${\cal X}$ over $S$. We will show that
there exists a unique projective heat operator on ${\cal L}$
over $S$, see \cite{W}.
We first remark that in this case the map $(**)$ of
\ref{heat+symbol} is an isomorphism. This implies that
$\Gamma(p^{-1}(U), {\rm Diff}^{(1)}_{{\cal X}/S}({\cal L}) )={\cal O}_S(U)$
for any $U\subset S$. Hence, by the discussions in
\ref{heat+symbol} we get a unique heat operator as
soon as we have a (symbol) map $\rho$ satisfying $(**)$.
We have $\mu_{\cal O}=0$ in this situation, as we can let
elements of $S^2{\cal T}_{{\cal X}/S}$ act by translation invariant
operators (see \cite[1.20]{W}).
Hence, $\mu_{\cal L}(-)= - [{\cal L}] \cup (-) $. Now we use that
$$p_\ast (\otimes^2{\cal T}_{{\cal X}/S})
\stackrel{[{\cal L}] \cup} {\longrightarrow}
R^1p_\ast{\cal T}_{{\cal X}/S}\cong o^\ast {\cal T}_{{\cal X}/S}\otimes R^1p_\ast{\cal O}_{\cal X}$$
is an isomorphism. Consider $\rho:=([{\cal L}] \cup)^{-1} \circ \kappa_{{\cal X}/S}$.
It is well known that this is a map into $p_\ast S^2{\cal T}_{{\cal X}/S}$
and it solves $(*)$ by definition; it is of course the
unique solution to $(*)$. The unique
projective heat operator $\bar D=\bar D_{{\cal L}}$ with $\rho=\rho_{\bar D}$
is called the projective heat operator associated to $({\cal X}/S,{\cal L})$.
We make the remark that if ${\cal L}' = {\cal L}^{\otimes n}$,
then we have $ \rho_{\bar D'} = ({1/n}) \rho_{\bar D}$.
It is known that these projective heat operators are
projectively flat. This can be seen in a number of ways;
one way is to show that $p_\ast({\rm Diff}^{(3)}_{{\cal X}/S}({\cal L}))={\cal O}_S$
in this case, compare \ref{heat+flat}.
Uniqueness of the construction of $\bar D$ implies that
$\bar D$ commutes (projectively) with the action of the theta-group
${\cal G}({\cal L})$. In particular the projective monodromy
group of the (projectively flat) connection $\bar \nabla(\bar D)$
on $(p_\ast {\cal L})_0$ normalizes the action of ${\cal G}(L)$.
\subsubsection{Heat operators and functoriality}\label{heat+funct}
Let ${\cal X}\to S$ and ${\cal L}$ as above, and let $D$ be a heat operator on
${\cal L}$ over $S$. Note that for any morphism of smooth schemes
$\varphi:S'\to S$ there is a pullback heat operator
$D'=\varphi^\ast(D)$ on the pull back ${\cal L}'$ on ${\cal X}'=S'\times_S{\cal X}$.
We leave it to the reader to give the precise definition.
This
pullback preserves flatness. If
$\varphi^\ast(p_\ast{\cal L})\cong p'_\ast{\cal L}'$, then the connection
$\nabla(D')$ is the pullback of the connection $\nabla(D)$.
In addition pullback is well defined for
projective heat operators and preserves projective flatness.
If $\psi:{\cal U}\to {\cal X}$ is an \'etale morphism of schemes
over $S$, then the heat operator $D$ induces a
heat operator $\psi^\ast(D)$ on $\psi^\ast({\cal L})$
over $S$.
\subsubsection{Heat operators and compatibility}\label{heat+comp}
Suppose we have a second smooth surjective morphism $q:{\cal Y}\to S$
and that we have a morphism $f: {\cal Y} \to {\cal X}$ over $S$. We will
assume that $f$ is a submersion. This means that for any point
$y\in {\cal Y}({\bf C})$ there is a coordinate patch $(V, t_1, \ldots, t_r,
x_1,\ldots,x_c, y_1,\ldots, y_m)$ of $f(y)$ in ${\cal X}$ such that
$(f^{-1}(V), t_1, \ldots, t_r, y_1=y_1\circ f,\ldots, y_m=y_m\circ f)$
is a coordinate patch for $y$ on ${\cal Y}$ and $x_i\circ f=0$.
Let ${\cal L}$ be a line bundle on ${\cal X}$ as before.
Let us say that a (projective) heat operator $D$ on ${\cal L}$
over $S$ is {\it (weakly) compatible} with $f:{\cal Y}\to {\cal X}$ if
whenever we have a coordinate patch $(V, t_1, \ldots, t_r,
x_1,\ldots,x_c, y_1,\ldots, y_m)$ as above, and a trivialization
${\cal L}|_V\cong {\cal O}_V$, then the operator $D(\partial/\partial t_1)$
has the following symbol:
$$ \rho_D({\partial\over\partial t_1})= \sum\nolimits_{i\leq j=1}^m f_{ij}
{\partial^2\over\partial y_i\partial y_j} + \sum\nolimits_{i=1}^c x_i\;
\Xi_i$$
for certain $f_{ij}\in {\cal O}_{\cal X}(V)$ and $\Xi_i\in S^2{\cal T}_{{\cal X}/S}(V)$.
More precisely, this means there exists a
$\rho=\rho_{f,D} : q^\ast {\cal T}_S \to S^2{\cal T}_{{\cal Y}/S}$
such that the following diagram commutes:
$$\matrix{f^\ast(\rho_D) &:& f^\ast p^\ast {\cal T}_S & \longrightarrow &
f^\ast S^2{\cal T}_{{\cal X}/S}\cr
&&||&&\uparrow\cr
\rho_{f,D}& : & q^\ast {\cal T}_S&\longrightarrow & S^2{\cal T}_{{\cal Y}/S}\cr}$$
Suppose that $D'$ is a heat operator on $f^\ast{\cal L}$ on ${\cal Y}$
over $S$. We say that $D'$ is {\it (weakly) compatible} with
$D$ if $D$ is (weakly) compatible with $f$ and we have
$\rho_{D'}=\rho_{f,D}$. In this case it is not true in general
that $D(\theta)(s)|_{\cal Y}=D'(\theta)(s|_{\cal Y})$.
Assume that $D$ is compatible with $f$. Let us write
$N_{\cal Y}{\cal X}$ for the generalized normal bundle of ${\cal Y}$
in ${\cal X}$, i.e., $N_{\cal Y}{\cal X}={\rm Coker}({\cal T}_{{\cal Y}/S}\to f^\ast{\cal T}_{{\cal X}/S})$.
Choose a local coordinate patch
$(f^{-1}(V), t_1, \ldots, t_r, y_1=y_1\circ f,\ldots, y_m=y_m\circ f)$
as above. Write the operator $D(\partial/\partial t_1)$ in the form
$$D({\partial\over\partial t_1})= f + {\partial\over\partial t_1} +
\sum\nolimits_{i=1}^c f_i {\partial\over\partial x_i} +
\sum\nolimits_{i=1}^m g_i {\partial\over\partial y_i} +
\sum\nolimits_{i\leq j=1}^m f_{ij} {\partial^2\over\partial y_i\partial
y_j}
+ \sum\nolimits_{i=1}^c x_i\; \Xi_i, $$
for certain local functions $f, f_i, g_i, f_{ij}$ and second order
operators $\Xi_i$. It can be seen (and we leave this to the
reader) that the class of $\sum f_i {\partial\over\partial x_i}$
in $N_{\cal Y}{\cal X} (f^{-1}(V))$ is independent of the choice of the
local trivialization and coordinates. The upshot of this is that
if $D$ is compatible with ${\cal Y}\to {\cal X}$, then there is a
first order symbol
$$ \sigma_{f,D} : q^\ast{\cal T}_S \longrightarrow N_{\cal Y}{\cal X}.$$
We say that $D$ is {\it strictly compatible} with $f: {\cal Y} \to {\cal X}$
if $D$ is compatible with $f$ and the symbol $\sigma_{f,D}$ is zero.
We remark that this notion is well defined for a projective heat
operator as well.
Assume that $D$ is strictly compatible with ${\cal Y}\to{\cal X}$.
It follows from the local description of our compatibility of
restrictions that $D$ induces a heat operator $D_{\cal Y}$
on the invertible ${\cal O}_{\cal Y}$-module $f^\ast{\cal L}$ over $S$.
It is characterized by the property
$D(\theta)(s)|_{\cal Y}=D_{\cal Y}(\theta)(s|_{\cal Y})$ for any
local section $\theta$ of ${\cal T}_S$ and any local section
$s$ of ${\cal L}$.
It is clear that the symbol of $D_{\cal Y}$ is equal to
$\rho_{f,D}$, hence that $D_{\cal Y}$ is compatible with $D$.
We say that a heat operator $D'$ on $f^\ast{\cal L}$ is {\it strictly
compatible} with $D$ if $D$ is strictly compatible with
$f:{\cal Y}\to {\cal X}$ and $D'=D_{\cal Y}$. In this case the restriction
map
$$ p_\ast {\cal L} \longrightarrow q_\ast f^\ast {\cal L}$$
is horizontal for the connections induced by $D$ and $D'=D_{\cal Y}$.
Note $D_{\cal Y}$ is flat if $D$ is flat.
Similar remarks hold for projective heat operators and projective
flatness.
\subsection{Moduli of bundles}\label{modbun}
\subsubsection{The symbol $\rho_{Hitchin}$ in terms of
moduli of bundles}\label{Hitchin-symbol}
In this subsection we explain how to get a symbol map $\rho$
as in \ref{heat+symbol} in the case of the relative moduli space
of rank 2 bundles of a family of curves.
Let $C$ be a smooth projective curve over $\mathop{\rm Spec}({\bf C})$, and
let $E$ be a stable invertible ${\cal O}_C$-module of rank 2
with ${\rm det}(E)\cong {\cal O}_C$. This gives a point
$[E]\in {\cal M}_C({\bf C})$. There are canonical identifications
$$
T_{[E]}{\cal M}_C=H^1(C,{{\cal E}{\it nd}}_0(E))\qquad{\rm and}\quad
T_{[E]}^*{\cal M}_C=H^0(C,{{\cal E}{\it nd}}_0(E)\otimes \Omega^1_C)
$$
where ${{\cal E}{\it nd}}_0(E)$ denotes the sheaf of endomorphisms of
$E$ with trace zero. A cotangent vector
$\phi\in T^*_{[E]}{\cal M}_C$
thus corresponds to a homomorphism of ${\cal O}_C$-modules
$\phi : E\rightarrow E\otimes \Omega^1_C$.
(Note that this is a Higgs field on the bundle $E$.)
Composing and taking the trace gives a symmetric bilinear pairing
$$
\matrix{T_{[E]}^*{\cal M}_C \times T_{[E]}^*{\cal M}_C &
\longrightarrow & H^0(C,(\Omega^1_C)^{\otimes 2})\cr
(\phi , \psi) & \longmapsto & {\rm Trace}(\phi\circ\psi).\cr}
$$
We dualize this and use Serre duality to obtain a map
$$
\rho_{C,E}:H^1(C,T_C)\longrightarrow S^2T_{[E]}{\cal M}_C.
$$
Let us make the observation that the construction above can be
performed in families. Let $\pi:{\cal C} \to S$ a smooth projective family
of curves over the scheme $S$ smooth over $\mathop{\rm Spec}({\bf C})$. Let
$p : {\cal M} \to S$ be the associated family of moduli spaces of
rank 2 semi-stable bundles with
trivial determinant up to $S$-equivalence.
We remark that $p$ is a flat projective morphism, not smooth in general.
However, the open part of stable bundles ${\cal M}^s\subset {\cal M}$
is smooth over $S$. We denote this smooth morphism
by $p^s: {\cal M}^s\to S$.
Now let $T\to S$ be a morphism of finite type. We denote by
an index ${}_T$ base change to $T$.
Let ${\cal E}$ be a locally free sheaf of rank 2 on ${\cal C}_T$.
For a point $0\in T({\bf C})$ we put $C={\cal C}_0$ and $E={\cal E}_0={\cal E}|_C$.
Suppose that for any $0$ we have (a) ${\rm det}(E)\cong {\cal O}_C$, and
(b) the bundle $E$ is stable. In this case we get an $S$-morphism
$t: T\to {\cal M}^s$. Analogously to the above
we have a canonical isomorphism
$$ t^\ast(\Omega^1_{{\cal M}^s/S})\cong
(\pi_T)_\ast({{\cal E}{\it nd}}_0({\cal E})\otimes_{{\cal O}_{{\cal C}_T}}\Omega^1_{{\cal C}_T/T}). $$
Using the same pairing and dualities
as above we get an ${\cal O}_T$-linear map
$$ R^1\pi_\ast {\cal T}_{{\cal C}_T/T} \longrightarrow
S^2\Big(t^\ast({\cal T}^1_{{\cal M}^s/S})\Big).$$
If we compose this with the pullback to $T$ of the Kodaira-Spencer map
${\cal T}_S\to R^1\pi_\ast {\cal T}_{{\cal C}/S}$ of the family ${\cal C}\to S$, then we
get a map
$$
\rho_{{\cal E}/{\cal C}_T} : {\cal T}_S\otimes{\cal O}_T \longrightarrow
S^2\Big(t^\ast({\cal T}^1_{{\cal M}^s/S})\Big).
$$
If there existed a universal bundle ${\cal E}^{univ}$ over ${\cal M}^s$,
then we would get a ``symbol''
$$
\rho^s_{Hitchin} = \rho_{{\cal E}^{univ}}: (p^s)^\ast {\cal T}_S \to
S^2{\cal T}_{{\cal M}^s/S}.
$$
Although the universal bundle does not exist, the symbol
$\rho^s_{Hitchin}$ does: it is the unique ${\cal O}_{{\cal M}^s}$-linear
map such that, whenever ${\cal E}/{\cal C}_T$ is given, the pullback
of $\rho_{Hitchin}$ to $T$ agrees with $\rho^s_{{\cal E}/{\cal C}_T}$.
Uniqueness and existence follows from the existence of ``universal''
bundles \'etale locally over ${\cal M}^s$ and the invariance
of $\rho_{{\cal E}/{\cal C}_T}$ under automorphisms of ${\cal E}$ over ${\cal C}_T$.
\subsubsection{Hitchin's results in genus at least 3}
Let us get back to our curve $C$ over $\mathop{\rm Spec}({\bf C})$. We see
from the above that any deformation $C_\epsilon$ of the curve will give a
tensor in $\Gamma({\cal M}_C^s, S^2{\cal T}_{{\cal M}_C})$. However, since ${\cal M}_C$
has points which do not correspond to stable bundles
(and these points are singular points of ${\cal M}_C$ for $g>2$)
such a deformation does not give a global tensor in
$H^0({\cal M}_C,S^2T_{{\cal M}_C})$. In \cite[Section 5]{Hi}, one can find a
sketch of a proof that the tensor extends if $g>2$ (or rank $>2$,
a case we do not even consider, although our present considerations
work in that case as well).
In the situation ${\cal C}\to S$ there is a canonical line bundle
${\cal L}$ on ${\cal M}$; it can be defined by the formula
$${\cal L} = {\rm det}\; R({pr}_2)_\ast {\cal E}^{univ},\
\quad pr_2 : {\cal C}\times_S {\cal M} \to {\cal M}.$$
It is shown in \cite[Theorem 3.6, Section 5]{Hi}, that if $g>2$ the tensor
$2/(2k+\lambda)\rho_{Hitchin}$ is the symbol of a unique projective
flat heat operator $\bar D_{Hitchin}$ on ${\cal L}^{\otimes k}$ over $S$.
(We remark that the extra factor 2 comes from our
way of normalizing symbols, see Subsection \ref{normalization}.)
In fact the arguments of \cite{Hi} prove the existence of this
heat operator over the moduli space of curves of genus $g>2$.
We will not use these results.
\subsection{The case of genus two curves}\label{g=2}
\subsubsection{Notation in genus 2 case}\label{notation-2}
We fix a scheme $S$ smooth over $\mathop{\rm Spec}({\bf C})$ and a
smooth projective morphism $\pi:{\cal C}\to S$, whose fibres
$C$ are curves of genus 2. We write $0\in S({\bf C})$ for
a typical point and $C={\cal C}_0$.
We introduce the following objects associated to the situation.
${\rm Pic}^1={\rm Pic}^1_{{\cal C}/S}\to S$ denotes the Picard scheme of
invertible ${\cal O}_{\cal C}$ bundles of relative degree 1 on ${\cal C}$
over $S$. There is a natural morphism ${\cal C}\to {\rm Pic}^1$ over $S$,
given by $P\in C({\bf C})$ maps to $[{\cal O}_C(P)]$ in
$({\rm Pic}^1)_0({\bf C})={\rm Pic}^1(C)$.
The image of this morphism is a relative divisor
$\Theta^1=\Theta^1_{{\cal C}/S}$
on ${\rm Pic}^1$ over $S$; of course $\Theta^1\cong {\cal C}$ as $S$-schemes.
Let us denote $\alpha : {\rm Pic}^1\to S$ the structural morphism.
Note that
$\alpha_\ast{\cal O}_{{\rm Pic}^1}(2\Theta^1)$
is a locally free ${\cal O}_S$-module of rank $4$ on $S$. We put
${\bf P}=|2\Theta^1|$ equal
to the projective space of lines in this locally free sheaf.
More precisely, we define
$$
p : {\bf P} = {\bf P}\bigg(
{\cal H}{\it om}_{{\cal O}_S}\Big(\alpha_\ast{\cal O}_{{\rm Pic}^1}(2\Theta^1)\;,\;
{\cal O}_S\Big)\bigg) \longrightarrow S,
$$
see \cite[page 162]{Ha} for notation used. For a point $0\in S({\bf C})$
we put $P={\bf P}_0$.
Let $T\to S$ be a morphism of finite type, and let
${\cal E}$ be a locally free sheaf of ${\cal O}_{{\cal C}_T}$-modules of rank 2.
For a point $0\in T({\bf C})$ we put $C={\cal C}_0$ and $E={\cal E}_0={\cal E}|_C$.
Suppose that for any $0$ we have
\vspace{.5 \baselineskip}
(a) $\quad {\rm det}(E)\cong {\cal O}_C$,
(b)$\quad$ the bundle $E$ is {\it semi}-stable.
\vspace{.5 \baselineskip}
(Compare with Subsection
\ref{Hitchin-symbol}.) We recall the relative divisor
${\cal D}_{\cal E}\subset {\rm Pic}^1_T$ associated to ${\cal E}$. Set-theoretically
it has the following description:
$$ D_E := {\cal D}_{\cal E} \cap {\rm Pic}^1(C) = \{ [L]\in {\rm Pic}^1(C) :
\dim H^1(C, E\otimes L) \geq 1\}$$
To construct ${\cal D}_{\cal E}$ as a closed subscheme
we may work \'etale locally on $T$. Hence we may assume that
a Poincar\'e line bundle ${\cal L}$ on ${\cal C}_T\times_T {\rm Pic}^1_T$ exists,
i.e., which has relative degree 1 for
$p_2:{\cal C}_T\times_T {\rm Pic}^1_T\to {\rm Pic}^1_T$ and induces
${\rm id}: {\rm Pic}^1_T \to {\rm Pic}^1_T$. We consider the line bundle
$$ {\cal N}=\Big({\rm det} R(p_2)_\ast \Big( p_1^\ast({\cal E})
\otimes {\cal L} \Big)\Big)^{\otimes -1}$$
on ${\rm Pic}^1_T$. This line bundle has a natural section (the theta
function) $\theta({\cal E})$, which extends the section $1$ on the
open schematically dense subscheme $U\subset {\rm Pic}^1_T$ over
which the complex $R(p_2)_\ast \Big( p_1^\ast({\cal E})
\otimes {\cal L} \Big)$ is trivial. We define ${\cal D}_{\cal E}$ to be the
zero set of the section $\theta({\cal E})$; it is also the largest
closed subscheme of ${\rm Pic}^1_T$ over which the coherent
${\cal O}_{{\rm Pic}^1_T}$-module $R^1(p_2)_\ast \Big( p_1^\ast({\cal E})
\otimes {\cal L} \Big)$ has rank $\geq 1$ (i.e., defined in terms of
a fitting ideal of this sheaf).
One can prove that ${\cal N}\otimes {\cal O}(-2\Theta^1_T)$ is isomorphic to
the pullback of an invertible ${\cal O}_T$-module on $T$.
Hence the divisor ${\cal D}_{\cal E}$ defines an $S$-morphism of $T$
into ${\bf P}$:
$$ \varphi_{(T,{\cal E})} : T \longrightarrow {\bf P}.$$
These constructions define therefore a transformation of
the stack of semi-stable rank 2 bundles with trivial determinant
on ${\cal C}$ over $S$ towards the scheme ${\bf P}$. It turns out that
this defines an isomorphism of the coarse moduli scheme
$$ {\cal M}_{\cal C} \longrightarrow {\bf P}$$
towards ${\bf P}$, see \cite{NR}.
A remark about the natural determinant bundle ${\cal L}$ on ${\cal M}_{\cal C}$
(see \cite[page 360]{Hi}).
The relative Picard group of ${\bf P}$ over $S$ is ${\bf Z}$ and
is generated by ${\cal O}_{\bf P}(1)$. Hence it is clear that
${\cal L} \cong {\cal O}_{\bf P}(n)\otimes p^\ast ({\cal N})$ for some
invertible ${\cal O}_S$-module ${\cal N}$ on $S$. The integer $n$
may be determined as follows. We know by \cite[page 360]{Hi}
that ${\cal O}_P(n)^{-\lambda} \cong K_P$. The integer $\lambda=4$
in this case, hence $n=1$. We are going to define a
projective heat operator on ${\cal L}^{\otimes k}$ over $S$, but we have seen
in Subsection \ref{heat+change} that this is the same
as defining a projective heat operator on ${\cal O}_{\bf P}(k)$.
Hence we will work with the line bundle ${\cal O}_{\bf P}(k)$ from now on.
(Note that as both ${\cal O}_{\bf P}(1)$ and ${\cal L}$ are defined
on the moduli stack, they must be related by a line bundle
coming from the moduli stack ${\cal M}_2$ of curves of genus 2;
however ${\rm Pic}({\cal M}_2)$ is rather small.)
Let ${\rm Pic}^0={\rm Pic}^0_{{\cal C}/S}\to S$ denote the Picard scheme of
invertible ${\cal O}_{\cal C}$ bundles of relative degree 0 on ${\cal C}$
over $S$. There is a natural $S$-morphism
$$
f : {\rm Pic}^0 \longrightarrow {\bf P}
$$
which in terms of the moduli-interpretations of both spaces
can be defined as follows:
$$
{\rm Pic}^0(C) \ni [L] \longmapsto [L\oplus L^{ -1}]\in {\bf P}({\bf C}).
$$
It is clear that the divisor $D_{E}$ associated to
$E=L\oplus L^{ -1}$ is equal to
$D_E= (\Theta^1_C + [L]) \cup (\Theta^1_C + [L^{-1}]) \subset {\rm Pic}^1(C)$.
This implies readily that $f^\ast {\cal O}_{\bf P}(1) \cong
{\cal O}_{{\rm Pic}^0}(2\Theta^0)$,
in fact $f^\ast{\cal O}_{\bf P}(1)$ may serve as the definition of
${\cal O}_{{\rm Pic}^0}(2\Theta^0)$ on ${\rm Pic}^0$.
This induces an isomorpism ${\bf P}\cong |{\cal O}_{{\rm Pic}^0}(2\Theta^0)|^*$ which
identifies $f$ with the natural map. Thus $f$ factors over
${\cal K}:={\rm Pic}^0/\langle\pm1 \rangle$, the relative Kummer surface,
and
defines
a closed immersion (over $S$):
$$
{\cal K}\hookrightarrow {\bf P}.
$$
Any $E$ on $C$ that is semi-stable but not stable is an extension
of the form $0\to L\to E\to L^{-1}\to 0$, with $L$ of degree zero.
This bundle is $S$-equivalent to the bundle $L\oplus L^{-1}$.
Thus the open subscheme ${\bf P}^s\subset {\bf P}$ is equal to the
complement of the image of $f: {\rm Pic}^0 \to {\bf P}$.
\subsubsection{Automorphisms}\label{autos}
The group scheme
${\cal H}={\rm Pic}^0[2]$ of 2-torsion points of ${\rm Pic}^0$ over $S$
acts on
the schemes ${\rm Pic}^1$,
Indeed, if ${\cal E}$ over ${\cal C}_T$ is a family of locally free
sheaves as above and if ${\cal A}$ is a line bundle on ${\cal C}_T$
which defines a 2-torsion point of ${\rm Pic}^0$, then
${\cal E}\otimes {\cal A}$ is another family of locally free sheaves
satisfying (a) and (b) of \ref{notation-2}. This defines an action
${\cal H}\times_S {\bf P}\to {\bf P}$ of ${\cal H}$ on ${\bf P}$, given
the identification of ${\bf P}$ as the moduli scheme. It is easy to
see that this action is induced from the natural
action of ${\cal H}$ on ${\rm Pic}^1$. Furthermore the morphism
$f : {\rm Pic}^0 \to {\bf P}$ is equivariant with respect to
this action.
Note that the action just defined does not lift to an
action of ${\cal H}$ on ${\cal O}_{\bf P}(1)$. Let us write ${\cal G}$ for
theta group of the relatively ample line bundle
${\cal O}_{{\rm Pic}^0}(2\Theta^0)$ on ${\rm Pic}^0$ over $S$; this
group scheme gives the Heisenberg group for any point
$0\in S({\bf C})$. We remark that ${\cal G}$ fits into the
exact sequence
$$
1\longrightarrow {\bf G}_{m,S} \longrightarrow
{\cal G} \longrightarrow {\cal H} \longrightarrow 0.
$$
There is a unique action of ${\cal G}$ on ${\cal O}_{\bf P}(1)$
which lifts the action of ${\cal H}$ on ${\bf P}$ and
agrees with the defining action ${\cal G}$ on $f^\ast{\cal O}_{\bf P}(1)$
over ${\rm Pic}^0$.
\subsubsection{Test families}
Let ${\cal E}$ be a rank 2 free ${\cal O}$-module on ${\cal C}_T$ satisfying
(a) and (b) of \ref{notation-2}. We will say that the pair
$(T, {\cal E})$ is a {\it test family} if the following conditions
are satisfied:
\begin{enumerate}
\item $T\to S$ is smooth,
\item all $E={\cal E}_0$ are simple vector bundles on $C={\cal C}_0$, and
\item the map ${\cal T}_{T/S} \longrightarrow
R^1{{\rm pr}_2}_\ast ({{\cal E}{\it nd}}_0({\cal E}))$ is an isomorphism.
\end{enumerate}
The last condition needs some clarification. The obstruction
for the locally free sheaf ${\cal E}$ to have a connection on ${\cal C}_T$
is an element in
$$H^1({\cal C}\times_ST, \Omega^1_{{\cal C}\times_ST}\otimes {{\cal E}{\it nd}}({\cal E})).$$
We can use the maps
$\Omega^1_{{\cal C}\times_ST} \to {\rm pr}_2^\ast\Omega^1_{T/S}$
and ${{\cal E}{\it nd}}({\cal E})\to {{\cal E}{\it nd}}_0({\cal E})$ to project this to a section of
$\Omega^1_{T/S}\otimes R^1{{\rm pr}_2}_\ast ({{\cal E}{\it nd}}_0({\cal E}))$.
Whence the map of condition 3. This condition means
that the deformation of $E={\cal E}_0$ for any $0\in T$ is
versal in the ``vertical direction''.
The following lemma is proved in the usual manner, using deformation
theory (no obstructions !) and Artin approximation.
\begin{lem}\label{existence}For any $0\in S({\bf C})$ and any simple
bundle $E$ on $C={\cal C}_0$ there exists a test family
$(T, {\cal E})$ such that $(E,C)$ occurs as one of its
fibres. \end{lem}
\begin{lem}\label{etale}Let $(T, {\cal E})$ be a test family. The
morphism $\varphi_{(T,{\cal E})} : T \to {\bf P}$ is
\'etale.\end{lem}
\begin{proof} Let $E$ be a simple rank 2 bundle on a smooth
projective genus two curve $C$ over $\mathop{\rm Spec}({\bf C})$ with
${\rm det}(E)={\cal O}_C$. Let $\eta\in H^1(C, {{\cal E}{\it nd}}_0(E))$. For
each invertible sheaf $L$ on $C$, with ${\rm deg}(L)=-1$
and given embedding $L\subset E$ consider the map
$${{\cal E}{\it nd}}_0(E) \longrightarrow {{\cal H}{\it om}}(L, E).$$
We have to show $(*)$: If for all $L\subset E$ as above $\eta$ maps to 0
in $H^1(C, {{\cal H}{\it om}}(L,E))$, then $\eta=0$.
Indeed, the condition $\eta\mapsto 0$ means that the trivial
deformation $L[\epsilon]$ of $L$ over
$C[\epsilon]=C\times \mathop{\rm Spec}({\bf C}[\epsilon])$ can be embedded
into the deformation of $E$ given by $\eta$. If this holds
for all $L\subset E$, then the divisor $D_E$ does not move,
and we want this to imply that the infinitesimal deformation of
$E$ is trivial.
Let us prove $(*)$. We will do this in the case that $E$
is not stable, as we already have the result in the stable case,
see Subsection \ref{Hitchin-symbol}. Thus we may assume the
bundle $E$ is simple but not stable: $E$ is a nontrivial
extension
$$ 0\longrightarrow A \longrightarrow E \longrightarrow A^{-1}
\longrightarrow 0,$$
with ${\rm deg}_C A=0$, $A^{\otimes 2}\not\cong {\cal O}_C$. In this
case it is easy to see that there exist three line bundles
$L_1\subset E$, $L_2\subset E$ and $L_3\subset E$ of degree $-1$ on $C$
such that the arrow towards $E$ in the following exact sequence
is surjective:
$$ 0\longrightarrow K \longrightarrow L_1\oplus L_2 \oplus L_3
\longrightarrow E \longrightarrow 0.$$
Here $K$ is just the kernel of the surjection.
For example we can take $L_i$ of the form $A^{-1}(-P_i)$ for
$i=1,2$ and $L_3$ of the form $A(-P_3)$.
(This is the only thing we will need in the rest of the
argument; it should be easy to establish this in the
case of a stable bundle $E$ also.) This property then holds for
$L_i\subset E$ sufficiently general also; choose $L_i$
sufficiently general. Note that
$K\cong L_1\otimes L_2\otimes L_3$ as ${\rm det}(E)={\cal O}_C$.
Suppose we have a deformation $E_\epsilon$ of $E$ given
by $\eta$ as above, i.e., we have $L_i[\epsilon]\to E_\epsilon$
lifting $L\to E$. This determines an exact sequence
$$ 0\longrightarrow K_\epsilon \longrightarrow
L_1[\epsilon]\oplus L_2[\epsilon] \oplus L_3[\epsilon]
\longrightarrow E_\epsilon \longrightarrow 0.$$
But ${\rm det} E_\epsilon\cong {\cal O}_{\cal C}[\epsilon]$,
hence $K_\epsilon\cong
L_1[\epsilon]\otimes L_2[\epsilon] \otimes L_3[\epsilon]
\cong (L_1\otimes L_2\otimes L_3)[\epsilon]$ is trivial
as well. Now note that by our general
position we have
$$ \dim H^0(C, {{\cal H}{\it om}}(K, L_1))=\dim H^0(C, L_2\otimes L_3)=1.\leqno{(1)}$$
Similar for $L_2$ and $L_3$. Therefore, up to a unit in
${\bf C}[\epsilon]$, there is only one map
$K[\epsilon]\to L_i[\epsilon]$, lifting $K\to L_i$.
Thus the sequence $(1)$ above is uniquely
determined up to isomorphism, hence $E_\epsilon\cong E[\epsilon]$
is constant, i.e., $\eta=0$.
\end{proof}
\subsection{Results} This lemma completes the preparations.
In the remainder of this section we prove the desired results on the
existence and the characterization of the connection.
\begin{thm} \label{symext}
The Hitchin symbol $\rho^s_{Hitchin}:p^\ast{\cal T}_S\rightarrow S^2{\cal T}_{{\bf P}^s/S}$
defined
in Subsection \ref{Hitchin-symbol} extends
to a symbol
$$
\rho_{Hitchin} : p^\ast {\cal T}_S
\longrightarrow S^2{\cal T}_{{\bf P}/S}.
$$
\end{thm}
\begin{proof} Let $(T, {\cal E})$ be a test family. We have the
morphism $\varphi=\varphi_{(T,{\cal E})} : T \to {\bf P}$ which is \'etale
and wich induces therefore an isomorphism
${\cal T}_{T/S}\to\varphi^\ast {\cal T}_{{\bf P}/S}$. On the other
hand we have the isomorphism ${\cal T}_{T/S} \to
R^1{{\rm pr}_2}_\ast ({{\cal E}{\it nd}}_0({\cal E}))$ of the test family $(T,{\cal E})$.
We also have the
pairing as in Subsection \ref{Hitchin-symbol}
$${{\rm pr}_2}_\ast ({{\cal E}{\it nd}}_0({\cal E})\otimes \Omega^1_{{\cal C}_T/T})
\otimes {{\rm pr}_2}_\ast ({{\cal E}{\it nd}}_0({\cal E})\otimes \Omega^1_{{\cal C}_T/T})
\longrightarrow {{\rm pr}_2}_\ast ((\Omega^1_{{\cal C}_T/T})^{\otimes 2}). $$
Note that ${{\rm pr}_2}_\ast ({{\cal E}{\it nd}}_0({\cal E})\otimes \Omega^1_{{\cal C}_T/T})$
is a locally free sheaf of ${\cal O}_T$-modules as ${\cal E}$ has simple fibres.
These maps and duality give us together with the Kodaira-Spencer map
a map
$$ {\cal T}_S \otimes {\cal O}_T \longrightarrow S^2\varphi^\ast {\cal T}_{{\bf P}/S}.$$
We remark that on the (nonempty) stable locus $T^s=\varphi^{-1}({\bf P}^s)$
this map is equal to the pullback of $\rho^s_{Hitchin}$.
Thus, by Lemma \ref{etale}, we see that we can extend
$\rho^s_{Hitchin}$ to any point
$x\in {\bf P}({\bf C})$ which is the image of some simple bundle.
Note that a semi-stable bundle $E$, which is a
nontrivial extension $0\to L\to E\to L^{-1}\to 0$
with $deg(L)=0$ is simple if and only if $L^{\otimes 2}\not\cong {\cal O}_C$.
It follows from Lemma \ref{existence} and the above that we can extend
$\rho^s_{Hitchin}$ to the
complement of the codimension 3 locus $f({\rm Pic}^0[2])$,
and hence by Hartog's theorem it extends.\end{proof}
\begin{prop}\label{comp+abelian}
Let $f : {\rm Pic}^0 \to {\bf P}$ be as in \ref{notation-2}, then $f$ is a submersion
on the locus ${\cal Y}:= {\rm Pic}^0 \setminus {\rm Pic}^0[2]$.
There is a
commutative diagram
$$
\matrix{f^\ast(\rho_{Hitchin}) &:& f^\ast p^\ast {\cal T}_S &
\longrightarrow & f^\ast S^2{\cal T}_{{\bf P}/S}\cr
&&||&&\uparrow\cr
4\rho_{Ab}|_{\cal Y}& : & {\cal T}_S\otimes {\cal O}_{\cal Y} &\longrightarrow
& S^2{\cal T}_{{\cal Y}/S}\cr}
$$
where $\rho_{Ab}$ is the symbol of the abelian heat operator
on ${\cal O}_{{\rm Pic}^0}(2\Theta^0)=f^\ast{\cal O}_{\bf P}(1)$.
\end{prop}
\begin{proof}
Consider once again
a simple bundle $E$, which is given as an
extension $ 0\to L\to E\to L^{-1}\to 0$ with $deg(L)=0$. Clearly, the kernel
of the map
$$
H^1(C, {{\cal E}{\it nd}}_0(E))\longrightarrow H^1(C, {{\cal H}{\it om}}(L, L^{-1}))
$$
represents deformations preserving the filtration on $E$,
i.e., tangent vectors along the Kummer surface. Dually, this
corresponds to the image of the map
$$ H^0(C, {{\cal H}{\it om}}(L^{-1},L)\otimes\Omega^1_C)\longrightarrow
H^0(C, {{\cal E}{\it nd}}_0(E)\otimes\Omega^1_C).$$
We use the Trace-form to identify ${{\cal E}{\it nd}}_0(E)$ with its dual.
Note that there is a canonical exact sequence
$$
0\longrightarrow {\cal O}_C \longrightarrow
{{\cal E}{\it nd}}_0(E)/L^{\otimes 2} \longrightarrow L^{\otimes -2}
\longrightarrow 0
$$
induced by the filtration $L\subset E$ on $E$.
Note that $\dim H^0(C, ({{\cal E}{\it nd}}_0(E)/L^{\otimes 2})\otimes \Omega^1_C)=2$,
since if it were $\geq 3$, then
$\dim H^0(C, {{\cal E}{\it nd}}_0(E)\otimes \Omega^1)\geq 4$ contrary to
the assumption that $E$ is simple. Thus we get the exact sequence
$$
H^0(C,\Omega^1)\cong H^0(C, ({{\cal E}{\it nd}}_0(E)/L^{\otimes 2})\otimes \Omega^1_C)
\ll\!\!\longleftarrow H^0(C, {{\cal E}{\it nd}}_0(E)\otimes\Omega^1_C) \hookleftarrow
H^0(C, L^{\otimes 2}\otimes \Omega^1_C)\cong {\bf C}.
$$
This is (canonically) dual to the sequence of cotangent spaces
$$ 0 \longrightarrow T_{[L]}{\rm Pic}^0 \longrightarrow
T_{f([L])}({\bf P}) \longrightarrow (N_{\cal Y} {\bf P})_{[L]} \longrightarrow 0.$$
We want to see that our symbols in the point $[L]$ lie in the
space $S^2T_{[L]}{\rm Pic}^0$.
We have to study the map
$$
\matrix{ H^0(C, {{\cal E}{\it nd}}_0(E)\otimes\Omega^1_C)\times
H^0(C, {{\cal E}{\it nd}}_0(E)\otimes\Omega^1_C) & \longrightarrow &
H^0(C, (\Omega^1_C)^{\otimes 2})\cr
X\otimes \eta \times Y\otimes \omega &\longmapsto
& {\rm Trace}(XY)\otimes \eta\omega.\cr}
$$
But the elements of $H^0(C, L^{\otimes 2}\otimes \Omega^1_C)$,
resp.\ those of $H^0(C, \Omega^1_C)$ locally
look like
$$\left( \matrix{ 0 & *\cr
0&0\cr}\right)\otimes \eta, \quad\hbox{ resp.\ }\quad
\left( \matrix{ 1 & 0\cr
0&-1\cr}\right)\otimes \omega.$$
Therefore the pairing just reduces to twice the multiplication
pairing $H^0(C,\Omega^1_C) \times H^0(C,\Omega^1_C) \to
H^0(C, (\Omega^1_C)^{\otimes 2})$.
This multiplication pairing, however, corresponds excactly
(via Kodaira-Spencer) to the symbol of the heat operator
on the theta-divisor of the Jacobian ${\rm Pic}^0$ of ${\cal C}$ over
$S$. Since we are looking at $2\Theta^0$, we get the
desired factor $4$, compare with Subsections
\ref{heat+abelian} and \ref{heat+symbol}.
\end{proof}
\begin{cor}\label{exisD}
The symbol $\rho_{Hitchin}$
is invariant under
the action of the group scheme ${\cal H}={\rm Pic}^0[2]$ on ${\bf P}$
and $S^2{\cal T}_{{\bf P}/S}$.
\end{cor}
\begin{proof}
This follows from the
observation that it is true for $\rho_{Ab|{\cal Y}}$ and thus for
$\rho_{Hitchin}|_{{\cal K}}$,
the Hitchin symbol restricted to the Kummer surface.
Further,
one uses the remark that there are no nonvanishing elements
of $H^0({\bf P}, S^2{\cal T}_{{\bf P}/S})$ which vanish on ${\cal K}$, that is
$H^0({\bf P},S^2{\cal T}_{{\bf P}/S}(-4))=0$.
\end{proof}
\begin{prop}\label{existence+heat}
There exists a unique projective heat operator $\bar D_{\lambda,k}$
(with $\lambda\in{\bf C}^\ast$ be a nonzero complex number)
on ${\cal O}_{\bf P}(k)$ over $S$ with the following properties:
\begin{enumerate}
\item the symbol of $\bar D_{\lambda,k}$ is equal to
${1\over 4\lambda}\rho_{Hitchin}$, and
\item the operator $\bar D_{\lambda,k}$ commutes (projectively) with
the action of ${\cal G}$ on ${\cal O}_{\bf P}(k)$.
\end{enumerate}
\end{prop}
\begin{proof}
This follows readily from the discussion in Subsection \ref{heat+symbol}.
Indeed, both obstructions mentioned there vanish in view of the
vanishing of $R^1p_\ast {\cal T}_{{\bf P}/S}$ and $R^1p_\ast{\cal O}_{\bf P}$. We have
a good way of choosing the elements $D(\theta)$: namely, we choose
them ${\cal G}$-invariantly. This is possible: first choose arbitrary lifts,
and then average over liftings of a full set of sections of ${\cal H}$
(this can be done \'etale locally over $S$). This determines the
$D(\theta)$ uniquely up to an element of $p^{-1}({\cal O}_S)$, as
$(p_\ast{\cal T}_{{\bf P}/S})^{{\cal H}}=(0)$. Hence we get the desired
projective heat operator.
\end{proof}
It follows from Proposition \ref{comp+abelian} that the heat
operators $\bar D_{\lambda,k}$ are compatible with the morphism
$f : {\cal Y} \to {\bf P}$. Also, for $k=1$ and $\lambda=1$, we see that
$\bar D_{1,1}$ is compatible with the abelian heat operator on
${\cal O}_{{\rm Pic}^0}(2\Theta^0)$ restricted to ${\cal Y}$ (this is one of the
reasons for the factor ${1\over 4}$, but see below also).
\subsubsection{Hitchin's connection for genus 2 and rank 2}
The title of this subsection is somewhat misleading. As mentioned
before, in the paper \cite{Hi} there is no definition of a heat
operator in the case of genus 2 and rank 2. However, we propose
the following definition.
\subsubsection{Definition.}
\label{H22}
The (projective) Hitchin connection on $p_\ast {\cal O}_{\bf P}(k)$
is the projective connection associated to the projective
heat operator ${\bf D}_k:=\bar{D}_{(k+2), k}$.
\
This definition makes sense for the following reason:
The symbol of the operator $\bar{D}_{(k+2), k}$ is equal
to ${2/(2k+4)}\rho_{Hitchin}$ which is equal to the symbol
that occurs in \cite[Theorem 3.6]{Hi} (the factor 2
comes from our way of normalizing symbols, see
Subsection \ref{normalization}).
Thus the only assumption we needed in order to get
${\bf D}_k$ was the assumption that it is compatible with the action
of ${\cal G}$, see Subsection \ref{existence+heat}.
\subsubsection{How to compute the heat operators?}
\label{crit}
To determine ${\bf D}_k$ we introduce heat operators $D_{\lambda,k}$
under the following assumptions:
\begin{enumerate}
\item The family of Kummer
surfaces ${\cal K}\subset {\bf P}$ is given by one equation
$F\in \Gamma({\bf P}, {\cal O}_{\bf P}(4))$.
\item We have chosen an integrable connection $\nabla_0$ on
${\cal F}:=p_\ast {\cal O}_{\bf P}(1)$ over $S$ which is equivariant for
the action of ${\cal G}$.
\end{enumerate}
We remark that both conditions can be satisfied on the members of
an open covering of $S$. Also the connection $\nabla_0$ in 2 is determined
uniquely up to the addition of a linear operator of the form
$\eta\cdot {\rm id}$ for some closed 1-form $\eta$ on $S$.
We remark that $\nabla_0$ induces a connection $\nabla_0$ on all the
locally
free sheaves $S^k{\cal F}=p_\ast {\cal O}_{\bf P}(k)$ over $S$, but these should not be
confused with the connections induced by the $D_{\lambda,k}$!
In addition $\nabla_0$ determines an integrable connection on
${\bf P}$ over $S$ and a lift of this to connections on
the sheaves ${\cal O}_{\bf P}(k)$ over ${\bf P}$. In other words we
have rigidified $({\bf P},{\cal O}_{\bf P}(k))$ over $S$.
There is a natural surjection
$$ S^2{\cal F}\otimes_{{\cal O}_S}S^2{\cal F}^\ast \longrightarrow p_\ast
S^2{\cal T}_{{\bf P}/S}.$$
which has a canonical splitting, given by decomposing
$S^2{\cal F}\otimes_{{\cal O}_S}S^2{\cal F}^\ast$ into irreducible ${\rm
GL}({\cal F})$-modules,
i.e., the unique ${\rm GL}({\cal F})$-equivariant splitting.
Thus we identify a section $X$ of $S^2{\cal T}_{{\bf P}/S}$ with a section
$X$ of $S^2{\cal F}\otimes_{{\cal O}_S}S^2{\cal F}^\ast$. Note that we can regard
sections of $S^2{\cal F}\otimes_{{\cal O}_S}S^2{\cal F}^\ast$ as sections of
${\rm Diff}^{(2)}_{{\bf P}/S}\big({\cal O}_{\bf P}(k)\big)$ in a natural manner,
by considering them as second order differential operators on the
affine 4-space $\underline{\mathop{\rm Spec}}(S^\ast{\cal F})$ invariant under the
scalar action of ${\bf G}_{m,S}$. We write $E$ for the
Euler-vector field, i.e., the section of ${\cal F}\otimes {\cal F}^\ast$ that
corresponds to the identity map of ${\cal F}$.
Assume we have a ${\cal G}$-invariant section
$$
X \in S^2{\cal F}\otimes_{{\cal O}_S}S^2{\cal F}^\ast\otimes_{{\cal O}_S}\Omega^1_S
$$
with the following property
$$
X \cdot F = \nabla_0(F) E \quad\hbox{in}\quad
S^5({\cal F})\otimes_{{\cal O}_S}{\cal F}^\ast\otimes_{{\cal O}_S}\Omega^1_S,\leqno{(Eq.)}
$$
where $X\cdot F$ denotes contracting once.
If we have such an $X$ we define a heat operator
$$
D_{\lambda,k}:p^\ast {\cal T}_S \longrightarrow {\rm Diff}^{(2)}_{\bf P}({\cal O}_{\bf P}(k))
$$
by the formula ($\theta$ a local section of ${\cal T}_S$)
$$ D_{\lambda,k}(\theta) = \theta - {1\over\lambda} X_{\theta,k} .$$
Here $X_{\theta,k}$ means the following: first contract $X$
with $\theta$ to get a section $X_\theta$ of
$S^2{\cal F}\otimes_{{\cal O}_S}S^2{\cal F}^\ast$ and then consider this as a
second order differential operator $X_{\theta,k}$ of ${\cal O}_{\bf P}(k)$ on $X$
over $S$ by the remarks above. Note that the term $\theta$ acts
on ${\cal O}_{\bf P}(k)$ using $\nabla_0$.
\begin{lem} \label{lemchar}
Let $D_{\lambda,k}$ be the heat operator defined above.
\begin{enumerate}
\item
The heat operator $D_{\lambda,k}$ commutes with the action of ${\cal G}$
on ${\cal O}_{{\bf P}}(k)$.
\item The
heat operators $D_{\lambda,k}$
are compatible with $f:{\cal Y}\to {\bf P}$. If $\lambda= k$
then $D_{k,k}$ is strictly compatible with $f:{\cal Y}\to {\bf P}$.
\end{enumerate}
\end{lem}
\begin{proof}
The second statement holds since $X$ is ${\cal G}$ invariant.
We now verify the last statement.
Note that $D_{k,k}$ is strictly
compatible with ${\cal K}\to {\bf P}$ if the operators $D_{k,k}(\theta)$
preserve the subsheaf
${\cal I}_{\cal K}\cdot{\cal O}_{\bf P}(k)$ of ${\cal O}_{\bf P}(k)$. This can be seen
by the description of strict compatiblility in terms of
local coordinates given in Subsection \ref{heat+comp}.
We may check this condition on the affine 4-space
$\underline{\mathop{\rm Spec}}(S^\ast{\cal F})$. Thus let $G\in S^{k-4}({\cal F})$
be a section, let $\theta$ be a local vector field
on $S$ and consider
$$ \nabla_{0,\theta}(GF) - {1\over k}X_\theta(GF)$$
We have to show that this is divisible by $F$.
We may assume that $G$ is horizontal for $\nabla_0$,
as we can find a horizontal basis for $S^{k-4}({\cal F})$
locally. Thus we get
$$ G\nabla_{0,\theta}(F) - {1\over k} X_\theta (GF)=
G\nabla_{0,\theta}(F) - {1\over k} X_\theta (G) F -
{1\over k}(X\cdot F)_\theta(G) - {1\over k} G X_\theta(F).$$
Here we have used a general formula for the application of a
second order operator like $X_\theta$ on a product like $FG$.
Note also that $X(F)$ is simply the contration of $X\cdot F$,
hence by $(Eq.)$ we get
$$ G\nabla_{0,\theta}(F) - {1\over k} X_\theta (G) F -
{1\over k} \nabla_{0,\theta}(F)E(G) -
{1\over k} G E(\nabla_{0,\theta}(F)) .$$
Next, we use that acting by $E$ on a homogeneous polynomial
gives degree times the polynomial:
$$ G\nabla_{0,\theta}(F) - {1\over k} X_\theta (G) F -
{1\over k}\nabla_{0, \theta}(F) (k-4)G -
{1\over k} G 4 \nabla_{0,\theta}(F) = - {1\over k} X_\theta (G) F$$
which is divisible by $F$. Thus $D_{k,k}$ is indeed
stricly compatible with $f : {\cal Y}\to {\bf P}$.
\end{proof}
\begin{thm}\label{269}
Let $X$ be a solution to the equation
$(Eq.)$ above and let $D_{\lambda,k}$ be the heat operators defined in this
subsection. Then
\begin{enumerate}
\item
The projective heat operator defined by $D_{\lambda,k}$
is the operator $\bar D_{\lambda,k}$
defined in Subsection \ref{existence+heat}.
Thus $D_{k+2,k}$ defines the Hitchin connection on ${\cal O}_{{\bf P}}(k)$.
\item
The heat operators
$D_{\lambda,k}$ are projectively flat for any $k$ and $\lambda$.
\item
For $\lambda=k$, the operator $D_{k,k}$ is strictly
compatible with the abelian projective heat operator on
${\cal O}_{{\rm Pic}^0}({\cal O}(2k\Theta^0))$ over an open part of ${\rm Pic}^0$.
\end{enumerate}
\end{thm}
\begin{proof} Consider first the case of $D_{k,k}$ as defined in
\ref{crit}. We have seen above that it is strictly compatible
with $f : {\cal Y} \to {\bf P}$. Therefore, by Subsection \ref{heat+comp}
it defines a heat operator $D_k$ on $f^\ast{\cal O}_{\bf P}(k)$ over ${\cal Y}$.
In view of Hartog's theorem this extends uniquely to a heat operator
$D_k$ of ${\cal O}_{{\rm Pic}^0}({\cal O}(2k\Theta^0))$ on ${\rm Pic}^0$ over $S$. We have
seen in Subsection \ref{heat+abelian} that there is a unique such
heat operator, whose symbol is $1/k$ times the symbol $\rho_{Ab}$
of $D_1$. Combining this with the assertions in Subsection
\ref{comp+abelian} we see that $(1/4k)\rho_{Hitchin}-\rho_{D_{k,k}}$
vanishes along ${\cal K}$ and hence is zero. This proves that $D_{k,k}$
agrees with the projective heat operator $\bar D_{k,k}$
defined in Subsection \ref{existence+heat}.
Now it follows from the transformation behaviour of the symbol
of $\bar D_{\lambda,k}$ and $D_{\lambda,k}$ that these agree
for arbitrary $\lambda$. Thus we get the agreement stated
in the theorem.
To see that these heat operators are projectively flat, we argue
as follows. Let $\theta, \theta'$ be two local commuting vector
fields on $S$. We have to see that the section
$[D_{\lambda,k}(\theta), D_{\lambda,k}(\theta')]$ of
${\rm Diff}^{(3)}_{{\bf P}/S}({\cal O}_{\bf P}(k))$ lies in the subsheaf
$p^{-1}({\cal O}_S)$. Again, for $\lambda=k$ this section
is zero when restricted to ${\cal K}$, and again this implies
that the section is zero in that case. Now, let us compute
$$[D_{\lambda,k}(\theta), D_{\lambda,k}(\theta')]=
[\theta - {1\over\lambda} X_{\theta,k},
\theta' - {1\over\lambda} X_{\theta',k}]
= {1\over \lambda}\Big([\theta, X_{\theta',k}] -
[\theta', X_{\theta,k}]\Big) +
{1\over \lambda^2} [X_{\theta,k}, X_{\theta',k}].$$
The index ${}_k$ is now superfuous, as we can see this as
an expression in $S^2{\cal F}\otimes_{{\cal O}_S} S^2{\cal F}^\ast \oplus
S^3{\cal F}\otimes_{{\cal O}_S} S^3{\cal F}^\ast$. We know that this expression
is zero for any $k$ with $\lambda=k$ considered as a
third order operator on ${\cal O}_{\bf P}(k)$. This implies that
both terms are zero, hence the expression is zero for
any $\lambda$ and any $k$. (We remark for the doubtful
reader that we will verify the flatness also by a direct computation
using the explicit description of the operator.)
\end{proof}
\section{The flat connection}\label{secheat}
\subsection{} We introduce certain families of flat connections on
(trivial) bundles over the configuration space ${\cal P}$ (cf.\ \ref{defP}).
These are defined by representations of the Lie algebra $so(2g+2)$.
Then we derive a convenient form for the equation of the family of Kummer
surfaces ${\cal K}\hookrightarrow{\bf P}$ (in fact for any $g$ we point out a
specific section $P_z$ of a trivial bundle over ${\cal P}$). In Theorem
\ref{thmcon}
we show that this equation is flat for one of our connections. This will
be important in identifying Hitchin's connection in the next section.
\subsection{Orthogonal groups}\label{ortgrp}
\subsubsection{The Lie algebra}
Let $Q=x_1^2+\ldots+x_{2g+2}^2$, then the (complex)
Lie algebra $so(Q)=so(2g+2)$ is:
$$
so(2g+2)=\{A\in End({\bf C}^{2g+2}):\; {}^tA+A=0\;\},\qquad{\rm let}\quad
F_{ij}:=2(E_{ij}-E_{ji})\;(\in\;so(2g+2)),
$$
where $E_{ij}$ is the matrix whose only non-zero
entry is a $1$ in the $(i,j)$-th position (the commutator of such
matrices is
$[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}$).
The alternating matrices $F_{ij}$
(with
$1\leq i<j\leq 2g+2$) are a basis of $so(2g+2)$. Note $F_{ij}=-F_{ji}$.
These matrices satisfy the relations:
$$
[F_{ij},\,F_{kl}]=
\left\{\begin{array}{rcl}
0&{\rm if}&i,\,j,\,k,\,l\quad\mbox{are distinct} ,\\
2F_{il}&{\rm if}& j=k.
\end{array}\right.
$$
\subsubsection{The universal envelopping algebra}
Recall that for a Lie algebra ${{\bf} g}$
the tensor algebra $T({{\bf} g})$ and the universal enveloping algebra
$U({{\bf} g})$
are defined by:
$$
T({{\bf} g}):={\bf C}\oplus{{\bf} g}\oplus {{\bf} g}\otimes{{\bf} g}\ldots,
\qquad
U({{\bf} g}):=T({{\bf} g})/I
$$
where $I$ is the ideal generated by all elements of the form
$x\otimes y-y\otimes x-[x,y]$ with $x,\,y\in {{\bf} g}$.
Lie algebra representations $\rho:{{\bf} g}\rightarrow End(V)$
(so $\rho([x,y])=\rho(x)\rho(y)-\rho(y)\rho(x)$ correspond
to representations $\tilde{\rho}:U({{\bf} g})\rightarrow
End(V)$ of associative algebras with $\rho(1)=id_V$. Given $\rho$ one
defines
$\tilde{\rho}(x_{i_1}\otimes x_{i_2}\ldots\otimes x_{i_k}):=
\rho(x_{i_1})\rho(x_{i_2})\ldots \rho(x_{i_k})$ which is well-defined
because we work modulo $I$.
\subsubsection{Definitions}\label{oneform}
We define elements in $U(so(2g+2))$ by:
$$
\Omega_{ij}:=F_{ij}\otimes F_{ij}\quad{\rm mod}\;I\qquad
(\in U(so(2g+2))),\qquad{\rm note}\quad \Omega_{ij}=\Omega_{ji}.
$$
In particular, for any Lie algebra representation $\rho:
so(2g+2)\rightarrow
W$ we now have endomorphisms
$\tilde{\rho}(\Omega_{ij})=\rho(F_{ij})^2:W\rightarrow W$
which satisfy the same commutation relations as the $\Omega_{ij}$.
For $\lambda\in{\bf C}^*$ we define an $U(so(2g+2))$-valued one form
$\omega_\lambda$
on ${\cal P}$:
$$
\omega_\lambda:=
\lambda^{-1} \left(\sum_{i\neq j}
{{{\Omega_{ij}} {\rm d}z_i}\over{z_i-z_j}}\right);\qquad
{\rm let}\quad \tilde{\rho}(\omega_\lambda):=
\lambda^{-1} \left(\sum_{i\neq j}{{\tilde{\rho}(\Omega_{ij}){\rm d} z_i} \over
{z_i-z_j}}\right)
$$
where $\tilde{\rho}:U(so(2g+2))\rightarrow End(W)$ is a representation.
Note $\tilde{\rho}(\omega_\lambda)\in \mbox{End}(W)\otimes\Omega^1_{\cal P}$.
The one form
$\omega_\lambda$
defines a connection on the trivial bundle $W\otimes_{\bf C}{\cal O}_{\cal P}$ by:
$$
\nabla^W_\lambda:W\otimes{\cal O}_{\cal P}\longrightarrow W\otimes \Omega^1_{\cal P},
\qquad fw\longmapsto w\otimes{\rm d}f-f\tilde{\rho}(\omega_\lambda)(w)
\qquad\qquad(f\in{\cal O}_{\cal P},\;w\in W).
$$
The covariant derivative of $f\otimes w$ with respect to the vector field
$\partial/\partial z_i$ on ${\cal P}$ is the composition of $\nabla^W_\lambda$
with contraction:
$$
\left(\nabla^W_\lambda(f\otimes w)\right)_{\partial/\partial z_i}=
w\otimes {\partial f\over\partial z_i} -\sum_{j,\,j\neq
i}\tilde{\rho}(\Omega_{ij})(w)\otimes{1\over z_i-z_j}\qquad\qquad
(\in W\otimes{\cal O}_{\cal P}).
$$
\begin{prop} \label{defcon}
Let $\rho:so(2g+2)\longrightarrow End(W)$ be a Lie algebra representation,
then $\nabla^W_\lambda$ is a flat connection
on $W\otimes{\cal O}_{\cal P}$ for any $\lambda\in{\bf C}^*$.
\end{prop}
\begin{proof}
Obviously $\nabla^W_\lambda$ is a connection.
It is well known (and easy to verify) that
the connection defined by $\omega$ is flat (so
${\rm d}\omega+\omega\wedge\omega=0$) if the following
{\it infinitesimal
pure braid relations} in $U(so(2g+2))$ are satisfied:
$$
[\Omega_{ij},\Omega_{kl}]=0,\qquad [\Omega_{ik},\Omega_{ij}+\Omega_{jk}]=0,
$$
where $i,\,j,\,k,\,l$ are distinct indices (cf.\ \cite{Ka}, Section XIX.2).
To check these relations we use that $X\otimes Y=Y\otimes X+[X,Y]$ in
$U(so)$.
The first relation is then obviously satisfied. We spell out the second.
Consider first:
$$
\begin{array}{rcl}
F_{ik}\otimes F_{ij}\otimes F_{ij}&=&
(F_{ij}\otimes F_{ik}+[F_{ik},F_{ij}])\otimes F_{ij}\\
&=&F_{ij}\otimes F_{ik}\otimes F_{ij}-F_{kj}\otimes F_{ij}\\
&=&F_{ij}\otimes(F_{ij}\otimes F_{ik}-F_{kj})+F_{jk}\otimes F_{ij}\\
&=&F_{ij}\otimes F_{ij}\otimes F_{ik}+
(F_{ij}\otimes F_{jk}+F_{jk}\otimes F_{ij}),
\end{array}
$$
so we have $[F_{ik},F_{ij}\otimes F_{ij}]=D_1$ with $D_1$
symmetric in the indices $i$ and $k$. Similarly we get:
$$
\begin{array}{rcl}
F_{ik}\otimes( F_{ik}\otimes F_{ij}\otimes F_{ij})&=&
F_{ik}\otimes( F_{ij}\otimes F_{ij}\otimes F_{ik}+D_1)\\
&=&(F_{ik}\otimes F_{ij}\otimes F_{ij})\otimes F_{ik}
+F_{ik}\otimes D_1\\
&=& F_{ij}\otimes F_{ij}\otimes F_{ik}\otimes F_{ik}+(D_1\otimes F_{ik}+
F_{ik}\otimes D_1).
\end{array}
$$
Thus $[\Omega_{ik},\Omega_{ij}]=D_1\otimes F_{ik}+
F_{ik}\otimes D_1$ which is antisymmeric in $i$ and $k$. Therefore
$$
[\Omega_{ik},\Omega_{ij}]=-[\Omega_{ki},\Omega_{kj}]=
-[\Omega_{ik},\Omega_{jk}]
$$
which proves the second infinitesimal braid relation.
\end{proof}
\subsubsection{Differential operators}\label{symcon}
Given a Lie algebra representation $\rho:{{\bf} g}\rightarrow {\rm End(S_1)}$
where $S_k\subset S:={\bf C}[\ldots,X_i,\ldots]$ is the subspace of homogeneous
polynomials of degree $k$, there is a convenient way to determine the
representation $\rho^{(k)}:g\rightarrow {\rm End}(S_k)$ induced by $\rho$.
If $\rho(A)=(a_{ij})$ w.r.t.\ the basis $X_i$ of $S_1$, then we
define
$$
L_A:=\sum_{i,j}a_{ij}X_i\partial_j\qquad(\in {\rm End}(S)),\qquad
{\rm with}\quad \partial_j(P):={\partial P\over\partial X_j}
$$
for $P\in S$. Then obviously $\rho(A)(P)=L_A(P)$ if $P$ is linear and the
Leibnitz rule shows that $\rho^{(k)}(A)=L_A:S_k\rightarrow S_k$ for all
$k$.
The composition (in ${\rm End}(S)$) of two operators is given by:
$$
(X_i\partial_j)\circ (X_k\partial_l)=
X_iX_k\partial_j\partial_l+\delta_{jk}X_i\partial_l,\qquad
{\rm thus}\quad
L_A^2=(1/2)\sigma(L_A^2)+L_{A^2},
$$
with symbol
$\sigma(L_A^2)=2\sum_{i,j,k,l}a_{ij}a_{kl}X_iX_k\partial_j\partial_l$
(see our convention \ref{normalization}).
Assume that we have $A_{ij}\in{{\bf} g}$ such that
$\Omega_{ij}=A_{ij}\otimes A_{ij}$ satisfy the infinitesimal braid
relations (cf.\ proof of the previous proposition) and that in the
representation $\rho$ we have $\rho(A_{ij})^2=\mu I$ with $\mu\in{\bf C}$.
Then we conclude that the operators
$$
\sigma(L_{A_{ij}}^2):S\longrightarrow S
$$
also satisfy the infinitesimal braid relations.
\subsection{The Kummer equation.}\label{kummer}
\subsubsection{Introduction}
We recall how one can obtain the equation of the Kummer
variety for a genus two curve, and more generally, how a hyperelliptic
curve ${\cal C}_z$ determines in a natural way a quartic polynomial
$P_z\in S^4V(\omega_{g+1})$ where
$V(\omega_{g+1})$ is a half spin representation of $so(2g+2)$.
The polynomial $P_z$ lies in the subrepresentation of $V(4\omega_{g+1})$
which also
occurs in $S^2(\wedge^{g+1}{\bf C}^{2g+2})$ where ${\bf C}^{2g+2}$ is the standard
representation of $so(2g+2)$. We will exploit this fact to verify that
$P_z$ is a flat section for one of the connections introduced in
Subsection \ref{ortgrp}.
\subsubsection{The orthogonal Grassmanian}
We define two quadratic forms on ${\bf C}^{2g+2}$ by:
$$
Q:\quad x_1^2+\ldots+x_{2g+2}^2,\qquad
Q_z:\quad z_1x_1^2+\dots+z_{2g+2}x_{2g+2}^2,
$$
and we use the same symbols to denote the corresponding quadrics in
${\bf P}^{2g+1}$. The quadric $Q$ has two rulings (families of linear ${\bf P}^g$'s
lying on it).
Each of these is parametrized by the orthogonal Grassmanian (spinor
variety)
denoted by $Gr_{SO}$.
The variety $Gr_{SO}$ smooth, projective of dimension
$\mbox{${1\over 2}$}g(g+1)$.
Let $V(\omega_{g+1})$ be a half spin representation of $so(2g+2)$.
There is an embedding
$$
\phi:Gr_{SO}\longrightarrow {\bf P} V(\omega_{g+1})^*,\qquad {\rm let}\quad
{\cal O}_{Gr_{SO}}(1):=\phi^*{\cal O}_{{\bf P} V}(1).
$$
in fact, $\phi(Gr_{SO})$ is
the orbit of the highest weight vector.
The map $\phi$ is equivariant for the action of the spin group
$\widetilde{SO}$
(which
acts through a half-spin representation on $V(\omega_{g+1})$).
It induces isomorphisms:
$$
H^0(Gr_{SO},{\cal O}(n))=V(n\omega_{g+1}).
$$
In case $g=2$, one has isomorphisms:
$$
\phi:Gr_{SO}\stackrel{\cong}{\longrightarrow} {\bf P}^3,\qquad
S_k=V(k\omega_{g+1})\qquad
{\rm and}\quad \widetilde{SO}_6\cong SL_4({\bf C})
$$
and the halfspin representation is identified
with the standard representation $SL_4({\bf C})$ on ${\bf C}^4$ (or its dual).
\subsubsection{The Pl\"ucker map}
Let $Gr_{SL}$ be the Grassmanian of $g$-dimensional subspaces of
${\bf P}^{2g+1}$, we denote the Pl\"ucker map by:
$$
p:Gr_{SL}\longrightarrow {\bf P} \wedge^{g+1}{\bf C}^{2g+2},\qquad
\langle v_1,\ldots,v_{g+1}\rangle \longmapsto v_1\wedge\ldots\wedge
v_{g+1}=\sum p_{i_1\ldots i_{g+1}}e_{i_1}\wedge \ldots \wedge e_{i_{g+1}}.
$$
The map $p$ is equivariant for the $SL(2g+2)$-action on both sides
and induces isomorphisms of $sl(2g+2)$-representations
$$
H^0(Gr_{SL},{\cal O}_{Gr_{SL}}(n))=V(n\lambda_{g+1}),\qquad
{\rm with}\quad
V(\lambda_{g+1})\cong \wedge^{g+1}{\bf C}^{2g+2}.
$$
As a representation of $so(2g+2)$ the space $\wedge^{g+1}{\bf C}^{2g+2}$ is
reducible, one has (cf.\ \cite{vG2} (3.7), \cite{FH}, $\S$19.2, Th.\ 19.2
and Remarks (ii)), with $V(\omega_g$ and $V(\omega_{g+1})$ the half spin
representations:
$$
\wedge^{g+1}{\bf C}^{2g+2}=V(2\omega_{g})\oplus V(2\omega_{g+1}).
$$
In particular, the irreducible component $V(4\omega_{g+1})$ of
$S^4V(\omega_{g+1})$ is also a component of $S^2(\wedge^{g+1}{\bf C}^{2g+2})$
(viewed as $so(2g+2)$ representation). Elements of $
S^2(\wedge^{g+1}{\bf C}^{2g+2})$ can be seen as quadratic forms in the
Pl\"ucker coordinates, when restricted to $Gr_{SO}$ they can be viewed as
(restrictions to $Gr_{SO}$ of) quartic polynomials on ${\bf P}
V(\omega_{g+1})^*$.
\subsubsection{Notation}\label{bsnot}
The following lemma gives this decomposition of $\wedge^{g+1}{\bf C}^{2g+2}$
explicitly. Let $\{e_i\}$ be the standard basis of ${\bf C}^{2g+2}$. For any
$$
S\subset B:=\{1,\,2,\ldots,2g+2\},\qquad |S|=g+1,
$$
we write $S=\{i_1,\,i_2,\ldots,i_{g+1}\}$ with $i_1<i_2<\ldots<i_{g+1}$
and define
$$
e_S:=e_{i_1}\wedge\ldots\wedge e_{i_{g+1}}\qquad\in \wedge^{g+1}{\bf C}^{2g+2}.
$$
These elements give a basis of $\wedge^{g+1}{\bf C}^{2g+2}$.
For such an $S$ we let $S':=B-S$ be the complement of $S$ in $B$.
Writing $S'=\{j_1,\,j_2,\ldots,j_{g+1}\}$ with $j_1<j_2<\ldots<j_{g+1}$,
we define an element $\sigma_S$ in the symmetric group $S_{2g+2}$ by:
$$
\sigma_S(k):=i_k,\quad \sigma_S(g+1+k):=j_k.
$$
\begin{lem}\label{resp}
The two non-trivial $so(2g+2)$-invariant subspaces in
$\wedge^{g+1}{\bf C}^{2g+2}$
are:
$$
\langle\ldots, e_S+sgn(\sigma_S)i^{g+1}e_S',\ldots\rangle_{S\ni 1}\qquad
{\rm and}\qquad
\langle\ldots, e_S-sgn(\sigma_S)i^{g+1}e_S',\ldots\rangle_{S\ni 1},
$$
where $S$ runs over the subsets of $B$ with $g+1$ elements with $1\in S$.
\end{lem}
\begin{proof} (Cf.\ \cite{FH}, $\S$19.2 Remarks(iii).)
The quadratic form $Q$ defines an $so$-equivariant isomorphism
$$
B:\wedge^{g+1} {\bf C}^{2g+2}\longrightarrow (\wedge^{g+1} {\bf C}^{2g+2})^*=
\wedge^{g+1} ({\bf C}^{2g+2})^*,\qquad
e_S\longmapsto
\epsilon_S:=\epsilon_{i_1}\wedge\ldots\wedge \epsilon_{i_{g+1}}
$$
with $\{\ldots,\epsilon_j,\ldots\}$ the basis dual to the $e_i$.
There is also a canonical (in particular $so$-equivariant) isomorphism:
$$
C:\wedge^{g+1} {\bf C}^{2g+2}\longrightarrow (\wedge^{g+1} {\bf C}^{2g+2})^*,\qquad
\alpha\longmapsto [\beta\mapsto c_{\alpha,\beta}]
$$
with
$ c_{\alpha,\beta}\in{\bf C}$ defined by
$$
\alpha\wedge\beta=c_{\alpha,\beta}e_1\wedge e_2\wedge\ldots e_{2g+2}.
$$
Thus we have an isomorphism
$A:=B^{-1}\circ C:\wedge^{g+1} {\bf C}^{2g+2}\rightarrow\wedge^{g+1}
{\bf C}^{2g+2}$ whose
eigenspaces are $so$-invariant. These are the subspaces we have to
determine.
Note $e_S\wedge e_T\neq 0$ iff $T=S'$, the
complement of $S$. One easily verifies
that if $1\in S$ we have:
$$
e_S\wedge e_{S'}=
sgn(\sigma_S)e_1\wedge e_2\wedge\ldots e_{2g+2},\qquad
{\rm so}\quad C(e_S)=sgn(\sigma_S)\epsilon_{S'}.
$$
As $e_{S'}\wedge e_S=(-1)^{g+1}e_S\wedge e_{S'}$, we get
$C(e_{S'})=sgn(\sigma_S)(-1)^{g+1}\epsilon_S$ (with $1\in S$).
Therefore
$$
A(e_S)=sgn(\sigma_S)e_{S'},\qquad
A(e_{S'})=sgn(\sigma_S)(-1)^{g+1}e_S\qquad
{\rm if}\quad 1\in S.
$$
The eigenvalues of $A$ are thus $\pm i^{g+1}$
and the eigenvectors are $e_S\pm sgn(\sigma_S)i^{g+1}e_S'$.
\end{proof}
\subsubsection{A quartic polynomial}\label{defpz}
We consider the $g$-dimensional subspaces of ${\bf P}^{2g+1}$ which are tangent
to $Q_z$:
$$
\bar{B}_z:=\left\{{\bf P}^g\subset{\bf P}^{2g+1}:\;rank(Q_z\; \mbox{restricted
to}\;{\bf P}^{g})\leq g\;\right\}\qquad(\subset Gr_{SL}).
$$
The subvariety $\bar{B}_z$ is defined by a quadratic polynomial $P_z$
in the
Pl\"ucker coordinates, that is by a section in
$H^0(Gr_{SL},{\cal O}_{Gr_{SL}}(2))$
(\cite{vG2}, Thm 3):
$$
\bar{B}_z=Z(P_z),\qquad P_z\in
H^0(Gr_{SL},{\cal O}_{Gr_{SL}}(2))\subset S^2(\wedge^{g+1}{\bf C}^{2g+2}).
$$
This section is in fact the unique $SO_z$-invariant
in $H^0(Gr_{SL},{\cal O}_{Gr_{SL}}(2))$
and will be determined explicitly in the lemma below.
In case $g=2$ the quartic surface in ${\bf P} V$ defined by $P_z$
is the Kummer surface of the curve ${\cal C}_z$ (\cite{DR}).
\begin{lem}
The variety $\bar{B}_z$ is defined by
$$
P_z=\sum_S z_Se_S^2,
$$
where $S$ runs over the subsets with $g+1$ elements of
$B=\{1,\ldots,2g+2\}$
and
$$
z_S:=z_{i_1}z_{i_2}\ldots z_{i_{g+1}},\qquad
e_S=e_{i_1}\wedge\ldots\wedge e_{i_{g+1}}\qquad{\rm if}\quad
S=\{i_1,\ldots,i_{g+1}\}.
$$
\end{lem}
\begin{proof}
{}From \cite{vG2}, Theorem $3'$ and its proof we know that the
trivial representation of $so_z$ has multiplicity one in
$H^0(Gr_{SL},{\cal O}_{Gr_{SL}}(2))=V(2\lambda_{g+1})$.
This $sl_{2g+2}$ representation corresponds to the partition $\mu$:
$2g+2=2+2+\ldots+2$ and is thus realized as
$({\bf C}^{2g+2})^{\otimes 2g+2}c_\mu$ where $c_\mu$ is the Young symmetrizer,
one has in fact $V(2\lambda_{g+1})\subset S^2(\wedge^{g+1}{\bf C}^{2g+2})$
(see \cite{FH}, $\S$15.5, p.\ 233-236). We use the standard tableau as
in \cite{vG2} (and \cite{FH}) to construct $c_\mu$.
The trivial $so_z$ sub-representation is obtained as follows.
The quadratic form $Q_z$ defines the $so_z$-invariant tensor
$\sum_{i=1}^{2g+2}
z_ie_i\otimes e_i\in {\bf C}^{2g+2}\otimes {\bf C}^{2g+2}$, and we consider the
$g+1$-fold tensor product of this tensor, where we insert the $k$-th
factor
in the positions $k$ and $g+1+k$:
$$
\tau_z:=\sum_{i_1,\ldots,i_{g+1}=1}^{2g+2}
z_{i_1}z_{i_2}\ldots z_{i_{g+1}}e_{i_1}\otimes e_{i_2}\otimes\ldots
\otimes e_{i_{g+1}}\otimes e_{i_1}\otimes\ldots e_{i_{g+1}}\quad
(\in ({\bf C}^{2g+2})^{\otimes 2g+2}).
$$
Thus $\tau_z$ is $so_z$ invariant and it has the advantage that
$\tau_zc_\mu$ is easily determined:
$$
\tau_zc_\mu=(cst)\sum_S z_S\,e_S\cdot e_S,
$$
with $(cst)$ a non-zero integer and
where we sum over all subsets $S$ with $g+1$ elements of
$\{1,\ldots,2g+2\}$
(In fact, $c_\mu=a_\mu b_\mu$
and $a_\mu$ symmetrizes the indices $1,\,g+2$; $2,\,g+3$;
$\ldots$; $g+1,\,2g+2$ of a tensor (but $\tau_z$ is already symmetric in
these indices) and next $b_\mu$ antisymmetrizes the first $g+1$ and last
$g+1$ indices, giving the result above).
\end{proof}
\subsection{Verifying the contraction criterium}
\subsubsection{}\label{actf}
We need to determine the images of the $\Omega_{ij}$'s in the
representation
$S^2(\wedge^{g+1}{\bf C}^{2g+2})$.
In the standard representation ${\bf C}^{2g+2}$ of $so(2g+2)$,
with basis $e_i$ and dual basis $\epsilon_i$, we have:
$$
F_{ij}=2(e_i\otimes\epsilon_j-e_j\otimes\epsilon_i)\qquad
\in
({\bf C}^{2g+2})\otimes({\bf C}^{2g+2})^*=End({\bf C}^{2g+2}).
$$
On $\wedge^{g+1}{\bf C}^{2g+2}$, with basis
$e_S=e_{i_1}\wedge\ldots\wedge e_{i_{g+1}}$ as before, $F_{ij}$ acts as
$$
F_{ij}(e_{i_1}\wedge\ldots\wedge e_{i_{g+1}})=
F_{ij}(e_{i_1})\wedge e_{i_2}\wedge\ldots\wedge e_{i_{g+1}}+\ldots+
e_{i_1}\wedge\ldots\wedge e_{i_g}\wedge F_{ij}(e_{i_{g+1}}).
$$
Define a subset of $B=\{1,\ldots,2g+2\}$ by
$$
S+ij=(S\cup\{i,j\})-(S\cap\{i,j\}).
$$
Then
$F_{ij}(e_S)$ is, up to sign, $2e_{S+{ij}}$ if $|S\cap\{i,j\}|=1$
and is zero otherwise:
$$
F_{ij}=2\sum_{S,\;|S\cap\{i,j\}|=1} t_{S,ij}e_{S+ij}\otimes\epsilon_S
$$
with $t_{S,ij}= \pm 1$.
Note that if $|S\cap\{i,j\}|=1$ then $F_{ij}^2e_S=-4e_S$ so
$t_{S,ij}=-t_{S+ij,ij}$.
\begin{lem}\label{cont2}
Let $W$ be a complex vector space, let $D\in W\otimes W^*$ be a linear
differential operator and let $P\in S^kW$. Then the contraction of
$P$ with the symbol of $D^2$ is given by:
$$
\sigma(D^2)\cdot P=2D(P)D\qquad \in S^{k-1}W\otimes W^*.
$$
\end{lem}
\begin{proof}
We view elements of $W$ as linear forms in variables $X_a$. Then elements
of $W^*$ are linear operators with constant coefficients and
$D=\sum_af_a\partial_a$ with $f_a\in W$. The symbol of $D^2$ is then:
$$
\sigma(D^2)=\sum_{a,b} f_af_b\partial_a\partial_b\qquad\in S^2W\otimes
S^2W^*.
$$
Recall that we use the convention (see \ref{normalization}):
$$
\partial_a\partial_b:=
\partial_a\otimes\partial_b+\partial_b\otimes\partial_a
$$
for elements in $S^2W^*$. The contraction of a polynomial $P\in S^kW$
with $\partial_a\partial_b\in S^2W^*$ is thus:
$$
(\partial_a\partial_b)\cdot P=
(\partial_bP)\partial_a+(\partial_aP)\partial_b\qquad \in S^{k-1}W\otimes
W^*.
$$
Therefore
$$
\begin{array}{rcl} \sigma(D^2)\cdot P&=&
\sum_{a,b}f_af_b\left((\partial_bP)\partial_a+(\partial_aP)\partial_b\right)\\
&=&2\sum_af_a(\sum_bf_b(\partial_bP))\partial_a\\
&=&2D(P)D.
\end{array}
$$
\end{proof}
\begin{thm}\label{thmcon}
The composition, still denoted by $P_z$:
$$
{\cal P}\longrightarrow S^2\wedge^{g+1}{\bf C}^{2g+2}
\longrightarrow S^4V(\omega_{g+1}),\qquad
z\longmapsto P_z,
$$
with $P_z=\sum_Sz_Se_S^2$ (as in Subsection \ref{defpz}), satisfies the
differential equations:
$$
\left(\frac{1}{16}
\sum_{j\neq i}{\sigma\tilde{\rho}(\Omega_{ij})\over{z_i-z_j}}\right)
\cdot
P_z+(\partial_{z_i}P_z)E=0\qquad\qquad(1\leq i\leq 2g+2)
$$
with $E=\sum e_S\otimes\epsilon_S$ the Euler vector field and
$\tilde{\rho}:U(so(2g+2))\rightarrow {\rm End}(S^2(\wedge^{g+1}{\bf C}^{2g+2}))$.
\noindent
Thus $P_z$ is a horizontal section for the connection $\nabla^W_\lambda$
with $W=S^4V(\omega_{g+1})$ and $\lambda=-16$.
\end{thm}
\begin{proof}
For symmetry reasons it suffices to verify the equation for $i=1$.
By Lemma \ref{cont2}
$$
\sigma\tilde{\rho}(\Omega_{1j})\cdot e_S^2=
2\rho(F_{1j})(e_S^2)\rho(F_{1j})=4e_S\rho(F_{1j})(e_S)\rho(F_{1j}).
$$
To simplify notation we write $\Omega_{ij}$ for $\sigma\tilde{\rho}(\Omega_{ij})$
and $F_{ij}$ for $\rho(F_{ij})$ (as in \ref{actf}).
Then $F_{1j}(e_S)=0$ unless $|S\cap\{1,j\}|=1$.
Assume $1\in S$ and $j\not \in S$ (so $j\in B-S$) and consider
the contraction of $\Omega_{1j}$ with the term
$z_Se_S^2+z_{S+ij}e_{S+ij}^2$ from $P_z$:
$$
\begin{array}{rcl}
\Omega_{1j}\cdot (z_Se_S^2+z_{S+1j}e_{S+1j}^2)&=&
8(t_{S,1j}z_S+t_{S+1j,1j}z_{S+1j})e_Se_{S+1j}F_{1j}\\
&=&8t_{S,1j}(z_1z_{\bar S}-z_jz_{\bar{S}})e_Se_{S+1j}F_{1j}\\
&=&(z_1-z_j)\Omega_{1j}\cdot z_{\bar{S}} e_S^2,
\end{array}
$$
where ${\bar S}:=S-\{1\}$. Therefore we get:
$$
{\Omega_{1j}\over{z_1-z_j}}\cdot P_z=
\Omega_{1j}\cdot\left(\sum_{S\ni 1,\,j\not\in S} z_{\bar{S}} e_S^2\right).
$$
Summing this result over all $j$ and changing the order of summation gives:
$$
\begin{array}{rcl}
\left(\sum_{j\neq 1}{\Omega_{1j}\over{z_1-z_j}}\right)\cdot P_z&=&
\sum_{S\ni 1} z_{\bar{S}}\left(\sum_{j\in B-S} \Omega_{1j}\cdot
e_S^2\right)\\
&=&
8\sum_{S\ni 1} z_{\bar{S}}e_S\left(\sum_{j\in B-S} t_{S,1j}
e_{S+1j}F_{1j}\right)
\end{array}
$$
where, as before, $B=\{1,\ldots,2g+2\}$.
On the other hand, since $P_z=\sum z_Se_S^2$, we have:
$$
\partial_{z_1}P_z=\sum_{S\ni 1}z_{\bar{S}}e_S^2.
$$
Thus the theorem follows if we prove, for all $S$ with $1\in S$:
$$
2e_SE+\sum_{j\in B-S} t_{S,1j} e_{S+{1j}}F_{1j}=0.
$$
With the definition of $F_{1j}$ we find:
$$
\begin{array}{rcl}
\sum_{j\in B-S} t_{S,1j} e_{S+1j}F_{1j}&=&
\sum_{j\in B-S} t_{S,1j} e_{S+1j}\left(\sum_{T,\;|T\cap\{1,j\}|=1}
2t_{T,1j}e_{T+1j}\otimes\epsilon_T\right)\\
&=&
2\sum_{T}\left(\sum_{j\in B-S,\;|T\cap\{1,j\}|=1} t_{S,1j}t_{T,1j}
e_{S+1j}e_{T+1j}\right)\otimes \epsilon_T.
\end{array}
$$
Note that $e_SE=\sum_T e_Se_T\otimes \epsilon_T$ (with $T\in B,\;|T|=g+1)$.
Comparing coefficients of $\epsilon_T$,
it remains to prove that (for all $S\ni 1$ and all $T$):
$$
e_Se_T+\sum_{j\in B-S,\;|T\cap\{1,j\}|=1} t_{S,1j}t_{T,1j}
e_{S+1j}e_{T+1j}=0.
$$
We show that the relations for $1\in T$ follow from those with $1\not\in
T$.
In fact, we only want this relation in $V(4\omega_{g+1})$,
so we restrict ourselves to the subspace $V(2\omega_{g+1})\subset
\wedge^{g+1}{\bf C}^{2g+2}$. The kernel of the restriction map is
$\langle\ldots,e_T-sgn(e_T)i^{g+1}e_{T'},\ldots\rangle$ (or the other space
in Lemma \ref{resp}; the argument we give leads to the same conlusion in
both cases).
Consider a $T$ with $1\in T$ and $j\not\in T$. Then $1\not\in T+{1j}$ and
after restriction:
$$
e_T=sgn(\sigma_T)i^{g+1} e_{T'},\qquad
e_{T+1j}=sgn(\sigma_{(T+1j)'})i^{-(g+1)} e_{(T+1j)'}
$$
where $'$ stands for the complement in $B$. Since $(T+1j)'=T'+1j$,
substituting these relations and multiplying throughout by $i^{g+1}$ we
get:
$$
(-1)^{g+1}sgn(\sigma_T)e_Se_{T'}+
\sum_{j\in B-S,\;|T\cap\{1,j\}|=1} t_{S,1j}t_{T,1j}sgn(\sigma_{T'+1j})
e_{S+1j}e_{T'+1j}
$$
Next we observe that
$$
sgn(\sigma_{T'+1j})=(-1)^{g+1}t_{T,1j}t_{T',1j}sgn(\sigma_T),
$$
so it suffices to consider the relations with $1\not\in T$.
Let $T:=\{1=i_1,\ldots,i_{g+1}\},\;T':=\{j_1,\ldots,j_{g+1}\}$ with $j=j_k$
and $i_l<j<j_{l+1}$.
To get $\sigma_{T'+1j}$,
first apply the permutation $(1\;g+2)\ldots(g+1\;2g+2)$ to $B$, the sign
of this permutation is $(-1)^{g+1}$. Then apply $\sigma_T$. Next apply the
cyclic
permutation $(j_1\;\ldots j_{g+1}\;1)$ (with sign $(-1)^{g+1}$) and
finally apply the inverse of $(j_k\;\ldots j_{g+1}\;i_2\;\ldots i_l)$
(with sign
$(-1)^{g-k+l}$). The resulting permutation is $\sigma_{T'+1j}$ and has sign
$(-1)^{g+k+l}sgn(\sigma_T)$. On the other hand,
since $1\in T,\;j\not\in T$ we get
$F_{1j}(e_T)=-2(-1)^{l-1}e_{T+{1j}}$, so $t_{T,1j}=(-1)^{l}$ and
$F_{1j}(e_{T'})=+2(-1)^{k-1}e_{T'+1j}$, so $t_{T',1j}=(-1)^{k-1}$. This
gives
the formula for $sgn(\sigma_{T'+1j})$.
>{}From now on we consider only $T$'s with $1\not\in T$ and $S$ with $1\in
S$.
We show that the desired relation:
$$
e_Se_T+\sum_{i\in (B-S)\cap T} t_{S,1i}t_{T,1i} e_{S+1i}e_{T+1i}.
$$
is a Pl\"ucker relation. Thus it holds in
$H^0(Gr_{SL},{\cal O}(2))$ and therefore also upon restriction to
$Gr_{SO}\subset Gr_{SL}$.
Let $S=\{1=i_1,\ldots,i_{g+1}\}$ and
consider
the Zariski open subset $U\subset Gr_{SL}$ of $g+1$-dimensional
subspaces $W\subset{\bf C}^{2g+2}$ with
Pl\"ucker coordinate $p_{S}\neq 0$. Any such $W$
has a (unique) basis $\{\ldots,w_i,\ldots\}$ with:
$$
W=\langle w_1,\ldots,w_{g+1}\rangle,\qquad
{\rm and}\quad
(w_k)_{i_l}=\delta_{kl},\qquad
(S=\{1=i_1,\ldots,i_{g+1}\},\;i_1<\ldots<i_{g+1})
$$
where $\delta_{kl}$ is Kronecker's delta.
Let $M$ be the $(g+1)\times(2g+2)$ matrix whose rows are the $w_i$.
Let $M_i$ be the $i$-th column of $M_W$ (note $M_{i_j}=f_j$,
the $j$-th standard basis vector of ${\bf C}^{g+1}$).
Then the Pl\"ucker coordinate $p_{k_1,\ldots,k_{g+1}}$ is the determinant
of the $(g+1)\times (g+1)$ submatrix of $M$ whose $j$-th column is
$M_{k_j}$,
we write:
$$
p_{k_1,\ldots,k_{g+1}}=det(M_{k_1},\ldots,M_{k_{g+1}})\qquad
{\rm with}\quad k_1<\ldots<k_{g+1}.
$$
Let $j\not\in S$, with $i_l<j<i_{l+1}$. Then $p_{S+1j}=(-1)^{l-1}(M_j)_1$
since
$$
det(f_{i_2},\!\ldots,f_{i_l},M_j,f_{i_{l+1}},\!\ldots,f_{g+1})
\!=(-1)^{l-1}det(M_j,f_{i_2},\!\ldots,f_{i_l},f_{i_{l+1}},\!\ldots,f_{g+1})
=\!(-1)^{l-1}(M_j)_1.
$$
As earlier, since $1\in S$ we have $t_{S,1j}=-(-1)^{l-1}$, hence
$p_{S+1j}=-t_{S,1j}(M_j)_1$.
Let $T=\{j_1,\ldots,j_{g+1}\}$, let $T\cap S'=\{a_1,\ldots,a_q\}$
and let $j=j_n=a_k$.
Then $p_T=det(M_{j_1},\ldots,M_{j_{g+1}})$ is, up to sign,
the determinant of a $q\times q$
submatrix $N$ of the $(g+1)\times q$ matrix with columns
$M_{a_1},\ldots,M_{a_q}$, we write $p_T=t\cdot det(N)$ with $t=\pm 1$.
Then
$$
\begin{array}{rcl}
p_{T+1j}&=&det(f_1,M_{j_1},\ldots,\widehat{M_{j_n}},\ldots,M_{j_{g+1}})\\
&=&
(-1)^{j_n-1}det(M_{j_1},\ldots,M_{j_{n-1}},f_1,M_{j_{n+1}},\ldots,M_{j_{g+1}})\\
&=&(-1)^{j_n-1}(-1)^{k+1}t\cdot det(N^{1k}),
\end{array}
$$
where $N^{1k}$ is the $(q-1)\times(q-1)$ submatrix of $N$ obtained by
deleting the first row and $k$-th column.
Since $1\not\in T$, we have $t_{T,1j}=(-1)^{j_n-1}$ and thus
$p_{T+1j}=t(-1)^{k+1}det(N^{1k})$.
Substituting these expressions for $p_{S+1j}$ and $p_{T+1j}$ we get:
$$
1\cdot t\cdot det(N)+\sum_{j\in \{a_1,\ldots,a_q\}}
t(-1)^k(M_j)_1det(N^{1k})
$$
which is zero in virue of a well known formula for the determinant.
\end{proof}
\section{The Heisenberg group and the Spin representation} \label{Hspin}
\subsection{}
We recall the basic facts on the Heisenberg group and we discuss the
projective representation of its automorphism group.
In Subsection \ref{spin}
we relate the Heisenberg group (in its
irreducible $2^g$-dimensional representation) with the (half) spin
representation of the orthogonal group.
Combining this with previous results, we can finally write down
Hitchin's connection in Subsection \ref{hitcon}.
\subsection{The Heisenberg group}
\label{Hgroup}
\subsubsection{Definitions}\label{Hdefs}
We introduce a variant of the Heisenberg group (cf.\ Subsection
\ref{autos}).
For any positive integer $g$ we define
a (finite) Heisenberg group $G$ by:
$$
G=G_g:=\{(t,x):\;t\in{\bf C},\;t^4=1,\quad
x=(\xi,\xi')\in {\bf F}_2^g\times {\bf F}_2^g\}
$$
with identity element $(1,0)$ and multiplication law:
$$
(t,(\xi,\xi'))(s,(\eta,\eta')):=(ts(-1)^{\xi\eta'},\xi+\eta,\xi'+\eta')
$$
with $\xi\eta':=\sum_{i=1}^g\xi_i\eta'_i$. The group $G$ has $2^{2g+2}$
elements, it is non-abelian, in fact its center is $\{(t,0)\}$.
The inverse of $(t,x)$ can be found as follows:
$$
(t, (\xi,\xi'))(s, (\xi,\xi'))=((-1)^{\xi\xi'}ts,(0,0)),\qquad
{\rm thus}\quad (t, x)^{-1}=((-1)^{\xi\xi'}t^{-1},x),
$$
So for any non-zero $x$ there is are elements $(t,x)\in G$
of order 2 (and also of order 4).
We define a symplectic form on ${\bf F}_2^{2g}$ by:
$$
E(x,y)=\xi\eta'+\eta\xi',\qquad{\rm with}\quad
x=(\xi,\xi'),\;y=(\eta,\eta')
$$
thus $E(x,x)=0$ for all $x$ but $E$ is non-degenerate.
Note that $E$ is related to the commutator in $G$:
$$
(t,x)(s,y)(t,x)^{-1}(s,y)^{-1}=((-1)^{E(x,y)},0).
$$
\subsubsection{Representations.}\label{repH}
Let $V$ be the $2^g$ dimensional vector space of complex valued functions
${\bf F}^g_2\rightarrow {\bf C}$. It has a standard basis consisting of
$\delta$-functions
$$
X_\sigma:{\bf F}^g_2\longrightarrow {\bf C},\qquad X_\sigma(\sigma)=1,\quad
X_\sigma(\rho)=0\quad{\rm if}\;\sigma\neq\rho.
$$
The Heisenberg group $G$ has a representation $U$ on
$V$,
the Schr\"odinger
representation:
$$
(U(t,(\xi,\xi'))f)(\sigma):=t(-1)^{\sigma\xi'}f(\sigma+\xi),\qquad
{\rm thus}\quad
U(t,(\xi,\xi'))X_\sigma=t(-1)^{(\sigma+\xi)\xi'}X_{\sigma+\xi}.
$$
For every $x\in{\bf F}_2^{2g}-\{0\}$ choose a $(t_x,x)\in G$ of order
two. We define:
$$
U_x:=U(t_x,x)\qquad (\in GL(V)),\qquad{\rm so}\quad U_x^2=I
$$
and define $U_0=I$. Then $Im(U)=\{tU_x:\;t^4=1,\;x\in{\bf F}^{2g}_2\}$.
\subsubsection{Automorphisms.}
We define a subgroup of $Aut(G)$ by:
$$
A(G):=\left\{\phi\in
Aut(G):\;\phi((t,0))=(t,0)\quad\forall t\right\}.
$$
The elements of $A(G)$ are the automorphisms which are the
identity when restricted to the center of $G$. For $\phi\in
A(G)$ and $(t,x)\in G$ we can then write:
$$
\phi(t,x):=(f_\phi(x)t,M_\phi(x)),\qquad{\rm with} \quad
M_\phi:{\bf F}_2^{2g}\longrightarrow {\bf F}_2^{2g},\quad
f_\phi:{\bf F}_2^{2g}\longrightarrow {\bf C}^*
$$
(note $\phi(t,x)=\phi(t,0)\phi(1,x)=(t,0)\phi(1,x)$).
The map $M:A(G)\rightarrow Aut({\bf F}_2^{2g}),\;\phi\mapsto M_\phi$
is a homomorphism.
Assume that $\phi\in\ker(M)$, thus $M_\phi$ is
the identity.
Then one verifies that $f_\phi$ is a homomorphism so
we must have $f_\phi(x)=(-1)^{E(x,y)}$ for some $y=y_\phi\in{\bf F}_2^{2g}$.
But then $\phi$ is an interior automorphism since also
$$
Int_y:(t,x)\longmapsto (1,y)(t,x)(1,y)^{-1}=((-1)^{E(x,y)}t,x).
$$
Thus $\ker(M)\cong G/Center(G)\cong{\bf F}^{2g}_2$.
Since automorphisms preserve commutators in $G$,
the image of $M$ lies in
$Sp(2g,{\bf F}_2)=Sp({\bf F}_2^{2g},E)$. There is an exact sequence:
$$
0\longrightarrow {\bf F}^{2g}_2\longrightarrow A(G)
\stackrel{M}{\longrightarrow}Sp(2g,{\bf F}_2)\longrightarrow 0.
$$
(See Theorem \ref{twine} below for the surjectivity of $M$.)
\subsubsection{A projective representation.}\label{projrep}
The Schr\"odinger representation $U$
of $G$ on $V$ is the unique
irreducible representation of $G$ in which $(t,0)$ acts by
mutiplication by $t$. Given $\phi\in A(G)$, the representation
$U\circ\phi$ enjoys the same property. Hence by Schur's lemma we get
a linear map, unique up to scalar multiple,
$$
\tilde{T}_\phi:V\longrightarrow V,\qquad{\rm with}\quad
\tilde{T}_\phi U(h)=U(\phi(h))\tilde{T}_\phi,
$$
for all $h\in G$.
In this way we get a projective representation of $A(G)$:
$$
\tilde{T}:A(G)\longrightarrow PGL(V),\qquad
\phi\longmapsto \tilde{T}_\phi.
$$
Note that we may take $\tilde{T}_\phi=U_y$ when $\phi=Int_y$.
On the other hand, as $U:G\rightarrow GL(V)$ is injective
we have $Im(U)\cong G$. Any $T\in GL(V)$ which normalizes $Im(U)$
thus
defines an automorphism $\phi_T$ of $G$ (if $TU(h)T^{-1}=U(h')$
then $\phi_T(h):=h'$). Since the center of $G$ acts by scalar
multiples of the
identity, we have $\phi_T\in A(G)$. Thus we get an exact sequence
$$
0\longrightarrow {\bf C}^*\longrightarrow Normalizer_{GL(V)}(Im(U))
\longrightarrow A(G)\longrightarrow 0.
$$
\subsubsection{Definition}
For $x=(\xi,\xi')\in {\bf F}_2^{2g}-\{0\}$, we define the transvection
$$
T_x:{\bf F}_2^{2g}\longrightarrow {\bf F}_2^{2g},\qquad
y\longmapsto y+E(y,x)x.
$$
Then $T_x\in Sp(2g,{\bf F}_2)$ and
the (finite) symplectic group $Sp(2g,{\bf F}_2)$ is generated
by transvections. Note that the transvections are involutions: $T^2_x=1$.
\begin{thm}\label{twine}
For $x\in {\bf F}^2_{2g}-\{0\}$ let
$$
\tilde{T}_x:=U_x+iI,\qquad{\rm with}\quad i^2=-1.
$$
Then $\tilde{T}_x\in A(G)$ and
$$
M(\tilde{T}_x)=T_x\qquad\mbox{ that is:}\qquad
\tilde{T}_xU_y\tilde{T}_x^{-1}=t_{x,y}U_{T_x(y)},
$$
for all $y\in{\bf F}_2^{2g}$ and some $t_{x,y}\in {\bf C}$ with $t^4_{x,y}=1$.
Therefore the homomorphism $M:A(G)\rightarrow Sp(2g,{\bf F}_2)$ is
surjective.
Moreover, in $A(G)$ we have
$\tilde{T}_x^2=Int_x$.
\end{thm}
\begin{proof}
Since $U^2_x=I$, the eigenvalues of $U_x$ are $\pm 1$. Thus $U_x+iI$ is
invertible (its inverse is $(1/2)(U_x-iI)$). In case $E(x,y)=0$, $U_x$ and
$U_y$ commute and $T_x(y)=y$, thus the relation holds. In case $E(x,y)=1$,
$U_xU_y=-U_yU_x$ and $T_x(y)=T_{x+y}$, and the relation holds because:
$$
\begin{array}{rcl}
\tilde{T}_xU_y\tilde{T}_x^{-1}&=&
(1/2)\left(U_xU_yU_x-i(U_xU_y-U_yU_x)+U_y\right)\\
&=&
(1/2)(-U_y-2iU_xU_y+U_y)\\
&=& \pm U_{x+y}\qquad(\in Im(U)),
\end{array}
$$
in fact, $U_{x+y}$ and $-iU_xU_y\in Im(U)$ differ by a scalar multiple $t$
and since both elements have order two, $t=\pm1$.
Thus $\tilde{T}_x$ normalizes $Im(U)$
and defines an element of $A(G)$ indicated by the same symbol.
The relation shows that $M(\tilde{T}_x)=T_x$.
Since the $T_x$ generate $Sp(2g,{\bf F}_2)$ the map
$M$ is surjective.
Finally $(U_x+iI)^2=U_x^2+2iU_x-I=2iU_x$, and conjugation by $U_x$ induces
$Int_x$, which proves the last statement.
\end{proof}
\subsubsection{Example}\label{extt}
A particular case
is when $x=(0,\xi')$ with $\xi'=(1,0,\ldots,0)\in{\bf F}_2^g$. Then on the
basis
$X_{(0,\ldots,0)},\ldots,X_{(0,\tau)},\ldots,X_{(1,0,\ldots,0)},\ldots,
X_{(1,\tau)},\ldots$ with $\tau\in{\bf F}^{g-1}_2$
we have:
$$
U_x=\pmatrix{I&0\cr 0&-I},\qquad
\tilde{T}_x':=\mbox{$1\over{1+i}$}\tilde{T}_x=\pmatrix{I&0\cr 0&iI},
$$
note that $\tilde{T}_x'$ and $\tilde{T}_x$ define the same element of
$A(G)$.
\subsection{Notation}
\subsubsection{}\label{defB}
In dealing with hyperelliptic curves and the half-spin representation of
the orthogonal group, the following (classical) notation for points in
${\bf F}_2^{2g}$ is convenient (\cite{DO}, VIII.3;
\cite{M}).
Let
$$
B:=\{1,\,2,\ldots,2g+2\},\qquad{\rm then}\quad
F_B:=\left\{f:B\rightarrow {\bf F}_2:\;\sum_{b\in B}f(b)=0\right\}\Big/
\{0,[b\mapsto 1]_{b\in B}\}.
$$
is an ${\bf F}_2$-vector space of dimension $2g$.
For a subset $T\subset B$ with an even number elements we denote
by $x_T\in F_B$ (or simply $T$)
the element defined by the function $f$ with $f(b)=1$
iff $b\in T$. Note that $x_T=x_{T'}$ when $T'$ is the complement of
$T$ in $B$, moreover
$$
x_T+x_S=x_R\qquad{\rm with}\quad R=T+ S:=(T\cup S)-(T\cap S).
$$
\subsubsection{}\label{notiso}
We fix the following isomorphism of ${\bf F}_2$-vector
spaces, and identify them in this way in the remainder
of the paper
(\cite{DO}, VIII.3, Lemma 2, but note we interchanged
$(e_i,0)\leftrightarrow (0,e_i)$):
$$
F_B\stackrel{\cong}{\longrightarrow} {\bf F}_2^{g}\times{\bf F}^{g}_2,\qquad
x_{\{2i-1,\,2i\}}=(0,e_i),\quad
x_{\{2i,\,2i+1,\ldots,2g+1\}}=(e_i,0)\qquad (1\leq i\leq g).
$$
The symplectic form $E$ on ${\bf F}_2^{g}\times{\bf F}^{g}_2$
can be now be easily computed on $F_B$ by:
$$
E(x_T,x_S)=|T\cap S| \;\;{\rm mod}\,2,\qquad
{\rm thus}\quad
E(x_{ij},x_{kl})=
\left\{\begin{array}{rcl}
0&{\rm if}&i,\,j,\,k,\,l\quad\mbox{are distinct} ,\\
1&{\rm if}& i<j=k<l
\end{array}\right.
$$
where we write $x_{ij}$ for $x_{\{i,j\}}$ etc. For $g=1$ we have
$$
x_{12}=x_{34}=(1,0),
\quad x_{13}=x_{24}=(1,1),\quad
x_{14}=x_{23}=(0,1).
$$
For $g=2$ one has 15 non-zero points $x_{ij}=x_{klmn}$ when
$\{i,j,k,l,m,n\}=B$:
$$
x_{12}=x_{3456}=
((0,0),(1,0)),\quad
x_{2345}=x_{16}=
((1,0),(0,0)),\quad
x_{26}=x_{1345}=
((1,0),(1,0)).
$$
\subsection{The Spin representation}\label{spin}
\subsubsection{}
The two half spin representations
of $so(2g+2)$ are each realized on a $2^g$-dimensional vector space.
We recall, using the Clifford algebra, how they can be constructed
using the Heisenberg group.
We will have to consider the Heisenberg group
$G_{g+1}$
which acts on a $2^{g+1}$-dimensional vector space.
Elements of order two in the Heisenberg group will define the spin
representation of $so(2g+2)$. Restriction to suitable subspaces will give
the half spin representations and their relation with $G_g$ (a
subquotient of $G_{g+1}$).
This relation between $so(2g+2)$ and the Heisenberg group is used in
\ref{explop} to prove the Heisenberg invariance of the flat connections
introduced in \ref{defcon}.
\subsubsection{}
To accomodate both $G_g$ and $G_{g+1}$
we extend the construction of $\S$\ref{defB}. The inclusion
$$
B:=\{1,\ldots,2g+2\}\hookrightarrow B^\sharp:=\{1,\ldots,2g+4\}
$$
and
extension by zero of functions on $B$ to functions on $B^\sharp$ induces
$$
\tilde{F}_B:=
\left\{f:B\rightarrow{\bf F}_2:\sum_{b\in B}f(b)=0\right\}\;\hookrightarrow \;
\tilde{F}_{B^\sharp}:=
\left\{f:B\rightarrow{\bf F}_2:\sum_{b'\in B^\sharp}f(b')=0\right\}.
$$
The function $g'\in \tilde{F}_{B^\sharp}$ with $g'(b')=1$
(all $b'\in B^\sharp$) does not lie in
$\tilde{F}_B$, thus
$$
\tilde{F}_B
\hookrightarrow
F_{B^\sharp}:=\tilde{F}_{B^\sharp}/\{0,g'\}.
$$
The image in $F_{B^\sharp}$ of the function $g
\;(\in \tilde{F}_{B})$ with $g(b)=1$
(all $b\in B$) is the element
$$
p':=
x_{\{1,2,\ldots,2g+2\}}
=x_{\{2g+3,2g+4\}}\in F_{B^\sharp}.
$$
Using the definition of the symplectic form on $F_{B^\sharp}$
($\S$\ref{notiso}), which we denote by $E^\sharp$, one finds that:
$$
\tilde{F}_B=(p')^\perp,\qquad{\rm with}\quad
p'^\perp:=\{x\in F_{B^\sharp}:\;E^\sharp(x,p')=0\;\}\quad
{\rm and}\quad F_B=(p')^\perp/\{0,p'\}.
$$
{}From $\S$\ref{notiso} we have an identification:
$$
F_{B^\sharp}={\bf F}^{g+1}\times{\bf F}^{g+1},\qquad
p'=(0,(1,1,\ldots,1))\qquad(\in F_{B^\sharp}).
$$
The Heisenberg group defined by this identification
(cf.\ $\S$\ref{Hdefs}) will be
denoted by $G^\sharp$, its Schr\"odinger representation by $U^\sharp$ (on the
$2^{g+1}$-dimensional vector space
$V^\sharp:=\{f:{\bf F}^{g+1}_2\rightarrow{\bf C}\}$
with basis of $\delta$-functions $Y_{\sigma'}$, $\sigma'\in{\bf F}_2^{g+1}$).
For any $x'\in (p')^\perp\;(\subset F_{B^\sharp})$,
the maps $U^\sharp_{p'}$ and $U^\sharp_{x'}$
commute. Therefore the $U^\sharp_{x'}$ with $x'\in (p')^\perp$
act on the two eigenspaces of $U^\sharp_{p'}$
which are:
$$
V^\sharp_+:=
\langle\ldots,Y_{\sigma'},\ldots\rangle_{\sigma_1'+\ldots+\sigma'_{g+1}=0},
\qquad
V^\sharp_-:=
\langle\ldots,Y_{\sigma'},\ldots\rangle_{\sigma_1'+\ldots+\sigma'_{g+1}=1}.
$$
A quotient map from $(p')^\perp\;(\subset
({\bf F}_2^g\times{\bf F}_2)\times ({\bf F}_2^g\times{\bf F}_2))$ to $F_B={\bf F}_2^g\times
{\bf F}_2^g$
with kernel $\{0,p'\}$ is:
$$
(p')^\perp\longrightarrow F_B=(p')^\perp/\{0,p'\}\cong {\bf F}_2^{2g},\qquad
x':=((a,a_{g+1}),(b,b_{g+1}))\longmapsto x:=(a,\bar{b})
$$
with $\bar{b}=(b_1+b_{g+1},\ldots,b_g+b_{g+1})$.
We define an isomorphism of vector spaces:
$$
V^\sharp_+\longrightarrow V,\qquad
Y_{\sigma_1,\ldots,\sigma_g,\sigma_{g+1}}\longmapsto
X_{\sigma_1,\ldots,\sigma_g}.
$$
\begin{lem}
For any $x'\in (p')^\perp$ mapping to $x\in F_B$
there is a commutative diagram:
$$
\begin{array}{rcl}
V^\sharp_+&\stackrel{\cong}{\longrightarrow}&V\\
U^\sharp(t,x')\,\Big\downarrow&&\Big\downarrow\, U(t,x)\\
V^\sharp_+&\stackrel{\cong}{\longrightarrow}&V
\end{array}
$$
Moreover, if $1\leq j,k\leq 2g+1$,
then restriction of $U^\sharp_{x_{jk}}$ to $V^\sharp_+\cong V$ is given by
$\pm U_{x_{ij}}$.
\end{lem}
\begin{proof}
The two compositions along the square are (with $a,\,b,\,\sigma\in{\bf F}_2^g$):
$$
Y_{\sigma,\sigma_{g+1}}\longmapsto
X_\sigma\longmapsto (t,(a,\bar{b}))X_\sigma=
t(-1)^{(a+\sigma)\bar{b}}X_{a+\sigma},\qquad{\rm and}
$$
$$
\begin{array}{rcl}
Y_{\sigma,\sigma_{g+1}}&\longmapsto&
(t,(a,a_{g+1}),(b,b_{g+1}))Y_{\sigma,\sigma_{g+1}}\\
&=&
t(-1)^{(a+\sigma)b+(a_{g+1}+\sigma_{g+1})b_{g+1}}
Y_{a+\sigma_g,a_{g+1}+\sigma_{g+1}}\\
&\longmapsto&
t(-1)^{(a+\sigma)b+(a_{g+1}+\sigma_{g+1})b_{g+1}}X_{a+\sigma}.
\end{array}
$$
Recall $E^\sharp(x',p')=0$, so
$a_{g+1}=\sum_{i=1}^ga_i$. As $Y_{\sigma,\sigma_{g+1}}\in V^\sharp_+$
we also have
$\sigma_{g+1}=\sum_{i=1}^g\sigma_i$, thus:
$$
(a+\sigma)b+(a_{g+1}+\sigma_{g+1})b_{g+1}=
\left(\sum_{i=1}^g(a_i+\sigma_i)b_i\right)+
\left(\sum_{i=1}^g(a_i+\sigma_i)b_{g+1}\right)=(a+\sigma)\bar{b},
$$
which shows that the diagram commutes.
The last statement follows from the fact that such a $x'=x_{jk}$ lies in
$(p')^\perp\;(\subset F_{B^\sharp})$ and that the homomorphism
$x'\mapsto x\;(\in F_B)$ maps $x_{jk}$ to $x_{jk}$ (use \ref{notiso}).
Since $U^\sharp_{x_{jk}}=U^\sharp(t',x'),\;U_{x_{jk}}=U(t,x)$
for some $t',\,t\in{\bf C}$ choosen such that each transformation
has order two, the statement
follows from the commutativity of the diagram.
\end{proof}
\subsubsection{The Clifford algebra}
The Clifford algebra $C(Q)$ of the quadratic form
$Q=x_1^2+\ldots+x_{2g+2}^2$ on
$V_Q={\bf C}^{2g+2}$ is the quotient of the tensor algebra $T(V_Q)$ by the
two-sided ideal $I$ generated by the elements $v\otimes v-Q(v)$ for $v\in
V_Q$ (\cite{FH}, $\S$20.1):
$$
C(Q):=T(V_Q)\Big/I=
({\bf C}\,\oplus\, V_Q\,\oplus\,V_Q\!\otimes\! V_Q\oplus\ldots)\Big/
(\ldots,-Q(v)+v\otimes v,\ldots)_{v\in V_Q}.
$$
Let $e_1,\ldots,e_{2g+2}$ be the standard basis vectors of $V_Q$, then
the $C(Q)$ is generated by ${\bf C}$ and the $e_j$ with relations:
$$
e_j\cdot e_j=1,\qquad e_j\cdot e_k+e_k\cdot e_j=0\quad(j\neq k),
$$
here $\cdot$ stands for the product induced by $\otimes$ on $C(Q)$.
The (associative) algebra $C(Q)$ becomes a Lie algebra by defining, as
usual,
the Lie bracket to be
$[x,y]:=x\cdot y-y\cdot x$.
The following proposition relates the Heisenberg group, the Clifford
algebra and the spin representation of the Lie algebra $so(2g+2)$.
\subsubsection{Proposition.} \label{halfspin}
With the notation as above we have:
\begin{enumerate}
\item
The ${\bf C}$-linear map
$$
\gamma^\sharp:C(Q)\longrightarrow \mbox{End}(V^\sharp),\qquad
\lambda\mapsto\lambda I,\quad
e_k\mapsto U^\sharp_{x_{k,\,2g+4}}\qquad(\lambda\in{\bf C},\;\;k=1,\ldots,2g+2)
$$
defines an isomorphism of (associative ${\bf C}$-) algebras.
\item
The linear map:
$$
\rho_s^\sharp:so(2g+2)\longrightarrow
\mbox{End}(V^\sharp)\cong_{\gamma^\sharp} C(Q),\qquad
F_{jk}\longmapsto U^\sharp_{x_{j,2g+4}}U^\sharp_{x_{k,2g+4}}
\;(=\gamma^\sharp(e_j\cdot e_k))
$$
is an injective homomorphism of Lie algebras.
\item
The subspace $V^\sharp_+$ of $V^\sharp$ is invariant under the action of
$so(2g+2)$. The Lie algebra representation
$$
\rho_s:so(2g+2)\longrightarrow \mbox{End}(V^\sharp_+)\cong
\mbox{End}(V),\qquad
\rho_s(x):=\left.\rho_s^\sharp(x)_{\phantom{y}}\!\right|_{V^\sharp_+}
$$
is an (irreducible) half spin representation of $so(2g+2)$.
In particular, $V\cong V(\omega_{g+1})$.
\item
We have
$$
\rho_s(F_{jk})=\pm iU_{x_{jk}},\qquad(F_{jk}\in so(2g+2),\quad i^2=-1),
$$
where the $U_{x_{jk}}\;(\in \mbox{End}(V))$ with $x_{jk}\in {\bf F}_2^{2g}$
are in the Schr\"odinger
representation of the Heisenberg group $G$.
\end{enumerate}
\begin{proof}
By definition we have $(U^\sharp_{x'})^2=I$ for all $x'\in{\bf F}_2^{2g+2}$.
Moreover,
$E^\sharp(x_{\{k,2g+4\}},x_{\{l,2g+4\}})=1$ when $k\neq l$ so
the corresponding maps anti-commute:
$U^{\sharp}_{x_{k,2g+4}}U^\sharp_{x_{l,2g+4}}=
-U^{\sharp}_{x_{l,2g+4}}U^{\sharp}_{x_{k,2g+4}}$.
Then the map $\gamma^\sharp$ preserves the relations in $C(Q)$ and is thus an
algebra homomorphism.
It is surjective because the matrices $U^\sharp_{x'}$, where $x'$
runs over ${\bf F}_2^{g+1}\times{\bf F}_2^{g+1}$ are a basis of ${\rm End}(V^\sharp)$
(the Schr\"odinger representation being irreducible) and any $U^\sharp_{x'}$
is a product (up to scalar multiple) of suitable $U^\sharp_{x_{j,2g+4}}$.
Therefore $\gamma^\sharp$ is
an isomorphism since both algebras have the same dimension.
\noindent
(2)$\quad$
This is worked out in \cite{FH}, Lemma 20.7.
With their notations and our $Q$, we have
$$
\phi:\wedge^2V_Q\stackrel{\cong}{\longrightarrow} so(2g+2),\qquad
e_j\wedge e_k\longmapsto F_{jk},
$$
$$
\psi:\wedge^2V_Q\longrightarrow C(Q),\qquad
e_j\wedge e_k\longmapsto e_j\cdot e_k\qquad(j\neq k),
\qquad{\rm and}\quad
\rho_s^\sharp=\psi\circ \phi^{-1}.
$$
\noindent
(3)$\quad$
By definition of $\rho_s^\sharp$ and the fact that $U^\sharp$ is a
representation we have:
$$
\rho_s^\sharp(F_{jk})=U^\sharp_{x_{j,2g+4}}U^\sharp_{x_{k,2g+4}}=
c_{jk}U^\sharp_{x_{jk}},
$$
with
$1\leq j,k\leq 2g+2$ and some $c_{jk}\in{\bf C}^*$.
Thus the $\rho_s^\sharp(F_{jk})$ commute with $U^\sharp_{p'}$ in
${\rm End}(V^\sharp)$. Therefore we obtain a Lie algebra representation on
$V^\sharp_+$
cf.\ \cite{FH}, Prop.\ 20.15 (and identify their $End(\wedge^{even}W)$
with our $End(V^\sharp_+)$), where also the irreducibility is proved.
\noindent
(4)$\quad$We have $\rho_s^\sharp(F_{jk})=
U^\sharp_{x_{j,2g+4}}U^\sharp_{x_{k,2g+4}}$, thus
$\rho_s^\sharp(F_{jk})^2=-I$ since
these two elements anti-commute. Therefore $\rho_s^\sharp(F_{jk})=\pm i
U^\sharp_{x_{jk}}$, which acts as $\pm iU_{x_{jk}}$ on $V^\sharp_+$.
\end{proof}
\subsection{The Hitchin connection}\label{hitcon}
\subsubsection{} \label{PPdescent}
The symmetric group $S_{2g+2}$ acts on ${\cal P}$ by permuting the coordinates,
the quotient wil be denoted by ${\overline{\cP}}$.
The fundamental group of ${\cal P}$ is the pure braid group and
$\pi_1({\overline{\cP}})=B_{2g+2}$, the Braid group.
In case $g=2$, we have $S_6\cong Sp(4,{\bf F}_2)$, in fact there is a
surjective map (which factors over the mapping class group)
$$
\pi_1(\overline{{\cal P}})=B_{6}\longrightarrow Sp(4,{\bf Z}),
$$
the kernel of the composition $B_{6}\rightarrow Sp(4,{\bf Z})\rightarrow
Sp(4,{\bf F}_2)$ is $\pi_1({\cal P})$.
>{}From the theory of theta functions we know that the groupscheme ${\cal G}$
is trivialized on the cover $\tilde{{\cal P}}$ of ${\cal P}$ defined by the kernel
of the composition:
$$
\phi:\pi_1(\overline{{\cal P}})\longrightarrow
Sp(4,{\bf Z})\longrightarrow Sp(4,{\bf Z})/\Gamma_2(2,4)\cong A(G).
$$
Here we use Igusa's notation:
$$
\Gamma_2(2,4):=\left\{\pmatrix{I+2A&2B\cr 2C&I+2D}\in Sp(4,{\bf Z}):\;
diagonal(B)\equiv diagonal(C)\equiv (0,0)\;{\rm mod}\;2\right\}.
$$
Thus we consider the following diagram of etale Galois
coverings:
$$
{{\widetilde{\cP}}}\longrightarrow {\cal P}\longrightarrow \overline{{\cal P}}:={\cal P}/S_{6},
$$
and the corresponding exact sequence of covering groups:
$$
0\longrightarrow {\bf F}_2^{2g}\longrightarrow A(G)\longrightarrow S_6
\longrightarrow 0.
$$
Recall that we have defined a projective representation
$\tilde{T}:A(G)\rightarrow PGL(V)$ in \ref{projrep}.
This induces projective representations on each $S^kV$.
\begin{thm}\label{thmhitcon}
With the notation as in \ref{PPdescent}
(except for $S_k$) and $g=2$ we have:
\begin{enumerate}
\item
There is a line bundle ${\cal N}$ on ${{\widetilde{\cP}}}$
such that the pull-back of $p_*{\cal L}^{\otimes k}$ is isomorphic
to $S_k\otimes_{\bf C}{\cal N}$.
Here $S_k=S^kV(\omega_{g+1})$
and $V(\omega_{g+1})$ is a half spin representation of $so(6)\cong sl(4)$
(which is the standard representation (or its dual) of $sl(4)$).
\item\label{hits}
The Hitchin connection on the pull-back
of ${p}_*{\cal L}^{\otimes k}$ to ${\widetilde{\cP}}$
is given by the pull-back to ${\widetilde{\cP}}$ of
the connection on $S_k\otimes_{\bf C}{\cal O}_{{\bf P}}$ defined by the one form
$$
{-1\over{16(k+2)}}\sum_{i,j\,i\neq j}
{{\sigma\tilde{\rho_s}(\Omega_{ij}){\rm d} z_i}\over{z_i-z_j}}
$$
with $\tilde{\rho_s}:U(so(2g+2))\rightarrow{\rm End}(S_1)$ the half spin
representation (then the $\sigma\tilde{\rho_s}(\Omega_{ij})$ give
endomorphisms of each $S_k$, cf.\ \ref{symcon}).
\item \label{hitbp}
The Hitchin connection on the bundle $S_k\otimes{\cal O}_{\widetilde{\cP}}$ over ${\widetilde{\cP}}$
descends to a projective flat connection on the the bundle
$(S_k\otimes{\cal O}_{\widetilde{\cP}})/A(G)$ over ${\overline{\cP}}$, where $A(G)$ acts on $S_k$ via
the $k$-th symmetric power of its projective representation on $S_1$
and $A(G)$ acts on ${\widetilde{\cP}}$ as $Gal({\widetilde{\cP}}/{\overline{\cP}})$.
This descent gives the
kth Hitchin connection of the natural descend of the family
of curves described in 1.4.
\end{enumerate}
\end{thm}
\begin{proof}
For the first part it suffices to show that the pull-back of
$p:{\bf P}\rightarrow{\cal P}$ to $\tilde{{\cal P}}$ is trivial.
Since ${\cal G}$ is isomorphic to the constant group scheme $G$ (cf.\
\ref{Hdefs} (with here $t\in{\bf C}^*$)), the uniqueness of the Schr\"odinger
representation gives the global triviallization.
We already observed that the line bundle ${\cal N}$ does not interfere with the
projective connections.
\noindent
(2)$\quad$ We use Theorem \ref{269} so we must verify $(Eq.)$ from \ref{crit}.
The covariant derivatives given are defined by the heat operator
$$
X=-
{1\over{16(k+2)}}\sum_{j\neq i}
{{\sigma\tilde{\rho_s}(\Omega_{ij})}}\otimes{{{\rm d}z_i}\over{z_i-z_j}}
\qquad(\in (S_2\otimes_{\bf C} S_2^*)\otimes\Omega_{\cal P}^1).
$$
The Heisenberg-invariance of this operator follows from the facts that
$\Omega_{ij}=F_{ij}\otimes F_{ij}$, that $\rho_s(F_{kj})=\pm iU_{x_{jk}}$
with $i^2=-1$
and that $U_yU_xU_y^{-1}=\pm U_x$.
Since $S^4(\tilde\rho_s)$ is a subrepresentation of
$\tilde\rho:so(2g+2)\rightarrow {\rm End} S^2(\wedge^2{\bf C}^{2g+2})$
we have
$\sigma\tilde{\rho_s}(\Omega_{ij})=\sigma\tilde{\rho}(\Omega_{ij})$ on
$S_4$.
For the connection $\nabla_0$ on $\tilde{p}_*{\cal O}_{{\bf P}}(1)=
S_1\otimes {\cal O}_{\tilde{P}}$
we simply take $\nabla_0(fw)=w\otimes {\rm d} f$.
Then the equation $(Eq.)$ is exactly the statement of Theorem \ref{thmcon}.
\noindent
(3)$\quad$
We have a smooth projective family
of genus 2 curves over the scheme $\bar{\cal P}$ given by
$$
y^2 = f(x) = x^6+a_1 x^5 +a^2 x^4 + ... + a_6 = \prod_i (x-z_i),
$$
where $a_i$ is the $i$-th symmetric function of the $z_j$.
Thus by the general theory of Section 2 we have a ${\bf P}^3$-bundle
$\bar p : \bar {\bf P} \longrightarrow \bar{\cal P} $
and a projective Hitchin connection $\bar{\bf D}_k$ on $\bar p_\ast {\cal O}(k)$.
Let $\varphi : \tilde{\cal P} \longrightarrow \bar{\cal P}$ be
the natural map. Then there is an isomorphism
$$
(*) \qquad\qquad \varphi^* ( \bar {\bf P} ) = {\bf P}^3 \times \tilde{\cal P}
$$
which is compatible with the action of $A(G)$ (which lies over the action
of $A(G)$ on $\tilde{\cal P}$) on both sides: this isomorphism is given by the
trivialization of the action of the theta-group scheme ${\cal G}$ which was
mentioned earlier. The isomorphism (*) induces an isomorphism
$$
(**) \qquad\qquad \varphi^* ( {\bar p}_*O(k) ) \cong S_k \otimes O_{\tilde{\cal P}}
$$
which is compatible with the natural projective action of $A(G)$
on the LHS (via ${\cal G}$) and the action of $A(G)$ on the RHS via the $Sym^k$
of its natural representation on $S_1$. The Heat operator given by $X$
(as in proof of \ref{thmhitcon}.\ref{hits}, see \ref{crit})
on the LHS of (*) is VIA (*) compatible
with
the pullback of the Hitchin heat operator of the LHS of (*). But both heat
operators
are invariant under the action of $A(G)$. Thus the projective connections
on the RHS and LHS of (**) agree and are compatible with the projective
action of $A(G)$ on both sides. In other words the natural descent
datum on the LHS of (**) agrees with the descent datum on the RHS
of (**).
\end{proof}
\subsubsection{Examples}\label{explop}
We give some examples of the $\rho_s(\Omega_{ij})$'s in case $g=2$.
We identify $V=S_1$, with its standard basis $X_\sigma,\;
\sigma\in{\bf F}_2^2$. A linear map with matrix $(a_{kl})$ is then also given
by the linear differential operator $\sum_{kl}a_{kl}X_k\partial_l$ (since
$\partial_l(X_m)=0$ for $l\neq m$ and is $1$ if $l=m$).
Recall from
\ref{notiso} that (for $g=2$):
$$
x_{12}=((0,0),(1,0)),\qquad{\rm so}\quad
U_{x_{12}}=X_{00}\partial_{00}+X_{01}\partial_{01}-
X_{10}\partial_{10}-X_{11}\partial_{11}.
$$
Similarly, we had $x_{16}=((1,0),(0,0))$ and $x_{26}=((1,0),(1,0))$ and so:
$$
U_{x_{16}}=X_{10}\partial_{00}+X_{11}\partial_{10}+
X_{00}\partial_{10}+X_{01}\partial_{10},\quad
U_{x_{26}}=i(-X_{10}\partial_{00}-X_{11}\partial_{10}+
X_{00}\partial_{10}+X_{01}\partial_{10}).
$$
Since $U_{x_{jk}}=\pm i\rho_s(F_{jk})$ and $U_{x_{jk}}^2=I$
we get (cf.\ \ref{symcon}):
$$
\sigma(\tilde{\rho_s}(\Omega_{12}))=
-2(X_{00}^2\partial_{00}^2+X_{01}^2\partial_{01}^2+
X_{10}^2\partial_{10}^2+X_{11}^2\partial_{11}^2+
2X_{00}X_{01}\partial_{00}\partial_{01}-2X_{10}X_{11}\partial_{10}\partial_{11}),
$$
\subsection{Local Monodromy}\label{localmon}
\subsubsection{Introduction}
We want to obtain some information on the representation of the mapping
class group for genus two curves defined by Hitchin's connection. This
representation has been studied, for arbitrary $g$ and $k$,
by Moore and Seiberg \cite{MS},
Kohno \cite{K2} and in the case $k=2$ by Wright \cite{Wr}.
In some sense, we merely find
(weaker, local) results which do agree with their results.
\subsubsection{The method} \label{monex}
We recall how to determine the local monodromy on the trival bundle
${\cal O}_{\cal P}\otimes_{\bf C} W$ with a connection $\nabla$.
We consider a holomorphic map
$$
\phi:D^\ast\longrightarrow {\cal P},\qquad{\rm with}\quad
D^*:=\{t\in{\bf C}-\{0\}:\;|t|<\epsilon\,\}.
$$
(for small, positive $\epsilon$) and pull-back the
one form $\omega\in {\rm End}(W)\otimes\Omega^1_{\cal P}$
which defines the connection (so $\nabla(fw)=w{\rm d} f-f\omega(w)$. We write
$$
\phi^*\omega=
\left({R\over t}+A(t)\right){\rm d}t,\qquad R\in {\rm End}(W)
$$
with $A(t)$ holomorphic for $t=0$, and $R$ is called the residue at $t=0$.
We identify $\pi_1(D^*)={\bf Z}$; $1\in{\bf Z}$ represents
a small circle traversed in anti-clockwise direction.
The eigenvalues of the monodromy of $1\in\pi_1(D^*)$
are the
$exp(-2\pi i\mu_j)$
were the $\mu_i$ are the eigenvalues of $R$ on $W$.
In case the monodromy transformations are semi-simple, the local monodromy
is conjugated to $exp(-2\pi iR)$ (cf. \cite{D}, p. 54).
However we could not prove the semi-simplicity
(but it seems to be known to the physicists).
To get the eigenvalues of the monodromy of the Hitchin connection
for a $\gamma \in\pi_1({{\overline{\cP}}})$,
let $n$ be the order of its image in
$Gal({\cal P}/{\overline{\cP}})$ and let $\bar{\gamma}$ be its image in
$A(G)=Gal({{\widetilde{\cP}}}/{{\overline{\cP}}})$. We choose a $\phi:D^*\rightarrow {\cal P}$
in such a way that $\phi_*(1)=\gamma^n\;(\in \pi_1({\cal P}))$.
Then these eigenvalues are the eigenvalues of the matrix $exp(
\mbox{$-2\pi i\over n$}R)
\tilde{T}_{\bar{\gamma}}$, with $R$ determined as above for the
corresponding connection on ${\cal P}$ (cf.\ \cite{K}).
Since $\tilde{T}$ is a projective representation
(on $S_1$ and thus on any $S_k$), the set of eigenvalues are only defined up to
multiplication by one non-zero constant. This corresponds to the fact
that we only have a projectively flat connection.
\subsubsection{Non-seperating vanishing cycle}\label{nsep}
Let $\gamma\in\pi_1(\bar{{\cal P}})$ such that $\gamma^2=\phi_*(1)$ with
$$
\phi:D^\ast\longrightarrow {\cal P},\qquad
t\longmapsto (t+z_2,z_2,\ldots,z_6)
$$
where we fix distinct $z_j\in{\bf C}$. (Then $\gamma$ corresponds to a Dehn
twist in a simple non-seperating loop in the mapping class group.)
The residue of $\phi^*\omega$ for
the connection on $S_k\otimes{\cal O}_{\cal P}$ corresponding to
the Hitchin connection on $p_*{\cal L}^{\otimes k}$ is
(cf.\ \ref{thmhitcon} and \ref{explop}):
$
\mbox{${-1\over {16(k+2)}}$}\sigma\tilde{\rho_s}(\Omega_{12}),
$
$$
-\tilde{\rho_s}(\Omega_{12})=
R_{12}:=
\left((X_{(0,0)}\partial_{(0,0)}+X_{(0,1)}\partial_{(0,1)})-
(X_{(1,0)}\partial_{(1,0)}+X_{(1,1)}\partial_{(1,1)})\right)^2
$$
\begin{lem}\label{spec12}
The eigenvalues of of the monodromy of $\gamma$ as in \ref{nsep}
on $S_k$ are:
$$
\lambda_c:=exp(-2\pi i \mbox{${c(c+1)}\over {k+2}$}),
\qquad
{\rm mult}(\lambda_c)=(k-2c+1)(2c+1)
$$
(up to multiplication by a non-zero constant independent of $c$),
with $2c\in{\bf Z}$ and $0\leq c\leq \mbox{$k\over 2$}$.
\end{lem}
\begin{proof}
Since the eigenvalues are only determined up to a constant and since
the difference between $2R_{12}$ and
$\sigma(R_{12})$ is a multiple of the
Euler vector
field, which acts by multiplication by $k$ on $S_k$, it suffices to consider
the eigenvalues of $R_{12}$.
For a monomial in $S_k$ we have:
$$
R_{12}(X^{l_0}_{(0,0)}X^{l_1}_{(0,1)}
X^{m_0}_{(1,0)}X^{m_1}_{(1,1)})=
=((l_0+l_1)-(m_0+m_1))^2(X^{l_0}_{(0,0)}X^{l_1}_{(0,1)}
X^{m_0}_{(1,0)}X^{m_1}_{(1,1)})
$$
Thus each monomial is an eigenvector.
Let $b:=m_0+m_1$.
Since the monomial has degree $k$, its eigenvalue is
$(k-2b)^2$. The multiplicity of the eigenvalue is $(k-b+1)(b+1)$
(the dimension of
the space of homogeneous polynomials in $2$ variables of degree $c$ is
$c+1$).
Thus the eigenvalue $2(k-2b)^2$ of $2R_{12}$
has multiplicity $(k-b+1)(b+1)$.
The image $\bar{\gamma}$ of $\gamma$ in $A(G)$ is
$\tilde{T}_{x_{12}}$ which acts on such a monomial by
(cf.\ \ref{extt}):
$$
\tilde{T}_{x_{12}}(X^{l_0}_{(0,0)}X^{l_1}_{(0,1)}
X^{m_0}_{(1,0)}X^{m_1}_{(1,1)})
=
i^b(X^{l_0}_{(0,0)}X^{l_1}_{(0,1)}
X^{m_0}_{(1,0)}X^{m_1}_{(1,1)}).
$$
The eigenvalues of $\gamma$ on $S_k$
are then, with $c:=b/2$:
$$
\begin{array}{rcl}
exp({-2\pi i\over 2}
\left(\mbox{$+1\over {16(k+2)}$}{2(k-2b)^2}+\mbox{$b\over
4$}\right))
&=& exp(-2\pi i\left(\mbox{${k^2-8ck+16c^2+8ck+16c}\over
{16(k+2)}$}\right))\\
&=&exp(-2\pi i\left(\mbox{${k^2}\over {16(k+2)}$}\right))\,
exp(-2\pi i\left(\mbox{${c(c+1)}\over {k+2}$}\right)).
\end{array}
$$
\end{proof}
\subsubsection{Seperating vanishing cycle}
Now we consider a $\gamma\in\pi_1(\bar{{\cal P}})$ such that
$\gamma=\phi_*(1)$ with
$$
\phi:D^*\longrightarrow {\cal P},\qquad
t\longmapsto (tz_1,tz_2,tz_3,z_4,z_5,z_6),
$$
with distinct nonzero $z_i$'s.
The residue in $t=0$ of the connection on $S_k\otimes{\cal P}$ corresponding
to Hitchin's connection is
$$
\mbox{$-1\over{16(k+2)}$}R_{123}, \qquad{\rm with}\quad
R_{123}:=2\sigma(\tilde{\rho_s}(\Omega_{12}+\Omega_{13}+\Omega_{23})).
$$
\begin{lem} \label{xq} Let $g=2$.
\begin{enumerate}
\item
There are constants $\lambda_k\in{\bf Q}$ such that (in ${\rm End}(S_k)$):
$$
R_{123}=16QX_Q+\lambda_kI
\quad{\rm with}\quad
QX_Q=(X_{00}X_{10}-X_{01}X_{11})
(\partial_{00}\partial_{10}-\partial_{01}\partial_{11}).
$$
\item
We define a subspace of $S_k$ by:
$$
V_k:=Kernel(X_Q:=\partial_0\partial_1-\partial_2\partial_3:\,
S_k\longrightarrow S_{k-2}).
$$
Then the vector space $S_k$ is a direct sum:
$$
S_k=V_k\oplus QV_{k-2}\oplus Q^2V_{k-4}\oplus\ldots
$$
Moreover, each subspace $Q^lV_{k-2l}$ is an eigenspace of $R_Q$
with eigenvalue:
$$
\lambda_l=
l(k-l+1)\qquad
{\rm and}\quad \dim V_{k-2l}=(k-2l+1)^2.
$$
\item
The eigenvalues of of the monodromy of $\gamma$ on $S_k$ are
$$
\lambda_l:=exp(-2\pi i \mbox{${l(l+1)}\over
{k+2}$})\qquad
mult(\lambda_l)=(k-2l+1)^2
$$
(up to multiplication by a constant independent of $l$),
with $l\in{\bf Z}$ and $0\leq l\leq k/2$.
\end{enumerate}
\end{lem}
\begin{proof}
Recall $x_{12}$ corresponds
to $((0,0),(1,0))\in{\bf F}^2_2\times{\bf F}^2_2$ and
$x_{23}=x_{\{2,3,4,5\}}+
x_{\{4,5\}}$ corresponds to $((1,1),(0,0))$
thus $x_{13}=x_{12}+x_{23}$ corresponds to
$((1,1),(1,0))$.
Since $(1,1)(1,0)=1$, we have a
$+$ sign in front of $\Omega_{13}$ below.
Let $\tilde{\rho_s}(\Omega_{00})$
be the square of the Euler vector field (which acts
as $k^2I$ on $S_k$).
Then:
$$
\begin{array}{rcr}
\tilde{\rho_s}(\Omega_{00})&=&+(X_{00}\partial_{00}+X_{01}\partial_{01}+
X_{10}\partial_{10}+X_{11}\partial_{11})^2,\\
\tilde{\rho_s}(\Omega_{12})&=&-(X_{00}\partial_{00}+X_{01}\partial_{01}-
X_{10}\partial_{10}-X_{11}\partial_{11})^2,\\
\tilde{\rho_s}(\Omega_{23})&=&-(X_{11}\partial_{00}+X_{10}\partial_{01}+
X_{01}\partial_{10}+X_{00}\partial_{11})^2,\\
\tilde{\rho_s}(\Omega_{13})&=&+(X_{11}\partial_{00}+X_{10}\partial_{01}-
X_{01}\partial_{10}-X_{00}\partial_{11})^2.
\end{array}
$$
The sum of these operators, minus the
(degree one) parts which act as constants on each $S_k$, is $4QX_Q$.
Remembering the factor $2$ in $R_{123}$ and
in our symbols, we find the first statement.
\noindent
(2)$\quad$
This is well-known. Let $f\in V_{n}$, so $f$ is
homogeneous of degree $n$ and
$X_Q(f)=0$. Then an elementary
computation gives for $l\geq 1$ (with variables $X_i$ and
$Q=X_0X_1-X_2X_3$):
$$
\begin{array}{cl}
&(\partial_0\partial_1-\partial_2\partial_3)(fQ^l)\\
=&(\partial_0 f)(\partial_1 Q^l)+(\partial_0 Q^l)(\partial_1 f)-
(\partial_2 f)(\partial_3 Q^l)-(\partial_2 Q^l)(\partial_3 f)+
f(\partial_0\partial_1-\partial_2\partial_3)(Q^l)
\\
=&lQ^{l-1}(X_0\partial_0+X_1\partial_1+X_2\partial_2+X_3\partial_3)(f)+
l(l+1)fQ^{l-1}
\\=& l(n+l+1)fQ^{l-1}.
\end{array}
$$
Writing $n=k-2l$ we get, for all integers $l$ with $0\leq 2l\leq k$:
$$
QX_Q(fQ^l)=l(k-l+1))fQ^l,\qquad f\in V_{k-2l}.
$$
Hence each $V_{k-2l}Q^l$ is an eigenspace for $R_Q$. By induction (the
cases
$k=0,\,1$ being trivial) we may
assume that $S_{k-2}=\oplus S_{k-2-2l}Q^{l}$. Therefore $QS_{k-2}=
\oplus Q^{l+1}S_{k-2(l+1)}\subset S_k$ is a direct sum of eigenspaces of
$R_Q$
and none of the eigenvalues of $R_Q$ on this subspace is zero
($k+l(l+1)>0$ for $k\geq 2,\, l\geq 0$). Thus $\ker(R_Q)\cap
QS_{k-2}=\{0\}$ and $R_Q$ induces an isomorphism on
$QS_{k-2}$. Therefore $S_k=Ker (R_Q)\oplus
Im(R_Q)=\oplus Q^lS_{k-2l}$.
The dimension of $V_k$ is then $\dim S_k-\dim
S_{k-2}={{k+3}\choose 3}-{{k+1}\choose 3}=(k+1)^2$.
\noindent
(3)$\quad$
The image of $\gamma$ in $A(G)$ is
trivial.
The eigenvalues of $R_{123}=8QX_Q$
on $S_k$ are $16l(k-l+1)$. Thus the eigenvalues of $\gamma$ are (see
\ref{monex}):
$$
\begin{array}{rcl}
exp(-2\pi i\mbox{${-1}\over{16(k+2)}$} 16l(k-l+1))
&{=}&
exp(-2\pi i\left(\mbox{${l^2-l-l(k+2)+2l}\over{k+2}$}\right))\\
&=&
exp(-2\pi i \mbox{${l(l+1)}\over{k+2}$}).
\end{array}
$$
\end{proof}
\subsubsection{Kohno's results}
The monodromy of the Hitchin connection has been studied by Moore-Seiberg
and Kohno (among others), we will relate the results above to
those contained in \cite{K2} for $g=2$.
Let $\gamma$ be a tri-valent graph with two edges, so there are three
edges meeting
in each vertex.
For $k\in{\bf Z}_{\geq 0}$ we define a finite set $b_k(\gamma)$ of functions
on the edges of $\gamma$ by:
$$
f:Edges(\gamma)\longrightarrow \{0,\mbox{$1\over 2$},1,
\ldots,\mbox{$k\over 2$}\}
\quad{\rm with}\quad
\left\{\begin{array}{r}
f(e_i)+f(e_j)+f(e_k)\leq k,\\
|f(e_i)-f(e_j)|\leq f(e_k)\leq f(e_i)+f(e_j),\\
f(e_i)+f(e_j)+f(e_k)\in{\bf Z},
\end{array}
\right.
$$
for any three edges $e_i,\,e_j,\;e_k$ meeting in a vertex.
The Verlinde space
$V_k$ is the ${\bf C}$-vector space with basis
$b_k(\gamma)$, it has the same dimension as $S_k$.
The graph $\gamma$ corresponds to a pants decomposition of a genus two
Riemann surface. Each vertex is a pant, homeomorphic to ${\bf P}^1$ minus 3
points,
the edges correspond to these points, which in turn correspond to
`vanishing cycles' on the Riemann surface.
The mapping class group has a projective representation on $V_k$.
The Dehn twist in a vanishing cycle corresponding to an edge
$e$ of $\gamma$ acts, up to a scalar multiple, conjugated to the diagonal
matrix with entries (cf.\ \cite{K2}, p.\ 217; p.\ 214, (2-1))
$$
exp(-2\pi i f(e)(f(e)+1)/(k+2)\qquad(f\in b_k(\gamma)).
$$
Using this recipe we find the same results as before (see Lemmas
\ref{spec12}
and \ref{xq}):
\begin{lem}\label{tmon} With the notation as above:
\begin{enumerate}
\item
The eigenvalues of the Dehn twist in a non-seperating cycle on $V_k$
are
$exp(-2\pi i v(v+1)/(k+2))$ with multiplicity
$(k-2v+1)(2v+1)$ where $v\in \{0,\mbox{$1\over 2$},1,
\ldots,\mbox{$k\over 2$}\}$.
\item
The eigenvalues of a Dehn twist in a seperating cycle on $V_k$
are $exp(-2\pi il(l+1)/(k+2))$ with multiplicity
$(k-2l+1)^2$ where $l$ is an integer with $0\leq l\leq k/2$.
\end{enumerate}
\end{lem}
\begin{proof}
Consider the graph $\gamma$ with two vertices joined by three edges
$e_1,\,e_2,\,e_3$.
We first determine the set $b_k(\gamma)$. The edge $e_1$ corresponds to
a non-seperating cycle and we determine $f(e_1)$ for all $f\in
b_k(\gamma)$, this will give the first result.
Writing $f_i:=f(e_i)$ for $f\in b_k(\gamma)$, the conditions are the same
in each of the two edges. Assume that we have $f_1\leq f_2\leq f_3$.
The conditions on the $f_i$ are then equivalent to the following 3
conditions:
$$
f_1+f_2+f_3\leq k,\qquad f_3\leq f_1+f_2,\qquad f_1+f_2+f_3\in{\bf Z},\qquad
f_i\in\{0,\mbox{$1\over 2$},1,
\ldots,\mbox{$k\over 2$}\}
$$
In case $f_1+f_2+f_3=l\;(\leq k)$ and $f_3>l/2$, the second condition
implies
$f_1+f_2>l/2$, a contradiction. Thus the set $b_k(\gamma)$ is a disjoint
union of sets $c_l(\gamma)$ for $l\in{\bf Z}$ and $0\leq l\leq k$:
$$
c_l(\gamma):=\{f\in b_k(\gamma):\;f_1+f_2+f_3=l,\quad f_i\leq l/2\;\},
\qquad
b_k(\gamma)=\stackrel{.}{\cup}_{0\leq l\leq k} c_l(\gamma).
$$
Next we observe that if $f\in c_l(\gamma)$ then $f_1+f_2+f_3=l$ implies
$f_1+f_2=l-f_3\geq l/2$ (since $f_3\leq l/2$), thus certainly $f_3\leq
f_1+f_2$, so the second condition is always fulfilled.
It remains to determine the number of triples $(f_1,f_2,f_3)$ with
$f_i\in \{0,\mbox{$1\over 2$},1,\ldots,\mbox{$l\over 2$}\}$
and $f_1+f_2+f_3=l$. Equivalently, we have to find the number of monomials
$X^aY^bZ^c$ with $a+b+c=2l$ and $0\leq a,\,b,\,c\leq l$.
If such a monimial with $a+b+c=2l$ does not satifisfy $0\leq a,\,b,\,c\leq
l$, then at most one of $a,\,b,\,c$ can be $>l$, say $c$, and then
$a+b=2l-c$ which gives $(2l-c)+1$ possible couples $a,\,b$. Varying $c$
between $l+1$ and $2l$ we get $l+(l-1)+\ldots+1=(1/2)l(l+1)$ monomials
with $c>l$ and thus there are $(3/2)l(l+1)$ monomials of degree $2l$
which do not satisfy the condition $0\leq a,\,b,\,c\leq l$.
Then
$$
|c_l(\gamma)|=\left({{2l+2}\atop 2}\right)-(3/2)l(l+1)={l+2\choose 2},
\quad{\rm hence}\quad
|b_k(\gamma)|=\sum_{l=0}^k {l+2\choose 2}=\left({{k+3}\atop 3}\right),
$$
so indeed $|b_k(\gamma)|=\dim S_k$.
Next we determine the number of triples $(v,f_2,f_3)\in
b_k(\gamma)=\stackrel{.}{\cup}c_l(\gamma)$.
First we will count such triples in $c_l(\gamma)$.
Let $a:=2v\in{\bf Z}_{\geq 0}$ with $a\leq l$, then
there are $2l-a+1$ couples $(b,c)$ with
$a+b+c=2l$. Of these, only $a+1$ have $b,\,c\leq l$
(they are $(l-a,l),\;(l-a+1,l-1),\ldots,(l,l-a)$).
This gives $a+1$ such triples in $c_l(\gamma)$ provided $f_1=a/2\leq l/2$.
Thus $v$ occurs for $l/2=v,\,v+1/2,\ldots,k/2$, that is for $k-2v+1$
values of $l/2$.
Thus there are $(k-2v+1)(2v+1)$ triples $(v,f_2,f_3)$.
\noindent
(2)$\quad$
We now consider the case of a seperating vanishing cycle.
The recipe we folow is not explicitly given in \cite{K2}, but is similar
to the one used above and gives a result that agrees with
Lemma \ref{xq}.
Let $\delta$ be the graph with two vertices $v_1,\,v_2$
and three edges $e_1,\,e_2,\,e_3$ such that
begin and end of $e_i$ is $v_i$ and where $e_3$ connects $v_1$ and $v_2$.
Thus the edge $e_3$ corresponds to a seperating vanishing cycle.
As before, let $f_i:=f(e_i)$. The second set of inequalities on the $f(e_i)$
reduce to
$$
f_3\leq 2f_1,\qquad f_3\leq 2f_2;\qquad{\rm moreover}\quad
f_1+f_1+f_3\in{\bf Z}\Rightarrow
f_3\in {\bf Z}\cap\{0,1/2,\ldots,k/2\},
$$
this is the third condition for the vertex $v_1$. Finally we have
$2f_i+f_3\leq k$ so $f_3/2\leq f_i\leq (k-f_3)/2$ for $i=1,\,2$.
Thus given $f_3$, we find $(k-2f_3)+1)^2$
possibilities for couples $f_1,\,f_2$ ($f_i\in\{
{{f_3}\over2},{{f_3+1}\over2},\ldots,
{{k-f_3}\over2}\}$).
We observe that for fixed $f_1$, $f_3$ may have the values
$0,1,\ldots,2f_1$,
correspondingly $f_2$ has $k+1,\; k+1-2,\ldots, k+1-4f_1$ different
values.
Thus the multiplicity of $f_1$ is, again,
$(2f_1+1)((k+1)-2(1+2+\ldots+2f_1)=(2f_1+1)(k+1-2f_1)$.
\end{proof}
\subsubsection{The cases $k=1,\;2$}
The case $k=1$ is particularly easy since the operators $\Omega_{ij}$ are
homogeneous of degree two and thus act as zero on $S_1$. The monodromy
representation then factors over the projective representation of
$A(G)$.
In case $k=2$, the space $S_k$ is a direct sum of 10 distinct, one dimensional
$G$-representations (this is easy to verify, see also \cite{vG2}).
On each eigenspace, any $\sigma\tilde{\rho_s}(\Omega_{ij})$ acts as
scalar multiplication by an integer. A (local) basis of flat sections
of $S_2\otimes{\cal P}$ is then given by 10 functions of the type
$(\prod(z_i-z_j)^{r_{ijm}})Q_m$ with $r_{ij}\in{\bf Q}$ and $Q_m\in S_2$ is a
basis of the eigenspace.
These eigenspaces
are permuted by the projective representation of $A(G)$.
|
1997-01-09T10:33:40 | 9701 | alg-geom/9701005 | fr | https://arxiv.org/abs/alg-geom/9701005 | [
"alg-geom",
"math.AG"
] | alg-geom/9701005 | Jean-Marc Drezet | J.-M. Drezet | Quotients par des groupes non r\'eductifs et vari\'et\'es de modules de
complexes | 46 pages, in French, LaTeX | International Journal of Mathematics Vol. 9 No 7 (1998) , 769-819 | null | null | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ | I give a way to construct moduli spaces of complexes, as quotients of open
subsets of the space of all complexes by the product of the groups of
automorphisms of the terms.
| [
{
"version": "v1",
"created": "Thu, 9 Jan 1997 09:32:39 GMT"
},
{
"version": "v2",
"created": "Tue, 2 Jun 2015 08:56:58 GMT"
}
] | 2015-06-03T00:00:00 | [
[
"Drezet",
"J. -M.",
""
]
] | alg-geom | \section{Introduction}
\subsection{Le probl\`eme des quotients alg\'ebriques par des groupes non
r\'eductifs}
Soit $G$ un groupe alg\'ebrique lin\'eaire de radical unipotent $H$. On
suppose qu'il existe un sous-groupe alg\'ebrique r\'eductif \m{G_{red}} de $G$
dont l'inclusion dans $G$ induit un isomorphisme \ \m{G_{red}\simeq G/H}.
Soit $Y$ une vari\'et\'e alg\'ebrique projective munie d'une action
alg\'ebrique de $G$, et $L$ un $G$-fibr\'e en droites tr\`es ample sur $Y$.
On dit qu'un point $y$ de $Y$ est {\em semi-stable} relativement \`a $L$
s'il existe un entier \ \m{k>0} \ et une section $G$-invariante $s$ de \m{L^k}
telle que \ \m{s(y)\not =0}. Soit \m{Y^{ss}(L)} l'ouvert $G$-invariant de Y
constitu\'e des points $G$-semi-stables relativement \`a $L$. La construction
d'un {\em bon quotient} \m{Y^{ss}(L)//G} est possible dans le cas o\`u
\ \m{H=\lbrace 0\rbrace}, le groupe $G$ \'etant dans ce cas r\'eductif (cf \cite{mumf},
\cite{news}). Le cas o\`u $G$ n'est pas r\'eductif est plus difficile, et
a \'et\'e abord\'e par A. Fauntleroy dans \cite{faunt}. On doit consid\'erer
un ouvert $G$-invariant plus petit de \m{Y^{ss}(L)//G} (en imposant des
conditions suppl\'ementaires qui d\'ependent essentiellement de l'action de
$H$) et les quotients obtenus sont en g\'en\'eral seulement des {\em quotients
cat\'egoriques}. De plus, la d\'efinition de l'ouvert \`a quotienter est peu
explicite. Ces restrictions s'expliquent sans doute par la grande
g\'en\'eralit\'e des probl\`emes trait\'es dans \cite{faunt}. On propose ici
une d\'efinition l\'eg\`erement diff\'erente de la semi-stabilit\'e :
\bigskip
\begin{defin}
On dit qu'un point $y$ de $Y$ est {\em $G$-semi-stable} (resp.
{\em $G$-stable})
relativement \`a $L$ si tout point de l'orbite \m{Hy} est
\m{G_{red}}-semi-stable
(resp. \m{G_{red}}-stable) relativement \`a $L$ (vu comme un
\m{G_{red}}-fibr\'e en droites).
\end{defin}
Il est clair que les points $G$-semi-stables au sens de cette d\'efinition le
sont aussi au sens pr\'ec\'edent. Cette d\'efinition me semble plus explicite,
car les points $G_{red}$-semi-stables peuvent en g\'en\'eral \^etre
d\'etermin\'es \`a l'aide de crit\`eres num\'eriques (cf. \cite{mumf}).
\bigskip
\subsection{Espaces de complexes}
On s'int\'eresse dans cet article \`a un type particulier d'action. Soient $X$
une vari\'et\'e alg\'e-\break brique projective, \m{p\geq 1} \ un entier,
\m{n_0,\ldots,n_p} des entiers positifs, et pour \ \m{0\leq i\leq p},
\m{1\leq j\leq n_i}, \ \m{{\cal E}^{(i)}_j} un faisceau coh\'erent sur $X$ et
\m{M^{(i)}_j} un espace vectoriel non nul de dimension finie. On pose, pour
\m{0\leq i\leq p}
$${\cal E}_i \ = \ \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq j\leq n_i}({\cal E}^{(i)}_j\otimes M^{(i)}_j).$$
On suppose que les faisceaux \m{{\cal E}^{(i)}_j} sont simples, et que
$$\mathop{\rm Hom}\nolimits({\cal E}^{(i)}_j, {\cal E}^{(i')}_{j'}) \ = \ \lbrace 0\rbrace$$
si \ \m{i>i'}, ou \ \m{i=i'}, \m{j>j'}. Soit \m{{\cal W}} la vari\'et\'e des
complexes
$${\cal E}_0\longrightarrow{\cal E}_1\longrightarrow\ldots\longrightarrow{\cal E}_p,$$
sur laquelle op\`ere le groupe alg\'ebrique
$$G \ = \ \mathop{\rm Aut}\nolimits({\cal E}_0)\times\ldots\times\mathop{\rm Aut}\nolimits({\cal E}_p).$$
Le sous-groupe unipotent $H$ est constitu\'e des \m{(g_0,\ldots,g_p)} tels que
pour tous $i$, $j$, la composante
$${\cal E}^{(i)}_j\otimes M^{(i)}_j\longrightarrow{\cal E}^{(i)}_j\otimes M^{(i)}_j$$
de \m{g_i} soit l'identit\'e. Le sous-groupe r\'eductif \m{G_{red}} est
constitu\'e des \m{(g_0,\ldots,g_p)} tels que pour tous $i$,$j$ ont ait
$$g_i({\cal E}^{(i)}_j\otimes M^{(i)}_j)\subset{\cal E}^{(i)}_j\otimes M^{(i)}_j.$$
On a un isomorphisme
$$G_{red} \ \simeq \ \prod_{0\leq i\leq p,1\leq j\leq n_i} GL(M^{(i)}_j).$$
Si on veut retrouver une action sur une vari\'et\'e projective, il convient de
consid\'erer plut\^ot la vari\'et\'e projective \m{\projx{{\cal W}}} sur laquelle
op\`ere le groupe \m{G/\cx{*}}.
L'action de \m{G_{red}} sur \m{{\cal W}} est un cas particulier des actions
\'etudi\'ees par A. King dans \cite{king}. Une lin\'earisation de l'action de
\m{G/\cx{*}} sur \m{\projx{{\cal W}}} est d\'efinie par une suite
\noindent
\ \m{\Lambda=(\lambda_{ij})_{0\leq i\leq p,1\leq j\leq n_i}} \ de nombres
rationnels non nuls telle que
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{0\leq i\leq p,1\leq j\leq n_i}\lambda_{ij}\dim(M^{(i)}_j) \ = \ 0.$$
On appelle $\Lambda$ une {\em polarisation} de l'action de \m{G} sur \m{{\cal W}}.
Un complexe
$${\cal E}_0 \ \hfl{f_0}{} \ {\cal E}_1 \ \hfl{f_1}{} \ \ldots \ \hfl{f_{p-1}}{} \ {\cal E}_p$$
est \m{G_{red}}-semi-stable (resp. \m{G_{red}}-stable) relativement \`a
\m{\Lambda} si et seulement si pour tous sous-espaces vectoriels
$${M'}^{(i)}_j \ \subset M^{(i)}_j,$$
avec \ \m{({M'}^{(i)}_j)\not = (\lbrace 0\rbrace)} ou \m{( M^{(i)}_j)}, tels que
$$f_i(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq j\leq n_i}({\cal E}^{(i)}_j\otimes {M'}^{(i)}_j)) \ \subset
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq j\leq n_{i+1}}({\cal E}^{(i+1)}_j\otimes {M'}^{(i+1)}_j))$$
pour \ \m{0\leq i<p}, on a
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{0\leq i\leq p,1\leq j\leq n_i}\lambda_{ij}\dim({M'}^{(i)}_j)
\ \leq \ 0 \ \ \ \ {\rm (resp. } \ < \ {\rm )}.$$
Le probl\`eme de la construction de quotients par $G$ d'ouverts de ${\cal W}$
a \'et\'e abord\'e dans \cite{dr_tr} et \cite{dr2}, dans le cas \ \m{p=1}
(les complexes sont alors en fait des morphismes).
On va donner ici une m\'ethode de construction de quotients par $G$ d'ouverts
$G$-invariants de ${\cal W}$, qui est une g\'en\'eralisation de la m\'ethode
employ\'ee dans \cite{dr_tr}.
On donnera en particulier des exemples de quotients
par $G$ de l'ouvert des points $G$-semi-stables. On ne peut pas obtenir en
g\'en\'eral des {\em bons quotients}, mais ce qu'on appelle des {\em
quasi-bons quotients} (cf. chapitre 2).
\subsection{Mutations constructives}
Pour construire des bons quotients par $G$ d'ouverts $G$-invariants de ${\cal W}$,
on introduit une nouvelle vari\'et\'e de complexes \m{{\cal W}'}, sur laquelle le
groupe \m{G'} qui agit est r\'eductif. On tente ensuite d'\'etablir une relation
entre les quotients par \m{G'} d'ouverts \m{G'}-invariants de \m{{\cal W}'} et les
quotients par $G$ d'ouverts $G$-invariants de ${\cal W}$.
La m\'ethode est bas\'ee sur le r\'esultat suivant (cf. \paragra~\hskip -2pt 3) : soient
\m{{\cal E}}, ${\cal F}$, ${\cal G}$, $\Gamma$ des faisceaux coh\'erents sur $X$ et $M$ un
espace vectoriel de dimension finie. On suppose que le morphisme d'\'evaluation
$$\eva{\Gamma}{{\cal G}}$$
est surjectif. Soit \m{{\cal E}'} son noyau. On suppose aussi que la composition
$$\mathop{\rm Hom}\nolimits({\cal E},\Gamma)\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal G})\longrightarrow\mathop{\rm Hom}\nolimits({\cal E},{\cal G})$$
est surjective. Soit
$$(1) \ \ \ \ \ \ {\cal E} \ \hfl{A}{} \ (\Gamma\otimes M)\oplus{\cal G} \ \hfl{B}{} \ {\cal F}$$
un complexe. Alors on peut associer \`a \m{(1)} un complexe
$$(2) \ \ \ \ \ \
{\cal E}\oplus{\cal E}' \ \hfl{\alpha}{} \ \Gamma\otimes N \ \hfl{\beta}{} \ {\cal F},$$
avec
$$N \ = \ \mathop{\rm Hom}\nolimits(\Gamma,{\cal G})\oplus M.$$
et tel que
$$\ker(\alpha)\simeq\ker(A), \ \ \
\ker(\beta)/\mathop{\rm Im}\nolimits(\alpha)\simeq\ker(B)/\mathop{\rm Im}\nolimits(A), \ \ \
\mathop{\rm coker}\nolimits(\beta)\simeq\mathop{\rm coker}\nolimits(\alpha).$$
La r\'eciproque est aussi vraie, si on part d'un complexe \m{(2)} tel que
$\alpha$ induise une injection
$$\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)\longrightarrow N,$$
et les deux transformations sont inverses l'une de l'autre (\`a l'action
pr\`es des groupes d'automorphismes des complexes). Le passage de \m{(1)} \`a
\m{(2)} consiste \`a effectuer une {\em mutation \`a gauche} de ${\cal G}$. Cette
notion a \'et\'e introduite dans l'\'etude des {\em fibr\'es exceptionnels } (cf. \cite{dr_lp},
\cite{dr1}, \cite{go_ru}). On notera \m{{\cal W}_0} (resp. \m{{\cal W}'_0}) l'espace des
complexes \m{(1)} (resp. \m{(2)}), et \m{G_0} (resp. \m{G'_0}) le groupe
alg\'ebrique agissant sur \m{{\cal W}_0} (resp. \m{{\cal W}'_0}).
La transformation qui fait passer de \m{(1)} \`a \m{(2)} est purement
formelle (cf. \paragra~\hskip -2pt 3.2). Le complexe \m{(2)} associ\'e \`a \m{(1)} n'est pas en
g\'en\'eral unique, mais sa \m{G'_0}-orbite l'est, et ne d\'epend que de la
\m{G_0}-orbite de \m{(1)}. On obtient ainsi (sous certaines hypoth\`eses) une
bijection
$${\cal W}_0/G_0 \ \simeq U_0/G'_0,$$
\m{U_0} d\'esignant l'ouvert de \m{{\cal W}'_0} constitu\'e des complexes \m{(2)} tels
que \m{\alpha} induise une injection \ \m{\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)\longrightarrow N}. Cette
correspondance est {\em alg\'ebrique} dans le sens suivant : si un ouvert
invariant d'un des espaces de complexes admet un quasi-bon quotient, l'ouvert
correspondant de l'autre c\^ot\'e admet aussi un quasi-bon quotient, et les deux
quotients sont isomorphes. La bijection pr\'ec\'edente est en fait un
{\em quasi-isomorphisme fort} (cf. chapitre 2). Cela entraine que certaines
propri\'et\'es v\'erifi\'ees par un quasi-bon quotient d'un ouvert invariant
d'un des espaces de complexes seront automatiquement v\'erifi\'ees le
quotient de l'ouvert correspondant de l'autre espace de complexes (cf. \paragra~\hskip -2pt
2.3, concernant la descente sur les quotients de fibr\'es vectoriels).
On a aussi une notion similaire de {\em mutation \`a droite}.
Pour appliquer ce qui pr\'ec\`ede aux vari\'et\'es de complexes de type
$${\cal E}_0\longrightarrow{\cal E}_1\longrightarrow\ldots\longrightarrow{\cal E}_p$$
on proc\`ede de la fa\c con suivante : on \'ecrit le dernier terme
$${\cal E}_p \ = (\Gamma\otimes M)\oplus{\cal G},$$
avec
$$\Gamma={\cal E}^{(p)}_1, \ \ M=M^{(p)}_1, \ \
{\cal G}=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_p}({\cal E}^{(p)}_j\otimes M^{(p)}_j).$$
Dans ce cas, on a
$${\cal E}' \ = \ \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_p}({{\cal E}'}^{(p)}_j\otimes M^{(p)}_j),$$
\m{{{\cal E}'}^{(p)}_j} d\'esignant le noyau du morphisme d'\'evaluation
$$\eva{{\cal E}^{(p)}_1}{{\cal E}^{(p)}_j}$$
(suppos\'e surjectif). Les complexes obtenus par mutation sont du type
$${\cal E}_0\longrightarrow{\cal E}_1\longrightarrow\ldots\longrightarrow{\cal E}'_{p-1}\longrightarrow{\cal E}^{(p)}_1\otimes N,$$
avec
$$N \ = \ M^{(p)}_1\oplus\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_p}
(\mathop{\rm Hom}\nolimits({\cal E}^{(p)}_1, {\cal E}^{(p)}_j)\otimes M^{(p)}_j)\biggr)$$
et
$${\cal E}'_{p-1} \ = \ {\cal E}_{p-1}\oplus\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_p}
({{\cal E}'}^{(p)}_j\otimes M^{(p)}_j)\biggr).$$
On peut continuer en \'ecrivant
$${\cal E}'_{p-1} \ = (\Gamma\otimes M)\oplus{\cal G},$$
avec
$$\Gamma={\cal E}^{(p-1)}_1, \ \ M=M^{(p-1)}_1, \ \
{\cal G}=\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_{p-1}}({\cal E}^{(p-1)}_j\otimes M^{(p-1)}_j)\biggr)\oplus
\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_p}({{\cal E}'}^{(p)}_j\otimes M^{(p)}_j)\biggr).$$
On peut ainsi proc\'eder \`a
$$q \ = \mathop{\hbox{$\displaystyle\sum$}}\limits_{0\leq i\leq p}n_i$$
mutations successives, et on obtient finalement un complexe du type
$${\cal F}_0\otimes N_0\longrightarrow\ldots\longrightarrow{\cal F}_q\otimes N_q,$$
o\`u le groupe \m{G'} qui op\`ere est r\'eductif. Il existe un autre chemin
possible, en effectuant des mutations \`a droite en partant du terme de gauche.
Supposons fix\'ee une polarisation \m{\Lambda} de l'action de $G$ sur ${\cal W}$. On
peut alors d\'efinir naturellement une polarisation \m{\Lambda'} de l'action
de \m{G'} sur l'espace \m{{\cal W}'} des complexes pr\'e-\break c\'edents. Il reste
\`a \'etudier les relations qu'il y a entre la \m{G}-(semi-)stabilit\'e des
complexes de \m{{\cal W}} relativement \`a \m{\Lambda}, et la
\m{G'}-(semi-)stabilit\'e
des complexes de \m{{\cal W}'} relativement \`a \m{\Lambda'}. Il est toujours vrai
que si la mutation est \m{G'}-(semi-)stable relativement \`a \m{\Lambda'}, le
complexe d'origine est \m{G}-(semi-)stable relativement \`a \m{\Lambda}. La
r\'eciproque est vraie si on impose des conditions \`a \m{\Lambda}. Il faut
ensuite montrer que tous les complexes \m{G'}-semi-stables de \m{{\cal W}'} sont
(\`a l'action de \m{G'} pr\`es) des mutations de complexes de ${\cal W}$. Cela n'est
vrai que si on impose encore d'autres conditions \`a \m{\Lambda}. On obtient
alors l'existence d'un quasi-bon quotient projectif de l'ouvert des points
$G$-semi-stables de ${\cal W}$. En consid\'erant les mutations \`a droite, on
obtient g\'en\'eralement d'autres valeurs de $\Lambda$ pour lesquelles il
existe un quasi-bon quotient projectif.
\subsection{Vari\'et\'es de modules de complexes}
On consid\`ere dans le \paragra~\hskip -2pt 4 des complexes du type
$$(3) \ \ \ \ \ \
{\cal E}_1\otimes L_1\longrightarrow({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2)\longrightarrow{\cal G}_1\otimes N_1,$$
o\`u \m{{\cal E}_1}, \m{{\cal F}_1}, \m{{\cal F}_2}, \m{{\cal G}_1} sont des faisceaux coh\'erents
simples sur $X$, et \m{L_1}, \m{M_1}, \m{M_2}, \m{N_1} des espaces vectoriels de
dimension finie. Quelques hypoth\`eses doivent \^etre faites, notamment que
le morphisme d'\'evaluation
$$\eva{{\cal F}_1}{{\cal F}_2}$$
est surjectif. On note \m{{\cal H}_1} son noyau. En effectuant une premi\`ere
mutation \`a gauche on associe au complexe \m{(3)} un complexe
$$(4) \ \ \ \ \ \
({\cal E}_1\otimes L_1)\oplus({\cal H}_1\otimes M_2)\longrightarrow{\cal F}_1\otimes P_1\longrightarrow{\cal G}_1\otimes N_1,$$
avec
$$P_1 \ = \ (\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2)\oplus M_1.$$
On suppose ensuite que le morphisme d'\'evaluation
$$\eva{{\cal E}_1}{{\cal H}_1}$$
est surjectif. Soit \m{{\cal K}_1} son noyau. En effectuant une seconde mutation
on obtient un complexe
$$(5) \ \ \ \ \ \
{\cal K}_1\otimes M_2\longrightarrow{\cal E}_1\otimes Q_1\longrightarrow{\cal F}_1\otimes P_1\longrightarrow{\cal G}_1\otimes N_1,$$
avec
$$Q_1 \ = \ (\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M_2)\oplus L_1,$$
et ici le groupe qui op\`ere est r\'eductif.
On en d\'eduit dans le th\'eor\`eme 4.4 l'existence de quasi-bons quotients
projectifs d'ou-\break
verts de complexes $G$-semi-stables de type \m{(3)} (pour
certaines polarisations).
Dans le \paragra~\hskip -2pt 4.5 on \'etudie le cas des complexes
$${\cal O}(-2)\longrightarrow({\cal O}(-1)\otimes M_1)\oplus{\cal O}\longrightarrow{\cal O}(1)$$
sur \m{\proj{n}}, \m{M_1} \'etant un espace vectoriel tel que \
\m{0<\dim(M_1)<n+1}. Dans le cas de \m{\proj{2}} et \ \m{\dim(M_1)=3}, on
obtient trois types de quotients distincts, dont deux lisses. Un des quotients
lisses peut \^etre obtenu de mani\`ere \'el\'ementaire. Le second est non
trivial.
\subsection{Vari\'et\'es de modules de morphismes}
Dans le \paragra~\hskip -2pt 5 on consid\`ere des morphismes du type
$$({\cal E}_1\otimes M_1)\oplus({\cal E}_2\otimes M_2)\longrightarrow{\cal F}_1\otimes N_1,$$
o\`u \m{{\cal E}_1}, \m{{\cal E}_2}, \m{{\cal F}_1} sont des faisceaux coh\'erents simples sur
$X$, et \m{M_1}, \m{M_2}, \m{N_1} des espaces vectoriels de dimension finie.
On rappelle dans les \paragra~\hskip -2pt 5.1 et 5.2 la construction de vari\'et\'es de
modules de morphismes $G$-semi-stables de ce type (pour certaines polarisations)
effectu\'ee dans \cite{dr_tr}. On proc\`ede simplement ici \`a une seule
mutation \`a gauche pour obtenir une action d'un groupe r\'eductif. Les
quotients obtenus par cette m\'ethode sont des bons quotients projectifs,
et les ouverts
correspondant aux morphismes $G$-stables sont des quotients g\'eom\'etriques.
Dans le \paragra~\hskip -2pt 5.3 on emploie des mutations \`a droite. Il faut alors deux
mutations successives pour obtenir une action d'un groupe r\'eductif. On
obtient des quasi-bons quotients de l'ouvert des morphismes $G$-semi-stables,
pour
d'autres polarisations qu'avec la m\'ethode pr\'ec\'edente.
Dans la \paragra~\hskip -2pt 5.4 on donne des exemples de constructions de vari\'et\'es de
modules de morphismes au moyen des m\'ethodes de \cite{dr_tr}, \cite{dr2}
et du \paragra~\hskip -2pt 5.3. On donne un cas o\`u il n'y a pas de quotient
g\'eom\'etrique de l'ouvert des points stables.
\subsection{Mutations non constructives}
D'autres sortes de mutations ont \'et\'e d\'efinies dans \cite{dr2}. On
pourrait les appeler des {\em mutations non constructives}. Dans \cite{dr2}
elles sont appliqu\'ees \`a des morphismes, mais on peut sans difficult\'e
\'etendre leur d\'efinition aux complexes. Elles peuvent aussi servir \`a
cons-\break truire des bons quotients d'ouverts de points $G$-semi-stables, pour
d'autres polarisations que celles qui sont accessibles par les m\'ethodes
d\'ecrites ici. La d\'efinition des mutations non constructives est bas\'ee sur
le r\'esultat suivant : soient \m{{\cal E}}, \m{{\cal E}'}, \m{\Gamma}, \m{{\cal G}} et \m{{\cal F}}
des faisceaux coh\'erents sur $X$. On suppose que le morphisme canonique
$$\coeva{{\cal E}'}{\Gamma}$$
est injectif et on note \m{{\cal F}_0} son conoyau. On suppose aussi que
$$\mathop{\rm Ext}\nolimits^1({\cal F}_0,{\cal F})=\mathop{\rm Ext}\nolimits^1({\cal E},\Gamma)=\mathop{\rm Ext}\nolimits^1(\Gamma,{\cal F})=\lbrace 0\rbrace.$$
Soient $M$ un espace vectoriel de dimension finie et
$${\cal E}\oplus{\cal E}' \ \hfl{A}{} \ (\Gamma\otimes M)\oplus{\cal F} \ \hfl{B}{} \ {\cal G}$$
un complexe tel que l'application lin\'eaire
$$\lambda : \mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\longrightarrow M$$
d\'eduite de $A$ soit surjective. Alors il existe un complexe
$${\cal E}\oplus(\Gamma\otimes\ker(\lambda)) \ \hfl{\alpha}{} \ {\cal F}_0\oplus{\cal F} \
\hfl{\beta}{} \ {\cal G}$$
tel que
$$\ker(\alpha)\simeq\ker(A), \ \ \
\ker(\beta)/\mathop{\rm Im}\nolimits(\alpha)\simeq\ker(B)/\mathop{\rm Im}\nolimits(A), \ \ \
\mathop{\rm coker}\nolimits(\beta)\simeq\mathop{\rm coker}\nolimits(B).$$
Ce r\'esultat admet aussi une r\'eciproque.
Le diff\'erence essentielle entre les mutations constructives et les
mutations non cons-\break tructives est la suivante : dans le premier cas, on
effectue la mutation d'une paire de faisceaux situ\'es dans le m\^eme
terme du complexe, et dans le second cas on effectue la mutation d'une paire
de faisceaux situ\'es dans des termes adjacents du complexe.
\vfil\eject
\section{Quotients alg\'ebriques}
\subsection{Quasi-bons quotients}
Soit $G$ un groupe alg\'ebrique. On appelle {\em $G$-espace} une vari\'et\'e
alg\'ebrique $X$ munie d'une action alg\'ebrique de $G$. Rappelons qu'on
appelle {\em bon quotient} de $X$ par $G$ un morphisme
$$\pi : X\longrightarrow M$$
(o\`u $M$ est une vari\'et\'e alg\'ebrique) tel que :
\noindent (i) Le morphisme $\pi$ est $G$-invariant, affine, ouvert et surjectif.
\noindent (ii) Si $U$ est un ouvert de $M$, alors on a
$${\cal O}(U) \ \simeq \ {\cal O}(\pi^{-1}(U))^G.$$
\noindent (iii) Si $F_1$, $F_2$ sont des sous-vari\'et\'es ferm\'ees
$G$-invariantes disjointes de $X$, alors \m{\pi(F_1)} et \m{\pi(F_2)} sont des
sous-vari\'et\'es disjointes de $M$.
\medskip
Cette d\'efinition est particuli\`erement
bien adapt\'ee \`a l'\'etude des actions de groupes alg\'ebriques r\'eductifs,
car on sait que dans ce cas il existe toujours un bon quotient si $X$ est
affine. On utilisera une notion l\'eg\`erement diff\'erente :
\bigskip
\begin{defin}
On appelle {\em quasi-bon quotient} de $X$ par $G$ un morphisme
$$\pi : X\longrightarrow M$$
(o\`u $M$ est une vari\'et\'e alg\'ebrique) qui est $G$-invariant, ouvert et
surjectif et tel que les conditions (ii) et (iii) pr\'ec\'edentes soient
v\'erifi\'ees.
\end{defin}
\bigskip
On dit parfois par abus de language que $M$ est le quasi-bon quotient de $X$
par $G$. On note comme dans le cas des bons quotients
$$M \ = \ X//G.$$
Il est clair qu'un bon quotient est un quasi-bon quotient.
\bigskip
\subsection{Quasi-isomorphismes}
Soient $G$, $G'$ des groupes alg\'ebriques, $X$ un $G$-espace et $X'$ un
\m{G'}-espace.
\bigskip
\begin{defin}
1 - On appelle {\em quasi-morphisme} de $X$ vers $X'$ une application
$$\phi : X/G\longrightarrow X'/G'$$
telle que pour tout point $x$ de $X$ il existe un ouvert de Zariski $U$ de $X$
contenant $x$ et un morphisme \ \m{U\longrightarrow X'} \ induisant $\phi$.
\medskip
\noindent 2 - On appelle {\em quasi-morphisme fort} de $X$ vers $X'$ la
donn\'ee d'un {\em quasi-morphisme}
$$\phi : X/G\longrightarrow X'/G',$$
d'un recouvrement ouvert \m{(U_i)_{i\in I}} de X, et d'une famille
\m{(\phi_i)_{i\in I}} de rel\`evements de \m{\phi} :
$$\phi_i : U_i\longrightarrow X',$$
telle que :
\noindent (i) pour tout \m{j\in I} et \m{g\in G}, le morphisme
$$gU_j\longrightarrow X'$$
$$x\longmapsto \phi_j(g^{-1}x)$$
appartient \`a la famille \m{(\phi_i)}.
\noindent (ii) Pour tous \ \m{i,j\in I}, et tout \ \m{x\in U_i\cap U_j} \
il existe un voisinage $V$ de $x$ dans \m{U_i\cap U_j} et un morphisme \
\m{\lambda_{ij} : V\longrightarrow G'} \ tel que
$${\phi_j}_{\mid V} \ = \ \lambda_{ij}{\phi_i}_{\mid V},$$
\noindent (iii) Pour tout \ \m{i\in I} \ et \ \m{x\in U_i}, il existe un
voisinage $V$ de \m{(e,x)} dans \ \m{G\times U_i} \ et un morphisme
$$\gamma : V\longrightarrow G'$$
tel que \ \m{\gamma(e,x)=e} \ et que pour tous \ \m{(g,y)\in V}, on
ait \ \m{gy\in U_i} \ et
$$\phi_i(gy)=\gamma(g,y)\phi_i(y).$$
\end{defin}
\bigskip
On appelle {\em quasi-isomorphisme} de $X$ vers $X'$ une bijection
$$\phi : X/G\longrightarrow X'/G'$$
qui est un quasi-morphisme ainsi que son inverse.
\bigskip
\begin{defin}
Soit \ \m{\sigma = (\phi : X/G\longrightarrow X'/G', (\phi_i : U_i\longrightarrow X'))} \ un
quasi-morphisme fort. On appelle {\em carte} de \m{\sigma} un rel\`evement
local
\ \m{f : U\longrightarrow X'} de \m{\phi}
($U$ \'etant un ouvert non vide de $X$) tel que pour tout
\m{x\in U} les propri\'et\'es suivantes soient v\'erifi\'ees :
\noindent (i) Pour tout \m{i\in I} il existe un voisinage $V$ de $x$ dans
$U\cap U_i$ et un morphisme
\m{\lambda_{i} : V\longrightarrow G'} \ tel que
$${f}_{\mid V} \ = \ \lambda_{i}{\phi_i}_{\mid V},$$
\noindent (ii) Il existe un voisinage $V$ de
\m{(e,x)} dans \ \m{G\times U} \ et un morphisme
$$\gamma : V\longrightarrow G'$$
tel que \ \m{\gamma(e,x)=e} \ et que pour tous \ \m{(g,y)\in V}, on
ait \ \m{gy\in U} \ et
$$\phi_i(gy)=\gamma(g,y)\phi(y).$$
\end{defin}
\bigskip
On dit que deux quasi-morphismes forts de $X$ dans \m{X'} sont {\em
\'equivalents} si les cartes de l'un sont aussi des cartes de l'autre.
\bigskip
Il est clair que la composition de deux quasi-morphismes en est un. On va
d\'efinir ce qu'on entend par {\em composition} de deux quasi-morphismes forts.
Soient $G$, \m{G'}, \m{G''} des groupes alg\'ebriques, \m{X} un $G$-espace,
\m{X'} un \m{G'}-espace et \m{X''} un \m{G''}-espace. Soient
$$\sigma = (\phi : X/G\longrightarrow X'/G', (\phi_i : U_i\longrightarrow X')_{i\in I}),$$
$$\sigma' = (\phi' : X'/G\longrightarrow X''/G'', (\phi'_j : U'_j\longrightarrow X'')_{j\in J})$$
des quasi-morphismes forts. Soient
$$\psi=\phi'\circ\phi : X/G\longrightarrow X''/G'',$$
et pour \m{i\in I}, \m{j\in J}, \ \m{V_{ij}=U_i\cap\phi^{-1}(U'_j)}, et
$$\psi_{ij} = \phi'_j\circ\phi_i : V_{ij}\longrightarrow X''.$$
On pose
$$\tau = (\psi : X/G\longrightarrow X''/G'',
(\psi_{i,j} : V_{ij}\longrightarrow X'')_{(i,j)\in I\times J}).$$
\bigskip
\begin{xlemm}
$\tau$ est un quasi-morphisme fort de $X$ dans $X'$.
\end{xlemm}
\bigskip
\noindent{\em D\'emonstration}. Soit \ \m{x\in V_{ij}}. Pour \m{(g,y)} au voisinage convenable de
\m{(e,x)}, on a
$$\psi_{ij}(gy) = \phi'_j\circ\phi_i(gy)=\phi'_j(\gamma_i(g,y)\phi_i(y))
=\gamma'_j(\gamma_i(g,y),\phi_i(y))\psi_{ij}(y),$$
\m{\gamma_i} (resp. \m{\gamma'_j}) \'etant un morphisme \`a valeurs dans
\m{G'} (resp. \m{G''}) d\'efini sur un voisinage de \m{(e,x)} (resp.
\m{(e,\phi_i(x))}). En posant
$$\gamma_{ij}(g,y) = \gamma'_j(\gamma_i(g,y),\phi_i(y))$$
on obtient donc
$$\psi_{ij}(gy) = \gamma_{ij}(g,y)\psi_{ij}(y).$$
D'autre part, soient \ \m{i,i'\in I},\m{j,j'\in J} \ et \
\m{x\in V_{i,j}\cap V_{i',j'}}. Avec des notations \'evidentes, on a, pour $y$
dans un voisinage de $x$
$$\phi'_{j'}\circ\phi_{i'}(y)=\phi'_{j'}(\lambda_{ii'}(y)\phi_i(y)).$$
Posons
$$g'_0=\lambda_{ii'}(x)\in G'.$$
Alors on a
$$\phi'_{j'}(\lambda_{ii'}(y)\phi_i(y))=\gamma'_{j'}(\lambda_{ii'}(y){g'_0}^{-1},
g'_0\phi_i(y))\phi'_{j'}(g_0\phi_i(y))=\gamma'_{j'}(\lambda_{ii'}(y){g'_0}^{-1},
g'_0\phi_i(y))\phi'_{k}(\phi_i(y))$$
pour un $k$ convenable dans $J$. On obtient finalement
$$\phi'_{j'}\circ\phi_{i'}(y)=\gamma'_{j'}(\lambda_{ii'}(y){g'_0}^{-1},
g'_0\phi_i(y))\lambda'_{kj}(\phi_i(y))\phi'_j\circ\phi_i(y),$$
c'est-\`a-dire
$$\phi'_{j'}\circ\phi_{i'}(y)=\theta_{i'j',ij}(y)\phi'_{j}\circ\phi_{i}(y),$$
avec
$$\theta_{i'j',ij}(y)=\gamma'_{j'}(\lambda_{ii'}(y){g'_0}^{-1},
g'_0\phi_i(y))\lambda'_{kj}(\phi_i(y)).$$
\noindent$\Box$
\bigskip
On d\'efinit de mani\`ere \'evidente le quasi-morphisme fort {\em identit\'e}
\m{I_X} de $X$ dans $X$. Il est clair que la composition \`a droite ou \`a
gauche d'un quasi-morphisme fort avec \m{I_X} donne un quasi-morphisme fort
\'equivalent. On d\'efinit un {\em quasi-isomorphisme fort} $\sigma$ de $X$
dans \m{X'} comme \'etant un quasi-morphisme fort de $X$ dans \m{X'} tel qu'il
existe un quasi-morphisme fort \m{\sigma'} de \m{X'} dans $X$ tel que
\m{\sigma\circ\sigma'} (resp. \m{\sigma'\circ\sigma}) soit \'equivalent \`a
\m{I_{X'}} (resp. \m{I_X}).
\bigskip
\noindent{\bf Exemple : } 1 - Supposons qu'il existe un morphisme
$$\phi : X\longrightarrow X'$$
compatible avec un morphisme de groupes alg\'ebriques
$$\alpha : G\longrightarrow G',$$
c'est-\`a-dire que pour tous \m{g\in G}, \m{x\in X}, on a \
\m{\phi(gx)=\alpha(x)\phi(x)}. Soit
$$\ov{\phi} : X/G\longrightarrow X'/G'$$
l'application d\'eduite de \m{\phi}. Alors \m{(\ov{\phi}, \phi)} est un
quasi-morphisme fort de $X$ dans \m{X'}. C'est l'exemple le plus simple.
\medskip
\noindent 2 - Supposons qu'il existe un quasi-isomorphisme fort \
\m{\sigma : X\longrightarrow X'}, et soit $U$ un ouvert $G$-invariant de $X$.
Soit \m{U'} l'ouvert
\m{G'}-invariant correspondant de \m{X'}. Alors \m{\sigma} induit un
quasi-isomorphisme fort \ \m{U\longrightarrow U'}.
\bigskip
Il est clair que si
$$\phi : X/G\longrightarrow X'/G'$$
est un quasi-isomorphisme (non n\'ecessairement fort),
\m{\phi} induit une bijection entre l'en-\break
semble des ouverts (resp. ferm\'es) $G$-invariants de $X$ et
l'ensemble des ouverts (resp. ferm\'es) \m{G'}-invariants de \m{X'}. Si $U$
est un ouvert $G$-invariant de $X$, et \m{U'} l'ouvert \m{G'}-invariant
correspondant de \m{X'}, \m{\phi} induit un isomorphisme d'anneaux
$${\cal O}(U)^G \ \simeq \ {\cal O}(U')^{G'}.$$
Plus g\'en\'eralement, si $Y$ est une vari\'et\'e alg\'ebrique, les morphismes
$G$-invariants \ \m{X\longrightarrow Y} \ s'identifient de mani\`ere \'evidente aux
morphismes \m{G'}-invariants \ \m{X'\longrightarrow Y}.
\bigskip
\begin{xprop}
Si $X$ et $X'$ sont quasi-isomorphes, il existe un quasi-bon quotient de
$X$ par $G$ si et seulement si il existe un quasi-bon quotient de \m{X'} par
\m{G'}, et dans ce cas les deux quotients sont isomorphes.
\end{xprop}
\bigskip
\noindent{\em D\'emonstration}. Supposons qu'il existe un quasi-bon quotient
$$\pi : X\longrightarrow M$$
de $X$ par $G$. Soit
$$\phi : X/G\longrightarrow X'/G'$$
un quasi-isomorphisme. Puisque \m{\pi} est $G$-invariant, il d\'efinit un
morphisme \m{G'}-invariant
$$\pi' : X'\longrightarrow M.$$
Il est imm\'ediat que c'est un quasi-bon quotient de \m{X'} par \m{G'}.
$\Box$
\bigskip
\noindent{\bf Remarque : } La proposition 2.2 est vraie pour des bons quotients
si on suppose en plus que les vari\'et\'es $X$ et \m{X'} sont affines et si
tout ouvert d'une de ces vari\'et\'es est le compl\'ementaire d'une
hypersurface (c'est le cas par exemple lorsque $X$ et \m{X'} sont
{\em factorielles}, c'est-\`a-dire que leurs anneaux de fonctions r\'eguli\`eres
sont factoriels). En effet dans ce cas les ouverts affines $G$-invariants
(resp. \m{G'}-invariants) de $X$ (resp. \m{X'}) sont alors les
compl\'ementaires des hypersurfaces $G$-invariantes
(resp. \m{G'}-invariantes), et il est ais\'e de voir que via \m{\phi} les
hypersurfaces invariantes de $X$ correspondent exactement \`a celles de
\m{X'}. C'est ce qui se produit dans \cite{dr2}, o\`u $X$ et \m{X'} sont des
ouverts d'espaces affines.
\subsection{Lemme de descente}
On montre ici que la notion de quasi-isomorphisme fort pr\'eserve une
propri\'et\'e importan-\break te : la
possibilit\'e de descendre au quotient des $G$-fibr\'es ad\'equats sur $X$.
Rappelons qu'un {\em $G$-fibr\'e vectoriel} sur $X$ est un fibr\'e vectoriel
alg\'ebrique $F$ sur $X$ muni d'une action alg\'ebrique lin\'eaire de $G$, au
dessus de l'action de $G$ sur $X$. S'il existe un quasi-bon quotient
\ \m{\pi : X\longrightarrow M}, on dit que $F$ {\em descend} \`a $M$ s'il existe un fibr\'e
vectoriel alg\'ebrique $E$ sur $M$ et un isomorphisme de $G$-fibr\'es
$$F \ \simeq \ \pi^*(E).$$
\bigskip
\begin{defin}
On dit qu'un $G$-fibr\'e vectoriel $F$ sur $X$ est {\em admissible} si pour
tout point $x$ de $X$, le stabilisateur de $x$ dans $G$ agit trivialement sur
$F_x$.
\end{defin}
\bigskip
Il est clair que s'il existe un quasi-bon quotient \ \m{M=X//G}, et si $F$
descend \`a $M$, alors $F$ est admissible.
On d\'emontre dans \cite{dr_na} le r\'esultat suivant :
\bigskip
\begin{xlemm}
{\em (Lemme de descente)} Si le groupe $G$ est r\'eductif et s'il existe un bon
quotient \ \m{\pi : X\longrightarrow M}, tout $G$-fibr\'e admissible sur $X$ descend \`a
$M$.
\end{xlemm}
\bigskip
Une justification de la notion de quasi-isomorphisme fort est le r\'esultat
suivant :
\bigskip
\begin{xprop}
On suppose donn\'e un quasi-isomorphisme fort de $X$ dans \m{X'}. Soit $r$
un entier. Alors
\noindent 1 - Il existe une bijection canonique entre l'ensemble des classes
d'isomorphisme de $G$-fibr\'es vectoriels admissibles de rang $r$ sur $X$ et
l'ensemble des classes d'isomorphisme de \m{G'}-fibr\'es vectoriels
admissibles de rang $r$ sur \m{X'}.
\noindent 2 - On suppose qu'il existe un quasi-bon quotient
$$M \ = \ X//G \ = \ X'//G'.$$
Alors un $G$-fibr\'e vectoriel admissible sur $X$ descend \`a $M$ si et
seulement si le \m{G'}--fibr\'e vectoriel admissible correspondant sur \m{X'}
descend \`a $M$, et les fibr\'es vectoriels associ\'es sur $M$ sont les
m\^emes.
\end{xprop}
\bigskip
\noindent{\em D\'emonstration}.
Soit \ \m{(\phi : X/G\longrightarrow X'/G', (\phi_i : U_i\longrightarrow X'))} \ le quasi-isomorphisme
fort.
Soit $F'$ un $G'$-fibr\'e vectoriel admissible sur \m{X'}. On va lui associer
un $G$-fibr\'e vectoriel admissible $F$ sur $X$. On commence par construire
$F$ sans sa structure alg\'ebrique. Soit \ \m{\epsilon : X\longrightarrow X'} \ une
application au dessus de \m{\phi}. On pose
$$F_\epsilon = \epsilon^*(F').$$
Si \ \m{\epsilon' : X\longrightarrow X'} \ est une autre application au dessus de \m{\phi},
il existe une application
$$\theta : X\longrightarrow G'$$
telle que \ \m{\epsilon'=\theta.\epsilon}. Puisque $F'$ est un $G'$-fibr\'e
on obtient un isomorphisme
$$F_\epsilon \simeq F_{\epsilon'}$$
qui en \m{x\in X} est la multiplication par \m{\theta(x)}
$$F'_{\epsilon(x)}\longrightarrow F'_{\epsilon'(x)}.$$
Puisque $F'$ est admissible, l'isomorphisme pr\'ec\'edent est ind\'ependant du
choix de \m{\epsilon}. On peut donc d\'efinir sans ambiguit\'e
$$F \ = \ F_\epsilon.$$
D\'efinissons maintenant l'action de $G$ sur $F$. Soient \m{g\in G}, \m{x\in X}.
Alors \m{\epsilon(x)} et \m{\epsilon(gx)} sont dans la m\^eme \m{G'}-orbite,
donc on a un isomorphisme canonique
$$F_x=F'_{\epsilon(x)}\longrightarrow F'_{\epsilon(gx)}=F_{gx},$$
qui ne d\'epend pas de $\epsilon$. Il est clair qu'on d\'efinit ainsi une
action de $G$ sur $F$, et que le stabilisateur de $x$ agit trivialement sur
\m{F_x}.
On d\'efinit maintenant la structure alg\'ebrique sur $F$. Soient \m{i\in I}
et \m{x\in U_i}. Soit $V$ un voisinage de \m{\phi_i(x)} tel qu'on ait une
trivialisation
$$F'_{\mid V} \ \simeq {\cal O}_V\otimes\cx{r}.$$
On d\'efinit sur \ \m{U=U_i\cap\phi_i^{-1}(V)} \ une trivialisation locale
$$F_U \ \simeq \phi^*(F'_{\mid V}) \simeq {\cal O}_U\otimes\cx{r}.$$
Il faut v\'erifier que ces trivialisations se recollent alg\'ebriquement. Cela
d\'ecoule imm\'e-\break diatement de la condition (ii) de la d\'efinition d'un
quasi-morphisme fort. Le fait que l'action de $G$ est alg\'ebrique d\'ecoule
de la condition (iii).
Les autres assertions sont imm\'ediates et laiss\'ees au lecteur.
\noindent$\Box$
\bigskip
Si $G$ est r\'eductif et s'il existe un bon quotient \ \m{M=X//G},
on en d\'eduit
que tout \m{G'}-espace fortement quasi-isomorphe \`a $X$ v\'erifie le
lemme de descente, c'est-\`a-dire que tout $G'$-fibr\'e vectoriel admissible
sur $X'$ descend \`a $M$.
\vfil\eject
\section{Mutations constructives}
\subsection{Mutations constructives en termes de faisceaux}
\subsubsection{Un exemple simple}
Soient \m{{\cal E}}, \m{{\cal F}}, \m{{\cal G}}, \m{\Gamma} des faisceaux coh\'erents sur une
vari\'et\'e projective \m{X}, \m{M} un espace vectoriel de dimension
finie non nul. On suppose que le morphisme d'\'evaluation
$$\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal G})\longrightarrow{\cal G}$$
est surjectif. Soit \m{{\cal E}'} son noyau. On suppose aussi que la
composition
$$c : \mathop{\rm Hom}\nolimits({\cal E},\Gamma)\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal G})\longrightarrow\mathop{\rm Hom}\nolimits({\cal E},{\cal G})$$
est surjective.
\bigskip
\begin{xprop}
Soit
$${\cal E}\ \hfl{A}{}\ (\Gamma\otimes M)\oplus{\cal G}\ \hfl{B}{}\ {\cal F}$$
un complexe, avec \m{A} injectif et \m{B} surjectif. Alors il existe un
complexe
$${\cal E}\oplus{\cal E}'\ \hfl{\alpha}{}\ \Gamma\otimes N\ \hfl{\beta}{}\ {\cal F},$$
avec
$$N=\mathop{\rm Hom}\nolimits(\Gamma,{\cal G})\oplus M,$$
\m{\alpha} \'etant injectif, \m{\beta} surjectif, et
$$\ker(\beta)/\mathop{\rm Im}\nolimits(\alpha) \ \simeq \ \ker(B)/\mathop{\rm Im}\nolimits(A).$$
\end{xprop}
\medskip
\noindent{\em D\'emonstration}. Le morphisme
$$\alpha : {\cal E}\oplus{\cal E}'\longrightarrow\Gamma\otimes (M\oplus\mathop{\rm Hom}\nolimits(\Gamma,{\cal G}))$$
est la somme d'un morphisme
$${\cal E}\longrightarrow\Gamma\otimes (M\oplus\mathop{\rm Hom}\nolimits(\Gamma,{\cal G}))$$
provenant de \m{A} (qui existe car \m{c} est surjective), et de l'inclusion
$${\cal E}'\longrightarrow\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal G}).$$
On a un diagramme commutatif avec colonnes exactes :
\bigskip
\begin{picture}(360,230)
\put(135,220){$0$}
\put(280,220){$0$}
\put(133,170){${\cal E}'$}
\put(280,170){${\cal E}'$}
\put(120,120){${\cal E}'\oplus{\cal E}$}
\put(230,120){$\Gamma\otimes(M\oplus\mathop{\rm Hom}\nolimits(\Gamma,{\cal G}))$}
\put(135,70){${\cal E}$}
\put(260,70){$(\Gamma\otimes M)\oplus{\cal G}$}
\put(135,20){$0$}
\put(280,20){$0$}
\put(187,126){$\alpha$}
\put(195,76){$A$}
\put(137,214){\vector(0,-1){30}}
\put(137,164){\vector(0,-1){30}}
\put(137,114){\vector(0,-1){30}}
\put(137,64){\vector(0,-1){25}}
\put(282,214){\vector(0,-1){30}}
\put(282,164){\vector(0,-1){30}}
\put(282,114){\vector(0,-1){30}}
\put(282,64){\vector(0,-1){30}}
\put(145,172){\line(1,0){129}}
\put(145,174){\line(1,0){129}}
\put(157,123){\vector(1,0){70}}
\put(145,73){\vector(1,0){112}}
\end{picture}
\bigskip
On en d\'eduit que \m{\alpha} est injectif et que \ \m{\mathop{\rm coker}\nolimits(\alpha) \
\simeq \mathop{\rm coker}\nolimits(A)}. On d\'efinit le morphisme \m{\beta} par le carr\'e
commutatif
\bigskip
\begin{picture}(360,90)
\put(105,80){$\Gamma\otimes(M\oplus\mathop{\rm Hom}\nolimits(\Gamma,{\cal G}))$}
\put(280,80){${\cal F}$}
\put(120,30){$(\Gamma\otimes M)\oplus{\cal G}$}
\put(280,30){${\cal F}$}
\put(242,86){$\beta$}
\put(230,36){$B$}
\put(223,83){\vector(1,0){47}}
\put(200,33){\vector(1,0){70}}
\put(145,74){\vector(0,-1){30}}
\put(284,74){\line(0,-1){30}}
\put(286,74){\line(0,-1){30}}
\end{picture}
\bigskip
On en d\'eduit que \m{\beta} est surjectif, \m{\beta\circ\alpha=0} \ et
$$\ker(\beta)/\mathop{\rm Im}\nolimits(\alpha) \ \simeq \ \ker(B)/\mathop{\rm Im}\nolimits(A).$$
$\Box$
\bigskip
On a bien s\^ur une r\'eciproque :
\begin{xprop}
Soit
$${\cal E}\oplus{\cal E}'\ \hfl{\alpha}{}\ \Gamma\otimes N\ \hfl{\beta}{}\ {\cal F}$$
un complexe, avec \m{\alpha} injectif et \m{\beta} surjectif. On suppose
que l'application lin\'eaire
$$\lambda : \mathop{\rm Hom}\nolimits({\cal E}',\Gamma)\longrightarrow N$$
d\'eduite de \m{\alpha} est injective. Alors il existe un complexe
$${\cal E}\ \hfl{A}{}\ (\Gamma\otimes M)\oplus{\cal G}\ \hfl{B}{}\ {\cal F}$$
avec \ \m{M=\mathop{\rm coker}\nolimits(\lambda)}, \m{A} \'etant injectif, \m{B} surjectif et
$$\ker(B)/\mathop{\rm Im}\nolimits(A) \ \simeq \ \ker(\beta)/\mathop{\rm Im}\nolimits(\alpha).$$
\end{xprop}
\noindent{\em D\'emonstration}. Analogue \`a la proposition pr\'ec\'edente. $\Box$
\bigskip
\subsubsection{Le cas g\'en\'eral}
On se place dans la situation du \paragra~\hskip -2pt 3.1. Soient \m{{\cal U}}, \m{{\cal V}}, \m{{\cal G}_0}
des faisceaux coh\'erents sur \m{X}. On d\'emontre comme la proposition 3.1
la
\bigskip
\begin{xprop}
1 - Soit
$$0\longrightarrow{\cal U}\longrightarrow{\cal E}\ \hfl{A}{}\ (\Gamma\otimes M)\oplus{\cal G}\oplus{\cal G}_0\ \hfl{B}{}\
{\cal F}\longrightarrow{\cal V}\longrightarrow 0$$
un complexe, exact en \m{{\cal U}}, \m{{\cal E}}, \m{{\cal F}}, \m{{\cal V}}. Alors il existe un
complexe
$$0\longrightarrow{\cal U}\longrightarrow{\cal E}\oplus{\cal E}'\ \hfl{\alpha}{}\ (\Gamma\otimes N)\oplus{\cal G}_0\
\hfl{\beta}{}\ {\cal F}\longrightarrow{\cal V}\longrightarrow 0,$$
avec
$$N = \mathop{\rm Hom}\nolimits(\Gamma,{\cal G})\oplus M,$$
exact sauf au plus en \ \m{(\Gamma\otimes N)\oplus{\cal G}_0}, et tel que
$$\ker(\beta)/\mathop{\rm Im}\nolimits(\alpha) \ \simeq \ \ker(B)/\mathop{\rm Im}\nolimits(A).$$
\medskip
\noindent 2 - R\'eciproquement, si \m{N} est un espace vectoriel, et si on
a un complexe du second type exact en \m{{\cal U}}, \m{{\cal E}\oplus{\cal E}'}, \m{{\cal F}},
\m{{\cal V}}, tel que que l'application lin\'eaire
$$\lambda : \mathop{\rm Hom}\nolimits({\cal E}',\Gamma)\longrightarrow N$$
d\'eduite de \m{\alpha} soit injective. Alors il existe un complexe du
premier type, avec
$$M=\mathop{\rm coker}\nolimits(\lambda),$$
exact en \m{{\cal U}}, \m{{\cal E}}, \m{{\cal F}}, \m{{\cal V}}, et tel que
$$\ker(B)/\mathop{\rm Im}\nolimits(A) \ \simeq \ \ker(\beta)/\mathop{\rm Im}\nolimits(\alpha).$$
\end{xprop}
\vfill\eject
\subsection{Mutations constructives abstraites}
On d\'ecrit ici de mani\`ere abstraite la situation de la proposition 3.1
(sans tenir compte de l'injectivit\'e de \m{A} et \m{\alpha} et de la
surjectivit\'e de \m{B} et \m{\beta}). Il est possible de faire la
m\^eme chose dans le cas plus g\'en\'eral de la proposition 3.3.
On \'etudie l'action de certains
groupes d'automorphismes sur l'espace de tous les complexes, et on
\'etudie la relation entre les orbites des deux types de complexes.
Des hypoth\`eses suppl\'ementaires sont faites dans cette version
abstraite, par exemple on suppose que \m{\Gamma} est simple.
\bigskip
\subsubsection{Espaces de complexes de type 1 (version simplifi\'ee)}
\noindent{\bf 3.2.1.1 }{\it D\'efinition}
\medskip
\noindent
Soient \m{Z_1}, \m{Z_2}, \m{Z_3}, \m{Z_4} , \m{H}, \m{T}, \m{M} des espaces
vectoriels de dimension finie. On pose
$$W_C = (Z_1\otimes M)\oplus Z_2\oplus (Z_3\otimes M^*)\oplus Z_4.$$
Soient
$$\sigma : Z_1\otimes H\longrightarrow Z_2,$$
$$\sigma' : H\otimes Z_4\longrightarrow Z_3,$$
$$\tau : Z_1\otimes Z_3\longrightarrow T,$$
$$\tau' : Z_2\otimes Z_4\longrightarrow T$$
des applications lin\'eaires. On suppose que \m{\sigma} est surjective, et
que \m{\sigma'} induit une inclusion \ \m{Z_4\subset H^*\otimes Z_3}. On
suppose aussi que le diagramme suivant est commutatif :
$$(D) \ \ \ \ \ \ \diagram{
Z_1\otimes H\otimes Z_4 & \hfl{\sigma\otimes I_{Z_4}}{} & Z_2\otimes Z_4\cr
\vfl{I_{Z_1}\otimes\sigma'}{} & & \vfl{}{\tau'}\cr
Z_1\otimes Z_3 & \hfl{\tau}{} & T\cr}$$
Soit \ \m{Q_C\subset W_C} \ l'ensemble des points
\m{(\phi_1,z_2,\phi_3,z_4)} tels que
$$\tau(\pline{\phi_1,\phi_3})+\tau'(z_2\otimes z_4)=0,$$
\m{\pline{\phi_1,\phi_3}} d\'esignant l'image de \m{\phi_1\otimes\phi_3} par la
contraction de \m{M}
$$Z_1\otimes M\otimes Z_3\otimes M^*\longrightarrow Z_1\otimes Z_3.$$
Soient \m{G_L}, $G_R$ et $G_0$ des groupes. On suppose que :
\begin{itemize}
\item[] $G_L$ op\`ere lin\'eairement \`a droite sur $Z_1$, $Z_2$,
$T$.
\item[] $G_R$ op\`ere lin\'eairement \`a gauche sur $Z_3$, $Z_4$,
$T$.
\item[] $G_0$ op\`ere lin\'eairement \`a gauche sur $Z_2$, $H$, et
lin\'eairement \`a droite sur $Z_4$.
\end{itemize}
On suppose que ces actions sont compatibles entre elles et avec \m{\sigma},
\m{\sigma'}, \m{\tau}, \m{\tau'}. Par exemple, on a
$$\sigma(z_1g_L\otimes h)=\sigma(z_1\otimes h)g_L,
\sigma'(g_0h\otimes z_4) = \sigma(h\otimes z_4g_0),
(g_Rt)g_L = g_R(tg_L),$$
si \ \m{z_1\in Z_1}, \m{z_4\in Z_4}, \m{h\in H}, \m{t\in T}, \m{g_L\in G_L},
\m{g_R\in G_R} et \m{g_0\in G_0}.
\bigskip
\begin{defin}
La donn\'ee \m{\Theta} de \m{Z_1}, \m{Z_2}, \m{Z_3}, \m{Z_4}, \m{H}, \m{T},
\m{M},
\m{\sigma}, \m{\sigma'}, \m{\tau}, \m{\tau'}, et des actions de \m{G_L},
\m{G_R} et \m{G_0} s'appelle un {\em espace abstrait de complexes de type 1},
et \m{Q_C} est l'{\em espace total} de \m{\Theta}.
\end{defin}
\bigskip
\noindent{\bf 3.2.1.2 }{\it Dictionnaire}
\medskip
\noindent Dans la situation du \paragra~\hskip -2pt 3.1, on a
$$Z_1 = \mathop{\rm Hom}\nolimits({\cal E},\Gamma), \ \ Z_2 = \mathop{\rm Hom}\nolimits({\cal E},{\cal G}),$$
$$Z_3 = \mathop{\rm Hom}\nolimits(\Gamma,{\cal F}), \ \ Z_4 = \mathop{\rm Hom}\nolimits({\cal G},{\cal F}),$$
$$T = \mathop{\rm Hom}\nolimits({\cal E},{\cal F}), \ \ H = \mathop{\rm Hom}\nolimits(\Gamma,{\cal G}),$$
les applications \m{\sigma}, \m{\sigma'},\m{\tau},\m{\tau'} sont les
compositions, et
$$G_L=\mathop{\rm Aut}\nolimits({\cal E}), \ \ G_R=\mathop{\rm Aut}\nolimits({\cal F}), \ \ G_0=\mathop{\rm Aut}\nolimits({\cal G}).$$
\bigskip
\noindent{\bf 3.2.1.3 }{\it Groupes associ\'es}
\medskip
\noindent
Les groupes \m{G_L^{op}} et \m{G_R} agissent \`a gauche de mani\`ere
\'evidente sur \m{W_C}, et \m{Q_C} est invariant par ces groupes. Soit
\m{G_1} le groupe constitu\'e des matrices
$$\pmatrix{g_M & 0 \cr \phi & g_0\cr}$$
avec \ \m{g_M\in GL(M)}, \m{g_0\in G_0}, \m{\phi\in M^*\otimes H} (la loi de
composition est \'evidente). Le groupe \m{G_1} agit lin\'eairement \`a
gauche sur \m{W_C} : cette action provient d'une action \`a gauche sur
\ \m{(Z_1\otimes M)\oplus Z_2} \ et d'une action \`a droite sur
\ \m{(Z_3\otimes M^*)\oplus Z_4} : si \ \m{(\phi_1,z_2,\phi_3,z_4)\in W_C} \ et
\ \m{g_1\in G_1}, on a
$$g_1(\phi_1,z_2,\phi_3,z_4) = (g_1(\phi_1,z_2), (\phi_3,z_4)g_1^{-1}).$$
L'action \`a gauche de \m{G_1} sur \ \m{(Z_1\otimes M)\oplus Z_2} \ est :
$$\pmatrix{g_M & 0 \cr \phi & g_0\cr}\pmatrix{\phi_1\cr z_2} =
\pmatrix{g_M\phi_1\cr \sigma(\pline{\phi,\phi_1})+g_0z_2}.$$
L'action \`a droite de \m{G_1} sur \ \m{(Z_3\otimes M^*)\oplus Z_4} \ est :
$$(\phi_3,z_4)\pmatrix{g_M & 0 \cr \phi & g_0\cr} =
(\phi_3g_M+(I_M\otimes \sigma')(\phi\otimes z_4), z_4g_0).$$
En utilisant la commutativit\'e du diagramme \m{(D)}, on montre ais\'ement
que \m{Q_C} est \m{G_1}-invariant. Plus g\'en\'eralement, on montre que
l'application
$$W_C\longrightarrow T$$
$$(\phi_1,z_2,\phi_3,z_4)\longmapsto \tau(\pline{\phi_1,\phi_3})+
\tau'(z_2\otimes z_4)$$
est \m{G_1}-invariante. Dans la situation du \paragra~\hskip -2pt 2.2.1.2, on a
\ \m{G_1=\mathop{\rm Aut}\nolimits((\Gamma\otimes M)\oplus{\cal G})}.
On pose
$$G \ \ = \ \ G_L^{op}\times G_1\times G_R,$$
qui agit \`a gauche sur \m{W_C} et \m{Q_C}.
\bigskip
\subsubsection{Espaces de complexes de type 2}
\noindent{\bf 3.2.2.1 }{\it D\'efinition}
\medskip
\noindent Soient \m{Z_1}, \m{Y_2}, \m{T_2}, \m{Z_3}, \m{T}, \m{K}, \m{N} des
espaces vectoriels de dimension finie, et
$$W'_C = (Z_1\otimes N)\oplus(Y_2\otimes N)\oplus(Z_3\otimes N^*).$$
Soient
$$\nu : K\otimes Y_2\longrightarrow Z_1,$$
$$\nu' : K\otimes T_2\longrightarrow T,$$
$$\lambda : Y_2\otimes Z_3\longrightarrow T_2,$$
$$\tau : Z_1\otimes Z_3\longrightarrow T$$
des applications lin\'eraires. On suppose que \m{\nu} induit une
inclusion \ \m{K\subset Z_1\otimes Y_2^*} \ et que \m{\lambda} est surjective.
On suppose aussi que le diagramme suivant est commutatif :
$$(D') \ \ \ \ \ \ \diagram{
K\otimes Y_2\otimes Z_3 & \hfl{\nu\otimes I_{Z_3}}{} & Z_1\otimes Z_3\cr
\vfl{I_K\otimes\lambda}{} & & \vfl{}{\tau}\cr
K\otimes T_2 & \hfl{\nu'}{} & T}$$
Soit \ \m{Q'_C\subset W'_C} \ l'ensemble des points
\m{(\psi_1,\psi_2,\psi_3)} tels que
$$\tau(\pline{\psi_1,\psi_3}) \ = \lambda(\pline{\psi_2,\psi_3}) \ = \ 0,$$
o\`u \m{\pline{\ \ }} d\'esigne la contraction de \m{N}.
Soient \m{G_L}, \m{G_0}, \m{G_R} des groupes. On suppose que
\begin{itemize}
\item[] $G_L$ op\`ere lin\'eairement \`a droite sur $T$, $Z_1$ et $K$,
\item[] $G_R$ op\`ere lin\'eairement \`a gauche sur $T$, $Z_3$ et $T_2$,
\item[] $G_0$ op\`ere lin\'eairement \`a gauche sur $K$ et
lin\'eairement \`a droite sur $T_2$ et $Y_2$.
\end{itemize}
On suppose comme pour les complexes de type 1 que les actions des groupes
sont compatibles entre elles et avec les applications \m{\nu}, \m{\nu'},
\m{\lambda} et \m{\tau}.
\bigskip
\begin{defin}
La donn\'ee \m{\Theta'} de \m{Z_1}, \m{Y_2}, \m{T_2}, \m{Z_3}, \m{T}, \m{K},
\m{N}, \m{\nu}, \m{\nu'}, \m{\lambda}, \m{\tau} et des actions de
\m{G_L}, \m{G_0}, \m{G_R} s'appelle un {\em espace abstrait de complexes
de type 2}, et \m{Q'_C} est {\em l'espace total de } \m{\Theta'}.
\end{defin}
\bigskip
\noindent{\bf 3.2.2.2 }{\it Dictionnaire}
\medskip
\noindent Dans la situation du \paragra~\hskip -2pt 3.1, on a
$$Z_1=\mathop{\rm Hom}\nolimits({\cal E},\Gamma), \ Y_2=\mathop{\rm Hom}\nolimits({\cal E}',\Gamma),$$
$$Z_3 = \mathop{\rm Hom}\nolimits(\Gamma,{\cal F}), \ \ T_2=\mathop{\rm Hom}\nolimits({\cal E}',{\cal F}),$$
$$T=\mathop{\rm Hom}\nolimits({\cal E},{\cal F}), \ \ K=\mathop{\rm Hom}\nolimits({\cal E},{\cal E}'),$$
les applications \m{\nu}, \m{\nu'}, \m{\lambda}, \m{\tau} sont les
compositions et
$$G_L=\mathop{\rm Aut}\nolimits({\cal E}), \ G_R=\mathop{\rm Aut}\nolimits({\cal F}), \ G_0=\mathop{\rm Aut}\nolimits({\cal E}').$$
\bigskip
\noindent{\bf 3.2.2.3 }{\it Groupes associ\'es}
\medskip
\noindent
Les groupes \m{G_L^{op}} et \m{G_R} agissent \`a gauche de mani\`ere
\'evidente sur \m{W'_C}, et \m{Q'_C} est invariant par ces groupes. Soit
\m{G'_1} le groupe constitu\'e des matrices
$$\pmatrix{g_L & 0\cr k & g_0}$$
avec \ \m{g_L\in G_L}, \m{g_0\in G_0}, \m{k\in K} (la loi de composition
est \'evidente). Alors \m{G'_1} agit \`a droite sur \ \m{Z_1\oplus Y_2} :
$$(z_1,y_2)\pmatrix{g_L & 0\cr k & g_0}=
(z_1g_L+\nu(k\otimes y_2),y_2g_0).$$
Dans la situation du \paragra~\hskip -2pt 3.2.2.2, on a \ \m{G'_1=\mathop{\rm Aut}\nolimits({\cal E}\oplus{\cal E}')}.
On en d\'eduit une action \`a gauche de
$$G' \ = \ GL(N)\times {G'_1}^{op}\times G_R$$
sur \m{W'_C}. On v\'erifie comme dans le cas des complexes de type 1 que
\m{Q'_C} est \m{G'}-invariant.
\bigskip
\subsubsection{Mutations \ 1 \m{\Longrightarrow} 2}
\noindent{\bf 3.2.3.1 }{\it Mutations d'espaces abstraits de complexes}
\medskip
\noindent
On consid\`ere l'espace abstrait de complexes de type 1 \m{\Theta} du
\paragra~\hskip -2pt 3.2.1. On va en d\'eduire \m{\Theta'}, espace abstrait de complexes de
type 2.
Les espaces vectoriels \m{Z_1}, \m{Z_3}, \m{T} de \m{\Theta'} sont les
m\^emes que ceux de \m{\Theta}. On prend
$$N=M\oplus H, \ \ Y_2=H^*, \ \ T_2=(H^*\otimes Z_3)/Z_4, \ \
K=\ker(\sigma)\subset Z_1\otimes H.$$
L'application \m{\tau} de \m{\Theta'} est la m\^eme que celle de \m{\Theta}.
\noindent
L'application \ \m{\lambda : Y_2\otimes Z_3\longrightarrow T_2} \ est la projection \
\m{H^*\otimes Z_3\longrightarrow (H^*\otimes Z_3)/Z_4}.
\noindent
L'application \ \m{\nu : K\otimes Y_2\longrightarrow Z_1} \ est la compos\'ee
$$K\otimes Y_2\subset Z_1\otimes H\otimes Y_2 = Z_1\otimes H\otimes H^*\longrightarrow Z_1.$$
\noindent Pour d\'efinir \ \m{\nu' : K\otimes T_2\longrightarrow T} \ on part du
diagramme commutatif suivant, d\'eduit de \m{(D')}
$$\diagram{Z_1\otimes H\otimes Z_4 & \hfl{\sigma\otimes I_{Z_4}}{} & Z_2\otimes Z_4\cr
\vfl{\alpha}{} & & \vfl{}{\tau'}\cr
Z_1\otimes H\otimes Z_3\otimes H^* & \hfl{\tau\otimes tr}{} & T\cr}$$
(\m{tr} d\'esignant la trace \ \m{H\otimes H^*\longrightarrow\cx{}} \ et \m{\alpha}
provenant de l'inclusion \ \m{Z_4\subset Z_3\otimes H^*} \ d\'eduite de
\m{\sigma'}). Il en d\'ecoule que
$$(\tau\otimes tr)\circ\alpha(\ker(\sigma)\otimes Z_4) \ = \ \lbrace 0\rbrace .$$
Donc \ \m{\tau\otimes tr} \ induit une application lin\'eaire
$$\ker(\sigma)\otimes((Z_3\otimes H^*)/Z_4) \ = \ K\otimes T_2\longrightarrow T$$
qui est par d\'efinition \m{\nu'}.
La commutativit\'e de \m{(D')} se v\'erifie ais\'ement.
Les groupes \m{G_R}, \m{G_L}, \m{G_0} de \m{\Theta'} sont les m\^emes
que ceux de \m{\Theta} et leurs actions sont \'evidentes. L'espace
abstrait de complexes de type 2 \m{\Theta'} est ainsi compl\`etement
d\'efini. On notera
$$\Theta' \ = \ D_0(\Theta).$$
\bigskip
\noindent{\bf 3.2.3.2 }{\it Mutations de complexes}
\medskip
\noindent
Soit \ \m{(\phi_1,z_2,\phi_3,z_4)\in Q_C}. On va en d\'eduire une orbite
\ \m{G'.(\psi_1,\psi_2,\psi_3)} \ de \m{Q'_C}.
\begin{itemize}
\item[--] D\'efinition de \ $\psi_1\in Z_1\otimes N$ : on
prend \ $\psi_0\in Z_1\otimes H$ \ tel que \ $\sigma(\psi_0)=z_2$, et
$$\psi_1=\psi_0+\phi_1 \ \in \ (Z_1\otimes H)\oplus(Z_1\otimes M)=Z_1\otimes N.$$
\item[--] D\'efinition de \ $\psi_2\in Y_2\otimes N$ : on prend
$$\psi_2=I_H \ \in \ H^*\otimes H\subset H^*\otimes(M\oplus H)=Y_2\otimes N.$$
\item[--] D\'efinition de \ $\psi_3\in Z_3\otimes N^*$ : on prend
$$\psi_3=\phi_3+z_4 \ \in \ (Z_3\otimes M^*)\oplus Z_4\subset
(Z_3\otimes M^*)\oplus (Z_3\otimes H^*)=Z_3\otimes N^*.$$
\end{itemize}
\medskip
On v\'erifie ais\'ement que \m{(\psi_1,\psi_2,\psi_3)} est un \'el\'ement de
\m{Q_C} d\'efini \`a l'action pr\`es du sous-groupe de \m{G'_1} isomorphe
\`a \m{K}, constitu\'e des matrices
$$\pmatrix{1 & 0\cr k & 1}, \ \ \ k\in K.$$
On a donc d\'efini une application
$$\ov{D_0} : Q_C\longrightarrow Q'_C/G' .$$
\bigskip
\begin{xlemm}
Si \ \m{x\in Q_C} \ et \ \m{g\in G}, alors \ \m{\ov{D_0}(gx)=\ov{D_0}(x)}.
Donc \m{\ov{D_0}} induit une appli-\break cation
$$D_0 : Q_C/G\longrightarrow Q'_C/G'.$$
\end{xlemm}
\medskip
V\'erification imm\'ediate. $\Box$
\bigskip
\subsubsection{Mutations \ 2 \m{\Longrightarrow} 1}
\noindent{\bf 3.2.4.1 }{\it Mutations d'espaces abstraits de complexes}
\medskip
\noindent
On consid\`ere l'espace abstrait de complexes de type 2 \m{\Theta'} du
\paragra~\hskip -2pt 3.2.2. On va en d\'eduire \m{\Theta}, espace abstrait de complexes
de type 1. On doit supposer que
$$\dim(N)\geq\dim(Y_2).$$
On note \m{{W'}^0_C} le sous-ensemble \m{G'}-invariant de \m{W'_C}
constitu\'e des \m{(\psi_1,\psi_2,\psi_3)} tels que \
\m{\psi_2 : Y_2^*\longrightarrow N} \ soit injective. Soit \ \m{{Q'}^0_C=
Q'_C\cap {W'}^0_C}.
On d\'efinit maintenant \m{\Theta}. Les espaces vectoriels \m{Z_1}, \m{Z_3},
\m{T} de \m{\Theta} sont les m\^emes que ceux de \m{\Theta'}. On prend pour
\m{M} un espace vectoriel de dimension \ \m{\dim(N)-\dim(Y_2)}, et
$$H=Y_2^*, \ \ Z_2=(Z_1\otimes Y_2^*)/K, \ \ Z_4=\ker(\lambda)\subset
Y_2\otimes Z_3.$$
\noindent L'application \m{\tau} de \m{\Theta} est la m\^eme que celle de
\m{\Theta'}.
\noindent L'application \ \m{\sigma : Z_1\otimes H\longrightarrow Z_2} \ est la projection
\ \m{Z_1\otimes Y_2^*\longrightarrow (Z_1\otimes Y_2^*)/K}.
\noindent L'application \ \m{\sigma' : H\otimes Z_4\longrightarrow Z_3} \ est la restriction
de
$$tr\otimes I_{Z_3} : H\otimes H^*\otimes Z_3\longrightarrow Z_3.$$
\noindent Pour d\'efinir \ \m{\tau' : Z_2\otimes Z_4\longrightarrow T} \ on part du
diagramme commutatif suivant d\'eduit de \m{(D')}
$$\diagram{K\otimes Y_2\otimes Z_3 & \hfl{I_K\otimes\lambda}{} & K\otimes T_2\cr
\vfl{\ov{\nu}\otimes I_{Y_2\otimes Z_3}}{} & & \vfl{}{\nu'}\cr
Z_1\otimes Y_2^*\otimes Y_2\otimes Z_3 & \hfl{\tau\otimes tr}{} & T\cr}$$
\m{\ov{\nu}} d\'esignant l'inclusion \ \m{K\subset Z_1\otimes Y_2^*} \ d\'eduite
de \m{\nu}. On a donc
$$(\tau\otimes tr)\circ(\ov{\nu}\otimes I_{Y_2\otimes Z_3})(K\otimes \ker(\lambda)=\lbrace 0\rbrace,$$
et \ \m{\tau\otimes tr} \ induit donc
$$\tau' : ((Z_1\otimes Y_2^*)/K)\otimes\ker(\lambda)=Z_2\otimes Z_4\longrightarrow T.$$
La commutativit\'e du diagramme \m{(D)} se v\'erifie ais\'ement.
Les groupes \m{G_0}, \m{G_L} et \m{G_R} de \m{\Theta} sont les m\^emes que
ceux de \m{\Theta'} et leurs actions sont \'evidentes. L'espace abstrait de
complexes de type 1 \m{\Theta} est ainsi compl\`etement d\'efini. On notera
$$\Theta \ = \ D'_0(\Theta').$$
\bigskip
\begin{xprop}
On a \ \m{D_0\circ D'_0(\Theta')=\Theta'} \ et \
\m{D'_0\circ D_0(\Theta)=\Theta}.
\end{xprop}
\medskip
Imm\'ediat. $\Box$
\bigskip
\noindent{\bf 3.2.4.2 }{\it Mutations de complexes}
\medskip
\noindent
Soit \ \m{(\psi_1,\psi_2,\psi_3)\in {Q'}^0_C}. On va en d\'eduire une
orbite \ \m{G.(\phi_1,z_2,\phi_3,z_4)} \ de \m{Q_C}.
\begin{itemize}
\item[--] D\'efinition de \ $\phi_1\in Z_1\otimes M$ : on fixe d'abord un
isomorphisme entre $M$ et un suppl\'ementaire de l'image de $\psi_2$ dans
$N$ :
$$N \ = \ Y_2^*\oplus M.$$
On prend pour $\phi_1$ la composante de $\psi_1$ dans \ $Z_1\otimes M$.
\item[--] D\'efinition de \ $z_2\in Z_2$ : on prend la projection sur \
$(Z_1\otimes Y_2^*)/K$ \ de la composante de $\psi_2$ dans \ $Z_1\otimes Y_2^*$.
\item[--] D\'efinition de \ $\phi_3\in Z_3\in M^*$ : on prend la composante
de $\psi_3$ dans \ $Z_3\otimes M^*$.
\item[--] D\'efinition de \ $z_4\in Z_4$ : on prend \
$z_4=\pline{\psi_2,\psi_3}\in\ker(\lambda).$
\end{itemize}
\medskip
On v\'erifie ais\'ement que \m{(\phi_1,z_2,\phi_3,z_4)} est un \'el\'ement
de \m{Q_C} d\'efini \`a l'action pr\`es du groupe \m{G_1}.
On a donc d\'efini une application
$$\ov{D'_0} : {Q'}^0_C\longrightarrow Q_C/G .$$
\bigskip
\begin{xlemm}
Si \ \m{x\in {Q'}^0_C} \ et \ \m{g\in G'}, alors
\ \m{\ov{D'_0}(gx)=\ov{D'_0}(x)}.
Donc \m{\ov{D'_0}} induit une appli-\break cation
$$D'_0 : {Q'}^0_C/G'\longrightarrow Q_C/G.$$
\end{xlemm}
\medskip
V\'erification imm\'ediate. $\Box$
\bigskip
\subsubsection{Th\'eor\`emes d'isomorphisme}
\begin{xtheo}
On a \ \m{D'_0\circ D_0 = I_{Q_C/G}} \ et \
\m{D_0\circ D'_0 = I_{{Q'}^0_C/G'}}.
\end{xtheo}
\medskip
V\'erification imm\'ediate. $\Box$
\bigskip
On a donc obtenu une bijection canonique
$$Q_C/G \ \simeq {Q'}^0_C/G'.$$
\medskip
On suppose maintenant que les groupes sont alg\'ebriques, ainsi que leurs
actions sur les espaces vectoriels dont il est question. On a alors :
\bigskip
\begin{xtheo}
Il existe un quasi-isomorphisme fort canonique de \m{Q_C} vers \m{{Q'}^0_C},
au dessus de l'isomorphisme pr\'ec\'edent.
\end{xtheo}
\bigskip
\noindent{\em D\'emonstration}.
Le quasi-isomorphisme fort \ \m{Q_C\longrightarrow {Q'}^0_C} \ est d\'efini par une
seule carte, obtenue en utilisant une section de \m{\sigma}. Le
quasi-isomorphisme inverse est d\'efini par des cartes ind\'ex\'ees sur
la grassmannienne \m{Gr_0} des sous-espaces vectoriels de $N$ de dimension
\'egale \`a celle de $M$. L'ouvert correspondant \`a \ \m{M_0\in Gr_0} \
est l'ensemble des points \m{(\psi_1,\psi_2,\psi_3)} de \m{{Q'}^0_C} tels que
l'image de \ \m{\psi_2 : Y_2^*\longrightarrow N} \ ne rencontre pas \m{M_0}. Les
v\'erifications (fastidieuses) sont laiss\'ees au lecteur.
$\Box$
\bigskip
On peut donner une version abstraite de la proposition 3.3, et obtenir des
r\'esultats analogues aux th\'eor\`emes 3.7 et 3.8.
\subsection{Application aux espaces de complexes}
Soient $X$
une vari\'et\'e alg\'ebrique projective, \m{p\geq 1} \ un entier,
\m{n_0,\ldots,n_p} des entiers positifs, et pour \ \m{0\leq i\leq p},
\m{1\leq j\leq n_i}, \ \m{{\cal E}^{(i)}_j} un faisceau coh\'erent sur $X$ et
\m{M^{(i)}_j} un espace vectoriel non nul de dimension finie. On pose, pour
\m{0\leq i\leq p}
$${\cal E}_i \ = \ \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq j\leq n_i}({\cal E}^{(i)}_j\otimes M^{(i)}_j).$$
On suppose que les faisceaux \m{{\cal E}^{(i)}_j} sont simples, et que
$$\mathop{\rm Hom}\nolimits({\cal E}^{(i)}_j, {\cal E}^{(i')}_{j'}) \ = \ \lbrace 0\rbrace$$
si \ \m{i>i'}, ou \ \m{i=i'}, \m{j>j'}. Soit \m{Q_C} la vari\'et\'e des
complexes
$${\cal E}_0\longrightarrow{\cal E}_1\longrightarrow\ldots\longrightarrow{\cal E}_p.$$
On pose \ \m{{\cal E}_{i}=0} \ si \m{i<0} ou \m{i>p}.
Soit \m{i_0} un entier, avec \ \m{0\leq i_0\leq p}. On pose
$${\cal E}={\cal E}_{i_0-1}, \ \ \Gamma={\cal E}^{(i_0)}_1, \ \ M=M^{(i_0)}_1, \ \
{\cal G}=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq j\leq n_i}({\cal E}^{(i_0)}_j\otimes M^{(i_0)}_j), \ \
{\cal F}={\cal E}_{i_0+1},$$
de telle sorte que les complexes pr\'ec\'edents se mettent sous la forme
$$\cdots\longrightarrow{\cal E}_{i_0-2}\longrightarrow{\cal E}\longrightarrow(\Gamma\otimes M)\oplus{\cal G}\longrightarrow{\cal F}\longrightarrow{\cal E}_{i_0+2}\longrightarrow
\cdots$$
On suppose que les conditions du \paragra~\hskip -2pt 3.1.1 sont v\'erifi\'ees, ce qui
d\'efinit le faisceau \m{{\cal E}'}. Soit \m{Q'_C} la vari\'et\'e des
complexes du type
$$\cdots\longrightarrow{\cal E}_{i_0-2}\longrightarrow{\cal E}\oplus{\cal E}'\longrightarrow\Gamma\otimes N\longrightarrow{\cal F}\longrightarrow{\cal E}_{i_0+2}\longrightarrow
\cdots$$
et \m{{Q'}^0_C} l'ouvert de \m{Q'_C} constitu\'e des complexes tels que le
morphisme \ \m{{\cal E}'\longrightarrow\Gamma\otimes N} \ induise une inclusion
\ \m{\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\subset N}.
Le groupe \m{G} (resp. \m{G'}) op\'erant sur \m{Q_C} (resp. \m{Q'_C}) est
le produit des groupes d'automorphismes des termes des complexes. On
g\'en\'eralise sans difficult\'es les r\'esultats du \paragra~\hskip -2pt 3.2 \`a ce cas et
on d\'efinit de mani\`ere \'evidente les mutations de complexes de \m{Q_C} ou
\m{{Q'}^0_C} (qui sont des complexes de \m{{Q'}^0_C} et \m{Q_C} respectivement),
et on obtient l'analogue du th\'eor\`eme 3.8 pour les vari\'et\'es de
complexes.
\vfill\eject
\section{Vari\'et\'es de modules de complexes}
La construction des vari\'et\'es de modules de morphismes de \cite{dr_tr}
est une application du th\'eor\`eme 3.8. On donne ici une autre
application de ce th\'eor\`eme \`a la construction de vari\'et\'es de
modules de complexes. On travaille dans le language des faisceaux, ce
qui donne des d\'emonstrations plus explicites. Il est \'evidemment
possible de faire une version abstraite de la construction des
vari\'et\'es de modules, comme dans \cite{dr_tr}.
Soient \m{{\cal E}_1}, \m{{\cal F}_1}, \m{{\cal F}_2}, \m{{\cal G}_1} des faisceaux coh\'erents
sur une vari\'et\'e projective \m{X}, et \m{L_1}, \m{M_1}, \m{M_2}, \m{N_1}
des espaces vectoriels de dimension finie. On s'int\'eresse \`a des
complexes du type
$${\cal E}_1\otimes L_1\longrightarrow ({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2)\longrightarrow {\cal G}_1\otimes N_1.$$
On fait les hypoth\`eses suivantes :
\begin{itemize}
\item[--] Les faisceaux ${\cal E}_1$, ${\cal F}_1$, ${\cal F}_2$ et ${\cal G}_1$ sont simples, et
\ $\mathop{\rm Hom}\nolimits({\cal F}_2,{\cal F}_1)=\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal E}_1)=\lbrace 0\rbrace.$
\item[--] Le morphisme canonique
$${\cal F}_1\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\longrightarrow{\cal F}_2$$
est surjectif. On note \m{{\cal H}_1} son noyau.
\item[--] Le morphisme canonique
$${\cal E}_1\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\longrightarrow{\cal H}_1$$
est surjectif. On note \m{{\cal K}_1} son noyau.
\item[--] On a
$$\mathop{\rm Ext}\nolimits^1({\cal F}_2,{\cal E}_1)=\mathop{\rm Ext}\nolimits^1({\cal E}_1,{\cal H}_1)=\mathop{\rm Ext}\nolimits^1({\cal F}_2,{\cal F}_1)=\lbrace 0\rbrace,$$
$$\mathop{\rm Ext}\nolimits^1({\cal F}_2,{\cal F}_1)=\mathop{\rm Ext}\nolimits^1({\cal H}_1,{\cal H}_1)=\mathop{\rm Ext}\nolimits^1({\cal H}_1,{\cal E}_1)=\lbrace 0\rbrace.$$
\end{itemize}
Cette derni\`ere hypoth\`ese entraine que les faisceaux \m{{\cal H}_1} et
\m{{\cal K}_1} sont simples.
On se trouve dans la situation
de la proposition 3.1 (avec \ \m{{\cal E}={\cal E}_1\otimes L_1},
\m{\Gamma={\cal F}_1}, \m{M=M_1}, \m{{\cal G}={\cal F}_2\otimes M_2} et \m{{\cal F}={\cal G}_1\otimes N_1}). On
note \m{Q_C} la vari\'et\'e des complexes du type pr\'ec\'edent. Elle peut
donc \^etre vue comme l'espace total d'un espace abstrait de complexes de
type 1. Le groupe \m{G} op\'erant sur \m{Q_C} est
$$G \ = \ GL(L_1)\times \mathop{\rm Aut}\nolimits(({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2))\times
GL(N_1).$$
On peut voir \ \m{\mathop{\rm Aut}\nolimits(({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2))} \ comme constitu\'e
de matrices du type
$$\pmatrix{g_1 & 0\cr\phi & g_2\cr},$$
avec \ \m{g_1\in GL(M_1)}, \m{g_2\in GL(M_2)} \ et \ \m{\phi\in
\mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)^*\otimes M_1,M_2)}.
On pose
$$a=\dim(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)),\ \ \ b=\dim(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)).$$
\subsection{Notions de (semi-)stabilit\'e et vari\'et\'es de modules de
complexes}
Le groupe $G$ poss\`ede deux sous-groupes importants : le premier est le
sous-groupe normal unipotent maximal \'evident $H$, isomorphe au groupe
additif
\noindent
\m{\mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)^*\otimes M_1,M_2))}. Le second est le sous-groupe
r\'eductif
$$G_{red} \ = \ GL(L_1)\times GL(M_1)\times GL(M_2)\times GL(N_1),$$
dont l'inclusion dans $G$ induit un isomorphisme \
\m{G_{red}\simeq G/H}.
L'action de \m{G_{red}} est un cas particulier des actions \'etudi\'ees
dans \cite{king}. On consid\`ere l'espace vectoriel
$$W \ = \ \mathop{\rm Hom}\nolimits({\cal E}_1\otimes L_1,({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2))\times
\mathop{\rm Hom}\nolimits(({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2),{\cal G}_1\otimes N_1),$$
dont $Q_C$ est une sous-vari\'et\'e ferm\'ee. L'action de $G$ sur \m{Q_C}
s\'etend de mani\`ere \'evidente \`a $W$.
Soient \m{\lambda_1}, \m{\mu_1}, \m{\mu_2}, \m{\nu_1} des nombres rationnels
non nuls tels que
$$\lambda_1\dim(L_1)+\mu_1\dim(M_1)+\mu_2\dim(M_2)+\nu_1\dim(N_1)=0.$$
\bigskip
\begin{defin}
Un point \m{(\phi,\psi)} de $W$ est dit {\em $G_{red}$-semi-stable} (resp.
{\em $G_{red}$-stable}) relativement \`a \m{(\lambda_1,\mu_1,\mu_2,\nu_1)} si
pour tous sous-espaces vectoriels \ \m{L'_1\subset L_1}, \m{M'_1\subset M_1},
\m{M'_2\subset M_2}, \m{N'_1\subset N_1}, avec \
\m{(L'_1,M'_1,M'_2,N'_1)\not = (L_1,M_1,M_2,N_1)} \ ou
\m{(\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace)}, tels que
$$\phi({\cal E}_1\otimes L'_1)\subset ({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M'_2), \ \ \
\psi(({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M'_2))\subset {\cal G}_1\otimes N'_1,$$
on a
$$\lambda_1\dim(L'_1)+\mu_1\dim(M'_1)+\mu_2\dim(M'_2)+\nu_1\dim(N'_1)\leq 0
\ \ \ {\rm (resp. \ \ } \ < \ {\rm )}.$$
\end{defin}
\bigskip
On dit que \m{(\lambda_1, \mu_1, \mu_2, \nu_1)} est une {\em polarisation}
de l'action de $G$ sur $W$ (ou \m{Q_C}).
On note \m{W^{ss}_{red}} (resp. \m{W^s_{red}}) l'ouvert de \m{W} constitu\'e
des points \m{G_{red}}-semi-stables (resp. \m{G_{red}}-stables). Soient
\ \m{Q^{ss}_{C,red}=W^{ss}_{red}\cap Q_C},
\m{Q^s_{C,red}=W^s_{red}\cap Q_C}. D'apr\`es \cite{king}, il existe un
bon quotient \ \m{W^{ss}_{red}//G} \ et un quotient g\'eom\'etrique
lisse \m{W^s_{red}/G}. Par cons\'equent il existe aussi un bon quotient
\ \m{Q^{ss}_{C,red}//G}. Mais ce n'est pas le quotient que nous
recherchons.
\bigskip
\begin{defin}
Un point \m{x} de $W$ est dit {\em $G$-semi-stable} (resp.
{\em $G$-stable}) relativement \`a \m{(\lambda_1,\mu_1,\mu_2,\nu_1)} si
tous les points de l'orbite \m{H.x} sont $G_{red}$-semi-stables (resp.
$G_{red}$-stables).
\end{defin}
\bigskip
On note \m{W^{ss}} (resp. \m{W^s}) l'ouvert de \m{W} constitu\'e
des points \m{G}-semi-stables (resp. \m{G}-stables). Soient
\ \m{Q^{ss}_{C}=W^{ss}\cap Q_C}, \m{Q^s_{C}=W^s\cap Q_C}. On cherche
\`a prouver l'existence de quasi-bons quotients \ \m{Q^{ss}_C//G}. De tels
quotients seront appel\'es des {\em vari\'et\'es de modules de complexes}.
\medskip
On montre ais\'ement que si \m{W^{s}} est non vide on doit avoir
$$\lambda_1>0, \ \ \nu_1<0, \ \ \mu_1\dim(M_1)+\nu_1\dim(N_1)<0, \ \
\mu_2\dim(M_2)+\nu_1\dim(N_1)<0.$$
On supposera par la suite que ces in\'egalit\'es sont v\'erifi\'ees.
\subsection{Mutations et polarisations associ\'ees}
\subsubsection{La premi\`ere mutation}
\noindent{\bf 3.2.1.1 - }{\it D\'efinition}
\medskip
Soit
$$(*) \ \ \ \ \ \ \ {\cal E}_1\otimes L_1 \ \hfl{(f_1,f_2)}{} \
({\cal F}_1\otimes M_1)\oplus({\cal F}_2\otimes M_2) \ \hfl{(g_1,g_2)}{} \ {\cal G}_1\otimes N_1.$$
un complexe. Une premi\`ere mutation donne un complexe
$$(**) \ \ \ \ \ \ \ ({\cal E}_1\otimes L_1)\oplus({\cal H}_1\otimes M_2) \
\hfl{(\phi_1,\phi_2)}{} \
{\cal F}_1\otimes P_1\ \hfl{\phi}{} \ {\cal G}_1\otimes N_1,$$
avec
$$P_1=(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2)\oplus M_1.$$
Le groupe op\'erant sur la vari\'et\'e \m{Q'_C} de ces complexes est
$$G'=\mathop{\rm Aut}\nolimits(({\cal E}_1\otimes L_1)\oplus({\cal H}_1\otimes M_2))\times GL(P_1)\times GL(N_1).$$
Le groupe \ \m{\mathop{\rm Aut}\nolimits(({\cal E}_1\otimes L_1)\oplus({\cal H}_1\otimes M_2))} \ est constitu\'e de
matrices
$$\pmatrix{g_1 & 0\cr \phi & g_2\cr},$$
avec \ \m{g_1\in GL(L_1)}, \m{g_2\in GL(M_2)},
\m{\phi\in\mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)^*\otimes L_1,M_2)} (on a
\noindent
\m{\mathop{\rm Hom}\nolimits({\cal H}_1,{\cal E}_1)=\lbrace 0\rbrace}, \`a cause du fait que
\m{\mathop{\rm Ext}\nolimits^1({\cal F}_2,{\cal E}_1)=\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal E}_1)=\lbrace 0\rbrace}). Le complexe \m{(**)} n'est
pas unique, mais sa $G'$-orbite l'est.
D\'ecrivons maintenant une mutation du complexe \m{(*)}. Le morphisme
$$\phi_1 : {\cal E}_1\otimes L_1\longrightarrow {\cal F}_1\otimes P_1$$
est la somme de
$$f_1 : {\cal E}_1\otimes L_1\longrightarrow {\cal F}_1\otimes M_1$$
et d'un rel\`evement de \ \m{f_2 : {\cal E}_1\otimes L_1\longrightarrow {\cal F}_2\otimes M_2} \ en un
morphisme
\noindent \m{{\cal E}_1\otimes L_1\longrightarrow {\cal F}_1\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2}. Un tel
rel\`evement est possible car
\noindent \m{\mathop{\rm Ext}\nolimits^1({\cal E}_1,{\cal H}_1)=\lbrace 0\rbrace}. Le morphisme
$$\phi_2 : {\cal H}_1\otimes M_2\longrightarrow{\cal F}_1\otimes P_1$$
est \'egal \`a \ \m{\sigma\otimes I_{M_2}}, o\`u \m{\sigma} est l'inclusion \
\m{{\cal H}_1\longrightarrow{\cal F}_1\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)}. Le morphisme \m{\phi} est \'egal \`a
\m{g_1} sur \m{{\cal F}_1\otimes M_1}, et sur \m{{\cal F}_1\otimes \mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2}, c'est
la compos\'ee
$${\cal F}_1\otimes \mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2\longrightarrow {\cal F}_2\otimes M_2 \ \hfl{g_2}{} \ {\cal G}_1\otimes N_1.$$
\vfill\eject
\noindent{\bf 3.2.1.2 - }{\it (Semi-)stabilit\'e dans \m{Q'_C}}
\medskip
On d\'efinit comme pour les complexes de type \m{(**)} une notion de
(semi-)stabilit\'e, d\'ependant d'une suite \m{(\alpha_1, \alpha_2,\beta_1,
\gamma_1)} de nombres rationnels telle que
$$\alpha_1\dim(L_1)+\alpha_2\dim(M_2)+\beta_1\dim(P_1)+\gamma_1\dim(N_1)
=0.$$
Soient \m{G'_{red}} les sous-groupe r\'eductif canonique de $G'$,
et $H'$ le sous-groupe normal unipotent maximal isomorphe au groupe additif
\m{\mathop{\rm Hom}\nolimits(L_1\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1),M_2)}.
La \m{G'_{red}}-\break\hbox{(semi-)}stabilit\'e relativement \`a
\m{(\alpha_1,\alpha_2,\beta_1,\gamma_1)}
est encore un cas particulier des actions
\'etudi\'ees dans \cite{king} : le complexe \m{(**)} est
\m{G'_{red}}-semi-stable (resp. \m{G'_{red}}-stable) si pour tous
sous-espaces vectoriels \ \m{L'_1\subset L_1}, \m{M'_2\subset M_2},
\m{P'_1\subset P_1}, \m{N'_1\subset N_1}, avec
\noindent \m{(L'_1,M'_2,P'_1,N'_1)\not = (L_1,M_2,P_1,N_1)} \ ou
\m{(\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace)}, tels que
$$(\phi_1,\phi_2)(({\cal E}_1\otimes L'_1)\oplus({\cal H}_1\otimes M'_2))\subset{\cal F}_1\otimes P'_1,
\ \ \ \phi({\cal F}_1\otimes P'_1)\subset{\cal G}_1\otimes N'_1,$$
on a
$$\alpha_1\dim(M'_1)+\alpha_2\dim(M'_2)+\beta_1\dim(P'_1)+
\gamma_1\dim(N'_1)\leq 0 \ \ \ \ {\rm(resp. \ } \ < \ {\rm )}.$$
Le complexe \m{(**)} est $G'$-semi-stable (resp. \m{G'}-stable) si tous les
points de sa $H'$-orbite sont \m{G'_{red}}-semi-stables (resp.
\m{G'_{red}}-stables).
Soient \ \m{L'_1\subset L_1}, \m{M'_1\subset M_1}, \m{M'_2\subset M_2},
\m{N'_1\subset N_1} \ des sous-espaces vectoriels tels que
$$(f_1,f_2)({\cal E}_1\otimes L'_1)\subset ({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M'_2), \ \ \
(g_1,g_2)(({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M'_2))\subset {\cal G}_1\otimes N'_1.$$
On pose
$$P'_1=(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M'_2)\oplus M'_1.$$
Le rel\`evement de \m{f_2} peut \^etre choisi de telle sorte
qu'il envoie \m{{\cal E}_1\otimes L'_1} dans
\noindent\m{{\cal F}_1\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M'_2}. On a alors
$$(\phi_1,\phi_2)(({\cal E}_1\otimes L'_1)\oplus({\cal H}_1\otimes M'_2))\subset{\cal F}_1\otimes P'_1,
\ \ \ \phi({\cal F}_1\otimes P'_1)\subset{\cal G}_1\otimes N'_1.$$
Soit \m{(\lambda_1,\mu_1,\mu_2,\nu_1)} une polarisation de l'action de $G$
sur \m{Q_C}. On pose
$$\alpha_1=\lambda_1, \ \ \alpha_2=\mu_2-a\mu_1, \ \ \beta_1=\mu_1,
\ \ \gamma_1=\nu_1.$$
On d\'eduit imm\'ediatement de ce qui pr\'ec\`ede la
\bigskip
\begin{xprop}
Si le complexe \m{(**)} est \m{G'_{red}}-semi-stable
(resp. \m{G'_{red}}-stable) relativement \`a \m{(\alpha_1,\alpha_2,\beta_1,
\gamma_1)}, alors \m{(*)} est \m{G}-semi-stable (resp. \m{G}-stable)
relativement \`a
\noindent \m{(\lambda_1,\mu_1,\mu_2,\nu_1)}.
\end{xprop}
\bigskip
On voit ais\'ement que si \m{Q'^s_C} est non vide, alors on a
$$\mu_2 \ > \ a\mu_1.$$
\subsubsection{La seconde mutation}
\medskip
\noindent{\bf 3.2.2.1 - }{\it D\'efinition}
\medskip
Une seconde mutation donne, partant du complexe \m{(**)},
un complexe du type
$$(***) \ \ \ \ \ \ \ {\cal K}_1\otimes M_2 \ \hfl{\psi}{} \ {\cal E}_1\otimes Q_1
\ \hfl{\psi'}{} \ {\cal F}_1\otimes P_1 \ \hfl{\phi}{} \ {\cal G}_1\otimes N_1,$$
avec
$$Q_1=(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M_2)\oplus L_1.$$
Le groupe op\'erant sur la vari\'et\'e \m{Q''_C} de ces complexes est
$$G''=GL(M_2)\times GL(Q_1)\times GL(P_1)\times GL(N_1),$$
qui est r\'eductif. Notons qu'ici la
mutation est uniquement d\'etermin\'ee (\`a partir de \m{(**)}).
D\'ecrivons maintenant une mutation du complexe \m{(**)}.
Le morphisme \m{\psi}
est \'egal \`a \m{\sigma'\otimes I_{M_2}}, o\`u \m{\sigma'} est l'inclusion \
\m{{\cal K}_1\subset{\cal E}_1\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)}. Posons
$$\psi' \ = \ \pmatrix{\psi'_{11} & \psi'_{12}\cr
\psi'_{21} & \psi'_{22}\cr},$$
relativement aux d\'ecompositions
$$P_1=(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2)\oplus M_1, \ \ \
Q_1=(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M_2)\oplus L_1.$$
On a
$$\pmatrix{\psi'_{12}\cr \psi'_{22}\cr} \ = \ \phi_1.$$
Le morphisme \m{\psi'_{11}} provient de l'application canonique
$$\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\longrightarrow\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2),$$
compte tenu de l'isomorphisme canonique \ \m{\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\simeq
\mathop{\rm Hom}\nolimits({\cal H}_1,{\cal F}_1)^*}. On a enfin \ \m{\psi'_{21}=0}.
\bigskip
\noindent{\bf 3.2.2.2 - }{\it (Semi-)stabilit\'e dans \m{Q''_C}}
\medskip
On d\'efinit comme pour les complexes de type \m{(*)} une notion de
(semi-)stabilit\'e, d\'ependant d'une suite \m{(\delta,\epsilon,\theta,
\rho)} de nombres rationnels telle que
$$\delta\dim(M_2)+\epsilon\dim(Q_1)+\theta\dim(P_1)+\rho\dim(N_1)=0.$$
La \m{G''}-(semi-)stabilit\'e relativement \`a \m{(\delta,
\epsilon,\theta,\rho)} est encore un cas particulier des actions
\'etudi\'ees dans \cite{king} : le complexe \m{(***)} est
\m{G''}-semi-stable (resp. \m{G''}-stable) si pour tous
sous-espaces vectoriels \ \m{M'_2\subset M_2}, \m{Q'_1\subset Q_1},
\m{P'_1\subset P_1}, \m{N'_1\subset N_1}, avec
\noindent\m{(M'_2,Q'_1,P'_1,N'_1)\not = (M_2,Q_1,P_1,N_1)} \ ou
\m{(\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace)}, tels que
$$\psi({\cal K}_1\otimes M'_2)\subset{\cal E}_1\otimes Q'_1, \ \psi'({\cal E}_1\otimes Q'_1)\subset
{\cal F}_1\otimes P'_1, \ \phi({\cal F}_1\otimes P'_1)\subset{\cal G}_1\otimes N'_1,$$
on a
$$\delta\dim(M'_2)+\epsilon\dim(Q'_1)+\theta\dim(P'_1)+
\rho\dim(N'_1)\leq 0 \ \ \ \ {\rm(resp. \ } \ < \ {\rm )}.$$
Soient \ \m{L'_1\subset L_1}, \m{M'_1\subset M_1}, \m{M'_2\subset M_2},
\m{N'_1\subset N_1} \ des sous-espaces vectoriels tels que
$$(f_1,f_2)({\cal E}_1\otimes L'_1)\subset ({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M'_2), \ \ \
(g_1,g_2)(({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M'_2))\subset {\cal G}_1\otimes N'_1.$$
On pose
$$P'_1=(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M'_2)\oplus M'_1, \ \
Q'_1=(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M'_2)\oplus L'_1.$$
Comme dans le \paragra~\hskip -2pt 3.2.1.2, le rel\`evement de \m{f_2} peut \^etre choisi
de telle sorte qu'il envoie \m{{\cal E}_1\otimes L'_1} dans
\m{{\cal F}_1\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M'_2}. On a alors
$$\psi({\cal K}_1\otimes M'_2)\subset{\cal E}_1\otimes Q'_1, \ \psi'({\cal E}_1\otimes Q'_1)\subset
{\cal F}_1\otimes P'_1, \ \phi({\cal F}_1\otimes P'_1)\subset{\cal G}_1\otimes N'_1.$$
Soit \m{(\lambda_1,\mu_1,\mu_2,\nu_1)} une polarisation de l'action de $G$
sur \m{Q_C}. On pose
$$\delta=\mu_2-a\mu_1-b\lambda_1, \ \ \epsilon=\lambda_1, \ \ \theta=\mu_1,
\ \ \rho=\nu_1.$$
On d\'eduit imm\'ediatement de ce qui pr\'ec\`ede la
\bigskip
\begin{xprop}
Si le complexe \m{(***)} est \m{G''}-semi-stable
(resp. \m{G''}-stable) relativement \`a \m{(\delta,\epsilon,\theta,
\rho)}, alors \m{(*)} est \m{G}-semi-stable (resp. \m{G}-stable)
relativement \`a \m{(\lambda_1,\mu_1,\mu_2,\nu_1)}.
\end{xprop}
On voit ais\'ement que si \m{Q''^s_C} est non vide, alors on a
$$\mu_2 \ > \ a\mu_1+b\lambda_1.$$
Notons que si \m{Q_C^s} est non vide, cette condition est plus forte
que celle que l'on avait trouv\'ee en supposant que \m{Q'^s_C} est non
vide.
\bigskip
\subsection{Cas d'\'equivalence des (semi-)stabilit\'es et construction des
vari\'et\'es de modules}
\subsubsection{D\'efinitions de constantes}
On d\'efinit ici des constantes qui interviendront par la suite. Elles
ont d\'ej\`a \'et\'e d\'efinies et utilis\'ees sans \cite{dr_tr}.
On consid\`ere l'application canonique
$$\tau : \mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\longrightarrow\mathop{\rm Hom}\nolimits({\cal F}_2,{\cal G}_1)^*.$$
Pour tout entier positif $k$, soit
$$\tau_k = \tau\otimes I_{\scx{k}} :
\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*\otimes(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes\cx{k})
\longrightarrow\mathop{\rm Hom}\nolimits({\cal F}_2,{\cal G}_1)^*\otimes\cx{k}.$$
Soit \m{G_k} l'ensemble des sous-espaces vectoriels propres \m{K} de
\m{\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes\cx{k}} tels que pour tout sous-espace vectoriel
propre \ \m{V\subset\cx{k}}, \m{K} ne soit pas contenu dans
\noindent\m{\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes V}. On pose
$$c_1(k)= \mathop{\hbox{$\sup$}}\limits_{K\in{G_k}}(\q{\mathop{\rm codim}\nolimits(
\tau_k(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes K))}{\mathop{\rm codim}\nolimits(K)}).$$
On d\'efinit de m\^eme \m{c_2(k)}, qui correspond \`a l'application
canonique
$$\tau' : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\longrightarrow\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2),$$
provenant de l'isomorphisme canonique \
\m{\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\simeq\mathop{\rm Hom}\nolimits({\cal H}_1,{\cal F}_1)^*}.
Il est clair qu'on a
$$c_1(k+1)\geq c_1(k), \ \ c_2(k+1)\geq c_2(k).$$
\subsubsection{Equivalence des (semi-)stabilit\'es}
\begin{xprop}
On suppose que
$$\mu_2 \ \geq \ (b-\q{a}{c_2(m_2)})\lambda_1-ac_1(m_2)\nu_1.$$
Alors le complexe \m{(*)} est \m{G}-semi-stable relativement \`a
\m{(\lambda_1,\mu_1,\mu_2,\nu_1)} si et seulement si \m{(***)} est
\m{G''}-semi-stable relativement \`a \m{(\delta,\epsilon,\theta,\rho)}.
\end{xprop}
\bigskip
\noindent{\em D\'emonstration}. D'apr\`es la proposition 4.2, il suffit de montrer que si \m{(***)}
n'est pas \m{G''}-semi-stable, \m{(*)} n'est pas \m{G}-semi-stable.
On suppose que le complexe \m{(***)} n'est pas semi-stable.
Soient \ \m{M'_2\subset M_2}, \m{Q'_1\subset Q_1}, \m{P'_1\subset P_1},
\m{N'_1\subset N_1} \ des sous-espaces vectoriels, de dimensions respectives
\m{m'_2}, \m{q'_1}, \m{p'_1}, \m{n'_1}, tels que
$$\psi({\cal K}_1\otimes M'_2)\subset{\cal E}_1\otimes Q'_1, \ \
\psi'({\cal E}_1\otimes Q'_1)\subset{\cal F}_1\otimes P'_1, \ \ \phi({\cal F}_1\otimes P'_1)\subset
{\cal G}_1\otimes N'_1,$$
et
$$s \ = \ \delta m'_1+\epsilon q'_1+\theta p'_1+\rho n'_1 \ > \ 0.$$
En faisant agir le sous-groupe unipotent $H$ de $G$ on se ram\`ene au cas
o\`u
$$P'_1 \ = \ X\oplus M'_1,$$
\m{X} \'etant un sous-espace vectoriel de \ \m{\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2}.
En changeant \'eventuellement le rel\`evement de \m{f_2} servant \`a
d\'efinir \m{\phi_1}, on se ram\`ene au cas o\`u
$$Q'_1 \ = \ Y\oplus L'_1,$$
\m{Y} \'etant un sous-espace vectoriel de \ \m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M_2}.
Soit \m{M''_2} le plus petit sous-espace vectoriel de \m{M_2} tel que
$$Y\subset \mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M''_2.$$
Montrons qu'on peut aussi supposer que \m{M''_2} est le plus petit
sous-espace vectoriel de \m{M_2} tel que
$$X\subset \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M''_2.$$
D'abord on a \ \m{X\subset \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M''_2} : cela d\'ecoule du fait
que pour toute droite \m{L} de \m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)}, la
restriction de \m{\tau'} \`a \ \m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes L} \ est non nulle
(ceci r\'esultant du fait que \ \m{{\cal H}_1\subset{\cal F}_1\otimes\mathop{\rm Hom}\nolimits({\cal H}_1,{\cal F}_1)^*}.
D'autre part on peut, puisque \ \m{\epsilon=\lambda_1>0}, remplacer \m{X}
par \ \m{{\psi'_{11}}^{-1}(Y)\cap(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1)\otimes M''_2)}, ce qui
assure la minimalit\'e de \m{M''_2}.
Soit
$$N''_1 \ = \ \phi((\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M''_2)\oplus M'_1).$$
Alors on a
$$(f_1,f_2)({\cal E}_1\otimes L'_1)\subset ({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M''_2), \ \ \
(g_1,g_2)(({\cal F}_1\otimes M'_1)\oplus({\cal F}_2\otimes M''_2))\subset {\cal G}_1\otimes N''_1.$$
Posons \ \m{l'_1=\dim(L'_1)}, \m{m'_1=\dim(M'_1)}, \m{n''_1=\dim(N''_1)},
\m{m''_2=\dim(M''_2)} \ et
$$t \ = \ \lambda_1l'_1+\mu_1m'_1+\mu_2m''_2+\nu_1n''_1.$$
Alors on a
$$t \ = s + \delta(m''_2-m'_2)
+\epsilon(bm''_2-x)+\theta(am''_2-y)+\rho(n''_1-n'_1).$$
Pour montrer que \m{(*)} n'est pas semi-stable, il suffit donc de montrer que
$$u \ = \ \delta(m''_2-m'_2)+\epsilon(bm''_2-x)+\theta(am''_2-y)
+\rho(n''_1-n'_1) \ \geq \ 0.$$
C'est \'evident si \ \m{am''_2-y=0}, car alors \ \m{n''_1-n'_1=0}. On
peut donc supposer que \ \m{am''_2-y>0}, ce qui entraine \
\m{c_2(m''_2)>0}. On a alors
$$m''_2-m'_2\geq\q{am''_2-y}{a}, \ \
bm''_2-x\geq\q{am''_2-y}{c_2(m''_2)}, \ \
n''_1-n'_1\leq c_1(m''_2)(am''_2-y).$$
Donc
$$u \ \geq (am''_2-y)(\q{\delta}{a}+\q{\epsilon}{c_2(m''_2)}+\theta
+\rho c_1(m''_2)),$$
c'est-\`a-dire
$$u \ \geq (am''_2-y)(\q{\mu_2}{a}-(\q{b}{a}-\q{1}{c_2(m''_2)})\lambda_1
+\nu_1c_1(m''_2)).$$
On en d\'eduit la proposition. $\Box$
\bigskip
\noindent{\bf Remarque : } En g\'en\'eral, on a
$$b-\q{a}{c_2(m_2)} \ \leq \ 0.$$
C'est le cas par exemple si \m{\tau'} est stable pour l'action du groupe
r\'eductif
\noindent \m{SL(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1))\times SL(\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2))}.
\subsection{Projectivit\'e des vari\'et\'es de modules}
\subsubsection{D\'efinition d'une constante}
Soient
$$\tau'' : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\longrightarrow
\mathop{\rm Hom}\nolimits({\cal F}_2,{\cal G}_1)^*$$
l'application lin\'eaire d\'efinie par :
$$\tau''=\tau\circ(\tau'\otimes I_{\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*}),$$
et pour tout entier positif \m{k},
$$\tau''_k=\tau''\otimes I_{\scx{k}}.$$
Soit \m{G''_k} l'ensemble des sous-espaces vectoriels propres \m{K} de
\ \m{\mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes\cx{k}} \ tels que pour tout sous-espace
vectoriel propre $V$ de \m{\cx{k}}, $K$ ne soit pas contenu dans
\noindent\m{\mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes V}. On pose
$$c(k)= \mathop{\hbox{$\sup$}}\limits_{K\in{G''_k}}(\q{\mathop{\rm codim}\nolimits(
\tau''_k(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*\otimes K))}{\mathop{\rm codim}\nolimits(K)}).$$
On montre ais\'ement qu'on a
$$c(k) \ \leq \ c_1(k)c_2(k).$$
\subsubsection{Projectivit\'e}
\begin{xtheo}
Il existe un quasi-bon quotient \ \m{Q^{ss}_C//G}, qui est une vari\'et\'e
projective, si les conditions suivantes sont r\'ealis\'ees :
\noindent 1 - On a
$$\mu_2-a\nu_1c_1(m_2)>0, \ \ \mu_2-a\mu_1-b\lambda_1>0.$$
\noindent 2 - Si \ \m{\mu_1 < 0} \ et \ \m{\lambda_1+\nu_1c_(m_2)+
\mu_1c_2(m_2)<0}, alors
$$\mu_2-a\mu_1+b\nu_1c(m_2)+b\mu_1c_2(m_2)\geq 0.$$
\end{xtheo}
\bigskip
\noindent{\em D\'emonstration}.
Le bon quotient \m{{Q''}^{ss}_C//G} existe et est projectif, car \m{G''}
est r\'eductif. D'apr\`es le th\'eor\`eme 3.8 et la remarque qui suit la
proposition 4.3, le quasi-bon quotient \m{Q^{ss}_C//G} existe et est projectif
si \m{{Q''}^{ss}_C} est contenu dans l'ouvert
$U$ de \m{Q''_C} contenant les orbites des mutations des points de \m{Q_C}.
Compte tenu des deux \'etapes de mutations, on va caract\'eriser les
complexes
$${\cal K}_1\otimes M_2 \ \hfl{\psi}{} \ {\cal E}_1\otimes Q_1
\ \hfl{\psi'}{} \ {\cal F}_1\otimes P_1 \ \hfl{\phi}{} \ {\cal G}_1\otimes N_1$$
non contenus dans $U$. Soit
$$f : \mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes M_2\longrightarrow Q_1$$
l'application lin\'eaire d\'eduite de \m{\psi}. L'application lin\'eaire
$$g : \mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes M_2\longrightarrow P_1$$
d\'eduite de \m{\psi} et \m{\psi'} se factorise de la fa\c con suivante :
$$\mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes M_2 \ \hfl{\tau'}{} \
\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)\otimes M_2 \ \hfl{f'}{} \ P_1,$$
compte tenu des isomorphismes
$$\mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\simeq \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal H}_1), \ \ \
\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal F}_2)^*\simeq \mathop{\rm Hom}\nolimits({\cal H}_1,{\cal F}_1).$$
Alors, \m{(\psi,\psi',\phi)} n'est pas dans $U$ si et seulement si
une des deux propri\'et\'es suivantes est v\'erifi\'ee:
\begin{itemize}
\item[--] L'application lin\'eaire $f$ n'est pas injective.
\item[--] L'application lin\'eaire $f$ est injective, et $f'$ n'est pas
injective.
\end{itemize}
\bigskip
Supposons que $f$ n'est pas
injective. Soit \m{M'_2} le plus petit sous-espace vectoriel
de \m{M_2} tel que
$$\ker(f) \ \subset \ \mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes M'_2.$$
Soient
$$d=\dim(\ker(f)), \ \ p = \dim(\ker(\tau')\otimes M'_2 \ + \
\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\ker(f)), \ \ m'_2=\dim(M'_2).$$
Alors on a
$$bm'_2\dim(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1))-p \ = \
\mathop{\rm codim}\nolimits((\tau'\otimes I_{M'_2})(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\ker(f))).$$
Il en d\'ecoule que
$$bm'_2\dim(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1))-p \ \leq \ c_2(m'_2)(bm'_2-d).$$
Soit
$$F : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes Q_1\longrightarrow P_1$$
l'application lin\'eaire d\'eduite de \m{\psi'}, et
$$G : \mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*\otimes P_1\longrightarrow N_1$$
l'application lin\'eaire d\'eduite de \m{\phi}. Posons
$$Q'_1=\mathop{\rm Im}\nolimits(f), \ \ P'_1=F(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes Q'_1).$$
Alors on a
$$\dim(Q'_1)=bm'_2\dim(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1))-p.$$
Il r\'esulte de la factorisation pr\'ec\'edente de $g$ qu'on a
$$\dim(P'_1)\leq c_2(m'_2)(bm'_2-d).$$
De m\^eme, l'application lin\'eaire
$$\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal K}_1,{\cal E}_1)^*\otimes M_2\longrightarrow
N_1$$
d\'eduite de \m{\psi}, \m{\psi'}, \m{\phi} se factorise par
\ \m{\tau''\otimes I_{M_2}}, et il en d\'ecoule que
$$\dim(N'_1)\leq c(m'_2)(bm'_2-d).$$
Posons
$$x = \delta\dim(M'_2)+\epsilon\dim(Q'_1)+\theta\dim(P'_1)+\rho\dim(N'_1).$$
Il faut montrer que \ \m{x > 0}, ce qui montrera que \m{(\psi,\psi',\phi)}
n'est pas semi-stable. On a
$$x \geq (\mu_2-a\mu_1-b\lambda_1)m'_2+(\lambda_1+\nu_1c(m'_2))(bm'_2-d)
+\mu_1\dim(P'_1).$$
Il d\'ecoule ais\'ement des conditions 2- et 3- du th\'eor\`eme que
\ \m{x > 0}.
Si $f$ est injective, mais pas $f'$, on d\'eduit de m\^eme de la condition
1- du th\'eor\`eme que \m{(\psi,\psi',\phi)} n'est pas semi-stable.
$\Box$
\subsection{Exemple}
Soient \ \m{n\geq 2} \ un entier et $V$ un espace vectoriel de dimension
n+1. On va \'etudier des complexes sur \ \m{\proj{n}=\projx{V}}, du type
$${\cal O}(-2)\longrightarrow ({\cal O}(-1)\otimes M_1)\oplus{\cal O}\longrightarrow{\cal O}(1),$$
o\`u $M_1$ est un espace vectoriel tel que \ \m{0<\dim(M_1)\leq n+1}.
On a
$$c_1(1)=0, \ \ c_2(1)=\q{2}{n+2}, \ \ c(1)=0.$$
Compte tenu du th\'eor\`eme 4.4, en supposant que \ \m{\lambda_1=1}, on
obtient un quasi-bon quotient projectif d\`es que \m{\mu_1}, \m{\mu_2} sont
positifs, \m{\nu_1} n\'egatif, et
$$\mu_2>(n+1)\mu_1+\q{n(n+1)}{2}.$$
Pour ces valeurs de la polarisation, les quotients sont tous les m\^emes,
On n'obtient donc dans ce cas qu'un seul quasi-bon quotient projectif.
Traitons plus pr\'ecis\'ement le cas de \m{\proj{2}}, avec \ \m{m_1=3}.
On montre ais\'ement que \m{Q_C} est irr\'eductible. En cas d'existence
de points stables, les quotients seront donc des vari\'et\'es
irr\'eductibles de dimension 6.
Un complexe
$${\cal O}(-2)\longrightarrow({\cal O}(-1)\otimes\cx{3})\oplus{\cal O}\longrightarrow{\cal O}(1)$$
\'equivaut \`a une paire
$$((z_1,z_2,z_3,q_0),(q_1,q_2,q_3,z_0)),$$
o\`u \ \m{z_i\in V^*}, \m{q_i\in S^2V^*}, et
$$q_0z_0+q_1z_1+q_2z_2+q_3z_3=0.$$
L'ouvert de \m{Q_C} o\`u \m{z_1}, \m{z_2}, \m{z_3} sont lin\'eairement
ind\'ependants est non vide. Pour un complexe dans cet ouvert, on peut se
ramener, en faisant agir le sous-groupe unipotent maximal, au cas o\`u
\ \m{q_0=0}. Il en d\'ecoule imm\'ediatement que s'il existe des complexes
stables, on doit avoir
$$\mu_2 \ > \ 0.$$
Il n'existe en fait que trois quotients projectifs distincts, qui
correspondent aux cas o\`u \m{\mu_1} est n\'egatif, nul, ou positif.
Dans les deux cas o\`u \m{\mu_1} est non nul, la semi-stabilit\'e d'un
complexe entraine sa stabilit\'e.
Le cas o\`u \ \m{\mu_1>0} \ est celui qu'on peut traiter en appliquant
directement le th\'eor\`eme 4.4. Dans ce cas un complexe $x$ est stable
seulement si le morphisme de gauche est non nul, \m{z_0\not = 0}, et si
pour tout complexe dans la $H$-orbite de $x$, d\'efini par la paire
$$((z_1,z_2,z_3,q'_0),(q'_1,q'_2,q'_3,z_0)),$$
\m{q'_1}, \m{q'_2} et \m{q'_3} sont lin\'eairement ind\'ependants.
Le quotient existe et est projectif. Il est non vide, car si
\m{(z_1,z_2,z_3)} est une base de \m{V^*}, le complexe d\'efini par
$$((z_1,z_2,z_3),(z_3^2,-z_2z_3,z_2^2-z_1z_3,z_1))$$
est stable.
Le cas o\`u \ \m{\mu_1<0} \ ne peut pas \^etre trait\'e directement
(mais on peut le faire en consid\'erant les complexes duaux). Dans ce cas,
un complexe d\'efini par
$$((z_1,z_2,z_3,q_0),(q_1,q_2,q_3,z_0))$$
est stable si et seulement si \m{(z_1,z_2,z_3)} est une base de $V^*$.
On peut donner une description compl\`ete du quotient, qui est isomorphe
\`a la vari\'et\'e $X$ suivante : soient \m{(z_1,z_2,z_3)} une base de
\m{V^*}, $E$ le sous-espace vectoriel de \ \m{S^2V^*\otimes\cx{3}} \
constitu\'e des triplets \m{(q_1,q_2,q_3)} tels que
$$z_1q_1+z_2q_2+z_3q_3=0,$$
$H'$ le sous-espace vectoriel de \ \m{V^*\otimes\cx{3}} \ constitu\'e des
triplets \m{(\phi_1,\phi_2,\phi_3)} tels que
$$\phi_1z_1+\phi_2z_2+\phi_3z_3=0.$$
On a un morphisme injectif de fibr\'es vectoriels sur \
\m{\proj{2}=\projx{V^*}} :
$$\Phi : {\cal O}(-1)\otimes H'\longrightarrow{\cal O}\otimes E$$
$$z_0\otimes(\phi_1,\phi_2,\phi_3)\longmapsto (z_0\phi_1,z_0\phi_2,z_0\phi_3)$$
On prend alors
$$X \ = \ \projx{\mathop{\rm coker}\nolimits(\Phi))}.$$
\vfill\eject
\section{Vari\'et\'es de modules de morphismes}
Soient \m{{\cal E}_1}, \m{{\cal E}_2}, \m{{\cal F}_1} des faisceaux coh\'erents sur une
vari\'et\'e projective $X$, et \m{M_1}, \m{M_2}, \m{N_1} des espaces
vectoriels de dimension finie. On pose
$$a=\dim(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)), \ \ m_1=\dim(M_1), \ \ m_2=\dim(M_2),
\ n_1=\dim(N_1).$$
On s'int\'eresse \`a des morphismes du
type
$$(*) \ \ \ \ \ \ \
({\cal E}_1\otimes M_1)\oplus({\cal E}_2\otimes M_2)\ \hfl{(f_1,f_2)}{} \ {\cal F}_1\otimes N_1.$$
On fait les hypoth\`eses suivantes :
\begin{itemize}
\item[--] Les faisceaux ${\cal E}_1$, ${\cal E}_2$, ${\cal F}_1$ sont simples, et
$$\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal E}_1)=\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal E}_1)=\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal E}_2)=\lbrace 0\rbrace.$$
\item[--] Le morphisme canonique
$${\cal E}_1\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\longrightarrow{\cal E}_2$$
est surjectif. On note ${\cal H}_1$ son noyau.
\item[--] On a \ $\mathop{\rm Ext}\nolimits^1({\cal E}_2,{\cal E}_1)=\lbrace 0\rbrace$, ce qui entraine un
isomorphisme canonique
$$\mathop{\rm Hom}\nolimits({\cal H}_1,{\cal E}_1)\simeq\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*.$$
\end{itemize}
\bigskip
On se trouve dans la situation de la proposition 3.1, avec \ \m{{\cal E}=0},
\m{\Gamma={\cal E}_1}, \m{M=M_1}, \m{{\cal G}={\cal E}_2\otimes M_2}, \m{{\cal F}={\cal F}_1\otimes N_1}. On a
dans ce cas
$$Q_C=W_C=\mathop{\rm Hom}\nolimits(({\cal E}_1\otimes M_1)\oplus({\cal E}_2\otimes M_2),{\cal F}_1\otimes N_1).$$
Le groupe $G$ op\'erant sur $W_C$ est
$$G \ = \ \mathop{\rm Aut}\nolimits(({\cal E}_1\otimes M_1)\oplus({\cal E}_2\otimes M_2))\times GL(N_1).$$
On peut voit \ \m{\mathop{\rm Aut}\nolimits(({\cal E}_1\otimes M_1)\oplus({\cal E}_2\otimes M_2))} \ comme
constitu\'e de matrices du type
$$\pmatrix{g_1 & 0 \cr \phi & g_2\cr},$$
avec \ \m{g_1\in GL(M_1)}, \m{g_2\in GL(M_2)} \ et \
\m{\phi\in\mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M_1,M_2)}.
Les mutations des morphismes \m{(*)} sont d\'ecrites au \paragra~\hskip -2pt 5.2. On les
appelle des {\em mutations directes} car elles nous
ram\`enent imm\'ediatement
\`a une action d'un groupe r\'eductif, et permettent de construire les
vari\'et\'es de modules de morphismes. C'est la m\'ethode appliqu\'ee dans
\cite{dr_tr} (les r\'esultats sont rappel\'es dans le \paragra~\hskip -2pt 5.2).
On peut appliquer la proposition 3.1 d'une autre mani\`ere. C'est
simple \`a voir si \m{{\cal E}_1}, \m{{\cal E}_2} et \m{{\cal F}_1} sont localement libres :
les complexes de type \m{(*)} sont \'equivalents \`a des complexes du
type
$${\cal F}_1^*\otimes N_1^*\longrightarrow ({\cal E}_2^*\otimes M_2^*)\oplus({\cal E}_1^*\otimes M_1^*),$$
et on applique la proposition 2.1 en prenant \ \m{{\cal E}={\cal F}_1^*\otimes N_1^*},
\m{\Gamma={\cal E}_2}, \m{M=M_2^*},
\noindent\m{{\cal G}={\cal E}_1^*\otimes M_1^*}, \m{{\cal F}=0}. Les
mutations obtenues sont des morphismes de type (*) (avec d'autres
faisceaux et d'autres espaces vectoriels). On peut ensuite appliquer la
mutation directe \`a ces morphismes, et on obtient de nouveaux quotients
(cf. \paragra~\hskip -2pt 5.3).
\subsection{Notions de (semi-)stabilit\'e et vari\'et\'es de modules de
mor-\break phismes}
Le groupe $G$ poss\`ede deux sous-groupes importants : le premier est le
sous-groupe normal unipotent maximal \'evident $H$, isomorphe au groupe
additif
\noindent
\m{\mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M_1,M_2))}. Le second est le sous-groupe
r\'eductif
$$G_{red} \ = \ GL(M_1)\times GL(M_2)\times GL(N_1),$$
dont l'inclusion dans $G$ induit un isomorphisme \
\m{G_{red}\simeq G/H}.
L'action de \m{G_{red}} est un cas particulier des actions \'etudi\'ees
dans \cite{king}. Soient \m{\lambda_1}, \m{\lambda_2}, \m{\mu_1}
des nombres rationnels non nuls tels que
$$\lambda_1\dim(M_1)+\lambda_2\dim(M_2)-\mu_1\dim(N_1)=0.$$
\bigskip
\begin{defin}
Un point \m{(f_1,f_2)} de $W_C$ est dit {\em $G_{red}$-semi-stable} (resp.
{\em $G_{red}$-stable}) relativement \`a \m{(\lambda_1,\lambda_2,\mu_1)} si
pour tous sous-espaces vectoriels \ \m{M'_1\subset M_1},
\m{M'_2\subset M_2}, \m{N'_1\subset N_1}, avec \
\m{(M'_1,M'_2,N'_1)\not = (M_1,M_2,N_1)} \ ou \m{(\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace)}, tels que
$$f_1({\cal E}_1\otimes M'_1)\subset {\cal F}_1\otimes N'_1, \ \ \
f_2({\cal E}_2\otimes M'_2)\subset {\cal F}_1\otimes N'_1,$$
on a
$$\lambda_1\dim(L'_1)+\lambda_2\dim(M'_2)-\mu_1\dim(N'_1)\leq 0
\ \ \ {\rm (resp. \ \ } \ < \ {\rm )}.$$
\end{defin}
\bigskip
On dit que \m{(\lambda_1, \lambda_2, \mu_1)} est une {\em polarisation}
de l'action de $G$ sur $W_C$. On note \m{W^{ss}_{C,red}} (resp.
\m{W^s_{C,red}}) l'ouvert de \m{W_C} constitu\'e
des points \m{G_{red}}-semi-stables (resp. \m{G_{red}}-stables). D'apr\`es
\cite{king}, il existe un bon quotient \ \m{W^{ss}_{red,C}//G} \ et un
quotient g\'eom\'etrique lisse \m{W^s_{C,red}/G}. Mais ce ne sont pas les
quotients que nous recherchons.
\bigskip
\begin{defin}
Un point \m{x} de $W_C$ est dit {\em $G$-semi-stable} (resp.
{\em $G$-stable}) relativement \`a \m{(\lambda_1,\lambda_2,\mu_1)} si
tous les points de l'orbite \m{H.x} sont $G_{red}$-semi-stables (resp.
$G_{red}$-stables).
\end{defin}
\bigskip
On note \m{W_C^{ss}} (resp. \m{W_C^s}) l'ouvert de \m{W_C} constitu\'e
des points \m{G}-semi-stables (resp. \m{G}-stables). On cherche
\`a prouver l'existence de bons quotients \ \m{W_C^{ss}//G}. De tels
quotients seront appel\'es des {\em vari\'et\'es de modules de complexes}.
\medskip
On montre ais\'ement que si \m{W^{s}} est non vide on doit avoir
$$\lambda_1>0, \ \ \lambda_2>0, \ \ \mu_1>0.$$
On supposera par la suite que ces in\'egalit\'es sont v\'erifi\'ees.
On peut alors {\em normaliser} la polarisation, c'est-\`a-dire supposer
que
$$\lambda_1\dim(M_1)+\lambda_2\dim(M_2)=1, \ \ \mu_1=\q{1}{\dim(M_1)}.$$
\subsection{Construction des vari\'et\'es de modules par les
mutations di-\break rectes}
La mutation de \m{(*)} est un complexe
$$(**) \ \ \ \ \ \ \ \
{\cal H}_1\otimes M_2\ \hfl{\phi_1}{} \ {\cal E}_1\otimes P_1\ \hfl{\phi_2}{} \ {\cal F}_1\otimes N_1,$$
avec
$$P_1 \ = \ (\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes M_2)\oplus M_1.$$
Le morphisme \m{\phi_1} est d\'efini par l'inclusion
$$\mathop{\rm Hom}\nolimits({\cal H}_1,{\cal E}_1)^*\otimes M_2=\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes M_2 \ \subset \ P_1.$$
Le morphisme \m{\phi_2} provient d'une application lin\'eaire
$$F : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes P_1\longrightarrow N_1,$$
qui est \'egale \`a celle d\'eduite de \m{f_1} sur \
\m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes M_1}. Sur l'autre facteur, $F$ est la compos\'ee
$$\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes M_2\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes M_2
\longrightarrow N_1,$$
la premi\`ere application provenant de la composition
$$\sigma : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1),$$
et la seconde de \m{f_2}.
Le groupe op\'erant sur la vari\'et\'e \m{Q'_C} des complexes \m{(**)} est
$$G' \ = \ GL(M_2)\times GL(P_1)\times GL(N_1),$$
qui est r\'eductif. Une notion de (semi-)stabilit\'e pour les points de
\m{Q'_C} est d\'efinie par une suite \m{(\alpha,\beta,\gamma)} de nombres
rationnels non nuls telle que
$$\alpha\dim(M_2)+\beta\dim(P_1)+\gamma\dim(N_1) = 0.$$
Un complexe \m{(\phi'_1,\phi'_2)} de type \m{(**)}
est semi-stable (resp.stable) relativement \`a \m{(\alpha,\beta,\gamma)} si
et seulement si pour tous sous-espaces vectoriels
$$M'_2\subset M_2, \ \ P'_1\subset P_1, \ \ N'_1\subset N_1,$$
avec \ \m{(M'_2,P'_1,N'_1)\not = (M_2,P_1,N_1)} \ ou \m{(\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace)},
tels que
$$\phi'_1({\cal H}_1\otimes M'_2)\subset{\cal E}_1\otimes P'_1, \ \
\phi'_2({\cal E}_1\otimes P'_1)\subset{\cal F}_1\otimes N'_1,$$
on a
$$\alpha\dim(M'_2)+\beta\dim(P'_1)+\gamma\dim(N'_1)\leq 0 \ \ \
\ {\rm (resp. \ } < \ {\rm )}.$$
S'il existe des complexes stables on doit avoir
$$\alpha > 0, \ \ \gamma>0.$$
Soient \ \m{M'_1\subset M_1}, \m{M'_2\subset M_2}, \m{N'_1\subset N_1} \
des sous-espaces vectoriels tels que
$$f_1({\cal E}_1\otimes M'_1)\subset{\cal F}_1\otimes N'_1, \ \
f_2({\cal E}_2\otimes M'_2)\subset{\cal F}_1\otimes N'_1.$$
On pose
$$P'_1 \ = \ (\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes M'_2)\oplus M'_1.$$
On a alors
$$\phi_1({\cal H}_1\otimes M'_2)\subset{\cal E}_1\otimes P'_1, \ \
\phi_2({\cal E}_1\otimes P'_1)\subset{\cal F}_1\otimes N'_1.$$
On suppose que
$$\alpha=\lambda_2-a\lambda_1, \ \beta=\lambda_1, \ \gamma=-\mu_1.$$
On a alors
$$\alpha\dim(M'_2)+\beta\dim(P'_1)+\gamma\dim(N'_1) \ = \
\lambda_1\dim(M'_1)+\lambda_2\dim(M'_2)-\mu_1\dim(N'_1).$$
On en d\'eduit imm\'ediatement la
\bigskip
\begin{xprop}
Si le complexe \m{(**)} est semi-stable (resp. stable) relativement \`a
\noindent
\m{(\alpha,\beta,\gamma)}, le complexe (*) est $G$-semi-stable (resp.
$G$-stable) relativement \`a \m{(\lambda_1,\lambda_2,\mu_1)}.
\end{xprop}
\bigskip
Pour continuer on utilise une constante analogue \`a celles qui ont \'et\'e
d\'efinies au \paragra~\hskip -2pt 4.3.1. On consid\`ere l'application canonique
$$\tau : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*.$$
Pour tout entier positif $k$, soit
$$\tau_k = \tau\otimes I_{\scx{k}} :
\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\cx{k})
\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes\cx{k}.$$
Soit \m{G_k} l'ensemble des sous-espaces vectoriels propres \m{K} de
\m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\cx{k}} tels que pour tout sous-espace vectoriel
propre \ \m{V\subset\cx{k}}, \m{K} ne soit pas contenu dans
\noindent\m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes V}. On pose
$$c_0(k)= \mathop{\hbox{$\sup$}}\limits_{K\in{G_k}}(\q{\mathop{\rm codim}\nolimits(
\tau_k(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes K))}{\mathop{\rm codim}\nolimits(K)}).$$
On d\'emontre dans \cite{dr_tr} le
\bigskip
\begin{xtheo}
Si
$$\q{\lambda_2}{\lambda_1}>a, \ \ \ \lambda_2>\q{a}{n_1}c_0(m_2),$$
il existe un bon quotient projectif \ \m{W_C^{ss}//G}, et un quotient
g\'eom\'etrique \ \m{W_C^{s}/G}, qui est un ouvert de \ \m{W_C^{ss}//G}.
\end{xtheo}
\bigskip
On montre en fait que sous les hypoth\`eses du th\'eor\`eme, la
r\'eciproque de la proposition 5.1 est vraie. On peut conclure en
utilisant le th\'eor\`eme 3.8.
\subsection{Mutations indirectes}
\subsubsection{D\'efinition et premi\`eres propri\'et\'es}
On utilise maintenant la deuxi\`eme mani\`ere d'effectuer des mutations
constructives.\break
Comme indiqu\'e au d\'ebut du \paragra~\hskip -2pt 5, c'est plus facile \`a
voir si \m{{\cal E}_1}, \m{{\cal E}_2} et \m{{\cal F}_1} sont localement libres. Mais les
constructions \'etant purement formelles, il n'est pas n\'ecessaire de faire
cette supposition. On fait les hypoth\`eses suppl\'ementaires suivantes :
\begin{itemize}
\item[--] Le morphisme canonique
$${\cal E}_1\longrightarrow{\cal E}_2\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*$$
est injectif. Soit ${\cal G}_1$ son conoyau.
\item[---] La composition
$$\sigma : \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)$$
est surjective.
\item[--] On a \ $\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_1)=\lbrace 0\rbrace$.
\end{itemize}
\bigskip
\noindent Cette derni\`ere hypoth\`ese n'est pas strictement indispensable,
on le verra plus loin.
On a, puisque \ \m{\mathop{\rm Ext}\nolimits^1({\cal E}_2,{\cal E}_1)=\lbrace 0\rbrace}, un
isomorphisme canonique
$$\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal G}_1) \ \simeq \ \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*.$$
Une mutation d'un morphisme \m{(*)} est un morphisme
$$(***) \ \ \ \ \ \ \
{\cal E}_2\otimes Q_1 \ \hfl{(\psi_1,\psi_2)}{} \ ({\cal G}_1\otimes M_1)\oplus({\cal F}_1\otimes N_1),$$
avec
$$Q_1 \ = \ (\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M_1)\oplus M_2.$$
Le morphisme \m{\psi_1} est nul sur \ \m{{\cal E}_2\otimes M_2}, et sur l'autre facteur
c'est
$$ev\otimes I_{M_1} : {\cal E}_2\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M_1 =
{\cal E}_2\otimes\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal G}_1)\otimes M_1\longrightarrow{\cal G}_1\otimes M_1,$$
\m{ev} d\'esignant le morphisme d'\'evaluation. Le morphisme \m{\psi_2} est
\'egal \`a \m{f_2} sur
\noindent \m{\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes M_2}, et sur l'autre facteur
c'est un rel\`evement de \m{f_1}, qui existe \`a cause de la seconde
hypoth\`ese suppl\'ementaire. Remarquons que \m{\psi_2} est d\'efini \`a un
\'el\'ement pr\`es de \m{\mathop{\rm Hom}\nolimits({\cal G}_1\otimes M_1, {\cal F}_1\otimes N_1)}.
Soit \m{W'_C} l'espace vectoriel des morphismes de type \m{(***)}, \m{G'} le
groupe op\'erant sur \m{W'_C} :
$$G' \ = \ GL(Q_1)\times\mathop{\rm Aut}\nolimits(({\cal G}_1\otimes M_1)\oplus({\cal F}_1\otimes N_1)).$$
Si on omet la derni\`ere hypoth\`ese suppl\'ementaire, il faut remplacer
\m{G'} par un groupe plus petit. Soient \m{H'} le sous-groupe normal unipotent
maximal de \m{G'}, isomorphe au groupe additif
\m{\mathop{\rm Hom}\nolimits({\cal G}_1\otimes M_1, {\cal F}_1\otimes N_1)}, et \m{G'_{red}} le sous-groupe r\'eductif :
$$G'_{red} \ = \ GL(Q_1)\times GL(M_1)\times GL(N_1).$$
Soient \m{\nu_1}, \m{\nu_2} des nombres rationnels positifs tels que
$$\nu_1\dim(N_1)+\nu_2\dim(M_1) = 1.$$
Alors \m{(1/\dim(Q_1),\nu_2,\nu_1)} d\'efinit une notion de semi-stabilit\'e
pour l'action de \m{G'} sur \m{W'_C}. Rappelons qu'un point
\m{(\psi'_1,\psi'_2)}
de \m{W'_C} est {\em \m{G'_{red}}-semi-stable}
(resp. {\em \m{G'_{red}}-stable})
relativement \`a \m{(1/\dim(Q_1),\nu_2,\nu_1)} si et seulement si pour tous
sous-espaces vectoriels
$$Q'_1\subset Q_1, \ \ M'_1\subset M_1, \ \ N'_1\subset N_1,$$
avec \ \m{(Q'_1,M'_1,N'_1)\not = (Q_1,M_1,N_1)} \ ou \m{(\lbrace 0\rbrace,\lbrace 0\rbrace,\lbrace 0\rbrace)},
tels que
$$\psi'_1({\cal E}_2\otimes Q'_1)\subset{\cal G}_1\otimes M'_1, \ \
\psi'_2({\cal E}_2\otimes Q'_1)\subset{\cal F}_1\otimes N'_1,$$
on a
$$\q{\dim(Q'_1)}{\dim(Q_1)} - \nu_2\dim(M'_1)-\nu_1\dim(N'_1)\leq 0,
\ \ \ \ {\rm (resp. \ } \ < \ {\rm)}.$$
On dit que \m{(\psi'_1,\psi'_2)} est {\em \m{G'}-semi-stable} (resp.
{\em \m{G'}-stable}) si tous les points de sa \m{H'}-orbite sont
\m{G'_{red}}-semi-stables (resp. \m{G'_{red}}-stables).
Soient \ \m{M'_1\subset M_1}, \m{M'_2\subset M_2}, \m{N'_1\subset N_1} \ des
sous-espaces vectoriels tels que
$$f_1({\cal E}_1\otimes M'_1)\subset{\cal F}_1\otimes N'_1, \ \
f_2({\cal E}_2\otimes M'_2)\subset{\cal F}_1\otimes N'_1.$$
On pose
$$Q'_1 \ = \ (\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M'_1)\oplus M'_2.$$
Il est clair qu'on peut choisir la mutation \m{(***)} de telle sorte que
$$\psi_1({\cal E}_2\otimes Q'_1)\subset{\cal G}_1\otimes M'_1, \ \
\psi_2({\cal E}_2\otimes Q'_1)\subset{\cal F}_1\otimes N'_1.$$
Posons
$$\nu_1=\q{1}{\dim(Q_1)n_1\lambda_2}, \ \
\nu_2=\q{a\lambda_2-\lambda_1}{\dim(Q_1)\lambda_2},$$
de telle sorte qu'on a
$$\q{\dim(Q'_1)}{\dim(Q_1)} - \nu_2\dim(M'_1)-\nu_1\dim(N'_1)=
\q{\lambda_1\dim(M'_1)+\lambda_2\dim(M'_2)-\dim(N'_1)/n_1}
{\lambda_2\dim(Q_1)}.$$
Alors on a d'apr\`es ce qui pr\'ec\`ede la
\bigskip
\begin{xprop}
Si le morphisme \m{(***)} est \m{G'}-semi-stable (resp. \m{G'}-stable)
relativement \`a \m{(1/\dim(Q_1),\nu_2,\nu_1)}, alors le morphisme \m{(*)} est
\m{G}-semi-stable (resp. \m{G}-stable) relativement \`a
\m{(\lambda_1,\lambda_2,1/\dim(N_1))}.
\end{xprop}
\bigskip
S'il existe des morphismes \m{G'}-stables dans \m{W'_C}, on a \ \m{\nu_1>0},
c'est-\`a-dire
$$a\lambda_2-\lambda_1>0.$$
On supposera que c'est le cas dans toute la suite.
\subsubsection{Cas d'\'equivalence des (semi-)stabilit\'es}
Soit
$$\sigma : \mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*$$
une application lin\'eaire dont la composition avec la transpos\'ee de la
composition
$$\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)$$
est l'identit\'e de \m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*}.
On d\'efinit ici des constantes analogues \`a celles du \paragra~\hskip -2pt 4.3.1.
Pour tout entier positif $k$, soit
$$\sigma_k = \sigma\otimes I_{\scx{k}} :
\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes\cx{k})
\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes\cx{k}.$$
Soit \m{G_k} l'ensemble des sous-espaces vectoriels propres \m{K} de
\m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes\cx{k}} tels que pour tout sous-espace vectoriel
propre \ \m{V\subset\cx{k}}, \m{K} ne soit pas contenu dans
\noindent\m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes V}. On pose
$$c'_0(k)= \mathop{\hbox{$\sup$}}\limits_{K\in{G_k}}(\q{\mathop{\rm codim}\nolimits(
\sigma_k(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes K))}{\mathop{\rm codim}\nolimits(K)}).$$
Il est clair qu'on a
$$c'_0(k+1)\geq c'_0(k).$$
\medskip
\noindent{\bf Remarque : } les constantes pr\'ec\'edentes d\'ependent aussi
a priori du choix de \m{\sigma}.
\bigskip
\begin{xprop}
On suppose que
$$\lambda_2\geq \q{c'_0(m_1)}{n_1}.$$
Si le morphisme \m{(*)} est \m{G}-semi-stable
relativement \`a \m{(\lambda_1,\lambda_2,1/\dim(N_1))}, alors le mor-\break
phisme
\m{(***)} est \m{G'}-semi-stable relativement \`a \m{(1/\dim(Q_1),\nu_2,\nu_1)}.
\end{xprop}
\bigskip
\noindent{\em D\'emonstration}.
Soient \ \m{Q'_1\subset Q_1}, \m{M'_1\subset M_1}, \m{N'_1\subset N_1} \ des
sous-espaces vectoriels tels que
$$\psi_1({\cal E}_2\otimes Q'_1)\subset {\cal G}_1\otimes M'_1, \ \
\psi_2({\cal E}_2\otimes Q'_1)\subset {\cal F}_1\otimes N'_1$$
et
$$X = \q{\dim(Q'_1)}{\dim(Q_1)}-\nu_2\dim(M'_1)-\nu_1\dim(N'_1)>0.$$
En faisant agir le sous-groupe unipotent $H$ de $G$, on peut supposer que
\m{Q'_1} est de la forme
$$Q'_1 \ = \ K\oplus M'_2,$$
avec \ \m{K\subset\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M_1} \ et \ \m{M'_2\subset M_2}.
On peut supposer que \m{M'_1} est le plus petit sous-espace vectoriel de \m{M_1}
tel que \ \m{K\subset\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M'_1}. Soit \m{N''_1} l'image de
\ \m{\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes M'_1} \ par l'application lin\'eaire
$$\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes M_1\longrightarrow N_1$$
d\'eduite de \m{f_1}. Alors on a
$$\dim(N''_1)-\dim(N'_1)\leq c'_0(m_1)(a\dim(M'_1)-\dim(K)).$$
On a
$$f_1({\cal E}_1\otimes M'_1)\subset {\cal F}_1\otimes N''_1, \ \ \
f_2({\cal E}_2\otimes M'_2)\subset {\cal F}_1\otimes N''_1.$$
On va montrer que
$$X'= \lambda_1\dim(M'_1)+\lambda_2\dim(M'_2)-\q{\dim(N''_1)}{n_1} > 0,$$
ce qui prouvera que \m{(*)} n'est pas semi-stable. On a
$$\q{X'}{\dim(Q_1)\lambda_2} \ = \ X + \q{a\dim(M'_1)-\dim(K)}{\dim(Q_1)}
-\nu_1(\dim(N''_1)-\dim(N'_1)),$$
donc
$$\q{X'}{\dim(Q_1)\lambda_2} \ > \
(\q{1}{\dim(Q_1)}-\nu_1c'_0(m_1))(a\dim(M'_1)-\dim(K)),$$
c'est-\`a-dire
$$X' \ > \ (\lambda_2-\q{c'_0(m_1)}{n_1})(a\dim(M'_1)-\dim(K)) \geq 0.$$
$\Box$
\bigskip
\subsubsection{Egalit\'e des quotients et projectivit\'e}
On se place dans l'hypoth\`ese du th\'eor\`eme 5.5. On peut se poser la question
de l'\'egalit\'e des quotients \ \m{{W'_C}^{ss}//G'} \ et \ \m{W_C^{ss}//G}.
Soit \m{W'_0} l'ouvert de \m{W'_C} constitu\'e des morphismes
\m{(\psi'_1,\psi'_2)} tels que \m{\psi'_1} induise une surjection
$$F : Q_1\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)^*\otimes M_1.$$
Cet ouvert est donc constitu\'e des morphismes dont un \'el\'ement de la
$G'$-orbite peut \^etre obtenu comme mutation d'un \'el\'ement de $W_C$.
Si \ \m{(\psi'_1,\psi'_2)\in W'_C\backslash W'_0}, il existe un sous-espace
vectoriel \ \m{Q'_1\subset Q_1} \ de dimension \ \m{m_2+1} \ contenu dans
\m{\ker(F)}. On a
$$\psi'_1({\cal E}_2\otimes Q'_1)=0.$$
Il en d\'ecoule que si \m{(\psi'_1,\psi'_2)} est \m{G'}-semi-stable
relativement \`a \m{(1/\dim(Q_1,),\nu_2,\nu_1)}, on doit avoir
$$\q{m_2+1}{\dim(Q_1)}-\nu_1n_1\geq 0,$$
c'est-\`a-dire
$$\lambda_2 \ \leq \q{1}{m_2+1}.$$
On en d\'eduit le
\bigskip
\begin{xtheo}
On suppose que
$$\lambda_2\geq \q{c'_0(m_1)}{n_1} \ \ \ \ {\rm et} \ \ \ \
\lambda_2>\q{1}{m_2+1}.$$
S'il existe un quasi-bon quotient \ \m{{W'_C}^{ss}//G'} \ pour la polarisation
\m{(1/\dim(Q_1,),\nu_2,\nu_1)}, il existe un quasi-bon quotient \
\m{W_C^{ss}//G} \ pour la polarisation
\m{(\lambda_1,\lambda_2,1/n_1)}, et il est isomorphe \`a \ \m{{W'_C}^{ss}//G'}.
\end{xtheo}
\bigskip
\subsubsection{Cas d'existence d'un quasi-bon quotient projectif}
On fait ici la synth\`ese du th\'eor\`eme 5.2, du th\'eor\`eme 7.6 de
\cite{dr2} et des r\'esultats pr\'ec\'edents.
Pour pouvoir employer le th\'eor\`eme 7.6 de \cite{dr2}, il faut faire les
hypoth\`eses suppl\'ementai-\break res suivantes :
\begin{itemize}
\item[--] Les morphismes
$$\coeva{{\cal E}_1}{{\cal F}_1}, \ \ \coeva{{\cal E}_2}{{\cal F}_1}$$
sont injectifs. On note \ ${\cal H}_1$, ${\cal H}_2$ leurs conoyaux respectifs.
\item[--] On a \ \ $\mathop{\rm Ext}\nolimits^1({\cal F}_1,{\cal E}_1)=\mathop{\rm Ext}\nolimits^1({\cal F}_1,{\cal E}_2)=\lbrace 0\rbrace$.
\end{itemize}
\bigskip
Soit
$$({\cal E}_1\otimes M_1)\oplus({\cal E}_2\otimes M_2)\longrightarrow{\cal F}_1\otimes N_1$$
un morphisme tel que l'application lin\'eaire associ\'ee
$$\phi : (\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)^*\otimes M_1)\oplus(\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes M_2)\longrightarrow N_1$$
soit surjective. On d\'efinit dans \cite{dr2} un autre type de mutation
associant au morphisme pr\'ec\'edent un morphisme
$${\cal F}_1\otimes\ker(\phi)\longrightarrow({\cal H}_1\otimes M_1)\oplus({\cal H}_2\otimes M_2).$$
En appliquant le th\'eor\`eme 5.2 \`a ces nouveaux morphismes, on obtient de
nouveaux cas d'existence de quasi-bons quotients.
On aura besoin de deux types de constantes suppl\'ementaires. Soit $k$ un
entier positif. On note \m{c_1(k)} la constante analogue \`a \m{c_0(k)},
obtenue en consid\'erant l'application lin\'eaire
$$\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)\otimes(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\cx{k})\longrightarrow \mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)\otimes\cx{k}$$
au lieu de \m{\tau_k} (cf. \paragra~\hskip -2pt 5.2). On note \m{c_2(k)} la constante analogue
\`a \m{c_0(k)}, obtenue en consid\'erant l'application lin\'eaire
$$\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)^*\otimes(\ker(\sigma)\otimes\cx{k})\longrightarrow\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal E}_2)\otimes\cx{k}.$$
On pose
$$h_{11}=\dim(\mathop{\rm Hom}\nolimits({\cal E}_1,{\cal F}_1)), \ \ h_{12}=\dim(\mathop{\rm Hom}\nolimits({\cal E}_2,{\cal F}_1)), \ \
a'=ah_{12}-h_{11}.$$
Toutes m\'ethodes confondues, on obtient le
\bigskip
\begin{xtheo}
Il existe un quasi-bon quotient projectif \ \m{W^{ss}_C//G} \ dans les cas
suivants :
\noindent 1 - On a
$$\q{\lambda_2}{\lambda_1} > a, \ \ \ \ \lambda_2 > \q{a}{n_1}c_0(m_2).$$
\noindent 2 - On a
$$\lambda_1 < \q{h_{11}}{n_1}, \ \ \ \ \lambda_2 < \q{h_{12}}{n_1}, \ \ \ \
a\lambda_2-\lambda_1 > \q{a'}{n_1}, \ \ h_{11}-\lambda_1n_1\geq c_1(m_1)a.$$
\noindent 3 - On a
$$\lambda_2\geq\q{c'_0(m_1)}{n_1}, \ \ \ \ \lambda_2 > \q{1}{m_2+1}, \ \ \ \
a\lambda_2-\lambda_1 \ > \ a'.{\rm Max}(c_2(m_1)\lambda_2, \q{1}{n_1}).$$
\end{xtheo}
On peut montrer que dans la situation du th\'eor\`eme 5.6, l'ouvert du
quotient correspondant aux points stables est lisse.
\vfill\eject
\subsection{Exemples}
Si \m{(\lambda_1,\lambda_2,1/n_1)} est une polarisation de l'action de $G$
sur \m{W_C}, on notera
$$\rho \ = \ \q{\lambda_2}{\lambda_1}.$$
Notons que la polarisation est enti\`erement d\'etermin\'ee par \m{\rho}, ou
par un des nombres \m{\lambda_1}, \m{\lambda_2}.
Dans les exemples qui vont suivre, on utilise implicitement des calculs de
constantes \m{c_0(k)} ou \m{c'_0(k)} qui proviennent de \cite{dr_tr},
\subsubsection{Exemple 1}
Soit $n$ un entier, avec \ \m{n\geq 2}. On consid\`ere des morphismes
$$({\cal O}(-2)\otimes\cx{2})\oplus{\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{n+1}$$
sur \m{\proj{n}}. Cet exemple a d\'ej\`a \'et\'e trait\'e dans \cite{dr_tr},
pour \ \m{n=2}. L'application directe du th\'eor\`eme 5.2 montre qu'il existe
un bon quotient projectif \ \m{W_C^{ss}//G} \ si
$$\rho \ > \ n+1.$$
En utilisant des mutations indirectes on se ram\`ene \`a des morphismes
$${\cal O}(-1)\otimes\cx{2n+3}\longrightarrow(Q(-1)\otimes\cx{2})\oplus({\cal O}\otimes\cx{n+1}).$$
L'application des th\'eor\`emes 5.2 et 5.5 permet de montrer qu'il existe
un quasi-bon quotient projectif \ \m{W_C^{ss}//G} \ d\`es que
$$\rho \ > \ 2 + \q{2}{n}.$$
Si \ \m{n\geq 4}, on obtient ainsi des vari\'et\'es de modules de morphismes
suppl\'ementaires.
\subsubsection{Exemple 2}
On consid\`ere les morphismes
$$(f_1,f_2) : {\cal O}(-2)\oplus{\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{n+2}$$
sur \proj{n}. C'est un exemple d\'ej\`a donn\'e dans \cite{dr2}.
On sait construire des bons quotients (en utilisant le th\'eor\`eme 5.2))
d\`es que
$$\rho > n+1.$$
Mais dans ce cas le quotient est vide. En effet, il existe toujours un
sous-espace vectoriel \ \m{H\subset\cx{n+2}} \ de dimension \m{n+1} tel
que \ \m{\mathop{\rm Im}\nolimits(f_2)\subset {\cal O}\otimes H}. On doit donc avoir, si \m{(f_1,f_2)}
est \m{G}-semi-stable relativement \`a \m{(\lambda_1,\lambda_2,1/(n+2))},
$$\lambda_2 - \q{n+1}{n+2} \leq 0,$$
c'est-\`a-dire \ \m{\rho\leq n+1}. L'application des th\'eor\`emes 5.5 et
5.2 permet de construire un quasi-bon quotient projectif \
\m{W^{ss}_C//G} \ d\`es que
$$\rho \ > 1.$$
On am\'eliore l\'eg\`erement le r\'esultat de \cite{dr2} dans le cas o\`u $n$
est impair.
Les valeurs {\em singuli\`eres} de \m{\rho} sont par d\'efinition celles
pour lesquelles la \m{G}-semi-stabilit\'e n'implique pas la
\m{G}-stabilit\'e. Ces valeurs sont exactement les nombres
$$\rho_k = \q{k}{n+2-k}$$
pour \m{1\leq k\leq n+1} . Dans ce cas un morphisme \m{(\phi_1,\phi_2)}
$G$-semi-stable non $G$-stable est construit de la fa\c con suivante :
on consid\`ere un sous-espace vectoriel \ \m{H\subset\cx{n+2}} \ de
dimension \m{k}, et on prend pour \m{\phi_2} un morphisme tel que
\ \m{\mathop{\rm Im}\nolimits(\phi_2)\subset{\cal O}\otimes H} \ et que $H$ soit le plus petit
sous-espace vectoriel ayant cette propri\'et\'e. On prend pour \m{\phi_1}
un morphisme tel que l'application lin\'eaire induite
$$H^0({\cal O}(2))^*\longrightarrow\cx{n+2}$$
soit surjective. Toutes le polarisations telles que \m{\rho} soit situ\'e entre
\m{\rho_k} et \m{\rho_{k+1}} donnent la m\^eme notion de (semi-)stabilit\'e.
Notons \m{M_k} le quotient obtenu pour \ \m{\rho=\rho_k}, et \m{M^0_k} celui
obtenu pour \ \m{\rho_{k-1}<\rho<\rho_k}. On sait donc construire \m{M_k} et
\m{M^0_k} pour \ \m{\lbrack\q{n}{2}\rbrack+2\leq k\leq n+1}.
Les quotients \m{M^0_k} construits dans \cite{dr2}, c'est-\`a-dire pour \
\m{\lbrack\q{n+3}{2}\rbrack+1\leq k\leq n+1}, sont des quotients
g\'eom\'etriques.
\subsubsection{Exemple 3}
C'est une g\'en\'eralisation de l'exemple pr\'ec\'edent. Soit \ \m{n\geq 2} \
un entier. On consid\`ere sur \proj{n} les morphismes
$$(f_1,f_2) : {\cal O}(-2)\oplus{\cal O}(-1)\longrightarrow{\cal O}\otimes N_1.$$
\bigskip
\subsubsubsection{5.4.3.1}{Conditions d'existence de points stables}
\medskip
Pour qu'il existe des points stables (pour au moins une polarisation), on doit
avoir
$$n_1=\dim(N_1) \ \leq \ \q{(n+1)(n+2)}{2}+n.$$
Si \ \m{n_1\geq n+1}, il existe toujours un sous-espace vectoriel \m{N'_1}
de \m{N_1} de dimension \m{n+1} tel que l'image de \m{f_2} soit contenue dans
\ \m{{\cal O}\otimes N'_1}. Il en d\'ecoule que s'il existe des points stables
relativement \`a la polarisation d\'efinie par \m{\lambda_2}, on doit avoir
$$\lambda_2 \ < \ \q{n+1}{n_1}.$$
Si c'est le cas on montre ais\'ement qu'il existe toujours des morphismes
stables.
Les valeurs de \m{\lambda_2} pour lesquelles il existe des morphismes
semi-stables non stables, ainsi que des morphismes stables, sont les
$$\alpha_k = \q{k}{n_1}, \ \ \ 1\leq k\leq \ {\rm Inf}(n,n_1-1).$$
Posons \ \m{m=1+{\rm Inf}(n,n_1-1)} et
$$\alpha_0=0, \ \ \ \alpha_m= \ {\rm Inf}(1,\q{n+1}{n_1}).$$
Si \ \m{1\leq k\leq \ {\rm Inf}(n,n_1-1)}, et \ \m{\alpha_{k-1}<\lambda_2
<\alpha_k}, le morphisme \m{(f_1,f_2)} est stable si et seulement si
\begin{itemize}
\item[--] 1 - Si l'image de $f_2$ est contenue dans
\ ${\cal O}\otimes N'_1\subset{\cal O}\otimes N_1$, on a \ $\dim(N'_1)\geq k$.
\item[--] Pour tout \ $(f'_1,f'_2)\in H.(f_1,f_2)$, si l'image de $f'_1$ est
contenue dans \ ${\cal O}\otimes N'_1\subset{\cal O}\otimes N_1$, on a \
$\dim(N'_1)\geq n_1-k+1$.
\end{itemize}
\bigskip
\subsubsubsection{5.4.3.2}{Les constantes}
\medskip
Pour appliquer le th\'eor\`eme 5.6 on a besoin des constantes \m{c_0(1)},
\m{c'_0(1)}, \m{c_1(1)} et \m{c_2(1)}. On calcule ais\'ement que
$$c_0(1)=0, \ \ c'_0(1)=c_1(1)=\q{n+1}{2}, \ \ c_2(1)=\q{2n}{n^2+n-2}.$$
\bigskip
\subsubsubsection{5.4.3.3}{Application du th\'eor\`eme 5.6, 1-}
\medskip
On obtient un quotient si
$$\q{\lambda_2}{\lambda_1} \ > \ n+1,$$
c'est-\`a-dire si
$$\lambda_2 \ > \ \q{n+1}{n+2}.$$
On obtient la seule vari\'et\'e de modules \m{M_{n_1}} si \ \m{n_1\leq n+1},
et aucune si \ \m{n_1>n+1}.
\bigskip
\subsubsubsection{5.4.3.4}{Application du th\'eor\`eme 5.6, 2-}
\medskip
On doit avoir
$$\lambda_2 \ > \ \q{1}{n+2}+\q{n(n+1)}{2n_1(n+2)} \ \ \ \ {\rm si } \
n_1\leq n+1,$$
$$\lambda_2 \ > \ 1-\q{n+1}{2n_1} \ \ \ \ \ {\rm si } \ n_1>n+1.$$
Dans le premier cas, on obtient la construction de \m{M_k} si
$$k \ > \q{n_1}{n+2}+\q{n(n+1)}{2(n+2)}.$$
Ceci donne des vari\'et\'es de modules suppl\'ementaires si
$$n_1 \ \geq \ \lbrack\q{n}{2}\rbrack + 2.$$
Dans le second cas, puisque \ \m{\lambda_2<\q{n+1}{n_1}}, on doit avoir
$$n_1<\q{3}{2}(n+1).$$
On obtient alors les vari\'et\'es de modules \m{M_k} pour
$$n_1-\q{n+1}{2} \ < \ k \ \leq \ n+1.$$
\bigskip
\subsubsubsection{5.4.3.5}{Application du th\'eor\`eme 5.6, 3-}
\medskip
On obtient les m\^emes vari\'et\'es de modules que pr\'ec\'edemment si
\ \m{n_1\leq n+1}.
Si \ \m{n_1>n+1}, on sait construire le quotient si
$$\lambda_2 \ > \ \q{1}{2}+\q{n}{2n^2-4}.$$
On sait dans ce cas construire \m{M_k} pour
$$n_1(\q{1}{2}+\q{n}{2n^2-4}) \ < \ k \ \leq n+1.$$
On obtient donc d'autres vari\'et\'es de modules de morphismes
si \ \m{n+3\leq n_1<2n} \ et si $n$ est assez grand.
\bigskip
\subsubsubsection{5.4.3.6}{Polarisations pathologiques. Cas o\`u
\ \m{\lambda_2<1/2}}
\medskip
On suppose que \ \m{n_1} est pair : \ \m{n_1=2p}, et que \ \m{n_1\leq 2n+2}.
Soit \m{(z_1,\ldots,z_{n+1})} une base de \m{H^0({\cal O}(1))}. On consid\`ere le
morphisme
$$(f_1,f_2) : {\cal O}(-2)\oplus{\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{2p}$$
o\`u \m{f_2}, \m{f_1} sont d\'efinis respectivement par les matrices
$$\pmatrix{z_1\cr .\cr .\cr .\cr z_p\cr 0\cr 0\cr .\cr .\cr .\cr 0}, \ \ \
\pmatrix{z_2^2\cr .\cr .\cr .\cr z_{p+1}^2\cr z_1^2\cr z_1z_2\cr .\cr .\cr .\cr
z_1z_p}.$$
Alors, si \ \m{\lambda_2<1/2}, il est ais\'e de voir que \m{(f_1,f_2)} est
stable. Cependant son stabilisateur dans $G$ n'est pas r\'eduit \`a \m{\cx{*}}.
Il ne peut donc pas y avoir de quotient g\'eom\'etrique de \m{W^{s}_C} par
$G$ dans ce cas.
\bigskip
\subsubsubsection{5.4.3.7}{Polarisations pathologiques. Cas o\`u
\ \m{\lambda_2=1/2}}
\medskip
On suppose que $n$ est
pair et \ \m{n_1=n+2}. Soient \m{K_1}, \m{K_2} des sous-espaces vectoriels de
\m{H^0({\cal O}(1))} de dimension \m{\q{n+2}{2}}, et \m{D} une droite de
\m{H^0({\cal O}(1))}. Soient
$$(z_1,\ldots,z_{\q{n+2}{2}}), \ \ (z'_1,\ldots,z'_{\q{n+2}{2}}), \ \ z$$
des bases de \m{K_1}, \m{K_2} et $D$ respectivement. Alors l'\'el\'ement
de \m{W_C} d\'efini par les matrices
$$\pmatrix{z_1\cr .\cr .\cr .\cr z_{\q{n+2}{2}}\cr 0\cr .\cr .\cr .\cr
0\cr} \ \ , \ \ \pmatrix{0\cr .\cr .\cr .\cr 0\cr zz'_1\cr .\cr .\cr .\cr
zz'_{\q{n+2}{2}}\cr}$$
est semi-stable et sa $G$-orbite est ferm\'ee et ne
d\'epend que de \m{K_1}, \m{K_2} et $D$. On la note \m{\phi(K_1,K_2,D)}.
Remarquons que si \ \m{(K'_1,K'_2,D')\not = (K_1,K_2,D)}, on a
$$\phi(K'_1,K'_2,D') \ \not = \ \phi(K_1,K_2,D).$$
Une mutation de \m{\phi(K_1,K_2,D)} dans \m{W'_C} est un morphisme
$$\psi : {\cal O}\otimes\cx{n+2}\longrightarrow Q(-1)\oplus({\cal O}\otimes\cx{n+2}).$$
L'adh\'erence de sa \m{G'}-orbite contient le morphisme somme directe des
morphismes
$$\psi_1 : {\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{n+2}, \ \ \psi_2 : {\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{n+2},
\ \psi_3 : {\cal O}(-1)\otimes\cx{n}\longrightarrow Q(-1)$$
d\'efinis respectivement par \m{K_1}, \m{K_2}, \m{D}. La $G'$-orbite de
ce morphisme est not\'ee \break\m{\psi(K_1,K_2,D)}. Notons qu'elle est contenue
dans le compl\'ementaire dans \m{{W'}^{ss}_C} de l'ouvert contitu\'e
des orbites des morphismes mutations de morphismes de \m{W^{ss}_C}. On
a
$$\psi(K_1,K_2,D) \ = \ \psi(K_2,K_1,D).$$
Ceci prouve qu'on ne peut pas obtenir par des mutations indirectes un quotient
s\'eparant les orbites de \m{\phi(K_1,K_2,D)} et \m{\phi(K_2,K_1,D)}.
\vfill\eject
|
1997-01-30T14:16:12 | 9701 | alg-geom/9701019 | en | https://arxiv.org/abs/alg-geom/9701019 | [
"alg-geom",
"math.AG"
] | alg-geom/9701019 | Arnaud Beauville | Arnaud Beauville | Counting rational curves on K3 surfaces | Plain TeX, 11 pages | null | null | null | null | The aim of these notes is to explain the remarkable formula found by Yau and
Zaslow to express the number of rational curves on a K3 surface. Projective K3
surfaces fall into countably many families F(g) (g>0); a surface in F(g) admits
a g-dimensional linear system of curves of genus g. Such a system contains a
positive number, say n(g), of rational (highly singular) curves. The formula is
\sum n(g) q^g = q/D((q), where D(q) = q \prod (1-q^n)^{24} is the well-known
modular form of weight 12.
| [
{
"version": "v1",
"created": "Thu, 30 Jan 1997 13:16:03 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Beauville",
"Arnaud",
""
]
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\newlabel{e}{1.1
\centerline{\bf Counting rational curves on K3 surfaces}
\smallskip
\smallskip \centerline{Arnaud {\pc BEAUVILLE\note{1}{Partially supported by the
European HCM project ``Algebraic Geometry in Europe" (AGE).}}}
\smallskip
\centerline{Version 0}
\vskip1.2cm
{\bf Introduction}
\smallskip
\par\hskip 1truecm\relax The aim of these notes is to explain the remarkable formula found by Yau
and Zaslow [Y-Z] to express the number of rational curves on a K3 surface.
Projective K3 surfaces fall into countably many
families $({\cal F}_g)_{g\ge1}$; a surface in ${\cal F}_g$
admits a
$g$\kern-1.5pt - dimensional linear system of curves of genus $g$. A na\"{\i}ve count of
constants suggests that such a system will contain a positive number, say
$n(g)$, of rational (highly singular) curves. The formula is
$$\sum_{g\ge0}n(g)q^g= {q\over \Delta (q)}\ ,$$
where $\Delta (q)=q\prod_{n\ge1}(1-q^n)^{24}$ is the well-known modular form
of weight $12$, and we put by convention $n(0)=1$.
\par\hskip 1truecm\relax To explain the idea in a nutshell, take the case
$g=1$. We are thus looking at K3 surfaces with an elliptic
fibration $f:S\rightarrow {\bf P}^1$, and we are asking for the number of
singular fibres. The (topological) Euler-Poincar\'e characteristic of a fibre
$C_t$ is $0$ if $C_t$ is smooth, $1$ if it is a rational curve with one
node, $2$ if it has a cusp, etc. From the standard properties of the
Euler-Poincar\'e characteristic, we get
$\displaystyle e(S)=\sum_t e(C_t)$; hence $n(1)=e(S)=24$, and this number
counts
nodal rational curves with multiplicity $1$, cuspidal rational curves with
multiplicity $2$, etc. \par\hskip 1truecm\relax The idea of Yau and Zaslow is to generalize this
approach to any genus. Let $S$ be a K3 surface with a $g$\kern-1.5pt - dimensional linear
system $\Pi$ of curves of genus $g$. The role of $f$ will be played by the
morphism $\bar{\cal J}{\cal C} \rightarrow \Pi$ whose fibre over a point
$t\in\Pi$
is the compactified Jacobian $\bar JC_t$. To apply the same method, we would
like to prove the following facts:
\par\hskip 0.5cm\relax 1) The Euler-Poincar\'e characteristic $e(\bar{\cal J}{\cal C})$ is the
coefficient of $q^g$ in the Taylor expansion of $q/\Delta(q)$.
\par\hskip 0.5cm\relax 2) $e(\bar JC_t)=0$ if $C_t$ is not rational.
\par\hskip 0.5cm\relax 3) $e(\bar JC_t)=1$ if $C_t$ is a rational curve with nodes as only
singularities. Moreover $e(\bar JC_t)$ is positive when $C_t$ is rational, and
can
be computed in terms of the singularities of $C_t$.
\par\hskip 0.5cm\relax 4) For a generic K3 surface $S$ in ${\cal F}_g$, all rational curves in
$\Pi $
are nodal.
\par\hskip 1truecm\relax The first statement is proved in \S 1, by comparing with
the Euler-Poincar\'e characteristic of the Hilbert scheme $S^{[g]}$ which has
been
computed by G\"ottsche. The assertion 2) is proved in \S 2. The situation for
3)
is less satisfactory: though I can express $e(\bar JC)$, for a rational curve
$C$,
in terms of a local invariant of the singularities of $C$, and compute this
local
invariant in a number of cases, at this moment I am
not able to prove that it is always positive.
Finally 4) seems to be still open, despite recent progress by Xi Chen.
\par\hskip 1truecm\relax The outcome (see Cor.\ \ref{conc}) is that the coefficient of $q^g$ in
$q/\Delta (q)$ counts the rational curves in $\Pi$ with a certain multiplicity,
which is $1$ for a nodal curve and can be computed explicitely in many cases;
the
two missing points are the positivity of this multiplicity, and the fact that
only
nodal curves occur on a generic surface in ${\cal F}_g$.
\section{The compactified relative Jacobian }
\global\def\currenvir{subsection\label{e} Let $X$ be a complex variety; we denote by $e(X)$ its
Euler-Poincar\'e characteristic, defined by $e(X)=\sum_p(-1)^p\mathop{\rm dim}\nolimits_{\bf
Q}H_c^p(X,{\bf Q})$. Recall that this invariant is additive, i.e.\ satisfies
$e(X)=e(U)+e(X\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}
U)$ whenever $U$ is an open subset of $X$.
\global\def\currenvir{subsection We consider a projective K3 surface $S$
with a complete linear system $(C_t)_{t\in \Pi}$ of curves of genus $g\ge1$ (so
$\Pi $ is a projective space of dimension $g$). We will assume that {\it all
the
curves $C_t$ are integral} (i.e.\ irreducible and reduced). This is a
simplifying
assumption, which can probably be removed at the cost of various technical
complications. It is of course satisfied if the class of $C_t$ generates
$\mathop{\rm Pic}\nolimits(S)$. \par\hskip 1truecm\relax Let ${\cal C}\rightarrow \Pi $ be the morphism with fibre
$C_t$
over $t\in\Pi$. For each integer $d\in {\bf Z}$, we denote by $\bar{\cal
J}{\cal C}=\cprod_{d\in{\bf Z}}^{}\bar{\cal J}^d{\cal C}$ the compactified
Picard scheme of this family. $\bar{\cal J}^d{\cal C}$ is a projective
variety of dimension $2g$, which parameterizes pairs $(C_t,{\cal L})$ where
$t\in\Pi $ and
${\cal L}$ is a torsion
free, rank $1$ coherent sheaf on $C_t$ of degree $d$ (i.e.\ with $\chi ({\cal
L})=d+1-g$). According to Mukai ([M], example 0.5), $\bar{\cal J}^d{\cal C}$ can be viewed as a
connected component of the moduli space of simple sheaves on $S$, and
therefore is smooth, and admits a (holomorphic) symplectic
structure.
\par\hskip 1truecm\relax The simplest symplectic varieties associated to the K3 surface $S$ are
the Hilbert schemes $S^{[d]}$, which parameterize finite subschemes of
length $d$ of $S$. The birational comparison of the symplectic varieties
$\bar{\cal J}^d{\cal C}$,
for various values of
$d$, with $S^{[g]}$ is an interesting problem, about which not much seems to be
known. There is one easy case:
\th Proposition
\enonce The compactified Jacobian $\bar{\cal J}^g{\cal C}$ is birationally isomorphic
to~$S^{[g]}$.
\endth
{\it Proof}: Let $U$ be the open subset of $\bar{\cal J}^g{\cal C}$ consisting of pairs
$(C_t,L)$ where $L$ is invertible and $\mathop{\rm dim}\nolimits H^0(C_t,L)=1$. To such a pair
corresponds a unique effective Cartier divisor $D$ on $C_t$ of degree $g$,
which
can be viewed as a length $g$ subscheme of $S$; since
$\mathop{\rm dim}\nolimits H^0(C_t,{\cal O}_{C_t}(D))=1$ it is contained in a unique curve
of $\Pi$, namely $C_t$. This provides an isomorphism betwen $U$ and the open
subset $V$ of $S^{[g]}$ parameterizing finite subschemes of $S$ contained in a
unique curve of $\Pi$ and defining a Cartier divisor in this curve. \cqfd
\smallskip
\th Corollary
\enonce Write $\displaystyle {q\over \Delta(q)}=\sum_{g\ge0}e(g)\,q^g$. Then
$e(\bar{\cal J}^g{\cal C})=e(g)$.
\endth
{\it Proof}: We can either use a recent result of Batyrev and Kontsevich [?]
saying that two birationnally equivalent projective Calabi-Yau manifolds have
the
same Betti numbers, or a more precise result of Huybrechts [H]:
two birationally equivalent
projective symplectic manifolds which are isomorphic in codimension $2$ are
diffeomorphic (note that the open subsets $U$ and $V$ appearing in the above
proof
have complements of codimension $\ge2$). It remains to apply G\"ottsche's
formula
$e(S^{[g]})=e(g)$ [G]. \cqfd
\section{The compactified Jacobian of a non-rational curve}\smallskip
\par\hskip 1truecm\relax Let $C$ be an integral curve.
By a {\it rank $1$ sheaf} on $C$ I will mean
a torsion free, rank $1$ coherent sheaf.
The rank $1$ sheaves ${\cal L}$ on
$C$ of degree $d$ are parameterized by the
compactified Jacobian
$\bar J^dC$. If $L$ is an invertible sheaf of degree $d$ on $C$, the map
${\cal L}\mapsto {\cal L}\otimes L$ is an isomorphism of $\bar J C$ onto $\bar
J^dC$, so we can restrict our study to degree $0$ sheaves.
\par\hskip 1truecm\relax Let ${\cal L}\in \bar JC$; the endomorphism ring of ${\cal L}$ is an
${\cal O}_C$\kern-1.5pt - subalgebra of the sheaf of rational functions on $C$. It is
finitely generated as a ${\cal O}_C$\kern-1.5pt - module, hence contained in ${\cal
O}_{\widetilde{C}}$. It is thus of the form ${\cal O}_{C'}$, where
$f:C'\rightarrow C$ is some partial normalization of $C$. The sheaf ${\cal L}$
is
a ${\cal O}_{C'}$\kern-1.5pt - module, which amounts to say that it is the direct image
of a
rank $1$ sheaf ${\cal L}'$ on $C'$.
\th Lemma
\enonce Let $L\in JC$. Then ${\cal L}\otimes L$ is isomorphic to ${\cal L}$ if
and
only if $f^*L$ is trivial.
\endth\label{free}
{\it Proof}: The sheaf ${\cal L}\otimes L$ is isomorphic to $f_*({\cal
L}'\otimes f^*L)$, hence to ${\cal L}$ if $f^*L$ is trivial.
On the other hand we have
$${\cal H}om_{{\cal O}_C}({\cal L},{\cal L}\otimes L)\cong {\cal E}nd_{{\cal
O}_C}({\cal L})\otimes_{{\cal O}_C}L\cong f_*{\cal O}_{C'}\otimes L\cong
f_*f^*L\ ,$$
so if $f^*L$ is non-trivial, the space $\mathop{\rm Hom}\nolimits({\cal L},{\cal L}\otimes L)$ is
zero, and ${\cal L}\otimes L$ cannot be isomorphic to ${\cal L}$. \cqfd
\smallskip
\th Proposition
\enonce Let $C$ be an integral curve whose normalization $\widetilde{C}$ has
genus $\ge1$. Then $e(\bar J^d C)=0$.
\endth\label{g}
{\it Proof}: We have an exact sequence
$$0\rightarrow G\longrightarrow JC\longrightarrow J\widetilde{C}\rightarrow
0\ ,$$
where $G$ is a product of additive and multiplicative groups. In particular,
$G$ is a divisible group, hence this exact sequence splits as a sequence of
abelian groups. For each integer $n$, we can therefore find a subgroup of order
$n$
in
$JC$ which maps injectively into $J\widetilde{C}$.
By Lemma
\ref{free}, this group acts freely on $\bar JC$, which implies that $n$ divides
$e(\bar JC)$; since this holds for any $n$ the Proposition follows. \cqfd
\smallskip
\th Corollary
\enonce Write $\displaystyle {q\over \Delta(q)}=\sum_{g\ge0}e(g)\,q^g$; let
$\Pi_{\rm rat}\i\Pi$ be the (finite) subset of rational curves. Then
$\displaystyle e(g)=\sum_{t\in\Pi_{\rm rat}}e(\bar JC_t)$.
\endth\label{conc}
{\it Proof}: We first make a general observation: let $f:X\rightarrow Y$ be a
surjective morphism of complex algebraic varieties whose fibres have Euler
characteristic $0$; then $e(X)=0$. This is well known (and easy) if $f$ is a
locally trivial fibration; the general case follows using (\ref{e}), because
there
exists a stratification of $Y$ such that $f$ is locally trivial above each
stratum
[V]. \par\hskip 1truecm\relax The set
$\Pi_{\rm rat}$ is finite because otherwise it would contain a curve, so $S$
would be
ruled. Consider the morphism $p:\bar{\cal J}^g{\cal C}\rightarrow \Pi$ above $\Pi\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Pi_{\rm
rat}$;
by the above remark, we have $e(p^{-1} (\Pi\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Pi_{\rm rat}))=0$, hence the
result using again (\ref{e}). \cqfd \medskip
\par\hskip 1truecm\relax In other words, $e(g)$ counts the
number of rational curves with multiplicity, the multiplicity of a curve $C$
being
$e(\bar JC)$. In the next two sections we will try to show that this is indeed
a
reasonable notion of multiplicity (with only partial success, as explained in
the
introduction).
\section{The compactified Jacobian of a rational curve}
\th Lemma
\enonce Let $f:C'\rightarrow C$ be a partial normalization of $C$. The morphism
$f_*:\bar JC'\rightarrow \bar JC$ is a closed embedding.
\endth\label{embed}
{\it Proof}: Let ${\cal L},{\cal M}$ be two rank $1$ sheaves on $C'$.
We claim that any ${\cal O}_C$\kern-1.5pt - homo\-morphism $u:f_*{\cal L} \rightarrow
f_*{\cal M}$ is actually $f_*{\cal O}_{C'}$\kern-1.5pt - linear. Let $U$ be a Zariski
open
subset of $C$, $\varphi\in \Gamma (U,f_*{\cal O}_{C'}) $, $s\in \Gamma
(U,f_*{\cal
L})$; the rational function $\varphi $ can be written as $a/b$, with $a,b\in
\Gamma (U,{\cal O}_{C})$ and $b\not=0$. Then the element $u(\varphi s)-\varphi
u(s)$ of $\Gamma (U,f_*{\cal M})$ is killed by $b$, hence is zero since
$f_*{\cal
M}$ is torsion-free.
\par\hskip 1truecm\relax Therefore if $f_*{\cal L}$ and $f_*{\cal M}$ are isomorphic as ${\cal
O}_C$\kern-1.5pt - modules, they are also isomorphic as $f_*{\cal O}_{C'}$\kern-1.5pt - modules,
which
means that ${\cal L}$ and ${\cal M}$ are isomorphic: this proves the
injectivity
of $f_*$ (which would be enough for our purpose). Now if $S$ is any base
scheme,
the same argument applies to sheaves ${\cal L}$, ${\cal M}$ on $C\times S$,
flat
over $S$, whose restrictions to each fibre $C\times\{s\}$ are torsion free rank
$1$ (observe that a local section $b$ of ${\cal O}_C$ is ${\cal M}$\kern-1.5pt -
regular because it is on each fibre, and ${\cal M}$ is flat over $S$). This
proves that $f_*$ is a monomorphism; since it is proper, it is a closed
embedding.
\cqfd
\medskip
\global\def\currenvir{subsection Recall that the curve $C$ is said to
be {\it unibranch} if its normalization $\widetilde{C}\rightarrow C$ is a
homeomorphism. Any curve $C$ admits a unibranch partial normalization $\check
\pi:
\check C\rightarrow C$ which is minimal, in the sense that any unibranch
partial normalization $C'\rightarrow C$ factors through $\check \pi$. To see
this, let
${\cal C}$ be the conductor of $C$, and let $\widetilde{\Sigma }$ be the
inverse
image in $\widetilde{C}$ of the singular locus $\Sigma \in C$. The
finite-dimensional $k$\kern-1.5pt - algebra $A:={\cal O}_{\widetilde{C}}/{\cal C}$ is a
product of local rings $(A_{x})_{x\in\widetilde{\Sigma }}$; let
$(e_x)_{x\in\widetilde{\Sigma }}$ be the corresponding idempotent elements of
$A$.
A sheaf of algebras ${\cal O}_{C'}$ with ${\cal O}_{C}\i{\cal O}_{C'}\i{\cal
O}_{\widetilde{C}}$ is unibranch if and only if ${\cal O}_{C'}/{\cal C}$
contains
each $e_x$, or equivalently ${\cal O}_{C'}$ contains the classes $e_x+{\cal
C}$
for each $x\in\widetilde{\Sigma }$; clearly there is a smallest such algebra,
namely the algebra ${\cal O}_{\check C}$ generated by ${\cal O}_C$ and the
classes $e_x+{\cal C}$. The completion of the local ring of $\check C$ at a
point
$y$ is the image of $\widehat{\cal O}_{C,\check \pi(y)}$ in $\widehat{\cal
O}_{\widetilde{C},y}$.
\smallskip
\th Proposition
\enonce With the above notation, $e(\bar JC)=e(\bar J\check C)$.
\endth \label{elag}
{\it Proof}: In view of Prop.\ \ref{g}, we may suppose that $\widetilde{C}$ is
rational. As
before we denote by
$\Sigma $ the singular locus of $C$, and by $\widetilde{\Sigma }$ its inverse
image
in $\check C$.
The cohomology exact sequence associated to the short exact sequence
$$1\rightarrow {\cal O}_C^*\longrightarrow{\cal O}_{\widetilde{C}}^*
\longrightarrow {\cal O}_{\widetilde{C}}^*/{\cal O}_C^*\rightarrow 1$$
provides a bijective homomorphism (actually an isomorphism of algebraic groups)
${\cal O}_{\widetilde{C}}^*/{\cal O}_C^*\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} JC$.
\par\hskip 1truecm\relax The evaluation maps
${\cal O}_{\widetilde{C}}^*\rightarrow ({\bf C}^*)^{\widetilde{\Sigma}} $ and
${\cal O}_C^*\rightarrow ({\bf C}^*)^{\Sigma} $ give rise to a surjective
homomorphism ${\cal O}_{\widetilde{C}}^*/{\cal O}_C^*\rightarrow ({\bf
C}^*)^{\widetilde{\Sigma}}/({\bf C}^*)^{\Sigma}$; its kernel is unipotent,
i.e.\
isomorphic to a vector space. If $n$ is any integer $\ge\mathop{\rm Card}\nolimits(\widetilde{\Sigma
})$, it follows that we can find a section $\varphi$ of ${\cal
O}_{\widetilde{C}}^*$ in a neighborhood of $\widetilde{\Sigma} $ such that the
numbers $\varphi(\tilde x)$ for $\tilde x\in\widetilde{\Sigma}$ are all
distinct,
but $\varphi ^n$ belongs to ${\cal O}_C$. Let $L$ be the line bundle on $JC$
associated to the class of $\varphi $ in ${\cal O}_{\widetilde{C}}^*/{\cal
O}_C^*$.
\par\hskip 1truecm\relax Let $U$ be the complement of $\check \pi_*(\bar J\check C)$ in $\bar JC$;
according to \ref{e} and Lemma \ref{embed}, our assertion is equivalent to
$e(U)=0$. We claim that the line bundle $L$ acts freely on $U$; since the order
of
$L$ in $JC$ is finite and arbitrary large, this will finish the proof. Let
${\cal
L}\in U$, and let $C'$ be the partial normalization of $C$ such that ${\cal
E}nd({\cal L})={\cal O}_{C'}$; by definition of $U$, $C'$ is not unibranch,
hence
there are two points of $\widetilde{\Sigma}$ mapping to the same point of $C'$;
this implies that the function $\varphi$ does not belong to ${\cal O}_{C'}^*$.
{}From the commutative diagram
$$\diagram{\widetilde{\cal O}_{\widetilde{C}}^*/{\cal O}_C^* &\hfl{\raise
-3mm\hbox{$\sim$}}{} &JC\cr \vfl{}{}& &\vfl{}{}\cr {\cal
O}_{\widetilde{C}}^*/{\cal O}_{C'}^* &\hfl{\sim}{} & JC' }$$
we conclude that the pull back of $L$ to $JC'$ is non-trivial; by Lemma
\ref{free} this implies that ${\cal L}\otimes L$ is not isomorphic to ${\cal
L}$.
\cqfd
\th Corollary
\enonce For a rational nodal curve $C$, we have $e(\bar JC)=1$. \cqfd
\endth
\medskip
\rem{Remark}\label{An} Consider a rational curve $C$ whose singularities are
all
of type $A_{2l-1}$, i.e.\ locally defined by an equation $u^2-v^{2l}=0$.
Locally around such a singularity, the curve $C$ is the union of two smooth
branches with a high order contact, so by \ref{elag} $e(\bar JC)$ is equal to
$1$. The fact that some highly singular curves count with multiplicity one
looks
rather surprising. The case $g=2$ provides a (modest) confirmation: the surface
$S$ is a double covering of ${\bf P}^2$ branched along a sextic curve $B $; the
curves $C_t$ are the inverse images of the lines in ${\bf P}^2$, and they
become
rational when the line is bitangent to $B$. We get an $A_3$\kern-1.5pt - singularity when
the line has a contact of order $4$; thus our assertion in this case follows
from
the (certainly classical) fact that a line with a fourth order contact counts
as a
simple bitangent.
\medskip
\global\def\currenvir{subsection Prop.\ \ref{elag} reduces the computation of the invariant $e(\bar
J
C)$ to the case of a unibranch (rational) curve. To understand this invariant
we
will use a construction of Rego ([R], see also [G-P]). For each $x\in C$, we
put
$\delta_x=\mathop{\rm dim}\nolimits {\cal O}_{\widetilde{C},x}/{\cal O}_{C,x}$ and we denote by
${\cal
C}$ the ideal ${\cal O}_{\widetilde{C}}(-\sum_x (2\delta_x)\, x)$; it is
contained
in the conductor of $C$ (but the inclusion is strict unless $C$ is Gorenstein).
\par\hskip 1truecm\relax For $x\in C$, we denote by $A_x$
and
$\widetilde{A}_x$ the finite dimensional algebras
${\cal O}_{C,x}/{\cal C}_x$ and ${\cal O}_{\widetilde{C},x}/{\cal C}_x$. Let
${\bf G}(\delta_x,\widetilde{A}_x)$ be the Grassmannian of codimension
$\delta_x$ subspaces of
$\widetilde{A}_x$, and ${\bf G}_x$ the closed
subvariety of
${\bf G}(\delta_x,\widetilde{A}_x)$ consisting of sub-$\!A_x$\kern-1.5pt - modules.
We can also view ${\bf
G}_x$ as parameterizing the sub-$\!{\cal O}_{C,x}$\kern-1.5pt - modules
${\cal L}_x$ of codimension $\delta_x$ in
${\cal O}_{\widetilde{C},x}$, because any such sub-module contains ${\cal C}_x$
([G-P], lemma 1.1 (iv)).
Since
${\cal O}_{\widetilde{C}}/{\cal C}$ is a skyscraper sheaf with fibre
$\widetilde{A}_x$ at
$x$, the product $\pprod_{x\in\Sigma }^{}{\bf
G}_x$ parameterizes sub-$\!{\cal O}_C$\kern-1.5pt - modules ${\cal L}\i
{\cal O}_{\widetilde{C}}$ such that $\mathop{\rm dim}\nolimits {\cal O}_{\widetilde{C},x}/{\cal
L}_x=\delta_x$ for all $x$.
This implies $\chi ({\cal O}_{\widetilde{C}}/{\cal L})=\sum_x \delta_x
=\chi ({\cal O}_{\widetilde{C}}/{\cal O}_C)$, hence ${\cal L}\in \bar JC$.
We have thus defined a morphism $e:\pprod_{x\in\Sigma }^{}{\bf
G}_x\rightarrow \bar JC$.
\th Proposition
\enonce The map $e$ is a homeomorphism.
\endth\label{homeo}
\par\hskip 1truecm\relax Note that $e$ is not an isomorphism, already when $C$ is a rational curve
with one ordinary cusp $s$: the Grassmannian ${\bf
G}_s$ is isomorphic to ${\bf P}^1$, while $\bar JC$ is
isomorphic to $C$.
\par\hskip 1truecm\relax Since we are dealing with compact varieties, it suffices to prove
that
$e$ is bijective.\smallskip
{\it Injectivity}: Let ${\cal L}$, ${\cal M}$ be two sub-$\!{\cal O}_C$\kern-1.5pt -
modules
of ${\cal O}_{\widetilde{C}}$ containing ${\cal C}$. If ${\cal L}$ and ${\cal
M}$
give the same element in $\bar JC$, there exists a rational function $\varphi$
on
$\widetilde{C}$ such that ${\cal M}=\varphi{\cal L}$. But the equalities $\mathop{\rm dim}\nolimits
{\cal O}_{\widetilde{C},x}/{\cal M}_x = \mathop{\rm dim}\nolimits {\cal O}_{\widetilde{C},x}/{\cal
L}_x=\mathop{\rm dim}\nolimits \varphi_x{\cal O}_{\widetilde{C},x}/{\cal M}_x$ imply $\varphi_x
{\cal
O}_{\widetilde{C},x}={\cal O}_{\widetilde{C},x}$ for all $x$, which means that
$\varphi$ is constant.\smallskip {\it Surjectivity}: Let
$f:\widetilde{C}\rightarrow C$ be the normalization morphism, and
${\cal L}\in \bar JC$.
Let us denote by
$\widetilde{\cal L}$ the line bundle on $\widetilde{C}$ quotient
of $f^*{\cal L}$ by its torsion subsheaf. We
claim that its degree is
$\le 0$: we have an exact sequence
$$0\rightarrow {\cal L}\longrightarrow f_*\widetilde{\cal
L}\longrightarrow {\cal T}\rightarrow 0$$
where ${\cal T}$ is a skyscrapersheaf supported on the singular locus of $C$,
such
that $\mathop{\rm dim}\nolimits{\cal T}_x\le\delta_x$ for all $x\in
C$ ([G-P], lemma 1.1); this implies $\chi(\widetilde{\cal L})
-\chi({\cal L})\le$ $ \chi({\cal O}_{\widetilde{C}})-\chi({\cal O}_C)$,
from which the required inequality follows. Since $\widetilde{C}$ is rational,
it
follows that $\widetilde{\cal L}^{-1}$ admits a global section whose zero set
is
contained in $\Sigma $. \par\hskip 1truecm\relax Because of the canonical isomorphisms
$$\mathop{\rm Hom}\nolimits_{{\cal
O}_C}({\cal L},{\cal O}_{\widetilde{C}})\cong \mathop{\rm Hom}\nolimits_{{\cal
O}_{\widetilde{C}}}(f^*{\cal L},{\cal O}_{\widetilde{C}}) \cong \mathop{\rm Hom}\nolimits_{{\cal
O}_{\widetilde{C}}}(\widetilde{\cal L},{\cal O}_{\widetilde{C}})\ ,$$
we conclude that there exists a homomorphism $i:{\cal L} \rightarrow {\cal
O}_{\widetilde{C}}$ which is bijective outside $\Sigma$.
Put $n_x=\mathop{\rm dim}\nolimits {\cal O}_{\widetilde{C},x}/i({\cal L}_x)$ for each
$x\in\Sigma$.
Since $$\sum_{x\in\Sigma}n_x=\mathop{\rm dim}\nolimits {\cal
O}_{\widetilde{C}}/i({\cal L})=\chi( {\cal O}_{\widetilde{C}})-\chi({\cal
L})=g=\sum_{x\in\Sigma}
\delta_x\ ,$$
there exists a rational function $\varphi$ on $\widetilde{C}$ with divisor
$\sum_x(\delta_x-n_x)\,x$. Replacing ${\cal L}$ by $\varphi{\cal L}$, we may
assume
$n_x=\delta_x$ for all $x$, which means that ${\cal L}$ belongs to the image
of
$e$. \cqfd
\medskip
\par\hskip 1truecm\relax The variety ${\bf G}_x$ depends only on the local ring ${\cal O}$
of $C$ at $x$ (even only on its completion); we will also denote it by ${\bf
G}_{\cal O}$. Recall that ${\bf G}_{\cal O}$ parameterizes the sub-$\!{\cal
O}$\kern-1.5pt -
modules $L$ of the normalization $\widetilde{\cal O}$ of ${\cal O}$ with $\mathop{\rm dim}\nolimits
\widetilde{\cal O}/L= \mathop{\rm dim}\nolimits \widetilde{\cal O}/{\cal O}$. We put $\varepsilon
(x)=e({\bf G}_x)$ (or
$\varepsilon ({\cal O})=e({\bf G}_{\cal O})$). The above Proposition gives:
\th Proposition
\enonce Let $C$ be a rational unibranch curve; then $e(\bar J C)=\pprod_{x\in
C}^{}\varepsilon (x)$.~\cqfd\endth\label{prod}
\par\hskip 1truecm\relax Of course $\varepsilon (x)$ is equal to $1$ for a smooth point, so we
could as
well consider the product over the singular locus $\Sigma $ of $C$. Note that
in
view of Prop.\ \ref{elag}, we may define $\varepsilon (x)$ for a non-unibranch
singularity by taking the product of the $\varepsilon $\kern-1.5pt - invariants of each
branch; Prop.\ \ref{prod} remains valid.
\section{Examples}
\global\def\currenvir{subsection {\it Singularities with ${\bf C}^*$\kern-1.5pt - action}\label{C*}
\par\hskip 1truecm\relax Assume that the local, unibranch ring ${\cal O}$ admits a ${\bf C}^*$\kern-1.5pt -
action. This action extends to its completion, so we will assume that
${\cal O}$ is complete. The ${\bf C}^*$\kern-1.5pt - action also extends to the
normalization
$\widetilde{\cal O}$ of ${\cal O}$, and there exists a local coordinate
$t\in\widetilde{\cal O}$ such that the line ${\bf C}t$ is preserved (this is
because the pro-algebraic group $\mathop{\rm Aut}\nolimits(\widetilde{\cal O})$ is an extension of
${\bf C}^*$ by a pro-unipotent group, hence all subgroups of
$\mathop{\rm Aut}\nolimits(\widetilde{\cal
O})$ isomorphic to ${\bf C}^*$ are conjugate). It follows that the graded
subring
${\cal O}$ is associated to a semi-group $\Gamma \i{\bf N}$, i.e.\ ${\cal O}$
is
the ring ${\bf C}[[\Gamma ]]$ of formal series $\displaystyle \sum_{\gamma
\in\Gamma }a_\gamma t^\gamma $. \par\hskip 1truecm\relax The ${\bf C}^*$\kern-1.5pt - actions on ${\cal O}$
and
$\widetilde{\cal O}$ give rise to a ${\bf C}^*$\kern-1.5pt - action on
${\bf G}_{\cal O}$. The fixed points of this action are the submodules of
$\widetilde{\cal O}$ which are graded, that is of the form ${\bf C}[[\Delta
]]$,
where $\Delta $ is a subset of ${\bf N}$; the condition $\mathop{\rm dim}\nolimits \widetilde{\cal
O}/{\bf C}[[\Delta ]]=\mathop{\rm dim}\nolimits \widetilde{\cal O}/{\cal O}$ means $\mathop{\rm Card}\nolimits({\bf
N}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Delta )= \mathop{\rm Card}\nolimits({\bf N}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Gamma) $, and the condition that ${\bf
C}[[\Delta ]]$ is a ${\cal O}$\kern-1.5pt - module means $\Gamma +\Delta \i\Delta $. The
first condition already implies that there are only finitely many such fixed
points. According to [B], the number of these fixed points is equal to $e({\bf
G}_{\cal O})$. We conclude: \th Proposition
\enonce Let $\Gamma \i{\bf N}$ be a semi-group with finite complement. The
number
$\varepsilon ({\bf C}[[\Gamma ]])$ is equal to the number of subsets $\Delta
\i{\bf N}$ such that $\Gamma +\Delta \i\Delta $ and $\mathop{\rm Card}\nolimits({\bf N}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Delta
)=
\mathop{\rm Card}\nolimits({\bf N}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Gamma) $. \cqfd
\endth\label{\Gamma}
\par\hskip 1truecm\relax I do not know whether there exists a closed formula computing this number,
say in terms of a minimal set of generators of $\Gamma $. This turns out to be
the
case in the situation we were originally interested in, namely planar
singularities. The semi-group $\Gamma $ is then generated by two coprime
integers
$p$ and $q$, which means that the local ring ${\cal O}$ is of the form ${\bf
C}[[u,v]]/(u^p-v^q)$.
\th Proposition
\enonce Let $p,q$ be two coprime integers. Then $$ \varepsilon ({\bf
C}[[u,v]]/(u^p-v^q))= {1\over p+q}{p+q\choose p}\ .$$
\endth\label{pq}
{\it Proof}: The following proof has been shown to me by P. Colmez.
\par\hskip 1truecm\relax (\ref{pq}.1) We first observe that if a subset $\Delta$ satisfies $\Gamma
+\Delta \i\Delta $, all its translates $n+\Delta$ $(n\in{\bf Z})$ contained in
${\bf N}$ have the same property; moreover, among all these translates, there
is
exactly one
with $\mathop{\rm Card}\nolimits({\bf N}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Delta ) = \mathop{\rm Card}\nolimits({\bf N}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\Gamma)$. Thus the
number we want to compute is the cardinal of the set ${\cal D}$ of subsets
$\Delta\i{\bf N}$ such that $\Gamma
+\Delta \i\Delta $, modulo the identification of a subset and its translates.
\smallskip
\par\hskip 1truecm\relax (\ref{pq}.2) For such a subset $\Delta$, let us introduce the
generating function
$\displaystyle F_\Delta (T)= \sum_{\delta\in\Delta }T^\delta\in {\bf
Z}[[T]]$. Since
$p+\Delta\i\Delta $, we can write, in a unique way, $ \Delta
=\union_{i=1}^p (a(i)+p{\bf N})$ ; then
$\displaystyle\ (1-T^p)\,F_\Delta (T)=\sum_{i=1}^p T^{a(i)}$ .
Writing similarly $ \Delta =$ $\union_{j=1}^q (b(j)+q{\bf
N})$, we get $\displaystyle\ (1-T^q)\,F_\Delta (T)=\sum_{j=1}^q T^{b(j)}$ . Put
$a(j)=b(j-p)+p$ for $p+1\le j\le p+q$;
the equality
$\displaystyle\ (1-T^p)\sum_{j=p+1}^{p+q} T^{a(j)-p}=(1-T^q)\sum_{i=1}^p
T^{a(i)}\ $ reads
$$\sum_{i=1}^{p+q} T^{a(i)} = \sum_{i=1}^p T^{a(i)+q} +
\sum_{j=p+1}^{p+q} T^{a(j)-p}\quad.\leqno (\ref{pq}\ {\it a})$$
\par\hskip 1truecm\relax Conversely, given a function $a:[1,p+q]\rightarrow {\bf N}$ satisfying
(\ref{pq} {\it a}), the set $ \Delta =$ $\union_{i=1}^p (a(i)+p{\bf N})$ is
equal
to $\union_{j=p+1}^{p+q}(a(j)-p+q{\bf N})$, and therefore satisfies $\Gamma
+\Delta \i \Delta $ (note that (\ref{pq}~{\it a}) implies that the classes
(mod.\
$p$) of the $a(i)$'s, for $1\le i\le p$, are all distinct).
\par\hskip 1truecm\relax The equality (\ref{pq} {\it a}) means that there
exists a permutation $\sigma \in{\fam\gothfam S}_{p+q}$ such that $a(\sigma i)$ is
equal
to $a(i)+q$ if $i\le p$ and to $a(i)-p$ if $i>p$. This implies that $a(\sigma
^m(i))$ is of the form $a(i)+\alpha q-\beta p$, with $\alpha ,\beta \in{\bf N}$
and $\alpha +\beta =m$; since $p$ and $q$ are coprime, it follows that $\sigma
$
is of order $p+q$, i.e.\ is a circular permutation. It also follows that the
numbers $a(i)$ are all distinct; hence the permutation $\sigma $ is uniquely
determined. Let $\tau $ be a permutation such that $\tau \sigma \tau^{-1}$ is
the
permutation $i\mapsto i+1$ (mod.\ $p+q)$, and let $S_\Delta =\tau ([1,p])$.
Replacing $a$ by $a\kern 1pt{\scriptstyle\circ}\kern 1pt\tau^{-1} $, our
function $a$ satisfies
$$a(i+1)=\cases{
a(i)+q & if $i\in S_\Delta $,\cr
a(i)-p & if $i\notin S_\Delta $\ .}\leqno(\ref{pq}\ {\it b})$$
Since $\tau $ is determined up to right multiplication by a power of $\sigma $,
the
set $S_\Delta \i[1,p+q]$ is well determined up to a translation (mod.\ $p+q)$.
Note that replacing $\Delta$ by $n+\Delta$ amounts to add the constant value
$n$ to
the function
$a$, hence does not change $S_\Delta$.
\par\hskip 1truecm\relax (\ref{pq}.3) Conversely, let us start from a subset $S\i[1,p+q]$ with $p$
elements. We
define inductively a function $a_S$ on $[1,p+q]$ by the relations (\ref{pq}
{\it
b}), giving to $a_S(1)$ an arbitrary value, large enough so that $a_S$
takes its values in ${\bf N}$. By construction the function $a_S$ satisfies
(\ref{pq} {\it b}), so by (\ref{pq}.2) the subset
$ \Delta_S =\union_{s\in S}^{} (a_S(s)+p{\bf N})$ satisfies $\Gamma
+\Delta_S \i\Delta_S $.
\par\hskip 1truecm\relax An easy computation gives $a_{S+1}(i+1)=a_S(i)$ and therefore
$\Delta _{S+1}=\Delta _S$. Let ${\cal S}$ be the set of subsets of $[1,p+q]$
with
$p$ elements, modulo translation; the maps
$\Delta \mapsto S_\Delta $ from ${\cal D}$ to ${\cal S}$ and $S\mapsto
\Delta_S $ from ${\cal S}$ to ${\cal D}$ are inverse of each other. Since
$\displaystyle \mathop{\rm Card}\nolimits({\cal S})={1\over
p+q}{p+q\choose p}$, the Proposition follows. \cqfd
\bigskip
\global\def\currenvir{subsection {\it Simple singularities}
\par\hskip 1truecm\relax We now consider the case where the singularities of $C$ are {\it simple},
i.e.\
of $A,D,E$ type. The local ring of
such a singularity has only finitely many isomorphism classes of torsion free
rank
$1$ modules, and this property characterizes these singularities among all
plane
curves singularities [G-K].
\th Proposition \enonce Let ${\cal O}$ be the local ring of a
simple singularity. Then $\varepsilon ({\cal O})$ is the number of isomorphism
classes of torsion free rank $1\ {\cal O}$\kern-1.5pt - modules. It is given by:
\par\hskip 1truecm\relax -- $\varepsilon ({\cal O})=l+1$ if ${\cal O}$ is of type $A_{2l}\,;$ \par\hskip 1truecm\relax
--
$\varepsilon ({\cal O})=1$ if ${\cal O}$ is of type $A_{2l+1}\,;$ \par\hskip 1truecm\relax --
$\varepsilon ({\cal O})=1$ if ${\cal O}$ is of type $D_{2l}\ (l\ge2)\,;$ \par\hskip 1truecm\relax
--
$\varepsilon ({\cal O})=l$ if ${\cal O}$ is of type $D_{2l+1}\ (l\ge2)\,;$ \par\hskip 1truecm\relax
--
$\varepsilon ({\cal O})=5$ if ${\cal O}$ is of type $E_6\,;$ \par\hskip 1truecm\relax --
$\varepsilon
({\cal O})=2$ if ${\cal O}$ is of type $E_7\,;$
\par\hskip 1truecm\relax -- $\varepsilon ({\cal O})=7$ if ${\cal O}$ is of type $E_8$.
\endth\label{rat}
{\it Proof}: Let $C$ be a rational curve with only one simple singularity, with
local ring ${\cal O}$; the action of $JC$ on $\bar JC$ has finitely many
orbits,
corresponding to the different isomorphism classes of rank $1\ {\cal O}$\kern-1.5pt -
modules. Since each orbit is an affine space, its Euler characteristic is $1$,
hence by (\ref{e}) $ \varepsilon ({\cal O})=e(\bar JC)$ is equal to the number
of
these orbits.
\par\hskip 1truecm\relax If ${\cal O}$ is unibranch, its completion is of the form
${\bf C}[[u,v]]/(u^p-v^q)$,
with $p=2$, $q=2l+1$ for the type $A_{2l}$, $p=3$, $q=4$ for the type $E_6$
and
$p=3$, $q=5$ for the type $E_8$; in these cases the result follows from
\ref{pq}.
We have
already observed that $\varepsilon =1$ for a $A_{2l+1}$ singularity (Remark
\ref{An}). A $D_l$ singularity is the union of a $A_{l-3}$ branch and a
transversal smooth branch, hence the result by \ref{elag}. Finally an $E_7$
singularity is the union of an ordinary cusp and its tangent, hence has
$\varepsilon =2$. \cqfd
\smallskip
\rem{Remark} Let ${\cal D}$ be the set of graded sub-$\!{\cal O}$\kern-1.5pt - modules
$L\i\widetilde{\cal O}$ with $\mathop{\rm dim}\nolimits \widetilde{\cal O}/L=$ $\mathop{\rm dim}\nolimits \widetilde{\cal
O}/{\cal O}$. Two modules $L$, $M$ in ${\cal D}$ are isomorphic if and only if
$M=t^nL$ for some
$n\in{\bf Z}$, but the dimension condition forces $n=1$. It follows that {\it
each
torsion free rank $1\ {\cal O}$\kern-1.5pt - module is isomorphic to exactly one element
of}
${\cal D}$.
It is quite easy that way to write down the list of isomorphism classes of
rank $1\ {\cal O}$\kern-1.5pt - modules (which is of course well-known, see e.g.\ [G-K]).
For
instance if ${\cal O}$ is of type $E_8$, we get the following
modules (with the notation of \ref{C*}):\par
${\cal O}$, ${\cal O}t+{\cal O}t^8$, ${\cal O}t^2+{\cal O}t^6$, ${\cal O}t^2+
{\cal O}t^4$, ${\cal O}t^3+{\cal O}t^4$, ${\cal O}t^3+{\cal O}t^5+{\cal O}t^7$,
$\widetilde{\cal O}t^4$.
\vskip2cm
\centerline{ REFERENCES} \vglue15pt\baselineskip12.8pt
\def\num#1{\smallskip \item{\hbox to\parindent{\enskip [#1]\hfill}}}
\parindent=1.3cm
\num{B} A.\ {\pc BIALYNICKI-BIRULA}: {\sl On fixed point schemes of actions of
multiplicative and additive groups}. Topology {\bf 12}, 99-103 (1973).
\num{G} L.\ {\pc G\"OTTSCHE}: {\sl The Betti numbers of the Hilbert scheme of
points on a smooth projective surface}. Math.\ Ann.\ {\bf 86}, 193-207 (1990).
\num{G-K} G.-M.\ {\pc GREUEL}, H.\ {\pc KN\"ORRER}: {\sl Einfache
Kurvensingularit\"aten und torsionfreie Moduln}. Math.\ Ann.\ {\bf 270},
417-425
(1985). \num{G-P} G.-M.\ {\pc GREUEL}, G.\ {\pc PFISTER}: {\sl Moduli spaces
for
torsion free modules on curve singularities, I}. J.\ Algebraic Geometry {\bf
2},
81-135 (1993).
\num{H} D. {\pc HUYBRECHTS}: {\sl Birational symplectic manifolds and their
deformations}. Preprint alg-geom/9601015.
\num{M} S.\ {\pc MUKAI}: {\sl Symplectic structure of the moduli
space of sheaves on an abelian or K3 surface}. Invent.\ math.\ {\bf 77},
101-116
(1984).
\num{R} C. J. {\pc REGO}: {\sl The compactified Jacobian}. Ann.\ scient.\
\'Ec.\
Norm.\ Sup.\ {\bf 13}, 211-223 (1980).
\num{V} J.-L.\ {\pc VERDIER}: {\sl Stratifications de Whitney et th\'eor\`eme
de
Bertini-Sard}. Invent.\ math.\ {\bf 36}, 295-312 (1976).
\num{Y-Z} S.-T.\ {\pc YAU}, E.\ {\pc ZASLOW}: {\sl BPS states, string duality,
and
nodal curves on K3}. Preprint hep-th/9512121.
\vskip1cm
\def\pc#1{\eightrm#1\sixrm}
\hfill\vtop{\eightrm\hbox to 5cm{\hfill Arnaud {\pc BEAUVILLE}\hfill}
\hbox to 5cm{\hfill DMI -- \'Ecole Normale
Sup\'erieure\hfill} \hbox to 5cm{\hfill (URA 762 du CNRS)\hfill}
\hbox to 5cm{\hfill 45 rue d'Ulm\hfill}
\hbox to 5cm{\hfill F-75230 {\pc PARIS} Cedex 05\hfill}}
\end
|
1997-01-30T22:19:54 | 9701 | alg-geom/9701004 | en | https://arxiv.org/abs/alg-geom/9701004 | [
"alg-geom",
"math.AG"
] | alg-geom/9701004 | Daisuke Matsushita | D. Matsushita | Simultaneous minimal model of homogeneous toric deformation | LaTeX2e, 8 pages with no figures, [email protected] | null | null | null | null | For a flat family of Du Val singularities, we can take a simultaneous
resolution after finite base change. It is an interesting problem to consider
this analogy for a flat family of higher dimensional canonical singularities.
In this note, we consider an existence of simultaneous terminalization for K.
Altmann's homogeneous toric deformation whose central fibre is an affine
Gorenstein toric singularity. We obtain examples that there are no simultaneous
terminalization even after finite base change and give a sufficient condition
for an existence of simultaneous terminalization. Some examples of
4-dimensional flop are obtained as an application.
| [
{
"version": "v1",
"created": "Thu, 9 Jan 1997 08:40:53 GMT"
},
{
"version": "v2",
"created": "Fri, 10 Jan 1997 02:24:23 GMT"
},
{
"version": "v3",
"created": "Thu, 30 Jan 1997 21:14:59 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Matsushita",
"D.",
""
]
] | alg-geom | \section{Introduction}
For a flat family of surfaces $f : X \to S$,
it is an interesting problem exists
a birational morphism $\tau : \widetilde{X} \to X$ such that
\begin{enumerate}
\item $f \circ \tau$ is a flat morphism
\item The fibre $\widetilde{X}_s := (f\circ \tau)^{-1}(s)$
$(s \in S)$ is the minimal resolution of
$X_s$.
\end{enumerate}
If there exists such $\tau$,
we say $f$ admits a simultaneous resolution.
Let $f : X \to S$ be a flat morphism whose fibres have
only Du Val singularities. Then
Brieskorn \cite{brie1,brie2} and Tyurina\cite{tyur} show
that
there exists a finite surjective morphism $S' \to S$ and
a flat morphism
$f' : X\times_{S}S' \to S'$ admits
a simultaneous resolution.
It is a key fact for Minimal Model Theory.
Thus it is natural to consider an analogy for
higher dimensional singularity.
According to Minimal Model theory,
it is suitable to consider an existence of
``simultaneous terminalization'' for
a flat family of higher dimensional singularity.
\noindent
{\sc Definition 1. \quad}
For a flat morphism $f : X \to S$ whose
fibres are $n$-folds $(n \ge 3)$, we say $f$ admits a
simultaneous terminalization if there exists a birational
morphism $\tau : \widetilde{X} \to X$ which satisfies the
following conditions:
\begin{enumerate}
\item $f \circ \tau$ is a flat morphism.
\item The fibre $\widetilde{X}_s := (f\circ \tau)^{-1}(s)$ $(s \in S)$
has only terminal singularity.
\item $K_{\widetilde{X}_s}$ is $\tau$-nef.
\end{enumerate}
Recently
K.~Altmann constructed in
\cite[Definition 3.1]{homogeneous}
affine toric varieties which are called
``homogeneous toric deformation''.
This toric varieties
can be described
many flat families of toric singularities such as
versal deformation space of an isolated Gorenstein toric singularity
\cite[Theorem 8.1]{alt}.
In this note, using toric Minimal Model Theory,
we investigate the existence of simultaneous terminalization of
Gorenstein homogeneous toric deformation which is a
homogeneous toric deformation whose central fibre is
an affine Gorenstein toric variety.
Our main results are as follows:
\begin{thm}
There exists a
Gorenstein homogeneous toric deformation whose fibre
is $n$-dimensional $(n \ge 3)$ and which
admits no simultaneous
terminalization even after finite base change.
\end{thm}
We consider the condition of an existence of simultaneous
terminalization, and obtain the following results:
\begin{thm}
A Gorenstein homogeneous toric deformation $f : X \to {\mathbb C}^{m}$
admits a simultaneous terminalization
if $X$ has a crepant desingularization.
\end{thm}
We recall
the definition of
homogeneous toric deformation
by K.~Altmann, and Gorenstein homogeneous
toric deformation in section 2.
Theorems 1 and 2 will be proved in Sections 3 and 4, respectively.
Some examples of 4-dimensional flops
are obtained as an application.
\section{Homogeneous toric deformation}
\noindent
First we introduce the definition homogeneous
toric deformation by K.~Altmann.
\noindent
{\sc Definition 2. \quad} A flat morphism $f : X \to {\mathbb C}^{m}$
is called a homogeneous toric deformation if the following conditions are
satisfied:
\begin{enumerate}
\item $X := {\rm Spec}{\mathbb C}[\sigma^{\vee}\cap M]$
is an affine toric variety.
\item $f$ is defined by $m$ equations such that
$x^{r_i} - x^{r_0}$ where $r_i \in \sigma^{\vee} \cap M$,
$(0 \le i \le m)$.
\item Let $L := \oplus^{i = m}_{i = 1}{\mathbb Z}(r_{i} - r_{0})$
be the sublattice of $M$.
The central fibre $Y := f^{-1}(0,\cdots , 0)$ is isomorphic to
${\rm Spec}{\mathbb C}[\bar{\sigma}^{\vee}\cap \bar{M}]$ where
$\bar{\sigma} = \sigma \cap L^{\perp}$ and $\bar{M} := M/L$.
\item $i : Y \to X$ sends the
closed orbit in $Y$ isomorphically onto the closed orbit in
$X$.
\end{enumerate}
In this note, we consider a homogeneous toric deformation
with some additional conditions:
\noindent
{\sc Definition 3. \quad}
We call homogeneous toric deformation $f: X \to {\mathbb C}^{m}$ a
Gorenstein homogeneous toric deformation if it
satisfies the following two conditions:
\begin{enumerate}
\item $Y$ is an affine Gorenstein singularity.
\item The restriction $-r_i$, $(0 \le i \le m)$ to $\bar{M}$
determines $K_Y$.
\end{enumerate}
{\sc Examples.}
\begin{enumerate}
\item Most plain example is $f : {\mathbb C}^{2}(x,y) \to {\mathbb C}(t)$.
In this case, $f$ is defined by $x - y = t$.
\item Let $g : {\cal X} \to S$ be a versal deformation space of
$A_n$ singularity. It can be described as follows:
\[
{\cal X} = (xy + z^{n+1} + t_1 z^{n-1} + \cdots + t_{n-1}z + t_n = 0)
\subset {\mathbb C}^{n+3} \to {\mathbb C}^{n}(t_1 , \cdots t_n).
\]
Let $\alpha_{i}$, $(0 \le i \le n)$ be elementary symmetric
polynomials on ${\mathbb C}^{n+1}(s_0 , \cdots s_n)$ and
$H$ a hyperplane such that $\sum_{i=0}^{n}s_i = 0$.
We take a base change by
$\alpha_i (s_0 , \cdots , s_{n})=t_i :
H \to {\mathbb C}^{n}$,
\[
\begin{CD}
{\cal X}\times_{{\mathbb C}^{n}} H @>>> {\cal X} \\
@V{f}VV @VV{g}V\\
H @>>> {\mathbb C}^{n}.
\end{CD}
\]
Then ${\cal X}\times_{{\mathbb C}^{n}} H$ can be described
\[
(xy + \prod_{i=0}^{n}(z + s_i ) = 0)\wedge (\sum_{i=0}^{n}s_i = 0)
\subset {\mathbb C}^{n+4}(x,y,z,s_0 , \cdots , s_n).
\]
Let $u_0 := \sum_{i=0}^{n}s_i$,
$u_i : = s_i - s_0$, $(1 \le i \le n)$
and $z_i := z + s_i$. By using
this coordinate, we can describe
\[
{\cal X} \times_{{\mathbb C}^{n}} H = (xy + \prod_{i=0}^{n}z_i = 0)
\subset {\mathbb C}^{n+3}(x,y,z_0 , \cdots , z_n)
\]
and $f = (z_1 - z_0 , \cdots , z_n - z_0 )$.
Thus $f : {\cal X}\times_{{\mathbb C}^{n}}H
\to H$ is a Gorenstein homogeneous toric
deformation.
\item Let $g : {\cal X} \to {\cal M}$ be a versal deformation space
of $n$-dimensional $(n \ge 3)$
isolated Gorenstein toric singularity. K.~Altmann
shows in \cite[Theorem 8.1]{alt}
that for every irreducible component ${\cal S}$ of ${\cal M}$,
\[
\begin{CD}
X= {\cal X}\times_{{\cal S}_{\text{red}}}{\cal M} @>>> {\cal X} \\
@V{f}VV @VV{g}V \\
{\cal S}_{\text{red}} @>>> {\cal M}
\end{CD}
\]
the pull back
$f : X \to {\cal S}_{\text{red}}$ is a Gorenstein
homogeneous toric deformation.
\end{enumerate}
\section{Simultaneous terminalization for Gorenstein homogeneous
toric deformation}
In this section, we give a proof of Theorem 1.
Let $Y$ be a hypersurface in a quotient space such that
$$
Y = (h := x_1 \cdots x_p - x_{p+1} \cdots x_{n+1} = 0) \subset
{\mathbb C}^{n+1}/G
$$
where $G \cong {\mathbb Z}/n{\mathbb Z}$.
The action of $G$
on ${\mathbb C}^{n+1}$ is as follows:
$$
(x_1 , \cdots , x_{n+1}) \to
(\zeta^{a_1}x_1 , \cdots , \zeta^{a_{n+1}}x_{n+1}), \quad (0 \le a_i < l)
$$
where $\zeta$ is a $l$-th root of unity.
We assume that
${\mathbb C}^{n+1}/G$ has only Gorenstein terminal singularities.
Moreover we assume that
$$
\sum_{i=1}^{n+1} a_i \le l + \max \{\sum_{i=1}^{p} a_i ,
\sum_{i=p+1}^{n+1} a_i \}.
$$
Then $Y$ has canonical singularities by \cite[Theorem 4.6]{youngperson}.
Let $X := (h = t) \subset {\mathbb C}^{n+1}/G \times {\mathbb C}(t)$
and $f :X \to {\mathbb C}(t)$ projection. Then a general fibre
of $f$ has only ${\mathbb Q}$-factorial terminal singularities
because the total space
${\mathbb C}^{n+1}/G \times {\mathbb C}$ has only ${\mathbb Q}$-factorial
terminal singularities.
Suppose
that there exists a simultaneous terminalization after some
finite base change. Let
$( h = t^m) \subset {\mathbb C}^{n+1}/G \times {\mathbb C}$ be
a finite base change of $X$ and $ \tau : {\cal X} \to (h = t^m)$
a simultaneous terminalization.
We consider the following diagram:
$$
\begin{array}{ccccc}
(h = t) \subset {\mathbb C}^{n+2}
& \leftarrow &
(h = t^m) \subset {\mathbb C}^{n+2}&
\stackrel{\tau'}{\leftarrow} & {\cal X'} \\
\downarrow & & \downarrow & & \downarrow \\
X = (h = t)
\subset {\mathbb C}^{n+1}/G \times {\mathbb C}
& \leftarrow &
(h = t^m)
\subset {\mathbb C}^{n+1}/G \times {\mathbb C}
& \stackrel{\tau}{\leftarrow} & {\cal X}
\end{array},
$$
where ${\cal X}' =
{\cal X} \times_{{\mathbb C}^{n+1}/G \times {\mathbb C}} {\mathbb C}^{n+2}$.
Because general fibres of $f : X \to {\mathbb C}$ have only
${\mathbb Q}$-factorial terminal singularities, the codimension of
exceptional sets of $\tau$ is greater than two. Thus $\tau$ and
$\tau'$ are small birational morphisms. But
$(h =t^m )\subset {\mathbb C}^{n+2}$ has only hypersurface singularities
whose singular locus has codimension four.
Thus it is ${\mathbb Q}$-factorial,
a contradiction. \hspace*{\fill} $\Box$
\section{Simultaneous minimal model of Gorenstein homogeneous
toric deformation}
\begin{thm}
Let $f : X := {\rm Spec}{\mathbb C}[\sigma^{\vee}\cap M]
\to {\mathbb C}^{m}$ be a Gorenstein homogeneous
toric deformation and
$\tau : \widetilde{X} \to X$ a toric minimal model of $X$.
Assume that $\dim X = n+m$. Then
\begin{enumerate}
\item $f \circ \tau : \widetilde{X} \to {\mathbb C}^{m}$ is a flat morphism.
\item $K_{\widetilde{X}_t}$ is $\tau$-nef
\item $\widetilde{X}_t$ has only canonical complete intersection
singularities in quotient space such that
\[
(F_i - F_0 = 0) \subset
{\mathbb C}^{n+m}/G, \quad (1 \le i \le m)
\]
where $G$ is an abelian group acting on ${\mathbb C}^{n+m}$
diagonally, ${\mathbb C}^{n+m}/G$ has only Gorenstein
terminal singularities
and $F_i$ are invariant monomials of the action of $G$.
The monomials $F_i$ can be written as
\begin{eqnarray*}
F_i &=& \prod_{j=p_i + 1}^{p_{i+1}} x_j \quad (0 \le i \le m) \\
&& 0 = p_0 < p_1 < p_2 < \cdots < p_m < p_{m+1} = n+m
\end{eqnarray*}
where $x_j$ is the j-th coordinate of ${\mathbb C}^{n+m}$.
\item If $\widetilde{X}$ is smooth, $\widetilde{X}_t$ has only
terminal singularities.
\end{enumerate}
\end{thm}
{\sc Remark. \quad} If $\dim X = 2+m$ ( i.e. $f$ is 2-dimensional fibration ),
then we can write $F_i$, $(1 \le i \le m)$ as
$$
F_i = x_{i+1} \quad (0 \le i \le m-1) , \quad F_m = x_{m}x_{m+1}
$$
by changing indices if necessary. Because $F_i$ are invariant monomials
under the action of $G$, the action of each element of $g \in G$
is nontrivial only $x_m$, $x_{m+1}$ axises.
But ${\mathbb C}^{2+m}/G$
has only Gorenstein terminal singularity, the action of $G$ must be
trivial. Thus each fibre of $f\circ \tau$ is smooth, and $\tau$ gives a
simultaneous resolution.
\noindent
{\sc Proof of Theorem 3. \quad}
By K.~Altmann \cite[Theorem 3.5 , Remark 3.6]{homogeneous},
we can state the construction of $\sigma$ as follows:
\begin{enumerate}
\item $\sigma$ can be written as $\sigma = {\mathbb R}_{\ge 0}P$ where
$P$ is an $(n+m-1)$-dimensional polygon
such that
\begin{eqnarray*}
P &:=& {\rm Conv}(\cup_{i=0}^{m}R_i \times e_i) \\
&&\text{where $R_i \times e_i =
\{(x_1 , \cdots , x_{n-1}, 0 , \cdots , 1 , \cdots ,0)
\in {\mathbb R}^{n+m}|
(x_1 , \cdots , x_{n-1}) \in R_i \}$}
\end{eqnarray*}
and $R_i$, $(0 \le i \le m)$ are integral polytopes in ${\mathbb R}^{n-1}$.
\item
$f$ can be written as $(x^{r_i} - x^{r_0})$, $(1 \le i \le m)$ where
$r_i : N_{{\mathbb R}} = {\mathbb R}^{n+m} \to {\mathbb R}$
is the $n+i$-th projection.
\end{enumerate}
Thus, all primitive one dimensional generators of $\sigma$
are contained in
the hyperplane defined by $r_0 + \cdots + r_m = 1$.
Let $\tau : \widetilde{X} \to X$ be a toric minimal model of $X$
and $\sigma = \cup \sigma_{\lambda}$ is the corresponding cone
decomposition. Then these cones satisfy the following
three conditions:
\begin{enumerate}
\item $\sigma_{\lambda}$ is a simplex.
\item Let $ k_1 ,\cdots , k_{n+m} $ be
one dimensional primitive generators of $\sigma_{\lambda}$.
Then all $k_i$ are contained in the hypersurface
defined by $r_0 + \cdots + r_m = 1$.
\item The polytope
$$
\Delta_{\lambda} := \sum_{i=0}^{n+m} \alpha_i k_i, \quad
\sum_{i=0}^{n+m} \alpha_i \le 1, \quad \alpha_i \ge 0
$$
contains no lattice points except its vertices.
\end{enumerate}
Let $X_{\lambda}:= {\rm Spec}{\mathbb C}[\sigma_{\lambda}^{\vee}\cap M]$
and $k_{i}^{\vee}$, $(1 \le i \le n+m)$ be the dual vectors of $k_{i}$.
Then, by (1), $X_{\lambda}$ can be written
as follows:
$$
X_{\lambda} \cong {\mathbb C}^{n+m}/G
$$
where
$G := N/\oplus_{i=1}^{n+m}{\mathbb Z}k_i$ and the action of $G$
is diagonal.
Because $k_j$ are contained in the hypersurface defined by
$r_0 + \cdots + r_m = 1$ and
$\langle r_i , k_j \rangle\ge 0$ ($r_i \in \sigma^{\vee}$),
$$
\left\{
\begin{array}{cc}
\langle r_i , k_j \rangle = 1 & \text{for $p_i < j \le p_{i+1}$} \\
\langle r_i , k_j \rangle = 0 & \text{other $j$}
\end{array}
\right.
$$
where $0=p_0 < p_1 < p_2 < \cdots < p_{m} < p_{m+1} = n+m$.
Thus we can write
$$
x^{r_i} = \prod_{j=p_{i}+1}^{p_{i+1}} x_{j}
$$
where $x_j = x^{k^{\vee}_{j}}$ is the j-th coordinate of ${\mathbb C}^{n+m}$.
The monomials $x^{r_i}$ are invariant under the action of $G$,
because $r_i \in \sigma_{\lambda}^{\vee}\cap M$. Thus
if we set $F_i = x^{r_i}$, proofs of theorem (1), (2) and
(3) are completed. Finally, we show (4). Since $X_{\lambda}$ is
smooth, a general fibre of $f\circ \tau$ is smooth.
We study a singularity of central fibre.
Because $X_{\lambda}$ is smooth
we may set $k_i = e^{i}$ $(1 \le i \le n+m)$ where
$e^{i}$ are the standard basis of ${\mathbb Z}^{n+m}$. Then
$r_i$ can be written
$$
r_i = (e^{p_i + 1})^{\vee} + \cdots + (e^{p_{i+1}})^{\vee}.
$$
Thus a generator of the cone of central fibre $f\circ \tau$
can be written
$$
e^{s_1} + \cdots + e^{s_m}, \quad p_i < s_i \le p_{i+1},
$$
by the definition of homogeneous toric deformation.
So all generators of this cone contains hypercube of ${\mathbb Z}^{n+m}$,
hence central fibre has only terminal singularities.
\hspace*{\fill} $\Box$
\noindent
A toric minimal model of $X$ is not unique, and which
can be joined by flops each other. We can obtain
some examples of 4-dimensional flops as an application.
\noindent
{\sc Example. \quad}
There are 4-dimensional flops which satisfy the following diagram:
$$
\begin{array}{ccc}
{\mathbb P}(1,a,b) \subset Z & - \to & {\mathbb P}(1,a,b)
\subset Z^{+} \\
\phi \searrow & & \swarrow \phi^{+} \\
& W &
\end{array}
$$
where $\phi({\mathbb P}(1,a,b)) = \phi^{+}({\mathbb P}(1,a,b)) = {\rm pt}$.
\noindent
{\sc Remark.} \quad In the case $a = b = 1$, it is known as ``Mukai
transformation''.
\noindent
{\sc Construction of Example. \quad}
Let $\bar{\sigma}$ be a 3-dimensional cone whose primitive generators
are
$$
\bar{\sigma} = \langle (1,0,1),(0,1,1),(-a,-b,1),(1,1,1),(-a,1-b,1),
(1-a,-b,1) \rangle \subset {\mathbb R}^{3}.
$$
We consider a deformation of toric singularity
$Y := {\rm Spec}{\mathbb C}[\bar{\sigma}^{\vee} \cap \bar{M}]$.
The polytope $Q := \bar{\sigma}\cap
\{(x,y,z) \in {\mathbb R}^{3} | z=1\}$ has
Minkowski decomposition $R_0 + R_1 + R_2$ where
$$
R_0 = \langle (1,0),(0,0) \rangle, \quad
R_1 = \langle (0,1),(0,0) \rangle, \quad
R_2 = \langle (-a,-b),(0,0) \rangle.
$$
Thus by \cite[Remark 3.6]{homogeneous} the corresponding toric homogeneous
deformation space of $Y$ is a toric variety
$X := {\rm Spec}{\mathbb C}[\sigma^{\vee} \cap M]$ where
$$
\sigma := {\mathbb R}_{\ge 0}\langle e_1 , e_2 , e_3 , e_4 , e_5 , e_6 \rangle,
$$
$$
\begin{array}{ccc}
e_1 := (1,0,1,0,0), & e_2 := (0,0,1,0,0), & e_3 := (0,1,0,1,0) \\
e_4 := (0,0,0,1,0), & e_5 := (-a,-b,0,0,1), & e_6 := (0,0,0,0,1).
\end{array}
$$
We construct two different crepant resolutions
of $X$.
Because $ae_1 -ae_2 + be_3 - be_4 + e_5 - e_6 = 0$,
by Reid\cite[Lemma 3.2]{reid}, there are
two cone decompositions of $\sigma$ such that
$$
\sigma = \langle e_2 , e_4 , e_6, e_1 , e_3 \rangle \cup
\langle e_2 , e_4 , e_6, e_3 , e_5 \rangle \cup
\langle e_2 , e_4 , e_6, e_5 , e_1 \rangle
$$
and
$$
\sigma = \langle e_1 , e_3 , e_5 , e_2 , e_4 \rangle \cup
\langle e_1 , e_3 , e_5 , e_4 , e_6 \rangle \cup
\langle e_1 , e_3 , e_5 , e_6 , e_2 \rangle.
$$
Let $\tilde{Z}$ and
$\tilde{Z^{+}}$ be the toric varieties corresponding
to above cone decompositions.
Then the exceptional sets of $\phi : \tilde{Z} \to X$ and
$\phi^{+} : \tilde{Z^{+}} \to X$ are isomorphic
to ${\mathbb P}(1,a,b)$ and $\phi({\mathbb P}(1,a,b))
= \phi^{+}({\mathbb P}(1,a,b)) = pt$.
There is a diagram
$$
\begin{array}{ccccc}
\tilde{Z} & \stackrel{\phi}{\to} & X &
\stackrel{\phi^{+}}{\leftarrow} & \tilde{Z^{+}} \\
f \downarrow & & \downarrow & & \downarrow f^{+} \\
{\mathbb C}^2 & = & {\mathbb C}^{2} & = & {\mathbb C}^{2}.
\end{array}
$$
Then the exceptional set of $\phi$ and $\phi^{+}$ are contained in a fibre
of $f$ and $f^{+}$ respectively, since $\phi({\mathbb P}(1,a,b))
= \phi^{+}({\mathbb P}(1,a,b)) = pt$.
Let $\imath : {\mathbb C} \to {\mathbb C}^{2}$ be the diagonal map.
We set $Z$, $Z^{+}$ and
$W$ the pull back by $\imath$ of
$\tilde{Z} \to {\mathbb C}^{2}$,
$\tilde{Z^{+}} \to {\mathbb C}^{2}$ and
$X \to {\mathbb C}^{2}$ respectively.
\hspace*{\fill} $\Box$
|
1992-05-22T10:58:07 | 9205 | alg-geom/9205007 | en | https://arxiv.org/abs/alg-geom/9205007 | [
"alg-geom",
"math.AG"
] | alg-geom/9205007 | Subhashis Nag | Subhashis Nag | On the Tangent Space to the Universal Teichmuller Space | 25 pages. latex paper | null | null | null | null | We find a remarkably simple relationship between the following two models of
the tangent space to the Universal Teichm\"uller Space: (1) The real-analytic
model consisting of Zygmund class vector fields on the unit circle; (2) The
complex-analytic model comprising 1-parameter families of schlicht functions on
the exterior of the unit disc which allow quasiconformal extension. Indeed, the
Fourier coefficients of the vector field in (1) turn out to be essentially the
same as (the first variations of) the corresponding power series coefficients
in (2). These identities have many applications; in particular, to conformal
welding, to the almost complex structure of Teichm\"uller space, to study of
the Weil-Petersson metric, to variational formulas for period matrices, etc.
These utilities are explored.
| [
{
"version": "v1",
"created": "Fri, 22 May 1992 11:15:05 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Nag",
"Subhashis",
""
]
] | alg-geom | \section{Introduction}
Let $ \Delta $ denote the open unit disc, and $ S^{1} = \partial \Delta $.
Two classic models of the Universal Teichm\" uller Space $T(1) = T(\Delta)$ are
well-known (see [6],[7]) : \\
\noindent (a) the real-analytic model containing all (M\" obius-normalised)
quasisymmetric homeomorphisms of the unit circle $S^1$ ; \\
\noindent (b) the complex-analytic model comprising all (normalized) schlicht
functions on the exterior of the disc :
$$
\Delta^\star = \{ z \in \widehat{{\bf C}}:\;\mid z\mid > 1 \} = \widehat{{\bf
C}} - (\Delta \cup S^1)
$$
\noindent which allow quasiconformal extension to the whole of $\widehat{{\bf
C}}$ (the Riemann Sphere).
\par The relationship between them is via the rather mysterious operation
called ``Conformal Welding" (see [5], and below). Nevertheless, at the
infinitesimal level, the above models have an amazingly simple relationship
that forms the basis for this paper. Indeed, the $k^{th}$ {\it{Fourier
Coefficient}} of the vector field representing a tangent vector in model (a),
and the (first variation of) the $k^{th}$ {{\it{power series
coefficient}} representing the {\it{same}} tangent vector in model (b),
turn out to be just ($\sqrt{-1}$ times) complex conjugates of each other. See
Theorem 1 below.
It seems that this unexpectedly simple relationship has several interesting
consequences.
\begin{enumerate}
\item It allows a description of the tangent space to $T(\Delta)$ by ``Zygmund
class power series". (Section 3)
\item It provides immediate proof for the remarkable fact that the Hilbert
transform on Zygmund class vector fields on $S^1$ represents the almost complex
structure on $T(\Delta)$. (Section 4)
\item It provides a simple excplicit formula for the derivative of the
conformal welding map. (Section 5)
\item The infinite-dimensional Weil-Petersson metric on (the ``smooth
points" of) $T(\Delta)$ that was found by the present another in [10] Part II
gets a new expression. \\
(Section 6)
\item We get a formula for the derivative of the infinite-dimensional
period mapping studied by us in [8], [9] in terms of power series variations.
This relates to formulas claimed in [4]. (Section 7)
\end{enumerate}
\center${\star~~~~~\star~~~~~\star}$
\bf Acknowledgement}
\par I would like to thank Clifford Earle and Dennis Sullivan for helpful
communications. In particular, C.Earle suggested some fairly explicit examples
that led me to my general results.
\section{Teichm\" uller Theory}
\par The universal Teichm\" uller space $ T(\Delta) $ is a holomorphically
homogeneous complex Banach manifold that serves as the universal ambient space
where all the Teichm\" uller spaces (of arbitrary Fuchsian groups) lie
holomorphically embedded.
\par As usual, we set the stage by introducing the chief actor -- namely the
space of (proper) Beltrami coefficients $ L^{\infty}(\Delta)_{1} $ ; it is the
open unit ball in the complex Banach space of $L^{\infty}$ functions on the
unit disc $\Delta$. The principal construction is to solve the Beltrami
equation.
$$
w_{\bar{z}} = \mu w_{z} \eqno(1)
$$
\noindent for any $\mu \in L^{\infty} (\Delta)_{1}$. The two
above-mentioned models of Teichm\" uller space correspond to discussing two
pertinent solutions for (1) :
\noindent {\bf (a)} $w_{\mu}$ - theory : The quasiconformal hemeomorphism of
{{\bf C}} which is $\mu$-conformal (i.e. solves (1)) in $\Delta$, fixes $\pm 1$
and $-i$, and keeps $\Delta$ and $\Delta^{\star}$ (= exterior of $\Delta$) both
invariant. This $w_{\mu}$ is obtained by applying the existence and uniqueness
theorem of Ahlfors-Bers (for (1)) to the Beltrami coefficient which is $\mu$ on
$\Delta$ and extended to $\Delta^{\star}$ by reflection $(\tilde{\mu}
(1/\bar{z}) = \overline{\mu(z)} z^{2}/\bar{z}^{2}$ for $z \in \Delta)$.
\noindent {\bf (b)} $w^{\mu}$ - theory : The quasiconformal homeomorphism on
{\bf C}, fixing $0,1,\infty$, which is $\mu$-conformal on $\Delta$ and
conformal on $\Delta^{\star}$. $w^{\mu}$ is obtained by applying the
Ahlfors-Bers theorem to the Beltrami coefficient which is $\mu$ on $\Delta$ and
zero on $\Delta^{\star}$.
\par The fact is that $w_{\mu}$ depends only real analytically on $\mu$,
whereas $w^{\mu}$ depends complex-analytically on $\mu$. We therefore obtain
two standard models ({\bf{(a)}} and {\bf{(b)}} below) of the universal
Teichm\" uller space, $T(\Delta)$.
Define the universal Teichm\" uller space :
$$
T(\Delta) = {L}^{\infty} (\Delta)_{1} / \sim . \eqno(2)
$$
\noindent Here $\mu \sim \nu$ if and only if $w_{\mu} = w_{\mu}$ on $\partial
\Delta = S^{1}$, and that happens if and only if the conformal mappings
$w^{\mu}$ and $w^{\nu}$ coincide on $\Delta^{\star} \cup S^{1}$.
We let
$$
\Phi : L^{\infty}(\Delta)_{1} \longrightarrow T(\Delta) \eqno(3)
$$
\noindent denote the quotient (``Bers") projection. $T(\Delta)$ inherits its
canonical structure as a complex Banach manifold from the complex structure of
$L^\infty(\Delta)_{1}$ ; namely, $\Phi$ becomes a {\sl{holomorphic
submersion}}.
The derivative of $\Phi$ at $\mu = 0$ :
$$
d_{0} \Phi : L^\infty(\Delta) \longrightarrow T_{\sl{O}} T(\Delta) \eqno(4)
$$
\noindent is a complex-linear surjection whose kernel is the space $N$ of
``infinitesimally trivial Beltrami coefficients''.
$$
N = \{ \mu \epsilon L^{\infty}(\Delta) : \int \int_{\Delta} \mu \phi = 0~~
\mbox{ for all}~~~\phi \in A(\Delta) \} \eqno(5)
$$
\noindent where $A(\Delta)$ is the Banach space of integrable $(L^1)$
holomorphic functions on the disc. Thus, the tangent space at the origin
$(\sl{O}= \Phi(0))$ of $T(\Delta)$ is $L^{\infty}(\Delta)/N$.
\par See Ahlfors[2], Lehto[6], and Nag[7] for this material and for what
follows.
\par It is now clear that to $\mu \epsilon L^{\infty}(\Delta)_{1}$ we can
associate the {\it{quasisymmetric homeomorphism}}
$$
f_{\mu} = w_{\mu} \mid_{S^{1}} \eqno(6)
$$
\noindent as representing the Teichm\" uller point $[\mu]$ in version
{\bf{(a)}} of $T(\Delta)$. Indeed $T(\Delta)_{(a)}$ is the homogeneous space :
$$
\begin{array}{llcl} {\bf{(a)}} & T(\Delta) & = & \mbox{{Homeo$_{q.s.}$}
$(S^{1})$ / {M\"ob} $(S^1)$} \\
& & = & \{ \mbox{quasisymmetric homeomorphisms of $S^1$ fixing $\pm 1$ and
$-i$} \}
\end{array}
$$
Alternatively, $[\mu]$ is represented by the univalent function
$$
f^\mu = w^\mu \mid_{\Delta^{\star}} \eqno(7)
$$
\noindent on $\Delta^{\star}$, in version {\bf (b)} of $T(\Delta)$. A more
natural choice of the univalent function representing $[\mu]$ is to use a
different normalisation for the solution $w^{\mu}$ (since we have the freedom
to post-compose by a M\" obius transformation). In fact, let
$$
W^{\mu} = M^\mu o w^\mu \eqno(8)
$$
\noindent where $M^\mu$ is the unique M\" obius transformation so that the
univalent function (representing $[\mu]$) :
$$
F^\mu = W^\mu \mid_{\Delta^{\star}} \eqno(9)
$$
\noindent has the properties: \\
\noindent (i) $F^\mu$ has a simple pole of residue 1 at $\infty$ \\
\noindent (ii) $(F^{\mu}(\zeta) - \zeta) \rightarrow 0$ as $\zeta \rightarrow
\infty$.
\noindent Namely, the expansion of $F^\mu$ in $\Delta^{\star}$ is of the form :
$$
F^\mu(\zeta) = \zeta (1 + {{c_{2}} \over {\zeta^{2}}} + {{c_{3}} \over
{\zeta^{3}}} + \ldots) \eqno(10)
$$
\noindent Let us note that the original ($0,1,\infty$ fixing) normalisation
gives an expansion of the form :
$$
f^\mu(\zeta) = \zeta (a + {{b_{1}} \over {\zeta}} + {{b_{2}} \over
{\zeta^{2}}} + {{b_{3}} \over {\zeta^{3}}} \ldots) \eqno(11)
$$
\noindent and the M\" obius transformation $M^\mu$ must be $M^{\mu }(w) = w/a -
b_{1}/a$. Since $(a,b_{1},b_{2},..)$ depend holomorphically on $\mu$, we see
that $(c_{2},c_{3},...)$ also depend holomorphically on $\mu$. Thus, our
complex-analytic version $T(\Delta)_{(b)}$ of the universal Teichm\" uller
space is :
$$
\begin{array}{lrcl}
{\bf{(b)}} & T(\Delta) & = & \left\{{\rm Univalent\;
functions \;in \;\Delta^{\star}\; with\; power \;series \;of\; the\; form\;
(10),}\right. \\
& & & \left. {\rm allowing\; quasiconformal\; extension\; to\; the\; whole\;
plane }\right\}.
\end{array}
$$
\noindent $T(\Delta)_{(b)}$ is simply a\,\,``pre-Schwarzian-derivative''\,\,
version of the Bers embedding of Teichm\" uller space.
\par It is worth remarking here that the criteria that an expansion of the form
(10) represents an univalent function, and that it allows quasiconformal
extension, can be written down solely in terms of the coefficients $c_{k}$,
(using the Grunsky inequalities etc.). See Pommerenke [11]. Thus
$T(\Delta)_{(b)}$ can be thought of as a certain space of sequences
$(c_{2},c_{3},...)$, and its tangent space will be given the concomitant
description below.
\noindent {\underline{Tangent space to the real-analytic model}} : Since
$T(\Delta)$ is a homogeneous space (see version {\bf (a)}) for which the right
translation (by any fixed quasisymmetric homeomorphism) acts as a biholomorphic
automorphism, it is enough in all that follows to restrict attention to the
tangent space at a single point -- the origin ({\sl{O}} = class of the
identity homeomorphism) -- of $T(\Delta)$.
\par Given any $\mu \in L^\infty (\Delta)$, the tangent vector $d_{0}\Phi(\mu)$
is represented by the real vector field $V[\mu] = \dot{w}[\mu] {{\partial}
\over {\partial z}}$ on the circle that produces the 1-parameter flow $w_{t
\mu}$ of quasisymmetric homeomorphisms:
$$
w_{t \mu}(z) = z + t \dot{w} [\mu](z) + o(t) \eqno(12)
$$
\par The vector field becomes in the $\theta$-coordinate :
$$
V[\mu] = \dot{w}[\mu](z) {{\partial} \over {\partial z}} =
u(e^{i \theta}) {{\partial} \over {\partial \theta}} ,
$$
\noindent where,
$$
u(e^{i \theta}) = {{\dot{w}[\mu](e^{i \theta})} \over {i e^{i \theta}}}~.
\eqno(13)
$$
By our normalization, $u$ vanishes at $1,-1$ and $-i$.
In Gardiner-Sullivan $[3, Section8]$ the precise class of vector fields
arising from such quasisymmetric flows is determined as the Zygmund
$\Lambda$ class. They have delineated the theory on the upper half-plane
$U$; we adapt that result to the disc using the M\" obius transformation
$$
T(z) = {{z-i} \over {z+i}}~~,~~T : U \longrightarrow \Delta~~.
\eqno(14)
$$
\noindent We point out that $(0,1,\infty)$ go to $(-1,-i,1)$ respectively.
Notice that the corresponding identification of the real line to $S^1$ is given
by
$$
x = - cot {{\theta} \over {2}}~~, {\rm or}~~,~~ e^{i \theta} = {{x-i} \over
{x+i}}~~. \eqno(15)
$$
\noindent The continuous vector field~~$u(e^{i \theta}) {{\partial} \over
{\partial \theta}}$~~becomes, on~~{\bf R},~~$F(x) {{\partial} \over {\partial
x}}$~~ with
$$
F(x) = {{1} \over {2}} (x^2 + 1) u{({{x-i} \over {x+i}})}~~. \eqno(16)
$$
\noindent Conversely,
$$
u(e^{i \theta}) = {{2 F(x)} \over {x^2 +1}}~~. \eqno(17)
$$
\noindent Since $u$ vanishes at (-1,-i,1), we see
$$
F(0) = F(1) = 0 \mbox{ and}~~{{F(x)} \over {x^2+1}} \rightarrow 0\;{\rm
as}\;x \rightarrow \infty~. \eqno(18)
$$
\noindent Introduce (following Zygmund [13]),
$$
\begin{array}{lll}
\Lambda ({\bf R}) & = & \{ F : {\bf R}
\rightarrow {\bf R}; F {\rm is\; continuous,\; satisfying\; normalizations\;
(18);} \\
& & {\rm and}, \mid F(x+t) + F(x-t) -2F(x) \mid \leq B\mid t\mid {\rm for\;
some}\; B, \\
& & {\rm for\; all}\; x\; {\rm and}\; t\; {\rm real}. \}
\end{array}
\eqno(19)
$$
\noindent $\Lambda({\bf R})$ is a (non-separable) Banach space under the
Zygmund norm -- which is the best constant $B$ for $F$. Namely,
$$
|| F || = {\sup_{x,t}} \left \vert {{F(x+t) + F(x-t) - 2F(x)} \over {t}}
\right \vert \eqno(20)
$$
\noindent In [3] it is shown that $\Lambda ({{\bf R}})$ comprises precisely the
vector fields for quasisymmetric flows on ${{\bf R}}$. Hence, the tangent
space
to version {\bf (a)} of $T(\Delta)$ becomes :
$$
T_{{\sl O}} T(\Delta)_{{(a)}} = \left\{ \begin{array}{lll} u(e^{i \theta})
{{\partial} \over {\partial \theta}} : &
(i) & u : S^1 \rightarrow {{\bf R}}~~\mbox{is continuous}~, \\
& & \mbox{vanishing at} (1,-1,-i) ; \\
& (ii) & F_{u}(x) = {{1} \over {2}} (x^2+1) u{({{x-i} \over {x+i}})}~~\mbox{is
in}~~\Lambda({{\bf R}})
\end{array} \right\}
\eqno(21)$$
\noindent {\bf Remark.} The normalization by M\" obius corresponds to adding
an arbitrary $sl(2,{\bf R})$ vector field, $(ce^{i \theta} + \bar{c}e^{-i
\theta} + b) {{\partial} \over {\partial \theta}}~~,~(c \in {{\bf C}}, b \in
{{\bf R}})$, to $u$. On the real line this is exactly adding an arbitrary real
quadratic polynomial to $F(x)$. These operations allow us to enforce the
3-point normalization in each description.
We will say a continuous function $u : S^1 \rightarrow$ {{\bf R}} is in
{\it{the Zygmund class}} $\Lambda(S^1)$ {\it{on the circle}}, if, after adding
the requisite $(ce^{i \theta} + \bar{c}e^{i \theta} + b)$ to normalize $u$, the
function satisfies (21). [Can we find a characterization of $\Lambda(S^1)$ in
terms of the decay properties of the Fourier coefficients?]
\underline{Tangent space to the complex analytic model} :
A tangent vector at {\sl{O}} (the identity mapping) to $T(\Delta)_{(b)}$
corresponds to a 1-parameter family $F_{t}$ of univalent functions (each
allowing quasiconformal extension):
$$
F_{t}(\zeta) = \zeta (1 + {{c_{2}(t)} \over {\zeta^{2}}} + {{c_{3}(t)}
\over {\zeta^{3}}} + \ldots)~,~~{\rm in}~~\mid \zeta \mid > 1~, \eqno(22)
$$
\noindent with $c_{k}(t) = t \dot{c_{k}}(0) + o(t)~, k = 2,3,4, \ldots$. The
sequences $\{ \dot{c}_{k}(0)~,~k \geq 2 \}$ arising this way correspond
uniquely to the tangent vectors.
\par Our theorem below will allow us to say precisely which sequences occur
(See Corollary 1).
\section{The Promised Relationship}
\par The principal ingredient in the stew is, of course, the infinitesimal
theory for solutions of the Beltrami equation.
\par For any $\nu \in L^{\infty}({{\bf C}})$ let $w^{t\nu}$ be the
quasiconformal homeomorphism of the plane, fixing $0,1,\infty$, and having
complex dilatation (i.e., Beltrami coefficient) $t \nu$ ; (t small complex).
Then, (see, for example, Ahlfors [2, p.104]), uniformly on compact $\zeta$-sets
we have
$$
\begin{array}{rcl}
w^{t \nu} (\zeta) & = & \zeta + t \dot{f}(\zeta) + o(t)~, (t \rightarrow o) \\
\dot{f}(\zeta) & = & - {{\zeta(\zeta -1)} \over {\pi}} \int\!\int_{\bf C} \quad
{{\nu (z)} \over {z(z-1)(z-\zeta)}} dx dy
\end{array}
\eqno(23)
$$
\par For version {\bf (b)} considerations, apply this to
$$
\nu = \left\{ \begin{array}{ll}
\mu & \mbox{on $\Delta$} \\
0 & \mbox{on $\Delta^{\star}$}
\end{array}\right.
\eqno(24)$$
\noindent We see that
$$
{{\partial} \over {\partial t}} \left\vert_{t=0} {(f^{t \mu} (\zeta))} = -
{{\zeta(\zeta -1)} \over {\pi}} \int\!\int_{\Delta} \quad {{\mu (z)} \over
{z(z-1)(z- \zeta)}} dx dy~,\;\mid \zeta \mid > 1~, \right.
\eqno(25)
$$
\noindent with the univalent functions $f^{t \mu}$ as in (11) above. Expand
$(z- \zeta)^{-1}$ in powers of $\zeta^{-1}$~, collect terms, and compare with
$$
f^{t \mu} (\zeta) = \zeta \left( a(t) + {{b_{1}(t)} \over {\zeta}} +
{{b_{2}(t)}
\over {\zeta^{2}}} + \ldots \right)~~. \eqno(26)
$$
\noindent One obtains (dot represents ${{\partial} \over {\partial t}}$) :
$$
\begin{array}{lcl}
\dot{a}(0) & = & {{1} \over {\pi}} \int\!\int_{\Delta} {{\mu (z)} \over
{z(z-1)}} dx dy \\
\dot{b}_{k}(0) & = & {{1} \over {\pi}} \int\int_{\Delta}~~{\mu (z) z^{k-2}}
dx dy~~,~~k \geq 1
\end{array}
\eqno(27)$$
\noindent The associated normalized univalent functions
$$
F^{t \mu}(\zeta) = \zeta \left( 1 + {{c_{2}(t)} \over {\zeta^{2}}} +
{{c_{3}(t)} \over {\zeta^{3}}} + \ldots \right)~~, \eqno(28)
$$
\noindent have coefficients $c_{k}(t) = b_{k}(t)/a(t)$. Consequently, we
derive easily (since $a(0) = 1~, b_{k}(0) = 0$ ) :
$$
\dot{c}_{k}(0) = \dot{b}_{k}(0) = {{1} \over {\pi}} \int\int_{\Delta}~~{\mu
(z) z^{k-2}} dx dy~~,~~k \geq 2 \eqno(29)
$$
\par Our aim is to compare these formulas with the Fourier coefficients of the
vector field $V[\mu]$ corresponding to the same $\mu$ in version {\bf (a)}.
Applying (23) to
$$
\nu = \left\{ \begin{array}{ll}
\mu & \mbox{on $\Delta$} \\
\tilde{\mu} & \mbox{(obtained by ``reflection'' of $\mu$) on $\Delta^{\star}$},
\end{array}
\right. \eqno(30)
$$
\noindent and keeping track of the normalizations, one gets (compare p.134 of
[10, Part II]) :
$$
w_{t \mu}(\zeta) = \zeta + t \dot{w} [\mu](\zeta) + o(t)~~, t
\rightarrow 0,
$$
$$
\begin{array}{lcll}
\dot{w}[\mu](\zeta) & = & - {{(\zeta -1)(\zeta +1)(\zeta +i)}
\over {\pi}} & \left\{ \int\!\int_{\Delta}\quad {{\mu(z)} \over
{(z-1)(z+1)(z+i)(z-\zeta)}} dxdy \right. \\
& & & + \left. i {\int\!\int_{\Delta}} \quad {{\overline{\mu(z)}} \over
{(\bar{z}-1)(\bar{z}+1)(\bar{z}-i)(1-\zeta \bar{z})}} dx\,dy \right\}
\end{array}
\eqno(31)
$$
\noindent Now we want to expand in Fourier series the vector field $V[\mu]$ :
$$
u(e^{i \theta}) = {{\dot{w}[\mu](e^{i \theta})} \over {ie^{i \theta}}} =
\sum_{k=-\infty}^{\infty} a_{k} e^{ik \theta} \eqno(32)
$$
\noindent Since $u$ is real valued, one knows $a_{-k} = \bar{a}_{k}, \quad k
\geq 1$. Calculating the $a_{k}$ from (31) one derives, after taking care of
some remarkable simplifications, (to which I drew attention in [10] also), that
$$
a_{-k} = - {{i} \over {\pi}} \int\!\int_{\Delta} \quad \mu(z) z^{k-2} dx \,
dy,\quad k \geq 2~~.
\eqno(33)
$$
\noindent (The remark after (21) shows that $a_{o}$ and $a_{1}$ do not matter
owing to the $sl(2,{{\bf R}})$ normalization.)
{\bf Theorem 1.}
The tangent vector to $T(\Delta)$ represented by $\mu \in L^\infty (\Delta)$,
corresponds to the Fourier expansion
$$
u(e^{i \theta}) = \sum_{k = -\infty}^{\infty} a_{k} e^{ik\theta} \quad
{\mbox{in version {\bf (a)}}}~~.
$$
\noindent The same $\mu$ corresponds to the 1-parameter family of schlicht
functions
$$
F^{t \mu} (\zeta) = \zeta \left( 1 + {{t \dot{c}_{2}(0)} \over
{\zeta^{2}}} + {{t \dot{c}_{3}(0)} \over {\zeta^{3}}} + \ldots \right) + o(t)
$$
\noindent in version {\bf (b)}. The identities
$$
{\fbox{ $\dot{c}_{k}(0) = i a_{-k} = i \bar{a}_{k}$, \quad {\mbox{for every}}
\quad {$k \geq 2$}}}
\eqno(34)
$$
\noindent hold.
{\bf Proof :} Compare (29) with (33).
We now have the promised precise description of the complex-analytic tangent
space.
\noindent{\bf Corollary 1} As we saw at the end of Section 2, a tangent vector
to $T(\Delta)_{(b)}$ is determined by a complex sequence $\left(
\dot{c}_{2}(0), \dot{c}_{3}(0), \ldots \right)$. Precisely those sequences
$(\gamma_{2}, \gamma_{3}, \ldots)$ occur for which the function
$$
u(e^{i \theta}) = i \sum_{k=2}^{\infty} \bar{\gamma}_{k} e^{ik\theta} -i
\sum_{k=2}^{\infty} \gamma_{k} e^{-ik\theta} \eqno(35)
$$
\noindent is in the Zygmund class on $S^1$.
{\bf{Harmonic (Bers') Beltrami Coefficients }}:
Introduce the Banach space of Nehari-bounded holomorphic ``quadratic
differentials''
$$
B(\Delta) = \left\{ \phi \in {Hol}(\Delta) : || \phi (z) (1- \mid z
\mid^2)^2 ||_\infty < \infty \right\} \eqno(36)
$$
\noindent To every $\phi \in B(\Delta)$ we associate the $L^\infty$ function
$$
\nu_{\phi} = \overline{\phi(z)} (1-\mid z \mid^2)^2~~{\rm on}~~\Delta.
\eqno(37)
$$
\noindent Foundational results of Teichm\" uller theory guarantee that the
Beltrami coefficients $\{\nu_{\phi} : \phi \in B(\Delta) \}$ form a
complementary subspace to the kernel $N$ (see equation (5)) of the map
$d_{0}\Phi$. Thus, this space of {\sl{harmonic Beltrami coefficients}} :
$$
H = \{ \nu \in L^{\infty} (\Delta) : \nu \mbox{ is of the form (37) for some}~~
\phi \in B(\Delta) \} \eqno(38)
$$
\noindent is isomorphic to the tangent space $T_{\sl{O}} T(\Delta)$. In fact,
this remains true for the Teichm\" uller space $T(G)$ of any Fuchsian group $G$
acting on $\Delta$, simply by replacing $B(\Delta)$ by the subspace
$B(\Delta,G)$ comprising those functions which are quadratic differentials for
$G$. See Ahlfors [2, Chapter 6] and Nag[7, Chapter 3] for all this.
Therefore the tangent vector associated to an arbitrary $\mu$ is also
represented by a unique Beltrami form of harmonic type (37). For harmonic
$\mu$ we get a beautifully simple formula for the Fourier coefficients, and
hence using (34) also for the power series coefficients, representing that
tangent vector $d_{0} \Phi(\mu)$.
{\bf Proposition 1.} Let $\mu = \overline{\phi(z)} (1-\mid z \mid^2)^2$ on
$\Delta$ with $\phi \in B(\Delta)$~,
$$
\phi(z) = h_o + h_1z + h_2z^2 + \ldots, in \mid z \mid < 1~~. \eqno(39)
$$
\noindent The relevant Fourier coefficients $a_{k}$ of the corresponding
Zygmund class vector field $V[\mu]$ on $S^1$ are
$$
\bar{a}_{-k} = a_k = {{2i} \over {(k^3-k)}} h_{k-2}~~, {\rm for}~~k \geq 2~~.
\eqno(40)
$$
{\bf Proof :} Compute using formula (33) above.
{\bf Remark 1.} In the presence of a Fuchsian group $G,~\phi$ is a (2,0)-form
for $G$ and the vector field $V[\mu]$ is also $G$-invariant. That imposes
conditions on the coefficients $h_k$ and $a_k$ respectively, which interact
closely with the relationship (40) exhibited above.
{\bf Remark 2.} The Proposition above can be utilised to analyse why Bers
coordinates are geodesic for the Weil-Petersson metric (Section 6). (Vide
Ahlfors[1] and later work of Royden and Wolpert.)
{\bf{Explicit family of examples}} :
Here is a computable family of examples for which $w^{t \mu}$ can be explicitly
determined, and hence our result can be checked. These examples are a modified
form of some that were suggested to me by Clifford Earle.
Look at $\mu \in L^\infty (\Delta)$ given by
$$
\mu(z) = -n z^2 \bar{z}^{n-1} \eqno(41)
$$
\noindent Here $n \geq 3$ ; ($n=2$ works also, with minor changes).
The vector field $V[\mu]$ on $S^1$ has Fourier coefficients (from
(33)) as exhibited :
$$
a_{k} = \left \{ \begin{array}{ll}
$-i$ & \mbox{for $k=n-1$} \\
$i$ & \mbox{for $k=1-n$} \\
0 & \mbox{for any other $k \geq 2$ or $\leq -2$.}
\end{array} \right.
\eqno(42)
$$
\noindent The interesting thing is that we can explicitly describe $w^{t\mu}$
for these $\mu$, for all complex $t$ satisfying $\mid t \mid < {1\over n}~$.
Indeed, $w^{t\mu}(\zeta) = f^{t\mu}(\zeta)/(1+t)$, where :
$$
f^{t \mu}(\zeta) = \left\{ \begin{array}{ll}
\zeta \left( 1 + {{t} \over {\zeta^{n-1}}} \right)^{-1} & \mbox{on $\mid
\zeta \mid \geq 1$} \\
\left( {{1} \over {\zeta}} + t \bar{\zeta}^{n} \right)^{-1} & \mbox{on $\mid
\zeta \mid \leq 1$}
\end{array} \right.
\eqno(43)
$$
\noindent It is not hard to check that $f^{t \mu}$ is quasiconformal on {{\bf
C}}, and that its complex dilatation on $\Delta$ is $t \mu$~. The $\{ \mid
\zeta \mid \geq 1 \}$ portion in (43) represents the 1-parameter family of
schlicht functions, and we can write down immediately :
$$
\dot{c}_{k}(0) = \left\{ \begin{array}{ll}
$-1$ & \mbox{for $k=n-1$} \\
0 & \mbox{for any other $k \geq 2$.}
\end{array} \right.
\eqno(44)
$$
\noindent This corroborates Theorem 1.
{\bf Remark.} In constructing these examples, it is, of course, the
quasiconformal homeomorphisms (43) that were written down first; (41) and
(42) were derived from it.
\section{The Almost Complex Structure}
Using Theorem 1 we get an immediate proof of the fascinating fact that the
almost complex structure of $T(\Delta)$ transmutes to the operation of Hilbert
transform on Zygmund class vector fields on $S^1$. {\sl{Namely, we want to
prove that the vector field}} $V[\mu]$ (equation (13)) {\sl{is related to}}
$V[i \mu]$ {\sl{as a pair of conjugate Fourier series}}.
But the tangent vector represented by $\mu$ in the complex-analytic description
$T(\Delta)_{(b)}$ corresponds to a sequence $(\dot{c}_{2}(0), \dot{c}_{3}(0),
\ldots )$, as explained. Since the $c_{k}$ are holomorphic in $\mu$, the
tangent vector represented by $i \mu$ corresponds to $(i \dot{c}_{2}(0),
i \dot{c}_{3}(0), \ldots)$. The relation (34) of Theorem 1 immediately shows
that the $k^{th}$ Fourier coefficient of $V[i \mu]$ is $-i.sgn(k)$ times the
$k^{th}$ Fourier coefficient of $V[\mu]$. We are through.
{\bf Remark 1.} The Hilbert transform description of the complex structure
on the tangent space of the Teichm\" uller space was first pointed out by
S.Kerckhoff. A proof of this was important for our previous work, and appeared
in [10, Part I].
{\bf Remark 2.} The result above gives an independent proof of the fact that
conjugation of Fourier series preserves the Zygmund class $\Lambda(S^1)$. That
was an old theorem of Zygmund [13].
\section{Conformal Welding and its derivative}
The Teichm\" uller point $[\mu]$ in $T(\Delta)_{(b)}$ is the univalent function
$F^{\mu}$ [or, equivalently its image quasidisc $F^{\mu}(\Delta^{\star})$].
The same $[\mu]$ appears in $T(\Delta)_{(a)}$ as the quasisymmetric
homeomorphism $w_{\mu}$ on $S^{1}$. The relating map is the ``Conformal
Welding"
$$
{\bf W} : T(\Delta)_{(b)} \longrightarrow T(\Delta)_{(a)}~~.
$$
\noindent Namely, given a simply connected Jordan region $D$ on
$\widehat{{\bf{C}}}$ one looks at any Riemann mapping $\rho$ of $\Delta$ onto
$D^{\star}$(=exterior of $D$) and also a Riemann mapping $\sigma$ of
$\Delta^{\star}$ onto $D$. Both $\rho$ and $\sigma$ extend continuously to the
boundaries to provide two homeomorphisms of $S^{1}$ onto the Jordan curve
$\partial D$. We define the welding homeomorphism :
$$
{\bf W} (D) = \rho^{-1} o \sigma : S^1 \rightarrow S^1~~. \eqno(45)
$$
\noindent We can normalize by a M\" obius transformation so that {\bf W}$(D)$
fixes $1,-1,-i$.
Clearly, since $\rho = w^{\mu} o w^{-1}_{\mu}$ on $\Delta$, and $\sigma =
w^\mu$ on $\Delta^{\star}$, work as Riemann maps, we see that $T(\Delta)_{(b)}$
and $T(\Delta)_{(a)}$ are indeed related by this fundamental operation of
conformal welding.
{\bf Theorem 2.} The derivative at the origin of $T(\Delta)$ to the map {\bf W}
is the linear isomorphism :
$$
d_{\sl{O}}{\bf W} : \{ \left( \dot{c}_{2}(0), \dot{c}_{3}(0), \ldots \right)
\} \rightarrow {\rm Zygmund\; class}\; \Lambda(S^1) \eqno(46)
$$
\noindent sending $(\gamma_2, \gamma_3, \ldots)$ to the vector field $u(e^{i
\theta}) {{\partial} \over {\partial \theta}}$, where
$$
u(e^{i \theta}) = i \sum_{k=2}^{\infty} \bar{\gamma}_{k} e^{ik\theta} -i
\sum_{k=2}^{\infty} {\gamma}_{k} e^{-ik\theta}
\eqno(47)
$$
{\bf Proof :} Follows from Theorem 1 and the remarks above.
{\bf Remark.} Conformal welding has been studied by many authors even for
domains more general than quasidiscs. See, for example,
Katznelson-Nag-Sullivan[5]. The derivative formula should now be extended to
the larger context.
\section{Diff($S^1$)/M\" ob($S^1$) inside $T(\Delta)$}
As usual, let Diff$(S^1)$ denote the infinite dimensional Lie group of
orientation preserving $C^\infty$ diffeomorphisms of $S^1$. The complex-
analytic homogeneous space (see Witten [12])
$$
M = {\rm Diff} (S^1) / {\rm Mob}(S^1)
\eqno(48)
$$
\noindent injects holomorphically into $T(\Delta)_{(a)}$. This was proved in
[10, Part I]. The submanifold $M$ comprises the ``smooth points" of
$T(\Delta)$ ; in fact, in version (b), the points from $M$ are those quasidiscs
$F^{\mu}(\Delta^{\star})$ whose boundary curves are $C^\infty$.
$M$, together with its modular group translates, foliates $T(\Delta)$ -- and
the fundamental Kirillov-Kostant K\" ahler (sympletic) form (Witten[12]) exists
on each leaf of the foliation. Up to an overall scaling this homogeneous K\"
ahler metric gives the following pairing $g$ on the tangent space at {\sl{O}}
to $M$ :
$$
g(V,W) = Re \left[ \sum_{k=2}^{\infty} a_{k} \bar{b}_{k} (k^3-k) \right]
\eqno(49)
$$
$$
\begin{array}{lclcl}
V & = & \sum_{2}^{\infty} a_{k} e^{ik \theta} & + & \sum_{2}^{\infty}
\bar{a}_{k} e^{-ik \theta}, \\
W & = & \sum_{2}^{\infty} b_{k} e^{ik \theta} & + & \sum_{2}^{\infty}
\bar{b}_{k} e^{-ik \theta},
\end{array}
$$
\noindent represent two smooth vector fields on $S^1$.
The metric $g$ on $M$ was proved by this author [10, Part II] to be {\sl{the
(regularized) Weil-Petersson metric (WP) of universal Teichm\" uller space}}.
Theorem 1 allows us to express the pairing for $g = WP$ in terms of 1-parameter
flows of schlicht functions.
{\bf Theorem 3.} Let $F_{t}(\zeta)$ and $G_{t}(\zeta)$ denote two curves
through origin in $T(\Delta)_{(b)}$ of the form (22), representing two tangent
vectors, say $\dot{F}$ and $\dot{G}$. Then the Weil-Petersson pairing assigns
the inner product
$$
WP(\dot{F},\dot{G}) = -Re \left[\sum_{k=2}^{\infty} \overline{\dot{c}_{k}(0)}
\dot{d}_k(0) (k^3-k) \right], \eqno(50)
$$
\noindent where $c_k(t)$ and $d_k(t)$ are the power series coefficients for the
schlicht functions $F_t$ and $G_t$ respectively, in equation (22). The series
above converges absolutely whenever the corresponding Zygmund class functions
(see equations (35) or (47)) are $C^{3/2 + \epsilon}$.
{\bf Proof:} Combine (34) with (49) and the results of [10, Part II].
\section{Variational Formula for the Period Mapping}
Recently in [8],[9] the author has studied a generalisation of the classical
period mappings to the infinite dimensional context of universal Teichm\" uller
space. Indeed, there is a natural equivariant, holomorphic and K\" ahler-
isometric immersion
$$
\prod : M \longrightarrow D_{\infty} \eqno(51)
$$
\noindent of M = Diff($S^1$)/M\" ob($S^1$) into the infinite dimensional
version, $D_{\infty}$, of the Siegel disc. $D_{\infty}$ is a complex manifold
comprising certain complex symmetric Hilbert-Schmidt ${(Z_{+}\times Z_{+})}$
matrices. $\prod$ qualifies as a generalised period matrix map, and its
variation satisfies a Rauch-type formula. See [9]. An extension of $\prod$ to
{\sl{all}} of $T(\Delta)$ is being worked on by this author using ideas
communicated by Dennis Sullivan.
In the work cited above we proved that for any $\mu$ in $L^\infty(\Delta)$, the
$(rs)^{th}$-entry of the period matrix satisfies $(r,s \geq 1)$
$$
\prod ([t \mu])_{rs} = t \sqrt{-rs}\> a_{-(r+s)} + O(t^2) \eqno(52)
$$
\noindent as $t \rightarrow 0$. Here $a_k$ as usual denotes the Fourier
coefficients of the vector field represented by $\mu$ (equation (32)).
By Theorem 1 we see that the variational formula above may be written
$$
\prod ([t \mu])_{rs} = \sqrt{rs}\> c_{r+s}(t) + O(t^2) \eqno(53)
$$
\noindent where $c_k(t)$ are the power series coefficients appearing in (22)
for the schlicht functions $F^{t \mu}$. Equation (53) may be compared with
the formula [(30) in their paper] claimed by Hong-Rajeev[4] in just this
setting.
\center${\star~~~~~\star~~~~~\star}$
\baselineskip = 24pt
\begin{center}
{\bf Appendix}\footnote{Appendix to the paper ``On the Tangent
Space to the Universal Teichm\" uller Space'' by SUBHASHIS NAG.} \\
{\bf{A FUGUE ON AHLFORS [1], AHLFORS-WEILL AND OUR RESULTS}}
\end{center}
The intriguing formulas that Ahlfors exhibited (more than thirty years
ago!) in his paper [1], do have a surprising way of cropping up in
various contexts in later work of many authors. Upon reading the present
article, Prof.Clifford Earle has pointed out to me that the formulas
(1.20) and (1.21) in Section 1.6 of Ahlfors [1] could be used to derive an
elegant proof of our main theorem (equation (34)) above.
The crucial thing is to verify our formulas (29) and (33) (in Section 3
above). As explained with equations (37) - (38), every Beltrami form
$\mu$ is uniquely the sum of an infinitesimally trivial one (equation
(5)) and a harmonic form (equation (5)) and a harmonic form (equation
(37)). Thus the formulas need to be checked only for these two types of
forms. Of course, all the relevant quantities are zero for
infinitesimally trivial forms. {\it The gist of the matter is that for
harmonic Beltrami forms the Ahlfors-Weill section implies formula}
(29), {\it whereas} (1.21) {\it of Ahlfors} [1] {\it implies formula}
(33)!
{\bf Ahlfors-Weill and formula (29):} \hspace{.5cm} Let
$$
\mu = \overline{\phi(z)} (1-|z|^{2})^{2} \eqno(54)
$$
\noindent be a harmonic Beltrami form on $\Delta$ with $\phi$ as in
(39). The Ahlfors-Weill theorem tells us explicitly the schlicht
function $w^{t\mu}$ on $\Delta^{\star}$ (for small t) and hence allows
us to compute the variation of the power series coefficients,
$\dot{c}_{k}(0)$. We refer to Section 3.8 of Nag[7] -- especially
3.8.6 -- for that result.
In fact, let $v_{1}$ and $v_{2}$ be linearly independent solutions in
the unit disc of the differential equation:
$$
v'' = \phi v \eqno(55)
$$
\noindent normalized so that $v_{1}(0) = v'_{2}(0) = 1$ and $v_{2}(0) =
v'_{1}(0) = 0$. Then Ahlfors-Weill tells us that (up to possibly a M\"
obius transformation)
$$
w^{t\mu} (\zeta) = \bar{v}_{1} (1/\bar{\zeta}) / \bar{v}_{2}
(1/\bar{\zeta}) \ {\rm for} \ |\zeta| > 1\,. \eqno(56)
$$
Solving (55) for $v_{1}$ and $v_{2}$ by the method of indeterminate
coefficients yields:
$$
v_{1}(z) = 1 + t \sum_{k=2}^{\infty} {{h_{k-2}} \over {k(k-1)}} z^{k}
\quad + o(t) \eqno(57)
$$
$$
v_{2}(z) = z + t \sum_{k=2}^{\infty} {{h_{k-2}} \over {k(k+1)}} z^{k+1}
+ o(t) \eqno(58)
$$
\noindent Substituting these in (56) we deduce quickly that
$$
\dot{c}_{k}(0) = {{2} \over {(k^{3}-k)}} \overline{h_{k-2}}\,, {\rm for}\;
k \geq 2\,. \eqno(59)
$$
\noindent For the harmonic form $\mu$, formula (59) is exactly formula
(29).
\noindent{\bf Remark:} In 3.8.5 of Nag[7] a new proof of the
Ahlfors-Weill theorem was given using an idea of Royden. The
calculations made there are closely relevant to proving (29) {\it
directly} for harmonic $\mu$ -- without passing to series expansions.
{\bf Ahlfors[1] and formula (33):} Formula (1.21) in Ahlfors [1] in our
notation reads:
$$
\dot{w}[\mu] = \bar{\Phi}'' (1-|z|^{2})^{2} + 2 \bar{\Phi}' z(1-|z|^{2})
+ 2 \bar{\Phi}z^{2} -2 \Phi \eqno(60)
$$
\noindent valid for $|z| \leq 1$, where $\mu$ is the harmonic Beltrami
form (54). Here $\Phi$ (holomorphic in $\Delta$) is related to $\phi$
(of (54)) by
$$
\Phi''' = \phi \eqno(61)
$$
\noindent See Ahlfors[1] formula (1.20) for this. In order to calculate
the Fourier coefficients $a_{k}$, defined as in (32) above, we only
require (60) on the circle $|z| = 1$. The first two terms of (60)
therefore drop off, and a straightforward computation produces:
$$
a_{k} = {{2i} \over {(k^{3}-k)}} h_{k-2}\,, {\rm for}\; k \geq 2.
\eqno(62)
$$
\noindent But this is exactly formula (40), which is formula (33) for
harmonic Beltrami forms.
Consequently, comparing (59) with (62) proves the complete result.
\newpage
|
1992-05-19T17:15:07 | 9205 | alg-geom/9205006 | en | https://arxiv.org/abs/alg-geom/9205006 | [
"alg-geom",
"math.AG"
] | alg-geom/9205006 | null | Anna Maria Bigatti | Upper Bounds for the Betti Numbers of a given Hilbert Function | 18 pages, plain tex | null | null | null | null | From a Macaulay's paper it follows that a lex-segment ideal has the greatest
number of generators (the 0-th Betti number $\b_0$) among all the homogeneous
ideals with the same Hilbert function. In this paper we prove that this fact
extends to every Betti number, in the sense that all the Betti numbers of a
minimal free resolution of a lex segment ideal are bigger than or equal to the
ones of any homogeneous ideal with the same Hilbert function.
| [
{
"version": "v1",
"created": "Tue, 19 May 1992 16:14:00 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bigatti",
"Anna Maria",
""
]
] | alg-geom | 4{0}
\ \vskip1true in
\centerline{UPPER BOUNDS FOR THE BETTI NUMBERS}
\centerline{OF A GIVEN HILBERT FUNCTION}
\hfill\break\acapo
\centerline{\bf Anna Maria Bigatti}
\hfill\break
\centerline{Dipartimento di Matematica dell'Universit\`a di Genova}
\centerline{Via L.B. Alberti 4, I-16132 Genova Italy}
\centerline{E-mail [email protected]}
\hfill\break\acapo
Let $R:=k[X_1,\dots,X_N]$ be the polynomial ring in $N$ indeterminates over a
field $k$ of characteristic 0 with $\deg(X_i)=1$ for $i=1,\dots,N$, and let
$I$ be a homogeneous ideal of $R$. The Hilbert function of $I$ is the function
from $\bf N$ to $\bf N$ which associates to every natural number $d$ the dimension
of $I_d$ as a $k$-vectorspace.
$I$ has an essentially unique minimal graded free resolution
$$
0 \longrightarrow L_m
{\buildrel d_m\over\longrightarrow} L_{m-1}
{\buildrel d_{m-1}\over\longrightarrow} \dots
{\buildrel d_2\over\longrightarrow} L_1
{\buildrel d_1\over\longrightarrow} L_0
{\buildrel d_0\over\longrightarrow} I
\longrightarrow 0
$$
which is characterized, among the free graded
resolutions, by the condition
$$
d_q(L_q) \subseteq (X_1,\dots,X_N)L_{q-1}\hbox{\ \ \ } \ \forall~
q\ge 1
$$
And therefore the Betti numbers, which are defined by
$$\beta_q(I):={\rm rank}L_q$$
are invariants of $I$.
{}From Macaulay [M] (see also Robbiano [R]) it follows that a lex-segment ideal
has the greatest number of generators (the 0-th Betti number $\beta_0$) among all
the homogeneous ideals with the same Hilbert function.
In this paper we prove that this fact extends to every Betti number, in the
sense that
all the Betti numbers of a lex segment ideal are bigger than or equal to
the ones of any homogeneous ideal with the same Hilbert function.
Section 1 gives some useful notation and definitions and many simple
properties of Borel normed sets.
In Section 2 a Theorem is derived (Theorem 2.1) which is our main
tool in comparing lex-segment and Borel normed sets.
In Section 3, using a result due to Eliahou and Kervaire [E-K], we compare
lex-segment and Borel normed ideals, and then, using some results
due to Galligo [Ga] and M\"oller-Mora [M-M], we compare
lex-segment and homogeneous ideals.
Section 4 gives the formula which computes the Betti numbers of the lex-segment
ideal, given its Hibert function, and these are the sharp upper bounds for the
Betti numbers of any homogeneous ideal with the same Hilbert function.
\hfill\break
{\bf 1.Some remarks on Borel normed sets.}
\def4{1}
\bigskip
\ProclaimPar
{\bf Notations.} Let
${\bf X}_N$ denote the set of indeterminates $\{X_1,\dots,X_N\}$; then
$({\bf X}_N)^D$ indicates the set of all monomials of degree $D$ in ${\bf X}_N$.
\hfill\break
Let ${\bf S}$ be a
subset of $({\bf X}_N)^D$; then ${\bf X}_N{\bf S}$ denotes the multiples of
${\bf S}$ of degree $D+1$, i.e. ${\bf X}_N{\bf S}=\cup_{T\in {\bf S}} \{X_1T,\dots,X_NT \}$.
\hfill\break
If $T=X_1^{t_1}\dots X_N^{t_N}$, then we denote by
$m(T)$ := $\max\{i \;|\; t_i>0\}$, i.e. the largest index of the
indeterminates actually occuring in $T$.
{\bf\ProclaimPar Definition.}
A set of monomials ${\bf S}\subseteq({\bf X}_N)^D$ is {\bf Borel normed} if
$
T\in {\bf S}$ implies $X_i{T\over X_j}\in {\bf S}$ for all $j$ such that $X_j$
divides $T$ and for all $i<j$.
{\bf\ProclaimPar Definition.}
On $({\bf X}_N)^D$ we will use the {\bf lexicographic order}, i.e.
if $T=X_1^{t_1}\dots X_N^{t_N}$ and $T'=X_1^{s_1}\dots X_N^{s_N}$ are two
monomials
in $({\bf X}_N)^D$ we will say that $T>T'$ if $t_1=s_1,\dots,t_{i-1}=s_{i-1}$ and
$t_i>s_i$.
\hfill\break
Note that it is a total ordering and then there exists the minimum of every
subset of $({\bf X}_N)^D$.
\Lemma 1
Let ${\bf S}$ be a Borel normed set.
\hfill\break
Then $X_i(\min {\bf S}) \in{\bf X}_N({\bf S}\backslash \{\min {\bf S} \})$ \hbox{ $\Longleftrightarrow$ } $i<m(\min {\bf S})$.
{\it\ProclaimPar Proof.}
Let $T:=\min {\bf S}$.
\CasePar
{}
\item {`\hbox{ $\Longrightarrow$ }'}:
If $X_iT\in{\bf X}_N({\bf S}\backslash \{T\})$ then
$X_iT=X_jT'$ where $T'\in {\bf S}\backslash \{T\}$
\hfill\break
Thus $T'>T$, hence $i<j$.
\hfill\break
On the other hand $X_j$ divides $T$
and so $j\le m(T)$.
\hfill\break
Then $i<m(T)$.
\item {`\hbox{ $\Longleftarrow$ }'}:
If $i<m(T)$ then,
since ${\bf S}$ is Borel normed, $T':=X_i{T\over X_{m(T)}}\in {\bf S} \backslash \{T \}$.
Hence $X_iT=X_{m(T)}T'\in {\bf X}_N({\bf S}\backslash \{T\})$.\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\Prop 2
Let ${\bf S}$ be a Borel normed set.
\hfill\break
Then
${\bf X}_N {\bf S}=\cup _{T\in {\bf S}}\{X_{m(T)}T,\dots, X_NT\}$ and this is a disjoint union
(i.e. $\{X_{m(T)}T,\dots, X_NT\} \cap
\{X_{m(T')}T',\dots, X_NT'\} = $\O$ $ $\ \forall~ T'\ne T$).
{\it\ProclaimPar Proof.}
By induction on the cardinality of the set:
\CasePar {}
\item {$|{\bf S}|=1$:}
Since ${\bf S}$ is Borel normed it follows that ${\bf S}= \{ X_1^D \} $.
Then ${\bf X}_N {\bf S}=\{X_1X_1^D,\dots,X_NX_1^D\}$.
\item {$|{\bf S}|>1$:}
Let the thesis be true if the cardinality is smaller then $|{\bf S}|$.
\hfill\break
Let $T':=\min {\bf S}$.
Then
${\bf X}_N {\bf S}=\{X_1T',\dots,X_NT'\} \cup {\bf X}_N({\bf S}\backslash\{T'\})$.
But from Lemma 1.1
we have $X_iT'\in {\bf X}_N({\bf S}\backslash\{T'\})$ $\ \forall~ i<m(T')$.
Hence
${\bf X}_N {\bf S}=\{X_{m(T')}T',\dots,X_NT' \} \cup {\bf X}_N({\bf S}\backslash\{T'\})$.
\hfill\break
Note that ${\bf S} \backslash \{ T' \} $ is a Borel normed set and
then, by the inductive hypothesis:
${\bf X}_N ( {\bf S} \backslash \{ T' \} ) =
\cup _{T\in {\bf S}\backslash \{ T' \} }\{X_{m(T)}T,\dots, X_NT\}$.
\hfill\break
Therefore
${\bf X}_N {\bf S}=\cup _{ T\in {\bf S} } \{ X_{m(T)}T,\dots, X_NT \}$.
\hfill\break
Moreover from Lemma 1.1 we have that if $i\ge m(T')$ then $X_iT'\not\in
{\bf X}_N({\bf S}\backslash\{T'\})$
and so
$\{X_{m(T')}T',\dots, X_NT'\}\cap{\bf X}_N({\bf S}\backslash\{T'\})= $\O$ $.
\hfill\break
So, in particular,
$\{X_{m(T')}T',\dots,X_NT'\}\cap\{X_{m(T)}T,\dots,X_NT\}= $\O$ $ $\ \forall~
T>T'$ .
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Definition.} Define
$$ m_i({\bf S}) := \big| \ \{ T\in{\bf S} \;|\; m(T)=i \} \ \big|$$
i.e. the number of the elements of ${\bf S}$ which ``finish'' with $X_i$,
and similary
$$ m_{\le i}({\bf S}) := \big| \ \{ T\in{\bf S} \;|\; m(T)\le i \} \ \big|$$
i.e. the number of the elements of ${\bf S}$ in the first $i$ indeterminates.
\Prop 3 Let ${\bf S}$ be a Borel normed set, then
\CasePar
{}
\item {i)} $m_i({\bf X}_N{\bf S})=m_{\le i}({\bf S})$.
\item {ii)} $|{\bf X}_N{\bf S}|=\sum_{i=1}^N m_{\le i}({\bf S})$.
{\it\ProclaimPar Proof.}
\CasePar
{}
\item {i)} From Proposition 1.2 we have that
$\cup _{T\in {\bf S}} \{ X_{m(T)}T,\dots, X_N T\}$
is a disjoint union.
Then,
in such a representation of ${\bf X}_N {\bf S}$
every monomial $T$ with $m(T)=i$
can be uniquely expressed as
$X_iT'$ where $T'\in{\bf S}$ and
$i\ge m(T')$.
Therefore there exists a 1-1 correspondence between the monomials $T$ in
${\bf X}_N{\bf S}$
with $m(T)=i$ and the monomials $T'$ in ${\bf S}$ with $m(T')\le i$.
Hence
$m_i({\bf X}_N{\bf S})=m_{\le i}({\bf S})$.
\item {ii)} $|{\bf X}_N{\bf S}|=\sum_{i=1}^N m_i({\bf X}_N{\bf S})=\sum_{i=1}^N m_{\le i}({\bf S})$.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Definition.} A set of monomials ${\bf S}\subseteq({\bf X}_N)^D$ is {\bf lex-segment} if
$$
T\in {\bf S} \hbox{ and } \ T'>T \Longrightarrow T' \in {\bf S}
$$
or equivalently ${\bf S}$ is a lex-segment set if and only if ${\bf S}=\{ T \;|\;
T\ge\min
{\bf S} \}$.
{\bf\ProclaimPar Remark.} ${\bf S}$ is a lex-segment set \hbox{ $\Longrightarrow$ } ${\bf S}$ is a Borel normed set.
{\bf\ProclaimPar Definition.} We can uniquely decompose ${\bf S}$, with respect
to $X_N$, as follows
$$
{\bf S}= \ {\bf S}_0 \ \cup \ X_N{\bf S}_1 \ \cup \ X_N^2{\bf S}_2 \ \cup \dots \cup \ X_N^D{\bf S}_D
\ $$
where the ${\bf S}_d$'s are sets of monomials in $N-1$ indeterminates.
\hfill\break
More precisely $${\bf S}_d\subseteq ({\bf X}_{N-1})^{D-d}$$
{\bf\ProclaimPar Remark.} It is easy to see that
$\ \forall~ i<N \hbox{\ \ \ } m_{\le i}({\bf S})=m_{\le i}({\bf S}_0)$.
\hfill\break
In particular, $m_{\le N-1}({\bf S})=|{\bf S}_0|$.
\Prop 4
\CasePar{}
\item {i)}
Let ${\bf S}$ be a Borel normed (lex-segment) set in $N$ indeterminates. For every
$d$ \ ${\bf S}_d$ is a Borel normed (lex-segment) set in $N-1$ indeterminates.
\item {ii)}
If ${\bf S}$ is a Borel normed (lex-segment) set in $({\bf X}_N)^D$ then
${\bf X}_N {\bf S}$ is a Borel normed (lex-segment) set in $({\bf X}_N)^{D+1}$.
{\it\ProclaimPar Proof.} Easy exercise.
\Lemma 5 ${\bf S}$ is a Borel normed set \hbox{ $\Longleftrightarrow$ }
${\bf S}_d$ is a Borel normed set $\ \forall~ d$, \ and \
${\bf X}_{N-1} {\bf S}_d\subseteq {\bf S}_{d-1}$ $\ \forall~ d>0$.
{\it\ProclaimPar Proof.}
\CasePar{}
\item{`\hbox{ $\Longrightarrow$ }':}
${\bf S}_d$ is Borel normed set follows from Proposition 1.4.i.
\hfill\break
Then it remains to prove that ${\bf X}_{N-1} {\bf S}_d\subseteq {\bf S}_{d-1}$ $\ \forall~ d>0$.
\hfill\break
Let $T\in {\bf S}_d$ i.e. $X_N^dT\in {\bf S}$.
Since ${\bf S}$ is Borel normed we have:
$ X_N^{d-1} X_i T = X_i{X_N^d T\over X_N}\in {\bf S}$ $\ \forall~ i<N$.
\hfill\break
Hence
$X_i T \in {\bf S}_{d-1}$ $\ \forall~ i<N$.
Thus ${\bf X}_{N-1} {\bf S}_d\subseteq {\bf S}_{d-1}$.
\item{`\hbox{ $\Longleftarrow$ }':}
We need to prove:
$T\in{\bf S}$ \hbox{ $\Longrightarrow$ } $X_i{T\over X_j}\in {\bf S}$ $\ \forall~ i<j$ and $X_j\big| T$:
\hfill\break
Let $T = X_N^d T'$ where $T'\in {\bf S}_d$. Then:
\hfill\break
Since ${\bf S}_d$ is a Borel normed set
it follows that
\hfill\break
$X_i{ T \over X_j} = X_N^d \Big( X_i{ T' \over X_j}\Big) \in {\bf S}$
\ $\ \forall~ i<j<N$, $X_j\big| T$.
\hfill\break
Since ${\bf X}_{N-1} {\bf S}_d\subseteq {\bf S}_{d-1}$ $\ \forall~ d>0$ (and then $X_N\big| T$)
it follows that
\hfill\break
$X_i{ T \over X_N} = X_N^{d-1}( X_i T')\ \in {\bf S}$ \ $\ \forall~ i<N$.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Definition.} Given a set ${\bf S}\subseteq ({\bf X}_N)^D$ we can uniquely define a
corresponding {\bf lex-segment with respect to $X_N$} (denoted
${\bf S}{^*}$) as follows:
\hfill\break
Recall that ${\bf S}_d\subseteq({\bf X}_{N-1})^{D-d}$ then denote by ${\bf S}{^*}_d$
the lex-segment set in $({\bf X}_{N-1})^{D-d}$ with $|{\bf S}{^*}_d|=|{\bf S}_d|$,
i.e. the set of the greatest $|{\bf S}_d|$ monomials in $({\bf X}_{N-1})^{D-d}$.
\hfill\break
Then ${\bf S}{^*}$ :=
$ \ X_N^0 {\bf S}{^*}_0 \ \cup \dots \cup \ X_N^D {\bf S}{^*}_D $.
{\bf\ProclaimPar Remark.} $m_{\le N-1}({\bf S})=|{\bf S}_0|=|{\bf S}{^*}_0|=m_{\le N-1}({\bf S}{^*})$.
\Lemma 6
Let ${\bf S}$ be a Borel normed set.
If $m_{\le i}({\bf S}{^*}_d)\le m_{\le i}({\bf S}_d)$ $\ \forall~ i\le N-1$ and $\ \forall~
d$,
then ${\bf S}{^*}$ is a Borel normed set.
{\it\ProclaimPar Proof.}
By Lemma 1.5 it sufficies to show that
${\bf S}{^*} _d$ is a Borel normed set $\ \forall~ d$
\ and \
${\bf X}_{N-1}{\bf S}{^*} _d\subseteq {\bf S}{^*} _{d-1}$ $\ \forall~ d>0$.
\hfill\break
The fact that ${\bf S}{^*}_d$ is a Borel normed set is obvious since
${\bf S}{^*}_d$
is a lex-segment set.
\hfill\break
It remains to prove ${\bf X}_{N-1}{\bf S}{^*} _d\subseteq {\bf S}{^*} _{d-1}$:
\hfill\break
{}From Proposition 1.3.ii it follows that
for every Borel normed set ${\bf S}$ \
$|{\bf X}_N{\bf S}|=\sum_{i=1}^N m_{\le i}({\bf S})$.
But, by hypothesis we have
\hfill\break
$|{\bf X}_{N-1}{\bf S}{^*} _d| =
\sum_{i=1}^{N-1} m_{\le i}({\bf S}{^*}_d)\le
\sum_{i=1}^{N-1} m_{\le i}({\bf S} _d) =
|{\bf X}_{N-1}{\bf S}_d|$ $\ \forall~ d$.
Since ${\bf S}$ is Borel normed,
it follows from Lemma 1.5 ${\bf X}_{N-1}{\bf S}_d \subseteq {\bf S}_{d-1}$ $\ \forall~ d>0$.
Hence
$$|{\bf X}_{N-1}{\bf S}{^*}_d|\le|{\bf X}_{N-1}{\bf S}_d|\le|{\bf S}_{d-1}|=|{\bf S}{^*}_{d-1}|
\ \ \forall~ d$$
Thus, since ${\bf X}_{N-1}{\bf S}{^*}_d$ and ${\bf S}{^*}_{d-1}$ are lex-segments sets
(Proposition 1.4), we have
$${\bf X}_{N-1}{\bf S}{^*}_d\subseteq {\bf S}{^*}_{d-1} \ \ \forall~ d$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Remark.} We will see (in Theorem 2.1) that the hypothesis of this Lemma
are always verified. Thus for every Borel normed set ${\bf S}$ we have that
${\bf S}{^*}$ is Borel normed.
{\bf\ProclaimPar Definition.} Let $T=(X_1^{t_1},\dots, X_N^{t_N})\in ({\bf X}_N)^D$. Define the
{\bf corresponding monomial} $\ch T $ in $({\bf X}_{N-1})^D$ as follows:
$$\ch T :=(X_1^{\ch t_1 },\dots, X_{N-1}^{\ch t_{N-1} })$$
$$\hbox{ where \hbox{\ \ \ } }
\ch t_i :=t_i \hbox{\ \ \ } \ \forall~ i<N-1 \hbox{ \hbox{\ \ \ } and \hbox{\ \ \ } } \ch t_{N-1}
:=t_{N-1}+t_N$$
or equivalently
$$\ch T := T \left( { X_{N-1} \over X_N } \right)^{t_N} $$
\Lemma 7
\CasePar
{}
\item {i)}
If $T,T'\in ({\bf X}_N)^D$ \ $T\le T'$
then $\ch T \le \ch T' $.
\item {ii)}
Let ${\bf S}$ be a Borel normed set
then \ $\ch {\min {\bf S}} = \min {\bf S}_0$.
{\it\ProclaimPar Proof.}
\CasePar
{}
\item {i)}
Let $T=(X_1^{t_1},\dots, X_N^{t_N})$, $T'=(X_1^{s_1},\dots,X_N^{s_N})$ and
$T<T'$, then we have $t_i=s_i$ $\ \forall~ i<j$ \ and \ $t_j<s_j$.
\hfill\break
Note that $j\ne N$ since otherwise $D=\deg T = \sum t_i <\sum s_i =\deg
T'=D$.
\hfill\break
Then:
$\ch t_i =t_i=s_i=\ch s_i $ $\ \forall~ i< j$ \ and, (two cases)
\hfill\break
if $j=N-1$:
$\ch t_{N-1} =\deg T-\sum_{i=1} ^{N-2} t_i=\ch s_{N-1} $ \
then $\ch T =\ch T' $;
\hfill\break
if $j<N-1$:
$\ch t_j =t_j < s_j = \ch s_j $ \ then $\ch T <\ch T' $.
\item {ii)}
Obviously $\min {\bf S}\le \min {\bf S}_0$.
\hfill\break
Hence from i) it follows that $\ch {\min {\bf S}}
\le \ch {\min {\bf S}_0} = \min {\bf S}_0$.
\hfill\break
On the other hand, since ${\bf S}$ is Borel normed, $\ch {\min{\bf S}} \in {\bf S}_0$.
\hfill\break
Thus $\ch {\min{\bf S}} = \min {\bf S}_0$.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\hfill\break
{\bf 2. Comparisons between lex-segment and Borel normed sets.}
\def4 {2}
\bigskip
\Thm 1 Let ${\bf L}$ be a lex-segment set and ${\bf B}$ a Borel normed set in
$({\bf X}_N)^D$ such that $|{\bf L}|\le|{\bf B}|$. Then
$$m_{\le i}({\bf L})\le m_{\le i}({\bf B}) \hbox{\ \ \ } \hbox{\ \ \ } \ \forall~ i=1,\dots,N$$
{\it\ProclaimPar Proof.}
By induction on the number of indeterminates:
\CasePar {}
\item {$N=2$:} $m_{\le 2}({\bf L}) =|{\bf L}|\le|{\bf B}|= m_{\le 2}({\bf B})$ \hbox{\ \ \ } and \hbox{\ \ \ }
$m_{\le 1}({\bf L}) = 1 = m_{\le 1}({\bf B})$.
\item {$N>2$:}
Inductive hypothesis: let the thesis be true in $N-1$
indeterminates, and then study for all $i=1\dots N$ the relations between
$m_{\le i}({\bf L})$ and $m_{\le i}({\bf B})$.
\itemitem { $i=N$:}
$m_{\le N}({\bf L})=|{\bf L}|\le|{\bf B}|=m_{\le N}({\bf B})$.
\itemitem {$i=N-1$:}
We need to prove:
$$m_{\le N-1}({\bf L})\le m_{\le N-1}({\bf B})$$
i.e.
$$|{\bf L}_0|\le |{\bf B}_0|$$
From the definition of the lex-segment with respect to $X_N$ we have
$|{\bf B}_0|=|{\bf B}{^*}_0|$. So it will be enough to prove
$$|{\bf L}_0|\le |{\bf B}{^*}_0|.$$
\hfill\break {}
Now ${\bf B}{^*}_d$ and ${\bf B}_d$ are, $\ \forall~ d$, respectively
lex-segment and Borel normed sets of monomials in $N-1$ indeterminates with
the
same cardinality. Then by the inductive hypothesis it follows
$$m_{\le i}({\bf B}{^*}_d)\le m_{\le i}({\bf B}_d) \hbox{\ \ \ }
\ \forall~ i=1,\dots , N-1 \ \ \forall~ d$$
Then, since ${\bf B}$ is Borel normed, it follows from Lemma 1.6 that
${\bf B}{^*}$ is Borel normed.
\hfill\break
Now recall the definition of corresponding monomial in $({\bf X}_{N-1})^D$
and
consider
$\ch {\min{\bf B}{^*}} $ and $ \ch {\min{\bf L}} $.
\hfill\break
Note that
$\min {\bf L} \ge \min {\bf B}{^*} $ (otherwise, since ${\bf L}$ is lex-segment,
${\bf B}{^*} \subset {\bf L}$ and then
$|{\bf B}|=|{\bf B}{^*}|<|{\bf L}|$), therefore from Lemma 1.7.i
$$ \ch {\min{\bf L}} \ge \ch {\min{\bf B}{^*}} $$
It follows from Lemma 1.7.ii that,
since ${\bf L}$ and ${\bf B}{^*}$ are Borel normed
$$\ch {\min{\bf L}} = \min {\bf L}_0 \hbox{\ \ \ } , \hbox{\ \ \ }
\ch {\min{\bf B}{^*}} = \min {\bf B}{^*}_0 $$
and then
$$ \min {\bf L}_0 \ge \min {\bf B}{^*}_0 $$
Moreover, since ${\bf B}{^*}$ is lex-segment w.r.to $X_N$, we have that
${\bf B}{^*}_0$ is a lex-segment set in $({\bf X}_{N-1})^D$.
From these facts it follows that
$$ {\bf L}_0 \subseteq {\bf B}{^*}_0$$
Hence
$$ |{\bf L}_0| \le |{\bf B}{^*}_0| $$
\itemitem {$i<N-1$:}
From the case $i=N-1$ we have $|{\bf L}_0|\le|{\bf B}_0|$
where ${\bf L}_0$ and ${\bf B}_0$ are respectively lex-segment
and Borel normed sets in $N-1$ indeterminates.
By the inductive hypothesis
$$m_{\le i}({\bf L}_0)\le m_{\le i}({\bf B}_0)
\ \ \ \forall~ i=1,\dots,N-1$$
So
$$m_{\le i}({\bf L}) = m_{\le i}({\bf L}_0) \le m_{\le i}({\bf B}_0) = m_{\le i}({\bf B})
\ \ \ \forall~ i<N-1$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\Cor 2 $|{\bf L}|=|{\bf B}|$ \hbox{ $\Longrightarrow$ } $|{\bf X}_N {\bf L}|\le|{\bf X}_N {\bf B}|$.
{\it\ProclaimPar Proof.}
$|{\bf X}_N {\bf L}| = \sum_{i=1}^N m_{\le i}({\bf L}) \le
\sum_{i=1}^N m_{\le i}({\bf B}) = |{\bf X}_N {\bf B}|$.\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Definition.} Let ${\bf S}$ be any set of monomials, then define
$$ b_q({\bf S}) := \sum_{T\in {\bf S}}\bin (m(T)-1 q)$$
\Prop 3 Let ${\bf B}\subseteq({\bf X}_N)^D$ be a Borel normed set, then
$$b_q({\bf B})=\bin (N-1 q)|{\bf B}|-\sum_{i=1}^{N-1}
\left[ m_{\le i}({\bf B})\bin (i-1 q-1) \right]$$
{\it\ProclaimPar Proof.}
$$
\sum_{T\in {\bf B}}\bin (m(T)-1 q) \ = \
\sum_{i=1}^N \left[m_i({\bf B})\bin (i-1 q)\right]=
$$ $$
=\sum_{i=1}^N
\left[\Big(m_{\le i}({\bf B})-m_{\le i-1}({\bf B})\Big)\bin (i-1 q)\right] =
$$ $$
=\sum_{i=1}^N \left[m_{\le i}({\bf B}) \bin (i-1 q)\right] -
\sum_{i=0}^{N-1} \left[m_{\le i}({\bf B}) \bin (i q)\right] =
$$ $$
=\bin (N-1 q) m_{\le N}({\bf B}) +
\sum_{i=1}^{N-1}
\left[m_{\le i}({\bf B}) \left(\bin (i-1 q) - \bin (i q)\right)\right] =
$$ $$
=\bin (N-1 q)|{\bf B}|-
\sum_{i=1}^{N-1} \left[m_{\le i}({\bf B})\bin (i-1 q-1)\right]
$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\Cor 4 Let ${\bf L}$ be a lex-segment set and ${\bf B}$ a Borel normed set in
$({\bf X}_N)^D$ such that \ $|{\bf L}|=|{\bf B}|$, then:
\CasePar {}
\item {i)} $b_q({\bf L}) \ge b_q({\bf B})$;
\item {ii)} $b_q({\bf X}_N {\bf L}) \le b_q({\bf X}_N {\bf B})$.
{\it\ProclaimPar Proof.}
{}From Theorem 2.1 we have $m_{\le i}({\bf L})\le m_{\le i}({\bf B})$ $\ \forall~
i=1,\dots,N$.
\CasePar {Then:}
\item {i)}
$$b_q({\bf L})= \bin (N-1 q)|{\bf L}|-\sum_{i=1}^{N-1}
\left[m_{\le i}({\bf L})\bin(i-1 q-1)\right]\ge$$
$$\ge\bin (N-1 q)|{\bf B}|-\sum_{i=1}^{N-1}\left[m_{\le i}({\bf B})
\bin(i-1 q-1)\right] = b_q({\bf B})$$
\item{ii)} Recall from Proposition 1.3.i that if ${\bf S}$ is a Borel normed set
then $m_i({\bf X}_N{\bf S})=m_{\le i}({\bf S})$. Thus:
\hfill\break
$$b_q({\bf X}_N {\bf L})=
\sum_{T\in {\bf X}_N{\bf L}} \bin(m(T)-1 q)=
\sum_{i=1}^N m_{i}({\bf X}_N{\bf L})\bin(i-1 q)=$$
$$=\sum_{i=1}^N m_{\le i}({\bf L})\bin(i-1 q)
\le
\sum_{i=1}^N m_{\le i}({\bf B})\bin(i-1 q)=
b_q({\bf X}_N {\bf B})$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\hfill\break
{\bf 3. Comparisons between lex-segment and homogeneous ideals.}
\def4 {3}
\bigskip
{\bf\ProclaimPar Definition.} Let $I$ be a monomial ideal in $k[X_1,\dots,X_N]$. Then we denote by
${\bf G}(I)$ the minimal system of generators of $I$, i.e. the set of all
monomials in $I$ which are not proper multiples of any monomial in $I$, and
by
${\bf G}_k(I_d)$ the basis of $I_d$ as a $k$-vectorial space.
{\bf\ProclaimPar Definition.} A monomial ideal $I$ in $k[X_1,\dots,X_N]$ is called:
\hfill\break
i) {\bf Lex-segment} if ${\bf G}_k(I_d)$ is a lex-segment set $\ \forall~ d$;
\hfill\break
ii) {\bf Borel normed} if $T\in I$
\hbox{ $\Longrightarrow$ } $X_i{T\over X_j}\in I$ $\ \forall~ i<j$ such that $X_j\big| T$,
or equivalently if ${\bf G}_k(I_d)$ is a Borel normed set $\ \forall~ d$;
\hfill\break
iii) {\bf Stable} if $T\in I$ \hbox{ $\Longrightarrow$ } $X_i{T\over X_{M(T)}}\in I$.
{\bf\ProclaimPar Remark.} If $I$ is a monomial ideal. Then
\hfill\break
$I$ is lex-segment \hbox{ $\Longrightarrow$ } $I$ is Borel normed \hbox{ $\Longrightarrow$ } $I$ is stable.
\Thm 1.Eliahou-Kervaire(1987) Let $I$ be a stable ideal, then
$$\beta_q(I)=\sum_{T\in {\bf G}(I)}\bin (m(T)-1 q)$$
\Cor 2 Let $I$ be a stable ideal.
\hfill\break
Then
$\beta_q(I) = \sum_{d>0} \big[ b_q({\bf G}_k(I_d)) - b_q({\bf X}_N {\bf G}_k(I_{d-1})) \big]$.
{\it\ProclaimPar Proof.}
{}From Theorem 3.1 we have
$\beta_q(I)=b_q({\bf G}(I))=\sum_{d>0}b_q\Big(\big({\bf G}(I)\big)_d\Big)$.
Then, since
$\big( {\bf G}(I) \big)_d ={\bf G}_k(I_d)\backslash \{ X_N {\bf G}_k(I_{d-1}) \}$,
the thesis follows.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\Cor 3 Let $I^{\bf L}$ be a lex-segment ideal and $I^{\bf B}$ a Borel normed
ideal with the same Hilbert function, then
$$\beta_q(I^{\bf L})\ge \beta_q(I^{\bf B})$$
{\it\ProclaimPar Proof.}
Note that for all $d$
${\bf G}_k(I^{\bf L}_d)$ and ${\bf X}_N {\bf G}_k(I^{\bf L}_d)$ are lex-segment sets,
and
${\bf G}_k(I^{\bf B}_d)$ and ${\bf X}_N {\bf G}_k(I^{\bf B}_d)$ are Borel normed sets.
\hfill\break
Moreover $I^{\bf L}$ and $I^{\bf B}$ have the same Hilbert function, i.e. $\ \forall~ d$
$$|{\bf G}_k(I^{\bf L}_d)|=H_{I^{\bf L}}(d)=H_{I^{\bf B}}(d)=|{\bf G}_k(I^{\bf B}_d)|$$
{}From Corollary 2.4, we then have
$$b_q({\bf G}_k(I^{\bf L}_d)) \ge b_q({\bf G}_k(I^{\bf B}_d))
\hbox{\ and \ }
b_q({\bf X}_N {\bf G}_k(I^{\bf L}_d)) \le b_q({\bf X}_N {\bf G}_k(I^{\bf B}_d)) \ \ \forall~ d$$
and, from Corollary 3.2
$$\beta_q( {\bf G}(I^{\bf L}) ) \ =\
\sum_{d>0} \left[ b_q\left( {\bf G}_k(I^{\bf L}_d)) \right) -
b_q\left( {\bf G}_k(I^{\bf L}_{d-1})) \right)\right]
\ge $$
$$ \ge
\sum_{d>0} \left[ b_q\left( {\bf G}_k(I^{\bf B}_d)) \right) -
b_q\left( {\bf G}_k(I^{\bf B}_{d-1})) \right)\right]=
\beta_q( {\bf G}(I^{\bf B}) )$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Remark.}
Note that for every Borel normed ideal $I$ there exists a lex
segment ideal with the same Hilbert function as that of $I$.
In fact
let ${\bf S}_d$ be the lex-segment set in $({\bf X}_N)^d$ with $|{\bf S}_d|=H_I(d)$.
Then,
from Corollary 2.2,
$$|{\bf X}_N{\bf S}_d| \le |{\bf X}_N {\bf G}_k(I^{\bf B})|$$
Since $I^{\bf B}$ is an ideal we have
${\bf X}_N{\bf G}_k(I^{\bf B})_d\subseteq{\bf G}_k(I^{\bf B})_{d+1}$.
Thus
$$|{\bf X}_N{\bf S}_d|\le|{\bf X}_N{\bf G}_k(I^{\bf B})_d| \le H_I(d+1)=|{\bf S}_{d+1}|$$
Since ${\bf X}_N{\bf S}_d$ and ${\bf S}_{d+1}$ are lex segments we get
${\bf X}_N{\bf S}_d\subseteq{\bf S}_{d+1}$.\hfill\break
Hence we can consider the ${\bf S}_d$'s as the basis of the part in degree $d$ of an
ideal that is lex-segment and has the same Hilbert function of $I$.
\Thm 4.Galligo(1974)
\hfill\break
Let $I$ be a homogeneous ideal in
$k[X_1,\dots,X_N]$ and let $\sigma$ a term-ordering. There exists a Zariski
open subset $U\subseteq GL(N)$ such that for every $g\in U$, $Lt_\sigma(g(I))$ is
invariant under the action of the Borel subgroup $B(N)$ of $GL(N)$. In
particular, if $char(k)=0$, then $Lt_\sigma(g(I))$ is Borel normed.
{\bf\ProclaimPar Remark.} In this way we can obtain for every homogeneous ideal $I$, an ideal
$I^{\bf B}$
with the same Hilbert function and the same Betti numbers as those of $I$, and
such that $Lt_\sigma(I^{\bf B})$ is Borel normed.
\Thm 5.Macaulay(1927)
\hfill\break
Let $I$ be a homogeneous ideal in
$k[X_1,\dots,X_N]$ and let $\sigma$ a term-ordering. Then
$$H_{I}=H_{Lt_\sigma(I)}$$
{\bf\ProclaimPar Remark.} Let $I$ be a homogeneous ideal in
$k[X_1,\dots,X_N]$. Then there exists a lex
segment ideal with the same Hilbert function as that of $I$.
\hfill\break
In fact let $I^{\bf B}$ be the ideal obtained from $I$ by a generic
change of coordinates (Theorem 3.4). We have that $Lt_\sigma (I^{\bf B})$ is a Borel
normed ideal and hence there exists a lex
segment ideal with Hilbert function $H_{Lt_\sigma(I^B)}=H_{I^B}$
(Theorem 3.5).
\Thm 6.M\"oller-Mora(1983)
\hfill\break
Let $I$ be a homogeneous ideal in
$k[X_1,\dots,X_N]$ and let $\sigma$ a term-ordering. Then
$$\beta_{I}\le\beta_{Lt_\sigma(I)}$$
\Thm 7 Let $I$ be a homogeneous ideal and let $I^{\bf L}$ be the lex-segment ideal
with the same Hilbert function as that of $I$.
Then for all $q$
$$\beta_q(I^{\bf L})\ge\beta_q(I)$$
{\it\ProclaimPar Proof.} Let $I^{\bf B}$ be the ideal obtained by Theorem 3.4. Then
$$
H_{I^{\bf L}} = H_I = H_{I^B}
\hbox{ \hbox{\ \ \ } and \hbox{\ \ \ } }
\beta_q \big( I^{\bf B} \big)=\beta_q(I)
$$
{}From Macaulay's Theorem it follows that
$$ H_{I^B} = H_{Lt_\sigma( I^{\bf B} )}$$
Then, from Corollary 3.3
$$\beta_q(I^{\bf L})\ge \beta_q \big(Lt_\sigma(I^{\bf B})\big)$$
{}From M\"oller-Mora's Theorem
$$\beta_q \big(Lt_\sigma(I^{\bf B})\big) \ge \beta_q \big( I^{\bf B} \big)$$
and then
$$\beta_q(I^{\bf L})\ge\beta_q(I)$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\hfill\break
{\bf 4. Upper Bounds for Betti Numbers.}
\def4 {4}
\bigskip
\Thm 1
Let $I$ be a Borel normed ideal and, with abuse of notation, let $I_d$ denote
${\bf G}_k(I_d)$. Then
$$\beta_q(I) =$$
$$=
\bin (N-1 q)|I_D|-\sum_{i=1}^{N-1} m_{\le i}(I_D)\bin (i-1 q-1)
- \sum_{d=1}^{D-1} \left[ \
\sum_{i=1}^{N-1} \left[m_{\le i}(I_d) \bin(i q) \right]\ \right]
$$
Where $D$ is the largest degree of a generator of $I$.
{\it\ProclaimPar Proof.}
{}From Corollary 3.2 it follows that
$$
\beta_q(I)=
\sum_{d=1}^D \left[b_q( I_d ) - b_q( {\bf X}_N I_{d-1} )\right]
=
$$
$$
=
b_q( I_D )
+ \sum_{d=1}^{D-1} \left[ b_q( I_d )\right]
- \sum_{d=0}^{D-1} \left[ b_q( {\bf X}_N I_d )\right]
=
$$
$$
=
b_q( I_D )
+ \sum_{d=1}^{D-1} \left[ b_q(I_d)-b_q( {\bf X}_N I_d)\right]=
$$
$$
=
b_q( I_D )
+ \sum_{d=1}^{D-1} \left[ \
\sum_{i=1}^N \left[m_i( I_d ) \bin(i-1 q) \right]
-\sum_{i=1}^N \left[m_i({\bf X}_N I_d) \bin(i-1 q) \right] \ \right]
$$
Since $I_d$ is a Borel normed set it follows from Proposition 1.3.i
that $m_i({\bf X}_N I_d)=m_{\le i}(I_d)$, $\ \forall~ d$.
Then $\beta_q(I) =$
$$
=
b_q( I_D )
+ \sum_{d=1}^{D-1} \left[ \
\sum_{i=1}^N \left[( m_i(I_d) - m_{\le i}(I_d)) \bin(i-1 q) \right]
\ \right]=
$$
$$
=
b_q( I_D )
- \sum_{d=1}^{D-1} \left[ \
\sum_{i=1}^N \left[m_{\le i-1}(I_d) \bin(i-1 q) \right]\ \right]=
$$
$$
=
b_q( I_D )
- \sum_{d=1}^{D-1} \left[ \
\sum_{i=1}^{N-1} \left[m_{\le i}(I_d) \bin(i q) \right]\ \right]=
$$
$$
=
\bin (N-1 q)|I_D|-\sum_{i=1}^{N-1} m_{\le i}(I_D)\bin (i-1 q-1)
- \sum_{d=1}^{D-1} \left[ \
\sum_{i=1}^{N-1} \left[m_{\le i}(I_d) \bin(i q) \right]\ \right]
$$
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
{\bf\ProclaimPar Definition.} It is well known (see Robbiano [R]) that, if $h$ and $n$ are positive
integers, then $h$ can be written uniquely in the form
$$h= \bin(h(n) n) + \bin (h(n-1) n-1) + \dots + \bin (h(i) i)$$
where
$h(n) >h(n-1) > \dots > n(i) \ge i \ge 1$.
\hfill\break
This unique expression is called {\bf binomial expansion}
of $h$ in base $n$ and it is denoted by $h_n$, and define
$$\big( h_n \big) ^s_t :=
\bin(h(n)+s n+t) + \bin (h(n-1)+s n-1+t) + \dots + \bin (h(i)+s i+t)$$
\medskip
The particular significance of the binomial
expansion of the values of the Hilbert function becomes apparent when we attend
to write an explicit formula which computes the Betti numbers of a
lex-segment ideal:
\hfill\break
Let ${\bf S}$ be a lex-segment set in $({\bf X}_N)^D$ and
let $d$ be the largest integer such that
$X_1^{D-d} X_N^d\in{\bf S}$.
\hfill\break
Since ${\bf S}$ is a lex-segment set, ${\bf S}$ contains all the monomials
$$X_1^{D-d} \{ X_1,\dots,X_N \} ^d$$
The number of these elements is $\bin({N+d-1} {N-1})$ which is exactly
the first binomial in the binomial expansion of $H(D)$ in base $N-1$.
\hfill\break
The set of the remaining monomials of ${\bf S}$ is strictly contained in
$$X_1^{D-d-1} \{ X_2,\dots,X_N \} ^{d+1}$$
Thus, we can think of it as a lex-segment set (strictly contained) in
$\{ X_2,\dots,X_N \} ^{d+1}$.
So, repeating the reasoning, we obtain the
whole binomial expansion.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\Prop 2.(Macaulay) Let $I$ be a lex-segment ideal. Then
$$|{\bf X}_N {\bf G}_k (I_D)|=(H_I(D)_{N-1})^1$$
{\it\ProclaimPar Proof.}
As we saw before,
the first binomial of the binomial expansion of $H(D)$ in base $N-1$, $\bin
(N+d-1 N-1)$, represents the number of monomials in $\{ X_1,\dots, X_N \}^D$.
Thus the multiples of these elements are a set with
$\bin (N+{(d+1)}-1 N-1)$ elements.
\hfill\break
And so on.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\Prop 3 Let $I$ be a lex-segment ideal. Then
$$m_{\le i}(I_d)=\Big( H(d)_{N-1} \Big)^{-(N-i)}_{-(N-i)}$$
(Where $\bin(h n):=0$ if $n<0$).
{\it\ProclaimPar Proof.}
As before,
$\bin (N+d-1 N-1)$ is the number of monomials
in $\{ X_1,\dots, X_N \}^D$.
Among these, the elements which use only the first $i$ indeterminates number
$\bin({i+d-1} {i-1})$ \ i.e.
$$\bin({N+d-1-(N-i)} {N-1-(N-i)})$$
And so on.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\medskip
{\bf\ProclaimPar Remark.} If $I$ is a homogeneous ideal we can calculate the largest degree of a
generator of the lex-segment ideal with the same Hilbert function. In fact,
Green [Gr] proved that $D+1$ is the smallest integer greater then the
maximum degree of a generator of $I$ for which $H_I(D)^1= H_I(D+1)$.
\hfill\break
Hence Theorem 4.1 and Proposition 4.3 give a formula which computes the Betti
numbers of a lex-segment ideal.
\hfill\break
They are sharp upper bounds for homogeneous
ideals with the same Hilbert function.
\hfill\break
In particular, to count the first syzigies, it is possible to give a simpler
formula.
\Cor 4
$\beta_1(I)=$
$$(N-1)H(D)-\big( H(D)_{N-1}\big)_{-1} +
\sum_{d=1}^{D-1}
\bigg[(N-1)(H(d)_{N-1})_{-1} -(H(d)_{N-1})_{-2} \bigg]$$
{\it\ProclaimPar Proof.}
$b_1({\bf S})$
=
$\sum_{i=1}^Nm_ i({\bf S}) \big( i-1 \big)$
=
$\sum_{i=1}^N \big( m_{\le i}({\bf S}) - m_{\le i-1}({\bf S}) \big) \big( i-1 \big)$
=
$\sum_{i=1}^N \bigg[ m_{\le i}({\bf S}) \big( i-1 \big) \bigg]
- \sum_{i=1}^{N-1} \bigg[ m_{\le i}({\bf S}) i \bigg]$
=
$(N-1)|{\bf S}| - \sum_{i=1}^{N-1} m_{\le i}({\bf S})$.
\hfill\break
{}From Proposition 4.3 it is easy to see that
$\sum_{i=1}^{N-1} m_{\le i}({\bf S})=(|{\bf S}|_{N-1})_{-1}$.
Then
$b_1({\bf S})=(N-1)|{\bf S}| -(|{\bf S}|_{N-1})_{-1}$.
\hfill\break
{}From Proposition 4.2 it follows that
$|{\bf X}_N{\bf S}|=(|{\bf S}|_{N-1})^1$ and from Corollary 3.2 that
$\beta_1(I) = \sum_{d>0} \big[ b_1({\bf G}_k(I_d)) - b_1({\bf X}_N {\bf G}_k(I_{d-1})) \big]$.
\hfill\break
Hence
$$\beta_1=$$
$$=\sum_{d=1}^D
[(N-1)|I_d|-(|I_d|_{N-1})_{-1}-((N-1)|{\bf X}_NI_{d-1}|-(|{\bf X}_NI_{d-1}|_{N-1})_{-1})]
=$$
$$=\sum_{d=1}^D
[(N-1)H(d)-(H(d)_{N-1})_{-1}-$$
$$((N-1)(H(d-1)_{N-1})^1-(H(d-1)_{N-1})^1_{-1})]$$
The thesis follows easily.
\hfill\vrule height5 true pt width 5 true pt depth 0 pt
\hfill\break
\centerline{ACKNOWLEGMENTS}
\medskip
Sincere thanks go to Prof. L.Robbiano and Prof. T.Mora for their useful
suggestions.
\eject
\hfill\break
\centerline{REFERENCES}
\bigskip
\CasePar {}
\item{[B-C-R]}
A.M. Bigatti, M. Caboara, L. Robbiano:
On the Computation of Hilbert-Poincar\'e Series.
AAECC {\bf 2} (1991) 21-33.
\item{[E-K]}
S. Eliahou, M. Kervaire:
Minimal Resolution of Some Monomial Ideals.
J. Algebra {\bf 129} (1990) 1-25
\item{[Ga]}
A. Galligo:
A propos du th\'eor\`eme de pr\'eparation de Weierstrass,
Functions de Plusieurs Variables Complexes.
Lecture Notes in Mathemathics
{\bf 409}, Berlin, Heidelberg, New York: Springer (1974), 543-579.
\item{[Gr]}
M. Green:
Restriction of linear series to hyperplanes, and some results of Macaulay
and Gotzmann, Algebraic Curves and Projective Geometry Proceedings, Trento
(1988) Springer Lecture Notes in Mathematics {\bf 1389}
\item{[M]}
F.S. Macaulay:
Some properties of enumeration in the theory of modular system.
Proc. London Math. Soc. {\bf 26} (1927), 531-555.
\item{[M-M]}
H.M. M\"oller, T. Mora:
New Constructive methods in classical ideal theory.
J. Algebra {\bf 100} (1986), 138-178.
\item{[R]}
L. Robbiano:
Introduction to the Theory of Hilbert Function. Queen's Papers in Pure and
Applied Mathematics {85} (1990) B1-B26.
\end
|
1996-03-18T06:20:07 | 9601 | alg-geom/9601007 | fr | https://arxiv.org/abs/alg-geom/9601007 | [
"alg-geom",
"math.AG"
] | alg-geom/9601007 | null | Nicole Mestrano | Sur les espaces de modules des fibres vectoriels de rang deux sur des
hypersurfaces de P^3 | LaTeX, Dans cette version on tient compte du fait (qui m'a ete
communique' par C. Walter) que la proposition 2.4 de la premiere version a
deja ete demontree par Rao. D'autre part l'exposition est amelioree | null | null | null | null | Soit $X$ une hypersurface lisse de degr\'e $\delta \geq 4$ dans $\pp ^3$
telle que $Pic(X)=Z$. On d\'esigne par $M_X(c_2)$ l'espace de modules des
fibr\'es vectoriels sur $X$ de classes de Chern $c_1 = 0$ et $c_2$,
semi-stables par rapport au diviseur hyperplan. Nous contribuons ici \`a la
recherche de diff\'erentes composantes irr\'eductibles pour $c_2$ petit.
On prouve que pour tout entier $c_2 \geq \delta ^3/4 - \delta ^2/2$ l'espace
des modules $M_X(c_2)$ contient une composante irr\'eductible r\'eduite de
dimension attendue. En utilisant un r\'esultat de O'Grady, on en d\'eduit que
pour tout entier $c_2$ tel que
$$ {1/4}(\delta ^3-2\delta ^2) \leq c_2 \leq
{1/3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3) $$ l'espace $M_X(c_2)$
poss\`ede au moins deux composantes irr\'eductibles de dimensions distinctes.
Pour $\delta \geq 27$, de tels entiers existent! D'autre part, on prouve que
pour $c_2 > {1/12}(13 \delta ^3 - 24 \delta ^2 + 8 \delta)$, le fibr\'e
g\'en\'eral de la bonne composante de $M_X(c_2)$ que nous construisons a
cohomologie naturelle. Plus g\'en\'eralement, on d\'emontre que pour toute
surface projective lisse $X$ telle que le fibr\'e canonique soit de la forme
$K_X = \Oo_X(k)$ o\`u $\Oo_X(1)$ est un fibr\'e ample, alors si $c_2$ est
suffisamment grand, le fibr\'e g\'en\'eral de l'unique composante de $M_X(c_2)$
a la cohomologie naturelle.
| [
{
"version": "v1",
"created": "Mon, 8 Jan 1996 10:08:07 GMT"
},
{
"version": "v2",
"created": "Wed, 13 Mar 1996 15:32:17 GMT"
},
{
"version": "v3",
"created": "Fri, 15 Mar 1996 08:57:07 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Mestrano",
"Nicole",
""
]
] | alg-geom | \section*{Sur les espaces de modules des
fibr\'es vectoriels de rang deux sur des hypersurfaces
de ${\bf P} ^3$}
\auteur
Soit $X$ une hypersurface lisse de degr\'e $\delta \geq 4$ dans ${\bf P} ^3$
telle que $Pic(X)\cong {\bf Z}$.
On d\'esigne par $M_X(c_2)$ l'espace de modules des fibr\'es vectoriels
sur $X$ de rang $2$, de classes de
Chern $c_1 = 0 $ et $c_2$, semi-stables par rapport au diviseur hyperplan
(ou, ce qui est
\'equivalent, par rapport \`a n'importe quel diviseur ample) \cite{Gieseker}.
Il est bien connu (Donaldson \cite{Donaldson}, Taubes \cite{Taubes},
Gieseker-Li \cite{GiLiJDG}, O'Grady \cite{OGrady}) que lorsque $c_2$
est suffisamment grand, l'espace $M_X(c_2)$
est r\'eduit et irr\'eductible de dimension attendue.
Nous contribuons ici \`a la recherche, lorsque $c_2$ est petit,
de diff\'erentes composantes irr\'eductibles.
On montre (corollaire \ref{L})
que pour tout entier $c_2 \geq \delta ^3/4 - \delta ^2/2 $
l'espace des modules $M_X(c_2)$ contient une composante
irr\'eductible r\'eduite de dimension attendue.
En utilisant un r\'esultat de O'Grady, on en d\'eduit
(corollaire \ref{M}) que pour tout entier $c_2$ tel que
$
\frac{1}{4}(\delta ^3-2\delta ^2)
\leq c_2 <
\frac{1}{3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3)
$
l'espace
$M_X(c_2)$ poss\`ede au moins deux composantes irr\'eductibles
de dimensions distinctes.
(Pour $\delta \geq 27$, de tels entiers existent!)
On peut obtenir un r\'esultat analogue pour les fibr\'es vectoriels
avec $c_1 = 1$
(voir remarque \ref{R1}).
D'autre part, on prouve (th\'eor\`eme \ref{A1}) que pour $c_2 >
\frac{1}{12}(13 \delta
^3 - 24 \delta ^2 + 8 \delta)$, le fibr\'e g\'en\'eral de la bonne
composante de
$M_X(c_2)$ que nous construisons a la cohomologie naturelle.
\newline
Plus g\'en\'eralement, on d\'emontre (corollaire \ref{V}) que pour toute
surface
projective lisse $X$ telle
que le fibr\'e canonique $K_X$ est de la forme $K_X = {\cal O} _X(k)$
pour un entier $k\in {\bf Z}$ o\`u ${\cal O} _X(1)$ est un fibr\'e ample,
si
$c_2$ est suffisamment grand alors, le fibr\'e g\'en\'eral de
l'unique composante de $M_X(c_2)$ a la
cohomologie naturelle o\`u $M_X(c_2)$ est l'espace de modules des
fibr\'es stables (par rapport \`a ${\cal O} _X(1)$) de classes de Chern
$c_1=0$ dans $Pic (X)$ et $c_2$.
Les fibr\'es vectoriels que nous construisons ici, et
qui sont dans une composante irr\'eductible r\'eduite de dimension attendue,
sont des extensions
de ${\cal J} _{P/X}(\sigma )$ par ${\cal O} _X(-\sigma )$ o\`u $P$ est un sous
sch\'ema de dimension z\'ero en position sp\'eciale dans $X$.
On prend pour $P$ des intersections compl\`etes de $X$ avec des courbes
de ${\bf P} ^3$ dont les id\'eaux ont les r\'esolutions les plus simples.
(Ces courbes ont \'et\'e \'etudi\'ees
par de nombreux auteurs e.g.
\cite{Ellia}, \cite{Floystad}, \cite{Rao}, \cite{Walter}.)
C'est un grand plaisir de remercier Carlos Simpson
pour de nombreuses discussions et de lui d\'edier cet article.
\numero{Pr\'eliminaires}
Sauf mention du contraire,
$X$ d\'esigne une hypersurface lisse de degr\'e $\delta \geq 4$ dans ${\bf P} ^3$
telle que $Pic(X)\cong {\bf Z}$.
\begin{proposition}
\label{A}
Soient $\sigma $ un entier positif et $C\subset {\bf P} ^3$ une courbe lisse et
irr\'eductible de degr\'e $d$ telle que $H^1(C,{\cal O} _C(2\sigma -4))\neq 0$.
Quitte \`a translater $C$ par une hom\'eographie g\'en\'erale
de ${\bf P} ^3$, si on pose
$P:= C\cap X$ alors il existe un fibr\'e vectoriel $E$ sur $X$
qui s'ins\'ere dans la suite
exacte $(\ast )$ suivante:
\vspace*{.5cm}
\newline
$(\ast )$
\vspace*{-1cm}
\newline
$$
0\rightarrow {\cal O} _{X}(-\sigma )
\rightarrow E \rightarrow {\cal J} _{P/X} (\sigma ) \rightarrow 0.
$$
On a $c_1(E)=0$ et $c_2(E)= \delta (d-\sigma ^2)$.
\end{proposition}
{\em D\'em:}
On sait (cf \cite{RHSRS}, Theorem 4.1) que la donn\'ee d'une courbe
$C\subset {\bf P} ^3$
de degr\'e $d$, munie
d'une section non-nulle $\eta \in H^0(C,\omega _C(4-2 \sigma ))$
est \'equivaute \`a celle
d'un faisceau reflexif ${\cal F}$ sur ${\bf P} ^3$
de classes de Chern $c_1({\cal F})=2 \sigma$ et $c_2({\cal F})= d$
s'ins\'erant dans la suite exacte
$$
0 \rightarrow {\cal O} _{{\bf P} ^3}
\rightarrow {\cal F} \rightarrow {\cal J} _{C/{\bf P} ^3}(2 \sigma ) \rightarrow 0.
$$
L'ensemble des points o\`u ${\cal F}$
n'est pas localement libre est le sous sch\`ema
$Z$ de $C$ des z\'eros de la section $\eta$.
Soit $f : {\bf P} ^3 \rightarrow {\bf P} ^3$ une hom\'eographie
de ${\bf P} ^3$ telle que $X$ ne rencontre pas $f^{-1}(Z)$.
On remplace $C$ par $f^{-1}(C)$ et on pose
$E:= f^{*}F(- \sigma)|_X$.
\hfill $\Box$\vspace{.1in}
On \'etablit quelques propri\'et\'es de $E$.
\begin{lemme}
\label{B}
Soit $C\subset {\bf P} ^3$ une courbe irr\'eductible lisse telle que
l'intersection $P := X\cap C$ soit transverse.
\newline
Si $H^0({\bf P} ^3 ,{\cal J} _{C/{\bf P} ^3}(\tau ))= H^1({\bf P} ^3 ,{\cal J} _{C/{\bf P} ^3}
(\tau -\delta ))=0$
alors $H^0({\bf P} ^3 ,{\cal J} _{P/X}(\tau ))=0$.
\end{lemme}
{\em D\'em.}
Soient $u\in H^0({\cal J} _{P/X}(\tau ))$ et
$v \in H^0({\bf P} ^3, {\cal O} _{{\bf P} ^3}(\tau ))$ un relev\'e de $u$
consid\'er\'e
comme une section de ${\cal O} _{X}(\tau )$.
La restriction $v|_C$ est une section de ${\cal O} _C(\tau )$ qui s'annulle sur
$P$
donc, la suite exacte
$$
0\rightarrow {\cal O} _{{\bf P} ^3}(\tau -\delta )
\rightarrow {\cal O} _{{\bf P} ^3}(\tau ) \rightarrow
{\cal O} _{X}(\tau ) \rightarrow 0
$$
restreinte \`a $C$ prouve qu'il
existe un \'el\'ement $w\in H^0(C,{\cal O} _C(\tau -\delta ))$ tel que $fw=v|_C$
o\`u $f\in H^0({\bf P} ^3, {\cal O} _{{\bf P} ^3}(\delta ))$ est l'\'equation de $X$.
Par hypoth\`ese
$H^1({\cal J} _{C/{\bf P} ^3}(\tau -\delta ))=0$, donc
$w$ s'\'etend en une section $w'\in H^0({\bf P} ^3, {\cal O} _{{\bf P} ^3}(\tau -\delta ))$.
On a $(v-fw' )|_C = 0$. L'hypoth\`ese $H^0({\cal J} _{C/{\bf P} ^3}(\tau ))=0$ implique
$v-fw'=0$, et par cons\'equent $u = v|_X$=0.
\hfill $\Box$\vspace{.1in}
\begin{corollaire}
\label{C}
Dans la construction la proposition \ref{A}, si on a
$\tau < \sigma$ et
\newline
$H^0({\bf P} ^3 ,{\cal J} _{C/{\bf P} ^3}(\sigma + \tau ))=
H^1({\bf P} ^3 ,{\cal J} _{C/{\bf P} ^3}(\sigma + \tau -\delta ))=0$,
alors $H^0(X, E(\tau ))=0$.
\end{corollaire}
{\em D\'em.}
On applique le lemme pr\'ec\'edent en utilisant la suite exacte $(\ast )$.
\hfill $\Box$\vspace{.1in}
\begin{corollaire}
\label{D}
Dans la construction de la proposition \ref{A}, si
\newline
$H^0({\bf P} ^3 ,{\cal J} _{C/{\bf P} ^3}(\sigma ))=
H^1({\bf P} ^3 ,{\cal J} _{C/{\bf P} ^3}(\sigma -\delta ))=0$
alors $E$ est stable.
\end{corollaire}
{\em D\'em.}
Puisque $E$ est un fibr\'e de rang $2$ sur une surface
dont le groupe de Picard est de rang un, il suffit de voir que
$H^0(X, E)=0$. pour cela, on applique
le corollaire pr\'ec\'edent avec $\tau = 0$.
\hfill $\Box$\vspace{.1in}
\subnumero{Th\'eorie locale}
Nous rappelons ici quelques d\'efinitions et observations habituelles.
Soit $[E]$ un point de l'espace des modules $ M_{X}^{st}(c_2)$ des
faisceaux sans
torsion stables de classes de Chern $c_1 =0$ et $c_2$,
repr\'esent\'e par le faisceau $E$.
Le morphisme de multiplication
$$
H^j(X, \Omega ^2_X) \rightarrow Ext^j(E,E\otimes \Omega ^2_X)
$$
donne, par dualit\'e de Serre (cf \cite{LePotier-Drezet} Proposition 1.2)
le morphisme {\em trace}
$$
Ext^i(E,E) \rightarrow H^i(X, {\cal O} _X).
$$
Comme le premier morphisme est injectif pour $j=0$, le morphisme trace
est surjectif pour
$i=2$. Le noyau $Ext^i(E,E)^o\subset Ext^i(E,E)$
du morphisme trace s'appelle la partie {\em sans
trace}.
L'espace tangent en $[E]$ \`a $M_X^{st}(c_2)$ est l'espace vectoriel
$Ext^1(E,E)^o$.
D'autre part,
le germe \'etale de l'espace des modules $M_X(c_2)$ en $[E]$ est
isomorphe au germe \`a l'origine du sous-sch\'ema de $Ext^1(E,E)^o$
d\'efini par un
morphisme (nonlin\'eaire) $\Phi :Ext^1(E,E)^o \rightarrow Ext^2(E,E)^o$.
Ceci est
classiquement bien connu. Une d\'emonstration purement alg\'ebrique est
indiqu\'ee par J. Le Potier dans ses notes \cite{LePotierLuminy}.
La {\em dimension attendue} de $M_X(c_2)$ en $[E]$ est par
d\'efinition la diff\'erence
$$
{\rm dim\; att} := {\rm dim} \; Ext^1(E,E)^o - {\rm dim} \; Ext^2(E,E)^o
$$
Le th\'eor\`eme de Riemann-Roch (avec l'hypothese $c_1=0$) fournit la valeur
$$
{\rm dim\; att} _E = 4c_2 - 3 \begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X).
$$
On a $Ext^2(E,E)^o=0$ si et seulement si l'espace
$M_{X}(c_2)$ est lisse de dimension ${\rm dim\; att}$ au point $[E]$.
Dans ce cas, on dit que l'espace des modules est {\em bon} en $[E]$.
Une composante irr\'eductible non vide $Y\subset M_{X}(c_2)$
est {\em bonne} si l'espace des modules est bon au point g\'en\'erique de
$Y$ ce qui \'equivaut \`a dire que $Y$ est r\'eduite de dimension
\'egale \`a
la dimension attendue.
Nous utiliserons la proposition bien connue suivante (cf \cite{OGradyInvent}
ou \cite{GiLiJDG}).
\begin{proposition}
\label{E}
S'il existe une bonne composante de $M_{X}(c_2)$, alors il
existe une bonne composante de $M_{X}(c_2+1)$.
\end{proposition}
{\em D\'em.}
Soit $E'$ un bon fibr\'e stable avec $c_1(E')=0$ et $c_2(E')=c_2$.
Soit ${\cal G}$ le faisceau gratte-ciel de rang $1$ au-dessus d'un
point $x\in X$. Soit $E_0$ le
noyau d'une surjection g\'en\'erale $E' \rightarrow {\cal G}$.
Alors $E_0$ est un faisceau coh\'erent sans torsion
de classes de Chern $c_1(E_0)=0$ et $c_2(E_0)=c_2 +1$.
Il a les m\^emes sous faisceaux que $E'$ donc il est stable.
\newline
Montrons que $E_0$ est bon c'est-\`a-dire que le morphisme trace
$t: Ext^2(E_0,E_0) \rightarrow H^2(X,{\cal O} _X)$ est injectif.
\newline
En appliquant le foncteur $Ext(E', \ast )$ \`a
la suite exacte
$
0 \rightarrow E_0 \rightarrow E' \rightarrow {\cal G} \rightarrow 0
$
et en remarquant que $Ext^1( E', {\cal G}) = 0 $ (car $E'$ est
localement libre et ${\cal G}$ est un faisceau gratte ciel)
on obtient l'injection
$f: Ext^2( E',E_0) \rightarrow Ext^2( E',E')$.
Par hypoth\`ese $E'$ est bon, donc le morphisme trace
$g: Ext^2(E',E') \rightarrow H^2(X,{\cal O} _X)$ est injectif.
Soit maintenant
$h: Ext^2(E',E_0) \rightarrow Ext^2(E_0,E_0)$
le morphisme obtenu en appliquant le foncteur $Ext( \ast ,E_0)$.
Comme $Ext^3( {\cal G} ,E_0) = 0 $, ce morphisme est surjectif.
On en conclut que $t$ est injectif en remarquant que
$t \circ h = g \circ f$.
\newline
Pour terminer la d\'emonstration de cette proposition,
il suffit maintenant de prouver que $E_0$ se d\'eforme en un
fibr\'e vectoriel $E$. En effet, par platitude, $E$ aura pour
classes de Chern $c_1(E)=0$ et $c_2(E)=c_2 +1$
et il sera stable et
bon puisque ces propri\'et\'es sont des propri\'et\'es ouvertes.
\newline
On raisonne par l'absurde.
\newline
Supposons que pour toute d\'eformation
${\cal E}$ de $E_0$ param\'etr\'ee par une courbe $T$
de point g\'en\'eral $\eta$ et de point sp\'ecial
$s$ avec ${\cal E}_s = E_0$, le faisceau ${\cal E}_{\eta}$ ne
soit pas locallement libre.
Soit ${\cal E}$ une telle d\'eformation.
D\'esignons par ${\cal E}^{\ast \ast}$
le double dual du faisceau ${\cal E}$ et par ${\cal M}$ le
conoyau de l'injection de ${\cal E}$ dans ${\cal E}^{\ast \ast}$.
On a la suite exacte
$
0 \rightarrow {\cal E} \rightarrow {\cal E}^{\ast \ast} \rightarrow {\cal M} \rightarrow 0.
$
On a l'inclusion $E_0 \subset ({\cal E}^{\ast \ast})_s $ donc,
en remarquant que $E' = E_0 ^{\ast \ast}$, on voit que
$E' \subset ({\cal E}^{\ast \ast})_{s} ^{\ast \ast}$. Or ce sont deux
faisceaux localement libre \'egaux hors codimension $2$,
d'o\`u $E' = ({\cal E}^{\ast \ast})_{s} ^{\ast \ast}$ et on a la double inclusion
$E_0 \subset ({\cal E}^{\ast \ast})_s \subset E'$.
Par hypoth\`ese $longueur (E'/E_0) =1$,
donc $ longueur (E'/ ({\cal E}^{\ast \ast})_s) + longueur ({\cal M}_s) = 1$.
Par semicontinuit\'e, $longueur ({\cal M} _{\eta}) \leq 1$ mais
on a suppos\'e que ${\cal E}_{\eta}$ n'est pas locallement libre, donc
$longueur ({\cal M} _{\eta}) = 1$. Par semi-continuit\'e, on en d\'eduit
$longueur ({\cal M}_s) = 1$.
Par suite $longueur (E'/ ({\cal E}^{\ast \ast})_s) = 0$
i.e. $E' = ({\cal E}^{\ast \ast})_s$ et ${\cal E}^{\ast \ast}$ est une d\'eformation de
$E'$.
Soit $U\subset M_X^{st}(c_2)$ un voisinage ouvert de $E'$ de dimension
$4c_2 - 3 \begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X)$.
Soit $Z _1\subset M_X^{st}(c_2 +1)$ l'ensemble des points correspondant
aux faisceaux
$F$ avec $longeur(F^{\ast\ast}/F)\geq 2$ et $Z_2\subset M_X^{st}(c_2+1)$
l'ensemble des
points correspondant aux faisceaux $F$ avec
\newline
$longeur (F^{\ast \ast}/F)=1$ et
$F^{\ast\ast} \not\in U$. Les sous-ensembles $Z_1$
et $Z_2$ sont constructibles.
L'argument ci-dessus montre que, pour $i \in \{ 1,2 \}$, la fermeture de
$Z_i$ ne contient pas $E_0$. En effet si $E_0$
\'etait dans la fermeture de $Z_i$ alors, il
existerait une d\'eformation
${\cal E}$ de $E_0$ param\'etr\'ee par une courbe $T$
de point sp\'ecial
$s$ avec ${\cal E}_s = E_0$ et de point g\'en\'eral $\eta$
avec ${\cal E}_{\eta} \in Z_i$. D'apr\'es l'argument ci-dessus:
\newline
(i) \, $ longueur({\cal E}^{\ast \ast}_{\eta}/{\cal E}_{\eta}) = 1$ donc
${\cal E}_{\eta} \not\in Z_1$ d'o\`u la contradiction pour $i=1$.
\newline
(ii) \, ${\cal E}^{\ast \ast}$ est une d\'eformation de
$E'$, donc ${\cal E}^{\ast \ast}_{\eta} \in U$
et on ne peut pas avoir ${\cal E}_{\eta} \in Z_2$;
d'o\`u la contradiction pour $i=2$.
L'ensemble $Z_1\cup Z_2$ ne contenant pas $E_0$ dans sa fermeture,
son compl\'ementaire dans $M_X^{st}(c_2 +1)$,
contient un voisinage ouvert $V$ de $E_0$.
Quitte \`a restreindre $V$, puisque $E_0$ ne se d\'eforme pas en faisceau
localement
libre (par hypoth\`ese absurde), on peut supposer que les points de $V$
repr\'esentent des faisceaux $F$ non localement libres. Par suite,
si $F$ est repr\'esent\'e par un point de $V$, alors
$longeur(F^{\ast\ast}/F)=1$ et $F^{\ast\ast} \subset U$.
Les faisceaux $F$ s'ins\`erent donc dans des suites exactes de la forme
$$
0\rightarrow F \rightarrow F^{\ast\ast} \rightarrow M\rightarrow 0
$$
o\`u $M$ est un faisceau gratte-ciel de longeur $1$.
La dimension de l'ensemble de ces faisceaux $F$ est donc
inf\'erieure ou \'egale \`a
$ 3+ dim(U) = 3 + 4c_2 - 3\begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X)$.
Ceci nous fournit la contradiction cherch\'ee puisqu'on
sait que la
dimension de $M_X^{st}(c_2+1)$ en tout point est au moins
\'egale \`a $4(c_2+1) - 3\begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X)$.
\hfill $\Box$\vspace{.1in}
\begin{lemme}
Soit $E$ un fibr\'e vectoriel de rang $2$ et d\'eterminant trivial. On a
$$
Ext^i(E,E)^o = H^i(X, Sym ^2(E)).
$$
\end{lemme}
{\em D\'em.}
Comme $E$ est un fibr\'e on a $Ext ^i(E,E)^o= H^i (X,End ^o(E))$ o\`u $End
^o(E)$ est le noyau du morphisme trace $End(E)\rightarrow {\cal O} _X$.
D'autre part, $E$ \'etant de rang $2$ et de d\'eterminant trivial,
$E\cong E^{\ast}$, et par suite,
$End (E)= E^{\ast}\otimes E \cong E \otimes E$. Via cet
isomorphisme le morphisme trace devient (au signe pr\`es)
$$
E\otimes E \rightarrow \bigwedge ^2E \cong {\cal O} _X.
$$
Le noyau du morphisme trace est donc le noyau du morphisme $E\otimes E
\rightarrow \bigwedge ^2E$, c'est-\`a-dire $Sym ^2(E)$.
\hfill $\Box$\vspace{.1in}
\begin{lemme}
Soit $E$ un fibr\'e s'ins\'erant dans la suite exacte $(\ast )$.
Alors on a la suite exacte
\vspace*{.5cm}
\newline
$(\ast \ast )$
\vspace*{-1cm}
\newline
$$
0 \rightarrow E(-\sigma ) \rightarrow Sym ^2(E) \rightarrow
{\cal J} ^2_{P/X}(2\sigma )\rightarrow 0
$$
\end{lemme}
{\em D\'em.}
En tensorisant la suite exacte $(\ast )$ par $E$ on obtient le morphisme
$$
0 \rightarrow E(-\sigma ) \rightarrow E\otimes E
$$
qui, compos\'e avec la projection $E\otimes E\rightarrow Sym ^2(E)$
donne le morphisme
\newline
$f : E(-\sigma ) \rightarrow Sym ^2(E).$
C'est un exercice d'alg\`ebre lin\'eaire de voir que $f$
est injective au dessus du point g\'en\'erique, il s'ensuit que $f$
est une injection de faisceaux.
Soit $g: Sym ^2(E)
{\rightarrow} {\cal O} _X(2\sigma )$
le morphisme induit par la suite exacte $(\ast )$, en consid\`erant
${\cal J} _{P/X}(\sigma ) $ comme un sous faisceau de ${\cal O} _X(\sigma )$.
On a la suite suivante:
$$
0 \rightarrow E(-\sigma ) \stackrel{f}{\rightarrow} Sym ^2(E)
\stackrel{g}{\rightarrow} {\cal O} _X(2\sigma ).
$$
(C'est encore un exercice d'alg\`ebre lin\'eaire de voir que
la suite est exacte au dessus de l'ouvert compl\'ementaire de $P$,
on en d\'eduit
l'exactitude sur $X$ en utilisant le fait que
toute section de $E(-\sigma )$ d\'efinie hors
codimension $2$ s'\'etend.)
L'image du morphisme naturel $h:
{\cal J} _{P/X}(\sigma )\otimes {\cal J} _{P/X}(\sigma ) \rightarrow {\cal O} _X(2\sigma )
$
est le faisceau d'id\'eaux ${\cal J} ^2_{P/X}(2\sigma )$.
Donc, le diagramme commutatif suivant:
$$
\begin{array}{ccc}
E\otimes E & \rightarrow & {\cal J} _{P/X}(\sigma )\otimes {\cal J} _{P/X}(\sigma ) \\
\downarrow & & \downarrow {}^{{}_h}\\
Sym ^2(E) & \stackrel{g}{\rightarrow} & {\cal O} _X(2\sigma )
\end{array}
$$
o\`u les fl\`eches verticale \`a gauche et horizontale en haut sont
surjectives montre que l'image du morphisme $g$ est le faisceau
${\cal J} ^2_{P/X}(2\sigma )$.
\hfill $\Box$\vspace{.1in}
\begin{proposition}
\label{F}
Soient $\sigma $ un entier positif
et $P\subset X$ un sous-sch\'ema fini r\'eduit tels que:
\newline
i)\, \,\, $\delta -4 < 2\sigma$
\newline
ii)\, \, $H^0(X,{\cal J} _{P/X}(\delta - 4))= 0$, et
\newline
iii)\, $H^0(X, {\cal J} ^2_{P/X}(2\sigma + \delta - 4))=0.$
\newline
Alors tout fibr\'e vectoriel qui est extension de ${\cal J} _{P/X}(\sigma )$ par
${\cal O} _X(-\sigma )$ est bon.
\end{proposition}
{\em D\'em.}
Soit $E$ un tel fibr\'e. D'apr\`es le lemme 1.6 et, par
dualit\'e il suffit de montrer que $H^0(Sym^2(E)\otimes
{\cal O} _X(\delta -4))=0$ (car
$E$ \'etant autodual, $Sym ^2(E)$ l'est aussi).
En tensorisant par ${\cal O} _X(\delta -4)$
la suite exacte $(\ast \ast )$ du lemme ci-dessus,
on obtient la suite exacte:
$$
0 \rightarrow E (\delta -4-\sigma ) \rightarrow Sym ^2(E) \otimes
{\cal O} _X(\delta -4)\rightarrow {\cal J}
_{P/X}^2(2\sigma + \delta - 4)\rightarrow 0.
$$
On est alors ramen\'es, gr\^ace \`a l'hypoth\`ese
$(iii)$, \`a prouver que $H^0(E(\delta -4-\sigma ))=0$.
Pour ceci, on tensorise par ${\cal O} _X(\delta -4-\sigma )$
la suite exacte $( \ast )$.
Les hypoth\`ese $(i)$ et $(ii)$ permettent de conclure.
\hfill $\Box$\vspace{.1in}
\subnumero{Construction via des points en position g\'en\'erale
(d'apr\`es O'Grady)}
\begin{proposition}
\label{G}
Soit $\delta \geq 14$ un entier.
Alors, pour tout entier $c_2$ avec
$$
\frac{1}{6}(\delta ^3 - 7 \delta )
< c_2 <
\frac{1}{3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3),
$$
l'espace de modules $M_X(c_2)$
contient une composante irr\'eductible de dimension plus grande
que la dimension
attendue.
\end{proposition}
{\em D\'em.}
C'est un r\'esultat d'O'Grady (cf \cite{OGrady} Proposition 3.33)
appliqu\'e ici aux
hypersurfaces de ${\bf P} ^3$. La condition $\delta \geq 14$
assure l' existence de tels entiers. La composante irr\'eductible obtenue ici
est celle qui contient les extensions de ${\cal J} _{P/X}(\sigma )$ par
${\cal O} _X(-\sigma )$
en prenant $ \sigma = 1$ et $P$ en position g\'en\'erale.
\hfill $\Box$\vspace{.1in}
\numero{Construction de fibr\'es via certaines courbes de ${\bf P} ^3$}
Soit $C$ une courbe localement intersection compl\`ete dans ${\bf P} ^3$.
Rappelons les notations habituelles suivantes:
$$
s(C):= {\rm max} \{ s' \in {\bf Z} \; ; \;\; \forall s'' < s',\;\;
H^0 ({\bf P} ^3, {\cal J} _{C/{\bf P}
^3}(s'') =0\} , $$
et
$$
e(C) := {\rm max} \{ e' \in {\bf Z} \; ; \;\; H^1 (C, {\cal O} _C(e') \neq 0 \} .
$$
Posons de plus:
$$
t(C):= {\rm max} \{ t' \in {\bf Z} \; ; \;\; \forall t'' < t',\;\;
H^1 ({\bf P} ^3, {\cal J} _{C/{\bf P}
^3}(t'') =0\}
$$
\begin{remarque}
\label{H}
Pour tout $e' \leq e(C)$ on a
$H^1 (C, {\cal O} _C(e')) \neq 0$. Notons aussi que
$t(C)$ est toujours au moins \'egal \`a $0$. (En fait, les
courbes que l'on va utiliser auront $t(C)= \infty$.)
\end{remarque}
\begin{lemme}
\label{I}
Soit $C\subset {\bf P} ^3$ une courbe irr\'eductible lisse telle que
l'intersection $P:= C\cap X$ soit transverse. Si
$$
H^1({\bf P} ^3, {\cal J} _{C/{\bf P} ^3}(n-\delta ))=
H^0({\bf P} ^3, {\cal J} _{C/{\bf P} ^3}^2(n))=
H^0(C, N^{\ast}_{C/{\bf P} ^3}(n-\delta ))=0
$$
alors
$$
H^0(X, {\cal J} _{P/X}^2(n))=0.
$$
\end{lemme}
{\em D\'emonstration}
Soit $f\in H^0(X, {\cal J} _{P/X}^2(n))$ consid\'er\'ee comme
section de ${\cal O} _X(n)$.
La suite exacte \vspace*{.5cm}
\newline
$(\ast \ast \ast )$
\vspace*{-1cm}
\newline
$$
0 \rightarrow {\cal O} _{{\bf P} ^3}(n-\delta ) \rightarrow {\cal O} _{{\bf P} ^3}(n)
\rightarrow {\cal O} _X(n) \rightarrow 0
$$
et le
fait que $H^1({\bf P} ^3, {\cal O} _{{\bf P} ^3}(n-\delta ))=0$ impliquent
que $f$ s'\'etend en une section $g\in H^0({\bf P} ^3,
{\cal O} _{{\bf P} ^3}(n))$.
\newline
La restriction $g|_C$ est une section de ${\cal O} _C(n )$ qui s'annule sur $P$
donc c'est une section de ${\cal O} _C(n-\delta )$
(cf. la suite exacte $
0 \rightarrow {\cal O} _{C}(n-\delta ) \rightarrow {\cal O} _{C}(n)
\rightarrow {\cal O} _P(n) \rightarrow 0
$
obtenue en restreignant $(\ast \ast \ast)$ \`a $C$).
\newline
Or la suite exacte
$$
0 \rightarrow {\cal J} _{C/ {\bf P} ^3}(n-\delta ) \rightarrow {\cal O} _{{\bf P} ^3}(n-\delta )
\rightarrow {\cal O} _{C}(n-\delta ) \rightarrow 0
$$
et l'hypoth\`ese
$H^1({\bf P} ^3, {\cal J} _{C/{\bf P} ^3}(n-\delta ))=0$
assurent l'existence d'une section
\newline
$h\in H^0({\bf P} ^3,
{\cal O} _{{\bf P} ^3}(n-\delta )) $ telle que
$h|_C = g|_C$ (consid\'er\'ees comme
section de ${\cal O} _{C}(n-\delta )$).
\newline
Donc, quitte \`a remplacer $g$ par $g - h'$
o\`u $h'$ est l'image de $h$ par l'injection
\linebreak
$
0 \rightarrow {\cal O} _{{\bf P} ^3}(n-\delta ) \rightarrow {\cal O} _{{\bf P} ^3}(n)$,
on peut supposer que
$g|_C=0$ i.e. $g \in H^0(X, {\cal J} _{C/{\bf P} ^3}(n))$
(tout en gardant l'hypoth\`ese $g|_X=f$ i.e. $g|_X$ s'annule
deux fois le long de $P = C \cap X$).
\newline
Soit $g'$ la d\'eriv\'e normale de $g$ le long
de $C$ c'est-\`a-dire l'image de $g$ dans \linebreak
$H^0(X, {\cal J} _{C/{\bf P} ^3}(n) / {\cal J} _{C/{\bf P} ^3}^2(n)) =
H^0(X, N^{\ast}_{C/{\bf P} ^3}(n))$.
Comme $C$ et $X$ se coupent transversalement on a:
$$
{\cal J} _{P/X}/{\cal J} ^2_{P/X} = ({\cal J} _{C/{\bf P} ^3}/{\cal J} ^2_{C/{\bf P} ^3} )
\otimes _{{\cal O} _{{\bf P} ^3}}
{\cal O} _X
= ({\cal J} _{C/{\bf P} ^3}/{\cal J} ^2_{C/{\bf P} ^3} )\otimes _{{\cal O} _C}
{\cal O} _P = N^{\ast}_{C/{\bf P} ^3}|_P.
$$
\newline
Donc l'hypoth\`ese $g|_X=f \in H^0(X, {\cal J} _{P/X}^2(n))$ donne
$g'|_P = 0$.
\newline
Par suite,
$g' \in H^0(C, N^{\ast}_{C/{\bf P} ^3}(n-\delta ))$.
\newline
Or on a suppos\'e $ H^0(C, N^{\ast}_{C/{\bf P} ^3}(n-\delta ))=0$.
Donc $g'=0$, d'ou
$g\in H^0({\bf P} ^3, {\cal J} _{C/{\bf P} ^3}^2(n))$.
\newline
On a aussi suppos\'e $H^0({\bf P} ^3, {\cal J} _{C/{\bf P} ^3}^2(n))=0$
donc $g=0$ d'o\`u $f=0$.
\hfill $\Box$\vspace{.1in}
\begin{proposition}
\label{J}
Soit $\sigma >0$ et $C\subset {\bf P} ^3$ une courbe irr\'eductible
lisse de degr\'e
$d$. Quitte \`a translater $C$ par une hom\'eographie
g\'en\'erale de ${\bf P} ^3$, si on pose $P:= C\cap X$et si
\newline
a)\, $2\sigma - 4 \leq e(C)$;
\newline
b)\, $\sigma < s(C)$ et $\sigma - \delta < t(C)$;
\newline
c)\, $\delta - 4 < 2 \sigma $;
\newline
d)\, $\delta - 4 < s(C) $;
\newline
e)\, $2\sigma - 4 \leq t(C)$;
\newline
f)\, $H^0({\bf P} ^3, {\cal J}^2_{C/{\bf P} ^3}(2\sigma + \delta - 4))=0$; et
\newline
g)\, $H^0(C, N^{\ast}_{C/{\bf P} ^3}(2\sigma - 4))=0$;
\newline
alors il existe un fibr\'e vectoriel $E$ extension de
${\cal J} _{P/X}(\sigma )$ par
${\cal O} _X(-\sigma )$. De plus $E$ est stable, bon et a pour classes de Chern
$c_1(E)=0$ et $c_2(E)= \delta (d-\sigma ^2)$.
\end{proposition}
{\em D\'emonstration}
L'existence du fibr\'e $E$ et le calcul de ses
classes de Chern se d\'eduisent de la proposition \ref{A},
gr\^ace \`a l'hypoth\`ese $(a)$. La stabilit\'e est cons\'equence
du corollaire \ref{D} via l'hypoth\`ese $(b)$.
Les hypoth\`eses $(e)$, $(f)$ et $(g)$ permettent d'utiliser le
lemme \ref{I} pour en d\'eduire
$H^0(X, {\cal J} _{P/X}^2(2\sigma + \delta - 4))=0$,
puis la proposition \ref{F} pour en d\'eduire (gr\^ace \`a $(c)$ et $(d)$)
que $E$ est {\em bon}.
\hfill $\Box$\vspace{.1in}
\bigskip
Pour obtenir, pour de nombreux entiers $c_2$, une {\em bonne} composante
irr\'eductible de $M_X(c_2)$ de dimension attendue, il reste \`a
d\'eterminer des courbes
$C$ qui nous donneront une grande famille d'entiers $\sigma$
v\'erifiant les conditions de la proposition 2.3.
\begin{proposition}
\label{K}
Pour tout entier $s \geq 1$, il existe
une courbe irr\'eductible lisse $C$ de ${\bf P}^3$ avec
$s(C)= s$, $t(C)=\infty$ et $e(C) = s-3$
telle que
$H^0(C, N^{\ast}_{C/{\bf P} ^3}(\tau ))=0$ pour $\tau < s$ et
$H^0({\bf P}^3, {\cal J} _{C/{\bf P} ^3}^2(\tau ))=0$ pour $\tau < 2s$.
\end{proposition}
\noindent
{\em D\'emonstration:}
Les {\em courbes d\'eterminentielles} sont les courbes $C$ dont la
r\'esolution
de l'id\'eal est de la forme
$$
0\rightarrow {\cal O} _{{\bf P} ^3}(-s-1) ^s \rightarrow {\cal O} _{{\bf P} ^3}(-s)^{s+1}
\rightarrow {\cal J} _{C/{\bf P} ^3} \rightarrow 0
$$
pour un entier $s$.
Floystad (cf \cite{Floystad}) et, ind\'ependamment,
Walter (cf \cite{Walter}) ont
d\'emontr\'e que
de telles courbes lisses existent pour tout $s\geq 1$. Il est facile de voir
que $s(C)= s$, $t(C)=\infty$ et $e(C) = s-3$ (cf
\cite{Floystad} ou \cite{Walter}). Rao
montre ( cf \cite{Rao} 1.12) d'une part que
$H^0(C, N^{\ast}_{C/{\bf P} ^3}(\tau ))=0$ pour $\tau < s$, et, d'autre part,
que l'id\'eal ${\cal J} ^2_{C/{\bf P} ^3}$ a la r\'esolution
$$
0\rightarrow {\cal O} _{{\bf P} ^3}(-2s-2) ^a \rightarrow
{\cal O} _{{\bf P} ^3}(-2s-1) ^b
\rightarrow {\cal O} _{{\bf P} ^3}(-2s)^c \rightarrow {\cal J} ^2_{C/{\bf P} ^3} \rightarrow 0.
$$
En utilisant le
fait que $H^i({\bf P} ^3, {\cal O} _{{\bf P} ^3}(m))=0$ pour $i=1,2$,
on en d\'eduit facilement
que $H^0({\bf P}^3, {\cal J} _{C/{\bf P} ^3}^2(\tau ))=0$ pour $\tau < 2s$.
\hfill $\Box$\vspace{.1in}
\begin{remarque}
\label{R20}{\bf :}
\end{remarque}
Dans une premi\`ere version de cet article, nous avons donn\'e un argument
g\'eom\'etrique pour prouver les propri\'et\'es
$H^0(C, N^{\ast}_{C/{\bf P} ^3}(\tau ))=0$ pour $\tau < s$ et
$H^0({\bf P}^3, {\cal J} _{C/{\bf P} ^3}^2(\tau ))=0$ pour $\tau < 2s$. Cet argument
\'etait bas\'e sur une courbe d\'eterminentielle sp\'eciale
singuli\`ere qui est une r\'eunion de droites (``stick figure'')
d\'ej\`a utilis\'ee par Fl{\o}ystad et Walter, avec un argument
combinatoire pour
les annulations. Walter m'a ensuite fait remarquer que ces
propri\'et\'es ont
\'et\'e prouv\'ees par Rao.
\smallskip
\begin{corollaire}
\label{L}
Soit $X$ une hypersurface de degr\'e $\delta \geq 4$ dans ${\bf P}^3$.
Pour tout entier $c$ avec $c\geq \delta ^3/4 - \delta ^2/2 $
il existe un bon fibr\'e stable sur $X$
avec $c_1=0$ et $c_2=c$.
(Si $\delta$ est impair le r\'esultat reste vrai pour
$c\geq \delta ^3/4 - \delta ^2 + \frac{3}{4}\delta $.)
\end{corollaire}
\noindent
{\em D\'emonstration}
D'apr\`es la proposition \ref{E}, il suffit de construire un bon
fibr\'e stable
avec $c_1=0$ et $c_2 = \frac{1}{4}\delta ^2 (\delta - 2)$ si
$\delta$ est pair
(resp. $c_2 = \frac{1}{4}\delta (\delta -1)(\delta -3)$ si
$\delta$ est impair).
Pour cela posons
$
s= \delta -3 $ si $\delta $ est impair et $ s = \delta -2 $
si $\delta $ est pair.
Soit $\sigma = s/2$.
Soit $C$ une courbe comme dans la proposition \ref{K} On v\'erifie
facilement que les conditions de la proposition \ref{J} sont
satisfaites et on obtient
un bon fibr\'e stable $E$ ayant pour classes de Chern $c_1(E)=0$ et
$$
c_2(E)= (s(s+1)/2 - \sigma ^2)\delta
$$
$$
= (\sigma (2\sigma + 1) - \sigma ^2)\delta
$$
$$
= \delta \sigma (\sigma +1).
$$
En particulier, si $\delta$ est pair on a
$$
c_2(E)= \frac{1}{4}\delta ^2 (\delta - 2),
$$
et si $\delta $ est impair on a
$$
c_2(E) = \frac{1}{4}\delta (\delta -1)(\delta -3).
$$
\hfill $\Box$\vspace{.1in}
\begin{corollaire}
\label{M}
Pour tout entier $c_2$
dans l'intervalle non vide suivant
$$
\frac{1}{4}(\delta ^3-2\delta ^2)
\leq c_2 <
\frac{1}{3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3)
\;\;\; \mbox {si} \; \; \delta \geq 28 \;\; \mbox{est un entier pair}
$$
ou
$$
\frac{1}{4}(\delta ^3-4\delta ^2 + 3\delta )
\leq c_2 <
\frac{1}{3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3)
\;\;\; \mbox {si} \;\; \delta \geq 21 \;\; \mbox{est un entier impair}
$$
l'espace des modules $M_X(c_2)$
contient une composante irr\'eductible
de dimension \'egale \`a la dimension
attendue et une autre
de dimension plus grande que la dimension
attendue.
\end{corollaire}
{\em D\'emonstration} C'est une cons\'equence directe
de la proposition \ref{G} et du corollaire \ref{L}.
La condition $\delta \geq 28$ (resp. $\delta \geq 21$)
assure l'existence d'entiers compris entre
$$
\frac{1}{4}(\delta ^3-2\delta ^2)
\;\;\; \mbox{et} \;\;\;
\frac{1}{3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3)
$$
resp.
$$
\frac{1}{4}(\delta ^3-4\delta ^2 + 3\delta )
\;\;\; \mbox{et} \;\;\;
\frac{1}{3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3).
$$
\hfill $\Box$\vspace{.1in}
\begin{remarque}
\label{R1}
{\bf :}
\end{remarque}
Nous avons utilis\'e la construction d'O'Grady dans le cas
o\`u elle donne des fibr\'es
{\em stables}. On peut de m\^eme construire des extensions
de ${\cal J} _{P/X}$ par ${\cal O} _X$
(donc $H=0$ suivant les notations de \cite{OGrady}). On obtient
alors une famille de
fibr\'es {\em semi-stables} de dimension plus grande que la
dimension attendue quand
$$ \frac{1}{6}(\delta ^3 - 6 \delta ^2 + 11 \delta ) < c_2 <
\frac{1}{3}(\delta ^3 - 6 \delta ^2 + 11 \delta -3).
$$
D'o\`u l'existence de deux composantes dans l'espace
des modules des faisceaux
sans torsion, l'une bonne et contenant
des fibr\'es stables, l'autre de
dimension plus grande que la dimension attendue contenant
des fibr\'es semistables (mais on
ne sait pas si elle contient des fibr\'es stables), pour
$$
\frac{\delta ^3}{4} - \frac{\delta ^2}{2} \leq c_2 <
\frac{1}{3}(\delta ^3 - 6 \delta ^2 + 11 \delta -3)
\;\;\;\;\; \mbox{si} \;\; \delta \;\; \mbox{est pair}
$$
et
$$
\frac{\delta ^3}{4} - \delta ^2 +\frac{3}{4}\delta \leq c_2 <
\frac{1}{3}(\delta ^3 - 6 \delta ^2 + 11 \delta -3)
\;\;\;\;\; \mbox{si} \;\; \delta \;\; \mbox{est impair}.
$$
Ces intervalles sont non-vides pour $\delta \geq 16$
si $\delta $ est pair et pour $\delta \geq 9$
si $\delta$ est impair.
\begin{remarque}
\label{R2}{\bf :}
\end{remarque}
Pour simplifier la r\'edaction nous n'avons trait\'e que le cas
$c_1 = 0$ (c'est-\`a-dire, quitte \`a tensoriser par un fibr\'e en droites
ad\'equat, le cas $c_1$ pair). La d\'emonstration s'adapte au cas
$c_1 = 1$ (c'est-\`a-dire au cas $c_1$ impair) et on obtient
l'existence de deux composantes irr\'eductibles de
$M_X(1,c_2)$, l'une de dimension plus grande
que la dimension attendue et l'autre bonne (toutes deux
contenant des fibr\'es
stables), pour
$$
\frac{1}{4}\delta (\delta - 1)^2 \leq c_2 <
\frac{1}{6}(2\delta ^3 - 15 \delta ^2 + 37 \delta -6)
\;\;\;\;\; \mbox{si} \;\; \delta \;\; \mbox{est impair}
$$
et
$$
\frac{1}{4}\delta (\delta - 2)^2 \leq c_2 <
\frac{1}{6}(2\delta ^3 - 15 \delta ^2 + 37 \delta -6)
\;\;\;\;\; \mbox{si} \;\; \delta \;\; \mbox{est pair} .
$$
Ces intervalles sont non-vides pour $\delta \geq 21$ si $\delta $ est
impair et
pour $\delta \geq 14$
si $\delta$ est pair.
\numero{Fibr\'es \`a cohomologie naturelle}
Soit $X$ une surface projective lisse telle que le fibr\'e
canonique $K_X$ soit de la forme $K_X = {\cal O} _X(k)$
pour un entier $k\in {\bf Z}$ o\`u ${\cal O} _X(1)$ est un fibr\'e ample.
Soit $M_X(c_2)$ l'espace des modules des fibr\'es stables
(par rapport \`a ${\cal O} _X(1)$)
avec $c_1=0$ dans $Pic (X)$ et $c_2$ donn\'e.
\begin{definition}
{\bf :}
\end{definition}
Un fibr\'e $E$ sur $X$ a la {\em cohomologie naturelle} (cf \cite{AHRH}) si
pour tout $n$ l'un au plus des groupes
$H^i(E(n))$ est non nul
pour $i=0,1,2$.
\newline
Plus g\'en\'eralement soit $\beta \geq k/2$ un entier. On dira qu'un
fibr\'e $E$ sur $X$ a la {\em cohomologie $\beta$-naturelle} si
pour tout $n \geq \beta$ l'un au plus des groupes
$H^i(E(n))$ est non nul pour $i=0,1,2$.
\begin{remarque}
{\bf :}
\end{remarque}
Pour tout fibr\'e $E$ avec $c_1(E) = 0$ dans $Pic(X)$, la dualit\'e de Serre
et la condition $K_X = {\cal O} _X(k)$, donnent l'\'egalit\'e
$h^2(X,E(n)) = h^0(X,E(k -n))$ et $h^1(X,E(n)) = h^1(X,E(k -n))$.
Il suffit donc d'\'etudier la cohomologie des $E(n)$
pour les entiers $n \geq \frac{k}{2}$.
Si de plus $E$ a la {\em cohomologie naturelle} alors
pour $n \geq \frac{k}{2}$ on a $ h^2(E(n)) = 0$ et la cohomologie
des $E(n)$ se
d\'eduit de la caract\'eristique d'Euler $\begin{array}{c} \rule[-.12in]{0in}{.2in (E(n)) =
2\begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X (n)) -c_2$.
Enfin on peut
remarquer que si $k$ est pair et $\begin{array}{c} \rule[-.12in]{0in}{.2in (E(\frac{k}{2}))>0$ alors $E$
ne peut pas
avoir la cohomologie naturelle.
Nous \'etudions maintenant le comportement de la fonction de Hilbert de
$E$ quand on
applique la construction de proposition \ref{E} pour passer de $c_2$ \`a
$c_2+1$.
\smallskip
\begin{notations}
{\bf :}
\end{notations}
Soit $c_2$ un entier tel qu'il existe un bon fibr\'e
vectoriel stable $E'$ sur $X$ avec $c_1(E')=0$ et $c_2(E')=c_2$.
Soit ${\cal G}$ le faisceau gratte-ciel de rang $1$ au-dessus d'un point
$x\in X$. Soit $E_0$ le
noyau d'une surjection g\'en\'erale $E' \rightarrow {\cal G}$. Soit $E$
un fibr\'e stable et
bon qui est une d\'eformation g\'en\'erale de $E_0$.
\smallskip
\begin{proposition}
\label{T}
Soit $\beta \geq k/2$ un entier.
Si $E'$ a la {\em cohomologie $\beta$-naturelle}, alors $E$ aussi.
\end{proposition}
{\em D\'em.} Soit $n \geq \beta \geq k/2$ un entier.
Notons tout d'abord l'\'egalit\'e $H^2(E'(n)) = 0$. En effet,
si $H^2(E'(n)) \neq 0$, alors, par dualit\'e de Serre,
$H^0(E'(k-n)) \neq 0$ donc,
$\forall m \geq k-n \; , \; H^0(E'(m)) \neq 0$.
En particulier $H^0(E'(n)) \neq 0$
et ceci est impossible si $E'$ a la cohomologie $\beta$-naturelle.
De la suite exacte
$$
0 \rightarrow E_0 \rightarrow E' \rightarrow {\cal G} \rightarrow 0
$$
et par semi continuit\'e, on en d\'eduit $H^2(E(n)) = 0$.
\newline
De m\^eme, si $H^0(E'(n)) = 0$ alors $H^0(E(n)) = 0$.
\newline
Enfin, si $H^0(E'(n)) \neq 0$, alors d'une part $H^1(E'(n)) = 0$
et, d'autre part, pour une surjection g\'en\'erale le morphisme
$$
H^0(E'(n)) \rightarrow H^0({\cal G} (n))= {\bf C}
$$
est surjectif. Donc, $H^1(E(n)) = 0$.
\hfill $\Box$\vspace{.1in}
\begin{notations}
{\bf :}
\end{notations}
On fixe maintenant un entier $c'_2$ tel qu'il existe
un bon fibr\'e vectoriel stable $E'$ sur $X$ avec $c_1(E')=0$ et
$c_2(E')= c'_2$. (C'est toujours possible, il suffit d'avoir
$c'_2$ assez grand.)
\newline
Soit $\beta $ un entier $\geq \frac{k}{2}$ tel que $E'$
a la cohomologie $\beta$-naturelle. (Un tel $\beta $
existe car $H^1(E'(n))=H^2(E'(n))=0$ pour $n$ assez grand
d'apr\'es les th\'eor\`emes d'annulations de Serre.)
\newline
Pour tout entier $c_2\geq c'_2$ d\'esignons par $M_X'(c_2)$ la composante
irr\'eductible de $M_X(c_2)$ obtenue \`a partir de $E'$ via la construction
d'augmentation de $c_2$ (it\'er\'ee $c_2-c'_2$ fois) de la proposition
\ref{E}.
\begin{proposition}
\label{U}
Soit $E \in M_X'(c_2)$ un fibr\'e ayant
la cohomologie $\beta$-naturelle.
Si $\begin{array}{c} \rule[-.12in]{0in}{.2in (E(\beta ))<0$, alors $E$ a la cohomologie naturelle.
\end{proposition}
{\em D\'em.} Soit $n$ un entier. On doit
\'etudier la cohomologie de $E(n)$.
Par dualit\'e de Serre et puisque $E$ a
la cohomologie $\beta$-naturelle, on peut supposer
$k/2 \leq n < \beta$.
\newline
L'hypoth\`ese
$\begin{array}{c} \rule[-.12in]{0in}{.2in (E(\beta )) < 0 $ donne $H^1 (E(\beta )) \neq 0 $
et donc, $H^0 (E(\beta )) = 0 $. On en d\'eduit:
$$
\forall m \leq \beta \; , \; H^0 (E(m)) = 0.
$$
D'o\`u
$H^0 (E(n)) = 0$ et, par le m\^eme argument que pr\'ec\'edemment,
$H^2 (E(n)) = 0$.
\hfill $\Box$\vspace{.1in}
\smallskip
\begin{corollaire}
\label{V}
On pose $\gamma := 2 \begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X(\beta ))$.
Si $c_2 > \gamma $ alors le fibr\'e g\'en\'eral $E$ de la composante
$M'_X(c_2)$ a la cohomologie naturelle.
En particulier quand $c_2\gg 0$ le fibr\'e g\'en\'eral de l'unique
composante de $M_X(c_2)$ a la cohomologie naturelle.
\end{corollaire}
{\em D\'em.}
D'apr\`es le choix de $\beta$ et la proposition \ref{T},
le fibr\'e g\'en\'eral de
$M'_X(c_2)$ a la cohomologie $\beta$-naturelle. L'hypoth\`ese $c_2(E)>
\gamma $
donne $\begin{array}{c} \rule[-.12in]{0in}{.2in (E(\beta ))= 2\begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X(\beta ))-c_2 < 0$, donc $E$ a la
cohomologie naturelle d'apr\`es la proposition \ref{U}.
\hfill $\Box$\vspace{.1in}
\bigskip
Revenons au cas o\`u $X$ est une hypersurface lisse de degr\'e
$\delta$ dans ${\bf P}^3$, avec $Pic (X)={\bf Z}$. On a alors
$K_X = {\cal O} _X(\delta -4)$ si ${\cal O} _X(1)$ est le fibr\'e hyperplan.
La composante $M'_X(c_2)$ est celle que nous avons construite.
Les propositions suivantes permettent de d\'eterminer $\gamma$.
\begin{proposition}
\label{R}
Soit $E'$ le fibr\'e vectoriel construit en $2.6$ avec,
suivant la parit\'e de $\delta$,
$c_2(E')=\delta ^3/4-\delta ^2/2$
ou $c_2(E')=\frac{\delta ^3}{4} -\delta ^2 +
\frac{3\delta }{4}$.
Soit $\beta $ le plus petit entier $> \frac{3}{2} \delta -4$.
Alors, $E'$ a la cohomologie $\beta$-naturelle.
\end{proposition}
{\em D\'em.}
Nous allons prouver que pour tout entier $n> \frac{3}{2}\delta -4$ on a
$h^1(E'(n))= h^2(E'(n))=0$.
Par construction et d'apr\`es le corollaire $1.3$, on a $h^0(E'(n))=0$ pour
$n< (\delta - 3)/2 $. Par dualit\'e, on en d\'eduit
$h^2(E'(n))=0$ pour $n> (\delta - 4)/2$.
D'autre part, on sait que pour les courbes $C$
consid\'er\'ees ici, $H^1({\cal J} _{C/{\bf P} ^3}(m))=0$ pour tout $m$.
Donc la suite exacte
$$
0\rightarrow {\cal J} _{C/{\bf P} ^3}(\sigma +n-\delta ) \rightarrow
{\cal J} _{C/{\bf P} ^3}(\sigma +n) \rightarrow
{\cal J} _{P/X}(\sigma +n) \rightarrow 0,
$$
donne l'implication:
$$
H^2({\cal J} _{C/{\bf P} ^3}(\sigma + n-\delta ))=0 \; \Rightarrow \;
H^1(E'(n))=0
$$
(en effet,
d'apr\`es la suite exacte $(\ast )$ on a
$h^1(E'(n)) \leq h^1({\cal J} _{P/X}(\sigma + n))$).
De plus, la r\'esolution de l'id\'eal homog\`ene de $C$ donne la
r\'esolution de
faisceaux
$$
0\rightarrow {\cal O} _{{\bf P} ^3}(-s-1)^s \rightarrow {\cal O} _{{\bf P} ^3}(-s)^{s+1}
\rightarrow {\cal J} _{C/{\bf P} ^3}\rightarrow 0,
$$
d'o\`u l'implication:
$$
H^3({\cal O} _{{\bf P} ^3}(\sigma + n -\delta -s-1))=0 \; \Rightarrow \; H^1(E'(n))=0,
$$
ou encore:
$$
\sigma + n -\delta -s-1 \geq -3 \; \Rightarrow \; H^1(E'(n))=0,
$$
Rappelons que $s= \delta - 2$ ou $\delta - 3$ et $\sigma = s/2$.
On en d\'eduit facilement que, pour $n> \frac{3}{2}\delta -4$,
on a $
\sigma + n -\delta -s-1 \geq -3$.
\hfill $\Box$\vspace{.1in}
\begin{theoreme}
\label{A1}
Pour tout entier $c_2 > \frac{1}{12}(13 \delta ^3 - 24
\delta ^2 + 8 \delta )$,
le fibr\'e g\'en\'eral de la bonne composante de
$M_X(c_2)$ que nous avons construite a la cohomologie naturelle.
\end{theoreme}
{\em D\'em:}
D'apr\`es la proposition \ref{R} on a $\beta \leq
\frac{3}{2}\delta - 3$.
Soit $P(t)$ le polyn\^ome tel que
\newline
$P(n) = 2\begin{array}{c} \rule[-.12in]{0in}{.2in ({\cal O} _X(n))$.
Pour $t\geq k/2$, $P(t)$ est une fonction croissante donc
$$
\gamma := P(\beta ) \leq P( \frac{3}{2}\delta - 3).
$$
On conclut en calculant:
$$
P( \frac{3}{2}\delta - 3)=
\frac{1}{12}(13 \delta ^3 - 24 \delta ^2 + 8 \delta ).
$$
\hfill $\Box$\vspace{.1in}
|
1996-01-29T06:20:16 | 9601 | alg-geom/9601025 | en | https://arxiv.org/abs/alg-geom/9601025 | [
"alg-geom",
"math.AG"
] | alg-geom/9601025 | Pawel Gajer | Pawel Gajer | Geometry of Deligne cohomology | 51 pages, uses XY-pic, author-supplied PostScript and DVI files
available at http://www.math.tamu.edu/~pawel.gajer/gdc.html . LaTeX2e | null | 10.1007/s002220050118 | null | null | It is well known that degree two Deligne cohomology groups can be identified
with groups of isomorphism classes of holomorphic line bundles with
connections. There is also a geometric description of degree three Deligne
cohomology, due to J-L. Brylinski and P. Deligne, in terms of gerbes with
connective structures and curvings. This paper gives a geometric interpretation
of Deligne cohomology of all degrees, in terms of equivalence classes of higher
line bundles with $k$-connections. It is also shown that the classical
Abel-Jacobi isomorphism $Pic^0(X) \cong J(X)$ generalizes to the isomorphism
between groups of equivalence classes of topologically trivial 1-holomorphic
higher line bundles with $k$-connections and Griffiths intermediate Jacobians.
| [
{
"version": "v1",
"created": "Fri, 26 Jan 1996 15:17:07 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Gajer",
"Pawel",
""
]
] | alg-geom | \section{Differentiable structures on $EB^sG$ and $B^{s+1}G$}
In this section we define for any abelian Lie group $G$ a
differentiable space structure on the spaces $EB^sG$ and $B^{s+1}G$
for every $s\geq 1$. For the definition and basic properties of the
geometric bar construction we refer the reader to Appendix~B.
Let $M.$ be a simplicial smooth manifold. That is, $M.$ consists of a
family $\{M_n\}_{n\in \ensuremath{\mathbb{N}}\xspace}$ of smooth manifolds, together with smooth
maps
\[\partial_i:M_n\longrightarrow M_{n-1},\quad s_i:M_n\longrightarrow M_{n+1},\]
where $i=0,1, \dots , n$, satisfying the identities
\begin{gather*}
\partial_i\partial_j = \partial_{j-1}\partial_i \quad\text{for}\quad i<j,\\ s_is_j =
s_{j+1}s_i \quad\text{for}\quad i\leq j,\\ \partial_is_j =
\begin{cases}
s_{j-1}\partial_i \quad\text{for}\quad i<j,\\ \text{id}|_{M_n}
\quad\text{for}\quad i=j,j+1\\ s_j\partial_{i-1} \quad\text{for}\quad
i>j+1
\end{cases}
\end{gather*}
The geometric realization $|M.|$ of $M.$ is the quotient space of the
disjoint union
\[\coprod_{n\geq 0} \ensuremath{\Delta}\xspace^n\times M_n \]
with respect to the equivalence relation $\sim$ generated by the
relations
\begin{gather*}
(\partial^ix,m) \sim (x,\partial_im) \quad \text{for}\quad (x,m)\in
\ensuremath{\Delta}\xspace^{n-1}\times M_{n},\\ (s^ix,m) \sim (x,s_im) \quad \text{for}\quad
(x,m)\in \ensuremath{\Delta}\xspace^{n+1}\times M_{n},
\end{gather*}
where the maps $\partial^i : \ensuremath{\Delta}\xspace^{n-1} \rightarrow \ensuremath{\Delta}\xspace^n$ and $s^i: \ensuremath{\Delta}\xspace^{n+1} \rightarrow
\ensuremath{\Delta}\xspace^n$ are defined in the baricentric coordinates by
\begin{align*}
\partial^i(\seq[n-1]{x}) &= (\seq[i-1]{x},0,x_{i},\, \ldots \,
,x_{n-1}),\\ s^i(\seq[n+1]{x}) &=
(\seq[i-1]{x},x_i+x_{i+1},x_{i+2},\, \ldots \, ,x_{n+1}).
\end{align*}
A {\em differentiable space structure} on the geometric realization
$|M.|$ of $M.$ consists of the class of all smooth \ensuremath{\mathbb{R}}\xspace-valued functions
on $|M.|$. We say that a function $f:|M.|\rightarrow \ensuremath{\mathbb{R}}\xspace$ is {\em smooth} if
the composition
\[\begin{CD}
\coprod\limits_{n \geq 0} \Delta^n \times M_n @>q>>|M.| @>f>> \ensuremath{\mathbb{R}}\xspace
\end{CD}\]
is smooth\footnote{A map $g:\Delta^n \times M_n\rightarrow \ensuremath{\mathbb{R}}\xspace$ is smooth at
$x\in \partial\Delta^n \times M_n$ if there is an open neighborhood of
$x$ in $H^n\times M_n$, where $H^n=\{(\seq{x})\in\ensuremath{\mathbb{R}}\xspace^{n+1}\, | \,
\sum x_i=1\}$, and a smooth function $\tilde{g}:U\rightarrow \ensuremath{\mathbb{R}}\xspace$ which
restricted to $U\cap (\Delta^n \times M_n)$ coincides with $g$.},
where $q$ is the quotient map. Equivalently, a smooth \ensuremath{\mathbb{R}}\xspace-valued
function on $|M.|$ is given by a family of smooth maps $f^n :\Delta^n
\times M_n \rightarrow \ensuremath{\mathbb{R}}\xspace$ such that for every $n\geq 0$ the following two
diagrams commute
\[{\diagram
{\ensuremath{\Delta}\xspace^n\times M_n} \rto^-{f^n}& {\ensuremath{\mathbb{R}}\xspace} && {\ensuremath{\Delta}\xspace^n\times M_n}
\rto^-{f^n}& {\ensuremath{\mathbb{R}}\xspace} \\ {\ensuremath{\Delta}\xspace^{n-1}\times M_n} \uto^{\partial^i\times
id}\rto_-{id\times\partial_i}& {\ensuremath{\Delta}\xspace^{n-1}\times M_{n-1}}
\uto_{f^{n-1}}&& {\ensuremath{\Delta}\xspace^{n+1}\times M_n} \uto^{s^i\times
id}\rto_-{id\times s_i}& {\ensuremath{\Delta}\xspace^{n+1}\times M_{n+1}} \uto_{f^{n+1}}&
\enddiagram}\]
Let $M$ and $N$ are differentiable spaces (that is $M$ and $N$ are
spaces equipped with the appropriately defined classes of smooth
functions). Then a map $f:M\rightarrow N$ is called a {\em smooth map} if for
every smooth function $g:N\rightarrow \ensuremath{\mathbb{R}}\xspace$ the composition $g\circ f:M\rightarrow \ensuremath{\mathbb{R}}\xspace$
is a smooth map.
It is easy to see that if $f. : M.\rightarrow N.$ is a simplicial smooth map
between simplicial smooth manifolds $M.$ and $N.$, then $f.$ induces a
smooth map $|f.|: |M.|\rightarrow |N.|$ between the geometric realizations of
$M.$ and $N.$ respectively.
\begin{Exa}\label{e1.2}
With every Lie group $G$ there are associate simplicial smooth
manifolds $G.$, $EG.$, and $BG.$ with simplicial smooth maps
\[ G. \longrightarrow EG. \longrightarrow BG. \]
whose geometric realizations give a universal principal $G$-bundle
\[ G \longrightarrow EG \longrightarrow BG \]
Thus, the inclusion $G \longrightarrow EG$ and the projection $EG \longrightarrow BG$ are
smooth maps.
\end{Exa}
{\Le \label{BV} Let $V$ be a vector space over a field $k$. Then $EV$
and $BV$, taken with respect to the additive group structure of $V$,
are $k$-vector spaces with respect to the following multiplication
by scalars
$$\begin{array}{ll} k\times EV \rightarrow EV,& \hspace{.5cm} c \cdot \bigl\lvert
t_1, \cdots, t_n, v_0[v_1|\, \cdots \, |v_n]\bigr\rvert = \bigl\lvert t_1, \cdots,
t_n, cv_0[cv_1|\, \cdots \, |cv_n]\bigr\rvert \\ k\times BV \rightarrow BV,&
\hspace{.5cm} c \cdot \bigl\lvert t_1, \cdots, t_n, [v_1|\, \cdots \,
|v_n]\bigr\rvert = \bigl\lvert t_1, \cdots, t_n, [cv_1|\, \cdots \, |cv_n]\bigr\rvert
\end{array}$$
Moreover, the projection $EV \rightarrow BV$ is a linear map. }
The proof of Lemma~\ref{BV} is an easy exercise which we leave for the
reader.
\begin{Exa}\label{e1.0}
Let $V$ be a separable \ensuremath{\mathbb{C}}\xspace-vector space. It is easy to see that the
homomorphism
\[l:EV \longrightarrow V, \quad\quad l(|\seq{x}, \seq{v} |) =
\sum\limits_{i=0}^n x_iv_i\] is a splitting of the short exact sequence\xspace
\[0\longrightarrow V \longrightarrow EV \longrightarrow BV \longrightarrow 0\]
We will show that $l:EV \longrightarrow V$ is a smooth map.
Let $\seq{e},\ldots$ be a base of $V$ and let $\pi_n:V\rightarrow \ensuremath{\mathbb{C}}\xspace$ be the
projection on the subspace span by $e_n$. To prove smoothness of $l:EV
\longrightarrow V$ it is enough to show that for every $k\geq 0$ the composition
\[\begin{CD}
EV @>l>> V @>\pi_k>> \ensuremath{\mathbb{C}}\xspace
\end{CD} \]
is smooth. But
\[ \pi_k(l(|\seq{x}, \seq{v} |)) =
\pi_k\bigl(\sum\limits_{i=0}^n x_iv_i\bigr) =
\bigl<\sum\limits_{i=0}^n x_iv_i, e_k \bigr> = \sum\limits_{i=0}^n
x_i<v_i,e_k>
\]
is a smooth map on $EV$. Hence $l:EV \longrightarrow V$ is smooth.
\end{Exa}
Let $G$ be an abelian Lie group. A differentiable space structure on
$EB^sG$ and $B^{s+1}G$, for $s\geq 1$, is defined by the following
inductive procedure.
Suppose we have a notion of a smooth function on $B^{s}G$ as well as
on each product $\ensuremath{\Delta}\xspace^k\times (B^{s}G)^m$ for $k,m\geq 0$. Then
$f:EB^{s}G\rightarrow \ensuremath{\mathbb{R}}\xspace$ is smooth if the composition
\[\begin{CD}
\coprod\limits_{n \geq 0} \Delta^n \times (B^{s}G)^{n+1}@>q_E>> EB^{s}G
@>f>> \ensuremath{\mathbb{R}}\xspace
\end{CD}\]
is smooth and $f:B^{s+1}G\rightarrow \ensuremath{\mathbb{R}}\xspace$ is smooth if the composition
\[\begin{CD}
\coprod\limits_{n \geq 0} \Delta^n \times (B^{s}G)^{n}@>q_B>> B^{s+1}G
@>f>> \ensuremath{\mathbb{R}}\xspace
\end{CD}\]
is smooth. A function $f:\ensuremath{\Delta}\xspace^k\times (B^{s+1}G)^m\rightarrow \ensuremath{\mathbb{R}}\xspace$ is smooth if
the composition
\[\begin{CD}
\ensuremath{\Delta}\xspace^k\times(\coprod\limits_{n \geq 0} \Delta^n \times (B^{s}G)^{n})^m
@>\text{id}\times (q_B)^m>> \ensuremath{\Delta}\xspace^k\times (B^{s+1}G)^m @>f>> \ensuremath{\mathbb{R}}\xspace
\end{CD}\]
is smooth.
Directly from the above definition of differentiable structures on
$B^{s+1}G$ and $EB^sG$ it follows that all maps in the short exact sequence\xspace
\[0\longrightarrow B^sG \longrightarrow EB^sG \longrightarrow B^{s+1}G\longrightarrow 0 \]
are smooth. It is also not difficult to see that the map
\[B^{s}G\times B^{s}G \longrightarrow B^{s}G,\quad (g,h)\mapsto gh^{-1} \]
is smooth.
A group $G$ carring a differentiable space structure so that the map
\[G\times G\rightarrow G,\quad (g,h)\mapsto gh^{-1}\]
is smooth is called a {\em differentiable group}.
\begin{Exa}\label{e1.4}
\label{p6} For every differentiable group $G$, there is a smooth deformational
retraction $r: EG \times I \longrightarrow EG$ of $EG$ to $e \in EG$ which is a
minor modification of the standard contraction from \cite{milg-bar}.
In particular, if $G=B^s\ensuremath{\mathbb{C}^\ast}\xspace$, then for every $s\geq 1$ there is a
smooth deformational retraction $r: EB^s\ensuremath{\mathbb{C}^\ast}\xspace \times I \longrightarrow EB^s\ensuremath{\mathbb{C}^\ast}\xspace$.
The map $r$ is represented by the family of maps
\[r_n: (EG)_n \times I \longrightarrow (EG)_{n+1},
\]
where
\[ (EG)_n = q_E\bigl(\coprod\limits_{i\leq n} \Delta^i\times
G^{i+1}\bigr)\subset EG,
\]
and $r_n$ is defined by the formula
\[ r_n(|t_1,\dots, t_n, h_0[h_1|\dots |h_n]|, t) =
\bigl\lvert \Phi(0,t), \Phi(t_1,t), \dots, \Phi(t_n,t), [h_0|h_1|\dots
|h_n]\bigr\rvert,
\]
\noindent where $\Phi : [0,1]^2 \rightarrow [0,1]$ is the composition
$$\Phi(x,t) = \phi(\min(1,x+t))$$ \noindent with $\phi : [0,1]\rightarrow [0,1]$
being a smooth nondecreasing function so that $\phi(0) = 0$ and
$\phi(1) = 1$.
The contraction $r: EG \times I \longrightarrow EG$ is a smooth map, because for
every smooth function $g: EG \rightarrow \ensuremath{\mathbb{R}}\xspace$ the diagram
$${\diagram {(\Delta^n\times G^{n+1})\times I}
\rrto^-{\tilde{r}_n}\dto_{q_n \times id}& & {\Delta^{n+1}\times
G^{n+2}} \dto^{q_E} \drto^{g^{n+1}} & \\ {EG \times I} \rrto_r& &
{EG} \rto_g& {\ensuremath{\mathbb{R}}\xspace} \enddiagram}$$ \noindent commutes, where
$$\tilde{r}_n:(\Delta^n\times G^{n+1})\times I \longrightarrow \Delta^{n+1}\times
G^{n+2}$$ \noindent is a smooth map defined by the formula
$$\tilde{r}_n(t_1,\dots, t_n, h_0, h_1, \dots, h_n, t) =
\bigl(\Phi(0,t), \Phi(t_1,t), \dots, \Phi(t_n,t), e, h_0, h_1, \dots,
h_n\bigr).$$
\end{Exa}
\begin{Le}
Suppose $M$ is a smooth manifold and $G$ is a differentiable group.
If $f:M\rightarrow BG$ is a map so that for every $x\in M$ there is an open
neighborhood $U$ of $x$ in $M$ so that $f$ restricted to $U$ is of
the form
\[f=\bigl\lvert f_0, f_1, \ldots ,f_n, [g_1|\ldots |g_n]\bigr\rvert \]
with $f_0, f_1, \ldots ,f_n, g_1, \ldots ,g_n$ being smooth maps, then
$f$ is a smooth map.
\end{Le}
\noindent{\bf Proof. \ } Suppose $g: BG\rightarrow \ensuremath{\mathbb{R}}\xspace$ is a smooth map. Thus, for every $n\geq
1$ the composition
\[ \ensuremath{\Delta}\xspace^n\times G^n \overset{q_B}{\longrightarrow} BG \overset{g}{\longrightarrow} \ensuremath{\mathbb{R}}\xspace \]
is smooth. Consider the commutative diagram
\[{\diagram
{M} \rrto^f\drto_{\bar{f}}&& {BG}\rto^g& {\ensuremath{\mathbb{R}}\xspace}\\ & {\ensuremath{\Delta}\xspace^n\times
G^n}\urto_{q_B}& \enddiagram}\] where $\bar{f}=(f_0, f_1, \ldots
,f_n, g_1, \ldots ,g_n)$. Since both $\bar{f}$ and $q_B\circ g$ are
smooth, the composition $f\circ g = \bar{f}\circ q_B\circ g$ is smooth
as well. Thus $f:M\rightarrow BG$ is a smooth map. \qed
\section{Bar resolutions of sheaves}
The key to the geometric interpretations of the cohomology groups from
Theorems A, B, C, and D is the following construction of a bar resolution
of a sheaf.
Let $G$ be an abelian group. The composition of the short exact sequences\xspace
\begin{gather}\label{basicses}
0\longrightarrow B^nG \longrightarrow EB^nG \longrightarrow B^{n+1}G \longrightarrow 0
\end{gather}
\noindent induces the long exact sequence\xspace
\begin{gather}\label{basicles}
0\longrightarrow G \longrightarrow EG \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EBG \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^2G
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \, \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^nG
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\end{gather}
\noindent where for every $n\geq 0$ the homomorphism
\[ \ensuremath{\sigma}\xspace : EB^nG \longrightarrow EB^{n+1}G \]
\noindent is the composition
\[ EB^nG \longrightarrow B^{n+1}G \longrightarrow EB^{n+1}G \]
of the surjection $EB^nG \rightarrow B^{n+1}G$ and the monomorphism $B^{n+1}G
\rightarrow EB^{n+1}G$.
If $G$ is an abelian Lie (or differentiable) group, then, as we saw in
Example~\ref{e1.2}, the short exact sequence\xspace \eqref{basicses} is a {\em smooth}
$B^sG$-extension of $B^{s+1}G$ (that is both $ B^sG \longrightarrow EB^sG$ and
$EB^sG \longrightarrow B^{s+1}G$ are smooth homomorphisms). Hence, the long exact sequence\xspace
\eqref{basicles} induces the long exact sequence\xspace of sheaves
\begin{gather}\label{basiclesofsh}
0\longrightarrow \u{G} \longrightarrow \u{EG} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EBG}
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EB^2G} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EB^nG} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\end{gather}
which will be called the {\em bar resolution} of the sheaf $\u{G}$.
\begin{Prop}\label{acyclicprop}
The sequence \eqref{basiclesofsh} is an acyclic resolution of the
sheaf \u{G}.
\end{Prop}
\noindent{\bf Proof. \ } It is enough to show that for every differentiable group $G$
the group $H^i(\u{EG})$ is trivial, for every $i>0$.
Recall, that a sheaf $\cl{F}$ on $X$ is {\em soft} if for every closed
subset $Z$ of $X$ the restriction map $\cl{F}(X) \rightarrow \cl{F}(Z)$ is a
surjection. If $X$ is a paracompact space and $\cl{F}$ is a soft sheaf
on $X$, then $H^i(X;\cl{F})\cong 0$ for all $i>0$ (see
\cite[Theorem~1.4.6]{bry-greenbook} ).
{\Le\label{acyclic} For every differentiable group $G$ the sheaf
$\u{EG}$ is soft.}
\noindent{\bf Proof. \ } Let $Z$ be a close subset of $M$ and let $\sigma_Z$ be a
section of $\u{EG}$ over $Z$. By the definition of a section of a
sheaf over a closed set there is an open set $U \supset Z$ and an
extension $\sigma_U$ of $\sigma_Z$ to $U$. Since $M$ is paracompact,
there is a neighborhood\xspace $V$ of $Z$ such that $\bar{V} \subset U$. The
extension $\sigma$ of $\sigma_U$ (and hence also $\sigma_Z$) to a
global section of $\u{EG}$ is given by the formula
\[\sigma(x) = r(\sigma_U(x), \psi(x)),
\]
where $r: EG \times I \rightarrow EG$ is the deformational retraction from
Example~\ref{e1.2}.B and $\psi: M \rightarrow [0,1]$ is a smooth function
equal to 1 on $V$ and equal to 0 on $M - U$. \qed
A bar resolution of the sheaf $\ensuremath{\mathcal{A}}\xspace^k_M$ of germs of smooth differential
$k$-forms on $M$ is constructed as follows. Let $\ensuremath{\Lambda}\xspace^kT^\ast M$ be the
$k$th exterior power of the cotangent bundle $T^\ast M$ of $M$ and let
$E\ensuremath{\Lambda}\xspace^kT^\ast M$ and $B\ensuremath{\Lambda}\xspace^kT^\ast M$ be the associated with
$\ensuremath{\Lambda}\xspace^kT^\ast M$ bundles with fibers over $x\in M$ equal to
$E(\ensuremath{\Lambda}\xspace^kT^\ast_x M)$ and $B(\ensuremath{\Lambda}\xspace^kT^\ast_x M)$ respectively. The groups
$E(\ensuremath{\Lambda}\xspace^kT^\ast_x M)$ and $B(\ensuremath{\Lambda}\xspace^kT^\ast_x M)$ carry vector spaces
structures as in Lemma~\ref{BV}.
Let $E\ensuremath{\mathcal{A}}\xspace^k_M$ and $B\ensuremath{\mathcal{A}}\xspace^k_M$ be the sheaves of germs of smooth
sections of the vector bundles $E\ensuremath{\Lambda}\xspace^kT^\ast M$ and $B\ensuremath{\Lambda}\xspace^kT^\ast M$
respectively. A section \ensuremath{\alpha}\xspace of the sheaf $E\ensuremath{\mathcal{A}}\xspace^k_M$ over $U\subset M$
is of the form
\[ \ensuremath{\alpha}\xspace = \bigl\lvert f_0, \, \ldots \, ,f_n, \ensuremath{\alpha}\xspace_0, \, \ldots \, ,\ensuremath{\alpha}\xspace_n\bigr\rvert, \]
and a section $\ensuremath{\beta}\xspace$ of the sheaf $B\ensuremath{\mathcal{A}}\xspace^k_M$ over $U$ is of the form
\[ \ensuremath{\beta}\xspace = \bigl\lvert f_0, \, \ldots \, ,f_n, [\ensuremath{\beta}\xspace_0:\; \cdots \; :\ensuremath{\beta}\xspace_n]\bigr\rvert, \]
where $\ensuremath{\alpha}\xspace_0, \, \ldots \, , \ensuremath{\alpha}\xspace_n, \ensuremath{\beta}\xspace_0, \, \ldots \, , \ensuremath{\beta}\xspace_n$ are
smooth differential $k$-forms on $U$ and $\{f_i\}_{i=0}^n$ is a smooth
partition of unity on $U$. The group of section of the sheaf
$E\ensuremath{\mathcal{A}}\xspace^k_M$ over an open set $U\subset M$ will be denoted by
$\ensuremath{\Gamma}\xspace(U,E\ensuremath{\mathcal{A}}\xspace^k_M)$. Similarly, $\ensuremath{\Gamma}\xspace(U,B\ensuremath{\mathcal{A}}\xspace^k_M)$ stands for the group
of sections of $B\ensuremath{\mathcal{A}}\xspace^k_M$ over $U$.
Since the sequence of vector bundles
\[ 0\longrightarrow \ensuremath{\Lambda}\xspace^kT^\ast M \longrightarrow E\ensuremath{\Lambda}\xspace^kT^\ast M \longrightarrow
B\ensuremath{\Lambda}\xspace^kT^\ast M \longrightarrow 0 \] is exact, the sequence of the groups
\[ 0\longrightarrow \ensuremath{\Gamma}\xspace(U,\ensuremath{\mathcal{A}}\xspace^k_M) \longrightarrow \ensuremath{\Gamma}\xspace(U,E\ensuremath{\mathcal{A}}\xspace^k_M) \longrightarrow \ensuremath{\Gamma}\xspace(U,B\ensuremath{\mathcal{A}}\xspace^k_M) \longrightarrow 0
\]
is exact, for every open subset $U$ of $M$. Hence, the sequence of
sheaves
\[ 0\longrightarrow \ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow E\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow B\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow 0 \]
is exact. Similarly, if $EB^{s-1}\ensuremath{\mathcal{A}}\xspace^k_M$ and $B^s\ensuremath{\mathcal{A}}\xspace^k_M$ are the
sheaves of smooth sections of the vector bundles $EB^{s-1}\ensuremath{\Lambda}\xspace^kT^\ast
M$ and $B^s\ensuremath{\Lambda}\xspace^kT^\ast M$ respectively, then the sequence of sheaves
\[ 0\longrightarrow B^{s-1}\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow EB^{s-1}\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow B^s\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow 0 \]
is exact. The composition of these sequences induces a long exact sequence\xspace
\begin{gather}\label{difffbarres}
0\longrightarrow \ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow E\ensuremath{\mathcal{A}}\xspace^k_M \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB\ensuremath{\mathcal{A}}\xspace^k_M
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}\, \cdots \, \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^s\ensuremath{\mathcal{A}}\xspace^k_M
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots
\end{gather}
where
\[ \ensuremath{\sigma}\xspace: EB^s\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow EB^{s+1}\ensuremath{\mathcal{A}}\xspace^k_M \]
is the composition
\[ EB^s\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow B^{s+1}\ensuremath{\mathcal{A}}\xspace^k_M \longrightarrow EB^{s+1}\ensuremath{\mathcal{A}}\xspace^k_M \]
The sequence \eqref{difffbarres} will be called the {\em bar
resolution} of the sheaf $\ensuremath{\mathcal{A}}\xspace^k_M$.
A bar resolution of an arbitrary sheaf \cl{F} on a space $X$, which is
not necessarily a smooth manifold, can be defined as follows.
Let $E\cl{F}$ and $B\cl{F}$ be the sheaves associated with the
presheaves
\[ U \mapsto E(\cl{F}(U)) \qquad\text{and}\qquad U \mapsto B(\cl{F}(U))\]
respectively. Since the stalks of $E\cl{F}$ and $B\cl{F}$ at $x\in X$
are $E(\cl{F}_x)$ and $B(\cl{F}_x)$ respectively, where $\cl{F}_x$ is
the stalk of the sheaf \cl{F} at $x$, and the sequence
\[ 0\longrightarrow \cl{F}_x \longrightarrow E(\cl{F}_x) \longrightarrow B(\cl{F}_x) \longrightarrow 0 \]
is exact, the sequence of sheaves
\[ 0\longrightarrow \cl{F} \longrightarrow E\cl{F} \longrightarrow B\cl{F} \longrightarrow 0 \]
is exact. Iterating the above bar constructions we get for every
$s\geq 1$ the sheaves $EB^{s-1}\cl{F}$ and $B^s\cl{F}$ so that the
sequence
\[ 0\longrightarrow B^{s-1}\cl{F} \longrightarrow EB^{s-1}\cl{F} \longrightarrow B^s\cl{F} \longrightarrow 0 \]
is exact. The composition of these sequences gives the {\em bar
resolution of \cl{F}}
\begin{gather*}
0\longrightarrow \cl{F} \longrightarrow E\cl{F} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB\cl{F}
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^2\cl{F} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^n\cl{F} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\end{gather*}
The complex of sheaves
\begin{gather*}
\cl{B}^\ast(\cl{F}):\qquad E\cl{F} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB\cl{F}
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^2\cl{F} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} EB^n\cl{F} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\end{gather*}
will be called the {\em bar complex of \cl{F}}.
An easy modification of the proof of Lemma~\ref{acyclic} shows that
the bar resolution of \cl{F} is an acyclic resolution of \cl{F}.
Therefore, the cohomology of \cl{F} is equal to the cohomology of the
cochain complex
\begin{gather*}
\ensuremath{\Gamma}\xspace(M,E\cl{F}) \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \ensuremath{\Gamma}\xspace(M,EB\cl{F})
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \ensuremath{\Gamma}\xspace(M,EB^2\cl{F}) \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots
\, \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \ensuremath{\Gamma}\xspace(M,EB^n\cl{F}) \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \,
\cdots \,
\end{gather*}
The above complex will be called the {\em bar cochain complex of
\cl{F}} and we will denote it by $C_B^\ast(\cl{F})$.
Note, that the above construction applied to \u{G} and $\ensuremath{\mathcal{A}}\xspace^{k}_M$
produces resolutions of \u{G} and $\ensuremath{\mathcal{A}}\xspace^{k}_M$ that do not coincide
with the resolutions \eqref{basiclesofsh} and \eqref{difffbarres}. In
the sequel, when referring to bar resolutions of \u{G} or $\ensuremath{\mathcal{A}}\xspace^{k}_M$
we will always mean the resolutions \eqref{basiclesofsh} or
\eqref{difffbarres} respectively.
The bar cochain complex $C_B^\ast(\u{G})$ of the sheaf \u{G} is
of the form
\[\cinf{EG}\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}
\cinf{EBG}\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}\, \cdots \, \overset{\ensuremath{\sigma}\xspace}{\longrightarrow}
\cinf{EB^nG}\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}\, \dots \] \noindent where
\[ \ensuremath{\sigma}\xspace : \cinf{EB^nG}\longrightarrow \cinf{EB^{n+1}G}\]
is the composition
\[ \cinf{EB^nG}\overset{\pi_\ast}{\longrightarrow}
\cinf{B^{n+1}G}\overset{i_\ast}{\longrightarrow} \cinf{EB^{n+1}G} \] \noindent with
$\pi: E(B^nG) \rightarrow B(B^nG)=B^{n+1}G$ being the projection map of the
universal principal $B^nG$-bundle and $i: B^{n+1}G \rightarrow EB^{n+1}G$
being the inclusion of the fiber into the total space of the universal
principal $B^{n+1}G$-bundle.
In a sense, the bar cochain complex of the sheaf \u{\ensuremath{\mathbb{Z}}\xspace}
\[\cinf{E\ensuremath{\mathbb{Z}}\xspace}\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}
\cinf{EB\ensuremath{\mathbb{Z}}\xspace}\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}\, \cdots \, \overset{\ensuremath{\sigma}\xspace}{\longrightarrow}
\cinf{EB^n\ensuremath{\mathbb{Z}}\xspace}\overset{\ensuremath{\sigma}\xspace}{\longrightarrow}\, \dots \] can be thought of as a
smooth version of Karoubi's complex (see \cite{kar-ftnc})
\[\operatorname{Map}(X, AG(\ensuremath{\mathbb{D}}\xspace^1))\overset{\ensuremath{\sigma}\xspace_\ast}{\longrightarrow} \operatorname{Map}(X, AG(\ensuremath{\mathbb{D}}\xspace^2))
\overset{\ensuremath{\sigma}\xspace_\ast}{\longrightarrow} \, \cdots \, \overset{\ensuremath{\sigma}\xspace_\ast}{\longrightarrow} \operatorname{Map}(X,
AG(\ensuremath{\mathbb{D}}\xspace^n)) \overset{\ensuremath{\sigma}\xspace_\ast}{\longrightarrow} \, \cdots \] of the topological
non-commutative differential forms on a space $X$, where $AG(\ensuremath{\mathbb{D}}\xspace^n)$ is
the free abelian group on the disk $\ensuremath{\mathbb{D}}\xspace^n$, and
\[\ensuremath{\sigma}\xspace_\ast : \operatorname{Map}(X, AG(\ensuremath{\mathbb{D}}\xspace^n)) \longrightarrow \operatorname{Map}(X, AG(\ensuremath{\mathbb{D}}\xspace^{n+1}))\]
is a homomorphism induced by the composition of maps
\[ \ensuremath{\mathbb{D}}\xspace^n \longrightarrow \ensuremath{\mathbb{D}}\xspace^n/\partial\ensuremath{\mathbb{D}}\xspace^n = S^n = \partial \ensuremath{\mathbb{D}}\xspace^{n+1} \hookrightarrow \ensuremath{\mathbb{D}}\xspace^{n+1}. \]
{}From the functoriality of the geometric bar construction it follows
that the bar resolution of sheaves is functorial as well. Moreover,
since for every short exact sequence\xspace of topological groups
\[ 0\longrightarrow K \longrightarrow G \longrightarrow H \longrightarrow 0 \]
the sequences
\begin{gather*}
0\longrightarrow EK \longrightarrow EG \longrightarrow EH \longrightarrow 0\\ \intertext{and} 0\longrightarrow BK \longrightarrow BG
\longrightarrow BH \longrightarrow 0
\end{gather*}
are exact, every short exact sequence\xspace of sheaves
\[ 0\longrightarrow \cl{E} \longrightarrow \cl{F} \longrightarrow \cl{G} \longrightarrow 0 \]
induces a short exact sequence\xspace of complexes of sheaves
\[ 0\longrightarrow \cl{B}^\ast(\cl{E}) \longrightarrow \cl{B}^\ast(\cl{F})
\longrightarrow \cl{B}^\ast(\cl{G}) \longrightarrow 0 \] Hence, every complex of sheaves
$\cl{F}^\ast$ has an acyclic resolution given by the total complex
$\operatorname{Tot}^\ast(\cl{B}^\ast(\cl{F}^\ast))$ associated with the double
complex $\cl{B}^\ast(\cl{F}^\ast)$. The cohomology of $\cl{F}^\ast$ is
equal to the cohomology of the total cochain complex
$\operatorname{Tot}^\ast(C_B^\ast(\cl{F}^\ast))$.
\begin{Exa}
The double cochain complex $C_B^\ast(\ensuremath{\mathcal{A}}\xspace^\ast_M)$ of the de Rham
complex
\[\ensuremath{\mathcal{A}}\xspace^\ast_M:\qquad \ensuremath{\mathcal{A}}\xspace^0_M \overset{d}{\longrightarrow} \ensuremath{\mathcal{A}}\xspace^1_M
\overset{d}{\longrightarrow} \ensuremath{\mathcal{A}}\xspace^2_M \overset{d}{\longrightarrow} \, \cdots \,
\overset{d}{\longrightarrow} \ensuremath{\mathcal{A}}\xspace^n_M \overset{d}{\longrightarrow} \cdots\] is given by the
diagram {\small
\[{\diagram
& & &\\ {\ensuremath{\Gamma}\xspace(M, E\ensuremath{\mathcal{A}}\xspace^2_M)}\uto^-d\rto^\ensuremath{\sigma}\xspace& & & \\ {\ensuremath{\Gamma}\xspace(M,
E\ensuremath{\mathcal{A}}\xspace^1_M)}\uto^-d\rto^\ensuremath{\sigma}\xspace& {\ensuremath{\Gamma}\xspace(M,EB\ensuremath{\mathcal{A}}\xspace^1_M)}\uto^-d\rto^-\ensuremath{\sigma}\xspace& \\
{\ensuremath{\Gamma}\xspace(M,E\ensuremath{\mathcal{A}}\xspace^0_M)}\uto^-d\rto^\ensuremath{\sigma}\xspace& {\ensuremath{\Gamma}\xspace(M,EB\ensuremath{\mathcal{A}}\xspace^0_M)}\uto^-d\rto^\ensuremath{\sigma}\xspace&
{\ensuremath{\Gamma}\xspace(M,EB^2\ensuremath{\mathcal{A}}\xspace^0_M)}\uto^-d\rto^-\ensuremath{\sigma}\xspace &\\ \enddiagram} \]} Note that
the $n$th column of this double complex is the complex of global
sections of the acyclic resolution
\[EB^{n-1}\ensuremath{\mathcal{A}}\xspace^0_M\rightarrow EB^{n-1}\ensuremath{\mathcal{A}}\xspace^1_M\rightarrow \cdots \rightarrow EB^{n-1}\ensuremath{\mathcal{A}}\xspace^s_M\rightarrow
\cdots\] of the sheaf $\u{EB^{n-1}\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace$, where $\ensuremath{\mathbb{R}}\xspace^\ensuremath{\delta}\xspace$ is the
group \ensuremath{\mathbb{R}}\xspace taken with the discrete topology. Therefore, the total
complex of $C_B^\ast(\ensuremath{\mathcal{A}}\xspace^\ast_M)$ is an acyclic resolution of the de
Rham complex of $M$ and the bar complex
\[\u{E\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace \longrightarrow \u{EB\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace \longrightarrow \u{EB^2\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace \longrightarrow\cdots \longrightarrow
\u{EB^s\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace\longrightarrow \cdots
\]
of $\u{\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace$. Thus, the bar complex of the de Rham complex
$\cl{A}^\ast_M$ of $M$ plays a similar role to the \v{C}ech complex of
$\cl{A}^\ast_M$ inducing an isomorphis between the de Rham cohomology
of $M$ and the sheaf cohomology of the constant sheaf $\u{\ensuremath{\mathbb{R}}\xspace}^\ensuremath{\delta}\xspace$.
\end{Exa}
\begin{Rem}
There is a close relationship between the bar and \v{C}ech cochain
complexes of the sheaf $\ensuremath{\mathcal{A}}\xspace^k_M$. Actually, every smooth partition
of unity $\{f_i\}_{i\in I}$ subordinated to an open covering
$\cl{U}=\{U_i\}_{i\in I}$ of a manifold $M$ induces a cochain
homomorphism
\[ \ensuremath{\varphi}\xspace^\ast:\check{C}^\ast(\cl{U},\ensuremath{\mathcal{A}}\xspace^k_M) \longrightarrow C_B^\ast(\ensuremath{\mathcal{A}}\xspace^k_M) \]
so that
\[ \ensuremath{\varphi}\xspace^p : \check{C}^p(\cl{U},\ensuremath{\mathcal{A}}\xspace^k_M) \longrightarrow C_B^p(\ensuremath{\mathcal{A}}\xspace^k_M)=
EB^p\ensuremath{\mathcal{A}}\xspace^k_M(M)\] is the composition
\[ \check{C}^p(\cl{U},\ensuremath{\mathcal{A}}\xspace^k_M)\longrightarrow \check{C}^{p-1}(\cl{U},B\ensuremath{\mathcal{A}}\xspace^k_M)\longrightarrow
\, \cdots \, \longrightarrow \check{C}^0(\cl{U},B^p\ensuremath{\mathcal{A}}\xspace^k_M)\longrightarrow EB^p\ensuremath{\mathcal{A}}\xspace^k_M(M)\]
where for $r>0$ the homomorphism
\[ \ensuremath{\varphi}\xspace^{r,s} : \check{C}^r (\cl{U}, B^{s}\ensuremath{\mathcal{A}}\xspace^k_M)
\longrightarrow \check{C}^{r-1} (\cl{U}, B^{s+1}\ensuremath{\mathcal{A}}\xspace^k_M) \] is defined for $\xi =
\{ \xi_{\seq[r]{i}} \in B^s\ensuremath{\mathcal{A}}\xspace^k_M(\bigcap\limits_{j=0}^r U_{i_j}) \}$
by the formula
\[ \ensuremath{\varphi}\xspace^{r,s}(\xi)_{\seq[r-1]{i}} =
\bigl\lvert f_{l_0}, \, \ldots \, ,f_{l_n}, [\xi_{l_0,\seq[r-1]{i}}: \, \cdots
\, :\xi_{l_n,\seq[r-1]{i}}]\bigr\rvert \] and the homomorphism
\[ \ensuremath{\varphi}\xspace^{0,p}:\check{C}^0(\cl{U},B^p\ensuremath{\mathcal{A}}\xspace^k_M)\longrightarrow EB^p\ensuremath{\mathcal{A}}\xspace^k_M(M)\]
is given by
\[ \ensuremath{\varphi}\xspace^{0,p}(\{\xi_i\}) =
\bigl\lvert f_{l_0}, \, \ldots \, ,f_{l_n}, [\xi_{l_0}:\, \cdots \, :
\xi_{l_n}]\bigr\rvert. \]
\end{Rem}
\section{Smooth principal $B^s\ensuremath{\mathbb{C}}\xspace^{\ast}$-bundles}
In this section we will show that if $M$ is a smooth manifold, then
the group $H^k(M;\ensuremath{\mathbb{Z}}\xspace)$ can be identified with the group of isomorphism
classes of smooth principal $B^{k-2}S^1, B^{k-2}\ensuremath{\mathbb{C}^\ast}\xspace$, or $B^{k-1}\ensuremath{\mathbb{Z}}\xspace$
bundles over $M$.
Let $G$ be an abelian Lie group. A principal $B^sG$-bundle $E \rightarrow M$
over a smooth manifold $M$ is {\em smooth} if the transition functions
of this bundle are smooth.
The proof of the following proposition, essentially due to tom Dieck
\cite{tDie-klass}, shows an explicit formula for a classifying map of
a smooth principal bundle in terms of its transition functions.
{\Prop \label{smoothpr} Let $G$ be a differentiable group. Then for
every smooth principal $G$-bundle $\pi :E \rightarrow M$ there is a smooth
map $\ensuremath{\varphi}\xspace:M \rightarrow BG$ such that $E \rightarrow M$ is the pull-back of the
universal principal $G$-bundle by $\ensuremath{\varphi}\xspace$. }
{\Co \label{smoothco} For every smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E
\rightarrow M$ there is a smooth map $\ensuremath{\varphi}\xspace:M \rightarrow B^{s+1}\ensuremath{\mathbb{C}^\ast}\xspace$ such that $E
\rightarrow M$ is the pull-back of the universal principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle
by $\ensuremath{\varphi}\xspace$. }
\noindent{\bf Proof of Proposition~\ref{smoothpr}.} Let $\cl{U}
=\{U_i\}_{i\in I}$ be an open covering of $M$ so that for every $i\in
I$ there is a trivialization $$\ensuremath{\psi}\xspace_i : \pi^{-1}(U_i) \longrightarrow U_i\times
G.$$
Define $g_i : E \rightarrow G$ by the formula
$$g_i(x)=
\begin{cases}
\text{pr}_2(\ensuremath{\psi}\xspace_i(x))& \text{for}\quad x\in \pi^{-1}(U_i)\\ e &
\text{for}\quad x\notin \pi^{-1}(U_i)
\end{cases}$$
\noindent where $e$ is the neutral element of $G$ and $\text{pr}_2 :
U_i\times G \rightarrow G$ is the projection on the second factor.
Let $\{f_i\}_{i\in I}$ be a partition of unity subordinated to the
covering $\cl{U}$ and let \linebreak $\bar{\ensuremath{\varphi}\xspace}:E \rightarrow EG$ be the map
$$\bar{\ensuremath{\varphi}\xspace}(y) = \bigl\lvert f_{i_0}(\pi(y)), f_{i_1}(\pi(y)), \, \ldots \, ,
f_{i_n}(\pi(y)), g_{i_0}(y), g_{i_1}(y), \, \ldots \, ,
g_{i_n}(y)\bigr\rvert,$$ \noindent where $\seq{i}$ are the indices so that for each
$i\in \{\seq{i}\}\quad f_i(\pi(y))\neq 0$. It is easy to see that
$\bar{\ensuremath{\varphi}\xspace}$ is a $G$-equivariant map and hence it induces a morphism
of principal $G$-bundles
$${\diagram {E} \rto^{\bar{\ensuremath{\varphi}\xspace}}\dto_{\pi}& {EG}\dto \\ {M}
\rto_{\ensuremath{\varphi}\xspace}& {BG} \enddiagram}$$ \noindent where the restriction of $\ensuremath{\varphi}\xspace : M
\rightarrow BG$ to $U_j \subset M$ is given by the formula
$$\ensuremath{\varphi}\xspace (x) = \bigl\lvert f_{i_0}(x), f_{i_1}(x), \, \ldots \, , f_{i_n}(x),
[g_{i_0}(\ensuremath{\sigma}\xspace(x)): g_{i_1}(\ensuremath{\sigma}\xspace(x)): \, \cdots \, :
g_{i_n}(\ensuremath{\sigma}\xspace(x))]\bigr\rvert,$$ \noindent where $\ensuremath{\sigma}\xspace: U_j\rightarrow \pi^{-1}(U_j)$ is a
smooth section of the restriction $\pi^{-1}(U_j) \rightarrow U_j$ of $\pi:E\rightarrow
M$ to $\pi^{-1}(U_j)$. Note that $\ensuremath{\varphi}\xspace(x)$ does not depend on the
choice of the section $\ensuremath{\sigma}\xspace$ because $g_i$s are $G$-equivariant maps.
In the non-homogeneous coordinates
\[ \ensuremath{\varphi}\xspace (x) = \bigl\lvert f_{i_0}(x), f_{i_1}(x),\, \ldots \, ,f_{i_{n}}(x),
[g_{i_0i_1}(x)|g_{i_1i_2}(x)|\, \cdots \,
|g_{i_{n-1}i_n}(x)]\bigr\rvert,
\]
where
\[ g_{ij}(x) = (g_i(\ensuremath{\sigma}\xspace(x)))^{-1}\cdot g_j(\ensuremath{\sigma}\xspace(x)) \]
are the transition functions of the bundle $E\rightarrow M$ associated with
the open covering of $M$ by the sets $\{x\in M\, | \, f_i(x)>0\}$.
Since $g_i$ is smooth on $\operatorname{supp} (f_i)\cap U_j$ for every $i,j\in I$
and \ensuremath{\sigma}\xspace is smooth on $U_j$, the map \ensuremath{\varphi}\xspace is smooth on $U_j$ for every
$j\in I$ and hence \ensuremath{\varphi}\xspace is smooth on $M$. T. tom Dieck showed in
\cite{tDie-klass} that $\ensuremath{\varphi}\xspace : M \rightarrow BG$ is the classifying map of the
bundle $\pi : E \rightarrow M$ (tom Dieck works in the setting of Milnor's bar
construction, but all he does extends easily to the context of
Milgram's bar construction). \qed
\begin{Exa}\label{e1.1}
The isomorphism $H^2(\ensuremath{\mathbb{R}}\xspace^3 - 0,\ensuremath{\mathbb{Z}}\xspace) \cong H^1(\ensuremath{\mathbb{R}}\xspace^3 -
0,\underline{\ensuremath{\mathbb{C}^\ast}\xspace}_{\ensuremath{\mathbb{R}}\xspace^3 - 0})$ implies that every element of
$H^2(\ensuremath{\mathbb{R}}\xspace^3 - 0,\ensuremath{\mathbb{Z}}\xspace)$ corresponds to a unique isomorphism class of a
smooth principal \ensuremath{\mathbb{C}^\ast}\xspace-bundle over $\ensuremath{\mathbb{R}}\xspace^3 - 0$. Let $L$ be a smooth
principal \ensuremath{\mathbb{C}^\ast}\xspace-bundle over $\ensuremath{\mathbb{R}}\xspace^3 - 0$ representing a generator of
$H^2(\ensuremath{\mathbb{R}}\xspace^3 - 0,\ensuremath{\mathbb{Z}}\xspace) \cong \ensuremath{\mathbb{Z}}\xspace$. The proof of Proposition~\ref{smoothpr}
shows how to describe a smooth classifying map $\ensuremath{\psi}\xspace_L:(\ensuremath{\mathbb{R}}\xspace^3 -0) \rightarrow
B\ensuremath{\mathbb{C}^\ast}\xspace$ of $L$ in terms of some transition functions of $L$. Let $S^3
= \ensuremath{\mathbb{R}}\xspace^3 \cup \{\infty\}$ and consider the open subsets $U_0 =\ensuremath{\mathbb{R}}\xspace^3$
and $U_\infty = S^3-\{0\}$ of $S^3$. Since $U_0\cap U_\infty = \ensuremath{\mathbb{R}}\xspace^3
- 0$ we can think of the classifying map $\ensuremath{\psi}\xspace_L:(\ensuremath{\mathbb{R}}\xspace^3 -0) \rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace$
as a transition function of a smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $BL$
over $S^3$. From the proof of Proposition~\ref{H(M,Z)} it follows
that the isomorphism class of $BL$ corresponds to the generator of
$H^3(S^3,\ensuremath{\mathbb{Z}}\xspace)$. Let $\widetilde{BL}$ be the pull-back of $BL$ by the
standard retraction $(\ensuremath{\mathbb{R}}\xspace^4 - 0) \rightarrow S^3$. The classifying map of the
bundle $\widetilde{BL}$ can be identified with a transition function
of a smooth principal $B^2\ensuremath{\mathbb{C}^\ast}\xspace$-bundle over $S^4$, representing a
generator of $H^4(S^4,\ensuremath{\mathbb{Z}}\xspace)$. Iterating the above procedure we get a
family of smooth principal $B^k\ensuremath{\mathbb{C}^\ast}\xspace$-bundle over $S^k$, representing
generators of the groups $H^k(S^k,\ensuremath{\mathbb{Z}}\xspace)$ for $k\geq 2$.
\end{Exa}
\begin{Prop} \label{H(M,Z)} Let $G$ be one of the group $S^1, \ensuremath{\mathbb{C}^\ast}\xspace$,
or $B\ensuremath{\mathbb{Z}}\xspace$. Then for every smooth manifold $M$ and every $p\geq 2$ the
group $H^p(M,\ensuremath{\mathbb{Z}}\xspace)$\ is isomorphic to:
\renewcommand{\theenumi}{(\roman{enumi})}
\begin{enumerate}
\item the group $L(B^{p-2}G)_M$ of isomorphism classes of {\em smooth}
principal $B^{p-2}G$-bundles over $M$.
\item the group $[M, B^{p-1}G]^\infty$ of smooth homotopy classes of
smooth maps from $M$ to $B^{p-1}G$.
\end{enumerate}
\end{Prop}
\noindent{\bf Proof of part (i) of Proposition~\ref{H(M,Z)}.} Since for any
abelian differentiable group $G$ the group of isomorphism classes of
smooth principal $G$-bundles over $M$ is isomorphic to $H^1(\u{G})$,
we have to prove that there is an isomorphism
\[ H^p(M;\ensuremath{\mathbb{Z}}\xspace) \cong H^1(\u{B^{p-2}G}). \]
Consider the cohomology long exact sequence\xspace
\begin{gather*}
\rightarrow \ensuremath{\mathbb{H}}\xspace^{p-1}(\u{EB^{<p-2}G})\rightarrow H^1(\u{B^{p-2}G})\rightarrow H^p(M;\ensuremath{\mathbb{Z}}\xspace)\rightarrow
\ensuremath{\mathbb{H}}\xspace^{p}(\u{EB^{<p-2}G})\rightarrow
\end{gather*}
associated with the generalized exponential sequence
\begin{gather}\label{expseq1}
0\longrightarrow \u{\ensuremath{\mathbb{Z}}\xspace} \longrightarrow \u{EB^{<p-2}G} \longrightarrow \u{B^{p-2}G}[-p+2] \longrightarrow 0
\end{gather}
where \u{EB^{<p-2}G} is the complex
\begin{gather*}
\u{H} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EG} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EBG}
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EB^2G} \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EB^{p-3}G}
\end{gather*}
with \u{H} being equal to $\ensuremath{\mathbb{R}}\xspace, \ensuremath{\mathbb{C}}\xspace$, or $E\ensuremath{\mathbb{Z}}\xspace$ for $G=S^1, \ensuremath{\mathbb{C}^\ast}\xspace$, or
$B\ensuremath{\mathbb{Z}}\xspace$ respectively.
Since for every $s\geq 0$ the sheaf \u{EB^sG} is acyclic, the
cohomology of the complex \u{EB^{<p-2}G} is equal to the cohomology of
the cochain complex
\begin{gather*}
H \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EG}(M) \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EBG}(M)
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EB^2G}(M) \overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \, \cdots \,
\overset{\ensuremath{\sigma}\xspace}{\longrightarrow} \u{EB^{p-3}G}(M)
\end{gather*}
of the groups of global sections of the components of \u{EB^{<p-2}G}.
Therefore, for every $q>p-2$
\[ \ensuremath{\mathbb{H}}\xspace^{q}(\u{EB^{<p-2}G})\cong H^q(\u{EB^{<p-2}G}(M))\cong 0.\]
Hence, the coboundary homomorphism
\[H^1(\u{B^{p-2}G}) \longrightarrow H^p(M;\ensuremath{\mathbb{Z}}\xspace) \]
in the cohomology long exact sequence\xspace associated with \eqref{expseq1} is an
isomorphism. \qed
\begin{Rem}\label{rem1}
Let $G$ be an arbitrary abelian Lie group. Replacing in the proof of
Proposition~\ref{H(M,Z)} the sequence \eqref{expseq1} be the
appropriate short exact sequence\xspace associated with the bar resolution of \u{G}, we
would get an isomorphism between $H^p(\u{G})$ and the group
$L(B^{p-2}G)_M$ of isomorphism classes of {\em smooth} principal
$B^{p-2}G$-bundles over $M$.
\end{Rem}
Part (ii) of Proposition~\ref{H(M,Z)} is a straightforward consequence
of the following lemma.
\begin{Le}\label{classification1}
Let $G$ be a differentiable group. Then the group of isomorphism
classes of smooth principal $G$-bundles over $M$ is isomorphic to
the group $[M, BG]^\infty$ of smooth homotopy classes of smooth maps
from $M$ to $BG$.
\end{Le}
\noindent{\bf Proof. } Let $G$ be a differentiable group. We will show that
there is an isomorphism
\[ [M, BG]^\infty \cong H^1(\u{G}). \]
The beginning of the cohomology long exact sequence\xspace associated with the short exact sequence\xspace
\[ 0 \longrightarrow \u{G} \longrightarrow \u{EG} \longrightarrow \u{BG} \longrightarrow 0 \]
\noindent is of the form
\[ \, \cdots \, \longrightarrow C^\infty(M,EG) \overset{\pi_\ast}{\longrightarrow}
C^\infty(M,BG) \longrightarrow H^1(\u{G}) \longrightarrow H^1(\u{EG}) \longrightarrow \, \cdots \]
Since $H^1(\u{EG}) \cong 0$, we have the isomorphism
\[ \dfrac{C^\infty(M,BG)}{\pi_\ast C^\infty(M,EG)} \cong
H^1(\u{G}). \] The image $\pi_\ast C^\infty(M,EG)$ of the group
$C^\infty(M,EG)$ in $C^\infty(M,BG)$ consists of those smooth maps
from $M$ to $BG$ that lift to maps from $M$ to $EG$. It is easy to see
that $f:M\rightarrow BG$ has a lift to $\tilde{f}: M \rightarrow EG$ if and only if
$f$ is smooth homotopic to a constant map. Hence
\[ \dfrac{C^\infty(M,BG)}{\pi_\ast C^\infty(M,EG)} \cong [M,
BG]^\infty \] \qed
\begin{Rem}
Let $G$ be a topological group. Replacing in the proof of
Lemma~\ref{classification1} the sheaves of smooth maps on $M$ by
sheaves of continues maps on some space $X$, we get an isomorphism
between the group of isomorphism classes of principal $G$-bundles
over $X$ and the group $[X,BG]$ of homotopy classes of maps from $X$
to $BG$.
\end{Rem}
\section{Flat Connections on principal $B^s\ensuremath{\mathbb{C}}\xspace^{\ast}$-bundles}
In this section we show that for every $s\geq 1$ the group
$B^s\ensuremath{\mathbb{C}}\xspace^{\ast}$ is equipped with the canonical
$B^s\ensuremath{\mathbb{C}}\xspace^{\ast}$-equivariant $B^s\ensuremath{\mathbb{C}}\xspace$-valued connection 1-form
$B^s(z^{-1}dz)$. By analogy with the Lie group case, the form
$B^s(z^{-1}dz)$ is used to define connections on smooth principal
$B^s\ensuremath{\mathbb{C}}\xspace^{\ast}$-bundles. We show that for $q>p$ the smooth Deligne
cohomology group $\HD{p}{q}$ is isomorphic to the group of isomorphism
classes of smooth principal $B^{p-2}\ensuremath{\mathbb{C}}\xspace^{\ast}$-bundles with flat
connections. Moreover, we prove Theorem~B.
\subsection{The canonical connection 1-forms on $B^s\ensuremath{\mathbb{C}}\xspace^{\ast}$ and
$EB^s\ensuremath{\mathbb{C}}\xspace^{\ast}$}
Let $M.$ be a simplicial smooth manifold. A {\em smooth $p$-form}
$\alpha$ on the geometric realization $|M.|$ of $M.$ is a family
$\{\alpha^n\}$ of differential $p$-forms $\alpha^n$ on $\Delta^n\times
M_n$ satisfying for every $0\leq i \leq n$ the following compatibility
conditions
\begin{gather}
(\partial^i\times id )^{\ast}\alpha^n = (id \times \partial_i
)^{\ast}\alpha^{n-1}\label{compcond1}\\ (s^i\times id
)^{\ast}\alpha^n = (id \times s_i
)^{\ast}\alpha^{n+1}\label{compcond2}
\end{gather}
\noindent where $\partial^i\times id,\, id \times \partial_i,\, s^i\times id$, and
$id \times s_i$ are the maps
$${\diagram {\ensuremath{\Delta}\xspace^{n-1}\times M_{n-1}} & {\ensuremath{\Delta}\xspace^{n-1}\times M_n}
\lto_-{id\times \partial_i}\rto^{\partial^i\times id}& {\ensuremath{\Delta}\xspace^n\times M_n}\\
{\ensuremath{\Delta}\xspace^{n+1}\times M_{n+1}} & {\ensuremath{\Delta}\xspace^{n+1}\times M_n} \lto_-{id\times
s_i}\rto^{s^i\times id}& {\ensuremath{\Delta}\xspace^n\times M_n} \enddiagram}$$ \noindent with
$\partial^i$ and $s^i$ being the coface and the codegeneracy maps on
$\Delta^n$s and $\partial_i$, $s_i$ being the face and the degeneracy maps
on $M_n$s.
\begin{Exa} \label{e2.1}\end{Exa}
\begin{enumerate}
\item Let $G$ be a Lie group and let $g^{-1}dg$ be the canonical
\ensuremath{\mathbf{g}}\xspace-valued connection 1-form on $G$, where \ensuremath{\mathbf{g}}\xspace is the Lie algebra
of $G$. The total space $EG$ of the universal principal $G$-bundle
$EG\rightarrow BG$ carries a smooth \ensuremath{\mathbf{g}}\xspace-valued form \go so that \go
evaluated at $\bigl\lvert\seq{x},\seq{g}\bigr\rvert$ is
\[ x_0g_{0}^{-1}dg_{0} + x_1g_{1}^{-1}dg_{1} + \, \cdots \, +
x_ng_{n}^{-1}dg_{n},
\]
where \seq{x} are the barycentric coordinates in $\ensuremath{\Delta}\xspace^n$ and
$g_{i}^{-1}dg_{i} = \pi_i^\ast(g^{-1}dg)$ for the projection $\pi_i:
G^{n+1}\rightarrow G$ on the $i$th factor.
\item The canonical connection 1-form $E(z^{-1}dz)$ on $E\ensuremath{\mathbb{C}^\ast}\xspace$ is
defined by the family of $E\ensuremath{\mathbb{C}}\xspace$-valued 1-forms $E(z^{-1}dz)^n$ on
$\Delta^n\times (\ensuremath{\mathbb{C}^\ast}\xspace)^{n+1}$ such that $E(z^{-1}dz)^n$ evaluated
on a vector $(v_\ensuremath{\Delta}\xspace, v_0, \dots, v_n)$ at a point $|t_1,\, \ldots
\, , t_n, z_0[z_1|\, \cdots \, |z_n]|$ is given by the formula
\[\bigl\lvert t_1, \, \ldots \, , t_n, z_0^{-1}v_0[z_1^{-1}v_1| \, \cdots \,
|z_n^{-1}v_n]\bigr\rvert.\]
In the sequel we will use the notation
\[E(z^{-1}dz)^n_{|t_1, \, \ldots \, , t_n, z_0[z_1|\, \cdots \, |z_n]|}=
\bigl\lvert t_1, \, \ldots \, , t_n, z_0^{-1}dz_0[z_1^{-1}dz_1| \, \cdots \,
|z_n^{-1}dz_n]\bigr\rvert.
\]
Similarly, the canonical connection 1-form $B(z^{-1}dz)$ on $B\ensuremath{\mathbb{C}^\ast}\xspace$ is
defined by the family of $B\ensuremath{\mathbb{C}}\xspace$-valued 1-forms $B(z^{-1}dz)^n$ on
$\Delta^n\times (\ensuremath{\mathbb{C}^\ast}\xspace)^n$, where
\[B(z^{-1}dz)^n_{|t_1, \, \ldots \, , t_n,
[z_1|\, \cdots \, |z_n]|}= \bigl\lvert t_1, \, \ldots \, , t_n,
[z_1^{-1}dz_1| \, \cdots \, |z_n^{-1}dz_n]\bigr\rvert.\] The compatibility
conditions \eqref{compcond1}, \eqref{compcond2} are easy to check
calculations. It is also easy to see that $E(z^{-1}dz)$ is a
$E\ensuremath{\mathbb{C}^\ast}\xspace$-equivariant 1-form and $B(z^{-1}dz)$ is a $B\ensuremath{\mathbb{C}^\ast}\xspace$-equivariant
1-form.
\end{enumerate}
Let $G$ be an abelian Lie group. A smooth $p$-form on $EB^sG$ and
$B^{s+1}G$, for $s\geq 1$, is defined by the following inductive
procedure.
Suppose we have defined smooth $p$-forms on $B^{s}G$ as well as on
each product $\ensuremath{\Delta}\xspace^k\times (B^{s}G)^m$ for $k,m\geq 0$. Then a smooth
$p$-form \ensuremath{\alpha}\xspace on $EB^sG$ consists of a family of $p$-forms $\ensuremath{\alpha}\xspace^n$ on
$\ensuremath{\Delta}\xspace^n\times (B^{s}G)^{n+1}$ satisfying the compatibility conditions
\eqref{compcond1} and \eqref{compcond2}. Similarly, a smooth $p$-form
\ensuremath{\alpha}\xspace on $B^{s+1}G$ consists of a family of $p$-forms $\ensuremath{\alpha}\xspace^n$ on
$\ensuremath{\Delta}\xspace^n\times (B^{s}G)^{n+1}$ satisfying the compatibility conditions
\eqref{compcond1} and \eqref{compcond2}. A smooth $p$-form \ensuremath{\alpha}\xspace on
$\ensuremath{\Delta}\xspace^k\times (B^{s+1}G)^m$ consists of a family of $p$-forms $\ensuremath{\alpha}\xspace^n$
on $\ensuremath{\Delta}\xspace^k\times(\ensuremath{\Delta}\xspace^n\times (B^{s}G)^{n+1})^m$ satisfying the
compatibility conditions
\begin{gather*}
(id_{\ensuremath{\Delta}\xspace^k}\times (\partial^i\times id)^m )^{\ast}\alpha^n =
(id_{\ensuremath{\Delta}\xspace^k}\times (id \times \partial_i)^m )^{\ast}\alpha^{n-1}\\
(id_{\ensuremath{\Delta}\xspace^k}\times (s^i\times id)^m )^{\ast}\alpha^n =
(id_{\ensuremath{\Delta}\xspace^k}\times (id \times s_i)^m)^{\ast}\alpha^{n+1}
\end{gather*}
\begin{Exa}\label{e4.2}
Now, for every $s> 0$ we are going to construct $EB^{s-1}\ensuremath{\mathbb{C}}\xspace$-valued
differential 1-form $EB^{s-1}(z^{-1}dz)$ on $EB^{s-1}\ensuremath{\mathbb{C}^\ast}\xspace$ and
$B^s\ensuremath{\mathbb{C}}\xspace$-valued differential 1-form $B^s(z^{-1}dz)$ on $B^s\ensuremath{\mathbb{C}^\ast}\xspace$.
Note that from Lemma~\ref{BV} we know that for every $s> 0$ the
groups $EB^{s-1}\ensuremath{\mathbb{C}}\xspace$ and $B^s\ensuremath{\mathbb{C}}\xspace$ are $\ensuremath{\mathbb{C}}\xspace$-vector spaces. Thus, it
make sense to talk about $EB^{s-1}\ensuremath{\mathbb{C}}\xspace$ or $B^s\ensuremath{\mathbb{C}}\xspace$-valued differential
forms.
The canonical connection 1-form $EB^{s-1}(z^{-1}dz)$ on
$EB^{s-1}\ensuremath{\mathbb{C}^\ast}\xspace$ is a 1-form on $EB^{s-1}\ensuremath{\mathbb{C}^\ast}\xspace$ so that
$EB^{s-1}(z^{-1}dz)$ evaluated at $|t_1, \, \, \ldots \, \, , t_n,
g_0[g_1|\, \cdots \, |g_n]|$ is given by the inductive formula
\[\bigl\lvert t_1, \, \ldots \, , t_n, B^{s-1}(g_0^{-1}dg_0)[B^{s-1}(g_1^{-1}dg_1)|
\, \cdots \, |B^{s-1}(g_n^{-1}dg_n)]\bigr\rvert,\] The canonical connection
1-form $B^s(z^{-1}dz)$ on $B^s\ensuremath{\mathbb{C}^\ast}\xspace$ is a 1-form on $B^s\ensuremath{\mathbb{C}^\ast}\xspace$ so that
$B^s(z^{-1}dz)$ evaluated at $|t_1, \, \ldots \, , t_n, [g_1|\, \cdots
\, |g_n]|$ is given by
\[\bigl\lvert t_1, \, \ldots \, , t_n, [B^{s-1}(g_1^{-1}dg_1)|
\, \cdots \, |B^{s-1}(g_n^{-1}dg_n)]\bigr\rvert, \] where $g_0, g_1, \, \ldots
\, , g_n \in B^{s-1}\ensuremath{\mathbb{C}^\ast}\xspace$ and $B^{s-1}(g_i^{-1}dg_i)$ is the canonical
connection 1-form $B^{s-1}(z^{-1}dz)$ on $B^{s-1}\ensuremath{\mathbb{C}^\ast}\xspace$ evaluated at
$g_i$.
\end{Exa}
\subsection{Connections on principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles}
A {\em connection} on a smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E\rightarrow M$ is
a collection $\{\omega_i\in \ensuremath{\Gamma}\xspace(U_i,B^s\A{1})\}_{i\in I}$ of
$B^s\ensuremath{\mathbb{C}}\xspace$-valued 1-forms, for some open covering $\{U_i\}_{i\in I}$ of
$M$, such that for every $i,j\in I$ so that $U_i\cap U_j \neq
\emptyset$
\[\omega_i -\omega_j =g_{ij}^{\ast}B^s(z^{-1}dz),\]
where $g_{ij}: U_i\cap U_j \rightarrow B^s\ensuremath{\mathbb{C}^\ast}\xspace$ is a
transition function of the bundle $E\rightarrow M$.
Equivalently, a {\em connection} on a smooth principal
$B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E\rightarrow M$ is given by a $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-equivariant global
section of the sheaf $B^s\ensuremath{\mathcal{A}}\xspace^1_{E,\ensuremath{\mathbb{C}}\xspace}$.
The pull-back $g_{ij}^{\ast}B^s(z^{-1}dz)$ can be described explicitly
by the formula
$$g_{ij}^{\ast}B^s(z^{-1}dz) = \operatorname{dlog} (g_{ij}),$$
\noindent where $\operatorname{dlog} (g_{ij})$ \noindent is defined by the induction on $s$ as
follows. For any smooth function $f:U \rightarrow B^s\ensuremath{\mathbb{C}^\ast}\xspace$ given locally by the formula
\begin{gather*}
f(x) = \bigl\lvert t_1(x), \, \ldots \, , t_n(x), [f_1(x)| \, \cdots \, |f_n(x)]\bigr\rvert,
\end{gather*}
\noindent where $f_i(x): U \rightarrow B^{s-1}\ensuremath{\mathbb{C}}\xspace$, we define $\operatorname{dlog} f \in
\ensuremath{\Gamma}\xspace(U,B^s\A{1})$ by
\begin{gather*}\label{dlog}
\operatorname{dlog} f(x) = \bigl\lvert t_1(x), \, \ldots \, , t_n(x), [\operatorname{dlog} f_1(x)| \,
\cdots \, |\operatorname{dlog} f_n(x)]\bigr\rvert.
\end{gather*}
It is easy to see that if $f:U \rightarrow B^s\ensuremath{\mathbb{C}^\ast}\xspace$ and $\operatorname{dlog} (f) =0$, then
$f:U \rightarrow B^s(\ensuremath{\mathbb{C}^\ast}\xspace)^{\delta}$, where $(\ensuremath{\mathbb{C}}\xspace^{\ast})^{\delta}$ is the group
$\ensuremath{\mathbb{C}^\ast}\xspace$ with the discreet topology.
\begin{Exa}\label{e2.2}\end{Exa}
\begin{enumerate}
\item It is easy to see that the differential 1-form $\go$
from Example~\ref{e2.1} is $G$-equivariant. Hence, it is a
connection 1-form on the universal principal $G$-bundle $EG \rightarrow BG$.
We will call it {\em the canonical connection 1-form} of $EG \rightarrow BG$.
For $G=\ensuremath{\mathbb{C}^\ast}\xspace$ the form $\go$ evaluated at
$|x_0, x_1, \, \ldots \, , x_n, z_0, z_1, \, \ldots \, , z_n|$
is given by the formula
$$\go_{|x_0, x_1, \, \ldots \, , x_n, z_0, z_1, \, \ldots \, , z_n|}=
\sum_{i=0}^n x_i \frac{dz_i}{z_i}.$$
Note that $\go$ is the composition $l\circ E\go$, where $E\go$ is
the canonical $E\ensuremath{\mathbb{C}}\xspace$-valued connection 1-form on $E\ensuremath{\mathbb{C}^\ast}\xspace$ and $l: E\ensuremath{\mathbb{C}}\xspace
\rightarrow \ensuremath{\mathbb{C}}\xspace$ is the splitting of the short exact sequence\xspace
$$0 \longrightarrow \ensuremath{\mathbb{C}}\xspace \longrightarrow E\ensuremath{\mathbb{C}}\xspace \longrightarrow B\ensuremath{\mathbb{C}}\xspace \longrightarrow 0$$
given by the formula
\begin{gather}\label{splitting}
l(|x_0, x_1, \, \ldots \, , x_n, z_0, z_1, \, \ldots \, , z_n|) =
\sum_{i=0}^n x_i z_i.
\end{gather}
\item The canonical connection 1-form on the universal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle
$$B^s\ensuremath{\mathbb{C}^\ast}\xspace \longrightarrow EB^s\ensuremath{\mathbb{C}^\ast}\xspace \longrightarrow B^{s+1}\ensuremath{\mathbb{C}^\ast}\xspace$$
\noindent can be defined as the composition $l\circ EB^s(z^{-1}dz)$, where
$l:EB^s\ensuremath{\mathbb{C}}\xspace \rightarrow B^s\ensuremath{\mathbb{C}}\xspace$ is the splitting of the short exact sequence\xspace
$$0 \longrightarrow B^s\ensuremath{\mathbb{C}}\xspace \longrightarrow EB^s\ensuremath{\mathbb{C}}\xspace \longrightarrow B^{s+1}\ensuremath{\mathbb{C}}\xspace \longrightarrow 0$$
\noindent given by the formula \eqref{splitting}, where now $z_i \in B^s\ensuremath{\mathbb{C}}\xspace$.
\item From Corollary~\ref{smoothco} and the above example it follows that
every smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle carries a connection.
In particular, the smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle over $S^3$ from
Example~\ref{e1.1} can be equipped with the connection 1-form $\go$
that is the pull-back of the canonical connection 1-form $l\circ E\go$
on $E\ensuremath{\mathbb{C}^\ast}\xspace \rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace$. Following J-L. Brylinski and P. Deligne (see
\cite[Chapter~7]{bry-greenbook}) one can interpret \go as the Dirac
monopole.
\end{enumerate}
The ordinary exterior derivative $d: \cl{A}^k_{M,\mathbb{C}} \rightarrow
\cl{A}^{k+1}_{M,\mathbb{C}}$ has an extension
\[ d:B^s\cl{A}^k_{M,\mathbb{C}} \longrightarrow
B^s\cl{A}^{k+1}_{M,\mathbb{C}} \]
\noindent defined inductively by the formula
\[ d\bigl(\bigl\lvert t_1(x), \, \ldots \, , t_n(x), [\alpha_1(x)|\, \cdots \,
|\alpha_n(x)]\bigr\rvert\bigr)=
\bigl\lvert t_1(x), \, \ldots \, , t_n(x), [d\alpha_1(x)|\, \cdots \,
|d\alpha_n(x)]\bigr\rvert. \]
Similarly, we define
$$d: EB^s\cl{A}^k_{M,\mathbb{C}} \longrightarrow EB^s\cl{A}^{k+1}_{M,\mathbb{C}}.$$
Note that $d \bigl(B^s(z^{-1}dz)\bigr) =0$, and hence,
\[d\omega_i -d\omega_j =dg_{ij}^{\ast}(B^s(z^{-1}dz)) =
g_{ij}^{\ast}(d B^s(z^{-1}dz)) =
g_{ij}^{\ast}(B^s(d(z^{-1}dz))) =0.
\]
Thus, the family $\{d\omega_i\}$ defines a global section \ensuremath{\Omega}\xspace of the
sheaf $B^s\A{2}$, which is by definition the {\em curvature} of the
connection $\{\omega_i\}$.
\noindent {\bf Proof of Theorem B. } The exponential short exact sequence\xspace
\begin{gather}\label{exp}
0\longrightarrow \u{\ensuremath{\mathbb{Z}}\xspace} \longrightarrow \u{\ensuremath{\mathbb{C}}\xspace}^\ensuremath{\delta}\xspace \longrightarrow \u{(\ensuremath{\mathbb{C}^\ast}\xspace)}^\ensuremath{\delta}\xspace \longrightarrow 0
\end{gather}
induces the short exact sequence\xspace of the bar cochain complexes
\[ 0\longrightarrow C_B^\ast(\u{\ensuremath{\mathbb{Z}}\xspace}) \longrightarrow C_B^\ast(\u{\ensuremath{\mathbb{C}}\xspace}^\ensuremath{\delta}\xspace)
\longrightarrow C_B^\ast(\u{(\ensuremath{\mathbb{C}^\ast}\xspace)}^\ensuremath{\delta}\xspace) \longrightarrow 0 \]
For every abelian Lie group $G$, the group $H^p(M;G)$ is isomorphic to
the group $H^p(C_B^\ast(\u{G}))$, which in turn can be identified with the
group $L(M, B^{p-2}G)$ of isomorphism classes of smooth principal
$B^{p-2}G$-bundles over $M$ (see Remark~\ref{rem1}). Therefore, the
cohomology long exact sequence\xspace associated with the exponential short exact sequence\xspace \eqref{exp}
induces a commutative diagram {\small
\[{\diagram
\rto & {H^{p-1}(M;\ensuremath{\mathbb{Z}}\xspace)} \rto\dto^-{\cong}&
{H^{p-1}(M;\ensuremath{\mathbb{C}}\xspace)} \rto\dto^-{\cong}&
{H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace)} \rto\dto^-{\cong}&
{H^{p}(M;\ensuremath{\mathbb{Z}}\xspace)} \rto\dto^-{\cong}& \\
\rto & {L(M, B^{p-2}\ensuremath{\mathbb{Z}}\xspace)} \rto&
{L(M, B^{p-2}\ensuremath{\mathbb{C}}\xspace^\ensuremath{\delta}\xspace)} \rto&
{L(M, B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace)} \rto^-f&
{L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace)} \rto&
\enddiagram} \]}
\noindent where the isomorphism $H^{p}(M;\ensuremath{\mathbb{Z}}\xspace) \rightarrow L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace)$ is the
composition of isomorphisms
\[ H^{p}(M;\ensuremath{\mathbb{Z}}\xspace) \longrightarrow L(M, B^{p-1}\ensuremath{\mathbb{Z}}\xspace) \longrightarrow L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace) \]
\noindent and
\[ f : L(M, B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace) \longrightarrow L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace) \]
is the forgetful homomorphism induced by the homomorphism
$B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace \rightarrow B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$.
It is easy to see that the above diagram induces the following
commutative diagram
\[{\diagram
{0} \rto & {\frac{H^{p-1}(M;\ensuremath{\mathbb{C}}\xspace)}{H^{p-1}(M;\ensuremath{\mathbb{Z}}\xspace)_{TF}}} \rto\dto^-{\cong}&
{H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace)} \rto\dto^-{\cong}&
{\operatorname{Tors} H^{p}(M;\ensuremath{\mathbb{Z}}\xspace)} \rto\dto^-{\cong}& {0} \\
{0} \rto & {\ker (f)} \rto&
{L(M, B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace)} \rto^-f&
{\operatorname{im} (f)} \rto& {0}
\enddiagram} \]
\noindent whose rows are exact sequences.
In order to finish the proof of Theorem~B, we have to show the for
every $q>p$ there is an isomorphism
\begin{gather}\label{basiciso}
\HD{p}{q} \cong H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace)
\end{gather}
\noindent and that the group $L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,\nabla^{\text{flat}})$ of
isomorphism classes of flat connections on smooth principal
$B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundles over $M$ is isomorphic to
$L(M, B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace)$.
The isomorphism \eqref{basiciso} follows from the fact that there is a
quasi-isomorphism
\begin{gather}\label{quasi-iso}
{\diagram
{\u{\ensuremath{\mathbb{Z}}\xspace(q)}} \rto\dto& {\A{0}} \rto^d\dto^{\alpha}&
{\A{1}}\rto^d\dto^{(2\pi\sqrt{-1})^{1-q}} &
{\dots} \rto^d& {\A{q-1}}\dto^{(2\pi\sqrt{-1})^{1-q}}\\
{0} \rto & {\us{\ensuremath{\mathbb{C}}\xspace}} \rto^\operatorname{dlog}& {\A{1}} \rto^d& {\dots}
\rto^d& {\A{q-1}}
\enddiagram}
\end{gather}
where $\alpha(f) = \exp((2\pi\sqrt{-1})^{1-q}\cdot f)$, between the
smooth Deligne complex $\ensuremath{\mathbb{Z}}\xspace(q)^{\infty}_D$ and the complex
$\mathcal{A}^{<q}_{M,\mathbb{C}}(\operatorname{dlog})[-1]$, where
\begin{align*}
\begin{CD}
\quad &\mathcal{A}^{<q}_{M,\mathbb{C}}(\operatorname{dlog}): \qquad
\us{\ensuremath{\mathbb{C}}\xspace}& @>\operatorname{dlog}>>
\A{1} @>\text{d}>>
\, \cdots \, @>\text{d}>> \A{q-1}\\
\intertext{is the truncation of the complex}
\quad &\mathcal{A}^{\ast}_{M,\mathbb{C}}(\operatorname{dlog}): \qquad
\us{\ensuremath{\mathbb{C}}\xspace}& @>\operatorname{dlog}>>
\A{1} @>\text{d}>>
\, \cdots \, @>\text{d}>> \A{q-1}@>\text{d}>> \, \cdots \,
\end{CD}
\end{align*}
\noindent which is an acyclic resolution of the constant sheaf $\ensuremath{\mathbb{C}^\ast}\xspace_M$. Therefore,
for every $q>p$ there are isomorphisms
$$\HD{p}{q} \cong \mathbb{H}^{p-1}(\mathcal{A}^{<q}_{M,\mathbb{C}}(\operatorname{dlog}))\cong
\mathbb{H}^{p-1}(\mathcal{A}^{\ast}_{M,\mathbb{C}}(\operatorname{dlog}))\cong
H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace).$$
The isomorphism
\[ L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,\nabla^{\text{flat}}) \cong
L(M, B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace) \]
\noindent is a consequence of the following lemma.
{\Le \label{flat1} There is a one-to-one correspondence between flat
connections on a smooth principal $B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E\rightarrow M$ and
reductions of the structure group of $E\rightarrow M$ to $B^{p-2}(\ensuremath{\mathbb{C}^\ast}\xspace)^\ensuremath{\delta}\xspace$.}
\noindent{\bf Proof. \ } Let $E\rightarrow M$ be a smooth principal $B^{s}\ensuremath{\mathbb{C}}\xspace^{\ast}$-bundle with
a flat connection. We will show that the structure group of $E\rightarrow M$
can be reduced to $B^{s}(\ensuremath{\mathbb{C}}\xspace^{\ast})^{\delta}$ or equivalently, that
$E\rightarrow M$ has transition functions $\tilde{g}_{ij}: U_{ij} \rightarrow
B^{s}(\ensuremath{\mathbb{C}}\xspace^{\ast})^{\delta}$.
Let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open covering of $M$
consisting of contractible subsets of $M$ and let $\{g_{ij}: U_{ij}
\rightarrow B^{s}\ensuremath{\mathbb{C}}\xspace^{\ast}\}_{i,j\in I}$ be a family of transition functions of $E\rightarrow
M$. Suppose, $\{\omega_i\in \ensuremath{\Gamma}\xspace(U_i,B^s\A{1})\}_{i\in I}$ is a flat
connection on $E\rightarrow M$. That is every $\omega_i$ is a closed form and
for every $i,j$ so that $U_i \cap U_j \neq \emptyset$
\[\omega_i - \omega_j =\operatorname{dlog} g_{ij}.\]
It is easy to see (using the Poincare Lemma and the induction on $s$)
that if $d \omega_i = 0$, then there is a $B^s\ensuremath{\mathbb{C}}\xspace$-valued function
$f_i$ such that $df_i =\omega_i$.
For any smooth function $f:U\rightarrow B^s\ensuremath{\mathbb{C}}\xspace$ we define, by the induction
on $s$, the $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-valued function $\exp(f)$. If $f:U\rightarrow
B^s\ensuremath{\mathbb{C}}\xspace$ is given locally by the formula
$$f(x) = \bigl\lvert t_1(x), \, \ldots \, , t_n(x), [g_1(x)| \, \cdots \,
|g_n(x)]\bigr\rvert,$$
\noindent where $g_i: U\rightarrow B^{s-1}\ensuremath{\mathbb{C}}\xspace$, then
$$\exp f(x) = \bigl\lvert t_1(x), \, \ldots \, , t_n(x), [\exp g_1(x)| \, \cdots \,
|\exp
g_n(x)]\bigr\rvert.$$
Since $\operatorname{dlog} (\exp f) = df$,
\[\operatorname{dlog} g_{ij} = \omega_i - \omega_j = d(f_i - f_j) = \operatorname{dlog}
(\exp (f_i - f_j)) = -\operatorname{dlog} (\delta(\exp f)_{ij}).
\]
Therefore, for $\tilde{g}_{ij} = g_{ij} +\delta(\exp f)_{ij}$
\[\operatorname{dlog} \tilde{g}_{ij} = 0\]
\noindent and hence $\tilde{g}_{ij}: U_{ij}\rightarrow B^{s}(\ensuremath{\mathbb{C}}\xspace^{\ast})^{\delta}$.
The family $\{\tilde{g}_{ij}\}$ gives the required transition
functions of $E\rightarrow M$.
Now suppose, $E\rightarrow M$ is a principal $B^{s}\ensuremath{\mathbb{C}}\xspace^{\ast}$-bundle with
transition functions $g_{ij}: U_{ij}\rightarrow B^{s}(\ensuremath{\mathbb{C}}\xspace^{\ast})^{\delta}$. A
flat connection on $E\rightarrow M$ is given by the family $\{\omega_i\}$ of
trivial (tautologicly equal to zero) 1-forms. Obviously, $d\omega_i=0$
and $\omega_i - \omega_j = 0 = \operatorname{dlog} g_{ij}$. $\hfill\Box$
\section{$k$-connections on principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles}
In this section we define $k$-connections, $k=1,\ldots ,s+1$, and
scalar curvatures on smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles and prove
Theorems A and C. In particular, we show that for $p\geq 2$ the group
$\HD{p}{p}$ is isomorphic to the group of equivalence classes of
smooth principal $B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with $k$-connections for
$k=1,\ldots ,p-1$.
By definition, a {\em 1-connection} on smooth principal
$B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle is a connection on this bundle. To motivate a
definition of a $k$-connection for $k\geq 2$, we will first
reformulate the standard definition of a connection on smooth
principal $\ensuremath{\mathbb{C}^\ast}\xspace$-bundle.
A smooth principal $\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E\rightarrow B$ is given either by a family
\[ \{g_{ij}: U_i\cap U_j \rightarrow \ensuremath{\mathbb{C}^\ast}\xspace\}_{i,j\in I} \]
of transition functions associated with an open covering
$\cl{U}=\{U_i\}_{i\in I}$ of $M$ or by a smooth map $g:M\rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace$ so
that $E\rightarrow B = g^\ast (E\ensuremath{\mathbb{C}^\ast}\xspace \rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace)$. The map $g$ can be described
in terms of the transition functions $\{g_{ij}\}$ by the formula
\[ g(x) = \bigl\lvert f_{i_0}(x), f_{i_1}(x),\ldots ,f_{i_{n}}(x),
[g_{i_0i_1}(x)|g_{i_1i_2}(x)|\; \cdots \;
|g_{i_{n-1}i_n}(x)]\bigr\rvert,
\]
wher $\{f_i\}_{i\in I}$ is a partition of unity subordinated to the
covering $\cl{U}$ (see the proof of Proposition~\ref{smoothpr}).
Classically, a connection on $E\rightarrow B$ is given by a family of 1-forms
$\{\go_i \in \ensuremath{\Gamma}\xspace(U_i,A^{1}_\ensuremath{\mathbb{C}}\xspace)\}_{i\in I}$ so that
\[ \go_i - \go_j = \operatorname{dlog} g_{ij},\qquad \text{on}\quad U_i\cap U_j\neq
\emptyset\]
Alternatively, in terms of the map $g:M\rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace$, a connection on
$E\rightarrow B$ is a global section -\go of the sheaf $E\A{1}$ so that
$\pi_\ast \go = \operatorname{dlog} g$, where $\pi_\ast : \ensuremath{\Gamma}\xspace(M,E\A{1})\rightarrow
\ensuremath{\Gamma}\xspace(M,B\A{1})$ is the homomorphism induced by the morphism of sheaves
$\pi : E\A{1}\rightarrow B\A{1}$.
Indeed, if \go is given by
\[ \go(x) = \bigl\lvert f'_{i_0}(x), f'_{i_1}(x),\, \ldots \, , f'_{i_n}(x),
\go_{i_0}(x), \go_{i_1}(x),\, \ldots \, , \go_{i_n}(x)\bigr\rvert, \] then
\[ \pi_\ast\go(x) = \bigl\lvert f'_{i_0}(x), f'_{i_1}(x),\, \ldots\, ,
f'_{i_n}(x), [\go_{i_0}(x): \go_{i_1}(x):\; \cdots \; :
\go_{i_n}(x)]\bigr\rvert,
\]
and in the non-homogeneous coordinates
\[ \pi_\ast\go(x) = \bigl\lvert f'_{i_0}(x), \ldots ,f'_{i_{n}}(x),
[\go_{i_1}(x)-\go_{i_0}(x)|\; \cdots \;
|\go_{i_n}(x)-\go_{i_{n-1}}(x)]\bigr\rvert. \]
Thus, the condition $\pi_\ast \go = \operatorname{dlog} g$ is equivalent to the
system of equations
\[\begin{cases}
f'_i = f_i\\ \go_j - \go_i = \operatorname{dlog} g_{ij}
\end{cases}\]
where the second equation holds for all $x\in M$ so that $f_i(x)\neq
0$ and $f_j(x)\neq 0$.
Since $\{f_i\}_{i\in I}$ is a partition of unity on $M$, the sets
$U_i=\{x\in M\, |\; f_i(x)\neq 0\}$ form an open covering of $M$ and
the family of 1-forms $\{-\go_i \in \ensuremath{\Gamma}\xspace(U_i,\cl{A}^{1}_\ensuremath{\mathbb{C}}\xspace)\}_{i\in I}$
determines a connection on a smooth principal \ensuremath{\mathbb{C}^\ast}\xspace-bundle induced by
the map $g:M\rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace$.
In other words, the group $\tilde{L}(M,\ensuremath{\mathbb{C}^\ast}\xspace,\nabla)$ of smooth principal
\ensuremath{\mathbb{C}^\ast}\xspace-bundles with connections over $M$ is the pull-back
\[\begin{CD}
\tilde{L}(M,\ensuremath{\mathbb{C}^\ast}\xspace, \nabla) @>>> \ensuremath{\Gamma}\xspace(M,E\A{1})\\
@VVV @VV{\pi_\ast}V \\ \cinf{B\ensuremath{\mathbb{C}^\ast}\xspace} @>\operatorname{dlog}>>
\ensuremath{\Gamma}\xspace(M,B\A{1})
\end{CD}\]
of the projection $\pi_\ast : \ensuremath{\Gamma}\xspace(M,E\A{1})\rightarrow \ensuremath{\Gamma}\xspace(M,B\A{1})$ by the
homomorphism
\[\operatorname{dlog}:\cinf{B\ensuremath{\mathbb{C}^\ast}\xspace}\longrightarrow \ensuremath{\Gamma}\xspace(M,B\A{1}). \]
The group $L(M, \ensuremath{\mathbb{C}^\ast}\xspace,\nabla)$ of the isomorphism classes of smooth
principal \ensuremath{\mathbb{C}^\ast}\xspace-bundles over $M$ is the quotient of $\tilde{L}(M,
\ensuremath{\mathbb{C}^\ast}\xspace,\nabla)$ by the action of $\cinf{E\ensuremath{\mathbb{C}^\ast}\xspace}$ given by the formula
\[f\cdot (g,\go) = (g+\pi_\ast(f),\go+\operatorname{dlog}(f)),\]
where $\pi_\ast:\cinf{E\ensuremath{\mathbb{C}^\ast}\xspace}\rightarrow \cinf{B\ensuremath{\mathbb{C}^\ast}\xspace}$ is the homomorphism
induced by the projection $\pi:E\ensuremath{\mathbb{C}^\ast}\xspace\rightarrow B\ensuremath{\mathbb{C}^\ast}\xspace$.
Essentially the same as above arguments show that a connection on a
smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle induced by a map $g:M\rightarrow
B^{s+1}\ensuremath{\mathbb{C}^\ast}\xspace$ is given by a global section -\go of the sheaf $EB^s\A{1}$
so that $\operatorname{dlog} g = \pi_\ast \go$, where
\[\pi_\ast: \ensuremath{\Gamma}\xspace(M,EB^s\A{1}) \longrightarrow \ensuremath{\Gamma}\xspace(M,B^{s+1}\A{1}).\]
Moreover, two pairs $(g,\go), (g',\go')\in \cinf{B^{s+1}\ensuremath{\mathbb{C}^\ast}\xspace}\oplus
\ensuremath{\Gamma}\xspace(M,EB^s\A{1})$ determine isomorphic smooth principal
$B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with connections if and only if there is a smooth
map $h:M\rightarrow EB^s\ensuremath{\mathbb{C}^\ast}\xspace$ so that
\begin{gather*}
(g,\go) = (g'+\pi_\ast h, \go'+\operatorname{dlog} h).
\end{gather*}
Now, we are going to define a 2-connection of the isomorphism class
$[E,\go]$ of a smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E\rightarrow M$ with a
connection \go.
Let $E\rightarrow M$ be a smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle induced from the
universal principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $EB\ensuremath{\mathbb{C}^\ast}\xspace\rightarrow B^2\ensuremath{\mathbb{C}^\ast}\xspace$ by a map
$g:M\rightarrow B^2\ensuremath{\mathbb{C}^\ast}\xspace$ and let $\go\in \ensuremath{\Gamma}\xspace(M, EB\A{1})$ be a connection on
$E\rightarrow M$. That is, $\pi_\ast\go = \operatorname{dlog} g$, where
$\pi_\ast:\ensuremath{\Gamma}\xspace(M,EB\A{1})\rightarrow \ensuremath{\Gamma}\xspace(M,B^2\A{1})$ is the homomorphism
induced by the morphism of sheaves $\pi:EB\A{1}\rightarrow B^2\A{1}$.
The curvature $-d\go$ of the connection -\go is a global section of the
sheaf $B\A{2}$, because the sequence
\[0\longrightarrow \ensuremath{\Gamma}\xspace(M,B\A{2}) \overset{i_\ast}{\longrightarrow} \ensuremath{\Gamma}\xspace(M,EB\A{2})
\overset{\pi_\ast}{\longrightarrow} \ensuremath{\Gamma}\xspace(M,B^2\A{2}) \longrightarrow 0 \]
is exact and $\pi_\ast (d\go) = d(\pi_\ast\go) = d(\operatorname{dlog} g) = 0$.
If $(g',\go')$ determines an isomorphic to $(E,\go)$ smooth principal
$B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle with a connection, then $d\go =d(\go'+\operatorname{dlog} h) =
d\go'$, and hence a curvature determines a homomorphism
\[ d : L (M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla) \longrightarrow \ensuremath{\Gamma}\xspace(M,B\A{2}), \]
where $L (M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla)$ is the group of isomorphism classes of
smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with connections over $M$.
Consider the following pull-back diagram
\[ \begin{CD}
\tilde{L}(M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1, \nabla_2) @>>> \ensuremath{\Gamma}\xspace(M,E\A{2})\\
@VVV @VV{-\pi_\ast}V \\ L(M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla) @>d>>
\ensuremath{\Gamma}\xspace(M,B\A{2})
\end{CD}
\]
The group $\tilde{L}(M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1, \nabla_2)$ consists of elements
$([g,\go_1],\go_2)$, where $g:M\rightarrow B^2\ensuremath{\mathbb{C}^\ast}\xspace$ is a smooth map,
$[g,\go_1]$ is the isomorphism class of a smooth principal
$B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $g^\ast(EB\ensuremath{\mathbb{C}^\ast}\xspace \rightarrow B^2\ensuremath{\mathbb{C}^\ast}\xspace)$ with a connection $-\go_1$,
and $\go_2$ is a global section of the sheaf $E\A{2}$ so that
$-\pi_\ast (\go_2) = d\go_1$. The equation $-\pi_\ast \go_2 = d\go_1$ is
an analogue of the connection condition $-\pi_\ast \go = \operatorname{dlog} g$,
therefore we will refer to $\go_2$ as a {\em 2-connection} of the
equivalence class $[g,\go_1]$ of the pair $(g,\go_1)$.
Note that there is an action
\[ \ensuremath{\Gamma}\xspace(M,E\A{1}) \times \tilde{L}(M, B\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1, \nabla_2)
\longrightarrow \tilde{L}(M, B\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1, \nabla_2) \] of $\ensuremath{\Gamma}\xspace(M,E\A{1})$ on
$\tilde{L}(M, B\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1, \nabla_2)$ given by
\[\ensuremath{\alpha}\xspace \cdot ([g,\go_1],\go_2) = ([g,\go_1-\ensuremath{\sigma}\xspace(\ensuremath{\alpha}\xspace)],\go_2+d\ensuremath{\alpha}\xspace),\]
where \ensuremath{\sigma}\xspace is the composition
\[ \ensuremath{\Gamma}\xspace(M,E\A{1}) \overset{\pi_\ast}{\longrightarrow} \ensuremath{\Gamma}\xspace(M,B\A{1}) \overset{i_\ast}{\longrightarrow}
\ensuremath{\Gamma}\xspace(M,EB\A{1}). \] The quotient
\[L(M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1, \nabla_2) = \dfrac{\tilde{L}(M, B\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1,
\nabla_2)}{\ensuremath{\Gamma}\xspace(M,E\A{1})} \] will be called the group of equivalence
classes of smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with 1 and 2-connections
over $M$.
For $s\geq 1$, a group
\[L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1,\nabla_2,\ldots , \nabla_{s+1}) =
L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,\{\nabla_i\}_{i=1}^{s+1})\] of equivalence classes of smooth
principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with k-connections, $k=1,2,\ldots, s+1$,
over $M$ will be defined by the following inductive procedure.
Suppose, we have already constructed the group $L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace, \{
\nabla_i\}_{i=1}^k)$ of equivalence classes $[g,\go_1, \ldots ,\go_k]$ of
smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with $j$-connections, for $1\leq
j\leq k<s+1$, over $M$. Then the group $L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,
\{\nabla_i\}_{i=1}^{k+1})$ is defined as follows.
Consider the pull-back diagram
\[ \begin{CD}
\tilde{L}(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1, \ldots ,\nabla_{k+1}) @>>>
\ensuremath{\Gamma}\xspace(M,EB^{s-k}\A{k+1})\\ @VVV @VV{(-1)^k\pi_\ast}V \\
L (M, B^s\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1,\ldots , \nabla_k) @>d>> \ensuremath{\Gamma}\xspace(M,B^{s-k+1}\A{k+1}),
\end{CD}
\]
where $d([g,\go_1, \ldots ,\go_k])=d\go_k$. There is an action
\[ \ensuremath{\Gamma}\xspace(M,EB^{s-k}\A{k}) \times
\tilde{L}(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1, \ldots ,\nabla_{k+1}) \longrightarrow \tilde{L}(M,
B^s\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1, \ldots ,\nabla_{k+1})
\]
of $\ensuremath{\Gamma}\xspace(M,EB^{s-k}\A{k})$ on $\tilde{L}(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,\nabla_1, \ldots
,\nabla_{k+1})$ given by
\[ \ensuremath{\alpha}\xspace \cdot ([g,\go_1,\ldots ,\go_k],\go_{k+1}) =
([g,\go_1,\ldots ,\go_{k-1}, \go_k+(-1)^k\ensuremath{\sigma}\xspace(\ensuremath{\alpha}\xspace)],\go_{k+1}+d\ensuremath{\alpha}\xspace), \]
where \ensuremath{\sigma}\xspace is the composition
\[\ensuremath{\Gamma}\xspace(M,EB^{s-k}\A{k}) \overset{\pi_\ast}{\longrightarrow}
\ensuremath{\Gamma}\xspace(M,B^{s-k+1}\A{k}) \overset{i_\ast}{\longrightarrow}
\ensuremath{\Gamma}\xspace(M,EB^{s-k+1}\A{k}). \]
We set
\[ L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace, \nabla_1, \ldots ,\nabla_{k+1}) = \dfrac{\tilde{L}(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,
\nabla_1, \ldots ,\nabla_{k+1})}{ \ensuremath{\Gamma}\xspace(M,EB^{s-k}\A{k})}. \]
The form $(-1)^{k+1}\omega_{k+1}$, where $\go_{k+1}$ is the component
of an element $([g,\go_1,\ldots ,\go_k],\go_{k+1})$ of $\tilde{L}(M,
B^s\ensuremath{\mathbb{C}^\ast}\xspace,\{\nabla_i\}_{i=1}^{k+1})$ is called a {\em $(k+1)$-connection}
of $[g,\go_1,\ldots ,\go_k]$. The image of $([g,\go_1,\ldots
,\go_k],\go_{k+1})$ in $L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,\{\nabla_i\}_{i=1}^{k+1})$ will be
denoted by $[g,\go_1,\ldots ,\go_{k+1}]$.
Iterating the above procedure we get the group $L(M,
B^s\ensuremath{\mathbb{C}^\ast}\xspace,\{\nabla_i\}_{i=1}^{s+1})$ of equivalence classes of smooth
principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with $k$-connections, $k=1, \ldots ,s+1$.
\begin{Prop}\label{prop3.2}
For every $p\geq 2$ there is an isomorphism
\[ \HD{p}{p} \quad \cong \quad L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1})
\]
\end{Prop}
\noindent{\bf Proof. \ } Consider the double complex
\[ \begin{CD}
@AAdA @AAdA @AAdA\\ \ensuremath{\Gamma}\xspace(M,E\A{2}) @>>\ensuremath{\sigma}\xspace> \ensuremath{\Gamma}\xspace(M,EB\A{2}) @>>\ensuremath{\sigma}\xspace>
\ensuremath{\Gamma}\xspace(M,EB^2\A{2}) @>>\ensuremath{\sigma}\xspace> \\ @AAdA @AAdA @AAdA\\ \ensuremath{\Gamma}\xspace(M,E\A{1})
@>>\ensuremath{\sigma}\xspace> \ensuremath{\Gamma}\xspace(M,EB\A{1}) @>>\ensuremath{\sigma}\xspace> \ensuremath{\Gamma}\xspace(M,EB^2\A{1}) @>>\ensuremath{\sigma}\xspace> \\
@AA{\operatorname{dlog}}A @AA{\operatorname{dlog}}A @AA{\operatorname{dlog}}A\\ \cinf{E\ensuremath{\mathbb{C}^\ast}\xspace} @>>\ensuremath{\sigma}\xspace>
\cinf{EB\ensuremath{\mathbb{C}^\ast}\xspace} @>>\ensuremath{\sigma}\xspace> \cinf{EB^2\ensuremath{\mathbb{C}^\ast}\xspace} @>>\ensuremath{\sigma}\xspace>
\end{CD} \]
of the bar cochain complexes of the components of $\A{\ast}(\operatorname{dlog})$.
There is a sequence of isomorphisms
\[ \HD{p}{p} \cong \ensuremath{\mathbb{H}}\xspace^{p-1}(\A{<p}(\operatorname{dlog}))\cong
H^{p-1}(\text{Tot}^\ast(B^{\ast,<p}_M),D), \] where
$(\text{Tot}^\ast(B^{\ast,<p}_M),D)$ is the total complex of the
double complex $B^{\ast,<p}_M = \{B^{n,s}_M\}_{s<p}$ defined as
follows
\begin{gather*}
\operatorname{Tot}^m(B^{\ast,<p}_M) = \bigoplus\limits_{n+s=m,
s<p}B^{n,s}_M\\[6pt] B^{n,s}_M =
\begin{cases}
C^\infty(M, EB^n\ensuremath{\mathbb{C}^\ast}\xspace) &\text{for}\quad s=0,n\geq 0\\
\ensuremath{\Gamma}\xspace(M,EB^n\A{s}) &\text{for}\quad s>0,n\geq 0\\
0 &\text{for}\quad s<0 \thickspace \text{or}\thickspace
n< 0
\end{cases} \\
\intertext{and} D: \operatorname{Tot}^m(B^{\ast,<p}_M) \longrightarrow
\operatorname{Tot}^{m+1}(B^{\ast,<p}_M)\\ \intertext{is so that} D =
\begin{cases}
\operatorname{dlog} +\ensuremath{\sigma}\xspace &\text{on}\quad B^{n,0}_M\\ d+ (-1)^s\ensuremath{\sigma}\xspace
&\text{on}\quad B^{n,s}_M \quad \text{for $s>0$.}
\end{cases}
\end{gather*}
A $(p-1)$-cocycle in $(\text{Tot}^\ast(B^{\ast,<p}_M),D)$ is a
sequence $(g,\go_1,\ldots ,\go_{p-1})$, where $g\in
\cinf{EB^{p-1}\ensuremath{\mathbb{C}^\ast}\xspace}$ and $\go_i\in \ensuremath{\Gamma}\xspace(M,EB^{p-i-1}\A{i})$ so that
\[\begin{cases}
\ensuremath{\sigma}\xspace(g)=0&\\ \operatorname{dlog} g = \ensuremath{\sigma}\xspace(\go_1)&\\ d\go_i = (-1)^{i}\ensuremath{\sigma}\xspace(\go_{i+1})&
\text{for}\; 1\leq i\leq p-2
\end{cases}
\]
The condition $\ensuremath{\sigma}\xspace(g)=0$ means that $g$ is a smooth map from $M$ to
$B^{p-1}\ensuremath{\mathbb{C}^\ast}\xspace$, the condition $\operatorname{dlog} g = \ensuremath{\sigma}\xspace(\go_1)$ means that
$-\go_1$ is a connection on the smooth principal $B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundle
over $M$ induced by $g$, and the conditions $d\go_i =
(-1)^{i}\ensuremath{\sigma}\xspace(\go_{i+1})$ mean that $(-1)^{i+1}\go_{i+1}$ is a
$(i+1)$-connection of the sequence $(g,\go_1, \ldots ,\go_i)$. It is
easy to see that two cocycles $(g,\go_1,\ldots ,\go_{p-1})$ and
$(g',\go'_1,\ldots ,\go'_{p-1})$ are cohomologous in
$(\text{Tot}^\ast(B^{\ast,<p}_M),D)$ if and only if the corresponding
principal bundles with connections are equivalent in $L(M,
B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1})$. Thus we get an isomorphism
\[ H^{p-1}(\text{Tot}^\ast(B^{\ast,<p}_M),D) \cong L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,
\{\nabla_i\}_{i=1}^{p-1}). \] \qed
The rest of the section is devoted to proofs of Theorems A and C.
\noindent {\bf Proof of Theorem C.} Let us start from a definition of a
scalar curvature.
The {\em scalar curvature} of the element $[g,\go_1, \go_2, \ldots,
\go_{p-1}]$ of $L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1})$ is the
\ensuremath{\mathbb{C}}\xspace-valued $p$-form $(-1)^{p-1}d\go_{p-1}$. Note, that a priori
$d\go_{p-1}$ is a a global section of the sheaf $E\A{p}$, because
$\go_{p-1} \in \ensuremath{\Gamma}\xspace(M,E\A{p-1})$. But $\pi_\ast (\go_{p-1}) =
d\go_{p-2}$, and hence, $\pi_\ast (d\go_{p-1}) = d(\pi_\ast \go_{p-1})
=d(d\go_{p-2}) = 0$. Therefore, $d\go_{p-1} \in A^p_\ensuremath{\mathbb{C}}\xspace (M)$.
Actually, $d\go_{p-1}$ is a closed (but not necessarily exact)
\ensuremath{\mathbb{C}}\xspace-valued $p$-form, because locally it is exact.
Thus, a scalar curvature induces a homomorphism
\[ s: L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1}) \longrightarrow
A^p_\ensuremath{\mathbb{C}}\xspace (M)_{cl}, \] \noindent where $A^p_\ensuremath{\mathbb{C}}\xspace (M)_{cl}$ is the group of
\ensuremath{\mathbb{C}}\xspace-valued closed $p$-forms on $M$.
The form $d\go_{p-1}$ is a $p$-cocycle in $\operatorname{Tot}^\ast(B^{\ast,\ast}_M)$
which is cohomologous to zero in this complex, because
$d\go_{p-1}=D(g,\go, \go_2, \ldots ,\go_{p-1})$.
Since $\operatorname{Tot}^\ast(B^{\ast,\ast}_M)$ is the acyclic resolution of
$\A{\ast}(\operatorname{dlog})$, which in turn is a resolution of the constant sheaf
of the group $\ensuremath{\mathbb{C}^\ast}\xspace$, the image of $d\go_{p-1}$ in
\[ H^p(\operatorname{Tot}^\ast(B^{\ast,\ast}_M)) \cong
H^p(\A{\ast}(\operatorname{dlog})) \cong H^p(M;\ensuremath{\mathbb{C}^\ast}\xspace) \] \noindent is zero. Therefore,
because the diagram
\[{\diagram
{A^p_\ensuremath{\mathbb{C}}\xspace (M)_{cl}} \drto\rrto&& {H^{p}(M;\ensuremath{\mathbb{C}^\ast}\xspace)}\\ &
{H^{p}(M;\ensuremath{\mathbb{C}}\xspace)}\urto & \enddiagram} \] \noindent commutes and the
sequence
\[ 0\longrightarrow H^{p}(M;\ensuremath{\mathbb{Z}}\xspace)_{TF} \longrightarrow H^{p}(M;\ensuremath{\mathbb{C}}\xspace) \longrightarrow H^{p}(M;\ensuremath{\mathbb{C}^\ast}\xspace) \]
\noindent is exact, the cohomology class of $d\go_{p-1}$ in $H^p(M;\ensuremath{\mathbb{C}}\xspace)$
belongs to the image of $H^p(M;\ensuremath{\mathbb{Z}}\xspace)$ in $H^p(M;\ensuremath{\mathbb{C}}\xspace)$. That is
$d\go_{p-1}$ is a closed form with integral periods. Thus, we showed
that the image $\operatorname{im} (s)$ of the scalar curvature homomorphism
\[ s: L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1}) \longrightarrow
A^p_\ensuremath{\mathbb{C}}\xspace (M)_{cl} \] is contained in the group $A^p_\ensuremath{\mathbb{C}}\xspace (M)_0$ of
\ensuremath{\mathbb{C}}\xspace-valued closed $p$-forms with integral periods on $M$.
Consider the following ``scalar curvature diagram''
\[{\diagram
{0} \rto& {L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \nabla^{\text{flat}})} \rto\dto & {L(M,
B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1})} \rto^-s\dto& {\operatorname{im}(s)}
\rto\dto^-i& {0}\\ {0} \rto& {\H{p-1}{C^{\ast}}} \rto& {\HD{p}{p}}
\rto& {A^{p}_{\ensuremath{\mathbb{C}}\xspace}(M)_0} \rto& {0} \enddiagram}\] \noindent where the first
vertical arrow is the isomorphisms from Theorem~B the second vertical
arrow is the isomorphisms from Propositions~\ref{prop3.2}, and $i:
\operatorname{im}(s) \rightarrow A^{p}_{\ensuremath{\mathbb{C}}\xspace}(M)_0$ is the inclusion homomorphism.
The lower row short exact sequence\xspace of the scalar curvature diagram is obtained from
the cohomology long exact sequence\xspace
\[ 0\longrightarrow H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace) \longrightarrow \ensuremath{\mathbb{H}}\xspace^{p-1}(\A{<p}(\operatorname{dlog}))
\overset{d}{\longrightarrow} A^p_\ensuremath{\mathbb{C}}\xspace (M)_{cl} \longrightarrow H^{p}(M;\ensuremath{\mathbb{C}^\ast}\xspace) \longrightarrow \] \noindent
associated with the short exact sequence\xspace of sheaves
\[ 0\longrightarrow \ensuremath{\mathbb{C}^\ast}\xspace_M \longrightarrow \A{<p}(\operatorname{dlog}) \overset{d}{\longrightarrow}
(\A{p})_{cl}[-p+1] \longrightarrow 0 \]
In order to prove the exactness of the upper row of the scalar
curvature diagram one has to show that the kernel $\ker (s)$ of the
scalar curvature homomorphism $s$ coincides with the group $L(M,
B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,\nabla^{\text{flat}})$ of isomorphism classes of smooth
principal $B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with flat connections over $M$.
If $(g,\go_1, \ldots ,\go_{p-1})$ is a $(p-1)$-cocycle in
$\operatorname{Tot}^\ast(B^{\ast,<p}_M)$, then the condition $d\go_{p-1}=0$ holds if
and only if $(g,\go_1, \ldots ,\go_{p-1})$ is a $(p-1)$-cocycle in
$\operatorname{Tot}^\ast(B^{\ast,\ast}_M)$. That is $(g,\go_1, \ldots ,\go_{p-1})$
represents an element of the group $H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace)$. By Theorem~B the
group $H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace)$ is isomorphic to $L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,
\nabla^{\text{flat}})$. Hence
\[ \ker(s) \cong H^{p-1}(M;\ensuremath{\mathbb{C}^\ast}\xspace) \cong L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,
\nabla^{\text{flat}}). \]
It is easy to see that the scalar curvature diagram commutes.
Therefore, from 5-lemma it follows that the inclusion $i: \operatorname{im}(s) \rightarrow
A^{p}_{\ensuremath{\mathbb{C}}\xspace}(M)_0$ is an isomorphism. This finishes the proof of
Theorem~C. \qed
\noindent {\bf Proof of Theorem A.} First we are going to show that there is
a commutative diagram
\[ \begin{CD}
\HD{p}{p} @>>> \H{p}{Z}\\ @VV{\cong}V @VV{\cong}V \\ L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace,
\{\nabla_i\}_{i=1}^{s+1}) @>>> L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace)
\end{CD} \]
with the vertical arrows being the isomorphisms from Propositions
\ref{prop3.2} and \ref{H(M,Z)}.
For every $s\geq 1$ there is the forgetful homomorphism
\[ \ensuremath{\varphi}\xspace^L : L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{s+1}) \longrightarrow
L(M, B^s\ensuremath{\mathbb{C}^\ast}\xspace)
\]
that sends the element $[E,\go_1, \go_2, \ldots,\go_{s+1}]$ of $L(M,
B^s\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{s+1})$ to the isomorphism class of the
bundle $E$. The homomorphism $\ensuremath{\varphi}\xspace^L$ is surjective, because every
smooth principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle carries a connection and for every
$i\geq 1$ the homomorphism
\[ \pi_\ast : \ensuremath{\Gamma}\xspace(M,EB^{s-i}\A{i}) \longrightarrow \ensuremath{\Gamma}\xspace(M,B^{s-i+1}\A{i}) \]
is surjective.
If $(g, \go_1, \ldots ,\go_{p-1})$ is a cocycle of
$\operatorname{Tot}^\ast(B^{\ast,<p}_M)$, then the assignment
\[ (g, \go_1, \ldots ,\go_{p-1}) \mapsto \{g_{ij}\}, \]
where
\[ g(x) = \bigl\lvert t_{i_1}(x), t_{i_2}(x),\ldots ,t_{i_{n}}(x),
[g_{i_0i_1}(x)|g_{i_1i_2}(x)|\ldots |g_{i_{n-1}i_n}(x)]\bigr\rvert \]
induces a homomorphism
\[ \tilde{\ensuremath{\varphi}\xspace}^H : \ensuremath{\mathbb{H}}\xspace^{p-1}(\A{<p}(\operatorname{dlog})) \longrightarrow H^1(\u{B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace})
\]
so that the diagram
\[ \begin{CD}
\ensuremath{\mathbb{H}}\xspace^{p-1}(\A{<p}(\operatorname{dlog})) @>>{\tilde{\ensuremath{\varphi}\xspace}^H}> H^1(\u{B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace}) \\
@VV{\cong}V @VV{\cong}V \\ L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,
\{\nabla_i\}_{i=1}^{p-1}) @>>{\ensuremath{\varphi}\xspace^L}> L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace)
\end{CD} \]
\noindent commutes.
Composing $\tilde{\ensuremath{\varphi}\xspace}^H$ with the isomorphisms
\begin{gather*}
\HD{p}{p} \longrightarrow \ensuremath{\mathbb{H}}\xspace^{p-1}(\A{<p}(\operatorname{dlog}))\\
\intertext{and}
H^1(\u{B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace}) \longrightarrow \H{p}{Z}
\end{gather*}
\noindent we get the homomorphism
\[ \ensuremath{\varphi}\xspace^H : \HD{p}{p} \longrightarrow \H{p}{Z} \]
\noindent so that the diagram
\[ \begin{CD}
\HD{p}{p} @>>{\ensuremath{\varphi}\xspace^H}> \H{p}{Z} \\ @VV{\cong}V @VV{\cong}V \\
L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1}) @>>{\ensuremath{\varphi}\xspace^L}> L(M,
B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace)
\end{CD}\]
\noindent commutes.
To finish the proof of Theorem~A we have to show that there is a
commutative diagram
\[{\diagram
{0} \rto& {\ker(\ensuremath{\varphi}\xspace^L)} \rto\dto^-{\cong} & {L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace,
\{\nabla_i\}_{i=1}^{p-1})} \rto^-{\ensuremath{\varphi}\xspace^L}\dto^-{\cong}& {L(M,
B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace)} \rto\dto^-{\cong}& {0}\\ {0} \rto&
{\frac{A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)}{A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)_0}} \rto& {\HD{p}{p}} \rto&
{\H{p}{Z}} \rto& {0} \enddiagram}\] \noindent with exact rows and the
vertical arrows being isomorphisms.
We have already shown that the right square of the above diagram is
commutative. Exactness of the upper row is obvious. Exactness of the
lower row short exact sequence\xspace is derived from the cohomology long exact sequence\xspace associated with the
short exact sequence\xspace
\[ 0 \longrightarrow \A{<p}[-1] \longrightarrow \ensuremath{\mathbb{Z}}\xspace(p)^\infty_D \longrightarrow \u{\ensuremath{\mathbb{Z}}\xspace(p)} \longrightarrow 0 \]
For details the reader is referred to the proof of Theorem~1.5.3 in
\cite{bry-greenbook}.
Now we will show that there is a homomorphism $\ker(\ensuremath{\varphi}\xspace^L) \longrightarrow
A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)/A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)_0$.
Suppose $(g,\go_1, \go_2, \ldots, \go_{p-1})$ represents an element
\ensuremath{\Lambda}\xspace of $L(M, B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1})$ which is in the
kernel of the homomorphism $\ensuremath{\varphi}\xspace^L$. That is, $g$ is a smooth map from
$M$ to $B^{p-1}\ensuremath{\mathbb{C}^\ast}\xspace$ inducing a smooth principal $B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundle
isomorphic to the trivial $B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundle over $M$. Equivalently,
$g$ is homotopic to a constant map. Hence, it has a lift to a map $h$
from $M$ into $EB^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$. That is, $\pi_\ast h =g$. Therefore, the
cocycle
\[(g,\go_1, \go_2, \ldots, \go_{p-1}) = (\pi_\ast h,\go_1, \go_2, \ldots,
\go_{p-1})\] is cohomologous to a cocycle
\[ (0, \go_1-\operatorname{dlog} h, \go_2, \ldots ,\go_{p-1}) =
(0, \go'_1, \go_2, \ldots ,\go_{p-1}) \]
Since the rows in the double complex $B^{\ast, \ast}_M$ are exact
(everywhere except at the zero level), there is $\ensuremath{\beta}\xspace_1\in
\ensuremath{\Gamma}\xspace(M,EB^{p-3}\A{1})$ so that $\ensuremath{\sigma}\xspace(\ensuremath{\beta}\xspace_1) = \go'_1$. Hence, the
sequence $(0, \go'_1, \go_2,\ldots, \go_{p-1})$ is cohomologous to the
sequence $(0, 0, \go_2+d\ensuremath{\beta}\xspace_1,\ldots, \go_{p-1})$.
Iterating the above process we get a representative of \ensuremath{\Lambda}\xspace which is of
the form $(0,0 ,0 ,\ldots,0 ,\go_{p-1}')$. Since $\pi_\ast
(\go_{p-1}')=0$, $\go_{p-1}'$ is actually a \ensuremath{\mathbb{C}}\xspace-valued $(p-1)$-form on
$M$.
If $(0,0 ,0 ,\ldots,0 ,\go_{p-1}'')$ is another representative of \ensuremath{\Lambda}\xspace,
then there is $(\seq[p-2]{\ensuremath{\beta}\xspace})\in \operatorname{Tot}^{p-2}(B^{\ast, \ast}_M)$ so
that
\[ (0, 0 ,0 ,\ldots,0 ,\go_{p-1}')-(0, 0 ,0 ,\ldots,0 ,\go_{p-1}'') =
D(\seq[p-2]{\ensuremath{\beta}\xspace}). \] The above equality means that $(\seq[p-2]{\ensuremath{\beta}\xspace})$
is a cocycle in $\operatorname{Tot}^{p-2}(B^{\ast, <p-1}_M)$ whose scalar curvature
is $\go_{p-1}'-\go_{p-1}''$. From Theorem~C we know that scalar
curvatures are closed forms with integral periods. Therefore, we get a
homomorphism
\begin{gather*}
\ker(\ensuremath{\varphi}\xspace^L) \longrightarrow A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)/A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)_0 \\ [0, 0 ,0
,\ldots,0 ,\go_{p-1}] \mapsto [\go_{p-1}],
\end{gather*}
\noindent where $[\go_{p-1}]$ is the class of the form $\go_{p-1}$ in the
quotient $A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)/A^{p-1}_{\ensuremath{\mathbb{C}}\xspace}(M)_0$. It is easy to see that
this homomorphism makes the right square of the diagram of
Theorem~A commutes. Hence, by 5-lemma, it is an isomorphism. \qed
\section{Holomorphic Deligne cohomology}
\label{sec:hol_case}
In this section we define holomorphic principal \bcs{s}-bundles and
holomorphic $k$-connections on them and prove Theorem~D.
A smooth map $f: X \rightarrow B^n\ensuremath{\mathbb{C}^\ast}\xspace$ is called a \emph{holomorphic map} if
$\bar{\partial} f =0$, where for
\begin{gather*}
f(x) = \bigl\lvert t_1(x), \, \ldots \, , t_n(x), [f_1(x)| \, \cdots \,
|f_n(x)]\bigr\rvert,
\end{gather*}
$\bar{\partial} f$ is defined by the analogous to $df$ inductive formula
\begin{gather*}
\bar{\partial} f(x) = \bigl\lvert t_1(x), \, \ldots \, , t_n(x), [\bar{\partial} f_1(x)| \,
\cdots \, |\bar{\partial} f_n(x)]\bigr\rvert.
\end{gather*}
In a similar way we define $EB^n\ensuremath{\mathbb{C}^\ast}\xspace$-valued holomorphic maps.
A smooth principal \bcs{n}-bundle is called a \emph{holomorphic
principal \bcs{n}-bundle} if its transition functions are
holomorphic maps. It is easy to see that if $f: X \rightarrow B^{n+1}\ensuremath{\mathbb{C}^\ast}\xspace$ is
a holomorphic map, then the induced by $f$ principal \bcs{n}-bundle
over $X$ is a holomorphic principal \bcs{n}-bundle. There is also an
inverse to the above statement.
\begin{Prop}\label{holpr}
For every holomorphic principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E \rightarrow M$ there is
a holomorphic map $f:M \rightarrow B^{s+1}\ensuremath{\mathbb{C}^\ast}\xspace$ such that $E \rightarrow M$ is the
pull-back of the universal principal $B^s\ensuremath{\mathbb{C}^\ast}\xspace$-bundle by $f$.
\end{Prop}
The proof of Proposition~\ref{holpr} is essentially a similar as the
proof of Proposition~\ref{smoothpr}.
Let $B^n\cl{O}_X^\ast$ and $EB^n\cl{O}_X^\ast$ be the sheaves of germs
of $B^n\ensuremath{\mathbb{C}^\ast}\xspace$ and $EB^n\ensuremath{\mathbb{C}^\ast}\xspace$-valued holomorphic maps on $X$. The
composition of the short exact sequences\xspace
\[0\longrightarrow B^n\cl{O}_X^\ast \longrightarrow EB^n\cl{O}_X^\ast \longrightarrow
B^{n+1}\cl{O}_X^\ast \longrightarrow 0\]
gives the bar resolution
\[E\ensuremath{\mathcal{O}^\ast_X}\xspace \longrightarrow EB\ensuremath{\mathcal{O}^\ast_X}\xspace \longrightarrow EB^2\ensuremath{\mathcal{O}^\ast_X}\xspace \longrightarrow \cdots \]
of the sheaf $\ensuremath{\mathcal{O}^\ast_X}\xspace$ of non-vanishing holomorphic functions on $X$.
Let \OO{r} be the sheaf of holomorphic $r$-forms on $X$ and let
$\cl{A}^{r,s}_X$ be the sheaf of smooth $(r,s)$-forms on $X$. The
sheaf $EB^n\OO{r}$ is the kernel of the sheaf morphism
\[\bar{\partial} : EB^n\cl{A}^{r,0}_X \longrightarrow EB^n\cl{A}^{r,1}_X,\]
which assignes to a local section
\[\bigl\lvert t_1(x), \, \ldots \, , t_n(x), \alpha_1(x)\, \ldots \,
\alpha_n(x)\bigr\rvert
\]
of the sheaf $EB^n\cl{A}^{r,0}_X$ the section
\[\bigl\lvert t_1(x), \, \ldots \, , t_n(x), \bar{\partial}\alpha_1(x)\, \ldots
\,\bar{\partial}\alpha_n(x)\bigr\rvert
\]
of the sheaf $EB^n\cl{A}^{r,1}_X$. In the same way we define the sheaf
$B^n\OO{r}$. The composition of the short exact sequences\xspace
\[0\longrightarrow B^n\OO{r} \longrightarrow EB^n\OO{r} \longrightarrow B^{n+1}\OO{r}\longrightarrow 0\]
gives the bar resolution
\[E\OO{r} \longrightarrow EB\OO{r} \longrightarrow EB^2\OO{r} \longrightarrow \cdots \]
of the sheaf \OO{r}.
\begin{Le}\label{ebnle}
For every $n\geq 0$ the sheaves $EB^n\ensuremath{\mathcal{O}^\ast_X}\xspace$ and $EB^n\OO{r}$ are soft.
\end{Le}
The proof of Lemma~\ref{ebnle} is essentially the same as the proof of
Lemma~\ref{acyclic}.
We will denote by $L^{hol}(X, B^{r}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{q-1})$
the group of equivalence classes of holomorphic principal
\bcs{r}-bundles over $X$ with $k$-connections, for $k=1, 2,
\ldots, q-1$, which is defined by replacing everywhere in the
definition of the group of equivalence classes of smooth principal
\bcs{r}-bundles with $k$-connections, the word ``smooth'' by the
word ``holomorphic''.
\noindent \textbf{Proof of Theorem D.} Let $\Omega^{<q}_X(\operatorname{dlog})$ be the
complex
\[ \ensuremath{\mathcal{O}^\ast_X}\xspace \overset{\operatorname{dlog}}{\longrightarrow} \OO{1} \overset{\partial}{\longrightarrow} \cdots
\overset{\partial}{\longrightarrow} \OO{q-1}\]
with \ensuremath{\mathcal{O}^\ast_X}\xspace placed in degree zero. There is a quasi-isomorphism between
$\Omega^{<q}_X(\operatorname{dlog})[-1]$ and the Deligne complex $\ensuremath{\mathbb{Z}}\xspace(q)_D$, which is a
holomorphic analogue of the quasi-isomorphism \eqref{quasi-iso}. Thus
\[\ensuremath{\mathbb{H}}\xspace^{r}(X,\ensuremath{\mathbb{Z}}\xspace(q)_D) \cong \ensuremath{\mathbb{H}}\xspace^{r-1}(\Omega^{<q}_X(\operatorname{dlog})).\]
Consider the bar resolution $\cl{B}(\Omega^{<q}_X(\operatorname{dlog}))$
\[{\diagram
& & & &\\ E\OO{2}\uto^{\partial}\rto^{\ensuremath{\sigma}\xspace}&
EB\OO{2}\uto^{\partial}\rto^{\ensuremath{\sigma}\xspace}& & &\\ E\OO{1}\uto^{\partial}\rto^{\ensuremath{\sigma}\xspace}&
EB\OO{1}\uto^{\partial}\rto^{\ensuremath{\sigma}\xspace}& EB^2\OO{1}\uto^{\partial}\rto^{\ensuremath{\sigma}\xspace} & &\\
E\ensuremath{\mathcal{O}^\ast_X}\xspace \uto^{\operatorname{dlog}}\rto^{\ensuremath{\sigma}\xspace}& EB\ensuremath{\mathcal{O}^\ast_X}\xspace \uto^{\operatorname{dlog}}\rto^{\ensuremath{\sigma}\xspace}&
EB^2\ensuremath{\mathcal{O}^\ast_X}\xspace \uto^{\operatorname{dlog}}\rto^{\ensuremath{\sigma}\xspace}& EB^3\ensuremath{\mathcal{O}^\ast_X}\xspace \uto^{\operatorname{dlog}}\rto^{\ensuremath{\sigma}\xspace}&
\enddiagram}
\]
of the complex $\Omega^{<q}_X(\operatorname{dlog})$. Since this is an acyclic
reolution there is an isomorphism
\[\ensuremath{\mathbb{H}}\xspace^{r-1}(\Omega^{<q}_X(\operatorname{dlog}))\cong
H^{r-1}(\operatorname{Tot}^\ast(B^{\ast,<q}_X)),
\]
where $B^{\ast,<q}_X$ is the global sections complex associated with
$\cl{B}(\Omega^{<q}_X(\operatorname{dlog}))$ and $\operatorname{Tot}^\ast(B^{\ast,<q}_X))$ is the
total complex of $B^{\ast,<q}_X$.
A $(r-1)$-cocycle in $\text{Tot}^\ast(B^{\ast,<q}_X)$ is a
sequence $(g,\go_1,\ldots ,\go_{q-1})$, where $g\in
\ensuremath{\Gamma}\xspace(X, EB^{r-1}\ensuremath{\mathcal{O}^\ast_X}\xspace)$ and $\go_i\in \ensuremath{\Gamma}\xspace(X,EB^{r-i-1}\OO{i})$ so that
\[\begin{cases}
\ensuremath{\sigma}\xspace(g)=0&\\ \operatorname{dlog} g = \ensuremath{\sigma}\xspace(\go_1)&\\ \partial\go_i = (-1)^i\ensuremath{\sigma}\xspace(\go_{i+1})&
\text{for}\; 1\leq i\leq r-2
\end{cases}\]
The condition $\ensuremath{\sigma}\xspace(g)=0$ means that $g$ is a holomorphic map from
$X$ to $B^{r-1}\ensuremath{\mathbb{C}^\ast}\xspace$, the condition $\operatorname{dlog} g = \ensuremath{\sigma}\xspace(\go_1)$ means
that $-\go_1$ is a connection on the smooth principal
$B^{p-2}\ensuremath{\mathbb{C}^\ast}\xspace$-bundle over $X$ induced by $g$, and the conditions
$\partial\go_i = (-1)^i\ensuremath{\sigma}\xspace(\go_{i+1})$ mean that $(-1)^{i+1}\go_{i+1}$
is a $(i+1)$-connection of the sequence $(g,\go_1, \ldots ,\go_i)$.
Two cocycles $(g,\go_1,\ldots ,\go_{q-1})$ and $(g',\go'_1,\ldots
,\go'_{q-1})$ are cohomologous in $\text{Tot}^\ast(B^{\ast,<q}_X)$
if and only if the corresponding principal bundles with connections
are equivalent. This gives us an isomorphism
\[ H^{r-1}(\text{Tot}^\ast(B^{\ast,<q}_X)) \cong L^{hol}(X, B^{r-2}\ensuremath{\mathbb{C}^\ast}\xspace,
\{\nabla_i\}_{i=1}^{q-1}). \]
In order to get the commutative diagram from Theorem~D consider the
bar resolution
\begin{gather}\label{holbarres}
0\longrightarrow \cl{B}(\Omega^{<p}_X[-1])\longrightarrow \cl{B}(\ensuremath{\mathbb{Z}}\xspace(p)_D) \longrightarrow
\cl{B}(\underline{\ensuremath{\mathbb{Z}}\xspace(p)}_X) \longrightarrow 0
\end{gather}
the short exact sequence\xspace
\[0\longrightarrow \Omega^{<p}_X[-1] \longrightarrow \ensuremath{\mathbb{Z}}\xspace(p)_D \longrightarrow
\underline{\ensuremath{\mathbb{Z}}\xspace(p)}_X \longrightarrow 0\]
Since
\[\ensuremath{\mathbb{H}}\xspace^{2p}(\operatorname{Tot}^\ast(\cl{B}(\ensuremath{\mathbb{Z}}\xspace(p)_D))) \cong L^{hol}(X,
B^{r}\ensuremath{\mathbb{C}^\ast}\xspace, \{\nabla_i\}_{i=1}^{p-1}), \]
and
\[\ensuremath{\mathbb{H}}\xspace^{2p}(\operatorname{Tot}^\ast(\cl{B}(\underline{\ensuremath{\mathbb{Z}}\xspace(p)}_X))) \cong L(X,
B^{r}\ensuremath{\mathbb{C}^\ast}\xspace). \]
The hypercohomology long exact sequence\xspace associated with \eqref{holbarres} induces the
lower short exact sequence\xspace of the diagram from Theorem~D. The quasi-isomorphisms
\begin{gather*}
\Omega^{<p}_X[-1] \longrightarrow \operatorname{Tot}^\ast(\cl{B}(\Omega^{<p}_X[-1]))\\
\ensuremath{\mathbb{Z}}\xspace(p)_D \longrightarrow \operatorname{Tot}^\ast(\cl{B}(\ensuremath{\mathbb{Z}}\xspace(p)_D))\\
\underline{\ensuremath{\mathbb{Z}}\xspace(p)}_X \longrightarrow \operatorname{Tot}^\ast(\cl{B}(\underline{\ensuremath{\mathbb{Z}}\xspace(p)}_X))
\end{gather*}
induce the vertical isomorphisms in this diagram. \qed
\setcounter{section}{1}
\setcounter{Th}{0}
\renewcommand{\thesection}{\Alph{section}}
\section*{Appendix A\\Principal Bundles, Topological Extensions, and Gerbs}
\label{sec:appA}
In this appendix we show that there is an isomorphism between the
group of isomorphism classes of smooth (or holomorphic) principal
\bcs{}-bundles over a manifold $M$ (or a complex projective variety
$X$) and the group of equivalence classes of smooth (or holomorphic)
gerbes bound by \us{\ensuremath{\mathbb{C}}\xspace} (or \ensuremath{\mathcal{O}^\ast_X}\xspace). This isomorphism is induced by a
construction, described in \cite{bry-greenbook}, which assigns to a
principal $G$-bundle $\pi: E\rightarrow B$ and a topological central extension
\[1\longrightarrow C \longrightarrow K \longrightarrow G\longrightarrow 1\]
a sheaf of goupoids $\cl{G}_\pi$ measuring the
obstruction to the existence of a reduction of the structure group of
$\pi: E\rightarrow B$ to $K$ (see pp. 171-172 in \cite{bry-greenbook}). In
the case of smooth (or holomorphic) principal \bcs{}-bundles and the
extension
\[0\longrightarrow \ensuremath{\mathbb{C}^\ast}\xspace \longrightarrow E\ensuremath{\mathbb{C}^\ast}\xspace \longrightarrow \bcs{}\longrightarrow 0\]
the gerbe $\cl{G}_\pi$ is equivalent to the gerbe of sections of the
bundle. We will also describe a procedure which assigns to connection
on a principal \bcs{}-bundle a connective structure on the associated
gerbe (see pp. 169-170 in \cite{bry-greenbook}).
Let us start with a definition of a gerbe. A {\em gerbe} on a space $X$
is a sheaf of categories \cl{C} on $X$ (for the precise definition of
a sheaf of categories see Chapter~5 in \cite{bry-greenbook})
satisfying the following three conditions
\begin{itemize}
\item For every open subset $U\subset X$ the category $\cl{C}(U)$ is a
groupoid, that is, every morphism is invertible.
\item Each point $x\in X$ has a neighborhood $U_x$ for which
$\cl{C}(U_x)$ is non-empty.
\item Any two objects $P_1$ and $P_2$ of $\cl{C}(U)$ are locally
isomorphic. This means that each $x\in U$ has a neighborhood $V$ such
that the restrictions of $P_1$ and $P_2$ to $V$ are isomorphic.
\end{itemize}
A gerbe \cl{C} is said to be {\em bound} by a sheaf \cl{A} of abelian
groups on $X$, if for every open set $U\subset X$ and every object $P$
of $\cl{C}(U)$ there is an isomorphism of sheaves
\[\alpha: \underline{\text{Aut}}(P) \longrightarrow \cl{A}|_U,\]
where $\cl{A}|_U$ is the restriction of the sheaf \cl{A} to $U$, and
$\underline{\text{Aut}}(P)$ is the sheaf of authomorphisms of $P$ so
that for an open subset $V$ of $U$ the group
$\underline{\text{Aut}}(P)(V)$ is the group of authomorphisms of the
restriction $r_V(P)$ of $P$ to $V$. Such an isomorphism is supposed
to commute with morphisms of \cl{C} and must be compatible with
restriction to smaller open sets.
Two gerbes \cl{C} and \cl{D} bound by \cl{A} on a manifold $M$ are
\emph{equivalent} if the following two conditions are satisfied.
\begin{itemize}
\item For every open subset $U$ of $M$ there is an equivalence of
categories $\phi(U): \cl{C}(U) \rightarrow \cl{D}(U)$ so that for every
object $P$ of $\cl{C}(U)$ there is a commutative diagram
\[{\diagram
\text{Aut}_{\cl{C}(U)}(P) \rrto^-{\phi(U)} \drto_{\alpha_\cl{C}}&&
\text{Aut}_{\cl{D}(U)}(P) \dlto^{\alpha_\cl{D}} \\
& \ensuremath{\Gamma}\xspace(U,\cl{A}) &
\enddiagram}\]
\item For every pair of open subsets $V, U$ of $M$ so that $V \subset
U$ there is an invertible natural transformation
\[\beta: \phi(U)\circ r_{\cl{D}} \longrightarrow r_{\cl{C}}\circ \phi(V),\]
where
\[r_{\cl{C}}: \cl{C}(U) \longrightarrow \cl{C}(V), \qquad r_{\cl{D}}: \cl{D}(U)
\longrightarrow \cl{D}(V),\] are the restriction natural transformations. It is
required that for a triple of open set $V \subset U\subset W $ in $M$
some compatibility conditions are satisfied (see p. 200 in
\cite{bry-greenbook}).
\end{itemize}
With every principal $G$-bundle $\pi: E\rightarrow B$ and every central
extension of topological groups
\[1\longrightarrow C \longrightarrow K \longrightarrow G\longrightarrow 1\]
we can associate a gerbe $\cl{G}_\pi$ bound by \u{C} on $B$.
The gerbe $\cl{G}_\pi$ is derived from the sheaf of sections of the
bundle $\pi: E\rightarrow B$. For every open subset $U$ of $B$ the objects
and morphisms of $\cl{G}_\pi(U)$ are defined as follows.
Every section $s: U\rightarrow \pi^{-1}(U)$ of $\pi^{-1}(U)\rightarrow U$ can be
identified with a $G$-equivariant map
\[t_s: \pi^{-1}(U)\longrightarrow G\]
so that for every $\xi \in \pi^{-1}(U)$ we have $t_s(\xi)\cdot
s(\pi(\xi)) =\xi$. Let $E_s \rightarrow \pi^{-1}(U)$ be the pull-back of
principal $C$-bundle $K\rightarrow G$ from $G$ to $\pi^{-1}(U)$, by the map
$t_s: \pi^{-1}(U) \rightarrow G$. It is clear that the composition
$\pi\circ\pi_s : E_s \rightarrow U$ is a principal $K$-bundle, and hence a
reduction of the structure group of $\pi^{-1}(U)\rightarrow U$ to $K$. The
objects of $\cl{G}_\pi(U)$ are pairs $(E,f)$ of principal $K$-bundles
$\tilde{\pi}: E\rightarrow U$ and principal $C$-bundles $f: E \rightarrow \pi^{-1}(U)$
so that the diagram
\[{\diagram
E \rrto^-f \drto_{\tilde{\pi}}&&
\pi^{-1}(U) \dlto^\pi \\
& U &
\enddiagram}\]
\noindent commutes. A morphism from $(E,f)$ to $(E',f')$ is a morphism of
principal $K$-bundles $g: E\rightarrow E'$ so that the diagram
\[{\diagram
E \rrto^g\drto_f&& E'\dlto^{f'}\\
& \pi^{-1}(U) &
\enddiagram}
\]
commutes. The above condition implies that the group of authomorphisms
of any object $(E,f)$ of $\cl{G}_\pi(U)$ is the group of maps from $U$
to $C$, which is the section of the sheaf \u{C} over $U$. Thus
$\cl{G}_\pi$ is the gerbe bound by \u{C}.
Note, that the gerbe $\cl{G}_\pi$ has a global section if and only if
there is a reduction of the structure group of $\pi: E\rightarrow B$ to $K$.
In particular, if $\pi: E\rightarrow B$ is a principal \bcs{}-bundle, and our
extension is the universal extension
\[0\longrightarrow \ensuremath{\mathbb{C}^\ast}\xspace \longrightarrow E\ensuremath{\mathbb{C}^\ast}\xspace \longrightarrow \bcs{}\longrightarrow 0\]
then the associated with $\pi: E\rightarrow B$ gerbe $\cl{G}_\pi$ measures the
obstruction for the existence of a reduction of the structure group of
$\pi: E\rightarrow B$ to $E\ensuremath{\mathbb{C}^\ast}\xspace$. Since $E\ensuremath{\mathbb{C}^\ast}\xspace$ is contractible, every
principal $E\ensuremath{\mathbb{C}^\ast}\xspace$-bundle is trivial. Thus, the gerbe $\cl{G}_\pi$ has a
global section if and only if $\pi: E\rightarrow B$ is a trivial \bcs{}-bundle.
The same property has the gerbe $\cl{S}_\pi$ of local sections of the
bundle $\pi: E\rightarrow B$, which is defined as follows. For every open
subset $U$ of $M$ the objects of $\cl{S}_\pi(U)$ are sections of $\pi:
E\rightarrow B$ over $U$. Every local section $s: U\rightarrow\pi^{-1}(U)$ of $\pi:
E\rightarrow B$ induces a \bcs{}-equivariant map $t_s: \pi^{-1}(U)\rightarrow
\bcs{}$, which in turn gives a map $\tau_s = s\circ t_s: U \rightarrow
\bcs{}$. Let $L_s$ be the principal \ensuremath{\mathbb{C}^\ast}\xspace-bundle over $U$ induced by
the map $\tau_s$. A morphism between the objects $s,s'\in
\cl{S}_\pi(U)$ is a morphism $L_s\rightarrow L_{s'}$ of the corresponding
principal \ensuremath{\mathbb{C}^\ast}\xspace-bundles. It is clear that $\cl{S}_\pi$ is a gerbe bound
by \us{\ensuremath{\mathbb{C}}\xspace}. It is not a difficult exercise to see that the natural
transformation $\cl{S}_\pi(U)\rightarrow \cl{G}_\pi(U)$ sending a section $s$
to the pull-back $E_s$ of the universal principal \ensuremath{\mathbb{C}^\ast}\xspace-bundle by $t_s$
is an equivalence of categories that extends to an equivalence of
gerbes $\cl{S}_\pi\rightarrow \cl{G}_\pi$.
The following theorem is an easy consequence of Theorem~H (see the
introduction) and Theorem~5.2.8 from \cite{bry-greenbook}.
\begin{Th}
A map which sends to the isomorphism class of a principal
$B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $\pi: E\rightarrow B$ the equivalence class of the gerbe of
section $\cl{S}_\pi$ of $\pi: E\rightarrow B$ induces an isomorphism between
the group of isomorphism classes of principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundles and
the group of equivalence classes of gerbes bound by \ensuremath{\mathbb{C}^\ast}\xspace.
\end{Th}
Let \cl{G} be a gerbe on $M$ bound by \us{\ensuremath{\mathbb{C}}\xspace}. A {\em connective
structure} on \cl{G} is an assignment to each object $P$ in
$\cl{G}(U)$ a \A{1}-torsor $\cl{Co}_P$ on $U$. That is
$\cl{Co}_P$ is a sheaf with an action of \A{1} on $\cl{Co}_P$ such that
every point has a neighborhood\xspace $U$ with the property that for each open set
$V\subset U$ the group $\cl{Co}_P(V)$ is a principal homogeneous space
under the group $\ensuremath{\Gamma}\xspace(V,\A{1})$. The assignment $P \mapsto \cl{Co}_P(U)$
should be functorial with respect to restriction of $U$ to smaller
open set and should be so that for any morphism $\ensuremath{\psi}\xspace:P\rightarrow Q$ of objects
of $\cl{G}(U)$ (necessarily an isomorphism since \cl{G} is a gerbe),
there is an isomorphism $\ensuremath{\psi}\xspace_\ast: \cl{Co}_P(U) \rightarrow \cl{Co}_Q(U)$ of
\A{1}-torsors, which is compatible with composition of morphisms in
$\cl{G}(U)$ and also compatible with restrictions to smaller open
sets. If \ensuremath{\psi}\xspace is an automorphism of $P$ induced by a \ensuremath{\mathbb{C}^\ast}\xspace-valued
function $g$, we require that $\ensuremath{\psi}\xspace_\ast$ be the automorphism $\nabla
\mapsto \nabla -\frac{dg}{g}$ of the \A{1}-torsor $\cl{Co}_P(U)$. In a
similar way one can define a holomorphic connective structure on a
holomorphic gerbe bound by \ensuremath{\mathcal{O}^\ast_X}\xspace.
A connection $\go$ on a smooth principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $\pi:E\rightarrow M$
induces the following connective structure on $\cl{G}_\pi$. Let $U$ be
an open subset of $M$ so that $\cl{G}_\pi (U)$ is non-empty and let
$\go_U$ be the restriction of $\go$ to $\pi^{-1}(U)$. To every element
$(E,f)$ of $\cl{G}_\pi (U)$ we assign a set $\cl{Co}_E^\omega(U)$ of
connections on $E$ compatible with \go. That is $\tilde{\go} \in
\cl{Co}_E^\omega(U)$ if $q\circ \go = f^\ast \go$, where $q: E\ensuremath{\mathbb{C}}\xspace \rightarrow
B\ensuremath{\mathbb{C}}\xspace$ and $f: E \rightarrow \pi^{-1}(U)$ is the principal \ensuremath{\mathbb{C}^\ast}\xspace-bundle. It is
easy to see that the assignment $\omega \mapsto \cl{Co}^\omega$ is a
connective structure on $\cl{G}_\pi$ (for detail see pp. 169-170 in
\cite{bry-greenbook}). The equivalence of gerbes $\cl{S}_\pi\rightarrow
\cl{G}_\pi$ can be used to pull-back the connective structure from
$\cl{G}_\pi$ to $\cl{S}_\pi$. A similar to the above construction
assigns to a holomorphic connection on a holomorphic principal
$B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $E\rightarrow X$ a holomorphic connective structure on the
associated with $E\rightarrow X$ holomorphic gerbe.
\begin{Th}
A map which sends to the isomorphism class of a principal
$B\ensuremath{\mathbb{C}^\ast}\xspace$-bundle $\pi: E\rightarrow B$ with a connection $\omega$ the
equivalence class of the gerbe of section $\cl{S}_\pi$ of $\pi: E\rightarrow
B$ with the connective structure on $\cl{S}_\pi$ induced by $\omega$
induces an isomorphism between the group of isomorphism classes of
principal $B\ensuremath{\mathbb{C}^\ast}\xspace$-bundles with connection and the group of
equivalence classes of gerbes bound by \ensuremath{\mathbb{C}^\ast}\xspace with connective
structures.
\end{Th}
We leave the proof of this theorem as an exercise for the reader.
\setcounter{section}{2}
\section*{Appendix B\\The Geometric Bar Construction}
The objective of this appendix is twofold. First, we define and review
basic properties of the geometric bar construction. Second, we explain
how the geometric bar construction can be derived from the projective
space construction. Our basic references for the geometric bar
construction are \cite{milg-bar} and the survey paper \cite{sta-hsp}.
The geometric bar construction assigns to every topological group $G$
a sequence of principal $G$-bundles $E_n\rightarrow B_n$
\[{\diagram
{G} \dto\rdouble& {E_1}\dto\rto|<\hole|<<\ahook&{E_2}\dto\rto|<\hole|<<\ahook&{\; \cdots\;}
\rto|<\hole|<<\ahook&{E_n}\dto\rto|<\hole|<<\ahook&{\; \cdots} \\ {pt} \rdouble& {B_1}\rto|<\hole|<<\ahook& {B_2}
\rto|<\hole|<<\ahook&{\; \cdots \;} \rto|<\hole|<<\ahook&{B_n} \rto|<\hole|<<\ahook&{\; \cdots} \enddiagram}\] so that
for every $n\geq 0$ the space $E_n$ is contractible in $E_{n+1}$. The
universal principal $G$-bundle $EG \rightarrow BG$ is the union
$\bigcup_{n\geq 1} E_n \rightarrow \bigcup_{n\geq 1} B_n$ taken with the weak
topology.
If $G$ is an abelian topological group, then $EG$ and $BG$ are abelian
topological groups and the projection $EG\rightarrow BG$ is a continuous
homomorphism with $G$ as the kernel.
The geometric bar construction is functorial and it preserves
products. That is, every continuous homomorphism $f:G\rightarrow H$ induces
continuous maps $Ef: EG\rightarrow EH$ and $Bf: BG\rightarrow BH$, which are
homomorphisms for $G$ abelian, and
\[E(G\times H) = EG\times EH\qquad B(G\times H) = BG\times BH,\]
where each product is taken with the compactly generated
topology\footnote{To every topological space $X$ one can assign a
space $(X,k)$ with compactly generated topology so that a set is
open in $(X,k)$ if and only if its intersection with every compact
subset of $X$ is open.}.
If $G$ is a countable CW-group\footnote{A topological group $G$ is
called a countable CW-group if it is a countable CW-complex so that
the map $g\mapsto g^{-1}$ of $G$ into itself and the product map
$G\times G \rightarrow G$ are both cellular (that is, they carry the
$k$-skeleton into the $k$-skeleton).}, then $EG$ and $BG$ are
countable CW-complexes. In this appendix $G$ is a countable CW-group.
Actually, in the main body of the paper $G$ is \ensuremath{\mathbb{Z}}\xspace, \ensuremath{\mathbb{C}}\xspace, \ensuremath{\mathbb{C}^\ast}\xspace, $S^1$, or
the abelian group of a separable \ensuremath{\mathbb{C}}\xspace-vector space. A techincal
advantage of working with countable CW-groups is that on the spaces
appearing in the definitions of $EG$ and $BG$ one can take the
product, versus compactly generated, topology.
The archetypes of the geometric bar construction are infinite real,
complex, and quaternionic projective spaces. Actually, Milnor found a
construction that associates with every topological group $G$ a
principal $G$-bundle $E_\ensuremath{\Delta}\xspace G \rightarrow B_\ensuremath{\Delta}\xspace G$, which is a limit of a
sequence of principal $G$-bundles $(E_\ensuremath{\Delta}\xspace G)_n \rightarrow (B_\ensuremath{\Delta}\xspace G)_n$, so
that for $G=S^0, S^1$, and $S^3$ the bundle $(E_\ensuremath{\Delta}\xspace G)_n \rightarrow (B_\ensuremath{\Delta}\xspace
G)_n$ is isomorphic to $S^n\rightarrow \ensuremath{\mathbb{R}}\xspace\P^n, S^{2n+1}\rightarrow \ensuremath{\mathbb{C}}\xspace\P^n$, and
$S^{4n+3}\rightarrow \ensuremath{\mathbb{H}}\xspace\P^n$ respectively.
A drawback of Milnor's constraction is that for $G$ being an abelian
topological group the spaces $E_\ensuremath{\Delta}\xspace G$ and $B_\ensuremath{\Delta}\xspace G$ are not abelian
groups, so the construction cannot be iterated. The geometric bar
construction is a ``normalized version'' of Milnor's construction that
fixes this problem.
There are several approaches to geometric bar construction (for a
survey on this subject see \cite{sta-hsp}). Usually, the spaces $EG$
and $BG$ are defined as the quotients of the disjoint unions
$\coprod_{n\geq 0} \ensuremath{\Delta}\xspace^n\times G^{n+1}$ and $\coprod_{n\geq
0}\ensuremath{\Delta}\xspace^n\times G^{n}$ respectively, by cetrain equivalence relations.
To explain the geometric meaning of these relations we preceded the
formal definition of geometric bar construction with the Milnor and
the Dold-Lashof constructions \cite{miln-cub2}, \cite{dol&las-pqf}.
\subsection{The unnormalized geometric bar construction.}
Let $G$ be a countable CW-group. The {\em join} $G\ast G$ is the
quotient of the product $\ensuremath{\Delta}\xspace^1\times G\times G$ of the standard
1-simplex
\[\ensuremath{\Delta}\xspace^1 = \{(x_0,x_1)\in \ensuremath{\mathbb{R}}\xspace^2 |\; x_0,x_1\geq 0,\quad
x_0+x_1=1\}\] with $G\times G$ by the equivalence relation
\[(0,1,g_0,g_1)\sim (0,1,e,g_1),\qquad (1,0,g_0,g_1) \sim (1,0,g_0,e)
\]
where $e$ is the neutral element of $G$. The equivalence class of the
sequence $(x_0,x_1,g_0,g_1)$ will be denoted by $x_0g_0\oplus x_1g_1$.
Let $I=[0,1]$. The homeomorphism
\[ \ensuremath{\Delta}\xspace^1\times G\times G\longrightarrow G\times I\times G,\quad
(x_0,x_1,g_0,g_1)\mapsto (g_0, x_1, g_1) \] induces a homeomorphisms
between $G\ast G$ and the quotient of the product $G\times C(G)$ of
$G$ with the cone $C(G)=I\times G/0\times G$ by the equivalence
relation
\[ (g_0,[1,g_1]) \sim (e,[1,g_1]) \]
where $[t,g]$ is the image of the pair $(t,g)$ in $C(G)$.
The $(n+1)$-fold join
\[ G\ast (n+1) \ast G = \underbrace{G\ast (G\ast \, \cdots \, (G\ast
G)}_{(n+1)\; \text{times}} \, \cdots \, ) \] which we will also
denote by $(E_\ensuremath{\Delta}\xspace G)_n$, can be identified with the quotient of the
product $\ensuremath{\Delta}\xspace^n\times G^{n+1}$ of the standard simplex
\[ \ensuremath{\Delta}\xspace^n = \{ (\seq{x})\in \ensuremath{\mathbb{R}}\xspace^{n+1} |\; x_i\geq 0, \quad
\sum\limits_{i=0}^n x_i =1 \} \] and the $(n+1)$-fold product
$G^{n+1}$ of $G$ with itself, by the equivalence relation
\begin{align*}
&(\seq[i-1]{x}, 0, x_{i+1}, \, \ldots \, , x_n, \seq[i]{g}, \,
\ldots \, , g_n) \sim \\ \sim &(\seq[i-1]{x}, 0, x_{i+1}, \, \ldots
\, , x_n, g_0, \, \ldots \, ,e, \, \ldots \, , g_n)
\end{align*}
Actually, if we denote by $x_0g_0\oplus \, \cdots \, \oplus x_ng_n$
the equivalence class of the sequence $(\seq{x},\seq{g})$, then the
homeomorphism between $(E_\ensuremath{\Delta}\xspace G)_n$ and the quotient $(\ensuremath{\Delta}\xspace^n\times
G^{n+1})/\sim$ is given by
\begin{gather*}
x_0g_0\oplus (1-x_0)\Bigl(x_1g_1\oplus (1-x_1)\bigl( \, \cdots \,
(x_{n-1}g_{n-1}\oplus (1-x_{n-1})g_n) \, \cdots \,
\bigr)\Bigr)\mapsto\\ \mapsto x_0g_0\oplus (1-x_0)x_1g_1\oplus \, \cdots
\, \oplus \prod\limits_{i=0}^{n-1}(1-x_i)g_n
\end{gather*}
On the other hand, $(E_\ensuremath{\Delta}\xspace G)_n = G\ast (E_\ensuremath{\Delta}\xspace G)_{n-1}$ can be
identified with the quotient of $G\times C((E_\ensuremath{\Delta}\xspace G)_{n-1})$ by the
quivalence relation
\[ (g,[1,x]) \sim (e,[1,x]). \]
Note that there are inclusions
\[{\diagram
{(E_\ensuremath{\Delta}\xspace G)_{n-1}} \rto|<\hole|<<\ahook_-i& {C((E_\ensuremath{\Delta}\xspace G)_{n-1})} \rto|<\hole|<<\ahook_-j& {(E_\ensuremath{\Delta}\xspace
G)_{n}} \enddiagram}\] given by $i(y)=[1,y]$ and $j([t,y]) =
(1-t)e\oplus ty$.
Let
\[ E_\ensuremath{\Delta}\xspace G = \bigcup_{n\geq 2} (E_\ensuremath{\Delta}\xspace G)_n = \bigcup_{n\geq 2}
C((E_\ensuremath{\Delta}\xspace G)_n). \]
Since $E_\ensuremath{\Delta}\xspace G$ is the union of cones, it is contractible.
There is a free action of $G$ on $(E_\ensuremath{\Delta}\xspace G)_n$ given by
\begin{gather}\label{action}
g\cdot (x_0g_0\oplus \, \cdots \, \oplus x_ng_n) = x_0(g g_0)\oplus
\, \cdots \, \oplus x_n(g g_n)
\end{gather}
The orbit space of this action is denoted by $(B_\ensuremath{\Delta}\xspace G)_n$. For
example, $(B_\ensuremath{\Delta}\xspace G)_0$ is a single point and $(B_\ensuremath{\Delta}\xspace G)_1$ is the
suspension of $G$.
Since the actions of $G$ on $(E_\ensuremath{\Delta}\xspace G)_n$ and $(E_\ensuremath{\Delta}\xspace G)_{n+1}$ are
compatible with the embedding $(E_\ensuremath{\Delta}\xspace G)_n \subset (E_\ensuremath{\Delta}\xspace G)_{n+1}$,
there is a free action of $G$ on $E_\ensuremath{\Delta}\xspace G$. The quotient space $(E_\ensuremath{\Delta}\xspace
G)/G$ is denoted by $B_\ensuremath{\Delta}\xspace G$ and the natural map $E_\ensuremath{\Delta}\xspace G \rightarrow B_\ensuremath{\Delta}\xspace
G$ is Milnor's universal principal $G$-bundle.
\begin{Exa}\label{appex1}
For $G=S^0 = \ensuremath{\mathbb{Z}}\xspace/2\ensuremath{\mathbb{Z}}\xspace=\{\pm 1\}$ there are homeomorphisms
\[(E_\ensuremath{\Delta}\xspace S^0)_n \cong S^n,\qquad (B_\ensuremath{\Delta}\xspace S^0)_n \cong \ensuremath{\mathbb{R}}\xspace\P^n \]
induced by the map
\[(E_\ensuremath{\Delta}\xspace S^0)_n\ni x_0\ensuremath{\alpha}\xspace_0\oplus \, \cdots \, \oplus x_n\ensuremath{\alpha}\xspace_n \mapsto
(\ensuremath{\alpha}\xspace_0\sqrt{x_0}, \, \ldots \, , \ensuremath{\alpha}\xspace_n\sqrt{x_n}) \in S^n \]
Similarly, for $G=S^1=U(1), S^3=SU(2)$, or \ensuremath{\mathbb{C}^\ast}\xspace there are the
following homeomorphisms
\begin{align*}
(E_\ensuremath{\Delta}\xspace S^1)_n &\cong S^{2n+1},& (B_\ensuremath{\Delta}\xspace S^1)_n &\cong\ensuremath{\mathbb{C}}\xspace\P^n\\ (E_\ensuremath{\Delta}\xspace
S^3)_n &\cong S^{4n+3},& (B_\ensuremath{\Delta}\xspace S^3)_n &\cong\ensuremath{\mathbb{H}}\xspace\P^n\\ (E_\ensuremath{\Delta}\xspace
\ensuremath{\mathbb{C}^\ast}\xspace)_n &\cong S^{2n+1}\times (\ensuremath{\mathbb{R}}\xspace_{+})^{n+1},& (B_\ensuremath{\Delta}\xspace\ensuremath{\mathbb{C}^\ast}\xspace)_n
&\cong\ensuremath{\mathbb{C}}\xspace\P^n\times S^n_{+}
\end{align*}
where $\ensuremath{\mathbb{R}}\xspace_{+}$ is the set of positive real numbers and $S^n_{+}$ is
the intersection $(\ensuremath{\mathbb{R}}\xspace_{+})^{n+1}\cap S^n$.
\end{Exa}
Sometimes, it is convenient to replace the diagonal action
\eqref{action} of $G$ on $(E_\ensuremath{\Delta}\xspace G)_n$ by the action of $G$ on the
first component of $(E_\ensuremath{\Delta}\xspace G)_n$. This can be done by introducing a
{\em non-homogeneous coordinates} on $(E_\ensuremath{\Delta}\xspace G)_n$
\[ \bigl\lvert \seq{x},h_0[h_1| \, \cdots \, |h_n]\bigr\rvert_\ensuremath{\Delta}\xspace =
x_0g_0\oplus x_1(g_0^{-1}g_1)\oplus \, \cdots \, \oplus x_n
(g_{n-1}^{-1}g_n)\] where $(\seq{g})\in G^{n+1}$, $h_0 =g_0$, and $h_i
= g_{i-1}^{-1}g_i$ for $i>0$.
The non-homogeneous coordinates on $(E_\ensuremath{\Delta}\xspace G)_n$ lead to yet another
model of $(E_\ensuremath{\Delta}\xspace G)_n$, due to Dold and Lashof \cite{dol&las-pqf}.
For example, in the 2-fold join $G\ast G$ the relations
\[ 0g_0\oplus 1g_1 = 0e\oplus 1g_1,\qquad 1g_0\oplus 0g_1 = 1g_0\oplus
0e \] correspond, in the non-homogeneous coordinates, to the relations
\[ \bigl\lvert 0,1,h_0[h_1]\bigr\rvert_\ensuremath{\Delta}\xspace = \bigl\lvert 0,1,e[h_0h_1]\bigr\rvert_\ensuremath{\Delta}\xspace,\quad
\bigl\lvert 1,0, h_0[h_1]\bigr\rvert_\ensuremath{\Delta}\xspace = \bigl\lvert 1,0, h_0[e]\bigr\rvert_\ensuremath{\Delta}\xspace \]
Thus, the symbols $\bigl\lvert x_0, x_1 , h_0[h_1]\bigr\rvert_\ensuremath{\Delta}\xspace$ can be identified
with the points of the space
\[DL(G)=G\times C(G)\cup_\mu G =
(G\times C(G)\sqcup G)/\sim \] where $\mu:G\times G \rightarrow G$ is the
group operation in $G$ and $\sim$ is an equivalence relation
identifying $(h_0,[1,h_1])$ with $\mu(h_0,h_1)=h_0h_1$.
Note, that there is an action of $G$ on $DL(G)$ given by
\[g\cdot (\bigl\lvert x_0, x_1 , h_0[h_1]\bigr\rvert_\ensuremath{\Delta}\xspace) = \bigl\lvert x_0, x_1 ,
gh_0[h_1]\bigr\rvert_\ensuremath{\Delta}\xspace\] and hence we can apply the above construction to
$DL(G)$.
In general, to any space $E$ with a $G$-action $\mu:G\times E\rightarrow E$ we
can associate the space
\[ DL(E) = G\times C(E)\cup_\mu G = (G\times C(E)\sqcup G)/\sim \]
where $\sim$ is an equivalence relation identifying $(h,[1,x])$ with
$\mu(h,x)$, and the action
\[ G\times DL(E)\rightarrow DL(E), \qquad g\cdot (h|t|x) = (gh)|t|x \]
where $h|t|x$ is the equivalence class of the sequence $(h,[t,x])\in
G\times C(E)$ in $DL(E)$.
The spaces $DL(E)$ and $G\ast E$ are $G$-equivariantly homeomorphic to
each other with the $G$-equivariantly homeomorphisms given by
\begin{align*}
&DL(E) \rightarrow G\ast E, & h|t|y &\mapsto th \oplus (1-t)(hy)\\ &G\ast E\rightarrow
DL(E), & x_0h\oplus x_1y &\mapsto h|x_0|h^{-1}y
\end{align*}
Therefore, the bundles $DL(E)\rightarrow DL(B)$ and $G\ast E \rightarrow (G\ast E)/G$
are isomorphic.
Applying $n$ times the Dold-Lashof construction to a topological group
$G$, we get a principal $G$-bundle $DL^n(G)\rightarrow DL^n(G)/G$ which is
isomorphic to the bundle $(E_\ensuremath{\Delta}\xspace G)_n \rightarrow (B_\ensuremath{\Delta}\xspace G)_n$.
\subsection{The geometric bar construction.} In general, the spaces
$E_\ensuremath{\Delta}\xspace G$ and $B_\ensuremath{\Delta}\xspace G$ have not group structure, but for $G$
abelian, some quotients of these spaces are groups.
The appropriate quotients are obtained by replacing the cone $C(E)$ in
the Dold-Lashof construction $DL(E)$ by the reduced cone
\[ \ensuremath{\tilde{C}}(E) = (I\times E)/(0\times E \cup I\times e) \]
where $e$ is a base point of $E$. For example, for $(E,e) = (G,e)$
where $G$ is a topological group with the neutral element $e$ we
define
\[ \widetilde{DL}(G) = G\times \ensuremath{\tilde{C}}(G)\cup_\mu G \]
The space $\widetilde{DL}(G)$ is a quotient of $DL(G)$ by the equivalence
relation
\[ h|t|e = h|0|e. \]
The group action of $G$ on $DL(G)$ decents to a group action of $G$
on $\widetilde{DL}(G)$. Thus, we can iterate this construction getting for
every $n\geq 1$ a space $\widetilde{DL}^n(G)$ with a free action of $G$ on
itself. We set $(EG)_n = \widetilde{DL}^n(G)$ and $(BG)_n = \widetilde{DL}^n(G)/G$.
It is easy to see that $(EG)_n$ is the quotient of the disjoint union
$\coprod\limits_{m=0}^n \ensuremath{\Delta}\xspace^m\times G^{m+1}$
by the equivalence relations
\begin{multline*}
(\seq[m]{x}, \seq[m]{g}) \sim\\
\sim \begin{cases}
(\seq[i]{x}+x_{i+1}, \, \ldots \, ,x_m, g_0, \, \ldots \, ,\hat{g_i}, \,
\ldots \,
,g_m)&\text{for $g_i=g_{i+1}$ or $x_i=0, \; 0\leq i<m$}\\
(\seq[m-1]{x}+x_{m},\seq[m-1]{g})&\text{for $g_{m-1}=g_{m}$ or
$x_m=0$}
\end{cases}
\end{multline*}
In the non-homogeneous coordinates on $(EG)_n$ the above relations
take the form
\begin{multline*}
(t_1, \, \ldots \, , t_m, h_0[h_1| \, \cdots \, |h_m]) \sim\\
\sim \begin{cases}
(t_2, \, \ldots \, , t_m, h_0h_1[h_2| \, \cdots \, |h_m])&\text{for $t_1=0$
or
$h_0=e$} \\
(t_1, \, \ldots \, , \hat{t_i}, \dots , t_m, h_0[h_1| \, \cdots \,
|h_ih_{i+1}|
\, \cdots \, |h_m])&
\text{for $t_{i}=t_{i+1}$ or $h_i=e$} \\
(t_1, \, \ldots \, , t_{m-1}, h_0[h_1| \, \cdots \, |h_{m-1}])& \text{for
$t_m=1$
or $h_m=e$}
\end{cases}
\end{multline*}
where $0\leq t_1 \leq t_2 \leq \, \cdots \, \leq t_m \leq 1$ are
non-homogeneous coordinates on $\ensuremath{\Delta}\xspace^n$ related with the baricentric
coordinated $\seq{x}$ on $\ensuremath{\Delta}\xspace^n$ by the formula
\[ t_i = x_0+x_1+ \, \cdots \, + x_{i-1}. \]
The equivalence class of a sequence $(\seq[m]{x}, \seq[m]{g})$ will be
denoted by $\bigl\lvert \seq[m]{x}, \seq[m]{g} \bigr\rvert$ and the equivalence class
of a sequence $(t_1, \, \ldots \, , t_m, h_0[h_1| \, \cdots \, |h_m])$ will
be
denoted by $\bigl\lvert t_1, \, \ldots \, , t_m, h_0[h_1| \, \cdots \, |h_m]\bigr\rvert$.
The space $EG$ is the quotient of the disjoint union
$\coprod\limits_{m=0}^\infty \ensuremath{\Delta}\xspace^m\times G^{m+1}$
by the above equivalence relations.
Similarly, $(BG)_n$ is the quotient of the disjoint union
$\coprod\limits_{m=0}^n \ensuremath{\Delta}\xspace^m\times G^{m}$
by the equivalence relations
\begin{multline*}
(\seq[m]{x}, [g_0: \, \cdots \, :g_m]) \sim\\
\sim \begin{cases}
(\seq[i]{x}+x_{i+1}, \, \ldots \, ,x_m, [g_0: \, \cdots \, :\hat{g_i}: \,
\cdots \,
:g_m])&\text{for $g_i=g_{i+1}$ or $x_i=0, 0\leq i<m$}\\
(\seq[m-1]{x}+x_{m},[g_0: \, \cdots \, :g_{m-1}])&\text{for $g_{m-1}=g_{m}$ or
$x_m=0$}
\end{cases}
\end{multline*}
where $[g_0: \, \cdots \, :g_m]$ is the equivalence class of the sequence
$(\seq{g})\in G^{m+1}$ by the equivalence relation
\[ (\seq{g}) \sim (gg_0, gg_1, \, \ldots \, ,gg_m) \]
for any $g\in G$.
In the non-homogeneous coordinates on $(BG)_n$ the above relations
take the form
\begin{multline*}
(t_1, \, \ldots \, , t_m, [h_1| \, \cdots \, |h_m]) \sim\\
\sim \begin{cases}
(t_2, \, \ldots \, , t_m, [h_2| \, \cdots \, |h_m])&\text{for $t_1=0$ or
$h_0=e$} \\
(t_1, \, \ldots \, , \hat{t_i}, \dots , t_m, [h_1| \, \cdots \, |h_ih_{i+1}|
\, \cdots \, |h_m])&
\text{for $t_{i}=t_{i+1}$ or $h_i=e$} \\
(t_1, \, \ldots \, , t_{m-1}, [h_1| \, \cdots \, |h_{m-1}])& \text{for
$t_m=1$
or $h_m=e$}
\end{cases}
\end{multline*}
The equivalence class of a sequence $(\seq[m]{x},[g_0: \, \cdots \, :g_m] )$
will be
denoted by $\bigl\lvert \seq[m]{x}, [g_0: \, \cdots \, :g_m] \bigr\rvert$ and the equivalence
class
of a sequence $(t_1, \, \ldots \, , t_m, [h_1| \, \cdots \, |h_m])$ will be
denoted by $\bigl\lvert t_1, \, \ldots \, , t_m, [h_1| \, \cdots \, |h_m]\bigr\rvert$.
The space $BG$ is the quotient of the disjoint union
$\coprod\limits_{m=0}^\infty \ensuremath{\Delta}\xspace^m\times G^{m}$
by the above equivalence relations.
The projection $EG \rightarrow BG$ is given by the formula
\[ \bigl\lvert \seq[m]{x}, \seq[m]{g} \bigr\rvert \mapsto \bigl\lvert \seq[m]{x}, [g_0: \, \cdots \,
:g_m] \bigr\rvert\]
or in the non-homogeneous coordinates by
\[\bigl\lvert t_1, \, \ldots \, , t_m, h_0[h_1| \, \cdots \, |h_m]\bigr\rvert \mapsto
\bigl\lvert t_1, \, \ldots \, , t_m, [h_1| \, \cdots \, |h_m]\bigr\rvert \]
Sometimes it is convenient to write the elements of $EG$ and $BG$ in
the form
\[\bigl\lvert \seq{m}{x}, h_0[h_1| \, \cdots \, |h_m]\bigr\rvert\]
and
\[\bigl\lvert \seq{m}{x}, [h_1| \, \cdots \, |h_m]\bigr\rvert\]
respectively, which is mixture of the baricentric coordinates on
$\ensuremath{\Delta}\xspace^m$ and homogeneous coordinates on $G^{m+1}$ or $G^{m}$.
\subsection{A simpicial description of the geometric bar construction.}
\label{app3}
The above definitions of $EG$ and $BG$ can be interpreted in terms of
geometric realizations of some simplicial objects. Actually, to every
topological group $G$ one can assign simplicial topological groups
$EG.$ and $BG.$ defined as follows.
$EG_n =G^{n+1}$, the face homomorphisms
$\partial_i:EG_n\rightarrow EG_{n-1}$ are given by the formula
\[ \partial_i(\seq{g}) = (\seq[i-1]{g} ,\widehat{g_{i}},g_{i+1}, \, \ldots \,
,g_n)\] or in the non-homogeneous coordinates by
\begin{multline*}
\partial_i(h_0[h_1| \, \cdots \, |h_n])=
\begin{cases}
h_0h_1[ h_2|h_3| \, \cdots \, |h_n] & \text{for} \quad i=0\\{}
h_0[h_1| \, \cdots \, |h_i\cdot h_{i+1}| \dots |h_n] & \text{for} \quad 0< i
< n\\{}
h_0[h_1| \, \cdots \, |h_{n-1}]& \text{for} \quad i=n
\end{cases}
\end{multline*}
The degeneracy homomorphism $s_i: EG_n\rightarrow EG_{n+1}$ are
given by the formula
\[ s_i(\seq{g}) = (\seq[i-1]{g}, g_i, g_i, g_{i+1}, \, \ldots \, ,g_n)\]
or in the non-homogeneous coordinates by
\begin{multline*}
s_i(h_0[h_1| \, \cdots \, |h_n]) = \begin{cases}
h_0[e|h_1| \, \cdots \, |h_n] & \text{for} \quad i=0\\{}
h_0[h_1| \, \cdots \, |h_i| e| h_{i+1}| \dots |h_n] & \text{for} \quad 0< i
< n\\{}
h_0[h_1| \, \cdots \, |h_n|e]& \text{for} \quad i=n
\end{cases}
\end{multline*}
$EG_n =G^{n}$, the face homomorphisms $\partial_i:BG_n\rightarrow
BG_{n-1}$ in the homogeneous coordinates on $G^{n} = G^{n+1}/G$ is
given by the formula
\[\partial_i([g_0: \, \cdots \, : g_n]) = [g_0: \, \cdots \, g_{i-1}:
\widehat{g_i}:\, \cdots \,
:g_n]\]
or in the non-homogeneous coordinates by
\begin{multline*}
\partial_i([h_1| \, \cdots \, |h_n]) =
\begin{cases}
[h_2|h_3| \, \cdots \, |h_n] & \text{for} \quad i=0\\{}
[h_1| \, \cdots \, |h_i\cdot h_{i+1}| \dots |h_n] & \text{for} \quad 0< i
< n\\{}
[h_1| \, \cdots \, |h_{n-1}]& \text{for} \quad i=n
\end{cases}
\end{multline*}
The degeneracy homomorphisms $s_i: BG_n\rightarrow BG_{n+1}$ are
given by the formula
\[ s_i([g_0: \, \cdots \, : g_n]) = [g_0: \, \cdots \, g_{i-1}: g_i: g_i :\,
\cdots \,
:g_n]\]
or in the non-homogeneous coordinates by
\begin{multline*}
s_i([h_1| \, \cdots \, |h_n]) =
\begin{cases}
[e|h_1| \, \cdots \, |h_n] & \text{for} \quad i=0\\{}
[h_1| \, \cdots \, |h_i| e| h_{i+1}| \dots |h_n] & \text{for} \quad 0< i
< n\\{}
[h_1| \, \cdots \, |h_n|e]& \text{for} \quad i=n
\end{cases}
\end{multline*}
The geometric realization $|EG.|$ of the simplicial space
$EG.$ is by definition the quotient space of the infinite
disjoint union $\coprod\limits_{n=0}^\infty \ensuremath{\Delta}\xspace^n\times G^{n+1}$
by the equivalence relations
\begin{gather*}
(\partial^ix,\bar{g}) \sim (x,\partial_i\bar{g}) \quad \text{for}\quad
(x,\bar{g})\in \ensuremath{\Delta}\xspace^{n-1}\times G^{n+1}\\
(s^ix,\bar{g}) \sim (x,s_i\bar{g}) \quad \text{for}\quad
(x,\bar{g})\in \ensuremath{\Delta}\xspace^{n+1}\times G^{n+1}
\end{gather*}
where the maps $\partial^i : \ensuremath{\Delta}\xspace^{n-1} \rightarrow \ensuremath{\Delta}\xspace^n$ and $s^i: \ensuremath{\Delta}\xspace^{n+1}
\rightarrow \ensuremath{\Delta}\xspace^n$ are defined in the baricentric coordinates by
\begin{align*}
\partial^i(\seq{x}) &= (\seq[i-1]{x},0,x_{i},\, \ldots \, ,x_n)\\
s^i(\seq{x}) &= (\seq[i-1]{x},x_i+x_{i+1},x_{i+2},\, \ldots \, ,x_n)
\end{align*}
and in the non-homogeneous coordinates by
\begin{align*}
\partial^i(\seq[n+1]{t}) &= (\seq[i]{t},t_i,t_{i+1},\, \ldots \, ,t_{n+1})\\
s^i(\seq[n+1]{t}) &= (\seq[i]{t},\widehat{t_{i+1}},t_{i+2},\, \ldots \,
,t_{n+1})
\end{align*}
Similarly, the geometric realization $|BG.|$ of the simplicial space
$BG.$ is the quotient space of the disjoint union
$\coprod\limits_{n=0}^\infty\ensuremath{\Delta}\xspace^n\times G^{n}$
by the equivalence relations
\begin{gather*}
(\partial^ix,\bar{g}) \sim (x,\partial_i\bar{g}) \quad \text{for}\quad
(x,\bar{g})\in \ensuremath{\Delta}\xspace^{n-1}\times G^{n}\\{}
(s^ix,\bar{g}) \sim (x,s_i\bar{g}) \quad \text{for}\quad
(x,\bar{g})\in \ensuremath{\Delta}\xspace^{n+1}\times G^{n}
\end{gather*}
\subsection{Group structures on $EG$ and $BG$.}
The usefulness of the non-homogeneous coordinates $t_1, \, \ldots \, ,
t_{n}$ on $\ensuremath{\Delta}\xspace^n$ comes from the fact that they supply a very simple
formula
\begin{gather*}
(t_1, \, \ldots \, , t_{n}) \times (t_{n+1}, \, \ldots \, ,t_{n+m+1}) \mapsto
(t_{\ensuremath{\sigma}\xspace(1)}, t_{\ensuremath{\sigma}\xspace(2)}, \, \ldots \, , t_{\ensuremath{\sigma}\xspace(n+m+1)})
\end{gather*}
for a homeomorphism pairing
\[\ensuremath{\Delta}\xspace^n \times \ensuremath{\Delta}\xspace^{m} \longrightarrow \ensuremath{\Delta}\xspace^{n+m} \]
where \ensuremath{\sigma}\xspace is a permutation of the set $\{1,2, \, \ldots \, ,n+m+1\}$ such
that
\[ t_{\ensuremath{\sigma}\xspace(1)} \leq t_{\ensuremath{\sigma}\xspace(2)} \leq \, \ldots \, \leq t_{\ensuremath{\sigma}\xspace(n+m+1)}.\]
Using this pairing we can define, for $G$ an abelian topological
group, commutative, associative, and continuous pairings
\begin{multline*}
\bigl\lvert t_1, \, \ldots \, , t_{n} , h[h_1| \, \cdots \, |h_n]\bigr\rvert +
\bigl\lvert t_{n+1},\, \ldots \, ,t_{n+m+1}, h'[h_{n+1}|\, \cdots \, |h_{n+m+1}]\bigr\rvert
=\\
= \bigl\lvert t_{\ensuremath{\sigma}\xspace(1)}, \, \ldots \, ,t_{\ensuremath{\sigma}\xspace(n+m)}, h\cdot h'[h_{\ensuremath{\sigma}\xspace(1)}| \, \cdots
\, |
h_{\ensuremath{\sigma}\xspace(n+m+1)}]\bigr\rvert
\end{multline*}
and
\begin{multline*}
\bigl\lvert t_1, \, \ldots \, , t_{n} , [h_1| \, \cdots \, |h_n]\bigr\rvert +
\bigl\lvert t_{n+1},\, \ldots \, ,t_{n+m+1}, [h_{n+1}|\, \cdots \, |h_{n+m+1}]\bigr\rvert =\\
= \bigl\lvert t_{\ensuremath{\sigma}\xspace(1)}, \, \ldots \, ,t_{\ensuremath{\sigma}\xspace(n+m+1)}, [h_{\ensuremath{\sigma}\xspace(1)}| \, \cdots \, |
h_{\ensuremath{\sigma}\xspace(n+m+1)}]\bigr\rvert
\end{multline*}
\noindent on $EG$ and $BG$ respectively, which induce group structure on
these spaces \cite{milg-bar}.
The above group parings can be interpreted as the compositions
\begin{gather*}
EG\times EG = |EG.|\times |EG.|
\overset{\bar{\ensuremath{\varphi}\xspace}}{\longrightarrow}
|EG.\times EG.| \overset{\bar{\ensuremath{\psi}\xspace}}{\longrightarrow}
|E(G\times G).| \overset{|\bar{\mu}|}{\longrightarrow}|EG.|=EG\\
BG\times BG = |BG.|\times |BG.|\overset{\ensuremath{\varphi}\xspace}{\longrightarrow}
|BG.\times BG.| \overset{\ensuremath{\psi}\xspace}{\longrightarrow}
|B(G\times G).| \overset{|\mu|}{\longrightarrow}|BG.|=BG
\end{gather*}
where $\bar{\ensuremath{\varphi}\xspace}$ and $\ensuremath{\varphi}\xspace$ are the commutativity of geometric
realization and product operations homeomorphisms, $\bar{\ensuremath{\psi}\xspace}$ and
$\ensuremath{\psi}\xspace$ are induced by the maps
\begin{gather*}
EG_n \times EG_n \longrightarrow EG_n\\
(\seq{g})\times (\seq{g'}) \mapsto ((g_0,g'_0), \, \ldots \, ,(g_n,g'_n))\\
\intertext{and}
BG_n \times BG_n \longrightarrow BG_n\\
[g_0:\, \cdots\, :g_n]\times [g'_0:\, \cdots\, :g'_n]
\mapsto [(g_0,g'_0):\, \cdots\, :(g_n,g'_n)]
\end{gather*}
respectively, and $|\bar{\mu}|, |\mu|$ are induced by the simplicial
morphisms
\begin{gather*}
\bar{\mu}: E(G\times G). \longrightarrow EG.\\
\bar{\mu}((g_0,g'_0), \, \ldots \, ,(g_n,g'_n)) = (g_0g'_0, \, \ldots \,
,g_ng'_n)\\
\intertext{and}
\mu: B(G\times G). \longrightarrow BG.\\
\mu([(g_0,g'_0):\, \cdots\, :(g_n,g'_n)]) = [g_0g'_0:\, \cdots\,
:g_ng'_n]
\end{gather*}
which are well defined only when the multiplication pairing
$\mu:G\times G \rightarrow G$ is a homomorphism, or equivalently, when $G$ is
an abelian group.
|
1996-01-22T01:52:11 | 9601 | alg-geom/9601009 | en | https://arxiv.org/abs/alg-geom/9601009 | [
"alg-geom",
"math.AG"
] | alg-geom/9601009 | Elizabeth Gasparim | Elizabeth Gasparim | Holomorphic Rank Two Vector Bundles on Blow-ups | Latex2e. University of New Mexico, Ph.D. Thesis | null | null | null | null | In this paper we study holomorphic rank two vector bundles on the blow up of
$ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck
says that any holomorphic vector bundle on the projective plane ${\bf P}^1$
splits into a sum of line bundles. If $E$ is a holomorphic vector bundle over
the blow up of $ {\bf C}^2$ at the origin, then the restriction of $E$ to the
exceptional divisor is a vector bundle over ${\bf P}^1$ and therefore splits.
Moreover we assume that $E$ is a rank two bundle that has zero first Chern
class. Hence its restriction to the exceptional divisor is of the form $ {\cal
O}(j) \oplus {\cal O}(-j) $ for some integer $j.$ We denote by ${\cal M}_j$ the
moduli space of equivalence classes (under holomorphic isomorphisms) of rank
two holomorphic vector bundles on the blow up of $ {\bf C}^2$ at the origin
whose restriction to the exceptional divisor is $ {\cal O}(j) \oplus {\cal
O}(-j) .$
| [
{
"version": "v1",
"created": "Fri, 12 Jan 1996 16:18:50 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Gasparim",
"Elizabeth",
""
]
] | alg-geom | \section*{ Introduction}
\thispagestyle{empty}
Vector bundles on complex surfaces have been
extensively studied by means of several different methods.
See for example the books of
Kobayashi [12] and Okonek, Schneider, Spindler [13].
Stable holomorphic bundles on a K\"ahler surface
correspond by a theorem of Donaldson [5]
to irreducible anti-self-dual connections on the surface.
This result connects the study of holomorphic
vector bundles with moduli space of instantons.
{}From the point of view of the study of instantons,
vector bundles on the blow up of ${\bf P}^2$
appear in Hurtubise's paper on Instantons and Jumping Lines
[10] and in Boyer-Hurtubise-Milgram-Mann [1] in
their proof of the Atiyah-Jones conjecture.
Subsequently Hurtubise and Milgram [11]
proved an extended version of the
Atiyah-Jones conjecture for ruled surfaces, by means of studying the
structure of holomorphic bundles on ruled
surfaces.
\\
To motivate our study of holomorphic vector bundles
on blow-ups from a different stand point
we mention a fundamental result on the classification
of rational surfaces, see Griffiths and Harris [8].
\\
{ \bf Theorem} : Every rational surface is obtained by blowing up
points on either ${\bf P}^2$ or on a rational ruled
surface.
\\
The previous theorem
suggests that
the understanding
of vector bundles on rational surfaces
depends on the analysis of the behavior of
vector bundles under blow-ups.
A large amount of work has been done on vector
bundles on ${\bf P}^2$ ( see for example
the book by Okonek, Schneider, Spindler [13]).
In a sense we can also say that vector bundles on ruled surfaces
are well understood ( see Brosius [2] [3],
Qin [14], Hurtubise and Milgram [11]).
Some examples of work on moduli spaces of
holomorphic vector bundles on blow-ups
are the papers by by Freedman and Morgan [6][7],
Brussee [4], and Qin [15].
\\
The blow-up of a point on a surface is a local
operation in the sense that one blows-up the point
inside one of its coordinate neighborhoods.
Roughly speaking we may see the ``difference''
between moduli spaces of bundles
on a rational surface and moduli spaces of
bundles on one of its minimal models by studying bundles
on the blow up of ${\bf C}^2.$
\\
In this work we concentrate on the study
of bundles on blow-ups in the local sense,
that is in a neighborhood of the exceptional divisor.
Our approach is quite concrete, as we
give bundles explictly by their transition matrices
and present the moduli spaces as quotients of
a vector space ${\bf C}^n$ by an equivalence relation.
\\
In Section 3 we construct a canonical form of transition matrix
for rank two bundles on the blow up of ${\bf C}^2.$ Namely, we prove
the following:
\\
\vspace {5 mm}
\noindent{\bf Theorem 2.1}: Let $E$ be a
holomorphic rank two vector bundle on $ \widetilde{\bf C}^2 $ with
zero first Chern class and
let j be the integer that satisfies
$E_{\ell} \simeq {\cal O}(j) \oplus {\cal O}(-j). $
(Where $E_{\ell}$ is the restriction of $E$ to
the exceptional divisor.)
Then $E$ has a transition matrix
of the form
\\
$$\left(\matrix {z^j & p \cr 0 & z^{-j} \cr }\right)$$
from $U$ to $V,$ where
\\
$$p = \sum_{i = 1}^{2j-2} \sum_{l = i-j+1}^{j-1}p_{il}z^lu^i.$$
In particular $p$ depends on a finite number of parameters.
\\
\vspace{5 mm}
\\
We then define the moduli space ${\cal M}_j$ as
the space of equivalence classes of such bundles having
restriction ${\cal O}(j) \oplus {\cal O}(-j)$ to the
exceptional divisor, modulo holomorphic equivalence.
It follows immediately from our canonical form
of a transition matrix that:
\\
\vspace{5 mm}
\noindent{\bf Corollary 2.3}: ${\cal M}_0$ consists of a single point.
\\
\vspace{5 mm}
\\
\noindent{\bf Corollary 2.5}: ${\cal M}_1$ consists of a single point.
\\
\vspace{5 mm}
\\
In Section 4 we continue the study of ${\cal M}_j$
for $j \ge 2.$
To do this we analyze the problem of when two holomorphic
bundles in ${\cal M}_j$ are isomorphic.a simple characterization on
the first formal neighborhood
of the exceptional divisor.
\\
\vspace{ 5 mm}
\noindent{\bf Proposition 3.3}
On the first formal neighborhood,
two holomorphic bundles $E^{(1)}$ and $E^{(1)\prime}$ with
transition matrices
$$\left(\matrix{z^j & p_1 \cr 0 & z^{-j} \cr}\right)$$
and
$$\left(\matrix{z^j & p^\prime_1 \cr 0 & z^{-j} \cr}\right)$$
respectively are isomorphic iff
$p^\prime_1 = \lambda p_1$ for some
$\lambda \in {\bf C} - \{0\}$.
\\
\vspace {5 mm}
\\
Once one passes the first formal neighborhood,
the holomorphic equivalences become more intricate.
In 5.1 we give a detailed description of
${\cal M}_2.$ Topologicaly, we have:
\\
\vspace{5 mm}
\\
\noindent{\bf Theorem 4.2}: The moduli space ${\cal M}_2$ is
homeomorphic to
the union ${\bf P}^1 \cup \{p,q\},$
of a complex projective plane ${\bf P}^1$ and two points
with a basis of open sets given by
$${\cal U} \cup \{p,U : U \in {\cal U} - \phi \} \cup
\{p,q,U : U \in {\cal U} - \phi \} $$
where ${\cal U}$ is a basis for the standard topology on ${\bf P}^1.$
\\
\vspace{5 mm}
\\
In 5.2 we describe ${\cal M}_3$ and in 5.3 we give the generic
description of ${\cal M}_j.$ Our general results are:
\\
\vspace {5 mm}
\noindent{\bf Theorem 4.4}
The generic set of the moduli space ${\cal M}_j$ is a complex
projective space of dimension $2j-3$ minus a closed
subvariety of complex codimension bigger than or equal to two.
\\
\vspace{5 mm}
\\
\noindent{\bf Remark 4.6}: The moduli space ${\cal M}_j$ also contains
complex projective spaces of every dimension smaller
than $2j-3, $ each minus some closed subvariety.
\\
\vspace {5 mm}
\\
\noindent{\bf Remark 4.7}: If we give ${\cal M}_j$ the topology
induced
from ${\bf C}^N$, then ${\cal M}_j$ is not a Hausdorff space.
For example, the direct sum bundle given by
$\left(\matrix{ z^j & 0 \cr 0 & z^{-j} \cr
}\right)$ is arbitrarily close to any other bundle.
\\
\vspace {5 mm}
\\
\noindent{\bf Remark 4.8}: Note that the word generic here is used
in the sense that the moduli space ${\cal M}_j$
consists of subsets out of
which ${\bf P}^{2j-3} $ is the subset of highest
dimension.
\\
\vspace{5mm}
\\
Finally in Section 6 we give some examples
of the result of building up bundles on the
blow up of a compact surface using our
canonical form of a transition matrix for a neighborhood of the
exceptional divisor.
\\
\vspace{5mm}
\\
Note: This is a quite long file, so I am only sending the "introduction."
If anyone wants the whole file, be welcome to write to
[email protected]
\\
\end{document}
|
1996-01-22T06:20:16 | 9601 | alg-geom/9601020 | en | https://arxiv.org/abs/alg-geom/9601020 | [
"alg-geom",
"math.AG"
] | alg-geom/9601020 | Christoph Lossen | Gert-Martin Greuel, Christoph Lossen and Eugenii Shustin | Geometry of families of nodal curves on rational surfaces | AMSLaTeX v 1.2 | null | null | null | null | Let P^2_r be the projective plane blown up at r generic points. Denote by
E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the
exceptional divisors of the blown-up points on P^2_r respectively. We consider
the variety V of all irreducible curves C in |dE_0-d_1E_1-...-d_rE_r| with k
nodes as the only singularities and give asymptotically nearly optimal
sufficient conditions for its smoothness, irreducibility and non-emptyness.
Moreover, we extend our conditions for the smoothness and the irreducibility on
families of reducible curves. For r<10 we give the complete answer concerning
the existence of nodal curves in V.
| [
{
"version": "v1",
"created": "Fri, 19 Jan 1996 14:43:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Greuel",
"Gert-Martin",
""
],
[
"Lossen",
"Christoph",
""
],
[
"Shustin",
"Eugenii",
""
]
] | alg-geom | \section*{Introduction}
We deal with the following general problem: given a
smooth rational surface $S$ and a divisor $D$ on $S$, when is the variety
$V_{irr}(D,k)$ of nodal irreducible curves in the complete linear system $|D|$
with a fixed number $k$ of nodes non--empty, when non--singular and when
irreducible? For \mbox{$S=\P^2$}, these questions are completely answered by
the classical result of F.~Severi (\cite{Sev}), stating that the variety
$V_{irr}(dH,k)$ of
irreducible curves of degree $d$ having $k$ nodes is non--empty and smooth
exactly if $$0\leq k \leq \frac{(d-1)(d-2)}{2}\:,$$
and the result of J.~Harris (\cite{Har}), stating that $V_{irr}(dH,k)$ is
always irreducible.\\
A modification of Severi's method did lead to a sufficient
(smoothness--)criterion for general smooth rational surfaces $S$
(\cite{Ta1,Nob}): let \mbox{$C_0\subset S$} be a smooth
irreducible curve, let \mbox{$C\in |C_0|$} be a reduced (nodal) curve with
precisely $k$ nodes, such that \mbox{$C=C_1\cup \ldots \cup C_s$}, $C_i$
irreducible and
\begin{equation}
\label{0.1}
K_S\cdot C_i < 0
\end{equation}
for each \mbox{$1\leq i\leq s$}, then the variety $V_{irr}(C_0,k)$ of
irreducible curves in $|C_0|$ having precisely $k$ nodes is smooth (see
\cite{Ta2,GrM,GrK,GrL} for generalizations to other surfaces). Moreover,
in those cases each node of $C$ can be smoothed independently.\\
In this paper, we concentrate on the case \mbox{$S=\P^2_r$}, the projective
plane blown up at $r$ {\em generic} points \mbox{$p_1,\ldots,p_r$}. Let
\mbox{$E_0,E_1,\ldots ,E_r$} denote the strict transform of a generic
straight line on $\P^2$ and the exceptional divisors of the blown--up points on
$\P^2_r$, respectively. Then for an irreducible nodal curve
\mbox{$C\in|dE_0-\sum_{i=1}^{r} d_iE_i|\,$} the condition (\ref{0.1}) reads as
\begin{equation}
\label{0.2}
3d > \sum_{i=1}^r d_i\:.
\end{equation}
In the blown--down situation, such a curve $C$ corresponds to a plane curve of
degree $d$ having (not necessarily ordinary) $d_i$--fold points at $p_i$,
\mbox{$1\leq i\leq r$}, and \mbox{$k'\leq k$} nodes outside. For the variety of
irreducible plane curves of fixed topological (or analytic) type, E.~Shustin
gives in (\cite{Sh2}) an asymptotically improved sufficient condition for
smoothness and irreducibility:
$$
\alpha d^2+o(d^2) > \sum \sigma(S_i)\:,
$$
where $\sigma$ denotes some positive invariant of the singular points. In our
case, \mbox{keeping} $k$, $d$ and the $d_i$ \mbox{$(1\leq i \leq r)$} fixed,
certainly $k'$ and
the topological types of the multiple points may vary. Nevertheless, we shall
obtain sufficient conditions for the smoothness and the
irreducibility of the same type, that is, with the same
exponent in $d$. Moreover, we should like to emphasize that we can extend them
on families of reducible curves
- --- in contrast to A.~Tannenbaum's result for K3-surfaces (cf.~\cite{Ta2}).\\
In section 3, we shall give a complete answer for the existence problem in case
of \mbox{$r\leq 9$} blown--up points (Theorems \ref{4.1A} and \ref{4.1B}). For
\mbox{$r\geq 10$}, we obtain an {\em exponentially optimal} sufficient
condition (Corollary \ref{3.1.4}), that is, of the same exponent in
$d$ as the known restrictions for the existence of the corresponding plane
curves with $d_i$--fold singularities $S_i$ \mbox{$(1\leq i \leq r)$} and $k'$
nodes \mbox{$S_{r+1},\ldots,S_{r+k'}$} (from
Pl\"ucker formulae to inequalities by Varchenko \cite{Var} and Ivinskis
\cite{Ivi,HiF}). These restrictions are of type
$$
\alpha_2 d^2+\alpha_1 d+\alpha_0 > \sum_{i=1}^{r+k'}
\sigma(S_i)\;\;\;\;\;\;\;\;\;\; (\alpha_2=\mbox{const}>0)
$$
with $\sigma$ some positive invariant depending, at most, quadratically on $d$.
Our result improves for the given situation the only previously known
(general) existence criterion (in \cite{Sh1}):
$$ \frac{(d+3)^2}{2} \geq \sum_{i=1}^{r+k'} (\mu(S_i)+4)(\mu(S_i)+5)\:, $$
which is not exponentially optimal since the right--hand side may be of order
four in $d$.
For the proof, we combine a modification of the method of
A.~Hirschowitz in \cite{Hir} and the smoothing of nodes (cf.~\cite{Ta1}).
\section*{Notation and terminology}
Throughout this article we consider all objects to be defined over an
algebraically closed field {\bf $K$} of characteristic zero. We use the
following notations:
\begin{itemize}
\itemsep0.1cm
\item $\P^2_r$ --- the projective plane blown up at r generic
points \mbox{$p_1,\ldots,p_r$}.
\item $\frak{m}_{z_\nu}$ --- the maximal ideal in the local ring
${\mathcal O}_{\P^2_r,z_{\nu} }$, \mbox{$z_{\nu}\in \P^2_r$}.
\item $E_0$ --- the strict transform of a generic straight line (in $\P^2$).
\item $E_i$ \mbox{$(1\leq i\leq r)$} --- the exceptional divisor of the
blown--up point $p_i$ on $\P^2_r$.
\item \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ ---
the variety of all irreducible curves $C$ in the linear system
\mbox{$|dE_0-\sum_{i=1}^{r} d_iE_i|\,$}
having k nodes as their only singularities.
\end{itemize}
Furthermore, for a reduced nodal curve \mbox{$C\subset \P^2_r$},
\mbox{$\,C=C_1\cup\ldots\cup C_s$} ($C_i$ irreducible), having precisely $k$
nodes as their only singularities and a divisor $D$ on $\P^2_r$, we denote:
\begin{itemize}
\item \mbox{$V(|D|;C)$} --- the variety of all reduced curves
\mbox{$\tilde{C} =\tilde{C}_1 \cup\ldots\cup \tilde{C}_s $} in the linear
system
$|D|$ with precisely $k$ nodes as their only singularities, whose components
$\tilde{C}_i $ have the same type (that is, are in the same linear system and
have the same number of nodes) as the components $C_i$ (\mbox{$1\leq i\leq
s$}) of $C$.
\end{itemize}
\section{Smoothness}
\subsection{Formulation of the result}
\setcounter{equation}{0}
For $\P^2_r\,$, \mbox{$r\leq 8\,$}, condition (\ref{0.1}) is fulfilled for each
irreducible curve $C$, hence \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ is always smooth. In case $r=9$ and
\mbox{$C\in V_{irr}(d;\,d_1,\ldots,d_9;\,k)$}, (\ref{0.1}) reads
$$ 3d > \sum_{i=1}^9 d_i\:, $$
which is satisfied exactly if $C$ is not the (unique) smooth cubic through
\mbox{$p_1,\ldots, p_9$}. Thus \mbox{$V_{irr}(d;\,d_1,\ldots,d_9;\,k)$} is
always smooth, too.
In this section we shall prove:
\begin{theorem}[Smoothness Theorem] \label{1.1.1}
Let \mbox{$r\geq 10$}, and let the positive integers
\mbox{$d;\,d_1,\ldots,d_r$} satisfy the two (smoothness) conditions
\renewcommand{\arraystretch}{1.5}
\begin{eqnarray}
\left[ \; \sqrt{2k}\;\, \right] & < & \frac{d}{2} +3-\frac{\sqrt{2} }{2}
\sqrt{\sum_{i=1}^r (d_i+2)^2} \label{Bed1} \\
\left[ \; \sqrt{2k}\;\, \right] & < & d+3-\sqrt{2}\sqrt{2+\sum_{i=1}^r
(d_i+2)(d_i+1)} \:\:.
\label{Bed2}
\end{eqnarray}
Then \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ is smooth and has the T--property (that is, each germ of \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ is
a transversal intersection of germs of equisingular strata corresponding to the
$k$ nodes).
Let \mbox{$C \subset \P^2_r$} be a reduced nodal curve, \mbox{$C\in
|D|$}. If, for each irreducible component \mbox{$C_{\nu} \in
|d^{(\nu)}E_0-\sum_{i=1}^{r} d^{(\nu)}_iE_i|\,$} of $C$ (having precisely
$k^{(\nu)}$ nodes), the two smoothness conditions are fulfilled, then
\mbox{$V(|D|;C)$} is smooth and has the T--property.
\end{theorem}
\subsection{Vanishing criteria} \label{1.2}
\setcounter{equation}{0}
We introduce the vanishing criteria which we
shall mainly use
in the proof of the Smoothness and the Irreducibility Theorem (in the next
paragraph):\\
For a curve \mbox{$C\in \,$\mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}} and a subset \mbox{$\Sigma_0 \subset$
Sing$\,C$} we define
the sheaf ${\mathcal T} ^1_{C,\Sigma_0}$ to be the skyscraper sheaf concentrated at
\mbox{$\mbox{Sing}\,C=\{z_1,\ldots ,z_k\}$} with stalks
\renewcommand{\arraystretch}{1.3}
$$\left( {\mathcal T} ^1_{C,\Sigma_0} \right) _{z_\nu} := \Bigg\{
\begin{array}{ccl}
{\mathcal O}_{\P^2_r,z_\nu}/\frak{m}_{z_\nu} & \mbox{for } & z_\nu \in
\mbox{Sing}\,C-\Sigma_0 \\
{\mathcal O}_{\P^2_r,z_\nu}/\frak{m}_{z_\nu}^2 & \mbox{for } & z_\nu \in
\Sigma_0 \:\:.\\
\end{array}
$$
Furthermore, put
$$ {\mathcal N}_{C/\P^2_r}^{\Sigma_0} :=\mbox{Ker} \left({\mathcal N}_{C/\P^2_r}\longrightarrow
{\mathcal T}^1_{C,\Sigma_0} \right) .$$
\renewcommand{\labelenumi}{(\Alph{enumi})}
\begin{proposition} \label{1.2.1}
Let \mbox{$C\in |dE_0-\sum_{i=1}^r d_iE_i|$} be a reduced nodal curve having
$k$ nodes as its only singularities,
\mbox{$\tilde{C}\sim dE_0-\sum_{i=1}^{r} \tilde{d_i}E_i\,$},
where \mbox{$\tilde{d_i}\geq d_i$} for \mbox{$1\leq i\leq r$}, and
\mbox{$\Sigma_0 \subset$
Sing$\,C=\{z_1,\ldots z_k\}$}. Moreover, let $\tilde{H}$ be a reduced
curve whose local equations map to $0\in\left( {\mathcal T} ^1_{C,\Sigma_0} \right)
_{z_\nu}$ for $1\leq \nu \leq k$.\\
Then $H^1(C,{\mathcal N}_{C/\P^2_r}^{\Sigma_0})$ vanishes, if the following
conditions are satisfied:
\begin{enumerate}
\itemsep0.2cm
\item \label{A} $\;H^1(\P^2_r,{\mathcal O}_{\P^2_r}(C))=0$
\item \label{B} $\;H^1(\P^2_r,{\mathcal O}_{\P^2_r}(\tilde{C}-\tilde{H}))=0$
\item \label{C}
$\;H^1(\tilde{H},\tilde{{\mathcal N}}_{\tilde{H}/\P^2_r}^{\Sigma_0})=0$
\end{enumerate}
where $\;\tilde{{\mathcal N}}_{\tilde{H}/\P^2_r}^{\Sigma_0}:= \mbox{Ker}
\left({\mathcal O}_{\P^2_r}(\tilde{C})\otimes {\mathcal O}_{\tilde{H}} \longrightarrow
{\mathcal T}^1_{C,\Sigma_0} \right)\,$.
\end{proposition}
\begin{proof}
We have an exact sequence
$$ \ldots \rightarrow H^1(\P^2_r,{\mathcal O}_{\P^2_r}(C))\rightarrow
H^1(C,{\mathcal N}_{C/\P^2_r} )\rightarrow H^2(\P^2_r,{\mathcal O}_{\P^2_r})\rightarrow
\ldots
$$
where, by Serre duality,
$H^2(\P^2_r,{\mathcal O}_{\P^2_r})=H^0(\P^2_r,K_{\P^2_r})=0\,$. Hence, the statement of
the proposition follows immediately from the following commutative diagram with
exact columns and diagonal
\unitlength0.7cm
\begin{picture}(18,8)
\put(0.3,7.0){\makebox(3.2,0.8){$\vdots$}}
\put(3.0,6.8){\makebox(7.0,0.8){$H^0({\mathcal O}_{\P^2_r}(C))\supset
H^0({\mathcal O}_{\P^2_r}(\tilde{C}))$}}
\put(9.7,7.0){\makebox(3.2,0.8){$\vdots$}}
\put(1.9,6.7){\vector(0,-1){0.6}}
\put(4.1,6.7){\vector(-2,-1){1.25}}
\put(9.0,6.7){\vector(2,-1){1.25}}
\put(11.3,6.7){\vector(0,-1){0.6}}
\put(0.3,5.2){\makebox(3.2,0.8){$H^0({\mathcal N}_{C/\P^2_r})$}}
\put(9.7,5.2){\makebox(3.2,0.8){$H^0({\mathcal O}_{\P^2_r}(\tilde{C})\!\otimes\!
{\mathcal O}_{\tilde{H}})$}}
\put(1.9,5.1){\vector(0,-1){0.6}}
\put(11.3,5.1){\vector(0,-1){0.6}}
\put(12.2,5.1){\vector(2,-1){1.25}}
\put(0.3,3.6){\makebox(3.2,0.8){$H^0({\mathcal T}^1_{C,\Sigma_0})$}}
\put(3.6,3.94){\line(1,0){6.1}}
\put(3.6,4.06){\line(1,0){6.1}}
\put(9.7,3.6){\makebox(3.2,0.8){$H^0({\mathcal T}^1_{C,\Sigma_0})$}}
\put(13.8,3.6){\makebox(3.2,0.8){$H^1({\mathcal O}_{\P^2_r}(\tilde{C}\!
-\!\tilde{H}))$}}
\put(15.47,3.4){\line(0,-1){0.3}}
\put(15.33,3.4){\line(0,-1){0.3}}
\put(13.8,2.3){\makebox(3.2,0.8){$0$}}
\put(1.9,3.5){\vector(0,-1){0.6}}
\put(11.3,3.5){\vector(0,-1){0.6}}
\put(0.3,1.9){\makebox(3.2,0.8){$H^1({\mathcal N}_{C/\P^2_r}^{\Sigma_0})$}}
\put(9.7,1.9){\makebox(3.2,0.8){$
H^1(\tilde{{\mathcal N}}_{\tilde{H}/\P^2_r}^{\Sigma_0})$}}
\put(1.9,1.8){\vector(0,-1){0.6}}
\put(11.37,1.8){\line(0,-1){0.3}}
\put(11.23,1.8){\line(0,-1){0.3}}
\put(9.7,0.7){\makebox(3.2,0.8){$0$}}
\put(0.3,0.3){\makebox(3.2,0.8){$H^1({\mathcal N}_{C/\P^2_r})$}}
\put(3.6,0.3){\makebox(0.6,0.8){$=\;0$}}
\put(16.0,0.0){\makebox(1.0,0.8){\qed}}
\end{picture}
\renewcommand{\qed}{}\end{proof}
In the following we shall obtain the vanishing properties (A) -- (C) by
applying two well--known criteria:
\begin{proposition}[Hirschowitz-Criterion, \cite{Hir}] \label{Hirschowitz}
Let \mbox{$C\sim dE_0-\sum_{i=1}^r d_iE_i$}, where $d\,,d_1,\ldots d_r$ are
non--negative integers satisfying
\begin{eqnarray} \label{1.2.3}
\sum_{i=1}^r \frac{d_i(d_i+1)}{2} & < & \left[\frac{(d+3)^2}{4}\right]\:,
\end{eqnarray}
then $H^1(\P^2_r,{\mathcal O}_{\P^2_r} (C))=0$.
In a more special situation, let $S_r^I$ be the projective plane blown up at
r points \mbox{$p_1,\ldots ,p_r$} where $p_i$, \mbox{$i\in I$}, lie on a
line, and all the
other points are in generic position. Let $C$ be as above such that
\begin{eqnarray} \label{1.2.3b}
\sum_{i\in I} d_i & \leq & d+1
\end{eqnarray}
and condition $($\ref{1.2.3}$)$ holds, then \mbox{$H^1(S_r^I,{\mathcal O}_{S^I_r}
(C))=0$}.
\end{proposition}
\begin{proposition}[\cite{GrK}] \label{Greuel} Let $S$
be a smooth surface,
\mbox{$C\subset S$} a compact reduced curve, ${\mathcal F} $ a torsionfree coherent
${\mathcal O}_C$-module which has rank 1 on each irreducible component $C_i$ of $C$
\mbox{$(1\leq i\leq s)$}. Then \mbox{$H^1(C,{\mathcal F})=0$} if for \mbox{$1\leq
i\leq s$}
\begin{equation}
\label{1.2.5}
\chi (\overline{{\mathcal F} \otimes {\mathcal O}_{C_i}}) > \chi (\omega_C \otimes
{\mathcal O}_{C_i}) - \mbox{isod}_{C_i}({\mathcal F}, {\mathcal O}_C)\:.
\end{equation}
Here $\overline{\raisebox{0.2cm}{\ \ \ } }$ denotes reduction modulo torsion,
$\omega_C $ the
dualizing sheaf and the isomorphism defect $\mbox{isod}_{C_i}({\mathcal F}, {\mathcal O}_C)$ is
defined to be the sum of all
$$ \:\mbox{isod}_{C_i,x}({\mathcal F}, {\mathcal O}_C):=\mbox{min}\,(\mbox{dim}_{\,{\Bbb C}}\:
\mbox{coker}\,(\varphi
_{C_i}:\,(\overline{{\mathcal F} \otimes {\mathcal O}_{C_i}})_x \rightarrow {\mathcal O}_{C_i,x}))\,,$$
$x\in C_i\,$, where the minimum is taken over all $\varphi_{C_i}$ which are
induced by local homomorphisms \mbox{$\,\varphi :\,{\mathcal F}_x\rightarrow
{\mathcal O}_{C,x}$}.
\end{proposition}
\begin{corollary} \label{1.2.6}
If, in the situation of Proposition \ref{1.2.1}, we have for each irreducible
component $\tilde{H_i}\,$ \mbox{$(1\leq i\leq s)$} of $\tilde{H} $
\begin{eqnarray}
\label{1.2.7}
&& \tilde{H}_i\cdot (\tilde{C}\!-\!\tilde{H}\!-\!K_{\P^2_r})\; >\;
\#\,(\mbox{Sing}\,C\cap \tilde{H}_i)\;:= \! \sum_{z\in
\mbox{\footnotesize Sing}\,C} \!\mbox{multiplicity}\,(\tilde{H}_i,z)
\end{eqnarray}
then $H^1(\tilde{H},\tilde{{\mathcal N}}_{\tilde{H}/\P^2_r}^{\Sigma_0})\,$ vanishes.
\end{corollary}
\begin{proof}
Applying the Riemann--Roch--Theorem and the adjunction formula,
condition (\ref{1.2.5}) reads
$$\mbox{deg} \,(\overline{\tilde{\kn}_{\tilde{H}/\P^2_r}^{\Sigma_0} \otimes
{\mathcal O}_{\tilde{H}_i}})\;>\;(K_{\P^2_r}+\tilde{H})\cdot \tilde{H}_i -
\mbox{isod}_{\tilde{H}_i}(\tilde{\kn}_{\tilde{H}/\P^2_r}^{\Sigma_0}, {\mathcal O}_{\tilde{H}})\:. $$
The exact sequence \mbox{$\;0\rightarrow \tilde{\kn}_{\tilde{H}/\P^2_r}^{\Sigma_0} \rightarrow
{\mathcal O}_{\P^2_r}(\tilde{C}) \otimes
{\mathcal O}_{\tilde{H}} \rightarrow {\mathcal T}^1_{C,\Sigma_0} \rightarrow 0\;$} implies
$$ \mbox{deg}\,(\overline{\tilde{\kn}_{\tilde{H}/\P^2_r}^{\Sigma_0} \otimes {\mathcal O}_{\tilde{H}_i}})\:=\:
\mbox{deg}\,({\mathcal O}_{\P^2_r}(\tilde{C}) \otimes {\mathcal O}_{\tilde{H}_i}) -
\chi\,({\mathcal T}^1_{C,\Sigma_0} \otimes {\mathcal O}_{\tilde{H}_i})\:. $$
Finally, an easy consideration shows that
\begin{eqnarray*}
\lefteqn{\chi\,({\mathcal T}^1_{C,\Sigma_0} \otimes
{\mathcal O}_{\tilde{H}_i})-\mbox{isod}_{\tilde{H}_i}(\tilde{\kn}_{\tilde{H}/\P^2_r}^{\Sigma_0},
{\mathcal O}_{\tilde{H}}) } \hspace{1.0cm}\\
& & = \sum_{z\in \,\mbox{\footnotesize Sing}\,C}
dim_{{\Bbb C}}({\mathcal T}^1_{C,\Sigma_0}
\otimes {\mathcal O}_{\tilde{H}_i})_z-isod_{C_i}(\tilde{\kn}_{\tilde{H}/\P^2_r}^{\Sigma_0},{\mathcal O}_{\tilde{H}})\\
& & \leq \#\,(\mbox{Sing}\:C\:\cap \: \tilde{H}_i)\:. \qed
\end{eqnarray*}
\renewcommand{\qed}{}\end{proof}
\subsection{Proof of the Smoothness Theorem}
\setcounter{equation}{0}
Following (\cite{GrK}, Theorem 6.1), it is sufficient
to show that the first
cohomology group $H^1(C,{\mathcal N}_{C/\P^2_r}^{\;\emptyset})$ of the sheaf
$$ {\mathcal N}_{C/\P^2_r}^{\;\emptyset}=\mbox{Ker}\,({\mathcal N}_{C/\P^2_r}\longrightarrow
{\mathcal T}^1_{C} )$$
vanishes, where ${\mathcal T}^1_{C}$ denotes the skyscraper sheaf concentrated
in the singular set \mbox{Sing$\,C=\{z_1,\ldots,z_k\}$} with stalk in
$z_\nu$
$$({\mathcal T}^1_{C})_{z_{\nu}} =
{\mathcal O}_{\P^2_r,z_{\nu}}/\frak{m}_{z_{\nu}}\:.$$
If the reduced curve \mbox{$C\subset \P^2_r$} decomposes as \mbox{$C=C'\cup
C''$}, then we can consider the exact sequence
$$ 0\longrightarrow {\mathcal N}_{C'/\P^2_r}\oplus {\mathcal N}_{C''/\P^2_r}
\stackrel{\alpha}{\longrightarrow} {\mathcal N}_{C/\P^2_r} \longrightarrow {\mathcal O}_{C'\cap
C''} \longrightarrow 0\,,$$
$\alpha$ being induced by \mbox{$\mbox{id}_1\otimes G+F\otimes \mbox{id}_2$},
where $F$ (resp.~$G$) denotes a (local) equation of $C'$ (resp.~$C''$). Since
$C'$ and $C''$ intersect only in nodes, $\alpha$ maps precisely
\mbox{${\mathcal N}_{C'/\P^2_r}^{\;\emptyset}\oplus {\mathcal N}_{C''/\P^2_r}^{\;\emptyset}$} to
\mbox{${\mathcal N}_{C/\P^2_r}^{\;\emptyset}$}, and the statement of the theorem follows
immediately (by induction) from the vanishing statement in the irreducible
case.
First, we have to make some easy considerations about exceptional curves:
\begin{lemma} \label{1.3.1}
Let \mbox{$H\in |hE_0-\sum_{i=1}^r h_iE_i|$} be an irreducible curve, then
$$ \frac{1}{h^2} \sum_{i=1}^r h_i^2\; \leq\;1+\frac{1}{h}\;\leq 2\:.$$
\end{lemma}
\begin{proof}
The r blown--up points \mbox{$p_1,\ldots,p_r$} are chosen generically. Hence,
the existence of an
irreducible curve \mbox{$H \in |hE_0-\sum_{i=1}^r h_iE_i|$}, that is of an
irreducible curve \mbox{$\bar{H} \subset \P^2$} passing through the $p_i$ with
multiplicity $h_i$ (\mbox{$1\leq i\leq r$}), implies for an additional point
$p\:\!'\!\!_{\nu}\not\in H$, close to $p_{\nu}$ with \mbox{$h_{\nu }\geq 1$},
the existence of a
curve \mbox{$ \bar{H'} \subset \P^2 $} passing through $p_i$ with
multiplicity $h_i$ (\mbox{$i\not= \nu$}), through $p_{\nu}$ with multiplicity
$h_{\nu}-1$ and through the additional point $p\:\!'\!\!_{\nu}$.\\
We can assume \mbox{$\,h\geq h_1\geq h_2,\ldots \geq h_r>0\,$} and obtain by
B\'ezout's theorem:
$$h_1^2+\ldots +h_{r-1}^2+h_r(h_r-1) \leq h^2\:.$$
The above statement follows immediately.
\end{proof}
\begin{remark} \label{1.3.2}
We call an irreducible curve \mbox{$H \in |hE_0-\sum_{i=1}^r h_iE_i|$} an {\em
exceptional curve}, if $$\:\sum_{i=1}^r h_i^2 > h^2\:.$$ Applying
B\'ezout's
theorem, it is clear that for fixed data $h$, $h_i$ (\mbox{$1\leq i\leq r$})
there is at most one such exceptional curve $H$; hence for fixed degree $h$
there are only finitely many exceptional curves. For example, for $h=1$ the
exceptional curves are just the lines connecting two of the blown--up points.
\end{remark}
We divide \mbox{Sing$\,C=\Sigma_1 \cup \Sigma_2 $} where $\Sigma_2 $ denotes
the set
of all nodes lying on the ex\-cep\-tional divisors $E_i$ \mbox{$(1\leq i\leq
r)$}. Let
\mbox{$ H\in |hE_0-\sum_{i=1}^r h_iE_i|$} be a (reduced) curve of minimal
degree passing through $\Sigma_1 $. Such a curve exists (at least) for each $h$
fulfilling
$ h(h+3)/2 \geq k $,
hence we can suppose \mbox{$h\leq [\sqrt{2k} ]\,$}. Moreover define
$$ \tilde{H}:=H \cup E_1\cup \ldots \cup E_r\:\in \:|hE_0-\sum_{i=1}^r
(h_i-1)E_i| $$
and let \mbox{$\tilde{C} \sim dE_0-\sum_{i=1}^r \tilde{d}_iE_i$} with
\mbox{$\tilde{d}_i :=
\mbox{max}\,\{d_i,\,h_i+[\frac{d_i}{2}]-1\}$}. Applying Proposition \ref{1.2.1}
we have to check three conditions:
\begin{enumerate}
\item By the Hirschowitz-Criterion (\ref{Hirschowitz})
$\:H^1(\P_r^2,{\mathcal O}_{\P^2_r}(C))$ vanishes, because (\ref{Bed2}) implies
(\ref{1.2.3}).
\item The same criterion gives the vanishing of
$\,H^1(\P^2_r,{\mathcal O}_{\P^2_r}(\tilde{C}-\tilde{H}))$, because
\begin{eqnarray*}
\sqrt{\frac{(d-h+3)^2}{4}} & \geq & \frac{d+3-[\sqrt{2k}]}{2}
\:\,\stackrel{\mbox{\footnotesize(\ref{Bed2})} }{>}
\:\,\sqrt{1+\sum\nolimits_{i=1}^r
\frac{(d_i+1)(d_i+2)}{2}}\\
& \geq & \sqrt{1+\sum\nolimits_{i=1}^r
\frac{(\tilde{d}_i-h_i+1)(\tilde{d}_i-h_i+2)}{2}}\:.\\
\end{eqnarray*}
\item Applying Corollary \ref{1.2.6}, we have to check condition (\ref{1.2.7})
for
each irreducible component \mbox{$H_{\nu}\in |h^{(\nu)}-\sum_{i=1}^r h^{(\nu
)}_iE_i|\,$}, \mbox{$1\leq \nu \leq s$}, of $H$ and each exceptional
divisor
$E_i$, \mbox{$1\leq i\leq r\:$}:
\begin{eqnarray*}
E_i\cdot (\tilde{C}-\tilde{H}-K_{\P_r^2} )-\#(\mbox{Sing}\,C \cap
E_i) &\stackrel{\mbox{\footnotesize B\'ezout}}{\geq} & \tilde{d}_i -
(h_i-1)+1-\left[\frac{d_i}{2}\right]\\
& > & 0
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{\hspace*{-0.55cm}H_{\nu}\cdot (\tilde{C}-\tilde{H} -K_{\P^2_r}) -
\#(\mbox{Sing}\,C
\cap H_{\nu} ) } \hspace{-0.8cm}\\
& \stackrel{\mbox{\footnotesize B\'ezout}}{\geq} &
\!\!\!h^{(\nu)}(d\!-\!h\!+\!3)-\! \sum_{i=1}^r
h_i^{(\nu)}(\tilde{d}_i\!-\!(h_i\!-\!1)\!+\!1)-\!\left[\frac{h^{(\nu)}d\!-\!
\sum_{i=1}^r h_i^{(\nu)} d_i}{2}\right]\\
& \stackrel{\mbox{\footnotesize Cauchy}}{\geq} &
\!\!\!h^{(\nu)}\,\left(\frac{d}{2}+3-[\sqrt{2k}]\right)-\sqrt{
\sum\nolimits_{i=1}^r
\big(h_i^{(\nu)}\big)^2} \:\sqrt{\sum\nolimits_{i=1}^r
\frac{(d_i+2)^2}{4}}\\
& \stackrel{\mbox{\footnotesize (\ref{1.3.1})}}{\geq} &
\!\!\!h^{(\nu)}\,\left(\frac{d}{2}+3-[\sqrt{2k}]-\frac{\sqrt{2}}{2}\,\sqrt{
\sum\nolimits_{i=1}^r (d_i+2)^2}\right)
\;\;\stackrel{\mbox{\footnotesize (\ref{Bed1})}}{>} \;\; 0 \;\;\qed
\end{eqnarray*}
\end{enumerate}
\section{Irreducibility}
\subsection{Formulation of the result}
\setcounter{equation}{0}
For $\P^2_1$, the projective plane blown up at one point $p_1$, Z.~Ran shows in
\cite{Ran} that the variety of all irreducible nodal curves
\mbox{$C\in|dE_0-d_1E_1|$}
having exactly k nodes, none of them lying on the exceptional divisor $E_1$, is
irreducible. Using the smoothness of $V_{irr}(d;d_1;k)$, one can easily deduce
its irreducibility. The aim of this section is to prove the following
irreducibility criterion for \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ (\mbox{$r\geq2$}):
\begin{theorem}[Irreducibility Theorem] \label{2.1.1}
Let \mbox{$r\geq 2$}, and let the positive integers \mbox{$d;\,d_1,\ldots,d_r$}
satisfy the two (irreducibility) conditions
\renewcommand{\arraystretch}{1.5}
\begin{eqnarray}
\left[ \; \sqrt{2k}\;\, \right] & < &
\frac{d}{4}+1-\frac{1}{4}\sqrt{\sum_{i=1}^r d_i^2} \label{2.1.2} \\
\left[ \; \sqrt{2k}\;\, \right] & < &
\frac{d}{2}+1-\frac{\sqrt{2}}{2}\sqrt{\sum_{i=1}^r (d_i+2)^2}\:. \label{2.1.3}
\end{eqnarray}
Then \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ is (smooth and) irreducible.
Let \mbox{$C \subset \P^2_r$} be a reduced nodal curve, \mbox{$C\in
|D|$}. If for each irreducible component \mbox{$C_{\nu} \in
|d^{(\nu) }E_0-\sum_{i=1}^{r} d^{(\nu)}_iE_i|\,$}, \mbox{$1\leq \nu\leq
n$},
of $C$ (having precisely $k^{(\nu)}$ nodes) the variety \mbox{$V_{irr}(d^{(\nu)
};\,d^{(\nu)}_1,\ldots,d^{(\nu)}_r;\,k^{(\nu)})$} is smooth and irreducible,
then \mbox{$V(|D|;C)$} is irreducible.
\end{theorem}
The main idea of our proof is as follows. We show that for an irreducible
curve \mbox{$C\in|dE_0-\sum_{i=1}^{r} d_iE_i|\,$} in an open dense subset
\mbox{$\tilde{V} \subset$ \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}} the cohomology group
$H^1(C,{\mathcal N}_{C/\P^2_r}^{\mbox{\scriptsize \it Sing$\,$C}})$ vanishes
(cf.~Section \ref{1.2}), especially, that the conditions imposed by fixing the
$k$ singular points are independent. It follows that the restricted morphism
$$
\begin{array}{cccl}
\pi_{\tilde{V}}: & \tilde{V} & \longrightarrow & \mbox{Sym}^k(\P^2_r)\\
& C & \longmapsto & \mbox{Sing}\, C \\
\end{array}
$$
is dominant, its fibres are all equidimensional and irreducible as open subsets
of the linear system
\mbox{$H^0 ( \P^2_r,\mbox{Ker}\,({\mathcal O}(dE_0\!-\!\sum
d_iE_i)\rightarrow
{\mathcal T}^1_{C,\mbox{\scriptsize \it Sing$\,$C}})\:\!)$}. Hence
$\tilde{V}$ is irreducible, which implies the irreducibility of \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}.
The second statement is a consequence of the fact that for a fixed
reduced nodal curve $C$ as above, each generic member $\tilde{C}$ of a
component of
\mbox{$V(|D|;C)$} decomposes into components $\tilde{C}_{\nu }$, \mbox{$1\leq
\nu \leq
n$}, which are generic elements of \mbox{$V_{irr}(d^{(\nu)};\,d^{(\nu)
}_1,\ldots,d^{(\nu)}_r;\,k^{(\nu)})$}. Hence there is a well--defined dominant
morphism
$$\prod_{\nu =1}^n U_{\nu} \longrightarrow V(|D|;C)$$
where $U_{\nu}$ is open dense in \mbox{$V_{irr}(d^{(\nu)};\,d^{(\nu)
}_1,\ldots,d^{(\nu)}_r;\,k^{(\nu)})$}.
\subsection{Proof of the Irreducibility Theorem}
\setcounter{equation}{0}
We start the proof defining the subset $\tilde{V}$ of
\mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ as the set of all irreducible curves $C$ in \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ having
the subsequent properties:
\begin{enumerate}
\itemsep0.1cm
\item[(a)] \mbox{$\mbox{Sing}\,C \cap E_i =\emptyset \:$} for
\mbox{$i=1,\ldots ,r.$}
\item[(b)] If $E$ is an exceptional curve of
degree $e\leq 2k\,$ then \mbox{$\,\mbox{Sing}\,C \cap E =\emptyset $}.
\item[(c)] The $k$ nodes of $C$ are in general position, that is, if
\mbox{$H\in
|hE_0-\sum_{i=1}^r h_iE_i|\,$} is a curve containing Sing$\,C$,
\mbox{$H_{\nu}\in
|h^{(\nu)}E_0-\sum_{i=1}^r h_i^{(\nu)}E_i|\,$} is an irreducible component of
$H$, then
\mbox{$h^{(\nu)}(h^{(\nu)}\!+3)/2\geq k_{\nu} := \#(\mbox{Sing}\,C\cap
H_{\nu})\:.$}
\end{enumerate}
\begin{remark} \label{2.2.1}
Condition (c) implies the existence of an irreducible curve among all
curves \mbox{$H\in |hE_0-\sum_{i=1}^r h_iE_i|\,$} of minimal degree
containing $\mbox{Sing}\,C$:\\
Assume a curve \mbox{$\,H=H_1\cup H_2\cup \tilde{H}\,$} of minimal degree $h$
decomposes \mbox{($h^{(1)}\!\leq h^{(2)}$)}, then we know that \mbox{$H_1\cup
H_2$} contains at most
\begin{eqnarray*}
\lefteqn{\frac{h^{(1)}(h^{(1)}\!+3)}{2} + \frac{h^{(2)}(h^{(2)}\!+3)}{2} =
}\hspace{0.5cm}\\
& &\frac{(h^{(1)}+h^{(2)}\!-1)(h^{(1)}+h^{(2)}\!+2)}{2}-
\frac{h^{(1)}(h^{(2)}\!-2)+h^{(2)}(h^{(1)}\!-2)}{2}+1
\end{eqnarray*}
nodes of $C$. The degree of $H$ being minimal, we conclude that either
\mbox{$h^{(1)}\!=1$} or \mbox{$h^{(1)}\!=h^{(2)}\!=2$}. But in
these cases,
using the obvious constructions and Bertini's theorem, we can show the
existence of an irreducible curve of degree \mbox{$h^{(1)}\!+h^{(2)}$}, which
contains the nodes of $C$ lying on \mbox{$H_1\cup H_2$}.
\end{remark}
\begin{lemma} \label{2.2.2}
\mbox{$\,\tilde{V}\subset$ \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}} is an open dense subset.
\end{lemma}
\begin{proof}
The openess being obvious, it is enough to show that there are no obstructions
for (locally) moving singular points of \mbox{$C\in$ \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}} in a prescribed
position (such that the conditions (a)--(c) are satisfied). Again, we divide
\mbox{Sing$\,C=\Sigma_1\cup \Sigma_2$} where $\Sigma_2$ denotes the set of all
nodes
lying on the exceptional divisors $E_i$ \mbox{$(1\leq i\leq r$)} and start
moving nodes away from the exceptional divisors (such that
finally \mbox{$\Sigma_2=\emptyset$)}:\\
Let $z$ be a node of $C$ on $E_{i_0}$, we have to show that for
\mbox{$\Sigma_0:=\mbox{Sing}\,C-\{z\}$} the cohomology group
$H^1(C,{\mathcal N}_{C/\P^2_r}^{\Sigma_0})$ vanishes: indeed, from the commutative
diagram
$$
\renewcommand{\arraystretch}{1.0}
\begin{array}{ccccccccc}
&&&& 0 \\
&&&& \downarrow \\
&&&& {\mathcal O}_{\P^2_r} \\
&&&& \downarrow \\
&&&& {\mathcal O}_{\P^2_r}(C) & \longrightarrow & {\mathcal T} ^1_{C,\Sigma_0} & \longrightarrow
& 0 \\
&&&& \downarrow & & \| \\
0 & \longrightarrow & {\mathcal N}_{C/\P^2_r}^{\Sigma_0} & \longrightarrow &
{\mathcal N}_{C/\P^2_r} & \longrightarrow & {\mathcal T} ^1_{C,\Sigma_0} & \longrightarrow &
0\:\, \\
&&&& \downarrow \\
&&&& 0 \\
\end{array}
$$
we can conclude the required surjectivity of
$ H^0(\P^2_r,{\mathcal O}_{\P^2_r}(C))\rightarrow H^0(\P^2_r,{\mathcal T}
^1_{C,\Sigma_0})\:. $
As above, we denote by \mbox{$H\in
|hE_0-\sum_{i=1}^r h_iE_i|\,$} a curve of minimal degree \mbox{$h\leq
[\sqrt{2k}\,]\,$}
passing through $\Sigma_1$. There are two cases to consider:
{\em Case 1} : $\,\{z\}\not\subset H$
Define $L$ to be the strict transform of a straight line
\mbox{$\overline{L}\subset \P^2$} through
$p_{i_0}$ with tangent direction corresponding to $z$ and denote
\mbox{$J:=\left\{\:j\:|\:\{p_j\}\subset \overline{L}\,\right\}\supset
\left\{i_0\right\}\:\:$} (the genericity of the blown--up points implies that
\mbox{$\#J\leq 2$}). Consider the curve \mbox{$\tilde{H}:=H\cup E_1\cup \ldots
\cup E_r\cup
L \,\in\,|(h+1)E_0-\sum_{i=1}^r (\tilde{h}_i-1)E_i|$}, where
$$
\renewcommand{\arraystretch}{1.1}
\tilde{h}_i:=\bigg\{
\begin{array}{c}
h_i+1\\
h_i
\end{array}
\:\mbox{for}\:
\begin{array}{c}
i\in J\\
i\notin J\:.
\end{array}
$$
Moreover, let \mbox{$\,\tilde{C}\sim dE_0-\sum_{i=1}^r \tilde{d}_iE_i\,$} with
$$\tilde{d}_i:=\mbox{max}\,\left\{d_i,\tilde{h}_i+\left[\frac{d_i}{2}\right]
- -1\right\}\:.$$
According to Proposition \ref{1.2.1} we have three conditions, sufficient
for the vanishing of $H^1(C,{\mathcal N}_{C/\P^2_r}^{\Sigma_0})\,$:
as above, the Hirschowitz--Criterion (\ref{Hirschowitz}) together with
(\ref{2.1.3}) guarantees the \mbox{vanishing} property (A). (B) follows in the
same manner, because
\begin{eqnarray*}
\sqrt{\frac{(d\!-\!h\!+\!2)^2}{4} } & \geq &
\frac{d\!+\!2\!-\!\left[\sqrt{2k}\right]}{2} \:\:
\stackrel{\mbox{\footnotesize (\ref{2.1.3})}}{>} \:\:
\sqrt{\sum\nolimits_{i=1}^r \frac{(d_i\!+\!2)^2}{2}} \\
&\geq &\sqrt{1+\sum\nolimits_{i=1}^r
\frac{(\tilde{d}_i\!-\!\tilde{h}_i\!+\!1)(\tilde{d}_i\!-\!\tilde{h}_i
\!+\!2)}{2} }\:.
\end{eqnarray*}
Finally, property (C) is an immediate consequence of Corollary \ref{1.2.6},
knowing
that for each exceptional divisor $E_i$ \mbox{$(1\leq i\leq r)$} we have
$$
E_i\cdot (\tilde{C}-\tilde{H}-K_{\P_r^2} )-\#(\mbox{Sing}\,C \cap
E_i) \stackrel{\mbox{\footnotesize B\'ezout}}{\geq} \tilde{d}_i -
(\tilde{h}_i-1)+1-\left[\frac{d_i}{2}\right]
\;\:>\;\: 0
$$
and that for each irreducible component \mbox{$H_{\nu}\in
|h^{(\nu)}-\sum_{i=1}^r h^{(\nu)}_iE_i|\,$} of
\mbox{$L\cup H$}
\begin{eqnarray*}
\lefteqn{H_{\nu}\cdot (\tilde{C}-\tilde{H} -K_{\P^2_r}) -
\#(\mbox{Sing}\,C
\cap H_{\nu} ) } \hspace{0.2cm}\\
& \stackrel{\mbox{\footnotesize B\'ezout}}{\geq} &
h^{(\nu)}\,(d\!-\!(h\!+\!1)\!+\!3)- \sum_{i=1}^r
h_i^{(\nu)}\,(\tilde{d}_i\!-\!(\tilde{h}_i\!-\!1)\!+\!1)-
\left[\frac{h^{(\nu)}d\!-\!\sum h_i^{(\nu)} d_i}{2}\right]\\
& \stackrel{\mbox{\footnotesize Cauchy}}{\geq} &
h^{(\nu)}\,\left(\frac{d}{2}+2-[\sqrt{2k}]\right)-\sqrt{ \sum\nolimits_{i=1}^r
\big(h_i^{(\nu)}\big)^2} \:\sqrt{\sum\nolimits_{i=1}^r
\frac{(d_i+2)^2}{4}}\\
& \stackrel{\mbox{\footnotesize (\ref{1.3.1})}}{\geq} &
h^{(\nu)}\,\left(\frac{d}{2}+2-[\sqrt{2k}]-\frac{\sqrt{2}}{2}\,\sqrt{
\sum\nolimits_{i=1}^r (d_i+2)^2}\right)\:\:
\stackrel{\mbox{\footnotesize (\ref{2.1.3})}}{>} \:\: 0\:.
\end{eqnarray*}
{\em Case 2} : $\,\{z\}\subset H$
In this case we can omit the additional component $L$ in the definition of
$\tilde{H}$ and, proceeding as in case 1, we obtain again the vanishing of
$H^1(C,{\mathcal N}_{C/\P^2_r}^{\Sigma_0})\,$.
Now, we can assume \mbox{Sing$\,C=\Sigma_1$} (that is, there are no nodes of
$C$ on the exceptional divisors $E_i$) and we go on moving, subsequently, nodes
away from exceptional curves:\\
Assume \mbox{$E\in |eE_0-\sum_{i=1}^r e_iE_i|\,$} to be an irreducible curve
satisfying \mbox{$\,\sum_{i=1}^r e_i^2>e^2\,$}, let \mbox{$\,z\in \Sigma_1$} be
a node of $C$ on $E$
and denote \mbox{$\Sigma_0:=\mbox{Sing}\,C-\{z\}\,$}. As above, we construct a
curve
$$\tilde{H}:=H\cup \tilde{L}\,\in \:|(h+1)E_0-\sum_{i=1}^r h_iE_i|$$
where \mbox{$\tilde{L}\not\subset H$} is a line in $\P^2$ containing none of
the blown--up points such that \mbox{$\tilde{L}\cap
\,\mbox{Sing}\,C=\{z\}$}. Moreover, we consider \mbox{$\,\tilde{C}\sim
dE_0-\sum_{i=1}^r \tilde{d}_iE_i\,$} with
$$\,\tilde{d}_i:=\mbox{max}\,\left\{d_i,h_i+\left[\frac{d_i}{2}
\right]\right\}\,,$$ and the above reasoning gives again
\mbox{$H^1(C,{\mathcal N}_{C/\P^2_r}^{\Sigma_0})=0\,$}.
By Remark \ref{1.3.2}, we can end up with a curve \mbox{$C\in$
\mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}} close to the original one having properties (a) and (b). It
remains to move the nodes in general position to obtain a curve \mbox{$C\in
\tilde{V}\,$}:
Again, we choose a (not necessarily irreducible) curve \mbox{$H\in
|hE_0\!-\!\sum_{i=1}^r h_iE_i|\,$} of minimal degree
containing \mbox{$\Sigma_1=\mbox{Sing}\,C\,$}. Assume $H$ decomposes into
irreducible components \mbox{$H_{\nu}\in |h^{(\nu)}\!-\!\sum_{i=1}^r
h^{(\nu)}_iE_i|\,$ $(1\leq \nu \leq s)$} and assume that there are more than
\mbox{$\,h^{(\nu)}(h^{(\nu)}\!+3)/2+1\,$} nodes on one component $H_{\nu}$. We
show that we
can move \mbox{$\,h^{(\nu)}(h^{(\nu)}\!+3)/2+1\,$} of them in general
position:
Define \mbox{$\tilde{H}:=H\cup G$} where \mbox{$G\in |gE_0-\sum_{i=1}^r
g_iE_i|$} is a curve
not containing $H_{\nu}$ through the selected
\mbox{$\,h^{(\nu)}(h^{(\nu)}\!+3)/2+1\,$} nodes. Such a curve exists for each
$g$ satisfying
$$ \frac{g(g+3)}{2} -\left(\frac{h^{(\nu)}(h^{(\nu)}\!+3)}{2}+1\right)\:>\:
\frac{(g-h^{(\nu)})(g-h^{(\nu)}+3)}{2} $$
(the right--hand side is the dimension of the linear system of curves
containing $H_{\nu}$). Hence, we can suppose \mbox{$g=h^{(\nu)}\!+2\,$}, and
\mbox{$\,h^{(\nu)}(h^{(\nu)}\!+3)/2\,+1\leq k\,$} \mbox{implies} \mbox{
$h^{(\nu)}\leq [\sqrt{2k}]-1\,$}. Proceeding as before, we have to prove the
vanishing of \mbox{$H^1(\P^2_r,{\mathcal O}_{\P^2_r}(C-\tilde{H}))$}
respectively $H^1(\tilde{H},\tilde{{\mathcal N}}_{\tilde{H}/\P^2_r}^{\Sigma_0})\,$. The
first
is an immediate consequence of the Hirschowitz--Criterion (\ref{Hirschowitz}),
because
\arraycolsep0.0cm
\begin{eqnarray*}
\sqrt{\frac{(d\!-\!h\!-\!g\!+\!3)^2}{4}} & \geq &\frac{d\!-\!2[\sqrt{2k}]\!+
\!2}{2} \\
& \stackrel{\mbox{\footnotesize (\ref{2.1.3})}}{>} &
\sqrt{\sum\nolimits_{i=1}^r
\!\frac{(d_i\!+\!2)^2}{2}}\geq \sqrt{1\!+\!\sum\nolimits_{i=1}^r
\!\frac{(d_i\!-\!h_i\!-\!g_i)(d_i\!-\!h_i\!-\!g_i\!+\!1)}{2}}
\end{eqnarray*}
while the second results from Corollary \ref{1.2.6}, knowing that $H_{\nu}$
$(1\leq \nu \leq s)$ and the components of $G$ are no exceptional curves:
\begin{eqnarray*}
\lefteqn{H_{\nu}\cdot (C-\tilde{H} -K_{\P^2_r}) - \#(\mbox{Sing}\,C \cap
H_{\nu} )}\hspace{0.6cm}\\
& \stackrel{\mbox{\footnotesize B\'ezout}}{\geq} &
h^{(\nu)}\,(d\!-\!h\!-\!g\!+\!3)- \sum_{i=1}^r
h_i^{(\nu)}\,(d_i\!-\!h_i\!-\!g_i\!+\!1)-
\left[\frac{h^{(\nu)}d\!-\!\sum_{i=1}^r h_i^{(\nu)} d_i}{2}\right]\\
& \stackrel{\mbox{\footnotesize Cauchy}}{\geq} &
h^{(\nu)}\,\left(\frac{d}{2}+2-2[\sqrt{2k}]-\frac{\sqrt{\sum\nolimits_{i=1}^r
d_i^2}}{2}\right)\:\:
\stackrel{\mbox{\footnotesize (\ref{2.1.2})}}{>} \:\: 0
\end{eqnarray*}
and the same holds for each irreducible component $G_{\nu}$ of G in place of
$H_{\nu}$. \qed\\
\renewcommand{\qed}{}\end{proof}
\begin{lemma} \label{2.2.3}
Let $\,C\in \tilde{V}$, then $H^1(C,{\mathcal N}^{\mbox{\scriptsize \it
Sing$\,$C}}_{C/\P^2_r})=0\,$.
\end{lemma}
\begin{proof}
By Remark \ref{2.2.1}, we can choose an irreducible curve
\mbox{$H\in |hE_0-\sum_{i=1}^r h_iE_i|\,$} of degree \mbox{$h=[\sqrt{2k}]$}
through all nodes of
$C$. Moreover, there is a curve \mbox{$G\not\supset H$}, \mbox{$G\in
|gE_0-\sum_{i=1}^r
g_iE_i|$}, such that \mbox{Sing$\,C\subset G\cap H$}, for each $g$ satisfying
$$ \frac{g(g+3)}{2}-k\:>\:\frac{(g-h)(g-h+3)}{2}\:\:, $$
hence, especially for \mbox{$g=h+1\,$}. Let \mbox{$\tilde{H}:=H\cup
G$}. Applying proposition \ref{1.2.1} as before, we conclude the vanishing
of $H^1(C,{\mathcal N}^{\mbox{\scriptsize\it Sing$\,$C}}_{C/\P^2_r})\,$. Indeed, the
above inequalities hold again, since neither $H$ nor the irreducible
components $G_{\nu}$ of $G$ are exceptional curves.
\end{proof}
\section{Existence}
\subsection{Formulation of the result}
\setcounter{equation}{0}
We shall treat
the problem of the existence of nodal curves in $\P^2_r$ for \mbox{$r\leq 9$}
and \mbox{$r\geq 10$}
separately. If \mbox{$r\leq 9$}, Theorems \ref{4.1A} and \ref{4.1B} will
give the complete answer,
while for \mbox{$r\geq 10$}, we obtain an asymptotically nearly optimal
sufficient criterion (Theorem \ref{3.1.1}).
\begin{theorem}[Existence Theorem A] \label{4.1A}
Let \mbox{$r=1$} then \mbox{$V_{irr}(d;d_1;k)\neq \emptyset$} if and only if
\mbox{$(d_1\leq d\!-\!1$} or \mbox{$d=d_1=1)$} and $$0 \leq k
\leq \frac{(d-1)(d-2)-d_1(d_1-1)}{2}.$$
Let $r=2$ and $d,d_1,d_2$ be positive integers then
\mbox{$V_{irr}(d;d_1,d_2;k)\neq \emptyset$} if and only if $$0 \leq k
\leq \frac{(d-1)(d-2)-d_1(d_1-1)-d_2(d_2-1)}{2}$$ and either
\mbox{$(d_1+d_2\leq d)$} or \mbox{$(d=d_1=d_2=1)$}.
\end{theorem}
Let \mbox{$3\leq r\leq 9$}, then we define two \mbox{$(r\!+\!1)$}--tuples
\mbox{$(d;d_1,\ldots,d_r)$} and
\mbox{$(\tilde{d};\tilde{d}_1,\ldots,\tilde{d}_r)$} of non--negative integers
to be
{\em equivalent}, if there is a finite sequence of Cremona maps and
a permutation $\sigma$ transforming \mbox{$(d;d_1,\ldots,d_r)$} to
\mbox{$(\tilde{d};\tilde{d}_{\sigma(1)},\ldots,\tilde{d}_{\sigma(r)})$} . Here,
by a {\em Cremona map}, we denote a mapping
$$
\begin{array}{rccc}
\Sigma_{j,m,n} : & {\Bbb Z}^{r+1}&\longrightarrow & {\Bbb Z}^{r+1}\\
&(d;d_1,\ldots,d_r)& \mapsto & (d';d'_1,\ldots,d'_r)
\end{array}
$$
$$
\begin{array}{ll}
\mbox{with} & d'=2d-d_j-d_m-d_n\,,\;\;\;d'_i=d_i\,\;\mbox{ for each }
\;i\not\in\{j,m,n\},\\
& d'_j=d-d_m-d_n\,, \;\,d'_m = d-d_j-d_n\,\;\mbox{ and }\;\,d'_n=d-d_j-d_m\,.
\end{array}
$$
Such a Cremona map corresponds to the standard Cremona transformation in
${\P}^2$ inducing the base change in Pic($\P^2_r$):
\begin{equation}
\label{4.6}
\left\{
\begin{array}{lcl}
E'_0 & = & 2E_0-E_j-E_m-E_n\\
E'_j & = & E_0-E_m-E_n\\
E'_m & = & E_0-E_j-E_n\\
E'_n & = & E_0-E_j-E_m\\
E'_i & = & E_i\;\;\mbox{ for each }\;i\not\in\{j,m,n\}.
\end{array}
\right.
\end{equation}
Due to the generality of the blown--up points, such a transformation maps
generic elements in \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ to elements in
\mbox{$V_{irr}(d';d'_1,\ldots,d'_r;k)$} supposing
\mbox{$d,d',d_i,d'_i$} \mbox{$(1\leq i\leq r)$} to be non--negative. Since
\mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ is smooth, we deduce that its non--emptyness is equivalent to the
existence of a curve in \mbox{$V_{irr}(d';d'_1,\ldots,d'_r;k)$}.
An (ordered) tuple \mbox{$(d;d_1,\ldots,d_r)\in {\Bbb N}^{r+1}$}, \mbox{$d_1\geq
d_2\geq \ldots \geq d_r$}, is called {\em minimal}, if it satisfies the
(minimality) condition
\begin{equation}
\max_{\#\{j,m,n\}=3} (d_j\!+\!d_m\!+\!d_n)\: =\: d_1+d_2+d_3 \:\leq\:
d. \label{minimal}
\end{equation}
\begin{theorem}[Existence Theorem B] \label{4.1B}
Let \mbox{$3\leq r\leq 9$} and positive integers \mbox{$d\geq d_1\geq
\ldots\geq d_r$} satisfy the conditions
\begin{eqnarray}
\sum_{i=1}^r d_i & \leq & 3d-1 \label{4.2}\\
\sum_{i=1}^r d_i(d_i-1) & \leq & (d-1)(d-2) \label{4.3}
\end{eqnarray}
then \mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ $\neq \emptyset$ if and only if
\begin{eqnarray*}
0\;\:\leq\;\: k &\leq &\frac{(d\!-\!1)(d\!-\!2)}{2}-\sum_{i=1}^r
\frac{d_i(d_i-1)}{2}
\end{eqnarray*}
and \mbox{$(d;d_1,\ldots,d_r)$} is equivalent to a minimal tuple
\mbox{$(\tilde{d};\tilde{d}_1,\ldots,\tilde{d}_r)$} of non--negative integers
or to the tuple \mbox{$(1;1,1,0,\ldots,0)$}.\\
\end{theorem}
\begin{remark} \label{4.4} \noindent
\begin{enumerate}
\item Condition (\ref{4.2}) is necessary in the following
sense. By B\'ezout's Theorem and the generality of the blown--up points, the
only type of an irreducible curve not
satisfying (\ref{4.2}) is the smooth cubic through the 9 generic points.
\item For \mbox{$3\leq r\leq 8$} the tuple \mbox{$(d;d_1,\ldots,d_r)$} is
equivalent to a minimal one exactly if the following conditions are satisfied
\begin{eqnarray*}
d & \geq & d_1+d_2 \\
2d & \geq &d_1+d_2+d_3+d_4+d_5 \\
3d & \geq &2d_1+d_2+d_3+d_4+d_5+d_6+d_7 \\
4d & \geq &2d_1+2d_2+2d_3+d_4+d_5+d_6+d_7+d_8 \\
5d & \geq &2d_1+2d_2+2d_3+2d_4+2d_5+2d_6+d_7+d_8 \\
6d & \geq &3d_1+2d_2+2d_3+2d_4+2d_5+2d_6+2d_7+2d_8 \\
\end{eqnarray*}
\vspace{-0.7cm}
\item The exceptional case \mbox{$(d;d_1,\ldots,d_r)\sim (1;1,1,0,\ldots,0)$}
corresponds exactly to the exceptional curves with data
\begin{center}
\begin{tabular}{lp{7.5cm}}
$(2;1,1,1,1,1)$ & the conic through 5 of the generic points\\
$(3;2,1,1,1,1,1,1)$ & the cubic through 7 of the generic points having a node
at one of them\\
$(4;2,2,2,1,1,1,1,1)$ & the quartic through 8 generic points having
nodes at three of them\\
$(5;2,2,2,2,2,2,1,1)$ & the quintic through all 8 generic points having nodes
at 6 of them\\
$(6;3,2,2,2,2,2,2,2)$ & the sixtic having nodes at 7 of the generic points
and a triple point at the remaining one
\end{tabular}
\end{center}
\end{enumerate}
\end{remark}
For the proof, using Cremona transformations, we already saw that we can reduce
the existence problem to the case of minimal data.
Due to the independence of node smoothings in the case \mbox{$r\leq 9$},
it will then be enough to construct only rational curves, i.e.~with
$$ k=\frac{(d-1)(d-2)}{2}-\sum_{i=1}^r \frac{d_i(d_i-1)}{2} $$
nodes, in case (\ref{4.2}), (\ref{4.3}) and the minimality condition
(\ref{minimal}) are satisfied.
\begin{theorem}[Existence Theorem C] \label{3.1.1}
Let \mbox{$r\geq 10$}, and let the positive integers
\mbox{$d,d';\,d_1,\ldots,d_r$}
satisfy \mbox{$d\geq d'$} and
\begin{eqnarray}
\frac{{d'}^2+6d'-1}{4} -\left[\frac{d'}{2}\right] & > & \sum_{i=1}^r
\frac{d_i(d_i+1)}{2}\:. \label{3.1.2}
\end{eqnarray}
Then for any integer $k$ such that
\begin{eqnarray}
0\;\:\leq\;\: k &\leq &\frac{(d\!-\!1)(d\!-\!2)}{2}-\frac{d'(d'\!-\!1)}{2}\:,
\label{3.1.3}
\end{eqnarray}
there exists a reduced irreducible curve $C$ in the linear system
\mbox{$|dE_0-\sum_{i=1}^r d_iE_i|$} on $\P^2_r$, having k nodes as its only
singularities, that is, \mbox{\mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ $\neq \emptyset$}.
\end{theorem}
We prove this in two steps: first, we shall prove the existence of a
non--singular curve in such linear systems on $\P^2_r$ by means of some
modification of the Hirschowitz--criterion (\ref{Hirschowitz}), afterwards, we
obtain the required nodal curves by a suitable deformation of the union of the
previous curve with generic straight lines.
\begin{corollary} \label{3.1.4}
If \mbox{$r\geq 10$} and positive integers \mbox{$d;\,d_1,\ldots,d_r$} satisfy
$$ d\geq \sqrt{2}\sqrt{\sum_{i=1}^r d_i(d_i+1)}\:\:,$$
then, for any non--negative integer
$$ k\leq \frac{(d\!-\!1)(d\!-\!2)}{2}-\sum_{i=1}^r d_i(d_i+1)\:,$$
\mbox{$V_{irr}(d;\,d_1,\ldots,d_r;\,k)$}\ $\neq \emptyset$.
\end{corollary}
This easily follows from Theorem \ref{3.1.1}, because
\mbox{$\,d':=\sqrt{2}\sqrt{\smash[b]{\sum_{i=1}^r d_i(d_i+1)}}\;$} satisfies
$$ \frac{{d'}^2+6d'-1}{4}-\frac{d'}{2} \:>\: \sum_{i=1}^r
\frac{d_i(d_i+1)}{2}\:. $$
\subsection{Existence of nodal curves on $\P^2_r$, $r\leq9$} \label{4}
\setcounter{equation}{0}
As mentioned before, we can restrict to minimal tuples
\mbox{$(d;d_1,\ldots,d_r)$} and
construct only rational curves. In the case \mbox{$r\leq 8$} the statement is,
probably, known. Nevertheless, we provide here both, the proof for
\mbox{$r\leq8$} and \mbox{$r=9$}.
\subsubsection{Assume that \mbox{$r=1$}}
Clearly, B\'ezout's Theorem implies \mbox{$d_1\leq d\!-\!1$} for all
curves \mbox{$C\in V_{irr}(d;d_1;k)$}, which are not the strict transform of a
line in $\P^2$ through $p_1$.
We take the union of $d_1$ distinct straight lines in $\P^2$ through $p_1$ and
\mbox{$d-d_1$} more generic straight lines in $\P^2$, lift this curve to
$\P^2_1$ and get a reduced curve in the linear system \mbox{$|dE_0-d_1E_1|$}
with
$$\frac{(d\!-\!d_1)(d\!-\!d_1\!-\!1)}{2} +
d_1(d\!-\!d_1)=\frac{(d\!-\!1)(d\!-\!2)}{2}-\frac{d_1(d_1\!-\!1)}{2} +d\!-\!1$$
nodes. After smoothing $d-1$ intersection points, we obtain the desired
irreducible rational curve.
\subsubsection{Assume that \mbox{$r=2$}}
Again, by B\'ezout's Theorem, \mbox{$d_1+d_2\leq d$} with the exception of the
strict transform of the line through $p_1$ and $p_2$. We consider in $\P^2$
the union of $d_1$ distinct straight lines through $p_1$, $d_2$ distinct
straight lines through $p_2$ and \mbox{$d-d_1-d_2$} additional generic
straight lines. The strict transform in $\P^2_2$ of this curve is a reduced
curve in the linear system \mbox{$|dE_0-d_1E_1-d_2E_2|$} with
\begin{eqnarray*}
\lefteqn{d_1d_2+(d_1\!+\!d_2)(d\!-\!d_1\!-\!d_2)
+\frac{(d\!-\!d_1\!-\!d_2)(d\!-\!d_1\!-\!d_2\!-\!1)}{2}}\hspace{1.5cm}\\
& = &\frac{(d\!-\!1)(d\!-\!2)}{2}-\frac{d_1(d_1\!-\!1)}{2}
- -\frac{d_2(d_2\!-\!1)}{2} +d\!-\!1
\end{eqnarray*}
nodes. Again one smooths \mbox{$d-1$} nodes to obtain an irreducible rational
curve.
\subsubsection{Assume that \mbox{$r=3$}}
Due to the minimality condition (\ref{minimal}), we can proceed as follows: in
the plane we choose the union of $d_i$ distinct straight lines through $p_i$,
\mbox{$i=1,2,3$}, and \mbox{$d-d_1-d_2-d_3$} generic straight lines. After
smoothing \mbox{$d-1$} nodes of its strict transform in $\P^2_3$, as above, we
end up with the desired rational nodal curve.
\subsubsection{Assume that \mbox{$r=4$}}
By induction on $d$, we shall show that the minimality condition
(\ref{minimal}) is sufficient for the existence of a rational irreducible
(nodal) curve
in \mbox{$|dE_0-\sum_{i=1}^4 d_iE_i|$}.
In case \mbox{$d\leq 4$} the only tuples which are not covered by the
(preceding) cases (\mbox{$r\leq 3$}) are \mbox{$(d;1,1,1,1)$}, \mbox{$d\in
\{3,4\}$}, and \mbox{$(4;2,1,1,1)$}, hence the statement is trivial. If
\mbox{$d\geq 5$} we have
$$\max_{\#\{j,m,n\}=3} ((d_j\!-\!1)+(d_m\!-\!1)+(d_n\!-\!1))\leq d-2
$$
thus, by the induction assumption, there is an irreducible rational (nodal)
curve $C$ in the linear system
\mbox{$|(d\!-\!2)E_0-\sum_{i=1}^4 (d_i\!-\!1)E_i|$}. As is well--known
(cf.~e.g.\cite{Wae}), a
generic (smooth rational) curve $C'$ in the one--dimensional base--point--free
linear system \mbox{$|2E_0-\sum_{i=1}^4 E_i|$} intersects $C$
transversally. Finally, smoothing one intersection point in the
union of $C$ and $C'$ completes the induction step.
\subsubsection{Assume that \mbox{$r=5$}}
We can proceed as in the case \mbox{$r=4$} with the only exception that in the
induction step $C$ will be an irreducible rational nodal curve in
\mbox{$|(d\!-\!2)E_0-d_1E_1-\sum_{i=2}^5 (d_i\!-\!1)E_i|$}, and $C'$ has to be
chosen as a generic smooth curve in \mbox{$|2E_0-\sum_{i=2}^5 E_i|$}. Thereby,
obviously, we have to treat the case
\mbox{$(d;d_1,\ldots,d_5)=(d;d\!-\!2,1,1,1,1)$} separately, because
there is no irreducible curve in the linear system
\mbox{$|(d\!-\!2)E_0-(d\!-\!2)E_1|$}. In this case, we have to choose $C$ as
the union of \mbox{$d-2$} generic lines through $p_1$ and to smooth
\mbox{$d-2$} intersection points of $C$ and $C'$.
\subsubsection{Assume that \mbox{$r=6$}}
Again, we construct inductively irreducible rational (nodal) curves only
supposing that (\ref{minimal}) holds. First, we consider separately the case
\mbox{$d_1=\ldots=d_6=1$}, where in the induction step a generic line has to be
added (and one intersection point smoothed). Then,
supposing \mbox{$d_1\geq 2$}, in case \mbox{$d\leq 4$} the only
(additionally) possible tuple is \mbox{$(4;2,1,1,1,1,1)$}, where the statement
is trivial. For the induction step, we know, that
$$ \max \left\{(d_1\!-\!2)+(d_2\!-\!1)+(d_3\!-\!1),\:(d_2\!-\!1)+(d_3\!-\!1)+
(d_4\!-\!1) \right\} \leq d-3\:. $$
Thereby, we can construct the desired curve by smoothing one intersection point
in the union of an irreducible nodal rational curve
$$C\in |(d\!-\!3)E_0-(d_1\!-\!2)E_1-\sum_{i=2}^6 (d_i\!-\!1)E_i|$$
and a generic curve in the one--dimensional (base--point--free) linear system
$$ |3E_0-2E_1-E_2-E_3-E_4-E_5-E_6|\:.$$
\subsubsection{Assume that \mbox{$r=7$}}
Changing \mbox{$(d;d_1,\ldots,d_r)$} to
\mbox{$(d\!-\!3;d_1\!-\!1,\ldots,d_7\!-\!1)$} leaves
the minimality condition intact. Hence, in the induction step (\mbox{$d\geq
5$}), we consider the family ${\mathcal F}$ of curves \mbox{$C\cup C'$}, where $C$ is
a generic rational nodal curve in
$$ |(d\!-\!3)E_0-\sum_{i=1}^7 (d_i\!-\!1)E_i| $$
and $C'$ is a generic rational nodal curve in \mbox{$|3E_0-E_1-\ldots
-E_7|$}. First, we consider the
only case where $C$ cannot be supposed to be irreducible, namely
\mbox{$(d;d_1,\ldots,d_7)=(d;d\!-\!2,1,1,1,1,1,1)$}. In this situation, we
proceed as in case \mbox{$r=5$}, take $C$ as the union of \mbox{$d-3$} generic
lines through $p_1$ and smooth \mbox{$d-3$} intersection points of $C$ and
$C'$. Now, we assume $C$ to be irreducible. If
\mbox{$d=5$}, then $C$ is a generic curve in one of the following
base--point--free linear systems
$$|2E_0|\:,\;\;\;|2E_0-E_1|\:,\;\;\;|2E_0-E_1-E_2|\:,$$
whence a generic member of ${\mathcal F} $ turns out to be a nodal curve.
It remains to prove that for \mbox{$d\geq 6$} a generic member $\tilde{C} $ of
the family ${\mathcal F}$ is a nodal curve. Indeed, for the canonical divisor
$K_{\P^2_7}$ we have \mbox{$(K_{\P^2_7}\cdot C) < 0$} and
\mbox{$(K_{\P^2_7}\cdot C')<0$}, hence by (\cite{Nob}, Theorem 3.10)
\begin{eqnarray*}
\mbox{dim }{\mathcal F} & = & \left(\frac{(d\!-\!3)d}{2}-\frac{(d\!-\!4)(d\!-\!5)}{2}
- -\sum_{i=1}^7 (d_i\!-\!1)\right) + \left(\frac{3\cdot6}{2}-1-7\right) + 1\\
& = & \left(\frac{d(d\!+3)}{2}-\sum_{i=1}^7 \frac{d_i(d_i\!+\!1)}{2}\right)-
\left(\frac{(d\!-\!1)(d\!-\!2)}{2}-\sum_{i=1}^7
\frac{d_i(d_i\!-\!1)}{2}\right)\\
& = & \mbox{dim }|\tilde{C}| - \Delta
\end{eqnarray*}
where $\Delta$ denotes the ``virtual'' number of nodes of $\tilde{C}$. On the
other hand, the family of rational curves in \mbox{$|3E_0-E_1-\ldots -E_7|$}
has no base point. Hence, the only
possibility for a non--nodal singularity of $\tilde{C}$ is a tangency point of
$C$, $C'$ with smooth branches. But \mbox{$(K_{\P^2_7}\cdot C')=-2<-1$} and
applying (\cite{Nob}, Theorem 3.12), in this case we would have had
$$ \mbox{dim }{\mathcal F} < \mbox{dim }|\tilde{C}| - \Delta \:.$$
Finally, smoothing one node in $\tilde{C}$ we get the desired curve.
\subsubsection{Assume that \mbox{$r=8$}}
Once again, the case \mbox{$d\leq 4$} is trivial and we suppose \mbox{$d\geq
5$}. In the induction step, we proceed as in the case \mbox{$r=7$}: We define
${\mathcal F}$ to be the family of curves \mbox{$C\cup C'$}, where C is a generic
rational nodal curve in
$$ |(d\!-\!3)E_0-\sum_{i=1}^8 (d_i\!-\!1)E_i| $$
and $C'$ is a generic rational nodal curve in \mbox{$|3E_0-E_1-\ldots
-E_8|$}. In the case \mbox{$(d;d_1,\ldots,d_8)=(d;d\!-\!2,1,1,1,1,1,1,1)$},
we can repeat the above construction completely. Hence, by the induction
assumption, we can assume $C$ to be irreducible. If \mbox{$d=5$}, as in the
case \mbox{$r=7$}, a generic member $\tilde{C}$ of ${\mathcal F} $ is a nodal curve. If
\mbox{$d=6$} $C$, $C'$ can be chosen as the strict
transforms of two distinct (irreducible) plane rational cubics through $8$
generic points, hence \mbox{$C\cup C'$} is nodal. Let \mbox{$d\geq 7$}. Since
the minimality condition implies
$$ (K_{\P^2_8}\cdot C) = -3(d-3)+\sum_{i=1}^8 (d_i-1) \leq -3d +1
+\left[\frac{8}{3} d\right]<-1\:,$$
by (\cite{Nob}, Theorem 3.10), for dimension reasons the family of rational
nodal curves \mbox{$C\in |(d\!-\!3)E_0-\sum_{i=1}^8 (d_i\!-\!1)E_i| $} cannot
have base points. Hereby, again, the only possibility for a non--nodal
singularity of $\tilde{C}$ is a tangency point with smooth branches. Counting
dimensions as above, we see that such a singularity may not occur, and we
complete the construction as before.
\subsubsection{Assume that \mbox{$r=9$}}
By condition (\ref{4.2}) we have only to consider curves of degree \mbox{$d\geq
4$}. We split the induction step into three parts depending on the shape of
$d$:
If \mbox{$d=3m+2$}, \mbox{$m\geq 1$}, then the minimality condition
(\ref{minimal}) implies
$$ \sum_{i=1}^9 d_i\leq 3d-3 \;\;\;\mbox{ and }\;\;\;(d_1>d_3 \;\;\mbox{ or
}\;\;d_1+d_2+d_3\leq d-1).$$
In each case, changing \mbox{$(d;d_1,\ldots,d_9)$} to
\mbox{$(d\!-\!1;d_1\!-\!1,d_2,\ldots,d_9)$} preserves both, (\ref{4.2}) and the
minimality condition. Hence, we can assume the existence of an irreducible
rational nodal curve $C$ in the linear system
$$|(d\!-\!1)E_0-(d_1\!-\!1)E_1-\sum_{i=2}^9 d_iE_i|\:.$$
Adding a generic line through $p_1$ and smoothing one intersection point, we
obtain the desired curve.
If \mbox{$d=3m+1$}, \mbox{$m\geq 1$}, the only case where we have to use
another construction is
$$ (d;d_1,d_2,\ldots,d_9)=(3m+1;m+1,m,\ldots,m)\:. $$
We shall apply the following lemma, which will be proven at the end of this
section:
\setcounter{equation}{9}
\begin{lemma}
\label{4.18}
Let $L_{ij}$ be the (unique) line in the linear system \mbox{$|E_0-E_i-E_j|$},
\mbox{$1\leq i < j \leq 9$}. For any \mbox{$m\geq 1$} there exists an
irreducible rational nodal
curve $$F_m\in |3mE_0-mE_1-\ldots -mE_8-(m\!-\!1)E_9|\:,$$
which meets every line $L_{ij}$, \mbox{$1\leq i<j\leq 9$}, transversally and
only at non--singular points.
\end{lemma}
Taking the curve $F_m$ from this lemma and applying the base change (\ref{4.6})
in Pic($\P^2_9$) with \mbox{$(j,m,n)=(1,2,9)$}, one easily sees that $F_m$
belongs to the linear system
$$|(3m+1)E'_0-(m+1)E'_1-(m+1)E'_2-mE_3-\ldots-mE_8-mE'_9|\:,$$
and is transversal to \mbox{$E'_2=L_{19}$}. Hence, we get the desired
irreducible rational nodal curve in
$$|(3m+1)E'_0-(m+1)E'_1-mE'_2-mE_3-\ldots-mE_8-mE'_9|\:,$$
by smoothing one intersection point of the nodal curve \mbox{$F_m\cup E'_2$}.
If \mbox{$d=3m$}, then we have to consider three possibilities
\begin{itemize}
\item \mbox{$d_1+d_2+d_3\leq d-1$}
\item \mbox{$d_1>d_3$} and \mbox{$d_1+\ldots +d_9\leq 3d-3$}
\item $(3m;m,m,m,d_4,\ldots,d_9)$, where \mbox{$m\geq d_4\geq \ldots \geq
d_9$}, \mbox{$d_9\leq m-1$}
\end{itemize}
Clearly, to get the latter curves, it is enough to prove Lemma
\ref{4.18}. In the first two cases we can proceed as in the situation
\mbox{$d=3m+2$}.
\begin{proof}[Proof of Lemma \ref{4.18}]
We divide our reasoning into several steps.
{\em Step 1.} First, we shall show the following. There is a hypersurface
\mbox{$S\subset (\P^2)^9$} such that on the surface $\P^2(p_1,\ldots,p_9)$
(which is the plane blown up at \mbox{$p_1,\ldots,p_9$}) the linear system
\begin{equation}
\label{4.19}
|3mE_0-(m\!+\!1)E_1-mE_2-\ldots-mE_8-(m\!-\!1)E_9|
\end{equation}
is non--empty if and only if \mbox{$(p_1,\ldots,p_9)\in S$}.\\
Indeed, applying successively the base change (\ref{4.6}) with
\begin{equation}
\label{4.20}
(j,m,n)=(1,2,3),\;(4,5,6),\;(7,8,9),
\end{equation}
respectively, we transform the system (\ref{4.19}) into the system
$$|3(m\!-\!1)E'_0-mE'_1-(m\!-\!1)E'_2-\ldots-(m\!-\!1)E'_8-(m\!-\!2)E'_9|\:.$$
After $m\!-\!1$ such steps we end up with a system of type
$$|3E'_0-2E'_1-E'_2-\ldots-E'_8|$$
whose non--emptiness, evidently, imposes one condition on the nine blown--up
points. Namely, in the latter representation the condition means that the
blown--up points \mbox{$p'_2,\ldots,p'_8$} are distinct points on a plane
cubic $C_3$
with a singularity at $p'_1$. On the other hand, choosing an irreducible cubic
$C_3$ with a singularity at a point $p'_1$, arbitrary distinct points
\mbox{$p'_2,\ldots,p'_8$} on $C_3$, and \mbox{$p'_9\not\in C_3$}, the inverse
process gives us an irreducible curve in the system (\ref{4.19}), which is
unique by B\'ezout's Theorem.
{\em Step 2.} In the previous notation, let us specialize the points
$p'_1,p'_4,p'_9$ on a line \mbox{$L'\subset \P^2$}. Applying the base change
(\ref{4.20}) in the inverse order to \mbox{$\P^2(p'_1,\ldots, p'_9)$} and
blowing down the new exceptional curves, we obtain an irreducible plane sextic
curve with a triple point $p''_1$, double points \mbox{$p''_2,\ldots,p''_8$}
and a non--singular point $p''_9$. Since the applied operation is a
composition of three Cremona transformations of $\P^2$ with the fundamental
points
\mbox{$(p'_7,p'_8,p'_9)$},\mbox{$(p'_4,p'_5,p'_6)$},\mbox{$(p'_1,p'_2,p'_3)$},
respectively, the
points $p''_1,p''_4,p''_9$ lie on a straight line \mbox{$L''\subset \P^2$},
which corresponds to the strict transform of $L'$. We continue in such a
manner until we get an irreducible plane curve $C_{3m}(\underline{p})$ of
degree $3m$ with an
\mbox{$(m\!+\!1)$}--fold point $p_1$, $m$--fold points \mbox{$p_2,\ldots,p_8$}
and an \mbox{$(m\!-\!1)$}--fold point $p_9$, such that $p_1,p_4,p_9$ lie on a
straight line $L$. By B\'ezout's Theorem, $L$ meets $C_{3m}(\underline{p} )$
transversally at $p_1,p_4,p_9$. Now, fixing $p'_i$, \mbox{$i\neq 4$}, we vary
the point $p'_4$ along $C_3$. The above construction will give us a
one--parametric family of sets \mbox{$\{p_1,\ldots, p_9\}$}, thereby a
one--parametric (continuous) family of curves
\mbox{$C_{3m}(p_1,\ldots,p_9)$}. Generically, the line $L$ will split into
three lines $(p_1p_4),(p_4p_9),(p_1p_9)$, which intersect
\mbox{$C_{3m}(\underline{p} )$} transversally. \\
Varying the numbering of \mbox{$p'_2,\ldots, p'_8$}, respectively repeating
the previous reasoning with the initial specialization of the points
$p'_2,p'_5,p'_8$ on a straight line, we obtain, finally, that for a generic
element \mbox{$\underline{p} =(p_1,\ldots,p_9)\in S$}, the strict transform of
$C_{3m}(\underline{p} )$ intersects each $L_{ij}$, \mbox{$1\leq i<j\leq 9$}
transversally. Since the base change simply interchanges lines $L_{ij}$ with
exceptional divisors $E_s$, we can claim that the intersection with each
exceptional divisor \mbox{$E_1,\ldots,E_9$} is transversal as well.
{\em Step 3.} The previous statement means, in particular, that for a generic
$\underline{p} \in S$ the curve \mbox{$C_{3m}(\underline{p} )\subset \P^2$}
has nine ordinary multiple points and
meets every line $(p_ip_j)$ transversally. Let us denote by ${\mathcal G}$ the germ at
$C_{3m}(\underline{p} )$ of the family of plane rational curves of degree $3m$,
having ordinary singular points in a neighbourhood of \mbox{$p_2,\ldots,p_9$}
with the same multiplicities \mbox{$m,\ldots,m,m-1$}, respectively, and having
a point of multiplicity at least $m$ in a neighbourhood $U$ of $p_1$. Clearly,
${\mathcal G}$ is the intersection of a germ $\Sigma_{1} $ of the equisingular stratum
in $|{\mathcal O}_{\P^2} (3m)|$, corresponding to the ordinary singular points
\mbox{$p_2,\ldots,p_9$}, and a germ $\Sigma_{2} $ of the following family of
curves of degree $3m$. A curve in $\Sigma_{2} $ has a point of multiplicity at
least $m$ in a neighbourhood $U$ of $p_1$ and the sum of $\sigma$--invariants
in $U$ is equal to $\frac{1}{2} (m\!+\!1)m$. It is not difficult to see that
$\Sigma_{2} $ is the union of $m\!+\!1$ smooth germs, such that their
intersection $\Sigma_3$ consists of curves having an ordinary
\mbox{$(m\!+\!1)$}--fold
point in $U$, and a curve in \mbox{$\Sigma_{2}\setminus \Sigma_{3}$} has in $U$
an ordinary $m$--fold point and $m$ nodes (geometrically, such a deformation
looks as if one of the local branches of $C_{3m}(\underline{p} )$ at $p_1$
moves away from the multiple point). The classical smoothness criteria say that
$\Sigma_{1} $ is smooth, and
$$ \mbox{codim}_{|{\mathcal O}_{\P^2}(3m)|} \Sigma_{1} \leq 7\left(\frac{(m\!+\!1)m}{2}
- 2 \right) + \left(\frac{m(m\!-\!1)}{2} -
2 \right) = 4m^2+3m-16\:. $$
Since, evidently,
$$ \mbox{codim}_{|{\mathcal O}_{\P^2}(3m)|} \Sigma_{2} \leq \frac{(m\!+\!1)m}{2} -2+m=
\frac{m^2}{2}+\frac{3m}{2}-2\:,$$
we obtain
$$ \mbox{codim}_{|{\mathcal O}_{\P^2}(3m)|} {\mathcal G} \leq
\frac{9m^2}{2}+\frac{9m}{2}-18\:.$$
On the other hand, we have
$$ \mbox{dim}\:(\Sigma_{1}\cap \Sigma_{3})= \mbox{dim}\:{\mathcal S} = 17 < 18 \leq
\mbox{dim}\:(\Sigma_{1}\cap \Sigma_{2})\:.$$
That means, there exists a rational plane curve of degree $3m$ with $8$
ordinary $m$--fold points, one ordinary \mbox{$(m\!-\!1)$}--fold point and,
additionally, $m$ nodes. Moreover,
this curve intersects transversally with a straight line through any
two of the $9$ multiple points. Blowing up these $9$ points, we get the desired
curve \mbox{$F_m \subset \P^2_9$}.
\end{proof}
\subsection{Plane curves with generic multiple points} \label{3.2}
\setcounter{equation}{0}
It will be convenient for us to deal here with plane curves having ordinary
multiple points instead of non--singular curves on the blown--up plane. For
abuse of language, we shall use the notation: given an ordered set
\mbox{$\underline{p}=\{p_1,\ldots,p_r\}$} of distinct points in $\P^2$ and an
integral vector \mbox{$\underline{d}=(d_1,\ldots,d_r)$}, by
$S_d(\underline{p},\underline{d})$ we shall denote the set of reduced
irreducible curves of degree $d$ which have ordinary singular points at
\mbox{$p_1,\ldots,p_r$} of multiplicities
\mbox{$d_1,\ldots,d_r$}, respectively, as their only singularities.
\begin{lemma}
\label{3.2.1}
Let \mbox{$p_1,\ldots,p_r$}, \mbox{$r\geq 1$}, be distinct generic points in
$\P^2$. Then, for any positive integers \mbox{$d;d_1,\ldots,d_r$} satisfying
\begin{eqnarray}
\frac{d^2+6d-1}{4}-\left[\frac{d}{2}\right] & > & \sum_{i=1}^r
\frac{d_i(d_i+1)}{2}\:, \label{3.2.2}
\end{eqnarray}
there exists a curve \mbox{$F_d\in S_d(\underline{p},\underline{d})$}.
\end{lemma}
\begin{proof}
Following \cite{Hir}, we shall prove a more general statement. Let $I$ be a
subset in \mbox{$\{1,\ldots,r\}$}, let the points $p_i$, \mbox{$i\in I$}, lie
on a straight
line $G$, and let the points $p_i$, \mbox{$i\not\in I$}, be in general position
outside $G$. If condition (\ref{3.2.2}) and
$$ \sum_{i\in I} d_i \leq d $$
hold, then there exists a curve \mbox{$F_d\in
S_d(\underline{p},\underline{d})$}, which is transversal to the line $G$.
We shall use induction on $d$. If \mbox{$d=2$} then (\ref{3.2.2}) reads
$2 \geq \sum_{i=1}^r d_i(d_i+1)/2$,
so the only possibilities are: \mbox{$\,(r\!=\!1,d_1\!=\!1)$} or
\mbox{$(r\!=\!2,d_1\!=\!d_2\!=\!1)$}, when the required curves do exist. Assume
\mbox{$d\geq 3$}.
{\em Step 1}: Assume that \mbox{$I=\{1,\ldots,r\}$}, \mbox{$\,\sum_{i=1}^r
d_i\leq d\,$}.
If \mbox{$r=1$} then, by (\ref{3.2.2}), \mbox{$d_1\!<d$}, hence one obtains the
equation of the desired curve in the form
$$ F(x,y)\::=\sum_{d_1\leq i+j\leq d} A_{ij}x^iy^j\:,$$
with \mbox{$p_1=(0,0)$} and generic coefficients $A_{ij}$. If \mbox{$r>1$} then
one can obtain the desired curve as a generic member of the linear family of
all curves with equations
$$ \lambda'C'\prod_{i=1}^r \prod_{j=1}^{d_i} L'\!\!_{ij}+
\lambda''C''\prod_{i=1}^r \prod_{j=1}^{d_i}
L''\!\!\!_{ij}\:,\;\;\;(\lambda',\lambda'')\in \P^1\:,
$$
where for any \mbox{$1\leq i \leq r$}, we take
distinct generic straight lines $L'\!\!_{ij}$, $L''\!\!\!_{ij}$ through $p_i$,
\mbox{$1\leq j\leq d_i$}, and $C'$, $C''$ are distinct generic curves of degree
\mbox{$d-\sum_{i=1}^r d_i$}.
{\em Step 2}: Assume that \mbox{$\,\sum_{i=1}^r
d_i> d\:$} and \mbox{$\,\frac{d+2}{2} \leq \sum_{i\in I} d_i \leq d\:$}.
Put \mbox{$\underline{\tilde{d}}:=(\tilde{d}_1,\ldots,\tilde{d}_r)$}, where
\mbox{$\tilde{d}_i=d_i-1$}, \mbox{$i\in I$}, and \mbox{$\tilde{d}_i=d_i$},
\mbox{$i\not\in I$}. Then
\begin{eqnarray}
\sum_{i=1}^r \frac{\tilde{d}_i(\tilde{d}_i\!+\!1)}{2} & = & \sum_{i=1}^r
\frac{d_i(d_i\!+\!1)}{2} - \sum_{i\in I} d_i \:\stackrel{\mbox{\footnotesize
(\ref{3.2.2})}}{<}\:
\frac{d^2\!+\!6d\!-\!1}{4}-\left[\frac{d}{2}\right]-\frac{d\!+\!2}{2}
\nonumber\\
& \leq &
\frac{(d\!-\!1)^2+6(d\!-\!1)-1}{4}-\left[\frac{d\!-\!1}{2}\right]\:,
\label{3.2.3}
\end{eqnarray}
hence, by the induction assumption, there exists a curve \mbox{$F_{d-1}\in
S_{d-1}(\underline{p},\underline{\tilde{d}})$}, transversal to the line
$G$. Put \mbox{$\,q:=d-\sum_{i\in I}d_i\,$} and fix \mbox{$q\!+\!1$} distinct
generic points \mbox{$z_1,\ldots,z_{q+1}$} on $G$ outside $F_{d-1}$. Since
$$ \sum_{i\in I} d_i+q+1=d+1\:,$$
we have
\begin{eqnarray}
\sum_{i=1}^r \frac{d_i(d_i+1)}{2}+q+1 & < &
\frac{d^2\!+\!6d\!-\!1}{4}-\left[\frac{d}{2}\right]+(d\!+\!1\!-\!\sum_{i\in I}
d_i ) \nonumber\\
& \leq & \frac{d^2\!+\!6d\!-\!1}{4}-\left[\frac{d}{2}\right]
+\frac{d}{2}\:\:\:\leq\:\:\:\left[\frac{(d\!+\!3)^2}{4}\right]\:,\nonumber
\end{eqnarray}
and, according to the Hirschowitz--Criterion (\ref{Hirschowitz}),
$$ h^1(\P^2,{\mathcal J}(d))=0 $$
where ${\mathcal J}\subset {\mathcal O}_{\P^2}$ is the ideal sheaf defined by
$${\mathcal J}_{p_i}=(\frak{m}_{p_i})^{d_i}\,,\;\;i=1,\ldots,r\,,\;\;\;
{\mathcal J}_{z_j}=\frak{m}_{z_j}\,,\;\;j=1,\ldots,q+1\:.
$$
That means,
\begin{eqnarray*}
h^0(\P^2,{\mathcal J}(d)) & = &
h^0(\P^2,{\mathcal O}_{\P^2}(d))-\sum_{i=1}^r\mbox{dim}\,{\mathcal O}_{\P^2,p_i}/
(\frak{m}_{p_i})^{d_i}-\sum_{j=1}^{q+1} \mbox{dim}\,{\mathcal O}_{\P^2,z_j}/
\frak{m}_{z_j} \\
& = & \frac{(d\!+\!1)(d\!+\!2)}{2}\,-\,\sum_{i=1}^r
\frac{d_i(d_i\!+\!1)}{2}-q-1\:,
\end{eqnarray*}
or, in other words, the
$$ m:=\sum_{i=1}^r \frac{d_i(d_i\!+\!1)}{2}+q+1$$
linear conditions on curves of degree $d$, imposed by the multiple points
\mbox{$p_1,\ldots,p_r,z_1,\ldots,z_{q+1}$} are independent. We write these
conditions as linear equations \mbox{$\Lambda_j(F)=0$}, \mbox{$1\leq j\leq m$},
in the coefficients of a curve $F$ of degree $d$, such that
\mbox{$\Lambda_m(F)=0$} expresses the passage of $F$ through $z_{q+1}$. Due to
the above independence, we find a curve \mbox{$F\in H^0({\mathcal O}_{\P^2}(d))$}
satisfying
$$\Lambda_1(F)=\ldots=\Lambda_{m-1}(F)=0\,,\;\;\;\Lambda_m(F)=1\:.$$
Let us consider the linear family \mbox{$\,\lambda F_{d-1}G+\mu F\,$},
\mbox{$(\lambda ,\mu)\in \P^1$}. By Bertini's \mbox{Theorem} and by the
construction of $F_{d-1}$ and $F$, the
generic member $F_{\lambda ,\mu }$ of this family is irreducible and belongs to
$S_d(\underline{p},\underline{d})$. The only thing we should show, is the
transversality of $F_{\lambda ,\mu }$ and $G$. By construction, $F_{\lambda
,\mu }$ has multiplicities $d_i$
at $p_i$, \mbox{$i\in I$}, respectively, and contains \mbox{$q=d-\sum_{i\in I}
d_i$} extra points \mbox{$z_1,\ldots,z_q$} on $G$, hence, clearly,
$F_{\lambda ,\mu }$ and $G$ meet transversally.
{\em Step 3}: Assume that \mbox{$\:\sum_{i=1}^r
d_i> d\:$}, \mbox{$\;\,d_1\geq \frac{d+2}{2}\;$} and \mbox{$\;\sum_{i\in I} d_i
\leq \frac{d+1}{2}\:$}.
Define \mbox{$\underline{\tilde{d}}:=(d_1-1,d_2,\ldots,d_r)$}. As is
(\ref{3.2.3}), we obtain
$$
\sum_{i=1}^r \frac{\tilde{d}_i(\tilde{d}_i\!+\!1)}{2} <
\frac{(d\!-\!1)^2+6(d\!-\!1)-1}{4}-\left[\frac{d\!-\!1}{2}\right]\:,
$$
hence, there exists a curve \mbox{$F_{d-1} \in
S_{d-1}(\underline{p},\underline{\tilde{d}})$}, transversal to $G$.
Note that
\begin{eqnarray*}
\mbox{dim}\,S_{d-1}(\underline{p},\underline{\tilde{d}})
& \geq & \frac{(d\!-\!1)(d\!+\!2)}{2}-\frac{(d_1\!-\!1)d_1}{2}-\sum_{i=2}^r
\frac{d_i(d_i\!+\!1)}{2} \\
& > & \frac{(d\!-\!1)(d\!+\!2)}{2}-\frac{(d\!-\!1)^2+6(d\!-\!1)-1}{4}+
\left[\frac{d\!-\!1}{2}\right] \\
& = & \frac{d^2-2d+2}{4}+\left[\frac{d\!-\!1}{2}\right]\:\:\geq \:\:2\:,
\end{eqnarray*}
as \mbox{$d\geq3$}. Therefore we find a curve \mbox{$\tilde{F}_{d-1}\in
S_{d-1}(\underline{p},\underline{\tilde{d}})$}, linearly independent of
$F_{d-1}$. Consider the linear family
$$\lambda F_{d-1}L+\mu \tilde{F}_{d-1} \tilde{L}\,,\;\;\;(\lambda, \mu)\in
\P^1\,,$$
where $L$, $\tilde{L}$ are distinct generic straight lines through $p_1$. Then,
by Bertini's Theorem a generic member of this family is irreducible, belongs to
$S_d(\underline{p},\underline{d})$ and is transversal to $G$.
{\em Step 4}: Assume that $d$ is odd and \mbox{$\;\sum_{i\in I} d_i
=\frac{d+1}{2}=d_m:=\mbox{max}\,\{d_i\,|\:i\not\in I\,\}\,$}.
First, we show that $\#(I)>1$. Indeed, otherwise, we had
$$ 0\,\stackrel{\mbox{\footnotesize (\ref{3.2.2})}}{<}\,
\frac{d^2\!+\!6d\!-\!1}{4}-\frac{d\!-\!1}{2}
- -\frac{d\!+\!1}{2}
\left(\frac{d\!+\!1}{2}+1\right)\: =\: -\frac{1}{2}\:.
$$
Choose \mbox{$j\neq k \in I$}. Since
\mbox{$d_m\!+\min\,\{d_j,\,d_k\}\geq\frac{d+2}{2}\,$} and
\mbox{$d_m\!+\max\,\{d_j,\,d_k\}\leq d\,$}, once again as in
(\ref{3.2.3}), we obtain for
$$
\begin{array}{lll}
\underline{\tilde{d}}=(\tilde{d}_1,\ldots,\tilde{d}_r)\,, &
\tilde{d}_i:=d_i\,\mbox{ for } i\not\in
\{m,j\}\,, & \tilde{d}_i:=d_i\!-\!1\,
\mbox{ for } i\in \{m,j\} \\
\underline{\tilde{d'}\!}=(\tilde{d}'\;\!\!\!_1,\ldots,
\tilde{d}'\;\!\!\!_r)\,,&
\tilde{d}'\;\!\!\!_i:=d_i\,\mbox{ for } i\not\in
\{m,k\}\,, & \tilde{d}'\;\!\!\!_i:=d_i\!-\!1\,
\mbox{ for } i\in \{m,k\}
\end{array}
$$
the existence of curves \mbox{$\tilde{F}_{d-1}\in
S_{d-1}(\underline{p},\underline{\tilde{d}}\,)$},
\mbox{$\,\tilde{F}'\!\!_{d-1}\in
S_{d-1}(\underline{p},\underline{\tilde{d'}\!}\:)$}, transversal to $G$. Let
$\tilde{G}$ be the straight line through $p_m$, $p_j$, and $\tilde{G'}$ be the
straight line through $p_m$, $p_k$. Consider the linear family
$$ \tilde{\lambda}\tilde{F}_{d-1}\tilde{G} + \tilde{\lambda
'}\tilde{F}'\!\!_{d-1}\tilde{G'},\;\;\;(\tilde{\lambda},\tilde{\lambda
'})\in \P^1\:.$$
By Bertini's Theorem, a generic member $F$ of this family has the only singular
points \mbox{$p_1,\ldots,p_r$} of multiplicities \mbox{$d_1,\ldots,d_r$},
respectively, and is transversal to $G$. Since
\mbox{$\tilde{F}_{d-1}\tilde{G}$} has ordinary
singularities at $p_i$, \mbox{$i\not\in \{j,m\}$}, and
\mbox{$\tilde{F}'\!\!_{d-1}\tilde{G'}$} has ordinary
singularities at $p_i$, \mbox{$i\not\in \{k,m\}$}, $F$ has ordinary
singularities at $p_i$, \mbox{$i\neq m$}. At the point $p_m$, the curve
\mbox{$\tilde{F}_{d-1}\tilde{G}$} has at most one multiple tangent, which
should coincide with $\tilde{G}$, on the other hand
\mbox{$\tilde{F}'\!\!_{d-1}\tilde{G'}$} has at most one multiple tangent, which
should coincide with \mbox{$\tilde{G'}\neq \tilde{G}$}. Therefore, $F$ has no
multiple tangent at $p_m$. Finally we have to show that $F$ is irreducible,
but this follows immediately from Bertini's Theorem: indeed, the only
possibility for $F$ to be reducible is \mbox{$F=F_{d-1}L$},
where $F_{d-1}$ is an irreducible curve of degree \mbox{$d\!-\!1$} and $L$ is a
straight line, which must vary from $\tilde{G}$ to $\tilde{G'}$ as
$(\tilde{\lambda},\tilde{\lambda'})$ runs through $\P^1$, which is impossible,
because $F_{d-1}$ must have multiplicity $d_j$ at $p_j$ and multiplicity $d_k$
at $p_k$, while $\tilde{F}_{d-1}$ has multiplicity \mbox{$d_j-1$} at $p_j$.
{\em Step 5}: Assume that \mbox{$\:\sum_{i=1}^r
d_i> d\:$}, \mbox{$\;\sum_{i\in I} d_i\leq
\min\,\{\frac{d+1}{2},\,d\!-\!d_m$\}} and \mbox{$\;d_m\leq \frac{d+1}{2}\,$}.
(Again, $d_m$ denotes \mbox{$\max\,\{d_i\,|\:i\not\in I\}$}.) In this case, we
specialize the point $p_m$ to a generic point in
\mbox{$G\setminus\{p_i\,|\:i\in I\}$}, and we end up with one of the cases
2--5.
Finally, consider for each $r$--tuple \mbox{($p_1,\ldots,p_r$)} of points in
$\P^2$ the non--empty linear system of curves with multiplicity at least $d_i$
at $p_i$ (\mbox{$1\leq i\leq r$}). Then in each of the occurring cases the
Hirschowitz criterion (\ref{Hirschowitz}) implies the non--speciality. Hence
such linear systems are equidimensional and give an irreducible variety. But
the condition to be irreducible with only given non--degenerate multiple points
is open (it is described by inequalities); hence the existence of such a curve
in a more special situation (as considered above) implies the existence in the
original situation.
\end{proof}
\subsection{Construction of nodal curves} \label{3.3}
\setcounter{equation}{0}
Given \mbox{$k,d,d',d_1,\ldots,d_r$} satisfying conditions (\ref{3.1.2}) and
(\ref{3.1.3}), we shall construct a reduced irreducible plane curve of degree
$d$ with ordinary singular points \mbox{$p_1,\ldots,p_r$} of multiplicities
\mbox{$d_1,\ldots,d_r$}, respectively, and with $k$ additional nodes as its
only singularities.
Let us fix distinct generic points \mbox{$p_1,\ldots,p_r\in \P^2$}, take an
irreducible curve
\mbox{$\Phi\in S_{d'}(\underline{p},\underline{d})$}, and put
$$F:=\Phi \prod_{i=1}^{d-d'} L_i\:,$$
where $L_1,\ldots,L_{d-d'}$ are distinct generic straight lines. This curve $F$
has ordinary singular points \mbox{$p_1,\ldots,p_r$} of multiplicities
\mbox{$d_1,\ldots,d_r$}, respectively, and \mbox{$m:=d(d-1)/2-d'(d'-1)/2$}
nodes \mbox{$z_1,\ldots,z_m$} as its only singularities. We shall show that it
is possible to smooth prescribed nodes keeping the given ordinary singularities
and the rest of nodes, and thus prove Theorem \ref{3.1.1}.
As we have seen
above, we can deduce the required independence of the deformations of the nodes
from
\begin{eqnarray}
H^1(F,{\mathcal N}^{\,'}\!\!\!_{F/\P^2})=0\:,\label{3.3.1}
\end{eqnarray}
\mbox{$ {\mathcal N}^{\,'}\!\!\!_{F/\P^2}:=\mbox{Ker}\,({\mathcal N}_{F/\P^2}\rightarrow
{\mathcal T}_F)\:,$} where ${\mathcal T}_F$ denotes the skyscraper sheaf on $\P^2$ concentrated
at \mbox{$p_1,\ldots,p_r$}, \mbox{$z_1,\ldots,z_m$} defined by
$$
{\mathcal T}_{F,p_i}:={\mathcal O}_{\P^2,p_i}/(\frak{m}_{p_i})^{d_i}\,,
\;\:1\leq i\leq r\,,\:\:\;\;
{\mathcal T}_{F,z_j}:={\mathcal O}_{\P^2,z_j}/\frak{m}_{z_j}\,,\;\:1\leq j\leq m\,.$$
We prove (\ref{3.3.1}) by induction on $d$. If \mbox{$d=d'$}, then the
vanishing of
$ H^1(F,{\mathcal N}^{\,'}\!\!\!_{F/\P^2}) $ is provided by the Hirschowitz--Criterion
(\ref{Hirschowitz}), because (\ref{3.1.2}) implies
$$\left[\frac{(d'\!+3)^2}{4}\right] > \sum_{i=1}^r\frac{d_i(d_i+1)}{2}\:.$$
Assume that \mbox{$d>d'$}. Then denote
$$ \tilde{F}:=\Phi\prod_{i=1}^{d-d'-1} L_i \,,$$
that is,
\mbox{$F=\tilde{F}L_{d-d'}$}.
Let \mbox{$\tilde{F}\cap
L_{d-d'}:=\{z_1,\ldots z_l\}$}, then ${\mathcal T}_{\tilde{F}}$ denotes the restriction
of ${\mathcal T}_F$ on \mbox{$\{p_1,\ldots,p_r,z_{l+1},\ldots,z_m\}$}. Consider the
exact sequence
\renewcommand{\arraystretch}{1.0}
$$
\begin{array}{ccccccccc}
0&\longrightarrow &{\mathcal O}_{\tilde{F}}(d-1) \oplus {\mathcal O}_{L_{d-d'}}(1)
&\stackrel{\alpha}{\longrightarrow} & {\mathcal O}_F(d)
&\longrightarrow &{\mathcal O}_{\tilde{F}\cap L_{d-d'}} & \longrightarrow & 0\\
& & \:\| & & \| \\
& & \:\:{\mathcal N}_{\tilde{F}/\P^2}\oplus {\mathcal O}_{\P^1}(1) & & {\mathcal N}_{F/\P^2} \\
\end{array}
$$
where $\alpha$ is given by \mbox{$\,\mbox{id}_1\otimes
L_{d-d'}+\tilde{F}\otimes \mbox{id}_2\,$}. Clearly, $\alpha $ maps
\mbox{${\mathcal N}^{\,'}\!\!\!_{\tilde{F}/\P^2}\oplus {\mathcal O}_{\P^1}(1)$} to
${\mathcal N}^{\,'}\!\!\!_{F/\P^2}$. Therefore, (\ref{3.3.1})
follows from the induction assumption
$$ H^1(\tilde{F},{\mathcal N}^{\,'}\!\!\!_{\tilde{F}/\P^2})=0\:, $$
and we are finished.
|
1996-01-15T06:20:20 | 9601 | alg-geom/9601010 | en | https://arxiv.org/abs/alg-geom/9601010 | [
"alg-geom",
"math.AG"
] | alg-geom/9601010 | Kai Behrend | K. Behrend and B. Fantechi | The intrinsic normal cone | LaTeX, Postscript file available at
http://www.math.ubc.ca/people/faculty/behrend/inc.ps | null | 10.1007/s002220050136 | null | null | We suggest a construction of virtual fundamental classes of certain types of
moduli spaces.
| [
{
"version": "v1",
"created": "Mon, 15 Jan 1996 03:53:32 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Behrend",
"K.",
""
],
[
"Fantechi",
"B.",
""
]
] | alg-geom | \section{Introduction}
Moduli spaces in algebraic geometry often have an expected dimension
at each point, which is a lower bound for the dimension at that
point. For instance, the moduli space of smooth, complex projective
$n$-dimensional varieties with ample canonical class has expected
dimension $h^1(V,T_V)-h^2(V,T_V)$ at a point $[V]$. In general, the
expected dimension will vary with the point; however, in some
significant cases it will stay constant on connected components. In
the previous example, this is the case if $n\le 2$, for then the
expected dimension is $-\chi(V,T_V)$. In some cases the dimension
coincides with the expected dimension, in others it does so under some
genericity assumptions. However, it can happen that there is no way to
get a space of the expected dimension; it is also possible that
special cases with bigger dimension are easier to understand and to
deal with than the generic case.
When we have a moduli space $X$ which has a well-defined expected
dimension, it can be useful to be able to construct in its Chow ring a
class of the expected dimension. The main examples we have in mind are
Donaldson theory (with $X$ the moduli space of torsion-free,
semi-stable sheaves on a surface) and the Gromov-Witten invariants
(with $X$ the moduli space of stable maps from curves of genus $g$ to
a fixed projective variety). In this paper we deal with the problem
of defining such a class in a very general set-up; the construction is
divided into two steps.
First, given any Deligne-Mumford stack $X$, we associate to it an
algebraic stack ${\frak C}_X$ over $X$ of pure dimension zero, its {\em
intrinsic normal cone}. This has nothing to do with $X$ being a moduli
space; it is just an intrinsic invariant, whose structure is related
to the singularities of $X$ (see for instance Proposition
\ref{colci}).
Then, we define the concept of an obstruction theory and of a perfect
obstruction theory for $X$. To say that $X$ has an obstruction theory
means, very roughly speaking, that we are given locally on $X$ an
(equivalence class of) morphisms of vector bundles such that at each
point the kernel of the induced linear map of vector spaces is the
tangent space to $X$, and the cokernel is a space of
obstructions. Usually, if $X$ is a moduli space then it has an
obstruction theory, and if this is perfect then the expected dimension
is constant on $X$. Once we are given an obstruction theory, with the
additional (technical) assumption that it admits a global resolution,
we can define a virtual fundamental class of the expected dimension.
An application of the results of this work is contained in a paper
\cite{gwi} by the first author. There Gromov-Witten invariants are
constructed for any genus, any target variety and the axioms listed in
\cite{BM} are verified.
We now give a more detailed outline of the contents of the paper. In
the first section we recall what we need about cones and we introduce
the notion of cone stacks over a Deligne-Mumford stack $X$. These are
Artin stacks which are locally the quotient of a cone by a vector
bundle acting on it. We call a cone {\em abelian }if it is defined as
$\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits {\cal F}$, where ${\cal F}$ is a coherent sheaf on $X$. Every cone is
contained as a closed subcone in a minimal abelian one, which we call
its {\em abelian hull}. The notions of being abelian and of abelian hull
generalize immediately to cone stacks.
In the second section we
construct, for a complex $E^{\scriptscriptstyle\bullet}$ in the derived category $D(\O_X)$
which satisfies some suitable assumptions (which we call Condition
($\star$), see Definition \ref{dost}), an associated abelian cone
stack $h^1/h^0((E^{\scriptscriptstyle\bullet}\dual))$. In particular the cotangent complex
$L_X^{\scriptscriptstyle\bullet}$ of $X$ satisfies Condition ($\star$), so we can define the
abelian cone stack ${\frak N}_X:=h^1/h^0((L_X^{\scriptscriptstyle\bullet}\dual))$, the {\em
intrinsic normal sheaf}.
The name is motivated in the third section, where ${\frak N}_X$
is constructed more directly as follows: \'etale locally on $X$, embed
an open set $U$ of $X$ in a smooth scheme $W$, and take the stack
quotient of the normal sheaf (viewed as abelian cone) $N_{U/W}$ by the
natural action of $T_W|_U$. One can glue these abelian cone stacks
together to get ${\frak N}_X$. The intrinsic normal cone ${\frak C}_X$ is the closed
subcone stack of ${\frak N}_X$ defined by replacing
$N_{U/W}$ by the normal cone $C_{U/W}$ in the previous construction.
In the fourth section we describe the relationship between the intrinsic normal
sheaf of a Deligne-Mumford stack $X$ and the deformations of affine
$X$-schemes, showing in particular that ${\frak N}_X$ carries obstructions for such
deformations. With this motivation, we introduce the notion of
obstruction theory for $X$. This is an object $E^{\scriptscriptstyle\bullet}$ in the
derived category together with a morphism $E^{\scriptscriptstyle\bullet}\to L_X^{\scriptscriptstyle\bullet}$,
satisfying Condition ($\star$) and such that the induced
map ${\frak N}_X\to h^1/h^0((E^{\scriptscriptstyle\bullet}\dual))$ is a closed immersion.
An obstruction theory $E^{\scriptscriptstyle\bullet}$ is called perfect if
${\frak E}=h^1/h^0((E^{\scriptscriptstyle\bullet}\dual))$ is smooth over $X$. So we have a vector
bundle stack ${\frak E}$ with a closed subcone stack ${\frak C}_X$, and to define
the virtual fundamental class of $X$ with respect to $E^{\scriptscriptstyle\bullet}$ we simply
intersect ${\frak C}_X$ with the zero section of ${\frak E}$. This construction
requires Chow groups for Artin stacks, which we do not have at our
disposal. There are several ways around this problem. We choose to
assume that $E^{\scriptscriptstyle\bullet}$ is globally given by a homomorphism of vector
bundles $F^{-1}\to F^0$. Then ${\frak C}_X$ gives rise to a cone $C$ in
$F_1=\dual{F^{-1}}$ and we intersect $C$ with the zero section of
$F_1$.
Another approach, suggested by Kontsevich \cite{K}, is via virtual
structure sheaves (see Remark~\ref{vss}). The drawback of that
approach is that it requires a Riemann-Roch theorem for
Deligne-Mumford stacks, for which we do not know a reference.
In the sixth section we give some examples of how this construction
can be applied in some standard moduli problems. We consider the
following cases: a fiber of a morphism between smooth algebraic
stacks, the scheme of morphisms between two given projective schemes,
a moduli space for Gorenstein projective varieties.
In the seventh section we give a relative version of the intrinsic
normal cone and sheaf ${\frak C}_{X/Y}$ and ${\frak N}_{X/Y}$ for a morphism $X\to
Y$ with unramified diagonal of algebraic stacks; we are mostly
interested in the case where $Y$ is smooth and pure-dimensional, which
preserves many good properties of the absolute case (e.g., ${\frak C}_{X/Y}$
is pure-dimensional). This is not needed in this paper, but will be
applied by the first author to give an algebraic definition of
Gromov-Witten classes for smooth projective varieties.
The starting point for this work was a talk by Jun Li at the AMS
Summer Institute on Algebraic Geometry, Santa Cruz 1995, where he
reported on joint work in progress with G.~Tian. Their construction,
in the complex analytic context, is based on the existence of the
Kuranishi map; by using it they define, under suitable assumptions, a
pure-dimensional cone in some bundle and get classes of the expected
dimension by intersecting with the zero section.
Our construction owes its existence to theirs; we started by trying to
understand and reformulate their results in an algebraic way, and
found stacks to be a convenient, intrinsic language. In our opinion
the introduction of stacks is very natural, and it seems almost
surprising that the intrinsic normal cone was not defined before. We
find it important to separate the construction of the cone, which can
be carried out for any Deligne-Mumford stack, from its embedding in a
vector bundle stack. We work completely in an algebraic context; of
course the whole paper could be rewritten without changes over the
category of analytic spaces.
\smallskip\noindent
{\em Acknowledgments. }This work was started in the inspiring
atmosphere of the Santa Cruz conference. A significant part of it was done
during
the authors' stay at the Max-Planck-Institut f\"ur Mathematik in Bonn, to which
both authors are grateful for hospitality and support. The second author is a
member of GNSAGA of CNR.
\subsection{Notations and Conventions}
Unless otherwise mentioned, we work over a fixed ground field $k$.
An {\em algebraic stack }is an algebraic stack over $k$ in the sense
of \cite{vdas} or \cite{laumon}. Unless mentioned otherwise, we assume
all algebraic stacks (in particular all algebraic spaces and all
schemes) to be quasi-separated and locally of finite type over $k$.
A {\em Deligne-Mumford stack }is an algebraic stack in the sense of
\cite{DM}, in other words an algebraic stack with unramified
diagonal. For a Deligne-Mumford stack $X$ we denote by $X_{\mbox{\tiny fl}}$ the big
fppf-site and by $X_{\mbox{\tiny \'{e}t}}$ the small \'etale site of $X$. The associated
topoi of sheaves are denoted by the same symbols.
Recall that a complex of sheaves of modules is {\em of perfect
amplitude contained in $[a,b]$}, where $a,b\in{\Bbb Z}$, if, locally, it is
isomorphic (in the derived category) to a complex $E^a\to\ldots\to E^b$
of locally free sheaves of finite rank.
\section{Cones and Cone Stacks}
\subsection{Cones}
To fix notation we recall some basic facts about cones.
Let $X$ be a Deligne-Mumford stack. Let
\[S=\bigoplus_{i\geq0}S^i\]
be a graded quasi-coherent sheaf of $\O_X$-algebras such that
$S^0=\O_X$, $S^1$ is coherent and $S$ is generated locally by
$S^1$. Then the affine $X$-scheme $C=\mathop{\rm Spec}\nolimits S$ is called a {\em cone
}over $X$. A {\em morphism }of cones over $X$ is an $X$-morphism
induced by a graded morphism of graded sheaves of $\O_X$-algebras. A
{\em closed subcone }is the image of a closed immersion of cones. If
\[\begin{array}{ccc}
& & C_2 \\
& & \downarrow \\ C_1 & \longrightarrow & C_3
\end{array}\] is a diagram of cones over $X$, the fibered product
$C_1\times_{C_3}C_2$ is a cone over $X$.
Every cone $C\to X$ has a section $0:X\to C$, called the {\em vertex
}of $C$, and an ${\Bbb A}^1$-action (or a multiplicative contraction onto
the vertex), that is a morphism
\[\gamma:{\Bbb A}^1\times C\longrightarrow C\]
such that
\begin{enumerate}
\item \[\comtri{C}{(1,\mathop{\rm id})}{{\Bbb A}^1\times C}{\mathop{\rm id}}{\gamma}{C}\] commutes,
\item \[\comtri{C}{(0,\mathop{\rm id})}{{\Bbb A}^1\times C}{0}{\gamma}{C}\] commutes,
\item \[\comdia{{\Bbb A}^1\times{\Bbb A}^1\times C} {\mathop{\rm id}\times\gamma}
{{\Bbb A}^1\times C} {m\times\mathop{\rm id}}{}{\gamma} {{\Bbb A}^1\times C}{\gamma}{C}\]
commutes, where $m:{\Bbb A}^1\times{\Bbb A}^1\to{\Bbb A}^1$ is multiplication,
$m(x,y)=xy$.
\end{enumerate}
The vertex of $C$ is induced by the augmentation $S\to S^0$, the
${\Bbb A}^1$-action is given by the grading of $S$. In fact, the morphism
$S\to S[x]$ giving rise to $\gamma$ maps $s\in S^i$ to $sx^i$.
Note that a morphism of cones is just a morphism respecting $0$ and
$\gamma$.
\subsection{Abelian Cones}
If ${\cal F}$ is a coherent $\O_X$-module we get an associated cone
\[C({\cal F})=\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits({\cal F}).\]
For any $X$-scheme $T$ we have
\[C({\cal F})(T)=\mathop{\rm Hom}\nolimits({\cal F}_T,\O_T),\]
so $C({\cal F})$ is a group scheme over $X$. We call a cone of this form an
{\em abelian cone}. A fibered product of abelian cones is an abelian
cone. If $E$ is a vector bundle over $X$, then $E=C({\cal E}^{\vee})$,
where ${\cal E}$ is the coherent $O_X$-module of sections of $E$ and
${\cal E}^{\vee}$ its dual.
Any cone $C=\mathop{\rm Spec}\nolimits\bigoplus_{i\geq0}S^i$ is canonically a closed
subcone of an abelian cone $A(C)=\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits S^1$, called the {\em
associated abelian cone }or the {\em abelian hull }of $C$. The abelian
hull is a vector bundle if and only if $S^1$ is locally free. Any
morphism of cones $\phi:C\rightarrow D$ induces a morphism
$A(\phi):A(C)\rightarrow A(D)$, extending $\phi$. Thus $A$ defines a
functor from cones to abelian cones called {\em abelianization}. Note
that $\phi$ is a closed immersion if and only if $A(\phi)$ is.
\begin{lem}
A cone $C$ over $X$ is a vector bundle if and only if it is smooth over $X$.
\end{lem}
\begin{pf} Let $C=\mathop{\rm Spec}\nolimits\bigoplus_{i\geq0}S^i$, and assume that $C\to X$ has
constant relative dimension $r$. Then $S^1=0^*\Omega_{C/X}$ is a rank $r$
vector
bundle. $C$ is a closed subcone of $A(C)=(S^1)^\vee$, hence by dimension
reasons
$C=A(C)$.
\end{pf}
If $E$ and $F$ are abelian cones over $X$, then any morphism of cones
$\phi:E\rightarrow F$ is a morphism of $X$-group schemes. If $E$ and
$F$ are vector bundles, then $\phi$ is a morphism of vector bundles.
\begin{example} If $X\rightarrow Y$ is a closed immersion with ideal sheaf
${\cal I}$, then
$$\bigoplus_{n\geq0}{\cal I}^n/{\cal I}^{n+1}$$ is a sheaf of $\O_X$-algebras and
$$C_{X/Y}=\mathop{\rm Spec}\nolimits
\bigoplus_{n\geq0}{\cal I}^n/{\cal I}^{n+1}$$ is a cone over $X$, called the {\em normal
cone }of $X$ in $Y$. The associated abelian cone $N_{X/Y}=\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits{\cal I}/{\cal I}^2$
is
also called the {\em normal sheaf }of
$X$ in $Y$.
More generally, any local immersion of Deligne-Mumford stacks has a normal cone
whose abelian hull is its normal sheaf (see \cite{vistoli}, definition 1.20).
\end{example}
\subsection{Exact Sequences of Cones}
\begin{defn} A sequence of cone morphisms $$
\ses E{i}C{}D$$
is {\em exact} if $E$ is a vector bundle and locally over $X$ there is a
morphism of cones $C\to E$ splitting $i$ and inducing an isomorphism $C\to
E\times D$.
\end{defn}
\begin{rmk} Given a short exact sequence $$
\ses {{\cal F}'}{}{\cal F}{}{\cal E}$$
of coherent sheaves on $X$, with ${\cal E}$ locally free, then $$
\ses {C({\cal E})}{}{C({\cal F}')}{}{C({\cal F})}$$
is exact, and conversely (see \cite{fulton}, Example 4.1.7).
\end{rmk}
\begin{lem} \label{ssmc}
Let $C\to D$ be a smooth, surjective morphism of cones, and let
$E=C\times_{D,0}X$; then the sequence $$\ses E{}C{}D$$ is exact.
\end{lem}
\begin{pf}
Write $C=\mathop{\rm Spec}\nolimits\bigoplus S^i$, $D=\mathop{\rm Spec}\nolimits\bigoplus S^{\prime i}$. We
start by proving that $$
\ses E{}{A(C)}{}{A(D)}$$is exact.
By base change we may assume $S^{\prime i}=0$ for $i\ge 2$. The cone
$E=\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits{\cal E}$ is a vector bundle because it is smooth. On the other
hand, $E=\mathop{\rm Spec}\nolimits\bigoplus(S^i/S^{\prime 1}S^{i-1})$. As $C\to D$ is
smooth and surjective, $S^1\to S^{\prime 1}$ is injective. So we get
an exact sequence $$
\ses{S^1}{}{S^{\prime 1}}{}{\cal E}.$$
To complete the proof, remark that $C\to A(C)\times_{A(D)}D$ is a
closed immersion, and both these schemes are smooth of the same
relative dimension over $C$.
\end{pf}
\subsection{$E$-Cones}
If $E$ is a vector bundle and $d:E\rightarrow C$ a morphism of cones, we say
that {\em $C$ is an
$E$-cone}, if $C$ is invariant under the action of $E$ on $A(C)$. We denote the
induced action of
$E$ on $C$ by
\begin{eqnarray*} E\times C & \longrightarrow & C \\ (\nu,\gamma) & \longmapsto
& d\nu+\gamma\quad.
\end{eqnarray*}
A {\em morphism }$\phi$ from an $E$-cone $C$ to an $F$-cone $D$ (or a
{\em morphism of vector bundle cones}) is a
commutative diagram of cones
\[\comdia{E}{d}{C}{\phi}{}{\phi}{F}{d}{D.}\] If
$\phi:(E,d,C)\rightarrow(F,d,D)$
and $\psi:(E,d,C)\rightarrow(F,d,D)$ are morphisms, we call them {\em
homotopic}, if there exists a morphism of cones $k:C\rightarrow F$, such that
\begin{enumerate}
\item $kd=\psi-\phi$,
\item $dk=\psi-\phi$.
\end{enumerate} Here the second condition is to be interpreted as saying that
$\phi+dk=\psi$. (More precisely, we say that $k$ is a {\em homotopy }from
$\phi$
to $\psi$.)
\begin{rmk} A sequence of cone morphisms with $E$ a vector bundle$$
\ses E{i}C{}D$$ is exact if and only if $C$ is an $E$-cone, $C\to D$ is
surjective, and the diagram $$\comdia{E\times
C}{\sigma}{C}{p}{}{\phi}{C}{\phi}{D}$$ is cartesian, where
$p$ is the projection and
$\sigma$ the action.
\end{rmk}
\begin{prop} \label{lcc}
Let $(C,0,\gamma)$ and $(D,0,\gamma)$ be algebraic $X$-spaces with
sections and ${\Bbb A}^1$-actions and let $\phi:C\rightarrow D$ be an
${\Bbb A}^1$-equivariant $X$-morphism, which is smooth and surjective. Let
$E=C\times_{D,0}X$. Then $C$ is an $E$-cone over $X$ if and only if
$D$ is a cone over $X$. Moreover, $C$ is abelian (a vector bundle) if
and only if $D$ is.
\end{prop}
\begin{pf}
Let us first assume that $C$ is an abelian cone, $C=\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits{\cal F}$.
The morphism $E\rightarrow C$ gives rise to
${\cal F}\rightarrow\dual{{\cal E}}$, where ${\cal E}$ is the coherent $\O_X$-modules
of sections of $E$. Note that ${\cal F}\rightarrow\dual{{\cal E}}$ is an
epimorphism, since $E\rightarrow C$ is injective. Let ${\cal G}$ be the
kernel, so that
\[\ses{{\cal G}}{}{{\cal F}}{}{\dual{{\cal E}}}\] is a short exact sequence. Then
\[\ses{E}{}{C}{}{C({\cal G})}\] is a short exact sequence of abelian cones over $X$,
so $D\cong C({\cal G})$ and so $D$ is an abelian cone.
In general, $C\subset A(C)$ is defined by a homogeneous sheaf of ideals
${\cal I}\subset\mathop{\rm Sym}\nolimits {\cal S}^1$, where ${\cal S}=\bigoplus{\cal S}^i$ and $C=\mathop{\rm Spec}\nolimits{\cal S}$. Let
${\cal F}={\cal S}^1$ and let ${\cal G}$ as above be the kernel of
${\cal F}\rightarrow\dual{{\cal E}}$. Let ${\cal J}={\cal I}\cap\mathop{\rm Sym}\nolimits{\cal G}$, which is a homogeneous
sheaf of ideals in
$\mathop{\rm Sym}\nolimits{\cal G}$, so $C'=\mathop{\rm Spec}\nolimits\mathop{\rm Sym}\nolimits{\cal G}/{\cal J}$ is a cone over $X$. By construction, $C'$
is
the scheme theoretic image of $C$ in $C({\cal G})$. Hence $C'$ is the quotient of
$C$
by $E$ and so $C'\cong D$ and
$D$ is a cone.
Now for the converse. The claim is local in $X$. So since $D$ is affine over
$X$
we may assume that
$C=D\times E$ as $X$-schemes with ${\Bbb A}^1$-action. Then we are done.
\end{pf}
\subsection{Cone Stacks}
Let $X$ be, as above, a Deligne-Mumford stack over $k$. We need to
define the 2-category of algebraic stacks with ${\Bbb A}^1$-action over
$X$.
\begin{defn}
Let ${\frak C}$ be an algebraic stack over $X$, together with a section
$0:X\to{\frak C}$. An {\em ${\Bbb A}^1$-action }on $({\frak C},0)$ is given by a
morphism of $X$-stacks $$\gamma:{\Bbb A}^1\times{\frak C}\longrightarrow{\frak C}$$ and
three 2-isomorphisms $\theta_1$, $\theta_0$ and $\theta_{\gamma}$
between the 1-morphisms in the following diagrams.
\begin{enumerate}
\item \[\comtri{{\frak C}}{(1,\mathop{\rm id})}{{\Bbb A}^1\times {\frak C}}{\mathop{\rm id}}{\gamma}{{\frak C}}\]
and $\theta_1:\mathop{\rm id}\to\gamma\mathbin{{\scriptstyle\circ}}(1,\mathop{\rm id})$.
\item \[\comtri{{\frak C}}{(0,\mathop{\rm id})}{{\Bbb A}^1\times {\frak C}}{0}{\gamma}{{\frak C}}\]
and $\theta_0:0\to\gamma\mathbin{{\scriptstyle\circ}}(0,\mathop{\rm id})$.
\item \[\comdia{{\Bbb A}^1\times{\Bbb A}^1\times {\frak C}} {\mathop{\rm id}\times\gamma}
{{\Bbb A}^1\times {\frak C}} {m\times\mathop{\rm id}}{}{\gamma} {{\Bbb A}^1\times {\frak C}}{\gamma}{{\frak C}}\]
and
$\theta_{\gamma}:\gamma\mathbin{{\scriptstyle\circ}}(m\times\mathop{\rm id})\to\gamma\mathbin{{\scriptstyle\circ}}(\mathop{\rm id}\times\gamma)$.
\end{enumerate}
The 2-isomorphisms $\theta_1$, $\theta_0$ and $\theta_{\gamma}$ are
required to satisfy certain compatibilities which we leave to the
reader to make explicit (see also Section~1.4 in Expos\'e~XVIII of
\cite{sga4}, where a similar problem, the definition of Picard stacks,
is dealt with).
Let $({\frak C},0,\gamma)$ and $({\frak D},0,\gamma)$ be $X$-stacks with sections
and ${\Bbb A}^1$-actions. Then an {\em ${\Bbb A}^1$-equivariant morphism }
$\phi:{\frak C}\to{\frak D}$ is a triple $(\phi,\eta_0,\eta_{\gamma})$, where
$\phi:{\frak C}\to{\frak D}$ is a morphism of algebraic $X$-stacks and $\eta_0$
and $\eta_{\gamma}$ are 2-isomorphisms between the morphisms in the
following diagrams.
\begin{enumerate}
\item\begin{equation}\label{na1}
\comtri{X}{0}{{\frak C}}{0}{\phi}{{\frak D}}
\end{equation}
and $\eta_0:0\to\phi\mathbin{{\scriptstyle\circ}} 0$.
\item \begin{equation}\label{na2}
\comdia{{\Bbb A}^1\times{\frak C}}{\mathop{\rm id}\times\phi}{{\Bbb A}^1\times{\frak D}}
{\gamma}{}{\gamma} {{\frak C}}{\phi}{{\frak D}}
\end{equation}
and $\eta_{\gamma}:\phi\mathbin{{\scriptstyle\circ}}\gamma\to\gamma\mathbin{{\scriptstyle\circ}}(\mathop{\rm id}\times\phi)$.
\end{enumerate}
Again, the 2-isomorphisms have to satisfy certain compatibilities we
leave to the reader to spell out.
Finally, let $(\phi,\eta_0,\eta_{\gamma}):{\frak C}\to{\frak D}$ and
$(\psi,\eta'_0,\eta'_{\gamma}):{\frak C}\to{\frak D}$ be two ${\Bbb A}^1$-equivariant
morphisms. An {\em ${\Bbb A}^1$-equivariant isomorphism
}$\zeta:\phi\to\psi$ is a 2-isomorphism $\zeta:\phi\to\psi$ such that
the diagrams (notation compatible with (\ref{na1}) and (\ref{na2}))
\begin{enumerate}
\item \[\comtri{0}{\eta_0}{\phi\comp0}
{\eta'_0}{\zeta\comp0}{\psi\comp0} \]
\item \[\comdia{\phi\mathbin{{\scriptstyle\circ}}\gamma}{}{\gamma\mathbin{{\scriptstyle\circ}}(\mathop{\rm id}\times\phi)}
{\zeta\mathbin{{\scriptstyle\circ}}\gamma}{}{\gamma\mathbin{{\scriptstyle\circ}}(\mathop{\rm id}\times\zeta)}
{\psi\mathbin{{\scriptstyle\circ}}\gamma}{}{\gamma\mathbin{{\scriptstyle\circ}}(\mathop{\rm id}\times\psi)} \]
\end{enumerate}
commute.
\end{defn}
If $C$ is an $E$-cone, then since $E$ acts on $C$, we may form the
stack quotient of $C$ by $E$ over $X$, denoted $[C/E]$. For an
$X$-scheme $T$, the groupoid of sections of $[C/E]$ over $T$ is the
category of pairs $(P,f)$, where $P$ is an $E$-torsor (a principal
homogeneous $E$-bundle) over $T$ and $f:P\rightarrow C$ is an
$E$-equivariant morphism.
The $X$-stack $[C/E]$ comes with a section $0:X\to[C,E]$ and an
${\Bbb A}^1$-action $\gamma:{\Bbb A}^1\times[C/E]\to[C/E]$. The section $0$ is
given by the pair $(E_T,0)$ over every $X$-scheme $T$; here $E_T$ is
the trivial $E$-bundle on $T$ and $0:E_T\to C$ is the vertex morphism.
The ${\Bbb A}^1$-action of $\alpha\in{\Bbb A}^1(T)=\O_T(T)$ on the category
$[C/E](T)$ is given by $\alpha\cdot(P,f)=(\alpha P,\alpha f)$, where
$\alpha P=P\times_{E,\alpha} E$ and $\alpha f:P\times_{E,\alpha} E\to
C$ is given by $[p,\nu]\mapsto \alpha f(p)+d(\nu)$.
If $\phi:(E,C)\to(F,D)$ is a morphism of vector bundle cones we get an
induced ${\Bbb A}^1$-equivariant morphism $\tilde{\phi}:[C/E]\to[D/F]$. A
homotopy $k:\phi\to\psi$ gives rise to an ${\Bbb A}^1$-equivariant
2-isomorphism $\tilde{k}:\tilde{\phi}\to\tilde{\psi}$ of
${\Bbb A}^1$-equivariant morphism of stacks with ${\Bbb A}^1$-action. (See
Section~\ref{sohh} where these constructions are made explicit in a
similar case.)
\begin{lem}\label{lo2i}
Let $\phi,\psi:(E,C)\to(F,D)$ be morphisms and
$\zeta:\tilde{\phi}\to\tilde{\psi}$ an ${\Bbb A}^1$-equivariant
2-isomorphism between the associated ${\Bbb A}^1$-equivariant morphisms
$[C/E]\to [D/F]$. Then $\zeta=\tilde{k}$, for a unique
homotopy $k:\phi\to\psi$.
\end{lem}
\begin{pf}
We indicate how to construct $k:C\to F$. Given a section $c\in C(T)$
of $C$ over the $X$-scheme $T$, we consider the induced object
$(E_T,c)$ of $[C/E](T)$. The associated $F_T$-torsors
$E_T\times_{E_T,\phi^0}F_T$ and $E_T\times_{E_T,\psi^0}F_T$ are
trivial, so that $\phi(T)(E_T,c)$ is a section of $F$ over $T$. This
section we define to be $k(c)$.
\end{pf}
\begin{prop} \label{cics}
Let $C$ be an $E$-cone and $D$ an $F$-cone. Let
$\phi:(E,C)\rightarrow(F,D)$ be a morphism. If the diagram
\[\comdia{E}{}{C}{}{}{}{F}{}{D}\] is cartesian and $F\times C\rightarrow
D;(\mu,\gamma)\mapsto d\mu+\phi(\gamma)$ is surjective, then
$[C/E]\rightarrow[D/F]$ is an isomorphism of algebraic $X$-stacks with
${\Bbb A}^1$-action.
\end{prop}
\begin{pf}
Similar to the proof of Proposition~\ref{tpifs} below.
\end{pf}
\begin{defn} \label{docs}
We call an algebraic stack $({\frak C},0,\gamma)$ over $X$ with section and
${\Bbb A}^1$-action a {\em cone stack}, if, locally with respect to the
\'etale topology on $X$, there exists a cone $C$ over $X$ and an
${\Bbb A}^1$-equivariant morphism $C\to{\frak C}$ that is smooth and surjective.
The morphism $C\rightarrow{\frak C}$, or by abuse of language $C$, is called
a {\em local presentation }of ${\frak C}$. The section $0:X\to{\frak C}$ is called
the {\em vertex }of ${\frak C}$.
Let ${\frak C}$ and ${\frak D}$ be cone stacks over $X$. A {\em morphism of cone
stacks } $\phi:{\frak C}\rightarrow{\frak D}$ is an ${\Bbb A}^1$-equivariant morphism
of algebraic $X$-stacks.
A {\em 2-isomorphism of cone stacks } is just an ${\Bbb A}^1$-equivariant
2-isomorphism.
\end{defn}
If $C\rightarrow{\frak C}$ is a presentation of ${\frak C}$, and
$E=C\times_{{\frak C},0}X$, then $C$ is an $E$-cone and ${\frak C}\cong[C/E]$
as stacks with ${\Bbb A}^1$-action (use Lemma~\ref{ssmc} and
Proposition~\ref{lcc}).
If $\phi:{\frak C}\to{\frak D}$ is a morphism of cone stacks, then, locally with
respect to the \'etale topology on $X$, $\phi$ is
${\Bbb A}^1$-equivariantly isomorphic to $[C/E]\to [D/F]$, where $E\to F$
is a morphism of vector bundles over $X$ and $C\to D$ is a
morphism from the $E$-cone $C$ to the $F$-cone $D$.
A 2-isomorphism of cone stacks $\zeta:\phi\to\psi$, where
$\phi,\psi:{\frak C}\to{\frak D}$, is locally over $X$ given by a homotopy of
morphisms of vector bundle cones. More precisely, one can find local
presentations ${\frak C}\cong[C/E]$ and ${\frak D}\cong[D/F]$ such that both
$\phi$ and $\psi$ are induced by morphisms of vector bundle cones
$\overline{\phi},\overline{\psi}:(E,C)\to(F,D)$ and under these identifications
$\zeta$ comes from a homotopy from $\overline{\phi}$ to $\overline{\psi}$. This
follows from Lemma~\ref{lo2i}.
\begin{rmk}
Let ${\frak C}$ be a cone stack over $X$. By Proposition~\ref{lcc} the
fibered product over ${\frak C}$ of any two local presentations is again a
local presentation. Moreover, if ${\frak C}$ is a representable cone stack
over $X$, then ${\frak C}$ is a cone. Every fibered product of cone stacks
is a cone stack.
\end{rmk}
\begin{examples}
All cones are cone stacks and all morphisms of cones are morphisms of
cone stacks. For a vector bundle $E$ on $X$, the classifying stack
$BE$ is a cone stack. Every homomorphism of vector bundles
$\phi:E\rightarrow F$ gives rise to a morphism of cone stacks.
\end{examples}
\begin{defn}
A cone stack ${\frak C}$ over $X$ is called {\em abelian}, if, locally in
$X$, one can find presentations $C\rightarrow{\frak C}$, where $C$ is an
abelian cone. A cone stack is a {\em vector bundle stack}, if one can
find such local presentations such that $C$ is a vector bundle. If
${\frak C}$ is abelian (a vector bundle stack), then for every local
presentation $C\rightarrow{\frak C}$ the cone $C$ will be abelian (a vector
bundle).
\end{defn}
\begin{prop}
Every cone stack is a closed subcone stack of an abelian cone
stack. There exists a universal such abelian cone stack. It is called
the {\em associated abelian cone stack }or the {\em abelian hull}.
\end{prop}
\begin{pf}
Just glue the stacks obtained from the abelian hulls of local
presentations.
\end{pf}
\begin{defn} \label{dics}
Let ${\frak E}$ be a vector bundle stack and ${\frak E}\to{\frak C}$ a morphism of cone
stacks. We say that ${\frak C}$ is an {\em ${\frak E}$-cone stack}, if ${\frak E}\to{\frak C}$
is locally isomorphic (as a morphism of cone stacks, i.e.\
${\Bbb A}^1$-equivariantly) to the morphism $[E_1/E_0]\to[C/F]$ coming from
a commutative diagram
\[\comdia{E_0}{}{F}{}{}{}{E_1}{}{C,}\]
where $C$ is both an $E_1$- and an $F$-cone.
\end{defn}
If ${\frak C}$ is an ${\frak E}$-cone stack, then there exists a natural morphism
${\frak E}\times{\frak C}\to{\frak C}$ coming from the action $E_1\times C\to C$ in a
local presentation of ${\frak E}\to{\frak C}$ as above. We call
${\frak E}\times{\frak C}\to{\frak C}$ the {\em action } of ${\frak E}$ on ${\frak C}$.
\begin{defn} \label{dsescs}
Let ${\frak E}\to{\frak C}\to{\frak D}$ be a sequence of morphisms of cone stacks, where
${\frak C}$ is an ${\frak E}$-cone stack. If
\begin{enumerate}
\item ${\frak C}\to{\frak D}$ is a smooth epimorphism,
\item the diagram
\[\comdia{{\frak E}\times{\frak C}}{\sigma}{{\frak C}}{p}{}{}{{\frak C}}{}{{\frak D}}\]
(where $p$ is the projection and $\sigma$ the action) is cartesian,
\end{enumerate}
we call $0\to{\frak E}\to{\frak C}\to{\frak D}\to 0$ a {\em short exact sequence }of
cone stacks. Note that this is equivalent to ${\frak C}$ being locally
isomorphic to ${\frak E}\times {\frak D}$.\end{defn}
\begin{prop}\label{csescs}
The sequence ${\frak E}\to{\frak C}\to{\frak D}$ of morphisms of cone stacks is exact if
and only if locally in $X$ there exist commutative diagrams
\[\begin{array}{ccccccccc}
0 & \longrightarrow & E_0 & \longrightarrow & F & \longrightarrow & G
& \longrightarrow & 0 \\
& &\downarrow & & \downarrow & & \downarrow & & \\
0 &\longrightarrow & E_1 & \longrightarrow & C & \longrightarrow & D &
\longrightarrow& 0,
\end{array}\]
where the top row is a short exact sequence of vector bundles and the
bottom row is a short exact sequence of cones, such that
${\frak E}\to{\frak C}\to{\frak D}$ is isomorphic to $[E_1/E_0]\to[C/F]\to[D/G]$.
\end{prop}
\begin{pf}
The statement is local on $X$. To prove the only if part we can assume
${\frak C}={\frak E}\times {\frak D}$, and then it's trivial. To prove the if part, note
that both short exact sequences are locally split.
\end{pf}
\section{Stacks of the Form $h^1/h^0$} \label{sohh}
\subsection{The General Theory}
We shall review here some aspects of the theory of Picard stacks
developed by Deligne in Section~1.4 of Expos\'e~XVIII in \cite{sga4}.
For the precise definition of Picard stack see [ibid.]. Roughly
speaking, a Picard stack is a stack together with an `addition'
operation, that is both associative and commutative. An example would
be the stack of torsors under a commutative group sheaf.
Let $X$ be a topos and $d:E^0\rightarrow E^1$ a homomorphism of
abelian sheaves on $X$, which we shall consider as a complex of
abelian sheaves on $X$. Via $d$, the abelian sheaf $E^0$ acts on $E^1$
and we may consider the stack-theoretic quotient of this action,
denoted
\[h^1/h^0(E^{\scriptscriptstyle\bullet})=[E^1/E^0],\]
which is a Picard stack on $X$. (See also [ibid.] 1.4.11, where
$h^1/h^0(E^{\scriptscriptstyle\bullet})$ is denoted $\mbox{ch}(E^{\scriptscriptstyle\bullet})$.) For an object
$U\in\mathop{\rm ob} X$ the groupoid $h^1/h^0(E^{\scriptscriptstyle\bullet})(U)$ of sections of
$h^1/h^0(E^{\scriptscriptstyle\bullet})$ over $U$ is the category of pairs $(P,f)$, where $P$
is an $E^0$-torsor (principal homogeneous $E^0$-bundle) over $U$ and
$f:P\rightarrow E^1{ \mid } U$ is an $E^0$-equivariant morphism of
sheaves on $U$.
Now if $d:F^0\rightarrow F^1$ is another homomorphism of abelian
sheaves on $X$ and $\phi:E^{\scriptscriptstyle\bullet}\rightarrow F^{\scriptscriptstyle\bullet}$ is a homomorphism of
homomorphisms (or in other words a homomorphism of complexes), then we
get an induced morphism of Picard stacks (an additive morphism in the
terminology of [ibid.])
\[h^1/h^0(\phi):h^1/h^0(E^{\scriptscriptstyle\bullet})\longrightarrow h^1/h^0(F^{\scriptscriptstyle\bullet}).\] For an object
$U\in\mathop{\rm ob} X$ the functor $h^1/h^0(\phi)(U)$ maps the pair $(P,f)$ to the pair
$(P\times_{E^0,\phi^0}F^0,\phi^1(f))$, where $\phi^1(f)$ denotes the map
\begin{eqnarray*}
\phi^1(f):P\times_{E^0}F^0 & \longrightarrow & F^1 \\ {[p,\nu]} & \longmapsto &
\phi^1(f(p))+d(\nu).
\end{eqnarray*}
Now, if $\psi:E^{\scriptscriptstyle\bullet}\rightarrow F^{\scriptscriptstyle\bullet}$ is another homomorphism of complexes and
$k:\phi\rightarrow\psi$ is a homotopy, i.e.\ a homomorphism of abelian sheaves
$k:E^1\rightarrow F^0$, such that
\begin{enumerate}
\item $kd=\psi^0-\phi^0$,
\item $dk=\psi^1-\phi^1$,
\end{enumerate} then we get an induced isomorphism
$\theta:h^1/h^0(\phi)\rightarrow h^1/h^0(\psi)$ of morphisms of Picard
stacks from $h^1/h^0(E^{\scriptscriptstyle\bullet})$ to $h^1/h^0(F^{\scriptscriptstyle\bullet})$. If $U\in\mathop{\rm ob} X$ is an
object, then $\theta(U)$ is a natural transformation of functors from
$h^1/h^0(\phi)(U)$ to $h^1/h^0(\psi)(U)$. For an object $(P,f)$ of
$h^1/h^0(E^{\scriptscriptstyle\bullet})(U)$ the morphism $\theta(U)(P,f)$ is a morphism from
$h^1/h^0(\phi)(U)(P,f)$ to $h^1/h^0(\psi)(U)(P,f)$ in the category
$h^1/h^0(F^{\scriptscriptstyle\bullet})(U)$. In fact, $\theta(U)(P,f)$ is the isomorphism of
$F^0{ \mid } U$-torsors
\begin{eqnarray}
\theta(U)(P,f):P\times_{E^0,\phi^0}F^0 & \longrightarrow &
P\times_{E^0,\psi^0}F^0\\ {[p,\nu]} & \longmapsto & [p,kf(p)+\nu],\nonumber
\end{eqnarray} such that the diagram of $F^0{ \mid } U$-sheaves
\[\inversecomtri{P\times_{E^0,\phi^0}F^0}{\theta(U)(P,f)}{\phi^1(f)
}{P\times_{E^0,\psi^0}F^0}
{\psi^1(f)}{F^1}\] commutes.
\begin{prop} \label{tpifs}
Let $\phi:E^{\scriptscriptstyle\bullet}\rightarrow F^{\scriptscriptstyle\bullet}$ be a homomorphism of homomorphisms
of abelian sheaves on $X$, as above. If $\phi$ induces isomorphisms on
kernels and cokernels (i.e.\ if $\phi$ is a quasi-isomorphism), then
$h^1/h^0(\phi):h^1/h^0(E^{\scriptscriptstyle\bullet})\rightarrow h^1/h^0(F^{\scriptscriptstyle\bullet})$ is an
isomorphism of Picard stacks over $X$.
\end{prop}
\begin{pf}
First let us treat the case that $\phi$ is a homotopy equivalence.
Then, in fact, any homotopy inverse of $\phi$ will provide an inverse
to $h^1/h^0(\phi)$, by the above remarks.
As a second case, let us assume that $\phi^{\scriptscriptstyle\bullet}:E^{\scriptscriptstyle\bullet}\rightarrow F^{\scriptscriptstyle\bullet}$ is an
epimorphism (i.e.\
$\phi^0$ and $\phi^1$ are epimorphisms). In this case $E^1\rightarrow[F^1/F^0]$
is an epimorphism, so for $[E^1/E^0]$ to be isomorphic to $[F^1/F^0]$, it is
necessary and sufficient that
\[\comdia{E^0\times E^1}{d+\mathop{\rm id}}{E^1}{\mathop{\rm pr}\nolimits}{}{}{E^1}{}{[F^1/F^0]}\] be cartesian.
This quickly reduces to proving that
\[\comdia{E^1\times E^0}{}{E^1}{}{}{}{E^1\times F^0}{}{F^1}\] is cartesian,
which, in turn, is equivalent to
\[\comdia{E^0}{}{E^1}{}{}{}{F^0}{}{F^1}\] being cartesian, which is a
consequence of the assumptions.
Finally, let us note that a general $\phi$ factors as a homotopy equivalence
followed by an epimorphism. To see this consider $E^{\scriptscriptstyle\bullet}\oplus F^0$, which is
homotopy equivalent to $E^{\scriptscriptstyle\bullet}$. Define a homomorphism $\psi:E^{\scriptscriptstyle\bullet}\oplus
F^0\rightarrow F^{\scriptscriptstyle\bullet}$ by $\psi^0(\nu,\mu)=\phi^0(\nu)+\mu$ and
$\psi^1(\chi,\mu)=\phi^1(\chi)+d(\mu)$. Then $\psi$ is surjective and
$\phi=\psi\mathbin{{\scriptstyle\circ}} i$, where
$i:E^{\scriptscriptstyle\bullet}\rightarrow E^{\scriptscriptstyle\bullet}\oplus F^0$ is given by $i=\mathop{\rm id}\oplus 0$.
\end{pf}
If $E^{\scriptscriptstyle\bullet}$ is a complex of arbitrary length of abelian sheaves on $X$, let
\begin{eqnarray*} Z^i(E^{\scriptscriptstyle\bullet}) & = & \ker(E^i\rightarrow E^{i+1}) \\ C^i(E^{\scriptscriptstyle\bullet}) &
= & \mathop{\rm cok}(E^{i-1}\rightarrow E^i).
\end{eqnarray*} The complex $E^{\scriptscriptstyle\bullet}$ induces a homomorphism
\[\tau_{[0,1]} E^{\scriptscriptstyle\bullet}=[C^0(E^{\scriptscriptstyle\bullet})\rightarrow Z^1(E^{\scriptscriptstyle\bullet})]\] and we let
$h^1/h^0(E^{\scriptscriptstyle\bullet})=h^1/h^0(\tau_{[0,1]}E^{\scriptscriptstyle\bullet})$.
Now let $\O_X$ be a sheaf of rings on $X$ and $C(\O_X)$, $K(\O_X)$ and
$D(\O_X)$
the category of complexes of $\O_X$-modules, the category of complexes of
$\O_X$-modules up to homotopy and the derived category of the category
$\mathop{\rm Mod}\nolimits(\O_X)$ of $\O_X$-modules, respectively. Let
$\phi:E^{\scriptscriptstyle\bullet}\rightarrow F^{\scriptscriptstyle\bullet}$ be a morphism in $D(\O_X)$. Let
\[\begin{array}{ccc} H^{\scriptscriptstyle\bullet} & \stackrel{\psi}{\longrightarrow} & F^{\scriptscriptstyle\bullet} \\
\ldiag{\alpha} & & \\ E^{\scriptscriptstyle\bullet} & &
\end{array}\] be a diagram in $C(\O_X)$ giving rise to $\phi$, where $\alpha$
is a quasi-isomorphism. We get an induced diagram of Picard stacks
\[\begin{array}{ccc} h^1/h^0(H^{\scriptscriptstyle\bullet}) & \stackrel{h^1/h^0(\psi)}{\longrightarrow}
& h^1/h^0(F^{\scriptscriptstyle\bullet}) \\
\ldiag{h^1/h^0(\alpha)} & & \\ h^1/h^0(E^{\scriptscriptstyle\bullet}), & &
\end{array}\] where $h^1/h^0(\alpha)$ is an isomorphism by
Proposition~\ref{tpifs}. Choosing an inverse of
$h^1/h^0(\alpha)$ induces a morphism
\[h^1/h^0(E^{\scriptscriptstyle\bullet})\longrightarrow h^1/h^0(F^{\scriptscriptstyle\bullet}).\] One checks that different
choices of $(\alpha, H^{\scriptscriptstyle\bullet},\psi)$ and $h^1/h^0(\alpha)^{-1}$ give rise
to isomorphic morphisms $h^1/h^0(E^{\scriptscriptstyle\bullet})\rightarrow
h^1/h^0(F^{\scriptscriptstyle\bullet})$. This proves in particular that if $E^{\scriptscriptstyle\bullet}$ and $F^{\scriptscriptstyle\bullet}$
are isomorphic in $D(\O_X)$, then the Picard $X$-stacks
$h^1/h^0(E^{\scriptscriptstyle\bullet})$ and $h^1/h^0(F^{\scriptscriptstyle\bullet})$ are isomorphic.
\begin{example}
If $d:E^0\rightarrow E^1$ is a monomorphism then
$h^1/h^0(E^{\scriptscriptstyle\bullet})=\mathop{\rm cok}(d)$ is a sheaf over $X$.
If $d:E^0\rightarrow E^1$ is an epimorphism then
$h^1/h^0(E^{\scriptscriptstyle\bullet})=B\ker(d)$ is a gerbe over $X$.
\end{example}
\begin{lem} \label{lmh}
1. Let $\phi,\psi:E^{\scriptscriptstyle\bullet}\to F^{\scriptscriptstyle\bullet}$ be two morphisms in $D(\O_X)$. Then,
if for some choice of $h^1/h^0(\phi)$ and $h^1/h^0(\psi)$ we have
$h^1/h^0(\phi)\cong h^1/h^0(\psi)$ as morphisms of Picard stacks, then
$\phi=\psi$.
2. Let $0(E,F)$ be the zero morphism $0(E,F):h^1/h^0(E^{\scriptscriptstyle\bullet})\to
h^1/h^0(F^{\scriptscriptstyle\bullet})$. Then $\mathop{\rm Aut}\nolimits(0(E,F))=\mathop{\rm Hom}\nolimits^{-1}_{D(\O_X)}(E^{\scriptscriptstyle\bullet},F^{\scriptscriptstyle\bullet})$.
\end{lem}
\begin{pf}
These are similar to Lemma~\ref{lo2i}. See also [ibid.].
\end{pf}
\subsection{Application to Schemes}
Let $X$ be a Deligne-Mumford stack. Consider the morphism of topoi
\[v:X_{\mbox{\tiny fl}}\longrightarrow X_{\mbox{\tiny \'{e}t}}.\] The functor $v_{\ast}$ restricts a sheaf on the
big fppf-site to the small \'etale site and its left adjoint $v^{-1}$ extends
the embedding of the \'etale site into the flat site.
Let $\O_{X_{\mbox{\tiny fl}}}$ and $\O_{X_{\mbox{\tiny \'{e}t}}}$ denote the sheaves of rings induced by $\O_X$
on $X_{\mbox{\tiny fl}}$ and
$X_{\mbox{\tiny \'{e}t}}$, respectively. There is a canonical morphism of sheaves of rings
$v^{-1}\O_{X_{\mbox{\tiny \'{e}t}}}\rightarrow\O_{X_{\mbox{\tiny fl}}}$, so that we have a morphism of ringed
topoi
$$v:(X_{\mbox{\tiny fl}},\O_{X_{\mbox{\tiny fl}}})\rightarrow (X_{{\mbox{\tiny \'{e}t}}},\O_{X_{\mbox{\tiny \'{e}t}}}).$$ The induced functor
{}from $\mathop{\rm Mod}\nolimits(\O_{X_{\mbox{\tiny \'{e}t}}})$ to $\mathop{\rm Mod}\nolimits(\O_{X_{\mbox{\tiny fl}}})$ will be denoted by $v^{\ast}$:
\[v^{\ast}(M)=v^{-1}M\otimes_{v^{-1}\O_{X_{\mbox{\tiny \'{e}t}}}}\O_{X_{\mbox{\tiny fl}}}.\] Since
$\mathop{\rm Mod}\nolimits(\O_{X_{\mbox{\tiny \'{e}t}}})$ has enough flat modules we may derive the right exact
functor
$v^{\ast}$ to get the functor $Lv^{\ast}:D^-(\O_{X_{\mbox{\tiny \'{e}t}}})\rightarrow
D^-(\O_{X_{\mbox{\tiny fl}}})$. To abbreviate notation, we write $M_{\mbox{\tiny fl}}^{\scriptscriptstyle\bullet}=Lv^{\ast} M^{\scriptscriptstyle\bullet}$
for
$M^{\scriptscriptstyle\bullet}\in\mathop{\rm ob} D^-(\O_{X_{\mbox{\tiny \'{e}t}}})$.
We shall also need to consider the functor
\[R\mathop{\rm {\mit{ \hH\! om}}}\nolimits({{\,\cdot\,}},\O_{X_{\mbox{\tiny fl}}}):D^-(\O_{X_{\mbox{\tiny fl}}})\longrightarrow
D^+(\O_{X_{\mbox{\tiny fl}}}).\] It is defined using an injective resolution
$\O_{X_{\mbox{\tiny fl}}}\stackrel{\sim}{\rightarrow}{\cal I}^{\scriptscriptstyle\bullet}$ of $\O_{X_{\mbox{\tiny fl}}}$, i.e.\
\[R\mathop{\rm {\mit{ \hH\! om}}}\nolimits(M^{\scriptscriptstyle\bullet},\O_{X_{\mbox{\tiny fl}}})=\mathop{\rm tot}\mathop{\rm {\mit{ \hH\! om}}}\nolimits(M^{\scriptscriptstyle\bullet},{\cal I}^{\scriptscriptstyle\bullet}),\] but if $M^{\scriptscriptstyle\bullet}$
happens to have a projective resolution ${\cal P}^{\scriptscriptstyle\bullet}\stackrel{\sim}{\rightarrow} M^{\scriptscriptstyle\bullet}$, then we have
\[R\mathop{\rm {\mit{ \hH\! om}}}\nolimits(M^{\scriptscriptstyle\bullet},\O_{X_{\mbox{\tiny fl}}})\cong\mathop{\rm {\mit{ \hH\! om}}}\nolimits({\cal P}^{\scriptscriptstyle\bullet},\O_{X{\mbox{\tiny fl}}}).\] We shall
abbreviate notation by writing
\[\dual{M^{\scriptscriptstyle\bullet}}=R\mathop{\rm {\mit{ \hH\! om}}}\nolimits(M^{\scriptscriptstyle\bullet},\O_{X_{\mbox{\tiny fl}}}).\]
We will be interested in the stack $h^1/h^0(\dual{(M^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})$ associated to
an object
$M^{\scriptscriptstyle\bullet}\in\mathop{\rm ob} D^-(\O_{X_{\mbox{\tiny \'{e}t}}})$. Note that for such $M^{\scriptscriptstyle\bullet}\in\mathop{\rm ob} D^-(\O_{X_{\mbox{\tiny \'{e}t}}})$
we have
\[h^1/h^0(\dual{(M^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})\cong h^1/h^0(\dual{(\tau_{\geq-1}M^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})}).\]
\begin{defn} \label{dost} We say that an object $L^{\scriptscriptstyle\bullet}$ of
$D(\O_{X_{\mbox{\tiny \'{e}t}}})$ {\em satisfies Condition~}($\star$) if
\begin{enumerate}
\item $h^i(L^{\scriptscriptstyle\bullet})=0$ for all $i>0$,
\item $h^i(L^{\scriptscriptstyle\bullet})$ is coherent, for $i=0,-1$.
\end{enumerate}
\end{defn}
\begin{prop} \label{has} Let $L^{\scriptscriptstyle\bullet}\in\mathop{\rm ob} D(\O_{X_{\mbox{\tiny \'{e}t}}})$ satisfy
Condition~{\rm (}$\star${\rm )}. Then the $X$-stack
$h^1/h^0(\dual{(L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})$ is an algebraic $X$-stack, in fact an abelian
cone
stack over $X$. Moreover, if $L^{\scriptscriptstyle\bullet}$ is of perfect amplitude contained in
$[-1,0]$, then
$h^1/h^0(\dual{(L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})$ is a vector bundle stack.
\end{prop}
\begin{pf} The claim is local in $X$ (with respect to the \'etale topology), so
we may assume that $L^{\scriptscriptstyle\bullet}$ has a free resolution, or that $L^{\scriptscriptstyle\bullet}$ itself
consists of free $\O_X$-modules. We may also assume that
$L^i=0$, for all $i>0$ and that $L^0$ and $L^{-1}$ have finite rank. Then
$L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}}$ is given by
$L^{\scriptscriptstyle\bullet}$ itself, since a free sheaf is flat, and $\dual{(L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})}$ is given
by
$\dual{L}^{\scriptscriptstyle\bullet}$, taking duals component-wise, since a free module is projective.
Thus
\[h^1/h^0(\dual{(L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})=[Z^1(\dual{L}^{\scriptscriptstyle\bullet})/\dual{L}^0],\] which is the
cone stack given by the homomorphism of abelian cones $\dual{L}^0\rightarrow
Z^1(\dual{L}^{\scriptscriptstyle\bullet})=C(C^{-1}(L^{\scriptscriptstyle\bullet}))$.
If $L^{\scriptscriptstyle\bullet}$ is of perfect amplitude contained in $[-1,0]$, then we may assume
that in addition to the above assumptions $L^i=0$, for all $i\leq-2$. Then
$Z^1(\dual{L}^{\scriptscriptstyle\bullet})=\dual{L}^1$ is a vector bundle.
\end{pf}
So if $\phi:E^{\scriptscriptstyle\bullet}\rightarrow L^{\scriptscriptstyle\bullet}$ is a homomorphism in $D(\O_{X_{\mbox{\tiny \'{e}t}}})$, where
$E^{\scriptscriptstyle\bullet}$ and
$L^{\scriptscriptstyle\bullet}$ satisfy ($\star$), then we get an induced morphism of algebraic stacks
\[\dual{\phi}:h^1/h^0(\dual{(L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})\longrightarrow
h^1/h^0(\dual{(E^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})}).\]
\begin{prop}\label{rmcs}
The morphism $\dual{\phi}$ is a morphism of abelian cone stacks.
Moreover, $h^0(\phi)$ is surjective, if and only if $\dual{\phi}$ is
representable.
\end{prop}
\begin{pf}
The fact that $\dual{\phi}$ is a morphism of abelian cone stacks is
immediate from the definition. The second question is local in $X$, so
we may assume that $E^{\scriptscriptstyle\bullet}$ and $L^{\scriptscriptstyle\bullet}$ are complexes of free
$\O_X$-modules and that $E^i=L^i=0$, for $i>0$, and that $L^0$,
$L^{-1}$, $E^0$ and $E^{-1}$ are of finite rank. Consider the
commutative diagram
\[\comdia{C^{-1}(E^{\scriptscriptstyle\bullet})}{}{E^0}{}{}{}{C^{-1}(L^{\scriptscriptstyle\bullet})}{}{L^0}\] of coherent
sheaves on $X$. Let
$F$ be the fibered product
\[\comdia{F}{}{E^0}{}{}{}{C^{-1}(L^{\scriptscriptstyle\bullet})}{}{L^0.}\] The fact that $h^0(\phi)$ is
surjective, is equivalent to saying that the sequence
\[\ses{F}{}{E^0\oplus C^{-1}(L^{\scriptscriptstyle\bullet})}{}{L^0}\]
is exact. Since $L^0$ is free, we get an
induced exact sequence of cones
\[\ses{\dual{L}^0}{}{\dual{E}^0\oplus Z^1(\dual{L}^{\scriptscriptstyle\bullet})}{}{C(F)}.\] Hence
by Proposition~\ref{cics} we have
\[[Z^1(\dual{L}^{\scriptscriptstyle\bullet})/\dual{L}^0]\cong[C(F)/\dual{E}^0].\]
In particular the diagram
\[\comdia{C(F)}{}{Z^1(\dual{E}^{\scriptscriptstyle\bullet})}{}{}{}{h^1/h^0(\dual{(L^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})}{}
{h^1/h^0(\dual{(E^{\scriptscriptstyle\bullet}_{\mbox{\tiny fl}})})}\] is cartesian, hence $\dual{\phi}$ is
representable.
For the converse, note that $\dual{\phi}$ representable implies that
$\dual{L}^0\to\dual{E}^0\times Z^1(\dual{L}^{\scriptscriptstyle\bullet})$ is a closed
immersion, which implies that $E^0\oplus C^{-1}(L^{\scriptscriptstyle\bullet})\to L^0$ is an
epimorphism.
\end{pf}
\begin{prop}\label{isoiso}
The morphism $\dual{\phi}$ is a closed immersion if and only if
$h^0(\phi)$ is an isomorphism and $h^{-1}(\phi)$ is surjective.
Moreover, $\dual{\phi}$ is an isomorphism if and only if $h^0(\phi)$
and $h^{-1}(\phi)$ are.
\end{prop} \begin{pf}
Following the previous argument, $\dual{\phi}$ is a closed immersion
if and only if $C(F)\to Z^1(\dual{E}^{\scriptscriptstyle\bullet})$ is. This is equivalent to
$C^{-1}(E^{\scriptscriptstyle\bullet})\to F$ being surjective. A simple diagram chase shows
that this is equivalent to $h^0(\phi)$ being an isomorphism and
$h^{-1}(\phi)$ being surjective. The `moreover' follows similarly.
\end{pf}
\begin{prop} \label{dtscs}
Let
\[\dt{E^{\scriptscriptstyle\bullet}}{}{F^{\scriptscriptstyle\bullet}}{}{G^{\scriptscriptstyle\bullet}}{}\]
be a distinguished triangle in $D(\O_{X_{\mbox{\tiny \'{e}t}}})$, where $E^{\scriptscriptstyle\bullet}$ and
$F^{\scriptscriptstyle\bullet}$ satisfy ($\star$) and $G^{\scriptscriptstyle\bullet}$ is of perfect amplitude
contained in $[-1,0]$. Then the induced sequence
\[h^1/h^0(\dual{G})\longrightarrow h^1/h^0(\dual{F})\longrightarrow
h^1/h^0(\dual{E})\]
is a short exact sequence of abelian cone stacks over $X$.
\end{prop}
\begin{pf} The question is local, so assume that
$E^i$ and $F^i$ are $0$ for $i>0$ and vector bundles for $i=0,-1$,
and that
$G^i=F^i\oplus E^{i+1}$. We have to prove that $$
\ses {[Z^1(\dual{G})/\dual{G}^0]} {} {[Z^1(\dual{F})/\dual{F}^0]}{}
{[Z^1(\dual{E})/\dual{E}^0]}$$
is a short exact sequence of cone stacks.
By Proposition \ref{csescs}, it is enough to prove that the exact sequence of
sheaves
$$
\ses {C^{-1}(E^{\scriptscriptstyle\bullet})} {} {C^{-1}(F^{\scriptscriptstyle\bullet})\oplus E^0}{} {C^{-1}(G^{\scriptscriptstyle\bullet})}$$
is exact. This is then a straightforward verification.
\end{pf}
\section{The Intrinsic Normal Cone} \label{stinc}
\subsection{Normal Cones}
Normal cones have the following functorial property. Consider a commutative
diagram of (arbitrary) algebraic $k$-stacks
\begin{equation} \label{cdci}
\comdia{X'}{j}{Y'}{u}{}{v}{X}{i}{Y,}
\end{equation} where $i$ and $j$ are local immersions. Then there is
a natural morphism of cones over $X'$
\[\alpha:C_{X'/Y'}\longrightarrow u^{\ast} C_{X/Y}.\] If (\ref{cdci}) is
cartesian, then $\alpha$ is a closed immersion. If, moreover, $v$ is
flat, then $\alpha$ is an isomorphism.
\begin{prop} \label{lonc} Consider a commutative diagram of Deligne-Mumford
stacks
\[\comtri{X}{i'}{Y'}{i}{f}{Y,}\] where $i$ and $i'$ are local immersions
and
$f$ is smooth. Then the sequence of morphisms of cones over $X$
\begin{equation} \label{sesc} {i'}^{\ast}
T_{Y'/Y}\stackrel{\beta}{\longrightarrow}C_{X/Y'}\stackrel{
\alpha}{\longrightarrow}C_{X/Y},
\end{equation}
where the maps $\alpha$ and $\beta$ are the natural ones, is exact.
\end{prop}
\begin{pf} The question is local, so we can assume that $X$, $Y$ and $Y'$ are
schemes and that $i'$ and $i$ are immersions. This is then Example 4.2.6 in
\cite{fulton}.
\end{pf}
\begin{lem}\label{nci}
Let
\[U \stackrel{f}{\longrightarrow} M \] be a local immersion of affine
$k$-schemes of finite type, where $M$ is smooth over $k$. Then the normal cone
$C_{U/M}\hookrightarrow N_{U/M}$ is invariant under the action of $f^{\ast} T_M$
on
$N_{U/M}$. In other words, $C_{U/M}$ is an $f^{\ast} T_M$-cone.
\end{lem}
\begin{pf}
Let $p_i:M\times M\rightarrow M$, $i=1,2$, be the two projections.
Each one gives rise to a commutative diagram
\[\comtri{U}{\Delta f}{M\times M}{f}{p_i}{M,}\]
and hence to an exact sequence
\[\ses{f^{\ast} T_M}{j_i}{N_{U/M\times M}}{{p_i}_{\ast}}{N_{U/M}}\]
of abelian cones on $U$.
The diagonal gives rise to the commutative diagram
\[\comtri{U}{f}{M}{\Delta f}{\Delta}{M\times M}\]
and hence to a homomorphism
\[N_{U/M}\stackrel{s}{\longrightarrow} N_{U/M\times M}\]
of abelian cones on $U$.
Now $s$ is a section of both ${p_1}_{\ast}$ and ${p_2}_{\ast}$. Using
$(j_1,{p_1}_{\ast})$ we make the identification
\begin{equation} \label{nci1}
N_{U/M \times M}=f^{\ast} T_M\times N_{U/M}.
\end{equation}
Then ${p_2}_{\ast}$ is identified with the action of $f^{\ast} T_M$ on
$N_{U/M}$. Since the same functorialities of normal sheaves used so
far are enjoyed by normal cones, we get that under the
identification~(\ref{nci1}) the subcone $C_{U/M\times M}\subset
N_{U/M\times M}$ corresponds to $f^{\ast} T_M\times C_{U/M}$ and the
action ${p_2}_{\ast}:f^{\ast} T_M\times N_{U/M}\rightarrow N_{U/M}$
restricts to ${p_2}_{\ast}:f^{\ast} T_M\times C_{U/M}\rightarrow C_{U/M}$.
\end{pf}
The following is not used until Section~\ref{otafc}.
Consider the diagram~(\ref{cdci}), assume it is cartesian and assume
that $v$ is a regular local immersion. Assume also that $Y$ is smooth
of constant dimension. Let $C=C_{X/Y}$ and $N=N_{Y'/Y}$. Then we get
an induced cartesian diagram
\begin{equation} \label{bdnc}
\begin{array}{ccccc}
N\times_YC & \longrightarrow & u^{\ast} C & \longrightarrow & C \\
\ldiag{} & & \ldiag{} & & \rdiag{} \\
j^{\ast} N & \longrightarrow & X' & \stackrel{u}{\longrightarrow} & X \\
\ldiag{} & & \ldiag{j} & & \rdiag{i} \\
N & \stackrel{\rho}{\longrightarrow} & Y' &
\stackrel{v}{\longrightarrow} & Y.
\end{array}
\end{equation}
If $Y$ is a scheme, Vistoli constructed in \cite{vistoli} a canonical
rational equivalence $\beta(Y',X)\in W_{\ast}(N\times_YC)$ such that
\[\partial\beta(Y',X)=[C_{u^{\ast} C/C}]-[\rho^{\ast} C_{X'/Y'}].\]
\begin{note}
Let $0:u^{\ast} C\rightarrow N\times_YC$ be the zero section. Then
$$0^{!}[C_{u^{\ast} C/C}]=v^{!}[C]\in A_{\ast}(u^{\ast} C),$$ by the
definition of $v^{!}$. On the other hand, $$0^{!}[\rho^{\ast}
C_{X'/Y'}]=0^{!}\rho^{!}[C_{X'/Y'}]=[C_{X'/Y'}]\in A_{\ast}(u^{\ast}
C).$$ So the existence of Vistoli's rational equivalence implies that
$$v^{!}[C]=[C_{X'/Y'}].$$
\end{note}
\begin{prop} \label{vrqsb}
Vistoli's rational equivalence commutes with any smooth base change
$\phi:Y_1\rightarrow Y$. More precisely, if we denote by a subscript
$({{\,\cdot\,}})_1$ the base change via $\phi$ of any object in
(\ref{bdnc}), then
$$\phi^{\ast}\beta(Y',X)=\beta(Y'_1,X_1)\in W_{\ast}(N_1\times_{Y_1}C_1).$$
\end{prop}
\begin{pf}
If $\phi$ is \'etale, this is Lemma 4.6(ii) in \cite{vistoli}. Vistoli's proof
is based on the fact that the following commute with \'etale base change:
blowing up a scheme along a closed subscheme; normalization; order of
a Cartier divisor along an irreducible Weil divisor on a reduced,
equidimensional
scheme. But all these operations do in fact commute with smooth base change.
\end{pf}
A first consequence of this proposition is that we may drop the
assumption that $Y$ be a scheme. We get $\beta(Y',X)\in
W_{\ast}(N\times_YC)$ for any situation~(\ref{bdnc}). The consequence
$v^{!}[C]=[C_{X'/Y'}]$ holds if $Y$ (and hence all other stacks in
(\ref{bdnc})) is of Deligne-Mumford type.
Now let us assume that $i:X\rightarrow Y$ factors as
\[\comtri{X}{\tilde{\imath}}{\tilde{Y}}{i}{\pi}{Y,}\]
where $\tilde{\imath}$ is another local immersion and $\pi$ is of
relative Deligne-Mumford type (i.e.\ has unramified diagonal) and is
smooth of constant fiber dimension. Then we construct the cartesian
diagram
\[\comdia{\tilde{Y}'}{\tilde{v}}{\tilde{Y}}{}{}{\pi}{Y'}{v}{Y}\]
and over
\[\comdia{X'}{u}{X}{\tilde{\jmath}}{}{\tilde{\imath}}{\tilde{Y}'}{\tilde{u}}
{\tilde{Y}}\]
we construct the analogue of (\ref{bdnc}):
\begin{equation} \label{bdnct}
\begin{array}{ccccc}
N\times_Y\tilde{C} & \longrightarrow & u^{\ast} \tilde{C} &
\longrightarrow & \tilde{C} \\
\ldiag{} & & \ldiag{} & & \rdiag{} \\
j^{\ast} N & \longrightarrow & X' & \stackrel{u}{\longrightarrow} & X \\
\ldiag{} & & \ldiag{\tilde{\jmath}} & & \rdiag{\tilde{\imath}} \\
\pi^{\ast} N & \stackrel{\tilde{\rho}}{\longrightarrow} & \tilde{Y}' &
\stackrel{\tilde{v}}{\longrightarrow} & \tilde{Y},
\end{array}
\end{equation}
i.e.\ $\tilde{C}=C_{X/\tilde{Y}}$. Diagrams~(\ref{bdnc})
and~(\ref{bdnct}) may be fused into one large diagram
\begin{equation} \label{bdnctt}
\begin{array}{ccccc}
N\times_Y\tilde{C} & \longrightarrow & u^{\ast} \tilde{C} &
\longrightarrow & \tilde{C} \\
\ldiag{} & & \ldiag{} & & \rdiag{\alpha} \\
N\times_YC & \longrightarrow & u^{\ast} C & \longrightarrow & C \\
\ldiag{} & & \ldiag{} & & \rdiag{} \\
j^{\ast} N & \longrightarrow & X' & \stackrel{u}{\longrightarrow} & X \\
\ldiag{} & & \ldiag{\tilde{\jmath}} & & \rdiag{\tilde{\imath}} \\
\pi^{\ast} N & \stackrel{\tilde{\rho}}{\longrightarrow} & \tilde{Y}' &
\stackrel{\tilde{v}}{\longrightarrow} & \tilde{Y} \\
\ldiag{} & & \ldiag{} & & \rdiag{\pi} \\
N & \stackrel{\rho}{\longrightarrow} & Y' &
\stackrel{v}{\longrightarrow} & Y.
\end{array}
\end{equation}
By Proposition~\ref{lonc} the morphism $\tilde{C}\rightarrow C$ is a
$T_{\tilde{Y}/Y}\times_{\tilde{Y}}C$-bundle.
\begin{prop} \label{vcgt}
We have $\alpha^{\ast}(\beta(Y',X))=\beta(\tilde{Y}',X)$ in
$W_{\ast}(N\times_Y\tilde{C})$.
\end{prop}
\begin{pf}
By the compatibilities of $\beta$ proved in \cite{vistoli} we reduce
to the case that $\tilde{Y}={\Bbb A}^n_Y$, $\pi:{\Bbb A}^n_Y\rightarrow Y$ is a
relative affine $n$-space and $\tilde{\imath}:Y\to{\Bbb A}^n_Y$ is the zero
section. Then one checks that Vistoli's construction commutes with
$\pi$.
\end{pf}
\begin{prop} \label{vrqi}
In the situation of Diagram~(\ref{bdnc}) assume that $Y$ is of
Deligne-Mumford type. Vistoli's rational equivalence $\beta(Y',X)\in
W_{\ast}(N\times_YC)$ is invariant under the natural action of $j^{\ast}
N\times_Y T_Y$ on $N\times_YC$.
\end{prop}
\begin{pf}
The vector bundle $i^{\ast} T_Y$ acts on the $X$-cone $C$ by
Lemma~\ref{nci}. Pulling back from $X$ to $j^{\ast} N$ gives the natural
action of $j^{\ast} N\times_Y T_Y$ on $N\times_YC$. Using the
construction of the proof of Lemma~\ref{nci} the claim follows from
Proposition~\ref{vcgt} applied to $\tilde{Y}=Y\times Y$ and
$\tilde{\imath}=\Delta\mathbin{{\scriptstyle\circ}} i:X\to Y\times Y$.
\end{pf}
\subsection{The Intrinsic Normal Cone}
Let $X$ be a Deligne-Mumford stack, locally of finite type over $k$. Let
$L_X^{\scriptscriptstyle\bullet}$ be the cotangent complex of $X$ relative to $k$. Then $L_X^{\scriptscriptstyle\bullet}\in\mathop{\rm ob}
D(\O_{X{\mbox{\tiny \'{e}t}}})$ and $L_X^{\scriptscriptstyle\bullet}$ satisfies ($\star$).
\begin{defn}
We denote the algebraic stack
$h^1/h^0(\dual{((L_X^{\scriptscriptstyle\bullet})_{\mbox{\tiny fl}})})$ by ${\frak N}_X$ and call it the {\em
intrinsic normal sheaf }of $X$.
\end{defn}
We shall now construct the intrinsic normal cone as a closed subcone
stack of ${\frak N}_X$.
\begin{defn} A {\em local embedding }of $X$ is a diagram
\[\begin{array}{ccc} U & \stackrel{f}{\longrightarrow} & M \\
\ldiag{i} & & \\ X & &\quad,
\end{array}\] where
\begin{enumerate}
\item $U$ is an affine $k$-scheme of finite type,
\item $i:U\rightarrow X$ is an \'etale morphism,
\item $M$ is a smooth affine $k$-scheme of finite type,
\item $f:U\rightarrow M$ is a local immersion.
\end{enumerate} By abuse of language we call the pair $(U,M)$ a local embedding
of $X$.
A morphism of local embeddings $\phi:(U',M')\rightarrow(U,M)$ is a pair of
morphisms
$\phi_U:U'\rightarrow U$ and $\phi_M:M'\rightarrow M$ such that
\begin{enumerate}
\item $\phi_U$ is an \'etale $X$-morphism,
\item $\phi_M$ is a smooth morphism such that
\[\comdia{U'}{f'}{M'}{\phi_U}{}{\phi_M}{U}{f}{M}\] commutes.
\end{enumerate}
\end{defn}
If $(U',M')$ and $(U,M)$ are local embeddings of $X$, then
$(U'\times_X U,M'\times M)$ is naturally a local embedding of $X$
which we call the {\em product }of $(U',M')$ and $(U,M)$, even though
it may not be the direct product of $(U',M')$ and $(U,M)$ in the
category of local embeddings of $X$.
Let
\[\begin{array}{ccc} U & \stackrel{f}{\longrightarrow} & M \\
\ldiag{i} & & \\ X & &\quad
\end{array}\]
be a local embedding of $X$. Let $I/I^2$ be the conormal sheaf of $U$
in $M$. There is a natural homomorphism of coherent $\O_U$-modules
$I/I^2\rightarrow f^{\ast}\Omega_M$. Moreover, there exists a natural
homomorphism
\[\phi:L_X^{\scriptscriptstyle\bullet}{ \mid } U\longrightarrow[I/I^2\rightarrow f^{\ast}\Omega_M] \] in
$D(\O_{U_{\mbox{\tiny \'{e}t}}})$, where we think of $[I/I^2\rightarrow f^{\ast}\Omega_M]$
as a complex concentrated in degrees $-1$ and $0$. Moreover, $\phi$
induces an isomorphism on $h^{-1}$ and $h^0$ (see \cite{Ill},
Chapitre~III, Corollaire~3.1.3). Hence by Proposition~\ref{isoiso} we
get an induced isomorphism of cone stacks
\[\dual{\phi}:[N_{U/M}/f^{\ast} T_M]\longrightarrow i^{\ast} {\frak N}_X,\]
where $T_M$ is the tangent bundle of $M$ and $N_{U/M}$ is the normal
sheaf of the local embedding $f$. In other words, $N_{U/M}$ is a local
presentation of the abelian cone stack ${\frak N}_X$.
If $\chi:(U',M')\rightarrow(U,M)$ is a morphism of local embeddings we get an
induced commutative diagram
\[\comdia{I/I^2{ \mid } U'}{}{{f}^{\ast}\Omega_{M}{ \mid }
U'}{}{}{}{I'/I'^2}{ }{{f'}^{\ast}\Omega_{M'}\quad,}\] in other words a
homomorphism
\[\tilde{\chi}:[I/I^2\rightarrow{f}^{\ast}\Omega_{M}]{ \mid }
U'\longrightarrow[I'/I'^2\rightarrow{f'}^{\ast}\Omega_{M'}]
\quad.\] We have
$\tilde{\chi}\mathbin{{\scriptstyle\circ}}\phi{ \mid } U'=\phi'$ in $D(\O_{U'_{\mbox{\tiny \'{e}t}}})$, because of
the naturality of $\phi$. Thus the induced morphism
\[\dual{\tilde{\chi}}:[N_{U'/M'}/{f'}^{\ast}
T_{M'}]\longrightarrow[N_{U/M}/f^{\ast} T_M]{ \mid } U'
\] is compatible with the isomorphisms to
${\frak N}_X$. Note that, in particular,
$\dual{\tilde{\chi}}$ is an isomorphism of cone stacks over $U'$.
Recall Lemma~\ref{nci}. Let $\chi:(U',M')\rightarrow(U,M)$ be a morphism of
local embeddings. Then we get an induced morphism from the ${f'}^{\ast}
T_{M'}$-cone $C_{U'/M'}$ to the $f^{\ast} T_M{ \mid } U'$-cone $C_{U/M}{ \mid } U'$.
Note that the kernel of ${f'}^{\ast} T_{M'}\rightarrow f^{\ast} T_M{ \mid } U'$ is
${f'}^{\ast} T_{M'/M}$.
\begin{lem} The pair $(C_{U/M}\hookrightarrow N_{U/M}){ \mid } U'$ is the
quotient of $(C_{U'/M'}\hookrightarrow N_{U'/M'})$ by the action of ${f'}^{\ast}
T_{M'/M}$.
\end{lem}
\begin{pf} This follows immediately from Proposition~\ref{lonc}.
\end{pf}
\begin{cor} The isomorphism
\[\dual{\tilde{\chi}}:[N_{U'/M'}/{f'}^{\ast}
T_{M'}]\longrightarrow[N_{U/M}/f^{\ast} T_M]{ \mid } U'
\] identifies the closed subcone stack $[C_{U'/M'}/{f'}^{\ast} T_{M'}]$
with the closed subcone stack $[C_{U/M}/f^{\ast} T_M]{ \mid } U'$.
\end{cor}
By this corollary, there exists a unique closed subcone stack
${\frak C}_X\hookrightarrow {\frak N}_X$, such that for every local embedding
$(U,M)$ of $X$ we have ${\frak C}_X{ \mid } U=[C_{U/M}/f^{\ast} T_M]$, or in
other words that
\[\comdia{C_{U/M}}{}{N_{U/M}}{}{}{}{{\frak C}_X}{}{{\frak N}_X}\] is cartesian.
\begin{defn}
The cone stack ${\frak C}_X$ is called the {\em intrinsic normal cone }of
$X$.
\end{defn}
\begin{them}
The intrinsic normal cone ${\frak C}_X$ is of pure dimension zero. Its
abelian hull is ${\frak N}_X$.
\end{them}
\begin{pf} The second claim follows because the normal sheaf is the abelian
hull of the normal cone, for any local embedding. To prove the claim
about the dimension of ${\frak C}_X$, consider a local embedding $(U,M)$ of
$X$, giving rise to the local presentation $C_{U/M}$ of
${\frak C}_X$. Assume that $M$ is of pure dimension. We then have a
cartesian and cocartesian diagram of $U$-stacks
\[\comdia{f^{\ast} T_M\times C_{U/M}}{}{C_{U/M}}{}{}{}{C_{U/M}}{}{[C_{U/M}/f^{\ast}
T_M].}\] Thus ${C_{U/M}}{\rightarrow}{[C_{U/M}/f^{\ast} T_M]}$ is a
smooth epimorphism of relative dimension $\dim M$. So since $C_{U/M}$
is of pure dimension $\dim M$ (see \cite{fulton}, B.6.6) the stack
$[C_{U/M}/f^{\ast} T_M]$ has pure dimension $\dim M-\dim M=0$.
\end{pf}
\begin{rmk}
One may construct ${\frak N}_X$ by simply gluing the various stacks
$[N_{U/M}/f^{\ast} T_M]$, coming from the local embeddings of $X$. So
one doesn't need the construction preceding Proposition~\ref{has} to
define the intrinsic normal sheaf and the intrinsic normal cone. But
for objects $E^{\scriptscriptstyle\bullet}$ of $D^-(\O_{X_{\mbox{\tiny \'{e}t}}})$ satisfying ($\star$) other
than $L_X^{\scriptscriptstyle\bullet}$, we could not prove that such gluing works a
priori. The problem is, that in general one does not have such a nice
distinguished class of local resolutions of $E^{\scriptscriptstyle\bullet}$ (like the one
coming from local embeddings for $L_X^{\scriptscriptstyle\bullet}$). In general, local (free)
resolutions of $E^{\scriptscriptstyle\bullet}$ are only compatible up to homotopy.
\end{rmk}
\subsection{Basic Properties}
\begin{prop}[Local Complete Intersections] \label{colci}
The following are equivalent.
\begin{enumerate}
\item \label{colci1} $X$ is a local complete intersection,
\item \label{colci2} ${\frak C}_X$ is a vector bundle stack,
\item \label{colci3} ${\frak C}_X={\frak N}_X$.
\end{enumerate}
If, for example, $X$ is smooth, we have ${\frak C}_X={\frak N}_X=BT_X$.
\end{prop}
\begin{pf}
(\ref{colci1})$\Longrightarrow$(\ref{colci3}).
If $X$ is a local complete intersection, then local embeddings of $X$
are regular immersions, but for regular immersions normal cone and
normal sheaf coincide.
(\ref{colci3})$\Longrightarrow$(\ref{colci2}).
If for a local embedding normal cone and normal sheaf coincide, then it is a
regular immersion. Thus $X$ is a local complete intersection so that
${\frak N}_X$ is a vector bundle stack.
(\ref{colci2})$\Longrightarrow$(\ref{colci1}).
If ${\frak C}_X$ is a vector bundle stack it is equal to its abelian hull.
Hence ${\frak C}_X={\frak N}_X$ and $X$ is a local complete intersection.
\end{pf}
\begin{prop}[Products] \label{prod}
Let $X$ and $Y$ be Deligne-Mumford stacks of finite type over
$k$. Then
\[{\frak N}_{X\times Y}={\frak N}_{X}\times
{\frak N}_{Y}\] and \[{\frak C}_{X\times Y}={\frak C}_X\times {\frak C}_Y.\]
\end{prop}
\begin{pf} If $X\subset V$ and $Y\subset W$ are affine schemes, it is easy to
check that there is a natural isomorphism $C_{X/V}\times
C_{Y/W}\to C_{X\times Y/V\times W}$, compatible with \'etale base change; the
same is true if we replace the normal cone by the normal sheaf.
If $C$ is an $E$-cone and $D$ is an $F$-cone, then $C\times D$ is an $E\times
F$-cone and there is a canonical isomorphism of cone stacks $[C/E]\times
[D/F]\to [C\times D/E\times F]$.
Putting together this remarks and verifying that the canonical isomorphisms
glue completes the proof.
\end{pf}
\begin{prop}[Pullback] \label{fshecs}
Let $f:X\to Y$ be a local complete intersection morphism. Then we have
a natural short exact sequence of cone stacks
\[{\frak N}_{X/Y}\longrightarrow{\frak C}_X\longrightarrow f^{\ast}{\frak C}_Y\]
over $X$, where ${\frak N}_{X/Y}=h^1/h^0(T_{X/Y}^{\scriptscriptstyle\bullet})$.
\end{prop}
\begin{pf}
We have a distinguished triangle in $D(\O_{X_{\mbox{\tiny \'{e}t}}})$
\[\dt{f^{\ast} L_Y}{}{L_X}{}{L_{X/Y}}{},\]
and $L_{X/Y}$ is of perfect amplitude contained in $[-1,0]$. So by
Proposition~\ref{dtscs} we have a short exact sequence of abelian cone
stacks \[{\frak N}_{X/Y}\longrightarrow{\frak N}_X\longrightarrow f^{\ast}{\frak N}_Y\] on
$X$. So the claim is local in $X$ and we may assume that we have a
diagram
\[\begin{array}{ccccc}
X & \stackrel{i}{\longrightarrow} & M'' & \longrightarrow & M' \\
& \searrow & \downarrow & & \downarrow \\
& & Y & \longrightarrow & M,
\end{array}\]
where the square is cartesian, the vertical maps are smooth, the
horizontal maps are local immersions, $i$ is regular and $M$ is
smooth. Then we have a morphism of short exact sequences of cones on
$X$:
\[\begin{array}{ccccc}
i^{\ast} T_{M''/Y} & \longrightarrow & T_{M'}{ \mid } X & \longrightarrow
& T_M{ \mid } X \\
\downarrow & & \downarrow & & \downarrow \\
N_{X/M''} & \longrightarrow & C_{X/M'} & \longrightarrow &
C_{Y/M}{ \mid } X.
\end{array}\]
This is a local presentation for the short exact sequence
\[{\frak N}_{X/Y}\longrightarrow{\frak C}_X\longrightarrow f^{\ast}{\frak C}_Y\]
of cone stacks.
\end{pf}
\section{Obstruction Theory}
\subsection{The Intrinsic Normal Sheaf as Obstruction}
A closed immersion $T\to \overline T$ of schemes is called a {\em
square-zero extension }with ideal sheaf $J$ if $J$ is the ideal sheaf
of $T$ in $\overline T$ and $J^2=0$.
Let $X$ be a Deligne-Mumford stack, ${\frak N}_X$ its intrinsic normal
sheaf. Let $T\to\overline T$ be a square zero extension with ideal sheaf $J$
and $g:T\to X$ a morphism. By the functorialities of the cotangent
complex we have a canonical homomorphism
\begin{equation} \label{idkwtp}
g^{\ast} L_X^{\scriptscriptstyle\bullet}\longrightarrow L_T^{\scriptscriptstyle\bullet}\longrightarrow L^{\scriptscriptstyle\bullet}_{T/\overline T}
\end{equation}
in $D(\O_{T_{\mbox{\tiny \'{e}t}}})$.
Since $\tau_{\geq-1}L^{\scriptscriptstyle\bullet}_{T/\overline T}=J[1]$, this homomorphism may be
considered as an element $\omega(g)$ of $\mathop{\rm Ext}\nolimits^1(g^{\ast} L_X^{\scriptscriptstyle\bullet},J)$.
Recall the following basic facts of deformation theory. An extension
$\overline{g}:\overline{T}\to X$ of $g$ exists if and only if $\omega(g)=0$ and if
$\omega(g)=0$ the extensions form a torsor under $\mathop{\rm Ext}\nolimits^0(g^{\ast}
L^{\scriptscriptstyle\bullet}_X,J)=\mathop{\rm Hom}\nolimits(g^{\ast} \Omega_X,J)$.
These facts can be interpreted in terms of the intrinsic normal sheaf
${\frak N}_X$ of $X$. To do this, note that (\ref{idkwtp}) gives rise to a
morphism
\[h^1/h^0(L^{\scriptscriptstyle\bullet}_{T/\overline T})\longrightarrow h^1/h^0(g^{\ast} L^{\scriptscriptstyle\bullet}_X)\]
of cone stacks over $T$. Since $h^1/h^0(L^{\scriptscriptstyle\bullet}_{T/\overline T})=C(J)$ and
$h^1/h^0(g^{\ast} L^{\scriptscriptstyle\bullet}_X)=g^{\ast}{\frak N}_X$ we have constructed a morphism
$ob(g):C(J)\to g^{\ast}{\frak N}_X$. We also consider the morphism
$0(g):C(J)\to g^{\ast}{\frak N}_X$ given as the composition of $C(J)\to X$
with the vertex of $g^{\ast}{\frak N}_X$. By $\underline{\mathop{\rm Hom}\nolimits}(ob(g),0(g))$ we shall
denote the sheaf of 2-isomorphisms of cone stacks from $ob(g)$ to
$0(g)$, restricted to $T_{\mbox{\tiny \'{e}t}}$.
Given a square zero extension $T\to\overline T$ and a morphism $g:T\to X$,
we denote the set of extensions $\overline{g}:\overline{T}\to X$ of $g$ by
$\mathop{\rm Ext}\nolimits(g,\overline{T})$. These extensions in fact form a sheaf on $T_{\mbox{\tiny \'{e}t}}$
which we shall denote $\underline{\mathop{\rm Ext}\nolimits}(g,\overline{T})$.
\begin{prop} \label{ince}
There is a canonical isomorphism
\[\underline{\mathop{\rm Ext}\nolimits}(g,\overline{T})\stackrel{\textstyle\sim}{\longrightarrow}\underline{\mathop{\rm Hom}\nolimits}_{\O_T}(ob(g),0(g))\]
of sheaves on $T_{\mbox{\tiny \'{e}t}}$. In particular, extensions of $g$ to $\overline T$
exist, if and only if $ob(g)$ is ${\Bbb A}^1$-equivariantly isomorphic to
$0(g)$.
\end{prop}
\begin{pf}
Locally, we may embed $X$ into a smooth scheme $M$ and call the
embedding $i:X\to M$, the conormal sheaf $I/I^2$. Then there always
exist local extensions $h:\overline{T}\to M$ of $i\mathbin{{\scriptstyle\circ}} g:T\to M$.
\[\comdia{T}{}{\overline T}{g}{}{h}{X}{i}{M}\]
Any such $h$ gives rise to a homomorphism $h^\sharp:g^{\ast} I/I^2\to
J$, and hence to a realization of $ob(g)$ as the morphism of cone
stacks induced by the homomorphism of complexes
\[h^\sharp:g^{\ast}[I/I^2\to i^{\ast}\Omega_M]\longrightarrow[J\to0].\]
Note that if $\tilde{h}$ is another such extension, the difference
between $h$ and $\tilde h$ induces a homomorphism $g^{\ast} i^{\ast}
\Omega_M\to J$, which is in fact a homotopy from $h^\sharp$ to
$\tilde{h}^\sharp$.
Now let $\overline{g}:\overline{T}\to X$ be an extension of $g$. Then $(i\mathbin{{\scriptstyle\circ}}\overline
g)^\sharp=0$, so that we get a homotopy from any local $h^\sharp$ as
above to $0$, or in other words a local ${\Bbb A}^1$-equivariant
isomorphism from $ob(g)$ to $0(g)$. Since these local isomorphisms
glue, we get the required map
\[\underline{\mathop{\rm Ext}\nolimits}(g,\overline{T})\longrightarrow\underline{\mathop{\rm Hom}\nolimits}(ob(g),0(g)).\]
To construct the inverse, let $\theta:ob(g)\to0(g)$ be a 2-isomorphism
of cone stacks. Note that $\theta$ defines for every local $h$ as
above an extension of $h^\sharp$ to $\overline{h}^\sharp:i^{\ast}\Omega_M\to J$
(use Lemma~\ref{lo2i}). Changing $h$ by $\overline{h}^\sharp$ defines
$h':\overline{T}\to M$ such that $(h')^\sharp=0$. Thus $h'$ factors through
$X$, and in fact these locally defined $h'$ glue to give the required
extension $\overline{g}:\overline{T}\to X$.
\end{pf}
\begin{prop}
There is a canonical isomorphism
\[\underline{\mathop{\rm Aut}\nolimits}(0(g))\stackrel{\textstyle\sim}{\longrightarrow}\mathop{\rm {\mit{ \hH\! om}}}\nolimits(g^{\ast}\Omega_X,J)\]
of sheaves on $T_{\mbox{\tiny \'{e}t}}$.
\end{prop}
\begin{pf}
Again, Lemma~\ref{lo2i} shows that the automorphisms of $0(g)$ are
(locally) the homomorphisms from $g^{\ast} i^{\ast} \Omega_M$ to $J$
vanishing on $g^{\ast} I/I^2$. The exact sequence
\[I/I^2\longrightarrow i^{\ast} \Omega_M\longrightarrow
\Omega_X\longrightarrow 0\]
finishes the proof. See also Lemma~\ref{lmh}.
\end{pf}
\begin{cor}
The sheaf $\underline{\mathop{\rm Hom}\nolimits}(\mathop{\rm ob}(g),0(g))$ is a formal
$\mathop{\rm {\mit{ \hH\! om}}}\nolimits(g^{\ast}\Omega_X,J)$-torsor. So if $ob(g)\cong0(g)$, the set
$\mathop{\rm Hom}\nolimits(ob(g),0(g))$ is a torsor under the group
$\mathop{\rm Hom}\nolimits(g^{\ast}\Omega_X,J)$.
\end{cor}
\begin{note}
Combining this with Proposition~\ref{ince} gives that $\mathop{\rm Ext}\nolimits(g,\overline T)$
is a $\mathop{\rm Hom}\nolimits(g^{\ast}\Omega,J)$-torsor if the obstruction vanishes,
reproving this fact from deformation theory alluded to above.
\end{note}
\subsection{Obstruction Theories}
\begin{defn} \label{doot} Let $E^{\scriptscriptstyle\bullet}\in\mathop{\rm ob} D(\O_{X_{\mbox{\tiny \'{e}t}}})$ satisfy
($\star$) (see Definition~\ref{dost}). Then a homomorphism
$\phi:E^{\scriptscriptstyle\bullet}\rightarrow L_X^{\scriptscriptstyle\bullet}$ in $D(\O_{X_{\mbox{\tiny \'{e}t}}})$ is called an {\em
obstruction theory }for $X$, if $h^0(\phi)$ is an isomorphism and
$h^{-1}(\phi)$ is surjective. By abuse of language we also say that
$E^{\scriptscriptstyle\bullet}$ is an obstruction theory for $X$.
\end{defn}
\begin{note}
By Proposition~\ref{isoiso} the homomorphism $\phi:E^{\scriptscriptstyle\bullet}\to L^{\scriptscriptstyle\bullet}_X$
is an obstruction theory if and only if
\[\dual{\phi}:{\frak N}_X\longrightarrow{\frak E}\]
is a closed immersion, where ${\frak E}= h^1/h^0(\dual{(E_{\mbox{\tiny fl}}^{\scriptscriptstyle\bullet})})$. So
if $E^{\scriptscriptstyle\bullet}$ is an obstruction theory and ${\frak C}_X\subset {\frak N}_X$ is the
intrinsic normal cone of $X$, then $\dual{\phi}({\frak C}_X)$ is a closed
subcone stack of ${\frak E}$ of pure dimension zero. We sometimes call
$\dual{\phi}({\frak C}_X)$ the {\em obstruction cone }of the obstruction
theory $\phi:E^{\scriptscriptstyle\bullet}\rightarrow L_X^{\scriptscriptstyle\bullet}$.
\end{note}
Let $E^{\scriptscriptstyle\bullet}\in\mathop{\rm ob} E(\O_{X_{\mbox{\tiny \'{e}t}}})$ satisfy ($\star$) and let
$\phi:E^{\scriptscriptstyle\bullet}\to L^{\scriptscriptstyle\bullet}_X$ be a homomorphism. Let
${\frak E}=h^1/h^0(\dual{(E_{\mbox{\tiny fl}}^{\scriptscriptstyle\bullet})})$ and $\dual{\phi}:{\frak N}_X\to{\frak E}$ the
induced morphism of cone stacks. If $T\to\overline T$ is a square zero
extension of $k$-schemes with ideal sheaf $J$ and $g:T\to X$ is a
morphism, then we denote by $\phi^{\ast}\omega(g)$ the image of the
obstruction $\omega(g)\in\mathop{\rm Ext}\nolimits^1(g^{\ast} L_X^{\scriptscriptstyle\bullet},J)$ in $\mathop{\rm Ext}\nolimits^1(g^{\ast}
E^{\scriptscriptstyle\bullet},J)$ and by $\dual{\phi}(ob(g))$ the composition
$$C(J)\stackrel{ob(g)}{\longrightarrow}
g^{\ast}{\frak N}_X \stackrel{g^{\ast}\dual{\phi}}{\longrightarrow}g^{\ast}{\frak E}$$
of morphisms of cone stacks over $T$.
\begin{them}\label{ontoh1}
The following are equivalent.
\begin{enumerate}
\item \label{ont1} $\phi:E^{\scriptscriptstyle\bullet}\to L_X^{\scriptscriptstyle\bullet}$ is an obstruction theory.
\item \label{ont2}$\dual{\phi}:{\frak N}_X\to{\frak E}$ is a closed immersion of
cone stacks over $X$.
\item \label{ont3}For any $(T,\overline T, g)$ as above, the obstruction
$\phi^{\ast}(\omega(g))\in\mathop{\rm Ext}\nolimits^1(g^{\ast} E^{\scriptscriptstyle\bullet},J)$ vanishes if and only
if an extension $\overline{g}$ of $g$ to $\overline T$ exists; and if
$\phi^{\ast}(\omega(g))=0$, then the extensions form a torsor under
$\mathop{\rm Ext}\nolimits^0(g^{\ast} E^{\scriptscriptstyle\bullet},J)=\mathop{\rm Hom}\nolimits(g^{\ast} h^0(E^{\scriptscriptstyle\bullet}),J)$.
\item \label{ont4}For any $(T,\overline T, g)$ as above, the sheaf of
extensions $\underline{\mathop{\rm Ext}\nolimits}(g,\overline T)$ is isomorphic to the sheaf
$\underline{\mathop{\rm Hom}\nolimits}(\dual{\phi}(ob(g)),0)$ of ${\Bbb A}^1$-equivariant
isomorphism from $\dual{\phi}(ob(g)):C(J)\to g^{\ast}{\frak E}$
to the vertex $0:C(J)\to
g^{\ast} {\frak E}$.
\end{enumerate}
\end{them}
\begin{pf}
The equivalence of (\ref{ont1}) and (\ref{ont2}) has already been
noted. In view of Proposition~\ref{ince} it is clear that
(\ref{ont2}) implies (\ref{ont4}). The implication
(\ref{ont4})$\Rightarrow$(\ref{ont3}) follows from Lemma~\ref{lmh}. So
let us prove that (\ref{ont3}) implies (\ref{ont1}).
To prove that $h^0(\phi)$ is an isomorphism we can assume that
$X=\mathop{\rm Spec}\nolimits R$ is an affine scheme (as the statement is local); let $A$
be any $R$-algebra, $M$ any $A$-module. Let $T=\spec A$, $\overline T=\spec
(A\oplus M)$, where the ring structure is given by
$(a,m)(a',m')=(aa',am'+a'm)$. Let $g:T\to X$ be the morphism induced
by the $R$-algebra structure of $A$. Then $g$ extends to $\overline T$, so
there is a bijection $\mathop{\rm Hom}\nolimits(h^0(L_X^{\scriptscriptstyle\bullet})\otimes A,M)\to
\mathop{\rm Hom}\nolimits(h^0(E^{\scriptscriptstyle\bullet})\otimes A,M)$.
This implies easily that $h^0(\phi)$ is an isomorphism.
The fact that $h^{-1}(\phi)$ is surjective is local in the \'etale
topology (and only depends on $\tau_{\ge -1}E^{\scriptscriptstyle\bullet}$). Assume therefore
that $X$ is an affine scheme, $i:X\to W$ a closed embedding in a
smooth affine scheme $W$, and let $I$ be the ideal of $X$ in $W$. We
can assume that $E^0=f^*\Omega_W$ (see the proof of
\ref{rmcs}), that $E^{-1}$ is a coherent sheaf, and that $E^i=0$ for $i\ne
0,-1$.
We have to prove that $E^{-1}\to I/I^2$ is surjective; let $M$ be its
image. Let $T=X$, $\tilde M\subset I$ the inverse image of $M$, and
$\overline T\subset W$ the subscheme defined by $\tilde M$; let $g:T\to X$
be the identity. We can extend $g$ to the inclusion $\tilde g:\overline T\to
W$. Let $\pi:I/I^2\to I/\tilde M$ be the natural projection. By
assumption $\pi$ factors via $E^0$ if and only if $g$ extends to a map
$\overline T\to X$, if and only if $\pi\circ
\phi^{-1}:E^{-1}\to I/\tilde M$ factors via $E^0$. As
$\pi\circ\phi^{-1}$ is the zero map, it certainly factors. Therefore
$\pi$ also factors. Consider now the commutative diagram with exact
rows
\[\begin{array}{ccccccc}
E^{-1} & \longrightarrow & E^0 & \longrightarrow & h^0(E^{\scriptscriptstyle\bullet})
& \longrightarrow & 0 \\
\ldiag{\phi} & & \Vert & & \Vert & & \\
I/I^2 & \longrightarrow & E^0 & \longrightarrow & h^0(E^{\scriptscriptstyle\bullet}) &
\longrightarrow & 0.
\end{array}\]
By an easy diagram chasing argument, the fact that $\pi$ factors via $E^0$
together with $\pi\circ\phi^{-1}=0$ implies $\pi=0$, hence
$\phi^{-1}:E^{-1}\to I/I^2$ is surjective.
\end{pf}
\noprint{
A square-zero extension will be called {\em curvilinear }if it is isomorphic
to $\spec K[t]/t^r\to \spec K[t]/t^{r+s}$, with $K$ a field.
\begin{lem} Let $T\to \overline T$ be a curvilinear extension, $g:T\to X$ a
morphism. Then $\mathop{\rm ob}(g)$ factors via ${\frak C}_X$. Conversely, if ${\frak D}$ is a closed
subcone stack of ${\frak N}_X$ such that for every curvilinear extension $T\to
\overline T$ and every morphism $g:T\to X$ the morphism $ob(g)$ factors via ${\frak D}$,
then ${\frak D}$ contains ${\frak C}_X$.
\end{lem}
\begin{pf} The first statement is local, so assume that $X$ is an affine
scheme embedded as closed subscheme in a smooth scheme $W$, with ideal sheaf
${\cal I}$. Let $\hat T=\spec K[[t]]$; extend $g$ to a morphism
$\hat g:\hat T\to W$. (I am not ready with it yet).
\end{pf}
}
\newcommand{\hbox{\sl Sets}}{\hbox{\sl Sets}}
\newcommand{\hbox{\sl Art}}{\hbox{\sl Art}}
\newcommand{\hbox{\sl ann}\,}{\hbox{\sl ann}\,}
\newcommand{h^1/h^0(T_X\com)}{h^1/h^0(T_X^{\scriptscriptstyle\bullet})}
\newcommand{\ol{ N}_p}{\overline{ N}_p}
\newcommand{\ol{ C}_p}{\overline{ C}_p}
\subsection{Obstructions for Small Extensions}
Let $\hbox{\sl Art}$ be the category of local Artinian $k$-algebras with residue
field $k$. A {\sl small extension} will be a surjective morphism $A'\to A$ in
$\hbox{\sl Art}$ with kernel $J$ isomorphic to $k$. A {\sl semi-small} extension is one
with
kernel isomorphic to a $k$-vector space as an $A'$-algebra.
Let $F:\hbox{\sl Art} \to \hbox{\sl Sets}$ be a pro-representable covariant functor (in the
sense of \cite{schlessinger}). An {\em obstruction space} for $F$ is a
set $k$-vector space $T^2$ and, for any semi-small extension $A'\to A$
with kernel $J$, an exact sequence $$ F(A')\longrightarrow
F(A)\stackrel{ob}{\longrightarrow} T^2\otimes J.$$ This means that,
for all $\xi\in F(A)$, $\xi$ is in the image of $F(A')$ if and only if
$ob(\xi)=0$. It is also required that $ob$ is functorial in the
obvious sense (see \cite{kawamata}). We say that $v\in T^2$ {\em
obstructs a small extension }$A'\to A$ if $ob(\xi)=v\otimes w$ for
some $\xi\in F(A)$ and some nonzero $w\in J$.
Let $X$ be a Deligne-Mumford stack, $p\in X$ a fixed point with
residue field $k$. Let $h_p:\hbox{\sl Art}\to \hbox{\sl Sets}$ be the covariant functor
associating to an object $A$ of $\hbox{\sl Art}$ the set of morphisms $\mathop{\rm Spec}\nolimits
A\to X$ sending the closed point to $p$. The functor $h_p$ is
pro-representable, and it is unchanged if we replace $X$ by any \'etale
open neighborhood of $p$.
Let $N_p=p^*{\frak N}_X$, and let $\ol{ N}_p$ be the coarse moduli space of
$N_p$. Note that $\ol{ N}_p =T^1_{X,p}/T^0_{X,p}$, so that $\ol{ N}_p$ is in
fact a $k$-vector space. Here $T^i_{X,p}=h^i(p^{\ast}
T_X^{\scriptscriptstyle\bullet})=\dual{h^i(p^{\ast} L_X^{\scriptscriptstyle\bullet})}$ are the `higher tangent spaces'
of $X$ at $p$. Let $\ol{ C}_p\subset \ol{ N}_p$ be the subcone coarsely
representing $p^{\ast}{\frak C}_X$. Proposition~\ref{ince} implies that $\ol{ N}_p$
is an obstruction space for $h_X$. The following is probably known but
we include a proof for lack of a suitable reference; it is a version
of Theorem~\ref{ontoh1} for semi-small extensions.
\begin{lem} The space $\ol{ N}_p$ is a universal
obstruction space for $h_p$; that is, for any other obstruction space $T^2$,
there is a unique injection
$N_p\to T^2$ compatible with the obstruction maps.
\end{lem}
\begin{pf} Let $(U,W)$ be a local embedding
for $X$ near
$p$. Assume that $W=\mathop{\rm Spec}\nolimits P$, $U=\mathop{\rm Spec}\nolimits R=\mathop{\rm Spec}\nolimits P/I$; let ${\frak m}$ be the maximal
ideal of $p$ in $P$, and assume that $I\subset {\frak m}^2$. In this case
$\ol{ N}_p=\dual{(I/{\frak m} I)}$.
If $n$ is sufficiently large, the natural map
$I/{\frak m} I\to (I+{\frak m}^n)/({\frak m} I+{\frak m}^n)$ is an isomorphism; choose such an $n$. Let
$A'_n\to A_n$ be the extension $P/({\frak m} I+{\frak m}^n)\to P/(I+{\frak m}^n)$, and let
$\xi_n\in h_p(A_n)$ be the natural quotient map. Then if $T^2$ is any
obstruction space, the obstruction to $\xi_n$ gives a linear map $I/{\frak m} I\to
T^2$ which must be injective. It is easy to check by functoriality that taking
a
different $n$ does not change the map. But given any semi-small extension
$A'\to
A$, there is always an extension of the type $A'_n\to A_n$ mapping to it, so
one
can apply functoriality again.
\end{pf}
\begin{prop}
Every $v\in \ol{ N}_p$ obstructs
some small extension; it obstructs some small curvilinear extension
if and only if
$v\in\ol{ C}_p$.
\end{prop}
\begin{pf}
Let $v\in \ol{ N}_p$, and view it as a linear map $I\to k$ having ${\frak m} I$ in the
kernel; we prove first that $v$ is an obstruction for some small extension. Let
$L=\ker v$, and choose $n$ sufficiently large, so that $L+{\frak m}^n\ne I+{\frak m}^n$.
Let
$A=P/I+{\frak m}^n$, and $A=P/L+{\frak m}^n$; choose $\xi:R\to A$ to be the natural
surjection. Let $J=\ker (A'\to A)$; $J$ is naturally isomorphic to $I/L$. Then
$\mathop{\rm ob}_\xi:I/{\frak m} I\to J$ is the obvious map, and the image of the dual map in
$\ol{ N}_p$ is the vector space generated by $v$.
Choose a set of generators $f_1,\ldots,f_r$ of
$I$ inducing a basis for $I/{\frak m} I$. This defines a map $f:W\to {\Bbb A}^r$ such
that
$U$ is the fiber over the origin. Then $\ol{ C}_p$ is the normal cone to the image of
$W$ in ${\Bbb A}^r$. The proof then follows the argument of Proposition 20.2 in
\cite{harris}.
\end{pf}
\section{Obstruction Theories and Fundamental Classes} \label{otafc}
\subsection{Virtual Fundamental Classes}
As usual, let $X$ be a Deligne-Mumford stack over $k$.
\begin{defn} We call an obstruction theory $E^{\scriptscriptstyle\bullet}\rightarrow L_X^{\scriptscriptstyle\bullet}$ {\em
perfect}, if $E^{\scriptscriptstyle\bullet}$ is of perfect amplitude contained in $[-1,0]$.
\end{defn}
Now assume that $X$ is separated (or, more generally, satisfies the condition
of
Vistoli in
\cite{vistoli}). We shall denote by $A_k(X)$ the rational Chow group of cycles
of dimension $k$ on
$X$ modulo rational equivalence tensored with ${\Bbb Q}$ (see [ibid]). We shall also
use the corresponding bivariant groups $A^k(X\rightarrow Y)$, for morphisms
$X\rightarrow Y$ of separated Deligne-Mumford stacks.
Let $E^{\scriptscriptstyle\bullet}$ be a perfect obstruction theory for $X$, and let
${\frak C}_X\hookrightarrow h^1/h^0(\dual{E})$ be the intrinsic normal
cone. We call $\mathop{\rm rk} E^{\scriptscriptstyle\bullet}$ the {\em virtual dimension }of $X$ with
respect to the obstruction theory $E^{\scriptscriptstyle\bullet}$. Recall that $\mathop{\rm rk} E^{\scriptscriptstyle\bullet}=\dim
E^0-\dim E^{-1}$, if locally $E^{\scriptscriptstyle\bullet}$ is written as a complex of vector
bundles $[E^{-1}\rightarrow E^0]$. This is a well-defined locally
constant function on $X$. We shall assume that the virtual dimension
of $X$ with respect to $E^{\scriptscriptstyle\bullet}$ is constant, equal to $n$.
To construct the {\em virtual fundamental class }$[X,E^{\scriptscriptstyle\bullet}]\in A_n(X)$
of $X$ with respect to the obstruction theory $E^{\scriptscriptstyle\bullet}$, we would like
to simply intersect the intrinsic normal cone ${\frak C}_X$ with the vertex
(zero section) of $h^1/h^0(\dual{E})$. Since $h^1/h^0(\dual{E})$ is
smooth of relative dimension $-n$ over $X$, the codimension of $X$ in
$h^1/h^0(\dual{E})$ is $-n$, so that the dimension of the intersection
of ${\frak C}_X$ with $X$ is $0-(-n)=n$. Unfortunately, this construction would
require Chow groups for Artin stacks, which we do not have at our
disposal. This is why we shall make the assumption that $E^{\scriptscriptstyle\bullet}$ has
global resolutions.
\begin{defn} Let $F^{\scriptscriptstyle\bullet}=[F^{-1}\rightarrow F^0]$ be a homomorphism of vector
bundles on $X$ considered as a complex of $\O_X$-modules concentrated in
degrees
$-1$ and $0$. An isomorphism $F^{\scriptscriptstyle\bullet}\rightarrow E^{\scriptscriptstyle\bullet}$ in $D(\O_{X_{\mbox{\tiny \'{e}t}}})$ is
called a {\em global resolution }of $E^{\scriptscriptstyle\bullet}$.
\end{defn}
Let $F^{\scriptscriptstyle\bullet}$ be a global resolution of $E^{\scriptscriptstyle\bullet}$. Then
\[h^1/h^0(\dual{E})=[\dual{F^{-1}}/\dual{F^0}],\] so that $F_1=\dual{F^{-1}}$
is
a (global) presentation of $h^1/h^0(\dual{E})$. Let $C(F^{\scriptscriptstyle\bullet})$ be the fibered
product
\[\comdia{C(F^{\scriptscriptstyle\bullet})}{}{F_1}{}{}{}{{\frak C}_X}{}{h^1/h^0(\dual{E}).}\]
Then $C(F^{\scriptscriptstyle\bullet})$ is a closed subcone of the vector bundle $F_1$. We
define the {\em virtual fundamental class }$[X,E^{\scriptscriptstyle\bullet}]$ to be the
intersection of $C(F^{\scriptscriptstyle\bullet})$ with the zero section of $F_1$. Note that
$C(F^{\scriptscriptstyle\bullet})\rightarrow {\frak C}_X$ is smooth of relative dimension $\mathop{\rm rk} F_0$
(where $F_0=\dual{F^0}$), so that $C(F^{\scriptscriptstyle\bullet})$ has pure dimension $\mathop{\rm rk}
F_0$ and $[X,E^{\scriptscriptstyle\bullet}]$ then has degree
\[\mathop{\rm rk} F_0-\mathop{\rm rk} F_1=\mathop{\rm rk} E^{\scriptscriptstyle\bullet} =n.\]
\begin{prop} \label{vfcigr}
The virtual fundamental class $[X,E^{\scriptscriptstyle\bullet}]$ is independent of the global
resolution $F^{\scriptscriptstyle\bullet}$ used to construct it.
\end{prop}
\begin{pf} Let $H^{\scriptscriptstyle\bullet}$ be another global resolution of $E^{\scriptscriptstyle\bullet}$. Without loss of
generality assume that
$H^{\scriptscriptstyle\bullet}\rightarrow E^{\scriptscriptstyle\bullet}$ and $F^{\scriptscriptstyle\bullet}\rightarrow E^{\scriptscriptstyle\bullet}$ are given by morphisms of
complexes. Then we get an induced homomorphism $H^0\oplus F^0\rightarrow E^0$.
So by constructing the cartesian diagram
\[\comdia{K^{-1}}{}{H^0\oplus F^0}{}{}{}{E^{-1}}{}{E^0,}\] and letting
$K^0=H^0\oplus F^0$, we get a global resolution $K^{\scriptscriptstyle\bullet}$ of $E^{\scriptscriptstyle\bullet}$ such that
both
$H^{\scriptscriptstyle\bullet}$ and $F^{\scriptscriptstyle\bullet}$ map to $K^{\scriptscriptstyle\bullet}$ by a strict monomorphism. So it suffices to
compare $F^{\scriptscriptstyle\bullet}$ with
$K^{\scriptscriptstyle\bullet}$. Dually, we have an epimorphism $K_1\rightarrow F_1$. Consider the
diagram
\[\begin{array}{ccccc} X & \stackrel{0}{\longrightarrow} & C(H^{\scriptscriptstyle\bullet}) &
\longrightarrow & C(F^{\scriptscriptstyle\bullet}) \\
\ldiag{} & & \ldiag{} & & \rdiag{} \\ X & \stackrel{0}{\longrightarrow} & K_1 &
\stackrel{\alpha}{\longrightarrow} & F_1,
\end{array}\] in which both squares are cartesian. Note that $\alpha$ is
smooth.
The virtual fundamental class using $F^{\scriptscriptstyle\bullet}$ is equal to
\[(\alpha\mathbin{{\scriptstyle\circ}}
0)^{!}[C(F^{\scriptscriptstyle\bullet})]=0^{!}\alpha^{!}[C(F^{\scriptscriptstyle\bullet})]=0^{!}[C(H^{\scriptscriptstyle\bullet})],\]
which is the virtual fundamental class using $H^{\scriptscriptstyle\bullet}$.
\end{pf}
\begin{example} If $X$ is a complete intersection, then $L_X^{\scriptscriptstyle\bullet}$ is of
perfect amplitude contained in
$[-1,0]$, so that $L_X^{\scriptscriptstyle\bullet}$ itself is a perfect obstruction theory.
Any embedding of $X$ into a smooth Deligne-Mumford stack gives rise to
a global resolution of $L_X^{\scriptscriptstyle\bullet}$.The virtual fundamental class
$[X,L_X^{\scriptscriptstyle\bullet}]$ thus obtained is equal to $[X]$, the `usual' fundamental
class.
\end{example}
\begin{numrmk}[Virtual Structure Sheaves] \label{vss}
Let $X$ be a Deligne-Mumford stack and let ${\frak C}\hookrightarrow{\frak E}$ be
a closed subcone stack of a vector bundle stack. Then we define a
graded commutative sheaf of coherent $\O_X$-algebras $\O_{({\frak C},{\frak E})}$
as follows.
If ${\frak E}\cong[E_1/E_0]$, then ${\frak C}$ induces a cone $C$ in $E_1$ and we
set
\[\O_{({\frak C},{\frak E})}^i=\mathop{\rm {\mit{ \tT\! or}}}\nolimits_i^{\O_{E_1}}(\O_C,\O_X),\]
where we think of $\O_X$ as an $\O_{E_1}$-algebra via the zero section
of $E_1$. Standard arguments show that
\[\O_{({\frak C},{\frak E})}=\bigoplus_i \O_{({\frak C},{\frak E})}^i\]
is independent of the choice of presentation ${\frak E}\cong[E_1/E_0]$.
Hence the locally defined sheaves glue, giving rise to a globally
defined sheaf.
If ${\frak C}={\frak C}_X$, $E^{\scriptscriptstyle\bullet}$ is a perfect obstruction theory of $X$ and
${\frak E}=h^1/h^0(\dual{E^{\scriptscriptstyle\bullet}})$, we call $\O_{({\frak C},{\frak E})}$ the {\em virtual
structure sheaf }of $X$ with respect to the obstruction theory
$E^{\scriptscriptstyle\bullet}$, denoted $\O_{(X,E^{\scriptscriptstyle\bullet})}$. This seems to be the virtual
structure sheaf proposed by Kontsevich in
\cite{K}.
If one has on $X$ a homological Chern character
$\tau:K_0(X)\rightarrow A_{\ast}(X)$ one can define the virtual
fundamental class of $X$ with respect to $E^{\scriptscriptstyle\bullet}$ by
\[[X,E^{\scriptscriptstyle\bullet}]=\mathop{\rm td}(E^{\scriptscriptstyle\bullet})\cap\tau(\O_{(X,E^{\scriptscriptstyle\bullet})}).\]
This agrees with the above definition using global resolutions if they
exist. In the absence of a general Riemann Roch theorem, we rather
assume the existence of global resolutions.
\end{numrmk}
\subsection{Basic Properties}
\begin{prop}[No obstructions] \label{ehs} If $E^{\scriptscriptstyle\bullet}$ is
perfect, $h^0(E^{\scriptscriptstyle\bullet})$ is locally free and $h^1({E^{\scriptscriptstyle\bullet}})=0$, then
$X$ is smooth, the virtual dimension of $X$ with respect to $E^{\scriptscriptstyle\bullet}$ is $\dim X$
and
the virtual fundamental class $[X,E^{\scriptscriptstyle\bullet}]$ is just $[X]$, the usual
fundamental class. {\nolinebreak $\Box$}
\end{prop}
\begin{prop}[Locally free obstructions] \label{xself} Let $X$ be
smooth and $E^{\scriptscriptstyle\bullet}$ a perfect
obstruction theory for $X$. If $h^0(E^{\scriptscriptstyle\bullet})$ is locally free (or
equivalently $h^1(\dual{E^{\scriptscriptstyle\bullet}})$ is locally free) then the virtual
fundamental class is
\[[X,E^{\scriptscriptstyle\bullet}]=c_r(h^1(\dual{E^{\scriptscriptstyle\bullet}}))\cdot[X],\]
where $r=\mathop{\rm rk} h^1(\dual{E^{\scriptscriptstyle\bullet}})$.
\end{prop}
\begin{pf}
To see this, note that if $F^{\scriptscriptstyle\bullet}$ is a global resolution of $E^{\scriptscriptstyle\bullet}$,
then $C(F^{\scriptscriptstyle\bullet})=\mathop{\rm im}(F_0\rightarrow F_1)$.
\end{pf}
\begin{prop}[Products]
Let $E\to L_X$ be a perfect obstruction theory for $X$ and $F\to L_Y$
a perfect obstruction theory for $Y$. Then $L_{X\times Y}=L_X\boxplus
L_Y$. The induced homomorphism $E\boxplus F\to L_X\boxplus L_Y$ is a
perfect obstruction theory for $X\times Y$. If $E$ and $F$ have global
resolutions, then so does $E\boxplus F$ and we have
\[[X\times Y,E\boxplus F]=[X,E]\times[Y,F]\]
in $A_{\mathop{\rm rk} E+\mathop{\rm rk} F}(X\times Y)$.
\end{prop}
\begin{pf}
The statement about cotangent complexes is \cite{Ill}, Chapitre II, Corollaire
3.11.
To prove the rest, use Proposition~\ref{prod}.
\end{pf}
Consider a cartesian diagram of Deligne-Mumford stacks
\begin{equation}\label{asd}
\comdia{X'}{u}{X}{g}{}{f}{Y'}{v}{Y,}
\end{equation}
where $v$ is a local complete intersection morphism.
Let $E\to L_X$ and $F\to L_{X'}$ be perfect obstruction theories for
$X$ and $X'$, respectively.
\begin{defn} A {\em compatibility datum }(relative to $v$) for $E$ and
$F$ is a triple
$(\phi,\psi,\chi)$ of morphisms in $D(\O_{X'})$ giving rise to a
morphism of distinguished triangles
\[\begin{array}{ccccccc}
u^{\ast} E & \stackrel{\phi}{\longrightarrow} & F &
\stackrel{\psi}{\longrightarrow} & g^{\ast} L_{Y'/Y} &
\stackrel{\chi}{\longrightarrow} & u^{\ast} E[1] \\
\ldiag{} & & \ldiag{} & & \ldiag{} & & \rdiag{} \\
u^{\ast} L_X & {\longrightarrow} & L_{X'} &
{\longrightarrow} & L_{X'/X} &
{\longrightarrow} & u^{\ast} L_X[1] .
\end{array}\]
Given a compatibility datum, we call $E$ and $F$ {\em compatible
}(over $v$).
\end{defn}
Assume that $E$ and $F$ are endowed with such a compatibility datum.
Then we get (Proposition~\ref{dtscs}) a short exact sequence of vector
bundle stacks
\[g^{\ast} h^1/h^0(T_{Y'/Y}^{\scriptscriptstyle\bullet})\longrightarrow
h^1/h^0(\dual{F})\longrightarrow u^{\ast} h^1/h^0(\dual{E})\]
which we shall abbreviate by
\[g^{\ast}{\frak N}_{Y'/Y} \longrightarrow{\frak F}\stackrel{\phi}{\longrightarrow}
u^{\ast}{\frak E}.\]
If $v$ is a regular local immersion, then ${\frak N}_{Y'/Y}=N_{Y'/Y}$ is the
normal bundle of $Y'$ in $Y$. Its pullback to $X'$ we shall denote by
$N$.
\begin{lem} \label{ture}
If $Y$ and $Y'$ are smooth and $v$ a regular local immersion, then
there is a (canonical) rational equivalence $\beta(Y',X)\in
W_{\ast}(N\times{\frak F})$ such that
\[\partial\beta(Y',X)=[\phi^{\ast}
C_{u^{\ast}{\frak C}_X/{\frak C}_X}]-[N\times{\frak C}_{X'}].\]
\end{lem}
\begin{pf}
Let $X\to M$ be a local embedding, where $M$ is smooth. We get an
induced cartesian diagram
\[\comdia{X'}{}{X}{}{}{}{Y'\times M}{}{Y\times M,}\]
which we enlarge to
\[
\begin{array}{ccccc}
N\times_XC & \longrightarrow & u^{\ast} C & \longrightarrow & C \\
\ldiag{} & & \ldiag{} & & \rdiag{} \\
N & \longrightarrow & X' & \stackrel{u}{\longrightarrow} & X \\
\ldiag{} & & \ldiag{j} & & \rdiag{i} \\
N_{Y'/Y}\times M & \stackrel{\rho}{\longrightarrow} & Y'\times M &
\stackrel{v}{\longrightarrow} & Y\times M,
\end{array}
\]
where $C$ is the normal cone of $X$ in $Y\times M$.
As in Section~\ref{stinc} we have a canonical rational equivalence
$\beta(Y'\times M,X)\in W_{\ast}(N\times_X C)$ such that
\[\partial\beta(Y'\times M,X)=[C_{u^{\ast} C/C}]-[N\times C_{X'/Y'\times M}].\]
By Proposition~\ref{vrqi} $\beta(Y'\times M,X)$ is invariant under the
action of $N\times u^{\ast} i^{\ast} T_{Y\times M} $ on $N\times_X C$. So
it descends to $N\times_X{\frak C}_X$. In particular, $\beta(Y'\times M,X)$
is invariant under the subsheaf $N\times j^{\ast} T_{Y'\times M}$ and
thus descends to $N\times[u^{\ast} C/j^{\ast} T_{Y'\times M}]$. Note that
$[u^{\ast} C/j^{\ast} T_{Y'\times M}]={\frak F}\times_{{\frak E}}{\frak C}_X$, which is a
closed subcone stack of ${\frak F}$. So pushing forward via this closed
immersion, we get a rational equivalence on $N\times{\frak F}$ which we
denote by $\beta(Y',X)$. We have
\[\partial\beta(Y',X)=[\phi^{\ast}
C_{u^{\ast}{\frak C}_X/{\frak C}_X}]-[N\times{\frak C}_{X'}]\]
as required.
Now use Proposition~\ref{vcgt} to show that $\beta(Y',X)$ does not
depend on the choice of the local embedding $X\to M$. So even if no
global embedding exists, the locally defined rational equivalences
glue, proving the lemma.
\end{pf}
\begin{prop}[Functoriality] \label{fcrp}
Let $E$ and $F$ be compatible perfect obstruction theories, as above.
If $E$ and $F$ have global resolutions then
\[v^{!}[X,E]=[X',F]\]
holds in the following cases.
\begin{enumerate}
\item \label{fcrp1} $v$ is smooth,
\item \label{fcrp2} $Y'$ and $Y$ are smooth.
\end{enumerate}
\end{prop}
\begin{pf}
First note that one may choose global resolutions $[E_0\rightarrow
E_1]$ of $\dual{E}$ and $[F_0\to F_1]$ of $\dual{F}$ together with a
pair of epimorphisms $\phi_0:F_0\to u^{\ast} E_0$ and $\phi_1:F_1\to
u^{\ast} E_1$ such that
\[\comdia{F_0}{\phi_0}{u^{\ast} E_0}{}{}{}{F_1}{\phi_1}{u^{\ast} E_1}\]
commutes. Letting $G_i$ be the kernel of $\phi_i$ we get a short exact
sequence of homomorphisms of vector bundles
\[\begin{array}{ccccccccc}
0 & \longrightarrow & G_0 & \longrightarrow & F_0 & \longrightarrow &
u^{\ast} E_0 & \longrightarrow & 0\\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \longrightarrow & G_1 & \longrightarrow & F_1 & \longrightarrow &
u^{\ast} E_1 & \longrightarrow & 0.
\end{array}\]
The induced short exact sequence
\[[G_1/G_0]\longrightarrow [F_1/F_0]\longrightarrow [u^{\ast} E_1/u^{\ast}
E_0]\] of vector bundle stacks is isomorphic to
$g^{\ast}{\frak N}_{Y'/Y}\to{\frak F}\to {\frak E}$. We let $C_1={\frak C}_X\times_{{\frak E}}E_1$ and
$D_1={\frak C}_{X'}\times_{{\frak F}}F_1$. Then $[X,E]=0_{E_1}^{!}[C_1]$ and
$[X',F]=0_{F_1}^{!}[D_1]$, where $0_{E_1}$ and $0_{F_1}$ are the zero
sections of $E_1$ and $F_1$, respectively.
If $v$ is smooth, then by Proposition~\ref{fshecs} the diagram
\[\comdia{{\frak C}_{X'}}{}{u^{\ast}{\frak C}_X}{}{}{}{{\frak F}}{}{u^{\ast}{\frak E}}\]
is cartesian, which implies that
\[\comdia{D_1}{}{u^{\ast} C_1}{}{}{}{F_1}{}{u^{\ast} E_1}\]
is cartesian. Hence $0_{u^{\ast} E_1}^{!}[u^{\ast} C_1]=0_{
F_1}^{!}[D_1]$ and we have
\begin{eqnarray*}
v^{!}[X,E] & = & v^{!} 0_{E_1}^{!}[C_1] \\
& = & 0_{u^{\ast} E_1}^{!}[u^{\ast} C_1] \\
& = & 0_{F_1}^{!}[D_1] \\
& = & [X',F].
\end{eqnarray*}
If $Y'$ and $Y$ are smooth, let us first treat the case that $v$ is a
regular local immersion. Then we may choose $F_1$ as the fibered
product
\[\comdia{F_1}{}{E_1}{}{}{}{{\frak F}}{\phi}{{\frak E}.}\]
Lifting the rational equivalence $\beta(Y',X)$ of Lemma~\ref{ture} to
$N\times F_1$ we get that
\[[N\times D_1]=\phi^{\ast}[C_{u^{\ast} C_1/C_1}]\]
in $A_{\ast}(N\times F_1)$. Then we have
\begin{eqnarray*}
[X',F_1] & = & 0_{F_1}^{!}[D_1] \\
& = & 0_{N\times F_1}^{!}[N\times D_1] \\
& = & 0_{N\times F_1}^{!}\phi^{\ast}[C_{u^{\ast} C_1/C_1}] \\
& = & 0_{N\times u^{\ast} E_1}^{!}[C_{u^{\ast} C_1/C_1}] \\
& = & 0_{^{\ast} E_1}^{!} v^{!}[C_1] \\
& = & v^{!} 0_{E_1}^{!} [C_1] \\
& = & v^{!} [X,E].
\end{eqnarray*}
In the general case factor $v$ as
$$Y'\stackrel{\Gamma_v}{\longrightarrow} Y'\times Y
\stackrel{p}{\longrightarrow} Y.$$ Then Diagram~\ref{asd} factors as
\[\begin{array}{ccccc}
X' & \longrightarrow & Y'\times X & \longrightarrow & X \\
\downarrow & & \downarrow & & \downarrow \\
Y' & \stackrel{\Gamma_v}{\longrightarrow} & Y'\times Y &
\stackrel{p}{\longrightarrow} & Y.
\end{array}\]
Since $Y'$ is smooth it has a canonical obstruction theory, namely
$\Omega_{Y'}$. As obstruction theory on $Y'\times X$ take
$\Omega_{Y'}\boxplus E$. Then $\Omega_{Y'}\boxplus E$ is compatible
with $E$ over $p$ and $F$ is compatible with $\Omega_{Y'}\boxplus E$
over $\Gamma_v$. So combining Cases~(\ref{fcrp1}) and~(\ref{fcrp2})
yields the result.
\end{pf}
\section{Examples}
\subsection{The Basic Example}
Assume that
\[\comdia{X}{j}{V}{g}{}{f}{Y}{i}{W}\]
is a cartesian diagram of schemes, that $V$ and $W$ are smooth and
that $i$ is a regular embedding. Let $E^{\scriptscriptstyle\bullet}$ be the complex
$[g^{\ast}\dual{N_{Y/W}}\to j^*\Omega_V]$ (in degrees $-1$ and $0$),
where the map is given by pulling back to $X$ and composing
$\dual{N_{Y/W}}\to i^*\Omega_W$ with $f^*\Omega_W\to \Omega_V$. The complex
$E^{\scriptscriptstyle\bullet}$ has a natural morphism to $L_X^{\scriptscriptstyle\bullet}$, induced by $g^*L_Y^{\scriptscriptstyle\bullet}\to
L_X^{\scriptscriptstyle\bullet}$ and $j^*L_V^{\scriptscriptstyle\bullet}\to L_X^{\scriptscriptstyle\bullet}$ (note that $E^{\scriptscriptstyle\bullet}$ is the
cokernel of $g^*i^*L_W^{\scriptscriptstyle\bullet}\to j^*L_V^{\scriptscriptstyle\bullet}\oplus g^*L_Y^{\scriptscriptstyle\bullet}$, where the
first component is the negative of the canonical map).
This makes $E^{\scriptscriptstyle\bullet}$ into a perfect obstruction theory for $X$; the
virtual fundamental class $[X,E^{\scriptscriptstyle\bullet}]$ is just $i^{!}[V]$ as defined
in \cite{fulton}, p.~98. The construction also works in case $X$, $Y$,
$V$ and $W$ are assumed to be just Deligne-Mumford stacks.
\subsection{Fibers of a Morphism between Smooth Stacks}
Let $f:V\rightarrow W$ be a morphism of algebraic stacks. We shall
assume that $V$ and $W$ are smooth over $k$ and that $f$ has
unramified diagonal, so that $V$ is a relative Deligne-Mumford stack
over $W$. Let $w:\mathop{\rm Spec}\nolimits k\rightarrow W$ be a $k$-valued point of $W$
and let $X$ be the fiber of $f$ over $w$. In this situation $X$ has an
obstruction theory as follows.
Choose a smooth morphism $\tilde W\to W$, with $\tilde W$ a scheme,
and a lifting $\tilde w:\mathop{\rm Spec}\nolimits k\to \tilde W$ of $w$ (assume $k$
algebraically closed). Let $\tilde V$ be the fiber product
$V\times_W\tilde W$; by the assumptions $\tilde V$ is a smooth
Deligne-Mumford stack. Then $X$ is isomorphic to the fiber over
$\tilde w$ of $\tilde V\to \tilde W$, hence it has an obstruction
theory as above.
To check that the obstruction theory so defined does not depend on the
choices made, it is enough to compare two different ones induced by a
smooth morphism of schemes $\tilde W'\to \tilde W$; this is then a
straightforward verification. Similarly, one generalizes to the case
of arbitrary ground field $k$.
See Example~\ref{rfe} for an alternative construction.
\newcommand{\omega}{\omega}
\newcommand{\mathop{{\rm Ext}}\nolimits}{\mathop{{\rm Ext}}\nolimits}
\subsection{Moduli Stacks of Projective Varieties}
Let $M$ and $X$ be Deligne-Mumford stacks. Let $p:M\to X$ be a flat,
relatively
Gorenstein projective morphism: by this we mean that it has constant relative
dimension and that the relative dualizing complex $\omega^{\scriptscriptstyle\bullet}_{M/X}$ is a line
bundle
$\omega$.
If $G^{\scriptscriptstyle\bullet}\in D^+(\O_X)$, we have $p^{!} G^{\scriptscriptstyle\bullet}=p^{\ast} G^{\scriptscriptstyle\bullet}\otimes \omega$. So
for any complex $F^{\scriptscriptstyle\bullet}\in D^-(\O_M)$ we have natural isomorphisms
\[\mathop{{\rm Ext}}\nolimits^k_{\O_M}(F^{\scriptscriptstyle\bullet},p^{\ast} G^{\scriptscriptstyle\bullet})\to
\mathop{{\rm Ext}}\nolimits^k_{\O_M}(F^{\scriptscriptstyle\bullet}\otimes \omega,p^{!} G^{\scriptscriptstyle\bullet})\to
\mathop{{\rm Ext}}\nolimits^k_{\O_X}(Rp_{\ast} (F^{\scriptscriptstyle\bullet} \otimes \omega), G^{\scriptscriptstyle\bullet}).
\]
In particular, the Kodaira-Spencer map
$L_{M/X}\to p^*L_X[1]$ induces a map
$E^{\scriptscriptstyle\bullet}\to L^{\scriptscriptstyle\bullet}_X$ (well-defined up to homotopy). Define the complex
$E^{\scriptscriptstyle\bullet}$ on
$X$ to be
$Rp_*(L^{\scriptscriptstyle\bullet}_{M/X}\otimes \omega)[-1]$.
\begin{prop} Let $p:M\to X$ be a flat, projective, relatively Gorenstein
morphism of
Deligne-Mumford stacks, and assume that the family $M$ is universal at every
point
of $X$ (e.g., $X$ is an open set in a fine moduli space and $M$ is the
universal
family). Then $E^{\scriptscriptstyle\bullet}\to L_X^{\scriptscriptstyle\bullet}$ is an obstruction theory for $X$.
\end{prop}
\begin{pf}
Let $T$ be a scheme, $f:T\to X$ a morphism, and consider the cartesian
diagram \[
\comdia{N}{g}{M}{q}{}{p}{T}{f}{X.}\]
If $T\to \bar T$ is a square zero extension with ideal sheaf ${\cal J}$, the
obstruction to extending $N$ to a flat family over $\bar T$ lies in
$\mathop{{\rm Ext}}\nolimits^2(L^{\scriptscriptstyle\bullet}_{N/T},q^*{\cal J})$, and the extensions, if they exist, are a torsor
under $\mathop{{\rm Ext}}\nolimits^1(L^{\scriptscriptstyle\bullet}_{N/T},q^*{\cal J})$.
Now $L^{\scriptscriptstyle\bullet}_{N/T}=g^*L^{\scriptscriptstyle\bullet}_{M/X}$ because $p$ is flat, hence
$$\mathop{{\rm Ext}}\nolimits^k_{\O_N}(L^{\scriptscriptstyle\bullet}_{N/T},q^*{\cal J})=\mathop{{\rm Ext}}\nolimits^k_{\O_M}(L^{\scriptscriptstyle\bullet}_{M/X},Rg_*q^*{\cal J})=
\mathop{{\rm Ext}}\nolimits^k_{\O_M}(L^{\scriptscriptstyle\bullet}_{M/X},p^*Rf_*{\cal J}).$$
By the
previous
argument,
$$\mathop{{\rm Ext}}\nolimits^k_{\O_M}(L^{\scriptscriptstyle\bullet}_{M/X},p^*Rf_*{\cal J})=\mathop{{\rm Ext}}\nolimits^{k-1}_{\O_X}(E^{\scriptscriptstyle\bullet},Rf_*{\cal J})=
\mathop{{\rm Ext}}\nolimits^{k-1}_{\O_T}(f^*E^{\scriptscriptstyle\bullet},{\cal J}).$$
Assume now that $X$ is an open subset of a fine moduli space,
that is the family $M$ is universal at every point. This implies that
the fibers of $p$ have finite and reduced automorphism group, hence
$E^{\scriptscriptstyle\bullet}$ satisfies ($\star$).
The map $E^{\scriptscriptstyle\bullet}\to L_X^{\scriptscriptstyle\bullet}$ induces morphisms
$$\phi_k:\mathop{{\rm Ext}}\nolimits^k_{\O_N}(L^{\scriptscriptstyle\bullet}_{N/T},q^*{\cal J})=\mathop{{\rm Ext}}\nolimits^{k-1}_{\O_T}(f^*E^{\scriptscriptstyle\bullet},{\cal J})
\to\mathop{{\rm Ext}}\nolimits^{k-1}_{\O_T}(f^*L^{\scriptscriptstyle\bullet}_{X},{\cal J})$$ and the fact that $X$ is a moduli
space
implies that
$\phi_1$ is an isomorphism
and $\phi_2$ is
injective. By Theorem \ref{ontoh1}, this implies that $E^{\scriptscriptstyle\bullet}$
is an obstruction theory for $X$.
\end{pf}
\begin{rmk} If $p$ is smooth of relative dimension $\le 2$, then $E^{\scriptscriptstyle\bullet}$ is a
perfect
obstruction theory.
\end{rmk}
\subsection{Spaces of Morphisms}
Let $C$ and $V$ be projective $k$-schemes. Let $X=\mathop{\rm Mor}\nolimits(C,V)$ be the $k$-scheme
of morphisms from $C$ to $V$ (see \cite{fgaIV}). Let $f:C\times X\rightarrow V$
be the universal morphism and $\pi:C\times X\rightarrow X$ the
projection. By the functorial properties of the cotangent complex we
get a homomorphism
\[f^{\ast} L^{\scriptscriptstyle\bullet}_V\longrightarrow L^{\scriptscriptstyle\bullet}_{C\times X}\longrightarrow
L^{\scriptscriptstyle\bullet}_{C\times X/C}\]
and a homomorphism
\[\pi^{\ast} L^{\scriptscriptstyle\bullet}_X\longrightarrow L^{\scriptscriptstyle\bullet}_{C\times X/C}.\]
The latter is an isomorphism so that we get an induced homomorphism
\[e:f^{\ast} L_V^{\scriptscriptstyle\bullet}\longrightarrow\pi^{\ast} L_X^{\scriptscriptstyle\bullet}.\] Assume that $C$ has a
dualizing complex $\omega_C$. Then we get a homomorphism
\[e\otimes\omega_C:f^{\ast} L_V^{\scriptscriptstyle\bullet}\mathop{\displaystyle\stackrel{L}{\otimes}}\omega_C\longrightarrow\pi^{\ast}
L_X^{\scriptscriptstyle\bullet}\mathop{\displaystyle\stackrel{L}{\otimes}}\omega_C=\pi^{!} L_X^{\scriptscriptstyle\bullet}\] and by adjunction a homomorphism
\[\pi_{\ast}(e\otimes\omega_C):R\pi_{\ast}(f^{\ast}
L_V^{\scriptscriptstyle\bullet}\mathop{\displaystyle\stackrel{L}{\otimes}}\omega_C)\longrightarrow L_X^{\scriptscriptstyle\bullet}.\] By duality we have
\[R\pi_{\ast}(f^{\ast} L_V^{\scriptscriptstyle\bullet}\mathop{\displaystyle\stackrel{L}{\otimes}}\omega_C)=\dual{(R\pi_{\ast}(f^{\ast} T_V^{\scriptscriptstyle\bullet}))}.\]
Let us denote the resulting homomorphism by
\[\dual{\pi_{\ast}(\dual{e})}:\dual{(R\pi_{\ast}(f^{\ast} T_V^{\scriptscriptstyle\bullet}))}\longrightarrow
L_X^{\scriptscriptstyle\bullet}.\]
\begin{prop} \label{feot} Assume that $C$ is Gorenstein. Then
the homomorphism
$\phi:=\dual{\pi_{\ast}(\dual{e})}$ is an obstruction theory for $X$. If $C$ is a
curve and $V$ is smooth then this obstruction theory is perfect.
\end{prop}
\begin{pf} Let $T$ be an affine scheme, $g:T\to X$ a
morphism, ${\cal J}$ a coherent sheaf on $T$; let $p:C\times T\to T$ be the
projection, $h:C\times T\to V$ the morphism induced by $g$.
By an argument analogous to that in the previous example, we get
$$\mathop{{\rm Ext}}\nolimits^k_{\O_{C\times T}}(h^*L_V^{\scriptscriptstyle\bullet},p^*{\cal J})=
\mathop{{\rm Ext}}\nolimits^k_{\O_C}(g^*E^{\scriptscriptstyle\bullet},{\cal J}).$$
Apply now Theorem \ref{ontoh1}, more precisely the equivalence between
(1) and (3). Choose any square zero extension $\bar T$ of $T$ with
ideal sheaf ${\cal J}$. Then $g$ extends to $\bar g:\bar T\to X$ if and
only if $h$ extends to $\bar h:C\times \bar T\to V$, if and only if
$\phi^*\omega(g)$ is zero in $\mathop{{\rm Ext}}\nolimits^1_{\O_{C\times
T}}(h^*L_V^{\scriptscriptstyle\bullet},p^*{\cal J})$. The extensions, if they exist, form a torsor
under $\mathop{\rm Hom}\nolimits_{\O_{C\times T}}(h^*L_V^{\scriptscriptstyle\bullet},p^*{\cal J})$.
\end{pf}
\section{The Relative Case}
\subsection{Bivariant Theory for Artin Stacks}
For what follows, we need a little bivariant intersection theory for
algebraic stacks that are not necessarily of Deligne-Mumford type.
For simplicity, let us assume that $f:X\to Y$ is a morphism of
algebraic $k$-stacks which is representable. This assumption implies
that whenever
\[\comdia{X'}{}{Y'}{}{}{}{X}{f}{Y}\]
is a cartesian diagram and $Y'$ is a Deligne-Mumford stack satisfying
the condition needed to define its Chow group (see \cite{vistoli}),
then $X'$ is of the same type. The following remarks can be
generalized to any morphism $f$ satisfying this property, e.g.\ any
$f$ which has finite unramified diagonal.
For such an $f:X\to Y$ we define bivariant groups $A^{\ast}(X\to Y)$ by
using the same definition as Definition~5.1 in \cite{vistoli}. Then
just as in [ibid.] one proves that the elements of $A^{\ast}(X\to Y)$
act on Chow groups of Deligne-Mumford stacks.
The same definition as [ibid.] Definition~3.10 applies in case $f:X\to
Y$ is a regular local immersion, and defines a canonical element
$[f]\in A^{\ast}(X\to Y)$ whose action on cycle classes is denoted by
$f^{!}$. This is justified, since Theorems~3.11,~3.12, and~3.13 from
[ibid.] hold with the same proofs in this more general context. In
fact, $[f]$ even commutes with the Gysin morphism for any other local
regular immersion of algebraic stacks.
Similarly, if $f:X\to Y$ is flat, flat pullback of cycles defines a
canonical orientation $[f]\in A^{\ast}(X\to Y)$.
\subsection{The Relative Intrinsic Normal Cone}
We shall now replace the base $\mathop{\rm Spec}\nolimits k$ by an arbitrary smooth (or
more generally pure dimensional, but always of constant dimension)
algebraic $k$-stack $Y$ (not necessarily of Deligne-Mumford type). We
shall consider algebraic stacks $X$ over $Y$ which are of relative
Deligne-Mumford type over $Y$, i.e.\ such that the diagonal
$X\rightarrow X\times_Y X$ is unramified. This assures that
$h^i(L_{X/Y}^{\scriptscriptstyle\bullet})=0$, for all $i>0$ (i.e.\ $h^1(L_{X/Y}^{\scriptscriptstyle\bullet})=0$), so
that $L_{X/Y}$ satisfies Condition~($\star$).
The {\em relative intrinsic normal sheaf }${\frak N}_{X/Y}$ is defined as
$${\frak N}_{X/Y}=h^1/h^0(T_{X/Y}^{\scriptscriptstyle\bullet}).$$ Using local embeddings of $X$ into
schemes smooth over $Y$, we construct as in the absolute case a
subcone stack ${\frak C}_{X/Y}\subset{\frak N}_{X/Y}$ called the {\em relative
intrinsic normal cone }of $X$ over $Y$. If $n=\dim Y$, then
${\frak C}_{X/Y}$ is of pure dimension $n$.
The definition of a {\em relative obstruction theory }is the same as
Definition~\ref{doot}, with $L_X^{\scriptscriptstyle\bullet}$ replaced by $L_{X/Y}^{\scriptscriptstyle\bullet}$. As in
the absolute case the relative intrinsic normal cone embeds as a
closed subcone stack of a vector bundle stack
$${\frak C}_{X/Y}\subset h^1/h^0(\dual{E}),$$ if $E$ is a perfect relative
obstruction theory. (Note that `perfect' means `absolutely perfect'.)
So let $E$ be a perfect obstruction theory for $X$ over $Y$ admitting
global resolutions.
If $X$ is a separated Deligne-Mumford stack then we get a virtual
fundamental class $[X,E^{\scriptscriptstyle\bullet}]\in A_{n+\mathop{\rm rk} E}(X)$ by `intersecting
${\frak C}_X$ with the vertex of $h^1/h^0(\dual{E})$' as in the discussion
preceding Proposition~\ref{vfcigr}.
Consider the following diagram, where $Y$ and $Y'$ are smooth of
constant dimension, $v$ has finite unramified diagonal and $X$ and
$X'$ are separated Deligne-Mumford stacks.
\begin{equation}\label{dfrcpb}
\comdia{X'}{u}{X}{}{}{}{Y'}{v}{Y}
\end{equation}
\begin{prop} \label{ncfprc}
There is a natural morphism
$$\alpha:{\frak C}_{X'/Y'}\longrightarrow{\frak C}_{X/Y}\times_YY'.$$ If
(\ref{dfrcpb}) is cartesian, then $\alpha$ is a closed immersion. If,
moreover, $v$ is flat, then
$\alpha$ is an isomorphism.
\end{prop}
\begin{pf}
Both statements follow immediately from the corresponding
properties of normal cones for schemes.
\end{pf}
\begin{prop}[Pullback]\label{pull}
Let $E\rightarrow L_{X/Y}$ be a perfect obstruction theory for $X$
over $Y$. If (\ref{dfrcpb}) is cartesian then $u^{\ast} E$ is a perfect
obstruction theory for $X'$ over $Y'$. If $E$ has global resolutions
so does $u^{\ast} E$ and for the induced virtual fundamental classes we
have
\[v^{!}[X,E]=[X',u^{\ast} E],\]
at least in the following cases.
\begin{enumerate}
\item $v$ is flat,
\item $v$ is a regular local immersion.
\end{enumerate}
\end{prop}
\begin{pf}
Let $E^{-1}\to E^0$ be a global resolution of $E^{\scriptscriptstyle\bullet}$ and $C$ the cone
induced by ${\frak C}_{X/Y}$ in $E_1$. Let $u^{\ast} E_i=E'_i$, and $D$ the
cone induced by ${\frak C}_{X'/Y'}$ in $E'_1$.
If $v$ is flat we have ${\frak C}_{X'/Y'}=v^{\ast}{\frak C}_{X/Y}$ and hence
$D=v^{\ast} C$ by Proposition~\ref{ncfprc} and the statement follows
>from the fact that $v^{!}$ is a bivariant class; in this case that
$v^{!}$ commutes with $0^{!}_{E_1}$, where $0:X\to E_1$ is the zero
section.
If $v$ is a regular local immersion, let $N=N_{Y'/Y}$ and use
Vistoli's rational equivalence
\[\beta(Y',X)\in W_{\ast}(N\times_Y C)\]
(see Proposition~\ref{vrqsb}) to prove that $v^{!}[C]=[D]$. Then
proceed as before.
\end{pf}
The following are relative versions of the basic properties of virtual
fundamental classes from Section~\ref{otafc}.
\begin{prop}[Locally free obstructions] \label{rulfc}
Let $E^{\scriptscriptstyle\bullet}$ be a perfect relative obstruction theory for $X$ over $Y$
such that $h^{0}(E^{\scriptscriptstyle\bullet})$ is locally free. Assume that $E^{\scriptscriptstyle\bullet}$ has
global resolutions and $X$ is a separated Deligne-Mumford stack, so
that the virtual fundamental class $[X,E^{\scriptscriptstyle\bullet}]$ exists.
\begin{enumerate}
\item If $h^{-1}(E^{\scriptscriptstyle\bullet})=0$, then $X$ is smooth over $Y$ and
$[X,E^{\scriptscriptstyle\bullet}]=[X]$.
\item If $X$ is smooth over $Y$, then $h^1(\dual{E})$ is locally free and
$[X,E^{\scriptscriptstyle\bullet}]=c_r(h^1(\dual{E}))\cdot[X]$, where $r=\mathop{\rm rk} h^1(\dual{E})$.
\end{enumerate}
\end{prop}
\begin{pf} The proofs are the same as in the absolute case (Propositions
\ref{ehs}and \ref{xself}).
\end{pf}
\begin{prop}[Products] \label{rvopf}
Let $E$ be a perfect relative obstruction theory for $X$ over $Y$ and
$F$ a perfect relative obstruction theory for $X'$ over $Y'$. Then
$E\boxplus F$ is a perfect relative obstruction theory for $X\times
X'$ over $Y\times Y'$. If $E$ and $F$ have global resolutions and $X$
and $X'$ are separated Deligne-Mumford stacks, then $E\boxplus F$ has
global resolutions and $X\times X'$ is a separated Deligne-Mumford
stack and we have
\[[X\times X',E\boxplus F]=[X,E]\times[X',F]\]
in $A_{\dim Y+\dim Y'+\mathop{\rm rk} E+\mathop{\rm rk} F}(X\times X')$.
\end{prop}
Let $E$ be a perfect relative obstruction theory for $X$ over $Y$ and
$F$ a perfect relative obstruction theory for $X'$ over $Y$. Let
$v:Z'\rightarrow Z$ be a local complete intersection morphism of
$Y$-stacks that have finite unramified diagonal over $Y$. Let there be
given a cartesian diagram
\[\comdia{X'}{u}{X}{g}{}{f}{Z'}{v}{Z}\]
of $Y$-stacks. Then $E$ and $F$ are {\em compatible over $v$} if there
exists a homomorphism of distinguished triangles
\[\begin{array}{ccccccc}
u^{\ast} E & {\longrightarrow} & F &
{\longrightarrow} & g^{\ast} L_{Z'/Z} &
{\longrightarrow} & u^{\ast} E[1] \\
\ldiag{} & & \ldiag{} & & \ldiag{} & & \rdiag{} \\
u^{\ast} L_{X/Y} & {\longrightarrow} & L_{X'/Y} &
{\longrightarrow} & L_{X'/X} &
{\longrightarrow} & u^{\ast} L_{X/Y}[1] .
\end{array}\]
in $D(\O_{X'})$.
\begin{prop}[Functoriality]\label{rvofp}
If $E$ and $F$ are compatible over $v$, then
\[v^{!}[X,E]=[X',F],\]
at least if $v$ is smooth or $Z'$ and $Z$ are smooth over $Y$.
\end{prop}
\begin{pf}
The proof is the same as that of Proposition~\ref{fcrp}.
\end{pf}
\begin{numex} \label{rfe}
Consider a cartesian diagram
$$\comdia{X}{j}{V}{g}{}{h}{Y}{i}{W}$$
of algebraic stacks, where $i$ and $j$ are local immersions and $h$
has unramified diagonal. We have a canonical homomorphism
$$\phi:j^{\ast} L_{V/W}\longrightarrow L_{X/Y},$$
which makes $j^{\ast} L_{V/W}$ a relative obstruction theory for $X$
over $Y$. To see this, it suffices to prove that
$h^{-1}(F^{\scriptscriptstyle\bullet})=h^0(F^{\scriptscriptstyle\bullet})=0$, where $F^{\scriptscriptstyle\bullet}$ is the cone of $\phi$. But
$F^{\scriptscriptstyle\bullet}$ is isomorphic to the cone of the homomorphism
$$g^{\ast} L_{Y/W}\longrightarrow L_{X/V},$$ so this is indeed true.
Now if $V$ and $W$ are smooth, then $h^i(L_{V/W})=0$ for all
$i\not=-1,0$ and $j^{\ast} L_{V/W}$ is a perfect obstruction theory. In
particular, we get a virtual fundamental class
$$[X,j^{\ast} L_{V/W}]\in A_{\dim Y+\dim V-\dim W}(X),$$
if $Y$ is pure dimensional, $j^{\ast} L_{V/W}$ has global resolutions
and $X$ is a separated Deligne-Mumford stack.
If, in addition, $i$ is a regular local immersion with normal bundle
$N_{Y/W}$, the normal cone $C_{X/V}$ of $X$ in $V$ is a closed subcone
of $g^{\ast} N_{Y/W}$ and intersecting it with the zero section $0$ of
$g^{\ast} N_{Y/X}$ gives a class
$$0^{!}[C_{X/V}]\in A_{\dim Y+\dim V-\dim W}(X).$$
The proof that
$$0^{!}[C_{X/V}]=[X,j^{\ast} L_{V/W}]$$
is similar to the proof of Proposition~\ref{pull}.
\end{numex}
|
1996-01-16T06:20:21 | 9601 | alg-geom/9601012 | en | https://arxiv.org/abs/alg-geom/9601012 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9601012 | Maarten Bergvelt | M.J. Bergvelt and J.M. Rabin | Super Curves, their Jacobians, and super KP equations | AMSLaTeX v 1.2 | null | null | null | null | We study the geometry and cohomology of algebraic super curves, using a new
contour integral for holomorphic differentials. For a class of super curves
(``generic SKP curves'') we define a period matrix. We show that the odd part
of the period matrix controls the cohomology of the dual curve. The Jacobian of
a generic SKP curve is a smooth supermanifold; it is principally polarized,
hence projective, if the even part of the period matrix is symmetric. In
general symmetry is not guaranteed by the Riemann bilinear equations for our
contour integration, so it remains open whether Jacobians are always projective
or carry theta functions.
These results on generic SKP curves are applied to the study of
algebro-geometric solutions of the super KP hierarchy. The tau function is
shown to be, essentially, a meromorphic section of a line bundle with trivial
Chern class on the Jacobian, rationally expressible in terms of super theta
functions when these exist. Also we relate the tau function and the Baker
function for this hierarchy, using a generalization of Cramer's rule to the
supercase.
| [
{
"version": "v1",
"created": "Mon, 15 Jan 1996 18:27:48 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bergvelt",
"M. J.",
""
],
[
"Rabin",
"J. M.",
""
]
] | alg-geom | \section{Introduction.}
In this paper we study algebraic super curves with a view towards
applications
to super Kadomtsev-Petviashvili hierarchies (SKP). We deal from the
start with
super curves $X$ over a nonreduced ground ring $\Lambda$, i.e., our
curves
carry global nilpotent constants. This has as an advantage, compared
to super
curves over
the complex numbers $\mathbb C$, that our curves can be nonsplit, but
this
comes at the price of some technical complications. The main problem
is that
the cohomology groups of coherent sheaves on our curves should be
thought of as
finitely generated modules over the ground ring $\Lambda$, instead of
vector
spaces over $\mathbb C$. In general these modules are of course not
free.
Still we have in this situation Serre duality, as explained in
Appendix
\ref{app:dualSerredual}, the dualizing sheaf being the (relative)
berezinian
sheaf $\Berx$. In applications to SKP there occurs a natural class of
super
curves that we call generic SKP curves. For these curves the most
important
sheaves, the structure sheaf and the dualizing sheaf, have free
cohomology. In
the later part of the paper we concentrate on these curves.
Super curves exhibit a remarkable duality uncovered in
\cite{DoRoSc:SupModSpaces}.
The projectivized cotangent bundle of any $N=1$ super curve has the
structure of an $N=2$ super Riemann surface (SRS), and super curves
come in
dual pairs $X,\hat{X}$ whose associated $N=2$ SRSs coincide.
Further, the ($\Lambda$-valued) points of a super curve can
be identified with the effective divisors of degree 1 on its dual.
Ordinary
$N=1$ SRSs, widely studied in the context of super string theory, are
self dual
under this duality. By the resulting identification of points with
divisors they enjoy many of the properties that distinguish Riemann
surfaces from higher-dimensional varieties.
By exploiting the duality we extend this good behaviour to all
super curves.
In particular we define for all super curves a contour integration
for sections
of $\Berx$, the holomorphic differentials in this situation. The
endpoints of a
super contour turn out to be not $\Lambda$-points of our super
curve, but
rather irreducible divisors, i.e., $\Lambda$-points
on the dual curve! For SRSs these notions are the same and our
integration is a
generalization of the procedure already known for SRSs. We use this
to prove
Riemann bilinear relations, connecting in this situation periods of
holomorphic
differentials on our curve $X$ with those on its dual curve.
In case the cohomology of the structure sheaf is free, e.g. if $X$ is
a generic
SKP curve, we can define a period matrix and use this to define the
Jacobian
of $X$ as the quotient of a super vector space by a lattice generated
by the
period matrix. In this case the Jacobian is a smooth supermanifold. A
key
question is whether the Jacobian of a generic SKP curve admits ample
line
bundles (and hence embeddings in projective super space),
whose sections could serve as the super analog of theta functions.
We show that the symmetry of the even part of the period matrix
(together with the automatic positivity of the imaginary part of the
reduced matrix) is sufficient for this, and construct the super
theta functions in this case.
We derive some geometric necessary and sufficient conditions for
this symmetry to hold, but it is not an automatic consequence of the
Riemann bilinear period relations in this super context.
Neither do we know an explicit example in which the symmetry fails.
The usual proof that symmetry of the period matrix is necessary for
existence of a (principal) polarization also fails because crucial
aspects of Hodge theory, particularly the Hodge decomposition of
cohomology, do not hold for supertori.
The motivation for writing this paper was our wish to generalize
the theory of the
algebro-geometric solutions to the KP hierarchy of nonlinear PDEs,
as described in \cite{SeWi:LpGrpKdV} and references
therein, to the closest supersymmetric analog, the ``Jacobian"
super KP hierarchy of Mulase and Rabin \cite{Mu:Jac,Ra:GeomSKP}.
In the super KP case the geometric data leading to a solution include
a super
curve $X$ and a line bundle $\mathcal L$ with vanishing cohomology
groups over
$X$. For such a line bundle to exist the super curve $X$ must have a
structure
sheaf $\mathcal O_X$ such that the associated split sheaf ${\mathcal O}_{X}\spl$, obtained
by
putting the global nilpotent constants in $\Lambda$ equal to zero, is
a direct
sum ${\mathcal O}_{X}\spl={\mathcal O}_{X}^{\text{red}}\oplus \mathcal N$, where ${\mathcal O}_{X}^{\text{red}}$ is the
structure sheaf
of the underlying classical curve $X^{\text{red}}$ and $\mathcal N$
is an
invertible ${\mathcal O}_{X}^{\text{red}}$-sheaf of degree zero. We call such an $X$ an SKP
curve, and
if moreover $\mathcal N$ is not isomorphic to ${\mathcal O}_{X}^{\text{red}}$ we call $X$ a
generic
SKP curve.
The Jacobian SKP hierarchy describes linear
flows $\mathcal L(t_i)$ on the Jacobian of $X$ (with even and odd
flow
parameters).
The other known SKP hierarchies, of Manin--Radul
\cite{ManRad:SusyextKP} and
Kac--van de
Leur \cite{KavdL:SuperBoson} , describe flows on the universal
Jacobian over
the moduli space
of super curves, in which $X$ as well as $\mathcal L$ vary with the
$t_i$
\cite{Ra:GeomSKP}.
These are outside the scope of this paper, although we hope to
return to them elsewhere.
As in the non-super case, the basic objects in the theory are the
(even and odd) Baker functions, which are sections of $\mathcal
L(t_i)$
holomorphic except for a single simple pole, and a tau
function which is a section of the super determinant (Berezinian)
bundle over a super Grassmannian $\mathcal S\text{gr}$.
In contrast to the non-super case, we show that the Berezinian
bundle has trivial Chern class, reflecting the fact that the
Berezinian is a ratio of ordinary determinants.
The super tau function descends, essentially, to $\operatorname{Jac}(X)$ as a section
of a
bundle with trivial Chern class also, and can be rationally expressed
in
terms of super theta functions when these exist (its reduced part is
a ratio of ordinary tau functions).
We also obtain a formula for the even and odd Baker functions in
terms of the tau function, confirming that one must know the tau
function for the more general Kac--van de Leur flows to compute the
Baker functions for even the Jacobian flows in this way, cf.
\cite{DoSc:SuperGrass,Takama:GrassmannSKP}.
For this we need a slight extension of Cramer's rule for solving
linear equations in even and odd variables, which is developed in an
Appendix via the theory of quasideterminants.
In another Appendix we use the Baker functions found in
\cite{Ra:SupElliptic} for Jacobian flow in the case of super elliptic
curves to
compute the corresponding tau function.
Among the problems remaining open we mention the following.
First, to obtain a sharp criterion for when a super Jacobian admits
ample line bundles --- perhaps always?
Second, the fact that generic SKP curves have free cohomology is a
helpful simplification which allows us to represent their period
maps by matrices and results in their Jacobians being smooth
supermanifolds.
However, our results should generalize to arbitrary super curves with
more singular Jacobians.
Finally, one should study the geometry of the universal Jacobian and
extend our analysis to the SKP system of Kac--van de Leur.
\section{Super curves and their Jacobians.}
\subsection{Super curves.}
Fix a Grassmann algebra $\Lambda$ over $\mathbb C$; for instance we
could take
$\Lambda=\mathbb C\,[\beta_1,\beta_2,\dots,\beta_n]$, the polynomial
algebra
generated by $n$ odd indeterminates. Let $(\bullet,\Lambda)$
be the super scheme $\operatorname{Spec} \Lambda $, with underlying topological
space a
single point $\bullet$.
A smooth compact connected complex super curve over $\Lambda$ of
dimension
$(1|N)$ is a pair $(X,\mathcal O_X)$, where $X$ is a topological space and
$\mathcal O_X$ is a
sheaf of super commutative $\Lambda$-algebras
over $X$, equipped with a structure morphism $(X,\mathcal O_X)\to
(\bullet,\Lambda)$,
such that
\begin{enumerate}
\item $(X,{\mathcal O}_{X}^{\text{red}})$ is a smooth compact connected complex curve,
algebraic or
holomorphic, depending on the category one is working in. Here
${\mathcal O}_{X}^{\text{red}}$ is the
reduced sheaf of $\mathbb C$-algebras on $X$ obtained by quotienting
out the
nilpotents in the structure sheaf $\mathcal O_X$,
\item For suitable open sets $U_\alpha\subset X$ and suitable
linearly
independent odd elements $\theta_\alpha^i$ of $\mathcal O_X(U_\alpha)$ we have
$$
\mathcal O_X(U_\alpha)={\mathcal O}_{X}^{\text{red}}\otimes
\Lambda[\theta_\alpha^1,\theta_\alpha^2,\dots,\theta_\alpha^N].
$$
\end{enumerate}
The $U_\alpha$\rq s above are called coordinate neighborhoods of
$(X,\mathcal O_X)$ and
$(z_\alpha,\theta_\alpha^1,\theta_\alpha^2,\dots,\theta_\alpha^N)$
are called
local coordinates for $(X,\mathcal O_X)$, if $z_\alpha$ (mod nilpotents) is a
local
coordinate for $(X,{\mathcal O}_{X}^{\text{red}})$. On overlaps of coordinate neighborhoods
$U_\alpha\cap U_\beta$ we have
\begin{equation}\label{eq:coordchange}
\begin{split}
z_\beta &= F_{\beta\alpha}(z_\alpha,\theta_\alpha^j),\\
\theta_\beta^i &= \Psi_{\beta\alpha}^i(z_\alpha,\theta_\alpha^j).
\end{split}
\end{equation}
Here the $F_{\beta\alpha}$ are even functions and
$\Psi_{\beta\alpha}^i$ odd
ones, holomorphic or algebraic depending on the category we are
using.
\begin{exmpl}\label{exmpl:split} A special case is formed by the {\it
split}
super curves. For $N=1$ they are given by transition functions
\begin{equation}\label{eq:splitcoordchange}
\begin{split}
z_\beta &= f_{\beta\alpha}(z_\alpha),\\
\theta_\beta &= \theta_\alpha B_{\beta\alpha}(z_\alpha),
\end{split}
\end{equation}
with $f_{\beta\alpha}(z_\alpha),B_{\beta\alpha}(z_\alpha)$ even
holomorphic (or
algebraic) functions that are independent of the nilpotent constants
in
$\Lambda$.
So in this case the $f_{\beta\alpha}$ are the transition functions
for ${\mathcal O}_{X}^{\text{red}}$
and $\mathcal O_X={\mathcal O}_{X}^{\text{red}}\otimes\Lambda \mid \mathcal N\otimes \Lambda$, where
$\mathcal
N$ is the ${\mathcal O}_{X}^{\text{red}}$-module with transition functions
$B_{\beta\alpha}(z_\alpha)$. Here and henceforth we denote by a
vertical $\mid$
a direct sum of free $\Lambda$-modules, with on the left an evenly
generated
summand and on the right an odd one.
To any super curve $(X,\mathcal O_X)$ there is canonically associated a split
curve
$(X,{\mathcal O}_{X}\spl)$ over $\mathbb C$: just take
${\mathcal O}_{X}\spl=\mathcal O_X\otimes_{\Lambda}
\Lambda/\mathfrak{m}=\mathcal O_X/\mathfrak{m}\mathcal O_X$, with
$\mathfrak m=\langle \beta_1,\dots,\beta_n\rangle$ the maximal ideal
of
nilpotents in $\Lambda$. There is a functor from the category of
$\mathcal O_X$-modules
to the category of ${\mathcal O}_{X}\spl$-modules that associates to a sheaf
$\mathcal F$
the {\it associated split sheaf} $\mathcal F^{\text{split}}=\mathcal
F/\mathfrak {m}\mathcal F$.\qed\end{exmpl}
\smallskip
A $\Lambda$-point of a super curve $(X,\mathcal O_X)$ is a morphism
$\phi:(\bullet,\Lambda)\to (X,\mathcal O_X)$ such that the composition with
the structural morphism \linebreak $(X,\mathcal O_X)\to(\bullet,\Lambda)$ is
the
identity (of $(\bullet,\Lambda)$). Locally, in an open set
$U_\alpha$
containing $\phi(\bullet)$, a $\Lambda$-point is given by specifying
the
images under the even $\Lambda$-homo\-mor\-phism
$\phi^\sharp:\mathcal O_X(U_\alpha)\to
\Lambda$
of the local coordinates:
$p_\alpha=\phi^\sharp(z_\alpha),\pi^i_\alpha
=\phi^\sharp( \theta^i_\alpha)$ . The local parameters
$(p_\alpha,\pi^i_\alpha)$ of a $\Lambda$-point transform precisely as
the
coordinates do, see (\ref{eq:coordchange}). By quotienting out
nilpotents in a
$\Lambda$-point $(p_\alpha,\pi^i_\alpha)$ we obtain a complex number
$p_\alpha^{\text{red}}$, the coordinate of the reduced point of
$(X,{\mathcal O}_{X}^{\text{red}})$
corresponding to the $\Lambda$-point $(p_\alpha,\pi^i_\alpha)$.
\subsection{Duality and $N=2$ curves.}\label{ss:DualN=2curves}
Our main interest is the theory of $N=1$ super curves but as a
valuable tool
for the study of these curves we make use of $N=2$ curves as well in
this
paper. Indeed, as is well known,
\cite{DoRoSc:SupModSpaces,Schwarz:SuperanalogsSYMPLCONT}, one can
associate in
a canonical way to an $N=1$ curve an (untwisted) super conformal
$N=2$ curve,
as we will now recall. The introduction of the super conformal $N=2$
curve
clarifies the whole theory of $N=1$ super curves.
Let from now on $(X,\mathcal O_X)$ be an $N=1$ super curve. Any invertible
sheaf
$\mathcal E$ for $(X,\mathcal O_X)$ and any extension
of $\mathcal E$ by the structure sheaf:
\begin{equation*}
0\rightarrow \mathcal O_X\rightarrow \hat{\mathcal E}\rightarrow \mathcal
E\rightarrow
0,\label{eq:extension}
\end{equation*}
defines in the obvious way an $N=2$ super curve $(X,\hat{\mathcal
E})$. It has
local coordinates $(z_\alpha,\theta_\alpha,\rho_\alpha)$, where
$(z_\alpha,\theta_\alpha)$ are local coordinates for $(X,\mathcal O_X)$. On
overlaps we
will have
\begin{equation}\label{eq:coordchangen=2}
\begin{split}
z_\beta &= F_{\beta\alpha}(z_\alpha,\theta_\alpha),\\
\theta_\beta &= \Psi_{\beta\alpha}(z_\alpha,\theta_\alpha),\\
\rho_\beta &= H_{\beta\alpha}(z_\alpha,\theta_\alpha)
\rho_\alpha + \phi_{\beta\alpha}
(z_\alpha,\theta_\alpha).
\end{split}
\end{equation}
(So $H_{\beta\alpha}(z_\alpha,\theta_\alpha)$ is the transition
function for
the generators of the invertible sheaf $\mathcal E$.) We want to
choose
the extension (\ref{eq:extension}) such that $(X,\hat{\mathcal E})$
is {\it
super conformal}, in the sense that the local differential form
$\omega_\alpha=dz_\alpha- d\theta_\alpha \rho_\alpha$ is globally
defined up
to a scale factor. Now
$$
\omega_\beta =dz_\beta- d\theta_\beta \rho_\beta
=dz_\alpha
(\frac{\partial F}{\partial z_\alpha} - \frac{\partial
\Psi}{\partial
z_\alpha}\rho_\beta) -
d\theta_\alpha
(-\frac{\partial F}{\partial \theta_\alpha} + \frac{\partial
\Psi}{\partial
\theta_\alpha}\rho_\beta).
$$
(Here we suppress the subscripts on $F$ and $\Psi$, as we will do
below.) We
see that for $\hat{\mathcal E}$ to be super conformal we need
$$
\rho_\alpha=\frac{(-\frac{\partial F}{\partial \theta_\alpha} +
\frac{\partial
\Psi}{\partial \theta_\alpha}\rho_\beta)}{
(\frac{\partial F}{\partial z_\alpha} - \frac{\partial \Psi}{\partial
z_\alpha}\rho_\beta)},
$$
or
\begin{equation}\label{eq:transformrho}
\rho_\beta=\frac{
(\frac{\partial F}{\partial \theta_\alpha} + \frac{\partial
F}{\partial
z_\alpha}\rho_\alpha)}
{(\frac{\partial \Psi}{\partial \theta_\alpha} - \frac{\partial
\Psi}{\partial
z_\alpha}\rho_\alpha)}.
\end{equation}
Conversely one checks that if (\ref{eq:transformrho}) holds for all
overlaps
the cocycle condition is satisfied and that we obtain in this manner
an $N=2$
super curve. To show that this super curve is an extension as in
(\ref{eq:extension}), it is useful to note that
(\ref{eq:transformrho}) can
also be written as
\begin{equation}\label{eq:transformrho2}
\rho_\beta=\ber\begin{pmatrix}\partial_z F&\partial_z \Psi\\
\partial_\theta F&\partial_\theta\Psi
\end{pmatrix}
\rho_\alpha+\frac{\partial_\theta F}{\partial_\theta\Psi}.
\end{equation}
The homomorphism $\ber$ is defined in Appendix
\ref{app:Lineqsupercat}, see
\eqref{eq:defberber*}.
Recall that the local generators $f_\alpha$ of the dualizing sheaf
(see
Appendix \ref{app:dualSerredual}) $\mathcal{B}\text{er}_X$ of
$(X,\mathcal O_X)$
transform as
\begin{equation}\label{eq:transfBer}
f_\beta=\ber\begin{pmatrix}\partial_z F&\partial_z \Psi\\
\partial_\theta F&\partial_\theta\Psi
\end{pmatrix}f_\alpha.
\end{equation}
If we denote by $\Cox$ the structure sheaf of the super conformal
$N=2$ super
curve just constructed, we see that we have an exact sequence
\begin{equation}
0\rightarrow\mathcal O_X\rightarrow\Cox\rightarrow
\mathcal{B}\text{er}_X\rightarrow0.
\label{eq:extberbystruct}\end{equation}
$\Cox$ is the only extension of $\Berx$ by the structure sheaf that
is super
conformal. This sequence is {\it trivial} if it isomorphic
(\cite{HiSt:HomAlg})
to a split sequence.
\begin{defn}\label{def:projected}
A super curve is called {\it{projected} }if there is a cover of $X$
such that
the transition functions $F_{\beta\alpha}$ in (\ref{eq:coordchange})
are
independent of the odd coordinates $\theta_\alpha^j$.
\end{defn}
For projected curves we have a
projection morphism $(X,\mathcal O_X)\to (X,{\mathcal O}_{X}^{\text{red}}\otimes \Lambda)$
corresponding to the
sheaf inclusion ${\mathcal O}_{X}^{\text{red}}\otimes \Lambda\to \mathcal O_X$.
This inclusion can be defined only for projected curves.
A projected super curve has a $\Cox$ that is a trivial extension but
the
converse is not true, as we will see when we discuss super Riemann
surfaces in
subsection
\ref{ss:SRS}.
The relation between projectedness of $(X,\mathcal O_X)$ and the triviality of
the
extension defining $(X,\Cox)$ is discussed in detail in subsection
\ref{ss:Symmperiodmatrices}.
\begin{exmpl}\label{exmpl:splitber}
If $(X,\mathcal O_X)$ is split, (\ref{eq:transfBer}) becomes
\begin{equation}\label{eq:transfBersplit}
f_\beta=\frac{\partial_z f_{\beta\alpha}}{B_{\beta\alpha}} f_\alpha.
\end{equation}
This means that in this case $\Berx=\mathcal K\mathcal N\inv\otimes
\mathcal O_X=\mathcal K\mathcal N\inv\otimes\Lambda\mid \mathcal K\otimes
\Lambda$,
where $\mathcal K$ is the canonical sheaf for ${\mathcal O}_{X}^{\text{red}}$.
Split curves are projected and the sequence (\ref{eq:extberbystruct})
becomes
trivial. As an ${\mathcal O}_{X}^{\text{red}}$-module we have $\Cox=({\mathcal O}_{X}^{\text{red}} \oplus \mathcal
K)\otimes\Lambda\mid (\mathcal N\oplus \mathcal K\mathcal
N\inv)\otimes
\Lambda$.
\qed\end{exmpl}
\smallskip
The map $\Cox\to \Berx$ is locally described by the differential
operator
$\Dc^\alpha=\partial_{\rho_\alpha}$.
Indeed, the operator $\Dc^\alpha$ transforms homogeneously,
$\Dc^\beta=\ber\begin{pmatrix}\partial_z F&\partial_z \Psi\\
\partial_\theta F&\partial_\theta\Psi
\end{pmatrix}\inv \Dc^\alpha$, so this
defines a
global $(0\mid 1)$ dimensional distribution $\Dc$ and the quotient of
$(X,
\mathcal {CO}_X)$ by this distribution is precisely $(X,\mathcal O_X)$.
Now the distribution $\Dc$ annihilates the 1-form $\omega$ used to
find $\Cox$.
This form locally looks like $\omega_\alpha=dz_\alpha
-d\theta_\alpha
\rho_\alpha$ and its kernel is generated by $\Dc^\alpha$ and a
second
operator
$\hat{D}_{\mathcal C}^\alpha=\partial_{\theta_\alpha}+
\rho_\alpha\partial_{z_\alpha}$. (The
operators that we call $\Dc$ and $\hat{D}_{\mathcal C}$ are in the literature
also denoted by $D^+$ and $D^-$, cf. \cite{DoRoSc:SupModSpaces})
To study the result of ``quotienting by the distribution $\hat{D}_{\mathcal C}$''
we
introduce in each coordinate neighborhood $U_\alpha$ new coordinates:
\begin{equation*}
\begin{split}
\hat z_\alpha &= z_\alpha-\theta_\alpha\rho_\alpha,\\
\hat \theta_\alpha &=\theta_\alpha ,\\
\hat \rho_\alpha &= \rho_\alpha.
\end{split}
\end{equation*}
In the sequel we will drop the hats $\hat {}$ on $\theta$ and $\rho$,
hopefully
not causing too much confusion.
In these new coordinates we have
$$
\hat{D}_{\mathcal C}^\alpha= \partial_{\theta_\alpha},\quad
\Dc^\alpha=\partial_{\rho_\alpha}+ \theta_\alpha\partial_{\hat
z_\alpha}.
$$
So the kernel of $\hat{D}_{\mathcal C}$ consists locally of functions of
$\hat z_\alpha,\rho_\alpha$. To see that this makes global sense we
observe
that
\begin{equation}\label{eq:coordtransfdualcurve}
\begin{split}
\hat z_\beta &= F(\hat z_\alpha,\rho_\alpha) + \frac{DF
(\hat z_\alpha,\rho_\alpha)}{D\Psi(\hat
z_\alpha,\rho_\alpha)}\Psi(\hat
z_\alpha,\rho_\alpha),\\
\rho_\beta &=\frac{DF(\hat
z_\alpha,\rho_\alpha)}{D\Psi(\hat z_\alpha,\rho_\alpha)},
\end{split}
\end{equation}
where $D=\partial_\theta +\theta\partial_z$. The details of this
somewhat
unpleasant calculation are left to the reader. From
(\ref{eq:coordtransfdualcurve}) we see that $\Cox$ contains the
structure sheaf
$\hat{\mathcal O}_X$ of another $N=1$ super curve: $\hat{\mathcal O}_X$ is the sheaf of
$\Lambda$-algebras locally generated by $\hat z_\alpha,\rho_\alpha$.
We call
$\hat X=(X,\hat{\mathcal O}_X)$ the {\em{dual curve}} of $(X,\mathcal O_X)$. We have
\begin{equation}\label{eq:extberbystructdual}
0\rightarrow\hat{\mathcal O}_X\rightarrow\Cox
\overset{\hat{D}_{\mathcal C}}\rightarrow\mathcal B\text{{\^e}r}_{{X}}\rightarrow0,
\end{equation}
where $\mathcal B\text{{\^e}r}_{{X}}$ is the dualizing sheaf of the dual curve. One can
show that
the dual curve of the dual curve is the original curve, thereby
justifying the
terminology.
\begin{exmpl}\label{exmpl:dualsplit}
We continue the discussion of split curves. In this case
(\ref{eq:coordtransfdualcurve}) becomes
\begin{equation}\label{eq:coordtransfdualcurvesplit}
\begin{split}
\hat z_\beta &= f(\hat z_\alpha),\\
\rho_\beta &=\frac{\partial_{\hat z}f(\hat
z_\alpha)}{B(\hat z_\alpha)}\rho_\alpha,
\end{split}
\end{equation}
So the dual split curve is $\hat{\mathcal O}_X^{\text{split}}={\mathcal O}_{X}^{\text{red}}\otimes\Lambda\mid
\mathcal
K\mathcal N\inv \otimes \Lambda$. The Berezinian sheaf for the dual
split curve
has generators that satisfy
\begin{equation}\label{eq:transfBersplitdual}
\hat f_\beta=B(z_\alpha) \hat f_\alpha.
\end{equation}
This means that $\mathcal B\text{{\^e}r}_{{X}}=\mathcal N\otimes \hat{\mathcal O}_X=\mathcal N\otimes
\Lambda\mid \mathcal K\otimes \Lambda$. \qed
\end{exmpl}
\smallskip
A very useful geometric interpretation of the dual curve exists, cf.
\cite{DoRoSc:SupModSpaces,Schwarz:SuperanalogsSYMPLCONT}: the points
(i.e., the
$\Lambda$-points) of the dual curve correspond precisely to the
irreducible
divisors of the original curve and vice versa, as we will presently
discuss. In
subsection \ref{ss:IntegraOnSupCurve} we will see that irreducible
divisors
are the limits that occur in contour integration on a super curve.
An irreducible divisor (for $\mathcal O_X$) is locally given by an even
function
$P_\alpha=z_\alpha-\hat z_\alpha -
\theta_\alpha \rho_\alpha\in \mathcal O_X(U_\alpha)$, where $\hat z_\alpha$
and
$\rho_\alpha$ are now respectively even and odd constants, i.e.,
elements of
$\Lambda$. Two divisors $P_\alpha,P_\beta$ defined on coordinate
neighborhoods
$U_\alpha$ and $U_\beta$, respectively,
are said to correspond to each other on the overlap if
\begin{equation}
P_\beta(z_\beta,\theta_\beta)=P_\alpha(z_\alpha,\theta_\alpha)
g(z_\alpha,\theta_\alpha), \quad g(z_\alpha,\theta_\alpha)\in
\mathcal O^\times_{X,\text{ev}}(U_\alpha\cap U_\beta).\label{eq:corresponddiv}
\end{equation}
(If $R$ is a ring (or sheaf of rings) ${R}^\times$ is the set of
invertible
elements.)
\begin{lem}\label{lem:roots} Let $(U, \mathcal O(U))$ be a $(1\mid
1)$
dimensional super domain with coordinates $(z,\theta)$ and let
$f(z,\theta)\in
\mathcal O(U)$. Then, with $D=\partial_\theta+\theta\partial_z$,
\begin{equation*}
f(z,\theta)=(z-\hat z -\theta
\rho)g(z,\theta)\quad\Leftrightarrow\quad f(\hat
z,\rho)=0,Df(\hat z,\rho)=0,
\end{equation*}
for $g(z,\theta)$ in $\mathcal O(U)$.
\end{lem}
Applying Lemma \ref{lem:roots} to (\ref{eq:corresponddiv}) we find
\begin{equation*}
\begin{split}
P_\beta(F(\hat z_\alpha,\rho_\alpha),
\Psi(\hat z_\alpha,\rho_\alpha))
&= F(\hat z_\alpha,\rho_\alpha)-\hat z_\beta-
\Psi(\hat z_\alpha,\rho_\alpha)\rho_\beta=0,\\
DP_\beta(F(\hat z_\alpha,\rho_\alpha),
\Psi(\hat z_\alpha,\rho_\alpha))
&=DF(\hat z_\alpha,\rho_\alpha)-
D\Psi(\hat z_\alpha,\rho_\alpha)\rho_\beta=0.
\end{split}
\end{equation*}
{}From this one sees that the parameters $(\hat
z_\alpha,\rho_\alpha)$ in the
local expression for an irreducible divisor transform as in
(\ref{eq:coordtransfdualcurve}), so they are $\Lambda$-points of the
dual
curve.
The $N=2$ super conformal super curve canonically associated to a
super curve
has a structure sheaf $\Cox$ that comes equipped with two sheaf maps
$\Dc$ and
$\hat{D}_{\mathcal C}$ with kernels the structure sheaves $\mathcal O_X$ and $\hat{\mathcal O}_X$ of the
original
super curve and its dual. The intersection of the kernels is the
constant sheaf
$\Lambda$. The images of these maps are the dualizing sheaves
$\Berx$ and $\mathcal B\text{{\^e}r}_{{X}}$. In fact we can restrict $\Dc,\hat{D}_{\mathcal C} $ to the
subsheaves
$\hat{\mathcal O}_X$ and $\mathcal O_X$, respectively, without changing the images. This
gives us
exact sequences
\begin{equation}\label{eq:DandDhatseq}
\begin{split}
0\rightarrow \Lambda\rightarrow &\mathcal O_X\overset{\hat
D}\rightarrow\mathcal B\text{{\^e}r}_{{X}}\rightarrow0,\\
0\rightarrow \Lambda\rightarrow
&\hat{\mathcal O}_X\overset{D}\rightarrow\Berx\rightarrow0,
\end{split}
\end{equation}
with $D=\Dc |_{\hat{\mathcal O}_X}$ and $\hat D=\hat{D}_{\mathcal C}|_{\mathcal O_X}$. Just as the sheaf
maps
$\Dc,\hat{D}_{\mathcal C}$ have local expressions as differential operators, also
their
restrictions are locally expressible in terms of differential
operators: if
$\{f_\alpha(z_\alpha,\theta_\alpha)\}$ is a section of $\mathcal O_X$ then the
corresponding section $\{(\hat{D}_{\mathcal C} f_\alpha)(\hat
z_\alpha,\rho_\alpha)\}$ of
$\mathcal B\text{{\^e}r}_{{X}}$ is given by
$$
\hat D f_\alpha(\hat
z_\alpha,\rho_\alpha)=
[(\partial_\theta+\theta\partial_z)f_\alpha]|_{z_\alpha=\hat
z_\alpha,\theta_\alpha=\rho_\alpha}.
$$
Similarly, if $\{\hat f_\alpha(\hat z_\alpha,\rho_\alpha)\}$ is a
section of
$\hat{\mathcal O}_X$ then the corresponding section of $\Berx$ is
$$
{D}\hat{f}_\alpha(
z_\alpha,\theta_\alpha)=[(\partial_\rho+\rho\partial_{\hat
z})\hat f_\alpha]|_{\hat z_\alpha=
z_\alpha,\rho_\alpha=\theta_\alpha}.
$$
We summarize the relationships between the various sheaves and sheaf
maps in
the following commutative diagram (of sheaves of $\Lambda$-algebras):
\begin{equation}
\begin{CD}\label{eq:bigcd}
{} @. 0 @. 0 @. {} @. {}
\\
@. @VVV @VVV @. @.
\\
0 @>>> \Lambda@>>>\hat{\mathcal O}_X@>{ D}>>\Berx @>>>0 \\
@. @VVV @VVV @\vert @.
\\
0 @>>> \mathcal O_X@>>>\Cox@>{\Dc}>>\Berx @>>>0 \\
@. @V\hat{D}VV @V\hat{D}_{\mathcal C} VV @.
@. \\
{} @. \mathcal B\text{{\^e}r}_{{X}} @= \mathcal B\text{{\^e}r}_{{X}} @.{}
@.{}\\
@. @VVV @VVV @. @.
\\
{} @. 0 @. 0 @. {} @. {}
\\
\end{CD}
\end{equation}
\medskip
We conclude this subsection with the remark that the dualizing sheaf
${\mathcal B\text{er}(\Cox)}$ of the super conformal super curve
$(X,\Cox)$
associated to a super curve $(X,\mathcal O_X)$ is trivial, making $(X,\Cox)$ a
super
analog of an elliptic curve or a Calabi-Yau manifold, cf.
\cite{DistNelson:SemiRigidSGra}. In fact, this statement is true for
any $N=2$
super curve $(X,\mathcal E)$ where $\mathcal E$ is an extension of
$\Berx$ by
the structure sheaf: if we have
\begin{equation*}
0 \to \mathcal O_X \to{\mathcal E} \to \Berx \to 0,
\end{equation*}
then $\mathcal E$ has local generators
$(z_\alpha,\theta_\alpha,\rho_\alpha)$
on $U_\alpha$, and on overlaps we get
\begin{equation} \label{eq:generalextensionberbystruct}
\begin{split}
z_\beta &= F_{\beta\alpha}(z_\alpha,\theta_\alpha),\\
\theta_\beta &= \Psi_{\beta\alpha}(z_\alpha,\theta_\alpha),\\
\rho_\beta &=
\Phi_{\beta\alpha}(z_\alpha,\theta_\alpha,\rho_\alpha)=
\ber(J(z,\theta))\rho_\alpha + \phi_{\beta\alpha}
(z_\alpha,\theta_\alpha),
\end{split}
\end{equation}
where $\ber(J(z,\theta))$ is the Berezinian of the super Jacobian
matrix of the
change of $(z,\theta)$ coordinates; this is precisely the transition
function
for $\Berx$, see \eqref{eq:transfBer}. Then the super Jacobian
matrix
\begin{multline*}
J(z,\theta,\rho)=\ber
\begin{pmatrix}
\partial_z F &\partial_z \Psi &\partial_z
\Phi\\
\partial_\theta F &\partial_\theta \Psi
&\partial_\theta \Phi\\
\partial_\rho F &\partial_\rho \Psi
&\partial_\rho \Phi\\
\end{pmatrix}=
\ber
\begin{pmatrix}
\partial_z F &\partial_z \Psi &\partial_z
\Phi\\
\partial_\theta F &\partial_\theta \Psi
&\partial_\theta \Phi\\
0 &0
&\partial_\rho \Phi\\
\end{pmatrix}= \\
=\ber( J(z,\theta))/\partial_\rho \Phi=1,\quad
\end{multline*}
for all overlaps $U_\alpha\cap U_\beta$, and therefore $(X,\mathcal
E)$ has
trivial dualizing sheaf.
\subsection{Super Riemann surfaces.}\label{ss:SRS}
In this subsection we briefly discuss a special class of $N=1$ super
curves,
the super Riemann surfaces (SRS). This class of curves is studied
widely in the
literature because of its applications in super string theory, see
e.g.,
\cite{Fried:NoteString2DCFT,GidNelson:GeomSRS,LebrRoth:ModuliSRS,%
CraneRabin:SRSuniTeichm}. (Also the term
$\text{SUSY}_1$ curve is used,
\cite{Manin:GaugeFieldTheoryComplexGeom,Manin:Topicsnoncomgeom}, or
super
conformal manifold, \cite{RoSchVor:GeomSupConf}.)
{}From our point of view super Riemann surfaces are special because
irreducible
divisors and $\Lambda$-points can be identified and because there is
a
differential operator taking functions to sections of the dualizing
sheaf. Both
facts simplify the theory considerably. However, by systematically
using the
duality of
the $N=2$ super conformal curve one can extend results previously
obtained
solely for super Riemann surfaces to arbitrary super curves.
In the previous subsection we have seen that every $N=1$ super curve
$(X,\mathcal O_X)$
has
a dual curve $(X,\hat{\mathcal O}_X)$. Of course it can happen that the
transition
functions of
$(X,\mathcal O_X)$ are identical to those of the dual curve $(X,\hat{\mathcal O}_X)$. This
occurs if
the transition functions satisfy
\begin{equation}\label{eq:SRScondition}
DF(z_\alpha,\theta_\alpha)=
\Psi(z_\alpha,\theta_\alpha)D\Psi(z_\alpha,\theta_\alpha).
\end{equation}
If (\ref{eq:SRScondition}) holds then the operator
$D_\alpha=\partial_{\theta_\alpha}+\theta_\alpha\partial_{z_\alpha}$
transforms
as
\begin{equation}\label{eq:transfDSRS}
D_\beta=(D\Psi)\inv D_\alpha
\end{equation}
So in the situation of (\ref{eq:SRScondition}) the super curve
$(X,\mathcal O_X)$ carries a $(0\mid 1)$ dimensional distribution $D$ such
that $D^2$ is
nowhere vanishing (in fact $D^2=\partial_z$). A super curve carrying
such a
distribution is called a ($N=1$) super Riemann surface. Equivalently
an $N=1$
super Riemann surface is a ($N=1$) super curve that carries an odd
global
differential operator with nowhere vanishing square that takes values
in some
invertible sheaf.
Recall the Berezinian that occurs in the transformation law for
generators of
$\Berx$, (\ref{eq:transfBer}). It can be written in general as
$$
\ber\begin{pmatrix} \partial_z F &\partial_z \Psi\\
\partial_\theta F &\partial_\theta\Psi
\end{pmatrix}=D(\frac{DF}{D\Psi}).
$$
Therefore if (\ref{eq:SRScondition}) holds we have
$\ber\begin{pmatrix}\partial_z F&\partial_z \Psi\\
\partial_\theta F&\partial_\theta\Psi
\end{pmatrix}=D\Psi$ so (\ref{eq:transfDSRS})
tells us that $D$ takes values in the dualizing sheaf $\Berx$.
So super Riemann surfaces are self dual, as probably first noted in
\cite{DoRoSc:SupModSpaces}. More generally, the question then arises
what
happens if the curves $(X,\mathcal O_X)$ and $(X,\hat{\mathcal O}_X)$ are isomorphic, but
a priori
not with identical transition functions. We claim that also in this
case
the curve $(X,\mathcal O_X)$ is a super Riemann surface.
Indeed, the operator $\hat{D}_{\mathcal C}$ restricted to $\mathcal O_X$ takes values in
the
dualizing sheaf $\mathcal B\text{{\^e}r}_{{X}}$ of $\hat{\mathcal O}_X$, as we have seen above. Using
the
isomorphism we can think of $\hat{D}_{\mathcal C}$ as a differential operator
taking values
in a sheaf isomorphic to the dualizing sheaf $\Berx$ on $\mathcal O_X$. Since
$\hat{D}_{\mathcal C}^2$
does not vanish we see that $(X,\mathcal O_X)$ is a super Riemann surface. Now
it is
known (and easy to see) that for any super Riemann surface there are
coordinates such that (\ref{eq:SRScondition}) holds. In these
coordinates the
transition functions of $(X,\mathcal O_X)$ and $(X,\hat{\mathcal O}_X)$ are in fact equal.
The $N=2$ super conformal curve $(X,\Cox)$ associated to a SRS
$(X,\mathcal O_X)$ is
very
simple. Recall that $\Cox$ is an extension
\begin{equation*}
0\to\mathcal O_X\to\Cox\overset\epsilon\to\Berx \to 0.
\end{equation*}
where locally $\epsilon(z)=\epsilon(\theta)=0$ and
$\epsilon(\rho)=f$, with
$f$ a local generator of $\Berx$. For SRS there is a splitting
$e:\Berx\to\Cox$, given locally by $e(f)=\rho-\theta$. One needs to
use the
definition of a SRS to check that this definition makes global sense,
i.e.,
that $\rho-\theta$ transforms as a section of $\Berx$; for this see
\cite{Ra:oldnew}. In other words for a SRS the associated $N=2$ curve
has a
split structure sheaf:
$$
\Cox=\mathcal O_X\oplus \Berx.
$$
Note that not all SRS's are projected, so there are examples where
$\Cox$ is a
trivial extension but where $(X,\mathcal O_X)$ is not projected.
\subsection{Integration on super curves.}\label{ss:IntegraOnSupCurve}
Let us first recall the classical situation. On an ordinary Riemann
surface
$(X,{\mathcal O}_{X}^{\text{red}})$ we can integrate a holomorphic 1-form $\omega$ along a
contour
connecting two points $p$ and $q$ on $X$.
If the contour connecting $p$ and $q$
lies in a single, simply connected, coordinate neighborhood
$U_\alpha$ with
local coordinate $z_\alpha$ we can write
$\omega=d f_\alpha$, with $f_\alpha\in {\mathcal O}_{X}^{\text{red}}(U_\alpha)$ determined
up to a
constant. The points $p,q$ are described by the irreducible divisors
$z_\alpha-p_\alpha$ and $z_\alpha-q_\alpha$. Then we calculate the
integral of
$\omega$ along the contour by
$\int_p^q\omega=f_\alpha(q_\alpha)-f_\alpha(p_\alpha)$.
Suppose next that $p$ and $q$ are in different coordinate
neighborhoods
$U_\alpha$ and $U_\beta$, with coordinates $z_\alpha,z_\beta$ related
by
$z_\beta=F(z_\alpha)$ on overlaps. Assume furthermore that the
contour
connecting them contains a point $r \in U_\alpha\cap U_\beta$. Then
we can
write $\omega=df_\alpha$ on $U_\alpha$, and $\omega=df_\beta$ on
$U_\beta$,
with $f_\alpha(z_\alpha)=f_\beta(F(z_\alpha))+ c_{\alpha\beta}$ on
overlaps,
where $c_{\alpha\beta}$ is locally constant on $U_\alpha\cap
U_\beta$. The
intermediate point $r$ can be described by two (corresponding)
irreducible
divisors $z_\alpha-r_\alpha$ and $z_\beta-r_\beta$. Then
$\int_p^q\omega=\int_p^r\omega+\int_r^q\omega=f_\beta(q_\beta)-
f_\beta(r_\beta)+f_\alpha(r_\alpha) -f_\alpha(p_\alpha)$. This is
independent
of the intermediate point because the parameter $r_\alpha$ in the
irreducible
divisor $z_\alpha-r_\alpha$ transforms as a $\mathbb C\,$-point of
the curve:
we have $r_\beta=F(r_\alpha)$, and $f_\alpha(r_\alpha)-
f_\beta(r_\beta)=c_{\alpha\beta}$; therefore we can replace $r$ by
any other
intermediate point in the same connected component of $U_\alpha\cap
U_\beta$.
If $p$ and $q$ are not in adjacent coordinate neighborhoods we need
to
introduce more intermediate points.
So there are three crucial facts in the construction of the contour
integral of
holomorphic 1-forms on a Riemann surface: the parameter in an
irreducible
divisor transforms as a point, $d$ is an operator that produces from
a
function on $X$ a section of the dualizing sheaf on $X$, and the
kernel of the
operator $d$ consists of the constants. We will find analogs of all
three facts
for super curves.
We have seen that for an $N=1$ super curve in general the parameters
in an
irreducible divisor correspond to a $\Lambda$-point of the dual
curve. Also the
sheaf map $D$ acting on the dual curve maps sections of $\hat{\mathcal O}_X$ to
sections of
$\Berx$, see (\ref{eq:bigcd}).
This suggests that we define a {\it (super) contour}
$\Gamma=(\gamma,P,Q)$
on $(X,\mathcal O_X)$ as an ordinary contour $\gamma$ on the underlying
topological
space $X$, together with two irreducible divisors $P$ and $Q$ for
$(X,\mathcal O_X)$
such that the reduced divisors of $P$ and $Q$ are the endpoints of
$\gamma$. So
if
$$
P=z-\hat p-\theta\hat \pi,\quad Q=z-\hat q-\theta \hat\chi,
$$
then the corresponding $\Lambda$-points on the dual curve
$(X,\hat{\mathcal O}_X)$ are
$(\hat p,\hat \pi)$, $(\hat q,\hat \chi)$, and $z=\hat
p^{\text{red}}$ and
$z=\hat q^{\text{red}}$ are the equations for the endpoints of the
curve
$\gamma$. Then we define the integral of a section
$\{\omega_\alpha={D}\hat
f_\alpha\}$ of the dualizing sheaf on $(X,\mathcal O_X)$ along $\Gamma$ by
$$
\int_P^Q \omega=\int_P^Q D \hat f=\hat f(\hat q,\hat \chi)-\hat
f(\hat p,\hat
\pi).
$$
Here we assume that the contour connecting $P$ and $Q$ lies in a
single simply
connected open set. If the contour traverses various open sets we
need to
choose intermediate divisors on the contour, as before.
A super contour $\Gamma$ is called {\it closed} if it is of
the form $\Gamma=(\gamma,P,P)$, with the underlying
contour $\gamma $ closed in the usual sense. Observe
that the integral over $\Gamma$ is independent of the choice of $P$,
so
we will omit reference to it.
The contour integration on $N=1$ super curves introduced here seems
to be new;
it is a nontrivial generalization of the contour integral on super
Riemann
surfaces, as described for instance in
\cite{Fried:NoteString2DCFT,McArthur:LineintegralsSRS,Rog:ContourSRS}.
For
closed contours it agrees with the
integration theory described in
\cite{GaiKhudShvar:IntegrationSurSuperSpce,Khud:BVformoddsympl}.
We can also understand this integration procedure in terms of the
contour
integral on the
$N=2$ super conformal super curve $(X,\mathcal{CO}_X)$, introduced by
Cohn,
\cite{Cohn:N=2SRS}. To this end
define on
$\Cox(U)\oplus\Cox(U)$ the sheaf map $(\Dc,\hat{D}_{\mathcal C} )$ by the local
componentwise
action of the differential operators $\Dc^\alpha$ and $\hat{D}_{\mathcal C}^\alpha$
as
before. Then the square of the operator $(\Dc,\hat{D}_{\mathcal C} )$ vanishes and
the
Poincar\'e Lemma holds for $(\Dc,\hat{D}_{\mathcal C})$:
\begin{lem}\label{lem:poincare}
Let $U$ be a simply connected open set on $X$ and let $(f,g)\in
\Cox(U)\oplus\Cox(U)$ such that $(\Dc,\hat{D}_{\mathcal C} )(f,g)=0$. Then there is
an
element $H\in \Cox(U)$, unique up to an additive constant, such that
$$(f,g)=(\Dc H,\hat{D}_{\mathcal C} H).$$
\end{lem}
Let then $\mathcal M(U)\subset \Cox(U)\oplus\Cox(U)$ be the subsheaf
of
$(\Dc,\hat{D}_{\mathcal C} )$-closed sections. Note that a section of
$\mathcal M$ looks in $U_\alpha$ like
$(f_\alpha,g_\alpha)=(f(z_\alpha,\theta_\alpha),g(\hat
z_\alpha,\rho_\alpha))$
and furthermore $f$ is a section of $\Berx$ and $g$ is a section of
$\mathcal B\text{{\^e}r}_{{X}}$.
This means that $\mathcal M$ globalizes to the direct sum
$\Berx\oplus
\mathcal B\text{{\^e}r}_{{X}}$.
So we get an exact sequence of sheaves:
$$
0\to \Lambda\to \mathcal{CO}_X\overset(\Dc,\hat{D}_{\mathcal C})\to
\mathcal M\rightarrow0.
$$
Now the sections of $\mathcal M$ are the objects on $\Cox$ that can
be
integrated. A contour for $\Cox$ is a triple $(\gamma, \mathcal
CP,\mathcal
CQ)$ where $\mathcal CP,\mathcal CQ$ are two $\Lambda$-points of
$(X,\mathcal{CO}_X)$ with as reduced points the endpoints of the
contour
$\gamma$. Assume that the contour lies in a single simply connected
open set
$U$. If $\omega\in \mathcal M(U)$ then we can write $\omega=(\Dc
H,\hat{D}_{\mathcal C}
H)$ for some $H\in \Cox(U)$ and we put $\int_{\mathcal CP}^{\mathcal
CQ}\omega=H(\mathcal CQ)-H(\mathcal CP)$. Extension to more
complicated
contours as before.
Now start with a section $\{s_\alpha\}$ of $\Berx$ on $(X,\mathcal O_X)$. We
can lift it
to the section $\{(s_\alpha,0)\}$ of $\mathcal M $. In particular
there is a
section $\{H_\alpha\}$ of
$\mathcal{CO}_X$ such that $s_\alpha=\Dc H_\alpha$, $\hat{D}_{\mathcal C}
H_\alpha=0$. This
means that $\{H_\alpha\}$ is in fact a section of the subsheaf
$\hat{\mathcal O}_X$. So in
specifying the $\Lambda$-points of $(X,\mathcal{CO}_X)$ on the ends
of the
contour we have the freedom to shift along the fiber of the
projection $\hat
\pi:(X,\Cox)\to (X,\hat{\mathcal O}_X)$. In other words we only need to specify
$\Lambda$-points of the dual curve, or, equivalently, irreducible
divisors on
the original curve.
Therefore we can define the integral of a section $s=\{s_\alpha\}$ of
$\mathcal
B\text{er}_X$ along a contour with $P,Q$ two irreducible divisors
for
$(X,\mathcal O_X)$ at the end point as follows. We choose two $\Lambda$-points
$\mathcal
CP$ and $\mathcal CQ$ of $(X,\Cox)$ that project to the
$\Lambda$-points of
$(X,\hat{\mathcal{O}}_X)$ corresponding to $P,Q$. Then $\int_P^Q
s=H(\mathcal
CQ)-H(\mathcal CP)$
if $s_\alpha=\Dc H$ and $\hat{D}_{\mathcal C} H=0$. Again we are assuming here that
the
contour lies in a simply connected region and extend for the general
case using
intermediate points. One checks that this procedure of integrating a
section of
the dualizing sheaf on $(X,\mathcal O_X)$ using integration on $\Cox$ is the
same as we
had defined before.
\subsection{Integration on the universal cover.}
We consider from now on only holomorphic (compact, connected, $N=1$)
super
curves $(X,\mathcal O_X)$ of genus $g>1$. We fix a point $x_0\in X$ and
1-cycles $A_i,B_i, i=1,\dots ,g$ through $x_0$ with intersection
$A_i\cdot B_j=\delta_{ij}$, $A_i\cdot A_j=B_i\cdot B_j=0$ as usual.
Then the
fundamental group $\pi_1(X,x_0)$ is generated by the classes
$a_i,b_i$
corresponding to the loops $A_i,B_i$, subject solely to the relation
$a_1b_1a_1\inv b_1\inv a_2b_2a_2\inv b_2\inv \dots a_gb_ga_g\inv
b_g\inv=e$.
The universal cover of the super curve $(X,\mathcal O_X)$ is the open
superdisk
$\done=(D,\mathcal O_{\done})$ of dimension $(1\mid1)$, where
$D=\{z\in \mathbb
C\mid |z|<1\}$ and
$\mathcal O_{\done}=\mathcal O_D\otimes_{\mathbb C}\Lambda[\theta]$,
with
$\mathcal O_D$ the usual sheaf of holomorphic functions on the unit
disk. The
group $G$ of covering transformations of $(D,\mathcal O_{\done})\to
(X,\mathcal O_X)$ is
isomorphic to $\Pi_1(X,x_0)$ and each covering transformation $g$ is
determined
by its action on the global coordinates $(z,\theta)$ of $\done$.
Introduce
super holomorphic functions by
$$
F_g(z,\theta):=g\inv\cdot z , \quad\Psi_g(z,\theta):=g\inv\cdot
\theta.
$$
If $P_p$ is a $\Lambda$-point of $\done$, i.e., a homomorphism
$\mathcal O_{\done}\to \Lambda$, determined by $z\mapsto
z_P\in \Lambda_0,\theta \mapsto \theta_P\in \Lambda_1$, then
the action of $g$ in the covering group is defined by
$g\cdot P_p(f)=P_p(g\inv\cdot f)$. Then $z_P\mapsto
F_g(z_P,\theta_P)$ and
$\theta_P\mapsto \Psi_g(z_P,\theta_P)$. So
$\Lambda$-points transform as the coordinates under the covering
group.
Next consider irreducible divisors $P_d=z-\hat z_1
-\theta\hat\theta_1$,
$Q_d=z-\hat z_2 -\theta\hat\theta_2$. We say that $g\cdot P_d=Q_d$
as
divisors if we have the identity
$g\inv Q_d=P_d h(z,\theta)$ as holomorphic functions for some
invertible $h(z,\theta)$. By the same calculation as the one
following Lemma
\ref{lem:roots} we find that
\begin{equation*}
\begin{split}
\hat z_2&= F_g(\hat z_1, \hat \theta_1) + \frac {DF_g(\hat z_1, \hat
\theta_1)}{D\Psi_g(\hat z_1, \hat \theta_1)}\Psi_g(\hat z_1, \hat
\theta_1),\\
\hat \theta_2&= \frac {DF_g(\hat z_1, \hat \theta_1)}{D\Psi_g(\hat
z_1, \hat
\theta_1)}.
\end{split}
\end{equation*}
So irreducible divisors transform with the dual action, compare with
(\ref{eq:coordtransfdualcurve}).
There is a parallel theory for the dual curve: we have a covering
$(D,\mathcal O_{\done})\to (X,\hat{\mathcal O}_X)$, with covering group $\hat G$.
The dual covering group $\hat G$ is isomorphic to $G$ by a
distinguished
isomorphism: $g$ and $\hat g$ are identified if they give the same
transformation of the reduced disk. Their action on functions is in
general
different, however, unless we are dealing with a super Riemann
surface. In
fact, since duality interchanges irreducible divisors and
$\Lambda$-points on
the curve and its dual we see that the action
of $\hat g$ on the coordinates is dual to the transformation of
$g$:
\begin{equation*}
\begin{split}
\hat g\inv\cdot z&= F_g(z,\theta) + \frac {DF_g(z,\theta )}{D\Psi_g(
z ,
\theta )}\Psi_g( z , \theta ),\\
\hat g\inv\cdot\theta &= \frac {DF_g( z , \theta)}{D\Psi_g( z ,
\theta )}.
\end{split}
\end{equation*}
A function $f$ on $(X,\mathcal O_X)$ lifts to a function that is invariant
under the
covering group $G$ and similarly $\hat f$, a function on
$(X,\hat{\mathcal O}_X)$, lifts to a function that is invariant under the dual
covering
group $ \hat G$. An irreducible divisor or a $\Lambda$-point on
$(X,\mathcal O_X)$ lifts
to an infinite set of divisors or points, one for each point on the
underlying
disk above the corresponding reduced point of $X$.
Let as before $x_0$ be a point on $X$ and $d_0$ a point on the disk
lying over
$x_0$.
Let $\gamma$ be a contour for integration on $(X,\mathcal O_X)$, so $\gamma$
consists of
a contour on
$X$ and two irreducible divisors at the endpoints.
The contour lifts to a unique contour on the disk starting at $d_0$
and the
irreducible divisors lift to unique
irreducible divisors for $(D,\mathcal O_{\done})$ that reduce to
$d_0$ and the
endpoint on the disk, respectively.
Also we can pull back sections of $\Berx$ to $(D,\mathcal O_{\done})$
and
calculate integrals on $(X,\mathcal O_X)$ by
lifting to $(D,\mathcal O_{\done})$.
Since $D$ is simply connected this is a great simplification.
For instance any integral over a closed contour is zero.
Similar considerations apply to the $N=2$ curve $(X,\Cox)$ and its
universal covering space $D^{1|2}$ and covering group $\mathcal G$.
Of course $D^{1|2}$ is the $N=2$ curve canonically associated to the
$N=1$
curve
$D^{1|1}$ as in subsection \ref{ss:DualN=2curves}, and the lifts of
$f \in \mathcal O_X$
to $D^{1|2}$ via
either $(X,\Cox)$ or $D^{1|1}$ as intermediate space coincide.
\subsection{Sheaf cohomology for super
curves.}\label{ss:Sheafcohomology}
Our super curves are in fact families of curves over the base scheme
$(\bullet,\Lambda)$, with $\Lambda$ the Grassmann algebra of
nilpotent
constants. This means that for any coherent sheaf the cohomology
groups are
finitely generated $\Lambda$-modules, but they are not necessarily
free. This
means in particular that standard classical theorems, like the
Riemann-Roch
theorem, do not hold in general in our situation. (See for instance
\cite{Hodg:ProblFieldSRS}.)
The basic facts about sheaf cohomology of families of super curves
are
completely parallel to the classical theory (explained for instance
in
\cite{Kempf:AbInt}). For a coherent locally free sheaf $\mathcal L$
there
exist $\Lambda$-homo\-mor\-phisms $\alpha:F\to G$, with $F, G$ free
finite rank
$\Lambda$-modules, that calculate the cohomology. More precisely, for
every
$\Lambda$-module $M$ we have an exact sequence
\begin{equation}\label{eq:calccohom}
0\to H^0(X,\mathcal L\otimes M)\to F\otimes M\overset{\alpha\otimes
1_{M}}
\to G\otimes M\to H^1(X,\mathcal L\otimes M)\to0.
\end{equation}
Recall from Example \ref {exmpl:split} that for any sheaf of
$\mathcal O_X$-modules
$\mathcal F$ we have an associated split sheaf $\mathcal
F^{\text{split}}=\mathcal F\otimes_\Lambda \Lambda/\mathfrak m$.
Therefore, if we choose $M=\Lambda/\mathfrak m$, the sequence
(\ref{eq:calccohom}) calculates the cohomology groups of the split
sheaf
$\mathcal L\spl$. (These cohomology groups are
$\mathbb Z_2$-graded vector spaces over $\Lambda/\mathfrak m=\mathbb
C$.)
Without loss of generality one can choose the homomorphism
$\alpha:F\to G$ such
that $\alpha\spl=\alpha\otimes 1_{\Lambda/\mathfrak m}$ is
identically zero.
This means that
$H^0(X,\mathcal L)$ (respectively $H^1(X,\mathcal L)$) is a submodule
(resp. a
quotient module) of a free $\Lambda$-module of rank $\dim
H^0(X,\mathcal
L\spl)$ (resp. of rank $\dim H^1(X,\mathcal L\spl)$).
We are interested in the question when the $H^i(X,\mathcal L)$ are
free. The
idea is to check this by an inductive procedure, starting with the
free
cohomology of $\mathcal L\spl$.
We have for every $j=1,\dots,n-1$ the split exact sequence
\begin{equation}\label{eq:seqlambda}
0\to \mathfrak m^j/\mathfrak m^{j+1}\to \Lambda/\mathfrak m^{j+1}\to
\Lambda/\mathfrak m^j\to 0.
\end{equation}
Since $\mathfrak m^j/\mathfrak m^{j+1}\otimes_\Lambda \mathcal
L=\mathfrak
m^j/\mathfrak m^{j+1}\otimes_{\mathbb C} \mathcal L\spl$,
$\Lambda/\mathfrak
m^i\otimes_\Lambda \mathcal L=\mathcal L/\mathfrak m^i \mathcal L$
and
$\mathcal L$ is flat over $\Lambda$ we obtain by tensoring with
$\mathcal L$
and taking cohomology the exact sequence ($\Lambda^j=\mathfrak
m^j/\mathfrak
m^{j+1}$)
\begin{equation}\label{eq:longexactcohom}
\begin{aligned}
0 &\to\Lambda^j\otimes_{\mathbb C}H^0(X,\mathcal L\spl)&
&\to H^0(X, \mathcal L/\mathfrak m^{j+1}\mathcal L) &
&\to H^0(X, \mathcal L/\mathfrak m^{j}\mathcal L) &
&\overset{q^j}\to \\
&\overset{q^j}\to \Lambda^j\otimes_{\mathbb C}H^1(X,\mathcal L\spl)
&
&\to H^1(X, \mathcal L/\mathfrak m^{j+1}\mathcal L) &
&\to H^1(X, \mathcal L/\mathfrak m^{j}\mathcal L) &
&\to 0.
\end{aligned}
\end{equation}
If $H^0(X, \mathcal L/\mathfrak m^{j}\mathcal L)$ and $H^1(X,
\mathcal
L/\mathfrak m^{j}\mathcal L)$ are free $\Lambda/\mathfrak
m^j$-modules, then
the module
$H^0(X, \mathcal L/\mathfrak m^{j+1}\mathcal L)$ is free over
$\Lambda/\mathfrak m^{j+1}$ iff the connecting map $q^j$ in
(\ref{eq:longexactcohom}) is zero iff $H^1(X, \mathcal L/\mathfrak
m^{j+1}\mathcal L)$ is free as $\Lambda/\mathfrak m^{j+1}$-module
(see \cite{Kempf:AbInt}, Lemma 10.4).
The relation between the connecting homomorphisms $q^j$ and the
homomorphism $\alpha$ that calculates cohomology is as follows: if we
assume as
above $\alpha\spl$ is zero then $q^1=\alpha\otimes
1_{\Lambda/\mathfrak m^2}$.
More generally, if $q^1=q^2=\dots=q^{j-1}=0$
then $q^j=\alpha\otimes 1_{\Lambda/{\mathfrak m}^{j+1}}$.
More concretely, we can assume that
$\alpha$ is a matrix of size rank $G\times \text{rank }F$ and the
$q^j$ are
quotients of this matrix by $\mathfrak m^{j+1}$. Then the cohomology
of
$\mathcal L$ is the kernel and cokernel of the matrix $\alpha$, and
the
cohomology is free iff $\alpha$ is identically zero.
If now $\mathcal L$ is an invertible sheaf, $\mathcal{L}\spl$ obeys a
super
Riemann-Roch relation and in case of free cohomology we get ($h^i=
\text{rank } H^i$):
\begin{equation} \label{superRR}
h^0(X,\mathcal{L}) - h^1(X,\mathcal{L}) = (\deg \mathcal{L} + 1-g\mid
\deg \mathcal{L} + \deg \mathcal{N} + 1-g),
\end{equation}
where $\mathcal O_X\spl={\mathcal O}_{X}^{\text{red}}\mid \mathcal N$. We can relate by Serre
duality the
cohomology groups of $\mathcal L$ and $\mathcal L^*\otimes \Berx$,
see Appendix
\ref{app:dualSerredual}. In particular, $H^0(X,\mathcal L^*\otimes
\Berx)$ is
free iff $H^1(X,\mathcal L)$ is.
We summarize the discussion in this subsection in the following
theorem.
\begin{thm}\label{thm:freeness}
Let $\mathcal L$ be an invertible $\mathcal O_X$-sheaf. Then $H^0(X,\mathcal
L)$
(respectively $H^1(X,\mathcal L)$) is a submodule (respectively a
quotient
module) of a free $\Lambda$-module of rank $\dim H^0(X,\mathcal
L\spl)$
(respectively of rank $\dim H^1(X,\mathcal L\spl)$). Furthermore
\begin{align*}
H^0(X,\mathcal L)\text{ is a free $\Lambda$-module}
&\Longleftrightarrow
H^1(X,\mathcal L)\text{ is free},\\
&\Longleftrightarrow H^0(X,\mathcal L^*\otimes\Berx)\text{ is
free},\\
&\Longleftrightarrow H^1(X,\mathcal L^*\otimes\Berx)\text{ is free},
\end{align*}
in which case the rank of $H^i(X,\mathcal L)$ is equal to $\dim
H^i(X,\mathcal
L\spl)$.
\end{thm}
\subsection{Generic SKP curves.}
\begin{defn} An { \it SKP curve} is a super curve $(X,\mathcal O_X)$ such
that the
split sheaf ${\mathcal O}_{X}\spl$ is of the form
$$
{\mathcal O}_{X}\spl={\mathcal O}_{X}^{\text{red}}\mid \mathcal N,
$$
where $\mathcal N$ is an invertible ${\mathcal O}_{X}^{\text{red}}$-module of degree zero.
If
$\mathcal N\ne{\mathcal O}_{X}^{\text{red}}$ then $(X,\mathcal O_X)$ is called a {\it generic SKP
curve}.
\qed
\end{defn}
We will discuss in subsection \ref{ss:Krichever} a Krichever map
that
associates to an invertible sheaf on a super curve $(X,\mathcal O_X)$ (and
additional
data) a point $W$ of an infinite super Grassmannian. If this point
$W$ belongs
to the {\it big cell} (to be defined below) we obtain a solution of
the super
KP hierarchy. For $W$ to belong to the big cell it is necessary that
$(X,\mathcal O_X)$
is an SKP curve. The generic SKP curves enjoy simple cohomological
properties.
\begin{thm}\label{thm:cohomcurve}
Let $(X,\mathcal O_X)$ be a generic SKP curve. Then the cohomology groups of
the
sheaves
$\mathcal O_X, \Berx$ are free $\Lambda$-modules. More precisely:
\begin{alignat*}{2}
H^0(X,\mathcal O_X) &= \Lambda \mid 0,
&\qquad H^1(X,\mathcal O_X)&= \Lambda^g\mid \Lambda ^{g-1},\\
H^0(X,\Berx) &= \Lambda^{g-1} \mid \Lambda^g,
&\qquad H^1(X,\Berx)&= 0\mid\Lambda.
\end{alignat*}
\end{thm}
\begin{proof}
Since $\mathcal N$ has no global sections, $H^0(X, {\mathcal O}_{X}\spl)=\mathbb
C\mid 0$
consists of the constants only.
Now by definition of a curve over $(\bullet,\Lambda)$ we have an
inclusion
$0\to\Lambda\to \mathcal O_X$, so $H^0(X,\mathcal O_X)$ contains at least the constants
$\Lambda$. By Theorem \ref{thm:freeness} then $H^0(X,\mathcal O_X)$ must be
equal to
$\Lambda\mid 0$. Again using Theorem \ref{thm:freeness} then also
$H^1(X, \mathcal O_X)$
and the cohomology of $\Berx$ will be free, and the rest of the
theorem follows
from the properties of the split sheaves, see Examples
\ref{exmpl:split},
\ref{exmpl:splitber}.
\end{proof}
\begin{rem}
It is not true that all invertible $\mathcal O_X$-sheaves for a generic SKP
curve have
free cohomology. For instance, consider a sheaf $\mathcal L$ with
$\mathcal
L\spl={\mathcal O}_{X}\spl$, but $\mathcal L\neq \mathcal O_X$. Then, for a covering
$\{U_\alpha\}$
of $X$, the transition functions of $\mathcal L$ will have the form
$g_{\alpha\beta}=1 + f_{\alpha\beta}(z,\theta)$, with
$f_{\alpha\beta}(z,\theta)=0$ modulo the maximal ideal $\mathfrak m$
of
$\Lambda$. Let then $I\subset\Lambda$ be the ideal of elements that
annihilate
all $f_{\alpha\beta}$. Then we have $H^0(X,\mathcal L)=I$ and is in
particular
not free.
\qed
\end{rem}
\subsection{Riemann bilinear relations.}
Let us call sections of $\Berx$ and $\mathcal B\text{{\^e}r}_{{X}}$ holomorphic
differentials (on
$(X,\mathcal O_X)$ and $(X,\hat{\mathcal O}_X)$ respectively).
We will in this subsection introduce analogs of the classical
bilinear
relations for holomorphic differentials.
\begin{thm}
\label{thm:bilinear}
Let $(X,\mathcal O_X)$ be a super curve and let $\omega$, $\hat \omega$ be
holomorphic
differentials on $(X,\mathcal O_X)$ and $(X,\hat{\mathcal O}_X)$ respectively. Let
$\{a_i,b_i\}$ be a standard symplectic basis for $H_1(X,\mathbb Z)$.
Then
$$
\sum_{i=1}^g\oint_{a_i}\omega\oint_{b_i} \hat
\omega=\sum_{i=1}^g\oint_{a_i}\hat\omega\oint_{b_i} \omega.
$$
\end{thm}
Note that we think here of closed contours on the underlying
topological space
$X$ as closed super contours on either $(X,\mathcal O_X)$ or on $(X,\hat{\mathcal O}_X)$.
\begin{proof}
The argument is clearest using the $N=2$ curve $(X,\Cox)$ and its
universal
covering superdisk $D^{1\mid2}$;
this way only one universal covering group $\mathcal G$ appears
instead of both
$G$ and $\hat{G}$.
Choose any holomorphic differentials $\omega$ on $X$ and
$\hat{\omega}$ on
$\hat{X}$, and lift them
to sections $(\omega,0)$ and $(0,\hat{\omega})$ of $\mathcal M$ on
$(X,\Cox)$.
Lifting further to $D^{1\mid2}$, let $\Omega$ be an antiderivative of
$(0,\hat{\omega})$, so that
$(\Dc\Omega,\hat{D}_{\mathcal C}\Omega) = (0,\hat{\omega})$.
The crucial point is that $(\Omega\omega,0)$ is itself a section of
$\mathcal
M$, because $\Dc(\Omega\omega)=0$.
This could not have been achieved using only differentials from $X$.
As per the standard argument, we integrate this object around the
polygon
obtained by cutting open $(X,\Cox)$.
To form this polygon, fix arbitrarily one vertex $P$ (a
$\Lambda$-point of the
$N=2$ disk $D^{1\mid2}$) and let the other vertices be
$a_1^{-1}P,\;b_1^{-1}a_1^{-1}P,\ldots,a_gb_g^{-1}a_g^{-1}\cdots
b_1a_1b_1^{-1}a_1^{-1}P$, where $a_i,b_i$ are
the generating elements of $\mathcal G$.
The vertices are the endpoints of super contours whose reduced
contours are the
sides of the usual polygon bounding a
fundamental region for $\mathcal G$.
These contours project down to any of $X,\hat{X},(X,\Cox)$ as closed
loops
generating the homology; integrating
a differential lifted from any of these spaces along a side of our
polygon will
yield the corresponding period.
Labeling the sides of the polygon with generators of $\mathcal G$ as
usual,
neighborhoods of the sides labeled $a_i$
are identified with each other by $b_i$ and vice versa.
Then we have
\begin{multline*}
0 = \oint \Omega(\omega,0) =
\sum_{i=1}^g \left[ \int_{a_i} \Omega(\omega,0) - \int_{a'_i}
\Omega(\omega,0)
\right]
+ \\ +\sum_{i=1}^g \left[ \int_{b_i} \Omega(\omega,0) - \int_{b'_i}
\Omega(\omega,0) \right].
\end{multline*}
In the first sum, the two integrals are related by the change of
variables
given by $b_i$; the differential
$(\omega,0)$ is invariant under this covering transformation while
$\Omega$
changes by the
$b_i$-period of $\hat{\omega}$.
The second sum is simplified in the same manner, with the result
\begin{equation*}
\sum_{i=1}^g \left[ \int_{ {a}_i}\omega \int_{b_i} \hat{\omega} -
\int_{a_i}
\hat{\omega} \int_{{b}_i} \omega \right] = 0.
\end{equation*}
\end{proof}
\subsection{The period map and cohomology.}
The commutative diagram \eqref{eq:bigcd} gives a commutative diagram
in
cohomology that partly reads:
\begin{equation}
\begin{CD}\label{eq:bigcdcohom}
{} @. H^0(X,\mathcal B\text{{\^e}r}_{{X}}) @=
H^0(X,\mathcal B\text{{\^e}r}_{{X}}) \\
@. @V\operatorname{p\hat er}VV @V{\hat q}VV
\\
H^0(X,\Berx) @>{\operatorname{per}}>> H^1(X,\Lambda)
@>{\operatorname{r\hat ep}}>> H^1(X,\hat{\mathcal O}_X) \\
@| @V\operatorname{rep}VV @. \\
H^0(X,\Berx) @>{q}>>H^1(X,\mathcal O_X)@.
\end{CD}
\end{equation}
Let $\{a_i,b_i; i=1,\dots, g\}$ be a symplectic basis for
$H_1(X,\mathbb Z)$
and let $\{a_i^*,b_i^*; i=1,\dots, g\}$ be a dual basis for
$H^1(X,\mathbb Z)$
and also for $H^1(X,\Lambda)$. We will use Serre
duality (see Appendix \ref{appss:SerredualSupermanifold}) to identify
$H^1(X,\mathcal O_X)$ and $H^1(X,\hat{\mathcal O}_X)$ with the duals of $H^0(X, \Berx)$
and $H^0(X,
\mathcal B\text{{\^e}r}_{{X}})$.
\begin{lem}\label{lem:perrep}
The maps $\operatorname{per}$, $\operatorname{p\hat er}$,
$\operatorname{rep}$
and $\operatorname{r\hat ep}$ are explicitly given by
\begin{align*}
\operatorname{per}(\omega) &=\sum_{i=1}^g(\oint_{a_i} \omega)
a_i^*
+\sum_{i=1}^g(\oint_{b_i} \omega) b_i^*,\\
\operatorname{p \hat er}(\hat\omega) &=\sum_{i=1}^g(\oint_{a_i}
\hat\omega)
a_i^* +\sum_{i=1}^g(\oint_{b_i} \hat\omega) b_i^*,\\
\operatorname{rep}(\sigma)(\omega)
&=\sum_{i=1}^g\alpha_i(\oint_{b_i}
\omega)-\sum_{i=1}^g \beta_i(\oint_{a_i} \omega),\\
\operatorname{r\hat ep} (\sigma)(\hat\omega)
&=\sum_{i=1}^g\alpha_i(\oint_{b_i}
\hat\omega) -\sum_{i=1}^g \beta_i(\oint_{a_i} \hat\omega),
\end{align*}
where $\omega\in H^0(X,\Berx)$, $\hat\omega\in H^0(X,\mathcal B\text{{\^e}r}_{{X}})$ and
$\sigma=
\sum_{i=1}^g \alpha_i a_i^*+\beta_i b_i^*\in H^1(X,\Lambda)$.
\end{lem}
\noindent If we introduce
a basis $\{\omega_\alpha,\alpha=1,\dots,g-1\mid w_j,j=1,\dots,g\}$ of
$H^0(X,\Berx)$ we obtain the {\it period matrix} associated to
$\operatorname{per}$:
$$
\Pi=\begin{pmatrix}
\oint_{a_i}\omega_\alpha&\oint_{a_i}w_j\\
\oint_{b_i}\omega_\alpha&\oint_{b_i}w_j
\end{pmatrix},
$$
where $i,j$ run from $1$ to $g$ and $\alpha$ runs from $1$ to $g-1$.
For the split curve we have $H^0(X,\hat{\mathcal O}_X^{\text{split}})=\mathbb C\mid\mathbb
C^{g-1}$
and the map
$$
D:H^0(X,\hat{\mathcal O}_X^{\text{split}})\to H^0(X,\Berxsplit)
$$
has as image a $g-1$ dimensional, even subspace of exact
differentials. For
these elements the periods vanish, and one finds the reduction mod
$\mathfrak
m$ of $\Pi$ is given by
$$
\Pi\spl=\begin{pmatrix}
0&\Pi\red(a)\\
0&\Pi\red(b)
\end{pmatrix},
$$
where $\Pi\red=\begin{pmatrix}\Pi\red(a)\\ \Pi\red(b)\end{pmatrix}$
is the
classical period matrix of the underlying curve $(X,{\mathcal O}_{X}^{\text{red}})$. By
classical
results we can choose the basis of holomorphic differentials on the
reduced
curve so that $\Pi\red(a)=1_g$. From this it follows that we can also
choose in
$H^0(X,\Berx)$ a basis such that $\oint_{a_i}w_j=\delta_{ij}$ and so
that the
period matrix takes the form
\begin{equation}\label{eq:normperiodmatrix}
\Pi=\begin{pmatrix}
0&1_g\\
Z_{o}&Z_e
\end{pmatrix}.
\end{equation}
Note that $\Pi$ is not uniquely determined by the conditions we have
imposed:
we are still allowed to change $\Pi\mapsto \Pi^\prime=
\begin{pmatrix}
0&1_g\\
Z_{o}G&Z_e + Z_o \Gamma
\end{pmatrix}
$, corresponding to a change of basis of $H^0(X,\Berx)$ by an even
invertible
matrix
$\begin{pmatrix}
G&\Gamma\\
0&1_g
\end{pmatrix}$ of size $g-1\mid g\times g-1\mid g$.
Using the same basis we see that $\operatorname{rep}$ has matrix
$$
\begin{pmatrix}
\oint_{b_i}\omega_\alpha&-\oint_{a_i}\omega_\alpha\\
\oint_{b_i}w_j&-\oint_{a_i}w_j
\end{pmatrix}=\Pi^tI=\begin{pmatrix}
Z_o^t&0\\
Z_e^t&-I_g
\end{pmatrix}, \quad I=\begin{pmatrix}
0&-1_g\\
1_g&0
\end{pmatrix}.
$$
Again this matrix is not entirely determined by our choices.
{}From the commutativity of the diagram \eqref{eq:bigcdcohom} we see
that
the matrix of the map $q$ is given by
\begin{equation}\label{eq:matrixconnectinghom}
Q=\Pi^t I \Pi=\begin{pmatrix}
0&Z_o^t\\
-Z_o&Z_e^t-Z_e
\end{pmatrix}.
\end{equation}
In general, the structure sheaf $\hat{\mathcal O}_X$ and dualizing sheaf
$\mathcal B\text{{\^e}r}_{{X}}$ of the
dual curve will not have free cohomology, so that we cannot represent
the maps
$\operatorname{rep}$, $\operatorname{r\hat ep}$ and $\hat q$ by
explicit
matrices.
The nonfreeness of the cohomology of $\hat{\mathcal O}_X$ and $\mathcal B\text{{\^e}r}_{{X}}$
is determined by the odd component $Z_o$ of the period matrix, see
\eqref{eq:normperiodmatrix}.
Recall that $\hat{\mathcal O}_X^{\text{split}}={\mathcal O}_{X}^{\text{red}} \mid \mathcal K\mathcal N\inv$ and
$\Berxhatsplit=\mathcal N\mid \mathcal K$
(see Example \ref{exmpl:dualsplit}) and hence
\begin{alignat*}{2}
H^0(X,\hat{\mathcal O}_X^{\text{split}})&= \mathbb C\mid \mathbb C^{g-1}, &
H^1(X,\hat{\mathcal O}_X^{\text{split}})&=
\mathbb C^g\mid 0,\\
H^0(X,\Berxhatsplit)&= 0\mid \mathbb C^{g}, & H^1(X,\Berxhatsplit)&=
\mathbb
C^{g-1}\mid \mathbb C.
\end{alignat*}
{}From the diagram \eqref{eq:bigcd} we extract in cohomology, using
that
the map $H^1(X, \Berx) \to H^2(X,\Lambda)=\Lambda\mid 0$ is an (odd!)
isomorphism and $H^0(X, \Lambda)=\Lambda$,
\begin{equation}\label{eq:period-seq}
0 \to \frac{H^0(X, \hat{\mathcal O}_X)}{\Lambda}\overset{D}\to H^0(X, \Berx)
\overset
{\operatorname{per}}\to H^1(X,\Lambda) \to H^1(X, \hat{\mathcal O}_X) \to 0
\end{equation}
so that the period map
has as kernel $H^0(X, \hat{\mathcal O}_X)$ mod constants and as cokernel $H^1(X,
\hat{\mathcal O}_X)$.
Therefore $\operatorname{per}$ is essentially one of the
homomorphisms that calculate cohomology introduced in
subsection \ref{ss:Sheafcohomology}. We can even be more
explicit: if $\{\omega_\alpha\mid w_j\}$ is the (partially)
normalized basis
of
holomorphic differentials as above the homomorphism $\operatorname{per}$ maps
the submodule generated by the $w_j$ isomorphically to a free rank
$g$ summand of $H^1(X,\Lambda)$.
This is irrelevant for the calculation of cohomology,
so we can replace the sequence \eqref{eq:period-seq} by
\begin{equation}\label{eq:period-seqsimple}
0 \to H^0(X, \hat{\mathcal O}_X)/\Lambda \overset{D}\to \Lambda^{g-1}
\overset{Z_o}\to
\Lambda^g \to H^1(X, \hat{\mathcal O}_X) \to 0,
\end{equation}
and $H^0(X, \hat{\mathcal O}_X)$ mod constants is the kernel of $Z_o$, whereas
the cokernel
of $Z_o$ is $H^1(X, \hat{\mathcal O}_X)$.
Similarly, the cohomology of $\mathcal B\text{{\^e}r}_{{X}}$ is calculated by the
sequence
\begin{multline}\label{eq:repiod-seq}
0 \to H^0(X, \mathcal B\text{{\^e}r}_{{X}}) \overset{\operatorname{p\hat er}}\to H^1(X,
\Lambda)
\overset{\operatorname{rep}}\to H^0(X,\Berx)^* \\
\to H^1(X, \mathcal B\text{{\^e}r}_{{X}}) \to \Lambda\to 0
\end{multline}
The image of a holomorphic differential $\hat\omega$ in
$H^1(X,\Lambda)$ is
then a vector
$\operatorname{p\hat er}(\hat\omega)=\begin{pmatrix} a(\hat\omega)\\
b(\hat\omega)\end{pmatrix}$, where $a(\hat\omega)$ and
$b(\hat\omega)$ are the
vectors of $a$ respectively $b$ periods of $\hat\omega$. By
exactness of
\eqref{eq:repiod-seq} we have $\operatorname{rep}\circ
\operatorname{p\hat
er}=0$, or, using bases,
$$
\begin{pmatrix}
Z_o^t&0\\
Z_e^t&-I_g
\end{pmatrix}\begin{pmatrix} a(\hat\omega)\\
b(\hat\omega)\end{pmatrix}=0
$$
This means that the vector $b(\hat\omega)$ of $b$ periods is
(uniquely)
determined by the $a$ periods: $b(\hat\omega)= Z_e^t a(\hat\omega)$,
and the
vector of $a$ periods is constrained by the equation $Z_o^t
a(\hat\omega)=0$.
The submodule of $H^1(X, \Lambda)$ generated by the elements $b_i^*$
maps under
$\operatorname {rep}$ isomorphically to a free rank $0\mid g$ summand
of
$H^1(X,\Berx)^*$, so that for the calculation of cohomology we can
simplify
\eqref{eq:repiod-seq} to
\begin{equation}\label{eq:repiod-seqsimple}
0 \to H^0(X, \mathcal B\text{{\^e}r}_{{X}}) \to \Lambda^{g} \overset{Z_o^t}\to
\Lambda^{g-1}
\to H^1(X, \mathcal B\text{{\^e}r}_{{X}}) \to \Lambda\to 0.
\end{equation}
We summarize the results on the cohomology of the dual curve in the
following
Theorem.
\begin{thm}
Let $(X,\mathcal O_X)$ be a generic SKP curve with odd period matrix $Z_o$.
Then
$$
H^0(X,\hat{\mathcal O}_X)/\Lambda\simeq \operatorname{Ker}(Z_o),\quad
H^1(X,\hat{\mathcal O}_X)\simeq\operatorname{Coker}(Z_o).
$$
Furthermore $H^0(X,\mathcal B\text{{\^e}r}_{{X}})\simeq \operatorname{Ker}(Z_o^t)$ and
$\operatorname{Coker}(Z_o^t)$ is a submodule of $H^1(X,\mathcal B\text{{\^e}r}_{{X}}) $
such that
$$H^1(X,\mathcal B\text{{\^e}r}_{{X}})/\operatorname{Coker}(Z_o^t)\simeq\Lambda.
$$
\end{thm}
\subsection{$\Cox$ as extension of $\Berx$.}
We discuss in this subsection, for generic SKP curves, the structure
of $\Cox$
as extension of $\Berx$ and the relation with free cohomology and the
projectedness of the curve $(X,\mathcal O_X)$.
{}From the sequence (\ref{eq:extberbystruct}) that defines $\Cox$ we
obtain in
cohomology
\begin{gather}\label{eq:cohomn=2}
\begin{aligned}
0 &\to H^0(X,\mathcal O_X)& &\to H^0(X, \Cox) & &\to H^0(X, \Berx) &
&\overset{q}\to
\\
{}&\overset{q}\to H^1(X,\mathcal O_X)&&\to H^1(X, \Cox) &&\to H^1(X,
\Berx)&&\to 0
\end{aligned}
\end{gather}
The cohomology of the sheaves $\mathcal O_X, \Berx$ is given by Theorem
\ref{thm:cohomcurve}. By Theorem \ref{thm:freeness} (or its extension
to rank
two sheaves) $H^0(X,\Cox)$ is a submodule of
a $\Lambda^{g+1}\mid\Lambda^{g-1}$ and $H^1(X,\Cox)$ is a quotient of
a $\Lambda^{g+1}\mid \Lambda^{g-1}$. We see from this that the
cohomology of
$\Cox$ is free if and only if $q$ is the zero map.
To describe the map $q$ in more detail we need to recall some facts
about
principal parts and extensions, (see e.g., \cite{Kempf:AbInt}). For
any
invertible sheaf $\mathcal L$ let $\underline { \mathcal
Rat}(\mathcal L)$ and
$\underline { \mathcal Prin}(\mathcal L)$ denote the sheaves of
rational
sections and principal parts for $\mathcal L$ and denote by $\mathcal
Rat(\mathcal L)$ and
$ \mathcal Prin(\mathcal L)$ their $\Lambda$-modules of global
sections. Then
the cohomology of $\mathcal L$ is calculated by
$$
0\to H^0(X,\mathcal L)\to \mathcal Rat(\mathcal L)\to
\mathcal Prin(\mathcal L)\to H^1(X,\mathcal L)\to 0.
$$
In particular we can represent a class $\alpha\in H^1(X,\mathcal L)$
as a
principal part $p=\sum p_{x}$, where $p_{x}\in \mathcal Rat(\mathcal
L)/\mathcal L_{x}$, for $x\in X$.
If $\alpha\in H^1(X,\mathcal L)$ and $\omega\in H^0(X,\mathcal M)$,
for some
other invertible sheaf $\mathcal M$, then we can define the {\it cup
product}
$\omega\cup \alpha$ by representing $\alpha$ by a principal part $p$
and
calculating the principal part $\omega p= \sum \omega_{x}p_{x}$ in
$\mathcal
Prin(\mathcal M\otimes \mathcal L)$; the image of $\omega p$ in
$H^1(X,
\mathcal M\otimes \mathcal L)$ is then by definition $\omega\cup
\alpha$.
We want to understand the kernel of the cup product with $\omega \in
H^0(X,
\mathcal M)$ in case $\omega$ is odd and free (i.e., linearly
independent over
$\Lambda$). In this case there will be for any invertible sheaf
$\mathcal L$
sections that are immediately annihilated by $\omega$; let therefore
$\operatorname{Ann}(\mathcal L,\omega)\subset \mathcal L$ be the subsheaf of such
sections.
Putting $\mathcal L_\omega=\mathcal L/\operatorname{Ann}(\mathcal L,\omega)$, we
get, because
$\omega^2=0$, the exact sequence
\begin
{equation}
\label{eq:cupsequence}
0\to \mathcal L_\omega\overset{\omega}\to \operatorname{Ann}(\mathcal L\otimes
\mathcal
M,\omega)\to Q\to 0
\end
{equation}
Locally, in an open set $U_\alpha\subset X$, we have $\mathcal
L(U_\alpha)=\mathcal O_X(U_\alpha)l_\alpha$, $\mathcal
M(U_\alpha)=\mathcal O_X(U_\alpha)m_\alpha$ and we write
$\omega=\omega_\alpha(z,\theta)m_\alpha$, with
$\omega_\alpha=\phi_\alpha+\theta_\alpha f_\alpha$. Then
$f_{\alpha}^{\text{red}}$ is a regular function on $U_\alpha$ with
some divisor
of zeros $D_f=\sum n_i q_i$. Some of the $q_i$ may also be zeros of
(the lowest
order part of) $\phi_\alpha$ and there will be a maximal
$g_\alpha(z,\theta) \in \mathcal O_X(U_\alpha)_{\bar 0}$ (here
$\mathcal O_X(U_\alpha)_{\bar 0}$
is the module of even sections) such that
$$
\omega_\alpha(z,\theta)=\omega_\alpha(z,\theta)^\prime
g_\alpha(z,\theta)
$$
with $g_\alpha^{\text{red}}$ a regular function with divisor of zeros
$D_g$ (on
$U_\alpha$) satisfying $0\le D_g\le D_f$.
Then $\operatorname{Ann}(\mathcal L\otimes \mathcal M,\omega)(U_\alpha)$ is
generated by
$\omega_\alpha(z,\theta)^\prime l_\alpha\otimes m_\alpha$ and we see
that
$Q$ is a torsion sheaf: $Q$ is killed by the invertible sheaf
generated locally
by the even invertible rational function $g_\alpha(z,\theta)$. Let
$D_\omega=\{(g_\alpha(z,\theta),U_\alpha)\}$ be the corresponding
Cartier
divisor. Then we have an isomorphism
$$
\operatorname{Ann}(\mathcal L\otimes \mathcal M,\omega)\to \mathcal L(D_\omega),
\quad
\omega_\alpha(z,\theta)^\prime l_\alpha\otimes m_\alpha\mapsto
l_\alpha\otimes
1/g_\alpha(z,\theta)
$$
The sequence \eqref{eq:cupsequence} is equivalent to
$$
0\to \mathcal L\to \mathcal L(D_\omega)\to \mathcal
L(D_\omega)|_{D_\omega}\to
0.
$$
Now the cup product with $\omega$ gives a map $H^1(\mathcal
L_\omega)\to H^1(X,
\operatorname{Ann}(\mathcal L\otimes \mathcal M,\omega))$ with kernel the image of
the
natural map $\phi:H^0(X,Q)\to H^1(\mathcal L_\omega)$. Identifying
$H^0(X,Q)$
with $H^0(X, \mathcal L(D_\omega)|_{D_\omega})$, we see that $\phi$
is the
composition
$$
H^0(X,\mathcal L(D_\omega)|_{D_\omega})\to \mathcal Prin (\mathcal
L)\to
H^1(X,\mathcal L).
$$
Therefore the kernel of $\omega\cup$ consists of those $\alpha\in
H^1(X,\mathcal L)$ that have a representative $p\in
\mathcal{P}{rin}(\mathcal
L)$ such that $\omega p$ has zero principal part, i.e., the poles in
$p$ are
compensated by the zeros in $\omega$.
Extensions of the form (\ref{eq:extberbystruct})
are classified by $\delta\in H^1(X,\Berx^*)$: we think of
$\Cox$ as a subsheaf of $\underline {\mathcal Rat}(\mathcal O_X\oplus \Berx)$
consisting
on an open set $U$ of pairs $(f, \omega)$ where $\omega\in \Berx(U)$
and $f$ a
rational function such that the principal part $\bar f$ is equal to
$\omega p$,
for $p\in \mathcal Prin(\Berx^*)$; then $\delta\in H^1(X,\Berx^*)$ is
the class
of $p$. It is then easy to see that the connecting map $q:
H^0(X,\Berx)\to
H^1(X,\mathcal O_X)$ is cup product by the extension class $\delta$:
$q(\omega)=\omega\cup \delta$. The class $\delta$ is ly represented
by the
{\it \v Cech} cocycle
\begin{equation}\label{eq:cechcocycle}
\phi_{\beta\alpha} = \partial_\theta F_{\beta\alpha} /
\partial_\theta \Psi_{\beta\alpha}\in \Berx^*(U_\alpha\cap U_\beta),
\end{equation}
from \eqref{eq:coordchangen=2}, \eqref{eq:transformrho2}.
\begin{lem}\label{lem:q=0iffexttriv}
Let $(X,\mathcal O_X)$ be a generic SKP curve. Then the connecting
homomorphism $q:
H^0(X,\Berx)\to H^1(X,\mathcal O_X)$ in \eqref{eq:cohomn=2} is the zero map
iff the
extension \eqref{eq:extberbystruct} is trivial. In particular the
cohomology of
$\Cox$ is free iff the extension is trivial.
\end{lem}
\begin{proof}It is clear that if the extension is trivial the
connecting map
$q$ is trivial.
From the explicit form, in particular the $\theta$ independence, of
the
cocycle we see that it is not immediately killed by multiplication by
an odd
free section $\omega$ of $\Berx$, i.e., the cocycle is not zero in
the
cohomology group $H^1(X,\Berx^*)/\operatorname{Ann}(\Berx^*,\omega)$ if it is
nonzero in
$H^1(X, \Berx^*)$. The split sheaf $\Berxsplit$ is $\mathcal
K\mathcal
N\inv\mid \mathcal K$. An odd free section $\omega$ of $\Berx$
therefore has an
associated divisor $D_\omega$ as constructed above with reduced
support
included in the divisor of a section of $\mathcal K$ on the
underlying curve.
Now $q(\omega)=\omega\cup \delta$ is zero if the zeros of $\omega$
cancel the
poles occuring in the principal part $p$ representing $\delta$. But
by
classical results the complete linear system of $\mathcal K$ has no
base
points, i.e.,
there is no point on $X$ where where all global sections of $\mathcal
K$
vanish. This means that wherever the poles of $p$ occur, there will
be a
section $\omega$ of $\Berx$ that does not vanish there. So $q$ being
zero on
all odd generators of $H^0(X,\Berx)$ implies that the extension is
trivial. A
fortiori if $q$ is the zero map the extension will also be trivial.
\end{proof}
The extension given by the cocycle \eqref{eq:cechcocycle} is trivial
if
\begin{equation} \label{eq:phitriv}
\phi_{\beta\alpha}(z_\alpha) = \sigma_\beta(z_\beta,\theta_\beta) -
H_{\beta\alpha}(z_\alpha,\theta_\alpha)
\sigma_\alpha(z_\alpha,\theta_\alpha)
\end{equation}
for some 1-cochain $\sigma_\alpha \in\Berx^*(U_\alpha)$.
In that case, a splitting $e: \Berx \rightarrow \Cox$ is obtained by
$e(f_\alpha) = \rho_\alpha - \sigma_\alpha(z_\alpha,\theta_\alpha)$.
\begin{thm} \label{thm:split=proj}
For a generic SKP curve $(X,\mathcal O_X)$, $\Cox$ is a trivial extension of
$\Berx$ iff $(X,\mathcal O_X)$ is projected.
\end{thm}
\begin{proof}
We have already observed (in subsection \ref{ss:DualN=2curves}) that
$X$
projected implies
$\phi_{\beta\alpha} = 0$ in a projected atlas, making the extension
trivial.
Now suppose, if possible, that the extension is trivial but that $X$
is not projected.
Write the transition functions of $X$ in the form
\begin{equation*}
z_\beta = f_{\beta\alpha}(z_\alpha) + \theta_\alpha
\eta_{\beta\alpha}(z_\alpha), \;\;\;\;
\theta_\beta = \psi_{\beta\alpha}(z_\alpha) + \theta_\alpha
B_{\beta\alpha}(z_\alpha)
\end{equation*}
and assume that the atlas has been chosen so that
$\eta_{\beta\alpha}$ vanishes to the highest possible (odd) order
$n$ in nilpotents.
That is, $\eta_{\beta\alpha} = 0$ mod $\mathfrak m^n$, but not mod
$\mathfrak m^{n+2}$.
Writing also $\sigma_\alpha(z_\alpha,\theta_\alpha) =
\chi_\alpha(z_\alpha) + \theta_\alpha h_\alpha(z_\alpha)$ and
substituting in (\ref{eq:phitriv}) yields two conditions.
From the $\theta_\alpha$-independence of $\phi_{\beta\alpha}$ one
finds that $h_\alpha$ mod $\mathfrak m^{n+1}$ is a global section
of ${\mathcal K}^{-1}{\mathcal N}^2$.
Since $X$ is a generic SKP curve, there are no such sections and
$h_\alpha=0$ to this order.
Using this, the second condition becomes,
\begin{equation*}
\eta_{\beta\alpha} = B_{\beta\alpha} \chi_\beta(f_{\beta\alpha}) -
f'_{\beta\alpha} \chi_\alpha \;\;\;
{\text{mod }} {\mathfrak m}^{n+2}.
\end{equation*}
This condition implies that the coordinate change
$\tilde{z}_\alpha = z_\alpha - \theta_\alpha \chi_\alpha$
will make $\eta_{\beta\alpha}$ vanish to higher order than $n$, a
contradiction.
\end{proof}
To lowest order in nilpotents, the cocycle conditions for the
transition functions of $X$ imply that
$\eta_{\beta\alpha}/B_{\beta\alpha}$ is a cocycle for
$H^1(X,\mathcal{NK}^{-1})$, while $\psi_{\beta\alpha}$ is a
cocycle for $H^1(X,\mathcal N^{-1})$.
This implies that the projected $X$'s have codimension
$(0 \mid 3g-3)$ in the moduli space of generic SKP curves, which
has dimension $(4g-3 \mid 4g-4)$ (see
\cite{Vain:DeformSupSpacShe}).
The proof of Theorem \ref{thm:split=proj} generalizes to higher order
in
nilpotents the fact that at lowest order $\phi_{\beta\alpha}$ is a
cocycle in
$H^1(X,\mathcal{NK}^{-1} \mid \mathcal N^2 \mathcal K^{-1})$.
\subsection{$\Cox$ as extension of $\mathcal B\text{{\^e}r}_{{X}}$ and symmetric period
matrices}
\label{ss:Symmperiodmatrices}
One can equally view $\Cox$ as an extension of $\mathcal B\text{{\^e}r}_{{X}}$ by
$\hat{\mathcal O}_X$.
Obviously, if $(X,\hat{\mathcal O}_X)$ is projected this extension is trivial,
but the converse no longer holds.
In the proof of Theorem \ref{thm:split=proj} there is now the
possibility that
$h_\alpha \neq 0$.
(Recall from subsection \ref {ss:SRS} that for $X$ a SRS, a splitting
of the
extension was
universally given by $\chi_\alpha=0,h_\alpha=-1$.)
One can see that this extension is not always trivial, however, by
constructing examples with $\psi_{\beta\alpha} = 0$ and
$\phi_{\beta\alpha}$ a nontrivial class.
(We are now refering to an atlas for $(X,\hat{\mathcal O}_X)$.)
In this subsection we will exhibit a connection between the structure
of $\Cox$
as extension of $\mathcal B\text{{\^e}r}_{{X}}$ and the symmetry
of the component $Z_e$ of the period matrix, see
\eqref{eq:normperiodmatrix}.
By classical results $Z_e^{\text{red}}$ is symmetric.
However, there seems to be no reason that $Z_e$ is symmetric in
general.
\begin{thm} \label{thm:Zsym&noZo=proj}
Let $(X,\mathcal O_X)$ be a generic SKP curve and $Z_e,Z_o$ its (partially)
normalized
period matrices (as in \eqref{eq:normperiodmatrix}).
Then we have $Z_e$ symmetric and $Z_o=0$
iff $(X,\mathcal O_X)$ is projected.
\end{thm}
\begin{proof}
This follows immediately from Theorem \ref{thm:split=proj}, Lemma
\ref{lem:q=0iffexttriv} and the explicit form
\eqref{eq:matrixconnectinghom} of
the connecting homomorphism $q$.
\end{proof}
Recall the exact sequence
$$
0\to \Lambda\to \Cox \overset{(\Dc,\hat{D}_{\mathcal C} )}\to \mathcal M
\rightarrow 0,
$$
where $\mathcal M=\Berx\oplus\mathcal B\text{{\^e}r}_{{X}}$ is the sheaf of objects that
can be
integrated on $\Cox$.
The corresponding cohomology sequence is in part
\begin{equation*}
0 \to \Lambda \to H^0(X,\Cox)
\stackrel{(\Dc,\hat{D}_{\mathcal C})}{\longrightarrow}
H^0(X,{\mathcal M}) \stackrel{\operatorname{cper}}{\longrightarrow}
H^1(X,\Lambda)
\end{equation*}
where $\operatorname{cper}(\omega,\hat\omega)=\{\sigma\mapsto
\int_\sigma
[\omega+\hat\omega]\}$. So we see that we can identify
$H^0(X,\Cox)/\Lambda$
with
pairs $(\omega,\hat{\omega})$ of differentials with opposite periods.
Now let $(\omega,\hat{\omega})$ be such a pair.
$\omega$ can be written in terms of the basis of $H^0(X,\Berx)$ in
the form
\begin{equation*}
\omega = \sum a_i(\omega) w_i + \sum A_\alpha \omega_\alpha,
\end{equation*}
where $a_i(\omega)$ denote the a-periods and $A_\alpha$ are
other constants uniquely determined by $\omega$.
Then the vector of b-periods of $\omega$ will be
\begin{equation*}
b(\omega) = Z_e a(\omega) + Z_o A.
\end{equation*}
Since these coincide with minus the b-periods of $\hat{\omega}$,
which are
$b(\hat{\omega}) = Z_e^t a(\hat\omega)=-Z_e^t a(\omega)$, we obtain
for each
such pair of
differentials a relation
\begin{equation} \label{eq:basicZrelation}
(Z_e - Z_e^t) a(\omega) + Z_o A = 0.
\end{equation}
We have a sequence analogous to \eqref{eq:cohomn=2} for $\Cox$ as
extension
of $\mathcal B\text{{\^e}r}_{{X}}$ and a connecting map $\hat q$ for this situation.
\begin{thm} \label{thm:Zsym->dualexttriv}
Let $(X,\mathcal O_X)$ be a generic SKP curve and $Z_e,Z_o$ its normalized
period matrices.
If $Z_e$ is symmetric, then
$\hat q$ is the zero map.
\end{thm}
\begin{proof}
Assuming that $Z_e=Z_e^t$, we determine the set of pairs
$(\omega,\hat{\omega})$ with opposite periods.
The a-periods of $\hat{\omega}$ can be chosen freely from the kernel
of $Z_o^t$.
According to (\ref{eq:basicZrelation}), any $\omega$ chosen to match
these a-periods will also have matching b-periods iff $A_\alpha$
belongs to the kernel of $Z_o$.
Therefore, $H^0(X,\Cox)$ mod constants can be identified with
$\text{Ker} Z_o \oplus \text{Ker} Z_o^t$, which is precisely
$H^0(X,\hat{\mathcal O}_X)/\Lambda \oplus H^0(X,\mathcal B\text{{\^e}r}_{{X}})$.
In this case $\hat q$ is the zero map.
\end{proof}
In general it seems that $\hat q=0$ will not imply that the extension
$\Cox$ of
$\mathcal B\text{{\^e}r}_{{X}}$ is trivial, as in Lemma \ref{lem:q=0iffexttriv} for the
extension
of $\Berx$ by $\mathcal O_X$. Also it seems that $Z_e=Z_e^t$ cannot be
deduced from
(\ref{eq:basicZrelation}) as long as
the a-periods are constrained to the kernel of $Z_o^t$.
\subsection{Moduli of invertible sheaves.}\label{ss:modinvertible
sheaves}
In this subsection we will discuss some facts about invertible
sheaves on super
curves and their moduli spaces, see also
\cite{RoSchVor:GeomSupConf,GidNelson:LinebSRS}.
An invertible sheaf on $(X,\mathcal O_X)$ is determined by transition
functions
$g_{\alpha\beta}$ on overlaps $U_\alpha\cap U_\beta$, and so
isomorphism
classes of invertible sheaves are classified by the cohomology group
$H^1(X,\mathcal O^\times_{X,\text{ev}})$.
The degree of an invertible sheaf $\mathcal L$ is the degree of the
underlying
reduced sheaf $\mathcal L^{\text{red}}$, with transition functions
$g_{\alpha\beta}^{\text{red}}$. Let $\Pic^0(X)$ denote the group of
degree zero
invertible sheaves on $(X,\mathcal O_X)$.
The exponential sheaf sequence
\begin{equation}\label{expsequence}
0\to \mathbb Z\to \mathcal O_{X,\text{ev} } \overset{\exp(2\pi i \times \cdot )}\to
\mathcal O^\times_{X,\text{ev}}\to 0
\end{equation}
reduces mod nilpotents to the usual exponential sequence for ${\mathcal O}_{X}^{\text{red}}$
and we
see that $\Pic^0(X)=H^1(X,\mathcal O_{X,\text{ev} } )/H^1(X,\mathbb Z)$.
If $(X,\mathcal O_X)$ is a generic SKP curve $H^1(X,\mathcal O_X)$ is a free rank
$g\mid g-1$
$\Lambda$-module and the map $H^1(X,\mathbb Z)\to H^1(X,\mathcal O_X)$ is the
restriction of the map $H^1(X,\Lambda)\to H^1(X,\mathcal O_X)$, which is dual
to
the map $\operatorname{per}$ of Lemma \ref{lem:perrep}. So with respect to a
suitable basis
$H^1(X,\mathbb Z)\to H^1(X,\mathcal O_X)$ is described by the transpose of
the period
matrix \eqref{eq:normperiodmatrix}. This implies that the image of
$H^1(X,\mathbb Z)$ is generated by $2g$ elements that are linearly
independent
over the real part $\Lambda_\Re$ of $\Lambda$ (see Appendix
\ref{ss:realstrconj} for the definition of $\Lambda_\Re$). The
elements of the
quotient
$\Pic^0(X)=H^1(X,\mathcal O_{X,\text{ev} } )/H^1(X,\mathbb Z)$ are the
$\Lambda$-points of a super torus of dimension $(g\mid g-1)$. Each
component of
$\operatorname{Pic}(X)$ is then isomorphic as a supermanifold to this
supertorus.
In general, however, $H^1(X,\mathcal O_X)$ is not free, nor is the image of
$H^1(X,\mathbb Z)$ generated by $2g$ independent vectors. It seems an
interesting question to understand $\Pic^0(X)$ in this generality.
For any supercurve $(X,\mathcal O_X)$ we define the Jacobian by
$$\operatorname{Jac}(X)=H^0(X,\Berx)^*_{\text{odd}}/H_1(X,\mathbb Z),$$
where elements of
$H_1(X,\mathbb Z)$ act by odd linear functionals on holomorphic
differentials
from $H^0(X,\Berx)$ by integration over 1-cycles.
We have, as discussed in Appendix \ref{app:dualSerredual}, a pairing
of
$\Lambda$-modules
\begin{equation}\label{eq:lambdapairing}
H^1(X,\mathcal O_X)\times H^0(X,\Berx)\to \Lambda.
\end{equation}
As we will discuss in more detail in subsection
\ref{ss:effdivisorpoinc}
invertible sheaves are also described by divisor classes. We use this
in the
following Theorem.
\begin{thm}
The pairing (\ref{eq:lambdapairing}) induces an isomorphism of the
identity
component $\Pic^0(X)$ with the Jacobian $\operatorname{Jac}(X)$ given by the usual
Abel map: a bundle $\mathcal L \in \Pic^0(X)$ with divisor $P-Q$
corresponds to
the class of
linear functionals $\int_Q^P$, modulo the action of $H_1(X,\mathbb
Z)$ by
addition of cycles to
the path from $Q$ to $P$.
\end{thm}
\begin{proof}
Let $\mathcal L \in \Pic^0(X)$ have divisor $P-Q$, with the reduced
points
$P^{\text{red}}$ and $Q^{\text{red}}$ contained
in a single chart $U_0$ of a good cover of $X$.
If $P=z-p-\theta\pi$ and $Q=z-q-\theta\xi$,
this bundle has a canonical section equal to unity in every other
chart, and
equal to
$$
\frac{z-p-\theta\pi}{z-q-\theta\xi} = \frac{z-p}{z-q} -
\frac{\theta\pi}{z-q} +
\theta\xi \frac{z-p}{(z-q)^2}
$$
in $U_0$.
In the covering space $H^1(X,\mathcal O_{X,\text{ev} } )$ of $\Pic^0(X)$, with covering group
$H^1(X,\mathbb Z)$, $\mathcal L$ lifts to a discrete
set of cocycles given by the logarithms of the transition functions
of
$\mathcal L$ in the chart overlaps, namely
$$
a_{0i} = \frac{1}{2\pi i}[\log (z-p) - \log (z-q) -
\frac{\theta\pi}{z-p} +
\frac{\theta\xi}{z-q}]
$$
in $U_0 \cap U_i$, and zero in other overlaps.
The covering group acts by changing the choice of branches for the
logarithms.
We now fix the particular choice for which the branch cut $C$ from
$Q$ to $P$
lies entirely in $U_0$ and meets no other
$U_i$.
Under the Dolbeault isomorphism, this cocycle corresponds to a
$(0,1)$ form
most conveniently represented by the current
$\bar{\partial}a_i$ in $U_i$, where $a_{ij} = a_i - a_j$ and
$\bar{\partial} = d\bar{z} \partial_{\bar{z}} + d\bar{\theta}
\partial_{\bar{\theta}}$.
It is supported on the branch cut $C$, and we can take $a_i = 0$ for
$i \ne 0$.
The pairing (\ref{eq:lambdapairing}) now associates to this the
linear
functional on $H^0(X,\Berx)$
which sends $\omega \in H^0(X,\Berx)$, written as $f(z) + \theta
\phi(z)$ in
$U_0$, to \cite{HaskeWells:Serreduality}
$$
\int_X i(\partial_{\bar{z}}) \bar{\partial}a_0 \, \omega \bar{\theta}
\,[dz\,
d\bar{z}\, d\bar{\theta} \, d\theta] =
\int_X (\partial_{\bar{z}}a_0) \omega dz \,d\bar{z}\, d\theta.
$$
By the definition of the derivative of a current
\cite{GrHa:PrincAlgGeom} and
Stokes' theorem this can be rewritten
\begin{multline*}
- \int_{\partial(X-C)} dz \int d\theta \, a_0 (f+\theta\phi)= \\
= - \frac{1}{2\pi i} \int_{\partial(X-C)} dz \{[\log(z-p) -
\log(z-q)]\phi +
[\frac{\xi}{z-q} - \frac{\pi}{z-p}]f\},
\end{multline*}
where $\partial(X-C)$ denotes the limit of a small contour enclosing
$C$.
Using the residue theorem and the discontinuity of the logarithms
across the
cut, this evaluates to
$$ \int_C \phi \, dz + \pi f(p) - \xi f(q) = \int_Q^P \omega .$$
By linearity of the pairing (\ref{eq:lambdapairing}),
we can extend this correspondence to arbitrary bundles of degree zero
by taking
sums of divisors of the form $P_i-Q_i$.
In particular, the divisor $(P-Q) + (P_1-P) + (P_2-P_1) + \cdots +
(P_n-P_{n-1}) + (P-P_n)$ is equivalent to $P-Q$,
but if the contour $PP_1P_2 \cdots P_nP$ represents a nontrivial
homology class
then the corresponding linear
functionals $\int_Q^P$ differ by addition of this cycle to the
integration
contour.
This shows that the action of $H_1(X,\mathbb Z)$ specified in the
definition of
$\operatorname{Jac}(X)$ is the correct one.
\end{proof}
\subsection{Effective divisors and Poincar\'e sheaf for generic SKP
curves.}
\label{ss:effdivisorpoinc}
Another description of invertible sheaves is given by divisor
classes.
Recall that a divisor $D\in \Divx$ is a global section of
the sheaf $\mathcal Rat_{\text{ev}}^\times(X)/\mathcal O^\times_{X,\text{ev}}$, so $D$, up to equivalence, is
given by a
collection
$(f_\alpha,U_\alpha)$ where the $f_\alpha$ are even
invertible rational functions that are on overlaps related by
an element of $\mathcal O^\times_{X,\text{ev}}(U_\alpha\cap U_\beta)$. Each
$f_\alpha$ reduces mod nilpotents to a nonzero rational
function $f_\alpha^{\text{red}}$ on the reduced curve, so
that $D$ determines a divisor $D^{\text{red}}$. Then the {\it
degree} of $D$ is the usual degree of its reduction
$D^{\text{red}}$. We have a mapping $\mathcal Rat_{\text{ev}}^\times(X)\to \Divx$,
$f\mapsto (f)$, and elements $(f)$ of the image are called
{\it principal}. Two even invertible rational functions $f_1,
f_2$ give rise to the same divisor iff $f_1=kf_2$ where $k\in
H^0(X,\mathcal O^\times_{X,\text{ev}})$. So if $(X,\mathcal O_X)$ is a generic
SKP curve $k$ is just an even invertible element of $\Lambda$
but in general more exotic possibilities for $k$ exist. A
divisor $D$ is {\it effective}, notation $D\ge 0$, if all
$f_\alpha\in \mathcal O_{X,\text{ev} } (U_\alpha)$. An invertible $\mathcal O_X$-module
$\mathcal L$ can be thought of as a submodule of rank $1|
0$ of the constant sheaf $\mathcal Rat(X)$. If $\mathcal
L(U_\alpha)=\mathcal O_X(U_\alpha)e_\alpha$, then
$e_\alpha\in \mathcal Rat_{\text{ev}}^\times(X)$ and $\mathcal L$ determines the
divisor $D=\{(f_\alpha=e_\alpha\inv,U_\alpha)\}$. Conversely
any divisor $D$ determines an invertible sheaf $\mathcal O_X(D)$ (in
$\mathcal Rat(X)$) with local generators $e_\alpha=f_\alpha\inv$. Two
divisors $D_1=\{(f^{(1)}_\alpha,U_\alpha)\}$ and
$D_2=\{(f^{(2)}_\alpha,U_\alpha)\}$ give rise to equivalent
invertible sheaves iff they are {\it linearly equivalent},
i.e., $D_1=D_2+(f)$ for some element $f$ of $\mathcal Rat_{\text{ev}}^\times(X)$, or
more explicitly iff $f^{(1)}_\alpha=ff^{(2)}_\alpha$ for all
$\alpha$. If $f\in \mathcal Rat_{\text{ev}}^\times(X)$ is a global section of an
invertible sheaf $\mathcal L=\mathcal O_X(D)$ then $D+(f)\ge 0$ and
vice versa. The {\it complete linear system} $|D|=|\mathcal O_X(D)|$
of a divisor (or of the corresponding invertible sheaf) is
the set of all effective divisors linearly equivalent to $D$. So
we see that if $\mathcal L=\mathcal O_X(D)$ then
$$
|D|\simeq H^0(X,\mathcal
L)_{\text{ev}}^\times/H^0(X,\mathcal O_X)_{\text{ev}}^\times.
$$
In case the cohomology of $\mathcal L$ is free of rank
$p+1\mid q$ and $H^0(X,\mathcal O_X)$ is just the constants
$\Lambda\mid 0$, the complete linear system $|D|$ is (the set
of $\Lambda$-points of) a super projective space $\mathbb
P^{p\mid q}_\Lambda$. In particular, if $(X,\mathcal O_X)$ is a
generic SKP curve and the degree $d$ of $\mathcal L$ is $\ge
2g-1$ the first cohomology of $\mathcal L$ vanishes, the
zeroth cohomology is free of rank $d +1-g\mid d+1-g$ and
$|D|\simeq \mathbb P_\Lambda^{d-g\mid d+1-g}$.
Let $\hat X=(X,\hat{\mathcal O}_X)$ be the dual curve and denote by
$\hatxd$ the $d$-fold symmetric product of $\hat X$, see
\cite{DomPerHerRuiSanchSal:Superdiv}. This smooth
supermanifold of dimension $(d\mid d)$ parametrizes
effective divisors of degree $d$ on $(X,\mathcal O_X)$. We have a map
(called {\it Abelian sum}) $A:\hatxd\to \operatorname{Pic}^d(X)$ sending an
effective divisor $D$ to the corresponding invertible sheaf
$\mathcal O_X(D)$. An invertible sheaf $\mathcal L$ is in the image of
$A$ iff $\mathcal L$ has a even invertible global section: if
$D\in \hatxd$ and $\mathcal L=A(D)$ then the fiber of $A$ at
$\mathcal L$ is the complete linear system $|D|$. If the
degree $d$ of $\mathcal L$ is at least $2g-1$ $H^1(X,\mathcal L)$ is
zero and
hence the cohomology
of $\mathcal L$ is free. So in that case $A$ is
surjective and the fibers of $A$ are all projective spaces $\mathbb
P^{d-g\mid d+1-g}$ and $A$ is in fact a fibration.
The symmetric product $\hatxd$ is a universal parameter space
for effective divisors of degree $d$. This is studied in
detail by Dom\'\i nguez P\'erez et al.
\cite{DomPerHerRuiSanchSal:Superdiv}; we will summarize some
of their results and refer to their paper for more details.
(In fact they consider curves over a field, but the theory is
not significantly different for curves over $\Lambda$.) A
{\it family of effective divisors } of degree $d$ on
$X$ parametrized by a super scheme $S$ is a pair $(S,D_S)$,
where $D_S$ is a Cartier divisor on $X\times_\Lambda S$ such
that for any morphism $\phi:T\to S$ the induced map
$(1\times\phi)^*\mathcal O_{X\times S}(-D_S)\to
(1\times\phi)^*\mathcal O_{X\times S}$ is injective and such
that for any $s\in S$ the restriction of $D_S$ to $X\times
\{s\}\simeq X$ is an effective divisor of degree $d$. For example, in
$X\times \hatxd$ there is a canonical divisor $\Delta^{(d)}$
such if $p_D$ is any $\Lambda$-point of $\hatxd$
corresponding to a divisor $D$ then the restriction of
$\Delta^{(d)}$ to $X\times \{p_D\}\simeq X$ is just $D$. Then
$(\hatxd, \Delta^{(d)}) $ is universal in the sense that for
any family $(S,D_S)$ there is a unique morphism $\Psi:S\to
\hatxd$ such that $D_S=\Psi^*\Delta^{(d)}$.
A {\it family of invertible sheaves } of degree $d$ on $X$
parametrized by a super scheme $S$ is a pair $(S,\mathcal
L_S)$, where $\mathcal L_S$ is an invertible sheaf on
$X\times_\Lambda S$ such that for any $s\in S$ the
restriction of $\mathcal L_S$ to $X\times \{s\}$ is a sheaf
of degree $d$ on $X$. For example, $(\hatxd,\mathcal
O_{X\times \hatxd}(\Delta^{(d)})$ is a family of invertible
sheaves of degree $d$. Two families $(S,\mathcal L_1)$,
$(S,\mathcal L_2)$ are equivalent if $\mathcal L_1=\mathcal
L_2\otimes \pi_S^* \mathcal N$, where $\pi_S:X\times S\to S$
is the canonical projection and $\mathcal N$ is an invertible
sheaf on $S$. For example, fix a point $x$ of $X$;
then$(\hatxd,\mathcal O_{X\times \hatxd}(\Delta^{(d)}))$ is
equivalent to $(\hatxd,\mathcal R_{x}))$, where $\mathcal
R_x=\mathcal O_{X\times \hatxd}(\Delta^{(d)})\otimes
\pi_{\hatxd}^*[\mathcal O_{X\times
\hatxd}(\Delta^{(d)})|_{\{x\}\times\hatxd})]\inv$. The family
$(\hatxd,\mathcal R_x)$ is normalized: it has the property
that $\mathcal R_x$ restricted to $\{x\}\times \hatxd$ is
canonically trivial. Now consider the mapping $(1\times
A):X\times \hatxd\to X\times\operatorname{Pic}^d(X)$ and the direct image
$\mathcal P^{(d)}_x=(1\times A)_* \mathcal R_x$.
\begin{thm}
Let $(X,\mathcal O_X)$ be a generic SKP curve. Let $d\ge
2g-1$. Then $\mathcal P^{(d)}_x$ is a Poincar\'e sheaf on
$X\times\operatorname{Pic}^d(X)$, i.e., $(\operatorname{Pic}^d(X), \mathcal P^{(d)}_x)$
is a family of invertible sheaves of degree $d$ that is
universal in the sense that for any family $(S,\mathcal L)$ of
degree $d$ invertible sheaves there is a unique morphism
$\phi:S\to \operatorname{Pic}^d(X)$ so that $\mathcal L=\phi^*\mathcal
P^{(d)}_x$. Furthermore $\mathcal P^{(d)}_x$ is normalized so
that the restriction to $\{x\}\times \operatorname{Pic}^d(X)$ is
canonically trivial.
\end{thm}
\subsection{Berezinian bundles.}\label{subs:Berbundles}
We continue with the study of a generic SKP curve
$(X,\mathcal O_X)$; we fix an integer $n$ and write $\mathcal P$ for
$\mathcal P^n_X$, the Poincar\'e sheaf on $X\times
\operatorname{Pic}^n(X)$. Let $\mathcal L_s$ be an invertible sheaf corresponding
to $s\in \operatorname{Pic}^n(X)$. The cohomology groups $H^i(X,\mathcal
L_s)$ will vary as $s$ varies over $\operatorname{Pic}^n(X)$ and can in
general be nonfree, as we have seen. Even if the cohomology
groups are free $\Lambda$-modules their ranks will jump.
Still it is possible to define an invertible sheaf $\Ber$ over
$\operatorname{Pic}^n(X)$ with fiber at $s$ the line
$$
\ber(H^0(X,\mathcal L_s))\otimes \ber^*(H^1(X,\mathcal L_s)),
$$
in case $\mathcal L_s$ has free cohomology. Here $\ber(M)$ for
a free rank $d\mid \delta$ $\Lambda$-module with basis
$\{f_1,\dots,f_d,\phi_1,\dots,\phi_\delta\}$ is the rank 1
$\Lambda$-module with generator
$B[f_1,\dots,f_d,\phi_1,\dots,\phi_\delta]$. If we are given
another basis
$\{f^\prime_1,\dots,f^\prime_d,\phi^\prime_1,\dots,\phi^\prime
_\delta\}=g\cdot \{f_1,\dots,f_d,\phi_1,\dots,\phi_\delta\}$,
with $g\in Gl(d\mid\delta,\Lambda)$, we have the relation
$$
B[f^\prime_1,\dots,f^\prime_d,\phi^\prime_1,\dots,\phi^\prime_\delta]
=\ber(g)B[f_1,\dots,f_d,\phi_1,\dots,\phi_\delta].
$$
Similarly $\berdual(M)$ is defined using the inverse
homomorphism $\berdual$. Here $\ber$ and $\berdual$ are the
group homomorphisms defined in (\ref{eq:defberber*}).
The invertible sheaf $\mathcal L_s$ is obtained from the
Poincar\'e sheaf via $i_s^*\mathcal P$. We can reformulate
this somewhat differently: $\mathcal P$ is an $\mathcal
O_{\operatorname{Pic}^n(X)}$-module and for every $\Lambda$-point $s$ of
$\operatorname{Pic}^n(X)$, via the homomorphism $s^\sharp:\mathcal
O_{\operatorname{Pic}^n(X)}\to \Lambda$, also $\Lambda$ becomes an
$\mathcal O_{\operatorname{Pic}^n(X)}$-module, denoted by $\Lambda_s$. Then
$\mathcal L_s=i_s^*\mathcal P=\mathcal P\otimes_{\mathcal
O_{\operatorname{Pic}^n(X)}} \Lambda_s$. It was Grothendieck's idea to
study the cohomology of $\mathcal P\otimes M$ for arbitrary
$\mathcal O_{\operatorname{Pic}^n(X)}$-modules $M$. We refer to Kempf
(\cite{Kempf:AbInt}) for an excellent discussion and more
details on these matters.
The basic fact is that, given the Poincar\'e bundle $\mathcal
P$ on $X\times \operatorname{Pic}^n(X)$, there is a homomorphism
$\alpha:\mathcal F\to \mathcal G$ of locally free coherent
sheaves on $\operatorname{Pic}^n(X)$ such that we get for any sheaf of
$\mathcal O_{\operatorname{Pic}^n(X)}$-modules $M$ an exact sequence
\begin{gather*}
\begin{aligned}
0 \to H^0(X\times \operatorname{Pic}^n(X),\mathcal P\otimes M)\to \mathcal
F\otimes
M\overset
{\alpha\times
1_M}\to \mathcal G\otimes M\to \\
\to H^1(X\times \operatorname{Pic}^n(X),
\mathcal P\otimes M)\to 0.
\end{aligned}
\end{gather*}
The proof of this is the same as for the analogous statement in the
classical case, see \cite{Kempf:AbInt}.
Now $\mathcal F$ and $\mathcal G$ are locally free, so for small
enough open
sets $U$ on $\operatorname{Pic}^n(X)$ one can define
$\ber(\mathcal F(U))$ and $\ber^*(\mathcal G(U))$. This
globalizes to invertible sheaves $\ber(\mathcal F)$ and
$\ber^*(\mathcal G)$. Next we form the ``Berezinian
of the cohomology of $\mathcal P$'' by defining
$\Ber=\ber(\mathcal F)\otimes \ber^*(\mathcal G)$. Finally one
proves, as in Soul\'e, \cite{Soul:Arakelov}, VI.2, Lemma 1,
that $\Ber$ does not depend, up to isomorphism, on the choice
of homomorphism $\alpha:\mathcal F\to \mathcal G$.
\begin{thm}\label{thm:1ChernBertriv} The first Chern class of
the $\Ber$ bundle is zero. \end{thm}
We will prove this theorem
in subsection \ref{ss:ChernclassBeronPic}, after the
introduction of the infinite super Grassmannian and the
Krichever map. The topological triviality of the $\Ber$ bundle
is a fundamental difference from the situation of classical
curves: there the determinant bundle on $\operatorname{Pic}$ is ample.
Next we consider the special case of $n=g-1$. In this case,
because of Riemann-Roch (\ref{superRR}) $\mathcal F$ and
$\mathcal G$ have the same rank. Indeed, locally
$\alpha(U):\mathcal F(U)\to \mathcal G(U)$ is given, after
choosing bases, by a matrix over $\mathcal O_{\operatorname{Pic}^n(X)}(U)$ of
size $d\mid\delta \times e\mid \epsilon$, say. If we fix a
$\Lambda$-point $s$ in $U$ we get a homomorphism
$\alpha(U)_s:\mathcal F(U)\otimes \Lambda_s\to \mathcal
G(U)\otimes\Lambda_s$ represented by a matrix over $\Lambda$.
The kernel and cokernel are the cohomology groups of
$\mathcal L_s$ and these have the same rank by Riemann-Roch. On the
other hand if the kernel and cokernel of a matrix over
$\Lambda$ are free we have rank of kernel $-$ rank of
cokernel= $d-e\mid\delta- \epsilon=0\mid 0$. So $\alpha(U)$ is a
square
matrix. This allows us to
define a map
$$
\ber(\alpha):\ber(\mathcal F)\to \ber (\mathcal G).
$$
But this is a (non-holomorphic!) section of $\ber^*(\mathcal
F)\otimes \ber(\mathcal G)$, i.e., of the dual Berezinian
bundle $\mathcal P^*$ on $\operatorname{Pic}^{g-1}$, because of the non-polynomial
(rational) character of the Berezinian. This section $\ber(\alpha)$
is
essential for the definition of the $\tau$-function in subsection
\ref{ss:Bakerf-fullsuperH-tau}.
\subsection{Bundles on the Jacobian; theta functions}
We continue with $X$ being a generic SKP curve.
Super theta functions will be defined as holomorphic sections
of certain ample bundles on $J=\text{Jac}(X)$, when such bundles
exist. (As usual, the existence of ample invertible sheaves is
necessary and
sufficient for projective embeddability.)
Given one such bundle, all others with the same Chern
class $c_1$ are obtained by tensor product with bundles having
trivial Chern class, so we begin by determining these, that
is, computing $\text{Pic}^0(J)$.
As we briefly discussed in subsection \ref{ss:modinvertible sheaves}
$J$ is the
quotient of the affine super space
$V={\mathbb A}^{g|g-1} = \operatorname{Spec}
\Lambda[z_1,\ldots,z_g,\eta_1,\ldots,\eta_{g-1}]$
by the lattice $L$ generated by the columns of the transposed
period matrix:
\begin{equation}\label{eq:latticegen}
\begin{aligned}
\lambda_i: & \quad z_j \rightarrow z_j + \delta_{ij}, &\quad
&\eta_\alpha \rightarrow\eta_\alpha,\\
\lambda_{i+g}: &\quad z_j \rightarrow z_j+(Z_e)_{ij}, &\quad
&\eta_\alpha \rightarrow \eta_\alpha + (Z_o)_{i
\alpha}, \quad
i=1,2,\ldots,g.
\end{aligned}
\end{equation}
We will often omit the parity labels $e,o$ on $Z$, since the index
structure makes clear which is meant.
Any line bundle $\mathcal L$ on such a supertorus $J$ lifts to a
trivial bundle on the covering space $V$.
A section of $\mathcal L$
lifts to a function on which the translations $\lambda_i$ act
by multiplication by certain invertible holomorphic functions,
the {\it multipliers} of $\mathcal L$. We can factor the quotient map
$V \rightarrow J$ through the cylinder $V/L_0$, where $L_0$ is
the subgroup of $L$ generated by the first $g$ $\lambda_i$
only. Since holomorphic line bundles on a cylinder are
trivial, this means that the multipliers for $L_0$ can always
be taken as unity. We have $\text{Pic}^0(J) \cong
H^1(J,\mathcal{O}_{\text{ev}})/H^1(J,\mathbb{Z})$. It is very
convenient to compute the numerator as the group cohomology
$H^1(L,\mathcal{O}_{\text{ev}})$ of $L$ acting on the even
functions on the covering space $V$, in part because the
cocycles for this complex are precisely (the logarithms of)
the multipliers. For the basics of group cohomology, see for
example \cite{Si:ArithEllipticCurves,Mum:AbelVar}. In
particular, factoring out the subgroup $L_0$ reduces our
problem to computing $H^1(L/L_0,\mathcal{O}^{L_0})$, the
cohomology of the quotient group acting on the $L_0$-invariant
functions.
A 1-cochain for this complex assigns to each generator of
$L/L_0$ an even function (log of the multiplier) invariant
under each shift $z_j \rightarrow z_j + 1$,
\begin{equation*}
\lambda_{i+g} \mapsto F^i(z,\eta) = \sum_{\vec{n}}
F_{\vec{n}}^i(\eta) e^{2 \pi i \vec{n} \cdot \vec{z}}.
\end{equation*}
It is a cocycle if the multiplier induced for
every sum $\lambda_{i+g} + \lambda_{j+g}$ is independent of the
order of addition,
which amounts to the symmetry of the matrix $\Delta_i F^j$ giving
the change in $F^j$ under the action of $\lambda_{i+g}$:
\begin{multline*}
F^i(z_k + Z_{jk},\eta_\alpha + Z_{j \alpha}) - F^i(z_k,\eta_\alpha)
=\\
F^j(z_k + Z_{ik},\eta_\alpha + Z_{i \alpha}) - F^j(z_k,\eta_\alpha),
\end{multline*}
or, in terms of Fourier coefficients,
\begin{equation*}
F_{\vec{n}}^i(\eta_\alpha + Z_{j \alpha})
e^{2\pi i\sum_k n_k Z_{jk}} - F_{\vec{n}}^i(\eta) =
F_{\vec{n}}^j(\eta_\alpha + Z_{i \alpha}) e^{2\pi i\sum_k
n_k Z_{ik}} - F_{\vec{n}}^j(\eta).
\end{equation*}
One does
not have to allow for an integer ambiguity in the logarithms
of the multipliers in these equations, precisely because we
are considering bundles with vanishing Chern class. The
coboundaries are of the form,
\begin{equation*}
\lambda_{i+g}
\mapsto A(z,\eta) - A(z_k + Z_{ik},
\eta_\alpha + Z_{i \alpha})
\end{equation*}
for a single function $A$, that is,
those cocycles for which
\begin{equation*}
F_{\vec{n}}^i(\eta)
= A_{\vec{n}}(\eta) - A_{\vec{n}}(\eta_\alpha + Z_{i
\alpha}) e^{2\pi i\sum_k n_k Z_{ik}}.
\end{equation*}
This equation has the form,
\begin{equation*}
F_{\vec{n}}^i(\eta)
= A_{\vec{n}}(\eta)
( 1 - e^{2\pi i\sum_k n_k Z_{ik}}) + O(Z_o).
\end{equation*}
The point now is that, by the linear independence of the columns of
$Z_e^{\text{red}}$, for any $\vec{n} \ne \vec{0}$ there is some
choice of $i$
for which the
reduced part of the exponential in the last
equation differs from unity.
This ensures that, for this $i$, the equation is solvable for
$A_{\vec{n}}$, first to zeroth order in $Z_o$ and then to all orders
by iteration.
Adding this coboundary to the cocycle produces one for which
$F_{\vec{n}}^i = 0$, whereupon the cocycle conditions imply
$F_{\vec{n}}^j = 0$ for all $\vec{n} \ne \vec{0}$ and all $j$ as
well.
Thus the only potentially nontrivial cocycles are independent
of $z_i$.
In the simplest case, when the odd period matrix $Z_o=0$,
all such cocycles are indeed nontrivial, and we have an analog of
the classical fact that bundles of trivial Chern class are specified
by $g$ constant multipliers.
Here a cocycle is specified by giving $g$
even elements $F^i_{\vec{0}}(\eta)$ in the exterior algebra
$\Lambda[\eta_{\alpha}]$ (elements of $H^0(J,\mathcal O_J$)),
leading to $\dim \text{Pic}^0(J) =
g^{2^{g-2}} \mid g^{2^{g-2}}$ (the number of $\eta_{\alpha}$
is $g-1$).
In general, when $Z_o \neq 0$, not all cochains
specified in this way will be cocycles, and some cocycles
will be trivial:
$\text{Pic}^0(J)$ will be smaller, and in general not a
supermanifold.
As to the existence of ample line bundles, let us examine in
the super case the classical arguments
leading to the necessary and sufficient Riemann conditions
\cite{GrHa:PrincAlgGeom,LaBirk:ComplAbVar}.
The Chern class of a very ample bundle is represented in de Rham
cohomology by a $(1,1)$ form obtained as the pullback of the
Chern class of the hyperplane bundle via a projective embedding.
We can introduce real even coordinates $x_i,i=1,\ldots,2g$ for $J$
dual to
the basis $\lambda_i$ of the lattice $L$, meaning that
$x_j \rightarrow x_j + \delta_{ij}$ under the action of
$\lambda_i$.
The associated real odd coordinates $\xi_\alpha,\alpha=1,\ldots,2g-2$
can be taken
to be globally defined because every real supermanifold is split.
The relation between the real and complex coordinates can be taken
to be
\begin{eqnarray*}
z_j & = & x_j + \sum_{i=1}^g Z_{ij} x_{i+g}, \; j=1,\ldots,g, \\
\eta_\alpha & = & \xi_\alpha + i \xi_{\alpha + g-1} +
\sum_{i=1}^g Z_{i \alpha} x_{i+g}, \; \alpha = 1,\ldots,g-1.
\end{eqnarray*}
The de Rham cohomology is isomorphic to that of the reduced torus
and can be represented by translation-invariant forms in the
$dx_i$.
The Chern class represented by a form
$\sum_{i=1}^g \delta_i\, dx_i\, dx_{i+g}$
is called a polarization of type $\Delta =
\text{diag}(\delta_1,\ldots,\delta_g)$ with elementary divisors the
positive integers $\delta_i$.
We consider principal polarizations $\delta_i=1$ only, because
nontrivial nonprincipal polarizations generically do not exist,
even on the reduced torus \cite{Lef:ThmCorrAlgCurv}.
Furthermore, a nonprincipal polarization is always obtained by
pullback
of a principal one from another supertorus whose lattice $L'$
contains $L$ as a sublattice of finite index
\cite{GrHa:PrincAlgGeom}.
Reexpressing the Chern form in complex coordinates, the standard
calculations lead to the usual Riemann condition $Z_e = Z_e^t$ to
obtain a $(1,1)$ form.
Together with the positivity of the imaginary part of the reduced
matrix, the symmetry of $Z_e$ (in some basis) is necessary and
sufficient
for the existence of a $(1,1)$ form
with constant coefficients representing the Chern class.
This can be viewed as the cocycle condition, symmetry of
$\Delta_iF^j$, for the usual multipliers of a theta bundle,
$F^j = -2\pi i z_j$.
The usual argument that the $(1,1)$ form representing the Chern
class can always be taken to have constant coefficients depends
on Hodge theory, particularly the Hodge decomposition of
cohomology, for a K\"ahler manifold such as a torus.
This does not hold in general for a supertorus with $Z_o \neq 0$.
For example, $H_{\text{dR}}^1(J)$ is generated by the $2g$ 1-forms
$dx_i$,
whereas $H^{1,0}(J)$ contains the $g \mid g-1$ nontrivial forms
$dz_i,\,d\eta_\alpha$, with certain nilpotent multiples of the
latter being trivial.
Indeed, since by (\ref{eq:latticegen}) $\eta_\alpha$ is defined
modulo entries of column $\alpha$ of $Z_o$, $\epsilon\eta_\alpha$
is a global function and $\epsilon d \eta_\alpha$ is exact when
$\epsilon \in \Lambda$ annihilates these entries.
Thus, $H^{1,0}(J)$ cannot be a direct summand in
$H^1_{\text{dR}}(J)$.
Correspondingly, some $\eta$-dependent multipliers
$F^j = -2\pi i z_j + \cdots$ may satisfy the cocycle condition
and give ample line bundles.
We do not know a simple necessary condition for a Jacobian to
admit such polarizations.
When $Z_e$ is symmetric, we can construct theta functions
explicitly.
Consider first the trivial case with $Z_o=0$ as well.
Then the standard Riemann theta function $\Theta(z;Z_e)$
gives a super theta function on $\operatorname{Jac}(X)$, where
$\Theta(z;Z_e)$ is defined by Taylor expansion in the nilpotent part
of
$Z_e$ as usual.
It has of course the usual multipliers,
\begin{equation} \label{eq:thetafactors}
\Theta(z_j + \delta_{ij};Z_e) = \Theta(z_j;Z_e),\;\;\;
\Theta(z_j + Z_{ij};Z_e) = e^{-\pi i (2z_i + Z_{ii})}
\Theta(z_j;Z_e).
\end{equation}
Multiplication of $\Theta(z;Z_e)$ by any monomial in the odd
coordinates
$\eta_{\alpha}$ gives another, even or odd,
theta function having the same multipliers, whereas
translation of the argument $z$ by polynomials in the
$\eta_{\alpha}$ leads to the multipliers for another bundle
with the same
Chern class.
In the general case with $Z_o \neq 0$, theta functions with the
standard multipliers can be constructed as follows.
Such functions must obey
\begin{align*}
H(z_j + \delta_{ij},\eta_{\alpha};Z) &= H(z_j,\eta_{\alpha};Z),\\
H(z_j + Z_{ij},\eta_{\alpha} + Z_{i \alpha};Z) &=
e^{-\pi i (2z_i + Z_{ii})} H(z_j,\eta_{\alpha};Z).
\end{align*}
The function $\Theta(z;Z_e)$ is a trivial example independent of
$\eta$;
to obtain others one checks that
when $H$ satisfies these relations then so does
$$H_\alpha = \left( \eta_{\alpha} + \frac{1}{2\pi i} \sum_k Z_{k
\alpha}
\frac{\partial}{\partial z_k} \right) H.$$
Applying this operator repeatedly one constructs super theta
functions
$\Theta_{\alpha \cdots \gamma}$ reducing to
$\eta_\alpha \cdots \eta_\gamma \Theta(z;Z_e)$
when $Z_o=0$.
``Translated" theta functions which are sections of other bundles
having the same Chern class can be obtained by literally
translating the arguments of these only in the simplest cases.
Constant shifts in the multiplier exponents $F^j$ can be achieved by
constant shifts of the arguments $z_j$.
Shifts linear in the $\eta_\alpha$ are obtained by
$z_j \rightarrow z_j + \eta_\alpha \Gamma_{\alpha j}$,
which is a change in the chosen basis of holomorphic differentials
on $X$, see the discussion after \eqref{eq:normperiodmatrix}.
The resulting theta functions have the new period matrix
$Z_e + Z_o\Gamma$.
More generally, translated theta functions
can be obtained by the usual method of determining their
Fourier coefficients from the recursion relations following from the
desired multipliers.
We do not know an explicit expression for them in terms of
conventional
theta functions.
It is easy to see that any meromorphic function $F$ on the Jacobian
can be rationally expressed in terms of the theta functions we have
defined.
Expand $F(z,\eta) = \sum_{IJ} \beta_I \eta_J F_{IJ}(z)$ in the
generators of
$\Lambda[\eta_{\alpha}]$, with multi-indices $I,J$.
Then the zeroth-order term $F_{00}$ is a meromorphic
function on the reduced Jacobian, hence a rational expression
in ordinary theta functions.
Using $Z_e$ as the period matrix argument of these theta functions
gives a meromorphic function
on the Jacobian itself, whose reduction agrees with $F_{00}$.
Subtract this expression from $F$ to get a meromorphic function on
the Jacobian
whose zeroth-order term vanishes,
and continue inductively, first in $J$, then in $I$.
For example, $F_{0\alpha}$ is equal, to lowest order in the
$\beta$'s, to a rational expression in theta functions of which one
numerator factor is a $\Theta_\alpha$.
Subtracting this expression removes the corresponding term in $F$
while only modifying other terms of higher order in $\beta$'s.
\section{Super Grassmannian, $\tau$-function and Baker
function.}
\subsection{Super Grassmannians.}
In this subsection we will introduce an infinite super Grassmannian
and related
constructions. The infinite Grassmannian of Sato
(\cite{Sa:KPinfDymGr}) or of
Segal-Wilson (\cite
{SeWi:LpGrpKdV}) consists (essentially) of ``half infinite
dimensional'' vector
subspaces $W$ of an infinite dimensional vector space $H$ such that
the
projection on a fixed subspace $H_-$ has finite dimensional kernel
and
cokernel. In the super category we replace this by the super
Grassmannian of
free ``half infinite rank'' $\Lambda$-modules of an infinite rank
free
$\Lambda$-module $H$ such that the kernel and cokernel of the
projection on
$H_-$ are a submodule respectively a quotient module of a free finite
rank
$\Lambda$-module. In \cite{Schwarz:FermStringModSpa} a similar
construction can
be found, but it seems that there $\Lambda=\mathbb C$ is taken as is
also the
case in \cite{Mu:Jac}. This is too restrictive for our purposes
involving
algebraic super curves over nonreduced base ring $\Lambda$.
Let $\Linfinf$ be the free $\Lambda$-module $\Lambda[z,z\inv,\theta]$
with $z$
an even and $\theta$ an odd variable. Introduce the notation
\begin{equation}\label{eq:basisLinfinf}
e_i=z^i,\quad e_{i-\frac12}=z^i\theta,\quad i\in \mathbb Z.
\end{equation}
We will think of an element $h=\sum_{i=-N}^\infty h_i e_i$, $h_i\in
\Lambda$
of $\Linfinf$ not only as a series in $z,\theta$ but also as an
infinite
column vector:
$$
h=(\dots,0,\dots,
h_{-1},h_{-\frac12},h_0,h_{\frac12},h_1,\dots,0,\dots)^t
$$
Introduce on $\Linfinf$ an odd Hermitian product
\begin{multline}
\langle f(z,\theta),g(z,\theta)\rangle=
\frac1{2\pi i}\oint
\frac{dz}{z}d\theta\overline{f(z,\theta)}g(z,\theta)=\\
=\frac1{2\pi i}\oint (\overline{f_{\bar 0}}g_{\bar
1}+\overline{f_{\bar
1}}g_{\bar 0})\frac{dz}{z},
\end{multline}
where $\overline{f(z,\theta)}$ is the extension of the complex
conjugation of
$\Lambda$ (see Appendix \ref{ss:realstrconj}) to $\Linfinf$ by
$\overline{z}=z\inv$ and $\overline{\theta}=\theta$, and
$f(z,\theta)=f_{\bar
0}+\theta f_{\bar 1}$, and similarly for $g$. Let $H$ be the
completion of
$\Linfinf$ with respect to the Hermitian inner product.
We have a decomposition $H=H_-\oplus H_+$, where $H_-$ is the closure
of the
subspace spanned by $e_i$ for $i\le0$, and $H_+$ is the closure of
the space
spanned by $e_i$ with $i>0$, for $i\in \halfz$.
The super Grassmannian $\mathcal S\text{gr}$ is the collection of all free closed
$\Lambda$-modules $W\subset H$ such that the projection $\pi_-:W\to
H_-$ is
super Fredholm, i.e., the kernel and cokernel are a submodule
respectively a
quotient module of a free finite rank $\Lambda$-module.
\begin{exmpl} Let $W$ be the closure of the subspace generated by
$\delta +z, \theta$ and $z^i, z^i\theta$ for $i\le -1$, for $\delta$
a
nilpotent even constant. Let $A\subset \Lambda$ be the ideal of
annihilators of
$\delta$. Then $W$ is free and the kernel of $\pi_-$ is
$A (\delta +z) \subset \Lambda (\delta+z)$ and the cokernel is
isomorphic to
$\Lambda/\Lambda\delta$. \qed
\end{exmpl}
Let $I$ be the subset $\{i\in \halfz\mid i\le 0\}$.
We consider matrices with coefficients in $\Lambda$ of size
$\halfz\times I$:
$$
\mathcal W=(W_{ij}) \quad \text{where } i\in \halfz,\ j\in I.
$$
An even matrix of this type is called an {\it admissible frame} for
$W\in \mathcal S\text{gr}$
if the closure of the subspace spanned by the columns of $\mathcal W$
is $W$
and if moreover in the decomposition
$\mathcal W=\begin{pmatrix} W_-\\W_+\end{pmatrix}$ induced by
$H=H_-\oplus H_+$
the operator $W_-:H_-\to H_-$ differs from the identity by an
operator of super
trace class and $W_+:H_-\to H_+$ is compact.
Let $Gl(H_-)$ be the group of invertible maps $1+X:H_-\to H_-$ with
$X$ super
trace class. Then the super frame bundle $\mathcal S\text{fr}$, the collection of
all pairs
$(W,\mathcal W)$ with $\mathcal W$ an admissible frame for $W\in
\mathcal S\text{gr}$, is a
principal $Gl(H_-)$ bundle over the super Grassmannian. Elements of
$Gl(H_-)$
have a well defined berezinian, see
\cite{ Schwarz:FermStringModSpa} for some details. This allows us
to define
two associated line bundles $\Bersgr$ and $\Bersgrdual$ on $\mathcal S\text{gr}$.
More
explicitly, an element of $\Bersgr$ is an equivalence class of
triples
$(W,\mathcal W, \lambda)$, with $\mathcal W$ a frame for $W$,
$\lambda\in
\Lambda$; here $(W,\mathcal Wg, \lambda)$ and $(W,\mathcal W,
\ber(g)\lambda)$
are equivalent for $g\in Gl(H_-)$. For $\Bersgrdual$ replace
$\ber(g)$ by
$\berdual(g)$. For simplicity we shall write $(\mathcal W,\lambda)$
for
$(W,\mathcal W, \lambda)$, as $\mathcal W$ determines $W$ uniquely.
The two bundles $\Bersgr$ and $\Bersgrdual$ each have a canonical
section.
Let $\mathcal W$ be a frame for $W\in \mathcal S\text{gr}$ and write $\mathcal
W=\begin{pmatrix}W_-\\W_+\end{pmatrix}$ as above. Then
\begin{equation}\label{eq:defsigma*}
\sigma(W)=(\mathcal W, \ber(W_-)),\quad \sigma^*(W)=(\mathcal W,
\berdual(W_-)),
\end{equation}
are sections of $\Bersgrdual$ and $\Bersgr$, respectively. It is a
regrettable
fact of life that neither of these sections is holomorphic; indeed
there are no
global sections to $\Bersgr$ or $\Bersgrdual$ at all, see
\cite{Manin:GaugeFieldTheoryComplexGeom}. This is a major difference
between
classical geometry and super geometry.
\subsection{The Chern class of $\Bersgr$ and the
$gl_{\infty\mid\infty}$
cocycle.}
First we summarize some facts about complex supermanifolds that are
entirely
analogous to similar facts for ordinary complex manifolds. Then we
apply this
to the super Grassmannian, following the treatment in
\cite{PrSe:LpGrps} of the
classical case.
Let $M$ be a complex supermanifold. The Chern class of an invertible
sheaf
$\mathcal L$ on $M$ is an element $c_1(\mathcal L)\in H^2(M,\mathbb
Z)$. By
the sheaf inclusion $\mathbb Z\to \Lambda$ and the de Rham theorem
$H^2(M,\Lambda)\simeq H^2_{\text{dR}}(M)$ we can represent
$c_1(\mathcal L)$ by
a closed two form on $M$. On the other hand, if $\nabla:\mathcal L\to
\mathcal
L\otimes \mathcal A^1$, with $\mathcal A^1$ the sheaf of smooth
1-forms, is a
connection compatible with the complex structure, the curvature $F$
of $\nabla$
is also a two form. By the usual proof (see e.g.,
\cite{GrHa:PrincAlgGeom}) we
find that $c_1(\mathcal L) $ and $F$ are equal, up to a factor of
$i/2\pi$.
We can locally calculate the curvature on an invertible sheaf
$\mathcal L$ by
introducing a Hermitian metric $\langle\,, \rangle$ on it: if $s,t\in
\mathcal
L(U)$ then $\langle s, t\rangle(m)$ is a smooth function in $m\in U$
taking
values in $\Lambda$, linear in $t$ and satisfying $\langle s,
t\rangle(m)=\overline{\langle t, s\rangle(m)}$. Choose a local
generator $e$
of $\mathcal L$ and let $h=\langle e, e\rangle$. The curvature is
then
$F=\bar\partial \partial \log h$, with $\partial=\sum
dz_i\frac{d}{dz_i}+\sum
d\theta_\alpha \frac{\partial}{\partial \theta_\alpha}$ and
$\bar\partial$
defined by a similar formula.
Now consider the invertible sheaf $\Bersgr$ on $\mathcal S\text{gr}$. If
$s=(\mathcal W,
\lambda)$ is a section the square length is defined to be $\langle
s,s\rangle=
\bar \lambda \lambda\ber (\mathcal W^H \mathcal W)$, where
superscript ${}^H$
indicates conjugate transpose. Of course, this metric is not defined
everywhere
on $\mathcal S\text{gr}$ because of the rational character of $\ber$,
but we are interested in a neighborhood of the point $W_0$ with
standard frame
$\mathcal W_0=\begin{pmatrix} 1_{H_-}\\ 0\end{pmatrix}$ where there
is no
problem. The tangent space at $W_0$ can be identified with the space
of maps
$H_-\to H_+$, or, more concretely, by matrices with the columns
indexed by
$I=\{i\in \halfz\mid i\le 0\}$ and with rows indexed by the
complement of $I$.
Let $x,y$ be two tangent vectors at $W_0$. Then the curvature at
$W_0$ is
calculated to be
\begin{equation}\label{eq:curvature}
F(x,y)=\bar\partial\partial \log h (x,y)= Str(x^Hy-y^Hx),
\end{equation}
where we take as local generator $e=\sigma$, the section defined by
\eqref{eq:defsigma*}, so that $h=\langle \sigma, \sigma\rangle$.
We can map the tangent space at $W_0$ to the Lie super algebra
$gl_{\infty\mid
\infty}(\Lambda)$ via $x\mapsto \begin{pmatrix}
0&-x^H\\x&0\end{pmatrix}$. Here
$gl_{\infty\mid \infty}(\Lambda)$ is the Lie super algebra
corresponding to the
Lie super group $Gl_{\infty\mid \infty}(\Lambda)$ of infinite even
invertible
matrices $g$ (indexed by $\halfz$) with block decompostion
$\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $b,c$ compact and $a,d$
super
Fredholm. We see that (\ref{eq:curvature}) is the pullback under this
map of
the cocycle on $gl_{\infty\mid \infty}(\Lambda)$ (see also
\cite{KavdL:SuperBoson}) given by
\begin{equation}\label{eq:cocycle}
\begin{aligned}
c:gl_{\infty\mid \infty}(\Lambda)\times gl_{\infty\mid
\infty}(\Lambda)&\to
\quad\quad\quad\Lambda\\
(X,Y)\quad \quad\quad&\mapsto\quad\frac14 \Str(J[J,X][J,Y]),
\end{aligned}
\end{equation}
where $J=\begin{pmatrix} 1_{H_-}&0\\0&-1_{H_+}\end{pmatrix}$. In
terms of the
block decomposition of $X,Y$ we have
$$
c(X,Y)= \Str(c_Xb_Y-b_Xc_Y).
$$
The natural action of $Gl_{\infty\mid \infty}(\Lambda)$ on $\mathcal S\text{gr}$
lifts to
a projective action on $\Bersgr$; the cocycle $c$ describes
infinitesimally the
obstruction for this projective action to be a real action. Indeed,
if
$g_1=\exp(f_1),g_2=\exp(f_2)$ and $g_3=g_1g_2$ are all in the open
set of
$Gl_{\infty\mid \infty}(\Lambda)$ where the ${}_{--}$ blocks $a_i$
are
invertible, the action on a point of $\Bersgr$ is given by
\begin{equation} \label{eq:lift}
g_i\circ(\mathcal W,\lambda)=(g_i\mathcal Wa_i\inv,\lambda).
\end{equation}
(One checks as in \cite{SeWi:LpGrpKdV} that if $\mathcal W$ is an
admissible
basis then so is $g\mathcal Wg_{--}\inv$.)
Then we have
$$
g_1\circ g_2 \circ(\mathcal
W,\lambda)=\exp[c(f_1,f_2)]g_3\circ(\mathcal
W,\lambda).
$$
We can also introduce the {\it projective multiplier } $C(g_1,g_2)$
for
elements $g_1$ and $g_2$ that commute in
$Gl_{\infty\mid\infty}(\Lambda)$:
\begin{equation}\label{eq:projmult}
g_1\circ g_2\circ g_1\inv\circ g_2\inv (\mathcal
W,\lambda)=C(g_1,g_2)(\mathcal
W,\lambda),
\end{equation}
where $C(g_1,g_2)=\exp[S(f_1,f_2)]$ if $g_i=\exp(f_i)$ and
\begin{equation}\label{eq:logprojmult}
S(f_1,f_2)=\Str([f_1,f_2]).
\end{equation}
We will in subsection \ref{ss:ChernclassBeronPic} use the projective
multiplier
to show that the Chern class of the Berezinian bundle on $\Pic^0(X)$ is
trivial.
\subsection{The Jacobian super Heisenberg algebra.}
In the theory of the KP hierarchy an important role is played by a
certain
Abelian subalgebra of the infinite matrix algebra and its universal
central
extension, loosely referred to as the (principal) Heisenberg
subalgebra. In
this subsection we introduce one of the possible analogs of this
algebra in
the super case.
Let the {\it Jacobian super Heisenberg algebra} be the
$\Lambda$-algebra
$\mathcal J\text{Heis}=\Lambda[z,z\inv,\theta]$. Of course, this is as a
$\Lambda$-module the
same as $\Linfinf$ but now we allow multiplication of elements. When
convenient
we will identify the two; in particular we will use the basis
$\{e_i\}$ of
(\ref{eq:basisLinfinf}) also for $\mathcal J\text{Heis}$. We think of elements of
$\mathcal J\text{Heis}$ as
infinite matrices in $gl_{\infty\mid\infty}(\Lambda)$: if $E_{ij}$ is
the
elementary matrix with all entries zero except for the $ij$th entry
which is 1,
then
$$
e_i=\sum_{n\in\mathbb Z} E_{n+i,n}+E_{n+i-\frac12,n-\frac12},\quad
e_{i-\frac12}=
\sum_{n\in\mathbb Z} E_{n+i-\frac12,n}.
$$
We have a decomposition $\mathcal J\text{Heis}=\mathcal J\text{Heis}_-\oplus\mathcal J\text{Heis}_+$ in
subalgebras
$\mathcal J\text{Heis}_-=z\inv\Lambda[z\inv,\theta]$ and
$\mathcal J\text{Heis}_+=\Lambda[z,\theta]$.
Elements of $\mathcal J\text{Heis}_+$ correspond to lower triangular matrices and
elements of
$\mathcal J\text{Heis}_-$ to upper triangular ones. By exponentiation we obtain
from
$\mathcal J\text{Heis}_-$ and $\mathcal J\text{Heis}_+$ two subgroups $G_-$ and $G_+$ of
$Gl_{\infty\mid\infty}(\Lambda)$, generated by
$$
g_\pm(t)=\exp(\sum_{i\in\pm I} t_i e_i),
$$
where $t_i\in \Lambda$ is homogeneous of the same parity as $e_i$
(and $t_i$ is
zero for almost all $i$, say).
For an element $g=\begin{pmatrix} a&b\\c&d\end{pmatrix}$ of $G_+$
the block $b$ vanishes, whereas if $g\in G_-$ the block $c=0$. In
either case
the diagonal block $a$ is invertible and we can lift the action of
either $G_-$
or $G_+$ to a (potentially projective) action on $\Bersgr$ and
$\Bersgrdual$,
via (\ref{eq:lift}). Since the cocycle (\ref{eq:cocycle}) is zero
when
restricted to both $\mathcal J\text{Heis}_-$ and $\mathcal J\text{Heis}_+$ we get an honest action
of the
Abelian groups $G_\pm$ on $\Bersgr$ and $\Bersgrdual$, just as in the
classical
case.
In contrast with the classical case, however, as was pointed out in
\cite{Schwarz:FermStringModSpa}, the actions of $G_-$ and $G_+$ on
the line
bundles $\Bersgr$ and $\Bersgrdual$ mutually commute. This follows
from the
following Lemma.
\begin{lem}\label{lem:comactionJheis}
Let $g_\pm\in G_\pm$ and write $a_\pm=\exp(f_\pm)$, with $f_\pm \in
gl(H_-)$.
Then
$$
\Str_{H_-}([f_-,f_+])=0,
$$
so that the actions of $g_-$ and $g_+$ on $\Bersgr$ and $\Bersgrdual$
commute.
\end{lem}
\begin{proof}
The elements $f_\pm$ act on $H_-$ by multiplication by an element of
$\mathcal J\text{Heis}_\pm$, followed by projection on $H_-$ if necessary. So write
$f_\pm=\pi_{H_-}\circ \sum_{i>0} c^\pm_iz^{\pm i}
+\gamma^\pm_{i}z^{\pm
i}\theta$. To find the supertrace we need to calculate the projection
on
the rank $1\mid 0$ and $0\mid 1$ submodules of $H_-$ generated by
$z^{-i}$ and
$z^{-i}\theta$:
\begin{align*}
f_+ f_- z^{-k}|_{\Lambda z^{-k}}&=
f_+(\sum_{i>0}c_i^-z^{-i-k})|_{\Lambda
z^{-k}}=(\sum_{i>0}c^+_ic^-_i)z^{-k},\\
f_+ f_- z^{-k}\theta|_{\Lambda z^{-k}\theta}&=
(\sum_{i>0}c^+_ic^-_i)z^{-k}\theta,\\
f_- f_+ z^{-k}|_{\Lambda z^{-k}}&=
f_-(\sum_{i=1}^ka_i^+z^{i-k})|_{\Lambda
z^{-k}}=(\sum_{i=1}^k c^+_ic^-_i)z^{-k},\\
f_- f_+ z^{-k}\theta|_{\Lambda z^{-k}\theta}&=
f_-(\sum_{i=1}^kc_i^+z^{i-k}\theta)|_{\Lambda
z^{-k}\theta}=(\sum_{i=1}^k
c^+_ic^-_i)z^{-k}\theta.
\end{align*}
Since the super trace is the difference of the traces of the
restrictions to
the even and odd submodules we see that $\Str([f_+,f_-])=0$ so that,
by (\ref
{eq:projmult},\ref{eq:logprojmult}), the actions of $G_\pm$ on
$\Bersgr$
commute.
\end{proof}
\subsection{Baker functions, the full super Heisenberg algebra, and
$\tau$-functions.}\label{ss:Bakerf-fullsuperH-tau}
We define $W\in \mathcal S\text{gr}$ to be {\it in the big cell} if it has an
admissible frame
$\mathcal W^{(0)}$ of the form
\[\mathcal{W}^{\,(0)}=\begin{pmatrix}\ddots
&\vdots&\vdots&\vdots&\vdots\\
\dots &1&0&0&0\\
\dots &0&1&0&0\\
\dots &0&0&1&0\\
\dots &0&0&0&1\\
***&*&*&*&*\\
\end{pmatrix},
\]
i.e. $(\mathcal W^{\,(0)})_-$ is the identity matrix. Note that the
canonical
sections $\sigma$ and $\sigma^*$ do not vanish, nor blow up, at a
point in the
big cell.
If $\mathcal{W}$ is any frame of a point $W$ in the big cell we
can calculate the
standard frame $\mathcal{W}^{\,(0)}$ through quotients of
Berezinians of minors
of $\mathcal{W}$. Indeed, if we put $A=\mathcal W_-$ then the
maximal minor $A$ of $\mathcal W$ is invertible and we have
\begin{equation}\label{eq:connw0w}
\mathcal{W}^{\,(0)}A=\mathcal{W}.
\end{equation}
Write $\mathcal{W}^{\,(0)}=\sum w_{ij}^{(0)}E_{ij}$. Then we
can solve \thetag{\ref{eq:connw0w}} by Cramer's rule,
\thetag{\ref{eq:supercramer}}, to find for
$i>0,j\le 0$:
\begin{equation*}
w_{ij}^{(0)}=\begin{cases}
\ber\,(A_j(r_i))/
\ber\,(A)&\text{if $j\in \mathbb Z$},\\
\berdual \,(A_j(r_i))/
\berdual \,(A)&\text{if $j\in \mathbb Z+\frac
12$}.
\end{cases}
\end{equation*}
Here $A_j(r_i)$ is the matrix obtained from $A$ by replacing
the $j$th row by $r_i$, the $i$th row of $\mathcal W$. In
particular the even and odd ``Baker vectors'' of $W$, i.e.
the zeroth and $-\frac12$th column of $\mathcal W^{\,(0)}$, are
given by
\begin{equation}\label{eq:bakervector}
\begin{split}
w_{\bar 0}&=e_0+\sum_{\frac{i>0}
{i\in \frac 12 \mathbb Z}}\frac{\ber\,(
A_0(r_i))}{\ber\,(A)}e_i,\\
w_{\bar 1}&=e_{-\frac 12}+\sum_{\frac{i>0}
{ i\in \frac 12 \mathbb Z}}\frac{\berdual \,(
A_{-\frac 12}(r_i))}{\berdual \,(A)}e_i
\end{split}
\end{equation}
The corresponding ``Baker functions'' are obtained by using
$e_i=z^{i}$, $e_{i-\frac12}=z^{i}\theta$.
Then \thetag{
\ref{eq:bakervector}} reads
\begin{equation}\label{eq:bakerfunction}
\begin{split}
w_{\bar 0}(z,\theta)&=1 + \sum_{i>0}
z^{i}\frac { \ber\,( A_0(r_i))+
\ber\,( A_0(r_{i-\frac 12}))\theta
}
{\ber\,(A)},\\
w_{\bar 1}(z,\theta)&=\theta + \sum_{i>0}
z^{i}\frac{ \berdual \,(A_{-\frac12}(r_i))+
\berdual
\,(A_{-\frac12}(r_{i-\frac12}))\theta}
{\berdual \,(A)}.
\end{split}
\end{equation}
Here and henceforth (unless otherwise noted) the summations
run over (subsets of) the integers.
The full super Heisenberg algebra $\mathcal S\text{Heis}$ is the extension
$\mathcal J\text{Heis}[\frac{d}{d\theta}]=\Lambda[z,z\inv,\theta][\frac{d}{d\theta}]$
. This
is, just as the Jacobian super Heisenberg algebra, a possible analog
of the
principal Heisenberg of the infinite matrix algebra used in the
standard KP
hierarchy, see \cite{KavdL:SuperBoson}. $\mathcal S\text{Heis}$ is non--Abelian and
the
restriction of the cocycle \thetag{\ref{eq:cocycle}} to it is
nontrivial, in
contrast to the subalgebra $\mathcal J\text{Heis}$.
$\mathcal S\text{Heis}$ acts in the obvious way on $\Linfinf$ and we can represent
it by
infinite matrices from $gl_{\infty\mid\infty}(\Lambda)$. Introduce a
basis for
$\mathcal S\text{Heis}$ by
\begin{alignat*}{2}
\lambda(n)&=z^{-n}(1-\theta\frac{d}{d\theta})=\sum_{k\in \mathbb
Z}E_{k,k+n},&\quad
f(n)&=z^{-n}\frac{d}{d\theta}=\sum_{k\in \mathbb Z}E_{k,k+n-\frac
12},\\
\mu(n)&=z^{-n}\theta\frac{d}{d\theta}=\sum_{k\in \mathbb
Z}E_{k-\frac12,k-\frac12+n},&\quad
e(n)&=z^{-n}\theta=\sum_{k\in \mathbb Z}E_{k-\frac12,k+n}.
\end{alignat*}
We can rewrite the Baker functions as quotients of
Berezinians, using $\mathcal S\text{Heis}$. To this end define the following even
invertible
matrices (over the ring $\mathbb \Lambda[u,\phi,\frac{\partial}
{\partial\phi}]$):
\begin{align*}
Q_{\bar 0}(u,\phi)&=1+\sum_{n=1}^\infty
{u^n}[\lambda(n)+f(n)\phi],\\
Q_{\bar 1}(u,\phi)&=1+\sum_{n=1}^\infty
{u^n}[{\mu(n)+e(n)\frac{\partial} {\partial\phi}}],
\end{align*}
where $u$, resp $\phi$, is an even, resp. odd, variable. We can let
these
matrices act on $H$ and obtain in this way infinite vectors over the
ring
$\Lambda[u,\phi,\frac{\partial}{\partial\phi}]$. Also we can let
these matrices
act on an admissible frame and obtain a matrix over
$\Lambda[u,\phi,\frac{\partial}{\partial\phi}]$.
\begin{lem} Let $w_{\bar 0}(u,\phi)$ and $w_{\bar 1}(u,\phi)$ be the
even and
odd Baker functions of a point $W$ in the big cell. For any frame
$\mathcal W$
of $W$ we have:
\begin{equation*}
w_{\bar 0}(u,\phi)=\frac{\ber\,([Q_{\bar
0}(u,\phi)\mathcal W]_-)}
{\ber\,(A)},\quad
w_{\bar 1}(u,\phi)=\frac{\berdual \,([Q_{\bar
1}(u,\phi)\mathcal W]_-)\phi}
{\berdual \,(A)},
\end{equation*}
with $A=\mathcal W_-$.
\end{lem}
\begin{proof}
Let $r_i$, $r_{i,\bar 0}$ and $r_{i,\bar 1}$, be respectively the
$i$th
row of $\mathcal W$, $Q_{\bar 0}(u,\phi)\mathcal W$ and of
$Q_{\bar 1}(u,\phi)\mathcal W$. Then one
calculates that for $i\in \mathbb Z$ we have $\quad r_{i-\frac 12,
\bar 0} =
r_{i-\frac 12}$, and $r_{i, \bar 1}= r_{i}$ and :
\begin{equation}\label{eq:tildewinw}
\begin{aligned}
r_{i,\bar 0}
&=r_i+\sum_{k\ge 1}{u^{k}}({r_{i+k}+r_{i+k-\frac12}\phi}),
& & \\
&=r_i+u( r_{i+1,\bar 0}+ r_{i+\frac 12,\bar 0}\phi),\\
r_{i-\frac12,\bar 1}
&=r_{i-\frac12}+\sum_{k\ge 1} {u^{k}}(r_{i+k-\frac12}
+r_{i+k}\frac{\partial}{\partial \phi}),
& &\quad \\
&=r_{i-\frac12}+u ( r_{i+1-\frac 12,\bar 1}+
r_{i+1,\bar 1}\frac{\partial}{\partial \phi}).
\end{aligned}
\end{equation}
Let $X$ be an even matrix. Because of the multiplicative
property of Berezinians we can add multiples of a row to another row
of $X$
without changing $\ber\,(X)$ and
$\berdual(X)$. Using such row operations we see, using
\thetag{\ref{eq:tildewinw}}, that
\begin{align*}
\ber\,([Q_{\bar 0}(u,\phi)\mathcal W]_-)
&=\ber\,(A_0(r_{0,\bar 0})),\\
\berdual \,([Q_{\bar 1}(u,\phi)\mathcal W]_-)
&=\berdual\,(A_{-\frac 12}( r_{-\frac 12,\bar 1})).
\end{align*}
Now $\ber$ is linear in even rows, and $\berdual $ in odd
rows, so by \thetag{\ref{eq:tildewinw}} we find
\begin{multline*}
\ber\,([Q_{\bar 0}(u,\phi)\mathcal
W]_-)=\ber\,(A)+
\\\sum_{i>0 }u^{i}[\ber\,(A_0(r_i))+\ber\,(A_0(r_{i-\frac12}))\phi],
\end{multline*}
and
\begin{multline*}
\berdual \,([Q_{\bar1}(u,\phi)\mathcal W]_-) =\berdual \,(A)+\\
+\sum_{i>0}\,u^{i}[\berdual\,(A_{-\frac12}(r_{i-\frac12}))+
\berdual\,(A_{-\frac12}(r_i))\frac{\partial}
{\partial\phi}] .
\end{multline*}
Comparing with \thetag{\ref{eq:bakerfunction}} proves the
lemma.
\end{proof}
We now consider the flow on $\mathcal S\text{gr}$ generated by the negative part of
the
Jacobian Heisenberg algebra: define
\begin{equation}\label{expfactor}
\gamma(t)=\exp(\sum_{{i>0}} t_i z^{-i}+ t_{i-\frac12}z^{-i}\theta),
\quad t_i
\in \Lambda_{\text{ev}},t_{i-\frac12}\in\Lambda_{\text{odd}}
\end{equation}
and put for $W\in \mathcal S\text{gr}$:
$$W(t)=\gamma(t)\inv W.$$
The $\tau$-functions associated to a point $W$ in the big cell are
then
functions on $\mathcal J\text{Heis}_{-,\text{ev}}$:
\begin{equation}\label{eq:tau}
\tau_W(t),\tau^*_W(t):\mathcal J\text{Heis}_{-,\text{ev}}\to \Lambda\cup \{\infty\}
\end{equation}
given by
\begin{align}\label{eq:taudef}
\tau_W(t)
&=\frac{\sigma(\gamma(t)\inv W)}{\gamma(t)\inv \sigma (W)}=
\frac{\ber( [\gamma(t)\inv\circ \mathcal
W]_-)}{\ber([\mathcal
W]_-)},\\
\tau_W^*(t)
&=\frac{\sigma^*(\gamma(t)\inv W)}{\gamma(t)\inv \sigma^*
(W)}=
\frac{\berdual( [\gamma(t)\inv\circ \mathcal
W]_-)}{\berdual([\mathcal
W]_-)}.
\end{align}
Here $\sigma$ and $\sigma^*$ are the sections of $\Bersgrdual$ and
$\Bersgr$
defined in \thetag{\ref{eq:defsigma*}} and $\gamma\inv\in
Gl_{\infty\mid\infty}(\Lambda)$ acts via \thetag{\ref{eq:lift}} on
$\Bersgr$
and $\Bersgrdual$.
The Baker function of $W$ becomes now a function on
$\mathcal J\text{Heis}_{-,\text{ev}}$, and
we have
an expression in terms of a quotient of (shifted) $\tau$-functions:
\begin{equation} \label{eq:Bakertauquotient}
w_{\bar 0}(t;u,\phi)=\frac{\tau_W(t;Q_{\bar 0})}{\tau_W(t)},\quad
w_{\bar 1}(t;u,\phi)=\frac{\tau_W^*(t;Q_{\bar 1})}{\tau^*_W(t)},
\end{equation}
where
$$
\tau_W(t;Q_{\bar 0})=\frac{\ber([Q_{\bar0}\mathcal
W]_-)}{\ber(\mathcal W_-)},
\quad
\tau^*_W(t;Q_{\bar 1})=\frac{\berdual([Q_{\bar1}\mathcal
W]_-)\phi}{\berdual(\mathcal W_-)}.
$$
Note that even if we are only interested in the Jacobian Heisenberg
flows the
full Heisenberg flows automatically appear in the theory if we
express the
Baker functions in terms of the $\tau$ functions.
In principle we could also consider the flows on $\mathcal S\text{gr}$ generated by
the full
super Heisenberg algebra $\mathcal S\text{Heis}$. However, since $\mathcal S\text{Heis}$ is
non--Abelian the
interpretation of these flows is less clear and therefore we leave
the
discussion of these matters to another occasion.
\section{The Krichever map and algebro-geometric solutions}
\subsection{The Krichever map.}\label{ss:Krichever}
Consider now a set of geometric data
$(X,P,(z,\theta),\mathcal{L},t)$, where:
\begin{itemize}
\item $X$ is a generic SKP curve as before.
\item $P$ is an irreducible divisor on $X$, so that $P^{\text{red}}$
is a single point of the underlying Riemann surface $X^{\text{red}}$.
\item $(z,\theta)$ are local coordinates on $X$ near $P$, so that
$P$ is defined by the equation $z=0$.
\item $\mathcal{L}$ is an invertible sheaf on $X$.
\item $t$ is a trivialization of $\mathcal{L}$ in a neighborhood of
$P$,
say $U_P = \{| z^{\text{red}}|<1\}$.
\end{itemize}
We will associate to this data a point of the super Grassmannian
$\mathcal S\text{gr}$.
For studying meromorphic sections of $\mathcal L$ we have the exact
sequence
\begin{equation}
0 \rightarrow \mathcal{L} \overset{\text{inc}}{\rightarrow}
\mathcal{L}(P)
\overset{\text{res}}{\rightarrow} \mathcal{L}_{P^{\text{red}}} \cong
\Lambda |
\Lambda \rightarrow 0,
\end{equation}
which gives
\begin{equation} \label{polesequence}
H^0(\mathcal{L}) \hookrightarrow H^0(\mathcal{L}(P)) \rightarrow
\Lambda | \Lambda \rightarrow H^1(\mathcal{L}) \rightarrow
H^1(\mathcal{L}(P)) \rightarrow 0,
\end{equation}
where the residue is the pair of coefficients of $z^{-1}$ and $\theta
z^{-1}$
in the Laurent expansion.
Let $\mathcal{L}(*P) = \lim_{n \rightarrow \infty} \mathcal{L}(nP)$
be
the sheaf of sections of $\mathcal{L}$ holomorphic except possibly
for a pole of arbitrary order at $P$.
The {\it Krichever map} associates to a set of geometric data
as above the $\Lambda$-module of formal Laurent series
$W = z\ t[H^0(X,\mathcal{L}(*P))]$, which will be viewed as a
submodule of $H$.
In \cite{MuRa:SupKrich,Ra:GeomSKP} the concern was expressed that $W$
might not be freely generated, and hence not an element of $\mathcal S\text{gr}$ as
we have defined it.
However,
\begin{thm} $H^0(X,\mathcal{L}(*P))$ is a freely generated
$\Lambda$-module, and $W \in \mathcal S\text{gr}$.
Further, $W$ belongs to the big cell if the geometric data satisfy
$H^0(X,\mathcal{L}) = H^1(X,\mathcal{L}) = 0$, which happens
generically if $\deg \mathcal{L} = g-1$.
\end{thm}
\begin{proof}
Assume first that $H^0(X,\mathcal{L}) = H^1(X,\mathcal{L}) = 0$.
Then the sequence \thetag{\ref{polesequence}} applied to
$\mathcal{L}$ gives
\begin{equation}
0 \rightarrow H^0(\mathcal{L}(P)) \rightarrow \Lambda | \Lambda
\rightarrow 0 \rightarrow H^1(\mathcal{L}(P)) \rightarrow 0,
\end{equation}
so that $H^0(X,\mathcal{L}(P))$ is freely generated by an even and
odd section having principal parts $z^{-1}$ and $\theta z^{-1}$, and
$H^1(X,\mathcal{L}(P))$ is still zero.
Applying the same sequence inductively to $\mathcal{L}(nP)$ shows
that
$H^0(X,\mathcal{L}(*P))$ is freely generated by one even and one odd
section of each positive pole order.
So $W$ is obtainable from $H_-$ by multiplication by a lower
triangular
invertible matrix, and $W$ belongs to the big cell of $\mathcal S\text{gr}$.
We also have $H^i(\mathcal{L}\spl) = 0, i=0,1$, from Theorem
\ref{thm:freeness}.
And, by the super Riemann-Roch Theorem \thetag{\ref{superRR}},
$\deg \mathcal{L} = \deg \mathcal{L}\spl = g-1$.
Moreover, by semicontinuity, in $\operatorname{Pic}^{g-1}(X)$ the cohomology groups
$H^i(\mathcal{L})$ can only get larger on Zariski
closed subsets, so generically they are zero.
Now consider the general situation in which $H^i(\mathcal{L})$ may
not be zero.
Still, by twisting,
$H^1(\mathcal{L}(nP)) = H^1(\mathcal{L}\spl(nP)) = 0$ for $n$
sufficiently
large.
Then, by the previous argument, $H^0(\mathcal{L}(*P))$ has
non purely nilpotent elements with poles of order $n+1$ and higher;
the worry
is
that one may only be able to find nilpotent generators for
$H^0(\mathcal{L}(nP))$.
So take $f \in H^0(\mathcal{L}(nP))$ of order $k$ in nilpotents:
its image in $H^0(\mathcal{L}(nP)/\mathfrak m^k)$ is zero, but its
image
$\hat{f}$ in
$H^0(\mathcal{L}(nP)/\mathfrak m^{k+1})$ is nonzero
and also lies in $\Lambda^k H^0(\mathcal{L}\spl(nP))$.
Then $\hat{f}$ can be identified with a sum of elements $f_a$
of $H^0(\mathcal{L}\spl(nP))$ with coefficients from $\Lambda^k$.
By the extension sequence \thetag{\ref{eq:longexactcohom}}, each
$f_a$ can
be extended order by order in nilpotents to an element of
$H^0(\mathcal{L}(nP))$ which is not purely nilpotent.
So we can write the order $k$ element $f$ as a $\Lambda$-linear
combination of
not purely nilpotent elements of $H^0(\mathcal{L}(nP))$, modulo an
element of
order
$k+1$. Induction on $k$ shows then that any element of
$H^0(\mathcal{L}(nP))$
is a
$\Lambda$-linear combination of not purely nilpotent elements of
$H^0(\mathcal{L}(nP))$. So there
exists a set of non-nilpotent
elements which span $H^0(\mathcal{L}(nP))$ over $\Lambda$.
A linearly independent subset of these completes a basis for
$H^0(\mathcal{L}(*P))$.
\end{proof}
\subsection{The Chern class of the Ber bundle on
$\Pic^0(X)$.}\label{ss:ChernclassBeronPic}
By the arguments of the previous subsection we have, in case $W\in
\mathcal S\text{gr}$ is
obtained by the Krichever map from geometric data $(X, P, \mathcal L,
(z,\theta), t)$, an exact sequence of $\Lambda$-modules:
\begin{equation}\label{eq:seqcohomW}
0\to H^0(X,\mathcal L)\to W\to H_-\to H^1(X,\mathcal L)\to 0.
\end{equation}
We can interpret the Ber bundle $\Bersgr$ in terms of this sequence
as follows.
Let $M$ be a free $\Lambda$-module, possibly of infinite rank, and
let
$B=\{\mu\}$ be a collection of bases for $M$ such that any two bases
$\mu,\mu^\prime\in B$ are related by $\mu^\prime=\mu T$ where $T\in
\operatorname{Aut}(M)$ has a well defined Berezinian. Then we
associate to the
pair $(M,B)$ a free rank $(1\mid 0)$ module $\ber(M)$ with generator
$b(\mu)$
for any $\mu\in B$ with identification $b(\mu^\prime)=\ber(T)b(\mu)$.
The fiber of $\Bersgr$ at $W$ can then be interpreted as $\ber(W)$,
using the
collection of admissible bases as $B$ in the above definition.
Similarly we can
construct on $\mathcal S\text{gr}$ a line bundle with fiber at $W$ the module
$\ber(H_-)$.
Clearly this bundle is trivial, so we can, even better, think of
$\Bersgr$ as
having fiber $\ber(W)\otimes \berdual(H_-)$. But by the properties of
the
Berezinian we get from \eqref{eq:seqcohomW}
$$
\ber(W)\otimes \berdual(H_-)=\ber(H^0(X,\mathcal L))\otimes
\berdual(H^1(X,\mathcal L)).
$$
Now we have seen in subsection \ref{subs:Berbundles} that the Ber
bundle
$\Ber(\Pic^0(X))$ on $\Pic^0(X)$ has the same fiber, with the difference
that there
we were dealing with bundles of degree 0 and here $\mathcal L$ has
degree
$g-1$.
For fixed $(X,P,(z,\theta))$ the collection $M$ consisting of
Krichever data
$(X,P,(z,\theta),\mathcal L,t)$ forms a supermanifold and we have two
morphisms
$$
i:M\to \mathcal S\text{gr},\quad p:M\to \Pic^0(X)
$$
where $i$ is the Krichever map and $p$ is the projection from
Krichever data to
the line bundle $\mathcal L$. (Here we identify $\operatorname{Pic}^n(X)$ with
$\Pic^0(X)$ via
the invertible sheaf $\mathcal O_X(-nP)$.) Then we see that
$i^*(\Bersgr)\simeq
p^*(\Ber(\Pic^0(X)))$. This fact allows us to prove Theorem \ref
{thm:1ChernBertriv}.
Note first that we have a surjective map
\begin{equation}\label{eq:surjecttoH1}
\mathcal J\text{Heis}_-\to H^1(X,\mathcal O_X).
\end{equation}
Indeed, let $X=U_0\cup U_P$ be an open cover where
$U_0=X-P^{\text{red}}$ and
$U_P$ is a suitable disk around $P^{\text{red}}$. Then if $[a]\in
H^1(X,\mathcal O_X)$
is represented by $a\in\mathcal O_X(U_0\cap U_P)$ we can write, using the
local
coordinates on $U_P$,
$a=a_P+\sum_{i>0} a_i z^{-i}+\alpha_i z^{-i}\theta$, with $a_P\in
\mathcal O_X(U_P)$.
Then
$a-a_P=\sum a_i z^{-i}+\alpha_i z^{-i}\theta\in \mathcal J\text{Heis}_-$ and
$[a]=[a-a_P]$.
Now the tangent space to any point $\mathcal L\in\Pic^0(X)$ can be
identified with
$H^1(X,\mathcal O_X)$ and so we have a surjective map from $\mathcal J\text{Heis}_-$ to the
tangent
space of $\Pic^0(X)$. Note secondly that a change of trivialization of
$\mathcal
L$, given by $t\mapsto t^\prime$, corresponds to multiplication of
the point
$W\in \mathcal S\text{gr}$ by an element $a_0+\alpha_0\theta +\sum_{i>0}a_i z^{
i}+\alpha_i
z^{i}\theta$ of the group corresponding to $\mathcal J\text{Heis}_+$. From these two
facts we
conclude that there is a surjective map from $\mathcal J\text{Heis}$ to the tangent
space to
the image of the Krichever map $i:M\to \mathcal S\text{gr}$ at any point
$W=W(X,P,(z,\theta),\mathcal L,t)$. Now the first Chern class of
$\Bersgr$ is
calculated from the cocycle \eqref{eq:cocycle} on
$gl_{\infty\mid\infty}(\Lambda)$ and it follows from Lemma
\ref{lem:comactionJheis} that the restriction of this cocycle to
$\mathcal J\text{Heis}$ is
identically zero. This implies that
$$
i^*(c_1[\Bersgr])=p^*(c_1[\Ber(\Pic^0(X))])=0.
$$
But the map $p:M\to \Pic^0(X)$ is surjective, so we finally find that
$c_1(\Ber(\Pic^0(X)))=0$ and $\Ber(\Pic^0(X))$ is topologically trivial,
proving
Theorem \ref{thm:1ChernBertriv}.
\subsection{Algebro-geometric tau and Baker functions.}
\label{ss:AlgebrogeometrictauBaker}
We consider geometric data mapping to $W$ in the big cell of $\mathcal S\text{gr}$,
so that $\deg \mathcal{L} = g-1$.
As discussed in Section 3, we can associate to $W$ both a tau
function
and a Baker function.
A system of super KP flows on $\mathcal S\text{gr}$ applied to $W$ produces an orbit
corresponding
to a family of deformations of the original geometric data.
The simplest system of super KP flows, the ``Jacobian" system of
Mulase and
Rabin
\cite{Mu:Jac,Ra:GeomSKP}, deforms the geometric data by
moving $\mathcal{L}$ in $\text{Pic}^{g-1}(X)$.
Solutions to this system for $X$ a super elliptic curve were
obtained in terms of super theta functions in \cite{Ra:SupElliptic}.
On the basis of the ordinary KP theory, cf. \cite{SeWi:LpGrpKdV},
section 9,
we might expect that in general the tau and Baker
functions for this family can be given explicitly as functions of the
flow
parameters by means
of the super theta functions (when these exist) on the Jacobian of
$X$.
We now discuss the extent to which this is possible.
Recall from \eqref{eq:surjecttoH1}
that we have a surjection from $\mathcal J\text{Heis}_{-,\text{ev}}$ to the
cohomology group
$H^1(X,\mathcal O_{X,\text{ev} } )$. By exponentiation we obtain a map from
$\mathcal J\text{Heis}_{-,\text{ev}}$
to $\Pic^0(X)$ and these maps fit together in a diagram
\begin{equation}
\begin{CD}\label{eq:cdtautheta}
{} @. {} @. {} @. 0 @. {}
\\
@. @. @. @VVV @.
\\
{} @. 0 @. {} @. H^1(X,\mathbb Z)@.
\\
@. @VVV @. @VVV @.
\\
0 @>>> K_0 @>>>\mathcal J\text{Heis}_{-,\text{ev}}@>>> H^1(X,\mathcal O_{X,\text{ev} } ) @>>>0
\\
@. @VVV @\vert @VVV @. \\
0 @>>> K @>>>\mathcal J\text{Heis}_{-,\text{ev}}@>>>\Pic^0(X) @>>>0 \\
@. @VVV @. @VVV @.
\\
{} @. K/K_0 @. {} @.0 @.{}\\
@. @VVV @. @. @.
\\
{} @. 0 @. {} @. {} @. {}
\\
\end{CD}
\end{equation}
Here $K_0$ is the $\Lambda_{\text{ev}}$-submodule of elements $f$ of
$\mathcal J\text{Heis}_{-,\text{ev}}$ that split as $f=f_0 + f_P$, with $f_0\in
\mathcal O_X(U_0)$ and
$f_P\in \mathcal O_X(U_P)$ and $K$ is the Abelian subgroup (not submodule!) of
elements
$k$ of $\mathcal J\text{Heis}_{-,\text{ev}}$ that after exponentiation factorize:
$e^k=\phi_k
e^{k_p}$, with $\phi_k\in \mathcal O_X(U_0)^\times$ and $k_P\in \mathcal O_X(U_P)$.
{}From the
Snake Lemma it then follows that $H^1(X,\mathbb Z)\simeq K/K_0$. So a
function
$\hat F$ on $\mathcal J\text{Heis}_{-,\text{ev}}$ descends to a function $ F$ on
$H^1(X,\mathcal O_{X,\text{ev} } )$ if it is invariant under $K_0$.
The automorphic behavior of such a function $ F$ with respect to the
lattice
$H^1(X,\mathbb Z)$ translates into behaviour of
$\hat{F}$ under shifts by elements of $K$.
In particular we consider the function $\tau_W$ associated to a point
$W$ in
the big cell of $\mathcal S\text{gr}$, see \eqref{eq:tau} and \eqref{eq:taudef}.
This is a function on $\mathcal J\text{Heis}_{-,\text{ev}}$ and, because of Lemma
\ref{lem:comactionJheis}, we see by an easy adaptation of the proof
of Lemma
9.5 in \cite{SeWi:LpGrpKdV} that
$$
\tau_W(f+k)=\tau_W(f)\tau_W(k),\quad f\in \mathcal J\text{Heis}_{-,\text{ev}}, k\in
K.
$$
In particular we obtain by restriction a homomorphism
$$
\tau_W:K_0\to \Lambda^\times_{\text{ev}}.
$$
Let $\eta:K_0\to \Lambda_{\text{ev}}$ be a homomorphism such that
$\tau_W(k_0)=e^{\eta(k_0)}$, for all $k_0\in K_0$. Then we can define
a new
function
$$
\hat \tau_1(f)=\tau_W(f)e^{-\eta(f)}.
$$
Then $\hat \tau_1(k_0)=1$, but still we have
\begin{equation}\label{eq:multtau}
\hat \tau_1(f+k)=\hat \tau_1(f)\hat \tau_1(k),
\end{equation}
so that $\hat \tau_1$ descends to a function $\tau_1$ on
$H^1(X,\mathcal O_{X,\text{ev} } )$. From
\eqref{eq:multtau} we see that $\tau_1$ corresponds to a
(meromorphic) section
of a line bundle on $\Pic^0(X)$ with trivial Chern class.
A suitable ratio of translated theta functions gives a section of
this same bundle, so that $\tau_1$ is expressed as this ratio times
a meromorphic function, the latter being rationally expressible
in terms of super theta functions.
Then the modified tau function $\tau_1$ is rationally expressed in
terms of
super theta functions.
The even Baker function $w^W_{\bar 0}(z,\theta)$ associated to the
point $W$
is just
the even section of $\mathcal L$ holomorphic except for a pole $1/z$
at $P$.
Such a section can be specified by its restrictions to the charts
$U_0$ and $U_P$.
The Jacobian super KP flows act by multiplying the transition
function of
$\mathcal L$ across
the boundary of $U_P$ by a factor
$\gamma(t)$ as in \eqref{expfactor}.
The corresponding action on the associated point $W$ of $\mathcal S\text{gr}$ is
generated by
the matrices $\lambda(n)+\mu(n)$ and
$f(n)$ of Section 3; the remaining matrices generate deformations of
the curve
$X$
and enter the Kac--van de Leur SKP flows.
Then $w^{W(t)}_{\bar 0} / w^W_{\bar 0}$ is a section of the bundle
with
transition function \thetag{\ref{expfactor}}.
Equivalently, it is a meromorphic function on $U_0$ which extends
into $U_P$
except for an
essential singularity of the form \thetag{\ref{expfactor}}, having
zeros at the
divisor of $\mathcal L(t)$
and poles at the divisor of $\mathcal L$.
By analogy with the ``Russian formula" of ordinary KP theory, such a
function
would be expressed in the form
\begin{equation} \label{Russian}
\exp
[
\sum_{k=1}^{\infty}
\int_{(0,0)}^{(z,\theta)}
(t_k \hat{\psi}_k + t_{k-\frac12} \hat{E}_k)
+ c(t)
]
\end{equation}
times a ratio of theta functions providing the zeros and poles.
Here $\hat{\psi}_k$ and $\hat{E}_k$ are differentials on $\hat{X}$,
with vanishing $a$-periods and holomorphic except for the behavior
near $P$,
\begin{equation}
\hat{\psi}_k \sim \hat{D}(z^{-k}) = -k\rho \hat{z}^{-(k+1)},\;\;\;\;
\hat{E}_k \sim \hat{D}(\theta z^{-k}) = \hat{z}^{-k}.
\end{equation}
The constant $c(t)$ is linear in the flow parameters.
In addition to the symmetry of the period matrix, we have to require
the
existence of these differentials.
This requires that they exist in the split case, and then that these
split
differentials extend through
the sequence \thetag{\ref{eq:longexactcohom}}.
In the split case, the odd differentials $\hat{\psi}_k$ are just
$\theta$ times
the ordinary differentials
on the reduced curve which appear in the Russian formula (and they do
extend).
However, the even differentials $\hat{E}_k$ are sections of $\mathcal
N$, which
is of degree zero
and nontrivial, with $h^1 = g-1$.
Consequently, when $g>1$ there will be Weierstrass gaps in the list
of pole
orders of these differentials.
This means that the odd flow parameters corresponding to the missing
differentials
must be set to zero in order for the Baker function to assume the
``Russian"
form.
Even then, however, the function given by the Russian formula will
generically
behave as $1 + \alpha\theta + \mathcal O(z)$ for
$z \rightarrow 0$, rather than the correct $1 + O(z)$ for
$w^{W(t)}_{\bar 0}$
containing no $\theta/z$ pole.
In \cite{Ra:SupElliptic} this was dealt with by including a term $\xi
\hat{E}_0$ in the exponential, taking
$\partial_\xi$ to construct a section with a pure $\theta/z$ pole,
and
subtracting off the appropriate multiple of this.
In general, however, no such $\hat{E}_0$ will exist.
These difficulties are understandable in view of the relations
(\ref{eq:Bakertauquotient}) which require that the tau function
be known for the full set of K-vdL flows in order to compute
the Baker functions for even the Jacobian flows.
Since the dependence of the tau function on the non-Jacobian flows is
likely to
be far more complicated than
our super theta functions, it is unlikely that the
Baker functions can be expressed in terms of them.
|
1996-01-08T06:20:35 | 9601 | alg-geom/9601005 | en | https://arxiv.org/abs/alg-geom/9601005 | [
"alg-geom",
"dg-ga",
"hep-th",
"math.AG",
"math.DG"
] | alg-geom/9601005 | Yongbin Ruan | Yongbin Ruan, Gang Tian | Higher genus symplectic invariants and sigma model coupled with gravity | LaTeX | null | 10.1007/s002220050192 | null | null | We define higher genus Gromov-Witten invariants and establish a mathematical
theory of sigma model coupled with gravity over any semi-positive symplectic
manifolds. As applications, we verify the stablizing conjecture of symplectic
4-manifolds for simply connected elliptic surfaces and construct smooth
6-manifolds admitting infinitely many deformation classes of symplectic
structures.
| [
{
"version": "v1",
"created": "Sat, 6 Jan 1996 20:20:16 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Ruan",
"Yongbin",
""
],
[
"Tian",
"Gang",
""
]
] | alg-geom | \section{ Introduction}
This paper is a continuation of our previous paper \cite{RT}.
In \cite{RT}, among other things, we build up the mathematical
foundation of quantum cohomology ring on semi-positive symplectic manifolds.
We also defined higher genus symplectic invariants without gravity
(topological sigma model) in terms of inhomogeneous
holomorphic maps from a fixed Riemann surface,
and proved the composition law they satisfy.
Topological gravity, proposed by Witten, concerns the intersection
theory of the moduli space of marked Riemann surfaces. Based
on the physical intuition, Witten suggested a relation between
those intersection numbers and the KdV hierarchy.
This relation was clarified by Kontsevich (cf. \cite{Ko}). However,
the mathematical, as well as physical, phenomenon will become much more
interesting
if the topological sigma model
is coupled with the topological gravity.
In fact, in \cite{W2} Witten proposed an approach to
the topological sigma model coupled with gravity,
and made a very important conjecture
on the basic feature of this new model. The purpose of
this paper is to establish a mathematical foundation for the
theory of topological sigma model coupled with topological gravity over
any semi-positive symplectic manifolds. This new theory also provides
many more new geometric examples of the topological field
theory coupled with gravity.
For each semi-positive symplectic manifold $V$, we can associate a
topological sigma model with gravity, or
simply a topological field theory coupled with gravity.
This theory begins with the GW-invariants
$$ \Psi ^V_{(A,g,k)} : H_*(\overline {\cal M}_{g,k}, {\bf Q})
\times H_*(V, {\bf Z})^k\mapsto {\bf Q},$$
for any $A\in H_2(V, {\bf Z})$ and $2g +k \ge 3$. Here
$\overline {\cal M}_{g,k}$ is the Deligne-Mumford compactification
of the moduli space of genus $g$ Riemann surfaces with $k$
marked points. The GW-invariants are multilinear and
supersymmetric on $H_*(V,{\bf Z})^k$.
At first, we will rigorously define the GW-invariant $\Psi^V$
on semi-positive symplectic manifolds (cf. section 2).
From the analytic point of view,
it is the most convenient to use the inhomogeneous
holomorphic maps from Riemann
surfaces in $\overline {\cal M}_{g,k}$, though
other equivalent formulations may be possible, such as using stable maps
and establishing a more sophisticated intersection theory.
An inhomogeneous holomorphic map is a solution of an
inhomogeneous Cauchy-Riemann equation (cf. Section 2).
Putting aside technical details for the time being, we can intuitively
define the GW-invariants (cf. Section 2 for details) as follows:
let $V$ be any symplectic manifold and $A\in H_2(V, {\bf Z})$. For any
homology classes $[K] \in H_*(\overline {\cal M}_{g,k}, {\bf Q})$ and
$\alpha _i\in H_*(V, {\bf Z})$, represented by cycles $K$, $A_i$,
respectively, we define
$\Psi^V_{(A,g,k)}([K]; \alpha _1, \cdots, \alpha _k)$ to be
the number of tuples $(\Sigma ; x_1, \cdots, x_k; f)$
with appropriate sign, satisfying:
$\Sigma \in K$, $f: \Sigma \mapsto V$ solves a given inhomogeneous
Cauchy-Riemann equation, and $f(x_i) \in A_i$, whenever
$$\sum cod(A_i)+cod(K)= 2c_1(V)(A)+2(3-n)(g-1)+2k;
\eqno(1.1)$$
We simply put $\Psi^V_{(A,g,k)}([K]; \alpha _1, \cdots, \alpha _k)$
to be zero if (1.1) is not satisfied.
This approach towards defining new invariants has been used
before in many cases (cf. [Do], [Gr], [R], [R3], [RT], [W1]).
For symplectic 4-manifolds, using unperturbed holomorphic maps,
the first author already defined the invariant $\Psi$ in the very important
case that
$k=0$ and $[K] = \overline {\cal M}_{g,0}$.
However, in each case, there are specified difficulties
to be overcome. Using the techniques we developed in [RT], we
will first prove
\vskip 0.1in
\noindent
{\bf Theorem A (Theorem 2.14.)} {\it If $V$ is a semi-positive symplectic
manifold,
the GW-invariant $\Psi^V_{(A,g,k)}$ can be well defined
for any $g, k \ge 0$ with $2g + k \ge 3$. Moreover,
this $\Psi^V$ depends only the symplectic structure of $V$.}
\vskip 0.1in
A symplectic manifold $V$ is semi-positive if
it is compact and there is no $J$-holomorphic
map $f: S^2 \mapsto V$ such that $3-n \le \int_{S^2} f^*c_1(V) < 0$,
where $J$ is any given compatible almost complex structure on $V$.
In particular, any algebraic manifold of dimension $\le 3$ is
semi-positive in this sense, also any algebraic manifold
$V$ with $c_1(V) \ge 0$ is semi-positive.
One new consequence of our theorem, which was not
obvious at all to physicists based on
mathematically unjustified path integrals,
is that the invariant $\Psi^V$ is a symplectic invariant. The path
integral starts from a Lagrangian. The Lagrangian for sigma model or sigma
model coupled with gravity is valid
for any almost complex manifolds (symplectic
or not). There was a speculation that its correlation functions will be
the invariants of homotopy class of almost complex structures. This is in fact
false. Our invariants are symplectic invariants rather than the invariants of
almost complex structures. In particular, they can distinguish different
symplectic
manifolds with the same homotopy class of almost complex structures (see
section 5 or \cite{R}, \cite{R1}).
One of fundamental properties of a topological field theory is the
axiom on the decomposition of correlation functions. In our case,
the GW-invariants serve as the correlation functions. Therefore, in order
to make them more useful, or at least to construct a correct model for
the topological
field theory, we need to verify that our invariants satisfy
the composition law.
The composition law
governs how the GW-invariants change
during the degeneration of stable curves. Its classical cousin in enumerative
algebraic geometry is the degeneration method, which was only derived
in very special cases. The classical degeneration method never became a
general theory as neat as the composition law describes.
One reason might be that the classical counting of holomorphic
curves, particularly of higher genus,
does not obey the composition laws predicted by physicists, even for the
projective plane ${\bf P}^2$. Namely the way of counting was not good.
In [RT], we found the correct counting in terms of inhomogeneous
holomorphic maps and established the composition law at least for
the mixed invariants, corresponding to the $\sigma$-models
without gravity. Based on the same techniques developed in [RT], we
are also able to prove the composition law for all GW-invariants.
Assume $g=g_1+g_2$ and $k=k_1+k_2$ with $2g_i + k_i \ge 3$.
Fix a decomposition $S=S_1\cup S_2$ of $\{1,\cdots , k\}$ with
$|S_i|= k_i$. Then there is a canonical embedding
$\theta _S: \overline {\cal M}_{g_1,k_1+1}\times \overline {\cal M}_{g_2,k_2+1}
\mapsto \overline {\cal M}_{g,k}$, which assigns to marked
curves $(\Sigma _i; x_1^i,\cdots ,x_{k_1+1}^i)$ ($i=1,2$), their
union $\Sigma _1\cup \Sigma _2$ with $x^1_{k_1+1}$ identified to
$x^2_{k_2+1}$ and remaining points renumbered by $\{1,\cdots,k\}$ according to
$S$.
There is another natural map $\mu : \overline {\cal M}_{g-1, k+2}
\mapsto \overline {\cal M}_{g,k}$ by gluing together the last two marked
points.
Choose a homogeneous
basis $\{\beta _b\}_{1\le b\le L}$ of $H_*(V,{\bf Z})$ modulo
torsion. Let $(\eta _{ab})$ be its intersection matrix. Note that
$\eta _{ab} = \beta _a \cdot \beta _b =0$ if the dimensions of
$\beta _a$ and $\beta _b$ are not complementary to each other.
Put $(\eta ^{ab})$ to be the inverse of $(\eta _{ab})$.
Now we can state the composition law, which consists of
two formulas.
\vskip 0.1in
\noindent
{\bf Theorem B. (Theorem 2.10)} {\it Let $[K_i] \in H_*(\overline {\cal M}_{g_i,
k_i+1}, {\bf Q})$ $(i=1,2)$ and $[K_0] \in H_*(\overline {\cal M}_{g-1,
k +2}, {\bf Q})$. For any $\alpha _1,\cdots,\alpha _k$ in $H_*(V,{\bf Z})$.
Then we have}
$$\begin{array}{rl}
&\Psi ^V_{(A,g,k)}(\theta _{S*}[K_1\times K_2];\{\alpha _i\})\\
=~& \sum \limits _{A=A_1+A_2} \sum \limits_{a,b}
\Psi ^V_{(A_1,g_1,k_1+1)}([K_1];\{\alpha _{i}\}_{i\le k_1}, \beta _a)
\eta ^{ab}
\Psi ^V_{(A_2,g_2,k_2+1)}([K_2];\beta _b,
\{\alpha _{j}\}_{j> k_1}) \\
\end{array}
\leqno (1.5)
$$
$$
\Psi ^V_{(A,g,k)}(\mu_*[K_0];\alpha _1,\cdots, \alpha _k)
=\sum _{a,b} \Psi ^V_{(A,g-1,k+2)}([ K_0];\alpha _1,\cdots, \alpha _k,
\beta _a,\beta _b) \eta ^{ab}\leqno (1.6)
$$
\vskip 0.1in
There is a natural map $\pi:
\overline{{\cal M}}_{g,k}\rightarrow \overline{{\cal M}}_{g, k-1}$ as follows: For
$(\Sigma, x_1, \cdots, x_k)\in \overline{{\cal M}}_{g,k}$, if $x_k$ is not in any
rational component of $\Sigma$ which contains only three special points,
then we define
$$\pi(\Sigma, x_1, \cdots, x_k)=(\Sigma, x_1, \cdots, x_{k-1}),$$
where a distinguished point of $\Sigma$ is either a singular point or a
marked point. If $x_k$ is in one of such rational components, we contract
this component and obtain a stable curve $(\Sigma', x_1, \cdots, x_{k-1})$ in
$\overline{{\cal M}}_{g, k-1}$, and define $\pi(\Sigma, x_1, \cdots, x_k)=(\Sigma',
x_1, \cdots, x_{k-1}).$
Clearly, $\pi$ is continuous. One should be aware that
there are two exceptional cases $(g,k)=(0,3), (1,1)$ where $\pi$ is not well
defined. Associated with $\pi$,
we have two {\em k-reduction formulas} for $\Psi^V_{(A, g, k)}$.
\vskip 0.1in
\noindent
{\bf Proposition C (Theorem 2.15). }{\it Suppose that $(g,k)
\neq (0,3),(1,1)$.
\vskip 0.1in
\noindent
(1) For any $\alpha _1, \cdots , \alpha _{k-1}$ in $H_*(V, {\bf Z})$,
we have}
$$\Psi ^V_{(A,g,k)}([K]; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~
\Psi ^V_{(A,g,k-1)}([\pi (K)]; \alpha _1, \cdots,\alpha _{k-1})
\leqno (3.3)$$
\vskip 0.1in
\noindent
(2) Let $\alpha _k$ be in $H_{2n-2}(V, {\bf Z})$, then
$$\Psi ^V_{(A,g,k)}([\pi^{-1}(K)]; \alpha _1,
\cdots,\alpha _{k-1}, \alpha _k)~=~\alpha^* _k (A)
\Psi ^V_{(A,g,k-1)}([K]; \alpha _1, \cdots,\alpha _{k-1})
\leqno (3.4)$$
where $ \alpha^* _k$ is the Poincare dual of $\alpha _k$.
\vskip 0.1in
\noindent
In order to formulate the generalized
Witten conjecture in terms of our invariants, we need to introduce
special cycles in $\overline {\cal M}_{g,k}$. Let
$\pi : \overline{{\cal U}_{g,k}}\rightarrow \overline{{\cal M}}_{g,k}$ be
the universal family of stable curves of genus $g$ and $k$ marked points.
Each marked point gives rise to a section $\sigma _i$ ($1\le i \le k$)
of this fibration. Following Witten,
we let ${\cal L}_i$ be the pull-back of the relative
cotangent sheave of $\pi:
\overline{{\cal U}_{g,k}}\rightarrow \overline{{\cal M}}_{g,k}$ by $\sigma_i$.
Then we put $W_{d_1, \cdots , d_k}$ to be
the Poincare dual of the cohomology class
$c_1({\cal L}_1)^{d_1}\cup c_1({\cal L}_2)^{d_2}\cdots\cup c_1({\cal L}_k)^{d_k}$.
We call these $W_{d_1, \cdots , d_k}$ Witten cycles.
For convenience, as Witten did, we use
$$<\tau_{d_1, \alpha_1}, \tau_{d_2, \alpha_2}, \cdots,
\tau_{d_k, \alpha_k}>_{g,k}$$
to denote the GW-invariants $\Psi_{(A,g,k)}([W_{d_1, \cdots , d_k}];
\alpha_1, \cdots, \alpha_k)$.
Following Witten, we introduce potential functions
$$F_g=\sum_A\sum_{n_{r, \alpha}}\prod_{r, \alpha}
\frac{(t^{\alpha}_r)^{n_{r,
\alpha}}}{n_{r, \alpha}!}
<\prod_{r, \alpha}\tau^{n_{r, \alpha}}_{r, \alpha}>q^A,~~~g=0, 1,2, \cdots.$$
where $q^A$ is an element of Novikov ring \cite{MS}, \cite{RT} (section 8).
We further define
$$F^V=\sum_{g\geq 0} F_g.$$
One of fundamental problems on $F^V$, even to physicists,
is to find
the complete set of equations $F^V$ satisfies.
In Section 6, imitating the arguments of Witten in [W2], we
will prove (cf. Lemma 6.1, 6.2)
\vskip 0.1in
\noindent
{\bf Theorem C. }{\it $F^V$ satisfies the generalized string
equation
$${\partial F^V\over \partial t^1_0} = {1\over 2} \eta _{ab} t^a_0 t^b_0
+ \sum \limits_{i=0}^\infty \sum \limits _{a} t^a_{i+1}
{\partial F^V\over \partial t^a_i}.\leqno(1.7)$$
$F_g$ satisfies the dilation equation
$$\frac{\partial F_g}{\partial t^1_1}=(2g-2+\sum_{i=1}^{\infty}\sum_{a}t^a_i
\frac{\partial }{\partial t^a_i})F_g+\frac{\chi(V)}{24}\delta_{g,1},
\leqno(1.8)
$$
where $\chi(V)$ is the Euler characteristic of $V$.}
\vskip 0.1in
In general, Witten suggested
$$U = {\partial ^{2} F^V \over \partial t _{0, 1}
\partial t _{0,\sigma}},~
U' = {\partial ^{3} F^V \over \partial t _{0, 1}^2
\partial t _{0,\sigma}},~\cdots,~U^{(l)}_\sigma =
{\partial ^{l+2} F^V \over \partial t _{0, 1}^{l+1}
\partial t _{0,\sigma}},~~~~{\rm for~}~ l \ge 0
$$
We will regard $U^{(l)}$ to be of degree $l$. By a differential
function of degree $k$ we mean
a function $G(U, U', U'', \cdots)$ of degree $k$
in that sense. In particular,
any function of form $G(U)$ is of degree zero, and
$(U')^2$ has degree two.
\vskip 0.1in
\noindent
{\bf Generalized Witten Conjecture:} {\it For every $g\geq 0$, there are
differential functions
$G_{m,\alpha; n, \beta}(U_{\alpha}, {U_{\alpha}}',
{U_{\alpha}}'', \cdots)$ of degree $2g$ such that
$${\partial ^2 F_g \over \partial
\tau_{m, \alpha} \partial
\tau_{n, \beta}}= G_{m,\alpha; n, \beta}(U_{\alpha},
{U_{\alpha}}', {U_{\alpha}}'', \cdots)$$
up to terms of genus $g$.}
\vskip 0.1in
This conjecture was affirmed in case $V=pt$ by Kontsevich \cite{Ko}.
When $g=0$, it is a consequence of
the associativity equation proved in \cite{RT}. But
the general case is still open.
We call $\Psi^V_{(A,g,k)}([\overline {\cal M}_{g,k}]; \cdots)$
primitive GW-invariants of genus $g$. Those invariants
correspond to
the enumerative invariants of counting genus $g$ holomorphic
curves passing through generic $k$ cycles
in enumerative algebraic geometry.
\vskip 0.1in
\noindent
{\bf Corollary E (Proposition 6.5). }
{\it For genus $\leq 1$, the Witten invariants $<>$ can be
reduced to primitive GW-invariants.}
\vskip 0.1in
In general, we conjecture that all the Witten invariants
can be derived from primitive GW-invariants.
Our invariant can be also applied to studying topology of symplectic
manifolds. As an example, we will verify the {\it Stabilizing conjecture}
of the first author in the case of simply connected elliptic surfaces.
The conjecture claims: {\it Suppose that $X$, $Y$ are two simply connected
homeomorphic symplectic 4-manifolds. Then $X$, $Y$
are diffeomorphic if and only if
$X\times S^2$, $Y\times S^2$ are deformation equivalent
as symplectic manifolds.} He also verified this conjecture
for certain complex surfaces (cf. [R1]). By calculating our invariants for
the product of simply connected
elliptic surfaces with $S^2$, we will prove that
\vskip 0.1in
\noindent
{\bf Theorem F: (Theorem 5.1)} {\it The stabilizing
conjecture holds for simply connected elliptic
surfaces.}
\vskip 0.1in
It has been an interesting question in symplectic topology to find how many
different deformation classes of symplectic structures with the same
tamed almost complex structures (up to a homotopy) could exist on a fixed
smooth
manifold. In \cite{R1}, for any positive integer $n$, the first author
constructed examples admitting at least $n$-many
different deformation classes. Using our calculation of
GW-invariant, we can produce examples with infinitely many deformation
classes of symplectic structures.
\vskip 0.1in
\noindent
{\bf Proposition H: (Proposition 5.4) }{\it Let $X$ be the blow-up of a simply
connected elliptic
surface at one point. Then, the smooth 6-manifold $X\times S^2$ admits
infinitely many
deformation classes of symplectic structures with the same tamed almost complex
structure up to a homotopy.}
\vskip 0.1in
This paper is organized as follows. We will define the invariants and state
the basic properties (including composition law) of our invariants in section
2. The section 3 is a technical section where we will prove the various
results about the compactification and transversality. All the results
stated in section 2 will be proved in section 4.
We will discuss the applications to the stabilizing conjecture and the
Witten conjecture in section 5, 6.
Some of the results in this paper have been lectured by us in last few
years. Also, the main results of this paper were announced
in the paper [T] of the second author published in the proceeding
of the first "Current developments in Mathematics", Boston, May, 1995.
All the basic techniques were developed in [RT].
The first author wish to thank S. K. Donaldson
who suggested the example of Section 5 to him.
\section{Higher genus symplectic invariants and composition law}
In this section, we construct the higher genus symplectic invariants. Its
physical counterparts are the correlation functions of topological sigma model
coupled with gravity. Some important cases of
these invariants were
first studied
for symplectic 4-manifolds in \cite{R3} and also discussed
in [RT]. The construction here is similar
to that of \cite{RT}.
First of all, let's introduce the inhomogeneous Cauchy-Riemann equation, which
plays a central role in \cite{RT}. Compared to that of
\cite{RT}, we
would like to define the inhomogeneous term varying continuously as we vary
the complex structures of the Riemann surfaces. This makes the construction
more complicated.
Let $(V, \omega)$ be a symplectic manifold and $J$ be a
tamed almost complex structure. Let ${\cal M}_{g, k}$ be the moduli space
of genus $g$ Riemann surfaces with $k$-marked points and $\overline{{\cal M}}_{g, k}$
be the Deligne-Mumford compactification. Suppose that
$$\pi: \overline{{\cal U}}_{g,k}\rightarrow\overline{{\cal M}}_{g,k}$$
is the universal curve. Both $\overline{{\cal U}}_{g,k}$ and $\overline{{\cal M}}_{g,k}$
are projective varieties. Unfortunately, it is well-known that $\overline{{\cal U}}_{
g,k}$ is not a universal family. Namely, if $\Sigma\in \overline{{\cal M}}_{g,k}$ has
a nontrivial automorphism, then $\pi^{-1}(\Sigma)=\Sigma/Aut(\Sigma)$ instead
of $\Sigma$. We can not directly define the inhomogeneous term over
$\overline{{\cal U}}_{g,k}$. However, this problem can be overcome by constructing
some finite covers of $\overline{{\cal M}}_{g,k}$.
\vskip 0.1in
\noindent
{\bf Definition 2.1: }{\it A finite connected cover $p_{\mu}:\overline{{\cal M}}^{
\mu}_{g,k}\rightarrow
\overline{{\cal M}}_{g,k}$ is a good cover if $\overline{{\cal M}}^{\mu}_{g,k}$ is a normal
projective variety with
quotient singularity such that there is universal family
$$\pi_{{\cal M}}:\overline{{\cal U}}^{\mu}_{g,k}\rightarrow \overline{{\cal M}}^{\mu}_{g,k},$$
i.e., for each $b\in \overline{{\cal M}}^{\mu}_{g,k}, \pi^{-1}_{{\cal M}}(b)$ is a stable
Riemann surface isomorphic to $p_{\mu}(b)$. Furthermore, we have following
commutative diagram
$$\begin{array}{lccc}
p_{\mu}& \overline{{\cal U}}^{\mu}_{g,k}&\rightarrow & \overline{{\cal U}}_{g,k}\\
& \downarrow \pi_{{\cal M}} && \downarrow\pi\\
p_{\mu}: & \overline{{\cal M}}^{\mu}_{g,k}& \rightarrow &\overline{{\cal M}}_{g,k}
\end{array}$$
It is clear that $\overline{{\cal U}}^{\mu}_{g,k}$ is projective and unique.}
\vskip 0.1in
To simplify the notation, we will not distinguish $b$ with $p_{\mu}(b)$ without
any confusion. As we mentioned, the problem for $\overline{{\cal M}}_{g,k}$ is that
some elements have nontrivial automorphism groups. One can resolve this problem
by taking finite cover locally. Hence, locally one always has a universal
family. Mumford proved \cite{Mu} that such local covers can be glued together
to form a global finite cover. Moreover, it can be
explicit constructed via level $m$-structure. We will not give any detail of
level $m$-structure. The idea is to fix a basis of $H^1$ with
the coefficient in ${\bf Z}_m$. Then, there is no automorphism of stable Riemann
surface preserving the fixed basis. We refer
the reader to \cite{Mu} for the detail.
Let $p_{\mu}: \overline{{\cal M}}^{\mu}_{g,k}\rightarrow \overline{{\cal M}}_{g,k}$ be a
good
cover. Suppose that
$$\phi_{\mu}:\overline{{\cal U}}^{\mu}_{g,k}\rightarrow {\bf P}^{N}$$
is a projective embedding.
There are two
relative tangent bundles over ${\bf P}^{N} \times V$ with respect to $h_i$ ($i=1,2$)
, where $h_i$ is the projection from ${\bf P}^{N}\times V$ to
its $i$-th factor. A
section $\nu$ of $Hom(h^*_1 T{\bf P}^{N}, h^*_2 TV)$ is said to be anti-$J$-linear
if for any tangent vector $v$ in $T{\bf P}^N$,
$$ \nu(j_{{\bf P}^N}(v)) ~=~ - J (\nu(v))
\leqno (2.1)$$
where $j_{{\bf P}^N} $ is the complex structure on ${\bf P}^N$.
Usually, we call such a $\nu$ an inhomogeneous term. We will often drop $h_i$
from the notation without any confusion.
\vskip 0.1in
\noindent
{\bf Definition 2.2.} {\it Let $\nu$ be an inhomogeneous term.
A $(J,\nu)$-perturbed holomorphic map, or simply a $(J,\nu)$-map, is
a smooth map $f:\Sigma\rightarrow V$ satisfying the inhomogeneous
Cauchy-Riemann
equation
$$(\bar{\partial}_J f)(x)=
\nu(\phi_{\mu}(x), f(x)),
\leqno (2.2)$$
where $\bar{\partial}_J$ denotes the differential operator
$d + J\cdot d\cdot j_\Sigma$.}
\vskip 0.1in
Let ${\cal M}_{g,k,\kappa}$ be the subset of ${\cal M}_{g,k}$ with automorphism group
$\kappa$. We will use $I$ to denote the trivial automorphism group.
Since ${\cal M}_{g,k,I}$ is smooth, without the loss of generality, we can assume
that ${\cal M}^{\mu}_{g,k,I}$ is smooth, where ${\cal M}^{\mu}_{g,k,
\kappa}=p^{-1}_{\mu}({\cal M}_{g,k, \kappa})$. Let ${\cal U}^{\mu }_{
g,k, I}$ be the preimage of ${\cal M}^{\mu}_{g,k, I}$.
We denote by ${\cal M}_A(\mu, g,k, J, \nu)_I$ the moduli space of
$(J, \nu)$-perturbed
holomorphic maps from $(\Sigma, x_1, \cdots, x_k)\in {\cal M}^{\mu }_{g,k,I}$ into
$V$. There
are some important topological properties as follows.
Let
$\pi: {\cal U}_A(\mu, g,k, J, \nu)_I\rightarrow {\cal M}_A(\mu, g,k, J, \nu)_I$ be the
universal family of
curves, i.e.,
$$\pi^{-1}(f, \Sigma, \{x_i\})=\Sigma.$$
We can define the evaluation
map
$$e_A(g,k): {\cal U}_A(\mu, g,k, J, \nu)_I\rightarrow V$$
by
$$e_A(g,k)(f, \Sigma, \{x_i\}, y)=f(y).\leqno(2.3)$$
Each marked point $x_i$ defines a section
$$\sigma_i: {\cal M}_A(\mu,g,k, J,\nu)_I\rightarrow {\cal U}_A(\mu, g,k, J, \nu)_I$$
by
$$\sigma_i((f,\Sigma, \{x_i\}))=x_i.\leqno(2.4)$$
The composition
$$e_i=e_A(g,k)\circ \sigma_i: {\cal M}_A(\mu, g,k, J,\nu)_I\rightarrow V.$$
Let
$$\Xi^A_{g,k}=\prod^{k}_{i=1}e_A(g,k)\circ \sigma_i: {\cal M}_A(g,k, J,\nu)_I
\rightarrow V^k.$$
Evidently, we have a map $\Upsilon_A: {\cal M}_A(\mu, g,k, J, \nu)_I\rightarrow
{\cal M}^{\mu }_{g,k,I}$
by assigning each $(J, \nu)$-map to its domain. Together, we get a smooth map
$$\Upsilon_A \times \Xi^A_{g,k}:{\cal M}_A(\mu, g,k, J, \nu)_I\rightarrow {\cal M}^{\mu
}_{g,k,I}\times V^k.$$
In general, ${\cal M}_A(\mu, g,k, J, \nu)_I$ is not compact. However, there
is a natural compactification $\overline{{\cal M}}_A(\mu, g,k, J, \nu)_I$, which we
call
GU-compactification (cf. Section 3). For our purpose, we also need to consider
certain quotient $\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)_I$ of $\overline{{\cal M}}_A(
\mu, g,k, J, \nu)_I$.
\vskip 0.1in
\noindent
{\bf Proposition 2.3.} {\it Suppose that $(V, \omega)$ is a semi-positive
symplectic manifold. Then, there is a Baire set of second category-${\cal H}$ among
all the smooth pairs $(J, \nu)$ such that for any $(J, \nu)\in {\cal H}$
\vskip 0.1in
\noindent
(1) ${\cal M}_A(\mu, g,k,J, \nu)_I$ is a smooth, oriented manifold of real dimension
$$2c_1(V)(A)+2(3-n)(g-1)+2k;$$
\vskip 0.1in
\noindent
(2) $\Upsilon_A$ and $\Xi^A_{g,k}$ extends to continuous maps, still
denoted by the same symbols, from $\overline{{\cal M}}^r_A
(\mu, g,k, J, \nu)_I$ to $\overline{{\cal M}}^{\mu}_{g,k}$ and $V^k$,
respectively;
\vskip 0.1in
\noindent
(3) The boundary $\Upsilon_A\times \Xi^A_{g,k} \left (
\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)_I\backslash {\cal M}_A(\mu,g,k,J, \nu)_I\right )$
is of real
codimension at least two.
We call that $(J, \nu)$ is generic if Proposition 2.3 is satisfied.}
\vskip 0.1in
The proof of this proposition is the main topic of Section 3.
One can construct natural cohomology classes over $\overline
{\cal M}_A(\mu, g,k, J, \nu)_I$ by pulling
back of the cohomology classes of $\overline{{\cal M}}^{\mu}_{g,k}\times V^k$ through
$\Upsilon _A
\times \Xi^A_{g,k}$. Then, our invariants can be defined as the paring of the
cup products of those natural cohomology classes against the fundamental class
of $\overline{{\cal M}}_A(\mu, g,k, J, \nu)_I$. The existence of such a
fundamental class is a much more difficult problem, which will be discussed in
\cite{RT1}. Here, we choose to avoid this problem by considering the
intersection theory as we did in \cite{RT}.
Let $\{\alpha_i\}_{1\leq i\leq k}$ be integral homology classes of $V$. Each
$\alpha_i$ can be represented by a so called pseudo-submanifold $(P,f)$. A
pseudo-submanifold is a pair $(P, f)$, where $P$ is a finite
simplicial complex of dimension
$d_i=deg(\alpha_i)$
such that $P^{top}=P-P_{d_i-2}$ ($P_{d_i-2}-(d_i-2)$
skeleton) is a smooth manifold and $f: P\rightarrow V$ is piecewise linear
with respect to a triangulation of $V$ and smooth over $P^{top}$ in the usual
sense. Any two such pseudo-submanifolds representing the same homology class
are the boundary of a pseudo-submanifold cobordism in the usual sense. We refer
to the Section 4 for details. We choose pseudo-submanifolds $(Y_i, F_i)$ to
represent
$\alpha_i$. Let
$$Y=\prod^k_{i=1} Y_i, F=\prod^k_{i=1} F_i.$$
We define $Y^{top}=\prod^k_{i=1} Y^{top}_i.$
Clearly, $(Y, F)$ represents $\prod^k_{i=1}\alpha_i\in H_*(V^k, {\bf Z})$.
In general, not every
integral homology class of $\overline{{\cal M}}^{\mu}_{g,k}$ can be represented by a
pseudo-submanifold. However, some of its multiples does.
Therefore, the homology
classes represented by pseudo-submanifolds generate the rational homology
$H_*(\overline{{\cal M}}^{\mu}_{g,k}, {\bf Q})$. We say
that a pseudo-manifold $(G, K)$ is {\em in the general position} if
$K(G^{top})\subset {\cal M}^{\mu }_{g,k,I}$ has codimension
at least two in $K(G^{top})$.
Let $(G, K)$ be such a pseudo-submanifold in
${\cal M}^{\mu}_{g,k,I}$. Suppose
$$\sum^k_{i=1} (2n-d_i)+(6g-6+2k-deg(G))=2c_1(V)(A)+2(3-n)(g-1)+2k.
\leqno(2.5)$$
Then, we can choose a small perturbation of $F_i, K$ such
that $K\times F$ is transverse to $\Upsilon_A\times \Xi^A_{g,k}$ as the PL-maps
with respect to some triangulation of $V$ and as the smooth maps over $Y^{top}
\times G^{top}$. By the dimension counting, we can show that
$$\begin{array}{rl}
&Im(K\times F )\cap Im(\Upsilon_A\times \Xi^A_{g,k}
(\overline{{\cal M}}_A(\mu,g,k, J, \nu)_I-
{\cal M}_A(\mu,g,k, J, \nu)_I))=\emptyset;\\
&K\times F (Y\times G -Y^{top}\times G^{top})\cap
\Upsilon_A\times \Xi^A_{g,k} =\emptyset.\\
\end{array}
\leqno(2.6)$$
Let $\Delta\subset ({\cal M}^{\mu}_{g,k,I}\times V^k) \times ({\cal M}^{\mu }_{g,k,I}\times
V^k)$ be the diagonal. Then,
$$(\Upsilon_A\times \Xi^A_{g,k}\times K\times F)^{-1}(\Delta)\subset
{\cal M}_A(\mu,g,k, J, \nu)_I \times Y^{top}\times G^{top} \leqno(2.7)$$
is a zero-dimensional smooth submanifold. Following from (2.6), it is compact
and hence
finite. Suppose that
$$(\Upsilon_A\times \Xi^A_{g,k}\times K\times F)^{-1}(\Delta)=\{(f_1, s_1),
\cdots,
(f_m, s_m)\} \leqno(2.8)$$
where $f_i\in {\cal M}_A(\mu,g,k,J,\nu)_I$ and $s_i$ represents other factors. For
each
$(f_i, s_i)$, we define a number $\epsilon(f_i, s_i)=\pm 1$ as follows: We
define $\epsilon(f_i, s_i)=+1$ if the orientation, induced by the Jacobian of
$\Upsilon_A\times \Xi^A_{g,k}\times K\times F$ and the orientation of
${\cal M}_A(\mu,g,k, J, \nu)_I \times Y^{top}\times G^{top}$ at $(f_i, s_i)$,
together with the orientation
of $\Delta$ matches the orientation of $({\cal M}^{\mu }_{g,k,I}\times V^k)
\times ({\cal M}^{\mu}_{g,k,I}\times V^k)$. Otherwise, we define $\epsilon(f_i,
s_i)=-1$. Now, we define
$$\Psi^V_{(A, g, k,\mu)}(K; \alpha_1, \cdots, \alpha_k)=\sum^m_{i=1}
\epsilon(f_i,s_i).$$
To Justify our notation, we will show
\vskip 0.1in
\noindent
{\bf Proposition 2.4.} {\it
\vskip 0.in
\noindent
(1) $\Psi^V_{(A, g, k,\mu)}(K; \alpha_1, \cdots, \alpha_k)$
is independent of $J, \nu$, the pseudo-submanifold representatives
$(Y_i, F_i)$.
\vskip 0.1in
\noindent
(2) $\Psi^V_{(A, g, k,\mu)}(K; \alpha_1, \cdots, \alpha_k)$ is independent of
semi-positive deformations of $\omega$.
\vskip 0.1in
\noindent
(3) If $(G', K')$ is another pseudo-submanifold which is in the general
position
and represents the same homology class as that of $(G, K)$,
$$\Psi^V_{(A, g, k,\mu)}(K; \alpha_1, \cdots, \alpha_k)=\Psi^V_{(A, g, k,\mu)}
(K'; \alpha_1, \cdots, \alpha_k).$$}
\vskip 0.1in
Therefore, we can write $\Psi^V_{(A, g, k,\mu)}([K]; \alpha_1, \cdots,
\alpha_k)$
for $\Psi^V_{(A, g, k,\mu)}(K; \alpha_1, \cdots, \alpha_k)$, where
$[K]$ denotes the homology
class represented by the cycle $K$.
We postpone the proof of Proposition 2.4 to Section 4.
\vskip 0.1in
\noindent
{\bf Proposition 2.5. }{\it $\Psi^V_{(A, g, k,\mu)}(K; \alpha_1, \cdots,
\alpha_k)$ is independent of the embedding $\phi_{\mu}$. Hence, $\Psi_{(A,
g,k, \mu)}$ is a symplectic invariant}
\vskip 0.1in
\noindent
{\bf Proof: } Suppose that
$$\tilde{\phi}_{\mu}: \overline{{\cal U}}_{\mu}\rightarrow {\bf P}^{N'}$$
is a different projective embedding. Then,
$$\phi_{\mu}\times \tilde{\phi}_{\mu}: \overline{{\cal U}}_{\mu}\rightarrow
{\bf P}^{N}\times {\bf P}^{N'}.$$
One can consider the inhomogeneous term $\bar{\nu}\in \overline{Hom}_J(T(
{\bf P}^N\times {\bf P}^{N'}), TV)$ where $
\overline{Hom}_J$ means anti-complex linear homomorphism. Moreover, one can
use $(J, \bar{\nu})$ to define invariant in the same fashion and prove that the
invariant is independent of $(J, \bar{\nu})$ using the same proof of
Proposition 2.4. Let $\nu$, $\tilde{\nu}$ be the inhomogeneous term defined
through the embedding
$\phi_{\mu}$ and $\tilde{\phi}_{\mu}$. Notes that
$$T({\bf P}^N\times {\bf P}^{N})= T({\bf P}^N)\times T({\bf P}^{N'}).$$
Therefore, we can view both $\nu$ and $\tilde{\nu}$ as the sections of
$\overline{Hom}_J(T({\bf P}^N\times {\bf P}^{N'}), TV)$, where we view that $\nu$ maps
the factor $T({\bf P}^{N'})$ to zero and $\tilde{\nu}$ maps the first factor to
zero. Let's denote them
by $\bar{\nu}$ and $\bar{\tilde{\nu}}$. We observe that if $\nu(\tilde{
\nu})$ is generic, so is $\bar{\nu}(\bar{\tilde{\nu}})$. It follows from the
definition that
we have the same invariant $\Psi$ using $(J, \nu)$ or $(J, \bar{\nu})$. In
the same way, we have the same invariant using $(J, \tilde{\nu})$ or
$(J, \bar{\tilde{\nu}})$.
As we mentioned, by repeating the proof of Proposition 2.4, we can show
that the invariant defined by $(J, \bar{\nu})$ is the same as the invariant
defined by $(J, \bar{\tilde{\nu}})$. Then, we finish the proof.
\vskip 0.1in
\noindent
{\bf Remark 2.6: }{\it We don't have to restrict ourself to projective
embedding. In
fact, we can embed $\overline{{\cal U}}^{\mu}_{g,k}$ into any smooth complex manifold
and define the inhomogeneous term in the same fashion. The proof of Proposition
2.5 shows that the resulting invariant is independent of such an embedding.}
\vskip 0.1in
For the convenience, we define $\Psi^V_{(A, g, k, \mu)}(K, \alpha_1, \cdots,
\alpha_k)=0$ if (2.5) is not satisfied.
Let's collect some properties of $\Psi^V_{(A, g, k,\mu)}.$ The
following proposition essentially
follows from the definition. We will omit its proof.
\vskip 0.1in
\noindent
{\bf Proposition 2.7.}{\it
\vskip 0.1in
\noindent
(1) $\Psi^V_{(A, g, k,\mu)}=0$, if either $\omega(A)< 0$ or $c_1(V)(A)+(3-n)(g
-1)<0$, in particular, $\Psi_{(0,g,k,\mu)}=0$ for any $g\geq 2$ and $n\geq 4$.
\vskip 0.1in
\noindent
(2) $\Psi^V_{(A, g, k,\mu)}$ is multilinear and supersymmetry on $H_*(V, {\bf Z})^k$
with respect to the ${\bf Z}_2$-grading by even and odd degrees.}
\vskip 0.1in
We have established
the symplectic invariant
$$ \Psi ^V_{(A,g,k,\mu)} : H_*(\overline{{\cal M}}^{\mu}_{g,k} , {\bf Q})
\times H_*(V, {\bf Z})^k\mapsto {\bf Q},$$
for any $A\in H_2(V, {\bf Z})$ and $2g +k \ge 3$.
Now we discuss other more interesting properties of $\Psi^V_{(A, g, k,\mu)}$
associated with the structures of $\overline{{\cal M}}_{g,k}$.
As we mentioned in the introduction, there is a natural map
$$\pi: \overline{{\cal M}}_{g,k}\rightarrow \overline{{\cal M}}_{g, k-1} \leqno(2.9)$$
by forgetting the last
marked point and contracting the unstable rational component.
One should be aware that
there are two exceptional cases $(g,k)=(0,3), (1,1)$ where $\pi$ is not well
defined. Suppose $\overline{{\cal M}}^{\mu}_{g,k}\rightarrow \overline{{\cal M}}_{g,k}$ is
a good
cover constructing through level-$m$ structure. Then, one can observe that
$\overline{{\cal M}}^{\mu}_{g,k+1}=\pi^*\overline{{\cal M}}^{\mu}_{g,k}$ is a good cover of
$\overline{{\cal M}}_{g, k+1}$.
Let
$$\pi_{\mu}: \overline{{\cal M}}^{\mu}_{g, k+1}\rightarrow
\overline{{\cal M}}^{\mu}_{g,k}.$$
Then, $\pi_{\mu}$ induces a map on the universal families
(still denoted by $\pi_{\mu}$.)
$$\begin{array}{clc}
\overline{{\cal U}}^{\mu}_{g,k+1}&\stackrel{\pi_{\mu}}{\rightarrow}&\overline{{\cal U}}^{\mu}_{g,k}\\
\downarrow \pi_{{\cal M}}& & \downarrow \pi_{{\cal M}}\\
\overline{{\cal M}}^{\mu}_{g,k+1}&\stackrel{\pi_{\mu}}{\rightarrow} &
\overline{{\cal M}}^{\mu}_{g, k}
\end{array}$$
Let $b\in \overline{{\cal M}}^{\mu}_{g, k+1}$ and $\Sigma_b$ be the underline stable
Riemann
surface.
Clearly,
$$\pi_{\mu}: \Sigma_b\rightarrow \Sigma_{\pi_{\mu}(b)}$$
is precisely $\pi$ defined in (2.9).
Associated with $\pi$,
we have two {\em k-reduction formulas} for $\Psi^V_{(A, g, k, \mu)}$.
\vskip 0.1in
\noindent
{\bf Proposition 2.8. }{\it Suppose that $(g,k)\neq (0,3),(1,1)$. Furthermore,
suppose that $\overline{{\cal M}}^{\mu}_{g,k+1}$ and $\overline{{\cal M}}^{\mu}_{g,k}$ are
defined as above.
\vskip 0.1in
\noindent
(1) For any $\alpha _1, \cdots , \alpha _{k-1}$ in $H_*(V, {\bf Z})$,
we have}
$$\Psi ^V_{(A,g,k,\mu)}([K]; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~
\Psi ^V_{(A,g,k-1,\mu)}((\pi_{\mu})_*[K]; \alpha _1, \cdots,\alpha _{k-1})
\leqno (2.10)$$
\vskip 0.1in
\noindent
(2) Let $\alpha _k$ be in $H_{2n-2}(V, {\bf Z})$, then
$$\Psi ^V_{(A,g,k,\mu)}([K]; \alpha _1,
\cdots,\alpha _{k-1}, \alpha _k)~=~\alpha^* _k (A)
\Psi ^V_{(A,g,k-1,\mu)}([(\pi_{\mu})^{-1}(K)]; \alpha _1, \cdots,\alpha _{k-1})
\leqno (2.11)$$
where $ \alpha^* _k$ is the Poincare dual of $\alpha _k$.
\vskip 0.1in
\noindent
{\bf Proof:} The proof is similar to that of Proposition 2.5. Let
$$\phi^k_{\mu}: \overline{{\cal U}}^{\mu}_{g,k}\rightarrow {\bf P}^N; \ \phi^{k-1}_{\mu}:
\overline{{\cal U}}^{\mu}_{g, k-1}\rightarrow {\bf P}^{N'}$$
be projective embedding and $\nu_k, \nu_{k-1}$ be inhomogeneous terms over
${\bf P}^N$ or ${\bf P}^{N'}$ respectively. Consider embedding
$$\phi^k_{\mu}\times (\phi^{k-1}_{\mu}\circ \pi_{\mu}):
\overline{{\cal U}}^{\mu}_{g,k}\rightarrow {\bf P}^N\times {\bf P}^{N'}. \leqno(2.12)$$
As in Remark 2.6, we can define the invariant $\bar{\Psi}$ using
$\phi^k_{\mu}\times
(\phi^{k-1}_{\mu}\circ \pi_{\mu})$ and the inhomogeneous terms over ${\bf P}^N\times
{\bf P}^{N'}$ in the same fashion. One can show that such an invariant is
independent of inhomogeneous term. Furthermore, we can view $\nu_k, \nu_{k-1}$
(denoted by $\bar{\nu}_k, \bar{\nu}_{k-1}$) as inhomogeneous terms over
${\bf P}^N\times {\bf P}^{N'}$ as in the proof of Proposition 2.5. Clearly, using
$(J, \bar{\nu}_k)$, we have
$$\bar{\Psi}^V_{(A,g,k,\mu)}([K]; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~
\Psi^V_{(A,g,k,\mu)}([K]; \alpha _1, \cdots,\alpha _{k-1}, [V]).\leqno(2.13)$$
Using $(J, \bar{\nu}_{k-1})$, we claim that
$$\bar{\Psi}^V_{(A,g,k,\mu)}([K]; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~
\Psi^V_{(A,g,k-1,\mu)}(\pi_*[K]; \alpha _1, \cdots,\alpha
_{k-1}).\leqno(2.14)$$
Suppose that $(J, \nu_{k-1})$ is generic. We claim that
$(J, \bar{\nu}_{k-1})$ satisfies the Proposition 2.2.
The Proposition 2.3 (1)
and (2) are obvious. A remark is required for (3). In the proof of (3) in the
next section, the idea is to stratify $\overline{{\cal M}}^r_A(\mu, g, k, J, \nu)_I-
{\cal M}_A(\mu, g, k, J, \nu)_I$ and show that each strata is of real codimension
at least 2. In the proof, we use the fact that $\nu$ is generic over each
component of stable Riemann surface called the principal components. Note that
$\bar{\nu}_{k-1}$ is zero over the rational components $\pi_{\mu}$ contracts.
So it is not generic. However, we can simply treat these rational components as
bubble components (see Definition 3.6), and construct another quotient space
$$\overline{{\cal M}}^r_A\rightarrow \overline{{\cal M}}^{rr}_A$$
in the same way as we construct $\overline{{\cal M}}^r_A$. Then, the proof of
Proposition 2.3 shows that
$$\overline{{\cal M}}^{rr}_A(\mu, g, k, J, \nu)_I-{\cal M}_A(\mu, g, k, J, \nu)_I
\leqno(2.15)$$
is of real codimension at least 2.
Once $(J, \bar{\nu}_{k-1})$ satisfies the Proposition 2.3, we can choose
$(Y_i, F_i), (G, K)$ to satisfy (2.6), (2.7). Then,
$$\bar{\Psi}_{(A, g,k,\mu)}(K; \alpha_1, \cdots, \alpha_k)=\sum^m_{i=1}
\epsilon(f_i, s_i),
$$
where $(f_i, s_i)$ is given in (2.7). Suppose that $s_i=(\Sigma, x_1, \cdots,
x_k, y_1, \cdots, y_{k-1}, y_k).$ Then,
$$(\Sigma, x_1, \cdots,x_k)\in {\cal M}^{\mu}_{g,k,I}$$
and
$$\pi(\Sigma, x_1, \cdots, x_k)=(\Sigma, x_1, \cdots, x_{k-1}).$$
Clearly, $f\in {\cal M}_A(\mu, g,k, J, \bar{\nu}_{k-1})_I$ can also be viewed as an
element of
${\cal M}_A(\mu, g,k-1, J, \nu)_I$.
If $Y_k=V, F_k=Id$, let
$$\bar{s}_i=(\Sigma, x_1, \cdots, x_{k-1}, y_1, \cdots, y_{k-1}).$$
Clearly
$$(\Upsilon_A \times \Xi_{g,k-1}\times \pi_{\mu}(K)\times
\prod^{k-1}_{i=1}F_i)^{-1}(\Delta)=
\{(f_1, \bar{s}_1), \cdots, (f_m, \bar{s}_m)\}. \leqno(2.16)$$
Furthermore, it is easy to check that
$$\epsilon(f_i, s_i)=\epsilon(f_i, \bar{s}_i).$$
Therefore,
$$\bar{\Psi} ^V_{(A,g,k,\mu)}([K]; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~
\Psi ^V_{(A,g,k-1,\mu)}((\pi _{\mu})_*[K]; \alpha _1, \cdots,\alpha _{k-1}).$$
Here, we use the fact that $(G, \pi_{\mu}\circ K)$ represents the homology
class $(\pi _{\mu})_*[K]$.
The proof of (2) is similar.
Next, we discuss the composition law. Roughly speaking, the composition law
governs the change of $\Psi$ under the surgery of Riemann surfaces.
Compared to $k$-reduction formula, one
can view the composition law as $g$-reduction formula,
i.e., the reduction of
genus. As we mentioned in the introduction, its classical cousin in
enumerative algebraic geometry is the
degeneration formula, which was only derived individually in very special
cases. One technical reason is that it is very difficult to have a good
deformation theory in algebraic
geometry. But our $\Psi$ is just a symplectic invariant. The counterpart of
deformation theory in symplectic category can be realized by
a {\em gluing theorem}. It
has been established by the authors in \cite{RT}.
Recall that in the definition of $\Psi_{(A,g,k, \mu)}([K]; \alpha_1, \cdots,
\alpha_k)$, we require that $K$ does not
lie in the boundary of $\overline{{\cal M}}^{\mu}_{g,k}$.
We first remove this technical assumption.
\vskip 0.1in
\noindent
{\bf Proposition 2.9.} {\it $\Psi_{(A,g,k, \mu)}(K; \alpha_1,
\cdots, \alpha_k)$ can be defined when $K$ is in the boundary
of $\overline{{\cal M}}^{\mu}_{g,k}$. Namely, $K\subset Im \theta_S$
or $K\subset Im \bar{\mu}$, where $\theta_S, \bar{\mu}$ are defined below.
Furthermore, $\Psi_{(A,g,k, \mu)}(K; \alpha_1,
\cdots, \alpha_k)$ depends only on the homology class represented by $(G, K)$.
Then, we extend $\Psi_{(A, g,k)}([K]; \cdots)$ for any $[K]\in H_*(\overline{
{\cal M}}^{\mu}_{g,k}, {\bf Q})$ by the linearity.}
\vskip 0.1in
The proof follows basically from the gluing theorem in \cite{RT},
Theorem 6.1. The details will appear in Section 4.
Assume $g=g_1+g_2$ and $k=k_1+k_2$ with $2g_i + k_i \ge 3$.
Fix a decomposition $S=S_1\cup S_2$ of $\{1,\cdots , k\}$ with
$|S_i|= k_i$. Recall that
$\theta _S: \overline {\cal M}_{g_1,k_1+1}\times \overline {\cal M}_{g_2,k_2+1}
\mapsto \overline {\cal M}_{g,k}$, which assigns to marked
curves $(\Sigma _i; x_1^i,\cdots ,x_{k_1+1}^i)$ ($i=1,2$), their
union $\Sigma _1\cup \Sigma _2$ with $x^1_{k_1+1}$ identified to
$x^2_1$ and remaining points renumbered by $\{1,\cdots,k\}$ according to $S$.
Suppose that $\overline{{\cal M}}^{\mu}_{g_1, k_1+1}, \overline{{\cal M}}^{\mu}_{g_2,
k_2+1}, \overline{{\cal M}}^{\mu}_{g,k}$ are
good covers over $\overline {\cal M}_{g_1,k_1+1}, \overline {\cal M}_{g_2,k_2+1},
\overline {\cal M}_{g,k}$ such that
$$\theta^*_S\overline{{\cal M}}^{\mu}_{g,k}= \overline{{\cal M}}^{\mu}_{g_1, k_1+1}\times
\overline{{\cal M}}^{\mu}_{g_2, k_2+1}.
\leqno(2.17)$$
Such good covers can be constructed using the level-$n$ structure of $\overline
{{\cal M}}_{g,k}$.
We have another natural map defined in the introduction
$\mu : \overline {\cal M}_{g-1, k+2}
\mapsto \overline {\cal M}_{g,k}$ by gluing together the last two marked
points.
Let
$$\overline{{\cal M}}^{m_{\mu}}_{g-1, k+2}=\mu^*\overline{{\cal M}}^{\mu}_{g,k}, \
\bar{\mu}: \overline{{\cal M}}^{m_{\mu}}_{g-1, k+2} \rightarrow
\overline{{\cal M}}^{\mu}_{g,k}.\leqno(2.18)$$
Choose a homogeneous
basis $\{\beta _b\}_{1\le b\le L}$ of $H_*(V,{\bf Z})$ modulo
torsion. Let $(\eta _{ab})$ be its intersection matrix. Note that
$\eta _{ab} = \beta _a \cdot \beta _b =0$ if the dimensions of
$\beta _a$ and $\beta _b$ are not complementary to each other.
Put $(\eta ^{ab})$ to be the inverse of $(\eta _{ab})$.
Now we can state the composition law, which consists of two formulas.
\vskip 0.1in
\noindent
{\bf Theorem 2.10 }. {\it Let $[K_i] \in H_*(\overline {\cal M}^{\mu}_{g_i,
k_i+1}, {\bf Q})$ $(i=1,2)$ and $[K_0] \in H_*(\overline {\cal M}^{m_{\mu}}_{g-1,
k +2}, {\bf Q})$. Suppose that $\overline{{\cal M}}^{\mu}_{g_1, k_1+1}, \overline{{\cal M}}^{
\mu}_{g_2, k_2+1},
\overline{{\cal M}}^{\mu}_{g,k}, \overline{{\cal M}}^{m_{\mu}}_{g-1, k+2}$ are defined as
(2.17), (2.18). For any $\alpha _1,
\cdots,\alpha _k$ in $H_*(V,{\bf Z})$.
Then we have}
$$\begin{array}{rl}
&\Psi ^V_{(A,g,k, \mu)}(\theta _{S*}[K_1\times K_2];\{\alpha _i\})\\
=~& \sum \limits _{A=A_1+A_2} \sum \limits_{a,b}
\Psi ^V_{(A_1,g_1,k_1+1, \mu)}([K_1];\{\alpha _{i}\}_{i\le k}, \beta _a)
\eta ^{ab}
\Psi ^V_{(A_2,g_2,k_2+1, \mu)}([K_2];\beta _b,
\{\alpha _{j}\}_{j> k}) \\
\end{array}
\leqno (2.19)
$$
$$
\Psi ^V_{(A,g,k, \mu)}(\bar{\mu}_*[K_0];\alpha _1,\cdots, \alpha _k)
=\sum _{a,b} \Psi ^V_{(A,g-1,k+2, m_{\mu})}([ K_0];\alpha _1,\cdots, \alpha _k,
\beta _a,\beta _b) \eta ^{ab}\leqno (2.20)
$$
\vskip 0.1in
The proof of composition law essentially follows from Proposition 2.9 by some
topological arguments. We
postpone it to Section 4.
So far, we are working on the covers $\overline{{\cal M}}^{\mu}_{g,k}$. To define
invariants over $\overline{{\cal M}}_{g,k}$, we introduce following classical
notion in algebraic topology to relate homology of $\overline{{\cal M}}^{\mu}_{g,k}$
to homology of $\overline{{\cal M}}_{g,k}$.
\vskip 0.1in
\noindent
{\bf Definition 2.11. }{\it Suppose that $f: M\rightarrow N$ be a continuous
map such that both $M$ and $N$ have Poincare duality. Then we define transfer
map
$$f_{!}: H_*(M)\rightarrow H_*(N)$$
by $f_{!}(\alpha)=PD_{M}\circ f^*\circ PD^{-1}_N(\alpha)$, where $PD_{M},
PD_{N}$
are Poincare duality maps. Furthermore, transfer maps compose functorially. }
\vskip 0.1in
For our case, we use ${\bf Q}$ as coefficients since Poincare duality only holds
over rational coefficient. If $M$ is a finite cover of $N$, it is easy to
observe that
$$f_*f_{!}(\alpha)=\lambda\alpha,$$
where $\lambda$ is the order of covers.
Suppose that $\lambda^{\mu}_{g,k}$ is the order of cover for
$p_{\mu}: \overline{{\cal M}}^{\mu}_{g,k}\rightarrow \overline{{\cal M}}_{g,k}$.
\vskip 0.1in
\noindent
{\bf Definition 2.12. }{\it For any $[K]\in H_*(\overline{M}_{g,k}, {\bf Q})$ and
$\{\alpha_i\}\in H_*(V, {\bf Z})$, define
$$\Psi_{(A,g,k)}([K];
\{\alpha_i\})=\frac{1}{\lambda^{\mu}_{g,k}}\Psi_{(A,g,k,\mu)}((p_{\mu})_{!}([K]); \{\alpha_i\}). \leqno(2.21)$$}
\vskip 0.1in
\noindent
{\bf Lemma 2.13.}{\it $\Psi_{(A,g,k)}([K]; \{\alpha_i\})$ is independent of
$\overline{{\cal M}}^{\mu}_{g,k}$.}
\vskip 0.1in
\noindent
{\bf Proof: } Consider the fiber product
$$\overline{{\cal M}}=\overline{{\cal M}}^{\mu}_{g,k}\times_{\overline{{\cal M}}_{g,k}}\overline{{\cal M}}^{\mu'}_{
g,k}.$$
Let
$$p^1: \overline{{\cal M}}\rightarrow \overline{{\cal M}}^{\mu}_{g,k}; \ p^2: \overline{
{\cal M}}\rightarrow \overline{{\cal M}}^{\mu'}_{g,k}$$
be projections. Then, we can pull back the universal family $\overline{{\cal U}}^{
\mu}_{g,k}, \overline{{\cal U}}^{\mu'}_{g,k}$ by $p^1, p^2$. Both are obviously the
universal family over
$\overline{{\cal M}}$. By the uniqueness of universal family, they must be the same.
In other
words, let $\overline{{\cal U}}$ be the universal family over $\overline{{\cal M}}$.
$$\overline{{\cal U}}=(p^1)^*\overline{{\cal U}}^{\mu}_{g,k}=(p^2)^*\overline{{\cal U}}^{\mu'}_{g,k}. \leqno(2.22).$$
Let
$$\phi_{\mu}: \overline{{\cal U}}^{\mu}_{g,k}\rightarrow {\bf P}^N; \phi_{\mu'}:
\overline{{\cal U}}^{\mu'}_{g,k}\rightarrow {\bf P}^{N'}$$
be projective embedding. We have a natural
embedding
$$\phi=\phi_{\mu}\times \phi_{\mu'}: \overline{{\cal U}}\rightarrow
{\bf P}^N\times{\bf P}^{N'}.
\leqno(2.23)$$
By Remark 2.6, we can use this embedding to define inhomogeneous term and
define
the analogue of $\Psi$. Furthermore, we can show that such invariant is
independent of the choice of a generic choice of inhomogeneous term. Let's
denote this invariant as $\tilde{\Psi}_{(A, g,k)}$.
We claim that
$$\tilde{\Psi}_{(A, g,k)}((p^1p_{\mu})_{!}([K]), \{\alpha_i\})=\lambda^{\mu'}_{
g,k}\Psi_{(A,g,k,\mu)}((p_{\mu})_{!}([K]), \{\alpha_i\})\leqno(2.24)$$
and
$$\tilde{\Psi}_{(A, g,k)}((p^2p_{\mu'})_{!}([K]), \{\alpha_i\})=\lambda^{\mu}_{
g,k}\Psi_{(A,g,k,\mu')}((p_{\mu'})_{!}([K]), \{\alpha_i\}).\leqno(2.25)$$
For any inhomogeneous term $\nu$ over ${\bf P}^N$, we can again view it as an
inhomogeneous term over ${\bf P}^N\times {\bf P}^{N'}$ and denote it by $\bar{\nu}$.
If $(J, \nu)$ is generic, so is $(J, \bar{\nu})$. Choose $(H, L)$ to represent
$(p^1p_{\mu})_{!}([K])$ and together with $Y$ satisfying (2.6), (2.7). Then,
$(H, p^1\circ L)$ represents $p^1_*((p^1p_{\mu})_{!}([K]))=\lambda^{\mu'}_{g,k}
(p_{\mu})_{!}(
[K])$ since the order of cover $p^1: \overline{{\cal M}}\rightarrow \overline{{\cal M}}^{
\mu}_{g,k}$ is
$\lambda^{\mu'}_{g,k}$. It is straightforward to check that
$$(\Upsilon_A\times \Xi^A_{g,k}\times F \times L)^{-1}(\Delta)=(\Upsilon_A
\times \Xi^A_{g,k}\times F \times p^1\circ L)^{-1}(\Delta).$$
Furthermore, the orientation matches. Then,
$$\begin{array}{rcl}
\tilde{\Psi}_{(A, g,k)}((p^1p_{\mu})_{!}([K]),
\{\alpha_i\})&=&\Psi_{(A,g,k,\mu)}(p_*(p^1p_{\mu})_{!}([K]), \{\alpha_i\})\\
&=&\lambda^{\mu'}_{g,k}\Psi_{(A,g,k,\mu)}((p_{\mu})_{!}([K]), \{\alpha_i\})
\end{array}.\leqno(2.26)$$
It is the same argument to show that
$$\tilde{\Psi}_{(A, g,k)}((p^2p_{\mu'})_{!}([K]), \{\alpha_i\})=\lambda^{\mu}_{
g,k}
\Psi_{(A,g,k,\mu')}((p_{\mu'})_{!}([K]), \{\alpha_i\}).\leqno(2.27)$$
On the other hand, $p^1p_{\mu}=p^2p_{\mu'}$. Therefore,
$$\frac{1}{\lambda^{\mu}_{g,k}}\Psi_{(A,g,k,\mu)}((p_{\mu})_{!}([K]),
\{\alpha_i\})
=\frac{1}{\lambda^{\mu'}_{g,k}}\Psi_{(A,g,k,\mu')}((p_{\mu'})_{!}([K]),
\{\alpha_i\}).\leqno(2.28)$$
This finishes the proof.
\vskip 0.1in
\noindent
{\bf Proposition 2.14. }{\it
\vskip 0.1in
\noindent
(1) $\Psi^V_{(A, g, k)}(K, \alpha_1, \dots, \alpha_k)$ is
a symplectic invariant.
\vskip 0.1in
\noindent
(2) $\Psi^V_{(A, g, k)}(K, \alpha_1, \dots, \alpha_k)$ is independent of
semi-positive deformations of $\omega$.
\vskip 0.1in
\noindent
(3) $\Psi^V_{(A, g, k)}=0$, if either $\omega(A)\leq 0$ or $C_1(V)(A)+(3-n)(g
-1)<0$, in particular, $\Psi_{(0,g,k)}=0$ for any $g\geq 2$ and $n\geq 4$.
\vskip 0.1in
\noindent
(4) $\Psi^V_{(A, g, k)}=0$ is multilinear and supersymmetry on $H_*(V, {\bf Z})^k$
with respect to the ${\bf Z}_2$-grading by even and odd degrees.}
\vskip 0.1in
The proof follows from the Proposition 2.4,
2.7 and the Definition 2.12.
\vskip 0.1in
\noindent
{\bf Proposition 2.15. }{it Suppose that $(g,k)
\neq (0,3),(1,1)$.
\vskip 0.1in
\noindent
(1) For any $\alpha _1, \cdots , \alpha _{k-1}$ in $H_*(V, {\bf Z})$,
we have}
$$\Psi ^V_{(A,g,k)}(K; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~
\Psi ^V_{(A,g,k-1)}([\pi (K)]; \alpha _1, \cdots,\alpha _{k-1})
\leqno (2.29)$$
\vskip 0.1in
(2) Let $\alpha _k$ be in $H_{2n-2}(V, {\bf Z})$, then
$$\Psi ^V_{(A,g,k)}([\pi^{-1}(K)]; \alpha _1,
\cdots,\alpha _{k-1}, \alpha _k)~=~\alpha^* _k (A)
\Psi ^V_{(A,g,k-1)}(K; \alpha _1, \cdots,\alpha _{k-1})
\leqno (2.30)$$
where $ \alpha^* _k$ is the Poincare dual of $\alpha _k$.
\vskip 0.1in
\noindent
{\bf Proof: } Notes that $\overline{{\cal M}}^{\mu}_{g,
k+1}=\pi^*\overline{{\cal M}}^{\mu}_{g, k}$.
Geometrically, $(p_{\mu})_{!}([K])$ is represented by $(p_{\mu})^{-1}(K)$.
By the construction of ${\cal M}^{\mu}_{g, k+1}$,
$$\pi_{\mu}(p_{\mu})^{-1}(K)=(p_{\mu})^{-1}(\pi_{\mu}(K)). \leqno(2.31)$$
Hence,
$$(\pi_{\mu})_*(p_{\mu})_{!}([K])=(p_{\mu})_{!}(\pi_{\mu})_*([K]).
\leqno(2.32)$$
Then, (1) follows from Proposition 2.8 and (2.32).
For (2), $[\pi^{-1}((p_{\mu})_{!}([K]))]=\pi_{!}((p_{\mu})_{!}([K]))$. By the
natality of transfer map,
$$(p_{\mu})_{!}\pi_{!}([K])=(p_{\mu}\pi)_{!}([K])=(\pi_{\mu}p_{\mu})_{!}([K])=
(\pi_{\mu})_{!}(p_{\mu})_{!}([K]). \leqno(2.33)$$
Then, (2) follows from the Proposition 2.8(2) and (2.33).
\vskip 0.1in
\noindent
{\bf Theorem 2.16 (Composition Law).} {\it Let $[K_i] \in H_*(\overline
{\cal M}_{g_i,
k_i+1}, {\bf Q})$ $(i=1,2)$ and $[K_0] \in H_*(\overline {\cal M}_{g-1,
k +2}, {\bf Q})$. For any $\alpha _1,\cdots,\alpha _k$ in $H_*(V,{\bf Z})$.
Then we have}
$$\begin{array}{rl}
&\Psi ^V_{(A,g,k)}(\theta _{S*}[K_1\times K_2];\{\alpha _i\})\\
=~& \sum \limits _{A=A_1+A_2} \sum \limits_{a,b}
\Psi ^V_{(A_1,g_1,k_1+1)}([K_1];\{\alpha _{i}\}_{i\le k}, \beta _a)
\eta ^{ab}
\Psi ^V_{(A_2,g_2,k_2+1)}([K_2];\beta _b,
\{\alpha _{j}\}_{j> k}) \\
\end{array}
\leqno (2.34)
$$
$$
\Psi ^V_{(A,g,k)}(\mu_*[K_0];\alpha _1,\cdots, \alpha _k)
=\sum _{a,b} \Psi ^V_{(A,g-1,k+2)}([ K_0];\alpha _1,\cdots, \alpha _k,
\beta _a,\beta _b) \eta ^{ab}\leqno (2.35)
$$
\vskip 0.1in
\noindent
{\bf Proof: } Let
$$p^{\mu}_{g, k}: \overline{{\cal M}}^{\mu}_{g,k}\rightarrow \overline{{\cal M}}_{g,k}, \
p^{m_{\mu}}_{g-1, k+2}: \overline{{\cal M}}^{m_{\mu}}_{g-1. k+2}\rightarrow
\overline{{\cal M}}_{g-1,
k+2}. \leqno(2.36)$$
By the Proposition 2.10,
$$\begin{array}{rl}
&\Psi ^V_{(A,g,k)}(\theta _{S*}[K_1\times K_2];\{\alpha _i\})\\
=~&\frac{1}{\lambda^{\mu}_{g,k}}\Psi ^V_{(A,g,k, \mu)}(\theta _{S*}(p^{\mu}_{g,
k})_{!}[K_1\times K_2];\{\alpha _i\})\\
=~&\frac{1}{\lambda^{\mu}_{g_1, k_1+1}\lambda^{\mu}_{g_2, k_2+1}}\Psi
^V_{(A,g,k, \mu)}(\theta _{S*}(p^{\mu}_{
g_1, k_1+1})_{!}([K_1])\times (p^{\mu}_{g_2, k_2+1})_{!}([K_2]);\{\alpha
_i\})\\
=~&\sum \limits _{A=A_1+A_2} \sum \limits_{a,b}
\frac{1}{\lambda^{\mu}_{g_1, k_1+1}}\Psi ^V_{(A_1,g_1,k_1+1,
\mu)}((p^{\mu}_{g_1,k_1+1})_{!}[K_1];\{\alpha _{i}\}_{i\le k}, \beta _a) \\
&\eta ^{ab} \frac{1}{\lambda^{\mu}_{g_2, k_2+1}}
\Psi ^V_{(A_2,g_2,k_2+1, \mu)}(p^{\mu}_{g_2, k_2+1})_{!}[K_2];\beta _b,
\{\alpha _{j}\}_{j> k}) \\
=~&\sum \limits _{A=A_1+A_2} \sum \limits_{a,b}
\Psi ^V_{(A_1,g_1,k_1+1)}([K_1];\{\alpha _{i}\}_{i\le k}, \beta _a)
\eta ^{ab}
\Psi ^V_{(A_2,g_2,k_2+1)}([K_2];\beta _b,
\{\alpha _{j}\}_{j> k})
\end{array}
\leqno(2.37)
$$
It is the same argument for (2).
\vskip 0.1in
\noindent
{\bf Remark 2.17: } The mixed invariant $\Phi _{(A,\omega ,g)}$ in [RT] can
be identified with certain $\Psi$ by choosing appropriate cycles $[K]$.
More precisely, for any $k,l\ge 0$ with $2 g + k \ge 3$, put
$K_{k,l}$ to be the closure of the cycle
$K^0_{k,l}$ in $\overline {\cal M}_{g,k+l}$, where
$K^0_{k,l}$ is the set of all
$(\Sigma , x_1,\cdots, x_{k+l})$ in ${\cal M}_{g,k+l}$
with $(\Sigma , x_1,\cdots,x_k)$ being a fixed point in ${\cal M}_{g,k}$.
Then
$$\Psi ^V_{(A,g,k+l)}(K_{k,l}; \alpha_1,\cdots ,\alpha _{k+l})
=\Phi_{(A,\omega ,g)}(\alpha _1,\cdots, \alpha _k |
\alpha _{k+1},\cdots ,\alpha _{k+l})\leqno(2.38)$$
It follows from the Proposition 2.5 that
$\Phi_{(A,\omega ,g)}(\alpha _1,\cdots |\cdots ,\alpha _{k+l})$ is $0$ if
$\dim (\alpha _{k+l}) > 2n -2$ (cf. [RT]).
\vskip 0.1in
Recall that $\Psi$ is only defined for so call ``generic $(J, \nu)\in {\cal H}$''.
Following from \cite{RT}, we can relax this genericity condition as follows:
\vskip 0.1in
\noindent
{\bf Definition 2.18. }{\it We call $(J, \nu)$ to be $A$-good if the following
two conditions are satisfied.
\vskip 0.1in
\noindent
(1). The set $\{ f\in {\cal M}_A(g,k, J,\nu)_I| Coker(Jdf \oplus D_f)\neq 0\}$ is
of real codimension 2, where $D_f$ is the linearization of the inhomogeneous
Cauchy-Riemann equation for $(J, \nu)$-maps at $f$.
\vskip 0.1in
\noindent
(2). $\overline{{\cal M}}^r_A(g,k,J,\nu)_I-{\cal M}_A(g,k, J, \nu)_I$ is of real
codimension
2.}
\vskip 0.in
One can see from the construction that $\Psi$ is well defined if $(J,\nu)$ is
$A$-good.
\vskip 0.1in
\noindent
{\bf Remark 2.19 (Relation of $\Psi$ to enumerative function):} One of main
applications of the composition law of
the genus-0 GW-invariant $\Psi^V_{(A,0,k)}$ is
to compute the enumerative invariants of rational curves in
complex homogeneous
spaces. In \cite{RT}, Lemma 10.1, the authors proved the equivalence
of $\Psi^V_{(A,0,k)}$ with the enumerative function of
rational curves on complex Grassmannian manifolds.
More precisely, we showed
that for Grassmannian manifolds,
the integrable complex structure $J_0$ with zero perturbation term satisfies
the condition of Proposition 2.3, i.e., $(J_0, 0)\in {\cal H}$.
Therefore, one doesn't
need to deform the integrable complex structure or add any perturbation terms.
The integrable complex structure is already $A$-good for calculating the
invariant $\Psi^V_{(A,0,k)}$. Then, by the definition, $\Psi^V_{(A,0,k)}$ is
an enumerative function.
The same has been proved by
Jun Li and the second author in [LT] for any complex
homogeneous manifolds (also see \cite{CF} for an alternative proof
for Flag manifolds).
In contrast to the genus 0 case, $\Psi^V_{(A,g,k)}$($g\geq 1$) is not the same
as the enumerative function even for the projective plane ${\bf P}^2$.
For example, a
simple computation through the composition law will yield the mixed
invariant
$$\Psi^{{\bf P}^2}_{([L], 1, 3)}([K_{3,0}]; [L], [L], [L])=3, \Psi^{{\bf P}^2}_{(3[L],
1, 9)}([K_{1,8}]; [L], [pt], \cdots, [pt])=27,$$
where $L$ is a line in ${\bf P}^2$ and $pt$ is a point.
The first number is supposed to represent the
number of degree 1 elliptic curves with fixed $j$-invariant mapping three
marked points to three distinct lines. But it is well-known that there is no
degree 1 elliptic curves at all in ${\bf P}^2$. The second number is supposed to
represent the number of degree three elliptic curves with fixed
$j$-invariant passing through 8 points. It is known in classical algebraic
geometry that such a number should be 12. What happen was that for the
integrable complex structure $J_0$ on ${\bf P}^2$, the boundary of the Gromov-
Uhlenbeck compactification $\overline{{\cal M}}_A(1,k, J_0, 0)-{\cal M}_A(1,k, J_0, 0)_I$
has a component whose dimension is larger than the dimension of
${\cal M}_A(1,k,J_0,0)$ itself.
Such a component consists of the maps from the union of
an elliptic curve and
a rational curve to ${\bf P}^2$, which map the elliptic curve to a point.
The effect of considering the inhomogeneous Cauchy Riemann equation
$\bar{\partial}_J f=\nu$ instead of the
homogeneous Cauchy Riemann equation is to
perturb away all those maps. In the process, we creat finitely many
$(J, \nu)$-maps, which provides the correct account of the contribution of
the component described above. Only adding those contributions, our invariants
will satisfy the composition law, while
the classical enumerative invariants
do not.
In fact, the composition law of the mixed invariant we proved in \cite{RT}
computes
all the mixed invariants of any genus for ${\bf P}^{N}$ and many other Fano
manifolds.
It is still a
major problem how to use it to compute the enumerative invariants. Recall that
to define the mixed invariants, we fix the complex structure on the Riemann
surfaces. If we allow the complex structure of Riemann surfaces to vary, it is
not clear how the composition law
will even help to compute the enumerative invariants.
\section{Compactification and Transversality}
In this section, we discuss the structures of the
moduli space ${\cal M}_A(\mu, g,k, J, \nu)$ and the certain quotient of its
Gromov-Uhlenbeck compactifications
$\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)$. The problems
we will discuss here are smoothness, the orientation of
${\cal M}_A(\mu, g,k, J,
\nu)$ and the stratification of $\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)$.
By the natural
of those questions, this section is rather technical. For the readers who only
wish to get a sense of big picture, one can skip over this section. On the
other hand, if reader wish to have a good understanding about the results in
this paper, the properties we discuss in this section are crucial or the
proof of all the results in Section 2, which will be provided in the Section 4.
This section is roughly divided as two parts. In the first part, we prove the
smoothness of ${\cal M}_A(\mu, g,k, J, \nu)$ and construct its canonical orientation.
The idea of their proof is quite standard.
For the smoothness, the basic
tool is the Sard-Small Transversality Theorem.
We refer the readers to \cite{M1},
\cite{R3}, \cite{M2} in the case of the Cauchy-Riemann equation.
In both \cite{R3} and \cite{M2}, the argument relies on
a cumbersome norm on the space of tamed
almost complex structures defined by
Floer. Because the case (for the
inhomogeneous Cauchy-Riemann equation) here didn't appear
in the literature, we will include an outlined proof. For the
orientation, the genus 0 case was also due to McDuff \cite{M1}. But the
treatment we follow is that of the first author \cite{R}
(see also 4.12 of \cite{RT}),
which was in the spirit of Donaldson's treatment
of the orientation problem
in the gauge theory. The
second part will be devoted to prove Proposition 2.3,
which is similar to
Section 3, 4, \cite{RT}.
The idea of applying the Sard-Smale Transversality Theorem
to the moduli problem was due to Freed-Uhlenbeck \cite{FU}. Recall that the
Sards-Smale Theorem says that if $X, Y$ are Banach manifolds and
${\cal F}': X\rightarrow Y$ is
a Fredholm map of index $k$, then the set $Y_{reg}$ of regular values of ${\cal F}$
is of the Baire second category, provided that ${\cal F}$ is sufficiently
differentiable. Recall that
$y\in Y$ is called a regular value, if the derivative
$D{\cal F}(x): T_x X\rightarrow T_y Y$ is surjective at any $x$ with
${\cal F}(x)=y$. It
then follows from the Implicit Function Theorem that ${\cal F}^{-1}(y)$ is
$k$-dimensional manifold for every $y\in Y_{reg}$.
One obvious problem is that ${\cal M}^{\mu}_{g,k}$ may not be smooth.
But it can be stratified by smooth manifolds. Notes that ${\cal M}_{g,k}$
has a stratification parameterized by
the automorphism group of Riemann surfaces. Namely, one can write
$${\cal M}_{g,k}=\sum_{\alpha\in I} {\bf T}^{\kappa}_{g,k}, \leqno(3.1)$$
where each smooth strata ${\bf T}^{\kappa}_{g,k}$ consists of
the stable Riemann surfaces of a fixed
automorphism group $\kappa$. Without the loss of generality, we can assume that
$${\cal M}^{\mu}_{g,k}=\sum_{\alpha\in I} {\bf T}^{\mu, \kappa}_{g,k}, \leqno(3.2)$$
where ${\bf T}^{\mu, \kappa}_{g,k}=p^{-1}_{\mu}({\bf T}^{\kappa}_{g,k})$ is smooth.
Using the same arguments as in the smooth case, we will
establish the transversality theory for each stratum
${\bf T}^{\mu,\kappa}_{g,k}$. It is rather straightforward.
The precise structure of ${\bf T}^{\mu,\kappa}_{g,k}$ is not needed. One
only has to know that each ${\bf T}^{\mu, \kappa}_{g,k}$ is smooth.
Let ${\cal M}_A(\mu, g,k, J,\nu)_\kappa$ consist of $(J, \nu)$-maps $f$ such
that the domain of $f$ has automorphism group $\kappa$.
We shall prove that
\vskip 0.1in
\noindent
{\bf Theorem 3.1. }{\it There is a set ${\cal H}_{reg}$ of
Baire second category among
all the smooth pairs $(J, \nu)$ such that for any $(J, \nu)$,
${\cal M}_A(\mu, g,k, J, \nu)_\kappa$ is a smooth manifold of dimension $2c_1(V)(A)
-2n ( g-1)+\dim {\bf T}^{\mu,\kappa}_{g,k}$.}
\vskip 0.1in
Fix a smooth topological surface $\Sigma_g$ of genus $g$. Our basic
topological object is
$$Map_A(\Sigma_g, V)=\{f:\Sigma_g\rightarrow V \mbox{ such that $f$ is smooth
and } f_*[\Sigma_g]=A\}. $$
To apply the Sard-Smale theorem, we need to put some Sobolev norm on
$Map_A(\Sigma_g, V)$, so that it has a structure of Banach manifold.
To specify a Sobolev norm, we choose a smooth family of metrics on $\Sigma_g$,
parameterized by the elements of ${\bf T}^{\mu, \kappa}_{g,k}$. For example, one can
choose a projective embedding
$$\phi_{\mu}: \overline{{\cal U}}^{\mu}_{g,k}\rightarrow {\bf P}^N$$
in section 2 and consider the restriction of Fubini-Study metric on
$\phi_{\mu}\pi^{-1}_{{\cal M}}(j)$ for each $j\in {\bf T}^{\mu, \kappa}_{g,k}$.
Then for each $j\in {\bf T}^{\mu, \kappa}_{g,k}$, $
j$ defines a Sobolev
$L^p_{m, j}$-norm on $Map_A(\Sigma_g, V).$ Its completion under this norm is a
smooth
Banach manifold if $pm>2$. We shall also use $j$ to denote the complex
structure of the underlying Riemann surface. When $j$ varies in ${\bf T}^{\mu,
\kappa}_{g,k}$,
$$\chi^{p,m}_{(A,\kappa,g,k)}=\bigcup_{j\in {\bf T}^{\mu,\kappa}_{g,k}}L^p_{m, j}
(Map_A(\Sigma_g, V))\times \{j\} \leqno(3.3)$$
is a smooth Banach manifold as well since ${\bf T}^{\mu, \kappa}$ is smooth.
Obviously,
there
is a map $\chi^{p,m}_{(A,\kappa,g,k)}\rightarrow {\bf T}^{\mu, \kappa}_{g,k}$. Let
${\cal H}^l$ be the completion of ${\cal H}$
the space of all smooth pairs $(J, \nu)$ under
$C^{l}$-topology. Then, ${\cal H}^l$ is
a smooth Banach manifold. Consider the
{\em universal moduli space}
$${\cal M}^l_A(\kappa,g,k)=\{(f,j, J, \nu)\in \chi^{m,p}_{(A,\kappa,g,k)}\times {\cal H}^l;
\bar{\partial}_Jf(x)=\nu(
\phi(x), f(x))\}.\leqno(3.4)$$
When $p>2, 1\leq m\leq l$, by the elliptic regularity, ${\cal M}_A^l(\kappa,g,k)$ is
independent of $m, p$.
\vskip 0.1in
\noindent
{\bf Proposition 3.2. }{\it For every $A\in H_2(V, {\bf Z})$ and $g\geq 0, l\geq 1$,
the universal moduli space ${\cal M}_A^l(\kappa,g,k)$ is a smooth Banach manifold.}
\vskip 0.1in
\noindent
{\bf Proof:} There is an infinite dimensional vector bundle
$${\cal E}^{m-1,p}_{(f,j,J, \nu)}
\rightarrow \chi^{m,p}_{(A,\kappa,g,k)}\times {\cal H}^l, \leqno(3.5)$$
where the fiber ${\cal E}^{m-1,p}_{(f,j,J, \nu)}=W^{m-1,p}(\Lambda^{0,1}_{
j}T^*\Sigma_g\otimes_J f^*TV).$ The perturbed holomorphic equation
defines a section of this bundle by
$${\cal F}: \chi^{m,p}_{(A,\kappa,g,k)}\times
{\cal H}^l\rightarrow {\cal E}^{m-1,p}_{(A,\kappa ,g,k)},\
F(f,j, J, \nu)(x)=\bar{
\partial}_Jf(x)-\nu(\phi(x), f(x)).\leqno(3.6)$$
Notice that the definition of $\bar{\partial}_J$ depends on the complex
structure $j$ on $\Sigma_g$. Then, it is enough to show that
${\cal F}$ is transverse to the zero section. Suppose $\Sigma_j=(\Sigma, x_1, x_2,
\dots, x_k).$ Let $T\Sigma_j=T\Sigma\otimes^k_{i=1} {\cal O}(-x_i)$. Then,
$T_j {\bf T}^{\mu, \kappa}_{g,k}= H^{0,1}_{(\kappa,j)}(T\Sigma_j)$-space of $\kappa$
invariant (0,1)-forms. Notice that one can also identity
$$H^{0,1}_{(\kappa,j)}(T\Sigma_j)=(H^{0,1}_{(\kappa,j)}(T\Sigma)\oplus^k_{i=1}
T_{x_i}\Sigma)^{\kappa},$$
where $(.)^{\kappa}$ means the $\kappa$-invariant subspace.
Let ${\cal F}(f, j, J, \nu)=0$. We have
$$\begin{array}{rl}
&T_{(f, j)}\chi^{m,p}_{(A,\kappa,g,k)}=W^{m,p}(\Lambda^0f^*TV)\oplus H^{0,1}_{(
\kappa,j)}(T\Sigma_j);\\
& T_{(J, \nu)}{\cal H}^l=C^l(End(TV, J))\oplus C^l(\overline{
Hom}_J(T{\bf P}^N, TV)),\\
\end{array}
\leqno(3.7)$$
where $End(TV, J)=\{Y: TV\rightarrow TV; YJ+JY=0\}$.
Furthermore, $\overline{Hom}_J(T{\bf P}^N, TV)$ is
the space of anti-complex linear
homomorphism with respect to the complex structure.
There is a natural identification
$$\overline{Hom}_J(T{\bf P}^N, TV)|_{\Gamma_f}=\Omega^{0,1}_J(f^*TV),\leqno(3.8)$$
where $\Gamma_f\subset \Sigma_g\times V\subset {\bf P}^N\times V$ is the graph of
$f$.
Now, we shall show the surjectivity of the differential
$$\begin{array}{rl}
&D{\cal F}(f, j, J, \nu):W^{m,p}(\Lambda^0f^*TV)\oplus H^{0,1}_{(\kappa, j)}
(T\Sigma_j)\oplus C^l(End(TV, J))\oplus C^l(\overline{
Hom}_J(T {\bf P}^{N}, TV))\\
&~~~~\longrightarrow W^{m-1,p}(\Lambda^{0,1}_{j}T^*\Sigma\otimes_J f^*TV).\\
\end{array}
\leqno(3.9)$$
An easy computation yields
$$D{\cal F}(f, j, J, \nu)(\xi, s,Y,X)=D_{f}\xi+J\circ df\circ
s+ f^*Y\circ df
\circ j-X|_{\Gamma_f}, \leqno(3.10)$$
where
$$D_f: W^{m,p}(\Lambda^0f^*TV)\rightarrow W^{m-1,p}(\Lambda^{0,1}_{j, J}
(f^*TV)) \leqno(3.11) $$
is the linearization of the Cauchy-Riemann equation at $f$.
By the elliptic regularity theory, a $(J, \nu)$-map $f$ is in $W^{l+1, q}$ for
any $q > 0$ if
$(J, \nu)$ is in $C^l$. It follows that $D_f$ in (3.11) is well-defined.
Moreover, it follows that the cokernel of $D_f$ is contained
in $C^l(\Lambda^{0,1}_{j, J}(f^*TV))$. Since $D_f$ is elliptic,
its cokernel is of finite dimension.
However, the map
$$X|_{\Gamma_f}: C^l(\overline{Hom}_J(T{\bf P}^N, TV)\rightarrow C^l(
\Lambda^{0,1}_{j, J}(f^*TV))\leqno(3.12)$$
is surjective (here we also use the fact that $f$ is in the space
$C^{l}$). Therefore, by (3.10), $D{\cal F}(f,j, J, \nu)$ is surjective.
\vskip 0.1in
By the Implicit Function Theorem, we conclude that the
universal moduli space ${\cal M}^l_A(\kappa, g)$ is a smooth Banach manifold.
\vskip 0.1in
\noindent
{\bf Proof of Theorem 3.1:} Based on Proposition 3.2, the proof is just
a standard application of the Sard-Smale Theorem.
For the reader's convenience, we outline the
arguments here.
Consider the projection
$$\pi: {\cal M}^l_A(\kappa, g, k)\rightarrow {\cal H}^l\leqno(3.13)$$
as a map between the Banach manifolds. The tangent space $T_{(f,j, J, \nu)}
{\cal M}^l_A(\kappa, g,k)$ consists of $(\xi, s, Y, X)$ such that
$$D_{f}\xi+J\circ df\circ s+f^*Y\circ
df\circ j-X|_{\Gamma_f}=0. \leqno(3.14)$$
The derivative
$$d\pi: T_{(f,j, J, \nu)}{\cal M}^l_A(\mu, g)\rightarrow T_{(J,\nu)}{\cal H}^l$$
is just the projection to $(Y, X)$ factors. One can show that
$d\pi$ is a Fredholm operator whose kernel is isomorphic to the kernel of
$D_f\oplus J\circ df$ and has the same index as that of
$D_f\oplus J\circ df$, where
$$J\circ df: H^{0,1}_{(\kappa, j)}( T\Sigma_j)\rightarrow W^{m-1,
p}(\Lambda^{0,1}_{j, J}(f^*TV)). \leqno(3.15)$$
Hence, the operator $d\pi$ is onto precisely when
$D_f\oplus J\circ df$ is onto for any $(J, \nu)$-map $(f, j)$
in ${\cal M}^l_A(\mu, g,k,J, \nu)_{\kappa}=
\pi^{-1}((J, \nu)$. In other words,
$${\cal H}^l_{reg}=\{(J, \nu)\in{\cal H}^l; D_f\oplus J\circ df
\mbox{ is onto for all } (f, j)
\in {\cal M}^l_A(\mu, g,k,J, \nu)_\kappa\}\leqno(3.16)$$
is precisely the set of the regular values of $\pi$. By the
Sard-Smale
Theorem, this set is of the second category. Thus we have proved that
${\cal H}^l_{reg}$ is dense in ${\cal H}^l$ with respect $C^l$-topology.
Then one can easily deduce that ${\cal H}_{reg}$ is of the second category in
${\cal H}$ with respect to $C^{\infty}$ topology.
Let ${\bf T}^{\mu,\kappa}_{g,k}=\bigcup_{K=1}^{\infty} N_K$, where
$N_K$ is compact and
$N_K\subset N_{K+1}$.
Consider the set ${\cal H}_{reg, K}\subset {\cal H}$ of all smooth $(J, \nu)$ with the
property that the operator $D_f\oplus J\circ df$ is onto for
any $(f, j)$ satisfying: $||df||_
{L^{\infty}}< K$ and $j\in N_K$. Clearly,
$${\cal H}_{reg}=\bigcap_{K>0}{\cal H}_{reg, K}.\leqno(3.17)$$
Similarly, we can define ${\cal H}_{reg, K}^l$.
We claim that ${\cal H}_{reg, K}$ is open and dense in ${\cal H}$ with respect to the
$C^{\infty}-topology$.
The openness is clear. It remains
to prove that ${\cal H}_{reg, K}$ is dense in ${\cal H}$ with respect to
$C^{\infty}$-topology. Note that ${\cal H}_{reg, K}={\cal H}^l_{reg, K}\cap {\cal H}$.
Then ${\cal H}^l_{reg, K}$ is open in ${\cal H}^l$ with respect to
$C^l$-topology. Since ${\cal H}^l_{reg}\subset {\cal H}^l_{
reg, K}$ and ${\cal H}^l_{reg}$ is dense in ${\cal H}^l$
with respect to
$C^l$-topology, so is ${\cal H}^l_{reg, K}$.
It follows that ${\cal H}_{reg, K}$ is dense in ${\cal H}$ with respect
to $C^{\infty}$-topology. Notice that
${\cal H}_{reg}$ is an intersection of countable
open dense subsets, so it is of
second category.
The dimension formula follows from the Riemann-Roch Theorem.
\vskip 0.1in
\noindent
{\bf Theorem 3.3. }{\it For any $(J, \nu), (J',\nu')\in {\cal H}_{reg}$, there is
a second category set of paths ${\cal H}_{((J,\nu),(J', \nu'))}$ connecting
$(J, \nu), (J',\nu')$ among the set of all such smooth paths such that for
any path $(J_t, \nu_t)\in {\cal H}_{((J,\nu),(J', \nu'))}$
$${\cal M}_A(\mu, g,k,(J_t,\nu_t))_\kappa=\bigcup_{t\in[0,1]}
{\cal M}_A(\mu, g,k,J_t,\nu_t)_\kappa\times
\{t\}\leqno(3.18)$$
is a smooth cobordism.}
\vskip 0.1in
The proof is identical to that of 3.1. We omit it.
\vskip 0.1in
\noindent
{\bf Remark 3.4:} In the case of homogeneous Cauchy-Riemann equation ($\nu=0$),
one can use the Teichmuller space ${\bf T}_{g,k}$ ( which is smooth) in the place of
${\bf T}^{\mu, \kappa}_{g,k}$. Moreover, there is no need to consider finite covers.
Let ${\bf T}^*_A( g,k,J,0)$ be the set of
$(f, \lambda)\in Map_A(\Sigma_g, V)\times {\bf T}_{g,k} $
such that $f$ is a $(J,0)$ map for the complex
structure induced by $\lambda$ but not a multiple cover of another $(J,0)$
map. The same argument implies that there is ${\cal H}_{reg}$ of second
category among all the tamed almost complex structure such that for any $J\in
{\cal H}_{reg}$, ${\bf T}^*_A( g,k,J,0)$ is smooth. The mapping class group $G_g$ acts
freely on ${\bf T}^*_A(g,k,J,0)$. Hence,
$${\bf T}^*_A(g,k,J,0)/G_g\subset {\cal M}_A( g,k,J,0) \leqno(3.19)$$
is smooth. In our case, the inhomogeneous Cauchy-Riemann equation is not
preserved under the action of mapping class group. Therefore, we have to
consider the smoothness for each strata of ${\cal M}_g$. On the other hand, our
inhomogeneous equation can handle the multiple covered map, which can not be
handled by the homogeneous equation except dimension 4 \cite{R3}.
\vskip 0.1in
Next, we construct the canonical orientation of ${\cal M}_A(\mu, g,k, J,\nu)_\kappa$
and ${\cal M}_A(\mu, g,k, (J_t,\nu_t))_\kappa$.
The construction is identical to that
in \cite{R} (3.3.1) and \cite{RT} (4.12).
\vskip 0.1in
\noindent
{\bf Theorem 3.5. }{\it There is a canonical orientation over ${\cal M}_A(\mu, g,k,J,
\nu)_\kappa$ and ${\cal M}_A(\mu, g,k,(J_t,\nu_t))_\kappa$.}
\vskip 0.1in
\noindent
{\bf Proof:}
Recall that the linearization of ${\cal F}$ at $(f,j)\in {\cal M}_A(\mu, g,k,
J,\nu)_\kappa$ is
$$D_f\oplus J\cdot df: \Omega^0(f^*TV)\times H^{0,1}_{\kappa,j}(T\Sigma_j)
\rightarrow \Omega^{0,1}(f^*TV).$$
The tangent space $T_{f,j} {\cal M}_A(\mu, g,k, J,\nu)_\kappa=Ker (D_f\oplus J\cdot
df).$ Its determinant is
$$det(T{\cal M}_A(\mu, g,k, J,\nu)_\kappa)=det (D_f\oplus J\cdot df),$$
which is defined over
$Map_A(\Sigma, V)\times {\bf T}^{\mu, \kappa}_{g,k}$. As usual, an orientation of
${\cal M}_A(\mu, g,k, J,\nu)_\kappa$ is just a nowhere
vanishing section of $det(T{\cal M}_A(\mu, g,k, J,\nu)_\kappa)$ up to
multiplication by positive
functions. We shall omit `` up to multiplication by positive functions'' if
there is no
confusion. Therefore, to construct a
canonical orientation of ${\cal M}_A(\mu, g,k, J,\nu)_\kappa$,
it is enough to construct a
canonical section of $det (D_f\oplus J\cdot df)$ over the
whole $Map_A(\Sigma, V)\times
{\bf T}_{\kappa, g}$. We define
$$D_f^J = {1\over 2} ( D_f - J\cdot D_f \cdot J ) \leqno(3.20)$$
Clearly, it is $J$-linear. Moreover, we have
$$D_f=D_f^J + Z_f, \leqno(3.21)$$
where $Z_f$ is
the zero order term. Let
$$D_{f,t}=D^J_f +tZ_f.\leqno(3.22)$$
Then, $det(D_{f, t}\oplus J \cdot df)$ is isomorphic to $det(D_{f, 0}\oplus J
\cdot df)$.
Hence, $det(D_f\oplus J\cdot df)$ is
isomorphic to $det(D^J_f\oplus J\cdot df)$.
On the other hand, both $ker (
D^J_f\oplus J\cdot df)$ and
$coker (D^J_f\oplus J\cdot df)$ are complex vector spaces.
Therefore, there is a canonical
section of the determinant line bundle $det(D_f\oplus id)$
corresponding this
complex structure.
Similarly, one can construct a canonical orientation on
${\cal M}_A(\mu, g,k, (J_t,\nu_t))_{\kappa}$.
\vskip 0.1in
As the
oriented manifolds, we have
$${\cal M}_A(\mu, g,k, J_0, \nu_0)_\kappa \times \{0\}\bigcup {\cal M}_A(\mu, g,k,
J_1,\nu_1)_\kappa
\times \{1\}={\cal M}_A(\mu, g,k, J_0, \nu_0)_\kappa\bigcup -{\cal M}_A(\mu,
g,k,J_1,\nu_1)_\kappa,$$
where ``-'' means the opposite orientation. Let
$${\cal M}_A(\mu, g,k, J,\nu)=\bigcup_\kappa {\cal M}_A(\mu, g,k, J,\nu)_\kappa.$$
In the second half of this section, we focus on
the compactification of ${{\cal M}}_A(\mu, g,k,J,\nu)$.
\vskip 0.1in
\noindent
{\bf Definition 3.6 ([PW], [Ye], [Ko]). }{\it Let $(\Sigma, \{x_i\})$ be a
stable Riemann surface.
A stable map (associated with $(\Sigma, \{x_i\})$) is an equivalence class
of continuous maps $f$ from $\Sigma'$ to
$V$ which are smooth at smooth points of $\Sigma'$,
where the domain $\Sigma'$ is obtained by joining chains of ${\bf P}^1$'s
at some double points of $\Sigma$ to separate the two components, and then
attaching some
trees of ${\bf P}^1$'s. We call components of $\Sigma$ {\em principal components}
and others {\em bubble components}. Furthermore,
\begin{description}
\item[(1)] If we attach a tree of ${\bf P}^1$ at a marked point $x_i$, then $x_i$
will
be replaced by a point different from intersection points on some component of
the tree. Otherwise, the marked
points do not change.
\item[(2)] The singularities of $\Sigma'$ are normal crossing and there are at
most two components intersecting at one point.
\item[(3)] If the restriction of $f$ on a bubble component is constant, then
it has
at least three special points (intersection points or marked points). We call
this component {\em a ghost bubble} \cite{PW}.
\item[(4)] For each principal component, the restriction of
$f$ is a $(J,\nu)$-map.
\item[(5)] The restriction of $f$ to
each bubble is $J$-holomorphic.
\end{description}
Two such maps are equivalent if one is the composition of the other with
an automorphism of bubble components fixing
the special points.}
\vskip 0.1in
Evidently, the equivalence relation is trivial unless some bubble component
has one or two special points. The restriction of $f$ to each component
carries a homology class. We shall use $[f]$ to denote the summation of
all those homology classes.
\vskip 0.1in
\noindent
{\bf Remark 3.7: } The terminology {\em stable maps} was first used by
Kontsevich and Manin in
\cite{KM}.
It had appeared before
in Parker-Wolfson-Ye's proof of Gromov-Uhlenbeck compactness
theorem under the name {\it cusp curves} \cite{PW}, \cite{Ye}. Later, it was
introduced to algebraic geometry by Kontsevich and Manin and known more
commonly as stable maps. Here, we follow their terminology.
\vskip 0.1in
Then Theorem 3.1 of \cite{RT} (see also \cite{PW} and \cite{Ye}) can be
restated as follows:
\vskip 0.1in
\noindent
{\bf Theorem 3.8. }{\it Let $f_m\in {\cal M}_A(\mu, g,k, J, \nu)$. Suppose that the
domain $(\Sigma_m, \{x^m_i\})$ of $f$ converges to a stable Riemann surface
$(\Sigma, \{x_i\})$ in the sense of Deligne-Mumford. Then, there is a
subsequence $\{f_{m_t}\}$ ``weakly converging'' to a stable map $f$(
associated with $(\Sigma, \{x_i\})$) such that $[f]=A$.
Here, by the weak convergence, we mean that the
image of $f_{m_t}$
converges to the image of $f$ in the Hausdorff topology.}
\vskip 0.1in
Strictly speaking, Proposition 3.1 of \cite{RT} only proves the version of
Theorem 3.7 without marked points. But one can easily keep track of marked
points in the proof and deduce Theorem 3.7 as we stated.
We denote the space of stable maps with fundamental class $A$
by $\overline{{\cal M}}_A
(\mu, g,k,J,\nu)$. Clearly,
$$\overline{{\cal M}}_A(\mu, g,k,J,\nu)\supset {\cal M}_A(\mu, g,k,J,\nu). \leqno(3.23)$$
One can easily deduce from Theorem 3.7 the following:
\vskip 0.1in
\noindent
{\bf Corollary 3.9. }{\it $\overline{{\cal M}}_A(\mu, g,k,J,\nu)$ is compact
in the Hausdorff topology. Moreover,
the evaluation map $e_i$ extends to a continuous map from
$\overline {{\cal M}} _A(\mu, g,k,J,\nu)$.}
\vskip 0.1in
We shall call $\overline {{\cal M}}_A(\mu, g,k,J,\nu)$ GU-compactification of
${\cal M}_A(\mu, g,k,J,\nu)$, since Gromov and Uhlenbeck first studied
the compactness problem for harmonic maps and pseudo-holomorphic
curves.
\vskip 0.1in
\noindent
{\bf Definition 3.10. }{\it We call $f$ a reduced GU-map if $f$ satisfies (1),
(4),(5) of Definition 3.6. Furthermore, it satisfies:
\begin{description}
\item[(2')] The singularities of $\Sigma'$ are of normal crossing, but it
could have three or more components intersecting at one point;
\item[(3')] There are no ghost bubbles;
\item[(6)] There are no bubble components which are multiple covering maps;
\item[(7)] There are no subtrees of the bubbles whose components have
the same image.
\end{description}}
\vskip 0.1in
For any stable map, we can define a GU-map (with possibly different
fundamental class) as follows: (i) we
collapse the ghost bubbles; (ii) we replace each multiple covered bubble
component by its
image; (iii) we collapse each subtree of the bubbles whose components have the
same image.
Clearly, this reduction may destroy the property
(2) of the Definition 3.6, but still preserve the property (2').
Also in this reduction, the fundamental class may change.
Define
$\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)$ to be the quotient of $\overline{{\cal M}}_A(\mu,
g,k,J,
\nu)$ by the above reduction. Furthermore, we define the topology
on $\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)$ as the quotient topology.
By the definition, $\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)$ is a union of
GU-maps
with possibly different fundamental classes. We will prove the following
structure theorem.
\vskip 0.1in
\noindent
{\bf Theorem 3.11. }{\it Let $(V, \omega)$ be a semi-positive symplectic
manifold. There is a set ${\cal H}_{reg}$ of Baire second category among all the
smooth pairs $(J, \nu)$ such that for any $(J,\nu)\in {\cal H}_{reg}$,
$\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)-{\cal M}_A(\mu, g,k,
J, \nu)_I$ consists of finitely many strata, such that
each stratum is a smooth manifold of
real codimension at least 2.}
\vskip 0.1in
\noindent
{\bf Proof of Proposition 2.3.} (1) follows from Theorem
3.1. (2) is obvious (cf. Corollary 3.9).
By the construction of $\overline{
{\cal M}}^r_A(\mu, g,k,J,\nu)$, both $\Upsilon$ and $\Xi^A_{g,k} $ descend to
$\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)$. Then (3) follows from Theorem 3.11.
\vskip 0.1in
\noindent
{\bf Remark 3.12:} One may ask whether or not
$\overline{{\cal M}}^r_A(\mu, g,k, J,\nu)$ carries a fundamental class.
This is indeed the case if $\overline{{\cal M}}^r_A(\mu, g,k, J,\nu)$ admits a real
analytic structure. These will be established in
\cite{RT1} by
more delicate analysis.
Then one can directly use the GU-compactification
to prove Proposition 2.3.
\vskip 0.1in
In the rest of this section, we outline the proof of Theorem 3.11. The
proof is identical to Section 4 of \cite{RT}. We refer the readers to
\cite{RT} for certain details.
First we shall decompose $\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)-{\cal M}_A(\mu, g,k,J,
\nu)$
into strata. A stratum is the set of GU-maps (possibly with
total homology class different from $A$) satisfying:
(1) their domains with
marked points are of
the same homeomorphic
type; (2) Each connected component carries a fixed homology class.
Furthermore, for technical
reasons, we need to specify those bubble components, which have the
same image even though
they may not be adjacent to each other, and their intersection points having
the
same image. Therefore,
the strata of $\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)$ are indexed by data:
(i) homeomorphism type of the domain of GU-maps with marked points; (ii) a
homology class associated to each
component; (iii) a specification of components with the same image and their
intersection points with the same image. We
denote by $D$ a
set of those three data. Let ${\cal D}_{g,k}$ be the collection of
such $D$'s. Note that when we drop
the multiplicity from a multiple covering
map, we change the homology class.
However it is still $A$-admissible in the following sense:
\vskip 0.1in
\noindent
{\bf Definition 3.13.} {\it Let $D$ be given as above. We
define $[D]$ to be the sum of homology classes of components in (ii).
Let $P_1, \cdots, P_o$ be principal components and $B_1,
\cdots, B_p$ be bubble components of $D$. We say that
$D$ is called $A$-admissible if there
are positive integers $b_1, \cdots, b_k$ such that
$$A=\sum^o_1 [P_i]+\sum^p_1 b_j [B_j]\leqno(3.24)$$
where $[P_i]$, $[B_j]$ are the homology classes of $P_i$, $B_j$.
We say that $D$ is $(J, \nu)$-effective if every principal
component can be represented
by a $(J, \nu)$-map and every bubble component can be represented by a
$J$-holomorphic map. }
\vskip 0.1in
We will always denote by $\Sigma _i$ the domain of the $(J,\nu)$-map
representing $P_i$. Let ${\cal D}^{J, \nu}_{g,k}\subset {\cal D}_{g,k}$
be the set of $A$-admissible,
$(J, \nu)$-effective $D$.
\vskip 0.1in
\noindent
{\bf Lemma 3.14.} {\it The set ${\cal D}^{J, \nu}_{g,k}$ is finite.}
\vskip 0.1in
\noindent
This is the analogue of Lemmas 4.5 of \cite{RT} and a simple
corollary of the Gromov-Uhlenbeck compactness theorem (cf. Theorem 3.8).
The presence of marked points
doesn't affect the proof at all. We omit it.
One can consider ${\cal D}^{J, \nu}_{g,k}$ as the set of indices of strata.
For each $D\in {\cal D}^{J, \nu}_{g,k}$, let ${\cal M}_{D}(\mu, g,k, J, \nu)$ be the space
of
GU-maps such that the homeomorphism type of its domain with marked
points,
homology class of each component, and components and their intersection points
which have the same image are specified by $D$.
The following lemma can be deduced from
the definition.
\vskip 0.1in
\noindent
{\bf Lemma 3.15. }{\it
$$\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)=\bigcup_{D\in {\cal D}^{J,\nu}_{g,k}} {\cal M}_D(\mu,
g,k,J,\nu).\leqno(3.25)$$
}
\vskip 0.1in
By the definition, each $D$ is associated with a stable Riemann surface
$\Sigma_{D, r}$, which can be obtained by contracting all the bubble
components. Recall that for each principal component we have to fix the
automorphism
group preserving the special points to make the transversality arguments work
(Theorem 3.1). Here we use
$\bar {\kappa}$ to denote the $o$-tuple $(\kappa _1, \cdots, \kappa_o)$,
where $\kappa_i$ is the
automorphism group of the principal component $P_i$ fixing the special points
(marked points and intersection points) of $\Sigma_{D, r}$.
Then,
$${\cal M}_D(\mu, g,k,J,\nu)=\bigcup_{\bar {\kappa}}{\cal M}_D(\mu,
g,k,J,\nu)_{\bar{\kappa}},\leqno(3.26)$$
where ${\cal M}_D(\mu, g,k,J,\nu)_{\bar{\kappa}}$ consists of all maps in
${\cal M}_D(\mu, g,k,J,\nu)$ whose $i^{\rm th}$-principal component has
the automorphism group $\kappa_i$. Next we prove the smoothness of
${\cal M}_D(\mu, g,k,J,\nu)_{\bar{\kappa}}$.
First we make another reduction by identifying the domains of those bubble
components which have the same image,
and change the homology class accordingly. Furthermore, we identify the
corresponding intersection points with the same image. Suppose that the
resulting new domain and homology class of each component are specified
by $\bar{D}$. This process may destroy the tree structure and
creat some cycles in the domain.
The total homology class
may also change. However, it remains to be $A$-admissible.
Given such $D$ and $\bar{D}$, we can identify
${\cal M}_D(\mu, g,k,J,\nu)_{\bar{\kappa}}$ with
the space of $(J, \nu)$-maps whose
domain, homology class of each component are specified by $\bar{D}$ and the
automorphism group of its principal components are specified by $\bar
{\kappa}$.
Let us denote this space by ${\cal M}_{\bar D}(\mu, g,k,J,\nu)_{\bar{\kappa}}$. Then,
$${\cal M}_D(\mu, g,k,J,\nu)_{\bar{\kappa}}=\{\cal M}_{\bar D}(\mu,
g,k,J,\nu)_{\bar{\kappa}}.\leqno(3.27)$$
For each $f$ in ${\cal M}_{\bar D}(\mu, g,k,J,\nu)_{\bar{\kappa}}$,
the bubble components have different images. As before,
let $P_1, \cdots, P_o$
be the
principal components and let $B_1, \cdots, B_p$ be
the bubble components.
Let $\Sigma _{\bar D}$ be the domain of maps in the stratum
${\cal M}_{\bar{D}}(\mu, g,k, J, \nu)_{\bar {\kappa}}$. This is a union
of $\Sigma _i$ (genus $g_i$) and some $S^2$'s intersecting each other
according to the intersection pattern given by $D$. Let $h_i$ be the number of
intersection points on the component
$P_i$. Note that we count a self-intersection point twice. Here, the
intersection points between the components are the points in
their domain, not in their image.
Similarly,
let $h^j$ be the number of intersection points on the bubble component
$B_j$. Let $k_i(k^j)$ be the number of marked points on $P_i$-component (bubble
component $B_j$), which are different from intersection points. Notice that
$k^j=0$ or $1$. Moreover,
$$\sum k_i +\sum k^j\leq 2k.\leqno(3.28)$$
It may not be equal to $2k$ because the collapsing of the ghost bubbles
containing a marked point forces the marked point to lie on the intersection.
Let ${\cal M}^*_{[B_j]}(S^2, J,0)\subset
{\cal M}_{[B_j]}(S^2, J,0)$ be the space of non-multiple covering maps and
$${\cal M}^*_{[B_j]}(S^2, h^j+k^j, J, 0)={\cal M}^*_{[B_j]}(S^2, J,0)\times \overline{
S^2}^{h^j+k^j}$$
where $\overline{S^2}^{h^j+k^j}$ is the set of distinct $h^j+k^j$-tuple points
of $S^2$.
Consider
$$
\tilde{{\cal M}}_{\bar{D}}(\mu, g,k, J, \nu)_{ \bar {\kappa}}
= \{f: \Sigma _{\bar D}
\rightarrow V ~|~
f_{P_i}\in {\cal M}_{[P_i]}(\mu, g_i, h_i+k_i, J, \nu_i)_{\kappa_i},
Im(f_{B_j})\neq Im(f_{B_{j'}}) ~~\mbox{ if }
j\neq j' \}
\leqno(3.29)
$$
For each bubble component, there is a parameterization group $G=PSL_2$.
Therefore, $G^{p_{\bar{D}}}$ acts on $\tilde{{\cal M}}_{\bar{D}}(\mu, g, J, \nu)_{
\bar {\kappa}}$, where $p_{\bar{D}}$ is the number of bubble components.
Then ${\cal M}_{\bar D}(\mu, g,J,\nu)_{\bar{\kappa}}=
\tilde{{\cal M}}_{\bar{D}}(\mu, g,k, J,\nu)_{ \bar {\kappa}}/G^{p_{\bar{D}}}$.
Clearly,
$$\tilde{{\cal M}}_{\bar D}(\mu, g,k,J,\nu)_{\bar{\kappa}} \subset \prod {\cal M}_{[P_i]}
(\mu, g_i, h_i+k_i, J, \nu_i)_{\kappa_i}
\times \prod {\cal M}^*_{[B_j]}(S^2, h^j+k^j, J,0),\leqno(3.30)$$
whose components intersecting each other according to the intersection pattern
given by $\bar{D}$.
Consider the evaluation map
$$e_{\bar D}:
\prod {\cal M}_{[P_i]}(\mu, g,h_i+k_i, J,\nu)_{{\kappa _i}}\times \prod
{\cal M}^*_{[B_j]}(S^2, h^j+k^j, J,0)\mapsto \prod
V^{h_i}\times \prod V^{h^j}=V^{h_{\bar{D}}},\leqno(3.31)$$
where $h_{\bar{D}}=\sum h_i +\sum h^j$, and $e_{\bar{D}}$ is defined as
follows:
We first define
$$\begin{array}{rl}
&e_{P_i}:{\cal M}_{[P_i]}(\mu, g, h_i+k_i, J,\nu)_{{\kappa _i}}\rightarrow V^{h_i}\\
&e_{P_i}(f, x_1, \cdots, x_{h_i}, x_{h_i+1}\cdots x_{h_i+k_i})=(f(x_1), \cdots,
f(x_{h_i}))
\end{array}
\leqno(3.32)$$
For each $B_j$, we define $e_{B_j}: {\cal M}^*_{[B_j]}(S^2, h^j+k^j, J,0)
\rightarrow
V^{h^j}$ by
$$e_{B_j}(f, y_1, \cdots, y_{h^j}, y_{h^j+1}\cdot y_{h^j+k^j})=(f(y_1), \cdots,
f(y_{h^j})).
\leqno(3.33)$$
Then we define $e_{\bar{D}}=\prod e_{P_i}\times \prod e_{B_j}$.
Recall that if $M, N$ are
submanifolds of $X$, $M\cap N$ can be interpreted as $M\times N\cap \Delta$,
where $\Delta$ is the diagonal of $X\times X$. This means that we can realize
any intersection pattern by constructing a ``diagonal'' in the product.
Let us construct a submanifold $\Delta_{\bar{D}}\subset V^h$ which
plays the role of the diagonal. Let $z_1,\cdots,z_{t_{\bar{D}}}$ be all
the intersection
points. For each $z_s$, let
$$I_s=\{P_{i_1}, \cdots, P_{i_q}, B_{j_1}, \cdots,
B_{j_r}\}$$
be the set of components which intersect at $z_s$.
Now we will construct a product $V_s$ of $V$ such that
its diagonal describes the intersection at
$z_s$. This is done as follows: we allocate one or two factors from
each of $V^{h_{i_1}}, \cdots,
V^{h_{i_q}}$, according to whether or not $z_s$
is a self-intersection point of the corresponding principal component.
We allocate one
factor from each of
$V^{h^{j_1}}, \cdots, V^{h^{j_r}}$.
Here $V^{h_i}$ or $V^{h^j}$
are the image of $e_{P_i}$ or $e_{B_j}$.
Then, we take the product of
those factors and denote it by $V_s$.
Let $\Delta_s$ be the diagonal of $V_s$. Then the product
$\Delta_{\bar{D}}=\Delta_1\times \cdots \times
\Delta_{t_{\bar{D}}}\subset V^{h_{\bar{D}}}$ is the diagonal to realize the
intersection pattern between the components of $\bar{D}$.
Then
$e^{-1}_{\bar{D}}(\Delta_{\bar{D}})\supset \tilde{{\cal M}}_{\bar D}
(\mu, g,k, J, \nu)$. But
they may not be equal because we require that bubble
components have different image. But $\tilde{{\cal M}}_{\bar D}
(\mu, g,k, J, \nu)$ is an open subset.
Moreover, the group $G^{p_{\bar{D}}}$ acts
on $e^{-1}_{\bar D}(\Delta_{\bar{D}})$.
\vskip 0.1in
\noindent
{\bf Theorem 3.16.} {\it There is a set ${\cal H}_{reg}$ of Baire second category
among all the smooth pairs $(J,\nu)$ such that for any $(J, \nu)\in
{\cal H}_{reg}$, ${\cal M}_{\bar D}(\mu, g,k,J,\nu)_{\bar{\kappa}}$ is a
smooth manifold of dimension
$$\sum(2c_1(V)(P_i)+2n(1-g_i))+\sum \dim {\bf T}^{\mu, \kappa}_{g_i,0}+\sum
(2c_1(V)(B_j)+2n-6)+2h_{\bar{D}}+\sum k_i+\sum
k^j-2n(h_{\bar{D}}-t_{\bar{D}}),$$
where $g_i$ is the genus of $\Sigma_i$ and $t_{\bar{D}}$ is the number of
intersection points of $
\bar{D}$.
Moreover, for any $(J,\nu)$ and $(J', \nu')$ of ${\cal H}_{reg}$, there is
a second category set of paths ${\cal H}_{((J,\nu), (J', \nu'))}$ connecting
$(J,\nu)$ and $(J', \nu')$ among all the smooth paths such that for any path
$(J_t, \nu_t)\in {\cal H}_{((J,\nu), (J', \nu'))}$,
$$\bigcup_{t\in [0,1]}{\cal M}_{\bar D}(\mu, g,k,J,\nu)_{\bar{\kappa}}$$
is a smooth cobordism of one dimension higher.}
\vskip 0.1in
By the construction of $\bar{D}$, it is evident that $t_{\bar{D}}\leq t_D$ and
$h_{\bar{D}}\leq h_D$. But, $h_{\bar{D}}-t_{\bar{D}}=h_D-t_D$. Therefore,
\vskip 0.1in
\noindent
{\bf Corollary 3.17.} {\it Under the conditions of Theorem 3.16,
the dimension of $ {\cal M}_{\bar D}(\mu, g,k,J,\nu)_{\bar{\kappa}}$ is less than or
equal to
$$\sum(2c_1(V)(P_i)+2n(1-g_i))+\sum \dim {\bf T}^{\mu, \kappa}_{g_i,0}+2k+\sum
(2c_1(V)(B_j)+
2n-6)+2h_{D}-2n(h_{D}-t_{D}).$$
}
\vskip 0.1in
\noindent
{\bf Proof of Theorem 3.16:} The idea of the proof
is similar to that in the proof of
Lemma 4.8-4.11 in \cite{M1}.
Also many arguments are the same as those in the proof of Theorem
4.7 in \cite{RT}. But we will avoid the Floer's
norm on the space of almost complex structures as we did before.
First of all, $\tilde{{\cal M}}_{
\bar{D}}(\mu, g,k,J,\nu)_{\bar{\kappa }}$ is an open subset of
$e^{-1}_{\bar{D}}(\Delta_{
\bar{D}})$. Hence for the purpose of proving smoothness, we can assume that
$$\tilde{{\cal M}}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar{\kappa }}=
e^{-1}_{\bar{D}}(\Delta_{
\bar{D}})$$
to simplify the argument.
Suppose $p>2,m\geq 1$. Following (3.3),
$$\chi^{p,m}_{\bar {\kappa}, \bar{D}}=\prod_{i=1}\bigcup_{j\in
{\bf T}^{\mu,\kappa}_{g_i,
h_i+k_i}} L^p_{m,j}
(Map_{[P_i]}(\Sigma_i, V))\times \{j\} \times \prod_{s=1}L^p_k (Map_{[B_s]}
({\bf P}^1, V))\times (\overline{S^2})^{h^j+k^j}\leqno(3.34)$$
is a smooth Banach manifold. Then, we define
$${\cal M}^l(\bar{D}, \bar {\kappa})=\{(f,j,J,\nu)\in \chi^{p,m}_{\bar
{\kappa},\bar{D}}\times
{\cal H}^l ; \bar{\partial}_J f(x)=\nu(\phi(x), f(x))\},\leqno(3.35)$$
where $m\leq l$ and the equation
$$\bar{\partial}_J f(x)=\nu(\phi(x), f(x))$$
means the $(J,\nu)$-holomorphic equation for each $P_i$ component and
$J$-holomorphic equation for each $[B_s]$ component.
It is not hard to observe that
$${\cal M}^l(\bar{D}, \bar {\kappa})=\bigcup_{(J,\nu)\in {\cal H}^l}(\prod_1 {\cal M}^l_{[P_i]}(
\mu, g_i,h_i+k_i, J,\nu)_{{\kappa _i}} \times \prod_1 ({\cal M}^*)^l_{[B_s]}(S^2,h^j
+k^j, J,0).
\leqno(3.36)$$
\vskip 0.1in
\noindent
{\bf Lemma 3.18. }{\it ${\cal M}^l(\bar{D}, \bar {\kappa})$
is smooth Banach manifold.}
\vskip 0.1in
\noindent
{\bf Proof of Lemma 3.18: } ${\cal M}^l(\bar{D}, \bar {\kappa})$
is just the analogue of
$\Theta(\Delta_{{\cal J}})$ in Lemma 4.9 of \cite{RT}.
The proof of Lemma 3.18
is identical to that of Lemma 4.9 in \cite{RT}. We omit it.
\vskip 0.2in
Note that the evaluation map extends
$$e_{\bar{D}}: {\cal M}^l(\bar{D}, \bar {\kappa})\rightarrow
V^{h_{\bar{D}}}.\leqno(3.37)$$
We define
$$\tilde{{\cal M}}^l_{\bar{D}}(\mu, g, \bar
{\kappa})=e^{-1}_{\bar{D}}(\Delta_{\bar{D}})\leqno(
3.38)$$
\vskip 0.1in
\noindent
{\bf Lemma 3.19. }{\it $\tilde{{\cal M}}^l_{\bar{D}}(\mu, g, \bar {\kappa})$
is a smooth Banach manifold.}
\vskip 0.1in
Its proof is identical to the proof of Theorem 4.7 of \cite{RT}. We omit it.
Then, the rest of proof of Theorem 3.16 is similar to that of Theorem 3.1.
We sketch the argument.
Consider the projection
$$\pi: \tilde{{\cal M}}^l_{\bar{D}}(\mu, g, \bar {\kappa})\rightarrow {\cal H}^l.\leqno(3.
39)$$
Then,
$$\tilde{{\cal M}}^l_{\bar{D}}(\mu, g,k,J,\nu)_{\bar
{\kappa}}=(\pi)^{-1}((J,\nu)).\leqno(3.40)$$
Then $\pi$ is a Fredholm map between two Banach manifolds. It follows from
the Sard-Smale
Transversality Theorem that the set
$${\cal H}^l_{reg}=\{(J, \nu)\in{\cal H}^l\,|\, d\pi \mbox{ is onto for all } f\in
{\cal M}^l_{\bar{D}}
(\mu, g,k,J,\nu)_{\bar {\kappa}}\}$$
is of Baire second category. Let
$${\cal H}_{reg}=\bigcap_l {\cal H}^l_{reg}.$$
Then,
$$\tilde{{\cal M}}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa }}=\bigcap_l
{\cal M}^l_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa }}.
\leqno(3.41)$$
As in the proof of Theorem 3.1, we can deduce that ${\cal H}_{reg}$ is
of Baire second category. We leave it to the readers as an exercise.
Then, for any
$(J,\nu)\in {\cal H}_{reg}$,
$$\tilde{{\cal M}}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa }}=\pi^{-1}((J, \nu))$$
is a smooth manifold. Since $G^{p_{\bar{D}}}$ acts freely on
$\tilde{{\cal M}}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa }}$,
$${\cal M}_{\bar D}(\mu, g,k,J,\nu)_{\bar {\kappa }}=\tilde{{\cal M}}_{\bar{D}}(\mu, g,k,
J,\nu)_{\bar{\kappa}}/G^{p_{\bar{D}}}$$
is smooth.
An routine counting argument yields the dimension formula.
The proof of the second part of Theorem 3.16 is identical.
\vskip 0.1in
Recall that if we contract all the bubble components of $D$, we obtain the
stable Riemann surface $\Sigma _{D, r}$ in the sense of Deligne-Mumford.
An interesting special case of Theorem 3.18 is
\vskip 0.1in
\noindent
{\bf Corollary 3.20.} {\it If $\Sigma _D$ has no bubble components at all,
i.e., $\Sigma _{D}=\Sigma_{D,r}$,
${\cal M}_{\Sigma_D}(\mu, g,k, J, \nu)_{\bar {\kappa}}$
is smooth for a generic
$(J, \nu)$. Moreover, for a generic $(J_t, \nu_t)$,
$\bigcup_t {\cal M}_{D}{(\mu, g,k,J_t,\nu _t)_{\bar {\kappa }}}
\times \{t\}=\bigcup_t {\cal M}_{\Sigma_D}(\mu, g,k,J_t,\nu _t)_{\bar {\kappa
}}\times \{t\}$ is a smooth cobordism.
Here, the word ``generic'' means that it is in a set of Baire second
category.}
\vskip 0.1in
Next, we compute the codimension of ${\cal M}_{\bar D}(\mu, g,k,J,\nu)_{{\kappa
_i}}$. First
\vskip 0.1in
\noindent
{\bf Proposition 3.21.} {\it Suppose that $(V, \omega)$ is a semi-positive
symplectic
manifold. Let ${\cal M}_{\Sigma_{D,r}}\subset \overline{{\cal M}}_{g,k}$
be the set of stable Riemann surfaces such that their homeomorphism types are
specified by $\Sigma_{D, r}$. Then,
$$\dim {\cal M}_{\bar{D}}(\mu, g,k,J,\nu )_{\bar {\kappa }}\leq
2c_1(V)(A)+2n(1-g)+\dim {\cal M}_{
\Sigma_{D,r}}-2p_D,\leqno(3.42)$$
where $p_D$ is the number of bubble components of $D$(not $\bar{D}$)}
\vskip 0.1in
\noindent
{\bf Proof:} By Corollary 3.17,
the dimension of ${\cal M}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa }}$ is less than
or equal to
$$\sum_i (2c_1(V)([P_i])+2n(1- g_i))+2k+\sum_i \dim {\bf T}^{\mu, \kappa_i}_{g_i,0}+
\sum_j (
2c_1(V)(B_j)+(2n-6))+2h_{D}-2n(h_{D}-t_{D})$$
$$=2c_1(V)([\bar{D}])+2n\sum_i(1-g_i)+2k+\sum \dim_i {\bf T}^{\mu, \kappa_i}_{g_i,0}
+(2n-6)p_{\bar{D}}+2h_D-2n(h_{D}-t_{D})$$
For a generic $J$,
$$2c_1(V)(B_j)+2n-6=\dim {\cal M}^*_{[B_j]}(S^2, J,0)/PSL_2\geq 0. \leqno(3.43)$$
If some bubble component $B_j$ happens
to be the image of two or more bubble
components of $D$, by adding $2c_1(V)(B_j)+2n-6$ to the dimension formula,
$$\dim {\cal M}_{\bar{D}}(\mu, g, k,J, \nu)\leq 2c_1(V)([D])+2n\sum_i(1- g_i)+2k+
\sum_i \dim {\bf T}^{\mu, \kappa}_{g_i,0}+(2n-6)p_{D}+2h_{D}-2n(h_{D}-t_{D}).$$
Since $(V, \omega)$ is semi-positive, $c_1(V)(B_j)\geq 0$ for a generic
$J$. Since $D$ is $A$-admissible, $c_1(V)([D])\leq c_1(V)(A)$. Let
$$\lambda_D=(2n-6)p_D+2h_D-2n(h_D-t_D).\leqno(3.44)$$
Then, Proposition 4.13 of $\cite{RT}$ implies that
$$\lambda_D\leq \lambda_{\Sigma_D}-2p_D. \leqno(3.45)$$
Note that in \cite{RT}, we use $k_D$ to
denote the number of bubble component
instead of $p_D$ we used in this paper
(here $k$ was used to denote the number
of marked points). Therefore,
$$dim {\cal M}_{\bar{D}}(\mu, g,k,J,\nu )_{\bar {\kappa }}
\leq \sum_i 2c_1(V)(A)+2n\sum_i(1- g_i)+2k+
\sum_i \dim {\bf T}^{\mu, \kappa_i}_{g_i,0}+
2h_{\Sigma_D}-2n(h_{\Sigma_{D}}-t_{\Sigma_{D}})-2p_D.
$$
Since $\Sigma_{D,r}$ is homeomorphic to a stable curve, $t_{\Sigma_D}$
is the number of double points and $h_{\Sigma_D}=2t_{\Sigma_D}$. An easy
inductive argument (Proposition of 4.13, \cite{RT}) shows that
$$o-t_{\Sigma_D}-\sum_i g_i=1-g.\leqno(3.46)$$
It is easy to observe that
$$dim {\cal M}_{\Sigma_{D,r}}=2k+\sum_i \dim
{\bf T}^{\mu,\kappa_i}_{g_i,0}+2h_{\Sigma_D}\leq \dim {\cal M}_{\Sigma_D} \leqno(3.47)$$
and equal iff all the $\kappa_i$ are trivial.
\vskip 0.1in
\noindent
{\bf Proof of Theorem 3.11 (Structure Theorem):} It follows from Lemma 3.14,
3.15, Theorem 3.16, Proposition 3.21.
\section{Proof of Proposition 2.4, 2.9 and 2.10}
In this section, we first establish the existence
of the GW-invariants $\Psi^V$ (Proposition 2.4) and
its independence from various parameters.
Hence, $\Psi^V$ is a symplectic invariant. Then, we will prove Proposition
2.9 and 2.10.
Technically speaking, this section is the analogue of
section 5 and 7 of \cite{RT}.
We shall repeatly use the word ``generic'' to
mean something belonging to a set of Baire second category.
First of all, we extend the definition of
pseudo-submanifolds to the singular
space with quotient singularities such has $\overline{{\cal M}}^{\mu}_{g,k}$.
Furthermore, we also need to consider the transversality theory of such
pseudo-submanifolds. It is well-known that over the rational coefficient ${\bf Q}$,
the usual theory for the smooth manifolds extends to the singular space with
quotient singularities, where the Poincare duality holds over ${\bf Q}$.
\vskip .1in
\noindent
{\bf Definition 4.1: }{\it An n-dimensional finite simplicial complex $P$ is
called an abstract pseudo-manifold if $P^{top}=P-P_{n-2}$ ($n-2$ skeleton)
is an open smooth oriented $n$-dimensional manifold.
$P$ is called an abstract
pseudo-manifold with boundary if $P^{top}$ is a
$n$-dimensional oriented smooth
manifold with boundary $\partial P^{top}$.
Let $\partial P=\overline{\partial
P^{top}}$. Then $\partial P \cap P_{n-2}$ is a subcomplex of dimension less
than or equal to $n-3$. Let $V$ be a stratified PL-manifold such that each
stratum is even dimensional and its triangulation is compatible with the
stratification. A pseudo-submanifold is a pair $(P, f)$, where $P$ is
an abstract pseudo-manifold and $f: P\rightarrow V$ is a piece-wise linear
map (PL) with respect to some triangulation of $V$. Furthermore, we require
that $f$ maps $P^{top}$ into one stratum and smooth.
A pseudo-submanifold
cobordism between pseudo-submanifolds
$(P, f), (Q, h)$ is a pair $(K, H)$ such
that $K$ is an abstract pseudo-manifold
with boundary with $\partial K= P\cup
-Q$ and $H$ is PL with respect to some
triangulation of $V$ and smooth over
$K^{top}$ in the sense that $H$ maps the $K^{top}$
smoothly in one stratum or maps
the interior $K^o$ of $K^{top}$ to
one stratum and $\partial K^{top}$ to the
lower strata.
Moreover, $H|_{P\cup -Q}=f\cup - h$,
where $-$ means the opposite orientation.}
\vskip .1in
Furthermore, we have the following lemma on transversality.
\vskip 0.1in
\noindent
{\bf Lemma 4.2. } {\it Let $V$ be a stratified PL-manifold with fundamental
class $[V]$ such that the Poincare duality holds over the rational coefficient.
Then, for each homology class $\alpha$, there exists $p$ and
a pseudo-submanifold representative $(P,f)$
of a homology class $p(\alpha)$. Furthermore,
if $h_i: X_i \rightarrow V$ be
smooth maps from
smooth manifolds $X_i$ to smooth strata of $V$, then there is a small
perturbation $\tilde{f}: P
\rightarrow V$ such that $\tilde{f}$ is transverse to each $h_i$, i.e.,
$\tilde{f}$ is
transverse to $h_i$ as a PL map and transverse over $P^{top}$ as a smooth map.}
\vskip 0.1in
\noindent
{\bf Proof:} This lemma is the consequence of
standard transversality results
in PL topology \cite{Mc}(Theorem 5.2). One remark is that if $V$ is
a smooth manifold, this lemma holds for any $\alpha$. Otherwise, the lemma
holds for those class of the form $[V]\cap \beta^*$ for a cohomology class
$\beta^*$ (Theorem 5.2 of \cite{Mc}). Then, the lemma follows from the
assumption
that the Poincare Duality holds over the rational coefficient ${\bf Q}$.
\vskip 0.1in
Recall that in (2.4), we have defined the evaluation map
$$e_i: {\cal M}_A(\mu, g,k, J,\nu)_\kappa\rightarrow V.\leqno(4.1)$$
$e_i$ extends obviously to each
${\cal M}_{\bar D}(\mu, g,k,J,\nu )_{\bar {\kappa }}$ (Lemma 3.15).
We shall still denote it by $e_i$. Furthermore,
$$\Upsilon: {\cal M}_A(\mu, g,k, J,\nu)_\kappa\rightarrow
{\cal M}^{\mu}_{g,k,\kappa},\leqno(4.2)$$
which takes the domain of maps in ${\cal M}_A(\mu, g,k, J,\nu)_\kappa$ (section 2)
extends over
$$\Upsilon: {\cal M}_{\bar D}(\mu, g,k,J,\nu )_{\bar {\kappa }}\rightarrow {\cal M}^{\mu}_{
\Sigma_{D,r}, \bar{\kappa}}$$
as well. There is an obvious
version of the maps $e_i$ and $\Upsilon$ for the corbordisms $\bigcup_t
{\cal M}_A(\mu, g,k,J_t,\nu_t)_\kappa \times
\{t\}$ and
$\bigcup_t {\cal M}_{\bar D}(\mu, g,k,J_t,\nu_t )_{\bar {\kappa }}\times \{t\}$. We
denote them by $e_i^{(t)}$ and $\Upsilon^{(t)}$. Then, we
define
$$(\Xi^A_{g,k})^{(t)}=\prod_i e_i^{(t)}.\leqno(4.3)$$
\vskip 0.1in
\noindent
{\bf Definition 4.3. }{\it Let $(P_i, f_i)$ be a pseudo-submanifold of $V$.
We say that $(P_i, f_i)$ is transverse to $e_i$ (hence $\Xi^A_{g,k}$) if
$(P_i,f_i)$ is transverse to
$e_i$ as the maps from ${\cal M}_A(\mu, g,k, J,\nu)_\kappa$ and their extensions over
${\cal M}_{\bar D}(\mu, g,k,J,\nu )_{\bar {\kappa }}$ for each $D\in
{\cal D}^{J,\nu}_{g,k}$ in the sense of
Lemma 4.2. We say that $(P_i, f_i)$ is transverse to $e_i^{(t)}$ if it is
transverse to $e_i^{(t)}$ (hence $(\Xi^A_{g,k})^{(t)}$) as the maps from
$\bigcup_t {\cal M}_A(\mu, g,k,J_t,\nu_t )_{\bar {\kappa }}\times
\{t\}$ and their
extensions over $\bigcup_t
{\cal M}_{\bar D}(\mu, g,k,J_t,\nu_t )_{\bar {\kappa }}\times \{t\}$ for each
$D\in {\cal D}^{J,\nu}_{g,k}$ in the sense of Lemma 4.2. Similarly, we say
that a pseudo-submanifold $(G, K)$ of $\overline{{\cal M}}^{\mu}_{g,k}$ is
transverse to $\Upsilon$ (or $\Upsilon^{(t)}$) if
it transverse to them as the maps from ${\cal M}_A(\mu, g,k, J,\nu)_\kappa$ and its
extensions
over ${\cal M}_{\bar D}(\mu, g,k,J,\nu )_{\bar {\kappa }}$ (or from $\bigcup_t
{\cal M}_A(\mu, g,k,J_t,\nu_t )_{\bar {\kappa }}
\times \{t\}$ and its extensions over $\bigcup_t
{\cal M}_{\bar D}(\mu, g,k,J_t,\nu_t )_{\bar {\kappa }}
\times \{t\}$) respectively.}
\vskip 0.1in
Let us recall the construction of section 2.
Let $\alpha_i$ be integral homology
classes of $V$. We choose pseudo-submanifolds $(Y_i, F_i)$ to represent
$\alpha_i$. Let
$$Y=\prod^k_{i=1} Y_i, F=\prod^k_{i=1} F_i.$$
Then, $Y^{top}=\prod^k_{i=1} Y^{top}_i.$
Clearly, $(Y, F)$ represents $\prod^k_{i=1}\alpha_i\in H_*(V^k, {\bf Z})$.
Note that
$\overline{{\cal M}}^{\mu}_{g,k}$ may not be smooth, but the Poincare Duality holds
over
rational coefficients. Hence, we can assume that each
homology class can be represented by a pseudo-submanifold and the
corresponding
transversality holds as long as we work over $H_*(\overline{{\cal M}}^{\mu}_{g,k},
{\bf Q})$.
Let $(G, K)$ be a pseudo-submanifold in
$\overline{{\cal M}}^{\mu}_{g,k}$ and first we assume that $(G, K)$ is {\em in
general position } i.e., $K(G^{top})\subset {\cal M}^{\mu}_{g,k, I}$, where $I$
represents trivial automorphism group.
\vskip 0.1in
\noindent
{\bf Lemma 4.4. }{\it By choosing small perturbations
if necessary, we have that
$K\times F$ is transverse to $\Upsilon\times \Xi^A_{g,k}$ as the PL-maps
with respect to some triangulation of $V$ and as the smooth maps over $Y^{top}
\times G^{top}$.}
\vskip 0.1in
\noindent
{\bf Proof: } This follows obviously from Lemma 4.2.
\vskip 0.1in
\noindent
{\bf Corollary 4.5. }{\it Suppose that
$$\sum^k_{i=1} (2n-d_i)+(6g-6+2k-deg(\mu, g))=2c_1(V)(A)+2(3-n)(\mu,
g-1)+2k.\leqno(4.4)$$
and $F_i, K$ satisfy the statement of Lemma 4.4. Then,
$$\begin{array}{rl}
&K\times F \cap \Upsilon\times \Xi^A_{g,k}(\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)-
{\cal M}_A(\mu, g,k, J, \nu))_I=\emptyset;\\
& K\times F (Y\times G -Y^{top}\times G^{top})\cap
\Upsilon\times \Xi^A_{g,k} =\emptyset.\\
\end{array}
\leqno(4.5)$$}
{\bf Proof: } By Lemma 4.4, $K\times F$ is transverse to $\Upsilon\times
\Xi^A_{g,k}$. To prove
$$K\times F \cap \Upsilon\times \Xi^A_{g,k}
(\overline{{\cal M}}^r_A(\mu, g,k, J, \nu)-
{\cal M}_A(\mu, g,k, J, \nu)_I)=\emptyset,$$
by (3.25), (3.26),
$$\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)=\bigcup_{D\in {\cal D}^{J,\nu}_{g,k}}\bigcup_{
\bar {\kappa}}{\cal M}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa}}.$$
By the Proposition 3.20, except the main component ${\cal M}_A(\mu, g,k, J, \nu)_I$,
all other components ${\cal M}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa}}$ are
smooth manifolds of dimension
$$\leq 2c_1(V)(A)+2(3-n)(g-1)+2k-2 \leqno(4.6)$$
Since $K\times F$ is transverse to
${\cal M}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa}}$, then
$$\dim (K\times F \cap \Upsilon\times \Xi^A_{g,k}
({\cal M}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa}}))\leq -2.\leqno(4.7)$$
Hence, it is empty. Therefore,
$$K\times F\cap \Upsilon\times \Xi^A_{g,k}({\cal M}_{D}(\mu, g,k,J,\nu)_{\bar
{\kappa}}))=K\times F \cap \Upsilon\times \Xi^A_{g,k}
({\cal M}_{\bar{D}}(\mu, g,k,J,\nu)_{\bar {\kappa}}))=\emptyset.\leqno(4.8)$$
$$K\times F (Y\times G -Y^{top}\times G^{top})\cap \Upsilon\times \Xi^A_{g,k}
=\emptyset\leqno(4.9)$$
follows from a similar dimension counting argument. We leave it to the readers.
Now we adopt the notations from section 2. By (2.7),
$$(\Upsilon\times \Xi^A_{g,k}\times K\times F)^{-1}(\Delta)\subset
{\cal M}_A(\mu, g,k, J, \nu)_I \times Y^{top}\times G^{top} \leqno(4.10)$$
is a zero-dimensional smooth submanifold.
\vskip 0.1in
\noindent
{\bf Lemma 4.6. }{\it $(\Upsilon\times \Xi^A_{g,k}\times K\times F)^{-1}
(\Delta)$ is compact and hence finite.}
\vskip 0.1in
\noindent
{\bf Proof: }{ Suppose that there is a sequence of distinct elements
$$(f_s, X_s, x_s)\in (\Upsilon\times \Xi^A_{g,k}\times K\times F)^{-1}
(\Delta).$$
By taking a subsequence, we can assume that
$$f_s\rightarrow f\in \overline{{\cal M}}^r_A(\mu, g,k,J,\nu)$$
and
$$(X_s, x_s)\rightarrow (X, x)\in Y\times G.$$
However, $(\Upsilon\times \Xi^A_{g,k}\times K\times F)^{-1}(\Delta)$ is smooth.
Thus, either
$$(f,X,x)\in K\times F \cap \Upsilon\times \Xi^A_{g,k}(\overline{{\cal M}}^r_A(\mu,
g,
k, J, \nu)-{\cal M}_A(\mu, g,k, J, \nu)_I)=\emptyset\leqno(4.11)$$
or
$$(f, X, x)\in K\times F (Y\times G -Y^{top}\times G^{top})\cap \Upsilon\times
\Xi^A_{g,k}
=\emptyset.\leqno(4.12)$$
In both cases, we have a contradiction.
Once the Lemma 4.6 is proved, as in section 2, we can define
$$\Psi^V_{(A,g,k, \mu)}(K, \alpha_1, \cdots, \alpha_k)$$
as the algebraic sum of $(\Upsilon\times \Xi^A_{g,k}\times K\times F)^{-1}
(\Delta)$. To emphasis the dependence on $(J, \nu)$ at this
moment, we define
$$Z(A,\mu, g,k, J,\nu,F,K)=(\Upsilon\times \Xi^A_{g,k}\times K\times F)^{-1}
(\Delta)\leqno(4.13)$$
Sometimes (for example (4.29)), we also use $Y, G$ in the place of $F, K$
in $Z(\cdots)$, if there is no confusion.
To abuse the notation, we denote its algebraic sum by
$|Z(A,\mu,g,k, J,\nu,F,K,)|$.
Next we have to prove Proposition 2.4. The proof
will be divided into a series of Lemmas:
\vskip 0.1in
\noindent
{\bf Lemma 4.7. }{\it $|Z(A,\mu,g,k,J,\nu,F,K)|$ is independent of the
representative
$(Y,F)$ and $(G,K)$, whenever $(G,K)$ is in general position.}
\vskip 0.1in
\noindent
{\bf Proof: } Suppose that $(Y',F'),(G' ,K')$ are other representatives such
that $(G', K')$ is in general position. There are corbordisms $(Q,H)$ and
$(L, P)$ such that
$$\partial(Q)=Y\cup -Y', \ H|_{\partial(Q)}=F\cup -F'; \partial(L)=G\cup -G',
\ P|_{\partial(L)}=K\cup -K'\leqno(4.14)$$
Let's first work on $(Q, H)$.
By choosing a small perturbation of $H$ relative to $\partial(Q)$ if
necessary, we can assume that $H\times K$ is transverse to
$\Upsilon \times \Xi^A_{g,k}$. Then, by counting dimensions as
one did in the proof of Lemma 4.4, one can show that
$$\begin{array}{rl}
&H\times K\cap \Upsilon \times \Xi^A_{g,k}(\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)-
{\cal M}_A(\mu, g,k,J,\nu)_I)=\emptyset;\\
&\ H\times K(Q\times K-Q^{top}\times K^{top})\cap
\Upsilon \times \Xi^A_{g,k}=\emptyset.\\
\end{array} \leqno(4.15)$$
Then, it follows from the same argument as in
the proof of Corollary 4.5 that
$$Z(A,\mu,g,k,J,\nu, H,K)=(\Upsilon \times \Xi^A_{g,k}\times H\times
P)^{-1}(\Delta)\subset {\cal M}_A(\mu, g,k,
J,\nu)_I\times Q^{top}\times K^{top}\leqno(4.16)$$
is a compact, smooth, oriented $1$-manifold with boundary
$$\partial(Z(A,\mu,g,k,J,\nu, H,K))=Z(A,\mu,g,k,J,\nu, F,K)\bigcup
-Z(A,\mu,g,k,J,\nu, F',K).\leqno(4.17)$$
Hence,
$$|Z(A,\mu,g,k,J,\nu,F,K)|=|Z(A,\mu,g,k,J,\nu,F',K)|.\leqno(4.18)$$
Next, we fix a $(Y,F)$ and consider $(L, P)$. By choosing a small perturbation
of $P$ relative to $\partial(L)$,
we can assume that $(L,P)$ is in general position and $F\times P$ is
transverse
to $\Upsilon\times \Xi^A_{g,k}$. Repeating the previous argument, we can
show that
$$|Z(A,\mu,g,k,J,\nu,F,K)|=|Z(A,\mu,g,k,J,\nu,F,K')|.\leqno(4.19)$$
\vskip 0.1in
\noindent
{\bf Lemma 4.8. }{\it $|Z(A,\mu,g,k,J,\nu,F,K)|$ is independent of generic
$(J,\nu)$.}
\vskip 0.1in
\noindent
{\bf Proof: }Let $(J', \nu')$ be another generic pair. Choose a generic path
$(J_t, {\nu}_t)$ connecting $(J,\nu)$ to $(J', \nu')$, such
that
$${\cal M}_{\bar{D}}(\mu, g,k, (J_t),(\nu_t))_{\bar {\kappa}}=\bigcup_t
{\cal M}_{\bar{D}}(\mu, g,k, J_t, \nu_t)_{\bar {\kappa}} \times \{t\}$$
is a smooth, oriented cobordism between
${\cal M}_{\bar{D}}(\mu, g,k, J, \nu)_{\bar {\kappa}}$
and ${\cal M}_{\bar{D}}(\mu, g,k, J', \nu')_{\bar {\kappa}}$.
By Lemma 4.2 and choosing a small perturbation if necessary,
we can assume that $F\times
K$ is transverse to $\Upsilon^{(t)}\times (\Xi^A_{g,k})^{(t)}$, where the
choices of $F, K$ do not affect our result by the Lemma 4.7. Then, a dimension
accounting argument shows that
$$K\times F\cap \Upsilon^{(t)} \times (\Xi^A_{g,k})^{(t)}
(\overline{{\cal M}}^r(\mu, g,k,
(J_t),(\nu_t))-{\cal M}_A(\mu, g,k,(J_t), (\nu_t))_I)=\emptyset,$$
$$ K\times F(Y\times G-Y^{top}\times G^{top})\cap \Upsilon^{(t)} \times
(\Xi^A_{g,k})^{(t)}=\emptyset. \leqno(4.20)$$
Then,
$$\begin{array}{rl}
&Z(A,\mu,g,k, (J_t), (\nu_t), F,K)=(\Upsilon^{(t)}
\times (\Xi^A_{g,k})^{(t)}\times K\times F)^{-1}(\Delta)\\
\subset &{\cal M}_A(\mu, g,k, (J_t), (\nu_t))_I
\times Y^{top}\times G^{top}\\
\end{array}
\leqno(4.21)$$
is a compact, smooth, oriented $1$-manifold with boundary. Moreover,
$$\partial(Z(A,\mu,g,k, (J_t), (\nu_t), F,K))=Z(A,\mu,g,k, J, \nu, F,K)\cup
-Z(A,\mu,g,
k, J, \nu', F,K).$$
Hence,
$$|Z(A,\mu,g,k,J',\nu',F,K)|=|Z(A,\mu,g,k,J,\nu,F,K)|.\leqno(4.22)$$
\vskip 0.1in
\noindent
{\bf Lemma 4.9. } {\it $|Z(A,\mu,g,k,J,\nu,F,K)|$ is independent of
semi-positive
symplectic deformations.}
\vskip 0.1in
\noindent
{\bf Proof:} Let $\omega_t$ be a family of semi-positive symplectic deformation
of $\omega_0=\omega$. Then, we can choose a generic path $(J_t, \nu_t)$
such that $J_t$ is $\omega_t$-tamed. Then, this lemma follows from the
same arguments as the proof of Lemma 4.7.
\vskip 0.1in
Next, we prove Proposition 2.9, which gives the analytic foundation
for the composition law (Theorem 2.10). Then,
the composition law can be easily
derived from Proposition 2.9 by using an observation of Witten.
First, we extend the definition of $\Psi$ to the case
that $(G,K)$ is in the
boundary, say $K(G)\subset \overline{{\cal M}}^{\mu}_{\Sigma}$, where
$\overline{{\cal M}}^{\mu}_{\Sigma}$ is one stratum of $\overline{{\cal M}}^{\mu}_{g,k}$.
If $(G,K)$ has more than
two components lying in different strata, the corresponding GW-invariant is
just the sum of the GW-invariants of each component. Therefore, without
loss of generality, we can assume that $(G,K)$ is in the one stratum of
$\overline{{\cal M}}^{\mu}_{g,k}$. Note that $\overline{{\cal M}}^{\mu}_{\Sigma}$ is also a
PL-manifold with
local quotient singularities where the
Poincare Duality holds over the
rational coefficient. Therefore, we can replace $\overline{{\cal M}}^{\mu}_{g,k}$ by
$\overline{{\cal M}}^{\mu}_{\Sigma}$ in our construction of $\Psi$. In this case, the
proper moduli space
is $\overline{{\cal M}}^r_{\Sigma}(\mu, g,k,J,\nu)$. Note that we still have
the restriction of $\Xi^A_{g,k}$, $\Upsilon$
to $\overline{{\cal M}}^r_{\Sigma}(\mu,
g,k,J,\nu)$. We shall denote it by $\Xi^A_{\Sigma}$ and $\Upsilon_{\Sigma}$.
Then, we can choose a generic $J,\nu$ and $F$ and $K$ in general position
(inside $\overline{{\cal M}}^{\mu}_{\Sigma}$),i.e., $K(G^{top})\subset {\cal M}^{\mu}_{
\Sigma, \bar{I}}$, where $\bar{I}$ means the trivial automorphism group for
each component. Repeating the
construction of Lemma 4.4, 4.5,
we can define
$$\Psi^V_{(A,g,k, \mu)}(K, \alpha_1, \cdots, \alpha_k)$$
as the algebraic sum of
$$Z(A, \overline{{\cal M}}^{\mu}_{\Sigma}, J,\nu, F,K)=
(\Upsilon_{\Sigma}\times \Xi^A_{\Sigma}\times K\times F)^{-1}(\Delta)\subset
{\cal M}_{\Sigma}(\mu, g,k,J,\nu)_{\bar{I}}\times Y^{top}\times
G^{top}\leqno(4.23).$$
Before proving Proposition 2.9, we need the following family version of
the gluing theorem in \cite{RT}.
Let $U\subset \bar{U}\subset{\cal M}^{\mu}_{\Sigma, \bar{I}}$ be
an oriented submanifold (not necessarily closed) and
$$\phi: U\times [0, \epsilon) \rightarrow \overline{{\cal M}}^{\mu}_{g,k}$$
be a diffeomorphism such that $\phi(U\times \{0\})=U$ and if $t\neq 0$,
$\phi(U\times \{t\})\subset
{\cal M}^{\mu}_{g,k, I}$.
Let
$$U_t=\phi(U\times \{t\}).\leqno(4.24)$$
Define
$${\cal M}_{A}(U_0, J, \nu)=(\Upsilon)^{-1}(U_0)\cap {\cal M}_{A,\Sigma}(\mu, g,k,J,\nu)_{
\bar{I}}.$$
For each $t\neq 0$, we define
$${\cal M}_{A}(U_t, J, \nu)=(\Upsilon)^{-1}(U_t)\cap {\cal M}_{A}(\mu, g,k,J,\nu)_I.
\leqno(4.25)$$
Fix a generic $(J,\nu)$ and $\phi$ such that for each t, ${\cal M}_{A}(U_t, J, \nu)$
is a smooth manifold of dimension
$$2c_1(V)(A)+2n(1-g)+\dim U_t.\leqno(4.26)$$
Then, the following is a slight modification of Theorem 6.1, [RT].
We leave the details to the readers.
\vskip 0.1in
\noindent
{\bf Gluing Theorem: }{\it \it Let $f_0$ be any map in
${\cal M} _A ( U _0 , J , \nu )$.
Then there is
a continuous family of injective maps $T _t$ from
${\cal W}$ into ${\cal M} _A ( U _t , J , \nu )$, where $t$
is small and ${\cal W}$ is a neighborhood of $f_0$ in
${\cal M} _A ( U _0 , J , \nu )$,
such that (1) for any $f$ in ${\cal W}$, as $t$ goes to zero,
$T _t(f)$ converges to $f$ in $C^0$-topology on $\Sigma _0$ and
in $C^3$-topology outside
the singular set of $\Sigma _0$; (2) there are $\epsilon , \delta > 0$
satisfying : if $f^\prime$ is in ${\cal M} _A ( U_t , J , \nu )$
and
$$d_V(f^\prime(x), f_0(y)) \leq \epsilon, ~~~{\rm whenever}~x \in \Sigma _t
\in U_t,$$
where $d_V$ is the distance functions
of a $J$-invariant metric $h_V$ on $V$ and then $f^\prime$ is
in $T_t({\cal W})$. Moreover, for each $t$, $T_t$ is an orientation-preserving
smooth map from
${\cal W}$ into ${\cal M} _A ( U _t , J , \nu )$.}
\vskip 0.1in
\noindent
{\bf Proof of Proposition 2.9: } If $(G',K')$ is another pseudo-submanifold
representing the same homology class, we can assume
that there is a pseudo-manifold cobordism $(L, P)$ between $(G, K)$ and
$(G', K')$.
Without loss of the generality, we may assume that $(G', K')$ is
in general position. By choosing a small perturbation relative to the
boundary if necessary, we can further assume that
$$P(L^o)\subset {\cal M}^{\mu}_{g,k, I}. \leqno(4.27)$$
Again, by counting dimensions, we can show that
$$F\times P \bigcap \Upsilon\times \Xi^A_{g,k}
(\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)
-{\cal M}_A(\mu, g,k, J, \nu)_I\cup {\cal M}_{\Sigma}(\mu,
g,k,J,\nu)_{\bar{I}})=\emptyset,$$
$$F\times P(Y\times L-Y^{top}\times L^{top})=\emptyset. \leqno(4.28)$$
Then
$$\begin{array}{rl}
&Z(A,\mu,g,k,J,\nu, F, L)=
(\Upsilon\times \Xi^A_{g,k}\times F\times P)^{-1}(\Delta)\\
\subset &{\cal M}_A(\mu, g,k, J, \nu)_I\cup {\cal M}_{\Sigma}(\mu, g,k,J,\nu)_{\bar{I}}
\times Y^{top}\times L^{top}.\\
\end{array}
\leqno(4.29)$$
Since $L^{top}$ is a manifold with boundary, we can choose an open subset
$$\tilde{L}\subset \overline{\tilde{L}}\subset L^{top}\leqno(4.30)$$
such that
$$Z(A,\mu,g,k,J,\nu, F, L)\subset {\cal M}_A(\mu, g,k, J, \nu)_I\cup
{\cal M}_{\Sigma}(\mu,g,k,J,\nu)_{\bar{I}}\times
Y^{top}\times \tilde{L},\leqno(4.31)$$
$$\partial{\tilde{L}}=U\cup U'$$
where $U\subset G^{top}, U'\subset (G')^{top}.$ Furthermore, there is a
uniform constant $\epsilon$ such that there are collars
$$U\times [0, \epsilon)\subset \tilde{L};~~~ U'\times [0, \epsilon)\subset
\tilde{L}.\leqno(4.32)$$
Thus,
$$Z(A,\mu,g,k,J, \nu, F, \tilde{L})=Z(A,\mu,g,k,J,\nu, F, L).\leqno(4.33)$$
Let $U_t=U\times\{t\}$ and $U'_s=U'\times \{s\}.$
It follows from
an ordinary cobordism argument, which is identical to that of Lemma 4.6,
that
$Z(A, g,k,J,\nu, \tilde{L}-U\times [0,\epsilon)\cup U'\times [0,\epsilon))$
is a smooth, compact, oriented 1-manifold with boundary
$$\begin{array}{rl}
&\partial(Z(A, g,k,J,\nu, \tilde{L}-U\times [0,\epsilon)\cup U'\times [0,
\epsilon)))\\
=&Z(A,\mu,g,k,J,\nu, F,
U_{\epsilon})\cup - Z(A,\mu,g,k, J,\nu, F, U'_{\epsilon}).\\
\end{array}
\leqno(4.34)$$
Therefore,
$$|Z(A,\mu,g,k,J,\nu, F, U_{\epsilon})|=|Z(A,\mu,g,k,J,\nu, F,
U'_{\epsilon})|.\leqno(
4.35)$$
Obviously, we can use any $U_t, U'_s$ in the place of $U_{\epsilon}, U'_{
\epsilon}$. Then, we show
$$|Z(A,\mu,g,k,J,\nu, F, U_t)|=|Z(A,\mu,g,k,J,\nu, F, U'_{s})|\leqno(4.36).$$
Next, we study the behavior of $Z(A,\mu,g,k,J,\nu,U_t)$ as $t\rightarrow 0$.
By the Gromov-Uhlenbeck
compactness theorem, and by taking a subsequence if necessary,
we may assume that any sequence $f_t\in Z(A,\mu,g,k,J,\nu,U_t)$ converges to a
$$f\in \overline{{\cal M}}_{\Sigma}(\mu, g,k,J,\nu)_{\bar{I}}\times Y\times K.
\leqno(4.37)$$
By (4.28),
$$f\in {\cal M}_{\Sigma}(\mu, g,k,J, \nu)_{\bar{I}}\times Y^{top}\times
K^{top}.\leqno(4.38
)$$
Hence, $f\in Z(A,\mu,g,k,J,\nu, U_0)$ by (4.31). On the other hand, by
the gluing
theorem, for $t$ small, there is a bijective map
$$T_t: Z(A,\mu,g,k,J,\nu, U_0)\rightarrow Z(A,\mu,g,k,J,\nu, U_t)\leqno(4.39)$$
preserving the orientation. Therefore,
$$|Z(A,\mu,g,k,J,\nu, U_0)|=|Z(A,\mu,g,k,J,\nu, U_t)|. \leqno(4.40)$$
Since $(G',K')$ is in general position, we can allow $s=0$ in the (4.36).
Therefore,
$$|Z(A, g,k, J,\nu, U_0)|=|Z(A,\mu,g,k,J,\nu,U'_0)|. \leqno(4.41)$$
Thus, we finish the proof of Proposition 2.9.
\vskip 0.1in
\noindent
{\bf Proof of Theorem 2.10. } Let's consider the first case where we have the
embedding
$$\theta_S: \overline{{\cal M}}_{g_1, k_1+1}\times \overline{{\cal M}}_{g_2, k_2+1}$$
by identifying the $k_1+1$-th marked point of the first component with the
$1$-st marked point of the second component. Suppose that the marked points
in the first component are $\{x_1, \cdots, x_{k_1}, y_1\}$ and the marked
points in the second component are
$\{y_2, x_{k_1+1}, \cdots, x_{k_1+k_2}\}$.
The map $\theta_S$ identifies $y_1$ with $ y_2$ and gives
rise to a stable curve of
genus $g_1+g_2$ with marked points $\{x_1, \cdots x_{k_1+k_2}\}$.
By (2.17), for their finite covers,
$$\theta_S^*\overline{{\cal M}}^{\mu}_{g,k}=\overline{{\cal M}}^{\mu}_{g_1, k_1+1}\times
\overline{{\cal M}}^{\mu}_{g_2, k_2+1}.$$
Let $[K_i]\in H_*(\overline{{\cal M}}^{\mu}_{g_i,k_i+1}, {\bf Q})$ represented by the
pseudo-submanifold $K_i$. By Proposition 2.9,
$$\Psi^V_{A,g,k, \mu}([\theta_S(K_1\times K_2)]; \{\alpha_i\})$$
doesn't depend on the particular representative of $[\theta_S(K_1\times K_2)]$.
In particular, it could have a representative,
which is in general position. This is very
important in the later applications. Here, we choose a representative
$\theta_S(K_1\times K_2)$. Let
$$\overline{{\cal M}}^{\mu}_{\Sigma}=\theta_S
(\overline{{\cal M}}^{\mu}_{g_1, k_1+1}\times \overline{
{\cal M}}^{\mu}_{g_2, k_2+1})\leqno(4.42)$$
Then,
$$Z(A, \mu, g,k,J,\nu, F, \theta_S(K_1\times K_2))=Z(A, \theta_S(\overline{{\cal M}}^
{\mu}_{g_1, k_1+1}\times \overline{{\cal M}}^{\mu}_{g_2, k_2+1}), J, \nu, F,
\theta_S(K_1\times K_2)).
\leqno(4.43)$$
Recall (4.23) that
$$Z(A,\overline{{\cal M}}^{\mu}_{\Sigma},J,\nu, F, \theta_S(K_1\times
K_2))\leqno(4.44)$$
$$\mbox{\hskip 0.5in} =(\Upsilon_{\Sigma}\times \Xi^A_{\Sigma}\times F\times
\theta_S(K_1\times K_2))^{-1}(\Delta)\subset {\cal M}_{\Sigma}(\mu, g,k,J,\nu)_{
\bar{I}}
\times Y^{top}\times \theta_S(K_1\times K_2)^{top}.$$
But any element of ${\cal M}_{\Sigma}(\mu, g,k,J,\nu)_{\bar{I}}$ is a pair
$$(f_1, f_2)\in {\cal M}_{A_1}^{\mu}
(\mu, g_1,k_1+1,J,\nu)_I\times {\cal M}^{\mu}_{A_2}(\mu,g_2,k_2+1,
J,\nu)_I; ~~f_1(y_1)=f_2(y_2)\leqno(4.45)$$
with $A_1+A_2=A$. Let $e^{A_1}_{y_1}$ be the evaluation map of $y_1$'s and
$e^{A_2}_{y_2}$ be the evaluation map of $y_2$'s. Then,
$${\cal M}_{\Sigma}(\mu, g,k,J,\nu)_{\bar{I}}=\bigcup_{A_1+A_2=A}
(e^{A_1}_{y_1}\times
e^{A_2}_{y_2})^{-1}(\delta), \leqno(4.46)$$
where $\delta$ is the ordinary diagonal in $V\times V$.
Using this decomposition and switching the order of the components
appropriately, we have
$$\bigcup_{A_1+A_2=A}\Upsilon_{A_1}\times \Upsilon_{A_2}\times \Xi^{A_1}_{g_1,
k_1}\times \Xi^{A_2}_{g_2, k_2}\times K_1\times K_2\times F \times
e^{A_1}_{y_1}
\times e^{A_2}_{y_2},\leqno(4.47)$$
which maps the space
$$\bigcup_{A_1+A_2=A}{\cal M}_{A_1} (\mu, g_1, k_1+1, J, \nu)_I\times
{\cal M}_{A_2}(\mu, g_2, k_2+1, J, \nu)_I\times K^{top}_1\times K^{top}_2\times
Y^{top}
$$
into
$$\mbox{\hskip 1in} {\cal M}^{\mu}_{\Sigma_1,I}\times {\cal M}^{\mu}_{\Sigma_2,I}\times
V^{k_1}\times
V^{k_2}\times {\cal M}^{\mu}_{\Sigma_1,I}\times
{\cal M}^{\mu}_{\Sigma_2,I}\times V^{k_1+k_2}\times V \times V.$$
We will denote this map
by $\Upsilon_{\Sigma}\times \Xi^A_{\Sigma}\times F\times \theta_S(K_1\times
K_2)$. Let
$$\Delta_{A_1, A_2}\subset ({\cal M}^{\mu}_{\Sigma_1,I}\times {\cal M}^{\mu}_{\Sigma_2,I}
\times V^{k_1}\times V^{k_2})\times ( {\cal M}^{\mu}_{\Sigma_1,I}\times
{\cal M}^{\mu}_{\Sigma_2,I}\times V^{k_1+k_2})\leqno(4.48)$$
be the diagonal.
Define
$$\begin{array}{rl}
&Z(A_1, A_2, g_1, g_2, k_1, k_2, \delta, J,\nu)\\
=&(\Upsilon_{A_1}\times
\Upsilon_{A_2}\times \Xi^{A_1}_{g_1, k_1}\times \Xi^{A_2}_{g_2, k_2}\times
K_1\times K_2\times F \times e^{A_1}_{y_1}\times e^{A_2}_{y_2})^{-1}(
\Delta_{A_1, A_2}\times \delta).
\end{array}
\leqno (4.49) $$
Then,
$$Z(A,\overline{{\cal M}}^{\mu}_{\Sigma},J,\nu, F, \theta_S(K_1\times
K_2))=\bigcup_{A_1+
A_2=A}Z(A_1, A_2, g_1, g_2, k_1, k_2, \delta, J,\nu)\leqno(4.50)$$
$$=\bigcup_{A_1+A_2=A}(\Upsilon_{A_1}\times
\Upsilon_{A_2}\times \Xi^{A_1}_{g_1, k_1}\times \Xi^{A_2}_{g_2, k_2}\times
K_1\times K_2\times F \times e^{A_1}_{k_1+1}\times e^{A_2}_{k_2+1})^{-1}(
\Delta_{A_1, A_2}\times \delta). $$
A corbordism argument as that of Lemma 4.6 shows that
$|Z(A_1, A_2, g_1, g_2,
k_1, k_2, \delta, J,\nu)|$ only depends on the homology class of
$[\delta]\in H_*(V\times V, {\bf Z})$. Choose a homogeneous basis $\{\beta_b\}_{1
\leq b\leq L}$ of $H_*(V, {\bf Z})$ modulo torsion. Let $(\eta_{ab})$ be its
intersection matrix. Note that $\eta_{ab}=\beta_a.\beta_b=0$ if the dimension
of $\beta_a$ and $\beta_b$ are not complementary to each other. Let
$(\eta^{ab})$ be the inverse of $(\eta_{ab})$. Then,
$$[\delta]=\sum_{a,b} \eta^{ab}\beta_a\otimes \beta_b. \leqno(4.51)$$
Choose a pseudo-submanifold representing $\beta_a$ (still denoted it by
$\beta_a$). Then, one observes that
$$|Z(A_1, A_2, g_1, g_2, k_1, k_2, \delta, J,\nu)|\leqno(4.52)$$
$$=\sum_{a,b} \eta^{ab}|Z(A_1, \mu, g_1, k_1, \beta_a, J,\nu, \prod_{i\leq
k_1}
Y_i, K_1)||Z(A_2, \mu, g_2, k_2, \beta_2, J,\nu, \prod_{j>k_1} Y_j, K_2)|.$$
Together with (4.50), it yields the first formula of Theorem 2.10
$$\begin{array}{rl}
&\Psi ^V_{(A,g,k,\mu)}(\theta _{S*}[K_1\times K_2];\{\alpha _i\})\\
=~& \sum \limits _{A=A_1+A_2} \sum \limits_{a,b}
\Psi ^V_{(A_1,g_1,k_1+1, \mu)}([K_1];\{\alpha _{i}\}_{i\le k}, \beta _a)
\eta ^{ab}
\Psi ^V_{(A_2,g_2,k_2+1,\mu)}([K_2];\beta _b,
\{\alpha _{j}\}_{j> k}) \\
\end{array}
\leqno (4.53)
$$
The second formula can be derived in the similar fashion. Here, we have an
embedding
$$\mu: \overline{{\cal M}}_{g-1, k+2}\rightarrow \overline{{\cal M}}_{g, k}\leqno(4.54)$$
by gluing last two marked points $x_{k+1}, x_{k+2}$. By (2.18), $\mu$ induces
a map on the corresponding finite covers
$$\bar{\mu}: \overline{{\cal M}}^{m_{\mu}}_{g-1, k+2}\rightarrow
\overline{{\cal M}}^{\mu}_{
g, k}.$$
We choose
a representative
$\bar{\mu}(K)$. Let
$$\overline{{\cal M}}^{\mu}_{\Sigma}=\bar{\mu}(\overline{{\cal M}}^{m_{\mu}}_{g,
k+2}).\leqno(4.55)$$
Notes that
$${\cal M}_{\Sigma}(\mu, g,k,I,J,\nu)=(e_{k+1}\times e_{k+2})^{-1}(\delta).$$
It implies that
$$Z(A,\mu, g,k,J,\nu, F, \mu(K))=Z(A, \overline{{\cal M}}^{\mu}_{\Sigma}, J,\nu, F,
\mu(K))\leqno(4.56)$$
$$=Z(A, m_{\mu}, g-1, k+2, \delta, J, \nu, F, K).$$
By (4.52),
$$|Z(A,\mu, g,k,J,\nu, F, \mu(K))|=\sum_{a,b} \eta^{ab}|Z(A, m_{\mu}, g-1,
k+2, J,\nu,
Y\times \beta_a\times \beta_b, K)|.\leqno(4.57)$$
It yields the second formula
$$\Psi ^V_{(A,g,k, \mu)}(\bar{\mu}_*[K_0];\alpha _1,\cdots, \alpha _k)
=\sum _{a,b} \Psi ^V_{(A,g-1,k+2, m_{\mu})}([ K_0];\alpha _1,\cdots, \alpha _k,
\beta _a,\beta _b) \eta ^{ab}\leqno (4.58)
$$
We finish the proof of Theorem 2.10.
\section{Stabilizing Conjecture}
One of the initial motivations for studying the
GW-invariants is to use it to distinguish nondeformation equivalent symplectic
manifolds. For example, the first author had successfully calculated the
genus-$0$ GW-invariant in \cite{R1} to produce a large number of diffeomorphic,
non-symplectic-deformation equivalent symplectic manifolds, whose
existence were unknown in symplectic topology before. During the course of work
in \cite{R1}, the first author observed a correspondence
between the
differentiable category of symplectic 4-manifolds $X$ and the symplectic
deformation category of its stabilized manifold $X\times {\bf P}^1$. It can be
summarized as the following stabilizing conjecture:
\vskip 0.1in
\noindent
{\bf Stabilizing Conjecture \cite{R1}: }{\it Suppose that $X,Y$ are two simply
connected
homeomorphic symplectic 4-manifolds. $X, Y$ are diffeomorphic iff the
stabilized manifolds $X\times {\bf P}^1,
Y\times {\bf P}^1$ with the product symplectic structures are deformation
equivalent.}
\vskip 0.1in
It has been demonstrated by the
Donaldson theory that the differentiable structure of smooth
4-manifolds is a delicate problem. However, by the results of Freedman, the
two simply connected 4-manifolds $X, Y$ are
homeomorphic if and only if $X\times {\bf P}^1, Y\times {\bf P}^1$ are diffeomorphic.
Therefore, the delicate problem about the differentiable structures of smooth
4-manifolds will disappear after the stabilizing process. On the other hand,
the stabilizing conjecture can be viewed as an analogy of Freedman's theorem
between
the smooth and the symplectic category.
The first pair of examples supporting the
conjecture were constructed in
\cite{R1}, where $X$ is the blow-up of ${\bf P}^2$ at 8-points and $Y$ is a Barlow
surface. Furthermore, the first author also verified the conjecture for the
cases: (1). $X$ is rational, $Y$ is irrational; (2). $X, Y$ are irrational but
have different number of $(-1)$ curves. Since then, a lot of more evidences
supporting the stabilizing Conjecture have been discovered.
Note that the first Chern class is an obvious symplectic deformation
invariant. The stabilizing conjecture implies that the first Chern class
of a simply connected symplectic 4-manifold is a differentiable invariant,
which was only proved
recently by Taubes \cite{T1}. Recently, the first author was informed by
Donaldson that his results on the existence of
symplectic submanifolds imply that if
$X\times {\bf P}^1, Y\times {\bf P}^1$ are symplectic deformation equivalent, then some
branched covers of $X, Y$ are diffeomorphic. In this section, we will compute
some higher genus GW-invariants to prove the stabilizing conjecture for the
case of simply connected elliptic surfaces $E^n_{p,q}$. The examples
$E^n_{p,q}\times {\bf P}^1$ were suggested to the first author by Donaldson.
Let's recall the construction of $E^n_{p,q}$. Let $E^1$ be the blow-up
of ${\bf P}^2$ at generic 9 points, and let $E^n$ be the fiber connected
sum of $n$ copies of $E^1$. Then $E^n_{p,q}$ can be obtained from $E^n$ by
logarithmic transformations alone two smooth fibers with multiplicity $p$ and
$q$. Note that
$E^n_{p,q}$ is simply connected if and only if $p,q$ are coprime.
Moreover, the Euler number $\chi(E^n_{p,q})=12n$, and hence $n$ is a
topological number.
\vskip 0.1in
\noindent
{\bf Theorem 5.1. }{\it $E^n_{p,q}\times {\bf P}^1, E^n_{p',q'}\times {\bf P}^1$ with
product symplectic structures are symplectic deformation equivalent if and only
if $(p,q)=(p', q')$.}
\vskip 0.1in
Combining with known results about the smooth classification of $E^n_{p,q}$ by
\cite{Ba}, \cite{FM}, \cite{MO}, \cite{MM}, we can prove
\vskip 0.1in
\noindent
{\bf Corollary 5.2. }{\it The stabilizing conjecture holds for $E^n_{p,q}.$}
\vskip 0.1in
Let $F_p, F_q$ be two multiple fibers and $F$ be a general fiber. Let
$A_p=[F_p], A_q=[F_q]$. It is known that $A_p=\frac{[F]}{p}, A_q=\frac{[F]}{q}
$. The primitive class $A=[F]/pq$. Another piece of topological information is
that the canonical class $K$ is Poincare dual to
$$(n-2)F+(p-1)F_p+(q-1)F_q=((n-2)pq+(p-1)p+(q-1)q)A.
\leqno(5.1)$$
Then, the Theorem 5.2 follows from the following calculation
\vskip 0.1in
\noindent
{\bf Proposition 5.3. }{\it
$$\Psi^{E^n_{p,q}\times {\bf P}^1}_{(mA,1,1)}(\overline{{\cal M}}_{1,1}; \alpha)=
\left\{ \begin{array}{ll}
2q(A\cdot \alpha) ;& m=q (mA=A_p), \\
2p(A\cdot \alpha); &m=p (mA=A_q), \\
0; & m\neq p,q \mbox{ and } m<pq,
\end{array}\right. \leqno(5.2) $$
where $\alpha$ is a 4-dimensional homology class. In particular,
$$\Psi^{E^n_{p,q}\times {\bf P}^1}_{(mA,1,1)}(\overline{{\cal M}}_{1,1}; \cdot)\neq 0
\mbox{ for } m=p,q.$$}
\vskip 0.1in
\noindent
{\bf Proof: } By the deformation theory of elliptic surfaces, we can
choose a complex structure $J_0$ on $E^n_{p,q}$
such that all singular fibers are
nodal elliptic curves. Furthermore, we can assume that the complex structures
of multiple fibers are generic,
i.e., whose $j$-invariants are not 0 or 1728.
We shall choose $\nu=0$. Therefore, there is no need to consider finite covers.
We shall drop $\mu$ from the notation.
Let $j_0$ be the standard complex structure on ${\bf P}^1$.
Let's describe $\overline{{\cal M}}_{mA}(1,1,J_0\times j_0, 0)$ for $m< pq$. For
any $f\in \overline{{\cal M}}_{mA}(1,1,J_0\times j_0, 0)$, the image $im(f)$ is a
connected effective holomorphic
curve.
Namely, $im(f)=\sum_i a_i C_i$ where $a_i>0$ and $C_i$ are irreducible
components. Note that
$$mA=\sum_i a_i [C_i].\leqno(5.3)$$
For the product complex
structure $J_0\times j_0$, $C_i=(C^1_i, C^2_i)$ where $C^1_1\subset E^n_{p,q},
C^2_i\subset {\bf P}^1$. $C^2_i$ can be realized as a holomorphic map from either
an elliptic curve or a rational curve to ${\bf P}^1$. Hence, $[C^2_i]=p_i [{\bf P}^1]$
for $p_i\geq 0$. By (5.3), $p_i=0$ and $C^2_i=\{x\}$ for some $x\in {\bf P}^1$.
Since
$im(f)$ is connected. We can write
$$im(f)=(\sum a_i C^1_i)\times \{x\},\leqno(5.4)$$
where $\sum a_i C^1_i$ is a connected effective curve. By our assumption on
singular fibers, each $C^1_i$ is
either a multi-section or a fiber. A multi-section has positive
intersection with a general fiber. A fiber has zero intersection with general
fiber. Since $mA\cdot [F]=0$, this implies that each $C^1_i$ is a fiber,
i.e., a general fiber, a singular fiber or a multiple fiber.
Since $m<pq$ and
a singular fiber has the same homology class of a general fiber, $C^1_i$ can
only be
a multiple fiber. Since $im(f)$ is connected, therefore, $im(f)$ is either
$F_p\times \{x\}$ or $F_q\times \{x\}$. In particular,
$$\overline{{\cal M}}_{mA}(1,1,J_0, 0)=\emptyset \mbox{ for }m\neq p,q \mbox{ and }
m<pq. \leqno(5.5)$$
Obviously, $(J_0, 0)$ is $mA$-good for such $m$'s. Hence
$$\Psi^{E^n_{p,q}\times {\bf P}^1}_{(mA,1,1)}(\overline{{\cal M}}_{1,1}; \alpha)=
0 \mbox{ for } m\neq p,q \mbox{ and } m<pq.\leqno(5.6
)$$
Fix a marked point $y_o$,
$$\overline{{\cal M}}_{A_p}(1,1,J_0\times j_0, 0)=\{f: F_p\rightarrow E^n_{p,q}
\times {\bf P}^1 ~|~ im(f)=F_p\times \{x\} \}/{\bf Z}_2$$
$$=Aut(F_p)/{\bf Z}_2\times {\bf P}^1=F_p\times {\bf P}^1,\leqno(5.7)$$
because a general element of $\overline{{\cal M}}_{1,1}$ has automorphism
group ${\bf Z}_2$.
Recall the definition of $A_p$-goodness (Definition 2.18). Since
$$\overline{{\cal M}}^r_{A_p}(1,1,J_0\times j_0)={\cal M}_{A_p}(1,1,J_0\times j_0,0).
\leqno(5.8)$$
Definition 2.18, (2) is automatically satisfied. Unfortunately,
Definition 2.18, (1) is not satisfied.
This can also be viewed from the fact that the virtual dimension
$$2c_1(V)(mA)+(3-n)(g-1)+2=2,$$
but we have a moduli space of real dimension 4. For each $f\in {\cal M}_{A_p}(1,1,
J_0\times j_0, 0)$, the normal bundle
$$N_f (E^n_{p,q}\times {\bf P}^1)=N_{F_p}(E^n_{p,q})\otimes T_x {\bf P}^1, \leqno(5.9)$$
where $im(f)=F_p\times \{x\}$. It is known that $N_{F_p}(E^n_{p,q})$ is a
torsion element of order $p$. Hence,
$$H^1(N_f)=T_x {\bf P}^1.\leqno(5.10)$$
Furthermore, since $f$ is an embedding, we have a short exact exact sequence
$$0\rightarrow TF_{p}\rightarrow T_f E^{n}_{p,q}\times {\bf P}^1\rightarrow N_f
\rightarrow 0. \leqno(5.11)$$
It induces a long exact sequence of cohomologies
$$ H^1(F_p)\rightarrow H^1(T_f E^{n}_{p,q}\times {\bf P}^1)\rightarrow
H^1(N_f)\rightarrow 0.\leqno(5.12) $$
Hence, the obstruction space (3.11), (3.17)
$$Coker(D_f\oplus J\cdot df)=H^1(N_f)=T_x {\bf P}^1.\leqno(5.13)$$
The obstruction bundle
$$COK= \pi_2^*T{\bf P}^1,\leqno(5.14)$$
i.e., the pull-back of tangent bundle of ${\bf P}^1$.
Now we need to use
following result which is an analogue of Proposition 5.7 in \cite{R2} for genus
zero invariants. The proofs are identical. We shall
adapt the notation of Lemma 4.6.
\vskip 0.1in
\noindent
{\bf Proposition 5.4. }{\it Suppose that $(J_0, \nu_0)$ is not $A$-good,
but satisfies the following hypotheses:
$$(1).\ K\times F\cap \Upsilon_A\times \Xi^A_{g,k}(\overline{{\cal M}}^r_{A}(g,k,
J_0,
\nu_0)-{\cal M}_A (g,k, J_0,\nu_0))=\emptyset\leqno(5.15)$$
and hence $Z(A,g,k,J,\nu, F, K)$ {\rm (cf. (4.13))} is compact.
(2). $\dim Coker(D_f\oplus J\cdot df)$ is constant for all $f\in
Z(A,g,k,J,\nu,F,K)$ and
$Z(A,g,k,J,\nu,F,K)$ is a smooth manifold with the dimension
$ \dim Coker(D_f\oplus J\cdot df)$. For any generic $(J,\nu)$ sufficiently
close to
$(J_0,\nu_0)$, $Z(A,g,k,J,\nu,F,K)$ is oriented cobordant to the zero set of a
transverse section of the
obstruction bundle $COK$. Hence it is dual to the Euler class of the
obstruction bundle $COK$.}
\vskip 0.1in
Now we continue the the proof of
Proposition 5.3. Note that $H_4(E^n_{p,q}\times
{\bf P}^1)=H_2(E^n_{p,q})\otimes H_2({\bf P}^1)$. Without the loss of generality, suppose
that $\alpha=\beta\otimes [{\bf P}^1]$. Choose a smooth surface $Y\subset
E^n_{p,q}$ such that $Y$ represents $\beta$ and intersects $F_p$ transversely.
Then,
$$Z(A,1,1,J_0,0,Y,{\cal M}_{1,1})=\bigcup_{y\in F_p\cap Y} \{y\}\times {\bf P}^1.
\leqno(5.16)$$
Then, by Proposition 5.4,
$$\Psi^{E^n_{p,q}\times{\bf P}^1}_{(A_p,1,1)}(\overline{{\cal M}}_{1,1}, \alpha)=
(A_p\cdot \alpha) e(T{\bf P}^1)
=2q(A_0\cdot \alpha).\leqno(5.17)$$
The same proof yields that
$$\Psi^{E^n_{p,q}\times{\bf P}^1}_{(A_q,1,1)}(\overline{{\cal M}}_{1,1}, \alpha)=
2p(A_0\cdot \alpha).\leqno(5.18)$$
We finish the proof of Proposition 5.3.
\vskip 0.1in
\noindent
{\bf Proof of Theorem 5.1: } First of all, if $(p,q)=(p',q')$, $E^n_{p,q},
E^n_{p',q'}$ were known to be complex deformation equivalent
as K\"ahler surfaces regardless where
we perform the logarithmic transform.
It follows that $E^n_{p,q}\times {\bf P}^1$ and $ E^{n}_{p',q'}\times {\bf P}^1$ with
product symplectic structures are deformation equivalent.
Conversely, suppose that $E^n_{p,q}, E^n_{p',q'}$ are
symplectic deformation equivalent. Then, there is a diffeomorphism
$$F: E^n_{p,q}\times {\bf P}^1\rightarrow E^n_{p',q'}\times {\bf P}^1\leqno(5.19)$$
such that
$$\Psi^{E^n_{p',q'}\times{\bf P}^1}_{(F_*(A),1,1)}(\overline{{\cal M}}_{1,1}, F_*(
\alpha))=\Psi^{E^n_{p,q}\times{\bf P}^1}_{(A,1,1)}(\overline{{\cal M}}_{1,1}, \alpha),
\leqno(5.20)$$
and
$$F^*c_i(E^n_{p',q'}\times{\bf P}^1)=c_i (E^n_{p,q}\times{\bf P}^1); F^*p_1(E^n_{p',q'}
\times{\bf P}^1)=p_1(E^n_{p,q}\times{\bf P}^1). \leqno(5.21)$$
Let $e_0\in H^2({\bf P}^1, {\bf Z})$ be the positive generator. First, we claim
$$F^*(e_0)=e_0.\leqno(5.22)$$
Suppose that
$F^*(e_0)=ne_0 + \beta$ for $\beta\in H^2(E^n_{p,q}, {\bf Z})$. Note that the first
Pontrjagan class $p_1(E^n_{p,q}\times {\bf P}^1)=p_1(E^n_{p,q})\neq 0$ and $p_1(
E^n_{p',q'}\times {\bf P}^1)=p_1(E^n_{p'q'})\neq 0$. Let $\gamma(E^n_{p,q})\in
H^4(E^n_{p,q}, {\bf Z})$ be such that $\gamma(E^n_{p,q})[E^n_{p,q}]=1$. Define
$\gamma(E^n_{p',q'})$ in the same way. Then $p_1(E^n_{p,q})$ is a
nonzero multiple of $\gamma(E^n_{p,q})$ and $p_1(E^n_{p',q'})$ is a non-zero
multiple of $\gamma(E^n_{p',q'})$. Thus, $F^*\gamma(E^n_{p',q'})=\gamma(
E^n_{p,q})$. Then,
$$\begin{array}{rl}
1&=(\gamma(E^n_{p',q'})\cup e_0)[E^n_{p',q'}\times {\bf P}^1]=F^*(\gamma(E^n_{p',
q'})\cup e_0)[E^n_{p,q}\times {\bf P}^1]\\
=&\gamma(E^n_{p,q})\cup (ne_0 +\beta)[
E^n_{p,q}\times {\bf P}^1]=n.\\
\end{array}
\leqno(5.23)$$
Hence $n=1$. Furthermore, $F^*(e_0^2)=0$. Then
$(e_0+\beta)^2=2e_0
\beta +\beta^2=0$. Therefore, $2e_0\beta=0$
and $\beta^2=0$, consequently, $\beta=0$.
$$c_1(E^{n}_{p,q}\times {\bf P}^1)=c_1(E^{n}_{p,q}) +2e_0.\leqno(5.24)$$
By (5.21), (5.22),
$$F^*(c_1(E^n_{p',q'}))=c_1(E^n_{p,q}).\leqno(5.25)$$
However,
$F$ sends primitive classes to primitive classes, so $F^*(A^*)=A^*$,
where $A^*$ is the Poincare dual of $A$. Hence,
$F_*(A)=A$ and
$$\Psi^{E^n_{p',q'}\times{\bf P}^1}_{(F_*(nA),1,1)}(\overline{{\cal M}}_{1,1}, F_*(
\alpha))=\Psi^{E^n_{p',q'}\times{\bf P}^1}_{(nA,1,1)}(\overline{{\cal M}}_{1,1}, F_*(
\alpha)).\leqno(5.26)$$
Suppose that $q<p$ and $q'< p'$.
Then, $A_p(=qA)$ and $A_q(=pA)$ are the first and the second class of
$\{nA\}$ such that
$\Psi$ is nonzero and so are $A_{p'}$ and $A_{q'}$. Hence
$$F_*(A_p)=A_{p'}, F_*(A_q)=A_{q'}.\leqno(5.27)$$
This implies that
$$p=p', q=q'.$$
We finish the proof of Theorem 5.1.
Even though $E^n_{p,q}\times S^2$ are diffeomorphic to each other, they may
have different first Chern classes. This problem can be resolved by blowing
up $E^n_{p, q}$ at one point. Namely, if $E^n_{p,q}\#\bar{{\bf P}}^2$ is a
blow-up of $E^n_{p,q}$ at one point, $E^n_{p,q}\#\bar{{\bf P}}^2$ are diffeomorphic
to each other and have the same first Chern class up to a diffeomorphism. By
a theorem of Wall, they have the same almost complex structure up to a
homotopy. Furthermore, we can choose the blow-up loci away
from multiple fibers. All the calculations in Theorem 5.3 and Theorem 5.1
can be carried through without change. Then, we show that
\vskip 0.1in
\noindent
{\bf Proposition 5.5. }{\it Let $X$ be the blow-up of a simple connected
elliptic surface. Then, the smooth 6-manifold $X\times S^2$ admits infinitely
many deformation classes of symplectic structures with the same tamed almost
complex structure up to a homotopy.}
\vskip 0.1in
\section {The Generalized Witten Conjecture}
In this section,
we formulate a conjecture on the structure of our invariants. This
conjecture was originated by Witten in [W2], but he used
path integrals, which are not well accepted by mathematicians.
Our only contribution here is put his arguments on a rigorous
mathematical footing. During the course of discussions, we
also use the results obtained in this paper to
derive several other equations of the generating function rigorously.
Those equations were known to physicists \cite{W2}, \cite{Ho} in the physical
context. Most arguments in this section are due to Witten.
As before, we denote by $\overline {\cal U}_{g,k}$ the universal curve
over $\overline {\cal M}_{g,k}$. Then each marked point $x_i$ gives rise to
a section, still denoted by $x_i$, of the fibration
$\overline {\cal U}_{g,k} \mapsto
\overline {\cal M}_{g,k}$. If ${\cal K}_{{\cal U} |{\cal M}}$ denotes the cotangent
bundle to fibers of this fibration, we put ${\cal L}_{(i)} = x_i^* (
{\cal K}_{{\cal U} |{\cal M}})$. Following Witten, we put
$$\langle \tau _{d_1,\alpha _1}\tau_{d_2,\alpha _2}\cdots \tau _{d_k,\alpha _k}
\rangle _g (q) ~=~\sum _{A \in H_2(V,{\bf Z})}
\Psi^V_{(A,g,k)}([K_{d_1,\cdots,d_k}];
\{\alpha _i\}) \, q^A \leqno (6.1)$$
where $\alpha _i \in H_*(V,{\bf Q})$ and $[K_{d_1,\cdots,d_k}]$ is the Poincare dual
of $c_1({\cal L}_{(1)})^{d_1} \cup \cdots \cup
c_1({\cal L}_{(k)})^{d_k}$ and $q$ is an element of Novikov ring.
Symbolically, $\tau _{d,\alpha}$'s denote ``quantum field theory operators''.
For simplicity, we only consider homology classes
of even degree. Choose a
basis $\{\beta_a\}_{1\le a\le N}$ of
$H_{*,\rm even}(V, {\bf Z})$ modulo torsions. We introduce formal variables $t_r^a$,
where $r= 0, 1, 2, \cdots$ and $1\le a \le N$. Witten's generating
function (cf. [W2]) is now simply defined to be
$$F^V(t^a_r ; q) =
\langle e^{\sum _{r,a} t^a_r \tau _{r, \beta _a}} \rangle (q)
=\sum _{n_{r,a}} \prod _{r,a} {(t^a_r)^{n_{r,a}} \over {n_{r,a}}!}
\left \langle \prod _{r,a} \tau _{r,\beta _a}^{n_{r,a}}
\right \rangle (q)
\leqno (6.2)$$
where $n_{r,a}$ are arbitrary collections of nonnegative integers,
almost all zero, labeled by $r, a$. The summation in (6.2) is over all
values of the genus $g$ and all homotopy classes $A$ of $(J,\nu)$-maps.
Sometimes, we write $F_g^V$ to be the part of $F^V$ involving only
GW-invariants of genus $g$. It is clear that $F^V$ is a generalization
of the prepotential $\Phi^V = F^V_0$ of genus 0 invariants
(cf. \cite{RT}, section 9). Indeed this generalized function contains
more information on the underlying manifold, for instance, using
Taubes' theorem \cite{T2}, one observes (cf. [T])
that for a minimal algebraic surface
$V$ of general type,
$$F^V(t^a_r;q) ~=~F^V(t^a_r; 0) + q^{K_V} e^{\tau _{0,0}} + \cdots,\leqno (
6.3)$$
while $\Phi^V$ depends only the intersection form of $V$.
One of Witten's goals is to find out the equations which $F^V$ satisfies.
The case that $V$ is a point corresponds to
the topological gravity, where
$F^V$ is governed by a complete set of solution- the KdV hierarchy,
conjectured by Witten ([W2]) and
clarified by Kontsevich ([Ko]). In general,
it is not clear what is (or there
exists at all) the
complete set of equations which
$F^V$ satisfies, though there are partial results for $V={\bf C} P^1$ (see \cite{EY}
). However, in [W2], Witten made a conjecture on $F^V$, which we will
describe in this section.
First of all, we have obtained several important
recursion formulas in section
2 about $\Psi$. In general, we can always rewrite a recursion formula as a
differential equation of the generating function. Assume that $\beta _1 = [V]$.
Following Witten's
arguments in [W2], one can deduce from (2.15)
that $F^V$ satisfies the generalized string equation:
$${\partial F^V\over \partial t^1_0} = {1\over 2} \eta _{ab} t^a_0 t^b_0
+ \sum \limits_{i=0}^\infty \sum \limits _{a} t^a_{i+1}
{\partial F^V\over \partial t^a_i}
\leqno (6.4)$$
For the reader's convenience, we reproduce the arguments here.
\vskip 0.1in
\noindent
{\bf Lemma 6.1. }{\it Suppose that $(V, \omega)$ is a
semi-positive symplectic
manifold. Then, the generating function $F^V$ satisfies
the generalized string
equation
$${\partial F^V\over \partial t^1_0} = {1\over 2} \eta _{ab} t^a_0 t^b_0
+ \sum \limits_{i=0}^\infty \sum \limits _{a} t^a_{i+1}
{\partial F^V\over \partial t^a_i}$$}
{\bf Proof: } By (2.15),
$$\Psi^V_{(A,g,k+1)}([K_{d_0,d_1, \cdots, d_k}]; [V], \alpha_1, \cdots,
\alpha_k)
=\Psi^V_{(A,g,k)}([\pi(K_{d_0,d_1, \cdots, d_k})]; \alpha_1, \cdots,
\alpha_k),\leqno(6.5)$$
where for convenience, we choose to forget the first marked point instead of
the last marked point as in Proposition 2.15.
We choose $d_0=0$,
i.e., $c_1({\cal L}_{(1)})^{d_0}=1$. Next, let us find
$[\pi(k_{d_0,d_1, \cdots, d_k})]$ for
$(g,k) \not= (0,2), (1,0)$.
Let's use ${\cal L}'_{(j)}$ to denote the
corresponding line bundle over $\overline{{\cal M}}_{g,k}$. Then, it is natural to
compare ${\cal L}_{(j)}$ with $\pi^*{\cal L}'_{(j)}$. It was known in algebraic
geometry that
$${\cal L}_{(j)}=\pi^*{\cal L}'_{(j)}+D_j, \leqno(6.6)$$
where $D_i$ is the divisor consisting of the stable curves where $x_0, x_i$ are
in a rational component with only three special points. Hence,
$$c_1({\cal L}_{(j)})^m=c_1(\pi^*{\cal L}'_{(j)})^m+D_j\sum_{i=1}^{m-1}c_1({\cal L}_{(j)}
)^ic_1(\pi^*{\cal L}'_{(j)})^{m-i-1}.\leqno(6.7)$$
Furthermore,
$$c_1({\cal L}_{(j)})\cap [D_j]=0; [D_i]\cap [D_j]=0 \mbox{ for } i\neq
j.\leqno(6.8)$$
Therefore,
$$c_1({\cal L}_{(j)})^m=c_1(\pi^*{\cal L}'_{(j)})^m+D_jc_1(\pi^*{\cal L}'_{(j)})^{m-
1}\leqno(6.9)$$
and
$$\begin{array}{rl}
&c_1({\cal L}_{(1)})^{d_1}\cup\cdots\cup c_1({\cal L}_{(d_k)})^{d_k}\\
=~&(\pi^*{\cal L}'_{(1)}
)^{d_1}\cup \cdots\cup (\pi^*{\cal L}'_{(1)})^{d_k}+\sum_{j=1}^k ([D_j]\cap
\bigcup_{i=1}^n c_1(\pi^*{\cal L}_{(i)})^{d_i-\delta_{ij}}).\\
\end{array}
\leqno(6.10)$$
Note that $[\pi(D_j)]=[\overline{{\cal M}}_{g,k}].$ Then,
$$\Psi^V_{(A,g,k)}([\pi(K_{d_0,d_1, \cdots, d_k})]; \alpha_1, \cdots,
\alpha_k)\leqno(6.11)$$
$$=\Psi^V_{(A,g,k)}([K_{d_1, \cdots, d_k}]; \alpha_1, \cdots,
\alpha_k)+\sum_{j=1}^k\Psi^V_{(A,g,k)}([K_{d_1, \cdots, d_j-1, \cdots, d_k}];
\alpha_1, \cdots, \alpha_k). $$
For the dimension reason, the first term is zero. Therefore, if
$(g,k) \not= (0,2), (1,0)$, we have
$$<\tau_{0,1}\prod_{i=1}^k\tau_{d_i, \alpha_i}>=\sum_{j=1}^k<\prod_{i=1}^k
\tau_{d_i-\delta_{i,j}, \alpha_i}>, \leqno(6.12)$$
where we simply define $\tau_{r, \alpha}=0$ for $r<0$.
There are two exceptional
cases for the previous arguments, namely, $g=0, k=2$ and $g=1, k=0$.
In those special cases, forgetting one
marked point will result in a unstable curve. For the special case
$g=0, k=2$, since $\overline{{\cal M}}_{0,3}=pt$, the only non-zero term is
$$\Psi^V_{(0, 0,3)}([\overline{{\cal M}}_{0,3}]; [V], \beta_a, \beta_b).$$
Moreover, one can show that
$$\Psi^V_{(0, 0,3)}([\overline{{\cal M}}_{0,3}]; [V], \beta_a, \beta_b)=\eta_{a,b}.
\leqno(6.13)$$
In the case that $g=1, k=0$, for the dimension reason, we have that
$$\Psi^V_{(A,1,1)}([\overline{{\cal M}}_{1,1}]; [V])=0.\leqno(6.14)$$
Therefore, we have an equation
$$<\tau_{0,1}\prod_{i=1}^k\tau_{d_i, \alpha_i}>=\sum_{j=1}^k<\prod_{i=1}^k
\tau_{d_i-\delta_{i,j}, \alpha_i}>+\delta_{k,2}\delta_{d_1,0}\delta_{d_2,0}
\eta_{a_1,a_2}, \leqno(6.15)$$
The corresponding equation for
the generating function is the generalized string
equation
$${\partial F^V\over \partial t^1_0} = {1\over 2} \eta _{ab} t^a_0 t^b_0
+ \sum \limits_{i=0}^\infty \sum \limits _{a} t^a_{i+1}
{\partial F^V\over \partial t^a_i}
$$
We can choose $d_0=1$ and obtain another equation for $F^V$.
\vskip 0.1in
\noindent
{\bf Lemma 6.2. }{\it $F_g$ satisfies dilation equation
$$\frac{\partial F_g}{\partial t^1_1}=(2g-2+\sum_{i=1}^{\infty}\sum_{a}t^a_i
\frac{\partial }{\partial t^a_i})F_g+\frac{\chi(V)}{24}\delta_{g,1},
\leqno(6.16)$$
where $\chi(V)$ is the Euler characteristic of $V$.}
\vskip 0.1in
\noindent
{\bf Proof: } We choose $d_0=1$. Repeating the analysis we just have, we get
$$c_1({\cal L}_{(0)})\bigcup_{j=1}^k c_1({\cal L}_{(i)})^{d_i}=c_1({\cal L}_{(0)})\bigcup_{j=1}^k
c_1({\cal L}'_{(i)})^{d_i}.\leqno(6.17)$$
On the another hand, one has a natural identification
$$\alpha: \overline{{\cal M}}_{g,k+1}\cong \overline{{\cal U}}_{g,k}.\leqno(6.18)$$
Furthermore,
$${\cal L}_{(0)}=\alpha^*({\cal K}_{{\cal U}|{\cal M}})\otimes^n_{j=1}{\cal O}(D_j).\leqno(6.19)$$
Note that $\pi[{\cal K}_{{\cal U}|{\cal M}})]=2g-2$.
Therefore, modulo the exceptional case we have
$$<\tau_{1,1}\prod_{i=1}^k\tau_{d_i, \alpha_i}>_g=(2g-2-n)<
\prod_{i=1}^k\tau_{d_i, \alpha_i}>_g.\leqno(6.20)$$
Since $\overline{{\cal M}}_{0,3}=pt$ and $c_1({\cal L}_{(0)})$ is a nontrivial class, the
contribution from the exceptional case $g=0, k=2$ is zero. The exceptional
case $g=1,k=0$ corresponds to
$$\Psi_{(A,1,1)}([K_1]; [V]). \leqno(6.21)$$
For the dimension reason, $A$ has to be zero. Moreover,
$$\dim \overline{{\cal M}}_{1,1}=1; [K_1]=\frac{1}{24} \{pt\}.\leqno(6.22)$$
Now we fix a generic element $(\Sigma_1, x)\in \overline{{\cal M}}_{1,1}$ and
let $J_0$ be the any almost complex structure. Then
$$K\times F \cap \Upsilon_0\times \Xi^0_{1,1}=\{f: (\Sigma, x)\rightarrow V |
Im(f)=pt\}=V.\leqno(6.23)$$
Furthermore, $Coker(D_f\oplus J\cdot df)$ can be canonically identified with
$T_{Im(f)}V$.
Therefore, $(J_0,0)$ satisfies the requirement of Proposition 5.4.
Hence, by Proposition 5.4,
$$\Psi_{(0,1,1)}([pt], [V])=e(TV)=\chi(V). \leqno(6.24)$$
Therefore, the contribution from the exceptional case is
$$\frac{1}{24}\chi(V) \leqno(6.25)$$
and
$$<\tau_{1,1}\prod_{i=1}^k\tau_{d_i, \alpha_i}>_g=(2g-2-k)<\prod_{i=1}^k
\tau_{d_i,
\alpha_i}>_g+\frac{1}{24}\chi(V)\delta_{g,1}\delta_{k,0}.\leqno(6.26)$$
In terms of the generating function, this is equivalent to
the following differential equation
$$\frac{\partial F_g}{\partial t^1_1}=(2g-2+\sum_{i=1}^{\infty}\sum_{a}t^a_i
\frac{\partial }{\partial t^a_i})F_g+\frac{\chi(V)}{24}\delta_{g,1}, \leqno(
6.27)$$
which coincides with the formula derived by Hori \cite{Ho} using a different
definition.
In the dilation equation, we have a unpleasant term $2g-2$ to prevent us to
write
it as equation of $F^V$. As pointed out by Witten, there is
a dimension constraint
$$c_1(V)(A)+(3-n)(g-1)+k=\sum_{i}(d_i+cod(\beta_{a_i})), \leqno(6.28)$$
can be rewritten as an equation
$$(\sum_i\sum_a(i-1+q_a)t^a_i\frac{\partial}{\partial
t^a_i}-c_1(A)-(3-n)(g-1))F_g=0,
\leqno(6.29)$$
where $2q_a=cod(\beta_a)$.
Combining the above equations, one can deduce
\vskip 0.1in
\noindent
{\bf Corollary 6.3. }{\it When $c_1=0$, $F^V$ satisfies dilation equation
$$\frac{\partial F^V}{\partial t^1_1}=\sum_{i=1}^{\infty}\sum_a(\frac{2}{3-n}
(i-1+q_a)+1)t^a_i \frac{\partial F^V}{\partial t^a_i}+\frac{\chi(V)}{24}.
\leqno(6.29)$$}
\vskip 0.1in
Similarly, we can also use (2.15) to derive the equations (for $d_0=0,1$) of
the generating function.
Following Witten, one can introduce
$$U = {\partial ^{2} F^V \over \partial t _{0, 1}
\partial t _{0,\sigma}},~
U' = {\partial ^{3} F^V \over \partial t _{0, 1}^2
\partial t _{0,\sigma}},~\cdots,~U^{(l)}_\sigma =
{\partial ^{l+2} F^V \over \partial t _{0, 1}^{l+1}
\partial t _{0,\sigma}},~~~~{\rm for~}~ l \ge 0
\leqno (6.30)
$$
We will regard $U^{(l)}$ to be of degree $l$. By a differential
function of degree $k$ we mean
a function $G(U, U', U'', \cdots)$ of degree in that sense. In particular,
any function of the form $G(U)$ is of degree zero, and
$(U')^2$ has degree two.
\vskip 0.1in
\noindent
{\bf Witten Conjecture.} {\it For every $g\ge 0$, there are
differential functions $G_{m,a,n,b}(U_\sigma, U_\sigma ', \cdots )$
of degree $2g$ such that
$${\partial ^{2} F_g \over \partial t _{m, a}
\partial t _{n,b}} = G_{m,a,n, b} (U_\sigma, U_\sigma ', \cdots)
\leqno(6.31)
$$
up to and including terms of genus $g$.}
\vskip 0.1in
It was pointed out by Witten (cf. [W2])
that the composition law implies
$${\partial ^3 F_0 \over \partial t_{d_1,a_1}\partial t_{d_2,a_2}
\partial t _{d_3,a_3}}= \sum _{a,b}
{\partial ^2 F_0 \over \partial t_{d_1-1,a_1}\partial t_{0,a}}
\eta ^{ab}
{\partial ^3 F_0 \over \partial t_{0,b}\partial t_{d_2,a_2}
\partial t _{d_3,a_3}}\leqno (6.32)$$
and consequently, the conjecture for $g=0$.
Recall that in the genus 0 case, WDVV equation is a direct consequence of the
composition law. In the higher genus case, it is unclear if the composition
law is helpful at all to derive the equation and solve Witten's conjecture.
Here, we state a closely related conjecture.
\vskip 0.1in
\noindent
{\bf Conjecture 6.4. }{\it $\langle \tau _{d_1,\alpha _1}\tau_{d_2,\alpha
_2}\cdots \tau _{d_k,\alpha _k}
\rangle _g$ can be reduced to enumerative invariants $\Psi^V_{(A,g,k)}(
\overline{{\cal M}}_{g,k}; \cdots)$.}
\vskip 0.1in
\noindent
{\bf Proposition 6.5. }{\it Conjecture 6.4 holds for $g\leq 1$}
\vskip 0.1in
A special case
that $g=1$ and $V ={\bf C} P^1$ was checked in [W2].
\vskip 0.1in
\noindent
{\bf Proof:} First we assume that any $c_1({\cal L}_{(i)})$ is Poincare
dual to a divisor in ${\overline {\cal M}}_{g,k} \backslash {{\cal M}}_{g,k}$ for
$g \le 1$. Then any cycle $[K_{d_1, \cdots , d_k}]$ can be represented
by a cycle in the boundary ${\overline {\cal M}}_{g,k} \backslash {{\cal M}}_{g,k}$
so long as $d_1 + \cdots + d_k > 0$. It follows from the composition
law that $\langle \tau _{d_1,\alpha _1}\tau_{d_2,\alpha _2}\cdots \tau
_{d_k,\alpha _k}\rangle _g$ can be computed in terms of
$\langle \tau _{d_1,\alpha _1}\tau_{d_2,\alpha _2}\cdots \tau _{d_l,\alpha _l}
\rangle _{g'}$ with either $l < k$ or $g' < g$. Then the proposition
follows from a standard induction.
Next we check our assumption stated at the beginning.
Given any two points $x_1, x_2$ in $S^2$, one can construct
a canonical meromorphic section
$$s_{x_1, x_2} = {(x_1 - x_2) d z \over (z-x_1) (z-x_2)}.$$
This section has simple poles at $x_1, x_2$. Moreover, for
any $\sigma \in SL(2, {\bf C} )$,
$\sigma ^* s_{\sigma (x_1), \sigma (x_2)} = s_{x_1, x_2}$.
The moduli space ${\cal M}_{0,k}$ ($k \ge 3$) is the quotient
of $(S^2 )^k \backslash \Delta_k$ by $SL(2, {\bf C} )$, where
$SL(2, {\bf C})$
acts on $(S^2)^k$ diagonally, and $\Delta $ denotes the set
of $(x_1, \cdots, x_k)$ with $x_i = x_j$ for some $i, j$.
Notice that the universal family ${\cal U} _{0,k}$ is
biholomorphic to $S^2 \times {\cal M} _{0, k}$. Then by the
invariance of $s_{x_1, x_2}$ under $SL(2, {\bf C})$,
one can define a section
section $s$ of the relative cotangent bundle over ${\cal U} _{0,k}$,
such that $s(z; x_1, \cdots , x_k) = s_{x_1, x_2}(z)$.
For any $i \ge 3$, this $s$ restricts to an nonvanishing
section $s_i$ of ${\cal L}_{(i)}$
over ${\cal M} _{0,k}$, i.e., $s_i (x_1, \cdots , x_k) = s(x_i)$.
Clearly, each $s_i$ extends to be a meromorphic section
on ${\cal M} _{0,k}$. Therefore,
$c_1({\cal L}_{(i)})$ must be Poincare
dual to a divisor in ${\overline {\cal M}}_{0,k} \backslash {{\cal M}}_{0,k}$
for $i\ge 3$. Similarly, one can also show this for $i\le 3$.
Now let $g=1$. Note that each torus $T$ is a branched covering of $S^2$
of degree. There are four branched points, say $x_1$, $x_2$, $_3$, $x_4$.
Conversely, given any $x_1$, $x_2$, $_3$, $x_4$, one can construct
a branched covering $T$ with those as branched points. The resulting
torus $T$ is uniquely determined by the orbit
of $(x_1, \cdots, x_4)$ in $(S^2)^4$ by $SL(2, {\bf C})$.
Let $\pi: T \mapsto S^2$ be the branched
covering map. Then
$\pi^*(s_{x_1, x_2} s_{x_3, x_4})$
defines a nonvanishing section $s_T$ of $K_T^2$ over $T$.
Using the invariance of $s_T$ under $SL(2, {\bf C})$, we can easily construct
a nonvanishing section of the relative canonical bundle over
${\cal U}_{1,1}$, which can be extended meromorphically to ${\overline {\cal U}}_{1,1}$.
It follows that any $c_1({\cal L}_{(i)})$ is Poincare
dual to a divisor in ${\overline {\cal M}}_{1,k} \backslash {{\cal M}}_{1,k}$
for any $k \ge 1$.
|
1996-01-22T01:52:45 | 9601 | alg-geom/9601017 | en | https://arxiv.org/abs/alg-geom/9601017 | [
"alg-geom",
"math.AG"
] | alg-geom/9601017 | Shihoko Ishii | Shihoko Ishii (Tokyo Institute of Technology) | The canonical modifications by weighted blow-ups | AMSLaTeX, 16 pages | null | null | null | null | In this paper we give a criterion for an isolated, hypersurface singularity
of dimension $n\ (\geq 2)$ to have the canonical modification by means of a
suitable weighted blow-up. Then we give a counter example to the following
conjecture by Reid-Watanabe: For a 3-dimensional, isolated, non-canonical,
log-canonical singularity $(X,x)$ of embedded dimension 4, there exists an
embedding $(X,x)\subset ({\bC}^4, 0)$ and a weight ${\bw}=(w_0,w_1,\ldots
,w_n)$, such that the ${\bw}$-blow-up gives the canonical modification of
$(X,x)$.
| [
{
"version": "v1",
"created": "Thu, 18 Jan 1996 02:06:32 GMT"
},
{
"version": "v2",
"created": "Thu, 18 Jan 1996 02:22:14 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ishii",
"Shihoko",
"",
"Tokyo Institute of Technology"
]
] | alg-geom | \section{Introduction}
\label{introduction}
\begin{say}
\label{def of canonical modification}
Throughout this paper
all varieties are defined over the complex number field ${\Bbb C}$.
The canonical modification of a singularity $(X,x)$ is a partial
resolution $\varphi : Y \to X$ such that $Y$ admits at worst
canonical singularities, and the canonical divisor $K_Y$ is
relatively ample with respect to $\varphi$.
If a canonical modification exists,
then it is unique up to isomorphisms over $X$.
It is well known that it exists if the minimal model conjecture holds.
For a 2-dimensional singularity $(X,x)$,
the canonical modification is the RDP-resolution (\cite{L}).
For a 3-dimensional singularity $(X,x)$,
the canonical modification exists by the affirmative answer
(\cite{M}) to
the minimal model conjecture.
For higher dimensional singularities the existence is not generally
proved.
The motivation for writing this paper is the following conjecture by
Miles Reid
and Kimio Watanabe:
\end{say}
\begin{conj}
\label{R-W-conjecture}
For a 3-dimensional, isolated, non-canonical, log-canonical
singularity $(X,x)$
of embedded dimension 4,
there exists an embedding $(X,x)\subset ({{\Bbb C}}^4, 0)$ and a
weight ${{\bold w}}=(w_0,w_1,\ldots ,w_n)$ such that the
${{\bold w}}$-blow-up gives the canonical modification of $(X,x)$.
\end{conj}
Primarily,
both Reid and Watanabe brought up stronger versions in different ways:
%
Reid required the statement for elliptic singularities,
not only
for log-canonical singularities (Conjecture p.306, \cite{Reid});
Kimio Watanabe required it for all non-canonical,
log-canonical singularities defined by a non-degenerate
polynomial without replacing embedding to $({{\Bbb C}}^4, 0)$,
and he also required the weight ${\bold w}$ should be in 95-weights
listed by Yonemura \cite{Y} and Fletcher \cite{F}, which
give the weights of quasi-homogeneous simple K3-singularities
Tomari \cite{T} showed an affirmative answer for log-canonical
singuarities of special type.
Watanabe calculated many examples, and made a list of standard equations
of log-canonical singularities which admit the canonical modifications
by weighted blow-up with each weight of the 95's \cite{W2}.
\begin{say}
\label{purpose}
In this paper
we say that a weight is the canonical weight,
if it gives the weighted blow-up
which is the canonical modification.
We give a criterion
for an isolated, hypersurface singularity of dimension $n\ (\geq 2)$
to have the canonical weight in \S 2.
As a consequence,
for a non-canonical, log-canonical singularity
$(X,x) \subset ({{\Bbb C}}^{n+1},0)$
defined by a non-degenerate polynomial $f$
(definition cf. \cite{Kouchinirenco}),
a primitive vector ${\bold w} =(w_0,w_1,\ldots ,w_n)$
is the canonical weight
if and only if ${\bold w}$ is absolutely minimal (i.e. each coordinate
$w_i$ is the minimal integer) in the essential cone in the dual space
(Corollary \ref{log-canonical}).
We can see many singularities for which such vectors actually exist
(Corollary \ref{surface}$\sim $ Example \ref{typeT}).
But we also observe in \S 4
an example for which such a vector does not exist
and it turns out to be a counter example opposing
the Reid-Watanabe's conjecture.
In the other sections,
we prepare the formula for coefficients of divisors (in \S 1 )
and study deformations of isolated singularities (in \S 3).
\end{say}
{\bf Acknowledgments.}
The author would like to thank Professors Miles Reid and Kimio
Watanabe for introducing the conjecture,
stimulating discussions and providing many examples.
She also expresses her gratitude to Professor
Masataka Tomari
whose example helped to find an error of the first draft of
this paper.
She is also grateful to Professor Kei-ichi Watanabe
as well as other members of Waseda Tuesday Seminar,
for constructive comments and advice in preparation of this paper.
\section{Divisors on toric varieties}
\begin{say}
\label{coefficient of divisor}
Let $M$ be the free abelian group ${{\Bbb Z}}^r$ $(r\geq 1)$
and $N$ be the dual $Hom_{{\Bbb Z}}(M, {{\Bbb Z}})$.
We denote $M\otimes _{{\Bbb Z}}{{\Bbb R}}$ and $N\otimes_{{\Bbb Z}}{{\Bbb R}}$ by
${M_{{\Bbb R}}}$ and $N_{{\Bbb R}}$, respectively.
Then $N_{{\Bbb R}}=Hom_{{\Bbb R}}(M_{{\Bbb R}},{{\Bbb R}})$.
For a finite fan ${{\Delta}}$ in ${N_{{\Bbb R}}}$,
we construct the toric variety $V=T_N({{\Delta}})$.
Denote by ${{\Delta}}(1)$
the set of primitive vectors ${{\bold q}}=(q_1,\ldots , q_r)\in {{\Bbb N}}$
whose rays ${{\Bbb R}}_{\geq 0}{{\bold q}}$ belong to ${{\Delta}}$ as one-dimensional
cones.
For ${{\bold q}}\in {{\Delta}}(1)$,
denote by $D_{{\bold q}}$ the
corresponding divisor which is denoted by $\overline{orb\
{{\Bbb R}}_{\geq 0}{{\bold q}}}$ in \cite{Oda75}.
Denote by $U_{{\bold q}}$ the invariant affine open subset which contains
${orb\
{{\Bbb R}}_{\geq 0}{{\bold q}}}$ as the unique closed orbit.
Then $U_{{\bold q}}= {\rom Spec} {\Bbb C}[ {{\bold q}}^{\vee}\cap
M]$,
$U_{{\bold q}}\cap D_{{\bold q}}={orb\ {{\Bbb R}}_{\geq 0}{{\bold q}}}
={\rom Spec} {\Bbb C}[ {{\bold q}}^{\perp}\cap
M]$,
and $U_{{\bold q}}\cap D_{{\bold q}}$ is defined in $U_{{\bold q}}$
by the ideal ${\frak p}_{{\bold q}}$
which is generated by the elements ${\bold e }\in M$ with ${{\bold q}}
({\bold e})>0$.
Express ${\Bbb C}[M]$ by ${\Bbb C}[x^{{\bold a}}]_{{\bold a}\in M}$,
where $x^{{\bold a}}=x_1^{a_1}x_2^{a_2}\cdots x_r^{a_r}$
for ${\bold a}=(a_1,\ldots ,a_r)\in M$.
For convenience sake,
we write $x^{\bold a}\in f$,
if
$f=\sum_{{{\bold q}}\in M}{\alpha}_{{\bold q}}x^{{\bold q}},$
and ${\alpha}_{\bold a}\neq 0$.
\end{say}
\begin{defn}
\label{weight of divisor}
Under the notation above,
take ${{\bold q}}\in |{\Delta}|$.
(1) For a regular function $f$ on $T_N({\Delta})$,
define :
$${{\bold q}}(f):= \min\{{{\bold q}}({\bold a})|{\bold a}\in M,\
x^{\bold a}\in f\}$$.
(2) For a ${\Bbb Q}$-divisor
$D $ on $ T_N({\Delta})$
such that
$mD$ is defined by a regular function $f$ on $ T_N({\Delta})$,
define:
$${{\bold q}}(D):= \dfrac{1}{m}{{\bold q}}(f).$$
\end{defn}
\begin{prop}
\label{valuation of function}
Let $D\subset V $ be a ${\Bbb Q}$-principal divisor (i.e. $mD$ is defined by
a regular function on $V$).
Then $D$ is of the form:
$$D=D'+\sum_{{{\bold q}}\in {\Delta} (1)} {{\bold q}}(D)D_{{\bold q}},$$
where $D'$ is an effective divisor
which does not contain any $D_{{\bold q}}$.
\end{prop}
\begin{pf}
By the definition of ${\bold q}(D)$, we may assume that $D$ is defined
by a regular function $f$ on $V$.
Write $D=D'+ \sum_{{{\bold q}}\in {\Delta} (1)} m_{{\bold q}}D_{{\bold q}}$
such that $D'$ does not contain any $D_{{\bold q}}$
and $m_{{\bold q}}\geq 0$.
Let $\zeta$ be the defining function of
$D_{{\bold q}}$ on $U_{{\bold q}}$,
then $f={\zeta}^{m_{{\bold q}}}h$,
where $h\notin (\zeta)$.
Since ${{\bold q}}(f)=m_{{\bold q}}{{\bold q}}({\zeta})+{{\bold q}}(h)$ and ${\bold q}(h)=0$,
it is sufficient to prove that ${{\bold q}}(\zeta)=1$.
And this is clear,
because there is a vector ${\bold e}\in M$
such that ${{\bold q}}({\bold e})=1 $
and ${\zeta}$ is a generator of the ideal $\{g\in {\Bbb C}[{{\bold q}}^{\vee}\cap M]|
{\bold q}(g)>0\}$.
\end{pf}
\begin{say}
\label{simplicial}
Now we consider the case that ${\Delta}$ consists of all faces of a
simplicial cone $\sigma$ in $N_{{\Bbb R}}$.
Let ${{\bold a}}_1,\ldots , {{\bold a}}_r$ be the primitive vector of the
one dimensional faces of $\sigma \cap N$
and ${{\bold a}}^*_1,\ldots , {{\bold a}}^*_r$
be the dual system of $\{{{\bold a}}_i\}$'s
(i.e. ${{\bold a}}_j^*\in M_{{\Bbb R}}$ and ${{\bold a}}_i({{\bold a}}_j^*)
=\delta _{ij}$).
Denote by $\overline{N}$ the subgroup of $N$ generated by
$\{{{\bold a}}_i\}$.
Then the morphism $\pi : T_{\overline N}({\Delta}) \to T_N({\Delta})=V$
induced by $({\overline N},{\Delta}) \to (N, {\Delta})$ is the quotient
morphism of ${{\Bbb C}}^r$ by the finite group $N/{\overline N}$,
and each ${{\bold a}}_j^*$ belongs to
${\overline M}=Hom_{{\Bbb Z}}({\overline N}, {{\Bbb Z}})$.
Denote by $D_{{{\bold a}}_i}$ and ${\overline D_{{{\bold a}}_i}}$
the divisors on $T_N({\Delta})$ and
$T _{\overline N}({\Delta})={{\Bbb C}}^r$ respectively
which are
corresponding to ${{\bold a}}_i$.
Let $r_i$ be the ramification index of $\pi$ at
${\overline D_{{{\bold a}}_i}}$.
\end{say}
\begin{lem}
\label{dual}
Let ${{\Delta}}$ be as in \ref{simplicial}.
%
Then
it follows that
$${{\bold q}}(D_{{{\bold a}}_i})=
r_i{{\bold q} }({{\bold a}}_i^*)$$ for every ${\bold q}\in |{\Delta}|$ and every $i$.
\end{lem}
\begin{pf}
Let $D_{{{\bold a}}_i}$ be defined by $x_i=0$ on
$U_{{{\bold a}}_i} \subset V$.
Then $x_i=y_i^{r_i}$,
where the equation $y_i=0$ defines ${\overline D_{{{\bold a}}_i}}$ on
$T_{\overline N}({\Delta})$.
Therefore,
${{\bold q}}(D_{{{\bold a}}_i})={{\bold q}}(x_i)=r_i{{\bold q}}(y_i)$
and the last term equals $r_i {{\bold q} }({{\bold a}}_i^*)$,
because $y_i$ is the $i$-th coordinate function of
$T_{\overline N}({\Delta})={{\Bbb C}}^r$.
\end{pf}
\begin{prop}
\label{birational canonical}
Let ${\Delta}' $ be a finite subdivision of an arbitrary
finite fan ${\Delta}$ and
$\varphi :V'=T_N({\Delta}')\to V=T_N({\Delta})$ be the corresponding
birational
morphism.
Denote the divisor on $V'$ corresponding to ${\bold q} \in {{\Delta}}'(1)$
by $D'_{{\bold q}}$.
Assume that $K_{V}$ is ${{\Bbb Q}}$-principal.
Then
$$K_{V'}=\varphi^* K_V+ \sum _{{{\bold q}}\in {{\Delta}}' (1)-{{\Delta}} (1)}
({{\bold q} } (\varphi^*(\sum_{{{\bold t}}\in {{\Delta}} (1)}D_{{\bold t}}))-1)D'_{{\bold q}}.$$
If ${\Delta}$ is as in \ref{simplicial},
then
it follows that
$$K_{V'}=\varphi^*K_V+ \sum _{{{\bold q}}\in {{\Delta}}'(1)-{{\Delta}}(1)}
(\sum_{i=1}^r r_i{{\bold q}}({{\bold a}}_i^*)-1)D'_{{\bold q}}.$$
If moreover $\sigma$ is the positive quadrant in $N_{{\Bbb R}}$,
then
$$K_{V'}=\varphi^*K_V+ \sum _{{{\bold q}}\in {{\Delta}}'(1)-{{\Delta}}(1)}
({{\bold q}}({{\bold 1}})-1)D'_{{\bold q}},$$
where ${{\bold 1}}=(1,1,\ldots , 1) \in M$.
\end{prop}
\begin{pf}
For a toric variety,
the canonical divisor is represented by the sum of all toric
invariant divisors with coefficient $-1$.
Therefore
(1) $K_{V'} = -\sum _{{{\bold q}}\in {\Delta}' (1)-{\Delta} (1)}D'_{{\bold q}}-
\sum_{{{\bold t}}\in {\Delta} (1)}D'_{{\bold t}}$.
\newline
On the other hand, we represent $K_{V'}$ as
(2) $K_{V'} = \varphi^*K_V+
\sum_{{{\bold q}}\in {\Delta}' (1)-{\Delta} (1)}m_{{\bold q}}D'_{{\bold q}}$.
\newline
Substituting $K_V=-\sum_{{{\bold t}}\in {\Delta} (1)}D_{{\bold t}}$
into (2) and comparing (1) and (2),
we obtain the value of $m_{{\bold q}}$.
For the second and the third equalities,
note that ${\bold q}(\varphi ^* D)={\bold q}(D)$
for a ${\Bbb Q}$-principal divisor $D$ and apply \ref{dual}.
\end{pf}
\vskip 1truecm
Now we obtain the characterization of hypersurface singularities
by means of the
Newton diagram.
Parts of the following are stated in
\cite{HW}, \cite{W3} and \cite{Reid2}.
\begin{cor}
\label{characterization of log-canonical}
Let $(X,0) \subset ({{\Bbb C}}^{n+1}, 0)$ be an isolated
singularity defined by a polynomial $f$.
Denote by $\Gamma_{+}(f)$ and $\Gamma(f)$
Newton's diagram of $f$ and the union of the compact faces of it,
respectively.
Then the following hold:
(i) if $(X,0)$ is canonical, then ${{\bold 1}}=(1,1,\ldots , 1)\in
\Gamma_{+}(f)^o$,
where $\Gamma_{+}(f)^o$is the interior of $\Gamma_{+}(f)$;
(ii) if $(X,0)$ is log-canonical, then ${{\bold 1}}\in \Gamma_+(f)$.
\vskip .5truecm
If $f$ is non-degenerate, the following hold:
(iii) $(X,0)$ is canonical if and only if ${{\bold 1}}\in
\Gamma_{+}(f)^o$;
(iv) $(X,0)$ is non-canonical, log-canonical
if and only if ${{\bold 1}}\in \Gamma(f)$;
(v) $(X,0)$ is not log-canonical
if and only if ${{\bold 1}} \notin
\Gamma_{+}(f)$.
\end{cor}
\begin{pf}
Let $\sigma$ be the positive quadrant in $N_{{\Bbb R}}$
and ${\Delta}$ be the fan consisting of all faces of $\sigma$.
%
For a primitive ${{\bold q}}\in N\cap {\sigma} $ ,
take the subdivision ${\Delta ({\bold q})}$ of ${\Delta}$
consisting of all faces of $\sigma_i=\sum_{j\neq i}{\Bbb R} _{\geq 0}{{\bold e} _j}+
{\Bbb R} _{\geq 0}$, $i=0,\ldots , n$,
take the normalization
$\tilde X$ of
the proper transform
$\overline{X}\subset T_N({\Delta ({\bold q})})$ of $X$.
For the composite
$\psi:\tilde X \to \overline{X}
\stackrel{\varphi|_{\overline{X}}}\longrightarrow X$,
write the canonical divisor as follows:
$$K_{\tilde X}=\psi ^* K_X +\sum m_iE_i,$$
where $E_i$'s are the exceptional divisors of $\psi$.
On the other hand
$$K_{T_N({\Delta}({\bold q}))}+\overline{X}=
\varphi^*(K_{{{\Bbb C}^{n+1}}}+X)+({\bold q}({\bold 1})-1-{\bold q}(f))D_{{\bold q}}.$$
For the statement (i) (resp. (ii)),
it is sufficient to prove that if ${{\bold q}}({\bold 1})-1-{{\bold q}}(f)<0$
(resp. $<-1$),
then $m_i<0$
(resp. $m_i<-1$) for some $i$.
Since $T_{N}({\Delta ({\bold q})})$ has at worst ${{\Bbb Q}}$-factorial
log-terminal singularities and $K_X$ is linearly trivial,
we can apply the following lemma to a Weil divisor $\overline{X}
\subset T_{N}({\Delta ({\bold q})})$.
For the assertion of the case that $f$ is non-degenerate,
it is sufficient to prove the opposite implications in (i) and (ii).
For a non-singular subdivision ${\Delta}'$ of ${\Delta}$, on whose toric variety
the proper transform $X'$ is
non-singular and intersects transversally each orbit on $T_{N}({\Delta ({\bold q})})$
(for the existence of such ${\Delta}'$, cf. \cite{Kouchinirenco}
\cite{V}),
we have:
$$K_{X'}=\varphi ^*(K_X)+\sum_{{\bold q} \in {\Delta} '(1)-{\Delta}(1)}
({{\bold q}}({\bold 1})-1-{{\bold q}}(f))D'_{{\bold q}}|_{X'}$$
by \ref{valuation of function}, \ref{birational canonical}.
If ${\bold 1} \in {\Gamma_+(f)}^o$,
then ${\bold q}(f)<{\bold q}({{\bold 1}})$ for all ${\bold q} \in {\Delta} '(1)-{\Delta}(1)$,
which implies that $(X,0)$ is canonical.
If it is not log-canonical, then there exists ${\bold q}$ such that
${\bold q}(f)>{\bold q}({{\bold 1}})$,
which implies ${{\bold 1}} \notin
\Gamma_{+}(f)$.
%
\end{pf}
\begin{lem}
\label{normalization}
Let $Y\subset Z$ be an irreducible
Weil divisor on a normal variety $Z$.
Suppose Z admit at worst ${\Bbb Q}$-factorial log-terminal singularities.
Let $\tau :\tilde Y \to Y$ be the normalization.
Then:
\newline
(i) $Y$ is a Cohen-Macaulay variety;
\newline
(ii) $\omega_Y\simeq
(\omega_Z(Y)\otimes_{{\cal O}_Z}{\cal O}_Y)/{\cal T}$,
where ${\cal T}$ is the torsion submodule of
$\omega_Z(Y)\otimes_{{\cal O}_Z}{\cal O}_Y$;
\newline
(iii) if $\omega_Z(Y)\simeq {\cal O}_Z(-aD)$ $a\geq 1$ (resp.
$a>1$ ) for an effective divisor $D\subset Z$ such that
$\phi \neq D\cap Y \neq Y$,
then we have the canonical isomorphism
$\omega _{\tilde Y} \simeq {\cal O}_{\tilde Y}(-\sum_{i=1}^kb_iE_i)$
with $b_i\geq 1$ for divisors $E_i$ $(i=1,\ldots , k)$ such that
$\cup_{i=1}^kE_i\supset \tau^{-1}(D)$
(resp. in addition $b_i>1$ for some $i$ such that
$E_i\subset Supp(\tau^{-1}D)$).
\end{lem}
\begin{pf}
First, one can prove that every effective Weil divisor on $Z$ is a
Cohen-Macaulay variety in the same way as in 0.5 of \cite{IF},
because the covering constructed as in \cite{IF} has at worst
rational singularities in the present case too.
%
For the proof of (ii),
take the exact sequence:
$${\cal Hom}_{{\cal O}_Z}({\cal O}_Z ,\omega_Z)\to
{\cal Hom}_{{\cal O}_Z}({\cal O}_Z(-Y) ,\omega_Z)\to
{\cal Ext}^1_{{\cal O}_Z}({\cal O}_Y,\omega_Z)\to
{\cal Ext}^1_{{\cal O}_Z}({\cal O}_Z,\omega_Z)=0.
$$
Here ${\cal Ext}^1_{{\cal O}_Z}({\cal O}_Y,\omega_Z)=\omega_Y$,
because $Z$ is a Cohen-Macaulay variety.
So $\omega_Y$ is the image of
${\cal Hom}_{{\cal O}_Z}({\cal O}_Z(-Y) ,\omega_Z)=\omega_Z(Y)$,
and therefore also the image of $\omega_Z(Y)\otimes {\cal O}_Y$
which is isomorphic to $\omega_Y$ on general points of $Y$.
%
Since $\omega_Y$ is torsion-free,
it must be isomorphic to
$(\omega_Z(Y)\otimes_{{\cal O}_Z}{\cal O}_Y)/{\cal T}$
as desired in (ii).
Now,
since $\tau$ is finite,
we have the inclusion $\tau_*\omega_{\tilde Y}\hookrightarrow
\omega_Y$.
By (ii) and the assumption of (iii),
$\omega_Y$ is isomorphic to the defining ideal ${\cal J}$ of
a subscheme $aD\cap Y $ of $ Y$.
Therefore $\omega_{\tilde Y}\simeq {\cal O}_{\tilde Y}(-\sum b_iE_i)$,
$b_i> 0$ for divisors $E_i$ $(i=1,\ldots , k)$ such that
$\cup_{i=1}^kE_i\supset \tau^{-1}(D)$.
Next, assume $a>1$.
For the assertion,
we may replace $Y\subset Z$ with a small neighbourhood of a general
point on $D\cap Y$.
So we may assume that all $E_i$ are over $Supp(D|_Y)$ and $D|_Y$ is
irreducible.
If there is no $E_i\subset Supp(D)$ such that $b_i>1$,
then $\tau_*\omega_{\tilde Y}=\tau_*{\cal O}_{\tilde Y}(-\sum E_i)$
is a reduced ${\cal O}_Y$-ideal whose locus has the support on
$D\cap Y$.
On the other hand, ${\cal J}$ also has the locus with the support on
$D\cap Y$,
therefore they coincide.
By this equality $\tau_*\omega_{\tilde Y}=\omega_Y$, it follows that
$$\tau_*{\cal O}_{\tilde Y}={\cal Hom}_{\tau_*{\cal O}_{\tilde Y}}
(\tau_*{\omega}_{\tilde Y},\tau_*{\omega}_{\tilde Y})
\subset
{\cal Hom}_{{\cal O}_{ Y}}
({\omega}_{ Y},{\omega}_{Y})={\cal O}_Y,$$
where the left and right equalities follow from the fact that
$\tilde Y$
and $Y$ satisfy $S_2$-condition.
Now it follows that $\tau: \tilde Y \simeq Y$ is normal,
which induces
the contradiction to $a>1$.
\end{pf}
\begin{say}
\label{kappa}
For a normal isolated singularity $(X,x)$, we define an invariant
$\kappa_{\delta}(X,x)$(\cite{Iasym} )
by the growth order of the plurigenera $\delta_m$ $(m \in {{\Bbb N}})$
(\cite{W}).
In general, n-dimensional, normal, isolated singularitieis $(X,x)$ are
classified by the invariant $\kappa_{\delta}$ into (n+1)-classes:
$\kappa(X,x) = -\infty,\ 0, \ 1, \ \ldots , \ n-2, \ n$ (skipping $n-1$
curiously)
(\cite{Iasym}).
For hypersurface singularities,
the classes are only three :
$\kappa(X,x) = -\infty,\ 0,\ n$ (\cite{TW}).
A hypersurface singularity with $\kappa_{\delta}(X,x)=-\infty$
(resp.$ =0,\ n$)
is equivalent to the fact that $(X,x)$ is canonical (resp.
non-canonical-log-canonical, not log-canonical) (cf. \cite{IGor}).
Therefore \ref{characterization of log-canonical} also gives
the combinatoric characterization of non-degenerate hypersurface
singularities' classes by $\kappa_{\delta}$
\end{say}
\section{The weights which give the canonical modification}
\begin{say}
\label{quadrant}
Under the notation in \ref{simplicial},
put $r=n+1$ for $n\geq 2$ and number the elements of the basis
$\{{\bold e}_i\}$ from $i=0$ to $i=n$.
Let $\sigma=\sum _{i=0}^n{{\Bbb R}}_{\geq 0} {\bold e}_i$ be
the positive quadrant in $N_{{\Bbb R}}$,
and ${\Delta}=<\sigma>$ be the fan consisting of all faces of $\sigma$.
Denote Newton's diagram of a polynomial $f\in {{\Bbb C}}[x_0,\ldots ,x_n]$
and the union of its compact faces
by ${\Gamma_+(f)}$ and by ${\Gamma(f)}$ respectively.
\end{say}
\begin{defn}
\label{essential cone}
For a polynomial $f \in {{\Bbb C}}[x_0,\ldots ,x_n]$,
we define the essential cone as follows:
$${{C_{\bl}(f)}}:=\{ {\bold q} \in \sigma \subset N_{{\Bbb R}}|{\bold q}(f)-{\bold q}({\bold 1})\geq 0\}.$$
\end{defn}
\begin{rem}
\label{rem of essential cone}
(i) It is clear that if ${\bold 1} \in {\Gamma_+(f)} ^o$,
then ${C_{\bl}(f)} =\{ 0 \}$.
(ii) If ${\bold 1} \notin {\Gamma_+(f)}^o$, the essential cone $C_{\bf 1}(f)$ is
actually
the cone spanned by $\gamma_1^{\perp } ,\ldots , \gamma_r^{\perp } $,
where each $\gamma_i$ is an n-dimensional face of
$\Gamma_+(x_0\cdots x_n+f)$
which contains ${{\bold 1}}$.
Let $X$ be the divisor in ${\Bbb C} ^{n+1}=T_N({\Delta})$ defined by $f=0$.
If $X$ has an isolated singularity at the origin $0 \in {{\Bbb C}^{n+1}}$,
then every vector ${\bold q} \in {C_{\bl}(f)}-\{ 0\}$ has positive coordinates
$q_j$ for $j=0,1,\ldots , n$,
otherwise at least one $\gamma_i$ is parallel to one of the coordinate
axes which causes a contradiction to the isolatedness of the singularity
$(X,0)$.
(iii) in Def 3.3 of \cite{IGor} ,
we have the notion of an essential divisor of a resolution of a Gorenstein
singularity. Every 1-dimensional cone in the essential cone in
\ref{essential
cone} gives a component of the essential divisor in some resolution.
\end{rem}
\begin{defn}
\label{order}
(1) Let $C$ be a cone in $\sigma \subset N_{{\Bbb R}}$.
For ${\bold p} =(p_0,\ldots , p_n),\ \ {\bold q} = (q_0,\ldots ,q_n) \in C$,
we define
${\bold p}\leq {\bold q}$ if $p_i\leq q_i$ for every $i=0,\ldots ,n$.
We say that a primitive element ${\bold p} \in C\cap N-\{ 0\}$ is
absolutely minimal,
if ${\bold p}\leq {\bold q}$ for every primitive element
${\bold q} \in C\cap N-\{ 0\}$.
(2) For ${\bold p},\ {\bold q} \in {C_{\bl}(f)}$,
we define
${\bold p}\leq _f{\bold q}$ ,
if $p_i/({\bold p}(f)-{\bold p}({\bold 1})+1 )\leq q_i/({\bold q}(f)-{\bold q}({\bold 1})+1)$
for every $i=0,\ldots ,n$.
We define another order $\prec _f$ as follows:
${\bold p} \prec _f {\bold q}$ if
$p_i/({\bold p}(f) )\leq q_i/({\bold q}(f))$
for every $i=0,\ldots ,n$.
We say that a primitive element ${\bold p} \in {C_{\bl}(f)}\cap N-\{ 0\}$
is $f$-minimal,
if for every primitive element
${\bold q} \in {C_{\bl}(f)}\cap N-\{ 0\}$,
either ${\bold p}\leq _f{\bold q}$
or ${\bold p}\prec _f {\bold q}$ and ${\bold q}$ belongs to the interior of an
$n+1$-dimensional
cone of ${\Delta}({\bold p})$,
where the fan ${\Delta}({\bold p})$ consists of all faces of
$\sigma_i=\sum_{j\neq i}{\Bbb R}_{\geq 0}{\bold e}_j +{\Bbb R}_{\geq 0}{\bold p}
\subset N_{{\Bbb R}}$, $i=0,\ldots , n$.
\end{defn}
\begin{say}
\label{notation star}
For a primitive vector ${\bold p} \in \sigma\cap N-\{ 0\}$,
we have the star-shaped decomposition ${\Delta}({\bold p})$
by adding the ray ${\Bbb R}_{\geq 0}{{\bold p}}$ as in the definition above.
We denote the fan of all faces of $\sigma_i$ by ${\Delta}_i$.
Denote the proper transform of $X=\{f=0\}$ on
$T_N({\Delta}({\bold p}))$ by $X({\bold p})$.
The induced morphisms
$\varphi : T_N({\Delta}({\bold p}))\to T_N({\Delta})$,
$\varphi ':X({\bold p}) \to X$ are called weighted blow-ups with weight
${\bold p}$,
or simply ${\bold p}$-blow-ups of ${{\Bbb C}^{n+1}}$ and $X$ respectively.
%
Let $U_i$ be the invariant open subset
$U_{\sigma _i} \simeq T_N({\Delta}_i)$ of $T_N({\Delta}({\bold p}))$,
and $\varphi_i:U_i \to {{\Bbb C}^{n+1}}$ be the restriction of $\varphi$ onto
$U_i$.
Denote $X({\bold p})\cap U_i$ by $X_i$.
\end{say}
\begin{prop}
\label{prop star}
Under the notation in \ref{notation star},
let $\psi:\tilde{U_i}\to U_i$ be the birational morphism corresponding
to a finite subdivision $\Sigma_i$ of ${\Delta}_i$.
Denote the proper transform of $X_i$ by $\tilde{X_i}$,
then $$K_{\tilde{U_i}}+\tilde{X_i}=
\psi_i^*(K_{U_i}+X_i)
+\sum_{{\bold q}\in \Sigma_i(1)-{\Delta}_i(1)}(\dfrac{q_i}{p_i}
({\bold p} (f)-{\bold p}({\bold 1})+1)-({\bold q} (f)-{\bold q}({\bold 1})+1))D_{{\bold q}}.$$
\end{prop}
\begin{pf}
We can assume that $i=0$ without the loss of generality.
Let $\{{\bold a}_j\}_{j=0}^n$ be $\{{\bold p}, {\bold e} _1, \ldots ,{\bold e} _n\}$.
By \ref{valuation of function} and \ref{birational canonical},
it is sufficient to prove:
$$(1)\ \ \ \ \ \
\sum_{j=0}^nr_j{\bold q}({\bold a}^*_i)-{\bold q}(\psi^*X_0)-1=
\dfrac{q_0}{p_0}({\bold p} (f)-{\bold p}({\bold 1})
+1)-({\bold q} (f)-{\bold q}({\bold 1})+1))D_{{\bold q}}.$$
First, we can see that $r_j=1$ for every $j$.
In fact,
the quotient map $\pi:{{\Bbb C}^{n+1}}=T_{\overline{N}}({\Delta}_0)\to
T_{N}({\Delta}_0)=U_0$ is defined by the action of the cyclic group
generated by
$$\pmatrix
\epsilon & 0 & \ldots & \ldots & 0\\
0 & \epsilon^{-p_1} & 0 &\ldots & 0\\
0 & 0 &\epsilon^{-p_2} & 0 &\vdots\\
\vdots & \vdots &\vdots &\ddots &\vdots\\
0 & 0 & 0 &\hdots &\epsilon^{-p_n}
\endpmatrix ,$$
where $\epsilon$ is a primitive $p_0$-th root of unity.
Here it is easy to check that $\pi$ is etale in codimension one.
Next, since ${\bold a}_0^*={\bold p}^*=(1/p_0,0,\ldots ,0) $ and
${\bold a}_j^*={\bold e}_j^*=(-p_j/p_0,0,\ldots ,0,1,0,\ldots ,0)$
(j-th entry is 1) for $1\leq j \leq n$,
one obtains ${\bold q} (\sum {\bold a}_j^*)=
(1-p_1-\ldots -p_n)q_0/p_0+(q_1+\ldots +q_n)$.
On the other hand,
since $\varphi_0^*(X)=X_0+{\bold p} (f)D_{{\bold p}}$
by \ref{valuation of function},
it follows that
$$(2)\ \ \ \ \ \
{\bold q}(\psi^*X_0)={\bold q}(\psi^* \varphi_0^*(X))-
{\bold q}(\psi^*({\bold p}(f)D_{{\bold p}}))
={\bold q}(f)- {\bold p}(f){\bold q}({\bold p} ^*)
={\bold q}(f)- {\bold p}(f)q_0/p_0.$$
By substituting them into the left hand side of (1) we obtain the
equality (1).
\end{pf}
\vskip 1truecm
\begin{lem}
\label{normality}
Let $Y\subset Z$ be an irreducible Weil divisor on a variety
$Z$. Assume that $Z$ admits at worst ${\Bbb Q}$-factorial log-terminal
singularities.
Let $\psi:\tilde Y \to Y$ be a resolution of singularities on $Y$.
Assume $K_{\tilde Y}=\psi^*((K_Z+Y)|_{Y})+\sum_im_i E_i$
with $m_i>-1$ for all $i$,
where $E_i$'s are the exceptional divisors of $\psi$.
Then $Y$ is normal,
and $Y$ has at worst log-terminal singularities.
In particular,
if $m_i\geq 0$ for all $i$,
then $Y$ has at worst canonical singularities.
\end{lem}
\begin{pf}
First $Y$ is a Cohen-Macaulay variety as in \ref{normalization}.
Therefore it is sufficient to prove that $codim _YSing(Y)\geq 2$
by Serre's criterion.
Assume that $y\in Y $ is a general point of a component of $Sing( Y)$
of codimension one.
By replacing $Y$ with a small neighbourhood of $y$,
we may assume that $\psi$ is the normalization.
Claim that $\psi_*\omega_{\tilde Y}=\omega _Y$.
The inclusion $\subset$ is trivial.
For the proof of the opposite inclusion,
take an arbitrary $\theta\in \omega_Y$.
Then $\theta^r\in \omega_Z^{[r]}(rY)\otimes {\cal O}_Y$
for such $r$ that $\omega_Z^{[r]}(rY)$ is invertible,
because $\omega_Y=\omega_Z(Y)\otimes{\cal O}_Y/{\cal T}$
by (ii) of \ref{normalization}.
By the assumption of the lemma, one obtains:
$$\theta^r\in \omega_Z^{[r]}(rY)\otimes {\cal O}_Y\subset \psi_*\omega
_{\tilde Y}^r(-\sum rm_iE_i).$$
Hence for the valuation $\nu_i$ at each $E_i$,
$r\nu_i(\psi^*\theta)=\nu_i(\psi^*\theta^r)\geq rm_i >-r$.
Therefore $\nu_i(\psi^*\theta)\geq 0$ for every $E_i$,
which means that $\psi^*\theta\in \omega_{\tilde Y}$ as claimed.
By the same argument as in the proof of (iii) in \ref{normalization},
it follows that $Y$ is normal.
One can see also that $Y$ is ${\Bbb Q}$-Gorenstein,
because $\omega_Y^{[r]}=\omega_Z^{[r]}(rY)\otimes {\cal O}_Y$
is invertible for $r$ above.
\end{pf}
\vskip 1truecm
\begin{thm}
\label{main theorem}
Let $(X,0)\subset({\Bbb C} ^{n+1},0)$ be an isolated singularity defined
by a polynomial $f\in {\Bbb C}[x_0,\ldots , x_n]$.
For a primitive integral vector ${\bold p} =(p_0,\ldots ,p_n)$
such that $p_i>0$ for all $i$,
\newline
if
(i) ${\bold p}$ is the canonical weight, i.e.,
${\bold p}$-blow-up $\varphi :X({\bold p})\to X$ is
the canonical modification,
\newline
then
(ii) ${\bold p}$ is $f$-minimal in ${C_{\bl}(f)} \cap N-\{0\}$.
\vskip .5truecm
Suppose $f$ is non-degenerate,
then the converse (ii)$\Rightarrow$(i) also holds.
\end{thm}
\begin{pf}
We use the notaion in \ref{notation star}.
If the ${\bold p}$-blow-up $\varphi':{X({\bold p})}\to X$ is
the canonical modification,
then it follows that ${\bold p} \in {C_{\bl}(f)}$;
otherwise,
$K_{X({\bold p})}=\varphi^*K_X+({\bold p}({\bold 1})-1-{\bold p}(f))D_{{\bold p}}|_{X({\bold p})}$ with
${\bold p}({\bold 1})-1-{\bold p}(f)\geq 0$,
which shows a contradiction
that $(X,0)$ itself is a canonical singularity .
Let us prove that ${\bold p}$ is $f$-minimal.
If there exists ${\bold q} \in {C_{\bl}(f)} \cap N-\{0\}$
such that ${\bold q} \not\geq _f {\bold p}$,
then $\min\{\dfrac{q_i/({\bold q}(f)-{\bold q}({\bold 1})+1)}{p_i/({\bold p}(f)-{\bold p}({\bold 1})+1 )}
| i=0,\ldots , n\}
<1$.
Let $i=0$ attain the minimal value,
then it follows that ${\bold q}\in \sigma_0$,
because ${\bold q}$ is represented as
$\dfrac{q_0}{p_0}{\bold p}+ \sum _{i=1}^n(q_i-\dfrac{q_0}{p_0}p_i){\bold e}_i$
and its coefficients are all non-negative.
Taking the star-shaped subdivision ${\Delta}_0({\bold q})$ of ${\Delta}_0$
by adding a ray ${\Bbb R}_{\geq 0}{\bold q}$,
we have a birational morphism
$\psi: \tilde{U_0}:=T_N({\Delta}_0({\bold q})) \to U_0=T_N({\Delta}_0)$.
Denote the proper transform of $X_0$ by $\tilde{X_0}$,
then, by \ref{prop star}
$$(K_{\tilde{U_0}}+{\tilde{X_0}})|_{\tilde{X_0}}=
\psi^*(K_{X_0})
+(\dfrac{q_0}{p_0}
({\bold p} (f)-{\bold p}({\bold 1})+1)-({\bold q} (f)-{\bold q}({\bold 1})+1))D_{{\bold q}}|_{\tilde{X_0}}.$$
It follows that ${\tilde{X_0}}$ is normal.
In fact, for a resolution $\lambda:{\tilde{\tilde{X_0}}} \to {\tilde{X_0}}$,
denote
$K_{\tilde{\tilde{X_0}}}=\lambda^*((K_{\tilde{U_0}}+{\tilde{X_0}})|_{\tilde{
X_0}})
+\sum m_iE_i$,
then $K_{\tilde{\tilde{X_0}}}=\lambda^*\psi^*K_{X_0}+\sum (n_i+m_i)E_i$
where $n_i$ is the coefficient of $E_i$ in $\lambda^*((\dfrac{q_0}{p_0}
({\bold p} (f)-{\bold p}({\bold 1})+1)-({\bold q} (f)-{\bold q}({\bold 1})+1))D_{{\bold q}}|_{\tilde{X_0}})$
which is non-positive by the negativity of
$\dfrac{q_0}{p_0}
({\bold p} (f)-{\bold p}({\bold 1})+1)-({\bold q} (f)-{\bold q}({\bold 1})+1)$;
therefore if $m_i<0$ for some $i$,
then it contradicts to the fact that $X_0$ is canonical;
since $m_i \geq 0$ for all $i$,
by Lemma \ref{normality} $\tilde{X_0}$ is normal.
Now we obtain the partial resolution $\psi: \tilde{X_0} \to X_0$
with $K_{\tilde{X_0}}=
\psi^*(K_{X_0})
+(\dfrac{q_0}{p_0}
({\bold p} (f)-{\bold p}({\bold 1})+1)-({\bold q} (f)-{\bold q}({\bold 1})+1))D_{{\bold q}}|_{\tilde{X_0}}$.
If $D_{{\bold q}}\cap \tilde{X_0}\neq \phi$,
the coefficient of $D_{{\bold q}}|_{\tilde{X_0}}$ is negative
by the definition of ${\bold q}$,
which contradicts the hypothesis that $X_0$ has at
worst canonical singularities.
Therefore, $D_{{\bold q}}\cap \tilde X_0=\phi$
which happens if and only if $\psi(D_{{\bold q}})$ is a point away from
$X({\bold p})$, because $X({\bold p})$ is ample on $D_{{\bold p}}$.
It implies
${\bold q}(\psi ^*X_0)={\bold q}(f)-\dfrac{q_0}{p_0}{\bold p}(f)=0$
(c.f. (2) in the proof of \ref{prop star})
and ${\bold q}$ belongs to the interior of an $n+1$-dimensional
cone of ${\Delta}({\bold p})$.
By the minimality of $q_0/p_0$,
we obtain ${\bold p} \prec _f {\bold q}$.
Next suppose that $f$ is non-degenerate and ${\bold p}$ is
$f$-minimal in ${C_{\bl}(f)}\cap N-\{0\}$.
Take a non-singular subdivisioin ${\Delta}' $ of ${\Delta}({\bold p})$
such that the restriction of the corresponding morphism
$\psi:T_N({\Delta}')\to T_N({\Delta}({\bold p}))$ onto the proper transform
$X({\Delta}')$ of $X({\bold p})$ gives a resolution of $X({\bold p})$
such that every intersection of $X({\Delta}')$ and an orbit is transversal.
Let ${\Delta}'_i$ be the subdivision of ${\Delta}_i$ which is in ${\Delta}'$,
then $T_N({\Delta}')$ is covered by $T_N({\Delta}'_i)$'s
and the restriction $\psi_i: T_N({\Delta}'_i)\to U_i=T_N({\Delta}_i)$
of $\psi$ gives a resolution $X({\Delta}'_i):=X({\Delta}')\cap T_N({\Delta}'_i)\to X_i$
of each $X_i$.
Represent the canonical divisor on $X({\Delta}'_i)$ by
$$
K_{X({\Delta}'_i)}= \psi_i^*(K_{U_i}+X_i)|_{X({\Delta}'_i)}+\sum_{{\bold q}\in{\Delta}'_i(1)-
{\Delta}_i(1)}
m_{{\bold q}}(D_{{\bold q}}\cap {X({\Delta}'_i)})_{red}$$
If $D_{{\bold q}}\cap {X({\Delta}'_i)}\neq \phi$,
the both intersect generically transversally each other by the
construction of
$\psi$.
Therefore
$m_{{\bold q}}
=\dfrac{q_i}{p_i}
({\bold p} (f)-{\bold p}({\bold 1})+1)-({\bold q} (f)-{\bold q}({\bold 1})+1)$.
If ${\bold q}\notin {C_{\bl}(f)}$,
then ${\bold q} (f)-{\bold q}({\bold 1})+1\leq 0$
and therefore $m_{{\bold q}}>0$.
If ${\bold q}\in {C_{\bl}(f)}$ and $D_{{\bold q}}\cap {X({\Delta}'_i)}\neq \phi$,
then ${{\bold p}}\not\prec _f{{\bold q}}$.
In fact,
if ${{\bold p}} \prec _f{{\bold q}}$,
then ${{\bold q}} $ is in the interior of ${{\Delta}_i}$ for some $i$,
let it be ${{\Delta}_0}$,which implies $\psi_0(D_{{\bold q}})$ is a point.
We also have that ${{\bold q}}(f)-\dfrac{q_0}{p_0}{{\bold p}}(f)\leq 0$.
On the other hand,
${{\bold q}}(\psi_0^*X_0)={{\bold q}}(f)-\dfrac{q_0}{p_0}{{\bold p}}(f)\geq 0$.
Therefore ${{\bold q}}(\psi_0^*X_0)=0$, which shows that $X({\Delta}_0')\cap
D_{{\bold q}}=\phi$.
Thus it follows that $m_{{\bold q}}\geq 0$ by the absolute $f$-minimality of
${\bold p}$.
Now, by Lemma \ref{normality},
it follows that $X_i$'s have at worst canonical singularities.
The $\varphi$-ampleness of $K_{X({\bold p})}$ follows from
the $\varphi$-ampleness of $-D_{{\bold p}}|_{X({\bold p})}$ and
$K_{X({\bold p})}=\varphi^*K_X+({\bold p}({\bold 1})-{\bold p}(f)-1)D_{{\bold p}}|_{X({\bold p})}$,
where the coefficient of $D_{{\bold p}}$ is negative.
\end{pf}
\begin{cor}
\label{log-canonical}
Let $(X,0)\subset({{\Bbb C}^{n+1}},0)$ be an isolated, non-canonical,
log-canonical singularity defined by a polynomial $f\in {\Bbb C}[x_0,
\ldots , x_n]$.
For a primitive integral vector ${\bold p}=(p_0,\ldots ,p_n)$
such that $p_i> 0$ for all $i$,
\newline
if
(i) ${\bold p}$-blow-up $\varphi :X({\bold p})\to X$ is
the canonical modification,
\newline
then
(ii) ${\bold p}$ is absolutely minimal in ${C_{\bl}(f)} \cap N-\{0\}$.
\vskip .5truecm
Suppose $f$ is non-degenerate,
then the converse (ii)$\Rightarrow$(i) holds too.
\end{cor}
\begin{pf}
Since $(X,0)$ is log-canonical,
${\bold 1}\in{\Gamma_+(f)}$ by \ref{characterization of log-canonical}.
Then ${\bold q}(f)={\bold q}({\bold 1})$, for ${\bold q}\in {C_{\bl}(f)}$.
First,
two distinct primitive ${\bold p}$, ${\bold q} \in {C_{\bl}(f)}$,
neither ${\bold p} \prec _f {\bold q}$ nor ${\bold q} \prec _f {\bold p}$ hold.
In fact,
if ${\bold p} \prec _f {\bold q}$, then $p_i/{\bold p}({\bold 1})\leq q_i/{\bold q}({\bold 1})$
for every $i$, and moreover the equality holds for every $i$,
because,
by summing all these inequalities,
we obtain $1\leq 1$.
Hence ${\bold p}$ must coincide with ${\bold q}$.
On the other hand,
it is clear that $\geq_f$ is equivalent to $\geq$ and therefore
$f$-minimal is equivalent to absolutely minimal.
\end{pf}.
\begin{exmp}
(Tomari)
It is possible for a singularity to have more than one canonical weights.
In fact,
let $X\subset {\Bbb C} ^3$ be defined by $x_0^k+x_1^{k+1} +x_2^{k+1}=0$
$(k\geq 3)$,
then the weights $(1,1,1)$ and $(k+1, k, k)$ are both canonical weights.
\end{exmp}
\begin{say}
\label{various cor}
In the rest of this section a singularity $(X,0)\subset({{\Bbb C}^{n+1}},0)$
is assumed to be a non-canonical, log-canonical singularity
defined by a polynomial $f$.
In some cases,
one can easily see the existence of the absolutely minimal vector,
therefore one also sees the existence of the canonical modification for
these cases.
For 2-dimensional case,
singularities as above are either $\tilde{E_6}$ or $\tilde{E_7}$ or
$\tilde{E_8}$ or defined by equations of type:
$x_0x_1x_2+x_0^p+x_1^q+x_2^r=0$ with $\dfrac{1}{p}+
\dfrac{1}{q}+\dfrac{1}{r}<1$
by suitable coordinates transformations.
It is well known that there exist the canonical weights
for singularities defined by these equations.
The following corollary shows that one need not take a coordinate
transformation for 2-dimensional non-canonical,
log-canonical singularities to
admit the canonical weight.
\end{say}
\begin{cor}
\label{surface}
If $n=2$ and $f$ is non-degenerate,
there exists the absolutely minimal vector in ${C_{\bl}(f)}$ for every $f$
as in \ref{various cor}.
And the vector is either $(1,1,1)$ or $(3,2,1)$ or $(2,1,1)$.
\end{cor}
\begin{pf}
By the direct calculation, one can find the absolutely minimal vector
in ${C_{\bl}(f)}$ for each $f$.
\end{pf}
The next one was proved by Tomari
under a more general situation.
\begin{cor}
\label{Tomari's cor}
(\cite{T})
If ${{\bold 1}}$ is in the interior of an n-dimensional face $\gamma$
of $\Gamma(f)$,
which is equivalent to that the singularity $(X,0)$ is of Hodge type
$(0,n-1)$ (for the definition, cf. \cite{IGor}),
then the primitive vector ${{\bold p}}$ generating $\gamma^{\perp } $
gives the canonical modification $X({{\bold p}})\to X$.
\end{cor}
\begin{pf}
Since $ C_{\bf 1}(f)$ is of one dimension,
the primitive vector on it is clearly absolutely minimal.
It completes the proof of the non-degenerate case.
If $f$ is degenerate,
there may not exist toric embedded resolution.
But taking a resolution $\psi:Y\to X({\bold p})$,
$\varphi\psi$ is a resolution of
a log-canonical singularity of type $(0, n-1)$,
which yields that $K_Y=\psi^*\varphi^*(K_X)+\sum_i m_iE_i$
with the only one negative $m_i$.
By substituting
$(K_{T_N({\Delta}({\bold p}))}+X({\bold p}))|_{X({\bold p})}=
\varphi^*(K_X)-D_{{\bold p}}|_{X({\bold p})}$ into the equality above,
we can see that
the pair $X({\bold p})\subset T_N({\Delta}({\bold p}))$ satisfy the conditions of
\ref{normality}.
\end{pf}
\begin{cor}
If a non-degenerate polynomial $f$ is represented as $x_0\cdots x_n
+h(x_0,\ldots ,x_n)$,
where $deg\ h\geq n+1$,
then the blow-up by the maximal ideal of the origin is
the canonical modification.
\end{cor}
\begin{pf}
Since ${\Gamma_+(f)}$ is in the domain
$\{{\bold a}\in M_{{\Bbb R}}|a_0+\cdots +a_n \geq n+1\}$,
it follows that $(1,1,\ldots ,1)\in {C_{\bl}(f)}$ and clearly this is absolutely
minimal.
\end{pf}
\begin{cor}
\label{quasi-reduced}
If every vector ${\bold a}\in {\Gamma(f)}\cap M$ is quasi-reduced
(i.e. ${\bold a}=(a_0,\ldots , a_n)$ satisfies that $0\leq a_i\leq 1$
except for at most one $i$),
then there exists the absolutely minimal vector ${\bold p}$ in ${C_{\bl}(f)}$.
\end{cor}
\begin{pf}
A positive vector ${\bold q}\in N$ belongs to ${C_{\bl}(f)}$,
if and only if ${\bold q}({\bold a})\geq{\bold q}({\bold 1})$ for all ${\bold a}=(a_0,\ldots ,a_n)
\in {\Gamma(f)}\cap M$.
These inequalities are equivalent to the inequalities of the following
type:
$(a_i-1)q_i\geq\sum_{j\in \Lambda({\bold a})}q_j$,
where $a_i\geq 2$ and $\Lambda({\bold a})$ is the suitable subset of
$\{0,\ldots ,n\}$ such that $i\notin \Lambda({\bold a})$.
Let ${\bold p}=(p_0, \ldots ,p_n)$ and ${\bold q}=(q_0,\ldots ,q_n)$
belong to ${C_{\bl}(f)}$.
Define ${\bold r}=(r_0,\ldots ,r_n)$ by $r_i=min\{p_i,q_i\}$.
We show that ${\bold r}\in {C_{\bl}(f)}$.
For ${\bold a}\in{\Gamma(f)}\cap M$,
let $a_i\geq2$.
We can assume that $r_i=p_i$ by the definition of ${\bold r}$.
Then $(a_i-1)r_i=(a_i-1)p_i\geq
\sum_{j\in\Lambda({\bold a})}p_j\geq\sum_{j\in\Lambda({\bold a})}r_j$.
Hence ${\bold r}$ also satisfies ${\bold r}({\bold a})\geq{\bold r}({\bold 1})$
for all ${\bold a}\in {\Gamma(f)}\cap M$.
\end{pf}
\begin{exmp}
\label{typeT}
We say that $X$ is of type $T_{{\bold a}}$,
if it is defined by $f=x_0\cdots x_n+\sum x_i^{a_i}$ for
${\bold a}=(a_0,\ldots, a_n)$,
where $\sum_i1/a_i<1$.
Then $f$ satisfies the condition of \ref{quasi-reduced}
and therefore $X$ has a weight which gives the canonical modification.
The summary paper \cite{IB} contains the table of 3-dimensional
$T_{p,q,r,s}$-singularities $(X,0)$ with the absolutely minimal vectors
${\bold p}$.
All those weights are in the weights of 95-simple K3-singularities
listed in \cite{Y} which is bijective to the list of \cite{F}.
And therefore $T_{p,q,r,s}$-singularities have the same plurigenera
$\{\gamma_m\}$ with those of corresponding simple K3-singularities
(cf. \ref{deform 1}).
\end{exmp}
\vskip 1truecm
\section{Deformations and the simultaneous canonical modifications}
\begin{defn}
\label{simul cano}
Let $\pi :({\cal X},x) \to (C, 0)$ be a flat morphism
over a non-singular curve $C$.
A partial resolution $\Phi :{\cal Y}\to {\cal X}$ is called
the simultaneous canonical modification,
if the restriction
$\Phi_t :{\cal Y}_t\to {\cal X}_t$ is the canonical modification for
every $t \in C$,
where ${\cal X}_t=\pi ^{-1}(t)$ and ${\cal Y}_t=\Phi ^{-1}({\cal X}_t)$.
\end{defn}
\begin{prop}
\label{deform 1}
Let $(X,0)\subset ({{\Bbb C}^{n+1}}, 0)$ be an isolated, non-canonical,
log-canonical singularity defined by a polynomial $f$.
Assume that $X({\bold p})\to X$ is the canonical modification for a
positive integral vector ${\bold p}$.
Let $\{F_t\}_{t\in C}$ be a deformation of $f=F_0$ over a non-singular
curve $C$
such that $F_t$'s ($t\neq 0$) are non-degenerate and Newton's
diagrams $\Gamma_+(F_t)$ sit in the halfspace ${\bold 1}+{\bold p}^{\vee}$
of $M_{{\Bbb R}}$.
Then
the flat family $\pi:({\cal X},0)\to (C,0)$ defined by
$\{F_t\}_{t\in C}$
admits the simultaneous canonical modification and
$\gamma_m({\cal X}_t,0)$ is constant in $t\in C$ for every
$m\in {\Bbb N}$.
\end{prop}
\begin{pf}
By the assumption of $\{F_t\}_{t\in C}$,
${\bold p}({\bold 1})\leq {\bold p}(F_t)\leq {\bold p}(f)$.
Therefore ${\bold p} \in C_{{\bold 1}}(F_t) \subset {C_{\bl}(f)}$.
Since ${\bold p}$ is absolutely minimal in ${C_{\bl}(f)}$,
it is absolutely minimal in $C_{{\bold 1}}(F_t)$ for every $t\in C$,
which yields that ${\bold p}$ is the canonical weight for the singularities
defined by $F_t=0$.
On the other hand,
since $(X,0)$ is log-canonical, ${\bold 1}\in {\Gamma_+(f)}$ by
\ref{characterization of log-canonical}.
Hence ${\bold p}(f)={\bold p}({\bold 1})$ which implies also ${\bold p}({\bold 1})={\bold p}(F_t)$.
Take the morphism $\Phi:=\varphi\times id_C: {T_N({\Delta}({\bold p}))}
\times C \to {{\Bbb C}}^{n+1} \times C$,
where $\varphi : T_N({\Delta}({\bold p})) \to {{\Bbb C}}^{n+1}$
is the ${\bold p}$-blow-up.
Denote the proper transform of ${\cal X}$ in $T_N({\Delta}({\bold p}))
\times C $ by ${\cal Y}$,
then $\Phi^*{\cal X}={\cal Y}+{\bold p}(F_t)(D_{{\bold p}}\times C)$ for a
general $t\in C$,
where $D_{{\bold p}}$ is the corresponding divisor to ${\bold p}$ on
$T_N({\Delta}({\bold p}))$.
Since ${\bold p}(F_t)$ is constant for all $t\in C$,
${\cal Y}_t$'s are all irreducible, and therefore
these turn out to be the ${\bold p}$-blow-ups of
${\cal X}_t$,
which shows that ${\cal Y}\to {\cal X}$ is the simultaneous canonical
modification.
By Proposition 7 of \cite{Stevens},
${\cal Y}$ admits at worst canonical singularities, and,
on the other hand,
$K_{{\cal Y}/{\cal X}}=-(D_{{\bold p}}\times C)|_{\cal Y}$ is
$\Phi$-ample,
which means that ${\cal Y}\to {\cal X}$ is
the canonical modification
of ${\cal X}$.
Thus $\pi$ turns out to be an (FG)-deformation in terms of \cite{Isiml}.
By 1.11 of \cite{Isiml},
it follows that $\gamma_m({\cal X}_t, 0)$ is constant for all $t\in C$.
\end{pf}
\begin{prop}
\label{deform 2}
Let $(X,0)\subset ({{\Bbb C}^{n+1}}, 0)$ be an isolated, non-canonical,
log-canonical singularity defined by a polynomial $f$.
Assume that $X({\bold p})\to X$ is the canonical modification for a
positive integral vector ${\bold p}$.
If ${\bold p}/\sum_ip_i$ is the weight of a weighted-homogeneous
polynomial defining an isolated singularity
at the origin,
then $\gamma_m(X,0)=\gamma_m(Y,0)$,
where $(Y,0)\subset ({{\Bbb C}^{n+1}}, 0)$ is defined by a non-degenerate
weighted-homogeneous polynomial $g$ with the weight ${\bold p}/\sum_ip_i$.
Moreover there exists a flat deformation $\pi:({\cal X}, 0)\to
(C,0)$ of $(X,0)=({\cal X}_0, 0)$ over a non-singular curve $C$
with $({\cal X}_{\tau}, 0)\simeq (Y,0)$ for some $\tau\in C$
such that $\pi$ admits the simultaneous canonical modification.
\end{prop}
\begin{pf}
Let $F_t$ be $(1-t)f+tg$ for $t\in {\Bbb C}$.
Then, taking a suitable open subset $C\subset {\Bbb C}$ with $0, 1 \in C$,
it follows that $F_t$ $(t\neq 0 )$ defines a non-canonical, log-canonical
singularity of type $(0, n-1)$, because it is a small deformation
of such a singularity $\{g=0\}$ and ${\bold 1}\in \Gamma_+(F_t)$ (4.4 of
\cite{def},
2.2 of \cite{Isiml} and \ref{characterization of log-canonical}).
Hence by \ref{Tomari's cor}, ${\bold p}$ is the canonical weight for
$F_t$ $(t\neq 0)$.
Since ${\bold p}(F_t)={\bold p}({\bold 1})$ for all $t\in C$,
we can see that the deformation $\pi:{\cal X}\to C$ defined by
$\{F_t\}$
admits the simultaneous canonical modification ${\cal X}({\bold p})$
in the same way as in the proof of \ref{deform 1}.
Therefore
$\gamma_m(X,0)=\gamma_m({\cal X}_1,0)$.
\end{pf}
\begin{exmp}
(Watanabe)
One can see in \cite{W2}
95-examples of deformations such as in Proposition \ref{deform 2}.
For example,
let $X\subset {\Bbb C} ^4 $ be defined by
$f=x_0^2+x_1^3+x_2^7+x_3^{43+s}+x_0x_1x_2x_3=0$
$(s\geq 0)$
and $Y\subset {\Bbb C} ^4 $ by $g=x_0^2+x_1^3+x_2^7+x_3^{42}=0$.
Let ${\bold p}$ be $(21, 14, 6, 1)$,
then $X({\bold p}) \to X$ is the canonical modification and ${\bold p}/42$ is the
weight of
the quasi-homogeneous polynomial $g$.
One can construct a family $\{F_t\}$ connecting $f$ and $g$ as in the
proof of
Proposition \ref{deform 2}.
\end{exmp}
\section{A counter example to the conjecture}
\begin{say}
In this section
we show a counter example to the conjecture written in the introduction.
Let $f$ be the polynomial:
$x_0x_1x_2x_3+\alpha x_0^3 +\beta x_1^2x_2^2 +\gamma x_1^{a_1} +
\delta x_2^{a_2}+\epsilon x_3^{a_3} \in {\Bbb C}[x_0,\ldots ,x_3]$,
with $a_i\geq 6$ and $\alpha,\ \beta,\ \gamma, \ \delta, \ \epsilon \in
{\Bbb C}$ general.
Then $f$ is non-degenerate and defines an isolated, non-canonical,
log-canonical singularity $(X,0)$ at the origin by
\ref{characterization of log-canonical}.
The essential cone is as follows:
$${C_{\bl}(f)}=
\{{\bold q}\in \sigma |2q_0-q_1-q_2-q_3\geq 0,\ -q_0+q_1+q_2-q_3\geq 0,
\ (a_i-1)q_i-\sum_{j\neq i}q_j\geq 0,\ i=1,2,3\}$$
Here ${C_{\bl}(f)}$ has no absolutely minimal vector.
In fact,
it is easy to see that $(2,2,1,1) $ and $(2,1,2,1)$ belong to
${C_{\bl}(f)}$ but neither $(2,1,1,1)$ nor (1,1,1,1) does.
This shows that under these coordinates
there is no weighted blow-up which is the canonical
modification of $(X,0)$ by \ref{main theorem}.
In the following,
we prove the same statement under arbitrary coordinates.
\end{say}
\begin{lem}
If $ Y\to X$ is the canonical modification of $(X,0)$,
then $-K_Y^3>3/2$.
\end{lem}
\begin{pf}
We use the notation in \ref{quadrant} and \ref{notation star}.
Denote the fan consisting of all faces of the
positive quadrant in $N_{{\Bbb R}}$ by ${\Delta}$.
Let ${\bold q}$ be $(2,1,2,1)$,
and $\varphi:T_N({\Delta}({\bold q}))\to {\Bbb C}^4$,
$\varphi'=\varphi |_{X({\bold q})}:X({\bold q}) \to X$
be the ${\bold q}$-blow-ups of ${\Bbb C}^4$ and $X$ respectively
under the given coordinates.
First we prove that $X({\bold q})$ has log-terminal singularities.
For any resolution $\psi:\tilde X \to X({\bold q})$ of the singularities
on $X({\bold q})$,
we can write $K_{\tilde X}=\psi^*{\varphi'}^*(K_X)+\sum_i a_iE_i$
with $a_i\geq -1$ for all exceptional divisors $E_i$,
because $(X,0)$ is log-canonical.
On the other hand,
by \ref{valuation of function} and \ref{birational canonical},
$K_{T_N({\Delta}({\bold q}))}+X({\bold q})=\varphi^*(K_{{{\Bbb C}}^4}+X)
-D_{{\bold q}}$,
since ${\bold q}({\bold 1})-1-{\bold q}(f)=-1$.
Therefore if we write:
$K_{\tilde X}
=\psi^*((K_{T_N({\Delta}({\bold q}))}+X({\bold q}))|_{X({\bold q})})+\sum_jm_jE_j$,
then $m_j>-1$ for every exceptional divisor $E_j$ of $\psi$.
Hence, by \ref{normality},
$X({\bold q})$ has at worst log-terminal singularities.
Note that there is a non-canonical singularity,
because $m_j=-1/2$ for $E_j$ which corresponds to the vector
$(2,2,1,1)$.
Next construct a flat deformation $\pi:({\cal X},0)\to(C, 0)$
by $(1-t)f+t(x_0^3+x_1^6+x_2^3+x_3^6)$
as in \ref{deform 2}
so that $({\cal X}_0,0)\simeq (X,0)$, and $({\cal X}_1,0)$ is
defined by non-degenerate
weighted homogeneous polynomial $x_0^3+x_1^6+x_2^3+x_3^6$
with the weight
$\dfrac{1}{6}(2,1,2,1)$.
%
This deformation is proved to be an (FG)-deformation
(for the definition cf. \cite{Isiml}) as follows:
Let $\Phi:{\cal X}({\bold q})\to {\cal X}$ be
the restriction of $\varphi\times id_{{\Bbb C}}$ onto
the proper transform ${\cal X}({\bold q})$ of ${\cal X}$ in
$T_N({\Delta}({\bold q}))\times {\Bbb C}$;
since ${\bold q}(f)={\bold q}({\bold 1})={\bold q}(F_t)$ for $t\in C$,
${\cal X}({\bold q})_t$ is the ${\bold q}$-blow-up ${\cal X}_t({\bold q})$
of ${\cal X}_t$
for every $t\in C$
as in the proof of \ref{deform 1};
here ${\cal X}({\bold q})_t$ has at worst canonical singularities for
$t\neq 0$ by \ref{Tomari's cor}
and ${\cal X}({\bold q})_0=X({\bold q})$ has at worst log-terminal singularities as
proved above;
on the other hand, it is clear that $K_{{\cal X}({\bold q})}$ and
$K_{{ X}({\bold q})}$ are both ${\Bbb Q}$-Cartier divisors;
hence by Proposition 7 of \cite{Stevens},
${\cal X}({\bold q})$ admits at worst canonical singularities;
one can easily see that
$K_{{\cal X}({\bold q})}=-(D_{{\bold q}}\times C)|_{{\cal X}({\bold q})}$
which is $\varphi$-ample,
which shows that ${\cal X}$ admits the canonical modification
${\cal X}({\bold q})$ (i.e. $\pi$ is an (FG)-deformation).
Now we can apply the upper semi-continuity theorem on $\{\gamma_m\}$
(Theorem 1 of \cite{Isiml})
to our (FG)-deformation $\pi$.
Since $-K^3/3!$ of the canonical modification is the coefficient
of the leading term of a function $\gamma_m$ in $m$,
it follows
that $-K_Y^3\geq -K_{{\cal X}({\bold q})_t}^3=\sum q_i/\Pi q_i =3/2$
for $t\in C-\{0\}$.
Here the equality does not hold.
Because if it does,
$\pi$ would admit the simultaneous canonical modification
$\Psi:{\cal Y}\to {\cal X}$ by Corollary 1.11 on \cite{Isiml}.
Since the simultaneous canonical modification must be the canonical
modification of ${\cal X}$ by \cite{Stevens} again,
${\cal Y}$ would coincide with ${\cal X}({\bold q})$.
However ${\cal X}({\bold q})_0$ has a non-canonical singularity
as is seen above.
\end{pf}
\begin{say}
Now we assume that there are coordinates $y_0,\ldots ,y_3$ on ${\Bbb C}^4$
and a weight ${\bold p}=(p_0,\ldots ,p_3)$ such that the ${\bold p}$-blow-up
$X({{\bold p}})\to X$ under these coordinates gives the canonical
modification, and then will induce a contradiction.
Let $g(y)=0$ be the defining equation of $X$ under these coordinates.
By \ref{main theorem}, it follows that ${\bold p}(g)={\bold p}({\bold 1})$
and therefore $-K^3_{X({\bold p})}=\sum_ip_i/\Pi _i p_i>3/2$.
Now it is easy to prove that at least three of
the $p_i$'s must be 1.
Write the coordinates transformation as follows:
$$(T_i)\ \ \ \ \ \ \ \ \ \ \ x_i=\sum _{m\in
{\Bbb Z}_{\geq 0}^4}a_m^{(i)}y^m\ \
\ \ (a_m^{(i)}\in {\Bbb C}).$$
We may assume that the coefficient of $y_i$ in $(T_i)$ is not
zero for
each $i$ by reordering $\{y_i\}$'s.
Then $y_0^3\in g$ (see \ref{coefficient of divisor} for the notation),
since $x_0^3\in f$ and this is the unique monomial of degree 3 in $f$.
Therefore ${\bold p}(3,0,0,0)\geq {\bold p}({\bold 1})$ which means $p_0\geq 2$,
since ${\bold p}$ must be in ${C_{\bl}(g)}$ by \ref{main theorem}.
Then one obtains the fact that
$a_{0,1,0,0}^{(0)}=a_{0,0,1,0}^{(0)}=a_{0,0,0,1}^{(0)}
=0$,
otherwise $y_i^3\in g$, for $i=1,2,3$ which induce ${\bold p}(0,3,0,0)\geq
{\bold p}({\bold 1})$ and so on, therefore $3\geq p_0+3$ a contradiction.
One can also prove that $a_{0,0,1,0}^{(1)}= a_{0,1,0,0}^{(2)}=0$
in the same way.
Then it follows that $y_1^2y_2^2\in g$,
because this monomial comes from the term $x_1^2x_2^2 $ and is not
cancelled
by the contribution from other terms.
Hence ${\bold p}$ must satisfy ${\bold p}(0,2,2,0)\geq {\bold p}({\bold 1}) $
which is equivalent to $4\geq p_0+3$, a contradiction.
\end{say}
\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother
|
1996-01-25T06:20:22 | 9601 | alg-geom/9601021 | en | https://arxiv.org/abs/alg-geom/9601021 | [
"alg-geom",
"math.AG"
] | alg-geom/9601021 | null | Alexander Goncharov | Volumes of hyperbolic manifolds and mixed Tate motives | Better LaTeX file, no mathematical changes | null | null | null | null | Two different constructions of an invariant of an odd dimensional hyperbolic
manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The
volume of the manifold is equal to the value of the Borel regulator on that
element. The scissor congruence groups in non euclidian geometries are studied
and their relationship with algebraic K-theory of the field of complex numbers
is discussed.
| [
{
"version": "v1",
"created": "Fri, 19 Jan 1996 15:43:38 GMT"
},
{
"version": "v2",
"created": "Wed, 24 Jan 1996 14:33:40 GMT"
},
{
"version": "v3",
"created": "Wed, 24 Jan 1996 14:36:24 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Goncharov",
"Alexander",
""
]
] | alg-geom | \section{ Proof of the theorem 2.5}
{\bf 1. Some results on the $t$-structure on a triangulated
category}.
Recall that a $t$-structure on a triangulated category $\cal D$
is a pair of subcategories ${\cal D}^{\leq 0}$,${\cal D}^{\geq
1}$ satisfying the following
conditions:
1)$Hom_{\cal D}(X,Y) =0$ for all objects $X \in Ob
{\cal D}^{\leq 0}$ and $Y \in {\cal D}^{\geq 1}$.
2)For any $X \in Ob {\cal D}$ there exists an exact triangle
$$
X_{\leq 0} \longrightarrow X \longrightarrow X_{\geq 1}
\longrightarrow X_{\leq 0}[1]
$$
with $X_{\leq 0} \in {\cal D}^{\leq 0}$ and $X_{\geq 1} \in {\cal
D}^{\geq 1}$.
3)${\cal D}^{\leq 0} \subset {\cal D}^{\leq 1}$ and
${\cal D}^{\geq 1} \subset {\cal D}^{\leq 0}$
Here ${\cal D}^{\leq a}:= {\cal D}^{\leq 0}[-a]$, ${\cal D}^{\geq
a}:= {\cal D}^{\geq 0}[-a]$, ${\cal D}^{[a,b]}:= {\cal D}^{\leq
b}\cap {\cal D}^{\geq a}$ and ${\cal D}^{a}:= {\cal
D}^{[a,a]}$.
The exact triangle in (2) is defined uniquely up
to an isomorphism and depends functorially on $X$.
The subcategory ${\cal D}^{0}$ is called the heart of a
$t$-structure. It is an abelian category ([BBD]).
Let ${\cal A}$ and ${\cal B}$ be two sets of isomorphism classes
of objects in ${\cal D}$. Denote by
${\cal A} \ast {\cal B}$ the set of all objects $X$ in $Ob {\cal
D}$ which can be included into an exact triangle $A
\longrightarrow X \longrightarrow B \longrightarrow
A[1]$ with $A \in {\cal A}, B \in {\cal B}$.
\begin{lemma} \label{asso}
$({\cal A} \ast {\cal B}) \ast {\cal C} = {\cal A} \ast ({\cal B}
\ast {\cal C})$
\end{lemma}
{\bf Proof}. Follows from the octahedron lemma, see [BBD].
\begin{theorem} \label{trc}
a)Let $\cal D$ be a triangulated $\Bbb Q$-category
and ${\cal Q}$ be a full semisimple subcategory generating ${\cal D}$ as
a triangulated category. Suppose that for any two objects
$Q_1,Q_2 \in Ob {\cal Q}$ one has
\begin{equation} \label{negext}
Hom_{\cal D}^{-i}(Q_1,Q_2) = 0 \qquad i> 0
\end{equation}
Then there is canonical t-structure on ${\cal D}$ with the
abelian heart ${\cal M} = \cup{\cal Q} \ast {\cal Q} \ast ... \ast {\cal
Q}$.
b) If in addition $Hom_{\cal D}^{i}(Q_1,Q_2) = 0$ for
$i \geq 2$, then the
tensor category ${\cal D}$ is
equivalent to the derived
category of ${\cal M}$.
\end{theorem}
\begin{theorem} \label{trc1}
Let $\cal D$ be a triangulated tensor $\Bbb Q$-category
and ${\cal Q}$ be a full semisimple subcategory generated
by non isomorphic
objects $\Bbb Q(m)$, $m \in \Bbb Z$ such that $\Bbb Q(1)$
is invertible,
$\Bbb Q(m) = \Bbb Q(1)^{\otimes m}$, and
$$
Hom_{\cal D}^{i}( \Bbb Q(m),\Bbb Q(n)) = 0 \quad \mbox{if} \quad m > n, \qquad
Hom_{\cal D}( \Bbb Q(0),\Bbb Q(0)) = \Bbb Q
$$
Then the abelian heart ${\cal M}$
from the theorem (\ref{trc}) is a tensor category. It is equivalent to
the tensor category of finite dimensional
representations of a certain free negatively graded (pro)-Lie algebra.
\end{theorem}
Let ${\cal D}$ be a triangulated category and ${\cal M} \subset {\cal D}$
be a full subcategory.
\begin{theorem} \label{trc3}
${\cal M}$ is a heart for the unique bounded t-structure on
${\cal D}$ if and only if
i) ${\cal M}$ generates ${\cal D}$ as a triangulated category
ii) ${\cal M}$ is closed with respect to extensions
iii) $Hom_{\cal D}(X,Y[i]) = 0 \quad \mbox{for any} \quad X,Y \in
{\cal M}, \quad i<0 $
iv) ${\cal M} \ast {\cal M}[1] \subset {\cal M}[1] \ast {\cal M}$
\end{theorem}
For the proof of this theorem see [BBD] or [P]. Let me scetch
the construction of the $t$-structure on ${\cal D}$. We will use
the following
\begin{lemma} \label{dirs}
Let
$$
X \longrightarrow Y \longrightarrow Z
\stackrel{f}{\longrightarrow} X[1]
$$
be an exact triangle. Then $f=0$ if and only if $Y =X\oplus Z$
\end{lemma}
{\bf Proof}. Let us show that $f=0$ implies $Y =X\oplus Z$.
The composition of the identity morphism
$Z \stackrel{id}{\longrightarrow} Z$ with $f$
is zero, so one has a morphism
$g: Z \longrightarrow Y$ making the following diagram commutative:
$$
\begin{array}{ccccccc}
X & \stackrel{a}{\longrightarrow} & Y& \longrightarrow & Z &
\stackrel{f}{\longrightarrow}& X[1]\\
&&&&&&\\
&&&\nwarrow g& \uparrow id& \nearrow =0\\
&&&&&&\\
&&&&Z&&
\end{array}
$$
So there is a morphism $X \oplus Z
\longrightarrow X$. The universality property of this morphism
follows immediately from $f=0$.
Let ${\cal D}^{[a,b]}$ be the minimal full subcategory of ${\cal D}$
containing ${\cal M}[-i]$
for $a \leq i \leq b$ and closed under extensions. The
subcategories ${\cal D}^{\leq 0}$ and ${\cal D}^{\geq 1}$ are
defined similarly and satisfy 1) thanks to iii). Notice that
$Hom_{{\cal D}}({\cal M},{\cal M}[-i]) =0$ implies ${\cal
M} \ast {\cal M}[i+1] = {\cal M} \oplus {\cal M}[i+1]$ for $
i>0$ by lemma \ref{dirs}. So ${\cal M} \ast {\cal
M}[n] \subset {\cal M}[n] \ast {\cal M}$ for any $n>0$. Using
ii) and
lemma \ref{asso} we get ${\cal D}^{[a,b]} = {\cal M}[-a]\ast
{\cal M}[-a-1]\ast ... \ast {\cal M}[-b]$. This proves 2). It
remains to show ${\cal D}^{\leq 0} \cap {\cal D}^{\geq 1}
\subset {\cal D}^{[a,b]}$. If $X = Y_c \ast Y_{c+1} \ast ... \ast Y_d$
where $Y_i \in {\cal M}[-i] \in {\cal D}^{\geq a}$ then
$Hom(Y_c,X) =0$ for $c<a$ and so we get from the exact triangle
$$
Y_c \longrightarrow X \longrightarrow Y_{c+1} \ast ... \ast Y_d
\longrightarrow Y_c[1]
$$
that $Y_{c+1} \ast ... \ast Y_d = X \oplus Y_c[1]$. Now
$Hom_{{\cal D}}(Y_c[1],Y_{c+1} \ast ... \ast Y_d) =0$ by iii)
and so $Y_c
= 0$.
{\bf Proof of theorem\ref{trc}}. a).
Let us use theorem \ref{trc3}.
The properties i)-ii) are obvious. iii) is easy to prove
using (\ref{negext}), so we have to check
only iv). Using the associativity of $\ast$ operation we see that
one has to prove only that $Q_1 \ast Q_2[1] \subset Q_2[1] \ast Q_1
$ for any two simple objects in ${\cal Q}$. One has
$$
Q_1 \longrightarrow X \longrightarrow Q_2[1]
\stackrel{f}{\longrightarrow} Q_1[1]
$$
There are only two possibilities for
$f$:
1. $f$ is an isomorphism; then $X=0$.
2. $f=0$; then $X = Q_1 \oplus Q_2[1]$ by lemma \ref{dirs}.
{\bf Part b)}.
\begin{proposition} \label{ex}
a) Let ${\cal D}$ be a triangulated category and ${\cal M}$ be the
heart of a $t$-structure on ${\cal D}$. Suppose that
one has
\begin{equation} \label{ext2}
Hom^i_{{\cal D}}(X,Y)
=0 \quad \mbox{for any} \quad X,Y \in Ob{\cal M}, \quad i >1
\end{equation}
Then the category
$D^b({\cal M})$ is
equivalent to the category ${\cal D}$.
b) If we suppose in addition that the triangulated category ${\cal D}$ is
a subcategory of a derived category $D^b({\cal N})$ for some abelian category
${\cal N}$, then $D^b({\cal M})$ is equivalent to ${\cal D}$ as a
triangulated category.
\end{proposition}
{\bf Proof}. Let $\tilde {\cal D}$ be the full subcategory of
${\cal D}$
whose objects are direct sums $\oplus A_i[-i]$ where $A_i \in
{\cal D}^{-i}$,
i.e. $H^j_{{\cal M}}(A_i) =0$ for $j \not = -i$. There is a
canonical functor $i: \tilde {\cal D} \hookrightarrow {\cal D}$.
Then (\ref{ext2}) is a necessary and
sufficient condition for $i$ to be an equivalence of triangulated
categories.
Indeed, let us show that every object $Y$ in ${\cal D}$ is isomorphic
to $i(X)$ for some $X$. We may suppose $Y \in {\cal D}^{[0,n]}$ and will use
induction by $n$. Consider the exact triangle
$$
H^0_{{\cal M}}(Y) \stackrel{f}{\longrightarrow} Y \longrightarrow C
\longrightarrow H^0_{{\cal M}}(Y)[1]
$$
provided by the canonical morphism $f: H^0_{{\cal M}}(Y) \longrightarrow Y
$. Then $C \in {\cal D}^{[1,n]}$ and $H^0_{{\cal M}}(Y)[1] \in
{\cal D}^{-1}$,
so it is easy to see that the morphism $C
\longrightarrow H^0_{{\cal M}}(Y)[1]$ must equal to $0$ because
$Hom^{\geq 2}$ are zero. Therefore $Y = H^0_{{\cal M}}(Y) \oplus C$ by the
lemma
above.
Let us show that $D^b({\cal M})$ has cohomological dimension
one, i.e.
\begin{equation} \label{ext2!}
Hom^i_{D^b({\cal M})}(X,Y)
=0 \quad \mbox{for any} \quad X,Y \in Ob{\cal M}, \quad i >1
\end{equation}
Notice
that for any $X,Y \in {\cal M}$ one has
$$
Hom_{D^b({\cal M})}^i(X,Y) = Ext_Y^i(X,Y)
$$
where $Ext_Y^i(X,Y)$ is the Yoneda $Ext$-groups in ${\cal
M}$. Also
$Hom_{{\cal D}}^1(X,Y) = Ext_Y^1(X,Y)$.
One has $Ext^2_Y(X,Y) \subset Ext^2_{{\cal D}}(X,Y)$.
Therefore $Ext^2_{{\cal D}}(X,Y)=0$ implies $Ext^2_Y(X,Y) =0$.
Any
element in $Ext_Y^n$ can be represented as a product of certain
elements from $Ext_Y^1$ and $ Ext_Y^{n-1}$. So if $Ext^2_Y(X,Y) = 0$ for any 2
objects $X,Y$ in ${\cal
M}$, then $Ext^i_Y(X,Y) = 0$ for all $i \geq 2$. So we have an
equivalence of the categories. Part b) is proved using [BBD].
The proposition is proved.
The proof of the theorem (\ref{trc1}) is rather standard.
One shows that ${\cal M}$ is a mixed Tate category
(see [BD] or [G1] for the definitions) thanks to the conditions
imposed on $Hom$'s between $\Bbb Q(i)$'s; then the
Tannakian formalism leads to the theorem (\ref{trc1})
(see again [BD] or [G1]).
{\bf 2. An application: the abelian category of mixed Tate
motives over a number field}.
I will work with the category ${\cal D}{\cal M}_F$ of triangulated
mixed motives over a field $F$ from [V]. An object in ${\cal D}{\cal M}_F$
is a ``complex'' of regular (but not necessarily projective) varieties $X_1
\longrightarrow X_2
\longrightarrow ... \longrightarrow X_n$ where the morphisms are given by
finite
correspondences and the composition of any two successive morphisms is
zero.
A pair
$(\Bbb P^n \backslash Q,L)$ where $Q$ is a nondegenerate quadric and $L$
is a simplex provides an object $m(Q,L)$ in
${\cal D}{\cal M}_F$. Namely, for an subset set $I= \{j_1 < ... <j_{n-i}\}$
of $ \{0,1,...,n \}$ let $L(I):=
L_{j_1} \cap L_{j_2} \cap ... \cap
L_{j_{n-i}}$ be the corresponding $i$-dimensional face of $L$ and
$L^Q(I):=L(I) \backslash (Q \cap L(I))$. Let $L^Q(i):= \cup_{|I| =
n-i}L^Q(I)$. Then $m(Q,L):= Hom({\tilde m}(Q,L),\Bbb Q)$
Here $Hom$ is the inner Hom in the category ${\cal
D}{\cal M}_F$ and
$$
{\tilde m}(Q,L):= L^Q(0) \longrightarrow L^Q(1) \longrightarrow ...
\longrightarrow L^Q(n-1) \longrightarrow \Bbb P^n \backslash Q
$$
where the first group is sitting in degree $0$ and the
differentials decrease the
degree and given by the usual rules in the simplicial resolution.
There are the objects $\Bbb Q(n) \in {\cal D}{\cal M}_F$ satisfying
almost all
the needed properties including the relation with K-theory:
\begin{equation} \label{vvvvv}
Ext^i_{{\cal D}{\cal M}_F}(\Bbb Q(0), \Bbb Q(n)) =
gr^{\gamma}_nK_{2n-i}(F)\otimes \Bbb Q
\end{equation}
The formula above follows from the results of Bloch, Suslin and
Voevodsky; the key step is the relation of Higher Chow groups
and algebraic K-theory proved by Bloch (see [Bl3] and the moving
lemma in [Bl4]). For the relation between the Higher Chow groups
and motivic cohomology (i.e. the left hand side in (\ref{vvvvv})) see
[V], Proposition 4.2.9 (Higher Chow groups = Borel-Moore motivic homology)
and [V], Proposition 4.3.7 (duality for smooth varieties).
The only serious problem is
the Beilinson-Soul\'e vanishing conjecture which should guarantee
that the negative $Ext$'s are zero.
However if $F$ is a number field the vanishing conjecture follows from the
results of Borel and Beilinson ([B2], [Bo2]). So $Hom_{{\cal D}{\cal
M}}^{-i}(\Bbb Q, \Bbb Q(n))=0$ for all $i,n>0$. The category ${\cal D}{\cal
M}_F$ is a subcategory of the
derived category of ``sheaves with transfers'', see [V], so we
may apply part b) of the proposition 5.6.
Therefore we can apply the ``category
machine'' from s. 5.1 and get the abelian category ${\cal M}_T(F)$
of mixed Tate
motives over a number field $F$ with {\it all} the needed
properties. In particulary we have the Hopf algebra
${\cal A}_{\bullet}(F)$ provided by the Tannakian formalism.
For any embedding $\sigma: F \hookrightarrow \Bbb C$
there is the realisation functor $H_{\sigma}$
from the abelian category of mixed motives over a number field
$F$ to the abelian category of mixed Hodge Tate structures:
$$
H_{\sigma}: {\cal M}_T(F) \longrightarrow {\cal H}_T
$$
It follows from the Borel theorem (injectivity of
the regulator map on $K_{2n-1}(F)\otimes \Bbb Q$ for number fields) that
$\oplus_{\sigma} H_{\sigma}$ induces an injective map on
\begin{equation} \label{BeBo1}
\oplus_{\sigma} H_{\sigma}: Ext_{{\cal M}_T(F)}^1(\Bbb Q(0), \Bbb
Q(n)) \hookrightarrow \oplus_{\sigma}
Ext_{{\cal H}_T(F)}^1(\Bbb Q(0), \Bbb Q(n))
\end{equation}
{\bf 3. Proof of the theorem (\ref{Theorem 2.4}): the final step}.
The Hodge realisation provides a diagram
$$
\begin{array}{ccc} \label{codi1}
S(F)_{\bullet} & \stackrel{\Delta}{\longrightarrow} & S(F)_{\bullet} \otimes
S(F)_{\bullet}\\
&&\\
\downarrow&&\downarrow\\
&&\\
\oplus_{\sigma} {\cal H}_{\bullet }&
\stackrel{\nu}{\longrightarrow} &
\oplus_{\sigma}{\cal H}_{\bullet} \otimes {\cal H}_{\bullet}
\end{array}
$$
Here the vertical arrows are embeddings because (\ref{BeBo1}) is
injective. This diagram is commutative thanks to the main
results of the
chapter 4 (especially theorem 4.8).
Further, one has the diagram
$$
\begin{array}{ccc} \label{codi1}
S_{\bullet}(F) & \stackrel{\Delta}{\longrightarrow} & S_{\bullet}(F) \otimes
S_{\bullet}(F)\\
&&\\
\downarrow&&\downarrow\\
&&\\
{\cal A}_{\bullet}(F)& \stackrel{\Delta}{\longrightarrow} & {\cal
A}_{\bullet}(F) \otimes {\cal
A}_{\bullet}(F)\\
&&\\
\downarrow&&\downarrow\\
&&\\
\oplus_{\sigma} {\cal H}_{\bullet }&
\stackrel{\nu}{\longrightarrow} &
\oplus_{\sigma}{\cal H}_{\bullet} \otimes {\cal H}_{\bullet}
\end{array}
$$
where the composition of vertical arrows coincides with the
corresponding
vertical arrow in the previous diagram. (Abusing notations we
denoted the comultiplication in the two different Hopf
algebras by the same letter
$\Delta$).
This, together with
injectivity of vertical arrows in the bottom square
of the second diagram implies the
commutativity of the upper square of that diagram. Theorem
(\ref{Theorem 2.4}) follows from the commutativity of the upper square
in the second diagram. Indeed,
the kernel of the middle horisontal arrow coincides with
$K_{2n-1}(F)\otimes \Bbb Q$ and the Beilinson regulator comes
from the bottom square of that diagram.
{\bf Acknoledgements.} This work was essentially done during my stay in
MPI(Bonn)
MSRI(Berkeley) in 1992 and supported by NSF Grant DMS-9022140.
I am
grateful to
these institutions for hospitality and support. I was also
supported by the NSF Grant DMS-9500010 in 1995.
I am indebted to M.
Kontsevich, and D.B.Fuchs
for useful discussions and to L.Positselsky and V.Voevodsky for
helpful suggestions regarding
the last section. I am very gratefull to
A.Borel who read a preliminary version of this paper and pointed out many
misprints and some errors.
\vskip 3mm \noindent
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|
1996-01-18T06:20:14 | 9601 | alg-geom/9601015 | en | https://arxiv.org/abs/alg-geom/9601015 | [
"alg-geom",
"math.AG"
] | alg-geom/9601015 | Daniel Huybrechts | Daniel Huybrechts | Birational symplectic manifolds and their deformations | LaTex | null | null | null | null | The known counterexamples to the global Torelli theorem for
higher-dimensional hyperkahler manifolds are provided by birational manifolds.
We address the question whether two birational hyperkahler manifolds (i.e.
irreducible symplectic) manifolds always define non-separated points in the
moduli space of marked manifolds. An affirmative answer is given for the cases
of Mukai's elementary transformations and birational correspondences which are
isomorphic in codimension two. The techniques are applied to show that the
moduli spaces of rank two sheaves on a K3 surface are deformation equivalent to
appropriate Hilbert schemes.
| [
{
"version": "v1",
"created": "Wed, 17 Jan 1996 08:58:30 GMT"
},
{
"version": "v2",
"created": "Wed, 17 Jan 1996 09:39:36 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Huybrechts",
"Daniel",
""
]
] | alg-geom | \section{Introduction}
Compact complex manifolds $X^{2n}$ with holonomy group $Sp(n)$ can
algebraically be characterized as simply connected compact K\"ahler
manifolds with a
unique (up to scalars) holomorphic symplectic two-form (\cite{B}).
These manifolds,
which are higher-dimensional analogues of K3 surfaces, are called irreducible
symplectic.
Beauville was able to generalize the local Torelli theorem, one of the
fundamental
results in the theory of K3 surfaces, to all irreducible symplectic
manifolds. His
results show that there exists a (coarse) moduli space ${\cal M}$ of
marked
irreducible symplectic manifolds and that the period map
$$P:{\cal M}\to{\Bbb P}(\Gamma\otimes\hbox{\sym \char '103})$$
is \'etale over $Q\subset{\Bbb P}(\Gamma\otimes\hbox{\sym \char '103})$ -- an open subset of a
quadric defined
by $q(x)=0$ and $q(x+\bar x)>0$.
By definition, a marking is an isomorphism of lattices
$\sigma:H^2(X,\hbox{\sym \char '132})\cong\Gamma$,
where $H^2(X,\hbox{\sym \char '132})$ is endowed with the quadratic form defined in \cite{B}
and
$\Gamma$ is a fixed lattice.
For K3 surfaces the moduli space ${\cal M}$ consists of two connected
components which
can be identified by $(X,\sigma)\mapsto(X,-\sigma)$. The global Torelli
theorem
for K3 surfaces asserts that the period map $P$ restricted to either of
the two
components, say ${\cal M}_0$, is surjective and `almost injective'. More
precisely,
if $(X,\sigma)$ and $(X',\sigma')$ are two points in $P_0^{-1}(x)$, then
$(X,\sigma)$, $(X',\sigma')\in{\cal M}_0$ are non-separated and the underlying
$X$ and $X'$
are isomorphic K3 surfaces containing at least one $(-2)$-curve.
Furthermore, for
$x\in Q$ in the complement of the union of countably many proper closed
subsets
the fibre $P_0^{-1}(x)$ is a single point. In short, the failure of the
injectivity
of the period map $P_0$ is due to the non-separatedness of ${\cal M}_0$ and
two non-separated points are given by one K3 surface equipped with two
different
markings related by reflections orthogonal to $(-2)$-curves.
In the higher-dimensional situation, the global Torelli theorem does not
hold, i.e.
an isomorphism of Hodge structures $H^2(X,\hbox{\sym \char '132})\cong H^2(X',\hbox{\sym \char '132})$ compatible
with the quadratic forms does not imply $X\cong X'$.
In fact, for any two birational irreducible symplectic manifolds $X$ and
$X'$ one
finds markings $\sigma$ and $\sigma'$ such that $P(X,\sigma)=P(X',\sigma')$.
Due to
an example of Debarre, birational $X$ and $X'$ need
not be isomorphic in higher dimensions.
Although, only little evidence can be provided, we cannot resist to formulate
the following (cf. \cite{Mu2}):
{\bf Speculation} {\it (Global Torelli theorem) The period map $P_0$ is
almost
injective, i.e. two points $(X,\sigma)$ and $(X'\sigma')$ in the same fibre
of $P_0$
are non-separated in ${\cal M}_0$. In particular, $X$ and $X'$ are birational.}
The birationality of $X$ and $X'$ follows from \cite{MM}.\\
As the known counterexamples to the global Torelli theorem use birational
manifolds
$X$ and $X'$, the following conjecture can be regarded as a weaker version
of this speculation:
{\bf Conjecture} {\it Two irreducible symplectic
manifolds $X$ and $X'$ are birational if and only if they correspond
to non-separated points in
the moduli space.}
This paper proves the conjecture in two fairly general cases.
{\bf Theorem \ref{rho2}} {\it If $X$ and $X'$ are projective irreducible
symplectic
manifolds which are birational and isomorphic in codimension two
(cf. \ref{birat}), then the corresponding points
in the moduli space of symplectic manifolds are non-separated.}
Dropping the assumption on the codimension and the projectivity, but
restricting to
Mukai's elementary transformation, the most explicit birational
correspondence, one can prove
{\bf Theorem \ref{Mukaith}} {\it If $X'$ is the elementary transformation
of an irreducible
symplectic manifold $X$ along a smooth ${\Bbb P}_N$-bundle of codimension $N$,
then
$X$ and $X'$ correspond to non-separated points in the moduli space.}
Both results combined will be used in Sect. 5 to deduce the conjecture
for projective $X$ and $X'$ and birational correspondences which in
codimension two are
given by elementary transformations (cf. \ref{maincodtwo}).
Unfortunately, only few examples of irreducible symplectic manifolds
are knwon. Higher-dimensional examples were first described by
Beauville and Fujiki.
Starting with a K3 surface $S$, Beauville showed that the Hilbert schemes
$Hilb^n(S)$ of zero-dimensional subschemes are irreducible symplectic.\\
As shown by Mukai \cite{Mu1}, moduli spaces of stable sheaves on a K3
surface also
admit a (holomorphic) symplectic structure. That these spaces are
irreducible
symplectic, provided they are compact, was shown in \cite{GH} for
the rank two case and
in \cite{OG} in general. The idea in both approaches is to deform the
underlying K3 surface $S$ to a special K3 surface $S_0$, such that the
moduli space
of sheaves on $S_0$ is birational to the Hilbert scheme $Hilb^n(S_0)$.
As the moduli space of sheaves on $S_0$ is a deformation of the moduli
space
of sheaves on $S$, this shows that any smooth moduli space
is deformation equivalent to a manifold which is birational
to an irreducible symplectic manifold. This is enough to conclude that
the moduli spaces of higher rank sheaves are irreducible symplectic.\\
Proving this result \cite{GH}, we observed the following phenomenon.
Let $S$ be a K3 surface and
let $H$ and $H'$ be two different generic polarizations. Then the moduli
spaces
$X:=M_H$ and $X':=M_{H'}$ of $H$-stable, respectively $H'$-stable, sheaves,
which in general are not isomorphic, can be realized as the
special fibres of the same family, i.e.
equipped with appropriate markings they correspond to non-separated points
in the moduli space ${\cal M}$. This observation motivated the study of the
general question explained above. Moreover, since the birational
correspondence
between $M_H$ and $M_{H'}$ looks quite similar to the one between moduli
space and
Hilbert scheme on the special K3 surface $S_0$, we conjectured that moduli
spaces
of higher rank sheaves are deformation equivalent to Hilbert schemes
$Hilb^n(S)$.
The general results \ref{rho2} and \ref{Mukaith} do not cover this case,
since
the birational correspondence of moduli space and Hilbert scheme is not
an isomorphism
in codimension two. But using the result of Sect. 5 one can at least
prove the rank two
case.
{\bf Theorem \ref{moddefhilb}} {\it If $S$ is a K3 surface,
${\cal Q}\in{\rm Pic}(S)$ indivisible,
$2n:=4c_2-c_1^2({\cal Q})-6\geq10$ and $H$ a generic polarization,
then the moduli space $M_H({\cal Q},c_2)$ of $H$-stable rank two sheaves
$E$ with $\det(E)\cong{\cal Q}$ and $c_2(E)=c_2$ is deformation equivalent
to $Hilb^n(S)$.}\\
The assumption $2n\geq10$ is a technical condition, whereas the assumption on
the determinant and the polarization is needed to guarantee the smoothness of
the moduli space.
We believe that the same result can be proved for the rank $>2$ moduli
spaces,
as well. As there is evidence that our conjecture holds in general
and that
the higher rank case is an immediate consequence of it, we developed the
necessary
modification only in the rank two case.\\
Due to this result it seems that all known examples of irreducible
symplectic manifolds are either deformation
equivalent to some Hilbert scheme $Hilb^n(S)$, where $S$ is a K3 surface,
or to
a generalized Kummer variety $K^n(A)$, where $A$ is a two-dimensional torus.
{\bf Acknowldegements:} I had valuable and stimulating
discussions with many people.
Especially, I wish to thank A. Beauville, F. Bogomolov, P. Deligne,
B. Fantechi, R. Friedman, S. Keel, and E. Viehweg. I also wish to
thank M. Lehn who has read a first version of the paper.
Part of this work was done during the academic year 1994/95
while I was visiting the Institute for Advanced Study
(Princeton). I was financially supported by a grant from the DFG.
Support and hospitality of the Max-Planck-Institut f\"ur Mathematik (Bonn)
are also gratefully acknowledged.
\section{Preparations}
\refstepcounter{theorem}\label{sympl}{\bf \thetheorem. Symplectic manifolds.}
A complex manifold $X$ is called {\it symplectic} (in this paper!) if
there exists
a holomorphic two-form $\omega\in H^0(X,\Omega^2_X)$
which is non-degenerate at every point. Note that the existence of $\omega$
implies that the canonical bundle $K_X$ is trivial.
If $X$ is compact, then the symplectic structure is unique if and only if
$h^0(X,\Omega^2_X)=1$.
A simply connected compact K\"ahler manifold with a unqiue symplectic
structure
is called {\it irreducible symplectic}. By \cite{B} $X^{2n}$ is irreduible
symplectic
if and only if its holonomy is $Sp(n)$, i.e. it is irreducible
hyperk\"ahler.\\
For a compact irreducible symplectic K\"ahler
manifold Beauville introduced a
quadratic form on $H^2(X,\hbox{\sym \char '103})$ by
$$\alpha\mapsto\frac{n}{2}\int(\omega\bar\omega)^{n-1}\alpha^2+(1-n)
\int\omega^{n-1}\bar\omega^n\alpha\cdot\int\omega^n\bar\omega^{n-1}\alpha$$
where $\omega\in H^0(X,\Omega^2_X)=H^{2,0}$ is the symplectic
form. Using Hodge decomposition $\alpha=a\omega+\varphi+b\bar\omega$ with
$\varphi\in H^{1,1}(X)$ and assuming $\int(\omega\bar\omega)^{n}=1$
this form
can be written as
$\alpha\mapsto ab+(n/2)\int(\omega\bar\omega)^{n-1}\varphi$.
It turns out that this form is non-degenerate of index $(3,b_2-3)$.
Moreover, a positive multiple
of it is integral (cf. \cite{B}, \cite{Fuji}). The unique positive
multiple making it to a primitive integral form
is called the canonical form $q$ on $H^2(X,\hbox{\sym \char '103})$.
Using the weight-two Hodge structure endowed with this quadratic form
Beauville's
local Torelli theorem says that
${\cal X}_t\mapsto [H^{2,0}({\cal X}_t)]\in {\Bbb P}(H^2(X,\hbox{\sym \char '103}))$
induces a local isomorphism of the Kuranishi space $Def(X)$ with the
quadric in ${\Bbb P}(H^2(X,\hbox{\sym \char '103}))$ defined by
$q(\alpha)=0$.
\refstepcounter{theorem}\label{birat}{\bf \thetheorem. Birational
symplectic manifolds.}
Let $f:X\to X'$ be a birational map between two compact symplectic
manifolds and assume that the symplectic structure on $X$ is unique.
Then the largest open subset $U\subset X$ where $f$ is regular
satisfies ${\rm codim} (X\setminus U)\geq2$.
Moreover, one shows
$f|_U$ is an embedding:
Since $\omega_X$ is unique and $\hbox{\sym \char '103}=H^0(X,\Omega_X^2)=H^0(U,\Omega_U^2)$,
the pull-back
$f^*\omega_{X'}$ is a non-trivial multiple of $\omega_X$. Thus $f$ is
quasi-finite on $U$.
Since it is generically one-to-one, it is an embedding.
Note that, as a consequence, the symplectic structure on $X'$ is unique, too.
Thus, if $U\subset X$ and $U'\subset X'$ denote the maximal open subsets
where
$f$ and $f^{-1}$, respectively, are regular, then $U\cong U'$ and
${\rm codim} (X\setminus U)$,
${\rm codim} (X'\setminus U')\geq2$. A birational correspondence is by
definition an
{\it isomorphism in codimension two} if and only if ${\rm codim} (X\setminus U)$,
${\rm codim} (X'\setminus U')\geq3$. Recall, that a birational map
between two K3 surfaces
can always be extended to an isomorphism.\\
If $X$ is a projective manifold and $U\subset X$ is an open subset
with ${\rm codim} (X\setminus U)\geq2$, then the restriction defines an
isomorphism ${\rm Pic}(X)\cong{\rm Pic}(U)$.
In particular, for two birational projective manifolds
$X$ and $X'$ with unique symplectic structures one has
${\rm Pic}(X)\cong{\rm Pic}(U)\cong{\rm Pic}(U')\cong{\rm Pic}(X')$. The corresponding
line bundles on $X$ and $X'$ will usually be denoted by
$L$ and $L'$, or $M$ and $M'$. In particular, the Picard numbers
$\rho(X)$ and $\rho(X')$ are equal. Using the exponential sequence
one gets the same result for non-projective
$X$ and $X'$.\\
Frequently, we will use the following result due to Scheja [S].
If $E$ is a locally
free sheaf on $X$ and $U\subset X$ is an open subset, then
the restriction map $H^i(X,E)\to H^i(U,E|_U)$ is injective for
$i\leq{\rm codim} (X\setminus U)-1$ and bijective for
$i\leq{\rm codim} (X\setminus U)-2$.
In particular, this can be applied to the line bundles $L$ and $L'$. Thus,
$H^0(X,L)=H^0(U,L|_U)=H^0(U',L'|_{U'})=H^0(X',L')$ and if
${\rm codim}(X'\setminus U')\geq3$
we get $H^1(X,L)\subset H^1(X',L')$.\\
If $X$ and $X'$ are birational irreducible symplectic manifolds,
then there exists an isomorphism between their weight-two Hodge
structures compatible
with the canonical forms $q_X$ and $q_{X'}$ (\cite{Mu2}, \cite{OG}).
\refstepcounter{theorem}\label{defo}{\bf\thetheorem. Deformations.}
Any compact K\"ahler manifold $X$ with trivial canonical bundle $K_X$ has
unobstructed deformations, i.e. the base space of the Kuranishi family
$Def(X)$ is smooth. This is originally due to
Bogomolov, Tian and Todorov (\cite{Bo2}, \cite{Ti}, \cite{To}).
For an algebraic proof see \cite{R} and \cite{Ka}.\\
If $L$ is a line bundle on $X$, such that the cup-product
$c_1(L):H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ is surjective, then the deformations
of the pair $(X,L)$ are unobstructed as well. This follows from the fact
that the
infinitesimal deformations of $(X,L)$ are parametrized by $H^1(X,{\cal D}(L))$
and
the obstructions are contained in $H^2(X,{\cal D}(L))$. Here
${\cal D}(L)$ is the sheaf of differential operators of order $\leq1$ on $L$.
The symbol map
induces an exact sequence
$$\ses{{\cal O}_X}{{\cal D}(L)}{{\cal T}_X}$$
whose boundary map $H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ is the cup-product with
$c_1(L)$. In particular, $H^2(X,{\cal D}(L))\to H^2(X,{\cal T}_X)$ is injective.
Since $X$ is unobstructed, all obstructions of $(X,L)$ vanish.\\
All this can be applied to irreducible symplectic manifolds.
Using $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$ one finds that $Def(X)$ is
smooth of positive
dimension. Any small deformation of $X$ is again K\"ahler (cf. \cite{KS})
and irreducible symplectic.
In fact, any K\"ahler deformation of $X$ is irreducible symplectic \cite{B}.
Under
the isomorphism $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$ the kernel
of $c_1(L):H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)=\hbox{\sym \char '103}$ is identified with
the kernel of $q(c_1(L),~~):H^1(X,\Omega_X)\to\hbox{\sym \char '103}$ (cf. \cite{B}).
In particular, if
$L$ is non-trivial, then $c_1(L):H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ is
surjective and thus $Def(X,L)$ is a smooth hypersurface of $Def(X)$.
For the tangent space of $Def(X,L)$ we have
$T_0Def(X,L)\cong H^1(X,{\cal D}(L))\cong\ker(H^1(X,{\cal T}_X)
\rpfeil{5}{c_1(L)}H^2(X,{\cal O}_X))\cong
\ker(H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)\rpfeil{5}{q(c_1(L),~)}\hbox{\sym \char '103})$.
If $c_1(L)$ and $c_1(M)$ are linearly independent, then the
deformation spaces $Def(X,L)$ and $Def(X,M)$ intersect transversely.
\refstepcounter{theorem}\label{ms}{\bf\thetheorem. Moduli spaces.}
Due to
Beauville's local Torelli theorem one can easily construct a moduli
space ${\cal M}$
of marked irreducible symplectic manifolds. Here
a marking consists of an isomorphism of $H^2(X,\hbox{\sym \char '132})$ with
a fixed lattice compatible with the quadratic form $q$.
As for K3 surfaces the space of marked irreducible symplectic K\"ahler
manifolds is
smooth but non-separated. In contrast to the K3 surface case, the
moduli space ${\cal M}$
is in general not fine. This is due to the fact that higher-dimensional
irreducible symplectic manifolds permit automorphisms inducing the
identity on
$H^2(X,\hbox{\sym \char '132})$ (cf. \cite{B2}).\\
The quotient of ${\cal M}$ by the orthogonal group of $(H^2,q)$ is the moduli
space of unmarked manifolds, but this space
is not expected to have any reasonable analytic
structure. The theme of this paper is to prove statements like:
$X$ and $X'$ correspond to non-separated points in the moduli space.
Here, we usually refer to the moduli space of marked manifolds, though
this distinction does not really matter for our purposes. Explicitly,
this means that there are two one-dimensional deformations ${\cal X}\to S$ and
${\cal X}'\to S$ ($S$ is smooth),
which are isomorphic over $S\setminus\{0\}$ and the special fibres are
${\cal X}_0\cong X$ and ${\cal X}'_0\cong X'$.
\section{Elementary transformations}\label{Mukai}
An explicit birational correspondence between two symplectic manifolds
was introduced by Mukai \cite{Mu1}.
We briefly want to recall the construction.\\
Let $X$ be a complex manifold of dimension $2n$ which admits a holomorphic
everywhere non-degenerate two-form $\omega\in H^0(X,\Omega^2_X)$.
Furthermore,
let $P\subset X$ be a closed submanifold which itself is a projective
bundle $P={\Bbb P}(F)\rpfeil{5}{\phi}Y$. Here, $F$ is a rank-$(N+1)$ vector
bundle
on the manifold $Y$. Using the symplectic structure one can define the
elementary
transformation $X'$ of $X$ along $P$ as follows.\\
Since a projective space ${\Bbb P}_N$ does not admit any regular
two-form, the restriction of $\omega$ to any fibre of $\phi$ is trivial.
More is true, the relative tangent bundle ${\cal T}_\phi$ of $\phi$ is
orthogonal to
${\cal T}_P$ with respect to the restriction of $\omega$, i.e.
$\omega|_P:{\cal T}_\phi\times{\cal T}_P\to{\cal O}_P$
vanishes. Indeed,
this follows from the isomorphism $H^0(Y,\Omega_Y^2)\cong H^0(P,\Omega_P^2)$,
i.e. $\omega|_P$ is the pull-back of a two-form on $Y$. Thus the composition
of ${\cal T}_P\subset{\cal T}_X|_P$ with the isomorphism
${\cal T}_X|_P\cong\Omega_X|_P$ and the projection
$\Omega_X|_P\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow\Omega_P\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow\Omega_\phi$
vanishes. Hence $\omega$ induces a vector bundle homomorphism
${\cal N}_{P/X}\cong{\cal T}_X|_P/{\cal T}_P\to\Omega_\phi$.\\
Now let ${\rm codim}P=N$. Then both vector bundles ${\cal N}_{P/X}$ and
$\Omega_\phi$ are of
rank $N$ and,
since $\omega$ is non-degenerate, the homomorphism ${\cal N}_{P/X}\to\Omega_\phi$
is an isomorphism.\\
Let $\tilde X\to X$ denote the blow-up of $X$ in $P\subset X$ and let
$D\subset\tilde X$
be the exceptional divisor. The projection $D\to P$ is isomorphic to the
projective bundle ${\Bbb P}({\cal N}_{P/X})\cong{\Bbb P}(\Omega_\phi)\to P$.\\
The natural isomorphism of the incidence
variety $\{(x,H)|x\in H\}\subset{\Bbb P}_N\times{\Bbb P}_N^*$ as a projective bundle
over ${\Bbb P}_N$
with the projective bundle ${\Bbb P}(\Omega_{{\Bbb P}_N})\to{\Bbb P}_N$ can be generalized
to the relative situation, i.e.
there is a canonical embedding
$D={\Bbb P}(\Omega_\phi)\subset{\Bbb P}(F)\times_Y{\Bbb P}(F^*)$
compatible with the projection to ${\Bbb P}(F)$. The other projection
$D\to {\Bbb P}(F^*)$
is a projective bundle as well.
If ${\cal O}_{\tilde X}(D)$ restricts to ${\cal O}(-1)$ on every fibre of
$D\to{\Bbb P}(F^*)$ then there
exists a blow-down $\tilde X\to X'$ to a smooth manifold $X'$
such that $D\subset \tilde
X$ is the exceptional divisor and $D\to X'$ is the projection
$D\to{\Bbb P}(F^*)\subset X'$
(cf. \cite{FN}).
Adjunction formula shows that ${\cal O}_{\tilde X}(D)$ indeed
satisfies this condition.
\begin{definition} $X'=elm_PX$ is called the elementary transformation
of the symplectic
mani\-fold $X$ along the projective bundle $P$.
\end{definition}$~$
Mukai also shows that an elementary transformation $elm_PX$ of a
symplectic manifold
$X$ is again symplectic.\\
{\bf Example}\refstepcounter{theorem}\label{ex} {\bf\thetheorem} In
the case of a K3 surface $S$, which
is a two-dimensional symplectic manifold, and a $(-2)$-curve
$P={\Bbb P}_1\subset S$ one
obviously has $elm_PS\cong S$. The Hilbert scheme $X:=Hilb^n(S)$, which
is irreducible symplectic, then contains the projective space
${\Bbb P}_n\cong S^n(P)= Hilb^n(P)$. The elementary transformation of
$Hilb^n(S)$ along
this projective space is in general not isomorphic to $Hilb^n(S)$.
This is due
to an example of Debarre \cite{Deb}.
Though in his example the K3 surface $S$, and hence $X=Hilb^n(S)$,
is only K\"ahler,
it is expected that one can also find examples
$X\not\cong elm_PX$, where $X$ is projective. Also note that there are
examples where an elementary
transformation of $Hilb^n(S)$ is isomorphic to $Hilb^n(S)$ (\cite{B2}).
The following question was raised in \cite{Mu2}.\\
{\bf Question}\refstepcounter{theorem}\label{Qversion1} {\bf\thetheorem}
{\it Are the symplectic manifolds
$X$ and $X'=elm_PX$ deformation equivalent?}
We want to give an affirmative answer to this question in the case of
compact K\"ahler
manifolds.
\begin{theorem}\label{Mukaith} Let $X$ be a compact symplectic
K\"ahler manifold and let $P\subset X$ be a smooth
${\Bbb P}_N$-bundle of codimension $N$. Then there exist two smooth proper
families
${\cal X}\to S$ and ${\cal X}'\to S$ over a smooth and one-dimensional base $S$,
such that
${\cal X}$ and ${\cal X}'$ are isomorphic as families over $S\setminus \{0\}$
and the
fibres over $0\in S$ satisfy ${\cal X}_0\cong X$
and ${\cal X}'_0\cong X'\cong elm_PX$.
\end{theorem}
Note that the theorem is in fact stronger than what the original
question suggests.
The theorem shows that $X$ and $X'$ correspond to non-separated points
in the moduli space
of symplectic manifolds.
In particular, one has
\begin{corollary} The higher-weight Hodge structures of $X$ and $elm_PX$
are isomorphic.\hspace*{\fill}\hbox{$\Box$}
\end{corollary}
The following lemma is needed for the proof of the theorem.
Consider a deformation ${\cal X}\to S$ of $X$ and assume that $S$ is smooth and
one-dimensional. Let $v\in H^1(X,{\cal T}_X)$ be its Kodaira-Spencer class, i.e.
$\hbox{\sym \char '103}\cdot v$ is the image of the Kodaira-Spencer map
$T_0S\to H^1(X,{\cal T}_X)$. Furthermore, denote by $\bar v\in H^1(X,\Omega_X)$
the image of $v$ under the isomorphism $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$
induced
by the symplectic structure.
\begin{lemma}\label{normaltriv} Assume that $\bar v\in H^1(X,\Omega_X)$ is
a K\"ahler class. Then
the normal bundle ${\cal N}_{P/{\cal X}}$ is isomorphic
to $\phi^*F^*\otimes{\cal O}_\phi(-1)$.
\end{lemma}
{\it Proof:}
We certainly can assume that $Y$ is connected and hence
$H^1(P,{\cal N}_{P/X})\cong
H^1(P,\Omega_\phi)\cong H^0(Y,{\cal O}_Y)\cong\hbox{\sym \char '103}$.\\
By construction, the isomorphism ${\cal N}_{P/X}\cong \Omega_\phi$
commutes with the
projections ${\cal T}_X\to{\cal N}_{P/X}$, $\Omega_X\to\Omega_\phi$ and the
symplectic structure
${\cal T}_X\cong\Omega_X$. In particular, the image $\xi$ of $v$ under
$H^1(X,{\cal T}_X)\to H^1(P,{\cal N}_{P/X})$ is non-zero if and only if
$\bar v$ maps to a non-zero class under
$H^1(X,\Omega_X)\to H^1(P,\Omega_\phi)$. Since $\bar v$ is K\"ahler and
thus its
restriction to the fibres of $\phi$ non-trivial, one concludes that $\xi$
is the
extension class of the unique (up to scalars) non-trivial extension
of ${\cal O}_P$ by ${\cal N}_{P/X}\cong\Omega_\phi$. Thus it is isomorphic
to the
relative Euler sequence
$$\sesq{\Omega_\phi}{}{\phi^*F^*\otimes{\cal O}_\phi(-1)}{}{{\cal O}_P}.$$
Therefore, it suffices to show that $\xi$ is also the extension class
of the canonical sequence
$$\sesq{{\cal N}_{P/X}}{}{{\cal N}_{P/{\cal X}}}{}{{\cal N}_{X/{\cal X}}|_P},$$
where we use ${\cal N}_{X/{\cal X}}\cong{\cal O}_X$. This follows easily from
the definition of the Kodaira-Spencer class
$v$ as the extension class of
$$\sesq{{\cal T}_X}{}{{\cal T}_{\cal X}|_X}{}{{\cal N}_{X/{\cal X}}}.$$
\hspace*{\fill}\hbox{$\Box$}
{\bf Proof of \ref{Mukaith}:} By \ref{defo}
a one-dimensional deformation ${\cal X}\to S$ of
$X$ such that $\bar v$ is K\"ahler always exists.
Denote the blow-up of ${\cal X}$ in $P$ by
$\tilde {\cal X}\to{\cal X}$. By lemma \ref{normaltriv} the exceptional divisor
${\cal D}\to P$
is isomorphic to the projective bundle ${\Bbb P}(\phi^*F^*)\to P$. Obviously,
${\Bbb P}(\phi^*F^*)\cong{\Bbb P}(F)\times_Y{\Bbb P}(F^*)$. Now consider the second
projection
${\cal D}\cong{\Bbb P}(\phi^* F^*)\to{\Bbb P}(F^*)$. As before one checks that
${\cal O}_{\cal X}({\cal D})$ restricts to ${\cal O}(-1)$ on every fibre of this
projection, i.e. the condition of the Nakano-Fujiki criterion is satisfied.
Thus $\tilde{\cal X}$ can be blown-down to a smooth manifold ${\cal X}'$ such that
the exceptional divisor ${\cal D}$ is contracted to ${\Bbb P}(F^ *)$.
By the very construction ${\cal X}'\leftarrow\tilde {\cal X}\to{\cal X}$ is
compatible with $X'\leftarrow\tilde X\to X$, i.e. ${\cal X}'\to S$ is a smooth
proper family, isomorphic to ${\cal X}$ over $S\setminus\{0\}$, and its special
fibre
${\cal X}'_0$ is isomorphic to $X'$.\hspace*{\fill}\hbox{$\Box$}
\bigskip\\
Note that the two families ${\cal X}$ and ${\cal X}'$ are not isomorphic. In
particular,
one gets the well-known
\begin{corollary} If $X$ is a K3 surface with a $(-2)$-curve $P\subset X$,
then there
exist non-isomorphic families ${\cal X}, {\cal X}'\to S$ which are isomorphic over
$S\setminus\{0\}$
and ${\cal X}_0\cong{\cal X}'_0\cong X$.\hspace*{\fill}\hbox{$\Box$}
\end{corollary}
\section{Non-separated points in the moduli space}\label{general}
In this section we discuss other situations where birational symplectic
manifolds
present non-separated points in their moduli space.\\
Elementary transformations, dealt with previously, define very explicit
birational
correspondences between symplectic manifolds. But birational correspondences
encountered in the examples are usually more complicated.
This section is devoted to general birational
correspondences. The result is analogous to
\ref{Mukaith}, though we restrict to projective manifolds and
birational correspondences which are isomorphisms in codimension two.
Later (cf. Sect. \ref{codtwo}) the result will be generalized to the case
where in codimension two the birational correspondence is given by an
elementary
transformation.
Let us fix the following notations: $X$ and $X'$ denote irreducible
symplectic
manifolds which are isomorphic on the open sets $U\subset X$ and
$U'\subset X'$ (cf. \ref{birat}).
If $v$ is a class in $H^1(X,{\cal T}_X)$, then the symplectic structure
${\cal T}_X\cong\Omega_X$
induces a class $\bar v\in H^1(X,\Omega_X)$. The following proposition
does not make any assumptions
either on the projectivity of $X$ or on the codimension of $X\setminus U$.
It is not needed for the proof of the main theorem, but shows how
and to what extent the idea of Sect. 3 works in the general context.
\begin{proposition}\label{ampleKS} Let $S$ be smooth and one-dimensional
and let
${\cal X}\to S$ and ${\cal X}'\to S$ be deformations of ${\cal X}_0=X$ and ${\cal X}'_0=X'$,
respectively.
If ${\cal X}$ and ${\cal X}'$ are $S$-birational and the Kodaira-Spencer class $v$
of ${\cal X}\to S$
induces a class $\bar v\in H^1(X,\Omega_X)$ which is non-trivial on all
rational
curves in $X\setminus U$, then
${\cal X}|_{S\setminus\{0\}}\cong_S{\cal X}'|_{S\setminus\{0\}}$
(possibly after shrinking $S$ to an open neighbourhood of $0$).
\end{proposition}
\noindent{\bf Remarks }\refstepcounter{theorem}\label{RMtoampleKS}
{\bf\thetheorem}
{\it i)} $\bar v$ non-trivial on a rational curve means that the pull-back
of $\bar v\in H^2(X,\hbox{\sym \char '103})$ evaluated on the fundamental class
of such a curve is non-trivial.\\
{\it ii)} The condition on $v$ is satisfied if $\bar v$ is contained in
the cone spanned
(over ${\Bbb R}$) by classes which are ample on $X\setminus U$, e.g. if
$\bar v$ is ample.
Note that the rational curves could be singular and reducible.\\
{\it iii)} Whenever $X$ is projective there are deformations with
Kodaira-Spencer class
$v$ such that $\bar v$ is ample. The problem is to construct ${\cal X}'\to S$
simultaneously.
If the codimensions of $X\setminus U$ and $X'\setminus U'$ are at least
three, then the
isomorphisms
$H^1(X,{\cal T}_X)\cong H^1(U,{\cal T}_U)\cong H^1(U',{\cal T}_{U'})\cong
H^1(X',{\cal T}_{X'})$ suggest that
deformations of $X$ can be related to deformations of $X'$ via
the big open subsets
$U$ and $U'$. I don't know how to make this rigorous. In particular,
it is not clear
to me what deformations of $U$ should really mean.\\
{\it iv)} In the proof of \ref{Mukaith} the family ${\cal X}'\to S$ was
constructed explicitly
from ${\cal X}\to S$ as a blow-up followed by a blow-down. For the general
situation this approach seems to fail.\\
{\bf Proof of \ref{ampleKS}:} If the $S$-birational map ${\cal X}\to{\cal X}'$
does not extend
to an isomorphism ${\cal X}_t\cong{\cal X}'_t$ for generic $t$, then there exists a
surface ${\cal C}$ together with a flat morphism ${\cal C}\to S$ such that:\\
{\it i)} ${\cal C}$ is smooth and irreducible.\\
{\it ii)} For generic $t$ the fibre ${\cal C}_t$ is a disjoint union of
smooth rational
curves.\\
{\it iii)} There exists a finite $S$-morphism $\alpha:{\cal C}\to{\cal X}$ that
maps ${\cal C}_0$
to $X\setminus U$.\\
This follows from resolution of singularities:
By shrinking $S$ we can assume that there is a sequence of monoidal
transformations
${\cal Z}_n\to{\cal Z}_{n-1}\to...\to{\cal Z}_1\to{\cal X}'$ with smooth centers, which
either dominate $S$ or are contained in the fibre over $0\in S$, and
such that
there exists a morphism ${\cal Z}_n\to {\cal X}$ which resolves the birational
map
${\cal X}\to {\cal X}'$.
If ${\cal X}_t\to{\cal X}'_t$ does not extend to an isomorphism for generic $t$,
then at least one monoidal transformation ${\cal Z}_i\to{\cal Z}_{i-1}$ with
smooth center
$T_i$ dominating $S$ occurs. Let $i$ be
maximal with this property. Next one finds a morphism
$S'\to T_i$ from a smooth, irreducible curve $S'$ such that the composition
$S'\to T_i\to S$ is finite and smooth over $S\setminus\{0\}$.
Then ${\cal Z}_i\times_{{\cal Z}_{i-1}}S'\to S'$ is a projective bundle.
Since $i$ is maximal, we have
$({\cal Z}_i\times_{{\cal Z}_{i-1}}S')\times_S S\setminus\{0\}\subset{\cal Z}_n\times_SS'$,
Now pick a ${\Bbb P}_1$-bundle contained in
${\cal Z}_i\times_{{\cal Z}_{i-1}}S'\to S'$ such that its restriction
to $S'\times_SS\setminus\{0\}$ maps generically finite
to ${\cal X}$ under ${\cal Z}_n\to{\cal X}$.
The resolution of the closure of it in ${\cal Z}_n$ gives the surface ${\cal C}$.
Now we want to show how one can use the existence of ${\cal C}$ to derive a
contradiction.
First, we claim that the composition
$$T_tS\to H^1({\cal X}_t,{\cal T}_{{\cal X}_t})\cong
H^1({\cal X}_t,\Omega_{{\cal X}_t})\rpfeil{5}{\alpha_t^*}H^1({\cal C}_t,\Omega_{{\cal C}_t})$$
vanishes for generic $t$ (Here, the first map is the Kodaira-Spencer map and
the isomorphism is induced by the symplectic structure on ${\cal X}_t$).
This is a generalization of an argument explained in the proof of
\ref{normaltriv}.
One can either use deformation theory to show that the
existence of ${\cal C}\to S$ implies the vanishing of the obstruction to deform
${\cal C}_{t\not=0}\to{\cal X}_{t\ne0}$, which in turn gives the desired vanishing,
or one makes this
explicit by the following argument:
Note that we can assume ${\cal C}_{t\not=0}\cong{\Bbb P}_1$. Then, let
${\cal N}_t$ be the generalized normal sheaf of $\alpha_t$, i.e. the cokernel
of the injection ${\cal T}_{{\cal C}_t}\to\alpha_t^*{\cal T}_{{\cal X}_t}$.
Since for $t\ne0$ we know
${\cal T}_{{\cal C}_t}\cong{\cal T}_{{\Bbb P}_1}$ and ${\rm Hom }({\cal T}_{{\Bbb P}_1},\Omega_{{\Bbb P}_1})=0$,
the pull-back of the symplectic structure on ${\cal X}_t$
to ${\cal C}_t$ induces for $t\ne0$ a commutative diagram
$$\begin{array}{ccc}
\alpha_t^*{\cal T}_{{\cal X}_t}&\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow&{\cal N}_t\\
\downarrow&&\downarrow\\
\alpha_t^*\Omega_{{\cal X}_t}&\to&\Omega_{{\cal C}_t}\\
\end{array}$$
Thus, in order to show that $T_tS\to H^1({\cal C}_t,\Omega_{{\cal C}_t})$ is
trivial, it
is enough to prove that
$T_tS\to H^1({\cal X}_t,{\cal T}_{{\cal X}_t})\to H^1({\Bbb P}_1,\alpha_t^*{\cal T}_{{\cal X}_t})\to
H^1({\Bbb P}_1,{\cal N}_t)$
vanishes. The image of this map is spanned by the extension class of
$$\ses{{\cal N}_t}{\alpha_t^*({\cal T}_{{\cal X}}|_{{\cal X}_t})/{\cal T}_{{\cal C}_t}}
{\alpha_t^*({\cal T}_{\cal X}|_{{\cal X}_t}/{\cal T}_{{\cal X}_t})}$$
(cf. proof of \ref{normaltriv}). Since ${\cal N}_{{\cal C}_t/{\cal C}}$ together
with the natural inclusion
${\cal N}_{{\cal C}_t/{\cal C}}\subset\alpha_t^*({\cal T}_{\cal X}|_{{\cal X}_t})/{\cal T}_{{\cal C}_t}$
induced by ${\cal T}_{\cal C}\to\alpha^*{\cal T}_{\cal X}$ splits this sequence, we conclude
that $T_tS\to H^1({\Bbb P}_1,{\cal N}_t)$ is trivial.
Hence $T_tS\to H^1({\cal C}_t,\Omega_{{\cal C}_t})$ is trivial as well.\\
The Kodaira-Spencer map ${\cal T}_S\to R^1\pi_*{\cal T}_{{\cal X}/S}$ composed
with the isomorphism $R^1\pi_*{\cal T}_{{\cal X}/S}\cong R^1\pi_*\Omega_{{\cal X}/S}$
provides a global section of $R^1\pi_*\Omega_{{\cal X}/S}\otimes \Omega_S$.
Trivializing ${\cal T}_S$ we can think of it as an element
in $H^0(S,R^1\pi_*\Omega_{{\cal X}/S})$ or, using Hodge decomposition,
as a $C^\infty$-section of $R^2\pi_*\hbox{\sym \char '103}_{\cal X}\otimes{\cal O}_S$.
Moreover, making $S$ small enough we have $R^2\pi_*\hbox{\sym \char '103}_{\cal X}\cong
H^2({\cal X},\hbox{\sym \char '103})$.
Thus ${\cal T}_S\to R^1\pi_*{\cal T}_{{\cal X}/S}$ induces a $C^\infty$-map
$t\mapsto \bar v_t\in H^2({\cal X},\hbox{\sym \char '103})$. Restricting it to ${\cal C}$ we
get
$\bar w_t\in H^2({\cal C},\hbox{\sym \char '103})$. The vanishing we just proved implies
$\langle\bar w_t,[{\cal C}_t]\rangle=0$ for $t\ne0$.
Since also $t\mapsto[{\cal C}_t]\in H^2({\cal C},\hbox{\sym \char '103})$ is continous, we can
conclude
$\langle\bar w_0,[{\cal C}_0]\rangle=0$. This contradicts the assumption on
$\bar v|_{X\setminus U}$, since ${\cal C}_0$ as a
degeneration of rational curves is still rational, though singular,
reducible or
even non-reduced \cite{Sa}.\hspace*{\fill}\hbox{$\Box$}
If we in addition assume that $X'$ is projective, then birational
deformations of $X$
and $X'$ can be produced using the following proposition.
\begin{proposition}\label{firstlemma} Suppose $L'\in Pic(X')$ is very ample
and the corresponding line bundle $L\in{\rm Pic}(X)$ satisfies $H^1(X,L^n)=0$ for
$n>0$. Let ${\cal X}\to S$ be a deformation of $X={\cal X}_0$ over a smooth and
one-dimensional base $S$ and assume that there exists a line bundle ${\cal L}$ on
${\cal X}$ such that
${\cal L}_0:={\cal L}|_{{\cal X}_0}\cong L$. Then, replacing $S$ by an open neighbourhood
of $0\in S$
if necessary, there
exists a deformation ${\cal X}'\to S$ of ${\cal X}'_0=X'$ which is $S$-birational
to ${\cal X}$.
\end{proposition}
{\it Proof:} First, shrink $S$ to the open subset of points $t\in S$
such
that $H^1({\cal X}_t,{\cal L}_t)=0$.
Since $H^1(X,L)=0$, this is an open neighbourhood of $t=0$.
By base change theorem (cf. \cite{Ha}, III. 12.11) $h^0({\cal X}_t,{\cal L}_t)$
is constant
on $S$, since it can only jump at a point $t$ if $H^1({\cal X}_t,{\cal L}_t)\not=0$.
Hence $\pi_*{\cal L}$ is locally free on $S$ with fibre
$(\pi_*{\cal L})(t)=H^0({\cal X}_t,{\cal L}_t)$.\\
By the very ampleness of $L'$ the base locus $Bs(L)$ of $L$ is contained
in $X\setminus U$ and therefore of codimension at least $2$.
The set $\cup_{t\in S}Bs({\cal L}_t)$ is a closed subset of ${\cal X}$ and
hence ${\rm codim}_{{\cal X}_t} Bs({\cal L}_t)\geq2$ for $t$ in an open neighbourhood
of $t=0$
(semicontinuity of the fibre dimension).
Since $Bs({\cal L}_t^n)\subset Bs({\cal L}_t)$ we can assume that
${\rm codim}_{{\cal X}_t} Bs({\cal L}_t^n)\geq2$ for all $n>0$ and $t\in S$.\\
The rational maps $\phi_{|{\cal L}_t|}:{\cal X}_t - -\to{\Bbb P}(H^0({\cal X}_t,{\cal L}_t)^*)$,
defined by the complete linear system $|{\cal L}_t|$, glue to a rational $S$-map
$\phi:{\cal X} - - \to{\Bbb P}((\pi_*{\cal L})^*)$.
Then $\phi$ is regular at all points of
${\cal X}_t\setminus Bs({\cal L}_t)$ ($t\in S$).
Let ${\cal Z}$ be the scheme-theoretic closure of the graph $\Gamma_\phi$ of
$\phi$ in
${\cal X}\times_S{\Bbb P}((\pi_*{\cal L})^*)$, i.e. the closure of $\Gamma_\phi$ with
the reduced induced structure.\\
The projection $\varphi:{\cal Z}\to{\cal X}$ is isomorphic over every point of
${\cal X}_t\setminus Bs({\cal L}_t)$, $t\in S$.
Note that a fibre ${\cal Z}_t$ of ${\cal Z}$ over $t\in S$ does not necessarily
coincide with the closure
of the graph of $\phi_{|{\cal L}_t|}$.
However, since ${\cal X}$ has irreducible fibres and hence $\Gamma_\phi$,
the generic
fibre of ${\cal Z}\to S$ is irreducible as well. Thus,
shrinking to an open neighbourhood of $t=0$, we can assume
that ${\cal Z}_t$ is irreducible for $t\not=0$. In particular,
${\cal Z}_{t\not=0}$ equals the closure of the graph of $\phi_{|{\cal L}_t|}$ in
${\cal X}_t\times{\Bbb P}(H^0({\cal X}_t,{\cal L}_t)^*)$ at least set-theoretically.
Since ${\cal Z}$ is integral, i.e. irreducible and reduced, and $S$ is smooth and
one-dimensional, the dominant morphism ${\cal Z}\to S$ is flat (\cite{Ha},
III. 9.7.).\\
Now consider the other projection $\psi:{\cal Z}\to{\Bbb P}((\pi_*{\cal L})^*)$ and
denote its image by ${\cal X}'\subset{\Bbb P}((\pi_*{\cal L})^*)$. Strictly speaking,
${\cal X}'$ is the scheme-theoretic image of $\psi$ and since ${\cal Z}$ is reduced,
this
is the image with the reduced induced structure. Since ${\cal X}'$ then is
integral
and ${\cal X}'\to S$ is dominant, ${\cal X}'$ is flat over $S$.\\
Obviously, $X'$ is contained in ${\cal X}_0'$. To conclude that $X'={\cal X}_0'$ it
is enough to show that
$h^0(X',{\cal O}(n)|_{X'})\geq h^0({\cal X}_0',{\cal O}(n)|_{{\cal X}_0'})$ for $n\gg0$, where
${\cal O}(1)$ is the tautological ample line bundle on ${\Bbb P}(H^0({\cal X}_0,L)^*)$.
Since ${\cal O}(1)|_{X'}\cong L'$ and $h^0(X',L'^n)=h^0(X,L^n)$,
this is equivalent to $h^0(X,L^n)\geq h^0({\cal X}_0',{\cal O}(n)|_{{\cal X}_0'})$ for
$n\gg0$. For any $n$ there exists an open neighbourhood $S_n\subset S$ of
$0\in S$, such that $H^1({\cal X}_t,{\cal L}_t^n)=0$ for $t\in S_n$. This follows
from the vanishing
of $H^1(X,L^n)$ for all $n$. On the intersection
$\cap S_n\subset S$, which is the complement of countably many points,
all the cohomology groups $H^1({\cal X}_t,{\cal L}_t^n)$ vanish and therefore
$h^0({\cal X}_t,{\cal L}_t^n)=h^0(X,L^n)$.
Using this and the flatness of ${\cal X}'\to S$, the inequality
$h^0(X,L^n)\geq h^0({\cal X}_0',{\cal O}(n)|_{{\cal X}_0'})$ is equivalent to
$h^0({\cal X}_t,{\cal L}_t^n)\geq h^0({\cal X}_t',{\cal O}(n)|_{{\cal X}_t'})$ for $n\gg0$.
But the latter can be derived using the composition
$$H^0({\cal X}_t',{\cal O}(n)|_{{\cal X}_t'})\rpfeil{5}{\psi^*} H^0({\cal Z}_t,\psi^*{\cal O}(n))
\rpfeil{5}{i^*}
H^0({\cal X}_t\setminus Bs({\cal L}_t),{\cal L}_t^n)\cong H^0({\cal X}_t,{\cal L}_t^n).$$
Indeed, $\psi^*$ is injective since ${\cal Z}_t\to{\cal X}'_t$ is surjective, and
$i^*$ is injective, since
it is induced by the dense open embedding ${\cal X}_t\setminus
Bs({\cal L}_t)\subset{\cal Z}_t$
($t\not=0$). The last isomorphism is a consequence of
${\rm codim}({\cal X}_t\setminus Bs({\cal L}_t))\geq2$.
This shows that ${\cal X}'_0=X'$. Shrinking $S$ further we can also
assume that
${\cal X}'\to S$ is smooth (\cite{Ha},III. Ex. 10.2).\\
It remains to show the assertion on the
birationality. Let ${\cal Z}^*$ and ${\cal X}'^*$ denote the fibre products
${\cal Z}\times_S(S\setminus\{0\})$ and ${\cal X}'\times_S(S\setminus\{0\})$,
respectively.
Stein factorization decomposes $\psi:{\cal Z}^*\to{\cal X}'^*$ into a finite
morphism $f:{\cal Y}\to{\cal X}'^*$ and a morphism ${\cal Z}^*\to {\cal Y}$ with connected
fibres.
One first shows that $f:{\cal Y}\to{\cal X}'^*$ is in fact an isomorphism.
Since $f_t:{\cal Y}_t\to{\cal X}_t'$ is finite, the line bundle
$f_t^*{\cal O}(1)$ is ample. Thus $f_t^*{\cal O}(n)$ is very ample
for $n\gg0$. In order to prove that $f$ is an isomorphism, it is
therefore
enough to show that
$f_t^*:H^0({\cal X}_t',{\cal O}(n))\to H^0({\cal Y}_t,f_t^*{\cal O}(n))$ is
surjective. We argue as above: Consider
$$H^0({\cal X}_t',{\cal O}(n)|_{{\cal X}_t'})\rpfeil{5}{f_t^*} H^0({\cal Y}_t,f_t^*{\cal O}(n))
\hookrightarrow H^0({\cal Z}_t,\psi^*{\cal O}(n))\hookrightarrow H^0({\cal X}_t,{\cal L}_t^n)$$ and
use
$h^0({\cal X}'_t,{\cal O}(n)|_{{\cal X}'_t})=h^0({\cal X}_0',{\cal O}(n)|_{{\cal X}_0'})=h^0(X',L'^n)=h^0(X,L^n)=h^0({\cal X}_t,{\cal L}_t^n)$
for all $t\in\cap S_n$.
Hence $f_t^*$ is bijective for $t$ in the complement of countably many
points
and therefore ${\cal Y}\cong{\cal X}'^*$ after shrinking $S$.
Thus ${\cal Z}^*\to{\cal X}'^*$ has connected fibres. On the other hand
$\dim{\cal Z}_t=\dim{\cal X}_t=
\dim{\cal X}'_t$. Hence ${\cal Z}_t\to{\cal X}_t'$ is birational for $t\not=0$.
\hspace*{\fill}\hbox{$\Box$}
Note that the condition $H^1(X,L^n)=0$ is automatically satisfied if
${\rm codim} (X'\setminus U')\geq3$, i.e. if $X'$ and $X$ are isomorphic
in codimension two. Indeed,
$H^1(X,L^n)\subset H^1(U,L^n|_U)=H^1(U',L'^n_{U'})=H^1(X',L'^n)=0$
by Kodaira vanishing and \cite{Sch}. It is at this point that
the assumption on the codimension of $X\setminus U$ enters.\\
Also note that the existence of ${\cal L}$
implies $q(c_1(L),\bar v)=0$, where $\bar v\in H^1(X,\Omega_X)$ is
induced by the
Kodaira-Spencer class $v\in H^1(X,{\cal T}_X)$ of ${\cal X}\to S$ (cf. \ref{defo}).\\
Next, combining \ref{ampleKS} and \ref{firstlemma} we get
\begin{corollary}\label{orthog} Let $X$ and $X'$ be birational projective
irreducible symplectic manifolds isomorphic in codimension two. Assume
there exists a line bundle $L\in {\rm Pic}(X)$ and a class
$\bar v\in H^1(X,\Omega_X)$
such that:\\
-- The induced line bundle $L'\in {\rm Pic}(X')$ is ample.\\
-- The restriction of $\bar v$ to any rational curve in $X\setminus U$
is non-trivial.\\
-- $q(c_1(L),\bar v)=0$.\\
Then $X$ and $X'$ correspond to non-separated points in the moduli space.
\end{corollary}
{\it Proof:} By taking a high power of $L$ we can assume that $L'$ is very
ample.
Furthermore, $H^1(X,L^n)=H^1(X',L'^n)=0$ for $n>0$. The deformation space
$Def(X,L)$ of the pair $(X,L)$ is a smooth hypersurface of $Def(X)$.
Since $q(c_1(L),\bar v)=0$ and $T_0Def(X,L)\cong
\ker(H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)\rpfeil{5}{q(c_1(L),~)}\hbox{\sym \char '103})$
(cf. \ref{defo}), the class $v\in H^1(X,{\cal T}_X)$ is
tangent to $Def(X,L)$. Therefore, there exist a deformation
${\cal X}\to S$ over a smooth and one-dimensional base $S$
with Kodaira-Spencer class $v$ and a line bundle ${\cal L}$ on
${\cal X}$ such that ${\cal L}_0\cong L$. Then Proposition \ref{firstlemma} shows
that there
exists a deformation ${\cal X}'\to S$ of $X'$ which is $S$-birational to ${\cal X}$
and we conclude by Proposition \ref{ampleKS}.\hspace*{\fill}\hbox{$\Box$}
\noindent{\bf Remarks }\refstepcounter{theorem}\label{RMtoorthog}
{\bf\thetheorem}
{\it i)} If ${\Bbb P}_n\cong P\subset X$ is of codimension $n$, then $X$
and $X':=elm_PX$
satisfy the assumptions of the corollary provided they are projective.
Indeed, if $L'\in{\rm Pic}(X')$ is ample, then either there
exists an element $\bar v\in H^1(X,\Omega_X)$ orthogonal
to $c_1(L)$ or $X$ and $X'$ are isomorphic. The restriction $\pm\bar v|_P$
is either
ample, hence non-trivial on any rational curve in $P$, or zero.
In the latter case, change $\bar v$ and $L$ by a small rational multiple of
an ample divisor $H$ on $X$. Thus we get $\bar v_1:=\bar v+\beta c_1(H)$
and $L_1:=L+\gamma H$.
By adjusting $\beta$ and $\gamma$ we can assume $q(c_1(L_1'),\bar v_1)=0$
and $L_1'$
ample for small $\gamma$.
Obviously, $\bar v_1|_P\ne0$ and therefore $\bar v_1$ and $L_1$ satisfy
the conditions
of the corollary. Thus \ref{Mukaith} for elementary transformations along
a projective space can be seen as a corollary of \ref{orthog} if
$X$ and $X'$ are projective. Does \ref{orthog} work for general
elementary transformations?\\
{\it ii)} It is sometimes hard to check if $\bar v$ and $L$ satisfying
the conditions
of \ref{orthog} can be found.
I don't know the answer for the examples discussed in Sect. \ref{appl}.
Using \ref{firstlemma} one can in fact prove corollary \ref{orthog}
without the assumptions on $v$. The proof
relies on the fact that a compact Moishezon K\"ahler manifold is
projective.
It can be used to prove the following
\begin{lemma}\label{rho1} If $X$ and $X'$ are birational compact
irreducible symplectic
K\"ahler manifolds with $\rho(X)=\rho(X')=1$ and
$X'$ is projective, then $X\cong X'$.
\end{lemma}
{\it Proof:} $X$ is K\"ahler and Moishezon, hence projective. Thus,
if $L'$ is the ample
generator of ${\rm Pic}(X')$, then ${\rm Pic}(X)=\hbox{\sym \char '132}\cdot L$ and either $L$
or $L^*$ is ample. Since
$H^0(X,L^n)=H^0(X',L'^n)\ne0$ for $n\gg0$, one concludes that $L$
is ample and hence
$X\cong X'$.\hspace*{\fill}\hbox{$\Box$}
Note that the isomorphism can be chosen such that it extends the birational
map.
Here now is the main theorem of this paper.
\begin{theorem}\label{rho2} Let $X$ and $X'$ be projective irreducible
symplectic manifolds
which are birational and isomorphic in codimension two.
Then $X$ and $X'$ correspond to non-separated points in their moduli space.
\end{theorem}
{\it Proof:} Assume $X$ and $X'$ are not isomorphic. Then $\rho(X)\geq2$.
Let $L'$ be very ample on $X'$ and let $L$ be the associated line bundle
on $X$. Then $Def(X,L)\subset Def(X)$ is a smooth hypersurface of positive
dimension
$h^{1}(X,\Omega)-1$.
Since ${\rm Pic}(X)$ is countable and any line bundle $M\in {\rm Pic}(X)$ defines a
smooth
hypersurface $Def(X,M)$ intersecting $Def(X,L)$ transversely if
$M^n\not\cong L^m$ ($n\cdot m\ne0$) (\cite{B} and \ref{defo}), there exists
a generic smooth
and one-dimensional $S\subset Def(X,L)$ such that $S\cap Def(X,M)=\{0\}$
for all line bundles $M$ linearly independent of $L$. Let
$({\cal X},{\cal L})\to S$ be the associated deformation of $({\cal X}_0,{\cal L}_0)\cong(X,L)$.
Then $\rho({\cal X}_t)=1$ for general $t\in S$, i.e.
$t$ in the complement of countably many points. Now apply Proposition
\ref{firstlemma}.
We get a deformation ${\cal X}'\to S$ of ${\cal X}'_0\cong X'$ which is
$S$-birational to ${\cal X}$.
Moreover, the proof of \ref{firstlemma} shows that there is a line
bundle ${\cal L}'$ on ${\cal X}'$
such that ${\cal L}'_0\cong L'$. For small $t$ the fibre ${\cal X}_t$ is still
K\"ahler and
${\cal L}'_t$ is still ample on ${\cal X}'_t$.
Thus the lemma applies and shows ${\cal X}_t\cong{\cal X}'_t$ for general $t$
extending the
$S$-birational map ${\cal X}- -\to{\cal X}'$. Since the set of points $t\in S$,
where
${\cal X}_t- - \to{\cal X}'_t$ cannot be extended to an isomorphism is closed, we can
shrink $S$ such that ${\cal X}- - \to{\cal X}'$ is an isomorphism over
$S\setminus\{0\}$. \hspace*{\fill}\hbox{$\Box$}
We want to emphasize that the condition on the codimension
of $X\setminus U$ is only needed in order to apply \ref{firstlemma}.
If for the deformation ${\cal X}\to S$ considered in the proof the dimension
$h^0({\cal X}_t,{\cal L}_t^n)$ does not jump in $t=0$, then the argument goes through.
This will be discussed in length in the next section.\\
As an immediate consequence of the theorem we have the
\begin{corollary}\label{hodgenumbers} If $X$ and $X'$ are as in
theorem \ref{rho2}, then they
are diffeomorphic and their weight-$n$ Hodge structures are isomorphic for all
$n$.\hspace*{\fill}\hbox{$\Box$}
\end{corollary}
\section{The codimension two case}\label{codtwo}
As before, let $X$ and $X'$ be birational projective irreducible
symplectic manifolds. Let $L'\in{\rm Pic}(X')$ be an ample line bundle
and denote by $L\in{\rm Pic}(X)$ the corresponding line bundle on $X$.
The assumption on the codimension of $X\setminus U$ in theorem \ref{rho2}
was only needed to ensure $H^1(X,L^n)=0$ for $n>0$. If
${\rm cod}(X\setminus U)=2$,
then $H^1(X,L^n)$ is not necessarily zero. Indeed, consider an elementary
transformation of a four-dimensional manifold $X$ along a projective
plane ${\Bbb P}_2\subset X$. Then a standard calculation shows $H^1(X,L^n)\ne0$
if $n\geq2$. The vanishing $H^1(X,L^n)=0$ was only needed at one point
in the line
of arguments.
Namely, we used it in proposition
\ref{firstlemma} to conclude that $h^0({\cal X}_t,{\cal L}_t^n)\equiv const$
for a family ${\cal X}\to S$. One might hope that this holds for another
reason.
Indeed, if $X'=elm_PX$ is an elementary transformation in codimension two
and ${\cal X}\to S$ is as in \ref{Mukaith}, then $h^0({\cal X}_t,{\cal L}_t^n)\equiv const$.
To prove this use the family ${\cal X}'\to S$ constructed explicitly in the
proof of
\ref{Mukaith} and the equality
$h^0({\cal X}_t,{\cal L}_t^n)=h^0({\cal X}'_t,{{\cal L}'_t}^n)\equiv const$,
since $H^1(X',L'^n)=0$. For a general birational correspondence
the situation is more complicated, since we need
$h^0({\cal X}_t,{\cal L}_t)\equiv const $ in the first place in order to construct
${\cal X}'\to S$
(cf. \ref{firstlemma}).
First, we will show that under the above assumption ($L'$ ample)
the condition $h^0({\cal X}_t,{\cal L}^n_t)\equiv const$ holds true infinitesimally,
i.e.
for any deformation $\pi:({\cal X},{\cal L})\to S={\rm Spec} (k[\varepsilon])$
of $(X,L)$ the direct
image $\pi_*{\cal L}$ is locally free. This is not quite enough to prove
\ref{rho2}
in complete generality, but makes it highly plausible.\\
Under an additional assumption
(cf. \ref{ass}) one can in fact prove $h^0({\cal X}_t,{\cal L}^n_t)\equiv const$.
This is the second goal of the section and the result \ref{maincodtwo}
will be applied
in Sect. \ref{appl} to moduli space and Hilbert scheme on a K3 surface.
Consider $(X,L)$ as above and let $s$ be a global section of $L$. Then
there exists
a Kuranishi space $Def(X,L,s)$ of deformations of the triple $(X,L,s)$
together with the forgetful maps $Def(X,L,s)\to Def(X,L)\to Def(X)$.
The induced map between the tangent spaces of $Def(X,L,s)$ and $Def(X,L)$
is surjective for all $s$ if and only if for any deformation
$\pi:({\cal X},{\cal L})\to {\rm Spec}(k[\varepsilon])$ the direct image $\pi_*{\cal L}$
is locally free.
Therefore, in order to prove that $\pi_*{\cal L}$ is locally free we have
to describe the tangent spaces of $Def(X,L,s)$ and $Def(X,L)$ and the
homomorphism
between them. Note, if one could prove that
$Def(X,L,s)$ is smooth, the infinitesimal result
would immediately imply that $h^0({\cal X}_t,{\cal L}_t)\equiv const$.\\
We already know $T_0Def(X,L)\cong H^1(X,{\cal D}(L))$ (cf.
\ref{defo}).
\begin{proposition}\label{infconst} {\it i)} The Zariski tangent space
of $Def(X,L,s)$ is naturally isomorphic to the first hypercohomology
of the complex
$$\begin{array}{cccc}
{\cal D}(L,s):&{\cal D}(L)&\rpfeil{5}{s}&L\\
&D&\mapsto&D(s)\\
\end{array}$$
{\it ii)} The map between the Zariski tangent spaces
${\Bbb H}^1(X,{\cal D}(L,s))$ and $H^1(X,{\cal D}(L))$ is given by the
$E_1$-spectral sequence relating hypercohomology and cohomology.\\
{\it iii)} If $(X,L)$ and $(X',L')$ are as above, then
${\Bbb H}^1(X,{\cal D}(L,s))\to H^1(X,{\cal D}(L))$ is surjective.
\end{proposition}
{\it Proof:} {\it i)} and {\it ii)} are well-known (\cite{We}). For
{\it iii)}
we write down the beginning of the spectral sequence:
$$0\to \hbox{\sym \char '103}\to H^0(X,L)\to {\Bbb H}^1(X,{\cal D}(L,s))\to H^1(X,{\cal D}(L))\rpfeil{5}{s}
H^1(X,L).$$
Therefore, ${\Bbb H}^1(X,{\cal D}(L,s))\to H^1(X,{\cal D}(L))$ is surjective if and only
if $H^1(X,{\cal D}(L))\rpfeil{5}{s} H^1(X,L)$ vanishes.
Hence, we have to show that the pairing
$$\begin{array}{ccccc}
H^1(X,{\cal D}(L))&\otimes& H^0(X,L)&\to &H^1(X,L)\\
(D&,&s)&\mapsto&D(s)\\
\end{array}$$
is trivial.
Consider the injections $H^1(X,{\cal D}(L))\hookrightarrow H^1(U,{\cal D}(L_U))$ and
$H^1(X,{\cal D}(L'))\hookrightarrow H^1(U',{\cal D}(L'_{U'}))$. It suffices to
show that under the natural isomorphism
$ H^1(U,{\cal D}(L_U))\cong H^1(U',{\cal D}(L'_{U'}))$, given by
$L'_{U'}\cong L_{U}$, the two
spaces are identified. Indeed, if so then the commutative diagram
$$\begin{array}{ccccccc}
H^1(X,{\cal D}(L))&\otimes &H^0(X,L)&\to&H^1(X,L)&\hookrightarrow&H^1(U,L_U)\\
\downarrow\cong&&\downarrow\cong&&&&\downarrow\cong\\
H^1(X',{\cal D}(L'))&\otimes
&H^0(X',L')&\to&H^1(X',L')&\hookrightarrow&H^1(U',L'_{U'})\\
\end{array}$$
and the vanishing $H^1(X',L')=0$ prove the assertion.
In order to compare $H^1(X,{\cal D}(L))$ and $H^1(X',{\cal D}(L'))$ as
subspaces of
$H^1(U,{\cal D}(L_U))\cong H^1(U',{\cal D}(L'_{U'}))$ we make use of the exact sequence
$$\ses{{\cal O}_X}{{\cal D}(L)}{{\cal T}_X}.$$ Its cohomology sequence provides the
short exact
sequence
$$\ses{H^1(X,{\cal D}(L))}{H^1(X,{\cal T}_X)}{H^2(X,{\cal O}_X)}.$$
We first show that
the two subspaces $H^1(X,{\cal T}_X)\hookrightarrow H^1(U,{\cal T}_U)$ and
$H^1(X',{\cal T}_{X'})\hookrightarrow H^1(U',{\cal T}_{U'})$ are identified
under $H^1(U,{\cal T}_U)\cong H^1(U',{\cal T}_{U'})$. Using the symplectic structures
this is equivalent to the analogous statement for $\Omega_X$ and
$\Omega_{X'}$.
Let $X\leftarrow Z\rightarrow X'$ be a smooth resolution of the
birational
correspondence $U\cong U'$. Then
$H^{1,1}(X)\oplus\bigoplus_i\hbox{\sym \char '103}\cdot D_i\cong H^{1,1}(Z)
\cong H^{1,1}(X')\oplus\bigoplus_i\hbox{\sym \char '103}\cdot D_i$, where the $D_i$'s
are the exceptional
divisors. Since the $D_i$'s are trivial on $U\cong U'\subset Z$, the
induced isomorphism
$H^{1,1}(X)\cong H^{1,1}(X')$ is compatible with restriction.\\ To conclude
the proof
we have to show that under the identification of $H^1(X,{\cal T}_X)$ and
$H^1(X',{\cal T}_{X'})$ as subspaces of $H^1(U,{\cal T}_U)$ the
homomorphisms $c_1(L):H^1(X,{\cal T}_X)\to H^2(X,{\cal O}_X)$ and
$c_1(L'):H^1(X',{\cal T}_{X'})\to H^2(X',{\cal O}_{X'})$ have the same kernel.
Since $\ker(c_1(L)\cdot~)$ is identified with $\ker(q(c_1(L),~~))$
under the isomorphism $H^1(X,{\cal T}_X)\cong H^1(X,\Omega_X)$, this follows
immediately from the fact that $H^2(X,\hbox{\sym \char '103})\cong H^2(X',\hbox{\sym \char '103})$ respects
$q_X$ and $q_{X'}$
(cf. \cite{Mu2},\cite{OG}).\hspace*{\fill}\hbox{$\Box$}
The proposition gives evidence that $h^0({\cal X}_t,{\cal L}_t)\equiv const$ holds
in general.
In fact, I believe that the same technique should show the vanishing
of the higher obstructions to deform $(X,L,s)$,
but I don't know how to prove this.\\
In the examples it seems as if a birational correspondence between
irreducible symplectic
manifolds might be non-isomorphic in codimension two, but that in such
a case
the birational correspondence is in codimension two given by an elementary
transformation. Thus, it is not completely unlikely, that
the following assumption is always satisfied. For the
birational correspondence between the moduli space of rank two sheaves
and the Hilbert
scheme this is established in Sect. \ref{appl}.
{\bf Assumption} \refstepcounter{theorem}\label{ass}{\bf \thetheorem}
There exist open subsets
$U\subset V\subset X$ and $U'\subset V'\subset X'$ such that
${\rm codim}(X\setminus V),{\rm codim} (X'\setminus V')\geq3$
and $V':=elm_{V\setminus U}V$. In particular, we assume that
$P:=V\setminus U$ is a ${\Bbb P}_2$-bundle
${\Bbb P}(F)\to Y$ over a smooth not necessarily compact manifold $Y$.
If $X$ and $X'$ are
isomorphic in codimension two we set $U=V$ and $U'=V'$.
We are going to prove \ref{rho2} under this additional assumption
(without using \ref{infconst}).
First note that a modification of the proof of \ref{Mukaith}
immediately yields
\begin{corollary} Assume $X$ and $X'$ satisfy \ref{ass}. If
${\cal X}\to S$
is a deformation as in the proof of
\ref{Mukaith} (i.e. $\bar v$ is non-trivial on the
fibres of $P\to Y$), then there exists a smooth morphism
${\cal V}'\to S$ such that ${\cal V}'|_{S\setminus{0}}\cong{\cal X}_{S\setminus\{0\}}$
and
${\cal V}_0\cong V'$. \hspace*{\fill}\hbox{$\Box$}
\end{corollary}
It can be used to prove
\begin{proposition}\label{cod2th} Let $X$ and $X'$ be as before, in
particular
$L'$ ample, and assume that \ref{ass} is satisfied. If $({\cal X},{\cal L})\to S$ is a
deformation over a smooth and one-dimensional base $S$ such that the class
$\bar v\in H^1(X,\Omega_X)$ associated to the Kodaira-Spencer class is
non-trivial on the fibres of $P\to Y$, then
$h^0({\cal X}_t,{\cal L}_t)\equiv const$ in a neighbourhood of $t=0$.
\end{proposition}
Since replacing $L'$ by another ample line bundle (if necessary) ensures
that
the generic deformation ${\cal X}\to S$ in $Def(X,L)$ has a Kodaira-Spencer class
$v$ such that $\bar v$ is non-trivial on the fibres of $P \to Y$
(cf. \ref{RMtoorthog}),
the proposition immediately shows
\begin{corollary}\label{maincodtwo}
If $X$ and $X'$ are projective irreducible symplectic
manifolds such that $X'$ is an elementary transformation of $X$ in
codimension
two, i.e. \ref{ass} holds, then $X$ and $X'$ present non-separated points
in the moduli space.\hspace*{\fill}\hbox{$\Box$}
\end{corollary}
{\bf Proof of \ref{cod2th}:} Let $s$ be the local parameter of $S$ at
$0\in S$ and let $S_n$ denote the closed subspace
${\rm Spec}(k[s]/s^{n+1})\subset S$.
Furthermore, let ${\cal X}_n:={\cal X}\times_S S_n$ and ${\cal L}_n:{\cal L}|_{{\cal X}_n}$.
In order to show
that $h^0({\cal X}_t,{\cal L}_t)\equiv const$, it suffices to prove that for all
$n$ the restriction $H^0({\cal X}_n,{\cal L}_n)\to H^0({\cal X}_{n-1},{\cal L}_{n-1})$ is
urjective.
This will be achieved by comparing it with the analogous restriction maps
for the family ${\cal V}'\to S$. For this purpose we introduce the following
notations. ${\cal U}_n$ denotes the space $(U,{\cal O}_{{\cal X}_n}|_U)$ and is considered
as a deformation of $U$ over $S_n$. Analogously, let
${\cal V}'_n:={\cal V}'\times_SS_n$
and ${\cal U}'_n:=(U',{\cal O}_{{\cal V}'_n}|_{U'})$, which is isomorphic to ${\cal U}_n$.
The line bundle ${\cal L}$ induces a line bundle ${\cal L}'$
on ${\cal V}'$. Its restrictions to ${\cal V}'_n$ are denoted by ${\cal L}'_n$.
In particular ${\cal L}'_0$ is isomorphic to $L'|_{V'}$.
First, $H^0({\cal V}'_n,{\cal L}'_n)\to H^0({\cal V}'_{n-1},{\cal L}'_{n-1})$ is surjective
for all $n$.
Indeed, using the exact sequence
$$\ses{L'|_{V'}}{{\cal L}'_n}{{\cal L}'_{n-1}}$$
this follows from $H^1(V',L'|_{V'})=H^1(X',L')=0$. Next,
$H^0({\cal U}'_n,{\cal L}'_n|_{{\cal U}'_n})\to H^0({\cal U}'_{n-1},{\cal L}'_{n-1}|_{{\cal U}'_{n-1}})$
is surjective
and $H^0({\cal V}'_n,{\cal L}'_n)\to H^0({\cal U}'_n,{\cal L}'_n|_{{\cal U}'_n})$ is an isomorphism.
This is
proved by induction starting with $H^0(V', L'|_{V'})=H^0(U',L'|_{U'})$ and
the
commutative diagram
$$\begin{array}{cccccccc}
0\to&H^0(V',L'|_{V'})&\to&H^0({\cal V}'_n,{\cal L}'_n)&\to&
H^0({\cal V}'_{n-1},{\cal L}'_{n-1})&\to&0\\
&\downarrow\cong&&\downarrow&&\downarrow\cong&&\\
0\to&H^0(U',L'|_{U'})&\to&H^0({\cal U}'_n,{\cal L}'_n|_{{\cal U}'_n})&\to&
H^0({\cal U}'_{n-1},{\cal L}'_{n-1}|_{{\cal U}'_{n-1}})&\to&\\
\end{array}$$
The isomorphism $H^0({\cal U}'_n,{\cal L}'_n|_{{\cal U}'_n})\cong
H^0({\cal U}_n,{\cal L}_n|_{{\cal U}_n})$ and
a similar induction argument prove $H^0({\cal X}_n,{\cal L}_n)\cong H^0({\cal U}_n,{\cal L}_n)$
and $H^0({\cal X}_n,{\cal L}_n)\longrightarrow\!\!\!\!\!\!\!\!\longrightarrow H^0({\cal X}_{n-1},{\cal L}_{n-1})$. In the
analogous diagram one in addition has to use $H^1(X,L)\hookrightarrow
H^1(U,L|_U)$.\hspace*{\fill}\hbox{$\Box$}
\section{Application to moduli spaces of bundles on K3 surfaces}\label{appl}
We briefly recall some facts from \cite{GH} that are necessary for our
purposes.\\
Let $S$ be a K3 surface, let ${\cal Q}\in{\rm Pic}(S)$ be an indivisible line bundle
and let
$c_2\in\hbox{\sym \char '132}$ such that $2n:=4c_2-c_1^2({\cal Q})-6\geq10$.
Assume that $H$ is a generic polarization, i.e. an ample line bundle
such that a rank two sheaf $E$ with $\det(E)\cong{\cal Q}$ and $c_2(E)=c_2$ is
$H$-semi-stable
if and only if it is $H$-stable. Then the moduli space $M_H({\cal Q},c_2)$ of
$H$-stable rank-two sheaves with determinant ${\cal Q}$ and second Chern number
$c_2$ is
smooth and projective. By \cite{Mu1} the moduli space
$M_H({\cal Q},c_2)$ admits a symplectic structure.\\
Next, one finds a K3 surface $S_0$ such that ${\rm Pic}(S_0)\cong\hbox{\sym \char '132}\cdot H_0$,
where
$H_0$ is ample, and $H_0^2/2+3=n$. In \cite{GH} we showed that under all
these
assumptions the moduli space $M_H({\cal Q},c_2)$ is deformation equivalent to the
moduli space $M_{H_0}(H_0,n)$ of sheaves on $S_0$. In particular,
$M_H({\cal Q},c_2)$ is irreducible symplectic if and only if $M_{H_0}(H_0,n)$
is irreducible
symplectic. Moreover, both spaces have the same Hodge numbers.\\
In order to prove that
$M_{H_0}(H_0,n)$ is irreducible symplectic we used Serre correspondence to
relate
this space to the Hilbert scheme $Hilb^n(S_0)$. Roughly, the generic sheaf
$[E]\in M_{H_0}(H_0,n)$ admits exactly one global section and the zero set
of this
section
defines a point in $Hilb^n(S_0)$. To make this more explicit we consider
the
moduli space $N$ of $H_0$-stable
pairs $(E,s\in H^0(S_0,E))$, such that $\det(E)\cong H_0$ and
$c_2(E)=n$. The parameter in the stability condition for such pairs is chosen
very small and constant. As explained in \cite{GH}
the maps $(E,s)\mapsto Z(s)$ and $(E,s)\mapsto E$ define morphisms
$\varphi:N\to Hilb^n(S_0)$ and $\psi:N\to M_{H_0}(H_0,n)$, respectively.
For the fibers we have
$$\varphi^{-1}(Z)\cong{\Bbb P}({\rm Ext }^1(I_Z\otimes H_0,{\cal O}_{S_0}))$$
and
$$\psi^{-1}(E)\cong{\Bbb P}(H^0(S_0,E)).$$
Generically, $h^1(S_0,I_Z\otimes H_0)=1$ and $h^0(S_0,E)=1$.
Thus
$$X:=Hilb^n(S_0)\lpfeil{5}{\varphi} N\rpfeil{5}{\psi} M_{H_0}(H_0,n)=:X'$$
defines
a birational correspondence between irreducible symplectic manifolds.\\
Next, we want to show that $X\lpfeil{5}{\varphi} N\rpfeil{5}{\psi}X'$
satisfies
the assumption \ref{ass}.\\ Using the exact sequence
$$\ses{I_Z\otimes H_0}{H_0}{{\cal O}_Z},$$
the vanishing
$H^1(S_0,H_0)=0$ and $H^2_0/2+3=n$, one shows
$h^1(S_0,I_Z\otimes H_0)=1+h^0(S,I_Z\otimes H_0)$.
Therefore, $f:=\varphi\circ\psi^{-1}$ is regular at points $Z$ which are
not contained
in any divisor $D\in |H_0|$.\\
Let ${\cal D}\to |H_0|$ denote the family of divisors
parametrized by the complete linear system $|H_0|$ and let
$Hilb^n({\cal D})\to |H_0|$ be the relative Hilbert scheme.
Then $f$ is regular on the complement $U$ of the image of the natural map
$g: Hilb^n({\cal D})\to Hilb^n(S_0)=X$. Since $\dim Hilb^n({\cal D})=n+h^0(S_0,H_0)-1=
2n-2=\dim Hilb^n(S_0)-2$, the birational correspondence $f$ is not isomorphic
in codimension two.\\
Let ${\cal C}\to B\subset|H_0|$ denote the family of smooth curves.
The relative Hilbert scheme over $B$ is just the relative symmetric product
$S^n({\cal C}/B)\to B$, which factorizes naturally through
the relative Picard ${\rm Pic}^n({\cal C}/B)\to B$.\\
The fibre of
the factorization $\phi:S^n({\cal C}/B)\to {\rm Pic}^n({\cal C}/B)$ over a point
$L\in{\rm Pic}^n({\cal C}_t)$
is naturally isomorphic to ${\Bbb P}(H^0({\cal C}_t,L))$.
Note that by Riemann-Roch $\chi({\cal C}_t,L)=n-H^2_0/2=3$. Hence
$h^0({\cal C}_t,L)\geq3$.
Let $Y\subset {\rm Pic}^n({\cal C}/B)$ denote the open set of line bundle
$L\in{\rm Pic}^n({\cal C}_t)$
such that $h^0({\cal C}_t,L)=3$ and let $\phi:P:=\phi^{-1}(Y)\to Y$
be the induced
${\Bbb P}_2$-bundle.
\begin{proposition} {\it i)} The morphism $g:Hilb^n({\cal D})\to X$ restricted
to $P$ is an embedding.\\
{\it ii)} The union $V$ of $U$ and $g(P)$ is open and
${\rm codim} (X\setminus V)\geq3$.\\
{\it iii)} $V\lpfeil{5}{\varphi}\varphi^{-1}(V)\rpfeil{5}{\psi}
\psi(\varphi^{-1}(V))$
is an elementary transformation along the ${\Bbb P}_2$-bundle $P$.
\end{proposition}
{\it Proof:} There is a number of little things to check.\\
First, by our
assumption $n\geq5$ we have $H_0^2\geq4$. Thus we can apply a result
of Saint-Donat
(cf. \cite{Mo}) to conclude that $H_0$ is very ample. Hence
$B\subset |H_0|$ is dense.
Moreover, $Hilb^n({\cal C})$ is dense in $Hilb^n({\cal D})$
(cf. \cite{AIK}, Thm. 5).\\
Next, we show that $Y\subset{\rm Pic}^n({\cal C}/B)$ is non-empty and, therefore,
dense
in ${\rm Pic}^n({\cal C}/B)$.
Indeed, if $x_1,...,x_{n-2}$ are generic points in $S_0$, then there
is exactly one smooth curve $C\in |H_0|$ containing them all.
Let $x_{n-1}$ and $x_{n}$
be two more generic points on $C$ and let $Z:=\{x_1,...,x_n\}$. Then
$h^0(S_0,I_Z\otimes H_0)=1$ and hence $h^1(S_0,I_Z\otimes H_0)=2$. Using the
exact sequence
$$\ses{{\cal O}_{S_0}}{I_Z\otimes H_0}{{\cal O}_C(-Z)\otimes H_0}$$
we get
$h^1(C,{\cal O}_C(-Z)\otimes H_0)=h^1(S_0,I_Z\otimes H_0)+h^2(S_0,{\cal O}_{S_0})=3$
and therefore
$h^0(C,{\cal O}_C(Z))=3$.
Thus the line bundle $L:={\cal O}_C(Z)$ defines a point in $Y$. Note that one
could invoke
a result by Lazarsfeld \cite{La} to prove $Y\ne\emptyset$. His result
also shows that for the generic curve ${\cal C}_t$ the complemet
of $Y\cap{\rm Pic}^n({\cal C}_t)\subset{\rm Pic}^n({\cal C}_t)$ has at least codimension four.\\
Since $P$ is obviously smooth and any $Z\in {\rm Im}(g)$ satisfies
$h^0(S_0,I_Z\otimes H_0)=1$, the morphism $g$ is an embedding on $P$.\\
By definition $V$ is the intersection of the open set
$\{Z|h^0(S_0,I_Z\otimes H_0)\leq1\}$
and the complement of $g(Hilb^n({\cal D})\setminus Hilb^n({\cal C}))$.
Hence $V$ is open.
The assertion on the
codimension follows from $Y\ne\emptyset$ and the irreducibility of
$Hilb^n({\cal D})$ (cf. \cite{AIK}).\\
It remains to prove {\it iii)}. Here we make essential use of the
moduli space $N$.\\
Let $N_P$ denote $\varphi^{-1}(P)$. We first show that $\psi:N_P\to X'$
respects
the projective bundle $\phi:P\to Y$, i.e. a fibre of $\psi$ maps to a
fibre of $\phi$.
Indeed, if $[E]\in X'$ and $s_1,s_2\in H^0(S_0,E)$ are two linearly
independent global sections, then we have a diagram
$$\begin{array}{ccccccc}
&&&{\cal O}_{S_0}&&{\cal O}_{S_0}&\\
&&&s_1\downarrow~~~&&\tilde s_1\downarrow~~~&\\
&{\cal O}_{S_0}&\rpfeil{5}{s_2}&E&\rpfeil{5}{}&I_{Z(s_2)}\otimes H_0&\\
&=\downarrow&&\downarrow&&\downarrow&\\
&{\cal O}_{S_0}&\rpfeil{5}{\tilde s_2}&I_{Z(s_1)}\otimes H_0&\rpfeil{5}{}&{\cal H}&\\
\end{array}$$
Thus $\tilde s_1$ and $\tilde s_2$ vanish along the same curve
$C\in |H_0|$ and
${\cal H}\cong{\cal O}_C(-Z(s_i))\otimes H_0$ for $i=1,2$.
Hence ${\cal O}_C(Z(s_1))\cong {\cal O}_C(Z(s_2))$, i.e.
$\phi(\varphi(E,s_1))=\phi(\varphi(E,s_2))$.\\
This reduces assertion {\it iii)} to the following problem. Let
$L\in{\rm Pic}^n({\cal C}_t)\cap Y$, let $P_L:={\Bbb P}(H^0({\cal C}_t,L))\cong{\Bbb P}_2$
and let $N_{P_L}:=\varphi^{-1}(P_L)$, which is a ${\Bbb P}_1$-bundle over ${\Bbb P}_2$.
Identify
$P_L\lpfeil{5}{}N_{P_L}\rpfeil{5}{}\psi(N_{P_L})$ with
${\Bbb P}_2\lpfeil{5}{}{\Bbb P}(\Omega_{{\Bbb P}_2})\rpfeil{5}{}{\Bbb P}_2^*$ !!\\
The argument goes as follows. Any point $Z\in P_L$ gives an exact sequence
$$\begin{array}{ccc}
\ses{{\cal O}_{S_0}}{I_Z\otimes H_0}{&{\cal O}_{{\cal C}_t}(-Z)\otimes H_0 &}\\
&\cong L^*\otimes K_{{\cal C}_t}&\\
\end{array}$$
Now use the canonical isomorphisms
$${\Bbb P}(H^0({\cal C}_t,L))\cong {\Bbb P}(H^1({\cal C}_t,L^*\otimes K_{{\cal C}_t})^*)\cong
{\Bbb P}({\rm Ext }^1(L^*\otimes K_{{\cal C}_t},{\cal O}_{S_0}))$$
to obtain the exact sequence
$$\ses{q^*{\cal O}_{S_0}\otimes p^*{\cal O}_{P_L}(1)}{I_{{\cal Z}}\otimes
q^*H_0\otimes p^*{\cal O}_{P_L}(a)}
{q^*(L^*\otimes K_{{\cal C}_t})},$$
where $q$ and $p$ are the two projections from
$S_0\times P_L$ and $I_{\cal Z}$ is the ideal sheaf of the universal subscheme
${\cal Z}\subset S_0\times P_L$.
By restricting to $\{x\}\times P_L$, where $x\in S_0\setminus {\cal C}_t$,
we deduce
$a=1$. The push-forward under $p$ induces the exact sequence
$$\ses{R^1p_*(I_{\cal Z}\otimes q^*H_0)\otimes{\cal O}_{P_L}(1)}
{H^1({\cal C}_t,L^*\otimes K_{{\cal C}_t})\otimes {\cal O}_{P_L}}{{\cal O}_{P_L}(1)}.$$
Hence $R^1p_*(I_{\cal Z}\otimes q^*H_0)\cong \Omega_{P_L}$.
It is straightforward to identify $N_{P_L}\to P_L$ with
${\Bbb P}(R^1p_*(I_{\cal Z}\otimes q^*H_0)^*)$. Thus
$(N_{P_L}\to P_L)\cong ({\Bbb P}({\cal T}_{{\Bbb P}_2})\to{\Bbb P}_2)\cong
({\Bbb P}(\Omega_{{\Bbb P}_2})\to{\Bbb P}_2)$.\\
It remains to show that $\varphi:P_E:={\Bbb P}(H^0(S_0,E))=\psi^{-1}(E)\to P_L$
is a linear embedding.
On $P_E$ we have
$$\ses{{\cal O}}{q^*E\otimes p^*{\cal O}_{P_E}(1)}{(1\times\varphi)^*
(I_{\cal Z}\otimes q^*H_0)\otimes p^*{\cal O}_{P_E}(a)},$$
where by abuse of notation $q$ and $p$ are again the projections from
$S_0\times P_E$.
As above one finds $a=2$. Taking direct images we obtain
$$\ses{H^1(S_0,E)\otimes{\cal O}_{P_E}(1)}{\varphi^*(R^1p_*(I_{\cal Z}\otimes
q^*H_0))\otimes {\cal O}_{P_E}(2)}{{\cal O}_{P_E}},$$
i.e. $\ses{{\cal O}_{P_E}(1)}{\varphi^*\Omega_{P_L}\otimes{\cal O}_{P_E}(2)}
{{\cal O}_{P_E}}$. Thus
$\varphi^*{\cal O}_{P_L}(1)\cong{\cal O}_{P_E}(1)$.
\hspace*{\fill}\hbox{$\Box$}
{\bf Remark:} The identification $N_{P_L}\cong{\Bbb P}({\cal T}_{P_L})$ seems
to show that
the birational correspondence described by $N$ is not some kind of ``nested
elementary transformation'': It is only in the codimension two case where
one has ${\Bbb P}(\Omega_{P_L})\cong{\Bbb P}({\cal T}_{P_L})$.
Corollary \ref{maincodtwo} now immediately implies
\begin{corollary} If $S_0$ is a K3 surface with ${\rm Pic}(S_0)=\hbox{\sym \char '132}\cdot H_0$ and
$H_0^2\geq4$, then $M_{H_0}(H_0,n)$ and $Hilb^n(S_0)$ correspond to
non-separated points
in the moduli space of symplectic manifolds ($n=H_0^2/2+3$).\hspace*{\fill}\hbox{$\Box$}
\end{corollary}
Thus we can conclude
\begin{theorem}\label{moddefhilb} If $S$ is a K3 surface, ${\cal Q}\in{\rm Pic}(S)$
indivisible,
$2n:=4c_2-c_1^2({\cal Q})-6\geq10$ and $H$ a generic polarization,
then the moduli space $M_H({\cal Q},c_2)$ of $H$-stable rank two sheaves
$E$ with $\det(E)\cong{\cal Q}$ and $c_2(E)=c_2$ is deformation equivalent
to $Hilb^n(S)$.\hspace*{\fill}\hbox{$\Box$}
\end{theorem}
Note that in particular moduli space and Hilbert scheme are just different
complex structures on the same differentiable manifold.\\
{\bf Remark:} O'Grady works instead of $S_0$ with an elliptic surface and
shows
that every moduli space is deformation equivalent to a moduli space
on an elliptic surface \cite{OG}. The birational correpondence between
moduli space and Hilbert scheme on the elliptic surface is again
given by Serre correspondence. The picture there is slightly more complicated
than what we have encountered above. Nevertheless I believe, that
also in his situation
the assumptions \ref{ass} are satisfied and that moduli space and Hilbert
scheme
are deformation equivalent rank$>2$ as well.
{\footnotesize |
1996-01-19T06:20:19 | 9601 | alg-geom/9601019 | en | https://arxiv.org/abs/alg-geom/9601019 | [
"alg-geom",
"math.AG"
] | alg-geom/9601019 | Indranil Biswas | Indranil Biswas | Principle bundles admitting a holomorphic structure | AMSLatex | null | null | null | null | Let $M$ be a compact connected K\"ahler manifold and let ${\E}_{l-1}$ be the
smallest term in the Harder-Narasimhan filtration of its tangent bundle. Let
$G$ be an affine algebraic reductive group over $\C$. We prove the following
result: If $M$ satisfies the condition that $\deg (T/{\E}_{l-1}) \geq 0$, then
a holomorphic principal $G$-bundle $P$ on $M$ admitting a compatible
holomorphic connection is semistable. Moreover, if $\deg (T/{\E}_{l-1}) >0$,
then such a bundle $P$ actually admits a compatible flat $G$-connection.
| [
{
"version": "v1",
"created": "Thu, 18 Jan 1996 18:08:42 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Biswas",
"Indranil",
""
]
] | alg-geom | \section{Introduction}
Let $E$ be a holomorphic vector bundle over a
compact K\"ahler manifold $M$, such that $E$ admits a holomorphic
connection compatible with its
holomorphic structure. The following question is perhaps
due to S. Murakami \cite{M1}, \cite{M2}, \cite{M3} : does there
exists a flat connection on $E$ compatible with the holomorphic
structure ?
The same question can be posed for a more general principal
$G$ bundle where $G$ is a complex Lie group. Murakami constructed
examples of torus bundles over torus which admit holomorphic
connection but do not admit compatible flat connection; thus
providing a negative answer to the general question.
Let $M$ be a compact connected K\"ahler manifold of complex
dimension $d$, and let $\omega $
be a K\"ahler form on $M$. The degree
of a torsion-free ${{\cal O}}_M$-coherent sheaf $F$ is defined as
[Ko, Ch. V, (7.1)] $$\deg F \, := \, \int_Mc_1(F)\wedge
{\omega }^{d-1} \leqno{(1.1)}$$
Consider the Harder-Narasimhan filtration
of the holomorphic tangent bundle of $M$ [Ko, page 174, Ch. V,
Theorem 7.15] : $$0 \, = \, {{\cal E}}_0 \, \subset \,
{{\cal E}}_1 \, \subset \, {{\cal E}}_2
\, \subset \, {{\cal E}}_3 \, \subset \, \ldots \, \subset \,
{{\cal E}}_{l-1} \, \subset \, {{\cal E}}_l \, = \, T \leqno{(1.2)}$$
Let $G$ be a connected affine algebraic reductive group over ${\Bbb C}$.
Let $P$ be a holomorphic principal $G$ bundle on $M$ ({\it i.e.\/}\ the
transition functions of $P$ are holomorphic). Assume that $P$ admits
a holomorphic connection $D$ compatible with the holomorphic
structure. This means the following :
$D$ is a holomorphic $1$-form on $P$ with values in the Lie
algebra, $\frak g$, of $G$, such that $D$ is invariant for the
action of $G$ on $P$, and, when restricted to the fibers of
$P$, this form coincides
with the holomorphic Maurer-Cartan form. Using the natural
identification of the holomorphic tangent space of $P$ with
its real tangent space, the holomorphic connection $D$ gives
a $G$ connection on $P$.
A flat $G$ connection on $P$ is called compatible with
respect to the holomorphic structure if the $(1,0)$ component
of the connection form (which is a $\frak g$ valued $1$-form
on $P$) is actually a holomorphic $1$-form. This is equivalent
to the following : $M$ can be covered by open set $\{U_i\}$
such that over each $U_i$ the principal bundle $P$ admits
a holomorphic trivialization which is also constant (with
respect to the connection). Clearly the $(1,0)$ component
of such a flat connection compatible with the holomorphic
structure gives a holomorphic connection compatible with
the holomorphic structure.
Our aim here is to to prove the following theorem [Theorem 3.1] :
\medskip
\noindent {\bf Theorem A.}\,\ {\it If the K\"ahler manifold
$M$ satisfies the condition that $\deg (T/{{\cal E}}_{l-1}) \geq 0$, then
a holomorphic principal $G$-bundle $P$ on $M$ admitting
a compatible holomorphic connection is semistable. Moreover,
if $\deg (T/{{\cal E}}_{l-1}) >0$, then such a bundle $P$ actually
admits a compatible flat $G$-connection.}
\medskip
So, in particular, if $T$ is semistable with $\deg T >0$,
then a $G$-bundle $P$ on $M$ with a holomorphic connection
admits a compatible flat connection. In the case where
$G = GL(n, {\Bbb C})$, the above result was proved in \cite{Bi}.
The author is grateful to D. Akhiezer for some very useful
discussions.
\section{Preliminaries}
We continue with the notation of Section 1.
Let $D$ be a holomorphic connection on the holomorphic principal
$G$ bundle $P$ compatible with its holomorphic structure.
Let $Ad(P)$ denote the vector bundle on $M$ associated
to $P$ for the adjoint action of $G$ on its Lie algebra
$\frak g$. The holomorphic structure of $P$ will induce
a holomorphic structure on the vector bundle $Ad(P)$.
Let ${\overline{\partial}}_P$ denote the differential
operator of order one defining the holomorphic structure of
$Ad(P)$.
Let ${\Omega }^i_M$ denote the holomorphic vector bundle on $M$
given by the holomorphic $i$-forms.
The holomorphic connection $D$ on $P$ induces a holomorphic
connection on $Ad(P)$, which is again denoted by $D$. In other
words,
$$D \, : \, Ad (P) \, \longrightarrow \, Ad (P) \otimes {\Omega }^1_M$$
is a first order operator satisfying
the Leibniz
condition $D(f.s) = \partial f . s + fD(s)$, where $f$ is a smooth
function, such that the curvature,
$(D+ {\overline{\partial}}_P)^2$, of
the connection $D+{\overline{\partial}}_P$ is a
holomorphic section of ${\Omega }^2_M \otimes Ad (P)$. This condition
implies that the operator $D$ maps a holomorphic sections of
$Ad (P)$ to a holomorphic section of $Ad (P)\otimes {\Omega }^1_M$.
A vector bundle $E$ on $M$ is called {\it stable} (resp.
{\it semistable}) if for any ${{\cal O}}_M$-coherent proper subsheaf
$0 \neq F \subset E$, with $E/F$ torsion-free, the following
condition holds [Ko, Ch. V, \S 7] :
$$ \mu (F) := \deg F/{\rm rank}F \, < \mu (E) :=
\deg E/{\rm rank} E \, ~ \, ~\, ({\rm resp.}\,\, \mu (F) \, \leq \,
\mu (E))$$ A vector bundle is called {\it quasistable} if it is
a direct sum of stable bundles of same $\mu$ (slope).
\medskip
\noindent {\bf Lemma 2.1.}\,\, {\it If $\deg (T/{{\cal E}}_{l-1}) \geq 0$
then the vector bundle $Ad (P)$ on $M$ is semistable. Moreover,
if $\deg (T/{{\cal E}}_{l-1}) > 0$, then $Ad (P)$ is quasistable.}
\medskip
The proof of this lemma is actually contained in \cite{Bi}.
However, to be somewhat self-contained, we will give some
details of the proof.
\medskip
\noindent {\underline {Proof of Lemma 2.1.}}\, We will first
prove that if $\deg (T/{{\cal E}}_{l-1}) \geq 0$ then the bundle
$Ad (P)$ is semistable.
Suppose $Ad (P)$ is not semistable. In that case it has a nontrivial
Harder-Narasimhan filtration. Let
$$0 \, = \, V_0\, \subset \, V_1 \, \subset \, V_2 \, \subset \,
\ldots \, \subset \, V_{n-1} \, \subset \,
V_n \, =\, Ad (P) \leqno{(2.2)}$$
be the Harder-Narasimhan filtration of the vector bundle $Ad (P)$.
We restrict the domain of the operator $D$ (which gives the
holomorphic connection on $Ad (P)$) to
the subsheaf $V_1$, and consider the induced operator
$$D_1 \, : \, V_1 \, \longrightarrow \,
(Ad(P)/V_1)\otimes {\Omega }^1_M \leqno{(2.3)}$$ The Leibniz identity
for $D$ implies that $D_1(f.s) = f.D_1(s)$, {\it i.e.\/}\ $D_1$ is
${{\cal O}}_M$-linear.
The largest semistable subsheaf ({\it i.e.\/}\ the first nonzero term in the
Harder-Narasimhan filtration) of ${\Omega }^1_M$ is the kernel
of the surjective homomorphism
$$q_{l-1} \, :\, {\Omega }^1_M \, \longrightarrow \, {{\cal E}}^*_{l-1}$$
obtained by taking the dual of the inclusion in (1.2). Since
the tensor product of two semistable sheaves is again semistable
[MR, Remark 6.6 iii], the largest
semistable subsheaf of $(Ad (P)/V_1) \otimes {\Omega }^1_M$ is
$$W \, := \, (V_2/V_1)\otimes {\rm kernel}(q_{l-1})\leqno{(2.4)}$$
The general formula for the degree of a tensor product gives
$\mu (W) = \mu (V_2/V_1) + \mu ({\rm kernel}(q_{l-1}))$.
{}From the property of the Harder-Narasimhan filtration we have that
$\mu (V_2/V_1) < \mu (V_1)$. The assumption that
$\deg (T/{{\cal E}}_{l-1}) \geq 0$ implies that
$$\mu ({\rm kernel}(q_{l-1})) \, = \, - \mu (T/{{\cal E}}_{l-1}) \, \leq \,0$$
{}From these we conclude that
$$\mu (V_1) \, > \, \mu (W) \leqno{(2.5)}$$
Let $E$ denote the image of $D_1$ defined in (2.3). Assume
that $E$ is not the zero sheaf. Since
$V_1$ is semistable and $E$ is a quotient of $V$, we have
$\mu (E) \geq \mu (V_1)$. On the other hand, since $W$ is the
largest semistable subsheaf of $(Ad (P)/V_1)\otimes {\Omega }^1_M$,
we have $\mu (E) \leq \mu (W)$. But this contradicts (2.5). So
$E$ must be the zero sheaf, {\it i.e.\/}\ the homomorphism
$D_1$ in (2.3) must be the zero homomorphism. Thus we obtain that the
subsheaf $V_1$ is invariant under the connection $D$ on $Ad (P)$.
Any ${{\cal O}}_M$ coherent sheaf
with a holomorphic connection is a locally free ${{\cal O}}_M$ module
[B, p. 211, Proposition 1.7]. (This proposition in \cite{B}
is stated for integrable connections ($D$-modules), but the proof
uses only the Leibniz rule which is valid for a holomorphic
connection.)
Using the Chern-Weil construction of characteristic classes it is
easy to show that for a vector bundle $E$ on $M$, equipped with
a holomorphic connection, any Chern class $c_i(E) \in H^{2i}(M,{\Bbb Q})$,
$i\geq 1$, vanishes.
So we have $\deg V_1 = 0 = \deg Ad (P)$. If $V_1 \neq Ad(P)$
then $\mu (V_1) > \mu (Ad (P))$. So $Ad (P) = V_1$, {\it i.e.\/}\
the bundle $Ad (P)$ is semistable.
If $Ad (P)$ is stable then obviously
it is quasistable. Suppose that $Ad(P)$ is not stable.
Then there is a filtration [K, Ch. V, \S 7,
Theorem 7.18] $$0 \,= \,W_0 \, \subset \, W_1 \, \subset \,W_2 \,
\subset
\, \ldots \, \subset \, W_{n-1} \, \subset \, W_n \,
= \, Ad (P) \leqno{(2.6)}$$
such that $W_i/W_{i-1}$, $1\leq i \leq n$, is a stable
sheaf with $\mu (W_i/W_{i-1}) = \mu (Ad (P))$.
Now, as done in the proof of Proposition 3.4 of \cite{Bi},
using the given condition that $\deg (T/{{\cal E}}_{l-1})$ is
strictly positive, it is
possible to show that the filtration (2.6) splits, {\it i.e.\/}\
$$W_i \, = \, W_{i-1}\oplus (W_i/W_{i-1}) \leqno{(2.7)}$$
where $W_i/W_{i-1}$ is a locally free sheaf. The equality
(2.7) proves that $$Ad (P)\, =\, \sum_{i=1}^n W_i/W_{i-1}$$
This completes the proof of the lemma. $\hfill{\Box}$
\medskip
In the next section we will use Lemma 2.1 to construct
a flat connection on $P$.
\section{Existence of a flat connection}
Let $G$ be a connected affine algebraic reductive group over ${\Bbb C}$.
Let $P$ be a holomorphic principal $G$ bundle over $M$.
Let $i : U \longrightarrow M$ be the inclusion of an open subset
such that the complement $X-i(U)$ is an analytic subset of $X$ of
codimension at least two. For a ${{\cal O}}_U$ coherent sheaf $F$
on $U$, the direct image $i_*F$ is a ${{\cal O}}_M$ coherent sheaf.
The degree of $F$ is defined to be the degree of $i_*F$.
We will recall the definition of (semi)stability of $P$
([RR, Definition 4.7]).
Let $U\subset X$ be an open subset
with $X-U$ being an analytic set of codimension at least two.
Let $Q$ be a parabolic subgroup of $G$, and let $P'$
be a reduction of the structure group to $Q$
of the restriction of the principal $P$ to the open set $U$. The
principal bundle $P$ is said to be {\it stable}
(resp. {\it semistable}) if for any such $P'$, the degree of a
line bundle on $U$ associated to $P$ for any character
$\chi$ on $Q$ dominant with respect to a Borel subgroup
contained in $Q$, is strictly negative (resp. nonpositive).
Let $D$ be a holomorphic connection on $P$ (defined in Section 1).
The following theorem is a generalization of Lemma 2.1.
\medskip
\noindent {\bf Theorem 3.1.}\, {\it If $\deg (T/{{\cal E}}_{l-1}) \geq 0$
then $P$ is semistable. Moreover, if $\deg (T/{{\cal E}}_{l-1}) >0$,
then $P$ admits a flat $G$-connection compatible with
the holomorphic structure.}
\medskip
\noindent {\bf Proof.}\, Let $P' \subset P$ be a reduction
of structure group of $P$ to a maximal parabolic subgroup
$Q\subset G$. This reduction is given by a section, $\sigma $,
of the fiber bundle
$${\rho } \, : \, P/Q \, \longrightarrow \, U$$ Let $T_{\rm
rel}$ denote the relative tangent bundle for the map $\rho $.
{}From Lemma 2.1 of \cite{R} it
follows that in order to check that $P$ is semistable it is enough
to show that $\deg ({\sigma }^*T_{\rm rel}) \geq 0$.
The reduction $P'\subset P$ gives an injective homomorphism
$Ad(P') \longrightarrow Ad(P)$ of adjoint bundles on $M$.
The bundle ${\sigma }^*T_{\rm rel}$ on $M$ is the quotient
bundle $Ad(P)/Ad(P')$.
{}From Lemma 2.1 we know that if $\deg (T/{{\cal E}}_{l-1}) \geq 0$, the
adjoint bundle $Ad(P)$ is semistable. Since $G$ is reductive,
the Lie algebra $\frak g$ admits a nondegenerate
$G$ invariant bilinear form. This implies that $Ad(P) = Ad(P)^*$.
Hence $\deg Ad(P) =0$. Now the semistabilty of $Ad(P)$
implies that $\deg (Ad(P)/Ad(P')) \geq 0$. This proves that
the principal bundle $P$ is semistable.
If $\deg (T/{{\cal E}}_{l-1}) > 0$ then from Lemma 2.1 we know that
$Ad (P)$ is quasistable. So from the main theorem of
\cite{UY} it follows that the vector bundle $Ad(P)$ admits
a Hermitian-Yang-Mills connection. We will denote this
Hermitian-Yang-Mills connection
by $\nabla $. This connection $\nabla $ is unique (though the
Hermitian-Yang-Mills metric is not unique), and it is irreducible
if and only if $Ad(P)$ is stable.
We want to show that this connection $\nabla $ induces a connection
on the principal $G$ bundle $P$.
Let $Z_0$ denote the connected component of the
center of $G$ containing the identity element
(the center has finitely many components).
Define $$G_0 \, = \, G/Z_0$$ which is a semisimple group.
The group $G_0$ acts on the Lie algebra $\frak g$ (of $G$)
by conjugation and gives an homomorphism
$$\theta \, : \, G_0 \, \longrightarrow \,GL({\frak g})
\leqno{(3.2)}$$ which has a finite group as the kernel.
{}From a theorem of Chevalley [H, Theorem 11.2] we know that
there is a linear representation of the group $GL ({\frak g})$
$$\phi \, : \, GL({\frak g}) \, \longrightarrow \, GL(V)
\leqno{(3.3)}$$
in a vector space $V$ over ${\Bbb C}$ and a line $L$ in $V$ such that
$$\phi \circ \theta (G_0) \, =
\, \{g \in GL({\frak g}) \, \vert \,~\, {\phi }(g)(L) = L \}$$
Since $G_0$ is semisimple, it does not have
any nontrivial character. This implies that $\phi \circ \theta (G_0)$
fixes the line $L$ point-wise. Let $0\neq v \in L$ be a nonzero
vector. So the isotropy subgroup of $v$ for the action
of $GL({\frak g})$ on $V$ is precisely ${\phi \circ\theta }(G_0)$. It
is not difficult to see that the homomorphism $\phi $ can be
so chosen that it maps the center of the Lie algebra of
$GL({\frak g})$ into the center of the Lie algebra of $GL(V)$.
We will choose $\phi $ such that it satisfies this condition.
Let $q : G \longrightarrow G_0$ denote the obvious projection.
Let $P(G_0)$ denote the principal $G_0$ bundle on $M$ obtained by
extending the structure group of $P$ to $G_0$ using the
homomorphism $q$. The vector bundle $Ad(P)$, which can be
identified with a principal $GL({\frak g})$ bundle, is obtained by
extending the structure group of $P(G_0)$ to $GL({\frak g})$
using the homomorphism $\theta $ defined in (3.2).
Using the homomorphism $\phi $ we may extend the structure
group of $Ad(P)$ to $GL(V)$. The vector bundle on $M$ associated
to this principal $GL(V)$ bundle for the natural action of
$GL(V)$ on the vector space $V$ will be denoted by $E$.
The Hermitian-Yang-Mills metric on the vector bundle $Ad(P)$
gives a reduction of the structure group of $Ad(P)$ to
$U({\frak g})$, a maximal compact subgroup of $GL ({\frak g})$.
Since the image $\phi (U({\frak g}))$ is a compact
subgroup of $GL(V)$, it is contained in some
maximal compact subgroup of $GL(V)$. So the reduction of $Ad(P)$
to $U({\frak g})$ will induce a reduction of $E$ to a maximal
compact subgroup of $GL(V)$ ({\it i.e.\/}\ the vector bundle
$E$ will be equipped with a hermitian metric) such that the
connection on $E$ obtained by extending the connection
$\nabla $ (on $P({\frak g})$) is the hermitian connection (for the
hermitian metric on $E$).
Since the metric on $Ad(P)$ is a
Hermitian-Yang-Mills metric, the metric on $E$ obtained above
is also a Hermitian-Yang-Mills metric. Indeed, the
Hermitian-Yang-Mills condition of the connection on $Ad(P)$
implies that the curvature is a $2$-form on $M$ with values in the
center of the endomorphism bundle $Ad(P)^*\otimes Ad(P)$. Since $\phi $
maps the center of the Lie algebra of $GL({\frak g})$ into the the
center of the Lie algebra of $GL(V)$, the induced connection on
$E$ is a Hermitian-Yang-Mills connection. Let ${\nabla }'$ denote the
Hermitian-Yang-Mills connection on $E$ obtained this way.
Since ${\theta }(G_0)$ fixes the vector $v$, this vector $v$ will give
a nowhere zero section of $E$ (since $E$ is obtained by extending
the structure group of $P$ to $GL(V)$); let $s$
denote this section of $E$.
The holomorphic connection $D$ on $P$ will induce a holomorphic
connection on $E$. As we noted in Section 2, using
the Chern Weil construction it is easy to see that the existence
of a holomorphic connection on $E$ implies that any Chern class,
$c_i(E)$, $i\geq 1$, vanishes.
{}From [Ko, Ch. IV, Corollary 4.13] it follows that
Hermitian-Yang-Mills connection ${\nabla }'$
is a flat connection. Since ${\nabla }'$ is a flat unitary
connection, any holomorphic section
of $E$ must be a flat section for the connection ${\nabla }'$. Indeed,
the Laplacian of a flat unitary connection operator is twice the
Laplacian of the Dolbeault operator. In particular, the space of
harmonic sections for these two Laplacians coincide. Since any
holomorphic section is a harmonic section for the Dolbeault
Laplacian, it must be a flat section. So, in particular,
the section $s$ is flat.
Let $P({\frak g})$ denote the principal $GL({\frak g})$
obtained by extending the structure group of $P$ to
$GL({\frak g})$ using the homomorphism $\theta \circ q$. Since
$P({\frak g})$ is an extension, the fiber bundle,
$P({\frak g})/G$, with fiber $GL({\frak g})/G$ has a natural
section (which gives the reduction of the structure
group of $P({\frak g})$ to $G$). Let $\alpha $ denote this section
of $P({\frak g})/G$.
Since $\theta (G_0)$ fixes $v$ for the homomorphism $\phi $ in (3.3),
we have an embedding of the fiber bundle $P({\frak g})/G$
in the total space of $E$ (given by the orbit of $v$ for
the action of $GL({\frak g})$ using $\phi $).
It is easy to see that the image of the section $\alpha $ by the
above embedding of $P({\frak g})/G$ in $E$
is precisely the section $s$.
Now, since $s$ is a flat section for the connection ${\nabla }'$,
the connection $\nabla $ induces a $G_0$ connection on the principal
$G_0$ bundle $P(G_0)$ on $M$ as follows : Let $$p \, :\,
P(G_0) \, \longrightarrow \, P({\frak g})$$ denote the
holomorphic map induced by $\theta $ in (3.2). Take $x \in p (P(G_0))$,
and let $v \in T_xP({\frak g})$ be a horizontal vector for the
connection $\nabla $ on $P({\frak g})$. Let $w$ be the image
of $v$ by the differential of the map
$$P({\frak g})\, \longrightarrow \, E$$
induced by the homomorphism $\phi $ in (3.3). Since $s$ is flat,
the tangent vector $w$ lies in the submanifold of the total
space of $E$ given by the section $s$. But this implies that
the tangent vector $v$ lies in the image of $TP(G_0)$ under
the map given by the differential of $p$. Thus the connection
$\nabla $ on $P({\frak g})$ induces a connection on the principal
bundle $P(G_0)$ with structure group $G_0$. We will call
this connection on $P(G_0)$ as ${\nabla }_0$. Since $\nabla $ is a
flat connection, ${\nabla }_0$ is also flat.
The commutator subgroup
$G' \, := \,[G,G] \, \subset \,G$ is a semisimple group, and
the restriction of $q$ to $G'$ is a surjective homomorphism with a
finite kernel. So their Lie algebras are isomorphic. So $$G \,=
Z_0.G' \leqno{(3.4)}$$
with a finite intersection $\Gamma := Z_0\cap G'$.
The abelian Lie group $Z_0/\Gamma $ is a product of copies of
${{\Bbb C}}^*$, since $G$ is assumed to be affine. Let
$$f \, :\, G \,\longrightarrow \, Z_0/\Gamma $$ denote the
obvious projection (obtained from (3.4)).
Let $P(f)$ denote the principal $Z_0/\Gamma $ bundle on $M$ obtained
by extending the structure group of $P$ using the homomorphism
$f$.
The holomorphic connection $D$ on $P$ induces a holomorphic
connection on $P(f)$, which we will denote by $D(f)$.
Any holomorphic line bundle on $M$ admitting a holomorphic
connection actually admits a compatible flat connection. Indeed,
if $\partial$ is a holomorphic connection on a holomorphic
line bundle $L$ whose holomorphic structure is given by the
operator $\overline\partial$, then the curvature
$(\partial +{\overline{\partial}})^2$ is a holomorphic $2$-form
which is exact (since the cohomology class represented
by it is of the type $(1,1)$). So it is of the form
$\partial \beta$, where $\beta$ is a $(1,0)$-form. The new
connection $$\partial -\beta + {\overline{\partial}}$$ on $L$ is a
flat connection compatible with the holomorphic structure.
Recall that $Z_0/\Gamma $ is a product of copies of ${{\Bbb C}}^*$. In
view of the above remark, the the existence of the holomorphic
connection $D(f)$ implies that the principal $Z_0/\Gamma $ bundle
$P(f)$ admits a flat connection. Let ${\nabla }_1$ be a flat
connection on $P(f)$.
Since the exact sequence of the Lie algebras
$$0 \, \longrightarrow \, {\frak z}_0 \,\longrightarrow \,
{\frak g} \,\longrightarrow \,{\frak g}_0 \,\longrightarrow\,
0$$ has a natural splitting (given by the Lie algebra of $G'$),
the two flat connections ${\nabla }_0$ and ${\nabla }_1$ combine
together to induce a flat $G$ connection on $P$ as follows :
The horizontal subspace of the tangent space at a point $p \in P$
is defined to be the intersection of the inverse images of the
horizontal subspaces of $P(G_0)$ and $P(f)$ (horizontal
subspaces for the
flat connections ${\nabla }_0$ and ${\nabla }_1$ respectively) for the
obvious projections of $P$ onto $P(G_0)$ and $P(f)$ respectively.
The integrability of ${\nabla }_0$ and ${\nabla }_1$ will imply that
the connection on $P$ obtained above is actually flat.
This completes the proof of the theorem. $\hfill{\Box}$
|
1996-01-16T06:20:25 | 9601 | alg-geom/9601013 | en | https://arxiv.org/abs/alg-geom/9601013 | [
"alg-geom",
"math.AG"
] | alg-geom/9601013 | Youngook Choi | Youngook Choi | Severi Degrees in Cogenus 4 | TeX-Type: AMSLaTeX v 1.1, 11 pages | null | null | null | null | In this paper, we give closed-form formulae for Severi degrees in cogenus 3
and 4 using Ran's method. These formulae coincide with those of I. Vainsencher
and for cogenus 3 case, that of J. Harris and R. Pandharipande. Another result
of this paper is that we calculate the degree of the polynomial
$N(\pi,\delta,d)$ in $d$, which is the degree of the locus of curves $C$ in P^2
with degree d having $\delta$ nodes and $C \cap L$ is of type $\pi$ to fixed
line L. Using this result, we also calculate the coefficients of two leading
terms of Severi polynomial, $N(\delta,d)$.
| [
{
"version": "v1",
"created": "Mon, 15 Jan 1996 18:23:20 GMT"
},
{
"version": "v2",
"created": "Mon, 15 Jan 1996 21:48:42 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Choi",
"Youngook",
""
]
] | alg-geom | \section{Summary}
Let $\cal{P} (d)$
be the linear system of degree $d$ curves in $\bold P^2$.
Then $\cal{P} (d)$ is a projective space of dimension $\binom{d+2}{2}-1$.
The Severi variety $\cal{V} (\delta, d) \subset \cal{P} (d)$ is the subset
corresponding to reduced, nodal curves with $\delta$ nodes.
The ``well-known'' problem, whether $\cal{V} (\delta, d)$ is irreducible
or not, is solved affirmatively by Harris [H], and Ran [R1]. The next question
about the Severi variety is to find its degree, $\text{N}(\delta,d)$.
In the paper [R2],
Ran gave the recursive formulae. In this paper, we will give closed-form
formulae for cogenus 3 and 4 cases using his method.
These formulae coincide with
those of I. Vainsencher [V], and
for cogenus 3 case, that of J. Harris and R. Pandharipande [H-P].
Ran gave the formula of cogenus 2 case using his method in the
paper [R3], namely,
$$
\text{N}(2,d) = 3/2(d-1)(d-2)(3d^2-3d-11)
$$
Therefore, this paper is an extension of that paper.
One small
benefit of our approach is that we can calculate, as a special case, the
formulae of degree of the locus of all curves having less
than three nodes and some
tangency conditions to fixed line.
Another result of this paper is that we calculate the degree of
the polynomial N($\pi,\delta, d$) in $d$, which is
the degree of the locus of curves $C$ in $\bold P^2$ with degree $d$
and having $\delta$ nodes and $C \cap \text{L}$ is
{\it of type $\pi$} to fixed line L. Using this result, we also
calculate the coefficients of two leading terms
of Severi polynomial, N($\delta, d$). The results are :
for any $\delta$ and any partition $\pi = [\ell_1,...,\ell_n]$,
$$
\begin{array}{ll}
\text{deg} \ \text{N} (\pi,\delta,d) = 2\delta + \sum_{i=2}^n \ell_i
& (\text{Prop. 4.2}) \\
a_{2\delta}^{\delta} = {3^{\delta}\over{\delta !}}
& (\text{Cor. 4.3}) \\
a_{2\delta - 1}^{\delta} = -2\delta a_{2\delta}^{\delta}
={ {-2 \times 3^{\delta}}\over{(\delta-1)!}}
& (\text{Cor. 4.5}) \ ,
\end{array}
$$
\noindent
where $a_{\beta}^{\delta}$ is the coefficient of the degree $\beta$ term
of the polynomial N($\delta,d$). So,
$$
\text{N} (\delta,d) =
{3^{\delta}\over{\delta !}}d^{2\delta} -{{ 2 \times 3^{\delta}}
\over{(\delta -1)!}}{d^{2\delta - 1}} + \text{lower degree terms.}
$$
For $\delta \leq $ 6, these two coefficients coincide with those of the
polynomials of I. Vainsencher [V].
This paper is written under the guidence of Z. Ran. Without his advice, it
would be
impossible for me to write this paper. I deeply appreciate his advice.
\section{Set of divisors of $\bold P^1$}
In this section, we calculate the degree of set
of divisors of $\bold P^1$ because we need the degree of the locus of smooth curves
with tangency conditions to fixed line.
\begin{defn} A partition $\pi = [\ell_1,...,\ell_n]$ is a sequence
of nonnegative integers.
\end{defn}
\begin{defn} A divisor $D$ of $\bold P^1$ is of type $\pi$
if it has the form $D=\sum_{i=1}^n \sum_{j=1}^{\ell_i}iP_{ij}$
for some distinct points $P_{ij}$.
\end{defn}
\begin{rem}
\begin{enumerate}
\item Degree of a divisor $D$ of type $\pi$ is
$\sum_{i=1}^n i\ell_i$.
\item Let $\Gamma_\pi$ be the closure of the set of all divisors
in $\bold P^1$ of type $\pi$. If we identify the space of all divisors of degree
$N$ on $\bold P^1$ with $\bold P^N$, then $\Gamma_\pi$ is a subvariety of $\bold P^N$,
where $N=\sum_{i=1}^n i\ell_i$.
\item $\text{\rm dim} \ \Gamma_\pi = \sum_{i=1}^n \ell_i$.
\item $m(\pi) \buildrel \rm def \over = \prod i^{\ell_i}$
\item $n(\pi) \buildrel \rm def \over =
{(\sum \ell_i)! \over \ell_1! \dotsm \ell_n!}$
\end{enumerate}
\end{rem}
\begin{lem} $\text{\rm deg} \ \Gamma_\pi=m(\pi)n(\pi)$
\end{lem}
{\it Proof} \ \ \
Let $Q \in \bold P^1$, and $\text{N} = \sum_{i=1}^{n} i\ell_i $.
Let $H_Q=\{ D| D \ni Q, \ \text{\rm deg} \ D = \text{N}\}$. Then
$H_Q$ is a hyperplane in $\bold P^N$. Then
$\Gamma_\pi \cap H_Q = \sum_{i=1}^n i\Gamma_{\pi, i}$, where
$\Gamma_{\pi, i}= \{ D \in \Gamma_{\pi}| D \ni iQ, \ D \not\ni (i+1)Q \}$.
Therefore,
$\Gamma_{\pi} \cap H_{Q_1} \cap \dotsb \cap H_{Q_{\text{L}} }$,
where $\text{L}=\sum \ell_i$,
has $n(\pi)$ points (set theoretically) and each point has multiplicity
$m(\pi)$. Therefore, $deg \ \Gamma_{\pi}=n(\pi)m(\pi)$.
\begin{cor} Assume that $L$ is a line in $\bold P^2$.
Fix a partition $\pi=[a_1, \dotsc , a_n].$ Let $\cal{S} (\pi)$ be
the (locally closed) set of all smooth curves $C$ of degree $d$ in $\bold P^2$
such that $C \cap \text{L}$ is of type $\pi$,
Then $deg \ \overline {\cal{S} (\pi)} = n(\pi)m(\pi)$.
\end{cor}
\section{Recursion method}
In this section, we want to describe the method of calculating the
degree of $\overline{\cal{V} (\delta, d)}$. You can also consult
the paper [R2] or [R3].
\subsection{Degeneration of $\bold P^2$}
Let $S$ be the blow up of $\Bbb{C} \times \bold P^2$ at a point $(0, p)$,
and let $\pi^\prime : S \rightarrow \Bbb{C} \times \bold P^2$ be the
blowdown map and $\pi = \rho_2 \circ \pi^\prime : S \rightarrow \Bbb{C}$
be the composition map. Then
\begin{equation} \pi^{-1} (t) =
\begin{cases}
\bold P^2& \text{if $t \not= 0$}\\
\bold P^2 \cup \tilde{\bold P^2}& \text{if $t=0$} \ ,
\end{cases}
\end{equation}
\noindent
where $\tilde{\bold P^2}$ is the blow up of $\bold P^2$ at $p$.
In the case $\pi^{-1} (0) = \bold P^2 \cup \tilde{\bold P^2}, \
\text{E} \buildrel \rm def \over =
\bold P^2 \cap
\tilde{\bold P^2} $ called the axis is a line in $\bold P^2$,
and an exceptional divisor
in $\tilde{\bold P^2}$. This gives the degeneration of $\bold P^2$ to
$\bold P^2 \cup \tilde{\bold P^2}$.
\subsection{Degeneration of curve in $\bold P^2$}
As $\bold P^2$ is degenerated to $\bold P^2 \cup \tilde{\bold P^2}$,
a curve in $\bold P^2$ also degenerates to
$C_1 \cup C_2$, where $C_1 \subset \bold P^2$ and $C_2 \subset \tilde{\bold P^2}$.
The point is that the degenerated curve $C_1$
has smaller degree, therefore we can apply
recursion and the other part
$C_2$ has simple configuration, so it's easy to
count its dimension, degree etc.
Clearly, $\text{N} (\delta,d)
\buildrel \rm def \over = \deg \ \overline{\cal{V} (\delta, d)} $
is the same as the number of plane curves
with $\delta$ nodes and
through
$(\binom{d+2}{2}-1-\delta)$ generally given points.
Let's put
$\binom{d+1}{2}-1-\delta$ points into
$\bold P^2$ and $(d+1)$ points into $\tilde{\bold P^2}$, i.e,
$$
\begin{array}{ccccc}
\bold P^2 & \sim\longrightarrow & \bold P^2 & \cup & \tilde{\bold P^2} \\
\cup & & \cup & & \cup \\
C & \sim\longrightarrow & C_1 & \cup & C_2 \\
\cup & & \cup & & \cup \\
\binom{d+2}{2}-1-\delta \ \hbox{\rm{pts}}
& \sim\longrightarrow & \binom{d+1}{2}-1-\delta \ \hbox{\rm{pts}} & & (d+1) \ \hbox{\rm{pts}} \ ,
\end{array}
$$
\noindent
and think of all possible kinds
of degeneration of $\delta$ nodes.
Then, $\text{\rm deg} \ C_1 = d-1$, and $C_2$ is a smooth rational
curve plus several rulings (possibly multiple) [R3].
After all, the number of all possible curves
in $\bold P^2 \cup \tilde{\bold P^2}$
satisfying given conditions is $ \text{N} (\delta,d)$.
\subsection{Cogenus 3 case}
(Table 1.)
We fix the degree and the number of nodes to explain more clearly.
The way to get a
general formula is exactly the same, but much more computation.
Let $d=5$ (degree 5 curves) and $\delta =3$ (3 nodes). There is
a technical reason to assume that $d > \delta$. It makes computation easy.
Then we have 11 points on $\bold P^2$ and 6 points on
$\tilde{\bold P^2}$. The formula of [V] gives $\text{N} (3,5)=7915$.
\noindent
\vspace{-3 mm}
\begin{enumerate}
\item[(1)] Case A. \par
This case is easy. It's just $\text{N} (3,4) =
deg \ \overline{\cal{V} (3,4)}$ = 675. Generally, the degree of this locus
is $\text{N}_{3, d-1}$
\item[(2)] Case B. \par
$C_1 \cap \text{E} = C_2 \cap \text{E} = [4]$. This means that $C_1$ and E
meet in 4 distinct points. Deg $\ C_1 = 4$, but the condition of $C_1$
has 11 points + 2 nodes = 13 conditions, so one condition is needed.
One of the four points on $C_1 \cap \text{E}$ gives this condition.
So the number of all curves through 12 points (given 11 points + one
axis point) and having two nodes is $\text{N} {(2,4)}$ (generally,
$\text{N} {(2, d-1)}$). And
the others on $\text{E} \cap C_1$ give three conditions on $C_2$. $C_2$
has one node, so has one ruling. So the number of ways of choosing a ruling
is 6 + 3 (generally $(d+1)+(d-2)$). So degree of Case B is
$\text{N} (2,d-1) \times (2d-1)$.
\item[(3)] Case C and Case D. \par
The method is similar as that of B.
\end{enumerate}
\bigskip
Table 1. (Calculation of N(3,5), and
General Case N(3,$d$))
\newline
\noindent
\begin{tabular}{|c|c|c|c|c|r|l|} \hline
Case & \ M\ & N($C_1$) & N($C_2$) &
\ \ $\pi$ on E & d=5 & General Case \\
\hline
A & 1 & 3 & 0 & [d-1] & 675 & $1 \times \text{N} {(3,d-1)} \times 1$ \\
B & 1 & 2 & 1 & [d-1] & 2025 & $1 \times \text{N} {(2,d-1)} \times (2d-1)$ \\
C & 1 & 1 & 2 & [d-1] & 756 & $1 \times \text{N} {(1,d-1)}
\times (2d^2-5d+3)$\\
D & 1 & 0 & 3 & [d-1] & 35 & $1 \times 1 \times (4d^3-24d^2+47d-30)/3$\\
\hline
E & 2 & 2 & 0 & [d-3,1] & 2020 & $2 \times \text{N} {([d-3,1],2,d-1)}
\times 1$ \\
\hline
F & 2 & 1 & 1 & [d-3,1] & 1316 & $2 \times (6d^3-42d^2+90d-56)
\times (2d-3)$ \\ \cline{6-7}
& & & & & 200 & $2 \times (3d^2-12d+10) \times 2(d-3)$ \\
\hline
G & 2 & 0 & 2 & [d-3,1] & 24 & $2 \times 1 \times (4d^2-24d+32)$
\\ \cline{6-7}
& & & & & 60 & $2 \times 2(d-4) \times (2d^2-9d+10)$ \\
\hline
H & 3 & 1 & 0 & [d-4,0,1] & 405 & $3 \times \text{N} {([d-4,0,1],1,d-1)}
\times 1$ \\
\hline
I & 3 & 0 & 1 & [d-4,0,1] & 54 & $3 \times 3(d-4) \times (2d-4)$
\\ \cline{6-7}
& & & & & 9 & $3 \times 1 \times 3(d-4)$ \\
\hline
J & 4 & 1 & 0 & [d-5,2] & 320 & $4 \times \text{N} {([d-5,2],1,d-1)}
\times 1$ \\
\hline
K & 4 & 0 & 1 & [d-5,2] & 0 & $4 \times 2(d-4)(d-5)
\times (2d-5)$ \\ \cline{6-7}
& & & & & 0 & $4 \times 2(d-4) \times 2(d-5)$ \\
\hline
L & 4 & 0 & 0 & [d-5,0,0,1] & 16 & $4 \times 4(d-4) \times 1$\\
\hline
M & 6 & 0 & 0 & [d-6,1,1] & 0 & $6 \times 6(d-4)(d-5) \times 1$\\
\hline
N & 8 & 0 & 0 & [d-7,3] & 0 & $8 \times 4(d-4)(d-5)(d-6) \times 1$\\
\hline
Sum & & & & & 7915 & See *7 \\
\hline
\end{tabular}
\newline
*1. M means multiplicity.
\newline
*2. N($C_1$) means the number of nodes of $C_1$.
\newline
*3. N($C_2$) means the number of nodes of $C_2$.
\newline
*4. $\text{N} {(\delta,d)}$ = $deg \ \overline{\cal{V} (\delta,d)}$.
\newline
*5. $\text{N} {([\pi],\delta,d)}$ = degree of the closure of
the set of all curves $C$ of degree d having $\delta $
\newline
nodes
and $C \cap \text{L}$ is {\it of type} $\pi$ to fixed line L.
\newline
*6. In General Case, M $\times$ A $\times$ B means that
M is the multiplicity, A is the degree
\newline
of the locus of
curves $C_1$ in $\bold P^2$ and B is the degree of the locus of curves
$C_2$ in $\tilde{\bold P^2}$.
\newline
*7. N(3,d) = $9/2d^6-27d^5+9/2d^4+423/2d^3-229d^2-829/2d+525$.
\par
\newpage
\begin{enumerate}
\item[(4)] Case E. \par
Since $C_2$ is nonsingular, each point on $C_1 \cap \text{E} =
\pi = [2,1]$ (generally, [$d-3,1$]) gives a condition on $C_2$,
therefore the degree of
curves $C_2$ is just one. $C_1$ is the curve with two nodes
and one tangency
to E. So $C_1$ has exactly 14 conditions (11 points + 2 nodes +
1 tangency).
Therefore, the degree of Case E is the same as the degree
of the locus of all curves with two nodes and one tangency condition
to fixed line. The method of calculating this degree
is similar to that for the Severi variety $\cal{V} (2,4)$, except when
we degenerate $\bold P^2$ to $\bold P^2 \cup \tilde{\bold P^2}$,
we make the tangency on $C_1$ go to $\tilde{\bold P^2}$, i.e,
$$
\begin{array}{ccccc}
\bold P^2 & \sim\rightarrow & \bold P^2 & \cup & \tilde{\bold P^2} \\
\cup & & \cup & & \cup \\
C_1 & \sim\rightarrow & C_{11} & \cup & C_{12} \\
\cup & & \cup & & \cup \\
11 \ \text{pts} & \sim\rightarrow & 7 \ \text{pts} & & 4 \text{pts} + 1
\ \text{tangency to} \ \text{E} \ .
\end{array}
$$
Table 2 is the table of Case E{}.
The interesting subcategory is Case H1. This case doesn't happen
during calculation of the degree of the Severi variety $\cal{V} (2,d-1)$.
This case occurs when the
curve $C_1$ degenerates to $C_{12}$ which contains a double ruling.
\end{enumerate}
\vspace{-4 mm}
Table 2. (Calculation of $\text{N}{([d-3,1],2,d-1)}$)
\newline
\noindent
\begin {tabular}{|c|c|c|c|c|r|l|} \hline
Case & \ M\ & N($C_1$) & N($C_2$) &
$\pi$ on E & d=5 & General Case \\
\hline
A1 & 1 & 2 & 0 & [d-2] & 126 & $1 \times \text{N} {(2,d-2)}
\times 1 \times 2(d-2)$ \\
B1 & 1 & 1 & 1 & [d-2] & 280 & $1 \times \text{N} {(1,d-2)}
\times (2d-4) \times 2(d-3)$ \\
C1 & 1 & 0 & 2 & [d-2] & 20 & $1 \times 1 \times (2d^2-11d+15)
\times 2(d-4)$ \\
\hline
D1 & 2 & 1 & 0 & [d-4,1] & 432 & $2 \times \text{N} {([d-4,1],1,d-2)}
\times 1 \times 2(d-2)$ \\
\hline
E1 & 2 & 0 & 1 & [d-4,1] & 64 & $2 \times 2(d-4)
\times (2d-6) \times 2(d-3)$ \\
\cline{6-7}
& & & & & 16 & $2 \times 2(d-4) \times 2(d-3)$ \\
\hline
F1 & 3 & 0 & 0 & [d-5,0,1] & 54 & $3 \times 3(d-4) \times 1 \times 2(d-2)$ \\
\hline
G1 & 4 & 0 & 0 & [d-6,2] & 0 & $4 \times 2(d-4)(d-5) \times 1
\times 2(d-2)$ \\
\hline
H1 & 1 & 0 & 2 & [d-4,2] & 10 & $2(d-1)+2(d-4)$ \\
\hline
Sum & & & & & 1010 & $(9d^5-90d^4+300d^3-327d^2-76d+190)$ \\
\hline
\end{tabular}
\begin{enumerate}
\item[(5)] Case F. \par
Since $C_1$ has 13 conditions (11 points + 1 node + 1 tangent condition),
one more condition is needed. The point on $\text{E} \cap C_1$
gives one more condition.
Since the divisor $\text{E} \cap C_1$ has one multiple point, we have to
divide two cases to fix one point of that divisor.\par
\begin{enumerate}
\item We fix an ordinary point. \par
In this case, the condition of $C_1$ has 14 conditions. As in Case E, we do
the whole thing again, we get the formula
$6(d-3)^3+8(d-3)^2+2(d-3)+2(d-4)(2d-5) =
(6d^3-42d^2+90d-56)$. The number of ways of choosing a ruling is
$(2d-3)$, for the $(d+1)$ points in
$\tilde{\bold P^2}$ plus the $(d-4)$ points on axis $C_1 \cap E$.
(We can't choose the fixed point on E
which gives the condition on $C_1$ and tangent point.)
So in this case, the degree
is $(6d^3-42d^2+90d-56) \times (2d-3)$.
\item We fix the tangent point. \par
In this case, $C_1$ has 14 conditions.
(11 points + 1 node + 1 tangent condition
at fixed point.) As in Case E, we do the whole thing again, we get
$3(d-3)^2+4(d-3)+(2d-5) = (3d^2-12+10)$.
Since the tangent point is fixed on $C_1$,
this point is not fixed on $C_2$. This gives the
degree of the locus of rational smooth
curves of $C_2$. This is two since this
is the discriminant of the partition [0,1].
And the number of ways of choosing a ruling is just $(d-3)$,
for the $(d-3)$ points on
$C_1 \cap$ E (as in previous subcase,
we can't choose a tangent point. Also, we can't choose given inside points
because there are no ordinary fixed points on $C_1 \cap E$).
So in this case,
the degree is $(3d^2-12d+10) \times 2(d-3)$.
\end{enumerate}
\item[(6)] Case G. \par
Since $C_1$ has 12 conditions (11 points + 1 tangency on E),
two conditions are needed. Two points on
$C_1 \cap E$ give these conditions. But,
as in Case F, the tangency condition gives us two ways to
fix two points.
\begin{enumerate}
\item We fix one ordinary point and one tangent point. \par
Since $C_1$ has full conditions, such a curve exists and is unique.
The counting of degree of the locus of curves $C_2$ is a little tricky. If
we choose both of the two rulings in $(d-4)$ points on
the axis (as in Case F,
we can't choose the fixed ordinary point and tangent point.), then
the degree of the locus of rational smooth curves of $C_2$ is 4
since this is the discriminant of
the partition [1,1], and if we choose one ruling in axis point and one
ruling in $(d+1)$ points in $\tilde{\bold P^2}$,
then the degree of locus of rational
smooth curves is two (discriminant of [0,1]).
So in this case, the degree is
$4 \times \binom{d-4}{2} + 2(d+1)(d-4)$.
\item We fix two ordinary points. \par
The degree of the locus of curves $C_1$ is
$2(d-4)$ (discriminant of $[d-5,1]$),
and the number of ways of choosing a ruling is
$\{ \binom{d+1}{2} +(d-5)(d+1) + \binom{d-5}{2} \}$. So in this
case, the degree is $2(d-4) \times (2d^2-9d+10)$.
\end{enumerate}
\item[(7)] Case I, J, and K. \par
The method to calculate these cases is similar as those of
Case E, F, and G.
\item[(8)] Case L. \par
$C_2$ has no node, so all points on $E \cap C_1$ give conditions
on $C_2$, so just one such curve exists.
The degree of the locus of curves $C_1$ is
just the discrimant of the divisor of $C_1 \cap {\text{E}} $.
By Corollary 2.3, this is $4(d-4)$.
\item[(9)] Case M and N. \ \
The method is similar as that of Case L.
\end{enumerate}
\subsection{Cogenus 4}
The way to calculate in the case of cogenus 4 is exactly the
same as that of cogenus 3.
There are just more cases, so the only thing to do
is to be careful not to miss any. Case F involves
calculating the degree of the variety of all curves
with three nodes and one tangency to fixed line.
This counting method is the same as the case E of cogenus 3.
Table 3 and 4 are the calculations of low degrees.
Below each Table, we give the general formula of the degree of these cases.
Compared with Table 1, Table 4 has new cases (Cases O1, P1, and Q1).
These Cases correspond
to the cases having double rulings in $\tilde{\bold P^2}$.
\par
Table 3. (Calculation of N(4,5), and
N(4,6))
\newline
\noindent
\begin{tabular}{|c|c|c|c|c|r|r|} \hline
\ Case\ & Multiplicity & Nodes of $C_1$ & Nodes of $C_2$ &
\ \ \ $\pi$ on E & \ $d=5$ \ & \ $d=6$ \ \\
\hline
A & 1 & 4 & 0 & [d-1] & 666 & 36975 \\
B & 1 & 3 & 1 & [d-1] & 6075 & 87065 \\
C & 1 & 2 & 2 & [d-1] & 6300 & 39690 \\
D & 1 & 1 & 3 & [d-1] & 945 & 4032 \\
E & 1 & 0 & 4 & [d-1] & 15 & 70 \\
\hline
F & 2 & 3 & 0 & [d-3,1] & 4728 & 99160 \\
G & 2 & 2 & 1 & [d-3,1] & 10440 & 90708 \\
H & 2 & 1 & 2 & [d-3,1] & 1920 & 12800 \\
I & 2 & 0 & 3 & [d-3,1] & 0 & 224 \\
\hline
J & 3 & 2 & 0 & [d-4,0,1] & 2520 & 19170 \\
K & 3 & 1 & 1 & [d-4,0,1] & 1350 & 7002 \\
L & 3 & 0 & 2 & [d-4,0,1] & 0 & 252 \\
\hline
M & 4 & 2 & 0 & [d-5,2] & 1696 & 30240 \\
N & 4 & 1 & 1 & [d-5,2] & 0 & 5824 \\
O & 4 & 0 & 2 & [d-5,2] & 0 & 0 \\
P & 4 & 1 & 0 & [d-5,0,0,1] & 320 & 1344 \\
Q & 4 & 0 & 1 & [d-5,0,0,1] & 0 & 128 \\
\hline
R & 5 & 0 & 0 & [d-6,0,0,0,1] & 0 & 25 \\
\hline
S & 6 & 1 & 0 & [d-6,1,1] & 0 & 0 \\
T & 6 & 0 & 1 & [d-6,1,1] & 0 & 0 \\
\hline
U & 8 & 1 & 0 & [d-7,3] & 0 & 0 \\
V & 8 & 0 & 1 & [d-7,3] & 0 & 0 \\
W & 8 & 0 & 0 & [d-7,1,0,1] & 0 & 0 \\
\hline
X & 9 & 0 & 0 & [d-7,0,2] & 0 & 0 \\
\hline
Y & 12 & 0 & 0 & [d-8,2,1] & 0 & 0 \\
\hline
Z & 16 & 0 & 0 & [d-9,4] & 0 & 0 \\
\hline
Sum & & & & & 36975 & 437517 \\
\hline
\end{tabular}
\par
The general foumula for $\text{d} \geq 5$ is :
$$
\text{N}(4,d) = 27/8d^8-27d^7+1809/4d^5-642d^4-2529d^3+37881/8d^2+
18057/4d-8865
$$
Table 4. (Calculation of $\text{N} {([2,1],3,4)}$, and
$\text{N} {([3,1],3,5)}$)
\newline
\noindent
\begin{tabular}{|c|c|c|c|c|r|r|} \hline
\ Case \ & Multiplicity & Nodes of $C_1$ & Nodes of $C_2$ &
\ \ $\pi$ \ on E & \ $d=4$ \ & \ $d=5$ \ \\
\hline
A1 & 1 & 3 & 0 & [d-2] & 90 & 5400 \\
B1 & 1 & 2 & 1 & [d-2] & 504 & 10800 \\
C1 & 1 & 1 & 2 & [d-2] & 240 & 2268 \\
D1 & 1 & 0 & 3 & [d-2] & 0 & 40 \\
\hline
E1 & 2 & 2 & 0 & [d-4,1] & 360 & 16160 \\
F1 & 2 & 1 & 1 & [d-4,1] & 672 & 7968 \\
G1 & 2 & 0 & 2 & [d-4,1] & 0 & 240 \\
\hline
H1 & 3 & 1 & 0 & [d-5,0,1] & 378 & 3240 \\
I1 & 3 & 0 & 1 & [d-5,0,1] & 0 & 324 \\
\hline
J1 & 4 & 1 & 0 & [d-6,2] & 0 & 2560 \\
K1 & 4 & 0 & 1 & [d-6,2] & 0 & 128 \\
L1 & 4 & 0 & 0 & [d-6,0,0,1] & 0 & 0 \\
\hline
M1 & 6 & 0 & 0 & [d-7,1,1] & 0 & 0 \\
\hline
N1 & 8 & 0 & 0 & [d-8,3] & 0 & 0 \\
\hline
O1 & 1 & 1 & 2 & [d-3,1] & 96 & 344 \\
P1 & 1 & 0 & 3 & [d-3,1] & 24 & 60 \\
Q1 & 2 & 0 & 2 & [d-5,2] & 0 & 48 \\
\hline
Sum & & & & & 2364 & 49580 \\
\hline
\end{tabular}
\newline
Here, $\text{N} {(\pi,3,d-1)}$ is the degree of the locus
of all plane curves $C$ of degree
\newline
$d-1$
having 3 nodes and $C \cap \text{L}$
is {\it of type $\pi$} to fixed line L.
The general formula for $\text{d} \geq 4$ is :
$$
\begin{array}{l}
\text{N} {([d-3,1],3,d-1)}
=9d^7-126d^6+603d^5-891d^4-1118d^3+3223d^2+416d-2416 \\
\text{N} ([d-5,2],2,d-1)
=9d^6-135d^5+750d^4-1785d^3+1249d^2+1188d-1116 \\
\text{N} ([d-4,0,1],2,d-1)
=27/2d^5-297/2d^4+549d^3-1341/2d^2-357/2d+495 \\
\text{N} ([d-6,1,1],1,d-1)
=18d^4-234d^3+1038d^2-1734d+720 \\
\text{N} ([d-5,0,0,1],1,d-1)
=12d^3-96d^2+220d-120 \\
\text{N} ([d-7,3],1,d-1)
=4d^5-76d^4-540d^3-1732d^2+2304d-720.
\end{array}
$$
\section{Degree of Severi polynomial}
\begin{defn}
A Severi polynomial is a polynomial $\text{N} (\pi,\delta,d)$
which is the degree of the locus of curves $C$ in
$\bold P^2$ with degree d having $\delta$ nodes and
$C \cap \text{L}$ is {\it of type $\pi$} to fixed line L.
\end{defn}
\begin{rem}
The fact that $\text{N} (\pi,\delta,d)$ is a polynomial in d is
obvious because of the recursion formula of Ran [R2].
\end{rem}
\begin{pro}
Fix $\delta$ and $\pi = [\ell_1,...,\ell_n]$. Then
$$
\text{\rm deg N}(\pi,\delta,d) = 2\delta + \sum_{i=2}^{n} \ell_i.
$$
\end{pro}
{\it Proof} \ \ \
We use induction on $d$. $\text{N} (d)=\text{N} (\pi,\delta,d)$
is the number of curves in
$\bold P^2$ with degree $d$, having $\delta$ nodes, type $\pi$
conditions to fixed line and through given $n=(\binom{d+2}{2} - 1
- \delta - (\sum_{i=1}^{n} (i-1)\ell_i))$ points.
As in the previous section, we degenerate $\bold P^2$ into $\bold P^2 \cup
\tilde{\bold P^2}$, and put $n_1=(\binom{d+1}{2} -1 - \delta)$ points into $\bold P^2$
and $n_2=((d+1)-(\sum_{i=1}^{n} (i-1)\ell_i))$ points into $\tilde{\bold P^2}$, and
degenerate the tangency conditions into $\tilde{\bold P^2}$, i.e,
$$
\begin{array}{ccccc}
\bold P^2 & \sim\rightarrow & \bold P^2 & \cup & \tilde{\bold P^2} \\
\cup & & \cup & & \cup \\
C & \sim\rightarrow & C_1 & \cup & C_2 \\
\cup & & \cup & & \cup \\
n \ \text{points} & \sim\rightarrow & n_1 \ \text{points}
& & n_2 \ \text{points} + \text{tangency conditions to L} \ .
\end{array}
$$
Then $\text{N} (\pi,\delta,d)$ is the sum of the Severi polynomials of all
limit components. So, if we prove that the degree of each polynomial
is less than or equal to $2\delta + \sum_{i=2}^{n} \ell_i$, we are done.
Each polynomial of a limit component is the product of
the polynomial of the upper part of that component (the set of the
locus $C_1$) and the polynomial of the lower part (the set of the
locus $C_2$). First, look at the degree of the polynomial
of the lower part. This degree is the sum of the number of rulings,
$\delta_2$, and the degree of the polynomial of the locus
of the smooth curves with a divisor
{\it of type $\pi$} to fixed line, that is $\sum_{i=2}^{n} \ell_i$.
Second, look at the degree of the polynomial of the upper part.
$C_1$ has $\delta_1$ nodes and $C_1 \cap \text{E}$ is
{\it of type $\pi' = [\ell_1',...,\ell_m']$}
satisfying the equation
\begin{equation}
\delta_1 + \sum_{i=1}^{m} (i-1)\ell_i'
= \delta - \delta_2
\end{equation}
Since $C_1$ has degree $d-1$, by the induction, the polynomial of the
upper part has degree at most $2\delta_1 + \sum_{i=2}^{m} \ell_i'$. So the
degree of the polynomial of each component is less than or equal to
$\delta_2 + \sum_{i=2}^{n} \ell_i + 2\delta_1 + \sum_{i=2}^{m} \ell_i'$.
From the equation (2), we get
\begin{equation}
\delta_2 + \sum_{i=2}^{n} \ell_i + 2\delta_1 + \sum_{i=2}^{m} \ell_i'
\leq
2\delta + \sum_{i=2}^{n} \ell_i.
\end{equation}
The equality holds just for
the component which is the locus of curves such that
$C_1$ has $\delta$ nodes and
$C_2$ is a smooth rational curve.
\begin{cor}
Let $a_{2\delta}^{\delta}\ ( a_{2(\delta -1)}^{(\delta -1)})$ be the
coefficient of the degree $2\delta (2(\delta-1), respectively)$ term of the
Severi polynomial N($\delta$,d) (N($\delta-1$,d), respectively).
This is the leading coefficient of each polynomial by
the Proposition 4.2. Then
$$
a_{2\delta}^{\delta} \times 2\delta
= a_{2(\delta -1)}^{\delta -1} \times 6, \ \
hence \ \ a_{2\delta}^{\delta} = {3^{\delta}\over{\delta !}}.
$$
\end{cor}
{\it Proof} \ \ \
In the proof of Proposition 4.2, equality holds for just
one limit component. The Severi polynomial of this component
is N($\delta, d-1$). Let's find the limit components of degree one less than
the degree of Severi polynomial. By equation (3), there
exist exactly two such components. One such component, say $A_1$, consists of
curves such
that $C_1$ has ($\delta -1$) nodes and $C_2$ has one ruling, and
another component, say $A_2$, consists of
curves such that $C_1$ has ($\delta -1$) nodes
, $C_1 \cap \text{E}$ is {\it of type} [$d-3,1$] and $C_2$ is a
smooth rational curve. In Table 1(Table 3),
$ A_1$ is the component of Case B (Case B, respectively)
and $A_2$ is the component of Case E (Case F).
The coefficient of the degree $(2\delta -1)$ term of the polynomial of the
component $A_1$ is $2a_{2(\delta -1)}^{\delta -1}$. The coefficient of
the degree $(2\delta -1)$ term of the polynomial of the component $A_2$ is
$4a_{2(\delta -1)}^{\delta -1}$. So the sum is
$6a_{2(\delta -1)}^{\delta -1}$.
Therefore we get the following recursion
formula
$$
\text{N} (\delta,d) =
\text{N} (\delta,d-1) + 6a_{2(\delta -1)}^{\delta -1}d^{2\delta -1} \\
+ \text{lower degree terms}.
$$
Integrating, we get $a_{2\delta}^{\delta} \times (2\delta)
= 6 \times a_{2(\delta -1)}^{\delta -1}$.
\begin{rem}
$\text{N} (7,d) = {243\over{560}} d^{14} + \text{\rm lower degree terms.}
$
\end{rem}
\begin{cor}
Let $a_{2\delta -1}^{\delta}$ be the coefficient of the degree
$(2\delta -1)$ term of the polynomial N($\delta$,d). Then
$$
a_{2\delta -1}^{\delta} = - (2\delta) \times a_{2\delta}^{\delta},
\ \ hence \ \ a_{2\delta -1}^{\delta} =
{-2\times 3^{\delta}\over {(\delta -1)!}}.
$$
\end{cor}
{\it Proof} \ \ \
The way to prove this corollary is the same as that of corollary 4.3.
Let's find the components with the polynomial of
degree two less than the degree of Severi polynomial.
By equation (3), there exist exactly three such components.
First component, $B_1$, consists of curves such
that $C_1$ has ($\delta -2$) nodes and $C_2$
has two rulings. Second component, $B_2$, consists of curves such
that $C_1$ has ($\delta -2$)
nodes, $C_1 \cap \text{E}$ is {\it of type} [$d-3,1$] and $C_2$ has
one ruling. Third one, $B_3$, consists of curves such
that $C_1$ has ($\delta -2$) nodes, $C_1 \cap \text{E}$ is
{\it of type} [$d-5,2$] and $C_2$ is smooth.
In Table 1(Table 3), $B_1$ is the component of Case C (Case C, respectively)
and $B_2$ is the component of Case F (Case G, respectively) and $B_3$ is the
component of Case J (Case M, respectively).
Calculation of the coefficients of the degree $(2\delta -2)$ term of those
polynomials give $2a_{2(\delta -2)}^{\delta -2},
8a_{2(\delta -2)}^{\delta -2}, 8a_{2(\delta -2)}^{\delta -2}$
respectively. So the sum is $18a_{2(\delta -2)}^{\delta -2}$. The polynomial
of the limit components $A_1$ and $A_2$ also
have a term of this degree.
The coefficient of this degree term of polynomial of $A_1$ is $
-a_{2(\delta -1)}^{\delta -1} + 2(-2(\delta -1)
a_{2(\delta -1)}^{\delta -1} + a_{2\delta -3}^{\delta -1})$.
The coefficient of this degree term of polynomial of $A_2$ is
$-8a_{2(\delta -1)}^{\delta -1} +4(-4(\delta -1)
a_{2(\delta-1)}^{\delta -1} + a_{2\delta -3}^{\delta -1}) +
8a_{2(\delta-2)}^{\delta -2}
+16a_{2(\delta -2)}^{\delta -2}$.
Using $a_{2\delta -3}^{\delta -1} = -2(\delta -1)
a_{2\delta -2}^{\delta -1}$ (by induction on $\delta$)
and $a_{2(\delta -1)}^{\delta -1} \times 2(\delta -1)
= 6a_{2(\delta -2)}^{\delta -2}$
(by Corollary 4.3), we get that the coefficient of this degree term is
$-9a_{2(\delta-1)}^{\delta -1}(2\delta -1)$.
So, we get the following formula,
$$
\begin{array}{ll}
\text{N} (\delta,d) =&\text{N} (\delta,d-1)
+6a_{2(\delta -1)}^{\delta}d^{2\delta -1}
-9a_{2(\delta -1)}^{\delta -1}(\delta -2)d^{2\delta -2} \\
&+ \ \text{lower degree terms}
\end{array}
$$
Integrating, we get
$$
\text{N} (\delta,d) = a_{2\delta}^{\delta}d^{2\delta}
-(2\delta)a_{2\delta}^{\delta}d^{2\delta -1} + \text{lower degree terms.}
$$
\begin{rem}
$\text{N} (7,d) \ = \
{243\over{560}}d^{14} - {243\over{40}}d^{13} + \text{\rm lower degree terms.}
$
\end{rem}
|
1996-01-08T06:20:23 | 9601 | alg-geom/9601004 | en | https://arxiv.org/abs/alg-geom/9601004 | [
"alg-geom",
"math.AG"
] | alg-geom/9601004 | Tricia Pacelli | Patricia L. Pacelli | Uniform boundedness for rational points | Latex2e in latex 2.09 compatibility mode | null | null | null | null | We extend an earlier result by Dan Abramovich, showing that a conjecture of
S. Lang's implies the existence of a uniform bound on the number of
$K$-rational points over all smooth curves of genus $g$ defined over $K$, where
$K$ is any number field of fixed degree $d$, and $g$ is an integer greater than
1. The bound depends only on the genus $g$ and the degree of the number field
$K$.
| [
{
"version": "v1",
"created": "Fri, 5 Jan 1996 16:34:03 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pacelli",
"Patricia L.",
""
]
] | alg-geom | \section{Introduction}\label{intro}
According to a famous conjecture of S. Lang's, if $K$ is a number
field, then the set of $K$-rational points of any variety of general
type defined over $K$ is not dense in the Zariski topology. In a
recent paper, entitled {\it Uniformity of rational points}
(\cite{chm}), L. Caporaso, J. Harris, and B. Mazur show that this
conjecture implies the existence of a uniform bound on the number of
$K$-rational points over all smooth curves of genus $g$ defined over
$K$, for some fixed $g\geq 2$. Their bound depends on the genus $g$ and
on the number field $K$.
D. Abramovich has proved an extension of this result. In his paper,
\cite{a}, he proves that,
assuming Lang's conjecture, the bound $B(K,g)$ of \cite{chm} remains
bounded as $K$ varies over all quadratic number fields, or as $K$
varies over all quadratic extensions of a fixed number field.
It is this result of Abramovich's that we shall generalize in this
paper. We will prove that given a number field $K$, Lang's conjecture
implies the existence of a uniform bound on the number of $L$-rational
points over all smooth curves of a fixed genus $g>1$ defined over $L$,
as $L$ varies over all extensions of $K$ of degree $d$ for any
positive integer $d$. This bound will depend on $K$, $d$, and $g$,
but is independent of the actual number field $L$.
\begin{theorem}\label{mainth} Assume that Lang's conjecture regarding
varieties of general type is true. Let $g\geq 2$ and $d\geq 1$ be
integers, and let $K$ be a number field. Then there exists an integer
$B_K(d,g)$, which, for a given $K$ depends only $d$ and $g$, such that for any
extension $L$ of $K$ of degree $d$, and any curve $C$ of genus $g$
defined over $L$, it follows that $$\#C(L)\leq B_K(d,g).$$
\end{theorem}
By letting $K={\bf Q}$ we have the following:
\begin{corollary}\label{cor} Assume Lang's
conjecture is true. Let $g\geq 2$ and
$d\geq 1$ be integers.
Then there exists a bound $B(d,g)$, depending only on $d$ and $g$,
such that for any number field $L$ of degree $d$, and for any curve $C$
of genus $g$ defined over $L$, it follows
that
$$\#C(L)\leq B(d,g).$$
\end{corollary}
Since any extension $L$ of $K$ of degree $d$ is a number field of
some fixed degree $d'$, Theorem \ref{mainth} and Corollary \ref{cor}
are equivalent. We
state Theorem \ref{mainth} separately, however, as it might be interesting
to study the dependence of the bound on $K$ in later work, perhaps
to see what happens assuming Lang's so-called strong conjecture.
\subsection{Acknowledgements} I am extremely grateful
to Dan Abramovich for introducing me to this interesting problem. His
guidance and assistance have been invaluable. I would also like to
thank Emma Previato and Lucia Caporaso for listening to presentations
of this material, and Joseph Silverman for giving me advice and
comments. Finally, I owe a tremendous amount of thanks to Glenn
Stevens, for both the mathematics he has taught and the encouragement
he has provided.
\section{Definitions, Notation, and Ideas}
Lang's conjecture concerns varieties of general type. Recall the
definitions:
\begin{definition}\label{def1} A line bundle $L$ on a variety $Y$ is said to be
{\em big} if for high values of $k$, $H^0(Y,L)$ has enough sections to induce
a birational map to projective space.
\end{definition}
\begin{definition} A smooth projective variety $Y$ is of {\em general type} if
its dualizing sheaf $\omega_Y$ is big. An arbitrary
projective variety is of general type if a desingularization of it is.
\end{definition}
The main idea we will be working with is that given a family of curves
$X\to B$, we can study the symmetric $d$-th power of the
$n$-th fibered product (for some sufficiently large $n$),
$$Sym^d(X^n_B)={(X_B^n)}^d/S_d.$$
For our purposes we will take the family $X\to B$ that we
work with to be a {\it tautological family}.
This is a family
$X\to B$ of stable curves along with a finite surjective map
$\phi\colon B\to \overline{M_g}$, where $\overline{M_g}$ is
the
moduli space of smooth stable curves of genus $g$. The map $\phi$
is assumed to have the property that
$\phi(b) = [X_b]$ for all $b\in B$. See \cite{chm}, \S 5.1 for a proof that
such a family exists. The reason we work with a tautological
family is quite simple; our theorem asserts that there is a bound on the
number of rational points on any smooth curve of genus $g$ - hence we want a
family in which every such curve appears as a fiber, possibly after a
field extension of bounded degree.
Assume $d\geq1$ and $g\geq2$ to be fixed integers throughout
the remainder of the paper, where $d$ represents the degree of an extension
of number fields $K\subseteq L$, and $g$ represents the genus of the curves we
will be looking at. Following the notation in \cite{a}, write
$$Y_n = Sym^d(X_B^n).$$
Let $L$ be an extension of $K$ of degree $d$, and let $\sigma_1$,
\ldots,$\sigma_n$ be the $d$ embeddings of $L$ in $\overline{K}$
fixing $K$. Then
if $b\in B(L)$, and $(P_1, \ldots, P_n) \in X_b(L)$, we obtain a
$K$-rational point $y_{(P_1, \ldots, P_n)}$ of $Y_n$:
$$
y_{(P_1,\ldots, P_n)} = \{(P_1, \ldots, P_n)^{\sigma_1}, \ldots,
(P_1, \ldots, P_n)^{\sigma_d}\}
$$
\begin{definition} For a variety $V$ defined over $K$ and a field extension
$K\subset L$, let $V(L,K)$ be the set of those points lying
on the variety $V$ which are defined over $L$ but are not
defined over $K$ nor any other intermediate field between
$K$ and $L$.
\end{definition}
\begin{definition} Let $m\geq n$. Again following \cite{a}, we call a point
$y_{(P_1,\ldots,P_n)}\in Y_n(K)$
{\it m-prolongable}
if there exists a number field $L$, $[L\colon K] = d$, and a
{\it prolongation} $y_{(P_1,\ldots,P_m)}\in Y_m(K)$ such that
$P_i\in X(L,K)$ for all $1\leq i\leq m$, and if
$1\leq i\not= j\leq m$ then $P_i\not= P_j.$ In other words,
$y_{(P_1,\ldots,P_n)}\in Y_n$ is m-prolongable if the n points
$P_1,\ldots,P_n$ can be extended to m distinct
points $P_1,\ldots P_m \in X(L,K)$, producing a point
$y_{(P_1,\ldots,P_m)}\in Y_m.$
\end{definition}
We will call $E_n^{(m)}$ the set of $m$-prolongable points in $Y_n$, and
will denote by $F_n^{(m)}$ the Zariski closure $\overline{E_n^{(m)}}$.
\begin{lemma} For all $n\geq 1$, there exists an integer
$m(n)$ such that for all positive integers $k$,
$$F_n^{(m(n)+k)}=F_n^{(m(n))}.$$
\end{lemma}
{\bf Proof:} Since $F_n^{(m+1)}\subseteq F_n^{(m)}$, the $F_n^{(m)}$'s
form a decreasing sequence of closed sets in the noetherian space $Y_n$,
which must eventually stop. So for all $n$ there is an integer $m(n)$
such that $F_n^{(m(n)+k)}=F_n^{(m(n))}$ for all positive integers $k$.
Q.E.D.
To ease notation we will write $F_n$ for $F_n^{(m(n))}$.
Our goal is to show that each $F_n$ is empty. For each $n$, $F_n$ is
the closure of those $n$-tuples of distinct points defined over $L$
which can be extended to arbitrarily long $m$-tuples of distinct
points over the same field. If we show that each $F_n$ is empty, this
shows that there must be a bound of the desired type.
We want to look at some of the properties of the $F_n$'s, and study the
maps between them. The ideas in the remainder of this section are
generalizations of those
found in \cite{a}.
\begin{lemma} Suppose $n'>n$ and $I\subseteq \{1, 2, \ldots, n'\}$ is an
$n$-tuple. Then the projection $\pi_I :F_{n'}\to F_n$ is surjective.
\end{lemma}
{\bf Proof:} This is clearly true for the $E_n^{(m)}$'s by definition.
Thus the same holds for the $F_n$'s. Q.E.D.
We have a natural finite map $$\pi_{n,k} :F_{n+k}\to {(F_{n+1})}^k_{F_n}.$$
To see this, look at the case
where $k=2$. If we consider an element $y\in F_{n+2}$, then $y$ can be
written as:
$$y=\{ (P_{1,1}, \ldots, P_{n,1}, P_{n+1,1}, P_{n+2,1}), \ldots,
(P_{1,d}, \ldots, P_{n,d}, P_{n+1,d}, P_{n+2,d})\}$$
The map $\pi_{n,2}$ is defined by sending $y$ to the following two elements of
$F_{n+1}$:
$$\{(P_{1,1}, \ldots, P_{n,1}, P_{n+1,1}),\ldots,
(P_{1,d}, \ldots, P_{n,d}, P_{n+1,d})\}$$ and
$$\{(P_{1,1}, \ldots, P_{n,1}, P_{n+2,1}), \ldots,
(P_{1,d}, \ldots, P_{n,d}, P_{n+2,d})\}$$ These two elements together
form an element of ${(F_{n+1})}^2_{F_n}$, thus defining $\pi_{n,2}(y)$.
Similiarly, an induction argument shows
that $F_{n+k}$ maps finitely to ${(F_{n+1})}^k_{F_n}$.
Notice that by definition the $E_n$'s and the $F_n$'s are not contained in
the {\it big diagonal} in $Y_n$; by the big diagonal, we mean the set
of $n$-tuples for which at least $2$ entries agree. Consider the following
lemma from \cite{a}:
\begin{lemma}[see \cite{a}, Lemma 1]
Let $D\to Z$ be a generically finite morphism, and let
$\Delta_n$ be the big diagonal in $D_Z^n$. Then there exists an integer
$n$ for which the $n$-th fiber product of $D$ over $Z$,
$D_Z^n\backslash\Delta_n\to Z$, is not dominant.
\end{lemma}
If we let $D=F_{n+1}$, $Z=F_n$, then this lemma shows that if $y\in
F_n$ then the dimension of the fiber above $y$ in $F_{n+1}$ is at
least $1$. This means that any component of $F_n$ has a component of
$F_{n+1}$ above it of positive dimension. Moreover, the lemma shows
that if $y$ is in $F_n$ and if $k$ is a positive integer, then the
dimension of the fiber above $y$ in $F_{n+k}$ is at least $k$.
We want to perform an induction argument on the relative dimension of
$F_{n+1}$ over $F_n$ to show that $F_n$ is empty for all $n$. We know
that this dimension can't be greater that $d$, as this is the relative
dimension of $Y_{n+1}$ over $Y_n$, and we also know that it is at least $1$,
by the lemma above. What we will do is show that, for any $l$, if the relative
dimension of $F_{n+1}$ over $F_n$ is at least $l$ then it must be at
least $l+1$. Eventually we will get to $l=d$ where the argument must
stop, and we will have to conclude that each $F_n$ is empty.
Assume that for all $n$ and for all $y\in F_n$, the dimension of the
fiber above $y$ in $F_{n+1}$ is greater than or equal to $l$. Suppose
there exists an element $y\in F_n$ with the dimension of the fiber
above $y$ in $F_{n+1}$ exactly $l$. By the semicontinuity of the
fiber dimension of projective maps, there is an irreducible component
of $F_n$, call it $M_n$, with the properties that $M_n$ contains $y$,
and the general fibers in $F_{n+1}$ over $M_n$ have dimension equal to
$l$. It also follows by induction that for any integer $k$ the
dimensions of the
fibers in $F_{n+k}$ above $M_n$ have dimension equal to $kl$.
Consider the following diagram where, as you may recall, the top map
is finite and birational.
$$
\begin{array}{ccc}
F_{n+k} & \longrightarrow & {(F_{n+1})}^k_{F_n}\\
& & \\
& \searrow & \downarrow \\
& & \\
& & F_n \\
\end{array}
$$
Because the map $F_{n+k}\to {(F_{n+1})}^k_{F_n}$ is finite, there
exists at least one irreducible component $H_k$ of $F_{n+k}$
which dominates a component of ${(F_{n+1})}^k_{F_n}$; this
component of ${(F_{n+1})}^k_{F_n}$ dominates $M_n$ and has maximal relative
dimension $kl$ over $F_n$.
Later in this paper we will prove the following important proposition:
\begin{proposition}\label{prop} For large values of $k$ and $n$ every
component of the fiber product ${(F_{n+1})}^k_{F_n}$ of maximal
relative dimension is a variety of general type.
\end{proposition}
Therefore, for large $k$, $H_k$ dominates a variety of general type,
and because the map from $F_{n+k}$ to ${(F_{n+1})}^k_{F_n}$ is
birational, we can conclude that for large enough $k$ and $n$,
$F_{n+k}$ is also a variety of general type. Remember, however, that
by definition the set of rational points in $F_{n+k}$ is dense. Thus
Lang's conjecture combined with the work above implies a
contradiction. Thus we have discomvered that for all $n$ and for all
$y\in F_n$, the dimension of the fiber above $y$ in $F_{n+1}$ must be
at least $l+1$.
Continuing by induction on this relative dimension, we will be forced
to conclude that if Lang's conjecture is true, we have a contradiction
unless $F_{n+k}$ is empty for large values of $k$ and $n$. Since we
have surjective projections from $F_{n+1}\to F_n$ for all $n$,
however, this implies that all of the $F_n$'s must be empty, hence
proving Theorem \ref{mainth}.
The remainder of the paper will be devoted to proving Proposition
\ref{prop}.
\section{Background Lemmas and Definitions}
In definition \ref{def1}, we defined a big line bundle; now we provide an
extension of that definition which shall be quite useful.
\begin{definition} If $L$ is a line bundle on a variety $Z$ and $\cal J$ is
an ideal sheaf on $Z$, then we define $L\otimes \cal J$ to be { \em big} if
for high values of $k$, $H^0(L^{\otimes k}\otimes {\cal J}^k)$ induces
a birational map to projective space $P^N$ for some $N$.
\end{definition}
\begin{lemma}\label{curves.1} Assume $Z$ is a projective irreducible variety of
dim $l>0$, and that $$Z\subseteq C_1\times \cdots \times C_d$$ where the
$C_i$'s are irreducible projective curves. Suppose further that for some
$i\in \{1,\ldots,d\}$ the projection map $\pi_i :Z\to C_i$ is surjective.
Then there exists a set $J\subseteq \{1, \ldots, d\}$ such that $i\in J$,
$\# J=l$, and the projection map $$\pi_J :Z\to \prod_{j\in J}C_j$$
is surjective.
\end{lemma}
{\bf Proof:} We proceed with induction on $d$. If $d=1$, we simply
have $Z\subseteq C_1$; hence it must be that both $i=1$ and $l=1$. We
use the fact that the inclusion map from $Z$ to $C_1$ must be either
constant or surjectve, but as $Z$ has positive dimension, it can't be
constant. Therefore the lemma holds, with $J=\{1\}$.
Assume the result true for $d-1$ curves, and suppose that we have
$Z\subseteq C_1\times\cdots\times C_d$, with $Z$ surjecting onto $C_i$
for some $1\leq i\leq d$. Choose $I\subseteq\{1, \ldots, d\}$
to be a subset of cardinality $d-1$ with the property that $i\in I$,
and let $\pi_I$ be the projection map $$\pi_I : Z\to\prod_{i\in I}C_i.$$
Define $Z_I=\pi_I (Z)$. We know that $Z_I$ is
an irreducible projective variety, with dimension either $l$ or
$l-1$.
If $Z_I$ has dimension $l$, the induction hypothesis produces a
set $J\subseteq I$ such that $i\in J$,
$\# J=l$, and the projection map $$\pi_J :Z_I\to \prod_{j\in J}C_j$$
is surjective. Clearly $Z$
surjects to $Z_I$, so by composition we obtain a surjection
$$Z\to \prod_{j\in J}C_j.$$
Now suppose that $Z_I$ has dimension $l-1$. The induction hypothesis
gives a set $J\subseteq I$ such that $i\in J$, $\# J=l-1$, and the
projection map $$\pi_J :Z_I\to \prod_{j\in J}C_j$$ is
surjective. Let $k\in \{1, \ldots, d\}$ be the one element which is not
in the set $I$.
Consider $Z_I\times C_k$. A simple dimension argument
shows that this is equal to $Z$. Therefore we have a surjection
$$Z=Z_I\times C_k\to \prod_{j\in J}C_j\times C_k.$$
Therefore $Z$ surjects onto a product of $l$ curves, as desired.
Q.E.D.
\begin{lemma}\label{curves} Lemma \ref{curves.1} above still holds
if we replace
$Z\subseteq C_1\times\cdots\times C_d$ with a fiber product of
families of curves
$Z\subseteq C_1\times_B\cdots\times_B C_d$ where $B$ is a
projective variety, the $C_i$ form a family of curves over $B$, and that
for some $i$ the projection map from $Z$ the family $C_i$ is surjective.
\end{lemma}
{\bf Proof}: Let $\eta\in B$ be a generic point. Because $Z$ surjects to
each family $C_i$, it follows that each generic fiber of $Z$ surjects to
generic fibers of curves. In other words, we may apply lemma
\ref{curves.1} to the fiber $Z_{\eta}$. We obtain a set $J$ of size $l$
with $i\in J$ such that we have a surjective and generically finte
projection map
$$
\pi_J :Z_{\eta}\to \prod_{j\in J} C_{i,\eta}
$$
where the product above is a fiber product over $B$. Because generic points
and fibers are dense, we know that the projection map
$$
\pi_J : Z\to \prod_{j\in J} C_i
$$
is generically finite (where the above product is once again a fiber
product over $B$). The properness of the projection map implies the
surjectivity of $\pi_J$. Q.E.D.
\begin{lemma}\label{surject} Suppose we have two surjective maps
$g_i:Z_i\to B$ for $i=1,2$. Then the two natural maps
$f_i:Z_1 \times_B Z_2 \to Z_i$, $i=1,2$, are also surjective.
\end{lemma}
{\bf Proof:} To show that $f_1$ is surjective, choose an element
$z_1\in Z_1$. Let $b=g_1(z_1)$. Then there exists a $z_2\in Z_2$
such that $g_2(z_2)=b$, as $g_2$ is surjective. It follows that
$(z_1,z_2)\in Z_1\times_B Z_2$ and $f_1((z_1,z_2))=z_1$. A similiar
argument shows that the map $f_2$ is also surjective. Q.E.D.
\begin{lemma}\label{product} Let $Y\subseteq Z_1\times Z_2$
be a variety, and let
$f_i: Y\to Z_i$ be surjective projection maps for $i=1,2.$
Suppose $L_1$ and $L_2$ are big line bundles on $Z_1$ and
$Z_2$ respectively. Then the line bundle
$$
f_1^*L_1\otimes f_2^*L_2
$$
is big. Further, if ${\cal J}_1$ and ${\cal J}_2$ are
ideal sheaves on $Z_1$ and $Z_2$, such that each $L_i\otimes {\cal J}_i$
is big on $Z_i$, then it follows that
$$
f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2))
$$
is big on $Y$.
Finally, the above still holds if we replace $Y\subseteq Z_1\times Z_2$
by a variety $Y$ which maps generically finitely to $Z_1\times Z_2$,
$$
h:Y\to h(Y)\subseteq Z_1\times Z_2
$$
provided that the projections from $h(Y)$ to $Z_i$ are still surjective.
\end{lemma}
{\bf Proof:} The first statement is a consequence of the second; hence
we just prove the second.
By assumption, since each $L_i\otimes {\cal J}_i$ is big on $Z_i$,
there exists open sets $U_i\subseteq Z_i$ such that for large $k$, global
sections of $H^0(Z_i, L_i^k\otimes {\cal J}^k)$ separate points in
$U_i$. Let $U=f_1^{-1}(U_1)\cap f_2^{-1}(U_2)$, and let
$P=(P_1,P_2), Q=(Q_1,Q_2)\in U$. We claim that we can produce
sections of
$$
H^0(Z_1\times Z_2, f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2)))
$$
for large $k$ which separate $P$ and $Q$. In other words,
there is a section vanishing at $P$ and not at $Q$, and vice versa.
This will show that there are enough sections to induce a birational
map to projective space, hence this will suffice to show that
$$
f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2))
$$
is big.
We have the following map:
$$
f_1^*\otimes f_2^* :
H^0(L_1^{\otimes k}\otimes {{\cal J}_1}^k)\otimes
H^0(L_2^{\otimes k}\otimes {{\cal J}_2}^k)\to
H^0(f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2)))
$$
Therefore, sections of $H^0(Z_1\times Z_2,f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2)))$ have the form
$$
f_1^*(s_i)\otimes f_2^*(r_j)
$$
where $\{s_i\}$ and $\{r_j\}$ form bases for
$H^0(Z_1,L_1^{\otimes k}\otimes {\cal J}_1^k)$ and
$H^0(Z_2,L_2^{\otimes k}\otimes {\cal J}_2^k)$, respectively.
Because $P_1$ and $Q_1$ are elements of $U_1$, it follows that
there exists a section $s\in H^0(Z_1,L_1^{\otimes k}\otimes {\cal J}_1^k)$
such that $s(P_1)=0$ but $s(Q_1)\neq 0$. Similiarly, there exists
a section $r\in H^0(Z_2,L_2^{\otimes k}\otimes {\cal J}^k_2)$
such that $r(P_2)=0$, but $r(Q_2)\neq 0$. Consider the section
$f_1^*s\times f_2^*r$. It follows that
$$
f_1^*s\times f_2^*r(P)=(s(P_1),r(P_2))=0
$$ and
$$
f_1^*s\times f_2^*r(Q)=(s(Q_1),r(Q_2))\neq 0.
$$
Similiarly, we can find a section of
$H^0(Z_1\times Z_2, f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2)))$,
which vanishes on $Q$ but not on $P$. Therefore, for large $k$,
sections of
$H^0(Z_1\times Z_2, f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2)))$
generically separate points. Hence, we
have that
$$
f_1^*(L_1)\otimes f_2^*(L_2)\otimes
(f_1^{-1}({\cal J}_1)\cdot f_2^{-1}({\cal J}_2))
$$
is big.
We now prove the last statement of the lemma. Since
$h(Y)$ surjects both $Z_1$ and $Z_2$, the work above shows
that $f_1^*(L_1)\otimes f_2^*(L_2) \otimes (f_1^{-1}({\cal J}_1)\cdot
f_2^{-1}({\cal J}_2))$ is big on $h(Y)$. Now
$h:Y\to h(Y)$ is a generically finite surjective morphism; hence the
pullback of a big sheaf
on $h(Y)$ will be big on $Y$. Q.E.D.
\section{The Proof}
Recall that for a given $n$, $F_n$ is contained in $Y_n={Sym}^d(X_B^n)$.
Let $X_n={(X_B^n)}^d$. Then we have a map $\sigma : X_n\to Y_n$, namely the
quotient map given by action of $S_d$. Now we can look at the inverse image
of $F_n$ under $\sigma$, which is contained in $X_n$. Call it $G_n$.
These varieties $G_n$ will be central to our proof. We can think of the
$G_n$'s as forming a tower: we have projection maps
$\pi : G_{n+1} \to G_n$ such that
$$
\pi (G_{n+1})=G_n.
$$
Fix $n$.
Let $l$ be the relative dimension of $G_{n+1}$ over $G_n$.
Let $\phi$ be the projection of $G_n$ down to $B^d$, and for
$i=1, \ldots, d$, denote by $\pi_i$ the d projections from $B^d$ to $B$.
Notice that because the image of $G_n$ in ${(X_B^{n-1})}^d$ is $G_{n-1}$,
it follows that no matter which $n$ we start with the image
$\phi(G_n)$ in $B^d$ is the same.
Define
$B_i\subseteq B$ to be the image of an irreducible component of
$G_n$ in $B$ under the map
$\pi_i\circ \phi$. We shall see later that it is enough to restrict
our attention to one
component of $G_n$, hence, we do so now.
Choose $${\cal B}\subseteq B_1\times B_2\times
\cdots \times B_d$$ to be a component of $\phi(G_n)$
such that ${\cal B}$ surjects to each component
$B_i$. Let $\widetilde{\cal B}$ be a desingularization of
$\cal B$. By Hironaka's Theorem we know we can choose this desingularization
such that the discriminant locus is a divisor with normal crossings.
We have the following diagram:
$$\widetilde{\cal B}\to {\cal B}\subseteq B_1\times\ldots\times B_d
\subseteq B^d\to B$$
Notice that we have $d$ choices for the last map in the above diagram,
namely the $d$ projections $\pi_i$. Let $g_i : \widetilde{\cal B}\to B$
be the compostion of the above where $\pi_i$ is chosen as the last mapping.
Define families of curves
$$
C_i=g_i^* (X\to B)=X\times_B \widetilde{\cal B}.
$$
$$
\begin{array}{ccc}
C_i & \longrightarrow & X \\
& & \\
\downarrow & & \downarrow \\
& & \\
\widetilde{\cal B} & \stackrel{g_i}{\longrightarrow} & B \\
\end{array}
$$
We now pull back the varieties $G_n$ to $\widetilde{\cal B}$;
to simplify matters, we shall keep the same notation $G_n$.
We then also need to pull back the familes $C_i$ to $G_n$. Again, it
is easiest to abuse notation, and still refer to them as $C_i$. In
our new situation we have the following containment:
$$G_{n+1}\subseteq C_1\times_{G_n} \cdots\times_{G_n} C_d.$$
Our goal is to work with these varieties $G_n$ to show that
for large $n$, $F_n$ is of general type, which will lead to our
contradiction. Our main difficulty will be in avoiding the singularities
of $G_n$. To do this, we will eventually be working with products of
stable curves over the smooth base $\widetilde{{\cal B}}$. These
curves will have canonical singularities, which are easier to control.
We would like to have $G_{n+1}$ surject onto each of these families
of curves $C_i$. While this is not always so, we will show that
we may reduce to this case.
For $i=1, \ldots d$, define projection maps $p_i :G_{n+1}\to C_i$. To
illustrate the situation, assume for a moment that $G_{n+1}$ is
irreducible. Then for each $i$, $p_i(G_{n+1})$ is an irreducible
subvariety of $C_i$. Suppose that for one index $i$, the map is not
surjective; without loss of generality assume that $p_1(G_{n+1})=P$
where $P\subset C_i$ and has degree $k$ over $G_n$. Now we go up $k$
steps and use the fact that
$$G_{n+k+1}\subseteq {(G_{n+1})}^{k+1}_{G_n}.$$ In other words,
$$G_{n+k+1}\subseteq P^{k+1}_{G_n}\times_{G_n}C_2^{k+1}\times_{G_n}
\cdots\times_{G_n} C_d^{k+1}.$$
Since $P$ consists of $k$ points over $G_n$, the pigeon-hole principle says
that two of the coordinates of $P^{k+1}_{G_n}$ must agree, contradicting the
fact that $G_{n+k}$ is not contained in the big diagonal.
Thus if $G_{n+1}$ is irreducible, we see that it is contained inside a product
of $d$ families of curves fibered over $G_n$ and that it surjects onto each
family.
Now suppose that $G_{n+1}$ consists of $m$ irreducible components:
$$
G_{n+1}=V_1\cup V_2\cup\cdots\cup V_m\subseteq C_1^n\times_{G_n} \ldots
\times_{G_n} C_d^n.
$$
Define projection maps $p_{i,j}:V_i\to C_j$ for $1\leq i\leq m$ and
$1\leq j \leq d$. We want to show that at least one $V_i$ surjects onto all
$d$ curves. So suppose for contradiction that
$$
p_{1,j(1)}(V_1)=S_1, \ldots, p_{m,j(m)}(V_m)=S_m
$$
where each $S_i\subset C_{j(i)}$ maps generically finitely to $G_n$ with
degree $r_i$.
Therefore, we know
$$
G_{n+1}\subseteq (p_{1,1}(V_1)\times_{G_n}\cdots\times_{G_n}p_{1,d}(V_1))
\cup\cdots\cup
(p_{m,1}(V_m)\times_{G_n}\cdots\times_{G_n}p_{m,d}(V_m))
$$
and
$$
G_{n+k}\subseteq {(G_{n+1})}^k_{G_n}
$$
So $G_{n+k}$ is contained in the above union fibered $k$ times over $G_n$.
In taking elements of
$$(p_{1,1}(V_1)\times_{G_n}\cdots\times_{G_n}p_{1,d}(V_1))$$
we have only $r_1$ distinct choices for the $j(1)$-st component.
Continuing in this spirit, in taking an element of
$$(p_{m,1}(V_m)\times_{G_n}\cdots\times_{G_n}p_{m,d}(V_m))$$
we have only $r_m$ distinct choices for the $j(m)$-th entry.
Thus, if we take $k\geq r_1+\cdots+r_m$, an element of $G_{n+k}$
must have $2$ coordinates equal, contradicting the fact that
$G_{n+k}$ is not contained in the big diagonal.
Therefore we know that if $G_{n+1}$ at least one
component of $G_{n+1}$ surjects onto all $d$ families of curves $C_i$.
Suppose once again that $G_{n+1}=V_1\cup\cdots\cup V_m$. Define $W_1$ to
be the union of those $V_i$ which do not surject to each family $C_i$,
or are not dominant over $G_n$. Define $W_2$ to be the union of those
components which do surject onto each $C_i$; by our work above, $W_2$ is
not empty. Hence $$G_{n+1}\subseteq W_1\cup W_2$$
Choose a component $G$ of $G_{n+k}$ which is irreducible and dominant over
$G_n$ with relative dimension $kl$. We then have
$$
G\subseteq G_{n+k}\subseteq {(G_{n+1})}^k_{G_n}\subseteq
{(W_1\cup W_2)}^k_{G_n}.
$$
If we let $J_1$ and $J_2$ run over all subsets of $\{1, \ldots, k\}$ such
that $J_1\cup J_2=\{1, \ldots, k\}$, then it follows that
$$
G\subseteq \cup_{J_1,J_2}{(W_1)}^{J_1}_{G_n}\times_{G_n}{(W_2)}^{J_2}_{G_n}
$$
Because $W_1$ and $W_2$ consist of unions of distinct components,
and $G$ is a component, it
follows that for one choice of $J_1$ and $J_2$, say $J_{1,k}$ and
$J_{2,k}$ that
$$
G\subseteq {(W_1)}^{J_{1,k}}_{G_n}\times_{G_n}{(W_2)}^{J_{2,k}}_{G_n}
$$
Recall that $G_{n+k}$ is not contained in the big diagonal. Therefore,
applying the same reasoning as earlier, we conclude that there exists
an integer $r$ such that $\#J_{1,k}\leq r$ for all $k$. This is because,
by definition, the set $W_1$ consists of elements of $G_{n+1}$ which do
not surject to all of the families of curves $C_i$. Since we know that
$\#J_{1,k}+\#J_{2,k}=k$, it follows that $\#J_{2,k}\geq k-r$, and so
as $k$ grows, the size of the sets $J_{2,k}$ also grows. Since, by
definition, components of $W_2$ surject to each of the families $C_i$,
we can obtain a surjection
$$
{(W_2)}^{J_{2,k}}_{G_n}\to C_1^{k_1}\times_{G_n}\cdots\times_{G_n} C_d^{k_d}
$$
where we can make the exponents $k_i$ as large as we wish, by taking larger
values of $k$, since the size of $J_{2,k}$ grows with $k$.
Because $G$ is a component of $G_{n+1}$ of maximal dimension
which is dominant over $G_n$, it follows that
$G$ dominates a component $W$ of ${(W_2)}^{J_{2,k}}_{G_n}$. This component
$W$ is contained inside a product of, say, $c$ of the components of $W_2$.
As $k$ grows, so does $k-r$; for large
values of $k-r$ at least one of those $c$ components, call it $W'$,
will appear at
least $\kappa=\frac{k-r}{c}$ times, where $\kappa$ also grows with $k$.
Therefore $G$ dominates ${(W')}^\kappa_{G_n}$.
As $\kappa$ grows, ${(W')}^\kappa_{G_n}$ will be mapping surjectively to
higher and higher powers of the families of curves $C_i$. In the rest of
this paper we shall prove that
this is exactly what is needed to show that ${(W')}^\kappa_{G_n}$ is a
variety of general type. Moreover, we shall show that for $\kappa$ large
enough (i.e., for $k$ large enough) ${(W')}^{\kappa}_{G_n}$ modulo the
action of the symmetric group is a variety of general type.
Because $G$ dominates ${(W')}^\kappa_{G_n}$ we obtain a family of varieties
$$G\to {(W')}^\kappa_{G_n}.$$ Because $G_n$ is not contained in the fixed
point locus, over each general point in ${(W')}^\kappa_{G_n}$, the
fiber in $G$ consists of a product of curves of genus $g$ (as opposed to
a quotient of products of curves). In other words,
each fiber is a variety of general type, as is the base. We then utilize a
theorem of Viehweg (\cite{v}, Satz III)
to conclude that $G$ itself is a
variety of general type. Viehweg's theorem states that if $Z\to B$ is
a family of varieties of general type where the base $B$ is also of
general type, then it follows that $Z$ is of general type.
Let $F$ be the image of $G$ in $F_{n+k}$. Since $G$ is of maximal
dimension in $G_{n+k}$ it follows that $F$ is of maximal dimension
in $F_{n+k}$ Since ${(W')}^{\kappa}_{G_n}$ modulo the group
action is of general type, we may again apply Viehweg's theorem
as above to see that $F$ also is of general type, using that $F$
is a family of varieties of general type over the image of
${(W')}^{\kappa}_{G_n}$ modulo the group action. Therefore, we've
shown that for large $n+k$, a component of $F_{n+k}$ of maximal
dimension is a variety of general type, proving proposition \ref{prop}.
Our work above proves the following proposition.
\begin{proposition}\label{prop.2} In order to prove Proposition \ref{prop},
it suffices to prove it for the case where each $G_n$ is irreducible.
\end{proposition}
For the remainder of the paper,
we shall assume that for all large $n$ $G_{n+1}$ consists
of one irreducible component, which projects surjectively onto
each family of curves.
Because we are assuming that $G_{n+1}$ projects surjectively onto each
$C_i$, we may apply lemma \ref{curves}. We thus obtain $d$
generically finite maps $\pi_i$ from $G_{n+1}$ to a product of the
families of curves $C_i$ fibered over $G_n$. In other words, for each
$i=1,\ldots,d$ there exists a subset $J_i\subseteq \{1, \ldots, d\}$
such that $\#J_i=l$, $i\in J_i$, and
$$\pi_{J_i} :G_n\to \prod_{j\in J_i}C_j.$$
Recall, however, that we can project $G_n$ down to $\widetilde{\cal
B}$; this allows us to obtain $d$ generically finite surjective maps
from $G_{n+1}$ to a product of families of
curves fibered over $\widetilde{\cal B}$.
Since each $G_{n+k}$ is contained in a fiber
product of $G_{n+1}$ over $G_n$, we can apply the maps $\sigma_i$ to
each component of $G_{n+1}$ in this product, and we can produce a generically
finite map from $G_{n+k}$ to a product of powers of the $C_i$:
$$
G_{n+k}\to C_1^{k_1}\times_{\widetilde{\cal B}}\cdots
\times_{\widetilde{\cal B}} C_d^{k_d}.
$$
By taking $k$ large as necessary, we can increase the exponents
$k_i$, making them as large as we wish.
To ease notation, for large $n$ let
$$V_n=C_1^{k_1}\times_{\widetilde{\cal B}} \cdots
\times_{\widetilde{\cal B}} C_d^{k_d}.$$
So we have a generically finite map $G_n\to V_n$ where the exponents
appearing in $V_n$ can be made larger by increasing $n$.
Recall the statement of proposition \ref{prop}, which says that
for large values of $n$ and $k$, every component of
the fiber product ${(F_{n+1})}^k_{F_n}$ of maximal dimension is a variety
of general type. We are ready to begin in earnest the proof of this
proposition.
First, we prove two more lemmas which will be of assistance to us.
The first statement in the following lemma is a well known fact, but
we include it here, as we will utilize it later.
\begin{lemma}\label{ideal} Let $f:V\to B$ be a flat morphism of irreducible
projective varieties with irreducible general fiber. Let $L$ be a big line
bundle on $V$, and ${\cal J}\neq 0$ an ideal sheaf on $V$.
Let $$\pi_i:V_B^k\to V$$
be projections from the fiber product to the $i$-th factor. Define
$L_k$ and ${\cal J}_k$ by
$$L_k=\otimes_i \pi_i^*L$$ and $${\cal J}_k=\sum_i\pi_i^{-1}{\cal J}.$$ Then:
\begin{enumerate}
\item There exists an integer $m$ such that $L^{\otimes m}\otimes {\cal J}$ is
big on $V$;
\item For
high enough values of $k$, it follows that $$L_k\otimes {\cal J}_k$$ is big
on $V_B^k$.
\end{enumerate}
\end{lemma}
{\bf Proof:} Consider the following exact sequence of cohomology groups:
$$
0\to H^0(V,L^{\otimes m}\otimes {\cal J})\to H^0(V,L^{\otimes m})\to
H^0(V,L^{\otimes m}/L^{\otimes m}\otimes {\cal J})\to 0
$$
Because $L$ is big,
we know that the dimension of $H^0(V,L^{\otimes m})$ is $O(m^{dim V})$.
Since the zero set of a variety has a lower dimension, we have that
the dimension of $H^0(V,L^{\otimes m}/L^{\otimes m}\otimes {\cal J})$ is less
than the dimension of $H^0(V,L^{\otimes m})$. Therefore, the dimension
of $H^0(V,L^{\otimes m}/L^{\otimes m}\otimes {\cal J})$ is less than or equal
to $O(m^{dim V-1})$. By counting the dimensions in the exact sequence,
we can conclude that the dimension of $H^0(V,L^{\otimes m}\otimes {\cal J})$ is
greater than or equal to $O(m^{dim V}-m^{dim V-1})$. This growth in the
dimension of $H^0(V,L^{\otimes m}\otimes {\cal J})$ shows that there exists an
$m$ such that $L^{\otimes m}\otimes {\cal J}$ is big on $V$.
Because $\otimes \pi_i^*L^{\otimes m}=L_k^{\otimes m}$, we obtain a
map $$\otimes \pi_i^*H^0(V,L^{\otimes m})\to
H^0(V,\otimes\pi_i^*L^{\otimes m}).$$ The indicies $i$ range from
$1$ to $k$. The inclusions
$$
\otimes \pi_i^* H^0(V,L^{\otimes m}\otimes {\cal J})
\hookrightarrow \otimes \pi_i^* H^0(V,L^{\otimes m})
$$
and
$$
H^0(V^k_B,L_k^{\otimes m}\otimes {\cal J}_k^k)\hookrightarrow
H^0(V^k_B,L_k^{\otimes m})
$$
produce a commutative diagram inducing a map
$$
\otimes \pi_i^* H^0(V,L^{\otimes m}\otimes {\cal J}) \hookrightarrow
H^0(V^k_B,{L_k}^{\otimes m}\otimes {\cal J}_k^k).
$$
Notice also that if we choose
$k>m$, we also have the inclusion
$$
H^0(V^k_B,L_k^{\otimes m}\otimes {\cal J}_k^k)\hookrightarrow
H^0(V^k_B,L_k^{\otimes m}\otimes {\cal J}_k^m)
$$
since ${\cal J}_k^k\subseteq {\cal J}_k^m$.
$$
\begin{array}{ccccc}
\otimes_i\pi_i^*H^0(L^{\otimes m}) & \longrightarrow
& H^0(L_k^{\otimes m}) & & \\
& & & & \\
\uparrow & & \uparrow & & \\
& & & & \\
\otimes_i \pi_i^*H^0(L^{\otimes m}\otimes {\cal J}) & \longrightarrow &
H^0(L_k^{\otimes m}\otimes {\cal J}_k^k) & \longrightarrow &
H^0(L_k^{\otimes m}\otimes {\cal J}_k^m)\\
\end{array}
$$
As $L^{\otimes m}\otimes {\cal J}$ is big, we have a birational map from
$V$ to $P^n$, induced by $H^0(V, L^{\otimes m}\otimes {\cal J})$.
The same reasoning as in Lemma \ref{product} lets us conclude that
we can use the sections of $H^0(V,L^{\otimes m}\otimes {\cal J})$ to
generically separate points of $V_B^k$. Thus for large values of
$k$ it follows that $L_k\otimes {\cal J}_k$ is big on $V_B^k$.
Q.E.D.
\begin{lemma}\label{big} For large enough $n$, the relative dualizing sheaf
$$
\omega_{V_n/\widetilde{\cal B}}
$$
is big, and furthermore, the dualizing sheaf $$\omega_{V_n}$$ is also
big.
\end{lemma}
{\bf Proof:} Recall the following diagram:
$$\widetilde{\cal B}\to {\cal B} \hookrightarrow B_1\times \cdots \times
B_d\subseteq B^d\to B$$
Define $C_{i,0}$ to be the pullback of the family $X\to B$ to $B_i$.
Lemma 3.4 in \cite{chm} states that
the relative dualizing sheaf
$\omega_{C_{i,0}/B_i}$ is big.
Choose $n$ to be large, so that we have a generically finite map
$$
G_n\to C_1^{k_1}\times_{\widetilde{\cal B}}\cdots
\times_{\widetilde{\cal B}} C_d^{k_d}=V_n.
$$
Now for $i=1, \ldots, d$ consider the family
$$
{(C_{i,0})}_{B_i}^{k_i}\to B_i.
$$
Call
this map $f_i$. Notice that
$$
{(C_{i,0})}_{B_i}^{k_i}\subseteq
{(C_{i,0})}^{k_i}.
$$
Now, $C_{i,0}$ maps surjectively to $B_i$ by
definition. Hence lemma \ref{surject} implies that each projection
$$
{(C_{i,0})}_{B_i}^{k_i}\to C_{i,0}
$$
is surjective. Thus lemma \ref{product}
tells us that since $\omega_{C_{i,0}/B_i}$ is big on $C_{i,0}$, it
follows that $\omega_{f_i}$ is big on ${(C_{i,0})}_{B_i}^{k_i}$.
Let $\phi$ be the map
$$
\phi : {\cal B} \hookrightarrow B_1\times \cdots \times B_d.
$$
Let $V_{n,0}$ be the pullback under $\phi$ of the family
$$
{(C_{1,0})}^{k_1}_{B_1}\times\cdots\times {(C_{d,0})}^{k_d}_{B_d}\to
B_1\times\cdots\times B_d
$$
Thus
$$
V_{n,0}\hookrightarrow {(C_{1,0})}^{k_1}_{B_1}\times\cdots\times
{(C_{d,0})}^{k_d}_{B_d}
$$
We also have surjective projections
$$
V_{n,0}\to {(C_{i,0})}^{k_i}_{B_i}
$$
They are surjective because
$\cal B$ was chosen so as to project onto each component of
$B_1\times\ldots\times B_d$. Therefore, lemma \ref{product} tells us that
$\omega_{V_{n,0}/ {\cal B}}$
is big, since each $\omega_{f_i}$ is big on ${(C_{i,0})}^{k_i}_{B_i}$.
Let $\psi$ be the map
$$
\widetilde{\cal B}\to \cal B.
$$
We have that $V_n$ is the pullback of $V_{n,0}$ under $\psi$. Thus
we know that
$$
\omega_{V_n/ \widetilde{\cal B}}=\psi^*\omega_{V_{n,0}/\cal B}
$$
and since $\psi$ is a generically finite map, it follows that
$\omega_{V_n/ \widetilde{\cal B}}$
is big.
It remains to show that the dualizing sheaf $\omega_{V_n}$ is big.
Recall that
$$
\omega_{V_n}=\omega_{V_n/ \widetilde{\cal B}}\otimes
\omega_{\widetilde{\cal B}}
$$
Choose $\cal I$ to be an ideal on the base $\widetilde{\cal B}$ such that
there is an injection $${\cal I}\hookrightarrow \omega_{\widetilde{\cal B}}.$$
It will be sufficient to show that
$$\omega_{V_n/\widetilde{\cal B}}\otimes {\cal I}
$$
is big.
To do this, define $Z$ to be the restriction of $X^d$ to
$\widetilde{\cal B}$. In other words,
$$
Z=C_1\times_{\widetilde{\cal B}}\cdots\times_{\widetilde{\cal B}} C_d.
$$
By Lemma 3.4 in \cite{chm} we know that $\omega_{Z/\widetilde{\cal
B}}$ is big. Recall that by choosing $n$ large enough we can control
the size of the exponents in $V_n$, making them as large as we wish.
Therefore, for any integer $m$ there exists an integer $n_m$ such that
if $n> n_m$, all the $k_i$ are greater than $m$. Write
$$
k_i-m=r_i>0.
$$
Then we can rewrite
\begin{eqnarray*}
V_n & = & C_1^{r_1}\times_{\widetilde{\cal B}}\cdots
\times_{\widetilde{\cal B}}
C_d^{r_d}\times_{\widetilde{\cal B}} C_1^m\times_{\widetilde{\cal B}}
\cdots\times_{\widetilde{\cal B}}C_d^m\\
& = & C_1^{r_1}\times_{\widetilde{\cal B}}\cdots
\times_{\widetilde{\cal B}}
C_d^{r_d}\times_{\widetilde{\cal B}} Z^m_{\widetilde{\cal B}}\\
& = & V'\times_{\widetilde{\cal B}} Z^m_{\widetilde{\cal B}}\\
\end{eqnarray*}
where we define $V'$ to equal
$C_1^{r_1}\times_{\widetilde{\cal B}}\cdots
\times_{\widetilde{\cal B}}
C_d^{r_d}$.
The first part of this lemma shows that for large enough $n$,
$\omega_{V'/\widetilde{\cal B}}$ is big on $V'$. Lemma
\ref{ideal} states that, since $\omega_{Z/ \widetilde{\cal B}}$ is big on
$Z$, for large enough $m$,
$$
\omega_{Z^m_{\widetilde{\cal B}}}\otimes {\cal I}$$ is big on
$Z^m_{\widetilde {\cal B}}$.
We now apply Lemma \ref{product} to see that
$$
\omega_{V'/\widetilde{\cal B}}\otimes
(\omega_{Z^m_{\widetilde{\cal B}}}\otimes {\cal I})$$
is big on $V'\times_{\widetilde{\cal B}} Z^m_{\widetilde{\cal B}}$.
In other words,
$$
\omega_{V_n/\widetilde{\cal B}}\otimes {\cal I}$$ is big, as desired;
hence $\omega_{V_n}$ is big on $V_n$. Q.E.D.
We're almost ready to prove Proposition \ref{prop}. Let us set up
some notation first.
Let $\widetilde{G_{n+k}}$ be an equivariant desingularization of $G_{n+k}$.
In other words, the action of a subgroup of the symmetric group $S_d$ on
$\widetilde{\cal B}$ and $G_{n+k}$ lifts to $\widetilde{G_{n+k}}$.
Let $r$ be the map from
$\widetilde{G_{n+k}}$ to $G_{n+k}$ which gives the resolution of
singularities. Let $\Phi_{n+1}$ and $\Phi_{n+k}$ denote the set of points of
$G_{n+1}$ and $G_{n+k}$, respectively, which
are fixed by the group action.
Recall that we have a generically finite map, call it $\sigma_0$,
from $G_n$ to $V_n$:
$$
\sigma_0 :G_n\to V_n=C_1^{k_1}\times_{\widetilde{\cal B}} \cdots
\times_{\widetilde{\cal B}} C_d^{k_d}.
$$
For $i=1, \ldots, d$, denote by $\sigma_i$ the $d$ projections from $G_{n+1}$
to a product of $l$ families of curves, where $l$ is the relative dimension of
$G_{n+1}$ over $G_n$. Let $Z_i$ be the image of $G_{n+1}$ under $\sigma_i$.
We can write
$$
Z_i=C_{j_1(i)}\times_{\widetilde{\cal B}}\cdots\times_{\widetilde{\cal B}}
C_{j_l(i)}
$$
where one of $j_1(i), \ldots, j_l(i)$ is equal to $i$.
Define varieties $V_{n+1,i}$ by
$$
V_{n+1,i}=V_n\times_{\widetilde{\cal B}}Z_i.
$$
Then for any positive integer $k$, we have
$$
V_{n+k,i}=V_n\times_{\widetilde{\cal B}}
{(Z_i)}^k_{\widetilde{\cal B}}.
$$
Let $\pi_0$ be the map from $G_{n+k}$ to $G_n$ given by projection to the
first $n$ coordinates, and
for $j=1, \ldots, k$, define projection maps
$$
\pi_j:G_{n+k}\to G_{n+1}
$$
as follows: If $(P_1, \ldots, P_n, \ldots, P_{n+k})\in G_{n+k}$ then
$$
\pi_j((P_1, \ldots, P_n, \ldots, P_{n+k})=(P_1, \ldots, P_n, P_{n+j}).
$$
Recall that $G_{n+k}\subseteq {(G_{n+1})}^k_{G_n}$. We use this to define
maps
$$
(\sigma_0, \sigma_i, \ldots, \sigma_i):G_{n+k}\to V_{n+k,i}.
$$
They act in the following manner: $\sigma_0$ is applied to $G_n$, while the
$\sigma_i$ are applied to the copies of $G_{n+1}$. In other words,
if $P=(P_1, \ldots, P_n, P_{n+1}, \ldots, P_{n+k})\in G_{n+k}$, then
$$(\sigma_0, \sigma_i, \ldots, \sigma_i)(P)=
(\sigma_0(P_1, \ldots, P_n), \sigma_i(P_{n+1}), \ldots, \sigma_i(P_{n+k})).
$$
We have the following diagram:
$$
\begin{array}{ccc}
G_{n+k} & \stackrel{(\sigma_0, \sigma_i, \ldots, \sigma_i)}
{\longrightarrow} & V_{n+k,i}\\
& & \\
\downarrow \pi_j & & \downarrow\\
& & \\
G_{n+1} & \longrightarrow & V_{n+1,i}\\
& & \\
\downarrow & & \downarrow\\
& & \\
G_n & \stackrel{\sigma_0}{\longrightarrow} & V_n\\
\end{array}
$$
Recall from the proof of Lemma \ref{big} that we defined $Z$ to be the
restriction of $X^d$ to ${\widetilde{\cal B}}$.
$$
Z=C_1\times_{\widetilde{\cal B}}\cdots\times_{\widetilde{\cal B}}C_d.
$$
We know that $G_{n+1}$ maps to $Z$ and in fact, the $\sigma_i$'s
factor through this map. So let
$$
\tau :G_{n+1}\to W
$$
be the map from $G_{n+1}$ to $Z$, where $W$ is the image of $G_{n+1}$ in $Z$.
Let
$$
e:W\to Z
$$
be the inclusion map from $W$ to $Z$. Finally, define the projection maps
$$
\rho_i :Z\to Z_i
$$
where
$$
\rho_i (C_1\times_{\widetilde{\cal B}}\ldots\times_{\widetilde{\cal B}}C_d)
=C_{j_1(i)}\times_{\widetilde{\cal B}}\ldots\times_{\widetilde{\cal B}}
C_{j_l(i)}.
$$
We have the diagram:
$$
\begin{array}{ccccl}
& & W & & \\
& \nearrow & & \searrow e & \\
& & & & \\
G_{n+1} & & \longrightarrow & & Z\\
& & & & \\
& & \searrow & & \downarrow \rho_i\\
& & & & \\
& & & & Z_i\\
\end{array}
$$
An important fact to notice is that the image in $W$ of the fixed points
$\Phi_{n+1}$ is not all of $W$. Why is this? Well, let's look at what the
map $\tau :G_{n+1}\to W\subseteq Z$ does. Remember that
$G_{n+1}\subseteq {(X_B^{n+1})}^d$ and that $Z=X^d|_{\widetilde{\cal B}}$.
We can write a $K$-rational element, $g$, of $G_{n+1}$ as
$$
g=({(P_1, \ldots, P_{n+1})}^{\alpha_1}, \ldots,
{(P_1, \ldots, P_{n+1})}^{\alpha_d})
$$
where $\alpha_1, \ldots, \alpha_d$ are the embeddings of $L$ into
$\bar{K}$ fixing $K$. Then
$$
\tau (g)=({P_{n+1}}^{\alpha_1}, \ldots, {P_{n+1}}^{\alpha_d}).
$$
Therefore $\tau (g)$ is actually a Galois orbit. Recall that the
$P_{n+1}$'s are defined over $L$ but not any smaller field between $L$
and $K$. Hence all of ${P_{n+1}}^{\alpha_1}, \ldots, {P_{n+1}}^{\alpha_d}$
are distinct. Therefore, $\tau (G_{n+1})=W$ is the closure of a set of points
which are not fixed by the group action.
This fact that the fixed points do not surject to $W$ will soon be very
important.
We want to prove that every component of $F_{n+k}$ of maximal dimension is
a variety of general type. This means that if $\widetilde{F_{n+k}}$ is
a resolution of singularities of $F_{n+k}$, we need to show that
the dualizing sheaf $\omega_{\widetilde{F_{n+k}}}$ is big.
The main idea is to pull back sections of dualizing sheaves along our
various projections. We know that both $\omega_{V_n}$ and
$\omega_{V_{n+k,i}}$ are big for large $n$. ($\omega_{V_{n+k,i}}$
is big since the families of the curves $C_i$ appear in $V_{n+k,i}$ with
even larger exponents.) We would like to utilize the fact that
$\omega_{V_{n+k,i}}$ is big to pull back a lot of sections to
$\widetilde{G_{n+k}}$ which vanish along the fixed points of the group
action (for we will see that this is what suffices for $F_{n+k}$ to
be of general type). Unfortunately we can't pull back sections along
any one projection of $\widetilde{G_{n+k}}$ to $V_{n+k,i}$; the
sections pulled back in this way will not vanish along the fixed
points. To overcome this difficulty we will pull back sections via
all the various projections, tensor them together, and use our earlier
lemmas to obtain that $\omega_{\widetilde{G_{n+k}}}$ is big, with
lots of sections vanishing on the fixed points.
Recall that $G_{n+k}$ is contained in a fiber power of families of
stable curves over $\widetilde{\cal B}$, and, moreover,
$\widetilde{\cal B}$ was chosen to be smooth and irreducible, with the
property that its discriminant locus is a divisor of normal crossings.
We appeal to Lemma 3.3 of \cite{chm}, which states that this implies
that any singularities of $V_{n+k,i}$ are canonical. This means that
the singularities do not prohibit the extension of pluricanonical
sections from the smooth locus to any desingularization. Therefore we
may pull back sections of $V_{n+k,i}$ to $\widetilde{G_{n+k}}$. We
have the following injection:
$$
r^*(\sigma_0,\sigma_i, \ldots,\sigma_i)^*
\omega_{V_{n+k,i}}\hookrightarrow
\omega_{\widetilde{G_{n+k}}}
$$
and so
$$
r^*(\otimes_i((\sigma_0,\sigma_i, \ldots,\sigma_i))^*
\omega_{V_{n+k,i}}\hookrightarrow
(\omega_{\widetilde{G_{n+k}}})^{\otimes d}
$$
Now, the left-hand side above can be rewritten as:
$$
r^*(\pi_0^*\sigma_0^*\omega_{V_n})^{\otimes d}\otimes
r^*(\pi_1^*\otimes_i\sigma_i^*\omega_{Z_i/\widetilde{\cal B}})
\otimes\ldots\otimes
r^*(\pi_k^*\otimes_i\sigma_i^*\omega_{Z_i/\widetilde{\cal B}})
$$
Notice in the second diagram above that
$$
\sigma_i=\rho_i\circ e\circ \tau
$$
Hence,
$$
\sigma_i^*=\tau^* e^* \rho_i^*.
$$
Let's look at one of the last $k$ terms above (take some $j$ such
that $1\leq j\leq k$):
\begin{eqnarray*}
r^*(\pi_j^*\otimes_i\sigma_i^*\omega_{Z_i/\widetilde{\cal B}}) & = &
r^*(\pi_j^*(\otimes_i\tau^* e^* \rho_i^*\omega_{Z_i/\widetilde{\cal B}}))\\
& = & r^*(\pi_j^*(\otimes_i\tau^* e^*
(\omega_{C_{j_1(i)}/\widetilde{\cal B}}\otimes\cdots\otimes
\omega_{C_{j_l(i)}/\widetilde{\cal B}})))\\
& = & r^*(\pi_j^*\tau^* e^*(\otimes_i
(\omega_{C_{j_1(i)}/\widetilde{\cal B}}\otimes\cdots\otimes
\omega_{C_{j_l(i)}/\widetilde{\cal B}})))\\
& = & r^*(\pi_j^*\tau^*e^*(\omega_{C_1/\widetilde{\cal B}}^{\otimes l_1}
\otimes\cdots\otimes\omega_{C_d/\widetilde{\cal B}}^{\otimes l_d}))\\
\end{eqnarray*}
The exponents $l_i$ are defined by the last line above, and we are
guaranteed that each one is positive. We know that for each $i$,
one of the $j_k(i)$'s is equal to $i$; hence we when we tensor over all $i$
we know get each family $C_i$ appearing some positive number $l_i$ times.
Let $M$ denote the line bundle
$$
M=\omega_{C_1/\widetilde{\cal B}}^{\otimes l_1}
\otimes\cdots\otimes\omega_{C_d/\widetilde{\cal B}}^{\otimes l_d}.
$$
We claim that $M$ is a big line bundle on $Z$, and if we let $M_W$
denote $e^*M$, then, in fact, $M_W$ is big on $W$.
Recall that
$$
M=\otimes_i \rho_i^*\omega_{Z_i/\widetilde{\cal B}}.
$$
In other words,
$$
M=\omega_{C_1/\widetilde{\cal B}}^{\otimes l_1}\otimes\cdots\otimes
\omega_{C_d/\widetilde{\cal B}}^{\otimes l_d}
$$
where each $l_i$ is positive.
As in the proof of Lemma \ref{big}, let $C_{i,o}$ be the pullback of
$X\to B$ to $B_i\subseteq B$; recall that $\omega_{C_{i,0}/B_i}$ is big.
Moreover, using techniques from the proof of Lemma \ref{big}, we know
that $\omega_{{(C_{i,0})}^{l_i}_{B_i}}$ is big.
We have
$$
Z\subseteq C_{1,0}\times\cdots\times C_{d,0}\to B_1\times\cdots\times B_d
$$
and each $Z$ surjects to each $B_i$. Since each
$\omega_{C_{i,0}/B_i}$ is big, it follows that each
$\omega^{\otimes l_i}_{C_{i,0}/B_i}$ is also big. Therefore
applying Lemma \ref{product} we see that the pullback
of $\otimes_i\omega^{\otimes l_i}_{C_{i,0}/B_i}$ to $Z$ is big. This
pullback is equal to
$$\omega_{C_1/\widetilde{\cal B}}^{\otimes l_1}\otimes\cdots\otimes
\omega_{C_d/\widetilde{\cal B}}^{\otimes l_d}=M$$
so $M$ is big on $Z$. Now $W\subseteq Z$ and $W$ surjects to each
$C_i$ since $W$ is the image of $G_{n+1}$. Hence applying lemma
\ref{product} once again, we conclude that $M_W$ is big on $W$.
Putting all of the above together, we see that
$$
r^*(\pi_0^*\sigma_0^*\omega_{V_n})^d\otimes r^*(\pi_1^*\tau^* M_W)
\otimes\cdots\otimes r^*(\pi_k^*\tau_* M_W)\hookrightarrow
{\omega_{\widetilde{G_{n+k}}}}^{\otimes d}.
$$
Recall that the image of the fixed points $\Phi_{n+1}$ in $W$ is not all
of $W$. Therefore this image forms a proper subvariey
of $W$, and we obtain a non-zero ideal $\cal I$ of sections of
$M_W$ which vanish on this subvariety.
We now apply the following lemma, borrowed from
\cite{chm}.
\begin{lemma}\label{fixed}[see \cite{chm}, Lemma 4.1]
Suppose that $X$ is a projective variety with
$G$ a finite group of order $l$ acting on $X$. Let $\theta$ be an $m$
canonical form on $X$ which is invariant under the group action of $G$.
Then if $\theta$ vanishes to order $m(l-1)$ on the locus of all points
of $X$ fixed by the group action, then $\theta$ descends to smooth form
on $X/G$.
\end{lemma}
We use Lemma \ref{fixed} to show that
smooth pluricanonical forms on $\widetilde{G_{n+k}}$ descend to smooth
forms on $\widetilde{F_{n+k}}$ modulo the group action, if the forms vanish
to a prescribed order on the fixed point locus. Bear in mind that
$\widetilde{G_{n+k}}$ modulo the group action is nothing more than
$\widetilde{F_{n+k}}$, a desingularization of $F_{n+k}$. So to show that
$\omega_{\widetilde{F_{n+k}}}$ is big for large $k$, we will show that
$\omega_{\widetilde{G_{n+k}}}$ not only is big, but also has lots of
sections vanishing to arbitrarily high order on the fixed point locus.
By the first statement of Lemma \ref{ideal}, we know that for some $s$
the line bundle $M_W^{\otimes s}\otimes
{\cal I}$ is big.
Look at the following diagram:
$$
\begin{array}{ccccc}
\widetilde{G_{n+k}} & {r\atop\longrightarrow} & G_{n+k} &
\hookrightarrow & G_n\times W\times\cdots\times W \\
& & & & \\
& & & \searrow & \downarrow \\
& & & & \\
& & & & V_n\times W\times\cdots\times W \\
\end{array}
$$
Let ${\cal J}_\Phi$ be the locus of fixed points in $G_{n+k}$.
Then
$$
\sum_j \pi_j^{-1}\tau_j^{-1}{\cal J}\subset {\cal J}_\Phi
$$
and so
$$
\prod_j \pi_j^{-1}\tau_j^{-1}{\cal J}\subset{\cal J}^k_{\Phi}
$$
Therefore,
$$
r^*{(\pi_0^*\sigma_0^*\omega_{V_n})}^{\otimes ds}
\otimes r^*(\otimes_j\pi_j^*\tau^*M_W^{\otimes s}
\otimes(\prod_j \pi_j^{-1}\tau_j^{-1}{\cal J}))\hookrightarrow
\omega_{\widetilde{G_{n+k}}}^{\otimes ds} \otimes {\cal J}^k_\Phi
$$
We know that $\omega_{V_n}$ is big, and
as in the proof of lemma \ref{ideal}, as long as $k>s$,
$M_W^{\otimes s}
\otimes(\prod_j \pi_j^{-1}\tau_j^{-1}{\cal J}))$ is big.
Therefore, for $k>s$ the entire left hand side above is
big, so we have lots of sections of
$\omega_{\widetilde{G_{n+k}}}^{\otimes ds}$
vanishing to high order, and the proof is complete.
|
1996-01-24T06:20:12 | 9601 | alg-geom/9601022 | en | https://arxiv.org/abs/alg-geom/9601022 | [
"alg-geom",
"math.AG"
] | alg-geom/9601022 | Wilberd van der Kallen | Wilberd van der Kallen and Peter Magyar | The Space of Triangles, Vanishing Theorems, and Combinatorics | LaTeX Version 2.09 <7 Dec 1989>, 38 pages, author-supplied PostScript
file available at http://www.math.ruu.nl/people/vdkallen/triang.ps.gz | Journal of Algebra, Vol. 222 (1999) 17-50 | null | Universiteit Utrecht Mathematics preprint 939 | null | We consider compactifications of the space of triples of distinct points in
projective $n$-space. One such space is a singular variety of configurations of
points and lines; another is the smooth compactification of Fulton and
MacPherson; and a third is the triangle space of Schubert and Semple. We
compute the sections of line bundles on these spaces, and show that they are
equal as GL(n) representations to the generalized Schur modules associated to
``bad'' generalized Young diagrams with three rows (Borel-Weil theorem). On the
one hand, this yields Weyl-type character and dimension formulas for the Schur
modules; on the other, a combinatorial picture of the space of sections.
Cohomology vanishing theorems play a key role in our analysis.
| [
{
"version": "v1",
"created": "Tue, 23 Jan 1996 11:44:41 GMT"
}
] | 2007-06-06T00:00:00 | [
[
"van der Kallen",
"Wilberd",
""
],
[
"Magyar",
"Peter",
""
]
] | alg-geom | \section{Definitions}
\label{Definitions}
A {\em diagram} is a finite subset of
${\bf N} \times {\bf N}$.
Its elements $(i,j) \in D$ are called {\em squares}, and
the square $(i,j)$ is pictured in the
$i^{\mbox{th}}$ row and $j^{\mbox{th}}$ column.
We shall often think of $D$ as a sequence
$(C_1,C_2,\ldots,C_r)$ of columns $C_j \subset {\bf N}$.
{}Fix once and for all an integer $n\geq 3$.
We denote the interval $[1,n] = \{1,2,\ldots,n\}$.
We shall always write $G = GL(n,{\bf C})$, $B = $ the subgroup
of upper triangular matrices, and $T =$ the subgroup of diagonal
matrices. ${\bf C}^n$ is the defining representation of $G$.
We will assume our diagrams have at most $n$ rows:
$D \subset [1,n] \times {\bf N}$.
Let $\Sigma_D$ be the symmetric group permuting the squares of $D$,
and for any diagram $D$, let
$$
\mathop{\rm Col}(D) = \{\pi \in \Sigma_D \mid \pi(i,j) = (i',j) \ \exists
i'\}
$$
be the group permuting the squares of $D$
within each column, and we define $\mathop{\rm Row}(D)$ similarly
for rows.
Define the idempotents
${\alpha}_D$, $\beta_D$ in the group algebra
${\bf C}[\Sigma_D]$ by
$$
{\alpha_D} = {1 \over |\mathop{\rm Row} D|} \sum_{\pi \in \mathop{\rm Row} D} \pi, \ \ \ \
{\beta_D} = {1 \over |\mathop{\rm Col} D|} \sum_{\pi \in \mathop{\rm Col} D} \mathop{\rm sgn}(\pi) \pi ,
$$
where $\mathop{\rm sgn}(\pi)$ is the sign of the permutation.
Now, $G$ acts diagonally on the left of
the $|D|$-fold tensor product $({\bf C}^n)^{\otimes D}$,
and $\Sigma_D$ acts on the right by permuting the tensor factors:
$$
g(x_{t_1}, x_{t_2}, \ldots ) \pi =
(g x_{\pi t_1}, g x_{\pi t_2}, \ldots ) .
$$
These actions commute.
Define the {\em Schur module}
$$
S_D \stackrel{\rm def}{=} ({\bf C}^n)^{\otimes D} {\alpha_D} {\beta_D} \subset ({\bf C}^n)^{\otimes
D},
$$
a representation of $G$.
Interchanging two columns (or two rows) of the diagram
gives an isomorphic Schur module.
Now, let
$e_1, \ldots, e_n$ be the standard basis of ${\bf C}^n$,
and for $C\subset [1,n]$, define the coordinate subspace
$E_C = {\mbox{\rm Span}} (e_i \mid i\in C) \in \mbox{\rm Gr}(|C|,{\bf C}^n)$.
{}For a diagram $D = (C_1,\ldots,C_r)$,
let
$$
\mbox{\rm Gr}(D) = \mbox{\rm Gr}(|C_1|,{\bf C}^n) \times \cdots \times \mbox{\rm Gr}(|C_r|,{\bf C}^n)
\ ,
$$
and define the {\em configuration variety} as the closure
in $\mbox{\rm Gr}(D)$ of the $GL(n)$-orbit
of a configuration of coordinate subspaces:
$$
\FF_D = \mbox{closure } \left[ {G\cdot (E_{C_1},\ldots,E_{C_r})}
\right]
\subset \mbox{\rm Gr}(D) \ .
$$
This is clearly an irreducible subvariety.
We may also define the {\em inclusion variety}
$ { \cal I } _D \subset \mbox{\rm Gr}(D)$
by:
$$
{ \cal I } _D = \{ (V_1,\ldots,V_r) \in \mbox{\rm Gr}(D)
\mid C_i \subset C_j \Rightarrow V_i \subset V_j \} .
$$
We clearly have $ \FF_D \subset { \cal I } _D$.
Consider the Pl\"ucker line bundle
$ {\cal O} (1) = {\cal O} (1,\ldots, 1)$ on the
product of Grassmannians $\mbox{\rm Gr}(D)$.
We may define a line bundle $ \LL_D $ on $ \FF_D $ and $ { \cal I } _D$ as the
restriction of $ {\cal O} (1)$ to the subvarieties.
In case $D$ is a Young diagram $\{(i,j)\mid 1\leq i \leq \lambda _j\}$
for $ \lambda = ( \lambda _1 \geq \cdots \geq \lambda _n \geq 0)$, then
$ \FF_D = { \cal I } _D$ is a flag variety, and $ \LL_D $ is the Borel-Weil line
bundle whose sections are (the dual of) the irreducible Schur
module $S_D = S_{ \lambda }$.
In this paper, we will consider diagrams
$D$ with at most three rows:
that is,
all squares $(i,j) \in D$ have $i = 1$, 2, or 3.
We may consider
the diagram $D_3$ of the introduction
as universal:
it contains a column of each type, and
any three-rowed diagram can be specified
(up to the order of the columns)
by the multiplicity
$m_i \geq 0$, $i= 1,\ldots, 7$ of each column in $D_3$.
Note that the multiplicities $m_i$ will not affect the
varieties $ \FF_D $ or $ { \cal I } _D$ (provided all $m_i > 0$).
We will denote $ \FF_D = { \cal F } _{D_3} = { \FF_{3,n} }$ and
$ { \cal I } _D = { \cal I } _{D_3} = { \cal I } _{3,n}$,
both inside the product of Grassmannians:
$$
{ \FF_{3,n} } \subset { \cal I } _{3,n} \subset ({\bf P}^{n-1})^3 \times
\mbox{\rm Gr}(1,{\bf P}^{n-1})^3 \times \mbox{\rm Gr}(2,{\bf P}^{n-1}).
$$
The variety ${ \FF_{3,n} }$ is the closure of the $GL(n)$-orbit of the
coordinate 2-simplex in ${\bf P}^{n-1}$, and the inclusion variety is
$$
{ \cal I } _{3,n} =
\{\ (p_1, p_2, p_3, l_1, l_2, l_3, P) \ \mid \ p_i \subset l_j
\ \ \forall\ i\neq j, \ \ l_j \subset P \ \ \forall \ j \ \}.
$$
{}For a diagram $D$ defined by integers $m_i$,
$ \LL_D $ is the restriction of $ {\cal O} (m_1,\ldots,m_7)$ on the product
of
Grassmannians. This is a (very) ample line bundle
exactly when $m_i > 0$ for all $i$.
\section{The Singular space}
\subsection{Borel-Weil theorem}
Our first aim is to prove
\begin{thm}
\label{vanishing}
We have \\
(a) For any diagram $D$ with at most three rows, we have
$H^0({ \FF_{3,n} }, \LL_D ) = S_D^*$ and $H^i({ \FF_{3,n} }, \LL_D ) = 0$ for $i >0$. \\
(b) ${ \FF_{3,n} } = { \cal I } _{3,n}$ is a normal, irreducible variety,
and projectively normal with respect to $ \LL_D $.
\end{thm}
The proof will occupy the following sections. We start with the
elementary
parts.
\\[.3em]
{\bf Claim:\ } $ { \cal I } _{3,n}$ is irreducible of dimension
$3n - 3$, and hence equal to its subvariety ${ \FF_{3,n} }$.
\\[.3em]
It will suffice to prove this for $ { \cal I } _{3,3}$, since there is an
obvious fiber bundle
$$
\begin{array}{ccc}
{ \cal I } _{3,3} & \rightarrow & { \cal I } _{3,n} \\
& & \downarrow \\
& & \mbox{\rm Gr}(2,{\bf P}^{n-1})
\end{array}
$$
given by mapping a triangle to the
projective plane in which
it lies.
(The fiber is irreducible if and only if the whole bundle is, and
$\dim { \cal I } _{3,n} = \dim { \cal I } _{3,3} + 3(n - 3)$.)
Now, any triangle in $ { \cal I } _{3,3}$ can be deformed under the
$GL(3)$ action on ${\bf P}^2$ so as to approach a maximally degenerate
configuration, in which all three vertices and edges are identical.
{}Furthermore, $GL(3)$ acts transitively on this stratum
of degenerate triangles, so it suffices to check the irreducibility
in a neighborhood of a single degenerate triangle such as
$(E_1,E_1,E_1,E_{12},E_{12},E_{12})$,
for which the entries form a flag of coordinate subspaces of ${\bf P}^2$.
Nearby, we can give affine coordinates so that
a configuration in $({\bf P}^2)^3 \times \mbox{\rm Gr}(1,{\bf P}^2)^3$ is represented
by
$$
\left[
\left( \begin{array}{c} 1 \\ a_1 \\ b_1 \end{array} \right),
\left( \begin{array}{c} 1 \\ a_2 \\ b_2 \end{array} \right),
\left( \begin{array}{c} 1 \\ a_3 \\ b_3 \end{array} \right),
\left( \begin{array}{c} c_1 \\ d_1 \\ 1 \end{array} \right),
\left( \begin{array}{c} c_2 \\ d_2 \\ 1 \end{array} \right),
\left( \begin{array}{c} c_3 \\ d_3 \\ 1 \end{array} \right)
\right]
$$
subject to the six incidence conditions:
$$
\begin{array}{c}
c_2 + a_1 d_2 + b_1 = 0 \\
c_3 + a_1 d_3 + b_1 = 0
\end{array} \ \ \
\hfill
\begin{array}{c}
c_1 + a_2 d_1 + b_2 = 0 \\
c_3 + a_2 d_3 + b_2 = 0
\end{array}
\hfill \ \ \
\begin{array}{c}
c_1 + a_3 d_1 + b_3 = 0 \\
c_2 + a_3 d_2 + b_3 = 0
\end{array}
$$
By eliminating, we can reduce this to
the seven variables $a_1$, $a_2$, $a_3$, $d_1$, $d_2$, $d_3$, $c_1$
subject to the single equation
$$
(a_1 - a_2)(d_2 - d_3) + (a_3 - a_2)(d_1 - d_2) = 0 .
$$
A linear change of variables turns this into the product of an
affine space and a quadric, an irreducible variety of dimension 6.
This is what we wanted to show.
Therefore $ { \cal I } _{3,n}$ is irreducible and equal to ${ \FF_{3,n} }$.
\\[.5em]
{\bf Borel-Weil construction.}
\\[.5em]
Next, we recall from \cite{MaNW} the elementary construction connecting
the Schur module $S_D$ of a three-row diagram with the triangle space.
Let $V = {\bf C}^n$ and $U = V^*$ its dual space.
By definition, $S_D$ is the image of the composite map
$$
S_D = \mathop{\rm Im}\left[ V^{\otimes D} {\alpha_D}
\stackrel{\mbox{\small incl}}{\rightarrow}
V^{\otimes D}
\stackrel{{\beta_D}}{\rightarrow}
V^{\otimes D} {\beta_D}
\right].
$$
Taking dual spaces, we have
$$
S_D^* = \mathop{\rm Im}\left[ U^{\otimes D} {\beta_D}
\stackrel{\mbox{\small incl}}{\rightarrow}
U^{\otimes D}
\stackrel{{\alpha_D}}{\rightarrow}
U^{\otimes D} {\alpha_D}
\right].
$$
We translate this into geometric language as follows.
Consider the product space $({\bf P}^{n-1})^D$ as
all $|D|$-tuples of points
inscribed in the squares of $D$.
Define
$$
\phi : ({\bf P}^{n-1})^D \rightarrow \mbox{\rm Gr}(D)
$$
to be the rational map taking a $c$-tuple of vectors
in a column $C$ to the space in $\mbox{\rm Gr}(c,{\bf C}^n)$ which they span.
This is a rational map defined everywhere except a set of codimension
$\geq
2$, so
it induces maps of locally free coherent sheaves as if it were regular.
Also define the {\em row multidiagonal}
$\Delta^D({\bf P}^{n-1})$, the locus in $({\bf P}^{n-1})^D$
where all points in the
same row are equal: that is, for our $D = D_3$,
$$
\Delta^D({\bf P}^{n-1}) =
\left\{ \begin{array}{ccccccc}
p_1 & & & & p_1 & p_1 & p_1 \\
& p_2 & & p_2 & & p_2 & p_2 \\
& & p_3 & p_3 & p_3 & & p_3
\end{array}
\right\}
\subset ({\bf P}^{n-1})^D \ \ .
$$
The composite image of these maps is precisely the configuration
variety:
$$
\FF_D = \mbox{closure } \mathop{\rm Im} \left[
\Delta^D({\bf P}^{n-1})
\stackrel{\mbox{\small incl}}{\rightarrow}
({\bf P}^{n-1})^D
\stackrel{\phi}{\rightarrow} \mbox{\rm Gr}(D)
\right] .
$$
The above equation for the dual Schur module
now translates easily into
$$
\begin{array}{rcl}
S_D^* & = & \mathop{\rm Im}\left[
\begin{array}{rcl}
H^0(\mbox{\rm Gr}(D), \LL_D )
& \stackrel{\phi^*}{\rightarrow} &
H^0( ({\bf P}^{n-1})^D, {\cal O} (1) )
\\
& \stackrel{\mbox{\small rest}}{\rightarrow} &
H^0( \Delta^D({\bf P}^{n-1}), \mbox{rest}\, {\cal O} (1) )
\end{array}
\right]
\\
& & \\
& = &
\mathop{\rm Im} \left[
H^0(\mbox{\rm Gr}(D), \LL_D ) \stackrel{\mbox{\small rest}}{\rightarrow} H^0( \FF_D ,
\LL_D )
\right] \ ,
\end{array}
$$
where $\mbox{rest} = \mbox{incl}^*$.
This equation is true for an arbitrary diagram $D$.
If $ \FF_D = { \FF_{3,n} }$, the first equation in part (a) of the Theorem states
that
the above
restriction map is onto, so that the dual Schur module is just
equal to the sections of $ \LL_D $ over the triangle space $ \FF_D = { \FF_{3,n} }$.
If $D$ does not contain each column of $D_3$, then $ \FF_D \neq { \FF_{3,n} }$,
but there
is a surjective map ${ \FF_{3,n} } \rightarrow \FF_D $ given by forgetting the
data
associated
to the columns which do not appear in $D$.
We have the commutative diagram
$$
\begin{array}{ccc}
H^0(\mbox{\rm Gr}(D_3), \LL_D ) & \stackrel{\mbox{\small rest}}{\rightarrow} &
H^0({ \FF_{3,n} },
\LL_D ) \\
\downarrow & & \downarrow \\
H^0(\mbox{\rm Gr}(D), \LL_D ) & \stackrel{\mbox{\small rest}}{\rightarrow} & H^0( \FF_D ,
\LL_D )
\end{array}
$$
The vertical map between the sections over Grassmannians is clearly
an
isomorphism.
Hence in this case, we must show the surjectivity
of the top restriction map, {\em and} the bijectivity of the second
vertical map.
Thus, the first equation of part (a) reduces in general to
\begin{lem}
The natural map
$
H^0(\mbox{\rm Gr}(D), \LL_D ) {\rightarrow} H^0( \FF_D , \LL_D )
$
is surjective, and the natural map
$
H^0({ \FF_{3,n} }, \LL_D ) \rightarrow H^0( \FF_D , \LL_D )
$
is bijective.
\end{lem}
To prove these facts and the rest of the Theorem, we will need more
sophisticated techniques.
\subsection{Frobenius splitting}
The theory of Frobenius splittings invented by Mehta,
Ramanan, and Ramanathan
(\cite{MR}, \cite{RR}, \cite{R1},
\cite{vdK})
is a characteristic-$p$ technique for proving surjectivity
and vanishing results about coherent sheaves,
even in characteristic zero.
It is highly practical for dealing with homogeneous varieties
because one can work over the integers in a characteristic-free
way, and never consider the special features of characteristic-$p$
geometry. In fact, the method reduces to classical questions
about defining equations and canonical divisors of varieties.
Most of the theorem of the last section will follow immediately from
knowing
that the pair ${ \FF_{3,n} } \subset \mbox{\rm Gr}(D_3)$ is ``compatibly Frobenius split''
in any characteristic.
Given two algebraic varieties $Y \subset
X$ defined
over an algebraically closed
field $F$ of characteristic $p > 0$,
with $Y$ a closed subvariety of $X$,
we say that the pair $Y \subset X$ is
{\em compatibly Frobenius split} if: \\
(i) the $p^{th}$ power map $F: {\cal O} _X \rightarrow F_* {\cal O} _X$
has a splitting, i.e. an $ {\cal O} _X$-module morphism
$\phi: F_* {\cal O} _X \rightarrow {\cal O} _X$ such that $\phi F$ is the
identity; and \\
(ii) we have $\phi(F_* I) = I$, where
$I$ is the ideal sheaf of $Y$.
Because $ {\cal L} \otimes F_* {\cal O} _X=F_* {\cal L} ^p$ for any line bundle $ {\cal L} $,
a Frobenius splitting of $Y$ allows one to embed the
cohomology of an ample bundle on $Y$ into the cohomology
of its powers: $H^i(Y, {\cal L} ) \subset H^i(Y, {\cal L} ^{p^d})$
for all $d \geq 0$. Since the right-hand side becomes
zero for large $d$ by Serre vanishing,
the $H^i(Y, {\cal L} )$ itself must be zero ($i>0$).
If one can show this vanishing for reductions modulo
$p$ for all (or infinitely many) $p$,
then semi-continuity implies $H^i(X({\bf C}), {\cal L} ) = 0$ as well.
In fact, Mehta and Ramanathan prove the following
\begin{prop}\label{splitsur}
Let $X$ be a projective variety, $Y$ a closed subvariety,
and $ {\cal L} $ an ample line bundle on $X$.
If $Y \subset X$ is compatibly split, then
$H^i(Y, {\cal L} ) = 0$ for all $i >0$,
and the restriction map
$H^0(X, {\cal L} ) \rightarrow H^0(Y, {\cal L} )$ is surjective.
{}Furthermore, if $Y$ and $X$ are defined and projective over ${\bf Z}$
(and hence over any field), and they are compatibly split
over any field of positive characteristic,
then the above vanishing and surjectivity
statements also hold for all fields of characteristic zero.
\end{prop}
{}Frobenius splitting is also sufficient to establish
the normality of our varieties.
The main theorem of
Mehta and Srinivas \cite{MS} states that if $Y$ is a Frobenius-split
variety possessing a desingularization with connected fibers,
then $Y$ is normal. (Normality in all finite
characteristics implies normality in characteristic 0).
These strong properties of split varieties
will suffice to prove our Theorem,
provided we construct a compatible splitting.
This is rendered practical by a criterion that was made explicit
in
\cite{LMP}
in terms of a notion ``residually
normal crossing'', which we now recall.
A divisor $D$ defined by $f_0=0$ around a point $P$ on
a smooth affine variety $X$ of dimension $n$ has
{\em residually normal crossing} at $P$ if there exists a
system of parameters $\{x_1,\cdots, x_n\}$ and functions
$f_1,\cdots,f_{n-1}\in k[[x_1,\cdots, x_n]]=\widehat{ {\cal O} _P}$ such
that
$f_i=x_{i+1}f_{i+1}\ ({\rm mod}\ (x_1,\cdots, x_i)
)$ for $i=0,1,\cdots,n-1$, where $f_n=1$ (or a unit).
Now the criterion reads:
\begin{prop}
Let $X$ be a smooth projective variety of dimension $M$ over a
field of characteristic $p > 0$, and let $Z_1, \ldots, Z_M$
be irreducible closed subvarieties of codimension 1.
Suppose that there is a point $P \in X$ such that
$Z_1+ \cdots + Z_M$
has residually normal crossing at $P$.
{}Further suppose that there exists
a global section s of the anti-canonical bundle $K_X^{-1}$ such
that $\mathop{\rm div} s = Z_1 + \cdots + Z_M $.
Then the section $\sigma = s^{p-1}$ gives
a simultaneous compatible
splitting of $Z_1, \ldots, Z_M$ in $X$.
This is also a compatible splitting of
any variety obtained from $Z_1,\ldots,Z_M$ by repeatedly
taking intersections and irreducible
components.
\end{prop}
This works because
there is bijection between sections $s$
of $K_X^{-1+p}$
and $ {\cal O} _X$-module morphisms
$\phi: F_* {\cal O} _X \rightarrow {\cal O} _X$,
cf.\ appendix A3 of \cite{vdK}.
In the notation above, one shows that
$\phi$ induces a map $F_* \widehat{ {\cal O} _P}/(x_1,\ldots,x_i) \rightarrow
\widehat{ {\cal O} _P}/(x_1,\ldots,x_i)$
corresponding with the ``residue'' of $s$ along $x_1=\cdots=x_i=0$
whose divisor
is described by $f_i$.
Several other useful properties of Frobenius splittings can be found
in
\cite{R2}.
We will apply our theory first in the case $n=3$,
and then indicate the modifications necessary for general $n$.
Instead of directly splitting the pair
${ \FF_{3,n} } \subset \mbox{\rm Gr}(D_3)$, we will find it more convenient
to use an intermediate subspace, which for $n=3$ is just
the triple product of flag varieties $(G/B)^3$.
We embed ${ \FF_{3,3} } \subset (G/B)^3$ via the map
$(p_1, p_2, p_3, l_1, l_2, l_3, P)\mapsto ((p_1,l_2),(p_2,l_3),(p_3,l_1))$.
\begin{lem}
The pair ${ \FF_{3,3} } \subset (G/B)^3$ is compatibly Frobenius split
\end{lem}
\noindent {\bf Proof.}
We describe the divisor giving our Frobenius splitting on ${ \FF_{3,3} }$.
Given $(xB,yB,zB)\in (G/B)^3$, represented by matrices $x$, $y$,
$z$,
we consider the formula
\begin{eqnarray*}
\llap{$s$}&=&
\left| \begin{array}{ccc}
x_{11} & y_{11} & y_{12} \\
x_{21} & y_{21} & y_{22} \\
x_{31} & y_{31} & y_{32}
\end{array} \right|
\ \cdot \
\left| \begin{array}{ccc}
z_{11} & x_{11} & x_{12} \\
z_{21} & x_{21} & x_{22} \\
z_{31} & x_{31} & x_{32}
\end{array} \right|
\ \cdot \
\left| \begin{array}{ccc}
y_{11} & z_{11} & z_{12} \\
y_{21} & z_{21} & z_{22} \\
y_{31} & z_{31} & z_{32}
\end{array} \right|
\ \cdot \\&&
\left( \ \left| \begin{array}{cc}
x_{21} & x_{22} \\
x_{31} & x_{32}
\end{array} \right|
\cdot
\left| \begin{array}{cc}
y_{11} & y_{12} \\
y_{21} & y_{22}
\end{array} \right|
-
\left| \begin{array}{cc}
x_{11} & x_{12} \\
x_{21} & x_{22}
\end{array} \right|
\cdot
\left| \begin{array}{cc}
y_{21} & y_{22} \\
y_{31} & y_{32}
\end{array} \right|
\ \right)
\ \cdot \\&&
\left| \begin{array}{cc}
x_{11} & y_{11} \\
x_{31} & y_{31}
\end{array} \right|
\ \cdot \
\left| \begin{array}{cc}
z_{11} & z_{12} \\
z_{21} & z_{22}
\end{array} \right|
\ \cdot \
z_{3,1}
\end{eqnarray*}
Each factor is a section of some line bundle over $(G/B)^3$.
The first three factors describe the incidence relations $p_1\in
l_3$,
$p_3\in l_2$, $p_2\in l_1$ respectively. Their product is a section
of the pull-back to $(G/B)^3$ of the $ {\cal O} (1,\dots,1)$ bundle over
$({\bf P}^2)^3 \times \mbox{\rm Gr}(1,{\bf P}^2)^3$. (Just look which Pl\"ucker
coordinates are used.)
Similarly, the last four factors describe a section of that same
bundle so
that in total $s$ is a section of the anti-canonical bundle of
$(G/B)^3$,
which is the pullback of $ {\cal O} (2,\dots,2)$.
Choosing local coordinates
sensibly, one checks that $\mathop{\rm div} s$ has residually normal crossing
at
the totally degenerate triangle $(E_1,E_1,E_1,E_{12},E_{12},E_{12})$
and we thus have a splitting of $(G/B)^3$ compatible with the components
of
$\mathop{\rm div} s$ and with the intersection ${ \FF_{3,3} }= { \cal I } _{3,3}$ of the three
components
given by the first three factors of $s$.
One also computes that $s$ vanishes
to order five at the locus of totally degenerate triangles.
By \cite{LMP} this implies that the splitting extends to the blowup
of $(G/B)^3$ along that locus, compatibly with the proper transform
of
${ \FF_{3,3} }$.
\subsection{ Proof of Theorem \ref{vanishing} }
{\bf Proof for $n=3$.}
If $D$ contains
each of the columns of $D_3$ at least once,
then $ \LL_D $ is ample on $(G/B)^3$
and Proposition \ref{splitsur} implies
that $H^i( \FF_D , \LL_D )$ vanishes for $i>0$.
{}Furthermore,
$H^0((G/B)^3, \LL_D )\to H^0( \FF_D , \LL_D )$ is surjective,
and a
well-known fact
from representation theory \cite[II 14.20]{J} states
that the restriction map
$H^0(({\bf P}^2)^3 \times \mbox{\rm Gr}(1,{\bf P}^2)^3, {\cal L} )\to H^0((G/B)^3, {\cal L} )$
is surjective for any effective $ {\cal L} $.
Thus $H^0(\mbox{\rm Gr}(D), \LL_D ) \rightarrow H^0( \FF_D , \LL_D )$ is surjective,
and $H^0( \FF_D , \LL_D ) \cong S^*_D$ by the previous section,
proving part (a) of the Theorem in this case.
As for part (b), normality follows from the theorem
of Mehta and Srinivas, provided ${ \FF_{3,3} }$ possesses
a resolution of singularities with connected fibers.
We will construct two such resolutions in later sections.
The normality, together with the surjectivity of the restriction
map
above for any multiple of $D$,
is essentially equivalent to projective normality
by \cite{Hart}, Ch II, Ex 5.14(d),
given the projective normality of the Pl\"ucker embedding.
If $D$ does not contain all the columns of $D_3$, then we need a
strengthening of
Proposition \ref{splitsur}. Indeed Ramanathan has proved \cite[1.12]{R2}
that
instead of requiring $ {\cal L} $ to be ample it suffices
to have a subdivisor $E$ of $\mathop{\rm div} s$, not containing any component
of
$Y$,
so that $ {\cal L} ^p\otimes {\cal O} (E)$ is ample. In our case such an $E$ is
provided by the last four factors of $s$, when $ {\cal L} = \LL_D $.
We also need to show that $H^0( \FF_D , \LL_D )$ may be identified with
$H^0({ \FF_{3,3} }, \LL_D )$. For this one considers the projection map
from $({\bf P}^2)^3 \times \mbox{\rm Gr}(1,{\bf P}^2)^3$ to the subproduct
corresponding to the columns that do occur in $D$
(amongst the first six columns
of $D_3$). What one needs to know is that
$\pi_* {\cal O} _{{ \FF_{3,3} }} =
{\cal O} _{ \FF_D }$\, ,
where $\pi:{ \FF_{3,3} }\to \FF_D $ is the restriction of the projection map
to ${ \FF_{3,3} }= { \cal I } _{3,3}$.
Now the fibers of $\pi$
are connected and \cite[6.1.6 and A.1.5]{vdK} apply.
This finishes the proof for $n=3$.
\\[.5em]
\noindent {\bf Case of general $n$.}
\\[.5em]
{}For $n>3$ we need to repeat the argument
``fibered over $\mbox{\rm Gr}(2,{\bf P}^{n-1})$''.
An element of ${ \FF_{3,n} }$ may be represented
by a tuple $(M,x,y,z)$ where
$M$
is an $n \times 3$ matrix of rank three, whose columns span the plane
$P$ in ${\bf P}^{n-1}$, and $x$, $y$, $z$ are three by three matrices
as
before. The flags $((p_1,l_2)$, $(p_2,l_3)$, $(p_3,l_1))$
are described by the first two columns of
of $Mx$, $My$, $Mz$ respectively.
Let ${P_3}$ denote the stabilizer in $G$ of the plane $E_{123}$.
If $X$ is any ${P_3}$-space, we denote by $G\times^{P_3} X$ the associated
$G$-space fibered over $G/{P_3}=\mbox{\rm Gr}(2,{\bf P}^{n-1})$,
with fiber $X$ over the point $E_{123}$.
If we replace in the formula for $s$
each $x$ by $Mx$, each $y$ by $My$, each $z$ by $Mz$,
then we get a section of
the relative anti-canonical bundle of the fibration
$$
\begin{array}{ccc}
(GL(3)/B(3))^3 & \rightarrow & G\times^{P_3} (GL(3)/B(3))^3 \\
& & \downarrow \\
& & \mbox{\rm Gr}(2,{\bf P}^{n-1})
\end{array}
$$
Indeed it has the correct transformation properties under $G$ and
under
${P_3}$ and it restricts to our known
section of the anti-canonical bundle of the fiber over $E_{123}$.
Thus to get a section of the anti-canonical bundle of the total space,
one must still
multiply with a section of the pull-back of the anti-canonical
bundle of the base $\mbox{\rm Gr}(2,{\bf P}^{n-1})$.
There is a choice here. Let us take one for which one can easily
check
that it has residually normal crossing at $E_{123}$, to wit the
product
of
the $n$ Pl\"ucker coordinates (subdeterminants of $M$)
based on taking $3$ consecutive rows of $M$,
with the rows ordered cyclically.
The result of all this is a
section of the anti-canonical bundle of the total space
$G\times^{P_3} (GL(3)/B(3))^3$ that
gives us a splitting with the same virtues as in the case $n=3$.
In particular, it is compatible with ${ \FF_{3,n} }$.
(It suffices to check this in a
neighborhood of the fiber of $E_{123}$.)
The proof of the Theorem now goes through exactly as before, except
for one problem we still need to address: The analogue of the
``well-known fact from representation theory'' needs to be proved
now.
We need to show
that the map
$H^0(\mbox{\rm Gr}(D_3), {\cal L} )\to H^0(G\times^{P_3} (GL(3)/B(3))^3, {\cal L} )$
is surjective for
effective line bundles on $\mbox{\rm Gr}(D_3)$.
This is indeed a fact in representation
theory, for which we refer to the Appendix.
Theorem \ref{vanishing} is proved.
\subsection{Fixed points}
In order to gain further information about the
triangle space ${ \FF_{3,n} }$ and its line bundles, we will
use the method of Lefschetz: that is, to study the
fixed points of the torus of diagonal
matrices $T \subset GL(n)$ acting on our space.
The work of Atiyah, Bott, and others will then give
us precise formulas for the cohomologies of coherent
sheaves, expressed in terms of the
combinatorial data of the $T$-fixed points and their tangent vectors.
This technique applies only to smooth varieties,
so in subsequent sections
we will study desingularizations of the triangle space.
However, a desingularization map is an isomorphism on the
smooth locus of the variety, so
if a fixed point $\tau$ is a smooth point of ${ \FF_{3,n} }$,
then the local tangent data will be the same for ${ \FF_{3,n} }$ and
all desingularizations. We begin by
examining these smooth points.
We adopt the combinatorial framework for dealing with
general configuration varieties developed in \cite{MaNW},
\cite{MaSchub}. See also \cite{H}, Lect 16.
A $T$-fixed point of the Grassmannian $\mbox{\rm Gr}(c,{\bf C}^n)$
is a $c$-dimensional subspace spanned
by coordinate vectors $\{e_{i_1}, \ldots, e_{i_c}\}$,
a subset of
the standard basis $\{e_1,\ldots, e_n\}$ of ${\bf C}^n$.
That is, the fixed points are the spaces $E_I$ for
$I = \{i_1 \leq \cdots \leq i_c\} \subset [1,n]$.
Hence, we may index the fixed points of $ \FF_D $, for
any diagram $D$, by {\em column tabloids} $\tau$,
which are maps $\tau : D \rightarrow [1,n]$,
strictly increasing down each column, such that
for every inclusion of columns $C \subset C'$,
we have $\tau(C) \subset \tau(C')$.
(More precisely, these are the fixed points of $ { \cal I } _D$,
but in our case this is identical to $ \FF_D $.)
One may check for our $D = D_3$ that there are
$11 n(n-1)(n-2)$ such tabloids. As we shall see,
all of them are smooth points of ${ \FF_{3,n} }$ except those
corresponding to maximally degenerate triangles:
namely, the singular fixed points are of the form
$$
\tau_{ijk} = \left[
\begin{array}{ccccccc}
i & & & & i & i & i \\
& i & & i & & j & j \\
& & i & j & j & & k
\end{array}
\right] \ \ ,
\hspace{1in} \mbox{}
$$
for $i,j,k$ distinct integers in $[1,n]$.
{}For a space $V \in \mbox{\rm Gr}(c, {\bf C}^n)$, we may model the tangent
space as $T_V \mbox{\rm Gr}(c,{\bf C}^n) \cong \mbox{\rm Hom}_{{\bf C}}(V, {\bf C}^n/V )$.
(That is, the tangent bundle of the Grassmannian is isomorphic
to $\mbox{\rm Hom}$ of the tautological subbundle into the
tautological quotient bundle.) Furthermore, the incidence
variety
$$
{ \cal I } = \{(U,V) \in \mbox{\rm Gr}(c,{\bf C}^n) \times \mbox{\rm Gr}(c',{\bf C}^n) \mid U \subset
V \}
$$
has tangent space
$$
T_{(U,V)} { \cal I } = \{(\phi,\psi) \in \mbox{\rm Hom}_{{\bf C}}(U,{\bf C}^n/U)
\times \mbox{\rm Hom}_{{\bf C}}(V,{\bf C}^n/V) \mid \psi |_U = \phi \mbox{ mod }
V \} \ .
$$
{}From this, one can deduce as in \cite{MaNW}
\begin{lem}
The eigenvalues of $T$ near a smooth fixed point $\tau$
of ${ \FF_{3,n} }$
are all of the form $ \lambda ( \mbox{\rm diag}(x_1,\ldots,x_n) ) = x_i^{-1} x_j$
for
$i \neq j \in [1,n]$. The multiplicity $d_{ij}(\tau)$ of
$x_i^{-1} x_j$ is the number of connected components of the
following graph: vertices = $\{ \mbox{columns } C \mbox{ of }
D
\mid i\in \tau(C), \ j\not\in \tau(C) \}$,
edges = $\{(C,C') \mid C \subset C' \mbox{ or } C' \subset C \}$.
\end{lem}
{}For instance, consider the singular fixed point $\tau = \tau_{ijk}$
defined above.
Suppose $l,m \neq i,j,k$. Then the multiplicities
$d_{ab}(\tau)$ are given
by the table:
$$
\begin{array}{r@{\!}l}
\begin{array}{cc}&\\&d_{ab}\end{array}&
\begin{array}{ccccc}&&b & &\\i & j & k & l & m\end{array}\\
\begin{array}{cc}&i\\&j\\a&k\\&l\\&m\end{array}&
\begin{array}{|ccccc}\hline
0 & 3 & 1 & 1 & 1 \\
0 & 0 & 3 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}
\end{array} \ .
$$
Note that this makes a total of $3n-2$ eigenvectors,
whereas ${ \FF_{3,n} }$ is $3n-3$ dimensional. The $3n-2$ vectors
span the Zariski tangent space of ${ \FF_{3,n} } \subset \mbox{\rm Gr}(D_3)$
at the singular point.
These eigenvectors also correspond to $T$-stable curves
through $\tau$: that is,
$\Phi: {\bf C} \rightarrow { \FF_{3,n} }$, with $\Phi(0)=\tau$ and
$x\cdot \Phi(s) = \Phi( \lambda (x) s)$
for all $s \in {\bf C}$ and $x \in T$, and some eigenvalue character
$ \lambda $.
{}For all other tabloids, there are $3n-3$ eigenvectors, and the
fixed points are smooth points of ${ \FF_{3,n} }$. Now, the singular locus
of ${ \FF_{3,n} }$
is $GL(n)$-invariant, and every $GL(n)$-orbit which is not maximally
degenerate
approaches some $T$-fixed point which is not maximally degenerate,
so we may
conclude:
\begin{lem}
The singular locus of ${ \FF_{3,n} }$ consists of the maximally
degenerate triangles (for which all three points and all three
lines coincide).
\end{lem}
\section{The Schubert--Semple space}
Next we consider the smooth triangle space ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ first defined by
Schubert \cite{Schubert}, and given a modern construction
by Semple \cite{Semple}. See also Collino and Fulton \cite{CF}.
The points of this space may be thought of intuitively
as triangles with an extra piece of data: a circle passing
through the three vertices, and tangent to the corresponding
side if two vertices coincide. The circle is determined by the
triangle in all cases except the maximally degenerate triangles,
for which there is a ${\bf P}^1$ of compatible circles (including
radius zero and infinity).
The rigorous definition is as follows.
{}First, let $n=3$.
The space $Q$ of conic curves
in ${\bf P}^2$ can be identified as $Q = {\bf P}^*(\mbox{\rm Sym}\mbox{}^2 {\bf C}^3) \cong
{\bf P}^5$.
A projective plane in this ${\bf P}^5$ is called a {\em net}
of conics, and the space of nets is $\mbox{\rm Gr}(2, Q)$.
Let $ { \cal F } ^{\circ}_{3,3}$ be the general triangles in ${\bf P}^2$.
{}For any general triangle $\tau$, the conics
passing through its three vertices form a net $N_{\tau}$,
and we have an embedding
$$
\begin{array}{crcl}
\Phi: & { \cal F } ^{\circ}_{3,3} & \rightarrow &
{ \cal F } ^{\circ}_{3,3} \times \mbox{\rm Gr}(2, Q ) \\
& \tau & \mapsto & (\tau, N_{\tau})
\end{array}.
$$
The Schubert--Semple space is defined as the closure of the
image:
$$
{ \cal F } _{3,3}^{ { \cal S\hspace{-.2em}S } } = \mbox{closure } \mathop{\rm Im}(\Phi) \ \subset \
({\bf P}^2)^3 \times \mbox{\rm Gr}(1, {\bf P}^2)^3 \times \mbox{\rm Gr}(2,Q) \ .
$$
Now, for general $n$, we define ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ as a family of
such spaces with the plane varying in ${\bf P}^{n-1}$:
$$
\begin{array}{ccc}
{ \cal F } _{3,3}^{ { \cal S\hspace{-.2em}S } } & \rightarrow & { \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } } \\
& & \downarrow \\
& & \mbox{\rm Gr}(2,{\bf P}^{n-1})
\end{array}
$$
More formally,
$$
{ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } } = GL(n)\times^{P_3} { \cal F } _{3,3}^{ { \cal S\hspace{-.2em}S } }=
{GL(n) \times { \cal F } _{3,3}^{ { \cal S\hspace{-.2em}S } } \over {P_3}} \ .
$$
Here, ${P_3} \subset GL(n)$ is again the parabolic subgroup such
that
$GL(n)/{P_3} \cong \mbox{\rm Gr}(2, {\bf P}^{n-1})$, and ${P_3}$ acts on
$GL(n) \times { \cal F } _{3,3}^{ { \cal S\hspace{-.2em}S } }$ by
$p \cdot (g,t) = (gp, b(p^{-1}) t)$, $\ b:{P_3} \rightarrow GL(3)$
being the obvious homomorphism.
Semple shows that this is a smooth projective variety,
and the obvious projection ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } } \rightarrow { \FF_{3,n} }$ is an
isomorphism on the smooth locus of ${ \FF_{3,n} }$.
We wish to find the $T$-fixed points on ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$, as
well as their tangent eigenvectors.
{}For a point in ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ whose image is a smooth point of
${ \FF_{3,n} }$, this follows immediately from the results of the
previous section.
It remains to consider the fixed points of ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$
lying above the
singular fixed point
$$
\tau_{ijk} = (E_i,E_i,E_i,E_{ij},E_{ij},E_{ij},
E_{ijk}) \in { \FF_{3,n} } \ .
$$
The degenerate triangle $\tau_{ijk}$ lies in the plane
$E_{ijk}$, so that
$$
\mbox{\rm Sym}\mbox{}^2(E_{ijk})^* =
\langle u_i^2, u_j^2, u_k^2, u_i u_j, u_i u_k, u_j u_k \rangle \
,
$$
where $u_1, \ldots, u_n$ is the dual of the standard basis of ${\bf C}^n$.
A diagonal matrix $x =\mbox{\rm diag}(x_1,\ldots,x_n) \in T$ acts on
a monomial by the character
\mbox{$x \cdot u_i u_j = (x_i x_j)^{-1} u_i u_j$}.
According to \cite{CF}, p. 79, the fiber above $\tau_{ijk}$
consists of all nets
$N \in \mbox{\rm Gr}(2, {\bf P}^*(\mbox{\rm Sym}\mbox{}^2 E_{ijk}))$
such that
$$
\langle u_k^2, u_j u_k \rangle \subset
N \subset \langle u_k^2, u_j u_k, u_j^2, u_i u_k \rangle \ .
$$
This is a copy of ${\bf P}^1 \subset { \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$.
More precisely, it is isomorphic as a $T$-space to
${\bf P}({\bf C}_{x_j^{-2}} \oplus {\bf C}_{(x_i x_k)^{-1}})$,
where ${\bf C}_{ \lambda }$ is the one-dimensional representation of $T$
with character $ \lambda $.
There are exactly two nets in this fiber
which are fixed by $T$, namely
$$\eta_{ijk} = \langle u_k^2, u_j u_k, u_j^2 \rangle = [1:0]
$$
$$\zeta_{ijk} = \langle u_k^2, u_j u_k, u_i u_k \rangle = [0:1]
\ . $$
That is, each singular fixed point $\tau_{ijk} \in { \FF_{3,n} }$
splits into two isolated fixed points $\eta_{ijk}, \zeta_{ijk}
\in { \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$.
Now we find the eigenvalues of $T$ acting on the tangent
spaces of these fixed points.
{}First, consider the tangent vector at $\eta_{ijk}$
pointing along the
fiber above $\tau_{ijk}$.
The tangent space of this ${\bf P}^1$ at $\eta_{ijk} = [1:0]$ is
$$
\mbox{\rm Hom}({\bf C}_{x_j^{-2}}, {\bf C}_{(x_i x_k)^{-1}}) \cong
{\bf C}_{ x_j^{2} (x_i x_k)^{-1} } \ ,
$$
and the eigenvalue of our tangent vector is
$x_j^{2} (x_i x_k)^{-1}$.
Similarly, $(x_i x_k) x_j^{-2}$ is an eigenvalue of
$T$ acting on the tangent space at $\zeta_{ijk}$.
The other eigenvalues of the smooth tangent spaces
can all be found from examining the
Zariski tangent space of the singular
point $\tau_{ijk}$. For example, consider
the eigenvector
defined by
\begin{eqnarray*}
\phi \in T_{\tau_{ijk}} \mbox{\rm Gr}(D_3)& = &\mbox{\rm Hom}(E_i, {\bf C}^n / E_i)^3 \oplus
\mbox{\rm Hom}(E_{ij}, {\bf C}^n / E_{ij})^3 \oplus \\
&&\qquad\mbox{\rm Hom}(E_{ijk}, {\bf C}^n / E_{ijk})^3\ ,
\end{eqnarray*}
\begin{eqnarray*}
\phi& = &(\phi_{ij}, -\phi_{ij}, 0, 0 ,0,0,0) \ ,
\end{eqnarray*}
where $\phi_{ij}(e_l) = \delta_{il} e_j$.
This eigenvector has eigenvalue $x_i^{-1} x_j$
and points along the $T$-stable curve
$\Phi : {\bf C} \rightarrow { \FF_{3,n} }$
$$
\Phi(s) = (\Phi_{ij}(s), \Phi_{ij}(-s), E_i, E_{ij}, E_{ij}, E_{ij},
E_{ijk}) \ ,
$$
where $\Phi_{ij}:{\bf C} \rightarrow {\bf P}^{n-1}$,
$\Phi_{ij}(s) = E_i + s E_j$.
Now, $\Phi({\bf C} - 0)$ lies in the smooth locus
of ${ \FF_{3,n} }$, so it lifts uniquely to a $T$-stable curve
$\Phi^{ { \cal S\hspace{-.2em}S } }$ in ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$.
Clearly $\Phi^{ { \cal S\hspace{-.2em}S } }(0)$ is a fixed point above $\tau_{ijk}$,
and we may easily check that it is $\zeta_{ijk}$.
Differentiating $\Phi^{ { \cal S\hspace{-.2em}S } }$ at $s = 0$, we obtain a tangent vector
to $\zeta_{ijk}$ with eigenvalue $x_i^{-1} x_j$.
We may argue similarly for the other Zariski tangent vectors
which do not point along the singular locus of ${ \FF_{3,n} }$.
{}For a vector which {\em does} point along the singular locus,
for instance
$$
\phi = (\phi_{ij},\phi_{ij},\phi_{ij},0,0,0,0) \ ,
$$
the corresponding curve does not lift uniquely to
a $T$-stable curve in ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$.
In fact, it lifts in exactly two ways, one leading to
each point $\eta_{ijk}$, $\zeta_{ijk}$ above $\tau_{ijk}$.
Hence $\phi$ accounts for a tangent vector with eigenvalue
$x_i^{-1} x_j$ at {\em each} of the lifted fixed points.
Summarizing, we get:
\begin{lem}
\label{SS}
The eigenvalues at the fixed points in ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$
above $\tau_{ijk}$
are as follows:
$$
\begin{array}{cl}
\eta_{ijk} & \ \ \
x_i^{-1} x_k^{-1} x_j^{2}, \
x_i^{-1} x_j, \
x_i^{-1} x_k, \
x_j^{-1} x_k \mbox{ (3 times)}, \
x_i^{-1} x_l, \
x_j^{-1} x_l, \
x_k^{-1} x_l \\
& \\
\zeta_{ijk} & \ \ \
x_i x_k x_j^{-2}, \
x_i^{-1} x_j \mbox{ (3 times)}, \
x_i^{-1} x_k, \
x_j^{-1} x_k, \
x_i^{-1} x_l, \
x_j^{-1} x_l, \
x_k^{-1} x_l \ ,
\end{array}
$$
where $l$ runs over $[1,n] \setminus \{i,j,k\}$.
\end{lem}
\section{The Fulton--MacPherson Space}
We describe another desingularization
of ${ \FF_{3,n} }$, a very special case of
the construction of Fulton and MacPherson in \cite{FM}.
It seems this space ${ \FF_{3,n}^{\FM} }$ is isomorphic to ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ as an
abstract variety (\cite{FM}, p. 189),
but even if that is so, then it comes equipped with a different
map ${ \FF_{3,n}^{\FM} } \rightarrow { \FF_{3,n} }$ and a different $GL(n)$-action.
In particular, the $T$-fixed points of ${ \FF_{3,n}^{\FM} }$ are not isolated,
leading to a different type of local data
for our fixed-point formulas below.
Our analysis of the Fulton--MacPherson space
will also show that ${ \FF_{3,n} }$ has rational singularities.
\subsection{Strata}
Again, we first consider the case $n = 3$, and we will have a
fiber bundle ${ \FF_{3,3}^{\FM} } \rightarrow { \FF_{3,n}^{\FM} } \rightarrow \mbox{\rm Gr}(2, {\bf P}^{n-1})$.
It is shown in \cite{FM} how ${ \FF_{3,3}^{\FM} } = {\bf P}^2[3]$ can be constructed
as a union of 8 strata,
each consisting of
certain configurations of points and tangent vectors in
${\bf P}^2$.
{}For each stratum, there is
a natural $GL(3)$ action and an equivariant map to ${ \FF_{3,3} }$.
\begin{itemize}
\item
$D_{\emptyset}$, the configuration space
$({\bf P}^2)^3 \setminus \bigcup \Delta_{ij}$. \\
An open set in ${ \FF_{3,3}^{\FM} }$. \\
Triples $[p_1, p_2, p_3]$ of pairwise-distinct points. \\
A triple maps to the triangle
$$
(p_1,p_2,p_3, {\bf C} p_2 + {\bf C} p_3,
{\bf C} p_1 + {\bf C} p_3, {\bf C} p_1 + {\bf C} p_2) \ ,
$$
where ${\bf C} p + {\bf C} p'$ means the projective line through the points.
\item
$D_{12}$, corresponding to the diagonal
$\Delta_{12} \subset ({\bf P}^2)^3 $. \\
Codimension 1 in ${ \FF_{3,3}^{\FM} }$.\\
Configurations
$[p_{12} , p_3$, ${\bf C}^*v_3]$:
distinct points $p_{12}$ and $p_3$,
and a non-zero tangent vector
$v_3\in T_{p_{12}} {\bf P}^{n-1}$, up to
scaling of $v_3$. \\
Intuitively represents infinitesimally distinct points
$
(p_{12}
,\, p_{12} + v_3,\, p_3) \ .
$ \\
Maps to the triangle
$$
(\, p_{12}, p_{12}, p_3,\,
{\bf C} p_{12} + {\bf C} p_3,\, {\bf C} p_{12} + {\bf C} p_3, \,
{\bf C} p_{12} + {\bf C}
v_3\, ) \ .
$$
Similarly $D_{23}$, $D_{13}$.
\item
$D_{123}$, corresponding to the
total diagonal $\Delta_{123}$ of $({\bf P}^2)^3$.\\
Codimension 1 in ${ \FF_{3,3}^{\FM} }$. \\
Configurations
$[p_{123}, \, {\bf C}^*( v_1, v_2, v_3)]$: a point and three
non-zero tangent
vectors with $v_1 + v_2 + v_3 = 0$, up to
simultaneous scaling.\\
Intuitively represents
$
(p_{123
, p_{123}
+v_3, p_{123}
-v_2) \ .
$ \\
Maps to the triangle
$$
(\, p_{123}, p_{123}, p_{123}, \,
{\bf C} p_{123} + \C
v_1, \,
{\bf C} p_{123} + \C
v_2, \,
{\bf C} p_{123} + \C
v_3 \, ) \ .
$$
\item
$D_{123,12}$. \\
Codimension 2 in ${ \FF_{3,3}^{\FM} }$. \\
Configurations
$[p_{123}, {\bf C}^*v_{12}, {\bf C}^*v_3]$:
a point and two non-zero tangent vectors
up to scaling of each. \\
Intuitively represents
$
(\, p_{123} + v_{12} ,\,
p_{123} + v_{12} + v_3,\,
p_{123} \, ) \ ,
$ with $v_3$ infinitesimal compared to $v_{12}$, which is
itself already infinitesimal.\\
Maps to the triangle
$$
(\, p_{123}, p_{123}, p_{123}, \,
{\bf C} p_{123} + {\bf C} v_{12}, \,
{\bf C} p_{123} + {\bf C} v_{12}, \,
{\bf C} p_{123} + {\bf C} v_3 \, ) \ .
$$
Similarly $D_{123,23}$, $D_{123,13}$.
\end{itemize}
The closures of the codimension 1 strata are smooth divisors:
$\overline{D_{ij}} = D_{ij} \cup D_{123,ij}$, \ \
$\overline{D_{123}} = D_{123} \cup D_{123,12} \cup
D_{123,23} \cup D_{123,13}$,
and the intersections are transversal.
The above description is enough to define coordinate charts
for ${ \FF_{3,3}^{\FM} }$, gluing together the normal bundles of
the strata appropriately.
Also, the maps described above piece together into
a regular, equivariant desingularization $\pi: { \FF_{3,3}^{\FM} } \rightarrow
{ \FF_{3,3} }$.
The map is an isomorphism outside the singular locus of ${ \FF_{3,3} }$,
and the inverse image of a singular triangle
$(p,p,p,l,l,l)$ is the set of configurations:
$[p, {\bf C}^*(v_1,v_2,v_3)] \in D_{123}$ such that
$v_1$, $v_2$, $v_3$ are non-zero and parallel to $l$,
and $v_1 + v_2 + v_3 = 0$.
(There are also
three extra configurations
in $D_{123,12}$, $D_{123, 23}$, and $D_{123,13}$.)
Thus, the fiber $\pi^{-1}(p,p,p,l,l,l)$ is a projective line,
and it is easily seen that if $(p,p,p,l,l,l)$ is fixed by the torus
$T$,
then each point of this fiber is also fixed.
\subsection{Fixed lines}
Now let us consider the general ${ \FF_{3,n}^{\FM} }$, for which everything we have
said
carries over. In particular, ${ \FF_{3,n}^{\FM} }$ has two classes of fixed points:
the isolated ones, which map one-to-one to the
non-singular fixed triangles in
${ \FF_{3,n} }$, and the fibers over
the singular fixed triangles $\tau_{ijk}$,
$$
\PP^1_{ijk} = \pi^{-1}(\tau_{ijk}) \ .
$$
The tangent data at the isolated points is identical to that in
${ \FF_{3,n} }$.
{}For the $ \PP^1_{ijk} $, we will need to determine the types of
their normal bundles
as $T$-equivariant vector bundles over ${\bf P}^1$.
{}First, note that locally near $ \PP^1_{ijk} $, we have
$$
{ \FF_{3,n}^{\FM} } \cong { \FF_{3,3}^{\FM} }(E_{ijk}) \times T_{E_{ijk}} \mbox{\rm Gr}(2,{\bf P}^{n-1}) .
$$
That is, the normal bundles in the direction of the Grassmannian
are trivial, and we reduce to
${ \FF_{3,3}^{\FM} }(E_{ijk})$, the Fulton--MacPherson space relative to the
plane ${\bf P}^2 = {\bf P}(E_{ijk})$.
Now we will use the
alternative description (\cite{FM}, p. 196)
for ${ \FF_{3,3}^{\FM} }$ as a blowup of $({\bf P}^2)^3$,
first along the triple diagonal $\Delta_{123}$, then along the
proper transforms of the three partial diagonals $\Delta_{12}$,
$\Delta_{23}$,
$\Delta_{13}$.
Let ${\bf C}_{ab}$ denote the one-dimensional $T$-space
on which $\mbox{\rm diag}(x_1,\ldots,x_n)$ acts by the character $x_a^{-1}
x_b$.
We may take coordinates for a neighborhood
$U \subset ({\bf P}^2)^3 = {\bf P}(E_{ijk})^3$
near the fixed point $\tau_i = (E_i, E_i, E_i)$
so that, as $T$-spaces, we have
$$
\begin{array}{ccc}
U & \cong & \left( {\bf C}_{ij} \times {\bf C}_{ik} \right)^3 \\
\tau_i & \cong & (0,0,0,0,0,0) \\
\Delta_{12} & \cong & \{ (0,0,a,b,c,d) \} \\
\Delta_{23} & \cong & \{ (a,b,0,0,c,d) \} \\
\Delta_{13} & \cong & \{ (a,b,a,b,c,d) \} \\
\Delta_{123} & \cong & \{ (0,0,0,0,c,d) \} \ .
\end{array}
$$
In this blowup, there will be two $T$-fixed ${\bf P}^1$ above $\tau_i$
\
(namely, $\tau_{ijk}$ and $\tau_{ikj}$),
and we wish to determine their normal bundles.
Since all the centers of blowing up are products with the last
factor $({\bf C}_{ij} \times {\bf C}_{ik})$, the normal bundles will be
trivial
in these directions. Thus, we may reduce to
$$
\begin{array}{ccc}
U' & \cong & \left( {\bf C}_{ij} \times {\bf C}_{ik} \right)^2 \\
\Delta_{12}' & \cong & \{ (0,0,a,b) \} \\
\Delta_{23}' & \cong & \{ (a,b,0,0) \} \\
\Delta_{13}' & \cong & \{ (a,b,a,b) \} \\
\Delta_{123}' & \cong & \{ (0,0,0,0) \} \ .
\end{array}
$$
Performing the first blowup along $\Delta_{123}'$, we obtain:
$$
\begin{array}{ccc}
\mbox{\rm Bl}_{123} U' & \cong & {\cal O} (-1) \rightarrow
{\bf P}\left( ({\bf C}_{ij} \times {\bf C}_{ik})^2 \right)
\\
\tilde{\Delta}_{12}' & \cong & {\cal O} (-1) \rightarrow
\{\, [0:0:a:b] \, \} \\
\tilde{\Delta}_{23}' & \cong & {\cal O} (-1) \rightarrow
\{\, [a:b:0:0] \, \} \\
\tilde{\Delta}_{13}' & \cong & {\cal O} (-1) \rightarrow
\{ [a:b:a:b]\, \} \ .
\end{array}
$$
This has two $T$-fixed projective lines: let us focus on
one of them,
$ {\bf P}( {\bf C}_{ij}^2 ) = \{ [a:0:b:0] \}$. The normal bundle of a
line ${\bf P}^1
\subset {\bf P}^2$
is $ {\cal O} (1)$, so
restricting to a neighborhood $U''$ of our fixed line gives
$$
\begin{array}{ccc}
U'' & \cong & {\cal O} _{ij}(-1) \oplus 2 {\cal O} _{jk}(1) \rightarrow {\bf P}^1
\\
\Delta_{12}'' & \cong & \{\, (v,0,w) \, \} \rightarrow
[0:1] \\
\Delta_{23}'' & \cong & \{\, (v,w,0) \, \} \rightarrow
[1:0] \\
\Delta_{13}'' & \cong & \{\, (v,w,w) \, \} \rightarrow
[1:1] \ ,
\end{array}
$$
where $ {\cal O} _{ij}(m)$ indicates a line bundle over a $T$-fixed ${\bf P}^1$
with fibers of type ${\bf C}_{ij}$.
Now consider the next blowup, along $\Delta_{12}''$. This is
locally a product of
$ {\cal O} _{ij}(-1) \oplus {\cal O} _{jk}(1)$ and the
locus
$$
(\, 0 \rightarrow [0:1] \, )\ \subset \
(\, {\cal O} _{jk}(1) \rightarrow {\bf P}^1 \, ) \ ,
$$
so the blowup will not affect the first factors, and we may concentrate
on the last.
Thus, consider
the total space of the line bundle $ {\cal O} (m)$ over ${\bf P}^1$, and
blow up at a point on the zero-section.
It is easily seen in coordinates that
the normal bundle of the proper transform of the zero-section
is $ {\cal O} (m-1)$.
Thus, the second, third, and fourth blowups will transform
$ {\cal O} _{ij}(-1) \oplus 2 {\cal O} _{jk}(1)$ successively into
$$
{\cal O} _{ij}(-1) \oplus {\cal O} _{jk} \oplus {\cal O} _{jk}(1), \ \
{\cal O} _{ij}(-1) \oplus {\cal O} _{jk} \oplus {\cal O} _{jk}, \ \
{\cal O} _{ij}(-1) \oplus {\cal O} _{jk}(-1) \oplus {\cal O} _{jk}\ .
$$
Recalling the dimensions we dropped at the beginning, we obtain
our final answer.
\begin{lem}\label{bundle}
The normal bundle of the $T$-fixed
component $ \PP^1_{ijk} $ in ${ \FF_{3,n}^{\FM} }$ is
$$
{\cal O} _{ij}(-1) \oplus {\cal O} _{jk}(-1) \oplus {\cal O} _{jk} \oplus
{\cal O} _{ij} \oplus {\cal O} _{jk} \oplus
\sum_{l \in [1,n] \atop l \neq i,j,k}
\left( {\cal O} _{il} \oplus {\cal O} _{jl} \oplus {\cal O} _{kl} \right) \ ,
$$
where $ {\cal O} _{ab}(m)$ is the line bundle with Chern class $m$ and
fibers of character $x_a^{-1} x_b$.
\end{lem}
Recall that this describes not just the normal bundle, but an actual
open neighborhood of the $T$-fixed
component $ \PP^1_{ijk} $ in ${ \FF_{3,n}^{\FM} }$. It follows easily that the canonical
bundle is
trivial, as a line bundle, on that neighborhood.
As $GL(3)$ acts
transitively on the stratum
of degenerate triangles, we get
\begin{lem}\label{free}
The canonical bundle is trivial in a neighborhood of the fiber in
${ \FF_{3,n}^{\FM} }$
of any
singular point.
\end{lem}
\subsection{Rational singularities}
In later sections, it will be convenient to know that
our singular space
${ \FF_{3,n} }$ has rational singularities. Let us first recall Kempf's definition
\cite{K}.
A birational proper map $\pi:Y\to X$ is called a {\em rational resolution}\/
if
$Y$ is smooth, and \\
a. $\pi_* {\cal O} _Y= {\cal O} _X$ or, equivalently, $X$ is normal,\\
b. $R^i\pi_* {\cal O} _Y=0$ for $i>0$,\\
c. $R^i\pi_*K_Y=0$ for $i>0$.\\
The last condition is automatic in characteristic $0$.
One says that $X$ has {\em
rational singularities}\/ if there exists a rational resolution $\pi:Y\to
X$.
The usefulness of this notion lies in the
\begin{lem}
\label{cohom equal}
Let $\pi : Y \to X$ be a map satisfying conditions
(a)
and (b),
and $ {\cal L} $ a line bundle on $X$.
Then $H^i(X, {\cal L} ) = H^i(Y,\pi^* {\cal L} )$ for all $i$.
\end{lem}
This follows from the projection formula and
a degenerate case of the Leray spectral sequence
\cite[III, Ex. 8.1, 8.3]{H}. We will use the Lemma below
in the case of the triangle space and its desingularizations.
Now, we have seen that the singularity of ${ \FF_{3,n} }$ is that of the
cone over a quadric in ${\bf P}^3$, and it is well known that this
singularity is rational,
but we shall prove it directly from the definition.
\begin{prop}
The map $\pi:{ \FF_{3,n}^{\FM} }\to{ \FF_{3,n} }$ is a rational resolution, and so is
the map ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }\to { \FF_{3,n} }$.
\end{prop}
\noindent{\bf Proof.}
The target ${ \FF_{3,n} }$ of $\pi$ is normal by Theorem \ref{vanishing}.
{}For the second condition we use Grothendieck's theorem on formal
functions.
It tells us that we should try to
show that $H^i(\pi^{-1}(P)_m, {\cal O} / { \cal I } ^m)$ vanishes, where
$\pi^{-1}(P)_m$
is the \hbox{$m$-th} order neighborhood of the fiber ${\bf P}^1$
over a point $P$ of the
singular locus and $ { \cal I } $\/ is the ideal sheaf of this fiber.
By d\'evissage we only need to show that
$H^i(\pi^{-1}(P)_m, { \cal I } ^{m-1}/ { \cal I } ^m)=H^i({\bf P}^1, { \cal I } ^{m-1}/ { \cal I } ^m)$
vanishes for $m>0$. Now $ { \cal I } ^{m-1}/ { \cal I } ^m$ is just a power of the
conormal
bundle, so by the computation above (lemma \ref{bundle}),
it is a sum of line bundles with
nonnegative Chern class. The result follows.
{}From lemma \ref{free} one sees that the $R^i\pi_*K_{ \FF_{3,n}^{\FM} }$
are locally the same as the $R^i\pi_* {\cal O} _{ \FF_{3,n}^{\FM} }$, so they vanish too.
Alternatively, one checks that the Grauert-Riemenschneider vanishing
theorem
with Frobenius
splitting \cite{MvdK} applies. For this, observe that our splitting
of
${ \FF_{3,n} }$ gives one on the complement of the exceptional locus of $\pi$
in ${ \FF_{3,n}^{\FM} }$.
As this exceptional locus has codimension two the splitting extends
and in fact
our section $s$ of the anti-canonical bundle extends.
The divisor of the extended $s$ contains the proper transform of
the divisor
of the factor
$
s_4=\left( (- x_{2,2}x_{3,1} +
x_{2,1}x_{3,2} ) (- y_{1,2}y_{2,1} +
y_{1,1}y_{2,2})\right.-
\left(- x_{1,2}x_{2,1} +
x_{1,1}x_{2,2} )( - y_{2,2}y_{3,1} +
y_{2,1}y_{3,2} )\right)
$
of $s$, hence it contains the exceptional locus, as the equation
of the divisor of $s_4$
only puts constraints on two lines in a configuration, no
further restrictions on its points.
\\[.5em]
Before leaving the case of the $ { \cal F\hspace{-.2em}M } $ resolution let us note
that by lemma~\ref{free} there is a line bundle $\omega=\pi_*K_{ \FF_{3,n}^{\FM} }$
on ${ \FF_{3,n} }$ whose restriction to the smooth locus is the canonical bundle.
Now let $\pi$ denote the map ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }\to { \FF_{3,n} }$ instead.
The pull-back of $\omega$ to
${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ agrees with the canonical bundle outside the exceptional locus,
which
has codimension two again. It follows that the pull-back is isomorphic
with
the canonical bundle, so the analogue of lemma \ref{free} holds.
Thus
the vanishing of $R^i\pi_*K_{ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ is equivalent again
to the vanishing of $R^i\pi_* {\cal O} _{ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$.
To apply the Grauert-Riemenschneider vanishing theorem we now use
the
factor $s_5=( x_{3,1}y_{1,1} - x_{1,1}y_{3,1} )$, whose divisor
has a proper
transform containing the exceptional locus.
Another reason that the proposition also holds for the $ { \cal S\hspace{-.2em}S } $
resolution is that it locally looks the same as the $ { \cal F\hspace{-.2em}M } $
resolution.
If locally we
see the singularity as a product of an affine space and a cone over
a product of two projective lines, then it
clearly has an automorphism that interchanges these two lines.
One can pass between the
$ { \cal S\hspace{-.2em}S } $ resolution and the $ { \cal F\hspace{-.2em}M } $ resolution
by means of this local automorphism.
\begin{cor}
The desingularizations ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$, ${ \FF_{3,n}^{\FM} }$, and $\mbox{Bl}_{\mbox{sing
locus}}
{ \FF_{3,n} }$ are
all Frobenius split varieties in any characteristic.
\end{cor}
{}Finally, we remark that one can construct the blowup of
${ \FF_{3,n} }$ along its singular locus as the
fibered product of ${ \FF_{3,n}^{\FM} }$ and ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ over ${ \FF_{3,n} }$.
\section{Fixed-point formulas}
We apply equivariant fixed-point
theorems to the spaces of the preceding sections,
putting together all the fixed-point data we have accumulated.
This produces explicit formulas
for the $GL(n)$-character and dimension
of the Schur module $S_D$ of any 3-row diagram $D$.
The formulas are more complicated
than those of \cite{MaNW} for northwest diagrams, but
essentially similar.
We discuss the general fixed-point theorems in the first section,
and in the following ones give a summary of the results
in elementary language. We conclude by discussing
the possibility of drawing geometric implications from the
combinatorial formulas.
\subsection{General theory}
In what follows, $X$ is
a smooth projective variety
of dimension $M$ over ${\bf C}$,
$L \rightarrow X$ an algebraic line bundle,
and $T = ({\bf C}^*)^n$ a torus acting
on $X$ and $L$.
{\em Throughout this section, we also assume the vanishing
of the higher cohomology groups of $L$}:
$$
H^i(X,L) = 0 \mbox{ for all } i > 0 \ .
$$
The following formula is due to Atiyah and
Bott
\cite{AB}.
\begin{prop}
\label{AB thm}
Suppose the torus $T$
acts $X$ with isolated fixed points.
Then the character of $T$ acting on
the space of global sections of $L$
is given by:
$$
\mathop{\rm tr}(x \mid H^0(X,L)) =
\sum_{p \ \mbox{\tiny fixed}} {\mathop{\rm tr}(x \mid L|_p )
\over \det( \mathop{\rm id} -\, x \mid T^*_p X )},
$$
where $p$ runs over the fixed points of $T$, $L|_p$ denotes the
fiber of $L$ above $p$, and $T^*_p X$ is the cotangent space.
\end{prop}
We apply this to
$X = { \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ and $L = \pi^* \LL_D $.
By Lemma \ref{cohom equal} and Theorem~\ref{vanishing},
we have $H^0({ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }, \pi^* \LL_D ) = H^0({ \FF_{3,n} }, \LL_D ) = S^*_D$
and $H^i({ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }, \pi^* \LL_D ) = H^i({ \FF_{3,n} }, \LL_D ) = 0$ for $i>0$,
so the above Proposition gives us a character formula for the Schur
module in terms of the fixed-point data of Lemma \ref{SS}.
We write out the result in the next section.
{}For the $ { \cal F\hspace{-.2em}M } $ space, we need a more general
formula due to Atiyah, Bott, and Singer \cite{AS}.
It requires
the following characteristic
classes for a vector bundle $V$
over any smooth variety $Y$:
$$
{\cal U}^{ \lambda }(V) =
\prod_{i} {1 - \lambda ^{-1}\exp(-r_i)
\over 1 - \lambda ^{-1} }
$$
$$
{\cal T}(V) = \prod_{i} {r_i \over 1 - \exp(-r_i)} \ ,
$$
where $ \lambda $ is a character, and
$r_i$ are the Chern roots of the bundle $V$.
If $V = TY$, we denote ${\cal T}(V) = {\cal T}(Y)$.
\begin{prop}
Suppose ${\cal C}$ is the set of connected
components of the fixed set $X^T$ of $T$.
{}For each component $c \in {\cal C}$,
assume that each restriction
$L|_c$ is a trivial bundle.
Let $N(c) = \bigoplus_{ \lambda } N_{ \lambda }(c)$
denote the normal bundle with its
$T$-eigenspace decomposition.
Then the character of $T$ acting on
the space of global sections of $L$
is given by:
$$
\mathop{\rm tr}(x \mid H^0(X,L)) =
\sum_{c \in {\cal C}}
\left[
{tr(x \mid L|_c) \cdot \prod_{ \lambda } {\cal U}^{ \lambda }(N_{ \lambda }(c))(x)
\cdot {\cal T}(c)
\over
\det( \mathop{\rm id} -\, x \mid N^*(c) )}
\right](\mbox{\rm Fund }c) \ ,
$$
where the multiplication takes place in the cohomology
ring of the component $c$,
and $\mbox{\rm Fund }c$ denotes the fundamental
homology class.
\end{prop}
Applying this to $X = { \FF_{3,n}^{\FM} }$, \, $L = \pi^* \LL_D $, we
again get a formula for
the character of $S^*_D = H^0({ \FF_{3,n}^{\FM} }, \pi^* \LL_D )$,
this time in terms of the
data in Lemma~\ref{bundle}.
The next result we shall use
is based on the theorem of Hirzebruch-Riemann-Roch
\cite{AS}, combined with Bott's Residue Formula
\cite{Bott}, \cite{AB}, according to the method
of Ellingsrud and Stromme \cite{ES}.
\begin{prop}
Suppose the torus $T = {\bf C}^*$ is one-dimensional,
and acts with isolated fixed points.
Let $\mbox{\bf v} = 1$ in the Lie algebra ${\bf t} = {\bf C}$,
and at each $T$-fixed point $p$, let $b(p) = \mathop{\rm tr}(\mbox{\bf v} \mid L|_p
)$.
Denote the $\mbox{\bf v}$-eigenspace decomposition of the
tangent space by $T_p X = \oplus_{i = 1}^M {\bf C}_{r_i(p)} $,
where $r_i(p)$ are the integer eigenvalues.
Also, define the polynomial
$$
\mbox{RR}_M(b;r_1,\dots,r_M) = \mbox{\rm coeff at $U^M \!$ of }
\left( \exp(b U) \prod_{i = 1}^M {r_i U \over 1 -\exp(-r_i U)}
\right) \ ,
$$
where the right-hand side is considered as a Taylor series in the
formal variable $U$.
Then the dimension of the space of global sections of $L$ is
given by:
$$
\dim H^0(X,L) =
\sum_{p \ \mbox{\tiny fixed}}
{\mbox{RR}_M( b(p); r_1(p), \ldots, r_M(p) )
\over \det( \mbox{\bf v} \mid T_p X ) } \ .
$$
\end{prop}
We will consider $X = { \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ and take the $T$ in the Proposition
to be
${\bf C}^* \subset GL(n)$,\, $q \to \mbox{\rm diag}(q^{-1},q^{-2},\ldots,q^{-n})
$.
(This is the principal one-dimensional subtorus corresponding to
the
half-sum of positive roots.) Then the eigenvalue characters in Lemma
\ref{SS}
specialize to the subtorus, and give us the information required
to
compute the dimension of $S^*_D = H^0({ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }, \pi^* \LL_D )$. (We may
check
directly that the fixed points of
the subtorus are identical to those of
the large torus of all diagonal matrices.)
Let us also mention that for the smooth spaces ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ and
${ \FF_{3,n}^{\FM} }$, the theorem of Bialynicki-Birula \cite{BB}
gives cell decompositions of these spaces
using the fixed point data. Thus, one can
compute their singular cohomology groups and Chow groups
as is done in \cite{CF} and \cite{FM}.
\subsection{Character formulas}
\label{character}
{}First, we recall the necessary combinatorial constructions.
We specify a three-row diagram $D$ of squares in the plane
by assigning a multiplicity $m_C \geq 0$ to each column
of the ``universal three-row diagram''
$$
D_3 \ = \ \
\begin{array}{ccccccc}
m_1 & m_2 & m_3 & m_{1,2} & m_{2,3} & m_{1,3} & m_{1,2,3} \\
\Box & & & \Box & & \Box & \Box \\
& \Box & & \Box & \Box & & \Box \\
& & \Box & & \Box & \Box & \Box
\end{array}
$$
We define a {\em standard column tabloid} for $D$\,
with respect to $GL(n)$,
to be a filling (i.e. labeling)
of the squares of $D_3$ by integers in $\{1,\ldots,n\}$,
such that:\\
(i) the integers in each column are strictly increasing, and\\
(ii) if there is an inclusion $C \subset C'$ between two columns,
then all the numbers in the filling of $C$
also appear in the filling of $C'$.
The tabloids describe the fixed points of the
torus $T$ acting on the configuration variety ${ \FF_{3,n} }$.
Given a tabloid $\tau$ for $D$, define its generating monomial
$$
x^{ {\mathop{\rm wt}}\mbox{} (\tau)} = \prod_{(i,j) \in D_3} x_{\tau(i,j)}^{m_j} \ .
$$
That is, the power of $x_i$ is the number of times $i$ appears
in the filling $\tau$, counted with multiplicity.
Also, define integers $d_{ij}(\tau)$ to be the
number of connected components of the following graph:
the vertices are columns
$C$ of $D_3$ such that
$i$ appears in the filling of $C$, but $j$ does
not; the edges are $(C,C')$ such that $C \subset C'$ or $C' \subset
C$.
\\[1em]
Now, our formula is a sum of terms
corresponding to the column tabloids $\tau$ of $D$.
{}For the smooth tabloids,
the contribution ${\cal C}(\tau)$ is obtained
by the same formula as
in the northwest case discussed in previous works:
$$
{\cal C}(\tau) = { x^{ {\mathop{\rm wt}}\mbox{} (\tau)}
\over \prod_{i\neq j} (1-x_i^{-1} x_j)^{d_{ij}(\tau)} }
$$
However, for the tabloids where
the configuration variety is singular,
we substitute a special contribution
which can be defined in two ways, corresponding
to the two desingularizations ${ \FF_{3,n}^{ { \cal S\hspace{-.2em}S } } }$ and ${ \FF_{3,n}^{\FM} }$.
Surprisingly, these expressions reduce algebraically to another,
simpler form which does not appear to be associated with any
desingularization (c.f. section \ref{virtual}).
\begin{thm} The character of the Schur module $S_D$ for $GL(n)$
is
$$
{\mbox{\rm char}} _{S_D} = \sum_{\tau} {\cal C}(\tau) \ ,
$$
where ${\cal C}(\tau)$ are given in the table below.
\end{thm}
To get all tabloids from the types
shown in the table, one should take all permutations of
the first three and the second three columns.
There are $11 n (n-1)(n-2)$ tabloids altogether.
Set
$$
\Delta_{ijk} = \prod_{l \in [1,n] \atop l \neq i,j,k}
\left( 1 - {{x(l)}\over {x(i)}} \right) \,
\left( 1 - {{x(l)}\over {x(j)}} \right) \,
\left( 1 - {{x(l)}\over {x(k)}} \right) \ .
$$
\vfill
\mbox{}
\pagebreak
$$
\begin{array}{ccc}
\mbox{\large \mbox{Tabloid} $\tau$ }
&
\underline{\mbox{\large \mbox{Character contribution }
${\cal C}(\tau)$ } }
&
\underline{\mbox{\large Desing}}
\\
\overline{
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& j & & j & & j & j \\
& & k & k & k & & k
\\ & & & & & &
\end{array}
}}
&
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
\left( 1 - {{x(i)}\over {x(j)}} \right) \,\left( 1 - {{x(j)}\over
{x(i)}}
\right) \,
\left( 1 - {{x(i)}\over {x(k)}} \right) \,\left( 1 - {{x(j)}\over
{x(k)}}
\right) \,
\left( 1 - {{x(k)}\over {x(i)}} \right) \,\left( 1 - {{x(k)}\over
{x(j)}}
\right)
\Delta_{ijk}
}
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & j & j \\
& & j & j & j & & k
\\ & & & & & &
\end{array}
}
&
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
\left( 1 - {{x(i)}\over {x(j)}} \right) \,\left( 1 - {{x(j)}\over
{x(i)}}
\right) \,
\left( 1 - {{x(j)}\over {x(k)}} \right) \,{{\left( 1 - {{x(k)}\over
{x(i)}}
\right) }^2}\,
\left( 1 - {{x(k)}\over {x(j)}} \right)
\Delta_{ijk}
}
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & k & j \\
& & j & j & j & & k
\\ & & & & & &
\end{array}
}
&
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
\left( 1 - {{x(i)}\over {x(j)}} \right) \,{{\left( 1 - {{x(j)}\over
{x(i)}}
\right) }^2}\,
\left( 1 - {{x(k)}\over {x(i)}} \right) \,{{\left( 1 - {{x(k)}\over
{x(j)}}
\right) }^2}
\Delta_{ijk}
}
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & k & j \\
& & i & j & j & & k
\\ & & & & & &
\end{array}
}
&
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
{{\left( 1 - {{x(j)}\over {x(i)}} \right) }^2}\,\left( 1 - {{x(j)}\over
{x(k)}}
\right) \,
\left( 1 - {{x(k)}\over {x(i)}} \right) \,{{\left( 1 - {{x(k)}\over
{x(j)}}
\right) }^2}
\Delta_{ijk}
}
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & j & j \\
& & i & j & j & & k
\\ & & & & & &
\end{array}
}
&
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)}
\left( 1 - \left( { 1 - {x(i)\over x(j)} }\right)^{-1}
- \left( { 1 - {x(j)\over x(k)} }\right)^{-1}
\right)
\over
{{\left( 1 - {{x(j)}\over {x(i)}} \right) }^2}\,\left( 1 - {{x(k)}\over
{x(i)}}
\right) \,
{{\left( 1 - {{x(k)}\over {x(j)}} \right) }^2}
\Delta_{ijk}
}
& { \cal F\hspace{-.2em}M }
\\
&
=
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
{\left( 1 - {x(j)\over x(i)} \right)}^3\,
\left( 1 - {x(k)\over x(i)} \right) \,
\left( 1 - {x(k)\over x(j)} \right) \,
\left( 1 - {x(i)\,x(k) \over x(j)^2} \right)
\Delta_{ijk}
}
\ \ +
& { \cal S\hspace{-.2em}S }
\\
& \ \ \ \ \
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
\left( 1 - {x(j)\over x(i)} \right) \,
\left( 1 - {x(k)\over x(i)} \right) \,
{\left( 1 - {x(k)\over x(j)} \right)}^3\,
\left( 1 - {x(j)^2 \over x(i)\,x(k)} \right)
\Delta_{ijk}
}
\\[3em]
&
=
{x^{ {\mathop{\rm wt}}\mbox{} (\tau)} \over
{\left( 1 - {x(j)\over x(i)} \right)}^3\,
{\left( 1 - {x(k)\over x(j)} \right)}^3\,
\Delta_{ijk}
}
& \mbox{\bf ?}
\end{array}
$$
\pagebreak
\normalsize
\subsection{Dimension formula}
\label{dimension}
We compute the dimension of the $GL(n)$-module $S_D$.
Again, each tabloid gives a contribution, which is
the value of a certain multivariable polynomial
$\mbox{RR}$ at a sequence
of integers specific to the tabloid.
Let $M = 3n-3$, the dimension of the triangle space ${ \FF_{3,n} }$.
Define an $(M+1)$-variable polynomial, homogeneous of degree $M$,
by
$$
\mbox{RR}_M(b;r_1,\dots,r_M) = \mbox{coeff at $U^M \!$ of }
\left( \exp(bU) \prod_{i = 1}^M {r_i U \over 1 - \exp(-r_i U)}
\right) \ ,
$$
where the right side is understood as a Taylor series in $U$.
{}For example, for $n=3$, $M=6$, $\mbox{RR}_6(b;r_1,\ldots,r_6)$
is an irreducible 7-variable polynomial,
homogeneous of degree 6, with 567 terms.
However, since we will only
evaluate this polynomial at $(M+1)$-tuples
of small integers, this is
within the range of computer
calculations provided the column multiplicities
$m_j$ are not very large.
{}For the smooth tabloids, we again have a formula
for the contributions
in terms of the integers $d_{ij}(\tau)$.
Namely, define a multiset of integers
$r(\tau) = \{r_1, r_2, \ldots, r_{M}\}$ by inserting
the entry $i-j$ with multiplicity $d_{ij}(\tau)$
for each ordered pair $i, j \in [1,n]$.
That is, the total multiplicity of the integer $k$ in $r(\tau)$ is
$$
\sum_{i,j \in [1,n] \atop i-j = k} d_{ij}(\tau) \ .
$$
Let $b(\tau)$ be the sum of the entries of the tabloid $\tau$, counting
column multiplicities:
$$
b(\tau) = \sum_{(i,j) \in D} m_j\, \tau(i,j) \ .
$$
Then the contribution of $\tau$ to the dimension formula is
\begin{eqnarray*}
{\cal R}(\tau) & = & {\mbox{RR}_M(b(\tau); r(\tau)) \over \prod_{k
= 1}^M
r(\tau)_i} \\[.2em]
& = & {\mbox{RR}_M(b(\tau); r(\tau)) \over \prod_{i,j \in [1,n]} (i-j)}
\end{eqnarray*}
{}For the singular tabloids, we have only
an expression corresponding to the $ { \cal S\hspace{-.2em}S } $ desingularization,
as well as a simplified expression with no geometric explanation,
as before.
\begin{thm} The dimension of the Schur module $S_D$ for $GL(n)$
is
$$
\dim S_D = \sum_{\tau} {\cal R}(\tau) \ ,
$$
where ${\cal R}(\tau)$ are given in the table below.
\end{thm}
The entries in the table are terms of the form
${\cal R} = \mbox{RR}_M(b;r) / \prod r$ for integers $b$ and multisets
$r$.
Let $r'(i,j,k)$ be the standard multiset
with entries $i-l$, $j-l$, $k-l$ for each integer $l \in [1,n]$,
$l \neq i,j,k$.
(For example, if $n=4$, we have $r'(1,2,3) = \{ 1-4,\, 2-4,\,
3-4 \} =
\{-3,-2,-1\}$.)
\vfill
\mbox{}
\pagebreak
$$
\large
\begin{array}{c@{\!}c@{\!}c}
\mbox{Tabloid } \tau &
\underline{\mbox{Dimension contribution }{\cal R}(\tau) } &
\underline{\mbox{Desing}}\\
\overline{
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& j & & j & & j & j \\
& & k & k & k & & k
\\ & & & & & &
\end{array}
}}
&
{\mbox{RR}_M(b(\tau);\ j-i,\ i-j,\ k-i,\ k-j,\ i-k,\ j-k,\ r'(i,j,k)
)
\over (j-i)(i-j)(k-i)(k-j)(i-k)(j-k) \ \prod r'(i,j,k) }
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & j & j \\
& & j & j & j & & k
\\ & & & & & &
\end{array}
}
&
{\mbox{RR}_M(b(\tau);\ j-i,\ i-j,\ k-j,\ i-k,\ i-k,\ j-k,\ r'(i,j,k)
)
\over (j-i)(i-j)(k-j)(i-k)^2 (j-k) \ \prod r'(i,j,k) }
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & k & j \\
& & j & j & j & & k
\\ & & & & & &
\end{array}
}
&
{\mbox{RR}_M(b(\tau);\ j-i,\ i-j,\ i-j,\ i-k,\ j-k,\ j-k,\ r'(i,j,k)
)
\over (j-i)(i-j)^2 (i-k) (j-k)^2 \ \prod r'(i,j,k) }
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & k & j \\
& & i & j & j & & k
\\ & & & & & &
\end{array}
}
&
{\mbox{RR}_M(b(\tau);\ i-j,\ i-j,\ k-j,\ i-k,\ j-k,\ j-k,\ r'(i,j,k)
)
\over (i-j)^2 (k-j)(i-k)(j-k)^2 \ \prod r'(i,j,k) }
& \mbox{smooth}
\\
\underline{
\begin{array}{ccccccc}
& & & & & & \\
i & & & & i & i & i \\
& i & & i & & j & j \\
& & i & j & j & & k
\\ & & & & & &
\end{array}
}
&
\begin{array}{r}
{\mbox{RR}_M(b(\tau);\ i-j,\ i-j,\ i-j,\ i-k,\ j-k,\ 2j-i-k,\ r'(i,j,k)
)
\over (i-j)^3 (i-k)(j-k)(2j-i-k) \ \prod r'(i,j,k) } \ + \ \\[.4em]
{\mbox{RR}_M(b(\tau);\ i-j,\ i-k,\ j-k,\ j-k,\ j-k,\ i+k-2j,\ r'(i,j,k)
)
\over (i-j)(i-k)(j-k)^3 (i+k-2j) \ \prod r'(i,j,k) }
\end{array}
& { \cal S\hspace{-.2em}S }
\\[1em]
&
= {\mbox{RR}_M(b(\tau);\ i-j,\ i-j,\ i-j,\ j-k,\ j-k,\ j-k,\ r'(i,j,k)
)
\over (i-j)^3 (j-k)^3 \ \prod r'(i,j,k) }
& \mbox{\bf ?}
\end{array}
$$
\normalsize
\vfill \mbox{}
\pagebreak
\subsection{Virtual desingularization}
\label{virtual}
In the previous sections, we have drawn consequences about Schur
modules
from the geometry of the triangle space. However, one can imagine
reversing
this process.
{}For instance, consider the trivial line bundle over the triangle
space, $m_1 = m_2 = \cdots = m_7 = 0$. Its space of sections
is the
trivial Schur module,
with character 1. Hence, our character formulas for the $ { \cal F\hspace{-.2em}M } $ and
$ { \cal S\hspace{-.2em}S } $
desingularizations each give
ludicrously complicated expressions for 1.
There is a non-trivial contribution
${\cal C}(\tau)$ for each $T$-fixed point $\tau$ of the smooth space:
the numerators are reduced to 1, but the denominators still possess
a factor for each eigenvalue of the tangent space at the fixed point.
Because these eigenvalues determine a cell decomposition of the smooth
space, one can read off from these ``ludicrous formulas''
a great deal of geometric information about the desingularizations
of ${ \FF_{3,n} }$.
In fact, suppose we did not know of the existence of the $ { \cal S\hspace{-.2em}S } $
desingularization.
We could guess that there exists such a smooth space, with its cell
decomposition,
by finding a ludicrous formula for 1 with terms of Atiyah-Bott type.
This is not difficult:
let us start by assuming there exists a $GL(n)$-equivariant
desingularization of ${ \FF_{3,3} }$
with fibers of dimension 1 (the smallest possible), and with each
singular
fixed point lifting to only two fixed points in the smooth space.
The singular fibers must then be isomorphic to ${\bf P}^1$, since this
is the
only curve
possessing an appropriate torus action.
Now, the contributions
${\cal C}(\tau)$ from the
smooth tabloids of ${ \FF_{3,n} }$ are constrained to be the entries in the
table.
As for the singular tabloids, they must each correspond to two terms
in the
formula,
one for each lifted fixed point. We know most of their eigenvalues
from the
$T$-stable curves in ${ \FF_{3,n} }$: each will lift to either
two or one $T$-stable curves in the smooth space, depending on whether
or not it lies in the singular locus. Since there are six eigenvectors
at each fixed point of the smooth space, this leaves only one eigenvector
to determine at each of the two lifted fixed points.
Since the fiber is ${\bf P}^1$, these must be reciprocals of each other.
Now, the six singular tabloids lie in a single GL(n) orbit, so the
corresponding unknown eigenvalues are all images of each other.
Therefore, there is only one variable left unknown, which we can
solve
for in the ludicrous equation: over $\tau_{123}$ the value
must be $x_1 x_3 x_2^{-2}$, as in our formula.
Now, for the space of tetrahedra and higher-dimensional simplices
(corresponding to general diagrams with four or more rows), there
is no known explicit desingularization. One may hope to find evidence
of one by arguments like the above, combined with induction on the
dimension.
That is, an appropriate ludicrous formula for 1 may be considered
as a
combinatorial
or virtual desingularization.
{}Finally, let us point out one mystery in our results: the algebraic
simplification
of the character and dimension formulas, combining the complicated
contributions
given by our desingularizations into a single term of Atiyah-Bott
type.
In the above philosophy, this would mean that ${ \FF_{3,n} }$ is already ``virtually
smooth'',
with eigenvalues above $\tau_{ijk}$ equal to $x_i^{-1} x_j$ (three
times), and
$x_j^{-1} x_k$ (three times). Note that there can exist no actual
$G$-equivariant desingularization
of ${ \FF_{3,n} }$ for which the singular fixed points each lift uniquely,
since
this would
use up only six of the seven eigenvalues given by the $T$-stable
curves at $\tau_{ijk}$ (cf. Lemma \ref{SS}).
(Each of these curves must lift to at least one $T$-stable curve
in the
smooth space
and lead to some lifted fixed point.)
\section{Appendix: Restriction and induction}\label{resind}
We now prove a result from the representation theory of reductive
algebraic
groups, needed in the proof of theorem \ref{vanishing}.
Let $G$ be a connected reductive algebraic
group, $B$ a Borel subgroup, $P$ a parabolic subgroup
containing $B$.
Let us call a weight $\lambda$ {\em effective} if $\mbox{\rm ind}_B^G\lambda\neq0$.
(In \cite{J} an effective weight is called dominant, in \cite{vdK}
it is called anti-dominant.) For the notions of `induction', `good
filtration',
`excellent filtration', and the basic theorems concerning them we
refer to
\cite{J}, \cite{Mat}, \cite{vdK}.
\begin{lem}
Let $\lambda$ be effective and let $M$ be a $P$-module that has excellent
filtration (as a $B$-module).
Then $\mbox{\rm ind}_B^P(\lambda)\otimes M$ has an excellent filtration.
\end{lem}
\noindent {\bf Proof.} As
$\mbox{\rm ind}_B^P(\lambda)\otimes M=\mbox{\rm ind}_B^P(\lambda\otimes M)$
this follows from the main theorems
on excellent filtrations as collected
in \cite{vdK}.
\begin{prop}
{}For effective $\lambda_1$, \ldots, $\lambda_n$, the restriction
map
$$
\mbox{\rm res} :\mbox{\rm ind}_B^G(\lambda_1) \otimes \cdots \otimes \mbox{\rm ind}_B^G(\lambda_n)
\to
\mbox{\rm ind}_B^G(\mbox{\rm ind}_B^P(\lambda_1) \otimes \cdots \otimes \mbox{\rm ind}_B^P(\lambda_n))
$$
is surjective.
\end{prop}
\noindent {\bf Proof.}
It suffices to show that for each $i$ the kernel $\ker\phi_i$ of
the surjective map $\phi_i$:
$$\begin{array}{c}
\mbox{\rm ind}_B^G(\lambda_1)\otimes\cdots\otimes\mbox{\rm ind}_B^G(\lambda_{i-1})
\otimes\mbox{\rm ind}_B^G(\lambda_{i})\otimes\mbox{\rm ind}_B^P(\lambda_{i+1})
\otimes\cdots\otimes
\mbox{\rm ind}_B^P(\lambda_n)\\
\downarrow\\
\mbox{\rm ind}_B^G(\lambda_1)\otimes\cdots\otimes\mbox{\rm ind}_B^G(\lambda_{i-1})
\otimes\mbox{\rm ind}_B^P(\lambda_{i})\otimes\mbox{\rm ind}_B^P(\lambda_{i+1})
\otimes\cdots\otimes
\mbox{\rm ind}_B^P(\lambda_n)
\end{array}$$
is $\mbox{\rm ind}_B^G$-acyclic. Indeed $\mbox{\rm res}$ may be viewed as
$\mbox{\rm ind}_B^G(\phi_1)\circ\cdots\circ\mbox{\rm ind}_B^G(\phi_n)$. Now a module
$M$ is
$\mbox{\rm ind}_B^G$-acyclic
if and only if $k[G]\otimes M$ is $B$-acyclic, so that the result
follows
from the lemma and the
main theorems on excellent filtrations {\it etc.}
|
1996-01-23T06:20:10 | 9601 | alg-geom/9601002 | en | https://arxiv.org/abs/alg-geom/9601002 | [
"alg-geom",
"math.AG"
] | alg-geom/9601002 | Matei Toma | Matei Toma | Birational models for varieties of Poncelet curves | revised TeX-version, latex2e | null | null | null | null | We consider Poncelet pairs $(S,C)$, where $S$ is a smooth conic and $C$ is a
degree$-c$ plane curve having the Poncelet property with respect to $S$. We
prove that for $c>4$ the projection $(S,C)\mapsto C$ is generically one-to-one
and use this to describe a birational model of the variety of Poncelet curves
for $c$ odd.
| [
{
"version": "v1",
"created": "Wed, 3 Jan 1996 08:19:18 GMT"
},
{
"version": "v2",
"created": "Wed, 3 Jan 1996 10:57:44 GMT"
},
{
"version": "v3",
"created": "Mon, 22 Jan 1996 09:34:06 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Toma",
"Matei",
""
]
] | alg-geom | \section*{Introduction}
A pair $(S,C)$ consisting of a smooth conic $S$ and a curve $C$ of degree $c$
in the complex projective plane will be called a Poncelet pair if there exist
$c+1$ tangents to $S$ whose ${c+1\choose 2}$ intersection points lie on $C$.
$C$
will then be called Poncelet related to $S$ or simply a Poncelet curve.
A theorem of
Darboux (\cite{D}) states that in this case there is an infinity of such sets
of
$c+1$ tangents.
The Poncelet curves reappeared in a natural way in the study of stable vector
bundles on projective spaces, e.g. in \cite{B}, \cite{H}, \cite{BT}, \cite{NT}.
Motivated by these developments, Trautmann gave a modern description of their
geometry in \cite{T}.
In this paper we consider the variety $Pon_c$ of Poncelet curves of degree $c$.
It is the image of the space of Poncelet pairs $(S,C)$ through the second
projection. We show that for $c\ge 5$, this projection is birational. This
fails for $c\le 3$, simply because of dimension reasons, and has been proved
for $c=4$ by Le Potier via the study of divisors on the corresponding moduli
space of $rank -2$ semi-stable sheaves on $\Bbb P^2$, (\cite{L}).
As a corollary,
we get the rationality of $Pon_c$ for every $c$.
The generic injectivity of the above mentioned projection brings some evidence
for (and maybe could help proving) its generalizations to:
\begin{itemize}
\item Barth's morphism which associates to a stable $rank -2$ vector bundle of
type
$(0,c)$ on $\Bbb P^2$ its curve of jumping lines (\cite{B}), and
\item the restriction of the previous to Hulsbergen bundles. (The image will be
the variety of Darboux curves, cf. \cite{B}, \cite{ES}).
\end{itemize}
In the last paragraph we describe a projective birational model of $Pon_c$ when
$c$ is odd.
\section{Preliminaries}
For the proofs of the facts stated in this paragraph as well as for further
descriptions we refer the reader to \cite{T}.
Let $W$ be a 3-dimensional complex vector space, $S$ a smooth conic in $\Bbb
P^2=\Bbb P(W)$, $c$ a positive integer and $\Lambda$ a pencil of degree-$(c+1)$
divisors on $S$. We consider the intersection points of the tangents to $S$ at
the points of some member $D$ of
$\Lambda$. When $D$ moves in $\Lambda$, these intersection points describe a
curve $C=C(\Lambda)$ in $\Bbb P^2$. It is clear that the tangents at the base
points of $\Lambda$ will be components of $C$. We require that their
multiplicity in $C$ be equal to that of the base points in $D$. Then $C$ has
degree $c$, is called {\bf Poncelet related to} $S$ or simply a {\bf Poncelet
curve} and $(S,C)$ is called a {\bf Poncelet pair}.
That this definition is equivalent to that of the introduction follows from the
{\bf Theorem.}
{\it (Darboux) Let $S$ be a smooth conic and $C$ a curve of degree
$c$ in $\Bbb P^2$. If there exist $c+1$ tangents to $S$ such that their
intersection points lie on $C$, then $C$ is Poncelet related to $S$.
}
The base points of $\Lambda$ correspond to tangents to $S$ which are components
of $C(\Lambda)$ and the residual curve (obtained by eliminating these
components) is again Poncelet related to $S$ and associated to the pencil
obtained from $\Lambda$ by subtracting its base points.
For $\Lambda$ generic $C(\Lambda)$ is smooth.
Let's denote by $G_S(1,c+1)$ the Grassmannian of pencils of degree-$(c+1)$
effective divisors on $S$. The map
\begin{eqnarray}
G_S(1,c+1) &\to & \Bbb P(S^cW^*)=\Bbb P^{{c+2 \choose 2}-1}\nonumber\\
\Lambda &\mapsto & C(\Lambda)\nonumber
\end{eqnarray}
coincides with the Pl"ucker embedding (for a suitable choice of coordinates,
\cite{T}). Thus, there is a 1:1 correspondence between $G_S(1,c+1)$ and the
variety $Pon_{c,S}$ of curves $C$ which are Poncelet related to $S$.
Let $U$ be the open set of $\Bbb P(S^2W^*)$ which parameterizes smooth conics
in
$\Bbb P^2$, $I_U\subset U\times\Bbb P^2$ the tautological conic bundle
over $U$, $I^{(c+1)}_U$ the Hilbert scheme of degree-$(c+1)$ effective
divisors in the fibers of $I_U\to U$ (which is a $\Bbb P^{c+1}$-fibration over
$U$) and $G_U(1,c+1)$ the relative Grassmannian of lines in the fibers of
$I_U^{(c+1)}\to U$. We shall identify $G_U(1,c+1)$ with the subvariety
$Pon_{c,U}$ of Poncelet pairs in $U\times \Bbb P(S^cW^*)$. (The equations of
$Pon_{c,U}$ are given in \cite{T}).
We shall examine the second projection $pr_2: Pon_{c,U}\to\Bbb P(S^cW^*)$. Its
image is the variety of degree-$c$ Poncelet curves $Pon_c$.
We end this paragraph with some elementary remarks on Poncelet curves to be
used later:
{\it Remarks.} Let $S$ be a smooth conic.
\begin{enumerate}
\item[1.1] If $C$ is Poncelet related to $S$, then the only components of $C$
which may be multiple are tangent lines to $S$. (Apply Bertini's theorem to the
associated pencil, $\Lambda$.)
\item[1.2] If $C_1+C_2$ and $C_1+C_2'$ are two elements of $Pon_{c,S}$ such
that $C_2$ and $C_2'$ have no common component, then $C_1$ is Poncelet related
to $S$. (Start from a point on $C_1$, draw tangents to $S$ and continue this
procedure from the intersection points with $C_1$. The process stops after a
finite number of steps. If there exist intersection points of the drawn
tangents not on $C_1$, then these points lie on a common component of $C_2$ and
$C_2'$).
\item[1.3] Let $C_1, C_1+C_2$ be Poncelet related to $S$ with associated
pencils $\Lambda_1$ and $\Lambda$, respectively. If $\Lambda$ has no base
points
,
then the induced morphism $\varphi_\Lambda: S\to \Lambda^*\cong\Bbb P^1$
factorizes through $\varphi_{\Lambda'}:S\to\Lambda^{'*}$. In particular, $\deg
C_1+1$ divides $\deg(C_1+C_2)+1$.
\item[1.4] If $C$ is a singular conic Poncelet related to $S$, then one of its
components is tangent to $S$. (Follows from 1.3).
\end{enumerate}
\section{Generic injectivity of $pr_2$}
{\bf Theorem.}
{\it When $c\ge 5$, the projection $Pon_{c,U}\to Pon_c$ is
birational.}
{\it Remarks.}
\begin{enumerate}
\item[2.1] The corresponding statement for $c=4$ (and more) is proven in
\cite{L}.
\item[2.2] $Pon_{c,U}$ is rational (\cite{HN}, Prop.2.2), hence so is
$Pon_c$ too.
\end{enumerate}
The theorem follows by a dimension count out of the following
{\bf Proposition.}
{\it Let $S,S'$ be distinct smooth conics. Then
\begin{enumerate}
\item[(A)] for $c\ge 3$, $\dim (Pon_{c,S}\cap Pon_{c,S'})\le c$.
\item[(B)] $Pon_{5,S}\cap Pon_{5,S'}$ is at most 4-dimensional at points
represented by smooth quintics.
\end{enumerate}
}
Indeed, let us estimate the dimension of
\linebreak
$\tilde{G}_S:=\overline{pr_2^{-1} (Pon_{c,S})\backslash
G_S(1,c+1)}$. By part (A) of the Proposition,
$\dim \tilde{G}_S\le c+5$. Thus, for $c\ge 6$ we have $\dim pr_2(\tilde{G}_S)
\linebreak
< 2c=\dim Pon_{c,S}$ proving the theorem in this case.
When $c=5$, we use part (B) for a slightly modified $\tilde{G}_S$,
\linebreak
$\tilde{G}_S:=\overline{pr_2^{-1}(Reg_{c,S})\backslash G_S(1,c+1)}$, where
$Reg_{c,S}$ is the set of
\linebreak
smooth Poncelet curves related to $S$.
In order to prove the Proposition we look at the linear projection of
$Pon_{c,S}\cap Pon_{c,S'}$ with center $L_d:=\{\mbox{degree-$c$}$ curves in
$\Bbb P(W)$ allowing $d$ as a component$\} \subset \Bbb P(S^cW^*)$, for some
common tangent line $d$ to $S$ and $S'$.
Let $d$ be the tangent at $P$ to the smooth conic $S$. Then $Pon_{c,S}\cap L_d
\cong Pon_{c-1,S}$ corresponds to degree-$c$ pencils on $S$ having $P$ as base
point (or equivalently, to the Schubert variety on $G_S(1,c+1)$ of lines
contained in a hyperplane).
Let further $D$ be an element of $|{\cal O}_S(c+1)|$ containing $P$
and $L_D\subset
G_S(1,c+1)$ the Schubert variety of lines through $D$. Then $L_D\cong \Bbb P^c$
is linear in $\Bbb P(S^cW^*)$ and cuts on $L_d$ a $(c-1)$-dimensional linear
subspace, (the $L_{D-P}$ of $G_S(1,c)$). Thus $L_D\cup L_d$ spans a
$c$-dimensional linear subspace of $\Bbb P(S^cW^*)$.
Conversely, for every $c$-codimensional linear subspace $F$ which contains
$L_d$, these exists some $D$ as above such that $F\cap Pon_{c,S}=L_D\cup
(Pon_{c,S}\cap L_d)$.
For two distinct divisors $D,D'$ as above, $L_D$ and $L_{D'}$ cut themselves in
exactly one point lying on $L_d$. In particular, the projection of center $L_d$
displays $Pon_{c,S}\backslash L_d$ as a vector bundle over $|{\cal O}_S(c)|$.
Let now $V$ be an irreducible component of $Pon_{c,S}\cap Pon_{c,S'}$
not contained in $L_d$.
Let $\pi:V\backslash L_d\to|{\cal O}_S(c)|$ be the restriction to $V$
of the above
projection, $t:=\dim\pi(V\backslash L_d)$, and $f$ the dimension of a
generic fiber of $\pi$. Note that the fibers of $\pi$ are linear.
{\bf Lemma 2.1}
{\it When $c\ge 2$, up to finitely many exceptions, the fibers of
$\pi$ are at most $(c-1)$-dimensional.}
{\it Proof.} We choose a general hyperplane in $|{\cal O}_S(c)|$ and show that
the
fibers of $\pi$ over its points cannot be $c$-dimensional. Indeed, take $P_1$
on $S$ such that it doesn't belong to a common tangent to $S$ and $S'$.
The divisors containing $P_1$ form a hyperplane in $|{\cal O}_S(c)|$.
Take any such
divisor and add $P$ to it in order to obtain a divisor $D=P+P_1+P_2+\ldots
+P_c$ in $|O_S(c+1)|$.
Take further a pencil with base points $P_1,P_2,\ldots,P_{c-1}$ and containing
$D$. Its associated Poncelet curve is the union of the tangents at $P_1,\ldots,
P_{c-1}$ to $S$ and a line through the intersection point of $d$ with the
tangent to $S$ at $P_c$. By Remarks 1.2 and 1.3 not all such curves are
Poncelet related to $S'$.
\hfill$\Box$
\vspace{2ex}
{\it Proof of part (A) of the Proposition}
We argue by induction on $c$.
Let $c=3$ and suppose that $t+f>3$. Then $t\ge 2$ by Lemma 2.1. We pick then a
divisor $D=P+P_1+P_2+P_3$ on $S$ and look at the fiber of $\pi$ contained in
$L_D$.
We get a divisor $D'=P'+P'_1+P'_2+P'_3$ on $S'$ such that this fiber is exactly
$L_D\cap L_{D'}\backslash L_d$. ($L_{D'}$ is to be considered with respect to
$S'$).
Note that $P_1,P_2$ or $P_1,P'_2$ may be freely chosen so that this fiber be
non empty.
Denote by $d_i,d'_i$ the tangents at $P_i$ to $S$ and at $P'_i$ to $S'$,
respectively, $i\in\{1,2,3\}$.
We index our $P_i'-s$ so that $d\cap d_i=d\cap d'_i$.
Choose now $P_1,P_2'$ such that the point $d_1\cup d_2'$ does not lie on a
common tangent to $S$ and $S'$. In particular, it will not lie on both $d_3$
and $d'_3$, say not on $d'_3$.
The Poncelet curves in $L_{D'}\cap L_D$ touch $d'_2$ at its intersection points
with $d, d'_1$ and $d_3'$. (If one of the $P'_i$ had multiplicity $(k+1)$ in
$D'$ then the condition on the corresponding Poncelet curves would be to touch
$S$ at $P'_i$ with multiplicity $k$, by \cite{T}, Prop. 2.6). If we impose to
such a curve that $d_1$ be a component (which we may do since $f\ge 1$), we get
that $d'_2$ must be a component too. By Remark 1.4 the third component of our
cubic is tangent to both $S$ and $S'$, so $f=1$ and $t=3$. This allows a free
choice on $P_3'$ too and we get that the tangents through $P_2'$ and $P_3'$ to
$S'$ form a conic which is Poncelet related to $S$. But this contradicts
Remark 1.4.
Let now $c>3$.
{\bf Case (A1)} $t>\displaystyle\frac{c}{2}$, $f\ge 1$.\\
We choose as above $D=P+P_1+\ldots+P_c$ and get in the same way a
$D'=P'+P'_1+\ldots+P'_c$ such that
$L_D\cap L_{D'} \backslash L_s\neq \emptyset$.
In doing this $P_1,\ldots, P_t$ are free.
We claim that the choice may be done such that $d_1$ intersects
$d'_2,\ldots,d'_t$ away from the vertices of the polygon $d,d'_1,\ldots,d'_c$
(with the obvious extension of notation).
If this were not true then there would exist a point, say $P'_{t+1}$, in a
``bad position'', e.g. such that $d_1\cap d'_2\cap d'_{t+1}\neq \emptyset$.
We look
then at $P_2$ instead of $P_1$ and may find a second point $P'_{t+2}$, in a bad
position, and continue in this way. The assumption $t>\displaystyle\frac{c}{2}$ shows now
that our claim holds.
Next, from $f\ge 1$ it follows that we may degenerate a curve $C$ in our fiber,
so that $d_1$ and hence also $d'_2,\ldots,d'_t$ become its components. By
Remarks 1.2-4 the other components of $C$ are not all tangent lines to $S$,
otherwise we move the lines $d'_i$, $2\le i\le t$. In particular $t<c$.
If $f\ge 2$, then we may still move our curve $C$ (in its class), fact which
combined with Remark 1.3 shows the existence of a pair of degrees $(a,b)$ such
that $t-1\le a<b<c$ and $a+1$ divides $b+1$. But this would imply
$t\le\displaystyle\frac{c}{2}$, which proves that $f=1$ and $f+t\le c$.
{\bf Case (A2)} $f>\displaystyle\frac{c}{2}$\\
The proof is similar to that of Lemma 1.
We may first assume that the general $D=P+P_1+\ldots+P_c$ in the image of $\pi$
doesn't contain any tangency point of a common tangent to $S$ and $S'$
excepting $P$. Otherwise we restrict our attention to the subvariety of
Poncelet curves containing this tangent and apply the induction hypothesis.
In the fiber of $\pi$ over $D$ we require that the tangents
$d_1,\ldots,d_{f-1}$ at $P_1,\ldots,P_{f-1}$ to $S$ be components of the
Poncelet curves $C$. Then by Remarks 1.3 and 1.4 there exists a pair of degrees
$(a,b)$ with $f-1\le a<b\le c$ and $a+1$
divides $b+1$. This forces $f=\displaystyle\frac{c+1}{2}$ proving part (A) of the
Proposition.
\hfill$\Box$
{\it Proof of part (B) of the Proposition}\\
Keeping our previous notations we assume that $t+f=c=5$.
{\bf Case (B1)}, $t=5$.\\
Since in this case $\pi$ is dominant we find for any choice of points
$P_1,\ldots,P_c$ on $S$, an element in $L_D\cap V$, where $D=P+P_1+\ldots+P_c$.
Take $P_1$ on $S$ and $P'_1$ the corresponding point on $S'$ (draw the tangent
$d'_1$ from $d\cap d_1$ to $S'$). Choose further a point $Q$ on $d'_1$ and draw
tangents from $Q$ to $S$. Let these tangents be $d_2,d_3$. A curve in $L_D\cap
V$ has to contain $Q$ and the intersection points of $d'_1$ with
$d,d'_2,\ldots, d'_c$. Thus this curve has $d'_1$ as a component. In the same
way, choosing $d_4$ through $d'_2\cap d_1$ fixes $d'_2$ as a component of $C$.
We may still move our point $P_5$, and since $d'_1\cup d'_2$ is not in
$Pon_{2,S}$, we get by Remarks 1.2 and 1.3 degrees $a,b$ with $3\le a<b\le 6$
and $a+1\mid b+1$. This is absurd so case (B1) cannot occur.
{\bf Case (B2)}, $t=4$.\\
As in case (A1) we may fix the components $d_1,d'_2,\ldots,d'_t$ as we wish.
The last component of our Poncelet curve in not tangent to $S$ as already
remarked in (A1). Since we may consider $d'_2, d'_3$ fixed and move $d'_4$ we
get again a contradiction of Remarks 1.2 and 1.4. Case (B2) is thus excluded.
{\bf Case (B3)}, $t=3$, reduces itself to (A1).\\
{\bf Case (B4)} $t=2$.
We pick as usual two arbitrary tangent lines to $S$ and look at the curves in
$V$ containing them. They form an at least 1-dimensional subvariety. Since
these two tangents do not form a Poncelet conic with respect to $S'$ we get
degrees $3\le a<b\le c$ such that $a+1$ divides $b+1$. This implies $c\ge 7$
and excludes this case too.
{\bf Case (B5)}, t=1.\\
If we find some $D=P+P_1+\ldots+P_c$ in the ``image'' of $\pi$ containing two
points which don't lie on common tangents to $S$ and $S'$ we shall argue as in
(B4).
Let's assume then that $P_2,\ldots,P_c$ are fixed, all lying on common
tangents.
We have free choice on $P_1$. Imposing that $d_1$ become a component, we are
left with an $(f-1)$-dimensional family of curves $C$ in $L_{D-P_1}$, Poncelet
related to $S$, and such that $C+d_1$ belongs to $Pon_{c,S'}$.
Applying to this family the same
procedure with a different $P_1$, we get a 1-dimensional family of curves $C$
with the above
properties with respect to both choices of $P_1$. The curves $C$ do not
consist only of tangents to $S'$, since these would have to touch $S'$ only at
$P_2',\ldots,P_c'$. We thus get a contradiction of Remarks 1.2-4 when applied
to $Pon_{c,S'}$.
{\bf Case (B6)}, $t=0$.\\
In this case there exist a divisor $D=P+P_1+\ldots+P_c$ on $S$ such that
$V=L_D$. The argument from (B4) shows that
at most one point, $P_1$ say, is not on a common tangent to $S$ and $S'$. By
requiring that $d_3,\ldots, d_6$ be components we reduce ourselves to the case
$c=2=f$, $D=P+P_1+P_2$. But this is excluded by showing that $d_2$ has to be a
component of $C$.
If all points of $D$ are tangency points of common tangents to $S$ and $S'$,
then either two of them are multiple in $D$ or one of them has multiplicity
bigger than 3 in $D$. In both cases the associated Poncelet curves are all
singular by \cite{T}, Proposition 5.1.
\hfill$\Box$
\section{A projective birational model of $Pon_c$ for $c$ odd}
For odd $c$, $G_U(1,c+1)$ is the relative Grassmannian of 2-dimen\-sional
subspaces of a rank-$(c+2)$ vector bundle over $U$. This vector bundle and thus
also $G_U(1,c+1)$ may be extended to the whole of $\Bbb P(S^2W^*)$.
When $c\ge 5$ one obtains a birational map from this extended relative
Grassmannian to $\overline{Pon_c}$.
In this paragraph we describe the linear system which induces this birational
map and its base locus as a set. (Note that the knowledge of the full scheme -
structure of this base-locus would allow one to compute the degree of
$\overline{Pon_c}$). Our method is to consider the stable rank-two vector
bundles on $\Bbb P(W)$ associated to Poncelet curves and their curves of
jumping lines.
Throughout this paragraph $c$ is assumed to be odd and bigger than 4.
The extension of $G_U(1,c+1)$ to $\Bbb P(S^2 W^*)$ is obvious. Take $I\subset
\Bbb P(S^2W^*)\times \Bbb P(W)$ the incidence variety ``points of conics'',
$p_1:\Bbb P(S^2W^*)\times \Bbb P(W)\to \Bbb P(S^2W^*)$ the first projection,
${\cal V}:=p_{1,*}{\cal O}_I\left(0,\displaystyle\frac{c+1}{2}\right)$ and $G$ the relative
Grassmannian of 2-dimen\-sional subspaces in the fibers of ${\cal V}$. $G$ is
then a compactification of $G_U(1,c+1)$.
Consider now a Poncelet pair $(S,C)$ with $S$ smooth and $C=C(\Lambda)$ with
base-point-free pencil $\Lambda\subset\left|{\cal O}_S\left(\displaystyle\frac{c+1}{2}
\right)\right|$. (In this paragraph we use the notation
${\cal O}_S\left(\displaystyle\frac{c+1}{2}\right):={\cal O}_{\Bbb
P(W)}\left(\displaystyle\frac{c+1}{2}\right)\mid_S$). $\Lambda$ induces a surjective
morphism
$$
{\cal O}_{\Bbb P(W)}^2\to {\cal O}_S\left(\displaystyle\frac{c+1}{2}\right).
$$
Let $F$ be its kernel. Then $F$ is a stable rank-2 vector bundle on $\Bbb P(W)$
with $c_1(F)=-2$, $c_2(F)=c+1$. and its jumping lines are exactly the lines
joining the points of some divisor of $\Lambda$ (see \cite{T}, 4.1). Thus, the
curve $C'\subset \Bbb P(W^*)$ of jumping lines of $F$ is Poncelet related to
the conic $S'$ dual to $S$.
We shall make $F$ fit into a flat family of coherent rank-2 sheaves over $G$,
and examine the birational map associating the curves of jumping lines to the
fibers of this family.
We start with some Lemmata.
{\bf Lemma 3.1.}
{\it Let $0\to E'\to {\cal O}^N_{\Bbb P^2} \to E''\to 0$ be an
exact sequence with $E',E''$ coherent torsion-free, indecomposable sheaves on
$\Bbb P^2$ and $c_1(E')=-1$. Then $N\le 3$ and $E',E''$ are slope-stable.}
{\it Proof.}
$
E''$ torsion-free implies $E'$ locally-free.
If $E'$ were not slope-stable, there would exist a locally free subsheaf $F$ of
$E'$ with $c_1(F)=0$. Standard arguments on slope-stable vector bundles (cf.
\cite{K}, V 8.3) show that $F$ would be a direct summand in $E'$,
contradicting the
hypothesis.
Similarly, $E''$ must be slope-stable.
Let $r':=rank\; E'$, $r'':=rank\; E$.
Suppose $r'=1$. Then $E'={\cal O}_{\Bbb P^2}(-1)$ and the morphism ${\cal
O}_{\Bbb P^2}(-1)
\linebreak
\to {\cal O}^N_{\Bbb P^2}$ is induced by $N$ sections in
${\cal O}_{\Bbb P^2} (1)$. Since $\Gamma({\cal O}_{\Bbb P^2}(1))$ is
3-dimensional we may choose when $N>3$ a basis for ${\cal O}_{\Bbb P^2}(1)^N$
such that $N-3$ components of ${\cal O}_{\Bbb P^2}\to {\cal O}_{\Bbb
P^2}(1)^N$ vanish. But then $E''$ would split.
The case $r''=1$ is treated in the same way.
We are left with the situation $r'>1$, $r''>1$. But now the Bogomolov
inequality gives
$$
\renewcommand{\arraystretch}{2.5}
\begin{array}{l}
c_2(E')\ge \displaystyle\frac{r'-1}{2r'}\cdot c_1(E')^2>0\\
\\
c_2(E'')\ge \displaystyle\frac{r''-1}{2r''}\cdot c_1(E'')^2>0
\end{array}
$$
hence $c_2({\cal O}^N_{\Bbb P^2})=-1+c_2(E')+c_2(E'')>0$, a contradiction.
\hfill$\Box$
{\bf Lemma 3.2.}
{\it Let ${\cal K}$ be the kernel of the natural surjective morphism
$$
p_1^*{\cal V}\to {\cal O}_I\left(0,\displaystyle\frac{c+1}{2}\right)
$$
on $\Bbb P(S^2W^*)\times\Bbb P(W)$. Then ${\cal K}$ is locally free and
its restrictions to the fibers of $p_1$ are slope-stable.}
{\it Proof.} The restriction to a fiber of $p_1$ over some point $s$
representing a conic $S$ gives an exact sequence
$$
0\to{\cal K}\mid_{p^{-1}(s)}\to\Gamma\left({\cal O}_S
\left(\displaystyle\frac{c+1}{2}\right) \right)
\otimes {\cal O}_{\Bbb P(W)}\to {\cal O}_S\left(\displaystyle\frac{c+1}{2}\right)\to 0.
$$
It is enough to check that $K:={\cal K}\mid_{p^{-1}(s)}$ is locally free and
slope stable.
Checking the locally freeness is easy (compare $K$ to its double dual or
just use the fact that ${\cal K}$ appears from an elementary transformation!).
Suppose that $K$ were not slope-stable.
Since $c_1(K)=-2$, there would exist a destabilizing subsheaf $K'$ of $K$ with
$c_1(K')=-1$.
By Lemma 3.1. $K'$ would have at most rank 2, hence the rank of $K$ could not
exceed 4.
But rank $K=c+2$ and the Lemma follows from our assumption on $c$.
\hfill$\Box$
Let ${\cal S}$ and ${\cal Q}$ be the tautological sub- and quotient bundle
on the relative Grassmannian $G$ of 2-dimensional subspaces in the fibers of
${\cal V}$. We denote by ${\cal S}_{\Bbb P(W)}$, ${\cal Q}_{\Bbb P(W)}$,
${\cal K}_G$, ${\cal V}_{G\times\Bbb P(W)}$ the pullbacks to $G\times \Bbb
P(W)$ of ${\cal S, Q, K}$ and ${\cal V}$, respectively. (Subscript will be used
in the sequel to indicate pullbacks through obvious maps).
Denote further by $\alpha$ the composite morphism
$$
{\cal Q}_{\Bbb P(W)}^{\vee}\to {\cal V}_{G\times\Bbb P(W)}^{\vee}\to
{\cal K}_G^{\vee}
$$
on $G\times\Bbb P(W)$ and by ${\cal E}$ its cokernel.
{\it Remark.} One sees immediately that the fiber of ${\cal E}$ over a point of
$G$ represented by a Poncelet pair $(S,C(\Lambda))$ with $S$ smooth and
$\Lambda$ base-point-free is just the dual of the kernel $F$ of the natural
morphism
$$
{\cal O}_{\Bbb P^2(W)}^2\to{\cal O}_S\left(\displaystyle\frac{c+1}{2}\right).
$$
In particular, it is stable and locally free.
We claim that ${\cal E}$ is flat over $G$. This is a consequence of a local
flatness criterion (\cite{M}; 2.2.5) and the following
{\bf Lemma 3.3}
{\it The restrictions of $\alpha$ to the fibers of $G\times\Bbb
P(W)\to G$ are injective and their cokernels are stable as soon as they are
torsion-free.}
{\it Proof.} Fix a conic $S$ and a $c$-dimensional subspace $A$ of
$\Gamma
\linebreak
\left({\cal O}_S \left(\frac{c+1}{2}\right)\right)^* \cong \Gamma(\Bbb P(W),
K^{\vee})$. The slope-stability of $K$ and Lem\-ma 3.1 imply that
the natural morphism $s:A\otimes{\cal O}_{\Bbb P(W)}\to K^{\vee}$ is injective.
If Coker $s$ is torsion-free but not stable, it will admit a rank-1 subsheaf
$E'$ with torsion-free quotient and $c_1(E')=1$, $c_2(E')\le\frac{c-1}{2}$. In
particular, $\dim\: Ext^1(E',
{\cal O}_{\Bbb P(W)})\le\frac{c-1}{2}$. We get a
subsheaf $K'$ of $K^{\vee}$ with $c_1(K')=1$, $rank(K')=c+1$,
and which sits in an
exact sequence
$$
0\to A\otimes{\cal O}_{\Bbb P(W)}\to K'\to E'\to 0.
$$
This leads to a contradiction of the stability of $K$ in view of the following
simple fact of homological algebra:
{\bf Lemma 3.4}
{\it Let $A$ be a finite-dimensional $\Bbb C$-vector space,
${\cal C}$ a coherent sheaf on a compact complex manifold $X$ and
$$
0\to A^*\otimes{\cal O}_X\to {\cal B}\to {\cal C}\to 0
$$
an extension given by an element
$$
\eta\in Ext^1({\cal C}, A^*\otimes{\cal O}_X)\cong Hom\:(A, \: Ext^1({\cal
C},{\cal O}_X))
$$
such that $A=A'\oplus A''$ and $\eta\mid_{A''}=0$. Then
$$
\eta\mid_{A'}\in Hom\:(A', \: Ext^1({\cal C},{\cal O}_X))\cong Ext^1({\cal
C},A^{'*}\otimes{\cal O}_X)
$$
gives rise to a commutative diagram
$$
\xymatrix{
0 \ar[r] & A^{'*}\otimes {\cal O}_X \ar[d] \ar[r]
& {\cal B}'\ar[d] \ar[r]
& {\cal C} \ar[d] \ar[r]
& 0\\
0 \ar[r]
& A^*\otimes {\cal O}_X \ar[r]
& {\cal B} \ar[r]
& {\cal C} \ar[r]
& 0 }
$$
with injective vertical maps.}
\hfill$\Box$
Associating to a fiber of ${\cal E}$ over $G$ its divisor of jumping lines
defines a rational map
$G \to \Bbb P(S^cW) $ in the usual way. Indeed, we
consider the standard diagram
$$
\xymatrix{
\Bbb F \ar[d] \ar[r] & \Bbb P(W^*)\\
\Bbb P(W) & }
$$
where ${\Bbb F}$ is the incidence variety ``points on lines'', then take its
product with $G$
$$
\xymatrix{
G\times \Bbb F \ar[d]_p \ar[r]^q & G\times \Bbb P(W^*)\\
G\times \Bbb P(W) & }
$$
and look at the relative theta characteristic associated to ${\cal E}$,
$$
\Theta ({\cal E}):= R^1q_* p^*({\cal E}(-2)).
$$
One sees immediately that, for the torsion free fibers, ${\cal E}_g$ of ${\cal
E}$, the corresponding fibers of $\Theta({\cal E})$ are the usual
theta-characteristics associated to a stable sheaf on $\Bbb P^2$ with even
first Chern
class (cf. \cite{B}).
In particular, their supports are the curves of jumping lines
of the ${\cal E}_g-s$. Let $B$ be the subset of $G$ over which ${\cal E}$ has
nontrivial torsion in its fibers. Then we get a morphism
$$
G\backslash B\to \Bbb P(S^cW),
$$
(see \cite{Ma}).
From the defining sequence of ${\cal E}$ one obtains the following resolution
of $\Theta({\cal E})$ on $G\times \Bbb P(W^*)$
$$
0\to{\cal Q}^{\vee}_{\Bbb P(W^*)}(-1)\stackrel{\varphi}{\to}\left( R^{1}
p_{1*}{\cal K}^{\vee}\right)_{G\times\Bbb P(W^*)}\to \Theta({\cal E})\to 0.
$$
In order to obtain the ``support'' $Z$ of $\Theta({\cal E})$ one takes the
determinant of $\varphi$ and twists it correspondingly to get:
$$
0\to{\cal L}^{-1}\boxtimes {\cal O}_{\Bbb P(W^*)}(-c)\to{\cal O}_{G\times\Bbb
P(W^*)}\to{\cal O}_Z\to 0
$$
where
$$
{\cal L}:=\det {\cal Q}\otimes {\cal O}_{\Bbb P(S^2W^*)}
\left(-\displaystyle\frac{(c-1)(c-3)}{8}\right)_G.
$$
Notice that $Z$ is not flat over $G$. In fact, its fibers over $B$ are
2-dimensional. Thus the rational map $G \to \Bbb P(S^cW) $
above is given by a linear subsystem of $|{\cal L}|$ and has base locus $B$.
The
structural sheaf of this base locus may be recovered by taking the push-down
on $G$ of the previous exact sequence twisted by (-3). One obtains for it the
presentation
$$
{\cal L}^{-1}\otimes H^0({\cal O}_{\Bbb P(W^*)}(c))^*\to {\cal O}_G \to R^2
\pi_*({\cal O}_Z(-3))\to 0,
$$
where $\pi:G\times \Bbb P(W^*)\to G$ is the projection.
Finally we want to identify $B$.
Let $\psi:\Bbb P(W^*)\times \Bbb P(W^*)\to \Bbb P(S^2W^*)$ be the map induced
by the product, $p':\Bbb P(W^*)\times\Bbb P(W^*)\times \Bbb P(W)\to \Bbb
P(W^*)\times \Bbb P(W^*)$ the projection,
$$
I_i=\{(l_1,l_2,x)\in \Bbb P(W^*)\times \Bbb P(W^*)\times \Bbb P(W)\mid x\in
l_i\},\; i\in\{1,2\},
$$
the incidence varieties and ${\cal W}:=p_*'({\cal O}_{I_1}(0,-1,
\frac{c-1}{2}))$.
{\bf Proposition.}
{\it ${\cal W}$ is a subbundle of $\psi^*{\cal V}$ and the
natural map from the Grassmannian of 2-dimensional subspaces in the fibers of
${\cal W}$ to $G$ is an embedding whose image is $B$. In particular, $B$ is
smooth of dimension $c+5$.}
{\it Proof.} We examine when a fiber of ${\cal E}$ over $G$ admits nontrivial
torsion. Fix a conic $S$ and a 2-dimensional subspace $L$ of $\Gamma\left(
{\cal O}_S \left(\frac{c+1}{2}\right)\right)$. If $K$ and $E$ are the fibers
of ${\cal K}$ and ${\cal E}$ over the corresponding point of $G$ we have exact
sequences:
$$
\begin{array}{l}
0\to \left(\Gamma\left({\cal O}_S\left(\displaystyle\frac{c+1}{2}\right)\right)
\biggl/L\right)^* \otimes {\cal O}_{\Bbb P(W)}\to K^{\vee}\to E\to 0,\\
o\to E^{\vee}\to L\otimes {\cal O}_{\Bbb P(W)}\stackrel{\epsilon}{\to}{\cal
O}_S
\left(\displaystyle\frac{c+1}{2}\right)\to {\cal E}xt^1(E,{\cal O}_{\Bbb P(W)})\to 0.
\end{array}
$$
where ${\epsilon}$ is the evaluation morphism.
Since $K$ is slope-stable, $E$ may
only have purely 1-dimensional torsion, and this happens exactly when
${\cal E}xt^1(E,{\cal O}_{\Bbb P(W)})$
has 1-dimensional support or equivalently,
when ${\epsilon}$ fails to be ``generically surjective on some component of
$S$''. More precisely, if this is the case, then $S=S_1+S_2$ and the composite
morphism
$$
L\otimes {\cal O}_{\Bbb P(W)}\left(\displaystyle\frac{c+1}{2}\right)
\stackrel{\epsilon}{\to}
{\cal O}_S\left(\displaystyle\frac{c+1}{2}\right)\to{\cal O}_{S_2}
\left(\displaystyle\frac{c+1}{2}\right)
$$
vanishes. Then ${\epsilon}$ factors through ${\cal
O}_{S_1}\left(\displaystyle\frac{c+1}{2}\right)$ and the pullback of $L$ through $\psi$
will be a 2-dimensional subspace in a fiber of ${\cal W}$ over $\Bbb
P(W^*)\times \Bbb P(W^*)$.
Indeed, pushing the exact sequence
$$
0\to {\cal O}_{I_1}(0,-1,-1)\to\psi^*{\cal O}_I\to {\cal O}_{I_2}\to 0
$$
twisted by $\left(\frac{c+1}{2}\right)$ from $\Bbb P(W^*)\times\Bbb P(W^*)
\times\Bbb P(W)$ down to $\Bbb P(W^*)\times \Bbb P(W^*)$ gives
$$
0\to {\cal W}\to\psi^*{\cal V}\to p_*'({\cal
O}_{I_2}\left(0,0,\left(\frac{c+1}{2}\right)\right)\to 0.
$$
The claims of the Proposition are now easy to check; that the considered
morphism is an embedding follows e.g. from a computation of its Jacobian
matrix.
\hfill$\Box$
\markboth{}{}
\markright{}
{\footnotesize
\renewcommand{\baselinestretch}{0.3}
|
1996-01-08T06:20:19 | 9601 | alg-geom/9601003 | en | https://arxiv.org/abs/alg-geom/9601003 | [
"alg-geom",
"math.AG"
] | alg-geom/9601003 | Moriwaki Atsushi | Atsushi Moriwaki | A sharp slope inequality for general stable fibrations of curves | 18 pages (with 1 figure by epic.sty), AmSLaTeX version 1.2 (LaTeX2e) | null | null | null | null | Let M_g be the moduli space of stable curves of genus g >= 2. Let D_i be the
irreducible component of the boundary of M_g such that general points of D_i
correspond to stable curves with one node of type i. Let M_g^0 be the set of
stable curves that have at most one node of type i>0. Let d_i be the class of
D_i in Pic(M_g)_Q and h the Hodge class on M_g. In this paper, we will prove a
sharp slope inequality for general stable fibrations. Namely, if $C$ is a
complete curve on M_g^0, then ( (8g+4)h - g d_0 - \sum_{i=1}^{[g/2]} 4i(g-i)
d_i . C ) >= 0. As an application, we can prove effective Bogomolov's
conjecture for general stable fibrations.
| [
{
"version": "v1",
"created": "Fri, 5 Jan 1996 15:38:29 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Moriwaki",
"Atsushi",
""
]
] | alg-geom | \section{Introduction}
Let $k$ be an algebraically closed field.
Throughout this paper, we will fix this field $k$.
Let $X$ be a smooth projective surface over $k$,
$Y$ a smooth projective curve over $k$,
and $f : X \to Y$ a generically smooth semistable curve
over $Y$ of genus $g \geq 2$.
Let $P$ be a node of a singular fiber $X_y$ over $y$.
We can assign a number $i$ to the node $P$ in the following way.
Let $\iota : X'_y \to X_y$ be the partial normalization of $X_y$
at $P$. If $X'_y$ is connected, then $i=0$.
Otherwise, $i$ is the minimum of arithmetic genera of
two connected components of $X'_y$. We say the node $P$ of
the singular fiber $X_y$ is of type $i$.
We denote by $\delta_i$ the number of nodes of type $i$ in singular fibers.
If $\operatorname{char}(k) = 0$,
we know the following inequality due to Cornalba-Harris \cite{CH}
and Xiao \cite{Xi}:
\[
(8g+4)\deg(f_*(\omega_{X/Y})) \geq g \delta,
\]
where $\delta = \delta_0 + \delta_1 + \cdots + \delta_{\left[g/2\right]}$.
(By virtue of \cite{Mo3}, this holds even if $\operatorname{char}(k) > 0$.)
This inequality is actually sharp because we know
an example which attains equality of the above inequality.
When we consider a fibration with reducible fibers,
we can however observe that
$(8g+4)\deg(f_*(\omega_{X/Y}))$ is rather larger than $g \delta$.
According to the exact formula \cite[Proposition~4.7]{CH} for
hyperelliptic fibrations, we can guess a sharper inequality:
\[
(8g+4)\deg(f_*(\omega_{X/Y})) \geq g \delta_0 +
\sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i.
\]
In this paper, we would like to prove this sharper inequality
for general stable fibrations.
\begin{Theorem}[$\operatorname{char}(k) = 0$
\label{thm:sharp:slope:ineq:general:stable:curve}
Let $\bar{f} : \overline{X} \to Y$ be the stable model of
$f : X \to Y$. If every singular fiber of $\bar{f} : \overline{X} \to Y$
has at most one node of type $i > 0$, then
\[
(8g + 4) \deg(f_*(\omega_{X/Y})) \geq
g \delta_0 + \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i.
\]
In other words, we have the following.
Let $\overline{\mathcal{M}}_g$ be the moduli space of
stable curves of genus $g \geq 2$ over $k$.
Let $\Delta_i$ be the irreducible component of the boundary of
$\overline{\mathcal{M}}_g$ such that
general points of $\Delta_i$ correspond to stable curves with
one node of type $i$.
Let $\overline{\mathcal{M}}_g^0$ be the set of
stable curves that have
at most one node of type $i>0$.
Let $\delta_i$ be the class of $\Delta_i$ in
$\operatorname{Pic}(\overline{\mathcal{M}}_g) \otimes {\mathbb{Q}}$
and $\lambda$ the Hodge class on
$\overline{\mathcal{M}}_g$. Then,
for all complete curves $C$ on $\overline{\mathcal{M}}^0_g$,
\[
\left( (8g+4)\lambda - g \delta_0 -
\sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i
\ \cdot \ C \right) \geq 0.
\]
\end{Theorem}
An idea of the proof of
Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve} is as follow.
Let $E$ be the kernel of the natural homomorphism
$f^*f_*(\omega_{X/Y}) \to \omega_{X/Y}$. By \cite{PR},
$E$ is semistable on the geometric generic fiber $X_{\bar{\eta}}$ of $f$.
By the same idea as in \cite{Mo1},
we can apply Bogomolov inequality (Theorem~\ref{thm:B:ineq:fiber:space})
to $E$. This is however
insufficient to get the sharper inequality.
Actually we have only
Cornalba-Harris-Xiao inequality (cf. Remark~\ref{rem:C-H-X:ineq}).
For the sharper inequality, we need to modify $E$ along singular fibers,
namely, we change a compactification of $E_{\bar{\eta}}$ on
$X_{\bar{\eta}}$.
This modification can be done by a special
elementary transformation.
As an application of Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve},
we can show the following answer
for Bogomolov conjecture over function fields
(cf. Theorem~\ref{thm:bogomolov:function:fiels}).
First of all, we fix a notation.
Let $\bar{f} : \overline{X} \to Y$ be the stable model of
$f : X \to Y$. Let $X_y$ (resp. $\overline{X}_y$)
be a singular fiber of $f$ (resp. $\bar{f}$) over $y$, and
$S_y$ the set of nodes $P$ of $\overline{X}_y$ such that
$P$ is not an intersection of two irreducible components of $\overline{X}_y$.
Let $\pi : Z_y \to \overline{X}_y$ be the partial normalization of
$\overline{X}_y$ at each node in $S_y$.
We say $X_y$ is a chain of stable components if
the dual graph of $Z_y$ is homeomorphic to the interval $[0,1]$.
\begin{Theorem}[$\operatorname{char}(k) = 0$
We assume that $f$ is not smooth,
every singular fiber of $f$
is a chain of stable components, and one of the following conditions:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
the generic fiber of $f$ is hyperelliptic, or
\item
every singular fiber of the stable model $\bar{f} : \overline{X} \to Y$
has at most one node of type $i > 0$.
\end{enumerate}
Then Bogomolov conjecture holds for the generic fiber of $f$,
i.e., we have the following.
Let $K$ be the function field of $Y$, $C$ the generic fiber of $f$,
$\operatorname{Jac}(C)$ the Jacobian of $C$, and let $j : C(\overline{K})
\to \operatorname{Jac}(C)(\overline{K})$ be a morphism given by
$j(P) = \omega_C - P$. Then, $j(C(\overline{K}))$ is
discrete in terms of the semi-norm arising from the Neron-Tate height paring
on $\operatorname{Jac}(C)(\overline{K})$.
More precisely, the maximal radius of ball in which we have only
finitely many points coming from $C(\overline{K})$ via $j$ is
greater than or equal to
\[
\sqrt{\frac{(g-1)^2}{g(2g+1)}\left(
\frac{g-1}{3}\delta_0 + \sum_{i=1}^{\left[\frac{g}{2}\right]}
4i(g-i)\delta_i \right)}.
\]
\end{Theorem}
This is a generalization of the main result of \cite{Mo3} in the
case where $\operatorname{char}(k) = 0$.
For the proof of the above corollary, we will
calculate an invariant $e(G,D)$ for a metrized graph $G$
with a polarization $D$. Using this,
we have the following inequality, which is rather
weaker than the sharper inequality
(cf. Corollary~\ref{cor:e:for:semistable:chain}).
\begin{Proposition}[$\operatorname{char}(k) \geq 0$
If every singular fiber of $f$ is a chain of stable components,
then
\[
\left(\omega_{X/Y} \cdot \omega_{X/Y} \right) \geq
\frac{g-1}{3g} \delta_{0} +
\sum_{i=1}^{\left[\frac{g}{2}\right]}
\left( \frac{4 i (g-i)}{g} - 1 \right) \delta_{i}.
\]
Moreover, if the above inequality is strict,
then Bogomolov conjecture holds for the generic fiber of $f$.
\end{Proposition}
\section{Preliminaries}
\renewcommand{\theTheorem}{\arabic{section}.\arabic{subsection}.\arabic{Theorem}}
In this section, we will recall base loci of canonical linear systems of
reduced curves with only node singularities, and
Bogomolov inequality for semistable vector bundles.
\subsection{Canonical linear systems of
reduced curves with only node singularities}
\setcounter{Theorem}{0}
Let $C$ be a reduced projective curve over $k$ with only node
singularities.
Let $\{ z_1, \ldots, z_s \}$ be the set of singularities of $C$,
and $\pi : \widetilde{C} \to C$ the normalization of $C$.
We set $\pi^{-1}(z_i) = \{ x_i, y_i \}$ for each $i$.
The dualizing sheaf $\omega_C$ of $C$ is defined by
properties: (1) $\omega_C \subset
\pi_*(\Omega^1_{\widetilde{C}}(\sum_i x_i + y_i))$ and
(2) for an open set $U$ of $C$,
$t \in \rest{\omega_C}{U}$ if and only if $\operatorname{Res}_{x_i}(t) + \operatorname{Res}_{y_i}(t) = 0$
for all $i$ with $z_i \in U$.
Note that $\omega_C$ is an invertible sheaf on $C$, i.e.,
a local section $t \in \pi_*(\Omega_{\widetilde{C}}^1(x_i + y_i))_{z_i}$
with $\operatorname{Res}_{x_i}(t) + \operatorname{Res}_{y_i}(t) = 0$ and $\operatorname{Res}_{x_i}(t) \not= 0$
forms a local frame of $(\omega_{C})_{z_i}$.
The arithmetic genus of $C$, denoted by $p_a(C)$, is given by
$\dim_k H^0(C, \omega_C)$ ($=\dim_k H^1(C, {\mathcal{O}}_C)$).
\begin{Lemma}[$\operatorname{char}(k) \geq 0$
\label{lem:key:for:base:loci}
For smooth points $P_1, \ldots, P_n$ of $C$, let us consider
a homomorphism
\[
\phi : H^0(C, \omega_C(P_1 + \cdots + P_n)) \to k^n
\]
defined by $\phi(t) = (\operatorname{Res}_{P_1}(t), \ldots, \operatorname{Res}_{P_n}(t))$.
If $C$ is connected, then the kernel of $\phi$ is
$H^0(C, \omega_C)$ and the image of $\phi$ is
the subspace given by
$\{(a_1, \ldots, a_n) \in k^n \mid a_1 + \cdots + a_n = 0 \}$.
\end{Lemma}
{\sl Proof.}\quad
Let $\pi : \widetilde{C} \to C$, $x_i$, $y_i$ and $z_i$ be
the same as before.
Let $t$ be an element of $H^0(C, \omega_C(P_1 + \cdots + P_n))$
with $\phi(t) = 0$. Then, since $\operatorname{Res}_{P_j}(t) = 0$ for all $j$,
$t$ has no pole at each $P_j$, which implies $t \in H^0(C, \omega_C)$.
Thus, $\operatorname{Ker}(\phi) = H^0(C, \omega_C)$.
Next we would like to see $\sum_{j} \operatorname{Res}_{P_j}(t) = 0$
for all $t \in H^0(C, \omega_C(P_1 + \cdots + P_n))$.
By the definition of $\omega_C$,
$t \in H^0(\widetilde{C},
\Omega^1_{\widetilde{C}}(\sum_i (x_i + y_i) + P_1 + \cdots + P_n))$ and
$\operatorname{Res}_{x_i}(t) + \operatorname{Res}_{y_i}(t) = 0$
for all $i$. On the other hand, by Residue formula,
\[
\sum_{i=1}^s (\operatorname{Res}_{x_i}(t) + \operatorname{Res}_{y_i}(t)) + \sum_{j=1}^n
\operatorname{Res}_{P_j}(t) = 0.
\]
Thus, $\sum_{j=1}^n \operatorname{Res}_{P_j}(t) = 0$. Therefore, the image of $\phi$
is contained in $\{(a_1, \ldots, a_n) \in k^n \mid a_1 + \cdots + a_n = 0 \}$.
Since $C$ is connected, by Serre duality,
\[
\dim_k H^1(C, \omega_C(P_1 + \cdots + P_n)) =
\dim_k H^0(C, {\mathcal{O}}_C(-P_1 - \cdots -P_n)) = 0.
\]
Thus, using Riemann-Roch theorem, we have
\[
\dim_k H^0(C, \omega_{C}(P_1 + \cdots + P_n)) =
\dim_k H^0(C, \omega_{C}) + n - 1.
\]
Since the kernel of $\phi$ is
$H^0(C, \omega_{C})$, the above formula says us that
the dimension of the image of $\phi$ is $n-1$. Therefore,
we get the last assertion.
\QED
\begin{Corollary}[$\operatorname{char}(k) \geq 0$
\label{cor:gen:by:glo:irreducible}
Let $C$ be a reduced and irreducible projective curve over $k$ with
only node singularities. If $p_a(C) > 0$, then
$\omega_C$ is generated by global sections.
\end{Corollary}
{\sl Proof.}\quad
Let $s$ be the number of singularities of $C$ and
$g$ the genus of the normalization of $C$.
If $s=0$ and $g > 0$, or $s=1$ and $g = 0$, then
our assertion is trivial. We will prove it by induction on $s$.
Let $P$ be a node of $C$,
$h : C' \to C$ the partial normalization at $P$, and
$h^{-1}(P) = \{ Q, R \}$. Then, by hypothesis of induction,
$\omega_{C'}$ is generated by global sections and
$H^0(C' \omega_{C'}) \subset H^0(C, \omega_C)$.
Thus, $H^0(C, \omega_{C}) \otimes {\mathcal{O}}_C \to \omega_{C}$ is surjective
outside $P$. On the other hand, by Lemma~\ref{lem:key:for:base:loci},
there is a section $t$
of $H^0(C', \omega_{C'}(Q + R))$ such that $\operatorname{Res}_{Q}(t) + \operatorname{Res}_{R}(t) =0$
and $\operatorname{Res}_{Q}(t) \not= 0$. By the definition of $\omega_C$,
$t \in H^0(C, \omega_C)$ and $t$ generates $(\omega_C)_P$.
\QED
\bigskip
Let $C$ be a connected reduced projective curve over $k$
with only node singularities.
Let $P$ be a node of $C$ and $h : C' \to C$ the partial normalization
of $C$ at $P$. We say $P$ is a disconnecting node
if $C'$ is not connected.
Note that, if $C$ is semistable curve, then
a disconnected node is nothing more than a node of type $i > 0$.
An irreducible component $D$ of $C$ is said to be
of socket type if
$D$ is smooth and rational, and all nodes of $C$ on $D$ (i.e.
intersections on $D$ with other components) are disconnecting nodes.
The base locus of $|\omega_C|$, denoted by $\operatorname{Bs}(\omega_C)$,
is defined by the support of
$\operatorname{Coker}(H^0(C, \omega_C) \otimes {\mathcal{O}}_C \to \omega_C)$.
\begin{Proposition}[$\operatorname{char}(k) \geq 0$
\label{prop:base:locus:stable:curve}
Let $C$ be a connected reduced projective curve over $k$
with only node singularities,
$D_1, \cdots, D_r$ irreducible components of socket type, and
$DN_C$ the set of all disconnecting nodes of $C$. Then we have
the following.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$\operatorname{Bs}(\omega_C) = D_1 \cup \cdots \cup D_r \cup DN_C$.
\item
Let $P$ be a disconnecting node of $C$,
$h : C' \to C$ the partial normalization of $C$ at $P$,
$C_1$ and $C_2$ two connected components of $C'$.
Then,
\[
H^0(C_1, \omega_{C_1}) \oplus H^0(C_2, \omega_{C_2})
\overset{\sim}{\longrightarrow} H^0(C, \omega_C).
\]
\item
If $P$ is a disconnecting node not lying on
irreducible components of socket type,
then the image of $H^0(C, \omega_C) \otimes {\mathcal{O}}_{C,P} \to (\omega_{C})_{P}$
coincides with the image of $(\Omega^1_{C})_{P} \to (\omega_{C})_{P}$.
\end{enumerate}
\end{Proposition}
{\sl Proof.}\quad
In the following, we denote by $N_C$ the set of all nodes of $C$.
In order to see (1),
it is sufficient to show the following facts:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
If $P \in N_C$ and $P$ is a disconnecting node, then
$P \in \operatorname{Bs}(\omega_C)$.
\item
If $P \in N_C$ and $P$ is not a disconnecting node, then
$P \not\in \operatorname{Bs}(\omega_C)$.
\item
If $D$ is an irreducible component of socket type,
then $D \subset \operatorname{Bs}(\omega_C)$.
\item
If $D$ is an irreducible component not of socket type,
then $H^0(C, \omega_C) \otimes {\mathcal{O}}_C \to \omega_C$ is surjective
on $D \setminus N_C$.
\end{enumerate}
\medskip
(a) Let $h : C' \to C$ be the partial normalization of $C$ at $P$,
$C_1$ and $C_2$ two connected components of $C'$, and
$h^{-1}(P) = \{ Q, R \}$ with $Q \in C_1$ and $R \in C_2$.
Let $t$ be any section of $H^0(C, \omega_C)$.
Then, $\rest{t}{C_1}$ (resp. $\rest{t}{C_2}$) is a section
of $H^0(C_1, \omega_{C_1}(Q))$ (resp. $H^0(C_2, \omega_{C_2}(R))$).
Then, by Lemma~\ref{lem:key:for:base:loci},
$\operatorname{Res}_{Q}(\rest{t}{C_1}) = \operatorname{Res}_{R}(\rest{t}{C_2}) = 0$.
Thus, $\rest{t}{C_1} \in H^0(C_1, \omega_{C_1})$ and
$\rest{t}{C_2} \in H^0(C_2, \omega_{C_2})$. Hence, $t(P) = 0$.
Therefore, $P \in \operatorname{Bs}(\omega_C)$.
\medskip
(b) Let $h : C' \to C$ be the partial normalization of $C$ at $P$,
and $h^{-1}(P) = \{ Q, R \}$.
By Lemma~\ref{lem:key:for:base:loci}, there is a section $t$
of $H^0(C', \omega_{C'}(Q+R))$ such that $\operatorname{Res}_{Q}(t) + \operatorname{Res}_{R}(t) = 0$
and $\operatorname{Res}_{Q}(t) \not= 0$. Thus, $t$ is a section of
$H^0(C, \omega_C)$ and $t$ generates $(\omega_C)_P$.
\medskip
(c) Let $D \cap N_C = \{ Q_1, \ldots, Q_r \}$.
Let $t$ be any section of $H^0(C, \omega_C)$.
Then, $\rest{t}{D} \in H^0(D, \omega_D(Q_1 + \cdots + Q_r))$.
Since $Q_i$ is a disconnecting node, in the same way as in (a),
we can see that $\operatorname{Res}_{Q_i}(\rest{t}{D}) = 0$ for all $i$.
Thus, $\rest{t}{D}$ has no pole on $D$. Therefore,
$\rest{t}{D} = 0$ because $D \simeq {\mathbb{P}}^1$.
Hence, $D \subset \operatorname{Bs}(\omega_C)$.
\medskip
(d) If $p_a(D) > 0$, then our assertion is a consequence of
Corollary~\ref{cor:gen:by:glo:irreducible}.
Thus, we may assume that $D$ is smooth and rational.
Let $E$ be the closure of $C \setminus D$. Since
$D$ is not of socket type, there is a connected component $D'$
of $E$ with $\#(D \cap D') \geq 2$.
We set $D \cap D' = \{ Q_1, \ldots, Q_r \}$ ($r \geq 2$) and
$D'' = E \setminus D'$.
Let $R$ be any point of $D \setminus N_C$. We would like to see
$R \not\in \operatorname{Bs}(\omega_C)$. Since $\omega_D(Q_1 + \cdots + Q_r)$
is generated by global sections, there is a section
$t \in H^0(D, \omega_D(Q_1 + \cdots + Q_r))$ with $t(R) \not= 0$.
Since $\operatorname{Res}_{Q_1}(t) + \cdots + \operatorname{Res}_{Q_r}(t) = 0$,
by Lemma~\ref{lem:key:for:base:loci},
there is a section $t'$ of $H^0(D', \omega_{D'}(Q_1 + \cdots + Q_r))$
such that $\operatorname{Res}_{Q_i}(t') + \operatorname{Res}_{Q_i}(t) = 0$ for all $i$.
Moreover, let $t''$ be the zero form on $D''$.
Then, $t$, $t'$ and $t''$ give a section $s \in H^0(C, \omega_C)$
with $\rest{s}{D} = t$, $\rest{s}{D'} = t'$ and $\rest{s}{D''} = t''$.
Here $s(R) \not= 0$.
Thus, $s$ generates $(\omega_C)_R$.
Therefore, $R \not\in \operatorname{Bs}(\omega_C)$.
\medskip
(2) is obvious by the proof of (a).
\medskip
Finally, let us consider (3).
Let $h : C' \to C$ be the partial normalization of $C$ at $P$,
$C_1$ and $C_2$ two connected components of $C'$, and
$h^{-1}(P) = \{ Q, R \}$ with $Q \in C_1$ and $R \in C_2$.
Let $x$ (resp. $y$) be a local parameter of $C_1$ at $Q$ (resp.
$C_2$ at $R$).
The image of $H^0(C, \omega_C) \otimes {\mathcal{O}}_{C,P} \to (\omega_{C})_{P}$
is contained in the image of $(\Omega^1_{C})_{P} \to (\omega_{C})_{P}$
because $H^0(C, \omega_{C}) =
H^0(C_1, \omega_{C_1}) \oplus H^0(C_2,\omega_{C_2})$.
On the other hand,
since $P$ is not lying on irreducible components of socket type,
there are $t_1 \in H^0(C_1, \omega_{C_1})$ and
$t_2 \in H^0(C_2, \omega_{C_2})$ with $t_1(Q) \not= 0$
and $t_2(R) \not= 0$.
Locally, $t_1 = u_1 dx$ and $t_2 = u_2 dy$, where $u_1(Q) \not= 0$ and
$u_2(R) \not= 0$.
Thus, we have (3).
\QED
\subsection{Bogomolov inequality for fiber spaces}
\setcounter{Theorem}{0}
Let $X$ be a smooth projective surface over $k$ and
$E$ a torsion free sheaf of rank $r$ on $X$.
We set
\[
\delta(E) = 2 r c_2(E) - (r -1)c_1(E)^2.
\]
Then we have the following version of Bogomolov inequality
for fiber spaces.
\begin{Theorem}[$\operatorname{char}(k) = 0$
\label{thm:B:ineq:fiber:space}
Let $X$ be a smooth projective surface over $k$, $Y$
a smooth projective curve over $k$, and $f : X \to Y$
a surjective morphism with $f_*({\mathcal{O}}_X) = {\mathcal{O}}_Y$.
Let $E$ be a torsion free sheaf on $X$.
If $E$ is semistable on the geometric generic fiber $X_{\bar{\eta}}$
of $f$, then $\delta(E) \geq 0$.
\end{Theorem}
{\sl Proof.}\quad
Let $E^{\vee\vee}$ be the double dual of $E$. Then,
$E^{\vee\vee}$ is locally free, $c_1(E^{\vee\vee}) = c_1(E)$, and
$c_2(E^{\vee\vee}) = c_2(E) - \operatorname{length}(E^{\vee\vee}/E)$.
Thus, we may assume $E$ is locally free.
We assume $\delta(E) < 0$. Then, by Bogomolov instability theorem
(cf. \cite{Bog} or \cite{Mo0}),
there is a non-zero saturated subsheaf $G$ of $E$ such that, if
we set $D = (\operatorname{rk} E)c_1(G) - (\operatorname{rk} G) c_1(E)$, then
$(D^2) > 0$ and $(D \cdot H) > 0$ for some ample divisor $H$.
By Riemann-Roch theorem, we can see that there are
a positive number $n$ and an effective divisor $L$ such that
$nD$ is linearly equivalent to $L$.
Let $F$ be a general fiber of $f$. Then, $(L \cdot F) \geq 0$, which implies
that $(D \cdot F) \geq 0$. Here we claim that $(D \cdot F) > 0$.
To see this, we assume $(D \cdot F) = 0$. Then, $(L \cdot F) = 0$.
Thus, $L$ is a linear combination of irreducible components of fibers.
By Zariski lemma, this implies $(L^2) \leq 0$, which contradicts to
$(D^2) > 0$. Therefore, we get $(D \cdot F) > 0$.
On the other hand,
\[
(D \cdot F) = (\operatorname{rk} E) \deg(\rest{G}{F}) - (\operatorname{rk} G)\deg(\rest{E}{F}).
\]
Hence, $(D \cdot F) > 0$ means that $\rest{G}{F}$ is a destabilizing subsheaf
of
$\rest{E}{F}$. This contradicts to the assumption that
$E$ is semistable on the geometric generic fiber $X_{\bar{\eta}}$
of $f$.
\QED
\section{Proof of Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve}}
\renewcommand{\theTheorem}{\arabic{section}.\arabic{Theorem}}
To start the proof of Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve},
we need a lot of preparations.
Fix an integer $g \geq 2$ and a polynomial $P_g(n) = (6n-1)(g-1)$.
Let $H_g \subset \operatorname{Hilb}^{P_g}_{{\mathbb{P}}^{5g-6}}$ be a subscheme
of all tri-canonically embededded stable curves over $k$,
$Z_g \subset H_g \times {\mathbb{P}}^{5g-6}$
the universal tri-canonically embededded stable curves over $k$,
and $\pi : Z_g \to H_g$ the natural projection.
Let $S$ be the set of all points $x \in Z_g$ such that
$\pi$ is not smooth at $x$, and $\Delta = \pi(S)$.
Then, by \cite[Theorem~(1.6) and Corollary~(1.9)]{DM},
$Z_g$, $H_g$, and $S$ are smooth over $k$, $\Delta$ and $\pi^*(\Delta)$
are divisors with only normal crossings, and
$\rest{\pi}{S} : S \to \Delta$ is the normalization of $\Delta$.
Let $\Delta = \Delta_0 \cup \cdots \cup \Delta_{[g/2]}$
be the irreducible decomposition of $\Delta$ such that,
if $x \in \Delta_i \setminus \operatorname{Sing}(\Delta)$, then
$\pi^{-1}(x)$ is a stable curve with one node of type $i$.
Let $\delta_i(x)$ be the number of nodes of type $i$ on a fiber $\pi^{-1}(x)$
over $x \in Z_g$.
Since $\delta_i(x) = \operatorname{mult}_x(\Delta_i)$,
$\delta_i : H_g \to {\mathbb{Z}}$ is upper-semicontinuous.
Therefore, if we set
\[
H_g^0 = \{ x \in H_g \mid \text{$\pi^{-1}(x)$ has at most one node
of type $i > 0$} \},
\]
then $H_g^0$ is an open set.
In other words, $H_g^0 = H_g \setminus \operatorname{Sing}(\Delta_1 + \cdots
+\Delta_{[g/2]})$.
We set $Z_g^0 = \pi^{-1}(H_g^0)$,
$\Delta^0 = \Delta \cap H_g^0$,
$\Delta_i^0 = \Delta_i \cap H_g^0$,
$S^0 = \left(\rest{\pi}{S}\right)^{-1}(\Delta^0)$,
and $S_i^0 = \left(\rest{\pi}{S}\right)^{-1}(\Delta_i^0)$.
Then, for all $1 \leq i < j \leq [g/2]$,
$\Delta_i^0 \cap \Delta_j^0 = \emptyset$.
For $1 \leq i \leq [g/2]$,
let $\pi^{-1}(\Delta_i^0) = C_i^1 \cup C_i^2$ be
the irreducible decomposition
such that the generic fiber of $\rest{\pi}{C_i^1} : C_i^1 \to \Delta_i^0$
(resp. $\rest{\pi}{C_i^2} : C_i^2 \to \Delta_i^0$) is of genus $i$
(resp. $g-i$).
Then, $C_i^1 \cap C_i^2 = S_i^0$.
Moreover, we denote $S_1^0 \cup \cdots \cup S_{[g/2]}^0$ by
$S^0_{+}$.
\begin{Claim
\label{claim:image:can:sys}
$\operatorname{Supp}(\operatorname{Coker}(\pi^{*}\pi_{*}(\omega_{Z_g^0/H_g^0}) \to
\omega_{Z_g^0/H_g^0})) = S^0_{+}$ and, for $z \in S^0_{+}$,
the image of $\left(\pi^{*}\pi_{*}(\omega_{Z_g^0/H_g^0})\right)_z \to
\left(\omega_{Z_g^0/H_g^0}\right)_z$ is
$\left(\Omega^1_{Z_g^0/H_g^0}\right)_z$.
\end{Claim}
{\sl Proof.}\quad
The first assertion is a consequence of (1) of
Proposition~\ref{prop:base:locus:stable:curve}.
Let $F$ be the image of $\pi^{*}\pi_{*}(\omega_{Z_g^0/H_g^0}) \to
\omega_{Z_g^0/H_g^0}$ and $x = \pi(z)$.
As we see in (2) of Proposition~\ref{prop:base:locus:stable:curve},
any section $s$ of $\left(\pi_{*}(\omega_{Z_g^0/H_g^0})\right)_x$
has no pole on irreducible components of fibers around $z$, which means
$F_z \subseteq \left(\Omega^1_{Z_g^0/H_g^0}\right)_z$.
Therefore,
by (3) of Proposition~\ref{prop:base:locus:stable:curve},
we can conclude $F_z = \left(\Omega^1_{Z_g^0/H_g^0}\right)_z$.
\QED
We set
\[
E = \operatorname{Ker}\left(
\pi^{*}\pi_{*}(\omega_{Z_g^0/H_g^0}) \longrightarrow
\omega_{Z_g^0/H_g^0}\right).
\]
\begin{Claim
$E$ is locally free.
\end{Claim}
{\sl Proof.}\quad
Let $I_{S^0}$ (resp. $I_{S^0_{+}}$) be the defining ideal of $S^0$
(resp. $S^0_{+}$). It is well known that
$\Omega^1_{Z_g^0/H_g^0} = I_{S^0} \cdot \omega_{Z_g^0/H_g^0}$
(cf. \cite[Proof of Theorem~5.10]{Mu}).
Thus, by Claim~\ref{claim:image:can:sys},
the image of $\pi^{*}\pi_{*}(\omega_{Z_g^0/H_g^0}) \to
\omega_{Z_g^0/H_g^0}$ is $I_{S^0_{+}} \cdot \omega_{Z_g^0/H_g^0}$.
Since $S^0_{+}$ is smooth and of codimension $2$,
$\operatorname{\mathcal{E}\textsl{xt}}^i_{{\mathcal{O}}_{Z_g^0}}(I_{S^0_{+}}, {\mathcal{O}}_{Z_g^0}) = 0$ for $i \geq 2$.
Therefore, we can see that $\operatorname{\mathcal{E}\textsl{xt}}^i_{{\mathcal{O}}_{Z_g^0}}(E, {\mathcal{O}}_{Z_g^0}) = 0$
for $i \geq 1$. Hence, we have our claim.
\QED
\begin{Claim
For $1 \leq i \leq [g/2]$,
${\displaystyle
\pi_*(\omega_{C_i^1/\Delta_i^0}) \oplus \pi_*(\omega_{C_i^2/\Delta_i^0})
\overset{\sim}{\longrightarrow}
\pi_*(\omega_{C_i^1 \cup C_i^2/\Delta_i^0}).
}$
\end{Claim}
{\sl Proof.}\quad
Clearly we have the natural injection
\[
\pi_*(\omega_{C_i^1/\Delta_i^0}) \oplus \pi_*(\omega_{C_i^2/\Delta_i^0})
\longrightarrow \pi_*(\omega_{C_i^1 \cup C_i^2/\Delta_i^0}).
\]
For $x \in \Delta_i^0$, if we set $C_1 = \pi^{-1}(x) \cap C_i^1$ and
$C_2 = \pi^{-1}(x) \cap C_i^2$, then
by (2) of Proposition~\ref{prop:base:locus:stable:curve}
\[
H^0(C_1, \omega_{C_1}) \oplus H^0(C_2, \omega_{C_2}) =
H^0(\pi^{-1}(x), \omega_{\pi^{-1}(x)}).
\]
Thus, the above injection is bijective.
\QED
For $1 \leq i \leq [g/2]$ and $j= 1, 2$, let
\[
Q_i^j = \operatorname{Ker}\left(
\left(\rest{\pi}{C_i^j}\right)^*\left(\rest{\pi}{C_i^j}\right)_*
(\omega_{C_i^j/\Delta_i^0})
\longrightarrow \omega_{C_i^j/\Delta_i^0}\right).
\]
\begin{Claim
$Q_i^j$ is a locally free sheaf on $C_i^j$.
\end{Claim}
{\sl Proof.}\quad
The homomorphism
$\left(\rest{\pi}{C_i^j}\right)^*\left(\rest{\pi}{C_i^j}\right)_*
(\omega_{C_i^j/\Delta_i^0})
\to \omega_{C_i^j/\Delta_i^0}$ is surjective
by (1) of Proposition~\ref{prop:base:locus:stable:curve}.
Therefore, we can see our claim.
\QED
Using projection
$\pi_*(\omega_{C_i^1 \cup C_i^2/\Delta_i^0}) \to
\pi_*(\omega_{C_i^j/\Delta_i^0})$, we have the following commutative diagram:
\[
\begin{CD}
0 @>>> \rest{E}{C_i^j} @>>> \rest{\pi^*\pi_*(\omega_{Z_g^0/H_g^0})}{C_i^j}
@>>> \rest{\Omega^1_{Z_g^0/H_g^0}}{C_i^j} @>>> 0 \\
@. @VV{\alpha_i^j}V @VVV @VVV @. \\
0 @>>> Q_i^j @>>>
\left(\rest{\pi}{C_i^j}\right)^*\left(\rest{\pi}{C_i^j}\right)_*
(\omega_{C_i^j/\Delta_i^0})
@>>> \omega_{C_i^j/\Delta_i^0} @>>> 0,
\end{CD}
\]
where $\alpha_i^j : \rest{E}{C_i^j} \to Q_i^j$ is the induced
homomorphism.
Thus, we can give a homomorphism
\[
\phi_i : E \to \rest{E}{C_i^1} \oplus \rest{E}{C_i^2}
\overset{\alpha_i^1 \oplus \alpha_i^2}{\longrightarrow} Q_i^1 \oplus Q_i^2.
\]
\begin{Claim
$\phi_i$ is surjective.
\end{Claim}
{\sl Proof.}\quad
To show this claim, it is sufficient to see it on each fiber.
Let $C$ be a fiber of $\pi$ over $x \in \Delta_i^0$,
and $C = C^1 \cup C^2$ the irreducible decomposition with
$C^1 = C \cap C_i^1$ and $C^2 = C \cap C_i^2$.
For simplicity, we set $E_C = \rest{E}{C}$,
$Q^1 = \rest{Q_i^1}{C^1}$,
$Q^2 = \rest{Q_i^2}{C^2}$, i.e.,
$E_C$ is the kernel of
$H^0(C, \omega_C) \otimes {\mathcal{O}}_C \to \omega_C$ and
$Q^j$ is the kernel of $H^0(C^j, \omega_{C^j}) \otimes {\mathcal{O}}_{C^j}
\to \omega_{C^j}$.
Since $H^0(C, \omega_C) = H^0(C^1, \omega_{C^1}) \oplus
H^0(C^2, \omega_{C^2})$,
$Q^1 \oplus Q^2 \subset H^0(C, \omega_{C}) \otimes {\mathcal{O}}_C$.
On the other hand, $Q^1, Q^2 \subset E_C$. Thus, we get
$Q^1 \oplus Q^2 \subset E_C$, which shows us that
$E_C \to Q^1 \oplus Q^2$ is surjective.
\QED
Let
\[
F = \operatorname{Ker}\left( \bigoplus_{i=1}^{\left[\frac{g}{2}\right]} \phi_i
\ : \ E \longrightarrow
\bigoplus_{i=1}^{\left[\frac{g}{2}\right]} Q_i^1 \oplus Q_i^2 \right).
\]
\begin{Claim
$F$ is locally free.
\end{Claim}
{\sl Proof.}\quad
We set $F_1 = \operatorname{Ker}\left( E \to \bigoplus_{i=1}^{[g/2]} Q_i^1 \right)$.
Then, $F = \operatorname{Ker}\left( F_1 \to \bigoplus_{i=1}^{[g/2]} Q_i^2 \right)$.
$F_1$ is an elementary transformation of $E$.
Thus, by \cite{Ma}, $F_1$ is locally free. Since $\phi_i$ is surjective,
so is $\rest{F_1}{C_i^2} \to Q_i^2$. Therefore,
$F$ is locally free because $F$ is an elementary transformation of $F_1$.
\QED
For a vector bundle $G$ on $Z_g^0$,
we define $\delta(G) \in A_{\dim H_g^0 - 1}(H_g^0)$ by
\[
\delta(G) = \pi_*\left(2 \operatorname{rk}(G) c_2(G) - (\operatorname{rk}(G) - 1)c_1(G)^2\right).
\]
\begin{Claim
\label{claim:dis:of:E}
${\displaystyle \delta(E) = (8g+4)c_1(\pi_*(\omega_{Z_g^0/H_g^0})) -
g \Delta_0^0 -(3g-2) \sum_{i=1}^{\left[\frac{g}{2}\right]} \Delta_i^0}$.
\end{Claim}
{\sl Proof.}\quad
First of all, let us recall the Grothendieck-Riemann-Roch theorem.
Let $f : X \to Y$ be a proper morphism of smooth varieties over $k$.
The Grothendieck-Riemann-Roch theorem says that,
for any coherent sheaf $\mathcal{F}$ on $X$,
\[
\operatorname{ch}\left(\sum_{i} (-1)^i R^if_*(\mathcal{F})\right) =
f_*(\operatorname{ch}(\mathcal{F})\operatorname{td}(T_{X/Y})).
\]
We denote $\operatorname{ch}(\sum_{i} R^if_*(\mathcal{F}))$
by $\chi(X/Y, \mathcal{F})$, and $A_i(X)$-component of
$f_*(\operatorname{ch}(\mathcal{F})\operatorname{td}(T_{X/Y}))$
by $f_*(\operatorname{ch}(\mathcal{F})\operatorname{td}(T_{X/Y}))_{(i)}$.
Note that if $\dim(X/Y) = 1$, then
\[
f_*(\operatorname{ch}(\mathcal{F})\operatorname{td}(T_{X/Y}))_{(\dim Y)}
= f_* \left(c_1(\mathcal{F}) -
\operatorname{rk}(\mathcal{F}) \frac{\omega_{X/Y}}{2} \right)
\]
and
\begin{multline*}
f_*(\operatorname{ch}(\mathcal{F})\operatorname{td}(T_{X/Y}))_{(\dim Y - 1)}
= \\
f_* \left(
\frac{c_1(\mathcal{F}) \cdot (c_1(\mathcal{F}) - \omega_{X/Y})}{2}
- c_2(\mathcal{F}) + \operatorname{rk}(\mathcal{F})
\frac{\omega_{X/Y}^2 + c_2(\Omega^1_{X/Y})}{12}
\right).
\end{multline*}
Let us go back to the proof of our claim.
For simplicity, we denote $c_1(\pi_*(\omega_{Z_g^0/H_g^0}))$
and $\omega_{Z_g^0/H_g^0}$
by $\lambda$ and $\omega$ respectively.
Consider an exact sequence
\[
0 \to E \to \pi^*(\pi_*(\omega)) \to
I_{S^0_{+}} \cdot \omega \to 0.
\]
First of all, we have $c_1(E) = \pi^*(\lambda) - \omega$.
Moreover,
\[
\chi(Z_g^0/H_g^0, \pi^*(\pi_*(\omega))) = \chi(Z_g^0/H_g^0, E) +
\chi(Z_g^0/H_g^0, I_{S^0_{+}} \cdot \omega).
\]
Thus, using the Grothendieck-Riemann-Roch theorem, we can see
\[
\pi_*(c_2(E)) =
\pi_* \left( \omega \cdot \omega - \pi^*(\lambda) \cdot \omega -
S^0_{+} + c_2(\pi^*(\pi_*(\omega))) \right).
\]
Noting that $\pi_*(\pi^*(\lambda) \cdot \omega) = (2g-2)\lambda$,
$\pi_*(c_2(\pi^*(\pi_*(\omega)))) = 0$ and $\pi_*(S_i^0) = \Delta_i^0$,
the above implies
\[
\pi_*(c_2(E)) = \pi_*(\omega \cdot \omega) - (2g-2)\lambda -
\sum_{i=1}^{\left[\frac{g}{2}\right]} \Delta_i^0.
\]
Therefore, by virtue of Noether formula:
$\pi_*(\omega \cdot \omega) = 12 \lambda - \Delta^0$,
we can conclude our claim.
\QED
\begin{Claim
\label{claim:dis:of:F}
${\displaystyle \delta(F) = (8g+4) c_1(\pi_*(\omega_{Z_g^0/H_g^0})) -
g \Delta_0^0 - \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \Delta_i^0}$.
\end{Claim}
{\sl Proof.}\quad
Consider an exact sequence:
\[
0 \to F \to E \to \bigoplus_{i=1}^{\left[\frac{g}{2}\right]}
\left( Q_i^1 \oplus Q_i^2 \right)\to 0.
\]
First of all, we have
\begin{align*}
c_1(F) & = c_1(E) -
\sum_{i=1}^{\left[\frac{g}{2}\right]} (\operatorname{rk}(Q_i^1) C_i^1 + \operatorname{rk}(Q_i^2)C_i^2)
\\
& = c_1(E) - \sum_{i=1}^{\left[\frac{g}{2}\right]} ((i-1) C_i^1 + (i'-1)C_i^2),
\end{align*}
where $i' = g-i$.
Moreover, the above exact sequence gives rise to
\[
\chi(Z_g^0/H_g^0, E) = \chi(Z_g^0/H_g^0, F) +
\sum_{i=1}^{\left[\frac{g}{2}\right]} (\chi(C_i^1/\Delta_i^0, Q_i^1) +
\chi(C_i^2/\Delta_i^0, Q_i^2)),
\]
which, by Grothendieck-Riemann-Roch theorem, implies
{\allowdisplaybreaks
\begin{multline*}
\pi_*(c_2(F)) = \pi_*\left(c_2(E) +
\frac{c_1(F) \cdot (c_1(F) - \omega)}{2} -
\frac{c_1(E) \cdot (c_1(E) - \omega)}{2}\right) + \\
\sum_{i=1}^{\left[\frac{g}{2}\right]} \left\{ \left(\rest{\pi}{C_i^1}\right)_*
\left( c_1(Q_i^1) - \frac{(i-1)\omega_{C_i^1/\Delta_i^0}}{2} \right)
+ \left(\rest{\pi}{C_i^2}\right)_*
\left( c_1(Q_i^2) - \frac{(i'-1)\omega_{C_i^2/\Delta_i^0}}{2} \right)
\right\}.
\end{multline*}}
Thus, we have
{\allowdisplaybreaks
\begin{multline*}
\delta(F) = \delta(E) + \sum_{i=1}^{\left[\frac{g}{2}\right]}
\pi_*\left(((i-1)C_i^1 + (i'-1)C_i^2)^2\right) \\
+ \sum_{i=1}^{\left[\frac{g}{2}\right]}
\pi_* \left(((g-1) \omega - 2c_1(E))\cdot ((i-1) C_i^1 + (i'-1)C_i^2)\right)
\\
+ 2(g-1) \sum_{i=1}^{\left[\frac{g}{2}\right]} \left(\rest{\pi}{C_i^1}\right)_*
\left( c_1(Q_i^1) - \frac{(i-1)\omega_{C_i^1/\Delta_i^0}}{2} \right) \\
+ 2(g-1) \sum_{i=1}^{\left[\frac{g}{2}\right]} \left(\rest{\pi}{C_i^2}\right)_*
\left( c_1(Q_i^2) - \frac{(i'-1)\omega_{C_i^2/\Delta_i^0}}{2} \right).
\end{multline*}}
Therefore, using formulae:
{\allowdisplaybreaks
\[
\begin{cases}
\left(\rest{\pi}{C_i^1}\right)_*(c_1(Q_i^1)) = -2(i-1)\Delta_i^0, &
\left(\rest{\pi}{C_i^2}\right)_*(c_1(Q_i^2)) = -2(i'-1)\Delta_i^0, \\
& \\
\left(\rest{\pi}{C_i^1}\right)_*(\omega_{C_i^1/\Delta_i^0}) =
2(i-1)\Delta_i^0, &
\left(\rest{\pi}{C_i^2}\right)_*(\omega_{C_i^2/\Delta_i^0}) =
2(i'-1)\Delta_i^0, \\
& \\
\pi_*(c_1(E) \cdot C_i^1) = -(2(i-1) + 1)\Delta_i^0, &
\pi_*(c_1(E) \cdot C_i^2) = -(2(i'-1) + 1)\Delta_i^0, \\
& \\
\pi_*(\omega \cdot C_i^1) = (2(i-1) + 1)\Delta_i^0, &
\pi_*(\omega \cdot C_i^2) = (2(i'-1) + 1)\Delta_i^0, \\
& \\
\pi_*(C_i^1 \cdot C_i^2) = \Delta_i^0, &
\pi_*(C_i^j \cdot C_i^j) = -\Delta_i^0 \quad\text{for $j= 1, 2$}, \\
\end{cases}
\]
}
we can see that
\[
\delta(F) = \delta(E) + \sum_{i=1}^{\left[\frac{g}{2}\right]}
(3g-2 - 4ii') \Delta_i^0.
\]
Hence, by Claim~\ref{claim:dis:of:E}, we get
\[
\delta(F) = (8g+4) c_1(\pi_*(\omega_{Z_g^0/H_g^0})) -
g \Delta_0^0 - \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \Delta_i^0.
\]
\textbf{Proof of Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve}:}
\quad
Now we obtain everything to prove
Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve}.
Let $f : X \to Y$ be a semistable curve as in
Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve}.
Then, there is a morphism $h : Y \to H_g^0$ with
$Z_g^0 \times_{H_g^0} Y \simeq \overline{X}$.
Let $h' : X \to Z_g^0$ be the induced morphism.
Let us consider a vector bundle ${h'}^*(F)$ on $Y$.
By \cite{PR}, ${h'}^*(F)$ is semistable on the generic fiber of $f$.
Thus, $\delta({h'}^*(F)) \geq 0$ by Theorem~\ref{thm:B:ineq:fiber:space}.
On the other hand, by Claim~\ref{claim:dis:of:F},
\begin{align*}
\delta({h'}^*(F)) & = \deg(h^*((8g+4) c_1(\pi_*(\omega_{Z_g^0/H_g^0})) -
g \Delta_0^0 - \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \Delta_i^0)) \\
& = (8g+4)\deg(f_*(\omega_{X/Y})) - g \delta_0 -
\sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i.
\end{align*}
Thus, we have
\[
(8g+4)\deg(f_*(\omega_{X/Y})) \geq g \delta_0 +
\sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i.
\]
\QED
\bigskip
Let $\iota : Z_g^0 \to Z_g$ be the inclusion map.
If we set $\overline{F} = \iota_*(F)$, then
$\overline{F}$ is a reflexive coherent sheaf on $Z_g$ because
$\operatorname{codim}(Z_g \setminus Z_g^0) = 2$.
Using $\overline{F}$, we can slightly generalize
Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve}.
\begin{Theorem}[$\operatorname{char}(k) = 0$
Let $X$ be a smooth projective surface over $k$,
$Y$ a smooth projective curve over $k$,
and $f : X \to Y$ a semistable curve
of genus $g \geq 2$ over $Y$.
Let $h : Y \to H_g$ and $h' : X \to Z_g$ be the induced morphisms
such that the following diagram is commutative:
\[
\begin{CD}
X @>{h'}>> Z_g \\
@V{f}VV @VV{\pi}V \\
Y @>>{h}> H_g
\end{CD}
\]
If $\overline{F}$ is locally free along $h'(X)$, then
\[
(8g+4)\deg(f_*(\omega_{X/Y})) \geq g \delta_0 +
\sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i.
\]
\end{Theorem}
{\sl Proof.}\quad
Since $\operatorname{codim}(H_g \setminus H_g^0) = 2$, we have
\[
\delta(\overline{F}) = (8g+4) c_1(\pi_*(\omega_{Z_g/H_g})) -
g \Delta_0 - \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \Delta_i.
\]
Thus, we can conclude our theorem in the same way as before.
\QED
\begin{Remark
\label{rem:C-H-X:ineq}
Let $X$ be a smooth projective surface over $k$, $Y$
a smooth projective curve over $k$, and $f : X \to Y$
a surjective morphism with $f_*({\mathcal{O}}_X) = {\mathcal{O}}_Y$.
Then, in the same idea of Claim~\ref{claim:dis:of:E} or
\cite{Mo1}, we can see that, if $E$ is the kernel of
$f^*f_*(\omega_{X/Y}) \to \omega_{X/Y}$ and
$\omega_{X/Y}$ is $f$-nef,
\[
\delta(E) \leq g(\omega_{X/Y} \cdot \omega_{X/Y})
- 4(g-1)\deg(f_*(\omega_{X/Y})),
\]
where $g \geq 2$ is the genus of the generic fiber.
Thus, by \cite{PR} and Theorem~\ref{thm:B:ineq:fiber:space},
we can recover Cornalba-Harris-Xiao inequality:
\[
g (\omega_{X/Y} \cdot \omega_{X/Y}) \geq
4 (g-1) \deg(f_*(\omega_{X/Y})).
\]
\end{Remark}
\section{Calculation of invariants arising from Green functions}
\label{sec:cal:green:function}
\setlength{\unitlength}{.5in}
In this section, we would like to calculate an invariant
$e(G,D)$ for a metrized graph $G$ with a polarization $D$.
For details of metrized graphs, see \cite{Zh}.
Let $G$ be a connected metrized graph and
$D$ an ${\mathbb{R}}$-divisor on $G$.
If $\deg(D) \not= -2$, then
there are a unique measure $\mu_{(G,D)}$ on $G$ and
a unique function $g_{(G,D)}$ on $G \times G$
with the following properties.
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
${\displaystyle \int_{G} \mu_{(G,D)} = 1}$.
\item
$g_{(G,D)}(x, y)$ is symmetric and continuous on $G \times G$.
\item
For a fixed $x \in G$, $\Delta_y(g_{(G,D)}(x, y)) = \delta_x - \mu_{(G,D)}$.
\item
For a fixed $x \in G$, ${\displaystyle
\int_G g_{(G,D)}(x, y) \mu_{(G,D)}(y) = 0}$.
\item
$g_{(G,D)}(D, y) + g_{(G,D)}(y, y)$ is a constant for all $y \in G$.
\end{enumerate}
The constant $g_{(G,D)}(D, y) + g_{(G,D)}(y, y)$ is denoted by $c(G, D)$.
Further we set
\[
e(G, D) = 2\deg(D)c(G, D) - g_{(G,D)}(D, D).
\]
First of all, let's consider another expression of $e(G,D)$.
\begin{Lemma
\label{lem:formula:for:e}
Let $G$ be a connected metrized graph and $D$
an ${\mathbb{R}}$-divisor on $G$ with $\deg(D) \not= -2$.
Then, for any point $O$ of $G$,
\[
e(G,D) = (\deg(D) + 2) g_{(G,D)}(O, D) + r_G(O, D),
\]
where $r_G(P,Q)$ is the resistance between $P$ and $Q$ on $G$.
\end{Lemma}
{\sl Proof.}\quad
We set $D = \sum_i a_i P_i$. Then,
{\allowdisplaybreaks
\begin{align*}
e(G, D) & = 2 \deg(D) c(G,D) - \sum_i a_i g_{(G,D)}(D, P_i) \\
& = 2\deg(D) c(G,D) - \sum_i a_i \left( c(G,D) - g_{(G,D)}(P_i,P_i) \right) \\
& = \deg(D) c(G,D) + \sum_i a_i g_{(G,D)}(P_i,P_i)
\end{align*}}
Since we know
\[
r_G(P,Q) = g_{(G,D)}(P,P) - 2g_{(G,D)}(P,Q) + g_{(G,D)}(Q,Q)
\]
for all points $P,Q \in G$,
the above implies
{\allowdisplaybreaks
\begin{align*}
e(G,D) & = \deg(D)\left( g_{(G,D)}(O,O) + g_{(G,D)}(O,D) \right) \\
& \qquad + \sum_i a_i \left(
r_G(O,P_i) + 2g_{(G,D)}(O,P_i) - g_{(G,D)}(O,O) \right) \\
& = (\deg(D) + 2)g_{(G,D)}(O,D) + r_G(O,D).
\end{align*}}
\QED
\bigskip
Let $G_1$ and $G_2$ be metrized graphs.
Fix points $x_1 \in G_1$ and $x_2 \in G_2$.
The one point sum $G_1 \vee G_2$ with respect to $x_1$ and $x_2$,
defined by $G_1 \times \{x_2\} \cup \{x_1\} \times G_2$ in $G_1 \times G_2$,
is a metrized graph obtained by joining $x_1 \in G_1$ and $x_2 \in G_2$.
The joining point, which is $\{x_1\}\times\{x_2\}$ in $G_1 \times G_2$,
is denoted by $j(G_1 \vee G_2)$.
Any ${\mathbb{R}}$-divisor on $G_i$ ($i=1,2$) can be viewed as an ${\mathbb{R}}$-divisor on
$G_1 \vee G_2$.
\begin{Proposition
\label{prop:e:for:join:graph}
Let $G_1$ and $G_2$ be connected metrized graphs,
and $D_1$ and $D_2$ ${\mathbb{R}}$-divisors on $G_1$ and $G_2$ respectively
with $\deg(D_i) \not= -2$ \textup{(}$i=1,2$\textup{)}.
Let $G = G_1 \vee G_2$, $O = j(G_1 \vee G_2)$, and
$D = D_1 + D_2$ on $G_1 \vee G_2$. If $\deg(D_1 + D_2) \not= -2$, then
{\allowdisplaybreaks
\begin{multline*}
e(G, D) = e(G_1, D_1) + e(G_2, D_2) \\
+ \frac{2 \deg(D_2)(\deg(D_1) + 2)g_{(G_1, D_1)}(O,O) +
2 \deg(D_1)(\deg(D_2) + 2)g_{(G_2, D_2)}(O, O)}
{\deg(D_1) + \deg(D_2) + 2}.
\end{multline*}}
Moreover, if $P \in G_2$, then
{\allowdisplaybreaks
\begin{align*}
g_{(G, D)}(P,P) & =
\frac{\deg(D_1)}{\deg(D_1)+\deg(D_2)+2} r_{G_2}(O,P) \\
& \quad +\frac{\deg(D_2) + 2}
{\deg(D_1)+\deg(D_2)+2} g_{(G_2, D_2)}(P,P) \\
& \quad -\frac{\deg(D_1)(\deg(D_2)+2)}
{(\deg(D_1)+\deg(D_2)+2)^2}g_{(G_2,D_2)}(O,O) \\
& \quad +\frac{(\deg(D_1) +2)^2}
{(\deg(D_1) + \deg(D_2) + 2)^2}g_{(G_1, D_1)}(O,O).
\end{align*}}
\end{Proposition}
{\sl Proof.}\quad
For simplicity, we set $d_i = \deg(D_i)$ and $g_i = g_{(G_i,D_i)}(O,O)$
for $i=1,2$.
By \cite[Lemma~3.7]{Zh}, we have
\[
\mu_{(G,D)} = \frac{d_1+2}{d_1+d_2+2}\mu_{(G_1,D_1)} +
\frac{d_2+2}{d_1+d_2+2}\mu_{(G_2,D_2)} -
\frac{2}{d_1+d_2+2} \delta_O.
\]
Consider the following function on $G$:
\[
g(x) = \begin{cases}
{\displaystyle \frac{d_1+2}{d_1+d_2+2}g_{(G_1,D_1)}(O,x) +
\frac{(d_2+2)^2g_2 - d_2(d_1+2)g_1}{(d_1+d_2+2)^2}}
& \text{if $x \in G_1$}, \\
& \\
{\displaystyle \frac{d_2+2}{d_1+d_2+2}g_{(G_2,D_2)}(O,x) +
\frac{(d_1+2)^2g_1 - d_1(d_2+2)g_2}{(d_1+d_2+2)^2}}
& \text{if $x \in G_2$}. \\
\end{cases}
\]
Then, we can easily check that $g$ is continuous on $G$,
$\Delta(g) = \delta_O - \mu_{(G,D)}$ and $\int_G g \mu_{(G,D)} = 0$.
Thus, $g_{(G,D)}(O,x) = g(x)$. Therefore, by Lemma~\ref{lem:formula:for:e},
we get the first formula. Moreover, using
\[
g_{(G,D)}(P,P) -2g_{(G,D)}(O,P) +
g_{(G,D)}(O,O) = r_G(O,P) = r_{G_2}(O,P)
\]
and
\[
g_{(G_2,D_2)}(P,P) -2g_{(G_2,D_2)}(O,P) +
g_{(G_2,D_2)}(O,O) = r_{G_2}(O,P),
\]
we obtain the second formula.
\QED
\begin{Corollary
\label{cor:e:for:join:graph:circle}
Let $G$ be a connected metrized graph and $D$ an ${\mathbb{R}}$-divisor on $G$ with
$\deg(D) \not= -2$. Let $C$ be a circle of length $l$.
Then,
\[
e(G \vee C, D) = e(G, D) + \frac{\deg(D)}{3(\deg D + 2)} l.
\]
\end{Corollary}
{\sl Proof.}\quad
Let $O = j(G \vee C)$ and $t : C \to [0, l)$
a coordinate of $C$ with $t(O) = 0$.
Then, it is easy to see that
\[
\mu_{(C,0)} = \frac{dt}{l}\quad\text{and}\quad
g_{(C,0)}(O,x) = \frac{t(x)^2}{2l} - \frac{t(x)}{2} + \frac{l}{12}.
\]
Thus, we have this formula by Proposition~\ref{prop:e:for:join:graph}.
\QED
\bigskip
Next, let's consider $e(G,D)$ for a segment.
\begin{Lemma
\label{lem:e:for:1:segment}
Let $G$ be a segment of length $l$, and $P$ and $Q$
terminal points of $G$. Let $a$ and $b$ be real numbers
with $a + b \not= 0$, and $D$ an ${\mathbb{R}}$-divisor on $G$ given by
$D = (2a-1)P + (2b-1)Q$. Then,
\[
e(G, D) = \left(\frac{4ab}{a+b} - 1\right)l,\quad
g_{(G,D)}(P,P) = \frac{b^2}{(a+b)^2}l \quad\text{and}\quad
g_{(G,D)}(Q,Q) = \frac{a^2}{(a+b)^2}l.
\]
\end{Lemma}
{\sl Proof.}\quad
First of all, by \cite[Lemma~3.7]{Zh},
\[
\mu_{(G,D)} = \frac{1}{a+b}(a \delta_P + b \delta_Q).
\]
Let $t : G \to [0, l]$ be a coordinate of $G$.
We set
\[
f(x) = -\frac{b}{a+b}t(x) + \frac{b^2}{(a+b)^2}l.
\]
Then, $\Delta(f) = \delta_P - \mu$ and $\int_G f\mu = 0$.
Thus, $f(x) = g_{(G,D)}(x, P)$. Therefore,
\[
g_{(G,D)}(P, P) = \frac{b^2}{(a+b)^2} l
\qquad\text{and}\qquad
g_{(G,D)}(P, Q) = g_{(G,D)}(Q, P) = -\frac{ab}{(a+b)^2} l.
\]
In the same way, we can see that
\[
g_{(G,D)}(Q, Q) = \frac{a^2}{(a+b)^2} l.
\]
Therefore, we have
\[
e(G, D) = \left(\frac{4ab}{a+b} - 1\right)l.
\]
\QED
\bigskip
This lemma can be generalized as follows.
\begin{Proposition
\label{prop:e:for:n:segment}
Let $G_n$ be a metrized graph given by the following figure.
\begin{center}
\begin{picture}(9,2)
\put(1,1){\circle*{.25}}
\put(2,1){\circle*{.25}}
\put(3,1){\circle*{.25}}
\put(6,1){\circle*{.25}}
\put(7,1){\circle*{.25}}
\put(8,1){\circle*{.25}}
\thicklines
\dottedline{0.1}(3.8,1)(5.2,1)
\put(1,1){\line(1,0){1}}
\put(2,1){\line(1,0){1}}
\put(3,1){\line(1,0){0.8}}
\put(6,1){\line(-1,0){0.8}}
\put(6,1){\line(1,0){1}}
\put(7,1){\line(1,0){1}}
\put(0.8,1.3){$P_0$}
\put(1.8,1.3){$P_1$}
\put(2.8,1.3){$P_2$}
\put(5.8,1.3){$P_{n-2}$}
\put(6.8,1.3){$P_{n-1}$}
\put(7.8,1.3){$P_n$}
\put(1.5,0.65){$l_1$}
\put(2.5,0.65){$l_2$}
\put(6.3,0.65){$l_{n-1}$}
\put(7.5,0.65){$l_n$}
\end{picture}
\end{center}
Let $l_i$ be the length between $P_{i-1}$ and $P_i$.
Let
\[
D_n = (2a_0-1)P_0 + (2a_n -1)P_n + \sum_{i=1}^{n-1}2a_iP_i
\]
be an ${\mathbb{R}}$-divisor on $G$ with $a_i > 0$ for all $i$. Then, we have
\[
e(G_n,D_n) = \sum_{i=1}^n \left(
\frac{4(a_0 + \cdots + a_{i-1})(a_{i} + \cdots + a_n)}
{a_0 + \cdots + a_n} - 1 \right)l_i.
\]
\end{Proposition}
{\sl Proof.}\quad
We set $e_n = e(G_n, D_n)$ and $t_n = g_{(G_n,D_n)}(P_n,P_n)$.
We would like to prove
\[
e_n = \sum_{i=1}^n \left(
\frac{4(a_0 + \cdots + a_{i-1})(a_{i} + \cdots + a_n)}
{a_0 + \cdots + a_n} - 1 \right)l_i
\]
and
\[
t_n = \frac{{\displaystyle
\sum_{i=1}^n \left( a_0 + \cdots + a_{i-1} \right)^2 l_i}}
{(a_0 + \cdots + a_n)^2}.
\]
For this purpose, it is sufficient to show that
\[
t_{n+1} = \frac{(a_0 + \cdots + a_n)^2}{(a_0 + \cdots + a_n + a_{n+1})^2}
(t_n + l_{n+1})
\]
and
\[
e_{n+1} = e_n +
\frac{4a_{n+1}(a_0 + \cdots + a_n)}{a_0 + \cdots + a_n + a_{n+1}}t_n +
\left(
\frac{4 a_{n+1}(a_0 + \cdots + a_n)}{a_0 + \cdots + a_n + a_{n+1}} - 1
\right) l_{n+1}.
\]
Let $L$ be a segment of length $l_{n+1}$, and $Q$ and $P$
terminal points of $L$. Let $E$ be an ${\mathbb{R}}$-divisor on $L$ given by
$E = Q + (2a_{n+1}-1)P$.
Let's consider a one point sum $G_n \vee L$ obtained
by joining $P_n$ and $Q$. Then, $G_{n+1} = G_n \vee L$ and
$D_{n+1} = D_n + E$.
Thus, by Proposition~\ref{prop:e:for:join:graph} and
Lemma~\ref{lem:e:for:1:segment},
we have the above recursive equations.
\QED
\begin{Corollary}[$\operatorname{char}(k) \geq 0$
\label{cor:e:for:semistable:chain}
Let $X$ be a smooth projective surface over $k$,
$Y$ a smooth projective curve over $k$, and
$f : X \to Y$ a generically smooth semistable curve over $Y$
of genus $g \geq 2$.
Let $X_y$ be a singular fiber of $f$ over $y \in Y$.
Let $X_y = C_1 + \cdots + C_n$ be the irreducible decomposition of $X_y$.
Let $G_y$ be the metrized graph given by the configuration of $X_y$,
$v_i$ the vertex of $G_y$ corresponding to $C_i$, and
$\omega_y$ the divisor on $G_y$ defined by
$\omega_y = \sum_i (\omega_{X/Y} \cdot C_i) v_i$.
If $X_y$ is a chain of stable components, then
\[
e_y = e(G_y, \omega_y) = \frac{g-1}{3g} \delta_{0,y} +
\sum_{i=1}^{\left[\frac{g}{2}\right]}
\left( \frac{4 i (g-i)}{g} - 1 \right) \delta_{i,y},
\]
where $\delta_{i, y}$ is the number of nodes
of type $i$ in $X_y$.
In particular, if every singular fiber of $f$ is
a chain of stable components, then
\[
\left(\omega_{X/Y} \cdot \omega_{X/Y} \right) \geq
\frac{g-1}{3g} \delta_{0} +
\sum_{i=1}^{\left[\frac{g}{2}\right]}
\left( \frac{4 i (g-i)}{g} - 1 \right) \delta_{i}.
\]
Moreover, if the above inequality is strict,
then Bogomolov conjecture holds for the generic fiber of $f$.
\end{Corollary}
{\sl Proof.}\quad
Under our assumption, $G_y$ can be obtained by
performing one sum of one segment and
many circles. Thus, using Corollary~\ref{cor:e:for:join:graph:circle}
and Proposition~\ref{prop:e:for:n:segment},
we can see the formula for $e_y$.
For the last inequality, note that
$\left(\omega^a_{X/Y} \cdot \omega^a_{X/Y}\right)_a =
\left(\omega_{X/Y} \cdot \omega_{X/Y} \right) - \sum_{y} e_y$
and $\left(\omega^a_{X/Y} \cdot \omega^a_{X/Y}\right)_a \geq 0$
(cf. \cite{Zh} or \cite{Mo3}).
Furthermore,
if $\left(\omega^a_{X/Y} \cdot \omega^a_{X/Y}\right)_a > 0$,
then Bogomolov conjecture holds for the generic fiber of $f$. (cf. \cite{Zh})
\QED
\section{Bogomolov conjecture over function fields}
Let $X$ be a smooth projective surface over $k$,
$Y$ a smooth projective curve over $k$,
and $f : X \to Y$ a generically smooth semistable curve
of genus $g \geq 2$ over $Y$.
Let $K$ be the function field of $Y$, $\overline{K}$
the algebraic closure of $K$, and $C$ the generic fiber of $f$.
Let $j : C(\overline{K}) \to \operatorname{Jac}(C)(\overline{K})$ be a morphism
given by $j(x) = (2g-2)x - \omega_C$ and $\Vert\ \Vert_{NT}$
the semi-norm arising from the Neron-Tate height pairing on
$\operatorname{Jac}(C)(\overline{K})$.
We set
\[
B_C(P;r) = \left\{ x \in C(\overline{K}) \mid
\Vert j(x) - P \Vert_{NT} \leq r \right\}
\]
for $P \in \operatorname{Jac}(C)(\overline{K})$ and $r \geq 0$, and
\[
r_C(P) =
\begin{cases}
-\infty & \mbox{if $\#\left(B_C(P;0)\right) = \infty$}, \\
& \\
\sup \left\{ r \geq 0 \mid \#\left(B_C(P;r)\right) < \infty \right\} &
\mbox{otherwise}.
\end{cases}
\]
An effective version of Bogomolov conjecture claims the following.
\begin{Conjecture}[Effective Bogomolov conjecture
\label{conj:effective:bogomolov}
If $f$ is non-isotrivial, then
there is an effectively calculated positive number
$r_0$ with
\[
\inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq r_0.
\]
\end{Conjecture}
In \cite{Mo3} and \cite{Mo4}, we proved the following results.
\begin{enumerate}
\renewcommand{\labelenumi}{(\Alph{enumi})}
\item ($\operatorname{char}(k) \geq 0$)
If $f$ is non-isotrivial and the stable model of $f : X \to Y$
has only irreducible fibers, then
\[
\inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq
\begin{cases}
\sqrt{12(g-1)} & \mbox{if $f$ is smooth}, \\
& \\
{\displaystyle \sqrt{\frac{(g-1)^3}{3g(2g+1)}\delta_0}} & \mbox{otherwise}.
\end{cases}
\]
\item ($\operatorname{char}(k) \geq 0$)
If $f$ is non-isotrivial and $g = 2$, then
$f$ is not smooth and
\[
\inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq
\sqrt{\frac{2}{135} \delta_0 + \frac{2}{5} \delta_1}.
\]
(According to the exact calculations in \cite{Mo4},
$(\omega_{X/Y} \cdot \omega_{X/Y}) =
\frac{1}{5}\delta_0 + \frac{7}{5} \delta_1$ and
$\sum_y e_y \leq \frac{5}{27}\delta_0 + \delta_1$.)
\end{enumerate}
In this section, we would like to prove the following answer
as an application of our slope inequality.
\begin{Theorem}[$\operatorname{char}(k) = 0$
\label{thm:bogomolov:function:fiels}
We assume that $f$ is not smooth,
every singular fiber of $f$
is a chain of stable components, and one of the following conditions:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
the generic fiber of $f$ is hyperelliptic, or
\item
every singular fiber of the stable model of $f : X \to Y$
has at most one node of type $i>0$.
\end{enumerate}
Then we have
\[
\inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq
\sqrt{\frac{(g-1)^2}{g(2g+1)}\left(
\frac{g-1}{3}\delta_0 + \sum_{i=1}^{\left[\frac{g}{2}\right]}
4i(g-i)\delta_i \right)}.
\]
\end{Theorem}
{\sl Proof.}\quad
First of all, note the following fact
(cf. \cite[Theorem 5.6]{Zh},
\cite[orollary 2.3]{Mo3} or \cite[Theorem 2.1]{Mo4}).
If $(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a > 0$, then
\[
\inf_{P \in \operatorname{Jac}(C)(\overline{K})} r_C(P) \geq
\sqrt{(g-1)(\omega_{X/Y}^a \cdot \omega_{X/Y}^a)_a},
\]
where $(\ \cdot \ )_a$ is the admissible pairing.
By the definition of admissible pairing,
we can set
\[
\left(\omega_{X/Y}^a \cdot \omega_{X/Y}^a \right)_a =
\left(\omega_{X/Y} \cdot \omega_{X/Y} \right) - \sum_{y \in Y} e_y,
\]
where $e_y$ is $e(G_y, \omega_y)$
treated in \S\ref{sec:cal:green:function}.
Under our assumption, by \cite[Proposition~4.7]{CH} and
Theorem~\ref{thm:sharp:slope:ineq:general:stable:curve},
we have
\[
(8g + 4) \deg(f_*(\omega_{X/Y})) \geq
g \delta_0 + \sum_{i=1}^{\left[\frac{g}{2}\right]} 4i(g-i) \delta_i.
\]
Thus, using Noether formula, the above inequality implies
\[
(\omega_{X/Y} \cdot \omega_{X/Y}) \geq
\frac{g-1}{2g+1} \delta_0 + \sum_{i=1}^{\left[\frac{g}{2}\right]}
\left(\frac{12i(g-i)}{2g+1} - 1 \right) \delta_i.
\]
Moreover,
by Corollary~\ref{cor:e:for:semistable:chain}, we get
\[
\sum_{y} e_y = \frac{g-1}{3g} \delta_{0} +
\sum_{i=1}^{\left[\frac{g}{2}\right]}
\left( \frac{4i(g-i)}{g} - 1 \right) \delta_{i}.
\]
Thus, we have our theorem.
\QED
\bigskip
|
1996-05-16T02:27:14 | 9601 | alg-geom/9601006 | en | https://arxiv.org/abs/alg-geom/9601006 | [
"alg-geom",
"math.AG"
] | alg-geom/9601006 | Frank Sottile | Frank Sottile | Pieri's Formula Via Explicit Rational Equivalence | Revised version; 18 pages with one figure LaTeX 2e | Canad. J. Math., Vol. 49 (6), 1997 pp. 1281-1298 | null | null | null | We present a new geometric proof of Pieri's formula, exhibiting an explicit
chain of rational equivalences from a suitable sum of distinct Schubert
varieties to the intersection of a Schubert variety with a special Schubert
variety. The geometry of these rational equivalences indicate a link to a
combinatorial proof of Pieri's formula using Schensted insertion. This version
includes an example to illustrate the proof.
| [
{
"version": "v1",
"created": "Mon, 8 Jan 1996 00:11:09 GMT"
},
{
"version": "v2",
"created": "Wed, 15 May 1996 23:16:30 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sottile",
"Frank",
""
]
] | alg-geom | \section{Introduction}
Pieri's formula asserts that the product of a Schubert class and a special
Schubert class is a sum of certain other Schubert classes, each with
coefficient 1.
This determines the multiplicative structure of the Chow ring of a
Grassmann variety.
Pieri's formula also arises in algebraic combinatorics and
representation theory~\cite{Fulton_tableaux}, and has independent proofs in
each context.
One such proof~\cite{Fulton_tableaux} in combinatorics involves Schensted
insertion~\cite{Schensted}.
Its geometric proof (for example, in Hodge and
Pedoe~\cite{Hodge_Pedoe})
involves studying an intersection of three Schubert varieties and
invoking Poincar{\'e} duality to obtain the desired sum.
We present a new geometric proof of this formula,
explicitly describing a sequence of deformations (inducing
rational equivalence) that transform a general intersection of a
Schubert variety with a special Schubert variety into a sum of
distinct Schubert varieties.
The geometry of these deformations is quite interesting and their
form parallels the combinatorial proof of Pieri's formula using
Schensted insertion.
Let ${\bf G}_mV$ be the Grassmannian of
$m$-dimensional subspaces of an $n$-dimen\-sional vector
space $V$.
A decreasing sequence $\alpha$ of length $m$,
($n\geq \alpha_1>\cdots>\alpha_m\geq 1$),
and a complete flag ${F\!_{\DOT}}$ in $V$ together determine a
Schubert subvariety $\Omega_\alpha{F\!_{\DOT}}$ of ${\bf G}_mV$.
Special Schubert varieties $\Omega_L$ are those Schubert
varieties given by the single condition that an $m$-plane
intersect a given linear subspace $L$ non-trivially.
For any subscheme $X$ of ${\bf G}_mV$, let $[X]$ be the cycle
class of $X$ in the Chow ring of ${\bf G}_mV$.
Pieri's formula asserts
\begin{equation}\label{eq:pieri}
[\Omega_\alpha{F\!_{\DOT}}]\cdot[\Omega_L]\ =\
\sum [\Omega_\gamma{F\!_{\DOT}}],
\end{equation}
the sum over all sequences $\gamma$ with
$\gamma_1\geq \alpha_1>\gamma_2\geq
\cdots>\gamma_m\geq \alpha_m$ where
$\sum \gamma_i-\alpha_i$ is equal to the codimension $b$ of $\Omega_L$.
Let $\alpha *b$ denote this set of sequences
and let $Y_{\alpha,b}$ be the cycle
$\sum_{\gamma \in \alpha*b}\Omega_\gamma{F\!_{\DOT}}$.
Let $\mbox{\it Chow}\, {\bf G}_mV$ be the Chow variety of ${\bf G}_mV$
and let ${\cal G}\subset \mbox{\it Chow}\, {\bf G}_mV$
be the set of
cycles $\Omega_\alpha{F\!_{\DOT}} \bigcap \Omega_L$ for
all $L$ of a fixed dimension such that the intersection is
generically transverse.
Our proof involves a partial compactification of ${\cal G}$ in
$\mbox{\it Chow}\,{\bf G}_mV$ with $b+1$ rational strata, each an orbit of
the Borel subgroup of $GL(V)$ stabilizing ${F\!_{\DOT}}$, hence consisting
of isomorphic cycles.
The 0th stratum is dense in ${\cal G}$ and cycles in the $i$th stratum have
components
$X_\beta$ indexed by $\beta \in \alpha* i$, where the component
$X_\beta$ is a subvariety of $\Omega_\beta{F\!_{\DOT}}$.
Passing from one stratum to the next, each component $X_\beta$
deforms into some components of cycles in the next stratum.
The `history' of each component $\Omega_\gamma{F\!_{\DOT}}$ of
$Y_{\alpha,b}$ through this process gives a chain in the Bruhat
order of Schubert varieties, recording which component at each
stage gave rise to $\Omega_\gamma{F\!_{\DOT}}$.
This leads to the following interpretation of Pieri's formula:
The sum in (\ref{eq:pieri}) is over a certain set of chains in
the Bruhat order which begin at $\alpha$, the chain with endpoint
$\gamma$ recording the history of the cycle $\Omega_\gamma{F\!_{\DOT}}$
in the sequence of deformations.
In \S\ref{sec:schensted}, we show how this is similar to
a combinatorial proof of Pieri's formula based on Schensted
insertion.
In~\cite{sottile_real_lines}, these deformations were constructed in the
special case of ${\bf G}_2V$ and applied to obtain a completely geometric
understanding of intersections of Schubert subvarieties of
${\bf G}_2V$ in terms of explicit, multiplicity-free deformations.
This paper began as an effort to find similar constructions
for other Grassmannians, whose geometry is considerably more complicated
than that of ${\bf G}_2V$.
This proof of Pieri's formula is an initial step towards
understanding the structure of rational equivalence on these
Grassmann varieties in terms of the combinatorics of the Bruhat
order of the Schubert cellular decomposition.
A chain in the Bruhat order is a standard skew
tableau~\cite{Fulton_tableaux}.
Thus the Littlewood-Richardson rule for multiplying two Schubert
classes has an interpretation as a sum over certain chains
in the Bruhat order.
A (as yet unknown) geometric proof of the Littlewood-Richardson
rule for Grassmannians should provide an explanation for this,
similar to what we give for Pieri's formula.
In fact, we believe that all Schubert-type product formulas for
any Grassmannian or flag variety $X$ of any reductive group will
eventually be understood in terms of related combinatorics on
the Bruhat order on Schubert subvarieties on $X$,
perhaps with additional data giving rise to multiplicities.
Some of this picture is already known:
Both Chevalley's~\cite{Chevalley91} formula for multiplication
by hypersurface
Schubert classes and the Pieri-type formulas of
Boe-Hiller~\cite{Hiller_Boe} and
Pragacz-Ratajski~\cite{Pragacz_Ratajski_Pieri_Odd_I} for
Lagrangian and orthogonal Grassmannians are
similar to teh form of Pieri's formula (\ref{eq:pieri}), but each has
multiplicities depending upon root system data.
Formulas for multiplying arbitrary Schubert classes in maximal
Lagrangian
or orthogonal Grassmannians (\cite{Pragacz_S-Q},\cite{Stembridge_shifted})
are similar to the
Littlewood-Richardson formula, using combinatorics of the
lattice of shifted Young diagrams, the Bruhat order of these
varieties.
Known formulas for products in the ordinary flag manifold may also
interpreted in terms of chains in the Bruhat order
(\cite{sottile_pieri_schubert},\cite{bergeron_sottile_symmetry}).
It is only in characteristic zero that general subvarieties of a
Grassmannian intersect generically transversally.
Kleiman~\cite{Kleiman} proves this in characteristic zero and gives a
counterexample in positive characteristic.
In \S\ref{sec:geometry_of_intersections}, we work over an
arbitrary field and give a precise
determination (Theorem~\ref{thm:geometric_intersection}) of when
a special Schubert variety meets a fixed Schubert variety
generically transversally, and describe the components of such an
intersection.
The geometry of these components is interesting:
while not an intersection of Schubert varieties, each component is
`birationally fibred' over such an intersection, with Schubert
variety fibres.
Such cycles are the key to our proof of Pieri's formula
in \S\ref{sec:explicit_rational_equivalences};
they are the components of the intermediate cycles in the
deformations used to establish Pieri's formula.
\section{Geometry of Pieri-type
intersections}\label{sec:geometry_of_intersections}
\subsection{Preliminaries}
Let $k$ be a fixed, but arbitrary, field and $m\leq n$ positive
integers.
Let $V\simeq k^n$ be an $n$-dimensional vector space over $k$
and ${\bf G}_mV$ be the Grassmannian of $m$-planes in $V$.
A {\em complete flag} ${F\!_{\DOT}}$ in $V$ is a sequence of subspaces
$$
0=F_{n+1} \subset F_n\subset \cdots\subset F_2\subset F_1 = V
$$
of $V$ where $\dim F_j = n+1-j$.
Let $\Span{S}$ denote the linear span of a subset $S$ of $V$.
We let ${[n]\choose m}$ be the set of all $m$-element subsets of
$[n]:=\{1,2,\ldots,n\}$, considered as decreasing sequences
$\alpha$ of length $m$:
$n\geq\alpha_1>\alpha_2>\cdots>\alpha_m\geq 1$.
A complete flag ${F\!_{\DOT}}$ and a sequence $\alpha\in{[n]\choose m}$
together determine a Schubert (sub)variety of ${\bf G}_mV$,
$$
\Omega_\alpha{F\!_{\DOT}} \ :=\ \{H\in{\bf G}_mV\,|\,
\dim H\cap F_{\alpha_j}\geq j, \ 1\leq j\leq m\}.
$$
This variety has codimension $|\alpha|:= \sum \alpha_i-i$.
A {\em special Schubert variety} is the subvariety of all
$m$-planes $H$ which have a nontrivial intersection with
a single subspace $F_{m+s}$ in the flag, $\Omega_{m+s,m-1,\ldots,2,1}{F\!_{\DOT}}$.
We use a compact notation for special Schubert varieties.
Let $L:= F_{m+s}$, a subspace of dimension $n+1-m-s$, and
define
$$
\Omega_L \ :=\ \Omega_{m+s,m-1,\ldots,2,1}{F\!_{\DOT}}.
$$
Two subvarieties meet {\em generically transversally} if they
intersect trans\-versally along a dense subset of every
component of their intersection.
They meet {\em improperly} if the codimension of their
(non-empty) intersection is less than the sum of their
codimensions.
A subspace $L$ meets a flag ${F\!_{\DOT}}$ {\em properly} if it meets
each subspace $F_i$ properly.
To simplify some assertions and formulae, we adopt the convention
that if $\gamma$ is a decreasing sequence of length $m$ with
$\gamma_1>n$, then $\Omega_\gamma{F\!_{\DOT}} = \emptyset$.
Similarly, if the dimension of a subspace is asserted to be
negative, we intend that subspace to be $\{0\}$.
Also, $\dim \{0\} = -\infty$.
Let $\alpha\in{[n]\choose m}$ and $r$ be a positive integer.
Define $\alpha*r\subset{[n]\choose m}$ to be the set of those
$\beta\in{[n]\choose m}$ with
$\beta_1\geq\alpha_1>\beta_2\geq\cdots>\beta_m\geq\alpha_m$
and $|\beta|= |\alpha|+r$.
If $\beta\in \alpha*r$, define
$j(\alpha,\beta)$ to be the first index $i$ where $\beta_i$
differs from $\alpha_i$, $\min\{i\,|\,\beta_i>\alpha_i\}$.
For $1\leq j\leq m$, let $\delta^j$ be the Kroenecker delta, the
sequence with a 1 in the $j$th position and $0$'s elsewhere.
\subsection{The cycle $X_\beta(j,F\!\!_{\mbox{\bf .}},L)$}
\label{sec:intermediate_cycle}
Central to the geometry of Pieri-type
intersections are the components, $X_\beta(j,{F\!_{\DOT}},L)$, of
reducible intersections.
These subvarieties are also components of cycles
intermediate in deformations establishing Pieri's formula.
Let $\beta\in {[n]\choose m}$, $1\leq j\leq m$ be an integer, ${F\!_{\DOT}}$ a flag,
and $L$ a linear subspace in $V$.
Define
$$
X_\beta(j,{F\!_{\DOT}},L) \ :=\ \{H\in \Omega_\beta{F\!_{\DOT}}\,|\,
\dim H\cap F_{\beta_j}\cap L \geq 1\},
$$
a subvariety of $\Omega_\beta{F\!_{\DOT}}\bigcap \Omega_L$.
The following theorem gives precise conditions on $L$ and
${F\!_{\DOT}}$ which determine whether
$\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$ is improper, generically
transverse, or irreducible.
Moreover, it computes the components of the intersection in the
crucial case of a generically transverse intersection
with the maximal number of irreducible components.
\subsection{Theorem}\label{thm:geometric_intersection}
{\em
Let $\alpha\in {[n]\choose m}$, $s>0$, ${F\!_{\DOT}}$ be a complete
flag in $V$, and $L\in {\bf G}_{n+1-m-s}V$.
\begin{enumerate}
\item[(1)] Suppose $\dim F_{\alpha_j}\cap L > n+2-\alpha_j-j-s$
and $F_{\alpha_j}\cap L \neq \{0\}$, for some $1\leq j\leq m$.
Then
$\Omega_\alpha{F\!_{\DOT}} \bigcap \Omega_L$ is improper.
Otherwise, it is generically transverse.
\item[(2)] Suppose $\dim F_{\alpha_j}\cap L = n+2-\alpha_j-j-s$ for
each $1\leq j\leq m$. Let ${M_{\,\DOT}}$ be any flag satisfying
$M_{\alpha_j} = F_{\alpha_j}$ and
$M_{\alpha_j+1} \supset \Span{F_{\alpha_{j-1}}, F_{\alpha_j}\cap L}$,
for $1\leq j\leq m$.
Then $\Omega_\alpha {F\!_{\DOT}}$ meets $\Omega_L$ generically transversally, and
$$
\Omega_\alpha{F\!_{\DOT}} \bigcap \Omega_L\ =\
\sum_{\beta\in \alpha*1} X_\beta(j(\alpha,\beta),{M_{\,\DOT}},L).
$$
\item [(3)] Suppose $\dim F_{\alpha_j}\cap L< n+2-\alpha_j-j-s$
for each $1\leq j<m$ and
$F_{\alpha_m}$ meets $L$ properly, so that
$\dim F_{\alpha_m}\cap L = n+2-\alpha_m-m-s$.
Then $\Omega_\alpha{F\!_{\DOT}} \bigcap \Omega_L$ is irreducible.
\end{enumerate}
}
\smallskip
Note that $n+2-\alpha_j-j-s$, the critical dimension for
$F_{\alpha_j}\cap L$ in this theorem, exceeds the expected dimension of
$n+2-\alpha_j-m-s$ by $m-j$.
Thus, it is not necessary for ${F\!_{\DOT}}$ and $L$ to meet properly for
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$ to be generically transverse or
even irreducible.
However, it is necessary that $F_{\alpha_m}$ and $L$ meet properly.
We also see that, as the relative position of ${F\!_{\DOT}}$ and $L$ becomes more
degenerate, the intersection $\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$
`branches' into components, one for each
$j$ such that $\dim F_{\alpha_j}\cap L = n+2-\alpha_j-j-s$, and it
will attain excess intersection if
$\dim F_{\alpha_j}\cap L > n+2-\alpha_j-j-s$, for even one $j$.
\subsection{Remark}\label{remarkI}
In the situation of Theorem~\ref{thm:geometric_intersection}(2),
if $\beta\in \alpha*1$ and $j(\alpha,\beta) =1$,
then $\beta = \alpha+\delta^1$.
Suppose further that $M_{\alpha_1}\cap L= M_{\alpha_1+s}$.
Then
$$
X_\beta(1,{M_{\,\DOT}},L) \
=\ \Omega_{\alpha+s\delta^1}{M_{\,\DOT}}\ =\ \Omega_{\beta+(s-1)\delta^1}{M_{\,\DOT}},
$$
so we have
$$
\Omega_\alpha{F\!_{\DOT}} \bigcap \Omega_L\ =\
\sum_{\stackrel{\mbox{\scriptsize${\beta}{\in}{\alpha}{*}{1}$}}
{j(\alpha,\beta)=1}} \Omega_{\beta+(s-1)\delta^1
}{M_{\,\DOT}}\
+\sum_{\stackrel{\mbox{\scriptsize${\beta}{\in}{\alpha}{*}{1}$}}
{j(\alpha,\beta)>1}}
X_\beta(j(\alpha,\beta),{M_{\,\DOT}},L).
$$
We prove Theorem~\ref{thm:geometric_intersection} in
\S\ref{sec:proof_geometric_intersection}.
First, we study the varieties $X_\beta(j,{F\!_{\DOT}},L)$.
Let $\beta\in{[n]\choose m}$, ${F\!_{\DOT}}$ be a complete flag, and
$1\leq j\leq m$ an integer.
The map from $\Omega_\beta{F\!_{\DOT}}$ to ${\bf G}_jF_{\beta_j}$ given
by $H\longmapsto H\cap F_{\beta_j}$ is only defined on the dense
locus in $\Omega_\beta{F\!_{\DOT}}$ of those $H$ where $\dim H\cap
F_{\beta_j}=j$.
Resolving the ambiguity of this map gives the variety
$$
\widetilde{\Omega_\beta}^j\!{F\!_{\DOT}}\ :=\
\{(H,K) \in \Omega_\beta{F\!_{\DOT}} \times {\bf G}_jF_{\beta_j}
\,|\, K\subset H\mbox{\ \ and\ \ }
\dim K\cap F_{\beta_i}\geq i,
\ 1\leq i\leq j\}.
$$
In Lemma~\ref{lemma:rational_fibration}, we show that the
projection to ${\bf G}_jF_{\beta_j}$
realizes $\widetilde{\Omega_\beta}^j\!{F\!_{\DOT}}$ as a fibre bundle
with base and fibres themselves Schubert varieties.
The following definitions are needed to describe the base and
fibres.
Let $p$ be the first projection and $\pi$ the second.
For $K\subset V$, let ${F\!_{\DOT}}/K$ be the image of the flag ${F\!_{\DOT}}$
in $V/K$.
Let ${F\!_{\DOT}}|_{\beta_j}$ be the flag
$$
F_{\beta_j}\supset F_{\beta_j +1}\supset\cdots\supset F_n
$$
and $\beta|_j\in {[n+1-\beta_j]\choose j}$ the sequence
$$
\beta_1-\beta_j+1>\cdots>\beta_{j-1}-\beta_j+1>1=
(\beta|_j)_j.
$$
Unraveling this definition shows
$({F\!_{\DOT}}|_{\beta_j})_{(\beta|_j)_i}=F_{\beta_i}$, for $i\leq j$.
\subsection{Lemma.}\label{lemma:rational_fibration} {\em
Let $\beta\in {[n]\choose m}$, ${F\!_{\DOT}}$ be a flag, and
$1\leq j\leq m$.
Then $p$ is an isomorphism over the dense subset
$\{ H\in \Omega_\beta{F\!_{\DOT}}\, |\, \dim H\cap F_{\beta_j} = j\}$.
Also, $\pi$ exhibits
$\widetilde{\Omega_\beta}^j\!{F\!_{\DOT}}$ as a fibre bundle with
base $\Omega_{\beta|_j}{F\!_{\DOT}}|_{\beta_j}$ whose fibre over
$K\in\Omega_{\beta|_j}{F\!_{\DOT}}|_{\beta_j}$ is
the Schubert variety
$\Omega_{\beta_{j+1}\ldots\,\beta_m}{F\!_{\DOT}}/K\subset{\bf G}_{m-j}V/K$.
Moreover, each fibre of $\pi$ meets the locus where $p$
is an isomorphism.
}\smallskip
\noindent{\bf Proof:}
We describe the fibres of $\pi$.
Note that Schubert varieties
have a dual description:
$$
H\in \Omega_\beta{F\!_{\DOT}} \ \Longleftrightarrow\
\dim\frac{H}{H\cap F_{\beta_i}} \leq m-i,\ \mbox{\ for\ } 1\leq i\leq m.
$$
If $K\in \Omega_{\beta|_j}{F\!_{\DOT}}|_{\beta_j}$,
then $K\subset F_{\beta_j}\subset F_{\beta_i}$, for $i>j$.
Thus $\left( {F\!_{\DOT}}/K\right)_{\beta_i} = F_{\beta_i}/K$, for $i>j$.
Hence, if $H$ is in the fibre over $K$, then $H\in \Omega_\beta{F\!_{\DOT}}$ and
$K\subset H$, so
$$
\dim \frac{H/K}{H/K\cap \left( {F\!_{\DOT}}/K\right)_{\beta_i}}
\ =\ \dim\frac{H}{H\cap F_{\beta_i}} \leq m-i,\ \mbox{\ for\ }
j< i\leq m.
$$
Thus $H/K \in \Omega_{\beta_{j+1}\ldots\,\beta_m}{F\!_{\DOT}}/K$.
The reverse implication is similar
and the remaining assertions follow easily from the definitions.
\QED
Reformulating the definition of $X_\beta(j,{F\!_{\DOT}},L)$
in these terms gives a useful characterization.
\subsection{Corollary.}\label{cor:intermediate_cycles}
$
X_\beta(j,{F\!_{\DOT}},L) \ =\ p(\pi^{-1}(
\Omega_{\beta|_j}{F\!_{\DOT}}|_{\beta_j} \bigcap \Omega_{F_{\beta_j}\cap L})).
$
\smallskip
Since the fibres of $\pi$ meet the locus where $p$ is an isomorphism,
the map
$$
p \ :\ \pi^{-1}(
\Omega_{\beta|_j}{F\!_{\DOT}}|_{\beta_j} \bigcap \Omega_{F_{\beta_j}\cap L})
\longrightarrow X_\beta(j,{F\!_{\DOT}},L)
$$
is proper and birational.
Thus, while $X_\beta(j,{F\!_{\DOT}},L)$ is neither a Schubert variety nor an
intersection of Schubert varieties, it is `birationally fibred' over
an intersection of Schubert varieties with Schubert variety fibres,
and hence is intermediate between these extremes.
\subsection{Tangent spaces to Schubert varieties}
Let $H\in {\bf G}_m V$ and $K\in {\bf G}_{n-m}V$ be complementary
subspaces, so $H\cap K = \{0\}$.
The open set $U\subset {\bf G}_m V$ of those $H'$ with $H'\cap K = \{0\}$
is identified with $\mbox{Hom}(H,K)$ by
$\phi \in \mbox{Hom}(H,K) \mapsto \Gamma_\phi$, the graph of $\phi$ in
$H\oplus K =V$.
This shows the tangent space of ${\bf G}_mV$ at $H$, $T_H{\bf G}_mV$,
is equal to $\mbox{Hom}(H,V/H)$, since $K$ is canonically isomorphic to $V/H$.
The intersection of a Schubert variety $\Omega_\alpha{F\!_{\DOT}}$ containing $H$
with this open set $U$ can be used to determine whether $\Omega_\alpha{F\!_{\DOT}}$
is smooth at $H$ and its tangent space at $H$.
This gives the following description:
If $H\in {\bf G}_mV$ and $\dim H\cap F_{\alpha_j} = j$ for
$1\leq j\leq m$, then $\Omega_\alpha{F\!_{\DOT}}$ is smooth at $H$ and
$$
T_H\Omega_\alpha{F\!_{\DOT}}\ =\ \{ \phi\in \mbox{Hom}(H,V/H)\,|\,
\phi(H\cap F_{\alpha_j}) \subset (F_{\alpha_j} +H)/H,\ 1\leq j\leq m\}.
$$
Similarly, if $H\in {\bf G}_mV$,
$L\in {\bf G}_{n+1-m-s}V$, and $\dim H\cap L = 1$,
then $\Omega_L$ is smooth at $H$ and the tangent space of $\Omega_L$
at $H$ is
$$
T_H\Omega_L \ =\ \{\phi\in \mbox{Hom}(H,V/H)\,|\,
\phi(H\cap L)\subset (L+H)/H\}.
$$
Let $P$ be the subgroup of $GL(V)$ stabilizing the partial flag
$F_{\alpha_1}\subset F_{\alpha_2}\subset \cdots\subset F_{\alpha_m}$.
The orbit $P\cdot L'$ consists of those $L$ with
$\dim F_{\alpha_j}\cap L=\dim F_{\alpha_j}\cap L'$ for
$1\leq j\leq m$.
Similarly, $L\in \overline{P\cdot L'}$
if $\dim F_{\alpha_j}\cap L\geq \dim F_{\alpha_j}\cap L'$ for
$1\leq j\leq m$.
If
$P\cdot L= P\cdot L'$, then
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L
\simeq\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_{L'}$.
Thus $P$-orbits on ${\bf G}_{n+1-m-s}V$ determine the isomorphism type
of Pieri-type intersections.
\subsection{Lemma} \label{lemma:P-orbits}
{\em
Suppose that $L,L'\in{\bf G}_{n+1-m-s}V$ with
$L\in \overline{P\cdot L'}$.
Then
\begin{enumerate}
\item [(1)] $\dim\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L \geq
\dim\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_{L'}$.
\item [(2)] If $\,\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$ is
generically transverse, then
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_{L'}$ is generically transverse.
\item [(3)] If $\,\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$ is generically
transverse and
irreducible, then $\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_{L'}$ is generically
transverse and irreducible.
\end{enumerate}
}\smallskip
\noindent{\bf Proof:}
Let $\psi:{\bf P}^1 \rightarrow \overline{P\cdot L'}$ be a map with
$\psi(0)=L$ and $\psi({\bf P}^1) \cap (P\cdot L') \neq \emptyset$.
Then $\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_{\psi(t)}$ is isomorphic to
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_{L'}$, for any
$t\in \psi^{-1}(P\cdot L')$.
The lemma follows by considering the subvariety
of ${\bf P}^1\times {\bf G}_mV$ whose fibre over $t\in {\bf P}^1$
is $\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_{\psi(t)}$.
\QED
\subsection{\bf Proof of
Theorem~\ref{thm:geometric_intersection}:}
\label{sec:proof_geometric_intersection}
Let $\alpha\in{[n]\choose m}$, $s>0$, ${F\!_{\DOT}}$ be a complete flag, and
$L\in {\bf G}_{n+1-m-s}V$.
The conditions on $L$ in statement (2), that
$\dim F_{\alpha_j}\cap L = n+2-\alpha_j-j-s$ for each $j$, determine a
$P$-orbit,
which is the closure of any $P$-orbit $P\cdot L'$, where
$\dim F_{\alpha_j}\cap L' \leq n+2-\alpha_j-j-s$ for each $j$.
Thus (2) and Lemma~\ref{lemma:P-orbits}(2) together imply that if
$\dim F_{\alpha_j}\cap L \leq n+2-\alpha_j-j-s$ for each $j$, then
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$ is
generically transverse, proving the second part of (1).
For the first part of (1), suppose
$\dim F_{\alpha_j}\cap L > n+2-\alpha_j-j-s$ and let
$L':= F_{\alpha_j}\cap L\neq \{0\}$.
Then $L'$ has codimension at most $j+s-1$ in $F_{\alpha_j}$.
Hence
$\Omega_{\alpha|_j}{F\!_{\DOT}}|_{\alpha_j} \bigcap \Omega_{L'}\neq \emptyset$
and so has codimension in $\Omega_{\alpha|_j}{F\!_{\DOT}}|_{\alpha_j}$ at most that
of $\Omega_{L'}$ in ${\bf G}_jF_{\alpha_j}$, which is at most $s-1$.
Thus
$$
X_\alpha(j,{F\!_{\DOT}},L)\ = \
p (\pi^{-1}(\Omega_{\alpha|_j}{F\!_{\DOT}}|_{\alpha_j} \bigcap \Omega_{L'}))
$$
which has codimension less than $s$ in $\Omega_\alpha{F\!_{\DOT}}=
p (\pi^{-1}(\Omega_{\alpha|_j}{F\!_{\DOT}}|_{\alpha_j}))$.
Hence $\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$
is improper, as
$X_\alpha(j,{F\!_{\DOT}},L)\subset \Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$, proving (1).
\smallskip
Before proceeding with the rest of the proof, we make a computation.
Suppose $\dim F_{\alpha_j}\cap L \leq n+2-\alpha_j-j-s$ for
$1\leq j\leq m$ and $F_{\alpha_m}\cap L \not\subset F_{\alpha_{m-1}}$.
Then there exists $H\in \Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$ with
$\dim H\cap F_{\alpha_j}=j$ for $1\leq j\leq m$, $\dim H\cap L = 1$,
and $H\cap L\not\subset F_{\alpha_{m-1}}$:
Inductively choose linearly independent vectors $f_j\in F_{\alpha_j}$
for $1\leq j\leq m$ as follows.
Let $f_1\in F_{\alpha_1} - \{0\}$.
Then for $1<j<m$ suppose that $f_1,\ldots,f_{j-1}$ have been chosen.
Since $\dim F_{\alpha_j}$ exceeds
$$
\dim F_{\alpha_j}\cap \Span{L,f_1,\ldots,f_{j-1}}
\ \leq\ n+2-\alpha_j-j-s+(j-1)\ =\ n+1-\alpha_j-s
$$
and $F_{\alpha_{j-1}}\not\subset F_{\alpha_j}$, we can select a vector
$f_j$ in
$$
F_{\alpha_j}-\Span{L,f_1,\ldots,f_{j-1}}- F_{\alpha_{j-1}}.
$$
Let $f_m\in F_m\cap L-F_{\alpha_{m-1}}$, and let
$H := \Span{f_1,\ldots,f_m}$.
Then $H\in \Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$, $\dim H\cap F_{\alpha_j}=j$ for
$1\leq j\leq m$, $\dim H\cap L=1$, and $H\cap L\not\subset F_{\alpha_{m-1}}$.
Let $X_m^\circ$ be the set of all such $H$.
For $H\in X_m^\circ$,
$$
T_H\Omega_\alpha{F\!_{\DOT}}\bigcap T_H\Omega_L\ =\ \{ \phi\in T_H\Omega_\alpha{F\!_{\DOT}}
\,|\, \phi(H\cap L) \subset (F_{\alpha_m}\cap L +H)/H\}.
$$
This has codimension in $T_H\Omega_\alpha{F\!_{\DOT}}$ equal to
$\dim(F_{\alpha_m}+H) -
\dim(F_{\alpha_m}\cap L + H) = s$.
Thus $\Omega_\alpha{F\!_{\DOT}}$ and $\Omega_L$ meet transversally along
$X_m^\circ$.
\smallskip
We show (2).
Suppose $\dim F_{\alpha_j}\cap L = n+2-\alpha_j-s$ for each
$1\leq j\leq m$.
Let ${M_{\,\DOT}}$ be any flag satisfying $M_{\alpha_j}=F_{\alpha_j}$ and
$M_{\alpha_j +1}\supset\Span{F_{\alpha_{j-1}},\,F_{\alpha_j}\cap L}$,
for $1\leq j\leq m$.
Let $H\in \Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$.
Then there is some $1\leq j\leq m$ with
$H\cap L\cap F_{\alpha_j}\not\subset F_{\alpha_{j-1}}$.
Since $\dim H\cap F_{\alpha_{j-1}}\geq j-1$, we have
$\dim H\cap\Span{F_{\alpha_{j-1}}, F_{\alpha_j}\cap L}\geq j$
and so $\dim H\cap M_{\alpha_j+1}\geq j$.
Thus $H\in \Omega_{\alpha+\delta^j}{M_{\,\DOT}}$, if
$\alpha+\delta^j\in{[n]\choose m}$.
But this is the case, as $\alpha_j+1<\alpha_{j-1}$, for otherwise
dimensional considerations imply that
$L\cap F_{\alpha_j}=L\cap F_{\alpha_{j-1}}\subset F_{\alpha_{j-1}}$.
Let $\beta := \alpha+\delta^j \in \alpha*1$.
Then $j(\alpha,\beta) = j$
and $H\in X_\beta(j(\alpha,\beta),{M_{\,\DOT}},L)$, since
$H\in \Omega_\beta {M_{\,\DOT}}$ and $\dim H\cap L\cap M_{\beta_j}\geq 1$.
Conversely, if $\beta\in \alpha*1$, then
$\Omega_\beta{M_{\,\DOT}}\subset\Omega_\alpha{F\!_{\DOT}}$, so
$X_\beta(j(\alpha,\beta),{M_{\,\DOT}},L)\subset\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L$.
This shows
$$
\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L\ =\ \sum_{\beta\in \alpha*1}
X_\beta(j(\alpha,\beta),{M_{\,\DOT}},L).
$$
We claim this intersection is generically transverse.
Let $\beta\in \alpha*1$ and $j := j(\alpha,\beta)$.
Then
$X_\beta(j,{M_{\,\DOT}},L)$ has an open subset $X^\circ_j$
consisting of those $H$ with $\dim H\cap F_{\alpha_i} = i$ for
$1\leq i\leq m$, $\dim H\cap L= 1$, and $H\cap L\subset F_{\alpha_j}$
but $H\cap L\not\subset F_{\alpha_j-1}$.
As with $X^\circ_m$ above, $X^\circ_j$ is nonempty, so it is a dense open
subset of $X_\beta(j,{M_{\,\DOT}},L)$
For $H\in X^\circ_j$,
$$
T_H\Omega_\alpha{F\!_{\DOT}} \bigcap T_H\Omega_L \ = \
\{\phi\in T_H\Omega_\alpha{F\!_{\DOT}} \,|\,
\phi(H\cap L) \subset (L\cap F_{\alpha_j} +H)/H\}.
$$
Since
$\dim(F_{\alpha_j} +H)- \dim(L\cap F_{\alpha_j} +H)= s$,
this has codimension $s$
in $T_H\Omega_\alpha{F\!_{\DOT}}$, showing that $\Omega_\alpha{F\!_{\DOT}}$ and
$\Omega_L$ meet transversally along $X^\circ_j$, a dense
subset of $X_\beta(j(\alpha,\beta),{M_{\,\DOT}},L)$.%
\smallskip
By Lemma~\ref{lemma:P-orbits}(3), it suffices to prove
a special case of (3):
\begin{enumerate}
\item[(3)$'$] {\em
If $F_{\alpha_m}$ meets $L$ properly, and
for $1\leq j<m$, $\dim F_{\alpha_j}\cap L= n+2-\alpha_j-j-(s+1)$,
then $\Omega_\alpha{F\!_{\DOT}} \bigcap \Omega_L$ is irreducible.}
\end{enumerate}
These conditions imply $F_{\alpha_m}\cap L \not\subset F_{\alpha_{m-1}}$.
In the notation of \S \ref{remarkI}, let $L':=F_{\alpha_{m-1}}\cap L$,
${{F\!_{\DOT}}'} := {F\!_{\DOT}}|_{\alpha_{m-1}}$, and $\alpha':= \alpha|_{m-1}$.
Consider
$$
X_\alpha(m-1,{F\!_{\DOT}},L) \ =\ p(\pi^{-1}(
\Omega_{\alpha'}{{F\!_{\DOT}}'}\bigcap
\Omega_{L'})).
$$
For $j\leq m-1$,
\begin{eqnarray*}
\dim F_{\alpha_j}\cap L' &=& n+2-\alpha_j-j-(s+1)\\
&=& \dim F_{\alpha_{m-1}} +2 -\alpha'_j -j-(s+1),
\end{eqnarray*}
so
$L'$ and ${{F\!_{\DOT}}'}$ satisfy the conditions of (2) for the pair
$\alpha', s+1$.
Thus $\Omega_{\alpha'}{{F\!_{\DOT}}'}\bigcap \Omega_{L'}$
is generically transverse, which implies that
$X_\alpha(m-1,{F\!_{\DOT}},L)$ has codimension $s+1$ in
$\Omega_\alpha{F\!_{\DOT}}$ and hence is a proper subvariety of
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L-X_\alpha(m-1,{F\!_{\DOT}},L)$.
Since $X^\circ_m$ is dense in
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_L-X_\alpha(m-1,{F\!_{\DOT}},L)$, this completes
the proof of (3)$'$.
\QED
\section{Construction of explicit rational
equivalences}\label{sec:explicit_rational_equivalences}
Theorem~\ref{thm:geometric_intersection} shows that for $L$ in
a dense subset of ${\bf G}_{n+1-m-b}V$, the intersection
$\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$ is generically transverse
and irreducible.
We use Theorem~\ref{thm:geometric_intersection}(2)
to study such a cycle as $L$ `moves out of'
this set, ultimately deforming it into the cycle
$\sum_{\gamma\in \alpha*b}\Omega_\gamma{F\!_{\DOT}}$.
\subsection{Families and Chow varieties}\label{sec:Chow}
Suppose $\Sigma \subset ({\bf P}^1-\{0\}) \times {\bf G}_mV$
has equidimensional
fibres over ${\bf P}^1-\{0\}$.
Then its Zariski closure $\overline{\Sigma}$
in ${\bf P}^1\times{\bf G}_mV$ has equidimensional fibres over
${\bf P}^1$.
Denote the fibre of $\overline{\Sigma}$ over
$0$ by $ \lim_{t\rightarrow 0} \Sigma_t$,
where $\Sigma_t$ is the fibre of $\Sigma$ over $t\in {\bf P}^1-\{0\}$.
The association of a point $t$ of ${\bf P}^1$
to the fundamental cycle of the
fibre $\overline{\Sigma}_t$ determines a morphism
${\bf P}^1\rightarrow \mbox{\it Chow}\, {\bf G}_mV$.
Moreover, if $\Sigma$ is defined over $k$, then so
is the map ${\bf P}^1\rightarrow \mbox{\it Chow}\, {\bf G}_mV$
(\cite{Samuel}, \S I.9).
\subsection{The cycle $Y_{\alpha,r}(F\!\!_{\mbox{\bf .}},L)$}
\label{sec:Y_alpha,r}
In \S \ref{sec:intermediate_cycle}, we defined the components
$X_\beta(\alpha,{F\!_{\DOT}},L)$ of the cycles intermediate between
$\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$ and
$\sum_{\gamma\in \alpha*b}\Omega_\gamma{F\!_{\DOT}}$.
Here, we define those intermediate cycles,
$Y_{\alpha,r}({F\!_{\DOT}},L)$, which
are parameterized by subspaces $L$ in certain Schubert
cells $U_{\alpha,s}{F\!_{\DOT}}$ of ${\bf G}_{n+1-m-s}$.
Let $U_{\alpha,s}{F\!_{\DOT}}$ be the set of those
$L\in G_{n+1-m-s}V$ such that
\begin{enumerate}
\item[(1)] $F_{\alpha_1} \cap L = F_{\alpha_1+s}$, and
\item[(2)] $F_{\alpha_j}\cap L = F_{\alpha_j+1}\cap L$, and has dimension
$n+2-\alpha_j-j-s$, for $1\leq j\leq m$.
\end{enumerate}
These conditions are consistent and determine
$\dim F_i\cap L$ for $1\leq i\leq n$.
For example,
$$
\mbox{(\ref{sec:Y_alpha,r})}\hspace{.57in}
\alpha_j<i<\alpha_{j-1}\ \ \Longrightarrow\ \
\dim F_i\cap L \ =\ \dim F_i + 1-j-(s-1).
\hspace{.87in}
$$
Thus $U_{\alpha,s}{F\!_{\DOT}}$ is a single Schubert cell of $G_{n+1-m-s}V$.
Specifically, $U_{\alpha,s}{F\!_{\DOT}}$
is the dense cell of $\Omega_\beta{F\!_{\DOT}}$,
where $\beta\in{[n]\choose n+1-m-s}$ is defined as follows:
If $\alpha_1\leq n+1-s$, then
$\beta=[n]-\alpha-\{\alpha_1+1,\ldots,\alpha_1+s-1\}$.
Otherwise, $\beta$ is the smallest $n+1-m-s$ integers in
$[n]-\alpha$.
For $\beta\in \alpha*r$, recall that
$j(\alpha,\beta) = \min\{i\,|\, \alpha_i<\beta_i\}$.
If $L\in U_{\alpha,s}{F\!_{\DOT}}$, define the cycle
$$
Y_{\alpha,r}({F\!_{\DOT}},L)\ :=
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{r}$}}
{j(\alpha,\beta)=1}}
\Omega_{\beta+ (s-1)\delta^1}{F\!_{\DOT}}
\ +
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{r}$}}
{j(\alpha,\beta)>1}}
X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L).
$$
Let ${\cal G}_{\alpha,s,r}{F\!_{\DOT}} \subset \mbox{\it Chow}\, {\bf G}_mV$ be the set of
these cycles $Y_{\alpha,r}({F\!_{\DOT}},L)$ for
$L\in U_{\alpha,s}{F\!_{\DOT}}$.
Since $U_{\alpha,s}{F\!_{\DOT}}$ is a Schubert cell,
${\cal G}_{\alpha,s,r}{F\!_{\DOT}}$
is an orbit of the Borel subgroup
stabilizing ${F\!_{\DOT}}$ and hence is rational.
\subsection{Remark}\label{remarkII}
Suppose $L\in U_{\alpha,s}{F\!_{\DOT}}$, then by
Remark~\ref{remarkI},
\begin{eqnarray*}
\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L &=&
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{1}$}}
{j(\alpha,\beta)=1}}
\Omega_{\beta+ (s-1)\delta^1}{F\!_{\DOT}}\
+
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{1}$}}
{j(\alpha,\beta)>1}}
X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L)\\
&=& Y_{\alpha,1}({F\!_{\DOT}},L).
\end{eqnarray*}
The following lemma parameterizes our explicit rational
equivalences.
It is identical to Lemma~6.1 of~\cite{sottile_real_lines}.
\subsection{Lemma.}\label{lemma:limits_are_good}
{\em
Let $l\leq n$ and let
${M_{\,\DOT}}$ be a complete flag in $M\simeq k^n$.
Suppose $L_{\infty}$ is a hyperplane containing $M_l$ but not $M_{l-1}$.
Then there exists a pencil of hyperplanes $L_t$, for
$t\in {\bf P}^1$,
such that if $t\neq 0$, then $L_t$ contains $M_l$ but not $M_{l-1}$
and, for each $i\leq l-1$, the family of codimension $i+1$ planes
induced by $M_i\cap L_t$ for $t\neq 0$ has fibre $M_{i+1}$ over $0$.
}
\smallskip
\noindent{\bf Proof:}
Let $x_1,\ldots,x_n$ be a basis of $M^*$ such that
$L_{\infty}= \Span{x_{l-1}}^{\perp}$ and
$M_i = \Span{x_1,\ldots,x_{i-1}}^\perp$.
Let $e_1,\ldots,e_n$ be a basis for $M$ dual to $x_1,\ldots,x_n$
and define
$$
L_t \ :=\ \Span{M_l, te_j + e_{j+1}\,|\, 1\leq j\leq l-2}.
$$
For $t\neq 0$ and $1\leq i\leq l-1$,
$M_i \cap L_t = \Span{M_l, te_j + e_{j+1}\,|\, i\leq j\leq l-2}$
and so has dimension $n-i$.
The fibre of this family at $t=0$ is
$\Span{M_l, e_{j+1}\,|\, i\leq j\leq l-2} = M_{i+1}$.
\QED
\subsection{Theorem.} \label{thm:inductive_engine}{\em
Let $\alpha\in {[n]\choose m}$, s,r be positive integers and ${F\!_{\DOT}}$ a
flag in $V$.
Let $M\in U_{\alpha,s-1}{F\!_{\DOT}}$ and define ${M_{\,\DOT}}$ to be the flag in
$M$ consisting of the subspaces
in ${F\!_{\DOT}}\cap M$.
Let $L_\infty\subset M$ be any hyperplane containing $F_{\alpha_1+s}$
but not $F_{\alpha_1+s-1}$.
Suppose $L_t$ is the family of hyperplanes of $M$
given by Lemma~\ref{lemma:limits_are_good}.
Then
\begin{enumerate}
\item[(1)] For $t\neq 0$, $L_t \in U_{\alpha,s}{F\!_{\DOT}}$.
\item[(2)] ${\displaystyle \lim_{t\rightarrow 0} Y_{\alpha,r}({F\!_{\DOT}},L_t)
= Y_{\alpha, r+1}({F\!_{\DOT}},M)}$.
\end{enumerate}
}
\subsection{Theorem.}\label{thm:pieri} {\bf [Pieri's Formula]}
{\em Let $\alpha\in{[n]\choose m}$, ${F\!_{\DOT}}$ be a complete flag in $V$, and
$K\in{\bf G}_{n+1-m-b}V$ be a subspace which meets ${F\!_{\DOT}}$ properly.
Then the cycle $\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_K$, a generically
transverse intersection, is rationally equivalent to
$
\sum_{\gamma\in \alpha*b}\Omega_\gamma{F\!_{\DOT}}$.
Thus, in the Chow ring $A^*{\bf G}_mV$ of ${\bf G}_mV$,
$$
[\Omega_\alpha{F\!_{\DOT}}]\cdot[\Omega_K] \ =\
\sum_{\gamma\in \alpha*b}[\Omega_\gamma{F\!_{\DOT}}].
$$
Moreover, let ${\cal G}\subset \mbox{\it Chow}\, {\bf G}_mV$ be the set of cycles arising
as generically transverse intersections of the form
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_K$ for $K\in{\bf G}_{n+1-m-b}V$.
Then one may give $b+1$ explicit rational deformations inducing this
rational equivalence, where the cycles at the $i$th stage are of the form
$Y_{\alpha,i}({F\!_{\DOT}},M)$, with $M\in U_{\alpha,b+1-i}{F\!_{\DOT}}$, and all are
within $\overline{\cal G}$.
}\smallskip
The Borel subgroup $B$ of $GL(V)$ stabilizing ${F\!_{\DOT}}$ also
stabilizes ${\cal G}$ and the cycle
$\sum_{\gamma\in\alpha*b}\Omega_\gamma{F\!_{\DOT}}$
is the only $B$-stable cycle in this component of the Chow variety.
As Hirschowitz~\cite{Hirschowitz} observed,
this implies $\sum_{\gamma\in\alpha*b}\Omega_\gamma{F\!_{\DOT}}$ is in the Zariski
closure of ${\cal G}$.
Thus Theorem~\ref{thm:pieri} is an improvement in that
deformations inducing the rational equivalence are given explicitly.
\subsection{Proof of Pieri's formula using Theorem~3.5}\label{sec:pieri_proof}
Let $b>0$, and $\alpha\in {[n]\choose m}$.
For $1\leq i\leq b$, let $U_i := U_{\alpha,b+1-i}{F\!_{\DOT}}$
and ${\cal G}_i := {\cal G}_{\alpha,b+1-i,i}{F\!_{\DOT}}$.
Let $U_0\subset {\bf G}_{n+1-m-b}V$ be the (dense) set of those $L$
which meet $F_{\alpha_m}$ properly and for $1\leq j<m$,
$\dim F_{\alpha_j}\cap L< n+2-\alpha_j-j-b$.
By Theorem~\ref{thm:geometric_intersection},
if $L\in {\bf G}_{n+1-m-b}V$, then
$\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$ is generically transverse and
irreducible if and only if $L\in U_0$.
Let ${\cal G}_0\subset \mbox{\it Chow}\, {\bf G}_mV$ be the set of cycles
$\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_L$ for $L\in U_0$.
Let $L\in U_b$ and consider the cycle
$Y_{\alpha,b}({F\!_{\DOT}},L)\in {\cal G}_b$:
$$
Y_{\alpha,b}({F\!_{\DOT}},L)\ =
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{b}$}}
{j(\alpha,\beta)=1}}
\Omega_{\beta}{F\!_{\DOT}}
\ +
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{b}$}}
{j(\alpha,\beta)>1}}
X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L).
$$
We claim
$Y_{\alpha,b}({F\!_{\DOT}},L)=\sum_{\beta\in\alpha*b}\Omega_\beta{F\!_{\DOT}}$,
the cycle $Y_{\alpha,b}{F\!_{\DOT}}$ of the Introduction.
It suffices to show $X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L)=\Omega_\beta{F\!_{\DOT}}$
for $\beta\in \alpha*b$ with $j(\alpha,\beta)>1$.
Suppose $j = j(\alpha,\beta)>1$, then
$$
X_\beta(j,{F\!_{\DOT}},L) \ =\ p(\pi^{-1}(
\Omega_{\beta|_j}{F\!_{\DOT}}|_{\beta_j}\bigcap \Omega_{F_{\beta_j}\cap L})).
$$
By formula (\ref{sec:Y_alpha,r}),
$\dim F_{\beta_j}\cap L = \dim F_{\beta_j}-j+1$,
as $\alpha_j<\beta_j<\alpha_{j-1}$ and $s=1$.
So $\Omega_{F_{\beta_j}\cap L}$ is ${\bf G}_jF_{\beta_j}$,
since any $j$-plane in $F_{\beta_j}$ meets $F_{\beta_j}\cap L$
non-trivially.
Thus $X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L)=\Omega_\beta{F\!_{\DOT}}$, by the definition
of $p$ and $\pi$ in~\S\ref{remarkI}.
\smallskip
Let ${\cal G}\subset \mbox{\it Chow}\, {\bf G}_mV$ be the set of all cycles
$\Omega_\alpha{F\!_{\DOT}} \cap \Omega_L$, where
$L\in {\bf G}_{n+1-m-b}V$ and the intersection is generically transverse.
Then by Theorem~\ref{thm:geometric_intersection} and Remark~\ref{remarkII},
both ${\cal G}_0$ and ${\cal G}_1 $ are subsets of ${\cal G}$.
Arguing as in the proof of Lemma~\ref{lemma:P-orbits} shows
${\cal G}\subset \overline{{\cal G}_0}$.
Theorem~\ref{thm:inductive_engine} implies
${\cal G}_i\subset \overline{{\cal G}_{i-1}}$ for $2\leq i\leq b$,
so in particular, $Y_{\alpha,b}{F\!_{\DOT}}\in{\cal G}_b\subset \overline{\cal G}$.
Since ${\cal G}_0$ and hence $\overline{\cal G}$ is rational,
$Y_{\alpha,b}{F\!_{\DOT}}$ is rationally
equivalent to any cycle in ${\cal G}$, including
$\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_K$, proving Pieri's formula.
\smallskip
More explicitly, one may construct a sequence of parameterized rational
curves
$\phi_i: {\bf P}^1 \rightarrow \overline{\cal G}$ for
$1\leq i\leq b$ witnessing this
rational equivalence.
For $2\leq i\leq b$, select subspaces $M_i\in U_i$ and pencils
$L_{i,t}$ of hyperplanes of $M_i$ by downward induction on $i$ as follows:
Choose $M_b\in U_b$.
Given $M_i\in U_i$, let $L_{i,t}$
be a pencil of hyperplanes of $M_i$ as in
Theorem~\ref{thm:inductive_engine}, let $M_{i-1}: = L_{i,\infty}$, and
continue.
Then for each $i$, if $t\neq 0$, $L_{i,t}\in U_{i-1}$.
Define $\Sigma_i\subset {\bf P}^1 \times {\bf G}_mV$ to be the family whose
fibre over $t\in {\bf P}^1-\{0\}$ is the variety
$Y_{\alpha,i-1}({F\!_{\DOT}},L_{i,t})$.
Let
$\psi:{\bf P}^1\rightarrow \overline{U_0}={\bf G}_{n+1-m-s}V$ be a map with
$\psi(0) = M_1: = L_{2,\infty}$,
$\psi(\infty)=K$, and $\psi^{-1}(U_0)= {\bf P}^1-\{0\}$.
Let $\Sigma_1 \subset {\bf P}^1\times {\bf G}_mV$ be the family whose fibre
over $t\in {\bf P}^1$ is $\Omega_\alpha{F\!_{\DOT}}\bigcap\Omega_{\psi(t)}$,
a generically transverse intersection which is irreducible for $t\neq 0$, by
Theorem~\ref{thm:geometric_intersection}.
Then for $1\leq i\leq b$, $\Sigma_i\subset {\bf P}^1\times {\bf G}_mV$
is a family with equidimensional generically reduced fibres over ${\bf P}^1$.
For $1\leq i\leq b$, let
$\phi_i:{\bf P}^1 \rightarrow \overline{{\cal G}_{i-1}}$ be the
map associated to the family $\Sigma_i$, as in \S\ref{sec:Chow}.
Then $\phi_i(0) = \phi_{i+1}(\infty)\in {\cal G}_i$
and $\phi_i(t)\in {\cal G}_{i-1}$ for $t\neq 0$, by
Theorem~\ref{thm:inductive_engine}.
Thus these parameterized rational curves give a chain of
rational equivalences between
$\Omega_\alpha{F\!_{\DOT}}\bigcap \Omega_K$
and $Y_{\alpha,b}{F\!_{\DOT}}$.
\QED
Let $\beta\in \alpha*r$ and $\gamma\in \alpha*(r+1)$.
If $\gamma\in \beta*1$ with $j(\alpha,\gamma) = j(\beta,\gamma)$,
write $\beta \prec_\alpha\gamma$.
\subsection{Proof of Theorem 3.5}\label{sec:proof_of_deformations}
Let $t\neq 0$.
Recall that $L_t$ contains the subspace
$F_{\alpha_1 +s}$ of ${M_{\,\DOT}}$, but not $F_{\alpha_1 +s-1}$.
Since $M\in U_{\alpha,s-1}{F\!_{\DOT}}$, we have
$F_{\alpha_1}\cap M = F_{\alpha_1+s-1}$,
but $F_{\alpha_1}\cap L_t = F_{\alpha_1+s}$, thus
$F_i\cap L_t$ is a hyperplane of $F_i\cap M$ for any $i\leq \alpha_1$.
Then $L_t\in U_{\alpha,s}{F\!_{\DOT}}$, for $t\neq 0$, as
\begin{enumerate}
\item $F_{\alpha_1}\cap L_t=F_{\alpha_1+s}$.
\item For $1\leq j\leq m$, $F_{\alpha_j}\cap M = F_{\alpha_j+1}\cap M$.
So $F_{\alpha_j}\cap L_t = F_{\alpha_j+1}\cap L_t$.
Moreover, $\dim F_{\alpha_j}\cap L_t = \dim F_{\alpha_j}\cap M-1$,
which is $n+2-\alpha_j-j-s$.
\end{enumerate}
Suppose $t\neq 0$ and recall that
$$
Y_{\alpha,r}({F\!_{\DOT}},L_t)\ =
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{r}$}}
{j(\alpha,\beta)=1}}
\Omega_{\beta+ (s-1)\delta^1}{F\!_{\DOT}}
\ +
\sum_{\stackrel{\mbox{ \scriptsize ${\beta}{\in}{\alpha}{*}{r}$}}
{j(\alpha,\beta)>1}}
X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L_t).
$$
This defines a family $\Sigma\subset({\bf P}^1-\{0\})\times{\bf G}_mV$
with equidimensional (actually isomorphic) fibres over
${\bf P}^1-\{0\}$.
We establish Theorem~\ref{thm:inductive_engine}, showing the fibre of
$\overline{\Sigma}$ at 0 is $Y_{\alpha,r+1}({F\!_{\DOT}},M)$ by examining each
component of $Y_{\alpha,r}({F\!_{\DOT}},L_t)$ separately, then assembling the
result.
Let $\beta\in \alpha*r$.
Consider a component of $Y_{\alpha,r}({F\!_{\DOT}},L_t)$ in the first summand, so
$j(\alpha,\beta)=1$.
Then $\gamma := \beta +\delta^1$ is the unique sequence satisfying
$\beta \prec_\alpha \gamma$.
In this case,
$\Omega_{\beta+(s-1)\delta^1}{F\!_{\DOT}} =
\Omega_{\gamma+(s-2)\delta^1}{F\!_{\DOT}}$.
Now consider a component in the second sum, so $j=j(\alpha,\beta)>1$.
Let $\beta' := \beta|_j$,
${{F\!_{\DOT}}'} := {F\!_{\DOT}}|_{\beta_j}$, and $L'_t :=F_{\beta_j}\cap L_t$.
For $t\neq 0$, Corollary~\ref{cor:intermediate_cycles} gives
$$
X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L_t) \ = \ p(\pi^{-1}(
\Omega_{\beta'}{{F\!_{\DOT}}'}\bigcap \Omega_{L'_t})).
$$
As $\alpha_j<\beta_j<\alpha_{j-1}$, $\dim L_t'=\dim F_{\beta_j}+1-j-(s-1)$,
by formula (\ref{sec:Y_alpha,r}).
For $1\leq i<j$, $\beta_i = \alpha_i$ and so
$\dim L'_t\cap F_{\beta_i} = n+2-\beta_i -i-s$.
Thus, by Theorem~\ref{thm:geometric_intersection}(3),
$\Omega_{\beta'}{{F\!_{\DOT}}'}\bigcap \Omega_{L'_t}$ is
generically transverse and irreducible.
We study the `limit' of these cycles as $t\rightarrow 0$, in the sense of
\S\ref{sec:Chow}.
Define $L' := \lim_{t\rightarrow 0}L'_t
= \lim_{t\rightarrow 0} F_{\beta_j}\cap L_t$, which is
$F_{\beta_j +1}\cap M$, by Lemma~\ref{lemma:limits_are_good}.
Then
\begin{enumerate}
\item[(1)] $F_{\alpha_1}\cap L' = F_{\alpha_1}\cap M = F_{\alpha_1+s-1}$.
\item[(2)] For $1\leq i\leq j$, $F_{\beta_i}\cap L' = F_{\beta_i+1}\cap L'$.
This follows for $i=j$ because we have
$L'\subset F_{\beta_j+1}\subset F_{\beta_j}$ and
for $i<j$, because $\beta_i = \alpha_i$ and
$F_{\alpha_i}\cap M = F_{\alpha_i+1}\cap M$.
Moreover, for $1\leq i\leq j$,
$\dim F_{\beta_i}\cap L' =n+2-\beta_i-i-(s-1)$.
\end{enumerate}
Thus $L'\in U_{\beta',s-1}{{F\!_{\DOT}}'}$ so
$\Omega_{\beta'}{{F\!_{\DOT}}'} \bigcap \Omega_{L'}$ is generically
transverse, by Theorem~\ref{thm:geometric_intersection}(1).
So,
$$
\lim_{t\rightarrow 0} X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L_t) \ =\ p(\pi^{-1}(
\Omega_{\beta'}{{F\!_{\DOT}}'}\bigcap \Omega_{L'})).
$$
But $\Span{F_{\beta_{i-1}},F_{\beta_i}\cap L}
\subset F_{\beta_i +1}$, since $L'\in U_{\beta',s-1}{{F\!_{\DOT}}'}$.
By Remark~\ref{remarkI},
$$
\Omega_{\beta'}{{F\!_{\DOT}}'}\bigcap \Omega_{L'}
\ =\ \sum_{\stackrel{\mbox{\scriptsize
${\gamma'}{\in}{\beta'}{*}{1}$}}{j(\beta',\gamma')=1}}
\Omega_{\gamma'+(s-2)\delta^1}{F\!_{\DOT}} \ +
\sum_{\stackrel{\mbox{\scriptsize
${\gamma'}{\in}{\beta'}{*}{1}$}}{j(\beta',\gamma')>1}}
X_{\gamma'}(j(\beta',\gamma'),{{F\!_{\DOT}}'},L').
$$
And so
$
\lim_{t\rightarrow 0} X_\beta(j(\alpha,\beta),{F\!_{\DOT}},L_t)$
is the cycle
$$
\sum_{\stackrel{\mbox{\scriptsize
${\gamma'}{\in}{\beta'}{*}{1}$}}{j(\beta',\gamma')=1}}
p(\pi^{-1}(\Omega_{\gamma'+(s-2)\delta^1}{{F\!_{\DOT}}'})) \ +
\sum_{\stackrel{\mbox{\scriptsize
${\gamma'}{\in}{\beta'}{*}{1}$}}{j(\beta',\gamma')>1}}
p(\pi^{-1}(X_{\gamma'}(j(\beta',\gamma'),{{F\!_{\DOT}}'},L'))).
$$
We simplify this expression, beginning with the first sum.
Let $\gamma'\in \beta'*1$ satisfy $j(\beta',\gamma')=1$.
Then by Lemma~\ref{lemma:rational_fibration},
$p(\pi^{-1}(\Omega_{\gamma'+(s-2)\delta^1}{F\!_{\DOT}}))$ equals
$ \Omega_{\gamma+(s-2)\delta^1}{F\!_{\DOT}}$,
where $\gamma:=\beta+\delta^1$ is the unique sequence
with $\beta\prec_\alpha\gamma$ and
$j(\alpha,\gamma)=1$.
Consider terms in the second sum, those for which
$\gamma'\in \beta'*1$ with $j(\beta',\gamma')>1$.
Then $p(\pi^{-1}(X_{\gamma'}(j(\beta',\gamma'),{{F\!_{\DOT}}'},L')))$ is
the subvariety of $\Omega_\beta{F\!_{\DOT}}$ consisting of those
$H$ such that
there exists $ K\subset H$ with $\dim K=j$,
$K\in \Omega_{\gamma'}{{F\!_{\DOT}}'}$, and
$\dim K\cap F'_{\gamma'_{j(\beta',\gamma')}}\cap L' \geq 1$.
\smallskip
Let $\gamma := \beta+\delta^{j(\beta',\gamma')}$,
the unique sequence with $\beta\prec_\alpha \gamma$
and $j(\alpha,\gamma)=j(\beta',\gamma')$.
Then, as $\gamma_{j(\alpha,\gamma)}>\beta_j$,
the definition of ${{F\!_{\DOT}}'}$ implies
$F'_{\gamma'_{j(\beta',\gamma')}}=F_{j(\alpha,\gamma)}
\subset F_{\beta_j+1}$.
Since $L'=F_{\beta_j+1}\cap M$, we see that
$F'_{\gamma'_{j(\beta',\gamma')}}\cap L' =
F_{\gamma_{j(\alpha,\gamma)}}\cap M$.
Thus if
$$
H\ \in\ p(\pi^{-1}(X_{\gamma'}(j(\beta',\gamma'),{{F\!_{\DOT}}'},L'))),
$$
then $H\in \Omega_\gamma {F\!_{\DOT}}$ and
$\dim H\cap F_{\gamma_{j(\alpha,\gamma)}}\cap M\geq 1$, so
$H\in X_\gamma(j(\alpha,\gamma), {F\!_{\DOT}},M)$.
The reverse inclusion,
$$
X_\gamma(j(\alpha,\gamma), {F\!_{\DOT}},M)\ \subset\
p(\pi^{-1}(X_{\gamma'}(j(\beta',\gamma'),{{F\!_{\DOT}}'},L'))),
$$
is similar.
This shows that
$\lim_{t\rightarrow 0} X_{\beta}(j(\alpha,\beta),{F\!_{\DOT}},L_t)$
is the cycle
$$
(\ref{sec:proof_of_deformations})\hspace{1.1in}
\sum_{\stackrel{\mbox{\scriptsize${\beta}{\prec_\alpha}{\gamma}$}}
{j(\alpha,\gamma)=1}}
\Omega_{\gamma+ (s-2)\delta^1}{F\!_{\DOT}}
\ +
\sum_{\stackrel{\mbox{\scriptsize${\beta}{\prec_\alpha}{\gamma}$}}
{j(\alpha,\gamma)>1}}
X_\gamma(j(\alpha,\gamma),{F\!_{\DOT}},L).
\hspace{1.4in}
$$
The sets $\{\gamma \,|\, \beta\prec_\alpha\gamma\}$ for
$\beta\in \alpha*r$ partition
the set $\alpha*(r+1)$.
Thus
$$
\lim_{t\rightarrow 0}
Y_{\alpha,r}({F\!_{\DOT}},L_t)\ =
\sum_{\stackrel{\mbox{ \scriptsize ${\gamma}{\in}{\alpha}{*}{(r{+}1)}$}}
{j(\alpha,\gamma)=1}}
\Omega_{\gamma+ (s-2)\delta^1}{F\!_{\DOT}}
\ +
\sum_{\stackrel{\mbox{ \scriptsize ${\gamma}{\in}{\alpha}{*}{(r{+}1)}$}}
{j(\alpha,\gamma)>1}}
X_\beta(j(\alpha,\gamma),{F\!_{\DOT}},M),
$$
which is $Y_{\alpha,r+1}({F\!_{\DOT}},M)$.
\QED
\section{Link to Schensted insertion}\label{sec:schensted}
The set ${[n]\choose m}$ has a partial order, called the Bruhat order:
$\alpha\leq \beta$ if and only if
$\Omega_\beta{F\!_{\DOT}} \subset \Omega_\alpha{F\!_{\DOT}}$.
Combinatorially, this is $\alpha\leq \beta$ if $\alpha_i\leq \beta_i$ for
$1\leq i\leq m$.
We interpret the behavior of the components
$X_\beta(j(\alpha,\beta){F\!_{\DOT}},L)$ of
the intermediate varieties $Y_{\alpha,i-1}({F\!_{\DOT}},L)$
in our proof of Pieri's formula (\S\ref{sec:pieri_proof})
as the branching of a certain subtree of
${[n]\choose m}$ with root $\alpha$.
This tree arises similarly in the combinatorial proof of
Pieri's formula for Schur polynomials using Schensted
insertion given in~\cite{Fulton_tableaux}.
To simplify this discussion, assume further that $n>\alpha_1+b$.
Each rational equivalence of
\S\ref{sec:pieri_proof} is induced by a family $\Sigma_i$ over ${\bf P}^1$
with generic fibre in ${\cal G}_{i-1}$ and special
fibre in ${\cal G}_i$.
The components of cycles in ${\cal G}_{i-1}$ are indexed by
$\beta\in \alpha*(i-1)$, with
$\beta$th component $\Omega_{\beta+(b+1-i)\delta^1}{F\!_{\DOT}}$, if
$j(\alpha,\beta)=1$, and $X_{\beta}(j(\alpha,\beta),{F\!_{\DOT}},L)$
otherwise.
In passing to ${\cal G}_i$ via $\phi_{i}$, the component
$\Omega_{\beta+(b+1-i)\delta^1}{F\!_{\DOT}}$ is unchanged, but reindexed:
$\Omega_{\gamma+(b-i)\delta^1}{F\!_{\DOT}}$, where
$\gamma := \beta+\delta^1$ is the unique sequence with
$\beta\prec_\alpha\gamma$.
By equation (\ref{sec:proof_of_deformations}), the other components
become
$$
\sum_{\stackrel{\mbox{\scriptsize$\beta{\prec_\alpha}\gamma$}}
{j(\alpha,\gamma)=1}}
\Omega_{\gamma+ (b-i)\delta^1}{F\!_{\DOT}} \ +
\sum_{\stackrel{\mbox{\scriptsize$\beta{\prec_\alpha}\gamma$}}
{j(\alpha,\gamma)>1}}X_\gamma(j(\alpha,\gamma),{F\!_{\DOT}},M_i).
$$
Thus the component of the generic fibre of $\Sigma_i$
indexed by $\beta\in \alpha*(i-1)$
becomes a sum of
components indexed by $\{\gamma\,|\, \beta\prec_\alpha\gamma\}$
at the special fibre.
\smallskip
This suggest defining a tree ${\cal T}_{\alpha,b}$ whose branching
represents the `branching' of components of $Y_{\alpha,i-1}({F\!_{\DOT}},L)$ in
these deformations.
Let ${\cal T}_{\alpha,b}\subset {[n]\choose m}$ be the tree with
vertex set $\bigcup\{\alpha*i\,|\,0\leq i\leq b\}$ and covering relation
$\beta\prec_\alpha\gamma$.
This is a tree as $\alpha*i$ is partitioned by the sets
$\{\gamma \,|\, \beta\prec_\alpha\gamma\}$ for $\beta\in \alpha*(i-1)$.
For a decreasing $m$-sequence $\alpha$, let $\lambda(\alpha)$ be the partition
$(\alpha_1-m,\alpha_2-m+1,\ldots,\alpha_m-1)$.
The association $\alpha\longleftrightarrow \lambda(\alpha)$
gives an order isomorphism between the set of decreasing
$m$-sequences and the set of partitions of length at most $m$.
This transfers notions for sequences into
corresponding notions for partitions.
To a (semi-standard) Young tableau $T$ with entries among $1,\ldots,m$,
associate a monomial $x^T$ in the variables
$x_1,x_2\ldots,x_m$:
The exponent of $x_i$ in $x^T$ is the number of occurrences of
$i$ in $T$.
The Schur polynomial $s_\lambda$ is
$\sum x^T$, the sum over all tableaux $T$ of shape $\lambda$.
There is surjective homomorphism from the algebra of Schur polynomials
to the Chow ring of ${\bf G}_mV$ defined by:
$$
s_\lambda \longmapsto \left\{\begin{array}{ll}
[\Omega_\alpha{F\!_{\DOT}}]&\mbox{ \ if \ } \lambda = \lambda(\alpha)
\mbox{ for some }\alpha\in {[n]\choose m}\\
0&\mbox{ \ otherwise}\end{array}\right..
$$
Special Schur polynomials are indexed by partitions $(b,0,\ldots,0)$ with
a single row.
Schensted insertion gives a combinatorial proof of Pieri's
formula, providing a content-preserving bijection between the
set of pairs $(S,T)$ of tableaux where $S$ has shape $\lambda$ and
$T$ has shape $(b,0\ldots,0)$ and the set of all tableaux whose shape is in
$\lambda*b\,$:
Insert the reading word of $T$
into $S$.
The resulting tableau has shape $\mu\in \lambda*b$.
Let $\lambda = \lambda_0,\lambda_1,\ldots,\lambda_b=\mu$ be the sequence
of shapes resulting from the
insertion of successive entries of $T$ into $S$.
Since $T$ is a single row, it is a property
of the insertion algorithm that
$\lambda_i\prec_\lambda \lambda_{i+1}$,
and so this sequence is a chain in the tree ${\cal T}_{\alpha,b}$.
The totality of these insertions for all such pairs of tableaux
gives all chains in ${\cal T}_{\alpha,b}$.
Thus the `branching' of shapes during Schensted insertion
is identical to the branching of components
in the rational equivalences of \S\ref{sec:pieri_proof}.
We feel this relation to combinatorics is one of the more intriguing
aspects of our proof of Pieri's formula.
It leads us to speculate that similar ideas may yield a
geometric proof of the Littlewood-Richardson rule.
|
1996-01-15T06:20:25 | 9601 | alg-geom/9601011 | en | https://arxiv.org/abs/alg-geom/9601011 | [
"alg-geom",
"math.AG"
] | alg-geom/9601011 | Kai Behrend | K. Behrend | Gromov-Witten invariants in algebraic geometry | LaTeX, Postscript file available at
http://www.math.ubc.ca/people/faculty/behrend/gwag.ps | null | 10.1007/s002220050132 | null | null | Gromov-Witten invariants for arbitrary projective varieties and arbitrary
genus are constructed using the techniques from K. Behrend, B. Fantechi: The
intrinsic normal cone.
| [
{
"version": "v1",
"created": "Mon, 15 Jan 1996 04:00:17 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Behrend",
"K.",
""
]
] | alg-geom | \subsection{Introduction}
In \cite{BM} the problem of constructing the Gromov-Witten invariants
of a smooth projective variety $V$ was reduced to defining a `virtual
fundamental class'
$$[\overline{M}_{g,n}(V,\beta)]^{\mbox{\tiny virt}}\in A_{(1-g)(\dim
V-3)-\beta(\omega_V)+n}(\overline{M}_{g,n}(V,\beta))$$
in the Chow group of the algebraic stack
$$\overline{M}_{g,n}(V,\beta)$$
of stable maps of class $\beta\in H_2(V)$ from an $n$-marked prestable
curve of genus $g$ to $V$.
If $g=0$ and $V$ is convex (i.e.\ $H^1({\Bbb P}^1,f^{\ast} T_V)=0$, for all
$f:{\Bbb P}^1\to V$), then $\overline{M}_{0,n}(V,\beta)$ is smooth of the
expected dimension $\dim V-3-\beta(\omega_V)+n$ and the usual
fundamental class
$$[\overline{M}_{g,n}(V,\beta)]$$
will work. This was proved in \cite{BM}.
In this paper we treat the general case using the construction from
\cite{BF}. Recall from [ibid.] that virtual fundamental classes are
constructed using an {\em obstruction theory}, and the {\em intrinsic
normal cone}. The obstruction theory serves to give rise to a vector
bundle stack ${\frak E}$, into which the intrinsic normal cone ${\frak C}$ can be
embedded as a closed subcone stack. The virtual fundamental class is
then obtained by intersecting ${\frak C}$ with the zero section of ${\frak E}$.
In our context, this process works as follows. Let ${\frak M}_{g,n}$ be the
algebraic stack of $n$-marked prestable curves of genus $g$. This is
an algebraic stack, not of Deligne-Mumford (or even finite) type, but
smooth of dimension $3(g-1)+n$. There is a canonical morphism
$$\overline{M}_{g,n}(V,\beta) \to {\frak M}_{g,n},$$ given by forgetting the map,
retaining the curve (but not stabilizing). Then $\overline{M}_{g,n}(V,\beta)
\to {\frak M}_{g,n}$ is an open substack of a stack of morphisms, and as
such has a relative obstruction theory, which in this case is
$\dual{(R\pi_{\ast} f^{\ast} T_V)}$, where $\pi:C\to\overline{M}_{g,n}(V,\beta)$
is the universal curve and $f:C\to V$ is the universal stable
map. Saying that $\dual{(R\pi_{\ast} f^{\ast} T_V)}$is a relative
obstruction theory means that there is a homomorphism
$$\phi:\dual{(R\pi_{\ast} f^{\ast} T_V)} \longrightarrow
L^{\scriptscriptstyle\bullet}_{\overline{M}_{g,n}(V,\beta) / {\frak M}_{g,n}},$$ (where $L^{\scriptscriptstyle\bullet}$ is the
cotangent complex) such that $h^0(\phi)$ is an isomorphism and
$h^{-1}(\phi)$ is surjective.
The homomorphism $\phi$ induces a closed immersion
\[\dual{\phi}:{\frak N}_{\overline{M}_{g,n}(V,\beta) / {\frak M}_{g,n}}\longrightarrow
h^1/h^0(R\pi_{\ast} f^{\ast} T_V)$$ of abelian cone stacks (see \cite{BF})
over $\overline{M}_{g,n}(V,\beta)$, where ${\frak N}$ is the relative intrinsic
normal sheaf. The relative intrinsic normal cone
${\frak C}_{\overline{M}_{g,n}(V,\beta) / {\frak M}_{g,n}}$ is a closed subcone stack of
${\frak N}_{\overline{M}_{g,n}(V,\beta) / {\frak M}_{g,n}}$, and so we get a closed
immersion of cone stacks $${\frak C}_{\overline{M}_{g,n}(V,\beta) /
{\frak M}_{g,n}} \longrightarrow h^1/h^0(R\pi_{\ast} f^{\ast} T_V).$$ Now since
$R\pi_{\ast} f^{\ast} T_V$ has global resolutions (see
Proposition~\ref{egr}), we may intersect ${\frak C}_{\overline{M}_{g,n}(V,\beta) /
{\frak M}_{g,n}}$ with the zero section of the vector bundle stack
$h^1/h^0(R\pi_{\ast} f^{\ast} T_V)$ to get the virtual fundamental class
$[\overline{M}_{g,n}(V,\beta)]^{\mbox{\tiny virt}}$.
The fundamental axioms (see \cite{KM}) Gromov-Witten invariants need
to satisfy to deserve their name are reduced in \cite{BM} to five
basic compatibilities between the virtual fundamental classes. These
follow from the basic properties proved in \cite{BF}. The dimension
axiom, for example, follows from the basic fact that the intrinsic
normal cone always has dimension zero.
We also show that if $V=G/P$, for a reductive group $G$ and a parabolic
subgroup $P$, there is an alternative construction of the virtual
fundamental classes avoiding the intrinsic normal cone. We construct a
cone $C$ in the vector bundle $R^1\pi_{\ast}\O\otimes{\frak g}$ on
$\overline{M}_{g,n}(V,\beta)$, which may then be intersected with the zero
section of $R^1\pi_{\ast}\O\otimes{\frak g}$ to obtain the virtual fundamental
class. This cone $C$ is constructed as the normal cone of an embedding
of $\overline{M}_{g,n}(V,\beta)$ into a certain stack of principal
$P$-bundles (which is smooth, but not of Deligne-Mumford type).
\smallskip
A construction of Gromov-Witten invariants using a cone inside a vector
bundle has also been announced by J. Li and G. Tian. Their methods
differ from ours in that they use analytic methods, including the
Kuranishi map.
\smallskip
Most of this work was done during a stay at the Max-Planck-Institut
f\"ur Mathematik in Bonn, and I would like to take this opportunity to
acknowledge the hospitality and the wonderful atmosphere at the MPI\@.
I would also like to thank Professors G. Harder and Yu.\ Manin for fruitful
discussions about the constructions in this paper.
\subsection{Preliminaries on Prestable Curves}
Let $k$ be a field. We shall work over the category of locally noetherian
$k$-schemes (with the fppf-topology). For a modular graph $\tau$ (see
\cite{BM},
Definition~1.5) let ${\frak M}(\tau)$ denote the $k$-stack of $\tau$-marked prestable
curves (which are defined in \cite{BM}, Definition~2.6).
\begin{lem}\label{soss}
The algebraic $k$-stack $\overline{M}(\tau)$ of stable $\tau$-marked curves
is an open substack of ${\frak M}(\tau)$.
\end{lem}
\begin{pf}
Let ${\cal C}_v\rightarrow{\frak M}(\tau)$ be the universal curve
corresponding to the vertex $v\in V_{\tau}$. Let $\tilde{{\cal C}}_v$ be
the stabilization. Then $\overline{M}(\tau)$ is the substack of ${\frak M}(\tau)$
over which all $p_v:{\cal C}_v\to\tilde{{\cal C}}_v$ are isomorphisms. This is
open because the ${\cal C}_v$ are proper over ${\frak M}(\tau)$.
\end{pf}
Now consider a modular graph $\tau'$ obtained from $\tau$ by adding
some tails. We get an induced morphism of $k$-stacks
${\frak M}(\tau')\to{\frak M}(\tau)$ which simply forgets the markings
corresponding to the tails $S_{\tau'}-S_{\tau}$. If
$S_{\tau'}-S_{\tau}$ has cardinality 1, then
${\frak M}(\tau')\rightarrow{\frak M}(\tau)$ is a smooth
curve, hence representable and smooth of relative dimension 1. So by
induction, ${\frak M}(\tau')\to{\frak M}(\tau)$ is representable and smooth of
relative dimension $\#(S_{\tau'}-S_{\tau})$. By Lemma~\ref{soss} the
same is true for $\overline{M}(\tau')\to{\frak M}(\tau)$.
\begin{prop} The stack ${\frak M}(\tau)$ is a smooth algebraic $k$-stack of
dimension
\[\dim(\tau)=\#S_{\tau}-\#E_{\tau}-3\chi(\tau).\]
\end{prop}
\begin{pf} For the definition of $\dim(\tau)$ and $\chi(\tau)$ see \cite{BM},
Definitions~6.1 and~6.2.
Note that for every point of ${\frak M}(\tau)$ there exists a $\tau'$ as
above such that the induced morphism $\overline{M}(\tau')\to{\frak M}(\tau)$
contains this given point in its image. Thus
$\coprod_{\tau'}\overline{M}(\tau')$ is a presentation of ${\frak M}(\tau)$
showing that ${\frak M}(\tau)$ is algebraic.
\end{pf}
Now let $\tau^s$ be the stabilization of $\tau$. Stabilization defines
a morphism of algebraic $k$-stacks
$$s:{\frak M}(\tau)\longrightarrow\overline{M}(\tau^s).$$
If $\tau'$ is obtained as above by adjoining tails to $\tau$ such that
$\tau'$ is stable, we have a commutative diagram
\[\inversecomtri{\overline{M}(\tau')}{
}{\overline{M}(\phi)}{{\frak M}(\tau)}{s}{\overline{M}(\tau^s).}\]
Here $\phi:\tau'\to\tau^s$ is the canonical morphism of stable modular
graphs. In fact, one may define $s$ locally by using such diagrams.
\begin{prop}
The morphism $s:{\frak M}(\tau)\rightarrow \overline{M}(\tau^s)$ is flat.
\end{prop}
\begin{pf}
This follows by descent since the morphism $\overline{M}(\phi)$ for various
$\phi:\tau'\rightarrow \tau^s$ as above are flat.
\end{pf}
\subsection{The Virtual Fundamental Classes}
Over ${\frak M}(\tau)$ there is a family $({\cal C}_v)_{v\in V_{\tau}}$ of universal
curves, with sections
$x_i:{\frak M}(\tau)\rightarrow{\cal C}_{\partial\t(i)}$. Let ${\cal C}(\tau)\rightarrow{\frak M}(\tau)$
be the curve obtained from $\coprod_{v\in V\t}{\cal C}_v$ by identifying $x_i$ and
$x_j$, for every edge $\{i,j\}\in E\t$. The curve ${\cal C}(\tau)$ has markings
$x_i:{\frak M}(\tau)\rightarrow{\cal C}(\tau)$, for each $i\in S\t$. In fact,
${\cal C}(\tau)$ is a $\tilde{\tau}$-marked prestable curve, where $\tilde{\tau}$ is
the graph obtained from $\tau$ by contracting all edges of $\tau$. Let us
denote
the structure morphism by
\[\pi:{\cal C}(\tau)\longrightarrow{\frak M}(\tau).\] We shall also denote any base change
of $\pi$ by $\pi$.
Now let $V$ be a smooth projective $k$-variety, $(\tau,\beta)$ a stable
$V$-graph and let
$\mathop{\rm Mor}\nolimits_{{\frak M}(\tau)}(\tau,V)$ be the ${\frak M}(\tau)$-space of morphisms
{}from ${\cal C}(\tau)$ to
$V$. Denote the universal morphism by
\[f:{\cal C}(\tau)\times\mathop{\rm Mor}\nolimits_{{\frak M}(\tau)}(\tau,V)\longrightarrow V.\] By
\cite{fgaIV} the stack $\mathop{\rm Mor}\nolimits_{{\frak M}(\tau)}(\tau,V)$ is an algebraic
$k$-stack and the
structure morphism
\[\mathop{\rm Mor}\nolimits_{{\frak M}(\tau)}(\tau,V)\longrightarrow {\frak M}(\tau)\] is
representable.
\begin{prop}
The proper Deligne-Mumford stack $\overline{M}(V,\tau,\beta)$ of stable maps
is an open substack of $\mathop{\rm Mor}\nolimits_{{\frak M}(\tau)}(\tau,V)$.
\end{prop}
\begin{pf}
The set of points where stabilization is an isomorphism is open.
\end{pf}
To define the virtual fundamental class on $\overline{M}(V,\tau,\beta)$ we
consider the morphism $\overline{M}(V,\tau,\beta)\rightarrow{\frak M}(\tau)$ and
denote the relative intrinsic normal cone (see \cite{BF}) by
$${\frak C}(V,\tau,\beta)={\frak C}_{\overline{M}(V,\tau,\beta)/{\frak M}(\tau)}$$ The
intrinsic normal sheaf [ibid.] of $\overline{M}(V,\tau,\beta)$ over
${\frak M}(\tau)$ we shall denote by ${\frak N}(V,\tau,\beta)$.
By the relative version of \cite{BF} Proposition~6.2 we have a perfect
relative obstruction theory [ibid.]
\[\dual{\pi_{\ast}(\dual{e})}:\dual{R\pi_{\ast}(f^{\ast} T_V)}\longrightarrow
L_{\mathop{\rm Mor}\nolimits_{{\frak M}(\tau)}(\tau,V)/{\frak M}(\tau)}^{\scriptscriptstyle\bullet}.\] Restricting to the open
substack $\overline{M}(V,\tau,\beta)$ we get a perfect relative obstruction theory
\[\dual{\pi_{\ast}(\dual{e})}:\dual{R\pi_{\ast}(f^{\ast} T_V)}\longrightarrow
L_{\overline{M}(V,\tau,\beta)/{\frak M}(\tau)}^{\scriptscriptstyle\bullet},\]
which we shall also denote by $E^{\scriptscriptstyle\bullet}(V,\tau,\beta)$.
Thus ${\frak C}(V,\tau,\beta)$ is embedded as a closed subcone stack in the
vector bundle stack $${\frak E}(V,\tau,\beta)=h^1/h^0(R\pi_{\ast} f^{\ast}
T_V).$$
Note that the relative virtual
dimension of $\overline{M}(V,\tau,\beta)$ over ${\frak M}(\tau)$ with respect to the
obstruction theory $\dual{R\pi_{\ast}(f^{\ast} T_V)}$ is equal to
\begin{eqnarray*}
\mathop{\rm rk}\dual{R\pi_{\ast}(f^{\ast} T_V)} & = & \chi(f^{\ast} T_V) \\
& = & \deg f^{\ast} T_V+\dim V\cdot\chi({\cal C}(\tau)) \\
& = & \chi(\tau)\dim V-\beta(\tau)(\omega_V).
\end{eqnarray*} Essential is the following result.
\begin{prop} \label{egr}
Let $(C,x,f)$ be a stable map over $T$ to $V$, where $T$ is a finite
type algebraic $k$-stack. Let $E$ be a vector bundle on $C$. Then
$R\pi_{\ast} E$ has global resolutions, where $\pi:C\rightarrow T$ is the
structure map.
\end{prop}
\begin{pf}
Let $M$ be an ample invertible sheaf on $V$ and let
\[L=\omega_{C/T}(x_1+\ldots+x_n)\otimes f^{\ast} M^{\otimes 3}.\]
By Proposition~3.9 of \cite{BM} the sheaf $L$ is ample on the fibers
of $\pi$. So for sufficiently large $N$ we have that
\begin{enumerate}
\item $\pi^{\ast}\pi_{\ast}(E\otimes L^{\otimes N})\rightarrow E\otimes
L^{\otimes N} $ is surjective,
\item $R^1\pi_{\ast}(E\otimes L^{\otimes N})=0$,
\item for all $t\in T$ we have that $H^0(C_t,L^{\otimes -N}_t)=0$.
\end{enumerate}
Let $$F=\pi^{\ast}\pi_{\ast}(E\otimes L^{\otimes N})\otimes L^{\otimes
-N}$$ and let $H$ be the kernel of the map $F\to E$. Thus we have a
short exact sequence
\[\ses{H}{}{F}{}{E}\] of vector bundles on $C$. Note that for every
$t\in T$ we have
\begin{eqnarray*}
H^0(C_t,F) & = & H^0(C_t,\pi_{\ast}(E\otimes L^{\otimes N})_t\otimes L_t^{\otimes
-N}) \\ & = & H^0(C_t,L_t^{\otimes -N})\otimes\pi_{\ast}(E\otimes
L^{\otimes N})_t \\ & = & 0 \end{eqnarray*} and hence $H^0(C_t,H)=0$,
also. Therefore, $\pi_{\ast} H$ and $\pi_{\ast} F$ are zero and $R^1\pi_{\ast}
H$ and $R^1\pi_{\ast} F$ are locally free. This implies that $$R\pi_{\ast}
E\cong [R^1\pi_{\ast} H\rightarrow R^1\pi_{\ast} F].$$
\end{pf}
As shown in \cite{BF}, by Proposition~\ref{egr} the obstruction theory
$\dual{R\pi_{\ast}(f^{\ast} T_V)}$ gives rise to a virtual fundamental
class
\[[\overline{M}(V,\tau,\beta),\dual{R\pi_{\ast}(f^{\ast} T_V)}]\in
A_{\dim(V,\tau,\beta)}(\overline{M}(V,\tau,\beta)),\] since
\begin{eqnarray*}
\lefteqn{\dim{\frak M}(\tau) + \mathop{\rm rk} \dual{R\pi_{\ast}(f^{\ast} T_V)} }\\ & = &
\chi(\tau)(\dim V-3)-\beta(\tau)(\omega_V)+\#S\t-\#E\t \\
& = & \dim(V,\tau,\beta).
\end{eqnarray*}
(See Definition~6.2 in \cite{BM} for the definition of
$\dim(V,\tau,\beta)$.)
\begin{them} \label{ot} The system of virtual fundamental classes
\[J(V,\tau,\beta)=[\overline{M}(V,\tau,\beta),\dual{R\pi_{\ast}(f^{\ast} T_V)}]\] is an
orientation of $\overline{M}$ over ${\frak G}_s(V)$. If $V$ is convex, on the tree level
sub-category
${\frak T}_s(V)$, we get back the orientation of \cite{BM}, Theorem~7.5.
\end{them}
\begin{pf} If $V$ is convex and $\tau$ a forest, then $R^1\pi_{\ast}(f^{\ast}
T_V)=0$, so that the virtual fundamental class is the usual
fundamental class by \cite{BF} Proposition~7.3. Thus the virtual
fundamental class agrees with the orientation of \cite{BM},
Theorem~7.5. To check that $J$ is an orientation, we need to check the
five axioms listed in
\cite{BM}, Definition~7.1. This shall be done in the next Section.
\end{pf}
\begin{rmk} As shown in \cite{BM}, we get an associated system of
Gromov-Witten
classes for $V$.
\end{rmk}
\subsection{Checking the Axioms} \label{check}
\subsubsection{{\sc Axiom I.} Mapping to a point}
Let $\tau$ be a stable $V$-graph of class zero such that $|\tau|$ is
non-empty and connected. As noted in \cite{BM} Section~7 we have
\[\overline{M}(V,\tau,0)=V\times\overline{M}(\tau)\]
which is obviously smooth over ${\frak M}(\tau)$. In fact, the morphism
$\overline{M}(V,\tau,0)\to{\frak M}(\tau)$ is just the composition
\[V\times\overline{M}(\tau)\longrightarrow\overline{M}(\tau)\longrightarrow{\frak M}(\tau)\]
of projection followed by inclusion. If
$\tilde{\pi}:{\cal C}(\tau)\to\overline{M}{\tau}$ is the universal curve over
$\overline{M}(\tau)$, then ${\cal C}(V,\tau,0)=V\times{\cal C}(\tau)$ and
$\pi:{\cal C}(V,\tau,0)\to\overline{M}(V,\tau,0)$ is identified with
$\mathop{\rm id}\times\tilde{\pi}:V\times{\cal C}(\tau)\to V\times\overline{M}(\tau)$. Hence
\begin{eqnarray*}
R^1\pi_{\ast} f^{\ast} T_V&=&T_V\boxtimes
R^1\tilde{\pi}_{\ast}\O_{{\cal C}(\tau)}\\&=&{\cal T}^{(1)}\end{eqnarray*}
is locally free. So by \cite{BF} Proposition~7.3 we have
\begin{eqnarray*}
J(V,\tau,0) & = & c_{\mathop{\rm rk} R^1\pi_{\ast} f^{\ast} T_V}(R^1\pi_{\ast} f^{\ast}
T_V)\cdot[\overline{M}(V,\tau,0)]\\
& = & c_{g(\tau)\dim V}({\cal T}^{(1)})\cdot[\overline{M}(V,\tau,0)],
\end{eqnarray*}
which is Axiom~I.
\subsubsection{{\sc Axiom II.} Products}
Let $(\sigma,\alpha)$ and $(\tau,\beta)$ be stable $V$-graphs and
denote the `product' by $(\sigma\times\tau,\alpha\times\beta)$.
Note that
\[E^{\scriptscriptstyle\bullet}(V,\sigma\times\tau,\alpha\times\beta) =
E^{\scriptscriptstyle\bullet}(V,\sigma,\alpha)\boxplus E^{\scriptscriptstyle\bullet}(V,\tau,\beta),\]
so by \cite{BF} Proposition~7.4 we have
\begin{eqnarray*}
J(V,\sigma\times\tau,\alpha\times\beta) & = &
[\overline{M}(V,\sigma\times\tau,\alpha\times\beta),
E^{\scriptscriptstyle\bullet}(V,\sigma,\alpha)\boxplus E^{\scriptscriptstyle\bullet}(V,\tau,\beta)] \\
& = & [\overline{M}(V,\sigma,\alpha),E^{\scriptscriptstyle\bullet}(V,\sigma,\alpha)] \times
[\overline{M}(V,\tau,\beta),E^{\scriptscriptstyle\bullet}(V,\tau,\beta)] \\
& = & J(V,\sigma,\alpha)\times J(V,\tau,\beta),
\end{eqnarray*}
which is the product axiom.
\subsubsection{{\sc Axiom III.} Cutting Edges}
Use notation as in \cite{BM}, Section~7, modified as necessary to
avoid confusion. Let $\beta$ denote the $H_2(V)^+$-structure on both
$\sigma$ and $\tau$. Write ${\frak M}={\frak M}(\tau)={\frak M}(\sigma)$. Consider the
cartesian diagram
\[\comdia{\overline{M}(V,\sigma,\beta)}{\overline{M}(
\Phi)}{\overline{M}(V,\tau,\beta)}{ g}{}{ }{{\frak M}\times V}{ \Delta}{ {\frak M}\times
V\times V}\]
of stacks over ${\frak M}$. Let us show that the obstruction theories
$E^{\scriptscriptstyle\bullet}(V,\tau,\beta)$ and $E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)$ are compatible over
$\Delta$ (see \cite{BF}).
Over $\overline{M}(V,\sigma,\beta)$ let us consider the following two
curves. First the curve ${\cal C}={\cal C}(V,\sigma,\beta)$ obtained from the
universal curves $(C_v)_{v\in V_{\sigma}}$ by gluing according to the
edges of $\sigma$. Secondly, we have the curve ${\cal C}'$, which we obtain
from $(C_v)_{v\in V_{\sigma}}$ by gluing according to the
edges of $\tau$. In other words,
${\cal C}'=\overline{M}(\Phi)^{\ast}{\cal C}(V,\tau,\beta)$. Moreover, ${\cal C}$ is obtained
from ${\cal C}'$ by identifying the two sections $x_1$ and $x_2$ of ${\cal C}'$,
corresponding to the edge $\{i_1,i_2\}$ of $\sigma$ which is cut by
$\Phi$. Thus there is a structure morphism $p:{\cal C}'\to{\cal C}$ fitting into
the commutative diagram
\[\comtri{{\cal C}'}{p}{{\cal C}}{\pi'}{\pi}{\overline{M}(V,\sigma,\beta).}\]
We shall also use the diagram
\[\comtri{{\cal C}'}{p}{{\cal C}}{f'}{f}{V,}\]
where $f:{\cal C}\to V$ is the universal map. Let $x=p\mathbin{{\scriptstyle\circ}} x_1=p\mathbin{{\scriptstyle\circ}}
x_2$.
If $E$ is any locally free sheaf on ${\cal C}$, then for $i=1,2$ we have
the evaluation homomorphism
\[u_i:p^{\ast} E\longrightarrow {x_i}_{\ast}{x_i}^{\ast} p^{\ast} E = {x_i}_{\ast}
x^{\ast} E.\]
Applying $p_{\ast}$ we get
\[p_{\ast}(u_i):p_{\ast} p^{\ast} E\longrightarrow x_{\ast} x^{\ast} E.\]
Letting $u=p_{\ast}(u_2)-p_{\ast}(u_1)$ we have a short exact sequence
\[\ses{E}{}{p_{\ast} p^{\ast} E}{u}{x_{\ast} x^{\ast} E}\]
of coherent sheaves on ${\cal C}$. Applying $R\pi_{\ast}$ we get a
distinguished triangle
\[\dt{R\pi_{\ast} E}{}{ R\pi'_{\ast} p^{\ast} E}{R\pi_{\ast}(u)}{x^{\ast} E}{}\]
in $D(\O_{\overline{M}(V,\sigma,\beta)})$. Taking $E=f^{\ast} T_V$ we get the
distinguished triangle
\[\dt{R\pi_{\ast} f^{\ast} T_V}{}{ R\pi'_{\ast} {f'}^{\ast}
T_V}{R\pi_{\ast}(u)}{x^{\ast} f^{\ast} T_V}{},\]
or dually,
\begin{equation}\label{esita}
\dt{x^{\ast} f^{\ast} \Omega_V}{\dual{R\pi_{\ast}(u)}}{\dual{(R\pi'_{\ast}
{f'}^{\ast} T_V)}}{}{\dual{(R\pi_{\ast} f^{\ast} T_V)}}{}.
\end{equation}
Note that we have $E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)=\dual{(R\pi_{\ast} f^{\ast} T_V)}$
and $\overline{M}(\Phi)^{\ast}(E^{\scriptscriptstyle\bullet}(V,\tau,\beta))=\dual{(R\pi'_{\ast}{f'}^{\ast}
T_V)}$. Moreover, $L_{\Delta}^{\scriptscriptstyle\bullet}=\Omega_V[1]{ \mid }{\frak M}\times V$, so
that $g^{\ast} L_{\Delta}=x^{\ast} f^{\ast} \Omega_V[1]$, since $f\mathbin{{\scriptstyle\circ}}
x=p_V\mathbin{{\scriptstyle\circ}} g$. So (\ref{esita}) gives the distinguished triangle
\[{g^{\ast}
L_{\Delta}[-1]}\stackrel{\dual{R\pi_{\ast}(u)}}{\longrightarrow}
{\overline{M}(\Phi)^{\ast}
E^{\scriptscriptstyle\bullet}(V,\tau,\beta)}{\longrightarrow}{E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)}{\longrightarrow}
g^{\ast} L_{\Delta},\]
which we may shuffle around to give
\[\dt{\overline{M}(\Phi)^{\ast}
E^{\scriptscriptstyle\bullet}(V,\tau,\beta)}{}{E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)}{}{g^{\ast}
L_{\Delta}}{\dual{R\pi_{\ast}(-u)}}.\]
Now we have the obstruction morphisms $E^{\scriptscriptstyle\bullet}(V,\tau,\beta)\to
L^{\scriptscriptstyle\bullet}_{\overline{M}(V,\tau,\beta)/{\frak M}}$ and $E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)\to
L^{\scriptscriptstyle\bullet}_{\overline{M}(V,\sigma,\beta)/{\frak M}}$. Moreover, we have the natural
homomorphism $g^{\ast} L_{\Delta}\to L^{\scriptscriptstyle\bullet}_{\overline{M}(\Phi)}$. These give
rise to a homomorphism of distinguished triangles
\[\begin{array}{ccccccc}
\overline{M}(\Phi)^{\ast} E^{\scriptscriptstyle\bullet}(V,\tau,\beta) & \longrightarrow &
E^{\scriptscriptstyle\bullet}(V,\sigma,\beta) & \longrightarrow & g^{\ast} L_{\Delta} &
\stackrel{\dual{R\pi_{\ast}(-u)}}{\longrightarrow} & \overline{M}(\Phi)^{\ast}
E^{\scriptscriptstyle\bullet}(V,\tau,\beta)[1]\\
\downarrow & & \downarrow & &\downarrow & &\downarrow \\
\overline{M}(\Phi)^{\ast} L^{\scriptscriptstyle\bullet}_{\overline{M}(V,\tau,\beta)/{\frak M}} & \longrightarrow &
L^{\scriptscriptstyle\bullet}_{\overline{M}(V,\sigma,\beta)/{\frak M}} & \longrightarrow &
L^{\scriptscriptstyle\bullet}_{\overline{M}(\Phi)} & \longrightarrow & \overline{M}(\Phi)^{\ast}
L^{\scriptscriptstyle\bullet}_{\overline{M}(V,\tau,\beta)/{\frak M}}[1],
\end{array}\]
showing that $E^{\scriptscriptstyle\bullet}(V,\tau,\beta)$ and $E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)$ are
compatible over $\Delta$.
Hence by \cite{BF} Proposition~7.5 we have
\[\Delta^{!} J(V,\tau,\beta)=J(V,\sigma,\beta)\]
which is Axiom~III.
\subsubsection{{\sc Axiom IV.} Forgetting Tails}
Let us deal with the incomplete case, leaving the tripod losing cases
to the reader.
Letting ${\cal C}\to{\frak M}(\tau)$ be the universal curve corresponding to the
vertex $w\in V_{\tau}$ (notation from \cite{BM}, Section~7). We have a
cartesian diagram of algebraic $k$-stacks
\[\comdia{\overline{M}(V,\sigma,\beta)}{\overline{M}(\Phi)}{\overline{M}(V,\tau,\beta)}{
d}{ }{ }{{\cal C}}{}{{\frak M}(\tau).}\]
By \cite{BF} Proposition~7.2 we have
\[\overline{M}(\Phi)^{\ast}
J(V,\tau,\beta) = [\overline{M}(V,\sigma,\beta),\overline{M}(\Phi)^{\ast}
E^{\scriptscriptstyle\bullet}(V,\tau,\beta)].\]
Here the class on the right hand side is the virtual fundamental class
defined by the relative intrinsic normal cone of the morphism $d$ and
the relative obstruction theory $\overline{M}(\Phi)^{\ast}
E^{\scriptscriptstyle\bullet}(V,\tau,\beta)$.
Note that the structure morphism
$\overline{M}(V,\sigma,\beta)\to{\frak M}(\sigma)$ factors through
$d:\overline{M}(V,\sigma,\beta)\to{\cal C}$.
\[\comtri{\overline{M}(V,\sigma,\beta)}{d}{{\cal C}}{}{}{{\frak M}(\sigma)}\]
The morphism $d:\overline{M}(V,\sigma,\beta)\to{\cal C}$ associates to the stable
map $(C,x,h)$ the pair $((C',x'),y)$, where $(C',x',h')$ is the image
of $(C,x,h)$ under $\overline{M}(\Phi)$ and $(C',x')$ the underlying
$\tau$-marked prestable curve. Letting $x_f$ be the section of $C_v$
corresponding to the flag $f$, we obtain $(C',x',h')$ by forgetting
$x_f$ and stabilizing. Moreover, $y$ is the image of the forgotten
section $x_f$ in $C_w'$.
The morphism ${\cal C}\to{\frak M}(\sigma)$ associates to the pair $((C,x),y)$,
where $(C,x)$ is a $\tau$-marked prestable curve and $y$ a section of
$C_w$, the $\sigma$-marked prestable curve $(\tilde{C},\tilde{x})$
obtained as follows. For $v'\not=v$ we have $\tilde{C}_{v'}=C_{w'}$,
where $w'$ is the vertex of $\tau$ corresponding to $v'$. The curve
$(\tilde{C}_v,(\tilde{x}_j)_{j\in F_{\sigma}(v)})$ is obtained from
$((C_w,(x_j)_{j\in F_{\tau}(w)}),y)$ by `prestabilizing' (i.e.\
separating the special points) as in \cite{knudsen}, Definition~2.3.
\begin{lem}\label{loem}
The morphism ${\cal C}\to{\frak M}(\sigma)$ is \'etale.
\end{lem}
\begin{pf}
We will use the formal criterion for \'etaleness. Without loss of
generality assume that $w$ is the only vertex of $\tau$. So let
$((C,x),y)$ be a $\tau$-marked prestable curve with section over the
scheme $T$, $T\to T'$ a square zero extension and $(C',x')$ a
$\sigma$-marked prestable curve over $T'$ such that $(C',x'){ \mid } T$
is the prestabilization of $((C,x),y)$. We may assume that we may
choose additional sections $s$ of $C$ over $T$, making $(C,x,s)$ a
stable marked curve. Then we extend the sections $s$ to sections $s'$
of $C'$ over $T'$. Taking the stabilization of $(C',x',s')$ after
forgetting the section $x_f'$ gives an extension of $((C,x),y)$ to
$T'$ whose prestabilization is $(C',x')$.
\end{pf}
Consider the natural morphism
$p:{\cal C}(V,\sigma,\beta)\rightarrow\overline{M}(\Phi)^{\ast}{\cal C}(V,\tau)$, which
fits into the two commutative diagrams
\[\comtri{{\cal C}(V,\sigma,\beta)}{p}{\overline{M}(\Phi)^{\ast} {\cal C}(V,\tau,\beta)}
{\pi} {\pi'} {\overline{M}(V,\sigma,\beta)}\]
and
\[\comtri{{\cal C}(V,\sigma,\beta)}{p}{\overline{M}(\Phi)^{\ast} {\cal C}(V,\tau,\beta)}
{f}{f'}{\quad V\quad.}\]
Whenever $E$ is a locally free sheaf on
$\overline{M}(\Phi)^{\ast}{\cal C}(V,\tau,\beta)$ the canonical homomorphism
$E\rightarrow p_{\ast} p^{\ast} E$ is an isomorphism. Applying this
principle to $E={f'}^{\ast} T_V$ we get an isomorphism
\[{f'}^{\ast} T_V\longrightarrow p_{\ast} f^{\ast} T_V.\]
Applying $R\pi'_{\ast}$ to this, gives an isomorphism
\[R\pi'_{\ast} {f'}^{\ast} T_V\longrightarrow R\pi_{\ast} f^{\ast} T_V.\]
Noting that $R\pi'_{\ast}{f'}^{\ast} T_V=\overline{M}(\Phi)^{\ast}
E^{\scriptscriptstyle\bullet}(V,\tau,\beta)$ we get an isomorphism
\[\overline{M}(\Phi)^{\ast} E^{\scriptscriptstyle\bullet}(V,\tau,\beta)\longrightarrow
E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)\]
and whence an isomorphism
\[{\frak E}(V,\sigma,\beta)\longrightarrow\overline{M}(\Phi)^{\ast}
{\frak E}(V,\tau,\beta).\]
By \cite{BF} Proposition~7.1 there is a natural isomorphism
\[{\frak C}_{\overline{M}(V,\sigma,\beta)/{\cal C}}\longrightarrow\overline{M}(\Phi)^{\ast}
{\frak C}_{\overline{M}(V,\tau,\beta)/{\frak M}(\tau)}.\]
By Lemma~\ref{loem} we have a canonical isomorphism
\[{\frak C}_{\overline{M}(V,\sigma,\beta)/{\cal C}}\longrightarrow
{\frak C}_{\overline{M}(V,\sigma,\beta)/{\frak M}(\sigma)},\]
such that the diagram
\[\begin{array}{ccc}
{\frak C}_{\overline{M}(V,\sigma,\beta)/{\frak M}(\sigma)} & \stackrel{\textstyle\sim}{\longleftarrow} &
{\frak C}_{\overline{M}(V,\sigma,\beta)/{\cal C}} \\
\cap & & \cap\\
{\frak E}(V,\sigma,\beta) & \stackrel{\textstyle\sim}{\longrightarrow} & \overline{M}(\Phi)^{\ast}
{\frak E}(V,\tau,\beta)\end{array}\]
commutes. So finally, we have
\begin{eqnarray*}
\overline{M}(\Phi)^{\ast} J(V,\tau,\beta) & = & [\overline{M}(V,\sigma,\beta),
\overline{M}(\Phi)^{\ast} E^{\scriptscriptstyle\bullet}(V,\tau,\beta)]\\
& = & [\overline{M}(V,\sigma,\beta),E^{\scriptscriptstyle\bullet}(V,\sigma,\beta)]\\
& = & J(V,\sigma,\beta),
\end{eqnarray*}
which is Axiom~IV.
\subsubsection{{\sc Axiom V.} Isogenies}
Before we start with the proof, some general remarks. Let
$\Phi:\tau\to\sigma$ be an elementary contraction of stable modular
graphs, contracting the edge $\{f,\overline{f}\}$ of $\tau$. Let
$a:\tau\rightarrow\tau'$ and $b:\sigma\rightarrow\sigma'$ be
combinatorial morphisms of modular graphs identifying $\tau$ and
$\sigma$ as the stabilizations of $\tau'$ and $\sigma'$, respectively.
Finally, let $\Phi':\tau'\to\sigma'$ be as follows. We require
$\{a(f),a(\overline{f})\}$ to be an edge of $\tau'$ and
$\Phi':\tau'\to\sigma'$ to be the elementary contraction contracting
the edge $\{a(f),a(\overline{f})\}$. Moreover, we require $\Phi$ to be the
stabilization of $\Phi'$. To fix notation, denote the vertex onto
which $\Phi'$ contracts the edge $\{a(f),a(\overline{f})\}$ by $v_0\in
V_{\sigma'}$ and let $v_1=\partial_{\tau'}(a(f))$ and
$v_2=\partial_{\tau'}(a(\overline{f}))$.
In this situation we get a commutative diagram of algebraic stacks
\[\comdia{{\frak M}(\tau')}{{\frak M}(\Phi')}{{\frak M}(\sigma')}{ s}{}{
s}{\overline{M}(\tau)}{\overline{M}(\Phi)}{\overline{M}(\sigma).}\]
Define ${\frak P}$ to be the fibered product
\[\comdia{{\frak P}}{}{{\frak M}(\sigma')}{ }{}{
s}{\overline{M}(\tau)}{\overline{M}(\Phi)}{\overline{M}(\sigma).}\]
Consider the induced morphism $l:{\frak M}(\tau')\to{\frak P}$.
\begin{prop} \label{evfa}
We have $l_{\ast}[{\frak M}(\Phi')]=s^{\ast}[\overline{M}(\Phi)]$.
\end{prop}
\begin{pf}
First note that ${\frak M}(\tau')$ is irreducible, since ${\frak M}(\tau')$ is a
product of stacks of the form ${\frak M}_{g,n}$, which are irreducible since
the stacks $\overline{M}_{g,n}$ are. Moreover, ${\frak M}(\tau')\to{\frak P}$ is
surjective, so that ${\frak P}$ is irreducible, too.
Secondly, let us remark that there exist non-empty (hence dense) open
substacks ${\frak M}(\tau')^0\subset{\frak M}(\tau')$ and ${\frak P}^0\subset{\frak P}$ such
that $l$ induces an isomorphism $l^0:{\frak M}(\tau')^0\stackrel{\sim}{\rightarrow}{\frak P}^0$. In fact,
let ${\frak M}(\tau')^0$ be the open substack of ${\frak M}(\tau')$ characterized
by the requirement that the marked curves $C_{v_1}$ and $C_{v_2}$ be
stable. To construct ${\frak P}^0$, let ${\frak M}(\sigma')^0$ be the open
substack of ${\frak M}(\sigma')$ where the marked curve $C_{v_0}$ is stable.
Then set $${\frak P}^0=\overline{M}(\tau)\times_{\overline{M}(\sigma)}{\frak M}(\sigma')^0.$$
These facts imply the claim.
\end{pf}
Now let $(\Phi,m):\tau\to \sigma$ be an elementary isogeny of type
forgetting a tail. Let $f\in F_{\tau}$ be the forgotten tail. Let
$a:\tau\to\tau'$ and $b:\sigma\to\sigma'$ be as above. Finally, let
$\Phi':\tau'\to\sigma'$ be the `adjoint' of a combinatorial morphism
of graphs, such that there exists a tail map $m'$, a semigroup $A$ and
$A$-structures on $\tau'$ and $\sigma'$ making $(\Phi',m')$ the
elementary isogeny of stable $A$-graphs forgetting the tail $a(f)$.
Moreover, we require $\Phi$ to be the stabilization of $\Phi'$.
Let ${\frak P}$ be the fibered product
\[\comdia{{\frak P}}{}{{\frak M}(\sigma')}{}{}{s}{\overline{M}(\tau)}{}{\overline{M}(\sigma)}\]
and ${\cal C}$ the universal curve over ${\frak M}(\sigma')$ corresponding to
$w\in V_{\sigma'}$, where $w$ is the vertex of the forgotten tail. (If
$w$ does not exist, i.e.\ if $\Phi'$ is complete, then
${\cal C}={\frak M}(\sigma')$.) As
in the proof of Axiom~IV we have a morphism ${\cal C}\rightarrow{\frak M}(\tau')$
giving rise to a commutative diagram
\[\begin{array}{ccc}
{\cal C} & \stackrel{\pi'}{\longrightarrow} & {\frak M}(\sigma') \\
\ldiag{} & & \\
{\frak M}(\tau') & & \rdiag{s} \\
\ldiag{s} & & \\
\overline{M}(\tau) & \stackrel{\overline{M}(\Phi)}{\longrightarrow} &
\overline{M}(\sigma)
\end{array}\]
and hence to a morphism $l:{\cal C}\rightarrow{\frak P}$.
\begin{prop} \label{evfat}
We have $l_{\ast}[\pi']= s^{\ast} [\overline{M}(\Phi)]$.
\end{prop}
\begin{pf}
Again, ${\cal C}$ and ${\frak P}$ are irreducible and $l$ induces an isomorphism
$l^0:{\cal C}^0\to{\frak P}^0$, where ${\cal C}^0$ is the restriction of ${\cal C}$ to
${\frak M}(\sigma')^0$ and
${\frak P}^0=\overline{M}\times_{\overline{M}(\sigma)}{\frak M}(\sigma')^0$. Here
${\frak M}(\sigma')^0\subset{\frak M}(\sigma')$ is the open substack where $C_w$
is stable.
\end{pf}
Now let us prove Axiom~V.
According to \cite{BM}, Remark~7.2, it suffices to do this
for the case that $\Phi:\tau\to \sigma$ is an elementary isogeny,
$\#J=1$ and $(a_i,\tau_i,\Phi_i)_{i\in I}$ a pullback. So we shall use
notation as in the Definition of pullback (\cite{BM}, Definition~6.10).
We shall include the $H_2(V)^+$-structures on $\sigma'$ and $\tau_i$
($i\in I$) in the notation. They shall be denoted by $\beta'$ and
$\beta_i$ ($i\in I$), respectively.
The underlying graph of $(\tau_i,\beta_i)$ is the same for all $i\in
I$. Let us call it simply $\tau'$.
Let us first consider the case where $\Phi$ is a contraction.
\begin{lem} \label{tdic}
We have a cartesian diagram
\[\comdia{\displaystyle\coprod_{i\in I}\overline{M}(V,\tau',\beta_i)}
{}{\overline{M}(V,\sigma',\beta')} {}{}{}
{{\frak M}(\tau')}{{\frak M}(\Phi')}{{\frak M}(\sigma')}\]
of algebraic $k$-stacks. Moreover,
\[{\frak M}(\Phi')^{!} J(V,\sigma',\beta')=\sum_{i\in
I}J(V,\tau',\beta').\]
\end{lem}
\begin{pf}
The first fact follows immediately from the definitions. The second
fact is \cite{BF} Proposition~7.2.
\end{pf}
Axiom~V will follow by putting Lemma~\ref{tdic} and
Proposition~\ref{evfa} together as follows.
By Lemma~\ref{tdic} all squares in the following diagram are cartesian.
\[\begin{array}{ccccc}
{\displaystyle\coprod_{i\in I}\overline{M}(V,\tau',\beta_i)}&
\stackrel{h}{\longrightarrow} &
\overline{M}(\tau)\times_{\overline{M}(\sigma)}\overline{M}(V,\sigma',\beta') &
\longrightarrow &
\overline{M}(V,\sigma',\beta') \\
\ldiag{} & & \ldiag{} & & \rdiag{a} \\
{\frak M}(\tau') & \stackrel{l}{\longrightarrow} &
\overline{M}(\tau)\times_{\overline{M}(\sigma)}{\frak M}(\sigma') &
\longrightarrow & {\frak M}(\sigma')\\
& \sediag{s} & \ldiag{} & & \rdiag{s} \\
&&\overline{M}(\tau)& \stackrel{\overline{M}(\Phi)}{\longrightarrow} &
\overline{M}(\sigma) \end{array}\]
So we may calculate as follows.
\begin{eqnarray*}
\overline{M}(\Phi)^{!} J(V,\sigma',\beta') & = &
a^{\ast} s^{\ast}[\overline{M}(\Phi)]\cdot J(V,\sigma',\beta')\\
&=& a^{\ast} l_{\ast}[{\frak M}(\Phi')]\cdot J(V,\sigma',\beta')\\
\mbox{(by Proposition~\ref{evfa})}& & \\
&=& h_{\ast}{\frak M}(\Phi')^{!} J(V,\sigma',\beta')\\
&=& h_{\ast}\sum_{i\in I} J(V,\tau',\beta_i)
\end{eqnarray*}
by Lemma~\ref{tdic}. This is the context of Axiom~V.
The case that $\Phi$ is of type forgetting a tail is similar. Instead
of Lemma~\ref{tdic} one uses Axiom~IV, and Proposition~\ref{evfa} is
replaced by Proposition~\ref{evfat}.
This finishes the proof of
Axiom~V and hence the proof of Theorem~\ref{ot}.
\subsection{Homogeneous Spaces}
In the case where $V$ is a generalized flag variety, we can give a
more explicit construction of Gromov-Witten invariants as follows.
\subsubsection{Curves and Principal Bundles}
For a smooth algebraic $k$-group $G$ with Lie algebra ${\frak g}$, we denote
by
\[{\frak H}^1(\tau,G)\]
the $k$-stack of $G$-torsors on $\tau$-marked prestable curves. More
precisely, for a $k$-scheme $T$, the category
${\frak H}^1(\tau,G)(T)$ is the category of pairs $(C,E)$,
where $C=(C_v)_{v\in V\t}$ is a $\tau$-marked prestable curve over
$T$, giving rise to a morphism $f:T\rightarrow{\frak M}(\tau)$, and $E$ is a
$G$-torsor on $f^{\ast}{\cal C}(\tau)$.
Let $(C,E)$ be such a pair. Denote by $E_v$, for $v\in V\t$, the
$G$-bundle induced by $E$ on $C_v$. We call
\[\deg_v(E)=\deg(E_v)=\deg(E_v\times_{G,Ad}{\frak g})\]
the {\em degree }of $E$ at the vertex $v\in V\t$. The degree thus
defines a ${\Bbb Z}_{\geq0}$-structure on $\tau$, which is locally constant
on $T$. (See \cite{BM}, Definition~1.6, for ${\Bbb Z}_{\geq0}$-structures.)
In this way, we get for every ${\Bbb Z}_{\geq0}$-structure $\alpha$ on
$\tau$ an open and closed substack
${\frak H}^1_{\alpha}(\tau,G)
\subset{\frak H}^1(\tau,G)$,
the substack of $G$-torsors of degree $\alpha$.
\begin{prop} \label{h1sd}
For every ${\Bbb Z}_{\geq0}$-structure $\alpha$ on $\tau$ the stack
${\frak H}^1_{\alpha}(\tau,G)$ is an algebraic $k$-stack.
The canonical morphism
\[{\frak H}^1_{\alpha}(\tau,G)\longrightarrow{\frak M}(\tau)\]
is smooth of relative dimension
\[-\chi(\tau)\dim G-\alpha(\tau),\]
where $\alpha(\tau)=\sum_{v \in V\t}\alpha(v)$.
\end{prop}
\begin{pf}
To prove that ${\frak H}^1(\tau,G)$ is algebraic, choose a suitable
embedding $G\hookrightarrow GL_n$ to reduce the case of $G$-bundles to the
case of vector bundles, for which it is well-known. The smoothness of
${\frak H}^1(\tau,G)$ follows from the fact that
$H^2(C,E\times_{G,Ad}{\frak g})=0$ for any $G$-torsor $E$ on a $\tau$-marked
prestable curve $C$. The dimension of ${\frak H}^1(\tau,G)$ is equal to
\begin{eqnarray*}
-\chi(E\times_{G,Ad}{\frak g})& = &
-\deg(E\times_{G,Ad}{\frak g})-\chi(\O_C)\mathop{\rm rk}(E\times_{G,Ad}{\frak g}) \\
&=& -\alpha(\tau)-\chi(\tau)\dim G
\end{eqnarray*}
by Riemann-Roch.
\end{pf}
\subsubsection{Maps to $G/P$}
Now let $G$ be a reductive algebraic group over $k$ and $P$ a
parabolic subgroup of $G$. Then $G/P$ is a smooth projective variety
over $k$. Let us assume for simplicity that $G$ is split over $k$. The
morphism $G\rightarrow G/P$ is a principal $P$-bundle, which we shall
denote by $F$.
Let $U_1,\ldots,U_r$ be the elementary representations of $P$ over
$k$, $V_1,\ldots,V_r$ the corresponding vector bundles on $G/P$ and
$L_1,\ldots,L_r$ their determinants. For every $i=1,\ldots,r$ we have
\[V_i=F\times_PU_i.\]
Note that $\mathop{\rm Pic}\nolimits(G/P)\otimes{\Bbb Q}$ is spanned by $L_1,\ldots,L_r$ and
that $L_1^{-1}\otimes\ldots\times L_r^{-1}$ is ample.
Let $H_2(G/P)^+$ be the set of homomorphisms of abelian groups
$\psi:\mathop{\rm Pic}\nolimits(G/P)\rightarrow{\Bbb Z}$, which are non-negative on ample line
bundles. Then we get a canonical injection
\begin{eqnarray*}
H_2(G/P)^+ & \longrightarrow & ({\Bbb Z}_{\geq0})^r\\
\psi & \longmapsto & (\psi(L_1^{-1}),\ldots,\psi(L_r^{-1})).
\end{eqnarray*}
Using this injection we shall think of classes in $H_2(G/P)^+$ as
$r$-tuples of non-negative integers.
Let ${\frak g}$ and ${\frak p}$ be the Lie algebras of $G$ and $P$,
respectively. We will consider these only as adjoint representations,
ignoring the Lie algebra structure. Denote by ${\frak p}$ also the induced
vector bundle
\[F\times_{P,Ad}{\frak p}\]
on $G/P$. Evaluating on the inverse of its determinant defines a
morphism
\begin{eqnarray*}
\deg:H_2(G/P)^+ & \longrightarrow & {\Bbb Z}_{\geq0} \\
\psi & \longmapsto & \psi(\det({\frak p})^{-1}).
\end{eqnarray*}
This morphism has the property that $\deg(\psi)=0$ implies $\psi=0$.
\begin{rmk}
We have $\det{\frak p}\cong\omega_{G/P}$. In particular,
$\deg\psi=-\psi(\omega_{G/P})$.
\end{rmk}
Now fix an $H_2(G/P)^+$-graph $(\tau,\beta)$, with underlying modular
graph $\tau$. Let $(\tilde{\tau},\tilde{\beta})$ be the $H_2(G/P)^+$-graph
obtained by contracting all edges of $\tau$.
Consider the algebraic $k$-stacks ${\frak H}^1(\tau,G)$ and
${\frak H}^1(\tau,P)$. Since $G$ is reductive, any $G$-torsor
on a curve has degree zero, and thus
\[{\frak H}^1(\tau,G)\longrightarrow{\frak M}(\tau)\]
is smooth of relative dimension
\[-\chi(\tau)\dim G.\]
If $E$ is a $P$-torsor, then associated to $U_1,\ldots,U_r$ we have
associated vector bundles $E_i=E\times_PU_i$, for $i=1,\ldots,r$, and
thus we may associate to $E$ the {\em multi-degree }
\[\mathop{\rm\textstyle \mbox{mult-deg}}\nolimits(E)=(-\deg(E_1),\ldots,-\deg(E_r)).\]
Let ${\frak H}^1_{\beta}(\tau,P)$ be the open and closed
substack of ${\frak H}^1(\tau,P)$ of $P$-torsors whose
multi-degree is equal to $\beta$.
Let $\alpha=\deg\beta$ be the ${\Bbb Z}_{\geq0}$-structure on $\tau$
associated to $\beta$. Then we have
\[{\frak H}^1_{\beta}(\tau,P)
\subset{\frak H}^1_{-\alpha}(\tau,P),\]
so that by Proposition~\ref{h1sd} the stack
${\frak H}^1_{\beta}(\tau,P)$ is smooth of relative dimension
\[-\chi(\tau)\dim P-\beta(\tau)(\omega_{G/P})\]
over ${\frak M}(\tau)$.
Now let ${\frak M}(G/P,\tau,\beta)$ be the stack of maps from $\tau$-marked
prestable curves to $G/P$ of class $\beta$. More precisely, for a
$k$-scheme $T$, the objects of ${\frak M}(G/P,\tau,\beta)(T)$ are triples
$(C,x,f)$, where $(C,x)$ is a $\tau$-marked prestable curve over $T$
and $f=(f_v)_{v\in V\t}$ is a family of $k$-morphisms
$f_v:C_v\rightarrow G/P$ such that
\begin{enumerate}
\item for all $i\in F\t$ we have
$f_{\partial(i)}(x_i)=f_{\partial(j\t(i))}(x_{j\t(i)})$,
\item for all $v\in V\t$ we have ${f_v}_{\ast}[C_v]=\beta(v)$.
\end{enumerate}
\begin{rmk}
If $(\tau,\beta)$ is stable, then $\overline{M}(G/P,\tau,\beta)$ is an open
substack of ${\frak M}(G/P,\tau,\beta)$.
\end{rmk}
Note that $G^{V_{\tilde{\tau}}}$ acts on ${\frak M}(G/P,\tau,\beta)$ as
follows. An element $(g_w)_{w\in V_{\tilde{\tau}}}$ of
$G^{V_{\tilde{\tau}}}$ takes $(C,x,(f_v)_{v\in V\t})$ to
$(C,x,(g_{\phi(v)}\mathbin{{\scriptstyle\circ}} f_v)_{v\in V\t})$, where
$\phi:\tau\rightarrow\tilde{\tau}$ is the structure contraction. Let
$${\frak M}(G/P,\tau,\beta)/G^{V_{\tilde{\tau}}}$$ be the
stack-theoretic quotient of this action. This is an abuse of notation,
since this is a left and not a right action.
We shall let $G^{V_{\tilde{\tau}}}$ act trivially on ${\frak M}(\tau)$
and denote by
\[{\frak M}(\tau)/G^{V_{\tilde{\tau}}}\]
the quotient.
\begin{prop} \label{mcd}
There is a natural cartesian diagram of algebraic $k$-stacks
\[\comdia{{\frak M}(G/P,\tau,\beta)/G^{V_{\tilde{\tau}}}}{\kappa
}{{\frak H}^1_{\beta}(\tau,P)}{\eta}{
}{}{{\frak M}(\tau)/G^{V_{\tilde{\tau}}}}{
\iota}{{\frak H}^1(\tau,G).}\] The vertical maps are
representable, the horizontal maps are local immersions.
\end{prop}
\begin{pf}
This is essentially the fact that a map to $G/P$ is the same as a
principal $P$-bundle with a trivialization of the associated
$G$-bundle.
\end{pf}
The morphism $\iota$ is a local regular immersion with normal bundle
$R^1\pi_{\ast}\O\otimes{\frak g}$. Thus the normal cone $C(\tau,\beta)$ of
${\frak M}(G/P,\tau,\beta)/G^{V_{\tilde{\tau}}}$ in
${\frak H}^1_{\beta}(\tau,P)$ is a cone in
\[{\frak n}(\tau,\beta)=\eta^{\ast} R^1\pi_{\ast}\O\otimes{\frak g}.\]
Pulling back to ${\frak M}(G/P,\tau,\beta)$ and, if $(\tau,\beta)$ is
stable, to $\overline{M}(G/P,\tau,\beta)$ defines
$G^{V_{\tilde{\tau}}}$-equivariant cones, which we shall still
denote $C(\tau,\beta)$, inside equivariant vector bundles, which we
shall still denote by ${\frak n}(\tau,\beta)$.
Let us now assume that $(\tau,\beta)$ is stable. Then we may intersect
the cone $C(\tau,\beta)$ over $\overline{M}(G/P,\tau,\beta)$ with the zero
section of the vector bundle ${\frak n}(\tau,\beta)$, to define a cycle
class
\[J(\tau,\beta)\in A_{\dim(G/P,\tau,\beta)}(\overline{M}(G/P,\tau,\beta))\]
with rational coefficients. Note that $C(\tau,\beta)$ is pure of the
correct dimension, since it is constructed as a normal cone inside a
smooth stack of the correct dimension.
\begin{prop} \label{tmt}
The collection of cycle classes $J(\tau,\beta)$ is the orientation of
$\overline{M}$ over ${\frak G}_s(G/P)$ defined using the intrinsic normal cone.
\end{prop}
\begin{pf}
This follows from \cite{BF} Example~7.6, since
\[\dual{(R\pi_{\ast} f^{\ast} T_{G/P})}=\kappa^{\ast}
L^{\scriptscriptstyle\bullet}_{{\frak H}^1_{\beta}(\tau,P)/{\frak H}^1(\tau,G)}.\]
\end{pf}
\begin{rmk}
As a corollary we get that the orientation classes $J(\tau,\beta)$ are
$G^{V_{\tilde{\tau}}}$-invariant. The same is then true for the
Gromov-Witten invariants.
\end{rmk}
|
1992-02-24T14:08:27 | 9202 | alg-geom/9202015 | en | https://arxiv.org/abs/alg-geom/9202015 | [
"alg-geom",
"math.AG"
] | alg-geom/9202015 | Bert van Geemen | Elisabetta Colombo and Bert van Geemen | Note on curves in a Jacobian | 18 pages, Latex 2.09 | null | null | null | null | For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C
under multiplication by n on JC. We prove that the subgroup generated by these
cycles, in the Chow group modulo algebraic equivalence, has rank at most d-1,
where d is the gonality of C. We also discuss some general facts on the action
of n_* on the Chow groups.
| [
{
"version": "v1",
"created": "Mon, 17 Feb 1992 15:37:10 GMT"
},
{
"version": "v2",
"created": "Mon, 24 Feb 1992 13:08:00 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Colombo",
"Elisabetta",
""
],
[
"van Geemen",
"Bert",
""
]
] | alg-geom | \section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
-.2ex}{1.5ex plus .2ex}{\large\bf}
\def\secdef\empsubsection{\emppsubsection*}{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex
minus
-.2ex}{-1em}{\normalsize\bf}}
\let\emppsubsection\secdef\empsubsection{\emppsubsection*}
\def\empsubsection[#1]#2{\emppsubsection[#1]{#2\unskip}}
\def\secdef\empsubsection{\emppsubsection*}{\secdef\empsubsection{\emppsubsection*}}
\def\subsection{\secdef\empsubsection{\emppsubsection*}}
\makeatother
\newcommand{\newsubsubsection}%
{{\bf\refstepcounter{subsubsection}\thesubsubsection\ \ }
\renewcommand{\Huge}{\huge
\renewcommand{\theequation}{\thesubsubsection}
\makeatletter
\let\c@equation\c@subsubsection
\makeatother
\addtolength{\textheight}{2cm}
\addtolength{\topmargin}{-2cm}
\newcommand{{\msy B}}{{\msy B}}
\newcommand{{\msy C}}{{\msy C}}
\newcommand{{\msy E}}{{\msy E}}
\newcommand{{\msy F}}{{\msy F}}
\newcommand{{\msy H}}{{\msy H}}
\newcommand{{\msy N}}{{\msy N}}
\newcommand{{\msy P}}{{\msy P}}
\newcommand{{\msy Q}}{{\msy Q}}
\newcommand{{\msy R}}{{\msy R}}
\newcommand{{\msy S}}{{\msy S}}
\newcommand{{\msy Z}}{{\msy Z}}
\newcommand{\alpha}{\alpha}
\newcommand{\beta}{\beta}
\newcommand{\epsilon}{\epsilon}
\newcommand{\Gamma}{\Gamma}
\newcommand{\lambda}{\lambda}
\newcommand{\omega}{\omega}
\newcommand{\sigma}{\sigma}
\newcommand{\theta}{\theta}
\newcommand{\Theta}{\Theta}
\newcommand{\approx_{AJ}}{\approx_{AJ}}
\newcommand{\approx_{alg}}{\approx_{alg}}
\newtheorem{Proposition}[subsection]{Proposition}
\newtheorem{Theorem}[subsection]{Theorem}
\newtheorem{Lemma}[subsection]{Lemma}
\newtheorem{Corollary}[subsection]{Corollary}
\newtheorem{Remark}[subsection]{Remark}
\newcommand{{\unskip\nobreak\hfill\hbox{ $\Box$}\par}}{{\unskip\nobreak\hfill\hbox{ $\Box$}\par}}
\begin{document}
\title{Note on curves in a Jacobian}
\author{Elisabetta Colombo and Bert van Geemen}
\date{\mbox{}}
\maketitle
\section{Introduction}
\secdef\empsubsection{\emppsubsection*}{}
For an abelian variety $A$ over ${\msy C}$ and a cycle $\alpha\in CH_d(A)_{\msy Q}$ we
define a subspace $Z_\alpha$ of $CH_d(A)_{\msy Q}$ by:
$$
Z_\alpha:=\langle n_*\alpha:\;n\in{\msy Z} \rangle \quad\subset CH_d(J(C))_{{\msy Q}}.
$$
Results of Beauville imply that $Z_\alpha$ is finite dimensional (cf.\ \ref{short}
below). In case $A=J(C)$, the jacobian of a curve $C$, Ceresa has shown that
the cycle $C-C^-:=C-(-1)_*C\in Z_C$
is not algebraically equivalent to zero for generic $C$ of genus $g\geq 3$,
which implies that for such a curve $\dim_{\msy Q} Z_C\geq 2$.
In this note we investigate the subspace $Z_{W_m}$ of $CH_m(J(C))_{\msy Q}$, with
$W_m$ the image of the $m$-th symmetric power of $C$ in $J(C)$ (so $W_1=C$). To
simplify matters we will actually work modulo algebraic equivalence (rather
than linear equivalence, note that translates of a cycle are algebraically
equivalent).
\secdef\empsubsection{\emppsubsection*}{}
Let $Z_\alpha/\approx_{alg}$ be the image of $Z_\alpha\subset CH_d(A)_{\msy Q}$ in
$CH_d(A)_{{\msy Q}}/\approx_{alg}$.
A d-cycle $\alpha$ is Abel-Jacobi equivalent to zero, $\alpha\approx_{AJ} 0$, if $\alpha$ is
homologically equivalent to zero and its image in $J_d(A)$, the $d$-th
primitive Intermediate Jacobian of $A$, is zero. Recall that any curve of genus
$g$ is a cover of ${\msy P}^1$ for some $d\leq\frac{g+3}{2}$.
\begin{Theorem}\label{thm}
\begin{enumerate}
\item For any abelian variety $A$ and any $\alpha\in CH_d(A)$ we have:
$$
\dim_{\msy Q} \;(Z_\alpha/\approx_{AJ})\leq 2.
$$
\item For any curve of genus $g$ and $1\leq n\leq g-1$ we have:
$$
\dim_{\msy Q} \; (Z_{W_{g-n}}/\approx_{alg})\leq n.
$$
\item For a curve $C$ which is a $d:1$-cover of ${\msy P}^1$ we have:
$$
\dim_{\msy Q} \; (Z_C/\approx_{alg})\leq d-1.
$$
\end{enumerate}
\end{Theorem}
(We prove \ref{thm}.1 in \ref{short}, \ref{thm}.2 in \ref{ct} and
\ref{thm}.3 in \ref{recurs}.)
\secdef\empsubsection{\emppsubsection*}{}
Recall that Ceresa showed that the image of $W_m-W_m^-$ in $J_m(J(C))$ is
non-zero for generic $C$ of genus $g\geq 3$ and $1\leq m\leq g-2$. Therefore
\ref{thm}.1 and \ref{thm}.2 for $n=2$ are sharp.
In case $C$ is hyperelliptic, so $C$ is a 2:1 cover of ${\msy P}^1$, the cycles
$W_m$ and $W_m^-$ are however algebraically equivalent. Therefore \ref{thm}.3
is sharp for hyperelliptic curves ($d=2$) and generic trigonal curves ($d=3$).
In case $C$ is not hyperelliptic nor trigonal (like the generic curve of genus
$g\geq 5$), it would be interesting to know if \ref{thm}.3 is actually sharp.
Note that \ref{thm}.1 implies that one cannot use the Intermediate Jacobian
anymore to derive new algebraic relations among the cycles in $Z_C/\approx_{alg}$.
Recently M.Nori \cite{N} constructed cycles on complete intersections in
${\msy P}^N$ which are Abel-Jacobi equivalent to zero but not algebraically
equivalent to zero. There is thus the possibility that similar cycles can be
found on the Jacobian of a curve of genus $g\geq 5$.
A cycle in $Z_C$, for certain modular curves $C$, was investigated by B.H.Gross
and C.Schoen, \cite{GS}, esp.\ section 5, see also \ref{sgz}.
\secdef\empsubsection{\emppsubsection*}{} The inequalities \ref{thm}.1 and \ref{thm}.2 are consequences of
work of Beauville.
The main part of the paper deals with the proof of \ref{thm}.3. Recall that on
a smooth surface $S$ homological and algebraical equivalence for curves
coincide. Thus if we have map $\Phi:S\rightarrow J(C)$ and relation
$a_1[C_1]+\ldots +a_n[C_n]=0$ in $H^2(S,{\msy Q})$, we get $a_1\Phi_*C_1+\ldots
+a_n\Phi_*C_n\approx_{alg} 0$ in $J(C)$. We use this remark to obtain our result, the
main difficulty is of course to find suitable surfaces, curves in them and maps
to $J(C)$ and to determine Neron-Severi groups ($=Im(CH_1(S)\rightarrow
H^2(S,{\msy Q}))$) of the surfaces involved.
\secdef\empsubsection{\emppsubsection*}{} We are indebted to S.J.Edixhoven and C.Schoen for several helpful
discussions.
\section{General results}
\secdef\empsubsection{\emppsubsection*}{}
The effect of $n_*$ and $n^*$ on the Chow groups has been investigated by
Beauville (\cite{B1}, \cite{B2}), Dehninger and Murre \cite{DM} and K\"unnemann
\cite{K} using the Fourier transform on abelian varieties. Below we summarize
some of their results and derive the finite dimensionality of $Z_\alpha$ as
well as Thm \ref{thm}.1 and \ref{thm}.2.
\secdef\empsubsection{\emppsubsection*}{}
Let $A$ be a $g$-dimensional abelian variety, we will view $B:=A\times A$ as
an abelian scheme over $A$ using the projection on the first factor:
{\renewcommand{\arraystretch}{1.5}
$$
\begin{array}{c} B\\ \downarrow \\ A \end{array} \quad =\quad
\begin{array}{l} A\times A \\ \downarrow \pi_1 \\ A \end{array}
$$
}
For each integer $n$, we have an $s_n\in B(A)$ and a cycle $\Gamma_n\in
CH^g(B)$:
$$
s_n:A\longrightarrow B,\quad a\longmapsto (a,na),\qquad
\Gamma_n:=s_{n*} A\in CH^g(A\times A),
$$
and $\Gamma_n$ is, as the notation suggests, the graph of multiplication by $n$
on $A$. The cycle $\Gamma_n$ defines for each $i$ a map on $CH^i(A)$ which is
just $n_*$:
$$
n_*=\Gamma_n:CH^i(A)\longrightarrow CH^i(A),\quad
\alpha\longmapsto n_*\alpha=\pi_{2*}(\pi_1^*\alpha\cdot \Gamma_n).
$$
Next we introduce a relative version of the Pontryagin product $*$ on the Chow
groups of the $A$-scheme $B$ (cf. \cite{K}, (1.2)). Let $m_B:B\times_A
B\rightarrow B$ be the multiplication map, then:
$$
\alpha*\beta:=m_{B*}(\alpha\times_A \beta),\qquad \alpha,\beta\in CH^*(B).
$$
Since the relative dimension of $\Gamma_n$ over $A$ is $0$, the cycle
$\Gamma_n*\Gamma_m$ lies in $CH^g(B)$ and one has (\cite{K}, (1.3.4)):
$$
\Gamma_n * \Gamma_m =\Gamma_{n+m}\quad \in CH^g(A\times A)_{\msy Q},\qquad{\rm
so}\quad
\Gamma_n=\Gamma_1^{*n},
$$
where we write $\alpha^{*n}$ for the $n$-fold Pontryagin product of a cycle $\alpha$
with $n>0$ and we put $\alpha^{*0}:=\Gamma_0$. In \cite{K} 1.4.1 a generalization
of a theorem of Bloch is proved, which implies:
\begin{equation}\label{BG}
(\Gamma_1-\Gamma_0)^{*(2g+1)}=0\qquad (\in CH^g(B)_{\msy Q}).
\end{equation}
Using the ring structure on $CH^g(B)_{\msy Q}$ with product $*$ one can thus define
the following cycles $\pi_i$, $0\leq i\leq 2g$ in $CH^g(B)_{\msy Q}$:
$$
\pi_i:=\mbox{$\frac{1}{(2g-i)!}$}(\log \Gamma_1)^{*(2g-i)},
$$
with:
$$
\log \Gamma_1:=(\Gamma_1-\Gamma_0)-\mbox{$\frac{1}{2}$}(\Gamma_1-\Gamma_0)^{*2}+\ldots +
\mbox{$\frac{1}{2g}$}(\Gamma_1-\Gamma_0)^{*(2g)}.
$$
Let $\Delta=\Gamma_1\in CH^g(A\times A)_{\msy Q}$ the class of the diagonal, then
(\cite{K}, proof of 3.1.1):
\begin{equation}\label{del}
\Delta=\pi_0+\pi_1+\ldots + \pi_{2g},\qquad
\pi_i\pi_j=\pi_j\pi_i=\left\{\begin{array}{ll} \pi_i&{\rm if}\;\;i=j\\
0&{\rm if}\;\;i\neq j,
\end{array}\right.
\end{equation}
here $\alpha\beta$, for cycles $\alpha, \beta\in CH^g(A\times A)_{\msy Q}$, is their product
as correspondences: $\alpha\beta:=p_{13*}(p_{12}^*\alpha\cdot p_{23}^*\beta)$ with the
$p_{ij}:A^3\rightarrow A^2$ the projection to the $i,j$ factor. Moreover:
\begin{equation}\label{gp}
\Gamma_n\pi_{2g-i}=\pi_{2g-i}\Gamma_n=n^i\pi_{2g-i},
\end{equation}
(one has ${}^t\Gamma_n\pi_i=\pi_i{}^t\Gamma_n=n^i\pi_i$ and ${}^t\pi_i=\pi_{2g-i}$
(\cite{K}, 3.1.1), now take transposes).
\begin{Remark} We sketch how these results can be obtained from \ref{BG}.
Let $M\subset CH^g(B)_{\msy Q}$ be the subspace spanned by the $\Gamma_n$, $n\in{\msy Z}$.
Then \ref{BG} implies that $\dim_{\msy Q}\, M\leq 2g+1$ (use
$\Gamma_i*(\Gamma_1-\Gamma_0)^{*(2g+1)}=0$ for all $i\in {\msy Z}$).
Using the K\"{u}nneth formula, Poincar\'e duality,
one finds that the cohomology class of $\Gamma_n$ in
$$
H^{2g}(A\times A,{\msy Q})=\oplus H^{2g-i}(A,{\msy Q})\otimes H^i(A,{\msy Q})=\oplus
{\rm Hom}(H^i(A,{\msy Q}),H^i(A,{\msy Q}))
$$
is given by (note $\Gamma_n$ induces $n_*$):
$$
[\Gamma_n]=(n^{2g},n^{2g-1},\ldots ,n,1)\in \oplus_{i=0}^{2g}
{\rm Hom}(H^i(A,{\msy Q}),H^i(A,{\msy Q}))
$$
(where $n^{2g-i}$ in the $i$-th component means multiplication by $n^{2g-i}$ on
$H^i(A,{\msy Q})$). Therefore $\dim_{\msy Q}\,M/\sim_{hom} =2g+1$ and $\dim_{\msy Q}\, M=
2g+1$. Thus we have the (surprising) result that homological and linear
equivalence coincide in $M$. We can now define $\pi_i\in M$ by
$$
[\pi_i]:=(0,\ldots ,0,1,0,\ldots ,0)\in \;\oplus_i
{\rm Hom}(H^i(A,{\msy Q}),H^i(A,{\msy Q}))
$$
(1 in $i$-th spot) then \ref{del} and \ref{gp} follow. To express $\pi_i$ as
combination of
$\Gamma_0,\ldots ,\Gamma_{2g}$, note that the (${\msy Q}$-linear) ring homomorphism:
$$
{\msy Q}[X]\longrightarrow M\subset CH^g(B)_{\msy Q},\qquad X^i\longmapsto \Gamma_i=\Gamma_1^{*i}
$$
(with $*$ product on $CH^g(B)_{\msy Q}$) gives an isomorphism
${\msy Q}[X]/(X-1)^{2g+1}\cong M$. Since $\Gamma_n\Gamma_m=\Gamma_{nm}$ (product as
correspondences), $\pi_{2g-i}$ corresponds to a polynomial $f_{2g-i}$ with:
$$
f_{2g-i}(X^n)=n^if_{2g-i}(X)\qquad{\rm so}\quad
f_{2g-i}(X):=c_i(\log X)^i\;{\rm mod}\;(X-1)^{2g+1}
$$
and $c_i\in{\msy Q}$ can be determined with a little more work.
\end{Remark}
\secdef\empsubsection{\emppsubsection*}{} Since $\Delta=\Gamma_1:CH_d(A)\rightarrow CH_d(A)$ is the identity,
we get:
$$
CH_d(A)_{\msy Q}=\oplus CH_d(A)_{(i)},\qquad{\rm with}\quad
CH_d(A)_{(i)}:=\pi_{2g-i}CH_d(A)_{\msy Q}.
$$
and each $CH_d(A)_{(i)}$ is an eigenspace for the multiplication operators:
$$
n_*\alpha=n^i\alpha\qquad\forall n\in {\msy Z},\;\;\forall\alpha \in CH_d(A)_{(i)},
$$
a result which was first obtained by Beauville \cite{B2}. Moreover, he proves:
\begin{equation}\label{bb}
CH_d(A)_{(i)}\neq 0\qquad \Longrightarrow \quad d\leq i\leq d+g
\end{equation}
(and gives sharper bounds for some $d$ in prop.3 of \cite{B2}, it has been
conjectured that $CH_d(A)_{(i)}\neq 0\;\Leftrightarrow\; 2d\leq i\leq d+g$).
\begin{Proposition}\label{short}
Let A be an abelian variety and let $\alpha\in CH_d(A)_{\msy Q}$. Then:
$$
\dim_{\msy Q} Z_\alpha \leq g+1\qquad {\rm and}\quad \dim_{\msy Q} (Z_\alpha/\approx_{AJ}) \leq 2.
$$
Moreover, we have $\pi_i Z_\alpha \subset Z_\alpha$ for all $i$.
\end{Proposition}
{\bf Proof.}$\quad$We write $\alpha$ as a sum of weight vectors
($\alpha_{(i)}:=\pi_{2g-i}\alpha$):
$$
\alpha=\alpha_{(d)}+\alpha_{(d+1)}+\ldots +\alpha_{(g+d)},\qquad {\rm so}\quad
n_*\alpha_{(i)}=n^i\alpha.
$$
Taking $g$ distinct, non-zero, integers $n_j$, the determinant of the matrix
expressing the $n_{i*}\alpha\in Z_\alpha$ in terms of the $\alpha_i$ is a Vandermonde
determinant. Thus each $\alpha_{(i)}=\pi_{2g-i}\alpha\in Z_\alpha$ and $Z_\alpha$ is
spanned by the $\alpha_{(i)}$.
Since $n_*$ acts as $n^{2d}$ on $H^{2g-2d}(A,{\msy Q})$ and as $n^{2d+1}$ on
$J_d(A)$, the space $Z_\alpha/\approx_{AJ}$ is spanned by $\alpha_{(2d)}$ and
$\alpha_{(2d+1)}$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{} In this section, we fix an abel-jacobi map $C\hookrightarrow
J(C)$. Note that $W_d=\mbox{$\frac{1}{d!}$}C^{*d}$ (Pontryagin product on
$J(C)$). Let $\Theta\in CH^1(J(C))$ be a symmetric theta divisor, so $\Theta$ is a
translate of $W_{g-1}$. Let $\Theta^d\in CH_{g-d}(J(C))$ be the $d$-fold
intersection of $\Theta$.
\begin{Proposition}\label{ct}
For any $d,\;1\leq d\leq g-1$, we have $\Theta^d\in Z_{W_{g-d}}$, more
precisely:
$$
\pi_{2g-i}W_{g-d}\neq 0\;\Longrightarrow \; 2(g-d)\leq i\leq 2g-d,
\quad{\rm and}\quad
\pi_{2d} W_{g-d}= \mbox{$\frac{1}{d!}$}\Theta^d.
$$
Moreover we have:
$$
\dim_{\msy Q} \;(Z_{W_{g-d}}/\approx_{alg}) \leq d.
$$
\end{Proposition}
{\bf Proof.}$\quad$We first prove the case $d=g-1$. Since the map
$CH_1(C)_{\msy Q}\rightarrow H^{2g-2}(J(C),{\msy Q})$ factors over
$\pi_{2g-2}CH_1(J(C))_{\msy Q}$ (cf.\ the proof of \ref{short} or even better,
\cite{B2}, p.650) we know that $\pi_{2g-2}C\neq 0$. Then its Fourier transform
$F_{CH}(\pi_{2g-2}C)\in \pi_2CH_{g-1}(J(C))_{\msy Q}\;(\cong NS(J(C))_{\msy Q})$ (cf.
\cite{B2}, prop.1) is non-zero.
For a generic Jacobian $NS_{\msy Q}$ is one dimensional and thus
$\pi_2CH_{g-1}(J(C))_{\msy Q}={\msy Q}\Theta$ (\cite{B1}, prop.5 and \cite{B2}, prop.1).
By specializing, we have for all curves that $F_{CH}(\pi_{2g-2}C)\in {\msy Q}\Theta$.
From \cite{B1}, prop.5 we have
$\Theta=-F_{CH}(\mbox{$\frac{1}{(g-1)!}$}\Theta^{g-1})$,
so for a nonzero constant $c$:
$$
F_{CH}(\pi_{2g-2}C)=cF_{CH}(\mbox{$\frac{1}{(g-1)!}$}\Theta^{g-1})$$
and, by using $F_{CH}^2=(-1)^g\Gamma_{-1}$ and comparing cohomology classes, we
get:
$$
\pi_{2g-2}C=\mbox{$\frac{1}{(g-1)!}$}\Theta^{g-1}.
$$
Next we recall that $\pi_{2g-1}CH_1(A)=0$ for any abelian variety $A$
(\cite{B2}, prop.3), thus:
$$
C=C_{(2)}+\ldots + C_{g+1} \qquad {\rm with~} C_{(i)}=\pi_{2g-i}C,
$$
so $n_*C_{(i)}=n^iC_{(i)}$ and $C_{(2)}=\mbox{$\frac{1}{(g-1)!}$}\Theta^{g-1}$.
Therefore
\begin{equation} \label{W}
W_{g-d}=\mbox{$\frac{1}{(g-d)!}$}C^{*(g-d)}=
\mbox{$\frac{1}{(g-d)!}$}C_{(2)}^{*(g-d)}+Y
\end{equation}
with $Y$ a sum of cycles $C_{(i_1)}*\ldots *C_{(i_{g-d})}$ with all $i_j\geq 2$
and at least one $>2$. Therefore, using
$n_*(U*V)=(n_*U)*(n_*V)$, we get:
$\pi_{2d}W_{g-d}=\mbox{$\frac{1}{(g-d)!}$}C_{(2)}^{*(g-d)}$ and
$\pi_jW_{g-d}=0$ for $j>2d$.
By \cite{B1}, corr.2 of prop.5, $\Theta^d$ lies in the subspace spanned by
$(\Theta^{g-1})^{*(g-d)}$, so $C_{(2)}^{*(g-d)}=c\Theta^d$ for a non-zero
$c\in{\msy Q}$ and taking cohomology classes one finds $c=1$.
Finally, using \ref{bb} and \ref{W} we can write
$$
W_{g-d}=\alpha_{(2g-2d)}+\ldots +\alpha_{(2g-d)},\qquad{\rm with}\quad
\alpha_{(i)}:=\pi_{2g-i}W_{g-d},
$$
and since $\alpha_{(2g-d)}\approx_{alg} 0$ by \cite{B2}, prop.4a, we get $\dim_{\msy Q}
(Z_{W_{g-d}}/\approx_{alg}) \leq d$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{}\label{sgz}
In the paper \cite{GS} the following cycle (modulo $\approx_{alg}$) is considered:
$$
Z:=3_*C-3\cdot 2_*C +3C = (\Gamma_1-\Gamma_0)^{*3}C.
$$
This cycle is related to Ceresa's cycle in the following way:
\begin{Proposition}
We have:
$$
\pi_{2g-2}(Z)=\pi_{2g-2}(C-C^-)=0\qquad{\rm and}\quad
\pi_{2g-3}(Z)=3\pi_{2g-3}(C-C^-).
$$
In particular, $Z$ is abel-jacobi equivalent to $3(C-C^-)$.
\end{Proposition}
{\bf Proof.}$\quad$As we saw before (proof of \ref{ct}) we can write:
$$
C=C_{(2)}+C_{(3)}+\ldots \qquad {\rm with}\;C_{(i)}:=\pi_{2g-i}C.
$$
Since $n_*C_{(i)}=n^iC_{(i)}$ we have:
$$
C-C^-=2( C_{(3)}+0+C_{(5)}+\ldots ),\qquad Z=6C_{(3)}+36C_{(4)}+\ldots.
$$
Therefore $\pi_{2g-2}(C-C^-)=3\pi_{2g-2}(Z)=0$, so both cycles are
homologically equivalent to zero, and
$3\pi_{2g-3}(C-C^-)=\pi_{2g-3}(Z)=6C_{(3)}$. Since the abel-jacobi map factors
over $\pi_{2g-3}CH_1(J(C))$ we have that
$Z\approx_{AJ} 3(C-C^-)$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\section{Proof of 1.3.3}
\secdef\empsubsection{\emppsubsection*}{}
Let $C$ be a generic $d$:1 cover of ${\msy P}^1$, and denote by $g^1_d$ the
corresponding linear series.
For an integer $n$, $0<n<d$ we define a curve $G_n$ in the $n$-fold
$C^{(n)}$=Sym$^n(C)$:
$$
G_n=G_n(g^1_d)=\{x_1+x_2+\ldots+x_n\in C^{(n)}:\quad
x_1+x_2+\ldots+x_n<g^1_d\},
$$
see \cite{ACGH}, p.342 for the definition of the scheme-structure on $G_n$. We
define a surface by:
$$
S_n=S_n(g^1_d)=G_n\times C.
$$
\begin{Proposition}\label{NS}
Let $C$ be a generic $d$:1-covering of ${\msy P}^1$ of genus $g\geq 1$.
Then $S_n$ is a smooth, irreducible surface and
$$
\dim_{\msy Q} NS(S_n)_{\msy Q}=3.
$$
\end{Proposition}
{\bf Proof.}$\quad$Since $C$ is generic, the curve $G_n$ is irreducible
(consider the monodromy representation) and for a simple covering the
smoothness of $G_n$ follows from a local computation. Therefore also $S_n$ is
smooth and irreducible.
Note that for a product of 2 curves $S_n=G_n\times C$ we have that
$$
NS(S_n)_{\msy Q}={\msy Q} C\oplus {\msy Q} G_n\oplus {\rm Hom}(J(C),J(G_n))_{\msy Q}.
$$
Using Hodge structures, the last part are just the Hodge-cycles in
$H^1(C,{\msy Q})\otimes H^1(G_n,{\msy Q})$.
Since $J(C)$ and $Pic^0(C^{(n)})$ are isogeneous (their $H^1({\msy Q})$'s are
isomorphic Hodge structures), composing such an isogeny with the pull-back
$Pic^0(C^{(n)})$ $\rightarrow$ $J(G_n)$, we have a non-trivial map
$J(C)\rightarrow J(G_n)$. To show that $\dim_{\msy Q} {\rm Hom}(J(C),J(G_n))_{\msy Q}\leq
1$ we use a degeneration argument. First we recall that the Jacobian of $C$ is
simple and has in fact $End (J(C))={\msy Z}$.
In case $g=2$ this is clear since any genus 2 curve can be obtained by
deformation from a given $d$:1 cover and $C$ is generic. In case $g=3$ one
reasons similarly, taking the necessary care for the hyperelliptic curves.
In case $g(C)=g>3$, assume first that there is an elliptic curve in $J(C)$.
Specializing $C$ to a reducible curve with two components $C'$ and $C''$, both
of genus $\geq 2$, and themselves generic $d$:1-covers we obtain a
contradiction by induction. Assume now that there is no elliptic curve in
$J(C)$, then we specialize $C$ to $E\times C'$, with $E$ an elliptic curve and
$C'$ a generic $d$:1-covering of genus $g-1$. By induction again, $J(C')$ is
simple, which contradicts the existence of abelian subvarieties in $J(C)$. We
conclude that $J(C)$ is simple. Thus $End_{\msy Q} (J(C))$ is a division ring and
specializing again we find it must be ${\msy Q}$.
We may assume that the $g^1_d$ exhibits $C$ as a simple cover of ${\msy P}^1$, then
$G_n$ is a cover of ${\msy P}^1$ (of degree $({}^d_n)$) with only twofold
ramification points.
Letting two branch points coincide, we obtain a curve $\overline{C}$ with a
node, the normalization $C'$ of $ \overline{C}$ has genus $g-1$ and is again
exhibited as a generic $d:1$ cover of ${\msy P}^1$.
The curve $G_n$ acquires $({}^{d-2}_{n-1})$ nodes (since twice that number of
branch points coincide pairwise) and the normalization of that curve,
$\overline{G}_n$, is $G'_n$, the curve obtained from the $g^1_d$ on $C'$.
The (generalized) Jacobians of $\overline{C},\,\overline{G}_n$ are extensions
of the abelian varieties (of $\dim \geq 1$) $J(C'),\,J(G'_n)$ by multiplicative
groups.
Since the number of simple factors of $J(G_n')$ which are isogeneous to $J(C')$
is greater then or equal to the number of simple factors of $J(G_n)$ which are
isogeneous to $J(C)$, and $J(C),\, J(C')$ have $End_{\msy Q}={\msy Q}$ it follows:
$$
\dim_{\msy Q} {\rm Hom}(J(C),J(G_n))_{\msy Q}\; \leq \dim_{\msy Q} \;
{\rm Hom}(J(C'),J(G'_n))_{\msy Q}.
$$
Therefore it suffices to show that for a generic elliptic curve $C=E$ and a
generic $g^1_d$ on $E$ we have $\dim_{\msy Q} {\rm Hom}(E,J(E_n))_{\msy Q}\leq 1$ (with
$E_n=G_n(g^1_d)$ and $C=E$).
We argue again by induction, using degeneration. Note that for any $C$,
$G_1(g^1_d)\cong G_{d-1}(g^1_d)\cong C$. Since a generic elliptic curve has
${\rm Hom}(E,E)={\msy Z}$ the statement is true for $d\leq 3$ (and any $0<n<d$) and
since $E_1\cong E$ it is also true for $n=1$ (and any $d\geq 2$).
We fix $E$, a generic elliptic curve, but let the $g^1_d$ acquire a base point
$y\in E$. Then $E_n(g^1_d)$, $n\geq 2$, becomes a reducible curve $\bar{E_n}$,
having two components $E_n'=E_n(g^1_{d-1})$ and $E'_{n-1}=E_{n-1}(g^1_{d-1})$
which meet transversally in $N=({}^{d-2}_{n-1})$ points. Indeed, let the
divisor of the $g^1_{d-1}$ containing $y$ be $y+y_1+\ldots +y_{d-2}$, then the
points $y_{i_1}+\ldots +y_{i_{n-1}}\in E'_{n-1}$ and
$y+y_{i_1}+\ldots +y_{i_{n-1}}\in E'_{n}$ are identified.
Thus $J(\overline{E}_n)$ is an extension of $J(E'_n)\times J(E'_{n-1})$ by the
multiplicative group $({\msy C}^*)^{N-1}$, in fact there is an exact sequence:
$$
1\longrightarrow {\msy C}^*\stackrel{\Delta}{\longrightarrow} ({\msy C}^*)^N
\longrightarrow J(\overline{E}_n) \stackrel{\pi}{\longrightarrow} J(E'_n)\times
J(E'_{n-1}) \longrightarrow 0,
$$
where $\Delta$ is the diagonal embedding. We will write $\pi=(\pi_0,\pi_1)$.
We will prove that
$$
{\rm Hom}(E,J(\overline{E}_n))_{\msy Q} \longrightarrow {\rm Hom}(E,J(E'_n))_{\msy Q},
\qquad\tilde\phi\mapsto \pi_0\circ \tilde\phi
$$
is injective. By induction we may assume that $\dim{\rm Hom}(E,J(E'_n))_{\msy Q}
\leq 1$, thus also $\dim_{\msy Q} {\rm Hom}(E,J(\overline{E}_n))_{\msy Q}\leq 1$ and
since $\dim_{\msy Q} {\rm Hom}(E,J({E}_n))_{\msy Q} $ $\leq \dim_{\msy Q} {\rm Hom}
(E,J(\overline{E}_n))_{\msy Q}$ the assertion on the rank of the Neron-Severi group
follows.
Assume that $\tilde\phi\neq 0$, but $\pi_0\circ\tilde\phi=0$. Then
$\tilde\phi(E)\subset \bar{J}:=\pi_1^{-1}(J(E'_{n-1}))$. Pulling back this
$({\msy C}^*)^{N-1}$-bundle $\bar{J}$ over $J(E_{n-1}')$ to $E$ along
$\pi_1\circ\tilde\phi$, we obtain a $({\msy C}^*)^{N-1}$-bundle $\tilde{E}$ over
$E$.
The map $\tilde\phi:E\rightarrow \bar{J}$ gives a section of
$\tilde{E}\rightarrow E$, thus $\tilde{E}$ is a trivial
$({\msy C}^*)^{N-1}$-bundle.
We show that this gives the desired contradiction.
{\renewcommand{\arraystretch}{1.5}
\vspace{\baselineskip}
$$
\begin{array}{cccccccc}
1&\rightarrow & ({\msy C}^*)^{N-1} & \rightarrow &\bar{J}&\rightarrow &
J(E'_{n-1})&\rightarrow 0\\
\uparrow& &\uparrow & &\uparrow & &
\uparrow \pi_1\tilde\phi& \\
1&\rightarrow &({\msy C}^*)^{N-1}&\rightarrow &\tilde{E}&\rightarrow &
E&\rightarrow 0
\end{array}
$$
}
For any distinct $P,Q\in E'_n$ which are in $E'_n\cap E'_{n-1}$, the
$({\msy C}^*)^{N-1}$-bundle $\bar{J}$ over $J(E_{n-1}')$ has a quotient
${\msy C}^*$-bundle $\bar{J}_{PQ}$ whose extension class is $P-Q\in
Pic^0(E'_n)=Ext^1(J(E'_n),{\msy C}^*)$. By induction, there is a `unique' map in
${\rm Hom}(J(E'_{n-1}),E)$ which must thus be induced by $x_1+\ldots
+x_{n-1}\mapsto x_1+\ldots +x_{n-1}-(n-1)p,\; E'_{n-1}\rightarrow E$ for some
$p\in E$. Taking $P=y_1+\ldots +y_{n-1}$ and $Q=y_2+\ldots +y_{n-1}+y_n$, the
pull-back of $P-Q$ to $E=Pic^0(E)$ is $y_1-y_n$. Choosing the degeneration
suitably we may assume that $y_1-y_n$ is not a torsion point on $E$ and thus
$\bar{J}_{PQ}$ has a nontrivial pull-back to $E$, contradicting the fact that
$\tilde{E}$, the pull-back of $\bar{J}$ to $E$, is trivial.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{}
We define a curve $\Delta_n$ in the surface $S_n=G_n\times C$:
$$
\Delta_n:=\{(x_1+x_2+\ldots+x_n,p)\in S_n:\;p\in\{x_1,\ldots ,x_n\} \}
$$
and for $n+1\leq d$, another curve $H_n$ in $S_n$:
$$
H_n=\{(x_1+x_2\ldots+x_n,p)\in S_n:\;\; x_1+x_2+\ldots+x_n+p<g^1_d\}.
$$
Finally we define the map:
\begin{equation}
\Phi^n_{l,k}:S_n\longrightarrow J(C),
\end{equation}
$$
(x_1+x_2\ldots+x_n,p)\longmapsto l(x_1+x_2+\ldots+x_n)+kp-D_{nl+k},
$$
where $D_{nl+k}$ is some divisor of degree $nl+k$ on $C$. Finally we denote by
$$
u_n:C^{(n)}\longrightarrow J(C),\qquad D\longmapsto D_n,
$$ with $D_n$ some divisor of degree $n$, the Abel-Jacobi map on $C^{(n)}$. We
simply write $C$ for $u_{1*}C$.
\begin{Proposition} Let $C$ be a generic $d$:1 cover of ${\msy P}^1$ of genus $g\geq
1$. Then:
\begin{enumerate}
\item
$$
NS(S_n)_{\msy Q}=\langle C,G_n,\Delta_n\rangle.
$$
\item In $NS(S_n)$ we have:
\begin{equation}\label{eq2}
H_n=({}^d_{n})C+dG_{n}-\Delta_n.
\end{equation}
\item The image of $G_n$ in $J(C)$ is a combination of $C$, $2_*C,\ldots
,n_*C$:
\begin{equation}\label{eq3}
u_{n*}G_n\approx_{alg} ({}^d_{n-1})C-\mbox{$\frac{1}{2}$}({}^d_{n-2})2_*C+\ldots+
\mbox{$\frac{(-1)^{n-1}}{n}$}({}^d_0)n_*C.
\end{equation}
\end{enumerate}
\end{Proposition}
{\bf Proof.}$\quad$
The first part follows from the previous proposition and from the fact that in
the proof of \ref{eq2} we will see that $H_n$ can be uniquely expressed as a
combination of $C$, $G_n$ and $\Delta_n$.
The proof of 2 and 3 is by induction. In the case $n=1$, so $G_1\cong C$,
\ref{eq3} is trivial and for \ref{eq2} we have $S_1=C\times C$ and we must
show $H_1=d(C+G_1)-\Delta$. Note the following intersection numbers:
$$
H_1\cdot C=H_1\cdot G_1=d-1,\quad H_1\cdot \Delta=2(g+d-1)
$$
(use that the $g^1_d$ exhibits $C$ as $d$:1 cover of ${\msy P}^1$ with $2(g+d-1)$
simple ramification points), which imply the result.
Suppose that \ref{eq2} and \ref{eq3} are true for all $k<n$.
Note that $u_{n}(G_n)=\Phi^{n-1}_{1,1}(H_{n-1})$ and that map
$\Phi^{n-1}_{1,1}$ restricted to $H_{n-1}$ is $n$:1 so, using \ref{eq2} for
$n-1$, we have in $J(C)$:
\begin{equation}\label{eq4}
G_n\approx_{alg} \mbox{$\frac{1}{n}$}\left(
({}^d_{n-1})C+dG_{n-1}-(\Phi^{n-1}_{1,1})_*\Delta_{n-1}\right).
\end{equation}
where we write $G_k$ for $u_{k*}G_k$.
Next we observe that
\begin{equation}\label{fies}
(\Phi^{k}_{1,l})_*\Delta_k=(\Phi^{k-1}_{1,l+1})_*H_{k-1}
\qquad{\rm and}\qquad
(\Phi^{k-1}_{1,l+1})_*C=(l+1)_*C.
\end{equation}
Using \ref{eq2} in the cases $n-2,\,n-3,\ldots, 1$ we get:
\begin{eqnarray}\nonumber
(\Phi^{n-1}_{1,1})_*\Delta_{n-1}& \approx_{alg} &(\Phi^{n-2}_{1,2})_*H_{n-2}\\ \nonumber
&\approx_{alg} & ({}^d_{n-2})2_*C+dG_{n-2}-(\Phi^{n-3}_{1,3})_*H_{n-3} \\ \label{eq5}
&\approx_{alg} & ({}^d_{n-2})2_*C-({}^d_{n-3})3_*C+
d(G_{n-2}-G_{n-3})+(\Phi^{n-4}_{1,4})_*H_{n-4}\\ \nonumber
&\approx_{alg} & ({}^d_{n-2})2_*C-({}^d_{n-3})3_*C+\ldots+
(-1)^{n-2}({}^d_{0})n_*C+\\ \nonumber
& &+d(G_{n-2}-G_{n-3}\ldots+(-1)^{n-2}G_2+(-1)^{n-1}G_1)
\end{eqnarray}
(note that $(\Phi^{1}_{1,n-1})_*H_{1}\approx_{alg}({}^d_1)(n-1)_*C+dG_1-({}^d_0)n_*C$).
Substitute the expression for $(\Phi^{n-1}_{1,1})_*\Delta_{n-1}$ from \ref{eq5}
into \ref{eq4}:
\begin{eqnarray} \nonumber
\;\;G_n &\hbox{\rlap{$\approx_{alg}$}}& \\ \nonumber
& & \!\mbox{$\frac{1}{n}$}\left(({}^d_{n-1})C-({}^d_{n-2})2_*C\ldots +
(-1)^{n-2}({}^d_{1})(n-1)_*C+(-1)^{n-1}({}^d_{0})n_*C \right. \\ \label{Gn}
& &+\left. d(G_{n-1}-G_{n-2}+\ldots +(-1)^{n-3}G_2+(-1)^{n-2}G_1\right).
\end{eqnarray}
Using the formula \ref{eq3} for $G_k$, $k=1,2,\ldots ,n-1$ and the relation
$\sum_{k=0}^l(-1)^k({}^d_{l-k})=({}^{d-1}_l)$, (cf. \ref{bin1})
we get:
\begin{eqnarray}\label{Gs}
& & G_{n-1}-G_{n-2}+\ldots+(-1)^{n-2}G_1 \\ \nonumber
&\approx_{alg} &({}^{d-1}_{n-3})C-\mbox{$\frac{1}{2}$}({}^{d-1}_{n-4})2_*C+\ldots+
(-1)^{n-3}\mbox{$\frac{1}{n-1}$}({}^{d-1}_0)(n-1)_*C.
\end{eqnarray}
Substituting this in formula \ref{Gn} and using the identity:
$$
\mbox{$\frac{1}{n}$}[({}^d_{n-k})+\mbox{$\frac{d}{k}$}({}^{d-1}_{n-k+1})]=
\mbox{$\frac{1}{k}$}({}^d_{n-k})
$$
we obtain \ref{eq3}.
To obtain \ref{eq2}, note that by (1), there are $a,b,c\in {\msy Q}$ such that:
\begin{equation}\label{eq6}
H_{n}=aC+bG_n+c\Delta_n.
\end{equation}
To find them we compute the homology classes of the curves in \ref{eq6} in
$H_2(J(C),{\msy Q})$ and the intersection numbers of $H_n$ with $C$ and $G_n$.
For the generic curve $C$, the homology class $[B]$ of a curve $B$ in $J(C)$ is
a multiple of the class $[C]$ of $C$ and this multiple is
$\mbox{$\frac{1}{g}$}\Theta\cdot B$. We apply this to $B=G_n$. Using the map
$u:C^{(n)}\rightarrow J(C)$, we have $u_*(G_n)\cdot \Theta=u_*(G_n\cdot \theta)$
with $\theta$ the pull-back of $\Theta$ to $C^{(n)}$.
The homology class of $G_n$ in $H_2(C^{(n)},{\msy Q})$ is:
\begin{equation}
G_n=\sum_{k=0}^{n-1}
({}^{d-g-1}_k)\frac{x^k\theta^{n-k-1}}{(n-1-k)!},\qquad{\rm so}\quad
G_n\cdot\theta=\sum_{k=0}^{n-1} ({}^{d-g-1}_k)\frac{x^k\theta^{n-k}}{(n-1-k)!},
\end{equation}
cf.\ formula (3.2) on pag.342 of \cite{ACGH} (here $x$ is the class of the
divisor $C^{(n-1)}$ in $C^{(n)}$). From pag.343 of \cite{ACGH} one has:
$ u_*(x^{n-i}\theta^i)=({}^g_i)i![\Theta^g]/g! $
and since $[\Theta^g]/g!$ is the positive generator of $H^{2g}(J(C),{\msy Z})$, we
find:
\begin{eqnarray}
G_n\cdot\Theta &=&
\sum_{k=0}^{n-1} ({}^{d-g-1}_k)({}^g_{n-k})\frac{(n-k)!}{(n-1-k)!}\\ \nonumber
&=& g\sum_{k=0}^{n-1} ({}^{d-g-1}_k)({}^{g-1}_{n-k-1})\\ \nonumber
&=& g({}^{d-2}_{n-1}).
\end{eqnarray}
Therefore:
\begin{equation}
[G_k]=({}^{d-2}_{k-1})[C].
\end{equation}
Next we compute the homology class of $\Delta_n$. We use \ref{fies} for
$k=n,\;l=1$: $
\left(\Phi^n_{1,1}\right)_*\Delta_n=\left(\Phi^{n-1}_{1,2}\right)_*H_{n-1}
$.
By induction, similar to \ref{eq5} and using \ref{Gs} (with $n-1$ replaced by
$n$) we have:
\begin{eqnarray}
\left(\Phi^n_{1,1}\right)_*\Delta_n & \approx_{alg} &
d\left(G_{n-1}-G_{n-2}+\ldots(-1)^{n-3}G_2+(-1)^{n-2}G_1\right)+\\ \nonumber
&&+({}^d_{n-1})2_*C-({}^d_{n-2})3_*C+\ldots+(-1)^{n-1}({}^d_0)(n+1)_*C\\
\nonumber
&\approx_{alg} &
d\sum_{k=1}^{n-1}(-1)^{k-1}\mbox{$\frac{1}{k}$}({}^{d-1}_{n-1-k})k_*C+
\sum_{k=2}^{n+1} (-1)^{k} ({}^{d}_{n+1-k})k_*C.
\end{eqnarray}
Taking the homology classes in $J(C)$ we get:
$$
[\Phi^n_{1,1}(\Delta_n)]=\left(d\sum_{k=1}^{n-1}(-1)^{k-1}k({}^{d-1}_{n-1-k})+
\sum_{k=2}^{n+1}(-1)^{k}k^2({}^{d-1}_{n+1-k})\right)[C],
$$
Using \ref{bin3}:
$$
\sum_{k=2}^{n+1}(-1)^k({}^d_{n+1-k})k^2=({}^d_n)+
\sum_{k=0}^{n+1}(-1)^k({}^d_{n+1-k})k^2
=({}^d_n)+({}^{d-3}_{n-1})-({}^{d-3}_{n}),
$$
we find:
$$
[\Phi^n_{1,1}(\Delta_n)]=\left(d({}^{d-3}_{n-2})+({}^d_n)+
({}^{d-3}_{n-1})-({}^{d-3}_n)\right)[C]
$$
and because of
$$
({}^{d-3}_{n-1})-({}^{d-3}_{n})=d({}^{d-3}_{n-1})-(n+1)({}^{d-2}_n), \qquad
({}^{d-2}_{n-1})=({}^{d-3}_{n-1})+({}^{d-3}_{n-2})
$$
we finally obtain:
\begin{equation}\label{homD}
[\Phi^n_{1,1}(\Delta_n)]=\left(d({}^{d-2}_{n-2})-(n+1)({}^{d-2}_n)+({}^d_n)
\right)[C].
\end{equation}
Applying $\left(\Phi_{1,1}^n\right)_*$ to \ref{eq6} and taking homology classes
we get the
following equation for $a,b,c$:
$$
(n+1)({}^{d-2}_{n})=a+b({}^{d-2}_{n-1})+
c\left(d({}^{d-2}_{n-1})-(n+1)({}^{d-2}_n)+({}^d_n)\right).
$$
On the other hand, we have the following intersections numbers in $S_n$:
$$
C\cdot C=G_n\cdot G_n=0, ~~~C\cdot G_n=1,~~~C\cdot \Delta_n=n,~~~ G_n\cdot
\Delta_n =({}^{d-1}_{n-1})
$$
and
$$
H_n\cdot C=d-n,~~~H_n\cdot G_n=({}^{d-1}_n).
$$
Therefore, by intersecting \ref{eq6} with $C$ and $G_n$ respectively, we find
two more equations for $a,b,c$:
$$
d-n=b+cn,~~~~~~~({}^{d-1}_n)=a+c({}^{d-1}_{n-1}),
$$
and \ref{eq2} now follows.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{}
With these results it is easy to find many relations between the cycles
$n_*C$.
It is a little surprising that the ones we find are equivalent to $\pi_iC\approx_{alg}
0$ for $i<2g-d$ and $\pi_{2g-1}C\approx_{alg} 0$.
\begin{Proposition} \label{recurs}
Let $C$ be a $d$:1 covering of ${\msy P}^1$. Then:
$$
\pi_{2g-i} C \not\approx_{alg} 0 \quad\Longrightarrow\quad 2\leq i\leq d .
$$
The cycles $\pi_{2g-2}C,\ldots , \pi_{2g-d}C$ span $Z_C/\approx_{alg}$ and thus
$$
\dim_{\msy Q}\; (Z_C/ \approx_{alg}) \leq d-1.
$$
For any set $\{n_1,\ldots ,n_{d-1}\}$ of non-zero distinct integers the cycles
$n_{1*}C,\ldots ,n_{d-1*}C$ also span $Z_C/\approx_{alg}$.
\end{Proposition}
{\bf Proof.}$\quad$
First of all we show that for all $n\in{\msy Z}$ we have:
\begin{equation}\label{help}
P_nC:=(n+1)_*C-({}^d_{d-1})n_*C+\ldots (-1)^{d}({}^d_0)(n-d+1)_*C+F_d\approx_{alg} 0
\end{equation}
with $P_n=\Gamma_{n+1}-({}^d_{d-1})\Gamma_n+\ldots\in CH^g(A\times A)$ and with a cycle
$F_d$ depending on $d$ but not on $n$:
$$
F_d=d[-C+({}^{d-1}_{d-3})C^--\mbox{$\frac{1}{2}$}({}^{d-1}_{d-4})2_*C^- +
\ldots (-1)^{d-1}\mbox{$\frac{1}{d-2}$}({}^{d-1}_0)(d-2)_*C^-].
$$
This relation follows from the easily verified identity (for all $n\in{\msy Z}$):
$$
\left( \Phi^1_{1,n}\right)_* H_1 =\left( \Phi^{d-2}_{-1,n-1}\right)_*H_{d-2}.
$$
Indeed, the l.h.s is by \ref{eq2}:
$$
\left( \Phi^1_{1,n}\right)_* H_1
\approx_{alg} \left( \Phi^1_{1,n}\right)_* (dC+dG_1-\Delta_1)
\approx_{alg} dn_*C+dC-(n+1)_*C,
$$
while for the r.h.s. we use
$(\Phi^{k}_{-1,l})_*\Delta_k=(\Phi^{k-1}_{-1,l-1})_*H_{k-1})$:
\begin{eqnarray}
& & \left( \Phi^{d-2}_{-1,n-1}\right)_*H_{d-2}\\
\nonumber&\approx_{alg}&
({}^d_{d-2})(n-1)_*C+dG^-_{d-2}-\left(\Phi^{d-3}_{-1,n-2}\right)_*H_{d-3}\\
\nonumber &\approx_{alg} &
({}^d_{d-2})(n-1)_*C-({}^d_{d-3})(n-2)_*C+\ldots+
(-1)^{d-2}(n-d+1)_*C+ \\ \nonumber
\nonumber& & +d(G_{d-2}^- -G_{d-3}^- +\ldots+(-1)^{d-3}G_1^-)
\end{eqnarray}
(cf. also the proof of \ref{eq3}). From this \ref{help} follows by using
\ref{Gs}.
A more convenient set of relations is obtained from \ref{help} as follows:
\begin{eqnarray}\label{fin}
& & (P_n-P_{n-1})C \; \\ \nonumber
& = & (n+1)_*C-(d+1)n_*C+({}^{d+1}_{d-1})(n-1)_*C+\ldots
+(-1)^{d+1}(n-d)_*C \\ \label{bz}
& = & \left(\Gamma_{m+d+1}-({}^{d+1}_1)\Gamma_{m+d}+({}^{d+1}_2)\Gamma_{m+d-1}\ldots+
(-1)^{d+1} \Gamma_m\right)C \\ \nonumber
& \approx_{alg} & 0,
\end{eqnarray}
where we substituted: $n:=m+d$.
Relation \ref{bz} can be rewritten as:
\begin{equation}
\Gamma_m*(\Gamma_1-\Gamma_0)^{*(d+1)} C \approx_{alg} 0\qquad\qquad \forall m\in{\msy Z}.
\end{equation}
Next we look at the expansion of the $\pi_i$'s in $\Gamma_1-\Gamma_0$:
$$
\pi_i=\mbox{$\frac{1}{(2g-i)!}$}(\log \Gamma_1)^{*(2g-i)}=
$$
$$
= (\Gamma_1-\Gamma_0)^{*(2g-i)}*
\left(a_{2g-i}(\Gamma_1-\Gamma_0)^{*0} +\ldots + a_{2g}(\Gamma_1-\Gamma_0)^{*i}\right).
$$
Therefore $\pi_iC\approx_{alg} 0$ if $2g-i\geq d+1$ so $\pi_{2g-i}C\approx_{alg} 0$ if $i\geq
d+1$. Since $\pi_{2g}=\Gamma_0$ we also have $\pi_{2g}C=0$, and from \cite{B2},
prop.3 we know $C_{(1)}= 0$ thus:
$$
C\approx_{alg} C_{(2)}+C_{(3)}+ \ldots + C_{(d)} \qquad {\rm with}\qquad
C_{(i)}:=\pi_{2g-i}C
$$
and we conclude that $\dim_{\msy Q}\,( Z_C/\approx_{alg}) \leq d-1$. We can in fact obtain
$C_{(1)}\approx_{alg} 0$ from \ref{help}, because a somewhat tedious computation shows
that, for some $c\in {\msy Q}$:
$$
\Gamma_{-1}(\mbox{$\frac{1}{d-1}$}P_0 -\mbox{$\frac{1}{d}$}P_{-1})=c\Gamma_0+\log \Gamma_1
\qquad {\rm mod~} (\Gamma_1-\Gamma_0)^{*(d+1)}* CH^g(A\times A),
$$
here $\Gamma_{-1}\Gamma_{n}=\Gamma_{-n}$ is the product as correspondences, the equality is
in (a quotient of) the ring $CH^g(A\times A)$ with $*$-product. Since $\Gamma_0$
acts trivially on one cycles and $P_nC\approx_{alg} 0$ the statement follows.
In fact one shows that both sides are equal to, for some $c_1\in{\msy Q}$:
$$
\sum_{n=1}^{d} (-1)^{n-1}\mbox{$\frac{1}{n}$}({}^d_n)\Gamma_n+c_1\Gamma_0.
$$
(Comparing this expression with the one for $u_{d*}G_d$ in \ref{eq3} we see
that
$\pi_{2g-1}C\approx_{alg} u_{d*}G_d\approx_{alg} 0$ since $G_d\cong {\msy P}^1$ maps to a point in
$J(C)$, note that some care must be taken as we defined $G_n$ only for $n<d$).
Finally we observe that since $n_*C_{(i)}=n^iC_{(i)}$, the determinant of the
matrix expressing the $n_{i*}C$ in the $C_{(i)}$ is a Vandermonde determinant,
which is nonzero.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\begin{Proposition}
\begin{enumerate}
\item For a curve $C$ with $\pi_iC\approx_{alg} 0$ for $i\leq 2g-4$ (for example any
curve with a $g^1_3$) we have, with $C_{(i)}:=\pi_{2g-i}C$:
\begin{eqnarray}\nonumber
C_{(2)}&\approx_{alg} & \mbox{$\frac{1}{2}$}(C+C^-)\\ \nonumber
C_{(3)}&\approx_{alg} &\mbox{$\frac{1}{2}$}( C-C^-)
\end{eqnarray}
Furthermore: $n_*C\approx_{alg} \mbox{$\frac{n^3+n^2}{2}$}C\,
-\,\mbox{$\frac{n^3-n^2}{2}$}C^-$.
\item For a curve $C$ with $\pi_iC\approx_{alg} 0$ for $i\leq 2g-5$ (for example any
curve with a $g^1_4$) we have:
\begin{eqnarray}\nonumber
C_{(2)}&\approx_{alg} & \mbox{$\frac{-1}{12}$}(2_*C-12C-4C^-)\\ \nonumber
C_{(3)}&\approx_{alg} & \mbox{$\frac{1}{2}$}( C-C^-)\\ \nonumber
C_{(4)}&\approx_{alg} &\mbox{$\frac{1}{12}$}(2_*C-6C+2C^-).
\end{eqnarray}
\end{enumerate}
\end{Proposition}
{\bf Proof.}$\quad$We give the proof of the second statement, the first being
similar but easier. By assumption we have ($C_{(i)}:=\pi_{2g-i}C$):
$$
C\approx_{alg} C_{(2)}+C_{(3)}+C_{(4)}\qquad{\rm with}\quad n_*C_{(i)}=n^iC_{(i)}.
$$
Therefore:
\begin{eqnarray} \nonumber
C &\approx_{alg} & C_{(2)}+C_{(3)}+C_{(4)}\\ \nonumber
C^- &\approx_{alg} & C_{(2)}-C_{(3)}+C_{(4)}\\ \nonumber
2_*C &\approx_{alg} & 4C_{(2)}+8C_{(3)}+16C_{(4)}.
\end{eqnarray}
The result follows by straightforward linear algebra.
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\secdef\empsubsection{\emppsubsection*}{}
Note that if one specializes a curve $C$ with a $g^1_4$ to a trigonal curve,
then actually $C_{(4)}=\mbox{$\frac{1}{12}$}(2_*C-6C+2C^-)\approx_{alg} 0$. However, we
could not decide whether for a generic $4$:1 cover of ${\msy P}^1$ we have
$C_{(4)}\approx_{alg} 0$.
\section{Appendix}
\secdef\empsubsection{\emppsubsection*}{}
We recall some facts on binomial coefficients.
The binomial coefficients $({}^n_k)$ are defined (also for negative $n\in {\msy Z}$
!) as (cf. \cite{ACGH}, VIII.3)
$$
({}^n_k):=\frac{n(n-1) \cdots (n-k+1)}{k(k-1)\cdots 1}\quad (k> 0),\quad
({}^n_0):=1,\quad ({}^n_k):=0\quad (k<0).
$$
With this definition, they are the coefficients in the expansion of $(1+x)^n$:
$$
(1+x)^n=1+({}^n_1)x+\ldots +({}^n_k)x^k+\ldots=\sum_{k=0}^\infty ({}^n_k)x^k.
$$
Comparing the coefficients of $x^k$ in $(1+x)^n(1+x)^m=(1+x)^{n+m}$ one finds,
for all $n,m\in{\msy Z}$:
\begin{equation}\label{addbin}
\sum_i ({}^n_i)({}^m_{k-i})=(^{n+m}_{k}),
\end{equation}
\begin{Lemma} We have:
\begin{eqnarray}\label{bin1}
\sum_{i=0}^l (-1)^{i}({}^m_{l-i})&=&({}^{m-1}_l),\\ \label{bin2}
\sum_{i=0}^l (-1)^{i-1} i({}^m_{l-i})&=&({}^{m-2}_{l-1}),\\ \label{bin3}
\sum_{i=0}^l (-1)^i i^2({}^m_{l-i})&=&({}^{m-3}_{l-2})-(^{m-3}_{l-1}).
\end{eqnarray}
\end{Lemma}
\vskip0.5truecm
{\bf Proof.}$\quad$Using $({}^{-1}_k)=(-1)^k$ and \ref{addbin}, the first line
can be written as:
$$
\sum_{i=0}^l (-1)^{i}({}^m_{l-i})=\sum_{i=0}^l ({}^{-1}_i)({}^m_{l-i})=
({}^{m-1}_l).
$$
The second line follows in the same way, using $({}^{-2}_{k-1})=(-1)^{k-1}k$:
$$
\sum_{i=0}^l (-1)^{i-1} i({}^m_{l-i})=
\sum_{i=0}^l({}^{-2}_{i-1})({}^m_{l-i})=({}^{m-2}_{l-1}).
$$
For the last line we use $({}^{-3}_k)=(-1)^k(k+1)(k+2)/2$, so:
$$
(-1)^kk^2=2({}^{-3}_k)-3({}^{-2}_k)+({}^{-1}_k).
$$
Therefore:
\begin{eqnarray}\nonumber
\sum_{i=0}^l (-1)^i i^2({}^m_{l-i}) &=& 2\sum_{i=0}^l ({}^{-3}_i)({}^m_{l-i})
-3\sum_{i=0}^l ({}^{-2}_i) ({}^m_{l-i})+
\sum_{i=0}^l ({}^{-1}_i) ({}^m_{l-i})\\ \nonumber
&=& 2({}^{m-3}_l)-3({}^{m-2}_l)+({}^{m-1}_l).
\end{eqnarray}
Now use that:
$$
({}^{m-1}_l)=({}^{m-2}_{l-1})+({}^{m-2}_l)=
({}^{m-3}_{l-2})+2({}^{m-3}_{l-1})+({}^{m-3}_l),\quad{\rm and}
$$
$$
({}^{m-2}_l)=({}^{m-3}_{l-1})+({}^{m-3}_l).
$$
{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
|
1992-03-25T20:59:19 | 9202 | alg-geom/9202022 | en | https://arxiv.org/abs/alg-geom/9202022 | [
"alg-geom",
"math.AG"
] | alg-geom/9202022 | Richard Hain | Richard Hain | Classical Polylogarithms | 43 pages, LaTeX | null | null | null | null | This paper is an introduction to classical polylogarithms and is an expanded
version of a talk given by the author at the Motives conference. Topics covered
include, monodromy; the polylogarithm local systems; Bloch's constructions of
regulators using the dilogarithm; polylog locals systems as variations of mixed
Hodge structre; the polylogarithm quotient of the fundamental group of C -
{0,}. It is intended as background for understanding recent work of Beilinson,
Deligne, and Goncharov.
| [
{
"version": "v1",
"created": "Thu, 20 Feb 1992 22:14:54 GMT"
},
{
"version": "v2",
"created": "Wed, 25 Mar 1992 19:55:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hain",
"Richard",
""
]
] | alg-geom | \section{Introduction}
\label{intro}
This article is an introduction to classical polylogarithms. After
establishing some of their basic properties, we
present several examples of Spencer Bloch where the dilogarithm is
used to construct the second regulator. We also construct the
polylogarithm local systems and show that each underlies a Tate
variation of mixed Hodge structure \cite{deligne_letter}. We conclude
by giving an exposition of a motivic description of the polylogarithm
local systems. Most of the results in this paper were discovered by
Bloch, Deligne, Ramakrishnan, Suslin and Beilinson.
Let $k$ be a positive integer.
The $k$th {\it polylogarithm} $\ln_k x$ is defined by
\begin{equation}\label{series}
\ln_k x = \sum_{n=1}^\infty {x^n \over n^k}.
\end{equation}
This converges in the unit disk to a holomorphic
function.
The first polylogarithm, $\ln_1 x$, is just $- \log(1-x)$. The
second,
$$
\ln_2 x = \sum_{k=1}^\infty {x^n \over n^2}
$$
is called the {\it dilogarithm}, and was defined by Euler in 1768.
The higher polylogarithms were defined by Spence in 1809
(cf. \cite{lewin}).
It is believed that the $k$th regulator
$$
c_k : K_m(X) \to H_{\cal D}^{2k-m}(X,{\Bbb Z}(k))
$$
from the algebraic $K$-theory of a complex algebraic variety $X$ (and
therefore all varieties of finite type over ${\Bbb Q}$) to its
Deligne cohomology can be expressed in terms of the $k$th
polylogarithm.
If true, this would generalize
the classical fact that the logarithm occurs as the first Chern class
$$
c_1 : K_0(X) \to H^2(X,{\Bbb Z}(1))
$$
and its single valued cousin, $\log|\phantom{x}| : {\Bbb C}^\ast \to {\Bbb R}$,
occurs as the regulator
$$
c_1 : K_1({\Bbb C}) \approx {\Bbb C}^\ast \to {\Bbb R}
\approxH_{\cal D}^1({\rm spec}\, {\Bbb C},{\Bbb R}(1)).
$$
In the case where $X$ is ${\rm spec\, } {\Bbb C}$, this should mean
that some single valued cousin $D_k : {\Bbb C} - \{0,1\} \to {\Bbb R}$ of $\ln_k$
should represent a multiple of the Borel regulator element
$$
b_k \in H^{2k-1}(GL_k({\Bbb C})^\delta,{\Bbb R}),
$$
the cohomology class which gives rise to the regulator \cite{borel}
(cf. also \cite[\S 2.2]{ramakrishnan_survey}, \cite[\S 7.2]{D-H-Z}.
(Here, $GL_k({\Bbb C})^\delta$ denotes the general linear group viewed as
a discrete group.)
The cocycle condition would then be a
functional equation satisfied by $D_k$ which generalizes
the 3-term functional equation satisfied by $D_1 = \log
|\phantom{x}|$.
It is further believed that all of the rational $K$-theory of a field
$F$ should come from $F - \{0,1\}$, and that the relations should all
correspond to canonical functional equations satisfied by the $D_k$.
Such statements are often referred to as the Zagier conjecture.%
\footnote{Zagier's original conjecture \cite{zagier} asserted that
the
value at the positive integer $m$ of the Dedekind zeta function of a
number field $F$ could be expressed as a determinant of values of
$D_m$
at $F$ rational points of ${\Bbb P}^1 - \{0,1,\infty\}$.}
For example, for all fields, we may express the familiar fact
$$
K_1(F) = F^\times
$$
as
$$
K_1(F) = \left[\coprod_{x \in F - \{0,1\}} {\Bbb Z} \right]/{\cal R}
$$
where the relations $\cal R$ are generated by
$$
[x] - [xy] + [y] = 0\quad \hbox{and} \quad [x] + [x^{-1}] = 0,
$$
where $x,y\in F - \{0,1\}$ and $xy \neq 1$. These are the analogues of
the functional equations
$$
D_1(x) - D_1(xy) + D_1(y) = 0 \quad \hbox{and}
\quad D_1(x) + D_1(x^{-1}) = 0.
$$
Note also that the functional equation in this case is precisely the
condition that $D_1$ represent an element of $H^1(GL_1({\Bbb C}),{\Bbb R})$.
The corresponding story for the dilogarithm has been worked out by
Bloch \cite{bloch-irvine} and Suslin \cite{suslin}. We give an
account of this story in Section \ref{reg-k3}.
Useful references for basic material in this paper include
\cite{milnor}
and [this volume] for algebraic $K$-theory, \cite{lewin} for a
comprehensive
reference on classical aspects of polylogarithms,
\cite{hain-zucker_2}
for Tate variations of mixed Hodge structure, and \cite{geom} for
basic facts about iterated integrals and the mixed Hodge theory of
the
fundamental group. The book \cite{lewin_2} is a useful reference
for more recent developments. (Also, references to other articles in
this book.)
\noindent{\bf Notation:} The group of units of a ring $R$ will be
denoted
by $R^\times$. When $\Lambda$ is ${\Bbb Z}$, ${\Bbb Q}$, or ${\Bbb R}$, $\Lambda(k)$
will
denote the subgroup $(2\pi i)^k\Lambda$ of ${\Bbb C}$. It will also be
used to
denote the Hodge structure of type $(-k,-k)$ which has this abelian
group
as its lattice.
\section{Monodromy}
\label{monodromy}
An easy power series manipulation yields the formula
$$
\ln_k x = \int_0^x \ln_{k-1}z\, {dz \over z},
$$
where $x$ lies in the unit disk and $k\ge 2$.
It follows, by an induction argument, that each polylogarithm can be
analytically
continued to a multivalued holomorphic function on ${\Bbb C}-\{0,1\}$. In this
section we determine the monodromy of the polylogarithms. This was
first
computed by Ramakrishnan in \cite{ramakrishnan_monod}.
Set
$$
\omega_0 = {dz \over z} \hbox{ and } \omega_1 = {dz \over 1-z}.
$$
Let
$$
\omega = \pmatrix{
0 & \omega_1 & 0 & \cdots & 0 \cr
& \ddots & \omega_0 & \ddots & \vdots \cr
\vdots & & \ddots & \ddots & 0 \cr
& & & \ddots & \omega_0 \cr
0 & & \cdots & & 0 \cr }
\in H^0({\Bbb P}^1,\Omega^1_{{\Bbb P}^1}(\log\{0,1,\infty\}))\otimes gl_n({\Bbb C}).
$$
Consider the first order linear differential equation
\begin{equation}\label{diff-eqn}
d\lambda} %{\mbox{\boldmath$\lambda$} = \lambda} %{\mbox{\boldmath$\lambda$} \mbox{\boldmath$\omega$}
\end{equation}
where $\lambda} %{\mbox{\boldmath$\lambda$}$ is a possibly multivalued function ${\Bbb C} - \{0,1\} \to
{\Bbb C}^{n+1}$.
Denote the $k$th power of the standard logarithm $\log x = \int_1^x
\omega_0$
by $\log^k x$.
Let
$$
\Lambda(x) = \pmatrix{
1 & \ln_1 x & \ln_2 x & \cdots & \cdots & \ln_n x \cr
0 & 2\pi i & 2\pi i\log x & & \cdots & {2\pi i\over n!}\log^{n-1}x
\cr
0 & 0 & (2 \pi i)^2 & \ddots & \cdots & {(2\pi i)^2 \over
(n-1)!}\log^{n-1}x
\cr
\vdots & & \ddots & \ddots & & \vdots \cr
\vdots & & & 0 & (2\pi i)^{n-1} & (2\pi i)^{n-1}\log x \cr
0 & \cdots & \cdots & \cdots & 0 & (2\pi i)^n \cr } \in gl_{n+1}({\Bbb C}).
$$
More precisely,
$$
\Lambda_{j\, k}(x) = \cases{
\ln_k x & when $j=0$ and $k>0$; \cr
{(2\pi i)^j \over (k-j)!}\log^{k-j}x & when $j,k>0$; \cr
0 & when $k < j$.\cr}
$$
We will view this as a multivalued $gl_n({\Bbb C})$-valued function on
${\Bbb C} - \{0,1\}$. By the {\it principal branch} of $\Lambda(x)$ we shall
mean
the matrix-valued function on the disk $|x - 1/2|<1/2$ obtained
by
taking the standard branches of each of its entries on that disk.
(The
principal branch of $\ln_k$ on this disk is the one given by the
power series expansion (\ref{series}).)
\begin{proposition}\label{diffeq}
The function $\Lambda(x)$ is a fundamental solution of
(\ref{diff-eqn}).
That is,
$$
d\Lambda = \Lambda \mbox{\boldmath$\omega$}
$$
and $\Lambda(x)$ is non-singular for each $x\in {\Bbb C}-\{0,1\}$.\quad $\square$
\end{proposition}
If we analytically continue the principal branch of $\Lambda(x)$
about a loop
in ${\Bbb C}-\{0,1\}$ based at $1/2$, the resulting matrix of functions will
still
be a fundamental solution of (\ref{diff-eqn}). It follows that, for
each loop
$\gamma$ based at $1/2$, there is a matrix $M(\gamma)\in
GL_{n+1}({\Bbb C})$ such
that the analytic continuation of (the principal branch of)
$\Lambda(x)$
about $\gamma$ is $M(\gamma)\Lambda(x)$. For a pair of loops
$\alpha,\beta$ based at $1/2$, we have
$$
M(\alpha \beta) = M(\alpha) M(\beta).
$$
Since the value of $M(\gamma)$ depends only on the homotopy class of
$\gamma$,
we obtain a monodromy representation
\begin{equation}\label{mono-rep}
M : \pi_1({\Bbb C}-\{0,1\},1/2) \to GL_{n+1}({\Bbb C}).
\end{equation}
Let $\sigma_0,\sigma_1\in \pi_1({\Bbb C}-\{0,1\},1/2)$ be the loops defined
by
$$
\sigma_0(t) = e^{2 \pi i t}/2, \quad \sigma_1(t) = 1 - e^{2\pi i
t}/2,
\quad 0 \le t \le 1.
$$
These loops generate $\pi_1({\Bbb C}-\{0,1\},1/2)$.
\begin{proposition}{\rm \cite{ramakrishnan_monod}} \label{monod}
We have
$$
M(\sigma_0) = \left(
\begin{array}{c| c c c}
1 & 0 & \cdots & 0 \\ \hline
0 & & & \\
\vdots & & J & \\
0 & & & \\
\end{array}\right)
\hbox{ and } M(\sigma_1) = \left(
\begin{array}{c| r c c c}
1 & -1 & 0 & \cdots & 0 \\ \hline
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & & 0 \\
\vdots & \vdots & & \ddots & \\
0 & 0 & 0 & \cdots & 1\\
\end{array}\right)
$$
where
$$
J = \exp \pmatrix{
0 & 1 & 0 & \cdots & 0 \cr
& 0 & 1 & \ddots & \vdots \cr
\vdots & & \ddots & \ddots & 0 \cr
& & & \ddots & 1 \cr
0 & & \cdots & & 0 \cr }
$$
\end{proposition}
\noindent{\bf Proof.~} The monodromy around $\sigma_0$ is easy to calculate. Indeed,
since
the principal branch of each polylogarithm is single valued on the
disk $|z - 1/2| \le 1/2$, each is unchanged when continued along
$\sigma_0$.
The formula for $M(\sigma_0)$ follows as the analytic continuation of
$\log x$ about $\sigma_0$ is $\log x + 2\pi i$.
Since the principal branch of $\log x$ is defined in a neighbourhood
of
${\Bbb R}_+$, it follows that $\log x$ is invariant under analytic
continuation
along $\sigma_1$.
We compute the analytic continuation of $\ln_k$ along $\sigma_1$ by
induction. When $k=1$, $\ln_1 x$ changes to $\ln_1 x - 2 \pi i$.
Assume now that $k\ge 1$ and that the continuation of $\ln_k x$
along $\sigma_1$ is
$$
\ln_k x - {2 \pi i \over (k-1)!} \log^{k-1} x.
$$
Denote the integral of $f(z)dz$ along the straight line interval in
the
complex plane between the points $a$ and $b$ by
$$
\int_a^b f(z) dz.
$$
When $|x - 1/2| < 1/2$, we have
\begin{eqnarray}\label{first}
\ln_{k+1} x & = & \int_0^x \ln_k z {dz \over z} \nonumber \\
& = & \int_0^{1/2} \ln_k z {dz \over z} + \int_{1/2}^x \ln_k z {dz
\over z}.
\end{eqnarray}
The result of analytically continuing this along $\sigma_1$ is
\begin{equation}\label{second}
\int_0^{1/2} \ln_k z {dz \over z} + \int_{\sigma_1} \ln_k z {dz \over
z}
+ \int_{1/2}^x \ln_k z {dz \over z}.
\end{equation}
It follows from the inductive formula that the difference between
(\ref{first}) and (\ref{second}) is
\begin{equation}\label{difference}
\int_{\sigma_1} \ln_k z\, {dz \over z} -
{2 \pi i \over (k-1)!} \int_{1/2}^x \log^{k-1} z\, {dz \over z}.
\end{equation}
For each $\epsilon \in (0,1/2)$, the path which traverses the line
segment from $1/2$ to $1-\epsilon$, goes around the boundary of the
disk $|z-1| \le \epsilon$ in the positive direction, then returns
along the interval from $1-\epsilon$ to $1/2$ represents the homotopy
class $\sigma_1$. Again, using the inductive formula for the
monodromy
of $\ln_k x$ around $\sigma_1$, we have
$$
\int_{\sigma_1} \ln_k z\, {dz \over z} =
-{2 \pi i \over (k-1)!}\int_{1-\epsilon}^{1/2} \log^{k-1}z\,
{dz \over z} + \int_{-\pi}^\pi \ln_k(1+\epsilon e^{it})\,
{d(\epsilon e^{it})\over 1 + \epsilon e^{it}}.
$$
The inductive hypothesis also implies that $\ln_k x$ is bounded in a
neighbourhood of 1 when $k>1$, so that the last integral $\to 0$ as
$\epsilon \to 0$ for all $k$. (A separate argument is needed when
$k=1$.)
Combining this with (\ref{difference}), we see that $\ln_{k+1} x$
changes
by
$$
-{2 \pi i \over (k-1)!}
\left[\lim_{\epsilon \to 0}\int_{1-\epsilon}^{1/2} \log^{k-1}z\,
{dz \over z} + \int_{1/2}^x \log^{k-1}z\, {dz \over z} \right]
= -{2 \pi i \over k!} \log^k x
$$
when continued around $\sigma_1$. \quad $\square$
The monodromy calculation has several interesting consequences.
First,
even though it is does not make sense, in general, to talk about the
value of a multivalued function at a point, it does make sense to
talk about the value of $\ln_k$ at 1.
\begin{corollary} The value of $(k-1)!\ln_k 1$ is well defined modulo
${\Bbb Z}(n)$ and is congruent to $(k-1)!\zeta(k)$. \quad $\square$
\end{corollary}
The second important consequence of the monodromy calculation is the
rationality of the monodromy.
\begin{corollary}
The image of the monodromy representation (\ref{mono-rep}) is
contained
in $GL_{n+1}({\Bbb Q})$. \quad $\square$
\end{corollary}
The significance of this last result is that it implies that the
local
system over ${\Bbb C}-\{0,1\}$ which corresponds to the differential equation
(\ref{diff-eqn}) is defined over ${\Bbb Q}$. This local system is called
the
{\it $n$th polylogarithm local system}. These local systems fit
together
to form an inverse system of local systems whose limit we call the
{\it
polylogarithm local system}. We now describe these local systems in
detail.
Define a meromorphic connection $\nabla$ on the trivial bundle
\begin{equation}\label{bundle}
{\Bbb P}^1 \times {\Bbb C}^{n+1} \to {\Bbb P}^1
\end{equation}
by defining
$$
\nabla f = df - f\mbox{\boldmath$\omega$}
$$
where $f : {\Bbb C} - \{0,1\} \to {\Bbb C}^{n+1}$ is a section. This connection has
regular
singular points at $0,1$ and $\infty$, and is flat over ${\Bbb C}-\{0,1\}$ as
$\mbox{\boldmath$\omega$}$
satisfies the integrability condition
$$
d\mbox{\boldmath$\omega$} + \mbox{\boldmath$\omega$} \wedge \mbox{\boldmath$\omega$} = 0
$$
(equivalently, because the equation $\nabla f = 0$ is a system of
ordinary
differential equations). Let $\lambda} %{\mbox{\boldmath$\lambda$}_0,\lambda} %{\mbox{\boldmath$\lambda$}_1,\ldots,\lambda} %{\mbox{\boldmath$\lambda$}_n$ be
the rows of $\Lambda(x)$. Each of these satisfies (\ref{diff-eqn})
and is
therefore a flat section of (\ref{bundle}). Even though these are
multivalued, their ${\Bbb Q}$ linear span is well
defined as the monodromy representation is defined over ${\Bbb Q}$.
Suppose that $X$ is a smooth curve and that $\overline{X}$ is a smooth
completion
of $X$. Every flat bundle $E \to X$ has a {\it canonical
extension\/}
$\overline{E} \to \overline{X}$. Denote the local monodromy operator about a point
$p \in D := \overline{X} - X$ by $T_p$. When each $T_p$ is unipotent, the
canonical
extension is characterized by two properties. First, the meromorphic
extension $\overline{\nabla}$ of the connection to $\overline{E} \to \overline{X}$ has
logarithmic
singularities along $D$. That is,
$$
\overline{\nabla} : \overline{\cal E} \to
\overline{\cal E}\otimes_{{\cal O}_{\overline{X}}} \Omega^1_{\overline{X}} (\log D).
$$
Second, the residue of $\overline{\nabla}$ at each point of $D$ is nilpotent.
Denote the ${\Bbb Q}$ local system over ${\Bbb C}-\{0,1\}$ which corresponds to the
representation (\ref{mono-rep}) by ${\Bbb V}$.
Since $\mbox{\boldmath$\omega$}$ has nilpotent residue at 0,1 and $\infty$, we have:
\begin{proposition}\label{poly-loc-sys}
The canonical extension of the flat holomorphic vector bundle
${\Bbb V} \otimes_{\Bbb Q} {\cal O}_{{\Bbb C} -\{0,1\}}$ to ${\Bbb P}^1$ is the bundle (\ref{bundle})
with the connection $\nabla$ defined above. \quad $\square$
\end{proposition}
\section{The Bloch-Wigner function}
\label{bw_function}
Define
$$
D_2(x) = \Im \ln_2 x + \log |x| \arg(1-x)
$$
when $|x-1/2| < 1/2$ and where $\ln_2 x$, $\log x$, and $\arg (1-x)$
denote the principal branches of these functions in the disk
$|x-1/2| < 1/2$. An easy computation using the monodromy calculation
(\ref{monod}) shows that $D_2$ is invariant under continuation along
the generators $\sigma_0$ and $\sigma_1$ of $\pi_1({\Bbb C}-\{0,1\},1/2)$.
Consequently, the function $D_2$ extends to a single valued, real
analytic
function
$$
D_2 : {\Bbb C} - \{0,1\} \to {\Bbb R}.
$$
This is called the {\it Bloch-Wigner function\/}. If we define
$$
D_2(0) = D_2(1) = D_2(\infty) = 0
$$
then $D_2$ is a continuous function $D_2 : {\Bbb P}^1 \to {\Bbb R}$. The Bloch-%
Wigner function should be viewed as having the same relation to
$\ln_2$
as $D_1 := \log |\phantom{x}|$ bears to the logarithm.
The boundary of hyperbolic 3-space ${\Bbb H}^3$ is the Riemann sphere
${\Bbb P}^1$.
The group of orientation preserving isometries of ${\Bbb H}^3$ is
$PSL_2({\Bbb C})$.
The induced action on the boundary is just the standard action of
$PSL_2({\Bbb C})$ on ${\Bbb P}^1$ via fractional linear transformations.
Denote the ideal tetrahedron in ${\Bbb H}^3$ with vertices at
$a_0, a_1, a_2, a_3 \in {\Bbb P}^1$ by $\Delta(a_0,a_1,a_2,a_3)$. Since
the volume form of hyperbolic space is invariant under the action of
the isometry group,
$$
{\rm vol\, } \Delta(a_0,a_1,a_2,a_3) = {\rm vol\, } \Delta (\lambda,1,0,\infty).
$$
where $\lambda$ is the cross ratio $[a_0:a_1:a_2:a_3]$ of the
vertices.
The following result goes back to Lobachevsky (cf. \cite{milnor}).
A proof may be found in \cite[p.~172]{dupont-sah}.
\begin{theorem}\label{volume} For each $z \in {\Bbb P}^1$,
the volume of $\Delta (z,1,0,\infty)$ equals $D_2(z)$.
\end{theorem}
\begin{corollary}\label{funct_eqn_1}
If $a_0, a_1, a_2, a_3, a_4 \in {\Bbb P}^1$, then
$$
\sum_{j=0}^4 (-1)^j D_2([a_0:\cdots:\widehat{a_j}:\cdots:a_4]) = 0.
$$
Moreover, for all permutations $\sigma$ of $\{0,1,2,3\}$,
$$
D_2([a_{\sigma(0)}:\cdots:a_{\sigma(3)}]) =
{\rm sgn}(\sigma)D_2([a_0:\cdots:a_3]).
$$
\end{corollary}
\noindent{\bf Proof.~}
The first assertion follows as the ideal polyhedron $P$ with vertices
$a_0, a_1, a_2, a_3, a_4$ decomposes into a union of ideal tetrahedra
in 2 different ways. Viz.
$$
P = \Delta(a_1, a_2, a_3, a_4) \cup \Delta(a_0, a_1, a_3, a_4)
\cup \Delta(a_0, a_1, a_2, a_3)
$$
and
$$
P = \Delta(a_0, a_2, a_3, a_4) \cup \Delta(a_0, a_1, a_2, a_4)
$$
In each case the pieces intersect along 2-dimensional faces. The
first
assertion follows from Theorem \ref{volume} by comparing volumes.
The second follows as swapping the order of 2 vertices reverses the
orientation of the tetrahedron. \quad $\square$
\medskip
Taking the five points to be $y,x,1,0,\infty$, we obtain the usual
form
of the functional equation, which is the analogue for $D_2$ of the
Abel-Spence functional equation of $\ln_2$.
\begin{corollary}\label{funct_eqn_2}
If $y,x, 1, 0, \infty$ are distinct points of ${\Bbb P}^1$, then
$$
D_2(x) - D_2(y) + D_2(y/x) - D_2((1-y)/(1-x)) +
D_2((1-y^{-1})/(1-x^{-1})) = 0. \quad \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3
$$
\end{corollary}
Ramakrishnan \cite{ramakrishnan_bw} showed that all the
polylogarithms
have such single valued cousins. The essential point being that one
can use the
unipotence of the monodromy group and induction to kill off the
monodromy
of $\ln_k$. Zagier \cite{zagier} gave an explicit
formula for Ramakrishnan's functions. In general, there seems to be
no
canonical way to go from the multivalued polylogarithm to a single
valued
function. Goncharov \cite{goncharov} has modified Ramakrishnan's
trilogarithm and proved that his function satisfies a very natural
functional equation. His function is defined by
\begin{equation}\label{gonch_trilog}
D_3(x) = \Re\left[ \ln_3(x) - \log|x| \ln_2 x + (\log^2|x| \ln_1
x)/3\right].
\end{equation}
It is single valued and continuous on ${\Bbb P}^1$, and real analytic on
${\Bbb P}^1 - \{0,1,\infty\}$.
\section{The regulator $K_3(\protect{\Bbb C}) \to
\protect{\Bbb C}/\protect{\Bbb Q}(2)$}
\label{reg-k3}
The Deligne cohomology of ${\rm spec\, } {\Bbb C}$ is
$$
H_{\cal D}^m({\rm spec\, } {\Bbb C},\Lambda(k)) = \cases{
0 & $m \neq 1$; \cr
{\Bbb C}/\Lambda(k) & $m = 1$, \cr}
$$
where $\Lambda$ denotes ${\Bbb Z}$, ${\Bbb Q}$ or ${\Bbb R}$. So the only non-trivial
regulators for ${\rm spec\, } {\Bbb C}$ with values in Deligne cohomology are
$$
K_{2m-1}({\Bbb C}) \to H_{\cal D}^1({\rm spec\, } {\Bbb C}, \Lambda(m)) \approx {\Bbb C}/\Lambda(m)
$$
for each $m \ge 1$. The first regulators
$$
K_1({\Bbb C}) \approx {\Bbb C}^\ast \to {\Bbb C}/{\Bbb Z}(1)\quad\hbox{and}\quad
K_1({\Bbb C}) \approx {\Bbb C}^\ast \to {\Bbb C}/{\Bbb R}(1) \approx {\Bbb R}
$$
are given by the functions $\log$ and $\log|\phantom{x}|$,
respectively.
In this section we give an account of the construction of the second
regulators
$$
K_3({\Bbb C}) \to {\Bbb C}/{\Bbb Q}(2) \quad \hbox{and}
\quad K_3({\Bbb C}) \to {\Bbb C}/{\Bbb R}(2) \approx {\Bbb R}(1)
$$
using $\ln_2$ and $D_2$ respectively. These results go back to Bloch
and
Wigner (unpublished, cf. \cite{dupont-sah}). As is customary,
$GL_n({\Bbb C})^\delta$ signifies that $GL_n({\Bbb C})$ is viewed as a group with
the discrete topology.
\begin{lemma}\label{gen_posn}
The Bloch-Wigner function defines a canonical group cohomology class
$$
{\cal D}_2 \in H^3(GL_2({\Bbb C})^\delta,{\Bbb R}).
$$
\end{lemma}
\noindent{\bf Proof.~} Let $F$ be a field and $n\in {\Bbb N}$. For each $k\in{\Bbb N}$, define
$C_k(F,n)$ to be the free abelian group with basis ordered
$(k+1)$-tuples $(v_0,\ldots,v_k)$, of vectors $v_j$ of $F^n$ in
general
position. (I.e. , each $\min(n,k+1)$ of them are independent.) Define
a differential $\partial : C_k \to C_{k-1}$ by
$$
\partial : (v_0,\ldots,v_k) \mapsto
\sum_{j=0}^k (-1)^j (v_0,\ldots,\widehat{v_j},\ldots,v_k).
$$
When $F$ is infinite, the complex
$C_\bullet(F,n)$ is quasi-isomorphic to ${\Bbb Z}$. Since $GL_n(F)$ acts on
this complex, it is a resolution of the trivial module. So, for all
$GL_n(F)$ modules $M$, there is a natural map
$$
H^\bullet\left({\rm Hom}_{GL_n(F)}(C_\bullet(F,n),M)\right)
\to H^\bullet(GL_n(F),M),
$$
provided that $F$ is infinite.
Denote the point in ${\Bbb P}^{n-1}(F)$ determined by the non-zero vector
$v$ of $F^n$ by $[v]$.
Define a map $f : C_3({\Bbb C},2) \to {\Bbb R} $ by
$$
f(v_0,v_1,v_2,v_3) =
D_2\left(\left[[v_0]:[v_1]:[v_2]:[v_3]\right]\right).
$$
The cocycle condition is simply the functional equation
(\ref{funct_eqn_1}).
Denote the image of this cohomology class in
$H^3(GL_2({\Bbb C})^\delta,{\Bbb R})$
by ${\cal D}_2$. \quad $\square$
\medskip
A cohomology class $c \in H^m(GL_m(R),\Lambda)$ defines a map
$$
K_m( R) \to \Lambda.
$$
This map is obtained as the composite
$$
K_m(R) = \pi_m(BGL(R)^+) \to H_m(BGL(R)^+) \approx H_m(GL(R))
\mapright{c}\Lambda
$$
of the Hurewicz homomorphism with the $\Lambda$-valued functional
on homology induced by $c$.
\begin{theorem}\label{real_reg}
The map
$$
K_3({\Bbb C}) \to {\Bbb C}/{\Bbb R}(2)
$$
defined by $i{\cal D}_2$ equals the Beilinson regulator, and this map
equals
half of the Borel regulator.
\end{theorem}
\noindent{\bf Proof.~} The proof is an assemblage of results from the literature, and
we only give a sketch. First, the cocycle
$$
GL_2({\Bbb C})^4 \to {\Bbb C}/{\Bbb R}(2)
$$
induced by $iD_2$, composed with the cross ratio, represents a
continuous
cohomology class $i{\frak D}_2$, even though this cocycle is not itself
continuous. This apparently contradictory state of affairs arises
because
the cross ratio $\left({\Bbb P}^1\right)^4 \to {\Bbb P}^1$ is not everywhere
defined.
One can show (e.g. \cite{yang_thesis}) that the image of this class
under the natural map
$$
H_{\rm cts}^3(GL_2({\Bbb C}),{\Bbb C}/{\Bbb R}(2)) \to H^3(GL_2({\Bbb C})^\delta,{\Bbb C}/{\Bbb R}(2))
$$
is the class $i{\cal D}_2$ of (\ref{gen_posn}).
Next, since
$$
H_{\rm cts}^3(GL_2({\Bbb C}),{\Bbb C}/{\Bbb R}(2)) \approx H_{\rm cts}^3(GL_n({\Bbb C}),{\Bbb C}/{\Bbb R}(2))
\approx {\Bbb C}/{\Bbb R}(2)
$$
for all $n \ge 2$, and since this group is spanned by the continuous
cohomology class $\beta_2$ used to define the Borel regulator (cf.
\cite{ramakrishnan_survey}, \cite{D-H-Z}), it follows that
there is a real number $\lambda$ such that
$$
i {\frak D}_2 = \lambda \beta_2/2.
$$
Denote by $c_k$ the $k$th Beilinson Chern class of the universal flat
bundle over $BGL_n({\Bbb C})^\delta$. This is an element of
$$
H_{\cal D}^{2k}(BGL_n({\Bbb C})^\delta,{\Bbb R}(k)) \approx
H^{2k-1}(GL_n({\Bbb C}),{\Bbb C}/{\Bbb R}(k)).
$$
In \cite{D-H-Z} it is shown that, for all $n$ and $k$, the classes
$c_k$ and $\beta_k/2$ in
$$
H^k(GL_n({\Bbb C})^\delta,{\Bbb C}/{\Bbb R}(2))
$$
are equal. It follows that $i {\frak D}_2$ is $\lambda$ times the
Beilinson Chern class
$$
c_2 : K_3({\Bbb C}) \to {\Bbb C}/{\Bbb R}(2).
$$
Denote the rational $K$-theory $K_\bullet(R)\otimes {\Bbb Q}$ of a ring $R$
by $K_\bullet(R)_{\Bbb Q}$.
Since $BGL(R)^+$ is an H-space, the Hurewicz homomorphism
$$
K_m(R)_{\Bbb Q} := \pi_m(BGL(R)^+)\otimes {\Bbb Q} \to
H_m(BGL(R)^+,{\Bbb Q}) \approx H_m(GL(R),{\Bbb Q})
$$
is injective. Define the {\it rank filtration}
$$
r_1K_m(R) \subseteq r_2K_m(R) \subseteq r_3K_m(R) \subseteq \cdots
\subseteq K_m(R)_{\Bbb Q}
$$
of $K_m(R)_{\Bbb Q}$ by
$$
r_kK_m(R) = K_m(R)_{\Bbb Q} \cap
{\rm im}\left\{H_m(GL_k(R),{\Bbb Q}) \to H_m(GL(R))\right\}.
$$
Suslin \cite{suslin} has proved that for all infinite fields $F$
$$
K_m(F)_{\Bbb Q} = r_m K_m(F)
$$
and that
$$
K_m(F)_{\Bbb Q}/r_{m-1}K_m(F) \approx K_m^M(F)\otimes {\Bbb Q},
$$
where $K_\bullet^M(F)$ denotes the Milnor $K$-theory of $F$.
In particular, for all infinite fields $F$, there is a canonical
isomorphism
$$
K_3(F)_{\Bbb Q} = K_3^M(F)_{\Bbb Q} \oplus r_2K_3(F).
$$
Since all elements of $K_3^M$ of any field are decomposable, the
generalization of the Whitney sum formula to $K$-theory implies
that
$$
c_2 : K_3({\Bbb C}) \to {\Bbb C}/{\Bbb R}(2)
$$
vanishes on $K_3^M({\Bbb C})$. It follows that $c_2$ factors through
the projection onto $r_2K_3({\Bbb C})$.
$$
\matrix{
K_3({\Bbb C}) & \mapright{c_2} & {\Bbb C}/{\Bbb R}(2) \cr
\mapdownalt{{\rm proj}} & \nearrow & \cr
r_2K_3({\Bbb C}) & & \cr}
$$
Since $r_2K_3({\Bbb C})$ comes from $H_3(GL_2({\Bbb C}))$, the restriction of
$c_2$ to $r_2K_3({\Bbb C})$ is given by the class $i\lambda^{-1}{\frak D}_2$.
Now Dupont \cite{dupont} has proved that the $i{\frak D}_2$ equals the
Cheeger-Simons class
$$
\hat{c}_2 \in H^3(GL_2({\Bbb C})^\delta,{\Bbb C}/{\Bbb R}(2)).
$$
of the universal rank 2 flat bundle. But by the main result of
\cite{D-H-Z}, this class equals the Beilinson Chern class of the
universal flat bundle. It follows that $\lambda = 1$. \quad $\square$
Most of this story has been extended to the trilogarithm by
Goncharov \cite{goncharov} and Yang \cite{yang_thesis}. In
\cite{hain-macpherson} a canonical single valued trilogarithm
$$
S_3 : \left\{ \matrix
{\hbox{ ordered 6-tuples of points \hfil} \cr
\hbox{ in ${\Bbb P}^2$, no 3 on a line \hfil} \cr} \right\}/
\hbox{projective equivalence} \to {\Bbb R}
$$
was constructed. It satisfies the seven term functional equation
$$
\sum_{j=0}^6 (-1)^j S_3(a_0,\ldots,\widehat{a_j},\ldots,a_6) = 0
$$
where $a_0,\ldots,a_6$ are points in ${\Bbb P}^2$, no 3 of which lie on a
line. This equation is an obvious generalization of the 5-term
equation satisfied by $D_2$. As in (\ref{gen_posn}), this function
determines a class in $H^5(GL_3({\Bbb C}),{\Bbb R})$.
As in the case of the dilogarithm, the cocycle condition is precisely
the functional equation. Goncharov and Yang both
showed that this class in $H^5(GL_3({\Bbb C}),{\Bbb R})$ is a non-zero rational
multiple of the class which corresponds to the Beilinson Chern class
of the universal flat bundle over $BGL_3({\Bbb C})^\delta$. Goncharov
found an explicit formula for $S_3$ in terms of his single valued
version of the classical trilogarithm (\ref{gonch_trilog}). The
appropriate analogue of Suslin's theorem is not known for $K_5({\Bbb C})$.
However, Yang \cite{yang} proved the rank conjecture
for all $K$-groups of all number fields except ${\Bbb Q}$. Their
work enables one to find formulas for the regulator mapping
$$
c_3 : K_5(F) \to \left[{\Bbb C}/{\Bbb R}(3)\right]^{r_1 + r_2}
$$
for all number fields $F$ in terms of values of the trilogarithm of
\cite{hain-macpherson} in the case of Yang, and the classical
trilogarithm in the case of Goncharov.
The regulator $c_2 : K_3({\Bbb C}) \to {\Bbb C}/{\Bbb Q}(2)$ can be written in
terms of the multivalued dilogarithm. This work goes back to
unpublished
work of Bloch and Wigner (cf. \cite{dupont-sah}). The construction is
similar to, but more complicated than, the construction of the
regulator
given above. We only sketch the construction. More details can be
found
in \cite{dupont-sah}.
For $x$ in the disk $|x-1/2|< 1/2$, define
$$
\rho(x) = {1\over 2}\left[\log x \wedge \log(1-x) +
2\pi i \wedge {1 \over 2 \pi i}
\left( \ln_2(1-x) -\ln_2(x) - \pi^2/6\right)\right]
\in \Lambda^2_{\Bbb Z} {\Bbb C}.
$$
Here all functions are taken to be the principal branches. This
function
is single valued, and therefore extends to a single valued function
$$
\rho : {\Bbb C}-\{0,1\} \to \Lambda^2_{\Bbb Z} {\Bbb C}.
$$
It satisfies a generalization of the 5-term equation satisfied
by $D_2$. If $x,y \in {\Bbb C}- \{0,1\}$ and $x \neq y$, then
$$
\rho(x) - \rho(y) + \rho(y/x) - \rho((1-y)/(1-x)) +
\rho((1-y^{-1})/(1-x^{-1})) = 0.
$$
Define ${\cal P}(F)$ to be the free abelian group generated
by $F -\{0,1\}$ subject to the relations
$$
[x] - [y] + [y/x] - [(1-y)/(1-x)] + [(1-y^{-1})/(1-x^{-1})] = 0.
$$
This is often called the {\it scissors congruence group}, or the {\it
Bloch
group}. The function $\rho$ induces a map
$$
\rho : {\cal P}({\Bbb C}) \to \Lambda^2_{\Bbb Z}{\Bbb C}.
$$
There is a natural homomorphism
$$
H_3(SL_2(F)) \to {\cal P}(F)
$$
whose construction is similar to that of the homomorphism
$H_3(GL_2(F)) \to H_\bullet(C_\bullet(F,n))$
given in the proof of (\ref{gen_posn}).
If $F$ is algebraically closed of characteristic 0, then there is
an exact sequence
$$
0 \to {\Bbb Q}/{\Bbb Z} \to H_3(SL_2(F)) \to {\cal P}(F) \to \Lambda^2_{\Bbb Z} F^\times
\to K_2(F) \to 0.
$$
A proof can be found in the appendix of \cite{dupont-sah}. The map
${\cal P}(F) \to \Lambda^2_{\Bbb Z} F^\times$ takes the generator $[x]$ of
${\cal P}(F)$ to
$(1-x)\wedge x$. The most right hand map takes $x\wedge y$ to
$\{x,y\}$.
The kernel of the map
$$
\Lambda^2 \exp : \Lambda_{\Bbb Z}^2 {\Bbb C} \to \Lambda_{\Bbb Z}^2 {\Bbb C}^\ast
$$
is ${\Bbb C}/{\Bbb Q}(2)$, where this is included in $\Lambda_{\Bbb Z}^2 {\Bbb C}$ by taking
the
coset of $\lambda$ to $2\pi i \wedge (\lambda/2\pi i)$. Since the
diagram
$$
\matrix{
{\cal P}({\Bbb C}) & \to & \Lambda^2_{\Bbb Z} {\Bbb C}^\times \cr
\mapdownalt{\rho}& & \mapdown{id} \cr
\Lambda_{\Bbb Z}^2 {\Bbb C} & \mapright{\wedge^2\exp} & \Lambda_{\Bbb Z}^2 {\Bbb C}^\ast}
$$
commutes, $\rho$ induces a homomorphism
$$
H_3(SL_2({\Bbb C})) \to {\Bbb C}/{\Bbb Q}(2).
$$
By \cite{dupont}, this represents the second Cheeger-Simons class
$c_2$
of the universal flat bundle over $BSL_2({\Bbb C})^\delta$. By the main
result
of \cite{D-H-Z}, this equals the Beilinson Chern class of the
universal
flat bundle over $BSL_2({\Bbb C})^\delta$. As above, $c_2$ vanishes on
$K_3^M({\Bbb C})$ and the diagram
$$
\matrix{
K_3({\Bbb C}) & \mapright{c_2} & {\Bbb C}/{\Bbb Q}(2) \cr
\mapdownalt{{\rm proj}} & \mapne{\rho} & \cr
r_2K_3({\Bbb C}) & & \cr}
$$
commutes. One can show that the map
$$
H_3(SL_2(F),{\Bbb Q}) \to r_2K_3(F)
$$
is surjective. It follows that the map constructed above induces the
regulator on all of $K_3({\Bbb C})$. Alternatively, one can appeal to the
theorem of Suslin \cite{suslin_K3ind} which asserts that there are
natural isomorphisms
$$
H_3(SL_2(F),{\Bbb Q}) \approx r_2K_3(F)\approx
\ker \left\{{\cal P}(F) \to \Lambda^2_{\Bbb Z} F^\times \right\}\otimes {\Bbb Q}
$$
for all fields $F$. This last isomorphism says that all of the weight
2
part of $K_3$ of a field comes from ${\Bbb P}^1 - \{0,1,\infty\}$ and that
all the relations come from the functional equation of the
dilogarithm.
This generalizes the fact mentioned in the introduction that the
relations
in $K_1$ come from the functional equation of the logarithm. The
analogue of this statement for the weight 3 part of $K_5$ is not
known
at this time, although Goncharov \cite{goncharov} has made
significant
progress.
\section{Iterated integrals}
\label{it-ints}
At this stage it is convenient to introduce Chen's iterated integrals
\cite{chen} Basic references for this section are \cite{chen},
\cite{geom}.
Suppose that $M$ is a manifold and that $\row w 1 r$ are smooth
${\Bbb C}$-valued
1-forms on $M$.
For each piecewise smooth path $\gamma : [0,1] \to M$, we can define
$$
\int_\gamma \row w 1 r =
\int \cdots \int_{0 \le t_1 \le \cdots \le t_r \le 1}
f_1(t_1) \cdots f_r(t_r)\, dt_1 \ldots dt_r,
$$
where $\gamma^\ast w_j = f_j(t)\, dt$ for each $j$.
This can be viewed as a ${\Bbb C}$-valued function
$$
\int_\gamma \row w 1 r : PM \to {\Bbb C}
$$
on the the space of piecewise smooth paths in $M$.
When $r=1$, $\int_\gamma w$ is just the usual line integral.
An {\it iterated integral} is any function $PM \to {\Bbb C}$ which is a
linear
combination of constant functions and basic iterated integrals
$$
\int_\gamma \row w 1 r.
$$
Now let $M = {\Bbb C} - \{0,1\}$ and
$$
\omega_0 = {dz \over z}\hbox{ and } \omega_1 = {dz \over 1-z}.
$$
Then
$$
\ln_1 x = - \log(1-x) = \int_0^x \omega_1.
$$
By induction and the definition, we have, for all $k\ge 2$,
$$
\ln_k x = \int_0^x \ln_{k-1} z\, \omega_0 =
\int_0^x \omega_1\overbrace{\omega_0 \ldots \omega_0}^{k-1\rm \; times}.
$$
Here the path of integration must be chosen so that once it has left
0,
it never passes through 0 or 1 on its way to $x$.
The basic properties of iterated integrals are summarized in the
following proposition.
\begin{proposition}{\rm \cite{chen,geom}}
\label{props}
Suppose that $w_1, w_2, \ldots $ are ${\Bbb C}$-valued 1-forms on a
manifold $M$.
\begin{enumerate}
\item[(i)] The value of $\int_\gamma \row w 1 r$ is independent of
the
parameterization of $\gamma$.
\item[(ii)] If $\alpha,\beta : [0,1] \to M$ are composable paths
(i.e. $\alpha(1) = \beta(0)$), then
$$
\int_{\alpha \beta} \row w 1 r =
\sum_{j=0}^r \int_\alpha \row w 1 i \int_\beta \row w {i+1} r.
$$
Here, $\int_\gamma \row \phi 1 m$ is to be interpreted as 1 when
$m=0$.
\item[(iii)] For all paths $\gamma$,
$$
\int_{\gamma^{-1}} \row w 1 r = (-1)^r \int_\gamma \row w r 1.
$$
\item[(iv)] For all paths $\alpha$ in $M$,
$$
\int_\alpha \row w 1 r \int_\alpha \row w {r+1} {r+s} =
\sum_\sigma \int_\alpha \row w {\sigma(1)} {\sigma(r+s)},
$$
where $\sigma$ ranges over all shuffles of type $(r,s)$.
\end{enumerate}
\end{proposition}
\section{The Regulator $K_2(X) \to H^1(X,\protect{\Bbb C}^\ast)$}
\label{heisenberg}
This section is an exposition of Bloch's construction of the
regulator
$$
c_2 : K_2(X) \to H_{\cal D}^2(X,{\Bbb Z}(2))
$$
using the dilogarithm. We have freely incorporated the elegant
approaches of Deligne \cite{deligne_letter} and
Ramakrishnan \cite{ramakrishnan_heisenberg,ramakrishnan_survey}.
When $X$ is a curve, there is a natural isomorphism
$$
H^2(X,{\Bbb Z}(2)) \approx H^1(X,{\Bbb C}/{\Bbb Z}(2)).
$$
Identifying ${\Bbb C}/{\Bbb Z}(2)$ with ${\Bbb C}^\ast$ by the map
$\lambda \mapsto \exp[\lambda/2\pi i]$, we obtain a canonical
identification of $H_{\cal D}^2(X,{\Bbb Z}(2))$ with $H^1(X,{\Bbb C}^\ast)$, the group
of flat line bundles over $X$.
We set
$$
H_{\Bbb Z} = \pmatrix{
1 & {\Bbb Z}(1) & {\Bbb Z}(2) \cr
0 & 1 & {\Bbb Z}(1) \cr
0 & 0 & 1 \cr }
$$
and
$$
H_{\Bbb C}= \pmatrix{
1 & {\Bbb C} & {\Bbb C} \cr
0 & 1 & {\Bbb C} \cr
0 & 0 & 1 \cr }
$$
There is a natural bundle projection
\begin{equation}\label{heis-bundle}
H_{\Bbb Z}\backslash H_{\Bbb C} \to {\Bbb C}^\ast \times {\Bbb C}^\ast;
\end{equation}
it takes the coset of
$$
\pmatrix{
1 & u & w \cr
0 & 1 & v \cr
0 & 0 & 1 \cr}
$$
to $(e^u,e^v)$. It has fiber ${\Bbb C}/{\Bbb Z}(2)$, which we identify with
${\Bbb C}^\ast$
as above.
\begin{proposition}\label{connection}
There is a natural connection on this bundle with curvature
$(dx/x) \wedge (dy/y)/2 \pi i$, where $x$ and $y$ are the coordinates
in
${\Bbb C}^\ast \times {\Bbb C}^\ast$.
\end{proposition}
\noindent{\bf Proof.~} First consider the pullback of the bundle (\ref{heis-bundle})
to ${\Bbb C}\times{\Bbb C}$
\begin{equation}\label{lift}
{\Bbb Z}(2)\backslash H_{\Bbb C} \to {\Bbb C} \times {\Bbb C}
\end{equation}
along the map $(u,v) \mapsto (e^u,e^v)$. The map
$$
H_{\Bbb C} \to {\Bbb C}^\ast \times {\Bbb C} \times {\Bbb C}
$$
defined by
$$
\pmatrix{
1 & u & w \cr
0 & 1 & v \cr
0 & 0 & 1 \cr}
\mapsto (\exp(w/2\pi i), u,v)
$$
induces an isomorphism of ${\Bbb Z}(2)\backslash H_{\Bbb C}$ with
${\Bbb C}^\ast \times {\Bbb C} \times {\Bbb C}$ which commutes with the projections to
${\Bbb C}\times {\Bbb C}$. So the bundle (\ref{lift}) is trivial, and sections
of it can be identified with maps $\zeta : {\Bbb C} \times {\Bbb C} \to {\Bbb C}^\ast$.
Define a connection on this bundle by
$$
\nabla \zeta = d \zeta - \zeta udv/2 \pi i
$$
This connection is easily seen to be invariant under the left action
$$
(n,m) : (\zeta, u,v) \mapsto (e^{nv}\zeta, u + 2\pi i n, v + 2\pi i
m)
$$
of ${\Bbb Z}\times {\Bbb Z}$ on ${\Bbb C}^\ast \times {\Bbb C} \times {\Bbb C}$ induced by the left
action of $H_{\Bbb Z}$ on $H_{\Bbb C}$. It therefore descends to a connection on
the bundle (\ref{heis-bundle}). The connection form of the pullback
bundle is $udv/2\pi i$, from which it follows that its curvature is
$du\wedge dv/2\pi i$ and that the curvature of (\ref{heis-bundle}) is
$(dx/x) \wedge (dy/y)/2\pi i$. \quad $\square$
Now suppose that $X$ is a smooth curve over ${\Bbb C}$. Denote the function
field of $X$ by ${\Bbb C}(X)$ and the generic point ${\rm spec\, } {\Bbb C}(X)$ of $X$ by
$\eta_X$. We first define the regulator on $K_2(\eta_X) :=
K_2({\Bbb C}(X))$.
The Deligne cohomology of $\eta_X$ is defined by
$$
H_{\cal D}^m(\eta_X,\Lambda(k)) = \lim_{\to} H_{\cal D}^m(U,\Lambda(k))
$$
where the limit is taken over all Zariski open subsets $U$ of $X$. In
particular, we have
$$
H_{\cal D}^2(\eta_X,{\Bbb Z}(2)) = \lim_{\to} H^1(U,{\Bbb C}^\ast).
$$
This latter group is the group of flat line bundles at the generic
point
of $X$---elements of this group are flat line bundles defined on some
Zariski open subset of $X$, and two such are identified if they agree
on
a smaller open subset. The product is tensor product.
By Matsumoto's Theorem \cite{milnor_book}, $K_2(\eta_X)$ is generated
by symbols $\{f,g\}$, where $f,g \in {\Bbb C}(X)^\times$. The only
relations
which hold between these symbols are bilinearity
$$
\{f_1f_2,g\} = \{f_1,g\}\{f_2,g\},\quad \
\{f,g_1g_2\} = \{f,g_1\}\{f,g_2\}
$$
and the Steinberg relation
$$
\{1-f,f\} = 1
$$
whenever $f$ and $1-f$ are both in ${\Bbb C}(X)^\times$.
Now suppose that $\{f,g\}\in K_2(\eta_X)$. There is a Zariski open
subset $U$ of $X$ such that $f$ and $g$ are both defined and
invertible
on $U$. They therefore define a regular function
$$
(f,g) : U \to {\Bbb C}^\ast \times {\Bbb C}^\ast.
$$
The pullback of the line bundle $H_{\Bbb Z}\backslash H_{\Bbb C}$ to $U$ is flat
as it has curvature a multiple of $(df/f)\wedge (dg/g)$, which is
zero
as $U$ is a curve.
Denote it by $\langle f,g\rangle$. It is an element of
$H^1(U,{\Bbb C}^\ast)$,
and therefore of $H^1(\eta_X,{\Bbb C}^\ast)$.
\begin{remark}\label{natural}
Observe that this construction makes sense when $X$ is a Riemann
surface and ${\Bbb C}(X)$ denotes the field of meromorphic functions of
$X$.
It is natural with respect to holomorphic maps between Riemann
surfaces.
\end{remark}
\begin{proposition}\label{cocycle}
If $f,g$ are invertible functions on the Zariski open subset $U$ of
$X$, the monodromy of the flat bundle $\langle f,g \rangle$ about
a loop $\gamma$ based at $p\in U$ is
$$
I(f,g,\gamma) := \int_\gamma \dlog f \dlog g -
\log g(p) \int_\gamma \dlog f + \log f(p) \int_\gamma \dlog g
\in {\Bbb C}/{\Bbb Z}(2).
$$
\end{proposition}
\noindent{\bf Proof.~} We will deduce the result by computing the monodromy about
a loop $\gamma$ in ${\Bbb C}^\ast \times {\Bbb C}^\ast$. The assertion will then
follow by pulling back the answer to $U$. Let $\gamma$ be a path in
${\Bbb C}^\ast \times {\Bbb C}^\times$ which begins at $(x_0,y_0)$.
For $t \in [0,1]$, denote the path $s \mapsto \gamma(st)$ by
$\gamma_t$. The horizontal lift of $\gamma$ to $H_{\Bbb Z}\backslash H_{\Bbb C}$
which begins at the coset of
$$
\pmatrix{
1 & \log x_0 & w \cr
0 & 1 & \log y_0 \cr
0 & 0 & 1 }
$$
is
$$
t \mapsto H_{\Bbb Z}
\pmatrix{
1 & \log x_0 & w \cr
0 & 1 & \log y_0 \cr
0 & 0 & 1 }
\pmatrix{
1 & \int_{\gamma_t} \dlog x & \int_{\gamma_t} \dlog x \dlog y \cr
0 & 1 & \int_{\gamma_t} \dlog y \cr
0 & 0 & 1 \cr}.
$$
That is,
$$
t \mapsto \pmatrix{
1 & \log x_0 + \int_{\gamma_t} \dlog x & w +
\int_{\gamma_t} \dlog x \dlog y + \log x_0 \int_{\gamma_t} \dlog y
\cr
0 & 1 & \log y_0 + \int_{\gamma_t} \dlog y \cr
0 & 0 & 1 \cr}
$$
Now suppose that $\gamma$ is a loop. Then the endpoint of the
horizontal
lift of $\gamma$ is congruent to the matrix
$$
\pmatrix{
1 & -\int_\gamma \dlog x & \int_{\gamma} \dlog x \int_{\gamma} \dlog
y \cr
0 & 1 & -\int_\gamma \dlog y \cr
0 & 0 & 1 \cr}
\pmatrix{
1 & \log x_0 + \int_{\gamma} \dlog x & w +
\int_{\gamma} \dlog x \dlog y + \log x_0 \int_{\gamma} \dlog y \cr
0 & 1 & \log y_0 + \int_{\gamma} \dlog y \cr
0 & 0 & 1 \cr}
$$
$$
=
\pmatrix{
1 & \log x_0 & w + \int_{\gamma} \dlog x \dlog y +
\log x_0 \int_{\gamma} \dlog y - \log y_0 \int_\gamma \dlog x \cr
0 & 1 & \log y_0 \cr
0 & 0 & 1 \cr}
$$
modulo the left action of $H_{\Bbb Z}$. It follows that the holonomy about
$\gamma$ is
$$
\int_{\gamma} \dlog x \dlog y + \log x_0 \int_{\gamma} \dlog y -
\log y_0 \int_\gamma \dlog x \hbox{ mod }{{\Bbb Z}(2)}.
$$
The result follows by pulling the result back to $U$ along the map
$(f,g)$. \quad $\square$
\begin{proposition}\label{symbol}
If $f,f_1,f_2,g \in {\Bbb C}(X)^\times$, then
$$
\langle f_1 f_2, g \rangle = \langle f_1,g \rangle \langle f_2, g
\rangle
\quad \hbox{and}\quad \langle f,g \rangle^{-1} = \langle g,f \rangle.
$$
Moreover, if $f,1-f\in {\Bbb C}(X)^\ast$, then
$$
\langle 1-f,f \rangle = 1.
$$
\end{proposition}
\noindent{\bf Proof.~}
It is clear from properties of the logarithm and (\ref{props}) that
$$
I(f_1f_2,g, \gamma) = I(f_1,g,\gamma) + I(f_2,g,\gamma).
$$
It is not difficult to use (\ref{props}) to prove that
$$
I(f,g,\gamma) + I(g,f,\gamma) = 0.
$$
These imply the linearity and skew symmetry of the symbol
$\langle f,g \rangle$.
It suffices to prove the Steinberg relation in the universal case
where $U = {\Bbb C} - \{0,1\}$ and $f=x$. The line bundle
$\langle 1-x,x \rangle$ is trivial as a flat bundle if and only if it
has a flat section. The dilogarithm provides such a section.
Define $s : {\Bbb C} - \{0,1\} \to H_{\Bbb Z}\backslash H_{\Bbb C}$ by
$$
s(x) = H_{\Bbb Z}\pmatrix{
1 & \log(1-x) & - \ln_2 x \cr
0 & 1 & \log x \cr
0 & 0 & 1 \cr}
$$
This section is flat, so the Steinberg relation holds. \quad $\square$
\begin{theorem}{\rm{\cite{beilinson}}}
Taking $\{f,g\}$ to $\langle f,g \rangle$ defines a map
$$
K_2(\eta_X) \to H^1(\eta_X,{\Bbb C}^\ast)
$$
which is the Chern class $c_2$.
\end{theorem}
\noindent{\bf Proof.~} The first assertion is an easy consequence of Matsumoto's
description of $K_2$ and (\ref{symbol}). The second follows from the
fact
that the symbol $\{f,g\}$ is the cup product of
$f,g\in K_1(\eta_X)\approx {\Bbb C}(X)^\times$.
Properties of Chern classes then imply that
$$
c_2(\{f,g\}) = c_1(f) \cup c_1(g)
$$
where the right hand side is the cup product of
$$
c_1(f), c_2(g) \in H_{\cal D}^2(\eta_X,{\Bbb Z}(1)) \approx {\Bbb C}(X)^\ast.
$$
Under this isomorphism, $c_1$ is just the identity.
The formula for the cup product in Deligne cohomology implies that
$c_1(f)\cup c_1(g)$ is represented by the element of
$$
H_{\cal D}^2(\eta_X,{\Bbb Z}(2)) \approx H^1(\eta_X,{\Bbb C}/{\Bbb Z}(2))
$$
defined by
$$
\gamma \mapsto I(f,g,\gamma).\quad \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3
$$
We now globalize this construction. For each $x\in X$, there is map
$$
\delta_x : H^1(\eta_X,{\Bbb C}^\ast) \to {\Bbb C}^\ast.
$$
To define $\delta_x(l)$, represent $l$ by a flat line bundle $L \to
U$
over a Zariski open subset $U$ of $X$. Chose a small closed disk
$\overline{\Delta}$ in $X$, centered at $x$, such that $\overline{\Delta} - \{x\}$ is
contained in $U$. Define $\delta(l)$ to be the monodromy of $L$ about
the boundary of $\overline{\Delta}$.
Suppose that $\nu : F^\times \to {\Bbb Z}$ is a valuation on a field $F$.
Let $\cal O$ be the associated valuation ring (i.e., 0 and those
elements
of $F^\times$ with valuation $\ge 0$). Let ${\frak P}$ be the maximal
ideal of $\cal O$. The tame symbol of $f,g \in F^\times$ is defined
by
$$
(f,g)_\nu =
(-1)^{\nu(f)\nu(g)} {f^{\nu(g)}\over g^{\nu(f)}}\hbox{ mod } {\frak P}
$$
(See, e.g., \cite{milnor}.)
To each $x\in X$ associate the valuation which takes a function $f$
to its order $\nu_x(f)$ at $x$. In this way we associate a tame
symbol
$(\phantom{x},\phantom{x})_x$ to each $x\in X$.
\begin{proposition}\label{tame}
Suppose that $x \in X$. If $f,g \in {\Bbb C}(X)$, then
$$
\delta_x\langle f,g \rangle = (f,g)_x.
$$
\end{proposition}
\noindent{\bf Proof.~} Since both sides of the expression in the statement of the
proposition are skew symmetric and bilinear, we can reduce, with the
help of (\ref{natural}),
to the following 2 cases. First, if $z$ is a local holomorphic
parameter
about $x$, then we have to show that $\delta_x\langle z,z \rangle =
-1$.
Second, if $f$ is a unit in a neighbourhood of $x$, then
$$
\delta_x\langle f,g \rangle = f^{\nu_x(g)}(x)
$$
{}From (\ref{cocycle}) and (\ref{props})(ii) it follows that
$$
I(z,z,p) = \int_{\partial \Delta} \dlog z \dlog z = {1\over 2}
\left(\int_{\partial \Delta}\dlog z\right)^2
= {(2\pi i )^2 \over 2} \in {\Bbb C}/{\Bbb Z}(2),
$$
where $\Delta$ is a sufficiently small imbedded disk in $X$ centered
at $x$.
Under the standard isomorphism ${\Bbb C}/{\Bbb Z}(2) \approx {\Bbb C}^\ast$, this
corresponds to $e^{i\pi}= -1$. This proves the first assertion.
To prove the second, write $f = e^\phi$, where $\phi$ is holomorphic
in
a neighbourhood of $x$. By (\ref{cocycle}), we have
$$
I(f,g,p) = \int_{\partial \Delta} (\phi(z) - \phi(p))\dlog g
+ \phi(p)\int_{\partial \Delta}\dlog g =
\int_{\partial \Delta}\phi(z)\dlog g.
$$
By the Residue Theorem this equals $2 \pi i \nu_x(g)\phi(x)$.
So
$$
\delta_x \langle f,g \rangle = \exp\left(\nu_x(g)\phi(x)\right)=
f(x)^{\nu_x(g)}.
$$
This proves the second assertion. \quad $\square$
The following version of the Gysin sequence
is easily verified.
\begin{proposition}\label{gysin}
The sequence
$$
0\to H^1(X,{\Bbb C}^\ast) \to H^1(\eta_X,{\Bbb C}^\ast)
\mapright{\oplus \delta_x} \bigoplus_{x\in X}{\Bbb C}\ast
$$
is exact. \quad $\square$
\end{proposition}
The analogue of the Gysin sequence in algebraic $K$-theory is the
localization sequence (reference !). In our case, it asserts that
the sequence
$$
K_2(X) \to K_2(\eta_X)
\mapright{\oplus (\phantom{x},\phantom{x})_x}
\bigoplus_{x\in X} {\Bbb C}^\ast
$$
is exact. By (\ref{tame}), the diagram
$$
\matrix{ & &
K_2(X) & \to & K_2(\eta_X) &
\mapright{\oplus (\phantom{x},\phantom{x})_x} &
\bigoplus_{x\in X} {\Bbb C}^\ast \cr
& & & & \mapdown{c_2} & & \parallel \cr
0 & \to & H^1(X,{\Bbb C}^\ast) & \to & H^1(\eta_X,{\Bbb C}^\ast) &
\mapright{\oplus \delta_x}& \bigoplus_{x\in X}{\Bbb C}\ast\cr }
$$
commutes. Since the rows are exact, the Chern class
$c_2$ induces a map $K_2(X) \to H^1(X,{\Bbb C}^\ast)$ which must be the
Chern class by naturality.
This construction extends easily to give a description of the
regulator
$$
c_2 : K_2(X) \to H_{\cal D}^2(X,{\Bbb Z}(2))
$$
where $X$ is a smooth variety over ${\Bbb C}$. The construction proceeds in
the same way, except that the line bundles are no longer flat.
For a mixed Hodge structure $H$ denote
${\rm Hom}_{\rm Hodge}({\Bbb Z},H)$, the set of ``Hodge classes'' in $H$ of type
$(0,0)$, by $\Gamma H$.
\begin{proposition}
If $X$ is smooth over ${\Bbb C}$, then there is a natural isomorphism
between $H_{\cal D}^2(X,{\Bbb Z}(2))$ and the group which consists of the pairs
$(L,\nabla)$, where $L$ is a holomorphic line
bundles over $X$ and $\nabla$ is a holomorphic connection whose
curvature
times $2\pi i$ lies in $\Gamma H^2(X,{\Bbb Z}(2))$. \quad $\square$
\end{proposition}
\section{The polylogarithm variation of mixed Hodge structure}
\label{polylog_varn}
Let $X$ be a smooth complex algebraic curve and $\overline{X}$ a smooth
compactification of it. Let $D = \overline{X} -X$. Recall from
\cite{steenbrink-zucker}
that a {\it variation of mixed Hodge structure} over $X$ consists of
\begin{enumerate}
\item a ${\Bbb Q}$ local system ${\Bbb V} \to X$ which has a filtration by local
systems
$$
\cdots \subseteq {\Bbb W}_{l-1} \subseteq {\Bbb W}_l \subseteq {\Bbb W}_{l+1}
\subseteq \cdots
$$
which exhausts ${\Bbb V}$ and whose intersection is 0. We shall denote the
fiber
of ${\Bbb V}$ over $x\in X$ by $V_x$ and the fiber of ${\Bbb W}_l$ by $W_l V_x$.
We will
also assume that each local monodromy operator $T_P : V_P \to V_P$,
about
each $P \in D$, is unipotent.
\item a Hodge filtration
$$
\cdots \supseteq {\cal F}^{p-1}\supseteq {\cal F}^p \supseteq {\cal F}^{p+1}
\supseteq \cdots
$$
of the corresponding holomorphic vector bundle ${\cal V} := {\Bbb V} \otimes_{\Bbb Q}
{\cal O}_X$
by holomorphic sub-bundles. These are required to satisfy Griffiths'
transversality: If
$$
\nabla : {\cal V} \to {\cal V} \otimes_{{\cal O}_X} \Omega^1_X
$$
is the natural flat connection, then
$$
\nabla ({\cal F}^p) \subseteq {\cal F}^{p-1}\otimes_{{\cal O}_X} \Omega^1_X.
$$
Denote the fiber of ${\cal F}^p$ over $x\in X$ by $F^p V_x$.
\item For each $x\in X$, the filtrations $W_\bullet V_x$ and
$F^\bullet V_x$ define a mixed Hodge structure on $V_x$.
\item Denote Deligne's canonical extension of ${\cal V}$ to $\overline{X}$
by $\overline{\cV} \to \overline{X}$ \cite{deligne_diffeq}. The Hodge bundles ${\cal F}^p$
are
required to extend to holomorphic
sub-bundles $\overline{\cF}^p$ of $\overline{\cV}$. (Note that the weight bundles
${\cal W}_l := {\Bbb W}_l \otimes_{\Bbb Q} {\cal O}_X$ automatically extend to sub-bundles
$\overline{\cW}$ of $\overline{\cV}$ as they are flat.)
\item about each point $P\in D$, there is a relative weight
filtration
\cite{steenbrink-zucker}. This is an important condition which is
rather
technical in general. However, in the case where the global monodromy
representation
$$
\rho_x : \pi_1(X,x) \to GL(V_x)
$$
is unipotent, the condition reduces to the much simpler condition
$$
N_P(W_l V_x) \subseteq W_{l-2} V_x
$$
for each $P\in D$, where $N_P$ is the local monodromy logarithm
$$
N_P = {1 \over 2 \pi i} \log T_P.
$$
(See \cite{hain-zucker_1}.)
\end{enumerate}
\begin{theorem}\label{polylog-vmhs}
The $n$th polylogarithm local system underlies a good variation of
mixed
Hodge structure whose weight graded quotients are canonically
isomorphic
to ${\Bbb Q},{\Bbb Q}(1), \ldots, {\Bbb Q}(n)$.
\end{theorem}
\noindent{\bf Proof.~}
Let ${\Bbb V} \to {\Bbb C} - \{0,1\}$ be the $n$th polylogarithm local
system, and ${\cal V}$ the corresponding holomorphic vector bundle. By
(\ref{poly-loc-sys}), the canonical extension of this to ${\Bbb P}^1$ is
the
trivial bundle
$$
{\Bbb P}^1 \times {\Bbb C}^{n+1} \to {\Bbb P}^1.
$$
Denote the standard basis of ${\Bbb C}^{n+1}$ by
$e_0,e_1,\ldots, e_n$. The fiber $V_x$ is the ${\Bbb Q}$ linear span of
$\lambda} %{\mbox{\boldmath$\lambda$}_0(x), \lambda} %{\mbox{\boldmath$\lambda$}_1(x),\ldots, \lambda} %{\mbox{\boldmath$\lambda$}_n(x)$, the rows of $\Lambda(x)$.
Define
\begin{equation}\label{filt_1}
W_{-2l + 1}\, {\Bbb C}^{n+1} = W_{-2l}\, {\Bbb C}^{n+1} =
{\rm span}\left\{e_l,\ldots,e_n\right\}
\end{equation}
and
\begin{equation}\label{filt_2}
F^{-p}{\Bbb C}^{n+1} = {\rm span}\left\{e_0,\ldots,e_p\right\}.
\end{equation}
Define
$$
\overline{\cF}^p = {\Bbb P}^1 \times F^p{\Bbb C}^{n+1} \subseteq \overline{\cV}
\hbox{ and } \overline{\cW}_l = {\Bbb P}^1 \times W_l{\Bbb C}^{n+1} \subseteq \overline{\cV}.
$$
Observe that the weight filtration comes from a filtration defined
on ${\Bbb V}$:
$$
W_{-2l + 1}V_x = W_{-2l}V_x =
{\rm span}\left\{\lambda} %{\mbox{\boldmath$\lambda$}_l(x), \ldots, \lambda} %{\mbox{\boldmath$\lambda$}_n(x)\right\}\quad \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3
$$
Suppose that ${\Bbb V} \to X$ is a good variation of mixed Hodge structure
with unipotent monodromy about each point of $D = \overline{X} - X$. Let
$P \in D$. For each non-zero tangent vector $\vec{v}$ of $X$ at $p$, there
is a canonical mixed Hodge structure on $V_{\Bbb C}$, the fiber of $\overline{\cV}$
over $P$. This is called the {\it limit mixed Hodge structure
associated
to} $\vec{v}$. The Hodge and weight filtrations on ${\Bbb V}_{\Bbb C}$ are defined by
letting $F^p V_{\Bbb C}$ and $W_l V_{\Bbb C}$ be the fibers of $\overline{\cF}^p$ and
$\overline{\cW}_l$ over $P$, respectively. To construct the limit mixed Hodge
structure, we have to construct a rational form $V_{\Bbb Q}$ of $V_{\Bbb C}$
and show that the weight filtration defined above is the
complexification of a filtration of $V_{\Bbb Q}$.
To construct $V_{\Bbb Q}$, choose an imbedded closed disk $\overline{\Delta}$ in
$\overline{X}$ centered at $P$. Let $t$ be a holomorphic parameter in
$\overline{\Delta}$ such that $t(P) = 0$ and $|t|=1$ is $\partial \overline{\Delta}$.
By choosing the disk to be small enough, we may suppose that
$\overline{\Delta} -\{0\} \subseteq X$.
We first consider the case where $\vec{v} = \partial/\partial t$. Let
$x \in \overline{\Delta}$ be the point corresponding to $t=1$. Choose a
${\Bbb Q}$ basis $v_1,\ldots,v_m$ of $V_x$, the fiber of ${\Bbb V}$ over $x$.
Let $v_1(t),\ldots,v_m(t)$ be flat (possibly multivalued) sections
of ${\Bbb V}$ over $\overline{\Delta}^\ast$ which satisfy $v_j(1) = v_j$ for each
$j$. Let $T : V_x \to V_x$ be the local monodromy operator, and
$N= \log T/2 \pi i$ be the local monodromy logarithm. For each $j$,
define
$$
s_j(t) = t^{-N} v_j(t).
$$
Then each $s_j(t)$ is a single valued section of $\overline{\cV}$ over
$\overline{\Delta}^\ast$. In fact, by the construction of the canonical
extension $\overline{\cV} \to \overline{X}$, the $s_j$ comprise a local framing of
$\overline{\cV}$
over $\overline{\Delta}$. In particular, $s_1(0),\ldots,s_m(0)$ is a ${\Bbb C}$
basis of $V_{\Bbb C}$. Define the rational form $V_{\Bbb Q}$ of $V_{\Bbb C}$ which
corresponds to $\partial /\partial t$ to be the ${\Bbb Q}$-linear span of
$s_1(0),\ldots,s_m(0)$. By choosing the basis $v_1,\ldots,v_m$ of
$V_x$ to be adapted to its weight filtration, one can easily show
that the weight filtration of $V_{\Bbb C}$ is the complexification of a
filtration of $V_{\Bbb Q}$. The ${\Bbb Q}$ structure on $V_{\Bbb C}$ which corresponds
to the tangent vector $\vec{v} = \lambda \partial/\partial t$ is defined
to be
$$
V_{\Bbb Q}(\vec{v}) = \lambda^N V_{\Bbb Q}.
$$
It is not difficult to show that this rational structure depends only
on the tangent vector $\vec{v}$, and not on the choice of the parameter
$t$.
If the weight graded quotients of ${\Bbb V} \to X$ are constant as
variations of
Hodge structure (e.g. the polylogarithm variations), it is not
difficult
to show that $((V_{\Bbb Q}(\vec{v}),W_\bullet),(V_{\Bbb C}, F^\bullet))$ is a mixed
Hodge structure, and that $N: V \to V(-1)$ is a morphism of mixed
Hodge structures.
The following result is due to Deligne \cite{deligne_letter} in the
case
$n=2$. It is a straight forward computation using the monodromy
computation (\ref{monod}) and the procedure described above.
\begin{theorem}\label{limit}
Let $z$ be the natural coordinate function on ${\Bbb C}-\{0,1\}$.
The limit mixed Hodge structure on the $n$th polylogarithm
variation at the tangent vector $\partial /\partial z$
at 0 has rational structure spanned by the vectors
$$
\pmatrix{s_0 \cr s_1 \cr \vdots \cr s_n} = \left(
\begin{array}{c| c c c c}
1 & 0 & 0 & \cdots & 0 \\ \hline
0 & 2\pi i & 0 & \cdots & 0 \\
\vdots & 0 & \ddots & & \vdots \\
0 & \cdots & \cdots & 0 & (2\pi i)^n\\
\end{array}\right)
\pmatrix{e_0 \cr e_1 \cr \vdots \cr e_n}
$$
The limit mixed Hodge structure associated with the tangent vector
$-\partial/\partial z$ at 1 has rational structure spanned by the
vectors
$$
\pmatrix{s_0 \cr s_1 \cr \vdots \cr s_n} = \left(
\begin{array}{c| c c c c}
1 & 0 & \zeta(2) & \cdots & \zeta(n) \\ \hline
0 & 2\pi i & 0 & \cdots & 0 \\
\vdots & 0 & \ddots & & \vdots \\
0 & \cdots & \cdots & 0 & (2\pi i)^n\\
\end{array}\right)
\pmatrix{e_0 \cr e_1 \cr \vdots \cr e_n}
$$
where $\zeta(s)$ denotes the Riemann zeta function.
In both cases, the Hodge and weight filtrations are defined as in
(\ref{filt_1}) and (\ref{filt_2}).
\end{theorem}
\section{Mixed Hodge structure on $\pi_1$}
\label{fund_gp}
In this section we give the construction of the mixed Hodge structure
on the fundamental group $\pi_1(X,x)$ in the special case where $X$
is a smooth variety over ${\Bbb C}$ whose $H^1(X)$ is pure of weight 2.
This purity condition may be restated in several equivalent ways.
For example, $H^1(X)$ is pure of weight 2 if and only if one and
(and hence all) smooth completions $\overline{X}$ of $X$ have first Betti
number 0. In particular, all Zariski open subsets of a Grassmannians
have this property.
To put a mixed Hodge structure on $\pi_1(X,x)$, it is necessary to
linearize it. The first step is to replace the fundamental group
by its group algebra ${\Bbb Q}\pi_1(X,x)$. This object is not well enough
behaved, and we need to linearize it further, which we do by
completion.
Let $J$ be the augmentation ideal. That is, $J$ is the kernel of the
augmentation
$$
\epsilon : {\Bbb Q}\pi_1(X,x) \to {\Bbb Q}
$$
which takes each $g\in \pi_1(X,x)$ to 1. The powers of $J$ define a
topology on ${\Bbb Q}\pi_1(X,x)$, and we consider
$$
{\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;} = \lim_\rightarrow {\Bbb Q}\pi_1(X,x)/J^m,
$$
the $J$-adic completion.
The completed group ring is a complete Hopf algebra. That is, it is a
topological algebra and has a continuous algebra homomorphism
$$
\Delta : {\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;} \to
{\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;} \widehat\otimes {\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}
$$
Here $\widehat\otimes$ denotes completed tensor product.
This algebra homomorphism is induced by the usual diagonal
$$
{\Bbb Q}\pi_1(X,x) \to {\Bbb Q}\pi_1(X,x) \otimes {\Bbb Q}\pi_1(X,x)
$$
which takes each $g\in {\Bbb Q}\pi_1(X,x)$ to $g\otimes g$.
The Malcev Lie algebra ${\frak g}(X,x)$ associated to $\pi_1(X,x)$ is, by
definition, the set of primitive elements
$$
\left\{ X\in {\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;} : \Delta X =
1 \otimes X + X\otimes 1 \right\}
$$
of ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$.
This is a complete topological Lie algebra. The bracket of the
elements
$A,B$ of ${\frak g}$ is their commutator $AB - BA$, which is again
primitive.
The topology is induced from that of ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$. The
completed
group ring may be recovered from its primitive elements as its
completed
enveloping algebra $U~\widehat{\!}{\;} {\frak g}(X,x)$ (see \cite[Appendix A]{quillen}).
A mixed Hodge structure on ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$ is, by definition,
a compatible sequence of mixed Hodge structures
$$
\cdots \to {\Bbb Q}\pi_1(X,x)/J^3 \to {\Bbb Q}\pi_1(X,x)/J^2 \to
{\Bbb Q}\pi_1(X,x)/J \to {\Bbb Q} \to 0
$$
on the truncations of the group ring. Note that each of these is a
finite dimensional vector space.
If the product and diagonal of ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$ are morphisms
of mixed Hodge structure, then, as ${\frak g}(X,x)$ is the kernel of the
reduced diagonal
$$
\overline{\Delta} : {\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;} \to
\left[{\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}/{\Bbb Q}\right] \widehat\otimes
\left[{\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}/{\Bbb Q} \right] \approx J\widehat\otimes J,
$$
the Malcev Lie algebra inherits a mixed Hodge structure compatible
with
its Lie algebra structure. This mixed Hodge structure determines the
one on ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$ as ${\frak g}(X,x)$ generates ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$
topologically.
\begin{theorem}{\rm \cite{morgan},\cite{hain} }
If $(X,x)$ is a complex algebraic variety, then ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$
and
${\frak g}(X,x)$ have canonical mixed Hodge structures which are compatible
with their algebraic structures. \end{theorem}
It is important to note that if $H^1(X,{\Bbb Q})$ is non-trivial,
then this mixed Hodge structure depends non-trivially on the
basepoint
$x\in X$ (cf. \cite[\S\S6,7]{geom}).
We will sketch the construction of this mixed Hodge structure in the
case
when $H^1(X)$ is pure of weight 2. Write
$X = \overline{X} - D$, where $\overline{X}$ is smooth and complete, and $D$ is a divisor
with normal crossings in $\overline{X}$. For convenience, we set
$$
\Omega^\bullet(X) = H^0(\overline{X}, \Omega_{\overline{X}}^\bullet(\log D)).
$$
By results of Deligne \cite[(3.2.14)]{deligne_II}, every element of
$\Omega^\bullet(X)$ is closed and the obvious map
$$
\Omega^\bullet(X) \to H^\bullet(X,{\Bbb C})
$$
is injective. The tensor algebra
$$
T := \bigoplus_{n\in {\Bbb N}} \Omega^1(X)^{\otimes(-n)}
$$
on the dual of $\Omega^1(X)$
is a Hopf algebra; the diagonal is defined as the algebra
homomorphism
which takes each $X \in \Omega^1(X)^\ast$ to $1\otimes X + X\otimes
1$.
The ideal $({\rm im} \delta)$ in $T$ generated by the image
of the dual of the cup product
$$
\delta : \Omega^2(X)^\ast \to
\Lambda^2 \Omega^1(X)^\ast \subseteq \Omega^1(X)^{\otimes(-2)}
$$
is a Hopf ideal. It follows that
$$
A := T/({\rm im} \delta)
$$
is a Hopf algebra whose diagonal is induced from that of the tensor
algebra. The set of primitive elements of $A$ is
$$
PA = {\Bbb L}(\Omega^1(X)^\ast)/({\rm im} \delta),
$$
where ${\Bbb L}(V)$ denotes the free Lie algebra generated by the vector
space
$V$.
Denote the ideal generated by $\Omega^1(X)^\ast$ by $I$. The
powers of $I$ define a topology on $A$. Denote the $I$-adic
completion
of $A$ by $A~\widehat{\!}{\;}$. The set of primitive elements of $A~\widehat{\!}{\;}$ is the
$I$-adic completion of $PA$.
The following result is a special case of a theorem of K.-T. Chen
\cite[(3.5)]{chen}.
\begin{proposition}
For each $x\in X$ there is a canonical isomorphism
$$
\Theta_x : {\Bbb C}\pi_1(X,x)~\widehat{\!}{\;} \to A~\widehat{\!}{\;}
$$
of complete Hopf algebras.
\end{proposition}
\noindent{\bf Proof.~} Let $\omega \in \Omega^1(X) \otimes \Omega^1(X)^\ast$ be the
element which corresponds to the identity
$\Omega^1(X) \to \Omega^1(X)$. This can be viewed as an element of
$\Omega^1(X)\otimes PA~\widehat{\!}{\;}$. This form is integrable. That is,
$$
d\omega + \omega \wedge \omega = 0.
$$
It follows that the value of the $A$-valued iterated integral
$$
1 + \int\omega + \int \omega\omega + \int\omega\omega\omega + \cdots
$$
on each path in $X$ depends only on its homotopy class relative to
its end points (cf. \cite[\S3]{geom}, for example). It follows from
this and
(\ref{props})(ii) that
this map induces a well defined homomorphism from $\pi_1(X,x)$ into
the group of units of $A~\widehat{\!}{\;}$. This extends to an algebra
homomorphism
$$
{\Bbb C}\pi_1(X,x) \to A~\widehat{\!}{\;}.
$$
Since the augmentation ideal of ${\Bbb C}\pi_1(X,x)$ is mapped into the
ideal
$I$ of $A~\widehat{\!}{\;}$, it follows that this homomorphism extends to a
continuous algebra homomorphism
$$
\Theta_x : {\Bbb C}\pi_1(X,x)~\widehat{\!}{\;} \to A~\widehat{\!}{\;}.
$$
The property (\ref{props})(iv) implies that $\Theta_x$ commutes
with
the diagonals; that is, $\Theta_x$ is a Hopf algebra homomorphism.
The graded module associated to the filtration of
${\Bbb C}\pi_1(X,x)~\widehat{\!}{\;}$ by powers of $J$ is generated by $J/J^2$, and
this
is isomorphic to $H_1(X)$. Similarly, the graded module associated to
the filtration of $A~\widehat{\!}{\;}$ by the powers of $I$ is generated by
$I/I^2$,
which is also isomorphic to $H_1(X)$. The map $\Theta_x$ induces an
isomorphism $J/J^2 \approx I/I^2$. Since both algebras are complete,
this implies that $\Theta_x$ is surjective. One can show, without
too
much difficulty, that the map $J^2/J^3 \to I^2/I^3$ induced by
$\Theta_x$
is also an isomorphism (cf. \cite[(6.1)]{geom}). It is then
relatively
straightforward to show that
$\Theta_x$ must be injective. The idea is that $A~\widehat{\!}{\;}$ contains no
other
relations other than those that are consequences of the quadratic
ones,
while ${\Bbb C}\pi_1(X,x)~\widehat{\!}{\;}$ has at least these relations. Since
$\Theta_x$
is well defined, it must be an isomorphism. \quad $\square$
\medskip
The next step in constructing the mixed Hodge structure on
${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$ is to define the Hodge and weight filtrations.
We do this by defining them on $A~\widehat{\!}{\;}$ and transferring them to
${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$ via the isomorphism $\Theta_x$.
The ring $A$ is graded as the ideal $({\rm im} \delta)$ is graded. Write
$$
A = \bigoplus_{n\in {\Bbb N}} A_n
$$
where $A_n$ is the image of $\Omega^1(X)^{\otimes(-n)}$ in $A$.
The assumption that $H^1(X)$ be pure of weight 2 implies that it
is has Hodge type $(1,1)$. Consequently,
$\Omega^1(X)^\ast\approx H_1(X)$ has Hodge type $(-1,-1)$. It is
therefore
natural to define Hodge and weight filtrations on $A$ by
$$
F^{-p} A = \bigoplus_{n\le p} A_n
$$
and
$$
W_{-m} A = \bigoplus_{n\ge m/2}A_n
$$
Now define Hodge and weight filtrations on ${\Bbb Q}\pi_1(X,x)/J^l$ by
transferring the Hodge and weight filtrations from $A/I^l$ via
the isomorphism
$$
\Theta_x : {\Bbb Q}\pi_1(X,x)/J^l \to A/I^l.
$$
This data defines a mixed Hodge structure on
${\Bbb Q}\pi_1(X,x)/J^l$. To see this, first observe that
$$
Gr^W_m{\Bbb Q}\pi_1(X,x)/J^l = \cases{
J^r/J^{r+1} & when $m = -2r$ and $0\le r < l$;\cr
0 & otherwise.\cr}
$$
The Hodge filtration induced on $Gr^W_{-2p}$ satisfies
$$
F^{-p}\left[J^p/J^{p+1}\right] = J^p/J^{p+1},
\quad F^{-p+1}\left[J^p/J^{p+1}\right] = 0
$$
when $0\le p <l$. Because the weight filtration is defined over ${\Bbb Q}$,
it follows that these filtrations define a mixed Hodge structure on
${\Bbb Q}\pi_1(X,x)/J^l$ whose weight graded quotients are all of even
weight,
and where the $2p$th graded quotient is of type $(p,p)$. Since the
multiplication and comultiplication of $A$ preserve the
filtrations and are defined over ${\Bbb Q}$, they are morphisms of mixed
Hodge structure. It follows that ${\frak g}(X,x)$, endowed with the induced
filtrations, is a mixed Hodge structure.
We conclude this section by relating this mixed Hodge structure to
unipotent variations of mixed Hodge structure. A variation of
mixed Hodge structure over a smooth variety $X$ is good if its
restriction
to every curve satisfies the conditions in \S\ref{polylog_varn}. A
good
variation of mixed Hodge structure ${\Bbb V} \to X$ over a smooth variety
$X$
is {\it unipotent } if one (and hence all) monodromy representations
\begin{equation}\label{rep}
\rho_x : \pi_1(X,x) \to {\rm Aut} V_x
\end{equation}
are unipotent. This condition is equivalent to the condition that
each
of the variations of Hodge structure $Gr^W_m {\Bbb V}$ be constant.
The monodromy representation (\ref{rep}) induces a map
$$
\theta_x : {\Bbb Q}\pi_1(X,x) \to {\rm End} V_x.
$$
Since the representation is unipotent, there exists $l$ such that
$J^l$ is contained in $\ker \theta_x$. It follows that there is
an algebra homomorphism
\begin{equation}\label{ind_rep}
\theta_x : {\Bbb Q}\pi_1(X,x)/J^l \to {\rm End} V_x.
\end{equation}
Both sides of this last equation have natural mixed Hodge structures.
\begin{theorem}{\rm \cite{hain-zucker_1}}\label{morphism}
For each $ x\in X$, the representation (\ref{ind_rep}) is a morphism
of mixed Hodge structures.
\end{theorem}
Define the category of Hodge theoretic representations of
$\pi_1(X,x)$
to be the set of pairs $(V,\rho)$, where $V$ is a mixed Hodge
structure
and $\rho$ is a unipotent representation $\pi_1(X,x) \to {\rm Aut} V$
which induces a morphism of mixed Hodge structure
$$
\theta_x : {\Bbb Q}\pi_1(X,x)/J^l \to {\rm End} V
$$
for $l$ sufficiently large.
Theorem \ref{morphism} implies that taking the fiber at $x$ defines
a functor from the category of unipotent variations of mixed Hodge
structure over $X$ to the category of Hodge theoretic representations
of $\pi_1(X,x)$.
\begin{theorem}{\rm \cite{hain-zucker_1}}\label{equivalence}
This functor is an equivalence of categories.
\end{theorem}
The proofs of Theorems \ref{morphism} and \ref{equivalence} in the
case
when $H^1(X)$ is pure of weight 2 are considerably simpler than
in general case. (See \cite{hain-zucker_2} for a proof in this
case.)
A good variation of mixed Hodge structure ${\Bbb V}$ over a smooth variety
$X$ is called a {\it Tate variation of mixed Hodge structure} if all
of its weight graded quotients are constant and of even weight, and
if each of the variations $Gr^W_{2p}{\Bbb V}$ is of type $(p,p)$. The
polylogarithm
variations are examples of Tate variations of mixed Hodge structure.
Now suppose that $\overline{X}$ is any smooth compactification of $X$ where
$D = \overline{X} - X$ is a divisor with normal crossings in $\overline{X}$. Let
$\overline{\cV} \to \overline{X}$ be the canonical extension of ${\Bbb V}$ to $\overline{X}$. The
following
result is a simple consequence of Theorem \ref{morphism} and
\cite[(6.4)]{riem_hilb}.
\begin{theorem}\label{trivial}
If ${\Bbb V} \to X$ is a Tate variation of mixed Hodge structure, then its
canonical extension $\overline{\cV} \to \overline{X}$ is trivial as a holomorphic vector
bundle, so that there is a complex vector space $V$ and a bundle
isomorphism
$$
\matrix{
\overline{\cV} & \to & V\times \overline{X} \cr
\downarrow & & \downarrow \cr
\overline{X} & = & \overline{X} \cr}
$$
Moreover, there are filtrations $F^\bullet$ and $W_\bullet$ of $V$
such that the extended Hodge and weight bundles $\overline{\cF}^p$ and
$\overline{\cW}_l$ correspond to $F^p \times \overline{X}$ and $W_l \times \overline{X}$,
respectively, under the bundle isomorphism. \quad $\square$
\end{theorem}
One important example of a unipotent variation of mixed Hodge
structure over a smooth variety $X$ is the one whose fiber over
$x\in X$ is the truncated group ring ${\Bbb Q}\pi_1(X,x)/J^l$. This
is a good variation because the monodromy representation
$$
{\Bbb Q}\pi_1(X,x)/J^l \to {\rm Aut} {\Bbb Q}\pi_1(X,x)/J^l
$$
can be written in terms of left and right multiplication, and
is thus a morphism of mixed Hodge structure. Such variations form
an inverse system of variations, and we call the inverse limit
the {\it tautological variation} over $X$. In case when $H^1(X)$
has weight 2, this variation is a Tate variation of mixed Hodge
structure, and can be described explicitly. View $A~\widehat{\!}{\;}$ as a
subalgebra of ${\rm End}\, A~\widehat{\!}{\;}$ via the right regular representation.
\begin{proposition}\label{tautological}
If $H^1(X)$ is of weight 2, then the tautological variation over $X$
has canonical extension $A~\widehat{\!}{\;} \times \overline{X} \to \overline{X}$. The connection
form
of the canonical flat connection on this bundle is given by the
$PA~\widehat{\!}{\;}$ valued 1-form
$$
\omega \in \Omega^1(X) \otimes H_1(X) \subseteq
\Omega^1(X) \otimes PA~\widehat{\!}{\;}
\subseteq \Omega^1(X)\otimes {\rm End}\, A~\widehat{\!}{\;}
$$
which corresponds to the canonical isomorphism $\Omega^1(X)
\approx H^1(X)$. The extended Hodge and weight bundles
are $F^p A~\widehat{\!}{\;} \times \overline{X}$ and $W_l A~\widehat{\!}{\;} \times \overline{X}$.
\quad $\square$
\end{proposition}
\section{Hodge theoretic interpretation of regulators}
\label{hodge-theory}
In this section we give a Hodge theoretic interpretation of the
regulators constructed in Sections \ref{reg-k3} and \ref{heisenberg}.
These interpretations are due to Deligne \cite{deligne_letter}.
Throughout this section, $\Lambda$ will denote ${\Bbb Z},{\Bbb Q}$ or ${\Bbb R}$.
Suppose that $V=(V_\Lambda,(V_{\Lambda\otimes\Q},W_\bullet),(V_{\Bbb C},F^\bullet))$ is a
mixed Hodge structure where the underlying lattice $V_\Lambda$ is
torsion free. The ring
of endomorphisms ${\rm End} V$ has a mixed Hodge structure whose Hodge and
weight filtrations are defined by
$$
F^p {\rm End}_{\Bbb C} V =
\left\{ \phi \in {\rm End}_{\Bbb C} V : \phi(F^q V)\subseteq F^{p+q}V\right\}
$$
and
$$
W_l {\rm End}_{\Bbb Q} V =
\left\{ \phi \in {\rm End}_{\Lambda\otimes\Q} V : \phi(W_m V)\subseteq W_{m+l}V\right\}.
$$
Set ${\frak g} = W_{-1}{\rm End} V$. This is a nilpotent Lie algebra with a mixed
Hodge structure---the bracket being the commutator
$[\phi,\psi] = \phi\psi - \psi\phi$. The subspace $F^0{\frak g}$ is a Lie
sub-algebra. Denote the simply connected Lie groups which correspond
to ${\frak g}_{\Bbb C}$ and $F^0{\frak g}$ by $G$ and $F^0 G$, respectively. These are
unipotent subgroups of ${\rm Aut}_{\Bbb C} V$. Set
$G_\Lambda = G \cap {\rm End}_\Lambda V$. We view $G_\Lambda$ as acting on
the right of $V_\Lambda$.
For each $g\in G$, the triple
$(V_\Lambda g,(V_{\Lambda\otimes\Q} g,W_\bullet g),(V_{\Bbb C},F^\bullet))$
is a mixed Hodge structure whose weight graded quotients are
canonically
isomorphic to those of $V$. It is not difficult to show that every
mixed
Hodge structure with torsion free lattice and weight
graded quotients canonically isomorphic to those of $V$
can be constructed this way. More generally, we have the following
result
which is easily proved (cf. \cite{carlson}).
\begin{proposition}\label{moduli}
The set of $\Lambda$-mixed Hodge
structures whose weight graded quotients are canonically isomorphic
those of $V$ is naturally isomorphic to
$$
G_\Lambda\backslash G/F^0G.
$$
The double coset of $g\in G$ corresponds to the mixed Hodge structure
$$
V=(V_\Lambda g,(V_{\Lambda\otimes\Q} g,W_\bullet g),(V_{\Bbb C},F^\bullet)).\quad \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3
$$
\end{proposition}
This identification can be used to compute extension groups of mixed
Hodge
structures. Suppose that $A$ and $B$ are $\Lambda$-Hodge structures
whose underlying $\Lambda$ module is torsion free. Suppose that the
weight of $A$ is greater than that of $B$. If we take $V = A\oplus B$
then, for $R = \Lambda,{\Bbb C}$,
$$
G_R = {\rm Hom}_R(A,B)
$$
In this case the moduli space of mixed Hodge structures with weight
graded
quotients canonically isomorphic to $A$ and $B$ is the group
${\rm Ext}_{\cal H}^1(A,B)$
of extensions of $A$ by $B$ in the category ${\cal H}$ of mixed Hodge
structures.
Applying Proposition (\ref{moduli}) to the split mixed Hodge
structure
$A\oplus B$, we obtain the well known formula for ${\rm Ext}_{\cal H}^1$
(cf. \cite{carlson}).
\begin{proposition}\label{extensions}
With $A$ and $B$ as above, there is a canonical isomorphism
$$
{\rm Ext}_{\cal H}^1(A,B) \approx
{{\rm Hom}_{\Bbb C}(A,B) \over {\rm Hom}_\Lambda(A,B) + F^0{\rm Hom}_{\Bbb C}(A,B)}.
$$
\end{proposition}
An important special case is where $A={\Bbb Z}$, $B = {\Bbb Z}(n)$ and $n\ge 1$.
(Recall that ${\Bbb Z}(n)$ is the Hodge structure of type $(-n,-n)$ whose
lattice
is the subgroup $(2\pi i)^n{\Bbb Z}$ of ${\Bbb C}$.) In this case we have
$$
{\rm Ext}_{\cal H}^1({\Bbb Z},{\Bbb Z}(n)) \approx {\Bbb C}/{\Bbb Z}(n).
$$
Following through the construction, we see that
the mixed Hodge structure which corresponds to $\lambda \in {\Bbb C}/{\Bbb Z}(n)$
can be described as follows. Denote the standard basis of ${\Bbb C}^2$ by
$e_0,e_n$. These have type $(0,0),(-n,-n)$, respectively. The
Hodge and weight filtrations on ${\Bbb C}^2$ are defined by
$$
W_l {\Bbb C}^2 = {\rm span} \{e_j: -j \le l \}
$$
and
$$
F^p {\Bbb C}^2 = {\rm span} \{ e_j : -j \ge p\}.
$$
The mixed Hodge structure which corresponds to $\lambda$
has integral basis the two vectors
$$
\pmatrix{1 & \lambda \cr 0 & (2\pi i)^n \cr}
\pmatrix{e_0 \cr e_n \cr }.
$$
In particular, the extension of ${\Bbb Z}$ by ${\Bbb Z}(1)$ which corresponds to
$x \in {\Bbb C}^\ast \approx {\Bbb C}/{\Bbb Z}(1)$ has integral basis spanned by the
vectors
$$
\pmatrix{1 & \log x \cr 0 & 2\pi i \cr}
\pmatrix{e_0 \cr e_1 \cr }.
$$
A unipotent variation of mixed Hodge structure over a smooth variety
$X$
whose weight graded quotients are canonically isomorphic to ${\Bbb Z}$ and
${\Bbb Z}(m)$, $(m\ge 1)$ will determine a {\it classifying map}
$X \to {\rm Ext}^1_{\cal H}({\Bbb Z},{\Bbb Z}(m))$.
\begin{proposition}\label{ext_varns}
When $m >1$ the classifying map is constant. When $m = 1$, a
map
$$
X \to {\rm Ext}^1_{\cal H}({\Bbb Z},{\Bbb Z}(1)) \approx {\Bbb C}^\ast
$$
is the classifying map
of a good variation of mixed Hodge structure if and only if it is an
algebraic function on $X$.
\end{proposition}
\noindent{\bf Proof.~} In both cases, the canonical extension of the variation to
a good compactification $\overline{X}$ of $X$ is a trivial bundle
(\ref{trivial}), as are the extended Hodge and weight bundles. In the
first case, Griffiths' transversality forces the integral lattice to
be constant. In the second, the regularity of the connection of the
canonical extension corresponds to the classifying map $X \to
{\Bbb C}^\ast$
of the variation having poles at infinity. \quad $\square$
Since there is a canonical isomorphism
$$
{\rm Ext}_{\cal H}^1(\Lambda,\Lambda(m)) \approx {\Bbb C}/\Lambda(m),
$$
the regulator $K_m({\Bbb C}) \to {\Bbb C}/\Lambda(m)$
can then be interpreted as a map
$$
K_m({\Bbb C}) \to {\rm Ext}_{\cal H}^1(\Lambda,\Lambda(m)).
$$
A motivic description of this regulator in the case when $m=3$
is given in \cite{BGSV}.
The regulator
$$
c_2 : K_2(X) \to H_{\cal D}^2(X,{\Bbb Z}(2))
$$
also admits a Hodge theoretic interpretation. This time we take our
reference Hodge structure $V$ to be the direct sum of ${\Bbb Z}(0)$,
${\Bbb Z}(1)$
and ${\Bbb Z}(2)$. The moduli space of mixed Hodge structures whose weight
graded quotients are canonically isomorphic to these Hodge structures
is
$$
H_{\Bbb Z}\backslash H_{\Bbb C}.
$$
where $H$ denotes the Heisenberg group defined in \S\ref{heisenberg}.
The bundle projection
$$
H_{\Bbb Z} \backslash H_{\Bbb C} \to {\Bbb C}^\ast \times {\Bbb C}^\ast
$$
may be interpreted as the map which takes a mixed Hodge structure
$V \in H_{\Bbb Z}\backslash H_{\Bbb C}$ to
$$
(V/{\Bbb Z}(2),W_2V)\in {\rm Ext}^1_{\cal H}({\Bbb Z},{\Bbb Z}(1)) \times {\rm Ext}^1_{\cal H}({\Bbb Z}(1),{\Bbb Z}(2))
\approx {\Bbb C}^\ast \times {\Bbb C}^\ast.
$$
We next consider the problem of determining which maps
$X \to H_{\Bbb Z}\backslash H_{\Bbb C}$ classify variations of mixed Hodge
structure.
\begin{proposition}\label{classifying}
A function $f : X \to H_{\Bbb Z}\backslash H_{\Bbb C}$ is the classifying map
of a variation of mixed Hodge structure over $X$ with weight graded
quotients canonically isomorphic to ${\Bbb Z},{\Bbb Z}(1)$ and ${\Bbb Z}(2)$ if and
only if\begin{enumerate}
\item f is holomorphic;
\item the composite
$$
X \mapright{f} H_{\Bbb Z}\backslash H_{\Bbb C} \to {\Bbb C}^\ast \times {\Bbb C}^\ast
$$
of $f$ with the canonical projection is algebraic;
\item the map $f : X \to H_{\Bbb Z}\backslash H_{\Bbb C}$ is a flat section
of the bundle $H_{\Bbb Z}\backslash H_{\Bbb C}^\ast \to {\Bbb C}^\ast \times {\Bbb C}^\ast$.
\end{enumerate}
\end{proposition}
\noindent{\bf Proof.~} The first statement corresponds to the fact that the
connection
on the bundle ${\cal V} = {\Bbb V}\otimes_{\Bbb Z} {\cal O}_X$ is holomorphic. The second
follows from (\ref{ext_varns}) and the fact that if ${\Bbb V}$ is a
variation, then so are ${\Bbb V}/{\Bbb Z}(2)$ and $W_2{\Bbb V}$. The last condition
corresponds to Griffiths' transversality. One needs to use the fact
that the canonical extension of ${\Bbb V}$ to a good compactification $\overline{X}$
of $X$ is trivial, and that the extended Hodge and weight bundles are
also trivial (\ref{trivial}). \quad $\square$
This result allows us to give an interpretation of the regulator
$$
c_2 : K_2(X) \to H_{\cal D}^2(X,{\Bbb Z}(2))
$$
constructed in Section \ref{heisenberg}: If $f,g$ are invertible
functions on $X$, then $c_2(\{f,g\})$ is the obstruction to finding
a good variation of mixed Hodge structure ${\Bbb V}$ over $X$ with weight
graded quotients ${\Bbb Z},{\Bbb Z}(1),{\Bbb Z}(2)$ and whose subquotients ${\Bbb V}/{\Bbb Z}(2)$
and $W_{-2}{\Bbb V}$ are classified by
$$
f : X \to {\Bbb C}^\ast \approx {\rm Ext}^1_{\cal H}({\Bbb Z},{\Bbb Z}(1)) \hbox{ and }
g : X \to {\Bbb C}^\ast \approx {\rm Ext}^1_{\cal H}({\Bbb Z}(1),{\Bbb Z}(2)).
$$
More on extensions of variations of mixed Hodge structure can be
found
in \cite{carlson-hain} and \cite{alg_cycles}.
\section{The polylogarithm quotient of
$\pi_1(\protect{\Bbb P}^1-\{0,1,\infty\}$}
\label{polog_quot}
The polylogarithm quotient of the the fundamental group of
${\Bbb P}^1-\{0,1,\infty\}$ is the image of the monodromy representation
$$
\pi_1({\Bbb P}^1-\{0,1,\infty\},x) \to {\rm Aut} P_x
$$
where $P \to {\Bbb P}^1-\{0,1,\infty\}$ is the polylogarithm variation of
mixed Hodge structure.
Since the monodromy representation of the $n$th polylogarithm local
system is unipotent, it induces a representation
$$
{\Bbb Q}\pi_1({\Bbb C} - \{0,1\},x)/J^{n+1} \to gl_{n+1}({\Bbb C}).
$$
Denote the image of the composite
$$
{\frak g}(X,x) \to {\Bbb Q}\pi_1({\Bbb C} - \{0,1\},x)/J^{n+1} \to gl_{n+1}({\Bbb C})
$$
by ${\frak p}_n(x)$.
\begin{proposition}
For each $x\in {\Bbb C}-\{0,1\}$, ${\frak p}_n(x)$ has a natural mixed Hodge structure
compatible with its Lie algebra structure. The local system of the
${\frak p}_n(x)$ forms a good unipotent variation of mixed Hodge structure
over
${\Bbb C}-\{0,1\}$. Finally, these local systems form an inverse system of
variations of mixed Hodge structure.
\end{proposition}
\noindent{\bf Proof.~} The first assertion is an immediate consequence of
(\ref{polylog-vmhs}) and (\ref{morphism}). The second is a
consequence
of (\ref{tautological}). The last assertion is clear. \quad $\square$
By the construction given in \S\ref{fund_gp}, the Malcev Lie algebra
of $\pi_1({\Bbb C}-\{0,1\},x)$ is the completion of the free Lie algebra
generated
by $H_1({\Bbb C}-\{0,1\},{\Bbb C})\approx \Omega^1({\Bbb C}-\{0,1\})^\ast$. Let $X_0,X_1$ be
the
basis of $H_1({\Bbb C}-\{0,1\},{\Bbb Z})$ consisting of the homology classes of the
loops $\sigma_0$, $\sigma_1$ defined in \S\ref{monodromy}. This is
dual to the basis $\omega_0/2 \pi i$,
$- \omega_1/2\pi i$ of $\Omega^1({\Bbb C}-\{0,1\})$, where $\omega_0,\omega_1$
are the forms defined in
\S\ref{monodromy}. The completed group ring ${\Bbb C}\pi_1({\Bbb C}-\{0,1\},x)~\widehat{\!}{\;}$
is isomorphic to the completion ${\Bbb C}\langle\langle X_0,X_1
\rangle\rangle$
of the free associative algebra generated by $X_0,X_1$. The set of
primitive elements of ${\Bbb C}\langle\langle X_0,X_1 \rangle\rangle$ is
${\frak f} = {\Bbb L}(X_0,X_1)~\widehat{\!}{\;}$, the completion of the free Lie algebra
generated by $X_0$ and $X_1$. The isomorphism
$$
{\Bbb C}\pi_1({\Bbb C}-\{0,1\},x)~\widehat{\!}{\;} \to {\Bbb C}\langle\langle X_0,X_1 \rangle\rangle
$$
is induced by the map $\pi_1({\Bbb C}-\{0,1\},x)$ which takes $\gamma$ to
$$
1 + \int_\gamma \omega + \int_\gamma \omega\omega +
\int_\gamma \omega\omega\omega + \cdots
$$
where $\omega$ is the ${\frak f}$-valued 1-form
\begin{equation}\label{con_form}
\omega = \omega_0 X_0 - \omega_1 X_1.
\end{equation}
\begin{proposition}
The monodromy representation
$$
\pi_1({\Bbb C}-\{0,1\},x)\to gl_{n+1}({\Bbb C})
$$
of the polylogarithm local system is induced by the homomorphism
${\frak f} \to gl_{n+1}$ defined by
$$
X_0 \mapsto
\pmatrix{
0 & 0 & 0 & \cdots & 0 \cr
& 0 & 1 & \ddots & \vdots \cr
\vdots & & \ddots & \ddots & 0 \cr
& & & \ddots & 1 \cr
0 & & \cdots & & 0 \cr }
\quad X_1 \mapsto
\left(\begin{array}{c| r c c c}
0 & -1 & 0 & \cdots & 0 \\ \hline
0 & 0 & 0 & \cdots & 0 \\
0 & 0 & 0 & & 0 \\
\vdots & \vdots & & \ddots & \\
0 & 0 & 0 & \cdots & 0\\
\end{array}\right)
$$
\end{proposition}
\noindent{\bf Proof.~} This follows as the connection matrix of the polylogarithm
local
system is
$$
\pmatrix{
0 & \omega_1 & 0 & \cdots & 0 \cr
& \ddots & \omega_0 & \ddots & \vdots \cr
\vdots & & \ddots & \ddots & 0 \cr
& & & \ddots & \omega_0 \cr
0 & & \cdots & & 0 \cr }
$$
which is simply the ${\frak f}$-valued form $\omega$ (\ref{con_form})
composed the the homomorphism
${\frak f} \to gl_{n+1}({\Bbb C})$ defined in the statement of the proposition.
\quad $\square$
\begin{corollary}
The complex form of the polylogarithm quotient has presentation
$$
{\frak p} = {\Bbb L}(X_0,X_1)~\widehat{\!}{\;} /({\rm ad}(X_1)[{\Bbb L},{\Bbb L}])
$$
and a topological basis $\left\{{\rm ad} (X_0)^n(X_1): n\in {\Bbb N}\right\}$.
\quad $\square$
\end{corollary}
\section{Motivic Description of the Polylogarithm Variation}
In this section we give a motivic description of the polylogarithm
variations. This description goes back to Deligne.
First suppose that $X$ is a topological space. Denote the space of
paths
$\gamma : [0,1] \to X$ by $PX$. There is a canonical projection $PX
\to
X \times X$ which takes a path $\gamma$ to its endpoints
$(\gamma(0),\gamma(1))$. Denote the fiber of this map over $(x,y)$ by
$P_{x,y}X$.
The group $H_0(P_{x,y}X)$ is the free abelian group on the set of
homotopy
classes of paths in $X$ from $x$ to $y$. These form a local system
\begin{equation}\label{canon_sys}
\left\{H_0(P_{x,y}X)\right\}_{(x,y)} \to X \times X.
\end{equation}
When $x=y$ there is a canonical isomorphism
$H_0(P_{x,y}X;{\Bbb Q})\approx {\Bbb Q}\pi_1(X,x)$. This has a canonical
filtration
given by the powers of the augmentation ideal $J$. This filtration
extends
to a flat filtration of the local system (\ref{canon_sys}). Denote
the completion of $H_0(P_{x,y}X;{\Bbb Q})$ in the corresponding topology
by $H_0(P_{x,y}X;{\Bbb Q})~\widehat{\!}{\;}$.
\begin{theorem}\label{canon_varn}
{\rm \cite{hain-zucker_1}}
If $X$ is a smooth algebraic variety, the local system
$$
\left\{H_0(P_{x,y}X;{\Bbb Q})~\widehat{\!}{\;}\right\}_{(x,y)} \to X \times X
$$
is a good variation of mixed Hodge structure whose fiber over $(x,x)$
is
the canonical mixed Hodge structure on ${\Bbb Q}\pi_1(X,x)~\widehat{\!}{\;}$.
\end{theorem}
We call this the {\it canonical variation of mixed Hodge structure}
associated to $X$. Although this construction appears to be outside
the domain of algebraic geometry, it can be made motivic. There are
several equivalent ways of doing this. One can be found
in \cite{deligne:line}; the other is a construction in topology of a
cosimplicial model of $PX$ and the fibration $PX \to X \times X$.
It is called the {\it cobar
construction} and dates back to the paper \cite{adams} of F.~Adams.
It makes sense for varieties over any base, as was noted by
Wojtkowiak
\cite{wojtkowiak}. Briefly, it is the cosimplicial space
$$
P^\bullet = X^{\Delta[1]_\bullet}
$$
where $\Delta[1]_\bullet$ is the standard simplicial model of the
unit interval. The projection $PX \to X\times X$ corresponds to the
map $X^{\Delta[1]_\bullet} \to X^{\partial \Delta[1]_\bullet}= X^2$
induced by the inclusion of the boundary of the interval.
Since the set of $m$-simplices of
$\Delta[1]_\bullet$ is the set of order preserving maps
$\{0,1,\ldots,m\} \to \{0,1\}$, and since there are precisely
$m+2$ of these,
$$
P^m = X \times X^m \times X.
$$
The coface maps $P^m \to P^{m+1}$ are the various diagonals. Applying
the de~Rham functor to this cosimplicial space yields the bar
construction
on the de~Rham complex of $X$, which is essentially Chen's
complex of iterated integrals on $PX$ (\cite{chen},
see also \cite[\S\S 1,2]{hain}).
Suppose that $X = \overline{X}-D$ is a smooth curve and that $\vec{v}$ is a tangent
vector at a point $P\in D$. Suppose that $x\in X$. Denote by
$P_{\vec{v},x}X$ the set of paths $\gamma :[0,1] \to \overline{X}$ which have the
property that $\gamma'(0) = \vec{v}$, $\gamma(1) = x$ and
$\gamma(]0,1]) \subseteq X$. This space is easily seen to be homotopy
equivalent to
$P_{z,x}X$ where $z$ is a point in $X$ which is sufficiently close
to $P$ in the direction of $\vec{v}$. More generally, one can define
$P_{\vec{v}_1,\vec{v}_2}X$ where $\vec{v}_1$ and $\vec{v}_2$ are non-zero tangent vectors
to points of $D$. Deligne defines $\pi_1(X,\vec{v})$ to be the set of path
components of $P_{\vec{v},\vec{v}}X$. It is canonically isomorphic to
$\pi_1(X,z)$
when $z\in X$ is sufficiently close to $P$ in the direction of $\vec{v}$
(cf.\ \cite{deligne:line}).
It is useful to think of the limit mixed Hodge structure of the
local system
$$
\left\{H_0(P_{z,x}X;{\Bbb Q})~\widehat{\!}{\;}\right\}_{z\in X} \to X
$$
associated to $\vec{v}$ as a mixed Hodge structure on
$H_0(P_{\vec{v},x}X;{\Bbb Q})~\widehat{\!}{\;}$. Denote the limit mixed Hodge structure on
the Malcev Lie algebra of the polylogarithm quotient of
$\pi_1({\Bbb C}-\{0,1\},\vec{v})$ associated to the tangent vector $\vec{v}$ by ${\frak p}(\vec{v})$.
\begin{theorem}
Let $\vec{v}$ be the tangent vector $\partial /\partial z$ at $0\in {\Bbb P}^1$.
The polylogarithm local system is the quotient of the variation of
mixed Hodge structure
$$
\left\{H_0(P_{\vec{v},z}{\Bbb C}-\{0,1\};{\Bbb Q})~\widehat{\!}{\;}\right\}_{z\in {\Bbb C}-\{0,1\}}
\to {\Bbb C}-\{0,1\}
$$
whose fiber over $\vec{v}$ is the polylogarithm quotient ${\frak p}(\vec{v})$
of $\pi_1({\Bbb C}-\{0,1\},\vec{v})$.
\end{theorem}
Since both variations are isomorphic as local systems, to prove the
theorem it suffices to show that the fibers of of the two variations
over
one particular
point (or tangent vector) are isomorphic as mixed Hodge structures.
It is not difficult to show that the fiber of the
quotient of the canonical variation over the tangent vector
$\partial /\partial z$ at 0 is isomorphic to that of the polylog
variation,
which was calculated in (\ref{limit}).
\bibliographystyle{plain}
|
1992-03-23T18:47:09 | 9202 | alg-geom/9202018 | en | https://arxiv.org/abs/alg-geom/9202018 | [
"alg-geom",
"math.AG"
] | alg-geom/9202018 | Sheldon Katz | Sheldon Katz | Arithmetically Cohen-Macaulay Curves cut out by Quadrics | 6 pages, LaTeX Version 2.09 | null | null | null | null | Addressing a question of M. Stillman, it had been shown by Ein, Eisenbud, and
the author that in a projective space of dimension at most 5, every
arithmetically Cohen-Macaulay curve which is cut out by quadrics scheme-
theoretically also has its homogeneous ideal generated by quadrics. In this
note it is shown that this is not the case in higher dimensional spaces.
| [
{
"version": "v1",
"created": "Wed, 19 Feb 1992 14:47:26 GMT"
},
{
"version": "v2",
"created": "Mon, 23 Mar 1992 17:47:01 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Katz",
"Sheldon",
""
]
] | alg-geom | \section{Homogeneous and Scheme-Theoretic Generation by Quadrics}
Let $X$ be a projective variety. It is often of interest to know whether or
not
the homogeneous ideal of $X$ can be generated by quadrics, e.\hskip0em g.\ if
$X$ is a
general canonical curve. In such a case, $X$ is cut out scheme-theoretically
by
quadrics as well. It is usually easier to verify the scheme-theoretic
statement--- this amounts to ignoring the vertex of the affine cone over $X$.
\bigskip\noindent
{\bf Problem:} Let $C\subset\P r={\bf C}\P r$ be a smooth curve which is cut
out scheme theoretically by quadrics. Is the homogeneous
ideal of $C$ necessarily cut out by quadrics?
\bigskip
In \cite{EEK}, this problem was investigated. The answer is a resounding
\underbar{no}. A counterexample was found with $r=5$. However, positive
results
were found. The problem has an
affirmative answer for curves on scrolls, all curves with $r\le 4$, and
arithmetically Cohen-Macaulay curves which lie on projectively normal {\rm K}3
surfaces cut out by quadrics (this includes all arithmetically C-M curves with
$r=5$). This leads to a more precise question, which we could not answer:
\bigskip\noindent
{\bf Question:} Let $C\subset\P r={\bf C}\P r$ be a smooth arithmetically
Cohen-Macaulay
curve which is cut out scheme theoretically by quadrics. Is the homogeneous
ideal of $C$ necessarily cut out by quadrics?
\bigskip
It turns out that this question also has a negative answer.
\begin{fact}
\label{main}
Let $C\subset\P 7$ be a general degree 19 embedding of a general genus 12 curve
over an algebraically closed field of characteristic 0. Then $C$ is smooth and
arithmetically Cohen-Macaulay, $C$ is cut out scheme-theoretically by quadrics,
and the homogeneous ideal of $C$ is not cut out by quadrics.
\end{fact}
\section{Candidates for a Counterexample}
Let $C\subset\P r$ be an arithmetically Cohen-Macaulay\ curve of degree $d$ and genus $g$.
Assume in addition that $\mbox{${\cal O}$}_C(1)$ is non-special, i.\hskip0em e.\
$H^1(\mbox{${\cal O}$}_C(1))=0$. Then $d=g+r$.
It turns out that for certain values of $g$ and $r$, the homogeneous ideal of
such a curve $C$ {\em cannot\/} be cut out by quadrics, for simple dimension
reasons. Let $I$ denote the ideal sheaf of $C$. Then if
\begin{equation}
\label{ineq23}
(r+1)h^0(I(2))<h^0(I(3)),
\end{equation}
the natural map
\begin{displaymath}
H^0(I(2))\otimes H^0(\mbox{${\cal O}$}_{\P r}(1))\to H^0(I(3))
\end{displaymath}
{\em cannot\/} be surjective, so that the homogeneous ideal of $C$ cannot be
generated by quadrics. Using Riemann-Roch, (\ref{ineq23}) becomes
\begin{equation}
\label{glow}
g>\frac{r(r-2)}3.
\end{equation}
On the other hand, if we want $C$ to be scheme-theoretically cut out by
quadrics,
then we must have enough quadrics, i.\hskip0em e.
$$\pmatrix{r\cr
2}-g\ge r-1.$$
Equality holds if and only if $C$ is a complete intersection of $r-1$ quadrics;
but in this case the homogeneous ideal is cut out by quadrics as well.
This can be improved slightly: in \cite[Cor.\ 2.5]{EEK} it was shown that if
$C$
is cut out scheme theoretically by $r$ quadrics, then necessarily
\begin{equation}
\label{r2}
g=(r-1)d/2+
1-2^{r-1}.
\end{equation}
So if (\ref{r2})
does not hold, then
\begin{equation}
\label{gup}
g\le\frac{r^2-3r-2}2
\end{equation}
There are no counterexamples to the main question for $r\le 5$ \cite{EEK}.
Suppose that there is a non-special
counterexample with $r=6$. Then $g\ge 9$
by (\ref{glow}). Since (\ref{gup}) gives $g\le 8$, it follows that (\ref{r2})
holds,
and $g=9$. But then $d=15$, and a contradiction is reached.
Turning next to $r=7$, (\ref{glow}) gives $g\ge 12$, and (\ref{gup}) gives
$g\le 13$.
In the following section, we show that in fact the {\em general \/} curve of
degree 19 and genus 12 in \P7 is a counterexample.
\section{The counterexample}
Pick 22 general points $p_1,p_2,p_3,q_1,\ldots ,q_7,r_1,\ldots ,r_{12}$ in
$\P2$.
Let $C'$ be a general plane curve of degree 9 passing through the $p_i$ with
multiplicity 3, through the $q_i$ with multiplicity 2, and simply through the
$r_i$. The linear system $|L|$ of degree 7 curves passing doubly through the
$p_i$ and
simply through the $q_i$ and $r_i$ maps $C'$ birationally to a smooth
curve $C$ of
degree 19 and arithmetic genus 12 in $P^7$.
It is a simple matter to use {\sc Macaulay}\ \cite{BS} to construct such a curve. In
describing the calculation, I will informally say that a general curve has a
certain property, when I mean that the property is satisfied for an example
curve constructed using {\sc Macaulay}'s pseudo-random number generator. In fact, I
repeated
the construction several times with different pseudo-random coefficients, and
the properties mentioned below held in each instance. Thus, as expected, a
``general'' curve has been constructed.
{\sc Macaulay}'s pseudo-random number generator is used to construct 22 ``general'' points
in
$\P2_{{\bf F}_{31991}}$, and from this the
curve $C'$ (actually, there is no harm in supposing that the $p_i$ are
$(1,0,0),
(0,1,0),(0,0,1)$, to shorten computations). By calculating the Jacobian of
$C'$, it is checked that the singular
scheme of $C'$ has degree 19 as expected (triple points count at least 4
times).
Hence
$C'$ has the expected geometric genus 12. The equations of the image curve
$C$ can then be explicitly
calculated. $C$ is cut out ideal theoretically by 9 independent quadrics and 2
independent cubics, and has Hilbert function $(1+6t+12t^2)(1-t)^{-2}$. In
particular $C$ has arithmetic genus 12; being the image of the normalization of
$C'$ by the base point free system $|L|$ on the blowup of \P2, it follows that
$C$ is smooth. Let
${\tilde C}\subset \P2$ be the scheme cut out by the 9
quadrics
alone. Via {\sc Macaulay}, ${\tilde C}$ has degree 19 and arithmetic genus 12. It
follows easily that $C={\tilde C}$, i.\hskip0em e.\ $C$ is cut out
scheme-theoretically
by quadrics.
Next, to see that $C$ is arithmetically Cohen-Macaulay, note that $C$ is non-special since the
projective
dimension of the embedding system is 7, is linearly normal by construction,
and is quadratically normal by Riemann-Roch and $h^0(I_C(2))=9$ found by {\sc Macaulay}.
This
suffices to show that $C$ is arithmetically Cohen-Macaulay\ by \cite[P.\ 222]{ACGH} or the argument in
the proof of Theorem 1.2.7 in \cite{L}.
\bigskip\noindent
{\em Proof of Proposition~\ref{main}:\/} The key point is to show that the
conditions ``arithmetically Cohen-Macaulay'' and ``scheme-theoretically cut out by quadrics'' are dense.
A curve is arithmetically Cohen-Macaulay\ if and only if it is projectively normal. So $C$ is arithmetically Cohen-Macaulay\ if
and
only if $H^1(I_C(n))=0$ for all $n\ge 0$, where $I_C$ is the ideal sheaf of
$C$.
By \cite{GLP}, $H^1(I_C(n))=0$ for all $n\ge 13$, so there are only finitely
many cohomology groups that are required to vanish in addition. By upper
semicontinuity of $h^1(I_C(n))=\dim\ H^1(I_C(n))$, this is a Zariski open
condition in the Hilbert scheme.
As to the condition of being scheme-theoretically cut out by quadrics, we may
restrict to considering curves which are arithmetically Cohen-Macaulay. Let $V$ be the 9 dimensional
space
of quadrics containing $C$. Consider the maps
\begin{equation}
\label{Vspan}
V\otimes H^0(\mbox{${\cal O}$}_{\P7}(k))\to H^0(I_C(k+2))
\end{equation}
$V$ cuts out $C$ scheme-theoretically if and only if (\ref{Vspan}) is
surjective for some $k\ge 12$ (since $C$ is 14-regular by \cite{GLP}; a
smaller bound for effective $k$ can be given if desired). This is again an
open condition.
Finally, let ${\rm Hilb}_{19n-11}^0$ be the subset of the Hilbert scheme
parametrizing smooth, irreducible curves in \P7 of degree 19 and genus 12. It
is open in the Hilbert scheme by \cite[P.\ 99]{GIT}. ${\rm Hilb}_{19n-11}^0$
is
defined over Spec ${\bf Z}$ and is irreducible (its geometric fibers are
equidimensional and irreducible; this follows from the
irreducibility of ${\cal M}_{12}$ in arbitrary characteristic \cite{DM}, and
the
non-speciality of $|L|$).
Hence the set of smooth arithmetically Cohen-Macaulay\ curves scheme-theoretically cut out by quadrics
is
non-empty and open, hence dense, in ${\rm Hilb}_{19n-11}^0$. This completes
the
proof of Proposition~\ref{main}. \begin{flushright} QED \end{flushright}
\bigskip
It seems appropriate to conclude with some related questions.
In \cite{ACGH}, \cite[\S 3]{GL}, it was proven that a general linear system of
degree
$d\ge [(3g+4)/2]$ on a
curve $C$ of genus $g$ embeds $C$ as an arithmetically Cohen-Macaulay\ curve. Rather than looking for
a bound for {\em all} curves, instead one can ask:
\bigskip\noindent
Problem: Find the smallest possible $d(g)$ such that for all $d\ge d(g)$,
a general curve of genus $g$
admits a degree $d$ complete embedding which is arithmetically Cohen-Macaulay.
\bigskip\noindent
{\em Remark.} Suppose that $d\ge (2g+1+\sqrt{8g+1})/2$. Then the
general degree $d$ embedding of a general curve of genus $g$ is arithmetically Cohen-Macaulay\
\cite{BE}.
This bound is in fact sharp for {\em non-special} embeddings.
The inequality is just the solution of the inequality
$h^0(\mbox{${\cal O}$}_{\P r}(2))\ge h^0(\mbox{${\cal O}$}_C(2))$ for a general non-special embedding.
\bigskip
Similarly, one can ask
\bigskip\noindent
Problem: Find the smallest possible $d'(g)$ such that a general degree $d$
embedding of a general curve of genus $g$ is scheme theoretically cut out by
quadrics if $d\ge d'(g)$.
\bigskip
By work of Green and Lazarsfeld \cite[Prop.\ 2.4.2]{L}, \mbox{$d'(g)\le
[(3g+6)/2]$}, and Proposition~\ref{main} shows that this is not sharp.
\bigskip\noindent
Question: Is Proposition~\ref{main} true without restriction on the
characteristic? Is there a counterexample to the main question with $r=6$?
|
1992-02-12T01:47:04 | 9202 | alg-geom/9202007 | en | https://arxiv.org/abs/alg-geom/9202007 | [
"alg-geom",
"math.AG"
] | alg-geom/9202007 | Tadao Oda | Tadao Oda | The algebraic de Rham theorem for toric varieties | 19 pages, LaTeX. To appear in Tohoku Math. J | null | null | null | null | On an arbitrary toric variety, we introduce the logarithmic double complex,
which is essentially the same as the algebraic de Rham complex in the
nonsingular case, but which behaves much better in the singular case. Over the
field of complex numbers, we prove the toric analog of the algebraic de Rham
theorem which Grothendieck formulated and proved for general nonsingular
algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely,
for a finite simplicial fan which need not be complete, the complex cohomology
groups of the corresponding toric variety as an analytic space coincide with
the hypercohomology groups of the single complex associated to the logarithmic
double complex. They can then be described combinatorially as Ishida's
cohomology groups for the fan. We also prove vanishing theorems for Ishida's
cohomology groups. As a consequence, we deduce directly that the complex
cohomology groups vanish in odd degrees for toric varieties which correspond to
finite simplicial fans with full-dimensional convex support. In the particular
case of complete simplicial fans, we thus have a direct proof for an earlier
result of Danilov and the author.
| [
{
"version": "v1",
"created": "Wed, 12 Feb 1992 00:42:54 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Oda",
"Tadao",
""
]
] | alg-geom | \section*{Introduction}
Let $\Delta$ be a finite fan for a free ${\bf Z}$-module $N$ of rank $r$,
and denote by $X:=T_N\mathop{\rm emb}\nolimits(\Delta)$ the associated $r$-dimensional
toric variety over the field ${\bf C}$ of complex numbers.
We also denote by ${\cal X}:=X^{{\rm an}}$ the associated complex analytic space.
In Section~1, we introduce the {\em logarithmic double complex\/}
${\cal L}_X^{{\textstyle \cdot},{\textstyle \cdot}}$ of ${\cal O}_X$-modules. The associated single complex
${\cal L}_X^{{\textstyle \cdot}}$, called the {\em logarithmic complex\/} for $X$, turns
out to be a natural toric generalization of the algebraic de Rham complex
for general nonsingular algebraic varieties.
Indeed, when $X$ is a nonsingular toric variety, then ${\cal L}_X^{{\textstyle \cdot}}$
is canonically quasi-isomorphic to the algebraic de Rham complex
$\Omega_X^{{\textstyle \cdot}}$.
More generally, suppose $\Delta$ is a simplicial fan, so that $X$
has at worst quotient singularities. Then ${\cal L}_X^{{\textstyle \cdot}}$ is
canonically quasi-isomorphic to the complex $\widetilde{\Omega}_X^{{\textstyle \cdot}}$ of
${\cal O}_X$-modules consisting of the Zariski differential forms on $X$.
By Danilov's Poincar\'e lemma \cite{danilov}
(see also Ishida \cite[Prop.~2.1]{ishida_derham}), we thus see that the
associated complex analytic complex $({\cal L}_X^{{\textstyle \cdot}})^{{\rm an}}$ is
quasi-isomorphic to the constant sheaf ${\bf C}_{{\cal X}}$.
We would like to point out here that
according to Ishida, $({\cal L}_X^{{\textstyle \cdot}})^{{\rm an}}[2r]$ is quasi-isomorphic to
the {\em globally normalized dualizing complex\/} of ${\bf C}_{{\cal X}}$-modules
in the sense of Verdier \cite{verdier}.
In Section 2, we recall the definition of Ishida's $p$-th complex
$C^{{\textstyle \cdot}}(\Delta,\Lambda^p)$ of ${\bf Z}$-modules for $0\leq p\leq r$
and its cohomology groups $H^q(\Delta,\Lambda^p)$,
which were considered in Ishida \cite{ishida_dualizing} and
\cite{odabook} in different notation.
Section 3 is devoted to the algebraic de Rham theorem for the logarithmic
complex ${\cal L}_X^{{\textstyle \cdot}}$.
Without any condition on the fan $\Delta$, we first show that the
algebraic hypercohomology group ${\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})$ is
a direct sum of the scalar extensions $H^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf C}}$
to ${\bf C}$ of Ishida's cohomology groups for various $p$'s.
When $\Delta$ is simplicial but not necessarily complete, we mimic the
proof, due to Grothendieck \cite{grothendieck},
of the usual algebraic de Rham theorem
to show that the algebraic hypercohomology group
${\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})$ is canonically isomorphic to the
complex analytic hypercohomology group
${\bf H}^{{\textstyle \cdot}}({\cal X},({\cal L}_X^{{\textstyle \cdot}})^{{\rm an}})$.
Consequently, we have our main result
\[
H^l({\cal X},{\bf C})=\bigoplus_{p+q=l}H^q(\Delta,\Lambda^p)_{{\bf C}}
\qquad\mbox{for each}\quad l
\]
for $\Delta$ simplicial but not necessarily complete.
In Section 4, we prove various vanishing theorems for Ishida's
cohomology groups.
When $\Delta$ is simplicial and complete, we have a direct combinatorial
proof for
\[
H^q(\Delta,\Lambda^p)_{{\bf Q}}=0\qquad\mbox{for}\quad q\neq p.
\]
Hence $H^l({\cal X},{\bf C})=0$ for $l$ odd, while
\[
H^{2p}({\cal X},{\bf C})=H^p(\Delta,\Lambda^p)_{{\bf C}}\qquad\mbox{for}\quad
0\leq p\leq r.
\]
In the complete nonsingular case, this was proved earlier by means of the
usual algebraic de Rham theorem by Danilov \cite{danilov}
and \cite[Thm.\ 3.11]{odabook}.
When $\Delta$ is simplicial with support $|\Delta|$ convex of dimension $r$
but not necessarily complete, we continue to have the same vanishing
theorem
\[
H^q(\Delta,\Lambda^p)_{{\bf Q}}=0\qquad\mbox{for}\quad q\neq p,
\]
which turns out to be a special case of a more general vanishing theorem
due to Ishida. In view of our algebraic de Rham theorem in Section~3,
we continue to have $H^l({\cal X},{\bf C})=0$ for $l$ odd, while
\[
H^{2p}({\cal X},{\bf C})=H^p(\Delta,\Lambda^p)_{{\bf C}}\qquad\mbox{for}\quad
0\leq p\leq r
\]
when $\Delta$ is simplicial with $|\Delta|$ convex of dimension $r$
but not necessarily complete.
The intersection cohomology theory of Goresky and MacPherson
\cite{gmI}, \cite{gmII}, which was further developed by
Beilinson, Bernstein and Deligne \cite{bbd}, attaches to
singular spaces new topological invariants much better behaved than
ordinary cohomology or homology groups.
Since a toric variety is a simple-minded singular space described
in terms of a fan, it is natural to try to describe its intersection
cohomology group and, more generally its intersection complex,
in terms of the fan as well.
We believe that the logarithmic complex
will play a key role in describing
the intersection cohomology group
(especially with respect to the middle perversity) of toric varieties.
There have been earlier attemps in this direction by
J. N. Bernstein, A. G. Khovanskii and R. D. MacPherson
(see Stanley \cite{stanley_aspm}), Denef and Loeser \cite{denef_loeser},
Fieseler \cite{fieseler} and others. However, they depend either on
the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber
(see Beilinson, Bernstein and Deligne \cite{bbd}
as well as Goresky and MacPherson \cite{gm_map})
or on the purity theorem of Deligne and Gabber
(cf.\ Deligne \cite{deligne_weilII}, \cite{deligne_gabber}).
This problem turns out to be closely related to the problem of finding an
elementary proof of the strong Lefschetz theorem for toric varieties,
and hence of the elementary proof due to Stanley \cite{stanley} of the
``$g$-theorem'' for simplicial convex polytopes conjectured earlier by
McMullen \cite{mcmullen}.
(See also \cite{lefsch}, \cite{hydera},
Stanley \cite{stanley_local_h} and
McMullen \cite{mcmullen2}.)
We refer the reader to \cite{odabook}
for basic results on toric varieties used in this paper.
Thanks are due to Dr.~M.-N.~Ishida for stimulating discussion.
\section{Logarithmic double complex}
Let $\Delta$ be a finite fan for a free ${\bf Z}$-module $N$ of rank $r$,
and denote by $X:=T_N\mathop{\rm emb}\nolimits(\Delta)$ the associated $r$-dimensional
toric variety over the field ${\bf C}$ of complex numbers.
The complement $D:=X\setminus T_N$ of the algebraic torus
$T_N:=N\otimes_{{\bf Z}}{\bf C}^*\cong({\bf C}^*)^r$ is a Weil divisor on $X$ but is
not necessarily a Cartier divisor.
The dual ${\bf Z}$-module $M:=\mathop{\rm Hom}\nolimits_{{\bf Z}}(N,{\bf Z})$, with the canonical bilinear
pairing $\langle\phantom{m},\phantom{m}\rangle :M\times N\rightarrow{\bf Z}$,
is isomorphic to the character group of the algebraic torus $T_N$.
For each $m\in M$ we denote the corresponding character by
$t^m:T_N\rightarrow{\bf C}^*$ (which was denoted by ${\bf e}(m)$ in \cite{odabook}),
and identify ${\bf C}[M]:=\oplus_{m\in M}{\bf C} t^m$ with the group algebra of $M$
over ${\bf C}$. Hence $T_N$ is the group of ${\bf C}$-valued points of the
group scheme $\mathop{\rm Spec}\nolimits({\bf C}[M])$.
Each $n\in N$ gives rise to a ${\bf C}$-derivation $\delta_n$ of ${\bf C}[M]$
defined by $\delta_n(t^m):=\langle m,n\rangle t^m$. Consequently, we
have a canonical isomorphism to the Lie algebra
\[
{\bf C}\otimes_{{\bf Z}}N\stackrel{\sim}{\longrightarrow}\mathop{\rm Lie}\nolimits(T_N),
\qquad 1\otimes n\mapsto\delta_n,
\]
hence an ${\cal O}_X$-isomorphism
${\cal O}_X\otimes_{{\bf Z}}N\stackrel{\sim}{\rightarrow}\Theta_X(-\log D)$, where the
right hand side is the sheaf of germs of algebraic vector fields on $X$
with logarithmic zeros along the Weil divisor $D$. Its dual
$\Omega^1_X(\log D)$ is the sheaf of germs of algebraic $1$-forms with
logarithmic poles along $D$, and we get an ${\cal O}_X$-isomorphism
\[
{\cal O}_X\otimes_{{\bf Z}}M\stackrel{\sim}{\longrightarrow}\Omega^1_X(\log D),
\qquad 1\otimes m\mapsto \frac{dt^m}{t^m}.
\]
Taking the exterior product, we thus get an ${\cal O}_X$-isomorphism
${\cal O}_X\otimes_{{\bf Z}}\bigwedge^{{\textstyle \cdot}}M\stackrel{\sim}{\rightarrow}
\Omega^{{\textstyle \cdot}}_X(\log D)$. The exterior differentiation $d$ on the
right hand side corresponds to the operation on the left hand side
which sends a $p$-form $t^m\otimes m_1\wedge\cdots\wedge m_p$ to the
$(p+1)$-form $t^m\otimes m\wedge m_1\wedge\cdots\wedge m_p$
(cf. \cite[Chap.~3]{odabook}).
Recall that the set of $T_N$-orbits in $X$ is in one-to-one correspondence
with $\Delta$ by the map which sends each $\sigma\in\Delta$ to the
$T_N$-orbit
\[
\mathop{\rm orb}\nolimits(\sigma)=\mathop{\rm Spec}\nolimits({\bf C}[M\cap\sigma^{\perp}])=T_{N/{\bf Z}(N\cap\sigma)}.
\]
The closure $V(\sigma)$ in $X$ of $\mathop{\rm orb}\nolimits(\sigma)$ is known to be a toric
variety with respect to a fan for the ${\bf Z}$-module $N/{\bf Z}(N\cap\sigma)$.
Namely,
\[
V(\sigma)=
T_{N/{\bf Z}(N\cap\sigma)}\mathop{\rm emb}\nolimits(\{(\tau+(-\sigma))/{\bf R}\sigma\;\mid\;
\tau\in\mathop{\rm Star}\nolimits_{\sigma}(\Delta)\}),
\]
where ${\bf R}\sigma=\sigma+(-\sigma)$ is the smallest ${\bf R}$-subspace containing
$\sigma$ of $N_{{\bf R}}:=N\otimes_{{\bf Z}}{\bf R}$, while
$\mathop{\rm Star}\nolimits_{\sigma}(\Delta):=\{\tau\in\Delta\mid \tau\succ\sigma\}$.
Hence $D(\sigma):=V(\sigma)\setminus\mathop{\rm orb}\nolimits(\sigma)$ is a Weil divisor
on $V(\sigma)$. In particular, we have
$\mathop{\rm orb}\nolimits(\{0\})=T_N$, $D(\{0\})=D$ and $V(\{0\})=X$.
For each integer $q$ with $0\leq q\leq r$, denote
$\Delta(q):=\{\sigma\in\Delta\mid \dim\sigma=q\}$. For each pair of
integers $p$, $q$, let
\[
{\cal L}^{p,q}_X:=\bigoplus_{\sigma\in\Delta(q)}
\Omega^{p-q}_{V(\sigma)}(\log D(\sigma))
=\bigoplus_{\sigma\in\Delta(q)}{\cal O}_{V(\sigma)}\otimes_{{\bf Z}}
\bigwedge\nolimits^{p-q}(M\cap\sigma^{\perp})\qquad\mbox{if}\quad
0\leq q\leq p,
\]
and ${\cal L}^{p,q}=0$ otherwise.
$d_{\mbox{\rm {\scriptsize I}}}:{\cal L}^{p,q}_X\rightarrow{\cal L}^{p+1,q}_X$ is defined to be the direct
sum of the exterior differentiation for each $\sigma\in\Delta(q)$.
We define $d_{\mbox{\rm {\scriptsize II}}}:{\cal L}^{p,q}_X\rightarrow{\cal L}^{p,q+1}_X$ as follows:
The $(\sigma,\tau)$-component of $d_{\mbox{\rm {\scriptsize II}}}$ for $\sigma\in\Delta(q)$ and
$\tau\in\Delta(q+1)$ is defined to be zero when $\sigma$ is not a face of
$\tau$. On the other hand, if $\sigma$ is a face of $\tau$, then
a primitive element $n\in N$ is uniquely determined modulo $N\cap{\bf R}\sigma$
so that $\tau+(-\sigma)={\bf R}_{\geq 0}n+{\bf R}\sigma$.
$M\cap\tau^{\perp}$ is a ${\bf Z}$-submodule of corank one
in $M\cap\sigma^{\perp}$. The $(\sigma,\tau)$-component of $d_{\mbox{\rm {\scriptsize II}}}$ in
this case is then defined to be the tensor product of the restriction
homomorphism ${\cal O}_{V(\sigma)}\rightarrow{\cal O}_{V(\tau)}$ with the
interior product with respect to $n$. Namely, an element in the
$\sigma$-component of ${\cal L}^{p,q}_X$ of the form
\[
t^m\otimes m_1\wedge m_2\wedge\cdots\wedge m_{p-q},\qquad
m, m_1\in M\cap\sigma^{\perp},\quad
m_2,\ldots,m_{p-q}\in M\cap\tau^{\perp}
\]
is sent to $t^m\otimes\langle m_1,n\rangle m_2\wedge\cdots\wedge m_{p-q}$
if $m\in M\cap\tau^{\perp}$, and to 0 otherwise.
$d_{\mbox{\rm {\scriptsize II}}}$ is the Poincar\'e residue map.
$d_{\mbox{\rm {\scriptsize I}}}\circ d_{\mbox{\rm {\scriptsize I}}}=0$ and $d_{\mbox{\rm {\scriptsize I}}}\circ d_{\mbox{\rm {\scriptsize II}}}+d_{\mbox{\rm {\scriptsize II}}}\circ d_{\mbox{\rm {\scriptsize I}}}=0$
are obvious, while $d_{\mbox{\rm {\scriptsize II}}}\circ d_{\mbox{\rm {\scriptsize II}}}=0$
was shown in Ishida \cite[Lemma 1.4 and Prop. 1.6]{ishida_dualizing}.
Consequently, we get a double complex ${\cal L}^{{\textstyle \cdot},{\textstyle \cdot}}_X$
of ${\cal O}_X$-modules,
which we call the {\em logarithmic double complex\/} for the toric variety
$X$. The associated single complex is denoted by ${\cal L}^{{\textstyle \cdot}}_X$ and
is called the {\em logarithmic complex\/} for $X$.
For simplicity, we denote by ${\cal X}:=X^{{\rm an}}$ the complex analytic space
associated to the toric variety $X=T_N\mathop{\rm emb}\nolimits(\Delta)$.
\begin{Theorem} \label{thm_simplicialLdotan}
If $\Delta$ is a {\em simplicial\/} fan, then we have a quasi-isomorphism
${\bf C}_{{\cal X}}\simeq({\cal L}^{{\textstyle \cdot}}_X)^{{\rm an}}$.
\end{Theorem}
{\sc Proof.} \quad
It suffices to prove the assertion when $X$ is affine, and there is an
easy proof in that case. However, we here give a direct proof valid for
general $X$.
Let $\widetilde{\Omega}^{{\textstyle \cdot}}_X:=j_*\Omega^{{\textstyle \cdot}}_U$
be the complex of ${\cal O}_X$-modules consisting of the Zariski differential
forms on $X$, where $j:U\rightarrow X$ is the open immersion of the
smooth locus $U$ of $X$. By Danilov's Poincar\'e lemma
\cite{danilov} (see also Ishida \cite[Prop.~2.1]{ishida_derham}),
we have a quasi-isomorphism ${\bf C}_{{\cal X}}\simeq(\widetilde{\Omega}^{{\textstyle \cdot}}_X)^{{\rm an}}$
for an {\em arbitrary\/} fan $\Delta$. Furthermore, if $\Delta$ is
simplicial, then by \cite[Theorem 3.6]{odabook} we have a quasi-isomorphism
$\widetilde{\Omega}^p_X\simeq{\cal L}^{p,{\textstyle \cdot}}_X$ for each fixed $p$.
\hspace*{\fill}q.e.d.
\bigskip
{\sc Remark.} \quad
In \cite{odabook}, ${\cal L}^{p,{\textstyle \cdot}}_X$ was denoted by ${\cal K}^{{\textstyle \cdot}}(X;p)$.
As for an {\em arbitrary\/} fan $\Delta$ for $N\cong({\bf Z})^r$ which need not be
simplicial, Ishida has a proof for the following amazing result:
$({\cal L}^{{\textstyle \cdot}}_X)^{{\rm an}}[2r]$ is quasi-isomorphic to the {\em globally normalized
dualizing complex\/} ${\cal D}^{{\textstyle \cdot}}_{{\cal X}}$ of ${\bf C}_{{\cal X}}$-modules
in the sense of Verdier \cite{verdier}.
The point of this result of Ishida's lies in the fact that the dualizing
complex ${\cal D}^{{\textstyle \cdot}}_{{\cal X}}$ can be expressed in terms of a complex
comprizing of {\em algebraic\/} and {\em coherent\/} ${\cal O}_X$-modules.
Analogously, Ishida \cite[Theorem 3.3]{ishida_dualizing} and
\cite[Theorem 5.4]{ishida_derham} earlier showed ${\cal L}^{r,{\textstyle \cdot}}_X[r]$
to be quasi-isomorphic to the {\em globally normalized dualizing complex\/}
of ${\cal O}_X$-modules.
\bigskip
The following is a toric generalization of a result due to Grothendieck
and Deligne \cite[I, \S 3 and \S 6]{deligne_equadiff} on the complement
in a smooth variety of a divisor with normal crossings.
\begin{Proposition} \label{prop_Rj*CTN}
Let $j$ be the open immersion of $T_N$ into a toric variety $X$, and
denote by $j^{{\rm an}}:(T_N)^{{\rm an}}\rightarrow X^{{\rm an}}$ the corresponding open
immersion of complex analytic spaces. Then we have a canonical
quasi-isomorphism
\[
{\bf R} j^{{\rm an}}_*{\bf C}_{(T_N)^{{\rm an}}}\simeq\left(\Omega^{{\textstyle \cdot}}_X(\log D)\right)^{{\rm an}}
=({\cal O}_{X})^{{\rm an}}\otimes_{{\bf Z}}\bigwedge\nolimits^{{\textstyle \cdot}}M
\]
and canonical isomorphisms
\[
H^{{\textstyle \cdot}}((T_N)^{{\rm an}},{\bf C})={\bf H}^{{\textstyle \cdot}}(X,\Omega^{{\textstyle \cdot}}_X(\log D))
={\bf C}\otimes_{{\bf Z}}\bigwedge\nolimits^{{\textstyle \cdot}}M,
\]
where ${\bf H}^{{\textstyle \cdot}}(X,\Omega^{{\textstyle \cdot}}_X(\log D))$ is the hypercohomology
group of the complex $\Omega^{{\textstyle \cdot}}_X(\log D)$ consisting of ${\cal O}_X$-modules.
\end{Proposition}
{\sc Proof.} \quad
We may assume $X$ to be smooth, since
$\Omega^{{\textstyle \cdot}}_X(\log D)={\cal O}_X\otimes_{{\bf Z}}\bigwedge^{{\textstyle \cdot}}M$
and since toric singularities are rational so that
${\bf R} f_*{\cal O}_{X'}={\cal O}_X$ holds for any equivariant resolution
of singularities $f:X'\rightarrow X$
(see, for instance, \cite[Cor.\ 3.9]{odabook} and
Ishida \cite[Cor.\ 3.3]{ishida_derham}).
Consequently, $D:=X\setminus T_N$ is a divisor with simple normal
crossings on a smooth $X$ as in the situation dealt with by
Grothendieck and Deligne. Let us repeat their proof here for convenience.
For the proof of the first assertion, we may obviously assume $X$ to be affine
as well, hence $j^{{\rm an}}$ is a Stein morphism. By the Poincar\'e lemma,
we thus have
\[
{\bf R} j_*^{{\rm an}}{\bf C}_{(T_N)^{{\rm an}}}={\bf R} j_*^{{\rm an}}(\Omega^{{\textstyle \cdot}}_{T_N})^{{\rm an}}
=j_*^{{\rm an}}(\Omega^{{\textstyle \cdot}}_{T_N})^{{\rm an}}.
\]
The term on the extreme right hand side is canonically quasi-isomorphic
to $(\Omega^{{\textstyle \cdot}}_X(\log D))^{{\rm an}}$ by
Deligne \cite[Lemma 6.9]{deligne_equadiff}.
As for the second assertion, note that the equality
$H^{{\textstyle \cdot}}((T_N)^{{\rm an}},{\bf C})=\bigwedge^{{\textstyle \cdot}}M_{{\bf C}}$ itself is well-known,
where $M_{{\bf C}}:=M\otimes_{{\bf Z}}{\bf C}$.
To show the whole set of the equalities, we start with the consequence
\[
H^{{\textstyle \cdot}}((T_N)^{{\rm an}},{\bf C})={\bf H}^{{\textstyle \cdot}}(X^{{\rm an}},{\bf R} j_*^{{\rm an}}{\bf C}_{(T_N)^{{\rm an}}})
={\bf H}^{{\textstyle \cdot}}(X^{{\rm an}},(\Omega_X^{{\textstyle \cdot}}(\log D))^{{\rm an}})
\]
of the first assertion. The term on the extreme right hand side is canonically
isomorphic to the algebraic hypercohomology group
${\bf H}^{{\textstyle \cdot}}(X,\Omega^{{\textstyle \cdot}}_X(\log D))$ by
Deligne \cite[Thm.\ 6.2]{deligne_equadiff}.
Since $\Omega_X^{{\textstyle \cdot}}(\log D)={\cal O}_X\otimes_{{\bf Z}}\bigwedge^{{\textstyle \cdot}}M$
has ${\cal O}_X$-coherent components,
the algebraic hypercohomology group in question coincides with the cohomology
group of the single complex associated to the \v{C}ech double complex
$\check{C}^{{\textstyle \cdot}}({\cal U},\Omega_X^{{\textstyle \cdot}}(\log D))$ with respect
to the $T_N$-stable
affine open covering ${\cal U}:=\{U_{\sigma}\mid\sigma\in\Delta\}$
with $U_{\sigma}:=\mathop{\rm Spec}\nolimits({\bf C}[M\cap\sigma^{\vee}])$. Moreover,
$T_N$ has a canonical algebraic action on the \v{C}ech double complex,
which gives rise to an eigenspace decomposition
\[
\check{C}^{{\textstyle \cdot}}({\cal U},\Omega_X^{{\textstyle \cdot}}(\log D))=
\bigoplus_{m\in M}\check{C}^{{\textstyle \cdot}}({\cal U},\Omega_X^{{\textstyle \cdot}}(\log D))_m
\]
with respect to the characters $m\in M$ of $T_N$.
As a result, we have an eigenspace decomposition
\[
{\bf H}^{{\textstyle \cdot}}(X,\Omega_X^{{\textstyle \cdot}}(\log D))
=\bigoplus_{m\in M}{\bf H}^{{\textstyle \cdot}}(X,\Omega_X^{{\textstyle \cdot}}(\log D))_m
\]
for the hypercohomology group as well.
For $m\neq 0$ we have ${\bf H}^{{\textstyle \cdot}}(X,\Omega_X^{{\textstyle \cdot}}(\log D))_m=0$,
since the $m$-th component $d_m$ of the exterior differentiation
is the exterior multiplication by $m$, hence is exact.
On the other hand, for $m=0$ we have $d_0=0$, hence the \v{C}ech
double complex is a direct sum of the complexes
$\check{C}^{{\textstyle \cdot}}({\cal U},\Omega_X^p(\log D))_0$ for $0\leq p\leq r$.
Since $\Omega_X^p(\log D)={\cal O}_X\otimes_{{\bf Z}}\bigwedge^p M$ and since
\[
H^l(X,{\cal O}_X)_0=
\left\{\begin{array}{lll}
{\bf C} & & l=0 \\
0 & & l\neq 0. \\
\end{array}\right.
\]
as we recall later in Lemma \ref{lem_HVOV0}, we conclude that
\[
{\bf H}^{{\textstyle \cdot}}(X,\Omega_X^{{\textstyle \cdot}}(\log D))_0
=H^0(X,\Omega_X^{{\textstyle \cdot}}(\log D))_0=\bigwedge\nolimits^{{\textstyle \cdot}}M_{{\bf C}}.
\]
\hspace*{\fill}q.e.d.
\bigskip
Applying Proposition~\ref{prop_Rj*CTN} to the immersion
$j_{\sigma}:\mathop{\rm orb}\nolimits(\sigma)\rightarrow X$ for each $\sigma\in\Delta$, we
get the following:
\begin{Corollary} \label{cor_Ldotqan}
For each fixed $q$, we have a canonical quasi-isomorphism
\[
\left({\cal L}^{{\textstyle \cdot},q}_X\right)^{{\rm an}}\simeq
\bigoplus_{\sigma\in\Delta(q)}{\bf R}(j_{\sigma})^{{\rm an}}_*{\bf C}_{\mathop{\rm orb}\nolimits(\sigma)}[-q],
\]
where $[-q]$ denotes the degree shift to the right by $q$.
\end{Corollary}
\section{Ishida's complexes}
Let $\Delta$ be a finite fan for a free ${\bf Z}$-module $N$ of rank $r$,
and for $0\leq q\leq r$ denote
$\Delta(q):=\{\sigma\in\Delta\mid\dim\sigma=q\}$ as before.
For each integer $p$ with $0\leq p\leq r$, {\em Ishida's $p$-th complex\/}
$C^{{\textstyle \cdot}}(\Delta,\Lambda^p)$ of ${\bf Z}$-modules is defined as follows:
\[
C^q(\Delta,\Lambda^p):=\bigoplus_{\sigma\in\Delta(q)}
\bigwedge\nolimits^{p-q}(M\cap\sigma^{\perp})\qquad\mbox{if}\quad
0\leq q\leq p,
\]
and $C^q(\Delta,\Lambda^p)=0$ otherwise.
For $\sigma\in\Delta(q)$ and $\tau\in\Delta(q+1)$,
the $(\sigma,\tau)$-component of the coboundary map
\[
\delta:C^q(\Delta,\Lambda^p)=\bigoplus_{\sigma\in\Delta(q)}
\bigwedge\nolimits^{p-q}(M\cap\sigma^{\perp})\longrightarrow
C^{q+1}(\Delta,\Lambda^p)=\bigoplus_{\tau\in\Delta(q+1)}
\bigwedge\nolimits^{p-q-1}(M\cap\tau^{\perp})
\]
is defined to be $0$ if $\sigma$ is not a face of $\tau$.
On the other hand, if $\sigma$ is a face of $\tau$, then a primitive
element $n\in N$ is uniquely determined modulo $N\cap{\bf R}\sigma$ so that
$\tau+(-\sigma)={\bf R}_{\geq 0}n+{\bf R}\sigma$. The $(\sigma,\tau)$-component
of $\delta$ in this case is defined to be the interior product with
respect to this $n$. Namely, the element
$m_1\wedge m_2\wedge\cdots\wedge m_{p-q}$ with $m_1\in M\cap\sigma^{\perp}$
and $m_2,\ldots,m_{p-q}\in M\cap\tau^{\perp}$ is sent to
$\langle m_1,n\rangle m_2\wedge\cdots\wedge m_{p-q}$.
As Ishida \cite[Prop.\ 1.6]{ishida_dualizing} showed, $\delta\circ\delta=0$
holds so that $C^{{\textstyle \cdot}}(\Delta,\Lambda^p)$ is a complex of ${\bf Z}$-modules.
We denote its cohomology group by $H^{{\textstyle \cdot}}(\Delta,\Lambda^p)$.
We will be mainly concerned with their scalar extensions
$C^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf Q}}$, $C^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf C}}$,
$H^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf Q}}$, $H^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf C}}$
to ${\bf Q}$ and ${\bf C}$.
By definition, we have $H^q(\Delta,\Lambda^p)=0$ unless $0\leq q\leq p$.
\bigskip
{\sc Remark.} \quad
In \cite[\S 3.2]{odabook}, $C^{{\textstyle \cdot}}(\Delta,\Lambda^p)$
and $H^{{\textstyle \cdot}}(\Delta,\Lambda^p)$ were denoted by $C^{{\textstyle \cdot}}(\Delta;p)$ and
$H^{{\textstyle \cdot}}(\Delta;p)$, respectively.
As in \cite{lefsch}, we can define a similar complex
$C^{{\textstyle \cdot}}(\Pi,{\cal G}_p)$ of ${\bf R}$-vector spaces for a {\em simplicial\/}
polyhedral cone decomposition $\Pi$ of an ${\bf R}$-vector space
endowed with a marking for each one-dimensional cone in $\Pi$.
Note, however, that unless a lattice $N$ is given as in the case of a fan,
we cannot define the coboundary map in the case of a {\em non-simplicial\/}
convex polyhedral cone decomposition $\Pi$ even if it is endowed with a
marking. It is crucial that for a codimension one face $\sigma$ of
$\tau$ in a fan, a primitive element $n\in N$ is uniquely determined
modulo $N\cap{\bf R}\sigma$ so that $\tau+(-\sigma)={\bf R}_{\geq 0}n+{\bf R}\sigma$
holds as above, regardless of whether $\tau$ is simplicial or not.
\bigskip
The following result is slightly stronger than \cite[Lemma 3.7]{odabook},
and the proof is similar to that for \cite[Prop.\ 3.5]{lefsch},
which concerns analogous ${\bf R}$-coefficient cohomology groups for
the simplicial polyhedral cone decomposition consisting of the
faces of a simplicial cone in a finite dimensional ${\bf R}$-vector space:
\begin{Proposition} \label{prop_vanishing_simplicialcone}
Let $\pi$ be a {\em simplicial} rational polyhedral cone in $N_{{\bf R}}$.
Then for each $0\leq p\leq r$, the cohomology group of Ishida's $p$-th
complex for the
fan $\Gamma_{\pi}$ consisting of all the faces of $\pi$ satisfies
\[
H^q(\Gamma_{\pi},\Lambda^p)_{{\bf Q}}:=
H^q(\Gamma_{\pi},\Lambda^p)\otimes_{{\bf Z}}{\bf Q}=
\left\{
\begin{array}{lll}
\bigwedge^p(M_{{\bf Q}}\cap\pi^{\perp}) &\phantom{mm}& q=0 \\
0 & & q\neq 0, \\
\end{array}
\right.
\]
where $M_{{\bf Q}}:=M\otimes_{{\bf Z}}{\bf Q}$.
\end{Proposition}
\section{The algebraic de Rham theorem}
In this section, we denote by $X:=T_N\mathop{\rm emb}\nolimits(\Delta)$ the $r$-dimensional
toric variety over ${\bf C}$ corresponding to a finite fan $\Delta$ for
$N\cong{\bf Z}^r$.
For simplicity, we again denote the corresponding complex analytic space
by ${\cal X}:=X^{{\rm an}}$.
\begin{Proposition} \label{prop_alglog_ishida}
For an {\em arbitrary} fan $\Delta$ which need not be complete nor simplicial,
the hypercohomology group of the logarithmic complex
${\cal L}^{{\textstyle \cdot}}_X$ has a direct sum decomposition
\[
{\bf H}^l(X,{\cal L}^{{\textstyle \cdot}}_X)=\bigoplus_{p+q=l}H^q(\Delta,\Lambda^p)_{{\bf C}}
\qquad\mbox{for each $l$}.
\]
\end{Proposition}
{\sc Proof.} \quad
Consider the $T_N$-stable
affine open covering ${\cal U}:=\{U_{\sigma}\mid\sigma\in\Delta\}$
of $X$ with $U_{\sigma}:=\mathop{\rm Spec}\nolimits({\bf C}[M\cap\sigma^{\vee}])$.
We know that (cf.\ \cite{odabook})
\[
U_{\sigma_0}\cap U_{\sigma_1}\cap\cdots\cap U_{\sigma_q}
=U_{\sigma_0\cap\sigma_1\cap\cdots\cap\sigma_q}\qquad\mbox{for all}\quad
\sigma_0,\sigma_1,\ldots,\sigma_q\in\Delta.
\]
Since each component of ${\cal L}^{{\textstyle \cdot}}_X$ is a coherent ${\cal O}_X$-module,
the hypercohomology group ${\bf H}^{{\textstyle \cdot}}(X,{\cal L}^{{\textstyle \cdot}}_X)$ coincides with
the cohomology group of the single complex associated to the
\v{C}ech double complex $\check{C}^{{\textstyle \cdot}}({\cal U},{\cal L}^{{\textstyle \cdot}}_X)$ with
respect to the affine open covering ${\cal U}$.
Because of the canonical algebraic action of the algebraic torus $T_N$ on
$\check{C}^{{\textstyle \cdot}}({\cal U},{\cal L}^{{\textstyle \cdot}}_X)$, we have the eigenspace
decomposition
\[
\check{C}^{{\textstyle \cdot}}({\cal U},{\cal L}_X^{{\textstyle \cdot}})
=\bigoplus_{m\in M}\check{C}^{{\textstyle \cdot}}({\cal U},{\cal L}_X^{{\textstyle \cdot}})_m
\qquad\mbox{hence}\qquad
{\bf H}^{{\textstyle \cdot}}({\cal U},{\cal L}_X^{{\textstyle \cdot}})
=\bigoplus_{m\in M}{\bf H}^{{\textstyle \cdot}}({\cal U},{\cal L}_X^{{\textstyle \cdot}})_m
\]
with respect to the characters $m\in M$.
Let us consider the triple complex
\[
\check{C}_m^{{\textstyle \cdot},{\textstyle \cdot},{\textstyle \cdot}}:=
\check{C}^{{\textstyle \cdot}}({\cal U},{\cal L}_X^{{\textstyle \cdot},{\textstyle \cdot}})_m\qquad\mbox{with}\quad
\check{C}_m^{l,p,q}:=
\check{C}^{l}({\cal U},{\cal L}_X^{p,q})_m.
\]
The differential for $l$ is the $m$-th component $\check{\delta}_m$
of the \v{C}ech coboundary $\check{\delta}$, while those for $p$ and
$q$ are the $m$-th components $(d_{\mbox{\rm {\scriptsize I}}})_m$ and $(d_{\mbox{\rm {\scriptsize II}}})_m$ of
$d_{\mbox{\rm {\scriptsize I}}}$ and $d_{\mbox{\rm {\scriptsize II}}}$, respectively.
${\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})_m$ is the cohomology group of the
single complex associated to $\check{C}^{{\textstyle \cdot},{\textstyle \cdot},{\textstyle \cdot}}_m$.
For $m\neq 0$, we have ${\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})_m=0$, hence
${\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})={\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})_0$
is $T_N$-invariant.
Indeed, since $(d_{\mbox{\rm {\scriptsize I}}})_m$ is the exterior multiplication by $m$,
the complex $(\check{C}_m^{l,{\textstyle \cdot},q},(d_{\mbox{\rm {\scriptsize I}}})_m)$ is acyclic for
all $l$ and $q$.
Consequently, the cohomology group vanishes for
the single complex associated to the triple complex
$\check{C}^{{\textstyle \cdot},{\textstyle \cdot},{\textstyle \cdot}}_m$.
On the other hand, for $m=0$ we have $(d_{\mbox{\rm {\scriptsize I}}})_0=0$, hence
$\check{C}^{{\textstyle \cdot},{\textstyle \cdot},{\textstyle \cdot}}_0=
\bigoplus_p\check{C}^{{\textstyle \cdot},p,{\textstyle \cdot}}_0$ is a direct sum of double
complexes.
Since ${\cal L}_X^{p,q}$ is a direct sum of coherent sheaves of the form
${\cal O}_{V(\sigma)}$, we see by Lemma \ref{lem_HVOV0} below that
the cohomology group of the single complex associated to
$\check{C}^{{\textstyle \cdot},p,{\textstyle \cdot}}_0$ coincides with that of
$H^0(X,{\cal L}_X^{p,{\textstyle \cdot}})_0=C^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf C}}$.
Consequently, the cohomology group
${\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})={\bf H}^{{\textstyle \cdot}}(X,{\cal L}_X^{{\textstyle \cdot}})_0$
of the single complex associated to $\check{C}_0^{{\textstyle \cdot},{\textstyle \cdot},{\textstyle \cdot}}$
is the direct sum of $H^{{\textstyle \cdot}}(\Delta,\Lambda^p)_{{\bf C}}$ as claimed.
\hspace*{\fill}q.e.d.
\begin{Lemma} \label{lem_HVOV0}
For any toric variety $V$ with respect to an algebraic torus $T_N$,
the $T_N$-invariant part of the cohomology group for the structure sheaf
${\cal O}_V$ satisfies
\[
H^l(V,{\cal O}_V)_0=
\left\{\begin{array}{lll}
{\bf C} & & l=0 \\
0 & & l\neq 0. \\
\end{array}\right.
\]
\end{Lemma}
{\sc Proof.} \quad
Demazure and Danilov gave a general description of the eigenspace
$H^{{\textstyle \cdot}}(V,L)_m$, with respect to a character $m$, of the $T_N$-action on
the cohomology group of an equivariant line bundle $L$ on $V$ in terms of the
corresponding support function (see, for instance, \cite[Thm.\ 2.6]{odabook}).
The present lemma is the special case $L={\cal O}_V$ and $m=0$.
\hspace*{\fill}q.e.d.
\bigskip
We are now ready to state a generalization, in the toric context, of
the algebraic de Rham theorem due to Grothendieck \cite{grothendieck}.
An algebro-geometric proof valid in the case of complete nonsingular fans
can be found in Danilov \cite{danilov} and in \cite[Theorem 3.11]{odabook}.
\begin{Theorem} \label{thm_algderham}
{\rm (The algebraic de Rham theorem)}
For a {\em simplicial} fan $\Delta$ which need not be complete, we have the
following for each $l$:
\[
H^l({\cal X},{\bf C})={\bf H}^l({\cal X},({\cal L}^{{\textstyle \cdot}}_X)^{{\rm an}})
={\bf H}^l(X,{\cal L}^{{\textstyle \cdot}}_X)=\bigoplus_{p+q=l}H^q(\Delta,\Lambda^p)_{{\bf C}},
\]
where the second term from the left is the corresponding analytic
hypercohomology group.
\end{Theorem}
{\sc Proof.} \quad
The first equality follows from Theorem \ref{thm_simplicialLdotan},
while the third is a consequence of Proposition \ref{prop_alglog_ishida}.
To show the second equality, we mimic the proof due to
Grothendieck \cite{grothendieck}
in the case of the usual de Rham complex on smooth algebraic varieties.
There is a canonical homomorphism from the spectral sequence
\[
E_1^{p,q}:=H^q(X,{\cal L}_X^p)\Longrightarrow
{\bf H}^{p+q}(X,{\cal L}_X^{{\textstyle \cdot}})
\]
for the algebraic hypercohomology to the spectral sequence
\[
E_1^{p,q}:=H^q({\cal X},({\cal L}_X^p)^{{\rm an}})\Longrightarrow
{\bf H}^{p+q}({\cal X},({\cal L}_X^{{\textstyle \cdot}})^{{\rm an}})
\]
for the complex analytic hypercohomology.
To show that the homomorphism between the hypercohomology groups is an
isomorphism, we may assume $X$ to be affine, since the algebraic
(resp.\ complex analytic) hypercohomology group is the cohomology group
of the single complex associated to the \v{C}ech double complex
with respect to the affine open covering ${\cal U}$
(resp.\ the corresponding Stein open covering $({\cal U})^{{\rm an}}$)
as in the proof of Proposition \ref{prop_alglog_ishida}.
We thus assume $X=U_{\pi}:=\mathop{\rm Spec}\nolimits({\bf C}[M\cap\pi^{\vee}])$ for a simplicial
rational cone $\pi\subset N_{{\bf R}}$, hence $\Delta=\Gamma_{\pi}$.
By Propositions \ref{prop_alglog_ishida} and
\ref{prop_vanishing_simplicialcone}, we have
\[
{\bf H}^l(U_{\pi},{\cal L}^{{\textstyle \cdot}}_{U_{\pi}})
=\bigoplus_{p+q=l}H^q(\Gamma_{\pi},\Lambda^p)_{{\bf C}}
=\bigwedge\nolimits^l(M\cap\pi^{\perp})_{{\bf C}}
\]
for each $l$.
We are done in view of
\[
{\bf H}^l((U_{\pi})^{{\rm an}},({\cal L}^{{\textstyle \cdot}}_{U_{\pi}})^{{\rm an}})
=H^l((U_{\pi})^{{\rm an}},{\bf C})=\bigwedge\nolimits^l(M\cap\pi^{\perp})_{{\bf C}}
\]
by Theorem \ref{thm_simplicialLdotan} and Danilov \cite[Lemma 12.3]{danilov}.
\hspace*{\fill}q.e.d.
\bigskip
{\sc Remark.} \quad
Suppose $\Delta$ is an {\em arbitrary\/} finite fan which need not be
simplicial nor complete.
In view of the Verdier duality \cite{verdier} and Ishida's result mentioned
in the remark after Theorem \ref{thm_simplicialLdotan}, we have
\[
{\bf H}^l({\cal X},({\cal L}^{{\textstyle \cdot}}_X)^{{\rm an}})={\bf H}^l({\cal X},{\cal D}^{{\textstyle \cdot}}_{{\cal X}}[-2r])
=\mathop{\rm Hom}\nolimits_{{\bf C}}(H_c^{2r-l}({\cal X},{\bf C}),{\bf C}),
\]
where the extreme right hand side is the dual of the cohomology group with
compact support.
Consequently, if we can show
\[
\mathop{\rm Hom}\nolimits_{{\bf C}}(H_c^{2r-l}((U_{\pi})^{{\rm an}},{\bf C}),{\bf C})
=\bigoplus_{p+q=l}H^q(\Gamma_{\pi},\Lambda^p)_{{\bf C}}
\qquad\mbox{with}\quad U_{\pi}:=T_N\mathop{\rm emb}\nolimits(\Gamma_{\pi})
\]
for an arbitrary strongly convex rational polyhedral cone $\pi\subset N_{{\bf R}}$
which need not be simplicial, then Theorem \ref{thm_algderham}
could be generalized for an {\em arbitrary\/} fan in the form
\[
\mathop{\rm Hom}\nolimits_{{\bf C}}(H_c^{2r-l}({\cal X},{\bf C}),{\bf C})={\bf H}^l({\cal X},({\cal L}^{{\textstyle \cdot}}_X)^{{\rm an}})
={\bf H}^l(X,{\cal L}^{{\textstyle \cdot}}_X)
=\bigoplus_{p+q=l}H^q(\Delta,\Lambda^p)_{{\bf C}}.
\]
\bigskip
\section{Vanishing theorems}
Analogs of the following were proved in Danilov \cite{danilov} and
\cite[Theorem 3.11]{odabook}
for complete nonsingular fans by means of the algebraic de Rham theorem,
and then directly in \cite[Theorem 4.1]{lefsch} for
complete simplicial polyhedral cone decompositions endowed with markings:
\begin{Proposition} \label{prop_vanishing_complete}
Let $\Delta$ be a {\em simplicial} and {\em complete} fan for
$N\cong{\bf Z}^r$. Then for all $p$ with $0\leq p\leq r$ we have
\[
H^q(\Delta,\Lambda^p)_{{\bf Q}}=0\qquad\mbox{for}\quad q\neq p.
\]
Moreover for all $p$, we have a perfect pairing
\[
H^p(\Delta,\Lambda^p)_{{\bf Q}}\times H^{r-p}(\Delta,\Lambda^{r-p})_{{\bf Q}}
\longrightarrow H^r(\Delta,\Lambda^r)_{{\bf Q}}\cong\bigwedge\nolimits^r M_{{\bf Q}}.
\]
Consequently, the corresponding toric variety $X:=T_N\mathop{\rm emb}\nolimits(\Delta)$
satisfies $H^l(X^{{\rm an}},{\bf C})=0$ for odd $l$, while
\[
H^{2p}(X^{{\rm an}},{\bf C})=H^p(\Delta,\Lambda^p)_{{\bf C}}\qquad\mbox{for}\quad
0\leq p\leq r.
\]
\end{Proposition}
{\sc Proof.} \quad
We just indicate necessary modifications of the proof of
\cite[Theorem 4.1]{lefsch}.
We introduce a first quadrant double complex
$(K^{{\textstyle \cdot},{\textstyle \cdot}},d',d'')$ as follows:
For nonnegative integers $i$ and $j$, let
\[
K^{i,j}:=\bigoplus_{\varphi\in\Delta(r-i)}\;\;
\bigoplus_{\stackrel{\scriptstyle \sigma\in\Delta(j)}{\sigma\prec\varphi}}
\bigwedge\nolimits^{p-j}
\left(M\cap\sigma^{\perp}\right)
\otimes_{{\bf Z}}(\det\varphi)^{-1},
\]
where $\det\varphi:=\bigwedge^{\dim\varphi}(N\cap{\bf R}\varphi)$
is the orientation ${\bf Z}$-module of rank one, and
$(\det\varphi)^{-1}$ is its dual ${\bf Z}$-module.
If $\psi\in\Delta(k-1)$ is a facet of $\varphi\in\Delta(k)$, we have
mutually dual nonzero orientation ${\bf Z}$-homomorphisms
\[
\det\psi\longrightarrow\det\varphi\qquad\mbox{and}\qquad
(\det\varphi)^{-1}\longrightarrow(\det\psi)^{-1}
\]
in the following manner: A primitive element $n\in N$ is uniquely
determined modulo $N\cap{\bf R}\psi$ so that
$\varphi+(-\psi)={\bf R}_{\geq 0}n+{\bf R}\psi$. Then
$n_1\wedge\cdots\wedge n_{k-1}\in\det\psi$ is sent to
$n\wedge n_1\wedge\cdots\wedge n_{k-1}\in\det\varphi$.
For $\Delta(j)\ni\sigma\prec\varphi\in\Delta(r-i)$ and
$\Delta(j)\ni\sigma'\prec\psi\in\Delta(r-i-1)$, we define the
component of $d':K^{i,j}\rightarrow K^{i+1,j}$ with respect to
$(\varphi,\sigma)$ and $(\psi,\sigma')$ to be nonzero only when
$\varphi\succ\psi\succ\sigma=\sigma'$ and to be equal to
$(-1)^{j}$ times the dual orientation ${\bf Z}$-homomorphism
$(\det\varphi)^{-1}\rightarrow(\det\psi)^{-1}$ tensored with the
identity map for $\bigwedge^{p-j}(M\cap\sigma^{\perp})$.
On the other hand, for $\Delta(j)\ni\sigma\prec\varphi\in\Delta(r-i)$
and $\Delta(j+1)\ni\tau\prec\varphi'\in\Delta(r-i)$, we define the
component of $d'':K^{i,j}\rightarrow K^{i,j+1}$ with respect to
$(\varphi,\sigma)$ and $(\varphi',\tau)$ to be nonzero only when
$\varphi=\varphi'\succ\tau\succ\sigma$ and to be equal to the
tensor product of the homomorphism
$\bigwedge^{p-j}(M\cap\sigma^{\perp})\rightarrow
\bigwedge^{p-j-1}(M\cap\tau^{\perp})$ appearing in the definition of
Ishida's $p$-th complex, with the identity map for $(\det\varphi)^{-1}$.
It is easy to show that $(d')^2=(d'')^2=d'd''+d''d'=0$, hence
$K^{{\textstyle \cdot},{\textstyle \cdot}}$ is a double complex of ${\bf Z}$-modules.
Its scalar extension $K^{{\textstyle \cdot},{\textstyle \cdot}}_{{\bf Q}}$ to ${\bf Q}$ turns out to
satisfy
\[
H_{\mbox{\rm {\scriptsize I}}}^i(K^{{\textstyle \cdot},j}_{{\bf Q}})=
\left\{\begin{array}{lll}
C^j(\Delta,\Lambda^p)_{{\bf Q}} & & i=0 \\
0 & & i\neq 0, \\
\end{array}\right.
\]
since
\[
H^{r-i}(\mathop{\rm Star}\nolimits_{\sigma}(\Delta),{\bf Q})=
\left\{\begin{array}{lll}
{\bf Q} & & i=0 \\
0 & & i\neq 0 \\
\end{array}\right.
\]
for all $\sigma\in\Delta(j)$.
Hence we get
\[
H_{\mbox{\rm {\scriptsize II}}}^j(H_{\mbox{\rm {\scriptsize I}}}^i(K^{{\textstyle \cdot},{\textstyle \cdot}}_{{\bf Q}}))=
\left\{\begin{array}{lll}
H^j(\Delta,\Lambda^p)_{{\bf Q}} & & i=0 \\
0 & & i\neq 0. \\
\end{array}\right.
\]
Consequently, by one of the two spectral sequences for the double complex,
we see that the associated single complex $(K^{{\textstyle \cdot}}_{{\bf Q}},d'+d'')$
has the cohomology group
\[
H^k(K^{{\textstyle \cdot}}_{{\bf Q}})=H^k(\Delta,\Lambda^p)_{{\bf Q}}
\qquad\mbox{for all}\quad k.
\]
On the other hand, for each fixed $i$, we have an isomorphism of complexes
\[
K^{i,{\textstyle \cdot}}_{{\bf Q}}=\bigoplus_{\varphi\in\Delta(r-i)}
\left(C^{{\textstyle \cdot}}(\Gamma_{\varphi},\Lambda^p)\otimes_{{\bf Z}}
(\det\varphi)^{-1}\right)_{{\bf Q}},
\]
the coboundary map for the left hand side being $d''$, while that for
the right hand side is $\delta\otimes{\rm id}$.
Thus by Proposition \ref{prop_vanishing_simplicialcone}, we get
\[
H_{\mbox{\rm {\scriptsize II}}}^j(K^{i,{\textstyle \cdot}}_{{\bf Q}})=
\left\{\begin{array}{lll}
\bigoplus_{\varphi\in\Delta(r-i)}
\left((\bigwedge^p(M\cap\varphi^{\perp}))
\otimes_{{\bf Z}}(\det\varphi)^{-1}\right)_{{\bf Q}} & & j=0 \\
0 & & j\neq 0, \\
\end{array}\right.
\]
hence
\[
H_{\mbox{\rm {\scriptsize I}}}^i(H_{\mbox{\rm {\scriptsize II}}}^j(K^{{\textstyle \cdot},{\textstyle \cdot}}_{{\bf Q}}))=0
\qquad\mbox{for}\quad j\neq 0.
\]
Consequently, we have
\[
H^k(\Delta,\Lambda^p)_{{\bf Q}}=
H^k(K^{{\textstyle \cdot}}_{{\bf Q}})=H_{\mbox{\rm {\scriptsize I}}}^k(H_{\mbox{\rm {\scriptsize II}}}^0(K^{{\textstyle \cdot},{\textstyle \cdot}}_{{\bf Q}}))
\]
for all $k$.
The extreme left hand side is nonzero only when $0\leq k\leq p$,
while the extreme right hand side is nonzero only when $p\leq k$.
The asserted perfect pairing is a consequence of the canonical
identification, as in \cite[Thm.\ 4.1]{lefsch}, of the ${{\bf Q}}$-dual of
$H^p(\Delta,\Lambda^p)_{{\bf Q}}=H_{\mbox{\rm {\scriptsize I}}}^p(H_{\mbox{\rm {\scriptsize II}}}^0(K^{{\textstyle \cdot},{\textstyle \cdot}}_{{\bf Q}}))$
with $(\bigwedge\nolimits^r N_{{\bf Q}})\otimes_{{\bf Q}}
H^{r-p}(\Delta,\Lambda^{r-p})_{{\bf Q}}$.
The rest of the assertion follows from Theorem \ref{thm_algderham}.
\hspace*{\fill}q.e.d.
\bigskip
The following is an important generalization, due to Ishida, of our earlier
result stated as Corollary \ref{cor_vanishing_convexsupport} later.
\begin{Theorem} \label{thm_vanishing_ishida}
{\rm (Ishida)}
Let $\Delta$ be a finite simplicial fan for $N\cong{\bf Z}^r$ which may not be
complete.
If there exist a finite complete simplicial fan $\widetilde{\Delta}$ and
a $\rho\in\widetilde{\Delta}(1)$ such that
$\Delta=\widetilde{\Delta}\setminus\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$, then
\[
H^q(\Delta,\Lambda^p)_{{\bf Q}}=0\qquad\mbox{for all}\quad q\neq p.
\]
Consequently, the corresponding toric variety $X:=T_N\mathop{\rm emb}\nolimits(\Delta)$
satisfies $H^l(X^{{\rm an}},{\bf C})=0$ for odd $l$, while
\[
H^{2p}(X^{{\rm an}},{\bf C})=H^p(\Delta,\Lambda^p)_{{\bf C}}\qquad\mbox{for}\quad
0\leq p\leq r.
\]
\end{Theorem}
{\sc Proof.} \quad
Since $\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$ (resp.\ $\Delta$) is a star closed subset
(resp.\ a subcomplex) of $\widetilde{\Delta}$, we have an exact sequence of
complexes
\[
0\longrightarrow C^{{\textstyle \cdot}}(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p)
\longrightarrow C^{{\textstyle \cdot}}(\widetilde{\Delta},\Lambda^p)\longrightarrow
C^{{\textstyle \cdot}}(\Delta,\Lambda^p)\longrightarrow 0.
\]
Consider the projection $N\rightarrow\bar{N}:=N/{\bf Z}(N\cap\rho)$.
For each $\sigma\in\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$,
its image $\bar{\sigma}:=(\sigma+(-\rho))/{\bf R}\rho$ under the projection
$N_{{\bf R}}\rightarrow\bar{N}_{{\bf R}}$ is a strongly convex rational polyhedral cone,
and
\[
\bar{\Sigma}:=\{\bar{\sigma}\;\mid\;\sigma\in\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})\}
\]
is a finite complete simplicial fan for $\bar{N}$.
The dual of $\bar{N}$ is $\bar{M}=M\cap\rho^{\perp}$, hence
$M\cap\sigma^{\perp}=\bar{M}\cap\bar{\sigma}^{\perp}$ holds for each
$\sigma\in\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$. Consequently, we have an isomorphism
of complexes
\[
C^{{\textstyle \cdot}-1}(\bar{\Sigma},\Lambda^{p-1})\stackrel{\sim}{\longrightarrow}
C^{{\textstyle \cdot}}(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p).
\]
In view of Proposition \ref{prop_vanishing_complete} applied to
$\bar{\Sigma}$, we thus have $H^q(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p)_{{\bf Q}}=0$ for
$q\neq p$. Again by Proposition \ref{prop_vanishing_complete} applied
this time to $\widetilde{\Delta}$, we have $H^q(\widetilde{\Delta},\Lambda^p)_{{\bf Q}}=0$ for $q\neq p$.
Hence we get $H^q(\Delta,\Lambda^p)_{{\bf Q}}=0$ for $q\neq p-1, p$ as well
as a long exact sequence
\[
0\longrightarrow H^{p-1}(\Delta,\Lambda^p)_{{\bf Q}}\longrightarrow
H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p)_{{\bf Q}}\longrightarrow
H^p(\widetilde{\Delta},\Lambda^p)_{{\bf Q}}\longrightarrow H^p(\Delta,\Lambda^p)_{{\bf Q}}
\longrightarrow 0.
\]
In particular, the vanishing of $H^{p-1}(\Delta,\Lambda^p)_{{\bf Q}}$ is
equivalent to the injectivity of
\[
H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p)_{{\bf Q}}\rightarrow
H^p(\widetilde{\Delta},\Lambda^p)_{{\bf Q}}.
\]
To show the latter, we now introduce another finite complete simplicial fan
$\widetilde{\Phi}$ for $N$ containing $\rho$ such that
$\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi})=\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$.
This $\widetilde{\Phi}$ and $\Phi:=\widetilde{\Phi}\setminus\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi})$ turn out to be
easier to handle, and we will be able to show $H^q(\Phi,\Lambda^p)_{{\bf Q}}=0$
for $q\neq p$ in Lemma \ref{lem_vanishing_P1bundle} below.
Hence by the same argument as above applied to $\widetilde{\Phi}$ instead of
$\widetilde{\Delta}$ we have the injectivity of
\[
H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi}),\Lambda^p)_{{\bf Q}}\longrightarrow
H^p(\widetilde{\Phi},\Lambda^p)_{{\bf Q}}.
\]
Clearly, there exists a finite simplicial complete fan $\widetilde{\Delta}'$ for
$N$ which is a subdivision of both $\widetilde{\Delta}$ and $\widetilde{\Phi}$
such that $\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}')=\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})=\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi})$.
We claim the injectivity of the canonical homomorphisms
\[
H^p(\widetilde{\Delta},\Lambda^p)_{{\bf Q}}\longrightarrow H^p(\widetilde{\Delta}',\Lambda^p)_{{\bf Q}}
\qquad\mbox{and}\qquad
H^p(\widetilde{\Phi},\Lambda^p)_{{\bf Q}}\longrightarrow H^p(\widetilde{\Delta}',\Lambda^p)_{{\bf Q}}
\]
induced by the subdivisions. Consequently, we would get the injectivity of
\[
H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p)_{{\bf Q}}\rightarrow
H^p(\widetilde{\Delta},\Lambda^p)_{{\bf Q}},
\]
since the canonical homomorphism from
\[
H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi}),\Lambda^p)_{{\bf Q}}
=H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}),\Lambda^p)_{{\bf Q}}
=H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta}'),\Lambda^p)_{{\bf Q}}
\]
to $H^p(\widetilde{\Phi},\Lambda^p)_{{\bf Q}}$ is injective as above.
As for the proof of the above claim, it obviously suffices to prove the
injectivity of the scalar extension to ${\bf C}$ of the canonical homomorphisms
in question.
Let $\widetilde{X}$, $\widetilde{X}'$, $\widetilde{Y}$ be the complete toric varieties associated to the
fans $\widetilde{\Delta}$, $\widetilde{\Delta}'$, $\widetilde{\Phi}$, respectively, and denote by
$f:\widetilde{X}'\rightarrow\widetilde{X}$ and $g:\widetilde{X}'\rightarrow\widetilde{Y}$ the equivariant proper
birational morphisms associated to the subdivisions of the fans.
By Theorem \ref{thm_algderham} and
Proposition \ref{prop_vanishing_complete}, we have
\begin{eqnarray*}
H^p(\widetilde{\Delta},\Lambda^p)_{{\bf C}} &=& H^{2p}((\widetilde{X})^{{\rm an}},{\bf C}), \\
H^p(\widetilde{\Delta}',\Lambda^p)_{{\bf C}} &=& H^{2p}((\widetilde{X}')^{{\rm an}},{\bf C}),\\
H^p(\widetilde{\Phi},\Lambda^p)_{{\bf C}} &=& H^{2p}((\widetilde{Y})^{{\rm an}},{\bf C}).
\end{eqnarray*}
The canonical homomorphisms in question coincide with
\[
f^*:H^{2p}((\widetilde{X})^{{\rm an}},{\bf C})\rightarrow H^{2p}((\widetilde{X}')^{{\rm an}},{\bf C})
\qquad\mbox{and}\qquad
g^*:H^{2p}((\widetilde{Y})^{{\rm an}},{\bf C})\rightarrow H^{2p}((\widetilde{X}')^{{\rm an}},{\bf C}),
\]
which are well-known to be injective by
$f_*f^*={\rm id}$ and $g_*g^*={\rm id}$, where $f_*$ and $g_*$ are the direct images
\[
f_*:H^{2p}((\widetilde{X}')^{{\rm an}},{\bf C})\rightarrow H^{2p}((\widetilde{X})^{{\rm an}},{\bf C})
\qquad\mbox{and}\qquad
g_*:H^{2p}((\widetilde{X}')^{{\rm an}},{\bf C})\rightarrow H^{2p}((\widetilde{Y})^{{\rm an}},{\bf C}).
\]
Here is how we define the new complete simplicial fan $\widetilde{\Phi}$ for $N$
which satisfies the required properties mentioned above:
$\Delta^{\flat}:=\{\tau\in\Delta\mid \tau+\rho\in\widetilde{\Delta}\}$ is easily
seen to be a fan for $N$. It can be thought of as the
``boundary'' of $\Delta$ as well as the ``link'' of $\rho$ in $\widetilde{\Delta}$
so that $\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})=\{\tau+\rho\mid \tau\in\Delta^{\flat}\}$.
The projection $N\rightarrow\bar{N}$ induces a bijection from each
$\tau\in\Delta^{\flat}$ to its image $\bar{\tau}\in\bar{\Sigma}$.
The cone $-\rho$ in $N_{{\bf R}}$ is not contained
in $\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$ nor $\Delta^{\flat}$. Hence
\[
\widetilde{\Phi}:=\{\tau+\rho\;\mid\;\tau\in\Delta^{\flat}\}\;{\textstyle \coprod}\;
\Delta^{\flat}\;{\textstyle \coprod}\;
\{\tau+(-\rho)\;\mid\;\tau\in\Delta^{\flat}\}
\]
is a finite simplicial complete fan for $N$ satisfying
$\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi})=\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$.
Let $\Phi^{\flat}:=\Delta^{\flat}$ and
\[
\Phi:=\Phi^{\flat}\;{\textstyle \coprod}\;
\{\tau+(-\rho)\;\mid\;\tau\in\Phi^{\flat}\}.
\]
We clearly have $\Phi=\widetilde{\Phi}\setminus\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi})$ and
$\Phi^{\flat}=\{\tau\in\Phi\mid\tau+\rho\in\widetilde{\Phi}\}$.
We are done in view of Lemma \ref{lem_vanishing_P1bundle} below
and Theorem \ref{thm_algderham}
\hspace*{\fill}q.e.d.
\begin{Lemma} \label{lem_vanishing_P1bundle}
In the notation above, we have
$H^{{\textstyle \cdot}}(\Phi,\Lambda^p)_{{\bf Q}}=H^{{\textstyle \cdot}}(\bar{\Sigma},\Lambda^p)_{{\bf Q}}$.
In particular,
\[
H^q(\Phi,\Lambda^p)_{{\bf Q}}=0\qquad\mbox{for all}\quad q\neq p,
\]
hence the canonical homomorphism
$H^p(\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Phi}),\Lambda^p)_{{\bf Q}}\rightarrow H^p(\widetilde{\Phi},\Lambda^p)_{{\bf Q}}$
is injective.
\end{Lemma}
{\sc Proof.} \quad
As we mentioned above,
$\Phi^{\flat}=\Delta^{\flat}$ is in one-to-one correspondence with
$\bar{\Sigma}$ by the map which sends $\tau\in\Phi^{\flat}$ to its isomorphic
image under the projection $N_{{\bf R}}\rightarrow\bar{N}_{{\bf R}}=N_{{\bf R}}/{\bf R}\rho$.
On the other hand, $\Phi^{\flat}(q)$ for each $q$ is in one-to-one
correspondence with $\mathop{\rm Star}\nolimits_{-\rho}(\Phi)(q+1)$ by the map which sends
$\tau\in\Phi^{\flat}$ to $\tau+(-\rho)\in\mathop{\rm Star}\nolimits_{-\rho}(\Phi)$.
We have $\bar{M}\cap\bar{\tau}^{\perp}=M\cap(\tau+(-\rho))^{\perp}$ and an exact
sequence
\[
0\longrightarrow(\bar{M}\cap\bar{\tau}^{\perp})_{{\bf Q}}\longrightarrow
(M\cap\tau^{\perp})_{{\bf Q}}\longrightarrow{\bf Q}\longrightarrow 0,
\]
where the second arrow from the right is the interior product with the
unique primitive element $-n(\rho)$ of $N$ contained in the cone $-\rho$.
As a result, we have an exact sequence
\[
0\longrightarrow\bigwedge\nolimits^{p-q}(\bar{M}\cap\bar{\tau}^{\perp})_{{\bf Q}}
\longrightarrow\bigwedge\nolimits^{p-q}(M\cap\tau^{\perp})_{{\bf Q}}
\longrightarrow\bigwedge\nolimits^{p-q-1}(\bar{M}\cap\bar{\tau}^{\perp})_{{\bf Q}}
\longrightarrow 0
\]
for each $q$, hence an exact sequence of complexes
\[
0\longrightarrow C^{{\textstyle \cdot}}(\bar{\Sigma},\Lambda^p)_{{\bf Q}}\longrightarrow
C^{{\textstyle \cdot}}(\Phi^{\flat},\Lambda^p)_{{\bf Q}}\stackrel{\iota}{\longrightarrow}
C^{{\textstyle \cdot}}(\bar{\Sigma},\Lambda^{p-1})_{{\bf Q}}\longrightarrow 0.
\]
On the other hand, we have
\[
C^q(\Phi,\Lambda^p)=C^q(\Phi^{\flat},\Lambda^p)
\oplus C^q(\mathop{\rm Star}\nolimits_{-\rho}(\Phi),\Lambda^p)
=C^q(\Phi^{\flat},\Lambda^p)\oplus C^{q-1}(\bar{\Sigma},\Lambda^{p-1})
\]
for each $q$. We see easily that
$C^{{\textstyle \cdot}}(\Phi,\Lambda^p)_{{\bf Q}}$ coincides with
the mapping cone of the surjective homomorphism
$\iota:C^{{\textstyle \cdot}}(\Phi^{\flat},\Lambda^p)_{{\bf Q}}\rightarrow
C^{{\textstyle \cdot}}(\bar{\Sigma},\Lambda^{p-1})_{{\bf Q}}$, whose kernel is
$C^{{\textstyle \cdot}}(\bar{\Sigma},\Lambda^p)_{{\bf Q}}$ as we saw above. Consequently,
we have
\[
H^{{\textstyle \cdot}}(\Phi,\Lambda^p)_{{\bf Q}}=H^{{\textstyle \cdot}}(\bar{\Sigma},\Lambda^p)_{{\bf Q}}.
\]
In particular, we have
$H^q(\Phi,\Lambda^p)_{{\bf Q}}=H^q(\bar{\Sigma},\Lambda^p)_{{\bf Q}}=0$ for $q\neq p$
by Proposition \ref{prop_vanishing_complete}.
\hspace*{\fill}q.e.d.
\begin{Corollary} \label{cor_vanishing_convexsupport}
Let $\Delta$ be a finite simplicial fan for $N\cong{\bf Z}^r$ such that
its support $|\Delta|$ is convex of dimension $r$. Then
\[
H^q(\Delta,\Lambda^p)_{{\bf Q}}=0\qquad\mbox{for all}\quad q\neq p.
\]
Consequently, the corresponding toric variety $X:=T_N\mathop{\rm emb}\nolimits(\Delta)$
satisfies $H^l(X^{{\rm an}},{\bf C})=0$ for odd $l$, while
\[
H^{2p}(X^{{\rm an}},{\bf C})=H^p(\Delta,\Lambda^p)_{{\bf C}}\qquad\mbox{for}\quad
0\leq p\leq r.
\]
\end{Corollary}
{\sc Proof.} \quad
By Proposition \ref{prop_vanishing_complete}, we may assume that
$\Delta$ is not complete. Then there exists
a primitive element $n^{\circ}\in N$
such that $-n^{\circ}$ is contained in the interior of $|\Delta|$.
Let $\rho:={\bf R}_{\geq 0}n^{\circ}$.
Denote by $\Delta^{\flat}$ the subcomplex of $\Delta$ consisting of
those $\sigma\in\Delta$ which are contained in the boundary of the
convex cone $|\Delta|$. Obviously,
\[
\widetilde{\Delta}:=\Delta\;{\textstyle \coprod}\;
\{\sigma+\rho\;\mid\;\sigma\in\Delta^{\flat}\}
\]
is a finite complete simplicial fan for $N$ such that
$\Delta=\widetilde{\Delta}\setminus\mathop{\rm Star}\nolimits_{\rho}(\widetilde{\Delta})$. We are done by
Theorem \ref{thm_vanishing_ishida} and Theorem \ref{thm_algderham}.
\hspace*{\fill}q.e.d.
\bigskip
$\widetilde{\Phi}$, $\Phi$ and $\Phi^{\flat}$ appearing in the proof of
Theorem \ref{thm_vanishing_ishida}, Lemma \ref{lem_vanishing_P1bundle}
and Corollary \ref{cor_vanishing_convexsupport} are of independent
interest. Namely, let $n_0$ be the primitive element in $N$
such that $\rho={\bf R}_{\geq 0}(-n_0)$, and choose a splitting
$N\cong \bar{N}\oplus{\bf Z} n_0$. Then there exists a function
$\eta:\bar{N}_{{\bf R}}\rightarrow{\bf R}$, which is ${\bf Z}$-valued on $\bar{N}$
and piecewise linear with respect to the complete nonsingular fan
$\bar{\Sigma}$ for $\bar{N}$ so that, in terms of the graph
$g:\bar{N}_{{\bf R}}\rightarrow N_{{\bf R}}$ of $\eta$ defined by
$g(\bar{n}):=\bar{n}+\eta(\bar{n})n_0$ for each $\bar{n}\in\bar{N}_{{\bf R}}$, we have
\begin{eqnarray*}
\Phi^{\flat} &=& \{g(\bar{\sigma})\;\mid\;\bar{\sigma}\in\bar{\Sigma}\} \\
\Phi &=& \Phi^{\flat}
\coprod\{\tau+{\bf R}_{\geq 0}n_0\;\mid\;\tau\in\Phi^{\flat}\} \\
\widetilde{\Phi} &=& \Phi
\coprod\{\tau+{\bf R}_{\geq 0}(-n_0)\;\mid\;\tau\in\Phi^{\flat}\}. \\
\end{eqnarray*}
It is easy to see that
$T_N\mathop{\rm emb}\nolimits(\widetilde{\Phi})\rightarrow T_{\bar{N}}\mathop{\rm emb}\nolimits(\bar{\Sigma})$ is a ${\bf P}_1({\bf C})$-bundle,
while $T_N\mathop{\rm emb}\nolimits(\Phi)\rightarrow T_{\bar{N}}\mathop{\rm emb}\nolimits(\bar{\Sigma})$ and
$T_N\mathop{\rm emb}\nolimits(\Phi^{\flat})\rightarrow T_{\bar{N}}\mathop{\rm emb}\nolimits(\bar{\Sigma})$ are
the associated ${\bf C}$-bundle and the associated ${\bf C}^*$-bundle, respectively.
As in \cite[Thm.\ 4.3 and Prop.\ 4.4]{lefsch} in different notation
and in Park \cite{park}, \cite{park2}, $\eta$ determines an element
$\bar{\eta}\in H^1(\bar{\Sigma},\Lambda^1)$, the multiplication by which induces
a homomorphism
\[
H^{p-1}(\bar{\Sigma},\Lambda^{p-1})_{{\bf Q}}\longrightarrow
H^p(\bar{\Sigma},\Lambda^p)_{{\bf Q}}
\]
for each $p$.
Then we have the following:
\begin{Corollary} \label{cor_vanishing_C*bundle}
In the notation above, we have
\[
H^q(\Phi^{\flat},\Lambda^p)=0\qquad\mbox{for}\quad q\neq p-1,p.
\]
Moreover, $H^{p-1}(\Phi^{\flat},\Lambda^p)_{{\bf Q}}$
(resp.\ $H^p(\Phi^{\flat},\Lambda^p)_{{\bf Q}}$) coincides with the
kernel (resp.\ cokernel) of the homomorphism
\[
H^{p-1}(\bar{\Sigma},\Lambda^{p-1})_{{\bf Q}}\longrightarrow
H^p(\bar{\Sigma},\Lambda^p)_{{\bf Q}}
\]
induced by multiplication of the element $\bar{\eta}\in H^1(\bar{\Sigma},\Lambda^1)$.
\end{Corollary}
|
1992-02-13T09:59:47 | 9202 | alg-geom/9202009 | en | https://arxiv.org/abs/alg-geom/9202009 | [
"alg-geom",
"math.AG"
] | alg-geom/9202009 | Wilberd van der Kallen | V. B. Mehta and Wilberd van der Kallen | On a Grauert-Riemenschneider vanishing theorem for Frobenius split
varieties in characteristic p | 4 pages, LaTeX | Inventiones Mathematicae 108 (1992), 11-13 | 10.1007/BF02100595 | null | null | This paper is about sheaf cohomology for varieties (schemes) in
characteristic $p>0$. We assume the presence of a Frobenius splitting. (See
V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for
Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is
that a non-zero higher direct image under a proper map of the ideal sheaf of a
compatibly Frobenius split subvariety can not have a support whose inverse
image is contained in that subvariety. Earlier vanishing theorems for Frobenius
split varieties were based on direct limits and Serre's vanishing theorem, but
our theorem is based on inverse limits and Grothendieck's theorem on formal
functions. The result implies a Grauert--Riemenschneider type theorem.
| [
{
"version": "v1",
"created": "Thu, 13 Feb 1992 08:59:31 GMT"
}
] | 2009-10-22T00:00:00 | [
[
"Mehta",
"V. B.",
""
],
[
"van der Kallen",
"Wilberd",
""
]
] | alg-geom | \section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
-.2ex}{1.5ex plus .2ex}{\large\bf}
\def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex
minus
-.2ex}{-1em}{\normalsize\bf}}
\makeatother
\begin{document}
\slopp
\title{ On a Grauert-Riemenschneider
vanishing theorem for Frobenius split varieties in characteristic $p$.
}
\author{V. B. Mehta \and Wilberd van der Kallen}
\maketitle
\section{Introduction}
It is known that the Grauert-Riemenschneider vanishing theorem is not
valid
in characteristic $p$ (\cite{not}). Here we show that it may be
restored in the
presence of a suitable Frobenius splitting. The proof uses
interchanging two
projective limits, one involving iterated Frobenius maps,
cf.\ \cite{Boutot} and
\cite{HartshorneSpeiser}, the other coming from Grothendieck's theorem
on formal
functions. That leads to the following general vanishing theorem which
we then apply in the
situation of the Grauert-Riemenschneider theorem.
\begin{theor}\label{general}
Let $\pi :X\rightarrow Y$ be a proper morphism of schemes of finite type
over a perfect field
of characteristic $p>0$. Let $D$ be a closed
subscheme of $X$ with ideal sheaf
$\cal I$, let $E$ be a closed
subscheme of $Y$ and let $i\geq 0$ such that\\
1. $D$ contains the geometric points of $\pi^{-1}E$.\\
2. $R^i\pi_\ast(\cal I)$ vanishes off $E$.\\
3. $X$ is Frobenius split, compatibly with $D$.\\
Then $R^i\pi_\ast(\cal I)$ vanishes on all of $Y$.
\end{theor}
\begin{theor}[Grauert-Riemenschneider with Frobenius
splitting.]\label{special}
Let $\pi :X\rightarrow Y$ be a proper birational morphism of varieties
in characteristic $p>0$ such that:\\
1. $X$ is non-singular and there is $\sigma\in
H^0(X,K_X^{-1})=H^0(X,c_1(X)) $
such that $\sigma^{p-1}$ splits $X$. (cf.~\cite{MehtaRamanathan}.) \\
2. $D=\mathop{\rm div}(\sigma)$ contains the exceptional locus of $\pi$ set
theoretically.\\
Then $R^i\pi_\ast K_X=0$ for $i>0$.
\end{theor}
\begin{remark}
It will be clear from the proof that many variations on our
Grauert-Riemenschneider theorem are possible. For instance, one may
replace $D$ by some subdivisor which still contains the exceptional
locus, and thus replace $K_X$
in the conclusion by the new $\cal O_X(-D)$. Similarly, the
birationality
assumption may be weakened, as it is used only to conclude that
condition 2 of
\ref{general} is satisfied.
\end{remark}
\section{Proofs}
\subsection{Proof of \ref{special}.} We assume theorem \ref{general}.
For $E$ we take the image of the exceptional locus. Dualizing $\sigma$
we get a short exact sequence
$$0\rightarrow K_X \rightarrow \cal O_X \rightarrow \cal O_D
\rightarrow 0,
$$
so $K_X$ may be identified with the ideal sheaf $\cal I$ of $D$. That
$D$ is compatibly split is clear from local computations, cf.~Remark on
page 36 of
\cite{MehtaRamanathan}.{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
\begin{lemma}
Let $\cdots \rightarrow M_2 \rightarrow M_1 \rightarrow M_0$ be a
projective system of artinian
modules over some ring $R$, with transition maps $f^j_i:M_j\rightarrow
M_i$. If $f^i_0$
is nonzero for all $i$, then the projective limit is nonzero.
\end{lemma}
\begin{proof*}
Put $M^{\rm stab}_i=\bigcap_{j\geq i} f^j_i(M_j)$. Then
$M^{\rm stab}_i=f^k_i(M_k)$ for $k\gg0$.
So
$$ f^{i+1}_i(M^{\rm stab}_{i+1})=f^{i+1}_if^k_{i+1}(M_k)= f^k_i(M_k)=
M^{\rm stab}_i
$$
for $k\gg0$. Therefore we have a subsystem $(M^{\rm stab}_i)$ with
nonzero
surjective maps, whence the result.
\end{proof*}
\subsection{Proof of \ref{general}.}
We argue by contradiction.
We may assume $Y$ is affine, so that $R^i\pi_\ast(\cal I)$ equals
$H^i(X,\cal I)$. Choose an irreducible component, with generic point $y$ say,
of the support on $Y$ of
$H^i(X,\cal I)$, which we suppose to be nonzero. Observe that $y\in E$.
The Frobenius map $F$ as well as its
splitting act on the exact sequence of sheaves
$$0\rightarrow \cal I\rightarrow \cal O_X\rightarrow \cal
O_D\rightarrow 0.
$$
Therefore
the Frobenius and its iterates act by split injective endomorphisms,
$p$-linear over
$A=\Gamma (Y,\cal O_Y)$, on $H^i(X,\cal I)$,
and the same remains true after localisation and completion at $y$. Let
$R$ be a regular ring
of the form $L[[X_1,\ldots,X_m]]$ mapping onto $A^{\wedge}_y$, where
$L$ is a field of representatives in
the completed local ring $A^{\wedge}_y$.
In the projective system of artinian modules
$$\cdots R\otimes^{p^r} H^i(X,\cal I)^\wedge_y\rightarrow
R\otimes^{p^{r-1}} H^i(X,\cal I)^\wedge_y\rightarrow \cdots
$$ all maps towards $R\otimes^{p^{0}} H^i(X,\cal I)^\wedge_y=
H^i(X,\cal I)^\wedge_y $ are nonzero.
Here $R\otimes^{p^r}$ refers to base change along the $r$ times
iterated
Frobenius endomorphism of the regular ring $R$, and the projective
system
is thus the one defining the ``leveling''
$G(H^i(X,\cal I)^\wedge_y)$, in the sense of \cite{HartshorneSpeiser},
of $H^i(X,\cal I)^\wedge_y$
as an $R$ module. The projective limit is nonzero by the Lemma.
On the other hand, as $R$ is a finite free module over $R$ via $F^r$,
one
may also compute
$$G(H^i(X,\cal I)^\wedge_y)=\lim_{\leftarrow_r} R\otimes^{p^r}
H^i(X,\cal I)^\wedge_y$$ as follows
$$\lim_{\leftarrow_r} R\otimes^{p^r} H^i(X,\cal I)^\wedge_y=
\lim_{\leftarrow_r} R\otimes^{p^r}\lim_{\leftarrow_s} H^i(X_s,\cal
I_s)=$$ $$
\lim_{\leftarrow_r}\lim_{\leftarrow_s} R\otimes^{p^r} H^i(X_s,\cal
I_s)=\lim_{\leftarrow_s}\lim_{\leftarrow_r} R\otimes^{p^r} H^i(X_s,\cal
I_s)
$$ where $X_s$ and $\cal I_s$ are the usual thickenings from
Grothendieck's
theorem on formal functions. But by the Artin-Rees lemma the Frobenius
map acts
nilpotently on $\cal I_s$, (note that some power of $\cal I$
is contained in the pull back of the ideal sheaf of $E$), so
$\lim_{\leftarrow_r} R\otimes^{p^r} H^i(X_s,\cal I_s)$ vanishes.
But then $G(H^i(X,\cal I)^\wedge_y)$ is both nonzero and zero.{\unskip\nobreak\hfill\hbox{ $\Box$}\par}
|
1992-02-07T03:35:47 | 9202 | alg-geom/9202001 | en | https://arxiv.org/abs/alg-geom/9202001 | [
"alg-geom",
"math.AG"
] | alg-geom/9202001 | Sheldon Katz | Sheldon Katz | Rational Curves on Calabi-Yau Threefolds | 10 pages, AMSLaTeX v1.0 or 1.1. This is a survey talk given at the
May 1991 Workshop on Mirror Symmetry at MSRI | null | null | null | null | The point is to compare the mathematical meaning of the ``number of rational
curves on a Calabi-Yau threefold'' to the meaning ascribed to the same notion
by string theorists.
| [
{
"version": "v1",
"created": "Wed, 5 Feb 1992 17:01:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Katz",
"Sheldon",
""
]
] | alg-geom | \section{Rational Curves, Normal Bundles, Deformations}
Consider a Calabi-Yau threefold $X$ containing a smooth rational curve
\linebreak
$C\cong{\Bbb C}{\Bbb P}^1$. The normal bundle $N_{C/X}$ of $C$ in $X$ is
defined by the
exact sequence
\begin{equation}
\label{nbseq}
0\to T_C\to T_X|_C\to N_{C/X}\to 0.
\end{equation}
$N=N_{C/X}$ is a rank 2 vector bundle on $C$, so $N=\mbox{${\cal O}$}(a)\oplus\mbox{${\cal O}$}(b)$ for some
integers $a,b$. Now $c_1(T_C)=2$, and $c_1(T_X)=0$ by the Calabi-Yau
condition.
So the exact sequence (\ref{nbseq}) yields $c_1(N)=-2$, or $a+b=-2$. One
``expects'' $a=b=-1$ in the general case. This is because there is a moduli
space of deformations of the vector bundle $\mbox{${\cal O}$}(a)\oplus\mbox{${\cal O}$}(b)$, and the general
point of this moduli space is $\mbox{${\cal O}$}(-1)\oplus\mbox{${\cal O}$}(-1)$, no matter what $a$ and $b$
are, as long as $a+b=-2$.
Let \mbox{${\cal M}$}\ be the moduli space of rational curves in $X$. The tangent space to
\mbox{${\cal M}$}\ at
$C$ is given by $H^0(N)$ \cite[\S 12]{KS}. In other words, \mbox{${\cal M}$}\ may be locally
defined by finitely many equations in $\dim H^0(N)$ variables.
\bigskip\noindent
{\bf Definition.}
$C$ is {\em infinitesimally rigid} if $H^0(N)=0$.
\bigskip
Infinitesimal rigidity means that $C$ does not deform inside $X$, not even to
first order.
Note that $H^0(N)=0$ if and only if $a=b=-1$. Thus
\begin{itemize}
\item $C$ is infinitesimally rigid if and only if $a=b=-1$.
\item $C$ deforms, at least infinitesimally, if and only if $(a,b)\neq
(-1,-1)$.
\end{itemize}
\mbox{${\cal M}$}\ can split up into countably many irreducible components. For instance,
curves
with distinct homology classes in $X$ will lie in different components of \mbox{${\cal M}$}.
However, there will be at most finitely many components of \mbox{${\cal M}$}\ corresponding to
rational curves in a fixed homology class.
There certainly exist Calabi-Yau threefolds $X$ containing
positive dimensional families of rational
curves. For instance, the Fermat quintic threefold
$x_0^5+\ldots x_4^5=0$ contains the family of lines given parametrically in the
homogeneous coordinates $(u,v)$ of ${\Bbb P}^1$ by
$(u,-u,av,bv,cv)$, where $(a,b,c)$ are the parameters of the plane curve
$a^5+b^5+c^5=0$. However, suppose that all rational curves on $X$ have normal
bundle $\mbox{${\cal O}$}(-1)\oplus\mbox{${\cal O}$}(-1)$. Then since \mbox{${\cal M}$}\ consists entirely of
discrete points, the remarks above show that there would be only finitely many
rational
curves in $X$ in a fixed homology class. Since a quintic threefold has
$H_2(X,{\Bbb Z})\cong{\Bbb Z}$, the degree of a curve is essentially the same
as its
homology class. This discussion leads to Clemens' conjecture \cite{Ccon}:
\bigskip\noindent
{\bf Conjecture (Clemens).}
{\em A general quintic threefold contains only finitely many rational curves of
degree
$d$, for any $d\in {\Bbb Z}$. These curves are all infinitesimally rigid.}
\bigskip
Clemens' original constant count went as follows \cite{Chom}:
a rational curve of degree $d$ in
$\mbox{${\Bbb P}$}^4$ is given parametrically by 5 forms $\alpha_0(u,v),\ldots,\alpha_4(u,v)$, each
homogeneous of
degree $d$ in the homogeneous coordinates $(u:v)$ of $\mbox{${\Bbb P}$}^1$. These $\alpha_i$
depend on $5(d+1)$
parameters. On the
other hand, a quintic equation $F(x_0,\ldots,x_4)=0$ imposes the condition
$F(\alpha_0(u,v),\ldots,\alpha_4(u,v))\equiv 0$ for the parametric curve to be
contained in this
quintic threefold. This is a polynomial equation of degree $5d$ in $u$ and
$v$.
Since a general degree $5d$ polynomial $\sum a_{i}u^iv^{5d-i}$ has $5d+1$
coefficients, setting these equal to zero results in $5d+1$ equations among the
$5d+5$ parameters of the $\alpha_i$. If $F$ is {\em general}, it seems plausible
that
these equations should impose independent conditions, so that the solutions
should
depend on $5d+5-(5d+1)=4$ parameters. However, any curve has a 4-parameter
family of reparametrizations $(u,v)\mapsto (au+bv,cu+dv)$, so there are
actually
a zero dimensional, or finite number, of curves of the general $F=0$.
The conjecture is known to be true for $d\le 7$ \cite{Kfin}. For any $d$, it
can even be proven that there exists an infinitesimally rigid curve of degree
$d$ on a general $X$. Similar conjectures
can be stated for other Calabi-Yau threefolds.
There are many kinds of non-rational
curves which appear to occur in finite number on a general quintic
threefold. For instance,
elliptic cubic curves are all planar. The plane $P$ that one spans meets the
quintic in a quintic curve containing the cubic curve. The other component
must
be a conic curve. This sets up a 1-1 correspondence between elliptic cubics
and
conics on any quintic threefold. Hence the number of elliptic cubics on a
general
quintic must be the same as the number of conics, 609250 \cite{Kfin}.
Finiteness of
elliptic quartic curves has been proven by Vainsencher \cite{V}; the actual
number has not yet been computed.
On the other hand, there are infinitely many plane quartics on {\em any\/}
quintic
threefold: take any line in the quintic, and each of the infinitely many planes
containing the line must meet the quintic in the original line union a quartic.
If a curve has $N\cong\mbox{${\cal O}$}\oplus\mbox{${\cal O}$}(-2)$, then $C\subset X$ deforms to first
order. In fact, \nolinebreak since $H^0(N)$ is one-dimensional, there is a
family of curves on $X$ parametrized by a single variable $t$,
subject to the constraint $t^2=0$. In other words, start with a rational curve
given parametrically by forms $\alpha_0,\ldots ,\alpha_4$, homogeneous of degree $d$
in $u$ and $v$. Take a perturbation
$\alpha_i(u,v;t)=\alpha_i(u,v)+t\alpha_i'(u,v)$, still homogeneous in $(u,v)$.
Form the equation
$F(\alpha_0,\ldots a_4)=0$ and formally set $t^2=0$; the resulting equation has a
5 dimensional space of solutions for the $\alpha_i'$, which translates into a
unique
solution up to multiples and reparametrizations of $\mbox{${\Bbb P}$}^1$. The curve $C$
but may
or may not deform to second order. $C$ deforms to $n^{\mbox{th}}$ order for
all $n$ if and only if $C$ moves in a 1-parameter family. A pretty description
of the general situation is given in \cite{R}.
If $C$ deforms to
$n^{\mbox{th}}$ order, but not to $(n+1)^{\mbox{th}}$ order, then one sees
that
while $C$ is an isolated point in the moduli space of curves on $X$, it
more naturally is viewed as the solution to the equation $t^{n+1}=0$ in one
variable $t$. So $C$
should be viewed as a rational curve on $X$ with multiplicity $n+1$.
If a curve has $N\cong\mbox{${\cal O}$}(1)\oplus\mbox{${\cal O}$}(-3)$, $C$ has a 2 parameter space of
infinitesimal deformations, and the structure of \mbox{${\cal M}$}\ at $C$ is correspondingly
more
complicated. An example is given in the next section. The general situation
has
not yet been worked out.
\section{Counting Rational Curves}
In this section, a general procedure for calculating the number of smooth
rational curves of a
given type is described. Alternatively, a canonical definition of
this number can be given using the Hilbert scheme (this is what was done
by Ellingsrud and Str{\o}mme in their work on twisted cubics \cite{EScalc});
however, it is usually quite difficult to implement a calculation along
these lines.
Embed the Calabi-Yau threefold $X$ in a larger compact space
${\Bbb P}$ (which may be thought of as a projective space, a weighted
projective
space, or a product of such spaces). $\mbox{${\cal M}$}_\lambda$ will denote the moduli space
parametrizing smooth rational curves in ${\Bbb P}$ of a given topological type
or degree $\lambda$.
\mbox{${\rm Def}(X)$}\ denotes the irreducible component of $X$ in the moduli space of Calabi-Yau
manifolds
in ${\Bbb P}$ (here the K\"ahler structure is ignored). In other words, \mbox{${\rm Def}(X)$}\
parametrizes the deformations of $X$ in ${\Bbb P}$.
\begin{enumerate}
\item Find a compact moduli space $\bar {\mbox{${\cal M}$}_\lambda}$ containing $\mbox{${\cal M}$}_\lambda$ as a
dense open subset,
such that the points of $\bar{\mbox{${\cal M}$}_\lambda}-\mbox{${\cal M}$}_\lambda$ correspond to degenerate
curves of type $\lambda$. $\bar {\mbox{${\cal M}$}_\lambda}$ parametrizes degenerate deformations of
the smooth curve (not the mapping from ${\Bbb C}{\Bbb P}^1$ to $X$).
It is better for $\bar{\mbox{${\cal M}$}_\lambda}$ to be smooth.
\pagebreak
\item Find a rank $r=\dim(\mbox{${\cal M}$}_\lambda)$ vector bundle $\mbox{${\cal B}$}$ on
$\bar{\mbox{${\cal M}$}_\lambda}$ such that
\begin{enumerate}
\item To each $X'\in\mbox{${\rm Def}(X)$}$ there is a section $s_{X'}$ of \mbox{${\cal B}$}\ which vanishes at
$C\in\bar{\mbox{${\cal M}$}_\lambda}$ if and only if $C\subset X'$.
\item There exists an $X'\in\mbox{${\rm Def}(X)$}$ such that $s_{X'}(C)=0$ if and only if
$C\in \mbox{${\cal M}$}_\lambda$ and $C\subset X'$.
\item $C$ is an isolated zero of $s_{X'}$ with $\mbox{mult}_C(s_{X'})=1$ if
and only if
$N_{C/X}\cong\mbox{${\cal O}$}(-1)\oplus\mbox{${\cal O}$}(-1)$.
\end{enumerate}
\end{enumerate}
\bigskip\noindent
{\bf Working Definition.} The {\em number of smooth rational curves $n_\lambda$ of
type $\lambda$}
is given by the
$r^{\rm th}$ Chern class $c_r(\mbox{${\cal B}$})$.
\bigskip
Why is this a reasonable definition?
Suppose that an $X'\subset\mbox{${\rm Def}(X)$}$ can be found with the properties required above,
with the additional property that there are only finitely many curves of type
$\lambda$ on $X'$,
and that they are all infinitesimally rigid. Then it can be checked that the
number of
(possibly degenerate) curves
of type $\lambda$ on $X'$ is independent of the choice of $X'$ satisfying the above
properties, and is also equal to $c_r(\mbox{${\cal B}$})$. This last follows since
$c_{\mbox{rank}(E)}(E)$ always gives the homology class of the zero locus $Z$
of any
section of any bundle $E$ on any variety $Y$, whenever
$\dim(Z)=\dim(Y)-\mbox{rank}
(E)$. In our case, $0=\dim(\{{\rm lines}\})=\dim(\mbox{${\cal M}$}_\lambda)-\mbox{rank}(\mbox{${\cal B}$})$. In
essentially all known cases, the number of curves has been worked out by the
method of this working definition. Examples are given below.
There is a potential problem with this working definition. For
families of Calabi-Yau threefolds such that no threefold in the family contains
finitely many curves of given type, it may be that the ``definition'' depends
on the choice
of compactification and/or vector bundle, i.e. this is not well-defined. My
reason for almost calling this method a definition is that it {\em does\/}
give a finite number corresponding to an infinite family of curves, which
{\em is\/}
well-defined in the case that the Calabi-Yau threefold in question belongs to
a family containing some other Calabi-Yau threefold with only finitely many
rational curves of the type under consideration.
In the case where the general $X$ contains irreducible singular curves which
are
the images of maps from $\mbox{${\Bbb P}$}^1$, a separate
but similar procedure must be implemented to calculate these, since they give
rise to instanton corrections as well. For example,
there is a 6-parameter family of two-planes in $\mbox{${\Bbb P}$}^4$. A two plane $P$ meets a
quintic threefold $X$ in a plane quintic curve. For general $P$ and $X$,
this curve is a smooth genus 6 curve. But if the curve acquired 6 nodes, the
curve would be rational. This being 6 conditions on a 6 parameter family, one
expects that a general $X$ would contain finitely many 6-nodal rational plane
quintic curves, and it can be verified that this is indeed the case.
The problem is
that there is no way to deform a smooth rational curve to such a singular
curve --- the dimension of $H^1(\mbox{${\cal O}$}_C)$ is zero for a smooth rational curve $C$,
but positive for such singular curves, and this dimension is a deformation
invariant \cite[Theorem 9.9]{H}. For example, if one tried to deform a smooth
twisted cubic curve to a
singular cubic plane curve by projecting onto a plane, there would result an
``embedded point'' at the singularity, creating a sort of discontinuity in the
deformation process \cite[Ex.\ 9.8.4]{H}.
\bigskip\noindent
{\em Examples:}
\begin{enumerate}
\item Let $X\subset\mbox{${\Bbb P}$}^4$ be a quintic threefold. Take $\lambda=1$, so that we are
counting lines. Here $\mbox{${\cal M}$}_1=G(1,4)$ is the Grassmannian of lines in $\mbox{${\Bbb P}$}^4$ and
is
already compact, so take $\bar{\mbox{${\cal M}$}_1}=\mbox{${\cal M}$}_1$. Let $\mbox{${\cal B}$}=\mbox{Sym}^5(U^*)$, where
$U$, the universal bundle, is the rank 2 bundle on $\mbox{${\cal M}$}_1$ whose fiber over a
line
$L$ is the 2-dimensional subspace $V\subset{\Bbb C}^5$ yielding $L\subset\mbox{${\Bbb P}$}^4$
after
projectivization. Note that $\mbox{rank}(\mbox{${\cal B}$})=\dim(\mbox{${\cal M}$}_1)=6$.
$\mbox{${\rm Def}(X)$}\subset\mbox{${\Bbb P}$}(H^0(\mbox{${\cal O}$}_{\mbox{${\Bbb P}$}^4}(5)))$ is the subset of smooth
quintics. A quintic $X$ induces a section $s_X$ of \mbox{${\cal B}$}, since an equation for
$X$ is a
quintic form on ${\Bbb C}^5$, hence induces
a quintic form on $V$ for $V\subset{\Bbb C}^5$ corresponding to $L$. Clearly
$s_X(L)=0$ if and only if $L\subset X$. The above conditions are easily seen
to
hold. $c_6(\mbox{${\cal B}$})=2875$ is the number of lines on $X$. This
calculation is essentially the same as that done for cubics in
\cite[Thm.\ 1.3]{AK},
where dual notation is used, so that the $U^*$ used here becomes the universal
quotient bundle $Q$ in \cite{AK}. The number $2875$ agrees with the result of
Candelas et.\ al.\ \nopagebreak\cite{Ccalc}.
\bigskip
\item Continuing with the quintic, take $\lambda=2$. Any conic $C$ necessarily
spans a
unique 2-plane containing $C$. Let $G=G(2,4)$ be the Grassmannian of 2-planes
in
$\mbox{${\Bbb P}$}^4$, and let $U$ be the rank 3 universal bundle on $G$. Put
$\bar{\mbox{${\cal M}$}_2}=\mbox{${\Bbb P}$}(\mbox{Sym}^2(U^*))$ be the projective bundle over $G$ whose
fiber
over a plane $P$ is the projective space of conics in $P$. Clearly
$\mbox{${\cal M}$}_2\subset\bar{\mbox{${\cal M}$}_2}$ (but they are not equal; $\bar{\mbox{${\cal M}$}_2}$ also contains the
union of any two lines or a double line in any plane). Let
$\mbox{${\cal B}$}=\mbox{Sym}^5(U^*)/(\mbox{Sym}^3(U^*)\otimes\mbox{${\cal O}$}_{\mbox{${\Bbb P}$}}(-1))$ be the bundle on
$\bar{\mbox{${\cal M}$}_2}$ of quintic
forms on the 3 dimensional vector space $V\subset{\Bbb C}^5$, modulo those
which
factor as any cubic times the given conic. ($\mbox{${\cal O}$}_{\mbox{${\Bbb P}$}}(-1)$ is the line bundle
whose
fiber over a conic is the one dimensional vector space of equations for the
conic within its supporting
plane. The quotient is relative to the natural embedding
$\mbox{Sym}^3(U^*)\otimes\mbox{${\cal O}$}_{\mbox{${\Bbb P}$}}(-1)\to\mbox{Sym}^5(U^*)$ induced by
multiplication.) $\mbox{rank}(\mbox{${\cal B}$})=\dim(\bar{\mbox{${\cal M}$}_2})=11$.
$c_{11}(\mbox{${\cal B}$})=609250$. See \cite{Kfin} for more details, or \cite{CMW} for an
analogous computation in the case of quartics. The number $609250$ agrees
with
the result of
Candelas et.\ al.
\bigskip
\item Again consider the quintic, this time with $l=3$. In \cite{EScalc},
Ellingsrud and Str{\o}mme take $\bar{\mbox{${\cal M}$}_3}$ to be the closure of the locus of
smooth twisted cubics in $\mbox{${\Bbb P}$}^4$ inside
the Hilbert scheme. This space has dimension 16. $\mbox{${\cal B}$}$ is essentially the rank
16 bundle of degree 15 forms on $\mbox{${\Bbb P}$}^1$ induced from quintic polynomials in
$P^4$
by the the degree 3 parametrization of the cubic (it must be shown that this
makes sense for
degenerate twisted cubics as well). The equation of a general quintic gives a
section of $\mbox{${\cal B}$}$, vanishing precisely on the set of cubics contained in $X$.
$c_{16}(\mbox{${\cal B}$})=317206375$, again agreeing with
the result of
Candelas et.\ al.
\end{enumerate}
\bigskip
The key to the first two calculations are the Schubert calculus for calculating
in Grassmannians \cite[Ch.\ 1.5]{GH} and standard formulas for projective
bundles \cite[Appendix A.3]{H}. The third calculation is more intricate.
Regarding complete intersection Calabi-Yau manifolds, similar examples are
found in
\cite{Klin} for lines and \cite{SvS} for conics.
Note that the number of curves $c_r(\mbox{${\cal B}$})$ in no way depends on the choice
of $X$, even if $X$ is a degenerate Calabi-Yau threefold, or contains
infinitely many rational curves of type $\lambda$. It turns out that a natural
meaning can be assigned to this number. In fact, the moduli space of
curves of type $\lambda$ on $X$ splits up into ``distinguished varieties''
$Z_i$
\cite{F}, and a number, the {\em equivalence} of $Z_i$, can be assigned
to each distinguished variety (the number is 1 for each
$\mbox{${\cal O}$}(-1)\oplus
\mbox{${\cal O}$}(-1)$ curve). This number is precisely equal to the number of curves on $X$
in $Z_i$
which arise as limits of curves on $X'$ as $X'$ approaches $X$ in a 1-parameter
family \cite[Ch.\ 11]{F}.
\bigskip\noindent
{\em Examples:}
\begin{enumerate}
\item If a quintic threefold is a union of a hyperplane and a quartic, then the
quintic contains infinitely many lines and conics. However, in the case of
lines,
given a general 1-parameter family of quintics approaching this reducible
quintic,
$1275$ lines approach the hyperplane, and 1600 lines approach the quartic
\cite{Kdeg}.
So the infinite set of lines in the hyperplane ``count'' as $1275$, while
those in
the quartic count as $1600$. For counting conics, the $609250$ conics
distribute
themselves as $187250$ corresponding to the component of conics in the
hyperplane,
$258200$ corresponding to the component of conics in the quartic, and $163200$
corresponding to the component of conics which degenerate into a line in the
hyperplane union an intersecting line in the quartic \cite{Kit}. Note that
this
reducible conic lies in $\bar{\mbox{${\cal M}$}_2}-\mbox{${\cal M}$}_2$, and illustrates why $\mbox{${\cal M}$}_2$ itself is
insufficient for calculating numbers when there are infinitely many curves.
Most of these numbers have been calculated recently by Xian Wu \cite{W} using
a different method.
\item If a quintic threefold is a union of a quadric and a cubic, the lines
on the quadric count as $1300$, and the lines on the cubic count as $1575$
\cite{Kdeg,W2}. The conics on the quadric count as $215,950$, while the conics
on
the cubics count as $243900$ \cite{W2}. Presumably this implies that the conics
which
degenerate into a line in the quadric union an intersecting line in the cubic
count as $609250-(215950+243900)=149400$, but this has not been checked
directly yet.
\item There are infinitely many lines on the Fermat quintic threefold
$x_0^5+\ldots+x_4^5=0$. These divide up into 50 cones, a typical one being
the family of lines given parametrically in the
homogeneous coordinates $(u,v)$ of ${\Bbb P}^1$ by
$(u,-u,av,bv,cv)$, where $(a,b,c)$ satisfy $a^5+b^5+c^5=0$. Each of these count
as 20. There are also 375 special lines, a typical one being given by the
equations
$x_0+x_1=x_2+x_3=x_4=0$ (these lines were also noticed in \cite{DSWW}). These
lines $L$ count with multiplicity 5. Note that $50\cdot 20+375\cdot 5=
2875$ \cite{AKlin}. This example illustrates the potential complexity in
calculating the distinguished varieties
$Z_i$ --- some components can be embedded inside others. This may be
understood as well by looking at the moduli space of lines on the Fermat
quintic locally
at a line corresponding to a special line.
Since \linebreak $N_{L/X}\cong\mbox{${\cal O}$}(1)\oplus\mbox{${\cal O}$}(-3)$, $H^0(N_{L/X})$ has dimension
2, and the
moduli space of lines is a subset of a 2-dimensional space.
A calculation shows that it
is locally defined inside 2
dimensional $(x,y)$ space by the equations $x^2y^3=x^3y^2=0$. The $x$ and $y$
axes
correspond to lines on each of the 2 cones, each occurring with multiplicity
2; there
would be just one equation $x^2y^2=0$ if the special line corresponding to
$(0,0)$ played
no role; since this is not the case, it can be expected to have its own
contribution; i.\hskip0pt e.\ each special line is a distinguished
variety.
\item Examples for lines in complete intersection Calabi-Yau threefolds were
worked
out in \cite{Klin}.
\end{enumerate}
\bigskip
These numbers can also be calculated by intersection
theory techniques. Let $s(Z_i,\bar{\mbox{${\cal M}$}_\lambda})$ denote the {\em Segre class} of
$Z_i$
in $\bar{\mbox{${\cal M}$}_\lambda}$. If $Z_i$ is smooth, this is simply the formal inverse of the
total Chern class $1+c_1(N)+c_2(N)+\ldots$ of the normal bundle $N$ of $Z_i$
in $\bar{\mbox{${\cal M}$}_\lambda}$. Then if $Z_i$ is a connected component of the zero locus of
$s_X$, the equivalence of $Z_i$ is
the zero dimensional part of $c(\mbox{${\cal B}$})\cap s(Z_i,\bar{\mbox{${\cal M}$}_\lambda})$
\cite[Prop.\ 9.1.1]{F}.
If $Z_i$ is an irreducible component which is not a connected component, then
this
formula is no longer applicable. However, I have had recent success with a new
method that supplies ``correction terms'' to this formula. The method is
currently ad hoc (the most relevant success I have had is in calculating the
``number'' of
lines on a cubic surface which is a union of three planes), but I expect that
a more
systematic procedure can be developed.
This of course is a reflection of the situation in calculating instanton
corrections to the Yukawa couplings. If there is a continuous family of
instantons, then calculating the corrections will be more difficult. If the
parameter space for instantons is smooth, this should make the
difficulties more managable. If the
space is singular, the calculation is more difficult. I expect that the
calculation would be even more difficult if instantons occur in at least 2
families that intersect.
Of course, in the calculation of the Yukawa couplings via path integrals, there
is no mention of vector bundles on the moduli space. This indicates to
me that there should be a mathematical definition of the
equivalence of a distinguished variety that does not refer to an auxiliary
bundle. Along these lines, one theorem will
be stated without proof.
Let $Z$ be a $k$-dimensional unobstructed family of rational curves on a
Calabi-Yau threefold
$X$. There is the total space ${\cal Z}\subset Z\times X$ of the family, with
projection map $\pi:{\cal Z}\to Z$ such that $\pi^{-1}(z)$ is the curve in $X$
corresponding to $z$, for each $z\in Z$. Let $N$ be the
normal bundle of ${\cal Z}$ in $Z\times X$. Define the equivalence $e(Z)$ of
$Z$ to be the number $c_k(R^1\pi_*N)$. For example, if $k=0$, then the curve
is infinitesimally rigid, and $e(Z)=1$.
\bigskip\noindent
{\bf Theorem.} {\em Let $Z$ be an unobstructed family of rational curves of
type $\lambda$ on a
Calabi-Yau threefold $X$. Suppose that $X$ deforms to a Calabi-Yau threefold
containing only finitely many curves of type $\lambda$. Then precisely $e(Z)$ of
these curves (including multiplicity) approach curves of $Z$ as the Calabi-Yau
deforms to $X$}.
\bigskip
Some of the examples given earlier in this section can be redone via this
theorem. Also, as anticipated by \cite{Ccalc}, it can be calculated that a
factor of $1/{m^3}$ is introduced
by degree $m$ covers by a calculation similar in spirit to that found in
\cite{TFT}.
|
1992-02-26T03:26:22 | 9202 | alg-geom/9202026 | en | https://arxiv.org/abs/alg-geom/9202026 | [
"alg-geom",
"math.AG"
] | alg-geom/9202026 | David R. Morrison | David R. Morrison | Picard-Fuchs equations and mirror maps for hypersurfaces | 18 pp. (AmS-LaTeX 1.0 or 1.1) or 23 pp. (LaTeX) | Essays on Mirror Manifolds (S.-T. Yau, ed.), International Press,
Hong Kong, 1992, pp. 241-264 | null | null | null | We describe a strategy for computing Yukawa couplings and the mirror map,
based on the Picard-Fuchs equation. (Our strategy is a variant of the method
used by Candelas, de la Ossa, Green, and Parkes in the case of quintic
hypersurfaces.) We then explain a technique of Griffiths which can be used to
compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the
computation for four specific examples (including quintic hypersurfaces,
previously done by Candelas et al.). This yields predictions for the number of
rational curves of various degrees on certain hypersurfaces in weighted
projective spaces. Some of these predictions have been confirmed by classical
techniques in algebraic geometry.
| [
{
"version": "v1",
"created": "Wed, 26 Feb 1992 02:26:10 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Morrison",
"David R.",
""
]
] | alg-geom | \section*{Introduction}
The phenomenon of mirror symmetry dramatically caught the attention
of mathematicians with the recent work of
P.~Candelas, X.~C.~de la Ossa, P.~S.~Green, and L.~Parkes \cite{pair}.
Starting with a particular pair of ``mirror manifolds'',
calculating certain period integrals,
interpreting the results as Yukawa couplings, and then re-interpreting
those results in light of the ``mirror manifold'' phenomenon,
Candelas et al.\ were
able to give predictions for the numbers of rational curves of various
degrees on the general quintic threefold.
In fact, algebraic geometers have had a difficult time
verifying these predictions, but all
successful attempts to calculate the numbers of curves
have eventually confirmed the predictions.
What is so striking about this work
is that the calculation which predicts the
numbers of rational curves on quintic threefolds
is in reality a calculation about the
variation of Hodge structure on a completely {\em different\/} family of
Calabi-Yau threefolds. An asymptotic expansion is made of a function
which comes from that variation, and the coefficients in the
expansion are then used to predict numbers of rational curves.
In \cite{guide}, we interpreted the calculation of Candelas et
al.~\cite{pair}
in terms of variation of Hodge structure. Here we take a more down
to earth approach, and work directly with period integrals and their
properties. (This is perhaps closer in spirit to the original paper.)
We have found a way to modify the computational strategy employed in
\cite{pair}. Our modified method computes a bit less (there are
two unknown ``constants of integration''), but it
is easier to actually carry out the computation. We in
fact carry it out in three new examples. This leads to new predictions
about numbers of rational curves on certain Calabi-Yau threefolds.
Our strategy for computing Yukawa couplings is based on the Picard-Fuchs
equation for the periods of a one-parameter family of algebraic varieties.
We explain in sections 1 and 2 how this equation can be used to compute
Yukawa couplings and
the mirror map for a family of Calabi-Yau threefolds with
$h^{2,1}=1$. We then go on in section 3 to
review a method of Griffiths \cite{griffiths} for calculating
Picard-Fuchs equations of hypersurfaces. Related ideas have also been
introduced into the physics literature in
\cite{blok-var,cad-ferr,ferrara,lsw}.
In sections 4 and 5, we carry out the computation in four examples,
including the quintic hypersurface. The resulting
predictions about numbers of
rational curves are discussed in section 6.
\section{The Picard-Fuchs equation and monodromy}
Let $\bar\pi: \overline{\cal X}\to \overline C$
be a family of $n$-dimensional
projective algebraic varieties,
parameterized by
a compact Riemann surface $\overline C$. Let $C \subset \overline C$
be an open subset such that the induced family $\pi: {\cal X}\to C$
has smooth fibers. If we choose
topological $n$-cycles $\gamma_0,\dots,\gamma_{r-1}$ which give a basis
for the $n^{\text{th}}$
homology of one particular
fiber $X_0$, and choose a holomorphic $n$-form $\omega$ on $X_0$,
then the {\em periods} of $\omega$ are the integrals
\[ \int_{\gamma_0}\omega,\dots,\int_{\gamma_{r-1}}\omega. \]
Since the fibration $\pi: {\cal X}\to C$ is differentiably
locally trivial, a local trivialization can be used to extend
the cycles $\gamma_i$ from $X_0$ to cycles $\gamma_i(z)$ on $X_z$
which depend on
$z$, where $z$ is a local coordinate on $C$.
The holomorphic $n$-form $\omega$ can also be extended to a family of
$n$-forms $\omega(z)$ which depend on the parameter $z$. If this is
done in an algebraic way, then $\omega(z)$ extends to a meromorphic
family of
$n$-forms (i.e. poles are allowed) over the entire space $\overline{\cal X}$.
The cycles $\gamma_i(z)$ determine homology classes which
are locally constant in $z$.
However, an attempt to extend these cycles
globally will typically lead to monodromy:
for each closed path in $C$,
there will be some linear map $T$ represented by a matrix $T_{ij}$
such that transporting $\gamma_i$ along the path produces at the end
a cycle homologous to
$\sum T_{ij}\gamma_j$.
The same phenomenon will hold for the periods: for a globally
defined meromorphic family of $n$-forms $\omega(z)$, the local periods
$\int_{\gamma_i(z)}\omega(z)$ extend by analytic continuation to
multiple-valued functions of $z$, transforming according to the same
monodromy transformations
$T$ as do the homology classes of the cycles.
The periods $\int_{\gamma(z)}\omega(z)$
satisfy an ordinary differential equation called
the {\em Picard-Fuchs equation\/} of $\omega$.
The existence of this equation can be explained as follows.
Choose a local coordinate $z$
on some open set $U\subset C$, and consider the vector
\[ v_j(z) := [\frac{d^j}{dz^j}\int_{\gamma_0(z)}\omega(z),\dots,
\frac{d^j}{dz^j}\int_{\gamma_{r-1}(z)}\omega(z)] \in\Bbb C^r. \]
For generic values of the parameter $z$,
the dimensions
\[
d_j(z):=\dim(\operatorname{span}\{v_0(z),\dots,v_j(z)\})
\]
must be constant.
Since $d_j(z)\le r$, these spaces cannot continue to grow indefinitely.
There
will thus be a smallest $s$ such that
\[v_s(z)\in\operatorname{span}\{v_0(z),\dots,v_{s-1}(z)\}\]
(for generic $z$).
We can write
\[ v_s(z) =-\sum_{j=0}^{s-1}C_j(z)v_j(z) \]
with the coefficients $C_j(z)$ depending on $z$.
The {\em Picard-Fuchs equation},
satisfied by all the periods of $\omega(z)$, is then
\begin{equation} \label{picfuc}
\frac{d^sf}{dz^s}+\sum_{j=0}^{s-1}C_j(z)\frac{d^jf}{dz^j}=0.
\end{equation}
The precise form of the equation depends on both the local coordinate
$z$ on $C$,
and the choice of holomorphic form $\omega(z)$.
Note that the coefficients $C_j(z)$ may acquire singularities at special
values of $z$.
When we approach a point $P$
in $\overline C-C$, the Picard-Fuchs equation has (at worst)
a {\em regular singular point\/} at $P$ \cite{gr-bull,nkatz,deligne}.
If we choose a parameter $z$ which is
centered at $P$ (that is, $z=0$ at $P$), then the coefficients $C_j(z)$
in the Picard-Fuchs equation typically will have poles at $z=0$.
However, if we multiply the Picard-Fuchs operator
\begin{equation} \label{PFop}
\frac{d^s}{dz^s}+\sum_{j=0}^{s-1}C_j(z)\frac{d^j}{dz^j}
\end{equation}
by $z^s$ and rewrite the result in the form
\begin{equation} \label{logform}
(z\frac{d}{dz})^s+\sum_{j=0}^{s-1}B_j(z)(z\frac{d}{dz})^j
\end{equation}
then the new coefficients $B_j(z)$ are holomorphic functions of $z$.
(This is one of several equivalent definitions of ``regular singular point''.)
We call eq.~\eqref{logform} the {\em logarithmic form\/} of the Picard-Fuchs
operator.
The structure of ordinary differential equations with regular singular points
is a classical topic in differential equations: a convenient reference is
\cite{codlev}. We can rewrite eq.~\eqref{picfuc} as a system of first-order
equations, using the logarithmic form eq.~\eqref{logform}, as follows:
let
\begin{equation} \label{Az}
A(z) = \begin{bmatrix}
0 & 1 & & & \\
& 0 & 1 & & \\
& & \ddots & \ddots & \\
& & & 0 & 1 \\
-B_0(z) & -B_1(z) & \dots & \dots & -B_{s-1}(z)
\end{bmatrix}.
\end{equation}
Then solutions $f(z)$ to the equation eq.~\eqref{picfuc} are equivalent
to solution
vectors
\[w(z) = \begin{bmatrix} f(z) \\ z\frac d{dz}f(z) \\ \vdots \\
(z\frac d{dz})^{s-1}f(z) \end{bmatrix} \]
of the matrix equation
\begin{equation} \label{matrixeqn}
z\frac d{dz}w(z)=A(z)w(z).
\end{equation}
For a matrix equation such as eq.~\eqref{matrixeqn}, the facts are these
(see \cite{codlev}).
There is
a constant $s\times s$ matrix $R$ and a $s\times s$ matrix $S(z)$
of (single-valued)
functions of $z$, regular near $z=0$, such that
\[\Phi(z)=S(z) \cdot z^R \]
is a {\em fundamental matrix} for the system. This means that the
columns of
$\Phi(z)$ are a basis for the space of solutions at each nonsingular
point $z\ne0$.
The multiple-valuedness of the solutions has all been put into $R$, since
\[z^R:=e^{(log z)R} = I+(\log z)R+\frac{(\log z)^2}{2!}R^2+\cdots\]
is a multiple-valued matrix function of $z$. The local monodromy on
the solutions
given by analytic continuation along a path winding
once around $z=0$ in a counterclockwise direction is given
by $e^{2\pi iR}$ (with respect to the basis given by the columns of $\Phi$).
The matrix $R$ is by no means unique.
\begin{theorem}
Suppose that $z\frac d{dz}w(z)=A(z)w(z)$ is a system of ordinary
differential
equations with a regular singular point at $z=0$. Suppose that
distinct eigenvalues
of $A(0)$ do not differ by integers. Then there is a fundamental
matrix of
the form
\[\Phi(z)=S(z) \cdot z^{A(0)} \]
and $S(z)$ can be obtained as a power series
\[S(z)=S_0+S_1z+S_2z^2+\cdots\]
by recursively solving the equation
\[
z\frac d{dz}S(z)+S(z)\cdot A(0)=A(z)\cdot S(z)
\]
for the coefficient matrices $S_j$.
Moreover, any such series solution
converges in a neighborhood of $z=0$.
\end{theorem}
A proof can be found in \cite{codlev}, together with
methods for treating the case in which eigenvalues of
$A(0)$ {\em do} differ by integers.
We will be particularly interested in systems with
{\em unipotent monodromy}:
by definition, this means that $e^{2\pi iR}$ is a unipotent matrix,
so that
$(e^{2\pi iR}-I)^m\ne0$, $(e^{2\pi iR}-I)^{m+1}=0$ for some $m$
called the
{\em index}.
\begin{corollary}
Suppose that
$ (z\frac{d}{dz})^sf(z)+\sum_{j=0}^{s-1}B_j(z)(z\frac{d}{dz})^jf(z) $
is an ordinary differential equation with a regular singular point at $z=0$.
If $B_j(0)=0$ for all $j$, then the solutions of this equation have
unipotent
monodromy of index $s$.
\end{corollary}
The corollary follows by calculating with
eq.~\eqref{Az}, setting $z=0$ and $B_j(0)=0$
to produce
\[
e^{2\pi iA(0)}= \begin{bmatrix}
1 & 2\pi i & \frac{(2\pi i)^2}{2!} & \dots & \frac{(2\pi i)^{s-1}}{(s-1)!} \\
& 1 & 2\pi i & \dots & \frac{(2\pi i)^{s-2}}{(s-2)!} \\
& & \ddots & & \vdots \\
& & & 1 & 2\pi i \\
& & & & 1
\end{bmatrix}.\]
\section{Computing the mirror map}
Recall that a {\em Calabi-Yau manifold} is a compact K\"ahler manifold $X$
of complex dimension $n$ which has trivial canonical bundle,
such that the Hodge numbers $h^{k,0}$ vanish for $0<k<n$.
Thanks to a celebrated theorem of Yau \cite{yau}, every such manifold
admits Ricci-flat K\"ahler metrics.
Suppose now that $\pi: {\cal X}\to C$ is a family of Calabi-Yau threefolds
with $h^{2,1}(X)=1$, which is not a locally constant family.
The third cohomology group $H^3(X)$ has dimension
$r=4$. It follows that the Picard-Fuchs equation has order at most 4.
(In fact, it is not difficult to show
that it has order exactly 4.)
Let $z$ be a coordinate on $\overline C$ centered at a point
$P\in\overline C-C$. We say that
{\em $P$ is a point at which the monodromy is maximally
unipotent} if the monodromy is unipotent of index 4. As we have seen
in the corollary, if $B_j(0)=0$ in the logarithmic form of
the Picard-Fuchs equation, $z=0$ will be such a point. We will assume
for simplicity that our points of maximally unipotent monodromy have
this form, leaving appropriate modifications for the general case to
the reader.
We review the calculation of the Yukawa coupling, following \cite{pair}.
Let $\omega(z)$ be a family of $n$-forms, and let
\[W_k := \int_{X_z}\omega(z)\wedge\frac{d^k}{dz^k}\omega(z).\]
A fundamental principle from the theory of variation of Hodge structure
(cf.~\cite{transcendental}) implies that
$W_0$, $W_1$, and $W_2$ all vanish. The {\em Yukawa coupling} is the first
non-vanishing term $W_3$. Candelas et al.\ show that the Yukawa coupling
$W_3$ satisfies the differential equation
\[ \frac{dW_3(z)}{dz}=-\frac12C_3(z)W_3(z), \]
where $C_3(z)$ is a coefficient in the Picard-Fuchs equation \eqref{picfuc}.
The Yukawa coupling as defined clearly depends on the ``gauge'', that is,
on the choice of holomorphic $3$-form $\omega(z)$. If fact, if we alter
the gauge by $\omega(z)\mapsto f(z)\omega(z)$, then $W_k$ transforms
as
\[ W_k\mapsto f(z)\sum_{j=0}^k\binom{k}{j}\frac{d^jf(z)}{dz^j}W_{k-j}. \]
Since $W_0=W_1=W_2=0$, the change in the Yukawa coupling $W_3$ is
simply $W_3\mapsto f(z)^2W_3$.
The Yukawa coupling also depends on the choice of coordinate $z$,
and in fact is often denoted by $\kappa_{zzz}$.
If we change coordinates from $z$ to $w$, we must change the differentiation
operator from $d/dz$ to $d/dw$. The chain rule then
imples that
\[ \kappa_{www} = \left(\frac{dz}{dw}\right)^3\kappa_{zzz}.\]
Candelas et al.~\cite{pair} use physical arguments to set the gauge in this
calculation, and to find an appropriate (multiple-valued) parameter
$t$ with which to compute.
(The associated
differentiation operator $d/dt$ is single-valued.)
What will be important for us are
the following observations about their results.
The gauge used by Candelas et al.\ determines a family of meromorphic
$n$-forms $\widetilde\omega(z)$ with the property that the period function
\[\int_{\gamma}\widetilde\omega(z)\equiv1\]
for some cycle $\gamma$. Moreover, the parameter $t$ determined by Candelas
et al.\ is a parameter defined in an angular sector near $z=0$
which has two crucial
properties:
\begin{enumerate}
\item
If we analytically continue along
a simple loop around $z=0$ in the counterclockwise direction, $t$
becomes $t+1$.
(It will be convenient to also introduce $q=e^{2\pi it}$, which
remains single-valued near $z=0$.)
\item
There are cycles $\gamma_0$ and $\gamma_1$ such that
$\int_{\gamma_0}\omega(z)$
is single valued near $z=0$, and
\[t=\frac{\int_{\gamma_1}\omega(z)}{\int_{\gamma_0}\omega(z)}\]
in an angular sector near $z=0$.
\end{enumerate}
Each period function $\int_{\gamma}\omega(z)$ is a solution to the
Picard-Fuchs equation of the family. Translating the results of
the previous section into the present context, we obtain the following:
\begin{lemma}
Suppose that $z=0$ is a point of maximally unipotent monodromy
such that $B_j(0)=0$, where $B_j(z)$ are the coefficients in the
logarithmic form of the Picard-Fuchs equation. Then
\begin{enumerate}
\item
There is a period function for $\omega(z)$,
\[f_0(z):=\int_{\gamma_0}\omega(z)\]
which is single-valued near $z=0$. This period function is unique up to
multiplication by a constant.
(This implies that the cycle $\gamma_0$ is also unique up to a
constant multiple.)
In particular, the family of meromorphic $n$-forms
\[\widetilde\omega(z):=\frac{\omega(z)}{\int_{\gamma_0}\omega(z)}\]
will have the property that
\[\int_{\gamma}\widetilde\omega(z)\equiv1\]
for some $\gamma$, and it is the unique such family up to constant multiple.
\item
Fixing a choice of period function $f_0(z)$ as in part (1), there is a
period function
\[f_1(z):=\int_{\gamma_1}\omega(z)\]
such that $\varphi(z):=f_1(z)/f_0(z)$ transforms as
\[\varphi(z)\mapsto\varphi(z)+1\]
upon transport around $z=0$ in the counterclockwise direction.
The ratio $\varphi(z)$ is unique up to the addition of a constant.
\end{enumerate}
\end{lemma}
This, then, is our alternate strategy for computing the Yukawa coupling:
we find solutions of the Picard-Fuchs equation which have the properties
specified in the lemma, and we use those to fix the gauge and specify the
natural parameter, up to two unknown constants of integration.
\section{Picard-Fuchs equations for hypersurfaces}
We now review a method of Griffiths \cite{griffiths} for describing
the cohomology of a hypersurface, which can be used to determine
the Picard-Fuchs equation of a one-parameter family of hypersurfaces.
Calculations of this sort were earlier made by Dwork \cite[Sec. 8]{dwork}.
Griffiths' method was extended to the weighted projective case by
Steenbrink \cite{steen} and Dolgachev \cite{dolg}, who we follow.
We denote a weighted projective $n$-space by $\Bbb P^{(k_0,\dots,k_n)}$,
where $k_0,\dots,k_n$ are the weights of the variables $x_0,\dots,x_n$.
Weighted homogeneous polynomials can be identified with the aid of the
Euler vector field
\[
\theta=\sum k_jx_j\frac{\partial}{\partial x_j}
\]
which has the property that $\theta P=(\deg P)\cdot P$ for any weighted
homogeneous polynomial $P$. Contracting the volume from on $\Bbb C^{n+1}$
with $\theta$ produces the fundamental weighted homogeneous differential
form (of ``weight'' $k:=\sum k_j$)
\[
\Omega:=\sum_{j=0}^n(-1)^jk_jx_j\,
dx_0\wedge\dots\wedge\widehat{dx_j}\wedge\dots\wedge dx_n.
\]
Rational differentials of degree $n$ on $\Bbb P^{(k_0,\dots,k_n)}$
can be described as expressions $P\Omega/Q$, where $P$ and $Q$ are
weighted homogeneous polynomials with $\deg P+k=\deg Q$.
Suppose that $Q$ is a weighted homogeneous polynomial defining a quasismooth
hypersurface ${\cal Q}\subset \Bbb P^{(k_0,\dots,k_n)}$.
(That is, $Q=0$ defines a hypersurface in $\Bbb C^{n+1}$ which is smooth
away from the origin.) The middle cohomology of ${\cal Q}$ is then
described by means of differential forms with poles (of all orders)
along ${\cal Q}$. Each such form $P\Omega/Q^\ell$ is made into a
cohomology class by a ``residue'' construction: for an $(n-1)$-cycle
$\gamma$ on ${\cal Q}$, the tube over $\gamma$ (an $S^1$-bundle inside
the (complex) normal bundle of ${\cal Q}$) is an $n$-cycle $\Gamma$
on $\Bbb P^{(k_0,\dots,k_n)}$ disjoint from ${\cal Q}$. We can then
define the residue of $P\Omega/Q^\ell$ by
\[
\int_{\gamma}\operatorname{Res}_{\cal Q}\left(\frac{P\Omega}{Q^\ell}\right)=
\frac1{2\pi i}\int_{\Gamma}\frac{P\Omega}{Q^\ell}.
\]
Since altering $P\Omega/Q^\ell$ by an exact differential does not change
the value
of these integrals, we see that the cohomology of ${\cal Q}$ is
represented by
equivalence classes of rational differential forms $P\Omega/Q^\ell$
modulo exact forms.
Here is Griffiths' ``reduction of pole order'' calculation which shows
how to reduce modulo exact forms in practice. Let $Q$ and $A_j$
be weighted homogeneous polynomials, with $\deg Q=d$,
$\deg A_j =\ell d +k_j-k$. Define
\[
\varphi=\frac1{Q^\ell}\sum_{i<j}(k_ix_iA_j-k_jx_jA_i)
dx_0\wedge\dots\wedge\widehat{dx_i}\wedge
\dots\wedge\widehat{dx_j}\wedge\dots\wedge dx_n
\]
and then calculate
\begin{equation} \label{griffiths}
d\varphi=\frac
{\left(\ell\sum A_j\frac{\partial Q}{\partial x_j} -Q\sum\frac{\partial A_j}
{\partial x_j}\right)\Omega}{Q^{\ell+1}}
=\frac{\ell\sum A_j\frac{\partial Q}{\partial x_j}\Omega}{Q^{\ell+1}}
-\frac{\sum\frac{\partial A_j}
{\partial x_j}\Omega}{Q^\ell}.
\end{equation}
Thus, any form whose numerator lies
in the Jacobian ideal
$
J=(\partial Q/\partial x_0,
\dots,\partial Q/\partial X_n)
$
is equivalent (modulo exact forms)
to a form with smaller pole order.
This idea can be used to calculate Picard-Fuchs equations as follows.
The cycles $\Gamma$ do not change (in homology) when $z$ varies locally.
So we can differentiate under the integral sign
\[
\frac{d^k}{dz^k}\int_{\gamma}\operatorname{Res}_{\cal Q}\left(\frac{P\Omega}{Q^\ell}\right)=
\frac1{2\pi i}\int_{\Gamma}\frac{d^k}{dz^k}\left(\frac{P\Omega}{Q^\ell}\right)
\]
when $Q$ depends on a parameter $z$. (Note that $\Omega$ is independent
of $z$.) The Picard-Fuchs operator \eqref{PFop} will have the property
that
\[
\left( \frac{d^s}{dz^s}+\sum_{j=0}^{s-1}C_j(z)\frac{d^j}{dz^j} \right)
\left(\frac {P\Omega}Q\right)=d\varphi
\]
is an exact form. To find it, take successive $z$-derivatives of the
integrand
$P\Omega/Q$ and use the reduction of order of pole formula \cite{griffiths}
to determine a linear relation among those derivatives, modulo exact forms.
\section{Examples: Picard-Fuchs equations}
We will calculate the Picard-Fuchs equations for
certain one-parameter families of Calabi-Yau threefolds.
Our choice of families is motivated by
the mirror construction of Greene and Plesser \cite{greene-plesser}.
We choose weights $k_0,\dots,k_4$ with $k_0\ge k_1\ge\dots\ge k_4$
for a weighted projective 4-space
such that $d_j:=k/k_j$ is
an integer, where $k:=\sum k_j$.
We also assume that $\gcd\{k_j\ | \ j\ne j_0\}=1$ for every $j_0$.
These assumptions then imply that $k=\operatorname{lcm}\{d_j\}$.
Consider the pencil of hypersurfaces ${\cal Q}_\psi \subset
\Bbb P^{(k_0,\dots,k_4)}$ defined by $Q(x,\psi)=0$, where
\[
Q(x,\psi) := \sum_{j=0}^4 x_j^{d_j} - k\psi\prod_{j=0}^4 x_j .
\]
This pencil has a natural group of diagonal automorphisms preserving the
holomorphic 3-form. To define it,
let $\mugroup{m}$ denote the multiplicative group of $m^{\text{th}}$ roots
of unity (considered as a subgroup of $\Bbb C^\times$), and let
\[
G=(\mugroup{d_0}\times\dots\times\mugroup{d_4})/\mugroup{k},
\]
where we embed $\mugroup{k}$ in $\mugroup{d_0}\times\dots\times\mugroup{d_4}$
by
\[
\alpha\mapsto(\alpha^{k_0},\dots,\alpha^{k_4}).
\]
Note that since $\sum k_j=k$, the formula
\[
f(\alpha_0,\dots,\alpha_4)=(\prod \alpha_j)^{-1}
\]
determines a well-defined homomorphism $f:G\to\Bbb C^\times$.
Let $G_0=\ker(f)$.
We can regard $Q(x,\psi)=0$ as defining a hypersurface
${\cal Q}\subset\Bbb P^{(k_0,\dots,k_4)}\times\Bbb C$.
The group $G$ acts on $\Bbb P^{(k_0,\dots,k_4)}\times\Bbb C$ by
\[
(x_0,\dots,x_4;\psi)\mapsto(\alpha_0x_0,\dots,\alpha_4x_4;f(\alpha)\psi)
\]
for $\alpha=(\alpha_0,\dots,\alpha_4)\in G$. The polynomial $Q(x,\psi)$
is invariant under this action. Thus, the
action preserves ${\cal Q}$,
and maps ${\cal Q}_\psi$ isomorphically to ${\cal Q}_{f(\alpha)\psi}$.
It follows that the group $G_0$ acts on ${\cal Q}_\psi$ by automorphisms,
and that the induced action of
$G/G_0\cong\mugroup{k}$ establishes isomorphisms between
${\cal Q}_\psi/G_0$ and ${\cal Q}_{\lambda\psi}/G_0$ for
$\lambda\in\mugroup{k}$.
The quotient
space ${\cal Q}_\psi/G$ has only canonical singularities.
By a theorem of Markushevich \cite[Prop.~4]{markushevich} and
Roan \cite[Prop.~2]{roan1}, these singularities can be
resolved to give a Calabi-Yau manifold ${\cal W}_\psi$.
There are choices to be made in this resolution
process; we do not specify a choice.
By another theorem of Roan \cite[Lemma 4]{roan2},
any two resolutions differ by a sequence of flops.
Note that the differential form $\Omega$ from the previous section
transforms as $\Omega\mapsto(\prod \alpha_j)\Omega$ under the action
of $\alpha\in G$. Thus, the rational differential
\[
\omega_1=\frac{\psi\Omega}{Q(x,\psi)}
\]
is invariant under the action of $G$; we define
$\omega(\psi)=\operatorname{Res}_{{\cal Q}_\psi}(\omega_1)$.
Since the holomorphic 3-forms $\omega(\psi)$
on ${\cal Q}_\psi$ are invariant on $G_0$,
they induce holomorphic 3-forms on ${\cal W}_\psi$. Moreover, the
homology group $H_3({\cal W}_\psi)$ contains the $G_0$-invariant part
$H_3({\cal Q}_\psi)^{G_0}$ of the homology of ${\cal Q}_\psi$. If we know
that the dimensions of these spaces agree, then they will coincide
(at least for homology with coefficients in a field). In this case,
the periods of ${\cal W}_\psi$ can actually be computed as periods of
the holomorphic form $\omega(\psi)$
on ${\cal Q}_\psi$, over $G_0$-invariant cycles.
Thanks to the isomorphisms between ${\cal Q}_\psi$ and
${\cal Q}_{\lambda\psi}$
for $\lambda\in\mugroup{k}$ and the invariance of the rational differential
$\omega_1$ under $G$, these periods will be invariant under
$\psi\mapsto\lambda\psi$. In particular, they will be functions of
$z=\psi^{-k}$ alone.
It is likely that the resolutions ${\cal W}_\psi$
of ${\cal Q}_\psi/G_0$ could be chosen so
that the action of $G/G_0$ would lift to isomorphisms between
${\cal W}_\psi$ and ${\cal W}_{\lambda\psi}$. (We verified this in the case
of quintic hypersurfaces in \cite{guide}.) In this case, there would
be an actual family of Calabi-Yau threefolds for which $z$ served as
a parameter. It may be that such resolutions could be constructed by
finding an appropriate partial resolution of ${\cal Q}/G$. However,
we do not need the existence of
this family to describe the computation of the Yukawa
coupling.
\begin{table}[t]
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$k$ & $(k_0,\dots,k_4)$ & $Q(x,\psi)$ \\ \hline
5 & $(1,1,1,1,1)$ & $x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-5\psi x_0x_1x_2x_3x_4$ \\
6 & $(2,1,1,1,1)$ & $x_0^3+x_1^6+x_2^6+x_3^6+x_4^6-6\psi x_0x_1x_2x_3x_4$ \\
8 & $(4,1,1,1,1)$ & $x_0^2+x_1^8+x_2^8+x_3^8+x_4^8-8\psi x_0x_1x_2x_3x_4$ \\
10 & $(5,2,1,1,1)$ & $x_0^2+x_1^5+x_2^{10}+x_3^{10}+x_4^{10}
-10\psi x_0x_1x_2x_3x_4$ \\
\hline \end{tabular} \end{center}
\caption{The hypersurfaces.}
\label{tab1}
\end{table}
We will carry out the computation in four specific examples. These
come from the lists of
Candelas, Lynker and Schimmrigk
\cite{cls}; they found that there are
exactly four types of hypersurface in weighted projective four-space
which are Calabi-Yau threefolds with Picard number one.
The weights of the space are given in the second column of table~\ref{tab1}.
For each of
those cases, Greene and Plesser's mirror construction \cite{greene-plesser}
yields the family ${\cal W}_\psi$ which we have described above.
And Roan's formula \cite{roanpf} for the Betti numbers verifies that $b_3$
is indeed 4 (with $h^{2,1}=1$).
The remaining columns in table~\ref{tab1} show the value of $k$, and
give the equation $Q(x,\psi)$ explicitly.
We describe the $G_0$-invariant cohomology by means of the rational
differential forms
\[
\omega_\ell:=
\frac{
(-1)^{\ell-1}(\ell-1)!\,\psi^\ell(\prod x_i^{\ell-1})\Omega}{Q(x,\psi)^\ell}.
\]
These are chosen because of the evident $G$-invariance in the numerator;
the coefficients were adjusted so that the formula
\begin{equation} \label{derivs}
-\frac1k\psi\frac d{d\psi}\omega_\ell=-\frac{\ell}k\omega_\ell
+\omega_{\ell+1}
\end{equation}
would not be overly burdened with constants. We compute with the
differential
operator $-\frac1k\psi\frac d{d\psi}$ because it coincides with
$z\frac d{dz}$.
A basis for the $G_0$-invariant cohomology is then given by the residues of
$\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$. To compute the
Picard-Fuchs equation, we must find an expression for $\omega_5$
as a linear combination of $\omega_1,\dots,\omega_4$ modulo exact forms.
That expression, combined with \eqref{derivs}, will then yield the
desired differential equation.
We carried out this calculation using the Gr\"obner basis
algorithm \cite{buchberger},
modifying an implementation written in {\sc maple} by Yunliang Yu
(cf.~\cite{yu}). We first
calculated a Gr\"obner basis for the Jacobian ideal
$
J=(\partial Q/\partial x_0,\dots,\partial Q/\partial x_4)
$,
working in the ring $\Bbb C(\psi)[x_0,\dots,x_4]$ of polynomials whose
coefficients are rational functions of $\psi$. The reduction of pole
order was then achieved step by step as follows: given a form $\eta_\ell$,
the residue of a global form with a pole of order $\ell$, we
used the Gr\"obner basis to reduce the numerators of
both $\eta_\ell$ and $\omega_\ell$ to standard form. We could thus
determine a coefficient $\varepsilon_\ell\in\Bbb C(\psi)$
such that the numerator of
$\eta_\ell-\varepsilon_\ell\omega_\ell$ lies in $J$.
Another application of Gr\"obner basis reduction produced explicit
coefficients
\[
\eta_\ell-\varepsilon_\ell\omega_\ell=\sum A_{\ell j}\frac
{\partial Q}{\partial x_j}.
\]
Then the
Griffiths formula
\eqref{griffiths} determines forms $\varphi_\ell$
and $\eta_{\ell-1}$ such that
\[
\eta_\ell-\varepsilon_\ell\omega_\ell=d\varphi_\ell+\eta_{\ell-1},
\]
and $\eta_{\ell-1}$ has a pole of order $\ell-1$.
Beginning with $\eta_5=\omega_5$ and applying this procedure several times,
one finds
\[
\omega_5=\varepsilon_1\omega_1+\dots+\varepsilon_4\omega_4 +d\varphi.
\]
The results of this computation for our four examples
are summarized in table 2. The
coefficients $\varepsilon_\ell$ are in fact functions of $z=\psi^{-k}$
(as expected from our earlier discussion), and have been displayed as
such.
\begin{table}[t]
\renewcommand{\arraystretch}{1}
\begin{center}
\begin{tabular}{|c|cccc|} \hline
\rule{0pt}{14pt} $k$ & $\varepsilon_1$ & $\varepsilon_2$ & $\varepsilon_3$ &
$\varepsilon_4$ \\[4pt]
\hline
& & & & \\
5 & $\displaystyle\frac{1}{625(z-1)}$ &
$\displaystyle\frac{-3}{25(z-1)}$ & $\displaystyle\frac{1}{(z-1)}$ &
$\displaystyle\frac{-2}{(z-1)}$ \\
& & & & \\
6 & $\displaystyle\frac{1}{324(z-4)}$ & $\displaystyle\frac{-5}{18(z-4)}$ &
$\displaystyle\frac{-(z-50)}{18(z-4)}$
& $\displaystyle\frac{-(z+20)}{3(z-4)}$ \\
& & & & \\
8 & $\displaystyle\frac{1}{16(z-256)}$
& $\displaystyle\frac{-15(z+256)}{512(z-256)}$ &
$\displaystyle\frac{-5(3z-1280)}{64(z-256)}$
& $\displaystyle\frac{-(3z+1280)}{4(z-256)}$ \\
& & & & \\
10 & $\displaystyle\frac{5}{4(z-12500)}$
& $\displaystyle\frac{-(7z+37500)}{200(z-12500)}$ &
$\displaystyle\frac{-(7z-62500)}{20(z-12500)}$
& $\displaystyle\frac{-(z+12500)}{(z-12500)}$ \\
& & & & \\
\hline \end{tabular} \end{center}
\caption{The results of the Gr\"obner basis calculation.}
\label{tab2}
\end{table}
The differential equation for $[\omega_1,\dots,\omega_4]$ determined
by this procedure has the form
\[
z\frac d{dz}
\begin{bmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \\ \omega_4 \end{bmatrix}
=
\begin{bmatrix}
-\frac1k & 1 & 0 & 0 \\
0 & -\frac2k & 1 & 0 \\
0 & 0 & -\frac3k & 1 \\
\varepsilon_1 & \varepsilon_2 & \varepsilon_3 & \varepsilon_4-\frac4k
\end{bmatrix}
\begin{bmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \\ \omega_4 \end{bmatrix}.
\]
To calculate the Picard-Fuchs equation, we must change basis via
\[
{\renewcommand{\arraystretch}{1.5}
\begin{bmatrix} \omega_1 \\ z\frac d{dz}\omega_1 \\ (z\frac d{dz})^2\omega_1 \\
(z\frac d{dz})^3\omega_1 \end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-\frac1k & 1 & 0 & 0 \\
\frac1{k^2} & -\frac3k & 1 & 0 \\
-\frac1{k^3} & \frac7{k^2} & -\frac6k & 1
\end{bmatrix}
\begin{bmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \\ \omega_4 \end{bmatrix}.
}
\]
This determines an equation in the form \eqref{Az}, with
\begin{equation} \label{Beqn}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{rrr}
B_0(z) & = & -\varepsilon_1(z) - \frac1k\varepsilon_2(z) -
\frac2{k^2}\varepsilon_3(z)
-\frac6{k^3}\varepsilon_4(z)+\frac{24}{k^4} \\
B_1(z) & = & -\varepsilon_2(z)-\frac3k\varepsilon_3(z)-
\frac{11}{k^2}\varepsilon_4(z)
+\frac{50}{k^3} \\
B_2(z) & = & -\varepsilon_3(z)-\frac6k\varepsilon_4(z)+\frac{35}{k^2} \\
B_3(z) & = & -\varepsilon_4(z)+\frac{10}k.
\end{array}
}
\end{equation}
As can be directly verified in each of our cases, $B_j(0)=0$.
It
follows that the monodromy at $z=0$ is maximally unipotent.
(In the case of quintics ($k=5$), this had been shown in \cite{pair};
cf.~\cite{guide}.)
\section{Examples: Mirror maps}
We next compute the mirror maps for our four examples, based on their
Picard-Fuchs equations.
Expanding eqs.~\eqref{PFop} and \eqref{logform}, one finds that the
coefficient $C_3(z)$ coincides with $(6+B_3(z))/z$. Moreover, in
our four examples, a straightforward computation based on eq.~\eqref{Beqn}
and table~\ref{tab2} shows that
$B_3(z)=2z/(z-\lambda)$,
where $\lambda=1$, $4$, $256$, $12500$ when $k=5$, $6$, $8$, $10$,
respectively. Thus,
\[
C_3(z)=\frac{6+B_3(z)}{z}=\frac6z+\frac2{z-\lambda}.
\]
The Yukawa coupling $\kappa_{zzz}$ in the gauge $\omega(z)$ is
therefore given by
a function $W_3(z)$ which satisfies the differential equation
\[
\frac{dW_3(z)}{dz}=\left(\frac{-3}{z}+\frac{-1}{z-\lambda}\right)W_3(z).
\]
Thus, in the gauge $\omega(z)$ we have
\[
\kappa_{zzz}=\frac{c_1}{(2\pi i)^3z^3(z-\lambda)}.
\]
Here $c_1/(2\pi i)^3$ is the first ``constant of integration'':
we have introduced a factor of $(2\pi i)^3$ in order to simplify a
later formula.
In order to determine the natural gauge, we must find a solution
$f_0(z)$ of the
Picard-Fuchs equation which is regular near $z=0$. Using the corresponding
vector $w_0(z)$ of which $f_0(z)$ is the first component,
we want a solution to the vector equation
\begin{equation} \label{w0eqn}
z\frac d{dz}w_0(z)=A(z)w_0(z)
\end{equation}
which is regular near $z=0$. ($A(z)$ is given by
eqs.~\eqref{Az}, \eqref{Beqn},
and table~\ref{tab2}.)
This can be found
using power-series techniques, and there is a solution with $f_0(0)\ne0$
in each of our four cases. We normalize so that $f_0(0)=1$;
alternatively, we could have absorbed the leading term of $f_0(z)$ into the
constant of integration $c_1$.
As a result, the gauge-fixed value of $\kappa_{zzz}$ takes the form
\[
\kappa_{zzz}=\frac{c_1}{(2\pi i)^3z^3(z-\lambda)(f_0(z))^2},
\]
where the constant $c_1$ has yet to be determined.
We now search for the good parameter $t$. We should locate a second
solution $f_1(z)$, or its corresponding vector $w_1(z)$, which is
multiple-valued and has the correct monodromy properties.
The monodromy will be such that if we introduce
\[v(z):=2\pi iw_1(z)-(\log z)w_0(z)\]
and its first
component
\[g(z):=2\pi if_1(z)-(\log z)f_0(z),\]
then $v(z)$ will be single-valued and regular
near $z=0$. It is easy to calculate that the matrix equation
satisfied by $v(z)$ is
\begin{equation} \label{veqn}
z\frac d{dz}v(z)=A(z)v(z)-w_0(z).
\end{equation}
Solutions to this equation
can be found by power-series techniques. We normalize the solution
so that $g(0)=0$. The parameter $t$ is
then given by
\[
t=\frac1{2\pi i}\log c_2+\frac1{2\pi i}\log z+\frac{g(z)}{f_0(z)}
\]
($\frac1{2\pi i}\log c_2$ is the second ``constant of integration'')
and the associated parameter $q$ is
\[
q=e^{2\pi it}=c_2ze^{g/f_0}.
\]
Let us define
\[ \delta(z)=1+z\frac d{dz}\left(\frac{g(z)}{f_0(z)}\right), \]
so that
\[
\frac{dq}{dz}=c_2\delta(z)e^{g/f_0}.
\]
Then by the chain rule,
\[
\frac{dz}{dt}=\frac{dq/dt}{dq/dz}=\frac{2\pi iz}{\delta(z)}.
\]
It follows that the gauge-fixed value of $\kappa_{ttt}$ is
\[
\kappa_{ttt}=\left(\frac{dz}{dt}\right)^3\kappa_{zzz}
=\frac{c_1}{(\delta(z))^3(z-\lambda)(f_0(z))^2}.
\]
Finally we express this normalized $\kappa_{ttt}$ as a power series
in $q$. The constants $c_1$ and $c_2$ have
yet to be determined; however, we
can define
\begin{equation} \label{h0z}
h_0(z)=\frac{1}{(\delta(z))^3(z-\lambda)(f_0(z))^2}
\end{equation}
\begin{equation} \label{hjz}
h_j(z)=\frac1{\delta(z)e^{g/f_0}}\cdot\frac{dh_{j-1}(z)}{dz}
\end{equation}
and find that
\[
h_j(z)=\frac{(c_2)^j}{c_1}\left(\frac d{dq}\right)^j\kappa_{ttt},
\]
so that
\[
\kappa_{ttt}=\sum_{j=0}^{\infty} \frac{c_1}{(c_2)^j}\,
\frac{h_j(0)}{j!}\, q^j.
\]
\begin{proposition}
The numbers $h_j(0)$ are rational numbers.
\end{proposition}
\begin{pf}
The coefficient matrix $A(z)$ in the vector equation \eqref{w0eqn}
has entries in $\Bbb Q(z)$; if written out in power series, all
the power series coefficients will be rational numbers. Finding
a power series solution to \eqref{w0eqn} then involves solving linear
equations with rational coefficients at each step: the solutions
will be rational. Thus, $w_0(z)$ and $f_0(z)$ are power series in
$z$ with rational coefficients.
Similarly, $v(z)$ and $g(z)$ are power series with rational coefficients,
since they come from equation \eqref{veqn}. Furthermore, since
exponentiating a power series with rational coefficients (whose constant
term is zero) again gives a power series with rational coefficients,
$e^{g/f_0}$ and $\delta(z)$ are power series in $z$ with rational
coefficients.
But then by \eqref{h0z},
$h_0(z)$ is clearly a power series in $z$ with rational
coefficients; similarly for $h_j(z)$ by \eqref{hjz}. It follows that
each $h_j(0)$ is a rational number.
\end{pf}
\section{Choosing the constants and predicting the
numbers of rational curves}
Calabi-Yau threefolds with $h^{2,1}=1$ are conjectured to be the ``mirrors''
of other Calabi-Yau threefolds with $h^{1,1}=1$. In the four examples
we have considered, this mirror property can be realized by a construction
of Greene and Plesser \cite{greene-plesser}. The threefolds ${\cal W}_\psi$
are mirrors of threefolds ${\cal M}\subset\Bbb P^{(k_0,\dots,d_4)}$,
which are hypersurfaces of weighted degree $k=\sum k_j$. The Picard
group of ${\cal M}$ is cyclic, generated by some ample divisor $H$.
Mirror symmetry predicts that the $q$-expansion of the gauge-fixed
Yukawa coupling
\[
\kappa_{ttt}=a_0+a_1q+a_2q^2+\cdots
\]
will have integers as coefficients. Moreover, by a formula conjectured
in \cite{pair} and established in \cite{psa-drm}, if this $q$-expansion
is written in the form
\begin{equation} \label{formla2}
\kappa_{ttt} = n_0 +
\sum_{j=1}^\infty \frac{n_jj^3q^j}{1-q^j}
= n_0 + n_1q + (2^3n_2 + n_1)q^2 + \cdots .
\end{equation}
then the coefficients $n_j$ are also integers. The first term $n_0$
is predicted to coincide with $H^3$ (the absolute degree of ${\cal M}$),
and $n_j$ is predicted to be the number of rational curves $C$ on
${\cal M}$ with $C\cdot H=j$, assuming that all rational curves on ${\cal M}$
are disjoint and have normal bundle ${\cal O}(-1)\oplus{\cal O}(-1)$.
These two predictions can be used to choose the constants of integration
in our examples.
First, the absolute degree $d$ is the lowest order term which appears
in the polynomial $Q(x,\psi)$; to ensure that $n_0=d$ we must take
$c_1=-\lambda d$. Second, the formula~\eqref{formla2} puts very
strong divisibility constraints on the coefficients $a_j$, and it
seems likely that there will be a unique choice of $c_2$ which
satisfies all of these constraints.
\begin{table}[t]
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{|c|rrrrr|} \hline
$k$ & $n_0$ & $n_1$ & $n_2$ & $n_3$ & $n_4$ \\ \hline
5 & 5 & 2875 & 609250 & 317206375 & 242467530000 \\
6 & 3 & 7884 & 6028452 & 11900417220 & 34600752005688 \\
8 & 2 & 29504 & 128834912 & 1423720546880 & 23193056024793312 \\
10 & 2 & 462400 & 24431571200 & 3401788732948800 & 700309317702649312000 \\
\hline \end{tabular} \end{center}
\caption{The predicted numbers of curves.}
\label{tab3}
\end{table}
We have calculated the first 20 coefficients (using {\sc mathematica})
in each of our four examples. There does indeed appear to be a unique
choice for $c_2$ which produces integers for $n_1,\dots n_{20}$:
that choice turns out to be $c_2=k^{-k}$ in each of our examples.
Making this choice leads to the values for $n_j$
displayed in table~\ref{tab3}.
Table~\ref{tab3} therefore contains predictions about numbers of rational
curves on the weighted projective hypersurfaces. For a general
hypersurface in ${\cal M}\subset\Bbb P^{(k_0,\dots,d_4)}$ of
degree $k=\sum k_j$,
the prediction is that there should be $n_j$ rational curves $C$ with
$C\cdot H=j$, where $H$ generates $\operatorname{Pic}({\cal M})$.
The first line of
the table reproduces the predictions made by Candelas et al.\ about
quintic threefolds. Several of these have been verified: the number
of lines was known classically, the number of conics was computed by
Katz \cite{katz}, and the number of twisted cubics $n_3$ has recently
been computed by Ellingsrud and Str\o mme \cite{ell-str}---all of these
results agree with the predictions.
Of the remaining predictions in the table, we have only checked one.
Each hypersurface from the third family (the case $k=8$) can be regarded as
a double cover of $\Bbb P^3$ branched on a surface of degree 8.
The entry 29504 in the third line of the table can be interpreted as
follows: for a general surface of degree 8 in $\Bbb P^3$, there should
be 14752 lines which are 4-times tangent to the surface. (These lines
will then split into pairs of rational curves on the double cover.)
After we had obtained this number,
Steve Kleiman was kind enough to locate a
$19^{\text{th}}$-century formula of
Schubert \cite[Formula 21, p. 236]{schubert}, which
states that
the number of lines in $\Bbb P^3$ 4-times tangent to a general
surface of degree
$n$ is
\[ \frac1{12}n(n-4)(n-5)(n-6)(n-7)(n^3+6n^2+7n-30) .
\]
Substituting $n=8$, we find the predicted number 14752.
\section*{Acknowledgements}
Several of the ideas explained in this paper arose in conversations with
Sheldon Katz---it is a pleasure to acknowledge his contribution. I also
benefited greatly from the chance to interact with physicists which
was provided by the Mirror Symmetry Workshop. I wish to thank the M.S.R.I.
as well as the organizers and participants in the Workshop.
This work was partially supported by NSF grant DMS-9103827.
\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
|
2010-04-26T07:27:25 | 9202 | alg-geom/9202002 | en | https://arxiv.org/abs/alg-geom/9202002 | [
"alg-geom",
"math.AG"
] | alg-geom/9202002 | David R. Morrison | Sheldon Katz and David R. Morrison | Gorenstein Threefold Singularities with Small Resolutions via Invariant
Theory for Weyl Groups | 73 pages, AMSLaTeX v1.1 | J. Alg. Geom. 1 (1992) 449--530 | null | null | null | We classify simple flops on smooth threefolds, or equivalently, Gorenstein
threefold singularities with irreducible small resolution. There are only six
families of such singularities, distinguished by Koll{\'a}r's {\em length}
invariant. The method is to apply invariant theory to Pinkham's construction of
small resolutions. As a by-product, generators of the ring of invariants are
given for the standard action of the Weyl group of each of the irreducible root
systems.
| [
{
"version": "v1",
"created": "Wed, 5 Feb 1992 17:29:18 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Katz",
"Sheldon",
""
],
[
"Morrison",
"David R.",
""
]
] | alg-geom | \section*{}
A fundamental new type of birational modification which first occurs in
dimension three is the {\em simple flip}. This is a
birational map $Y \dasharrow Y^+$
which induces an isomorphism
$(Y - C) \cong (Y^+ - C^+),$
where $C$ and $C^+$ are smooth
rational curves such that $K_Y \cdot C < 0$ and $K_{Y^+} \cdot C^+ > 0$.
($Y$ and $Y^+$ should be allowed to have ``terminal" singularities.)
Mori's celebrated theorem \cite{[Mr]}
shows that these flips exist when numerically
expected.
A closely related type of modification is the {\em simple flop}.
This has a similar definition, except that $Y$ and $Y^+$
should be Gorenstein, with $K_Y \cdot C = K_{Y^+} \cdot C^+ = 0$.
(This is more than an analogy:
every flip has a branched double cover which is a flop,
and this construction was used in Mori's proof.)
For both flips and flops, the curves $C$ and $C^+$ can be contracted
to points (in $Y$ and $Y^+$, respectively), yielding the same normal
variety $X$. The birational map $Y \dasharrow Y^+$ can thus be
described in terms of the two contraction morphisms
$\pi\colon Y \to X$ and $\pi^+ \colon Y^+ \to X$.
In this paper, we study the case of simple flops
with $Y$ smooth, so that $\pi\colon Y \to X$ is an irreducible small
resolution of a Gorenstein threefold singularity
$P \in X$. (It is called ``small" because the exceptional set is
a curve rather than a divisor, and ``irreducible" because the curve has
only one component.)
In fact, the study of flops with $Y$ smooth
can be reduced to a study of Gorenstein
threefold singularities with small resolutions,
thanks to a theorem
of Reid \cite{[R]} (cf.\ also Koll\'ar \cite{[Kol]}) which produces a
second small resolution $\pi^+\colon Y^+ \to X$ out of the original
$\pi\colon Y \to X$.
Pinkham \cite{[P]} showed that in this situation, $X$ is Gorenstein if it is
merely assumed to be Cohen-Macaulay.
Early examples of small resolutions were constructed in an {\em ad hoc\/}
manner.
The first class of examples
is given by the singularities $x^2+y^2 + z^2 + t^{2k}=0$: a particularly
nice description of the associated simple flops
(with pictures!)\ can be found in a paper of
Reid \cite[\S 5]{[R]}.
These flops are exactly those for which the normal bundle of $C$ in $Y$
is $\O(-1) \oplus \O(-1)$ or $\O \oplus \O(-2)$.
A second example of simple flops, in which the normal bundle is
$\O(1) \oplus \O(-3)$, was found by
Laufer \cite{[L]}; variants of this example were also investigated by
the second author, Pinkham \cite{[P]}, and Reid \cite{[R]}.
Some other flops were studied in previous work of the authors
(\cite{[M]} and \cite{[K]}).
In general,
if $Y\to X$ is a small resolution of the isolated Gorenstein threefold
singularity $P\in X$, then by a lemma of Reid \cite{[R]}, the general
hyperplane section of $X$ through $P$ has a rational double point
at $P$, and the proper transform of that surface on $Y$ gives a
``partial resolution" of the rational double point.
Pinkham \cite{[P]} used this observation to give a construction which
includes all possible Gorenstein threefold
singularities with small resolutions. In the irreducible case, each such
singularity can be described by a map from the disk to a space
$\operatorname{PRes} (S,v)$ which parametrizes deformations which partially resolve.
The natural map $\operatorname{PRes} (S,v) \to \operatorname{Def} (S)$ to the deformation
space then gives a map from the disk to $\operatorname{Def} (S)$ which describes
the space $X$ as the pullback of a (semi-)universal family.
At first glance, Pinkham's construction appears to give a countable number
of families of Gorenstein threefold singularities with irreducible
small resolutions.
The discrete data which appear in the construction are the type
of the rational double point, together with a choice of component in the
exceptional divisor of the minimal resolution of that point.
However, the construction uses a {\em particular\/} hyperplane section through
$P$, and there is no guarantee that this hyperplane section is ``general".
(Examples for which it is {\em not\/} general were known to Pinkham at the time
he gave the construction.)
We have discovered that there are in fact only six families of
Gorenstein threefold singularities with irreducible
small resolutions. They can be
distinguished by a very simple invariant (the ``length") which was
introduced by Koll\'ar \cite[pp.~95, 96]{[CKM]} a few years ago. The precise
statement of our main theorem can be found in section 1.
Our methods do much more than simply characterize the six families:
our techniques can be used to calculate the map
$\operatorname{PRes} (S,v) \to \operatorname{Def} (S)$
quite explicitly in each of the six cases.
Composing this map with a general map from the disk to
$\operatorname{PRes} (S,v)$ describes the most general Gorenstein threefold with
irreducible small resolution of each type. For example, when the length
is 1, the
result is precisely the class of examples $x^2+y^2+z^2+t^{2k}=0$ mentioned
above.
\bigskip
In order to prove the main theorem, we need to solve a
fundamental problem: compute the singularity type of the
generic hyperplane section for irreducible small resolutions which
have been produced by Pinkham's construction.
The main tools we use to solve this problem are derived from the theory of
simultaneous resolution of rational double points, as developed by
Brieskorn \cite{[Bri0]},
\cite{[Bri]},
\cite{[Bri-Nice]} and Tyurina \cite{[T]}. In this theory, the
deformation space $\operatorname{Def} (S)$ is identified with a quotient
space $V/\frak W$, where $V$ is the complex root space and $\frak W$ is the
Weyl group of a certain root system $R$.
(The Dynkin diagram of this root system coincides with the dual
graph of the minimal resolution of the rational double point.)
A simultaneous resolution
of the semi-universal deformation over $V/\frak W$ is possible after making the
base change by $V \to V/\frak W$. Pinkham's construction of the space
$\operatorname{PRes} (S,v)$ included an identification of it with the quotient
space $V/\frak W_0$, where $\frak W_0$ is the Weyl group of a certain subsystem
of $R$. Thus, an understanding of the map $\operatorname{PRes} (S,v) \to \operatorname{Def} (S)$
can be obtained if one has an adequate understanding of the invariant
theory of the Weyl groups $\frak W$ and $\frak W_0$.
Our first theorem is a slight extension of the fundamental theorem on
simultaneous resolution. We need some additional information about
the loci in which certain curves in the exceptional set deform,
but most importantly, we need a version of the proof which provides
the simultaneous resolution in an explicitly computable form. With
a small amount of modification, the approach of Tyurina \cite{[T]} as amplified
by Pinkham \cite{[P-RDP]} provides us with what we need.
The theory of simultaneous resolution implies that the coefficients
of the defining polynomial of a semi-universal deformation serve as generators
of the algebra of $\frak W$-invariant polynomials on $V$. (By a theorem
of Coxeter \cite{[Cx]} and Chevalley \cite{[Chv]},
this is a free polynomial algebra.)
Our next theorems show that these generators are in an appropriate sense
unique up to ${\Bbb C}^*$-action,
and can be computed from the invariant theory alone. This is
an important step, since it allows us to recover the defining polynomials
of various semi-universal deformations directly from the invariant theory.
It has the effect of reducing our fundamental problem to a problem in
the invariant theory of Weyl groups.
The problem in invariant theory can be solved by hand when the Weyl
groups in question are those associated to the root systems $A_{n-1}$
or $D_n$, but in the cases of $E_6$, $E_7$, and $E_8$,
the invariant theory is not so well
understood. We computed the invariants in these cases with the
aid of the symbolic computing languages {\sc maple} and {\sc reduce},
running on a Macintosh II and on a Sun 3/60
workstation. We have also developed a number of tools for manipulating
these invariants, which enabled us to extract enough information
relating $\frak W$-invariants to $\frak W_0$-invariants to solve our fundamental
problem.
We describe our algorithm in sufficient detail that it could be
implemented in any symbolic manipulation language. We have, however,
been somewhat selective in displaying the calculated results in
the paper.
The implementations in {\sc maple} and {\sc reduce} were carried out
independently by the second and first authors, respectively,
and run on different machines.
We are happy to report that the two sets of
calculated results agree in every particular.
One of the by-products of the work described here is a
new explicit set of generators for the invariants of the Weyl groups
of $E_6$, $E_7$, and $E_8$. Following ideas of Tyurina,
these generators are obtained by computing
the anti-pluricanonical mappings for an appropriate family of del~Pezzo
surfaces, and putting the defining polynomials of the images into a
semi-universal form. This calculation was attempted in 1918 by
C. C. Bramble \cite{[Bra]} in the case of $E_7$, but our machine-aided
calculation
reveals that Bramble made a few errors.
A corrected version of Bramble's calculation appears in Appendix 2;
the corresponding calculation for $E_6$ is in Appendix 1.
As the final draft of this paper was being prepared, we learned of some
recent work of Shioda \cite{[Shioda]} who has also done an explicit
calculation of simultaneous resolutions using a quite
different approach.
J\'anos Koll\'ar has communicated to us another possible
approach to proving our
main theorem \cite{[Kpvt]}, using techniques of Clemens and Jim\'{e}nez
\cite{[J]} and an
analysis of higher order neighborhoods similar to that used by Mori for
extremal rays in \cite{[Mr]}.
The paper is organized as follows. Section 1 contains the statement
of the main theorem, and a discussion of all of the ingredients needed
to state it, including Pinkham's
construction and Koll\'ar's ``length" invariant.
In section 2, we set up the notation for root systems, and
we introduce the notion of {\em distinguished polynomials}, which provide an
efficient means for comparing invariants. In section 3, we establish
notation for the rational double points, and state the
simultaneous resolution theorem in the form we will need it. We prove
that theorem in sections 4 and 5, giving explicit constructions of
simultaneous resolutions.
In section 6, we show that the simultaneous resolutions we have
constructed are essentially unique, and we use this uniqueness to
analyze the defining polynomials of simultaneous partial resolutions.
In section 7, we use the distinguished polynomials to compare
invariants in different coordinate systems, and section 8 contains the proof of
the
main theorem, modulo some computer calculations. The computer-dependent
portions of the paper have been isolated in sections 9 and 10,
which describe the algorithm for explicitly putting the defining polynomials
of the simultaneous
resolutions for $E_6$, $E_7$, and $E_8$ into semi-universal form,
and for manipulating the
invariant polynomials to finish the proof of the main theorem.
We would like to thank Robert Bryant, Jonathan Wahl, and
Klaus Wirthm\"uller for helpful conversations, and Yunliang Yu for
sharing his {\sc maple}-to-\TeX\ package with us.
The first author would also like to thank the University of Bayreuth for a
stimulating working environment while part of this work was done.
We both gratefully acknowledge the facilities of Internet and {\sc bitnet},
which made our collaboration possible.
\section{Reid's lemma, Pinkham's construction, Koll\'ar's invariant, and
the statement of the main theorem.}
The analysis of Gorenstein threefold singularities with small resolutions
begins with the following lemma of Reid \cite[(1.1),(1.14)]{[R]}.
\begin{reidslemma}
Let $\pi\colon Y \to X$ be a resolution of an isolated
Gorenstein threefold singularity
$P \in X$. Suppose that the exceptional set of $\pi$ has pure
dimension 1. Let $X_0$ be a generic hyperplane section of $X$
which passes through $P$.
Then $X_0$ has a rational double point at $P$.
Moreover, if $X_0$ is any hyperplane section through $P$
with a rational double point,
and $Y_0$ is its proper transform, then $Y_0$ is normal,
and the minimal resolution
$Z_0 \to X_0$ factors through the induced map
$\pi |_{Y_0}\colon Y_0 \to X_0$.
\end{reidslemma}
Following Wahl \cite{[W]}, a map $Y_0 \to X_0$
through which the minimal resolution $Z_0 \to X_0$ factors
is called
a {\em partial resolution\/} of $X_0$ (provided that $Y_0$ is normal).
There is a natural graph
associated to such a map. Start with the dual graph $\Gamma$ of the
components of the exceptional divisor of the minimal resolution
$Z_0 \to X_0$.
The curves contracted by $Y_0 \to X_0$ correspond to vertices in the
graph which span a subgraph $\Gamma_0$;
we call $\Gamma_0 \subset \Gamma$ the {\em partial resolution graph\/}
of $\pi$.
Figure~\ref{figure1} below and figure~\ref{figure2}
in section 7 show some partial resolution graphs.
In both figures, the vertices
corresponding to $\Gamma_0$ are shown with an open circle $(\circ)$,
while those corresponding to $\Gamma - \Gamma_0$ are shown with a
closed circle $(\bullet)$.
Based on Reid's lemma, Pinkham \cite{[P]} gave a construction which
in principle describes
all Gorenstein threefold singularities with small resolutions.
Let ${\cal Y} \to \operatorname{Def} (Y_0)$ and
${\cal X} \to \operatorname{Def} (X_0)$ be semi-universal deformations of
$Y_0$ and $X_0$, respectively. By a theorem of Wahl
\cite[Theorem 1.4]{[Wahl]}, all deformations
of $Y_0$ blow down to give deformations of $X_0$, and there is a
natural map $\tau\colon \operatorname{Def} (Y_0) \to \operatorname{Def} (X_0)$. If we choose a local
equation $\{f=0\}$ for $X_0$ in a neighborhood $U$ of $P$ in $X$, then
$f$ can be regarded as a map $f\colon U \to \Delta$, where
$\Delta \subset {\Bbb C}$ is a small disk, such that
$X_0 \cap U = f^{-1}(0)$.
The fibers of $f$ give a deformation of $X_0$, so there is
a classifying map $\mu_f\colon \Delta \to \operatorname{Def} (X_0)$
which enables us to recover $U$. Similarly, the fibers of
$f \circ \pi$ give a deformation of $Y_0$, and the classifying
map $\mu_{f \circ \pi}\colon \Delta \to \operatorname{Def} (Y_0)$
satisfies $\tau \circ \mu_{f \circ \pi} = \mu_f$.
The only data which are really necessary to describe the map
$\tau\colon \operatorname{Def} (Y_0) \to \operatorname{Def} (X_0)$ are the type $S$ of the rational
double point ($S$ is one of $A_{n-1}$, $D_n$, $E_6$, $E_7$, or $E_8$)
and the subgraph $\Gamma_0 \subset \Gamma$. Having fixed such data, we denote
the map by $\tau\colon \operatorname{PRes} (S,\Gamma_0) \to \operatorname{Def} (S)$.
The space $\operatorname{PRes} (S,\Gamma_0)$ represents the functor of deformations of
a singularity of type $S$ which {\em p\/}artially {\em res\/}olve,
with partial resolution type specified by $\Gamma_0$.
When $\Gamma_0$ consists of a single vertex $v$, we abbreviate the
notation to $\operatorname{PRes} (S,v)$.
Pinkham's general construction then goes as follows. Fixing $S$ and
$\Gamma_0$ determines the map
$\tau\colon \operatorname{PRes} (S,\Gamma_0) \to \operatorname{Def} (S)$.
For any map $\nu\colon \Delta \to \operatorname{PRes} (S,\Gamma_0)$ there is an
induced map $\mu \mathrel{:=} \tau \circ \nu$.
Pulling back the semi-universal families by $\nu$ and $\mu$ gives
threefolds $Y \to X$. Pinkham shows that if $\nu$ is sufficiently
general, then $Y$ is smooth, $X$ is Gorenstein with an isolated
singular point, and $Y \to X$ is a small resolution. Reid's lemma
implies that all Gorenstein threefold
singularities with small resolutions arise in this way.
However, as Pinkham observed \cite[p.~367]{[P]},
``Although we have a construction for the singularities that arise, we
do not really have a classification. For example, the construction does not
explain how to get the generic hyperplane \dots".
In fact, the generic hyperplane section of $X$ can have a much simpler
singularity type than $S$ (the type of the hyperplane section which
was used to construct $Y \to X$).
The main theorem of this
paper will compute the singularity type of the
generic hyperplane section in the case that the
exceptional set of $\pi\colon Y \to X$ is irreducible.
\bigskip
There is a fundamental invariant of singularities $P \in X$ with an
irreducible small resolution
which was introduced by
Koll\'ar \cite[pp.~95, 96]{[CKM]}.
Let $\frak m_{P,X}$ denote the maximal ideal sheaf of $P$ in
$X$.
\begin{definition}
Let $\pi\colon Y\to X$ be an irreducible small resolution of an isolated
threefold singularity $P$. (That is, the exceptional set
$\pi^{-1}(P)$ is an irreducible curve $C$.)
The\/ {\em length\/} of $P$ is the length at the generic point of the
scheme supported on $C$ with structure sheaf ${\cal O}_Y/\pi^{-1}(\frak m_{P,X})$.
\end{definition}
\begin{remark} A different invariant, the\/ {\em normal bundle sequence\/}
of the exceptional curve, was introduced by Pinkham in \cite{[P]}. This is the
collection of normal bundles of the sequence of curves $C_1,\ldots $ beginning
with the
exceptional curve $C$, with $C_{i+1}$ being the negative section of the
exceptional divisor of the blowup along $C_i$. The sequence terminates when
the normal bundle becomes $\O(-1)\oplus\O(-1)$.
There are
five possible types of normal bundle sequences,
listed in \cite[p.~367]{[P]}.
The length and the normal bundle sequence measure a kind of ``thickness'' of
the singularity and exceptional curve, respectively. It can
be shown that these invariants are related as follows.
Length 1 corresponds to Pinkham's cases (1) or (2), length 2 corresponds to
case (3), lengths 3, 4, 6 each correspond to case (4),
and length 5 corresponds to
case (5).
\end{remark}
In the Gorenstein case,
if $X_0$ is any hyperplane section of $X$ with rational
double points, then the length of $X$ can also be computed from the
sheaf ${\cal O}_{Y_0}/\pi^{-1}(\frak m_{P,X_0})$.
The length therefore coincides with the multiplicity of the curve $C$ in
the maximal ideal cycle of the rational double point.
(The maximal ideal cycle is the same as Artin's {\em fundamental cycle\/}
\cite{[A]}.)
This property of the length puts a constraint on the possible deformations of
the hyperplane section. If we have a family $(X_0)_t$
of hyperplane sections
through $P$, then we can get different partial resolution graphs for different
values of $t$. However, the multiplicity of the curve $C$ in the maximal ideal
cycle must be independent of $t$, since it always coincides with the
length. This leads to the
following definition.
\begin{definition}
A partial resolution $Y_0 \to X_0$ with irreducible
exceptional set $C$ is\/ {\em primitive\/} if for
every nontrivial 1-parameter deformation of $Y_0 \to X_0$ for which
$C$ deforms, the
multiplicity of $C$ in the maximal ideal cycle at the generic point of the
family is strictly less than the multiplicity at the special point.
\end{definition}
The point of the definition is that if $Y\to X$ is an irreducible
small resolution with a
hyperplane section $X_0\subset X$ yielding a primitive $Y_0\to X_0$,
then
a 1-parameter deformation from $X_0$ to a general hyperplane section
through $P$ must be trivial. Thus,
$X_0$ has
the same singularity type as a general hyperplane section of $X$ through $P$.
The primitive partial resolution
graphs can be computed from the deformation theory of rational
double points. There are exactly six of them: the six graphs
shown in figure~\ref{figure1}.
The numbers labeling the vertices in the figure
are the multiplicities of the corresponding curves in
the maximal ideal cycle.
We omit the proof that these six graphs are the only primitive ones;
the proof is a fairly straightforward computation, but the result
is not logically necessary
for the purposes of this paper.
In fact, it can be obtained as a corollary of our main theorem.
\begin{figure}[t]
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\caption{}
\label{figure1}
\end{figure}
Note that the primitive partial resolution graphs shown
are uniquely determined by the multiplicity of
$C$ in the maximal ideal cycle. It
follows from our proof of the main theorem that every partial resolution graph
admits a deformation to a primitive graph through graphs for which $C$ has
constant multiplicity in the maximal ideal cycle.
\bigskip
We can now state our main theorem.
\begin{maintheorem}
The generic hyperplane section of
an isolated Gorenstein threefold singularity
which has an irreducible small resolution
defines one of the
primitive partial resolution graphs in figure~\ref{figure1}.
Conversely, given any
such primitive
partial resolution graph, there exists an
irreducible small resolution $Y\to X$
whose general hyperplane
section is described by that partial resolution graph.
\end{maintheorem}
It follows that there are exactly six basic types of simple flops on
smooth threefolds.
\begin{corollary}
The general hyperplane section of $X$ is uniquely determined
by the length of the singular point $P$.
\end{corollary}
Two special cases of this theorem were known previously. If $X$ has
some hyperplane section of type $A_n$, then the theorem asserts that
the general hyperplane section has type $A_1$. This was known to
several people (including Mori, Pinkham,
Shepherd-Barron and the second author) about
10 years ago, but has apparently never been published. And in \cite{[K]},
the first author
did the case in which $X$ has some hyperplane section of
type $D_4$.
\bigskip
The converse statement in the theorem follows from the discussion above:
Pinkham showed that examples exist for any partial resolution graph,
and as we have
observed, a hyperplane section leading to a primitive partial resolution
graph must be the
general hyperplane section. The first statement in the main theorem
will be proved in section 8.
\section{Root systems, Weyl groups,
and distinguished polynomials.}
In this section, we establish some notation for certain
root systems and their Weyl groups.
As is customary in algebraic
geometry, we take root systems to have
negative definite inner product;
aside from this, we follow the notation
of Bourbaki \cite{[Bo]} fairly closely for $A_{n-1}$
and $D_ n$, making some departures in
the case $E_ n$.
We begin with an inner product space
$\H ^ {n+1}$ ($n \ge 1$)
over a field $k$ of characteristic 0,
with orthogonal basis
$e_ 0 , e _ 1, \dots ,
e _ {n}$ such that
$\ip{e_ 0}{e_ 0} = 1$ and
$\ip{e _ i}{
e _ i} = -1$ for $i \ge 1$.
We let $e_0^*,e_1^*,\ldots,e_n^*$ be the dual basis of the dual space
$(\H ^ {n+1})^*$.
$\H ^ {n+1}$ contains the
lattice
\[\H ^ {n+1}_ {\Bbb Z} \mathrel{:=}
\{ x = \sum \xi_i e_i \in \H ^ {n+1} \ | \
\xi_i \in {\Bbb Z} \text{ and } \sum \xi_i \in 2 {\Bbb Z} \}.\]
We define three subspaces of
$\H ^ {n+1}$. $V_ {E_ n}$ is
the orthogonal complement of
the special vector $k \mathrel{:=} -3
e_ 0 + \sum _ {i=1}^ n e _ i$,
$V_ {D_ n}$ is the orthogonal
complement of the first basis vector
$e_ 0 $, and $V_ {A_ {n-1}}
= V_ {E_ n} \cap V_ {D_ n}$
is the orthogonal complement of the
subspace spanned by $k$
and $e_ 0 $.
The lattice $\H ^ {n+1}_ {\Bbb Z} $
induces (by intersection with
these subspaces) lattices $L_ {E_ n}$,
$L_ {D_ n}$, and $L_ {A_ {n-1}}$.
A {\em root\/} in one of these lattices
is an element of norm $-2$.
The set of roots in the lattice $L_ {E_ n}$ ($n \ge 3$),
$L_ {D_ n}$ ($n \ge 2$), or $L_ {A_ {n-1}}$ ($n \ge 1$)
is called a {\em root system\/}
of type ${E_ n}$, ${D_ n}$, or ${A_ {n-1}}$, respectively.
We denote this set of roots by $R_ {E_ n}$ (resp. $R_ {D_ n}$ or
$R_ {A_ {n-1}}$).
The lattice itself is called the {\em root lattice},
and the vector space $V_ {E_ n}$ (resp. $V_ {D_ n}$ or
$V_ {A_ {n-1}}$) is called the {\em root space over $k$}.
It is customary in Lie theory to take $k = {\Bbb R} $; here, we are
primarily interested in the case $k = {\Bbb C}$, and in that case
we refer to $V$ as the {\em complex root space}.
We have included a degenerate case $A_0$, as well as
two cases in which the root system is reducible:
$R_ {D_ 2} \cong R_ {A_ 1} \cup R_ {A_ 1}$, and
$R_ {E_ 3} \cong R_ {A_ 2} \cup R_ {A_ 1}$.
In addition, there are some isomorphisms among the irreducible ones:
$R_ {D_ 3} \cong R_ {A_ 3}$,
$R_ {E_ 4} \cong R_ {A_ 4}$, and
$R_ {E_ 5} \cong R_ {D_ 5}$.
When $n \ge 1$, the lattice $L_ {A_ {n-1}}$ can be
generated by the
{\em root basis\/} $v_ 1 , \dots ,
v_ {n-1}$, where
$v _ i \mathrel{:=} e _ i
- e _ {i+1}$. This root
basis has as its Dynkin diagram $\Gamma_{A_{n-1}}$:
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\end{center}
\noindent
Notice that in the degenerate case $A_0$, we have the empty root basis,
which forms
a basis for the zero vector space $V_{A_0}$.
When $n \ge 2$, adding the root $v _ n\mathrel{:=}
e _ {n-1} + e _ n$ to the set $v_1,\ldots,v_{n-1}$
produces a root basis
$v_ 1 , \dots , v_ {n-1}, v _ n$
of $L_ {D_ n}$, which has Dynkin diagram $\Gamma_{D_n}$:
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\end{center}
\noindent
Only the two end vertices $v_{n-1}$ and $v_n$ appear in the reducible
case $D_2$.
Finally, if $3 \le n \le 8$,
adding the root
$v _ 0 \mathrel{:=} e_ 0 -
e _ 1 - e _ 2 -
e _ 3$ to the set $v_1,\ldots,v_{n-1}$
produces a root basis $v _ 0,
v_ 1 , \dots , v_ {n-1}$
of $L_ {E_ n}$. This has Dynkin
diagram $\Gamma_{E_n}$:
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\end{center}
\noindent
In the reducible case $E_3$, this diagram consists of $v_1$ joined
to $v_2$ on the left, and $v_0$ below.
Many of our constructions for a root system $R$ will depend on its
{\em type\/} $S$, which is one of $A_{n-1}$, $D_n$, or $E_n$ in all
cases we consider. Constructions which depend on the type may fail to
be invariant under the isomorphisms of root systems
$R_ {D_ 3} \cong R_ {A_ 3}$,
$R_ {E_ 4} \cong R_ {A_ 4}$, and
$R_ {E_ 5} \cong R_ {D_ 5}$.
We single out certain linear functions on our root spaces.
Let $V$ be one of our root spaces $V_{A_{n-1}}$, $V_{D_n}$, or
$V_{E_n}$. We define the {\em distinguished functionals\/} on the
root space $V$ to be the $n$ functions $t_1,\ldots,t_n$ given by
\[t_i \mathrel{:=} (\frac13 e_0^* + e_i^*)|_V.\]
It is not difficult to express
the distinguished functionals
$t_1,\ldots,t_n$ in terms of the dual basis $v_{\alpha}^*, v_{\alpha + 1}^*,
\ldots, v_{\beta}^*$ of the root basis
$v_{\alpha}, v_{\alpha + 1}, \ldots, v_{\beta}$; the result is shown
in the third column of table~\ref{table12}. (In order to simplify
the notation, we have introduced a few extra $v_i^*$'s into the formulas,
which should be set equal to zero, as indicated in the second column of
the table.) In the fourth column of table~\ref{table12}, we have
solved for the dual
basis in terms of the distinguished functionals. A key step in doing
this is to let $s_1\mathrel{:=} t_1+\cdots+t_n$ be the sum of the distinguished
functionals, and to note that $s_1=0$ in the case of $A_{n-1}$.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|l|l|} \hline
& & & \\
$A_{n-1}$ & $v_0^*=0$ &
$t_i = - v_{i-1}^* + v_i^*$ &
$v_i^* = t_1 + \cdots + t_i$ \\
& $v_n^*=0$ & \quad ($1 \le i \le n$) & \quad ($1 \le i \le n-1$) \\
& & & \\ \hline
& & & \\
& & $t_i = - v_{i-1}^* + v_i^*$
& $v_i^* = s_1 - t_{i+1} - \cdots - t_n $ \\
& & \quad ($1 \le i \le n-2$) & \quad ($1 \le i \le n-2$) \\
& & & \\
$D_n$ & $v_0^* = 0$ & $t_{n-1} = - v_{n-2}^* + v_{n-1}^* + v_n^*$
& $v_{n-1}^* = \frac12 s_1 - t_n$ \\
& & & \\
& & $t_{n} = - v_{n-1}^* + v_n^*$
& $v_n^* = \frac12 s_1$ \\
& & & \\ \hline
& & & \\
& & $t_1 = - \frac23 v_0^* + v_1^* $
& $v_0^* = \frac{3}{n-9} s_1 $ \\
& & & \\
& & $t_2 = - \frac23 v_0^* - v_1^* + v_2^* $
& $v_1^* = \frac{2}{n-9} s_1 + t_1 $ \\
$E_n$ & $v_n^*=0$ & & \\
& & $t_3 = - \frac23 v_0^* - v_2^* + v_3^* $
& $v_2^* = \frac{4}{n-9} s_1 + t_1 + t_2 $ \\
& & & \\
& & $t_i = \frac13 v_0^* - v_{i-1}^* + v_i^* $
& $v_i^* = \frac{9-i}{n-9} s_1 + t_1 + \cdots + t_i $ \\
& & \quad ($4 \le i \le n$) & \quad ($3 \le i \le n-1$) \\
& & & \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{}
\label{table12}
\end{table}
The formulas in table~\ref{table12} imply that
the ring ${\Bbb C} [V]$ of polynomial functions on $V$ is generated
by $t_1,\ldots,t_n$, and is in fact isomorphic to
${\Bbb C} [t_1,\ldots,t_n]/I$, where $I = (t_1+\cdots+t_n)$ in case
$A_{n-1}$, and $I=(0)$ otherwise.
We collect the distinguished functionals into the {\em distinguished
polynomial\/}
\begin{equation}
\label{eqf1}
f_S(U;t)\mathrel{:=}\prod_{i=1}^n (U + t_i),
\end{equation}
where $S$ denotes the type of the root system
(one of $A_{n-1}$, $D_n$, or $E_n$)
and $t$ denotes $(t_1,\ldots,t_n)$.
If we let the symmetric group $\frak S_n$ act by permuting $\{t_1,\ldots,t_n\}$
in the usual way, then
the distinguished polynomial $f_S(U;t)$ lies in the subring
$({\Bbb C} [V]^{\frak S_n})[U]$ of polynomials
whose coefficients are invariant under $\frak S_n$. It can thus
be written in the form
\[
f_S(U;t)=U^n + \sum_{i=1}^n s_i U^{n-i},
\]
where
the coefficients $s_1,\ldots,s_n$ are the elementary symmetric
functions of the distinguished functionals $t_1,\ldots,t_n$.
(Note that $s_1$ is the sum of the distinguished functionals,
agreeing with the definition given above.)
However, the product expansion for $f_S(U;t)$ given in
equation (\ref{eqf1}) is only valid
in the larger ring $({\Bbb C} [V])[U]$.
It is important to observe that the construction of both
the distinguished functionals and the
distinguished polynomial depend on the
type of the root system having been identified as one of $A_{n-1}$,
$D_n$, or $E_n$.
For example, although $E_4$ and $A_4$
are isomorphic as root
systems, they have {\em different\/} distinguished functionals
and polynomials.
(Even the degrees of the distinguished polynomials are different.)
Notice also that when our definitions are applied to the degenerate case
$A_0$, the single distinguished functional $t_1$ is identically zero,
and the distinguished
polynomial is simply $f_{A_0}(U;t) = U \in {\Bbb C} [U]$.
We next describe the action of the Weyl group $\frak W$ of our
root system.
$\frak W$ is the subgroup of $\operatorname{Aut} (V^*)$ generated by the reflections
$r_v$ in the roots $v$ which belong to the root basis. We are particularly
interested in the invariant polynomials for the action of $\frak W$ on $V^*$.
A theorem of Coxeter \cite{[Cx]} and Chevalley \cite{[Chv]} guarantees
that the ring ${\Bbb C} [V]^{\frak W}$ of invariant polynomials is a free
polynomial algebra over ${\Bbb C}$. This implies that the quotient
space $V/\frak W$ is smooth.
The reflections $r_v \in \operatorname{Aut} (V^*)$ which generate $\frak W$ are
restrictions of reflections in $\operatorname{Aut} ((\H ^{n+1})^*)$, which
we also denote by $r_v$. The action of each $r_v$ on $(\H ^{n+1})^*$
is easily described. For example,
the reflections $r_{v_i}$ for $1 \le i \le n-1$
act on $(\H ^ {n+1})^*$
by fixing $e_0^*$ and mapping $e_j^*$
to $e_{\sigma (j)}^*$, where
$\sigma$ is the simple transposition $(i,i+1)$ of the set $\{1,\ldots,n\}$.
It follows that the action of these $r_{v_i}$ on $V^*$ maps
$t_j^*$ to $t_{\sigma(j)}^*$ in each of the three cases
$V=V_{A_{n-1}}$, $V=V_{D_n}$, and $V=V_{E_n}$.
The Weyl group $\frak W_ {A_ {n-1}}$ is the
group generated by the reflections
$r_ {v _ 1}, \dots , r_ {v _ {n-1}}$, and therefore coincides with
the symmetric
group $\frak S_n$.
Since $t_1 + \cdots + t_n = 0$, the
invariant polynomials for the action of $\frak W_{A_{n-1}}$ are generated by
the nonzero coefficients
$s_2,\ldots,s_n$ of the distinguished polynomial $f_{A_{n-1}}(U;t)$.
In the case of the Weyl group $\frak W_ {D_ n}$, the reflections
$r_{v_i}$ for $1 \le i \le n-1$ again act as permutations of
$t_1,\ldots,t_n$, although this time the sum $s_1$ is not necessarily
zero. $\frak W_ {D_ n}$ is generated by those reflections together
with $r_{v_n}$. The action of $r_{v_n}$ on $(\H ^ {n+1})^*$
is not hard to compute: we have
\[
r_ {v _ n}(e_i^*) =
\begin{cases}
\quad e_i^* & \text{if $0 \le i \le n-2$}, \\
- e_ n^* & \text{if $i=n-1$}, \\
- e_ {n-1}^* & \text{if $i=n$}.
\end{cases}
\]
It follows that the action on $V_{D_n}^*$ is given by
\[
r_ {v _ n}(t_i) =
\begin{cases}
\quad t_i & \text{if $1 \le i \le n-2$}, \\
- t_ n & \text{if $i=n-1$}, \\
- t_ {n-1} & \text{if $i=n$}.
\end{cases}
\]
The invariant polynomials for this action are generated by the
product
$t_1 \cdots t_n$,
together with the elementary
symmetric functions of the squares $t_1^2,\ldots,t_n^2$.
These invariants are captured by the constant term $s_n = f_{D_n}(0;t)$
of the distinguished polynomial, together with the coefficients of
a polynomial $g_{D_n}(Z;t)$
whose defining property is
\begin{equation}
\label{eqg1}
g_{D_n}(Z;t) \mathrel{:=} \prod_{i=1}^n (Z + t_i^2).
\end{equation}
(Note that $g_{D_n}(0;t) = f_{D_n}(0;t)^2 = s_n^2$, so its constant term is not
needed to generate the $\frak W$-invariants.)
We call $g_{D_n}(Z;t)$ the {\em second distinguished polynomial\/}; it is
only defined for root systems of type $D_n$.
The second distinguished polynomial $g_{D_n}(Z;t)$ lies in the ring
$({\Bbb C} [V]^{\frak W})[Z]$ of polynomials whose coefficients are $\frak W$-invariants,
although the defining property (\ref{eqg1}) only holds in the larger
ring $({\Bbb C} [V])[Z]$. The second distinguished polynomial could
also have been defined by the property
\begin{equation}
\label{eqg2}
g_{D_n}(- U^2;t) = f_{D_n}(U;t) \cdot f_{D_n}(-U;t)
\end{equation}
which holds in the auxiliary ring $({\Bbb C} [V]^{\frak S_n})[U]$. Following
Tyurina~\cite{[T]}, we can express this property directly in the subring
$({\Bbb C} [V]^{\frak S_n})[Z]$ of $({\Bbb C} [V])[Z]$ in the following way.
Collect terms of even and odd degree in the distinguished polynomial:
\begin{equation}
\label{eqg3a}
f_{D_n}(U;t) = U \cdot P_{D_n}(-U^2;t) + Q_{D_n}(-U^2;t).
\end{equation}
This defines two polynomials
\[P_{D_n}(Z;t), Q_{D_n}(Z;t) \in ({\Bbb C} [V]^{\frak S_n})[Z].\]
Then equation (\ref{eqg2}) can be rewritten as
\begin{equation}
\label{eqg3b}
g_{D_n}(Z;t) = Z \cdot P_{D_n}(Z;t)^2 + Q_{D_n}(Z;t)^2.
\end{equation}
We also record for later use the existence of a polynomial
$G_{D_n}(Z,U;t) \in ({\Bbb C} [V]^{\frak S_n})[Z,U]$ such that
\begin{equation}
\label{eqG1}
U \cdot P_{D_n}(Z;t) + Q_{D_n}(Z;t)
= (Z + U^2) \cdot G_{D_n}(Z,U;t) + f_{D_n}(U;t).
\end{equation}
Since $G_{D_n}(Z,U;t)$ has degree $n-2$ in $U$, if we define
\begin{equation}
\label{eqG2}
\widetilde{G}_{D_n}(Z,u,v;t) \mathrel{:=} v^{n-2} \cdot G_{D_n}(Z,u/v;t),
\end{equation}
then $\widetilde{G}_{D_n}(Z,u,v;t)$ is a polynomial which is
homogeneous in $u, v$.
Finally, in the case of the Weyl group $\frak W_ {E_ n}$, the reflections
$r_{v_i}$ for $1 \le i \le n-1$ still act as permutations of
$t_1,\ldots,t_n$, and again the sum $s_1$ is not necessarily
zero. $\frak W_ {E_ n}$ is generated by those reflections together
with $r_{v_0}$. The action of $r_{v_0}$ on $(\H ^ {n+1})^*$
is not hard to compute: we have
\[
r_ {v _ 0}(e_ i^*) =
\begin{cases}
e_ 0^* + (e_ 0^* + e_ 1^* + e_ 2^* + e_ 3^*) & \text{if $i=0$}, \\
e_ i^* - (e_ 0^* + e_ 1^* + e_ 2^* + e_ 3^*) & \text{if $1 \le i \le 3$}, \\
e_ i^* & \text{if $4 \le i \le n$}.
\end{cases}
\]
It follows that the action on $V_{E_n}^*$ is given by
\begin{equation}
\label{WE}
r_ {v _ 0}(t_ i) =
\begin{cases}
t_ i - \frac 23
(t_ 1 + t_ 2 + t_ 3) & \text{if $1 \le i \le 3$}, \\
t_ i + \frac 13
(t_ 1 + t_ 2 + t_ 3) & \text{if $4 \le i \le n$}.
\end{cases}
\end{equation}
The invariant polynomials for the
action of $\frak W_ {E_ n}$ are
not as simple to describe as those for the actions of
$\frak W_{A_{n-1}}$ and $\frak W_{D_n}$. In the reducible case $E_3$,
it is not hard to verify (using equation~(\ref{WE})) that the
polynomials
\begin{align}
\label{eqE3inv}
\begin{split}
&\varepsilon_2^{(1)} \mathrel{:=} s_1^2 \\
&\varepsilon_2^{(2)} \mathrel{:=} s_2 \\
&\varepsilon_3 \mathrel{:=} s_3 - \tfrac13 s_1 s_2 + \tfrac{2}{27} s_1^3
\end{split}
\end{align}
are invariant under $\frak W_{E_3}$, and generate the ring of invariants
${\Bbb C} [V_{E_3}]^{\frak W_{E_3}}$. In the cases of $E_4$ and $E_5$,
bases for the rings of
invariants can be calculated by using the isomorphisms
$R_ {E_ 4} \cong R_ {A_ 4}$ and
$R_ {E_ 5} \cong R_ {D_ 5}$.
We carry this out in the corollary to lemma~\ref{lem71} in section 7.
The invariant polynomials are extremely complicated in the classical
cases $E_6$, $E_7$, and $E_8$. Calculations of some of these were made
in the early 1950s by Coxeter \cite{[Cx]}, Frame \cite{[Fr]}, and
Racah \cite{[Rc]}, although published explicit formulas are limited
to the case of $E_6$.
One of the by-products
of the work discussed here is an
explicit description of a basis
for the invariant polynomials in these three cases.
(These invariants are of course polynomials in the symmetric functions
$s_1,\ldots,s_n$, since they are invariant under the subgroup
$\frak W_ {A_ {n-1}}$.) The basis we compute appears in Appendix 1 for
$E_6$ and in Appendix 2 for $E_7$. The polynomials in the basis
are too large to be worth writing down
in the case of $E_8$, but an algorithm for computing with those polynomials
is given in section 9.
\section{Rational double points and simultaneous resolution.}
Associated to each irreducible root system of type $S=A_{n-1}$, $D_n$,
or $E_n$
is a rational double point with the same name. The minimal
resolution of this complex
surface singularity has a dual graph which
is isomorphic to
the corresponding Dynkin diagram, and the singularity is
determined up to isomorphism by the diagram. For each singularity type,
we fix a representative for the isomorphism class: the hypersurface
in ${\Bbb C} ^3$ whose defining polynomial is given in the middle column of
table~\ref{table2A}. Notice
that as in section 2, our choices depend on the type $S$
rather than just on the isomorphism class.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|rcl|} \hline
& & & & \\
$S$ & Defining Polynomial & \multicolumn{3}{c|}{Action of $\l \in {\Bbb C} ^*$} \\
& & & & \\ \hline
& & & & \\
$A_{n-1}$ & $ - X Y + Z^n$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^kX,\l ^{n-k}Y,\l Z)$ \\
$(n \ge 2)$ & & & & \\
& & & & \\
$D_n$ & $ - X^2 - Y^2 Z + Z^{n-1}$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^{n-1}X,\l ^{n-2}Y,\l ^2Z)$\\
$(n \ge 3)$ & & & & \\
& & & & \\
$E_4$ & $ - X Y + Z^5$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^kX,\l ^{5-k}Y,\l Z)$\\
& & & & \\
$E_5$ & $ - X^2 - Y^2 Z + Z^4$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^4X,\l ^3Y,\l ^2Z)$\\
& & & & \\
$E_6$ & $ - X^2 - X Z^2 + Y^3$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^6X,\l ^4Y,\l ^3Z)$\\
& & & & \\
$E_7$ & $ - X^2 - Y^3 + 16 Y Z^3$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^9X,\l ^6Y,\l ^4Z)$\\
& & & & \\
$E_8$ & $ - X^2 + Y^3 - Z^5$ & $(X,Y,Z)$ & $\mapsto$ &
$(\l ^{15}X,\l ^{10}Y,\l ^6Z)$\\
& & & & \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{}
\label{table2A}
\end{table}
Each of our chosen representatives admits a ${\Bbb C} ^*$-action, as
indicated in the last column of
table~\ref{table2A}:
if ${\Bbb C} ^3$ is given the specified ${\Bbb C} ^*$-action then the
defining polynomial is {\em weighted homogeneous}, i.e., is an
eigenfunction for the ${\Bbb C} ^*$-action.
In the cases of $A_{n-1}$ and $E_4$,
there are several possible ${\Bbb C} ^*$-actions,
any one of which will suit our purposes.
By a theorem of Pinkham \cite{C*},
a singularity with a ${\Bbb C} ^*$-action admits a ${\Bbb C} ^*$-semi-universal
deformation. (Such a deformation is semi-universal for deformations
with a ${\Bbb C} ^*$-action, and is also semi-universal for arbitrary
deformations.)
For a surface in ${\Bbb C} ^3$ defined by a weighted homogeneous polynomial
$F$, this can be obtained as follows. Choose
weighted homogeneous polynomials $G_1,\ldots,G_r$ which descend to
a basis
of the vector space
${\Bbb C} [X,Y,Z]/(\frac{\partial F}{\partial X},
\frac{\partial F}{\partial Y}, \frac{\partial F}{\partial Z})$.
The polynomial
$\Phi \mathrel{:=} F + \mu_1 G_1 + \cdots + \mu_r G_r$
will then define a ${\Bbb C} ^*$-semi-universal deformation as a
hypersurface in ${\Bbb C} ^{3+r}$. The variable $\mu_i$ is given
the difference of the weights of $F$ and $G_i$ as its weight.
In the case of rational double points, even after fixing the defining
polynomial as in table~\ref{table2A}, there are still some choices to
be made, for there may be more than one
choice of weighted homogeneous polynomials which give a basis of
${\Bbb C} [X,Y,Z]/(\frac{\partial F}{\partial X},
\frac{\partial F}{\partial Y}, \frac{\partial F}{\partial Z})$.
We implicitly fix one such choice in
table~\ref{table2B}.
\begin{table}[t]
\begin{center}
$\begin{array}{|c|r@{\ + \ }l|} \hline
& \multicolumn{2}{c|}{} \\
$S$ & \multicolumn{2}{c|}{\text{Preferred Versal Form}} \\
& \multicolumn{2}{c|}{} \\ \hline
& \multicolumn{2}{c|}{} \\
A_{n-1} & - X Y + Z^n & \sum_{i=2}^{n}\a_i Z^{n-i} \\
(n \ge 2) & \multicolumn{2}{c|}{} \\
& \multicolumn{2}{c|}{} \\
D_n & \multicolumn{1}{r@{\ - \ }}{ X^2 + Y^2 Z - Z^{n-1} } &
\sum_{i=1}^{n-1}\d_{2i} Z^{n-i-1} + 2 \c_n Y \\
(n \ge 3) & \multicolumn{2}{c|}{} \\
& \multicolumn{2}{c|}{} \\
E_4 & - X Y + Z^5 & \varepsilon_2 Z^3 + \varepsilon_3 Z^2 + \varepsilon_4 Z + \varepsilon_5 \\
& \multicolumn{2}{c|}{} \\
E_5 & \multicolumn{1}{r@{\ - \ }}{ X^2 + Y^2 Z - Z^4 } &
\varepsilon_2 Z^3 - \varepsilon_4 Z^2 + 2 \varepsilon_5 Y - \varepsilon_6 Z - \varepsilon_8 \\
& \multicolumn{2}{c|}{} \\
E_6 & - X^2 - X Z^2 + Y^3 &
\varepsilon_2 Y Z^2 + \varepsilon_5 Y Z + \varepsilon_6 Z^2 + \varepsilon_8 Y \\
& &
\varepsilon_9 Z + \varepsilon_{12} \\
& \multicolumn{2}{c|}{} \\
E_7 & - X^2 - Y^3 + 16 Y Z^3 &
\varepsilon_2 Y^2 Z + \varepsilon_6 Y^2 + \varepsilon_8 Y Z + \varepsilon_{10} Z^2 \\
& &
\varepsilon_{12} Y
+ \varepsilon_{14} Z + \varepsilon_{18} \\
& \multicolumn{2}{c|}{} \\
E_8 & - X^2 + Y^3 - Z^5 &
\varepsilon_2 Y Z^3 + \varepsilon_8 Y Z^2 + \varepsilon_{12} Z^3 + \varepsilon_{14} Y Z \\
& &
\varepsilon_{18} Z^2 + \varepsilon_{20} Y + \varepsilon_{24} Z
+ \varepsilon_{30} \\
& \multicolumn{2}{c|}{} \\ \hline
\end{array}$
\end{center}
\medskip
\caption{}
\label{table2B}
\end{table}
\begin{definition}
The defining polynomial $\Phi_S$
of a ${\Bbb C} ^*$-semi-universal
deformation of a rational double point is said to be
in {\em preferred versal form\/} if it has
the form given in table~\ref{table2B}.
\end{definition}
We have chosen the notation
in table~\ref{table2B} to make the
${\Bbb C} ^*$-action explicit: the ${\Bbb C} ^*$-action on a
${\Bbb C} ^*$-semi-universal deformation whose defining polynomial is in
preferred versal form is determined by giving
$X$, $Y$, $Z$ the same eigenvalues as in table~\ref{table2A},
and giving each coefficient
with subscript $i$ the eigenvalue $\l ^i$.
\begin{remark}
Tables~\ref{table2A} and \ref{table2B}
are the first of several places in this paper where the normalization
chosen in the case of $E_7$ looks a bit peculiar. (The peculiarity
in this case is the coefficient ``$16$''.) In all of these cases, our
notation is chosen to match that of Bramble \cite{[Bra]}, so that
we may directly compare our results with his. The reader interested in
obtaining a defining polynomial of a ${\Bbb C} ^*$-semi-universal
deformation of $E_7$ of the more natural form $- X^2 - Y^3 + Y Z^3
+\varepsilon_2 Y^2 Z + \varepsilon_6 Y^2 + \varepsilon_8 Y Z + \varepsilon_{10} Z^2
+ \varepsilon_{12} Y + \varepsilon_{14} Z + \varepsilon_{18} $ may easily do so as follows. Start
with the defining polynomial of the ${\Bbb C} ^*$-semi-universal
deformation of $E_7$ in preferred versal form as given by table~\ref{table2B}
and Appendix 2 (which will be calculated later). Substituting $8X$ and $4Y$
for $X$ and $Y$ respectively, then dividing by 64 gives the desired defining
polynomial.
\end{remark}
Our first theorem is a slight refinement of the famous theorem on
simultaneous resolution of rational double points, due to
Brieskorn \cite{[Bri0]},
\cite{[Bri]},
\cite{[Bri-Nice]} and Tyurina \cite{[T]}.
(We follow Tyurina's approach, as amplified by Pinkham \cite{[P-RDP]}.)
Recall that a {\em simultaneous resolution\/} of a family
${\cal X} \to {\cal D}$ is a resolution of singularities of
${\cal X}$ which also resolves the singularities of each fiber of
the map ${\cal X} \to {\cal D}$. In general,
one can only hope to find a simultaneous
resolution after making a base change ${\cal R} \to {\cal D}$
and pulling back the original family to a family
${\cal X} \times_{{\cal D}} {\cal R}$.
Let $R$ be an irreducible root system of type $S=A_{n-1}$, $D_n$,
or $E_n$, in which a basis of simple roots $\{ v_i \}$
has been chosen. Let $V$ be the complex root space, let
$\frak W$ be the Weyl group acting on $V$ by reflections,
and let $\rho\colon V \to V/\frak W$ be the quotient by $\frak W$. Let $X_0$ be
the corresponding rational double point with ${\Bbb C} ^*$-action,
let $Z_0 \to X_0$ be the minimal resolution, and fix an identification
between the basis of simple roots $\{ v_i \}$
of $R$ and the irreducible exceptional
curves $\{ C_i \}$ on $Z_0$ which preserves the associated graphs.
For each positive root
$v = \sum a_i v_i$, the curve $C_v = \sum a_i C_i$ is an effective rational
curve of self-intersection $-2$ on $Z_0$.
The deformations of $Z_0$ to which $C_v$ lifts forms a natural subset
of all deformations of $Z_0$ (cf.\ \cite[\S 2]{[Wahl-disc]}).
We fix coordinates $X$, $Y$, $Z$ on ${\Bbb C} ^3$.
\begin{theorem}
Let $R$ be an irreducible root system of type $S$, and let
$\rho \colon V \to V/\frak W$ and
$Z_0 \to X_0$ be as above. Let ${\Bbb C} ^*$
act on the complex root space
$V$ via $v \mapsto \l \cdot v$ for $\l \in {\Bbb C} ^*$,
and give $V/\frak W$ the induced
${\Bbb C} ^*$-action. Then there is a ${\Bbb C} ^*$-semi-universal
deformation
${\cal X} \to V/\frak W$ of $X_0$ and a ${\Bbb C} ^*$-equivariant
simultaneous resolution ${\cal Z} \to {\cal X} \times_{\rho} V$
inducing $Z_0 \to X_0$ with the
following properties:
\begin{enumerate}
\item ${\cal X}$ can be embedded as
a hypersurface in ${\Bbb C} ^3 \times V/\frak W$ whose
defining polynomial $\Phi_S$ with respect to the coordinates $X$, $Y$, $Z$
is in preferred versal form.
\item The coefficients of the defining polynomial $\Phi_S$ are explicitly
computable functions on $V/\frak W$, which give a basis of ${\Bbb C} [V]^{\frak W}$.
\item For each positive root $v \in R$, the set of deformations of $Z_0$
in the family ${\cal Z}$ to which the curve $C_v$ lifts is parametrized
by the hyperplane $v^{\perp}$ (the orthogonal complement of $v$ in $V$).
\end{enumerate}
\end{theorem}
The two novelties in this statement are the phrase ``explicitly
computable", and the third part of the theorem (although the latter is
implicit in Slodowy's exposition \cite{[Slod]}, and in some of Looijenga's
work \cite{[Lj]}).
We will give a proof of this theorem, based on two constructions of
Tyurina \cite{[T]}, in the next two sections.
We have explicitly computed these coefficients as functions on $V/\frak W$,
and we give our results
in equation (\ref{eqA}) for $A_{n-1}$, equation (\ref{eqD}) for
$D_n$, equations (\ref{eqE4}) and (\ref{eqE5}) for $E_4$ and $E_5$
respectively,
and Appendices 1 and 2 for $E_6$ and $E_7$ respectively.
(In the last two cases, we ``clear denominators'' before displaying
the result.)
We have not attempted to write down the formulas for $E_8$, although
we give an algorithmic method of computing with them in section 9.
\section{Simultaneous resolutions for $A_{n-1}$ and $D_n$.}
In this section, we will prove theorem 1 in the cases of $A_{n-1}$ and $D_n$.
It is easy to see that the theorem then follows in cases $E_4$ and $E_5$,
since the ${\Bbb C} ^*$-semi-universal deformation of $E_4$ (resp.~$E_5$)
in table~\ref{table2B} coincides with the one for $A_4$ (resp.~$D_5$).
We postpone the cases of $E_6$, $E_7$, and $E_8$ to the next section.
We will construct the required
deformations ${\cal X}$ of
rational double points and their simultaneous resolutions ${\cal Z}$,
using the distinguished functionals
$t_1,\ldots,t_n$ (and the elementary symmetric functions
$s_1,\ldots,s_n$ in those distinguished functionals)
as the key ingredients in the construction.
The construction is due to Kas {\cite{[Kas]}} (for type $A_{n-1}$ only)
and Tyurina \cite{[T]}, both heavily influenced by work of Brieskorn
\cite{[Bri0]}.
\bigskip
We begin with the case of $A_{n-1}$.
Define functions $\a_i \in {\Bbb C} [V]^{\frak W}$ by
\begin{equation}
\label{eqA}
\a_i \mathrel{:=}
\begin{cases}
0 & \text{for $-1 \le i \le 1$}, \\
s_i & \text{for $\hphantom{-} 2 \le i \le n$}.
\end{cases}
\end{equation}
Since the Weyl group $\frak W$ coincides with the symmetric group
$\frak S_n$ and $s_1 = 0$ in ${\Bbb C} [V]$, it follows that
$\{ \a_i \}_{2 \le i \le n}$ is a basis of ${\Bbb C} [V]^{\frak W}$.
Define a hypersurface ${\cal X} \subset {\Bbb C} ^3 \times V/\frak W$ by means
of the polynomial
\[
\Phi_{A_{n-1}} \mathrel{:=} - X Y + Z^n + \sum_{i=2}^{n}\a_i Z^{n-i}
\]
which is in preferred versal form.
Notice that this
can also be written in the form
\[
\Phi_{A_{n-1}} = -XY + f_{A_{n-1}}(Z;t),
\]
and the functions $\a_i$ can be interpreted as the coefficients of
$f_{A_{n-1}}(Z;t)$.
Let $\rho\colon V \to V/\frak W$ be the quotient map.
To construct a simultaneous resolution of
${\cal X} \times_{\rho} V$, we recall that
in the ring $({\Bbb C} [V])[U]$, the distinguished polynomial can be
written in the factored form given in equation (\ref{eqf1}).
Therefore,
\[\rho^*(\Phi_{A_{n-1}}) = -XY + \prod_{i=1}^n (Z + t_i).\]
We construct a simultaneous resolution
${\cal Z} \to {\cal X} \times_{\rho} V$ by
taking the closure of
the graph of the morphism
\begin{align*}
{\cal X} \times_{\rho} V &\to (\P
^ 1)^{n-1} \\
(X,Y,Z,t_ 1,\ldots ,t_ n) &\mapsto [X,\prod_{\nu=1}^
i(Z+t_\nu)]_ i
\end{align*}
Assertions 1 and 2 of theorem 1 are clear, so we need only
verify assertion 3.
Let $(u_ k,v_ k)$ be homogeneous coordinates on the
$k$th $\P ^ 1$ arising from the resolution described above. Then
\begin{gather*}
-XY+\prod_{i=1}^ n(Z+t_ i)=0 \\
Xv_ j=u_ j\prod_{i=1}^ j(Z+t_ i)\quad (1\le
j\le n-1), \\
\prod_{i=k+1}^ j(Z+t_ i)u_ jv_ k=u_ kv_ j\quad
(1\le k<j\le
n-1)
\end{gather*}
are equations for ${\cal Z}$. Thinking of $V_{A_ {n-1}}$ as
a hyperplane in
${\Bbb C} ^ n$, these equations show that the $\pi$-exceptional
part of
the fiber of ${\cal Z}$
over $(t_
1,\ldots ,t_ n)=(0,\ldots ,0)$ is given as the union of the
curves
$C_ i,\ 1\le i\le
n-1$, where $C_ i$ is defined by $X=Y=Z=0$, $u_ j=0$ for
$j<i$, $v_ k=0$
for $k>i$.
All positive roots are
of the form $v=v_ i+\cdots + v_{j-1}$.
{}From this, it is easy to see that the locus where $C_ v=C_
i+
\cdots +C_{j-1}$ deforms is given
by $t_ i=t_ j$,
which is indeed the orthogonal complement of $v$. To see this,
note that if $t_ i=t_ j$, and are distinct from the other
$t_ k$, the
$\pi$-exceptional part of the fiber of ${\cal Z}$ over $(t_
1,\ldots ,
t_ n)$ is
given by $Z=-t_ i=-t_ j, u_ k=0\ (k<i),\ v_ k=0\ (k\ge
j),
u_ mv_ k\prod_{n=k+1}^ m(t_ n-t_ i)=u_ kv_ m,\
(i\le k<m<j)$.
For general parameter values, this is easily seen to be a $\P ^ 1$,
specializing to $C_ i+\cdots +C_{j-1}$ as the parameters
approach 0. Since
the genus is constant, this is a flat family. Since the general
fiber of this
family is irreducible, this is the only possible flat family over
the generic
point of $t_ i=t_ j$ which is contained in the exceptional
locus.
Since ${\cal Z}$
may be obtained from ${\cal X} \times_{\rho} V$ by successively blowing up
Weil divisors, the exceptional set lies entirely over the
discriminant
$\prod_{i<j}(t_ i-t_ j)$. Hence the locus over which any
(union of)
exceptional curves deform is contained in the locus where some
$t_ i$ equals
some $t_ j$. This proves the assertion.
\bigskip
Next we turn to the case of $D_n$. Define functions
$\c_n, \d_{2i} \in {\Bbb C} [V]^{\frak W}$ by
\begin{align}
\label{eqD}
\begin{split}
\gamma_n &\mathrel{:=} t_1
\cdots
t_n = s_n \\
\delta_{2i} &\mathrel{:=} \text{the $i^{\text{th}}$ elementary symmetric}
\\
& \qquad \text{function of $ t_1^2, \ldots, t_n^2$}
\end{split}
\end{align}
(Note that $\d_{2n} = (\c_n)^2$.) The functions
$\c_n, \d_2, \ldots, \d_{2n-2}$ generate ${\Bbb C} [V]^{\frak W}$. Moreover,
comparing (\ref{eqD}) with the definition of the second distinguished
polynomial in (\ref{eqg1}), we see that
\begin{align*}
g_{D_n}(Z;t) &= Z^n + \sum_{i=1}^{n-1} \d_{2i} Z^{n-i} + (\c_n)^2 \\
f_{D_n}(0;t) &= \c_n.
\end{align*}
Define a hypersurface ${\cal X} \subset {\Bbb C} ^3 \times V/\frak W$ by means
of the polynomial
\[
\Phi_{D_{n}} \mathrel{:=} X^2 + Y^2 Z -
Z^{n-1} - \sum_{i=1}^{n-1}\d_{2i} Z^{n-i-1} + 2 \c_n Y
\]
which is in preferred versal form.
Notice that this
can also be written in the form
\[
\Phi_{D_{n}} = X^2 + Y^2 Z - \frac{1}{Z}(g_{D_n}(Z;t)
- f_{D_n}(0;t)^2) + 2 f_{D_n}(0;t) Y.
\]
Let $\frak S_n \subset \frak W$ be the Weyl group of the subsystem of the
root system $R$ generated
by $v_1,\ldots,v_{n-1}$, which
acts on $V$ by permutations of the distinguished functionals
$t_1,\ldots,t_n$. The ring of invariant functions
${\Bbb C} [V]^{\frak S_n}$ is generated by $s_1,\ldots,s_n$.
In the ring $({\Bbb C} [V]^{\frak S_n})[Z]$, we can use the properties of
$g_{D_n}(Z;t)$ given in equations (\ref{eqg3a}) and (\ref{eqg3b}),
which are expressed in terms of the polynomials $P_{D_n}(Z;t)$
and $Q_{D_n}(Z;t)$. Now (\ref{eqg3a}) implies that $Q_{D_n}(0;t) = s_n$,
so we can also define
\begin{equation}
\label{eqS}
S_{D_n}(Z;t) \mathrel{:=} \frac{Q_{D_n}(Z;t) - s_n}{Z}.
\end{equation}
Let $\tau\colon V/\frak S_n \to V/\frak W$ be the natural map.
The first step in Tyurina's construction of the simultaneous resolution
is done most naturally on
${\cal X} \times_{\tau} V/\frak S_n$.
We abbreviate $P_{D_n}(Z;t)$, $Q_{D_n}(Z;t)$, and $S_{D_n}(Z;t)$ by
$P$, $Q$, and $S$, respectively. Using (\ref{eqg3a}), (\ref{eqg3b}),
(\ref{eqS}) and some algebraic tricks as in Tyurina \cite{[T]},
we find that the defining polynomial of ${\cal X} \times_{\tau} V/\frak S_n$
can be written as
\begin{align*}
\tau^*(\Phi_{D_ n}) &= X^2 + Y^2 Z
- \frac{1}{Z}(Z P^2 + (ZS + s_n)^2 - s_n^2)
+ 2 s_n Y \\
&= (X-P)(X+P)+(YZ+ZS+2s_n)(Y-S).
\end{align*}
Let ${\cal Y} \to {\cal X} \times_{\tau} V/\frak S_n$
be the blowup of the ideal
$(X-P,Y-S)$.
It is easy to see that the entire singular locus of ${\cal Y}$
is contained in the coordinate chart ${\cal Y}^0$ defined by
the substitution $X - P = (Y - S) U$. The defining polynomial in this chart
becomes
\begin{align*}
\Phi_{{\cal Y}^0} &= U \cdot ( 2P+(Y-S)U ) + (YZ+ZS+2s_n) \\
&= (Y-S)(Z+U^2)+2UP+2Q \\
&= (Y-S+2G)(Z+U^2) + 2f_{D_n}(U;t)
\end{align*}
(using the polynomial $G = G_{D_n}(Z,U;t)$ defined in equation (\ref{eqG1})).
Now let $\sigma\colon V \to V/\frak S_n$ be the natural quotient map.
Since $f_{D_n}(U;t)$ factors in $({\Bbb C} [V])[U]$, we have
\[
\sigma^*(\Phi_{{\cal Y}^0}) = (Y-S+2G)(Z+U^2) + 2 \prod_{i=1}^n (U+t_i),
\]
which after a change of coordinates is the same family used in the case
of $A_{n-1}$.
We can thus construct a resolution
${\cal Z} \to {\cal Y} \times_{\sigma} V$ (which also resolves
${\cal X} \times_{\rho} V$) by means of the construction used for $A_{n-1}$.
Again, we only need to verify assertion 3 from theorem 1.
Let $[u,v]$ be homogeneous coordinates on the $\P ^1$
of the first blow up for $D_n$ as described above, with $U=u/v$.
The homogeneous form of the equation
$\sigma^*(\Phi_{{\cal Y}^0})$ is
\[
(v^{n-2}(Y-S)+2{\widetilde
G})(v^2Z+u^2)+2\prod_{i=1}^n(u+t_iv)=0
\]
(using the homogeneous counterpart
${\widetilde G} = \widetilde{G}_{D_n}(Z,u,v;t)$
of $G$ which was defined in equation (\ref{eqG2})).
The $A_{n-1}$ singularity is then contained in the affine piece
$v=1$.
Let $[u_k,v_k]$ be homogeneous coordinates on the $k$th $\P ^1$ used in
resolving the $A_{n-1}$ singularity. Then we get as equations
for ${\cal Z}_{D_n}$:
\begin{gather*}
(v^2Z+u^2)v_{n-1}=vu_{n-1}(u+t_nv) \\
v^{n-2-l}(v^2Z+u^2)v_l=u_l\prod_{i=l+1}^n(u+t_iv)\quad (1\le
l\le n-2) \\
v^{k-j}v_ju_k=v_ku_j\prod_{i=j+1}^k(u+t_iv)\quad (1\le j<k\le
n-1).
\end{gather*}
These equations show that the $\pi$-exceptional part of the fiber
of ${\cal Z}$
over $(t_1,\ldots ,t_n)=(0,\ldots ,0)$ is given as the union of
the curves
$C_i, 1\le i\le n,$ where for $i<n$, $C_i$ is defined by
$X=Y=Z=0$, $u=0$, $u_j=0$
for $j>i$, $v_j=0$ for $j<i$ and $C_n$ is defined by $X=Y=Z=0$,
$uv_{n-1}=vu_{n-1}$, $v^{n-l-2}v_l=u^{n-l-2}u_l$ for $1\le l\le
n-2$,
$v^{k-j}v_ju_k=v_ku_j\prod_{i=j+1}^k(u+t_iv)$ for $1\le j<k\le
n-1$.
The positive roots are of five types: $r=v_ i+\cdots +
v_{j-1}\quad (i\le j\le n)$, $s=v_n$, $t=v_j+\cdots +v_{n-2}+v_n
\quad (j\le n-2)$,
$ u=v_j+\cdots +v_n\quad (j\le n-2)$,
$ v=v_ j+\cdots + v_{k-1}+2v_ k+\cdots
+2v_{n-2}+v_{n-1}+v_
n\quad (j<k\le n-2)$.
Exactly as in the $A_{n-1}$ case, it is easy to see from the
above construction
that the locus where $C_ r$ deforms
is given by $t_ i=t_ j$. For each of the remaining cases, we
exhibit the equations of a divisor in ${\cal Z}$ which defines a
flat family of generically irreducible curves over a hyperplane
in the
parameter space; the hyperplane will be given first.
\begin{gather*}
t_{n-1}+t_n=0 \tag*{{\bf s:}}\\
X=0,\ Y=-t_1\cdots t_{n-2},\ Z=-t_n^2 \\
(u-t_nv)v_{n-1}=vu_{n-1} \\
v_{n-2}=u_{n-2} \\
v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots (u+t_{n-2}v)\quad (1\le l\le
n-2) \\
v^{k-j}v_ju_k=v_ku_j\prod_{i=j+1}^k(u+t_iv)\quad (1\le j<k\le
n-1).
\end{gather*}
\begin{gather*}
t_j+t_n=0 \tag*{{\bf t:}}\\
X=0,\ Y=-t_1\cdots t_{j-1}t_{j+1}\cdots t_{n-1},\ Z=-t_n^2 \\
(u-t_nv)v_{n-1}=vu_{n-1} \\
(u-t_nv)v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots (u+t_{n-1}v)\quad
(j+1\le l\le n-2) \\
v^{n-j-2}v_j=u_j(u+t_{j+1}v)\cdots (u+t_{n-1}v) \\
v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots
(u+t_{j-1}v)(u+t_{j+1}v)\cdots
(u+t_{n-1}v)\quad (1\le l\le j) \\
v^{k-j}v_ju_k=v_ku_j\prod_{i=j+1}^k(u+t_iv)\quad (1\le j<k\le
n-1).
\end{gather*}
\begin{gather*}
t_j+t_{n-1}=0 \tag*{{\bf u:}} \\
X=0,\ Y=-t_1\cdots t_{j-1}t_{j+1}\cdots t_{n-2}t_n,\
Z=-t_{n-1}^2 \\
(u^2-t_{n-1}^2v^2)v_{n-1}=vu_{n-1}(u+t_nv) \\
(u-t_{n-1}v)v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots
(u+t_{n-2}v)(u+t_nv)\quad
(j+1\le l\le n-2) \\
v^{n-j-2}v_j=u_j(u+t_{j+1}v)\cdots (u+t_{n-2}v)(u+t_nv) \\
v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots
(u+t_{j-1}v)(u+t_{j+1}v)\cdots
(u+t_{n-2}v)(u+t_nv)\quad (1\le l\le j) \\
v^{k-j}v_ju_k=v_ku_j\prod_{i=j+1}^k(u+t_iv)\quad (1\le j<k\le
n-1).
\end{gather*}
\begin{gather*}
t_j+t_k=0 \tag*{{\bf v:}}\\
X=0,\ Y=-t_1\cdots t_{j-1}t_{j+1}\cdots t_{k-1}t_{k+1}\cdots
t_n,\
Z=-t_k^2 \\
(u^2-t_k^2v^2)v_{n-1}=vu_{n-1}(u+t_nv) \\
(u^2-t_k^2v^2)v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots (u+t_nv)\quad
(k\le l\le n-2) \\
(u-t_kv)v^{n-j-1}v_{j-1}=u_{j-1}(u+t_{j+1}v)\cdots (u+t_nv) \\
(u-t_kv)v^{n-l-2}v_l=u_l(u+t_{l+1}v)\cdots
(u+t_{j-1}v)(u+t_{j+1}v)\cdots
(u+t_nv)\quad (j\le l\le k-2) \\
v^{n-j-1}v_{j-1}=u_{j-1}(u+t_{j+1})\cdots
(u+t_{k-1})(u+t_{k+1})\cdots
(u+t_n)
\end{gather*}
\begin{multline*}
v^{n-l-2}v_l=
u_l(u+t_{l+1}v)\cdots
(u+t_{j-1}v)(u+t_{j+1}v)\cdots \\
\cdots
(u+t_{k-1}v)(u+t_{k+1}v)\cdots (u+t_nv)
\quad
(1\le l\le j-2)
\end{multline*}
\begin{gather*}
v^{k-j}v_ju_k=v_ku_j\prod_{i=j+1}^k(u+t_iv)\quad (1\le j<k\le
n-1).
\end{gather*}
For each of the families described above in the four cases $w=s,t,u,v,$ the
following statement holds. The general fiber is a $\P ^ 1$,
specializing to $C_w$ as the parameters approach 0. Since
the genus is constant, this is a flat family. Since the general
fiber of this
family is irreducible, this is the only possible flat family over
the generic
point of the corresponding hyperplane in the parameter space
which is contained
in the exceptional locus.
Since ${\cal Z}$
may be obtained from ${\cal X}_{D_n}$ by successively blowing up
Weil divisors, the exceptional set lies entirely over the
discriminant
$\prod_{i<j}(t_ i^2-t_ j^2)$. Hence the locus over which
any (union of)
exceptional curves deform is contained in the locus where some
$t_ i$ equals
plus or minus some $t_ j$. In each case, this is the desired
orthogonal
complement.
\section{Simultaneous resolutions for $E_6$, $E_7$, and $E_8$.}
In this section, we complete the proof of theorem 1,
treating the cases of $E_6$, $E_7$, and $E_8$. We use another
construction of Tyurina \cite{[T]}, one which had been anticipated by
Bramble \cite{[Bra]} in the case of $E_7$ in 1918. This construction
is discussed in considerable detail by Pinkham \cite{[P-RDP]},
whose approach
we follow, and also by M\'erindol \cite{Mer}. The strategy
in this case is to first build ${\cal Z}$, and then recover ${\cal X}$.
In fact, ${\cal Z}$ is constructed as an open subset of a relative
projective model $\bar{\cal Z} \to V$.
We begin with $\P ^2$
with homogeneous coordinates $[x, y, z]$, and let $C$ be the cuspidal
cubic with equation $x^3 = y z^2$. The smooth points of this
rational curve form an open set $C_0 \subset C$ isomorphic to the
affine line. The map $\eta\colon {\Bbb A} ^1 \to \P ^2$ defined by
$\eta(U) \mathrel{:=} [U,U^3,1]$ gives such an isomorphism.
Let $V$ be the root space of $E_n$, and let the distinguished functionals
$t_1,\ldots,t_n$ serve as coordinates on $V$.
Let $\bar{\cal Z}_0 = \P ^2 \times V$,
and ${\cal C}_0 = C \times V$.
Define $n$ sections $\sigma_j\colon V \to \bar{\cal Z}_0$, $1 \le j \le n$
as follows:
\[\sigma_j(t_1,\ldots,t_n) \mathrel{:=} (\eta(t_j),(t_1,\ldots,t_n)).\]
Now for $j = 1,\ldots,n$, let $\bar{\cal Z}_j$ be the blowup of $\bar{\cal
Z}_{j-1}$
along the proper transform of the section $\sigma_j$, and let ${\cal C}_j$
be the proper transform of ${\cal C}_{j-1}$ on $\bar{\cal Z}_j$. We use
$\bar{\cal Z}$ and ${\cal C}$ to denote $\bar{\cal Z}_n$ and ${\cal C}_n$,
respectively, and let $p\colon \bar{\cal Z} \to V$ be the natural map. The
fibers
of $p$ are all smooth surfaces.
As Pinkham points out, each fiber $\bar{Z}_x \mathrel{:=} p^{-1}(x)$
of $p$ is the blowup
of $\P ^2$ in a collection of points $\eta(t_1),\ldots,\eta(t_n)$
in ``almost general position"
in the sense of Demazure \cite{[D-RDP]}. In particular,
$\omega_{\bar{Z}_x}^{-1}$
is nef. We use the notation $\omega_{\bar{\cal Z}/V}^{- k}$ to stand for
$(\omega_{\bar{\cal Z}/V}^{-1})^{\otimes k}$. Let
\[\bar{\cal P} \mathrel{:=} \text{\bf Proj} _V \left( \bigoplus_{k \ge 0}
p_*(\omega_{\bar{\cal Z}/V}^{- k}) \right)\]
be the relative anti-canonical model. The fibers of $\bar{\cal P} \to V$
are ``generalized del Pezzo surfaces", that is, del Pezzo surfaces with
rational double points allowed.
Let ${\cal Z} \mathrel{:=} \bar{\cal Z} - {\cal C}$ and
${\cal P} \mathrel{:=} \bar{\cal P} - {\cal C}$.
(We have abused notation, and denoted
the image of ${\cal C}$ in $\bar{\cal P}$ again by ${\cal C}$.)
As Pinkham shows, the Weyl group $\frak W = \frak W_{E_n}$ acts
on ${\cal P}$ by Cremona transformations (and permutations
of $\{ \sigma_1,\ldots,\sigma_n \}$). In Pinkham's version of the
construction,
the parameter space is $(C_0)^n$ rather than $V$; Pinkham computes the
induced action of $\frak W$ on $(C_0)^n$ obtaining the
formulas on p.~196 of \cite{[P-RDP]}. Since those formulas
agree with our equation (\ref{WE}) which describes
the action of $\frak W$ on $V$ when the distinguished
functionals are used as coordinates,\footnote{This is why we
chose the distinguished functionals as we did.}
our identification of $V$ with
$(C_0)^n$ makes the map ${\cal P} \to V$ into a $\frak W$-equivariant map.
We define ${\cal X} \mathrel{:=} {\cal P}/\frak W.$
If we give $\P ^2$ and $V$ the linear ${\Bbb C} ^*$-actions defined by
\begin{align*}
[x,y,z] &\mapsto [\l x,\l^3y,z] \\
(t_1,\ldots,t_n) &\mapsto (\l t_1,\ldots,\l t_n)
\end{align*}
for $\l \in {\Bbb C} ^*$,
then the entire construction becomes ${\Bbb C} ^*$-equivariant.
Pinkham shows that ${\cal X} \to V/\frak W$ is a ${\Bbb C} ^*$-semi-universal
deformation of the central fiber $X_0$, and that
${\cal Z} \to {\cal P} = {\cal X} \times_{V/\frak W} V$ is a
${\Bbb C} ^*$-equivariant simultaneous resolution.
We need to verify the properties stated in theorem 1. The third
property is the easiest this time, since it was essentially checked
by Pinkham. For each $x \in V$, the singularities of the fiber
$\bar{P}_x$ correspond to the ``effective roots" of $\bar{Z}_x$. These
can be seen from the geometry of the set of points blown up: they
correspond to 2 points being infinitely near, 3 points being
collinear, 6 points being conconic, or (in the case of $E_8$) 8 points
lying on a nodal cubic with one of the points being the node. For each
possible ``effective root", Pinkham computes the equation of the
locus in $V$ in which the root is effective, giving the results in
a table on p.~193 of \cite{[P-RDP]}. We display those equations in
table~\ref{table44}, using our identification of $(C_0)^n$ with $V$. The other
column of the table gives the root $v$ in $V$ such that the equation
is proportional to the equation $v^{\perp} = 0$.
{\renewcommand{\arraystretch}{.6}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|} \hline
& \\
Equation & Root \\
& \\ \hline
& \\
$t_i - t_j = 0$ & $e_i - e_j$ \\
& \\
$t_i + t_j + t_k = 0$ & $e_0 - e_i - e_j - e_k$ \\
& \\
$\sum_{j=1}^6 t_{i_j} = 0$ & $2 e_0 - \sum_{j=1}^6 e_{i_j}$ \\
& \\
$2 t_{i_1} + \sum_{j=2}^7 t_{i_j} = 0$ &
$3 e_0 - 2 e_{i_1} - \sum_{j=2}^7 e_{i_j}$ \\
& \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{}
\label{table44}
\end{table}
}
To finish the proof of theorem 1, we must explain how to embed
${\cal X}$ into ${\Bbb C} ^3 \times V/\frak W$. We will actually embed
${\cal P}$ into ${\Bbb C} ^3 \times V$, and then note the $\frak W$-invariance
of our construction. And that embedding in turn will be a restriction
of a projective embedding of $\bar{\cal P}$.
To describe a projective embedding of $\bar{\cal P}$, we extend
Demazure's analysis \cite{[D-RDP]}
of anti-pluricanonical mappings to the case of
families of surfaces. (A similar extension in another context has been
made by M\'erindol \cite{Mer}.) For a single generalized del Pezzo surface
$\bar{P}_x$, Demazure found the
following.\footnote{The weighted projective spaces occurring in the
description are only implicit in Demazure's paper.}
In the case of $E_6$, the anti-canonical map embeds $\bar{P}_x$ into $\P ^3$,
and the image is a cubic surface with rational double points.
In the case of $E_7$, the anti-canonical map $\bar{P}_x \to \P ^2$ is
a finite map of degree 2, and the anti-bicanonical map embeds $\bar{P}_x$
into the weighted projective space $\P ^{(1,1,1,2)}$.
$\bar{P}_x$ can be described as a double cover of $\P ^2$ ramified along
a quartic curve. In the case of $E_8$, the anti-canonical system is a
pencil with a base point, and the anti-bicanonical map is again of degree
2, this time mapping to the weighted projective space
$\P ^{(1,1,2)}$ (which can be embedded as a
quadric cone in $\P ^3$). $\bar{P}_x$ is a double cover of
$\P ^{(1,1,2)}$, branched on
a curve of
graded\footnote{We use the term {\em graded degree\/} (rather than
{\em weighted degree\/}) for a polynomial
in a weighted projective space, to avoid confusion with the weights
under the background ${\Bbb C} ^*$-action.}
degree 6; the anti-tricanonical
map embeds $\bar{P}_x$ into $\P ^{(1,1,2,3)}$.
Our immediate goal is to show that these projective
embeddings can be described {\em globally\/} over $V$
as embeddings into $\P \times V$ for the appropriate weighted
projective space $\P $.
Then we will show that
the image in the case of $E_6$, and branch loci in the cases of $E_7$ and
$E_8$, can be similarly globally defined over $V$ by a single polynomial.
We will explain how to explicitly compute these polynomials in section 9;
here we simply show that they exist.
A related approach to constructing
projective embeddings of $\bar{\cal P}$ appears in \cite{Mer}.
\begin{lemma} \label{lem55}
\quad
\begin{enumerate}
\item
The sheaf $p_*(\omega_{\bar{\cal Z}/V}^{- k})$ is locally
free of rank
$h^0(\bar{Z}_x,\omega_{\bar{Z}_x}^{- k})
= 1 + \frac{(k^2+k)(9-n)}{2}$ in case $E_n$.
\item
In cases $E_7$ and $E_8$, the map
$\operatorname{Sym} ^2(p_*\omega_{\bar{\cal Z}/V}^{- 1})\to
p_*\omega_{\bar{\cal Z}/V}^{- 2}$ is injective as a morphism of vector
bundles, and its cokernel is locally free of rank 1.
\item
In the case of $E_8$, the natural map
$p_*\omega_{\bar{\cal Z}/V}^{- 1}\otimes
p_*\omega_{\bar{\cal Z}/V}^{- 2} \to
p_*\omega_{\bar{\cal Z}/V}^{- 3}$ has a cokernel which is
locally free of rank 1.
Its kernel coincides with
$\operatorname{Ker} (p_*\omega_{\bar{\cal Z}/V}^{- 1}\otimes
\operatorname{Sym} ^2(p_*\omega_{\bar{\cal Z}/V}^{- 1}) \to
\operatorname{Sym} ^3(p_*\omega_{\bar{\cal Z}/V}^{- 1}))$,
which is locally free of rank 2.
\end{enumerate}
\end{lemma}
\begin{pf}
(1) Since
$\omega_{\bar{Z}_x}^{-1}$
is nef, $H^1(\bar{Z}_x,\omega_{\bar{Z}_x}^{- k}) = 0$ for all $k \ge 0$.
Thus, each of the sheaves $p_*(\omega_{\bar{\cal Z}/V}^{- k})$ is locally
free. The rank follows from Riemann-Roch, since $c_1^2 = 9-n$ for $E_n$.
(2) According to Demazure \cite[V, Proposition 1b]{[D-RDP]}, the anticanonical
mapping in the case
of $E_7$ is 2-1. In particular, its image is not contained in any quadric,
hence the desired injectivity statement. In the case of $E_8$, the
anticanonical map maps to $\P ^1$, hence its image is not contained in a
quadric either. The rank 1 assertion follows from the ranks listed in the
table of the lemma.
(3) By Demazure [op.\ cit.], in the case
of $E_8$ the antibicanonical mapping is 2-1; in particular, its image is not
contained in a graded cubic hypersurface (thinking of the mapping as
factoring through the weighted projective space $\P ^{(1,1,2)}$). This
gives the equality of the two kernels, where
$\hbox{Sym}^2p_*\omega_{\bar{\cal Z}/V}^{- 1}$ is identified with its
image in $p_*\omega_{\bar{\cal Z}/V}^{- 2}$ by part 2 already proven.
The second mentioned kernel is easily computed to have rank 2 (the mapping is
surjective); from which it follows that the cokernel of the first mapping has
rank 1.
\end{pf}
\begin{lemma} \label{lem56}
There exist ${\Bbb C} ^*$-invariant subspaces
\begin{align*}
&L_1 \subset H^0(V,p_*(\omega_{\bar{\cal Z}/V}^{- 1})) \text{ of dimension
$10-n$,
in case $E_n$}, \\
&L'_2 \subset H^0(V,p_*(\omega_{\bar{\cal Z}/V}^{- 2})) \text{ of dimension 1,
in cases $E_7$ and $E_8$, and} \\
&L'_3 \subset H^0(V,p_*(\omega_{\bar{\cal Z}/V}^{- 3})) \text{ of dimension 1
in case $E_8$,}
\end{align*}
such that the natural maps
\begin{align*}
L_1 \otimes {\cal O}_V &\to p_*(\omega_{\bar{\cal Z}/V}^{- 1}) \\
L'_2 \otimes {\cal O}_V &\to
\operatorname{Coker} (\operatorname{Sym} ^2(p_*\omega_{\bar{\cal Z}/V}^{- 1})\to
p_*\omega_{\bar{\cal Z}/V}^{- 2}) \\
L'_3 \otimes {\cal O}_V &\to
\operatorname{Coker} (
p_*\omega_{\bar{\cal Z}/V}^{- 1}\otimes
p_*\omega_{\bar{\cal Z}/V}^{- 2} \to
p_*\omega_{\bar{\cal Z}/V}^{- 3})
\end{align*}
are ${\Bbb C} ^*$-equivariant isomorphisms of sheaves. In other words,
the targets of these maps are {\em trivial\/} locally
free sheaves.
\end{lemma}
\begin{pf}
The key ingredient is Quillen's affirmative answer \cite{[Q]}
to Serre's
conjecture that a finitely generated projective module over a polynomial ring
must be free. Since $p_*\omega_{\bar{\cal Z}/V}^{- 1}$ is coherent and $V$
is affine,
$M=H^0(V,p_*\omega_{\bar{\cal Z}/V}^{- 1})$ is a finitely generated module
over the polynomial algebra ${\Bbb C} [V]$. Since $M$ is also locally free, it is
in fact projective. (Cf.\ \cite[Chapter II, \S 5.2, Theorem 1]{[CA]}.)
Quillen's theorem
then implies that $M$ is free. This proves the triviality of the target of the
first map. The other two cases are similar; just observe that the sheaves on
the right hand side of the
arrows are coherent and locally free of rank 1
by lemma~\ref{lem55}.
Since all maps in question are ${\Bbb C} ^*$-equivariant, we may choose
${\Bbb C} ^*$-eigensections as generators of these trivial bundles.
\end{pf}
We let $L = \bigoplus L_k$ be the graded
${\Bbb C} [V]$-algebra generated by $L_1$ in the case of
$E_6$, by $L_1$ and $L'_2$ in the case of $E_7$, and by
$L_1$, $L'_2$, and $L'_3$
in the case of $E_8$. (We have $L_2 = \operatorname{Sym} ^2L_1 \oplus L'_2$
in cases $E_7$ and $E_8$, and
$L_3 = \operatorname{Sym} ^3L_1 \oplus (L_1 \otimes L'_2) \oplus L'_3$ in case $E_8$.)
The algebra $L$ has 4 generators, and
$\text{\bf Proj} _V(L)$
is a relative weighted projective space of dimension 3 over $V$,
isomorphic to
$\P ^{(1,1,1,1)} \times V$, $\P ^{(1,1,1,2)} \times V$,
or $\P ^{(1,1,2,3)} \times V$,
respectively. Moreover, there is a natural embedding of $\bar{\cal P}$
into $\text{\bf Proj} _V(L)$, where it forms a hypersurface.
We now show that $\bar{\cal P}$ can be defined by a weighted homogeneous
polynomial globally
over $V$.
\begin{lemma} \label{lem57}
There is a weighted homogeneous polynomial $\Phi_{E_n} \in L_k$ which
generates the ideal of $\bar{\cal P}$ in $\text{\bf Proj} _V(L)$,
where $k=3,4,6$ in cases $E_6$, $E_7$, $E_8$, respectively.
In the $E_7$ case, by an appropriate choice of $L'_2$ and a generator
$X_2$ of $L_2'$,
the polynomial takes the form $X_2^2-f_4$, for some
$f_4\in\hbox{Sym}^4(L_1)\subset
L_4$.
In the $E_8$ case, by an appropriate choice of $L'_3$ and a generator
$X_3$ of $L_3'$,
the polynomial takes the form
$X_3^2-f_6$, for some $f_6\in\hbox{Sym}^6(L_1)\oplus (\hbox{Sym}^4(L_1)\otimes
L_2')\oplus
(\hbox{Sym}^2(L_1)\otimes\hbox{Sym}^2(L_2'))\oplus\hbox{Sym}^3(L_2')\subset
L_6$.
\end{lemma}
\begin{pf}
By \cite[V.3]{[D-RDP]}, it follows in the case of
$E_6$ that each
$\bar{P}_x$ is a cubic. Hence the sheaf Ker$(\hbox{Sym}^3
p_*(\omega_{\bar{\cal Z}/V}^{- 1})\to p_*(\omega_{\bar{\cal Z}/V}^
{- 3}))$ is invertible. Since it is also coherent, being the kernel of
a morphism of coherent sheaves, its triviality follows from Quillen's theorem.
This yields a cubic with coefficients in ${\Bbb C} [V]$
which serves as a global defining polynomial for $\bar{\cal P}$.
In the case of $E_7$, pick a generator $X_2$ for $L'_2$, and define a
map
\[p_*(\hbox{Sym}^2\omega_{\bar{\cal Z}/V}^{- 1}) \oplus
p_*(\hbox{Sym}^4\omega_{\bar{\cal Z}/V}^{- 1}) \to
p_*(\omega_{\bar{\cal Z}/V}^{- 4})\]
by $(a,b) \mapsto aX_2+b$.
It follows from \cite[V.4]{[D-RDP]} that
this map is surjective on each fiber; by the triviality of the bundles,
it must be surjective on global sections as well. Thus, since
$(X_2)^2 \in H^0(V,p_*(\omega_{\bar{\cal Z}/V}^{- 4}))$,
there exist global sections
$a\in H^0(V,p_*(\hbox{Sym}^2\omega_{\bar{\cal Z}/V}^{- 1})),$
$b\in H^0(V,p_*(\hbox{Sym}^4\omega_{\bar{\cal Z}/V}^{- 1}))$
such that
$(X_2)^2=aX_2+b$. Since $p_*(\omega_{\bar{\cal Z}/V}^{- 1})$ is trivial,
it follows that $H^0(V,p_*(\hbox{Sym}^k\omega_{\bar{\cal Z}/V}^{- 1}))=
\hbox{Sym}^kH^0(V,p_*(\omega_{\bar{\cal Z}/V}^{- 1}))$, hence
$a$ and $b$ can be described as linear combinations of polynomials on $V$ times
monomials in a basis for $L_1$. We thus get
$\bar{\cal P}$ as defined by the graded quartic polynomial
$\Phi_{E_7} \mathrel{:=} -(X_2)^2+aX_2+b$
with coefficients in
${\Bbb C} [V]$. To put the defining polynomial into the form claimed,
we need only complete the square as in
\cite[V.4]{[D-RDP]}. This is tantamount to making a different choice for
$L_2'$.
The case of $E_8$ is similar.
Since $\bar{\cal P}$ is
invariant under the ${\Bbb C} ^*$-action, the defining polynomial $\Phi_{E_n}$ must
be weighted homogeneous.
\end{pf}
In order to calculate explicit generators for the algebra $L$,
we interpret global sections of
$p_*(\omega_{\bar{\cal Z}/V}^{- k})$ as being the defining polynomials
of curves of degree $3k$ in
$\P ^2 \times V$ with base conditions. The base conditions
state that the curve should pass through the zero-cycle
$\eta(t_1)+\cdots+\eta(t_n)$ with multiplicity
$k$. Choosing a basis for the space of such polynomials determines
a rational map $\pi\colon \P ^2 \times V \to \P ^N$, which can be
interpreted as the blowup of $\sigma_1,\ldots,\sigma_n$ followed
by the anti-$k$-canonical map of ${\cal Z}$.
To guarantee the $\frak W$-invariance of our defining polynomial $\Phi_{E_n}$
for $\bar{\cal P}$
(and hence obtain a defining polynomial for ${\cal X}$) we must choose our
generators of $L$ quite carefully.
\begin{definition}
Let $\frak m = (t_1,\ldots,t_n) \subset {\Bbb C} [V]$ be the maximal
ideal of the origin $0 \in V$.
We say that the weighted homogeneous polynomials
$X, Y, Z, W \in {\Bbb C} [V][x,y,z]$ form a {\em good generating set\/}
for $L$ if they generate $L$ and satisfy the defining conditions given in
table~\ref{table45}.
\end{definition}
The defining conditions given in table~\ref{table45} were obtained as follows.
Each element of $L_k$ when restricted to the central fiber
$\bar{Z}_0$ becomes an
element of $H^0(\bar{Z}_0,\omega_{\bar{Z}_0}^{- k})$.
In the case of $E_n$, the polynomials of
degree $3k$ which belong to that space are exactly the ones which have a zero
of order at least $nk$ at $U=0$ when they are pulled back via
$\eta\colon C_0 \to \P ^2$,
and whose partial derivatives with respect to $x$, $y$, and $z$
up through order $k-1$ have a zero of order at least $k$ when
{\em they\/} are pulled back via $\eta$. It is easy to see that these
conditions on the partial derivatives are superfluous when applied
to a monomial, since $n \ge 6$, which is more than the largest order of
vanishing at $U=0$ among $x$, $y$, and $z$ after pulling back via $\eta$.
A set of generators for the anti-pluricanonical ring of $\bar{Z}_0$ is
easy to find using this description. We have implicitly listed one such set
in
table~\ref{table45}, as the right-hand sides of congruences for $X$, $Y$,
$Z$, and $W$.
(The last column of the table includes congruences for $X$ derived from
the defining conditions in the cases of $E_7$ and $E_8$.)
The ${\Bbb C} ^*$-action preserves the central fiber $\bar{Z}_0$, so we
chose a generating set of polynomials which are weighted homogeneous on the
central fiber.\footnote{The coefficients in the case of $E_7$ were
chosen to match the notation of Bramble \cite{[Bra]}.}
We can see the weights of the generators from the information
given in table~\ref{table45}: they are $(9, 7, 6, 3)$ for
$({X},{Y},{Z},{W})$ in the case of $E_6$,
$(15, 9, 7, 3)$ for $({X},{Y},{Z},{W})$ in the case of $E_7$,
and $(24, 16, 9, 3)$ for $({X},{Y},{Z},{W})$
in the case of $E_8$.
Moreover, the last column of the table shows the leading order terms
of the polynomial $\Phi_{E_n}$ (determined by elimination from the defining
conditions), and
determines its weight as 21, 30, or 48, respectively.
{\renewcommand{\arraystretch}{.6}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|rcl|c|} \hline
& & & & \\
\multicolumn{1}{|c|}{} & \multicolumn{3}{c|}{Defining Conditions} &
\multicolumn{1}{c|}{Other Properties} \\
& & & & \\ \hline
& & & & \\
& ${W}$ & $=$ & $x^3 - y z^2$ & \\
& & & & \\
& ${Z}$ & $\equiv$ & $y^2 z\mod \frak m $ & \\
$E_6$ & & & &
${Y}^3 - {X} {Z}^2 - {X}^2 {W} \equiv 0\mod \frak m $ \\
& ${Y}$ & $\equiv$ & $x y^2\mod \frak m $ & \\
& & & & \\
& ${X}$ & $\equiv$ & $y^3\mod \frak m $ & \\
& & & & \\ \hline
& & & & \\
& ${W}$ & $=$ & $x^3 - y z^2$ & \\
& & & & \\
& ${Z}$ & $\equiv$ & $x y^2\mod \frak m $ &
${X} \equiv 8 y^5 z\mod \frak m $ \\
$E_7$ & & & & \\
& ${Y}$ & $\equiv$ & $4 y^3\mod \frak m $ &
$16 {Y} {Z}^3 - {X}^2 - {Y}^3 {W} \equiv 0\mod \frak m $ \\
& & & & \\
& ${X}$ & $=$ & $\frac13 \
\frac{\partial({Y},{Z},{W})}{\partial(x,y,z)}$ & \\
& & & & \\ \hline
& & & & \\
& ${W}$ & $=$ & $x^3 - y z^2$ & \\
& & & & \\
& ${Z}$ & $\equiv$ & $y^3\mod \frak m $ &
${X} \equiv y^8 z\mod \frak m $ \\
$E_8$ & & & & \\
& ${Y}$ & $\equiv$ & $x y^5\mod \frak m $ &
${Y}^3 - {X}^2 - {Z}^5 {W} \equiv 0\mod \frak m $ \\
& & & & \\
& ${X}$ & $=$ & $- \frac16 \
\frac{\partial({Y},{Z},{W})}{\partial(x,y,z)}$ & \\
& & & & \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{}
\label{table45}
\end{table}
}
\begin{proposition} \label{prop51}
There exists a good generating set $X, Y, Z, W$ for $L$ such that when the
defining polynomial $\Phi_{E_n} \in {\Bbb C} [V][X,Y,Z,W]$
of $\bar{\cal P} \subset \text{\bf Proj} _V(L)$ is restricted to the affine
chart $W=1$,
it gives
a ${\Bbb C} ^*$-semi-universal deformation of the rational double point
which is in preferred versal form.
\end{proposition}
\begin{pf}
We first claim that there exist good generating sets for
$L$. We can start with the polynomial $x^3-yz^2$ as one of the generators,
since it belongs to
$L$ in all three cases. On the central fiber $\bar{Z}_0$, the generator
$x^3-yz^2$ can be
extended to a generating set for the entire algebra
$\bigoplus H^0(\bar{Z}_0,\omega_{\bar{Z}_0}^{- k})$
as indicated in table~\ref{table45}: the additional generators are
$(y^2z, xy^2, y^3)$ in
case $E_6$, $(xy^2, 4y^3, 8y^5z)$ in case $E_7$, and
$(y^3,xy^5,y^8z)$ in case $E_8$.
Since each of the bundles involved is trivial, these generators on the
central fiber can be lifted
to generators of $L$ itself. The only thing left to show is that in the
cases of $E_7$ and $E_8$, we may use the Jacobian determinant as
the generator of top degree. To see this, we only need to note that
this Jacobian determinant does indeed belong to the algebra (since
it satisfies the base conditions), and its restriction mod $\frak m $
is $8y^5z$, resp.\ $y^8z$, as required.
Now let $\bar{X}, \bar{Y}, \bar{Z}, \bar{W}$ be a good generating set for
$L$, and let $\bar{\Phi}_{E_n}$ be the defining polynomial of $\bar{\cal P}$
with respect to this generating set. In cases $E_7$ and $E_8$, the
map determined by $\bar{Y}, \bar{Z}, \bar{W}$ expresses $\bar{\cal P}$
as a double cover of a weighted projective space.
Since $\bar{X}$ is the Jacobian determinant of
this map (up to a constant), it vanishes exactly on the ramification
locus of this double cover. It follows that $\bar{\Phi}_{E_n}$ takes the
form $-\bar{X}^2 + \widetilde{\Phi}_{E_n}(\bar{Y},\bar{Z},\bar{W})$ in these
two cases.
In all three cases, we know the weight $w$ of $\bar{\Phi}_{E_n}$,
and also its graded degree $d$ in the algebra $L$. Any monomial
$m$ in $\bar{X}, \bar{Y}, \bar{Z}, \bar{W}$ which appears in $\bar{\Phi}_{E_n}$
must
have graded degree $d$, and weight $w_m \le w$: the weight of the
coefficient of the monomial will then be $w-w_m$. Using
table~\ref{table45} to determine the leading order terms,
we can then write $\bar{\Phi}_{E_n}$ as follows, with undetermined
coefficients.\footnote{The two strange terms in the expression for
$\bar{\Phi}_{E_7}$ are yet another artifact of making our notation
match that of Bramble \cite{[Bra]}.}
(We adopt the convention that equation numbers which are followed by
$a$, $b$, or $c$ refer to the cases of $E_6$, $E_7$, or $E_8$, respectively.)
The notation is chosen so that the subscript on a coefficient shows
its weight.
\refstepcounter{equation}\label{eq11}
\begin{align*}
\begin{split}
\bar{\Phi}_{E_6} &=
- \bar{X}^2 \bar{W}
- \bar{X} \bar{Z}^2
+ \bar{Y}^3
+ \bar{\varphi}_1 \bar{Y}^2 \bar{Z}
+ \bar{\varphi}_2 \bar{X} \bar{Y} \bar{W}
+ \bar{\varepsilon}_2 \bar{Y} \bar{Z}^2
\\ &\qquad
+ \bar{\varphi}'_3 \bar{X} \bar{Z} \bar{W}
+ \bar{\varphi}''_3 \bar{Z}^3
+ \bar{\varphi}_4 \bar{Y}^2 \bar{W}
+ \bar{\varepsilon}_5 \bar{Y} \bar{Z} \bar{W}
+ \bar{\varphi}_6 \bar{X} \bar{W}^2
\\ &\qquad
+ \bar{\varepsilon}_6 \bar{Z}^2 \bar{W}
+ \bar{\varepsilon}_8 \bar{Y} \bar{W}^2
+ \bar{\varepsilon}_9 \bar{Z} \bar{W}^2
+ \bar{\varepsilon}_{12} \bar{W}^3
\end{split}\displaybreak[0]\tag{\ref{eq11}a}\\[1.5ex]
\begin{split}
\bar{\Phi}_{E_7} &=
- \bar{X}^2 - \bar{Y}^3 \bar{W}
+ 16 \bar{Y} \bar{Z}^3
+ \bar{\varepsilon}_2 \bar{Y}^2 \bar{Z} \bar{W}
+ \bar{\varphi}_2 (16 \bar{Z}^4 -
\makebox[0pt][l]{$
\bar{Y}^2 \bar{Z} \bar{W})
$}
\\ &\qquad
+ \bar{\varphi}_4 \bar{Y} \bar{Z}^2 \bar{W}
+ \bar{\varepsilon}_6 \bar{Y}^2 \bar{W}^2
+ \bar{\varphi}_6 (16 \bar{Z}^3 \bar{W} - \bar{Y}^2 \bar{W}^2)
\\ &\qquad
+ \bar{\varepsilon}_8 \bar{Y} \bar{Z} \bar{W}^2
+ \bar{\varepsilon}_{10} \bar{Z}^2 \bar{W}^2
+ \bar{\varepsilon}_{12} \bar{Y} \bar{W}^3
+ \bar{\varepsilon}_{14} \bar{Z} \bar{W}^3
+ \bar{\varepsilon}_{18} \bar{W}^4
\end{split}\displaybreak[0]\tag{\ref{eq11}b}\\[1.5ex]
\begin{split}
\bar{\Phi}_{E_8} &=
- \bar{X}^2 + \bar{Y}^3
- \bar{Z}^5 \bar{W}
+ \bar{\varepsilon}_2 \bar{Y} \bar{Z}^3 \bar{W}
+ \bar{\varphi}_4 \bar{Y}^2 \bar{Z} \bar{W}
+ \bar{\varphi}_6 \bar{Z}^4 \bar{W}^2
\\ &\qquad
+ \bar{\varepsilon}_8 \bar{Y} \bar{Z}^2 \bar{W}^2
+ \bar{\varphi}_{10} \bar{Y}^2 \bar{W}^2
+ \bar{\varepsilon}_{12} \bar{Z}^3 \bar{W}^3
+ \bar{\varepsilon}_{14} \bar{Y} \bar{Z} \bar{W}^3
\\ &\qquad
+ \bar{\varepsilon}_{18} \bar{Z}^2 \bar{W}^4
+ \bar{\varepsilon}_{20} \bar{Y} \bar{W}^4
+ \bar{\varepsilon}_{24} \bar{Z} \bar{W}^5
+ \bar{\varepsilon}_{30} \bar{W}^6
\end{split}\tag{\ref{eq11}c}
\end{align*}
Notice that this polynomial restricts to one in preferred versal form
in the affine
$W=1$ exactly when all of the $\bar{\varphi}_i$
terms vanish. We therefore wish to make a change of generating set
to ensure that this occurs.
Suppose that $X, Y, Z, W$ is another good generating set of
$L$. Since the mod $\frak m $ restriction of a good generating set is
determined by table~\ref{table45},
the change of generators must restrict to the identity mod $\frak m$.
Moreover (as follows from properties of the Jacobian determinant),
in cases $E_7$ and $E_8$ we have $\bar{X}=X$. Considering as
before the graded degrees of elements of the algebra $L$,
and the fact that each term in the equation expressing the
change of generators
must have a coefficient with nonnegative weight, we find that
the change of generators must take the form shown below, with
undetermined coefficients $\psi_i$.
\refstepcounter{equation}\label{eq23}
\begin{align*}
\begin{split}
\begin{tabular}{ccrcrcrcr}
$\bar{X}$ & $=$ & $X$ & + & $\psi_2 Y$ & + & $\psi'_3 Z$ & + & $\psi_6 W$ \\
$\bar{Y}$ & $=$ & & & $Y$ & + & $\psi_1 Z$ & + & $\psi_4 W$ \\
$\bar{Z}$ & $=$ & & & & & $Z$ & + & $\psi''_3 W$ \\
$\bar{W}$ & $=$ & & & & & & & $W$ \\
\end{tabular}
\end{split}\displaybreak[0]\tag{\ref{eq23}a}\\[1.5ex]
\begin{split}
\begin{tabular}{ccrcrcrcr}
$\bar{X}$ & $=$ & $X$ & & & & & & \\
$\bar{Y}$ & $=$ & & & $Y$ & + & $\psi_2 Z$ & + & $\psi_6 W$ \\
$\bar{Z}$ & $=$ & & & & & $Z$ & + & $\psi_4 W$ \\
$\bar{W}$ & $=$ & & & & & & & $W$ \\
\end{tabular}
\end{split}\displaybreak[0]\tag{\ref{eq23}b}\\[1.5ex]
\begin{split}
\begin{tabular}{ccrcrcrcr}
$\bar{X}$ & $=$ & $X$ & & & & & & \\
$\bar{Y}$ & $=$ & & & $Y$ & + & $\psi_4 Z W$ & + & $\psi_{10} W^2$ \\
$\bar{Z}$ & $=$ & & & & & $Z$ & + & $\psi_6 W$ \\
$\bar{W}$ & $=$ & & & & & & & $W$ \\
\end{tabular}
\end{split}\tag{\ref{eq23}c}
\end{align*}
To finish the proof, we substitute
equation (\ref{eq23})
into equation (\ref{eq11})
and collect the coefficients of the monomials
$\{Y^2 Z$, $XYW$, $ Z^3$, $XZW$, $Y^2 W$, $X W^2\}$ in case $E_6$,
$\{Z^4$, $Y Z^2 W$, $Z^3 W\}$ in case $E_7$, and
$\{Y^2 Z W$, $Z^4 W^2$, $Y^2 W^2\}$ in case $E_8$.
(These are the monomials which we cannot allow if we wish to achieve
preferred versal form.) Equating all such coefficients to zero gives
the following set of equations.
\refstepcounter{equation}\label{eq51}
\begin{align*}
\begin{split}
\begin{aligned}
0 &= { { 3 {\psi_{1}}}+ {\bar{\varphi}_{1}}} \\
0 &= {{- 2 {\psi_{2}}} +{\bar{\varphi}_{2}} } \\
0 &= { {- {\psi'_{3}}} + {\bar{\varphi}''_{3}} + { {\bar{\varepsilon}_{2}} {\psi_{1}}}
+ { {\psi_{1}}^{3}} + { {\bar{\varphi}_{1}} { {\psi_{1}}^{2}}}} \\
0 &= { {- 2 {\psi'_{3}}} {- 2 {\psi''_{3}}} +
{\bar{\varphi}'_{3}} + { {\bar{\varphi}_{2}} {\psi_{1}}} } \\
0 &= { 3 {\psi_{4}}}+ { { {\bar{\varphi}_{1}} {\psi''_{3}}} { - { {\psi_{2}}^{2}}}
+ { {\bar{\varphi}_{2}}
{\psi_{2}}}+ {\bar{\varphi}_{4}}} \\
0 &= {{- 2 {\psi_{6}}} + { {\bar{\varphi}_{2}} {\psi_{4}}}
+ { {\bar{\varphi}'_{3}} {\psi''_{3
}}} { - { {\psi''_{3}}^{2}}}+ {\bar{\varphi}_{6}}}
\end{aligned}
\end{split}\displaybreak[0]\tag{\ref{eq51}a}\\[3ex]
\begin{split}
\begin{aligned}
0 &= { { 16 {\psi_{2}}}+ { 16 {\bar{\varphi}_{2}}}} \\
0 &= { { 48 {\psi_{4}}} { - 3 { {\psi_{2}}^{2}}}
+ {\bar{\varphi}_{4}} + { 2 {\bar{\varepsilon}
_{2}} {\psi_{2}}} { - 2 {\bar{\varphi}_{2}} {\psi_{2}}}} \\
0 &= { { 16 {\psi_{6}}} + { 16 {\bar{\varphi}_{6}}} { - { {\psi_{2}}^{3}}}
+ { 48 {
\psi_{2}} {\psi_{4}}} + { {\bar{\varepsilon}_{2}} { {\psi_{2}}^{2}}} }\\
&\qquad {+ { 64 {\bar{\varphi}
_{2}} {\psi_{4}}} { - {\bar{\varphi}_{2}} { {\psi_{2}}^{2}}}
+ { {\bar{\varphi}_{4}} {
\psi_{2}}}}
\end{aligned}
\end{split}\displaybreak[0]\tag{\ref{eq51}b}\\[3ex]
\begin{split}
\begin{aligned}
0 &= { { 3 {\psi_{4}}}+ {\bar{\varphi}_{4}}} \\
0 &= { {- 5 {\psi_{6}}} + {\bar{\varphi}_{6}}
+ { {\bar{\varepsilon}_{2}} {\psi_{4}}}} \\
0 &= { { 3 {\psi_{10}}}+ {\bar{\varphi}_{10}}
+ { {\bar{\varphi}_{4}} {\psi_{6}}}}
\end{aligned}
\end{split}\tag{\ref{eq51}c}
\end{align*}
These equations are in a kind of triangular form, and it is clear
they can be solved for the unknown coefficients
$\psi_i$ in terms of the coefficients $\bar{\varepsilon}_i$ and $\bar{\varphi}_i$.
Making the corresponding change of generator produces the desired
generators for $L$.
\end{pf}
\begin{remark}
For the purposes of practical computation of these coefficients $\psi_i$
(which we carry out in section 9) it
is important to observe that these equations only depend
on a subset of the $\bar{\varepsilon}_i$ and $\bar{\varphi}_i$. Explicitly, in case
$E_6$, they depend on
$\{ \bar{\varphi}_1, \bar{\varphi}_2, \bar{\varepsilon}_2, \bar{\varphi}'_3, \bar{\varphi}''_3,
\bar{\varphi}_4, \bar{\varphi}_6 \}$;
in case $E_7$, on
$\{ \bar{\varepsilon}_2, \bar{\varphi}_2, \bar{\varphi}_4, \bar{\varphi}_6 \}$;
and in case $E_8$, on
$\{ \bar{\varepsilon}_2, \bar{\varphi}_4, \bar{\varphi}_6, \bar{\varphi}_{10} \}$.
\end{remark}
To complete the proof of theorem 1, we first observe that the equation
$W=0$ defines ${\cal C} \subset \bar{\cal P}$ in the generators we
are considering. What we still need to check is that the defining
polynomial $\Phi_{E_n}$ which we have found is invariant under the Weyl
group, and so can be used to define ${\cal X}$ as well as $\bar{\cal P}$.
This essentially follows from an argument of
Pinkham \cite[pp.\ 196-198]{[P-RDP]}, in the following way.
Let $\widetilde{\cal X} \to \operatorname{Def} (X_0)$ be the
${\Bbb C} ^*$-semi-universal deformation
given by the hypersurface in preferred versal form.
(The defining polynomial of that hypersurface is the polynomial
$\Phi_{E_n}$, with $W$ set equal to 1.)
Since ${\cal P}$ is a deformation of $X_0$ with ${\Bbb C} ^*$-action,
there is a natural ${\Bbb C} ^*$-equivariant map
$\rho\colon V \to \operatorname{Def} (X_0)$ such that
${\cal P} \cong \widetilde{\cal X} \times_{\rho} V$.
(The effect of proposition~\ref{prop51}
is to compute this map explicitly.) The isomorphism
${\cal P} \cong \widetilde{\cal X} \times_{\rho} V$
is in fact $\frak W$-equivariant, where $\frak W$ acts by Cremona transformations
on the left, and as the Weyl group on the right. Thus, $\rho$ factors
through a map $V/\frak W \to \operatorname{Def} (X_0)$. But the weights of the
${\Bbb C} ^*$-actions are the same for these two spaces, so the map
is an isomorphism. It follows that ${\cal P}/\frak W$ gives a semi-universal
deformation ${\cal X}$, and that $\Phi_{E_n}$ is $\frak W$-invariant.
This argument also implies that the $\frak W$-invariant functions $\varepsilon_i$
serve as generators of the ring ${\Bbb C} [V]^{\frak W}$ of Weyl group
invariants. Those generators (multiplied by appropriate integers)
are shown explicitly in cases $E_6$ and
$E_7$ in Appendices 1 and 2, respectively.
\section{Singularities of the simultaneous partial resolutions.}
In this section, we show that
the semi-universal deformations constructed in theorem 1 are essentially
unique (up to the
${\Bbb C} ^*$-action). We then use this uniqueness result to study
the singularities of simultaneous partial resolutions.
\begin{lemma} \label{lem61}
Let $V$ be the root space of an irreducible root system $R$. Give $V$
the linear ${\Bbb C} ^*$-action $x \mapsto \l \cdot x$ for all $x \in V$,
$\l \in {\Bbb C} ^*$.
Let $\gamma\colon V \to V$
be a ${\Bbb C} ^*$-equivariant map such that $\gamma (v^{\perp}) \subset
v^{\perp}$
for all $v \in R$. Then $\gamma = \lambda \cdot 1_V$ for some constant
$\lambda \in {\Bbb C} $.
\end{lemma}
\begin{pf}
We first note that $\gamma$ must be a linear map. For if we expand
$\gamma$ in a Taylor series at $0$ and compare coefficients in the
equation $\gamma(\l x) = \l \gamma (x)$, we see that all higher
order terms must vanish.
Let $\{ v_i \}$ be a basis of $V$ consisting of roots $v_i \in R$.
Define vectors $ v_j^{\vee} \in V$ by the property
$\ip{v_j^{\vee}}{v_i} = \delta_{ij}$. (The vectors $ v_j^{\vee}$
correspond to the elements of the dual basis $v_j^*$ of $V^*$
under the isomorphism $V \cong V^*$ determined by the inner product.)
Now ${\Bbb C} \cdot v_j^{\vee} = \bigcap_{i \ne j} v_i^{\perp}$. Thus
$v_j^{\vee} \in \bigcap_{i \ne j} v_i^{\perp}$, which implies that
$\gamma(v_j^{\vee}) \in \bigcap_{i \ne j} v_i^{\perp}$, and hence
$\gamma(v_j^{\vee}) = \lambda_j v_j^{\vee}$ for some constant $\lambda_j$.
If the vertex $v_j$ is connected to the vertex $v_k$ in the Dynkin
diagram, then $v_j + v_k$ must be a root as well. Let
$w_{jk} = v_j^{\vee} - v_k^{\vee}$. Then $w_{jk}$ lies in the space
$(v_j + v_k)^{\perp} \cap \bigcap_{i \ne j, k} v_i^{\perp}$,
and in fact it spans that space. Thus, since that space is preserved
by $\gamma$, we must have $\gamma(w_{jk}) = \lambda_{jk} w_{jk}$ for
some constant $\lambda_{jk}$. But then
\[\lambda_{jk} (v_j^{\vee} - v_k^{\vee}) = \gamma ( w_{jk})
= \gamma(v_j^{\vee}) - \gamma(v_k^{\vee})
= \lambda_j v_j^{\vee} - \lambda_k v_k^{\vee}.\]
It follows that $\lambda_j = \lambda_k = \lambda_{jk}$.
Thus, since $R$ has a connected Dynkin diagram, all the $\lambda_j$
must be equal to the same constant $\lambda$.
\end{pf}
Let $Z_0 \to X_0$ be the minimal
resolution of a rational double point of type $S$, and let ${\cal X} \to V/\frak W$
and
${\cal Z} \to {\cal X} \times_{V/\frak W} V$ be the deformation and
simultaneous resolution constructed in theorem 1 (which we fix
once and for all).
We call ${\cal X} \to V/\frak W$ the {\em standard deformation\/}
and ${\cal Z} \to {\cal X} \times_{V/\frak W} V$ the {\em standard
simultaneous resolution\/} of type $S$.
The coefficients
of the defining polynomial
$\Phi_S$ of ${\cal X}$ are specific functions
on the deformation space:
we call them the {\em standard coordinate functions\/} on $\operatorname{Def} (S)=V/\frak W$.
We call the standard coordinate function
of highest weight the {\em ``constant term"},
since it occurs as the constant term in the defining polynomial
of the hypersurface.
Notice that all of these definitions depend on identifying the type
as $S$.
\begin{theorem}
Let ${\cal X'} \to {\cal D}$ be a nontrivial
${\Bbb C} ^*$-equivariant deformation of $X_0$, and let
${\cal R}$
be a vector space with a linear ${\Bbb C} ^*$-action of pure weight one.
Suppose that
${\cal R} \to {\cal D}$ is a ${\Bbb C} ^*$-equivariant map, and
that there is a simultaneous resolution
${\cal Z'} \to {\cal X'} \times_{{\cal D}} {\cal R}$
inducing $Z_0 \to X_0$.
For each root $v \in R$, let ${\cal H}_v \subset {\cal R}$
be the locus in ${\cal R}$ parametrizing deformations of $Z_0$
to which $C_v$
lifts.
Suppose that $\alpha\colon {\cal R} \to V$ is a ${\Bbb C} ^*$-equivariant
surjective map such that $\alpha({\cal H}_v) \subset v^{\perp}$
for all $v \in R$. Then $\alpha$ descends to a map
$\bar{\alpha}\colon {\cal D} \to V/\frak W$, and
${\cal X'}$ is isomorphic to ${\cal X} \times_{\bar{\alpha}} {\cal D}$,
the pullback of the standard deformation via $\bar{\alpha}$.
\end{theorem}
\begin{pf}
Let ${\cal X} \to V/\frak W$ be the standard deformation, which is
${\Bbb C} ^*$-semi-universal by construction.
By a theorem of Pinkham \cite{C*}, since
${\cal X'} \to {\cal D}$ is a
${\Bbb C} ^*$-equivariant deformation of $X_0$,
there is a ${\Bbb C} ^*$-equivariant
map $\bar{\beta}\colon {\cal D} \to V/\frak W$ such that
${\cal X'}$ is isomorphic to the pullback
${\cal X} \times_{\bar{\beta}} {\cal D}$.
Since ${\cal X'}$ admits a simultaneous resolution after base
change to
${\cal R}$, there is a map $\beta\colon {\cal R} \to V$ which induces
$\bar{\beta}$.
Since the locus in $V$ parametrizing deformations of $Z_0$ to which
$C_v$ lifts is $v^{\perp}$ by construction, the functoriality of
that property of deformations implies that
$\beta({\cal H}_v) \subset v^{\perp}$
for all $v \in R$.
It will suffice to show that $\beta = \lambda \cdot \alpha$ for some
$\lambda \in {\Bbb C} $. For if that is the case, then $\lambda$ cannot
be zero since ${\cal X'} \to {\cal D}$ is a nontrivial deformation.
The desired map $\bar{\alpha}$ will then
simply be $\lambda^{-1} \cdot \bar{\beta}$,
and by using the action of $\lambda \in {\Bbb C} ^*$ on ${\cal X}$,
${\cal X} \times_{\bar{\alpha}} {\cal D}$ will be isomorphic to
${\cal X} \times_{\bar{\beta}} {\cal D}$, and therefore to ${\cal X'}$.
To show that $\beta = \lambda \cdot \alpha$, let $\{v_1,\ldots,v_r\}$
be a root basis of $V$, and define
\[{\cal K} = \bigcap_{i=1}^r {\cal H}_{v_i}.\]
Since $\bigcap_{i=1}^r {v_i}^{\perp} = (0)$, we
have ${\cal K} \subset \operatorname{Ker} \alpha$ and
${\cal K} \subset \operatorname{Ker} \beta$. There are thus
induced maps $\widetilde{\alpha}\colon {\cal R}/{\cal K} \to V$ and
$\widetilde{\beta}\colon {\cal R}/{\cal K} \to V$, and it will suffice to show
that $\widetilde{\beta} = \lambda \cdot \widetilde{\alpha}$.
Since $\alpha$ is
surjective, if we define $m \mathrel{:=} \dim {\cal R}$ then we have
\[m - r \le \dim {\cal K} \le \dim \operatorname{Ker} \alpha = m - r,\]
which implies that $\operatorname{Ker} \alpha = {\cal K}$. Thus,
$\widetilde{\alpha}$ is an isomorphism. Let
$\gamma \mathrel{:=} \widetilde{\beta} \circ \widetilde{\alpha}^{-1} \in \operatorname{Aut} V$.
$\gamma$ is clearly ${\Bbb C} ^*$-equivariant. And for
each $v \in R$ we have
\[\gamma(v^{\perp}) = \widetilde{\beta}({\cal H}_v \bmod{\cal K})
\subset v^{\perp}.\]
Therefore, $\gamma$ satisfies the hypotheses of the lemma, so
$\gamma = \lambda \cdot 1_V$ for some $\lambda$. It follows that
$\widetilde{\beta} = \lambda \cdot \widetilde{\alpha}$, as required.
\end{pf}
\begin{corollary}
If ${\cal X'} \to V/\frak W$ is any ${\Bbb C} ^*$-semi-universal deformation
parametrized by $V/\frak W$
such that the spaces ${\cal H}_v$ coincide with the
linear subspaces $v^{\perp} \subset V$, then ${\cal X'}$ is isomorphic
to the standard deformation ${\cal X}$. In particular, ${\cal X'}$
can be embedded
as a hypersurface in ${\Bbb C} ^3 \times V/\frak W$ with
defining polynomial given in table~\ref{table2B},
where the coefficients in the defining polynomial
are the standard coordinate functions on $V/\frak W$.
\end{corollary}
We define {\em standard coordinate functions\/} on $V/\frak W$ in two other
cases. The formulas for the standard coordinate functions on
$V_{D_n}/\frak W_{D_n}$
given in equation (\ref{eqD})
make sense even
when $n=2$, and generate ${\Bbb C} [V_{D_2}]^{\frak W_{D_2}}$; we call them
the {\em standard coordinate functions\/} on $V_{D_2}/\frak W_{D_2}$.
And in the case of $E_3$, we define the {\em standard coordinate functions\/}
on $V_{E_3}/\frak W_{E_3}$
to be the generators $\varepsilon_2^{(1)}$, $\varepsilon_2^{(2)}$, and $\varepsilon_3$ of
${\Bbb C} [V_{E_3}]^{\frak W_{E_3}}$ which are given in equation~(\ref{eqE3inv}).
In these cases, the functions do not directly have an interpretation
as coefficients of a semi-universal deformation. We relate the
standard coordinate functions on $V_{D_2}/\frak W_{D_2}$ to the deformation
theory in the next lemma.
A similar calculation could be done for $E_3$, but we omit it since
we do not need the result.
\begin{lemma} \label{lem72}
Let $f_{D_2}(U;t)$ be the distinguished polynomial
for $D_2$, let $\c_2 = f_{D_2}(0;t)$,
and let $g_{D_2}(Z;t) = Z^2 + \d_2 Z + (\c_2)^2$ be the
second distinguished polynomial,
so that $\c_2$ and $\d_2$ are the standard coordinate functions on
$V_{D_2}/\frak W_{D_2}$.
Then the ``constant terms" for
the two root systems of type $A_1$ which are the irreducible constituents
of $R_{D_2}$
are $-\frac14(\d_2-2\c_2)$ and $-\frac14(\d_2+2\c_2)$
respectively.
\end{lemma}
\begin{pf}
The irreducible constituents of $R_{D_2}$
are spanned by the root vectors $v_1$
and $v_2$, respectively.
Let $t'_1, t'_2$ be the distinguished functionals for the $A_1$ spanned
by $v_1$, and $t''_1, t''_2$ be those for $v_2$.
The ``constant terms" are then $t'_1 t'_2$ and $t''_1 t''_2$, respectively.
Moreover, the definition of distinguished functionals implies
\begin{alignat*}{2}
t'_1 &= v_1^*, & \qquad t'_2 &= -v_1^* \\
t''_1 &= v_2^*, & \qquad t''_2 &= -v_2^*.
\end{alignat*}
Now according to table~\ref{table12},
\begin{gather*}
v_1^* =\frac12s_1 - t_2 = \frac12(t_1-t_2)\\
v_2^* =\frac12s_1 = \frac12(t_1+t_2),
\end{gather*}
where $t_1$ and $t_2$ are the distinguished functionals for $D_2$.
Thus,
\begin{gather*}
t'_1 t'_2 = -(v_1^*)^2 = -\frac14(t_1-t_2)^2 = -\frac14(s_1^2-4s_2)\\
t''_1 t''_2 = -(v_2^*)^2 = -\frac14(t_1+t_2)^2 = -\frac14s_1^2.
\end{gather*}
If we
write $f_{D_2}(U;t) = U^2 + s_1 U + s_2$, then
$\c_2 = s_2$. Furthermore, $g_{D_2}(-U^2;t) = U^4 + (2s_2-s_1^2)U^2 + s_2^2,$
which implies that $\d_2=s_1^2-2s_2$.
It follows that
$t'_1 t'_2 = - \frac14(\d_2-2\c_2)$ and
$t''_1 t''_2 = - \frac14(\d_2+2\c_2)$.
\end{pf}
We now turn to the study of simultaneous partial resolutions.
Let $Y_0 \to X_0$ be a partial resolution of a singularity of type $S$,
and let $Z_0 \to Y_0$
be the minimal resolution. The ${\Bbb C} ^*$-action on $X_0$ lifts
to a ${\Bbb C} ^*$-action on $Y_0$.
As in section 1, there is an associated
partial resolution graph $\Gamma_0 \subset \Gamma$. We write
$\Gamma - \Gamma_0 = \bigcup \Gamma_i$ as a union of its connected
components, and let $\frak W_0 = \prod \frak W_i$ be the subgroup of $\frak W$ generated
by reflections corresponding to vertices of $\Gamma - \Gamma_0$.
(Such vertices are illustrated with closed circles ($\bullet $)
in figures~\ref{figure1} and \ref{figure2}.)
The components $\Gamma_i$ correspond to the singular
points $Q_i$ of $Y_0$.
Let ${\cal Z} \to {\cal X}$ be the standard simultaneous resolution of
type $S$. By the techniques of
\cite{[Wahl]}, deformations of $Z_0$ can be partially
blown down to deformations of $Y_0$, and ${\Bbb C} ^*$-actions are preserved
when this is done. Doing this universally gives a family
$\widehat{\cal Z} \to V$. Moreover, as Pinkham argues in \cite{[P]}, the map
$\widehat{\cal Z} \to V$ is $\frak W_i$-equivariant for each $i$, since
it provides a model for simultaneous resolution of $Q_i$.
(The argument uses a result of Burns and Wahl \cite{[BW]}, as
refined by Pinkham \cite{[P-Leop]}.)
Thus, if we let ${\cal Y} = \widehat{\cal Z}/\frak W_0$, then
${\cal Y} \to V/\frak W_0$ is a
${\Bbb C} ^*$-semi-universal deformation of $Y_0$.
We call this the {\em standard simultaneous partial resolution\/} of type
$(S,\Gamma_0)$.
Let $Y_0^{(i)}$ resp.\ ${\cal Y}^{(i)}$ be the union of all ${\Bbb C} ^*$-orbits
in $Y_0$ resp.\ ${\cal Y}$ whose closure contains $Q_i$. Then
$Y_0^{(i)}$ is a ${\Bbb C} ^*$-neighborhood of $Q_i \in Y_0$ and
${\cal Y}^{(i)} \to V/\frak W_0$ is a deformation of $Y_0^{(i)}$.
Now $Q_i \in Y_0^{(i)}$ is itself a rational double point, whose associated
root system $R_i$ is isomorphic to the subsystem of $R$ spanned by the
roots from $\Gamma_i$. If we specify the type of this rational double point
as one of $A_{n-1}$, $D_n$, or $E_n$, and identify the complex root space
of the root system $R_i$ with the subspace $V_i \subset V$ spanned by
the roots from $\Gamma_i$, then there is an associated standard deformation
${\cal X}_i \to V_i/\frak W_i$ of type $R_i$.
\begin{theorem}
Let $\operatorname{pr} _i\colon V/\frak W_0 \to V_i/\frak W_i$ be the map induced by the orthogonal
projection $V \to V_i$. Then ${\cal Y}^{(i)}$ is isomorphic to
${\cal X}_{i} \times_{\operatorname{pr} _i} V/\frak W_0$. In other words,
there is a neighborhood of $Q_i \in {\cal Y}$ which can be embedded
in ${\Bbb C} ^3 \times V/\frak W_0$ in such a way that the coefficients of
the defining polynomial are pullbacks via $\operatorname{pr} _i$ of the
standard coordinate functions on $V_i/\frak W_i$.
\end{theorem}
\begin{pf}
Let ${\cal Z}\to {\cal X}\times_\rho V$ be the standard
simultaneous resolution. Denoting the quotient map $V\to V/\frak W_0$ by
$\sigma$, there is a simultaneous resolution ${\cal Z}^{(i)}\to {\cal
Y}^{(i)}\times_{\sigma} V$, where ${\cal Z}^{(i)}$ is the union of all ${\bf
C}^*$-orbits in ${\cal Z}$ whose closure intersects some exceptional curve
lying
over $Q_i$. For each root $v\in
R_i$, the locus in $V$ parametrizing deformations of $Z_0$ to which $C_v$
lifts is precisely the orthogonal complement of $v$ in $V$. This space is
clearly mapped into the orthogonal complement of $v$ in $V_i$ by the orthogonal
projection $V\to V_i$. The theorem follows by applying theorem 2 to this
situation, using the root system $R_i$ in place of $R$, and the orthogonal
projection in place of $\alpha$.
\end{pf}
Consider now the case of an irreducible small resolution, so
that $\Gamma_0$ consists of a single vertex $v$. If
we write $\Gamma - \{v\} = \bigcup \Gamma_i$
as a union of its irreducible components, then
theorem 3 provides us with a natural isomorphism
$\operatorname{PRes} (S,v) \cong V/\frak W_0$, compatible with the
maps
$\operatorname{pr} _i\colon V/\frak W_0 \to V_i/\frak W_i$
which are induced by the orthogonal projections.
In fact, $\frak W_0$ fixes the
hyperplane $\operatorname{Ker} (v^*) \subset V$, and the projections $\operatorname{pr} _i$
induce an isomorphism $\operatorname{Ker} (v^*)/\frak W_0 \cong \bigoplus V_i/\frak W_i$.
Here, $v^*\in V^*$ is the element dual to $v$ in the basis dual to the root
basis.
We define standard coordinate functions on the partial resolution
spaces $\operatorname{PRes} (S,v)$
in the following way.
The definition depends on choosing a decomposition
$\Gamma - \{ v \} = \bigcup \Gamma^{(j)}$,
where this time each $\Gamma^{(j)}$ is a union of connected components
of $\Gamma - \{ v \}$. We assume that each corresponding root system
$R^{(j)}$ is either irreducible or of type $D_2$ or $E_3$; the definition
also depends on identifying the type of each $R^{(j)}$
as one of $A_{k-1}$, $D_k$, or $E_k$.
We define the {\em standard coordinate functions on
$\operatorname{PRes} (S,v) \cong V/\frak W_0$\/}
to be the linear functional $v^*$ together with the pullbacks of
the standard coordinate functions on the spaces $V^{(j)}/\frak W^{(j)}$ via the
mappings $V/\frak W_0 \to V^{(j)}/\frak W^{(j)}$ which are induced by
orthogonal projection.
These will be the coefficients of the defining polynomials of various
open sets on the partial resolution, except in the cases
where one of the constituents is a reducible root system of type
$D_2$ or $E_3$.
\section{Relations among distinguished polynomials.}
Let $R$ be a root system of type $S=A_{n-1}$, $D_n$, or $E_n$. The
complex root space $V_R$
has a root basis which can be written in the form
$v_{\alpha}, v_{\alpha + 1}, \ldots, v_{\beta}$, where $\alpha,
\alpha + 1, \ldots, \beta$ is a sequence of consecutive integers.
There is a natural dual basis $v_{\alpha}^*, v_{\alpha + 1}^*,
\ldots, v_{\beta}^*$ of the dual space $V^*$.
The basis element $v_k$ can be regarded as a vertex of the Dynkin
diagram $\Gamma_R$.
We fix $v_k$, and regard
the complement $\Gamma_R - \{v_k\}$ as forming two root systems, either of
which may
be empty or reducible: $R'$ is the part spanned by vertices
to the left of $v_k$ in $\Gamma_R - \{v_k\}$, and $R''$ is the part
spanned by vertices to the right.
(We use the orientation of the Dynkin diagrams displayed in section 2.)
This is admittedly ambiguous in a few cases, so to make everything
completely clear, we specify that when $(S,v_k) = (D_n,v_n)$ or $(E_n,v_0)$,
we take $R'$ to be spanned by $\Gamma_R - \{v_k\}$, and $R''$ to be empty.
We fix types $S'$ and $S''$ of the root systems $R'$ and $R''$,
as indicated in table~\ref{table4}. (We have omitted the case
$(S,v_k)=(D_n,v_{n-1})$, since we will not need it later.)
Having identified the type $S'$, there
are distinguished functionals $t_1',\ldots,t_{n'}'$, and
a distinguished polynomial $f_{S'}(U;t')$. The coefficients
of this polynomial are denoted by $s_i'$, and the standard coordinate
functions on $V'/\frak W'$ are denoted by $\a_i'$, $\c_i'$, $\d_i'$, or $\varepsilon_i'$, as
appropriate. We use analogous notation for $R''$, replacing `` $'$ ''
by `` $''$ '' throughout.
\begin{table}[t]
\begin{center}
$\begin{array}{|c|c|c|c|l|} \hline
& & & & \\
S & k & S' & S'' & \multicolumn{1}{c|}{\widetilde{v}_k} \\
& & & & \\ \hline
& & & & \\
A_{n-1} & \text{any} & A_{k-1} & A_{n-k-1} & v_k +
\sum_{i=1}^{k-1} \frac{i}{k} v'_i + \sum_{i=1}^{n-k-1}
\frac{n-k-i}{n-k} v''_i \\
& & & & \\
D_n & \le n-2 & A_{k-1} & D_{n-k} & v_k +
\sum_{i=1}^{k-1} \frac{i}{k} v'_i + \sum_{i=1}^{n-k-2} v''_i
+ \frac12 v''_{n-k-1} + \frac12 v''_{n-k} \\
& & & & \\
D_n & n & A_{n-1} & & v_n + \sum_{i=1}^{n-2} \frac{2i}{n} v'_i
+ \frac{n-2}{n} v'_{n-1} \\
& & & & \\
E_n & 0 & A_{n-1} & & v_0 + \frac{n-3}{n} v'_1 + \frac{2n-6}{n} v'_2 +
\sum_{i=3}^{n-1} \frac{3n-3i}{n} v'_i \\
& & & & \\
E_n & 1 & D_{n-1} & & v_1 + \sum_{i=1}^{n-3} \frac{i}{2} v'_i +
\frac{n-1}{4} v'_{n-2} + \frac{n-3}{4} v'_{n-1} \\
& & & & \\
E_n & 2 & A_1 & A_{n-2} & v_2 + \frac12 v'_1 + \frac{n-3}{n-1} v''_1 +
\sum_{i=2}^{n-2} \frac{2n-2i-2}{n-1} v''_i \\
& & & & \\
E_n & \ge 3 & E_k & A_{n-k-1} & v_k
+ \frac{3 v'_0 + 2 v'_1 + 4 v'_2 }{9-k}
+ \sum_{i=3}^{k-1} \frac{9-i}{9-k} v'_i
+ \sum_{i=1}^{n-k-1} \frac{n-k-i}{n-k} v''_i \\
& & & & \\ \hline
\end{array}$
\end{center}
\medskip
\caption{}
\label{table4}
\end{table}
By composing with the orthogonal projection maps $V_R \to V_{R'}$ and
$V_R \to V_{R''}$, we can regard the distinguished functionals on
$V_{R'}$ and $V_{R''}$ as linear functionals on $V_R$. We denote
them again by $t'_i$ and $t''_i$, suppressing
mention of the orthogonal projection maps for simplicity
of notation.
\begin{proposition} \label{prop71}
Let $v_k$ be a vertex of the Dynkin diagram $\Gamma_R$, and let
$\widetilde{v}_k \in V_R$ be the
vector specified in table~\ref{table4}. Then there is an
orthogonal direct sum decomposition
\[V_R = ({\Bbb C} \cdot\widetilde{v}_k) \oplus V_{R'} \oplus V_{R''},\]
and the dual space $V_R^*$ can be generated by the distinguished functionals
of $R'$ and $R''$ together with the functional $\mu_1\mathrel{:=} v_k^*$.
The distinguished functionals of $R$ are therefore linear
combinations of these generators, and those linear combinations can
be expressed by means of a relation among distinguished polynomials.
The relation involves $f_S(U;t)$, $f_{S'}(U;t')$, $f_{S''}(U;t'')$,
and $\mu_1$,
with some linear changes of variable and possible extra linear factors,
and is given explicitly
in table~\ref{tableAA}.
\end{proposition}
\begin{table}[b]
\begin{center}
\begin{tabular}{|c|c|rcl|} \hline
& & & & \\
$S$ & $k$ & & & \\
& & & & \\ \hline
& & & & \\
$A_{n-1}$ & any & $f_{A_{n-1}}(U;t)$ & = &
$f_{A_{k-1}}(U + \frac{1}{k} \mu_1;t') \cdot
f_{A_{n-k-1}}(U - \frac{1}{n-k} \mu_1;t'')$ \\
& & & & \\
$D_n$ & $ \le n-2$ & $f_{D_n}(U;t)$ & = &
$f_{A_{k-1}}(U + \frac{1}{k} \mu_1;t') \cdot
f_{D_{n-k}}(U;t'')$ \\
& & & & \\
$D_n$ & $n$ & $f_{D_n}(U;t)$ & = &
$f_{A_{n-1}}(U + \frac{2}{k} \mu_1;t')$ \\
& & & & \\
$E_n$ & $0$ & $f_{E_n}(U;t)$ & = &
$f_{A_{n-1}}(U - \frac{9-n}{3n} \mu_1;t')$ \\
& & & & \\
$E_n$ & $1$ & $(-1)^n \cdot f_{E_n}(U;t)$ & = &
$(- U + \frac13 \rho_1 - \frac{9-n}{6} \mu_1) \cdot
f_{D_{n-1}}(- U - \frac16 \rho_1 + \frac{9-n}{12} \mu_1;t')$ \\
& & & & \\
$E_n$ & $2$ & $f_{E_n}(U;t)$ & = &
$f_{A_{1}}(U + \frac23 \sigma_1 + \frac{9-n}{6n-6} \mu_1;t') \cdot
\frac{f_{A_{n-2}}(U - \frac13 \sigma_1 - \frac{9-n}{3n-3} \mu_1;t'')}
{(U - \frac43 \sigma_1 - \frac{9-n}{3n-3} \mu_1)}$ \\
& & & & \\
$E_n$ & $ \ge 3$ & $f_{E_n}(U;t)$ & = &
$f_{E_k}(U;t') \cdot f_{A_{n-k-1}}(U
- \frac{1}{9-k} \tau_1 - \frac{9-n}{(9-k)(n-k)} \mu_1;t'')$ \\
& & & & \\ \hline
\end{tabular}
\bigskip
{\it Notation:}
\medskip
\begin{tabular}{l}
$\mu_1$ denotes the coordinate function $v_k^*$ \\
$\rho_1$ denotes the coefficient of $U^{n-2}$ in $f_{D_{n-1}}(U;t')$ \\
$\sigma_1$ denotes a root of $f_{A_{n-2}}(U;t'')$ \\
$\tau_1$ denotes the coefficient of $U^{k-1}$ in $f_{E_k}(U;t')$
\end{tabular}
\end{center}
\bigskip
\caption{}
\label{tableAA}
\end{table}
\begin{pf}
We identify the root bases of $V_{R'}$ and $V_{R''}$
with subsets of the root basis of $V_{R}$ as follows. If $S \ne E_n$
or $k \ne 1, 2$
we identify $v'_i$ with $v_i$ for all
basis vectors of $V_{R'}$, and $v''_i$ with $v_{k+i}$ for all basis vectors
of $V_{R''}$.
If $S=E_n$
and $k=1$ or $2$, we use the identifications indicated in
figure~\ref{figure2}.
With this notation established, it is easy to
check that the vector $\widetilde{v}_k$ defined in table~\ref{table4}
is orthogonal to both $V_{R'}$ and $V_{R''}$.
Moreover, since $\Gamma_{R'}$ is disjoint from $\Gamma_{R''}$, the
spaces $V_{R'}$ and $V_{R''}$ are themselves
mutually perpendicular. The claimed orthogonal direct
sum decomposition follows.
The orthogonal direct sum decomposition can be regarded as a change
of basis from $\{v_i\}$ to
$\{v'_i\} \cup \{v''_i\} \cup \{\widetilde{v}_k\}$
in the space $V$. If we use maps $\sigma'$ and $\sigma''$ to describe
the identification of root bases, then this change of basis
can be written
\begin{align*}
v_i' & = v_{\sigma'(i)} \\
v_i'' & = v_{\sigma''(i)} \\
\widetilde{v}_k & = {\textstyle\sum a'_i v_i' + \sum a''_i v_i''},
\end{align*}
where the coefficients $a'_i$ and $a''_i$ are found in table~\ref{table4}.
(In all but two cases, $\sigma'(i)=i$ and $\sigma''(i)=k+i$.)
The corresponding change of dual basis
takes the form
\begin{align*}
v_{\sigma'(i)}^* & = a'_i \ \widetilde{v}_k^* + {v_i'}^* \\
v_{\sigma''(i)}^* & = a''_i \ \widetilde{v}_k^* + {v_i''}^* \\
v_{k}^* & = \widetilde{v}_k^*.
\end{align*}
It follows that $\mu_1 = v_{k}^* = \widetilde{v}_k^*$ can be used
along with the distinguished functionals on $V_{R'}$ and $V_{R''}$
to generate $V_R^*$.
To finish the proof, we must carry out the calculation which leads
to table~\ref{tableAA}. We will do this in a few cases, and leave
the remaining ones to the reader.
We first treat an easy case: the case $S = A_{n-1}$.
Using the fourth column of
table~\ref{table12} applied to $R'$ and $R''$,
we can write the change of basis as
\begin{alignat*}5
v_i^* & = & \tfrac{i}{k} &\widetilde{v}_k^* + {v_i'}^*
&& = & \tfrac{i}{k} &\mu_1 + t_{1}' + \cdots + t_i'
, \quad && 1 \le i \le k-1 \\
v_k^* & = & &\widetilde{v}_k^*
&& = & &\mu_1
\\
v_{k+i}^* & = & \tfrac{n-k-i}{n-k} &\widetilde{v}_k^* + {v_i''}^*
&& = & \tfrac{n-k-i}{n-k} &\mu_1 + t_{1}'' + \cdots + t_{i}''
, \quad && 1 \le i \le n-k-1
\end{alignat*}
Then using the third column of
table~\ref{table12} applied to $R$, we get
\begin{alignat*}2
t_i & = \tfrac{1}{k} \mu_1 + t_i'
, \quad && 1 \le i \le k
\\
t_{k+i} & = - \tfrac{1}{n-k} \mu_1 + t_i''
, \quad && k+1 \le k+i \le n.
\end{alignat*}
It follows that
\[f_{A_{n-1}}(U;t) = f_{A_{k-1}}(U + \tfrac{1}{k} \mu_1;t')
\cdot f_{A_{n-k-1}}(U - \tfrac{1}{n-k} \mu_1;t'').\]
\begin{figure}[t]
\begin{picture}(2.6,1)(1.9,.5)
\thicklines
\put(1.9,1){\circle{.075}}
\put(1.9375,1){\line(1,0){.4625}}
\put(2.4,1){\circle*{.075}}
\put(2.4,1){\line(1,0){.5}}
\put(2.9,1){\circle*{.075}}
\put(2.9,1){\line(0,-1){.5}}
\put(2.9,.5){\circle*{.075}}
\put(2.9,1){\line(1,0){.5}}
\put(3.4,1){\circle*{.075}}
\put(3.4,1){\line(1,0){.25}}
\put(3.75,1){\circle*{.02}}
\put(3.85,1){\circle*{.02}}
\put(3.95,1){\circle*{.02}}
\put(4.05,1){\line(1,0){.25}}
\put(4.3,1){\circle*{.075}}
\put(2.275,1.15){\makebox(.25,.25){$v'_{n-2}$}}
\put(2.775,1.15){\makebox(.25,.25){$v'_{n-3}$}}
\put(3.025,.375){\makebox(.25,.25){$v'_{n-1}$}}
\put(3.275,1.15){\makebox(.25,.25){$v'_{n-4}$}}
\put(4.175,1.15){\makebox(.25,.25){$v'_{1}$}}
\end{picture}
\hspace*{\fill}
\begin{picture}(2.6,1)(1.9,.5)
\thicklines
\put(1.9,1){\circle*{.075}}
\put(1.9,1){\line(1,0){.4625}}
\put(2.4,1){\circle{.075}}
\put(2.4375,1){\line(1,0){.4625}}
\put(2.9,1){\circle*{.075}}
\put(2.9,1){\line(0,-1){.5}}
\put(2.9,.5){\circle*{.075}}
\put(2.9,1){\line(1,0){.5}}
\put(3.4,1){\circle*{.075}}
\put(3.4,1){\line(1,0){.25}}
\put(3.75,1){\circle*{.02}}
\put(3.85,1){\circle*{.02}}
\put(3.95,1){\circle*{.02}}
\put(4.05,1){\line(1,0){.25}}
\put(4.3,1){\circle*{.075}}
\put(1.775,1.15){\makebox(.25,.25){$v'_1$}}
\put(2.775,1.15){\makebox(.25,.25){$v''_2$}}
\put(3.025,.375){\makebox(.25,.25){$v''_1$}}
\put(3.275,1.15){\makebox(.25,.25){$v''_3$}}
\put(4.175,1.15){\makebox(.25,.25){$v''_{n-2}$}}
\end{picture}
\caption{}
\label{figure2}
\end{figure}
We next treat the case of $R = E_n$, with $k=1$, which is displayed in the
left half of figure~\ref{figure2}. In this case,
\begin{alignat*}5
v_0^* & = & \tfrac{n-3}{4} &\widetilde{v}_1^* + {v_{n-1}'}^*
&& = & \tfrac{n-3}{4} &\mu_1 + \tfrac12 s_1'
\\
v_1^* & = & &\widetilde{v}_1^*
&& = & &\mu_1
\\
v_2^* & = & \tfrac{n-1}{4} &\widetilde{v}_1^* + {v_{n-2}'}^*
&& = & \tfrac{n-1}{4} &\mu_1 + \tfrac12 s_1' - t_{n-1}'
\\
v_{n-i}^* & = & \tfrac{i}{2} &\widetilde{v}_1^* + {v_i'}^*
&& = & \tfrac{i}{2} &\mu_1 + s_1' - t_{i+1}' - \cdots - t_{n-1}'
, \quad && 1 \le i \le n-3
\end{alignat*}
which implies
\begin{alignat*}2
t_1 & = \tfrac{9-n}{6} \mu_1 - \tfrac13 s_1'
\\
t_{i+1} & = \tfrac{n-9}{12} \mu_1 + \tfrac16 s_1' - t_{n-i}'
, \quad && 2 \le i+1 \le n.
\end{alignat*}
It follows that
\[f_{E_n}(U;t) =
(-1)^{n-1} \cdot (U + \tfrac{9-n}{6} \mu_1 - \tfrac13 s_1') \cdot
f_{D_{n-1}}(- U - \tfrac{n-9}{12} \mu_1 - \tfrac16 s_1';t'),\]
since the right-hand side is equal to
\begin{multline*}
(-1)^{n-1} \cdot (U + \tfrac{9-n}{6} \mu_1 - \tfrac13 s_1')
\cdot \prod (- U - \tfrac{n-9}{12} \mu_1 - \tfrac16 s_1' + t_j') \\
\begin{aligned}
& = (U + \tfrac{9-n}{6} \mu_1 - \tfrac13 s_1') \cdot
\prod (U + \tfrac{n-9}{12} \mu_1 + \tfrac16 s_1' - t_j') \\
& = \prod (U + t_i).
\end{aligned}
\end{multline*}
Finally, we treat the case $R = E_n$, $k = 2$, which is displayed in the
right half of figure~\ref{figure2}. In this case,
\begin{alignat*}5
v_0^* & = & \tfrac{n-3}{n-1} &\widetilde{v}_2^* + {v_1''}^*
&& = & \tfrac{n-3}{n-1} &\mu_1 + t_1''
\\
v_1^* & = & \tfrac12 &\widetilde{v}_2^* + {v_1'}^*
&& = & \tfrac12 &\mu_1 + t_1'
\\
v_2^* & = & &\widetilde{v}_2^*
&& = & &\mu_1
\\
v_{i+1}^* & = & \tfrac{2n-2i-2}{n-1} &\widetilde{v}_2^* + {v_i''}^*
&& = & \tfrac{2n-2i-2}{n-1} &\mu_1 + t_1'' + \cdots + t_i''
, \quad && 2 \le i \le n-2
\end{alignat*}
which implies
\begin{alignat*}2
t_i & = \tfrac{9-n}{6n-6} \mu_1 - \tfrac23 t_1'' + t_i'
, \quad && 1 \le i \le 2
\\
t_{i+1} & = \tfrac{n-9}{3n-3} \mu_1 + \tfrac13 t_1'' + t_{i}''
, \quad && 2 \le {i} \le n-1.
\end{alignat*}
(Notice that the functional
$\tfrac{n-9}{3n-3} \mu_1 + \tfrac13 t_1'' + t_1''$
is ``missing" here.)
It follows that
\[f_{E_n}(U;t) =
f_{A_1}(U + \tfrac{9-n}{6n-6} \mu_1 - \tfrac23 t_1'';t') \cdot
\frac{f_{A_{n-2}}(U + \tfrac{n-9}{3n-3} \mu_1 + \tfrac13 t_1'';t'')}
{(U + \tfrac{n-9}{3n-3} \mu_1 + \tfrac43 t_1'')}.\]
Since $-t''_1$ is a root of $f_{A_{n-1}}(U;t'')$, if we define
$\sigma_1\mathrel{:=} -t''_1$, the formula in the table follows.
All remaining cases are left to the reader.
\end{pf}
Proposition~\ref{prop71} provides a method for explicitly calculating
the map $\operatorname{PRes} (S,v_k) \to \operatorname{Def} (S)$ (which can also be written as
$V/\frak W_0 \to V/\frak W$). What we wish to calculate explicitly is the
map on coordinate rings ${\Bbb C} [V]^{\frak W} \subset {\Bbb C} [V]^{\frak W_0}$.
In other words, we want to express the standard coordinate functions
on $V/\frak W$ as polynomials in the standard coordinate functions on
$V/\frak W_0$.
Now each standard coordinate function $\varphi_j$ on $V/\frak W$ is a function
of $s_1,\ldots,s_n$ (the coefficients of the distinguished polynomial
$f_S(U;t)$). Using proposition~\ref{prop71},
these in turn are expressed as functions of the coefficients $s'_i$
and $s''_i$ of the distinguished polynomials of $f_{S'}(U;t')$
and $f_{S''}(U;t'')$, together
with $\mu_1$ (and possibly an auxiliary variable $\rho_1$, $\sigma_1$,
or $\tau_1$ which will eliminate itself in the end). The expression
for $\varphi_j$ in terms of these variables
is invariant under $\frak W_0$, and so can be expressed as a polynomial
in $\mu_1$ together with the pullbacks of the
standard coordinate functions on
$V'/\frak W'$ and $V''/\frak W''$. One approach to finding this polynomial expression
is the method of undetermined coefficients.
We carry this out for the cases of $A_{n-1}$ and $D_n$,
collecting the information we require in the form of certain
congruences. The part of the computation which we need in the $E_n$
cases will be stated in table~\ref{table-key1} in section 8, and verified
in section 10.
\begin{proposition} \label{prop72}
\quad
\begin{enumerate}
\item If $R = A_{n-1}$, then
$\a_{n-1} \equiv
\a'_{k-1} \a''_{n-k} + \a'_k \a''_{n-k-1}\mod{\mu_1}$, and
$\a_{n} \equiv \a'_k \a''_{n-k}\mod{\mu_1}$.
\item If $R = D_n$ and $k=1$, then
$\d_{2n-4} \equiv \d''_{2n-4}\mod{\mu_1}$.
\item If $R = D_n$ and $k \le n-2$, let $J_4$ denote the ideal generated by all
monomials of degree 4 in the standard coordinate functions on
$\operatorname{PRes} (D_n,v_k)$.
Then
$\c_n \equiv \a'_k \c''_{n-k}\mod{\mu_1}$, and
$\d_{2n-2} \equiv (\a'_k)^2 \d''_{2n-2k-2}\mod{J_4}$.
\item If $R = D_n$ and $k = n$, then
$\c_n \equiv \a'_{n}\mod{\mu_1}$.
\end{enumerate}
\end{proposition}
\begin{pf}
We will prove the third statement, and leave the others
(which are easier) to the reader.
When $R = D_n$ and $k \le n-2$ we have
\begin{align*}
f_{D_{n}}(U;t) & =
f_{A_{k-1}}(U + \tfrac{1}{k} \mu_1;t') \cdot
f_{D_{n-k}}(U;t'') \\
& \equiv f_{A_{k-1}}(U;t') \cdot
f_{D_{n-k}}(U;t'')\mod{\mu_1} \\
& \equiv \a'_k
\c''_{n-k}\mod{(\mu_1,U)}
\end{align*}
It follows that $\c_n \equiv \a'_k
\c''_{n-k}\mod{\mu_1}$. Moreover,
if we define $\widetilde{g}_{A_{k-1}}(Z;t')$ by
\[
\widetilde{g}_{A_{k-1}}(- U^2;t')
= f_{A_{k-1}}(U+\tfrac{1}{k}\mu_1;t') \cdot f_{A_{k-1}}(-U+\tfrac{1}{k}\mu_1;t')
\]
then
\begin{multline*}
\widetilde{g}_{A_{k-1}}(Z;t')
\equiv
({\a_{k-1}'}^2-2\a_{k-2}'\a_k'+
\tfrac{2}{k}\mu_1\a_{k-1}'\a_{k-2}')Z \\
+
({\a_k'}^2+\tfrac{2}{k}\mu_1\a_{k-1}'\a_k')\mod{(J_4,Z^2)}
\end{multline*}
while
\[
g_{D_{n-k}}(Z;t'') \equiv
\d''_{2n-2k-2} Z +(\c''_{n-k})^2\mod{Z^2}.
\]
Hence, if we multiply these congruences and retain only terms of degree
at most three in the standard coordinate functions
on $\operatorname{PRes} (D_n,v_k)$, we get
\[
g_{D_n}(Z;t) \equiv (\a'_{k})^2
\d''_{2n-2k-2} Z\mod{(J_4,Z^2)}.
\]
The congruence for $\d_{2n-2}$ follows.
\end{pf}
In order to effectively apply the last line of table~\ref{tableAA}
when $k=4$ or $5$,
we need to
compute the distinguished polynomials $f_{E_4}(U;t)$ and $f_{E_5}(U;t)$.
\begin{lemma} \label{lem71}
Let $t_1,\cdots,t_n$ be the distinguished functionals for $E_4$, resp.\
$E_5$, and let $\widetilde{t}_1,\cdots,\widetilde{t}_5$ be the distinguished
functionals for $A_4$, resp.\ $D_5$. If we identify these root systems
in such a way that $\widetilde{v}_i = v_i$ for $1 \le i \le n-1$ and
$\widetilde{v}_n = v_0$ then
\[f_{A_4}(U;\widetilde{t}) = (U + \frac35 s_1 ) \cdot
f_{E_4}(U - \frac25 s_1; t)\]
and
\[f_{D_5}(U;\widetilde{t}) = f_{E_5}(U - \frac12 s_1; t).\]
\end{lemma}
\begin{pf}
In the case of $E_4$, we calculate with table~\ref{table12}
as follows.
\begin{alignat*}6
\widetilde{t}_1 & = & &\widetilde{v}_1^* && = & &v_1^* && = &
- \tfrac25 &s_1 + t_1 \\
\widetilde{t}_2 & = & - \widetilde{v}_1^* &+ \widetilde{v}_2^* && = &
- v_1^* &+ v_2^* && = & - \tfrac25 &s_1 + t_2 \\
\widetilde{t}_3 & = & - \widetilde{v}_2^* &+ \widetilde{v}_3^* && = &
- v_2^* &+ v_3^* && = & - \tfrac25 &s_1 + t_3 \\
\widetilde{t}_4 & = & - \widetilde{v}_3^* &+ \widetilde{v}_4^* && = &
- v_3^* &+ v_0^* && = & \tfrac35 &s_1 - t_1 - t_2 - t_3 \\
\widetilde{t}_5 & = & - &\widetilde{v}_4^* && = &
- &v_0^* && = & \tfrac35 &s_1
\end{alignat*}
Now since $\frac35 s_1 - t_1 - t_2 - t_3 = - \frac25 s_1 + t_4$
we get
\[(U+\widetilde{t}_1)\cdots(U+\widetilde{t}_5)
=(U-\frac25s_1+t_1)\cdots(U-\frac25s_1+t_4)
\cdot (U+\frac35s_1),\]
and the first equation follows.
The case of $E_5$ is similar (and easier), and will be left to the
reader.
\end{pf}
\begin{corollary}
The standard coordinate functions on $\operatorname{Def} (E_4)$ and $\operatorname{Def} (E_5)$,
expressed in terms of
the elementary symmetric functions $s_i$ of their respective distinguished
functionals $t_i$, are given by
\begin{align}
\begin{split}
\varepsilon_2 &= s_2 - \tfrac35 s_1^2\\
\varepsilon_3 &= s_3 - \tfrac15 s_2s_1 + \tfrac{2}{25}s_1^3\\
\varepsilon_4 &= s_4 + \tfrac15 s_3s_1 - \tfrac{8}{25}s_2s_1^2 + \tfrac{12}{125}s_1^4\\
\varepsilon_5 &= \tfrac35 s_1s_4 - \tfrac{6}{25}s_3s_1^2 + \tfrac{12}{125}s_2s_1^3 -
\tfrac{72}{3125}s_1^5
\end{split}\label{eqE4}
\end{align}
in the case of $E_4$, and by
\begin{align}
\begin{split}
\varepsilon_2 &= -2s_2 + \tfrac54 s_1^2\\
\varepsilon_4 &= s_2^2 - 2s_2s_1^2 + \tfrac58 s_1^4 + s_1s_3 + 2s_4\\
\varepsilon_5 &= -\tfrac18 s_2s_1^3 + \tfrac{1}{32}s_1^5 + \tfrac14 s_1^2s_3 -
\tfrac12s_1s_4 +s_5\\
\varepsilon_6 &= \tfrac34 s_2^2s_1^2 - \tfrac34 s_2s_1^4 - s_2s_1
s_3 - 2s_2s_4 + \tfrac5{32} s_1^6 + \tfrac34 s_1^3s_3 + \tfrac12 s_1^2
s_4
\\ & \qquad
-
3s_1s_5 - s_3^2\\
\varepsilon_8 &= \tfrac3{16}s_2^2s_1^4 - \tfrac18 s_2s_1^6 - \tfrac12 s_2s_1^3s_3 +
3s_2s_1s_5 +
\tfrac5{256} s_1^8 + \tfrac3{16} s_1^5s_3
\\ & \qquad
-
\tfrac18 s_1^4s_4 - \tfrac12 s_1^3s_5 + \tfrac12 s_1^2s_3^2 - s_1s_4s_3 -
2s_5s_3 + s_4^2\\
\end{split}\label{eqE5}
\end{align}
in the case of $E_5$.
\end{corollary}
\begin{pf}
Write $f_{E_n}(U;t)=U^n+\sum_{i=0}^{n-1}s_iU^{n-i}$ for $n=4\hbox{ or }5$. Use
the formulas of the lemma to calculate
$f_{A_4}(U;\tilde{t})$ and $f_{D_5}(U;\tilde{t})$. In the case of $E_4$,
the
standard coordinate functions
$\varepsilon_i$ can be read off as the coefficients of $U^{5-i}$ in
$f_{A_4}$. In the case of $E_5$, form the second distinguished polynomial
$g_{D_5}$ via $g_{D_5}(-U^2;\tilde{t})=
f_{D_5}(U;\tilde{t})f_{D_5}(-U;\tilde{t})$. The standard coordinate
function $\varepsilon_{2i}$ can be read off as
the coefficient of $U^{5-i}$ in $g_{D_5}(U;\tilde{t})$, while $\varepsilon_5$ is simply
the
coefficient of $U^0$ in $f_{D_5}(U;\tilde{t})$.
\end{pf}
\section{Proof of the main theorem.}
In this section, we prove the main theorem, assuming the validity of
some results to be stated in table~\ref{table-key1}. The proof will be
complete
when we verify that table in section 10.
The partial resolution graphs shown in figure~\ref{figure1}
determine a singularity
type $S_{\ell}$ for each length $\ell$ between $1$ and $6$, which
we call the {\em associated type\/} of the length. Explicitly, this type is
\begin{center}
\begin{tabular}{|c|cccccc|} \hline
$\ell$ & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline
$S_{\ell}$ & $A_1$ & $D_4$ & $E_6$ & $E_7$ & $E_8$ & $E_8$ \\ \hline
\end{tabular}.
\end{center}
Figure~\ref{figure1}
(in section 1) illustrates the partial resolution graphs
$\{ v \} \subset \Gamma_{S_{\ell}}$, where $v$ is the vertex corresponding
to the unique component in the maximal ideal cycle of length $\ell$.
Our aim is to show that for $\pi \colon Y \to X$ of length $\ell$, the
singularity type of the general hyperplane section is $S_{\ell}$.
We say that the singularity type of a rational double point is {\em
at worst\/} $S$ if its dual graph is isomorphic to
a (proper or improper) subgraph
of $\Gamma_S$.
\begin{lemma} \label{lem82}
\quad
\begin{enumerate}
\item
The partial resolution graphs shown in figure~\ref{figure1} are primitive.
\item
Let $\pi\colon Y \to X$ be an irreducible small resolution
of an isolated Gorenstein threefold singularity, and let $\ell$ be
the length. Suppose that $X$ has a hyperplane section
whose singularity type is at worst $S_{\ell}$.
Then the generic hyperplane section defines the primitive partial
resolution graph corresponding to $S_{\ell}$ given in figure~\ref{figure1}.
\end{enumerate}
\end{lemma}
\begin{pf}
\quad
(1) For each $\ell$ between 1 and 6, let
$n(\ell)$
be the minimum $n$ such that there is a rational double point whose
dual graph has $n$ vertices, and at least one component has multiplicity
exactly $\ell$ in the maximal
ideal cycle.
Examining the maximal ideal cycles of the rational double points, it is
easy to see that for each $\ell$, there is a unique such rational
double point with $n(\ell)$ vertices, namely the one shown in
figure~\ref{figure1}.
Now for any nontrivial 1-parameter deformation of a rational double
point, the dual graph of the minimal resolution of the general fiber is
isomorphic to a proper subgraph of the dual graph of the special fiber.
It follows that each graph shown in figure~\ref{figure1}
is primitive: any proper
subgraph
will have fewer vertices, and so cannot have a component in its
maximal ideal cycle of multiplicity $\ell$.
(2)
Fix $\ell$. For the singularity of type $S_{\ell}$,
there is a unique component of multiplicity exactly $\ell$ in the
maximal ideal cycle. Moreover, no proper subgraph
of $\Gamma_{S_{\ell}}$ has any component
with multiplicity exactly $\ell$ in {\em its\/} maximal ideal cycle.
Thus, since the length is $\ell$, the partial
resolution graph determined by the given hyperplane section must
be of the type shown in figure~\ref{figure1}
(which indicates the unique component
of multiplicity $\ell$). On the other hand, since the graphs in
figure~\ref{figure1} are primitive, it follows that this is also the type of
the generic hyperplane section.
\end{pf}
Consider now $\pi\colon Y \to X$,
an irreducible small resolution
of an isolated Gorenstein threefold singularity $P \in X$, and
a hyperplane section $\{f=0\}$ which has a rational double point.
This determines a partial resolution graph $\{ v \} \subset \Gamma_S$, and
the length $\ell$ coincides with the multiplicity of $v$ in the maximal
ideal cycle.
There is a natural classifying map $\mu_f\colon \Delta \to \operatorname{Def} (S) = V/\frak W$
which allows us to recover a neighborhood of $P \in X$ as the pullback
of the standard deformation ${\cal X} \to V/\frak W$.
The map $\mu_f$ determines a discrete valuation
$\nu_f\colon {\Bbb C} [V]^{\frak W} \to {\Bbb Z} $
which is defined by
\[\nu_f(\varphi) = \text{order of vanishing at $0$ of $\mu^*_f(\varphi)$}.\]
Thanks to
lemma~\ref{lem82}, in order to prove the main theorem it suffices
to show that $X$ has some hyperplane section whose
singularity type is at worst $S_{\ell}$.
The following proposition shows how to use the
discrete valuation $\nu_f$ applied to the
standard coordinate functions on $\operatorname{Def} (S)$ (or in the $E_7$
case, to certain simple polynomial expressions in these functions)
to bound the
singularity type of the general hyperplane section.
\begin{proposition} \label{prop81}
Let $X$ be a threefold with an isolated rational Gorenstein
singular point $P$, and let $\{f=0\}$ be a hyperplane section
through $P$ with a rational double point of type $S$. Let
$\mu_f\colon \Delta \to \operatorname{Def} (S) = V/\frak W$ be the classifying map, and let
$\nu_f\colon {\Bbb C} [V]^{\frak W} \to {\Bbb Z} $ be the
associated discrete valuation.
\begin{enumerate}
\item
Suppose that $S=A_{n-1}$, and let $\{ \a_i \}$ be the standard coordinate
functions on $\operatorname{Def} (S)$. If $\nu_f(\a_{n-1})=1$ or $\nu_f(\a_n)=2$
then the general hyperplane section of $X$ has singularity type at worst
$A_1$.
\item
Suppose that $S=D_{n}$, and let $\{ \c_n,\ \d_i \}$ be the standard coordinate
functions on $\operatorname{Def} (S)$. If $\nu_f(\c_{n})=1$ or $\nu_f(\d_{2n-4})=1$
then the general hyperplane section of $X$ has singularity type at worst
$A_1$, while if $\nu_f(\c_{n})=2$ or $\nu_f(\d_{2n-2})=3$
then the general hyperplane section of $X$ has singularity type at worst
$D_4$.
\item
Suppose that $S=E_6$, $E_7$, or $E_8$, and let $\{\varepsilon_i\}$ be the
standard coordinate functions on $\operatorname{Def} (S)$.
Define
$\widetilde{\varepsilon}_i = \varepsilon_i$ if $S \ne E_7$ or $i \ne 12, 18$, and in
the case of $E_7$, define
\[\widetilde{\varepsilon}_{12} = \varepsilon_{12} + \frac13 \varepsilon_6^2
\quad \text{and} \quad
\widetilde{\varepsilon}_{18} = \varepsilon_{18} + \frac13\varepsilon_6\varepsilon_{12} + \frac{2}{27}\varepsilon_6^3.\]
Let $M_f$ be the set of monomials $T^dY^kZ^{\ell}$ such that
$\varepsilon_iY^kZ^{\ell}$ is one of the terms in the polynomial in preferred versal
form, and $\nu_f(\widetilde{e}_i)=d<\infty$.
If any of the monomials in $M_f$ are listed in the right half of
table~\ref{tableMONOrev}, then the general hyperplane
section of $X$ has singularity type at worst $S$, where $S$ is the
label on the leftmost column in the right half of the
table which contains some monomial from $M_f$.
{\parindent 1.5em
Moreover, if $X$ has an irreducible small resolution of
length $\ell$, then $M_f$ contains no monomials
to the left of the column whose label is the associated type $S_{\ell}$.
}
\end{enumerate}
\end{proposition}
The left half of table~\ref{tableMONOrev} has been included to make it
easier to find which monomials in $M_f$ come from which standard coordinate
functions $\varepsilon_i$. It is not actually necessary for the description of
the link between $M_f$ and the singularity type of the general hyperplane
section.
{\renewcommand{\arraystretch}{1.4}
\begin{table}[b]
\begin{center}
\begin{tabular}{|c|c|c||c|c|c|c|c|c|c|} \hline
$E_6$ & $E_7$ & $E_8$ & $A_0$ & $A_1$ & $A_2$ & $D_4$ & $D_k$ & $E_6$ &
$E_7$ \\ \hline
& & $\varepsilon_{8}$ & & & & & & & $T Y Z^2$
\\
& $\varepsilon_{6}$ & & & & & & & $T Y^2$ & \\
& & $\varepsilon_{12}$ & & & & & & $T Z^3$ & \\
$\varepsilon_{5}$ & $\varepsilon_{8}$ & $\varepsilon_{14}$ & & & & $T Y Z$ & & & $T^2 Y Z$
\\
$\varepsilon_{6}$ & $\varepsilon_{10}$ & $\varepsilon_{18}$ & & & & $T Z^2$ & & $T^2 Z^2$ & \\
$\varepsilon_{8}$ & $\widetilde{\varepsilon}_{12}$ & $\varepsilon_{20}$ & & $T Y$ & & & $T^2 Y$ &
& $T^3 Y$ \\
$\varepsilon_{9}$ & $\varepsilon_{14}$ & $\varepsilon_{24}$ & & $T Z$ & & $T^2 Z$ & & $T^3 Z$ & \\
$\varepsilon_{12}$ & $\widetilde{\varepsilon}_{18}$ & $\varepsilon_{30}$ & $T$ & & $T^2$ & & $T^3$ & $T^4$
& \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{}
\label{tableMONOrev}
\end{table}
}
\begin{pf}
The proof is based on the classification of rational double points
by means of their Newton polygons. A convenient reference for this is
\cite[(4.9)(3)]{[YPG]}. In brief, suppose that
$\{f=0\} \subset {\Bbb C} ^3$ has a
rational double
point at the origin. Then the type $S$ is determined by the defining
polynomial
$f$ as follows.
\begin{enumerate}
\item
$S=A_0$ (i.e. the surface is smooth at the origin) if and only if
$f$ contains a linear term. (Notice that $f$ contains no constant
term, since the origin lies on the surface.)
\item
$S=A_1$ if and only if the quadratic part $f_2$ of $f$ has rank 3.
\item
$S=A_{n-1}$, $n>2$ if and only if the quadratic part $f_2$ of $f$
has rank 2. The value of $n$ is determined by the higher order terms.
In particular, if the cubic part
$f_3$ of $f$ is nonzero and involves none of the variables appearing in
$f_2$, then $S=A_2$.
\item
If $f_2$ has rank 1, choose coordinates so that $f = x^2 + g(y,z)$,
and $g$ has no quadratic part. Note that $g_3$ is a homogeneous
cubic in 2 variables.
\begin{enumerate}
\item
$S=D_4$ if and only if the cubic part $g_3$ of $g$ has three distinct
linear factors.
\item
$S=D_n$, $n>4$ if and only if the cubic part $g_3$ of $g$ has two
distinct linear factors. (The value of $n$ is determined by the higher
order terms.)
\item
If $g_3$ has a unique linear factor, write $g_3 = h^3$.
\begin{enumerate}
\item
$S=E_6$ if and only if $h$ does not divide the quartic part $g_4$
of $g$.
\item
$S=E_7$ if and only if the quartic part $g_4$ of $g$ is divisible by
$h$ but not by $h^2$.
\item
$S=E_8$ otherwise.
\end{enumerate}
\end{enumerate}
\end{enumerate}
The threefold $X$ has defining polynomial $\mu_f^*(\Phi_S)$,
where $\Phi_S$ is the polynomial in preferred versal form of type $S$.
If $\varphi_i Y^k Z^{\ell}$ is a term in $\Phi_S$ and if $\nu_f(\varphi_i)=d<\infty$,
then the monomial $T^d Y^k Z^{\ell}$ appears in the defining polynomial of $X$
with
a nonzero coefficient. (Here, $T$ is the coordinate on the disk $\Delta$.)
In this way, we can analyze the low-degree terms
appearing in the defining polynomial of $X$ by using the set $M_f$.
We define the {\em leading terms\/} of $\Phi_S$ to be those
which have constant coefficients; there are 2 or 3 such terms.
For all other coefficients $\varphi_i$, we have $\nu_f(\varphi_i) \ge 1$.
Table~\ref{tableMONOrev} has been constructed so that all potential
low-degree terms (other than leading terms) are shown there.
Suppose first that $S=A_{n-1}$. The only possible monomial of degree 1
in the defining polynomial is $T$, and this occurs if and only if
$\nu_f(\a_n)=1$.
This is the condition for $X$ (and its general hyperplane section) to
be smooth at the origin; in this case, the singularity type is certainly
``at worst" $A_1$.
If $\nu_f(\a_n)>1$, we consider quadratic terms. The leading term of
degree 2 is $-XY$, while other potential quadratic terms must be
chosen from $\{TZ,T^2\}$. If at least one of those potential terms occurs
with a nonzero coefficient, then the rank of the quadratic part of
the defining polynomial is at least 3. And this implies that the quadratic
part of
the general hyperplane section will have rank 3, and so will have a
singularity of type $A_1$. But to guarantee that at least one of the
potential
terms occurs, we simply need $\nu_f(\a_{n-1})=1$ or $\nu_f(\a_n)=2$.
Suppose next that $S=D_n$. Again, the only possible monomial of degree 1
in the defining polynomial is $T$, and this occurs if and only if
$\nu_f(\d_{2n-2})=1$.
This is the condition for $X$ (and its general hyperplane section) to
be smooth at the origin; in this case, the singularity type is certainly
``at worst" $A_1$ or even $D_4$.
If $\nu_f(\d_{2n-2})>1$, we consider quadratic terms. The leading term of
degree 2 is $X^2$, while other potential quadratic terms must be
chosen from $\{TY,TZ,T^2\}$. If at least one of the terms $TY$, $TZ$ occurs
with a nonzero coefficient, then the rank of the quadratic part of
the defining polynomial is at least 3. And this implies that the quadratic
part of
the general hyperplane section will have rank 3, and so will have a
singularity of type $A_1$. But to guarantee that at least one of those
terms occurs, we simply need $\nu_f(\c_n)=1$ or $\nu_f(\d_{2n-4})=1$.
On the other hand, if neither of those terms occurs, yet $T^2$ occurs,
then the rank of the quadratic part is 2. (This happens when $\nu_f(\c_n)>1$,
$\nu_f(\d_{2n-4})>1$, and $\nu_f(\d_{2n-2})=2$.) Now the leading
term $Y^2Z$ also appears in our defining polynomial. Since this is a
term of degree 3 which involves neither of the variables
$X$, $T$ which appear in the quadratic part, there will be hyperplane
sections of type $A_2$. (For example, the hyperplane section defined
by $Y=Z$ will be of type $A_2$.)
It follows that the singularity type of the
general hyperplane section is at worst $A_2$,
and hence is certainly at worst $D_4$.
So we may assume that $\nu_f(\c_n)>1$,
$\nu_f(\d_{2n-4})>1$, and $\nu_f(\d_{2n-2})>2$. The defining polynomial can
then
be written in the form $X^2 + G(Y,Z,T)$, and the cubic part of $G$ takes
the form $G_3(Y,Z,T) = Y^2Z + T \cdot H(Y,Z,T)$. Moreover, the only
monomials which could appear in $T \cdot H(Y,Z,T)$ are $TZ^2$, $T^2Y$,
$T^2Z$, $T^3$. In order for the general hyperplane section of $G_3$
to fail to have 3 distinct linear factors, $G_3$ must be nonreduced.
Since $G_3=0$ defines a plane cubic, this implies that $G_3$ itself factors
in the form $H_1^2 H_2$, where $H_1$, $H_2$ are two
(possibly equal) linear polynomials.
Considering this factorization mod $T$, we see that it must take the
form
\[G_3(Y,Z,T) = (Y + \alpha T)^2 \cdot (Z + \beta T).\]
Moreover,
since $TY^2$ is not one of the monomials which can occur in $G_3$,
$\beta$ must in fact be 0. Thus, if the general hyperplane section
fails to have type $D_4$, $Z$ must divide $G_3$. The presence of
either of the monomials $T^2Y$ or $T^3$ will prevent this, and their
presence is guaranteed by the conditions
$\nu_f(\c_{n})=2$ and $\nu_f(\d_{2n-2})=3$, respectively. So when
either of these conditions holds, the general hyperplane section must
be of type $D_4$.
Suppose finally that $S=E_n$. We define $\widetilde{X}=X+\frac12Z^2$
in case $E_6$ and $\widetilde{X}=X$
in cases $E_7$ and $E_8$. Then the leading terms take the form
$- \widetilde{X}^2 +\frac14 Z^4 + Y^3$,
$-\widetilde{X}^2-Y^3+16YZ^3$, $-\widetilde{X}^2+Y^3-Z^5$, respectively.
The monomials in these leading terms together with the monomials in
$M_f$ will include all monomials of low degree in the defining polynomial
$\mu_f^*(\Phi_S)$.
The analysis of the cases with a linear part or with a quadratic part of
rank bigger than 1 proceeds almost exactly as in the case of $D_n$, using
$-\widetilde{X}^2$ in place of $X^2$ for the leading term of degree 2, and
$\pm Y^3$ for the leading term of degree 3. It yields the criteria
for having hyperplane sections of types $A_0$, $A_1$, and $A_2$ which
are stated in table~\ref{tableMONOrev}.
(The only remarks that need to be added to the argument given in the $D_n$
case concern the $E_7$ case, since we use two modified coefficients
$\widetilde{\varepsilon}_{12}$ and $\widetilde{\varepsilon}_{18}$ in that case. The remarks
(which follow from the defining formulas for the modified coefficients)
are that $\nu_f(\widetilde{\varepsilon}_{12})=1$ if and only if $\nu_f(\varepsilon_{12})=1$,
and that when $\nu_f(\varepsilon_{12})>1$, we have
$\nu_f(\widetilde{\varepsilon}_{18})=2$ if and only if $\nu_f(\varepsilon_{18})=2$.
Thus, the orders of vanishing which predict the presence of the monomials
$TY$ and $T^2$ are being calculated properly.)
So we may assume that none of the monomials $T^2$, $TY$, $TZ$, or $T$
appear in our defining polynomial. The defining polynomial can be written
in the form $-\widetilde{X}^2 + G(Y,Z,T)$, and this time
the cubic part of $G$ takes
the form $G_3(Y,Z,T) = \pm Y^3 + T \cdot H(Y,Z,T)$. As before,
the general hyperplane section will be of type $D_4$ unless
$G_3$ can be factored
in the form $\pm H_1^2 H_2$, where $H_1$, $H_2$ are two
(possibly equal) linear polynomials.
Considering this factorization mod $T$, we see that
it takes the form
\[\pm G_3(Y,Z,T) = (Y + \alpha T)^2 \cdot (Y + \beta T).\]
In particular,
for such a factorization to exist, $G_3$ must be a function of $Y$ and
$T$ alone. The presence of any of the monomials $TYZ$, $TZ^2$, or $T^2Z$
in $M_f$ prevents this, and forces the general hyperplane section to have
type $D_4$.
If none of those monomials is present in $M_f$, then $G_3$ is a homogeneous
binary cubic, and the general hyperplane section
has type $D_k$ unless $G_3$ is the cube of
a linear polynomial. (The type will be $D_4$ if there are three distinct
linear factors of $G_3$, and will be $D_k$, $k>4$ if there are only two.)
Now for $S \ne E_7$, the monomial $TY^2$ cannot occur in $G_3$ (as is
clear from table~\ref{tableMONOrev}). In this
case, if $G_3$ is a cube it must be $Y^3$, and the presence of either of
the monomials $T^2Y$ or $T^3$ in $M_f$ will prevent this from happening,
and lead to the general hyperplane section having type $D_k$.
The argument is more complicated in the case of
$E_7$.\footnote{The argument could have been simplified, eliminating the
use of $\widetilde{\varepsilon}_{12}$ and $\widetilde{\varepsilon}_{18}$, were it not
for our desire to match notation with Bramble \cite{[Bra]}.}
We define
$\widetilde{Y} = Y - \frac13 \varepsilon_6$, and note that
\[-\widetilde{Y}^3 + \widetilde{\varepsilon}_{12} \widetilde{Y} + \widetilde{\varepsilon}_{18}
= -Y^3 + \varepsilon_6 Y^2 + \varepsilon_{12} Y + \varepsilon_{18}.\]
In this case, if $G_3$ is a cube, then its cube root $H$ must be
the linear part
(with respect to $Y$, $T$)
of $- \widetilde{Y}$. Thus, if either
$\nu_f(\widetilde{\varepsilon}_{12})=2$ or $\nu_f(\widetilde{\varepsilon}_{18})=3$,
then $G_3$ cannot be a cube, and the singularity type must be $D_k$.
We now assume that $G_3$ is in fact a cube, and let $H$ be its cube
root. If $S=E_6$, then the general hyperplane section is at worst
$E_6$ (which is certainly at worst $E_7$), and we are finished.
Suppose instead that $S=E_7$ and $\nu_f(\varepsilon_6)=1$. If
$\varepsilon_6 \equiv \alpha T \mod{T^2}$, then $H = - Y + \frac13 \alpha T$.
The quartic part of $G$ includes the leading term $16YZ^3$. But since
the monomial $TZ^3$ cannot occur in $G$ and $\alpha \ne 0$,
it follows that the quartic part of $G$ cannot be divisible by $H$.
Thus, the general hyperplane section has type $E_6$.
We may therefore assume that either $S = E_7$ and $\nu_f(\varepsilon_6)>1$, or that
$S=E_8$. In
either case, the cube root $H$ is exactly $\pm Y$. This divides the
quartic part of the leading term (which is $16YZ^3$ or 0, respectively).
We can therefore identify which cases have general hyperplane
section $E_6$ or $E_7$ by finding in $M_f$ a monomial not divisible by $Y$,
or one not divisible by $Y^2$, respectively. This is exactly what
is done in the final two columns of table~\ref{tableMONOrev}.
To prove the last statement in the proposition,
note that all labels $L$ to the left of $S_{\ell}$
in table~\ref{tableMONOrev} correspond to singularities with the property that
the
maximum multiplicity
which occurs in the maximal ideal cycle for the singularity is strictly
less than $\ell$. In addition, any singularity whose type is at worst
$L$ has this same property. But if $X$ has an irreducible small
resolution of length $\ell$,
no such singularity can be the general hyperplane
section. Thus, there can be no monomials in columns to the left of that
labeled by $S_{\ell}$.
\end{pf}
\begin{lemma} \label{lem83}
Let $\pi\colon Y \to X$ be an irreducible small resolution
of an isolated Gorenstein threefold singularity, and let
$\{f=0\}$ define a hyperplane section with a rational double point.
Let $\mu_{f \circ \pi}\colon \Delta \to \operatorname{PRes} (S,v) = V/\frak W_0$
be the classifying map determined by $f$, and define a discrete valuation
$\nu_{f \circ \pi}\colon {\Bbb C} [V]^{\frak W_0} \to {\Bbb Z} $
by
\[\nu_{f \circ \pi}(\varphi) =
\text{order of vanishing at $0$ of $\mu^*_{f \circ \pi}(\varphi)$}.\]
If
$\varphi_i \in {\Bbb C} [V]^{\frak W_0}$ is any standard coordinate function on
$\operatorname{PRes} (S,v)$ , then
$\nu_{f \circ \pi}(\varphi_i) \ge 1$. If $\varphi_i$ is in fact a ``constant
term'', then $\nu_{f \circ \pi}(\varphi_i) = 1$.
\end{lemma}
\begin{pf}
The first assertion holds since $\mu_{f\circ\pi}(0)=0$. The
second holds since $Y$ is smooth at the singular point associated with $\varphi_i$.
\end{pf}
Consider again the inclusion of rings
${\Bbb C} [V]^{\frak W} \subset {\Bbb C} [V]^{\frak W_0}$,
which corresponds to the natural projection
$\sigma\colon \operatorname{PRes} (S,v) \to \operatorname{Def} (S)$.
The larger
ring ${\Bbb C} [V]^{\frak W_0}$ is a free polynomial ring generated by the standard
coordinate functions on $\operatorname{PRes} (S,v)$. Thus, each element of the smaller
ring ${\Bbb C} [V]^{\frak W}$ can be written as a polynomial in those standard
coordinate functions. Each monomial in such an expression has a
{\em degree\/} (in the standard coordinate functions
which are generating the ring)
as well as a {\em weight\/} under the background ${\Bbb C} ^*$-action.
For a fixed weight $i$ and degree $d$, we denote by $P_{i,d}$
the subspace of polynomials in
${\Bbb C} [V]^{\frak W_0}$ whose weight is $i$ and whose degree is less than
or equal to $d$.
\begin{lemma} \label{lem84}
Fix a partial resolution type $(S,v)$.
Let $\pi\colon Y \to X$ be an irreducible small resolution of an isolated
Gorenstein
threefold singular point, and let $\{f=0\}$ be a hyperplane section
with partial resolution type $(S,v)$. Let
$\nu_f\colon {\Bbb C} [V]^{\frak W} \to {\Bbb Z} $ be the
associated discrete valuation.
Suppose that $\varphi_i \in {\Bbb C} [V]^{\frak W}$ is a function
of weight $i$, and
$m_i \in P_{i,d}$ is a monomial of degree exactly $d$.
Suppose further that there is an ideal
$I \subset {\Bbb C} [V]^{\frak W_0}$ whose intersection with $P_{i,d}$ is $\{0\}$
such that
\[\varphi_i \equiv c \cdot m_i \mod I\]
for some nonzero constant $c$.
If $\nu_{f\circ\pi}(m_i) = d$, then $\nu_f(\varphi_i) = d$.
\end{lemma}
\begin{pf}
Consider the classifying map
$\mu_{f}\colon \Delta \to \operatorname{Def} (X_0) = \operatorname{Def} (S) = V/\frak W$
determining our threefold $X$, and its associated discrete valuation
$\nu_{f }\colon {\Bbb C} [V]^{\frak W} \to {\Bbb Z} $, as well as the related
map and valuation $\mu_{f \circ \pi}$ and $\nu_{f \circ \pi}$. Note that
$\nu_{f\circ\pi}$ extends $\nu_f$.
Suppose that $\varphi_i - c\cdot m_i$ has degree at most $d$.
Then it lies in $I \cap P_{i,d}$ and so must be $0$. Thus, we may
assume that
$\varphi_i-c\cdot m_i$ has degree strictly greater than $d$. It follows by
lemma~\ref{lem83} that
$\nu_{f \circ \pi}(\varphi_i - c\cdot m_i) > d$. Since
$\nu_{f \circ \pi}(c\cdot m_i) = d$, it follows that $\nu_{f }(\varphi_i) = d$.
\end{pf}
\begin{pf*}{Proof of the Main Theorem}
If $\pi\colon Y\to X$ is as above, we set about
showing that the orders of the standard coordinate functions on
$\operatorname{Def} (S)$ (or the
expressions
$\widetilde{\varepsilon}_i$ in the case of $E_7$) satisfy the relevant hypothesis from
proposition~\ref{prop81}.
We show that a
particular monomial occurs in the defining polynomial of $X$ with nonzero
coefficient,
then we use lemmas~\ref{lem83} and \ref{lem84} together with
proposition~\ref{prop81}
to conclude that the general hyperplane section of $X$ is as
claimed.
We assume henceforth that $(S,v_k)$ is {\em not\/} one of the pairs illustrated
in figure~\ref{figure1}.
This puts certain restrictions on $n$ and $k$ which we will exploit
without comment.
The first case to consider is $S=A_{n-1}$.
The standard coordinate functions on $\operatorname{PRes} (A_{n-1},v_k)$ are
$\mu_1, {\a'_2}, \ldots, {\a'_k}, {\a''_2}, \ldots, {\a''_{n-k}}$.
By proposition~\ref{prop72},
$\a_{n-1} \equiv
\a'_{k-1} \a''_{n-k} + \a'_k \a''_{n-k-1}\mod{\mu_1}$, and
$\a_{n} \equiv \a'_k \a''_{n-k}\mod{\mu_1}$.
Suppose that $k=1$ and $n \ge 2$. The length is 1, and
there is only one ``constant term" in this case:
$\a''_{n-1}$. (Note that there would be no constant terms whatsoever
in case $n=1$.) Since $\a'_0=1$ and $\a'_1=0$ by definition, we have
\[\a_{n-1} \equiv {\a''_{n-1}} \mod{(\mu_1)}.\]
Furthermore $P_{n-1,1} = {\Bbb C} \cdot \a''_{n-1}$, so its intersection with
the ideal $I = (\mu_1)$ is $\{0\}$. (Again we have used $n \ge 2$.)
By lemma~\ref{lem83}, $\nu_f(\a''_{n-1}) = 1$.
By lemma~\ref{lem84}, we conclude that
$\nu_f(\a_{n-1}) = 1$. By proposition~\ref{prop81}, the singularity type
of the general hyperplane
section is at worst $A_1$.
Suppose instead that $1 < k < n-1$. (We can omit the case $k=n-1$ by
symmetry.) In this case, the length is 1,
there are two ``constant terms"
$\a'_k$ and $\a''_{n-k}$, and we have
\[\a_{n} \equiv \a'_k \a''_{n-k}\mod{\mu_1}.\]
If $\varphi \in (\mu_1) \cap P_{n,2}$ then $\varphi = \mu_1 \psi$ for some
$\psi \in P_{n-1,1}$. But
since the maximum weight among the standard coordinate functions
on $\operatorname{PRes} (A_{n-1},v_k)$ is
$\max \{1,k,n-k\} \le n-2$, there can be no affine linear function
in these variables of weight $n-1$. Thus $\psi = 0$, so $\varphi = 0$ and
the intersection of $P_{n,2}$ with $(\mu_1)$ must be $\{0\}$.
By lemma~\ref{lem83}, $\nu_f(\a'_k\a''_{n-k})=2$.
By lemma~\ref{lem84}, we conclude that
$\nu_f(\a_{n}) = 2$. By proposition~\ref{prop81}, it follows that the
singularity type of the general
hyperplane section is at worst $A_1$.
The next case to consider is $S=D_{n}$, $k \le n-2$, with $n \ge 4$.
The standard coordinate functions on $\operatorname{PRes} (D_{n},v_k)$ are
$\mu_1, {\a'_2}, \ldots, {\a'_k},
{\d''_2}, \ldots, {\d''_{2n-2k-2}}, {\c''_{n-k}}$.
By proposition~\ref{prop72}, we have
$\c_n \equiv \a'_k \c''_{n-k}\mod{\mu_1}$, and
$\d_{2n-2} \equiv (\a'_k)^2 \d''_{2n-2k-2}\mod{J_4}$,
where $J_4$ is the ideal generated by all
monomials of degree 4 in the standard coordinate functions on
$\operatorname{PRes} (D_n,v_k)$.
Suppose that $k=1$. The length is 1 and
there is only one ``constant term" in this case:
$\d''_{2n-4}$. Moreover, proposition~\ref{prop72} provides us with
an additional congruence in this case:
\[\d_{2n-4} \equiv \d''_{2n-4}\mod{\mu_1}.\]
We have $P_{2n-4,1} = {\Bbb C} \cdot \d''_{2n-4}$, whose intersection with
the ideal $I=(\mu_1)$ is $\{0\}$.
By lemma~\ref{lem83}, $\nu_f(\d''_{2n-4})=1$.
By lemma~\ref{lem84}, we conclude that
$\nu_f(\d_{2n-4}) = 1$. By proposition~\ref{prop81}, the singularity
type of the general hyperplane
section is at worst $A_1$.
If $1< k< n-1$, the length is 2. Moreover, proposition~\ref{prop72} gives
\[\d_{2n-2}\equiv (\a_k')^2\d_{2n-2k-2}''\mod{J_4}.\]
It is clear from
considering degrees that $P_{2n-2,3}$ intersects $J_4$ trivially.
If $k < n-2$, then there are two constant terms $\a'_k$ and $\d''_{2n-2k-2}$.
By lemma~\ref{lem83}, $\nu_f((\a'_k)^2\d''_{2n-2k-2})=3$.
By lemma~\ref{lem84}, we conclude that
$\nu_f(\d_{2n-2}) = 3$. By proposition~\ref{prop81}, it follows that the
singularity type of the
general hyperplane section is at worst $D_4$.
If $k=n-2$, then by lemma~\ref{lem72} the constant terms are
$\a'_{n-2}$, $-\frac14(\d''_2-2\c''_2)$, and $-\frac14(\d''_2+2\c''_2)$.
According to lemma~\ref{lem83}, each of these has order 1 with
respect to the discrete valuation $\nu_f$; hence, at least one of
$\nu_f(\c''_2)$ and $\nu_f(\d''_2)$ is also equal to 1.
If $\nu_f(\d''_2)=1$, then $\nu_f((\a'_k)^2\d''_{2n-2k-2})=3$ and we
can use the same argument as in the case $k<n-2$ to conclude that
the general hyperplane section has type at worst $D_4$. On the other hand,
if $\nu_f(\c''_2)=1$, we use the additional congruence
\[\c_n \equiv \a'_{n-2} \c''_{2}\mod{\mu_1}\]
provided by proposition~\ref{prop72}.
We have $P_{n,2} \cap (\mu_1) = \{0\}$
since the maximum weight among standard coordinate functions
on $\operatorname{PRes} (D_n,v_{n-1})$ is $n-2$,
and we have $\nu_f(\a'_{n-2} \c''_{2}) = 2$. By lemma~\ref{lem84},
we conclude that
$\nu_f(\c_n) = 2$. By proposition~\ref{prop81}, it follows that the
singularity type of the general
hyperplane section is at worst $D_4$.
The third case to consider is $S=D_{n}$, $k = n$, in which the length is 1.
(By symmetry, we can omit
the case $k=n-1$.)
The standard coordinate functions on $\operatorname{PRes} (D_{n},v_n)$ are
$\mu_1, \a'_2, \ldots, \a'_n$, and $\a'_n$ is the unique ``constant term".
By proposition~\ref{prop72}, we have
\[\c_n \equiv \a'_{n}\mod{\mu_1},\]
and lemma~\ref{lem83} implies that $\nu_f(\a'_n)=1$.
Moreover, since $P_{n,1} = {\Bbb C} \cdot\a'_n$, its intersection with $(\mu_1)$
is $\{0\}$.
By lemma~\ref{lem84}, we conclude that
$\nu_f(\c_n) = 1$. By proposition~\ref{prop81}, it follows that the
singularity type of the general
hyperplane section is at worst $A_1$.
{\renewcommand{\arraystretch}{2}
\begin{table}[p]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
$(S,v_k)$ & length & $S'$ & $S''$ & congruence & monomial & type \\ \hline
\hline
$(E_6,v_0)$ & 2 & $A_5$ & -- &
$\varepsilon_6 \equiv - {\a_6}' \mod I$
& $TZ^2$ & $D_4$ \\ \hline
$(E_6,v_4)$ & 2 & $E_4$ & $A_1$ &
$\varepsilon_5 \equiv - {\varepsilon_5}' \mod I$
& $TYZ$ & $D_4$ \\ \hline
$(E_6,v_5)$ & 1 & $E_5$ & $A_0$ &
$\varepsilon_8 \equiv - \frac14 {\varepsilon_8}' \mod I$
& $TY$ & $A_1$ \\ \hline
\hline
$(E_7,v_0)$ & 2 & $A_6$ & -- &
$\varepsilon_{14} \equiv 64 ({\a_7}')^2 \mod I$
& $T^2Z$ & $D_4$ \\ \hline
$(E_7,v_1)$ & 2 & $D_6$ & -- &
$\varepsilon_{10} \equiv 16 {\d_{10}}' \mod I$
& $TZ^2$ & $D_4$ \\ \hline
$(E_7,v_2)$ & 3 & $A_1$ & $A_5$ &
$\varepsilon_{6} \equiv - 12 {\a_6}'' \mod I$
& $TY^2$ & $E_6$ \\ \hline
$(E_7,v_4)$ & 3 & $E_4$ & $A_2$ &
$\varepsilon_{10} \equiv 16 ({\varepsilon_5}')^2 \mod I$
& $T^2Z^2$ & $E_6$ \\ \hline
$(E_7,v_5)$ & 2 & $E_5$ & $A_1$ &
$\varepsilon_{8} \equiv - 4 {\varepsilon_8}' \mod I$
& $TYZ$ & $D_4$ \\ \hline
$(E_7,v_6)$ & 1 & $E_6$ & $A_0$ &
$\varepsilon_{12} \equiv 16 {\varepsilon_{12}}' \mod I$
& $TY$ & $A_1$ \\ \hline
\hline
$(E_8,v_0)$ & 3 & $A_7$ & -- &
$\varepsilon_{24} \equiv ({\a_8}')^3 \mod I$
& $T^3Z$ & $E_6$ \\ \hline
$(E_8,v_1)$ & 2 & $D_7$ & -- &
$\varepsilon_{24} \equiv
- \frac{1}{16} ({\d_{12}}')^2 \mod I$
& $T^2Z$ & $D_4$ \\ \hline
$(E_8,v_2)$ & 4 & $A_1$ & $A_6$ &
$\varepsilon_{14} \equiv ({\a_7}'')^2 \mod I$
& $T^2YZ$ & $E_7$ \\ \hline
$(E_8,v_5)$ & 4 & $E_5$ & $A_2$ &
$\varepsilon_{8} \equiv - \frac14 {\varepsilon_8}' \mod I$
& $TYZ^2$ & $E_7$ \\ \hline
$(E_8,v_6)$ & 3 & $E_6$ & $A_1$ &
$\varepsilon_{12} \equiv {\varepsilon_{12}}' \mod I$
& $TZ^3$ & $E_6$ \\ \hline
$(E_8,v_7)$ & 2 & $E_7$ & $A_0$ &
$\varepsilon_{18} \equiv
\frac{1}{64} {\varepsilon_{18}}' \mod I$
& $TZ^2$ & $D_4$ \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{Key computations}
\label{table-key1}
\end{table}
}
Finally, we consider the cases with $S=E_n$. Among the standard coordinate
functions
on $\operatorname{PRes} (S,v_k)$, let $\widetilde{\varphi}_N$ be
the ``constant term" of highest weight, say weight $N$.
(This is unique, since we are
avoiding the case $(S,v_k)=(E_6,v_3)$.) Let $I$ be the ideal
in ${\Bbb C} [V]^{\frak W}$ which is generated by all the standard coordinate
functions on $\operatorname{PRes} (S,v_k)$ other than $\widetilde{\varphi}_N$. We select a
standard
coordinate function $\varepsilon_i$ as indicated in table~\ref{table-key1},
and calculate it mod $I$ using the relations given in
proposition~\ref{prop71}. Table~\ref{table-key1} shows the results
of this calculation: we will describe the calculation itself in
section~10.
The calculated result takes the form
\[\varepsilon_i \equiv c \cdot (\widetilde{\varphi}_N)^d \mod{I}\]
for some nonzero constant $c$.
(The key point of the calculation is showing that this constant is
not 0.) Moreover, since $i=Nd$ and $N$ is the highest
weight among standard coordinate functions on $\operatorname{PRes} (S,v_k)$,
any monomial of weight $i$ other than $(\widetilde{\varphi}_N)^d$ must have degree
strictly greater than $d$. It follows that $I \cap P_{i,d} = \{0\}$
and thus by lemma~\ref{lem84}, $\nu_f(\varepsilon_i)=d$.
Let $M_f$ be the set of monomials from proposition~\ref{prop81}.
Since $\nu_f(\varepsilon_i)=d$, we conclude that $M_f$ contains the monomial
shown in the next-to-last column of table~\ref{table-key1}.
The label $L$ of the column in table~\ref{tableMONOrev} in which
that monomial appears has been reproduced in the last column of
table~\ref{table-key1}.
In each case, the label $L$ coincides with the type $S_{\ell}$ which
is associated with the length $\ell$. (The length itself is shown in the
second column.) Thus, by proposition~\ref{prop81},
the set $M_f$ can
contain no monomials to the left of the column labeled by $L$ in
table~\ref{tableMONOrev}, and the singularity type of
the general hyperplane section is at worst $L=S_{\ell}$.
The main theorem then follows from lemma~\ref{lem82}.
\end{pf*}
\section{The computation of preferred versal form in the $E_n$ cases.}
In this section, we explain how to explicitly compute a defining polynomial
in preferred versal form for the standard deformation
in the $E_n$ cases. The result will be a formula for
the standard coordinate functions $\varepsilon_i$ in terms of the elementary symmetric
functions
$s_1,\ldots,s_n$ of the distinguished functionals $t_1,\ldots,t_n$. One's
natural inclination is to expand everything completely as polynomials
in $s_1,\ldots,s_n$, and simply work with things in expanded form. For
this computation, however, that would not be a wise strategy:
the ``constant term" in the case of $E_8$ is a polynomial with
2462 terms.\footnote{It is possible that the coefficients of a few of
these terms may be zero.}
To keep things explicit, and yet in a compact format, we introduce two
notions. By a {\em set of substitution rules}, we mean a set of expressions
of the form $v_i = f_i(x_1,\ldots,x_k)$ which express certain variables
$v_i$ as polynomial functions of other variables $x_j$.
(This notion is more flexible than the notion of a ring homomorphism,
since the rings to which $v_i$ and $x_j$ belong do not need to be
specified until the substitution rules are used.)
We say that a set of
substitution rules ${\cal R}$ is given in {\em solve-list format\/} when it is
specified by three objects ${\cal R}'$, $P$, and $L$ which satisfy a certain
condition, as follows.
${\cal R}'$ is another set of substitution
rules, $P$ is a polynomial,
and $L$ is an ordered list, the {\em solve-list}, consisting of
pairs $(m_i,v_i)$ where $m_i$ is a monomial and $v_i$ is a variable.
(The substitution rules ${\cal R}'$ take the form
$w_{\alpha} = g_{\alpha}(y_1,\ldots,y_{\ell})$, where the $w_{\alpha}$ are
distinct from the $v_i$ but the $y_{\beta}$ may include some $v_i$'s.)
The condition which must be satisfied is this: if $c_i$ is the
coefficient of the monomial $m_i$ in the expression obtained by
substituting the rules ${\cal R}'$ into the polynomial $P$, then the
variable $v_i$ appears linearly in $c_i$ with a nonzero constant
coefficient, and does not appear in any $c_j$ with $j<i$.
(It is allowed,
however, that $v_i$ appear in $c_j$ with $j>i$, and it may even appear
in a nonlinear fashion.)
The algorithm for producing the rules ${\cal R}$ from the triple ${\cal R}'$,
$P$, $L$
is simple: compute the coefficients $c_i$, and for $i = 1, 2, \ldots ,$
successively solve the equations $c_i = 0$ for the variables $v_i$.
In solving the $i^{\text{th}}$ equation, one uses previously found values
of $v_j$ for $j<i$ to eliminate the variables $v_j$ from the expression
$c_i$.
We refer to this process as {\em expanding the solve-list}.
Specifying the rules ${\cal R}$ by means of ${\cal R}'$, $P$, and $L$ (without
actually
expanding the solve-list) can give a relatively compact representation of a
complicated set of
rules. Moreover, our later application of these explicit calculations
will be of the following form: calculate what happens when the rules
${\cal R}$ are restricted to a subspace which is parametrized in a simple
way.
In carrying out those applications, it will be much to our advantage to
begin by pulling back the ingredients of the solve-list format
(i.e. ${\cal R}'$, $P$, and $L$) to the parameter space,
and then computing the pullback of the rules ${\cal R}$ by directly
expanding the pulled-back solve-list.
This part of the computation is explained in
section 10.
We have already encountered (during the proof of proposition~\ref{prop51})
a set of substitution rules which is best
expressed in solve-list format. Let ${\cal R}_{\mu}$ denote the set of
substitution rules
given in equation (\ref{eq23}),
which
describes a change of generators in the algebra $L$. Then the substitution
${\cal R}_{\psi}$ (which describes the coefficients to use in ${\cal R}_{\mu}$
which will
produce the preferred versal form) can be given in solve-list format by
means of the substitution rules
${\cal R}_{\mu}$, the polynomial $\bar{\Phi}_{E_n}$, and the solve-list
\refstepcounter{equation}\label{eq29}
\begin{gather*}
\begin{tabular}{|c|c|c|c|c|c|} \hline
$Y^2Z$ & $XYW$ & $Z^3$ & $XZW$ & $Y^2W$ & $XW^2$ \\ \hline
$\psi_1$ & $\psi_2$ & $\psi'_3$ & $\psi''_3$ & $\psi_4$ & $\psi_6$ \\ \hline
\end{tabular}
\tag{\ref{eq29}a}\\[1.5ex]
\begin{tabular}{|c|c|c|} \hline
$Z^4$ & $YZ^2W$ & $Z^3W$ \\ \hline
$\psi_2$ & $\psi_4$ & $\psi_6$ \\ \hline
\end{tabular}
\tag{\ref{eq29}b}\\[1.5ex]
\begin{tabular}{|c|c|c|} \hline
$Y^2ZW$ & $Z^4W^2$ & $Y^2W^2$ \\ \hline
$\psi_4$ & $\psi_6$ & $\psi_{10}$ \\ \hline
\end{tabular}
\tag{\ref{eq29}c}
\end{gather*}
(We continue to use the convention that equation numbers which are followed by
$a$, $b$, or $c$ refer to the cases of $E_6$, $E_7$, or $E_8$, respectively.)
We computed the corresponding expressions $c_i$ explicitly in
equation (\ref{eq51}),
and verified that $\{ c_i=0\}$ has the appropriate triangular
form.
\bigskip
Throughout this section, the subscript on a variable
(when present)
indicates
its weight with respect to the ${\Bbb C} ^*$-action.
We retain the notation introduced in section 5.
The first step in our computation is to express the anti-pluricanonical
mappings explicitly in coordinates, and thereby obtain a good generating set
$\bar{X}$, $\bar{Y}$, $\bar{Z}$, $\bar{W}$ for $L$.
We identify these generators with polynomials in ${\Bbb C} [V][x,y,z]$
which satisfy certain base conditions.
The anti-canonical mapping (which corresponds to $L_1$)
is given by the cubics passing through the zero-cycle
$\eta(t_1)+\cdots+\eta(t_n)$. In other words, we want cubics
$F \in {\Bbb C} [V][x,y,z]$ such that $\Psi_n(U)$ divides $\eta^*(F)$,
where
$\Psi_n(U) \mathrel{:=} U^n - s_1 U^{n-1} + \dots + (-1)^n s_n$
is the monic polynomial of degree $n$
whose roots are $t_1,\ldots,t_n$.
It is not difficult
to find a basis for these in each case (by hand), and to make the basis
match the
first part of the normalizations established in table~\ref{table45}.
In the case of $E_6$, we get
\refstepcounter{equation}\label{eq1}
\begin{align*}
\begin{split}
\bar{W} & \mathrel{:=} x^3 - y z^2 \\
\bar{Z} & \mathrel{:=} y^ 2z
- s_ 1x^ 2y
+ s_ 2xyz
- s_ 3x^ 3
+ s_ 4x^2z
- s_ 5xz^ 2
+ s_ 6z^ 3 \\
\bar{Y} & \mathrel{:=} x y^ 2
- s_ 1y^ 2z
+ s_ 2x^ 2y
- s_ 3xyz
+ s_ 4x^3
- s_ 5x^ 2z
+ s_ 6xz^ 2 \\
\bar{X} & \mathrel{:=} y^ 3
+( s_ 2- s_ 1^ 2)xy^ 2
-( s_ 3- s_ 1 s_ 2)y^ 2z
+( s_ 4- s_ 1 s_ 3)x^ 2y
\\ &\qquad
-( s_ 5- s_ 1 s_ 4)xyz
+( s_ 6- s_ 1 s_ 5)x^ 3
+ s_ 1 s_ 6x^ 2z
\end{split}
\tag{\ref{eq1}a}
\end{align*}
which gives a good generating set for $L$, since $L$ is generated by
$L_1$ in this case.
In the case of $E_7$, a basis for the cubics
is given by the first three lines of equation~(\ref{eq1}b)
below.\footnote{This basis is chosen to match the one used by Bramble
\cite{[Bra]}; a simpler choice would have used $\bar{Y} - s_1^2 \bar{Z}$.}
This is completed to a good generating set for $L$ by using $\frac13$ of
the Jacobian determinant as the fourth generator.
\begin{align*}
\begin{split}
\bar{W} & \mathrel{:=} x^3
- y z^2 \\
\bar{Z} & \mathrel{:=} x y^ 2
- s_ 1y^ 2z
+ s_ 2x^ 2y
- s_ 3xyz
+ s_ 4x^
3
- s_ 5x^ 2z
+ s_ 6xz^ 2
-
\makebox[0pt][l]{$
s_7 z^3
$}
\\
\bar{Y} & \mathrel{:=} 4 y^ 3
+( 4 s_ 2
- 4 s_ 1^ 2
+ s_1^2)xy^ 2
-( 4 s_ 3
-
4 s_ 1 s_ 2
+ s_1^3)y^ 2z
+( 4 s_ 4
\\ &\qquad
- 4 s_ 1 s_ 3
+ s_1^2 s_2)x^ 2y
- ( 4 s_ 5
- 4 s_ 1 s_ 4
+ s_1^2 s_3)xyz
+(4 s_ 6
\\ &\qquad
- 4 s_ 1 s_ 5
+ s_1^2 s_4)
x^ 3
- (4 s_7
- 4 s_ 1 s_ 6
+ s_1^2 s_5) x^ 2z
+ (- 4 s_1 s_7
\\ &\qquad
+ s_1^2 s_6) x z^2
- s_1^2 s_7 z^3 \\
\bar{X} & \mathrel{:=} \tfrac13 \
\tfrac{\partial(\bar{Y},\bar{Z},\bar{W})}{\partial(x,y,z)}.
\end{split}
\tag{\ref{eq1}b}
\end{align*}
In the case of $E_8$, a basis for the cubics in $L_1$ is given by
\begin{align*}
\begin{split}
\bar{W} & \mathrel{:=} x^3
- y z^2 \\
\bar{Z} & \mathrel{:=} y^ 3
+( s_ 2
- s_ 1^ 2)xy^ 2
-( s_ 3
-
s_ 1 s_ 2)y^ 2z
+( s_ 4
- s_ 1 s_ 3)x^ 2y
\\ &\qquad
- ( s_ 5
- s_ 1 s_ 4)xyz
+ ( s_ 6
- s_ 1 s_ 5)
x^ 3
- (s_7
- s_ 1 s_ 6) x^ 2z
\\ &\qquad
+ (s_8
- s_1 s_7) x z^2
+ s_1 s_8 z^3.
\end{split}
\tag{\ref{eq1}c}
\end{align*}
A basis for the sextics which determine the anti-bicanonical map is then
given by quadratic expressions in these cubics (i.e. $\operatorname{Sym} ^2L_1$)
together with a new sextic $F$. To match the normalizations from
table~\ref{table45}, we assume that $F$ has weight 16 and satisfies
$F \equiv xy^5 \mod \frak m $.
We then need to ensure that $F$ has multiplicity 2 along the zero-cycle
$\eta(t_1)+\ldots+\eta(t_8)$. We do this by imposing two conditions
on $F$:
(i) $\eta^*(F) = \Psi_8(U)^2$ and
(ii) $\Psi_8(U)$ divides $\eta^*({\partial F}/{\partial x})$.
The first condition guarantees that $F$ meets $C$ with multiplicity at least
2 at each point in the zero-cycle. (Note that $F$ and $\Psi_8(U)^2$ are monic
of the same degree). The second guarantees that {\em one\/} of the partial
derivatives of $F$ vanishes at those points. But now by the chain rule,
\[\eta^*(\frac{\partial F}{\partial x}) + 3 U^2
\eta^*(\frac{\partial F}{\partial y})
= 2 \Psi_8(U)' \Psi_8(U).\]
It follows that $\Psi_8(U)$ divides $\eta^*(\partial F/\partial y)$ as well,
which implies that the other partial derivative vanishes at those points,
and therefore that $F$ has multiplicity 2 along the zero-cycle
$\eta(t_1)+\cdots+\eta(t_n)$.
Finding $F$ explicitly is now a matter of solving the equations in the
coefficients of $F$ implied by these conditions. The answer is not unique,
but still depends on 2 free parameters. We used {\sc maple} and {\sc reduce}
to solve the
equations, and made a choice for the free parameters which makes the
coefficients of $x^3y^3$ and $x^6$ both 0.
We use this polynomial $F$ as the third element $\bar{Y}$ of a good
generating set for $L$, and complete the good generating set
by using $- \frac16$ of
the Jacobian determinant as $\bar{X}$, that is,
\[ \bar{X} \mathrel{:=} - \frac16 \
\frac{\partial(\bar{Y},\bar{Z},\bar{W})}{\partial(x,y,z)}.\]
This good generating set is shown explicitly in Appendix 0;
in particular, the polynomial used as $\bar{Y}$ is displayed there
explicitly.
\bigskip
The second step of our computation is to compute the defining polynomial
$\bar{\Phi}_{E_n}$ of $\bar{\cal P} \subset \text{\bf Proj} _V(L)$ with respect to
the generating set $\bar{X}$, $\bar{Y}$, $\bar{Z}$, $\bar{W}$.
Such a polynomial was shown to exist in lemma~\ref{lem57}, and its
general form was given
in equation (\ref{eq11});
we must
compute the unknown coefficients which appear in that equation.
Thanks to the remark following proposition~\ref{prop51}, we only
need the coefficients of low weight at this stage of the computation.
Let $\pi\colon \P ^2 \times V \to \bar{\cal P} \subset \text{\bf Proj} _V(L)$
be the rational
map determined by the generating set
$\bar{X}$, $\bar{Y}$, $\bar{Z}$, $\bar{W}$.
Now $\bar{\Phi}_{E_n}$ vanishes on the image of $\pi$. Thus, if we write
$\bar{\Phi}_{E_n}$ with undetermined coefficients and
compute $\pi^*(\bar{\Phi}_{E_n})$,
every coefficient in the expression for $\pi^*(\bar{\Phi}_{E_n})$ must vanish.
This produces some equations for the undetermined coefficients.
\begin{proposition} \label{prop91}
Let $\bar{\cal R}_{\pi}$ be the set of substitution rules which describes
the map $\pi$ with respect to the coordinates
$\bar{X}$, $\bar{Y}$, $\bar{Z}$, $\bar{W}$, as given in equations
(\ref{eq1}a), (\ref{eq1}b)
and Appendix 0.
Then the coefficients
of low weight in the equation $\bar{\Phi}_{E_n}$ can be described by a set of
substitution rules ${\cal R}_{\nu}$, which are given in solve-list
format by means of the substitution rules
$\bar{\cal R}_{\pi}$, the polynomial $\bar{\Phi}_{E_n}$, and the solve-list
\refstepcounter{equation}\label{eq17}
\begin{gather*}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$x^2y^6z$ & $x^4y^5$ & $xy^6z^2$ & $x^3y^5z$ &
$y^6z^3$ & $x^5y^4$ & $x^4y^4z$ & $x^6y^3$ &
$x^3y^4z^2$
\\ \hline
$\bar{\varphi}_1$ & $\bar{\varphi}_2$ & $\bar{\varepsilon}_2$ & $\bar{\varphi}'_3$ &
$\bar{\varphi}''_3$ & $\bar{\varphi}_4$ & $\bar{\varepsilon}_5$ & $\bar{\varphi}_6$ &
$\bar{\varepsilon}_6$
\\ \hline
\end{tabular}
\tag{\ref{eq17}a}\\[1.5ex]
\begin{tabular}{|c|c|c|c|c|} \hline
$x^4 y^8$ & $x y^9 z^2$ & $x^5 y^7$ & $x^6 y^6$
& $x^3 y^7 z^2$ \\ \hline
$\bar{\varepsilon}_2$ & $\bar{\varphi}_2$ & $\bar{\varphi}_4$ & $\bar{\varepsilon}_6$
& $\bar{\varphi}_6$ \\ \hline
\end{tabular}
\tag{\ref{eq17}b}\\[1.5ex]
\begin{tabular}{|c|c|c|c|c|} \hline
$x^{4} y^{14}$ & $x^{5} y^{13}$ & $x^{6} y^{12}$ & $x^{7} y^{11}$
& $x^{8} y^{10}$ \\ \hline
$\bar{\varepsilon}_{2}$ & $\bar{\varphi}_{4}$ & $\bar{\varphi}_{6}$ & $\bar{\varepsilon}_{8}$
& $\bar{\varphi}_{10}$ \\ \hline
\end{tabular}
\tag{\ref{eq17}c}
\end{gather*}
\end{proposition}
\begin{pf}
We carry out the expansion specified by the solve-list format, but
work mod $\frak m $. The congruences in
table~\ref{table45} give substitutions for
$\bar{X}$, $\bar{Y}$, $\bar{Z}$, $\bar{W}$ which are valid mod $\frak m $.
If we make those substitutions into equation
(\ref{eq11})
and collect terms of low
degree in $z$ we obtain:
\begin{align*}
\begin{split}
\pi^*(\bar{\Phi}_{E_6}) &\equiv
\bar{\varphi}_1 x^2 y^6 z
+ \bar{\varphi}_2 x^4 y^5
+ (\bar{\varepsilon}_2 - \bar{\varphi}_2) x y^6 z^2
+ \bar{\varphi}'_3 x^3 y^5 z
+ (\bar{\varphi}''_3 - \bar{\varphi}'_3) y^6 z^3
\\ &\qquad
+ \bar{\varphi}_4 (x^5 y^4 - x^2 y^5 z^2)
+ \bar{\varepsilon}_5 (x^4 y^4 z - x y^5 z^3)
+ \bar{\varphi}_6 x^6 y^3
+ (\bar{\varepsilon}_6 - 2 \bar{\varphi}_6) x^3 y^4 z^2
\\ &\qquad
+ \bar{\varepsilon}_8 (x^7 y^2 - 2 x^4 y^3 z^2)
+ \bar{\varepsilon}_9 (x^6 y^2 z - 2 x^3 y^3 z^3)
+ \bar{\varepsilon}_{12} (x^9 - 3 x^6 y z^2) \\
&\qquad \mod{(\frak m ,z^4)}
\end{split}
\\[1.5ex] \begin{split}
\pi^*(\bar{\Phi}_{E_7}) &\equiv
16 \bar{\varepsilon}_2 x^4 y^8
+ (16 \bar{\varphi}_2 - 16 \bar{\varepsilon}_2) x y^9 z^2
+ \bar{\varphi}_4 (4 x^5 y^7 - 4 x^2 y^8 z^2)
+ 16 \bar{\varepsilon}_6 x^6 y^6
\\ &\qquad
+ (16 \bar{\varphi}_6 - 32 \bar{\varepsilon}_6) x^3 y^7 z^2
+ \bar{\varepsilon}_8 (4 x^7 y^5 - 8 x^4 y^6 z^2)
+ \bar{\varepsilon}_{10} (x^8 y^4 - 2 x^5 y^5 z^2)
\\ &\qquad
+ \bar{\varepsilon}_{12} (4 x^9 y^3 - 12 x^6 y^4 z^2)
+ \bar{\varepsilon}_{14} (x^{10} y^2 - 3 x^7 y^3 z^2)
+ \bar{\varepsilon}_{18} (x^{12} - 4 x^9 y z^2)
\\ &\qquad \mod{(\frak m ,z^3)} \end{split}
\\[1.5ex] \begin{split}
\pi^*(\bar{\Phi}_{E_8}) &\equiv
\bar{\varepsilon}_2 x^4 y^{14}
+ \bar{\varphi}_4 x^5 y^{13}
+ \bar{\varphi}_6 x^6 y^{12}
+ \bar{\varepsilon}_8 x^7 y^{11}
+ \bar{\varphi}_{10} x^8 y^{10}
+ \bar{\varepsilon}_{12} x^9 y^9
\\ &\qquad
+ \bar{\varepsilon}_{14} y^8 x^{10}
+ \bar{\varepsilon}_{18} x^{12} y^6
+ \bar{\varepsilon}_{20} x^{13} y^5
+ \bar{\varepsilon}_{24} x^{15} y^3
+ \bar{\varepsilon}_{30} x^{18}
\\ &\qquad \mod{(\frak m ,z)}.
\end{split}
\end{align*}
Since each equation which is to be solved is a coefficient of a monomial
in $x, y, z$, it is homogeneous (with respect to the
background ${\Bbb C} ^*$-action). For each such equation, the $\bar{\varepsilon}_i$'s
and $\bar{\varphi_i}$'s involved are either the ones given above, or ones of
strictly lower weight (since they are multiplied by nontrivial functions
of the $t_i$'s). Thus, if we proceed from lower weight to higher weight
in solving these equations, and use the leading order terms above as
a guide to the order in which equations of the same weight should be
solved, we arrive at the solve-lists stated in the proposition.
\end{pf}
The solve-lists given in proposition~\ref{prop91} can be extended to
determine the entire defining polynomial in these coordinates. We did this,
and
expanded the extended solve-lists using {\sc maple} and {\sc reduce}
in the cases of $E_6$ and $E_7$.
In the case of $E_6$, we obtained the expanded defining polynomial
\begin{align*}
\begin{split}
\bar{\Phi}_{E_6} &=
- \bar{X}^2 \bar{W} - \bar{X} \bar{Z}^2 + \bar{Y}^3
+ {s_{1}} \bar{Y}^2 \bar{Z}
- { {s_{2}}} \bar{X} \bar{Y} \bar{W}
+ 0 \bar{Y} \bar{Z}^2
- { {s_{3}}} \bar{X} \bar{Z} \bar{W}
\\ &\qquad
+ 0 \bar{Z}^3
- { {s_{4
}}} \bar{Y}^2 \bar{W}
+ ({ {s_{5}} { - {s_{1}} {s_{4}}}}) \bar{Y} \bar{Z} \bar{W}
+ ({ { 2 {s_{6}}} { - {s_{1}} {s
_{5}}}}) \bar{X} \bar{W}^2
+ 0 \bar{Z}^2 \bar{W}
\\ &\qquad
+ ({ { {s_{2}} {s_{6}} + { {
{s_{1}}^{2}} {s_{6}}} { - {s_{2}} {s_{1}} {s_{5}}} + { {s_{5}} {s_{3
}}}}}) \bar{Y} \bar{W}^2
+ ({ { {s_{2}} {s_{1
}} {s_{6}}} { - {s_{3}} {s_{6}}}}) \bar{Z} \bar{W}^2
\\ &\qquad
+ ({ { {s_{1}} {s_{6
}} {s_{2}} {s_{3}}} {- { {s_{6}}^{2}}} + { {s_{5}} {s_{1}} {s_{6}}} {
- { {s_{1}}^{2}} {s_{4}} {s_{6}}} { - { {s_{3}}^{2}} {s_{6}}}})
\bar{W}^3.
\end{split}
\end{align*}
In the case of $E_7$, the defining polynomial which we found
agrees with the one found by Bramble \cite[p.\ 357]{[Bra]}, and we have
not reproduced it here. The defining polynomial for $E_8$ is very large, and
we have not
attempted to write it down.
\bigskip
We can now describe a set of substitution rules ${\cal R}_{\pi}$
which describes the map $\pi$ with respect to the coordinates
$X$, $Y$, $Z$, $W$ as being a composition
\[{\cal R}_{\pi} = {\cal R}_{\mu}^{-1} \circ \bar{\cal R}_{\pi}
\circ {\cal R}_{\psi} \circ {\cal R}_{\nu},\]
where we have used the fact that the ${\cal R}_{\mu}$ as
given in equation (\ref{eq23})
can be solved for $X$, $Y$, $Z$, $W$ as functions of the other
variables, yielding ${\cal R}_{\mu}^{-1}$.
\begin{proposition} \label{prop92}
Let ${R}_{\pi}={\cal R}_{\mu}^{-1} \circ \bar{\cal R}_{\pi}
\circ {\cal R}_{\psi} \circ {\cal R}_{\nu}$
be the set of substitution rules which describes
the map $\pi$ with respect to the coordinates
${X}$, ${Y}$, ${Z}$, ${W}$. Then the coefficients
in the defining polynomial ${\Phi}_{E_n}$ can be described by a set of
substitution rules which are given in solve-list
format by means of the polynomial ${\Phi}_{E_n}$ (with undetermined
coefficients),
the substitution rules
${R}_{\pi}$, and the solve-list
\refstepcounter{equation}\label{eq41}
\begin{gather*}
\begin{tabular}{|c|c|c|c|c|c|} \hline
$xy^6z^2$ &
$x^4y^4z$ &
$x^3y^4z^2$ & $x^7y^2$ & $x^6y^2z$ & $x^6yz^2$
\\ \hline
$\varepsilon_2$ &
$\varepsilon_5$ &
$\varepsilon_6$ & $\varepsilon_8$ & $\varepsilon_9$ & $\varepsilon_{12}$
\\ \hline
\end{tabular}
\tag{\ref{eq41}a}\\[1.5ex]
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
$x^4 y^8$
& $x^6 y^6$ & $x^7 y^5$ & $x^8 y^4$ & $x^9 y^3$
& $x^{10} y^2$ & $x^{12}$ \\ \hline
$\varepsilon_2$
& $\varepsilon_6$ & $\varepsilon_8$ & $\varepsilon_{10}$ & $\varepsilon_{12}$
& $\varepsilon_{14}$ & $\varepsilon_{18}$ \\ \hline
\end{tabular}
\tag{\ref{eq41}b}\\[1.5ex]
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
$x^{4} y^{14}$ & $x^{7} y^{11}$
& $x^{9} y^{9}$ & $x^{10} y^{8}$ & $x^{12} y^{6}$
& $x^{13} y^{5}$ & $x^{15} y^{3}$ & $x^{18}$ \\ \hline
$\varepsilon_{2}$ & $\varepsilon_{8}$
& $\varepsilon_{12}$ & $\varepsilon_{14}$ & $\varepsilon_{18}$
& $\varepsilon_{20}$ & $\varepsilon_{24}$ & $\varepsilon_{30}$ \\ \hline
\end{tabular}
\tag{\ref{eq41}c}
\end{gather*}
\end{proposition}
\begin{pf}
The defining polynomial in preferred versal form
with undetermined coefficients is:
\refstepcounter{equation}\label{eq35}
\begin{align*}
\begin{split}
\Phi_{E_6} &=
- X^2 W - X Z^2 + Y^3
+ \varepsilon_2 Y Z^2 + \varepsilon_5 Y Z W + \varepsilon_6 Z^2 W + \varepsilon_8 Y W^2
\\ &\qquad
+ \varepsilon_9 Z W^2 + \varepsilon_{12} W^3
\end{split}
\tag{\ref{eq35}a}
\\[1.5ex] \begin{split}
\Phi_{E_7} &=
- X^2 - Y^3 W + 16 Y Z^3
+ \varepsilon_2 Y^2 Z W + \varepsilon_6 Y^2 W^2
+ \varepsilon_8 Y Z W^2
\\ &\qquad
+ \varepsilon_{10} Z^2 W^2 + \varepsilon_{12} Y W^3
+ \varepsilon_{14} Z W^3
+ \varepsilon_{18} W^4
\end{split}
\tag{\ref{eq35}b}
\\[1.5ex] \begin{split}
\Phi_{E_8} &=
- X^2 + Y^3 - Z^5 W
+ \varepsilon_2 Y Z^3 W + \varepsilon_8 Y Z^2 W^2 + \varepsilon_{12} Z^3 W^3
\\ &\qquad
+ \varepsilon_{14} Y Z W^3
+ \varepsilon_{18} Z^2 W^4
+ \varepsilon_{20} Y W^4
+ \varepsilon_{24} Z W^5
+ \varepsilon_{30} W^6.
\end{split}
\tag{\ref{eq35}c}
\end{align*}
As before, we carry out the procedure specified by the solve-list format,
working mod $\frak m $. The congruences in
table~\ref{table45} give substitutions for
${X}$, ${Y}$, ${Z}$, ${W}$ which are valid mod $\frak m $.
If we make those substitutions into the equation for $\Phi_{E_n}$ above
and collect terms of low
degree in $z$ we obtain:
\begin{align*}
\begin{split}
\pi^*({\Phi}_{E_6}) &\equiv
{\varepsilon}_2 x y^6 z^2
+ {\varepsilon}_5 x^4 y^4 z
+ {\varepsilon}_6 x^3 y^4 z^2
+ {\varepsilon}_8 (x^7 y^2 - 2 x^4 y^3 z^2)
\\ &\qquad
+ {\varepsilon}_9 x^6 y^2 z
+ {\varepsilon}_{12} (x^9 - 3 x^6 y z^2)
\\ &\qquad \mod{(\frak m ,z^3)}
\end{split}
\\[1.5ex] \begin{split}
\pi^*({\Phi}_{E_7}) &\equiv
16 {\varepsilon}_2 x^4 y^8
+ 16 {\varepsilon}_6 x^6 y^6
+ {\varepsilon}_8 4 x^7 y^5
+ {\varepsilon}_{10} x^8 y^4
\\ &\qquad
+ {\varepsilon}_{12} 4 x^9 y^3
+ {\varepsilon}_{14} x^{10} y^2
+ {\varepsilon}_{18} x^{12}
\\ &\qquad \mod{(\frak m ,z)} \end{split}
\\[1.5ex] \begin{split}
\pi^*({\Phi}_{E_8}) &\equiv
{\varepsilon}_2 x^4 y^{14}
+ {\varepsilon}_8 x^7 y^{11}
+ {\varepsilon}_{12} x^9 y^9
+ {\varepsilon}_{14} y^8 x^{10}
+ {\varepsilon}_{18} x^{12} y^6
\\ &\qquad
+ {\varepsilon}_{20} x^{13} y^5
+ {\varepsilon}_{24} x^{15} y^3
+ {\varepsilon}_{30} x^{18}
\\ &\qquad \mod{(\frak m ,z)}.
\end{split}
\end{align*}
The same argument used in the proof of proposition~\ref{prop91} shows that
if we proceed from lower weight to higher weight
in solving these equations,
we arrive at the solve-lists stated in the proposition.
\end{pf}
It is important to notice that the solve-lists for $E_7$ and $E_8$
given in proposition~\ref{prop92} do
not involve $z$; it is therefore possible (and desirable) to set $z=0$
at the beginning of any computation involving these solve-lists.
We have expanded these solve-lists using {\sc maple} and {\sc reduce}
in the cases of $E_6$ and $E_7$;
the results of this computation
are displayed in Appendices 1 and 2, respectively.
(The results for $E_7$ can be found, with some errors, in
Bramble \cite{[Bra]}; Appendix 2 gives the corrected results.)
The results for $E_8$ are too large to contemplate writing down.
We stress however that even in the cases of $E_6$ and $E_7$,
further calculations with these formulas are best done by leaving
them in solve-list format as long as possible, and only expanding
the solve-lists at the very end, after restricting to a suitable subspace.
\section{Restricted polynomials and
the main computation.}
In this section, we will justify the congruences stated in
table~\ref{table-key1}.
These congruences all take the form
$\varepsilon_i \equiv c \cdot (\widetilde{\varphi}_N)^d \mod I$.
We will in fact compute $\varepsilon_i$ modulo another
ideal $J \subset I$ which is generated by a certain subset of the set of
standard
coordinate functions, and note that our desired congruence is an
immediate consequence of this computation.
In order to describe the ideal $J$ and our computational method,
we must first describe in parametric form some subspaces
of the deformation spaces $\operatorname{Def} (S)$.
Let $R$ be a root system of type $S$, let $V_R$ be the complex root
space, and let $t_1,\ldots,t_n$ be the distinguished functionals,
with $s_1,\ldots,s_n$ their elementary symmetric functions.
Suppose that we are given a vector space $W$ and a monic
polynomial $r_S(U)$ of degree $n$ in $U$, whose coefficients lie in
the ring
${\Bbb C} [W]$ of polynomial functions on $W$.
This determines a map $\psi\colon W \to V_R/\frak S_n$ by means of the action on
polynomials
$\psi^*\colon{\Bbb C} [V_R]^{\frak S_n} \to {\Bbb C} [W]$
defined by
\[\psi^*(s_i) = \text{the coefficient of } U^{n-i} \text{ in } r_S(U).\]
In particular, the pullback of the distinguished polynomial is
$\psi^*(f_S(U;t)) = r_S(U)$. We call $r_S(U)$ the {\em restricted
polynomial\/} associated to $\psi$.
We wish to describe a particular case of this construction
in which the image of $\psi$ is defined
by an ideal $J_S$ which is generated by a subset of the standard
coordinate functions on $\operatorname{Def} (S)$. When we have done so, we will call the
generators of $J_S$ the {\em vanishing coordinates}, and refer to
coordinates on $W$ as {\em parameters}. We want $J_S$ to be as large
as possible, yet not to contain the ``constant term".
Equivalently, $W$ parametrizes a subspace of $\operatorname{Def} (S)$ on which many
of the standard coordinate functions (but not the ``constant term") vanish.
If $S=A_{n-1}$, let $J_{A_{n-1}}$ be the ideal
generated by all the standard
coordinate functions on $\operatorname{Def} (A_{n-1})$ other than the ``constant term".
Since the standard coordinate functions are the elementary symmetric
functions $s_i$ themselves,
it is easy to construct a restricted polynomial
for the ideal $J_{A_{n-1}}$. There are two natural choices:
we use either $r_{A_{n-1}}(U) = U^n + \l_n$, or
$r_{A_{n-1}}(U) = U^n - \l_1^n$. (We will
use the second form when we need an explicit
root $\l_1$ of the restricted polynomial.)
If $S=D_n$ and $n$ is even, we again let $J_{D_n}$ be the ideal
generated by all the standard
coordinate functions on $\operatorname{Def} (D_{n})$ other than the ``constant term".
The construction of a restricted polynomial in this case
is based on a special
factorization property: if we define
\[F(U) = U^n - \l_{n-1} U,\]
and
\[G(-U^2) = F(U) \cdot F(-U)\]
then
\[G(Z) = Z^n + \l_{n-1}^2 Z.\]
Thus, if we let $r_{D_{n}}(U) = F(U)$, then
the pullbacks via
$\psi^*$ of the standard coordinate functions will be the coefficients
of $G(Z)$, together with one coefficient of $F(U)$. It follows that
$\psi^*$ of all standard coordinate functions except for the
``constant term" vanish.
In the remaining cases, we can find an ideal $J_S$ and a
restricted polynomial as follows.
Begin with parameters $\l_1,\ldots,\l_n$, the initial restricted
polynomial
$U^n + \sum \l_i U^{n-i}$, and the map given by $\psi^*(s_i) = \l_i$.
The pullbacks of the standard coordinate functions $\varphi_j$ via $\psi^*$
can be computed
as functions of the $\l_i$. If the weight $j$ of a standard
coordinate function $\varphi_j$ is at most $n$,
then $\l_j$ appears in the formula for that
coordinate function with a nonzero constant coefficient.
(This must be checked case by case.) It follows that
if we set all such standard coordinate functions
equal to zero, we get a triangular
system of linear equations in a subset of the set $\{\l_i\}$ of parameters.
These equations can be derived from equations (\ref{eqD}),
(\ref{eqE4}), and
(\ref{eqE5}) in the cases of $D_7$,
$E_4$, and $E_5$ respectively, and from Appendices~1 and 2 in the
case of $E_6$ and $E_7$.
We used {\sc maple} and {\sc reduce} to solve the equations in those
5 cases; the resulting
restricted polynomials are shown in table~\ref{tableABC}.
If $\varphi_N$ is the ``constant term" (or for that matter any of the
standard coordinate functions on $\operatorname{Def} (S)$) it is a straightforward
matter to compute $\psi^*(\varphi_N)$, based on the description of the
mapping $\psi$ which is given by the coefficients of the restricted
polynomial as shown in table~\ref{tableABC}. For this purpose, one
again uses equations (\ref{eqD}), (\ref{eqE4}), and
(\ref{eqE5}) in the cases of $D_7$, $E_4$, and $E_5$, respectively.
We have carried out this computation using {\sc maple} and
{\sc reduce}, and displayed the answers in the last column of
table~\ref{tableABC}.
In the case of $E_6$ and $E_7$, it is more efficient to
perform this computation directly from the solve-list description
of the standard coordinate functions on $\operatorname{Def} (S)$
which was given in section 9.
(That is, we use the explicit description of $\psi^*(s_i)$ to substitute for
$s_i$ in the ingredients of the solve-list, and then solve the
resulting equations.)
When this is done, formulas $e_{12}(\l_1, \l_3, \l_4)$
and $e_{18}(\l_1, \l_3, \l_4,\l_5, \l_7)$ are obtained which express
the pullbacks of the respective ``constant terms" $\varepsilon_{12}$ and $\varepsilon_{18}$
in terms of the parameters. We calculated these formulas using
{\sc maple} and
{\sc reduce}, but as they are a bit long, we have not displayed them
in the table.
{\renewcommand{\arraystretch}{1.2}
\begin{table}[p]
\begin{center}
\begin{tabular}{|c|c|l|c|} \hline
& & & \\
$S$ & Vanishing &
\multicolumn{1}{c|}{Restricted Polynomial}
& ``Constant Term'' \\
& Coordinates & \multicolumn{1}{c|}{$r_S(U)$} & $\psi^*(\varphi_N)$ \\
& & & \\ \hline
& & & \\
$A_{n-1}$
& $\a_j$, $_{2 \le j \le n-1}$
& \multicolumn{1}{c|}{$U^n+\l_n$\ \ \ \text{or}\ \ \ $U^n - \l_1^n$}
& $\l_n$\ \ \ or\ \ \ $- \l_1^n$ \\
& & & \\
$D_n$ &
$\c_n$, $\d_{2j}$, & \multicolumn{1}{c|}{$U^n - \l_{n-1} U$}
& $\l_{n-1}^2$ \\
$_{n\ \text{even}}$ & $_{1 \le j \le n-2}$ & & \\
& & & \\
$D_7$
& $\d_2, \d_4, \d_6, \c_7$ &
$U^7 + \l_1 U^6 + \frac12 \l_1^2 U^5 + \l_3 U^4 +
(\l_1 \l_3
$ & $(\l_1 \l_5 - \frac12 \l_3 \l_1^3 + \frac{1}{16} \l_1^6
+ \frac12 \l_3^2)^2$ \\
& & $ -\ \frac18 \l_1^4) U^3 + \l_5 U^2
+
(\l_1 \l_5 - \frac12 \l_3 \l_1^3 $
& \\
& & $ +\ \frac{1}{16} \l_1^6
+ \frac12 \l_3^2) U$
& \\
& & & \\
$E_4$ & $\varepsilon_2$, $\varepsilon_3$, $\varepsilon_4$
& $U^4 + \l_1U^3 + \frac35 \l_1^2U^2 + \frac{1}{25}\l_1^3U + \frac{11}{125}
\l_1^4$ & $\frac{243}{3125}\l_1^5$ \\
& & & \\
$E_5$ & $\varepsilon_2, \varepsilon_4, \varepsilon_5$ &
$ U^5 + \l_1U^4 + \frac58 \l_1^2U^3 + \l_3U^2 + (
\frac{15}{128} \l_1^4 $
& ${ { \frac{ 2601 }{16384}}{ \l_{1}^{8}} + { \frac{ 9 }{4}}{ \l
_{3}^{2}} { \l_{1}^{2}} { - \frac{ 153 }{128}}{ \l_{1}^{5}} \l_{3
}}$ \\
& & $ -\ \frac12 \l_1\l_3)U + ( \frac{27}{256}\l_1^5 - \frac12 \l_1^2\l_3)$
& \\
& & & \\
$E_6$ & $\varepsilon_2, \varepsilon_5, \varepsilon_6$ &
$U^6 + \l_1 U^5 + \frac23 \l_1^2 U^4 + \l_3 U^3 + \l_4 U^2
$ & $e_{12}(\l_1, \l_3, \l_4)$ \\
& & $
+\ ( \frac13 \l_1 \l_4 - \frac13 \l_3 \l_1^2 + \frac{2}{27} \l_1^5) U
$ & \\
& & $
+\ ( \frac{5}{18} \l_1^2 \l_4 - \frac19 \l_3 \l_1^3 +
\frac{11}{486} \l_1^6 - \frac18 \l_3^2 )$
& \\
& & & \\
$E_7$ & $\varepsilon_2, \varepsilon_6$ &
$U^7 + \l_1 U^6 + \frac34 \l_1^2 U^5 + \l_3 U^4 + \l_4 U^3
$ & $e_{18}(\l_1, \l_3, \l_4,\l_5, \l_7)$ \\
& & $
+\ \l_5 U^2 + ( - \frac18 \l_3^2 + \frac{3}{64} \l_1^6 - \frac{3}{16} \l_3
\l_1^3
$ & \\
& & $
-\ \frac14 \l_1 \l_5 + \frac38 \l_1^2 \l_4 ) U
+ \l_7 $ & \\
& & & \\ \hline
\end{tabular}
\end{center}
\medskip
\caption{}
\label{tableABC}
\end{table}
}
We are now ready to explain how the congruences in table~\ref{table-key1} are
derived.
Let $(E_n,v_k)$ be one of the pairs considered in table~\ref{table-key1}.
The ``constant term" of highest weight $\widetilde{\varphi}_N$ is induced from the
projection onto a subspace $\widetilde{V}/\widetilde{\frak W}$ corresponding
to an irreducible subsystem $\widetilde{R}$ of the root system
$R_{E_n}$. (In all cases except $(E_7,v_2)$ and $(E_8,v_2)$, $R$
is the ``left part" $R'$; in those two cases,
$R$ is the ``right part" $R''$.)
As in section 8, let $I$ be the ideal
in ${\Bbb C} [V]^{\frak W}$ which is generated by all the standard coordinate
functions on $\operatorname{PRes} (E_n,v_k)$ other than $\widetilde{\varphi}_N$.
Let $J$ be the ideal in ${\Bbb C} [V]^{\frak W}$ which is generated by
$J_{\widetilde{R}}$, $\mu_1$, and all the standard coordinate functions
on $\operatorname{PRes} (E_n,v_k)$ which come from $R - \{v_k\} - \widetilde{R}$.
Since the ``constant term" of highest weight is associated to $\widetilde{R}$
but does not belong to $J_{\widetilde{R}}$, we have $J \subset I$.
We extend the map
$\widetilde{\psi}\colon\widetilde{W} \to \widetilde{V}/\frak S_{\widetilde{n}}$
to a map $\psi\colon \widetilde{W} \to V/(\frak S_{n'} \times \frak S_{n''})$
by simply composing it with the natural inclusion
$\widetilde{V}/\frak S_{\widetilde{n}} \subset
V/(\frak S_{n'} \times \frak S_{n''})$.
We need to compute $\varepsilon_i$ modulo $J$. Since $J$ vanishes on the
image of $\psi$, it suffices to compute $\psi^*(\varepsilon_i)$ in terms of
the pullbacks of the standard coordinate functions on $\operatorname{PRes} (E_n,v_k)$
via $\psi^*$.
The first step is to use proposition~\ref{prop71} and table~\ref{tableAA},
which relates the distinguished polynomial for $R$ to those for $R'$
and $R''$. Now $\psi^*(f_{S'}(U;t'))$ and $\psi^*(f_{S''}(U;t''))$
can be computed immediately:
one of them
is the restricted polynomial for $\widetilde{R}$, and the other one
is just a power of $U$ (since all of the corresponding standard coordinate
functions vanish when
pulled back via $\psi$). Table~\ref{tableAA} can then be used to compute
$\psi^*(f_S(U;t))$; we carry out this computation below.
We first consider the case $(E_n,v_0)$, in which the complementary root
system has type $A_{n-1}$. We have
$\psi^*(\mu_1)=0$ and
\[\psi^*(f_{A_{n-1}}(U;t'))=r_{A_{n-1}}(U)=U^n+\l_n.\]
(We use the first form of the restricted polynomial since we do not
need a root.) Thus, by table~\ref{tableAA},
$\psi^*(f_{E_n}(U;t))=U^n+\l_n$.
We next consider the case $(E_n,v_1)$, in which the complementary root
system has type $D_{n-1}$. Since
$\psi^*(f_{D_{n-1}}(U;t'))=r_{D_{n-1}}(U)$,
the coefficient $\psi^*(\rho_1)$ of $U^{n-2}$ in this polynomial is
0 when $n-1$ is even, and $\l_1$ when $n-1=7$. We also have
$\psi^*(\mu_1)=0$. Thus by table~\ref{tableAA},
in the case $(E_7,v_1)$
we get
\begin{align*}
\psi^*(f_{E_7}(U;t))
&= (-1)^7 \cdot (-U) \cdot \widetilde{\psi}^*(f_{D_6}(-U;t')) \\
&= U^7 + \l_5 U^2
\end{align*}
while in the case $(E_8,v_1)$ we get
\begin{align*}
\psi^*(f_{E_8}(U;t))
&= (-1)^8 \cdot (-U+\frac13\l_1)
\cdot \widetilde{\psi}^*(f_{D_7}(-U-\frac16\l_1;t')) \\
&= (-U+\frac13\l_1)
\cdot r_{D_7}(-U-\frac16\l_1).
\end{align*}
We next consider the case $(E_n,v_2)$, in which the complementary root
system has components of type $A_1$ and $A_{n-2}$. In this case,
$\psi^*(f_{A_{n-2}}(U;t''))=r_{A_{n-2}}(U)$,
which we write this time in the form $U^{n-1} - \l_1^{n-1}$ so that
$\psi^*(\sigma_1)=\l_1$ is a root of this polynomial.
Now
\[\frac{r_{A_{n-2}}(U)}{U-\l_1} = \sum_{i=0}^{n-2}\l_1^i\ U^{n-2-i}.\]
Moreover, $\psi^*(\mu_1)=0$, and
$\psi^*(f_{A_1}(U+\frac23\sigma_1;t'))=(U+\frac23\l_1)^2$.
Table~\ref{tableAA} then implies
\begin{align*}
\psi^*(f_{E_n}(U;t))
&= (U+\frac23\l_1)^2 \cdot
\sum_{i=0}^{n-2}\l_1^i\ (U-\frac13\l_1)^{n-2-i}
\end{align*}
Finally, we consider the case $(E_n,v_k)$ with $k \ge 4$,
in which the complementary root
system has components of type $E_k$ and $A_{n-k-1}$. In this case,
we have
$\psi^*(f_{E_k}(U;t'))=r_{E_k}(U)=U^k+\l_1U^{k-1}+\cdots$,
which implies that the coefficient $\psi^*(\tau_1)$ of
$U^{k-1}$ in this polynomial
is $\l_1$.
Since $\psi^*(\mu_1)=0$ and $\psi^*(f_{A_{n-k-1}}(U;t''))=U^{n-k}$,
table~\ref{tableAA} implies that
\begin{align*}
\psi^*(f_{E_n}(U;t))
&=
(U-\frac{1}{9-k}\l_1)^{n-k} \cdot r_{E_k}(U) .
\end{align*}
The second step in the computation of $\varepsilon_i$ modulo $J$ is to
use the coefficients of
the pulled-back distinguished polynomial $\psi^*(f_{E_n}(U;t))$
as ingredients for the solve-lists in section 9, and obtain (using
{\sc maple} and {\sc reduce}) a formula for $\psi^*(\varepsilon_i)$ in terms
of the parameters. Now we have already computed
$\psi^*(\widetilde{\varphi}_N)$ in terms of the parameters, as indicated
in table~\ref{tableABC}. So we simply need to compare the formulas
for $\psi^*(\varepsilon_i)$ and $\psi^*(\widetilde{\varphi}_N)^d$.
We illustrate this comparison
with an example which can be carried out by hand.
The case we consider is $(E_7,v_1)$, in which the complementary root
system has type $D_6$. We have
$\psi^*(f_{E_7}(U;t)) = U^7 + \l_5 U^2$, which implies that
$\psi^*(\varepsilon_{10})$ is computed by setting $s_5=\l_5$ and all
other $s_j=0$ in the formula for $\varepsilon_{10}$. The only term that then
remains is the term coming from $s_5^2$ in the original formula for
$\varepsilon_{10}$. Now inspection of Appendix 2 shows that the coefficient of
$s_5^2$ in the formula for $16\varepsilon_{10}$ is 256. Thus,
\[\psi^*(\varepsilon_{10}) = 16 \l_5^2 = 16 \psi^*({s_5})^2\]
which implies that
\[\varepsilon_{10} \equiv 16 s_5^2 \mod J,\]
as required.
To return to the general argument:
in all cases from table~\ref{table-key1} except $(E_8,v_1)$ and
$(E_8,v_7)$, the only monomial in the standard coordinate functions on
$\operatorname{PRes} (E_n,v_k)$ which
has weight $i$ and which does not pull back to zero under $\psi^*$
is $(\widetilde{\varphi}_N)^d$. Thus, in those cases it follows that
$\psi^*(\varepsilon_i)/\psi^*(\widetilde{\varphi}_N)^d$
is a constant. We calculated these constants
using {\sc maple} and {\sc reduce},
obtaining the values indicated in table~\ref{table-key1}.
This verifies the congruences
$\varepsilon_i \equiv c \cdot (\widetilde{\varphi}_N)^d \mod J$,
which suffices since $J \subset I$ in each case.
In the two remaining cases $(E_8,v_1)$ and
$(E_8,v_7)$, the congruences which hold modulo $J$ are
\begin{align}
\label{eqlast1}
\varepsilon_{24} &\equiv
0 \cdot (\d'_8)^3 + -\tfrac{1}{16}\cdot (\d'_{12})^2
\mod J, \quad \text{and}\\
\label{eqlast2}
\varepsilon_{18} &\equiv
-\tfrac{1}{3072} \cdot (\varepsilon'_8 \varepsilon'_{10})
+ \tfrac{1}{64}\cdot (\varepsilon'_{18})
\mod J,
\end{align}
respectively, and these imply the desired congruences modulo $I$.
To verify these congruences, we also need to calculate
\begin{gather*}
\psi^*(\d'_8)=
{ { { {\l_{1}}^{3}} {\l_{5}}} { - \tfrac{ 3 }{4}
}{ {\l_{1}}^{5}} {\l_{3}} + { \tfrac{ 3 }{2}}{ {\l_{1}}^{2}} { {\l_{3}}^{2}}
+ { \tfrac{ 5 }{64}}{
{\l_{1}}^{8}} { - 2 {\l_{3}} {\l_{5}}}},
\\
\psi^*(\varepsilon'_8)=e_8(\l_1, \l_3, \l_4,\l_5, \l_7),\\
\psi^*(\varepsilon'_{10})=e_{10}(\l_1, \l_3, \l_4,\l_5, \l_7).
\end{gather*}
The first of these formulas is obtained from equation (\ref{eqD}),
while the second and third lines refer to formulas which we have calculated
explicitly with {\sc maple} and {\sc reduce}
using the solve-list method, but do not display here.
(Notice that the calculations of $\d'_{12}$ and $\varepsilon'_{18}$
are indicated in table~\ref{tableABC}.)
Now the coefficients in equations (\ref{eqlast1}) and (\ref{eqlast2})
can be calculated with the method of undetermined coefficients.
That is, there will be some relation of the form
\begin{align*}
\psi^*(\varepsilon_{24}) &= c_1 \cdot \psi^*(\d'_8)^3
+ c_2 \cdot \psi^*(\d'_{12})^2
\mod J, \quad \text{or} \\
\psi^*(\varepsilon_{18}) &= c_1 \cdot \psi^*(\varepsilon'_8) \cdot \psi^*(\varepsilon'_{10})
+ c_2 \cdot \psi^*(\varepsilon'_{18})
\mod J,
\end{align*}
respectively. Substituting the calculated values of $\psi^*$
allowed us to solve (using {\sc maple} and {\sc reduce}) for
the undetermined coefficients $c_1$, $c_2$.
This completes the verification of table~\ref{table-key1}, and the
proof of the main theorem.
\bigskip
We would like to offer two pieces of advice to the ambitious reader who
wishes to duplicate our symbolic calculations. First, it is essential
when computing with solve-lists to keep them unexpanded as long
as possible. Even when a solve-list must be expanded, it may be
that all relevant information can be extracted by only {\em partially\/}
expanding the solve list, solving for a proper subset of the variables.
Second, the absence of $z$ from the monomials in the solve-lists
(\ref{eq17}c), (\ref{eq41}b), and (\ref{eq41}c)
means that $z$ can be set equal to $0$ before the expansion of
these solve-lists begins. This cuts down the size of the
computation tremendously.
For the less ambitious reader, {\sc maple} source files for all
calculations described in the paper are available upon request
(directed to the second author).
\newpage
\section*{Appendix 0. A good generating set in the case of $E_8$.}
\begin{align*}
\bar{W} &= x^3 - y z^2 \\
\bar{Z} &= y^ 3+( s_ 2- s_ 1^ 2)xy^ 2-( s_ 3-
s_ 1 s_ 2)y^ 2z+( s_ 4- s_ 1 s_ 3)x^ 2y
- ( s_ 5- s_ 1 s_ 4)xyz \\
&\quad
+ ( s_ 6- s_ 1 s_ 5)
x^ 3 - (s_7 - s_ 1 s_ 6) x^ 2z + (s_8 - s_1 s_7) x z^2 + s_1 s_8 z^3 \\
\bar{Y} &= x y^5
- 2 s_1 y^5 z
+ \left( s_1^2+ 2 s_2 \right) y^4 x^2
+ \left( - 2 s_3 - 2 s_1 s_2 \right) z y^4 x
+ \left( s_2^2 + 2 s_1 s_3
+ 2 s_4 \right) z^2 y^4 \\
&\quad
+ \left( - 2 s_1 s_4 - 2 s_5 - 2 s_3 s_2 \right) z y^3 x^2
+ \left( s_6 - 2 s_3 s_1^3 + 3 s_1^2 s_4
- s_2^3 - s_1^6 +
3 s_2 s_1^4 \right) y^2 x^4 \\
&\quad
+ \left( 2 s_1 s_5+ s_6 + 2 s_3 s_1^3+ s_2^3+ s_3^2
- 3 s_1^2 s_4 + 2 s_2 s_4
+ s_1^6 - 3 s_2 s_1^4 \right) z^2 y^3 x \\
&\quad
+ \left( 2 s_1 s_2 s_4
- 2 s_3 s_4 + s_5 s_1^2
- 2 s_1^2 s_2 s_3 - s_1^3 s_4 - s_2 s_1^5
- s_7
+ s_2^3 s_1 - s_1 s_6 + 2 s_2^2 s_1^3 \right. \\
&\quad \left.
+ s_3 s_1^4
- s_3 s_2^2 \right) z^3 y^3
+ \left( - 2 s_5 s_2 - s_7 + 2 s_1^2 s_2 s_3
+ s_3 s_2^2
- s_1 s_6 - s_2^3 s_1 - 2 s_2^2 s_1^3 \right. \\
&\quad \left.
+ s_1^3 s_4
- 2 s_1 s_2 s_4 + s_2 s_1^5 - s_3 s_1^4 - s_5 s_1^2 \right)
z y^2 x^3
+ \left( - s_4 s_2^2 - s_1 s_7 + 2 s_3 s_1 s_4 - s_4^2
\right. \\
&\quad \left.
- s_3^2 s_1^2 + 2 s_3 s_1^3 s_2 - s_8 - s_5 s_1^3
+ s_6 s_2
- s_4 s_2 s_1^2 + s_6 s_1^2 - s_3 s_1^5
+ s_4 s_1^4 + s_3 s_1 s_2^2 \right) y x^5 \\
&\quad
+ \left( 3 s_1 s_7 + s_4 s_2^2+ 3 s_8 + 2 s_3 s_5
+ s_3^2 s_1^2
+ s_6 s_2 - 2 s_3 s_1^3 s_2+ 2 s_4^2 + s_3 s_1^5
- s_4 s_1^4 \right. \\
&\quad \left.
- s_3 s_1 s_2^2 + s_4 s_2 s_1^2 + s_5 s_1^3
- s_6 s_1^2 - 2 s_3 s_1 s_4 \right) z^2 y^2 x^2
+ \left( s_5 s_2 s_1^2 - s_4 s_1 s_2^2 + s_1^2 s_4 s_3 \right. \\
&\quad \left.
- 2 s_1 s_4^2 + s_4 s_1^5
+ s_6 s_1^3 - s_6 s_3
+ s_1 s_8 - s_1^2 s_7 + s_5 s_2^2 - s_2 s_7
- 2 s_4 s_2 s_1^3 - s_5 s_1^4 \right) z y x^4 \\
&\quad
+ \left( s_4 s_1 s_2^2 - s_6 s_1^3 - s_5 s_2 s_1^2 - 3 s_1 s_8
- s_4 s_1^5 + 2 s_1 s_4^2 + s_1^2 s_7 - 2 s_5 s_4
- s_2 s_7 \right. \\
&\quad \left.
+ 2 s_4 s_2 s_1^3
- s_6 s_3 - s_5 s_2^2
- s_1^2 s_4 s_3 + s_5 s_1^4 \right) z^3 y^2 x
+ \left( s_2 s_8 - 2 s_5 s_1^3 s_2 - s_1 s_5 s_2^2 \right. \\
&\quad \left.
+ s_5 s_1^5
+ s_7 s_3 + s_2 s_6 s_1^2 - s_6 s_1^4 + s_1^3 s_7
+ s_2^2 s_6 + s_6 s_4 + s_3 s_5 s_1^2 - 2 s_1 s_4 s_5 \right. \\
&\quad \left.
- s_8 s_1^2 \right) z^4 y^2
+ \left( s_7 s_3 - s_2^2 s_6 + s_6 s_4 - s_1^3 s_7 - s_5 s_1^5
+ s_5^2 + s_2 s_8 + s_6 s_1^4
- s_3 s_5 s_1^2 \right. \\
&\quad \left.
+ 2 s_1 s_4 s_5 + 2 s_5 s_1^3 s_2 + s_1 s_5 s_2^2
- s_2 s_6 s_1^2 + s_8 s_1^2 \right) z^2 y x^3
+ \left( s_6 s_1^2 s_3
+ s_4 s_7 - 2 s_6 s_1^3 s_2 \right. \\
&\quad \left.
- s_7 s_1^4
- s_5 s_6 + s_6 s_1^5 - s_6 s_1 s_2^2 + s_8 s_1^3
+ s_7 s_2^2 - 2 s_6 s_1 s_4
+ s_3 s_8
+ s_2 s_7 s_1^2 \right) z x^5 \\
&\quad
+ \left( - s_5 s_6 - 3 s_4 s_7 + s_6 s_1 s_2^2 + 2 s_6 s_1^3 s_2
- s_8 s_1^3 - 3 s_3 s_8
- s_6 s_1^2 s_3 - s_6 s_1^5
+ 2 s_6 s_1 s_4 \right. \\
&\quad \left.
+ s_7 s_1^4 - s_2 s_7 s_1^2
- s_7 s_2^2 \right) z^3 y x^2
+ \left( 2 s_7 s_2 s_1^3
- s_1^2 s_7 s_3 + s_8 s_1^4+ s_6^2
+ s_5 s_7 - s_8 s_2 s_1^2 \right. \\
&\quad \left.
+ s_2^2 s_7 s_1 - s_4 s_8
+ 2 s_7 s_1 s_4 - s_7 s_1^5
- s_8 s_2^2 \right) z^2 x^4
+ \left( s_1^2 s_7 s_3 - s_2^2 s_7 s_1 + s_8 s_2 s_1^2 \right. \\
&\quad \left.
+ 3 s_4 s_8 + s_5 s_7 - 2 s_7 s_1 s_4
- 2 s_7 s_2 s_1^3
- s_8 s_1^4 + s_8 s_2^2 + s_7 s_1^5 \right) z^4 y x
+ \left( s_8 s_1 s_2^2 \right. \\
&\quad \left.
+ 2 s_1 s_8 s_4 + 2 s_8 s_1^3 s_2
- s_8 s_1^5
- s_3 s_8 s_1^2 - s_5 s_8 \right) z^5 y
+ \left( s_8 s_1^5 - 2 s_6 s_7 - s_8 s_1 s_2^2 \right. \\
&\quad \left.
- 2 s_1 s_8 s_4
- 2 s_8 s_1^3 s_2 - s_5 s_8
+ s_3 s_8 s_1^2 \right) z^3 x^3
+ \left( s_7^2 + 2 s_6 s_8 \right) z^4 x^2
- 2 s_7 s_8 x z^5
+ s_8^2 z^6 \\
\bar{X} &= - \frac16 \
\frac{\partial(\bar{Y},\bar{Z},\bar{W})}{\partial(x,y,z)}
\end{align*}
\section*{Appendix 1. Standard coordinates for $E_6$.}
\begin{align*}
6\,{\varepsilon_{2}} &= -2\,{s_{1}}^2 + 3\,s_{2}\\
81\,{\varepsilon_{5}} &= 4\,{s_{1}}^5 - 15\,{s_{1}}^3\,s_{2} + 27\,{s_{1}}^2\,
s_{3} - 27\,s_{1}\,s_{4} + 81\,s_{5}\\
1944\,{\varepsilon_{6}} &= -16\,{s_{1}}^6 + 72\,{s_{1}}^4\,s_{2} -
216\,{s_{1}}^3\,s_{3} + 27\,{s_{1}}^2\,{s_{2}}^2 + 216\,{s_{1}}^2\,s_{
4} \\&\qquad + 162\,s_{1}\,s_{2}\,s_{3} + 324\,s_{1}\,s_{5} - 81\,{s_{
2}}^3 + 324\,s_{2}\,s_{4} - 243\,{s_{3}}^2 - 1944\,s_{6} \\
34992\,{\varepsilon_{8}} &= -64\,{s_{1}}^8 + 384\,{s_{1}}^6\,s_{2} -
864\,{s_{1}}^5\,s_{3} - 324\,{s_{1}}^4\,{s_{2}}^2 + 864\,{s_{1}}^4\,s
_{4} \\&\qquad + 1944\,{s_{1}}^3\,s_{2}\,s_{3} - 2592\,{s_{1}}^3\,s_{5
} - 486\,{s_{1}}^2\,{s_{2}}^3 - 1944\,{s_{1}}^2\,s_{2}\,s_{4} - 2916\,
{s_{1}}^2\,{s_{3}}^2 \\&\qquad + 34992\,{s_{1}}^2\,s_{6} + 2916\,s_{1}
\,{s_{2}}^2\,s_{3} - 11664\,s_{1}\,s_{2}\,s_{5} + 5832\,s_{1}\,s_{3}\,
s_{4} - 729\,{s_{2}}^4 \\&\qquad + 5832\,{s_{2}}^2\,s_{4} - 4374\,s_{2
}\,{s_{3}}^2 + 17496\,s_{3}\,s_{5} - 11664\,{s_{4}}^2 \\
78732\,{\varepsilon_{9}} &= 64\,{s_{1}}^9 - 432\,{s_{1}}^7\,s_{2} +
1296\,{s_{1}}^6\,s_{3} + 324\,{s_{1}}^5\,{s_{2}}^2 - 1296\,{s_{1}}^5\,
s_{4} \\&\qquad - 3888\,{s_{1}}^4\,s_{2}\,s_{3} - 1944\,{s_{1}}^4\,s_{
5} + 1215\,{s_{1}}^3\,{s_{2}}^3 + 972\,{s_{1}}^3\,s_{2}\,s_{4} + 5832
\,{s_{1}}^3\,{s_{3}}^2 \\&\qquad - 14580\,{s_{1}}^3\,s_{6} - 2187\,{s
_{1}}^2\,{s_{2}}^2\,s_{3} + 17496\,{s_{1}}^2\,s_{2}\,s_{5} - 8748\,{s
_{1}}^2\,s_{3}\,s_{4} \\&\qquad + 2187\,s_{1}\,{s_{2}}^2\,s_{4} +
52488\,s_{1}\,s_{2}\,s_{6} - 8748\,s_{1}\,{s_{4}}^2 - 6561\,{s_{2}}^2
\,s_{5} - 78732\,s_{3}\,s_{6} \\&\qquad + 26244\,s_{4}\,s_{5}
\\
11337408\,{\varepsilon_{12}} &= -256\,{s_{1}}^{12} + 2304\,{s_{1}}^{
10}\,s_{2} - 6912\,{s_{1}}^9\,s_{3} - 4320\,{s_{1}}^8\,{s_{2}}^2
\\&\qquad + 6912\,{s_{1}}^8\,s_{4} + 36288\,{s_{1}}^7\,s_{2}\,s_{3}
+ 10368\,{s_{1}}^7\,s_{5} - 6480\,{s_{1}}^6\,{s_{2}}^3 \\&\qquad -
20736\,{s_{1}}^6\,s_{2}\,s_{4} - 54432\,{s_{1}}^6\,{s_{3}}^2 + 217728
\,{s_{1}}^6\,s_{6} - 11664\,{s_{1}}^5\,{s_{2}}^2\,s_{3} \\&\qquad -
186624\,{s_{1}}^5\,s_{2}\,s_{5} + 93312\,{s_{1}}^5\,s_{3}\,s_{4} +
10935\,{s_{1}}^4\,{s_{2}}^4 + 11664\,{s_{1}}^4\,{s_{2}}^2\,s_{4}
\\&\qquad + 104976\,{s_{1}}^4\,s_{2}\,{s_{3}}^2 - 1189728\,{s_{1}}^4
\,s_{2}\,s_{6} + 93312\,{s_{1}}^4\,{s_{4}}^2 - 78732\,{s_{1}}^3\,{s_{2
}}^3\,s_{3} \\&\qquad + 437400\,{s_{1}}^3\,{s_{2}}^2\,s_{5} - 209952\,
{s_{1}}^3\,s_{2}\,s_{3}\,s_{4} - 104976\,{s_{1}}^3\,{s_{3}}^3
\\&\qquad + 2729376\,{s_{1}}^3\,s_{3}\,s_{6} - 279936\,{s_{1}}^3\,s_{
4}\,s_{5} + 13122\,{s_{1}}^2\,{s_{2}}^5 + 196830\,{s_{1}}^2\,{s_{2}}^2
\,{s_{3}}^2 \\&\qquad + 629856\,{s_{1}}^2\,{s_{2}}^2\,s_{6} - 944784\,
{s_{1}}^2\,s_{2}\,s_{3}\,s_{5} - 209952\,{s_{1}}^2\,s_{2}\,{s_{4}}^2
\\&\qquad + 314928\,{s_{1}}^2\,{s_{3}}^2\,s_{4} - 7558272\,{s_{1}}^2
\,s_{4}\,s_{6} + 2834352\,{s_{1}}^2\,{s_{5}}^2 - 78732\,s_{1}\,{s_{2}}
^4\,s_{3} \\&\qquad + 314928\,s_{1}\,{s_{2}}^3\,s_{5} + 157464\,s_{1}
\,{s_{2}}^2\,s_{3}\,s_{4} - 236196\,s_{1}\,s_{2}\,{s_{3}}^3 \\&\qquad
+ 3779136\,s_{1}\,s_{2}\,s_{3}\,s_{6} - 1259712\,s_{1}\,s_{2}\,s_{4}
\,s_{5} - 472392\,s_{1}\,{s_{3}}^2\,s_{5} \\&\qquad + 629856\,s_{1}\,s
_{3}\,{s_{4}}^2 + 13122\,{s_{2}}^6 - 157464\,{s_{2}}^4\,s_{4} + 118098
\,{s_{2}}^3\,{s_{3}}^2 \\&\qquad - 472392\,{s_{2}}^2\,s_{3}\,s_{5} +
629856\,{s_{2}}^2\,{s_{4}}^2 - 472392\,s_{2}\,{s_{3}}^2\,s_{4} +
177147\,{s_{3}}^4 \\&\qquad - 2834352\,{s_{3}}^2\,s_{6} + 1889568\,s_{
3}\,s_{4}\,s_{5} - 839808\,{s_{4}}^3 \end{align*}
\newpage
{\samepage
\section*{Appendix 2. Standard coordinates for $E_7$.}
This Appendix gives the standard coordinate functions
$\varepsilon_i$ for $E_7$, and can also
serve as a correction to the formulas of Bramble \cite[pp.\ 358-360]{[Bra]}.
The $A_i$ which we calculate here are
integer multiples of the $\varepsilon_i$ which clear denominators; Bramble's paper
contains the same multiples. (Note that our $\varepsilon_i$ correspond to his
$\a_{ijk\ell}$.)
It is very impressive to observe
that Bramble, calculating by hand, was correct in the calculation of $A_2,
\ A_6,\ A_8$, and
$A_{10}$, and had only two incorrect coefficients for $A_{14}$. However,
the formulas for $A_{12}$ and $A_{18}$ from \cite{[Bra]} are mostly wrong.
\begin{align*}
A_2 &=
{\varepsilon_{2}} = 3\,{s_{1}}^2 - 4\,s_{2}\\
A_6 &=
48\,{\varepsilon_{6}} = 18\,{s_{1}}^6 - 72\,{s_{1}}^4\,s_{2} + 96\,{s
_{1}}^3\,s_{3} + 32\,{s_{1}}^2\,{s_{2}}^2 - 96\,{s_{1}}^2\,s_{4} -32
\,s_{1}\,s_{2}\,s_{3} \\&\qquad + 96\,s_{1}\,s_{5} - 64\,s_{2}\,s_{4}
+ 48\,{s_{3}}^2 + 384\,s_{6} \\
A_8 &=
48\,{\varepsilon_{8}} = -27\,{s_{1}}^8 + 144\,{s_{1}}^6\,s_{2} - 192
\,{s_{1}}^5\,s_{3} - 160\,{s_{1}}^4\,{s_{2}}^2 + 192\,{s_{1}}^4\,s_{4}
\\&\qquad + 320\,{s_{1}}^3\,s_{2}\,s_{3} - 192\,{s_{1}}^3\,s_{5} -
128\,{s_{1}}^2\,s_{2}\,s_{4} - 160\,{s_{1}}^2\,{s_{3}}^2 + 128\,s_{1}
\,s_{3}\,s_{4} \\&\qquad - 2304\,s_{1}\,s_{7} + 768\,s_{2}\,s_{6} +
384\,s_{3}\,s_{5} - 256\,{s_{4}}^2 \\
A_{10} &=
16\,{\varepsilon_{10}} = 3\,{s_{1}}^{10} - 20\,{s_{1}}^8\,s_{2} + 32
\,{s_{1}}^7\,s_{3} + 32\,{s_{1}}^6\,{s_{2}}^2 - 32\,{s_{1}}^6\,s_{4}
- 96\,{s_{1}}^5\,s_{2}\,s_{3} \\&\qquad + 32\,{s_{1}}^5\,s_{5} + 64\,
{s_{1}}^4\,s_{2}\,s_{4} + 80\,{s_{1}}^4\,{s_{3}}^2 - 128\,{s_{1}}^4\,s
_{6} - 128\,{s_{1}}^3\,s_{3}\,s_{4} \\&\qquad - 256\,{s_{1}}^3\,s_{7}
+ 256\,{s_{1}}^2\,s_{2}\,s_{6} + 128\,{s_{1}}^2\,s_{3}\,s_{5} + 512\,
s_{1}\,s_{2}\,s_{7} - 512\,s_{1}\,s_{3}\,s_{6} \\&\qquad - 1024\,s_{3}
\,s_{7} + 256\,{s_{5}}^2 \\
A_{12} &=
6912\,{\varepsilon_{12}} = -297\,{s_{1}}^{12} + 2376\,{s_{1}}^{10}\,
s_{2} - 3456\,{s_{1}}^9\,s_{3} - 5616\,{s_{1}}^8\,{s_{2}}^2 + 3456\,{s
_{1}}^8\,s_{4} \\&\qquad + 14976\,{s_{1}}^7\,s_{2}\,s_{3} - 3456\,{s_{
1}}^7\,s_{5} + 3328\,{s_{1}}^6\,{s_{2}}^3 - 11520\,{s_{1}}^6\,s_{2}\,s
_{4} \\&\qquad - 11520\,{s_{1}}^6\,{s_{3}}^2 - 6912\,{s_{1}}^6\,s_{6}
- 9984\,{s_{1}}^5\,{s_{2}}^2\,s_{3} + 11520\,{s_{1}}^5\,s_{2}\,s_{5}
\\&\qquad + 19584\,{s_{1}}^5\,s_{3}\,s_{4} - 20736\,{s_{1}}^5\,s_{7}
- 1536\,{s_{1}}^4\,{s_{2}}^2\,s_{4} + 13440\,{s_{1}}^4\,s_{2}\,{s_{3}
}^2 \\&\qquad + 34560\,{s_{1}}^4\,s_{2}\,s_{6} - 14976\,{s_{1}}^4\,s_{
3}\,s_{5} - 11520\,{s_{1}}^4\,{s_{4}}^2 + 3072\,{s_{1}}^3\,s_{2}\,s_{3
}\,s_{4} \\&\qquad + 55296\,{s_{1}}^3\,s_{2}\,s_{7} - 10240\,{s_{1}}^3
\,{s_{3}}^3 - 55296\,{s_{1}}^3\,s_{3}\,s_{6} + 18432\,{s_{1}}^3\,s_{4}
\,s_{5} \\&\qquad - 18432\,{s_{1}}^2\,{s_{2}}^2\,s_{6} - 9216\,{s_{1}}
^2\,s_{2}\,s_{3}\,s_{5} - 6144\,{s_{1}}^2\,s_{2}\,{s_{4}}^2 + 12288\,{
s_{1}}^2\,{s_{3}}^2\,s_{4} \\&\qquad - 55296\,{s_{1}}^2\,s_{3}\,s_{7}
+ 27648\,{s_{1}}^2\,{s_{5}}^2 - 110592\,s_{1}\,{s_{2}}^2\,s_{7} +
73728\,s_{1}\,s_{2}\,s_{3}\,s_{6} \\&\qquad - 18432\,s_{1}\,{s_{3}}^2
\,s_{5} + 6144\,s_{1}\,s_{3}\,{s_{4}}^2 + 221184\,s_{1}\,s_{4}\,s_{7}
- 110592\,s_{1}\,s_{5}\,s_{6} \\&\qquad + 55296\,s_{2}\,s_{3}\,s_{7}
+ 36864\,s_{2}\,s_{4}\,s_{6} - 55296\,{s_{3}}^2\,s_{6} + 18432\,s_{3}
\,s_{4}\,s_{5} \\&\qquad - 8192\,{s_{4}}^3 - 110592\,s_{5}\,s_{7} -
110592\,{s_{6}}^2 \\
A_{14} &=
768\,{\varepsilon_{14}} = 27\,{s_{1}}^{14} - 252\,{s_{1}}^{12}\,s_{2
} + 384\,{s_{1}}^{11}\,s_{3} + 752\,{s_{1}}^{10}\,{s_{2}}^2 - 384\,{s
_{1}}^{10}\,s_{4} \\&\qquad - 2176\,{s_{1}}^9\,s_{2}\,s_{3} + 384\,{s
_{1}}^9\,s_{5} - 704\,{s_{1}}^8\,{s_{2}}^3 + 1792\,{s_{1}}^8\,s_{2}\,s
_{4} + 1664\,{s_{1}}^8\,{s_{3}}^2
\displaybreak[0]
\\&\qquad - 768\,{s_{1}}^8\,s_{6} +
2816\,{s_{1}}^7\,{s_{2}}^2\,s_{3} - 1280\,{s_{1}}^7\,s_{2}\,s_{5} -
2944\,{s_{1}}^7\,s_{3}\,s_{4} \\&\qquad + 768\,{s_{1}}^7\,s_{7} - 1536
\,{s_{1}}^6\,{s_{2}}^2\,s_{4} - 3968\,{s_{1}}^6\,s_{2}\,{s_{3}}^2 +
2816\,{s_{1}}^6\,s_{2}\,s_{6} \\&\qquad + 2432\,{s_{1}}^6\,s_{3}\,s_{5
} + 1280\,{s_{1}}^6\,{s_{4}}^2 + 4608\,{s_{1}}^5\,s_{2}\,s_{3}\,s_{4}
- 2048\,{s_{1}}^5\,s_{2}\,s_{7} \\&\qquad + 2048\,{s_{1}}^5\,{s_{3}}^
3 - 5120\,{s_{1}}^5\,s_{3}\,s_{6} - 2048\,{s_{1}}^5\,s_{4}\,s_{5} -
1024\,{s_{1}}^4\,{s_{2}}^2\,s_{6} \\&\qquad - 512\,{s_{1}}^4\,s_{2}\,s
_{3}\,s_{5} - 1024\,{s_{1}}^4\,s_{2}\,{s_{4}}^2 - 4096\,{s_{1}}^4\,{s
_{3}}^2\,s_{4} + 8192\,{s_{1}}^4\,s_{3}\,s_{7} \\&\qquad + 8192\,{s_{1
}}^4\,s_{4}\,s_{6} - 1536\,{s_{1}}^4\,{s_{5}}^2 + 4096\,{s_{1}}^3\,{s
_{2}}^2\,s_{7} + 2048\,{s_{1}}^3\,{s_{3}}^2\,s_{5} \\&\qquad + 2048\,{
s_{1}}^3\,s_{3}\,{s_{4}}^2 + 16384\,{s_{1}}^3\,s_{4}\,s_{7} - 12288\,{
s_{1}}^3\,s_{5}\,s_{6} - 30720\,{s_{1}}^2\,s_{2}\,s_{3}\,s_{7}
\\&\qquad - 4096\,{s_{1}}^2\,s_{2}\,s_{4}\,s_{6} + 8192\,{s_{1}}^2\,s
_{2}\,{s_{5}}^2 - 2048\,{s_{1}}^2\,{s_{3}}^2\,s_{6} - 2048\,{s_{1}}^2
\,s_{3}\,s_{4}\,s_{5} \\&\qquad - 12288\,{s_{1}}^2\,s_{5}\,s_{7} +
12288\,{s_{1}}^2\,{s_{6}}^2 - 8192\,s_{1}\,s_{2}\,s_{4}\,s_{7} + 32768
\,s_{1}\,{s_{3}}^2\,s_{7} \\&\qquad + 8192\,s_{1}\,s_{3}\,s_{4}\,s_{6}
- 8192\,s_{1}\,s_{3}\,{s_{5}}^2 + 49152\,s_{1}\,s_{6}\,s_{7} - 24576
\,s_{2}\,s_{5}\,s_{7} \\&\qquad + 16384\,s_{3}\,s_{4}\,s_{7} - 24576\,
s_{3}\,s_{5}\,s_{6} + 8192\,s_{4}\,{s_{5}}^2 + 49152\,{s_{7}}^2
\\
A_{18} &=
9\cdot16^3\,{\varepsilon_{18}} = 63\,{s_{1}}^{18} - 756\,{s_{1}}^{16}\,s
_{2} + 1152\,{s_{1}}^{15}\,s_{3} + 3264\,{s_{1}}^{14}\,{s_{2}}^2 -
1152\,{s_{1}}^{14}\,s_{4} \\&\qquad - 9600\,{s_{1}}^{13}\,s_{2}\,s_{3}
+ 1152\,{s_{1}}^{13}\,s_{5} - 5888\,{s_{1}}^{12}\,{s_{2}}^3 + 8448\,{
s_{1}}^{12}\,s_{2}\,s_{4} \\&\qquad + 7488\,{s_{1}}^{12}\,{s_{3}}^2 +
24576\,{s_{1}}^{11}\,{s_{2}}^2\,s_{3} - 7680\,{s_{1}}^{11}\,s_{2}\,s_{
5} - 13824\,{s_{1}}^{11}\,s_{3}\,s_{4} \\&\qquad + 4608\,{s_{1}}^{11}
\,s_{7} + 3584\,{s_{1}}^{10}\,{s_{2}}^4 - 16896\,{s_{1}}^{10}\,{s_{2}}
^2\,s_{4} - 36864\,{s_{1}}^{10}\,s_{2}\,{s_{3}}^2 \\&\qquad - 3072\,{s
_{1}}^{10}\,s_{2}\,s_{6} + 12288\,{s_{1}}^{10}\,s_{3}\,s_{5} + 6912\,{
s_{1}}^{10}\,{s_{4}}^2 - 17920\,{s_{1}}^9\,{s_{2}}^3\,s_{3} \\&\qquad
+ 12800\,{s_{1}}^9\,{s_{2}}^2\,s_{5} + 53760\,{s_{1}}^9\,s_{2}\,s_{3}
\,s_{4} - 29184\,{s_{1}}^9\,s_{2}\,s_{7} + 20480\,{s_{1}}^9\,{s_{3}}^3
\\&\qquad + 7680\,{s_{1}}^9\,s_{3}\,s_{6} - 12288\,{s_{1}}^9\,s_{4}\,
s_{5} + 5120\,{s_{1}}^8\,{s_{2}}^3\,s_{4} + 37120\,{s_{1}}^8\,{s_{2}}^
2\,{s_{3}}^2 \\&\qquad + 13312\,{s_{1}}^8\,{s_{2}}^2\,s_{6} - 39424\,{
s_{1}}^8\,s_{2}\,s_{3}\,s_{5} - 21504\,{s_{1}}^8\,s_{2}\,{s_{4}}^2 -
49152\,{s_{1}}^8\,{s_{3}}^2\,s_{4} \\&\qquad + 52224\,{s_{1}}^8\,s_{3}
\,s_{7} + 24576\,{s_{1}}^8\,s_{4}\,s_{6} - 16128\,{s_{1}}^8\,{s_{5}}^2
- 20480\,{s_{1}}^7\,{s_{2}}^2\,s_{3}\,s_{4} \\&\qquad + 106496\,{s_{1
}}^7\,{s_{2}}^2\,s_{7} - 40960\,{s_{1}}^7\,s_{2}\,{s_{3}}^3 - 81920\,{
s_{1}}^7\,s_{2}\,s_{3}\,s_{6} \\&\qquad + 40960\,{s_{1}}^7\,s_{2}\,s_{
4}\,s_{5} + 45056\,{s_{1}}^7\,{s_{3}}^2\,s_{5} + 36864\,{s_{1}}^7\,s_{
3}\,{s_{4}}^2 - 24576\,{s_{1}}^7\,s_{4}\,s_{7} \\&\qquad - 8192\,{s_{1
}}^6\,{s_{2}}^3\,s_{6} - 4096\,{s_{1}}^6\,{s_{2}}^2\,s_{3}\,s_{5} -
8192\,{s_{1}}^6\,{s_{2}}^2\,{s_{4}}^2 \\&\qquad + 49152\,{s_{1}}^6\,s
_{2}\,{s_{3}}^2\,s_{4} - 299008\,{s_{1}}^6\,s_{2}\,s_{3}\,s_{7} -
90112\,{s_{1}}^6\,s_{2}\,s_{4}\,s_{6} \\&\qquad + 86016\,{s_{1}}^6\,s
_{2}\,{s_{5}}^2 + 20480\,{s_{1}}^6\,{s_{3}}^4 + 77824\,{s_{1}}^6\,{s_{
3}}^2\,s_{6} - 77824\,{s_{1}}^6\,s_{3}\,s_{4}\,s_{5} \\&\qquad - 8192
\,{s_{1}}^6\,{s_{4}}^3 + 73728\,{s_{1}}^6\,{s_{6}}^2 - 212992\,{s_{1}}
^5\,{s_{2}}^3\,s_{7} + 131072\,{s_{1}}^5\,{s_{2}}^2\,s_{3}\,s_{6}
\\&\qquad - 40960\,{s_{1}}^5\,s_{2}\,{s_{3}}^2\,s_{5} + 24576\,{s_{1}
}^5\,s_{2}\,s_{3}\,{s_{4}}^2 + 139264\,{s_{1}}^5\,s_{2}\,s_{4}\,s_{7}
\\&\qquad - 49152\,{s_{1}}^5\,s_{2}\,s_{5}\,s_{6} - 49152\,{s_{1}}^5
\,{s_{3}}^3\,s_{4} + 229376\,{s_{1}}^5\,{s_{3}}^2\,s_{7} \\&\qquad +
90112\,{s_{1}}^5\,s_{3}\,s_{4}\,s_{6} - 122880\,{s_{1}}^5\,s_{3}\,{s_{
5}}^2 + 32768\,{s_{1}}^5\,{s_{4}}^2\,s_{5}
\displaybreak[0]
\\&\qquad + 147456\,{s_{1}}
^5\,s_{6}\,s_{7} + 778240\,{s_{1}}^4\,{s_{2}}^2\,s_{3}\,s_{7} + 81920
\,{s_{1}}^4\,{s_{2}}^2\,s_{4}\,s_{6} \\&\qquad - 81920\,{s_{1}}^4\,{s
_{2}}^2\,{s_{5}}^2 - 286720\,{s_{1}}^4\,s_{2}\,{s_{3}}^2\,s_{6} +
40960\,{s_{1}}^4\,s_{2}\,s_{3}\,s_{4}\,s_{5} \\&\qquad - 32768\,{s_{1}
}^4\,s_{2}\,{s_{4}}^3 - 122880\,{s_{1}}^4\,s_{2}\,s_{5}\,s_{7} -
245760\,{s_{1}}^4\,s_{2}\,{s_{6}}^2 \\&\qquad + 81920\,{s_{1}}^4\,{s_{
3}}^3\,s_{5} - 409600\,{s_{1}}^4\,s_{3}\,s_{4}\,s_{7} + 122880\,{s_{1}
}^4\,s_{3}\,s_{5}\,s_{6} \\&\qquad - 131072\,{s_{1}}^4\,{s_{4}}^2\,s_{
6} + 122880\,{s_{1}}^4\,s_{4}\,{s_{5}}^2 + 147456\,{s_{1}}^4\,{s_{7}}^
2 \\&\qquad - 327680\,{s_{1}}^3\,{s_{2}}^2\,s_{4}\,s_{7} - 819200\,{s
_{1}}^3\,s_{2}\,{s_{3}}^2\,s_{7} + 163840\,{s_{1}}^3\,s_{2}\,s_{3}\,{s
_{5}}^2 \\&\qquad - 393216\,{s_{1}}^3\,s_{2}\,s_{6}\,s_{7} + 163840\,{
s_{1}}^3\,{s_{3}}^3\,s_{6} - 163840\,{s_{1}}^3\,{s_{3}}^2\,s_{4}\,s_{5
} \\&\qquad + 65536\,{s_{1}}^3\,s_{3}\,{s_{4}}^3 + 196608\,{s_{1}}^3\,
s_{3}\,s_{5}\,s_{7} + 393216\,{s_{1}}^3\,s_{3}\,{s_{6}}^2 \\&\qquad -
262144\,{s_{1}}^3\,{s_{4}}^2\,s_{7} + 393216\,{s_{1}}^3\,s_{4}\,s_{5}
\,s_{6} - 294912\,{s_{1}}^3\,{s_{5}}^3 \\&\qquad - 983040\,{s_{1}}^2\,
{s_{2}}^2\,s_{5}\,s_{7} + 589824\,{s_{1}}^2\,{s_{2}}^2\,{s_{6}}^2 +
1572864\,{s_{1}}^2\,s_{2}\,s_{3}\,s_{4}\,s_{7} \\&\qquad - 393216\,{s
_{1}}^2\,s_{2}\,s_{3}\,s_{5}\,s_{6} - 131072\,{s_{1}}^2\,s_{2}\,{s_{4}
}^2\,s_{6} + 131072\,{s_{1}}^2\,s_{2}\,s_{4}\,{s_{5}}^2 \\&\qquad -
393216\,{s_{1}}^2\,s_{2}\,{s_{7}}^2 + 327680\,{s_{1}}^2\,{s_{3}}^3\,s
_{7} - 131072\,{s_{1}}^2\,{s_{3}}^2\,s_{4}\,s_{6} \\&\qquad + 65536\,{
s_{1}}^2\,{s_{3}}^2\,{s_{5}}^2 - 65536\,{s_{1}}^2\,s_{3}\,{s_{4}}^2\,s
_{5} + 393216\,{s_{1}}^2\,s_{3}\,s_{6}\,s_{7} \\&\qquad + 393216\,{s_{
1}}^2\,s_{4}\,s_{5}\,s_{7} - 393216\,{s_{1}}^2\,s_{4}\,{s_{6}}^2 +
1179648\,s_{1}\,{s_{2}}^2\,s_{6}\,s_{7} \\&\qquad + 1572864\,s_{1}\,s
_{2}\,s_{3}\,s_{5}\,s_{7} - 1179648\,s_{1}\,s_{2}\,s_{3}\,{s_{6}}^2 -
262144\,s_{1}\,s_{2}\,{s_{4}}^2\,s_{7} \\&\qquad - 1441792\,s_{1}\,{s
_{3}}^2\,s_{4}\,s_{7} + 393216\,s_{1}\,{s_{3}}^2\,s_{5}\,s_{6} +
262144\,s_{1}\,s_{3}\,{s_{4}}^2\,s_{6} \\&\qquad - 131072\,s_{1}\,s_{3
}\,s_{4}\,{s_{5}}^2 + 393216\,s_{1}\,s_{3}\,{s_{7}}^2 - 1572864\,s_{1}
\,s_{4}\,s_{6}\,s_{7} \\&\qquad - 1179648\,s_{1}\,{s_{5}}^2\,s_{7} +
1179648\,s_{1}\,s_{5}\,{s_{6}}^2 + 589824\,{s_{2}}^2\,{s_{7}}^2
\\&\qquad - 1179648\,s_{2}\,s_{3}\,s_{6}\,s_{7} - 393216\,s_{2}\,s_{4
}\,s_{5}\,s_{7} + 589824\,{s_{3}}^2\,{s_{6}}^2 \\&\qquad + 524288\,s_{
3}\,{s_{4}}^2\,s_{7} - 393216\,s_{3}\,s_{4}\,s_{5}\,s_{6} + 65536\,{s
_{4}}^2\,{s_{5}}^2 - 1572864\,s_{4}\,{s_{7}}^2 \\&\qquad + 2359296\,s
_{5}\,s_{6}\,s_{7}
\end{align*}
}
\newpage
|
1992-02-11T00:25:33 | 9202 | alg-geom/9202004 | en | https://arxiv.org/abs/alg-geom/9202004 | [
"alg-geom",
"math.AG"
] | alg-geom/9202004 | David R. Morrison | David R. Morrison | Mirror symmetry and rational curves on quintic threefolds: a guide for
mathematicians | 26 pp., AmS-LaTeX (v1.0 or 1.1) | J. Amer. Math. Soc. 6 (1993) 223--247 | null | null | null | We give a mathematical account of a recent string theory calculation which
predicts the number of rational curves on the generic quintic threefold. Our
account involves the interpretation of Yukawa couplings in terms of variations
of Hodge structure, a new $q$-expansion principle for functions on the moduli
space of Calabi-Yau manifolds, and the ``mirror symmetry'' phenomenon recently
observed by string theorists.
| [
{
"version": "v1",
"created": "Mon, 10 Feb 1992 23:25:23 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Morrison",
"David R.",
""
]
] | alg-geom | \section*{Introduction}
There has been much recent excitement among mathematicians
about a calculation made by a group of string theorists
(P.~Candelas, X.~C.~de la Ossa, P.~S.~Green, and L.~Parkes \cite{pair})
which purports to give a count of the number of rational curves
of fixed degree on a general quintic threefold.
The calculation mixes arguments from string theory with arguments
from mathematics, and is generally quite difficult to follow for
mathematicians.
I believe that I now understand the essential
mathematical content of that calculation.
It is my purpose in this note to explain my understanding
in terms familiar to algebraic geometers.
What Candelas et al.\ actually calculate is a $q$-expansion of a
certain function determined by the variation of Hodge structure
of some {\em other\/} family of threefolds with trivial canonical
bundle. The ``mirror symmetry principle'' is then invoked to predict
that the Fourier coefficients in that expansion should be related
to the number of rational curves on a quintic threefold.
One mathematical surprise in this story is a new $q$-expansion principle
for functions on the moduli space of Calabi-Yau manifolds.
Near points on the boundary of moduli where the monodromy is ``maximally
unipotent,'' there turn out to be
natural coordinates in which to make $q$-expansions of functions.
In this paper, we will discuss these
$q$-expansions only
in the case of one-dimensional moduli spaces; the general
case will be treated elsewhere.
By focusing on this $q$-expansion principle, we place the computation
of \cite{pair} in a mathematically natural framework. Although there
remain certain dependencies on a choice of coordinates, the coordinates
used for calculation are canonically determined by the monodromy of
the periods, which is itself intrinsic. On the other hand, we have
removed some of the physical arguments which were
used in the original paper to help choose the coordinates appropriately.
The result may be that our presentation is less convincing to physicists.
\bigskip
The plan of the paper is as follows. In section
\ref{Yukawa}
we review variations of Hodge structure, and explain how to define
``Yukawa couplings'' in Hodge-theoretic terms.
(A discussion of Yukawa couplings along the same lines has
also been given by Cecotti \cite{cecotti1}, \cite{cecotti2}.)
In section
\ref{period}
we study the asymptotic behavior of the periods near points with maximally
unipotent monodromy. This is applied to find $q$-expansions
of the Yukawa couplings in section
\ref{qexp}.
In section \ref{mirror}
we attempt to describe mirror symmetry in geometric terms. In section
\ref{qm}
we turn to the main example (the family of ``quintic-mirrors''),
and in section
\ref{moonshine}
we explain how the mirror symmetry principle predicts from the
earlier calculations what the numbers of rational curves on quintic
threefolds should be. Several technical portions of the paper have
been banished to appendices.
We work throughout with algebraic varieties over the complex numbers,
which we often identify with complex manifolds (or complex analytic
spaces).
If $X$ is
a compact complex manifold and $p$, $q\ge 0$, we define
\[
H^{p,q}(X)=H^q(\Omega^p_X)=H^q(\Lambda ^p\Omega _X)
\]
where $\Omega _X$ is the holomorphic cotangent bundle of $X$.
(This is slightly non-standard.)
We extend this definition to the case $p<0$, $q\ge 0$ by
\[
H^{p,q}(X)=H^q(\Lambda ^{-p}\Theta_X),
\]
where $\Theta_X$ is the holomorphic tangent bundle of $X$.
(This is very non-standard.)
The dimension
of $H^{p,q}(X)$ is denoted by $h^{p,q}(X)$, or simply by $h^{p,q}$.
\section{Variations of Hodge structure arising from families of
Calabi-Yau manifolds}
\label{Yukawa}
Recall that a {\em Calabi-Yau manifold} is a compact K\"ahler manifold $X$
of complex dimension $n$ which has trivial canonical bundle,
such that the Hodge numbers $h^{k,0}$ vanish for $0<k<n$.
Thanks to a celebrated theorem of Yau \cite{yau}, every such manifold
admits Ricci-flat K\"ahler metrics.
Any non-zero section of the canonical bundle determines
isomorphisms
\[
H^i(\Theta_X) \cong H^{i}(\Omega^{n-1}_X).
\]
Thus, if $n>1$, then $X$ has no holomorphic vector fields. Moreover, the
tangent space to moduli $H^1(\Theta_X)$ has dimension
$h^{-1,1}=h^{n-1,1}$ and
the natural obstruction
space for the moduli problem $H^2(\Theta_X)$ has dimension $h^{n-1,2}$,
which is generally non-zero for $n>2$.
However,
the theorem of Bogomolov \cite{bogomolov}, Tian \cite{tian}, and
Todorov \cite{todorov}
says that the moduli problem is in fact unobstructed, and the moduli space
is therefore smooth of dimension $h^{n-1,1}$.
We review some facts about Hodge structures and their variation.
Good general references for this
are
Griffiths et al.\ \cite{transcendental}, and Schmid \cite{schmid}.
The $n^{\text{th}}$ cohomology group of $X$ has a {\em Hodge decomposition}
\[
H^n(X,\Bbb C) \cong \bigoplus\begin{Sb} p+q=n\\ p,q\ge 0 \end{Sb} H^{p,q}(X).
\]
(With our conventions, this follows from the Hodge theorem in de Rham
cohomology $H^n_{\text{DR}}(X) = \bigoplus H^{p,q}_{\bar\partial}(X)$
together with the Dolbeault isomorphism
$H^{p,q}_{\bar\partial}(X)\cong H^q(\Omega^p_X) = H^{p,q}(X)$.)
The Hodge decomposition can also be described by means of the {\em Hodge
filtration}
\[
F^p(X):=\bigoplus_{p\le p'\le n} H^{p',n-p'}(X);
\]
we then have $H^{p,n-p}(X)\cong F^p(X)/F^{p+1}(X)$.
The cup product on cohomology composed with evaluation on
the canonical orientation class of $X$
determines a bilinear map
\[
\ip{\,}{\,}: H^n(X,\Bbb Z) \times H^n(X,\Bbb Z) \to H^{2n}(X,\Bbb Z)
\overset{\cong}{\to} {\Bbb Z},
\]
called a {\em polarization}.
There is an associated {\em adjoint map}
\[
\operatorname{ad}_{\ip{\,}{\,}}: H^n(X,\Bbb Z) \to
\operatorname{Hom}(H^n(X,\Bbb Z),\Bbb Z)
\]
defined by
$
\operatorname{ad}_{\ip{\,}{\,}}(x)(y)=\ip{x}{y}
$.
After tensoring with ${\Bbb C}$ and invoking the Hodge decomposition,
the adjoint map induces isomorphisms
\[
\operatorname{ad}_{\ip{\,}{\,}}: H^{p,n-p}(X) \overset{\cong}{\to}
(H^{n-p,p}(X))^*.
\]
We now recall a construction which first arose in the study of
infinitesimal variations of Hodge structure by Carlson, M.~Green, Griffiths,
and Harris \cite{CGGH}.
The cup product determines a natural map
\begin{equation} \label{eqF}
H^1(\Theta_X)\to \bigoplus \operatorname{Hom}(H^{p,q}(X), H^{p-1,q+1}(X))
\end{equation}
called the {\em differential of the period map}. Iterates of this map
are symmetric in their variables; the {\em $n^{\text{th}}$ iterate
of the differential} is the induced map
\[
\operatorname{Sym}^n H^1(\Theta_X)\to\operatorname{Hom}(H^{n,0}(X), H^{0,n}(X)) .
\]
Using the canonical isomorphism $\operatorname{Hom}(H^{n,0}(X), H^{0,n}(X))=
(H^{n,0}(X))^*\otimes H^{0,n}(X)$
and the isomorphism
$H^{0,n}(X)\cong(H^{n,0}(X))^*$ induced by the adjoint,
we get a map
\begin{equation}
\label{eqB}
\operatorname{Sym}^nH^1(\Theta_X)\to(H^{n,0}(X)^*)^{\otimes 2}.
\end{equation}
We call this the {\em unnormalized Yukawa
coupling\/}\footnote{This particular Yukawa coupling
is probably only interesting in physics
if $n=3$. In dimension $n$, what is being computed here is the
``$n$-point Yukawa coupling.''}
of $X$.
If we choose an element
of $H^{n,0}(X)^{\otimes 2}$ and evaluate the map (\ref{eqB}) on that element,
we get a map
$
\operatorname{Sym}^nH^1(\Theta_X)\to\Bbb C
$
called a {\em normalized Yukawa coupling}. The ``normalization'' is the
choice of element of $H^{n,0}(X)^{\otimes 2}$.
\bigskip
We now analyze these constructions for a {\em family} of manifolds.
Suppose we are given a quasi-projective variety $C$ and a
smooth map $\pi: {\cal X}\to C$ whose fibers are
Calabi-Yau manifolds. Suppose also that this family
can be completed to a family of varieties
$\bar\pi: \overline{\cal X}\to \overline C$, where $\overline C$ is a
projective compactification of $C$. The fibers of $\bar\pi$ may degenerate
over the boundary $B:=\overline C - C$. We assume that $B$ is a divisor
with normal crossings on $\overline C$.
For any point $P\in C$ we denote the fiber $\pi^{-1}(P)$ of $\pi$
by $X_P$. The Kodaira-Spencer map
$
\rho:\Theta_{C,P}\to H^1(\Theta_{X_P})
$
maps the tangent space to $C$ at $P$ to the tangent space to moduli at
the point $[X_P]$. If $C$ is actually a moduli space for the fibers
of ${\cal X}$, the map $\rho$ will be an isomorphism.
The cohomology of the fibers of the map $\pi$ with coefficients in
${\Bbb Z}$ and ${\Bbb C}$ fit together into local
systems $R^n\pi_* {\Bbb Z}$ and $R^n\pi_* {\Bbb C}$ on $C$.
The Hodge filtration becomes
a filtration of the vector bundle
${\cal F}^0 := R^n\pi_* {\Bbb C} \otimes {\cal O}_C$ by
holomorphic subbundles:
\[
R^n\pi_* {\Bbb C} \otimes {\cal O}_C
= {\cal F}^0 \supset {\cal F}^1 \supset \cdots \supset
{\cal F}^{n-1} \supset
{\cal F}^n \supset (0).
\]
The vector bundle ${\cal F}^0$ has
a natural flat connection $\nabla: {\cal F}^0\to {\cal F}^0\otimes \Omega_C$
called the {\em Gau\ss-Manin connection},
whose horizontal sections determine the local system
$R^n\pi_* {\Bbb C}$.
The {\em Griffiths transversality property} says that
$
\nabla ( {\cal F}^p )\subset {\cal F}^{p-1} \otimes \Omega _C
$.
There is a natural extension of this setup over the boundary $B$,
which involves the sheaf $\Omega_{\overline C}(\log B)$ of
{\em logarithmic differentials}. (That sheaf is locally generated
by $\Omega_{\overline C}$ and all elements of the form $df/f$,
where $f=0$ is a local equation of a local component of $B$.) Although
the local system $R^n\pi_* {\Bbb C}$
cannot in general be extended across $B$ in a single-valued way, the
Hodge bundles ${\cal F}^p$ {\em do} have natural extensions to bundles
$\overline{\cal F}^p$ on $\overline C$. And the Gau\ss-Manin
connection $\nabla$ extends to a connection
$
\overline\nabla : \overline{\cal F}^0 \to \overline{\cal F}^0 \otimes
\Omega_{\overline C}(\log B)
$
which satisfies
$
\overline
\nabla ( \overline{\cal F}^p )\subset \overline{\cal F}^{p-1}
\otimes \Omega _{\overline C}(\log B)
$.
This restriction on the types of poles which $\overline\nabla$
may have along $B$ is equivalent to a requirement that
the connection $\nabla$ have
``regular singular points.''
The extended Gau\ss-Manin connection $\overline\nabla$ gives
rise to an ${\cal O}_{\overline C}$-linear map on the associated gradeds
\begin{equation} \label{eqD}
\widetilde\nabla: \overline{\cal F}^p/\overline{\cal F}^{p+1} \to
(\overline{\cal F}^{p-1}/\overline{\cal F}^{p}) \otimes
\Omega _{\overline C}(\log B).
\end{equation}
To make contact with the $n^{\text{th}}$ iterate of the differential
and the Yukawa coupling, we introduce the sheaf
$\Theta_{\overline C}({-\log B})$ of {\em vector fields with logarithmic
zeros},
which is the dual of $\Omega _{\overline C}(\log B)$.
The map (\ref{eqD}) then induces the bundle version of (\ref{eqF})
\[
\Theta_{\overline C}({-\log B})\to \bigoplus \mathop{\text{\it $\cal H$om}} (
\overline{\cal F}^p/\overline{\cal F}^{p+1} ,
\overline{\cal F}^{p-1}/\overline{\cal F}^{p}) .
\]
When this is iterated $n$ times, it produces a map
\begin{equation}
\label{eqC}
\operatorname{Sym}^n(\Theta_{\overline C}({-\log B}))\to
\mathop{\text{\it $\cal H$om}} (\overline{\cal F}^n ,
\overline{\cal F}^{0}/\overline{\cal F}^{1}).
\end{equation}
The polarizations fit together into a
bilinear map of local systems
\[
\ip{\,}{\,}: R^n\pi_*{\Bbb Z} \times R^n\pi_*{\Bbb Z} \to R^{2n}\pi_*{\Bbb Z}
\overset{\cong}{\to} {\Bbb Z}
\]
whose adjoint map induces an isomorphism
$
\operatorname{ad}_{\ip{\,}{\,}}: ({\cal F}^{0}/{\cal F}^{1}) \to
({\cal F}^{n})^*
$.
This extends to a map of bundles
\begin{equation} \label{eqG}
\operatorname{ad}_{\ip{\,}{\,}}: (\overline{\cal F}^{0}/\overline{\cal F}^{1}) \to
(\overline{\cal F}^{n})^*.
\end{equation}
Using the canonical isomorphism
$
\mathop{\text{\it $\cal H$om}} (\overline{\cal F}^n ,
\overline{\cal F}^{0}/\overline{\cal F}^{1}) =
(\overline{\cal F}^n)^* \otimes
(\overline{\cal F}^{0}/\overline{\cal F}^{1})
$
and composing the map (\ref{eqG}) with the map (\ref{eqC}),
we get
the {\em Yukawa map}
\[
\kappa: \operatorname{Sym}^n(\Theta_{\overline C}({-\log B})) \to
((\overline{\cal F}^{n})^*)^{\otimes 2}.
\]
If we also specify a section of
$(\overline{\cal F}^{n})^{\otimes 2}$,
we get a {\em normalized Yukawa map}
\[
\kappa^{\text{norm}}: \operatorname{Sym}^n(\Theta_{\overline C}({-\log B})) \to
{\cal O}_{\overline C}.
\]
Suppose that
$C$ is actually the moduli space for the fibers of ${\cal X}$, so that
$\rho$ is an isomorphism. If we compose $\rho^{-1}$
with a normalized Yukawa map $\kappa^{\text{norm}}$ we get
\[
\operatorname{Sym}^nH^1(\Theta_{X_P}) \overset{\rho^{-1}}{\longrightarrow}
\operatorname{Sym}^n(\Theta_{C,P}) \overset{\kappa^{\text{norm}}}{\longrightarrow}
{\cal O}_{C,P} = \Bbb C.
\]
In this way, we exactly recover the corresponding
normalized Yukawa coupling.
Candelas et al.\ \cite{pair}
typically compute the Yukawa coupling in local coordinates
(away from the boundary) as follows. Suppose that $\dim C = 1$,
and that $\psi$ is a local coordinate defined in an open set $U\subset C$.
There is an induced vector field $d/d\psi$, which is
a local section of $\Theta_U$.
Choose a section\footnote{To avoid confusion with the cotangent bundle,
we denote this section by $\omega$ rather than $\Omega$. However,
in appendix C below, we will revert to the notation $\Omega$ used in
\cite{pair}.}
$\omega$ of ${\cal F}^n$ over $U$, and define
\[
\kappa_{\psi\dots\psi}
=\kappa(\frac{d}{d\psi},\dots,\frac{d}{d\psi})\cdot\omega^2.
\]
(The number of $\psi$'s in the subscript is $n$.) This is a holomorphic
function on $U$. If we alter $\omega$
by the gauge transformation $\omega \mapsto f\omega$,
then the Yukawa coupling
transforms as $\kappa_{\psi\dots\psi} \mapsto f^2\kappa_{\psi\dots\psi}$.
``Normalizing the Yukawa map'' is the same thing as ``fixing the gauge.''
Our primary goal will be to compute the asymptotic behavior of the
Yukawa map $\kappa$ in a neighborhood of the boundary $B$.
\section{The asymptotic behavior of the periods}
\label{period}
For simplicity of exposition,
we now specialize to the case in which $C$ is a curve.
Let $P\in B$ be a boundary point, and let $T_P$ be the monodromy of the local
system $R^n\pi_*{\Bbb Z}$ around $P$.
We regard $T_P$ as an element of $\operatorname{Aut} H^n(X_{P'},\Bbb Z)$, where
$P'$ is a point near $P$; $T_P$ is determined by analytic continuation
along a path which goes once
around $P$ in the counterclockwise direction. By the monodromy theorem
\cite{monodromy}, $T_P$ is quasi-unipotent, which means that some power
$T_P^k$ is unipotent. Moreover, the index of unipotency is bounded:
we have $(T_P^k-I)^{n+1}=0$.
We say that {\em $P$ is a point at which the monodromy is maximally
unipotent} if the monodromy $T_P$ is unipotent, and if $(T_P-I)^n\ne 0$.
(Thus, the index of nilpotency of $T_P-I$ is maximal.)
Since $T_P-I$ is nilpotent, we can define the logarithm of the monodromy
$N=\log(T_P)\in\operatorname{Aut} H^n(X_{P'},\Bbb Q)$
by a finite power series
\[
\log(T_P)=(T_P-I)-\frac{(T_P-I)^2}2 +\dots +(-1)^{n+1}\frac{(T_P-I)^n}n.
\]
(Rational coefficients are needed in cohomology since
rational numbers appear in the power series.)
$N$ is also a nilpotent matrix, with the
same index of nilpotency as $T_P-I$.
\begin{lemma}
\label{lemma1}
Let $\pi: {\cal X}\to C$ be a one-parameter family of varieties with
$h^{n,0}=1$.
Let $P\in B=\overline C-C$ be a boundary point at which the monodromy
on $R^n\pi_*{\Bbb Z}$ is maximally unipotent
and let $N$ be the logarithm of the monodromy.
Then the image of $N^{n}$ is a $\Bbb Q$-vector space of dimension one,
and the image of $N^{n-1}$ is a $\Bbb Q$-vector space of dimension two.
\end{lemma}
We defer the proof of this lemma to appendix A.
\bigskip
We say that a basis $g_0$, $g_1$ of $(\Im N^{n-1})\otimes \Bbb C \subset
H^n(X_{P'},\Bbb C)$
is an
{\em adapted basis} if $g_0$ spans $(\Im N^n)\otimes \Bbb C$.
(We have extended scalars to $\Bbb C$ since certain computational
procedures lead more naturally to complex coefficients.)
If $g_0$, $g_1$ is an adapted basis for $(\Im N^{n-1})\otimes \Bbb C$, then
by Poincar\'e duality, there are homology classes
$\gamma_0, \gamma_1\in H_n(X_{P'},{\Bbb C})$
such that $\ip{g_j}{\alpha }=\int_{\gamma_j}\alpha $ for any
$\alpha \in H^n(X_{P'},{\Bbb C})$.
Here we denote the evaluation of cohomology classes on homology classes
by using an integral sign,
since that evaluation is often accomplished by integration.
\begin{proposition}
Let $\gamma_0, \gamma_1$ be the homology classes determined by an
adapted basis $g_0$, $g_1$ of $(\Im N^{n-1})\otimes \Bbb C$.
Define a constant $m$ by
$Ng_1=mg_0$.
Let $\overline U$ be a small neighborhood of $P$, and let $z$ be a
coordinate on $\overline U$ centered at $P$. Let $\omega$ be a non-zero
section of $\overline{\cal F}^n$ over $\overline U$.
Then
\begin{enumerate}
\item
$\int_{\gamma_0}\omega$ extends to a single-valued function on
$\overline U$.
\item
$\int_{\gamma_1}\omega$ is not single-valued. However, we can
write
\[
\frac{\frac1m\int_{\gamma_1}\omega}{\int_{\gamma_0}\omega}
= \frac{\log z}{2\pi i} + \text{ single valued function.}
\]
\end{enumerate}
\end{proposition}
\begin{pf}
Any $g\in H^n(X_{P'},{\Bbb C})$ can be extended to a section $g(z)$ of
the local system over $U=\overline U - P$, which may be multi-valued.
But by the nilpotent orbit theorem \cite{schmid},
$\exp(-\frac{\log z}{2\pi i}N)g(z)$ extends to a single-valued section.
Since $\omega$ is single-valued,
\[
\ip{\exp(-\frac{\log z}{2\pi i}N)g(z)}{\omega}
\]
will also be single-valued. Now $g_j\in (\Im N^{n-1})\otimes \Bbb C$
implies that $N^2g_j=0$ for $j=1$, $2$.
The series needed for $\exp$ in this case is thus rather simple:
\begin{eqnarray*}
\exp(-\frac{\log z}{2\pi i}N)g_j(z) &=& (I - \frac{\log z}{2\pi i}N)g_j(z)\\
&=& g_j(z) - \frac{\log z}{2\pi i}Ng_j(z).
\end{eqnarray*}
We conclude that
\[
\int_{\gamma_0}\omega = \ip{g_0(z)}{\omega}
\]
and
\[
\int_{\gamma_1}\omega - m\,\frac{\log z}{2\pi i}\int_{\gamma_0}\omega
=\ip{g_1(z) - \frac{\log z}{2\pi i}\,mg_0(z)}{\omega}
\]
are single-valued functions.
\end{pf}
\begin{corollary}
Let $\gamma_0, \gamma_1$ be the homology classes determined by an
adapted basis $g_0$, $g_1$ of $(\Im N^{n-1})\otimes \Bbb C$,
as in the proposition. The function
\[
t:= \frac{\frac1m\int_{\gamma_1}\omega}{\int_{\gamma_0}\omega}
\]
gives a natural parameter
on the universal cover $\widetilde U$
of $U$ called a {\em quasi-canonical parameter}, and
\[ q:=e^{2\pi it} \]
gives a natural coordinate on $\overline U$ called a {\em quasi-canonical
coordinate}. These are independent of
the choice of $\omega$.
We have
\[
\frac d{dt}=2\pi i\,q\,\frac d{dq},
\]
either of which serves as a local generator of the sheaf
$\Theta_{\overline C}({-\log B})$.
Moreover, under a change of adapted basis $(g_0,g_1)\mapsto (ag_0,bg_0+cg_1)$,
we have
\begin{eqnarray*}
m & \mapsto & \frac ca\,m, \\
t & \mapsto & t+\frac b{mc}, \text{\ and} \\
q & \mapsto & e^{2\pi ib/mc}q.
\end{eqnarray*}
Therefore, $t$ is uniquely determined up to an additive constant,
and $q$ is uniquely determined up to a multiplicative constant.
\end{corollary}
We can normalize further if we take the integral structure into
account. We call $g_0$, $g_1$ a {\em good integral basis} of $\Im N^{n-1}$
if $g_0$ is a generator of
$\Im N^n \cap H^n(X_{P'},{\Bbb Z})$, and $g_1$ is an indivisible element
of $H^n(X_{P'},{\Bbb Z})$ which can be written as
$g_1 = \frac1\lambda N^{n-1}g$ for some $\lambda >0$ and some
$g\in H^n(X_{P'},{\Bbb Z})$ such that $\ip{g_0}{g}=1$.
Notice that a good integral basis is an adapted basis.
The next lemma, which is based on some work of Friedman and Scattone
\cite{fs}, will be proved in appendix A.
\begin{lemma}
\label{lemma2}
Good integral bases exist. If $g_0$, $g_1$ and $g_0'$, $g_1'$
are good integral bases, then
\begin{eqnarray*}
g_1' & = & k\, g_0 + (-1)^\ell g_1 \\
g_0' & = & (-1)^\ell g_0,
\end{eqnarray*}
for some integers $k$ and $\ell$.
\end{lemma}
Since $(T-I)^2 =0$ on $\Im N^{n-1}$, we have the simple formula $N=T-I$
on that space. In particular, when restricted to
$\Im N^{n-1}$, the map $N$ is defined over the integers. Thus,
if $g_0$, $g_1$ is a good integral basis and we write $Ng_1=mg_0$,
then $m$ is an integer. Note that $m$ is independent of the choice
of good integral basis.
\begin{corollary}
Let $g_0$, $g_1$ be a good integral basis, and define
an integer $m$ by $Ng_1=mg_0$.
Then the quasi-canonical coordinate $q$ formed
from this basis is unique up to multiplication by an $|m|^{\text{th}}$
root of unity.
\end{corollary}
We call $q$ a {\em canonical coordinate\/} and $t$ a {\em canonical
parameter\/} under these circumstances.
These are actually unique if $|m|=1$; in this
case, we say that the monodromy is {\em small}.
\section{The $q$-expansion of the Yukawa coupling}
\label{qexp}
The first example of the construction of the previous section is
furnished by the classical theory of periods of elliptic curves.
Let $\pi:{\cal X}\to U$ be a family of smooth elliptic curves over
a punctured disk $U$ which can be completed to a family
$\bar\pi: \overline{\cal X}\to \overline U$ with a singular
fiber over the boundary point $P=\overline U - U$. The point $P$
is called a {\em cusp}.
Let $P'\in U$,
and suppose there is a symplectic
basis $\gamma_0$, $\gamma_1$ of the first homology
group $H_1(X_{P'},\Bbb Z)$ such that the monodromy $T_P$ acts as
\begin{eqnarray*}
T_P(\gamma_0) & = & \gamma_0 \\
T_P(\gamma_1) & = & \gamma_0 + \gamma_1.
\end{eqnarray*}
(The basis is {\em symplectic\/} if $\gamma_0\cap\gamma_1=1$.)
This easily implies that $P$ is a maximally unipotent boundary point,
that $\gamma_0$, $\gamma_1$ is the homology basis dual to a good
integral basis,
and that $m=1$.
For a fixed holomorphic one-form $\omega$ on $X_{P'}$, the numbers
$(\int_{\gamma_0}\omega, \int_{\gamma_1}\omega)$
were classically known
as the {\em periods\/} of the elliptic curve $X_{P'}$. By varying the
one-form, the periods can be normalized to take the form
$(1,\tau)$. The invariant way to formulate this is to define
\[
\tau=\frac{\int_{\gamma_1}\omega}{\int_{\gamma_0}\omega}.
\]
This function $\tau$ can be regarded as a map from the universal cover
$\widetilde U$ of $U$ to the upper half-plane $\Bbb H$. (The image lies
in the {\em upper\/} half-plane since the basis is symplectic.)
The monodromy transformation $T_P$ induces the map
\begin{equation} \label{fourier}
\tau\mapsto\tau+1.
\end{equation}
Thus, functions $f$ defined on $U$ pull back to functions $\widetilde f$
on $\widetilde U$
which are invariant under (\ref{fourier}). It follows that any such
function has a {\em Fourier series\/}
\[
\widetilde f(\tau) = \sum_{n\in\Bbb Z}a_ne^{2\pi in\tau}.
\]
If expressed in terms of the natural coordinate $q=e^{2\pi i\tau}$ on
$U$, this is called a {\em $q$-expansion}, and it takes the form
\[
f(q)=\sum_{n\in\Bbb Z}a_nq^n.
\]
If $f$ has a holomorphic extension across the cusp $P$, the only terms
appearing in this sum are those with $n\ge 0$.
\bigskip
What we have shown in section \ref{period} is that this classical
construction
generalizes to functions defined near a maximally unipotent boundary point
$P$
of a Calabi-Yau moduli space (at least when that space has dimension one).
Fix a good integral basis, which determines a canonical coordinate $q$
and a canonical parameter $t$.
The monodromy transformation $T_P$ acts on $t$
by $t\mapsto t+1$. Therefore, any function $f$ defined near $P$ which
is holomorphic at $P$
will have a $q$-expansion
\[
f(q)=\sum_{n= 0}^{\infty}a_nq^n,
\]
which can also be regarded as a Fourier series
\[
\widetilde f(t) = \sum_{n= 0}^{\infty}a_ne^{2\pi int}
\]
in $t$.
These expressions are unique if $|m|=1$, i.e., if the monodromy is small.
In order to obtain a $q$-expansion of the Yukawa coupling, we must
normalize that coupling. But there is a natural choice of normalization
determined by a good integral basis. To see this, note that
any good integral basis $g_0$, $g_1$ determines a section
$
(\int_{\gamma_0})^{-1} \in H^0(\overline U,(\overline{\cal F}^n))
$
by
\[
(\int_{\gamma_0})^{-1} := \frac{\omega}{\int_{\gamma_0}\omega}
\]
for any non-zero $\omega\in H^0(\overline U,\overline{\cal F}^n)$.
By lemma \ref{lemma2},
a change in good integral basis may change the sign of
$(\int_{\gamma_0})^{-1}$, but the induced section
$
(\int_{\gamma_0})^{-2}\in H^0(\overline U,(\overline{\cal F}^n)^{\otimes 2})
$
is independent of the choice of good integral basis.
We thus have a very natural normalization for the Yukawa map in $\overline U$.
We also have a natural parameter $t$ with which to compute, such that
$d/dt$ is a generator of $\Theta_{\overline U}({-\log B})$.
So we can define the {\em mathematically normalized
Yukawa coupling\/} $\kappa_{t\dots t}$
by the formula
\[
\kappa_{t\dots t}
=\kappa(\frac d{dt},\dots,\frac d{dt})\cdot(\int_{\gamma_0})^{-2}.
\]
This mathematically normalized Yukawa coupling
$\kappa_{t\dots t}$ is an intrinsically defined function on a neighborhood
of the boundary.
(It is canonically determined by our
choice of maximally unipotent boundary point; however, it could
conceivably change if the boundary point changes.)
The function $\kappa_{t\dots t}$ therefore has a $q$-expansion
\begin{equation}\label{qexpq}
\kappa_{t\dots t} = a_0+a_1q+a_2q^2+\cdots,
\end{equation}
which can also be regarded as a Fourier expansion in the
parameter $t$:
\begin{equation}\label{qexpt}
\kappa_{t\dots t} = a_0+a_1e^{2\pi it}+a_2e^{4\pi it}+\cdots.
\end{equation}
These expressions are unique if the monodromy is small.
\section{Mirror symmetry}
\label{mirror}
In this section I will attempt to outline the mirror symmetry
principle in mathematical terms, and describe some of the
mathematical evidence for it. I apologize to
physicists
for my
misrepresentations of their ideas,
and I apologize to
mathematicians for the vagueness
of my explanations.
Gepner \cite{gepner1} has conjectured that there is a one-to-one
correspondence between
$N=2$
superconformal field theories
with $c=3n$, and Calabi-Yau manifolds $X$ of dimension $n$
equipped with some
``extra structure'' $S$.
(This correspondence can be realized concretely in a number of
important cases using work of Greene, Vafa and Warner \cite{gvw},
Martinec \cite{martinec1}, \cite{martinec2}, and others.)
A precise geometric description of the
extra structure $S$ has not yet been given. It appears to involve
specifying a class in $U/\Gamma$, where $U\subset H^{1,1}(X)$
is some open set, and $\Gamma$ is some group of automorphisms of
$U$.
What {\em is} clear about
this extra structure is how to perturb it: first-order deformations
of $S$ correspond to elements of $H^{1,1}(X)$.
An instructive example is the case in which $X$ is
an elliptic
curve. In that case, as shown in \cite{dvv} and \cite{al},
one takes $U\subset H^{1,1}(X)\cong \Bbb C$
to be the upper half-plane, and $\Gamma=\operatorname{SL}(2,\Bbb Z)$.
Thus, the extra structure
$S$ represents a point in the $j$-line, or equivalently,
a choice of a {\em second} elliptic curve.
We specialize now to the case of dimension $n=3$. The space of first-order
deformations of the superconformal field theory can be decomposed
as\footnote{It has become common in the physics literature to use
$H^{2,1}(X)$ in place of $H^1(\Theta_X)$, largely because of the
success of Candelas \cite{candelas} and others in computing Yukawa
couplings on $H^{2,1}$. In order to get the correct answer in families,
however, we must return to the original analysis of Strominger and
Witten \cite{str-wit} and work with Yukawa couplings on $H^1(\Theta_X)$.
The point is that while $H^1(\Theta_X)$ and $H^{2,1}(X)$ are isomorphic
for a Calabi-Yau threefold, they are not {\em canonically\/} isomorphic.
This affects the bundles over the moduli space to which they belong.}
$H^1(\Theta_X) \oplus H^1(\Omega _X)$,
with
$H^1(\Theta_X)=H^{-1,1}(X)$ corresponding to first-order
deformations of the complex structure on $X$, and
$H^1(\Omega _X)=H^{1,1}(X)$ corresponding to first-order deformations of
the extra structure $S$.
These first-order deformations are called {\em marginal operators\/}
in the physics literature.
Specifying a superconformal field theory of this type also determines
cubic forms $\operatorname{Sym}^3H^{-1,1}(X)\to\Bbb C$ and
$\operatorname{Sym}^3H^{1,1}(X)\to\Bbb C$. The cubic form on $H^{-1,1}$ is the
Yukawa coupling described in section \ref{Yukawa}, normalized in
a way specified by the physical theory. From a
mathematical point of view, this is determined by the variation of Hodge
structure plus the choice of normalization.
This cubic form depends on the complex structure of $X$, but should be
independent of
the ``extra structure'' $S$.
The cubic form on $H^{1,1}$
lacks a precise geometric description at present.
By work of Dine, Seiberg, Wen, and Witten \cite{dsww2}
and Distler and Greene \cite{distler-greene},
it is known to have an expression of the form
\begin{equation}
\label{asymp}
\sum_{k=0}^\infty \sigma_k \, e^{-kR} ,
\end{equation}
where $R$ is a complex parameter which depends on the extra structure $S$.
The real part of $R$ is related to the ``radius'' in the physical theory
in such a way that $\Re R\to\infty$ is the ``large radius limit.''
The leading coefficient $\sigma_0$ is the natural topological product
$\operatorname{Sym}^3H^{1,1}(X)\to\Bbb C$.
(In other words, the cubic form on $H^{1,1}$ approaches the topological
product in the large radius limit.)
The higher coefficients $\sigma_k$ are
supposed to be related in some
well-defined way to the numbers of rational curves of various degrees
on the generic deformation of $X$ (assuming those numbers are finite).
One of the important unsolved problems in the theory is to determine this
relationship precisely.
As was first noticed by Dixon \cite[p.\ 118]{dixon}, and later developed
by Lerche, Vafa, and Warner \cite{lvw} and others, the identification
of one piece of the superconformal field theory with $H^{1,1}(X)$
and the other piece with $H^{-1,1}(X)\cong H^{2,1}(X)$ involves an arbitrary
choice, and the theory is also consistent with making the opposite choice.
Moreover, as we will describe below, there are examples in which the
Gepner correspondence can be realized for both choices.
But except in the very rare circumstance that
the Hodge numbers $h^{1,1}$ and $h^{2,1}=\dim H^{-1,1}$
coincide, changing the choice necessarily involves changing the
Calabi-Yau threefold $X$. The new threefold $\prim X$ will
have a completely different topology from the old: in fact, the
Hodge diamond is rotated by $90^\circ$ when passing from one
to the other.
This leads to a mathematical version of the mirror symmetry conjecture:
To each pair $(X,S)$ consisting
of a Calabi-Yau
threefold $X$ together with some extra structure $S$ there
should be associated a ``mirror pair''
$(\prim X,\prim S)$ which comes equipped with natural isomorphisms
$H^{-1,1}(X)\overset{\cong}{\to}H^{1,1}(\prim{X})$
and
$H^{1,1}(X)\overset{\cong}{\to}H^{-1,1}(\prim{X})$
that are compatible with the cubic forms.
Even in this rather imprecise\footnote{Among the things not properly
defined from a mathematical viewpoint, we must include
the normalization of the Yukawa coupling, the complex parameter $R$
(which depends on the ``extra
structure'' $S$), and the higher coefficients
$\sigma_k$.}
form, the conjecture as stated
is easily refuted:
There exist rigid Calabi-Yau threefolds, which have $h^{2,1}=0$
(see Schoen \cite{schoen}
for an example). Any mirror of such a threefold
would have $h^{1,1}=0$, and so could not be
K\"ahler.
A potentially correct version of the conjecture,
even less precise, begins: ``To most pairs
$(X,S)$, including almost all of interest in physics, there should be
associated\dots''.
It is tempting to speculate that the theory should
be extended to non-K\"ahler threefolds as in Reid's fantasy
\cite{nevertheless}, which might rescue the conjecture in its
original form.
Alternatively, Aspinwall and L\"utken \cite{al2} suggest that the
Gepner correspondence (and hence the mathematical version of
mirror symmetry) should only hold in the large radius
limit. Since no ``limits'' can be taken in the rigid case,
a mathematical mirror construction would not be expected there.
To be presented with a conjecture which has been only vaguely formulated
is unsettling to many mathematicians.
Nevertheless, the
mirror symmetry phenomenon appears to be quite widespread,
so it seems important to make further efforts to find a precise
formulation. In fact, there are at least
four major pieces of mathematically significant evidence for mirror symmetry.
\begin{enumerate}
\item[(i)]
Greene and Plesser \cite{greene-plesser} have studied a case in which
there are very solid physics arguments which tie the pair $(X,S)$
to the corresponding superconformal field theory (as predicted by Gepner).
The Calabi-Yau threefolds in question are desingularizations of
quotients of Fermat-type
weighted hypersurfaces by certain finite groups (including the trivial
group). For each pair $(X,S)$ of
this type, Greene and Plesser were able to find the corresponding
mirror pair $(\prim X,\prim S)$
by analyzing the associated superconformal field theories.
It turns out that the pairs are related by taking quotients:
$\prim X$ is a desingularization of $X/G$ for some symmetry group $G$.
By deformation arguments, the mirror symmetry phenomenon persists in
neighborhoods of $(X,S)$ and $(\prim X,\prim X)$.
Roan \cite{roanpf} subsequently gave a
direct mathematical proof that the predicted isomorphisms between
$H^{-1,1}$ and $H^{1,1}$ groups exist in this situation.
\item[(ii)]
Candelas, Lynker, and Schimmrigk \cite{cls} have computed the Hodge
numbers for a large class of Calabi-Yau threefolds which are
desingularizations of hypersurfaces in weighted
projective spaces. They put some extra constraints on the form of the
equation, and found about 6000 types of threefolds satisfying their
conditions.
The set of pairs $(h^{1,1},h^{2,1})$ obtained from these examples
is very nearly (but not precisely)
symmetric with respect to the interchange $h^{1,1}{\leftrightarrow}h^{2,1}$.
Since there is no {\em a priori} reason that the mirror of a desingularized
weighted hypersurface should again be a desingularized weighted
hypersurface, this is consistent with the conjecture and supports it
quite strongly.
\item[(iii)]
Aspinwall, L\"utken, and Ross \cite{alr} (see also \cite{al})
have carefully studied
a particular mirror pair $(X,S)$, $(\prim X,\prim S)$.
They put $X$ in a family ${\cal X} = \{X_t \}$ which has a degenerate limit
as $t$ approaches $0$. Some heuristics were used in choosing the
family $\cal X$, in an attempt to ensure that the limit as $t\to 0$
would correspond to the ``large radius limit'' for the mirror
($\prim X,\prim S)$. Aspinwall et al.\ then computed
the limiting behavior of
the cubic form on $H^{-1,1}(X_t)$, and showed that it coincides
with the topological product $\prim \sigma_0$ on
$H^{1,1}(\prim X)$,
as predicted by the conjecture. (Actually, there is a normalization
factor which was not computed, but the agreement is exact up to
this normalization.)
\item[(iv)]
The work of Candelas,
de la Ossa, P.~Green, and Parkes \cite{pair} being described in this paper
goes further, and computes the other coefficients in an asymptotic
expansion. This will be explained in more detail in the next two sections.
\end{enumerate}
\section{The quintic-mirror family}
\label{qm}
We now describe a certain one-parameter family of Calabi-Yau threefolds
constructed by Greene and Plesser \cite{greene-plesser}, as amplified
by Candelas et al.\ \cite{pair}.
Begin with the family of quintic threefolds
${\cal Q}_\psi = \{\vec{x}\in {\Bbb P}^4\ | \ p_\psi(\vec{x})=0\}$
defined by the polynomial
\[
p_\psi := \sum_{k=1}^5 x_k^5 - 5\psi\prod_{k=1}^5 x_k .
\]
Let \mbox{\boldmath ${\mu}_5$}\ be the multiplicative group of $5^{\text{th}}$ roots
of unity, and let
\[
\widetilde{G}:=\{\vec{\alpha}=(\alpha_1,\dots,\alpha_5)\in
(\mbox{\boldmath ${\mu}_5$})^5 \ | \
\prod_{k=1}^5 \alpha_k = 1\}
\]
act on ${\Bbb P}^4$ by $\vec{\alpha}: x_i\mapsto \alpha_i\cdot x_i$.
There is a ``diagonal'' subgroup of order $5$ which acts trivially; let
$G=\widetilde{G}/\{\mbox{diagonal}\}$
be the image of $\widetilde{G}$ in $\operatorname{Aut}({\Bbb P}^4)$. $G$ is a
group which is abstractly isomorphic to $({\Bbb Z}/5{\Bbb Z})^3$.
The action of $G$ preserves the threefold ${\cal Q}_\psi$; let
$\eta:{\cal Q}_\psi\to {\cal Q}_\psi/G$ denote the quotient map.
For each pair of distinct indices $i$, $j$, the set of 5 points
\[
S_{ij}:=\{x_i^5+x_j^5=0, x_\ell =0 \text{ for all }
\ell\ne i,j\}\subset {\cal Q}_\psi
\]
is preserved by $G$, and there
is a group $G_{ij}\subset G$ of order 25 which is the stabilizer
of each point in the set. The image $S_{ij}/G$
is a single point $p_{ij}\in {\cal Q}_\psi/G$. In addition, for each triple
of distinct indices $i$, $j$, $k$, the curve
\[
\widetilde C_{ijk}:=\{x_i^5+x_j^5+x_k^5=0, x_\ell =0 \text{ for all }
\ell \ne i,j,k \}\subset {\cal Q}_\psi
\]
is preserved by $G$. There is a subgroup $G_{ijk}\subset G$ of order
5 which is the stabilizer of every point in
$\widetilde C - \eta^{-1}(\{p_{ij},p_{jk},p_{ik}\})$.
The image $C_{ijk}=\widetilde C_{ijk}/G$ is a smooth curve in
${\cal Q}_\psi/G$.
The action of $G$ is free away from the curves $\widetilde C_{ijk}$.
The quotient
space ${\cal Q}_\psi/G$ has only canonical singularities.
At most points of $C_{ijk}$, the surface section of the singularity is a
rational
double point of type $A_4$, but at the points $p_{ij}$ the singularity
is more complicated:
three of the curves of $A_4$-singularities meet at each $p_{ij}$.
By a theorem of Markushevich \cite[Prop.~4]{markushevich} and
Roan \cite[Prop.~2]{roan1}, these singularities can be
resolved to give a Calabi-Yau manifold ${\cal W}_\psi$.
There are choices to be made in this resolution
process; we describe a particular
choice in appendix B.
(By another theorem of Roan \cite[Lemma 4]{roan2},
any two resolutions differ by a sequence of flops.)
For any $\alpha\in\mbox{\boldmath ${\mu}_5$}$,
there is a natural isomorphism between ${\cal Q}_{\alpha\psi}/G$ and
${\cal Q}_{\psi}/G$ induced by the map
\begin{equation}
\label{isomorphism}
(x_1,x_2,x_3,x_4,x_5)\mapsto(\alpha^{-1}x_1,x_2,x_3,x_4,x_5).
\end{equation}
This extends to an isomorphism between ${\cal W}_{\alpha\psi}$ and ${\cal
W}_\psi$,
provided that we have resolved singularities in a compatible way.
We verify in appendix B that the choices in the resolution can
be made in a sufficiently natural way that this isomorphism is
guaranteed to exist.
Thus, $\lambda:=\psi^5$ is a more natural parameter to use for our family.
We define the {\em quintic-mirror family} to be
\[
\{ {\cal W}_{\sqrt[5]{\lambda}} \} \to \{\lambda\}
\cong {\Bbb C}.
\]
This has a natural compactification to a family over ${\Bbb P}^1$,
with boundary $B={\Bbb P}^1-{\Bbb C}=\{\infty\}$.
The computation made by Candelas et al.\ \cite{pair} shows that
the monodromy at
$\infty$ is maximally unipotent,
and that $m=1$, i.e., that the monodromy is small.
(We explain in appendix C how this follows from \cite{pair}.)
The key computation in \cite{pair} is an explicit calculation of
the $q$-expansion of the mathematically normalized Yukawa coupling.
Candelas et al.\ find that the $q$-expansion begins:
\begin{equation} \label{formula1}
\kappa_{ttt} = 5 + 2875e^{2\pi it} + 4876875e^{4\pi it} + \cdots.
\end{equation}
In fact, they have computed at least 10 coefficients.
\section{Mirror moonshine?}
\label{moonshine}
Greene and Plesser \cite{greene-plesser}, using arguments from
superconformal field
theory, have identified the
family of quintic-mirrors $\{ {\cal W}_{\sqrt[5]{\lambda}} \}$
as the ``mirror'' of the family of smooth
quintic threefolds $\{ {\cal M}_z \}$. Note that
the Hodge numbers satisfy
\[\begin{array}{ll}
h^{1,1}({\cal M})=1 &
h^{2,1}({\cal M})=101\\
h^{1,1}({\cal W})=101 &
h^{2,1}({\cal W})=1.
\end{array}\]
According to the mirror symmetry conjecture,
varying the complex structure in the family $\{ {\cal W}_{\sqrt[5]{\lambda}}
\}$
should
correspond to varying the ``extra structure'' $S$ on a fixed smooth quintic
threefold ${\cal M}$. These are both one-parameter variations.
Candelas et al.\ \cite{pair}, arguing from physical principles,
propose an identification of the Yukawa coupling
of the quintic-mirrors
with the cubic form on $H^{1,1}({\cal M})$.
In terms of the mathematical framework established here, that identification
involves four assertions:
\begin{enumerate}
\item[(i)]
The isomorphism $H^{-1,1}({\cal W})\to H^{1,1}({\cal M})$ defined by
$d/dt\mapsto [H]$
(where $d/dt\in H^{-1,1}({\cal W})$ is the vector
field defined by the canonical parameter $t$, and $[H]\in H^{1,1}({\cal M})$
is the class
of a hyperplane section of ${\cal M}$) is the isomorphism
which is
predicted by the mathematical version of the mirror symmetry
conjecture.
\item[(ii)]
The mathematically normalized Yukawa coupling $\kappa_{ttt}$ on
$H^{-1,1}({\cal W})$ is
the correctly normalized coupling predicted by the physical theory.
\item[(iii)]
The parameter $R$ from the physical theory
coincides with $-2\pi it$, where $t$ is again the canonical
parameter.
Thus, the $q$-expansion of $\kappa_{ttt}$ in equation (\ref{qexpt}) will
coincide with the asymptotic expansion in $R$ given by equation (\ref{asymp}),
evaluated on the generator $H^{\otimes 3}$ of $\operatorname{Sym}^3H^{-1,1}({\cal M})$.
\item[(iv)]
There is an explicit formula for the coefficients $\sigma_k$, as described
below.
\end{enumerate}
To explain the formula for $\sigma_k$, let $n_k$ denote
the number of rational
curves of degree $k$ on the generic quintic threefold.
Candelas et al.\ propose the formula
\begin{equation} \label{formula2}
\kappa_{ttt} = 5 +
\sum_{k=1}^\infty \frac{n_kk^3e^{2\pi ikt}}{1-e^{2\pi ikt}}
= 5 + n_1e^{2\pi it} + (2^3n_2 + n_1)e^{4\pi it} + \cdots,
\end{equation}
which implicitly incorporates their
expressions
for the higher coefficients
(The first two expressions are $\sigma_1(H^{\otimes 3})=n_1$,
$\sigma_2(H^{\otimes 3})=2^3n_2 + n_1$.)
In the large radius limit $\Im t \to\infty$, the right hand side of
equation (\ref{formula2}) approaches $5$. This agrees with the
mirror symmetry conjecture,\footnote{This should not be taken as strong
evidence in favor of the conjecture, since the definitions have been
carefully designed to ensure that this limit would be correct.}
since the topological intersection form
on ${\cal M}$ is determined by its value on the standard generator $H$,
viz., $H^3=5$.
Moreover, by
comparing equations (\ref{formula1}) and (\ref{formula2}), we
can {\em predict\/} values for the numbers $n_k$. The first two
predictions are
$n_1=2875$, which was classically known
to be the number of lines on a quintic threefold, and $n_2=609250$,
which coincides with the number of conics on a quintic
threefold
computed by Katz \cite{katz}!
\raisebox{1.2ex}{\makebox[0pt][l]{\underline{\phantom
{Unfortunately, there seem to be difficulties with $n_3$.
}}}}
Unfortunately, there seem to be difficulties with $n_3$.
\quad Not any more!!
\bigskip
How was formula (\ref{formula2}) arrived at? I am told that the field
theory computation necessary to derive this formula can be
done in principle, but seems to be too hard to carry out in practice at
present. So Candelas et al.\ give a rough derivation of this formula
based on some assumptions.
Why do they believe the resulting formula to be correct? I quote from
\cite{pair}:
\begin{quote}
These numbers provide compelling evidence that our assumption about the
form of the prefactor is in fact correct. The evidence is not so much
that we obtain in this way the correct values for $n_1$ and $n_2$,
but rather that the coefficients in eq.~(\ref{formula1}) have remarkable
divisibility properties. For example asserting that the second coefficient
$4,876,875$ is of the form $2^3n_2+n_1$ requires that the result of
subtracting $n_1$ from the coefficient yields an integer that is divisible
by $2^3$. Similarly, the result of subtracting $n_1$ from the third
coefficient must yield an integer divisible by $3^3$. These conditions
become increasingly intricate for large $k$. It is therefore remarkable
that the $n_k$ calculated in this way turn out to be integers.
\end{quote}
I would add that it is equally remarkable that the coefficients in
eq.~(\ref{formula1}) themselves turn out to be integers: I know of
no proof that this is the case.
These arguments have
a rather numerological flavor.
I am reminded of the
numerological observations made by Thompson
\cite{numerology} and Conway and Norton \cite{monster}
about the $j$-function and the monster group.
At the time those papers were written, no connection between these
two mathematical objects was known. The $q$-expansion of the
$j$-function was known to have integer coefficients, and it was
observed that these integers were integral linear combinations
of the degrees of irreducible representations of the monster group.
This prompted much speculation about possible deep connections
between the two, but at the outset all such speculation had to
be characterized as ``moonshine'' (Conway and Norton's term).
The formal similarities to the present work should be clear:
a $q$-expansion
of some kind is found to have integer coefficients, and these integers
then appear to be integral linear combinations of another set of integers,
which
occur elsewhere in mathematics in a rather unexpected location.
Perhaps it is too much to hope that the eventual explanation will be
as pretty in this case.
\section*{Appendix A: Proofs of the monodromy lemmas}
Let
\[
W_0 \subset W_1 \subset \dots \subset W_{2n}=H^n(X_{P'},{\Bbb Q})
\]
be the monodromy weight filtration at $P$, and let
\[
F^0 \supset F^1 \supset \cdots \supset
F^{n-1} \supset
F^n \supset (0).
\]
be the limiting Hodge filtration at $P$.
(We refer the reader to \cite{transcendental} or \cite{schmid} for the
definitions.)
By a theorem of Schmid
\cite{schmid}, these induce a mixed Hodge structure on the cohomology.
Note that since $N^{n+1}=0$, we have $W_0=\Im N^n$.
Moreover, if $\ip{\ }{\ }$ denotes
the polarization on the cohomology, we have
\[
\ip{Nx}{y}=-\ip{x}{Ny}.
\]
Recall also that the polarization is symmetric or skew-symmetric,
depending on the dimension $n$:
\[
\ip{x}{y}=(-1)^n\ip{y}{x}
\]
\begin{pf*}{Proof of lemma \ref{lemma1}}
Since $W_{\textstyle\cdot}$
is the monodromy weight filtration, $N^n$ induces
an isomorphism
\begin{equation} \label{eq1}
N^n: W_{2n}/W_{2n-1} \to W_0.
\end{equation}
These spaces cannot be zero, since $(T_P-I)^n\ne 0$. On the other hand,
since $F^{n+1}=(0)$, the Hodge structure on $W_{2n}/W_{2n-1}$ must be
purely of type $(n,n)$. It follows that
$F^n/(F^n\cap W_{2n-1})=W_{2n}/W_{2n-1}$. But since $F^n$ is one-dimensional,
this can only happen if $F^n \subset W_{2n}-W_{2n-1}$, and
$W_{2n}/W_{2n-1}$ has dimension one. By the isomorphism (\ref{eq1}),
$W_0=\Im N^n$ has dimension one as well.
Next, note that $W_{2n-1}/W_{2n-2}$ has a Hodge structure with two types,
$(n,n-1)$ and $(n-1,n)$, each of which must determine a space of half
the total dimension. But since $F^n\cap W_{2n-1} = (0)$, nothing non-zero
can have type $(n,n-1)$. It follows that $W_{2n-1}/W_{2n-2}=(0)$,
and that $W_1/W_0=(0)$ as well (using the isomorphism induced by
$N^{n-1}$).
Thus, the image of $N^{n-1}$ comes entirely from the map
\[ N^{n-1}:W_{2n}\to W_2.\]
That this image is two-dimensional is easily seen:
$W_0$ is one-dimensional, and there is an isomorphism
\[ N^{n-1}:W_{2n}/W_{2n-1}\to (\Im N^{n-1})/W_0, \]
which shows that $(\Im N^{n-1})/W_0$ is also one-dimensional.
\end{pf*}
In order to prove lemma \ref{lemma2}, we must first prove
\begin{lemma}[Essentially due to Friedman and Scattone \cite{fs}]
\label{lemma3}
\mbox{}
\noindent
Good integral bases exist, and form bases of the two-dimensional
${\Bbb Q}$-vector space $\Im N^{n-1}$.
If $g_0$, $g_1=\frac1\lambda N^{n-1}g$ is a good integral basis, then
\begin{equation}
\label{nn-1}
\frac1\lambda N^{n-1}x= - \ip{g_1}{x}g_0 + \ip{g_0}{x}g_1
\end{equation}
for all $x\in H^n(X_{P'},{\Bbb Q})$.
\end{lemma}
\begin{pf} Choose either generator of $\Im N^n \cap H^n(X_{P'},{\Bbb Z})$
as $g_0$.
We claim that $(g_0)^\perp = W_{2n-2}$. Let $h\in W_{2n}-W_{2n-2}$,
so that $N^nh=ag_0$ with $a\ne 0$. Then for any $x$ we have
\[
\ip{N^nx}{h}=(-1)^n\ip{x}{N^nh}=(-1)^na\ip{x}{g_0}.
\]
Thus, $W_{2n-2}=\ker N^n \subset (g_0)^\perp$.
Since both $W_{2n-2}$ and $(g_0)^\perp$ are codimension one subspaces
of $W_{2n}$, they must be equal.
By Poincar\'e duality, the polarization on
$H^n(X_{P'},{\Bbb Z})$ is a unimodular pairing.
Thus, there exists an element $g\in H^n(X_{P'},{\Bbb Z})$ such that
$\ip{g_0}{g}=1$.
Since $g\not\in(g_0)^\perp$, neither $N^{n-1}g$ nor $N^n g$ is zero.
There is thus a unique positive rational number $\lambda$
such that $g_1 = \frac1\lambda N^{n-1}g$ is an indivisible element of
$H^n(X_{P'},{\Bbb Z})$.
It is clear that $g_0$, $g_1$ forms
a basis for the ${\Bbb Q}$-vector space $\Im N^{n-1}$.
We next claim that $\ip{g_1}{g}=\frac1\lambda \ip{N^{n-1}g}{g}=0$.
For on the one hand, moving the $N$'s
to the right side one at a time we have
\[
\ip{N^{n-1}g}{g}=(-1)^{n-1}\ip{g}{N^{n-1}g}
\]
while on the other hand, the symmetry of the polarization says that
\[
\ip{N^{n-1}g}{g}=(-1)^{n}\ip{g}{N^{n-1}g}.
\]
It follows that $\ip{N^{n-1}g}{g}=0$.
To prove equation (\ref{nn-1}), we first compute in general
\[
\ip{N^{n-1}x}{g}=(-1)^{n-1}\ip{x}{N^{n-1}g}=(-1)^{n-1}\ip{x}{\lambda g_1}
=-\lambda\ip{g_1}{x}.
\]
Now suppose that $x\in W_{2n-2}$.
Then $N^{n-1}x\in \Im N^n$, which implies that $N^{n-1}x=ag_0$ for some
$a$. Thus, in this case
\[
\ip{N^{n-1}x}{g} = \ip{ag_0}{g} = a,
\]
which implies that $a=-\lambda\ip{g_1}{x}$. Thus,
\[
\frac1\lambda N^{n-1}x=\frac1\lambda ag_0=\ip{g_1}{x}g_0
\]
and since $\ip{x}{g_0}=0$, the formula follows in this case.
To prove the formula in general, note that
\[
\ip{g_0}{x-\ip{g_0}{x}g\vphantom{N^{n-1}}}= 0
\]
for any $x$, so that $x-\ip{g_0}{x}g\in (g_0)^\perp=W_{2n-2}$.
Thus, applying the
previous case we find
\begin{eqnarray*}
\frac1\lambda N^{n-1}x &=& \frac1\lambda N^{n-1}(x-\ip{g_0}{x}g)
+ \frac1\lambda N^{n-1}(\ip{g_0}{x}g)\\
&=& -\ip{g_1}{(x-\ip{g_0}{x}g)\vphantom{N^{n-1}}}g_0
+ \ip{g_0}{x}\frac1\lambda N^{n-1}g\\
&=& -\left(\ip{g_1}{x}-\ip{g_0}{x}\ip{g_1}{g}\vphantom{N^{n-1}}\right)g_0
+ \ip{g_0}{x} g_1\\
&=& -\ip{g_1}{x}g_0 + \ip{g_0}{x}g_1
\end{eqnarray*}
since $\ip{g_1}{g}=0$.
\end{pf}
We can now prove lemma \ref{lemma2}.
\begin{pf*}{Proof of lemma \ref{lemma2}}
The only generators of $\Im N^n \cap H^n(X_{P'},{\Bbb Z})$ are $\pm g_0$,
so we must have $g_0'=(-1)^\ell g_0$ for some $\ell\in\Bbb Z$.
Write $g_1'=\frac1{\lambda'}N^{n-1}g'$ for some $g'$ with
$\ip{g_0'}{g'}=1$, and let $k=-\ip{g_1}{g'}\in\Bbb Z$. Then
by lemma \ref{lemma3},
\[
\frac1\lambda N^{n-1}g' = -\ip{g_1}{g'}g_0+\ip{g_0}{g'}g_1
= k\,g_0+(-1)^\ell g_1.
\]
Thus, $\frac1\lambda N^{n-1}g' \in H^n(X_{P'},{\Bbb Z})$.
We claim that it must be an indivisible element there. For if
$\frac1{\lambda\mu} N^{n-1}g'$ is integral for some $\mu\in\Bbb Z$
with $\mu>1$,
then reversing the roles of $g$ and $g'$ in the argument above
shows that $\frac1{\lambda\mu} N^{n-1} g$ is also integral, a
contradiction.
Thus, $g_1'=k\,g_0+(-1)^\ell g_1$.
\end{pf*}
\section*{Appendix B: Resolutions of certain quotient singularities}
In this appendix, we will verify that the singularities of the variety
${\cal Q}_\psi/G$ can be resolved in a natural way. The choices we make are
sufficiently natural that the isomorphism between ${\cal Q}_{\alpha\psi}/G$
and ${\cal Q}_\psi/G$ automatically lifts to an isomorphism between the
desingularizations.
We choose to follow the strategy outlined by Reid \cite{reid} for
resolving canonical threefold singularities. In brief, we perform
the following steps:
\begin{list}{}{\advance\leftmargin by 1.6em \advance\labelwidth by 1.6em}
\item[Step I:]
Blow up the ``non-cDV points'' of ${\cal Q}_\psi/G$. (These are exactly
the 10 points $p_{ij}\in {\cal Q}_\psi/G$ which are the images of
points in ${\cal Q}_\psi$ with stabilizer of order 25.)
\item[Step IIA:]
Blow up the singular locus. (It has pure dimension one.)
\item[Step IIB:]
Blow up the pure dimension one part of the singular locus. (60 isolated
singular points (lying over the $p_{ij}$)
were created by step IIA, and these are not to be
blown up yet.)
\item[Step III:]
Obtain a projective small resolution of the remaining 60 singular points
by blowing up the union of the proper transforms of the exceptional
divisors from step I.
\end{list}
Step III involved an additional choice, since Reid's strategy does not
specify how one should obtain small resolutions.
When stated in this form, it is clear that the process we have described
is sufficiently natural that it is preserved under any isomorphism.
It yields a projective (hence K\"ahler) variety ${\cal W}_\psi$ with trivial
canonical bundle.
In the remainder of this appendix, we will show that the process above
has the properties mentioned during its description, and that
it gives a resolution of singularities of ${\cal Q}_\psi/G$.
We first observe the effect of the process on the curve $C_{ijk}$,
away from the points $p_{ij}$, $p_{jk}$, $p_{ik}$.
Steps I and III are concentrated at
those special points (and their inverse images) and so these steps do
not affect $C_{ijk}$. Steps IIA and IIB simply blow up $C_{ijk}$ and
then the residual singular curve in the exceptional divisor.
But two blowups are precisely what is
required to resolve a rational double point of type $A_4$, as is
easily verified from its equation $xy+z^5=0$.
To verify that the process has the correct properties at the points $p_{ij}$,
we use the language of toroidal embeddings (see \cite{te} or \cite{oda}).
It suffices to consider the point $p_{45}$. Since $(x_1,x_2,x_3)$
serve as coordinates in a neighborhood of any of the points in
$\eta^{-1}(p_{45})$, the
singularity $p_{45}\in {\cal Q}_\psi/G$ is isomorphic to a neighborhood of
the origin in $\Bbb C^3/G_{45}$, where
$G_{45}\cong\{(\alpha_1,\alpha_2,\alpha_3)
\in(\mbox{\boldmath ${\mu}_5$})^3\ | \ \prod\alpha_k=1\}$
acts diagonally on $\Bbb C^3$.
Let $M$ be the lattice of $G_{45}$-invariant rational
monomials in $\Bbb C(x_1,x_2,x_3)$.
We embed $M$ in $\Bbb R^3$ by\footnote{This non-standard embedding is
chosen in order to make the coordinates of the dual lattice be integers.}
\[
M=\{(m_1,m_2,m_3)\in\Bbb R^3\ | \
x_1^{5m_1}x_2^{5m_2}x_3^{5m_3}\in\Bbb C(x_1,x_2,x_3)^{G_{45}}\}.
\]
It is easy to see that $\{(1,0,0),(0,1,0),(\frac15,\frac15,\frac15)\}$
is a basis of the lattice $M\subset\Bbb R^3$.
Let
\begin{eqnarray*}
N & = & \{\vec{n}\in\Bbb R^3\ | \ \vec{m}\cdot\vec{n}\in\Bbb Z
\text{\ for all\ }
\vec{m}\in M\} \\
& = & \{(n_1,n_2,n_3)\in\Bbb Z^3\ | \ n_1+n_2+n_3\equiv 0 \mod 5\}
\end{eqnarray*}
be the dual lattice, and let $\sigma\subset N_{\Bbb R}$ be the convex
cone generated by $(5,0,0)$, $(0,5,0)$, and $(0,0,5)$.
According to the theory of toroidal embeddings,
\[
\Bbb C^3/G_{45} = \operatorname{Spec}\Bbb C[x_1,x_2,x_3]^{G_{45}} = U_\sigma,
\]
where $U_\sigma$ is the toric variety associated to $\sigma$.
\begin{figure}[t]
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\put(12,6){\line(1,0){16}}
\put(28,6){\line(-2,3){8}}
\put(20,18){\line(-2,-3){8}}
\put(0,0){\line(2,1){12}}
\put(40,0){\line(-2,1){12}}
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\put(16,0){\line(2,3){12}}
\put(20,6){\line(-2,-1){12}}
\put(20,6){\line(2,-1){12}}
\put(32,12){\line(-1,0){24}}
\put(24,12){\line(2,-1){12}}
\put(24,12){\line(0,1){12}}
\put(12,18){\line(2,-3){12}}
\put(16,12){\line(0,1){12}}
\put(16,12){\line(-2,-1){12}}
\put(16,0){\circle*{1}}
\put(17,-3){\makebox(0,2)[r]{\scriptsize (3,0,2)}}
\put(24,0){\circle*{1}}
\put(23,-3){\makebox(0,2)[l]{\scriptsize (2,0,3)}}
\put(32,12){\circle*{1}}
\put(33,12){\makebox(0,0)[l]{\scriptsize (0,2,3)}}
\put(28,18){\circle*{1}}
\put(29,18){\makebox(0,0)[l]{\scriptsize (0,3,2)}}
\put(12,18){\circle*{1}}
\put(11,18){\makebox(0,0)[r]{\scriptsize (2,3,0)}}
\put(8,12){\circle*{1}}
\put(7,12){\makebox(0,0)[r]{\scriptsize (3,2,0)}}
\end{picture}
\hfill
\begin{picture}(40,30)
\put(20,-6){\makebox(0,0){Step III}}
\thinlines
\put(0,0){\line(1,0){40}}
\put(40,0){\line(-2,3){20}}
\put(20,30){\line(-2,-3){20}}
\put(4,6){\line(1,0){32}}
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\put(24,12){\line(0,1){12}}
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\put(16,12){\line(0,1){12}}
\put(16,12){\line(-2,-1){12}}
\end{picture}
\bigskip
\bigskip
}
\caption{The steps in the toroidal resolution. \protect\rule[-3ex]{0pt}{3ex}}
\label{fig1}
\end{figure}
Each blowup of $U_\sigma$ corresponds to a decomposition of $\sigma$
into a fan. The effects of the blowups in our process is illustrated
in figure \ref{fig1}, which depicts the intersection of the fan with
$\{(n_1,n_2,n_3)\in N_{\Bbb R}\ | \ \sum n_k=5\}$ after each step.
The exceptional divisors $D_{\vec{n}}$ of each blowup are indicated
by solid dots, labeled by the corresponding elements $\vec{n}\in N$.
(The fact that the stated blowups produce the illustrated decomposition
is a straightforward calculation with the toroidal embeddings.)
We can now see in detail what happens in our process. In
step I, we blow up $p_{45}$, and produce three exceptional divisors
$D_{(3,1,1)}$, $D_{(1,3,1)}$, and $D_{(1,1,3)}$.
The remaining singular locus at this stage consists of the original
three curves of $A_4$-singularities together with three new curves
of $A_1$-singularities: the intersections of pairs of exceptional
divisors. In step IIA, we blow up the union of these six curves, and
produce nine new exceptional divisors: one corresponding to each
curve of $A_1$-singularities (such as $D_{(2,1,2)}$), and two
corresponding to each curve of $A_4$-singularities (such as $D_{(4,0,1)}$
and $D_{(1,0,4)}$). The remaining singularities consist of six
isolated points (corresponding to the quadrilaterals in the figure)
and three curves: the intersections of the corresponding pairs of
exceptional divisors from the original $A_4$-singularities.
In step IIB, we blow up these three curves, producing six new
exceptional divisors, two for each curve
(such as $D_{(3,0,2)}$ and $D_{(2,0,3)}$).
This leaves the six isolated singular points; but blowing up the
proper transforms of $D_{(3,1,1)}$, $D_{(1,3,1)}$, and $D_{(1,1,3)}$
(which are now disjoint) in step III resolves those final singular points.
\section*{Appendix C: The monodromy of the quintic-mirrors}
In this appendix we will explain how to use the calculation of
Candelas et al.\ \cite{pair} to verify the monodromy statements
about the family of quintic-mirrors which we made in section \ref{qm}.
Candelas et al.\ begin by choosing an explicit
basis $\{A^1,A^2,B_1,B_2\}$
for the homology $H_3({\cal W}_\psi,\Bbb Z)$ of a quintic-mirror,
valid in some simply-connected region in $\{\psi\ | \ \psi^5\ne 0,1\}$
which includes the wedge
$\{\psi\ | \ 0<\arg\psi <2\pi/5\}$.
This basis is {\em symplectic}, i.e., ${A^a}\cap{B_b}={\delta^a}_b$
and ${A^a}\cap{A^b}={B_a}\cap{B_b}=0$.
The corresponding dual basis of $H^3({\cal W}_\psi,\Bbb Z)$
is denoted by $\{\alpha_1,\alpha_2,\beta^1,\beta^2\}$.
Fixing a particular holomorphic 3-form
$\Omega$ (which depends on $\psi$), we then get {\em period functions\/}
\[
z^a=\int_{A^a}\Omega ,\ \ \ \ {\cal G}_b=\int_{B_b}\Omega.
\]
These fit into a {\em period vector\/}
\[
\Pi = \left( \begin{array}{c} {\cal G}_1 \\ {\cal G}_2 \\ z^1 \\ z^2
\end{array} \right) .
\]
By doing some integrals, calculating the differential equation
satisfied by a period function, and manipulating certain hypergeometric
functions, the authors of \cite{pair} are able to obtain explicit formulas
for the four period functions. This allows them to calculate
the monodromy of the periods around various paths.
Notice that we are working in the $\psi$-plane at present. The family
$\{{\cal W}_\psi\}$ has singular fibers at $\psi=0$ and at $\psi=\alpha$
for all fifth roots of unity $\alpha$; there is also a singular fiber
over $\psi=\infty$. Candelas et al.\ calculate the monodromy on the
periods induced by transport around $\psi=1$, which they represent in
matrix form by $\Pi\to T\Pi$. They also compute, for $|\psi|<1$,
the effect on the
periods of the isomorphism ${\cal W}_{\alpha\psi} \cong {\cal W}_\psi$ given in
equation (\ref{isomorphism}), representing this by
$\Pi(\alpha\psi)=A\Pi(\psi)$.
We need to know the monodromy around $\infty$ in the $\lambda$-plane,
where $\lambda=\psi^5$. A moment's thought will convince the reader
that this is represented by
\[
\Pi\to (T^{-1}A^{-1})\Pi,
\]
and that $(AT)^{-5}$ describes the monodromy around $\infty$ in the
$\psi$-plane (as asserted in \cite{pair}).
Let $T_P=T^{-1}A^{-1}$.
The explicit calculations from \cite{pair} for the matrices $A$ and $T$ are:
\[
A=
\left( \begin{array}{rrrr}
-9 & -3 & 5 & 3 \\
0 & 1 & 0 & -1 \\
-20 & -5 & 11 & 5 \\
-15 & 5 & 8 & -4
\end{array} \right)
\ \ \ \
T=
\left( \begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)
\]
from which it easily follows that
\[
(\log (T_P))^2=
\left( \begin{array}{rrrr}
0 & 5 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
-10 & 0 & 5 & 0
\end{array} \right) ,
\ \ \ \
(\log (T_P))^3=
\left( \begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & -5 & 0 & 0
\end{array} \right) .
\]
In particular, the index of nilpotency
of $\log (T_P)$
is maximal.
(We note in passing that at
$\lambda=1$ the monodromy is represented by $T$, and since
$(T-I)^2=0$, the index is not maximal there. In addition, at
$\lambda=0$ the monodromy is represented by $A$. This monodromy
matrix is only quasi-unipotent, with $A^5=I$ unipotent; the index of $A^5$
is not maximal either. It follows that $\lambda=\infty$ is the only
possible boundary point with maximally unipotent monodromy.)
In order to construct a good integral basis $g_0$, $g_1$, we compute
\begin{eqnarray*}
(\log (T_P))^2(\int_{A^2}\Omega) & = &
-10\int_{B_1}\Omega + 5\int_{A^1}\Omega \\
(\log (T_P))^3(\int_{A^2}\Omega) & = &
-5\int_{B_2}\Omega .
\end{eqnarray*}
Using the relations
\begin{equation}
\label{relations}
\operatorname{ad}_{\ip{\,}{\,}}(\alpha_a)=\int_{B_a}\ ,\ \ \ \
\operatorname{ad}_{\ip{\,}{\,}}(\beta^b)=-\int_{A^b}\ ,
\end{equation}
this implies
\begin{eqnarray*}
(\log (T_P))^2(\beta^2) & = &
10\alpha_1+5\beta^1 \\
(\log (T_P))^3(\beta^2) & = &
5\alpha_2 .
\end{eqnarray*}
Thus, we may take $g_0=\alpha_2$. If we then choose $g=\beta^2$
so that $\ip{g_0}{g}=\ip{\alpha_2}{\beta^2}=1$, we get
$\lambda=5$ and $g_1=2\alpha_1+\beta^1$. It follows that
\[
(\log (T_P))(g_1) = \frac15(\log (T_P))^3(\beta^2)
=\alpha_2=g_0,
\]
which implies that $m=1$. Using the relations (\ref{relations}) again,
it follows that
$\gamma_0=B_2$, $\gamma_1=2B_1-A^1$. Thus,
$t=(\int_{2B^1-A_1}\Omega)/(\int_{A^2}\Omega)$.
We need to verify that our parameter $t$ is the same one used by
Candelas et al. Their parameter is defined in \cite[(5.9)]{pair}
by $t=w^1/w^2$, with
$w^1$ and $w^2$ determined by a pair of
equations
\[
\amalg = N\,\Pi,\ \ \ \ w^2={\cal G}_2,
\]
where\footnote{We
have taken the liberty of correcting a typographical error in $N$
when transcribing it from \cite{pair}.}
\[
\amalg=
\left( \begin{array}{c}
{\cal F}_1 \\ {\cal F}_2 \\ w^1 \\ w^2
\end{array} \right)
\text{\ \ and\ \ }
N=
\left( \begin{array}{rrrr}
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
2 & 0 & -1 & 0 \\
0 & 1 & 0 & 0
\end{array} \right)
\]
represent
a vector
$\amalg$ which is a sort of mirror analogue of the period vector $\Pi$,
and
a particular integral symplectic
matrix
$N$,
respectively.
(Sadly, in the published version of \cite{pair},
the symbols $\Pi$ and $\amalg$ were identified, making section 5.2 of that
paper difficult to read.)
It follows that
\[
t=\frac{w^1}{w^2} = \frac{2{\cal G}_1-z^1}{{\cal G}_2}
= \frac{\int_{2B^1-A_1}\Omega}{\int_{A^2}\Omega}
\]
as required.
\section*{Acknowledgements}
It is a pleasure to acknowledge helpful conversations and e-mail exchanges with
Paul Aspinwall,
Robert Bryant,
Philip Candelas,
Brian Greene,
Yujiro Kawamata,
Ronen Plesser,
Les Saper,
Chad Schoen,
and especially Sheldon Katz
as I was struggling to understand this material.
This work was partially supported by NSF grant DMS-9103827.
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1992-02-18T16:00:09 | 9202 | alg-geom/9202017 | en | https://arxiv.org/abs/alg-geom/9202017 | [
"alg-geom",
"math.AG"
] | alg-geom/9202017 | Temporary | G.Mikhalkin | Extensions of Rokhlin congruence for curves on surfaces | LaTeX, 6 pages | null | null | null | null | The subject of this paper is the problem of arrangement of a real nonsingular
algebraic curve on a real non-singular algebraic surface. This paper contains
new restrictions on this arrangement extending Rokhlin and
Kharlamov-Gudkov-Krakhnov congruences for curves on surfaces.
| [
{
"version": "v1",
"created": "Tue, 18 Feb 1992 14:56:04 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Mikhalkin",
"G.",
""
]
] | alg-geom | \section{Introduction}
If we fix a degree of a real nonsingular algebraic surface then in accordance
with Smith theory there is an upper bound for the total ${\bf Z}_2$-Betti
number of this
surface (the same thing applies to curves). The most interesting case according
to D.Hilbert is the case when the upper bound is reached (in this case the
surface is called an M-surface).
Let $A$ be a real nonsingular algebraic projective curve on a real
nonsingular
algebraic projective surface $B$. If $A$ is of even degree in $B$ then $A$
divides $B$ into
two parts $B_+$ and $B_-$ (corresponding to areas of $B$ where polynomial
determining $A$ is positive and negative).
Rokhlin congruence \cite{R} yields a congruence modulo 8 for Euler
characteristic
$\chi$ of $B_+$ provided that \begin{description}
\item[$(i)$]$B$ is an M-surface \item[$(ii)$] $A$ is an M-curve
\item[$(iii)$] $B_+$ lies wholly in one component of $B$ \item[$(iv)$] $
rk(in_*:H_1(B_+;{\bf Z}_2)\rightarrow H_1(B;{\bf Z}_2))=0 $ \item[$(v)$] if
the degree of polynomial determining
$A$ in $B$ is congruent to 2 modulo 4 then all components of $B$ containing no
components of $A$ are contractible in $P^q$ \footnote{{\em\underline {Remark}}.
Paper \cite{R} contains a mistake in the calculation of characteristic class
of
covering $Y\rightarrow {\bf C}B$. It leads to the omission (after
reformulation) of $(v)$ in assumptions of congruence. The proof given in
\cite{R}
really uses $(v)$.
} \end{description}
Kharlamov-Gudkov-Krakhnov \cite{Kh},\cite{GK} congruence yields congruence
modulo 8
for $\chi(B_+)$ under assumptions $(i)$,$(iii)$,$(v)$ and either assumption
that $A$ is an
(M-1)-curve and $rk(in_*)=0$ or assumption that $A$ is an M-curve and
$rk(in_*)=1$ and all
components of $A$ are ${\bf Z}_2$-homologically trivial. Recall that Rokhlin
congruence for surfaces yields congruence modulo 16 for $\chi(B)$ so a
congruence modulo 8 for $\chi(B_+)$ is equivalent to a congruence modulo 8
for $\chi(B_-)$.
One of the properties of M-surfaces (similar to the property of M-curves)
remarked by V.I.Arnold \cite{A}
is that a real M-surface is a characteristic surface
in its own complexification (for M-curves it means that a real M-curve divides
its own complexification since a complex curve is orientable).We shall
say that a real surface is of a characteristic type if it is a characteristic
surface in its own complexification.Note that the notion of characteristic
type of surfaces is analogous to the notion of type I of curves.
Consider at first the weakening of assumption $(i)$ in Rokhlin and
Kharlamov-Gudkov-Krakhnov congruences.Instead of $(i)$ we can only assume
that B is of characteristic type.
According to O.Ya.Viro \cite{V} there are some extra structures on real
surfaces
of characteristic type ,namely, $Pin_-$-structures and semiorientations or
relative semiorientations (semiorientation is the orientation up to the
reversing). In this paper we introduce another structure on surfaces of
characteristic type --- complex separation which is also determined by the
arrangement of a real surface in its complexification. The complex separation
is a natural separation of the set of components of a real surface of
characteristic type into two subsets.Note that the set of semiorientations
of a surface is an affine space over ${\bf Z}_2$-vector space of separations
of this surface.
We use the complex separation to weaken assumption $(iii)$. In theorem 1
instead
of $(iii)$ we assume only that $B_+$ lies in components of one class of
complex separation.
The further extension ,theorem 2, can be applied not only for curves of even
degree but also sometimes for curves of odd degree. Another weakening of
assumptions in theorem 2 is that components of a curve are not necessarily
${\bf Z}_2$-homologically trivial.
These extensions can be applied to curves on quadrics and cubics.
Theorem 1 together with an analogue of Arnold inequality for curves on cubics
gives a complete system of restrictions for real schemes of flexible curves
of degree 2 on cubics of characteristic type (see \cite{M1}).An application of
theorem 2 to curves on an ellipsoid gives a complete system of restrictions
for real schemes of flexible curves of degree 3 on an ellipsoid and reduces
the problem of classification of real schemes of flexible curves of degree 5
on an ellipsoid to the problem of the existence of two real schemes (see
\cite{M2}).
An application of theorem 2 to curves on a hyperboloid extends Matsuoka
congruences \cite{Ma} for curves with odd branches on a hyperboloid (see
\cite{M3}).
Applications of theorem 2 to empty curves on surfaces give restrictions for
surfaces involving complex separation of surfaces. Restrictions for curves
of even degree on surfaces can be obtained also by the application of these
restrictions for surfaces to the 2-sheeted covering of surface branched along
the curve , if we know the complex separation of this covering. This complex
separation is determined by complex orientation of the curve.For example in
this way one can obtain new congruences for complex orientations of curves on
a hyperboloid.
These applications and further extension of Rokhlin congruence for curves on
surfaces will be published separately in \cite{M3}. For example the assumption
that
the surface and the curve on the surface are complete intersections is quite
unnecessary ,but this assumption simplifies definition of number $c$ in
formulations of theorems. The formulations of these results were announced
in \cite{M2} as well as the formulations of results of the present paper.
The author is indebted to O.Ya.Viro for advices.
\section{Notations and formulations of main theorems}
Let the surface $B$ be the transversal intersection of hypersurface in
$P^q$ defined by equations $P_j(x_0,\ldots,x_q)=0 ,j=1,\ldots,s-1$ ;
${\bf C}B$ and ${\bf R}B$ be sets of complex and real points of $B$;
let $A$ be a nonsingular curve on $B$ defined by an equation
$P_s(x_0,\ldots,x_q)=0$ ,where $P_j $ is a real homogeneous polynomial
of degree $m_j$ $j=1,\ldots,s$, $q=s+1$; ${\bf C}A$ and ${\bf R}A$ be sets of
complex
and real points of $A$. Let $conj$ denote the involution of complex
conjugation.
Set $c$ to be equal to $\frac{\prod_{j=1}^{s-1} m_{j}}{4}$.If $m_s$ is
even then denote \{$x\in {\bf R}B | \pm P_s(x)\geq 0$\} by $B_{\pm}$ and
set $d$ to be equal to $rk(in_*:H_1(B_+;{\bf Z}_2)\rightarrow H_1({\bf R}B;{\bf
Z}_2))$.
A real algebraic variety is called an (M-$j$)-variety if its total ${\bf
Z}_2$-Betti number is less by $2j$ then total ${\bf Z}_2$-Betti number of
its complexification (Harnack-Smith inequality shows that $j\geq 0$). Let
$A$ be an (M-$k$)-curve. Let $D_M$ be the operator of Poincar\'{e} duality of
manifold $M$.We shall say that $B$ is a surface of characteristic type if
$[{\bf R}B]=D_{{\bf C}B}w_2({\bf C}B)\in H_2({\bf C}B;{\bf Z}_2)$
(as it is usual we denote by $[{\bf R}B]$ the element of
$H_2({\bf C}B;{\bf Z}_2)$ realized by ${\bf R}B$).
We shall say that $(B,A)$ is a pair of characteristic type if
$[{\bf R}B]+[{\bf C}A]+D_{{\bf C}B}(w_2({\bf C}B))=0\in H_2({\bf C}B;{\bf
Z}_2)$.It is said that $A$ is a curve of type I if $[{\bf R}A]=0\in H_1({\bf
C}A;{\bf Z}_2)$.It is said that $A$ is of even(odd) degree if
$[{\bf C}A]=0\in H_2({\bf C}B;{\bf Z}_2)$ (otherwise).
As it is usual we denote by $\sigma$ and $\chi$ the signature and the Euler
characteristic.By $\beta(q)$ we mean Brown invariant of ${\bf Z}_4$-valued
quadratic
form $q$.
\newtheorem{theorem}{Theorem}
\begin{theorem}
If $B$ is a surface of characteristic type then there is defined a natural
separation of surface ${\bf R}B$ into two closed surfaces $B_1$ and $B_2$
by the condition that $B_j,j=1,2$, is a characteristic surface in
${\bf C}B/conj$.Suppose that $m_s$ is even,$B_+\subset B_1$,
every component of ${\bf R}A$ is ${\bf Z}_2$-homologous to zero in ${\bf R}B$
and if $m_s\equiv 2\pmod{4}$ then suppose besides that $B_2$ is contractible
in ${\bf R}P^q$.
\begin{itemize}\begin{description} \item[a)] If $d+k=0$ then $\chi(B_+)\equiv
c\pmod{8}$.
\item[b)] If $d+k=1$ then $\chi(B_+)\equiv c\pm 1\pmod{8}$.
\item[c)] If $d+k=2$ and $\chi(B_+)\equiv c+4\pmod{8}$ then $A$ is of type I
and $B_+$ is orientable.
\item[d)] If $A$ is of type I and $B_+$ is orientable then $\chi(B_+)\equiv
c\pmod{4}$.
\end{description}\end{itemize}\end{theorem}
\begin{theorem}
If $(B,A)$ is a pair of characteristic type then there is defined a natural
separation of surface ${\bf R}B\setminus {\bf R}A$ into two surfaces
$B_1$ and $B_2$ with common boundary ${\bf R}A$ by the condition that
$B_j\cup{\bf C}A/conj$ is a characteristic surface in ${\bf C}B/conj$,
there is defined Guillou-Marin form $q_j$ on $H_1(B_j\cup{\bf C}A/conj;{\bf
Z}_2)$ and \begin{displaymath}
\chi(B_j)\equiv c+\frac{\chi({\bf R}B)-\sigma({\bf C}B)}{4}+\beta(q_j)\pmod{8}
.\end{displaymath}\end{theorem}
\section{Proof of theorem 2}
Consider the Smith exact sequence of double branched covering
$\pi:{\bf C}B\rightarrow {\bf C}B/conj$\begin{displaymath}
\stackrel{\beta_3}{\rightarrow}H_3({\bf C}B/conj,{\bf R}B;{\bf
Z}_2)\stackrel{\gamma_3}{\rightarrow}H_2({\bf R}B;{\bf Z}_2)\oplus H_2({\bf
C}B/conj,{\bf R}B;{\bf Z}_2)\stackrel{\alpha_2}{\rightarrow}H_2({\bf C}B;{\bf
Z}_2)\stackrel{\beta_2}{\rightarrow} .\end{displaymath}
Let $\phi$ denote the composite homomorphism \begin{displaymath} H_2({\bf
C}B/conj,{\bf R}B;{\bf Z}_2)\stackrel{0\oplus id}{\rightarrow}H_2({\bf R}B;{\bf
Z}_2)\oplus H_2({\bf C}B/conj,{\bf R}B;{\bf
Z}_2)\stackrel{\alpha_2}{\rightarrow}H_2({\bf C}B;{\bf Z}_2)
.\end{displaymath}
Let $j$ denote the inclusion map $({\bf C}B/conj,\emptyset)\rightarrow({\bf
C}B/conj,{\bf R}B)$.Recall that $\phi_*\circ j_*$ is equal to Hopf homomorphism
$\pi^!$. It is easy to deduce from the exactness of the Smith sequence that
$\phi$ is a monomorphism. Indeed, $\pi_1({\bf C}B)=0$ hence $\pi_1({\bf
C}B/conj)=0$ and $H_3({\bf C}B/conj;{\bf Z}_2)=0$.
Therefore boundary homomorphism $\partial:H_3({\bf C}B/conj,{\bf R}B;{\bf
Z}_2)\rightarrow H_2({\bf R}B;{\bf Z}_2)$ is a monomorphism.Therefore,since
$\partial$
is the first component of $\gamma_3$, $Im \gamma_3\cap(\{0\}\oplus H_2({\bf
C}B/conj,{\bf R}B;{\bf Z}_2))=0$ and $\phi$ is a monomorphism.
It is easy to check that \begin{displaymath}\pi^*w_2({\bf C}B/conj)=D_{{\bf
C}B}^{-1}[{\bf R}A]+w_2({\bf C}B) .\end{displaymath} Thus $\pi^!(D_{{\bf
C}B/conj}w_2({\bf C}B/conj))=[{\bf C}A]$,
therefore,because of the injectivity of $\phi$,we obtain that
\begin{displaymath} j_*D_{{\bf C}B/conj}w_2({\bf C}B/conj)=[{\bf C}A/conj,{\bf
R}A]\in H_2({\bf C}B/conj,{\bf R}B;{\bf Z}_2) .\end{displaymath} It means that
there exists a compact surface
$B_1\subset{\bf R}B$ with boundary ${\bf R}A$ such that
$B_1\cup{\bf C}A/conj$ is a characteristic surface in ${\bf C}B/conj$.
Surface ${\bf R}A$ is homologous to zero in ${\bf C}B/conj$
since ${\bf R}A$ is the set of branch points of $\pi$.
Set $B_2$ to be equal to the closure of $({\bf R}B\setminus B_1)$.
Then $B_2\cup {\bf C}A/conj$ is a characteristic surface ${\bf C}B/conj$,
$B_1\cup B_2={\bf R}B$,$B_1\cap B_2=\partial B_1=\partial B_2={\bf R}A$.
Let us prove the uniqueness of pair $\{B_1,B_2\}$.It is sufficient to prove
that the dimension of the kernel of inclusion homomorphism
$H_2({\bf R}B;{\bf Z}_2)\rightarrow H_2({\bf C}B/conj;{\bf Z}_2)$
is equal to 1. This follows from the equality
$\dim H_3({\bf C}B/conj,{\bf R}B;{\bf Z}_2)=1$ that can be deduced from
the exactness of the Smith sequence.
We apply now Guillou-Marin congruence \cite{GM} to pair $({\bf
C}B/conj,B_j\cup{\bf C}A/conj),j=1,2$\begin{displaymath}
\sigma({\bf C}B/conj)\equiv[B_j\cup{\bf C}A/conj]\circ[B_j\cup{\bf
C}A/conj]+2\beta(q_j)\pmod{16} .\end{displaymath}
Hirzebruch index theorem gives an equality $\sigma({\bf
C}B/conj)=\frac{\sigma({\bf C}B)-\chi({\bf R}B)}{2}$.To finish the proof note
that
$[B_j\cup{\bf C}A/conj]\circ[B_j\cup{\bf C}A/conj]=2c-2\chi(B_j)$
(the calculation is similar to Marin calculation in \cite{Marin}).
\section{Proof of the theorem 1}
Pair $(B,\emptyset)$ is of characteristic type since $A$ is of even degree
in $B$. Thus the first part of theorem 1 follows from theorem 2 --- there exist
a natural separation of $B$ into two surfaces $B_1$ and $B_2$ such that
$B_1$ and $B_2$ are characteristic surfaces in ${\bf C}B/conj$.
Let $V$ denote $B_+\cup{\bf C}A/conj$.Let $W$ denote $V\cup B_1$.
Recall that $B_+\cap B_2=\emptyset$ thus $V\cap B_2=\emptyset$.
\newtheorem{lemma}{Lemma}
\begin{lemma}
$[V]=0\in H_2({\bf C}B/conj;{\bf Z}_2)$\end{lemma}
{\em\underline{Proof}}. Since $A$ is of even degree in $B$, there exists a
2-sheeted covering $p:Y\rightarrow{\bf C}B$ branched along ${\bf C}A$.
Involution $conj$ can be lifted to involutions $T_+$ and $T_- :Y\rightarrow Y$
since ${\bf C}A$ is invariant under $conj$. It is easy to see using the
straight
algebraic construction of $p$ that $T_+$ and $T_-$ can be chosen in such
a way that the set of fixed points of $T_\pm$ is $p^{-1}(B_\pm)$.
Consider the diagram\begin{picture}(200,-100)(-50,0)
\put(0,0){$Y$}\put(60,0){${\bf C}B$}
\put(0,-40){$Y/T_-$}\put(60,-40){${\bf C}B/conj .$}
\put(10,4){\vector(1,0){46}}
\put(3,-3){\vector(0,-1){25}}
\put(67,-3){\vector(0,-1){25}}
\put(33,10){$p$}\put(71,-15){$\pi$}
\end{picture}\\[42pt]
This diagram can be expanded to a commutative one by map
$p^{'}:Y/T_\mp\rightarrow{\bf C}B/conj$.
It is easy to see that $p^{'}$ is a 2-sheeted covering branched along $V$.
Therefore $[V]=0\in H_2({\bf C}B/conj;{\bf Z}_2)$.
Using Lemma 1 we see that $W$ is a characteristic surface in ${\bf C}B/conj$
as well as $B_2$.We apply Guillou-Marin congruence to these two surfaces :
\begin{displaymath}
\sigma({\bf C}B/conj)\equiv[W]\circ[W]+2\beta(q_W)\equiv
2c-2\chi(B_+)-2\chi(B_2)+2\beta(q_W)\pmod{16}\end{displaymath}
\begin{displaymath}\sigma({\bf
C}B/conj)\equiv[B_2]\circ[B_2]+2\beta(q_{B_2})\equiv
-2\chi(B_2)+2\beta(q_{B_2})\pmod{16}\end{displaymath}
,where $q_W$ and $q_{B_2}$ are Guillou-Marin forms of $W$ and $B_2$.
Therefore \begin{displaymath} \chi(B_+)\equiv
c+\beta(q_W)-\beta(q_{B_2})\pmod{8} .\end{displaymath}
\begin{lemma}
$\forall x\in H_1(B_2;{\bf Z}_2), q_{B_2}(x)-q_W(x)=\left\{ \begin{array}{ll}
0 & \mbox{if $x$ is contractible in ${\bf R}P^q$} \\
\frac{m_s}{2} & \mbox{if $x$ is noncontractible in ${\bf R}P^q$ .}
\end{array} \right. $
\end{lemma}
{\em\underline{Proof}}. It follows from the definition of Guillou-Marin
form that values on $x$ of $q_{B_2}$ and $q_W$ are differed by linking
number of $x$ and $V$ in ${\bf C}B/conj$ that is equal to linking
number of $x$ and ${\bf C}A$ in ${\bf C}B$. The last linking number can be
calculated from the straight construction of a 2-sheeted covering
branched along ${\bf C}A$.
It was shown in \cite{KV} that Brown invariant of form $q$ on the union of two
surfaces with common boundary is equal to the sum of Brown invariants
of restrictions of $q$ on these surfaces in the case when $q$ vanishes on
the common boundary.
Now theorem 1 follows from this additivity of Brown invariant
and the classification of low-dimensional ${\bf Z}_4$-valued quadratic forms
(see\cite{KV}). Indeed, since every component of ${\bf R}A$ is homologous to 0
in ${\bf R}B$,
$\beta(q_W)=\beta(q_W{|_{{\bf C}A/conj}})+\beta(q_W|_{B_+})+\beta(q_W|_{B_2})$.
Lemma 2 shows that under assumptions of theorem 1
$\beta(q_W|_{B_2})=\beta(q_{B_2})$.
To complete the proof note that ranks of intersection forms on
$H_1(B_+;{\bf Z}_2)$ and $H_1({\bf C}A/conj;{\bf Z}_2)$ are equal to
$d$ and $k$ respectively.
|
1992-02-27T04:13:49 | 9202 | alg-geom/9202028 | en | https://arxiv.org/abs/alg-geom/9202028 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9202028 | Alexander A. Voronov | A. A. Voronov | Complex Divisors on Algebraic Curves and Some Applications to String
Theory | 8 pages, LaTeX | Contemp. Math. 131, part 3, 515-522 (1992) | null | null | null | These are notes of a talk to the International Conference on Algebra in honor
of A. I. Mal'tsev, Novosibirsk, USSR, 1989 (to appear in Contemporary
Mathematics). The concept of a divisor with complex coefficients on an
algebraic curve is introduced. We consider families of complex divisors, or,
equivalently, families of invertible sheaves and define Arakelov-type metrics
on some invertible sheaves produced from them on the base. We apply this
technique to obtain a formula for the measure on the moduli space that gives
tachyon correlators in string theory.
| [
{
"version": "v1",
"created": "Thu, 27 Feb 1992 03:13:24 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Voronov",
"A. A.",
""
]
] | alg-geom | \section{Complex Divisors}
Let $X$ be a complete complex curve of genus $g$, with a fixed ordered set
$\mbox{\g m} = \{Q_1, \ldots, Q_n\}$ of $n$ distinct points on $X$ and a closed disk
$B$, considered up to an isotopy in $X \setminus \mbox{\g m}$, such that $\mbox{\g m} \subset
B$. A {\em complex divisor\/} is a formal sum
\[ D = \sum_{P \in X} n_P \cdot P, \]
where
\[ n_P \in \left\{ \begin{array}{ll}
\mbox{\num C} & \mbox{for $P \in \mbox{\g m}$}, \\
\mbox{\num Z} & \mbox{otherwise},
\end{array}
\right.\]
\[ \deg D := \sum_{P \in X} n_P \; \in \mbox{\num Z}, \]
and only a finite number of $n_P \neq 0$. The corresponding {\em group of
complex divisors\/} is denoted by Div$\, (X,\mbox{\g m},B)$.
This definition may be not interesting itself, but it leads to a new class of
invertible sheaves over $X$.
\section{Multiple Valued Meromorphic Functions}
Let $p: \widetilde{X \setminus \mbox{\g m}} \rightarrow X \setminus \mbox{\g m}$ be the
universal covering with the complex structure lifted from the base. Denote by
$H$ the kernel of the natural epimorphism $\pi_1(X \setminus \mbox{\g m}) \rightarrow
\pi_1(X)$ determined by the embedding $X \setminus \mbox{\g m} \hookrightarrow X$:
\[1 \rightarrow H \rightarrow \pi_1(X \setminus \mbox{\g m}) \rightarrow \pi_1(X)
\rightarrow 1 .\]
We will call a holomorphic function $\phi$ on $\widetilde{X \setminus \mbox{\g m}}$
(more precisely, a section of the sheaf $p_*{\cal O}_{\widetilde{X \setminus
\mbox{\g m}}}$) a {\em multiple valued holomorpic function\/} on $X$, if
\begin{enumerate}
\item $\phi$ is $\pi_1(X \setminus B)$-invariant (i.e., $\phi$ is single
valued outside $B$),
\item for every $\sigma \in H$ \ $\phi^\sigma = f_\sigma \cdot \phi$, where
$\phi^\sigma(z) := \phi(\sigma z)$, and $f_\sigma$ is a constant ($f_\sigma$ is
called the {\em multiplicator\/}),
\item the branches of $\phi$, as the branches of a multiple valued analytic
function on $X$, have only removable singularities in \mbox{\g m}, that is for any
$Q_i \in \mbox{\g m}$ and any sequence $\{a_m\}$ in a domain of univalence in
$\widetilde{X \setminus \mbox{\g m}}$, such that $p(a_m) \rightarrow Q_i$ as $m
\rightarrow \infty$, there exists a limit $\lim_{m \rightarrow \infty}
\phi(a_m)$, depending only on $Q_i$:
\[ \phi(Q_i) := \lim_{m \rightarrow \infty} \phi(a_m) . \]
\end{enumerate}
Denote by $\cal O'$ the sheaf of holomorphic multiple valued functions on $X$.
Denote the corresponding sheaf of fields of fractions by $\cal M'$. We will
call sections of $\cal M'$ {\em multiple valued meromorphic functions\/} on
$X$. The following simple lemma describes the local behavior of such functions.
\begin{lm}
Let $z$ be a holomorphic coordinate on $X$ near $Q_i \in \mbox{\g m} \subset X$.
\begin{enumerate}
\item If $\phi \in \cal O'$, then either
\[ \phi(z) = z^A \cdot \sum^\infty_{j=0} \alpha_j z^j, \mbox{ where } 0 < \Re A
\leq 1, \]
or
\[ \phi(z) = \sum^\infty_{j=0} \alpha_j z^j. \]
\item If $\phi \in \cal M'$, then
\[ \phi(z) = z^A \cdot \sum^\infty_{j=n_0} \alpha_j z^j, \mbox{ where } 0 \leq
\Re A < 1. \Box \]
\end{enumerate}
\end{lm}
{\em Note}. One should remember that these expansions may also get monodromy at
other points $Q_i \in \mbox{\g m}$.
\smallskip \\
{\em Definition}. The number $A+n_0$ is called the order ord$_{Q_i} \phi$ of
the multiple valued holomorphic function $\phi$ at the singular point $Q_i$.
Let $\phi \in \Gamma(X, {\cal M}')$ be a globally defined multiple valued
holomorphic function on $X$. Then $\sum_{P \in X} \mbox{ord}_P \phi = 0$,
because d$\, \log \phi$ is a differential of the third kind on $X$ and the sum
of its residues vanishes.
\smallskip\\
{\em Definition}. A divisor of the type
\[
\mbox{div}\, \phi := \sum_{P \in X} \mbox{ord}_P \phi \cdot P \]
is called principal.
\smallskip\\
Define the {\em group\/} Cl$\,(X, \mbox{\g m}, B)$ {\em of classes of complex
divisors\/} as the quotient-group of the group Div$\, (X, \mbox{\g m}, B)$ by the
subgroup of principal divisors.
\section{Complex Divisors and Invertible Sheaves}
\begin{prop}
\begin{sloppypar}
\begin{enumerate}
\item The group {\rm Cl}$\,(X, \mbox{\g m}, B)$ is isomorphic to the group ${\rm
Cl}\,(X)$ of classes of ordinary (integral) divisors on $X$.
\item The group ${\rm Div}\,(X, \mbox{\g m}, B)$ is isomorphic to the group of
invertible $\cal O$-submodules in $\cal M'$.
\end{enumerate}
\end{sloppypar}
\end{prop}
{\em Proof}. Part 1 evidently follows from 2, so let us prove 2. Choose a
covering of $X$ with two open subsets $U_1 := \{ \mbox{a $\delta$-neighborhood
of $B$ for small $\delta > 0$}$, $U_2 := X \setminus B$, and given complex
divisor $D = \sum n_P \cdot P$ take a multiple valued meromorphic function
$f_1$ on $U_1$, such that ord$_P f_1 = n_P$ for $P \in U_1$, and a multiple
valued meromorphic function $f_2$ on $U_2$, such that ord$_P f_2 = n_P$ for $P
\in U_2$ and $f_2^{\gamma_i} = \exp (2 \pi \sqrt{-1} \cdot n_{Q_i}) \cdot
f_2,\; i = 1, \ldots, n$, where $\gamma_i$ is a loop in $B$ containing the
single point $Q_i$. Then $f_1 / f_2$ is a single valued nonzero holomorphic
function on $U_1 \cap U_2$, i.e., $f_1 / f_2 \in \Gamma(U_1 \cap U_2, {\cal
O}^*)$, and it determines an $\cal O$-submodule ${\cal O}(D)$ in $\cal M'$
having $f_2 / f_1$ as the glueing function. Thus, ${\cal O}(D)$ is an ordinary
invertible sheaf on $X$. $\Box$
\section{The Weil-Deligne Pairing}
\begin{sloppypar}
Let ${\cal L}_1$, ${\cal L}_2$ be two invertible $\cal O$-modules. [They may
well be $\cal O$-submodules in $\cal M'$]. Define a complex vector space
$\langle {\cal L}_1, {\cal L}_2 \rangle$ as the space generated by the
expressions
\begin{equation}
\langle l_1, l_2 \rangle , \label{wd}
\end{equation}
where $l_1$ and $l_2$ are single valued (i.e., having integral divisors)
meromorphic sections of ${\cal L}_1$ and ${\cal L}_2$, respectively, with
nonintersecting divisors. We place the following relations on the symbols
(\ref{wd}):
\[ \langle f \cdot l_1, l_2 \rangle = f({\rm div}\, l_2) \cdot \langle l_1, l_2
\rangle , \]
\[ \langle l_1, g \cdot l_2 \rangle = g({\rm div}\, l_1) \cdot \langle l_1,
l_2 \rangle , \]
where $f$ and $g$ are single valued meromorphic functions such that $f({\rm
div}\, l_2) := \prod_{P \in X} f(P)^{\mbox{\scriptsize ord}_P l_2} \neq 0,
\infty$ in the former formula and $g({\rm div}\, l_1) := \prod_{P \in X}
g(P)^{\mbox{\scriptsize ord}_P l_1} \neq 0, \infty$ in the latter. The
correctness of this definition is provided by Weil's reciprocity law:
\[ f({\rm div}\, g)= g({\rm div}\, f). \]
One can easily see that the space $\langle {\cal L}_1, {\cal L}_2 \rangle$ is a
one-dimensional complex vector space. We will call it the {\em Weil-Deligne
pairing\/} of ${\cal L}_1$ and ${\cal L}_2 $.
\end{sloppypar}
\section{The Arakelov-Deligne Metric}
Now, let ${\cal L}_1$ and ${\cal L}_2 $ be two Hermitian holomorphic line
bundles. Then one can define a natural Hermitian metric on the space $\langle
{\cal L}_1, {\cal L}_2 \rangle$ (cf.\ Deligne \cite{d:d}). That means that for
any two sinle valued sections $l_1$, $l_2$ of ${\cal L}_1$ and ${\cal L}_2 $
with nonintersecting divisors, there is defined a real number
\[ \parallel \langle l_1, l_2 \rangle \parallel \in \mbox{\num R} .\]
Below we define an analogous metric in a more general case, when $l_1$ and
$l_2$ are not necessarily single valued, but of degree 0. We will use this
construction in string theory later.
The definition is
\begin{equation}
\parallel \langle l_1, l_2 \rangle \parallel := \sqrt{ \prod_i
G^{\overline{n}_i}_{\mbox{\scriptsize div}\,l_2} (P_i) \cdot
G^{n_i}_{\overline{\mbox{\scriptsize div}\,l_2}} (P_i)}, \label{ad}
\end{equation}
where $\overline{\mbox{div}\,l_2}$ means the divisor with complex conjugated
coefficients, div$\, l_1 = \sum_i n_i P_i$ and $G_D (z) := \exp g_D(z)$,
$g_D(z)$ being the Green function of the divisor $D$, which is defined up to a
constant similar to the case of integral $D$ (for example, put $g_D (z) := \Re
\int_{z_0}^z \omega_D$, where $\omega_D$ is the differential of the third kind
associated with $D$, cf.\ Lang \cite{l}). The result does not depend on the
choice of Green function, because we assume deg$\, l_1 = \sum n_i = 0$.
Moreover, the obtained symbol $\parallel \langle \; , \; \rangle \parallel$ is
symmetric:
\[ \parallel \langle l_1, l_2 \rangle \parallel = \parallel \langle l_2, l_1
\rangle \parallel . \]
This can be observed from the formula
\begin{equation}
\parallel \langle l_1, l_2 \rangle \parallel = \sqrt{ \prod_i
G^{\overline{n}_i}_{\mbox{\scriptsize div}\,l_2} (P_i) \cdot \prod_j
G^{\overline{n}'_j}_{\mbox{\scriptsize div}\,l_1} (P'_j)}, \label{adsym}
\end{equation}
where div$\, l_2 = \sum_j n'_j P'_j$. In fact, $G_{\mbox{\scriptsize div}\,l_1}
(z) = \prod_i G^{n_i}_{P_i} (z)$ and $G_P(Q) = G_Q(P)$, so (\ref{adsym}) is
equivalent to (\ref{ad}). These arguments also imply the formula
\begin{equation}
\parallel \langle l_1, l_2 \rangle \parallel = \prod_{i,j} G^{\Re (n_i
\overline{n}'_j)}_{P_i} (P'_j) . \label{ad3}
\end{equation}
If $l_1$, $l_2$ and $k$ are sections of Hermitian line bundles ${\cal L}_1$ and
${\cal L}_2 $ and $\cal K$, then
\[ \parallel \langle l_1 \otimes l_2, k \rangle \parallel = \parallel \langle
l_1, k \rangle \parallel \cdot \parallel \langle l_2, k \rangle \parallel \]
whenever both sides are defined. There are some special properties of complex
divisors: if supp$\, D_1$, supp$\, D_2 \subset \mbox{\g m}$, then
\[ \parallel \langle \mbox{\bf 1}_{\alpha D_1}, \mbox{\bf 1}_{D_2} \rangle \parallel =
\parallel \langle \mbox{\bf 1}_{D_1}, \mbox{\bf 1}_{D_2} \rangle \parallel ^\alpha \mbox{ for
} \alpha \in \mbox{\num R} \]
and
\[ \parallel \langle \mbox{\bf 1}_{\alpha D_1}, \mbox{\bf 1}_{D_2} \rangle \parallel =
\parallel \langle \mbox{\bf 1}_{D_1}, \mbox{\bf 1}_{\overline{\alpha} D_2} \rangle \parallel
\mbox{ for } \alpha \in \mbox{\num C}. \]
Thereby, the symbol $\parallel \langle \; , \; \rangle \parallel$ is Hermitian.
More precisely, it is the modulus of the exponent of a Hermitian form on the
vector space of complex divisors of degree 0 with support in \mbox{\g m}. This
Hermitian form is easy to write out (cf.\ (\ref{ad3})):
\[ \sum_{i,j} n_i \, \overline{n}'_j \, g_{Q_i}(Q_j) . \]
\section{The Deligne-Riemann-Roch Theorem}
Let us consider an algebraic family of objects $(X, \mbox{\g m}, B)$ over a base $S$,
i.e., a smooth projective morphism $\pi: X \rightarrow S$ of smooth complex
algebraic varieties with fiber being a connected complex curve, \mbox{\g m}\ being the
disjoint union of $n$ regular sections of $\pi$ and $B$ varying continuously
along $S$. Let $D$ and $D'$ be two families of complex divisors on $X
\rightarrow S$, more generally, two invertible $\cal O$-submodules ${\cal L}_1$
and ${\cal L}_2$ in $\cal M'$. Suppose they are metrized as well as the sheaf
$\Omega$ of relative 1-differentials along the fibers of $\pi$. Then the
sheaves $\det \mbox{\num R} \pi_* \cal L$ (the determinant sheaf) and $\langle {\cal
L}_1, {\cal L}_2 \rangle$ (the Weil-Deligne sheaf, whose fiber over a single
curve $X_s$, $s \in S$, in the family is defined in Section~4) are defined.
$\det \mbox{\num R} \pi_* \cal L$ can be endowed with a Hermitian metric according to
Quillen (see \cite{d:d}), and $\langle {\cal L}_1, {\cal L}_2 \rangle$ is
metrized in Section~5. The following theorem is important for our string
applications.
\begin{th}[Deligne \cite{d:d}]
There is a canonical isometry
\[ \det \mbox{\num R} \pi_* ({\cal L})^2 \otimes \det \mbox{\num R} \pi_* ({\cal O})^{-2} =
\langle {\cal L} \otimes \Omega^* , {\cal L} \rangle . \Box \]
\end{th}
\section{String Applications}
The $g$-loop contribution to the string partition function can be reduced to
the integral
\[ Z_g := \int_{{\cal M}_g} \mbox{d}\pi_g \]
of the {\em Polyakov measure\/} d$\pi_g$ over the moduli space $ {\cal M}_g$ of
complete complex algebraic curves of genus $g$. The {\em Belavin-Knizhnik
theorem\/} represents d$\pi_g$ as the modulus squared
\[ {\rm d}\pi_g = \mu_g \wedge \overline{\mu}_g \]
of a {\em Mumford form\/} $\mu_g$, which is a section of the sheaf $\lambda_2
\otimes \lambda_1^{-13}$, where $\lambda_i := \det \mbox{\num R} \pi_* (\Omega^{\otimes
i})$, $\pi$ being the universal curve $\pi: X \rightarrow {\cal M}_g$.
The {\em tachyon scattering amplitude\/} is the integral
\[ A(g; {\bf p}_1, \ldots, {\bf p}_n) := \int_{{\cal M}_{g,n}}
\mbox{d}\pi_{g,n}, \]
where ${\cal M}_{g,n}$ is the moduli space of algebraic curves of genus $g$
with $n$ punctures and the measure $\mbox{d}\pi_{g,n}$ is expressed in terms of
determinants of Laplace operators and their Green functions. The vectors ${\bf
p}_i$ on which the amplitude depends are regarded as momentum vectors at the
scattering points, so they lie in the space-time of the critical dimension,
which we identify with $\mbox{\num C}^{13}$ endowed with the standard Hermitian metric.
These vectors satisfy the conditions:
\begin{enumerate}
\item $\sum_{i=1}^n {\bf p}_i = 0$ (the momentum conservation law).
\item The Hermitian square $({\bf p}_i, {\bf p}_i)$ is equal to 1 for every
$i$ (the mass of tachyon is $\sqrt{-1}$).
\end{enumerate}
Our application to string theory consists in proving the following analogue of
the Belavin-Knizhnik theorem for string amplitudes.
\begin{th}
\[{\rm d}\pi_{g,n} = \mu_{g,n,B} \wedge \overline{\mu}_{g,n,B} / \parallel
\mu_{g,n,B} \parallel^2, \]
where $\mu_{g,n,B}$ is a local holomorphic section of the Hermitian line bundle
$\lambda_2 \otimes \lambda_1^{-13} \otimes (\bigotimes_{\nu=1}^{13} \langle
{\cal O}(D^\nu), {\cal O}(D^\nu) \rangle )^{-1} $ over the moduli space ${\cal
M}_{g,n,B}$ of the data $(X, Q_1, \ldots, Q_n, B)$. Here $D^\nu := \sum_{i=1}^n
p_i^\nu \cdot Q_i$ is the complex divisor with the momentum components as
coefficients. The section $\mu_{g,n,B}$ is defined locally up to a holomorphic
factor. $\Box$
\end{th}
|
1992-02-20T22:06:45 | 9202 | alg-geom/9202021 | en | https://arxiv.org/abs/alg-geom/9202021 | [
"alg-geom",
"math.AG"
] | alg-geom/9202021 | null | Dave Bayer, Andre Galligo, Mike Stillman | Grobner bases and extension of scalars | 18 pages, LaTeX | null | null | null | null | This paper studies the behavior of Grobner bases with respect to extensions
of scalars. We prove that an extension of scalars commutes with taking Grobner
bases iff the extension is flat. We consider what information can be deduced
about fibers of a family, from the Grobner basis of the defining ideal of the
family itself. This information can be used to construct algorithms, such as to
find locii over which a map is finite, or an isomorphism.
| [
{
"version": "v1",
"created": "Thu, 20 Feb 1992 21:03:16 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bayer",
"Dave",
""
],
[
"Galligo",
"Andre",
""
],
[
"Stillman",
"Mike",
""
]
] | alg-geom | \section{Introduction}
\label{intro}
\setcounter{defn0}{0}
Let $A$ be a Noetherian commutative ring with identity, let $\mmbox{A[\x]} =
A[x_1,\ldots ,x_n]$ be a polynomial ring over $A$, and let $I \subset
\mmbox{A[\x]}$ be an ideal. Geometrically, $I$ defines a family of schemes over
the base scheme \Spec{A}; the fiber over each point $p \in \Spec{A}$
is a subscheme of the affine space ${\bf A}^n_{k(p)} =
\Spec{k(p)[\sliver{\bf x}\sliver]}$, where $k(p) = A_p/p_p$ is the residue field of $p$.
Let $>$ be a total order on the monomials of \mmbox{A[\x]}\ satisfying $\sliver{\bf x}\sliver^E >
\sliver{\bf x}\sliver^F \Rightarrow \sliver{\bf x}\sliver^G\sliver{\bf x}\sliver^E > \sliver{\bf x}\sliver^G\sliver{\bf x}\sliver^F$, and satisfying $x_i > 1$ for
each $i$. For $f \in \mmbox{A[\x]}$, define \iin{f} to be the initial (greatest)
term $c \sliver{\bf x}\sliver^E$ of $f$ with respect to the order $>$, where $c \in A$ is
nonzero. For $I \subset \mmbox{A[\x]}$, define the initial ideal $\iin{I}$ to be
the ideal $( \iin{f} \mid f \in I )$ generated by all initial terms of
elements of $I$. $\iin{I} \subset \mmbox{A[\x]}$ is generated by single terms of
the form $c \sliver{\bf x}\sliver^E$; we call such an ideal a monomial ideal.
$\{f_1,\ldots , f_r\} \subset I$ is a Gr\"{o}bner\ basis for $I$ if and only
if $\{\iin{f_1},\ldots , \iin{f_r}\}$ generates \iin{I}.
For an ideal $I$ in a ring of formal power series $A[[\sliver{\bf x}\sliver]]$, Hironaka
defined the corresponding notion (the standard basis of $I$) and
proved a generalized Weierstrass division theorem.The relation between
flatness and division was considered in \cite{hlt73} and later in
\cite{gal79} in order to obtain a presentation for the flattener of a
germ of an analytical morphism. The algebraic situation is slightly
different.
In this paper, we study the behavior of Gr\"{o}bner\ bases with respect to an
extension of scalars $A \rightarrow B$. When does a Gr\"{o}bner\ basis for
$I$ map to a Gr\"{o}bner\ basis for $I \mmbox{B[\x]}$? It suffices to have \iin{I}
generate \iin{I \mmbox{B[\x]}}; we focus on this condition. Taking $B = k(p)$
for $p \in \Spec{A}$, we consider the relationship between a Gr\"{o}bner\
basis for $I$, and the Gr\"{o}bner\ bases of the fibers of the family defined
by $I$. How much information about the fibers of this family can be
inferred from knowledge of \iin{I} alone?
Let $X \subset \Spec{A}$ be the support of the family defined by $I$.
A Gr\"{o}bner\ basis for $I$ encodes considerable information about this
family, even when $X$ is nonreduced or reducible. To interpret \iin{I}
in such situations, we work with its {\it coefficient ideals\,}: The
coefficient ideal for the monomial $\sliver{\bf x}\sliver^E$ vanishes on the support of
those fibers where $\sliver{\bf x}\sliver^E$ fails to belong to $\iin{I} \mmbox{k(p)[\x]}$. From this
point of view, a point $p \in \Spec{A}$ is \quotes{good} if each
coefficient ideal of \iin{I} defines a scheme which either avoids $p$,
or contains an open neighborhood of $p$ in $X$.
In \S 2, we study coefficient ideals of monomial ideals. In \S 3, we
prove that an extension of scalars commutes with taking the initial
ideal of any ideal $I$, if and only if the extension is flat. We then
prove that a Gr\"{o}bner\ basis for $I$ determines Gr\"{o}bner\ bases for the
localizations to dense open subsets of each isolated component of $X$.
We also prove that for this family, \iin{I} determines the fiber
initial ideals over \quotes{good} points, as defined above. These
results reveal that \iin{I} carries generic information for each
isolated component of $X$. In \S 4, we prove that if every point of
$X$ is \quotes{good}, then the family defined by $I$ is faithfully
flat over $X$. Faithful flatness imposes strong conditions on the
component structure of the total space of our family, so this result
has geometric applications, such as the removal of unwanted
components. In \S 5, we give two other applications of coefficient
ideals, describing the locus where a morphism of schemes is an
isomorphism, or a finite map.
If $A$ is a finitely generated $k$-algebra for a field $k$, then we
can write $A = k[a_1,\ldots,a_m]/J$ for some ideal $J$. We can
reformulate our problem as $A = k[\sliver{\bf a}\sliver]$, with $I \subset \mmbox{A[\x]}$ and $I
\cap A \supset J$. A Gr\"{o}bner\ basis for $I$ can then be computed by the
usual algorithm over a field, by combining orders $>_1$, $>_2$ into a
product order $$ \sliver{\bf a}\sliver^D \sliver{\bf x}\sliver^E > \sliver{\bf a}\sliver^F \sliver{\bf x}\sliver^G \ \Leftrightarrow \sliver{\bf x}\sliver^E >_1
\sliver{\bf x}\sliver^G, \mbox{\ or \ } \sliver{\bf x}\sliver^E = \sliver{\bf x}\sliver^G \mbox{\ and \ } \sliver{\bf a}\sliver^D >_2 \sliver{\bf a}\sliver^F.$$ In
this setting, $I$ defines a subscheme $Y \subset {\bf A}_k^{m+n}$
which projects to $X \subset {\bf A}_k^m$.
More generally, the computational relevance of this work depends on
our ability to compute in the base ring $A$. Specifically, $A$ needs
to be a ring where linear equations are solvable; see Trager, Gianni,
and Zacharias (\cite{gtz88}) for background material and references on
Gr\"{o}bner\ bases in this setting. Our paper continues their study of
families of Gr\"{o}bner\ bases; we would like to thank each of them, David
Eisenbud, and an anonymous referee, for many helpful conversations and
suggestions.
\section{Monomial Ideals}
\label{monoms}
\setcounter{defn0}{0}
Let $J \subset \mmbox{A[\x]}$ be a monomial ideal, i.e., an ideal generated by
single terms of the form $c \sliver{\bf x}\sliver^E$, with $c \in A$.
When $A$ is a field, $J$ is easily understood: its structure is
realized by the subset $L = \{ E \mid \sliver{\bf x}\sliver^E \in J \}$ of ${\bf N}^n$, where
${\bf N}$ denotes the nonnegative integers. ${\bf N}^n$ admits a natural partial
order $\leq$ defined by $E \leq F$ iff $\sliver{\bf x}\sliver^E$ divides $\sliver{\bf x}\sliver^F$. The
characteristic function of $L$ can be viewed as a poset homomorphism
from ${\bf N}^n$, ordered by $\leq$, to the set of ideals $\{(0),(1)\}$ of
$A$, ordered by inclusion.
To understand $J$ when $A$ is not a field, it is helpful to consider
the {\it coefficient ideals\,} of $J$: Define $J_E = J_{\sliver{\bf x}\sliver^E} = (c \in
A \mid c \sliver{\bf x}\sliver^E \in J)$. Alternatively, $J_E$ is the ideal quotient $(J
: \sliver{\bf x}\sliver^E) \cap A$. This construction defines a poset homomorphism $E
\mapsto J_E$ from ${\bf N}^n$, ordered by $\leq$, to the set of ideals of
$A$, ordered by inclusion. Conversely, any such poset homomorphism
determines a monomial ideal $J$, so we can think of this construction
as describing an equivalence of categories.
Viewing a monomial ideal as its collection of coefficient ideals is
thus a purely tautological change of perspective; any operation on $A$
can be viewed as acting on $J$ via the inclusion $J \subset \mmbox{A[\x]}$, or
equivalently as acting on the set of coefficient ideals of $J$. In
particular, if $v: A \rightarrow B$ is a ring homomorphism, then $v$
extends naturally to a homomorphism $v: \mmbox{A[\x]} \rightarrow \mmbox{B[\x]}$. The
image under $v$ of any ideal $I \subset \mmbox{A[\x]}$ generates the extension
ideal $I^e = I \mmbox{B[\x]}$. For monomial ideals, we have the immediate
proposition
\begin{prop}
\label{monoms:prop}
Let $v: A \rightarrow B$ be a ring homomorphism. Let $J B[\sliver{\bf x}\sliver]$ denote
the monomial ideal obtained by extension of scalars from $A$ to $B$,
and let $J_E B$ also be obtained by extension of scalars, for an
exponent $E$. Then $$J_E B = (J \mmbox{B[\x]})_E.$$
\mbox{~~~~\vrule height 1.2ex width .9ex depth .1ex}
\end{prop}
In particular, if $B$ is the residue field $k(p)$ of a prime ideal $p
\subset A$, then \propref{monoms:prop} asserts that $\sliver{\bf x}\sliver^E \in J \mmbox{k(p)[\x]}$
if and only if the point $p$ does not belong to the subscheme of
\Spec{A} defined by $J_E$. This subscheme is the support of the
$A$-module $A \sliver{\bf x}\sliver^E \subset \mmbox{A[\x]}/J$. The monomial $\sliver{\bf x}\sliver^E$ appears only
with a coefficient of zero in $J (A/J_E)[\sliver{\bf x}\sliver]$; $J_E$ is the
intersection of all ideals $K \subset A$ with the property that $(J
(A/K)[\sliver{\bf x}\sliver])_E = (0)$.
In other words, we can view $J$ as a family of monomial ideals over
\Spec{A}. The monomial ideals corresponding to each fiber of the
family defined by $J$ are defined over fields, and can be visualized
combinatorially: Each monomial $\sliver{\bf x}\sliver^E$ either belongs or does not
belong to a given fiber monomial ideal, which in turn is determined by
this data. The coefficient ideal $J_E$ defines the subscheme of
\Spec{A}\ over which $\sliver{\bf x}\sliver^E$ does not belong to the fiber monomial
ideals. $J$ is determined by these subschemes.
\begin{example}
\label{monoms:ex1}
Let $A = {\bf Z}$, let $\mmbox{A[\x]} = A[x,y]$, and let $J = (9x,\, 2y,\, x^2,\,
y^2)$. The coefficient ideals for $1$, $x$, $y$, $x^2$, $xy$, and
$y^2$ respectively are $(0)$, $(9)$, $(2)$, $(1)$, $(1)$, and $(1)$;
this is summarized in the diagram
\begin{center}
\vspace{.1 in}
\begin{tabular}{r|lll}
$y$ & $(1)$ \\
& $(2)$ & $(1)$ \\
& $(0)$ & $(9)$ & $(1)$ \\
\cline{2-4}
\multicolumn{4}{r}{$x$}
\end{tabular}
\end{center}
$J$ specializes to $(x,\, y)$ in each fiber over the open subset of
\Spec{{\bf Z}} which is the complement of the points $(2)$ and $(3)$. In
the fiber over $(2)$, $J$ specializes to $(x,\, y^2)$. Over the double
point at $(3)$, $J$ specializes to $(x^2,\, y)$.
$A$ is an integral domain. The union of the monomials with a nonzero
coefficient ideal spans the monomial ideal $(x,\, y)$, which occurs
generically. On the other hand, the union of the monomials with
coefficient ideal $(1)$ spans the monomial ideal $(x^2,\, xy,\, y^2)$.
There is no specialization which produces this monomial ideal, but it
is contained in every monomial ideal obtained by specialization.
\end{example}
\begin{example}
\label{monoms:ex2}
Modify the preceding example by taking $A = {\bf Z}/18{\bf Z}$. \Spec{A} is no
longer reduced or irreducible. The union of the monomials with nonzero
coefficient ideals again spans the monomial ideal $(x,\, y)$. There is
no specialization to a field which produces this monomial ideal; this
can happen whenever $A$ is not an integral domain.
\end{example}
\section{Initial Ideals}
\label{initial}
\setcounter{defn0}{0}
Let $I \subset \mmbox{A[\x]}$ be an ideal. For monomial ideals,
\propref{monoms:prop} asserts that the formation of coefficient ideals
commutes with extension of scalars. In contrast, the formation of the
initial ideal of an arbitrary ideal $I$ need not commute with
extension of scalars: If $v: A \rightarrow B$ is a ring homomorphism,
then it can happen that $\iin{I} \mmbox{B[\x]} \neq \iin{I \mmbox{B[\x]}}$. This is
because if $v$ maps the leading coefficient of a polynomial $f \in I$
to zero, then the first surviving term of the image of $f$ will
contribute to \iin{I \mmbox{B[\x]}}. When this happens, \iin{I \mmbox{B[\x]}} cannot be
predicted from knowledge of \iin{I} alone.
Let $\{ f_1,\ldots f_r \}$ be a Gr\"{o}bner\ basis for $I$. If $\iin{I} \mmbox{B[\x]} =
\iin{I \mmbox{B[\x]}}$, then the images $\{ v(f_1),\ldots v(f_r) \}$ form a
Gr\"{o}bner\ basis for $I$. We shall study the behavior of initial ideals
with respect to various hypotheses on $I$ and $B$; as a consequence we
obtain sufficient conditions for the construction of a Gr\"{o}bner\ basis for
$I$ to commute with the extension of scalars $v: A \rightarrow B$.
\begin{example}
\label{initial:ex1}
Let $A=k[a]$ for a field $k$, let $\mmbox{A[\x]}=A[x,y]$, and let $I = (ax-y)
\subset \mmbox{A[\x]}$. In this example, as in all subsequent examples involving
the variables $x$ and $y$, we use the lexicographic order extending $x
> y$. Choose a prime $p \subset A$, and let $k(p)$ be the residue
field $A_p/p_p$ of $p$. When $p \neq (a)$, $\iin{I} \mmbox{k(p)[\x]} = \iin{I
\mmbox{k(p)[\x]}} = (x)$. However, when $p = (a)$, $\iin{I} \mmbox{k(p)[\x]} = (0)$, but
$\iin{I \mmbox{k(p)[\x]}} = (y)$. Nevertheless, the image of $\{ ax-y \}$ is a
Gr\"{o}bner\ basis for $I \mmbox{k(p)[\x]}$.
$I$ defines a faithfully flat family over \Spec{A}, because $\mmbox{A[\x]}/I
\simeq A[x]$. The total space is given by the surface $a x - y = 0$,
and each fiber consists of a line through the origin in ${\bf A}^2_k$,
with slope $a$. \iin{I \mmbox{k(p)[\x]}} momentarily flips from $(x)$ to $(y)$ as
this slope passes through zero.
\end{example}
\begin{example}
\label{initial:ex2}
Let $A = k[a,b]$, let $\mmbox{A[\x]} = A[x,y]$, let $I = (ax^2+y,\, by^2+y+1 )$,
and let $B = A/(a,\, b)$. Then $\iin{I} \mmbox{B[\x]} = (0)$, but $\iin{I \mmbox{B[\x]}} =
(1)$. $\{ax^2+y,\, by^2+y+1\}$ is a Gr\"{o}bner\ basis for $I$, but its image
$\{y,\, y+1\}$ in $\mmbox{B[\x]}$ is not a Gr\"{o}bner\ basis for $I \mmbox{B[\x]}$.
\end{example}
\begin{example}
\label{initial:ex3}
Let $A = k[a]$, let $\mmbox{A[\x]} = A[x]$, let $I = (ax - 1)$, and let $p
\subset A$ be a prime. When $p \neq (a)$, $\iin{I} \mmbox{k(p)[\x]} = \iin{I \mmbox{k(p)[\x]}}
= (x)$. However, when $p = (a)$, $\iin{I} \mmbox{k(p)[\x]} = (0)$, but $\iin{I
\mmbox{k(p)[\x]}} = (1)$.
$I$ defines a flat family which is not faithfully flat: The fiber over
$a = 0$ is empty, so $(a) \subset A$ extends to the unit ideal in
$\mmbox{A[\x]}/I$.
\end{example}
As suggested by these examples, we do have an inclusion in one direction:
\begin{prop}
\label{initial:incprop}
For any ring homomorphism $v: A \rightarrow B$, and for any ideal $I
\subset \mmbox{A[\x]}$, we have $$\iin{I} \mmbox{B[\x]} \subset \iin{I \mmbox{B[\x]}}.$$
\end{prop}
\begin{proof}
It is enough to show that each generator of $\iin{I} \mmbox{B[\x]}$ also belongs
to $\iin{I \mmbox{B[\x]}}$. $\iin{I} \mmbox{B[\x]}$ is generated by $v(\iin{f})$ for $f
\in I$. For each $f \in I$, either \iin{f} maps to zero in \mmbox{B[\x]}, or
else $v(\iin{f}) = \iin{v(f)} \in \iin{I \mmbox{B[\x]}}$. \end{proof}
The following theorem asserts that taking initial ideals universally
commutes with an extension of scalars if and only if the extension is
flat. We apply the criterion for flatness given in \cite{mat86}, Thm.\
7.6, which asserts that $v: A \rightarrow B$ is flat iff the syzygies
in $B$ of a set of elements from $A$ can always be generated by
syzygies from $A$:
\begin{lemma}
\label{initial:flatlem}
Let $v: A \rightarrow B$ be a ring homomorphism. $v$ is flat if and
only if for each sequence $a_i \in A$ and $b_i \in B$ for $1 \leq i
\leq r$ so $$\sum_i b_i v(a_i) = 0,$$ then for some $s$ we can choose
$c_{ij} \in A$ and $d_j \in B$ for $1 \leq j \leq s$ so $$\sum_i
c_{ij} a_i = 0 \mbox{\ for each $j$, and\ } b_i = \sum_j d_j v(c_{ij})
\mbox{\ for each $i$.}$$
\mbox{~~~~\vrule height 1.2ex width .9ex depth .1ex}
\end{lemma}
\begin{thm}
\label{initial:thm}
Let $v: A \rightarrow B$ be a ring homomorphism. The following two
conditions are equivalent:
(a) for any number of variables $x_1, \ldots, x_n$, and for any ideal
$I \subset \mmbox{A[\x]}$, $\iin{I} \mmbox{B[\x]} = \iin{I \mmbox{B[\x]}}$;
(b) $B$ is a flat $A$-algebra.
\end{thm}
\begin{proof}
First, suppose that (b) holds, and let $I \subset \mmbox{A[\x]}$ be an ideal. We
need to show that $\iin{I \mmbox{B[\x]}} \subset \iin{I} \mmbox{B[\x]}$. Given $c \sliver{\bf x}\sliver^E \in
\iin{I \mmbox{B[\x]}}$, consider expressions of the form $c \sliver{\bf x}\sliver^E = \iin{\sum_i
b_i v(f_i)}$, where $b_i \in \mmbox{B[\x]}$ and $f_i \in I$. By expanding out
each $b_i$ and absorbing variables into each $f_i$, we need only
consider expressions for which each $b_i \in B$. Among all such
expressions, choose one for which the greatest monomial appearing in
any of the $f_i$ is minimal. We claim that this greatest monomial is
$\sliver{\bf x}\sliver^E$. Letting $c_i$ be the coefficient of $\sliver{\bf x}\sliver^E$ in each $f_i$, we
have $c = \sum_i b_i v(c_i)$, so $c \sliver{\bf x}\sliver^E \in \iin{I}\mmbox{B[\x]}$.
Suppose otherwise, that the greatest monomial of a minimal expression
is $\sliver{\bf x}\sliver^D > \sliver{\bf x}\sliver^E$. Let $a_i$ be the coefficient of $\sliver{\bf x}\sliver^D$ in each
$f_i$. Then $\sum_i b_i v(a_i) = 0$. Choosing $c_{ij}$, $d_j$ as in
\lemref{initial:flatlem}, define $g_j = \sum_i c_{ij} f_i \in I$ for
$1 \leq j \leq s$. Then $$\sum_j d_j v(g_j) = \sum_i b_i v(f_i),$$ and
$\iin{g_j} < \sliver{\bf x}\sliver^D$ for each $j$, contradicting the minimality of our
expression. This proves (a).
Now, suppose instead that (b) does not hold. Choose a sequence $a_i
\in A$ and $b_i \in B$ for $1 \leq i \leq r$ with $r$ minimal, so
$\sum_i b_i v(a_i) = 0$ but the $b_i$ cannot be expressed as in
\lemref{initial:flatlem}. We construct an example in two variables $x
> y$ for which $v$ does not commute with taking initial ideals: Let
$$f_i = a_i x^r + x^{r-i}y^i \mbox{\ \ for \ } 1 \leq i \leq r,$$ and
let $I \subset A[x,y]$ be defined by $ I = (f_1,\, \ldots,\, f_r)$.
Then $$\sum_i b_i v(f_i) = b_1 x^{r-1}y + \ldots + b_r y^r.$$
Moreover, for $c_{ij} \in A$, $$\sum_i c_{ij} a_i = 0
\;\Leftrightarrow\; c_{1j} x^{r-1}y + \ldots + c_{rj} y^r = \sum_i
c_{ij} f_i\in I.$$ Because the $b_i$ cannot be expressed as in
\lemref{initial:flatlem}, and because $r$ was chosen to be minimal,
already $b_1$ is not in the ideal generated by the images of all
$c_{1j}$ for $c_{ij}$ as above. Because $x^r > x^{r-1}y > \ldots >
y^r$, this ideal is the coefficient ideal of $\iin{I} \mmbox{B[\x]}$ with
respect to the monomial $x^{r-1}y$, so $$(\iin{I} \mmbox{B[\x]})_{x^{r-1}y} \neq
\iin{I \mmbox{B[\x]}}_{x^{r-1}y}.$$ This proves that (a) does not hold.
\end{proof}
The following corollary asserts that taking initial ideals commutes
with taking rings of fractions, and is due to Gianni, Trager, and
Zacharias (\cite{gtz88}, Prop.\ 3.4).
\begin{corollary}
\label{initial:gtzcor}
Let $I \subset \mmbox{A[\x]}$ be an ideal, and let $B = S^{-1}A$ for a
multiplicatively closed set $S \subset A$. Then $$ \iin{I} \mmbox{B[\x]} =
\iin{I \mmbox{B[\x]}}.$$
\end{corollary}
\begin{proof}
A ring of fractions is a flat extension (\cite{mat86}, Thm.\ 4.5).
\end{proof}
Suppose that we want to work with the extension $A/J$, for an ideal $J
\subset A$. If $J$ arises as the kernel of a map $A \rightarrow
S^{-1}A$ for some multiplicatively closed set $S \subset A$, then we
can apply \corref{initial:gtzcor} if we instead work with the
extension $S^{-1}A$. Viewing $S^{-1}A$ as a ring of fractions of
$A/J$, this extension retains generic information along the scheme
defined by $J$, but loses primes annihilated by elements of $S$. Such
primes can prevent the taking of initial ideals from commuting with
the extension $A/J$, as illustrated by the following example.
\begin{example}
\label{initial:fractex}
Let $A = k[a,b]/(ab)$, let $\mmbox{A[\x]} = A[x]$, let $I = (ax+1)$, and let $J
= (a)$. Then $\iin{I} = (ax,\, b)$. Taking $B = A/J$, we have $\iin{I}
\mmbox{B[\x]} = (b)$, and $\iin{I \mmbox{B[\x]}} = (1)$. Instead taking $B = A_b$, we have
$\iin{I} \mmbox{B[\x]} = \iin{I \mmbox{B[\x]}} = (1)$. Thus taking initial ideals
commutes with the extension $A_b$, but does not commute with the
extension $A/J$.
\Spec{A_b} differs from \Spec{A/J} only by the removal of the prime
$p = (b)$. This prime obstructs good behavior for the extension $A/J$:
$\iin{I} \mmbox{k(p)[\x]} = (0)$, and $\iin{I \mmbox{k(p)[\x]}} = (1)$.
\end{example}
Which kernels arise from taking rings of fractions? From the proof of
\cite{atma69}, Thm.\ 4.10, one sees that these kernels are precisely
the ideals $q \subset A$ which arise as the intersection of primary
components corresponding to an isolated set of associated primes of
$(0)$. For each such $q$, it is enough to choose a multiplicatively
closed subset $S \subset A$ which intersects \Ann{q}, for $B =
S^{-1}A/S^{-1}q$ to equal $S^{-1}A$.
More generally, we may wish to consider components of the subscheme
defined by $I \cap A$, when this ideal is nonzero. The following lemma
reduces us to the above setting.
\begin{lemma}
\label{initial:quolem}
Let $I \subset \mmbox{A[\x]}$ be an ideal, and let $B = A/(I \cap A)$. Then $$
\iin{I} \mmbox{B[\x]} = \iin{I \mmbox{B[\x]}}.$$
\end{lemma}
\begin{proof}
Let $v:A \rightarrow B$ be the quotient map. We need to show that
$\iin{I \mmbox{B[\x]}} \subset \iin{I} \mmbox{B[\x]}$. Given $c \sliver{\bf x}\sliver^E \in \iin{I \mmbox{B[\x]}}$,
choose $f \in I \mmbox{B[\x]}$ so $\iin{f} = c \sliver{\bf x}\sliver^E$. Among all $g \in I$ so
$v(g) = f $, choose one with miminal leading term \iin{g}. We claim
that $v(\iin{g}) = \iin{f}$, so $c \sliver{\bf x}\sliver^E \in \iin{I} \mmbox{B[\x]}$.
Suppose otherwise, that $\iin{g} = b \sliver{\bf x}\sliver^D$ with $\sliver{\bf x}\sliver^D > \sliver{\bf x}\sliver^E$, and
$v(b) = 0$. Then $b \in I \cap A$, so $b \sliver{\bf x}\sliver^D \in I$, and $g - b \sliver{\bf x}\sliver^D
\in I$ has image $f$. This element has a lower leading term than $g$,
contradicting the minimality of our choice for $g$.
\end{proof}
\begin{prop}
\label{initial:genericprop}
Let $I \subset \mmbox{A[\x]}$ be an ideal, let the ideal $q \subset A$ be an
isolated primary component of $I \cap A$, and let the ideal $p \subset
A$ be its associated minimal prime. Define $B = A_p/q_p$. Then
$$\iin{I} \mmbox{B[\x]} = \iin{I \mmbox{B[\x]}}.$$
\end{prop}
\begin{proof}
Let $q \cap \caps q2s$ be a minimal primary decomposition of $I \cap
A$, with associated primes $p$, $p_2, \ldots, p_s$. Then $\caps q2s
\not\subset p$, for otherwise we would have $q_i \subset p$ for some
$i$, and thus $p_i \subset p$, contradicting the minimality of $p$.
Choose an element $r \not\in p$ such that $r \in \caps q2s$. Then $rq
\subset q \cap \caps q2s = I \cap A$, so $r \in (I \cap A:q)$. Thus
$A_p/q_p = A_p/(I \cap A)_p$.
By \lemref{initial:quolem}, taking initial ideals commutes with taking
the quotient by $I \cap A$. By \corref{initial:gtzcor}, taking initial
ideals commutes with forming a ring of fractions. The proposition
follows by combining these results.
\end{proof}
\propref{initial:genericprop} affirms the utility of Gr\"{o}bner\ bases when
\Spec{A} is reducible: Enough information is encoded in such a basis
to determine the corresponding Gr\"{o}bner\ bases over dense open subsets of
each isolated component of \Spec{A/(I \cap A)}.
It is of interest computationally to be able to replace
multiplicatively closed sets by powers of a single element. In the
proof of \propref{initial:genericprop}, we have chosen an element $r$
which vanishes on every primary component of \Spec{A/(I \cap A)}
except the one defined by $q$. By construction, the product of $r$
with any element of $q$ vanishes everywhere. We observe that our
choice of a single element $r$ differs from the construction of single
elements to replace multiplicatively closed sets in \cite{gtz88},
Prop.\ 3.7:
\begin{example}
\label{initial:gtzex}
Let $A = k[a, b]/(a b)$, let $\mmbox{A[\x]} = A[x]$, and let $I =
\mbox{$(a(a-1)x,\, x^2)$}$. If $p$ is chosen to be the minimal prime
$(b) \subset A$, then $a \not\in p$ and $a \in \Ann{(b)}$. If we let
$r = a$, and let $B = A_r/p_r$, we have $$ \iin{I} \mmbox{B[\x]} = \iin{I \mmbox{B[\x]}} =
((a-1)x,\, x^2).$$ On the other hand, $$ \iin{I} \mmbox{k(p)[\x]} = \iin{I \mmbox{k(p)[\x]}} =
(x).$$ Following \cite{gtz88}, if we take $s = a (a-1)$, then $$ I
\mmbox{A_p[\x]} \cap \mmbox{A[\x]} = I A_s[\sliver{\bf x}\sliver] \cap \mmbox{A[\x]} = (x).$$ $r$ cannot replace $s$ in
this role.
\end{example}
In other words, the extension and contraction of an ideal with respect
to a local ring strips away all but generic behavior along the
corresponding prime, while it is possible to specialize a Gr\"{o}bner\ basis
in the sense of \propref{initial:genericprop} and still retain some
information about nongeneric behavior along the primary component.
\propref{initial:genericprop} makes no claims about the relationship
between initial ideals and their specializations to specific primes.
It can happen that no specialization to a prime is well-behaved, as is
illustrated by the following example.
\begin{example}
\label{initial:fuzzex}
We modify example \exref{initial:ex1}. Let $A=k[a,b]/(a^2)$, let
$\mmbox{A[\x]}=A[x,y]$, and let $I = (ax-y)$. For any prime $p \subset A$,
$\iin{I} \mmbox{k(p)[\x]} = (y^2)$, but $\iin{I \mmbox{k(p)[\x]}} = (y)$.
\end{example}
We would like to associate a set of monomials with each fiber of the
family defined by $I$, corresponding over each prime $p$ to the
monomials of \iin{I \mmbox{k(p)[\x]}}, and then assert that \iin{I} encodes enough
information to determine these sets generically, i.e. along a dense
open subscheme of the base. When the family has a nonreduced base, as
in \exref{initial:fuzzex}, what should we do over the fuzz? One feels
in this example that the fiber monomial ideals are generically $(x)$,
i.e. along the open set away from the subscheme cut out by $(a)$.
Alas, this open set is empty. This same phenomenon can be observed in
studying the ``open nature of flatness'', where for a nonreduced base
scheme, the open set along which a family is flat can be empty. One
could think of such open sets as being supported on the fuzz away from
a proper subscheme.
The following proposition characterizes those primes which are certain
to be well behaved with respect to specialization of a given Gr\"{o}bner\
basis.
\begin{prop}
\label{initial:whenprop}
Let $I \subset \mmbox{A[\x]}$ be an ideal, let $p \subset A$ be a prime, and let
$B = A_p/(I \cap A)_p$. If for each monomial $\sliver{\bf x}\sliver^E$, $\iin{I}_E B$ is
either $(0)$ or $(1)$, then $$\iin{I} \mmbox{k(p)[\x]} = \iin{I \mmbox{k(p)[\x]}}.$$
\end{prop}
\begin{proof}
If $I \cap A \not\subset p$, then $B$ is the zero ring, and $\iin{I}
\mmbox{k(p)[\x]} = \iin{I \mmbox{k(p)[\x]}} = (1)$. Otherwise, using \lemref{initial:quolem},
we can reduce to the case where $I \cap A = (0)$, so $B = A_p$. By
\corref{initial:gtzcor}, we know in any case that $$ \iin{I} \mmbox{A_p[\x]} =
\iin{I \mmbox{A_p[\x]}}.$$ Let $J = I \mmbox{A_p[\x]}$; we need to show that $$ \iin{J \mmbox{k(p)[\x]}}
\subset \iin{J} \mmbox{k(p)[\x]}.$$
Given $\sliver{\bf x}\sliver^E \in \iin{J \mmbox{k(p)[\x]}}$, choose $f \in J \mmbox{k(p)[\x]}$ so $\iin{f} =
\sliver{\bf x}\sliver^E$. Among all $g \in J$ with image $f$ in $k(p)[\sliver{\bf x}\sliver]$, choose one
with minimal leading term \iin{g}. We claim that $\iin{g} = (1+c)\sliver{\bf x}\sliver^E$
with $c \in p$, so $\sliver{\bf x}\sliver^E \in \iin{J}\, k(p)[\sliver{\bf x}\sliver]$.
Suppose otherwise, that $\iin{g} = c\sliver{\bf x}\sliver^D$ with $\sliver{\bf x}\sliver^D > \sliver{\bf x}\sliver^E$. Then $c
\in p$, and $\iin{I}_D A_p = (1)$. Choose $h \in J$ so $\iin{h} =
\sliver{\bf x}\sliver^D$. Then $g - ch$ also has image $f$ in $k(p)[\sliver{\bf x}\sliver]$, and has a lower
leading term than $g$, contradicting the minimality of our choice for
$g$.
\end{proof}
Let $X \subset \Spec{A}$ be the support of the family defined by $I$;
$X$ is cut out by $I \cap A$. Geometrically, the criterion of
\propref{initial:whenprop} is satisfied if the zero locus of each
coefficient ideal of \iin{I} either avoids the point $p$, or contains
an open neighborhood of $p$ in $X$.
This criterion is sufficient, but not necessary, for taking initial
ideals to commute with specialization. For example, if $I$ is a
monomial ideal, it can have arbitrary coefficient ideals, yet $\iin{I
\mmbox{k(p)[\x]}} = \iin{I} \mmbox{k(p)[\x]}$ for all primes $p$.
\section{Faithful Flatness}
\label{flat}
\setcounter{defn0}{0}
The criterion of \propref{initial:whenprop} gives a sufficient
condition for $\mmbox{A[\x]}/I$ to be faithfully flat over $A/(I \cap A)$. We
will apply the following criterion for faithful flatness; see
Matsumura \cite{mat86}, Atiyah and MacDonald \cite{atma69}, or
Bourbaki \cite{bou89}, for full expositions.
\begin{lemma}
\label{flat:lemma}
Let $M$ be an $A$-module. If for each prime $p \subset A$, $M_p$ is a
nontrivial, free $A_p$-module, then $M$ is faithfully flat over $A$.
\mbox{~~~~\vrule height 1.2ex width .9ex depth .1ex}
\end{lemma}
The following lemma will be used in two different proofs.
\begin{lemma}
\label{flat:nmlem}
Let $I \subset \mmbox{A[\x]}$ be an ideal, and define $M = \mmbox{A[\x]}/I$. Let $V = \{
\sliver{\bf x}\sliver^E \mid \iin{I}_E \neq (1) \}$, and let $N \subset M$ be the
$A$-submodule generated by $V$. Then $N = M$.
\end{lemma}
\begin{proof}
Suppose that $N \neq M$, and choose a nonzero element $f \in M/N$.
Among all $g \in \mmbox{A[\x]}$ with image $f$ in $M/N$, choose one with minimal
leading term \iin{g}. Let $\iin{g} = c\sliver{\bf x}\sliver^F$. If $\sliver{\bf x}\sliver^F \in V$, then
$c\sliver{\bf x}\sliver^F \in N$, so $g - c\sliver{\bf x}\sliver^F$ also represents $f$, contradicting the
minimality of our choice for $g$. On the other hand, if $\sliver{\bf x}\sliver^F \not\in
V$, then $\iin{I}_F = (1)$, so $\iin{h} = \sliver{\bf x}\sliver^F$ for some $h \in I$.
Then $g - ch$ also represents $f$, again contradicting the minimality
of our choice for $g$.
\end{proof}
\begin{prop}
\label{flat:prop1}
Let $I \subset \mmbox{A[\x]}$ be a proper ideal, and define $M = \mmbox{A[\x]}/I$. If for
each prime $p \subset A$ and for each monomial $\sliver{\bf x}\sliver^E$, $\iin{I}_E B$
is either $(0)$ or $(1)$ where $B = A_p/(I \cap A)_p$, then $M$ is a
faithfully flat $A/(I \cap A)$-module.
\end{prop}
\begin{proof}
Disregard primes $p \not\supset I \cap A$. Using
\lemref{initial:quolem}, we can reduce to the case where $I \cap A =
(0)$. We want to show that $M$ is a faithfully flat $A$-module.
Given a prime $p \subset A$, let $V = \{ \sliver{\bf x}\sliver^E \mid \iin{I}_E A_p = (0)
\}$, and let $N \subset M_p$ be the $A_p$-submodule generated by $V$.
$N = M_p$ by \lemref{flat:nmlem}; we claim that $N$ is nontrivial and
free. The result then follows from \lemref{flat:lemma}.
$N$ is nontrivial because $I \cap A = (0)$, so $\iin{I}_1 = (0)$, and
$1 \in V$. $N$ is free, because any relation among its generators
would be an element of $I \mmbox{A_p[\x]}$, whose lead term belongs to $V$. Since
$\iin{I \mmbox{A_p[\x]}}_E = \iin{I}_E A_p$ by \corref{initial:gtzcor}, this
would contradict the definition of $V$.
\end{proof}
\begin{corollary}
\label{flat:cor}
Given a Gr\"{o}bner\ basis $\{ f_1,\, \ldots,\, f_r \}$ for $I \subset \mmbox{A[\x]}$,
let $T$ denote the finite set of exponents $E$ for which $\iin{I}_E
\neq I \cap A$, and which occur as the exponent of some \iin{f_i}. If
$$s \in \bigcap_{E \in T}\; \{ \sqrt{(J^2:J)} \mid J = \iin{I}_E \}$$
and $s \not\in \sqrt{I \cap A}$, then $M_s$ is faithfully flat over
$A_s/(I \cap A)_s$.
\end{corollary}
\begin{proof}
We have the equality of sets
$$ \{ \iin{I}_E \mid \iin{I}_E \neq I \cap A \} \; =
\{ \sum_i \iin{I}_{E_i} \mid E_i \in T \; \mbox{for each} \; i \}. $$
Thus, we get the same intersection of ideals if we replace the index
set $T$ by the infinite set of exponents
$$\{ E \mid \iin{I}_E \neq I\cap A\}.$$
Reduce to the case where $I \cap A = (0)$. The second condition, that
$s$ is not nilpotent, insures that $A_s$ is not the zero ring.
For each $J = \iin{I}_E$, $(J^2:J)$ is supported on precisely those
primes $p$ so $J A_p$ is neither $(0)$ nor $(1)$: $$(J^2:J) A_p = (1)
\Leftrightarrow J A_p = J^2 A_p \Leftrightarrow J A_p = (0)
\mbox{\,or\,} (1).$$
\end{proof}
If the scheme defined by $I \cap A$ has any reduced components, then
such $s$ exist. \corref{flat:cor} remains true for $$s \in \bigcap_{E
\in T}\; \{ \sqrt{J} \mid J = \iin{I}_E \},$$ but nontrivial such $s$
need not exist when $I \cap A$ has more than one reduced component, as
is illustrated by the following example.
\begin{example}
\label{flat:redex}
Let $A = k[a,b,c,d]/((a,b) \cap (c,d))$, let $\mmbox{A[\x]} = A[x,y]$, let $I =
(ax+b, cy+d)$, and let $M = \mmbox{A[\x]}/I$. $\{ax+b, cy+d\}$ is a Gr\"{o}bner\ basis
for $I$, and we have $\iin{I}_x = (a)$, $\iin{I}_y = (c)$, $(a^2:a) =
(a,c,d)$, and $(c^2:c) = (a,b,c)$. For any $s \in (a,c,d) \cap (a,b,c)
= (a,c)$, and any prime $p \subset A$ such that $s \not\in p$,
$\iin{I}_x A_p$ and $\iin{I}_y A_p$ are each either $(0)$ or $(1)$.
Thus $M_s$ is faithfully flat over $A_s$ for any such $s$.
\Spec{A} consists of the union of the two planes $a = b = 0$ and $c =
d = 0$, meeting at a common origin. $\iin{I}_x = (a)$ vanishes
identically on the plane $a = b = 0$, and is nonzero away from the
line $a = 0$ on the plane $c = d = 0$. Thus $(a^2:a)$ is supported on
this line, which is the locus where $\iin{I}_x$ is locally neither
$(0)$ nor $(1)$. Analogous statements hold for $\iin{I}_y$.
In this example, $$\sqrt{\iin{I}_x} \cap \sqrt{\iin{I}_y} = (a) \cap
(c) = (0),$$ so we cannot simplify the criterion of \corref{flat:cor}.
\end{example}
Compare \propref{flat:prop1} with \propref{initial:whenprop}, and
\exref{initial:ex1}. Summarizing, if the set of monomials of $J =
\iin{I} \mmbox{k(p)[\x]}$ is locally constant as a function of $p$, then Gr\"{o}bner\
bases are well behaved with respect to specialization, and moreover,
$M$ is a faithfully flat $A$-module. However, away from the locus
where $J$ is locally constant, $M$ can remain faithfully flat over
$A$.
The next proposition gives some geometric consequences of faithful
flatness. Recall that a morphism of schemes $f: X \rightarrow Y$ is
said to be surjective if for every point $p \in Y$, there exists a
point $P \in X$ such that $f(P) = p$. A morphism of schemes is said to
be dominant if for every point $P \in X$, the induced map $f_P^\#:
{\cal O}_{Y,f(P)} \rightarrow {\cal O}_{X,P}$ is injective. While the first
condition is purely topological, the second condition considers the
effect of $f$ on the sheaves of rings ${\cal O}_X$, ${\cal O}_Y$, and does not
imply the first.
\begin{prop}
\label{flat:prop2}
Let $I \subset \mmbox{A[\x]}$ be an ideal, and let $X = \Spec{\mmbox{A[\x]}/I}$, $Y =
\Spec{A}$. If $M = \mmbox{A[\x]}/I$ is a faithfully flat $A$-module, then the
corresponding morphism of schemes $X \rightarrow Y$ is surjective and
dominant, and maps each primary component of $X$ to a primary
component of $Y$.
\end{prop}
\begin{proof}
Once a map is known to be flat, the surjectivity of $X \rightarrow Y$
is an equivalent definition of faithful flatness (\cite{mat86}, Thm.\
7.3; \cite{atma69}, Ch.\ 3., Ex.\ 16). Given any prime $P \subset M$,
let $p = P \cap A$. The local homomorphism $A_p \rightarrow M_P$ is
faithfully flat, and thus injective, because $M$ is flat over $A$
(\cite{atma69}, Ch.\ 3., Ex.\ 18). Thus $X \rightarrow Y$ is dominant.
It remains to prove that if $P$ is an associated prime of $(0)$ in
$M$, then $p = P \cap A$ is an associated prime of $(0)$ in $A$. This
follows from \cite{bou89}, Ch.\ 4, \S 2.6, Cor.\ 1 to Thm.\ 2.
\end{proof}
Combining \propref{flat:prop1} with \propref{flat:prop2}, Gr\"{o}bner\ bases
can be used to manipulate the component structure of the total space
of a family of schemes. In \cite{gtz88}, this problem is approached
differently, via rings of fractions. For example, in Cor.\ 3.8 of
\cite{gtz88}, for $A$ an integral domain with quotient field $K$, the
coefficients of \iin{I} are used to find an $s \in A$ so $I\, A_s[\sliver{\bf x}\sliver]
\cap \mmbox{A[\x]}$ computes $I\, K[\sliver{\bf x}\sliver] \cap \mmbox{A[\x]}$; the resulting ideal is then
the intersection of the components of $I$ which surject onto the base
\Spec{A}. In this setting, our choice of $s$ in \corref{flat:cor}
represents a modest improvement over their choice, and
\propref{flat:prop2} illuminates the connection between these
approaches.
When $M$ is finitely generated as an $A$-module, one could determine
the point set in \Spec{A} over which faithful flatness fails, by
studying a presentation matrix of $M$ as an $A$-module; the maximal
minors of this matrix generate one of the {\it Fitting ideals} of $M$.
The use of Gr\"{o}bner\ bases is more efficient than a brute force study of
these minors, as is evidenced by the following example:
\begin{example}
\label{flat:fitex}
This example is a modification of \exref{initial:ex1}. Let $A=k[a]$,
let $\mmbox{A[\x]}=A[x,y]$, and let $I = \mbox{$(ax+y,$}\, x^3,\, x^2 y,\, x
y^2,\, y^3) \subset \mmbox{A[\x]}$. The coefficient ideals of \iin{I} are given
by the following diagram:
\begin{center}
\begin{tabular}{r|llll}
$y$ & $(1)$ \\
& $(0)$ & $(1)$ \\
& $(0)$ & $(a)$ & $(1)$ \\
& $(0)$ & $(a)$ & $(a)$ & $(1)$ \\
\cline{2-5}
\multicolumn{5}{r}{$x$}
\end{tabular}
\end{center}
As an $A$-module, $M = \mmbox{A[\x]}/I$ is finitely generated by the set of
monomials having nonunit coefficient ideals, $\{x^2,\, x y,\, y^2,\,
x,\, y,\, 1\}$. These generators have as relations the multiples
$ax+y$,\, $x(ax+y)$, and $y(ax+y)$ of the ideal generator $ax+y$. We
organize this data into the following presentation matrix for $M$:
\begin{center}
\begin{tabular}{rl|ccccccl|}
\multicolumn{2}{l}{} &
$x^2$ & $x y$ & $y^2$ & $x$ & $y$ & $1$ &
\multicolumn{1}{l}{} \\
\cline{3-9}
$a x+y$ & & $0$ & $0$ & $0$ & $a$ & $1$ & $0$ & \\
$x(a x+y)$ & & $a$ & $1$ & $0$ & $0$ & $0$ & $0$ & \\
$y(a x+y)$ & & $0$ & $a$ & $1$ & $0$ & $0$ & $0$ & \\
\cline{3-9}
\end{tabular}
\end{center}
The $(x y,\, y^2,\, y)$-minor of this matrix is nonsingular, so $M$ is
flat over $A$. However, the leading nonzero minor, on columns $(x^2,\,
xy,\, x)$, has determinant $a^3$. This minor demonstrates that $M_a$
is flat over $A_a$, but leaves open the question of what happens over
$a = 0$.
The coefficient ideals of \iin{I} determine a locus away from which
this leading minor is nonsingular. Since one only needs to consider
coefficient ideals corresponding to minimal generators of \iin{I},
Gr\"{o}bner\ bases can be used to find this locus without explicitly
considering every row of the presentation matrix: a single element of
the Gr\"{o}bner\ basis stands in for many rows of the presentation matrix.
Note also that the coefficient ideals and the determinant give
different scheme structures for the set where this leading minor loses
rank. The coefficient ideals describe the support of the module
defined by the leading minor, while the determinant describes a
thicker scheme enjoying a universal property with respect to base
change; see \cite{eis89}, Ch.\ 10.
\end{example}
\section{Fibers}
\label{fibers}
\setcounter{defn0}{0}
The following pair of propositions concern the behavior of coefficient
ideals with respect to the geometry of the fibers of a family.
\begin{prop}
\label{fibers:isoprop}
Let $I \subset \mmbox{A[\x]}$ be an ideal, let $M = \mmbox{A[\x]}/I$, and let $p \subset
A$ be a prime ideal. The following two statements are equivalent:
(a) $f: A_p \rightarrow M_p$ is surjective;
(b) $\iin{I}_{x_i} A_p = (1)$ for each $i$.
\end{prop}
\begin{proof}
If $\iin{I}_{x_i} A_p = (1)$ for each $i$, then $M_p$ admits a
relation of the form $x_i - f_i(x_{i+1},\ldots,x_n)$, for each $i$.
This proves that $f$ is surjective. Conversely, if $f$ is surjective,
then $M_p$ admits a relation of the form $x_i - c_i$ with $c_i \in
A_p$, for each $i$. Thus, the corresponding coefficient ideals are
unit ideals.
\end{proof}
\begin{corollary}
\label{fibers:cor}
Let $I \subset \mmbox{A[\x]}$ be an ideal, let $X = \Spec{A/(I \cap A)}$, and
let $Y = \Spec{\mmbox{A[\x]}/I}$. If $I \cap A$ is a prime ideal, and each
coefficient ideal $\iin{I}_{x_i} \neq I \cap A$, then the induced
morphism of schemes $Y \rightarrow X$ is an isomorphism over a
nonempty open subset of $X$.
\mbox{~~~~\vrule height 1.2ex width .9ex depth .1ex}
\end{corollary}
Let $\iin{I}_{x_i^\infty}$ denote the stationary limit of the
ascending chain of coefficient ideals $\iin{I}_{x_i} \subset
\iin{I}_{x_i^2} \subset \ldots\,$. Each $\iin{I}_{x_i^\infty}$ can be
computed as the ideal generated by the leading coefficients of all
Gr\"{o}bner\ basis elements having a leading monomial of the form $x_i^e$.
\begin{prop}
\label{fibers:finprop}
Let $I \subset \mmbox{A[\x]}$ be an ideal, let $M = \mmbox{A[\x]}/I$, and let $p \in A$ be
a prime ideal. The following two statements are equivalent:
(a) $f: A_p \rightarrow M_p$ is a finite map;
(b) $\iin{I}_{x_i^\infty} A_p = (1)$ for each $i$.
\end{prop}
\begin{proof}
Suppose that (a) holds, so $M_p$ is a finite $A_p$-module. Fix $i$,
and let $N \subset M_p$ be the subalgebra generated by $x_i$. Then $N$
can be generated as an $A_p$-module by the finite set $\{ 1, x_i,
\ldots ,x_i^{e-1}\}$ for some $e$. $x_i^e$ can be expressed in terms
of these generators, yielding an expression in $I \mmbox{A_p[\x]}$ with leading
term $x_i^e$. Thus $\iin{I \mmbox{A_p[\x]}}_{x_i^e} = (1)$.
Conversely, assume (b). For each $i$, choose $e_i$ so $\iin{I
\mmbox{A_p[\x]}}_{x_i^e} = (1)$, and let $V$ denote the finite set of monomials
which do not belong to the ideal $(x_1^{e_1},\ldots,x_n^{e_n})$. If $N
\subset M_p$ is the $A_p$-submodule generated by $V$, then $N$ is
finitely generated, and $N = M_p$ by \lemref{flat:nmlem}.
\end{proof}
Geometrically, let $X = \Spec{\mmbox{A[\x]}/I}$, let $Y = \Spec{A/(I \cap A)}$,
and let $g: X \rightarrow Y$ be the morphism of schemes induced by the
inclusion $A/(I \cap A) \subset \mmbox{A[\x]}/I$. If we let $U \subset Y$ be the
complement of the union of the subschemes of $Y$ cut out by the
coefficient ideals $\iin{I}_{x_i}$, then \propref{fibers:isoprop}
asserts that $U$ is the largest open set with the property that
$g^{-1}(U) \rightarrow U$ is an isomorphism. If instead we let $U
\subset Y$ be the complement of the union of the subschemes of $Y$ cut
out by the coefficient ideals $\iin{I}_{x_i^\infty}$, then
\propref{fibers:finprop} asserts that $U$ is the largest open set with
the property that $g^{-1}(U) \rightarrow U$ is a finite morphism.
Thus, while Gr\"{o}bner\ bases can only generically detect faithful flatness,
they are capable of detecting precisely the locus where a morphism
restricts to an isomorphism, or to a finite map.
Detection of quasi-finite morphisms is more subtle; a family which
restricts to a finite family over an open subset of the base need not
be quasi-finite:
\begin{example}
\label{fibers:ex1}
Let $A=k[a, b]$, let $\mmbox{A[\x]}=A[x]$, let $I = (a x - b)$, and let $M =
\mmbox{A[\x]}/I$. The localization $M_a$ is finite over $A_a$. However, the
total space $a x - b = 0$ is irreducible, and consists of a line over
$a = b = 0$.
\end{example}
|
1992-02-18T04:03:26 | 9202 | alg-geom/9202016 | en | https://arxiv.org/abs/alg-geom/9202016 | [
"alg-geom",
"math.AG"
] | alg-geom/9202016 | Temporary | G.Mikhalkin | Congruences for real algebraic curves on an ellipsoid | LaTeX, 10 pages | null | null | null | null | The problem of arrangement of a real algebraic curve on a real algebraic
surface is related to the 16th Hilbert problem. We prove in this paper new
restrictions on arrangement of nonsingular real algebraic curves on an
ellipsoid. These restrictions are analogues of Gudkov-Rokhlin,
Gudkov-Krakhnov-Kharlamov, Kharlamov-Marin congruences for plane curves (see
e.g. \cite{V} or \cite{V1}). To prove our results we follow Marin approach
\cite{Marin} that is a study of the quotient space of a surface under the
complex conjugation. Note that the Rokhlin approach \cite{R} that is a study of
the 2-sheeted covering of the surface branched along the curve can not be
directly applied for a proof of Theorem \ref{Rokhlin} since the homology class
of a curve of Theorem \ref{Rokhlin} can not be divided by 2 hence such a
covering space does not exist.
| [
{
"version": "v1",
"created": "Tue, 18 Feb 1992 03:06:11 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Mikhalkin",
"G.",
""
]
] | alg-geom | \section{Formulations of results}
It is well-known that a complex quadric is isomorphic to ${\bf C} P^1\times
{\bf C} P^1$ and an algebraic curve on a quadric is defined by a bihomogeneous
polynomial of bidegree $(d,r)$.
If the curve is real and the quadric is an ellipsoid then $d=r$
and the curve can be represented as the intersection of an ellipsoid
and a surface of degree $d$ in ${\bf R} P^3$ (this is because a curve
of bidegree $(d,r), d\neq r$ can not be invariant under the involution
of the complex conjugation of ellipsoid).
Let $Q$ be an ellipsoid, ${\bf R} Q$ and ${\bf C} Q$ be the spaces of its real and
complex points; $A$ be a nonsingular real algebraic curve of bidegree
$(d,d)$ on $Q$; ${\bf R} A$ and ${\bf C} A$ be the spaces of real and complex points
of $A$.
Components of ${\bf R} A$ are called ovals and the number of ovals of ${\bf R} A$
is denoted by $l$.
${\bf R} A$ divides ${\bf R} Q$ into two parts with a common boundary.
Let $B_0$ and $B_1$ denote these parts in such a way that in the case
when $l$ is even congruence $\chi(B_0)\equiv 0\pmod{4}$ is correct
(as it is usual, $\chi$ denotes Euler characteristic).
According to F.Klein, $A$ is a curve of type I (II) if ${\bf C} A \setminus {\bf R} A$
is not connected (otherwise).
It is well-known (see \cite{R1}) that if $A$ is a curve of type I then
${\bf R} A$ has two natural opposite orientations.
We need the following definitions to formulate Theorem \ref{Fiedler}.
Let us choose one of two complex orientation of ${\bf R} A$ and
some orientation of $B_0$.
Oval $C$ of ${\bf R} A$ is called disorienting if these orientations induce
opposite orientations on it.
$C$ divides ${\bf R} Q$ into two disks $D$ and $D'$.
Let $x(D)$ be equal to $\chi(B_1\cap D)\bmod{2}$.
It is clear that if $l\equiv 0\pmod{2}$ then $x(D)=x(D')$;
in this case we set $x(C)$ to be equal to $x(D)$.
The main results of this paper are the following.
\newtheorem{th}{Theorem}
\begin{th}
\label{Rokhlin}
Let $d$ be an odd number.
\begin{itemize}
\begin{description}
\item[a)] If $A$ is an M-curve (i.e. $l=(d-1)^2+1$) then
\begin{displaymath}
\chi(B_0)\equiv\chi(B_1)\equiv\frac{d^2+1}{2}\pmod{8}
\end{displaymath}
\item[b)] If $A$ is an (M-1)-curve (i.e. $l=(d-1)^2$) then
\begin{displaymath}
\chi(B_0)\equiv\frac{d^2-1}{2}\pmod{8}
\end{displaymath}
\begin{displaymath}
\chi(B_1)\equiv\frac{d^2+3}{2}\pmod{8}
\end{displaymath}
\item[c)] If $A$ is an (M-2)-curve (i.e. $l=(d-1)^2-1$) and
\begin{displaymath}
\chi(B_0)\equiv\frac{d^2-7}{2}\pmod{8}
\end{displaymath}
then $A$ is of type I.
\item[d)] If $A$ is of type I then
\begin{displaymath}
\chi(B_0)\equiv\chi(B_1)\equiv1\pmod{4}
\end{displaymath}
\end{description}
\end{itemize}
\end{th}
\begin{th}
\label{Fiedler}
Let $d$ be an even number.
\begin{itemize}
\begin{description}
\item[a)] \label{af} If $A$ is an M-curve and if all components of $B_1$ have
even Euler
characteristics then
\begin{displaymath}
\chi(B_0)\equiv d^2\pmod{16}
\end{displaymath}
\begin{displaymath}
\chi(B_1)\equiv 2-d^2\pmod{16}
\end{displaymath}
\item[b)] \label{bf} Let $A$ be a curve of type I with chosen complex
orientation.
If there exist an orientation of $B_0$ such that $x(C)=0$ for every
disorienting oval $C$ then
\begin{displaymath}
\chi(B_0)\equiv d^2\pmod{8}
\end{displaymath}
\begin{displaymath}
\chi(B_1)\equiv 2-d^2\pmod{8}
\end{displaymath}
\end{description}
\end{itemize}
\end{th}
{\bf\noindent Remark:} Propositions d) of Theorem \ref{Rokhlin}
and b) of Theorem \ref{Fiedler} can be deduced from the formula of complex
orientations of Zvonilov \cite{Z}.
\section{Proof of Theorem \ref{Rokhlin}}
Let $W$ denote $B_0\cup{\bf C} A/conj$.
As it was shown in \cite{L} a manifold ${\bf C} Q/conj$ where $conj$
is the involution of complex conjugation is diffeomorphic to
${\overline{{\bf C} P^2}}$
It is easy to check that
\begin{equation}
\label{WW}
W\circ W=d^2-2\chi(B_0)\equiv 1\pmod{2}
\end{equation}
Hence, $W$ is a characteristic surface in ${\bf C} Q/conj$.
Let us apply Guillou-Marin congruence \cite{GM}
\begin{displaymath}
\sigma({\bf C} Q/conj)\equiv W\circ W+2\beta(q)\pmod{16}
\end{displaymath}
where $\sigma$ is signature, $\beta(q)$ is Brown invariant of quadratic
Guillou-Marin form \newline $q: H_1(W;{\bf Z}_2)\rightarrow {\bf Z}_4$ of surface $W$
in ${\bf C} Q/conj$.
This congruence, equality \ref{WW} and equality
$\sigma({\overline{{\bf C} P^2}})=-1$ follow that
\begin{displaymath}
\chi(B_0)\equiv\frac{d^2+1}{2}+\beta(q)\pmod{8}.
\end{displaymath}
The calculations of $\beta(q)$ below are similar to the calculations
in \cite{Marin}, \cite{KV}.
Let $L$ denote the subspace of $H_1(W;{\bf Z}_2)$ generated by classes realized
by ovals of ${\bf R} A$.
Let $U=L^{\perp}$ denote the orthogonal complement of $L$
with respect to the intersection form of $W$.
It is clear that $U^{\perp}=L\subset U$ and $q|_L=0$.
Therefore, $U$ is an informative subspace and $\beta(q)=\beta(q')$,
where $q'$ is the form on $U/U^{\perp}$ induced by $q$ (see \cite{KV}).
According to \cite{KV} one can get the following.
\begin{itemize}
\begin{description}
\item[a)] $dim(U/U^{\perp})=0$, hence $\beta(q')\equiv 0\pmod{8}$
\item[b)] $dim(U/U^{\perp})=1$, hence $\beta(q')\equiv\pm 1\pmod{8}$
\item[c)] $dim(U/U^{\perp})=2$, $\beta(q')\equiv 4\pmod{8}$, hence $q'$ is
even,
therefore $A$ is of type I
\item[d)] $q'$ is even, hence $\beta(q')\equiv 0\pmod{4}$.
\end{description}
\end{itemize}
\section{Lemma for Theorem 2}
Let $A_+$ be the closure of one of two components of ${\bf C} A\setminus{\bf R} A$.
Let $W_0$ denote $A_+\cup B_0$ and let $W_1$ denote $A_+\cup B_1$.
\newtheorem{lemma}{Lemma}
\begin{lemma}\label{ne0} Surface $W_1$ is not homologous to zero modulo 2 in
${\bf C} Q$
\end{lemma}
{\em{\underline{Proof}}} If $W_1$ is homologous to zero then $W_1$ is a
characteristic surface in ${\bf C} Q$ and we can apply Guillou-Marin congruence
\begin{displaymath}
0\equiv W_1\circ W_1+2\beta(q_1)\pmod{16}
\end{displaymath}
where $q_1$ is the Guillou-Marin form of surface $W_1$ in ${\bf C} Q$.
But $W_1\circ W_1=d^2-\chi(B_1)$ and $\beta(q_1)\equiv 0\pmod{2}$
since $dim H_1(W_1;{\bf Z}_2)\equiv \chi(W_1)\equiv 0\pmod{2}$,
note that $\chi(W_1)$ is even, because $W_1$ is the result of gluing
of two orientable surfaces along whole boundary.
Congruences above and $d^2\equiv 0\pmod{4}$ imply that $\chi(B_1)\equiv 0
\pmod{4}$ follow that $\chi(B_1)\equiv 0\pmod{4}$.
This is a contradiction to the choice of $B_1$.
\section{Proof of Theorem 2}
Let $e_1$ and $e_2$ denote the elements of $H_2({\bf C} Q)$ realized by generating
lines of quadric ${\bf C} Q$.
It is clear that $e_1$ and $e_2$ form a basis of $H_2({\bf C} Q)$,
$conj_*e_1=-e_2$, $conj_*e_2=-e_1$.
Let $(\alpha,\beta)$ denote $\alpha e_1+\beta e_2, \alpha,\beta \in {\bf Z}$.
Each of generating lines transversally intersects ${\bf R} Q$ at one point,
hence $[{\bf R} Q]\equiv (1,1)\pmod{2}$.
{}From relations $[W_j]-conj_*[W_j]\equiv [{\bf C} A]\equiv (d,d)\pmod{2},j=0,1$,
$[W_0]+[W_1]\equiv [{\bf R} Q]\equiv (1,1)\pmod{2}$ and Lemma \ref{ne0}
one can deduce that $[W_0]\equiv 0\pmod{2}$.
Therefore $W_0$ is a characteristic surface in ${\bf C} Q$ and there is
Guillou-Marin form $q_0$ on $H_1(W_0;{\bf Z}_2)$.
Value of $q_0$ on the element of $H_1(W_0;{\bf Z})$ realized by oval $C$
is equal to $x(C)$ (for the calculations see \cite{F}).
Let $L_0$ denote the subspace of $H_1(W_0;{\bf Z}_2)$ generated by such ovals C
that $x(C)=0$.
In the case \ref{af} we have $L_0=L_0^{\perp}$ therefore $\beta(q_0)\equiv 0
\pmod{8}$.
In the case \ref{bf} we have that form $q_0':L_0^{\perp}\rightarrow {\bf Z}_4$
induced by $q_0$ is even therefore $\beta(q_0)\equiv 0\pmod{4}$.
Now theorem 2 follows from Guillou-Marin congruence \cite{GM}
\begin{displaymath}
0\equiv d^2-\chi(B_0)+2\beta(q_0)\pmod{16}
\end{displaymath}
\section{Applications for the curves of bidegrees (3,3) and (5,5)}
Theorem \ref{Rokhlin} and Harnack inequality give a complete system
of restrictions for real schemes of flexible curves of bidegree (3,3)
(a definition of flexible curves on an ellipsoid is similar to the definition
given in \cite{V} for plane curves).
All real schemes realizable by flexible curves of bidegree (3,3) can be
realized by algebraic curves of bidegree (3,3)
(the classification of real schemes of algebraic curves of bidegree (3,3)
is $<$$\alpha$$>$, $\alpha\leq 5$ and $1\sqcup 1$$<$$1$$>$ -- see e.g.
\cite{Z}).
Real schemes of M-curves of bidegree (5,5) allowed by Theorem \ref{Rokhlin}
are $\alpha\sqcup 1$$<$$\beta$$>$$, \alpha+\beta=16,\beta\equiv 2\pmod{4}$ and
$\alpha\sqcup 1$$<$$\beta$$>$$\sqcup 1$$<$$\gamma$$>$$,\alpha+\beta+\gamma=15,
\alpha\equiv 1\pmod{4}$.
These schemes except might be $1\sqcup 1$$<$$6$$>$$\sqcup 1$$<$$8$$>$ and
$1\sqcup 1$$<$$5$$>$$\sqcup 1$$<$$9$$>$ can be constructed by smoothing
singularities of
images under birational transformations of appropriate plane M-curves
of degree 5 intersecting a real line at 5 different real points and
M-curves of degree 6 transversally intersecting a real line at 4 different real
points.
\section{An absence of congruences similar to Theorem 1
for curves of even bidegrees}
There is no congruence modulo 4 for each halves $B_1, B_2$ of a complement
of M-curve of even bidegree (because in this case $l\equiv 0\pmod{2}$,
hence $\chi(B_0)\equiv 0\pmod{2}$ and $\chi(B_0)\equiv\chi(B_1)+2\pmod{4}$).
A classification of the real schemes of curves of bidegree (4,4) shows
an absence of nontrivial congruences similar to Theorem 1 :
all schemes allowed by Harnack inequality and Bezout theorem are
realizable by algebraic curves - $<$$\alpha$$>$,$\alpha\leq 10$ and
$\alpha\sqcup 1$$<$$\beta$$>$, $\alpha+\beta\leq 9$
(see e.g.\cite{Z}).
Author is indebted to O.Ya.Viro, V.I.Zvonilov and V.M.Kharlamov for their
attention to the paper.
|
2000-03-24T16:04:43 | 9202 | alg-geom/9202023 | en | https://arxiv.org/abs/alg-geom/9202023 | [
"alg-geom",
"math.AG"
] | alg-geom/9202023 | Richard Hain | Johan Dupont, Richard Hain and Steven Zucker | Regulators and characteristic classes of flat bundles | 45 pages, Plain TeX. This will appear in "The Arithmetic and Geometry
of Algebraic Cycles", Proc. of CRM Summer School, Banff, AMS, 2000 | null | null | null | null | This is a substantial revision of the older version of this paper. The main
result of the old version (the equality, up to a factor of 2 of the Beilinson
and Borel regulators) is now a conjecture. The main results give equality of
Beilinson chern classes and Cheeger-Simons-Chern classes in various situations
such as for flat bundles over quasi projective varieties. We also prove the
equality (up to a factor of two) of the Borel regulator element and the
universal Cheeger-Simons Chern class.
| [
{
"version": "v1",
"created": "Thu, 20 Feb 1992 22:15:56 GMT"
},
{
"version": "v2",
"created": "Fri, 24 Mar 2000 15:04:43 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Dupont",
"Johan",
""
],
[
"Hain",
"Richard",
""
],
[
"Zucker",
"Steven",
""
]
] | alg-geom | \section{Introduction}
For each complex algebraic variety $X$, Deligne and Beilinson \cite{Be}
have defined cohomology groups
$$
H_{\mathcal D}^k (X, {\mathbb Z} (p)) \quad k, p \in {\mathbb N}
$$
which map to the usual singular cohomology groups:
\begin{equation}\label{1.1}
H_{\mathcal D}^k (X, {\mathbb Z} (p)) \to H^k (X, {\mathbb Z}(p)).
\end{equation}
(Here, ${\mathbb Z} (p)$ denotes the subgroup $(2\pi i)^p {\mathbb Z}$
of ${\mathbb C}$.) Beilinson \cite{Be} and Gillet \cite{G} have constructed
Chern classes for algebraic vector bundles $E \to X$
$$
c_p^B (E) \in H_{\mathcal D}^{2p} (X, {\mathbb Z} (p))
$$
whose images under (\ref{1.1}) are the usual Chern classes.
In a different
direction, Cheeger and Simons \cite{CS} have defined characteristic
classes
$$
\widehat{c}_p (E, \nabla) \in H^{2p-1} (M, {\mathbb C} /{\mathbb Z} (p))
$$
for vector bundles $E \to M$ endowed with a flat connection
$\nabla$ over a smooth manifold.
Flat complex vector bundles over an algebraic manifold $X$ can be made
algebraic;
we always want to take, as algebraic structure on the bundle, the unique one
with respect to
which the connection has regular singularities \cite[p.~98]{De1}.\footnote{If
$E$ is given
a priori as an algebraic bundle on $X$ with flat connection, it is possible
that the
two algebraic structures fail to coincide. See the paragraphs following
\ref{4.2.4}.}
We can then associate to such bundles the pair of characteristic classes
$$
\widehat{c}_p(E, \nabla) \in H^{2p-1} (X, {\mathbb C} /{\mathbb Z} (p)) \text{ and }
c_p^B \in H_{\mathcal D}^{2p} (X, {\mathbb Z} (p)).
$$
There is a natural homomorphism
\begin{equation}\label{1.2}
H^{2p-1} (X, {\mathbb C} /{\mathbb Z} (p)) \to H_{\mathcal D}^{2p} (X, {\mathbb Z}(p)).
\end{equation}
Our first result is:
\begin{bigtheorem}
\label{thm1}
If $E \to X$ is a flat algebraic vector bundle with regular singularities over
the algebraic
manifold $X$, then $c_p^B (E)$ is the image of $\widehat{c}_p(E,\nabla)$ under
(\ref{1.2}) for all $p \geq 1$.
\end{bigtheorem}
\noindent Previously known cases of this result are due to Bloch \cite{B}
(flat bundles
with unitary monodromy over a smooth projective base) and Soul\'e \cite{S}
(arbitrary
flat bundles over a smooth projective base). In \cite{Br}, there is a proof
of the
quasi-projective case of Theorem \ref{thm2} below that is similar to ours (in
\ref{6.2}) and
which invokes our Proposition \ref{6.1.4}.
Since both characteristic classes are functorial, the proof of
Theorem~1 would be rather straightforward if, given $E \to X$ as in the
theorem,
one could find a morphism of $X$ into some Grassmannian
$f : X \to G (n, {\mathbb C}^m)$ and a connection $\nabla$ on the universal
$n$-plane bundle over $G(n,{\mathbb C}^m)$ that classified {\it both} the bundle
$E$ and its flat connection. However, it is easy to see that this is
impossible in general; the context of flat vector bundles is just too rigid.
For example, if $X$ were compact and $E$ had
nontrivial monodromy, then $f$ could not be constant, implying that
$c_1 (E)$ is non-zero in $H^2 (X, {\mathbb C})$, which contradicts the
flatness of $E$. As we shall see, there are ways to evade this problem.
We first state a generalization of Theorem~\ref{thm1} that goes beyond the
case of
flat vector bundles, providing a sort of Chern-Weil theory for DB-cohomolgy
for algebraic
manifolds. This applies to algebraic vector bundles with what we call an
$F^1$-connection --- see \ref{4.2.3};
flat
connections with regular singularities at infinity are examples of
$F^1$-connections.
In order to state the result precisely, one has to first write $X={\overline X} -D$,
where ${\overline X}$
is compact and $D$ is a normal crossings divisor. Then one has to define
appropriate subgroups
$$
\widehat{H}^{2p}({\overline X}\log D,{\mathbb Z}(p))
$$
of $\widehat{H}^{2p}(X,{\mathbb Z}(p))$ and show that there is a natural map
\begin{equation}
\label{nat_map}
\widehat{H}^{2p}({\overline X}\log D,{\mathbb Z}(p)) \to H_{\cD}^{2p}(X,{\mathbb Z}(p))
\end{equation}
and that the Cheeger-Simons Chern classes bundles $E$ over $X$ with an
$F^1$-connection lie in this subgroup. This is done in \ref{4.F}.
\begin{bigtheorem}
\label{thm2} If $E \to X$ is an algebraic vector bundle with $F^1$-connection
over the
algebraic manifold $X$, then $c_p^B(E)$ is the image of $\widehat{c}_p(E,\nabla)$ under
(\ref{nat_map})
for all $p \geq 1$.
\end{bigtheorem}
We first observe that a necessary condition for Theorem~\ref{thm2} is that the
image of
$\widehat{c}_p(E,\nabla)$ in $H_{\cD}^{2p}(X,{\mathbb Z}(p))$ is the same for all
$F^1$-connections
$\nabla$ on $E$. This is verified {\it a priori},---though it appears later
in our
exposition (as \ref{6.1.4})---and it greatly simplifies the task.
The proof of Theorem \ref{thm2} in the quasi-projective case, given in
\ref{6.2}, goes
as follows. For any
algebraic extension ${\overline E}$ of $E$ to a compactification ${\overline X}$ of $X$, there
exists an ample line
bundle $L$ on ${\overline X}$ such that both ${\overline E} \otimes L$ and $L$ are
generated by global sections, hence, {\it are} pullbacks of universal
bundles by regular maps. Since the conclusion of Theorem~\ref{thm2} is a
tautology for the universal bundles,
we get the result by functoriality for ${\overline E} \otimes L$ and $L$; this
implies the same for ${\overline E}$ itself, hence for $E$.\footnote{This is,
in essence, the argument in \cite{S}, where ${\overline X} = X$; see also our
crucial \ref{6.1.4}. Note that this argument cannot be used in the
complex analytic setting, not even for compact manifolds.} The reduction from
algebraic manifolds to
the quasi-projective case
is fairly standard; see our \ref{40.5}.
We also formulate and prove the analogous result in the K\"ahler case in
\ref{10.1}.
In order to effect this, one has to make distinctions among the various
meromorphic
equivalence classes of compactifications, and then of vector bundle
extensions.
In the case that $X$ is algbraic, we have been tacitly using the obvious
choices,
viz.~the algebraic ones (cf.~\ref{conv}).
\medskip
{}From a different point of view, a trick \cite{DK} (see \ref{6.1.5})
can be used to construct a model (depending on $E \to X$ and its
connection) of the universal $n$-plane bundle $E_n \to BGL_n ({\mathbb C})$ in
the category of {\it simplicial} varieties, a connection on this bundle,
and a morphism
of simplicial varieties $X \to BGL_n({\mathbb C})$ that simultaneously classifies
the bundle and the connection. For this reason, it is both natural and
tempting to work in the category of simplicial varieties; Cheeger-Simons
classes, $DB$-cohomology and Beilinson Chern classes all extend directly
to the simplicial setting (cf.\ Appendix~\ref{simp}). We would then want
to view Theorem~\ref{thm2} as a special case of its simplicial analogue.
Unfortunately, there remain difficulties in trying to carry out this
approach; see Section~\ref{discussion}.
An abbreviated account of the above is given in \cite{Z}.
Another reason for working within the simplicial category is that
we can work with the ``universal case'' of a flat bundle. Denote the
general linear
group of complex $n \times n$ matrices, {\it endowed with the discrete
topology}, by $GL_n(\C)^\delta$. Its classifying space $BGL_n ({\mathbb C})^\delta$
classifies flat complex vector bundles of rank $n$. The Cheeger-Simons
classes of the universal flat bundle
$$
\widehat{c}_p \in H^{2p-1} (BGL_n ({\mathbb C})^\delta, {\mathbb C} /{\mathbb Z}(p))
$$
are the universal Cheeger-Simons classes.
To generalize a classical theorem of Dirichlet, Borel \cite{Bo} defined
canonical cohomology classes
$$
b_p \in H^{2p-1} (BGL ({\mathbb C})^\delta, {\mathbb C}/{\mathbb R}(p))
$$
which he used to define ``regulators'' $r_p : K_{2p-1} ({\mathbb C}) \to {\mathbb C}/{\mathbb R}(p)$
from Quillen's algebraic $K$-theory of
${\mathbb C}$ into ${\mathbb C}/{\mathbb R}(p) \cong H_{\cD}^1(\Spec {\mathbb C},{\mathbb R}(p))$. Denote by $x_p$ the
class in $H^{2p-1}(GL_n(\C),{\mathbb Z}(p))$ that transgresses to the universal Chern
class $c_p \in H^{2p}(BGL_n(\C),{\mathbb Z}(p))$. For each $n$, the cohomology class
$b_p$ restricts to the image in
$$
H^{2p-1}(GL_n(\C)^\delta,{\mathbb C}/{\mathbb R}(p)) \cong H^{2p-1}(BGL_n(\C)^\delta,{\mathbb C}/{\mathbb R}(p))
$$
of the element $\xi_p$ of $H_{\mathrm{cts}}^{2p-1}(GL_n(\C),{\mathbb C}/{\mathbb R}(p))$ that corresponds
to the element $x_p$ of $H^{2p-1}(GL_n(\C),{\mathbb R}(p))$, under the canonical
isomorphism
$$
H^{2p-1}(GL_n(\C),{\mathbb R}(p)) \mathop{\to}\limits^{\cong}
H_{\mathrm{cts}}^{2p-1}(GL_n(\C),{\mathbb C}/{\mathbb R}(p)).
$$
This construction and the relevant background is reviewed in
Sections \ref{7.1} and \ref{7.2}. Our second main result, proved
in Section~\ref{thm_3} is:
\begin{bigtheorem}
\label{thm4}
The image of $\widehat{c}_p$ in $H^{2p-1}(BGL({\mathbb C})^\delta, {\mathbb C}/{\mathbb R}(p))$
is $b_p/2$.
\end{bigtheorem}
Since $BGL_n({\mathbb C})^\delta$ is a simplicial variety, we also have the
Beilinson Chern classes of the universal flat bundle:
$$
c_p^B \in H_{\mathcal D}^{2p} (BGL_n ({\mathbb C})^\delta, {\mathbb Z} (p)) \cong
H^{2p-1}(BGL_n ({\mathbb C})^\delta, {\mathbb C} /{\mathbb Z} (p)).
$$
The homomorphism ${\mathbb C}/{\mathbb Z}(p) \to {\mathbb C}/{\mathbb R}(p)$ gives rise to a commutative
square:
\begin{equation*}
\begin{CD}
H^{2p}_{\mathcal D} (BGL ({\mathbb C})^\delta, {\mathbb Z} (p)) @>{\cong}>>
H^{2p-1} (BGL({\mathbb C})^\delta, {\mathbb C} /{\mathbb Z} (p)) \cr
@VVV @VVV \cr
H^{2p}_{\mathcal D} (BGL ({\mathbb C})^\delta, {\mathbb R} (p)) @>{\cong}>>
H^{2p-1} (BGL({\mathbb C})^\delta, {\mathbb C}/{\mathbb R}(p))
\end{CD}
\end{equation*}
We had hoped to prove that
$$
c_p^B = \widehat{c}_p \text{ in } H^{2p-1}(BGL_n({\mathbb C})^\delta,{\mathbb C}/{\mathbb Z}(p)).
$$
as a case of the simplicial version of Theorem~\ref{thm2}, which is
stated as \ref{6.1.1}.
Our approach in trying to prove
that, and some of the difficulties we encountered in our attempt, are
discussed
in Section~\ref{discussion}. Thus, we are obliged to
state the preceding as a conjecture:
\begin{conjecture}
\label{conj}
For all $p \geq 1$ and $n \in {\mathbb N} \cup\{\infty \}$,
$$
c_p^B = \widehat{c}_p \in H^{2p-1} (BGL_n({\mathbb C})^\delta,{\mathbb C}/{\mathbb Z}(p)).
$$
\end{conjecture}
This conjecture, if true, would imply that the universal Beilinson
Chern class for flat bundles is represented by half the
Borel regulator element. Combining Theorem~\ref{thm4} with the
conjecture yields the following conjectural refinement of Beilinson's
result \cite[A5.3]{Be} (see also the article by Rapoport in \cite{RSS})
which asserts that there is a non-zero rational constant $\lambda$ such
that $b_p \equiv \lambda c^B_p$ modulo the decomposable elements
$$
\left[ H^+(GL_n(\C)^\delta,{\mathbb R})\cdot H^+(GL_n(\C)^\delta,{\mathbb R})\right]\otimes {\mathbb C}/{\mathbb R}(p)
\subseteq H^{\dot}(GL({\mathbb C})^\delta,{\mathbb C}/{\mathbb R}(p)).
$$
He used this to prove that his regulators agreed with those of Borel up
to a non-zero rational constant. Theorem~\ref{thm4} yields as a corollary
the following strengthening of Beilinson's result .
\begin{bigprop}
\label{thm5} If Conjecture 4 is true, then the Borel regulator is two times
the Beilinson
regulator $K_{2p-1} ({\mathbb C}) \to {\mathbb C}/{\mathbb R}(p)$. Consequently, for all number
fields $F$, the Borel regulator and twice the Beilinson regulator
$$
K_{2p-1}(F) \to H_{\cD}^1(\Spec \, F, {\mathbb R}(p))
\cong \left[{\mathbb C}/{\mathbb R}(p)\right]^{d_p}
$$
are equal. Here
$$
d_p =
\begin{cases}
r_1 + r_2 & \text{$p$ odd,} \cr
r_2 & \text{$p$ even,}
\end{cases}
$$
where $r_1$ is the number of real embeddings of $F$, and $r_2$ is the
number of conjugate pairs of complex embeddings.
\end{bigprop}
We wish to stress that even if we considered only flat bundles in our
theorems, it would be necessary to consider Cheeger-Simons classes of
bundles with arbitrary connections \cite{CS}, as we make essential use
of the universal $n$-plane bundle which does not admit a flat connection.
By definition, these classes take values in the ring of differential
characters. We give a detailed account of Cheeger-Simons classes
and differential characters in Section~\ref{cs}, as we could not find
in the literature an exposition suitable for our purposes. One of the
original approaches to Cheeger-Simons classes uses the construction of a
universal connection by Narasimhan-Ramanan \cite{NR}. In Section~\ref{univ}
we give an alternative and more canonical approach to
universal connections. In Section~\ref{db_coho}, we review
Deligne-Beilinson cohomology, and prove Theorem~\ref{thm2}. Next,
Theorem~\ref{thm4}
is
proved in Section~\ref{thm_3}. Finally, we treat the issues and
traps involved in our attempt to prove Conjecture~\ref{conj} in
Section~\ref{discussion}.
We would like to thank the numerous mathematicians with whom we have had useful
discussions
related to this paper, especially Pierre Deligne, H\'el\`ene Esnault, Bernard
Shiffman, Vyacheslav Shokurov and
Christophe
Soul\'e. We are particularly grateful to the referee for his or her patience
over our
reluctance to recognize the relevance of the distinctions now
made
in \ref{8.9} and \ref{4.F}, even though this was known to the authors.
Hain and Zucker would also like to thank the Max-Planck-Institut f\"{u}r
Mathematik for
its hospitality and support during the fall of 1987. Hain would also like to
thank
the Mathematics Institute at \AA rhus for its support during several visits.
{\numberwithin{equation}{subsection}
\subsection{Conventions}\label{1.3}
To make all Chern classes compatible with those used in algebraic
geometry, we introduce the algebraic geometers' Tate twist. Denote by
${\mathbb Z}(p)$ the subgroup of ${\mathbb C}$ generated by $(2\pi i)^p$. For each
subgroup $\Lambda$ of ${\mathbb C}$, set
$$
\Lambda (p) = \Lambda \otimes_{\mathbb Z} {\mathbb Z} (p).
$$
The isomorphism ${\mathbb Z} \to {\mathbb Z} (p)$ that takes $1$ to $(2\pi i)^p$
induces a canonical isomorphism
\begin{equation}\label{1.3.1}
H^{\bullet} (X, {\mathbb Z}) \to H^{\bullet} (X, {\mathbb Z} (p)).
\end{equation}
In this paper, the $p^{\rm th}$ Chern class of a $GL_n ({\mathbb C})$
(equivalently, a complex $n$-plane) bundle over $X$ is the element
of $H^{2p} (X, {\mathbb Z} (p))$ which is the image under (\ref{1.3.1}) of
the usual topological Chern class as defined, for example, in
\cite{MS}.
Let
$$
C_k : {{\g\l}}_n ({\mathbb C}) \to {\mathbb C} \quad k = 0, \ldots, n
$$
be the $GL_n ({\mathbb C})$-invariant polynomials uniquely determined by
$$
\det (tI - A) = \sum_{k=0}^n C_k (A) t^{n-k}.
$$
Explicitly,
$$
C_k (A) = (-1)^k \tr \wedge^k A.
$$
If $E \to M$ is a complex vector bundle with connection $\nabla$ and
curvature $\Theta$, then it is well known (see \cite[p.~403]{GH})
that $C_k (\Theta)$ is a closed form that represents $c_k (E)$ in
$H^{2k} (M, {\mathbb C})$.
All varieties in this paper are defined over the field ${\mathbb C}$ of complex
numbers. We will work in the complex topology unless we explicitly
say otherwise. The complex of smooth, ${\mathbb C}$-valued forms on a manifold
$X$ will be denoted by $A^{\bullet}(X)$.
\section{Universal Connections}
\label{univ}
\subsection{Classical formulation}\label{2.1}
A construction of universal connections was first given in
\cite{NR}, where the universal bundle $U$ on a Grassmannian $G_{\mathbb C}
(n)$ (or equivalently, its frame bundle) is endowed with a connection
$\nabla^U$, such that any vector bundle with a unitary connection
$(E,\nabla)$ on the manifold $M$ is isomorphic to a pull-back of
$(U,\nabla^U)$ via a classifying mapping $g : M \to G_{\mathbb C} (n)$.
More precisely:
\begin{theorem}\label{2.1.1}{\cite[I\S 4]{NR}}
Fix positive integers $m$ and $n$. Then there is a positive integer
$\ell$ such that any complex $n$-plane bundle with a unitary connection
$(E, \nabla)$ on an $m$-dimensional manifold $M$ is the pull-back of
the universal one $(U, \nabla^U)$ on the Grassmannian $G(n,{\mathbb C}^\ell)$
of $n$-planes in ${\mathbb C}^\ell$. Moreover, the universal connections are
compatible with the inclusions
$G(n,{\mathbb C}^\ell)\hookrightarrow G(n, {\mathbb C}^{\ell+1})$.\endproof
\end{theorem}
\begin{remark}\label{2.1.2} The integer $\ell$ is given explicitly in
\cite{NR}. Note that it is necessarily larger than would be needed for
classifying bundles without connection.
Universal connections are constructed for principal bundles with
arbitrary connected structure group in \cite{NR}.
\end{remark}
\subsection{Alternate formulation}\label{2.2}
For our own purposes, it is better to have an alternate formulation
of universal connections. Let $P$ be a principal bundle, with structure
group $G$ (not assumed to be connected), over the manifold $Y$.
A connection on $P$ is, by definition, a $G$-equivariant lifting of
the tangent bundle $TY$ of $Y$ to $TP$ (the tangent bundle of $P$).
Consider, then, the diagram:
\begin{equation*}
\begin{CD}
TP @>>> TP/G @>p>> TY \cr
@VVV @V{\pi}VV @VVV \cr
P @>>> P/G @= Y
\end{CD}
\end{equation*}
In fact, $\pi$ is a vector bundle projection, and the left-hand
square is cartesian. Put
\begin{equation}\label{2.2.1}
\widetilde Y = \{ \alpha \in \Hom (TY, TP/G): p \circ \alpha =
\id_{TY}\}
\end{equation}
Let $q : \widetilde Y \to Y$ denote the natural projection. The
following is evident:
\begin{proposition}\label{2.2.2} The connections on $P$ are in
one-to-one correspondence with the cross-sections of $q$.
\end{proposition}
Let ${\mathfrak g}$ denote the Lie algebra of $G$. Associated to any connection
is its {\it connection $1$-form}:
$$
\omega \in A^1(P,\Ad({\mathfrak g})).
$$
In terms of the preceding, $\omega$ can be described as follows.
Let $\widetilde \alpha : TY \to TP$ be the horizontal lift associated
to $\alpha$, and let $\xi \in TP$. Then one has the simple formula:
\begin{equation}\label{2.2.3}
\omega(\xi) = \xi - \widetilde \alpha (p_*\xi).
\end{equation}
It is convenient to describe $\widetilde Y$ in terms of a local
trivialization of $P$. Thus, we replace $Y$ by a sufficiently small
open subset, {\it which we still call} $Y$. Then
\begin{equation}\label{2.2.4}
P \,\cong\, Y \times G,
\end{equation}
\begin{equation}\label{2.2.5}
TP/G \,\cong\, TY \times (TG/G) \,\cong \,TY \times {{\mathfrak g}},
\end{equation}
so any $\alpha$ in \ref{2.2.1} is determined by an element
$\overline \alpha \in \Hom (TY,{{\mathfrak g}})$. Thus we have:
\begin{proposition}\label{2.2.6}
$\widetilde Y$ is an affine-space bundle over $Y$, with fiber
$\Hom (T_y Y,{{\mathfrak g}})$ (for any $y \in Y$). In particular,
$\widetilde Y$ is a manifold of the same homotopy type as $Y$.
\end{proposition}
Given $f : M \to Y$, the pullback of $P$ is, in terms of \ref{2.2.4},
$$
f^{\ast}P \cong M \times G,
$$
so
\begin{equation}\label{2.2.7}
T(f^\ast P)/G \,\cong\, TM \times {{\mathfrak g}}.
\end{equation}
The pullback of the connection is represented by
$\overline \alpha \circ Tf \in \Hom (TM,{{\mathfrak g}})$.
Consider next the diagram
\begin{equation*}
\begin{CD}
@. q^{\ast}TP/G @= q^{\ast}TP/G @= q^{\ast}TP/G \cr
@. @AAA @. @VVV \cr
T\widetilde P @>>> T\widetilde P/G @>>> T\widetilde Y @>>>
q^{\ast} TY @>>> TY\cr
@VVV @VVV @VVV @VVV @VVV \cr
\widetilde P @>>> \widetilde P/G @= \widetilde Y
@= \widetilde Y @>q>> Y
\end{CD}
\end{equation*}
where $\widetilde P = q^\ast P$, the pullback
of $P$ along $q$. By construction, $\widetilde P$
has a tautological connection $\widetilde \nabla$, given by the
pullback of
\begin{equation}\label{2.2.8}
\widehat \beta \in \{ \beta \in \Hom (q^{\ast}TY, q^{\ast}TP/G):
\beta
\text{ projects to } \id_{q^{\ast}TY} \} ,
\end{equation}
where $\widehat \beta$ is defined by: if $\widetilde y \in
q^{-1}(y)$, $\widehat \beta (\widetilde y)$ is
$\widetilde y : T_yY \to (TP/G)_y$. When $P \cong Y \times G$,
\ref{2.2.8} is determined by
$\overline \beta \in \Hom(q^\ast TY, {{\mathfrak g}})$ (recall \ref{2.2.4}
and \ref{2.2.5}).
\begin{corollary}\label{2.2.9}
For any immersion of manifolds $f : M \to Y$ and connection
$\nabla$ on $f^\ast P$, there is a lifting
$\widetilde f : M \to \widetilde Y$ such that
$\widetilde f^\ast \widetilde \nabla = \nabla$.
\end{corollary}
\begin{proof} By \ref{2.2.2}, the connection $\nabla$
corresponds to a
cross-section $\sigma$ of
\begin{equation*}
\begin{CD}
\widetilde M @= \{\mu \in \Hom (TM, f^\ast TP/G):
f^\ast p \circ \mu = \id_{TM}\}\cr
@VVV \cr M
\end{CD}
\end{equation*}
Since $f$ is, by hypothesis, an immersion, the bundle mapping
$$
Tf : TM \to f^\ast TY
$$
is an injection. It follows that the natural mapping
\begin{equation*}
\begin{CD}
f^\ast \widetilde Y @= \widetilde Y \times_Y M \subset
\Hom (f^\ast TY, f^\ast TP/G)\cr
@VVrV \cr \widetilde M
\end{CD}
\end{equation*}
is surjective (it is induced by the dual of $Tf$). Any splitting
of $Tf$ determines a section $s$ of $r$. Pick one, and take
$\widetilde f = j \circ s \circ \sigma$ (see diagram below).
\begin{equation*}
\begin{CD}
\widetilde M @<r<< f^\ast \widetilde
Y @>j>> \widetilde Y\cr
@VVV @AA{\tilde{f}}A @VVV \cr
M @= M @>>f> Y
\end{CD}
\end{equation*}
\end{proof}
Taking $Y$ to be $G_{\mathbb C} (n)$, and $G = GL_n ({\mathbb C})$, we
obtain:
\begin{proposition}\label{2.2.10} The universal bundle
$\widetilde U = q^\ast U$ on $\widetilde {G}_{\mathbb C} (n)$, with its
tautological connection $\widetilde \nabla$, is universal for
connections on $GL_n({\mathbb C})$-bundles.
\end{proposition}
\begin{proof} Any vector bundle of rank $n$ on a manifold
can be
classified by an immersion into $G_{\mathbb C} (n)$. Now apply
\ref{2.2.9}.
\end{proof}
{}From the above interpretation of universal connections, one gets,
for free, the following useful fact.
\begin{corollary}\label{2.2.11}
If $\widetilde g_0, \widetilde g_1 : M \to \widetilde G_{\mathbb C} (n)$
are two immersions which classify the bundle with connection
$(E, \nabla)$. Then there is a piecewise-smooth homotopy
$$
\widetilde h : I \times M \to \widetilde G_{\mathbb C} (n).
$$
from $g_0$ to $g_1$ such that $\widetilde h^\ast \widetilde \nabla$
is the ``constant'' connection $d_t \oplus \nabla = p^\ast \nabla$ on
$p^\ast E$ ($p : I \times M \to M$ being the projection).
\end{corollary}
\begin{proof} Let $g_j : M \to G_{\mathbb C} (n)$ be the projection
$q \circ \widetilde g_j$ of $\widetilde g_j$. As $g_0$ and $g_1$
are necessarily homotopic, pick any homotopy $k$ from $g_0$ to $g_1$;
we may take $k:I \times M \to G_{\mathbb C} (n)$ to be an immersion. Let
$\widetilde k : I \times M \to \widetilde G_{\mathbb C} (n)$ be a lifting that
classifies $p^\ast (E,\nabla)$. This is, of course, a homotopy between
its ends, $\widetilde k_0$ and $\widetilde k_1$. By construction,
$$
q\circ \widetilde k_j = q\circ \widetilde g_j \quad (j = 0, 1),
$$
so $\widetilde k_j$ and $\widetilde g_j$ can be connected linearly in
the affine-space bundle $\widetilde G_{\mathbb C} (n)$. These homotopies
also classify the constant connection. By combining them with
$\widetilde k$, we obtain $\widetilde h$ as desired.
\end{proof}
\section{The Cheeger-Simons Chern Class}
\label{cs}
In this section we recall and elaborate on the definition and
properties of the Cheeger-Simons invariant, which can be interpreted
as a Chern classes in the ring of differential characters, associated
to a vector bundle with connection on a manifold \cite{CS}.
\subsection{Generalities}\label{3.1}
Let $M$ be a $C^\infty$ manifold, $S_{\bullet} (M)$ the complex
of $C^\infty$ singular chains on $M$ with integer coefficients, and
$Z_{\bullet} (M)$ the subgroup of cycles. For any abelian group $\Lambda$,
one has
\begin{equation}\label{3.1.1}
S^{\bullet} (M, \Lambda) = \Hom_{\mathbb Z} (S_{\bullet} (M),
\Lambda),
\end{equation}
the corresponding complex of $\Lambda$-valued smooth singular
cochains, whose coboundary operator we shall denote by $\delta$.
Let $A^{\bullet} (M)$ denote the complex of $C^\infty$ differential
forms on $M$ with complex coefficients (one could use real coefficients
here as well). An element $w \in A^k (M)$ determines a cochain
$$
c_w \in S^k (M, {\mathbb C})
$$
by the formula
\begin{equation}\label{3.1.2}
c_w (\sigma) = \int_{\Delta^k} \sigma^\ast w
\end{equation}
whenever $\sigma : \Delta^k \to M$ is a $C^\infty$ singular simplex.
This embeds $A^{\bullet} (M)$ as a subcomplex of
$S^{\bullet} (M, {\mathbb C})$. For any subgroup $\Lambda$ of ${\mathbb C}$,
we then get a morphism of complexes.
\begin{equation}\label{3.1.3}
\iota_\Lambda : A^{\bullet} (M) \to S^{\bullet} (M, {\mathbb C}/\Lambda),
\end{equation}
which is an injection whenever $\Lambda$ is totally disconnected.
\subsection{Differential characters}\label{3.2}
Following \cite{CS}, one makes the following:
\begin{definition}\label{3.2.1} The group of mod~$\Lambda$
{\it differential characters} of degree $k$ on $M$ is the group
$$
\widehat H^k (M, \Lambda) = \{ f, \alpha) \in
\Hom_{\mathbb Z}
(Z_{k-1} (M), {\mathbb C}/\Lambda) \oplus A^k (M): \delta f =
\iota_\Lambda \alpha, \text { and } d \alpha = 0 \}.
$$
\end{definition}
One sees that $\widehat H^{\bullet} (M, \Lambda)$ is a contravariant
functor of $M$; when $\Lambda$ is a ring there is a functorial
product \cite[(1.11)]{CS} that imparts a ring structure to the
differential characters.
\begin{remark}\label{3.2.2} (i) Though \cite{CS} would have us writing
$\widehat H^k (M, {\mathbb C} /\Lambda)$ in \ref{3.2.1}, we have introduced
the above change of notation for the sake of consistency
with that to be used in \S 5. We have also taken the liberty of
shifting by one the degree of a differential character from that of
\cite{CS}, so that it becomes compatible with other Chern classes.
This also makes the ring structure a graded one.
(ii) If $\Lambda$ is totally disconnected, then $\alpha$ above is
uniquely determined, \ref{3.1.3}, and the second condition of
\ref{3.2.1} is a consequence of the first.
\end{remark}
It is useful to understand \ref{3.2.1} in terms of conventional
homological algebra. We have:
\begin{para}\label{3.2.3}
(i) If $\overline f\in S^{k-1}(M, {\mathbb C}/\Lambda)$ is
any ${\mathbb Z}$-linear extension of $f$ to $S_{k-1} (M)$, then
$\delta \overline f = \iota_\Lambda \alpha$; in particular, if
$\iota (\Lambda) \alpha = 0$, then $\overline f$ is a cocycle.
(ii) If $\overline f_1$ and $\overline f_2$ are two extensions as in
(i), then $\overline f_2 - \overline f_1$ is a $({\mathbb C} / \Lambda)$-cocycle
vanishing on $Z_{k-1} (M)$ which, by a simple argument using the fact
that $C_{k-1} (M) = Z_{k-1} (M) \oplus F$, where $F$ is free, implies
that the class of $\overline f_2 - \overline f_1$
vanishes in $H^{k-1} (M, {\mathbb C} /\Lambda)$. We can therefore view
$\widehat H^k (M, \Lambda)$ as a subgroup of
$$
S^{k-1} (M, {\mathbb C} /\Lambda) /\delta S^{k-2} (M, {\mathbb C}/\Lambda)
$$
when $\Lambda$ is totally disconnected.
(iii) The form $\alpha$, representing zero in
$H^k (M, {\mathbb C} /\Lambda)$, has its periods (on $Z_k (M)$) in $\Lambda$.
\end{para}
{}From this, one obtains:
\begin{proposition}\label{3.2.4}
\textup{(i)} There is a canonical and functorial exact sequence
$$
0 \to H^{k-1} (M, {\mathbb C}/\Lambda) \to \widehat H^k (M, \Lambda )
\to A^k_{cl} (M, \Lambda) \to 0,
$$
where
$$
A_{cl}^k (M, \Lambda) = \{ \varphi \in A^k (M) : d\varphi = 0,
\text{ and all periods of $\varphi$ on } Z_k (M) \text{ lie in }
\Lambda \}.
$$
\textup{(ii)} \textup{ (cf.\ \cite[\S 4]{E})}
\begin{align*}
\widehat H^k (M, \Lambda) &\cong H^{k} (\cone
\{ A^{\geq k} (M)
\mathop {\to}\limits^{\iota_\Lambda} S^{\bullet} (M, {\mathbb C} /\Lambda)\} [-1])\cr
&= {\mathbb H}^{k-1} (M, \cone \{ {\mathcal A}_M^{\geq k} \to {\mathcal S}_M^{\bullet}({\mathbb C} /\Lambda) \} );
\end{align*}
here, ${\mathcal A}_M^{\bullet}$ and ${\mathcal S}_M^{\bullet} ({\mathbb C} /\Lambda)$ denote
the sheaves of $C^\infty$ forms and singular $({\mathbb C} /\Lambda)$-cochains
respectively. \endproof
\end{proposition}
\begin{corollary}\label{3.2.5}
If $H^{k-1}(M, {\mathbb C} /\Lambda) = 0$, then
$$
\widehat H^k (M, \Lambda )\cong A^k_{cl} (M, \Lambda).$$
\end{corollary}
\subsection{Cheeger-Simons classes.}\label{3.3}
Let $(E, \nabla)$ be a $C^\infty$ complex vector bundle with
connection on $M$. As in Section~\ref{1.3}, the image of the Chern
class
$$
c_p (E) \in H^{2p} (M, {\mathbb Z} (p))
$$
in $H^{2p} (M, {\mathbb C})$ is represented in de~Rham cohomology by the
polynomial $C_p (\Theta)$ in the curvature $\Theta$ of $\nabla$:
\begin{equation}\label{3.3.1}
C_p (\Theta) \in A^{2p}_{cl} (M, {\mathbb Z}(p)).
\end{equation}
As such, \ref{3.3.1} is functorial for pull-backs of bundles with
connection. The {\it Cheeger-Simons invariant} of $(E, \nabla)$
is a functorial lifting of $c_p (E, \nabla)$ to
$$
\widehat{c}_p (E, \nabla) \in \widehat H^{2p} (M, {\mathbb Z}(p))
$$
in \ref{3.2.4}(i). In terms of \ref{3.2.4}(ii), it become a
$p^{\rm th}$ Chern class with values in the group of differential
characters with ${\mathbb Z} (p)$ coefficients.
\begin{remark}\label{3.3.2} If $c_p (E, \nabla) = 0$ (e.g., if
$E$ is a flat bundle), then $\widehat{c}_p (E, \nabla)$ is an element of
$H^{2p-1} (M,{\mathbb C} /{\mathbb Z} (p))$.
\end{remark}
\subsection{Existence and uniqueness of $\widehat{c}_p$}\label{3.4}
The existence and uniqueness of the Cheeger-Simon invariants can be
deduced from the existence of universal connections (\S 2). By
functoriality, the invariant is completely determined by its value
on the universal connection $(\widetilde U, \widetilde \nabla)$ of
\ref{2.2.10}. On the other hand, because the odd-dimensional
cohomology of a Grassmannian is trivial, there is a unique
lifting $\widehat{c}_p (\widetilde U, \widetilde \nabla)$ of its Chern
form to a differential character (see \ref{3.2.4}(i)). Thus, at
most one Cheeger-Simons invariant can be defined: if
$\widetilde g : M \to \widetilde G_{\mathbb C}(n)$
is a smooth mapping that classifies $(E, \nabla)$, one {\it must} take
\begin{equation}\label{3.4.1}
\widehat{c}_p (E, \nabla) = \widetilde g^\ast \widehat{c} (\widetilde
U, \widetilde \nabla).
\end{equation}
One sees that \ref{3.4.1} actually gives a definition of $\widehat{c}_p$
by checking that it is, in fact, independent of the choice of $g$.
For this, let $g_0$ and $g_1$ be two classifying mappings, and
apply \ref{2.2.11} to produce a ``nice'' homotopy $\widetilde h$
between $\widetilde g_0$ and $\widetilde g_1$. Then the homotopy
formula (see \ref{3.A.13} in Appendix A) yields
\begin{multline}\label{3.4.2}
\widetilde g_1^\ast \widehat{c}_p (\widetilde U, \widetilde \nabla)
-
\widetilde g_0^\ast \widetilde c_p (\widetilde U, \widetilde \nabla)
- \delta (B \widetilde h^\ast \widehat{c}_p (\widetilde U, \widetilde
\nabla)) \\
= B (\widetilde h^\ast \delta \widehat{c}_p (\widetilde U, \widetilde
\nabla)) = B (\widetilde h^\ast c_p (\widetilde U, \widetilde
\nabla)) = 0,
\end{multline}
for $\widetilde h^\ast c_p (\widetilde U, \widetilde \nabla) \in
A^{2p} (I\times M)$ is pulled back from $M$, so is annihilated by the
fiber integration $B$.
\subsection{An intrinsic construction of $\widehat{c}_p$}\label{3.5}
There is a more intrinsic definition of $\widehat{c}_p (E, \nabla)$,
one that uses only functorial constructions on bundles and
connections \cite[\S 4]{CS}. (See also \cite[\S 3]{ch-sim}.) A truly
intrinsic construction is given in \ref{6.1.5}; that construction uses
simplicial methods.
Given integers $n \geq p \geq 1$, let $V_n^p$ denote the Stiefel
manifold of linearly independent $(n - p+1)$-tuples in ${\mathbb C}^n$. Then
$V_n^p$ has the homotopy type of its submanifold $U (n)/U (p-1)$ of
unitary $(n-p+1)$-frames. From this, one sees:
\begin{proposition}\label{3.5.1}\textup{(see \cite[(25.7)]{St})}
$$
\pi_i (V^p_n) =
\begin{cases}
0 & \text{for $i < 2p -1$,}\cr
{\mathbb Z} & \text{for $i = 2p-1$}.
\end{cases}
\leqno (i)
$$
\medskip
$(ii)$ $H_{2p-1} (V_n^p, {\mathbb Z}) \cong {\mathbb Z},$
and is generated by the homology class of
$U (p) / U (p-1) \cong S^{2p-1}$.
\end{proposition}
{}From a complex vector bundle $E$ rank $n$ on $M$, one can form the
associated {\it Stiefel bundle}
$$
\pi : V^p (E) \to M,
$$
with fiber $V_n^p$. (For instance, $V^1 (E)$ is just the frame
bundle of $E$.) By construction, $\pi^\ast E$ has $n - p+1$
tautological sections,
i.e., contains a (canonically) trivial bundle of rank $n - p+1$.
Therefore, the Chern classes of $\pi^\ast E$ are those of any
complementary $(p-1)$-plane bundle $W$. In particular,
\begin{equation}\label{3.5.2}
\pi^\ast c_p (E) =
c_p (\pi^\ast E) = 0 \text{ in } H^{2p}(V^p (E),{\mathbb Z}(p))
\end{equation}
Given any connection $\nabla$ on $E$, the Chern form $\pi^\ast c_p
(E,\nabla)$ is thus exact. The first goal is to achieve the exactness
in a functorial fashion.
\begin{para}\label{3.5.3}
For any $W$ as above, let $\nabla^W$ denote the unique connection on
$\pi^\ast E$ which satisfies both:
\begin{enumerate}
\item[(i)] $\nabla^W$ and $\pi^\ast \nabla$ agree on $W$,
\item[(ii)] the tautological sections are flat with respect to
$\nabla^W$.
\end{enumerate}
\end{para}
There is a functorial formula expressing the fact that the Chern forms
of two connections represent the same cohomology class
(see \cite[p.~48]{Ch}):
\begin{equation}\label{3.5.4}
c_p (E, \nabla_1) - c_p (E, \nabla_0) = d\eta_p,
\end{equation}
where
\begin{equation}\label{3.5.5}
\eta_p = p \int^1_0 \mu (\omega, \Theta_t, \ldots, \Theta_t)dt;
\end{equation}
here, $\mu$ is the $p$-linear symmetric form with
\begin{equation}\label{3.5.6}
\mu (\Theta, \ldots, \Theta) = C_p (\Theta),
\end{equation}
$\omega = \nabla_1 - \nabla_0$, and $\Theta_t$ is the curvature of
$$
\nabla_t = \nabla_0 + t\omega, ~~t \in [0, 1].
$$
\begin{observation}\label{3.5.7}
Let $\sigma_0, \sigma_1 : M \to \widetilde M$
be the sections corresponding to $\nabla_0, \nabla_1$ (recall
\ref{2.2.2}); let $h : I \times M \to \widetilde M$ classify
the connection
$$
\overline \nabla : d_t + (1-t) \nabla_0 + t \nabla_1.
$$
Then formula \ref{3.5.4} is just what comes out of the homotopy formula
\ref{3.A.8} when it is applied to the Chern form of the tautological
connection $\widetilde \nabla$ on $\widetilde M$, or equivalently,
when \ref{3.A.7} is applied to the Chern form of $\overline \nabla$.
To see this, one need only observe that the curvature $\Theta$ of $\overline
\nabla$
is given by
$$
\Theta = \Theta_t + dt \wedge \omega.
$$
\end{observation}
For $\nabla_0 = \nabla^W$ and $\nabla_1 = \pi^\ast \nabla$,
\ref{3.5.5} gives
\begin{equation}\label{3.5.8}
\pi^\ast c_p (E, \nabla) - c_p (\pi^\ast E, \nabla^W)
= d \eta^W.
\end{equation}
Since
$$
c_p (\pi^\ast E, \nabla^W) = 0 \quad \text { in } A^{2p} (V^p(E)),
$$
we can rewrite \ref{3.5.8} as:
\begin{equation}\label{3.5.9}
\pi^\ast c_p (E, \nabla) = d\eta^W.
\end{equation}
It remains to study the effect of changing $W$.
\begin{proposition}\label{3.5.10} There is a unique natural choice of
$$
\overline\eta (E, \nabla) \in A^{2p-1}(V^p (E))/dA^{2p-2} (V^p(E))
$$
such that $d\overline \eta (E, \nabla) = \pi^\ast c_p (E, \nabla)$.
\end{proposition}
\begin{proof} Let $F \cong V_n^p$ denote a fiber of $\pi$. From
\ref{3.5.1} and the Leray spectral sequence for $\pi$, one
deduces the exactness of:
$$
0 \to H^{2p-1} (M, {\mathbb Z}) \mathop {\to}\limits^{\pi^\ast} H^{2p-1}
(V^p (E), {\mathbb Z}) \to H^{2p-1} (F, {\mathbb Z}) \mathop {\to}\limits^\delta H^{2p} (M,
{\mathbb Z}).
$$
The image of one of the generators of $H^{2p-1} (F, {\mathbb Z}) \cong {\mathbb Z}$ under
$\delta$ is,
in fact, $c_p (E)$, which can be seen easily by considering the Serre
spectral sequence of the fibration. In particular, for the universal
bundle $U$ on $G_{\mathbb C} (n)$
\begin{equation}\label{3.5.11}
H^{2p-1}(V^p (U), {\mathbb Z}) \cong H^{2p-1}(G_{\mathbb C} (n),{\mathbb Z}) = 0.
\end{equation}
Given $\varphi$, the condition $d \eta = \varphi$ determines $\eta$
up to a {\it closed} form. From \ref{3.5.11}, a closed
$(2p-1)$-form on $V^p(U)$ is exact. It follows that
$\overline \eta (\widetilde U, \widetilde \nabla)$ is
uniquely determined. Therefore, there is at most one natural
construction of $\overline \eta (E, \nabla)$: it must be
$\widetilde g^\ast \overline \eta (\widetilde U, \widetilde \nabla)$
whenever $g : M \to G_{\mathbb C}(n)$ classifies $E$. The proof that this is
well-defined goes as in \ref{3.4.2}.
\end{proof}
As the preceding is contrary to the spirit with which we began
subsection~\ref{3.5}, we will show directly, after all, that $\eta^W$
(from \ref{3.5.4}) modulo exact forms, independent of $W$. Let $W_0$
and $W_1$ be two such. Consider the following connection on the pullback
of $E$ to $I \times I \times M$:
\begin{equation}\label{3.5.12}
d_s + d_t + (1-t) [(1 -s) \nabla^{W_0} + s \nabla^{W_1}] + t
(\pi^\ast
\nabla);
\end{equation}
it corresponds to a mapping
$$
I \times I \times M \to \widetilde M.
$$
Let
\begin{equation}\label{3.5.13}
B : A^{2p} (I \times I \times M) \to A^{2p-2} (M)
\end{equation}
denote the fiber integration, which is just the iteration of
\ref{3.A.1} in Appendix A; as such, we write
$B = B_2 \circ B_1$. From \ref{3.A.7}, it follows that
\begin{equation}\label{3.5.14}
d (B \varphi) = (B_2 \varphi_{1, t} - B_2 \varphi_{0, t}) - (B_1
\varphi_{1,s} - B_1 \varphi_{0, s})
\end{equation}
where $\varphi_{0, t}$ is the restriction of $\varphi$ to
$\{ 0 \} \times I \times M$, $\varphi_{1, s}$ to
$I \times \{ 1 \} \times M$, etc. Let $\varphi$ be the $p$th
Chern form of \ref{3.5.12}. Then, by \ref{3.5.7}
$$
B_1 \varphi_{1, s} = 0, ~~B_2 \varphi_{1, t} = \eta^{W_1}, \quad
B_2 \varphi_{0, t} = \eta^{W_0}
$$
This gives
$$
d (B\varphi) = \eta^{W_1} - \eta^{W_0} + B_1 \varphi_{0, s}.
$$
The desired conclusion follows from the realization that
$B_1 \varphi_{0, s}$ is the zero form. The reason for this is that
when $t = 0$, \ref{3.5.12} gives the linear interpolation between
$\nabla^{W_0}$ and $\nabla^{W_1}$, so $\nabla^{W_1} - \nabla^{W_0}$
is an endomorphism of $\pi^\ast E$ that is zero on the tautological
sub-bundle (so is of rank at most $p-1$). The same holds for the
curvature $\Theta_s$. Since $\mu$ in \ref{3.5.6}
is induced by polarization from trace $\wedge^p$, it follows that
$B_1 \varphi_{0, s} = 0$.
We can now complete the direct construction of $\widehat{c}_p(E,\nabla)$.
\begin{proposition}\label{3.5.15} There is a unique element
$\widehat \eta \in \widehat H^{2p} (M, {\mathbb Z}(p))$
with $\pi^\ast \widehat \eta = [\eta^W,\pi^\ast c_p (E, \nabla) ]$
(and $\delta \widehat \eta =\iota_{{\mathbb Z} (p)} c_p (E, \nabla)$).
\end{proposition}
\begin{proof} The uniqueness of $\widehat \eta$ is easy,
for it follows from the diagram
\begin{equation*}
\begin{matrix}
\scriptstyle 0 & \scriptstyle \to& \scriptstyle
H^{2p-1} (F, {\mathbb C} /{\mathbb Z} (p)) & \scriptstyle \to& \scriptstyle
\widehat H^{2p}(F, {\mathbb Z} (p))& \scriptstyle & \scriptstyle &
\scriptstyle & \scriptstyle \cr
& \scriptstyle & \scriptstyle \uparrow& \scriptstyle & \scriptstyle
\uparrow\cr
\scriptstyle 0& \scriptstyle \to & \scriptstyle
H^{2p-1}(V^p (E),{\mathbb C} /{\mathbb Z} (p))& \scriptstyle \to & \scriptstyle
\widehat H^{2p}(V^p(E),{\mathbb Z} (p))& \scriptstyle \to & \scriptstyle
A_{cl}^{2p} (V^p (E), {\mathbb Z} (p))& \scriptstyle \to& \scriptstyle 0\cr
& \scriptstyle & \scriptstyle \uparrow& \scriptstyle & \scriptstyle
\uparrow{\scriptstyle \pi^\ast}& \scriptstyle & \scriptstyle \uparrow&
\scriptstyle & \scriptstyle \cr
\scriptstyle 0& \scriptstyle \to& \scriptstyle
H^{2p-1}(M, {\mathbb C} /{\mathbb Z} (p))& \scriptstyle \to& \scriptstyle
\widehat H^{2p} (M,{\mathbb Z} (p))& \scriptstyle \to& \scriptstyle
A^{2p}_{cl}(M, {\mathbb Z} (p))& \scriptstyle \to& \scriptstyle 0\cr
& \scriptstyle & \scriptstyle \uparrow& \scriptstyle & \scriptstyle
& \scriptstyle & \scriptstyle \uparrow& \scriptstyle & \scriptstyle \cr
& \scriptstyle & \scriptstyle 0& \scriptstyle & \scriptstyle &
\scriptstyle & \scriptstyle 0& \scriptstyle & \scriptstyle
\end{matrix}
\end{equation*}
that $\pi^\ast$ is injective on differential characters. Likewise,
we argue that $[\eta^W]$ is in the image of $\pi^\ast$ by showing
that the restriction of $\eta^W$ to the fiber $F$ is the zero
differential character, or equivalently,
$$
\int_{S^{2p-1}} \eta^W \in {\mathbb Z} (p).
$$
(Recall that $d \eta^W = \pi^\ast c_p (\nabla, E)$ vanishes on $F$.)
A little bizarrely, we pass over what might have been a direct
calculation on the space $V_n^p$, the partial frames in the trivial
$n$-bundle over a point, and check it in the universal situation.
There, $S^{2p-1}$ bounds in $V^p (U)$, so we write
$S^{2p-1} = \partial \Gamma$. Then $Z = \pi_\ast \Gamma$ is a
cycle on the Grassmannian, with
$$
\int_{S^{2p-1}} \eta^W = \int_\Gamma \pi^\ast c_p (\nabla^U) =
\int_Z c_p (\nabla^U) \in {\mathbb Z} (p)
$$
This completes the proof of \ref{3.5.15}.
\end{proof}
With the preceding accomplished, we now make the:
\begin{definition}\label{3.5.16} The differential character
$\widehat \eta$ above is the {\it Cheeger-Simons Chern class}
of $(E, \nabla)$. We write $\widehat \eta = \widehat{c}_p (E, \nabla)$.
\end{definition}
\begin{remark}\label{3.5.17} By \ref{3.2.3}(ii), the Cheeger-Simons
class $\widehat{c}_p$ is the choice of an element
$$
y \in S^{2p-1}(M,{\mathbb C}/{\mathbb Z} (p))/\delta S^{2p-2}(M,{\mathbb C}/{\mathbb Z} (p))
$$
with $\delta y = \iota_{{\mathbb Z} (p)} c_p (E, \nabla)$. In terms of
the cone on $\iota_{{\mathbb Z} (p)}$ (recall \ref{3.2.4}(ii)), it is
represented by
$$
(c_p (E, \nabla), -y) \in A^{2p} (M) \oplus
S^{2p-1} (M, {\mathbb C} /{\mathbb Z}(p))/\delta S^{2p-2} (M, {\mathbb C} /{\mathbb Z} (p)),
$$
which is a cocycle by virtue of the calculation
$$
D (c_p (E, \nabla), -y) =
(-d c_p (E, \nabla), -\delta y + \iota_{{\mathbb Z} (p)} c_p (E, \nabla))
= (0, 0).
$$
\end{remark}
\section{Deligne-Beilinson Cohomology}
\label{db_coho}
In this section we recall the definition of Deligne-Beilinson cohomology,
and establish the precise relation with differential characters.
References on Deligne-Beilinson cohomology include \cite{Be} and \cite{EV}.
\subsection{Deligne-Beilinson (DB-) cohomology}\label{4.1}
Suppose that $Y$ is a smooth algebraic variety. By
resolution of singularities \cite{Hi1}, we can find a smooth completion ${\overline Y}$
of $Y$ such that ${\overline Y} - Y$ is a normal crossings
divisor $D$. Denote by $I_p$ the composite
$$
F^p A^{\bullet} ({\overline Y} \log D) \to
A^{\bullet} Y \mathop {\to}\limits^I S^{\bullet} (Y),
$$
where $F$ denotes the Hodge filtration of the de~Rham complex.
\begin{definition}\label{4.1.1} Suppose that $\Lambda$ is a subring
of $\mathbb R$ and that $p \in {\mathbb N}$. The {\it Deligne-Beilinson} (or
DB-) {\it cohomology} $H^{\bullet}_{\mathcal D} (Y, \Lambda (p))$ of
the variety $Y$ is the cohomology of the complex
$$
D^{\bullet} (Y, {\overline Y}; \Lambda (p)) =
\cone \{ F^p A^{\bullet} ( {\overline Y} \log D )
\mathop {\to}\limits^{I_p} S^{\bullet} (Y, {\mathbb C} /\Lambda
(p))\} [-1].
$$
\end{definition}
As the notation suggests, the DB-cohomology of a variety $Y$ is
independent of
the compactification ${\overline Y}$ chosen. This follows from
standard arguments (cf.\ \cite[(8.3.2)]{De2}).
As DB-cohomology is constructed from a cone, we have:
\begin{proposition}\label{4.1.2} For each smooth algebraic variety $Y$, there
is natural long exact sequence
\begin{multline}\label{les}
\ldots \to H^{k-1}(Y , {\mathbb C} /\Lambda (p)) \to
H_{\mathcal D}^k (Y, \Lambda (p)) \cr \to F^p H^k( Y ,{\mathbb C})
\to H^k ( Y, {\mathbb C} / \Lambda (p)) \to \ldots.
\end{multline}
\endproof
\end{proposition}
\begin{corollary}\label{4.1.3} For each variety
\begin{enumerate}
\item[(i)] For each variety $Y$, there is a natural homomorphism
$$
H^{k-1}(Y, {\mathbb C} /\Lambda (p)) \to H_{\mathcal D}^k (Y,\Lambda (p)).
$$
\item[(ii)] If $H^{2p-1}(Y, {\mathbb C} /\Lambda (p)) = 0$, there is a natural
isomorphism
$$
H_{\mathcal D}^{2p} (Y,\Lambda (p))\cong H^{p,p}(Y)\cap\, \image\{H^{2p}(Y,\Lambda (p))
\to H^{2p}(Y,{\mathbb C})\}.
$$
\end{enumerate}
\endproof
\end{corollary}
There are products defined in Deligne cohomology:
\begin{equation}\label{prod}
H_{\mathcal D}^k (Y,\Lambda (p)) \times H_{\mathcal D}^l (Y,\Lambda (q))\to H_{\mathcal D}^{k+l} (Y,
\Lambda (p+q))
\end{equation}
(see \cite[1.5.1]{Be}). In particular, $\bigoplus _p H_{\mathcal D}^{2p} (Y,\Lambda
(p))$ is a ring.
Beilinson \cite{Be} and Gillet \cite{G} have defined Chern classes
$$
c_p^B (E) \in H_{\mathcal D}^{2p} (Y, {\mathbb Z} (p))
$$
for vector bundles $E \to Y$ over a variety. The construction in
\cite[1.7]{Be},
is in the manner of \cite{Gd}: since the splitting principle
holds for
DB-cohomology, the existence of Chern classes in Deligne cohomology reduces to
the
existence of first Chern classes
$$
c_1 (L) \in H^2_{\mathcal D} (Y, {\mathbb Z} (1))
$$
for line bundles $L \to Y$.
\subsection{Differential characters and DB-cohomology}
\label{4.2}
Recall from \ref{3.2} that the
mod~$\Lambda$ differential characters of degree $k$ on a $C^{\infty}$
manifold $M$, $\widehat H^k (M, \Lambda)$, are represented by the
cocycles of degree $k-1$ in
$\cone\{ A^{\geq k}(M)\mathop{\to}\limits^{\iota_\Lambda}
S^{\bullet}(M,{\mathbb C}/\Lambda)\}.$
When $X$ is a {\it complex} manifold, of the form $X = {\overline X} - D$,
with $D$ a divisor with normal crossings ${\overline X}$, we can
incorporate the Hodge filtration and growth conditions
on $A^{{\bullet}}(X)$ to define subgroups of $\widehat H^k (X, \Lambda)$.
Specifically, take
\begin{equation}\label{4.2.1}
F^p\widehat H^k ({\overline X}\log D, \Lambda) = H^{k-1}
(\cone \{ F^p A^{\geq k} ({\overline X}\log D)
\mathop {\to}\limits^{\iota_\Lambda} S^{\bullet} (X, {\mathbb C} /\Lambda) \} )
\end{equation}
and
$$
\widehat{H}^k({\overline X}\log D,\Lambda) =
F^0\widehat{H}^k({\overline X}\log D,\Lambda).
$$
It is a subgroup of $\widehat H^k (X, \Lambda)$. When ${\overline X}$ is compact,
there is an exact sequence:
\begin{equation}\label{4.2.2}
0 \longrightarrow H^k(X,{\mathbb C}/\Lambda) \longrightarrow
\widehat H^k ({\overline X}\log D, \Lambda)
\longrightarrow F^p A_{cl}^k({\overline X}\log D,\Lambda) \longrightarrow 0
\end{equation}
where $A_{cl}^k({\overline X}\log D,\Lambda)$ is the set of closed elements of
$A^{\bullet}({\overline X}\log D)$ whose periods lie in $\Lambda$.
The reason for introducing (\ref{4.2.1}) comes from its similarity
to (\ref{4.1.2}). The cone in (\ref{4.2.1}) is a subset of the
cone in (\ref{4.1.1}); for $k = 2p$, one obtains, for $X$ algebraic and for
any $\Lambda$, a functorial exact sequence:
\begin{equation}\label{5.5.6}
F^p A^{2p-1}({\overline X} \log D) \rightarrow F^p\widehat H^{2p}
({\overline X} \log D,\Lambda)
\rightarrow H^{2p}_{\mathcal D} (X, \Lambda) \rightarrow 0.
\end{equation}
\subsection{Meromorphic equivalence}\label{8.9}
Though it may seem premature, it is helpful to leave the setting of algebraic
varieties and
algebraic vector bundle for awhile.
Let $X$ be a compactifiable complex manifold, in the sense that $X$
admits a
compactification $\overline X$ that is a compact manifold, and
for which
$D=\overline X - X$ is an analytic subvariety. We may then modify
$\overline X$
by blow-ups with smooth center \cite{Hi2} to make $D$ a divisor with
normal crossings.
It becomes necessary to divide the compactifications of $X$ into meromorphic
equivalence classes.
\begin{definition}\label{1}
Two compactifications $\overline X_1$ and $\overline X_2$ of $X$ are said to
be {\it
meromorphically equivalent} if there exists a compactification $\overline X_3$
of $X$
and morphisms of compactifications, i.e., extensions of the identity map of
$X$,
$\overline X_3\to\overline X_1$ and $\overline X_3\to\overline X_2$.
\end{definition}
\noindent The above is easily seen to be an equivalence relation.
We point out that (the underlying complex manifold of) a smooth algebraic
variety
can admit compactifications that are not meromorphically equivalent to the
algebraic
ones. A simple example of this is provided by \cite{CE}:
\begin{example}\label{CE}
Let $C$ be an elliptic curve. Then $X={\mathbb C}^{p+q}\times C$ admits non-algebraic
compactifications
${\overline X}_{p,q}$ ($p>0$, $0\le q\le p$) with the following properties:
\begin{enumerate}
\item[(i)] There is a principal $C$-fibration $\pi:{\overline X}_{p,q}\to\mathbb
P^p({\mathbb C})
\times\mathbb P^q({\mathbb C})$.
\item[(ii)] Every meromorphic function on ${\overline X}_{p,q}$ is a pullback from
$\mathbb
P^p({\mathbb C})\times\mathbb P^q({\mathbb C})$; the algebraic dimension of ${\overline X}_{p,q}$
is less
than its complex dimension, so it is not algebraic.
\item[(iii)] ${\overline X}_{p,q}$ is diffeomorphic to the product of spheres
$S^{2p+1}
\times S^{2q+1}$. It follows that $H^2({\overline X}_{p,q},{\mathbb C}) = 0$, and thus
${\overline X}_{p,q}$
is not K\"ahler.
\end{enumerate}
\end{example}
Furthermore, it is possible for a complex manifold to admit more than one
algebraic
structure; it can have inequivalent algebraic compactifications:
\begin{example}\label{2alg}
Let $C$ again be an elliptic curve. The {\it universal vector extension} of
$C$, as a
complex manifold, can be given Hodge-theoretically as follows. Put $H_{\mathbb Z} =
H^1(C,{\mathbb Z})$,
$H = H^1(C,{\mathbb C})$, and $F = F^1H^1(C,{\mathbb C})$. There is a short exact sequence of
abelian groups:
$$
0\to F\to H/H_{\mathbb Z}\to H/(H_{\mathbb Z} + F)\to 0,
$$
which is isomorphic to
$$
0\to {\mathbb C}\to({\mathbb C}^*)^2\to C\to 0.
$$
The associated $\mathbb P^1$-bundle:
\begin{equation*}
\begin{CD}
\mathbb P^1 @>>> {\overline X} \\
@. @VVV \\
{} @. C
\end{CD}
\end{equation*}
\noindent then gives an algebraic variety ${\overline X}$ that is a compactification
of $X = ({\mathbb C}^*)^2$.
However, ${\overline X}$ and $\mathbb P^1\times\mathbb P^1$ are not meromorphically
(birationally)
equivalent, for they have non-isomorphic function fields; only the latter one
is an
algebraic completion of the {\it algebraic} variety $X$.
\end{example}
We also recall the notion of a meromorphic mapping, from complex geometry.
Let $X$
and $Y$ be complex analytic varieties, and $\overline X$, $\overline Y$,
respective
partial compactifications.
\begin{definition}\label{2}
A map $f:X\to Y$ is said to be {\it meromorphic} with respect to $\overline X$
and $\overline Y$
(and one writes $f:\overline X \dashrightarrow \overline Y$) if the closure of
the
graph of $f$
in $\overline X \times \overline Y$ is a subvariety of $\overline X \times
\overline Y$.
\end{definition}
\noindent Because of the existence of resolutions of singularities for complex
analytic
varieties (\cite{Hi2} again), two compactifications of $X$ are meromorphically
equivalent
if and only if the identity map of $X$ defines a meromorphic map between them.
There is an obvious notion of the {\it extendability} of a holomorphic
vector bundle
$E$ on $X$ to ${\overline X}$.
We next recall the notion of meromorphic equivalence of bundle extensions; it
is presented in a
somewhat different, though equivalent, manner in \cite [p.~65ff]{De1}).
\begin{definition}\label{3}
Let ${\overline X}$ be a complex
manifold,
$D$ a divisor on ${\overline X}$, and put $X={\overline X} - D$. Let $E$ be a vector
bundle on
$X$. Two extensions ${\overline E}$ and ${\overline E}^\prime$ of $E$ to a vector bundle
on ${\overline X}$
are said to be {\it meromorphically equivalent} if, whenever $U$ is a nice
open
neighborhood of a point of $D$ in ${\overline X}$, on which both bundles are trivial,
the
change-of-frame matrix from one to the other (a priori holomorphic on $U\cap
X$) is
meromorphic (i.e., does not have essential singularities) along $U\cap D$.
\end{definition}
\begin{remark}\label{merom}
If $\mathfrak M$ is a meromorphic equivalence class of compactifications of
$X$, there are
corresponding notions of the $\mathfrak M$-extendability of $E$ and
$\mathfrak M$-meromorphic
equivalence classes of extensions of $E$.
\end{remark}
One can always regard a vector bundle as a complex manifold, forgetting the
linear structure.
The following is almost a tautology:
\begin{lemma}\label{meroeq} With notation as above,
\begin{enumerate}
\item[(i)] Two bundle extensions $\overline E$ and $\overline E^{\,\prime}$
are meromorphically
equivalent if and only if $\overline E$ and $\overline E^{\,\prime}$ are
equivalent
partial compactifications of $E$.
\item[(ii)] If $\overline E$ and $\overline E^{\,\prime}$ are meromorphically
equivalent,
then the meromorphic map $\overline E\dashrightarrow \overline E^{\,\prime}$
induces
a meromorphic
map $\mathbb P(\overline E) \dashrightarrow \mathbb P(\overline E^{\,\prime})$.
\end{enumerate}
\end{lemma}
\begin{remark}\label{conv}
When $X$ is an algebraic manifold, there is a canonical equivalence class of
compactifications
of $X$, namely the smooth algebraic completions. Algebraic vector bundles
admit algebraic
extensions to suitable completions, and this class of extensions is likewise
canonical.
We understand algebraic extensions of algebraic vector bundles on an algebraic
variety
if it is not specified otherwise.
\end{remark}
\subsection{$F^1$-connections}
\label{4.F}
Throughout, we let $X\subseteq {\overline X}$ be as in \ref{4.2}, and $E$ a vector
bundle on $X$.
\begin{definition}\label{4.2.3}
An {\it $F^1$-connection} on $E$ (relative to ${\overline X}$) is a connection for
which there
exists a vector bundle ${\overline E}$ over ${\overline X}$ such that
\begin{enumerate}
\item[(i)] the restriction of ${\overline E}$ to $X$ is $E$;
\item[(ii)] the local connection forms (with respect to holomorphic
frames of ${\overline E}$), and therefore also the curvature, lie in
$F^1 A^{\bullet} ({\overline X}\log D, \End({\overline E}))$.
\end{enumerate}
\end{definition}
\noindent We define a sort of ``universal $F^1$-connection'' in Appendix
\ref{4A}.
Note that
\ref{4.2.3} coincides with \ref{4.2.w} if and only if $X = {\overline X}$.
\begin{remark}\label{4.2.4}
This notion occurs in a well-known fact about Hermitian geometry (see
\cite[p.73]{GH}):
If $E$ is an Hermitian vector bundle over the {\it compact} complex manifold
$X$, there
exists a unique $F^1$-connection on $E$ with respect to which the Hermitian
metric on $E$
is horizontal. In particular, an extendable vector bundle always admits an
$F^1$-connection
(without singularities).
\end{remark}
The following is evident:
\begin{proposition}\label{4.2.5}
If ${\nabla}$ is an $F^1$-connection on $E$ (relative to ${\overline X}$), then
the Cheeger-Simons class $\widehat{c}_p (E, \nabla)$ is in the subgroup
$F^p\widehat H^{2p} ({\overline X}\log D, \Lambda)$ of
$\widehat H^{2p} (X, \Lambda)$.
\end{proposition}
To be able to talk sensibly about $F^1$-connections, one needs to know the
following:
\begin {proposition}\label{mero}
Let ${\overline X}$ be a compact complex manifold, $D$ a divisor with normal
crossings
on ${\overline X}$, and put $X={\overline X} - D$. Let $(E,\nabla)$ be a vector bundle
with connection on
$X$. Suppose that ${\overline E}$ and ${\overline E}^\prime$ are vector bundles on ${\overline X}$
with respect
to which $\nabla$ is an $F^1$-connection. Then ${\overline E}$ and ${\overline E}^\prime$
are
meromorphically equivalent.
\end{proposition}
\begin{proof}
The issue is fairly elementary. Let $U$ be open in ${\overline X}$. If $\omega$ is
the connection
form with respect to a frame of ${\overline E}$ on $U\subset {\overline X}$, and $\omega^
\prime$
is the connection form with respect to a frame of ${\overline E}^\prime$ on $U$,
suppose that
$\omega$ and $\omega^\prime$ have first order-poles\footnote{Since we will
be reducing
to curves in $U$, there is actually no need to assume that the poles are
logarithmic,
for the two notions coincide on a curve.} on $U\cap D$. Let $A$ be the matrix
expressing
the frame for ${\overline E}^\prime$ in terms of the one for ${\overline E}$ (thus $\omega^
\prime =
A\omega A^\- + (dA)A^\-$). We wish to conclude that the entries of $A$ must
be meromorphic
along $U\cap D$.
Like holomorphy, meromorphy can be detected by a curve test, i.e., a function
of several
variables is meromorphic if and only if its restriction to sufficiently many
curves is---e.g.,
if it is meromorphic in each variable separately---(see \cite{Sh};
cf.~\cite[II,\,4.1.1]{De1}).
Since connections are functorial, we may assume that $U$ is the unit disc in
${\mathbb C}$, and $D=\{0\}$.
In this case, the assertion is proved in \cite[II,\,1.19]{De1}.
\end{proof}
\begin{remark}\label{meron} Note that Prop.~\ref{mero} does not say that there
exists
${\overline E}$
for which the connection form is logarithmic; nor does it say that if such
${\overline E}$
exists, the connection form has logarithmic poles for all ${\overline E}^\prime$
meromorphically
equivalent to ${\overline E}$ (counterexamples abound). Also, there is no
contradiction
between Remark \ref{4.2.4} and Prop.~\ref{mero}.
\end{remark}
According to \cite[II, (5.2)]{De1}, given {\it any} compactification ${\overline X}$
of $X$,
a flat bundle $E$ on $X$ admits a vector bundle extension ${\overline E}$ to ${\overline X}$
with respect to which the flat connection can be
seen
to be an $F^1$-connection. In fact, according to \ref{mero} any two such
${\overline E}$ are
meromorphically equivalent. We point out that ${\overline E}$ can be determined from
constructions
that are local on ${\overline X}$ along $D$. Local connection forms are then to be
computed
on ${\overline X}$ in terms of local frames of ${\overline E}$. In particular, there is no
analogue of
\ref{mero} for equivalence classes of compactifications.
On the other hand, if one starts with an $F^1$-connection on an {\it algebraic}
vector bundle
on $X$, relative to an algebraic compactification ${\overline X}$, it does {\it not}
follow that
the meromorphic equivalence class of
extensions of
$E$ distinguished by \ref{mero} are the algebraic vector bundles on ${\overline X}$.
However,
the two are known to coincide in an important class of examples, namely the
flat bundles
``coming from algebraic geometry.'' By that, one means that the fibers of $E$
are
cohomology groups for a family of algebraic varieties over $X$, and $\nabla$
is the
Gauss-Manin connection. This result is commonly called {\it the regularity
theorem
in algebraic
geometry}. (See \cite{K} and its generalizations, e.g., \cite[\S 5]{SZ}.)
The following should shed some light on the issue.
\begin{example}\label{exme}
Let $j:X\hookrightarrow{\overline X}$ denote the inclusion. Given the holomorphic
vector
bundle $E$, the specification of ${\overline E}$ is equivalent to selecting a
locally-free
subsheaf (${\mathcal O}_{\overline X} ({\overline E} )$) of $j_*{\mathcal O}_X(E)$ on ${\overline X}$.
Suppose that $\dim\, X = 1$ (so ${\overline X}$ is automatically algebraic). Let
$\Delta\cong
U \subset {\overline X}$ be a disc, with coordinate $t$, for which the
restriction $j|_U$
is $\Delta^*\hookrightarrow\Delta$. Also, take $E$ to be a line bundle.
Restricting
to $U$, we consider the two inequivalent extensions of $E$, ${\overline E} = {\mathcal O}_\Delta$
and ${\overline E}^
\prime = {\mathcal O}_\Delta\cdot e^{t^\-}$. Let $\nabla$ be $\tfrac d{dt}$
(connection matrix
$\omega = 0$) and $\nabla^\prime = \nabla + \omega^\prime$, where $\omega^
\prime =
-t^{-2}dt$. Clearly, $\nabla$ is $F^1$ with respect to ${\overline E}$ (indeed
with respect
to $t^k{\overline E}$ for any $k\in{\mathbb Z}$); whereas $\nabla^\prime$, not $F^1$ with
respect to
${\overline E}$, {\it is} $F^1$ with respect to ${\overline E}^\prime$. (We leave it to the
reader to
squelch the possible misconception that the preceding contradicts GAGA.)
Note that
the preceding discussion is global when $X={\mathbb C} ^*$.
\end{example}
Fixing a meromorphic equivalence class $\mathfrak M$ of compactifications of
$X$,
we use (\ref{4.2.1}) to define
\begin{equation}\label{5.5.7}
F^p \widehat H^{2p}(X,\Lambda)_{\mathfrak M,\log} = \varinjlim F^p\widehat
H^{2p}
({\overline X} \log D,\Lambda),
\end{equation}
where the limit is taken over ${\overline X}\in \mathfrak M$;
and $F^p A^{2p-1}(X,\,\Lambda)_{\mathfrak M,\log}$ is defined analogously.
The following variant of \ref{4.2.3} is inevitable.
\begin{definition}\label{5.5.9}
An {\it $F^1$-connection} on $E$ {\it (relative to $\mathfrak M$)} is a
connection
that is $F^1$ relative to some member of $\mathfrak M$.
\end{definition}
\noindent Then \ref{4.2.5} yields immediately:
\begin{proposition}\label{5.5.10}
If ${\nabla}$ is an $F^1$-connection on $E$ relative to $\mathfrak M$,
then the Cheeger-Simons class
$\widehat{c}_p (E, \nabla)$ lies in the subgroup
$F^p\widehat H^{2p} (X, \Lambda)_{\mathfrak M,\log}$ of
$\widehat H^{2p} (X, \Lambda)$.
\end{proposition}
When $X$ is an algebraic manifold and $\mathfrak M$ is the [class generated by]
algebraic completions,
we drop the subscript ``$\mathfrak M$" in (\ref{5.5.7})--(\ref{5.5.10});
cf.~\ref{conv}.
\subsection{Proof of Theorem~\ref{thm2} (quasi-projective case)}
\label{6.2} We return to algebraic manifolds. First let $X$ be a smooth
projective variety (a fortiori compact), and $E$ an algebraic vector bundle on
$X$.
There exists an ample line bundle $L$ on $X$ such that both $L$ and
$E' = E \otimes L$ are generated by global sections, so that both
vector bundles are pullbacks of universal bundles on Grassmannians under
holomorphic
maps.
We get a pullback diagram:
$$
\begin{matrix}
E & \rightarrow & p_1^*U_n\otimes p_2^*U_1^{-1}\cr
\downarrow &&\downarrow\cr
X & \mathop {\to}\limits^g & G_{\mathbb C}(n) \times G_{\mathbb C}(1)
\end{matrix}
$$
in which $n$ denotes the rank of $E$, $p_1 \circ g$ classifies
$E'$, and $p_2 \circ g$ classifies $L$.
Also, choose any $F^1$-connection $\nabla^L$ on $L$. The latter,
together with any given $F^1$-connection $\nabla^E$ on $E$,
induces the tensor product $F^1$-connection:\footnote{
in terms of \ref{5.6.2}, the tensor product construction corresponds to
$
\Hom(T^{1,0}Y,{\mathfrak g}) \times \Hom(T^{1,0}Y,{\mathbb C}) \to \Hom(T^{1,0}Y,{\mathfrak g}).
$}
$$
\nabla^{E'}(e\otimes\ell)=\nabla^E e \otimes\ell+e\otimes\nabla^L\ell.
$$
Likewise, the tensor product of $\nabla^{E'}$ and the dual of
$\nabla^L$ is just $\nabla^E$ again (under the isomorphism
$E\cong E' \otimes L^{-1}$).
As the image in Deligne cohomology is independent of the $F^1$-connection
(see \ref{6.1.4}), we may assume that the connections are pullbacks of
$F^1$-connections
on $G_{\mathbb C}(n)$ and $G_{\mathbb C}(1)$, which exist by \ref{4.2.4}. Since $G_{\mathbb C}(n)\times
G_{\mathbb C}(1)$
satisfies the hypothesis of \ref{3.2.5} and \ref{4.1.3},\,(ii), the assertion
of Theorem~\ref{thm2}
holds for $p_1^*U_n\otimes p_2^*U_1^{-1}$, so by functoriality for $E$. This
completes
the proof when $X$ is compact.
We also use functoriality to get the assertion for non-compact $X$.
Specifically,
let ${\overline X}$ be an algebraic completion of the sort considered in \ref{4.2} for
which $E$
extends to an algebraic vector bundle ${\overline E}$ on ${\overline X}$. Then there is a
commutative diagram:
\begin{equation}
\label{quasi}
\begin{CD}
F^p\widehat H^{2p}({\overline X},{\mathbb Z}(p)) @>>> H^{2p}_{{\mathcal D}}({\overline X},{\mathbb Z}(p))\\
@VVV @VVV\\
F^p\widehat H^{2p}({\overline X}\log D,{\mathbb Z}(p)) @>>> H^{2p}_{{\mathcal D}}(X,{\mathbb Z}(p))
\end{CD}
\end{equation}
Let $\nabla^\prime$ be any $F^1$-connection on ${\overline E}$. By functoriality,
$\widehat c_p({\overline E},\nabla^\prime)$ restricts to
$\widehat c_p(E,\nabla^\prime)$ and $c^B_p({\overline E})$ maps to $c^B_p(E)$.
By \ref{6.1.4} again,
$\widehat c_p(E,\nabla^\prime)$ has the same image in $H^{2p}_{{\mathcal D}}(X,{\mathbb Z}(p))$
as $\widehat c_p(E,\nabla)$,
and we are done.
\subsection{General algebraic manifolds}\label{40.5}
We wish to generalize Theorem~\ref{thm2} to the context of general algebraic
manifolds, i.e., to ones that are not quasi-projective, so need not even be
K\"ahler. This is done by
reducing to the quasi-projective case as we next describe.
\begin{proposition}\label{CC}
Suppose that $f:Y\to X$ is a morphism of algebraic varieties satisfying:
\begin{enumerate}
\item[(i)] $f^*: H^{2p-1} (X,{\mathbb Z})/{tors}\to H^{2p-1} (Y,{\mathbb Z})/{tors}$ is
injective
with image a direct summand in the sense of mixed Hodge structures over ${\mathbb Z}$;
\item[(ii)] $f^*: H^{2p} (X,{\mathbb Z})_{\mathrm{tors}}\to H^{2p} (Y,{\mathbb Z}))_{\mathrm{tors}}$
is injective.
\end{enumerate}
Then $f^*: H^{2p}_{{\mathcal D}}(X,{\mathbb Z}(p)) \to H^{2p}_{{\mathcal D}}(Y,{\mathbb Z}(p))$
is injective.
\end{proposition}
\begin{proof} The exact sequence \ref{les} gives a commutative diagram with
exact rows:
\begin{equation}
\label{BB}
\begin{CD}
0 @>>> J^{\prime p} (X) @>>> H^{2p}_{{\mathcal D}}(X,{\mathbb Z}(p)) @>>> F^pH^{2p}(X,{\mathbb C})\\
@. @VVV @VVV @VVV\\
0 @>>> J^{\prime p}(Y) @>>> H^{2p}_{{\mathcal D}}(Y,{\mathbb Z}(p)) @>>> F^pH^{2p}(Y,{\mathbb C})
\end{CD}
\end{equation}
The following is standard:
\begin{lemma}\label{lemm} Given the commutative diagram with exact rows:
$$\CD
0 @>>> A @>>> B @>>> C \\
@. @V\alpha VV @V\beta VV @V\gamma VV \\
0 @>>> A' @>>> B'@>>> C'
\endCD
$$
a sufficient condition for the injectivity of $\beta$ is that $\alpha$ and
$\gamma$ be injective.
\end{lemma}
We apply the lemma twice.
The hypotheses imply that the rightmost vertical arrow in \ref{BB} is an
injection. Thus, to reach the conclusion,
it suffices by \ref{lemm} to show that leftmost arrow is injective. Recall
that $J^{\prime p}$ in the above is an extension by the intermediate Jacobian
$J^p$, having the functorial exact sequence
$$
0 \to J^p(X) \to J^{\prime p}(X) \to H^{2p}(X;{\mathbb Z}(p))_{\mathrm{tors}} \to 0.
$$
The hypothesis implies the injectivity of $J^p(X)\to J^p(Y)$ and of
$H^{2p}
(X;{\mathbb Z}(p))_{\mathrm{tors}} \to H^{2p}(Y;{\mathbb Z}(p))_{\mathrm{tors}}$. By \ref{lemm}, we are done.
\end{proof}
\begin{corollary}\label{FF}
Under the hypotheses of Proposition \ref{CC}, if the
conclusion of Theorem 2 holds for $Y$, then it also holds for $X$.
\end{corollary}
\begin{proof}
Consider the diagram that one gets by functoriality:
\begin{equation}
\begin{CD}
F^p\widehat H^{2p}(X) @>>> H^{2p}_{{\mathcal D}}(X,{\mathbb Z} (p))\\
@VVV @VVV\\
F^p\widehat H^{2p}(Y) @>>> H^{2p}_{{\mathcal D}}(Y,{\mathbb Z} (p))
\end{CD}
\end{equation}
Then $\widehat{c}_p(f^*E,\nabla)\in F^p\widehat H^{2p}(X)$ maps to
$c_p^B (f^*E) = f^*(c_p^B (E))$. Since the right vertical map is
injective by \ref{FF}, $\widehat{c}_p(E,\nabla)$ can only
map to $c_p^B (E)$ in $H^{2p}_{{\mathcal D}} (X,{\mathbb Z} (p))$.
\end{proof}
We next show that such $Y$ exist:
\begin {proposition}\label{DD}
\begin{enumerate}
\item[(i)] Given a smooth algebraic variety $X$, there is
a smooth quasi-projective variety $Y$ with surjective birational map $Y\to X$.
\item[(ii)] Under the conditions of (i), there is a direct sum decomposition
in the derived category of $X$:
$$
Rf_*{\mathbb Z}_Y \cong {\mathbb Z}_X \oplus \ker \tr(f).
$$
\item[(iii)] In particular,
$H^\bullet (Y,{\mathbb Z}) \cong H^\bullet (X,{\mathbb Z})\oplus \ker f_*$,
and the corresponding mixed Hodge structures are isomorphic over ${\mathbb Z}$.
\end{enumerate}
\end{proposition}
\begin {proof}
The argument for (i) was used already in \cite[II:(3.2)]{De2} for extending
Hodge theory to general algebraic varieties, and we sketch it here. By a
theorem of Nagata and the Chow Lemma, $X$ fits into a cartesian diagram
\begin{equation}
\begin{CD}
Y& \hookrightarrow & \overline Y\\
@VfVV @VV{\overline f}V\\
X& \hookrightarrow& \overline X
\end{CD}\label{AA}
\end{equation}
with $\overline X$ a completion of $X$, $\overline Y$ projective, and
$\overline f$ birational. (By resolution of singularities \cite{Hi1}, we may
assume
that $\overline X$ and $\overline Y$ are smooth, and $\overline X - X$ and
$\overline Y - Y$ are divisors with normal crossings.)
Statement (ii) is a topological assertion that follows from the identity
$\tr(f)\circ f^* = \mathbf 1_X$. Then (iii) follows immediately.
\end{proof}
Combining Propositions \ref{CC} and \ref{DD}, and keeping in mind our
convention \ref{conv}, we obtain:
\begin{theorem}\label{2a}
If $X$ is a non-singular complex algebraic variety, $E$ an algebraic vector
bundle on
$X$ that extends to an algebraic vector bundle on some completion of $X$, and
$\nabla$
an $F^1$-connection on $E$ (relative to said extension of $E$), then
$c_p^B (E)$
is the image of $\widehat{c}_p(E,\nabla)$ for all $p \geq 1$.
\end{theorem}
\begin{corollary}
The conclusion of Theorem~\ref{thm1} holds for all non-singular algebraic
varieties.
\end{corollary}
\subsection{Chern classes in $\mathfrak M$-DB-cohomology}\label{10.1}
We wish to define the analogue of $c_p^B(E)$ in the complex analytic setting.
Fix the meromorphic equivalence class $\mathfrak M$ of compactifications
$\overline X$ of $X$. We can use the formula in Def.~\ref{4.1.1} to define
$H^{\bullet}_{\mathcal D} (X,{\overline X}; \Lambda (p))$.
When ${\overline X}$ is K\"ahler,
this is seen to be independent of ${\overline X}\in\mathfrak
M$, for the same reason it was true in
\ref{4.1} (see \ref{hodge}), so we rename it:
\begin{definition}\label{MDB}
The $\mathfrak M$-DB-cohomology $H^{\bullet}_{\mathcal D} (X,\Lambda (p))_{\mathfrak M}$
of $X$ with coefficients in $\Lambda(p)$ is the common
value of $H^{\bullet}_{\mathcal D} (X,{\overline X};\Lambda (p))$ for ${\overline X}\in\mathfrak M$.
\end{definition}
Then (\ref{5.5.6}) gives rise to the exact sequence:
\begin{equation}\label{5.5.8}
F^p A^{2p-1}(X,\,\Lambda)_{\mathfrak M,\log}\rightarrow F^p \widehat H^{2p}(X,
\Lambda)_{
\mathfrak M,\log}\rightarrow H^{2p}_{\mathcal D} (X,\Lambda)_{\mathfrak M}\rightarrow 0.
\end{equation}
Next, fix a meromorphic equivalence class $\mathfrak E$ of extensions of $E$
to the members
of $\mathfrak M$ (note that $\mathfrak E$ subsumes $\mathfrak M$). We call
these the $\mathfrak E$-extensions
of the
vector bundle $E$ on $X$. We will construct Chern classes
\begin{equation}\label{CMBD}
c^B_p(E;\mathfrak E)\in H^{\bullet}_{\mathcal D} (X,{\mathbb Z} (p))_{\mathfrak M}
\end{equation}
in more or less the same way as in the algebraic setting. Choose ${\overline X}\in
\mathfrak M$ to
which $E$
extends to ${\overline E}\in\mathfrak E$, and let $j:X\hookrightarrow{\overline X}$ denote the
inclusion.
As usual, we consider first the case where $E$ is a line bundle.
Then there is a short exact sequence of complexes of sheaves on $\overline X$:
\begin{multline}\label{DBS}
0\to \cone\{{\mathbb Z}_{\overline X}(1)\to \mathcal O_{\overline X}\}[-1]
\to \cone\{Rj_*{\mathbb Z}_X(1)
\to \mathcal O_{\overline X}\}[-1]\cr
\to \cone\{{\mathbb Z}_{\overline X}(1)\to Rj_*{\mathbb Z}_X(1)\}\to 0.
\end{multline}
\noindent From this, we extract the following commutative diagram:
\begin{equation*}\label{DBH}
\begin{CD}
H_{\cD}^2(\overline X,\,{\mathbb Z}(1)) @>>> H_{\cD}^2(X,\,{\mathbb Z}(1))_{\mathfrak M} @. {} \\
@A\simeq A d\log A @AA c^B_1(\mathfrak E) A @. \\
H^1(\overline X,\mathcal O^*_{\overline X}) @>>> H^1(X,
\mathcal O^*_X)_{
\mathfrak E}\,\, @. \hookrightarrow H^1(X,\mathcal O^*_X),
\end{CD}
\end{equation*}
\noindent where $H^1(X,\mathcal O^*_X)_{\mathfrak E}$ denotes the
$\mathfrak E$-extendable line bundles on $X$.
Since any two extensions in $\mathfrak E$ differ by a divisor class supported
on $D$, we see that the Chern class $c_1^B(E;\mathfrak E)\in
H_{\cD}^2(
X,\,{\mathbb Z}(1))_{\mathfrak M}$ of an $\mathfrak E$-extendable line
bundle $E$
is well-defined.
To obtain the same for higher-rank $E$, one invokes the splitting
principle.
In the formulation of \cite[p.\,140,\,A1]{Gd}, one must know \ref{split}
below.
Because we will invoke a little Hodge theory in the argument, we must impose
a condition on $X$.
\begin{definition}\label{qck}
A complex manifold $X$ is said to be {\it quasi-c-K\"ahler} if it admits a
compactification
$\overline X$ that is a compact K\"ahler manifold, and for which $D=\overline
X - X$ is an
analytic subvariety.
\end{definition}
When $X$ is quasi-c-K\"ahler, one can then modify such $\overline X$ by a
sequence
of blow-ups with smooth center, a process that preserves the K\"ahler
condition, to make $D$ a divisor with normal crossings \cite{Hi2}.
Let
$\psi:\mathbb P(E)\to X$ be the projectivization of $E$, $L$ the
tautological line sub-bundle ${\mathcal O}(1)$ of $\psi^*(E)$, and
$\widetilde{\mathfrak M}$ the meromorphic equivalence
class of
$\{\mathbb P({\overline E}): {\overline E}\text{ an $\mathfrak E$-extension of } E\}$.
\begin{proposition}\label{split}
Let $X$ be quasi-c-K\"ahler, and $\mathfrak M$ a meromorphic equivalence class
of K\"ahler
compactifications. Let $\xi = c_1(L;\widetilde{\mathfrak M})$. Then there is
an isomorphism of
additive groups
$$
H^{\bullet}_{\mathcal D} (\mathbb P(E),{\mathbb Z} (p))_{\widetilde{\mathfrak M}}\cong \bigoplus_
{0\le j < r}
H^{\bullet}_{\mathcal D} (X,{\mathbb Z} (p))_{\mathfrak M}\cdot \xi^j,
$$
where $r$ is the rank of $E$; in other words, there is an injective map
$$
H^{\bullet}_{\mathcal D} (X,{\mathbb Z} (p))_{\mathfrak M}\to H^{\bullet}_{\mathcal D} (\mathbb P(E),{\mathbb Z} (p))_{
\widetilde{\mathfrak M}},
$$
and the right-hand side of the above is freely generated, as a module over
the
left-hand side, by $\{\xi^j: 0\le j < r\}$.
\end{proposition}
\begin{proof}
We want to reduce the assertion to more elementary cohomology for which one
already
knows the splitting principle. As in \ref{4.1.2}, there is a long exact
sequence:
\begin{multline}\label{4.1.2.A}
\dots\to H^{2p-1}(F^pA^{\bullet} ({\overline X}\log\,D))\to H^{2p-1}(X,{\mathbb C} /{\mathbb Z} (p))\cr
\to H^{2p}_{\mathcal D}
(X,{\mathbb Z} (p))_{\mathfrak M}\to H^{2p}(F^pA^{\bullet} ({\overline X}\log\,D))\to\dots
\end{multline}
Next, understanding $\xi$ to denote generically the Chern class of $L$ in any
cohomology
group,
multiply \ref{4.1.2.A} by $\xi^j$ ($0\le j < r$) and take the direct sum over
$j$ yields
\begin{equation}\label{4.1.3.A}
{\scriptsize
\setlength{\arraycolsep}{2pt}
\begin{array}{ccccc}
\oplus H^{2p-1-2j}(X,{\mathbb C} /{\mathbb Z} (p-j))\cdot\xi^j & \scriptstyle \to &
\oplus H^{2p-2j}_{\mathcal D} (X,{\mathbb Z} (p-j))_{\mathfrak M}
\cdot\xi^j & \scriptstyle \to & \oplus H^{2p-2j}(F^{p-j}A^{\bullet}({\overline X}\log\,D))
\xi^j \cr
\downarrow && \downarrow && \downarrow\cr
H^{2p-1}(\mathbb P(E),{\mathbb C} /{\mathbb Z} (p)) & \scriptstyle \to &
H^{2p}_{\mathcal D} (\mathbb P(E),{\mathbb Z} (p))_{\widetilde{\mathfrak M}} & \scriptstyle\to &
H^{2p}(F^pA^{\bullet} (\mathbb P({\overline E})\log\,\psi^*D))
\end{array}
}
\end{equation}
Now, we know the splitting principle for ${\mathbb Z}$-coefficients, hence also for
${\mathbb Z} (p)$- and
${\mathbb C}$-coefficients, and then also for $({\mathbb C} /{\mathbb Z} (p))$-coefficients. In other
words,
the left vertical arrow in the display \ref{4.1.3.A} is an isomorphism.
The easiest
way to deal with the one on the right is to use Hodge theory: there is, in
general, a canonical surjection
\begin{equation}\label{hodge}
H^{2k}(F^{k}A^{\bullet} ({\overline X}\log\,D))\rightarrow F^{k}H^{2k}(X,\,{\mathbb C} ),
\end{equation}
and this
is an isomorphism when ${\overline X}$ is K\"ahler. Thus, the right vertical arrow
is the
Hodge filtered version of the one for ${\mathbb C}$-coefficients, so it is an likewise
an isomorphism.
We conclude by the five-lemma that our assertion holds.
\end{proof}
The above determines \ref{CMBD}. We recall that $c_p(E;\mathfrak E)$ is, up
to a sign, the
coefficient in ${\mathbb H}^{2p}_{\mathcal D} (X,{\mathbb Z} (p))_{\mathfrak M}$ of $\xi^{r-p}$ in the
formula for
$\xi^r$ in terms of the additive decomposition given in \ref{split}.
Because of (\ref{6.1.4}), we obtain the generalization of Theorem \ref{thm2}
to the
K\"ahler case:
\begin{theorem}
Let $X$ be a quasi-c-K\"ahler manifold, $\mathfrak M$ an equivalence class of
K\"ahler
compactifications of $X$, $E$ an $\mathfrak M$-extendable vector
bundle on $X$,
and $\nabla$ an $F^1$-connection on $E$. Then $c_p^B (E;\mathfrak E)$ is the
image of
$\widehat{c}_p(E,\nabla)$ for all $p \geq 1$. Here, $\mathfrak E$ is the unique
equivalence
class of $\mathfrak M$-extensions of $E$ implied by \ref{mero}.
\end{theorem}
\begin{corollary}
The conclusion of Theorem 1 holds when $X$ is a quasi-c-K\"ahler manifolds
for the $\mathfrak M$-DB Chern classes of flat vector bundles that are
regular with respect to $\mathfrak E$.
\end{corollary}
\section{Proof of Theorem~\ref{thm4}}
\label{thm_3}
Throughout this section, the complexification ${\mathfrak g}\otimes {\mathbb C}$ of a
real Lie algebra ${\mathfrak g}$ will be denoted ${\mathfrak g}_{\mathbb C}$. In addition, we make
use of simplicial methods; all relevant definitions can be found in
Appendix~\ref{simp}.
\subsection{Continuous cohomology}\label{7.1} The continuous cohomology
$H_{\mathrm{cts}}^{\dot}(G,V)$ of a topological group $G$ with coefficients
in a real topological vector space $V$, on which $G$ acts, is
defined to be the cohomology of the complex $C_{\mathrm{cts}}^{\dot}(G,V)$,
whose elements in degree $m$ are $G$-equivariant continuous maps
$f: G^{m+1} \to V$ (with the usual coboundary map).
Now let $G$ be the real points of a reductive algebraic group defined
over ${\mathbb R}$, and $K$ a maximal compact subgroup. We recall that the
space $G/K$ is contractible, and thus $G$ and $K$ are of the same
homotopy type. Let ${\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}$ be the corresponding Cartan
decomposition of the Lie algebra of $G$. By van Est's Theorem~\cite{vE},
whenever $V$ is finite-dimensional with trivial $G$-action, there is
a canonical isomorphism
\begin{equation}\label{7.1.1}
H_{\mathrm{cts}}^{\dot}(G,V) \cong H^{\dot}({\mathfrak g},K) \otimes V.
\end{equation}
Here, the right-hand side is the relative Lie algebra cohomology,
and is canonically isomorphic to
$\left(\wedge ^{\dot} {\mathfrak p}^\ast\right)^K \otimes V$, the complex
of $G$-invariant $V$-valued differentials forms on $G/K$.
The isomorphism can be realized as follows. Fix a point $e$ of $G/K$,
and endow the latter with a $G$-invariant Riemannian metric. For
$(g_0,\ldots,g_m)\in G^{m+1}$, define $\Delta_e(g_0,\ldots,g_m)$ to
be the ``geodesic simplex'' in $G/K$ with vertices $g_0e,\ldots,g_me$;
it is constructed inductively as the cone swept out by all geodesics
from $g_0 e$ to $\Delta_e(g_1,\ldots,g_m)$. (Note that, in general,
the order of the vertices makes a difference, since the curvature
may not be constant.) Define a homomorphism
$$
\wedge^{\dot} {\mathfrak p}^\ast \to C_{\mathrm{cts}}^{\dot}(G,{\mathbb R})
$$
by taking an $m$-form $\omega$ to the function
\begin{equation}\label{7.1.2}
(g_0,\ldots,g_m) \mapsto
\int_{\Delta_e(g_0,\ldots,g_m)} \omega.
\end{equation}
This induces \ref{7.1.1}.
We now recall the standard trick for computing the continuous
cohomology of a reductive group. The compact real form of
${\mathfrak g}\otimes {\mathbb C}$ is the Lie algebra
${\mathfrak u} = {\mathfrak k} \, \oplus \, i{\mathfrak p} \subseteq {\mathfrak g}_{\mathbb C}$. Let $U$ be the
corresponding subgroup of
$G({\mathbb C})$. Since U is compact, there are canonical isomorphisms
$$
H^{\dot}(U/K,V) \cong H^{\dot}({\mathfrak u},K;V) \cong
\left(\wedge^{\dot}(i{\mathfrak p})^\ast\right)^K \otimes V.
$$
Composing these isomorphisms with the isomorphism
$$
\wedge ^{\dot}(i{\mathfrak p})^\ast \to \wedge ^{\dot} {\mathfrak p}^\ast
$$
induced by multiplication by $i$, we obtain an isomorphism
\begin{equation}\label{7.1.3}
H^{\dot}(U/K,{\mathbb C}) \mathop {\to}\limits ^{\sim}
H_{\mathrm{cts}}^{\dot}(G,{\mathbb C}),
\end{equation}
which carries $H^m(U/K,{\mathbb R}(p))$ onto the subspace $i^mH_{\mathrm{cts}}^m(G,{\mathbb R}(p))$
of $H_{\mathrm{cts}}^m(G,{\mathbb C})$.
\subsection{The Borel regulator elements}\label{7.2} We now take $G$
to be $GL_n(\C)$, viewed as a real group, and we take $K$ to be $U(n)$. In
this case, one identifies the complexification of $G$ as
$GL_n(\C) \times GL_n(\C)$; the natural map $G \to G_{\mathbb C}$ is the homomorphism
$$
\iota : GL_n(\C) \to GL_n(\C) \times GL_n(\C) ,
$$
with $\iota (g) = (g,\overline{g})$. The corresponding map on Lie algebras
\begin{equation}\label{7.2.1}
\gl_n(\C) \to \gl_n(\C) \oplus \gl_n(\C)
\end{equation}
takes $X$ to $(X,\overline{X})$; indeed, it is not hard to see that the
identification
\begin{equation}\label{7.2.2}
\gl_n(\C) \otimes _{\mathbb R} {\mathbb C} \cong \gl_n(\C) \oplus \gl_n(\C)
\end{equation}
is given by: for $X, Y \in \gl_n(\C)$,
\begin{equation}\label{7.2.3}
X\otimes 1 + Y\otimes i \mapsto \left(X + i Y, \overline X + i
\overline Y\right).
\end{equation}
It follows that the Lie algebra ${\mathfrak u}_n$ of $U(n)$ appears in
\ref{7.2.1} as the ``anti-diagonal"
$$
\{ (X, -^tX)~|~ X \in {\mathfrak u} _n \}.
$$
It is clear that the compact real form $U$ of $G_{\mathbb C}$ is
$U(n) \times U(n)$. The quotient space $U/K$ is therefore
$\left(U(n) \times U(n)\right)/ U(n)$, which we identify with
$U(n)$ via the map induced by inclusion as the first factor of the
product. Since $GL_n(\C)$ and $U(n)$ are homotopy-equivalent, we obtain
from \ref{7.1.3} an isomorphism
$$
H^{\dot}(GL_n(\C),{\mathbb C}) \cong H_{\mathrm{cts}}^{\dot}(GL_n(\C),{\mathbb C})
$$
which takes $H^m(GL_n(\C),{\mathbb R}(p))$ onto $i^mH_{\mathrm{cts}}^m(GL_n(\C),{\mathbb R}(p))$. This
yields a canonical isomorphism
\begin{equation}\label{7.2.4}
H^{2p-1}(GL_n(\C),{\mathbb R}(p)) \mathop {\longrightarrow}\limits^{\sim}
H_{\mathrm{cts}}^{2p-1}(GL_n(\C),{\mathbb C}/{\mathbb R}(p)).
\end{equation}
The real cohomology of $GL_n(\C)$ is an exterior algebra on generators
$x_1, \ldots, x_n$, where $x_p$ is in degree $2p -1$. We can choose
$x_p$ canonically by insisting that it be the unique (necessarily
primitive) class which transgresses, via
$$
d_{2p} : H^0(BGL_n(\C) ,H^{2p-1}(GL_n(\C))) \to H^{2p}(BGL_n(\C) ),
$$
to the universal Chern class $c_p \in H^{2p}(BGL_n(\C))$ in the Leray
spectral sequence associated to the universal $GL_n(\C)$-bundle; it
lies in
$$
W_{0}H^{2p-1}(GL_n(\C),{\mathbb Z}(p)),
$$
as ${\mathbb Z}(p)$ is of weight $-2p$. Define
$b_p \in H^{2p-1}(BGL_n(\C) ^\delta ,{\mathbb C}/{\mathbb R}(p))$ to be the image of $x_p$
under the composite of \ref{7.2.4} and the natural map
$$
H_{\mathrm{cts}}^{\dot}(GL_n(\C),{\mathbb C}/{\mathbb R}(p)) \to H_{\mathrm{cts}}^{\dot}(GL_n(\C) ^\delta ,{\mathbb C}/{\mathbb R}(p))
\cong H^{\dot}(BGL_n(\C) ^\delta ,{\mathbb C}/{\mathbb R}(p)).
$$
This is called the $p$-th {\it Borel regulator element}.\footnote{
Some prefer to define the Borel regulator element to be the class in
$H^{2p-1}(GL_n(\C)^\delta,{\mathbb C}/{\mathbb R}(p))$
that corresponds to the element $y_p$ of
$$
W_0 H^{2p-1}(GL_n(\C),{\mathbb Q}(p)) \cong W_0 H^{2p-1}(GL_p({\mathbb C}),{\mathbb Q}(p))
\cong H^{2p-1}({\mathbb C}^p - \{0\},{\mathbb Q}(p)) \cong {\mathbb Q}
$$
that takes the value 1 on the generator of $\pi_{2p-1}(GL_n({\mathbb C}))$.
Since $x_p = \pm (p-1)! y_p$ \cite[(24.5.2)]{Sch}, it follows that $y_p$
corresponds, up to a sign, to twice the degree $p$ part of the Chern
character
$$
ch_p : K_{2p-1}(F) \to H_{\cD}^1(\Spec F, {\mathbb R}(p)).
$$
for $F$ a number field or ${\mathbb R}$ or ${\mathbb C}$.}
\subsection{The Weil algebra}\label{7.3} Let $G$ be a real reductive
group and $K$ any compact subgroup. Denote the corresponding Lie
algebras by ${\mathfrak g}$ and ${\mathfrak k}$. Consider the {\it (real) Weil algebra}
\cite{C1}
\begin{equation}\label{7.3.1}
W({\mathfrak g}) = \wedge ^{\dot} {\mathfrak g}^\ast \otimes S^{\dot}({\mathfrak g}^\ast),
\end{equation}
a d.g.a. that is a ${\mathfrak g}$-module via the coadjoint action (here $S$ denotes
the symmetric algebra over ${\mathbb R}$). Denote the set of invariant polynomials
$S^{\dot}({\mathfrak g}^\ast)^G$, which is a subalgebra of $W({\mathfrak g})$, by $I^{\dot}(G)$.
Let $\omega \in A^1(P,{\mathfrak g})$ be a connection on a principal bundle
$P \to M$ over a (simplicial) manifold, which will be viewed as a map
${\mathfrak g}^\ast \to A^1(P)$. Then $\omega$ extends uniquely to a d.g.a.\
homomorphism
\begin{equation}\label{7.3.2}
k_P(\omega) : W({\mathfrak g}) \to A^{\dot}(P),
\end{equation}
such that for $\Phi \in S^p({\mathfrak g}^\ast)$,
$$
k_P(\omega)(\Phi) = \Phi(\Theta^p),
$$
where $\Theta$ denotes the curvature of the connection.
The {\it relative Weil algebra} $W({\mathfrak g},K)$ \cite{C2} is the subspace of
``$K$-basic" elements
\begin{equation}\label{7.3.3}
\left[\wedge^{\dot}\left({\mathfrak g}/{\mathfrak k}\right)^\ast \otimes
S^{\dot}({\mathfrak g}^\ast)\right]^K
\end{equation}
of $W({\mathfrak g})$. One recovers \ref{7.3.1} when $K$ is trivial; $W({\mathfrak g},K)$ is
a contravariant functor of pairs $({\mathfrak g} ,K)$, and it contains $I^{\dot}(G)$
for all $K$. A connection $\omega \in A^1(P,{\mathfrak g})$ on a principal bundle
$P \to M$ induces a d.g.a.\ homomorphism ({\it Chern-Weil homomorphism})
\begin{equation}\label{7.3.4}
k_P(\omega) : W({\mathfrak g},K) \to A^{\dot}(P/K)
\end{equation}
by restriction of \ref{7.3.2}. The natural map $W({\mathfrak g},K) \to I^{\dot}(K)$
(which factors through $I^{\dot}(G)$) is a quasi-isomorphism \cite{C2},
hence
$$
H^{\dot} (W({\mathfrak g} , K)) \cong H^{\dot} (K,{\mathbb R}).
$$
A consequence of this is that if $\Phi \in I^{\dot}(G)$ has the
property that its image in $I^{\dot}(K)$ is zero, then there exists
$T \in W({\mathfrak g},K)$ such that $dT = \Phi$. Note that when $K$ is
{\it maximal} compact, $K \hookrightarrow G$ is a homotopy equivalence,
so $P/K \to P/G = M$ is a homotopy equivalence, and \ref{7.3.4} induces
a map
$$
H^{\dot}(G,{\mathbb R}) \to H^{\dot}(M,{\mathbb R}).
$$
For a real vector space $V$, denote the Weil algebra $W({\mathfrak g})\otimes V$
with coefficients in $V$ by $W_V({\mathfrak g})$, and the corresponding Weil algebra
in the relative case by $W_V({\mathfrak g},K)$. In particular, we have the Weil
algebras $W_{\mathbb C}({\mathfrak g})$ (quasi-isomorphic to $A^{\dot} (EG) \cong {\mathbb C}$) and
$W_{\mathbb C}({\mathfrak g},K)$, as well as $W_{{\mathbb R}(p)}({\mathfrak g},K)$. When ${\mathfrak g}$ and ${\mathfrak k}$ are
{\it complex} Lie algebras, one can also define the complex Weil algebra
${\mathfrak W}({\mathfrak g},K)$ by doing the linear algebra in \ref{7.3.3}
over ${\mathbb C}$ instead of ${\mathbb R}$. We observe that when ${\mathfrak g}$ is a real Lie
algebra, $W_{\mathbb C}({\mathfrak g},K) \cong {\mathfrak W}({\mathfrak g}_{\mathbb C},K_{\mathbb C})$ as ${\mathbb C}$-algebras.
\subsection{The class $\tau_p$ and Cheeger-Simons classes}\label{7.4}
In this section, ${\mathfrak g}$ is $\gl_n(\C)$ viewed as a real Lie algebra, and
${\mathfrak k} = {\mathfrak u}_n$. Let
$$
C_p(X) = (-1)^p ~\tr (\wedge^p X),
$$
the invariant polynomial (of degree p) on $\gl_n(\C)$ which determines the
$p$th Chern class. We can express it as
$$
C_p(X) = P_p(X) + Q_p(X),
$$
where
$$
P_p(X) = \frac{1}{2}\left[C_p(X) + (-1)^p C_p(\overline{X})\right]\text{ and }
Q_p(X) = \frac{1}{2}\left[C_p(X) - (-1)^p C_p(\overline{X})\right]
$$
are elements of $S^p({\mathfrak g}^\ast)$. Note that $P_p$ is ${\mathbb R} (p)$-valued,
and $Q_p$ is ${\mathbb R} (p-1)$-valued. If $X\in {\mathfrak u}_n$, then $\overline{X} = -^t X$,
so $Q_p(X) = 0$. It follows from the discussion in the previous section
that there exists $T_p \in W_{{\mathbb R}(p)}(\gl_n(\C),U(n))$ such that $dT_p = Q_p$.
The connection on the universal flat bundle $P \to BGL_n(\C) ^\delta$ induces
a Chern-Weil homomorphism
$$
k : W_{\mathbb C}(\gl_n(\C),U(n)) \to A^{\dot}\left(P/U(n)\right).
$$
Denote $k(T_p) \in A^{2p-1}(P/U(n))$ by $\tau_p$; the curvature
$\Theta$ of $P$ is zero. We have
$$
d\tau_p = Q_p(\Theta) = 0,
$$
and thus $\tau_p$ defines a class in
$$
H^{2p-1}(P/U(n), {\mathbb C}) \cong H^{2p-1}(GL_n(\C)^\delta,{\mathbb C}).
$$
The first task is to show:
\begin{proposition}\label{7.4.1}
The class of $\tau_p$ in $H^{2p-1}(GL_n(\C)^\delta,{\mathbb C}/{\mathbb R}(p))$ is the
negative of the universal real Cheeger-Simons class $\widehat{c}_p$.
\end{proposition}
\begin{proof}
Let $E\to B$ be the model of the universal $GL_n(\C)$-bundle constructed
from $P \to BGL_n(\C)^\delta$ (over the standard model of $BGL_n(\C)^\delta$),
as in \ref{6.1.5}. Recall that $E$ has a canonical connection, induced
by that of $P$, and there is a tautological map $BGL_n(\C)^\delta \to B$
which classifies both the universal flat bundle and its connection.
Consider the Chern-Weil homomorphism
$$
k_E : W_{\mathbb C}(\gl_n(\C),U(n)) \to A^{\dot}\left(E/U(n)\right)
$$
associated to $E$ and its connection, writing $\tau_p^E$ for $k_E(T_p)$.
Since the polynomial $P_p$ is ${\mathbb R}(p)$-valued,
$$
C_p(\Theta _E) \equiv Q_p(\Theta _E) \mod {\mathbb R}(p));
$$
and since $dT_p = Q_p$, it follows that $d\tau_p^E = Q_p(\Theta _E)$,
so $(Q_p(\Theta _E), -\tau_p^E)$ represents the ``universal
Cheeger-Simons class"
$$
\widehat{c}_p \in \widehat{H}^{2p}(E/U(n),{\mathbb R}(p)),
$$
i.e., that of the pullback of $E \to B$ to $E/U(n) \simeq B$.
The naturality of Chern-Weil construction implies that the diagram
$$
\begin{matrix}
W_{\mathbb C}(\gl_n(\C),U(n)) & \mathop {\to}\limits^{k_E} &
A^{\dot}\left(E/U(n)\right)\cr
& \mathop {\searrow}\limits_k & \downarrow \cr
& & A^{\dot}\left(P/U(n)\right) \cr
\end{matrix}
$$
commutes. It follows that the Cheeger-Simons class of the pullback
of $P \to BGL_n(\C)^\delta$ to $P/U(n)$ is represented by
$(Q_p(\Theta ),-\tau_p) = (0,-\tau_p)$, and this gives the desired
assertion.
\end{proof}
\subsection{$\tau_p$ and $T_p$}\label{7.5}
The next step is to obtain a formula for the cocycle given by
$\tau_p$ in terms of $T_p$. For this we will need an explicit section
of the bundle $P/U(n) \to BGL_n(\C)^\delta$. It is more convenient to write
down a section of $EGL_n(\C)/U(n) \to BGL_n(\C)$ and pull it back to the
universal flat bundle. Actually, we will write down a section of the
bundle $\overline \pi : EG/K \to BG$, where $K$ is a maximal compact
subgroup of an arbitrary real reductive group $G$, following \cite{D1}.
The $m$-simplex $G^{m+1}$ of the usual model of $EG$ (see \ref{6.1.5},
and take $X_{\dot}$ to be a point) is identified with $G^{m+1}$ in
the {\it inhomogeneous}, or {\it reduced}, model via the map
$$
(g_0,g_1,\ldots,g_m) \mapsto (g_0,g_1g_0^{-1},\ldots,g_mg_{m-1}^{-1}).
$$
This map is $G$-equivariant with respect to the diagonal right
$G$-action on the left-hand side, and the right action of $G$ on the
first factor of the right-hand side. It follows that in the reduced
model, the space of $m$-simplices of $EG/K$ is $G/K \times G^m$. In
particular, when $G=K$, we have that the space of $m$-simplices of
the reduced model of $BG$ is $G^m$.
Suppose now that $G$ is reductive and that $K$ is a maximal compact
subgroup. For each $m$, define a map (which is a homotopy equivalence)
\begin{equation}\label{7.5.1}
s_m : G^m \times \Delta^m \to G/K \times G^m \times \Delta^m
\end{equation}
by
$$
\scriptstyle
\left((g_1,\ldots, g_m),(t_0,\ldots,t_m)\right) \mapsto
\left((\widetilde \Delta_e ((g_1,g_2g_1,\ldots,g_mg_{m-1}\cdots
g_1),(t_0,\ldots,t_m)),
(g_1,\ldots, g_m),(t_0,\ldots,t_m)\right),
$$
where $\widetilde \Delta_e : G^m \times \Delta^m \to G/K$ parametrizes
the geodesic simplices defined in \ref{7.1.2}.
\begin{proposition}{\cite[p.~241]{D1}}\label{7.5.2}
The maps \ref{7.5.1} are compatible with the face maps and
therefore induce a section $s: BG \to (EG)/K$ of $\overline \pi$.
\endproof
\end{proposition}
We now show that the cohomology class on $GL_n(\C)^\delta$ defined by
the continuous cohomology class corresponding to
$T_p \in \wedge^{2p-1}{\mathfrak p}^\ast$ equals the cohomology class defined by
$s^\ast\tau_p$. With respect to the inhomogeneous model
(see \ref{7.5.1}), the connection form of the universal flat bundle
is given locally by the Maurer-Cartan form in the vertical direction,
and is zero in the horizontal directions. Consequently, with respect
to the inhomogeneous model, the restiction of
$\tau_p$ to $GL_n(\C)/U(n)\times G^m \times \Delta^m$ satisfies
$$
\tau_p \in A^{2p-1}(GL_n(\C)/U(n)) \subseteq
A^{2p-1}(GL_n(\C)/U(n)\times G^m \times \Delta^m),
$$
and that it is the image of $T_p$ under the natural inclusion
$$
\wedge^{2p-1} {\mathfrak p}^\ast \hookrightarrow A^{2p-1}(GL_n(\C)/U(n)).
$$
It follows immediately that the value of the cocycle
$$
s^\ast\tau_p\in A^{2p-1}(BGL_n(\C)^\delta)
$$
on the simplex of $BGL_n(\C)$ with vertices $(g_0,\ldots,g_m)$ is given
by the integral
$$
\int_{\Delta_e(g_0,\ldots,g_m)} T_p.
$$
\subsection{$T_p$ and the Borel regulator element}\label{7.6}
In the final step, the idea is to use the complexification to show
that $T_p$ can be interpreted as a class that transgresses to the
$p$th Chern class of $E \to B$.
An invariant polynomial on $\gl_n(\C)$ defines, in an obvious way, an
invariant function on ${\mathfrak g}_{\mathbb C}$. Using \ref{7.2.2} to write the
elements of ${\mathfrak g}_{\mathbb C}$ as pairs $(X,Y)$, with $X,Y \in \gl_n(\C)$, we have
for $Q_p$ of (7.4):
\begin{equation}\label{7.6.1}
Q_p(X,Y) = \frac{1}{2}\left[C_p(X) - (-1)^p C_p(Y)\right].
\end{equation}
{}From this, we see that for all $X \in \gl_n(\C)$,
\begin{equation}\label{7.6.2}
C_p(X) = 2\,Q_p(X,0).
\end{equation}
Likewise, we can view $T_p$ (also from \ref{7.4}) as an element
of the complex Chern-Weil algebra ${\mathfrak W}({\mathfrak g}_{\mathbb C} , U(n)_{\mathbb C} )$,
which then determines elements of $W_{\mathbb C}({\mathfrak u}_n\oplus{\mathfrak u}_n,U(n))$ and
$$
W_{\mathbb C}({\mathfrak u}_n \oplus \{ 0\} ) \cong W_{\mathbb C}({\mathfrak u}_n),
$$
which we continue to denote $T_p$.
{}From \ref{7.6.2}, it follows that, in $W_{\mathbb C}({\mathfrak u}_n)$, the relation
$2\,dT_p = C_p$ holds. In other words, the invariant form on $U(n)$
defined by $2\,T_p$ represents a class which transgresses to the
$p$th Chern class. The corresponding class in
$$
\wedge ^{\dot} \left({\mathfrak u}_n\right)^\ast \mathop {\to} \limits^\sim
_{{pr_1}^\ast} \wedge ^{\dot} \left(({\mathfrak u}_n \oplus {\mathfrak u}_n)/{\mathfrak u}_n\right)^\ast
\cong \wedge ^{\dot} (i{\mathfrak p}^\ast)
$$
is the image of $2\,T_p$ in $\wedge^{2p-1}(i{\mathfrak p}^\ast)$.
By the definition given in \ref{7.2}, the Borel element is
represented by the differential form
$$
i^{2p-1}\left(2\,T_p|_{\wedge^{2p-1}i{\mathfrak p}}\right)
= (-1)^{2p-1}\left(2\,T_p|_{\wedge^{2p-1}{\mathfrak p}}\right) = -2\,T_p \in
\wedge^{2p-1} {\mathfrak p}^\ast.
$$
Combining the above with \ref{7.4} and \ref{7.5}, we have shown
that the Borel element is twice the Cheeger-Simons class, and
Theorem~3 is proved.
\section{Towards a Proof of Conjecture~4}
\label{discussion}
In this section we make use of simplicial methods. Again, see
Appendix~\ref{simp} for definitions and constructions.
\subsection{The goal}\label{8.1}
We would have preferred to have
Theorem~\ref{thm2} and Conjecture~\ref{conj}
as special cases of the simplicial
analogue of Theorem~\ref{thm2}, which we have not been able to prove:
\begin{statement}\label{6.1.1}
Suppose that $ \pi : {\overline P}_{\bullet} \rightarrow {\overline X}_{\bullet}$ is a morphism
in the category of smooth simplicial varieties, with ${\overline X}_{\bullet}$
complete, and that $D_{\dot}$ and $Q_{\dot} = \pi^{-1}(D_{\dot})$
and are divisors with normal crossings in
${\overline X}_{\bullet}$ and ${\overline P}_{\bullet}$, respectively. Let
$P_{\dot} = {\overline P}_{\bullet}- Q_{\dot}$ and
$X_{\dot} = {\overline X}_{\bullet} - D_{\dot}$. If the restriction
$P_{\dot} \rightarrow X_{\dot}$ of $\pi$ is a
principal $GL_n({\mathbb C})$-bundle with $F^1$-connection ---
i.e., the connection form $\omega$ satisfies
$$
\omega \in F^1 E^1(|{\overline P}_{\bullet}|\log Q_{\bullet},\End({\overline E}_{\bullet})),
$$
then the image of its Cheeger-Simons class
$\widehat{c}_p$ in $H_{\cD}^{2p}(X_{\dot} ,{\mathbb Z}(p))$ under the natural homomorphism
$$
F^p \widehat H^{2p} ({\overline X}_{\bullet}\log D,{\mathbb Z} (p))\longrightarrow
H_{\cD}^{2p}(X_{\dot}, {\mathbb Z}(p))
$$
is the Beilinson Chern class $c^B_p$.
\end{statement}
\begin{remark}\label{6.1.2}
Note that we are {\it not} assuming in the above that ${\overline P}_{\bullet}$ is
proper over ${\overline X}_{\bullet}$, but only that it is so ``in the horizontal
direction". In practice, $P_{\dot}$ would be the frame
bundle of a vector bundle $E_{\dot}$ on $X_{\dot}$, and ${\overline P}_{\bullet}$
the frame bundle of an extension ${\overline E}_{\dot}$ of $E_{\dot}$ to
${\overline X}_{\bullet}$.
\end{remark}
The following lemma, whose proof is quite direct, provides good evidence for
the conjecture.
(We retain the previous notation.)
\begin{lemma}\label{6.1.3}
Suppose that one has two connections $\nabla_0$ and $\nabla_1$
on $E_{\dot}$, and denote by $\Theta_0$ and $\Theta_1$ the
respective curvature forms. Write $\omega = \nabla_1 - \nabla_0$
and $\eta_p$ for the solution of
$$
d\eta_p = c_p(E_{\dot}, \nabla_1) - c_p(E_{\dot}, \nabla_0)
$$
given by \ref{3.5.5}. If
$$
\Theta_0, \Theta_1 \in F^1A^2(|{\overline X}_{\bullet}|\log D_{\dot},
\End({\overline E}_{\dot}))\,\, {\it and }\,\,
\omega \in F^1(A^1(|{\overline X}_{\bullet}|\log D_{\dot}, \End({\overline E}_{\dot}))),
$$
then
\begin{enumerate}
\item[(i)]$\eta_p \in F^p A^{2p-1}(|{\overline X}_{\bullet}|\log D_{\dot}))$;
\item[(ii)]$c_p(E_{\dot}, \nabla_0)$ and $c_p(E_{\dot},\nabla_1)$
represent the same cohomology class in
$$
H^{2p}(F^pA^{{\bullet}}(|{\overline X}_{\bullet}|\log D_{\dot}))=F^pH^{2p}(|X_{\dot}|,{\mathbb C}).
$$
\end{enumerate}
\end{lemma}
\begin{proof} This is a direct consequence of \ref{3.5.4} and
\ref{3.5.5}, as the assumptions imply that $\Theta_t$ is in $F^1$.
\end{proof}
The following ``refinement" of \ref{6.1.3}, which we will soon
prove, is a necessary condition for \ref{6.1.1}:
\begin{proposition}\label{6.1.4} Under the conditions of
\ref{6.1.3}, these two Cheeger-Simons classes have the same
image in $H_{\cD}^{2p}(X_{\dot},{\mathbb Z}(p))$.
\end{proposition}
The basic theme in any proof of \ref{6.1.1} is to reduce to the
universal case (where the assertion is a tautology; see
\ref{3.2.4}, and \ref{6.1.10} below), invoking functoriality.
We wanted to make use of the following device:
\begin{proposition}\label{6.1.5}
If $X_{\dot}$ is a simplicial manifold and
$P_{\dot} \rightarrow X_{\dot}$ is a principal
$GL_n({\mathbb C})$-bundle with connection given by
$$
\omega \in A^1(|P_{\dot}|, \Ad(\gl_n(\C))),
$$
then there exists a smooth bisimplicial variety $B_{\dot \dot}$ which
has the homotopy type of $BGL_n({\mathbb C})$ and a $GL_n({\mathbb C})$-principal bundle
$U_{\dot \dot} \rightarrow B_{\dot \dot}$ with connection
$$
\omega_U \in A^1(|U_{\dot \dot}|,\Ad(\gl_n(\C)))
$$
and a morphism of $GL_n({\mathbb C})$-bundles
$$
\begin{matrix}
P_{\dot} & \buildrel {G} \over \longrightarrow & U_{\dot \dot} \cr
\downarrow & & \downarrow \cr
X_{\dot} & \buildrel {g} \over \longrightarrow & B_{\dot \dot}
\end{matrix}
$$
such that $G^{\ast}\omega_U = \omega$. Moreover, if $X_{\bullet}$ is a smooth
simplicial variety such that
$$
\omega \in F^1A^1(|P_{\dot}|,\Ad(\gl_n(\C))),
$$
then we can choose $\omega_U$ to satisfy
$$
\omega_U \in A^{1,0}(|U_{\dot \dot}|,\Ad(\gl_n(\C))).
$$
\end{proposition}
\begin{proof} Consider the bisimplicial variety $U_{\dot \dot}$
whose simplicial variety of $m$ simplices is the simplicial variety
$(P_{\dot})^{m+1}$ with face maps
\begin{align*}
d_i : (P_{\dot})^{m+1} & \rightarrow (P_{\dot})^m
\qquad 0\le i \le m\cr
(u_0, \ldots, u_m) & \mapsto (u_0, \ldots, \hat{u}_i,
\ldots, u_m).
\end{align*}
The geometric realization of this simplicial variety is contractible
as it is the coskeleton of the trivial covering (cf.\ \cite{AM} or
\cite[p.~107]{Se}).
\footnote{This may be proved directly as follows. First show that the
geometric realization of the simplicial space is simply connected. This is
not difficult, as the fundamental group depends only on the 2-skeleton.
One then shows that this space has trivial integral homology, and is
therefore contractible, by standard arguments.} The free $GL_n({\mathbb C})$
action on $P_{\dot}$ induces a free $GL_n({\mathbb C})$ action on $U_{\dot \dot}$.
Let $B_{\dot \dot}$ be the quotient of $U_{\dot \dot}$. The resulting
principal bundle $U_{\dot \dot} \rightarrow B_{\dot \dot}$ is a model of
the universal $GL_n({\mathbb C})$-bundle.
Define a connection $\omega_U \in A^1(|U_{\dot \dot}|,\gl_n(\C))$ on the
bundle $|U_{\dot \dot}| \rightarrow |B_{\dot \dot}|$ by the compatible
family of 1-forms
$$
\sum_{j=0}^m t_j\pi^{\ast}_j \omega \in A^1(|P_{\dot}|^{m+1} \times
\Delta^m),
$$
where $(t_0, \ldots, t_m)$ are the barycentric coordinates of
$\Delta^m$, and $\pi_j : |P_{\dot}|^{m+1} \rightarrow |P_{\dot}|$
denote the canonical projections, $0 \leq j \leq m$.
The canonical isomorphism $P_{\dot} \rightarrow U_{{\bullet} 0}$
induces a $GL_n({\mathbb C})$-equivariant map
$G: P_{\dot} \rightarrow U_{\dot \dot}$, and therefore a map
$g: X_{\dot} \rightarrow B_{\dot \dot}$ such that the
pair of maps $(g,\, G)$ classifies the bundle and the connection.
The remaining assertion is easy to verify.
\end{proof}
\begin{remark}\label{6.1.7}
In other words, given a principal $GL_n({\mathbb C})$-bundle with connection
over a complete smooth simplicial variety , we can build a model of the
universal $GL_n({\mathbb C})$-bundle, and put a connection on this bundle, such that
a {\it fixed} mapping of our variety into this model of $BGL_n({\mathbb C})$
simultaneously classifies the bundle and its connection. Unfortunately the
above does not yield a proof of \ref{6.1.1}, for the
Maurer-Cartan
forms of $GL_n({\mathbb C})$, and hence the connection constructed in \ref{6.1.5}, are
not
logarithmic at infinity for $n>1$. This was our fundamental obstacle.
\end{remark}
However, \ref{6.1.5} does yield at once the following useful observation:
\begin{proposition}\label{6.1.6}
Suppose that $\nabla_1$ is a second connection on $E_{\dot}$ and that
$\widehat{c}_p (E_{\dot},\nabla_0)$ is represented by
$(c_p (E_{\dot}, \nabla_0), -y)$, as in \ref{3.5.17}. Let $\eta_p$
be as in \ref{3.5.5}. Then
$\widehat{c}_p (E_{\dot}, \nabla_1)$ is represented by
$$
(c_p (E_{\dot}, \nabla_1), -(y + \iota_{{\mathbb Z}(p)}(\eta_p ))) =
(c_p (E_{\dot}, \nabla_0), -y) + (d\eta_p, -\iota_{{\mathbb Z}(p)}(\eta_p ))).
$$
\end{proposition}
\begin{proof} By the functoriality of $c_p$, $\eta_p$ and
$\widehat{c}_p$, it suffices to check this ``universally", on the model of
$BGL_n({\mathbb C})$ from \ref{6.1.5}. But there, the Cheeger-Simons class
is completely determined by the Chern form (see \ref{3.4}), so it
comes down to the calculation:
$$
\delta (y + \iota_{{\mathbb Z}(p)}(\eta_p)) = \iota_{{\mathbb Z}(p)}(c_p
(E_{\dot} , \nabla_0)) +
\iota_{{\mathbb Z}(p)}(d \eta_p) = \iota_{{\mathbb Z}(p)}(c_p (E_{\dot} ,
\nabla_1)).
$$
\end{proof}
Using \ref{6.1.6}, we can now give:
\begin{proof}[Proof of \ref{6.1.4}] In the DB-complex of $X_{\dot}$, we
have that:
$$
(c_p (E, \nabla_0), -y)
- (c_p (E, \nabla_1), -(y + \iota_{{\mathbb Z}(p)}(\eta_p ))) =
(-d\eta_p , \iota_{{\mathbb Z}(p)}(\eta_p )) = D(\eta_p , 0),
$$
so \ref{6.1.4} follows.
\end{proof}
The simplicial version of \ref{4.2.1} is
\begin{equation}\label{6.1.8}
F^p\widehat H^k(|{\overline X}_{\bullet}|\log D_{{\bullet}}, \Lambda )
= H^{k-1}(\cone\{ F^pA^{\geq k}(|{\overline X}_{\bullet}|\log {\overline D}_{\bullet}) \rightarrow
S^{{\bullet}}(|X_{\dot} |, {\mathbb C} / \Lambda )\}).
\end{equation}
Taking $k = 2p$, and $\Lambda = {\mathbb Z}(p)$, and recalling the definitions in
Section~\ref{db_coho}, we obtain the following diagram, in which the rows
are exact:
\begin{equation}\label{6.1.9}
{\setlength{\arraycolsep}{4pt}
\begin{matrix}
\scriptstyle 0 & \scriptstyle \to &
\scriptstyle H^{2p-1}(|X_{\dot}|,{\mathbb C}/{\mathbb Z}(p)) &
\scriptstyle \to &
\scriptstyle F^p \widehat H^{2p}({\overline X}_{\bullet}\log D_{{\bullet}}, {\mathbb Z}(p) ) &
\scriptstyle \to & \scriptstyle
F^pA^{2p}(|{\overline X}_{\bullet}|\log D_{\dot},{\mathbb Z}(p))_{cl}
& \scriptstyle \to & \scriptstyle 0 \cr
& & || & & \scriptstyle \downarrow & & \scriptstyle \downarrow & & \cr
& & \scriptstyle H^{2p-1}(|X _{\dot}|,{\mathbb C} /{\mathbb Z}(p) ) & \scriptstyle \to
&\scriptstyle H_{\cD}^{2p}(X _{\dot} , {\mathbb Z}(p) ) & \scriptstyle \to &
\scriptstyle F^pH^{2p}(|X_{\dot}|,{\mathbb C}) & &
\end{matrix}
}
\end{equation}
We note that the following is easily read from \ref{6.1.9}:
\begin{proposition}\label{6.1.10} If $H^{2p-1}(|X _{\dot} |,{\mathbb C} /{\mathbb Z}(p))=0$,
then the assertion in \ref{6.1.1} holds for every holomorphic vector
bundle on $|X _{\dot} |$.
\end{proposition}
Next, consider the diagram
$$
\begin{matrix}
H_{\cD}^{2p}({\overline X}_{\bullet}, {\mathbb Z}(p)) & \longrightarrow & H_{\cD}^{2p}(X_{\dot},
{\mathbb Z}(p)) \cr
\uparrow & & \uparrow \cr
F^p \widehat H^{2p} ({\overline X}_{\bullet}, {\mathbb Z} (p)) & \longrightarrow &
F^p \widehat H^{2p} ({\overline X}_{\bullet} \log D_{{\bullet}}, {\mathbb Z} (p))
\end{matrix}
$$
Here, we would like to use \ref{6.1.4} to replace the given connection by
one that extends without singularity to the compactification, without
changing the image in Deligne-Beilinson cohomology; the desired
conclusion then follows by functoriality. This can always be done
when $X_{\dot}$ is just a single algebraic manifold $X = {\overline X} - D$.
Thus, the proof of Theorem~\ref{thm2} (the non-simplicial version of
\ref{6.1.1}) reduced to the compact case (see \ref{6.2}).
\begin{remark}\label{6.1.11}
It should be apparent that the preceding development goes through
verbatim to the setting of simplicial complex manifolds $X _{\dot}$
that are of the form ${\overline X}_{\bullet} - D_{\dot}$, with ${\overline X}_{\bullet}$
compact K\"{a}hler and $D_{\dot}$ a divisor with normal crossings in
${\overline X}_{\bullet}$. In fact, even the condition that ${\overline X}_{\bullet}$ be K\"{a}hler
can be dispensed with; one simply makes the distinction between the two
sides (isomorphic when ${\overline X}_{\bullet}$ is K\"{a}hler) of
$$
F^pH^{{\bullet}} (X_{\dot}, {\mathbb C} ) \longleftarrow H^{{\bullet}}
(F^pA^{\bullet} (|{\overline X}_{\bullet} |\log D_{\dot} ))
$$
and
$$
H^{{\bullet}} (X_{\dot}, {\mathbb C} )/F^pH^{{\bullet}} (X_{\dot}, {\mathbb C} )
\hookrightarrow
H^{{\bullet}} (A^{\bullet} (|{\overline X}_{\bullet} |\log D_{\dot} )/F^p),
$$
and takes the right-hand member wherever we have written the left.
\end{remark}
We present in Appendix D a couple of plausible techniques we tried in our
unsuccessful attempt to prove \ref{6.1.1}.
}
|
1995-02-22T06:20:21 | 9502 | alg-geom/9502022 | en | https://arxiv.org/abs/alg-geom/9502022 | [
"alg-geom",
"math.AG"
] | alg-geom/9502022 | Tyler J. Jarvis | Tyler J. Jarvis | Torsion-Free Sheaves and Moduli of Generalized Spin Curves | AMS-LaTeX, 31 pages, uses amscd.sty | null | null | null | null | This article treats compactifications of the space of generalized spin
curves. Generalized spin curves, or $r$-spin curves, are pairs $(X,L)$ with $X$
a smooth curve and $L$ a line bundle whose r-th tensor power is isomorphic to
the canonical bundle of $X.$ These are a natural generalization of $2$-spin
curves (algebraic curves with a theta-characteristic), which have been of
interest recently, in part because of their applications to fermionic string
theory. Three different compactifications over $\Bbb{Z}[1/r],$ all using
torsion-free sheaves, are constructed. All three yield algebraic stacks, one of
which is shown to have Gorenstein singularities that can be described
explicitly, and one of which is smooth. All three compactifications generalize
constructions of Deligne and Cornalba done for the case when $r=2.$
| [
{
"version": "v1",
"created": "Tue, 21 Feb 1995 20:45:01 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Jarvis",
"Tyler J.",
""
]
] | alg-geom | \section*{Introduction}
This article treats the compactification of the space of higher spin
curves, i.e. pairs $(X,{L})$ with ${L}$ an $\mbox{$r\th$}$ root of the
canonical bundle of $X.$ More precisely, for positive integers $r$
and $g,$ with $g > 2,$ $r$ dividing $2g-2,$ and for a flat family of
smooth curves $f:\mbox{${\cal X}$}
\rightarrow T,$ an {\em r-spin structure} on \mbox{${\cal X}$}\ is a line bundle
\mbox{${\cal L}$}\ such that $\mbox{${\cal L}$}^{\otimes r} \cong \omega_{\mbox{${\cal X}$}/T}.$
And an {\em r-spin curve} over $T$ is a flat family of smooth curves
with an $r$-spin structure. Now, for a fixed base scheme $S$ over
\mbox{${\Bbb Z}[1/r]$}, let \mbox{${\functor{Spin}}_{r,g}$}\ be the sheafification of the functor which takes an
$S$-scheme $T$ to the set of isomorphism classes of $r$-spin curves
over $T.$ A compactification of the space of spin curves is a space
(scheme or algebraic stack), which is proper over \mbox{$\overline{\mg}$}\ (the
Deligne-Mumford compactification of the space of curves), and whose
fibre over \mbox{${\cal M}_g$}\ represents, at least coarsely, the functor \mbox{${\functor{Spin}}_{r,g}$}.
It is possible (see \cite{thesis}) to compactify \mbox{${\functor{Spin}}_{r,g}$}\ using
geometric invariant theory. Namely, in the style of L. Caporaso
\cite{cap:thesis}, for a fixed $d>\! >0$ one can choose a subscheme of
the Hilbert scheme \mbox{${\text{Hilb}}^{dz-g}_{\pn}$}\ with a geometric quotient that coarsely
represents \mbox{${\functor{Spin}}_{r,g}$}. And using results of Gieseker (c.f.
\cite[Theorems 1.0.0 and 1.0.1]{gieseker}, ) one can show that the
semi-stable closure of the subscheme in \mbox{${\text{Hilb}}^{dz-g}_{\pn}$}\ has a categorical
quotient that provides a compactification. This compactification is
actually a subscheme of Caporaso's compactification of the relative
Picard scheme over \mbox{$\overline{\mg}$}.
The principle drawback to the GIT compactification is that it is not
obviously the solution to a moduli problem, and therefore it is
difficult to describe the resulting space and to make the construction
work over a general base, rather than only over algebraically closed
fields. Moreover, the GIT construction requires that one make some
arbitrary choices, and it is not clear that the resulting
compactification is completely independent of these choices.
Therefore, the approach we take here is to pose a moduli problem,
using torsion-free sheaves, and then show that the associated stack is
actually algebraic and that it does indeed compactify \mbox{${\functor{Spin}}_{r,g}$}.
We discuss three different moduli problems that provide
compactifications and describe some of their characteristics. The
naive approach would be to use a rank-one torsion-free sheaf $\mbox{${\cal E}$}$
with a suitable $\mbox{${\cal O}$}_{\mbox{${\cal X}$}}$-module homomorphism from $\mbox{${\cal E}$}^{\otimes r}$
to the canonical bundle. But this doesn't quite work, as the resulting
space is not separated. Some additional conditions on the cokernel of
the homomorhism are necesary to make the stack separated, and the
resulting moduli space is called the space of {\em quasi-spin} curves.
The moduli of quasi-spin curves is relatively easy to construct, but
is difficult to describe, due to the presence of nilpotent elements.
Two better moduli problems are {\em spin curves} and {\em pure spin
curves}. These are quasi-spin curves with some additional local
conditions. The local conditions require the use of log structures,
and thus the construction of the moduli space of spin curves and pure
spin curves is more difficult than for quasi-spin curves. But the
resulting spaces have well behaved singularities. In fact, the space
of pure spin curves is smooth over \mbox{$\overline{\mg}$}.
\subsection{Previous Results}
Work on this problem in the special case where the base $S$ is ${\Bbb
C}$ and $r=2$ has been done by M. Cornalba in \cite{corn:theta} and
over a more general base by P. Deligne in \cite{deligne:letter}. P.
Sipe and C.J. Earle have studied $\mbox{$r\th$}$ roots of the canonical
bundle on the universal Teichm\"uller curve (c.f.
\cite{sipe:roots2,sipe:roots} and \cite{sipe:roots3}). And
topological properties of the uncompactified moduli space of $2$-spin
curves have been studied in many places (e.g. \cite{har:spin2,lebrun}
).
\subsection{Overview}
In the first section we present some background on torsion-free
sheaves, some geometric motivation for their use, and some results of
Faltings on the local structure of torsion-free sheaves. In the
second section we define the first moduli problem---singular {\em
quasi-spin curves}---and make some local calculations that lead
naturally to another condition we impose later on quasi-spin curves to
get singular {\em spin curves}. In the third section we discuss how
to move from local to global structures using log-structures, and we
use them to formally define the spin curves and pure-spin curves. The
fourth and fifth sections treat deformation theory and isomorphisms,
respectively. The sixth section covers the construction of the
different compactifications, proves that they are algebraic stacks,
and discusses the nature of their singularities. And in the last
section we prove that all the constructions are proper over \mbox{$\overline{\mg}$},
and therefore are true compactifications.
\subsection{Notation and Conventions}
We will use the term {\em semi-stable curve} of genus $g$ to mean a
flat, proper morphism $ X \rightarrow T $ whose geometric fibres $X_t$
are reduced, connected, one-dimensional schemes, with only ordinary
double points, and with $\dim H^1(X_{t}, \mbox{${\cal O}$}_{X_{t}}) = g.$ A {\em
stable curve} is a semi-stable curve of genus greater than one, with
the additional property that any irreducible component which is
isomorphic to \mbox{${\Bbb P}^1$}\ meets the rest of the curve in at least three
points. Irreducible components of a semi-stable curve which are
isomorphic to \mbox{${\Bbb P}^1$}\ but meet the curve in only two points will be
called {\em exceptional} curves. By {\em line bundle} we mean an
invertible (locally free of rank one) coherent sheaf. An $r$-spin
structure on a smooth curve $X/T$ will be a line bundle $\mbox{${\cal L}$}$ such
that $\mbox{${\cal L}$}^{\otimes r}$ is isomorphic to the canonical bundle
$\omega_{X_{T}}.$ A smooth $r$-spin curve will be a smooth curve
$X/S$ with a spin structure.
\section{Torsion-Free Sheaves}
Compactifications of curve-line bundle pairs using geometric invariant
theory give boundary points that correspond to pairs $(X,L)$ with $X$ a
semi-stable curve having at most one exceptional curve (copy of \mbox{${\Bbb P}^1$}\
that intersects the remaining curve in at most two points) in each
chain of exceptional curves, and $L$ a line bundle of degree one on
each exceptional curve in $X.$ Contracting all the exceptional curves
makes the underlying curve stable, and the direct image of $L$ is a
torsion-free sheaf; namely, it has no associated primes of height one.
Furthermore the torsion-free sheaves on stable curves don't have the
problem of having infinite automorphism groups that the line bundles
on semi-stable curves have. It is therefore natural to expect that
torsion-free sheaves will be well-suited to the compactification of
the moduli of spin curves, and this is, in fact, the case.
To begin, we define torsion-free sheaves.
\begin{defn} By {\em relatively torsion-free sheaf} (or just
torsion-free sheaf) on a stable or semi-stable curve $f:\mbox{${\cal X}$}
\rightarrow T\ ,$ we mean a coherent sheaf \mbox{${\cal E}$}\ of
$\mbox{${\cal O}$}_{\mbox{${\cal X}$}}$-modules, which is of finite presentation and flat over
$T,$ with the additional property that on each fibre $\mbox{${\cal X}$}_{t} = \mbox{${\cal X}$}
\times_T \spec{k(t)}$ the induced $\mbox{${\cal E}$}_{t}$ has no associated primes
of height one. Of course, on the open set where $f$ is smooth, a
torsion-free sheaf is locally free.
\end{defn}
Our ultimate goal is to define and describe a notion of $r$-spin
structure for stable curves that corresponds to the previously defined
notion for smooth curves. Spin structures on a family of stable
curves $\mbox{${\cal X}$} @>>> T$ will be pairs $(\mbox{${\cal E}$},b)$ of a relatively
torsion-free sheaf $\mbox{${\cal E}$}$ and a morphism $b:\mbox{${\cal E}$}^{\otimes r} @>>>
\omega_{\mbox{${\cal X}$}/T}$ of $\mbox{${\cal O}$}_{\mbox{${\cal X}$}}$-modules, having certain properties that
we will describe later. But before we can define spin structures, we
need some general results about relatively torsion-free sheaves.
\subsection{General Properties of Torsion-Free Sheaves}
The following propositions give several basic but important properties of
torsion-free sheaves.
\begin{proposition} \label{exacttf} Given any
family of semi-stable curves $\mbox{${\cal X}$}/T$ and an exact sequence $$ 0
\rightarrow \mbox{${\cal E}$}' \rightarrow \mbox{${\cal E}$} \rightarrow \mbox{${\cal E}$}'' \rightarrow 0$$ of
coherent sheaves on $\mbox{${\cal X}$},$
\begin{enumerate}
\item If $\mbox{${\cal E}$}''$
is flat over $T$ and of finite presentation, and if $\mbox{${\cal E}$}$ is
relatively torsion-free, then $\mbox{${\cal E}$}'$ is also relatively torsion-free.
\item If $\mbox{${\cal E}$}'$ and $\mbox{${\cal E}$}''$ are relatively torsion-free, then $\mbox{${\cal E}$}$ is.
\end{enumerate}
\end{proposition}
\begin{pf}
That the sheaves in question are of finite presentation is
straightforward to check. It is enough to check that the sheaves are
torsion-free on each fibre, and the flatness of $\mbox{${\cal E}$}''$ over $T$ means that
the sequence is still exact after restriction to the fibres, where the
proposition is clear.
\end{pf}
\begin{proposition}
For any invertible sheaf $\mbox{${\cal L}$}$ and any relatively torsion-free sheaf
$\mbox{${\cal E}$}$ on $\mbox{${\cal X}$}/T,$ the sheaves $\mbox{${\ce\kern-.15em xt}$}_{\mbox{${\cal X}$}}^i (\mbox{${\cal E}$}, \mbox{${\cal L}$})$ are zero for all
$i >0.$
\end{proposition}
\begin{pf}
By \cite[7.3.1.1]{ega3} it is enough to check this on the individual
fibres, i.e. we may assume that $T$ is a field, and it is enough to
check at the stalk of a closed point ${\frak p}.$ But in this case
$$\mbox{${\ce\kern-.15em xt}$}_{X}^{i} (\mbox{${\cal E}$}, \mbox{${\cal L}$})_{{\frak p}} = \mbox{${\text{Ext}}$}^{i}_{\mbox{${\cal O}$}_{X,{\frak p}}}(\mbox{${\cal E}$}_{{\frak p}},
\mbox{${\cal L}$}_{{\frak p}}) = \mbox{${\text{Ext}}$}^{i}_{\mbox{${\cal O}$}_{X,{\frak p}}} (\mbox{${\cal E}$}_{{\frak p}}, \mbox{${\cal O}$}_{{\frak p}}).$$
And $X$ is Gorenstein, so these vanish for all $i >1.$ And in the case
$i =1,$ by duality theory (\cite[Theorem 6.3]{localcoho}), $$
\mbox{${\text{Ext}}$}^{1}_{\mbox{${\cal O}$}_{{\frak p}}} (\mbox{${\cal E}$}_{{\frak p}}, \mbox{${\cal O}$}_{{\frak p}}) @>{\sim}>>
\mbox{$\text{Hom}$}_{\mbox{${\cal O}$}_{x,{\frak p}}} (H^{0}_{\{{\frak p}\}} (\mbox{${\cal E}$}_{{\frak p}}), I)$$ for some
dualizing module $I.$ But $\mbox{${\cal E}$}_{{\frak p}}$ is torsion-free, so it has no
elements with support equal to $\{{\frak p}\},$ so
$H^{0}_{\{{\frak p}\}}(\mbox{${\cal E}$}_{{\frak p}})=0$ and thus also
$\mbox{${\text{Ext}}$}^{1}_{\mbox{${\cal O}$}_{X,{\frak p}}}(\mbox{${\cal E}$}_{{\frak p}}, \mbox{${\cal O}$}_{x, {\frak p}}) = 0.\qed$
\renewcommand{\qed}{}
\end{pf}
\begin{proposition}
If $\mbox{${\cal E}$}$ is relatively torsion-free, then for any invertible sheaf
$\mbox{${\cal L}$}$ the sheaf $\mbox{${\cal H}\kern-.15em om$}_{\mbox{${\cal X}$}} (\mbox{${\cal E}$}, \mbox{${\cal L}$})$ is
also relatively torsion free.
\end{proposition}
\begin{pf}
$\mbox{${\cal H}\kern-.15em om$} (-, \mbox{${\cal L}$})$ preserves flatness over $R$ and commutes with base
change, so it suffices to check the proposition over a field, where it
is clear.
\end{pf}
\begin{proposition}
Any relatively torsion-free sheaf $\mbox{${\cal E}$}$ is reflexive.
\end{proposition}
\begin{pf}
This follows from local duality.
\end{pf}
\subsection{Torsion-Free Sheaves on Semi-Stable Curves over a Field}
It is well known that the stalk $\mbox{${\cal F}$}_{{\frak p}}$ of a rank-one
torsion-free sheaf $\mbox{${\cal F}$}$ at a singular point ${\frak p}$ of a semi-stable
curve $X$ is isomorphic either to $\mbox{${\cal O}$}_{X,{\frak p}}$ or to the maximal
ideal $\frak m_{{\frak p}},$ which is isomorphic to the direct image $\pi_{*}
\mbox{${\cal O}$}_{\tilde{X},{\frak p}}$ of the normalization $\mbox{${\cal O}$}_{\tilde{X},{\frak p}}.$ In
particular, if the completion $\hat{\mbox{${\cal F}$}}_{{\frak p}}$ of the stalk at ${\frak p}$
is not free, then $\hat{\mbox{${\cal F}$}}_{{\frak p}} \cong xk[[x]]\oplus yk[[y]]$ over
$\hat{\mbox{${\cal O}$}}_{X,{\frak p}} \cong k[[x,y]]/xy,$ where $k$ is the residue field
$\mbox{${\cal O}$}_{X,{\frak p}}/\frak m_{{\frak p}}.$
The following are some simple but useful results which describe
torsion-free sheaves in terms of line bundles on the normalization of
the curve $X.$ Namely, if $\pi : X^{\nu} @>>> X $ is the normalization
of $X$ at one point $\bar {\frak p}$ of the singular set of $\mbox{${\cal F}$}$ (i.e.
where $\mbox{${\cal F}$}$ is not free), then the $\mbox{${\cal O}$}_{X^{\nu}}$-module $\pi^{*}
\mbox{${\cal F}$}$ has torsion elements, but modulo the torsion elements it is free
near the two points of $\pi^{-1}(\overline {\frak p}).$ For any
quasi-coherent sheaf $\mbox{${\cal G}$},$ define $\pi^{\natural} \mbox{${\cal G}$}$ to be the
torsion-free $\mbox{${\cal O}$}_{X^{\nu}}$-module $(\pi^{*}\mbox{${\cal G}$}/\text{torsion}).$
Straighforward checking yields the following proposition.
\begin{proposition}\label{equiv}
In the situation above, where $\pi$ is the normalization of $X$ at a
singularity of a rank-one, torsion-free sheaf $\mbox{${\cal F}$},$ the canonical map
$\mbox{${\cal F}$} \rightarrow \pi_{*} \pi^{\natural} \mbox{${\cal F}$}$ is an isomorphism.
\end{proposition}
As usual, define the degree of a sheaf \mbox{${\cal F}$}\ on a curve $Y$ over an
algebraically closed field to be $\deg (\mbox{${\cal F}$}) = \chi (\mbox{${\cal F}$}) -
\chi(\mbox{${\cal O}$}_{Y}).$ We are primarily interested in the case of relatively
torsion-free sheaves on stable curves, and in this case the degree of
$\mbox{${\cal E}$}$ is locally constant on the fibres since $\mbox{${\cal E}$}$ is flat over the
base. It is easy to see that this definition of degree corresponds to
the usual definition of degree if $\mbox{${\cal F}$}$ is a line bundle.
\begin{proposition} If $\mbox{${\cal F}$}$ is a rank-one torsion-free sheaf and
$\pi$ is, as above, the normalization of $X$ at a singular point
of $\mbox{${\cal F}$},$ then $\deg (\pi^{\natural} \mbox{${\cal F}$}) = \deg (\mbox{${\cal F}$}) -1.$
\end{proposition}
\begin{pf}
$R^i \pi_* (\mbox{${\cal F}$}) =0$ for all $i >0,$ so the Leray spectral sequence
degenerates, and $H^{q}(X, \mbox{${\cal F}$}) = H^{q} (Y, f_{*} \mbox{${\cal F}$})$ for all $q.$
Thus $\chi (\mbox{${\cal F}$}) = \chi (\pi^{\natural}\mbox{${\cal F}$}),$ and $\chi( \pi _{*}
\mbox{${\cal O}$} _{X^{\nu}}) = \chi (\mbox{${\cal O}$}_{X^{\nu}}).$ Taking Euler-Poincar\'e
characteristics of the exact sequence $ 0\rightarrow \mbox{${\cal O}$}_{X}
\rightarrow
\pi_{*} \mbox{${\cal O}$}_{X^{\nu}} \rightarrow k \rightarrow 0 $
gives $\chi (\pi_{*}\mbox{${\cal O}$}_{X^{\nu}})= \chi (\mbox{${\cal O}$}_{X}) + 1, $
and thus $\deg (\mbox{${\cal F}$}) = \deg (\pi^{\natural} \mbox{${\cal F}$} ) + 1.\qed$
\renewcommand{\qed}{}
\end{pf}
\begin{proposition}
If $ X^{\nu} @>{p}>> X$ is the normalization of $X$ at all the
singularities of $\mbox{${\cal F}$},$ then $p^{\natural} \mbox{${\cal F}$}$ is invertible and
$(p^{\natural} \mbox{${\cal F}$})^{\otimes r} \cong p^{\natural} (\mbox{${\cal F}$}^{\otimes r}).$
In fact, $ p^{\natural} \mbox{${\cal E}$}_1 \otimes p^{\natural} \mbox{${\cal E}$}_{2} \otimes
\dots \otimes p^{\natural} \mbox{${\cal E}$}_{n} = p^{\natural} (\mbox{${\cal E}$}_{1} \otimes
\dots \otimes \mbox{${\cal E}$}_{n})$ for any torsion-free sheaves $\mbox{${\cal E}$}_1, \mbox{${\cal E}$}_2,
\dots, \mbox{${\cal E}$}_n$ with singularities equal to the singularities of $\mbox{${\cal F}$}.$
\end{proposition}
This is, again, straighforward to check.
\begin{proposition}
$\pi^{\natural}$ is a covariant functor from coherent sheaves on $X$
to torsion-free sheaves on $X^{\nu}.$ And if ${\cal{TORF}}_{\nu}$ is
the category of rank-one torsion-free $\mbox{${\cal O}$}_X$-modules with
singularities exactly those which are normalized by $\pi,$ and
${\cal{PIC}}_{\nu}$ is the category of invertible
$\mbox{${\cal O}$}_{X^{\nu}}$-modules, then the categories ${\cal{TORF}}_{\nu}$ and
${\cal{PIC}}_{\nu}$ are equivalent via $\pi^{\natural}$ and $\pi_{*}.$
\end{proposition}
\begin{pf}
The first part is clear except perhaps the fact that $\pi^{\natural}$
commutes with composition of morphisms. But the effect of
$\pi^{\natural}$ applied to a morphism $ (\mbox{${\cal E}$} @>f>> \mbox{${\cal F}$})$ is induced
by the composition $ \pi^{*} \mbox{${\cal E}$} @>\pi^{*}f>> \pi^{*} \mbox{${\cal F}$} \rightarrow
\pi^{\natural} \mbox{${\cal F}$},$ which factors through $\pi^{\natural} \mbox{${\cal E}$},$ since
the target has no torsion. Moreover, $ \pi^{\natural}(f \circ g) $ is
given by the following commutative diagram.
\begin{center}
\setlength{\unitlength}{0.012500in}%
\begin{picture}(182,100)(80,677)
\put( 80,760){\makebox(0,0)[lb]{\smash{$\pi^{*}\mbox{${\cal E}$}$}}}
\put(160,760){\makebox(0,0)[lb]{\smash{$\pi^{*}\mbox{${\cal F}$}$}}}
\put(240,760){\makebox(0,0)[lb]{\smash{$\pi^{*}\mbox{${\cal G}$}$}}}
\thicklines
\put(100,765){\vector( 1, 0){ 55}}
\put(180,765){\vector( 1, 0){ 55}}
\put( 90,760){\vector( 0,-1){ 45}}
\put(170,760){\vector( 0,-1){ 45}}
\put(250,760){\vector( 0,-1){ 45}}
\put(100,705){\vector( 1, 0){ 55}}
\put(180,705){\vector( 1, 0){ 55}}
\put(100,699){\vector( 1,0){ 135}}
\put(160,690){\makebox(0,0)[lb]{\smash{$\scriptscriptstyle\pi^{\natural}
(f\circ g)$}}}
\put(115,710){\makebox(0,0)[lb]{\smash{$\scriptscriptstyle\pi^{\natural} f$}}}
\put(115,770){\makebox(0,0)[lb]{\smash{$\scriptscriptstyle\pi^{*} f$}}}
\put(195,770){\makebox(0,0)[lb]{\smash{$\scriptscriptstyle\pi^{*} g$}}}
\put(195,710){\makebox(0,0)[lb]{\smash{$\scriptscriptstyle\pi^{\natural} g$}}}
\put( 80,702){\makebox(0,0)[lb]{\smash{$\pi^{\natural}\mbox{${\cal E}$}$}}}
\put(160,702){\makebox(0,0)[lb]{\smash{$\pi^{\natural}\mbox{${\cal F}$}$}}}
\put(240,702){\makebox(0,0)[lb]{\smash{$\pi^{\natural}\mbox{${\cal G}$}$}}}
\end{picture}
\end{center}
Both sets of bottom arrows $ \pi^{\natural} (f \circ g)$ and
$\pi^{\natural} f \circ \pi^{\natural} g$ commute in the rectangle,
and the map $ \pi^{*} \rightarrow \pi^{\natural} $ is surjective, so
the bottom arrows must commute. The second part of the proposition is
clear by Proposition~\ref{equiv}.
\end{pf}
\subsection{Boundedness of Rank-One Torsion-Free Sheaves}
Since $X$ is projective, we can choose a very ample $ \mbox{${\cal O}$}(1) $ on $X.$
In general, we will write $\mbox{${\cal F}$}(m)$ for $\mbox{${\cal F}$} \otimes \mbox{${\cal O}$}(1)^{\otimes
m}.$ One of the more important facts about torsion-free sheaves of
rank one is that they form bounded families. In other words, the
following proposition holds. This will later be important in proving
that the functor of spin curves has a versal deformation.
\begin{proposition}
\label{thm:dsousa}
If \mbox{${\cal F}$} is a rank-one torsion-free sheaf on $X,$ there is an integer
$m_0$ depending only on the degree of \mbox{${\cal F}$}\ on each irreducible
component of $X,$ and on the genus of $X,$
such that for $m \geq m_0$ the following holds. \begin{enumerate}
\item $H^1(X,\mbox{${\cal F}$} (m)) = 0.$ \item $\mbox{${\cal F}$} (m)$ is generated by global
sections. \end{enumerate}
\end{proposition}
This is a straightforward generalization of D'Souza's propositions in
Section Three of \cite{dsou:jac}.
\subsection{Some Geometry}\label{geom}
The original motivation for studying torsion-free sheaves comes from
the fact that boundary points in the GIT compactification correspond
approximately to pairs $(X,\mbox{${\cal L}$})$ with $X$ a semi-stable curve, having
no more than one exceptional curve in each chain of exceptional curves
in any fibre, and $\mbox{${\cal L}$}$ is a line bundle with degree one on each of
the exceptional curves (c.f. \cite{cap:thesis,gieseker,thesis}).
Given a curve-bundle pair $(X,\mbox{${\cal L}$})$ of this sort, contracting all of
the exceptional curves, i.e. projection from $X$ to its stable model
$\rho: X @>>>\bar X$ makes $\rho_*\mbox{${\cal L}$}$ into a torsion-free sheaf. This
contraction and a related ``inverse'' action of blowing up certain
ideals warrant a more careful look.
Let $X/B$ be a stable curve over $B=\spec R,$ with $R$ a complete
local ring. The completion $A=\hat{\mbox{${\cal O}$}}_{X,{\frak p}} $ of the local ring
of $X$ at a point ${\frak p}$ in the special fibre is of the form $A=\mbox{$R[[x,y]]/(xy-\pi)$}$
for some choice of $\pi$ in the ring $R.$ If ${\frak p}$ is singular in the
special fibre, then $\pi$ is in the maximal ideal $\frak m$ of $R.$
Given two elements $p$ and $q$ of $R$ such that $pq=\pi,$ we will
construct a semi-stable curve with the properties mentioned above,
namely with no chains of exceptional curves of length greater than
one. Essentially, we want to undo the contraction to associate a
suitable semi-stable curve and line-bundle to each stable curve and
rank-one torsion-free sheaf.
First notice that if $p$ is not a zero divisor, blowing up the ideal
$I= (x,p)$ in $A$ to get $\tilde{X}_I :=
\proj_{X}(\oplus_{n}I^n)$ gives $$\tilde{X}_I \cong
\proj_{A}(A[P,\Xi]/(Px-p\Xi,Pq-\Xi y)).$$ Similarly, if $q$ is
not a zero divisor, blowing up the ideal $J= (y,q)$ gives
$$\tilde{X}_J \cong \proj_{A}(A [Q,Y]/(Qy-yQ, pQ-xY)).$$
And these are isomorphic via $P \mapsto Y,\ \Xi \mapsto Q.$
In general, define $\tilde{X} (p,q)$ to be the $X$-scheme
defined locally as $$\tilde{X} := \tilde{X} (p,q):=
\proj_{A}(A[P,\Xi]/(Px-p\Xi, Pq-\Xi y)) @>\rho>> X,$$
regardless of whether or not $p$ or $q$ is a zero divisor. It is
straighforward to check that the curve $\tilde{X}(p,q)$ is actually
semi-stable of the desired form (only one exceptional curve) and has
stable model equal to $X.$
Let $s=\frac{\Xi}{P},$ and let $U$ in $\tilde{X}$ be the open set
$U=\spec{A[s]/(x-ps, ys-q)}.$ Similarly, setting
$t=1/s=\frac{P}{\Xi}$ let $V$ in $\tilde{X}$ be the open set $V=
\spec{A[t]/(xt-p, y-qt)}.$ And finally, let $\tilde{A}$ be the
ring $ \tilde{A} = A[P,\Xi]/(p\Xi-xP,qP-\Xi y).$ The union of $U$ and
$V$ is all of $\tilde{X},$ and there are canonical line bundles on
$\tilde{X},$ namely $\mbox{${\cal O}$}_{\tilde{X}}(n) $ for all $n$ where
$$\mbox{${\cal O}$}_{\tilde{X}} (n) = \begin{cases} {P}^{n} \mbox{${\cal O}$}_U
\text{ on } U \\ {\Xi}^n \mbox{${\cal O}$}_V \text{ on } V
\end{cases}.$$
\begin{proposition}\label{tilden}
In the above construction of $\tilde{X}$ and $\mbox{${\cal L}$}:=
\mbox{${\cal O}$}_{\tilde{X}}(n)$ the following hold.
\begin{enumerate}
\item $n \geq -1 $ implies that $ \rho_{*} \mbox{${\cal L}$} $ is flat over
$R,$ it commutes with base change, and $ R^{1} \rho_{*} \mbox{${\cal L}$} =0.$
\item For $n \geq 0,$ $\Gamma (\tilde{X}, \mbox{${\cal L}$}) \cong \tilde{A}_n,$
the $n^{th}$ graded piece of $\tilde{A},$ and the natural map
$ \rho^{*} \rho_{*} \mbox{${\cal L}$} \rightarrow \mbox{${\cal L}$}$ is surjective.
\item $\rho_{*} \mbox{${\cal O}$}_{\tilde{X}} = \mbox{${\cal O}$}_X.$
\item $n=1$ implies $ \rho_{*} \mbox{${\cal L}$}$ is torsion-free of rank one.
\end{enumerate}
\end{proposition}
\begin{pf}
It suffices to consider the case $A = R[x,y]/(xy-\pi),$ and in this
case $\mbox{${\cal O}$}_U \cong R[s,y]/(sy-q),$ and $\mbox{${\cal O}$}_V \cong R[t,x]/(xt-p),$
both of which are flat over $R,$ and $\mbox{${\cal O}$}_{U \cap V}
\cong R[s,t]/(st-1).$
Now $\Gamma (\tilde{X}, \mbox{${\cal O}$}_{\tilde{X}}(n)) = \{(g,f) \in \mbox{${\cal O}$}_U \oplus
\mbox{${\cal O}$}_V | g = s^n f$ on $ \mbox{${\cal O}$}_{U \cap V} \}$ So $f$ and $g$ are of
the form $$ g = sg_+(s) + g_0 + yg_- (y) \quad \text{ and } \quad f = tf_- (t) + f_0 +
xf_+ (x)$$ with $g_+(s) \in R[s], g_- (y) \in R[y], f_- (t) \in R[t],$
and $f_+(x) \in R[x],$ and $$ sg_+ (s) + g_0 + tqg_- (tq) = s^n (tf_-
(t) + f_0 + spf_+ (sp)).$$ So $$sg_+ (s) + g_0 + tqg_- (tq) = s^{n-1}
f_- (t) + s^n f_0 + s^{n+1} pf_+ (sp),$$ and rewriting $sg_+ (s)$ as $
s^{n+1} \gamma (s) + s^n g_n + \dots + sg_1,$ with $g_i \in R,$ and
$\gamma (s) \in R[s],$ and $tf_+(t)$ as $t^{n+1} \phi (t) + t^{n}
f_{n} + t^{n-1} f_{n-1} + \dots + tf_1,$ with $\phi (t) \in R[t],$
gives $$ s^{n+1} \gamma (s)+ s^n g_n + \dots + sg_1 + g_0 + tqg_- (tq)
= t\phi (t) + f_n + \dots + s^{n-1} f_1 + s^n f_0 + s^{n+1} pf_+ (sp).$$
Now $s^{n+1} \gamma (s) = s^{n+1} pf_+ (sp),$ and even if $p$ or $q$
is a zero divisor, this implies that $\gamma(s) = pf_+ (sp),$ because
$\gamma(s)$ is an element of $R[s] \subseteq \mbox{${\cal O}$}_U $ and $\ann
(s^{n+1}) \cap R[s] = (0)$ (i.e. $R[s] \subseteq \mbox{${\cal O}$}_U \rightarrow
\mbox{${\cal O}$}_{U \cap V}$ is injective). Similarly, $g_{n-1} = f_i $ for $
0 \leq i \leq n$ and $\phi (t) = qg_- (tq).$
Several things are easy to see from this
formulation, namely
\begin{enumerate}
\item $\Gamma (\tilde X, \mbox{${\cal L}$})$ is free over $R,$ and hence
$\rho_{*} \mbox{${\cal L}$}$ is $R$-flat as long as $n \geq -1.$
\item $\rho_{*}\mbox{${\cal O}$}_{\tilde{X}} = \mbox{${\cal O}$}_{X}$
\item If $(g,f)$ is an element of
$\Gamma(\tilde{X}, \mbox{${\cal L}$}),$ then $g$ can be written as
$$g = s^n xf_+ (x) + s^n f_0 + s^{n-1} f_1 + \dots + sf_{n-1} + f_n +
yg_- (y)$$ in $\mbox{${\cal O}$}_U$ (with $ x=sp$), and $f$ can be written as $$f =
xf_+ (x) + f_0 + tf_1 + t^2 f_2 + \dots + t^n f_n + t^n yg_- (y)$$ in
$\mbox{${\cal O}$}_V$ (with $y = tq$). \end{enumerate}
The third fact shows that the element $$(f_0 + xf_+ (x))
{\Xi}^n + f_1 {\Xi}^{n-1} P + \dots + f_{n-1} \Xi {P}^{n-1} + (f_n +
yg_- (y)) {P}^n$$ in $\tilde{A}_n$ maps to $(g,f)$ in
$\Gamma(\tilde{X},\mbox{${\cal L}$}),$ and the natural homomorphism $\tilde{A}_n
\rightarrow \Gamma (\tilde{X}, \mbox{${\cal L}$})$ is surjective.
Moreover, it is easy to check that if we write $\tilde{A}_n$ as
$\tilde{A}_n = \{ (F_0 {\Xi}^n + F_1 {\Xi}^{n-1} P \dots + F_n
{P}^n)|F_i \in A\},$ we can assume that $F_0$ is in $R[x]$ and $F_n$
is in $R[y]$ and all the remaining terms are in $R$ (i.e. $F_i \in R$
for $0 < i < n$). And thus the homomorphism $\tilde{A}_n \rightarrow
\Gamma(\tilde{X}, \mbox{${\cal L}$})$ must actually be injective, hence an isomorphism.
Now since $R^1\rho_{*}$ is right exact, to see that it vanishes, it
suffices to check that it vanishes for each fibre. And $H^1 (\tilde{X}
\times_{X} x, \mbox{${\cal O}$}(n))$ is zero for all $x$ in $X,$ except possibly the
singular points of $X.$ But over a singular point $$H^1
(\mbox{${\cal O}$}(n)) \cong H^1 (\mbox{${\Bbb P}^1$}, \mbox{${\cal O}$}_{\mbox{${\Bbb P}^1$}} (n)) = 0.$$ And thus $R^1
\rho_{*}$ is zero, and $\rho_{*} \mbox{${\cal L}$}$ commutes with base change if $n
\geq -1.$
To show that $ \rho^{*} \rho_{*} \mbox{${\cal L}$} \rightarrow \mbox{${\cal L}$}$ is surjective,
note that this map is locally (on $U$) just the map taking $ \{ (g,f)
| g = s^{n} f\}
\otimes_{A} \mbox{${\cal O}$}_U$ to $ \mbox{${\cal O}$}_U,$ given by $(g,f) \otimes z \mapsto gz.$
So it suffices to show that there exists $(g,f) \in \rho_{*} \mbox{${\cal L}$}$ such
that $ g =1.$ But $ (g,f) = (1,t^{n})$ works as long as $t^{n} \in
\mbox{${\cal O}$}_V,$ i.e. if $ n \geq 0.$ A similar computation holds over $V,$ so
$\rho^{*} \rho_{*} \mbox{${\cal L}$} \rightarrow \mbox{${\cal L}$}$ is surjective.
To see that $\rho_{*} \mbox{${\cal L}$}$ is torsion-free, note that since it is flat
and commutes with base change, it suffices to check the case where $R$
is a field and $ p = q = \pi = 0.$ In this case $\mbox{${\cal O}$}_U = R[x,y]/sy$
and $\mbox{${\cal O}$}_v = R[t,x]/tx$ and global sections of $\mbox{${\cal O}$}(1)$ are $(g,f) \in
\mbox{${\cal O}$}_U \oplus \mbox{${\cal O}$}_V$ such that $g =sf.$ Moreover, $x(g,f)=(0,xf)$ and
$y(g,f) = (yg,0),$ hence if the ideal $(x,y)$ annihilates $(g,f),$
then $x(g,f) = y(g,f) = 0,$ and $xf=yg=0.$ Thus $f \in (t), g
\in (s),$ and this contradicts the fact that $g =sf$;
therefore, $ \rho_{*}
\mbox{${\cal O}$}(1)$ has no associated primes of height one and is torsion-free.
\end{pf}
\begin{lemma}[Cornalba \cite{corn:theta}]
If $\mbox{${\cal L}$}$ is a line bundle on a semi-stable curve $f:X\rightarrow T$
such that $\mbox{${\cal L}$} |_{E} \cong \mbox{${\cal O}$} _{E}$ for some exceptional component
$E$ of a special fibre $X_0,$ then there is an \'etale neighborhood
$T'$ of $0,$ such that $\mbox{${\cal L}$}|_T'$ on $X_{T'}$ is trivial in a
neighborhood of $E$ in $X_{T'}.$
\end{lemma}
\begin{pf}
There is an \'etale neighborhood $T''$ of $0$ in $T$ such that for
each irreducible component of the special fibre $X_{0}$ except $E,$
there is a section of $X''/T''$ that does not intersect $E$ but which
passes through the irreducible component (c.f. \cite[17.16.3]{ega4}).
Let $D$ be the divisor in $X$ corresponding to the sum of all these
sections, and note that $\mbox{${\cal L}$} (mD) |_{X_{0}}$ is generated by global
sections and has first cohomology group zero if $m$ is sufficiently
large.
So the natural map $ R^{1}f_{*}(\mbox{${\cal L}$}(mD)) \otimes k
\rightarrow R^1 f_* (X_{0},
\mbox{${\cal L}$}(mD) \otimes k) = 0$ is surjective, hence is an isomorphism, hence
$R^{1} f_{*} (\mbox{${\cal L}$}(mD))$ is zero on an open set $T'$ about $0$ in $T''$
(\cite[III.12.11]{H}). This implies that on $T'$ the map $
R^{0}f_{*}(\mbox{${\cal L}$}(mD))
\otimes k \rightarrow
\Gamma (X_{0}, \mbox{${\cal L}$} (mD) \otimes k)$ is also surjective.
And $\mbox{${\cal L}$}(mD)$ is generated by global sections. Thus $ \mbox{${\cal L}$}(mD)$ is
trivial on a neighborhood of $E,$ and on a sufficiently small
neighborhood of $E,$ $\mbox{${\cal L}$}(mD) \cong \mbox{${\cal L}$}.$
\end{pf}
As an immediate consequence of Cornalba's lemma we have the following
corollary.
\begin{corollary}
Proposition~\ref{tilden} holds for any line bundle $\mbox{${\cal L}$}$ which has
degree $1$ on the exceptional curve of the special fibre of
$\tilde{X}(p,q);$ namely,
\begin{enumerate}
\item $\rho_* \mbox{${\cal L}$}$ is flat over $R,$ it commutes with base change, and
$R^1 \rho_* \mbox{${\cal L}$} =0.$
\item $\Gamma(\tilde{X}, \mbox{${\cal L}$}) \cong \tilde{A}_1,$ and $\rho^* \rho_*
\mbox{${\cal L}$} \rightarrow \mbox{${\cal L}$}$ is surjective.
\item $\rho_* \mbox{${\cal L}$}$ is torsion-free of rank one.
\end{enumerate}
\end{corollary}
\subsection{Induced Maps}\label{indmap}
If $p$ and $q$ have the additional relations that $p^u = wq^v$ with
$u+v =r$ and $w
\in R^{\times},$ then there is a canonical map from
$ \mbox{${\cal O}$}_{\tilde{X}}(r)$ to $
\mbox{${\cal O}$}_{\tilde{X}}.$ Namely, on $U$ it is $ \mbox{${\cal O}$}(r) = (A[s]/ (ps-x, sy-q))
\cdot {P}^{\otimes r} $ maps to
$A[s] / (ps-x, sy-q)$ via ${P}^{\otimes r} \mapsto wy^{v}.$ And on $V$
it is $ \mbox{${\cal O}$} (r) = ( A[t]/ (p-xt, y-qt))
\cdot (\Xi)^{\otimes r}$ maps to $A [t] / (p-xt, y-qt)$ via
${\Xi}^{\otimes r}= s^{r} \cdot {P}^{\otimes r} \mapsto s^{r} wy^{v}
= x^{u}.$
The canonical map $ \rho^{*} \rho_* \mbox{${\cal O}$}(1) \rightarrow \mbox{${\cal O}$} (1) $
induces a map $ ( \rho^{*} \rho_* \mbox{${\cal O}$}(1))^{\otimes r} =
\rho^{*}(\rho_{*} \mbox{${\cal O}$}(1) ^{\otimes r})
\rightarrow \mbox{${\cal O}$}(1)^{\otimes r} = \mbox{${\cal O}$} (r),$ and the canonical
map $\mbox{${\cal O}$} (r) \rightarrow \mbox{${\cal O}$}_{\tilde{X}}$ gives a canonical map $
\rho^{*} (\rho_* \mbox{${\cal O}$}(1)) ^{\otimes r} \rightarrow \mbox{${\cal O}$}_{\tilde{X}}.$
And this induces a map on the push-forward by adjointness $$(\rho_*
\mbox{${\cal O}$}(1))^{\otimes r} \rightarrow \rho_* \mbox{${\cal O}$}_{\tilde{X}}= \
\mbox{${\cal O}$}_{X}.$$
When we define various generalizations of a spin structure, the
best-behaved ones will be those which are locally isomorphic to those
induced from the canonical map $\mbox{${\cal O}$}_{\tilde{X}}(r) @>>>
\mbox{${\cal O}$}_{\tilde{X}}.$
\subsection{Local Structure of Torsion-Free Sheaves}\label{falt}
As in the previous section, we work with a stable curve $X/B,$ where
the base $B$ is the spectrum of a complete local Noetherian ring $R$;
the completion of the ring $\mbox{${\cal O}$}_{X,{\frak p}}$ at a singular point ${\frak p}$ is
isomorphic to $A:=\mbox{$R[[x,y]]/(xy-\pi)$},$ and $\pi$ is an element of the maximal ideal
$\frak m$ of $R.$ A torsion-free sheaf $\mbox{${\cal E}$}$ corresponds to an $R$-flat
$A$-module, $E.$
Locally on $X,$ we can express a torsion-free sheaf obtained by
the contraction $\tilde X (p,q) @>>> X$ ($\mbox{${\cal E}$}$ is the
direct image of a line bundle $\mbox{${\cal L}$}$ of degree one on the exceptional
curve) in the following way. $$E \cong
\Gamma(\spec A, \pi_{*} \mbox{${\cal O}$}(1)) =
\tilde{A}_1=(A[\Xi,P]/(\Xi p-Px, \Xi y-Pq))_{1} =
\{f \Xi+gP|f,g \in A\}.$$ It is easy to see that we can assume $f$ is
in $R[[x]],$ and $g$ is in $R[[y]].$ And so the map $\rho_{*}\mbox{${\cal O}$} (1)
\rightarrow A^{\oplus 2},$ given by $(f\Xi+gP) \mapsto \left(
\begin{array}{c} fx+gp \\ fq+gy \end{array} \right)$ is a
well-defined homomorphism of $A$-modules. Even if $p$ and $q$ are
zero divisors, if $f$ is in $R[[x]]$ and $g$ is in $R[[y]],$ then
$fx+gp=0$ implies that $f=0.$ Similarly, $fq+yg=0$ implies that
$g=0.$ So the map is injective. We can, therefore, express $\rho_{*}
\mbox{${\cal O}$}(1) $ as the image of the $A$-homomorphism $$\alpha (p,q)= \left (
\begin{array}{cc} x & p \\ q & y \end{array} \right) : A^{\oplus 2}
@>>> A^{\oplus 2}.$$
A result of Faltings shows that every rank-one torsion-free sheaf is
of this form. Namely, let $E(p,q)$ be the image of $\alpha (p,q):
A^{\oplus 2} @>>> A^{\oplus 2},$ where $\alpha$ is the two-by-two
matrix $\left ( \begin{array} {cc} x & p \\ q & y \end{array}
\right),$ and $p$ and $q$ are, as before, elements of $R$ such that
$pq=\pi.$ We saw above that $E(p,q)$ is $R$-flat, and torsion-free.
When $p$ and $q$ are in $\frak m$ then $E(p,q)$ is a deformation of the
normalization of $A/\frak m A,$ i.e. of the unique (up to isomorphism)
non-free torsion-free sheaf $k[[x]] \oplus k[[y]]$ over the ring
$A/\frak m A \cong k[[x,y]]/xy.$ Faltings' result is the following.
\begin{theorem}[Faltings \cite{falt:torfree}]
Any reflexive $E$ of rank $1$ is isomorphic to an $E(p,q),$
for $p, q \in R$ with $ pq= \pi.$
\end{theorem}
The fact that
$\alpha \left( \begin{array}{c} y \\ 0 \end{array} \right) =
\alpha \left( \begin{array}{c} 0 \\ q\end{array} \right)$ and
$\alpha \left( \begin{array}{c} 0 \\ x \end{array} \right)=
\alpha \left( \begin{array}{c} p \\ 0 \end{array} \right)$
implies that for any $\alpha \left(
\begin{array}{c} f\\g \end{array} \right) \in E(p,q),$ we can assume
$f$ is in $R[[x]]$ and $g$ is in $R[[y]],$ and $E(p,q)$ is
$R$-isomorphic to $R[[x]] \oplus R[[y]]$ via the obvious
identification. Homomorphisms and isomorphisms of $E(p,q)$'s can be
described by their lifts to $A^{\oplus 2}.$ Namely, any morphism of
$A$-modules $E \rightarrow F,$ with $E$ relatively torsion-free, can
be lifted to a morphism from $A^{\oplus 2}$ to $F.$ And a homomorphism
from $E$ to $E'$ with $E$ and $E'$ both torsion-free lifts to an
endomorphism of $A^{\oplus 2}.$ More exactly, the following holds.
\begin{proposition}[Faltings \cite{falt:torfree}]
If $ p \equiv q \equiv 0 \mod \frak m,$ then $ E(p,q)$ is isomorphic to
$ E(p',q')$ if and only if there exist $u,v \in R^{\times}$ such that
$$ p'=upv^{-1}, \quad \text{ and } \quad q' = vqu^{-1}.$$ In this case the isomporhism
is induced by the ``constant'' map $\left ( \begin{array} {cc} u & 0
\\ 0 & v \end{array} \right): A^{\oplus 2} @>>> A^{\oplus 2}.$
Moreover, writing a homomorphism $\Phi$ in $\mbox{$\text{Hom}$}_{A} (E(p,q),
E(p',q'))$ as a lift to $A^{\oplus 2} @>>>A^{\oplus 2}$ given by
$\left(
\begin{array}{cc} \varphi_{+} & \psi_{+} \\ \psi_{-} & \varphi_{-}
\end{array}\right),$ $\varphi_+$ can be
taken to be in $R[[x]],$ $\varphi_-$ can be taken to be in $R[[y]],$
and the elements $\varphi_{+} (x)$ and $ \varphi_{-} (y)$ completely
determine $\psi_{+}$ and $\psi_{-}$ by the relations $$\psi_+ =
(\frac{p}{x}) (\varphi_+(x) - \varphi_+(0)) \quad \text{ and } \quad \psi_- =
(\frac{q}{y}) (\varphi_-(y) -
\varphi_- (0)),$$ and are subject to the condition that $$ p'
\varphi_{-} (0) =
\varphi_{+} (0) p, \quad \text{ and } \quad q' \varphi_{+} (0) = \varphi_{-} (0) q.$$
\end{proposition}
These results also hold for Henselian rings. Suppose now that $R$ is
the Henselisation of a local ring of finite type over a field or an
excellent Dedekind domain, $\frak m$ is the maximal ideal of $R,$ $ \pi
\in \frak m,$ and $A$ is the Henselisation of $R[x,y]/(x y-\pi)$ at
$\frak m + (x,y).$ As before, for each pair $p, q
\in R$ with $ p q = \pi,$ define $E(p,q),$ and the
theorem is
\begin{theorem}[Faltings]
\begin{enumerate}
\item Any torsion-free $E$ of rank one over $A$ is
isomorphic to $E(p,q)$ for $p,q \in R$ and $p q
= \pi.$
\item If $p,q$ are in $\frak m,$ then $E(p,q)$ and
$E(p',q') $ are isomorphic if and only if there exist $u,v \in
R^{\times}$ with $p'= upv^{-1}, q' = vqu^{-1}.$
\item Suppose $I \subseteq R$ is a nilpotent ideal. Modulo constant
automorphisms (given by $\left(\begin{array}{cc} u & 0\\ 0 & v
\end{array}\right)\in \text{GL}_2(R)$ with $up = pv, vq = qu$)
any automorphism of $E(p,q)/IE(p,q)$ lifts to an automorphism of
$E(p,q).$
\end{enumerate}
\end{theorem}
\section{Quasi-Spin Surves: Local Structure}
A spin structure on a stable curve should have the property that where
the torsion-free sheaf is free it is isomorphic to the canonical line
bundle $\omega.$ A natural object to study, therefore, is a triple
$(X,\mbox{${\cal E}$},b),$ with $X$ a stable curve, $\mbox{${\cal E}$}$ a relatively torsion-free
sheaf of rank one and degree $2g-2/r,$ and $b$ is a homomorphism of
$\mbox{${\cal O}$}_{X}$-modules $$b:\mbox{${\cal E}$}^{\otimes r} \rightarrow
\omega_{\mbox{${\cal X}$}/T},$$ which is an isomorphism on the open set where $\mbox{${\cal E}$}$
is locally free. Note that for smooth curves, such a triple is just
an $\mbox{$r\th$}$ root of the canonical bundle with an explicit isomorphism
of the \mbox{$r\th$} power of the bundle to the canonical bundle. Later we
will need a few more conditions on these triples to get the ``right''
generalization of a spin curve, but we begin with these alone.
\subsection{A-Linear Homomorphisms of Tensor Powers}
To better understand these triples we need to study $A$-linear maps
$b: E^{\otimes r} @>>> A$ of the \mbox{$r\th$}\ tensor power of a rank-one
torsion-free $A$-module $E.$ As before, $A$ is an \'etale neighborhood
of the closed point defined by $(x,y) + \frak m A$ in
$\spec{\mbox{$R[[x,y]]/(xy-\pi)$}}$ over the base ring $R,$ where $\frak m$
is a maximal ideal of $R$ containing $\pi, p,$ and $q.$
Any map $ b:E^{\otimes r} \rightarrow A$ that is $A$-linear, lifts to
a map $\tilde{b},$ thus
$$\begin{CD} A ^{\oplus 2^{r}} @>\tilde{b}>> A \\
@VV\alpha^{\otimes r}V @| \\ E^{\otimes r} @>b>> A
\end{CD}$$
Over $A[1/x]$ and over $A[1/y]$ the module $E$ is locally free, thus
over these rings any homomorphism $b:E^{\otimes r}
\rightarrow A$ will factor through $\text{Sym}^r(E),$ and its lift to
$(A^{\oplus 2})^{\otimes r}$ will factor through $\text{Sym}^r
(A^{\oplus r}).$ And since $A$ has no $(x,y)$-torsion, this holds in
general. So if $f$ and $g \in A^{\otimes 2}$ are defined as $\left (
\begin{array} {c} 1 \\ 0 \end{array} \right)$ and $\left (
\begin{array} {c} 0 \\ 1 \end{array} \right)$ respectively, we only need to
describe $b_i :=b(f^{r-i} \otimes g^i)$ for each $ 0 \leq i \leq r$ in
order to completely describe $b$ and $\tilde{b}.$ We will, therefore,
denote $\tilde{b} : A^{2^{r}} \rightarrow E^{\otimes r}
\rightarrow A$ by the vector $(b_0, b_1, \dots, b_r).$
Now, since $\alpha \left( \begin{array}{c} p\\0 \end{array} \right) =
\alpha \left( \begin{array}{c} 0\\x \end{array} \right)$ and $\alpha
\left( \begin{array}{c} 0\\q \end{array} \right) = \alpha \left(
\begin{array}{c} y\\0 \end{array} \right) ,$ we must have for all $i,$
$0 \leq i \leq r-1$
\begin{gather} \label{basicrelation}
pb_i =xb_{i+1} \quad \text{ and } \quad yb_i =qb_{i+1}.
\end{gather}
Now the fact that the map must be an isomorphism off of the singular
locus of the underlying curve means that $b: E(p,q)^{\otimes r} @>>>
A$ must be an isomorphism on $A[1/x]$ and $A[1/y].$ Over $A[1/x]$ the
map $\tilde{b} = (b_0,b_1,\dots,b_r)$ is completely determined by
$b_0,$ namely for any $i$ we have $x^ib_i = p^ib_0,$ and thus $$b_i =
b_0 p^i/x^i.$$ For $\tilde b$ to be surjective over $A[1/x]$ we must
have that $b_0$ is invertible in $A[1/x].$ Similarly, $b_r$ is
invertible in $A[1/y].$
On the special fibre, since $p$ and $q$ are in $\frak m,$ we
have $\tilde b \equiv (\bar b_0,0,0,\dots,0,\bar b_r)
\pmod{\frak m}.$ Here $\bar z$ denotes the image of $z$ modulo $\frak m.$
And $\bar b_0=x^u\bar\beta_0$ and $\bar b_r=y^v\bar\beta_r$ for some
$\bar\beta_0$ invertible in $(A/\frak m A)[1/x],$ but not in the ideal
$(x),$ and for some $\bar\beta_r$ invertible in $(A/\frak m A)[1/y],$
but not in $(y).$ This makes the length of the cokernel of $\bar b$
equal to $u+v-1.$ If $u$ (or similarly $v$) is zero, then over the
ring $A[1/y]$ the fact that $b_0 = b_r q^r/y^r \equiv 0
\pmod{\frak m}$ implies that $\bar\beta_0 = \bar b_0 \equiv 0
\pmod{\ann_{A/\frak m A}(y))},$ i.e. modulo $(x).$ But this
implies that $\bar\beta_0 = 0,$ and that is a contradiction. Hence we
have proven the following.
\begin{proposition} \label{b}
Any pair $(E,b)$ which is not free is of the form $(E(p,q),b)$ with
$p$ and $q$ in $\frak m,$ and $b$ lifts to $\tilde b=
(b_0,b_1,\dots,b_r),$ where, modulo the ideal $\frak m A$ we have $\bar
b_0=x^u\bar\beta_0$ and $\bar b_r=y^v\bar\beta_r$ for some
$\bar\beta_0$ invertible in $(A/\frak m A)[1/x],$ but not in the ideal
$(x),$ and for some $\bar\beta_r$ invertible in $(A/\frak m A)[1/y],$
but not in $(y).$ Moreover, $u$ and $v$ must both be at least one.
\end{proposition}
The final condition we will need on triples $(X, \mbox{${\cal E}$}, b)$ to generalize
spin curves is the condition that the two constants $u$ and $v$ in
Proposition~\ref{b} must sum to $r.$ In other words, the length of the
cokernel of $b$ at each singular point of each fibre must be $r-1$ if
the sheaf $\mbox{${\cal E}$}$ is not free there. Without this condition there would
be too many possible triples for the associated stack to be separated.
We now have the first generalization of spin structures on stable
curves.
\begin{defn}
A {\em quasi-spin structure} on a stable curve $X/T$ is a pair $(\mbox{${\cal E}$},b)$
where $\mbox{${\cal E}$}$ is relatively torsion-free of rank one, and $b$ is a
homomorphism of $\mbox{${\cal O}$}_{X}$-modules $$b:\mbox{${\cal E}$}^{\otimes r} \rightarrow
\omega_{X/T}$$ to the canonical dualizing sheaf, such that \begin{enumerate}
\item $\mbox{${\cal E}$}$ has degree $(2g-2)/r.$
\item $b$ is an isomorphism on the open set where $\mbox{${\cal E}$}$ is not free.
\item For each closed point $t$ of the base $T,$ and for each singular point
${\frak p}$ of the fibre $\mbox{${\cal X}$}_{t}$ where $\mbox{${\cal E}$}$ is not free, the length of
the cokernel of $b$ at ${\frak p}$ is $r-1.$
\end{enumerate}
\end{defn}
\begin{defn}
A {\em quasi-spin curve} is simply a stable curve with a quasi-spin
structure.
\end{defn}
In the special case that $r=2,$ the requirement that the cokernel of
the spin structure map be supported on the singular locus of $\mbox{${\cal E}$}$ is
enough to guarantee that the length of the cokernel is at least one at
all singular points. The condition on the total degree of $\mbox{${\cal E}$}$ can
be seen to guarantee that the length of the cokernel is at most (and
hence exactly) one at all singular points. Moreover, these conditions
are equivalent to the condition that the map $b$ induce an isomorphism
$\mbox{${\cal E}$} @>{\sim}>>
\mbox{$\text{Hom}$}_{\mbox{${\cal O}$}_{\mbox{${\cal X}$}}}(\mbox{${\cal E}$},\omega_{\mbox{${\cal X}$}})=\mbox{${\cal E}$}^{\vee}\otimes \omega_{\mbox{${\cal X}$}} .$
\subsection{Power Series Expansions}
Any $A$-linear map $ b=(b_{0}, \dots, b_{r})$ as above, over the complete
local ring $A=\hat{\mbox{${\cal O}$}}_{\mbox{${\cal X}$},{\frak p}}$ has a power series expansion $b_{i}
= \sum_{n\geq 0} b_{in} x^{n} +
\sum_{m>0} b_{i,-m} y^{m}.$ And the relations $p^i b_0 = x^i b_i$ and
$q^{r-i} b_r = y^{r-i} b_i$ imply that $$p^i b_{0,n+i} =b_{i,n}$$ for
$n \geq 0,$ and $$ b_{i,-m} = q^{r-i} b_{r, -m-(r-i)}$$ for $ m \geq
0.$ And in particular $$ p^{j} b_{0,j} = q^{r-j} b_{r, j-r}$$ for all
$ j,0 \leq j \leq r.$
Moreover, if $b$ induces a quasi-spin structure on the central fibre,
then there are $ u,v \in \Bbb Z^{+}$ such that $\pmod \frak m$
$$\bar{b}_{0} = x^{u} \bar{\beta}_{0} \quad \text{ and } \quad
\bar{\beta}_{0} \in (\bar{A}_{x})^{\times}.$$
This implies that $\bar{b}_{0} = \sum_{n \geq u} \bar{b}_{0,n} x^{n}$
with $
\bar{b}_{0,u} \neq 0,$ hence $b_{0,u} $ is not in $
\frak m$ and is invertible in $R.$
Similarly, $b_{r,-v} \in R^{\times}.$ So, in particular, $ p^{u}=q^{v}
b_{r,-v}/b_{0,u}.$ Letting $ w = b_{r,-v}/b_{0,u} \in A^{\times},$ we
have the relation $$ p^{u} = q^{v}w.$$
In the special case that $\pi$ is not a zero divisor, the relations $
p^{i} b_{0,i} = q^{r-i} b_{r, i-r}$ for $0 \leq i \leq u$ imply that
$$ b_{0,i} = w^{-1} p^{u-i} q^{u-i} b_{r,i-r} =
\frac{\pi^{u-i}}{w} b_{r, i-r}. $$ Similarly, $$ b_{r, i-r} = w
\pi^{i-u} b_{0, i}, \text{ for } u \leq i
\leq r.$$ But even when $\pi$ is a zero divisor $$b_{0,0} = \pi^u b_{r,
-r} , \quad b_{r,0} = \pi^v b_{0,r} , \quad b_{0,u} = w
b_{r, -v},$$ and $$b_{0,i}= \frac{\pi^{u-i}}{w} b_{r, i-r} + \sigma_i
\text{ for } 0 < i < u, \quad \text{ and } \quad b_{r, i-r} = w \pi^{i-u}b_{0,i} + \sigma_i
\text{ for } u<i < r.$$
The ``bad'' terms $\sigma_i$ are all nilpotent elements. On the one
hand, for any prime ideal ${\frak p} \in \spec R$ such that $p$ (and hence
$q$) is in ${\frak p},$ we have that $ b_i \equiv 0 \mod
{\frak p}$ for $ 0 <i<r.$ And $b_0 \equiv x^u \beta,$ and $b_r \equiv y^v
\gamma,$ with $\beta$ and $\gamma$ invertible elements of $R[[x]]$ and
$R[[y]]$ respectively. Accordingly, $$b_{0,i} \in {\frak p} \text{ for } 0
< i <u, \quad \text{ and } \quad b_{r, i-r} \in {\frak p} \text{ for } u <i < r,$$ and thus
$\sigma_i \in {\frak p}$ for $0 <i < r$ ($\sigma_u$ is obviously zero). On
the other hand, if $p$ (and therefore $q$) is not in ${\frak p},$ then $p$
and $q$ are not zero divisors in $R_{{\frak p}}$ and hence, as demonstrated
before, $\sigma_i \in {\frak p}$ for $ 0 <i <r .$ Thus $\sigma_i$ is
contained in the nilradical of $R$ for every $i.$ And in particular,
for any quasi-spin structure over a reduced, complete local ring the
relations
\begin{gather}\label{ssrelations}
b_{0,i} = \frac{\pi^{u-i}}{w} b_{r, i-r} \text{ for } 0 \leq i \leq
u \\ b_{r, i-r} = w \pi^{i-u} b_{0,i} \text{ for } r \geq i \geq
u \notag
\end{gather}
hold.
\begin{proposition}
When the relations (\ref{ssrelations}) hold, we can write $b_{0}$ and
$b_{r}$ as the following products: $$b_0=ax^u \quad \text{ and } \quad b_r = awy^v
\text{ for some } a \in A^{\times}.$$ In particular, given $u,v,$ and
$w,$ the fact that the specified relations (\ref{ssrelations}) hold
means that $b$ is completely determined by $a \in A^{\times}.$
\end{proposition}
\begin{pf}
$$ b_{0} = x^{u} \sum_{n \geq 0} b_{0, n+u} x^{n} +
1/w \sum_{u>m\geq 0} \pi^{u-m} b_{r, m-r} x^{m} + q^{r} \sum_{l>0}
b_{r,l-r} y^{l}$$ And thus $$ a = \sum_{n \geq 0} b_{0, n+u}
x^{n} + 1/w \sum_{m > 0 } b_{r, -m-v} y^{m}.$$ The calculation is
similar for $b_{r}.$
\end{pf}
\begin{proposition}
If the relations (\ref{ssrelations}) hold on $\tilde b,$ then
$b$ is actually the map induced on $E(p,q)^{\otimes r} = \left (
\rho_{*} \mbox{${\cal O}$}_{\tilde{X}(p,q)}(1) \right )^{\otimes r}$ as in
Section~\ref{indmap}. Moreover, the relations (\ref{ssrelations})
hold for (the lift to $A^{\oplus 2^r}$ of) the map induced from
$\mbox{${\cal O}$}_{\tilde{X}(p,q)}(1)$ for any $p$ and $q$ in $\frak m_R$ with
$pq=\pi.$
\end{proposition}
\begin{pf}
To see this out explicitly the map $$ (A^{2})^{\otimes r}
\tilde{@>>>} A^{2^{r}} @> \alpha^{r}>>(E(p,q))^{r}
\tilde{@>>\psi^{r}>} (\rho_{*} \mbox{${\cal O}$}(1))^{r} @>>\varphi> \rho_{*} \mbox{${\cal O}$}(r) @>>
\gamma> \mbox{${\cal O}$}_{X} = A.$$
The first map is just $$ \left (
\begin{array}{c} f_{1} \\ g_{1} \end{array} \right )\otimes \dots
\left ( \begin{array}{c} f_{r} \\ g_{r} \end{array} \right )
\mapsto \alpha \left ( \begin{array}{c} f_{1} \\ g_{1}
\end{array} \right ) \otimes \dots \otimes \alpha \left ( \begin{array}{c}
f_{r} \\ g_{r} \end{array} \right ).$$ The map $\psi$ is given by $
\psi : \alpha \left (
\begin{array}{c} f \\ g \end{array} \right ) \mapsto (sf+g, f+tg),$ so
$$\psi^{\otimes r} : \alpha \left ( \begin{array}{c} f_{1} \\ g_{1}
\end{array} \right ) \otimes \dots \otimes \alpha \left (
\begin{array}{c} f_{r} \\ g_{r} \end{array} \right ) \mapsto (sf_{1}
+ g_{1}, f_{1} + tg_{1}) \otimes \dots \otimes (sf_{r} + g_{r}, f_{r} +
tg_{r}).$$
The map $\varphi$ is given by
$ \varphi : (h_{1}, k_{1}) \otimes \dots \otimes (h_{r},
k_{r} ) \mapsto ( h_{1} h_{2} \dots h_{r}, k_{1} k_{2} \dots
k_{r})$
and $\gamma$ is $ \gamma: (h, k) \mapsto w y^{v} h = x^{u}k \in A.$
So the composite map is $$\left ( \begin{array}{c} f_{1} \\ g_{1}
\end{array} \right ) \otimes \dots \left ( \begin{array}{c} f_{r} \\
g_{r} \end{array} \right ) \mapsto x^{u} \prod_{1 \leq i \leq r}
(f_{i} + tg_{i}),$$ but this is just the map $$ b = (x^{u}, tx^{u}
\dots, t^{r} x^{u}) = ( x^{u}, px^{u-1}, \dots, p^{u}, p^{u}t, \dots,
p^{u} t^{v}). $$ And $ p^{u} = wq^{v},$ and $ qt=y,$ so $$ b = (x^{u},
px^{u-1}, \dots, p^{u}, wq^{v-1}y, \dots, w y^{v}).$$
This composite map depends only on the choice of isomorphism $
\Gamma(\tilde{X}, \mbox{${\cal O}$}_{\tilde{X}}) @>{\sim}>> A,$ and any
element $a$ in $A^{\times}$ induces an automorphism $A,$ so any
quasi-spin structure with the additional relations (\ref{ssrelations})
is actually an induced map.
\end{pf}
In the special case when $r=2$ every singularity has $u=1,$ and thus
$\sigma_0 =\sigma_1=\sigma_2=0.$ Therefore all quasi-spin structures
are actually locally isomorphic to an induced structure.
We can always map $ E(p,q) \tilde{\rightarrow} E(p',q')$ with $p' =
\lambda p, q' = \lambda^{-1} q.$ So if $ p^{u} = wq^{v},$ then
${p'}^{u} = \lambda^{u} p^{u} = \lambda^{u} wq^{v} = \lambda^{r}
w{q'}^{v}.$ So $ w \rightarrow \lambda^{r} w.$ And $w$ has an
\mbox{$r\th$} root in $k$ if the base ring
$R$ has its residue field $k$ algebraically closed, hence by the
step-by-step method the \mbox{$r\th$} root of $w$ will lift to all of $R,$ i.e.
if $\frak m I = 0$ there is an $i \in I$ such that $ w + i =
\lambda^{r},$ which implies that $(\lambda - \frac {i}{r
\lambda^{r-1}})^{r} = w.$ Thus we may assume that if the central fibre
has residue field $k =
\bar{k},$ then $w$ can be taken to be one, and, in general, $R$ has an
\'etale cover on which we can take $w$ to be one. Accordingly, all
$b$'s for which the relations (\ref{ssrelations}) hold are determined,
\'etale locally, by $ p,q,u,v,$ and an element of $A^{\times}.$
\subsection{Behavior of the Cokernel Under Deformation}
\label{sec:coker}
Because of the condition on the cokernel of the quasi-spin map
$b,$ we need to understand the way that the length of the cokernel
changes under deformation.
\begin{defn}
An $\mbox{${\cal O}$}_{X}$-linear map $b:\mbox{${\cal E}$}^{\otimes r} @>>> \omega$ from the \mbox{$r\th$}\
tensor power of a rank-one torsion-free sheaf $\mbox{${\cal E}$}$ to the canonical
bundle of a curve $X$ over $k$ is said to {\em have good cokernel} if
\begin{enumerate}
\item the cokernel is supported on the singularities of $X,$ and
\item for each point ${\frak p}$ of the support of the cokernel $C$ of $b$
$$ \mbox{${\text{length}}$}_{{\frak p}}C = r-1 .$$
\end{enumerate}
\end{defn}
\begin{lemma}
If the cokernel of $b$ is supported on the singular locus of $\mbox{${\cal E}$},$
then the property of having good cokernel is stable under
generization.
\end{lemma}
\begin{pf}
It is enough to consider the case where $R$ is a complete local ring,
$E \cong E(p,q)$ is an $A$-module, with $A=\mbox{$R[[x,y]]/(xy-\pi)$},$ and
$\tilde{b}=(b_0,\dots,b_r):A^{2^r} @>>>A$ is a lifting of the map
$b:E^{\otimes r} @>>> A.$ We can assume that on the special fibre
$\overline{b}_0 \in A/\frak m A$ equal to $x^{i}\overline{\beta}_0,$
with $\overline{\beta}_{0}$ an invertible element of $( A/\frak m).$
Similarly, $\overline{b}_r \in A/\frak m A$ with $\overline{b}_r =
y^{j} \overline{\beta}_r,$ and $\beta_{r}$ invertible in $A.$
Now for any map $R @>>> K$ of $R$ into a field, we have the following
possible cases.
\begin{enumerate}
\item $\pi$ does not map to zero in $K.$ In this case, the cokernel
is actually zero because \spec{A\otimes K} is regular. \item $\pi$
maps to zero, but at least one of $p$ and $q$ does not. In this case
again the cokernel of $b$ is zero. \item $\pi$ and $p$ and $q$ all
map to zero. This is the only interesting case. We have $A\otimes K
\cong K[[x,y]]/xy$ and $\tilde{b}_K = (b_0,0,\dots,0,b_r).$ Now
$$b_{0} = x^i \beta_0 + d_0 \quad \text{ and } \quad b_{r} = y^j \beta_r + d_r$$ with
$d_{0}=x^{l}\mbox{${\cal E}$}_{0}$ and $d_{r}=y^{m} \mbox{${\cal E}$} _{r},$ such that $\mbox{${\cal E}$} _{0}$
is in $\frak m A$ but not in $(x),$ and $\mbox{${\cal E}$} _{r}$ is in $\frak m A$ but
not in $(y).$ If, on the one hand, $l$ is larger than $i-1,$ then
$$x^{i}=b_0/(\beta_0 + x^{l-i}\mbox{${\cal E}$}_{0}).$$ The term in the denominator
is invertible because $\beta_0$ is invertible, and $d_0$ is in the
maximal ideal $\frak m A.$ If, on the other hand, $l$ is less than $i,$
then $$x^i = \frac{-x^l \mbox{${\cal E}$}_0 }{\beta_0},$$ and similarly for
$y^j.$ In either case $$K[[x,y]]/(xy,x^i,y^j) \cong
K < 1, x, x^2, \dots, x^{i-1}, y, y^2, \dots, y^{j-1}>$$ surjects onto
$$K[[x,y]]/(xy,b_0,b_r)=A \otimes_R K/\im{b}.$$
\end{enumerate}
So the length of the cokernel will be either zero (cases 1 and 2) or
bounded above by $ i+j-1=r-1$ (case 3).
Thus the length of the cokernel can only decrease under generization,
but the degree of $\mbox{${\cal E}$}_K$ on $\mbox{${\cal X}$}_K$ must be $(2g-2)/r = \deg
\theta^{\natural} \mbox{${\cal E}$} + \delta,$ where $\delta$ is the number of
singularities of $\mbox{${\cal X}$}_K,$ and $\theta: \mbox{${\cal X}$}^{\nu}_K \rightarrow \mbox{${\cal X}$}_K$
is the normalization of $\mbox{${\cal X}$}_K$ at the singularities of $\mbox{${\cal E}$}_K.$ On
the other hand, since the cokernel of $b$ is supported on the singular
set of $\mbox{${\cal E}$},$ we have that $\theta^{\natural} b$ factors
$$\theta^{\natural} \mbox{${\cal E}$}^{\otimes r}_K
@>{\sim}>> \theta^{*}
\omega_{\mbox{${\cal X}$}_K} (-\sum u_{{\frak p}}{\frak p}^+ -\sum v_{{\frak p}}{\frak p}^-) \hookrightarrow
\theta^{*} \omega_{\mbox{${\cal X}$}_K},$$ where the sum is taken over all ${\frak p}$ in the
singular set of $\mbox{${\cal E}$}_K,$ $\theta^{-1}({\frak p}) = \{{\frak p}^{\pm}\},$ and for
each ${\frak p},$ $u_{{\frak p}}+v_{{\frak p}} -1=\mbox{${\text{length}}$}_{{\frak p}} (\mbox{${\text{coker}}$}(b)) \leq r-1.$ So
$$\deg \mbox{${\cal E}$}_K = (2g-2)/r = \bigg(2g-2-\sum_{{\frak p}} (u_{{\frak p}} +
v_{{\frak p}})\bigg)/r +
\delta,$$ which will be strictly greater than $(2g-2)/r$ unless at each
singularity of $\mbox{${\cal E}$}_K$ the cokernel of $b$ has length $r-1.$ Thus the
property of having good cokernel is stable under generization.
\end{pf}
\begin{proposition}
\label{opencoker}
Given $ b: \mbox{${\cal E}$}^{\otimes r} \rightarrow \omega$ on $f:\mbox{${\cal X}$} @>>>T$ (with
the cokernel of $b$ supported on the discriminant locus) the set of $t
\in T$ such that $b_t$ has good cokernel is open in $T.$
In other
words, the functor of $T$-schemes $$F_{b} (T')= \begin{cases} \{1\} &
\text{if $b$ has good cokernel at every geometric point of $T'$} \\
\emptyset & \text{if there exists $\overline{t} \in T'$ where
$b$ does not have good cokernel}
\end{cases}$$
is an open subfunctor of the trivial functor $T' \mapsto \{1\}.$
\end{proposition}
\begin{pf}
It suffices to show the complement of the set is closed. And since
the previous lemma shows this complement is stable under
specialization, it suffices to show that the complement is
constructible. Let $P_{m}$ be the property of a geometric point
\mbox{${\overline{t}}$}\ of $T$ that $C:= \mbox{${\text{coker}}$} (b)$ has a point of its support over
\mbox{${\overline{t}}$}\ where $C$ has length $m.$ The set we want to show is
constructible is the set $\frak{T}_m :=\{ t \in T |
\mbox{${\overline{t}}$}
\text{ has } P_{m}\}.$ Actually, the set we are really looking
for is $$\bigcup\begin{Sb}{0 < m < r-1}\\ r - 1 < m < N \end{Sb}
\frak{T}_{m},$$ for some very large $N.$ $N$ can be taken to be
finite because the degree of the sheaves is fixed, and the sum over
all points in a given fibre of the length of the cokernel is bounded,
and this bound is determined by the number of singular points and the
degree. Moreover, the number of singular points is bounded as a
function of the genus of the underlying curve, so this number $N$ can
be chosen independently of the specific family.
Now to show constructibility we only need to consider one $m$ and one
irreducible component of the discriminant locus, say $D_{0},$ of $\mbox{${\cal X}$}$
and its image $\rho (D_{0} ) = T_{0},$ i.e. we only need to show that
$\frak{T}_m$ is constructible in $T_0.$ And it is enough to assume
$T_{0}$ is reduced and irreducible. Since $D_{0} $ is proper over
$T_{0},$ the semi-continuity theorem shows that $\frak{T}_m$ is
constructible for any $m \neq r-1.$
\end{pf}
\section{Local-to-Global Calculations}
\subsection{Log-Structures}
The well-behaved quasi-spin curves, i.e. those for which the relations
(\ref{ssrelations}) hold locally, also give a compactification of the
moduli space of spin curves, and their local moduli spaces are much
easier to describe than those of general quasi-spin curves. But in
order to formalize the notion of ``well-behaved'' we need to choose
local coordinates for the whole curve in such a way that our
constructions make sense globally.
From the deformation
theory of stable curves, we know that the complete local ring
$\hat{\mbox{${\cal O}$}}_{\mbox{${\cal X}$},x}$ over $\hat{\mbox{${\cal O}$}}_{T,t}$ is of the form
$\hat{\mbox{${\cal O}$}}_{\mbox{${\cal X}$},x} \cong \hat{\mbox{${\cal O}$}}_{T,t} [[x,y]]/(xy-\pi)$ for some $\pi
\in \hat{\mbox{${\cal O}$}}_{T,t}.$ And on some \'etale neighborhood $T'$ of $t,$
the induced curve $\mbox{${\cal X}$} \times_T T'$ has the
following additional structure: on an \'etale cover $\mbox{${\cal X}$}'$ of ${\mbox{${\cal X}$}}
\times_T T',$ there are
sections $x$ and $y$ in $\mbox{${\cal O}$}_{{\mbox{${\cal X}$}}'}$ such that
\begin{enumerate}
\item $xy=\pi \in \mbox{${\cal O}$}_{T,t}.$
\item The ideal generated by $x$ and $y$ has the discriminant
locus of $\mbox{${\cal X}$}/T$ as its associated closed subscheme.
\item The obvious homomorphism $\big(\mbox{${\cal O}$}_{T,t}
[x,y]/(xy-\pi)\big) \rightarrow
\mbox{${\cal O}$}_{{\mbox{${\cal X}$}}',x}$ induces an isomorphism on the completions
$\big(\hat{\mbox{${\cal O}$}}_{T,t} [[x,y]]/(xy-\pi)\big) @>{\sim}>>
\hat{\mbox{${\cal O}$}}_{{\mbox{${\cal X}$}}', x}.$
\end{enumerate}
Such a collection of data ($\mbox{${\cal X}$}',T',x,y,\pi$) is what we need locally.
But this data is not uniquely determined; it is only determined up to
the equivalence relation generated by the operations
\begin{enumerate}
\item pullback to \'etale covers.
\item change by units: namely $x' = \tilde{u}x, y'=\tilde{v}y, {\pi}'
= \tilde{w} \pi$ with $\tilde{u}, \tilde{v} \in \mbox{${\cal O}$}^*_{{\mbox{${\cal X}$}}'},$ and
$\tilde{u} \tilde{v} = \tilde{w} \in \mbox{${\cal O}$}^*_{T'}.$
\item switching branches: namely, interchanging $x$ and $y.$
\end{enumerate}
A log structure is a way of choosing these local data coherently.
\begin{defn}
A {\em log structure} for $\mbox{${\cal X}$}/T$ is given by \'etale covers $\mbox{${\cal X}$}'$
and $T'$ of $\mbox{${\cal X}$}$ and $T$ $$\begin{CD} \mbox{${\cal X}$} @<<< {\mbox{${\cal X}$}}' \\
@VVV @VVV \\ T @<<< T'
\end{CD}.$$ And for each irreducible component of the singular locus of
${\mbox{${\cal X}$}}'$ a choice of $\pi \in \mbox{${\cal O}$}_{T'}$ and a choice of $x$ and $y$ in
$\mbox{${\cal O}$}_{{\mbox{${\cal X}$}}'}$ with the three properties listed above, and with
descent data related to the equivalence relation. Namely, on ${\mbox{${\cal X}$}}''
= {\mbox{${\cal X}$}}' \times_{\mbox{${\cal X}$}} {\mbox{${\cal X}$}}'$ over $T'' = T' \times_T T'$ with
projection maps $pr_1$ and $pr_2,$ there are $1$-cocycles $u,v$ in
$\mbox{${\cal O}$}^*_{\mbox{${\cal X}$}''},$ and $w$ in $\mbox{${\cal O}$}^*_{T''}$ such that: $\text{pr}_2^*
(x) = u\text{pr}_1^*(x), \text{pr}_2^* (y) = v\text{pr}_1^* (y),$ and $uv
=w,$ and $\text{pr}_2^*(\pi) = w\text{pr}_1^* (\pi)$ with the cocycle
condition that on ${\mbox{${\cal X}$}}''' = {\mbox{${\cal X}$}}' \times_{\mbox{${\cal X}$}}{\mbox{${\cal X}$}}'
\times_{\mbox{${\cal X}$}}{\mbox{${\cal X}$}}',$ $u,v,$ and $w$ are all compatible with the
different projections, i.e. $ \text{pr}^*_{12}(u)\text{pr}^*_{23}(u) =
\text{pr}^*_{13}(u)$ and so forth.
\end{defn}
As in the local case, we also impose the equivalence relation on the
log structures generated by pullback to \'etale covers and by change
by units compatible with the descent data; namely, two log structures
$(\mbox{${\cal X}$}', T', x,y,\pi)$ and $(\mbox{${\cal X}$}', T', x', y', \pi ')$ are equivalent
if there exist $\tilde{u}, \tilde{v}$ in $\mbox{${\cal O}$}^*_{\mbox{${\cal X}$}'}$ and
$\tilde{w}$ in $\mbox{${\cal O}$}^*_{T'}$ such that $x' = \tilde{u}x,
y'= \tilde{v} x, \pi ' = \tilde{w} \pi, $ with $\tilde{u} \tilde{v} =
\tilde{w}$ and if $(u,v,w)$ and $(u',v',w')$ are the cocycles
corresponding to the two log structures, the units $\tilde{u},
\tilde{v} \quad \text{ and } \quad \tilde{w}$ must be compatible with them as
well, namely $u' = (\text{pr}^*_1 (\tilde{u})/\text{pr}^*_2
(\tilde{u})) u,$ and $ v' = (\text{pr}^*_1(\tilde{v})/\text{pr}^*_2
(\tilde{v}))v,$ and $ w' = (\text{pr}^*_1 (\tilde{w})/\text{pr}^*_2
(\tilde{w}))w.$
As discussed above, given any two log structures with distinguished
branches $(x)$ and $(y)$ we will have the relations $x' = \tilde{u} x$
and $y' = \tilde{v} y,$ etc. And they will be almost equivalent,
namely $\text{pr}^*_1 (x)
(u'-(\text{pr}^*_1(\tilde{u})/\text{pr}^*_2(\tilde{u}))u)=0$ and so
forth; thus if $\pi$ is not a zero divisor (and hence $x$ and $y$
also) all log structures are equivalent. In particular, since a
versal deformation of the curve has no zero divisors it has a unique
log structure.
Switching of branches $(x\mapsto y, y\mapsto x)$ and switching of
double points (i.e. interchange the different $\pi_i$) results in an
action of the $n^{\text{th}}$ symmetric group ($n$ is the number of
double points) and the group $(\Bbb Z/2\Bbb Z)^n$ on the log
structures. But for our purposes this is not a problem, namely we are
interested in expressing $\mbox{${\cal E}$}$ as an $E(p,q)$ and this switching just
interchanges $p$ and $q$ or the different $\pi_i.$ So given a log
structure on a stable curve, we can use the methods of Faltings to
describe rank-one torsion-free sheaves, namely any such sheaf $\mbox{${\cal E}$}$ is
isomorphic to an $E(p,q),$ and the results on homomorphisms and
isomorphisms still hold.
In general the choice of a log structure is unique up to the
automorphisms $x \mapsto ux,$ $y\mapsto vy,$ and $\pi
\mapsto w\pi,$ for $uv=w,$ but locally this might not be all of the
automorphisms of the henselization of the ring $R[x,y]/(xy-\pi).$ In
other words, on a curve $C \rightarrow B$ we might have different log
structures induced by different maps of $B$ to the versal deformation.
Nevertheless, we can get around this by considering the problem
globally instead. Namely let $\mbox{${\cal S}$}/\mbox{${\cal R}$} = \mbox{$\overline{\mg}$}$ be a presentation of
the stack of of stable curves, i.e. $\mbox{${\cal S}$}$ is \'etale over \mbox{$\overline{\mg}$}, and
$\mbox{${\cal R}$}$ is the \'etale equivalence relation (\mbox{${\stack{Isom}}$}). $\mbox{${\cal R}$}$ is smooth
and has no zero divisors, so the two pullbacks to $\mbox{${\cal R}$}$ of the
universal curve with its unique log structure over $\mbox{${\cal S}$}$ are
canonically isomorphic. Hence any curve over any base has a canonical
log structure induced by the unique log structure on the universal
curve over $\mbox{${\cal S}$}.$
Note that given a choice of $p$ and $q$ in $\mbox{${\cal O}$}_T,$ the descent data
for the canonical log structure determine gluing data for the various
blowings up. Thus the techniques of Section~\ref{geom} yield a
globally-defined semi-stable curve $\tilde{\mbox{${\cal X}$}}(p,q)$ over $\mbox{${\cal X}$},$ a
rank-one torsion-free sheaf $\mbox{${\cal E}$}(p,q) = P_{*} \mbox{${\cal O}$} (1)$ on $\mbox{${\cal X}$},$ and a
canonical map $b:\mbox{${\cal E}$}^{\otimes r} @>>> \mbox{${\cal M}$},$ for some line bundle
$\mbox{${\cal M}$}.$
\subsection{Spin Curves and Pure-Spin Curves}
We can now define our ``good'' quasi-spin curves using the canonical
log structure.
\begin{defn}
A {\em spin structure} on an arbitrary
stable curve $\mbox{${\cal X}$}/T$ is a pair $(\mbox{${\cal E}$}, b),$ where $\mbox{${\cal E}$}$ is
relatively torsion-free of rank one with degree $(2g-2)/r,$ and $b$ is
a morphism of $\mbox{${\cal O}$}_{X}$-modules $$b:\mbox{${\cal E}$}^{\otimes r} \rightarrow
\omega_{\mbox{${\cal X}$}/T},$$ which is an isomorphism on the open set where $\mbox{${\cal E}$}$
is locally free, and such that via the canonical log structure on
$\mbox{${\cal X}$}/T,$ the sheaf $\mbox{${\cal E}$}$ is isomorphic to $E(p,q)$ for some $p$ and
$q$ in $\mbox{${\cal O}$}_T$ with $pq=\pi,$ and the homomorphism $b:\mbox{${\cal E}$}^{\otimes r}
\rightarrow \omega$ is the canonical induced morphism.
\end{defn}
An even stronger condition that we can impose on the spin curves is
that $p$ and $q$ be such that $p=t^v$ and $q=t^u$ for some $t$ in
$\mbox{${\cal O}$}_T.$ Spin curves that have this property will be called {\em pure
spin curves.} Pure-spin curves also compactify the smooth spin curves,
and they have an especially well-behaved local structure, as we will
see later.
\section{Deformation Theory}\label{deftheory}
Given a spin structure $(\bar{\mbox{${\cal E}$}}, \bar{b})$ on a curve $\bar{\mbox{${\cal X}$}}$
over an Artin local ring $\bar{R}$ with residue field $k,$ and given a
deformation $R$ of $\bar{R},$ namely $\bar{R}= R/I$ with $I^2 =0,$ we
want to study deformations of $(\bar{\mbox{${\cal X}$}}, \bar{\mbox{${\cal E}$}}, \bar{b})$ to
spin curves and quasi-spin curves over $R.$
First, do this locally. For $\bar{A}:= \bar{R}[[x,y]]/(xy-\bar{\pi})$
and $(\bar{E},\bar{b}) =(E(\bar{p}, \bar{q}), \bar{b}),$ a spin
structure on $\bar{A},$ we want to lift $\bar{A}$ and $(\bar{E},\bar{b}).$
But any lift corresponds to a choice of $\beta,$ $P$ and $Q$ such that
$P^u=Q^v$ and an isomorphism $({\overline{E ({P}, {Q})}},\bar{\beta})
@>{\sim}>> (E(\bar{p},\bar{q}),\bar{b}).$ Here ${\overline{E
(P,Q)}}$ is the module $E(P,Q)/I\cdot E(P,Q) = E(\bar P,\bar Q)$
induced by pulling back $E(P,Q)$ along the canonical map $\spec R
\leftarrow \spec{\bar R},$ and the map $\bar{\beta}$ is the canonical map
induced from $\beta$ on $\overline{E(P,Q)}.$ By Faltings' theorem,
isomorphisms over $\overline{R}$ are of the form $\bar{\Phi} = \left
(\begin{array}{cc} \bar{\zeta} & 0 \\ 0 &
\bar{\xi} \end{array} \right) $ with $\bar{\zeta}, \bar{\xi} \in
\bar{R}^{\times}.$ These lift to isomorphisms $\Phi = \left(
\begin{array}{cc} \zeta & 0 \\ 0 & \xi
\end{array} \right) $
for lifts $\zeta,\xi \in R^{\times}$ of $\bar{\zeta}$ and
$\bar{\xi},$ and such lifts always exist in $R.$ Thus any local spin
structure is given simply by a choice of $P$ and $Q$ in $R$ such that
$\bar P = \bar p$ and $\bar Q = \bar q$ and a choice of $\beta$ such
that the induced map $\bar{\beta}$ on $E(\bar p,\bar q)^{\otimes r} =
\bar E^{\otimes r}$ differs from $\bar b$ only by an automorphism of
$\bar E.$ In particular, $\bar{\beta} = \bar{a} \bar{b}$ with
$\bar{a}$ in $\bar{A}^{\times},$ and thus $\beta$ is uniquely
determined by an element $a \in A^{\times},$ i.e.
$\beta=a(x^u,px^{u-1},\dots, y^v).$ This describes the local
deformations.
Any combination of local lifts will patch together into a global one.
This is due to the fact that if $\bar{\sigma}$ is a section of
$\mbox{${\cal O}$}^*_{\bar{\mbox{${\cal X}$}}}$ and $\gamma$ is a section of $\mbox{${\cal O}$}^*_{\mbox{${\cal X}$}}$
inducing $\bar{\gamma}$ in $\mbox{${\cal O}$}^*_{\bar{\mbox{${\cal X}$}}}$ such that
$\bar{\sigma}^r = \bar{\gamma},$ then $\bar{\sigma}$ lifts uniquely to
a section $\sigma$ of $\mbox{${\cal O}$}^*_{\mbox{${\cal X}$}}$ such that $\sigma^r = \gamma.$
This is easy to check. We can now use {\em fpqc} descent to lift
$(\bar{\mbox{${\cal X}$}}, \bar{\mbox{${\cal E}$}}, \bar{b}).$ Namely, a choice of $P$ and $Q$ for
each singularity of $\bar{\mbox{${\cal X}$}}$ still allows numerous choices of $\mbox{${\cal X}$}$
deforming $\bar{\mbox{${\cal X}$}}.$ And given such an $\mbox{${\cal X}$},$ we need to construct
a pair $(\mbox{${\cal E}$},b)$ extending $\bar{\mbox{${\cal E}$}}$ and $\bar{b}.$ On $U,$ the
complement of the discriminant locus, the line bundle $\bar{\mbox{${\cal E}$}}$
extends uniquely to a line bundle $\mbox{${\cal E}$}$ that is an \mbox{$r\th$}\ root of
$\omega.$ Given an extension $E(P,Q)$ at each singularity (i.e. at
$\spec{\hat{\mbox{${\cal O}$}}_{\mbox{${\cal X}$}, x_i}}$), we have a covering datum induced
by the unique lift of the covering datum on $\bar{\mbox{${\cal X}$}}$ that makes
$\bar{\mbox{${\cal E}$}}$ an \mbox{$r\th$}\ root of $\bar{\omega}.$ This datum is actually a
descent datum because of the uniqueness of \mbox{$r\th$}\ root lifts. And
since all {\em fpqc} descent data for coherent sheaves are effective,
we have the desired $(\mbox{${\cal E}$},b)$ on $\mbox{${\cal X}$}$ extending $(\bar{\mbox{${\cal E}$}},\bar{b}).$
In fact the universal deformation of a spin curve $(X,E, \beta)$ over
a field $k$ is the obvious formal spin curve $(\tilde{\mbox{${\cal X}$}}, \mbox{${\cal E}$}, b).$
Here $\tilde{\mbox{${\cal X}$}}$ is the pullback of the universal deformation $\mbox{${\cal X}$}
\rightarrow
\frak{o}_{k} [[t_1,\dots, t_{n}]]$ of the curve $X/k$ along the
homomorphism $$\frak{o}_k [[t_1, \dots, t_n]]
\rightarrow \frak{o}_k [[P_1, Q_1, \dots P_l, Q_l, t_{l+1}, \dots,
t_{n} ]]/(P_i^{u_i}-Q_i^{v_i})$$ via $t_i \mapsto P_iQ_i$ for $i
\leq l.$ $u_i$ and $v_i$ are determined by the map $\beta$ at each
singularity of $E$ on the central fibre.
A particularly useful corollary of this is the following.
\begin{proposition}\label{smoothdef}
All quasi-spin structures over a field have a deformation to a smooth
spin structure.
\end{proposition}
\section{Isomorphisms}
\subsection{Isomorphisms of Spin Structures over a Field}
Since spin structures, quasi-spin structures and pure-spin structures
are all the same over a field, the study of isomorphisms over a field
is fairly simple. Any two spin structures on $ X/k,$ say $(\mbox{${\cal E}$},b)$
and $(\mbox{${\cal E}$}', b'),$ which are singular at the same points and are the
same on $X^{\nu}$ ($X^{\nu} @>{\theta}>> X$ is the normalization of
$X$ at the singularities of $\mbox{${\cal E}$}$) via $\theta^{\natural},$ must be
isomorphic on $X.$ For a given $\mbox{${\cal E}$}$ on $X,$ any two spin structure
maps $b$ and $b'$ are the same if and only if $\theta^{\natural} b=
\theta^{\natural} b',$ and this is true if and only if $\mbox{${\text{length}}$} _{{\frak p}}
(\mbox{${\text{coker}}$} (\theta^{\natural} b))= \mbox{${\text{length}}$} _{{\frak p}} (\mbox{${\text{coker}}$} (\theta^{\natural}
b'))$ for all ${\frak p}$ in the inverse image under $\theta$ of each
singular point.
\begin{proposition}
Automorphisms of $(X,\mbox{${\cal E}$},b)$ that are trivial on $X$ are of the form $
\gamma =(\zeta_{1},\zeta_{2},\dots,\zeta_{l}),$ where for each $i,$
$\zeta_{i}^{r}=1,$ and each $\zeta_{i}$ corresponds to a connected
component $ X^{\nu}_{i}$ of the curve $X^{\nu}.$
In other words, if $U_r$ is the group of \mbox{$r\th$}-roots of unity in $k,$
and $\Gamma(X^{\nu})$ is the dual graph of $X^{\nu},$
then $$\mbox{${\functor{Aut}}$}_X(\mbox{${\cal E}$},b)= H^0(\Gamma(X^{\nu}), U_r).$$
\end{proposition}
\begin{pf}
$\theta : X^{\nu} \rightarrow X$ makes $ \mbox{${\cal E}$} \cong \theta_{*}
\theta^{\natural}
\mbox{${\cal E}$},$ and $ \theta^{\natural}$ and $ \theta _{*}$
induce an equivalence of the categories of torsion-free rank-one
$\mbox{${\cal O}$}_{X}$-{modules} which are singular at the double points normalized
by $\theta$ and invertible sheaves on $X^{\nu}.$ Therefore, it is
enough to study $
\theta^{\natural}(\mbox{${\cal E}$}).$ But automorphisms of line bundles on $X^{\nu}
$ are just given by $\nu$-tuples of $\zeta
\in k^{*}$; moreover, $\theta^{\natural}(\mbox{${\cal E}$})^{r}=1$ implies that
$\zeta^{r}_{i}=1$ for all $i.$
\end{pf}
Note that, in general, isomorphisms of $(\mbox{${\cal E}$},b)$ over $X$ must be
induced by isomorphisms of $\theta^{\natural} \mbox{${\cal E}$},$ and therefore are
always constant on each connected component of $X^{\nu}$; namely, at a
singularity they are of the form $\Phi = \left(
\begin{array}{cc} \varphi_{+} & 0 \\ 0 & \varphi_{-} \end{array}
\right),$ with $\varphi_{+}$ and $\varphi_{-}$ in the base field $k.$
\subsection{Isomorphisms of Families of Spin Structures}
We can also say something about isomorphisms of spin structures in
general. We have seen that over a field these are all constant, i.e.
at each singularity, any isomorphism of spin structures $$ \Phi : (E
(p,q), b)
\rightarrow (E (p',q'), b')$$ must be of the form $$\Phi = \left(
\begin{array}{cc} \varphi_+(0) & 0 \\ 0 & \varphi_-(0) \end{array}
\right).$$ Using the step-by-step method we can show that this is the
case over any complete local ring. We just need to show that if
$\frak m I =0$ for some ideal $I$ in $R,$ and if $\Phi$ is constant
over $R/I,$ then $\Phi$ is constant over $R.$ But this follows because
$\varphi_+ = \varphi_+(0) + xi_+(x),$ for some $i_+(x) \in IR[[x]],$
and $\psi_+ = pi_+ =0;$ and similarly $\varphi_- = \varphi_-(0) +
yi_-(y),$ for some $i_-(y) \in IR[[y]]$ and $\psi_- =0.$ Therefore,
$$\Phi = \left(
\begin{array}{cc} \varphi_+(0) + xi_+(x) & 0
\\ 0 & \varphi_- (0) + yi_-(y) \end{array} \right),$$
and $b' = a'(x^u, p'x^{u-1}, \dots, y^v)$ is mapped to $b,$ namely $$b'
\circ \Phi^r = a'(x^u \varphi^r_+, x^{u-1} p' \varphi^{r-1}_+
\varphi_-, \dots, \varphi^r_- y^v) =b=a(x^u, x^{u-1}p, \dots, y^v).$$
So comparing the $b_0$ and $b_r$ terms, $a' \varphi^r_+ = a =
a' \varphi^r_-,$ and $$\varphi^r_+ = (\varphi_+(0) + xi_+(x))^r= \varphi
_+(0)^r + rxi_+(x) \varphi_+(0)^{r-1}$$ $$ = \varphi^r_- = (\varphi_-(0)^r
+ yi_-(y))^r= \varphi_-(0)^r + ryi_-(y) \varphi_-(0)^{r-1}.$$
Equating terms with similar powers of $x$ and $y,$ and using the fact
that $\varphi_+(0)$ and $\varphi_-(0)$ are invertible, as is $r,$
we have $i_+(x) = i_-(y) = 0,$ and $\Phi$ is constant.
\subsection{Automorphisms of Families of Quasi-Spin Structures}
Isomorphisms of quasi-spin curves are harder to classify than those of
spin curves, but if we limit ourselves to automorphisms, we can
completely classify these.
Given a quasi-spin structure $(\mbox{${\cal E}$}, b)$ on $\mbox{${\cal X}$}/S,$ with $S$ local and
complete, we want to study $\mbox{${\functor{Aut}}$}_{X}(\mbox{${\cal E}$}, b).$ First, we study the
local structure, namely, automorphisms of $(E(p,q), b)$ over $ A =
R[[x,y]]/pq-xy,$ with $b=(b_{0}, \dots, b_{r}).$ Here again, $R$ is a
complete local ring with maximal ideal $\frak m.$ Now, given $ \Phi
\in \mbox{${\functor{Aut}}$}_{A} (E,b) ,$ we know $\Phi = \left( \begin{array}{cc}
\varphi_{+} & \psi_{+}
\\ \psi_{-} & \varphi_{-} \end{array} \right )$ with $ b\circ \Phi^{r}
= b,$ and $$\varphi_{+}(0) p = \varphi_{-}(0) p \quad \text{ and } \quad \varphi_{+}(0)
q = \varphi_{-} (0) q$$ $$ \varphi_{+} =
\varphi_{+}(0) + x \gamma_{+}, \psi_{+} = p \gamma_{+}, \quad \text{ and } \quad
\gamma_+ \in R[[x]]$$ $$
\varphi_{-} = \varphi_{-}(0) + y \gamma_{-}, \psi_{-}=q \gamma_{-},
\quad \text{ and } \quad \gamma_- \in R[[y]].$$
Now $b=(b_{0}, \dots, b_{r}) \equiv (\beta_{0} x^{i}, 0, 0, \dots, 0,
\beta_{r} y^{j})\bmod \frak m,$ $(i+j =r)$ and
$\beta_{r}, \beta_{0} \in
k^{*},$ and $\Phi \equiv \left( \begin{array}{cc}
\overline{\varphi}_{+} & 0
\\ 0 & \overline{\varphi}_{-} \end{array} \right ).$ By the results
of the previous section, $\varphi^r_+ \equiv 1 \equiv
\varphi^r_-.$ In fact, this will hold for the whole family, i.e. we
can replace congruence with equality.
\begin{proposition}
$\Phi = \left( \begin{array}{cc}\varphi_{+} & 0
\\ 0 & \varphi_{-} \end{array} \right )$ with $\varphi^{r}_{+} =1,
\varphi_{-}^{r} =1.$
\end{proposition}
\begin{pf}
Using the step-by-step method we can assume the claim is true
$\bmod I $ for some ideal $I$ with $\frak m I=0.$ So
$\varphi^{r}_{+} = 1 +i,$ which implies that $(\varphi_+ - \frac{i}{r
\varphi^{r-1}_{+}})^r =1.$ So $\varphi_{+} = \zeta + i_{+}$ for some
$i_+$ in $I \cdot R[[x]],$ with $\zeta^r =1.$ Similarly, $\varphi_{-}
= \xi+i_{-}$ for some $i_-$ in $I \cdot R[[y]],$ with $\zeta^r =1.$
Thus $\Phi = \left( \begin{array}{cc}\varphi_{+} & 0
\\ 0 & \varphi_{-} \end{array} \right )$ and $b \circ \Phi^{r} = b$
implies that $$b_{0} = b_{0} (\zeta+i)^{r} = b_{0} (1+r
\zeta^{r-1}i_{+}), \quad \text{ and } \quad b_{r} = b_{r} (1+r \xi^{r-1}i_{-}).$$ This
implies that $b_{0} r \zeta^{r-1} i_{+} = 0 = b_{r} r
\xi^{r-1}i_{-}.$ But $ b_{0} \equiv x^{i}
\overline{\beta}_{0} \bmod \frak m,$ with $ \overline{\beta}_{0} \in
k^{*},$ and since $i_+$ and $i_-$ annihilate $\frak m,$ this implies
that $b_{0} r \zeta^{r-1} i_{+} = x^{i} \beta_{0} (r \zeta^{r-1})
i_{+}$ for some $ \beta_{0} \in R^{\times}$ lifting $\bar{\beta}_0,$
and similarly for $b_r.$ Since $1/r \in R,$ $i_{\pm} =0,$ and
$\varphi^r_+= \varphi^r_-=1.$ Moreover, $\varphi_+$ and $\varphi_-$
are in $R.$
\end{pf}
Now note also that $ \xi p = \zeta p,$ $\xi q = \zeta q.$ So $ p(\xi -
\zeta) = q(\xi - \zeta) = 0.$ But if $ \gamma := \xi - \zeta ,$ then
$(\xi + \gamma)^{r} = 1,$ which implies that $$r \gamma \xi^{r-1} +
\left( \begin{array}{c} r \\ 2
\end{array} \right) \gamma^{2} \xi^{r-2} + \dots = 0.$$ And if
$(\gamma)$ is a proper ideal, then $\bmod (\gamma^{2})$ we get $r
\gamma \xi^{r-1} \equiv 0,$ i.e. if $1/r \in R,$ $ \gamma
\in (\gamma^{2}) ,$ which implies that $\gamma \in \bigcap_{n}
(\gamma^{n}) \subseteq \bigcap_{n} \frak m^{n} = 0.$ This implies that
$\gamma =0.$
So either
\begin{enumerate}
\item $\gamma$ is invertible, hence $ p$ and $ q $ are zero, or
\item $\gamma$ is zero and $ \varphi_+ = \varphi_-.$
\end{enumerate}
So at each singularity with at least one of $p $ and $q$ not zero,
$\mbox{${\functor{Aut}}$} (E(p,q),b) = U_{r} = \{\zeta \in R^{\times} | \zeta^{r}
=1\}.$ And thus all automorphisms of $(\mbox{${\cal E}$},b)$ are also in $U_{r}$ if
no singularities are of type $E(0,0).$ A singularity of type $(0,0)$
has automorphisms of type $(\xi, \zeta ) \in U_{r} \times U_{r}.$
Normalizing $X$ at each singularity of type $(0,0)$ to get $X^{\nu}$
shows that $\mbox{${\functor{Aut}}$} (\mbox{${\cal E}$}, b)$ will be of type $(\xi_{1} ,
\dots, \xi_{m}),$ where $m$ is equal to the number of connected
components of $X^{\nu}.$
\subsection{Properties of the Isom Functor}\label{isom}
For any two quasi-coherent sheaves $\mbox{${\cal E}$}$ and ${\mbox{${\cal E}$}}'$ on a curve $X/B$
the functors $\mbox{$\text{Hom}$} (\mbox{${\cal E}$}, {\mbox{${\cal E}$}}')$ and $\mbox{${\functor{Isom}}$}(\mbox{${\cal E}$},{\mbox{${\cal E}$}}')$ are
representable (cf. \cite[7.7.8 and 7.7.9]{ega3} and \cite{lau}). For
the $B$-scheme $V$ and map $ \Phi : \mbox{${\cal E}$}_{X_{V}}
\rightarrow {\mbox{${\cal E}$} '}_{X_{V}}$ on $ X_{V}$ which represent the functor
$\mbox{${\functor{Isom}}$} (\mbox{${\cal E}$}, \mbox{${\cal E}$}'),$ the condition that $ \Phi^{r}$ commutes with
$ b$ and $b'$ is clearly an open condition, and thus is representable
over $V.$ Moreover, the scheme representing the functor $T \mapsto
\mbox{${\stack{Isom}}$}_{X_{T}}((\mbox{${\cal E}$}_{T},{b_{T}}), ({\mbox{${\cal E}$} '}_{T}, {b'}_{T})),$ is
an open subscheme of $
\mbox{$\text{Hom}$}_{X_{T}}(\mbox{${\cal E}$}_{T}, {\mbox{${\cal E}$}'}_{T}),$ which is quasi-projective of
finite type. Thus we have the following proposition.
\begin{proposition}
For any two quasi-spin structures $(\mbox{${\cal E}$},b)$ and $({\mbox{${\cal E}$}}', b')$ over a
stable curve $X/B,$ the functor $T \mapsto
\mbox{${\functor{Isom}}$}_{X_{T}}((\mbox{${\cal E}$}_{T},{b_{T}}), ({\mbox{${\cal E}$} '}_{T}, {b'}_{T}))$ is
represented by a quasi-projective $B$-scheme of finite type.
\end{proposition}
The proof and proposition are also valid for isomorphisms of spin
structures and pure-spin structures. Moreover, because the \mbox{${\functor{Isom}}$}
functor for stable curves over $S$ is representable by a quasi-projective
$S$-scheme of finite type, we actually have that for
any two (quasi/pure) spin curves $\frak S/T$ and $\frak S'/T'$ the
functor $T \mapsto \mbox{${\functor{Isom}}$}_{T\times T'}(\frak S,\frak S')$ is also
representable by a quasi-projective $S$-scheme of finite type.
Not only is \mbox{${\functor{Isom}}$}\ representable, it is also unramified and finite,
as the next two propositions show.
\begin{proposition}
For any two quasi-spin curves (or spin curves or pure-spin curves)
$\frak{S}=(X,\mbox{${\cal E}$},b)/T$ and $\frak{S}'=(X', {\mbox{${\cal E}$}}', b')/T',$ the scheme
$\mbox{${\stack{Isom}}$}_{T \times T'} (pr^*_1 \frak{S}, pr^*_2 \frak{S}')$ is
unramified over $T \times T'.$
\end{proposition}
\begin{pf}
It suffices to show that for a ring $R$ with
square-zero ideal $I$ and for any two quasi-spin structures $(\mbox{${\cal E}$},b)$
and $({\mbox{${\cal E}$}}',b')$ on a stable curve $X$ over $R$ with two isomorphisms
from $(\mbox{${\cal E}$},b)$ to $({\mbox{${\cal E}$}}',b')$ which agree over $\bar{R} = R/I,$ the
two isomorphisms must then agree over $R.$ (We do not need to
consider isomorphisms of the underlying curve because the \mbox{${\functor{Isom}}$}\ functor
for stable curves is unramified.) Since \mbox{${\stack{Isom}}$}\ is a principal
homogeneous \mbox{${\stack{Aut}}$} -space, we are reduced to showing that any automorphism
of $(\mbox{${\cal E}$},b) $ which is the identity over $\bar{R}$ is the identity
over $R.$ But this follows easily from the fact that all
automorphisms of quasi-spin curves are constant and have \mbox{$r\th$}\ power
equal to the identity.
Therefore, \mbox{${\stack{Isom}}$}\ is unramified.
\end{pf}
Since \mbox{${\stack{Isom}}$}\ is of finite type and unramified, it is quasi-finite, so we
only need to check that it is proper to see that it is finite.
\begin{proposition}
For any two (quasi, pure) spin curves $\frak{S}=(X, \mbox{${\cal E}$}, b)/T$ and
$\frak{S'}=(X', {\mbox{${\cal E}$}}',b')/T',$ the scheme $\mbox{${\stack{Isom}}$}_{T \times T}
(pr_{1}^{*} \frak{S},$ $pr^{*}_{2} \frak{S}')$ is proper over $T
\times T.$
\end{proposition}
\begin{pf}
We use the valuative criterion. We must show that if we are given two
spin curves, quasi-spin curves, or pure-spin curves, $ \frak S=(X,
\mbox{${\cal E}$}, b)$ and $\frak{S}'=(X', \mbox{${\cal E}$}', b')$ both over $\spec R,$ where $R$
is a discrete valuation ring, and given an isomorphism $ \Phi_{\eta}
: {\frak S}_{\eta} \rightarrow {\frak S}'_{\eta}$ defined on the
generic fibres, then we can always extend $ \Phi_{\eta}$ to an
isomorphism $\Phi$ over all of $\spec R.$
We can also assume that $R$ is complete, and since for stable curves
the \mbox{${\functor{Isom}}$}\ functor is proper, we can assume that $X = X'.$ Now let
$Y$ be the $fpqc$ cover of $X$ given by $ Y = U \coprod \left(
\coprod_{{\frak p}} \spec{\hat{\mbox{${\cal O}$}}_{X,\frak{p}}}\right),$ with the union
being taken over all closed points ${\frak p}$ of the singular locus of the
special fibre of $X,$ and $U$ the smooth locus of $X.$
If $\Phi_{\eta}$ extends to all of $Y,$ then it will in fact be
constant on all intersections $\spec{\hat{\mbox{${\cal O}$}}_{X,{\frak p}}}
\times_{X} U,$ and these constant isomorphisms are uniquely determined by
$\Phi_{\eta},$ hence $\Phi_{Y}$ will descend to an extension of
$\Phi_{\eta}$ on $X.$
Thus we only need to consider the local situation; namely, about a
singular point of the special fibre. This is the case where $$A = \mbox{$R[[x,y]]/(xy-\pi)$}
, \quad \mbox{${\cal E}$} = E(p,q) \quad \text{ and } \quad \mbox{${\cal E}$}' = E (p', q')$$ with $pq = p'q'
= \pi.$ And we need to show that an isomorphism $\Phi_{\eta}$ on the
fibre over the field of fractions $K$ of $R$ extends to an isomorphism
on all of $A.$ $\Phi_{\eta}$ lifts to a map $ \tilde{\Phi}_{\eta}: (A
\otimes_{R} K) ^{\oplus 2} \rightarrow (A \otimes _{R} K) ^{\oplus 2},$
which induces the isomorphism $\Phi_{\eta}: E (p,q) \otimes_{R} K
\rightarrow E (p', q') \otimes_{R} K.$ Since $\Phi_{\eta} $ is
constant, $ \tilde{\Phi}_{\eta} $ is given as a matrix
$\tilde{\Phi}_{\eta} = \left ( \begin{array}{cc} \varphi_{+} & 0 \\ 0
& \varphi_{-} \end{array} \right) ,$ with $ \varphi_{\pm}
\in K.$ It suffices to show that $\varphi_{+}$ and $\varphi_{-}$ are
actually in $R.$ But to be an isomorphism,
$\tilde{\Phi}_{\eta}$ must be such that $ \tilde{b}' \circ
\tilde{\Phi}_{\eta}^{\otimes r} = \tilde{b}.$
And since $\tilde{b}= (b_{0}, b_{1}, \dots, b_{r})$ and $\tilde{b}'=
({b'_{0}},{b'_{1}}, \dots, {b'_{r}}),$ we have ${b'_{0}}
\varphi^{r}_{+} = b_{0}$ and $ {b'_{r}} \varphi_{-}^{r} = b_{r}.$ But
as we have seen, $b_{0}$ and ${b'}_{0}$ are both invertible in
$A[1/x],$ hence in $A[1/x]$ the constant $\varphi_{+}^{r} =
b_{0}/{b'}_{0} \in (\mbox{$R[[x,y]]/(xy-\pi)$}) [1/x],$ and thus $\varphi_{+}^{r}
\in R,$ similarly for $\varphi_{-}^{r}.$ And $R$ is normal, hence the
$\varphi_{\pm}$ are in $R.$ And so $\Phi_{\eta}$ extends to all of
$\spec A,$ and thus to all of $X.$
\end{pf}
\section{Construction of the Stacks}
Fix $S$ to be a scheme of finite type over a field or over an excellent
Dedekind domain with $r$ invertible in $S.$ These conditions are
necessary for us to be able to use Faltings' theorem from
Section~\ref{falt} and to be able to use the standard theorems on
algebraic stacks (see \cite{dav}).
We have two main functors to consider, namely \mbox{${{\functor{QSpin}}}_{r,g}$}\ and \mbox{${\overline{\functor{Spin}}}_{r,g}$}.
\mbox{${{\functor{QSpin}}}_{r,g}$}\ is the \'etale sheafification of the functor taking an
$S$-scheme $T$ to the set of isomorphism classes of quasi-spin curves
over $T.$ And \mbox{${\overline{\functor{Spin}}}_{r,g}$}\ is the subfunctor of \mbox{${{\functor{QSpin}}}_{r,g}$}\ induced by
restricting to spin curves instead of quasi-spin curves. Note that
for a quasi-spin structure the property of being a spin structure is
local on the curve in the \'etale topology; therefore, the property of
being a spin structure is independent of the choice of log structure.
We also will consider a third functor over \mbox{${\overline{\functor{Spin}}}_{r,g}$}\ given by restricting
to pure-spin curves, namely those curves which, locally in the \'etale
topology, have the form $E(p,q)$ with $p = t^v,$ $q = t^u,$ for some
$t$ in the base, and $u+v=r.$ In particular, this means that $\pi$ is
$t^r.$ Up to \'etale covers this condition is also independent of the
particular choice of log structure and of the particular choice of $p$
and $q.$ We call this functor \mbox{${{\functor{Pure}}}_{r,g}$}. Of course, spin structures,
quasi-spin structures, and pure-spin structures are all the same thing
if the underlying curve is smooth. And quasi-spin structures over a
reduced base (or if $\pi$ is not a zero divisor) are actually spin
structures.
The main result of this section is that \mbox{${{\functor{QSpin}}}_{r,g}$}, \mbox{${\overline{\functor{Spin}}}_{r,g}$}, and
\mbox{${{\functor{Pure}}}_{r,g}$}\ are all separated algebraic stacks, locally of finite type
over $\mbox{$\overline{\mg}$},$ the moduli space of stable curves, and \mbox{${\functor{Spin}}_{r,g}$}, the
moduli of smooth spin curves, is dense
in each of these.
The fact that \mbox{${\functor{Spin}}_{r,g}$}\ is dense in the stacks follows from
Proposition~{\ref{smoothdef}}. To prove that these stacks are
algebraic, we need to do the following (see, for example, \cite[pp.
15--23]{dav}, or \cite{lau}):
\begin{enumerate}
\item Prove that the functors are limit preserving.
\item Provide a smooth cover $U$ of the stack.
\item Prove that for a fixed family of curves \mbox{${\cal X}$}\ over $T$
the functor $\mbox{${\functor{Isom}}$}_{U_{T} \times U_{T}}(pr_{1}^*,pr_{2}^*)$ is
representable by a scheme (it is clearly a groupoid).
\item Prove that the stacks are separated by showing
that $\mbox{${\stack{Isom}}$}_{U_{T} \times U_{T}}(pr_{1}^*,pr_{2}^*)$ is actually proper
and finite over $U_T \times U_T.$
\end{enumerate}
The last two conditions follow from the results of Section~\ref{isom}.
For the first two we begin by considering the stack \mbox{${{\functor{QSpin}}}_{r,g}$}. Many
results on the other two stacks follow relatively easily from this
case. The fact that the stack \mbox{${{\functor{QSpin}}}_{r,g}$}\ is limit preserving is a
straightforward consequence of the following theorem of Grothendieck
and the fact that the the condition on the length of the cokernel is
an open condition (c.f. Proposition~\ref{opencoker}), hence limit
preserving.
\begin{theorem}[{\cite[8.5.2]{ega4}}]
Given a quasi-compact and quasi-separated scheme $S_0,$ and given a
projective system $\{S_{\gamma} \}$ of $S_0$-schemes, relatively
affine over $S_0,$ and quasi-coherent $\mbox{${\cal O}$}_{S_{\gamma}}$-modules
$\mbox{${\cal F}$}_{\gamma}$ and $\mbox{${\cal G}$}_{\gamma},$ with $ \mbox{${\cal F}$}_{\gamma}$ of finite
presentation, the canonical homomorphism of groups $$
\lim_{\rightarrow}(\mbox{$\text{Hom}$}_{S_{\gamma}}(\mbox{${\cal F}$}_{\gamma},\mbox{${\cal G}$}_{\gamma}))
\rightarrow \mbox{$\text{Hom}$}_{S}(\mbox{${\cal F}$},\mbox{${\cal G}$})$$ is an isomorphism. Here $S,$ $\mbox{${\cal F}$},$
and $\mbox{${\cal G}$}$ are the obvious limit objects.
\end{theorem}
All that remains is condition (2), i.e. to provide a smooth cover.
\subsection{A Smooth Cover of \mbox{${{\functor{QSpin}}}_{r,g}$}}
\begin{proposition}
Given a curve $X/B$ and an integer $N,$ sufficiently large, the
functor taking a $B$-scheme $T$ to the set of all triples
$(\mbox{${\cal E}$},b,(e_1, \dots, e_n))$ where $(\mbox{${\cal E}$},b)$ is a quasi-spin structure
on $X_T,$ and $(e_1, \dots, e_n)$ is a basis for the module $\Gamma
(X_T, \mbox{${\cal E}$}_T {\otimes}
\omega_T^{\otimes N})$ on
$X_T$ is representable.
\end{proposition}
\begin{pf}
Any quasi-spin structure $(\mbox{${\cal E}$},b)$ on $X$ must have total degree $=
\frac{1}{r}(2g-2),$ and on its normalization $ \theta: \tilde{X}
\rightarrow X$ $$ \theta^{\natural} \mbox{${\cal E}$}^{\otimes r} \cong \theta^*
\omega_{X/k} (-\sum (u_i {{\frak p}}^+_i + v_i {{\frak p}}^-_i)),$$ where the sum
is taken over all singularities $\{ {\frak p}_i \}$ of $\mbox{${\cal E}$}$ and
$\{\frak{p}_i^+,
\frak{p}_i^-\}$ are the inverse images of $\frak{p}_i$ via $\theta.$
In particular, for any given irreducible component $X_j$ of $X,$ we
have $$\deg_{X_j} (\theta^{\natural}
\mbox{${\cal E}$}) \geq
\frac{1}{r} (\deg_{X_j} (\omega_{X/k}) -r \delta_j),$$ where
$\delta_j$ is the number of singularities of $X$ in $X_j.$ Since
$\deg_{X_j}(\omega_{X/k})$ is always positive, and since the total
number of singularities in a stable curve of genus $g$ is bounded by
$3g-3,$ we have $$\deg_{X_i} \theta^{\natural}
\mbox{${\cal E}$} \geq 1-\delta_i \geq 2 -3g.$$
Thus, given a very ample line bundle $\mbox{${\cal O}$}(1)$ on $X,$ we can choose,
as in Theorem~\ref{thm:dsousa}, a fixed $m_0$ depending only on the
genus of $X$ such that $\mbox{${\cal E}$} \otimes \mbox{${\cal O}$}(m) $ is generated by global
sections and has vanishing higher cohomology for all $m \geq m_0.$
Fix, once and for all, an integer $N$ large enough so that $\omega^N$
is very ample and $\mbox{${\cal E}$} \otimes \omega^N$ has all the desired
properties. Now we can represent torsion-free rank-one sheaves with
bounded degree on each component by a subscheme of
$\mbox{${\text{Quot}}$}_{\mbox{${\cal O}$}_{X}^n/X/S}$ for some $n,$ i.e. there exists $ U_{1}
\hookrightarrow \mbox{${\text{Quot}}$}_{\mbox{${\cal O}$}_{X}^n/X/S}$ which represents the functor
$$ T \mapsto \{ \mbox{${\cal F}$} , ( e_{1} \dots e_{n})\},$$ where $ \mbox{${\cal F}$}$ is a
rank-one, torsion-free sheaf on $X_T$ with bounded degree on each
component, and $ (e_{1} \dots e_{n} )$ is a basis of $ \Gamma(X_{T},
\mbox{${\cal F}$} \otimes \omega^N)$ for $\omega^N$ sufficiently ample. So over
$X_1/U_1$ there is a universal pair $(\mbox{${\cal E}$}, ( e_{1} \dots e_{n})).$ And
to represent maps $ \mbox{${\cal E}$}^{\otimes r}
\rightarrow \omega _{X_{1}/U_{1}},$ take $$ V:= \Bbb V(\mbox{${\cal E}$}^{\otimes
r} \otimes \omega ^{-1}) := {\text{\bf Spec}}_{X_{1}} (
\text{Sym}_{\mbox{${\cal O}$}_{X_{1}}}(\mbox{${\cal E}$}^{\otimes r} \otimes \omega ^{-1})),$$
So that $\mbox{$\text{Hom}$}_{X_{1}}(Y,V) = \mbox{$\text{Hom}$} _Y(\mbox{${\cal E}$}^{\otimes r}_Y, \omega_Y).$
So letting $V_T:= V \times_{X_1} X_T = V
\times_{U_1}T,$ we get that
$$\mbox{$\text{Hom}$}_{X_{1}}(X_{T},V)=
\mbox{$\text{Hom}$}_{X_{T}}(X_{T},V_{T}) = (\prod_{X/S} V/X) (T)$$ is the functor we
want, and it is
representable because $X$ is flat and projective over $S$ (see \cite{fga}).
Now we have a universal triple $ \mbox{${\cal E}$}, ( e_{i} \dots e_{n}), b:
\mbox{${\cal E}$}^{\otimes r} \rightarrow
\omega$ on $ X_{2}/U_{2}$ representing all maps $ \mbox{${\cal E}$}^{\otimes r}
\rightarrow \omega,$ and the additional condition that the cokernel of
$b$ is supported on the singular locus of $X_{2}$ is also
representable; namely, it is just the condition that $b$ is an
isomorphism on the complement of the discriminant locus, and this is
an open condition. Finally, we need to represent the condition on the
cokernel, but this condition is open on the base, as proved in
Proposition~\ref{opencoker}.
\end{pf}
In general, for an arbitrary stable curve $\mbox{${\cal X}$}/T$ we have represented
by some scheme $U$ all quasi-spin structures $(\mbox{${\cal E}$},b)$ on $\mbox{${\cal X}$}_U$ such
that $\mbox{${\cal E}$} \otimes \omega^N$ can be expressed as a quotient of
$\mbox{${\cal O}$}^n_{\mbox{${\cal X}$}},$ together with a basis for the module $\Gamma(\mbox{${\cal X}$}, \mbox{${\cal E}$}
\otimes \omega^N).$ Moreover, at any closed point of $T$ all
quasi-spin structures on $\mbox{${\cal X}$} \times_T \spec{\mbox{${\cal O}$}_{T,t}}$ can be
expressed as such a quotient. Therefore, at each point $u$ of $U$ the
complete local ring $\hat{\mbox{${\cal O}$}}_{U,u}$ is a versal deformation of the
quasi-spin structure induced by $u.$ In particular, if the curve
$\mbox{${\cal X}$}$ we begin with is the universal curve over an \'etale cover $T$
of $\mbox{$\overline{\mg}$},$ the moduli stack of stable curves, then $U$ is a cover of
the stack \mbox{${{\functor{QSpin}}}_{r,g}$}.
\begin{proposition}
The scheme $U,$ which represents all quasi-spin structures $(\mbox{${\cal E}$},b)$
on the universal curve $\mbox{${\cal X}$}/T$ together with a choice $(e_1, e_2,
\dots, e_n)$ of global sections which generate $\mbox{${\cal E}$}
\otimes \omega^N,$ is smooth over the stack of quasi-spin
structures on the universal curve.
\end{proposition}
\begin{pf}
We have to show that if an affine $T$-scheme $Y=\spec B$ has a
square-zero ideal $I \subseteq B$ and a quasi-spin structure $(\mbox{${\cal E}$},b)$
on $X \times_T Y$ such that $(\mbox{${\cal E}$},b)$ restricted to $Y_0 = \spec{B/I}$
has a basis $(\bar{e}_1, \dots \bar{e}_n)$ for $\Gamma (X
\times_T Y_0, \bar{\mbox{${\cal E}$}} \otimes
\bar{\omega}^N),$ then $(\bar{e}_1, \dots, \bar{e}_n)$ lifts to a
basis of $\Gamma (X\times_T Y, {\mbox{${\cal E}$}} \otimes {\omega}^N)$ on $Y.$
Namely, it suffices to show that if $pr: X \times_T Y \rightarrow Y$
makes $pr_* (\bar{\mbox{${\cal E}$}} \otimes
\bar{\omega}^N)$ free on $Y_0,$ then the locally free sheaf $pr_* (\mbox{${\cal E}$}
\otimes \omega^N)$ is also free of the same rank as $pr_*(\bar{\mbox{${\cal E}$}}
\otimes \bar{\omega}^N).$ But this is clear because $\mbox{${\cal E}$} \otimes
\omega^N$ commutes with base change, and the exponential sequence $$ 0
\rightarrow \widetilde{M_n (I)}\rightarrow \mbox{${\cal G}$} l_n (\mbox{${\cal O}$}_Y) \rightarrow
\mbox{${\cal G}$} l_n (\mbox{${\cal O}$}_{Y_0}) \rightarrow 0 $$ shows that the kernel of the
homomorphism $H^1 (|Y_0|, \mbox{${\cal G}$} l_n (\mbox{${\cal O}$}_Y)) \rightarrow H^1 (|Y_0|,
\mbox{${\cal G}$} l_n (\mbox{${\cal O}$}_{Y_0}))$ is $H^1$ of the coherent sheaf
$\widetilde{M_n(I)}$ on an affine scheme, hence is zero. Therefore,
any rank $n,$ locally free sheaf on $Y$ which restricts to a free
sheaf on $Y_0$ must be free already.
\end{pf}
This completes the proof that the stack \mbox{${{\functor{QSpin}}}_{r,g}$}\ is algebraic.
\subsection{\mbox{${\overline{\functor{Spin}}}_{r,g}$}\ and \mbox{${{\functor{Pure}}}_{r,g}$}}
To construct a smooth cover of \mbox{${\overline{\functor{Spin}}}_{r,g}$}, we first take an arbitrary
smooth cover of \mbox{${{\functor{QSpin}}}_{r,g}$}, say $\frak{S}:U \rightarrow \mbox{${{\functor{QSpin}}}_{r,g}$},$ together
with its canonical log structure.
We can also assume that $U$ is affine. Thus all of our previous
descriptions of spin structures apply, and in particular, the
homomorphism $b: \mbox{${\cal E}$}^{\otimes r} \rightarrow \omega$ is given (\'etale
locally on $X/U$) as $b = (b_0, \dots, b_r).$ Also, we can still
write the $b_i$ as power series $$b_i = \sum_{n \geq 0} b_{i,n} x^n +
\sum_{m>0} b_{i,-m} y^m$$ with the same relations as before, and in
particular, up to suitable base extension and isomorphism of $E(p,q),$
$$p^u =q^v$$ and $$b_{0,i} = \pi^{u-i} b_{r,i-r} +
\sigma_i \text{ for } 0 < i \leq u \quad \text{ and } \quad b_{r,i} = \pi^{i-u} b_{0,i} +
\sigma_i \text{ for } u \leq i < r.$$
Moreover, $b$ is a spin structure if and only if $\sigma_i = 0$ for all
$i.$ So the closed subscheme $V$ defined by the ideal generated by the
$\sigma_i$ actually represents the condition that $b$ is a
spin-structure.
It is clear that the spin curve $ \frak{S}_V=(\mbox{${\cal X}$},\mbox{${\cal E}$},b)_V$ over $V$
makes $V$ a cover of \mbox{${\overline{\functor{Spin}}}_{r,g}$}. To see that it is smooth over
\mbox{${\overline{\functor{Spin}}}_{r,g}$}, note that for any spin curve $\frak{T}/T$ the scheme
$\mbox{${\stack{Isom}}$}_{\mbox{${{\functor{QSpin}}}_{r,g}$}} (\frak{S}/U, \frak{T}/T)$ is isomorphic to
$\mbox{${\stack{Isom}}$}_{\mbox{${\overline{\functor{Spin}}}_{r,g}$}} (\frak{S}/U, \frak{T}/T),$ which is isomorphic to
$\mbox{${\stack{Isom}}$}_{\mbox{${\overline{\functor{Spin}}}_{r,g}$}} (\frak{S}_V/V, \frak{T}/T).$ Hence smoothness of
$U$ over \mbox{${{\functor{QSpin}}}_{r,g}$}\ implies smoothness of $V$ over \mbox{${\overline{\functor{Spin}}}_{r,g}$}.
Alternately, we can consider, as in the construction of the versal
deformation of \mbox{${{\functor{QSpin}}}_{r,g}$}, a curve $\mbox{${\cal X}$}/T$ and a relatively torsion-free
sheaf $\mbox{${\cal E}$},$ so that the pair $(\mbox{${\cal X}$}/T, \mbox{${\cal E}$})$ is versal for stable
curves with rank-one, torsion-free sheaves with bounded degree on
each component. Taking the canonical log structure and constructing
the canonical induced map $\mbox{${\cal E}$} \rightarrow \rho_* \mbox{${\cal M}$},$ we can take the
scheme representing the property that $\mbox{${\cal M}$}$ is isomorphic to
$\omega_{X/T}$ to be our cover.
Since the property of being spin is independent of choice of log
structure, this is a cover of \mbox{${\overline{\functor{Spin}}}_{r,g}$}. Moreover, because it
represents all spin structure maps for $(\mbox{${\cal X}$}/T, \mbox{${\cal E}$}),$ it is smooth
over \mbox{${\overline{\functor{Spin}}}_{r,g}$}.
The representability of \mbox{${\functor{Isom}}$}, as well as the other properties (finite
and unramified), all follow from the case of \mbox{${{\functor{QSpin}}}_{r,g}$}, hence \mbox{${\overline{\functor{Spin}}}_{r,g}$}\
is an algebraic stack, locally of finite type over $S.$
Now to construct the stack \mbox{${{\functor{Pure}}}_{r,g}$}, take, as above, the cover $V$ of
\mbox{${\overline{\functor{Spin}}}_{r,g}$}\ together with its canonical log-structure on the universal
curve $\mbox{${\cal X}$}_V$ and isomorphisms $\mbox{${\cal E}$} \cong E(p,q)$ for $p,$ and $q$ in
$\mbox{${\cal O}$}_V.$ The condition that $\frak{S}_V$ is pure is representable by
the relatively affine $V$-scheme $W:={\text{\bf Spec}}_V (\mbox{${\cal O}$}_V
[\tau]/(p-\tau^v, q-\tau^u)).$ Again it is easy to verify that $W$ is
a smooth cover of \mbox{${{\functor{Pure}}}_{r,g}$}, and that \mbox{${{\functor{Pure}}}_{r,g}$}\ is algebraic. We
have proved the following theorem.
\begin{theorem}
\mbox{${{\functor{QSpin}}}_{r,g}$}, \mbox{${\overline{\functor{Spin}}}_{r,g}$}, and \mbox{${{\functor{Pure}}}_{r,g}$}\ all form separated algebraic stacks
of finite type over \mbox{$\overline{\mg}$}, and \mbox{${\functor{Spin}}_{r,g}$}\ is dense in each of these.
\end{theorem}
\subsection{Singularities and Smoothness of \mbox{${\overline{\functor{Spin}}}_{r,g}$}\ and \mbox{${{\functor{Pure}}}_{r,g}$}}
The deformation theory done in Section~\ref{deftheory} completely
describes the local structure of \mbox{${\overline{\functor{Spin}}}_{r,g}$}, and in fact it shows that
\mbox{${\overline{\functor{Spin}}}_{r,g}$}\ is relatively Gorenstein (i.e. \mbox{${\overline{\functor{Spin}}}_{r,g}$}\ is Gorenstein if
the base $S$ is), namely it is enough to check the completion of the
stalks for an \'etale cover, (c.f.
\cite[Theorem 18.3]{matsumura}), and these are of the form
$$\hat{\mbox{${\cal O}$}}_{S,s} [[P_1, Q_1, \dots, P_l, Q_l, t_{l+1}, \dots,
t_n]]/(P_i^{u_i}-Q_i^{v_i}).$$
Now it is enough to check the quotient $$\hat{\mbox{${\cal O}$}}_{S,s}/(a_{1},
a_{2}, \dots, a_{m}),$$ where $\{ a_{i} \}$ are any
$\hat{\mbox{${\cal O}$}}_{S,s}$-regular sequence. And taking $a_{1}= P_{1},
a_{2}=P_{2}, \dots a_{l} = P_{l}, a_{l+1}= t_{l+1}, \dots, a_{n} =
t_{n},$ we are reduced to the case of $
\hat{\mbox{${\cal O}$}}_{S,s}[[Q_{1}, Q_{2}, \dots Q_{l}]]/(Q_{i}^{v_{i}}).$
But this is Gorenstein because $\hat{\mbox{${\cal O}$}}_{S,s}[[Q_{1}, \dots,
Q_{l}]]$ is, and the ring in question is just $\hat{\mbox{${\cal O}$}}_{S,s}[[Q_{1},
\dots, Q_{l}]]$ modulo the regular sequence $ ( \{ Q_{i}^{v_{i}}\}).$
\mbox{${{\functor{Pure}}}_{r,g}$}\ provides a resolution of the singularities of \mbox{${\overline{\functor{Spin}}}_{r,g}$};
namely, \mbox{${{\functor{Pure}}}_{r,g}$}\ is smooth over $S$ because the completion of any of
its local rings is of the form $$ \hat{\mbox{${\cal O}$}}_{S,s} [[ \tau_1, \tau_2,
\dots, \tau_l, t_{l+1}, \dots, t_n]],$$ which is smooth over
$\hat{\mbox{${\cal O}$}}_{S,s}.$
\section{Compactness}
The goal of this section is to prove the properness of the stacks
\mbox{${{\functor{QSpin}}}_{r,g}$}, \mbox{${\overline{\functor{Spin}}}_{r,g}$}, and \mbox{${{\functor{Pure}}}_{r,g}$}. This is accomplished by studying the
boundary of these stacks, i.e. the degeneration of smooth spin curves
into spin structures on stable curves. To prove that the stacks are
proper we will use the valuative criterion and the fact that smooth
spin curves are dense (c.f. Proposition~\ref{smoothdef}) to justify
checking the valuative criterion only in the case that the generic
fibre is smooth (c.f. \cite[pg. 109]{dm}).
\subsection{Extending Spin-Structures and Line Bundles}
Given a complete, discrete valuation ring $R$ with field of quotients
$K,$ and a $K$-valued point $\eta$ of $\mbox{${{\functor{QSpin}}}_{r,g}$},$ corresponding to
$\frak S_{\eta} = (\mbox{${\cal X}$}_{\eta}, \mbox{${\cal E}$}_{\eta}, b_{\eta}),$ with
$\mbox{${\cal X}$}_{\eta}$ smooth over $K,$ we need to show that (up to finite
extension of $K$) there exists a quasi-spin curve $\frak S$ over $R,$
extending $\frak S_{\eta}.$ To this end, we construct a semi-stable
curve and line bundle which will give the desired extension when
contracted to its stable model, as in Section~\ref{geom}.
To begin, since $\mbox{$\overline{\mg}$}$ is proper, there is a stable curve $\mbox{${\cal X}$}$
extending $\mbox{${\cal X}$}_{\eta}$ over $R.$ Take a uniformizing parameter $t \in
R$ and map $R$ to itself via $t \mapsto t^r.$ Pulling back $\mbox{${\cal X}$}$ along
this map yields another (singular) curve $\mbox{${\cal X}$}_r.$ Resolving the
singularities of $\mbox{${\cal X}$}_r$ by blowing up yields a semi-stable curve
$\tilde{\mbox{${\cal X}$}}$ with generic fibre $\mbox{${\cal X}$}_{\eta}$ (up to a finite
extension of $K$) and special fibre having chains of $n_i r-1$
exceptional curves over each singularity of $\mbox{${\cal X}$}_r.$ Here $n_i$ is the
order of the corresponding singularity of $\mbox{${\cal X}$},$ namely $\mbox{${\cal X}$}$ has
local equation $ R[[x,y]]/xy-t^{n_i}.$
Now, since $\tilde{\mbox{${\cal X}$}}$ is regular, any line bundle on the generic
fibre will extend (but not uniquely) to the entire curve. In
particular, there is some line bundle $\mbox{${\cal L}$}$ on $\tilde{\mbox{${\cal X}$}}$ which
extends $\mbox{${\cal L}$}_{\eta}.$ It is well-known that in such a case, any two
line bundles which agree on the generic fibre differ only by Cartier
divisors supported on the special fibre. In other words, if
$\mbox{${\cal M}$}_{\eta} \cong \mbox{${\cal N}$}_{\eta}$ then $\mbox{${\cal M}$} \cong \mbox{${\cal N}$} \otimes \mbox{${\cal O}$} (\sum
a_i X_i),$ where $X_i$ are the irreducible components of the special
fibre of $\tilde{\mbox{${\cal X}$}},$ and $a_i$ are integers. In our case,
therefore, $\mbox{${\cal L}$}^{\otimes r} \cong \omega_{\tilde{\mbox{${\cal X}$}}}
\otimes \mbox{${\cal O}$} (\sum a_i X_i)$ for some integers $a_i.$
Of course, if $\mbox{${\cal L}$}$ extends $\mbox{${\cal L}$}_{\eta}$ then any line bundle of the
form $\mbox{${\cal L}$} \otimes \mbox{${\cal O}$} (\sum b_i X_i)$ also extends $\mbox{${\cal L}$}_{\eta}.$ The
following results show that there is a choice $\mbox{${\cal O}$} (\sum b_i X_i)$ so
that $\mbox{${\cal L}$}':= \mbox{${\cal L}$} \otimes \mbox{${\cal O}$} (\sum b_i X_i)$ is a line bundle with
degree zero on all but one exceptional curve per chain, has degree one
on the one remaining exceptional curve, and there exists an
$\mbox{${\cal O}$}_{\tilde{\mbox{${\cal X}$}}}$-module homomorphism $\beta:{\mbox{${\cal L}$}'}^{\otimes
r}\rightarrow \omega_{\tilde{\mbox{${\cal X}$}}}$ which is an isomorphism everywhere
except on the exceptional curves where $\mbox{${\cal L}$}'$ has degree one.
Contracting all the exceptional curves of $\tilde{\mbox{${\cal X}$}}$ induces a spin
curve on $\mbox{${\cal X}$}_r,$ and hence an $R$-valued point of \mbox{${{\functor{QSpin}}}_{r,g}$} extending
$\frak{S}_{\eta}.$
\begin{proposition}
Given $\mbox{${\cal L}$}$ on $\tilde{\mbox{${\cal X}$}}$ such that $\mbox{${\cal L}$}^{\otimes r} \cong
\omega_{\tilde{\mbox{${\cal X}$}}} \otimes \mbox{${\cal O}$}(\sum a_i X_i),$ the coefficients
$a_i$ which correspond to non-exceptional components of the special
fibre can all be assumed divisible by $r.$ In particular, the line
bundle $\displaystyle \mbox{${\cal L}$}':= \mbox{${\cal L}$} \otimes \mbox{${\cal O}$} \left ( -\frac{1}{r}
\!\!\! \sum
\begin{Sb} X_i \text{not} \\
\text{exceptional} \end{Sb} \!\!\! a_i X_i \right )$ has
${\mbox{${\cal L}$}'}^{\otimes r} \cong \omega \otimes \mbox{${\cal O}$} (\sum e_j E_j)$ where all
the $E_j$ are exceptional curves.
\end{proposition}
\begin{pf}
Basic intersection theory shows that, for any curve $X_j,$ the degree
of $\mbox{${\cal O}$}(\sum a_i X_i)$ on $X_j$ is $-a_j \delta_j + \sum a_i
\delta_{ij},$ where $\delta_j$ is the number of points in the
intersection of $X_j$ with the rest of the special fibre, and
$\delta_{ij}$ is the number of points in the intersection of $X_i$ and
$X_j.$ Now, on any given exceptional curve $E$ in a chain, with $E$
intersecting only two curves $C_1$ and $C_2,$ we have $\deg_E
(\omega(\sum a_i X_i)) = \deg_E(\mbox{${\cal O}$}(\sum a_i X_i)) = -2e+c_1+c_2,$
where $e, c_1,$ and $c_2$ are the coefficients in the sum $\sum a_i
X_i$ of $E, C_1,$ and $C_2$ respectively. Moreover,
$\deg_{X_i}(\omega(\sum a_i X_i)) = r \deg_{X_i} \mbox{${\cal L}$} \equiv 0
\pmod r$ for every $X_i.$ So, in particular, $c_1 + c_2 \equiv 2e
\pmod r.$ Now, given a chain of exceptional curves $E_1, \dots,
E_{nr-1},$ and the two non-exceptional curves $C$ and $D$ that the
chain joins, if their associated coefficients are $e_1, e_2, \dots,
e_{nr-1}, c, $ and $d,$ respectively, then we must have $e_2 \equiv
2e_1-c,$ $e_3 \equiv 2e_2-e_1\equiv 3e_1-2c$ and $e_i \equiv
ie_1-(i-1)c,$ so that $e_{nr-1} \equiv (nr-1)e_1-(nr-2)c$ and $d
\equiv nre_1 -(nr-1)c \equiv c.$ Therefore, since the special fibre is
connected, and since all of the non-exceptional curves are joined by
exceptional chains, all of the coefficients of the non-exceptional
curves are congruent to $c$ for some choice of $c.$ But since the
divisor $(\sum X_i)$ is trivial, we can assume that at least one of
the coefficients of a non-exceptional curve is zero, hence all of them
are congruent to zero $\pmod r,$ and thus $\mbox{${\cal L}$}':= \mbox{${\cal L}$} \otimes (-\sum
\!\!\! \begin{Sb} X_i \text{not} \\
\text{exceptional} \end{Sb}\!\!\! (a_i/r) X_i)$ is a line bundle extending
$\mbox{${\cal L}$}_{\eta}$ such that ${\mbox{${\cal L}$}'}^{\otimes r} \cong \omega(\sum e_i E_i)$
and the $E_i$ are all exceptional curves.
\end{pf}
\begin{proposition}
If $\mbox{${\cal L}$}$ is a line bundle on $\tilde{\mbox{${\cal X}$}}$ such that $\mbox{${\cal L}$}^{\otimes
r}\cong \omega(\sum e_iE_i),$ with all of the $E$'s exceptional curves
in the special fibre, then there is a choice of coefficients
$\{{e'}_i\}$ such that ${e'}_i \equiv e_i \pmod r$ for every $i,$ and
the degree of $\omega(\sum {e'}_i E_i)$ is zero on every exceptional
curve except perhaps one per chain, where it has degree $r.$ In
particular the bundle $$\mbox{${\cal L}$} ': = \mbox{${\cal L}$} \otimes \mbox{${\cal O}$}(\sum
(\frac{{e'}_i -e_i}{r}) E_i)$$ has degree zero on every exceptional
curve except perhaps one per chain, where it has degree one. And
${\mbox{${\cal L}$}'}^{\otimes r} \cong \omega(\sum {e'}_i E_i).$
\end{proposition}
\begin{pf}
Because $\mbox{${\cal L}$}^{\otimes r} \cong \omega(\sum e_i E_i),$ the degree of
$\mbox{${\cal O}$}(\sum e_i E_i)$ on each $E_i$ must be congruent to zero $\pmod r,$
and so for any particular chain $E_1, \dots E_{nr-1}$ we have $e_2
\equiv 2e_1,$ $e_3 \equiv 3e_1,$ and $e_i \equiv ie_1$ for each $i.$
Choose $0 \geq {e'}_1 > -r$ with ${e'}_1 \equiv e_1 \pmod r,$ and let
${e'}_i = ie'_1$ for $1 \leq i \leq n(r+e'_1).$ Choose ${e'}_i = ie'_1
+r$ for $n(r+e'_1) +1 \leq i \leq nr-1.$ This gives ${e'}_i \equiv e_i
\pmod r$ for all $i,$ $\deg_{E_j} \mbox{${\cal O}$}(\sum {e'}_i E_i) =0$ for all $j
\neq n(r+e'_1)$ and on $E_{nr+ne'_1},$ the degree is
$-2e'_{n(r+e'_1)}+e'_{n(r+e'_1)-1} +
e'_{n(r+e'_1)+1}=(-2(n(r+e'_1))+n(r+e'_1)-1+n(r+e'_1)+1)+r,$ which is
$r.$
\end{pf}
Note also that all of the ${e'}_i$ in the previous proposition were
negative, thus there is a canonical inclusion map $\omega(\sum {e'}_i
E_i)
\hookrightarrow \omega.$
\subsection{The Stacks are Proper}
By the results of the previous section, we have for any complete
discrete valuation ring $R$ with a smooth spin curve $\frak S_{\eta} =
(\mbox{${\cal X}$}_{\eta}, \mbox{${\cal L}$}_{\eta})$ over its generic point $\eta,$ an extension
of the spin curve to a curve/line-bundle pair $(\tilde{\mbox{${\cal X}$}},\mbox{${\cal L}$})$ over
$R$ (up to finite extension of the field of fractions) with the
following special properties.
\begin{itemize}
\item $\tilde{\mbox{${\cal X}$}}$ is semi-stable.
\item For any chain of
exceptional curves in the special fibre of $\tilde{\mbox{${\cal X}$}},$ the degree
of $\mbox{${\cal L}$}$ is zero on every exceptional curve in the chain except
perhaps one, where it has degree one.
\item $\mbox{${\cal L}$}^{\otimes r} \cong \omega_{\tilde{\mbox{${\cal X}$}}}
(\sum e_i E_i)$ with all of the ${e}_i$
negative.
\end{itemize}
This means there is a natural map $$ \omega(\sum {e}_i
E_i) \hookrightarrow \omega,$$ inducing a spin structure on the stable
model $\mbox{${\cal X}$}_r;$ namely, if
$\theta$ is the contraction $\tilde{\mbox{${\cal X}$}} \rightarrow \mbox{${\cal X}$}_r$ then
$\theta_* \mbox{${\cal L}$}$ is a rank-one torsion-free sheaf, and the map
$$\mbox{${\cal L}$}^{\otimes r} @>{\sim}>> \omega(\sum e_i E_i)\hookrightarrow
\omega_{\tilde{\mbox{${\cal X}$}}}$$ induces a spin structure map $$(\theta_*
\mbox{${\cal L}$})^{\otimes r} \rightarrow \omega_{\mbox{${\cal X}$}_r}.$$
Thus $\frak{S}_{\eta}$ extends to a spin structure $\frak{S}$ over all
of $R,$ and the valuative criterion holds. We have proven the
following theorem.
\begin{theorem}
The stack \mbox{${{\functor{QSpin}}}_{r,g}$} is proper over $S.$
\end{theorem}
Since \mbox{${\overline{\functor{Spin}}}_{r,g}$} is a closed subscheme of \mbox{${{\functor{QSpin}}}_{r,g}$}, and since it surjects
to \mbox{$\overline{\mg}$}, it is also a compactification of \mbox{${\functor{Spin}}_{r,g}$}\ over \mbox{$\overline{\mg}$}. And
since $\mbox{${{\functor{Pure}}}_{r,g}$} \rightarrow \mbox{$\overline{\mg}$}$ is surjective, and \mbox{${{\functor{Pure}}}_{r,g}$}\ is
proper over \mbox{${\overline{\functor{Spin}}}_{r,g}$}, the stack \mbox{${{\functor{Pure}}}_{r,g}$}\ is another compactification
of \mbox{${\functor{Spin}}_{r,g}$}\ over \mbox{$\overline{\mg}$}.
\section*{Conclusion}
We have constructed three algebraic stacks which compactify the moduli
space of spin curves. The stack of quasi-spin curves, which, in some
sense, is easiest to construct, is not as easy to describe as the
substack of spin curves, which has nice (Gorenstein) singularities.
And these singularities are resolved by the stack of pure-spin curves.
In the special case of $2$-spin curves, many of the difficulties
disappear. In particular, all three compactifications coincide.
Moreover, this compactification of $2$-spin curves can be shown to
agree with those of Cornalba \cite{corn:theta} and Deligne
\cite{deligne:letter}.
\section*{Acknowledgements}
I am grateful to all the people who have helped with this paper, in
particular to Gerd Faltings, for introducing me to moduli problems in
general and this problem in particular, and for his help and direction
throughout. Thanks also to B. Speiser, S. Zhang, N. Katz, R. Smith,
S. Voronov, A. Vaintrob, J.-F.~Burnol, and J. H. Conway for helpful
discussions, and to P. Deligne for providing me with a copy of his
letter to Yuri Manin.
\nocite{altman:kleiman,altman:kleiman2}
|
1995-02-17T06:20:19 | 9502 | alg-geom/9502016 | en | https://arxiv.org/abs/alg-geom/9502016 | [
"alg-geom",
"math.AG"
] | alg-geom/9502016 | Niels Lauritzen | Niels Lauritzen | Embeddings of homogeneous spaces in prime characteristics | 10 pages, AMS-LaTeX | null | null | Matematisk Institut, Aarhus Universitet, Preprint Series 3, 1995 | null | Let $G$ be a reductive linear algebraic group. The simplest example of a
projective homogeneous $G$-variety in characteristic $p$, not isomorphic to a
flag variety, is the divisor $x_0 y_0^p+x_1 y_1^p+x_2 y_2^p=0$ in $P^2\times
P^2$, which is $SL_3$ modulo a non-reduced stabilizer containing the upper
triangular matrices. In this paper embeddings of projective homogeneous spaces
viewed as $G/H$, where $H$ is any subgroup scheme containing a Borel subgroup,
are studied. We prove that $G/H$ can be identified with the orbit of the
highest weight line in the projective space over the simple $G$-representation
$L(\lambda)$ of a certain highest weight $\lambda$. This leads to some strange
embeddings especially in characteristic $2$, where we give an example in the
$C_4$-case lying on the boundary of Hartshorne's conjecture on complete
intersections. Finally we prove that ample line bundles on $G/H$ are very
ample. This gives a counterexample to Kodaira type vanishing with a very ample
line bundle, answering an old question of Raynaud.
| [
{
"version": "v1",
"created": "Thu, 16 Feb 1995 07:46:12 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Lauritzen",
"Niels",
""
]
] | alg-geom | \section{Preliminaries}
\label{charassumption}
Let $k$ be an algebraically closed field of prime
characteristic $p$. In the following, an algebraic variety $X$ is
assumed to be an
algebraic variety over $k$ and a morphism to be a
morphism of $k$-varieties. We let $X(A)=\operatorname{Mor}_k(\operatorname{Spec}\, A, X)$ denote
the set of $A$-points of $X$, where $A$ is a $k$-algebra.
Let $G$ be a simply connected and
semisimple algebraic group. We will assume that
$p>3$ if $G$ has a component of
type $G_2$ and $p>2$ if $G$ has a component of type $B_n$, $C_n$
or $F_4$.
\subsection{$G$-spaces}
A $G$-space is an algebraic variety $X$ endowed with
a morphism $G\times X\rightarrow X$ inducing
an action of $G(A)$ on $X(A)$ for all $k$-algebras $A$.
A $G$-space $X$ is called homogeneous if the action
$G(k)\times X(k)\rightarrow X(k)$ on $k$-points is
transitive. A $k$-point $x\in X(k)$ gives a natural
morphism $G\rightarrow X$. The fiber product $G_x=G\times_X\operatorname{Spec}(k)$
is easily seen to be a closed subgroup scheme of $G$. It is called
the stabilizer group scheme of $x$.
\subsection{The Frobenius subcover}
\label{fsubcover}
An algebraic variety $X$ gives rise to a new algebraic
variety $X^{(n)}$ with the same underlying point space
as $X$, but where the $k$-multiplication
is twisted via the ringhomomorphism: $a\mapsto \sqrt[p^n]{a}$. The
$n$-th order Frobenius homomorphism induces a natural morphism
$F^{n}_X: X\rightarrow X^{(n)}$. As $X$ is reduced, $\O_{X^{(n)}}$
can be identified with the $k$-subalgebra of $p^n$-th powers of
regular functions on $X$. We call $X^{(n)}$ the $n$-th Frobenius
subcover of $X$. Recall that $X$ is said to be defined over
${\Bbb F}_p$ if there exists an ${\Bbb F}_p$-variety $X'$, such that
$X\cong X'\times_{\operatorname{Spec}\, {\Bbb F}_p} \operatorname{Spec}\, k$. If $X$ is defined
over ${\Bbb F}_p$, then $X$ is isomorphic to $X^{(n)}$ (the isomorphism
is given locally by $f\otimes a\mapsto f\otimes a^{p^n}$, where
$a\in k$).
\subsection{The Frobenius kernel}
\label{fkernel}
Now $G^{(n)}$ is an algebraic group and $F^n_G:G\rightarrow G^{(n)}$
is a homomorphism of algebraic groups. The kernel of $F^n_G$
is called the $n$-th Frobenius kernel of $G$ and denoted $G_n$.
Let $X$ be a homogeneous $G$-space and $x$ a closed point of $X$. It
is easy to see that $X^{(n)}$ is a $G$-homogeneous space through
the homomorphism $F^n_G$. If $H=G_x$ then $G_{F^n_G(x)}=G_n\, H$.
\subsection{The diagonal action}
\label{product}
Now let $X$ and $Y$ be homogeneous $G$-spaces with distinguished
closed points $x$ and $y$ and let $H=G_x$ and $K=G_y$. The product
$X\times Y$ becomes a $G$-space through the diagonal action and
$G_{(x,y)}=H\cap K$.
\section{The unseparated incidence variety}
\label{incvar}
In this section we give a quite explicit geometric description (in
\ref{unsepdivisor}) of
certain projective homogeneous spaces for $\operatorname{SL}_n$ occuring
as divisors in $\P^n\times \P^n$.
\subsection{The incidence variety}
Let $n>1$ and $G=\operatorname{SL}_{n+1}(k)$. The natural action of $G$ on
$V=k^{n+1}$ makes $\P(V)$ and $\P(V^*)$ into homogeneous
spaces for $G$. We fix points $x_1\in \P(V)$ and $x_2\in \P(V^*)$,
such that $G_{x_1}=P_1$ and $G_{x_2}=P_2$
are appropriate parabolic subgroups containing the subgroup
of upper triangular matrices $B$ in $G$.
The orbit $Y$ of $(x_1, x_2)$ in $\P(V)\times \P(V^*)$ is a projective
homogeneous space for $G$ isomorphic to $G/P$, where $P=P_1\cap P_2$.
Notice that the points of $Y$ are just pairs of incident
lines and hyperplanes and that $Y=Z(s)$, where $s$ is
the section $x_0 y_0 +\dots + x_n y_n$ of $\O(1)\times\O(1)$.
\subsection{The unseparated incidence variety}
\label{unsepdivisor}
Let $X$ be the $G$-orbit of
of $(x_1, F^r(x_2))$ in $\P(V)\times \P(V^*)^{(r)}$.
By \ref{fkernel} and \ref{product},
$$
X\cong G/\tilde{P}
$$
where for any $k$-algebra $A$
$$
\tilde{P}(A)=\{\left(
\begin{array}{lllll}
* & * & * & \dots & * \\
0 & * & * & \dots & * \\
0 & * & * & \dots & * \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
0 & * & * & \dots & * \\
0 & a & \dots & a & *
\end{array}
\right)\in \operatorname{SL}_n(A) | a\in A,\, a^{p^n}=0\}
$$
There is a natural
equivariant morphism
$$
\varphi:\P(V)\times \P(V^*) \rightarrow \P(V)\times \P(V^*)^{(r)}
$$
and $X$ is the scheme theoretic image $\varphi(Y)$. The induced
morphism $\varphi:Y\rightarrow X$ is the natural morphism
$$
G/P\rightarrow G/\tilde{P}
$$
given by the inclusion $P\subseteq \tilde{P}$.
Since the sheaf of ideals of the scheme theoretic image is the kernel
of the comorphism of $Y\rightarrow \P(V)\times \P(V^*)^{(r)}$
(\cite{Hartshorne},
Exercise II.3.11 (d)), $X$ is the zero scheme of the section $\bar{s}=
x_0^{p^r} \bar{y_0} +\dots + x_n^{p^r} \bar{y_n}$ of
$\O(p^r)\times\bar{\O}(1)$.
Using the isomorphism from \ref{fsubcover}, we get that $X=Z(\bar{s})$
is isomorhic to its scheme theoretic image $Z(\tilde s)\subseteq
\P(V)\times \P(V^*)$, where
$$
\tilde s=x_0^{p^r} y_0 + \dots x_n^{p^r} y_n
$$
is a section of $\O(p^r)\times \O(1)$.
\subsection{Cohomology of effective line bundles}
Let $a,b \in {\Bbb Z}$. The restriction to $Y$ of the line bundle $
\O(a)\times\O(b)$ on $\P^n\times \P^n$ will be denoted $L(a, b)$. The
restriction to $X$ of the line bundle $\O(a)\times \bar{\O}(b)$ on
$\P^n\times (\P^n)^{(r)}$ will be denoted $L(a, \bar{b})$.
Notice that the isomorphism in \ref{fsubcover} maps $L(a, \bar{b})$ to
$L(a, b)$.
By \ref{unsepdivisor} there is an
exact sequence
$$
0\rightarrow \O(-p^r)\times \O(-1) \rightarrow \O_{\P^n\times \P^n}
\rightarrow \O_X\rightarrow 0
$$
For the line bundle $L=L(a, \bar{b})$ on $X$, we therefore get
the exact sequence
$$
0\rightarrow \O(a-p^r)\times \O(b-1) \rightarrow
\O(a)\times \O(b) \rightarrow L \rightarrow 0
$$
Now assume that $a,b\geq 0$ ($L(a,\bar{b})$ is effective). Then
tracing through the long exact sequence and using the K\"unneth formula,
we get $\H^i(X, L)=0$, if $1\leq i< n-1$ along with the following
exact sequences:
$$
0\rightarrow \H^0(\P^n, \O(a-p^r))\otimes \H^0(\P^n, \O(b-1))\rightarrow
\H^0(\P^n, \O(a))\otimes\H^0(\P^n, \O(b))\rightarrow \H^0(X, L) \rightarrow 0
$$
and
$$
0\rightarrow \H^{n-1}(X, L)\rightarrow
\H^n(\P^n, \O(a-p^r))\otimes \H^0(\P^n, \O(b-1))\rightarrow 0
$$
By Serre duality one has
$$
\H^{n-1}(X, L)\cong \H^0(\P^n, \O(p^r-a-n-1))\otimes \H^0(\P^n, \O(b-1))
$$
so that the higher cohomology of $L$ vanishes if $a>p^r-n-1$.
By the adjunction formula (\cite{Hartshorne}, Proposition II.8.20) we get
$\omega_X\cong \O(p^r-n-1)\times \O(-n)$.
A line bundle $L=L(a,\bar{b})$ on $X$ is ample if and and only if $a,b>0$.
Kodaira type vanishing (vanishing higher cohomology
for $L\otimes \omega_X$, where
$L$ is ample) for $X$ amounts to the fact that $a+p^r-n-1>p^r-n-1$, when
$a>0$.
The unseparated incidence variety admits a lifting to a flat ${\Bbb Z}$-scheme.
There are projective homogeneous spaces
for $SL_4$, which do not admit a lifting to a flat ${\Bbb Z}$-scheme
(\cite{HaLa:varunsep}, \S 6).
\section{Structure of projective homogeneous spaces}
A projective homogeneous $G$-space $X$ is determined through its
stabilizer group scheme $G_x$ at some closed point $x\in X$. Notice that
since $X$ is projective, Borel's fixed point theorem implies that
$G_x$ contains a Borel subgroup $B$. We introduce some more
notation.
Let $T$ be a maximal torus of $G$ contained in the Borel subgroup $B$. Denote
by $R=R(G, T)$ the roots of $G$ w.r.t. $T$.
Let the roots $R(B, T)$ of
$B$ be the positive roots $R^+$ in $R$ and $S\subseteq R^+$ the
simple roots of $R$. Let $X(T)$ be the characters of $T$ and $Y(T)$
the one parameter subgroups. The usual pairing
$X(T)\times Y(T)\rightarrow {\Bbb Z}$ is denoted $\<\cdot, \cdot\>$. The coroot
in $Y(T)$ corresponding to $\alpha\in R$ is denoted $\alpha^\vee$.
The
root subgroup corresponding to $\alpha\in R$ is denoted by $U_{\alpha}$.
Let $\{X_{\alpha}\}_{\alpha\in R}$,
$\{H_{\alpha}\}_{\alpha\in S}$ be a Chevalley basis for $\operatorname{Lie}(G)$.
The monomials
$$
\prod_{\alpha\in R^+} X_{-\alpha}^{(n'(\alpha))}\,
\prod_{\alpha\in S} \binom{H_\alpha}{m(\alpha)}\,
\prod_{\alpha\in R^+} X_{\alpha}^{(n(\alpha))}
$$
where $n'(\alpha), m(\alpha), n(\alpha)\in{\Bbb N}$, form
a basis for the $k$-algebra of distributions $\operatorname{Dist}(G)$ (\cite{Jantzen},
II.1.12).
Recall that a subgroup scheme $H\subseteq G$ is uniquely determined by
its subalgebra $\operatorname{Dist}(H)\subseteq \operatorname{Dist}(G)$. A $k$-basis
for $\operatorname{Dist}((U_{\alpha})_n)$ is given by $1, X_{\alpha}, X_{\alpha}^{(2)},
\dots, X_{\alpha}^{(p^n-1)}$.
\subsection{Parabolic subgroup schemes}
\label{parsub}
Let $\tilde P$ be a subgroup scheme containing $B$. Since $P=\tilde P_{\operatorname{{red}}}$
is a parabolic subgroup (the nil-radical is a Hopf ideal), it follows
that $\tilde P$ is a connected group scheme. In particular we get
for $\alpha\in R^-$ that $\operatorname{Dist}(\tilde P)\cap\operatorname{Dist}(U_\alpha)=
\operatorname{Dist}((U_\alpha)_{n_\alpha})$ for a suitable
$n_\alpha$, where $0\leq n_\alpha \leq \infty$ with the convention
$(U_\alpha)_\infty=U_\alpha$. The subalgebra $\operatorname{Dist}(\tilde P)$ is
determined completely by $(n_\alpha)_{\alpha\in R^-}$.
With the assumptions given in \S \ref{charassumption} on $p=\operatorname{char}\, k$
it follows by
(\cite{HaLa:varunsep}, Proposition 1.6) that $\operatorname{Dist}(\tilde P)$ is uniquely
determined by $(n_\alpha)_{\alpha\in-S}$.
One can construct $\tilde P\neq G$ as follows:
The maximal parabolic subgroup $P(\gamma)$
corresponding to a simple root $\gamma\in S$ is the
parabolic subgroup with roots generated by $S\setminus\{\gamma\}$.
The parabolic subgroup $P$ is the intersection
$P=P(\alpha_1)\cap\dots\cap P(\alpha_m)$ for certain
simple roots $S'=\{\alpha_1,\dots,\alpha_m\}\subseteq S$. It is easy
to see that
$\tilde P\subseteq G_n\,P(\alpha_i)$, for $n$ sufficiently big.
Let $n_i$ be the minimal $n$ with this property. Then
$$
\tilde P=G_{n_1}\, P(\alpha_1)\cap\dots\cap G_{n_m}\, P(\alpha_m)
$$
In the notation above $\operatorname{Dist}(\tilde P)$ is determined uniquely
by $n_{\alpha_1}=n_1,\dots, n_{\alpha_m}=n_m$ and $n_\alpha=\infty$ if
$\alpha\not\in S'$.
\subsection{The action of $G$ on $\P(L(\lambda))$}
Recall that the simple $G$-representations are parametrized by
dominant weights $X(T)_+$.
Let $L(\lambda)$ denote the simple $G$-representation associated
with $\lambda\in X(T)_+$. Then $L(\lambda)$ is generated by a $B$-stable
line of (highest) weight $\lambda$.
\begin{prop}
Let $S=\{\alpha_1,\dots, \alpha_l\}$ and
$\nu_p$ denote the $p$-adic valuation, such that $\nu_p(0)=\infty$.
Let $L(\lambda)$ be the simple representation of highest weight $\lambda\in
X(T)_+$ and $n_i=\nu_p(\<\lambda, \alpha_i^\vee\>)$.
Then
the stabilizer of the $B$-stable
line $x\in \P(L(\lambda))$ for the natural action of $G$ is
$$
G_x=G_{n_1} P(\alpha_1)\cap \dots\cap G_{n_l} P(\alpha_l)
$$
\end{prop}
\begin{pf}
Let $v$ be a generator for $x\in \P(L(\lambda))$. We compute the
algebra of distributions $\operatorname{Dist}(G_x)$. For the
induced action of $\operatorname{Dist}(G)$ on $L(\lambda)$ we have
$$
\operatorname{Dist}(G_x)=\{X\in \operatorname{Dist}(G) | X\, v=0\}
$$
By \ref{parsub} it suffices to show for a simple root $\alpha_i\in S$, that
$$
X_{-\alpha_i} \,v= X_{-\alpha_i}^{(2)}\, v
=\dots=X_{-\alpha_i}^{(p^{n_i}-1)}\,v=0
$$
and
$$
X_{-\alpha_i}^{(p^{n_i})}\, v\neq 0
$$
where the last condition is void in the case $n_i=\infty$.
Since $L(\lambda)$ is a highest weight module generated by $v$ and
$\alpha_i$ is a simple root, it suffices to prove that
$$
X_{\alpha_i}^{(n)} X_{-\alpha_i}^{(n)}\, v=0
$$
to conclude that $X_{-\alpha_i}^{(n)}\, v=0$.
In $\operatorname{Dist}(G)$ we have the following commutation formula for $\alpha\in R^+$:
$$
X_\alpha^{(m)} X_{-\alpha}^{(n)}=\sum_{j=0}^{\min(m,n)}
X_{-\alpha}^{(n-j)} \binom{H_\alpha-m-n+2j}{j} X_\alpha^{(m-j)}
$$
From this formula it follows that
$$
X_{\alpha_i}^{(n)} X_{-\alpha_i}^{(n)}\, v= \binom{\<\lambda, \alpha_i^\vee\>}
{n}\, v
$$
When $n_i=\infty$ it follows that $X_{\alpha_i}^{(n)}\, v=0$ for $n>0$. Assume
now that $n_i<\infty$.
Since $\binom{m}{n}\equiv 0\, (\operatorname{mod}\, p)$ if $0<n<p^r$
and $\not\equiv 0\,(\operatorname{mod}\, p)$ if $n=p^r$, when $\nu_p(m)=r$, the
result follows.
\end{pf}
\label{gaction}
\label{Llambda}
\subsection{Exceptional parabolic subgroup schemes}
\label{ex}
The action of $G$ on $\P(L(\lambda))$ gives a lot of examples of
exceptional parabolic subgroup schemes in characteristic $2$ -
parabolic subgroup schemes which are not the intersection of
thickenings of the maximal parabolic subgroups as in \S
\ref{parsub}. In this section $k$ is assumed to be of
characteristic $2$. Recall that the simple module $L(\lambda)$
is a quotient of the Weyl module $V(\lambda)$. The Weyl module
$V(\lambda)$ is the base extension to $k$ of the minimal admissible
${\Bbb Z}$-form in the simple representation of highest weight $\lambda$ for the
complex semisimple Lie algebra corresponding to $G$. Let $K(G)$ denote
the Grothendieck group of $G$. In the examples below, the decompositions
in $K(G)$ of Weyl modules were computed using Jantzen's sum
formula (\cite{Jantzen}, II.8). Example \ref{abuch} was discovered
using a computer program, developed by A.~Buch, for computations in modular
representation theory.
\begin{example}
Let $G$ be of type $B_2$ with positive roots $\alpha$, $\beta$,
$\alpha+\beta$ and $2\alpha+\beta$, where $\beta$ is the long simple root.
Let $\omega$ be the fundamental weight dual to $\beta$. In
$K(G)$ we have
$$
V(\omega)=L(\omega)+L(0)
$$
Let $v$ be a highest weight vector of $L(\omega)$.
To determine $G_x$, where $x=k\,v\in L(\omega)$,
we notice that $X_{-\alpha}\, v=0$, $X_{-\alpha-\beta}\, v=0$ (this
is because $0$ is not a weight of $L(\omega)$),
$X_{-\alpha-\beta}^{(2)}\, v\neq 0$, $X_{-2\alpha-\beta}\, v\neq 0$.
This means in the notation of \ref{parsub} that $\operatorname{Dist}(G_x)$ is given
by $n_{-\alpha}=0$,
$n_{-\beta}=\infty$, $n_{-\alpha-\beta}=1$,
$n_{-2\alpha-\beta}=0$.
\end{example}
\begin{example}
\label{abuch}
Let $G$ be of type $C_4$ with simple roots and fundamental dominant
weights numbered as below
\begin{picture}(150,40)(-125,10)
\put(50,20){\circle{3}}
\put(80,20){\circle{3}}
\put(110,20){\circle{3}}
\put(140,20){\circle{3}}
\put(47,10){$\alpha_1$}
\put(77,10){$\alpha_2$}
\put(107,10){$\alpha_3$}
\put(137,10){$\alpha_4$}
\put(47,30){$\omega_1$}
\put(77,30){$\omega_2$}
\put(107,30){$\omega_3$}
\put(137,30){$\omega_4$}
\put(51.5,19.5){\line(1,0){27}}
\put(81.5,19.5){\line(1,0){27}}
\put(111,21){\line(1,0){28}}
\put(111,18.3){\line(1,0){28}}
\put(123,17){$<$}
\end{picture}
\noindent
In $K(G)$ we have
$$
V(\omega_4)=L(\omega_4)+L(\omega_2)+L(0)
$$
and furthermore $\dim\, L(\omega_4)=16$, while $\dim\, V(\omega_4)=42$.
Let $v$ be a highest weight vector in $L(\omega_4)$ and $x=k\, v\in
\P(L(\omega_4))$. The stabilizer $G_x$ is given by $\operatorname{Dist}(G_x)$, which
is determined by the table
$$
\begin{array}{llll}
\alpha\in R^+ & n_{-\alpha} &
\alpha\in R^+ & n_{-\alpha}\\
1000 & \infty & 1100 & \infty \\
1110 & \infty & 0100 & \infty \\
0110 & \infty & 0010 & \infty \\
0001 & 0 & 0011 & 1 \\
0111 & 1 & 1111 & 1 \\
0021 & 0 & 0121 & 1 \\
1121 & 1 & 0221 & 0 \\
1221 & 1 & 2221 & 0
\end{array}
$$
The orbit $X=G/G_x$ of $x=[v]$ has dimension $10$ and we get an
example of a variety lying on the boundary of Hartshorne's
conjecture \cite{LazVen} ($10=\frac{2}{3}15$)
$$
X\hookrightarrow \P(L(\omega_4))=\P^{15}
$$
I do not know whether $X\subseteq \P^{15}$ is a complete intersection.
One may check in accordance with Zak's result \cite{LazVen} on linear
normality that the restriction map
$$
\H^0(\P^{15}, \O(1))\rightarrow \H^0(X, \O_X(1))
$$
is surjective.
\end{example}
\section{Line bundles}
\label{linebundles}
In this section we classify the line bundles on projective homogeneous
$G$-spaces following \cite{Lau:Linebundles}. When $G$ is simply connected, all
line bundles are homogeneous induced by a character on $G_x$.
\subsection{Characters}
Let $X$ be a projective homogeneous $G$-space. Suppose that $G_x$ is the
stabilizer group scheme at a closed point $x\in X(k)$. Let $B$ be the Borel
subgroup contained in $G_x$.
The character lattice $X(B)=X(T)$ is
$$
{\Bbb Z} \omega_{\alpha_1}+\dots+{\Bbb Z} \omega_{\alpha_l}
$$
where $\omega_{\alpha}$ is the fundamental dominant weight
associated with the simple root $\alpha\in S$.
The restriction homomorphism $X(H)\rightarrow X(T)$ is injective for any
subgroup scheme $H\supseteq T$.
Recall that for a maximal parabolic subgroup $P(\alpha)$, we have
$X(P(\alpha))={\Bbb Z}\, \omega_{\alpha}$.
\begin{lem}
\label{parchar}
Let $\alpha\in S$ be a simple root. Then
$$
X(G_n\, P(\alpha))={\Bbb Z} p^n \omega_{\alpha}
$$
\end{lem}
\begin{pf}
It follows from (\cite{Jantzen}, II.3.15, Remarks 2),
that a character on the $n$-th
Frobenius kernel $G_n$ has to be trivial. Now the first isomorphism
theorem for groups gives
\begin{eqnarray*}
X(G_n P(\alpha))&=& X(G_n P(\alpha)/G_n) = X(P(\alpha)/G_n\cap P(\alpha))\\
&=& X(P(\alpha))/P(\alpha)_n)=X(P(\alpha)^{(n)})=p^n
X(P(\alpha))\\
&=& {\Bbb Z} p^n \omega_{\alpha}
\end{eqnarray*}
\end{pf}
Let $x_\alpha:{\Bbb G}_a\rightarrow G$ be the root homomorphism associated
with $\alpha\in R$. There is a homomorphism (\cite{Jantzen}, p.~176)
$$
\varphi_{\alpha}:\operatorname{SL}_2\rightarrow G
$$
such that
$$
\varphi_{\alpha}\left(\begin{array}{ll} 1 & a \\ 0 & 1 \end{array}\right)=
x_{\alpha}(a)\,\,\text{and}\,\,
\varphi_{\alpha}\left(\begin{array}{ll} 1 & 0 \\ a & 1 \end{array}\right)=
x_{-\alpha}(a)
$$
and
$$
\alpha^\vee(t)=\varphi_\alpha\left(
\begin{array}{ll}
t & 0 \\
0 & t^{-1}
\end{array}
\right),\, t\in {\Bbb G}_m
$$
We are now ready to prove
\begin{prop}
Let $G_x=G_{n_1}\,P(\alpha_1)\cap\dots\cap G_{n_m}\, P(\alpha_m)$, where
$\alpha_1,\dots,\alpha_m\in S$. Then
$$
X(G_x)={\Bbb Z} p^{n_1} \omega_{\alpha_1} + \dots + {\Bbb Z} p^{n_m} \omega_{\alpha_m}
$$
\end{prop}
\begin{pf}
As $G_x\subseteq G_{n_i}\,P(\alpha_i),\ i=1, \dots,m$, we get by
lemma \ref{parchar}
$$
X(G_x)\supseteq {\Bbb Z} p^{n_1} \omega_{\alpha_1} +
\dots + {\Bbb Z} p^{n_m} \omega_{\alpha_m}
$$
Suppose on the other hand that $\lambda=a_1 \omega_{\alpha_1}+\dots
+a_m \omega_{\alpha_m}\in X(G_x)$. Then $\lambda\circ \varphi_{\alpha}$
is a character of $\tilde B\subseteq \operatorname{SL}_2(k)$, where for any
$k$-algebra $A$
$$
\tilde B (A)=\{
\left(
\begin{array}{ll}
* & * \\
a & *
\end{array}
\right)\in \operatorname{SL}_2(A) | a\in A,\, a^{p^n}=0
\}
$$
and $n=\<\lambda, \alpha^\vee\>$.
Therefore we get
$p^{n_i} | \<\lambda, \alpha^\vee\>$, so that $p^{n_i} | a_i$.
\end{pf}
\subsection{Ample line bundles}
\label{ample}
Let $\tilde P$ be a parabolic subgroup scheme and $\chi\in X(\tilde P)$. The
total space of the line bundle $L_{\tilde P}(\chi)$ induced by $\chi$ is
$G\times^{\tilde P} {\Bbb A}^1=G\times {\Bbb A}^1/\tilde P$, where $\tilde P$ acts on
$G\times {\Bbb A}^1$ through $h.(g, a)=(g\, h, \chi(h^{-1}) a)$.
The natural morphism
$$
G\times^{\tilde P}{\Bbb A}^1\rightarrow G/\tilde P
$$
is equivariant, when $G$ acts on $G\times^{\tilde P} {\Bbb A}^1$ through
left multiplication. Since $G$ is simply connected ($\operatorname{Pic} G=0$) every
line bundle is induced by a character.
Let $P=\tilde P_{\operatorname{{red}}}$.
A line bundle $L$ on $G/P$, where
$P=P(\alpha_1)\cap \dots \cap P(\alpha_r)$ for simple roots
$\alpha_1,\dots, \alpha_r\in S$, is induced by a character
$$
\chi\in X(P)={\Bbb Z}\omega_{\alpha_1}+\dots+{\Bbb Z}\omega_{\alpha_r}\subseteq X(B)=X(T)
$$
Then $L=L_P(\chi)$ is very ample on $G/P$ if and only if it is
ample on $G/P$ if and only if $\chi\in X(P)^{++}=\{\lambda\in X(P)|
\<\lambda, \alpha_1^\vee\>>0,\dots, \<\lambda, \alpha_r^\vee\>>0\}$.
It is easy to see that $f^* L_{\tilde P}(\chi)=L_P(\chi)$, where
$f:G/P\rightarrow G/\tilde P$ is the natural morphism.
\begin{thm}
Let $X=G/\tilde P$ be a projective homogeneous space such that
$$
\tilde P\cong G_{n_1}P(\alpha_1)\cap \dots \cap G_{n_r} P(\alpha_r)
$$
where $\alpha_1,\dots, \alpha_r\in S$ are simple roots and
$n_1,\dots, n_r$ integers $\geq 0$.
Let $\chi=a_1 p^{n_1} \omega_{\alpha_1} +\dots +
a_r p^{n_r} \omega_{\alpha_r}\in X(\tilde P)$.
Then $L_{\tilde P}(\chi)$ is very ample
on $X$ if and only if $L_P(\chi)=f^*L_{\tilde P}(\chi)$ is very
ample on $G/P$, where
$P=P(\alpha_1)\cap \dots \cap P(\alpha_r)$ and $f$ is the
natural morphism $f: G/P\rightarrow G/\tilde P$.
\end{thm}
\begin{pf}
Consider the natural diagram
$$
\begin{CD}
G/P @>>> G/P(\alpha_1)\times \dots \times G/P(\alpha_r)\\
@VfVV @VVV\\
G/\tilde P @>j>> (G/P(\alpha_1))^{(n_1)}\times \dots \times
(G/P(\alpha_r))^{(n_r)}
\end{CD}
$$
By \ref{fkernel} and \ref{product} it follows that $j$ is a closed
immersion. Since $(G/P(\alpha_i))^{(n_i)}=G/G_{n_i} P(\alpha_i)
\cong G/P(\alpha_i)$ it follows that ample line bundles on
$(G/P(\alpha_i))^{(n_i)}$ are very ample. Since the natural
morphism $G/P(\alpha_i)
\rightarrow G/G_{n_i} P(\alpha_i)$ is a finite surjective morphism
it follows (\cite{Hartshorne}, Exercise III.5.7 (d)) that
$L_{G_{n_i} P(\alpha_i)}(a_i\, p^{n_i} \omega_{\alpha_i})$ is
very ample if and only if $a_i>0$. By the Segre embedding we have
that
$$
L=L_{G_{n_1}P(\alpha_1)}(a_1\, p^{n_1} \omega_{\alpha_1})\times
\dots \times L_{G_{n_r}P(\alpha_r)}(a_r\, p^{n_r} \omega_{\alpha_r})
$$
is very ample. Now that $j$ is a closed immersion and
$j^* L=L_{\tilde P}(\chi)$ the result follows.
\end{pf}
\newpage
\bibliographystyle{amsplain}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1995-02-17T06:20:23 | 9502 | alg-geom/9502017 | en | https://arxiv.org/abs/alg-geom/9502017 | [
"alg-geom",
"math.AG"
] | alg-geom/9502017 | Thomas Bauer | W. Barth, Th. Bauer | Poncelet theorems | 20 pages, LaTeX | null | null | null | null | If there is one polygon inscribed into some smooth conic and circumscribed
about another one, then there are infinitely many such polygons. This is
Poncelet's theorem. The aim of this note is to collect some (mostly classical)
versions of this theorem, namely:
- Weyr's Poncelet theorem in $P_3$ (1870),
- Emch's theorem on circular series (1901),
- Gerbaldi's formula for the number of Poncelet pairs (1919),
- the Money-Coutts theorem on three circles (1971),
- the zig-zag theorem (1974),
- a (probably new) Poncelet theorem on three conics,
- a Poncelet formula for quadrics of revolution.
| [
{
"version": "v1",
"created": "Thu, 16 Feb 1995 11:15:19 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Barth",
"W.",
""
],
[
"Bauer",
"Th.",
""
]
] | alg-geom | \section{\@startsection{section}{1}{\z@}%
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\begin{document}
\title{Poncelet theorems}
\runningtitle{Poncelet theorems}
\author{W.\ Barth, Th.\ Bauer}
\date{ }
\oldmaketitle\thispagestyle{empty}
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
The aim of this note is to collect some more or less classical
theorems of Poncelet type and to provide them with short modern
proofs. Where classical geometers used elliptic functions
(or angular functions), we use elliptic curves (or degenerate elliptic
curves decomposing into two rational curves). In this way we
unify the geometry underlying these Poncelet type statements.
Our starting point is a space Poncelet theorem for
two quadrics in ${\rm I\!P}_3$ (Sect.\ \ref{section Weyr}).
This seems to have been observed
first by Weyr \cite{Weyr} p.\ 28, and amplified by Griffiths and
Harris \cite{GriHar77}.
The classical Poncelet theorem (cf.\ \cite{GriHar78a}),
a statement on two conics in the plane ${\rm I\!P}_2$,
follows from Weyr's space Poncelet theorem, if one of
the quadrics is taken to be a cone, see Sect.\ \ref{section Weyr}.
Gerbaldi \cite{Ger19} gave formulas counting the
number of conics in a pencil which are in Poncelet position
with respect to a fixed conic in the pencil. We show that
his formulas are simple consequences of the space
Poncelet theorem (Sect.\ \ref{section Gerbaldi}).
In Sect.\ \ref{section revolution}
we evaluate explicitly the space Poncelet
condition for two quadrics of revolution about the
same axis.
We show that theorems such as
Emch's theorem on circular series \cite{Emc01}
and (a complex-projective version of) the
'zig-zag' theorem \cite{BlaHowHow74}
can be understood
by considering torsion points on
elliptic curves
(see Sects.\ \ref{section Emch Steiner} and
\ref{section zig-zag}).
Further, we prove a Poncelet version of the Money-Coutts theorem
(Sect.\ \ref{section Money-Coutts}).
Although for most of the contents of this paper only the
presentation is new, it seems worth--while to us to consider
Poncelet type theorems from a modern geometric point of view.
In this spirit the equations of modular curves given in
\cite{BarMic93} have been transformed
by N. Hitchin \cite{Hitchin} into solutions of
the Painlev\'{e}--VI--equation. And E.\ Previato relates Poncelet
theorems to integrable Hamiltonian systems and billiards
\cite{Prev}.
\bigskip
{\em Conventions.} The base field always is the field
$\bbC$ of complex numbers.
If we mention circles, quadrics of revolution, or spheres, we
mean the corresponding varieties over $\bbC$.
\bigskip
{\em Acknowledgement. This research was supported by DFG contract
Ba 423/7-1 and EG contract SC1--0398--C(A).
}
\section{Weyr's Poncelet theorem in ${\rm I\!P}_3$}
\label{section Weyr}
Let $Q_1,Q_2\subset{\rm I\!P}_3$ be quadrics of ranks $\ge 3$ such that
their intersection curve $E=Q_1 \cap Q_2$ is either a smooth
elliptic curve or the union of two conics $C_1,C_2$ meeting in
two distinct points. We fix rulings $R_1$ on $Q_1$ and $R_2$ on $Q_2$.
\begin{theorem}[\cite{Weyr} p.\ 28, \cite{Hurw} p.\ 13]
\label{3-dim Poncelet}
Suppose that there exists a closed sequence of distinct lines
$L_1,\dots,L_{2n},L_{2n+1}=L_1$ such that the line
$L_i$ belongs to $R_1$ resp.\ $R_2$, if $i$ is odd resp.\ even,
and such that consecutive lines $L_i,L_{i+1}$ intersect each other.
Then there are such closed sequences of length $2n$ through
any point on $Q_1\cap Q_2$.
\end{theorem}
\startproof{\it Proof. }
The rulings $R_1,R_2$ define involutions $\iota_1,\iota_2$
on $E$
interchanging the two intersection points of $E$ with a line in
$R_1$ resp.\ $R_2$. Let $t:E\mathop{\longrightarrow}\limits E$ be the composition $\iota_2\iota_1$
and let $e:=L_{2n}\cap L_1$.
Then $L_{2k-1}$ is the unique line in $R_1$ through
$t^{k-1}e$ and
$L_{2k}$ is the unique line in $R_2$ through
$\iota_1t^{k-1}e$ for $k\ge 1$.
The closedness $L_{2n+1}=L_1$ implies $t^n(e)=e$.
First we consider the case that $E$ is smooth elliptic.
The involutions $\iota_1,\iota_2$ have fixed points, so their
composition $t$ is a translation on $E$.
But then $t^n(e)=e$ and the assumption that the lines $L_1,\dots,L_n$
are distinct
imply that $t$ is of order $n$, hence the
assertion.
Now assume $E=C_1\cup C_2$ is the union of two conics.
The involutions $\iota_1$ and $\iota_2$ interchange these
conics. So $t(C_i)=C_i, \, i=1,2$ and the two intersections
of $C_1$ and $C_2$ are fixed points of
$t$.
Therefore $t$ induces two automorphisms
$
t_i:\bbC^*\cong C_i\setminus(C_1\cap C_2)
\mathop{\longrightarrow}\limits C_i\setminus(C_1\cap C_2)\cong\bbC^*
$.
If $e$ lies on $C_1$, say, then
$t^n(e)=e$ implies that $t_1$ is the multiplication by a primitive
$n$-th root of unity, i.e.\ its order is $n$.
But then $t_2$ is of order $n$ as well, because of
$t_2=\iota_2 t_1^{-1}\iota_2$.
\nopagebreak\hspace*{\fill
Theorem \ref{3-dim Poncelet}
also holds for two different rulings $R_1$ and $R_2$
on the same smooth quadric $\simeq {\rm I\!P}_1 \times {\rm I\!P}_1$, if
$E$ is a curve on this quadric of bidegree $(2,2)$, either
smooth elliptic or the union of two rational curves meeting
in two distinct points.
The proof is literally the same.
We say that the pair of rulings $(R_1,R_2)$ {\it satisfies the
Poncelet-$n$-condition} if the automorphism $t:E\mathop{\longrightarrow}\limits E$ defined above
is of order $n$. A priori this is a property of the {\it ordered}
pair $(R_1,R_2)$. But the automorphism associated to $(R_2,R_1)$ is
just $t^{-1}$, thus:
\begin{assertion}
The pair of rulings $(R_1,R_2)$ satisfies the Poncelet-$n$-condition
if and only if $(R_2,R_1)$ does.
\end{assertion}
Already Hurwitz \cite{Hurw}, p.\ 13, observed that Theorem
\ref{3-dim Poncelet}
implies the usual Poncelet theorem in ${\rm I\!P}_2$.
In fact, if one of the quadrics,
$Q_1$, is smooth and the other one, $Q_2$, is a cone
with its top $P_0$ not on $E=Q_1 \cap Q_2$ and $E$ is
smooth elliptic, then
the three-dimensional Poncelet theorem is equivalent
to Poncelet's theorem for two conics in the plane. To see this,
we denote by $\pi_i:Q_i\mathop{\longrightarrow}\limits{\rm I\!P}_2$ the projections
from $P_0$. The morphism $\pi_1$ is of degree 2, ramified over a
smooth conic $C\subset{\rm I\!P}_2$. The image of $\pi_2$ is another
smooth conic $D\subset{\rm I\!P}_2$ in general position with respect
to $C$. We have:
\begin{proposition}\label{23}
The quadrics $Q_1$ and $Q_2$ are in Poncelet-$n$-position if and
only if the conic $C$ is $n$-inscribed into $D$.
\end{proposition}
We say: $C$ is $n$--inscribed into $D$, if there is a polygon
consisting of $n$ tangents to $C$ with its vertices on
$D$, \cite{BarMic93}.
\startproof{\it Proof. }
We choose an isomorphism $Q_1\cong{\rm I\!P}_1\times{\rm I\!P}_1$ such that
the ruling $R_1$ is parametrized by the first factor.
Let $\Delta = \pi^{-1}C \subset {\rm I\!P}_1 \times {\rm I\!P}_1$ be the
ramification curve. It is a curve of bidegree $(1,1)$
on ${\rm I\!P}_1 \times {\rm I\!P}_1$ and
induces an isomorphism between both copies of ${\rm I\!P}_1$
such that $\Delta$ is the diagonal.
We consider the
map
\begin{eqnarray*}
Q_1 &\mathop{\longrightarrow}\limits& {\rm I\!P}_2\times C^* \\
(a,b) &\mathop{\mapstochar\longrightarrow}\limits& (u,T)
\end{eqnarray*}
where $u:=\pi_1(a,b)$ and $T:=\pi_1(a\times{\rm I\!P}_1)$. So $T$ is the
tangent from $u$ to $C$ meeting $C$ in the point $\pi_1(a,a)$.
The projection $\pi_1$ restricts to a map $E\mathop{\longrightarrow}\limits D$ of degree 2.
Over a point $u\in D$ we have points $(a,b),(b,a)\in E$.
Now we determine the map on the pairs $(u,T)$ induced by $t$.
We start with a smooth point $P=(a,b)$ on $E$.
The line $L_1=a\times{\rm I\!P}_1\in R_1$ through
$P$ meets $E$ in another point $P'=(a,b')$. This means that we
pass from the pair $(u,T)$ to $(u',T)$, where $u':=\pi_1(a,b')$ is
the second intersection point of $T$ and $D$.
The image point $P'':= t(P)$ is the second intersection point of the
line $L_2\in R_2$ through $P'$ with $E$. Thus we have $P''=(b',a)$,
because points in the same ray of $Q_2$ lie in the same fibre of the
projection $\pi_2$. Passing from the point $(a,b')$ to $(b',a)$
means passing from the pair $(u',T)$ to $(u',T')$, where $T'$ is
the second tangent to $C$ through the point $u'$.
So we see that the map
$$
t:(u,T)\mathop{\mapstochar\longrightarrow}\limits (u',T')
$$
just describes the Poncelet process for the pair of conics $C,D$.
This implies the assertion.
\nopagebreak\hspace*{\fill
We should point out here, how Theorem \ref{3-dim Poncelet}
relates to the Poncelet theorem in space considered by
Griffiths and Harris in \cite{GriHar77} on finite polyhedra
both inscribed and circumscribing two smooth quadrics $Q_1$ and $Q_2$:
Such a polyhedron exists if and only if both of the pairs
$R_1,R_2$ and $R_1',R_2'$ satisfy a Poncelet condition in the sense
above, where $R_i,R_i'$ are the rulings on the dual quadrics
$Q_i^*$, $i=1,2$.
\section{Poncelet pairs in a pencil}
Let $Q_1,Q_2$ be quadrics of rank $\ge 3$ such that
the pencil $\lambda_1 Q_1+\lambda_2 Q_2$ generated by them is generic.
This means that its base locus $E = Q_1\cap Q_2$
is smooth, or equivalently that
the discriminant $d(\lambda_1,\lambda_2)=det(\lambda_1 Q_1+\lambda_2 Q_2)$
of
the pencil has no multiple roots.
The four roots of $d(\lambda_1,\lambda_2)$
correspond to the cones in the pencil,
i.e.\ to those quadrics carrying only one ruling.
Let $M\mathop{\longrightarrow}\limits{\rm I\!P}_1$ be the double cover of ${\rm I\!P}_1$ branched over the
roots of $d(\lambda_1,\lambda_2)$. The points of the elliptic curve $M$
can be identified with the rulings on the quadrics of the given pencil.
Each ruling $R$ in $M$ defines an involution $I_R:E\mathop{\longrightarrow}\limits E$ with fixed
points. Choosing an origin in $E$,
we can write $I_R$ as $x\mathop{\mapstochar\longrightarrow}\limits -x+a$
with a unique point $a\in E$, so we obtain a map
\begin{eqnarray*}
\Phi:M&\mathop{\longrightarrow}\limits& E \\
R &\mathop{\mapstochar\longrightarrow}\limits& a \ .
\end{eqnarray*}
\begin{proposition}
$\Phi$ is an isomorphism of groups (if the origin of $M$ is chosen
appropriately).
\end{proposition}
\startproof{\it Proof. }
It suffices to show that $\Phi$ is injective. So let $R_1,R_2\in M$
such that $\Phi(R_1)=\Phi(R_2)$, i.e.\ such that the involutions
$I_{R_1},I_{R_2}$ coincide. Let $P\in E$ be a point which is not
fixed under $I_{R_1}$ or $I_{R_2}$. Because of $I_{R_1}=I_{R_2}$
the lines
\begin{eqnarray*}
\overline{P,I_{R_1}(P)}\in R_1 \\
\overline{P,I_{R_2}(P)}\in R_2
\end{eqnarray*}
are equal. Varying the point $P$ in $E$ we conclude $R_1=R_2$.
\nopagebreak\hspace*{\fill
This gives the following corollary which will be applied in
Sect.\ \ref{section Gerbaldi}.
\begin{corollary}\label{torsion criterion}
a) Two rulings $R_1,R_2$ in the pencil satisfy the Poncelet-$n$-condition,
if and only if the point $R_2-R_1\in M$ is a primitive $n$-torsion
point.
b) For a fixed ruling $R_1\in M$ there are $T(n)$ rulings $R_2\in M$ such
that $R_1,R_2$ satisfy the Poncelet-$n$-condition.
Here $T(n)$ denotes the number of primitive $n$-torsion points on
an elliptic curve.
\end{corollary}
This corollary is independent of the choice of origin on $M$,
if one interprets $R_2-R_1$ as the translation on $M$ mapping $R_1$
to $R_2$.
In general the Poncelet property depends on the choice of the rulings
$R_1,R_2$ on $Q_1,Q_2$. However
if it holds for $R_1$ and $R_2$, then it also holds for
the complementary rulings $R_1'$ on $Q_1$ and $R_2'$ on $Q_2$.
In fact, $R_2-R_1=-(R_2'-R_1')$, independently of the choice of
origin on $M$. This is even true if one of the quadrics,
say $Q_2$ is a cone and $R_2'=R_2$:
\begin{assertion}
Let $Q_1$ be smooth and let $Q_2$ be a cone. We denote by
$R_1,R'_1$ the rulings on $Q_1$ and by $R_2$ the unique ruling
on $Q_2$. Then $R_1',R_2$ satisfy the Poncelet-$n$-condition
if and only if $R_1,R_2$ do.
\end{assertion}
\section{Gerbaldi's formula for the number of inscribed conics}
\label{section Gerbaldi}
Gerbaldi \cite{Ger19} considered the invariant of two conics
$C$ and $D \subset {\rm I\!P}_2$ vanishing if $C$ is $n$--inscribed
into $D$. Using elliptic functions and continued fractions
he showed:
\begin{theorem}[\cite{Ger19}, p.\ 103] \label{Gerb}
This invariant is of degree $\frac{1}{2} T(n)$ in $C$ and of degree
$\frac{1}{4} T(n)$ in $D$.
\end{theorem}
Here we do not want to make his assertion precise. Instead we
prove the following two assertions, together equivalent with
Theorem \ref{Gerb}. Ignorantly of Gerbaldi's paper they
were shown in \cite{BarMic93} using a rational elliptic surface.
\begin{theorem}\label{number inscribed}
Let $\lambda C+\mu D$, $(\lambda:\mu)\in{\rm I\!P}_1$,
be a generic pencil of conics in ${\rm I\!P}_2$.
Then the number of conics in the pencil, which are $n$-inscribed
into $D$ is $\frac12 T(n)$.
\end{theorem}
\startproof{\it Proof. }
1)
There is a smooth quadric $Q_1\subset{\rm I\!P}_3$ and a cone $Q_2\subset{\rm I\!P}_3$
such that
\begin{paritems}{iii)}
\item[i)]
the branch locus of the projection $Q_1\mathop{\longrightarrow}\limits{\rm I\!P}_2$ from the top $P_0$
of $Q_2$ is $C$,
\item[ii)]
the image of $Q_2$ under $P_0$-projection is $D$, and
\item[iii)]
the pencil generated by $Q_1$ and $Q_2$ is generic.
\end{paritems}
In fact, in suitable homogeneous coordinates
$x,y,t$ on ${\rm I\!P}_2$ and $x,y,z,t$ on ${\rm I\!P}_3$
the conics $C$ and $D$
are given by equations
$$
C: x^2+y^2+t^2 = 0 , \qquad
D: \alpha x^2+\beta y^2+\gamma t^2 = 0
$$
with $\alpha,\beta,\gamma\in\bbC$.
We choose $Q_2$ to be the cone
$$
Q_2: \alpha x^2+\beta y^2+\gamma t^2 = 0 \\
$$
with top $P_0=(0:0:1:0)$ and $Q_1$ to be the smooth quadric
$$
Q_1: x^2+y^2+z^2+t^2 = 0 \ .
$$
Now conditions i) and ii) are obviously satisfied. For condition iii)
note that the discriminant of the pencil $\lambda_1 Q_1+\lambda_2 Q_2$,
$$
d(\lambda_1,\lambda_2)=\lambda_1(\lambda_1+\lambda_2\alpha)
(\lambda_1+\lambda_2\beta)(\lambda_1+\lambda_2\gamma) \ ,
$$
has no multiple roots, since the pencil $\lambda C+\mu D$ is
generic by assumption.
2)
Let $Q_1,Q_2\subset{\rm I\!P}_3$ be quadrics with the properties i),ii),iii)
above. We claim that the branch locus of a smooth quadric
$Q_{\lambda_1,\lambda_2}=\lambda_1 Q_1+\lambda_2 Q_2$
under $P_0$-projection is a conic in the pencil $\lambda C+\mu D$.
It is enough to show that the branch divisors of the
$P_0$-projections $Q_{\lambda_1,\lambda_2}\mathop{\longrightarrow}\limits{\rm I\!P}_2$ vary in a pencil.
Now, a point $P\in Q_{\lambda_1,\lambda_2}$ is a branch point,
if the line $\overline{PP_0}$ touches $Q_{\lambda_1,\lambda_2}$
in $P$, i.e.\ if $P_0^t Q_{\lambda_1,\lambda_2} P=0$ (in matrix notation).
So the branch divisor is just the intersection of
$Q_{\lambda_1,\lambda_2}$ with the polar
$P_0^t Q_{\lambda_1,\lambda_2}$ of the point $P_0$.
But
$$
P_0^t Q_{\lambda_1,\lambda_2} = P_0^t(\lambda_1 Q_1+\lambda_2 Q_2)
= \lambda_1 P_0^t Q_1
$$
shows that this polar is the same for all quadrics in the pencil.
So the branch divisors vary in a pencil on this polar.
3) If a quadric $Q_{\lambda_1,\lambda_2}$ of the pencil is smooth,
then according to Proposition \ref{23} the quadrics
$Q_{\lambda_1,\lambda_2}$ and $Q_2$ are in Poncelet-$n$-position,
if and only if the branch locus of $Q_{\lambda_1,\lambda_2}$
is $n$-inscribed into $D$.
If $Q_{\lambda_1,\lambda_2}$ is a cone, then $Q_{\lambda_1,\lambda_2}$
and $Q_2$ are certainly not in Poncelet-$n$-position, because
their rulings are halfperiods on the elliptic curve $M$
parametrizing the rulings in the pencil.
We conclude that the number of conics in the pencil
$\lambda C+\mu D$, which are $n$-inscribed into $D$ equals the
number of quadrics $Q_{\lambda_1,\lambda_2}$ such that
$Q_{\lambda_1,\lambda_2}$ and $Q_2$ are in Poncelet-$n$-position.
But this number is just half the number of rulings $R\in M$
such that $R,R_2$ satisfy the Poncelet-$n$-condition.
Now the assertion of the theorem follows by
Corollary \ref{torsion criterion}.
\nopagebreak\hspace*{\fill
\begin{theorem}\label{number circumscribed}
Let $\lambda C + \mu D$ be a general pencil of conics in ${\rm I\!P}_2$.
Then the number of conics in this pencil, $n$--circumscribed
about $D$ is $\frac{1}{4} T(n)$.
\end{theorem}
\startproof{\it Proof. }
A conic $\lambda C+\mu D$ is $n$--circumscribed about $D$ if
and only if the dual conic $(\lambda C + \mu D)^*$ is
$n$--inscribed into $D^*$. The dual pencil
$(\lambda C + \mu D)^*$ is parametrized by a smooth
conic $\Gamma$ in the space ${\rm I\!P}_5$ of all conics.
Denote by $H \subset {\rm I\!P}_5$ the hypersurface parametrizing
conics $n$--circumscribed about $D$. By
Theorem \ref{number inscribed}
$$ \deg(H) = \frac{1}{2} \Gamma . H = \frac{1}{4} T(n).$$
This shows that the general pencil $\lambda C + \mu D$ meets
$H$ in $\frac{1}{4} T(n)$ points.
\nopagebreak\hspace*{\fill
\section{A Poncelet theorem on three conics}
\label{section three conics}
As an application of the three-dimensional Poncelet theorem
we now prove a Poncelet theorem on three conics.
Let $C,C_1,C_2\subset{\rm I\!P}_2$ be three smooth conics in a generic pencil.
Let $P_1$ be an arbitrary point on $C$. There are two tangent lines
to $C_1$ through $P_1$. We choose one of them, $T_1$ say, and define
$P_2$ to be its second point of intersection with $C$.
Next, we choose a tangent $T_2$ to $C_2$ through $P_2$.
We will now describe, how the data $P_1,T_1,T_2$ determine a sequence
$(T_i)_{i\ge 1}$ of lines such that for $i\ge 1$ the line $T_i$ is
tangent to $C_1$ resp.\ $C_2$, if $i$ is odd resp.\ even, and such
that the intersection points of consecutive lines $T_i,T_{i+1}$
lie on $C$.
To this end, we choose
(as in the proof of Theorem \ref{number inscribed})
smooth quadrics $Q_1,Q_2$ and a
cone $Q$ in a generic pencil in ${\rm I\!P}_3$ such that,
denoting the projection ${\rm I\!P}_3 - - \rightarrow{\rm I\!P}_2$
from the top of $Q$ by $\pi$,
\begin{paritems}{ii)}
\item[i)] the branch locus of $\pi\restr{Q_i}:Q_i\mathop{\longrightarrow}\limits{\rm I\!P}_2$ is $C_i$,
$i=1,2$, and
\item[ii)] the image of $Q$ under $\pi$ is $C$.
\end{paritems}
Then we have $P_1=\pi(e)$ for some point $e$ on the elliptic curve
$E=Q_1\cap Q_2$. The tangent $T_1$ is the image $\pi(L_1)$ of a line
$L_1$ on $Q_1$ through $e$. Let $R_1$ be the ruling on $Q_1$ containing
$L_1$ and let $\iota_1$ be the associated involution on $E$.
So $P_2=\pi(\iota_1e)$ and $T_2$ is the image $\pi(L_2)$ of a line
$L_2$ on $Q_2$ through $\iota_1e$. Let $R_2$ be the ruling on $Q_2$
containing $L_2$ and $\iota_2$ its associated involution.
Now the tangents $T_i$, $i\ge 3$, are defined to be the images
of the lines $L_i$, where for $k\ge 2$
\begin{eqnarray*}
L_{2k-1} &:=& \mbox{ the line in $R_1$ through the point
$(\iota_2\iota_1)^{k-1}e$} \\
L_{2k} &:=& \mbox{ the line in $R_2$ through the point
$\iota_1(\iota_2\iota_1)^{k-1}e$} \ .
\end{eqnarray*}
The intuition here is that the choice of the tangents $T_i$, $i\ge 3$,
is compatible with the choices made for $T_1$ and $T_2$.
Now suppose that the tangent sequence $(T_i)$
{\em closes after $n$ steps}, i.e.\ $T_{2n+1}=T_1$.
This occurs if and only if the translation $t := \iota_2\iota_1$
is of order $n$. Since this is independent of the starting point $P_1$,
we have:
\begin{theorem}
Suppose $C,C_1,C_2$ are smooth conics in a generic pencil in ${\rm I\!P}_2$
admitting a tangent sequence (in the sense above)
which closes after $n$ steps.
Then they admit such sequences starting with any point on $C$.
\end{theorem}
The considerations above show slightly more:
The existence of a closed tangent sequence is actually
a property of the pair
$C_1,C_2$ only, so it is not only independent of the starting point
on $C$, but even on the choice of $C$ within the pencil generated
by $C_1$ and $C_2$.
\section{The Poncelet condition for quadrics of revolution}
\label{section revolution}
The aim of this section is to give an explicit formula for the
Poncelet-$n$-condition on two quadrics of revolution.
To begin with, let $q,q'$ be quadrics in ${\rm I\!P}_1$, given by equations
$$
q(z,t)=az^2+bzt+ct^2, \qquad q'(z,t)=a'z^2+b'zt+c't^2 \ .
$$
The pair $(q,q')$ has three quasi-invariants under the
$\mathop{\rm PGL}\nolimits(2,\bbC)$-action:
the {\it discriminants} $D:=b^2-4ac$, $D':=b'^2-4a'c'$ and the
{\it jacobian} $J:=2(ac'+a'c)-bb'$.
We have:
\begin{assertion}
If $D,D'\ne 0$, then up to the $\mathop{\rm PGL}\nolimits(2,\bbC)$-action
the pair $(q,q')$ is uniquely determined
by its invariants $D,D'$ and $J$.
\end{assertion}
\startproof{\it Proof. }
Because of $D'\ne 0$ we may assume $q'(z,t)=zt$, i.e.\
$a'=0$, $c'=0$ and $b'=1$. Then we have $D'=1$ and $J=-b$.
So $b$ is determined by $J$ and the product $ac$ is determined
by $D=b^2-4ac$. For any $\alpha\in\bbC$ the projective transformation
$
(z:t)\mathop{\mapstochar\longrightarrow}\limits(\alpha z: \frac t{\alpha})
$
leaves the invariants $D,D',J$ unchanged. It transforms the
quadric $q$ into
$$
\alpha^2 a z^2 + bzt + \frac c{\alpha^2} t^2 \ ,
$$
so by a suitable choice of $\alpha$ we can change $a$ to any nonzero
value.
\nopagebreak\hspace*{\fill
In terms of the invariants $D,D',J$
the discriminant $d(\lambda,\mu)$
of the pencil $\lambda q+\mu q'$, $(\lambda:\mu)\in{\rm I\!P}_1$ reads
$$
d(\lambda,\mu)=
-\frac{\lambda^2}4 D+\frac{\lambda\mu}2 J - \frac{\mu^2}4 D' \ .
$$
\bigskip
Next we consider the quadrics
$$
Q_{a,b,c}:x^2+y^2=az^2+bzt+ct^2, \qquad a,b,c\in\bbC
$$
in ${\rm I\!P}_3$ with homogeneous coordinates $(x:y:z:t)$.
Such a quadric $Q_{a,b,c}$ is singular (a cone), if $b^2=4ac$.
The intersection curve of two quadrics
$Q_{a_1,b_1,c_1}$ and $Q_{a_2,b_2,c_2}$
consists of two circles
$$
C_j:x^2+y^2=a_1z_j^2+b_1z_jt_j+c_1t_j^2
$$
in the planes $zt_j=tz_j$, $j=1,2$, where
$(z_1:t_1)$ and $(z_2:t_2)$ are the roots of the equation
$$
(a_1-a_2)z^2+(b_1-b_2)zt+(c_1-c_2)t^2 = 0 \ .
$$
In case $(b_1-b_2)^2=4(a_1-a_2)(c_1-c_2)$ the two circles coincide,
i.e.\ the quadrics touch along one circle. Otherwise the circles
$C_1,C_2$ meet in the two points $(1:i:0:0),(i:1:0:0)$.
\bigskip
Now we give an explicit formula for the Poncelet-$n$-condition on two
quadrics $Q_{a_1,b_1,c_1}$ and $Q_{a_2,b_2,c_2}$. This is best
expressed in terms of the invariants of the binary quadrics
$q_{a_i,b_i,c_i}:=a_iz^2+b_izt+c_it^2$.
\begin{proposition}\label{Poncelet formula}
Let $Q_{a_1,b_1,c_1}$ and $Q_{a_2,b_2,c_2}$ be two quadrics as above
with reduced intersection
curve (two circles) and let $D_1,D_2,J_{12}$ be the
discriminants and the jacobian of the associated binary quadrics.
Then the following statements
are equivalent:
\begin{paritems}{ii)}
\item[i)] $Q_{a_1,b_1,c_1}$ and $Q_{a_2,b_2,c_2}$ are in
Poncelet-$n$-position.
\item[ii)] There is an integer $k$, $(k,n)=1$, such that
$$
\left(J_{12}(1+\cos\frac{2\pi k}n)+D_1+D_2\right)^2
= D_1D_2\left(1-\cos\frac{2\pi k}n\right)^2
$$
\end{paritems}
\end{proposition}
\startproof{\it Proof. }
Let $C_1,C_2$ be the circles of intersection of the two quadrics
and let $zt_j=tz_j$, $j=1,2$, be the associated planes.
It will be enough to consider the case that $t_1=t_2=1$.
We may then calculate in affine coordinates $x,y,z$.
To begin with, let
$$
P_1=(x_1,y_1,z_1),\quad P_2=(x_2,y_2,z_2),\quad P_1'=(x'_1,y'_1,z_1)
$$
be three points such that the line $\overline{P_1P_2}$ lies on $Q_1$
and $\overline{P_2P'_1}$ lies on $Q_2$. Now,
$\overline{P_1P_2}$ lies on $Q_1$ if and only if the points
$P_1,P_2$ belong to $Q_1$, i.e.\
$$
x_j^2+y_j^2=a_1z_j^2+b_1z_j+c_1
$$
and if the point $(P_1+P_2)/2$ lies on $Q_1$, i.e.\
$$
\left(\frac{x_1+x_2}2\right)^2+\left(\frac{y_1+y_2}2\right)^2
= a_1\left(\frac{z_1+z_2}2\right)^2+b_1\left(\frac{z_1+z_2}2\right)^2
+ c_1 \ .
$$
Because of $P_1,P_2\in Q_1$ this is equivalent to
\begin{equation}\label{eq 1}
x_1x_2+y_1y_2=a_1z_1z_2+\frac{b_1}2(z_1+z_2)+c_1 =: g_1 \ .
\end{equation}
Similarly the line $\overline{P_2P_1'}$ lies on $Q_2$ if and only if
\begin{equation}\label{eq 2}
x_1'x_2+y_1'y_2=a_2z_1z_2+\frac{b_2}2(z_1+z_2)+c_2 =: g_2 \ .
\end{equation}
Now we come to the Poncelet condition. The automorphisms of order
$n$ on $C_1\setminus\{(1:i:0:0),(i:1:0:0)\}\cong\bbC^*$ are
induced by the maps
\begin{equation}\label{eq 3}
\rho_k:
\left(
\begin{array}{cc}
x \\ y
\end{array}
\right)
\mathop{\mapstochar\longrightarrow}\limits
\left(
\begin{array}{rr}
\cos\tfrac{2\pi k}n&-\sin\tfrac{2\pi k}n \\
\sin\tfrac{2\pi k}n&\cos\tfrac{2\pi k}n
\end{array}
\right)
\left(
\begin{array}{cc}
x \\ y
\end{array}
\right)
\end{equation}
where $(k,n)=1$. So the two quadrics are in Poncelet-$n$-position
if and only if
$\rho_k(x_1,y_1)=(x_1',y_1')$
for some $k$ with $(k,n)=1$.
Without loss of generality we may assume $y_1=0$. Then
conditions \eqnref{eq 1}, \eqnref{eq 2} and \eqnref{eq 3} read
\begin{eqnarray}
x_1'&=&x_1\cos\tfrac{2\pi k}n \\
-y_1'&=&x_1\sin\tfrac{2\pi k}n \\
x_1 x_2&=&g_1 \label{eq x_2} \\
x_1'x_2+y_1'y_2&=&g_2
\end{eqnarray}
Inserting the first three equations into the last one we obtain
\begin{equation}\label{eq y_2}
g_1\cos\tfrac{2\pi k}n - x_1y_2\sin\tfrac{2\pi k}n = g_2 \ . \label{y_2}
\end{equation}
and
from the equations \eqnref{eq x_2} and \eqnref{eq y_2} we get
$$
x_2^2+y_2^2=\left(\frac{g_1}{x_1}\right)^2+
\left(
\frac{g_1\cos\tfrac{2\pi k}n-g_2}{x_1\sin\tfrac{2\pi k}n}
\right)^2 \ .
$$
Now, upon using the equation of $Q_1$ we can replace the left hand
side by $a_1z_2^2+b_1z_2+c_1$ and we can substitute $x_1^2$ on the
right hand side by $a_1z_1^2+b_1z_1+c_1$.
In this way
we arrive at the equation
\begin{equation}\label{eq ponc 1}
g_1^2-2g_1g_2\cos\tfrac{2\pi k}n+g_2^2=p\sin^2\tfrac{2\pi k}n
\end{equation}
where
$p:=(a_1z_1^2+b_1z_1+c_1)(a_1z_2^2+b_1z_2+c_1)$.
Now we can use that $z_1$ and $z_2$ are the roots of the
equation
$
(a_1-a_2)z^2+(b_1-b_2)z+(c_1-c_2)=0
$
to write $g_1,g_2$ and $p$ in terms of the coefficients
$a_1,b_1,c_1,a_2,b_2,c_2$.
This finally allows to express \eqnref{eq ponc 1} in the
invariants $D_1,D_2,J_{12}$. We omit the details.
\nopagebreak\hspace*{\fill
\section{Circles in the projective plane}
In the previous sections we considered Poncelet properties of
conics and quadrics related to the three-dimensional Poncelet theorem.
Now we turn to the study of closing theorems in circle geometry.
A {\em circle} is a conic $C\subset{\rm I\!P}_2$ passing through the two
circular points $(1:\pm i:0)$. Its equation is of the form
$$
a(x^2+y^2)+2bxz+2cyz+dz^2=0 \ ,
$$
where $(a:b:c:d)\in{\rm I\!P}_3$. The {\em discriminant} of the circle $C$ is
$$
\det\left(\begin{array}{ccc} a & 0 & b \\
0 & a & c \\
b & c & d
\end{array}\right)
= a(ad-b^2-c^2) \ .
$$
The quadratic form
$$
q(C) := ad-b^2-c^2
$$
is the basic invariant. A circle $C$ with $q(C)=0$ consists of two lines
each passing through a circular point. These are the {\em null-circles}.
If we have $a=0$, then $C$ decomposes into two lines, one of which
passes through the two circular points. In this case we
say that $C$ is a {\em line}.
The quadratic invariant $q$ associates with each circle $C$ two basic
surfaces in the space of circles ${\rm I\!P}_3$:
\subparagraph*{The polar plane.}
The polar plane of a circle $C$ with respect to
the quadric $q$ is
$$
q(C,C') = \tfrac12(ad'+da')-(bb'+cc') \ .
$$
\begin{proposition}[\cite{Ped57}, p.\ 32]
We have $q(C,C')=0$ if and only if the circles $C,C'$ intersect
orthogonally, i.e.\ if they have a point of intersection $P$ such that
the tangent vectors of $C$ and $C'$ in $P$ are orthogonal with respect to
the canonical bilinear form on $\bbC^2$.
\end{proposition}
\startproof{\it Proof. }
We may use affine coordinates $x,y$.
The tangent vector of $C$ at the point $(x,y)$ is
$$
\vect{\d Cx \\ \d Cy} = \vect{2ax+2b \\ 2ay+2c} \ ,
$$
so $C$ and $C'$ intersect orthogonally if and only if there is a point
$(x,y)$ such that the following equations are satisfied.
\begin{eqnarray*}
a(x^2+y^2)+2bx+2cy+d &=& 0 \\
a'(x^2+y^2)+2b'x+2c'y+d' &=& 0 \\
(2ax+2b)(2a'x+2b')+(2ay+2c)(2a'y+2c') &=& 0
\end{eqnarray*}
Elimination of $x,y$ from these equations yields $q(C,C')=0$, as claimed.
\nopagebreak\hspace*{\fill
\subparagraph*{The tangent cone.}
This is the variety of circles touching a given circle. We have:
\begin{proposition}[\cite{Ped57}, p.\ 32] \label{tangent cone}
a) The circles $C'$ touching a given circle $C$ are parametrized by the
quadric surface
$$
Q_{C}: q(C)q(C')-q(C,C')^2=0
$$
in the space of circles.
b) If $C$ is smooth, then $Q_{C}$ is a cone with top $C\in{\rm I\!P}_3$. If
$C$ is a null-circle, then $Q_{C}$ is a double plane.
\end{proposition}
\startproof{\it Proof. }
a)
The circles $C,C'$ touch if there is a point $(x,y)$ such that
\begin{eqnarray*}
C(x,y)=C'(x,y)=0 \quad\mbox{ and }\quad
dC(x,y)=\lambda\cdot dC'(x,y)
\end{eqnarray*}
for some $\lambda\in\bbC^*$. By eliminating $x,y$ and $\lambda$ from
these equations we find
\begin{eqnarray*}
&& -\tfrac14 a^2 d'^2+\tfrac12 ada'd'-adb'^2-adc'^2+abb'd'+acc'd'-
\tfrac14 d^2a'^2+bda'b'+cda'c'\\
&& -b^2a'd'+b^2c'^2-2bcb'c'-
c^2a'd'+c^2b'^2 = 0
\ ,
\end{eqnarray*}
which is equivalent to
$q(C)q(C')-q(C,C')^2=0$.
b)
An obvious calculation shows that $C=(a:b:c:d)$
is a singular point of $Q_C$.
Further, we find $\mathop{\rm rank}\nolimits Q_C=1$ if $q(C)=0$.
\nopagebreak\hspace*{\fill
Now we consider the variety of all circles $C$,
which touch two given (smooth) circles $C_1,C_2$. This is the
intersection curve of the two cones
\begin{eqnarray*}
Q_{C_1}: q(C_1)q(C)-q(C_1,C)^2&=&0 \\
Q_{C_2}: q(C_2)q(C)-q(C_2,C)^2&=&0 \ .
\end{eqnarray*}
\begin{proposition}\label{planes}
a) The intersection of the cones $Q_{C_1},Q_{C_2}$ consists of two
conics lying in the planes
$$
\Pi_{C_1,C_2}^{\pm}:=\sqrt{q(C_2)}q(C_1,C)\pm\sqrt{q(C_1)}q(C_2,C) \ .
$$
b) The intersection points of the two conics are the null-circles
associated to the intersection points of $C_1$ and $C_2$.
\end{proposition}
In the sequel the two conics above will be referred to as the two
{\em families of circles touching $C_1$ and $C_2$}.
\startproof{\it Proof. }
a) The cones $Q_{C_1},Q_{C_2}$ span the pencil of quadrics
$$
(\lambda q(C_1)+\mu q(C_2))q(C)-\lambda q(C_1,C)^2-\mu q(C_2,C)^2
\qquad ,(\lambda:\mu)\in{\rm I\!P}_1 \ .
$$
This pencil contains the quadric
$$
q(C_2)Q_{C_1}-q(C_1)Q_{C_2}=q(C_2)q(C_1,C)^2-q(C_1)q(C_2,C)^2 \ ,
$$
which splits in the two planes $\Pi_{C_1,C_2}^{\pm}$.
b) Let $C$ be a null-circle associated to $C_1,C_2$. It touches both
$C_1$ and $C_2$, so
$$
q(C_1)q(C)-q(C_1,C)^2=
q(C_2)q(C)-q(C_2,C)^2=0 \ .
$$
Because of $q(C)=0$ this implies
$q(C_1,C)=q(C_2,C)=0$, so $C$ lies on both planes
$\Pi_{C_1,C_2}^{\pm}$.
\nopagebreak\hspace*{\fill
We will also need the following degenerate case.
\begin{lemma}\label{touching}
If two circles $C_1,C_2$ touch each other, then the tangent cones
$Q_{C_1},Q_{C_2}$ touch along a line.
\end{lemma}
\startproof{\it Proof. }
We may assume that the circles have equations
$$
C_1: x^2+y^2+z^2=0 \quad\mbox{ and }\quad C_2:a(x^2+y^2)+2bxz+2cyz+dz^2=0 \ .
$$
Let $P$ be the
pencil of circles touching $C_1$ in its point of contact with $C_2$.
Thus $P$ is just the line joining the vertices
$p_1=(1:0:0:1)$ and $p_2=(a:b:c:d)$ of the two cones.
We calculate the tangent
planes of $Q_{C_1}$ and $Q_{C_2}$ in the points
$\lambda_1 p_1+\lambda_2 p_2$ of $P$:
\begin{eqnarray*}
Q_{C_1}(\lambda_1 p_1+\lambda_2 p_2)&=&\lambda_2Q_{C_1}p_2
= \tfrac{\lambda_2}4(d-a:-4b:-4c:a-d) \\
Q_{C_2}(\lambda_1 p_1+\lambda_2 p_2)&=&\lambda_1Q_{C_2}p_1
= \\
\rlap{$
\frac{\lambda_1}4(-d^2-2b^2-2c^2+ad:2b(a+d):
2c(a+d):
-a^2-2b^2-2c^2+ad)
$}\phantom{Q_{C_2}(\lambda_1 p_1+\lambda_2 p_2)}\hskip-0.5cm
\end{eqnarray*}
Now $C_1,C_2$ touch each other, i.e.\ $C_2\in Q_{C_1}$, i.e.\
$a^2+d^2+4b^2+4c^2-2ad=0$. This shows that
the tangent planes coincide along $P$.
\nopagebreak\hspace*{\fill
In Sect.\ \ref{section Money-Coutts}
we will have to work with the tangent cones of three given
circles $C_1,C_2,C_3$. The following property of their mutual
intersection will turn out to be crucial.
\begin{lemma}\label{common line}
If the
signs $\epsilon_{12},\epsilon_{13},\epsilon_{23}\in\{\pm1\}$ are chosen
such that
$\epsilon_{12}\epsilon_{13}\epsilon_{23}=-1$, then the planes
$
\Pi_{C_1,C_2}^{\epsilon_{12}},
\Pi_{C_1,C_3}^{\epsilon_{13}},
\Pi_{C_2,C_3}^{\epsilon_{23}}
$
have a line in common.
\end{lemma}
Notice: Changing the sign of a square root $\sqrt{q(C_i)}$
does not influence the condition
$\varepsilon_{12} \varepsilon_{13} \varepsilon_{23} = -1$.
\startproof{\it Proof. }
The pencil spanned by
$\Pi_{C_1,C_2}^{\epsilon_{12}}$ and
$\Pi_{C_1,C_3}^{\epsilon_{13}}$ contains the plane
$$
{\frac{\sqrt{q(C_2)}}{\sqrt{q(C_1)}}}\Pi_{C_1,C_3}^{\epsilon_{13}} -
{\frac{\sqrt{q(C_3)}}{\sqrt{q(C_1)}}}\Pi_{C_1,C_2}^{\epsilon_{12}}
=
\epsilon_{13}\sqrt{q(C_2)}q(C_3,C) -
\epsilon_{12}\sqrt{q(C_3)}q(C_2,C) \ ,
$$
which equals
$\Pi_{C_2,C_3}^{\epsilon_{23}}$, if
$\epsilon_{12}\epsilon_{13}\epsilon_{23}=-1$.
\nopagebreak\hspace*{\fill
\section{Emch's and Steiner's theorems on circular series}
\label{section Emch Steiner}
In this section we consider two classical theorems on circular series.
Our aim here is to show
that one obtains short proofs by considering an
elliptic resp.\ rational curve underlying the closing mechanism.
Let $C,C_1,C_2\subset{\rm I\!P}_2$ be smooth circles
in general position and let $F$ be one of the two families of circles
touching $C_1$ and $C_2$.
We start with a smooth circle $S_1\in F$ and choose
one of its points of intersection with
$C$, $P_1$ say.
The pair $(S_1,P_1)\in F\times C$ determines a second pair as
follows. There are two circles in $F$ through $P_1$, namely the points
of intersection of $F$ with the plane $\Pi_{P_1}$
in the space of circles consisting
of all circles through
$P_1$. Let $F\cap\Pi_{P_1}=\{S_1,S_2\}$.
The second circle $S_2$ in turn
intersects $C$ in $P_1$ and in a second point $P_2$. We set
$t(S_1,P_1) := (S_2,P_2)$.
\begin{theorem}[Emch \cite{Emc01}]
Suppose $t^n(S_1,P_1)=(S_1,P_1)$ holds for some pair $(S_1,P_1)$
such that $P_1\in C$.
Then this holds for any such pair.
\end{theorem}
\startproof{\it Proof. }
Consider the incidence curve
$$
E := \{(S,P)\in F\times C\mid P\in S\}
$$
Denoting by $\pi_1,\pi_2$ the projections onto $F$ resp.\ $C$, we
have for $S\in F$
$$
\pi_2\pi_1^{-1}(S) = S\cap C=Q+Q'+P+P' \ ,
$$
where $Q,Q'$ are the infinitely far points on $C$ and $P,P'\in C$.
For $P\in C$ we have
$\pi_1\pi_2^{-1}(P)=F\cap \Pi_P$,
where $\Pi_P$ is the plane in the space of circles consisting
of the circles through $P$.
Thus the curve $V$ is of bidegree $(4,2)$ and it decomposes as
$$
V=F\times \{Q\} + F\times\{Q'\} + E \ ,
$$
where $E$ is a curve of bidegree $(2,2)$.
The branch points of the restriction $\pi_1\restr E$ are the circles
$S\in F$ touching $C$. Since $C,C_1,C_2$ are in general position,
there are exactly four of them.
So $E$ is smooth elliptic by Lemma \ref{smooth elliptic} below.
The map $t$ is just the composition of the covering involutions of
$\pi_1\restr E$ and $\pi_2\restr E$, so it is a translation on $E$.
The assumption implies that it is of order $n$.
\nopagebreak\hspace*{\fill
\begin{lemma}\label{smooth elliptic}
Let $E\subset{\rm I\!P}_1\times{\rm I\!P}_1$ a curve of bidegree $(2,2)$. Assume
that the projections $\pi_1,\pi_2:E\mathop{\longrightarrow}\limits{\rm I\!P}_2$ are finite and that one of
them has at least four branch points. Then $E$ is smooth.
\end{lemma}
\startproof{\it Proof. }
Suppose to the contrary that $E$ is singular. First assume that $E$ is
reducible. Since the projections are finite, $E$ is then a sum of two
curves $E_1,E_2$ of bidegree $(1,1)$. By the adjunction formula
$E_1\cong E_2\cong{\rm I\!P}_1$. But then $\pi_1$ and $\pi_2$ have only two
branch points, namely the intersection points of $E_1$ and $E_2$, a
contradiction.
If $E$ is irreducible, then it has exactly one singularity and the
normalization $\widetilde E$ is a smooth rational curve. But then by
the Hurwitz formula the map $\widetilde E\mathop{\longrightarrow}\limits E\mathop{\longrightarrow}\limits{\rm I\!P}_1$ has only two branch
points, a contradiction again.
\nopagebreak\hspace*{\fill
Next we turn to Steiner's theorem. In contrast to the situation
of Emch's theorem here the relevant curve for the closing process
is rational.
Let $C_1,C_2\subset{\rm I\!P}_2$ be smooth circles in general position
and let $F$ be a family of circles touching $C_1$ and $C_2$.
\begin{theorem}[Steiner]
Suppose that
there is a closed sequence of distinct
circles $S_1,S_2,\dots,S_n,S_{n+1}=S_1$
in $F$ such that $S_i$ touches $S_{i+1}$ for $1\le i\le n$.
Then there are such sequences starting with any circle in $F$.
\end{theorem}
Steiner's theorem is a consequence of the following
\begin{proposition}
a) $V := \{(S_1,S_2)\in F\times F\mid S_1\in Q_{S_2} \}$
is a curve of bidegree $(4,4)$ and
$$
V = 2\cdot\Delta + V_1 + r^*V_1 \ ,
$$
where $\Delta\subset F\times F$ is the diagonal,
$V_1$ is a curve of bidegree $(1,1)$ and
$r:{\rm I\!P}_1\times{\rm I\!P}_1\mathop{\longrightarrow}\limits{\rm I\!P}_1\times{\rm I\!P}_1, (S_1,S_2)\mathop{\mapstochar\longrightarrow}\limits(S_2,S_1)$.
b) We have $V_1\setminus\Delta\cong r^*V_1\setminus\Delta
\cong\bbC^*$. Let
\begin{eqnarray*}
t:V_1\setminus\Delta &\mathop{\longrightarrow}\limits& V_1\setminus\Delta
\end{eqnarray*}
be defined by $t(S_1,S_2) := (S_2,S_3)$, where
$\pi_2^{-1}(S_2)=2(S_2,S_2)+(S_1,S_2)+(S_3,S_2)$.
Then $t$ is the multiplication by a nonzero constant.
\end{proposition}
\startproof{\it Proof. }
We may consider $V$ as a subvariety of ${\rm I\!P}_3\times{\rm I\!P}_3$, namely as the
intersection of the varieties $F\times{\rm I\!P}_3$, ${\rm I\!P}_3\times F$ and the
hypersurface $\{(S_1,S_2)\mid S_1\in Q_{S_2}\}$ of bidegree $(2,2)$.
Denoting by $\pi_1,\pi_2:V \mathop{\longrightarrow}\limits F$ the projections, we have for $S_1\in F$
$$
\pi_2\pi_1^{-1}(S_1) = Q_{S_1}\cap Q_{C_1} \restr{\Pi(F)} \ ,
$$
where $\Pi(F)\subset{\rm I\!P}_3$ is the plane containing $F$.
According to Lemma \ref{touching} the cones $Q_{S_1},Q_{C_1}$ touch
along a line, hence the restriction to $F$ of their intersection is
of the form
$$
Q_{S_1}\cap Q_{C_1} \restr{\Pi(F)} =2 S_1 + S_2 + S_2' \ ,
$$
where $S_2,S_2'\in F$. Therefore $V$ contains the diagonal $\Delta$
as a component of multiplicity two, the residual curve
$V'$ being of bidegree $(2,2)$.
Now let $S_0$ be one of the null-circles associated to $C_1$ and $C_2$,
i.e.\ $S_0$ consists of the two lines joining an intersection point $P$ of
$C_1,C_2$ with the infinitely far circle points. The quadric $Q_{S_0}$ is
then of rank 1, i.e.\ a double plane. It consists of the circles
through $P$. The set $\pi_2\pi_1^{-1}(S_0)$ contains the circles
through $P$ touching $C_1$ and $C_2$, so $\pi_1^{-1}(S_0)=\{(S_0,S_0)\}$.
We conclude that the two null-circles are branch points of both
projections $\pi_1,\pi_2$. Because of $p_a(V')=1$ this implies that
$V'$ is reducible. It consists of two curves $V_1,V_2$ of bidegree
$(1,1)$ meeting in the points $(S_0,S_0),(S_0',S_0')$, where $S_0,S_0'$
are the two null-circles associated to $C_1$ and $C_2$.
Because of $r^*V=V$, we have $V_2=r^*V_1$.
\nopagebreak\hspace*{\fill
\section{The zig-zag theorem}
\label{section zig-zag}
In this section we give a proof of (a complex-projective version of) the
zig-zag theorem \cite{BlaHowHow74}
based on the consideration of an elliptic curve in the product of two
circles.
For a point $P_0=(x_0:y_0:z_0:t_0)$ in ${\rm I\!P}_3$ and a complex number $r$
the quadric
$$
S_{P_0,r}: (xt_0-tx_0)^2+(yt_0-ty_0)^2+(zt_0-tz_0)^2=r^2t^2t_0^2
$$
is called the {\em sphere with center $P_0$ and radius $r$}.
Its intersection with the infinitely far plane $t=0$ is the conic
$x^2+y^2+z^2=0$. A {\em circle} in ${\rm I\!P}_3$ is a conic whose
infinitely far points lie on this conic.
\bigskip
Now let $C_1,C_2\subset{\rm I\!P}_3$ be smooth
circles and let a radius $r\in\bbC$ be fixed.
A {\em zig-zag} for $C_1,C_2$ is a sequence of points $P_1,P_2,\dots$ such
that for $i\ge 1$ the point $P_i$ lies on $C_1$ resp.\ $C_2$, if $i$ is
odd resp.\ even, and such that
$P_{i+1}\in S_{P_i,r}$. Consecutive points $P_i,P_{i+1}$
in a zig-zag are thought of as having constant ''distance'' $r$.
The zig-zag is said to {\em close after $n$ steps}, if $P_{2n+1}=P_1$.
\begin{theorem}[cf.\ \cite{BlaHowHow74}]
Let $C_1,C_2\subset{\rm I\!P}_3$ be a general pair of circles and let
$r\in\bbC$.
If the pair $C_1,C_2$ admits a zig-zag of distance $r$
which closes after $n$ steps, then
it admits such zig-zags starting with any point on $C_1$.
\end{theorem}
\startproof{\it Proof. }
1) We consider the curve
$$
V := \{(P_1,P_2)\in C_1\times C_2 \mid P_2\in S_{P_1,r}\} \ .
$$
It is the restriction of the hypersurface
$\{P_2\in S_{P_1,r}\}\subset{\rm I\!P}_3\times{\rm I\!P}_3$ of bidegree $(2,2)$,
so it is of bidegree $(4,4)$ in $C_1\times C_2$.
We denote by $\pi_i:V\mathop{\longrightarrow}\limits C_i$, $i=1,2$, the projections and by $Q_1,Q_1'$
resp.\ $Q_2,Q_2'$ the infinitely far points on $C_1$ resp.\ $C_2$.
For a point $P_1\in C_1$, different from $Q_1$ and $Q_1'$, we have
$$
\pi_2\pi_1^{-1}(P_1)=S_{P_1,r}\cap C_2=Q_2+Q_2'+P_2+P_2' \ ,
$$
where $P_2,P_2'\in C_2$. Therefore $V$ contains the lines
$$
C_1\times\{Q_2\}, \ C_1\times\{Q_2'\}
$$
and by the same reasoning also the lines
$$
\{Q_1\}\times C_2, \ \{Q_1'\}\times C_2 \ .
$$
The residual curve $E$ is of bidegree $(2,2)$.
2) Next we determine the branch points of the projection
$\pi_1\restr E:E\mathop{\longrightarrow}\limits C_1$ to show that $E$ is smooth elliptic.
Let us first consider the branch points of $\pi_1:V\mathop{\longrightarrow}\limits C_1$.
These are the points $P_1\in C_1$ such that the sphere $S_{P_1,r}$
touches $C_2$. We may assume $C_2$ to lie in the plane $z=0$
having equation $x^2+y^2+t^2=0$,
so $S_{P_1,r}$ intersects the plane of $C_2$
in the circle $S:=S_{P_1,r}(x,y,0,t)$. The sphere $S_{P_1,r}$
touches $C_2$ if and only if $S$ lies on the tangent cone of $C_2$.
Evaluating this condition by means of \eqnref{tangent cone} we find
\begin{equation}\label{branch point quartic}
-\tfrac14(x_1^2+y_1^2+z_1^2-r^2t_1^2)^2
+\tfrac12 t_1^2(x_1^2+y_1^2+z_1^2-r^2t_1^2)
-x_1^2t_1^2-y_1^2t_1^2-\tfrac14 t_1^4 = 0 \ .
\end{equation}
So the branch points of $\pi_1$
are just the intersection points of the circle
$C_1$ and the quartic surface $Y\subset{\rm I\!P}_3$ defined by
\eqnref{branch point quartic}.
One may think of $Y$ as the set of all points having ''distance''
$r$ from $C_2$.
Since $Y$ intersects the plane $t_1=0$ in the double conic
$(x_1^2+y_1^2+z_1^2)^2=0$, the points $Q_1,Q_1'$ are both contained
in $Y\cap C_1$ with multiplicity two. The four remaining points
of intersection are the branch points of $\pi_1\restr E:E\mathop{\longrightarrow}\limits C_1$.
For general $C_1$ these points are distinct, so $E$ is in fact
smooth elliptic.
3)
Having identified the underlying elliptic curve,
the proof now ends in the usual way:
We denote by
$\iota_1,\iota_2$ the covering involutions of
$\pi_1\restr E$ resp.\ $\pi_2\restr E$
and by $t := \iota_2\iota_1$ the associated translation on $E$.
The assumption that there is a zig-zag
which closes after $n$ steps implies that $t$ is of order $n$.
This proves the theorem.
\nopagebreak\hspace*{\fill
\section{The Money-Coutts theorem}
\label{section Money-Coutts}
Let $C_1,C_2,C_3\subset{\rm I\!P}_2$ be smooth circles in general position.
For each of the pairs $C_1,C_2$ and $C_2,C_3$
we choose one of the two families of circles touching both circles,
say $F_{1},F_{2}$ respectively.
We are interested in the pairs of circles $S_1\in F_1,S_2\in F_2$
such that $S_1$ touches $S_2$.
To begin with, we show:
\begin{proposition}\label{elliptic curve}
The curve
$$
V :=\{(S_1,S_2)\in F_{1}\times F_{2} \mid S_1\in Q_{S_2} \}
$$
is of bidegree $(4,4)$ and we have
$V = 2\widetilde\Delta+E$,
where $\widetilde\Delta\subset F_1\times F_2$ is a curve of bidegree $(1,1)$
and $E$ is a smooth elliptic curve.
\end{proposition}
\startproof{\it Proof. }
We may consider $V$ as the subvariety
$$
(F_1\times{\rm I\!P}_3)\cap({\rm I\!P}_3\times F_2)
\cap\{(S_1,S_2)\mid S_1\in Q_{S_2} \}
$$
of ${\rm I\!P}_3\times{\rm I\!P}_3$.
So for $S_1\in F_1$ we have
$$
\pi_2\pi_1^{-1}(S_1)=Q_{S_1}\cap Q_{C_2} \restr{\Pi(F_2)} \ ,
$$
where $\Pi(F_2)\subset{\rm I\!P}_3$ is the plane containing $F_2$ and
$\pi_i:V\mathop{\longrightarrow}\limits F_i$, $i=1,2$, are the projections. By Lemma \ref{touching}
the cones $Q_{S_1},Q_{C_1}$ touch along a line, so their intersection
restricts to $\Pi(F_2)$ as
$$
Q_{S_1}\cap Q_{C_2} \restr{\Pi(F_2)}=2S_1'+S_2+S_2' \ ,
$$
where $S_1'\in F_2$ lies on the line of contact and $S_2,S_2'\in F_2$ lie
on the residual conic. This shows that any line
$\{S_1\}\times F_2\subset F_1\times F_2$
touches $V$, hence $V$ must contain a curve $\widetilde\Delta$ of bidegree
$(1,1)$ as a component of multiplicity two. Next we consider the
residual curve $E$, which is of bidegree $(2,2)$ and we determine the
branch points of the projection $\pi_1\restr E:E\mathop{\longrightarrow}\limits F_1$.
$\bullet$ First let $S_1$ be one of the two null-circles associated to
$C_1$ and $C_2$. In this case the tangent quadric $Q_{S_1}$ is a double
plane.
Therefore
$
\pi_2\pi_1^{-1}(S_1)=Q_{S_1}\cap Q_{C_2} \restr{\Pi(F_2)}
$
consists of two points only, so $S_1$ is a branch point of
$\pi_1\restr E$.
$\bullet$ Next let $S_1$ be an Apollonius circle of $C_1,C_2,C_3$, i.e.\
a circle touching all three of them. Further suppose that $S_1$ lies on
$F_1$, but not on $F_2$. Since the tangent cones
$Q_{C_1},Q_{C_2},Q_{C_3}$ are in general position, there are exactly two
such circles. Since $S_1$ touches both $C_2$ and $C_3$, we have
\begin{eqnarray*}
Q_{S_1}\cap Q_{C_2}&=&2L_2+D_2 \\
Q_{S_1}\cap Q_{C_3}&=&2L_3+D_3 \ ,
\end{eqnarray*}
where $L_2,L_3\subset{\rm I\!P}_3$ are lines and $D_2,D_3\subset{\rm I\!P}_3$
are conics. Thus set-theoretically
$$
Q_{S_1}\cap Q_{C_2}\cap Q_{C_3}=
L_2\cap L_3+D_2\cap D_3+L_2\cap D_3+L_3\cap D_2
$$
consists of five points at most. We claim that $S_1$ is a branch point
of $\pi_1\restr E$. To this end we show that
only two of these points belong to $F_2$.
We use the notation and the statement of
Lemma \ref{common line}: The conics $D_2$ and $D_3$ lie in planes
$\Pi_{S_1,C_2}^{\epsilon_{12}}$ resp.\ $\Pi_{S_1,C_3}^{\epsilon_{13}}$, where
$\epsilon_{12},\epsilon_{13}\in\{\pm1\}$, so $D_2\cap
D_3\subset\Pi_{C_2,C_3}^{\epsilon_{23}}$
for $\epsilon_{23}:=-\epsilon_{12}\epsilon_{13}$. Then $L_2,L_3$ lie in
$\Pi_{S_1,C_2}^{-\epsilon_{12}}$ resp.\ $\Pi_{S_1,C_3}^{-\epsilon_{13}}$, so
$L_2\cap L_3\subset\Pi_{C_2,C_3}^{\epsilon_{23}}$ as well.
Since $S_1=L_2\cap L_3$ does not belong to $F_2$ by assumption, we find
\begin{eqnarray*}
Q_{S_1}\cap Q_{C_2}\restr{\Pi(F_2)} &=& L_2\cap D_3 + L_3\cap D_2 \\
Q_{S_1}\cap Q_{C_3}\restr{\Pi(F_2')} &=& L_2\cap L_3 + D_2\cap D_3 \ ,
\end{eqnarray*}
where $F_2'$ is the second family of circles touching $C_1$ and $C_2$.
This shows that $\pi_2\pi_1^{-1}(S_1)$ consists of the two points
$L_2\cap D_3,L_3\cap D_2$.
Applying Lemma \ref{smooth elliptic} we conclude that $E$ is a
smooth elliptic curve.
\nopagebreak\hspace*{\fill
As before, let $F_1,F_2$ be families of circles touching $C_1,C_2$ resp.\
$C_2,C_3$, and suppose $F_3$ is a family of circles touching $C_3$ and
$C_1$. By Proposition \ref{elliptic curve} we have three elliptic curves
\begin{eqnarray*}
E_1 &\subset& F_1\times F_2 \\
E_2 &\subset& F_2\times F_3 \\
E_3 &\subset& F_3\times F_1 \ .
\end{eqnarray*}
Let $\pi_1,\pi_1'$ resp.\ $\pi_2,\pi_2'$ resp.\ $\pi_3,\pi_3'$ denote the
projections onto the factors.
The next point we want to make is:
\begin{lemma}\label{same branch points}
Given $F_1$ and $F_2$, the family $F_3$ can be chosen in such a way that
the projections
$\pi_1',\pi_2$ as well as
$\pi_2',\pi_3$ and $\pi_3',\pi_1$
have the same branch points in $F_2$ resp.\ $F_3$ resp.\ $F_1$.
\end{lemma}
\startproof{\it Proof. }
According to Lemma \ref{common line} the family $F_3$ can be chosen such
that the planes $\Pi(F_1),\Pi(F_2),\Pi(F_3)$ have a line in common.
The branch points of $\pi_1'$ are the two null-circles of the pair
$C_1,C_2$ and the two Apollonius circles in $F_2$, which do not belong
to $F_1$. Because of our choice of $F_3$ these Apollonius circles do not
belong to $F_3$ either. This shows that the branch points of $\pi_1'$
coincide with those of $\pi_2$. The statement on the pairs
$\pi_2',\pi_3$ and $\pi_3',\pi_1$
follows in the same way.
\nopagebreak\hspace*{\fill
Now suppose that families $F_1,F_2,F_3$ have been chosen as above.
The Money-Coutts theorem on the circles $C_1,C_2,C_3$ can then be stated
as follows.
\begin{theorem}[Tyrrel-Powell \cite{TyrPow71}]
Suppose that circles
$S_1\in F_1$,
$S_2\in F_2$,
$S_3\in F_3$
and
$S_4\in F_1$
are chosen such that
$(S_1,S_2)\in E_1$,
$(S_2,S_3)\in E_2$ and
$(S_3,S_4)\in E_3$.
Then it is possible to choose circles
$S_5\in F_2$,
$S_6\in F_3$
and
$S_7\in F_1$
in such a way that again
$(S_4,S_5)\in E_1$,
$(S_5,S_6)\in E_2$ and
$(S_6,S_7)\in E_3$
and such that
$S_7$ coincides with $S_1$.
\end{theorem}
\startproof{\it Proof. }
According to
Lemma \ref{same branch points} the projections
$\pi_1',\pi_2$ resp.\
$\pi_2',\pi_3$ resp.\
$\pi_3',\pi_1$
have the same branch points, hence there are isomorphisms
$\varphi_1:E_1\mathop{\longrightarrow}\limits E_2$,
$\varphi_2:E_2\mathop{\longrightarrow}\limits E_3$
and
$\varphi_3:E_3\mathop{\longrightarrow}\limits E_1$
such that
$\pi_2\circ\varphi_1=\pi_1'$,
$\pi_3\circ\varphi_2=\pi_2'$
and
$\pi_1\circ\varphi_3=\pi_3'$.
We identify
$E_1=E_2=E_3=:E$ and $\pi_1'=\pi_2,\pi_2'=\pi_3$ by means of $\varphi_1$ and
$\varphi_2$.
In this way $\varphi_3$ is identified with an
automorphism $t$ of $E$ such that $\pi_1\circ t=\pi_3'$.
Since the elliptic curve $E$ is determined by the intersection
points of $C_1,C_2$ and two Apollonius circles of $C_1,C_2,C_3$,
for general $C_1,C_2,C_3$ the curve $E$ is general
as well. Therefore the automorphism $t$ is either a translation
or an involution.
By a suitable choice of $\varphi_1$ and $\varphi_2$ we can achieve that $t$ is
actually a translation.
We denote by $\iota_1,\iota_2,\iota_3$ the covering involutions of
$\pi_1,\pi_2,\pi_3$ on $E$.
We have $S_1=\pi_1(e)$ for some $e\in E$.
Then $S_2\in\pi_1'\pi_1^{-1}(S_1)$, which means
$S_2=\pi_1'(\alpha_1e)$ where $\alpha_1\in\{1,\iota_1\}$.
In the same way we proceed with $S_3$ and $S_4$ to get
$$
S_4=\pi_3'(\alpha_3\alpha_2\alpha_1e) =
\pi_1(t\alpha_3\alpha_2\alpha_1e) \ ,
$$
where $\alpha_i\in\{1,\iota_i\}$ for $1\le i\le 3$.
Now we determine the circles $S_5,S_6,S_7$ such that
$(S_4,S_5),(S_5,S_6),(S_6,S_7)\in E$. We find
$$
S_7=\pi_3'(\beta_3\beta_2\beta_1t\alpha_3\alpha_2\alpha_1e) =
\pi_1(t\beta_3\beta_2\beta_1t\alpha_3\alpha_2\alpha_1e) \ ,
$$
where $\beta_i\in\{1,\iota_i\}$ for $1\le i\le 3$.
Now we make our choice for $S_5,S_6,S_7$ resp.\ for
$\beta_1,\beta_2,\beta_3$ in the following way.
We let $\beta_2:=\alpha_2$, $\beta_3:=\alpha_3$ and we
choose $\beta_1$ such that the number of subscripts $i$, $1\le i\le 3$,
such that $\beta_i=\iota_i$ is odd.
With these choices we obtain
$$
t\beta_3\beta_2\beta_1t\alpha_3\alpha_2\alpha_1e =
\beta_3\beta_2\beta_1\alpha_3\alpha_2\alpha_1e =
(\alpha_3\alpha_2\beta_1)^2\beta_1\alpha_1e \ .
$$
Since the composition of an odd number of involutions is again an
involution, the latter expression equals $\beta_1\alpha_1e$,
so $S_7=\pi_1(\beta_1\alpha_1e)=\pi_1(e)=S_1$.
\nopagebreak\hspace*{\fill
Finally, we aim at a Poncelet-type statement
in the situation of the Money-Coutts theorem. So suppose that
$C_1,C_2,C_3$ are three circles as above and let families $F_1,F_2,F_3$
be chosen as before. As in the proof of the Money-Coutts theorem
the elliptic curve defining the contact relation
between circles of two families will be denoted by $E$.
Let $(S_i)_{i\ge 1}$ be a sequence of circles such that
$(S_i,S_{i+1})\in E$ for $i\ge 1$ and
$$
S_{3k+l}\in F_l \qquad\mbox{ for } k\ge 0 \mbox{ and } 1\le l\le 3 \ .
$$
If it were to happen that $S_{3n+1}=S_1$ for some integer $n\ge 1$, then
the sequence is said to {\em close after $n$ steps}.
\begin{theorem}
Suppose there are more than $2^{3n+2}$ sequences
$(S_i)_{i\ge 1}$ closing after $n$ steps.
Then there are infinitely many sequences
closing after $n$ steps,
starting with any given
circle $S'_1\in F_1$.
\end{theorem}
\startproof{\it Proof. }
Let $S_1=\pi_1(e)$, $e\in E$. The circles $S_i$, $i\ge 2$, are determined
by repeatedly choosing automorphisms in $\{1,\iota_1,\iota_2,\iota_3\}$.
So for $k\ge 1$ we have
\begin{eqnarray*}
S_{3k+1} &=&
\pi_1\(\prod_{i=1}^k t\alpha_3^{(i)}\alpha_2^{(i)}\alpha_1^{(i)}\) \\
S_{3k+2} &=&
\pi_1'\(\alpha_1^{(k+1)}\prod_{i=1}^k
t\alpha_3^{(i)}\alpha_2^{(i)}\alpha_1^{(i)}\) \\
S_{3k+3} &=&
\pi_2'\(\alpha_2^{(k+1)}\alpha_1^{(k+1)}\prod_{i=1}^k
t\alpha_3^{(i)}\alpha_2^{(i)}\alpha_1^{(i)}\) \ ,
\end{eqnarray*}
where $t:E\mathop{\longrightarrow}\limits E$ is a translation and
$\alpha_j^{(i)}\in\{1,\iota_j\}$ for $1\le j\le 3$ and $i\ge 1$.
By assumption the sequence $(S_i)$ closes after $n$ steps, i.e.\
$S_{3n+1}=S_1$, i.e.\ $\pi_1(\gamma e)=\pi_1(e)$, where $\gamma$ is the
automorphism
$\prod_{i=1}^n t\alpha_3^{(i)}\alpha_2^{(i)}\alpha_1^{(i)}$.
The map $\gamma$ is
an involution or a translation on $E$,
so replacing $e$ by $\iota_1 e$ we may assume that $\gamma$
is an involution.
We have $\gamma(e)=e$ or $\gamma(e)=\iota_1(e)$.
In the first case $e$ is one of the four fixed points of $\gamma$.
Since there are $2^{3n}$ choices for the automorphisms
$\alpha_j^{(i)}$, $1\le j\le 3$, $1\le i\le n$, this case occurs
for at most $2^{3n+2}$ sequences. In the second case
we conclude $\gamma=\iota_1$, so if we are given any circle $S_1'\in F_1$,
$S_1'=\pi(e')$, then we define a sequence $(S_1')_{i\ge 1}$ by means
of the automorphisms $\alpha_j^{(i)}$ and find
$$
S_{3n+1}=\pi_1(\gamma(e'))=\pi_1(\iota_1(e'))=\pi_1(e')=S_1 \ .
$$
\nopagebreak\hspace*{\fill
|
1995-02-20T06:20:27 | 9502 | alg-geom/9502020 | en | https://arxiv.org/abs/alg-geom/9502020 | [
"alg-geom",
"math.AG"
] | alg-geom/9502020 | Rahul Pandharipande | R. Pandharipande | A Compactification Over $\overline{M}_g$ Of The Universal Moduli Space
of Slope-Semistable Vector Bundles | JAMS, to appear, AMSLatex 64 pages | null | null | null | null | A projective moduli space of pairs (C,E) where E is a slope- semistable
torsion free sheaf of uniform rank on a Deligne- Mumford stable curve C is
constructed via G.I.T. There is a natural SL x SL action on the relative Quot
scheme over the universal curve of the Hilbert scheme of pluricanonical, genus
g curves. The G.I.T. quotient of this product action yields a functorial,
compact solution to the moduli problem of pairs (C,E). Basic properties of the
moduli space are studied. An alternative approach to the moduli problem of
pairs has been suggested by D. Gieseker and I Morrison and completed by L.
Caporaso in the rank 1 case. It is shown the contruction given here is
isomorphic to Caporaso's compactification in the rank 1 case.
| [
{
"version": "v1",
"created": "Fri, 17 Feb 1995 20:49:37 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{Introduction}
\subsection{Compactifications of Moduli Problems}
\label{cmp}
Initial statements of moduli problems in algebraic geometry often
do not yield compact moduli spaces. For example, the moduli space
$M_g$ of nonsingular, genus $g\geq 2$ curves is open. Compact
moduli spaces are desired for several reasons. Degeneration arguments
in moduli require compact spaces. Also, there are more techniques
available to study the
global geometry of compact
spaces. It is therefore valuable to find
natural compactifications of open moduli problems. In the case of
$M_g$, there is a remarkable compactification due to P. Deligne and D. Mumford.
A connected, reduced, nodal curve $C$ of arithmetic genus $g\geq 2$
is {\em Deligne-Mumford stable}
if each nonsingular rational component contains at least three nodes of $C$.
$\overline{M_g}$, the moduli space of Deligne-Mumford stable genus $g$ curves, is
compact and includes $M_g$ as a dense open set.
There is a natural notion of stability for a vector bundle $E$ on a nonsingular
curve $C$.
Let the slope $\mu$ be defined as follows:
$\mu(E)=degree(E)/rank(E)$.
$E$ is {\em slope-stable (slope-semistable)} if
\begin{equation}
\label{ahab}
\mu(F)< (\leq)\ \mu(E)
\end{equation}
for every proper subbundle
$F$ of $E$.
When the degree and rank are not coprime, the moduli space of slope-stable
bundles is open. $U_{C}(e,r)$, the moduli space of slope-semistable bundles of
degree
$e$ and rank $r$, is compact. An open set of $U_{C}(e,r)$ corresponds
bijectively
to isomorphism classes of stable bundles. In general, points of $U_C(e,r)$
correspond
to equivalence classes (see section (\ref{sfor}) ) of semistable bundles.
The moduli problem of pairs $(C,E)$ where $E$ is a slope-semistable
vector bundle on a
nonsingular curve $C$ can not be compact. No allowance is made for curves
that degenerate to nodal curves. A natural compactification of this
moduli problem of pairs is presented here.
\subsection{Compactification of the Moduli Problem of Pairs}
Let $\bold{C}$ be a fixed algebraically closed field. As before,
Let $M_g$ be the moduli space of nonsingular, complete, irreducible,
genus $g\geq2$ curves over the field $\bold{C}$.
For each $[C]\in M_g$, there
is a natural projective variety, $U_C(e,r)$, parametrizing
degree $e$,
rank $r$,
slope-semistable vector bundles (up to equivalence) on $C$.
For $g\geq2$,
let $U_g(e,r)$ be the set of equivalence classes of pairs $(C,E)$
where $[C] \in M_g$
and $E$ is a slope-semistable vector bundle on $C$
of the specified degree and rank.
A good compactification, K, of the moduli set of pairs $U_g(e,r)$
should satisfy at least the following conditions:
\begin{enumerate}
\item [(i.)] $K$ is a projective variety that functorially
parametrizes a class of geometric objects.
\item [(ii.)] $U_g(e,r)$ functorially corresponds to an open
dense subset of $K$.
\item [(iii.)] There exists a morphism $\eta: K \rightarrow \barr{M_g}$
such that the natural diagram commutes:
\begin{equation*}
\begin{CD}
U_g(e,r) @>>> K\\
@VVV @VV{\eta}V \\
M_g @>>> \barr{M_g}
\end{CD}
\end{equation*}
\item [(iv.)] For each $[C]\in M_g$, there exists a functorial
isomorphism $$\eta^{-1}([C]) \cong U_C(e,r)/Aut(C).$$
\end{enumerate}
The main result of this paper is the construction of a
projective variety $\overline{U_g(e,r)}$ that parametrizes
equivalence classes of slope-semistable, torsion free
sheaves on Deligne-Mumford stable, genus $g$ curves and
satisfies conditions (i-iv) above.
The definition of slope-semistability of torsion free sheaves
(due to C. Seshadri) is given in section (\ref{sfor}).
\subsection{The Method of Construction}
\label{moc}
An often successful approach to moduli constructions in algebraic
geometry involves two steps. In the first step, extra data
is added to rigidify the moduli problem. With the additional
data, the new moduli problem is solved by a Hilbert or Quot
scheme. In the second step, the extra data is removed by a group
quotient. Geometric Invariant Theory is used to study the
quotient problem in the category of algebraic schemes. In good cases,
the final quotient is the desired moduli space.
In order to rigidify the moduli problem of pairs, the following
data is added to $(C,E)$:
\begin{enumerate}
\item[(i.)] An isomorphism
$\bold{C}^{N+1} \stackrel {\sim}{\rightarrow} H^0(C, \omega_C^{10})$,
\item[(ii.)] An isomorphism
$\bold{C}^n \stackrel{\sim}{\rightarrow} H^0(C,E)$.
\end{enumerate}
Note $\omega_C$ is the canonical bundle of $C$.
The numerical invariants of the moduli problem of pairs are
the genus $g$, degree $e$, and rank $r$. The rigidified
problem should have no more numerical invariants. Hence,
$N$ and $n$ must be determined by $g$, $e$, and $r$.
Certainly, $N=10(2g-2)-g$ by Riemann-Roch. It is assumed
$H^1(E)=0$ and $E$ is generated by global sections.
In the end, it is checked these assumptions are consequences of the
stability condition for sufficiently high degree bundles.
We see $n=\chi(E)= e+r(1-g)$.
The isomorphism
of (i) canonically embeds $C$ in $\bold P^{N}=\bold P^{N}_{\bold C}$.
The isomorphism of (ii) exhibits $E$ as a canonical quotient
$$ \bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0.$$
The basic parameter spaces in algebraic geometry are Hilbert
and Quot schemes.
Subschemes of a fixed scheme $X$ are parametrized by Hilbert schemes
$Hilb(X)$.
Quotients of a fixed sheaf $F$ on $X$ are parametrized by Quot schemes
$Quot(X,F)$.
The rigidified curve $C$ can be parametrized by a Hilbert
scheme $H$ of curves in $\bold P^N$, and the rigidified bundle $E$ can
be parametrized by a Quot scheme $Quot(C,\bold{C}^n\otimes {\cal{O}}_C)$
of quotients on $C$.
In fact, Quot schemes can be defined in a relative context. Let $U_H$ be
the universal curve over the Hilbert scheme $H$.
The family of Quot schemes, $Quot(C,\bold{C}^n\otimes {\cal{O}}_C)$,
defined as $C\hookrightarrow \bold P^N$
varies in $H$ is simply the relative
Quot scheme of the universal curve over the Hilbert scheme:
$Quot(U_H\rightarrow H, \bold{C}^n\otimes {\cal{O}}_{U_H})$. This relative
Quot scheme is the parameter space of the rigidified pairs (up to
scalars).
The Quot scheme set up is discussed in detail in section (\ref{rqs}).
The actions of $GL_{N+1}(\bold{C})$ on $\bold{C}^{N+1}$ and $GL_{n}(\bold{C})$
on $\bold{C}^n$ yield an action of $GL_{N+1}\times GL_{n}$ on the
rigidified data.
There is an induced product action on the relative Quot scheme.
It is easily seen the scalar elements of the groups act
trivially on the Quot scheme. $\overline{U_g(e,r)}$ is constructed via the
quotient:
\begin{equation}
\label{tpg}
Quot(U_H\rightarrow H, \bold{C}^n\otimes {\cal{O}}_{U_{H}}) / SL_{N+1}\times
SL_n.
\end{equation}
There is a projection of the rigidified problem of
pairs $\{ (C,E)$ with isomorphisms (i) and (ii)$\}$ to a
rigidified moduli problem of curves $\{ C$ with isomorphism (i)$\}$.
The projection is $GL_{N+1}$-equivariant with respect to
the natural $GL_{N+1}$-action on the rigidified data of curves.
The Hilbert scheme $H$ is
a parameter space of the rigidified problem of curves (up to scalars).
By results of
Gieseker ([Gi]) reviewed in section (\ref{geese}),
the quotient $H/SL_{N+1}$ is
$\overline{M_g}$. A natural morphism $\overline{U_g(e,r)} \rightarrow \overline{M_g}$
is therefore obtained.
Gieseker's results require the choice in isomorphism (i) of at least
the $10$-canonical series.
The technical heart of the paper is the study of the Geometric
Invariant Theory problem (\ref{tpg}).
The method is divide and conquer. The action of
$SL_n$ alone is first studied. The $SL_n$-action is called the
fiberwise G.I.T. problem. It is solved in sections (\ref{puppy}) -
(\ref{lena}).
The action of $SL_{N+1}$ alone is then considered. There are
two pieces in the study of the $SL_{N+1}$ action. First, Gieseker's results
in [Gi] are used in an essential way. Second, the abstract
G.I.T. problem of
$SL_{N+1}$ acting on $\bold P(Z)\times \bold P(W)$ where $Z$, $W$
are representations of $SL_{N+1}$ is studied. If the linearization
is taken to be ${\cal{O}}_{\bold P(Z)}(k)\otimes {\cal{O}}_{\bold P(W)}(1)$ where
$k>>1$, there are elementary set theoretic
relationships between the stable and
unstable loci of $\bold P(Z)$ and $\bold P(Z)\times \bold P(W)$. These
relationships are determined in section (\ref{abgit}). Roughly speaking,
the abstract lemmas are used to import the invariants Gieseker
has determined in [Gi] to the problem at hand.
In section (\ref{conlo}), the
solution of fiberwise problem is combined with the study of the
$SL_{N+1}$-action
to solve the product G.I.T. problem (\ref{tpg}).
\subsection{Relationship with Past Results}
In [G-M],
D. Gieseker and I. Morrison propose
a different approach to a compactification over
$\overline{M_g}$ of the universal
moduli space of slope-semistable bundles.
The moduli problem of pairs is rigidified by adding only the
data of an isomorphism
$\bold{C}^n \rightarrow H^0(C,E)$.
By further assumptions on $E$,
an embedding into a Grassmanian is obtained
$$C \hookrightarrow \bold G(rank(E),
H^0(C,E)^*).$$ The rigidified data is thus parametrized by a
Hilbert scheme of a Grassmanian. The $GL_n$-quotient problem is
studied to obtain a moduli space of pairs.
Recent progress along this alternate path has been made by
D. Abramovich, L. Caporaso, and M. Teixidor ([A], [Ca], [T]).
A compactification, $\overline{P_{g,e}}$, of the universal
Picard variety
is constructed in [Ca]. There is a natural isomorphism
$$\nu: \overline{P_{g,e}} \rightarrow \overline{U_g(e,1)}.$$
This isomorphism is established in section (\ref{lcap}).
In the rank $2$ case, the approach of [G-M] yields
a compactification not equivalent to $\overline{ U_g(e,2)}\ $ ([A]).
The higher rank constructions of [G-M] have not been completed. They
are certainly expected to differ from $\overline{U_g(e,r)}$.
\subsection{Acknowledgements}
The results presented here constitute the author's 1994 Harvard
doctoral thesis.
It is a pleasure to
thank D. Abramovich and J. Harris for
introducing the author to
the higher rank compactification
problem. Conversations
with S. Mochizuki on issues
both theoretical and technical have been of enormous
aid. The author has also benefited from discussions
with L. Caporaso, I. Morrison, H. Shahrouz, and M. Thaddeus.
\section{The Quotient Construction}
\subsection{Definitions}
\label{sfor}
Let $C$ be a genus $g\geq2$, Deligne-Mumford stable
curve. A coherent sheaf $E$ on $C$ is
{\em torsion free} if $$\forall x\in C, \ \ depth_{{\cal{O}}_x}(E_x)=1,$$ or
equivalently, if there does not exist a subsheaf
$$0\rightarrow F \rightarrow E $$
such that $dim(Supp(F))=0$. Let
$$C=\bigcup_{1}^{q}C_i $$
where the curves $C_i$ are the irreducible components of $C$.
Let $\omega_i$ be the degree of the restriction of
the canonical bundle $\omega_C$ to $C_i$.
Let
$r_i$ be the the rank of $E$ at the generic point of $C_i$.
The {\em multirank} of $E$ is the $q$-tuple $(r_1, \ldots, r_q)$.
$E$ is of {\em uniform rank} $r$ if $r_i=r$ for each
$C_i$. If $E$ is of uniform rank $r$, define the degree of
$E$ by
$$e=\chi(E) - r(1-g).$$
A torsion free sheaf $E$ is defined
to be {\em slope-stable (slope-semistable)}
if for each nonzero, proper subsheaf
$$0 \rightarrow F \rightarrow E $$
with multirank $(s_1, \ldots, s_q)$, the following inequality holds:
\begin{equation}
\label{bhab}
{\chi(F)\over \sum_{1}^{q}s_i \omega_i} <
{\chi(E)\over \sum_{1}^{q}r_i\omega_i},
\end{equation}
respectively,
$$\paren{{\chi(F)\over \sum_{1}^{q}s_i \omega_i} \leq
{\chi(E)\over \sum_{1}^{q}r_i\omega_i}}.$$
The above is Seshadri's definition of slope-(semi)stability
in the case of canonical polarization. In case $E$ is a vector
bundle on a nonsingular curve $C$, the slope-(semi)stability
condition (\ref{ahab}) of section (\ref{cmp}) and condition (\ref{bhab})
above coincide.
A slope-semistable sheaf has a
Jordan-Holder filtration with slope-stable factors.
Two
slope-semistable sheaves are {\em equivalent} if
they possess the same set of Jordan-Holder factors.
Two equivalence classes are said to be {\em aut-equivalent} if
they differ by an automorphism of the underlying curve $C$.
For $g\geq2$ and each pair of integers $(e, \ r\geq1)$,
a projective variety $\overline{U_g(e,r)}$ and a
morphism
$$\eta: \overline{U_g(e,r)} \rightarrow \barr{M_g}$$
satisfying the following properties are constructed in
Theorem (\ref{priya}).
There is a functorial, bijective correspondence between the points
of $\overline{U_g(e,r)}$ and aut-equivalence classes of
slope-semistable, torsion free
sheaves of uniform rank $r$ and degree $e$ on
Deligne-Mumford stable curves of genus $g$.
The image of an aut-equivalence class under $\eta$ is the moduli
point of the underlying curve.
$\overline{U_g(e,r)}$ and $\eta$ will be
constructed via Geometric Invariant Theory. The G.I.T. problem
is described in sections (\ref{geese}-\ref{linto}).
The solution is developed in sections (\ref{puppy}-\ref{conlo}) of the
paper.
Basic properties of
$\overline{U_g(e,r)}$ are studied in section (\ref{laslo}).
In particular, the equivalence of $\overline{U_g(e,1)}$ and
$\overline{P_{g,e}}$ is established in section (\ref{ender}).
\subsection{Gieseker's Construction}
\label{geese}
We review Gieseker's beautiful construction of $\overline{M_g}$.
Fix a genus $g \geq 2$. Define:
$$ d= 10(2g-2) $$
$$ N=d-g .$$
Consider the Hilbert scheme $H_{g,d,N}$
of genus $g$, degree $d$, curves in
$\bold P^N_{\bold{C}}$. Let
$$H_g \subset H_{g,d,N}$$ denote the locus of nondegenerate,
$10$-canonical,
Deligne-Mumford stable curves of genus g. $H_g$ is naturally a
closed subscheme of the open locus of nondegenerate, reduced, nodal
curves. In fact,
$H_g$ is
a nonsingular, irreducible, quasi-projective variety ([Gi]).
The symmetries of $\bold P^N$ induce a natural $SL_{N+1}(\bold{C})$-action
on $H_g$.
D. Gieseker has studied the quotient $H_g/SL_{N+1}$ via
geometric invariant theory. It is shown in [Gi] that,
for suitable linearizations,
$H_g/SL_{N+1}$ exists as a G.I.T. quotient and is
isomorphic to $\barr{M_g}$.
\subsection{Relative Quot Schemes}
\label{rqs}
Let $U_H$ be the universal curve over
$H_g$. We have a closed immersion
$$ U_H \hookrightarrow H_g \times \bold P^N $$ and
two projections:
$$\mu : U_H \rightarrow H_g$$
$$\nu: U_H \rightarrow \bold P^N .$$
Let ${\cal{O}}_U$ be the structure sheaf of $U_H$.
The
Grothendieck relative Quot scheme is central to our
construction. We will be interested
in relative Quot schemes of the form
\begin{equation}
\label{qu}
Quot(\mu:U_H \rightarrow H_g,\ \bold{C}^n \otimes {\cal{O}}_U,\ \nu^*({\cal{O}}_{\bold P^N}(1)),\ f)
\end{equation}
where $f$ is a Hilbert polynomial
with respect to the $\mu$-relatively very ample line
bundle $ \nu^*({\cal{O}}_{\bold P^N}(1))$.
We denote the Quot scheme in (\ref{qu}) by
$Q_g(\mu,n,f)$.
We recall the basic properties of the Quot scheme.
There is a canonical projective morphism
$ \pi: Q_g(\mu,n,f) \rightarrow H_g.$
The fibered product $Q_g(\mu,n,f) \times_{H_g} U_H$
is equipped with two projections:
$$ \theta : Q_g(\mu,n,f) \times_{H_g} U_H \rightarrow Q_g(\mu,n,f)$$
$$ \phi : Q_g(\mu,n,f) \times_{H_g} U_H \rightarrow U_H $$
and a universal $\theta$-flat quotient
\begin{equation}
\label{uny}
\bold{C}^n \otimes {\cal{O}}_{Q\times U} \simeq \phi^* (\bold{C}^n \otimes {\cal{O}}_U) \rightarrow \cal{E}
\rightarrow 0.
\end{equation}
Let $\xi$ be a (closed) point of $Q_g(\mu,n,f)$.
The point $\pi(\xi) \in H_g$
corresponds to the $10$-canonical, Deligne-Mumford stable
curve $U_{\pi(\xi)}$. Restriction of the universal quotient
sequence (\ref{uny}) to $U_{\pi(\xi)}$ yields
a quotient
$$\bold{C}^n \otimes {\cal{O}}_{U_{\pi(\xi)}}\rightarrow \cal{E}_\xi \rightarrow 0$$ with
Hilbert polynomial
$$f(t)=\chi(\cal{E}_\xi\otimes {\cal{O}}_{U_{\pi(\xi)}}(t)).$$
The above is a functorial bijective correspondence between points
$\xi \in Q_g(\mu,n,f)$ and quotients
of $\bold{C}^n \otimes {\cal{O}}_{U_{\pi(\xi)}}$, $\pi(\xi) \in H_g$, with Hilbert polynomial
$f$.
\subsection{Group Actions}
Denote the natural actions of $SL_{N+1}$ on $H_g$ and
$U_H$ by:
\begin{equation*}
\begin{CD}
U_H\times SL_{N+1} @>{a_U}>> U_H\\
@VV{\mu\times id}V @VV{\mu}V \\
H_g \times SL_{N+1} @>{a_H}>> H_g
\end{CD}
\end{equation*}
Also define:
$$\overline{\mu}: U_H \times SL_{N+1}
\stackrel{\mu \times inv}{\longrightarrow} H_g \times SL_{N+1}
\stackrel{a_H} {\longrightarrow} H_g.$$
$$\overline{\pi}:Q_g(\mu,n,f) \times SL_{N+1}
\stackrel{\pi \times id}{\longrightarrow} H_g \times SL_{N+1}
\stackrel{a_H}{\longrightarrow} H_g.$$
There is a natural isomorphism between the two
fibered products
$$\big(Q_g(\mu,n,f) \times SL_{N+1}\big) \times _{H_g} U_H \simeq
Q_g(\mu,n,f) \times _{H_g} \big( U_H \times SL_{N+1}\big)$$
where the projection maps to $H_g$ are $(\overline{\pi}, \mu)$ and
$(\pi, \overline{\mu})$ in the first and second products
respectively. The inversion in the definition of $\overline{\mu}$ is
required for the isomorphism of the fibered products.
There is a natural commutative diagram
\begin{equation*}
\begin{CD}
U_H\times SL_{N+1} @>{\overline{a}_U}>> U_H\\
@VV{\overline{\mu}}V @VV{\mu}V \\
H_g @>{id}>> H_g
\end{CD}
\end{equation*}
where $$\overline{a}_U: U_H \times SL_{N+1}
\stackrel{id \times inv} {\longrightarrow} U_H \times SL_{N+1}
\stackrel{a_U} {\rightarrow} U_H.$$
We therefore obtain a natural map of schemes over $\bold{C}$:
$$\varrho:(Q_g(\mu,n,f) \times SL_{N+1}) \times _{H_g} U_H \rightarrow Q_g(\mu,n,f) \times_{H_g} U_H.$$
By the functorial properties of $Q_g(\mu,n,f)$, the
$\varrho$-pull-back of the universal quotient sequence (\ref{uny}) on
$Q_g(\mu,n,f) \times_{H_g} U_H$ yields a natural group action:
$$Q_g(\mu,n,f) \times SL_{N+1} \rightarrow Q_g(\mu,n,f) .$$
Hence,
the natural $SL_{N+1}$-action
on $H_g$ lifts naturally to $Q_g(\mu,n,f)$. There is
a natural $SL_n(\bold{C})$-action on $Q_g(\mu,n,f)$ induced by the
$SL_n(\bold{C})$-action on the tensor product $\bold{C}^n \otimes {\cal{O}}_U$. In fact,
the $SL_{N+1}$-action
and the $SL_n$-action commute on $Q_g(\mu,n,f)$.
The commutation is most easily seen in the explicit linearized
projective embedding developed below in section (\ref{said}).
Hence, there exists
a well-defined $SL_{N+1} \times SL_n$-action.
For suitable choices of
$n$, $f$, and linearization, a component of the
quotient $Q_g(\mu,n,f)/(SL_{N+1} \times SL_n)$
will be $\overline{U_g(e,r)}$.
\subsection{Relative Embeddings}
\label{van}
Following [Gr], a family of
relative projective embeddings of $Q_g(\mu,n,f)$ over $H_g$ is constructed.
Since the inclusion
$$Q_g(\mu,n,f) \times_{H_g} U_g \hookrightarrow Q_g(\mu,n,f) \times \bold P ^N$$
is a closed immersion,
the universal quotient $\cal{E}$ can be extended by zero to
$Q_g(\mu,n,f) \times \bold P^N$.
Let $${\theta}_{\bold P}: Q_g(\mu,n,f) \times\bold P^N \rightarrow Q_g(\mu,n,f)$$ be the
projection. The universal quotient sequence (\ref{uny})
induces the following sequence on $Q_g(\mu,n,f) \times \bold P^N$:
$$ 0 \rightarrow \cal{K} \rightarrow \bold{C}^n \otimes {\cal{O}}_{Q \times \bold P^N} \rightarrow \cal{E} \rightarrow 0 .$$
Since $\cal{E}$ and ${\cal{O}}_{Q \times \bold P^N}$ are ${\theta}_{\bold P}$-flat,
$\cal{K}$ is ${\theta}_{\bold P}$-flat. By the semicontinuity theorems for ${\theta}_{\bold P}$-flat,
coherent sheaves, there exists an integer $t_{\alpha}$ such that for
each $t >t_{\alpha}$ and each $\xi \in Q_g(\mu,n,f)$ :
\begin{equation}
\label{cat}
h^1(\bold P^N,\cal{K}_{\xi}\otimes {\cal{O}}_{\bold P^N}(t))=0
\end{equation}
\begin{equation}
\label{dog}
h^0(\bold P^N,\cal{E}_{\xi}\otimes {\cal{O}}_{\bold P^N}(t))=f(t)
\end{equation}
\begin{equation}
\label{skunk}
h^1(\bold P^N,\cal{E}_{\xi}\otimes {\cal{O}}_{\bold P^N}(t))=0
\end{equation}
\begin{equation}
\label{ppppp}
\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t)) \rightarrow H^0(\bold P^N, \cal{E}_{\xi}\otimes
{\cal{O}}_{\bold P^N}(t)) \rightarrow 0.
\end{equation}
The surjection of (\ref{ppppp})
follows from (\ref{cat}) and the
long exact cohomology sequence.
For each $t >t_{\alpha}$ there is a
well defined algebraic morphism (on points)
$$i_t: Q_g(\mu,n,f) \rightarrow \bold G(f(t),
(\bold{C}^n \otimes H^0(\bold P^N, {\cal{O}}_{\bold P^N}(t)))^*)$$
defined by sending $\xi \in Q_g(\mu,n,f)$ to the subspace
$$ H^0(\bold P^N, \cal{E}_\xi \otimes {\cal{O}}_{\bold P^N}(t))^* \subset
(\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t)))^*.$$
By the theorems of Cohomology and Base Change,
it follows from conditions (\ref{cat}-\ref{skunk})
there exists
a surjection
$$\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t))\otimes
{\cal{O}}_Q \simeq \theta_{\bold P *}(\bold{C}^n \otimes {\cal{O}}_{Q \times \bold P^N}(t))
\rightarrow \theta_{\bold P *}(\cal{E}\otimes {\cal{O}}_{\bold P^N}(t)) \rightarrow 0$$
where
$\theta_{\bold P *}(\cal{E}\otimes {\cal{O}}_{\bold P^N}(t))$ is a locally free, rank $f(t)$
quotient.
The universal property of the Grassmanian
defines $i_t$ as a morphism of schemes.
It is known that there exists an integer $t_{\beta}$ such that
for all $t > t_{\beta}$, the morphism $\pi\times i_t$ is a closed
embedding:
$$\pi \times i_t: Q_g(\mu,n,f) \rightarrow H_g \times \bold G(f(t),
(\bold{C}^n \otimes H^0(\bold P^N, {\cal{O}}_{\bold P^N}(t)))^*).$$
The morphisms $\pi \times i_t, \ t>t_{\beta}(g,n,f)$,
form a countable family of relative projective embeddings
of the Quot scheme $Q_g(\mu,n,f)$ over $H_g$.
\subsection{Gieseker's Linearization}
\label{said}
Since the Hilbert scheme $H_{d,g,N}$ is the Quot scheme
$$Quot(\bold P^N \rightarrow Spec(\bold C),\ {\cal{O}}_{\bold P^N}, {\cal{O}}_{\bold P^N}(1),\
h(s)=ds-g+1),$$
there
are closed embeddings for $s >s_{\alpha}$:
$$i'_s: H_{d,g,N} \rightarrow \bold G(h(s),
H^0(\bold P^N,{\cal{O}}_{\bold P^N}(s))^*).$$
By results of Gieseker, an integer $\overline{s}(g)$ can be chosen so
that the $SL_{N+1}$-linearized G.I.T. problem determined by $i'_{\overline{s}}$ has
two properties:
\begin{enumerate}
\item [(i.)] $H_g$ is contained in the stable locus.
\item [(ii.)] $H_g$ is closed in the semistable locus.
\end{enumerate}
In order to make use of (i) and (ii) above, we will only consider
immersions of the type $i'_{\overline{s}}$.
For each large $t$, there exists
an immersion:
$$i_{\overline{s},t}: Q_g(\mu,n,f) \rightarrow \bold G(h(\overline{s}),
H^0(\bold P^N,{\cal{O}}_{\bold P^N}(\overline{s}))^*)
\times
\bold G(f(t),
(\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t)))^*). $$
By the Pl\"ucker embeddings, we obtain
$$j_{\overline{s},t}: Q_g(\mu,n,f) \rightarrow
\bold P(\bigwedge^{h(\overline{s})}H^0(\bold P^N,{\cal{O}}_{\bold P^N}(\overline{s}))^*)
\times \bold P(\bigwedge^{f(t)}(\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t)))^*).$$
The fact that the $SL_{N+1}$ and $SL_{n}$-actions commute
on $Q_g(\mu,n,f)$ now follows from the observation that
the $SL_{N+1}$ and $SL_{n}$-actions commute on
$ \bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t))^*$.
\subsection{The G.I.T. Problem}
\label{linto}
Let $C$ be a Deligne-Mumford stable curve of genus $g\geq 2$.
For any coherent sheaf $F$ on $C$, it is not hard to see:
\begin{equation}
\label{oiler}
\chi(F\otimes \omega_C^t) = \chi(F) + (\sum_{1}^{q}s_i\omega_i)\cdot t
\end{equation}
where $(s_1,\ldots,s_q)$ is the multirank of $F$ ([Se]).
Equation (\ref{oiler}) and the slope inequalities
of section (\ref{sfor}) yield
a natural correspondence
$$(C,E) \rightarrow (C, E \otimes \omega_C^t)$$
between slope-semistable, uniform rank $r$, torsion free sheaves
of degrees $e$ and $e+rt(2g-2)$. Therefore, it suffices to construct
$\overline{U_g(e,r)}$ for $e>>0$.
The strategy for studying the G.I.T. quotient
$$Q_g(\mu, n,f)/(SL_{N+1} \times SL_n)$$ is as follows.
First a rank $r\geq1$ is chosen. Then the degree $e>e(g,r)$ is chosen
very large. The Hilbert polynomial
is determined by:
\begin{equation}
\label{goat}
f_{e,r}(t)= e+ r(1-g) + r10(2g-2)t.
\end{equation}
For $[C]\in H_g$,
$f_{e,r}(t)$ is
the Hilbert polynomial of degree $e$, uniform rank $r$, torsion free
sheaves on $C$ with respect to ${\cal{O}}_{\bold P^N}(1)$.
The integer $n$ is fixed by the Euler characteristic, $n=f_{e,r}(0)$.
As remarked in section (\ref{moc})
of the introduction, $n$ will equal $h^0(C,E)$
for semistable pairs.
Let
$$\hat{t}(g,r,e)\ =\ t_{\beta}(g,n=f_{e,r}(0), f_{e,r})$$
be the
constant defined in section (\ref{van}) for $Q_g(\mu, f_{e,r}(0), f_{e,r})$.
A very large $t>\hat{t}(g,r,e)$
is then chosen. Selecting $e$ and $t$ are the essential choices that
make the G.I.T. problem well-behaved. Finally, to
determine a linearization of the
$SL_{N+1}\times SL_{n}$-action on the image of $j_{\overline{s},t}$, weights
must be chosen on the two projective spaces. These are chosen
so that almost all the weight is on the first,
\begin{equation}
\label{ffact}
\bold P(\bigwedge^{h(\overline{s})}H^0(\bold P^N,{\cal{O}}_{\bold P^N}(\overline{s}))^*).
\end{equation}
Since $SL_n$ acts only on the second factor of the product, the weighting
is irrelevant to the G.I.T. problem for the $SL_n$-action alone. The $SL_n$
action is studied in sections (\ref{puppy})-(\ref{lena}).
Since $SL_{N+1}$ acts on both factors,
the weighting is very relevant to the $SL_{N+1}$- G.I.T. problem. General
results of section (\ref{abgit})
show that in the case of extreme weighting, information
on the stable and unstable loci of the $SL_{N+1}$-action on
first factor can be transferred to the $SL_{N+1}$-action on the
product of the factors. Gieseker's study of the $SL_{N+1}$-action on the
first factor (\ref{ffact}) can therefore be used. In section (\ref{conlo}),
knowledge
of the $SL_n$ and $SL_{N+1}$ G.I.T problems is combined to solve the
$SL_{N+1}\times SL_n$ G.I.T.
problem on $Q_g(\mu,n,f)$.
\section{The Fiberwise G.I.T. Problem}
\label{puppy}
\subsection{The Fiberwise Result}
The fiber of $\pi :Q_g(\mu,n,f) \rightarrow H_g$ over a point $[C] \in H_g$ is
the Quot scheme
$$Q_g(C,n,f)=Quot(C \rightarrow Spec(\bold{C}),\ \bold{C}^n \otimes {\cal{O}}_C ,\ \omega_C^{10},\ f).$$
For large $t$, the morphism $i_t$ embeds $Q_g(C,n,f$) in
$$\bold G(f(t), (\bold{C}^n \otimes Sym^t(H^0(C,\omega_C^{10})))^*).$$
The embedding $i_t$ yields an $SL_n$-linearized G.I.T. problem
on $Q_g(C,n,f).$ Before examining the global G.I.T. problem
for the construction of $\overline{U_g(e,r)},$ we will study this
fiberwise G.I.T. problem. The main result is:
\begin{tm}
\label{fred}
Let $g\geq2 $, $r>0$ be integers. There exist bounds $e(g,r)>r(g-1)$ and
$t(g,r,e)>\hat{t}(g,r,e)$ such that for each pair $e>e(g,r)$,
$t>t(g,r,e)$ and any $[C]\in H_g$, the following holds:
\noindent
A point $\ \xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$
corresponding to a
quotient $$ \bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
is G.I.T. stable (semistable) for the $SL_n$-linearization determined
by $i_t$ if and only if
$E$ is a slope-stable (slope-semistable), torsion free
sheaf on $C$ and
$$\psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$$
is an isomorphism.
\end{tm}
P. Newstead has informed the author that
a generalization of this fiberwise G.I.T. problem
has been solved recently by C. Simpson in [Si]. A slight twist in our
approach is the uniformity of bound needed for each
$[C] \in H_g$. The proof will be developed in many steps.
\subsection{The Numerical Criterion}
\label{nnmc}
Stability for a point $\xi$ in a linearized G.I.T. problem
can be checked by examining certain limits of $\xi$ along
$1$-parameter subgroups. This remarkable fact leads to the Numerical
Criterion. The most general form of the Numerical Criterion is presented in
section (\ref{nmc}). A more precise version for the fiberwise G.I.T. problem
is stated here.
Fix a vector space $Z$ with the
trivial $SL_n$-action. In the applications below,
$$Z \stackrel{\sim}{=} Sym^t(H^0(C,\omega_C^{10})).$$
Consider the linearized $SL_n$-action on $\bold G(k, (\bold{C}^n \otimes
Z)^*)$
obtained from the standard representation of $SL_n$ on $\bold{C}^n$.
Let $\xi \in \bold G(k, (\bold{C}^n \otimes Z)^*).$
The element $\xi$ corresponds to a
$k$-dimensional quotient
$$ \rho_{\xi}: \bold{C}^n \otimes Z \rightarrow K_{\xi}.$$
Let $\overline{v} =( v_1, \ldots ,v_n)$ be a basis of
$\bold{C}^n$ with integer
weights
$\overline{w} =(w(v_1), \ldots ,w(v_n) )$.
For combinatorial convenience,
the additional condition that the weights sum to zero
is avoided here.
The representation weights of the corresponding $1$-parameter subgroup
of $SL_n$ are given by rescaling: $e_i=w(v_i)- \sum_{i} w(v_i)/n$.
An element $a \in \bold{C}^n \otimes Z$ is
said to be
{\em $\overline{v}$-pure} if it lies in a subspace of the form
$v_i\otimes Z$. The weight, $w(a)$, of such an element is defined to be
$w(v_i)$.
The Numerical Criterion yields:
\begin{enumerate}
\item $\xi$ is unstable if and only if there exists a
basis $\overline{v}$ of $\bold{C}^n$ and weights
$\overline{w}$ with the
following property. For any $k$-tuple of $\overline{v}$-pure
elements $(a_1, \ldots, a_k ) $ such that
$(\rho_{\xi}(a_1), \ldots, \rho_{\xi}(a_k) )$ is
a basis of $K_{\xi}$, the inequality
$$\sum_{i=1}^{n} {w(v_i)\over n} < \sum_{j=1}^{k}{w(a_j)\over k} $$
is satisfied.
\item$\xi$ is stable (semistable) if and only if
for every basis $\overline{v}$ of $\bold{C}^n$ and any nonconstant weights
$\overline{w}$ the following holds. There exist
$\overline{v}$-pure
elements $(a_1, \ldots, a_k )$ such that
$( \rho_{\xi}(a_1), \ldots, \rho_{\xi}(a_k) )$ is
a basis of $K_{\xi}$ and
$$\sum_{i=1}^{n} {w(v_i)\over n} \ >(\geq) \ \sum_{j=1}^{k}{w(a_j)\over k}.$$
\end{enumerate}
See, for example, [N] or [M-F].
\subsection{Step I}
The instability arguments will use the following Lemma.
\begin{lm}
\label{archy}
Let $g\geq2$, $r>0$, $e>r(g-1)$ be integers, $[C]\in H_g$.
Suppose $\xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$ corresponds to a
quotient $$ \bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0.$$
Let $U\subset \bold{C}^n$ be a subspace. Let
$\psi(U\otimes H^0(C,{\cal{O}}_C))= W \subset H^0(C,E)$.
Let $G$ be the subsheaf of $E$ generated by $W$.
For any $t>\hat{t}(g,e,r)$ the following holds: if
\begin{equation}
\label{zara}
{dim(U) \over n} > {h^0(C, G\otimes \omega_C^{10t}) \over f_{e,r}(t)},
\end{equation}
then $\xi$ is G.I.T. unstable for the $SL_n$-linearization
determined by $i_t$.
\end{lm}
\begin{pf}
Let $u=dim(U)$.
Inequality (\ref{zara}) implies $0<u<n$. Let $\overline{v}$ be a basis
of $\bold{C}^n$ such that $(v_1,\ldots, v_u)$ is a basis of $u$.
Select weights as follows: $w(v_i)=0$ for $1\leq i \leq u$ and
$w(v_i)=1$ for $u+1\leq i \leq n$. We now use the Numerical Criterion
for the $SL_n$-action on
$\bold G(f_{e,r}(t),(\bold{C}^n \otimes Sym^t(H^0(C,\omega_C^{10})))^*)$. The element
$\xi$ corresponds to a quotient
$$\psi^t:\bold{C}^n \otimes Sym^t(H^0(C,\omega_C^{10})) \rightarrow H^0(C,E\otimes \omega_C^{10t})
\rightarrow 0.$$
Suppose $(a_1,\ldots, a_{f_{e,r}(t)})$ is a tuple of $\overline{v}$-pure
elements mapped by $\psi^t$ to a basis of $H^0(C,E\otimes \omega_C^{10t})$.
All $a_j$'s have weight $1$ except those contained in $U\otimes
Sym^t(H^0(C,\omega_C^{10}))$ which have weight $0$. The number
of $a_j$'s of weight $0$ is hence bounded by $h^0(C,G\otimes \omega_C^{10t})$.
Since
$$\sum_{i=1}^{n}{w(v_i)\over n}= 1-{u\over n} < 1-
{h^0(C,G\otimes \omega_C^{10t}) \over f_{e,r}(t)} \leq \sum_{j=1}^{f_{e,r}(t)}
{w(a_j)\over f_{e,r}(t)},$$
the Numerical Criterion implies $\xi$ is unstable.
\end{pf}
\begin{pr}
\label{wk1}
Let $g\geq2$, $r>0$ be integers. For each pair $e>r(g-1)$,
$t>\hat{t}(g,r,e)$ and any $[C]\in H_g$, the following holds:
\noindent
If $\xi\in Q_g(C,n=f_{e,r}(0) f_{e,r})$ corresponds to a quotient
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $\psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$ is not injective,
then $\xi$ is G.I.T. unstable for the $SL_n$-linearization determined
by $i_t$.
\end{pr}
\begin{pf}
Suppose $\psi$ is not injective.
Let $U\otimes H^0(C,{\cal{O}}_C)$ be the nontrivial kernel of $\psi$.
The assumptions of Lemma (\ref{archy}) are easily checked since
$W=0$ and $G$ is the zero sheaf. $\xi$ is G.I.T unstable by
Lemma (\ref{archy}).
\end{pf}
\section{Cohomology Bounds}
\label{bondy}
\subsection{The Bounds}
In order to further investigate the fiberwise $SL_n$-action,
we need to control the first cohomology in various ways.
\begin{lm}
\label{bat}
Let $g\geq 2$, $R>0$ be integers. There exists an
integer $b(g,R)$ with the following property. If
$E$ is a coherent sheaf
on a Deligne-Mumford stable, genus $g$ curve $C$ such that:
\begin{enumerate}
\item[(i.)] $E$ is generated
by global sections.
\item[(ii.)] $E$ has generic rank less than $R$ on each irreducible
component of $C$.
\end{enumerate}
Then
$h^1(C,E) < b(g,R).$
\end{lm}
\begin{pf}
Since $\omega_C$ is ample and of degree $2g-2$,
there is a bound $q(g)=2g-2$ on the number of components of $C$.
Since $E$ is generated by global sections and has bounded
rank, there exists an exact sequence:
$$ \bigoplus_{1}^{qR}{\cal{O}}_C \rightarrow E \rightarrow \tau \rightarrow 0 $$
where $Supp(\tau)$ is at most dimension zero.
Hence
$$ h^1(C,E) \leq qR \cdot h^1(C,{\cal{O}}_C).$$
Since $h^1(C,{\cal{O}}_C)=g$,
$\ b(g,R)= (2g-2)Rg +1$ will have the
required property.
\end{pf}
\begin{lm}
\label{cow}
Let $g\geq 2$, $R>0$, be integers. Let $E$ be a coherent
sheaf on a Deligne-Mumford stable, genus $g$ curve $C$
satisfying (i) and (ii) of Lemma (\ref{bat}).
Suppose $F$
is a subsheaf of $E$ generated by global sections.
Then
$$|\chi(F)| < | \chi(E)| +b(g,R) .$$
\end{lm}
\begin{pf}
By Lemma (\ref{bat}),
$h^1(C,F)<b(g,R)$. Therefore, $- b(g,R) < \chi(F)$. By
Lemma (\ref{bat}) applied to $E$,
$$\chi(F) \leq h^0(C,F) \leq h^0(C,E) < \chi(E) + b(g,r) \leq
|\chi(E)| + b(g,r).$$ The result follows.
\end{pf}
\begin{lm}
\label{cotton}
Let $g\geq 2$, $R>0$, $\chi$ be integers.
There exists an
integer $p(g,R,\chi)$ with the following
property. Let $E$ be any coherent
sheaf on any Deligne-Mumford stable, genus $g$ curve $C$
satisfying (i) and (ii) of Lemma (\ref{bat}) and satisfying
$\chi(E)=\chi$.
Let $F$
be any subsheaf of $E$ generated by $k$ global sections:
\begin{equation}
\label{eee}
\bold{C}^k \otimes {\cal{O}}_C \rightarrow F \rightarrow 0.
\end{equation}
Then
for all $t > p(g,R,\chi)$ :
\begin{enumerate}
\item[(i.)] $h^1(C,F\otimes \omega_C^{10t})=0.$
\item[(ii.)] $\bold{C}^k \otimes Sym^t(H^0(C,\omega_C^{10}))
\rightarrow H^0(C,F\otimes \omega_C^{10t}) \rightarrow 0.$
\end{enumerate}
\end{lm}
\begin{pf}
Let $C=\bigcup_{1}^{q}C_i$. Let
$\omega_i$ be the degree of $\omega_C$ restricted to $C_i$.
Let $(s_1, \ldots, s_q)$ be the multirank
of $F$. By (\ref{oiler}) of section (\ref{linto}), the Hilbert polynomial
of $F$ with respect to $\omega_C^{10}$ is:
$$\chi(F\otimes \omega_C^{10t})= \chi(F)+ (\sum_{1}^{q}s_i \omega_i)
\cdot 10t. $$
By Lemma (\ref{cow}), $|\chi(F)| < |\chi| +b(g,R)$.
Also
$$0\leq s_i <R,\ \ \ 1\leq q \leq 2g-2,\ \ \ 1\leq \omega_i \leq 2g-2.$$
Therefore the data $g$, $R$, and $\chi$
determine a finite collection of Hilbert polynomials
$$\{f_1, \ldots , f_m \}$$
that contains
the Hilbert polynomial of every allowed sheaf $F$.
The morphism (\ref{eee}) yields a
natural map $\psi: \bold{C}^k \rightarrow H^0(C,F).$
Let $$im(\psi) = V\subset H^0(C,F).$$
We note that (\ref{eee}) can be factored:
$$\bold{C}^k \otimes {\cal{O}}_C \rightarrow V\otimes {\cal{O}}_C \rightarrow F \rightarrow 0.$$
Since (ii) is surjective if and only if the analogous map in which
$\bold{C}^k$ is replaced by $V$ is surjective,
we can assume
$$k\leq h^0(C,F) \leq h^0(C,E) < |\chi| + b(g,R).$$
Suppose $F$
is a coherent sheaf on a Deligne-Mumford
stable, genus $g$ curve satisfying:
\begin{enumerate}
\item [(a.)] $F$ is generated by $k< |\chi|+ b(g,R)$
global sections: $\bold{C}^{k}\otimes {\cal{O}}_C \rightarrow F \rightarrow 0.$
\item [(b.)] $F$ has Hilbert polynomial
$f$ (with respect to $\omega_C^{10}$).
\end{enumerate}
Then there exists a integer ${\overline{t}}(g,k,f)$ such that
for all $t > {\overline{t}}(g,k,f)$ :
\begin{enumerate}
\item[(i.)] $h^1(C,F\otimes \omega_C^{10t})=0.$
\item[(ii.)] $\bold{C}^{k}\otimes Sym^t(H^0(C,\omega_C^{10}))
\rightarrow H^0(C,F\otimes \omega_C^{10t}) \rightarrow 0.$
\end{enumerate}
The
existence of ${\overline{t}}(g,k,f)$ follows from
statements (\ref{skunk}) and (\ref{ppppp}) of section
(\ref{van}) applied to the Quot scheme $Q_g(\mu,k,f)$.
Now let
$$p(g,R,\chi)=max \{\ {\overline{t}}(g,k,f_j)
\ | \ 1\leq k \leq |\chi|+b(g,R), \ 1\leq j \leq m \} .$$
It follows easily that $p(g,R,\chi)$ has the required property.
\end{pf}
\subsection{Step II} We apply these cohomology bounds along with the
Numerical Criterion in another simple case. First, two definitions:
\label{rdef}
Define $R(g,r)=r(2g-2)+1$.
If $E$ is a coherent sheaf on $C$ with multirank $(r_i)$ and
Hilbert polynomial $f_{e,r}$
with respect to $\omega_C^{10}$, then by
(\ref{oiler}) of section (\ref{linto}), $$\sum r_i \omega_i = r(2g-2).$$
Therefore, $r_i < R(g,r)$ for each $i$.
If $E$ is a coherent sheaf on $C$, there is canonical
sequence
$$0 \rightarrow \tau_E \rightarrow E \rightarrow E' \rightarrow 0$$
where $\tau_E$ is the {\em torsion subsheaf} of $E$ and $E'$ is torsion free.
\begin{pr}
\label{bob}
Let $g\geq2 $, $r>0$, $e>r(g-1)$ be integers. There exists a bound
$t_0(g,r,e)>\hat{t}(g,r,e)$ such that for each
$t > t_0(g,r,e)$, and $[C]\in H_g$ the following holds:
\noindent
If $\ \xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$ corresponds to
a quotient
\begin{equation}
\label{deer}
\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0
\end{equation}
where
$\psi\bigl( \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \bigr) \cap H^0(C,\tau_E) \neq 0$,
then $\xi$ is G.I.T. unstable for the $SL_n$-linearization
determined by $i_t$.
\end{pr}
\begin{pf}
Let $U\subset \bold{C}^n$ be a $1$ dimensional subspace such that
$$\psi(U\otimes H^0(C,{\cal{O}}_C))=W \subset H^0(C,\tau_E).$$ Let $G$ be
the subsheaf of $E$ generated by $W$.
For all $t$,
$$h^0(C, G\otimes \omega_C^{10t}) \leq
h^0(C,\tau_E \otimes \omega_C^{10t})=h^0(C,\tau_E).$$
By Lemma (\ref{bat}),$$h^0(C,\tau_E) \leq h^0(C,E) =\chi(E)+ h^1(C,E) <
f_{e,r}(0)+ b(g,R(g,r)).$$
There certainly
exists a $t_0(g,r,e) > \hat{t}(g,r,e)$ satisfying $\forall t > t_0(g,r,e)$,
$${1\over n} >
{f_{e,r}(0)+b(g,R(g,r)) \over f_{e,r}(t)}.$$
The Proposition is now a consequence of Lemma (\ref{archy}).
\end{pf}
\section{Slope-Unstable, Torsion Free Sheaves}
\label{ggh}
\subsection {Step III}
Propositions (\ref{wk1}) and (\ref{bob}) conclude G.I.T. instability from
certain undesirable properties of points in
$Q_g(C,n,f_{e,r})$. In this section, G.I.T. instability is
concluded from slope-instability in the case
where $\psi$ is an isomorphism and $E$ is torsion free.
The case where $\psi$ is not an isomorphism (and $E$ is arbitrary) is
analyzed in section (\ref{nassy}) where G.I.T.
instability is established.
The above results (for suitable choices of constants and linearizations) show
only points of $Q_g(C,n,f_{e,r})$ where $\psi$ is an isomorphism and
$E$ is a slope-semistable torsion free sheaf may be G.I.T. semistable.
The G.I.T.
(semi)stability results are established in section (\ref{lena}).
\begin{pr}
\label{kiwi}
Let $g\geq 2$, $r>0$ be integers. There exist bounds
$e_1(g,r)>r(g-1)$ and $t_1(g,r,e)>\hat{t}(g,r,e)$ such
that
for each pair $e>e_1(g,r)$, $t>t_1(g,r,e)$ and any
$[C]\in H_g$, the following holds:
\noindent
If $\ \xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$ corresponds to
a quotient
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where
$\psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$
is an isomorphism and
$E$ is a slope-unstable, torsion free sheaf,
then $\xi$ is G.I.T. unstable for the $SL_n$-linearization
determined by $i_t$.
\end{pr}
\subsection{Lemmas and Proof} The proof of Proposition (\ref{kiwi})
requires two Lemmas which are used to apply Lemma (\ref{archy}).
First a destabilizing subsheaf $F$ of $E$ is selected. $F$ determines
a filtration: $ W=H^0(C,F) \subset H^0(C,E)$. $H^0(C,E)$ is identified with
$\bold{C}^n$ by $\psi$. Let $U=\psi^{-1}(W)$.
If $F$ is generated by global sections,
the vanishing
theorems of section (\ref{bondy}) can be applied. Riemann-Roch then
shows the conditions of Lemma (\ref{archy})
for $U$, $W$ follow from the destabilizing
property of $F$ (Lemma (\ref{book})).
In fact, the vanishing argument is valid when
$F$ is generically generated by global sections. Lemma (\ref{water}) guarantees
that a destabilizing subsheaf $F$ generically generated by global sections
exists if $E$ is of high degree.
\begin{lm}
\label{water}
Let $g\geq2$, $r>0$ be integers.
There exists an integer $e_1(g,r)>r(g-1)$
such that for each
$e> e_1(g,r)$ and $[C]\in H_g$ the following holds:
\noindent
If $E$ be a slope-unstable, torsion free sheaf on
$C$ with Hilbert polynomial
$f_{e,r}$ (with respect to $\omega_C^{10}$), then there
exists a nonzero, proper destabilizing subsheaf $F$ of $E$ and an
exact sequence
\begin{equation}
\label{monk}
0 \rightarrow \overline{F} \rightarrow F \rightarrow \tau \rightarrow 0
\end{equation}
where $\overline{F}$ is generated by global sections
and $Supp(\tau)$ is at most dimension zero.
\end{lm}
\begin{pf}
Since $E$ is slope-unstable, there exists a nonzero, proper
destabilizing subsheaf,
$$0\rightarrow F \rightarrow E.$$
Let $C$ be the union of components $\{C_i \}$ where $1\leq i \leq q$.
and let $(s_i)$, $(r_i)$ be the multiranks
of $F$, $E$. Since $E$ is torsion free and $F$ is nonzero, the
multirank of $F$ is not identically zero.
Since the Hilbert polynomial of $E$ is
$f_{e,r}$, we see (by section (\ref{rdef})) $R(g,r)= r(2g-2)+1$
satisfies
$\forall i \ \ r_i < R(g,r)$.
$F$ can be chosen to have minimal multirank
in the following sense. If $F'$ is a nonzero
subsheaf of $F$ with
multirank $(s'_i )$ such that
$\exists j \ \ s'_j< s_j$ , then $F'$ is not destabilizing.
Let $\overline{F}$ be the subsheaf of $F$ generated by the
global sections $H^0(C,F)$.
Since $F$ is destabilizing:
$$ h^0(C,F) \geq \chi(F) > \chi(E)\cdot \paren{\sum s_i \omega_i\over r(2g-2)} =
(e+r(1-g)) \cdot \paren{\sum s_i \omega_i\over r(2g-2)}.$$
Hence if $e >r(g-1)$, $h^0(C,F)>0$ and $\overline{F}$ is nonzero.
We now assume $e > r(g-1)$.
Let $(\overline{s}_i )$ be the nontrivial
multirank of $\overline{F}$. The sequence
(\ref{monk}) has the required properties if and only if
$(\overline{s}_i )=(s_i )$. Suppose $\exists j$,
$\ \overline{s}_j < s_j$. Then $\overline{F}$ is not destabilizing, so
$$\chi(\overline{F}) \leq
(e+r(1-g)) \cdot \paren{\sum \overline{s}_i \omega_i\over r(2g-2)}.$$
We obtain
$$\chi(\overline{F}) < h^0(C,F) \cdot \paren{\sum \overline{s}_i \omega_i \over
\sum s_i \omega_i } \leq
h^0(C,F) \cdot \paren{r(2g-2)-1 \over r(2g-2)}.$$
The last inequality follows from the fact
$$0<\sum \overline{s}_i \omega_i < \sum s_i \omega_i \leq r(2g-2).$$
Since $\overline{F}$ is generated by global sections, Lemma (\ref{bat})
yields
$$h^1(C,\overline{F}) < b(g,R(g,r))=b.$$
We conclude,
$$h^0(C,\overline{F}) < b+ h^0(C,F) \cdot \paren{r(2g-2)-1\over r(2g-2)}.$$
Since $h^0(C,\overline{F}) =h^0(C,F)$ and
$$h^0(C,F) > {e+r(1-g)\over r(2g-2)}\ ,$$
we obtain the bound
$$\paren{e+r(1-g) \over r(2g-2)} \cdot \paren{ 1\over r(2g-2)}<
b.$$
Hence
$$e_1(g,r) = b(g,R(g,r)) \cdot (r^2(2g-2)^2)+r(g-1)$$
has the property required by the Lemma.
\end{pf}
\begin{lm}
\label{book}
Let $g \geq2$, $r>0$, $e>e_1(g,r)$ be integers. There
exists an integer $t_1(g,r,e)> \hat{t}(g,r,e)$ such that
for each
$t>t_1(g,r,e)$ and $[C]\in H_g$, the following holds:
\noindent
If $\xi \in Q_g(C, n=f_{e,r}(0),f_{e,r})$ corresponds to a
quotient
$$ \bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $E$ is a slope-unstable, torsion free sheaf on $C$,
then there exists a nonzero, proper subsheaf
$$0\rightarrow F \rightarrow E$$
such that
\begin{equation}
\label{crow}
{h^0(C,F)\over n}>
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .
\end{equation}
\end{lm}
\begin{pf} Let $t_1(g,r,e)>p(g,R(g,r),\chi=f_{e,r}(0))$ be
determined by Lemma (\ref{cotton}).
Take $F$ to be a nonzero, proper, destabilizing subsheaf of
$E$ for which there
exists a sequence
\begin{equation}
\label{god}
0 \rightarrow \overline{F} \rightarrow F \rightarrow \tau \rightarrow 0
\end{equation}
where $\overline{F}$ is generated by global sections and
$Supp(\tau)$ has dimension zero. Such $F$ exist by Lemma (\ref{water}). Since
$\overline{F}$ is a subsheaf of $E$ and is generated by global
sections, Lemma (\ref{cotton}) yields for any $t >t_1$,
$ h^1(C,\overline{F}\otimes \omega_C^{10t})=0$.
By the exact sequence in cohomology of (\ref{god}),
$ h^1(C,F\otimes \omega_C^{10t})=0$.
Let $(s_i)$ be the (nontrivial) multirank of $F$.
We have
$$h^0(C,F\otimes \omega_C^{10t})= \chi(F) + (\sum s_i \omega_i)10t,$$
$$f_{e,r}(t)= \chi(E) + r(2g-2)10t.$$
We obtain $$\chi(F) \cdot f_{e,r}(t)-
\chi(E) \cdot h^0(C,F\otimes \omega_C^{10t}) = $$
$$\chi(F) \cdot r(2g-2)10t - \chi(E) \cdot (\sum s_i \omega_i)10t >0.$$
The last inequality follows from the destabilizing property of $F$.
Hence
$$ {\chi(F)\over \chi(E)} >
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}.$$
Since $h^0(C,F) \geq \chi(F)$ and $\chi(E)=n$, the
Lemma is proven.
\end{pf}
We can now apply Lemma (\ref{archy}).
\begin{pf}[Of Proposition (\ref{kiwi})]
Let $e_1(g,r)$ be as in Lemma (\ref{water}). For $e>e_1(g,r)$,
let $t_1(g,r,e)$ be determined by Lemma (\ref{book}). Suppose
$t>t_1(g,r,e)$.
Let
$F$ be the subsheaf of $E$ determined by Lemma (\ref{book}).
Let $U\subset \bold{C}^n = \psi^{-1} (H^0(C,F))$.
Since $\psi$ is an isomorphism $dim(U)=h^0(C,F)$. Let $G$ be the
subsheaf generated by the global sections $H^0(C,F)$. Certainly
$$h^0(C, F\otimes \omega_C^{10t}) > h^0(C, G\otimes \omega_C^{10t}).$$
Lemmas (\ref{book}) and (\ref{archy}) are now sufficient to conclude
the desired G.I.T. instability.
\end{pf}
\section {Special, Torsion Bounded Sheaves}
\label{nassy}
\subsection{Lemmas}
As always, suppose $n=f_{e,r}(0)= \chi (E)$.
If $$\psi: \bold{C}^{n}\otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E) $$
is injective but not
an isomorphism, then $h^1(C,E)\neq 0$.
We now investigate this case and conclude G.I.T.
instability for the corresponding points of
$Q_g(C,n,f_{e,r})$. The strategy is the following. Since
$H^1(C,E)$ is dual to $Hom(E,\omega_C)$, the latter must be nonzero.
In Lemma (\ref{boy}),
The kernel of a nonzero element of $Hom(E, \omega_C)$ is analyzed
to produce a very destabilizing subsheaf $F$ of $E$. Lemma (\ref{archy})
is then applied as in section (\ref{ggh}).
In order to carry out the above plan,
the torsion of $E$ must be treated with care.
For any
coherent sheaf $E$ on $C$, let
$0 \rightarrow \tau_E \rightarrow E $
be the torsion subsheaf. $E$ is said to have {\em torsion bounded}
by $k$ if $\chi(\tau_E) < k.$
Let $R(g,r)=r(2g-2)+1$ as defined in section (\ref{rdef}).
\begin{lm}
\label{boy}
Let $g\geq2$, $r>0$ be integers. There exists an integer
$e_2(g,r)> r(g-1)$ such that for each
$e >e_2(g,r)$ and $[C]\in H_g$, the
following holds:
\noindent
If $E$ is a coherent sheaf
on $C$ with
Hilbert polynomial $f_{e,r}$ (with respect to $\omega_C^{10}$) satisfying
\begin{enumerate}
\item[(i.)] $h^1(C,E) \neq 0$
\item[(ii.)] $E$ has torsion bounded by $b(g,R(g,r))$,
\end{enumerate}
then there
exists a nonzero, proper subsheaf $F$ of $E$
with multirank $(s_i)$ not identically zero such that
\begin{enumerate}
\item [(i.)]$F$ is generated by global sections
\item [(ii.)]$${\chi(F) - b(g,R(g,r)) \over \sum s_i\omega_i}>
{\chi(E)\over r(2g-2)} +1\ .$$
\end{enumerate}
\end{lm}
\begin{pf}
Since by Serre duality $H^1(C,E)^* \cong Hom(E,\omega_C)$,
there exists a nonzero
morphism of coherent sheaves:
$$\sigma : E \rightarrow \omega_C .$$
We have
$0 \rightarrow \sigma(E) \rightarrow \omega_C$
where $\sigma(E) \neq 0$.
Since $\omega_C$ is torsion free, $\sigma(E)$ has multirank
not identically zero. Note
$$\chi(\sigma(E)) \leq h^0(C,\sigma(E)) \leq h^0(C,w_C)=g .$$
Consider the exact sequence:
$$0 \rightarrow K \rightarrow E \rightarrow \sigma(E) \rightarrow 0.$$
Since $\chi(K)= \chi(E) - \chi(\sigma(E))$,
$$\chi(K) \geq \chi(E) - g .$$
For $e>r(g-1) +g $, $\chi(K) >0$ and $K \neq 0$.
Let $F$ be the subsheaf generated by the global sections of $K$.
$\chi(K) >0$ implies $F \neq 0$. Let $b=b(g,R(g,r))$. We have
$$\chi(F)> h^0(C,F) - b = h^0(C,K)-b \geq \chi(K) -b
\geq \chi(E)-b-g .$$
For $e> r(g-1)+2b+g$, $\chi(F) > b$.
Now assume $e> r(g-1)+2b+g.$
By the bound on the torsion of $E$, $F$ is not contained
in $\tau_E$.
Let $(s_i)$ be the multirank of $F$. Since $F$ is not torsion,
the multirank is not identically zero. In fact,
since $\sigma(E)$ has multirank
not identically zero,
$$0< \sum s_i\omega_i < r(2g-2).$$
We conclude
\begin{eqnarray*}
{\chi(F)-b \over \sum s_i \omega_i} &>&
\paren{\chi(E)-2b-g \over r(2g-2)} \cdot \paren{r(2g-2)\over \sum s_i\omega_i} \\
& \geq &
\paren{\chi(E)-2b-g \over r(2g-2)} \cdot \paren{ r(2g-2)\over r(2g-2)-1} \ .
\end{eqnarray*}
For large $e$ depending only on $g$ and $r$,
$$\paren{\chi(E)-2b-g \over r(2g-2)} \cdot \paren{r(2g-2) \over r(2g-2)-1}
> {\chi(E) \over r(2g-2)} +1\ .$$
We omit the explicit bound.
\end{pf}
An analogue of Lemma (\ref{book}) is now proven.
\begin{lm}
\label{bag}
Let $g \geq2$, $r>0$, $e>e_2(g,r)$, be integers. There
exists an integer $t_2(g,r,e)> t_0(g,r,e)$ such
for each
$t > t_2(g,r,e)$ and $[C]\in H_g$, the following holds:
\noindent
If $\xi \in Q_g(C, n=f_{e,r}(0),f_{e,r})$
corresponds to a
quotient
$$ \bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $E$ is a coherent sheaf on $C$ satisfying
\begin{enumerate}
\item[(i.)] $ \psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E) $
is injective
\item[(ii.)] $h^1(C,E)\neq 0$
\item[(iii.)] $E$ has torsion bounded by $b(g,(R,g,r))$,
\end{enumerate}
then there exist a nonzero subspace $W \subset
\psi \Bigl( \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \Bigr)$
generating a nonzero, proper subsheaf
$0\rightarrow G \rightarrow E$
such that
\begin{equation}
\label{snow}
{dim(W)\over n} >
{h^0(C,G\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .
\end{equation}
\end{lm}
\begin{pf}
Let $F$ be the subsheaf of $E$ determined by Lemma (\ref{boy}).
Let $$W=im(\psi) \cap H^0(C,F) .$$
Since $$h^0(C,E)< \chi(E) + b(g,R(g,r)),$$
and $\psi$ is injective,
$$dim(W) > h^0(C,F) -b \geq \chi(F) -b.$$
Note by condition (ii) of $F$ in Lemma (\ref{boy}), $dim(W) > 0$.
Let $$t > p(g, R(g,r), \chi=f_{e,r}(0)).$$ Since
$F$ is generated by global sections,
$$h^1(C,F\otimes w_C^{10t})=0$$
by Lemma (\ref{cotton}).
We have
$$h^0(C,F\otimes \omega_C^{10t}) =\chi(F) + (\sum s_i\omega_i)10t,$$
$$f_{e,r}(t)=\chi(E) + r(2g-2)10t.$$
We compute
$$(\chi(F)-b)\cdot f_{e,r}(t)-
\chi(E) \cdot h^0(C,F\otimes \omega_C^{10t}) = $$
$$ (\chi(F)-b) \cdot r(2g-2)10t
-\chi(E) \cdot (\sum s_i\omega_i)10t -b\cdot\chi(E) >$$
$$ r(2g-2)\cdot (\sum s_i\omega_i) \cdot 10t - b \cdot \chi(E).$$
The last inequality follows from condition (ii) of $F$ in
Lemma (\ref{boy}).
If also $$t> b \cdot \chi(E) = b(g,R(g,r)) \cdot (e+r(1-g)),$$
then
$$ {\chi(F)-b \over \chi(E)} >
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .$$
Let $G$ be the subsheaf of $F$ generated by $W$.
Since $dim(W) > \chi(F) -b $, $n=\chi(E)$, and
$$h^0(C,F\otimes \omega_C^{10t}) \geq h^0(C,G\otimes \omega_C^{10t}),$$
the proof is complete.
\end{pf}
\subsection{Step IV}
\begin{pr}
\label{cake}
Let $g\geq 2$, $r>0$ be integers. There exist bounds
$e_2(g,r)>r(g-1)$ and $t_2(g,r,e)>t_0(g,r,e)$ such that
for each pair $e>e_2(g,r)$, $t>t_2(g,r,e)$ and any $[C]\in H_g$,
the following holds:
\noindent
If $\ \xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$ corresponds to
a quotient
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where
$h^1(E,C) \neq 0$,
then $\xi$ is G.I.T. unstable for the $SL_n$-linearization
determined by $i_t$.
\end{pr}
\begin{pf}
Let $e_2(g,r)$ be given by Lemma (\ref{boy}).
For $e>e_2(g,r)$, let $t_2(g,r,e)$ be given by
Lemma (\ref{bag}).
Let $$\psi : \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$$
be the map on global sections.
If $\psi$ has a nontrivial kernel, $\xi$ is unstable by
Proposition (\ref{wk1}). We can assume
$\psi$ is injective.
Note that $im(\psi)$ has codimension less than $b(g,R(g,r))$ in
$H^0(C,E)$. If
$0 \rightarrow \tau \rightarrow E $
is a torsion subsheaf such that $h^0(C,\tau)=\chi(\tau) \geq b(g,R(g,r))$,
then $$im(\psi) \cap H^0(\tau,C) \neq 0.$$
In this case, since $t>t_0(g,r,e)$,
$\xi$ is unstable by Proposition (\ref{bob}).
We can assume $E$ has torsion bounded by $b$.
We now can apply Lemma (\ref{bag}). Let
$W\subset im(\psi)$ be determined by Lemma (\ref{bag}).
Let $U=\psi^{-1}(W)$.
Since $\psi$ is injective, $dim(U)=dim(W)$.
Lemmas (\ref{bag}) and (\ref{archy}) now imply the desired
G.I.T. instability.
\end{pf}
\section{Slope-Semistable, Torsion Free Sheaves}
\label{lena}
\subsection{Step V}
Let $g \geq2$, $r>0$ be integers. Let
$$e > max(e_1(g,r), e_2(g,r)),$$
$$t> max(t_0(g,r,e), t_1(g,r,e), t_2(g,r,e)),$$
be determined by
Propositions (\ref{wk1}, \ref{bob}, \ref{kiwi}, \ref{cake}).
We now conclude the only
possible semistable points in the $SL_n$-linearized
G.I.T. problem determined
by
$$i_t: Q_g(C,n=f_{e,r}(0), f_{e,r}) \rightarrow
\bold G(f_{e,r}(t), (\bold{C}^n \otimes Sym^t(H^0(C,w_C^{10})))^*) $$
are elements $\xi\in Q_g(C,n,f_{e,r})$ that
correspond to quotients
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $$\psi:\bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$$
is an isomorphism and $E$ is a slope-semistable, torsion
free sheaf.
In order for $\xi$ to be semistable,
$\psi$ must be injective by Proposition (\ref{wk1}).
Surjectivity is equivalent to $h^1(C,E)=0$. By Proposition
(\ref{cake}), $\psi$ must be surjective. Since $\psi$ is
an isomorphism, $E$ must be torsion free by Proposition (\ref{bob}).
Finally, by Proposition (\ref{kiwi}), $E$ must be slope-semistable.
We now establish the converse.
\begin{pr}
\label{snake}
Let $g\geq 2$, $r>0$ be integers. There exist bounds
$e_3(g,r)>r(g-1)$ and $t_3(g,r,e)>\hat{t}(g,r,e)$
such that for each pair $e>e_3(g,r)$, $t>t_3(g,r,e)$ and any
$[C]\in H_g$, the following holds:
\noindent
If $\ \xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$ corresponds to
a quotient
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where
$$ \psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$$
is an isomorphism and $E$ is a slope-stable (slope-semistable),
torsion free sheaf,
then $\xi$ is a G.I.T. stable (semistable) point for the
$SL_n$-linearization determined by $i_t$.
\end{pr}
\subsection{Lemmas}
For the proof of G.I.T. (semi)stability, the fundamental step
is the inequality of Lemma (\ref{king}) for every subsheaf $F$ of
$E$ generated by global sections.
Note this is the reverse of the inequality required by
Lemma (\ref{archy}). Lemma (\ref{king}) follows by vanishing, Riemann-Roch,
and the slope-(semi)stability of $E$ when $F$ is nonspecial. In case
$h^1(C,F)\neq 0$, an analysis in Lemma (\ref{uinta})
utilizing $Hom(F,\omega)\neq 0$ yields the
required additional information. The Numerical Criterion of
section (\ref{nnmc}) and Lemma (\ref{king})
reduce the stability question to a purely
combinatorial result established in Lemma (\ref{queen}).
\begin{lm}
\label{uinta}
Let $g\geq 2$, $r>0$ be integers. Let $q$ be an integer.
There exists an integer
$e_3(g,r,q)$ such that for each
$e >e_3(g,r,q)$ and $[C]\in H_g$, the following holds:
\noindent
If $E$ is a slope-semistable, torsion free sheaf on
$C$ with Hilbert polynomial $f_{e,r}$ (with respect
to $\omega_C^{10}$) and
$$0 \rightarrow F \rightarrow E$$ is a nonzero subsheaf with multirank
$(s_i)$ satisfying
$h^1(C,F) \neq 0$, then:
\begin{equation}
\label{hot}
{\chi(F) +q \over \sum s_i\omega_i}
< {\chi(E) \over r(2g-2)}-1.
\end{equation}
\end{lm}
\begin{pf}
Since $h^1(C,F) \neq 0$, there exists a nontrivial morphism
$$\sigma: F \rightarrow \omega_C.$$
Consider the subsheaf
$0 \rightarrow \sigma(F) \rightarrow \omega_C $
where $\sigma(F) \neq 0$. By the proof of Lemma (\ref{boy}),
$\chi(\sigma(F)) \leq g$. Consider the exact sequence
$$0\rightarrow K \rightarrow F \rightarrow \sigma(F) \rightarrow 0.$$
If $K=0$, then
$$ {\chi(F) +q \over \sum s_i\omega_i}
< g+q. $$
Therefore, if
$$e> r(g-1) + r(2g-2)(g+q+1),$$ the case $K=0$ is
settled. Also, the case $\chi(F) \leq g$ is settled.
Now suppose $K\neq 0$ and $\chi(F)-g >0$.
We have $\chi(F)-g \leq \chi(K)$.
Let $(s'_i)$ be the nontrivial
multirank of $K$.
Since $\sigma(F)$ is of nontrivial multirank we have:
$$ 0 < \sum s'_i \omega_i < \sum s_i \omega_i \leq r(2g-2).$$
We obtain:
\begin{eqnarray*}
\paren{\chi(F) -g \over \sum s_i \omega_i} \cdot
\paren{r(2g-2) \over r(2g-2)-1}
& \leq & \paren{\chi(F) -g \over \sum s_i \omega_i } \cdot
\paren{\sum s_i\omega_i \over \sum s'_i\omega_i} \\
& \leq & {\chi(K) \over \sum s'_i \omega_i} \ \ .
\end{eqnarray*}
Using the slope-semistability of $E$ with respect to $K$,
we conclude:
$${\chi(F) -g \over \sum s_i \omega_i}\leq
\paren{\chi(E) \over r(2g-2)}
\cdot
\paren{r(2g-2)-1 \over r(2g-2)} \ .$$
It is now clear, for large $e$ depending
only on $g$ and $r$ and $q$, the inequality (\ref{hot})
is satisfied.
\end{pf}
\begin{lm}
\label{king}
Let $g\geq2$, $r> 0$. Let $b=b(g,R(g,r))$.
Let $e>e_3(g,r,b)>r(g-1)$.
There exists an integer
$t_3(g,r,e)> \hat{t}(g,r,e) $
such that for each
$t > t_3(g,r,e)$ and $[C]\in H_g$, the following holds:
\noindent
If $\xi \in Q_g(C,n=f_{e,r}(0), f_{e,r})$
corresponds to a quotient
$$ \bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $E$ is a torsion free, slope-semistable
sheaf on $C$ and
$0 \rightarrow F \rightarrow E$ is a nonzero, proper subsheaf generated by
global sections, then
$$ {h^0(C,F)\over n} \leq
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .$$
If E is slope-stable,
$$ {h^0(C,F)\over n} <
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .$$
\end{lm}
\begin{pf}
Suppose $t> p(g, R(g,r), \chi=f_{e,r}(0))$.
Let $(s_i)$ be the nontrivial multirank of
$F$. Since $F$ is generated by global sections,
the vanishing
guaranteed by Lemma (\ref{cotton}) yields
$$h^0(C,F\otimes \omega_C^{10t}) = \chi(F) + (\sum s_i\omega_i)10t.$$
The Hilbert polynomial can be expressed:
$$f_{e,r}(t) = \chi(E) + r(2g-2)10t.$$
First consider the case where $h^1(C,F)=0.$
Then $h^0(C,F) =\chi(F)$.
We compute
$$ \chi(F) \cdot f_{e,r}(t)-
\chi(E) \cdot h^0(C,F\otimes \omega_C^{10t}) =$$
$$\chi(F)\cdot r(2g-2)10t - \chi(E) \cdot (\sum s_i\omega_i)10t \ < (\leq) \ 0.$$
where $E$ is slope-stable, (slope-semistable). Hence
$$ {h^0(C,F)\over \chi(E)} =
{\chi(F)\over \chi(E)} < (\leq) \
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .$$
Since $n=\chi(E)$,
the nonspecial case is thus settled.
Now suppose $h^1(C,F)\neq0$.
Lemma (\ref{uinta}) now applies to $F$.
We compute
$$ (\chi(F)+b) \cdot f_{e,r}(t)-
\chi(E) \cdot h^0(C,F\otimes \omega_C^{10t}) =$$
$$(\chi(F)+b)\cdot r(2g-2)10t - \chi(E) \cdot (\sum s_i\omega_i)10t
+ b \cdot \chi(E) <$$
$$ -(\sum s_i \omega_i) \cdot r(2g-2) \cdot 10 t + b\cdot \chi(E).$$
For $t > b \cdot \chi(E) = b\cdot (e+r(1-g))$,
$$ {\chi(F)+b\over \chi(E)} <
{h^0(C,F\otimes \omega_C^{10t})\over f_{e,r}(t)}\ .$$
Since $h^0(C,F)<\chi(F)+b$,
$${h^0(C,F)\over \chi(E)} < {\chi(F)+b \over \chi(E)}$$
The proof is complete.
\end{pf}
We require a simple combinatorial Lemma.
\begin{lm}
\label{queen}
Let $n \geq 2$ be an integer.
Let $$W_1 \leq W_2 \leq \ldots \leq W_n , \ \ W_1 <W_n$$
be integers. Let $\{\alpha_i\}$, $ \{ \beta_i \} $ be rational numbers
such that
\begin{enumerate}
\item[(i.)] $\sum_{1}^{n}\beta_i= \sum_{1}^{n}\alpha_i$.
\item[(ii.)]
$\forall \ 1 \leq m \leq n-1$, $\ \ \sum_{1}^{m}\beta_i \ < (\leq)\
\sum_{1}^{m}\alpha_i$.
\end{enumerate}
Then:
$$\sum_{1}^{n}\beta_i \cdot W_i\ > (\geq) \ \sum_{1}^{n}\alpha_i \cdot W_i.$$
\end{lm}
\begin{pf}
Use discrete Abel summation:
$$\sum_{i=1}^{n}\beta_i \cdot W_i = (\sum_{i=1}^{n}\beta_i)\cdot W_n -
\sum_{m=1}^{n-1} \Bigg( (\sum_{1}^{m}\beta_i)\cdot (W_{m+1}-W_m) \Bigg)
\ \ > (\geq)$$
$$(\sum_{i=1}^{n}\alpha_i)\cdot W_n -
\sum_{m=1}^{n-1} \Bigg( (\sum_{1}^{m}\alpha_i)\cdot (W_{m+1}-W_m) \Bigg) =
\sum_{i=1}^{n}\alpha_i \cdot W_i.$$
The middle inequality follows from (i) and (ii) above.
\end{pf}
\subsection{Proof of Proposition (\ref{snake})}
\begin{pf}
Let $e_3(g,r)=e_3(g,r,b(g,R(g,r)))$
be determined by Lemma (\ref{uinta}). For $e>e_3(g,r)$, let
$t_3(g,r,e)> p(g,R(g,r), \chi=f_{e,r}(0))$
be given by Lemmas (\ref{king}) and (\ref{cotton}).
We will apply the Numerical Criterion to the linearized
$SL_n$-action on
$$\bold G(f_{e,r}(t), (\bold{C}^n \otimes Sym^t(H^0(C,\omega_C^{10})))^*).$$
The element $\xi$ corresponds to the quotient:
$$\psi^t:\bold{C}^n \otimes Sym^t(H^0(C,\omega_C^{10})) \rightarrow
H^0(C,E\otimes \omega_C^{10t}) \rightarrow 0.$$
Let $\overline{v}=(v_1, \ldots, v_n)$ be a basis of $\bold{C}^n$.
Let $(w(v_1),\ldots, w(v_n))$ be weights satisfying
$$ w(v_1) \leq w(v_2) \leq \ldots \leq w(v_n), \ \ w(v_1) < w(v_n).$$
To apply the Numerical Criterion for (semi)stability, an $f_{e,r}(t)$-tuple
of $\overline{v}$-pure
elements of $\bold{C}^n \otimes Sym^t(H^0(C, \omega_C^{10}))$ projecting to a basis
of $H^0(C, E\otimes \omega_C^{10})$ and satisfying the weight inequality (2)
of section (\ref{nnmc}) must be shown to exist.
For $1\leq i \leq n$,
let $F_i$ denote the subsheaf of $E$ generated by
$\psi(\bigoplus_{j=1}^{i} v_j\otimes H^0(C,{\cal{O}}_C)).$
By the surjectivity guaranteed by (ii) of Lemma (\ref{cotton}),
\begin{equation}
\label{brain}
\psi^t : \bigoplus_{j=1}^{i} v_j\otimes Sym^t(C,H^0(\omega_C^{10})) \rightarrow
H^0(C,F_i\otimes \omega_C^{10t}) \rightarrow 0.
\end{equation}
Define for $1\leq i \leq n$, $\ A_i= h^0(C,F_i\otimes \omega_C^{10t})$.
The required $f_{e,r}(t)$-tuple
$(a_1, a_2, \ldots, a_{f_{e,r}(t)})$ is constructed as
follows. Select elements $$(a_1, \ldots, a_{A_1})\in \
v_1 \otimes Sym^t(H^0(C,\omega_C^{10}))$$ such that
$(\psi^t(a_1), \ldots, \psi^t(a_{A_1}))$
determines a basis of $H^0(C,F_1\otimes \omega_C^{10t})$.
Select $$(a_{A_1+1}, \ldots , a_{A_2}) \in \
v_2 \otimes Sym^t(H^0(C,\omega_C^{10}))$$ such that
$(\psi^t(a_1), \ldots, \psi^t(a_{A_2}))$ determines a basis
of $H^0(C,F_2\otimes \omega_C^{10t})$. Continue selecting
$$(a_{A_i+1}, \ldots , a_{A_{i+1}}) \in \
v_{i+1} \otimes Sym^t(H^0(C,\omega_C^{10}))$$ such that
$(\psi^t(a_1), \ldots, \psi^t(a_{A_{i+1}}))$ determines a basis
of $H^0(C,F_{i+1}\otimes \omega_C^{10t})$. Note if $A_{i}=A_{i+1}$,
then $(\psi^t(a_1), \ldots, \psi^t(a_{A_i}))$ already
determines a basis of $H^0(C,F_{i+1} \otimes \omega_C^{10t})$ and
no elements of $v_{i+1}\otimes Sym^t(H^0(C,\omega_C^{10}))$
are chosen.
This selection
is possible by the surjectivity of (\ref{brain}).
Let $\alpha_1= A_1/ f_{e,r}(t)$ and $\alpha_i= (A_i-A_{i-1})/f_{e,r}(t)$ for
$2\leq i \leq n$. We have
$$\sum_{i=1}^{n}\alpha_iw(v_i) = \sum_{j=1}^{f_{e,r}(t)}{w(a_j)\over
f_{e,r}(t)}.$$
Let $\beta_i= (1/n)$.
Note $$\sum_{1}^{n}\beta_i=\sum_{1}^{n}\alpha_i = 1.$$
Let $1 \leq m \leq n-1$.
Since $\psi$ is an isomorphism, $m\leq h^0(C,F_m)$.
Suppose $F_m \neq E$.
Then Lemma (\ref{king}) yields
\begin{equation}
\label{dayo}
{m\over n} \ < (\leq) \ {A_m\over f_{e,r}(t)}\ .
\end{equation}
If $F_m$= $E$, then $A_m=f_{e,r}(t)$ and
the inequality (\ref{dayo}) holds trivially ($m\leq n-1$).
So for all $1\leq m \leq n-1$,
we have $$ \sum_{1}^{m}\beta_i\ < (\leq)\ \sum_{1}^{m}\alpha_i.$$
Lemma (\ref{queen}) yields
$$\sum_{i=1}^{n}{w(v_i)\over n} =
\sum_{1}^{n} \beta_i w(v_i) \ > (\geq)\ \sum_{1}^{n} \alpha_i w(v_i)
= \sum_{j=1}^{f_{e,r}(t)}{w(a_j)\over f_{e,r}(t)} .$$
By the Numerical Criterion, $\xi$ is G.I.T. stable (semistable).
\end{pf}
\subsection{Step VI}
Only one step remains in the proof of Theorem (\ref{fred}).
It must be checked that strict slope-semistability of $E$ implies
strict G.I.T. semistability.
\begin{lm}
\label{moon}
Let $g\geq 2$, $r>0$ be integers.
There exists an integer $e_4(g,r)$ such that for each
$e>e_4(g,r)$ and $[C]\in H_g$, the following holds:
\noindent
If $E$ is any slope-semistable, torsion free sheaf on a
$C$
with Hilbert polynomial $f_{e,r}$
(with respect to $\omega_C^{10t}$) and
$0 \rightarrow F \rightarrow E$ is
a nonzero subsheaf with multirank $(s_i)$ satisfying
\begin{equation}
\label{tart}
{\chi(F)\over \sum s_i \omega_i}=
{\chi(E) \over r(2g-2)},
\end{equation}
then
\begin{enumerate}
\item[(i.)] $h^1(C,F)=0.$
\item[(ii.)] $F$ is generated by global sections.
\end{enumerate}
\end{lm}
\begin{pf}
Suppose $F$ is a nonzero subsheaf of $E$ satisfying (\ref{tart}).
If $e>e_3(g,r,0)$, by Lemma (\ref{uinta}), $h^1(C,F)=0$.
Now let $x\in C$ be a point. We have an exact sequence
$$0 \rightarrow m_xF \rightarrow F \rightarrow {F\over m_xF} \rightarrow 0.$$
Since $F$ is torsion free and $C$ is nodal, it is not hard to show that
$$dim \paren{F\over m_xF} < 2\cdot R(g,r).$$
Since $(F/m_xF)$ is torsion, $m_xF$ has the same multirank
as $F$. Also $$\chi(m_xF) > \chi(F) - 2R.$$
By (\ref{tart}),
$${\chi(m_xF)+2R \over \sum s_i \omega_i} >
{\chi(E)\over r(2g-2)}. $$
If $e> e_3(g,r, 2R)$, $h^1(C,m_xF)=0$ by Lemma (\ref{uinta}).
In this case $F$ is generated by global sections.
We can therefore choose
$e_4(g,r) = e_3(g,r,2R)$.
\end{pf}
\begin{pr}
\label{snook}
Let $g\geq 2$, $r>0$ be integers. There exist bounds
$e_4(g,r)$ and $t_4(g,r,e)$ such that
for each pair $e>e_4(g,r)$, $t>t_4(g,r,e)$ and any
$[C]\in H_g$, the following holds:
\noindent
If $\xi \in Q_g(C,n=f_{e,r}(0),f_{e,r})$ corresponds to
a quotient
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where
$$ \psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$$
is an isomorphism and $E$ is a strictly slope-semistable,
torsion free sheaf,
then $\xi$ is a G.I.T. strictly semistable point for the
$SL_n$-linearization determined by $i_t$.
\end{pr}
\begin{pf}
Since $\xi$ is G.I.T. semistable for $e>e_3(g,r)$ and
$t>t_3(g,r,e)$, it suffices to find a semistabilizing
$1$-parameter subgroup. If $e>e_4(g,r)$, then, for any nonzero, proper
semistabilizing subsheaf
$0\rightarrow F \rightarrow E$,
we have $h^0(C,F)= \chi(F)$ and $F$ is generated
by global sections. It is now easy to see that
the flag $0\subset H^0(C,F) \subset H^0(C,E)$ with
weights $\{0,1 \}$ determines semistabilizing data for large
$t>t_4(g,r,e)$.
\end{pf}
We have now shown for the bounds:
$$ e(g,r) = max \{ e_i(g,r) \ | \ 1 \leq i \leq 4 \},$$
$$t(g,r,e) = max \{t_i(g,r,e) \ | \ 0 \leq i \leq 4 \}$$
the claim of Theorem (\ref{fred}) holds. This completes the proof of
Theorem (\ref{fred}).
By Lemma (\ref{moon}), each slope-semistable, torsion free
sheaf $E$ on $C$ with Hilbert polynomial $f_{e,r}$
appears in $Q_g(C,f_{e,r}(0),f_{e,r})^{SS}_{t}$ for $e>e(g,r)$,
$t>t(g,r,e)$. It is now
clear the $SL_n$-orbits of $Q_g(C,f_{e,r}(0), f_{e,r})^{SS}_{t}$
correspond exactly to the slope-semistable, torsion free
sheaves on $C$ with Hilbert polynomial $f_{e,r}$.
\subsection{Seshadri's Construction}
\label{sesh}
In [Se],
C. Seshadri has studied the $SL_n$-action on $Q_g(C,n=f_{e,r}(0),f_{e,r})$
via a covariant construction. For $e>>0$, he finds a G.I.T.
(semi)stable locus that coincides exactly with the
G.I.T. (semi)stable locus of Theorem (\ref{fred}). These results
appear in Theorem 18 of chapter 1 of [Se] for nonsingular curves and
Theorem 16 of chapter 6 for singular curves.
The
collapsing of semistable orbits is determined by the
Zariski topology. Seshadri shows that
\begin{enumerate}
\item[(i.)] If
$E_{t}$ is a flat family of
of slope-semistable, torsion free sheaves on $C$ such that the
Jordan-Holder factors of $E_{t\neq 0}$ are $\{ F_j \}$, then
the Jordan-Holder factors of $E_0$ are also $\{ F_j \}$.
\item[(ii.)] If $E$ is a
slope-semistable, torsion free sheaf on $C$ with
Jordan-Holder factors $\{ F_j \}$, then there exists a flat
family of slope-semistable, torsion free sheaves $E_t$
such that:
$$E_{t\neq 0} \cong E, \ \ \ E_0 \cong \bigoplus_{j} F_j\ \ .$$
\end{enumerate}
Statement (ii) is proven by constructing flat families
over extension groups.
It follows from these two results that
the points of our quotient
$$Q_g(C,n=f_{e,r}(0),f_{e,r})_{t}^{SS} /SL_n$$
for $e>e(g,r),\ t>t(g,r,e)$
naturally parametrize slope-semistable, torsion free sheaves
with Hilbert polynomial $f_{e,r}$ up to equivalence
given by Jordan-Holder factors.
\section{Two Results in Geometric Invariant Theory}
\label{abgit}
\subsection{Statements}
Let $V$, $Z$, and $W$ be finite dimensional $\bold{C}$-vector
spaces.
Consider two rational representations of $SL(V)$:
$$\zeta: SL(V) \rightarrow SL(Z) $$
$$\omega: SL(V) \rightarrow SL(W). $$
These representations define natural $SL(V)$-linearized actions
on $\bold P(Z)$ and $\bold P(W)$. There is an induced $SL(V)$-action
on the product $\bold P(Z) \times \bold P(W)$. Since
$$Pic(\bold P(Z) \times \bold P(W)) = \bold{Z} \oplus \bold{Z},$$
there is a $1$-parameter choice of linearization. For
$a,b \in \bold{N}^+$, let $[a,b]$ denote the linearization
given by the line bundle ${\cal{O}}_{\bold P(Z)}(a) \otimes
{\cal{O}}_{\bold P(W)}(b)$. Subscripts will be used to indicate
linearization. Let
$$\rho_Z : \bold P(Z) \times \bold P(W) \rightarrow \bold P(Z)$$
be the projection on the first factor.
\begin{pr}
\label{jack}
There exists an integer $k_S(\zeta,\omega)$ such that for all $k>k_S$:
$$\rho_Z^{-1}(\bold P(Z)^S) \subset (\bold P(Z) \times \bold P(W))_{[k,1]}^S.$$
\end{pr}
\begin{pr}
\label{jill}
There exists an integer $k_{SS}(\zeta,\omega)$ such that for all $k>k_{SS}$:
$$(\bold P(Z) \times \bold P(W))_{[k,1]}^{SS} \subset
\rho_Z^{-1}(\bold P(Z)^{SS}).$$
\end{pr}
D. Edidin has informed the author that Proposition (\ref{jack})
is essentially equivalent to Theorem 2.18 of [M-F].
\subsection{q-Stability}
\label{nmc}
Let
$\lambda: \bold{C}^* \rightarrow SL(V)$ be a $1$-parameter subgroup.
Let $dim(V)=a$.
It is well known there exists a basis $\overline{v}=(v_1, \ldots, v_a)$
of $V$ such that $\lambda$ takes the form
$$\lambda(t)(v_i)=t^{e_i} \cdot v_i, \ \ \ \ t\in \bold{C}^*.$$
Denote the tuple $(e_1, \ldots, e_a)$ by $\overline{e}$.
The exponents satisfy the determinant $1$ condition,
$\sum_{i=1}^{a}e_i=0$. Let $|\overline{e}|= max\{ |e_i| \}$.
For the representation $\zeta:SL(V) \rightarrow SL(Z)$, there
exists a basis $\overline{z}=(z_1, \ldots, z_b)$ such that
$\zeta \circ \lambda$ takes the form
$$\zeta \circ \lambda(t)(z_j)=t^{f_j} \cdot z_j,\ \ \ \ t\in \bold{C}^*.$$
Again, $\sum_{j=1}^{b} f_j=0.$ The pairs $\{\overline{v}, \overline{e} \}$ and
$\{\overline{z}, \overline{f} \}$ are said to be {\em diagonalizing data}
for $\lambda$ and $\zeta \circ \lambda$ respectively.
Let
$[z]\in \bold P(Z)$ correspond to the one dimensional subspace
of $Z$ spanned by $z\neq 0$. By the Mumford-Hilbert Numerical
Criterion, $[z]$ is a stable (semistable) point
for the $\zeta$-induced
linearization
on $\bold P (Z)$ if and only if for every $1$-parameter subgroup
$\lambda: \bold{C}^* \rightarrow SL(V)$, the following condition
holds: let $\{\overline{z}, \overline{f} \}$ be diagonalizing data for
$\zeta \circ \lambda$ and let $z=\sum_{j=1}^{b} \xi_j \cdot z_j$,
then there exists an index $j$ for which $\xi_j \neq 0$ and $f_j <0$
($f_j \leq 0$).
Let $q>0$ be a real number.
The point $[z]$ is defined to be {\em $q$-stable} for the
$\zeta$-induced linearization if and only if for every $1$-parameter
subgroup
$\lambda: \bold{C}^* \rightarrow SL(V)$ the following condition
holds: let $\{\overline{v}, \overline{e} \}$ and $\{\overline{z}, \overline{f} \}$
be diagonalizing data for $\lambda$ and
$\zeta \circ \lambda$ and let $z=\sum_{j=1}^{b} \xi_j \cdot z_j$,
then there exists an index $j$ for which $\xi_j \neq 0$ and
$f_j < -q \cdot |\overline{e}|.$
For $q>0$, let $\bold P(Z)^{qS}$ denote the $q$-stable locus
for the $\zeta$-induced linearization.
Proposition (\ref{jack}) will be established in two
steps:
\begin{lm}
\label{step1}
There exists $q(\zeta)>0$ such that $\bold P(Z)^{qS}=\bold P(Z)^S$.
\end{lm}
\begin{lm}
\label{step2}
For any $q>0$,
there exists an integer $k_{qS}(q,\omega)$ such that for all $k>k_{qS}$:
$$\rho_Z^{-1}(\bold P(Z)^{qS}) \subset (\bold P(Z) \times \bold P(W))_{[k,1]}^S.$$
\end{lm}
\noindent
Lemmas (\ref{step1}) and (\ref{step2}) certainly imply
Proposition (\ref{jack}).
\subsection{Proofs of Lemmas (\ref{step1}) and (\ref{step2})}
Let $U$ be a finite dimensional $\bold{Q}$-vector space.
Let $L=\{l_i \}$ be a finite set of elements of $U^*$.
The set $L$ is said to be a {\em stable configuration} if
$$\forall\ 0\neq u\in U, \ \ \exists i \ \ l_i(u)<0.$$
If $\overline{u}=(u_1,u_2,\ldots,u_d)$ is a basis of $U$, define
a norm $|\ |_{\overline{u}}: U\rightarrow \bold{Q}^{\geq 0}$ by
$$|u|_{\overline{u}}= max \{ |\gamma_i| \}\ \ \ \mbox{where }\ u=
\sum_{1}^{k}\gamma_iu_i .$$
\begin{lm}
\label{sam}
Suppose $L =\{l_i \}$ is a stable configuration in $U$.
Let $\overline{u}$ be a basis of $U$.
Then there exists $q>0$ depending upon $L$ and $\overline{u}$ such that
\begin{equation}
\label{drop}
\forall \ 0\neq u\in U, \ \ \exists i \ \ l_i(u)<-q\cdot |u|_{\overline{u}}.
\end{equation}
\end{lm}
\begin{pf}
Let $U \subset U_{\bold {R}} \cong U \otimes_{\bold{Q}} \bold{R} $.
Suppose there exists an element $0\neq u\in U_{\bold{R}}$ and a decomposition
$L=L' \cup L''$ satisfying:
\begin{enumerate}
\item[(i.)] $ \forall l\in L', \ \ l(u)=0.$
\item[(ii.)] $\forall l\in L'', \ \l(u)>0.$
\end{enumerate}
Since the locus $\{z\in U_{\bold{R}}\ | \ \forall l\in L',\ l(z)=0 \}$ is a
rational subspace and the locus
$\{ z\in U_{\bold{R}}\ | \ \forall l\in L'', \ l(z)>0 \}$ is
open, there must exist an element $0\neq \hat{u}\in U$ satisfying
(i) and (ii).
Since $L$ is a stable configuration in $U$, such
$\hat{u}$ do not exist.
It follows
$$\forall \ 0\neq u\in U_{\bold{R}}, \ \ \exists i \ \ l_i(u)< 0.$$
Let $S$ be the unit box in $U_{\bold{R}}$:
$S= \{ u\in U_{\bold{R}}\ | \ |u|_{\overline{u}}=1\}.$
Define a function $g: S \rightarrow \bold{R}^-$ by
$$g(s)= min\ \{l(s) \ |\ l\in L \}.$$
The function $g$ is continuous and strictly negative. Since
$S$ is compact, $g$ achieves a maximum value $-m$ on $S$ for some
$m>0$. The bound $q=m/2$ clearly satisfies (\ref{drop}).
\end{pf}
\begin{pf}[Of Lemma (\ref{step1})]
The proof consists of two simple pieces.
First, a basis $\overline{v}$ of $V$ is fixed. By applying Lemma (\ref{sam}),
it is shown there exists a $q>0$ such that the
stability of $[z]$ implies the $q$-stability inequality for all
$1$-parameter subgroups
subgroups of $SL(V)$ diagonal with respect $\overline{v}$.
Second, it is checked that this $q$ suffices for any selection of
basis.
Let $\overline{v}=(v_1, \ldots, v_a)$ be a basis of $V$.
Let $$U= \{(e_1,\ldots,e_a) \ |\ \ e_i\in \bold{Q}, \sum_{1}^{a}e_i=0 \}.$$
There exist linear functions $\{ l_1, \ldots , l_b \}$ on $U$ and
a basis $\overline{z}=(z_1, \ldots, z_b)$ of $Z$ satisfying the following:
if $\lambda : \bold{C}^* \rightarrow SL(V)$ is any $1$-parameter subgroup
with diagonalizing data $(\overline{v}, \overline{e})$, then the
diagonalizing data of $\zeta \circ \lambda$ is
$(\overline{z}, (l_1(\overline{e}), \ldots, l_b(\overline{e})))$.
Let $\{L_1, \ldots, L_{B} \}$ be the set of distinct
stable configurations in $\{l_1, \ldots ,l_b \}$. That is,
for all $1\leq J \leq B $,
$\ L_{J} \subset \{l_1, \ldots ,l_b \}$ and $L_{J}$ is
a stable configuration in $U$. Let $\overline{u}=(u_1, \ldots, u_{a-1})$
be a basis of $U$ of the following form:
$$u_1=(-1, 1,0,\ldots,0),\ \ldots\ ,\ u_{a-1}=(-1,0,\ldots,0,1).$$
Note $|\overline{e}| \leq a \cdot |\overline{e}|_{\overline{u}}$ for $\overline{e}\in U$.
By Lemma (\ref{sam}), there exists
$q_{J}>0$ such that (\ref{drop}) holds
for each stable configuration $L_{J}$.
Let $$q={1\over a} \cdot min\{q_{J} \}.$$
Suppose $[z]\in \bold P(Z)^S$. Let $z=\sum_{1}^{b} \xi_j z_j$.
By the Numerical Criterion, the stability of $[z]$ implies the
set $\{ l_j | \xi_j \neq 0 \}$ is a stable
configuration in $U$ equal to some $L_{J}$. For any $1$-parameter
subgroup with diagonalizing data
$({\overline{v}}, {\overline{e}})$,
the diagonalizing data of $\zeta \circ \lambda$ is
$(\overline{z}, (l_1(\overline{e}), \ldots, l_b(\overline{e})))$.
By (\ref{drop}), we see
there exists an $l_i\in L_{J}$ such that
$$ l_i(\overline{e})< -q_{J} \cdot |\overline{e}|_{\overline{u}}
\leq -q \cdot |\overline{e}|.$$
Suppose $\overline{v}'$ is another basis of $V$. Then, up to scalars, there
exists an element $\gamma \in SL(V)$ satisfying $\gamma (\overline{v})=
\overline{v}'$. It is now clear that
$$(\zeta(\gamma)(\overline{z}), (l_1(\overline{e}), \ldots, l_b(\overline{e})))$$
is diagonalizing data for $\zeta \circ \lambda$ where
$\lambda$ has diagonalizing data $(\overline{v}',\overline{e})$.
Since the set $\{ l_1, \ldots, l_b \}$ is independent of $\overline{v}$,
the above analysis is valid for any $1$-parameter subgroup.
We have shown that $[z]\in \bold P(Z)^{qS}.$
\end{pf}
\begin{lm}
\label{wally}
Let $\omega: SL(V) \rightarrow SL(W)$ be a rational representation. There
exists an $M_\omega > 0$ with the following property.
Let $\lambda: \bold{C}^* \rightarrow SL(V)$ be any $1$-parameter
subgroup. Let $(\overline{v}, \overline{e})$ and $(\overline{w},\overline{h})$
be diagonalizing data for $\lambda$ and $\omega \circ \lambda$.
Then $|h| < M_\omega \cdot |e|$.
\end{lm}
\begin{pf}
Let $\overline{v}$ be a basis of $V$. Let $U$ be as in the
proof of Lemma (\ref{step1}). There exist
linear functions $\{ l_1, \ldots, l_c \}$ on $U$ and a basis
$\overline{w}=(w_1, \ldots, w_c)$ of
$W$ satisfying the following: if $\lambda: \bold{C}^* \rightarrow
SL(V)$ is any 1-parameter subgroup with diagonalizing data
$(\overline{v}, \overline{e})$, then the diagonalizing data
of $\omega \circ \lambda$ is $(\overline{w}, (l_1(\overline{e}),
\ldots, l_c(\overline{e})))$. Choose $M_\omega$ so
$$\forall j, \ \ |l_j(\overline{e})| < M_\omega \cdot |\overline{e}|.$$
As in the proof of Lemma (\ref{step1}), the set of linear
functions does not depend on $\overline{v}$. The proof is complete.
\end{pf}
\begin{pf} [Of Lemma (\ref{step2})] It is clear that if an
element $[z]\in\bold P(Z)$ is $q$-stable for the $\zeta$-induced
linearization, then $[z^k]\in \bold P(Sym^k(Z))$ is
$kq$-stable for the $Sym^k(\zeta)$-induced
linearization. Let $M_\omega$ be determined by Lemma (\ref{wally})
for the representation $\omega$. Let $k_{qS}=M_\omega/q$.
We check for $k>k_{qS}$,
$$\rho_Z^{-1}(\bold P(Z)^{qS}) \subset (\bold P(Z) \times \bold P(W))_{[k,1]}^S.$$
The linearization $[k,1]$ corresponds to the embedding:
$$\bold P(Z) \times \bold P(W) \rightarrow \bold P(Sym^k(Z) \otimes W)$$
$$[z]\times [w] \rightarrow [z^k\otimes w].$$
Let $[z]\in \bold P(Z)^{qS}$ and $[w] \in \bold P(W)$. Let
$\lambda: \bold{C}^* \rightarrow SL(V)$ be any $1$-parameter subgroup.
Let $\overline{e}$ be the diagonalized exponents of $\lambda$.
Let $(\overline{z}^*, \overline{f}^*)$ and $(\overline{w}, \overline{h})$
be the diagonalizing data of $Sym^k(\zeta) \circ \lambda$ and
$\omega \circ \lambda$. Since $[z^k]$ is $kq$-stable for the
$Sym^k(\zeta)$-induced linearization, there exists an index
$\mu$ satisfying:
\begin{enumerate}
\item [(i.)] The basis element $z_\mu^*$ has a nonzero coefficient
in the expansion of $[z^k]$.
\item [(ii.)] $f_\mu^* < - kq \cdot |\overline{e}|
< -M_\omega \cdot |\overline{e}|$.
\end{enumerate}
Let $w_\nu$ be any basis element that has a nonzero
coefficient in the expansion of $w$. Note
$z_\mu^* \otimes w_\nu$ is an element of the
diagonalizing basis $\overline{z}^* \otimes \overline{w}$ of
$$(Sym^k(\zeta)\otimes \omega) \circ \lambda$$ having
nonzero coefficient in the expansion of $z^k\otimes w$.
The exponent corresponding to $z_\mu^* \otimes w_\nu$ is
simply $f_\mu^* + h_\nu$. Since
$$|h_\nu| \leq |\overline{h}| < M_\omega \cdot |e|,$$
condition (ii) above implies the exponent is
strictly negative. By the Numerical Criterion,
$[z^k\times w]$ is stable. The Lemma is proven.
\end{pf}
\subsection{Proof of Proposition (\ref{jill})}
Let $\zeta: SL(V) \rightarrow SL(Z)$
be a rational representation as above. An element $[z]\in \bold P(Z)$ is
{\em $(e_1, \ldots, e_a)$-unstable} for the $\zeta$-induced
linearization if there exists a destabilizing $1$-parameter subgroup
$\lambda: \bold{C}^* \rightarrow SL(V)$ with diagonalizing data
$(\overline{v}, \overline{e})$: if $(\overline{z}, \overline{f})$ is
diagonalizing data for $\zeta \circ \lambda$ and
$z= \sum_{1}^{b} \xi_j \cdot z_j$, then
$\xi_j\neq 0$ implies $f_j >0.$
Let $\bold P(Z)^{\overline{e}UN} \subset \bold P(Z)$ denote
$\overline{e}$-unstable locus.
{}From the Numerical Criterion, every unstable point is
$\overline{e}$-unstable for some $a$-tuple $\overline{e}=(e_1,\ldots, e_a)$. We need
a simple finiteness result:
\begin{lm}
\label{tom}
Consider the $\zeta$-linearized G.I.T. problem on $\bold P(Z)$.
There exists a finite set of
$a$-tuples, $\cal{P}$,
such that
$$\bigcup_{\overline{e}\in \cal{P}} \bold P(Z)^{\overline{e}UN} = \bold P(Z)^{UN}.$$
\end{lm}
\begin{pf}
We first show that
$\bold P(Z)^{\overline{e}UN}$ is a constructible
subset of $\bold P(Z)$. Fix a $1$-parameter subgroup
$\lambda: \bold{C}^* \rightarrow SL(V)$ with diagonalizing
data $(\overline{v}, \overline{e})$. Let $(\overline{z}, \overline{f})$
be diagonalizing data for $\zeta \circ \lambda$. Let $H$
be the projective subspace of $\bold P(Z)$ spanned by the set
$\{z_j| f_j>0 \}$. Certainly $H\subset \bold P(Z)^{\overline{e}UN}$.
Since every $1$-parameter subgroup of $SL(V)$ with
diagonalized exponents $\overline{e}$ is conjugate to
$\lambda$, we see the map:
$$\kappa: SL(V) \times H \rightarrow \bold P(Z)$$
defined by: $$\kappa(y,[z])=[\zeta(y)(z)]$$
is surjective onto $\bold P(Z)^{\overline{e}UN}$.
The unstable locus, $\bold P(Z)^{UN}$, is closed. Also,
$\bold P(Z)^{UN}$ is the countable union of the $\bold P(Z)^{\overline{e}UN}$. Over
an uncountable algebraically closed field, any algebraic variety
that is countable union of constructible subsets is
actually the union of finitely many of them. Therefore a
finite set of $a$-tuples, $\cal{P}$, with the
demanded property exists in the case $\bold{C}$ is uncountable.
There always exists a field extension $\bold{C} \rightarrow \bold{C}'$ where
$\bold{C}'$ is an uncountable algebraically closed field. By base
extension, $$\zeta_{\bold{C}'}:
SL_{\bold{C}'}(V\otimes_{\bold{C}} \bold{C}') \rightarrow
SL_{\bold{C}'}(Z\otimes_{\bold{C}} \bold{C}').$$
Since $\bold{C}$ is algebraically closed, it is easy to see that
the $\bold{C}$-valued closed points of
$\bold P_{\bold{C}'}(Z\otimes_{\bold{C}}
\bold{C}')^{\overline{e}UN}$ are simply the points of $\bold P_{\bold{C}}
(Z)^{\overline{e}UN}$. Hence, the assertion for $\bold{C}'$ implies the
assertion for $\bold{C}$. This settles the general case.
\end{pf}
\begin{pf}[Of Lemma (\ref{jill})] Let $\cal{P}$
be determined by Lemma (\ref{tom})
for the representation $\zeta$. Let
$M_\omega$ be determined by Lemma (\ref{wally}) for the representation
$\omega$. Let $N_\zeta$ satisfy
$$ \forall \overline{e}\in \cal{P}, \ \ \ N_\zeta > |\overline{e}|.$$
Let $k_{SS}=M_\omega \cdot N_\zeta$. Suppose
$k>k_{SS}$. For each element
$$[z]\times[w] \in \bold P(Z)^{UN} \times \bold P(W),$$
we must show that $[z^k\otimes w]$ is unstable for the
$Sym^k(\zeta)\otimes \omega$-induced linearization on
$\bold P(Sym^k(Z) \otimes W)$. Since $[z]\in \bold P(Z)^{UN}$,
there exists an $\overline{e}\in \cal{P}$ such that $[z]$ is $\overline{e}$-unstable
for the $\zeta$-induced linearization on $\bold P(Z)$.
Let $\lambda: \bold{C}^* \rightarrow SL(V)$ be a $1$-parameter
subgroup with diagonalized exponents $\overline{e}$ that
destabilizes $[z]$. Let $(\overline{z}, \overline{f})$ and $(\overline{w},
\overline{h})$ be diagonalizing data for $\zeta \circ \lambda$ and
$\omega \circ \lambda$. Let $z=\sum_{1}^{b}\xi_s\cdot z_s$ and
$w=\sum_{1}^{c}\sigma_t\cdot w_t$ be the basis expansions. Since
$\lambda$ destabilizes $[z]$, we see
\begin{equation}
\label{hah}
\xi_s \neq 0 \ \ \Rightarrow \ \ f_s>0.
\end{equation}
A diagonalizing basis of $Sym^k(\zeta) \circ \lambda$ can be
constructed by taking homogeneous monomials of degree $k$ in
$\overline{z}$. Denote this basis with the corresponding exponents by
$(\overline{z}^*, \overline{f}^*)$. Then $\overline{z}^*\otimes \overline{w}$
is a diagonalizing basis of
$$(Sym^k(\zeta)\otimes \omega) \circ \lambda.$$
We must show that every nonzero coefficient of the
expansion of $z^k\otimes w$ in the basis
$\overline{z}^*\otimes \overline{w}$ corresponds to a positive
exponent. Suppose the basis element $z_{s^*}^*\otimes w_t$ has
a nonzero coefficient. The element $z_{s^*}^*$ must
correspond to a homogeneous polynomial of degree $k$ in
those $z_s$ for which $\xi_s\neq0$. Therefore, by (\ref{hah}), the
exponent $f_{s^*}^*$ is not less than $k$.
The exponent corresponding to $z_{s^*}^*\otimes w_t$ is
$f_{s^*}^* + h_t$. Since
$$ |h_t| \leq |\overline{h}| < M_\omega \cdot |\overline{e}_p| <
M_\omega \cdot N_\zeta = k,$$
$$f_{s^*}^* + h_t>0.$$
The proof is complete.
\end{pf}
\section {The Construction of $\overline{U_g(e,r)}$}
\label{conlo}
\subsection{Uniform Rank}
Define $$Q_g^r(\mu,n,f_{e,r}) \subset Q_g(\mu,n,f_{e,r})$$
to be the subset corresponding to quotients
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $E$ has uniform rank $r$ on $C$.
Certainly $Q_g^r(\mu,n,f_{e,r})$ is $SL_{N+1}\times SL_n$ -invariant.
\begin{lm}
$Q_g^r(\mu,n,f_{e,r})$ is open and
closed in $Q_g(\mu,n, f_{e,r})$. ($Q_g^r(\mu,n,f_{e,r})$ is a union of
connected components).
\end{lm}
\begin{pf}
Let $\kappa: \cal{C} \rightarrow \cal{B}$ be a projective,
flat family of Deligne-Mumford
stable genus $g$ curves over an irreducible curve. Let
$\cal E$ be a $\kappa$-flat coherent sheaf of constant Hilbert polynomial
$f_{e,r}$ (with respect to $\omega_{\cal{C}/\cal{B}}^{10})$. Suppose
there exists a $b^*\in \cal{B}$ such that $\cal{E}_{b^*}$ has uniform rank
$r$ on $\cal{C}_{b^*}=C$. Let $\{ \cal{C}_i \}$ be the irreducible
components of $\cal{C}$. Since $\kappa:\cal{C}_i \rightarrow \cal{B}$
is surjective of relative dimension $1$,
each $\cal{C}_i$ contains a component of $C$. Since
the function $r(z)=dim_{k(z)}(\cal{E}\otimes k(z))$ is upper
semicontinuous
on $\cal{C}_i$,
there is an open set $U_i\subset \cal{C}_i$ where
$r(z) \leq r$. It follows there exists an
open set $U\subset \cal{B}$
such that $\forall b\in U$, the rank of $\cal{E}_b$ on each
component of $\cal{C}_b$ is
at most $r$. If $\exists b'$ such that $\cal{E}_{b'}$ is not of uniform
rank $r$, then (by semicontinuity) $\exists i $ so that $r(z)$ is strictly
less than $r$ on an open $W\subset\cal{C}_i$. For $b$ in the
nonempty intersection $U \cap \kappa(W)$, ranks of $\cal{E}_b$ are
at most $r$ on each component and strictly less
than $r$ on at least one component. By equation (\ref{oiler}) of
section (\ref{linto}),
$\cal{E}_b$ can not have Hilbert polynomial $f_{e,r}$. Thus
$\forall b \in \cal{B}$, $\cal{E}_b$ has uniform rank $r$.
The Lemma is proven.
\end{pf}
\subsection{Determination of the Semistable Locus}
Select $e>e(g,r)$ and $t>t(g,r,e)$. As usual, let $n=f_{e,r}(0)$.
Let $$Z=\bigwedge^{h(\overline{s})}H^0(\bold P^N,{\cal{O}}_{\bold P^N}(\overline{s}))^* ,$$
$$W= \bigwedge^{f_{e,r}(t)}(\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t)))^*.$$
Consider the immersion
$$j_{\overline{s},t}: Q_g^r(\mu,n,f_{e,r})
\rightarrow \bold P(\bigwedge^{h(\overline{s})}H^0(\bold P^N,{\cal{O}}_{\bold P^N}(\overline{s}))^*)
\times \bold P(\bigwedge^{f_{e,r}(t)}(\bold{C}^n \otimes H^0(\bold P^N,{\cal{O}}_{\bold P^N}(t)
))^*)$$
defined in section (\ref{said}). Recall $\overline{s}$ is the linearization
determined by Gieseker. There are three group actions to
examine. In what follows, the superscripts $\{ S',SS' \}$ will
denote stability and semistability with respect to the
$SL_{N+1}$-action. Similarly, $\{S'',SS''\}$ will correspond to the
$SL_n$-action, and $\{S,SS\}$ will correspond to the $SL_{N+1} \times
SL_n$-action.
The strategy for obtaining the desired $SL_{N+1}\times SL_n$-semistable
locus is as follows.
Consider first the $SL_{N+1}$-action. For
suitable linearization, it will be shown that
$Q_g^r(\mu,n,f_{e,r})$ is contained in the
$SL_{N+1}$-stable locus and is closed in $SL_{N+1}$-semistable locus.
This assertion is a consequence of Gieseker's conditions on $H_g$ (
(i), (ii) of section (\ref{said}) )
and the results of section (\ref{abgit}). Next,
$\overline{U_g(e,r)}$ defined as the G.I.T. quotient of
$Q_g^r(\mu,n,f_{e,r})^{SS}$ by $SL_{N+1}\times SL_{n}$.
$\overline{U_g(e,r)}$ is a projective variety. Finally,
in Proposition (\ref{lizzy}), it is shown that the $SL_n$ and
$SL_{N+1}\times SL_n$ semistable loci coincide on $Q_g^r(\mu,n,f_{e,r})$.
Similarly, the stable loci coincide. The results on the
fiberwise G.I.T. problem now yield a geometric identification
of the stable and semistable loci for the $SL_{N+1}\times SL_n$-
G.I.T. problem.
By Propositions (\ref{jack}) and (\ref{jill}),
an integer $k>\{ k_{S'},k_{SS'} \}$
can be found so that:
\begin{equation}
\label{one}
\rho_Z^{-1}(\bold P(Z)^{S'}) \subset (\bold P(Z) \times \bold P(W))_{[k,1]}^{S'},
\end{equation}
\begin{equation}
\label{two}
(\bold P(Z) \times \bold P(W))_{[k,1]}^{SS'} \subset
\rho_Z^{-1}(\bold P(Z)^{SS'}).
\end{equation}
By (i) of section (\ref{said}), $H_g \subset \bold P(Z)^{S'}$. Now
(\ref{one}) above yields
\begin{equation}
\label{skup}
H_g \times \bold P(W) \subset
(\bold P(Z) \times \bold P(W))_{[k,1]}^{S'}.
\end{equation}
Therefore,
\begin{equation}
\label{three}
Q_g^r(\mu,n,f_{e,r}) \subset (\bold P(Z) \times \bold P(W))_{[k,1]}^{S'}.
\end{equation}
By (ii) of section (\ref{said}), $H_g$ is closed in $\bold P(Z)^{SS'}$. Hence
$H_g\times \bold P(W)$ is closed in $\rho_Z^{-1}(\bold P(Z)^{SS'})$.
By
(\ref{two}) and the projectivity of
$Q_g^r(\mu,n,f_{e,r})$ over $H_g$:
$$Q_g^r(\mu,n,f_{e,r}) \mbox{ is closed in }
(\bold P(Z) \times \bold P(W))_{[k,1]}^{SS'}.$$
Since
$$(\bold P(Z) \times \bold P(W))_{[k,1]}^{SS} \subset
(\bold P(Z) \times \bold P(W))_{[k,1]}^{SS'},$$
it follows that
\begin{equation}
\label{four}
Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS} \mbox{ is closed in }
(\bold P(Z) \times \bold P(W))_{[k,1]}^{SS}.
\end{equation}
We define $$\overline{U_g(e,r)} = Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS} / (SL_{N+1}
\times SL_n).$$
By (\ref{four}), $\overline{U_g(e,r)}$ is a projective variety.
We now identify the locus $Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}$.
Certainly
$$Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{S}
\subset Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{S''} ,$$
$$Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}
\subset Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS''} .$$
In fact:
\begin{pr}
\label{lizzy}
There are two equalities:
$$Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{S} = Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{S''} ,$$
$$Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS} = Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS''} .$$
\end{pr}
\begin{pf}
We apply the Numerical Criterion.
Let
$$ \zeta^k: SL_{N+1} \times SL_n \rightarrow SL_{N+1} \rightarrow SL(Sym^k(Z)).$$
$$ {\omega}: SL_{N+1}\times SL_n \rightarrow SL(W) $$
denote the two representations.
Let $\xi \in
Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{S''}$. Recalling the morphisms defined
in section (\ref{van}),
$$j_{\overline{s},t}(\xi)= (\pi(\xi), i_t(\xi)).$$
Let $\lambda:\bold{C}^* \rightarrow SL_{N+1}\times SL_n $ be a nontrivial
$1$-parameter subgroup given by
$$\lambda_1 : \bold{C}^* \rightarrow SL_{N+1},$$
$$\lambda_2: \bold{C}^* \rightarrow SL_n.$$
Let $\{ m_i \}$ be a diagonalizing basis for
${\zeta}^k \circ \lambda$ with weights $\{ w(m_i) \}$. Let
$\{ n_j \}$ be a diagonalizing basis for
the $\bold{C}^* \times \bold{C}^*$ representation
${\omega }\circ (\lambda_1 \times \lambda_2)$.
Let $w_1(n_j)$ and $w_2(n_j)$ denote
the weights of the induced $\bold{C}^*$ representations
${\omega} \circ (\lambda_1 \times 1)$ and ${\omega} \circ
(1 \times \lambda_2)$.
The weights $\{w(n_j) \}$ of the $\bold{C}^*$ representation
${\omega}\circ \lambda$ are given by
$w(n_j)=w_1(n_j)+w_2(n_j).$ Finally let
$\{\overline{m}_i \}$ and $\{ \overline{n}_j \}$
denote the elements of the diagonalizing bases that appear
with nonzero coefficient in the expansions of
$\pi(\xi)$ and $i_t(\xi)$. There are three cases.
\begin{enumerate}
\item $\lambda_1=1$. Since $\xi$ is a stable point for the
$SL_n$-action, there is a $\overline{n}_j$ with $w_2(\overline{n}_j)<0.$
We see $$ w(\overline{m}_i \otimes \overline{n}_j)=w(\overline{m}_i) +
w_1(\overline{n}_j) + w_2(\overline{n}_j) =w_2(\overline{n}_j) <0 $$
for any $\overline{m}_i$.
\item $\lambda_2=1$. By (\ref{three}), $\xi$ is a stable point for the
$SL_{N+1}$-action. Hence there exists a pair $\overline{m}_i$, $\overline{n}_j$
so that
$$w(\overline{m}_i \otimes \overline{n}_j) =w(\overline{m}_i)+w_1(\overline{n}_j) <0.$$
\item $\lambda_1 \neq 1, \ \lambda_2 \neq 1$. Since $\xi$ is a stable
point for the
$SL_n$-action, there is a $\overline{n}_j$ with $w_2(\overline{n}_j)<0.$
By (\ref{skup}), $(\pi(\xi) \otimes \overline{n}_j)$ is a stable point
for the $SL_{N+1}$-action. Hence there exists an element $\overline{m}_i$
so that
$$w(\overline{m}_i)+w_1(\overline{n}_j) <0.$$
Therefore,
$$ w(\overline{m}_i \otimes \overline{n}_j)=w(\overline{m}_i) +
w_1(\overline{n}_j) + w_2(\overline{n}_j) <0.$$
\end{enumerate}
By the Numerical Criterion, $\xi\in Q_g^r(\mu,n,f_{e,r})_{[k,1]}^S.$
The proof for the semistable case is identical.
\end{pf}
By Theorem (\ref{fred}), we see the points of
$Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}$ correspond exactly to
quotients
$$\bold{C}^n \otimes {\cal{O}}_C \rightarrow E \rightarrow 0$$
where $E$ is slope-semistable, torsion free sheaf of uniform rank
$r$ on a $10$-canonical, Deligne-Mumford stable, genus $g$ curve
$C\subset\bold P^N$ and
$$\psi: \bold{C}^n \otimes H^0(C,{\cal{O}}_C) \rightarrow H^0(C,E)$$
is an isomorphism.
Similarly for $Q_g^r(\mu,n,f_{e,r})_{[k,1]}^S$.
We now examine orbit closures. Suppose
$\overline{\xi} \in Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}$ lies in the
$SL_{N+1} \times SL_n$-orbit closure
of of $\xi \in Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}$.
Let $$\gamma=(\gamma_1,\gamma_2): \bigtriangleup
-\{p\} \rightarrow SL_{N+1} \times SL_n$$
be a morphism of a nonsingular, pointed curve such that
$$Lim_{z\rightarrow p}(\gamma(z)\cdot\xi) = \overline{\xi}.$$
It follows that
$$Lim_{z\rightarrow p}(\gamma_1(z)\cdot \pi(\xi)) = \pi(\overline{\xi}).$$
Since $H_g \subset \bold P(Z)^{S}$, $\pi(\overline{\xi})$
lies in the $SL_{N+1}$-orbit of $ \pi(\xi)$. After a possible base
change, we can assume $\gamma_1$ extends over $p\in \bigtriangleup$ to
$$\overline{\gamma_1}: \bigtriangleup_p \rightarrow SL_{N+1}.$$
Since $Q_g^r(U_{\pi(\xi)}, n,f_{e,r})$ is projective,
$$\mu : \bigtriangleup - \{p \} \rightarrow Q_g^r(U_{\pi(\xi)}, n,f_{e,r})$$
defined by $\mu(z)= \gamma_2(z) \cdot \xi$ extends to
$\bigtriangleup_p$. Let
$\hat{\xi}= Lim_{z\rightarrow p}(\gamma_2(z)\cdot \xi)= \mu(p).$
By considering the map
$$\overline{\gamma_1}\cdot \mu:\bigtriangleup_p \rightarrow Q_g^r(\mu,n,f_{e,r})$$
defined by
$$\overline{\gamma_1}\cdot \mu(z)= \overline{\gamma_1}(z)\cdot \mu(z) $$
we obtain,
$$\overline{\gamma_1}(p) \cdot \hat{\xi}= \overline{\gamma_1}\cdot \mu (p)
= Lim_{z\rightarrow p}(\gamma_1(z)\cdot \gamma_2(z)\cdot \xi) =$$
$$Lim_{z\rightarrow p}(\gamma(z)\cdot \xi) =\overline{\xi}.$$
We have shown the $SL_{N+1}$-orbit of $\overline{\xi}$ meets
the $SL_n$-orbit closure of $\xi$. If $\xi$, $\overline{\xi}$ corresponds to
a slope-semistable, torsion free quotients $E$, $\overline{E}$ on
$C$, $\overline{C} \subset \bold P^N$, certainly $C$, $\overline{C}$ must be
projectively equivalent. The elements of the $SL_{N+1}$-orbit of
$\overline{\xi}$ that lie over $C$ are simply the images of $\overline{E}$
under automorphisms of $C$. Now from section (\ref{sesh}), we
conclude two semistable orbits $\xi$ and $\overline{\xi}$ are
identified in the quotient $\overline{U_g(e,r)}$ if and only if
$$\pi(\xi) \equiv \pi(\overline{\xi}) \equiv [C]$$ and
the corresponding semistable, torsion free quotient
sheaves $E$, $\overline{E}$ have Jordan-Holder factors that
differ by an automorphism of $C$. We see:
\begin{tm}
\label{priya}
$\overline{U_g(e,r)}$ parametrizes aut-equivalence classes of
slope-semistable, torsion free sheaves of uniform rank $r$
and degree $e$ on Deligne-Mumford stable curves of genus $g$.
\end{tm}
\noindent
Finally, since $$ Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS} \rightarrow H_g \rightarrow \barr{M_g}$$
is an $SL_{N+1}\times SL_{n}-$ invariant morphism, there exists
a map
$$\eta: \overline{U_g(e,r)} \rightarrow \barr{M_g}.$$
\section{Basic Properties of $\overline{U_g(e,r)}$}
\label{laslo}
\subsection{The Functor}
Let $\overline{\cal{U}_g(e,r)}$ be the functor that associates to
each scheme $S$ the set of equivalence classes of the following
data:
\begin{enumerate}
\item A flat family of Deligne-Mumford stable genus $g$ curves,
$\mu: \cal{C} \rightarrow S$.
\item A $\mu$-flat coherent sheaf $\cal{E}$ on $\cal{C}$ such that:
\begin{enumerate}
\item [(i.)] $\cal{E}$
is of constant Hilbert polynomial $f_{e,r}$ (with respect to
$\omega_{\cal{C}/S}^{10}$).
\item [(ii.)] $\cal{E}$ is a slope-semistable, torsion free
sheaf of uniform rank $r$ on each fiber.
\end{enumerate}
\end{enumerate}
Two such data sets are equivalent if there exists an $S$ isomorphism
$\phi: \cal{C} \rightarrow \cal{C}'$
and a line bundle $L$ on $S$ so that
$\cal{E} \cong \phi^*(\cal{E}') \otimes \mu^*L$.
\begin{tm}
\label{thwee}
There exists a natural transformation
$$\phi_U: \overline{\cal{U}_g(e,r)} \rightarrow Hom(*, \overline{U_g(e,r)}).$$
$\overline{U_g(e,r)}$ is universal in the following sense.
If $Z$ is a scheme and
$\phi_Z: \overline{\cal{U}_g(e,r)} \rightarrow Hom( *, Z)$ is
a natural transformation, there exists a unique morphism
$ \gamma: \overline{{U}_g(e,r)} \rightarrow Z $ such that
the transformations $\phi_Z$ and $\gamma\circ \phi_U$ are
equal.
\end{tm}
\begin{pf}
Let $e>e(g,r)$.
Let $\mu: \cal{C} \rightarrow S$ and $\cal{E}$ on $C$ satisfy (1) and (2)
above. Note $\mu_*(\omega_{\cal{C}/S}^{10})$ is a locally free
sheaf of rank $N+1= 10(2g-2)-g+1$. Since $\cal{E}_s$ is
nonspecial for each $s\in S$,
$\mu_*(\cal{E})$ is locally
free of rank $n=f_{e,r}(0)$. Choose an open cover
$\{W_i \}$ of $S$ trivializing both $\mu_*(\omega_{\cal{C}/S}^{10})$ and
$\mu_*(\cal{E})$.
Let $V_i=\mu^{-1}(W_i)$. For each $i$, we obtain isomorphisms:
\begin{equation}
\label{hoot}
\bold{C}^{N+1}\otimes {\cal{O}}_{W_i} \cong \mu_*(\omega_{\cal{C}/S}^{10})|_{W_i}
\end{equation}
\begin{equation}
\label{toot}
\bold{C}^{n}\otimes {\cal{O}}_{W_i} \cong \mu_*(\cal{E})|_{W_i}.
\end{equation}
These isomorphisms yield surjections:
$$ \bold{C}^{N+1}\otimes {\cal{O}}_{V_i} \cong
\mu^*\mu_*(\omega_{\cal{C}/S}^{10})|_{V_i} \rightarrow \omega_{\cal{C}/S}^{10}
|_{V_i} \rightarrow 0$$
$$ \bold{C}^n \otimes {\cal{O}}_{V_i} \cong
\mu^*\mu_*(\cal{E})|_{V_i} \rightarrow \cal{E}
|_{V_i} \rightarrow 0$$
The first surjection embeds $V_i$ in $W_i \times \bold P^N$.
By the universal property of the Quot scheme $Q_g(\mu,n,f_{e,r})$
and the second surjection,
there exists a map
$$\phi_i: W_i \rightarrow Q_g(\mu,n,f_{e,r}).$$
For $t>t(g,r,e)$, $\phi_i(W_i) \subset Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}$.
On the overlaps $W_i\cap W_j$, $\phi_i$ and
$\phi_j$ differ by a morphism
$$W_i \cap W_j \rightarrow PSL_{N+1} \times PSL_n$$ corresponding
to the choice of trivialization in (\ref{hoot}) and (\ref{toot}).
Hence there exists a well defined morphism
$$\phi: S \rightarrow \overline{U_g(e,r)}.$$
The functoriality of the universal property of the Quot scheme implies
$\phi$ is functorially associated to
the data $\cal{E}$ and $\mu:\cal{C} \rightarrow S$.
We have shown there exists a natural transformation
$$\phi_U:\overline{\cal{U}_g(e,r)} \rightarrow Hom( *, \overline{U_g(e,r)}).$$
Suppose
$\phi_Z:\overline{\cal{U}_g(e,r)} \rightarrow Hom( *, Z)$ is a
natural transformation. There exists a canonical
element of
$\delta \in \overline{\cal{U}_g(e,r)}(Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS})$
corresponding
to the universal family on the Quot scheme. The morphism
$$\phi_Z(\delta) :Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS} \rightarrow Z$$
is $SL_{N+1} \times SL_{n}$-invariant. Hence $\phi_Z(\delta)$
descends to $\gamma: \overline{U_g(e,r)}\rightarrow Z$.
The two transformations $\phi_Z$ and $ \gamma \circ \phi_U$
agree on $\delta$. Naturality now implies $\phi_Z= \gamma \circ \phi_U$.
\end{pf}
By previous considerations, there are natural transformations
$$t_l:\overline{\cal{U}_g(e,r)} \rightarrow \overline{\cal{U}_g(e+rl(2g-2),r)}$$
given by $\cal{E} \rightarrow \cal{E}\otimes \omega_{\cal{C}/S}^l$.
By Theorem (\ref{thwee}), these induce natural isomorphisms
$$t_l:\overline{U_g(e,r)} \rightarrow \overline{U_g(e+rl(2g-2),r)}.$$
The arguments in the above proof imply a useful Lemma:
\begin{lm}
\label{openy}
Let $\mu: \cal{C} \rightarrow S$ be a flat family of Deligne-Mumford
stable, genus $g\geq2$ curves. Let $\cal{E}$ be a $\mu$-flat
coherent sheaf
on $\cal{C}$. The condition that $\cal{E}_s $ is
a slope-semistable torsion free sheaf of uniform rank on $\cal{C}_s$ is
open on $S$.
\end{lm}
\begin{pf}
Suppose $\cal{E}_{{s_0}} $ is
a slope-semistable sheaf of uniform rank $r$ on $\cal{C}_{{s}_0}$
for some ${s}_0\in S$.
There exists a integer $m$ so :
\begin{enumerate}
\item [(i.)] $h^1(\cal{E}_s\otimes \omega_{\cal{C}_s}^m, \cal{C}_s)=0$ for all
$s\in S$.
\item [(ii.)] $\cal{E}_s\otimes \omega_{\cal{C}_s}^m$ is generated by global
section for all $s\in S$.
\item [(iii.)] $degree(\cal{E}_{s_0}\otimes \omega_{\cal{C}_{s_0}}^m) > e(g,r)$.
\end{enumerate}
It suffices to prove the Lemma
for $\cal{E} \otimes \omega_{\cal{C}/S}^m$. Let $f_{e,r}$ be
the Hilbert polynomial of $\cal{E} \otimes \omega_{\cal{C}/S}^m$.
By the proof of Theorem (\ref{thwee}), there
exists an open set $W \subset S$ containing $s_0$ and
a morphism $$\phi: W \rightarrow Q_g(\mu,n=f_{e,r}(0), f_{e,r})$$
such that $\cal{E}\otimes \omega_{\cal{C}/S}^m$ is isomorphic to
the $\phi$-pull back of the universal quotient.
Since $\phi({s_0}) \in Q_g^r(\mu,n,f_{e,r})_{[k,1]}^{SS}$ and
the latter is open, the Lemma is proven.
\end{pf}
\subsection{Deformations of Torsion Free Sheaves
and the Irreducibility of $\overline{U_g(e,r)}$}
We study deformation properties of uniform rank,
torsion free sheaves on nodal curves.
\begin{lm}
\label{d1}
Let
$\mu: S \rightarrow Spec(\bold{C}[t])$ be a $\mu$-flat, nonsingular
surface.
If $\cal{E}$ is a $\mu$-flat sheaf on $S$ such that the
restriction $\cal{E}/t\cal{E} =\cal{E}_0$
is torsion free, then $\cal{E}_0$ is locally free.
\end{lm}
\begin{pf}
Let $z\in \mu^{-1}(0)$. Since $S$ is a nonsingular surface, the
local ring ${\cal{O}}_{S,z}$ is regular of dimension $2$.
Consider the ${\cal{O}}_{S,z}$-module $\cal{E}_z$. Since $\cal{E}$
is $\mu$-flat, $t$ is a $\cal{E}_z$-regular element.
Since $\cal{E}_0=\cal{E}/t\cal{E}$ is torsion free,
$depth_{{\cal{O}}_{S,z}}(\cal{E}_z)\geq 2$. We have the
Auslander-Buchsbaum relation ([M]):
$$proj.\ dim_{{\cal{O}}_{S,z}}(\cal{E}_z) + depth_{{\cal{O}}_{S,z}}(\cal{E}_z) =
dim({\cal{O}}_{S,z})=2.$$
We conclude $\ proj.\ dim_{{\cal{O}}_{S,z}}(\cal{E}_z)=0$.
Hence $\cal{E}_z$ is free over ${\cal{O}}_{S,z}$. It follows that
$\cal{E}_0$ is locally free.
\end{pf}
Lemma (\ref{d1}) shows that it is not possible to deform a
torsion free, non-locally free sheaf on a nodal curve
to a locally free sheaf on a nonsingular curve if the deformations
at the nodes have local equations of the form
$(xy-t)\subset Spec(\bold{C}[x,y,t])$.
However, the next Lemma shows such deformations exist locally if the
deformations of the nodes have local equations of the form
$(xy-t^2)$. In Lemma (\ref{deform}), it is shown these local deformations
can be globalized.
\begin{lm}
\label{d2}
Let $S\subset Spec(\bold{C}[x,y,t])$ be the subscheme defined
by the ideal $(xy-t^2)$. Let
$\mu: S \rightarrow Spec(\bold{C}[t])$. Let $\zeta=(0,0,0)\in S$. There exists a
$\mu$-flat sheaf $\cal{E}$ on $S$ such that
$\cal{E}_{t\neq 0}$ is locally free and
$\cal{E}_0 \cong m_{\zeta}$ where $m_{\zeta}$ is the
maximal ideal defining $\zeta$ on $S_0$.
\end{lm}
\begin{pf}
There exists a section $L$ of $\mu$ defined by
the ideal $(x-t,y-t)$. Let $\cal{E}$ be the coherent
sheaf corresponding to this ideal. We have the
exact sequence:
\begin{equation}
\label{pddy}
0\rightarrow\cal{E} \rightarrow {\cal{O}}_S \rightarrow {\cal{O}}_L \rightarrow 0.
\end{equation}
Since ${\cal{O}}_S$ is torsion free over $\bold{C}[t]$,
so is $\cal{E}$. $\cal{E}$ is therefore $\mu$-flat.
Since ${\cal{O}}_L$ is $\mu$-flat, sequence (\ref{pddy})
remains exact after restriction to the special fiber. Hence
$\cal{E}_0\cong m_\zeta$.
\end{pf}
\begin{lm}
\label{deform}
Let $C$ be a Deligne-Mumford stable curve of genus $g\geq2$.
Let $E$ be a slope-semistable torsion free sheaf of uniform
rank $r$ on $C$. Then there exists a family
$\mu: \cal{C} \rightarrow \bigtriangleup_0$ and a $\mu$-flat coherent sheaf
$\cal{E}$ on $\cal{C}$ such that :
\begin{enumerate}
\item $\bigtriangleup_0$ is a pointed curve.
\item $\cal{C}_0 \cong C$, $\ \ \forall t\neq 0$ $\ \ \cal{C}_t$ is
a complete, nonsingular, irreducible genus $g$ curve.
\item $\cal{E}_0 \cong E$, $\ \ \forall t\neq 0$ $\ \ \cal{E}_t$ is
a slope-semistable torsion free sheaf of rank $r$.
\end{enumerate}
\end{lm}
\begin{pf}
Let $z\in C$ be a node.
Since $E$ is torsion free and of uniform rank $r$, it
follows from Propositions (2) and (3) of chapter (8) of
[Se]:
\begin{equation}
\label{locy}
E_z \cong \bigoplus_{1}^{a_z} {\cal{O}}_{C_z} \oplus \bigoplus _{a_z+1}^{r}
m_z
\end{equation}
where $m_z$ is the localization of the ideal of the node $z$.
To simplify the deformation arguments, let
$\bold{C}$ be the field of complex numbers.
Let $B(d)\subset \bold{C}^2$ be the open ball of radius $d$
with respect to the Euclidean norm;
let $B(d_1,d_2)\subset \bold{C}^2$ be the open annulus.
Disjoint open Euclidean neighborhoods, $z\in U_z\subset C$, can be
chosen for each node of $C$ satisfying:
\begin{enumerate}
\item[(i.)] $U_z$ is analytically
isomorphic to
$B(d_z) \cap (xy=0)\subset \bold{C}^2$.
\item[(ii.)]
$E|_{U_z}\cong \bigoplus_{1}^{a_z} {\cal{O}}_{U_z} \oplus \bigoplus _{a_z+1}^{r}
m_z$.
\end{enumerate}
Let $V_z\subset U_z$ be the closed neighborhood of radius $d_z/2$.
Let $W = C \setminus \cup V_z$.
A deformation of $C$ can be given the by the open cover:
$$\{W\times \bigtriangleup_0\}\cup \{Def_z |z \in C_{ns}\}$$
where $Def_z$ (to be defined below) is an open subset of
$$B(d_z)\times\bigtriangleup_0\ \cap\ (xy-t^2=0)\ \ \ \
\subset \ \ \bold{C}^2\times
\bigtriangleup_0$$
containing $(0,0,0)$. $Def_z$ is a local smoothing at the node $z$.
Define
$$K_z \ \ = \ \ B\paren{{d_z\over 2}, {d_z\over 2}+\epsilon_z}
\times
\bigtriangleup_0 \ \cap\ (xy=0)\ \ \subset \ \ \bold{C}^2\times
\bigtriangleup_0.$$
Note $B(r,s)$ is the
annulus.
Let $\mu$ denote the projection to $\bigtriangleup_0$.
For $\epsilon_z>0$ (small with respect to $\delta_z$)
and $|t|< \delta_z$, it is not hard
to find an isomorphism $\gamma_z$ commuting with $\mu$:
$$\gamma_z: K_z
\rightarrow D_z \ \ \subset B(d_z)\times\bigtriangleup_0\ \cap\ (xy-t^2=0)$$
such that
\begin{equation}
\label{duder}
B\paren{{d_z\over 3}}\times \bigtriangleup_0 \ \cap \ D_z = \emptyset
\end{equation}
and
$\gamma_z$ extends the identity on $t=0$.
Such a $\gamma_z$ can be constructed by considering
the holomorphic flow of a algebraic vector field on $(xy-t^2=0)$.
The space
$$B(d_z)\times\bigtriangleup_0 \ \cap\ (xy-t^2=0)\ \ \setminus \ D_z $$
is disconnected. Let $Def_z$ be the complement of the
component not containing $(0,0,0)$.
The isomorphism $\gamma_z$
determines a patching of $W\times \bigtriangleup_0$ and $Def_z$
along $K_z \simeq D_z$ in
the obvious manner. The constructed family satisfies claims
(1) and (2) of the Lemma.
$E_0$ can be extended trivially on $W\times \bigtriangleup_0$ to
$\cal{E}|_W$.
$\cal{E}|_{K_z}$ is trivial by condition (ii).
By Lemma (\ref{d2}), $m_z$ can be flatly extended to a line bundle $L_z$
on $Def_z$. By (\ref{duder}) and the construction of Lemma (\ref{d2}),
$L_z$ can be assumed to be
trivial on $D_z$. By patching
$$\bigoplus_{1}^{a_z} {\cal{O}}_{Def_z} \oplus \bigoplus _{a_z+1}^{r}
L_z$$
along $K_z\simeq D_z$,
$\cal{E}$ can be defined such that
$\cal{E}_0\cong E$ and $\cal{E}_{t\neq 0}$ is locally free.
Indeed, such a patching exists for $t=0$ by condition (ii).
The patching can be extended trivially along $K_z$ since
$$K_z= K_{z,t=0} \times \bigtriangleup_0.$$
Now condition (3) follows by Lemma (\ref{openy}).
For a general ground field, the
\'etale topology must be used.
\end{pf}
\begin{pr}
$\overline{U_g(e,r)}$ is an irreducible variety.
\end{pr}
\begin{pf}
Consider the morphism
$\pi_{SS}: Q^r_g(\mu,n=f_{e,r}(0),f_{e,r})_{[k,1]}^{SS} \rightarrow H_g$.
By Proposition (24) of chapter (1) of [Se], the
scheme
$$\pi_{SS}^{-1}([C])=Q_g(C,n, f_{e,r})^{SS}$$
is irreducible for each nonsingular curve $C$,
$[C]\in H_g$. Since the locus $H_g^0\subset H_g$ of nonsingular
curves is irreducible, $\pi_{SS}^{-1}(H_g^0)$ is irreducible.
By Lemma (\ref{deform}), $\pi_{SS}^{-1}(H_g^0)$ is dense in
$Q^r_g(\mu,n,f_{e,r})_{[k,1]}^{SS}$.
Finally, since there is a surjective morphism
$$Q^r_g(\mu,n,f_{e,r})_{[k,1]}^{SS} \rightarrow \overline{U_g(e,r)},$$
we conclude $\overline{U_g(e,r)}$ is irreducible.
\end{pf}
\subsection{The Normality of $\overline{U_g(e,1)}$}
We need an infinitesimal analogue of Lemma (\ref{d1}).
First, we establish some notation. Let
$$R= \bold{C}[[x,y]][\epsilon]/(xy-\epsilon,\epsilon^2).$$
Let $A= R/\epsilon R \cong \bold{C}[[x,y]]/(xy)$. Let
$m=(x,y)$ be the maximal ideal of $A$.
There is a natural, flat inclusion of rings
$\bold{C}[\epsilon]/(\epsilon^2) \rightarrow R$. For a
$\bold{C}[\epsilon]/(\epsilon^2)$-module $M$, let $*\epsilon$
denote the $M$ endomorphism given by multiplication by $\epsilon$.
\begin{lm}
\label{ey}
There does not exist a $\bold{C}[\epsilon]/(\epsilon^2)$-flat
$R$-module $E$ such that $$E/\epsilon E \cong m$$ as $A$-modules.
\end{lm}
\begin{pf}
Suppose such an $E$ exists. Let
$$\alpha: E \rightarrow E/\epsilon E \stackrel{\sim}{\rightarrow} m.$$
Let $e_x$, $e_y \in E$ satisfy $\alpha(e_x)=x$ and
$\alpha(e_y)=y$. We obtain a morphism
$$\beta : R\oplus R \rightarrow E $$
defined by $\beta(1,0)=e_x$ and $\beta(0,1)=e_y$.
Since $\epsilon$ is nilpotent,
$\beta$ is surjective by Nakayama's Lemma. We have an exact
sequence
$$0\rightarrow N \rightarrow R \oplus R \rightarrow E \rightarrow 0.$$
Since $E$
and $R\oplus R$ are $\bold{C}[\epsilon]/(\epsilon^2)$-flat,
$N$ is $\bold{C}[\epsilon]/(\epsilon^2)$-flat.
Hence, there is an {\em exact} sequence
\begin{equation}
\label{fyttf}
0 \rightarrow \epsilon N \rightarrow N \stackrel{* \epsilon}{\rightarrow} N
\end{equation}
obtained by tensoring
$0 \rightarrow (\epsilon) \rightarrow \bold{C}[\epsilon]/(\epsilon ^2)
\stackrel{* \epsilon}{\rightarrow} \bold{C}[\epsilon]/(\epsilon ^2)$
with $N$.
Since $\beta \bigl( (y,0) \bigr) \in \epsilon E$, there exists
an element $n=(y+ \epsilon f(x,y), \epsilon g(x,y))$
in $N$. Consider $x n \in N$:
$$xn = (xy+ \epsilon xf(x,y), \epsilon xg(x,y))=(\epsilon(1+
xf(x,y)), \epsilon x g(x,y)).$$
Note $\epsilon x n=0$.
By the exactness of (\ref{fyttf}), there exists an $\overline{n}\in N$
satisfying $\epsilon \overline{n}= xn$. Since
$R\oplus R$ is flat over $\bold{C}[\epsilon]/(\epsilon^2)$,
any such $\overline{n}$ must
be of the form
$$\overline{n}= (\ 1+xf(x,y) + \epsilon \hat{f}(x,y),\
xg(x,y)+\epsilon \hat{g}(x,y)\ ).$$
Since
$\alpha \circ \beta (\overline{n}) = x+x^2f(x,y) \neq 0$ in $m$,
$\ \overline{n}$ can not lie in $N$. We have a contradiction. No such $E$
can exist.
\end{pf}
To prove $\overline{U_g(e,1)}$ is normal, it suffices to
show $Q_g^1(\mu, n=f_{e,1}, f_{e,1})_{[k,1]}^{SS}$ is nonsingular.
The nonsingularity is established by computing the dimension of
$Q_g^1(\mu,n,f_{e,1})_{[k,1]}^{SS}$ and
then bounding the dimension of the Zariski tangent space at each point.
The Zariski tangent spaces are controlled by a study of the
differential $d\pi_{SS}$ where $\pi_{SS}$ is the canonical map
$$\pi_{SS}: Q_g^1(\mu, n, f_{e,1})_{[k,1]}^{SS} \rightarrow H_g.$$
\begin{lm}
$Q_g^1(\mu, n=f_{e,1}(0), f_{e,1})_{[k,1]}^{SS}$ is nonsingular.
\end{lm}
\begin{pf}
Consider the universal quotient sequence over
$Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS}$,
$$0 \rightarrow \cal{F} \rightarrow \bold{C}^n \otimes {\cal{O}}_{Q^{ss}\times U} \rightarrow \cal{E} \rightarrow 0.$$
Let
$\xi \in Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS}$
be a closed point and let $C=U_{\pi(\xi)}$.
$\xi$ corresponds to a quotient
\begin{equation}
\label{smooth}
0 \rightarrow \cal{F}_\xi \rightarrow \bold{C}^n \otimes {\cal{O}}_C \rightarrow \cal{E}_\xi \rightarrow 0.
\end{equation}
There is a natural identification of
the Zariski tangent space to $Q^1_g(C,n,f_{e,1})$ at $\xi$:
$$T_\xi(Q^1_g(C,n,f_{e,1})) \cong H^0(C,\underline{Hom}(\cal{F}_\xi,\cal{E}_\xi))$$
(see [Gr]).
If $h^1(C,\underline{Hom}(\cal{F}_\xi,\cal{E}_\xi))=0$
and the deformations of
\begin{equation}
\label{george}
\bold{C}^n \otimes {\cal{O}}_C \rightarrow \cal{E}_\xi \rightarrow 0
\end{equation}
are locally
unobstructed, then
$\xi$ is a nonsingular point of $Q^1_g(C,n,f_{e,1})$.
If $\cal{E}_\xi$ is locally free, the deformations of (\ref{george})
are locally unobstructed.
Sequence (\ref{smooth})
yields:
\begin{equation}
\label{exty}
0 \rightarrow \underline{Hom}(\cal{E}_\xi,\cal{E}_\xi) \rightarrow
\underline{Hom}(\bold{C}^n \otimes {\cal{O}}_C,\cal{E}_\xi)
\rightarrow \underline{Hom}(\cal{F}_\xi,\cal{E}_\xi) \rightarrow \underline{Ext}^1(\cal{E}_\xi,\cal{E}_\xi)
\rightarrow 0.
\end{equation}
Since $\underline{Ext}^1(\cal{E}_\xi,\cal{E}_\xi)$ is torsion and
$$h^1(C,\underline{Hom}(\bold{C}^n \otimes {\cal{O}}_C,\cal{E}_\xi)) =
n\cdot h^1(C,\cal{E}_\xi) =0,$$
we obtain $h^1(C,\underline{Hom}(\cal{F}_\xi,\cal{E}_\xi))=0$.
{}From (\ref{locy}), at each node $z\in C$, $\cal{E}_\xi$ is
either locally free or locally isomorphic to $m_z$.
Let $s$ be the number nodes where $\cal{E}_{\xi,z}\cong m_z$.
Using (\ref{exty}), we compute
$\chi(\underline{Hom}(\cal{F}_\xi,\cal{E}_\xi))$
in terms of $s$:
$$\chi(\underline{Hom}(\cal{F}_\xi,\cal{E}_\xi))= -\chi(\underline{Hom}(\cal{E}_\xi,\cal{E}_\xi))
+\chi( \underline{Hom}(\bold{C}^n \otimes {\cal{O}}_C,\cal{E}_\xi))
+\chi ( \underline{Ext}^1(\cal{E}_\xi,\cal{E}_\xi)).$$
It is clear
$\chi( \underline{Hom}(\bold{C}^n \otimes {\cal{O}}_C,\cal{E}_\xi))=n^2.$
Let $A=\bold{C}[[x,y]]/(xy)$ and $m=(x,y) \subset A$ as above.
It is not hard to establish:
\begin{equation}
\label{paul}
Ext_A^1(m,m)= \bold{C}^2,
\end{equation}
\begin{equation}
\label{doug}
0 \rightarrow A \stackrel{i}{\rightarrow} Hom_A(m,m) \rightarrow \bold{C} \rightarrow 0
\end{equation}
where $i$ is the natural inclusion.
Since $\underline{Ext}^1(\cal{E}_\xi,\cal{E}_\xi)$ is a torsion sheaf
supported at each $z\in C$ where $\cal{E}_{\xi,z}\cong m_z$ with
stalk (\ref{paul}),
$$\chi ( \underline{Ext}^1(\cal{E}_\xi,\cal{E}_\xi))= 2s.$$
There is a natural inclusion
$$0 \rightarrow {\cal{O}}_C \stackrel{i}{\rightarrow}
\underline{Hom}(\cal{E}_\xi,\cal{E}_\xi) \rightarrow \delta \rightarrow 0.$$
Since $\cal{E}_\xi$ is of rank $1$, $\delta$ is a torsion
sheaf supported at the nodes where $\cal{E}_\xi$ is not locally free.
At these nodes, $\delta$ can be determined locally by
(\ref{doug}). Hence
$$\chi(\underline{Hom}(\cal{E}_\xi,\cal{E}_\xi))= 1-g+s.$$
Summing the Euler characteristics yields:
$$h^0(C,\underline{Hom}(\cal{F}_\xi,\cal{E}_\xi))= g-1-s+ n^2 +2s= n^2-1+g+s.$$
If $C$ is a nonsingular curve,
$\cal{E}_\xi$ is locally free on $C$. The above results
show that $\xi$ is a nonsingular point of $Q^1_g(C,n,f_{e,1})$.
Thus $dim(Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS})= n^2-1+g+ dim(H_g)$.
Let $\xi \in Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS}$
be a closed point with the notation given above.
We examine the exact differential sequence:
$$0\rightarrow
T_\xi(\pi_{SS}^{-1}[C]) \rightarrow T_\xi(Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS})
\stackrel {d\pi_{SS}}{\rightarrow} T_{[C]}(H_g).$$
Recall $H_g$ is nonsingular.
By
Lemma (\ref{ey}) and the surjection of $T_{[C]}(H_g)$ onto
the miniversal deformation space of $C$,
$$dim( im (d\pi_{SS})) \leq dim(H_g)-s$$
($s$ is the number of nodes where $\cal{E}_\xi$ is not locally free).
By previous results:
$$dim(T_\xi(\pi_{SS}^{-1}[C]))= n^2-1+g+s.$$
It follows:
\begin{equation}
\label{zsde}
dim(T_\xi(Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS})) \leq (n^2-1+g+s)+(dim(H_g)-s)
\end{equation}
$$= dim(Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS}).$$
Equality must hold in (\ref{zsde}). $\xi$ is therefore
a nonsingular point of $Q^1_g(\mu,n,f_{e,1})_{[k,1]}^{SS}$.
\end{pf}
As a consequence, we obtain:
\begin{pr}
\label{popo}
$\overline{U_g(e,1)}$ is normal.
\end{pr}
\section{The Isomorphism Between $\overline{U_g(e,1)}$ and $\overline{P_{g,e}}$}
\label{lcap}
\subsection{A Review of $\overline{P_{g,e}}$}
\label{fofo}
In this section, the compactification of the
universal Picard variety of degree $e$
line bundles, $\overline{P_{g,e}}$, described in [Ca] is considered.
Let $e$ be large enough to guarantee
the existence and properties of $\overline{P_{g,e}}$ and $\overline{U_g(e,1)}$.
A natural isomorphism
$\nu : \overline{P_{g,e}} \rightarrow \overline{U_g(e,1)}$ will be constructed.
We follow the notation of [Ca], [Gi].
A Deligne-Mumford {\em quasi-stable}, genus $g$ curve $C$ is a
Deligne-Mumford semistable, genus $g$ curve with destabilizing
chains of length at most one.
Let $\psi:C\rightarrow C_s$ be the canonical
contraction to the Deligne-Mumford
stable model. For each
complete subcurve $D\subset C$, let $D^c=\overline{C \setminus D}$. Define:
$$k_D= \# (D \cap D^c).$$
Let $\omega_{C,D}$ be the degree of the canonical bundle
$\omega_C$ restricted to $D$. Let $L$ be a degree $e$ line bundle
on $C$.
Denote by $L_D$ the restriction of $L$ to $D$.
Let $e_D$ be the degree of
$L_D$. $L$ has
{\em (semi)stable multidegree} if for each complete, proper subcurve
$D\subset C$, the following holds:
\begin{equation}
\label{basinq}
e_D - e \cdot \paren{\omega_{C,D}\over 2g-2} \ \ \ (\leq) < \ \ \ k_D/2.
\end{equation}
Consider the Hilbert scheme $H_{g,e,M}$ of degree $e$, genus $g$
curves in $\bold P^M$ where $M=e-g+1$. In [Gi], it is shown there
exists an open locus
$Z_{g,e}\subset H_{g,e,M}$
parametrizing nondegenerate, Deligne-Mumford quasi-stable, genus
$g$ curves $C\subset \bold P^M$ satisfying:
\begin{enumerate}
\item[(i.)] $h^1(C, {\cal{O}}_C(1))= 0$.
\item[(ii.)] ${\cal{O}}_C(1)$ is of semistable multidegree on $C$.
\end{enumerate} In [Ca], $\overline{P_{g,e}}$ is constructed as the
G.I.T. quotient
$$\overline{P_{g,e}} \stackrel{\sim}{=} Z_{g,e}/ SL_{M+1}$$
(for a suitable linearization). $\overline{P_{g,e}}$ is a moduli space of
line bundles of semistable multidegree on Deligne-Mumford quasi-stable curves
(up to equivalence) compactifying the universal Picard variety.
The construction of the isomorphism $\nu: \overline{P_{g,e}} \rightarrow
\overline{U_g(e,1)}$ proceeds as follows. If $L$ is a very ample
line bundle of semistable multidegree on a Deligne-Mumford
quasi-stable curve $C$, then $\psi_*(L)$ is a slope-semistable
torsion free sheaf of uniform rank $1$ on $C_s$. This is the
result of Lemma (\ref{ms}). The map $\nu$ is constructed by globalizing
this correspondence.
There is a universal
curve
$$U_Z \hookrightarrow Z_{g,e}\times \bold P^M.$$
A deformation study shows $Z_{g,e}$ and $U_Z$ are nonsingular
quasi-projective varieties ([Ca], Lemma 2.2, p.609). There exists
a canonical contraction map $\psi:
U_Z \rightarrow U^s_Z$ over $Z_{g,e}$.
The map $\psi$ contracts each fiber of $U_Z$
over $Z_{g,e}$ to its Deligne-Mumford
stable model. $U^s_Z$ is a flat, projective family of Deligne-Mumford
stable curves over $Z_{g,e}$.
Let $L={\cal{O}}_{U_Z}(1)$ and
$\cal{E}= \psi_*(L)$. In Lemma (\ref{flaty}), $\cal{E}$ is shown to be a
flat family of slope-semistable torsion free sheaves of uniform rank $1$
and degree $e$ over $Z_{g,e}$. Care is required in establishing flatness.
The argument depends upon
Zariski's theorem on formal functions and the criterion of
Lemma (\ref{tpt}). By the universal property of $\overline{U_g(e,1)}$, $\cal{E}$
induces a map $\nu_Z: Z_{g,e} \rightarrow \overline{U_g(e,1)}$. Since $\nu_Z$ is
$SL_{M+1}$-invariant, a map $\nu: \overline{P_{g,e}} \rightarrow \overline{U_g(e,1)}$ is
obtained.
It remains to prove $\nu$ is an isomorphism. Since $\overline{U_g(e,1)}$ is
normal by Proposition (\ref{popo}), it suffices to show $\nu$ is bijective.
Surjectivity is clear. Injectivity is established in section (\ref{ender}).
\subsection{The Construction of $\nu$}
Multidegree (semi)stability corresponds to
slope-(semi)stability in the following manner:
\begin{lm}
\label{ms}
Let $C$ be a Deligne-Mumford quasi-stable curve.
If $L$ is a very ample, degree $e$ line bundle on $C$ of
(semi)stable multidegree, then
$E=\psi_*(L)$ is a slope-(semi)stable, torsion free
sheaf of uniform rank $1$ and degree $e$ on $C_s$. Also, if
$L$ is of strictly semistable multidegree, then $E$ is strictly
slope-semistable.
\end{lm}
First we need a simple technical result. For each complete
subcurve $D$ of $C$,
define the sheaf $F_D$ on $C$ by the sequence:
$$ 0 \rightarrow F_D \rightarrow L \rightarrow L_{D^c} \rightarrow 0.$$
$F_D$ is the subsheaf of sections of $L$ with support on $D$.
In fact, $F_D$ is exactly the subsheaf of sections of $L_D$
vanishing on $D \cap D^c$. Therefore $degree(F_D)=e_D-k_D$.
Note by Riemann-Roch, $\chi(F_D)=degree(F_D)+1-g_D$ where
$g_D$ is the arithmetic genus of $D$. We obtain,
\begin{equation}
\label{dami}
e_D= \chi(F_D) + g_D -1 +k_D.
\end{equation}
\begin{lm}
\label{tec}
\sloppy
Let $C$ be a Deligne-Mumford quasi-stable curve.
Let $L$ be a very ample line bundle of semistable multidegree.
For every complete subcurve $D\subset C$, $R^1\psi_*(F_D)=0$.
\end{lm}
\begin{pf}
A fiber of $\psi$ is either a point or a destabilizing
$\bold P^1$. By inequality (\ref{basinq}) and ampleness,
the restriction of $L$ to a destabilizing $\bold P^1$ is
${\cal{O}}_{\bold P^{1}}(1)$. Let $P$ be a destabilizing
$\bold P^1$ of $C$. There are five cases
\begin{enumerate}
\item [(i.)] $P\subset D^c$, $P\cap D =\emptyset$. Then $F_D|_P=0$.
\item [(ii.)] $P\subset D^c$, $P\cap D \neq \emptyset$.
Then $F_D|_P$ is torsion.
\item [(iii.)] $P\subset D$, $P \cap D^c = \emptyset$. Then $F_D|_P={\cal{O}}_P(1)$.
\item [(iv.)] $P\subset D$, $\#|P \cap D^c|=1$. Then
$F_D|_P= {\cal{O}}_P$.
\item [(v.)] $P\subset D$, $\# |P \cap D^c|=2$. Then $F_D|_P={\cal{O}}_P(-1)$.
\end{enumerate}
In each case, $h^1(P,F_D|_P)=0$.
The
vanishing of $h^1(P,F_D|_P)$
and the simple character of $\psi$ imply $R^1\psi_*(F_D)=0$.
\end{pf}
\begin{pf} [Of Lemma (\ref{ms})]
Let $C^1$ be the union of the destabilizing $\bold P^1 \ 's$ of
$C$. By Lemma (\ref{tec}), $R^1\phi_*(F_{C^1})=0$.
Since each destabilizing
$P$ in $C^1$ is of type (v) in the proof of
Lemma (\ref{tec}),
$F_{C^1}|_P={\cal{O}}_{P}(-1)$. It follows that
$\psi_*(F_{C^1}) = 0.$
By the long exact sequence associated to
$$0 \rightarrow F_{C^1} \rightarrow L \rightarrow L_{C^{1c}} \rightarrow 0,$$
it follows
$E \cong \psi_*( L_{C^{1c}})$. Since $L_{C^{1c}}$ is torsion
free on $C^{1c}$ and the morphism $C^{1c} \rightarrow C_s$ is finite,
$E$ is torsion free. Certainly, E is of uniform rank $1$.
Let $0 \rightarrow G \rightarrow E$
be a proper subsheaf. Let $D_s\subset C_s$ be the support of
$G$. If $D_s=C_s$, the inequality of slope-stability follows
trivially. We can assume $D_s$ is a complete, proper subcurve. Let
$D= \psi^{-1}(D_s)$. $D\subset C$ is a complete, proper subcurve.
It is clear that
$G$ is a subsheaf of $\psi_*(F_D)$ with torsion quotient. Therefore, it
suffices to check slope-(semi)stability for $\psi_*(F_D)$. Certainly
$$h^0(C_s,\psi_*(F_D)) = h^0(C,F_D).$$
By Lemma (\ref{tec}), $R^1\psi_*(F_D)=0$. Thus by a
degenerate Leray spectral sequence ([H], Ex. 8.1, p.252),
$$h^1(C_s,\psi_*(F_D)) = h^1(C,F_D).$$
Hence $\chi(F_D)=\chi(\psi_*(F_D))$.
Similarly $\chi(L)=\chi(\psi_*(L))$.
Inequality (\ref{basinq}) and equation (\ref{dami}) yield:
$$(\chi(F_D)+g_D-1+k_D) - (\chi(L)+g-1)
\cdot \paren{2g_D-2+k_D \over 2g-2}\ (\leq)
< \ k_D/2.$$
After some manipulation, we see
$${\chi(F_D) \over \omega_{D,C}} \ \ (\leq) < \ \ {\chi(L)\over 2g-2}\ .$$
The above results yield
$${\chi(\psi_*(F_D)) \over \omega_{D_s,C_s}}
\ \ (\leq) < \ \ {\chi(\psi_*(L)) \over 2g-2}\ .$$
Hence, $\psi_*(L)$ is slope-(semi)stable. The final claim about
strict semistability also follows from the proof.
\end{pf}
Let $\psi: U_Z \rightarrow U_Z^s$, $L={\cal{O}}_{U_Z}(1)$, and $\cal{E}=\psi_*(L)$ be
as defined in section (\ref{fofo}). We now establish that $\cal{E}$ is flat
over
$Z_{g,e}$. The vanishing of $R^1\psi_*(L)$ is proved by
Zariski's theorem on formal functions in Lemma (\ref{zyz}).
The flatness criterion of Lemma (\ref{tpt}) is
then applied to obtain the required flatness.
First we need an auxiliary result.
Let $[C]\in Z_{e,g}$ and
let $P\subset U_Z$ be a destabilizing
$\bold P^1$ of $C$. The
conormal bundle, $N_P^*$, of $P$ in $U_Z$ is locally free ($P$, $U_Z$ are
nonsingular).
Recall
a locally free sheaf $\bigoplus {\cal{O}}_{\bold P^1}(a_i)$ on
$\bold P^1$ is said to be non-negative if each
$a_i\geq 0$.
\begin{lm}
\label{tyt}
$N_P^*$ is non-negative.
\end{lm}
\begin{pf}
Let $T_U$ and $T_Z$ denote the
tangent sheaves of $U_Z$ and $Z_{e,g}$. Let $\rho:U_Z\rightarrow Z_{e,g}$
be the natural morphism. There is a differential map
$$d\rho: T_U \rightarrow \rho^*(T_Z).$$
Restriction to $P$ yields a (non-exact) sequence
$$0\rightarrow T_P \rightarrow T_U|_P \rightarrow \rho^*(T_Z)|_P .$$
Certainly, $\rho^*(T_Z)|_P \cong \bigoplus {\cal{O}}_P$.
We obtain a map
$$\alpha: N_P\rightarrow \bigoplus {\cal{O}}_P.$$
Let $\hat{P}\subset P$ be the locus of
nonsingular point of $C$. Since the morphism $\rho$ is smooth on $\hat{P}$,
$\alpha |_{ \hat{P}}$ is an isomorphism of sheaves. Since
$N_P$ is a torsion free sheaf, $\alpha$ must be an
injection of sheaves. It follows easily $N_P$ is
non-positive. Hence $N_P^*$ is non-negative.
\end{pf}
In fact, an examination of the deformation theory yields
$N_P^* \cong {\cal{O}}_P(1) \oplus {\cal{O}}_P(1) \oplus I$
where $I$ is a trivial bundle. We will need only
the non-negativity result.
\begin{lm}
\label{zyz}
$R^1\psi_*(L)=0.$
\end{lm}
\begin{pf}
Let $\zeta \in U_{Z}^s$. It suffices to prove
$$R^1\psi_*(L)_\zeta=0$$
in case $\zeta$ is a node of stable curve destabilized
in $U_Z$. Let $m$ be the ideal of $\zeta$.
$\psi^{-1}(m)$ is the ideal of the nonsingular
destabilizing $P=\bold P^1$.
Let $P_n$ denote the subscheme of $U_Z$ defined
by $\psi^{-1}(m^n) \cong \psi^{-1}(m)^n$. Let
$L_n$ be the restriction of $L$ to $P_n$.
By
Zariski's Theorem on formal functions:
$$R^1\psi_*(L)_\zeta\hat{\ } \ \ \cong \ \ \stackrel {lim}{\leftarrow}
H^1(P_n,L_n).$$
Since completion is faithfully flat for noetherian local
rings,
it suffices to show for each $n\geq 1$, $h^1(P_n,L_n)=0$. As above,
denote the conormal bundle of $P$ in $U_Z$ by
$N_P^*$. Since the varieties in question are nonsingular, there
is an isomorphism on $P$:
$$m^{n-1}/m^n \cong Sym^{n-1} (N_P^*).$$
Since the pair $(C,L_C)$ is of semistable multidegree and
$P$ is a destabilizing $\bold P^1$,
$L_1\cong {\cal{O}}_P(1)$. Hence $h^1(P_1, L_1)=0.$
There is an exact sequence on $P_n$ for each $n\geq 2$:
$$0 \rightarrow m^{n-1}/m^n \otimes L_n \rightarrow L_n \rightarrow L_{n-1} \rightarrow 0.$$
There is a natural identification
$$m^{n-1}/m^n\otimes L_n \cong Sym^{n-1}(N_P^*) \otimes_{{\cal{O}}_P} {\cal{O}}_P(1).$$
{}From the non-negativity of $N_P^*$ (Lemma (\ref{tyt})), we see
$$h^1(P_n,m^{n-1}/m^n \otimes L_n)=0.$$
By the induction hypothesis $$h^1(P_n,L_{n-1})=h^1(P_{n-1},L_{n-1})=0.$$
Thus $h^1(P_n,L_n)=0$.
The proof is complete.
\end{pf}
\begin{lm}
\label{tpt}
Let $\phi: B_1 \rightarrow B_2$ be a projective morphism of schemes over $A$.
If $F$ is a sheaf on $B_1$ flat over $A$ and
$\forall \ i\geq 1$, $\ R^i\phi_*(F)=0$, then
$\phi_*(F)$ is flat over $A$.
\end{lm}
\begin{pf}
We can assume $A$ and $B_2$ are affine and $B_1 \cong \bold P^k_{B_2}$.
Let $\cal{U}$ be the standard $k+1$ affine cover of $B_2$.
There is a Cech resolution computing the cohomology of
$F$ on $B_1$:
\begin{equation}
\label{cech}
0\rightarrow H^0(B_1,F) \rightarrow C^0(\cal{U},F) \rightarrow C^1(\cal{U},F) \rightarrow
\ldots \rightarrow C^k(\cal{U},F) \rightarrow 0.
\end{equation}
Since $R^i\phi_*(F)=0$ for $i\geq 1$, the resolution
(\ref{cech}) is exact. Since $F$ is $A$-flat, the
Cech modules $C^j(\cal{U},F)$ are all $A$-flat. Exactness
of (\ref{cech}) implies $H^0(B_1,F) \cong \phi_*(F)$ is
$A$-flat.
\end{pf}
\begin{lm}
\label{flaty}
$\cal{E}$ is a flat family of slope-semistable, torsion free
sheaves of uniform rank $1$ over $Z_{e,g}$.
\end{lm}
\begin{pf}
By Lemmas (\ref{zyz}) and (\ref{tpt}), $\cal{E}$ is
flat over $Z_{e,g}$.
Let $[C]\in Z_{e,g}$. We have a diagram:
\begin{equation}
\begin{CD}
C @>{i_C}>> U_Z \\
@VV{\psi_C}V @VV{\psi}V \\
C_s @>>{i_{C_s}}> U^s_Z
\end{CD}
\end{equation}
If $i_{C_s}^*(\cal{E}) \cong \psi_{C*}(i_C^*(L))$,
then the proof is complete by Lemma (\ref{ms}).
There is a natural morphism of sheaves
$$\gamma_C : i_{C_s}^*(\cal{E}) \rightarrow
\psi_{C*}(i_C^*(L)). $$
We first show $\gamma_C$ is a surjection. Since, by Lemma (\ref{ms}),
$\psi_{C*}(i_C^*(L))$ is a slope-semistable torsion free
sheaf of degree $e$, $\psi_{C*}(i_C^*(L))$ is generated by
global sections. There is a natural identification
$$H^0(C_s,\psi_{C*}(i_C^*(L)))\cong H^0(C,{\cal{O}}_C(1)).$$
By the nondegeneracy
of $C$ and the nonspeciality of ${\cal{O}}_C(1)$,
$H^0(C,{\cal{O}}_C(1))$ is canonically isomorphic
to $H^0(\bold P^M,{\cal{O}}_{\bold P^M}(1))$. These sections extend
over $U_Z$ and thus appear in $i_{C_s}^*(\cal{E})$.
Therefore, $\gamma_C$ is a surjection.
Since $\psi$ is an isomorphism except at
destabilizing $\bold P^{1} \ 's$ of $U_Z$,
the kernel of $\gamma_C$ is a torsion sheaf on $C_s$.
Flatness of
$\cal{E}$ over $Z_{e,g}$ implies
$\chi(i_{C_s}^*(\cal{E}))$ is independent of $[C]\in Z_{e,g}$.
By Lemma (\ref{ms}), $\chi(\psi_{C*}(i_C^*(L)))$ is independent
of $[C]\in Z_{e,g}$.
Over the open locus of nonsingular curves $[C]\in Z_{e,g}$,
$\psi$ is an isomorphism
thus:
\begin{equation}
\label{comb}
\chi(i_{C_s}^*(\cal{E})) =
\chi(\psi_{C*}(i_C^*(L)) ).
\end{equation}
By the above considerations, (\ref{comb})
holds for every $[C] \in Z_{e,g}$. Hence, the torsion kernel
of $\gamma_C$ must be zero.
$\gamma_C$ is an isomorphism. The proof is complete.
\end{pf}
By combining Lemma (\ref{flaty}) with Theorem (\ref{thwee}),
there exists a natural morphism $\nu_Z: Z_{e,g} \rightarrow
\overline{U_g(e,1)}$. Since $\nu_Z$ is $SL_{M+1}$-invariant, $\nu_Z$ descends to
$\nu: \overline{P_{g,e}} \rightarrow \overline{U_g(e,1)}$.
Certainly $\nu$ is surjective.
Since $\overline{U_g(e,1)}$ is normal, $\nu$ is an isomorphism
if and only if $\nu$ is injective. The injectivity of $\nu$ will
be established in section (\ref{ender}).
\subsection{Injectivity of $\nu$}
\label{ender}
Let $C$ be a Deligne-Mumford quasi-stable genus $g$ curve.
Let $D\subset C$ be a complete subcurve.
A node $z\in D$ is an {\em external} node of $D$ if
$z \in D^c$. $P\subset D$ is a {\em destabilizing $\bold P^1$ of
$D$} if $P$ is a destabilizing $\bold P^1$ of $C$.
A destabilizing
$\bold P^1$ of $D$ is {\em external} if
$P \cap D^c\neq \emptyset$.
Let $\hat{D}$ denote $D$ minus all the
external destabilizing $\bold P^1 \ 's$ of $D$.
$(C,L)$ is a {\em semistable pair} if $C$ is Deligne-Mumford
quasi-stable and
$L$ is a very ample line bundle of semistable
multirank. The semistable pairs
$(C,L)$ and $(C',L')$ are isomorphic if there exists an
isomorphism $\gamma:C\rightarrow C'$ such that $\gamma ^*(L')\cong L$.
A complete subcurve
$D\subset C$
is an {\em extremal subcurve} of the semistable pair
$(C,L)$ if equality holds in (\ref{basinq}):
$$e_D - e \cdot \paren{ \omega_{C,D} \over 2g-2} \ = \ k_D/2.$$
The semistable pair $(C,L)$ is said to be {\em maximal}
if the following condition is satisfied:
if $z \in C$ is an external node of an extremal subcurve,
$z$ is contained in a destabilizing $\bold P^1$.
Let $\psi: C \rightarrow C_s$ be the stable contraction.
By Lemma (\ref{ms}), $\psi_*(L)$ is a slope-semistable,
torsion free sheaf of uniform rank $1$. Let
$J(\psi_*(L))$ be the associated set of slope-stable Jordan-Holder
factors.
$(C,J)$ is a {\em Jordan-Holder pair} if $C$ is a Deligne-Mumford
stable curve and $J$ is a set of slope-stable, torsion free
sheaves. As before, the Jordan-Holder pairs $(C,J)$ and $(C',J')$
are isomorphic if there exists an isomorphism $\gamma:C \rightarrow C'$ such that
$\gamma ^*(J') \cong J$.
\begin{lm}
\label{det}
Let $(C,L)$, $(C',L')$ be maximal semistable pairs. If
$(C_s, J(\psi_*(L)))$ and $(C'_s, J(\psi'_*(L')))$ are
isomorphic Jordan-Holder pairs, then $(C,L)$ and $(C',L')$
are isomorphic semistable pairs.
\end{lm}
\begin{pf}
Consider a Jordan-Holder filtration of $E=\psi_*(L)$ on $C_s$:
$$0 = E_0 \subset E_1 \subset E_2 \subset \ldots \subset E_n=E.$$
Let $A_i=Supp(E_i)$. For each $1\leq i \leq n$,
$${\chi(E_i) \over w_{A_i,C_s}} = {\chi(E) \over 2g-2}.$$
By the proof of Lemma (\ref{ms}), we
see the $B_i=\psi^{-1}(A_i)$ are extremal subcurves of $(C,L)$
for $1\leq i\leq n$.
As before,
let $F_{B_i}$ be the subsheaf of sections of $L$ with support
on $B_i$. From the proof of Lemma (\ref{ms}),
it follows $E_i \cong \psi_*(F_{B_i})$. For $1\leq i \leq n$,
let $X_i= \overline{A_i \setminus A_{i-1}}$ and
$Y_i=\overline{B_i \setminus B_{i-1}}$. We see
$Supp(E_i/E_{i-1})=X_i$. If $z\in X_i$ is an internal node of $X_i$,
$z$ is destabilized by $\psi$ if and only if $(E_i/E_{i-1})$ is
locally isomorphic to $m_z$ at $z$. If $z\in X_i$ is an
external node, then there are two cases.
If $z\in A_{i-1}$, then $\psi^{-1}(z) \cong \bold{P}^1 \not\subset Y_i.$
If $z\in A_i^c$, then $\psi^{-1}(z) \cong \bold{P}^1 \subset Y_i.$
These conclusions follow from the maximality of $(C,L)$.
It is now clear $C_s$ and the Jordan-Holder factor $E_i/E_{i-1}$ determine
$\hat{Y_i}$ completely. Also, the $\hat{Y_i}$ are connected by
destabilizing $\bold P^1 \ 's$. We have shown
$(C_s, J)$ determines $C$ up to isomorphism.
We will show
below in Lemmas (\ref{mjk}-\ref{pushiso}) that $L_{\hat{Y_i}}$
is determined up to isomorphism by $E_i/E_{i-1}$.
Since the $\hat{Y}_i$ are connected by destabilizing ${\bold P^{1}}\ 's$, the
isomorphism class of the pair $(C,L)$ is determined by the
line bundles
$L_{\hat{Y_i}}$.
This completes the proof of the Lemma.
\end{pf}
\begin{lm}
\label{mjk}
There is an isomorphism
$\psi_*(L_{\hat{Y_i}}) \cong E_i/E_{i-1}.$
\end{lm}
\begin{pf}
We keep the notation of the previous Lemma.
Certainly $Supp(F_{B_i}/F_{B_{i-1}})=Y_i$.
Let $p_{i}$ be the divisor $B_{i-1}\cap Y_i \subset Y_i$.
Let $q_i$ be the divisor $ B_i^c\cap Y_i \subset Y_i$.
Since the points of $p_{i}$ lie on destabilizing $\bold P^1$'s joining
$Y_i$ to $Y_{i-1}$ and the points of $q_{i}$ lie on
destabilizing $\bold P^1$'s joining $Y_i$ to $Y_{i+1}$, we
note $p_i \cap q_i = \emptyset$.
There is a isomorphism $L_{Y_i} \cong F_{Y_i}\otimes {\cal{O}}_{Y_i} (p_i +q_i)$.
Since there is an exact sequence:
$$0 \rightarrow F_{Y_i} \rightarrow (F_{B_i}/F_{B_{i-1}}) \rightarrow p_i \rightarrow 0,$$
we see $F_{Y_i}= (F_{B_i}/F_{B_{i-1}})\otimes {\cal{O}}_{Y_i}(-p_i)$.
Thus $$L_{Y_i} \cong (F_{B_i}/F_{B_{i-1}})\otimes {\cal{O}}_{Y_i} (q_i).$$
Since $B_i$ is an extremal subcurve of $C$ and the pair $(C,L)$ is
maximal, we see $q_i$ lies on external destabilizing $\bold P^1 \ 's$
of $Y_{i}$. Hence $L_{\hat{Y_i}}$ is
isomorphic to
$(F_{B_i}/F_{B_{i-1}})_{\hat{Y_i}}$.
We have an exact sequence on $Y_i$:
$$0 \rightarrow I_{\hat{Y_i}} \otimes (F_{B_i}/F_{B_{i-1}}) \rightarrow
(F_{B_i}/F_{B_{i-1}}) \rightarrow (F_{B_i}/F_{B_{i-1}})_{\hat{Y_i}} \rightarrow 0.$$
Let $P$ be an external destabilizing $\bold P^1$ of $Y_i$.
If $P$ meets $B_{i-1}$ then $P\subset B_{i-1}$. Thus
each such $P$ meets $B_i^c$. It is now not hard to see
$ I_{\hat{Y_i}} \otimes (F_{B_i}/F_{B_{i-1}})$ restricts
to ${\cal{O}}_P(-1)$ on each such $P$. Therefore, by familiar arguments,
$$\psi_*( (F_{B_i}/F_{B_{i-1}})_{\hat{Y_i}}) \cong
\psi_*( F_{B_i}/F_{B_{i-1}} ).$$
By Lemma (\ref{tec}),
$R^1\psi_*(F_{B_{i-1}})=0$. Hence
$$\psi_*(F_{B_i}/F_{B_{i-1}})
\cong (\psi_*(F_{B_i})/\psi_*(F_{B_{i-1}})) \cong
E_i/E_{i-1}.$$
Following all the isomorphisms yields the Lemma.
\end{pf}
\begin{lm}
\label{pushiso}
Let $(C,L)$ be a semistable pair.
Let $D\subset C$ satisfying $\hat{D}=D$.
Then $L_D$ is determined up to isomorphism by $\psi_*(L_D)$.
\end{lm}
\begin{pf}
Let $D^1$ be the union of the
destabilizing $\bold P^1\ 's$ of $D$.
Note all these $\bold P^1 \ 's$ are internal. Let
$D'= \overline{D \setminus D^1}$ and
denote the restriction of $\psi$ to $D'$ by $\psi'$.
Consider the sequence on $D$:
$$0 \rightarrow I_{D'}\otimes L_D \rightarrow L_D \rightarrow L_{D'} \rightarrow 0.$$
Since $I_{D'}\otimes L_D$ restricts to ${\cal{O}}_{\bold P^1}(-1)$ on
each
destabilizing $\bold P^1$ of $D$, we see
$$ \psi_* (I_{D'}\otimes L_D)= R^1 \psi_* (I_{D'} \otimes L_D)=0.$$
Thus $\psi'_*(L_{D'})\cong
\psi_*(L_{D'}) \cong \psi_*(L_D)$. Since
$\psi'$ is a finite affine morphism,
$$\beta: \psi'\ ^*(\psi'_*(L_{D'})) \rightarrow L_{D'} \rightarrow 0.$$
Let $\tau $ be the torsion subsheaf of $\psi'\ ^*(\psi'_*(L_{D'}))$.
Since $\beta$ is generically an isomorphism and $L_{D'}$ is
torsion free on $D'$, we see
$$L_{D'} \cong (\psi'\ ^*(\psi'_*(L_{D'}))/ \tau).$$
We have shown that $L_{D'}$ is determined up to isomorphism by
$\psi_*(L_D)$. It is clear that $L_{D'}$
determines $L_D$ up to isomorphism.
\end{pf}
Let $\rho: Z_{e,g} \rightarrow \overline {P_{g,e}}$ be the
quotient map.
Let $\zeta \in \overline{P_{g,e}}$. It follows from
the results of [Ca] (Lemma 6.1, p.640)
that there exists a $[C]\in Z_{g,e}$ satisfying:
\begin{enumerate}
\item[(i.)]$\rho([C]) = \zeta$.
\item[(ii.)] $(C, L={\cal{O}}_C(1))$ is a maximal semistable pair.
\end{enumerate}
Let $\psi_C:C\rightarrow C_s$ be the stable contraction.
Let $E=\psi_*(L)$.
Let $J$ be the Jordan-Holder factors of $E$ on $C_s$. From the
definition of $\nu$,
$\nu(\zeta)$ is the element of $\overline{U_g(e,1)}$ corresponding to
the isomorphism class of the data $(C_s,J)$.
By Lemmas (\ref{mjk}-\ref{pushiso}), the isomorphism
class of $(C,L)$ is determined by the isomorphism class of
$(C_s, J)$. Therefore
$\nu$ is injective. By the previous discussion,
$\nu$ is an isomorphism.
\begin{tm} There is a natural isomorphism
$\nu: \overline{P_{g,e}} \rightarrow \overline{U_g(e,1)}$.
\end{tm}
\hbadness=9999
|
1996-03-31T05:43:58 | 9502 | alg-geom/9502006 | en | https://arxiv.org/abs/alg-geom/9502006 | [
"alg-geom",
"hep-th",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9502006 | Alexander A. Voronov | Takashi Kimura, Jim Stasheff, and Alexander A. Voronov | Homology of moduli spaces of curves and commutative homotopy algebras | Terminology corrections and minor changes made, 18 pages, AMS-LaTeX | null | null | null | null | We propose an explicit relation between the cohomology of compactified and
noncompactified moduli spaces of algebraic curves with punctures. This
relationship generalizes one between commutative algebras and Lie algebras
proposed by Lazard, Kontsevich and Ginzburg and Kapranov. More explicitly, we
show that the cohomology of one moduli space is the graph complex decorated
with the cohomology of the other, which in its turn generalizes Getzler's
result for the genus zero moduli. We also show that a certain class of algebras
over the chain operad of compactified moduli spaces has a natural structure of
a commutative homotopy algebra. This may be regarded as a lifting of the dot
product of Lian-Zuckerman's BV algebra structure to the cochain level, thus
complementing the results of an earlier paper hep-th/9307114 of the authors
regarding the BV bracket.
| [
{
"version": "v1",
"created": "Wed, 8 Feb 1995 22:42:01 GMT"
},
{
"version": "v2",
"created": "Thu, 13 Apr 1995 22:01:27 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kimura",
"Takashi",
""
],
[
"Stasheff",
"Jim",
""
],
[
"Voronov",
"Alexander A.",
""
]
] | alg-geom | \section{$C_\infty$ operad and $C_\infty$-algebras}
Let us begin with a definition of an operad.
\begin{df}
An {\sl operad ${ \cal O} = \{\,{ \cal O}(n)\,\}_{n\geq 0}$ with unit} is a
collection of objects (topological spaces, complexes, etc.) such that each
${ \cal O}(n)$ has an action of $S_n$, the permutation group on $n$ elements ($S_0$
is contains only the identity), and a collection of operations for $n\geq 1$
and $1\leq i\leq n$, ${ \cal O}(n)\times{ \cal O}(n')\,\to\,{ \cal O}(n+n'-1)$ given by
$(f,f')\,\mapsto\,f\circ_i f'$ satisfying
\begin{enumerate}
\item if $f\in{ \cal O}(n),$ $f'\in{ \cal O}(n')$, and $f''\in{ \cal O}(n'')$ where $1\leq i <
j \leq n$, $n', n''\geq 0,$ and $n\geq 2$ then
\begin{equation}
\label{eq:nosign}
(f\circ_i f')\circ_{j+n'-1} f'' = (-1)^{|f'| |f''|} (f\circ_j f'')\circ_i f'
\end{equation}
where signs on the right hand side should be ignored if ${ \cal O}$ is not an
operad of (graded) vector spaces,
\item if $f\in{ \cal O}(n),$ $f'\in{ \cal O}(n')$, and $f''\in{ \cal O}(n'')$ where $n,n'\geq
1,$ $n''\geq 0$, and $i = 1, \ldots, n$ and $j = 1, \ldots, n'$ then
\begin{equation*}
(f\circ_i f')\circ_{i+j-1} f'' = f\circ_i (f'\circ_j f''),
\end{equation*}
\item the composition maps are equivariant under the action of the
permutation groups,
\item there exists an element $\bf{I}$ in ${ \cal O}(1)$ called the {\sl
unit} such that for all $f$ in ${ \cal O}(n)$ and $i=1,\ldots,n$,
\begin{equation*}
\bf{I}\circ_1 f = f = f\circ_i \bf{I}
\end{equation*}
\end{enumerate}
\end{df}
By iterating $k$ composition maps, one obtains the more common form of the
operad composition, $\gamma\,:\,\Oo{k} \times \Oo{n_1} \times\cdots\times
\Oo{n_k}\,\to\,\Oo{n_1 + \cdots + n_k}.$
\begin{ex} The {\sl endomorphism operad $\End{V}:=\{\End{V}(n)\}$ of a
differential graded $k$-vector space $(V,Q)$} is defined to be $\End{V}(n) :=
\operatorname{Hom}_k(V^{\otimes n},V)$ where $S_n$ acts naturally upon $V^{\otimes n}$ and
the composition maps are given by
\begin{multline}
(f\circ_i f')(v_1\otimes \ldots \otimes v_{n+n'-1}) \\
=\, \pm
f(v_1\otimes\ldots\otimes v_{i-1}\otimes f'(v_i\otimes\ldots v_{i+n'})\otimes
v_{i+n'+1}\otimes v_{n+n'-1})
\end{multline}
for all $f$ in $\End{V}(n)$, $f'$ in $\End{V}(n')$ and $i = 1,\ldots, n$
for all $n,n'$, and the unit $e$ is just the identity map from $V$ to itself.
The $\pm$ is the sign which is obtained by sliding $f'$ through $v_1, \ldots,
v_{i-1}$. Let us denote the component of $\End{V}(n)$ with degree $g$ by
$\End{V}^g(n)$.
\end{ex}
Algebraic structures on a differential graded vector space $V$ are often
parametrized by an operad through the following notion.
\begin{df}
Let ${ \cal O}$ be an operad. A differential graded $k$-vector space,
$(V,Q)$, is said to be an {\sl ${ \cal O}$-algebra} if there is a morphism of
operads ${ \cal O}\,\to\,\End{V}$.
\end{df}
Notice that given an operad of topological spaces ${ \cal O}$, the singular chains
on ${ \cal O}$, $C_\bullet({ \cal O}):=\{C_\bullet({ \cal O}(n))\}$, naturally forms an operad
of differential graded vector spaces and, consequently, so do the homology
groups $H_\bullet({ \cal O}):=\{\,H_\bullet({ \cal O}(n))\,\}.$ Furthermore, if $(V,Q)$ is
an algebra over $C_\bullet({ \cal O})$, then $H_\bullet(V)$ is naturally an algebra
over $H_\bullet({ \cal O})$. However, there is more information in the
$C_\bullet({ \cal O})$-algebra, $(V,Q),$ than at the cohomology level. This
provides motivation for the study of algebraic structures up to homotopy, the
first of which we now recall.
\begin{df}
An {\sl $A_\infty$ (or strongly homotopy associative) algebra}, $V$, is a
complex $Q:V^g\,\to\,V^{g-1}$ endowed with a collection of $n$-ary (linear)
operations $\{\,m_n:V^{\otimes n}\,\to\,V\,\}_{n\geq 2}$ with $m_n$ having
degree $n-2$ satisfying
\begin{multline}
\label{eq:Aia}
Q (m_n(v_1,\ldots,v_n)) - (-1)^n \sum_{k=1}^n (-1)^{\epsilon(k)}
m_n(v_1,\ldots , Q v_k, \ldots, v_i) \\ = \sum_{r,s,k} (-1)^{k(s-1) + s n} \,
(m_r\circ_k m_s) (v_1,\ldots, v_n)
\end{multline}
where the summation runs over $r,s,k$ satisfying $r+s = n+1$, $ 1\leq k\leq r$,
$2\leq r < n$ for all $v_1,\ldots,v_n$ in $V$ and $n\geq 2$, and
$\epsilon(k)$ denotes the sign obtained by sliding $Q$ through $v_1,\ldots,
v_{k-1}$.
\end{df}
We note that $m_2$ is a multiplication, $m_3$ is an associating homotopy and
the further $m_k$'s are ``higher associating homotopies''.
In the context of (strong) homotopy algebras, the simplest definition of a
$C_{\infty}$-algebra, first appeared in work of Kadeishvili \cite{kad1, kad2} and then in
that of Smirnov \cite{smirn} (both of whom called them commutative
$A_\infty$-algebras), is:
\begin{df} A {\it $C_{\infty}$-algebra} is an $A_{\infty}$-algebra\ $(A, \{ m_n\})$ such that each map
$m_n:A^{\otimes n}\to A$ is a Harrison cochain, i.e. $m_n$ vanishes on the
sum of all $(p,q)$-shuffles for $p+q=n$, the sign of the shuffle coming
from the grading of $A$ shifted by $1$.
\end{df}
The single object equivalent definition (cf.\ \cite{SS, Q, jcm}) is:
\begin{df} A {\it $C_{\infty}$-algebra} is a graded vector space $A$ together with a
codifferential on the free Lie coalgebra cogenerated by $sA$,
which is isomorphic to $A$ with the grading shifted by $1$. (The
convention is that the shift is opposite to the degree of the differential;
since we will be working homologically at the operad level, $d$ is of
degree $-1$ while $s$ is of degree $+1$.)
\end{df}
This definition hints of the ancient Koszul duality (or adjointness) between
commutative algebras and Lie coalgebras (or Lie algebras and commutative
coalgebras) \cite{d,Q, jcm, SS}, which carries over to the operad level,
\cite{gk}.
Getzler and Jones \cite{gj} established the equivalence of the above
with the operadic definition of a $C_{\infty}$-algebra \ implicit in \cite{gk}.
\begin{df} A {\it $C_{\infty}$-algebra} is an algebra over the operad ${\cal Cobar} {\cal Lie^c}$.
\end{df}
The concept of a co-operad is defined dually to that of an operad \cite {gj},
so that if ${\cal K} = \{ {\cal K}(n) \}$ is a co-operad over a field $k$,
then the linear duals $Hom({\cal K}(n),k)$ form an operad. Conversely, if
${\cal O} = \{{\cal O}(n) \}$ is an operad with each ${\cal O}(n)$ finite
dimensional (or of finite type, i.e. graded and finite dimensional in each
grading), then the linear duals $Hom({\cal O}(n), k)$ form a co-operad. (A
co-operad is ``an operad with the arrows reversed''.)
The operad $\operatorname{{\cal Lie}} = \{\operatorname{{\cal Lie}} (n)\}$ \cite{hs} with $\operatorname{{\cal Lie}} (n)$ is finite
dimensional for each n. The co-operad $\operatorname{{\cal Lie^c}}$ has $\operatorname{{\cal Lie^c}} (n) = \operatorname{Hom} (\operatorname{{\cal Lie}}
(n), k).$
${\cal Cobar} $ \cite{gj} is a functor from co-operads to operads. (In
\cite{gk}, the use of co-operads is avoided by assuming finite type for
operads and defining ${\cal Cobar}$ only for the linear duals of operads.) All
we need to know is:
For any co-operad ${\cal K}$, ${\cal Cobar}\ {\cal K}$ is an operad with
pieces indexed by trees, constructed as products of various ${\cal K}(n)$'s
according to a prescription determined by the tree. Our convention is that
trees have vertices all of valence $ > 1$, e.g. the {\sl corolla} with $n$
leaves, denoted by $\delta_n$, and one root has just one vertex. The piece of
${\cal Cobar} \ {\cal K}$ indexed by this corolla is just ${\cal K}(n)$ with a
shift in grading.
\begingroup
\input{psfig}
\begin{figure}[h]
\centerline{\psfig{figure=corolla.ps,height=1in}}
\caption{An $N$-corolla, $\delta_N$}
\label{corolla}
\end{figure}
\endgroup
To make a comparison between $C_{\infty}$-algebra s and $A_{\infty}$-algebra s at the operad level, recall
that the free Lie algebra ${\cal L}(x_1,\dots,x_n)$ can be realized as the
primitive subspace of $T(x_1,\dots,x_n)$ with respect to the unshuffle
coproduct \cite{Ree}. Dually, the free Lie coalgebra ${\cal
L}^c(x_1,\dots,x_n)$ can then be identified with the space of
indecomposables of the free associative coalgebra $T^c(x_1,\dots,x_n)$,
i.e. the {\it quotient\/} by the
image of the shuffle product. (See \cite{WM} for definitions which do not rely
on finite type and duality.) Then ${\cal Lie}^c (n)$ is defined as the {\it
quotient\/} of the free Lie coalgebra ${\cal L}^c(x_1,\dots,x_n)$ by those
multilinear functions which vanish whenever two arguments are equal.
An $A_{\infty}$-algebra\ is an algebra over the operad ${\cal Cobar} {\cal As}^c$ where
${\cal As}^c$
is the co-operad for associative coalgebras. ${\cal As}^c (n)$ can be
realized as the quotient of the tensor coalgebra $T^c (x_1, \dots, x_n)$
by those multilinear functions which vanish whenever two arguments are equal.
Since ${\cal Cobar} {\cal Lie^c} (n)$ has its corolla component equal to ${\cal Lie}^c (n)$
with a shift in grading, the structure map $m_n$ and its permutations are
given by ${\cal Lie}^c (n) \to \operatorname{Hom} (A^{\otimes n}, A)$. Interpretation of
$m_n$ as the structure map for an $A_{\infty}$-algebra\ means pulling back the map
${\cal Lie}^c (n) \to \operatorname{Hom} (A^{\otimes n}, A)$ to $ T^c (n) \to \operatorname{Hom}(A^{\otimes
n}, A)$, which guarantees that $m_n$ vanishes on the image of the shuffle
product.
Strictly speaking, an algebra over ${\cal As}$, i.e. a morphism ${\cal As}\to
End (V)$, does not determine a unique associative multiplication on $V$, but
rather two of them, since ${\cal As}(2) = k\lbrack S_2\rbrack.$ We make the
obvious choice corresponding to the identity element of $S_2$ to
determine an associative multiplication on $V$. For ${\cal Cobar} {\cal As}^c$,
we must make choices for each $m_n$, but we make the same choice,
corresponding to the identity element of $S_n$. (This choice is implicit in
the result of \cite{gk} that an $A_{\infty}$-algebra\ is the same as an algebra over
${\cal Cobar} {\cal As^c}.$) For ${\cal Cobar} {\cal Lie^c}$, we just take the equivalence class (mod the
shuffle product) of the above choice for ${\cal Cobar} {\cal As^c}$.
\section{Moduli Spaces of Punctured Spheres}
Let ${\cal M}_n$ be the moduli space of Riemann spheres with $n$ punctures. That
is, points in ${\cal M}_n$ consist of configurations of $n$ ordered points on
$\CP{1}$ with any two such configurations being identified if they are
related by a biholomorphic map. In other words,
$${\cal M}_n := ((\CP{1})^n \setminus \Delta)/ \operatorname{PSL}(2,\nc)$$
where $\Delta = \{(z_1,\dots , z_n) \linebreak[0] \in (\CP{1})^n \; |\; z_i =
z_j \; \text {\ for some \ } \; i \ne j\}$, the set of diagonals.
There is a compactification of ${\cal M}_n$ when $n\geq 3$ due to De\-ligne-Knud\-sen-Mum\-ford\
\cite{dl,kap,keel,kn} which is the {\sl moduli space of stable genus 0 curves
with $n$ punctures,} $\overline{{\cal M}}_n$.
Recall that a stable $n$ punctured complex curve of genus 0 is a connected
compact complex curve $C$ of genus 0 with $n$ punctures, such that it may
have ordinary double points away from the punctures, each irreducible
component of the curve $C$ is a projective line and the total number of
punctures and double points on each component of $C$ is at least 3. Both
${\cal M}_n$ and $\overline{{\cal M}}_n$ are smooth complex algebraic manifolds of complex
dimension $n-3$. The moduli space ${\cal M}_n$ of nonsingular curves is an open
submanifold in the projective manifold $\overline{{\cal M}}_n$. The complement is a divisor,
formed by all degenerate curves.
Let $\overline{{\cal M}}(1) := \{e_{\Bbb{C}}\}$ and $\overline{{\cal M}}(n) := \overline{{\cal M}}_{n+1}$ for $n\geq 2$, then the
set $\overline{{\cal M}}\, :=\, \{\,\overline{{\cal M}}(n)\,\}$ is naturally an operad of algebraic varieties
where the element $e_{\Bbb{C}}$ is defined to be a unit with respect to the operad
composition.\footnote{We include a unit in this manner for convenience.} The
permutation group on $n$ elements, $S_n$, acts on $\overline{{\cal M}}(n)$ by reordering the
first n-punctures. The composition maps $\gamma_i\, :\,
\overline{{\cal M}}(n)\,\times\,\overline{{\cal M}}(n')\,\to\,\overline{{\cal M}}(n+n'-1)$ for all $i = 1, \ldots, n$ and for
all $n,n'$ are defined by
$$(\Sigma,\Sigma')\,\mapsto\, \gamma_i(\Sigma,\Sigma')\, :=\, \Sigma \circ_i
\Sigma'$$
where $\Sigma\circ_i\Sigma'$ is obtained by attaching the $(n'+1)$st puncture
of $\Sigma'$ to the $i$th puncture of $\Sigma$ thereby creating a curve with
a new double point and the $(n+n'-1)$ punctures are ordered in the natural
way.
Each space $\overline{{\cal M}}(n)$ is naturally stratified by smooth, connected locally
closed algebraic subvarieties. Each stratum of $\overline{{\cal M}}(n)$ consists of those
points arising from $n$-punctured stable curves of a given topological type.
Since any stable curve can be obtained by attaching spheres together, the
combinatorics of this attaching process can be encoded in a tree. Therefore,
each stratum of $\overline{{\cal M}}(n)$ is naturally indexed by a tree (see Figure
\ref{strata}). This stratification gives rise to a filtration of $\overline{{\cal M}}(n)$:
$$ \F{-1}(n)\, =\, \emptyset \subseteq \F{0}(n) \subseteq \cdots \subseteq
\F{n-2}(n)\,=\, \overline{{\cal M}}(n)$$
where $\F{p}(n)$ is the disjoint union of all the strata of $\overline{{\cal M}}(n)$ with
complex dimension less than or equal to $p$. Furthermore, this filtration
is invariant under the action of the permutation group and it behaves
nicely with respect to operad composition, {\sl i.e.} for all
$i=1,\ldots, n$ and for all $n,n',p,p'$, we have
$$ \gamma_i\,:\,\F{p}(n) \times \F{p'}(n')\,\to\,\F{p+p'}(n+n'-1).$$
The filtration gives rise to a spectral sequence associated to each $\overline{{\cal M}}(n)$
which converges finitely to $H_\bullet(\overline{{\cal M}}(n))$. This spectral sequence is
known to degenerate at the $E^2$-term \cite{bg:1}. We show in
Section \ref{sec:filter} that any filtration which respects the operad
structure induces an operad structure on the $\Ee{r}$ term in its associated
spectral sequence for all $r\geq 0$. In particular, each $E^r$ term contains
a collection of suboperads. In our case, the only nontrivial suboperad is
the $q=0$ (``middle'') row of the $\Ee{1}$ term. The main result for our
purposes, due to Beilinson-Ginzburg \cite{bg:1}, cf.\ F.~Cohen
\cite{fc} and Schechtman-Varchenko \cite{sv}, is the following:
\begin{tth}
The ``middle'' row of the $\Ee{1}$ term of the spectral sequence associated
to the canonical filtration of $\overline{{\cal M}}(n)$
$$
0 @>>> H_{n-2} (\F{n-2}(n),\F{n-3}(n)) @>>> \dots @>>>
H_p(\F{p}(n),\F{p-1}(n)) @>>> \dots @>>> H_0(\F{0}(n)) @>>> 0, $$
is an operad isomorphic to ${\cal Cobar} {\cal Lie^c} (n)$, the $C_\infty$ operad.
\end{tth}
\begin{sloppypar}
This identification is quite explicit. Lefshetz duality gives an
isomorphism $H_p(F_p(n),F_{p-1}(n),{\Bbb{C}}) \simeq H^p(F_p(n) \setminus
F_{p-1}(n))$ but $F_p(n)\setminus F_{p-1}(n)$ is the disjoint union of
those strata in $\overline{{\cal M}}(n)$ with complex dimension $p$ each of which is
indexed by an $n$-tree with $n-2-p$ internal edges. If $T$ is a tree, then
let $S_T$ denote the strata associated to this tree, {\sl e.g.}\ see Figure
\ref{strata}. Each such tree $T$ is the iterated composition of corollas,
say $\delta_{n_1},\ldots, \delta_{n_k}$, and we have the isomorphism $S_T
\simeq S_{\delta_{n_1}}\times\cdots\times S_{\delta_{n_k}}.$ The stratum
associated to any corolla, $\delta_n,$ can be identified with ${\cal M}(n)$.
Therefore, $S_T \simeq {\cal M}(n_1)\times\cdots\times {\cal M}(n_k)$. Using the fact
that the cohomology of ${\cal M}(n)$ vanishes in degree above $n-2$, which
follows from a similar vanishing of homology of configuration spaces by
Arnold \cite{Ar}, we obtain the isomorphism $H^p(S_T) \simeq
H^{n_1-2}({\cal M}(n_1)) \otimes \cdots \otimes H^{n_k-2}({\cal M}(n_k))$. However,
$\operatorname{{\cal Lie}}(n)$ can be identified with $H_{n-2}({\cal M}(n))$ (with a shift in degree)
\cite{fc} and therefore, $H^{n-2}({\cal M}(n))$ can be identified with
$\operatorname{{\cal Lie^c}}(n)$. Finally, we obtain $H^p(S_T) \simeq \operatorname{{\cal Lie^c}}(n_1)\otimes \cdots
\otimes \operatorname{{\cal Lie^c}}(n_k)$ which is an element of ${\cal Cobar} {\cal Lie^c}(n)$ associated to
the tree $T$ with the proper element in $\operatorname{{\cal Lie^c}}$ decorating the
corresponding vertex of $T$.
\end{sloppypar}
\begingroup
\input{psfig}
\begin{figure}[h]
\centerline{\psfig{figure=strata.ps,height=2in}}
\caption{The composition map $\circ_1:\,\overline{{\cal M}}(3)\times\overline{{\cal M}}(2)\,\to\, \overline{{\cal M}}(4)$ and
the trees indexing the strata to which they belong}
\label{strata}
\end{figure}
\endgroup
\section{Moduli Spaces of Punctured Riemann Surfaces}
Here we make a generalization of the results of the previous section; we give
a complete description of the analogous spectral sequence in the case of moduli
spaces of Riemann surfaces of higher genera. In particular, we identify the
other rows of the $E^1$ term of the spectral sequence of the previous
section.
Let us restrict ourselves to those Riemann surfaces with genus $g$ and $n$
ordered punctures which have negative Euler characteristic, {\sl i.e.}\ $g \ge 2$
or $g=1,
n \ge 1$ or $g = 0 , n \ge 3$. Let ${\cal M}_{g,n}$ be the moduli space of genus
$g$ Riemann surfaces, that is, complex algebraic curves, with $n$ punctures
and let $\overline{{\cal M}}_{g,n}$ be its compactification due to
Deligne-Knudsen-Mumford. This is a smooth, complete stack of dimension
$\dim_{{\Bbb{C}}} \overline{{\cal M}}_{g,n} = 3g - 3 + n$. The compactified moduli parameterize
isomorphism classes of stable curves, ones which have a finite number of
singularities, which are double points, and such that each irreducible
component of genus 1 has at least one puncture or double point and each
irreducible component of genus 0 has at least three punctures or double
points. These conditions insure that each component of the complement of
the punctures and double points has negative Euler characteristic.
The spaces $\overline{{\cal M}}_{g,n}$ form a modular operad, see Getzler-Kapranov
\cite{modular}, that is, two kinds of operations are defined: attaching
two curves at punctures: $\overline{{\cal M}}_{g,n} \times \overline{{\cal M}}_{g',n'} \to \overline{{\cal M}}_{g+g',
n+n'-2}$ and gluing two punctures together on a single curve $\overline{{\cal M}}_{g,n}
\to \overline{{\cal M}}_{g+1,n-2}$. Furthermore, there is the action of the symmetric
group $S_n$ on $\overline{{\cal M}}_{g,n}$ which reorders the punctures.
Let $F_p = F_p(g,n) \subset \overline{{\cal M}}_{g,n}$ be the closed subspace (substack, in
fact) of dimension $p$ formed by stable curves with at least $\dim_{\Bbb{C}}\, (
\overline{{\cal M}}_{g,n}) - p = 3g-3+n -p$ double points. We obtain an ascending filtration
of the moduli space $\overline{{\cal M}}_{g,n}$:
\begin{equation*}
\F{-1}\, =\, \emptyset \subset \F{0} \subset \cdots \subset \F{3g-3 +
n}\,=\, \overline{{\cal M}}_{g,n}.
\end{equation*}
As in the genus zero case, this filtration behaves nicely with respect to the
modular operad operations:
\[
F_{p}(g,n) \times F_{p'} (g',n') \to F_{p+p'} (g+g', n+n'-2)
\]
corresponding to attaching two curves at punctures and
\[
F_p (g,n) \to F_{p} (g+1, n-2)
\]
corresponding to glueing together two punctures on a single curve.
Irreducible components (strata) $S_G$ of $F_p$ are indexed by {\it stable\/}
labeled $n$-graphs $G$ with $3g - 3 + n - p + 1$ vertices and with the
invariant $g(G)$, defined below, equal to the genus $g$. {\it Stable\/} refers
to graphs of the following kind. Each graph is connected, has a root vertex and
$n$ enumerated exterior edges, edges which are coincident with only one vertex
of the graph. Each vertex $v$ of the graph is labeled by a nonegative integer
$g(v)$, called the genus of a vertex. The stability condition means that any
vertex $v$ labeled by $g(v)=1$ should be coincident with at least one edge
(i.e., be of at least valence one) and each vertex $v$ with $g(v) = 0$ should
be of at least valence three. The invariant $g(G)$ is given by the formula
$g(G) = b_1(G) + \sum_v g(v)$, where $b_1(G)$ is the first Betti number of the
graph. Each component stratum $S_G$ is a quotient of the product of
uncompactified moduli
spaces (via the modular operad structure), the combinatorics of which are
neatly encoded in the graph:
\[
S_G = \left( \prod_{v \in G} {\cal M}_{g(v), n(v)} \right)/ \operatorname{Aut}(G)
\]
where the product is over all vertices of $G$ and $n(v)$ is the valence of the
vertex $v$ and $\operatorname{Aut}(G)$ is the automorphism group of a graph $G$ (a bijection on
vertices and edges, preserving the exterior edges, the labels of vertices and
the incidence relation).
Thus, we get a modular operad in the category of filtered varieties (stacks,
in fact) and therefore, applying the spectral sequence functor, we obtain
a modular operad of spectral sequences, see Section~\ref{sec:filter} for
related formalism. Its $(g,n)$-component can be described, as usual, in the
following way
\[
E^1_{p,q} = H_{p+q} (F_p, F_{p-1}, {\Bbb{C}}) = H^{p-q} (F_p \setminus F_{p-1},
{\Bbb{C}}),
\]
due to Poincar\'{e}-Lefschetz duality, with the differential $d^1:
H^{p-q} (F_p \setminus F_{p-1}) \to H^{p-q-1} (F_{p-1} \setminus
F_{p-2})$ being the Poincar\'{e} residue, and
\[
\bigoplus_{p+q = k} E^\infty_{p,q}
= H_{k} (\overline{{\cal M}}_{g,n}, {\Bbb{C}}).
\]
\begin{tth}
\begin{enumerate}
\item The $E^1$ term of the spectral sequence is naturally isomorphic to
the Feynman transform, see \cite{modular}, of the modular co-operad
$H^\bullet
({\cal M}_{g,n})$. Namely, $E_{p,q}^1 = 0$, unless $-p \le q \le p \le 3g-3+ n$,
when
\[
E_{p,q}^1 = \bigoplus_G \; \left( \bigoplus_{\sum k(v) = p - q} \quad
\bigotimes_{v \in G} H^{k(v)} ({\cal M}_{g(v), n(v)}) \right) ^{\operatorname{Aut}(G)},
\]
the first summation running over all stable labeled $n$-graphs $G$ with $g(G) =
g$ and $3g -3 + n -p +1$ vertices, the second over all functions $k(v) \in
{\Bbb{Z}}$ of vertices $ v \in G$ summing up to $p-q$. The differential is induced by
contracting internal edges in $G$, which corresponds to forming new double
points on a curve, and taking the Poincar\'{e} residue.
\item
When $g = 0$, the graphs $G$ have no loops and are labeled with zeroes,
i.e., just trees where all vertices have valence $\ge 3$. Then the $E^1$
term is nothing but the cobar construction of the co-operad $H^\bullet
({\cal M}_{0,n})$. Moreover, $E_{p,q}^1 = 0$ unless $0 \le q \le p \le n-3$, and
$E^2_{p,q} = E^\infty_{p,q} = 0$, except
\[
E^2_{p,p} = E^\infty_{p,p} = H_{2p} (\overline{{\cal M}}_{0,n}).
\]
\end{enumerate}
\end{tth}
\begin{rem}
The second part of the theorem, as well as of Theorem~\ref{dual} below, is
proved independently by Getzler. Just after we proved it, we came across his
two-day old preprint \cite{g:new} dedicated to this kind of duality between the
operads $H_\bullet ({\cal M}_{0,n})$ and $H_\bullet(\overline{{\cal M}}_{0,n})$.
In contrast to Getzler's proof, our proof below of the statement about
$E^2$ being concentrated on the diagonal uses the operad structure and
known results on the cohomology of $\overline{{\cal M}}_{0,n}$.
Getzler \cite{g:new}, on the other hand, uses the fact that the mixed Hodge
structure on the
cohomology of ${\cal M}_{0,n}$ is pure and that the operad $H_\bullet (\overline{{\cal M}}_{0,n})$ is
Koszul. The Hodge structure on
the cohomology of ${\cal M}_{g,n}$ is no longer pure for higher genera, but Mumford's
conjecture for the stable cohomology implies that the Hodge structure on the
stable cohomology of ${\cal M}_{g,n}$ should be pure.
\end{rem}
\begin{quest}
For $g > 0$, describe the locus of the $E^\infty$ term on the $(p, q)$ plane.
Is there a stable version of this theorem, that is, for large $g > 0$, where
the sequence degenerates at $E^2$ and the $E^2$ term is located on the diagonal
$p = q$? It would be interesting to find the proper notion of Koszulness for
the stable homology of the modular operad $H^{\operatorname{st}}_\bullet
(\overline{{\cal M}}_{g,n})$.
\end{quest}
\begin{pf}
The first part of the theorem is obvious after the description of the strata
$S_G$ above. In the second part, to show that $E_{p,q}^1 = 0$ for $q < 0$
notice that $H^{k(v)} ({\cal M}_{0,n(v)} ) = 0 $ for $k(v) > n (v) - 3$.
Denote by $\operatorname{ed} (G)$ the number of edges of the graph,
including the $n$ exterior edges and by $v(G)$ the number of vertices.
Adding the above inequalities together, we get $E_{p,q}^1 = 0$
for $p - q > \sum_v
(n(v) - 3) = 2 \operatorname{ed} (G) - n - 3 v(G) = 2 (n + v(G) -1) -n -3 v(G)
= n- 2 - v(G) = p$, that is, for $q < 0$.
The degeneration $E^2_{p,q} = E^\infty_{p,q}$ of the spectral sequence
for $g=0$ follows from the purity of the Hodge structure on $E^1$,
which is the sum of tensor products of the cohomologies of ${\cal M}_{0,n}$,
where the Hodge structure is pure, see \cite{bg:1}.
We will show the vanishing $E^2_{p,q} = 0 $ for $g= 0 $ and $p \ne q$
by using Keel's description \cite{keel} of the homology of
$\overline{{\cal M}}_{0,n}$, the operad structure and induction on $n$. $\overline{{\cal M}}_{0,3}$ is
a point and the statement is trivial. Assuming it is proved for $k \le
n$, let us prove it for $k = n+1$. Keel's description says that
$H_\bullet (\overline{{\cal M}}_{0,n})$ is generated as an intersection algebra by
$H_2$ so $H^1$ is zero. Thus the term $E^2_{1,0}$ must be zero and
hence the statement is true for $\overline{{\cal M}}_{0,4}$. Now let $E^r_{p,q} (n)$
refer to the spectral sequence for $\overline{{\cal M}}_{0,n}$. Except for the
fundamental class of $\overline{{\cal M}}_{0,n+1}$, which is in $E^2_{n-2, n-2}
(n+1)$, the rest of the terms $E^2_{p,q} (n+1)$ are the operad
compositions of $E^2_{p_i, q_i} (n_i)$'s for $n_i \le n$, where we
know $p_i = q_i$ by the induction assumption.
\end{pf}
The situation with the ``Koszul dual'' spectral sequence, the one
which converges to $H^{\bullet} ({\cal M}_{g,n}, {\Bbb{C}})$ and whose $E_1$ term
is formed by the cohomology of closed
strata is somewhat better. A similar theorem holds and moreover, the
spectral sequence degenerates at $E_2$ even for $g > 0$, due to the
purity of the Hodge structure on $H^{\bullet} (\overline{{\cal M}}_{g,n}, {\Bbb{C}})$, see
Deligne \cite{del:hodge}.
Let $X = \overline{{\cal M}}_{g,n}$, $U = {\cal M}_{g,n}$ and $D= X \setminus U$. Following
Deligne \cite{del:hodge}, consider the double complex $\Omega^{\bullet,
\bullet}_X (\log D)$, the smooth global $(p,q)$-forms on $X$ with at
most logarithmic singularities $\partial f/f$ along $D$, where $f=0$ is
a local equation of $D$. Its hyper(=total)cohomology is equal to $H^{p+q} (U,
{\Bbb{C}})$. Let $W_m$ be the subcomplex of the total complex generated by
products $\alpha \wedge
\partial f_{i_1}/ f_{i_1} \wedge \dots \wedge \partial f_{i_s}/
f_{i_s}$ for $s \le m$ and smooth $\alpha$, where each $f_i$ is a local
equation of an irreducible component of $D$. The $W^{m} = W_{-m}$'s
for $m \le 0$ define a decreasing filtration of the logarithmic double
complex. Consider the spectral sequence $E_r$ associated with this
filtration. With respect to this filtration, we have $E_1^{p,q} =
H^{2p+q} (\widetilde F_{3g- 3 +n + p})$, where $\widetilde F_s$ is the
disjoint union of irreducible components of $F_s$. The
sequence degenerates: $E_2 = E_\infty = H^{\bullet} ({\cal M}_{g,n}, {\Bbb{C}})$,
see \cite{del:hodge}.
In addition, for $g=0$, it is known from Arnold's explicit
description of cohomology classes of configuration spaces that all
nonzero cohomology classes in $H^{m} ({\cal M}_{0,n}, {\Bbb{C}})$ are
represented by $(m,0)$-forms with exactly $m$ logarithmic
singularities. Thus, $E^{p,q}_2 = E^{p,q}_\infty$ vanishes unless $p =
-m$ and $ 2p +q = 0$. It follows that $H^m ({\cal M}_{0,n}, {\Bbb{C}}) =
E^{-m,2m}_2$, where the Hodge structure is pure of weight $2m$, see
\cite{del:hodge}.
From these descriptions, we have the following theorem.
\begin{tth}
\label{dual}
\begin{enumerate}
\item The $E_1$ term of the spectral sequence is dual to
the Feynman transform of the modular co-operad $H^\bullet
(\overline{{\cal M}}_{g,n})$. Namely, $E^{p,q}_1 = 0$, unless $-3g +3 -n \le p \le
0$ and $-2p \le q \le 6g -6 + 2n$, when
\[
E^{p,q}_1 = \bigoplus_G \; \left( \bigoplus_{\sum k(v) = 2p + q} \quad
\bigotimes_{v \in G} H^{k(v)} (\overline{{\cal M}}_{g(v), n(v)}) \right) ^{\operatorname{Aut}(G)},
\]
the first summation running over all stable labeled $n$-graphs $G$ with $g(G) =
g$ and $- p +1$ vertices, the second over all functions $k(v) \in {\Bbb{Z}}$ of
vertices $ v \in G$ summing up to $2p +q$. The differential $d_1: E^{p,q}_1 \to
E^{p+1,q}_1$ is induced by contracting internal edges in $G$, which corresponds
to forming new double points on a curve. The $E_2$ term is equal to $E_\infty =
H^{\bullet} ({\cal M}_{g,n}, {\Bbb{C}})$.
\item
When $g = 0$, the graphs $G$ have no loops and are labeled with
zeroes, i.e., just trees where all vertices have valence $\ge 3$. Then
the $E_1$ term is nothing but the dual of the cobar construction of
the co-operad $H^\bullet (\overline{{\cal M}}_{0,n})$.
Moreover, $E_2^{p,q} =
E_\infty^{p,q} = 0$, except
\[
E_2^{p,-2p} = E_\infty^{p,-2p} = H^{-p} ({\cal M}_{0,n}), \quad 0 \le -p \le n-3.
\]
\end{enumerate}
\end{tth}
\section{Filtered Topological Gravity and $C_{\infty}$-algebra s}
\begin{df}
Let $\overline{{\cal M}}$ be the operad of the moduli space of stable curves of genus zero. A
{\sl topological gravity} consists of a differential graded complex vector
space $Q:V_g\,\to\, V_{g-1}$ and a collection $\{\omega_n\}$ such that
\begin{enumerate}
\item $\omega_n = \sum_{r=0}^{2n-4}\omega_n^r$ where $\omega_n^r$ belongs to
$\Omega(\overline{{\cal M}}(n))\otimes\End{V}(n)$ with bidegree $(r,-r)$.
\item $d\omega_n = Q\omega_n$
\item $\sigma^*\omega_n = \omega_n\circ \sigma$ for all $\sigma$ in $S_n$
where $\sigma^*$ denotes the pullback of the action of $S_n$ on $\overline{{\cal M}}(n)$ and
$\sigma$ on the right hand side indicates the action of $S_n$ on $V^{\otimes
n}$ and
\item $\gamma_i^*\omega_{n+n'-1} = \omega_n\circ_i\omega_n'$
for all $i=1,\ldots, n$ and all $n,n'$ where $\circ_i$ denotes composition in
the endomorphism operad and $\gamma_i^*$ is the pullback of the composition
map $\gamma_i:\overline{{\cal M}}(n)\times\overline{{\cal M}}(n')\,\to\,\overline{{\cal M}}(n+n'-1)$.
\end{enumerate}
A {\sl filtered topological gravity} is a topological gravity in which the
differential forms $\omega_n^p$ vanish for all $p > \dim_{\Bbb{C}} \overline{{\cal M}}(n) = n-2$.
\end{df}
Since $\overline{{\cal M}}$ is an operad of topological spaces, $(C_\bullet(\overline{{\cal M}}),\partial)$,
the singular chains on $\overline{{\cal M}}$, inherits the structure of an operad. If $(V,Q)$
is a topological gravity with differential forms $\{\,\omega_n\,\}$ then $V$
is an algebra over $C_\bullet(\overline{{\cal M}})$ where the morphism
$C_\bullet(\overline{{\cal M}}(n))\,\to\,\End{V}{n}$ is given by $c\,\mapsto\, \int_c
\omega_n$. The axioms of a topological gravity are formulated precisely so as
to insure that this map is a morphism of operads. This definition of
topological gravity was motivated by its origins in the physics
literature \cite{W}.
Let $(V,Q)$ be a filtered topological gravity with differential forms
$\{\omega_n\}$. Integration of $\{\omega_n\}$ makes $(V,Q)$ into a
$C_\bullet(\overline{{\cal M}})$ algebra with a morphism $\mu: C_\bullet(\overline{{\cal M}})\,\to\,
\End{V}$ such that $\mu(c) = 0$ for all $p$-chains $c$ on $\overline{{\cal M}}(n)$ with $p >
\dim_{\Bbb{C}} \overline{{\cal M}}(n) = n-2$.
\begin{tth}
Let $(V,Q)$ be a filtered topological gravity, let $V$ be filtered by
the degree in the complex and let $F = \{\,F(n)\,\}$ be the
filtration of $\overline{{\cal M}}$ arising from the canonical statification of $\overline{{\cal M}}$, then
$V$ is a $C_\bullet(\overline{{\cal M}})$-algebra preserving the filtration.
\end{tth}
\begin{pf}
The filtration of $\overline{{\cal M}}$ by $F$ induces a filtration on its singular chains,
{\sl i.e.}\ $\cdots \subseteq C_{p+q}(F_p(n))\subseteq
C_{p+q}(F_{p+1}(n))\subseteq \cdots.$ Similarly, $\End{V}$ is filtered by
$\cdots\subseteq F_p\,\End{V}^{p+q}(n) \subseteq F_p\,\End{V}^{p+q}(n)\subseteq
\cdots$. However, with the given choice of filtration degree, we see that
\begin{equation*}
F_p\,\End{V}(n)^{p+q}(n) = \cases 0, &\text{if $q\geq 1$}\\
\End{V}^{p+q}(n), &\text{if $q\leq 0$.}
\endcases
\end{equation*}
Therefore, the morphism $\mu:C_\bullet(\overline{{\cal M}})\,\to\,\End{V}$ is an algebra
preserving the filtrations if and only if $\mu(c) = 0$ when acting upon
elements in $C_{p+q}(F_p(n))$ for all $q\geq 1$. This is precisely the case
when $(V,Q)$ is a filtered topological gravity.
\end{pf}
\begin{crl}
A filtered topological gravity is a $C_{\infty}$-algebra.
\end{crl}
\begin{pf}
Let the $E^r$ terms in the spectral sequences associated to $C_\bullet(\overline{{\cal M}})$
and $\End{V}$ be denoted by $E^r$ and $E^r[V]$, respectively. Since $V$ is a
filtered $C_\bullet(\overline{{\cal M}})$-algebra, $E^1[V]$ is an algebra over the operad
$E^1$. However, since $E^1[V] \simeq \End{V}$ and $E^1[V]$ contains a
suboperad ${\cal D}^1_{0} = \oplus_{p} H_{p}(F_p(n),F_{p-1}(n))$ which is precisely
the $C_\infty$ operad, $V$ is a $C_{\infty}$-algebra.
\end{pf}
Notice that we only needed the fact that $\mu$ vanished when acting upon
chains that are greater than half the dimension of the corresponding moduli
space, thereby proving Theorem \ref{th:goal}.
\section{Filtrations of Operads and Algebras}
\label{sec:filter}
We will now study properties of operads with a filtration and algebras which
respect this filtration. We shall see that a filtered operad makes each
term in its associated spectral sequence into an operad. Furthermore, there
are natural suboperads in each term of the spectral sequence, one of which
is the $C_{\infty}$-algebra\ in the $E^1$ term of the spectral sequence to the moduli space
of stable curves. All of the results in this section can be naturally
extended to the moduli spaces of higher genus curves in a straightforward
way using the formalism of modular operads.
\begin{df}
Let ${ \cal O} = \{\,{ \cal O}(n)\,\}$ be an operad of complexes with differentials
$\partial : { \cal O}_p(n)\, \to \, { \cal O}_{p-1}(n)$. Let $F_p = \{\,F_p(n)
\,\}_{n\geq 1}$ be a filtration of ${ \cal O}(n)$ as complexes such that for all
$i=1,\ldots n$, the composition maps $\circ_i$ take $F_{p,q}(n)\otimes
F_{p',q'}(n')$ to $F_{p+p',q+q'}(n+n'-1)$ for all $p, p'$,
$q,q'$, and the filtration degree $q$ part of $F_p(n)$) is stable
under the action of the permutation group which commutes with the
differential. The collection $F$ is said to be a {\sl
filtration of the operad ${ \cal O}$.}
\end{df}
It is clear that a filtered operad can be defined in the category of
topological spaces as well and that such a filtration induces a filtration on
the associated operad of singular chains.
\begin{prop}
Let ${ \cal O}$ be an operad with filtration $F$ and let $E^r = \{\, E^r(n)\,\}$ be
the $E^r$ term in the associated spectral sequence then for all $r$, $E^r$
inherits the structure of an operad of complexes with a
differential $\partial^r:E^r_{p,q}(n)\,\to\,E^r_{p-r,q+r-1}(n)$ and
composition maps satisfying
\begin{equation*}
\circ_i:E^r_{p,q}(n) \otimes E^r_{p',q'}(n)\,\to\, E^r_{p+p',q+q'}(n + n' -1)
\end{equation*}
for all $i=1,\ldots, n$. In particular, $E^0_{p,q}(n) =
F_{p,p+q}(n)/F_{p-1,p+q}(n)$ and $E^1_{p,q}(n) = H_{p+q}(F_p(n),
F_{p-1}(n))$.
\end{prop}
\begin{pf}
Let $Z_{p,q}^r(n) = \{\,x\in F_{p,q}(n)\,|\,\partial x\in F_{p-r,q+r-1}(n)
\,\}$ then we can write (see, for example, \cite{lang})
\begin{equation*}
E^r_{p,q}(n) = \frac{Z^r_{p,q}(n)}{\partial Z^{r-1}_{p+(r-1), q- (r-1) +
1}(n) + Z^{r-1}_{p-1,q+1}(n)}
\end{equation*}
with the differential $\partial^r:E^r_{p,q}(n)\,\to\,E^r_{p-r,q+r-1}(n)$
induced from $\partial$. Keeping track of the composition maps, a computation
shows that the numerators assemble into an operad and the denominators into
an operad ideal so their quotient is an operad.
\end{pf}
Given a filtered operad, there is a natural collection of suboperads of each
$E^r$ term as shown by the following result.
\begin{prop}
Let ${ \cal O}$ be an operad with filtration $F$. For each $r$ and $k$, there
exists a suboperad of $E^r$, ${\cal D}^r_{k} = \{\,{\cal D}^r_{k}(n)\,\}_{n\geq 1}$
such that
\begin{equation*}
{\cal D}^r_{k}(n) = \bigoplus_{(r-1)p+rq = k(n-1)} E^r_{p,q}(n).
\end{equation*}
In particular, ${\cal D}^1_{0} = \{\,{\cal D}^1_{0}(n)\,\}$ is a suboperad of $E^1$ with
$n$th component ${\cal D}^1_{0}(n) = \bigoplus_p H_{p}(F_p(n),F_{p-1}(n))$.
\end{prop}
\begin{pf}
Each $E^r(n)$ is the union of subcomplexes $K^r_s(n) = \oplus_{(r-1) p + r q =
s} E^r_{p,q}(n)$ since the differential maps $\partial^r:E_{p,q}(n)\, \to\,
E_{p-r,q+r-1}(n)$. The composition maps take $\circ_i: K^r_s(n)\otimes
K^r_{s'}(n') \,\to\, K^r_{s+s'}(n+n'-1)$. Therefore, ${\displaystyle
{\cal D}^r_{k}(n)
\linebreak[1] = \linebreak[0] \bigoplus_{(r-1)p + rq = k(n-1)} K^r_{k(n-1)}(n)}
$
assemble to form an operad of complexes.
\end{pf}
In the case of $\overline{{\cal M}}$ with its canonical filtration, the suboperads ${\cal D}^1_{k}$
are all trivial for dimesional reasons
except when $k$ is $0$.
We now introduce a filtration on the endomorphism operad of $V$ by endowing
$V$ with an additional grading. Let $V$ be a complex with differential
$Q:V^g\,\to\,V^{g-1}$ and a second grading called the {\sl filtration
degree} denoted by $V_p$ such that $V$ is now a filtered complex. The
endomorphism operad $\End{V}$ inherits a natural filtration $F_p\,\End{V} =
\{\,F_p\,\End{V}(n)\,\}$ where $F_p\,\End{V}(n)$ is the space of maps in
$\End{V}(n)$ with filtration degree less than or equal to $p$. In this case,
the spectral sequence associated to this filtration makes each $E^r$ term
into an operad as well as the suboperads indicated above. Perhaps the
simplest filtration of $V$ is when the filtration degree is exactly the
degree in the complex. In this case, the associated spectral sequence
degenerates at the $E^2$ term, where $E^0 \simeq \End{V}$ with a zero
differential, $E^1 \simeq\End{V}$ with the differential $Q$, and $E^2 \simeq
H(\End{V})$.
\begin{df}
Let $V$ be a vector space graded by a filtration degree as above. We say that
an
${ \cal O}$-algebra, $V$, is a {\sl filtered ${ \cal O}$-algebra} if the morphism of
operads ${ \cal O}\,\to\, \End{V}$ preserves the filtrations.
\end{df}
If ${ \cal O}$ is an operad with filtration $F$ and $V$ a filtered ${ \cal O}$-algebra,
then the $E^r$ term of $\End{V}$ is an algebra over the $E^r$ term of
${ \cal O}$ for all $r\geq 0$.
In the case of the operad $\overline{{\cal M}}$ with its canonical filtration where $V$ is a
filtered $C_{\bullet}(\overline{{\cal M}})$-algebra with the filtration degree on $V$
equal to the degree of the complex, then $V$ is an algebra (with zero
differential) over the operad $E^0$, $V$ is an algebra (with $Q$
differential), over the operad $E^1$, and $H(V)$ is an algebra over $E^2$
(which is isomorphic to the operad $H_\bullet(\overline{{\cal M}})$) where we have used the
canonical morphism $H(\End{V})\,\to\,\End{H(V)}$ in the last step.
\begin{ack}
We would like to thank F.~Cohen, M.~Kontsevich, R.~Hain and
M.~Schlessinger for helpful discussions on the cohomology of moduli
spaces. We thank Gregg Zuckerman for his useful comments on Table~1. The third
author is very grateful to the hospitality of the
Weizmann Institute of Science, where part of the work has been done.
\end{ack}
\bibliographystyle{amsplain}
\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother
|
1995-02-16T06:20:18 | 9502 | alg-geom/9502012 | en | https://arxiv.org/abs/alg-geom/9502012 | [
"alg-geom",
"math.AG"
] | alg-geom/9502012 | Sandra Di Rocco | Sandra Di Rocco | k-very ample line bundles on Del Pezzo Surfaces | AMSLATEX file, 10 pages. To appear on Math. Nachrt | null | null | null | null | A $k$-very ample line bundle L on a Del Pezzo Surface is numerically
characterized. We find that the set of the exceptional curves and effective
divisors with selfintersection zero, $\cal S$, plays a very important rule in
checking the nefness and $k$-very ampleness of a line bundle on a Del pezzo
Surface. We show that a line bundle L is nef if and only if it is spanned and
that L is $k$-very ample if and only if the intersection with all the elements
of $\cal S$ is greater or equal than k. In the first part of the paper we give
a complete numerical description of the elements of $\cal S$ .
| [
{
"version": "v1",
"created": "Wed, 15 Feb 1995 09:12:38 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Di Rocco",
"Sandra",
""
]
] | alg-geom | \section{Introduction.}
Let L be a line bundle on a smooth connected projective surface. L is said
to be $k$-very ample
for an integer $k\geq 0$, if given any 0-dimensional subscheme $(Z,{\cal
O}_Z)$ of S with
$h^0({\cal O}_Z)=k+1$, the restriction map $H^0(L)\longrightarrow
H^0(L_{Z})$ is surjective. Notice
that L is spanned by its global section if and only if it is $0$-very ample
and that it is
very-ample if and only if it is 1-very ample. Using the Reider-type
criterion (theorem 2.3) it is possible to give an exact numerical
characterization of $k$-very ample line bundles on Surfaces whose Picard
group is fully
known. $k$-very ample line bundles on ${\bf P}^2$, on the Hirzebruch
surfaces ${\bf F_n}$ and on hyperelliptic surfaces
are indeed completely characterized in [2], [5],
[10].
This paper gives a numerical
characterization of $k$-very ample line bundles on Del Pezzo surfaces,
which improves the result
of Biancofiore and Ceresa in [7].
Let $S_r$ be a Del Pezzo surface of degree $9-r$, i.e. $-K_{S_r}$ is an
ample line bundle of degree
$9-r.$ We prove that:
\vspace{.20in}
{\em Let $L=al-\sum_1^rb_ie_i$ be a line bundle on $S_r$, $L\neq
-kK_{S_8}$, $L\neq -(k+1)K_{S_8}$ and
$L\neq -K_{S_7}$ when $k=1$, then $L$ is k-very-ample if and only if:\\
for $r=1$ $a\geq b_1+k$ and $b_1\geq k$ ;\\
for $r=2,3,4$, $b_1\geq b_2...\geq b_r\geq k$ and $a\geq b_i+b_j+k$, where
$i\neq j=1...r$;\\
for $r=5,6$, $b_1\geq ... \geq b_r\geq k$, $ a\geq b_i+b_j+k$, where $i\neq
j=1...r$, and
$ 2a\geq\sum_1^5b_{i_t}+k$;\\
for $r=7$, $b_1\geq ... \geq b_7\geq k$, $ a\geq b_i+b_j+k$, where $i\neq
j=1...r$,
$ 2a\geq\sum_1^5b_{i_t}+k$ and $ 3a\geq 2b_i+\sum_1^6 b_{j_t}+ k$;\\
for $r=8$, $b_1\geq ... \geq b_8\geq k$, $ a\geq b_i+b_j+k$ , $
2a\geq\sum_1^5b_{i_j}+k$ , \\
$ 3a\geq 2b_i+\sum_1^6 b_{j_t}+ k$, $4a\geq\sum_1^32b_i+\sum_1^5b_{j_t}+k$,
$5a\geq\sum_1^62b_{i_t}+b_j+b_k+k$
and $6a\geq 3b_i+\sum_1^72b_{j_t}+k$}.
\newpage
We find that the set of the exceptional curves and effective divisors with
selfintersection
zero, $\cal S$, plays a very important rule in checking the nefness and
$k$-very ampleness
of a line bundle on $S_r$.
We show that a line bundle L is nef if and only if it is spanned and that L
is $k$-very ample if and only if the intersection with all the elements of
$\cal S$ is greater or
equal than k. In the first part of the paper we give a complete numerical
description of the elements
of $\cal S$ that will allow us to prove the main theorem.\\
In [7] for L to be $k$-very ample it is required that $a\geq \sum_1^rb_i$
for any r. If we consider, for
example, the case $r=3$ Corollary 4.6 proves that it is enough to require
that $a\geq b_i+b_j$.\\
\noindent{\bf Acknowledgments.} I would like to thank my advisor, Andrew J.
Sommese, for suggesting this problem and for his help
throughout my studies. I would also like to thank Gianmario Besana for his
numerous helpful comments.
The author was supported by the C.N.R. of the Italian government.
\vspace{0.15in}
\section{Background material.}
\vspace{.18in}
\noindent{ \bf \sf Notation.}
\vspace{.15in}
Let S be a smooth connected projected surface defined over the complex
field {\bf C}.
$B_{P_1,...,P_r}(S)$ is the blowup of S at the points $P_1,...,P_r$;\\
Let L be a line bundle on $S$, we will use the following notation:\\
$h^i(L)$ is the dimension of $H^i(L)$;\\
$|L|$ is the complete linear system associated to L;\\
$K_S$ is the canonical bundle of S;\\
$L^2=d(L)$ is the degree of the line bundle L;\\
$g(L)$, the sectional genus of S, is the integer defined by the equality :\\
$2g(L)-2=L^2+K_SL$. If $L={\cal O}_S(D)$ for a divisor D we also write
$g(D)$ for $g(L)$ and if D is an
irreducible reduced curve on a surface S , $g(D)$ is the arithmetic genus of D.
\\ \\
L is said to be {\it numerically effective} (nef) if $LC\geq 0$ for every
curve $C$ on S and in this case
L is said to be {\it big} if $L^2>0$.\\
L is said to be Q-effective if, for some positive integer $n$, $nD$ is
effective.\\
L is said to be spanned if it is spanned by its global sections, i.e.\\
$h^0(L\bigotimes{\cal I}_P)=h^0(L)-1$ for every point on S.\\
L is said to be very ample if the map associated to $|L|$ embeds S in
$P^{h^0(L)-1}$, i.e.
$h^0(L\bigotimes{\cal I}_Z)
=h^0(L)-2$ for any $0$-dimensional subscheme Z of length 2 on S.\\
\newpage
A $(-1)$-curve $C$ on $S$, i.e. a rational curve of degree $-1$, will be
always called an exceptional
curve.
\vspace{.15in}
\noindent{ \bf \sf $k$-very ampleness. }
\begin{definition}
A line bundle is said to be {\bf k-very ample},
for an integer $k\geq 0$ if, given any 0-dimensional subscheme $(Z,{\cal
O}_Z)$ of S with
$h^0({\cal O}_Z)=k+1$, the restriction map $H^0(L)\longrightarrow
H^0(L_{Z})$ is surjective.
\end{definition}
Note that L is 0-very ample if and only if it is spanned by its global
section and L is 1-very ample if and only if
it is very ample.\\
The notion of $k$-very ampleness has a classical interpretation in terms of
associated secant mappings.
Let $S^{[k]}$ be the Hilbert scheme of 0-dimensional subschemes on S of
length k; a line bundle
L is $k$-very ample if an associated line bundle $L^{[k]}$, on $S^{[k]}$,
is very ample, see [3].\\
\begin{proposition}(see [4], 0.5.1) If L is a $k$-very ample line bundle on
a surface S, then $LC\geq k$ for
every effective curve C on S, with equality only if $C\cong {\bf P}^1$.
\end{proposition}
The following theorem will be the tool we will use to check the $k$-very
ampleness for line bundles:\\
\begin{theo}(see [3], Theorem 2.1)
Let M be a nef line bundle on a smooth surface S, such that $M^2\geq 4k+5$,
for $k\geq 0$. Then either $L=M+K_S$
is $k$-very ample or there exists an effective divisor D s.t M-2D is
Q-effective, D contains some
$0$-dimensional subscheme where the $k$-very ampleness fails and:
$$ MD-k-1\leq D^2<MD/2<k+1$$
\end{theo}
\vspace{.25in}
\noindent{ \bf \sf Del Pezzo surfaces.}
\vspace{.15in}
A surface is called a Del Pezzo surface if its anticanonical bundle $-K_S$
is ample.
If $K_S^2=s$ we will say that S is a Del Pezzo surface of degree s.
The degree turns out to be a very useful tool to classify Del Pezzo
surfaces. Indeed we have
the following classification:
\begin{proposition}(see [8])
Let S, s as before, then\\
a) $1\leq s\leq 9$;\\
b) $s=9$ if and only if $S=P^2$;\\
c) if s=8 then either $S=P^1\times P^1$ or $S=B_P(P^2)={\bf F_1}$;\\
d) if $1\leq r\leq 7$ then $S=B_{P_1,...,P_{9-s}}(P^2)$ where no three points
are on a line nor six points on a conic.
\end{proposition}
We will denote a Del Pezzo surface of degree s with $S_r$, where $r=9-s$.\\
Let $\pi :S_r\longrightarrow {\bf P}^2$ be the blowing up , as in d). It
follows that $Pic(S_r)={\bf Z}^{r+1}$ and it is generated by $\{l,e_1,...,
e_r\}$ where $l$ is the class of $\pi^*({\cal O}_{\bf P^2}(1))$, $e_i$ the
class of $\pi^{-1}(P_i)$, $l^2=1$, $e_1^2=-1$, $e_1e_j=0$ and $e_il=0$.\\
Therefore any line bundle on $S_r$ is of the form $al-\sum_1^rb_ie_i$ and
from the definition of blow up
it follows that $-K_S=3l-\sum_1^re_i$.\\
Since for $S_0$ and for $S_1=P^1\times P^1$ a complete characterization of
$k$-very ample line bundles
has been done in [2] and [5], from now on we'll focus our attention on
$S_r=B_{P_1,...,P_r}$ for
$r=1,...,8$, as in proposition 2.4 d).\\
We will use the following notation, according to [8]:\\
Let $I_r=\{L\in Pic(S_r)$ such that $L^2=-1$ and $LK_S=-1\}$ be the set of
exceptional curves
on $S_r$,
they are in fact irreducible effective divisors.
the cardinality of $I_r$ is finite for $1\leq r \leq 8$ and any $\xi\in
I_r$ can be expressed
as the pull back of a curve in {\bf P}$^2$ passing through the points blown up.
In the following table the number of exceptional curves in $S_r$ of all
possible
types it is shown
. For example if $r=6$, i.e $S_r$ is the cubic surface in $P^3$, the
table shows the 27 possible lines.
\vspace{.05in}
\begin{center}
\begin{tabular}{|l|} \hline
$type\backslash r$\\ \hline \hline
$(0;-1)$\\ \hline
$(1;1^2)$\\ \hline
$(2;1^5)$\\ \hline
$(3;2,1^6)$\\ \hline
$(4;2^3,1^5)$\\ \hline
$(5;2^6,1^2)$ \\ \hline
$(6;3,2^7)$ \\ \hline
\end{tabular}\hspace{.005pt}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
0 & 1 & 3 & 6 & 10 & 15 & 21 & 28 \\ \hline
0 & 0 & 0 & 0 & 1 & 6 & 21 & 56 \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 7 & 56 \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 56 \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 28 \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 \\ \hline
\end{tabular}
\end{center}
\vspace{.04in}
where $(a_0;a_1^{n_1},a_2^{n_2},...)$ is the class of the curve
$a_0l-\sum_1^{n_1}a_1e_{i_t}-\sum_1^{n_2}
a_2e_{j_t}-...$, i.e the pull back of a curve of degree $a_o$ in ${\bf
P}^2$ passing, with multiplicity
$a_i$ , through $n_i$ points blown up. \\
\begin {lemma}(see [8], Lemme 9)If D is an effective irreducible divisor on
$S_r$ such that
$DK_S=-1$ then D is an exceptional curve or $D=-K_{S_8}$.
\end{lemma}
\section{ Nefness on Del Pezzo surfaces}
Let $D=al-\sum_1^rb_ie_i\in Pic(S_r)$ s.t $D^2=0$, i.e. $a^2-\sum_1^rb_i^2=0$.
By the adjunction formula we have $g(D)=0$ and $K_SD=-2$ i.e.,
$3a-\sum_1^rb_i=2$. Using the inequality
$(\sum_1^rb_i)^2\leq r\sum_1^rb_i^2$ and:
$$\left\{\begin{array}{c}
\sum_1^r b_i = 3a-2 \\
\sum_1^r b_i^2 = a^2
\end{array}
\right. $$
we get $a\leq 11$.\\
Assuming $b_1\leq b_2\leq ... \leq b_r$, by numerical computations we
obtain that there is
a finite number of n+1-tuples $(a,b_1,b_2,...,b_r)$ , solutions of the
system above. In the
following table we collect the possible solutions and show the divisors
obtained:
\vspace{0.05in}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline
$a$ &$b_1$ & $b_2$ & $b_3$ & $b_4$ &$ b_5$ &$b_6$ & $b_7$ & $b_8$ &type \\
\hline\hline
1& 0& 0& 0 & 0 & 0 & 0 & 0 & 1&(1;1) \\ \hline
2& 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1&$(2;1^4)$ \\ \hline
3& 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2&$(3;1^5;2)$ \\ \hline
4 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 3&$(4;1^7;3)$ \\
& 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2&$(4;1^4;2^3)$ \\ \hline
5 & 0 & 1 & 2 & 2 & 2 & 2 & 2 & 2&$(5;1;2^6)$ \\
& 1 & 1 & 1 & 1 & 2 & 2 & 2 & 3&$(5;1^4;2^3;3)$ \\ \hline
6 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3&$(6;1^2;2^4;3^2)$ \\ \hline
7 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & 3&$(7;1;2^3;3^4)$ \\
& 2 & 2 & 2 & 2 & 2 & 2 & 3 & 4&$(7;2^6;3;4)$ \\ \hline
8 & 1 & 3 & 3 & 3 & 3 & 3 & 3 & 3&$(8;1;3^7)$ \\
& 2 & 2 & 2 & 3 & 3 & 3 & 3 & 4&$(8;2^3;3^4,4)$ \\ \hline
9 & 2 & 3 & 3 & 3 & 3 & 3 & 4 & 4&$(9;2,3^5;4^2)$ \\ \hline
10 & 3 & 3 & 3 & 3 & 4 & 4 & 4 & 4&$(10;3^4,4^4)$ \\ \hline
11 & 3 & 4 & 4 & 4 & 4 & 4 & 4 & 4&$(11;3;4^7)$ \\ \hline
\end {tabular}
\end{center}
\vspace{.04in}
where every row represents a solution.
Note that all the divisors obtained are sum of two exceptional divisors,
except for the case $ a=1 $.
The decomposition, for any value of $a$, it is shown in the next table :\\
\begin{center}
\begin{tabular}{|l|l|}\hline
a & type \\ \hline\hline
1 & $(1;1)$ \\ \hline
2 & $(1;1^2)$+$(1;1^2)$ \\ \hline
3 & $ (2;1^5)+(1;1^2)$ \\ \hline
4 & $ (3;2,1^6)+(1;1^2)$ \\ \hline
5 & $ (3;2,1^6)+(2;1^5)$ \\ \hline
6 & $(4;2^3,1^5)+(2;1^5)$ \\ \hline
7 & $(5;2^6,1^2)+(2;1^5)$ or $(6;3,2^7)+(1;1^2)$ \\ \hline
8 & $(5;2^6,1^2)+(3;2,1^6)$ or $(6;3,2^7)+(2;1^5)$ \\ \hline
9 & $(6;3,2^7)+(3;2,1^6)$ \\ \hline
10 & $(6;3,2^7)+(4;2^3,1^5)$ \\ \hline
11 & $(6;3,2^7)+(5;2^6,1^2)$ \\ \hline
\end{tabular}
\end{center}
\vspace{.08in}
The following proposition follows from the computations above:
\begin{proposition}Let $D$ be an effective divisor on $S_r$ with $D^2=0$,
then: \\
a) D is irreducible and $D=l-e_i$; \\
b) D is reducible and $D=\xi_1+\xi_2$, $\xi_1,\xi_2\in I_r$.
\end{proposition}
\begin{proposition}Let D be an effective divisor on $S_r$, then it is nef
and big, unless it is irreducible
and $D\in I_r$ or $D=l-e_i$, or it is reducible and it has at least one
exceptional curve as component.
\end{proposition}
\begin{pf} If S is a surface such that for every two divisors $A$, $B$,
with $A^2\geq0$ and $B^2
\geq 0$, $AB\geq 0$, then the same property is true for a surface obtained
by blowing up one
point. If $S$ is ${\bf P}^2$
the property is true, therefore it is true for $S_r$.
The only effective divisors on $S_r$ with negative self-intersection are
the elements of $I_r$. Thus If $D$ is an irreducible effective divisor, $
D^2>0$ unless
$D\in I_r$ or $D=l-e_i$ and $D\xi \geq 0$ for any $\xi\in I_r$, unless
$D=\xi\in I_r$.
If D is reducible then $D^2>0$ unless $D=\xi_1+\xi_2$ with $\xi_1,\xi_2\in
I_r$
and if $D\xi<0$ for some $\xi\in I_r$ then $\xi$ is one of its components.
\end{pf}
\begin{lemma}(see [8], Lemme 1) Let $L=al-\sum_i^rb_ie_i\in Pic(S_r)$. If
$a\geq -2$ and $d(L)\geq K_SL$
then L is effective.
\end{lemma}
\begin{theo} Let $L=al-\sum_1^3b_ie_i$ be a line bundle on $S_r$.
L is nef if and only if $L\xi\geq 0$ for any $\xi\in I_r$ and
$L(l-e_1)\geq 0$ for $r=1$ i.e.\\
for $r=1$, $a\geq b_1$ and $b_1\geq 0$ ;\\
for $r=2,3,4$, $b_1\geq b_2...\geq b_r\geq 0$ and $a\geq b_i+b_j$;\\
for $r=5,6$, $b_1\geq ... \geq b_r\geq 0$, $ a\geq b_i+b_j$, where $i\neq
j=1...r$ and
$ 2a\geq\sum_1^5b_{i_t}$;\\
for $r=7$, $b_1\geq ... \geq b_7\geq 0$, $ a\geq b_i+b_j$, where $i\neq
j=1...r$,
$ 2a\geq\sum_1^5b_{i_t}$ and $ 3a\geq 2b_i+\sum_1^6 b_{j_t}$;\\
for $r=8$, $b_1\geq ... \geq b_8\geq 0$, $ a\geq b_i+b_j$, where $i\neq
j=1...r$,
$ 2a\geq\sum_1^5b_{i_t}$, $ 3a\geq 2b_i+\sum_1^6 b_{j_t}$,
$4a\geq\sum_1^32b_{i_t}+\sum_1^5b_{j_t}$,
$5a\geq\sum_1^62b_{i_t}+b_{j_t}+b_k$ and \\ $6a\geq
3b_{i_t}+\sum_1^72b_{j_t}$.
\end{theo}
\begin{pf} If L is nef $L\xi\geq 0$ by definition.\\
Let L be a line bundle on $S_r$ s.t. $L\xi\geq 0$ for any $\xi\in I_r$ and
$L(l-e_1)\geq 0$ for $r=1$, then $a\geq 0$, $L^2\geq 0$ and $K_SL<L^2$ that
implies that L is effective on $S_r$.
Let $C$ be an effective divisor then, by proposition 3.2, $C$ is nef and
therefore $LC\geq 0$ unless
$C\in I_r$, $C=l-e_i$ or $C$ has an exceptional curve as component. If
$C\in I_r$
or has an exceptional curve as component $LC\geq 0$ by hypothesis.
If $C=l-e_i$ and $LC< 0$ then $r\geq 2$ and $a<b_i$; it follows that
$L\xi<0$ for $\xi=l-e_i-e_j$ that is impossible by
hypothesis.\end{pf}
\section{ $k$-very ampleness on Del Pezzo surfaces.}
Note that if $L=al-\sum_1^rb_ie_i$ is a $k$-very ample line bundle on $S_r$
then $LC\geq k$ for any
irreducible effective divisor C ; in particular $Ll\geq k$, $Le_i\geq k$
for $i=1...r$ and
$L\xi\geq k$ for any $\xi$ in $I_r$.\\
Let us assume $L\xi\geq k$ for any $\xi\in I_r$ and $l(l-e_i)\geq k$; that
implies L nef and big
$b_i\geq k$ for $i=1...r$ and
$a\geq 3k$, for $r\geq 2$. Therefore we can write $L=kL'+H$ where $L'$ is
an effective divisor with
all coefficients greater or equal then k and $H$ is an effective divisor
with all coefficients strictly
less than k.
\begin{lemma} Let $M=L-K_S$; if $(M-D)D\leq k+1$ for some effective divisor
D on S, then \\
a) $r\geq 2$ and either $D^2>0$ or $L\xi <k$ for some $\xi\in I_r$;\\
b) $r=1$ and either $D^2>0$ or $Le_1< k$ or $L(l-e_1)< k$.
\end{lemma}
\begin{pf} Assume $D^2<0$, then $D=\xi\in I_r$, $MD\leq k$ and $-K_SD\geq
1$, therefore $LD<k$.
Assume now $D^2=0$; if D is irreducible it is of the form $l-e_i$ with
$-K_SD=2$, that implies
$LD<k$. For $r\geq 2$, taking any exceptional curve of the form
$\xi=l-e_i-e_j$, we obtain $L\xi<k$.
If D is reducible then $r\geq 2$ and $D=\xi_1+\xi_2$ $\xi_1,\xi_2\in I_r$
which gives $L\xi_i<k$,
for i=1,2.
\end{pf}
\begin{lemma} Let $L=kL'+H$ be a line bundle on $S_r$ such that $L\xi\geq
k$ for any exceptional
curve $\xi\in I_r$, and $L(l-e_1)\geq k$ for $r=1$, and $M=L-K_{S_r}$,
then either $(L')^2>0$ or $r=8$, $L=k\xi+H$ or $L=k(\xi_i+
\xi_j)+H$ with $\xi,\xi_i,\xi_j$ of the
type $(5;2^6,1^2)$ or $(6;3,2^7)$ and there is no effective divisor $D$ on
$S_8$
such that $$ MD-k-1\leq D^2<MD/2<k+1$$.
\end{lemma}
\begin{pf} Assume $L'$ irreducible and $(L')^2\leq 0$. If $(L')^2=0$ then
$L'=l-e_i$ and therefore
$L(l-e_i-e_j)=H(l-e_i-e_j)<k$; since the coefficients of $H$ are strictly
less than k, that is
impossible by our setting. If $(L')^2<0$ then $L=k\xi+H$ where $\xi\in
I_r$ and for $r\leq 7$ we
always have $\xi(l-e_1-e_2)=0$ for any $\xi\in I_r$ and therefore
$L(l-e_1-e_2)=H(l-e_1-e_2)<k$,
impossible as before.
Assume now $r=8$ and $L=k\xi+H$ with $\xi$ of the type $(5;2^6,1^2)$ or
$(6;3,2^7)$,
then $H^2>0$ and $H\xi\leq 2k$ otherwise $L\xi<k$.
If $D$ were an effective divisor on $S_8$ s.t. $ MD-k-1\leq D^2<MD/2<k+1$
then $D^2>0$, by
lemma 4.1.
If $D^2=1$ then, since $(k\xi+H)^2\geq-k^2+1+4k^2=3k^2+1$ and $L^2D^2\leq
(LD)^2$, $LD>k$ and
$LD-K_SD-k-1>1=D^2$. If $D^2\geq 2$ then $LD>2k$, that implies
$k+1>D^2>2k+2-k-1$.
The case L irreducible, i.e $L=\xi_1+\xi_2$ with $\xi$ of the
type $(5;2^6,1^2)$ or $(6;3,2^7)$ is proved as before.\end{pf}
\begin{lemma}$M^2\geq 4k+5$ for any k unless $L=-kK_{S_8}$ or $k=1$ and
$L=-K_{S_7}$.
\end{lemma}\begin{pf} by Lemma 4.2 $(L')^2\geq 1$ and $L'H\geq 0$; note that
we
can also assume $-K_SL\geq 2$ unless r=8 and $L=-K_{S_8}$ in which case
$M^2\geq 4k+5$ for any k, unless $H\equiv 0$ and therefore $L=-kK_{S_8}$.
A numerical checking also
shows that $(-(k+1)K_{S_7})^2\leq 4k+4$ if and only if k=1. In all the
remaining cases we have
$M^2\geq k^2+6k+2\geq 4k+5 $.\end{pf}
Note that if L is a k-very-ample line bundle on a surface then $LE\geq k+2$
for any elliptic curve
E on S, therefore $-kK_{S_8}$ , $-(k+1)K_{S_8}$ cannot be $k$-very ample
and $-K_{S_7}$ cannot
be very ample.\\
\begin{theo} Let L be a line bundle on $S_r$ such that $L\xi\geq k$ for
any exceptional
curve $\xi\in I_r$, and $L(l-e_1)\geq k$ for $r=1$. L is
k-very-ample unless $L=-kK_{S_8}$, $L=-(k+1)K_{S_8}$ or $k=1$ and $L=-K_{S_7}$.
\end{theo}
\begin{pf} $M^2\geq 4k+5$ by Lemma 4.3 and L is nef and big by theorem
3.4, then L is k-very
ample unless there exist an effective divisor D on S such that $$
MD-k-1\leq D^2<MD/2<k+1.$$
Assume such D exists.
$(L')^2\geq 1$ by Lemma 4.2 and $D^2\geq 1$ by Lemma 4.1.
Therefore $L'D\geq 1$ and $-K_SL'\geq 2$ unless r=8, $L'=-K_{S_8}$
and $H$ is not 0, in which case we would get $MD\geq 2k+2$ that is
impossible. If $L'D=1$ then by the Hodge index theorem we should have
$D^2=(L')^2=1$ and $-K_SD\leq 2$. If $-K_SD=1$ then $D=K_{S_8}$ that is
impossible since $-L'K_S\geq 2$.
If $-K_SD=2$ we have $HD\leq 0$, i.e. $H^2\leq 0$ and $L^2\geq 3k^2-1$,
therefore $LD>2k$ and
$2k+2-k-1\leq 1$.
If $L'D\geq 2$ then $MD\geq 2k+2$, that is impossible.\end{pf}
\begin{corollary} Let L be a line bundle on $S_r$, s.t $L\neq -kK_{S_8}$,\\
$L\neq -(k+1)K_{S_8}$ and
$L\neq -K_{S_7}$ when $k=1$. L is $k$-very ample if and only if $L\xi\geq
k$, for any $\xi\in I_r$
and $L(l-e_1)\geq k$ for r=1.
\end{corollary}
Using the table of exceptional curves we easily obtain the following
numerical characterization:
\begin{corollary} Let $L=al-\sum_1^rb_ie_i$ be a line bundle on
$S_r$,$L\neq -kK_{S_8}$, \\$L\neq -(k+1)K_{S_8}$ and
$L\neq -K_{S_7}$ when $k=1$ then $L$ is k-very-ample if and only if:\\
for $r=1$, $a\geq b_1+k$ and $b_1\geq k$ ;\\
for $r=2,3,4$, $b_1\geq b_2...\geq b_r\geq k$ and $a\geq b_i+b_j+k$, where
$i\neq j=1...r$;\\
for $r=5,6$, $b_1\geq ... \geq b_r\geq k$, $ a\geq b_i+b_j+k$, where $i\neq
j=1...r$, and
$ 2a\geq\sum_1^5b_{i_t}+k$;\\
for $r=7$, $b_1\geq ... \geq b_7\geq k$, $ a\geq b_i+b_j+k$, where $i\neq
j=1...r$,
$ 2a\geq\sum_1^5b_{i_t}+k$ and $ 3a\geq 2b_i+\sum_1^6 b_{j_t}+ k$;\\
for $r=8$, $b_1\geq ... \geq b_8\geq k$, $ a\geq b_i+b_j+k$ ,$
2a\geq\sum_1^5b_{i_j}+k$ \\
$ 3a\geq 2b_i+\sum_1^6 b_{j_t}+ k$, $4a\geq\sum_1^32b_i+\sum_1^5b_{j_t}+k$,
$5a\geq\sum_1^62b_{i_t}+b_j+b_k+k$
and $6a\geq 3b_i+\sum_1^72b_{j_t}+k$.
\end{corollary}
Comparing Theorem 3.4 and Corollary 4.6 we can conclude that:
\begin{theo}A line bundle on a Del Pezzo surface is nef if and only if it
is spanned by its global sections.
\end{theo}
\section{observations}
{\sf \bf The Hirzebruch surface ${\bf F_1}$.} For $S_1={\bf F}_1$ a
different characterization is
often used, i.e. a line bundle L is
written as $L=a_0E_0+bf$, where $E_0$ and $f$ are equal respectively to
the minimal section and
the fiber of the morphism $S_1\longrightarrow {\bf P}^1$ and $ K_S=-2E_0
-3f$. A simple checking
shows that $E_0=e_1$ and $f=l-e_1$.\\
Therefore If $L\in Pic({\bf F}_1)$ is written as $a_0E_0+bf$
we can express it as $L=bl-(b-a_0)e_1$.
{}From Corollary 2.5 it follows that L is $k$-very ample if and only if
$a_0\geq k$ and $b\geq a_0+k$, same characterization as in
[5].\\
\\
{\bf \sf The adjoint bundle.} Let $L$ be a $k$-very ample line bundle on
$S_r$. Applying
Corollary 4.6 to the adjoint bundle $K_S+L=(a-3)-\sum_1^r(b_i-1)e_i$ we can
easily see that it is always $(k-1)$-very ample if
$r\geq 2$.
If $r=1$ it is $(k-1)$-very ample if $a\geq b_1+k+1$.\\
\\
{\bf\sf The degree.} If L is $k$-very ample, from [5, theorem 3.1] $
kK_S+L$ is nef and big unless $L=-kK_S$, with $K_S^2\geq 2$
for $k\geq 2$ and with $K_S^2\geq 3$ for $k=1$.
From [1, Corollary 3.31] it follows that $(kK_S+L)L\geq k+4$ for
$k\geq 2$ i.e. $$d(L)\geq k^2+3k+2$$ if $L\neq -kK_S$ and $k\geq 2$.
\bibliographystyle{abbrv}
|
1995-07-20T18:17:25 | 9502 | alg-geom/9502026 | en | https://arxiv.org/abs/alg-geom/9502026 | [
"alg-geom",
"math.AG"
] | alg-geom/9502026 | Robert Friedman | Robert Friedman and John W. Morgan | Algebraic surfaces and Seiberg-Witten invariants | null | null | null | null | null | In this revised version, we add some expository material and references and
make some minor corrections.
| [
{
"version": "v1",
"created": "Mon, 27 Feb 1995 19:46:16 GMT"
},
{
"version": "v2",
"created": "Tue, 9 May 1995 19:34:54 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Friedman",
"Robert",
""
],
[
"Morgan",
"John W.",
""
]
] | alg-geom | \section{1. Introduction.}
Donaldson theory has shown that there is a deep connection between the
4-manifold topology of a complex surface and its holomorphic geometry
\cite{5}, \cite{6}, \cite{8}, \cite{9}, \cite{14}, \cite{15}. Recently Seiberg
and Witten have introduced a new set of 4-manifold invariants
\cite{33}. These invariants have greatly clarified the structure of 4-manifolds
and have made it possible to prove various conjectures suggested by Donaldson
theory. The invariants take on an especially simple form for K\"ahler
surfaces, as realized by Witten \cite{34} and Kronheimer-Mrowka. For example,
their arguments show that, if $X$ is a minimal
algebraic surface of general type, then every orientation-preserving
self-diffeomorphism $f\: X \to X$ satisfies $f^*c_1(K_X) = \pm c_1(K_X)$, where
$K_X$, the canonical line bundle of $X$, is the line bundle whose local
holomorphic sections are holomorphic 2-forms. More generally, in case
$X$ is a K\"ahler surface with
$b_2^+
\geq 3$, the form of the invariants easily establishes some conjectures of
\cite{14}, that the only smoothly embedded 2-spheres of self-intersection $-1$
in $X$ represent the classes of exceptional curves, and that the pullback of
the
canonical class of the minimal model of $X$ is invariant up to sign under every
orientation-preserving diffeomorphism of $X$ (see Section 3 below). In this
paper we shall extend these results to cover the case of
$b_2^+ = 1$. Here, for a K\"ahler surface $X$, $b_2^+(X) = 2p_g(X) +1$, where
$p_g(X)$ is the number of linearly independent holomorphic 2-forms on $X$,
i\.e\.
$\dim H^0(X; K_X)$. More specifically, we shall prove:
\theorem{1.1} Let $X$ be a minimal surface of general type with $p_g(X) = 0$.
Let $\tilde X$ be a blowup of $X$ at $\ell$ distinct points, and let $E_1,
\dots, E_\ell$ be the exceptional curves on $\tilde X$. Finally let $K_0 \in
H^2(\tilde X; \Zee)$ be the pullback to $\tilde X$ of the canonical class of
the
minimal model $X$ of $\tilde X$. If $f$ is an orientation-preserving
self-diffeomorphism $f\: \tilde X \to \tilde X$, then for
all $i$ there is a $j$ such that $f^*[E_i] = \pm [E_j]$ and
$f^*K_0 =
\pm K_0$. More generally, let $X$ and $X'$ be two minimal
surfaces of general type satisfying the above hypotheses. Suppose that $\tilde
X$ and $\tilde{X'}$ are blowups of $X$ and $X'$ respectively at distinct
points,
that
$E_1, \dots, E_\ell$ and
$E_1', \dots E_m'$ are the exceptional curves on $\tilde X$ and $\tilde{X'}$
respectively and that $K_0$ and $K_0'$ are the pullbacks to $\tilde X$ and
$\tilde{X'}$ of the canonical classes of $X$ and $X'$. If
$f\: \tilde X\to \tilde{X'}$ is an orientation-preserving diffeomorphism, then
$\ell = m$, for every $i$ there exists a $j$ such that $f^*[E_i'] = \pm [E_j]$
and $f^*K_0' = \pm K_0$.
\endproclaim
More generally, we can replace embedded 2-spheres of square $-1$ in the above
theorem by more general negative definite 4-manifolds. Here, if $N$ is a
negative definite 4-manifold and $X$ is a K\"ahler surface which is
orientation-preserving diffeomorphic to $M\#N$ for some 4-manifold $M$, then
it is essentially a remark of Kotschick (see \cite{17}, \cite{20})
that
$N$ has no nontrivial finite covering spaces and in particular $H_1(N; \Zee) =
0$. Thus, if $N$ is a negative definite summand
of a K\"ahler manifold, $H_1( N; \Zee) = 0$ and $H^2(N; \Zee)$ is torsion free.
Again by a theorem of Donaldson \cite{7}, $H^2(N; \Zee)$ has a basis $\{n_1,
\dots, n_\ell\}$ such that $n_i^2 = -1$ and $n_i\cdot n_j = 0$ if $i\neq j$.
Such a basis is unique up to sign changes and permutation, and we will refer to
the $n_i$ as the {\sl exceptional classes\/} of $N$.
\theorem{1.2} Let $X$ be a minimal surface of general type with $p_g(X) = 0$.
Let $\tilde X$ be a blowup of $X$ at $\ell$ distinct points, and let $E_1,
\dots, E_\ell$ be the exceptional curves on $\tilde X$. Let $N$ be a closed
oriented negative definite $4$-manifold, and suppose that
$\{n_1,
\dots, n_k\}$ is a basis for $H^2(N; \Zee)$ such that $n_i^2 = -1$ for all $i$
and $n_i\cdot n_j = 0$ if $i\neq j$. If there is an orientation-preserving
diffeomorphism $\tilde X \to M \# N$, then, for every $i$, $n_i = \pm [E_j]$
for some $j$.
\endproclaim
Using the above theorems, the arguments of Witten, Kronheimer, and Mrowka in
case $p_g>0$, and standard material on algebraic surfaces (see e\.g\.
\cite{15}) we can deduce the following corollary:
\corollary{1.3} Let $X$ be a minimal K\"ahler surface which is not rational or
ruled, or equivalently such that the Kodaira dimension of $X$ is at least
zero. Then the conclusions of \rom{(1.1)} and \rom{(1.2)} hold for $X$.
\endproclaim
By the classification of surfaces, a K\"ahler surface $X$ which is not
rational or ruled either satisfies $p_g(X) >0$, $p_g(X) = 0$ and $X$ is
of general type, or $p_g(X) = 0$ and $X$ is elliptic. To establish the
corollary, the case $p_g >0$ is covered by the arguments of
Witten and Kronheimer-Mrowka. The case $p_g=0$ and $X$ of general type is
covered by the above theorems. There remains the case that $p_g=0$ and $X$
is elliptic. The case where $p_g(X) = 0$, $X$ is elliptic, and $b_1(X) = 0$
(essentially the case of Dolgachev surfaces) can be handled by arguments
similar
to the proof of the above theorems. In the remaining case $p_g(X) = 0$ and
$b_1(X) = 2$. In this paper, we shall just show by elementary methods that, in
the notation of (1.1), every orientation-preserving self-diffeomorphism
$f$ preserves $\pm K_0$ up to torsion, and similarly for the case of two
different surfaces $X$ and $X'$. In fact, one can also show that $\pm K_0$
itself is preserved.
Using the above results and the general theory of complex surfaces (not
necessarily K\"ahler), we can easily deduce that the plurigenera $P_n(X)$ of a
complex surface are smooth invariants:
\corollary{1.4} If $X$ and $X'$ are two diffeomorphic complex surfaces, then
$P_n(X) = P_n(X')$ for all $n\geq 1$.
\endproclaim
The method of proof of Theorem 1.1 also yields:
\corollary{1.5} Let $X$ be a K\"ahler surface, not necessarily minimal. If
there
exists a Riemannian metric of positive scalar curvature on $X$, then $X$ is
rational or ruled.
\endproclaim
{\it Acknowledgements\/}: It is a pleasure to thank Stefan Bauer, Ron
Fintushel, Tom Mrowka, Ron Stern, Cliff Taubes, Gang Tian, and Edward Witten
for many stimulating discussions about the new invariants. We are especially
grateful to Peter Kronheimer for showing us how to remove 2-torsion from the
statements of the results by taking differences of Spin${}^c$ structures.
\section{2. Seiberg-Witten invariants for K\"ahler metrics.}
Here we review the general theory of Seiberg-Witten invariants for K\"ahler
metrics. None of the results in this section are original, and most should
appear in \cite{10}, but we shall sketch some of the proofs for the sake of
clarity. Other expositions of this material and its consequences for K\"ahler
surfaces have appeared in
\cite{4}, \cite{24}. Let
$M$ be a a general closed oriented Riemannian 4-manifold with Riemannian metric
$g$. First we recall some of the properties of Spin${}^c$
structures on $M$. There is an exact sequence
$$\{1\} \to U(1) \to \operatorname{Spin}^c(4) \to SO(4) \to \{1\},$$
which realizes $\operatorname{Spin}^c(4)$ as a central extension of $SO(4)$. In
particular, by considering the exact sequence of cohomology sets associated
to this central extension, the set of Spin${}^c$ structures on
$M$ lifting the frame bundle (which is nonempty if and only if $w_2(M)$ is
the mod 2 reduction of an integral class) is a principal homogeneous space over
$H^1(M;\Cal U(1))$, where $\Cal U(1) = C^\infty(M)/\Bbb Z$ is the sheaf of
$C^\infty$ functions from $M$ to $U(1)$. Since
$H^1(M;C^\infty(M)/\Bbb Z) \cong H^2(M; \Bbb Z)$, given two
Spin${}^c$ structures $\xi _1, \xi _2$ on
$M$ lifting the frame bundle, their difference $\delta (\xi _1, \xi _2)$ is a
well-defined element of $H^2(M; \Bbb Z)$. In dimension four,
$\operatorname{Spin}^c(4)$ is the subgroup of $U(2) \times U(2)$ consisting
of pairs $(T_1,T_2)$ with $\det T_1 = \det T_2$. Thus there are two natural
homomorphisms $\operatorname{Spin}^c(4) \to U(2)$, and the corresponding
homomorphisms $\operatorname{Spin}^c(4) \to U(1)$ given by taking the
determinant agree. If $\xi$ is a Spin${}^c$ structure on $M$, there are two
associated $U(2)$ bundles $\Bbb S^{\pm}= \Bbb S^{\pm}(\xi)$, and $L = c_1(\Bbb
S^{\pm})$ is a complex line bundle which satisfies $c_1(L) \equiv w_2(M) \bmod
2$. Thus $L$ is characteristic, i\.e\. $c_1(L)$ has mod two reduction equal to
$w_2(M)$. We shall call $L$ the complex line bundle {\sl associated to\/}
$\xi$. Let
$(L, \xi)$ be a pair consisting of a characteristic complex line bundle
$L$ on $M$, and a
Spin${}^c$ structure $\xi$ whose associated line bundle is $L$. Of course, the
Spin${}^c$ structure determines the line bundle $L$, but we shall record both
since we shall primarily be interested in $L$ (and we shall sometimes omit the
$\xi$ from the notation). Conversely, $L$ determines $\xi$ up to 2-torsion in
$H^2(M; \Zee)$, and thus $L$ determines $\xi$ uniquely if there is no
2-torsion in $H^2(M; \Zee)$. In terms of the pairs
$(L_1,
\xi _1)$ and $(L_2, \xi _2)$, we have $2\delta (\xi _1, \xi _2) = c_1(L_1)
-c_1(
L_2)$. Thus we may define the difference $\dsize\frac{c_1(L_1) -c_1( L_2)}2=
\delta (\xi _1, \xi _2)$,
and this difference is well-defined in integral cohomology (not just modulo
2-torsion).
In case $X$ is a K\"ahler surface, or more generally a 4-manifold with an
almost complex structure, then there is a natural lifting of the reduction
of the structure group of the tangent bundle of $X$ to
$\operatorname{Spin}^c(4)$. Namely, we take the map $U(2) \to U(2) \times
U(2)$ defined by $(\rho _1, \rho _2)$, where $\rho _2 = \operatorname{Id}$
and
$$\rho _1(T) = \pmatrix 1 & 0\\ 0 & \det T \endpmatrix.$$
Thus $\Bbb S^+ = \underline \Cee \oplus K_X^{-1}$ and $\Bbb S^- = T_\Cee$,
where $K_X^{-1}$ is the inverse of the canonical line bundle of the almost
complex structure and $T_\Cee$ is the tangent bundle, viewed as a complex
2-plane bundle. In terms of the bundles of $(p,q)$-forms defined by the almost
complex structure, we may also write this as
$\Bbb S^+ = \Omega ^0_X \oplus \Omega ^{0,2}_X$ and $\Bbb S^- = \Omega
^{0,1}_X$. For this lift $L = K_X^{-1}$. If we replace $\xi$ by $\xi \otimes
\Xi$, where
$\Xi$ is a $C^\infty$ line bundle on $X$, then we replace $K_X^{-1}$ by $L=
\Xi^{\otimes 2} \otimes K_X^{-1}$. Thus, for a characteristic complex (but not
necessarily holomorphic) line bundle
$L$, a Spin${}^c$ structure for
$X$ is the same as the choice of a complex line bundle
$\Xi$ with
$\Xi
\otimes \Xi = L \otimes K_X$. In this case, given two such Spin${}^c$
structures,
the difference
$\dsize\frac{c_1(L_1) - c_1(L_2)}2$ is equal to
$c_1(\Xi _1) - c_1(\Xi _2)$. Of course, the obvious choices are $L= K_X^{-1}$
and $L = K_X$, with $\Xi$ the trivial bundle in the first case and $\Xi = K_X$
in the second. We shall refer to these two choices as the {\sl natural\/}
Spin${}^c$ structures.
Recall that the Seiberg-Witten equations associated to a {\rm Spin}${}^c$
structure $\xi$ on a Riemannian manifold $M$ are equations for a pair
$(A,\psi)$ where $A$ is a unitary connection of the determinant line
bundle $L$ of $\xi$ and $\psi$ is a section of the bundle $\Bbb S^+(\xi)$ of
plus
spinors.
The equations are
$$\align
F_A^+&=q(\psi)=\psi\otimes
\psi^*-\frac{|\psi|^2}{2}\operatorname{Id}\\
\dirac_A\psi&=0.
\endalign$$ For a generic
perturbation of these equations the moduli space of gauge equivalence classes
of
solutions forms a compact smooth manifold
${\Cal M}(L,\xi)$ which is oriented by a choice of orientation of
$H^1(M;{\Ar})$ and $H^2_+(M;{\Ar})$.
Furthermore, there is a natural two-dimensional cohomology class
$\mu\in H^2({\Cal M}(L,\xi))$ obtained by dividing the space of solutions
to the Seiberg-Witten equations by the group of based changes of
gauge, and taking the first Chern class of the $S^1$-bundle that arises.
Once we have the Seiberg-Witten moduli space, we can define the Seiberg-Witten
invariant for
$M$, which is a function
$SW _{M,g}$ which assigns an integer to each pair $(L, \xi)$ as above.
It suffices to evaluate $\mu ^{d/2}$ over the fundamental class of $\Cal
M(L, \xi)$, where $d = \dim \Cal M(L, \xi)$. By definition this integer is zero
if $d$ is odd. Strictly speaking, we also need to choose an orientation for
$H^1(X; \Ar) \oplus H^2_+(M; \Ar)$ to determine the sign of
$SW _{M,g}$, but we shall be a little careless on this point. If
$b_2^+(M) \geq 3$, then
$SW _{M,g}$ is independent of the choice of $g$,
whereas if
$b_2^+(M) = 1$, then $SW _{M,g}$ is only defined for generic $g$ and we
will describe the dependence of
$SW _{M,g}$ on $g$ more precisely later. If $SW _{M,g}(L, \xi)\neq 0$ we
shall
say that the pair $(L, \xi)$ (or $L$) is a {\sl basic class} (for $g$). This
can only happen in case the index of the corresponding linearized equations is
nonnegative. This index is
$$\frac14(L^2 -(2\chi (M) + 3\sigma (M)),$$
where $\chi (M)$ is
the Euler characteristic and $\sigma (M)$ is the signature, and we shall also
refer to this index as the {\sl index\/} of the basic class $L$. If
$M = X$ is a complex surface, then it follows from the Hirzebruch signature
formula that
$2\chi (X) + 3\sigma (X) = K_X^2$. Thus if $L$ is a basic class, then $L^2 \geq
K_X^2$. If all of the basic classes have index zero, then
$M$ is called {\sl of simple type} (for $g$). Finally we note that, even though
the function $SW$ is defined by perturbing the Seiberg-Witten equations, if the
unperturbed Seiberg-Witten equations have no solution for $(L, \xi)$, then $SW
_{M,g}(L, \xi) =0$. If on the other hand the solutions to the unperturbed
Seiberg-Witten equations are transverse in an appropriate sense, then we can
use
the moduli space for the unperturbed equations to calculate $SW _{M,g}(L,
\xi)$.
Recall that a solution $(A, \psi)$ to the SW equations is {\sl reducible\/}
if
and only if $\psi = 0$ and hence
$F_A^+ = 0$, where $F_A^+$ is the self-dual part of the curvature of a
connection on
$L$. In the case of a K\"ahler metric, $F_A^+ = 0$ implies that $A$ is a
$(1,1)$-connection (and thus $A$ defines a holomorphic structure on $L$) and
$c_1(L) \cdot \omega = 0$, where $\omega$ is the K\"ahler form. By
the Hodge index theorem, $L^2\leq 0$. Conversely, if
$L$ is a holomorphic line bundle and $c_1(L) \cdot \omega = 0$, then there
exists a $(1,1)$-connection $A$ on $L$ with $F_A^+ = 0$, giving a reducible
solution to the SW equations, and indeed in this case all solutions will be
reducible. We call a basic class $L$ {\sl reducible\/} if all solutions to the
corresponding SW equations are reducible, and {\sl irreducible\/} otherwise. Of
course, reducible solutions to the SW equations for $(L, \xi)$ do not
necessarily imply that $(L, \xi)$ is basic. For a generic K\"ahler metric, we
can assume that the K\"ahler form
$\omega$ is not orthogonal to any nontorsion class $L\in H^2(X; \Zee)$ with
$L^2
\geq K_X^2$. We shall call a K\"ahler metric whose associated K\"ahler form
$\omega$ satisfies this condition {\sl generic}. Thus for a generic K\"ahler
metric, there exist reducible basic classes $L$ only if
$c_1(L)$ is zero as an element of $H^2(X; \Ar)$. Of course, these will give
solutions of nonnegative index only if $0 = L^2 \geq K_X^2$.
In the K\"ahler case, we have the following criterion for the SW moduli
space to be nonempty and of nonnegative formal dimension \cite{34},
\cite{10}:
\proposition{2.1} Suppose that $X$ is a K\"ahler surface with K\"ahler form
$\omega$. Then the pairs $(L, \xi)$ of nonnegative index admitting irreducible
solutions to the SW equations are in one-to-one correspondence with
holomorphic characteristic line bundles $L$, together with a choice of a
holomorphic square root $(K_X\otimes L)^{1/2}$ for the line bundle $K_X\otimes
L$, satisfying:
\roster
\item"{(i)}" $L^2 \geq K_X^2$;
\item"{(ii)}" Either $H^0(X; (K_X\otimes L)^{1/2}) \neq 0$ and $\omega \cdot L
< 0$ or $H^0(X; (K_X\otimes L^{-1})^{1/2}) \neq 0$ and $\omega \cdot L
> 0$. \qed
\endroster
\endproclaim
Here the choice of a square root $(K_X\otimes L)^{1/2}$ for $K_X\otimes L$
naturally gives the square root $(K_X\otimes L^{-1})^{1/2} = (K_X\otimes
L)^{1/2} \otimes L^{-1}$ for $K_X\otimes L^{-1}$.
The idea behind the proof of (2.1) is that since $X$ has a complex structure,
$\Bbb S^+(\xi)$ splits as a sum of line bundles $(K_X\otimes L)^{1/2}$ and
$\Omega^{0,2}_X(( K_X\otimes L)^{1/2})$. Thus,
the spinor field $\psi$ decomposes
into components $(\alpha,\beta)$. The curvature part of the
Seiberg-Witten equations says that
$$\align
F^{0,2}
&=\bar\alpha\beta\\
(F_A^+)^{1,1} &=\frac{i}{2}(|\alpha|^2-|\beta|^2)\omega,
\endalign$$
where $\omega$ is the K\"ahler form. The Dirac equation for a K\"ahler surface
becomes
$$\bar\partial_A\alpha+\bar\partial^*_A\beta=0.$$
Applying $\bar\partial_A$ to this equation we get
$$\bar\partial_A\bar\partial_A\alpha
+\bar\partial_A\bar\partial^*_A\beta =0.$$
Equivalently,
$$F^{0,2}_A\cdot
\alpha+\bar\partial_A\bar\partial^*_A\beta =0.$$
Since $F^{0,2}_A=\bar\alpha\beta$, this equation
becomes
$$|\alpha|^2\beta+\bar\partial_A\bar\partial^*_A\beta =0.$$
Taking the $L^2$-inner product with $\beta$ yields
$$\int _X|\alpha|^2|\beta|^2+\|\bar\partial^*_A\beta\|^2_{L^2}=0.$$
It follows that $|\alpha|^2|\beta|^2$ and $\bar\partial^*_A\beta$ are zero.
Thus $F^{0,2}_A=\bar\alpha\beta =0$. This means that $A$ is a holomorphic
connection and so defines a holomorphic structure on $L$. Moreover
$\bar\partial^*_A\beta =0$ and so $\bar\partial_A\alpha =0$. Hence $\alpha$ is
a holomorphic section of $(K_X\otimes L)^{1/2}$ and $\bar\beta$ is a
holomorphic
section of
$(K_X\otimes L^{-1})^{1/2}$.
Since
$\alpha$ and $\bar \beta$ are holomorphic, they do not vanish
on any open subset unless they vanish identically. So either $\alpha = 0$ or
$\beta=0$. Furthermore,
$$\omega \cdot L = \int _X\omega \wedge \frac{i}{2\pi}F_A^+ = -\frac{1}{4\pi}
\int _X (|\alpha|^2 - |\beta|^2)\omega \wedge \omega,$$
and so
$\alpha$ is not zero if and only if $\omega \cdot L <0$ and
$\beta$ is not zero if and only if $\omega \cdot L >0$.
If $\alpha\neq 0$
then $\alpha$ is a nonzero holomorphic section of $(K_X\otimes L)^{1/2}$. If
$\beta\neq 0$, then $\bar\beta$ is a nonzero holomorphic section of
$(K_X\otimes L^{-1})^{1/2}$.
We have seen that the conditions listed in Proposition 2.1 are
necessary for a solution. Let us show that they are sufficient as
well. We shall just consider the case where $\omega \cdot L$ is
negative. The holomorphic structure on $L$ uniquely determines a connection
$A$, once we have chosen a hermitian metric on $L$, by choosing the unique
connection compatible with the holomorphic structure and the metric. Fix an
arbitrary hermitian metric on $L$. Given
$L$ and a nontrivial holomorphic section
$\alpha$ of
$(K_X\otimes L)^{1/2}$, we wish to change the metric on $L$ until the
curvature part of the Seiberg-Witten equation is satisfied.
If we think of varying the hermitian metric by $\exp\lambda$
for some real valued function $\lambda$, then the equation
we need to solve for $\lambda$ is
$$F_A^++(\bar\partial\partial
\lambda)^+=\frac{i}2 e^\lambda |\alpha|^2\omega.$$ For a $(1,1)$-form $\eta$,
$\eta ^+ = \Lambda \eta \cdot \omega$, where $\Lambda$ is contraction with
$\omega$. We take the pointwise contraction with the K\"ahler form
$\omega$ and obtain
$$\Delta \lambda -\frac{|\alpha|}{4}^2e^\lambda -\frac12*(iF_A^+\wedge
\omega)=0.$$ Here $\Delta$ is the negative definite Laplacian on functions (in
Euclidean space it would be $\sum _i\partial ^2/\partial x_i^2$). According to
results of Kazdan-Warner
\cite{18} (first applied in gauge theory to the vortex equation by Bradlow
\cite{3}), there is a unique solution
$\lambda$ to this equation provided that $\int _XiF_A^+\wedge \omega <0$, which
is just the condition that $\omega \cdot L$ is
negative. Thus, we have seen that for each non-trivial
holomorphic section of
$(K_X\otimes L)^{1/2}$ we can obtain a solution to the Seiberg-Witten
equations with the holomorphic section as the spinor field. This completes the
sketch of the proof of the proposition.
A gauge equivalence between two solutions $(A,\psi)$ and $(A',\psi')$
will be a holomorphic isomorphism between the holomorphic structures
$L$ and $L'$ determined by the two connections. It will also carry the
section $\psi$ to $\psi'$. Since the only holomorphic automorphisms of
a holomorphic bundle are multiplication by nonzero constant functions, this
implies that under the holomorphic identification $\psi$ and $\psi'$
define the same point of $\Pee H^0((K_X\otimes L)^{1/2})$. Conversely, it
is easy to check that two sections which agree modulo $\Cee ^*$ define gauge
equivalent solutions. Thus we may identify the moduli space to the
unperturbed equations with
$$\bigcup _L\Pee H^0((K_X\otimes L)^{1/2}),$$
where we think of $L$ as ranging over all holomorphic structures on a fixed
$C^\infty$ line bundle (the set of all such structures is isomorphic to the
complex torus
$\operatorname{Pic}^0X$). The moduli space may thus be identified with an
appropriate component of the Hilbert scheme of curves on $X$. Of course, there
will be another moduli space corresponding to
$L^{-1}$ as well.
We now divide the study of the
basic classes into two cases: the case where $X$ is minimal and the case where
$X$ is not minimal.
\medskip
{\bf The case of a minimal surface.} For a K\"ahler surface $X$ which is not
rational or ruled,
$X$ is minimal if and only if
$K_X$ is {\sl nef\/}: in other words, for every holomorphic curve $C$ on $X$,
$K_X\cdot C
\geq 0$. If $K_X$ is nef, then $K_X^2 \geq 0$. The case where $K_X$ is nef and
$K_X^2 >0$ is the case where $X$ is of general type. In this case,
since
$K_X^2 >0$, there are no reducible basic classes for any K\"ahler metric.
\proposition{2.2} With notation as above, suppose that $X$ is a minimal
surface of general type, i\.e\. suppose that $K_X$ is nef and that $K_X^2 >0$,
and that
$\omega$ is a generic K\"ahler metric. Then the only pairs
$(L, \xi)$ satisfying the conditions
\rom{(i)} and \rom{(ii)} are $L = K_X^{\pm 1}$, with the natural
\rom{Spin${}^c$} structures, i\.e\. the ones corresponding to the square root
$K_X$ of
$K_X\otimes K_X$ and the square root $0$ of $K_X \otimes K_X^{-1}$.
\endproclaim
\proof For the proof we use additive notation for holomorphic line bundles
(which
we could identify with divisor classes on $X$). After replacing
$L$ by $-L$, we may assume that $\omega \cdot L < 0$ and that $K_X+L$ is
effective. We have $(K_X+L) \cdot \omega \geq 0$, $L\cdot \omega < 0$, so
there is an $a\geq 1$ such that $(K_X+aL)\cdot \omega = 0$. By the Hodge index
theorem $(K_X+aL)^2 \leq 0$, with equality only if $K_X + aL$ is numerically
trivial. Thus
$$K_X^2 + 2a(K_X\cdot L) + a^2L^2 \leq 0.$$
On the other hand $K_X$ is nef, so that $(K_X+L)\cdot K_X \geq 0$.
Putting together
$$\align
2aK_X^2 + 2aK_X\cdot L &\geq 0;\\
K_X^2 + 2a(K_X\cdot L) + a^2L^2 &\leq 0,
\endalign$$
we obtain
$$(1-2a)K_X^2 + a^2L^2 \leq 0,$$
or in other words
$$L^2 \leq \frac{2a -1}{a^2}K_X^2.$$
But
$$\frac{2a -1}{a^2} = 1 - \fracwithdelims(){a-1}{a}^2 = 1 - \left(1 -
\frac1a\right)^2,$$
which is decreasing for $a\geq 1$. Thus
$$L^2 \leq \frac{2a -1}{a^2}K_X^2\leq K_X^2.$$
Since $L^2\geq K_X^2$, we have $L^2 = K_X^2$.
For equality to hold we must have $a=1$ and $K_X+L$ must be numerically
trivial.
In this case, $K_X+L$ has a section since $\dsize\frac{K_X+L}2$ has a section.
Thus
$K_X+L$ is the trivial divisor, so $L= -K_X$. Moreover $\dsize\frac{K_X+L}2$
is numerically trivial and has a section as well, so that it is trivial. Thus
the Spin${}^c$ structure corresponds to taking the trivial square root of
$K_X+L=0$.
\endproof
Essentially the same argument shows:
\proposition{2.3} With notation as above, suppose that $K_X$ is nef and
that $K_X^2 =0$. If $L$ is a line bundle satisfying the conditions
\rom{(i)} and \rom{(ii)}, then there exists a rational number $r\leq 1$ such
that $L$ is numerically equivalent to $\pm rK_X$. Moreover, in case $r= \pm
1$, then in fact $L =\pm K_X$, and the {\rm Spin}${}^c$ structures are again
the natural ones.
\qed
\endproclaim
Note that in all cases we have $L^2 = K_X^2$, in other words $X$ is of simple
type for $g$ if $X$ is minimal and $K_X$ is nef. Of course, so far we have not
actually shown that there are any basic classes. But in case
$L = \pm K_X$, the value of $SW$ is $\pm 1$. To determine the exact sign, we
need to make a choice of orientation for the moduli space.
The orientation convention we shall follow is this: To
orient the relevant determinant line bundle for a general $4$-manifold $M$, we
must choose an orientation for the vector space $H^1(M; \Ar) \oplus H^2_+(M;
\Ar)
\oplus H^0(M;
\Ar)$, by choosing orientations on $H^1(M; \Ar)$ and $H^2_+(M; \Ar)$ and using
the standard orientation on $H^0(M; \Ar)$. For a K\"ahler surface $X$,
$$ H^2_+(X; \Ar) \cong
\Ar \cdot \omega \oplus (H^{2,0}(X) \oplus H^{0,2}(X))_\Ar$$ and $H^1(X;
\Ar)\cong (H^{1,0}(X) \oplus H^{0,1}(X))_\Ar$. We choose the orientation
given by taking the standard orientation on $\Ar\cdot \omega$ and
using the isomorphism
$(H^{i,0}(X) \oplus H^{0,i}(X))_\Ar \to H^{0,i}(X)$ to transfer the
usual complex orientation on $ H^{0,i}(X)$ to $(H^{i,0}(X) \oplus
H^{0,i}(X))_\Ar$. We then
have the following result, which follows easily by considering the
linearization
of the SW equations
\cite{10}:
\proposition{2.4} For an arbitrary complex surface $X$, if $g$ is a K\"ahler
metric on
$X$ with K\"ahler form
$\omega$ and $\omega \cdot K_X >0$, then the value of
$SW _{X,g}$ on $-K_X$ for the natural {\rm Spin}${}^c$ structure is $1$ and
$SW_{X,g}(K_X)= (-1)^{q+ p_g}$ for the natural {\rm Spin}${}^c$ structure.
\endproclaim
\demo{Sketch of proof}
Let us consider the elliptic complex associated to the unique solution
for $L=-K_X$. The kernel of the Dirac operator is isomorphic to
$H^0(X;{\Cee})\oplus H^{0,2}(X)$. The cokernel of the Dirac operator
is
$H^{0,1}(X)$. Of course, $H^2_+(X;i{\Ar})$ is, as an oriented vector space,
isomorphic to $(i\Ar)\cdot \omega\oplus H^{0,2}(X)$.
Clifford multiplication by the solution $(\alpha,0)\in H^0\oplus
H^{0,2}$ induces an orientation-preserving isomorphism $H^1(X;i{\Ar})\to
H^{0,1}(X)$. If $Dq$ is the differential of the quadratic mapping $q$,
then $-Dq$ induces an orientation-preserving
mapping
$$H^{0,2}(X)\to H^{0,2}(X),$$ namely multiplication by
$-\bar\alpha$.
The map $Dq$ also induces a map
$$H^0(X;{\Cee})\to (i\Ar)\cdot \omega$$
which sends $\eta$ to $-i\operatorname{Re}\langle \alpha,\eta\rangle$.
Of course, the action of the stabilizer $S^1$ on the spin fields is by
the opposite of the complex orientation. Hence, the solution to the
equations modulo the action of the group of changes of gauge
is a single point. The equations are transverse at this
point and the orientation is plus one.
A similar computation shows that
$$SW_{X,g}(K_X) = (-1)^{q(X)+p_g(X)}.$$
In fact, for any smooth 4-manifold
$M$ and any Spin${}^c$ structure $\xi$ there is a naturally defined opposite
Spin${}^c$ structure
$-\xi$ and we have
$$SW_{M,g}(\xi)=(-1)^{(1-b_1(M)+b_2^+(M))/2}SW_{M,g}(-\xi).$$
\enddemo
To end our discussion of minimal K\"ahler surfaces, we consider the example of
elliptic surfaces. For simplicity, and because it is the most interesting case,
we shall just consider the case of simply connected elliptic surfaces, so that
linear, numerical, and homological equivalence are the same. In this case the
basic classes have a certain multiplicity which need not be one, but which we
shall not compute here. Let
$X$ be a simply connected elliptic surface, with $p_g(X) = p_g$. Then $X$ has
at
most two multiple fibers
$F_1$ and $F_2$, of multiplicities $m_1$ and $m_2$, say. From the canonical
bundle formula
$$K_X = (p_g-1)f + (m_1-1)F_1 + (m_2-1)F_2,$$
where $f$ is the class of a general
fiber:
$f= m_1F_1 = m_2F_2$. Let $D$ be a divisor which is a rational multiple of
$K_X$ and thus of the fiber $f$, say $D = rf$, and define $\deg D = r$.
For an arbitrary K\"ahler metric $\omega$, normalized so that $\omega \cdot
f=1$, we have $\deg D =
\omega \cdot D$. The basic classes correspond to line bundles
$L$ such that either $K_X + L = 2D$, where $D$ is effective and $\omega \cdot L
\leq 0$, i\.e\. $\dsize 0 \leq \deg D \leq \frac{\deg K_X}2$,
or $K_X - L = 2D$, where $D$ is effective and $\omega \cdot L
\geq 0$, i\.e\. again we have $\dsize 0 \leq \deg D \leq \frac{\deg K_X}2$.
The effective divisor $D$ on $X$ can be written as $af + bF_1 + cF_2$, where
$a\geq 0$, $0\leq b \leq m_1-1$, and $0\leq c \leq m_2 -1$. If $p_g >0$, $\deg
D \leq \deg K_X$ if and only if $a \leq p_g-1$. In this case it is clear that
$D$ is effective if and only if $K_X - D$ is effective. Thus in case $K_X + L =
2D$, where $D$ is effective and $\dsize 0 \leq \deg D \leq \frac{\deg K_X}2$,
we have
$$L = 2D - K_X = K_X - 2(K_X-D) = K_X - 2D',$$
where $D'$ is effective and $\dsize \frac{\deg K_X}2 \leq \deg D' \leq \deg
K_X$. Likewise if $K_X - L = 2D$, where $D$ is effective and $\omega \cdot L
\geq 0$, then $L= K_X - 2D$, where $D$ is effective and $\dsize 0 \leq \deg D
\leq \frac{\deg K_X}2$. Thus we see that in all cases the basic classes are
exactly the classes of the form $K_X - 2D$, where $D$ is effective and $0 \leq
\deg D \leq \deg K_X$. In other words, the basic classes are the classes
$(p_g-1-2a)f + (m_1 - 2b-1)F_1 + (m_2 -2c-1)F_2$ for $0\leq a \leq p_g-1$,
$0\leq b \leq m_1-1$, $0 \leq c\leq m_2-1$. These are exactly the
Kronheimer-Mrowka basic classes \cite{21}, \cite{11}, and one can show that the
appropriate multiplicity, up to sign, to attach to the class $K_X-2D$ with
$D =af + bF_1 + cF_2$, with $a \geq 0$, $0\leq b \leq m_1-1$, and $0\leq c
\leq
m_2 -1$, is
$\dsize \binom{p_g-1}{a}=\binom{h_++ h_-}{h_+}$, where $h_+ = h^0(D)-1$ and
$h_- =h^0(K_X-D)-1$.
A similar analysis holds for the case of an elliptic surface with $p_g =0$,
where
we must simply analyze the conditions on $L$ directly. For example, we obtain
the SW classes $L$ with nonnegative fiber degree (i\.e\. the $L$ such that $L =
rK_X$ with $r\geq 0$ in rational cohomology) by considering the effective
divisors $D$ such that $L=K_X - 2D$ has nonnegative fiber degree. In this case
$\dsize \frac{K_X-L}2 = D$ and the correct multiplicity, up to sign, is one.
\medskip
{\bf The case of a nonminimal surface.} It is enough to
consider the case where
$\tilde X
\to X$ is a single blowup.
\proposition{2.5} Let $X$ be a K\"ahler surface which is the blowup of a
surface for which the canonical bundle is nef and let
$g$ be a K\"ahler metric on $X$ with K\"ahler form $\omega$ such that $\omega$
is not orthogonal to any nontorsion class $L\in H^2(X; \Zee)$ with $L^2 =
K_X^2$.
Let
$\tilde X$ be the blowup of $X$ at a point, with $E$ the exceptional divisor,
and
consider a K\"ahler metric $\tilde g$ on
$\tilde X$ corresponding to the K\"ahler form $\tilde \omega = N\omega -E$, $N
\gg 0$. Then:
\roster
\item"{(i)}" Every basic class on $\tilde X$ for $\tilde g$ is irreducible.
\item"{(ii)}" If $(\tilde L, \tilde \xi)$ is a basic class on
$\tilde X$ for $\tilde g$, then either $\tilde L = L\pm E$, where
$L$ is an irreducible basic class on $X$ for $g$, or $\tilde L =L\pm E$, where
$L$ is a reducible basic class on $X$ and so the image of $L$ is zero in
$H^2(X; \Ar)$. If $\tilde L = L+ E$ is a basic class for $\tilde X$, then the
corresponding {\rm Spin}${}^c$ structure $\tilde \xi$, or in other words the
square root $\tilde \Xi$ of $\tilde L+K_{\tilde X} = L+K_X+ 2E$, is $\Xi + E$,
where
$\Xi$ is the square root of $L+K_X$ corresponding to the {\rm Spin}${}^c$
structure $\xi$ on $X$, and likewise if $\tilde L = L-E$, then $\tilde \Xi =
\Xi$. Moreover, the classes
$L\pm E$ where
$L$ is an irreducible basic class on $X$ for $g$ all give basic classes on
$\tilde X$ for $\tilde g$, and in this case
$$SW_{\tilde X, \tilde g}(L\pm E, \tilde \xi) = \pm SW_{X,g}(L, \xi).$$
\item"{(iii)}" If there is a basic class on $\tilde X$ of the form $L\pm E$,
where $L$ is a reducible basic class on $X$, then $X$ is minimal. Moreover, if
$K_X$ is torsion, then $L = \pm K_X$. In this case $\pm K_X \pm E$ are
basic classes on $\tilde X$ and $SW_{\tilde X, \tilde g}(\pm K_X \pm E) = \pm
1$ for the natural {\rm Spin}${}^c$ structures.
\item"{(iv)}" Let $\tilde X$ be a K\"ahler surface which is not rational or
ruled. Let $\tilde g$ be a K\"ahler metric on $\tilde X$ whose K\"ahler form
$\tilde \omega = N\omega - \sum _iE_i$, where the $E_i$ are the exceptional
curves on $\tilde X$, $\omega$ is the K\"ahler form of a generic K\"ahler
metric on the minimal model of $\tilde X$, and $N\gg 0$. For every basic class
$L$ on
$\tilde X$ for $\tilde g$, we have
$L^2=K_{\tilde X}^2$, so that
$\tilde X$ is of simple type for
$\tilde g$.
\endroster
\endproclaim
\proof First we prove (i). Clearly, for all $N \gg 0$, a K\"ahler metric with
K\"ahler form
$N\omega -E$ is generic. Thus the only possible reducible basic classes are
torsion. But a basic class is characteristic. If there were a torsion
characteristic element in
$H^2(\tilde X; \Zee)$, then every element of $H^2(\tilde X; \Zee)$ would have
even square. This contradicts the fact that $E^2 =-1$.
Next we consider (ii) and (iii). Suppose that $\tilde L$ is a basic class on
$\tilde X$. Possibly after replacing $\tilde L$ by
$-\tilde L$, we can assume that $\tilde L^2
\geq K_{\tilde X}^2 = K_X^2 -1$, $\dsize\frac{\tilde L + K_{\tilde X}}{2}$ is
effective, and $\tilde \omega \cdot \tilde L <0$. We have $K_{\tilde X} =
K_X+E$, and $\tilde L = L + aE$ for some characteristic $L\in H^2(X; \Zee)$ and
odd integer $a$. As $\dsize\frac{\tilde L + K_{\tilde X}}{2}$ is
effective, $\dsize\frac{ L + K_X}{2}$ is
effective as well. Since $\tilde
L^2 = L^2 - a^2
\geq K_X^2 -1$, and $a$ is odd, we have $L^2 \geq K_X^2$. As $\tilde \omega
\cdot
\tilde L <0$, we have
$N(\omega \cdot L) + a <0$ for all $N\gg 0$. Thus $\omega \cdot L \leq 0$.
First consider the case where
$\omega \cdot L <0$. Then the line bundle $L$ on $X$ satisfies (i) and (ii) of
(2.1). Conversely, starting with a $L$ on $X$ satisfying (i) and (ii) of
(2.1), the class
$\tilde L = L\pm E$ will satisfy (i) of (2.1). If say $L\cdot \omega <0$, then
$\tilde L
\cdot \tilde \omega <0$ provided that $N\gg 0$. Finally $\dsize\frac{K_{\tilde
X}
+\tilde L}2 = \frac{K_X+L}2$ or $\dsize \frac{K_X+L}2 +E$, and we have natural
isomorphisms
$$H^0(\tilde X;\frac{K_X+L}2 +E) \cong H^0(\tilde X;\frac{K_X+L}2) \cong
H^0(X;\frac{K_X+L}2),$$
where we have used the notation $H^0(X; D)$ to denote the group of sections of
$\scrO_X(D)$. Thus (ii) of (2.1) is satisfied as well, and we see that the
Spin${}^c$ structures are as claimed. We omit the argument that $SW_{\tilde X,
\tilde g}(L\pm E) =
\pm SW_{X,g}(L)$, where
$\tilde g$ and
$g$ are appropriate generic K\"ahler metrics on $\tilde X$ and
$X$ respectively. In case $L$ is irreducible, this result is established in the
general case via a general blowup formula in \cite{10}. Note that, in case $L=
K_X$, the main case of interest, $L+E = K_X+E = K_{\tilde X}$, and this case is
covered by (2.4) with $X$ replaced by $\tilde X$. The case $K_X-E$ then follows
by the naturality of the function $SW$, since $K_X-E= R^*(K_X+E)$, where $R\:
\tilde X \to \tilde X$ is the diffeomorphism corresponding to reflection in the
class $E \in H^2(\tilde X; \Zee)$.
Now consider the case where $\omega \cdot L=0$. By the hypothesis that the
metric is generic, $L$ is zero in rational cohomology. From $\tilde
L^2 = -a^2
\geq K_{\tilde X}^2 = K_X^2 -1$, we see that $K_X^2 \leq 1-a^2 \leq 0$. Since
$L$ is characteristic,
$X$ must in fact be minimal. Hence $K_X^2
\geq 0$ and so $K_X^2 =0$ and $a=\pm 1$. In this case $\tilde L^2 =
K_{\tilde X}^2$, so that all of the $\tilde L$ constructed in this way are of
index zero. We note that (ii) of (2.1) on $\tilde X$ is satisfied if and only
if $\dsize H^0(X;\frac{K_X+L}2) \neq 0$. Thus $L+K_X$ is effective. If $K_X$ is
torsion, then $L+K_X$ is also zero in rational cohomology. Thus it is the
trivial divisor, and so $L= -K_X$, and the Spin${}^c$ structure is the trivial
square root of $K_X+L$. It follows from (2.4) that $SW_{\tilde X, \tilde g}
( K_X + E) = \pm 1$, and arguments as in the irreducible case handle
show that $SW_{\tilde X,
\tilde g}(\pm K_X \pm E) =\pm 1$ also.
Lastly we prove (iv). If $X$ is minimal and $L$ is
irreducible, then (2.2) and (2.3) imply that
$L^2 = K_X^2$. If $L$ is reducible, then $X$ is minimal and $K_X^2 = 0$.
Since $L$ is torsion $L^2 = K_X^2 =0$ in this case as well. By induction on the
number of blowups, and using the above discussion for reducible
$L$, we may assume that
$\tilde X$ is the blowup of a surface $X$ at one point, where
$L^2=K_X^2$ for all basic classes
$L$ on $X$. We will show that the same is true for $\tilde X$. It suffices to
show that basic classes of the form $L + aE$, where $L$ is an irreducible class
on $X$ and $a$ is an odd integer, have square equal to $K_X^2 -1$. Since
$L^2 = K_X^2$ by induction on the number of blowups, $a^2 \leq 1$. Thus $a =
\pm 1$ and $\tilde L^2= K_{\tilde X}^2$.
\endproof
We note that using
blowups, we can calculate the invariant in case
the moduli space is singular because the
solutions are all reducible. For example, for a K\"ahler surface $X$ with a
generic metric $g$, suppose $X$ is a minimal surface and
$K_X$ is torsion. In this case the basic classes are exactly $\pm K_X$ with
the natural Spin${}^c$ structure, and the value of $SW_{g,X}$ on these classes
is $\pm 1$. This is analogous to the use of blowups in Donaldson theory to
define ``unstable" invariants, as in Chapter III, Section 8 of \cite{15}.
\section{3. The case where $\boldkey p_{\boldkey g}$ is nonzero.}
In this section we will describe the proofs of the results corresponding
to Theorems 1.1 and 1.2 in case
$p_g(X) \neq 0$. In this case $SW _{X,g}$ does not depend on the choice of
the metric, and the set of basic classes for $g$, which is independent of the
choice of $g$, is a diffeomorphism invariant of $X$. Let
$X$ be a minimal surface of general type, and let $\tilde X$ be a blowup of
$X$.
We may assume that $\tilde X$ is a blowup of $X$ at distinct points. If
$E_1, \dots, E_\ell$ are the exceptional classes of $\tilde X$, and $K_0$ is
the
pullback to $\tilde X$ of $K_X$, then the basic classes are $\pm K_0 + \sum
_{i=1}^\ell \pm E_i$, with the natural Spin${}^c$ structures. Consider the
subset of all expressions of the form
$\dsize\frac{\tilde L _1 - \tilde L _2}2$, where $\tilde L _1$ and $\tilde L
_2$
are distinct basic classes, where we have used the Spin${}^c$ structures to
define the square roots as integral cohomology classes. Here, if $\tilde L _1 =
\pm K_0 + \sum _{a\in A}E_a + \sum _{a\notin A}(-E_a)$, then the square root
$\tilde \Xi _1$ of $\tilde L _1\otimes K_{\tilde X}$ corresponding to the
choice
of Spin${}^c$ structure is
$$\tilde \Xi _1 = \cases K_0 + \sum _{a\in A}E_a, &\text{if $\tilde L_1 = K_0
+ \sum _{a\in A}E_a + \sum _{a\notin A}(-E_a)$,}\\
\sum _{a\in A}E_a, &\text{if $\tilde L_1 = -K_0
+ \sum _{a\in A}E_a + \sum _{a\notin A}(-E_a)$.}
\endcases$$
Thus the set of difference classes consists exactly of the elements
$\pm K_0$, $\pm K_0 + \sum _{i\in A} \pm E_i$, where $A$ is a proper subset of
$\{1, \dots, \ell\}$, or $\sum _{i\in A}\pm E_i$, where $A$ is here an
arbitrary subset of $\{1, \dots, \ell\}$. First we recover the classes $\pm
K_0$
as the two elements of maximal square in this collection. The
$E_i$ are then the elements of the collection orthogonal to $K_0$ of square
$-1$.
Thus $\pm K_0$ is preserved by every orientation-preserving
self-diffeomorphism of $\tilde X$, and evry such diffeomorphism induces a
permutation of the set $\{\pm E_1, \dots, \pm E_\ell\}$.
Similar results hold for orientation-preserving diffeomorphisms between two
surfaces. This establishes Theorem 1.1 in this case.
Note that, if we had only kept track of the line bundles $L$ in the basic
classes and not the Spin${}^c$ structures, we would only be able to prove
Theorem
1.1 modulo 2-torsion in case
$\tilde X$ is not minimal. This is because, if $2T = 0$ in $H^2(\tilde X;
\Zee)$, then the sets $\pm K_0 \pm E$ and $\pm (K_0 + T) \pm (E+T)$ are equal.
Now suppose that $N$ is a closed negative definite 4-manifold
such that $\tilde X$ is diffeomorphic to $M\#N$ for some $M$. As we have
seen, $H_1(N; \Zee) = 0$. Let
$\{n_1,
\dots, n_k\}$ be a basis for $H^2(N; \Zee)$ such that $n_i^2 = -1$ for all $i$
and $n_i\cdot n_j = 0$ if $i\neq j$. The blowup formula for basic classes
\cite{10} implies that $M$ is of simple type and that the basic classes for
$X$ are exactly of the form
$K +\sum _{i=1}^k\pm n_i$, where $K$ is a basic class for $M$. There is also
a formula for the corresponding Spin${}^c$ structures which is analogous to
(2.5)(ii). Thus
$n_i$ is of the form
$\dsize\frac{\tilde L_1 -
\tilde L_2}2$ for two distinct basic classes $\tilde L_1$ and $\tilde L_2$, and
$n_i\in \{\pm K_0, \pm K_0 + \sum _{i\in A'} \pm E_i, \sum _{i\in A}\pm E_i\}$,
where $A'$ denotes a proper subset of
$\{1, \dots, \ell\}$, and $A$ denotes an
arbitrary subset of $\{1, \dots, \ell\}$. Since $n_i^2 =-1$, the only
possibility
is $n_i = \pm E_j$ or $n_i = \pm K_0 + \sum _{i\in A}\pm E_i$,
where $\#(A) = K_0 ^2 +1$. On the other hand, the isometry given by reflection
in
$n_i$ fixes the set of basic classes $(L, \xi)$ and thus the set of differences
of basic classes. Thus the reflection must send $K_0$ to $\pm K_0$. We may
assume that $n_i = K_0 + \sum _{i\in A}\pm E_i$. Then reflection in $n_i$
applied to $K_0$ gives
$$K_0 + 2(K_0)^2(K_0 + \sum _{i\in A}\pm E_i).$$
Since $K_0^2 >0$, $K_0 + 2(K_0)^2(K_0 + \sum _{i\in A}\pm E_i)= (1+
2(K_0)^2)K_0 + 2(K_0)^2(\sum _{i\in A}\pm E_i)$ is never a difference of basic
class. This is a contradiction. Thus we must have
$n_i = \pm E_j$ for some $j$, proving Theorem 1.2 in this case.
Now let us extend the argument to handle the case $X$ is minimal and
$p_g(X) \neq 0$, but where
$K_X^2 = 0$. We again let $\tilde X$ be a blowup of $X$ with exceptional
classes
$E_i$ and denote by $K_0$ the image of $K_X$ in $H^2(\tilde X; \Zee)$. The
above
argument tells us how to recover the basic classes for
$X$ from the basic classes for $\tilde X$: they appear as the
differences $\dsize \frac{\tilde L_1- \tilde L_2}2$ of square zero, where the
$\tilde L_i$ range over the basic classes of $\tilde X$. Moreover
the classes
$\pm K_0$ are characterized among such classes by noting that, in rational
cohomology, all other classes $K$ can be written as $rK_0$ with $|r|< 1$, at
least if $K_0$ is not torsion, whereas if $K_0$ is torsion then the only basic
classes are $\pm K_0$. Thus we see that
$\pm K_0$ is preserved by every orientation preserving
self-diffeomorphism of
$\tilde X$, and similarly for diffeomorphisms between two surfaces.
We must now recover the classes of the exceptional curves, which again appear
as differences of basic classes, and are of square $-1$ orthogonal to $K_0$.
However in this case we have additional difference classes
$\dsize\frac{K_1 - K_2}2 \pm E_i$ of square
$-1$, which are also orthogonal to $K_0$. Moreover every difference class of
square $-1$ is of the form $\dsize\frac{K_1 - K_2}2 \pm E_i$, where $K_i \in
H^2(X; \Zee)$. Suppose that some cohomology class $\alpha$ is represented by
an
embedded
$2$-sphere, or more generally lies in
$H^2(N; \Zee)$, where $N$ is a negative definite 4-manifold such that $\tilde
X$ is diffeomorphic to $M\#N$ for some $M$. The blowup formula says that the
basic classes for $\tilde X$ are exactly of the form $\pm \alpha \pm L$ for
certain classes $L$ orthogonal to $\alpha$. In particular $\alpha$ is again an
difference class, and it has square $-1$. Thus $\alpha =
\dsize\frac{K_1 - K_2}2 \pm E_i$ for some $K_1, K_2 \in H^2(X; \Zee)$. Set $T =
\dsize\frac{K_1 - K_2}2$. After renumbering and sign
change we may assume that $\alpha = T+E_1$. Let
$K$ be an arbitrary basic class for $X$. Since $K+E_1 + \sum _{i>1}E_i$ is a
basic class for $\tilde X$, either $K+E_1 + \sum _{i>1}E_i = T+E_1+L$ or $K+E_1
+ \sum _{i>1}E_i = -(T+E_1)+L$ for some class $L$. We claim that we cannot have
$K+E_1 +
\sum _{i>1}E_i = -(T+E_1)+L$, for otherwise we would have
$$L = K+T+2E_1 + \sum _{i>1}E_i$$
and in this case
$$T+E_1+L = K+2T+3E_1 + \sum _{i>1}E_i$$
would be a basic class for $\tilde X$, which is clearly impossible.
Thus
$L= K-T + \sum _{i>1}E_i$. Since $ -(T+E_1)+L$ is also a basic class, we see
that
$$K-2T -E_1 + \sum _{i>1}E_i$$
is a basic class for $\tilde X$ whenever $K$ is a basic class for $X$. Now the
basic classes for $\tilde X$ are of the form $K' + \sum _i\pm E_i$, for $K'$ a
basic class on $X$. Since $T\in H^2(X; \Zee)$, the only such class
that can equal
$K-2T -E_1 +
\sum _{i>1}E_i$ is $K' -E_1 + \sum _{i>1}E_i$ for some basic class
$K'$ on
$X$. It follows that, if
$K$ is a basic class for
$X$, then $K-2T$ is also a basic class for $X$. Since there are
only finitely many basic classes, $T$ is torsion. Now applying the above to
$K=K_0$, we see that $K_0 -2T$ is a basic class which equals $K_0$ in rational
homology. By the last sentence in (2.3), it follows that
$2T = 0$. Finally, keeping track of the Spin${}^c$ structures in the blowup
formula for taking connected sum with a negative definite 4-manifold, one
checks that the difference class
$$\frac{(L+\alpha)-(-L+ \alpha)}2 = L = K_0-T + \sum _{i>1}E_i.$$
Since $T$ is torsion, it follows from (2.3) that the only way that such a class
can be of the form
$$\frac{K_1-K_2}2 + \sum _{i\in A} \pm E_i,$$
where $K_1$ and $K_2$ are basic classes on $X$, is if $K_1=K_0$ and
$K_2=-K_0$. In this case the difference $\dsize\frac{K_0-(-K_0)}2$ is equal to
$K_0$, and so $T=0$.
In general, using somewhat different methods, one can show the following (cf\.
\cite{15}, Chapter VI, Theorem 5.3 for the corresponding result in Donaldson
theory):
\proposition{3.1} Let $M$ be a closed oriented $4$-manifold with $b_2^+(M)
\geq 3$ and such that the set of basic classes for $M$ is nonempty. Suppose
that
$M$ is orientation-preserving diffeomorphic to $M_1
\#N_1$ and also to $M_2 \#N_2$, where the $N_i$ are negative
definite $4$-manifolds with $H_1(N_1) = H_1(N_2) = 0$. Let $n_1, \dots, n_r$ be
the exceptional classes for
$N_1$ and $n_1', \dots, n_s'$ the exceptional classes for $N_2$. Then, for
every
$i$, $1\leq i\leq s$, either there exists a $j$, $1\leq j \leq r$, such
that $n_i' = \pm n_j$ mod torsion, or $n_i'$ is orthogonal to the span of the
$n_j$. \qed
\endproclaim
However, without more knowledge about the nature of the basic classes as in the
case of a K\"ahler surface, the equality mod torsion in (3.1) seems to be the
optimal statement.
Finally let us show that the basic classes determine the plurigenera of $\tilde
X$. In all cases the basic classes determine $K_0^2$. If $K_0^2 > 0$, then
$\tilde X$ is of general type. It is well-known that, for $n\geq 2$,
$$P_n(\tilde X) = \frac{n(n-1)}2K_0^2 + \chi (\scrO_{\tilde X}).$$
Thus $P_n(\tilde X)$ is determined from the knowledge of $K_0^2$ for all $n\geq
2$, and $P_1(\tilde X) = p_g(\tilde X)$ is an oriented homotopy invariant
since $b_2^+(\tilde X) = 2p_g(\tilde X)+1$. Hence the plurigenera are
determined by $K_0^2$ as long as $K_0^2 >0$.
For
$K_0^2 = 0$,
$X$ is deformation equivalent to an elliptic surface and the plurigenera are
essentially determined from the knowledge of the multiple fibers (see
\cite{15}, Chapter I Proposition 3.22 for a more complete discussion). We will
deal here with the simply connected case, the case of at most two multiple
fibers of relatively prime multiplicity. Here the smooth classification of
elliptic surfaces with
$p_g \geq 1$ has been worked out in Donaldson theory; see
\cite{2}, \cite{23}, \cite{22}, \cite{12}, as well as \cite{21} and \cite{11}.
The general case may be reduced to the simply connected case (in the case of
finite cyclic fundamental group) or dealt with either by elementary arguments
involving the fundamental group (in case the fundamental group is not finite
cyclic), see
\cite{15} for this reduction. One could also use the basic classes to determine
the multiplicities in the general case along the lines of what we do here for
the
simply connected case.
Suppose that there are two multiple fibers $F_1$ and $F_2$, with relatively
prime multiplicities $m_1\leq m_2$. Here to handle all cases at once we will
also allow $m_1$ or both $m_1$ and $m_2$ to be 1. All the basic classes are of
the form $r\kappa$, where $\kappa$ is a primitive integral class and $r\in
\Bbb Q$. The largest value of $|r|$ is attained for $\pm K_X$, and it is
$(p_g+1)m_1m_2 -m_1 -m_2$. The next largest value is attained
for $L = \pm (K_X -2F_2)$, and it is
$$(p_g+1)m_1m_2 -m_1 -m_2- 2m_1,$$
provided that this number is not negative. Note that since $p_g \geq 1$, if
$m_1 \geq 2$, in other words if there are two multiple fibers, then as $m_1<
m_2$,
$$(p_g+1)m_1m_2 -m_1 -m_2- 2m_1 > 4m_2 - 4m_2 = 0.$$
Thus we determine $(p_g+1)m_1m_2 -m_1 -m_2$ and $m_1$, and so
$$((p_g+1)m_1-1)(m_2 -1) = (p_g+1)m_1m_2 -m_1 -m_2 -p_gm_1 + 1.$$
{}From the knowledge of $m_1$ and $((p_g+1)m_1-1)(m_2 -1)$ we may then
determine
$m_2$.
If $m_1 = 1$ then $(p_g+1)m_1m_2 -m_1 -m_2- 2m_1 = p_gm_2 -3 \geq 0$ provided
that $p_gm_2 \geq 3$. Except in these cases, we then find $m_1 = 1$ and can
solve for $m_2$ as before. The remaining cases are $p_g = 1$, $m_2 = 1$ or $2$
or $p_g = 2$ and $m_2 = 1$. For example if $p_g = 1$, the basic classes are
$\pm
K_0$ and $K_0$ is trivial if $m_2 = 1$ and nontrivial otherwise. Thus we can
distinguish these cases also. So we have determined the multiplicities of the
multiple fibers and thus the plurigenera.
\section{4. The case where $\boldkey p_{\boldkey g}$ is zero and $\boldkey X$
is of general type.}
In this section we consider the case where $p_g(X) = 0$, i\.e\. $b_2^+(X) = 1$,
and $X$ is of general type. If $X$ is of general type, then automatically
$b_1(X) = 0$ since
$\chi (\scrO_X) = 1 - q(X) + p_g(X) > 0$. For a $4$-manifold
$M$ with $b_2^+(M) = 1$, the function
$SW _{M,g}$ is no longer independent of the metric $g$, at least for $b_2(M)$
sufficiently large. For the purposes of this paper, we shall only consider
basic
classes of index zero. Equivalently we shall restrict the function $SW _{M,g}$
to
a function defined on characteristic line bundles $L$ with $L^2 = 2\chi (M) +
3\sigma (M)$. In this case, if
$b_2(M) = b\geq 10$, the orthogonal hyperplanes to characteristic cohomology
classes of square
$10-b$ divide
$$\Bbb H(M) = \{\, \alpha \in H^2(M; \Ar): \alpha ^2 = 1\,\}$$
into a set of chambers $\Cal C$, and we can define $SW _{M,\Cal C}$, for each
chamber $\Cal C$, as a function on pairs of characteristic line bundles and
Spin${}^c$ structures
$(L, \xi)$ with
$L^2 = 2\chi (M) + 3\sigma (M)$. Here given a metric
$g$, there is a unique associated self-dual harmonic 2-form $\eta$ for $g$, mod
nonzero scalars. For a generic metric $g$, $t\eta$ will lie in the interior of
a
chamber $\Cal C$ for an appropriate positive real number $t$, and we let $SW
_{M,\Cal C} = SW _{M,g}$ for every metric $g$ whose self-dual 2-form lies in
$\Ar^+\cdot \Cal C$. Changing
$\eta$ to $-\eta$ corresponds to changing the orientation of the SW moduli
space.
Thus
$$SW _{M, -\Cal C} = -SW _{M, \Cal C}.$$
This procedure defines $SW _{M, \Cal C}$ for every $\Cal C$ which contains
the self-dual harmonic 2-form of a Riemannian metric. However, we can define
$SW _{M, \Cal C}$ in general formally once we have a wall crossing formula
\cite{10}:
\proposition{4.1} Suppose $b_1(M) = 0$ and that $\Cal C_0$ and $\Cal C_1$ are
two
chambers whose boundaries intersect in an open subset of a wall $L^\perp$.
Suppose further that there is a path of metrics $\{\,g_t : t \in [0,1]\,\}$
such
that the self-dual harmonic $2$-form associated to $g_0$ lies in $\Cal C_0$ and
that the self-dual harmonic $2$-form associated to $g_1$ lies in $\Cal C_1$.
Then
$$SW _{M, \Cal C_1}(L',\xi) = \cases SW _{M, \Cal C_0}(L',\xi), &\text{if
$c_1(L')\neq \pm c_1(L)$ in $H^2(M; \Bbb R)$;}\\
SW _{M, \Cal C_0}(L', \xi) \pm 1, &\text{if $c_1(L')= \pm c_1(L)$ in $H^2(M;
\Bbb
R)$.}
\endcases$$
\endproclaim
Here the sign in the wall crossing formula depends on whether $L' \cdot \Cal
C_0$ is positive or negative, as well as on the general conventions we have
used
to orient the moduli space. Although we shall not need to know the sign
precisely, let us give the correct choice of sign in the case of interest to
us:
\claim{4.2} Suppose in the above situation that $X$ is a K\"ahler surface and
that our orientations are chosen so that the SW moduli space has its natural
complex orientation. Suppose further that $L$ is a wall of $\Cal C_+$ and $\Cal
C_-$ and that $L\cdot \Cal C_+ >0> L\cdot \Cal C_-$. Then
$$SW _{X, \Cal C_+}(L, \xi) = SW _{X, \Cal C_-}(L, \xi) -1.$$
\endproclaim
Let us show that the claim holds in a special situation. Suppose that $X$ is a
rational surface which is the blowup of $\Pee ^2$ at $d^2$ points which lie on
a smooth curve $C$ of degree $d \geq 4$. Thus $g(C) \geq 3$. We continue to
denote by
$C$ the proper transform of $C$ on $X$. On $X$, $C^2 = 0$ and so $K_X\cdot C +
C^2 = K_X\cdot C = 2g(C) -2 >0$. Let $H$ denote the pullback of the positive
generator of $H^2(\Pee ^2; \Zee)$ to $X$ and let the classes of the
exceptional curves be denoted by $E_1, \dots, E_{d^2}$. Let
$g$ be a K\"ahler metric on
$X$ with K\"ahler form $\omega _0$ equal to $NH - \sum _iE_i$, for $N\gg 0$.
Thus
as
$K_X = -3H + \sum _iE_i$, if $N\gg0$ $\omega _0 \cdot K_X <0$. It follows that
there are no holomorphic line bundles $L$ on $X$ with $L\cdot \omega _0\leq 0$
and
$\dsize \frac{K_X+L}2 $ effective (we would simultaneously have $(K_X+L) \cdot
\omega _0 <0$ and $(K_X+L)\cdot \omega _0 \geq 0$). Thus there are no basic
classes for the chamber $\Cal C_+$ containing $\omega _0$. Since $C^2=0$, the
curve $C$ has nonnegative intersection with every irreducible curve, so by the
Nakai-Moishezon criterion
$\omega = \omega _0 + tC$ is ample for every $t$. Using the fact that
$K_X\cdot C >0$, we can choose $t$ so large that $\omega \cdot K_X >0$. It
follows that $\pm K_X$ are basic classes for the chamber $\Cal C_-$ containing
$\omega$, and moreover, by (2.4),
$SW _{X, \Cal C_-}(-K_X) = 1$ for the natural Spin${}^c$ structure and the
natural choice of complex orientation. Now
$\Cal C_-$ and
$\Cal C_+$ are separated by the wall $(-K_X)^\perp$, and $-K_X\cdot \Cal C_+ >
0
> -K_X\cdot
\Cal C_-$. Thus $SW _{X, \Cal
C_+}(-K_X) = SW _{X, \Cal C_-}(-K_X) - 1$.
The sign in the general case can be computed by showing that there is a
universal sign associated to the wall-crossing formula and working it out in a
special case as above.
Using the wall crossing formula, we can define $SW _{M, \Cal C}$ for all
chambers $\Cal C$, and this definition
agrees with $SW _{M,g}$ in case $g$ is a generic
Riemannian metric whose associated harmonic self-dual 2-form lies in $\Cal C$.
A
characteristic vector
$L$ of the appropriate square for which $SW _{M, \Cal C}(L) \neq 0$ will be
called a {\sl basic class for the chamber $\Cal C$} (recall however that we
are only considering basic classes of index zero).
We consider now the following situation: $X$ is a minimal surface of general
type with $p_g(X) = 0$ and $b_1(X) = 0$.
Let $n = K_X^2$, so that $1\leq n\leq 9$. Let $\tilde X$ be a blowup of $X$ at
$\ell$ distinct points. Let $K_0$ be the pullback of $K_X$ in $\tilde X$ and
let $E_1, \dots, E_\ell$ be the exceptional curves.
\proposition{4.3} There is a unique chamber $\Cal C_0$ containing $tK_0$ in its
interior for some positive real number $t$. The chamber $\Cal C_0$ is
invariant under reflection by the classes $E_i$. The basic classes for
$\Cal C_0$ are then $\pm K_0 + \sum _{i=1}^\ell \pm E_i$, with the natural
{\rm Spin}${}^c$ structures.
\endstatement
\proof If $K_0$ lies on a wall, then $K_0$ is orthogonal to some
characteristic cohomology element $L = A + \sum _{i=1}^\ell m_iE_i$. Here $A\in
H^2(X; \Zee) \subseteq H^2(\tilde X; \Zee)$ is characteristic, hence nonzero,
and the $m_i$ are odd, hence nonzero. Since $K_0$ is perpendicular to $L$, we
have $A^2\leq 0$. Thus $L^2\leq -\sum _im_i^2 \leq -\ell$. But on the other
hand
$L^2 = n-\ell$, which is impossible. Thus $K_0$ lies on no wall, hence lies in
the interior of a chamber $\Cal C_0$. In this case, it follows from (2.2) and
from the blowup formula (2.5) that the basic classes are as claimed.
\endproof
\corollary{4.4} The chamber $\Cal C_0$ has the following properties:
\roster
\item"{(i)}" For all characteristic $L$ with $L^2 = K_{\tilde X}^2 = n-\ell$,
$SW _{\tilde X, \Cal C_0}(L, \xi)$ is zero or $\pm 1$. Moreover, if $(L, \xi)$
and $(L, \xi ')$ are two basic classes for $\Cal C_0$, then $\xi = \xi '$.
\item"{(ii)}" There are $2^{\ell +1}$ $(L, \xi)$ as in \rom{(i)} such that $SW
_{\tilde X, \Cal C_0}(L, \xi)=\pm 1$.
\item"{(iii)}" If $L_1\neq L_2$ are two classes for which $SW
_{\tilde X, \Cal C_0}(L_i, \xi _i)=\pm 1$, $i=1,2$, then
$$\fracwithdelims(){L_1+L_2}{2}^2 \leq n,$$
with equality holding for at least one such pair of $L_1$ and $L_2$.
\item"{(iv)}" For every $L_1$ and $L_2$ such that equality holds above, the
line through $L_1+L_2$ meets $\Cal C_0$. \qed
\endroster
\endproclaim
We now claim:
\proposition{4.5}
\roster
\item"{(i)}" Suppose that $H^2(X; \Bbb Z)$ has no $2$-torsion, or more
generally that the group of $2$-torsion elements of $H^2(X; \Bbb Z)$ is not
isomorphic to $\Bbb Z/2\Bbb Z$. If
$\Cal C$ is any chamber satisfying
\rom{(i)---(iii)} of
\rom{(4.4)}, then $\Cal C = \pm \Cal C_0$.
\item"{(ii)}" If the group of $2$-torsion elements of $H^2(X; \Bbb Z)$ is
isomorphic to $\Bbb Z/2\Bbb Z$, and $\Cal C$ is any chamber satisfying
\rom{(i)---(iii)} of
\rom{(4.4)}, then $(L, \xi)$ is a basic class for $\Cal C$ if and only if
there exists a $\xi '$ such that $(L, \xi')$ is a basic class for
$\Cal C_0$.
\endroster
\endproclaim
\proof Throughout this proof we shall identify classes in $H^2(\tilde X; \Zee)$
with their images in $H^2(\tilde X; \Ar)$, i\.e\. with their images mod
torsion, since we are only concerned with the chamber structure which lives
inside $H^2(\tilde X; \Ar)$. Let
$\Cal C$ be any chamber satisfying (i)---(iii) of (4.4). Since
$-\Cal C$ also satisfies these conditions, we can assume that $\Cal C$ and
$\Cal
C_0$ lie in the same component of $\Bbb H(\tilde X)$. Now let us show that we
may
assume that $K_0 + \sum _iE_i$ is positive on both $\Cal C_0$ and $\Cal C$:
\lemma{4.6} Suppose that $\Cal C$ is a chamber lying on the same component of
$\Bbb H(\tilde X)$ as $\Cal C_0$. Then, possibly after replacing $\Cal C$ by
$f^*\Cal C$, where
$f$ is an orientation-preserving self-diffeomorphism of $\tilde X$
corresponding
to reflection about some of the $E_i$'s, we may assume that $K_0 + \sum _iE_i$
is
positive on both $\Cal C_0$ and $\Cal C$. Thus $SW _{\tilde X,\Cal C}$ is not
identically zero for any chamber $\Cal C$. In particular $\tilde X$ is not
diffeomorphic to a rational surface and does not have a Riemannian metric of
positive scalar curvature.
\endproclaim
\proof Suppose that $x\in \Cal C$. Write $x = B + \sum _it_iE_i$, where $B\in
H^2(X;\Ar)$. After composing by a sequence of reflections in the $E_i$ (which
leave the chamber $\Cal C_0$ invariant) we may assume that $t_i \leq 0$ for all
$i$. Moreover
$x$ and
$K_0$ lie in the same component of the positive cone of $H^2(X; \Ar)$. Thus
$K_0\cdot x > 0$. It follows that $(K_0 + \sum _iE_i)\cdot x = K_0 \cdot x +
\sum _it_i >0$. Thus $K_0 + \sum _iE_i$ is
positive on $\Cal C$. By the wall-crossing formula, since $K_0 + \sum _iE_i$
does not separate $\Cal C_0$ from $\Cal C$, $SW_{\tilde X,\Cal C}(K_0 + \sum
_iE_i)\neq 0$. Hence $SW _{\tilde X,\Cal C}$ is not zero on any chamber. The
final statements are then clear.
\endproof
We note that the fact that no general type surface $\tilde X$ can be
diffeomorphic to a rational surface was proved via Donaldson theory in
\cite{16} (see also \cite{19}, \cite{27}, \cite{28}, \cite{31},
\cite{32}, \cite{30}, \cite{29}). Okonek and Teleman \cite{25} have
independently observed that one can use the method of (4.6) to show that no
surface of general type is diffeomorphic to a rational surface.
Returning to the situation where $\Cal C$ satisfies (i)---(iii) of (4.4), we
see
that we can assume that $SW _{\tilde X, \Cal C}(K_0 + \sum _iE_i) \neq 0$,
and that there exists an $x\in \Cal C$ of the form $B - \sum _ir_iE_i$, where
$B\in H^2(X;\Ar)$ and $r_i \geq 0$. Now consider the walls that
separate $\Cal C$ from $\Cal C_0$. Suppose that $L= C +
\sum _i(2s_i+1)E_i$ is such a wall, where $C\in H^2(X; \Zee)$, and suppose that
$SW _{\tilde X, \Cal C_0}(L, \xi) = 0$ for all choices of $\xi$. Thus $L$ is a
basic class for
$\Cal C$ but not for $\Cal C_0$. In this case we shall show that, possibly
after
modifying $L$, the condition (iii) of (4.4) does not hold, in other words
there is an average of classes for $\Cal C_0$ with square greater than $n$.
Since
$L$ is just defined up to sign, we may assume that $L\cdot K_0 >0$, and so
$L\cdot
\Cal C_0 > 0$. Thus
$L\cdot x<0$ for all
$x\in \Cal C$. We claim that we can replace $L = C+ \sum _i(2s_i+1)E_i$ by the
class $L' = C -\sum _i|2s_i+1|E_i$. Indeed,
$$x\cdot L' = (B\cdot C) -\sum _ir_i|2s_i+1| \leq (B\cdot C) -\sum
_ir_i(2s_i+1) = x\cdot L <0< K_0\cdot L = K_0\cdot L'.$$
Hence the class $L'$ also defines a wall which separates $\Cal C_0$ and $\Cal
C$. Replacing $L$ by $L'$, we can assume that $L= C- \sum
_i(2s_i+1)E_i$ where
$2s_i + 1 \geq 0$ and so $s_i \geq 0$. Moreover $L$ is a basic class for $\Cal
C$. It follows that $L^2 = n-\ell$ and so $C^2 = n-\ell + \sum _i(2s_i+1)^2$.
Since
$2s_i+1 \geq 1$, with equality only if $s_i=0$, we see that
$C^2 \geq n$,
with equality only if $s_i = 0$ for all $i$.
By assumption on the chamber $\Cal C$,
$((K_0 + \sum _iE_i)+ L)/2$ has square at most
$n$. If we calculate the square, however, we find:
$$\fracwithdelims(){(K_0 + \sum _iE_i)+ L}{2}^2 = \left(\frac{K_0 + C}2- \sum
_is_iE_i\right)^2= \frac{K_0^2 + C^2 +2(K_0\cdot C)}4 - \sum _is_i^2.$$
Now $K_0^2 = n$ and $C^2 = L^2 + \sum _i(2s_i+1)^2= n-\ell + \sum
_i(2s_i+1)^2\geq n$. By the Hodge index theorem,
$$|(K_0\cdot C)| = (K_0 \cdot C) \geq \sqrt{K_0^2}\sqrt{C^2}\geq
\sqrt{n}\sqrt{n},$$
with equality holding only if $K_0 = C$. Hence
$$\align
\frac{K_0^2 + C^2 +2(K_0\cdot C)}4 - \sum _is_i^2 &\geq \frac{n + n-\ell +
\sum _i(2s_i+1)^2 +2n}4 - \sum _is_i^2\\
&= \frac14(4n + 4\sum _is_i^2 + 4\sum _is_i) - \sum _is_i^2 \geq n +\sum _is_i.
\endalign$$
Thus the square of $((K_0 + \sum _iE_i)+ L)/2$ is greater than $n$ unless $s_i
=
0$ for all $i$ and $K_0 = C$. In this case $L= K_0 - \sum _iE_i$, which is
already a basic class for $\Cal C_0$. This contradicts the choice of $L$.
It follows that the only walls which can separate $\Cal C_0$ from $\Cal C$
are are of the form $L^\perp$, where $(L, \xi)$ is a basic class for $\Cal
C_0$.
First suppose that $H^2(X; \Bbb Z)$ has no $2$-torsion. In this case we shall
show that (ii) of (4.4) does not hold, in other words that $\ell$ decreases, if
the set of such walls is nonempty. In any case the basic classes for
$\Cal C$ must be a subset of the basic classes for $\Cal C_0$. The wall
crossing
formula then implies that, if there is such a wall, then there are fewer
classes
for
$\Cal C$ than for $\Cal C_0$. This contradicts our assumption on $\Cal C$.
(Note
that, if we did not know the sign in the wall crossing formula, we would still
be
able to conclude at this point that either $\Cal C$ had fewer classes than
$\Cal C_0$ or that there existed a basic class $L$ for $\Cal C$ for which
$SW _{\tilde X, \Cal C}(L) = \pm 2$. This would again contradict the
choice of $\Cal C$.)
If there is $2$-torsion in $H^2(X; \Bbb Z)$, then the wall crossing formula
implies that we lose the basic classes $(\pm L, \xi)$ as we cross the wall
$L^\perp$ but we gain new classes of the form $(\pm L, \xi')$ for $\xi' \neq
\xi$ (recall that for $\Cal C_0$ there is a unique $\xi$ such that $(L, \xi)$
is a basic class). If the group of $2$-torsion elements of $H^2(X; \Bbb Z)$ is
larger than $\Bbb Z/2\Bbb Z$, then there would be two distinct Spin${}^c$
structures $\xi _1\neq \xi _2$ such that $(L, \xi _1)$ and $(L, \xi _2)$ are
basic classes for $\Cal C$. Thus $\Cal C$ would violate (i) of (4.4). In the
remaining case it is clear that the basic classes for $\Cal C$ are exactly of
the form $(L, \xi')$, where $(L, \xi)$ is a basic class for $\Cal C_0$.
\endproof
Note that we do not as yet claim that the integer $n$ or equivalently $\ell$ is
specified by the 4-manifold $\tilde X$, in the sense that there might {\it a
priori\/} exist other positive integers $n'\leq 9$ and $\ell'$ with $n' - \ell
' = n-\ell$, and a chamber $\Cal C_0 '$ satisfying (i)---(iii) of (4.4) for
$n'$ and $\ell'$. In fact, although we shall not really need this, our
arguments
show that $\ell$ is the maximum over all possible $\ell '$ such that there
exists
a chamber $\Cal C_0 '$ satisfying (i)---(iii) of (4.4) for $\ell'$ and $n' =
K_{\tilde X}^2 + \ell$.
Let us now show how to deduce Theorems 1.1 and 1.2 from Proposition 4.5, at
least in the case where $p_g(X) = 0$ and $X$ is of general type. First suppose
that there is no $2$-torsion in $H^2(X; \Bbb Z)$ and that $f\: \tilde X \to
\tilde X$ is an orientation-preserving self-diffeomorphism. Then $f^*\Cal C_0$
has the same properties (i)---(iii) as
$\Cal C_0$, and so by (4.5) $f^*\Cal C_0 = \pm \Cal C_0$. In particular $f^*$
leaves invariant the basic classes for $\Cal C_0$. As in the case $p_g>0$ and
$K_0^2>0$ we see that $f^*$ preserves $\pm K_0$ and the span of the $E_i$. A
similar argument works in case the $2$-torsion subgroup of $H^2(X; \Bbb Z)$
is not isomorphic to $\Bbb Z/2\Bbb Z$. Finally, if the $2$-torsion subgroup of
$H^2(X; \Bbb Z)$ is isomorphic to $\Bbb Z/2\Bbb Z$ and $f\: \tilde X \to
\tilde X$ is an orientation-preserving self-diffeomorphism, then $f^*\Cal C_0$
has the same properties (i)---(iii) as
$\Cal C_0$, and so $f^*$
leaves invariant the set of $L$ such that there exists a $\xi$ with $(L,
\xi)$ a basic class for
$\Cal C_0$. Thus $f^*$ preserves $\pm 2K_0$ and so $f^*\Cal C_0 = \pm \Cal
C_0$. Again we may complete the argument as in the case $p_g >0$.
Next, let $N$ be a negative definite 4-manifold
such that $\tilde X$ is diffeomorphic to $M\#N$ for some 4-manifold $M$. Let
$x\in H^2(M; \Ar) \subseteq H^2(\tilde X; \Ar)$ satsify $x^2 =1$, and assume
that $x$ lies on no wall in $\Bbb H(M)$ or $\Bbb H(\tilde X)$. Let $\Cal C$ be
the chamber of $\Bbb H(\tilde X)$ containing $x$. Let $n_i$ be an exceptional
class of $N$ and let $R_i\: H^2(\tilde X; \Ar) \to H^2(\tilde X;
\Ar)$ be the reflection about $n_i$. Note that, as $H^2(N; \Zee)$ is torsion
free, $R_i$ acts on the set of Spin${}^c$ structures as well. Clearly
$R_i$ fixes
$\Cal C$. The blowup formula says that $R_i$ also fixes the set of basic
classes
for
$\Cal C$.
We claim that, in the above situation, the basic classes for $R_i(\Cal C_0)$
are
exactly those of the form $R_i(L)$, where $L$ is a basic class for $\Cal C_0$,
and more precisely that $SW _{\tilde X, R_i(\Cal C_0)}(R_i(L), R_i(\xi)) = SW
_{\tilde X,\Cal C_0}(L, \xi)$. (This would of course be clear if $n_i$ was
represented by a smoothly embedded 2-sphere.) Thus, arguing as above,
$R_i(\Cal C_0)$ satisfies (i)---(iii) of (4.4) and moreover
$R_i(\Cal C_0) =
\pm \Cal C_0$, indeed $R_i(\Cal C_0) = \Cal C_0$ since $R_i$ fixes the
components of $\Bbb H(\tilde X)$. It follows that the wall through $n_i$ passes
through $\Cal C_0$ for every $i$, and that (by induction on the number of
exceptional classes) we can choose an
$x\in
\Bbb H(M)$ whose image in $\Bbb H(\tilde X)$ lies in $\Cal C_0$. The proof
that every exceptional class $n_i$ must be equal to $\pm E_j$
for some $j$ then runs as in the case
$p_g(X)>0$ and $X$ of general type (i\.e\. $K_0^2 >0$). Thus it suffices to
show (we omit the Spin${}^c$ structures for notational simplicity):
\proposition{4.7} Let $Y=M\#N$ be an oriented smooth $4$-manifold such that
$b_2^+(M) =1$ and $N$ is negative definite. For an exceptional
class $n_i \in H^2(N; \Zee)$, let $R_i$ be the refection about $n_i$, viewed as
an automorphism of $H^2(Y; \Zee)$. Then for every chamber $\Cal C_0$ of $\Bbb
H(Y)$, we have:
$$SW _{Y, R_i(\Cal C_0)}(R_i(L)) = SW
_{Y,\Cal C_0}(L).$$
\endproclaim
\proof As above, let
$x\in H^2(M; \Ar) \subseteq H^2(Y; \Ar)$ satsify $x^2 =1$, and assume
that $x$ lies on no wall in $\Bbb H(M)$ or $\Bbb H(Y)$. Let $\Cal C$ be
the chamber of $\Bbb H(Y)$ containing $x$, so that the blowup formula holds for
the basic classes of $\Cal C$. To see that $SW _{Y, R_i(\Cal C_0)}(R_i(L)) = SW
_{Y,\Cal C_0}(L)$, we
may assume that $\Cal C$ and $\Cal C_0$ lie on the same component of $\Bbb
H(Y)$. Choose an $x_0 \in \Cal C_0$ and consider a generic path
$\gamma$ from
$x$ to $x_0$. Let $L_1, \dots, L_k$ be the walls crossing $\gamma$. Then if
$L\neq L_i$ for any $i$, the wall-crossing formula says that
$$SW _{Y,\Cal C_0}(L) =SW _{Y,\Cal C}(L).$$
If $L=L_\alpha$ for some $\alpha$, then
$$SW _{Y,\Cal C_0}(L_\alpha) =SW _{Y,\Cal
C}(L_\alpha)\pm 1.$$ Now consider the function $SW _{Y, R_i(\Cal
C_0)}$. The point
$R_i(x_0)$ lies in $R_i(\Cal C_0)$, the path $R_i(\gamma)$ joins $R_i(x)$ to
$R_i(x_0)$, and the walls crossed by $R_i(\gamma)$ are the walls
$R_i(L_\alpha)$. Note that the blowup formula implies that $L$ is a basic class
for $\Cal C$ if and only if $R_i(L)$ is a class for $\Cal C$, and indeed
$SW _{Y, \Cal C}(L) = SW _{Y, \Cal C}(R_i(L))$. Now to
calculate $SW _{Y, R_i(\Cal C_0)}(R_i(L))$, first assume that
$L\neq L_\alpha$ for any $\alpha$. Then
$$SW _{Y, R_i(\Cal C_0)}(R_i(L)) = SW _{Y, \Cal
C}(R_i(L)) = SW _{Y, \Cal C}(L) = SW _{Y,\Cal C_0}(L).$$
If $L= L_\alpha$, then
$$SW _{Y, R_i(\Cal C_0)}(R_i(L_\alpha)) = SW _{Y, \Cal
C}(R_i(L_\alpha)) \pm 1 = SW _{Y, \Cal C}(L_\alpha)\pm 1.$$
Now examination of the wall crossing formula says that
$$SW _{Y,\Cal C_0}(L_\alpha) = SW _{Y, \Cal
C}(L_\alpha)\pm 1,$$
and the sign must be the same as in the above formula. Putting this together we
see that $SW _{Y, R_i(\Cal C_0)}(R_i(L_\alpha)) = SW
_{Y,\Cal C_0}(L_\alpha)$, so the formula holds in all cases.
\endproof
Thus we have established Theorem 1.2. In particular, the cohomology
classes of embedded 2-spheres of self-intersection $-1$ span a sublattice of
$H^2(\tilde X; \Zee)$ of rank exactly $\ell$. It follows that if $\tilde{X'}$
is another surface of general type and $f\: \tilde{X'} \to
\tilde X$ is an orientation-preserving diffeomorphism, then $\tilde {X'}$ can
be
blown up at most
$\ell$ times from its minimal model. By symmetry
$\tilde X$ and $\tilde{X'}$ are blown up the same number of times, namely
$\ell$, from their minimal models. If $\Cal C_0'$ is the chamber on
$\tilde{X'}$
corresponding to $\Cal C_0$, it then follows that $f^*\Cal C_0' = \pm \Cal
C_0$. Theorem 1.1 is an immediate consequence.
\qed
\medskip
Finally let us deduce that if $Y$ is a complex surface diffeomorphic to $\tilde
X$, then $P_n(Y) = P_n(\tilde X)$ for all $n$. Note that $Y$ is K\"ahler since
$b_1(Y) =0$. Moreover $Y$ cannot be a rational surface by Lemma 4.6. We
will rule out the case where $Y$ is elliptic in the next section. Thus
$Y$ is again a surface of general type, and by Theorem 1.1 we may determine
$K_0^2$ for $Y$. It follows as in the case where $p_g>0$ that $P_n(Y) =
P_n(\tilde X)$ for all $n$.
We note that Theorem 1.1 has the following corollary:
\corollary{4.8} Let $\tilde X$ be a surface of general type with $p_g(\tilde X)
= 0$, and let
$D(\tilde X)$ be the image of the group of orientation-preserving
diffeomorphisms of
$\tilde X$ in the automorphism group of $H^2(\tilde X; \Zee)$. Then
$D(\tilde X)$ is finite.
\endproclaim
\proof Let $\phi \in D(\tilde X)$. Up to finite index we may assume that
$\phi(E_i) = E_i$ for all $i$ and that $\phi (K_0) = K_0$. Thus $\phi$ is
determined by its action on $K_0^\perp \cap H^2(X; \Zee)$. Since $K_0^\perp
\cap
H^2(X; \Zee)$ is negative definite, there are only finitely many automorphisms
of $K_0^\perp \cap H^2(X; \Zee)$. Hence there are only finitely many
possibilities for $\phi$.
\endproof
\section{5. The case where $\boldkey p_{\boldkey g}$ is zero and $\boldkey X$
is
not of general type.}
Suppose that $X$ is a minimal K\"ahler surface, not of general type, rational,
or ruled, such that
$p_g(X) = 0$. In this case $X$ is elliptic (possibly an Enriques or
hyperelliptic surface), $K_X^2=0$, and, since $\chi (\scrO_X) \geq 0$, either
$b_1(X) = 0$ and $X$ is elliptic or $b_1(X) = 2$. We shall first consider the
parts of the analysis of the topology of a blowup of $X$ which can be handled
either by elementary methods or by reduction to the case $p_g > 0$.
If $b_1(X) = 0$, then
$X$ is an elliptic surface, obtained from a rational elliptic surface by a
number of logarithmic transforms. Here $X$ is rational if there is just one
logarithmic transform and it is simply connected (a Dolgachev surface) if there
are two logarithmic transforms of relatively prime multiplicities. In all other
cases
$X$ has a finite covering space $Y$ which is an elliptic surface with $p_g \geq
1$. Let
$\tilde X$ be a blowup of $X$, and let $\tilde Y$ be the induced cover, which
is a blowup of $Y$. In this case, since Theorems 1.1 and 1.2 hold for
$\tilde Y$, they also hold for $\tilde X$ mod torsion. We can then determine
the plurigenera of $\tilde X$, either via elementary arguments involving the
fundamental group as in
\cite{15} if $\pi _1(X)$ is not finite cyclic or by reducing to the simply
connected case with $p_g \geq 1$ in case $\pi _1(X)$ is finite cyclic (but
nontrivial). Finally, since $\tilde Y$ has no Riemannian metric with positive
scalar curvature, the same is true for $\tilde X$.
Likewise, if $X$ is a
minimal K\"ahler surface with $p_g(X) = 0$ and $b_1(X) = 2$, or in other words
$q(X) = 1$, then $X$ is an elliptic surface with Euler number zero. In this
case either $X$ is ruled over an elliptic base or it is nonruled. If $X$ is not
ruled it is a logarithmic transform either of an elliptic surface without
singular or multiple fibers over a base curve of genus one, with nontrivial
holonomy (in other words, a hyperelliptic surface), or it is a logarithmic
transform of $E\times \Pee ^1$ so that the corresponding base orbifold is not
spherical (see \cite{15} for more description). In both of these cases,
provided that $X$ is not ruled, it again has a finite covering space $Y$ which
is an elliptic surface with $p_g \geq 1$. Again, we see that Theorems 1.1 and
1.2 hold for blowups of $X$, mod torsion. The determination of the plurigenera
then follows from the fundamental group as in \cite{15}, and the fact that
$\tilde X$ has no metric of positive scalar curvature, for every blowup $\tilde
X$ of $X$, follows as in the case where
$b_1(X) =0$ and $\pi _1(X)$ is not trivial.
The remaining case is the case of Dolgachev surfaces. This case was handled via
Donaldson theory in \cite{13}, \cite{1}, \cite{12} (see also \cite{26}). It
can
also be handled by the methods of this paper, as we now outline. Let $\tilde X$
be the blowup of a minimal Dolgachev surface $X$, and let $K_0$ be the image of
the canonical class of $X$ in
$H^2(\tilde X; \Zee)$. Suppose that $\tilde X$ is the blowup of the minimal
surface $X$ at $\ell$ distinct points. All of the basic classes for
$X$ are rational multiples $rK_0$ of $K_0$ with $|r| \leq 1$. We suppose that
there are
$d$ basic classes for $X$. It is easy to check that on each basic class of $X$
the value of $SW_X$, which does not depend on a choice of chamber, is $\pm 1$.
Now let $\Cal C_0$ be a chamber in $\Bbb H(\tilde X)$ which contains classes
of the form $\omega$, where $\omega$ is the K\"ahler form of a generic
K\"ahler metric on $X$. It follows that there are $d2^\ell$ basic classes for
$\Cal C_0$. They are exactly the classes $L + \sum _i\pm E_i$, where $L$ is a
basic class for
$X$ and the
$E_i$ are the exceptional classes on $\tilde X$, and the value of $SW_{\tilde
X,
\Cal C_0}$ on each such class is $\pm 1$. Moreover, for every average $\dsize
\frac{L_1+L_2}2$ of basic classes for $\Cal C_0$, we have $\dsize
\fracwithdelims(){L_1+L_2}{2}^2 \leq 0$, with equality holding for some pair of
basic classes $L_1 \neq \pm L_2$. Arguments very similar to those given in the
proof of Proposition 4.5 show that
$\pm \Cal C_0$ are the unique chambers with these properties. Thus every
orientation-preserving self-diffeomorphism $f\: \tilde X \to \tilde X$
satisfies $f^*\Cal C_0 = \pm \Cal C_0$, and if $n_i$ is an exceptional class
for a negative definite summand of $\tilde X$, then the reflection $R_i$ in
$n_i$ preserves $\Cal C_0$. Moreover, in the above two cases, we see that
both $f^*$ and $R_i$ permute the set of basic classes. In particular the wall
through $n_i$ passes through $\Cal C_0$ and $n_i$ is a difference class
for $\Cal C_0$. It follows then from the arguments in Section 3 for the case
$K_0^2 =0$ that $f^*$ preserves $\pm K_0$ and that $n_i = \pm
E_j$ for some $j$. In
particular there can be at most $\ell$ disjoint smoothly embedded 2-spheres
of self-intersection $-1$ on $\tilde X$. Thus $\tilde X$ cannot be
diffeomorphic to a blown up surface of general type. The basic classes for
$\Cal C_0$ determine the the basic classes for
$X$, by arguments similar to those in Section 3 for the case where $K_X^2 =0$.
Finally, the basic classes for $X$ determine the multiplicities of the
multiple fibers, by arguments along the lines of those given in Section 3 for
simply connected elliptic surfaces with $p_g\geq 0$. Here, in case $p_g =0$,
there are a few extra cases to consider. Lastly, the arguments used to prove
Lemma 4.6 also show that there is no chamber $\Cal C$ where the function
$SW_{\tilde X, \Cal C}$ is identically zero, and thus $\tilde X$ has no metric
of positive scalar curvature. Thus we have completed the proofs of Corollary
1.4 and Corollary 1.5.
Finally we note that one can modify the proofs of the results in Section 4 to
handle the non-simply connected elliptic surfaces with $p_g =0$, and replace
equality mod torsion with equality in the non simply connected
case. The main point is to handle wall crossings in case $b_1(X) = 2$. However,
we shall not give these arguments here.
\section{6. Some open questions.}
\noindent {\bf The non-K\"ahler case.} Can one generalize the above results to
the non-K\"ahler case? The structure of non-K\"ahler complex surfaces of
nonnegative Kodaira dimension is well-understood. In particular they are all
elliptic surfaces with odd first Betti number and Euler number zero. Elementary
methods \cite{15} show that all such surfaces are $K(\pi, 1)$'s and
hence that the classes of exceptional curves are preserved under
diffeomorphisms. It is also straightforward to show that the class of a
general fiber is preserved, and so $K_X$ mod torsion, and that the plurigenera
are diffeomorphism invariants. However, it does not seem possible to extend
these
methods to handle negative definite connected summands. On the other hand, the
analysis of the SW equations for K\"ahler or symplectic manifolds may admit a
straightforward generalization to this case (possibly under some assumptions
on the metric).
\medskip
\noindent {\bf Ruled surfaces.} First we note that the Seiberg-Witten theory
accounts for all the self-diffeomorphisms of a rational surface $X$. Let
$X$ be the blowup of $\Bbb P^2$ at $\ell$ distinct points. Inside
$\Bbb H(X)$ there is a distinguished convex set, the ``super $P$-cell"
$\bold S_0$ defined in \cite{13}. Its walls are certain characteristic elements
of $H^2(X; \Zee)$ of square $K_X^2$. In fact one can show that $\bold S_0$ is
a chamber for the set of walls defined by primitive characteristic elements of
square $K_X^2$. It is shown in
\cite{13} that an automorphism $\varphi$ of the lattice $H^2(X; \Zee)$ is the
image of an orientation-preserving self-diffeomorphism of $X$ if and only if
$\varphi (\bold S_0) = \pm \bold S_0$. This result can be established by
Seiberg-Witten theory as well. It is elementary to show that every $\varphi$
such that $\varphi (\bold S_0) = \pm \bold S_0$ is the image of a
diffeomorphism, and the difficult part of the argument is to show that, for
every
orientation-preserving self-diffeomorphism $\psi$ of $X$, $\psi ^* (\bold S_0)
=
\pm \bold S_0$. Let $\Cal C_0$ be the chamber of $\Bbb H(X)$, for the set of
all
walls defined by characteristic elements
of $H^2(X; \Zee)$ of square $K_X^2$, which contains $\omega _0$, where $\omega
_0$ is a K\"ahler metric on $\Bbb P^2$, or equivalently contains K\"ahler
metrics on $X$ with K\"ahler form a positive multiple of $N\omega _0 - \sum
_iE_i$. Thus $\Cal C_0$ contains the period points of metrics with positive
scalar curvature, and so $SW_{X, \Cal C_0}$ is identically zero (this also
follows from the blowup formula). Moreover $\pm\Cal C_0$ are the unique
chambers $\Cal C$ such that $SW_{X, \Cal C}$ is identically zero. Hence $\psi
^*\Cal C_0 = \pm \Cal C_0$. For simplicity assume that $\psi
^*\Cal C_0 = \Cal C_0$. Now the interior of $\Cal C_0$ is nonempty and is
contained in the interior of $\bold S_0$, since the walls defining $\bold S_0$
are a subset of the walls defining $\Cal C_0$ and $\omega _0 \in \Cal C_0\cap
\bold S_0$. Thus $\psi ^*(\bold S_0)$ is a super $P$-cell such that $\psi
^*(\bold S_0) \cap \bold S_0 \neq \emptyset$. It follows by Lemma 5.3(e) on
p\. 339 of \cite{13} that
$\psi ^*(\bold S_0) = \bold S_0$.
Similar arguments imply that, if $N$ is a negative definite summand of $X$ and
$n_i$ is an exceptional class for $N$, then the reflection in $n_i$ preserves
$\bold S_0$. Using this fact and the method of proof of \cite{16}, Theorem 1.7,
one can show that every negative definite summand $N$ of $X$ can be accounted
for in the following sense: if $X$ is orientation-preserving diffeomorphic to
$M\#N$, where $N$ is negative definite, then there is an
orientation-preserving diffeomorphism $f\: X \to \tilde Y$, where $\tilde Y$
is the blowup of a rational surface $Y$, such that $H^2(N; \Zee)$ corresponds
to the span of the exceptional classes of the blowup $\tilde Y \to Y$.
If one tries to extend the above results to surfaces $X$ which are (not
necessarily minimal) ruled surfaces over a nonrational base curve, most of the
results extend with elementary proofs. For example, let $f$ be the class of a
general fiber of $X$ and $E_1, \dots, E_\ell$ be the classes of the
exceptional curves. Then the cohomology classes of smoothly embedded
2-spheres of self-intersection zero are exactly the classes $nf, n\in \Zee$,
and the cohomology classes of smoothly embedded
2-spheres of self-intersection $-1$ are the classes $nf+E_i, n\in \Zee$.
Moreover every orientation-preserving self-diffeomorphism $\psi$ of $X$
satisfies $\psi ^*f = \pm f$, and indeed an automorphism $\varphi$ of $H^2(X;
\Zee)$ is the image of an orientation-preserving self-diffeomorphism if and
only
if $\psi ^*f = \pm f$. However, it is not clear how to generalize these results
to arbitrary negative definite summands. Such generalizations would
presumably follow from Seiberg-Witten theory provided that we have a better
understanding of the transition formula in case $b_1(X) \neq 0$. Working out
such transition formulas is of course an interesting problem in its own right.
\medskip
\noindent {\bf Questions seemingly inaccessible to Seiberg-Witten theory.} The
overall moral of the above is that the sum total of the basic classes gives us
information about the obvious invariants of a complex surface, the exceptional
curves and the pullback of the canonical class of the minimal model, and no
more. Presumably the same is true of Donaldson theory. One can ask if there is
more to the smooth topology of a complex surface than this. For example, there
exist minimal surfaces of general type, say $X_1$ and $X_2$, and an isometry
$\varphi \: H^2(X_1; \Zee) \to H^2(X_2; \Zee)$ such that $\varphi (K_{X_1}) =
K_{X_2}$, and such that $X_1$ and $X_2$ are not deformation equivalent. As
has been pointed out by Fintushel and Stern, certain pairs of Horikawa surfaces
are among the simplest examples of surfaces with this property. In particular
we
cannot distinguish such pairs
$X_1$ and
$X_2$ via the basic classes, and, at least in the case of Horikawa surfaces,
they
have the same Donaldson polynomials as well, as has been shown by the second
author and Z. Szab\'o. Can one find new smooth invariants which will show that
$X_1$ and
$X_2$ are not diffeomorphic?
In a similar vein, are there further restrictions
on self-diffeomorphisms $f$ of a K\"ahler surface $X$ of nonnegative Kodaira
dimension beyond the conditions $f^*K_0 = \pm K_0$ and $f^*E_i = \pm E_j$? For
example, if the algebraic geometry of $X$ imposes a fundamental asymmetry on
$X$, is this seen by the smooth topology? One example of this might be a
surface which is the double cover of $\Pee ^1\times \Pee ^1$ along an
asymmetric branch divisor in the linear system $|2af_1 + 2bf_2|$, where the
$f_i$ are the fibers of the two different projections and $a,b$ are positive
integers with $a\neq b$. Is it true that for general $a$ and $b$ every
orientation-preserving self-diffeomorphism of $X$ preserves the pullbacks to
$X$ of
$f_1$ and $f_2$ up to sign (and not just $K_X$ which is a positive combination
of the pullbacks)? Even for simply connected elliptic surfaces, there is a gap
of finite index between those isometries of $H^2(X; \Zee)$ known to arise from
self-diffeomorphisms and the restrictions placed on such isometries by
Donaldson theory or Seiberg-Witten theory (see for example \cite{15}, Chapter
II, Theorem 6.5 and \cite{13}). Can one close this
gap by constructing more diffeomorphisms, or can one find new invariants which
rule out the existence of such diffeomorphisms?
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\endRefs
\enddocument
|
1995-02-16T06:20:13 | 9502 | alg-geom/9502010 | en | https://arxiv.org/abs/alg-geom/9502010 | [
"alg-geom",
"math.AG"
] | alg-geom/9502010 | Gaitsgory Denis | D.Gaitsgory | Operads, Grothendieck topologies and Deformation theory | 13 pages, AmsTeX | null | null | null | null | The idea of the work is to find an invariant way to pass from deformation
theory to cohomology, which does not use any explicit cocycles. The appropriate
cohomology theory is based on considering sheaves on a certain site. An
advantage of the approach is that it enables one to give a simultanious
treatement to all types of algebras. As an application, we prove the PBW
theorem in cases where it is not yet known.
| [
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\topmatter
\title Operads, Grothendieck topologies \\
and deformation theory
\endtitle
\author Dennis Gaitsgory \endauthor
\address{School of Mathematical Sciences, Tel-Aviv University,
Ramat-Aviv, Israel} \endaddress
\email
gaitsgde\@math.tau.ac.il
\endemail
\endtopmatter
\heading 0. Introduction \endheading
\sbh{0.1}
Gerstenhaber's papers in the Annals showed that deformation theory of
associative algebras over a field is "controlled" by Hochschild cohomology.
The passage from deformations to cohomology is realised by
means of Hochschild cochains. This approach has two main drawbacks:
\sbh{1} It is impossible to generalize it to the case of
algebras over a ring, or more generally, to sheaves of algebras
over a scheme. This is because neither deformations are described
by cochains, nor cohomologies can be computed using bar-resolution.
\sbh{2} Although deformations and cohomology are
invariant objects they are connected by choosing some specific resolution.
\sbh{0.2}
Our aim in the present work is to define an appropriate cohomology
theory and to find an invariant way to pass from deformations to
cohomology.
The initial idea, which goes back, probably, to Quillen, is to describe
deformations of an algebraic object (e.g. associative algebra) by means of
"resolving" it by free objects of the same type (in our example, free
associative algebras). In principal, all the results can be formulated
already on the level of our initial algebra $A$ and a free algebra $B$
mapping onto it. However, the picture is much easier to grasp,
when we consider the category of all algebras over $A$. These algebras
form a site, and cohomologies that we are looking for are just cohomologies
of certain sheaves on this site. This is the main idea of the paper.
An advantage of this approach is that we can treat in the same framework
algebras of all types, i.e. algebras over an arbitrary operad.
\sbh{0.3}
Let us describe briefly the contents of the paper.
In sections 1 and 2 we describe the formalism of operads, algebras
over operads and modules over them. Our presentation is inspired
by some ideas of A.Beilinson [6] and is very close to that of [1].
The essential difference is that we are using the language of
\pm categories.
In section 3 we introduce the site $C(A)$ and study the connection
between the category of sheaves on this site and the category
of $A$-modules. In paricular, we introduce the notion of the cotangent
complex of an algebra. The site $C(A)$ was introduced
first in [1]. However, one can think of deformation theory
(e.g. Theorem 4.2) as giving a hint how to define correct cohomologies:
just look at deformations of the corresponding object.
In section 4 we study deformations of an algebra over
an operad. Cohomology classes that arise in deformation theory
are realised as classes of certain torsors and gerbes.
Finally, in section 5 we give an application of the theory
presented in the paper. We prove the \pbw for Lie algebras
which are flat modules over the ground ring. The idea of such an
approach to the \pbw belongs to J.Bernstein and has
been already realized in [6].
\sbh{Acknowledgements}
The author would like to thank A.Joseph, J.Bernstein, V.Hinich, S.Shnider,
L.Breen, A.Braverman and R.Bezrukavnikov for interesting and
stimulating discussions.
\heading 1. Pseudo-monoidal categories and operads \endheading
\subheading{1.0}
We are working over a fixed ground ring $R$ and all categories are assumed
to be $R$-linear. If $C$ is a gategory, $C^o$ will denote the
opposite category.
\subheading{1.1 Pseudo-monoidal categories}
Let $C$ be a category. A pseudo-monoidal structure on it is a collecion
of functors $\for$:
$\fs:\CSo\times C\to \Rm$ equipped with the
following additional data (composition maps):
If $\st$, for each element $\bt$ of $\CT$ we are given a natural
transformation
between two functors $\CSo\times C\to \Rm$:
$${\bigotimes}_{t\in T}\fst(\underset{s\in S_t}\to\times A_s,B_t)
\otimes\ft(\underset{t\in T}\to\times
B_t,D)
\longrightarrow \fs(\underset{s\in S}\to\times A_s,D),$$
($S_t$ denotes here the preimage of $t\in T$)
with $F_1^C(A,B)=Hom(A,B)$ (subscript $1$ means one element set)
such that these natural transformations are compatible with respect to
compositions of partitions in the obvious sense.
\subheading{1.1.1 Example}
If $C$ is a strictly symmetric monoidal category, we endow it with a
pseudo-monoidal structure by setting
$$\fs(\underset{s\in S}\to\times A_s,B)=Hom(\otimes_{s\in S}A_s,B)$$
where the above natural transformations are just composition maps.
\sbh{1.1.2 Variant}
As in usual monoidal categories, one can require existence of a unit
object in a \pm category. This means that there must be an object
$\un\in C$ such that $\for$, for each $\underset{s\in S}\to\times
A_s\in C^{\times S}$ and for each $B\in C$, $F^C_{S\cup 1}(1\underset{s\in S}
\to\times A_s,B)$ is canonically isomorphic to $F^C_S(\underset{s\in S}
\to\times A_s,B)$. Such \pm categories will be called unital.
\subheading{1.1.3}
A pseudo-monoidal functor between two pseudo-monoidal
categories $C_1$ and $C_2$ is a (covariant) functor $G: C_1\to C_2$
equipped with a
natural transformation $F^{C_1}_S\to F^{C_2}_S\circ G$ which
is compatible with composition maps $\for$.
Pseudo-monoidal natural transformations are defined in a similar way.
\subheading{1.2 Operads}
Operad is by definition a \pm category $O$ which has essentially
one object.
\subheading{1.2.1}
Equivalently,
one can view operads as the following linear algebra data:
\roster
\item A collection of $\Rm$ \for denoted by $O(S)$ (thought of
as $\fs(A^{\times S},A)$ for $A\in C$)
\item A distinguished element $1\in O(1)$
\item If $\st$ there is a composition map \newline
$$O(T)\underset{t\in T}\to\bigotimes O(S_t)\to O(S)$$
\endroster
that satisfies
\roster
\item The composition $1\otimes O(S)\to O(1)\otimes O(S)\to O(S)$ \newline
is the identity map $\for$.
\item Composition maps are compatible with compositions of partitions.
\endroster
\sbh{1.2.2 Variant}
We define unital operads as unital \pm categories having at most
one isomorphism class of objects distinct from $\un$. It is
not difficult to work out this definition also in linear
algebra terms.
\sbh{1.2.3}
If $R\to R'$ is a ring homomorphism, to any operad over $R$
one can assign an operad over $R'$ by taking
tensor products over $R$ with $R'$. We will denote them
by same letters when no confusion can occur.
\subheading{1.3 Algebras over an operad}
In order to simplify the exposition we will consider
only algebras of $\Rm$.
In principal, one can define algebras over an operad in any strictly
symmetric monoidal category and develop deformation theory for them.
An $O$-algebra (of $\Rm$), or equivalently, an algebra over $O$
is by definition a \pm functor $O\to \Rm$.
Morphisms between $O$-algebras are defined
to be \pm natural transformations between such functors.
\subheading{1.3.1}
In linear agebra terms, an $O$-algebra is
an $R$-module $A$ which \newline
$\for$ is endowed with a map $$O(S)\otimes A^{\otimes S}\to A$$
such that if $\st$ the square
$$ \CD
O(T)\Ost\otimes A^{\otimes S} @>>> O(S)\otimes A^{\otimes S} \\
@VVV @VVV \\
O(T)\otimes A^{\otimes T} @>>> A
\endCD $$
is commutative.
\subheading{1.3.2 Examples}
\subheading{1}
Set $\for$, $O(S)=R$ . This is called $O^{com,ass}$. Algebras over it are
commutative and associative algebras.
\subheading{2}
Set \for, $O(S)=R^{\text{all bijections:\{1,2,...,$\vert S\vert$\}}\to S}$ with
an obvious definition of composition maps. This operad (denoted $O^{ass}$)
corresponds to associative algebras.
\sbh{3}
One can define in the same manner unital operads $O^{comm,ass,1},O^{ass,1}$
and they will correspond to unital algebras.
\sbh{4}
In a similar way one defines operads $O^{Lie}$, $O^{Poisson}$, etc.
\sbh{1.3.3 Variant}
A \pm functor from an operad $O$ to the category of
graded $\Rm$ (morphisms in this last category are
homogeneous of degree 0) will be called a graded $O$-algebra.
\subheading{1.4 Free O-algebras}
In this subsection we fix an operad $O$.
\proclaim{Lemma}
The forgetful functor ($O-algebras\to R-modules)$ admits a left adjoint.
\endproclaim
\demo{Proof}
For an $R$-module $V$ we will construct an $O$-algebra $Free(V)$:
\subheading{Construction}
$$Free(V)=\oplus_i(Free_i(V))\text{, where }Free_i(V)=(V^{\times T}
\otimes O(T))_{S^T}$$
Here $T=\{1,2,...,i\}$ and $S^T$ is the group
of permutations of the set $T$.
It is not difficult to see that $V\to Free(V)$ is the adjoint functor
we looked for.
\enddemo
\sbh{Remark}
Free $O$-algebras satisfy usual properties; e.g. if $V$ is projective
as an $R$-module then any surjection onto $Free(V)$ admits a section.
\heading 2. Modules over an algebra over an operad \endheading
\subheading{2.0}
If $A$ is an algebra over an operad $O$ we will introduce
the notion of a module over it. Our definition is motivated by a
suggestion of A.Beilinson. As it was said earlier we are dealing
only with algebras of $\Rm$
and hence modules over them will also lie in the category
of $\Rm$,
although they can be defined in a more general context.
\subheading{2.1}
Let $C$ be a \pm category and let $\Tau$ be another category. We say
that $C$ acts on $\Tau$ if $\for$ and $s\in S$ we are given a functor
$$\fss:\Csso\times{\Tau}^o\times\Tau\to\Rm$$
with $F^{\Tau}_{1,1}(\tau,\tau')=Hom(\tau,\tau')$ such that for $\sst$,
for each $\underset{t\neq t'}\to\times B_t\in C^{\times T-t'}$
and for each $\tau\in\Tau$ we
are given a natural transformation between
two functors $\Csso\times{\Tau}^o\times\Tau\to\Rm$ (composition maps): from the
functor
$${\bigotimes}_{t\in T-t'}
\fst(\underset{s\in S_t}\to\times A_s,B_t)
\otimes\fsst(\underset{s\in S_{t'}-s'}\to\times A_s,{\tau}',\tau)
\otimes F^{\Tau}_{T,t'}(\underset{t\neq t'}\to\times B_t,
\tau,{\tau}'')$$ to the functor
$\fss(\underset{s\in S-s'}\to\times A_s,{\tau}',{\tau}'')$
\subheading{2.1.1 Examples}
\subheading{1}Any \pm category $C$ acts on itself
\subheading{2}Let $C$ be as in 1.1.1 and let $\Tau$ be a category equipped with
an action of $C$. Then when we consider $C$ as a \pm category
it will act on $\Tau$ in a natural way:
$$\fss(\underset{s\in S-s'}\to\times A_s,{\tau}_1,{\tau}_2)=
Hom(\underset{s\in S-s'}\to\otimes A_s({\tau}_1),{\tau}_2)$$
\subheading{2.1.2}
For two pairs $C_1,{\Tau}_1$ and $C_2,{\Tau}_2$ of a \pm category
and a category which it acts upon, one defines notions of \pm
functors and \pm natural transformations between \pm functors
as in 1.1.2.
\sbh{2.1.3 Variant}
One can modify the above definitions to the case of unital
\pm categories. Essentially, one needs that the unit object
$\un\in C$ "acts identically" on $\Tau$. In what follows there will
be no difference between operads and unital operads.
\subheading{2.2}
Let now $O$ be an operad and let $O$ act on a category $\Tau$.
We say that $\Tau$ is a model over $O$ if it has essentially
one object.
\subheading{2.2.1}
A model $\Tau$ can be thought of as the following linear algebra
data (analogously to 1.2 ):
\roster
\item A collection of $\Rm$ $\Tau(S,s')$ $\for$ and $s'\in S$
(they correspond to
$\fss(A^{\times(S-s')}\times\tau\times\tau)$ for $A\in O,\tau\in\Tau$)
\item A distinguished element $1\in\Tau(1,1)$
\item Composition maps $\for$ and $s'\in S$
$$\Tau(T,t')\underset{t\neq t'}\to\bigotimes O(S_t)\otimes \Tau(S_{t'},s')
\to\Tau(S,s')$$
\endroster
\subheading{2.3}
Let now $O$ be an operad and let $\Tau$ be a model over it.
A \pm functor from the pair $(O,\Tau)$ to the pair $(\Rm,\Rm)$
is called an $A$-module of type $\Tau$ for $A$ defined by the
underlying functor $O\to\Rm$. Morphisms between $A$-modules
of type $\Tau$ are defined to be natural transformations
between such functors. For any fixed $\Tau$ such modules form
an abelian category.
\subheading{2.3.1}
Again, we can describe $A$-modules of type $\Tau$ in linear
algebra terms:
An $A$-module is an $R$-module $M$ equipped with a system of maps
$\for$ and $s'\in S$
$$\Tau(S,s')\otimes A^{\otimes {S-s'}}\otimes M\to M$$ such that if
$\st$, the following diagram becomes commutative:
$$ \CD
\trrt\otimes\arss\otimes M @>>> \Tau(S,s)\otimes\arss\otimes M \\
@VVV @VVV \\
\Tau(T,t')\otimes A^{\otimes T-t'}\otimes M @>>> M
\endCD $$
\subheading{2.3.2 Example}
When we put $\Tau=O$, our definition coincides with that of
[2]: $\Tau(S,s')=O(S)$,
and for $O=O^{Lie}\text{ (resp. $O^{ass,comm}$)}$ we get
usual $A$-modules (resp. Lie-algebra representations), whereas
for $O=O^{ass}$ we get $A$-bimodules. In the case of the coresponding
unital operads we get modules acted on identically by the unit.
However, by means of varying $\Tau$ the above definition alows
to get modules with an additional structure.
\sbh{2.3.3 Variant}
A \pm functor from the pair $(O,\Tau)$ to the pair (graded $\Rm$, graded
$\Rm$) will be called a graded module over the corresponding graded
algebra. We have the functor $T=shift\ of\ grading$ on the category
of graded modules over a graded algebra.
\sbh{2.4 Free modules}
We will introduce the notion of a free module over
an algebra over an operad. In particular, this will
lead to the [2] construction of the universal enveloping
algebra of an algebra over an operad.
\sbh{2.4.1}
\proclaim{Lemma}
The forgetful functor from $A$-modules of type $\Tau$
to $R$-modules admits a left adjoint.
\endproclaim
\demo{Proof}
Let $U$ be an $R$-module. We will construct an $A$-module $F(U)$,
the free $A$-module on $U$, such that the functor $U\to F(U)$
is the adjoint functor we need. Let us observe first, that
it is sufficient to construct $F(R)$ because then we can
set $F(U)=F(R){\otimes}_R U$
\sbh{Construction}
Set $F'(R)=\oplus_{i=0,1,\dots} F'_i(R)$, where
$$F'_i(R)=(\Tau(\{0,1,\dots,i\},0)\otimes A^{i})_{S^i}.$$
For any $i$ and $j$ we have maps
$$\Tau(\{0,1,\dots,i\},0)\otimes\Tau(\{0,1,\dots,j\},0)\to
\Tau(\{0,1,\dots,i+j\},0)$$
which induce on $F'(R)$ a structure of an associative
algebra with a unit. Consider now for all $i,j$ (non-commutative)
diagrams of the type:
$$ \CD
\trrt\otimes\arss\otimes F'(R) @>>> \Tau(S,s)\otimes\arss\otimes F'(R) \\
@VVV @VVV \\
\Tau(T,t')\otimes A^{\otimes t-1}\otimes F'(R) @>>> F'(R)
\endCD $$
for $(S,s)'=(\{0,1,\dots,i\},0)$ and $(T,t')=(\{0,1,\dots,j\},0)$.
We define $F(R)$ as a quotient of $F'(R)$ by the ideal generated
by the images of $\phi_1-\phi_2$,
where $\phi_1$ and $\phi_2$ are two diagonal $(\searrow)$ maps
in the above diagrams. It is easy to see then, that
$F(R)$ constructed in this way satisfies the properties we need.
\enddemo
\sbh{2.4.2}
Put now $U=R$ and let us donote by $P_A$ the corresponding $A-$module $F(R)$:
$Hom_A(P_A,M)\simeq M$ as an $R$-module, for any $A$-module $M$.
Then $P_A$ has a natural structure of an associative algebra
with a unit, since $P_A\simeq End_A(P_A)$, and the category of $A-$
modules of type $\Tau$ is naturally equivalent to the category
of right $P_A$-modules. For $\Tau=O$ the algebra $P_A$ is
the universal enveloping algebra of $A$ in the terminology of [2].
\sbh{2.4.3}
Let now $B$ be another $O$-algebra and let us have
a homomorphism from $B$ to $A$. Then the obvious restriction
functor from the category of $A$-modules to the category
of $B$-modules admits a left adjoint, called the iduction
functor. Its existence is obvious from the equivalence
of categories of 2.4.2 and the fact that we have an associative
algebra homomorphism $P_B\to P_A$.
\subheading{2.5 Derivations}
{}From now on we will consider modules over an algebra over
an operad with $\Tau=O$ and we will call them just $A$-modules.
\sbh{2.5.1}
Let $A$ be an olgebra over an operad $O$ and let $M$ be an
$A$-module. \newline
An $R$-linear map $\phi:A\to M$ is said to be a derivation
(from $A$ to $M$) if \newline
$\for$ the diagram
$$ \CD
O(S)\otimes\as @>\sre({\phi}^{\otimes S-s}\otimes id)>> \sre(O(S,s)\otimes
A^{\otimes S-s}\otimes M) \\
@VVV @VVV
\\
A @>\phi>> M
\endCD $$
is commutative.
The set of all derivations from $A$ to $M$ will be denoted by
$\Omega (A,M)$
\subheading{2.5.2}
Suppose that $A$ is a free algebra $A=Free(V)$. Then $\Omega(A,M)=
Hom_R(V,M)$ for any $A$-module $M$.
\heading 3. Sheaves and cohomology \endheading
\subheading{3.0}
Starting with an $O$-algebra $A$, we will construct a site $C(A)$.
This definition appeared first in [1] where Quillen proved that
cohomologies of certain sheaves on this site provide correct
cohomology theories for $A$-modules (in the case of commutative,
associative and Lie algebras).
\subheading{3.1}
Let us consider the category $C(A)$ consisting
of $O$-algebras $B$ with a
homomorphism $B\to A$. This category possesses a fibered product
and we introduce a Grothendieck topology on it by declaring
epimorphisms to be the covering maps.
Thus we can consider sheaves on $C(A)$ and their cohomologies.
\subheading{3.2 Sheaves ${\Im}_M$}
Let an $M$ be an $A$-module. Then it is also a module over each
algebra $B\in C(A)$. We define a sheaf $\fm$ on $C(A)$ by setting
$$\Gamma(B,\fm)=\Omega(B,M)$$
Sheaf axioms are easily verified.
\sbh{3.2.1}
The following remark is due to essentially to [1]:
Each $\fm$ is representable by $A_M\in C(A)$, equal to
$A\oplus M$ with the natural algebra structure on it,
in other words $$\Gamma(B,\fm)=\text{algebra homomorphisms }B\to A\oplus M.$$
It is also not difficult to see that the functor $M\to\fm$ is fully
faithful.
\sbh{3.3}
Let us mention several properties of the
category $C(A)$.
\sbh{3.3.1}
If $E\to D$ is a covering in $C(A)$, then the functor \newline
$C(A)_D\text{ (objects of $C(A)$ over $D$)}\to\text
{descent data on $E$ with respect to $D$}$ \newline
is an equivalence of categories.
\sbh{3.3.2}
If $V$ is a projective $R$-module with an $R$-module map to $A$, then \newline
$Free(V)\in C(A)$ and the functor $\Im\to\Gamma(Free(V),\Im)$
is exact. This follows e.g. from Remark 1.4.
\sbh{3.3.3}
\proclaim{Proposition}
The functor $\Im$: ($A$-modules $\to$ sheaves) admits right and left
adjoint functors, $\RR$ and $\LL$ respectively. We have $\RR\circ\Im\simeq
\LL\circ\Im\simeq Id_{A-mod}$.
\endproclaim
\demo{Proof}
For each $X\in C(A)$ consider the sheaf $Const_X$ defined by
$$\Gamma(Y,Const_X)=\underset Hom(Y,X)\to\oplus R.$$
The sheaf $Const_X$ is defined uniquely by the following property:
$Hom(Const_X,S)=\Gamma(X,S)$ functorially in $S-$a sheaf over $C(A)$.
Let $B=Free(V)$. Then $$Hom(Const_{Free(V)},\fm)=\Gamma
(Free(V),\fm)=\Omega(Free(V),M)=Hom_R(V,M).$$
This fact together with 3.3.2 imply that the functor $\Im$ is exact.
In order to construct the functor $\LL$, it is sufficiant
to define the values of $\LL$ on sheaves of the form
$Const_B$ for free algebras $B=Free(V)$, because any sheaf over $C(A)$
is a quotient of a direct sum of such sheaves.
However, the above calculation shows that for these sheaves we can put
$\LL(Const_{Free(V)})=F(V)$ in the notation of 2.4.
To construct the functor $\RR$ we put for a sheaf $\FF$,
$\RR(\FF)=Hom(\Im(P_A),\FF)$ with the obvious structure
of a right module over $P_A$. Then we use 2.4.2.
The fact that $\RR\circ\Im\simeq\LL\circ\Im\simeq Id_{A-mod}$
follows from the full faithfulness of the functor $\Im$.
\enddemo
\sbh{3.3.4}
As always, the functor $\LL$ is right exact and the functor
$\RR$ is left exact and we can consider their left (resp. right)
derived functors $L^{\cdot}\LL$ (resp. $R^{\cdot}\RR$).
(The functor $\LL$ can be derived e.g. because
the sheaves $Const_{Free(V)}$ for projective $V$ are projective (!)
in the category of sheaves of $\Rm$.)
\sbh{3.4}
Let us apply the functor $L^{\cdot}\LL$ to the sheaf $R_A$.
We obtain an object $T_A$ in $D(A)$
(the derived category of $A$-modules). $T_A$ is called the
cotangent complex of $A$.
$$RHom(T_A,M)\simeq R\Gamma(A,{\Im}_M)$$
\sbh{3.4.1}
The fact that $\LL$ is right exact implies that
$$H^i(T_A)=\cases
0, & i>0 \cr
I_A, & i=0
\endcases$$
where $I_A$ is the $A$-module representing the functor
$M\to\Omega(A,M)$.
\sbh{3.4.2 Examples}
\sbh{1} If $O$ is the Lie operad $O^{Lie}$, $I_A$ canonically identifies
with the augmentation ideal of the universal enveloping
algebra.
\sbh{2} If $O=O^{ass,1}$, $I_A\simeq I=ker(A\otimes A\to A)$.
\sbh{3} If $O=O^{com,ass,1}$, $I_A\simeq I/I^2$, with
$I$ is as above.
\heading 4. Deformations \endheading
\sbh{4.0}
Results of this section are partially contained in [1],[3] and [4].
We decided to present them, since the formalism developed in the
preceeding sections seems to be a covenient tool for passage
from deformations to cohomology. By definition, we put
$H^i(A,M)=RHom^i(T_A,M)$.
\sbh{4.1}
Let $A$ be an $O$-algebra (over $R$).
An $i$-th level deformation of $A$ is an $O$ algebra
$A_i$ over $\rti$ (in the sense of 1.2.4) with an isomorphism
$\phi :A_i/t\cdot A_i\simeq A$ and such that
$$Tor_1^{\rti}(A_i,R)=0$$
In other words, we need that
$$ker(t:A_i\to A_i)=im(t^i:A_i\to A_i)\text{ identifies under
a natural map with } A$$
\sbh{4.1.1}
The category of $i$-th level deformations (morphisms are compatible
with $\phi$'s) is a groupoid denoted by $Deform^i(A)$. For each $i$ there
are functors from $Deform^{i+1}(A)$ to $Deform^i(A)$ (taking modulo $t^{i+1}$).
If $A_i\in Deform^i(A)$, the fiber $\Cai$ of $Deform^{i+1}(A)$ over $A_i$
will be called the category of prolongations of $A_i$.
\sbh{4.2}
\proclaim{Theorem}
\roster
\item
The category of $1$-st level deformations is equivalent
to the category $T(\fa)$
of $\fa$-torsors. In particular, $\pi_o(C_1)\simeq
H^1(A,A)$
\item
If $A_{i+1}\in\Cai$, $Aut(A_{i+1})$ as of an object of this
category is canonically isomorphic to $\Omega(A,A)$
\item
To each $A_i\in Deform^i(A)$ we can associate a gerbe $G_{A_i}$ over
$C(A)$ bounded by $\fa$ in such a way that $G_{A_i}$ is canonically
equivalent to $\Cai$.
In particular, this means that to each $A_i$ we can assign
an element in $H^2(A,A)$ that vanishes if and only if
$\Cai$ is nonempty. And if $\Cai$ is nonempty, its $\pi_o$
is a torsor over $H^1(A,A)$.
\endroster
\endproclaim
\demo{Proof}
\roster
\item
The functor $T:C(A)\to T(\fa)$ is given by:
$$\Gamma(B,T(A_1))=O-\text{algebras homomorhisms over R:}B\to A_1$$
Using 3.3.1 it is easy to show that it is an equivalence
of categories.
\item
This is a direct verification.
\item
We define the gerbe as follows:
$G_{A_i}(B)$ is the groupoid of $R[t]/t^{i+2}\cdot R[t]-O$ algebras $B_{i+2}$
with an isomorphism $B_{i+1}/t^{i+1}\cdot B_{i+1}\simeq
B\underset{A}\to\times A_i$ such that \newline
$ker(t^{i+1}:B_{i+1}\to B_{i+1})=im(t:B_{i+1}\to B_{i+1})$ and
identifies under a natural morphism with $A_i$.
Functors $G_{A_i}(B)\to G_{A_i}(D)$ for maps $D\to B$ are given
by taking fibre products.
It is easy to check that $G_{A_i}$ is indeed a gerbe bounded by
$\fa$ over $C(A)$ and that its fiber over $A$ is equivalent to $\Cai$.
\endroster
\enddemo
\sbh{4.2.1} To summarize, we have shown that the cohomology groups
$H^i(A,A)$ "control" the deformation theory of $A$.
\sbh{4.3}
For the remainder of this paper we restrict ourselves
to the case $O=O^{ass,1}$.
\sbh{4.3.1}
When $A$ is flat as an $R$-module, we have a theorem of Quillen [1]:
\proclaim{Theorem}
$H^i(T_A)=0$ for $i\neq 0$.
\endproclaim
By the cohomology long exact sequence of the triple
$$0\to I_A\to A\otimes A\to A\to 0$$ we get that in this case
$H^i(A,M)=\Ei(A,M)$ for any $A$-bimodule $M$ and $i\ge 1$.
\sbh{Variants}
\sbh{4.3.2}
If $A$ is an augmented algebra we can look for its
deformations in the class of augmented algebras.
Then Theorem 4.2 remains valid after replacing $H^{\cdot}(A,A)$ by
$H^{\cdot}(A,A_+)$, where $A_+$ denotes the augmentation ideal of $A$.
\sbh{4.3.3}
Suppose now that $A$ is a graded algebra. Then we will consider
the site $C(A)$ that corresponds to graded algebras over $A$.
If now $M$ is a graded $A$-bimodule, we introduce graded cohomology groups as
$(H^i(A,M))_j=H^i(A,T^j(M))$, with $T$ being the translation functor of 2.3.3.
A graded deformation of $A$ of $i$-th level is an algebra
$A_i$ as above endowed with a grading such that $deg(t)=1$.
Then the Theorem 4.2 is restated in the following way:
\roster
\item
Isomorphism classes of first level deformations are in 1-1
correspondence with the elements of $(H^1(A,A))_{-1}$
\item The automorphism group of a prolongation of an $i$-th
level deformation is naturally isomorphic to $\Omega(A,A)_{-i-1}$
\item The obstruction to the existence of a
prolongation of a given $i$-th level deformation
lies in $(H^2(A,A))_{-i-1}$
\item The set of isomorphism classes of prolongations of
a given $i$-th level deformation is a torsor over $(H^1(A,A))_{-i-1}$
\endroster
In the graded case Theorem 4.3.1 states that
$$(H^i(A,M))_j=(\Ei(A,M))_j$$ for any graded $A$-bimodule
$M$.
\sbh{4.3.3'}
Suppose now that we have a family $A_i$ of graded deformations
which are prolongations of one another. In this case we can form
an algebra
$$A_t=\text{elements of finite degree in }\underset{\longleftarrow}\to {lim}
(A_i)$$
This is an algebra over $R[t]$.
Consider its fiber at $t=1:A_1=A_t/(t-1)\cdot A_t$. This algebra will carry
a natural filtration and the associated graded algebra $gr(A_1)$
will be canonically isomorphic to $A$.
\sbh{4.3.4}
One, of course, can consider a combination of graded and augmented
situations. Statement of Theorem 4.2 will change correspondingly.
\heading 5. The Poincar\'e-Birkhoff-Witt theorem \endheading
\sbh{5.0}
In this section
we will give an application of the theory presented above. We will prove
the \pbw for Lie algebras over a ring $R$ which are flat
as $\Rm$.
\sbh{5.1}
Let us recall main definitions.
\sbh{5.1.1}
Let $\g$ be an $R$-module. Its $i$-th exterior power $\Lamda ^i(\g)$
is defined to be the subspace of $\g ^{\otimes i}$ spanned
by tensors of the form $Alt(g_1,g_2,\dots,g_i)$, where $Alt$
means alternating sum.
\sbh{5.1.2}
$S^{\cdot}(\g)$ will denote the symmetric algebra of $\g$ which
is by definition the quotient of the tensor algebra $T^{\cdot}(\g)$
by the ideal generated by $\Lamda ^2(\g)$. It is a graded algebra
with graded components denoted by $S^i(\g)$.
\sbh{5.1.3}
A Lie algebra structure on $\g$ is a map $[,]:\Lamda ^2(\g)\to\g$
such that the map $[,]\circ (id\otimes [,]-[,]\otimes id):\Lamda ^3
(\g)\to\g$ vanishes.
\sbh{5.1.4}
For a Lie algebra $\g$ its universal enveloping algebra $U(\g)$
is the quotient of the tensor algebra $T^{\cdot}(\g)$ by the
ideal generated by $\omega -[,](\omega)$ for $\omega\in\Lamda ^2(\g)$.
$U(\g)$ caries a natural filtration and there is a canonical
surjection $S^{\cdot}(\g)\to gr(U(\g))$.
\sbh{5.2 The \pbw }
\proclaim{Theorem}
Let $\g$ be a Lie algebra over $R$ which is flat as an $R$-module.
Then the canonical epimorphism $S(\g)\to gr(U(\g)$ is an
isomorphism.
\endproclaim
\sbh{5.2.1}
We will prove this theorem by constructing a graded deformation
as in 4.3.4 of $S(\g)$ in the class of augmented algebras. This idea
is borrowed from [5], where the \pbw is proved
for Koszul algebras also by means of deformation theory.
By 4.3.2 and 4.3.4 we know that in this case the
deformations of $A=S(\g)$ are controled by
$Ext_{A\otimes A}(A,A_{+})$'s.
\sbh{5.2.2}
\proclaim{Proposition}
Let $\g$ be a flat $R$-module.
Then
\roster
\item
$$(Ext_{A\otimes A}^2)_{-j}=\cases
0, & j>1 \cr
Hom_R(\Lamda^2(\g),\g) if & j=1
\endcases$$
\item
$$(Ext_{A\otimes A}^3)_{-j}=\cases
0, & j>2 \cr
Hom_R(\Lamda^3(\g),\g) if & j=2
\endcases $$
\endroster
\endproclaim
This proposition easily follows from the fact
that for flat $\g$-modules the Koszul complex
$$\dots\to S(\g)\otimes\Lamda^i(\g)\otimes S(\g)\to\dots\to
S(\g)\otimes\Lamda^2(\g)\otimes S(\g)\to S(\g)\otimes\g\otimes S(\g)
\to S(\g)$$
is exact.
\sbh{5.3}
\demo{Proof of the theorem}
We put $A=S(\g)$. This is a graded augmented algebra
and by a deformation we will mean a graded deformation in
the class of augmented algebras.
\sbh{Step 1}
Let us build a first level deformation of $A$ that corresponds to
the cohomology class of our $[,]:\Lamda ^2(\g)\to\g$ via 4.3.4
and 5.2.2. It is canonical up to a unique
isomorphism since $\Omega(A,(A_{+})_{-1}=0$.
Untwisting of the passage ( first level deformations $\to$
cohomology) shows that there exists a canonical $R$-linear
isomorphism $\phi:\g\to ((A_1)_{+})_1$ such that
$$\phi(x)\cdot\phi(y)-\phi(y)\cdot\phi(x)=t\phi([x,y])$$
\sbh{Step 2}
A direct verification shows that the obstruction to
the existense of a prolongation of this deformation as
a map $\Lamda ^3(\g)\to\g$ is given by the expression
$[,]\circ (id\otimes [,]-[,]\otimes id)$ which in turn vanishes by the
Jacobi identity.
So a prolongation exists and is unique up to a unique
isomorphism by 4.3.4. and 5.2.2.
\sbh{Step 3}
Again by 4.3.4 and 5.2.2 for any $i\geq 2$ an $i$-th level deformation
of $A$ can be prolonged in a unique up to a unique isomorphism
way and so we find ourselves in the situation of 4.3.3'.
\sbh{Step 4}
\proclaim{Claim}
There exists a canonical $R$-linear map $\phi':\g\to ((A_t)_{+})_1$ such that
$$\phi'(x)\cdot\phi'(y)-\phi'(y)\cdot\phi'(x)=t\phi'[x,y]$$
\endproclaim
Ideed, since $deg(t)=1$, there is a unique way to lift $\phi$
of Step 1 to a map $\g\to ((A_t)_{+})_1$.
Then, by the definition of $U(\g)$ there exists a map
$\phi':U(\g)\to A_1$ that prolongs the map $\phi'$ above.
\sbh{Step 5}
Let us consider the associated graded map
$gr(\phi'):gr(U(\g)\to gr(A_1)$ and let us also consider
the composition
$$S(\g)\underset{\text{by 5.1.4}}\to\to gr(U(\g))\underset
{\phi'}\to\to gr(A_1)\underset{\text{by 4.3.3'}}\to\simeq S(\g)$$
This composition is easily seen to be the identity map. This implies that
\roster
\item
$\phi'$ is an isomorphism between $U(\g)$ and $A_1$
\item
The \pbw.
\endroster
\enddemo
\Refs
\ref
\no 1
\by D.~Quillen
\paper On (co)-homology of commutative rings
\publ Proc. Symp. Pure. Math. \vol 17
\endref
\ref
\no 2
\by V.~Hinich and V.~Schechtman
\paper Homotopy Lie algebras
\publ I.~M.~Gel'gand Seminar \vol 2
\endref
\ref
\no 3
\by L.~Illusie
\book Complex cotangent et deformations
\publ Lect. Notes. in Math.
\endref
\ref
\no 4
\by M.~Schlessinger
\publ PhD Thesis, Harvard, 1965
\endref
\ref
\no 5
\by A.~Braverman and D.~Gaitsgory
\paper The PBW theorem for quadratic algebras of Koszul type
\publ preprint
\endref
\ref
\no 6
\by A.~Beilinson.
\publ private communications
\endref
\endRefs
\enddocument
|
1996-01-22T01:49:59 | 9502 | alg-geom/9502024 | en | https://arxiv.org/abs/alg-geom/9502024 | [
"alg-geom",
"math.AG"
] | alg-geom/9502024 | Yekutieli Amnon | Amnon Yekutieli | Traces and Differential Operators over Beilinson Completion Algebras | AMSLaTex, 37 pages. To appear in Comp. Math. (Hardcopies available on
request from the author.) | null | null | null | null | A Beilinson completion algebra (BCA) A is a complete semilocal algebra over a
perfect field k, whose residue fields are high dimensional local fields. In
addition A is a semi-topological algebra. The completion of the structure sheaf
of an algebraic k-variety along a saturated chain of points is the prototypical
example of a BCA. We single out two kinds of homomorphisms between BCAs:
morphisms and intensifications. The first kind includes residually finite local
homomorphisms, whereas the second kind is a sort of localization. We prove that
every BCA A has a dual module K(A), and these dual modules are contravariant
w.r.t. morphisms and covariant w.r.t. intensifications. For any
semi-topological A-module M we define its dual Dual_{A} M := Hom_{A}^{cont}(M,
K(A)). This duality operation has the remarkable property of respecting
differential operators: given a continuous DO D : M --> N, there is a dual DO
Dual_{A}(D) : Dual_{A} N --> Dual_{A} M. The results above are used (in a
subsequent paper) to construct the Grothendieck residue complex K_{X}^{.} on
any finite type k-scheme X, and to derive many of its properties.
| [
{
"version": "v1",
"created": "Sun, 26 Feb 1995 15:06:25 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Yekutieli",
"Amnon",
""
]
] | alg-geom | \section{Introduction}
In the short paper \cite{Be} A.\ Beilinson introduced a generalized
version of adeles, with values in any quasi-coherent sheaf on a noetherian
scheme $X$. In particular, taking the structure sheaf $\cal{O}_{X}$
one gets the cosimplicial ring of adeles
$\Bbb{A}^{{\textstyle \cdot}}(X, \cal{O}_{X})$. In each degree $n$,
$\Bbb{A}^{n}(X, \cal{O}_{X})$ is a subring (a ``restricted product'')
of the product of local factors
$\prod_{\xi} \cal{O}_{X, \xi}$. Here $\xi = (x_{0}, \ldots, x_{n})$
runs over all chains of length $n$ of points in $X$.
The Beilinson completion $\cal{O}_{X, \xi}$ is gotten by a process of
inverse and direct limits. For $n=0$, $\cal{O}_{X,(x_{0})}$
is simply the $\frak{m}$-adic completion of the local ring at $x_{0}$.
For applications to duality theory one is primarily interested in the
completion $\cal{O}_{X, \xi}$ along a {\em saturated chain} $\xi$.
As shown in \cite{Ye1}, the
semi-local ring $\cal{O}_{X, \xi}$ carries a natural topology, and
its residue fields carry rank $n$ valuations.
In the present paper we isolate the completion $\cal{O}_{X, \xi}$ from its
geometric environment,
and study it as a separate algebraic-topological object, which we call
a {\em Beilinson completion algebra} (BCA).
The methods used here belong to commutative algebra,
analysis and differential geometry. Our main results
have to do with {\em dual modules} of BCAs, their functorial behavior
and their interaction with differential operators. These results, in turn,
have some noteworthy applications to algebraic geometry (see \S 0.3).
One may view our paper partly as a continuation of the work of Lipman,
Kunz and others on explicit formulations of duality theory
(cf.\ \cite{Li1}, \cite{Li2}, \cite{Ku}, \cite{Hu1}, \cite{Hu2},
\cite{HK1}, \cite{HK2},
\cite{HS}, \cite{LS}, \cite{Hg}). Their work deals with {\em linear} aspects
of duality theory - construction of dualizing modules, trace maps etc.
To that we have little new to add in the present paper. The novelty of our
work is in establishing the {\em nonlinear} properties of duality theory.
We show how duality interacts with {\em differential phenomena}, such as
$\cal{D}$-modules and De Rham complexes.
Such results seem to have been beyond the reach of the methods
of commutative algebra used henceforth in this area.
In the remainder of the introduction we outline the content of the paper.
\subsection{Beilinson completion algebras}
Let $k$ be a fixed perfect base field.
A local BCA $A$ is a quotient of a ring
$F((\underline{s})) [\sqbr{ \underline{t} }] =
F(( s_{1}, \ldots, s_{m} )) [\sqbr{ t_{1}, \ldots, t_{n} }]$,
where $F$ is a finitely generated field extension of $k$, and
$F(( s_{1}, \ldots, s_{m} )) = F((s_{m})) \cdots ((s_{1}))$
is an iterated field of Laurent series. $A$ is a complete noetherian local
ring, and a
semi-topological (ST) $k$-algebra. On the residue field $A / \frak{m}$
there is a structure of $m$-dimensional topological local field (TLF).
(These terms are explained briefly in \S 1-2.) The surjection
$F((\underline{s})) [\sqbr{ \underline{t} }] \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} A$ is not part of the structure of $A$.
A general BCA is a finite product of local ones.
We are interested in two kinds of homomorphisms between BCAs. The first is
called a {\em morphism of BCAs}, and the second is called an {\em
intensification homomorphism}. Rather than defining these notions here
(this is done in \S 2-3), we demonstrate them by examples.
Let
$A := k(s)[\sqbr{ t }]$ and
$B := k(s)((t))$. These local BCAs arise geometrically: take
$X := \mathbf{A}^{2}_{k} = \operatorname{Spec} k \sqbr{ s,t }$ and
$x = (0), y = (t), z = (s,t) \in X$.
Then
$A \cong \cal{O}_{X, (y)}$ and
$B \cong \cal{O}_{X, (x, y)}$, the Beilinson completions of $\cal{O}_{X}$
along the chains $(y), (x,y)$ respectively. The inclusion
$A \rightarrow B$ is a morphism, which in ``cosimplicial'' notation is
$\partial^{+} : \cal{O}_{X, (y)} \rightarrow \cal{O}_{X, (x, y)}$.
Now let
$\hat{A} := k((s))[\sqbr{ t }] \cong \cal{O}_{X, (y, z)}$. Then
$A \rightarrow \hat{A}$ is an intensification homomorphism, which we also write
as
$\partial^{-} : \cal{O}_{X, (y)} \rightarrow \cal{O}_{X, (y, z)}$.
Whenever $A \rightarrow B$ is a morphism and $A \rightarrow \hat{A}$ is an intensification,
there is a BCA
$\hat{B} = B \otimes_{A}^{(\wedge)} \hat{A}$, a morphism
$\hat{A} \rightarrow \hat{B}$ and an intensification $B \rightarrow \hat{B}$. This
situation is called {\em intensification base change}. In our example,
$\hat{B} = k((s))((t)) \cong \cal{O}_{X, (x, y, z)}$.
BCAs and morphisms of BCAs constitute a category which is denoted by
$\mathsf{BCA}(k)$.
\subsection{The Results}
There are three main results in the paper. Their precise statement is in
the body of the paper, and what follows is only a sketch.
A {\em finite type ST module} $M$ over a BCA $A$ is a quotient of
$A^{n}$ for some $n$, with the quotient topology (so if $A / \frak{m}$ is
discrete, $M$ has the $\frak{m}$-adic topology.) The {\em fine topology}
on an $A$-module $M$ is characterized by the property that each finitely
generated submodule $M' \subset M$, with the subspace topology,
is of finite type.
(More on ST modules in \S 1.)
Given a TLF $K$ (i.e.\ a BCA which is a field), we denote by
$\omega(K)$ the top degree component of the separated algebra of
differentials
$\Omega_{K/k}^{{\textstyle \cdot}, \mathrm{sep}}$.
\bigskip \noindent
{\bf Theorem \ref{thm6.1}:}\ (Dual modules)\ Let $A$ be a local BCA and $M$
a finite type ST $A$-module. Then there is a {\em dual module}
$\operatorname{Dual}_{A} M$, enjoying the following properties. To any morphism
$\sigma: K \rightarrow A$ in $\mathsf{BCA}(k)$
with $K$ a field, there is a bijection
\[ \Psi_{\sigma}^{M} : \operatorname{Dual}_{A} M \stackrel{\simeq}{\rightarrow}
\operatorname{Hom}_{K; \sigma}^{\mathrm{cont}}(M, \omega(K)). \]
If $\sigma = \tau \circ f$ for some morphisms $f : K \rightarrow L$ and
$\tau : L \rightarrow A$, then
\[ \Psi_{\sigma}^{M}(\phi) = \operatorname{Res}_{L/K} \circ \Psi_{\tau}^{M}(\phi), \]
where
$\operatorname{Res}_{L/K} : \omega(L) \rightarrow \omega(K)$ is the residue on TLFs, see
\cite{Ye1} \S 2.4. If
$\sigma, \sigma' : K \rightarrow A$ are two pseudo coefficients fields (i.e.\
morphisms such that
$[A / \frak{m} : K] < \infty$) which are congruent modulo $\frak{m}$, then
the isomorphism
\[ \Psi_{\sigma, \sigma'}^{M} = \Psi_{\sigma'}^{M} \circ
(\Psi_{\sigma}^{M})^{-1} :
\operatorname{Hom}_{K; \sigma}^{\mathrm{cont}}(M, \omega(K)) \stackrel{\simeq}{\rightarrow}
\operatorname{Hom}_{K; \sigma'}^{\mathrm{cont}}(M, \omega(K)) \]
has an explicit formula in terms of ``Taylor expansions'' and differential
operators.
\bigskip
In particular for $M=A$ we set
$\cal{K}(A) := \operatorname{Dual}_{A} A$, with the fine topology.
$\cal{K}(A)$ is an injective hull of the residue field $A / \frak{m}$.
Note that for a field
$K$, $\cal{K}(K) = \omega(K)$.
If $M$ is any ST $A$-module we define
\[ \operatorname{Dual}_{A} M := \operatorname{Hom}_{A}^{\mathrm{cont}}(M, \cal{K}(A)) \]
with the $\operatorname{Hom}$ topology. (When $M$ is of finite type this is consistent
with Thm.\ \ref{thm6.1}.)
We show that given an intensification homomorphism
$v : A \rightarrow \hat{A}$ there is a continuous homomorphism of ST $A$-modules
\[ q_{\hat{A} / A}^{M} = q_{v}^{M} : \operatorname{Dual}_{A} M \rightarrow
\operatorname{Dual}_{\hat{A}} (\hat{A} \otimes_{A} M). \]
\bigskip \noindent
{\bf Theorem \ref{thm7.2}:}\ (Traces)\
Let $A \rightarrow B$ be a morphism in $\mathsf{BCA}(k)$. Then there exists a
continuous $A$-linear trace map
$\operatorname{Tr}_{B/A} : \cal{K}(B) \rightarrow \cal{K}(A)$. This trace is functorial:
$\operatorname{Tr}_{C/A} = \operatorname{Tr}_{B/A} \circ \operatorname{Tr}_{C/B}$. It induces a
bijection
\[ \cal{K}(B) \stackrel{\simeq}{\rightarrow} \operatorname{Hom}_{A}^{\mathrm{cont}}(B, \cal{K}(A)). \]
The trace
commutes with intensification base change: given an intensification
$A \rightarrow \hat{A}$, and letting
$\hat{B} := B \otimes_{A}^{(\wedge)} \hat{A}$, we have
\[ q_{\hat{A} / A} \circ \operatorname{Tr}_{B / A} =
\operatorname{Tr}_{\hat{B} / \hat{A}} \circ q_{\hat{B} / B}. \]
If $\sigma : K \rightarrow A$ is a morphism with $K$ a field, then
\[ \operatorname{Tr}_{A / K}(\phi) = \Psi_{\sigma}^{A}(\phi)(1) \in \omega(K) \]
for $\phi \in \cal{K}(A)$.
\bigskip \noindent
{\bf Theorem \ref{thm8.1}:}\ (Duals of continuous differential operators)\
Suppose $M, N$ are ST $A$-modules with the fine topologies and
$D : M \rightarrow N$ is a continuous DO. Then there is a continuous DO
\[ \operatorname{Dual}_{A}(D) : \operatorname{Dual}_{A} N \rightarrow \operatorname{Dual}_{A} M. \]
This operation is transitive in $D$ and compatible with intensification
base change $A \rightarrow \hat{A}$. $\operatorname{Dual}_{A}(D)$ is unique, has an explicit
description using the isomorphisms
$\Psi_{\sigma}^{M}, \Psi_{\sigma}^{N}$, and is the adjoint of
$D$ w.r.t.\ suitably defined residue pairings.
\subsection{Applications}
The primary application of our results, and the original motivation of
the paper, is the explicit construction of residue complexes on $k$-schemes.
This is carried out in \cite{Ye2}. The construction is extremely simple,
and we shall sketch it here. Suppose $X$ is a $k$-scheme of finite type
and $(x, y)$ is a saturated chain of points in it (i.e.\ $y$ is an immediate
specialization of $x$). There are natural homomorphisms
$\partial^{-} : \cal{O}_{X, (x)} \rightarrow \cal{O}_{X, (x, y)}$ and
$\partial^{+} : \cal{O}_{X, (y)} \rightarrow \cal{O}_{X, (x, y)}$, the first being
an intensification and the second a morphism (cf.\ example in \S 0.1 above).
According to Theorems \ref{thm6.1} and \ref{thm7.2} we get an
$\cal{O}_{X}$-linear homomorphism
\[ \delta_{(x, y)} : \cal{K}(\cal{O}_{X, (x)})
\exar{q_{\partial^{-}}} \cal{K}(\cal{O}_{X, (x, y)})
\exar{\operatorname{Tr}_{\partial^{+}}} \cal{K}(\cal{O}_{X, (y)}). \]
Considering $\cal{K}(\cal{O}_{X, (x)})$ as a skyscraper sheaf sitting on
$\{ x \}^{-}$, we define
\begin{eqnarray*}
\cal{K}^{{\textstyle \cdot}}_{X} & := & \bigoplus_{x \in X} \cal{K}(\cal{O}_{X, (x)}) \\
\delta_{X} & := & \sum_{(x, y)} \delta_{(x, y)}.
\end{eqnarray*}
Then $(\cal{K}^{{\textstyle \cdot}}_{X}, \delta_{X})$ is the residue complex on $X$
(cf.\ \cite{RD}, \cite{EZ}, \cite{Ye1} and \cite{SY}).
A special feature of this particular construction of $\cal{K}^{{\textstyle \cdot}}_{X}$
is that given a DO
$D : \cal{M} \rightarrow \cal{N}$ between $\cal{O}_{X}$-modules, there is a dual DO
\begin{equation} \label{eqn0.1}
\operatorname{Dual}_{X}(D) : \cal{H}om_{\cal{O}_{X}}(\cal{N}, \cal{K}^{{\textstyle \cdot}}_{X})
\rightarrow \cal{H}om_{\cal{O}_{X}}(\cal{M}, \cal{K}^{{\textstyle \cdot}}_{X})
\end{equation}
which is a homomorphism of complexes.
This implies that $\cal{K}^{{\textstyle \cdot}}_{X}$ is a complex of right
$\cal{D}_{X}$-modules. Conversely, $\cal{D}_{X}$ can be recovered from DOs
acting on $\cal{K}^{{\textstyle \cdot}}_{X}$.
Another consequence of (\ref{eqn0.1}) is that
$\cal{F}^{{\textstyle \cdot} {\textstyle \cdot}}_{X} :=
\cal{H}om_{\cal{O}_{X}}(\Omega^{{\textstyle \cdot}}_{X/k}, \cal{K}^{{\textstyle \cdot}}_{X})$
has a natural structure of double complex.
Using $\cal{F}^{{\textstyle \cdot} {\textstyle \cdot}}_{X}$ we are able to analyze the niveau
spectral sequence converging to
$\mathrm{H}^{\mathrm{DR}}_{{\textstyle \cdot}}(X)$, the algebraic De Rham homology of $X$.
\subsection{Plan of the paper} \blnk{1mm} \\
{\bf Section 1}: a quick review of semi-topological rings and modules, as
well as new facts on ST $\operatorname{Hom}$ modules.
\medskip \noindent
{\bf Section 2}: definition of BCAs and morphisms, including examples.
\medskip \noindent
{\bf Section 3}: definition of intensification homomorphisms, base change.
\medskip \noindent
{\bf Section 4}: general facts on continuous differential operators over ST
algebras; the Lie derivative.
\medskip \noindent
{\bf Section 5}: the structure of the ring of continuous DOs $\cal{D}(K)$
over a TLF $K$; $\omega(K)$ is a right $\cal{D}(K)$-module, and
the action is by adjunction in a suitable sense.
\medskip \noindent
{\bf Section 6}: existence of dual modules is proved.
\medskip \noindent
{\bf Section 7}: contravariance of dual modules w.r.t.\ morphisms is
proved (traces).
\medskip \noindent
{\bf Section 8}: the interaction between dual modules and DOs is examined,
leading to Thm. \ref{thm8.1} and a few corollaries.
\medskip \noindent
{\bf Acknowledgements.}\
I wish to thank J.\ Lipman for his continued interest in this work.
Also thanks to V.\ Lunts for very helpful remarks on differential
operators.
\section{Some Results on Semi-Topological Rings}
Let us recall some definitions and results from \cite{Ye1} \S 1.
A semi-topological (ST) ring is a ring $A$, with a linear topology on
its underlying additive group, such that
for all $a \in A$, left and right multiplication by $a$ are continuous maps
$\lambda_{a}, \rho_{a} : A \rightarrow A$. A
ST left $A$-module is an $A$-module $M$, whose underlying additive group is
linearly topologized, and such that for all $a \in A$ and $x \in M$,
the multiplication maps they define
$\lambda_{a} : M \rightarrow M$ and $\rho_{x} : A \rightarrow M$ are continuous. ST left
$A$-modules and continuous $A$-linear homomorphisms form a category, denoted
$\mathsf{STMod}(A)$. Similarly one defines ST right modules and bimodules.
Assume for simplicity that the ST ring $A$ is commutative. In
$\mathsf{STMod}(A)$ there are direct and inverse limits, and a tensor
product. Given a ST $A$-module $M$, the associated separated module
$M^{\operatorname{sep}} = M / \{ 0 \}^{-}$ is also a ST $A$-module.
The category $\mathsf{STMod}(A)$ is additive, but not abelian. An exact
sequence in it is, by definition, a sequence
$M' \exar{ \phi } M \exar{ \psi } M''$ which is exact in the untopologized
sense (i.e.\ in $\mathsf{Mod}(A)$), and such that both $\phi$ and $\psi$
are strict.
On any $A$-module
$M$ there is a finest topology making it into a ST module; it is called the
fine $A$-module topology. If $M$ has the fine topology, then for any ST
$A$-module $N$, one has
$\operatorname{Hom}_{A}^{\operatorname{cont}}(M,N) = \operatorname{Hom}_{A}(M,N)$,
and this in fact characterizes the fine topology.
Trivially, if $M$ has the fine topology, then so does $M^{\mathrm{sep}}$.
A free ST $A$-module is a free $A$-module with the fine topology. So $F$ is
free iff
$F \cong \bigoplus A$ with the $\bigoplus$ topology. A ST module $M$ has
the fine topology iff it admits a strict surjection $F \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} M$ with $F$
free.
\begin{dfn}
Let $M,N$ be ST $A$-modules. The (weak) $\operatorname{Hom}$ topology on the abelian
group
$\operatorname{Hom}_{A}^{\operatorname{cont}}(M,N)$ is the coarsest linear topology such that
for every $x \in M$, the map
$\rho_{x} : \operatorname{Hom}_{A}^{\operatorname{cont}}(M,N) \rightarrow N$, $\phi \mapsto \phi(x)$,
is continuous.
\end{dfn}
Unless otherwise specified, this is the topology we consider on
$\operatorname{Hom}_{A}^{\operatorname{cont}}(M,N)$. If $M$ has the fine topology, we shall
often drop the superscript ``$\operatorname{cont}$''.
\begin{rem}
A basis of neighborhoods of $0$ for the $\operatorname{Hom}$ topology is the
collection of open subgroups
$\{ V(F,U) \}$, where $F$ runs over the finite subsets of $M$, $U$ runs over
the open subgroups of $N$, and
$V(F,U) = \{ \phi\ |\ \phi(F) \subset U \}$. Such a topology is sometimes
called the weak topology (cf.\ \cite{Ko}). Usually, to obtain a duality one
needs a finer topology - the strong topology of \cite{Ko}, or the
compact-open topology of \cite{Mc}. In the present
paper duality is defined by indirect means, and for our purposes the
weak topology suffices (cf.\ Remark \ref{rem8.1}).
\end{rem}
The next lemma summarizes the properties of the $\operatorname{Hom}$ topology. Its
easy proof is left to the reader.
\begin{lem} \label{lem1.1}
Let $A$ be a commutative ST ring.
\begin{enumerate}
\item Let $\phi : M' \rightarrow M$ and $\psi : N \rightarrow N'$ be homomorphisms in
$\mathsf{STMod}(A)$. Then the induced homomorphism
$\operatorname{Hom}_{A}^{\operatorname{cont}}(M,N) \rightarrow \operatorname{Hom}_{A}^{\operatorname{cont}}(M',N')$
is continuous.
\item Let $M,N$ be ST $A$-modules. Then
$\operatorname{Hom}_{A}^{\operatorname{cont}}(M,N)$ is a ST $A$-module. \linebreak
$\operatorname{End}_{A}^{\operatorname{cont}}(M) = \operatorname{Hom}_{A}^{\operatorname{cont}}(M,M)$ is a ST
$A$-algebra, and $M$ is a ST left
$\operatorname{End}_{A}^{\operatorname{cont}}(M)$-module.
The natural bijection
$M \stackrel{\simeq}{\rightarrow} \operatorname{Hom}_{A}^{\operatorname{cont}}(A,M)$, $x \mapsto \rho_{x}$, is
an isomorphism of ST $A$-modules.
\item Suppose in \rom{(1)} $\phi$ is surjective and $\psi$ is a strict
monomorphism. Then
$\operatorname{Hom}_{A}^{\operatorname{cont}}(M,N) \rightarrow \operatorname{Hom}_{A}^{\operatorname{cont}}(M',N')$
is a strict monomorphism.
\item Let $(M_{\alpha})_{\alpha \in I}$ be a direct system in
$\mathsf{STMod}(A)$, with $I$ a directed set. Then for any ST $A$-module $N$
the natural map
\[ \lim_{\leftarrow \alpha} \operatorname{Hom}_{A}^{\operatorname{cont}}(M_{\alpha},N) \rightarrow
\operatorname{Hom}_{A}^{\operatorname{cont}}(\lim_{\alpha \rightarrow} M_{\alpha}, N) \]
is an isomorphism of ST $A$-modules.
\end{enumerate}
\end{lem}
{}From parts (1) and (2) of the lemma it follows that
$\operatorname{Hom}_{A}^{\operatorname{cont}}$ is an
additive bifunctor
$\mathsf{STMod}(A)^{\circ} \times \mathsf{STMod}(A) \rightarrow \mathsf{STMod}(A)$.
Tensor products are defined in $\mathsf{STMod}(A)$. The usual tensor
product $M \otimes_{A} N$ is given the finest linear topology s.t.\
the maps
$\rho_{y}: M \rightarrow M \otimes_{A} N$, $x' \mapsto x' \otimes y$ and
$\lambda_{x}: N \rightarrow M \otimes_{A} N$, $y' \mapsto x \otimes y'$ are all
continuous (see \cite{Ye1} Def.\ 1.2.11).
\begin{lem} \label{lem1.2}
\rom{(Adjunction)}\ Let $A,B$ be ST rings (not necessarily commutative), let
$L$ be a ST left $A$-module, $N$ a ST left $B$-module, and $M$ a ST
$B$-$A$-bimodule. Then
\[ \operatorname{Hom}_{B}^{\operatorname{cont}}(M \otimes_{A} L, N) \cong
\operatorname{Hom}_{A}^{\operatorname{cont}}(L, \operatorname{Hom}_{B}^{\operatorname{cont}}(M, N)) \]
as topological abelian groups.
\end{lem}
\begin{pf}
Immediate from the definitions of the $\operatorname{Hom}$ and $\otimes$
topologies.
\end{pf}
We say a homomorphism
$\phi: M \rightarrow N$ of topological abelian groups is dense if
$\phi(M) \subset N$ is (everywhere) dense.
\begin{lem} \label{lem1.3}
Suppose $A$ is a ST ring and $M \rightarrow \hat{M}$, $N \rightarrow \hat{N}$ are
continuous dense homomorphisms of ST $A$-modules. Then
$M \otimes_{A} N \rightarrow \hat{M} \otimes_{A} \hat{N}$ is dense.
\end{lem}
\begin{pf}
By transitivity of denseness it suffices to prove that
$M \otimes_{A} N \rightarrow \hat{M} \otimes_{A} N$ is dense. Choose a surjection
from a free module
$A^{(I)} = \bigoplus A$ onto $N$. This induces surjections
$M^{(I)} \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} M \otimes_{A} N$ and
$\hat{M}^{(I)} \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} \hat{M} \otimes_{A} N$. But according to \cite{Ye1}
Prop.\ 1.1.8 (c), $M^{(I)} \rightarrow \hat{M}^{(I)}$ is dense.
\end{pf}
\begin{dfn} \label{def1.1}
Let $A$ be a commutative noetherian ST ring. A ST $A$-module $M$ is called
of {\em finite type}
(resp.\ {\em cofinite type}, resp.\ {\em torsion type}) if it is
finitely generated (resp.\ it is artinian, resp.\
$\operatorname{Supp} M \subset \operatorname{Spec} A$ consists solely of maximal ideals),
and if it has the fine topology.
\end{dfn}
Denote the
full subcategories of $\mathsf{STMod}(A)$ consisting of finite type
(resp.\ cofinite type) modules by
$\mathsf{STMod}_{\operatorname{f}}(A)$ (resp.\ $\mathsf{STMod}_{\operatorname{cof}}(A)$).
Generalizing the Zariski and Artin-Rees properties for noetherian rings with
adic topologies, we make the following definition. Let us point out that
this definition is stronger then \cite{Ye1} Definition 3.2.10.
\begin{dfn} \label{def1.2}
Let $A$ be a noetherian commutative ST ring. $A$ is said to be a {\em
Zariski ST ring} if
\begin{enumerate}
\rmitem{i} Every ST $A$-module, which is either of finite type or of
torsion type, is separated.
\rmitem{ii} Every (continuous) $A$-linear homomorphism between two ST
$A$-modules, each either of finite type or of torsion type, is strict.
\end{enumerate}
\end{dfn}
\begin{prop} \label{prop1.1}
Let $A$ be a local Zariski ST ring, with maximal ideal ${\frak m}$. Assume that
$A \cong \lim_{\leftarrow i} A / {\frak m}^{i+1}$ as ST rings. Let $M,N$ be
ST $A$-modules.
\begin{enumerate}
\item If $M,N$ are both of finite type then so is
$\operatorname{Hom}_{A}^{\mathrm{cont}}(M,N)$.
\item If $M$ is of finite type and $N$ is of cofinite type then
$\operatorname{Hom}_{A}^{\mathrm{cont}}(M,N)$ is of cofinite type.
\item If $M,N$ are both of cofinite type then
$\operatorname{Hom}_{A}^{\mathrm{cont}}(M,N)$ is of finite type.
\end{enumerate}
\end{prop}
\begin{pf}
(1)\ Let $A^{r} \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} M$ be a surjection. By Lemma \ref{lem1.1}
(2) and (3),
$\operatorname{Hom}_{A}^{\mathrm{cont}}(M,N)$ \linebreak
$\hookrightarrow N^{r}$
is a strict monomorphism. Now use the
Zariski property to conclude that $\operatorname{Hom}_{A}(M,N)$ has the fine
topology.
\medskip \noindent (2)\ Like (1).
\medskip \noindent (3)\
Let $M_{i} := \operatorname{Hom}_{A}^{\mathrm{cont}}(A / {\frak m}_{i+1} ,M)$, so
$M_{i} \hookrightarrow M$ is strict, $M_{i}$ has the fine topology, and
$M = \lim_{i \rightarrow} M_{i}$. Similarly define $N_{i}$.
By part (4) of Lemma \ref{lem1.1},
\[ \operatorname{Hom}_{A}^{\mathrm{cont}}(M,N) =
\lim_{\leftarrow i} \operatorname{Hom}_{A}^{\mathrm{cont}}(M_{i}, N) =
\lim_{\leftarrow i} \operatorname{Hom}_{A}^{\mathrm{cont}}(M_{i}, N_{i}). \]
Now $M_{i}$ and
$N_{i}$ are of finite type, so we can we can use part (1) and
\cite{Ye1} Prop.\ 1.2.20.
\end{pf}
\begin{cor} \label{cor1.1}
\rom{(ST Version of Matlis Duality)}\
Let $A$ be as in the proposition. Suppose
$I$ is an injective hull of $A/{\frak m}$, endowed with the fine topology. Then
$\operatorname{Hom}_{A}^{\mathrm{cont}}(-,I)$ is an equivalence
\[ \mathsf{STMod}_{\operatorname{f}}(A)^{\circ} \leftrightarrow
\mathsf{STMod}_{\operatorname{cof}}(A)\ . \]
\end{cor}
\section{Definitions and Basic Properties of BCAs}
In this section $k$ is a fixed perfect field.
If $A$ is a ST $k$-algebra and
$\underline{t} = (t_{1}, \ldots, t_{n})$ is a sequence of indeterminates, we
denote by
$A [\sqbr{\underline{t}}] = A [\sqbr{t_{1}, \ldots, t_{n}}]$
the ring of formal power series, with the topology given by
\[ A [\sqbr{\underline{t}}] = \lim_{\leftarrow i} A \sqbr{\underline{t}} / (\underline{t})^{i},
\]
where for each $i$, $A \sqbr{ \underline{t} } / (\underline{t})^{i}$ has the fine
$A$-module topology.
The ring of Laurent series $A((t))$ is topologized by
\[ A((t)) = \lim_{j \rightarrow} t^{-j} A [\sqbr{t}], \]
and we define recursively
\[ A((\underline{t})) = A((t_{1}, \ldots, t_{n})) :=
A((t_{2}, \ldots, t_{n}))((t_{1})). \]
According to \cite{Ye1} \S 1.3,
$A [\sqbr{\underline{t}}]$ and $A((\underline{t}))$ are ST $k$-algebras.
A {\em topological local field} (TLF) over $k$ is a field $K$, together with
a topology, and valuation rings $\cal{O}_{i}$, $i=1, \ldots, n$,
such that the residue field $\kappa_{i}$ of $\cal{O}_{i}$ is the fraction
field of $\cal{O}_{i+1}$, and
$K = \operatorname{Frac}(\cal{O}_{1})$. These data are related by the existence of
a {\em parametrization}: an isomorphism
$K \cong F(( t_{1}, \ldots, t_{n} ))$ of ST $k$-algebras, s.t.\
$\cal{O}_{i} \cong F(( t_{i+1}, \ldots, t_{n} ))
[\sqbr{ t_{i} }]$. Here $F$ is a discrete field, and
$\Omega^{1}_{F/k}$ has finite rank. The number $n$ is the dimension of
the local field $K$.
Topological local fields constitute a category $\mathsf{TLF}(k)$.
For more details see \cite{Ye1} \S 2.1.
\begin{dfn} \label{dfn2.1}
A local {\em Beilinson completion algebra} (BCA) over $k$
is a commutative semi-topological local ring
$A$, together with a structure of topological local field on the
residue field $A/{\frak m}$. The following
condition must be satisfied: there exists a surjective homomorphism of
$k$-algebras
\[ F(( \underline{s} )) [\sqbr{ \underline{t} }] =
F(( s_{1}, \ldots, s_{m} )) [\sqbr{ t_{1}, \ldots, t_{n} }] \rightarrow A, \]
which is strict (topologically), and induces and isomorphism
of TLFs $F(( \underline{s} )) \cong A/ {\frak m}$. Such a surjection is called a
parametrization of $A$.
A Beilinson completion algebra is a finite product of local BCAs.
\end{dfn}
\begin{rem}
In greater generality one can define a BCA over any noetherian ring $R$,
to be any finite algebra over the $R$-algebra
$\Bbb{A}(\Xi, \cal{O}_{X}) = \prod_{\xi \in \Xi} \cal{O}_{X, \xi}$,
where $\Xi$ is a finite set of saturated chains in some finite type
$R$-scheme $X$, and $\Bbb{A}(- , -)$ is Beilinson's scheme theoretical
group of adeles. See \cite{Be}, \cite{Hr}, \cite{Ye1} and \cite{HY} for
the definition of adeles, and cf.\ Examples \ref{exa2.0} and \ref{exa2.1}
below.
\end{rem}
Observe that a Beilinson completion algebra $A$ is necessarily an
$\frak{r}$-adically complete, noetherian, semi-local ring, where
$\frak{r}$ is the Jacobson radical of $A$.
If $A$ is artinian, then in the terminology of \cite{Ye1}, it is a
cluster of TLFs (a CTLF).
For any ${\frak m} \in \operatorname{Max} A$
set $\operatorname{res.dim}_{{\frak m}} A := \operatorname{dim} A/{\frak m}$, the local field dimension.
We say that $A$ is equidimensional of dimension $n$ if
$\operatorname{res.dim}_{{\frak m}} A =n$ for all ${\frak m}$. In this case we set
$\operatorname{res.dim} A := n$, and
\begin{eqnarray*}
\cal{O}_{i}(A) & := & \prod_{{\frak m} \in \operatorname{Max} A} \cal{O}_{i}(A/{\frak m}) \\
\kappa_{i}(A) & := & \prod_{{\frak m} \in \operatorname{Max} A} \kappa_{i}(A/{\frak m}) \\
\end{eqnarray*}
for $1 \leq i \leq n$. Also we set
$\cal{O}_{0}(A) := A$ and
$\kappa_{0}(A) := A / \frak{r} = \prod A / {\frak m}$.
The motivating example is:
\begin{exa} \label{exa2.0}
Let $X$ be a scheme of finite type over $k$, and let
$\xi = (x_{0}, \ldots, x_{m})$ be a saturated chain in $X$. Then the
Beilinson completion $\cal{O}_{X, \xi}$ of the structure sheaf along $\xi$
is defined; see \cite{Ye1} \S 3.1.
We claim that $\cal{O}_{X, \xi}$ is an equidimensional BCA, of dimension
$m$. To see why, first choose a coefficient field
$\sigma: k(x_{0}) \rightarrow \hat{\cal{O}}_{X, x_{0}} = \cal{O}_{X, (x_{0})}$.
According to \cite{Ye1} Lemma 3.3.9, $\sigma$ extends to a lifting
$\sigma_{\xi} : k(\xi) = k(x_{0})_{\xi} \rightarrow \cal{O}_{X, \xi}$. Sending
$t_{1}, \ldots, t_{n}$ to generators of the maximal ideal
$\frak{m}_{x_{0}}$, we get a strict surjection
$k(\xi) [\sqbr{ t_{1}, \ldots, t_{n} }] \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} \cal{O}_{X, \xi}$. Finally,
according to \cite{Ye1} Prop.\ 3.3.6, $k(\xi)$ is a finite product of
TLFs, all of dimension $m$.
\end{exa}
\begin{exa} \label{exa2.1}
Consider a BCA
$A = F(( s_{1}, \ldots, s_{m} )) [\sqbr{ t_{1}, \ldots, t_{n} }]$. We
claim it is of the form $\cal{O}_{X, \xi}$. Choose an
integral $k$-scheme of finite type $Y$ such that $F = k(Y)$. Set
$X:= \mathbf{A}_{Y}^{n+m} = \mathbf{A}_{k}^{n+m} \times_{k} Y$, and let
$\xi=(x_{0}, \ldots, x_{m})$ be the saturated chain
$x_{i} := (t_{1}, \ldots, t_{n}, s_{1}, \ldots, s_{i})$, where
we write
$\mathbf{A}_{k}^{n+m} = \operatorname{Spec} k[\underline{s},\underline{t}]$. Then
$F((\underline{s}))[[\underline{t}]] \cong \cal{O}_{X,\xi}$ (cf.\ \cite{Ye1} Thm.\ 3.3.2
(c); it can be assumed that $Y$ is normal).
\end{exa}
Let $A$ be a local BCA of $\operatorname{res.dim}$ $n$. For every $1 \leq i \leq n$
there is a subring
$\cal{O}_{1, \ldots, i}(A) \subset A$ defined by
\[ \cal{O}_{1, \ldots, i}(A) \cong
A \times_{A / \frak{m}}
\cal{O}_{1}(A / \frak{m}) \times_{\kappa_{1}(A / \frak{m})} \cdots
\times_{\kappa_{i-1}(A / \frak{m})} \cal{O}_{i}(A / \frak{m}). \]
It is the largest subring of $A$ which projects onto $\kappa_{i}(A)$,
and it is actually the valuation ring of a rank $i$ valuation (hence
local). In \cite{Ye1} the notation
$\cal{O}(A)$ was used for $\cal{O}_{1, \ldots, n}(A)$.
\begin{dfn} \label{dfn2.2} (Morphisms)\
Let $A$ and $B$ be Beilinson completion algebras. A morphism $f: A \rightarrow B$
is a continuous $k$-algebra homomorphism, satisfying the following local
condition.
Given a maximal ideal $\frak{n} \subset B$, let ${\frak m} \subset A$ be the
unique maximal ideal such that $f^{-1}(\frak{n}) \subset {\frak m}$. Set
$i := \operatorname{res.dim} B_{\frak{n}} - \operatorname{res.dim} A_{\frak{m}}$,
which is assumed to be non-negative. Then
$f(A_{\frak{m}}) \subset \cal{O}_{1, \ldots, i}(B_{\frak{n}})$,
the induced homomorphism
$A_{\frak{m}} \rightarrow \kappa_{i}(B/\frak{n})$ sends $\frak{m}$ to $0$,
and
$A/{\frak m} \rightarrow \kappa_{i}(B/\frak{n})$ is a finite morphism of local
fields.
\end{dfn}
The composition of two morphisms is again a morphism, so we get a category,
which is denoted by $\mathsf{BCA}(k)$.
The number $i$ in the definition is called the relative residual dimension of
$f$ at $\frak{n}$, denoted $\operatorname{res.dim}_{\frak{n}} f$. If $f$ is
equidimensional we shall ommit the subscript $\frak{n}$. We call $f$ finite
if $B$ is a finitely generated $A$-module.
Observe that the full subcategory of $\mathsf{BCA}(k)$ consisting of fields
coincides with the full subcategory of $\mathsf{TLF}(k)$ consisting of
TLFs whose last residue field is finitely generated over $k$. (In
characteristic $0$ this is all of $\mathsf{TLF}(k)$.)
Here are some typical examples of morphisms of BCAs.
\begin{exa}
Let $A:= k[\sqbr{s}]$, $B:= k((s))[\sqbr{t}]$, and let $f: A \rightarrow B$ be the
inclusion. Then ${\frak m}=(s)$, $\frak{n} = (t)$, $\operatorname{res.dim}_{{\frak m}} A =0$,
$\operatorname{res.dim}_{\frak{n}} B =1$ and
$\operatorname{res.dim}_{\frak{n}} f = 1$.
\end{exa}
\begin{exa} \label{exa2.2}
Let $X$ be a finite type $k$-scheme, $\xi=(x,\ldots,y)$ a saturated chain in
$X$, $A:= \cal{O}_{X,(y)}$, $B:= \cal{O}_{X,\xi}$, and
$\partial^{+} : \cal{O}_{X,(y)} \rightarrow \cal{O}_{X,\xi}$ the coface map. Now
$\operatorname{res.dim} A = 0$, and
$\operatorname{res.dim} B = \operatorname{res.dim} \partial^{+}$ equals the length of $\xi$.
\end{exa}
\begin{exa}
Let $X,Y$ be finite type $k$-schemes, $f:X \rightarrow Y$ a
$k$-morphism, $y \in Y$ any point and $x$ a closed point in the fibre
$X_{y} := f^{-1}(y)$. Since $k(y) \rightarrow k(x)$ is finite,
$f^{*}: \cal{O}_{Y,(y)} \rightarrow \cal{O}_{X,(x)}$ is a morphism of BCAs, with
$\operatorname{res.dim} f^{*} = 0$.
\end{exa}
\begin{dfn}
Let $A$ be a local BCA over $k$, with maximal ideal ${\frak m}$. A
{\em coefficient field} (resp.\ {\em quasi coefficient field}, resp.\
{\em pseudo coefficient field}) for $A$ is a morphism
$\sigma: K \rightarrow A$ in $\mathsf{BCA}(k)$, with $K$ a field, and such that
the induced homomorphism
$K \rightarrow A/{\frak m}$ is bijective (resp.\ finite separable, resp.\ finite).
\end{dfn}
By definition, every local BCA has a coefficient field.
\begin{lem} \label{lem2.2}
Let $A$ be a local BCA over $k$, with maximal ideal ${\frak m}$. Then:
\begin{enumerate}
\rmitem{a} Suppose $A$ is artinian and $K \rightarrow A$ is a pseudo coefficient
field. Then $A$ has the fine $K$-module topology.
\rmitem{b} Letting $A_{i} := A / {\frak m}^{i+1}$,
the map $A \rightarrow \lim_{\leftarrow i} A_{i}$ is an isomorphism of ST
$k$-algebras.
\rmitem{c} Let $K \rightarrow A$ be a pseudo coefficient field, and let $M$ be a
torsion type ST $A$-module (see Def.\ \ref{def1.1}). Then $M$ is a free
ST $K$-module.
\rmitem{d} Suppose $\sigma: K \rightarrow A$ is a morphism of BCAs, with $K$ a
field. Then there exists a finite morphism
$f: L [\sqbr{ \underline{t} }] \rightarrow A$ extending $\sigma$, i.e.\
$\sigma: K \rightarrow L \rightarrow L [\sqbr{ \underline{t} }] \exar{f} A$.
\end{enumerate}
\end{lem}
\begin{pf}
(a)\ By \cite{Ye1} Prop.\ 2.2.2.
\medskip \noindent(b)\ This is true for $F((\underline{s}))[\sqbr{\underline{t}}]$
(by definition!) and hence, by \cite{Ye1} Prop.\ 1.2.20,
for every quotient $A$.
\medskip \noindent (c)\ Set
$M_{i} := \operatorname{Hom}_{A}(A_{i}, M)$, with the fine $A$-module topology.
According to \cite{Ye1} Cor.\ 1.2.6, $M \cong \lim_{i \rightarrow} M_{i}$. Now
$M_{i}$ is a ST $A_{i}$-module with the
fine topology. Since $A_{i}$ has the fine $K$-module topology, so does
$M_{i}$. Passing to the limit, $M$ has the fine $K$-module topology, so it
is a free ST $K$-module.
\medskip \noindent (d)\
According to \cite{Ye1} Cor.\ 2.1.19 we can find a finite morphism
$K(( \underline{s} )) = L \rightarrow A / \frak{m}$. As in the proof of ibid.\ Prop.\
2.2.2., this extends to a morphism
$L \rightarrow \lim_{\leftarrow i} A_{i} = A$, which we then extend to
$f: L [\sqbr{ \underline{t} }] \rightarrow A$ by sending the $t_{i}$ to generators of
the maximal ideal ideal $\frak{m}$.
\end{pf}
\begin{prop} \label{prop2.1}
Let $A$ be a BCA over $k$. Then:
\begin{enumerate}
\rmitem{a} If $f : A \rightarrow B$ is a finite morphism in $\mathsf{BCA}(k)$,
then $B$ has the fine $A$-module topology.
\rmitem{b} Conversely, if $B$ is a finite $A$-algebra, then $B$ admits
a unique structure of BCA s.t.\ $A \rightarrow B$ is a morphism of BCAs.
\rmitem{c} $A$ is a Zariski ST ring. Moreover, every finite type or torsion
type ST $A$-module is complete.
\end{enumerate}
\end{prop}
\begin{pf}
(a)\
Let $\frak{r} \subset A$ and $\frak{s} \subset B$ be the Jacobson radicals.
According to \cite{Ye1} Prop.\ 2.2.2 (b),
$B_{i} := B/ \frak{s}^{i+1}$ has the fine
$A_{i} := A / \frak{r}^{i+1}$-module topology, for
each $i \geq 0$. So $B_{i}$ also has the fine $A$-module
topology. Now use Lemma \ref{lem2.2} (b) and \cite{Ye1} Prop.\ 1.2.20.
\medskip \noindent (b)\ According to \cite{Ye1} Prop.\ 2.2.2 (c), this is
true for $A_{i} \rightarrow B_{i}$. Now use
$B \cong \lim_{\leftarrow i} B_{i}$.
\medskip \noindent (c)\
It suffices to consider $A = F((\underline{s}))[\sqbr{\underline{t}}]$.
By \cite{Ye1} Thm.\ 3.3.8, $A$ is a Zariski ST ring
in the sense of ibid.\ Def.\ 3.2.10. This means that every finite type ST
$A$-module is separated, and every homomorphism between two such modules
is strict.
Now consider two torsion type ST $A$-modules, $M$ and $N$. We may assume $A$
is local. Choose a pseudo coefficient field $K \rightarrow A$. Then $M,N$ are free ST
$K$-modules, and in particular they are separated and complete (cf.\
\cite{Ye1} Prop.\ 1.5). To prove that any homomorphism $\phi : M \rightarrow N$
is strict, we may assume it is injective. Then any $K$-linear splitting
$M \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} N$ is continuous, showing that $\phi$ is strict.
Finally, given a homomorphism $\phi : M \rightarrow N$, with $M,N$ either of
finite type or of torsion type, then the module
$\bar{M} := \phi(M)$, endowed with the fine topology, is a ST module of both
types. Therefore $M \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} \bar{M}$ and $\bar{M} \hookrightarrow N$ are both strict.
\end{pf}
\section{Intensification Base Change}
The operation of base change to be discussed in this subsection
is a generalization of the one in \cite{Ye1} \S 2.2.
The important notion is that of an intensification homomorphism
$u : A \rightarrow \hat{A}$ between two BCAs (Def.\ \ref{dfn3.2}). Differentially
$u$ is ``\'{e}tale``: the differential invariants of $\hat{A}$ descend
to $A$. From the point of view of valuations, $\hat{A}$ is like a completion
of $A$.
Again $k$ is a fixed perfect field.
\begin{dfn} \label{dfn3.1}
Let $A,\hat{A} \in \mathsf{BCA}(k)$ have Jacobson radicals
$\frak{r},\hat{\frak{r}}$
respectively, and let $u: A \rightarrow \hat{A}$ be a continuous $k$-algebra
homomorphism, with $u(\frak{r}) \subset \hat{\frak{r}}$.
\begin{enumerate}
\rmitem{a} $u$ is called {\em radically unramified} if
$\hat{\frak{r}} = \hat{A} \cdot u(\frak{r})$.
\rmitem{b} $u$ is called {\em finitely ramified} if
$\hat{A} / \hat{A} \cdot u(\frak{r})$ is artinian, and if for every
$\hat{{\frak m}} \in \operatorname{Max} \hat{A}$ lying over some
${\frak m} \in \operatorname{Max} A$, letting $n := \operatorname{res.dim} \hat{A} / \hat{\frak{m}}$,
the image of $(A/{\frak m})^{\times}$ in the rank $n$ valuation group of
$\hat{A} / \hat{{\frak m}}$
has finite index.
\end{enumerate}
\end{dfn}
\begin{prop} \label{prop3.1}
\rom{(Finitely Ramified Base Change)}\
Let $K, \hat{K}, A \in \mathsf{BCA}(k)$, with $A$ a local ring and
$K, \hat{K}$ fields. Suppose $f : K \rightarrow A$ is a morphism in
$\mathsf{BCA}(k)$ and $u: K \rightarrow \hat{K}$ is a finitely ramified homomorphism.
Then there exists a BCA $\hat{A}$, a morphism
$\hat{f} : \hat{K} \rightarrow \hat{A}$ in $\mathsf{BCA}(k)$, and a finitely
ramified homomorphism $v: A \rightarrow \hat{A}$, satisfying:
\begin{enumerate}
\rmitem{i} $v \circ f = \hat{f} \circ u$, and moreover the homomorphism
$A \otimes_{K} \hat{K} \rightarrow \hat{A}$ is dense.
\rmitem{ii} $\operatorname{res.dim} \hat{f} = \operatorname{res.dim} f$.
\rmitem{iii} Suppose $\hat{g} : \hat{K} \rightarrow \hat{C}$ is a morphism in
$\mathsf{BCA}(k)$, with $\hat{C}$ local, and let
$n:= \operatorname{res.dim} \hat{g} - \operatorname{res.dim} \hat{f}$. Suppose also
$w: A \rightarrow \hat{C}$ is a continuous homomorphism s.t.\
$w \circ f = \hat{g} \circ u$,
$w(A) \subset \cal{O}_{1, \ldots, n}(\hat{C})$, and
$A \rightarrow \kappa_{n}(\hat{C})$ is finitely ramified.
Then there exists a unique morphism
$\hat{h} : \hat{A} \rightarrow \hat{C}$ (of $\operatorname{res.dim}$ $n$)
in $\mathsf{BCA}(k)$, such that
$\hat{g} = \hat{h} \circ \hat{f}$ and
$w = \hat{h} \circ v$.
\end{enumerate}
\end{prop}
\begin{pf}
Choose a finite morphism
$K(( \underline{s} ))[\sqbr{ \underline{t} }] \rightarrow A$ (cf.\ Lemma \ref{lem2.2}), and set
\[ \hat{A} := A \otimes_{K(( \underline{s} ))[\sqbr{ \underline{t} }]}
\hat{K}(( \underline{s} ))[\sqbr{ \underline{t} }]. \]
$\hat{A}$ is a BCA by Prop.\ \ref{prop2.1}, and $\hat{f}, v$ are the
obvious maps.
Let us prove that $A \otimes_{K} \hat{K} \rightarrow \hat{A}$ is dense.
Denoting by
$K \sqbr{ \underline{s}, \underline{s}^{-1} }$ the ring of Laurent polynomials,
we have
$\hat{K} \otimes_{K} A \cong
\hat{K} \sqbr{ \underline{s}, \underline{s}^{-1}, \underline{t} }
\otimes_{K \sqbr{ \underline{s}, \underline{s}^{-1}, \underline{t} }} A$.
By \cite{Ye1} Lemma 1.3.9 the homomorphism
$\hat{K} \sqbr{ \underline{s}, \underline{s}^{-1} } \rightarrow \hat{K}(( \underline{s} ))$ is dense,
and a similar argument shows that so is
$\hat{K} \sqbr{ \underline{s}, \underline{s}^{-1}, \underline{t} } \rightarrow
\hat{K}(( \underline{s} ))[\sqbr{ \underline{t} }]$. Now use Lemma \ref{lem1.3}.
Finally, given $\hat{C}$, the arguments in the proof of \cite{Ye1}
Thm.\ 2.2.4 imply there is a morphism
$\hat{K}(( \underline{s} ))[\sqbr{ \underline{t} }] \rightarrow \hat{C}$, and tensoring with
$A$ we get
$\hat{h}: \hat{A} \rightarrow \hat{C}$. Uniqueness follows from the denseness of
$A \otimes_{K} \hat{K} \rightarrow \hat{A}$ .
\end{pf}
The algebra $\hat{A}$ in the proposition is unique (up to a unique
isomorphism). We shall denote it by
\begin{equation} \label{eqn3.2}
\hat{A} = A \otimes^{(\wedge)}_{K} \hat{K}\ .
\end{equation}
In contrast with the usual tensor product, this is not a symmetric
expression - we shall always put the algebra which is the range of the
finitely ramified homomorphism to the right.
In \cite{Ye1} \S 1.5 the notion of a topologically \'{e}tale homomorphism
relative to $k$ was defined. A homomorphism
$v: A \rightarrow \hat{A}$ in $\mathsf{STComAlg}(k)$, the category of commutative ST
$k$-algebras, is called topologically \'{e}tale
relative to $k$ if for any separated ST $\hat{A}$-module $\hat{M}$, any
continuous $k$-linear derivation $\partial : A \rightarrow \hat{M}$ has a
unique extension to a continuous derivation
$\hat{\partial} : \hat{A} \rightarrow \hat{M}$. Often we shall suppress the phrase
``relative to $k$''; this should not cause any confusion as we have no
notion of absolute topologically \'{e}tale homomorphism.
\begin{lem} \label{lem3.1} \mbox{ }
\begin{enumerate}
\rmitem{a} The homomorphism
$v: A \rightarrow \hat{A} = A \otimes^{(\wedge)}_{K} \hat{K}$ is flat.
\rmitem{b} If $u : K \rightarrow \hat{K}$ is topologically \'{e}tale relative to $k$,
then $v: A \rightarrow \hat{A}$ is topologically \'{e}tale and radically
unramified.
\end{enumerate}
\end{lem}
\begin{pf}
(a)\ We have
$\hat{A} \cong A \otimes_{K((\underline{s}))[\sqbr{\underline{t}}]}
\hat{K}((\underline{s}))[\sqbr{\underline{t}}]$.
According to \cite{CA} Ch.\ III \S 5.4 Prop.\ 4, the homomorphism
$K((\underline{s}))[\sqbr{\underline{t}}] \rightarrow \hat{K}((\underline{s}))[\sqbr{\underline{t}}]$
is flat; hence so is $A \rightarrow \hat{A}$.
\medskip \noindent (b)\
As in the proof of \cite{Ye1} Thm.\ 2.4.23,
$K((\underline{s}))[\sqbr{\underline{t}}] \rightarrow \hat{K}((\underline{s}))[\sqbr{\underline{t}}]$
is topologically \'{e}tale. By \cite{Ye1} Prop.\ 1.5.9 (b), so is
$A \rightarrow \hat{A}$. The ring
$\hat{A} / \hat{A} \cdot v({\frak m}) \cong A / {\frak m} \otimes_{K((\underline{s}))}
\hat{K}((\underline{s}))$
is reduced, since
$K((\underline{s})) \rightarrow \hat{K}((\underline{s}))$ is separable (cf.\ proof of \cite{Ye1}
Thm.\ 2.4.23). This shows that $\hat{A} \cdot v({\frak m})$ is the Jacobson
radical of $\hat{A}$.
\end{pf}
Let $A,\hat{A}$ be two local BCAs, with maximal ideals ${\frak m},\hat{{\frak m}}$
respectively. Suppose $v: A \rightarrow \hat{A}$ is a finitely ramified, radically
unramified homomorphism. Let $\sigma : K \rightarrow A$ be a pseudo coefficient field,
and assume there is some subfield
$\hat{K} \subset \hat{A} / \hat{{\frak m}}$ such that
$K \rightarrow \hat{K}$ is topologically \'{e}tale relative to $k$, and
$A/ {\frak m} \otimes_{K} \hat{K} \rightarrow \hat{A} / \hat{{\frak m}}$ is bijective.
Then $\hat{K} \rightarrow \hat{A} / \hat{{\frak m}}$ is finite, and
$K \rightarrow \hat{K}$ is finitely ramified. Also, this $\hat{K}$ is unique.
Since some
lifting $\hat{K} \rightarrow \hat{A}$ exists, there is a unique pseudo coefficient
field
\begin{equation} \label{eqn3.1}
\hat{\sigma} : \hat{K} \rightarrow \hat{A}
\end{equation}
extending $\sigma$ (cf.\ \cite{Ye1} formula (4.1.11)).
\begin{exa} \label{exa3.1}
If $v: A \rightarrow \hat{A}$ is topologically \'{e}tale and
$K \rightarrow A/{\frak m}$ is purely inseparable, then such a subfield $\hat{K}$ exists.
Indeed, we have
$\hat{A} / \hat{{\frak m}} = \hat{A} \otimes_{A} A/{\frak m}$, so
$A / {\frak m} \rightarrow \hat{A} / \hat{{\frak m}}$ is also topologically
\'{e}tale. If $\sigma$ is a coefficient field, the statement is trivial.
Otherwise, see \cite{Ye1} formula (4.1.10).
\end{exa}
We make
$\operatorname{gr}_{{\frak m}} A = \bigoplus_{i \geq 0} {\frak m}^{i} / {\frak m}^{i+1}$
into a graded ST ring by putting on
${\frak m}^{i} \subset A$ the subspace topology, and putting on
${\frak m}^{i} / {\frak m}^{i+1}$ the quotient topology. Similarly, for a ST $A$-module
$M$, $\operatorname{gr}_{{\frak m}} M$ is a graded ST $\operatorname{gr}_{{\frak m}} A$-module.
\begin{prop} \label{prop3.2}
In the situation above, suppose that $v : A \rightarrow \hat{A}$ is flat. Then:
\begin{enumerate}
\rmitem{a} For any finite type ST $A$-module $M$ which has finite length,
the canonical homomorphism
\[ \hat{K} \otimes_{K} M \rightarrow \hat{A} \otimes_{A} M \]
is an isomorphism of ST $\hat{K}$-modules.
\rmitem{b} The canonical morphism
$A \otimes^{(\wedge)}_{K} \hat{K} \rightarrow \hat{A}$ in $\mathsf{BCA}(k)$
is an isomorphism.
\rmitem{c} For any finite type ST $A$-module $M$,
the canonical homomorphism
\[ \hat{K} \otimes_{K} \operatorname{gr}_{{\frak m}} M \rightarrow \operatorname{gr}_{{\frak m}} (\hat{A}
\otimes_{A} M) \]
is an isomorphism of graded ST $\hat{K}$-modules.
\end{enumerate}
\end{prop}
\begin{pf}
(a)\ The proof is by induction on the length of $M$. For $M$ of length $1$,
we have by assumption
\[ \hat{K} \otimes_{K} M \cong \hat{K} \otimes_{K} A/{\frak m} \cong
\hat{A} / \hat{{\frak m}} \cong \hat{A} \otimes_{A} M\ . \]
Otherwise, we can find an exact sequence (of untopologized $A$-modules)
\[ M^{{\textstyle \cdot}} = ( 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 ) \]
which gives rise, by flatness, to a homomorphism of exact sequences
$\hat{K} \otimes_{K} M^{{\textstyle \cdot}} \rightarrow \hat{A} \otimes_{A} M^{{\textstyle \cdot}}$. By
induction and the Five Lemma, we conclude that
$\hat{K} \otimes_{K} M \cong \hat{A} \otimes_{A} M$. Since both modules have
the fine $\hat{K}$-module topologies, this is a homeomorphism.
\medskip \noindent (b)\
We have
$A \otimes_{K}^{(\wedge)} \hat{K} \cong \lim_{\leftarrow i}
A / {\frak m}^{i+1} \otimes_{K} \hat{K}$, and by Lemma \ref{lem2.2} (b),
$\hat{A} \cong \lim_{\leftarrow i} \hat{A} / \hat{{\frak m}}^{i+1}$.
Now use part (a) above, together with the isomorphism
$\hat{A} \otimes_{A} (A / {\frak m}^{i+1}) \cong \hat{A} / \hat{{\frak m}}^{i+1}$.
\medskip \noindent (c)\
By flatness and the fact that $A$ and $\hat{A}$ are Zariski ST rings, it
follows that
$\hat{A} \otimes_{A} {\frak m}^{i} M \cong {\frak m}^{i} (\hat{A} \otimes_{A} M)
\subset \hat{A} \otimes_{A} M$ as ST $\hat{A}$-modules. Therefore
$\hat{A} \otimes_{A} (\operatorname{gr}_{{\frak m}} M)_{i} \cong
\operatorname{gr}_{{\frak m}} (\hat{A} \otimes_{A} M)_{i}$
as ST $\hat{A} / \hat{{\frak m}}$-modules. Now use part (a).
\end{pf}
\begin{dfn} \label{dfn3.2} \rom{(Intensification)}\
Let $u: A \rightarrow \hat{A}$ be a continuous $k$-algebra homomorphism between
two BCAs. If $u$ is flat, finitely ramified, radically unramified and
topologically \'{e}tale relative to $k$, then $u$ is called an
{\em intensification} homomorphism.
\end{dfn}
\begin{exa} \label{exa3.2}
Let $X$ be a finite type $k$-scheme, $\xi=(x,\ldots,y)$ a saturated chain in
$X$, $A:= \cal{O}_{X,(x)}$, $B:= \cal{O}_{X,\xi}$, and
$\partial^{-} : \cal{O}_{X,(x)} \rightarrow \cal{O}_{X,\xi}$ the coface map. Then
$\partial^{-}$ is an intensification homomorphism (cf.\ Example
\ref{exa2.2}).
\end{exa}
\begin{thm} \label{thm3.1} \rom{(Intensification Base Change)}\
Let $A, \hat{A}, B$ be local BCAs, let $f : A \rightarrow B$ be a morphism
in $\mathsf{BCA}(k)$, and let $u : A \rightarrow \hat{A}$ be an intensification
homomorphism. Then there is a BCA
$\hat{B} = B \otimes_{A}^{(\wedge)} \hat{A}$, a morphism
$\hat{f} : \hat{A} \rightarrow \hat{B}$ and an intensification homomorphism
$v : B \rightarrow \hat{B}$, satisfying conditions \rom{(i) - (iii)} of Prop.\
\ref{prop3.1} \rom{(}but replacing the letters $K,A$ with $A,B$\rom{)}.
\end{thm}
\medskip \noindent
\begin{equation} \label{eqn3.3}
\setlength{\unitlength}{0.25mm}
\begin{array}{ccc}
B & \lrar{v} & \hat{B} \\
\luar{f} & & \luar{\hat{f}} \\
A & \lrar{u} & \hat{A}
\end{array}
\end{equation}
\begin{pf}
Choose a coefficient field $\sigma : K \rightarrow A$, and let
$\hat{\sigma} : \hat{K} = K \otimes_{A} \hat{A} \rightarrow \hat{A}$ be its unique
extension. So $\hat{A} \cong A \otimes_{K}^{(\wedge)} \hat{K}$. Set
$\hat{B} := B \otimes_{K}^{(\wedge)} \hat{K}$. We can find a surjective
morphism
$K[\sqbr{\underline{t}}] \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} A$, and it gives
$\hat{A} \cong A \otimes_{K[\sqbr{\underline{t}}]} \hat{K}[\sqbr{\underline{t}}] $.
The homomorphism
$\hat{K}\sqbr{\underline{t}} \rightarrow B$ extends uniquely to a morphism
$\hat{K}[\sqbr{\underline{t}}] \rightarrow B$ : define it inductively into
$\cal{O}_{i}(B)$, $i= \operatorname{res.dim} f, \ldots, 2, 1$. Hence
$\hat{f} : \hat{A} \rightarrow \hat{B}$ is also defined. The uniqueness of $\hat{B}$
is clear from its construction.
\end{pf}
\begin{exa}
Let
$A := k(s)[\sqbr{ t }]$,
$\hat{A} := k((s))[\sqbr{ t }]$ and
$B := k(s)((t))$, so the inclusion $A \rightarrow \hat{A}$
(resp.\ $A \rightarrow B$) is an intensification (resp.\ a morphism). We then have
\[ k(s)((t)) \otimes_{k(s)[\sqbr{ t }]}^{(\wedge)} k((s))[\sqbr{ t }]
\cong k((s))((t)). \]
\end{exa}
\begin{prop} \label{prop3.3} \rom{(Associativity)}\
Say
$C \leftarrow B \rightarrow \Hat{B} \leftarrow \Hat{A} \rightarrow \Hat{\Hat{A}}$
are BCAs and homomorphisms, where the ``$\leftarrow$'' are morphisms,
and the ``$\rightarrow$'' are intensifications. Then there is a canonical
isomorphism of BCAs
\[ (C \otimes_{B}^{(\wedge)} \Hat{B}) \otimes_{\Hat{A}}^{(\wedge)}
\Hat{\Hat{A}} \cong
C \otimes_{B}^{(\wedge)}
(\Hat{B} \otimes_{\Hat{A}}^{(\wedge)} \Hat{\Hat{A}}). \]
\end{prop}
\begin{pf}
Set
$\Hat{\Hat{B}} := \Hat{B} \otimes_{\Hat{A}}^{(\wedge)} \Hat{\Hat{A}}$
and
$\Hat{C} := C \otimes_{B}^{(\wedge)} \Hat{B}$.
By construction (cf.\ Prop.\ \ref{prop3.1}) we get an intensification
homomorphism
$\Hat{C} \rightarrow C \otimes_{B}^{(\wedge)} \Hat{\Hat{B}}$, and together with
the morphism
$\Hat{\Hat{A}} \rightarrow \Hat{\Hat{B}} \rightarrow
C \otimes_{B}^{(\wedge)} \Hat{\Hat{B}}$
we deduce, using Cor. \ref{thm3.1}, the existence of a morphism
$h: \Hat{C} \otimes_{\Hat{A}}^{(\wedge)} \Hat{\Hat{A}} \rightarrow
C \otimes_{B}^{(\wedge)} \Hat{\Hat{B}}$.
The same corollary says there is a morphism
$\Hat{\Hat{B}} \rightarrow \Hat{C} \otimes_{\Hat{A}}^{(\wedge)} \Hat{\Hat{A}}$,
and together with the intensification
$C \rightarrow \Hat{C} \otimes_{\Hat{A}}^{(\wedge)} \Hat{\Hat{A}}$
we get a morphism
$h': C \otimes_{B}^{(\wedge)} \Hat{\Hat{B}} \rightarrow
\Hat{C} \otimes_{\Hat{A}}^{(\wedge)} \Hat{\Hat{A}}$. Since the maps
from
$C \otimes_{B} \Hat{B} \otimes_{\Hat{A}} \Hat{\Hat{A}}$
to both these BCAs are dense, $h$ and $h'$ must be each other's inverse.
\end{pf}
\section{Continuous Differential Operators}
We begin with some general results on continuous differential
operators (DOs) over ST algebras. Let $k$ be a discrete commutative ring,
let $A$
be a commutative, separated, ST $k$-algebra, and let $M$ be a separated ST
$A$-module. For $n \geq 0$, the separated module of principal parts
$\cal{P}_{A/k}^{n,\operatorname{sep}}$ is the ST $A$-$A$-bimodule
$(A \otimes_{k} A / I^{n+1})^{\operatorname{sep}}$, where
$I := \operatorname{ker}(A \otimes_{k} A \rightarrow A)$.
Set
$\cal{P}_{A/k}^{n,\operatorname{sep}}(M) :=
(\cal{P}_{A/k}^{n,\operatorname{sep}} \otimes_{A} M)^{\operatorname{sep}}$, which is an
$A$-module by
$a \cdot (( 1 \otimes 1) \otimes x) = (a \otimes 1) \otimes x$.
The universal continuous DO of order $n$ is
$\operatorname{d}_{M}^{n} : M \rightarrow \cal{P}_{A/k}^{n,\operatorname{sep}}(M)$,
$\operatorname{d}_{M}^{n}(x) = (1 \otimes 1) \otimes x$
(see \cite{EGA} Ch.\ IV \S 16.8 and \cite{Ye1} \S 1.5).
For any separated ST $A$-module $N$, $\operatorname{d}_{M}^{n}$ induces a bijection
\[ \operatorname{Hom}_{A}^{\operatorname{cont}}(\cal{P}_{A/k}^{n,\operatorname{sep}}(M),N) \cong
\operatorname{Diff}_{A/k}^{n,\operatorname{cont}}(M,N)\ . \]
There are inclusions
\[ \operatorname{Diff}_{A/k}^{n,\operatorname{cont}}(M,N) \subset
\operatorname{Diff}_{A/k}^{\operatorname{cont}}(M,N) \subset
\operatorname{Hom}_{k}^{\operatorname{cont}}(M,N)\ . \]
$\operatorname{Diff}_{A/k}^{\operatorname{cont}}(M,N)$ is a filtered $A$-$A$-bimodule, where for
$D \in \operatorname{Diff}_{A/k}^{\operatorname{cont}}(M,N)$ and $a,b \in A$ we have
$a D b = a \circ D \circ b : M \rightarrow N$.
Denote the order of the DO $D$ by $\operatorname{ord}_{A}(D)$.
\begin{dfn} \label{4.1}
Given a separated ST $A$-module $M$, let
$\cal{D}(A;M) := \operatorname{Diff}_{A/k}^{\operatorname{cont}}(M,$ \linebreak
$ M)$,
which is a filtered $k$-algebra.
For $M=A$ we shall write simply
$\cal{D}(A) := \cal{D}(A;A)$.
\end{dfn}
Denote the left action of $\cal{D}(A;M)$ on $M$ by $D * x$,
for $D \in \cal{D}(A;M)$ and $x \in M$.
\begin{rem}
$\cal{D}(A)$ can be made into a ST $k$-algebra by giving it the subspace
topology w.r.t. the embedding
$\cal{D}(A) \subset \operatorname{End}_{k}^{\operatorname{cont}}(A)$.
However we shall not make use of this topology.
\end{rem}
\begin{lem} \label{lem4.3}
Assume that for some $n \geq 0$,
$\cal{P}_{A/k}^{n,\operatorname{sep}}$ is a finite type ST left $A$-module. If
$M$ is a finite type ST $A$-module, then so is
$\cal{P}_{A/k}^{n,\operatorname{sep}}(M)$.
\end{lem}
\begin{pf}
First note that
$\cal{P}_{A/k}^{n,\operatorname{sep}}$ is a commutative ST ring, admitting two
continuous $k$-algebra homomorphisms
$A \rightarrow \cal{P}_{A/k}^{n,\operatorname{sep}}$. By \cite{Ye1} Cor.\ 4.5,
$\cal{P}_{A/k}^{n,\operatorname{sep}} \otimes_{A} M$ is a finite type ST
$\cal{P}_{A/k}^{n,\operatorname{sep}}$-module. The left $A$-module structure on
$\cal{P}_{A/k}^{n,\operatorname{sep}}$ comes from the algebra homomorphism
$a \mapsto a \otimes 1$. From \cite{Ye1} Prop.\ 2.9 and our
assumption it follows that
$\cal{P}_{A/k}^{n,\operatorname{sep}} \otimes_{A} M$ has the fine $A$-module
topology. But then the same is true for
$\cal{P}_{A/k}^{n,\operatorname{sep}}(M) =
(\cal{P}_{A/k}^{n,\operatorname{sep}} \otimes_{A} M)^{\operatorname{sep}}$.
\end{pf}
Define
\[ \cal{T}(A) := \operatorname{Der}^{\operatorname{cont}}_{k}(A,A) = \operatorname{Hom}^{\operatorname{cont}}_{A}
(\Omega^{1,\operatorname{sep}}_{A/k},A)\ . \]
Corresponding to the decomposition
$\cal{P}_{A/k}^{1,\operatorname{sep}} = A \oplus \Omega^{1,\operatorname{sep}}_{A/k}$
we have
$A \oplus \cal{T}(A) = \cal{D}^{1}(A) \subset \cal{D}(A)$,
and just like in the discrete case, $\cal{T}(A)$ is a Lie algebra over $k$.
\begin{lem} \label{lem4.4}
Suppose $\hat{A}$ is another commutative, separated, ST $k$-algebra, and
$u: A \rightarrow \hat{A}$ is a topologically \'{e}tale homomorphism relative to $k$.
Then there is an induced homomorphism of filtered $k$-algebras
$\cal{D}(A) \rightarrow \cal{D}(\hat{A})$, sending an operator
$D : A \rightarrow A$ to its unique extension
$\hat{D} : \hat{A} \rightarrow \hat{A}$. More generally, if $M$ is a ST
$A$-module, there is a homomorphism
$\cal{D}(A; M^{\operatorname{sep}}) \rightarrow
\cal{D}(\hat{A}; (\hat{A} \otimes_{A} M)^{\operatorname{sep}})$.
\end{lem}
\begin{pf}
The existence and uniqueness of this ring homomorphism are immediate
consequences of \cite{Ye1} Thm.\ 1.5.11 (iv).
\end{pf}
The ring homomorphism
$\cal{D}(A) \rightarrow \cal{D}(\hat{A})$ restricts to a Lie algebra homomorphism
$\cal{T}(A) \rightarrow \cal{T}(\hat{A})$.
\begin{prop} \label{prop4.1}
Let $A,\hat{A}$ be separated ST $k$-algebras, and
let $u : A \rightarrow \hat{A}$ be a flat, topologically \'{e}tale homomorphism
relative to $k$. Assume that for every $n \geq 0$,
$\cal{P}_{A/k}^{n,\operatorname{sep}}$ is a finitely presented, finite type ST left
$A$-module.
Then the homomorphism
$\cal{D}(A) \rightarrow \cal{D}(\hat{A})$ induces an isomorphism of
filtered $\hat{A}$-$\cal{D}(A)$-bimodules
\[ \hat{A} \otimes_{A} \cal{D}(A) \stackrel{\simeq}{\rightarrow} \cal{D}(\hat{A})\ . \]
\end{prop}
\begin{pf}
Since
$\otimes$ commutes with $\lim_{\rightarrow}$, it suffices
to prove that for all $n \geq 0$,
$\hat{A} \otimes_{A} \cal{D}^{n}(A) \rightarrow \cal{D}^{n}(\hat{A})$
is bijective. The assumptions imply that
\[ \begin{array}{rcl}
\hat{A} \otimes_{A} \cal{D}^{n}(A) & = &
\hat{A} \otimes_{A}
\operatorname{Hom}^{\operatorname{cont}}_{A}(\cal{P}_{A/k}^{n,\operatorname{sep}},A) \\
& = & \operatorname{Hom}_{\hat{A}}^{\operatorname{cont}}
(\hat{A} \otimes_{A} \cal{P}_{A/k}^{n,\operatorname{sep}},\hat{A}) \\
& \cong & \operatorname{Hom}_{\hat{A}}^{\operatorname{cont}}
(\cal{P}_{\hat{A}/k}^{n,\operatorname{sep}},\hat{A}) \\
& = & \cal{D}^{n}(\hat{A}) .
\end{array} \]
\end{pf}
Now consider the separated algebra of differentials
$\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k}$, which is a graded ST $k$-algebra
(see \cite{Ye1} Def.\ 1.5.3). Then
\[ \cal{T}(\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k}) :=
\operatorname{Der}^{\operatorname{cont}}_{k}(\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k},
\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k}) \]
is a graded Lie algebra. For instance, the exterior
derivative $\operatorname{d}$ is an element of degree $1$ in
$\cal{T}(\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k})$.
We shall need a version of the Lie derivative for semi-topological
algebras (see \cite{Wa} \S 2.24 for the differentiable manifold version).
\begin{prop} \label{prop4.2} \rom{(Lie derivative)}\
Let $A$ be a separated ST $k$-algebra and let $\partial$ be a
continuous $k$-derivation of $A$. Then there exists a unique
continuous, degree $0$, $k$-linear derivation $\operatorname{L}_{\partial}$ of
$\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k}$, which extends $\partial$ and commutes with
$\mathrm{d}$. The map $\partial \mapsto {\rm L}_{\partial}$ is a
homomorphism of $k$-Lie algebras
$\cal{T}(A) \rightarrow \cal{T}(\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k})$, and is functorial
with respect to topologically \'{e}tale homomorphisms $A \rightarrow \hat{A}$ in
$\mathsf{STComAlg}(k)$.
\end{prop}
\begin{pf}
Let
$\partial \in \cal{T}(A) = \operatorname{Der}^{\operatorname{cont}}_{k}(A,A)$ be given.
Since $A$ is separated we get a continuous $A$-linear map
$\Omega^{1,\operatorname{sep}}_{A/k} \rightarrow A$, which extends by universality to a
continuous degree $-1$ derivation
$\iota_{\partial}: \Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k} \rightarrow
\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k}$,
the interior derivative.
Define $\operatorname{L}_{\partial} := \iota_{\partial} \circ \mathrm{d} +
\mathrm{d} \circ \iota_{\partial}$ (i.e. the graded commutator of
$\iota_{\partial}$ and $\mathrm{d}$). The properties of
$\operatorname{L}_{\partial}$
are easily deduced from its definition and the fact that
$\mathrm{d}^{2}=0$. To show uniqueness it suffices to consider
$\operatorname{L}_{\partial}(\alpha)$ for $\alpha=a$ or $\alpha=\mathrm{d}a$,
$a \in A$.
But $\operatorname{L}_{\partial}(a)=\partial(a)$ and
$\operatorname{L}_{\partial}(\mathrm{d}a)= \mathrm{d} (\operatorname{L}_{\partial}(a))=
\mathrm{d} \circ \partial(a)$.
Now let $\partial_{1},\partial_{2} \in \cal{T}(A)$.
Then $[\operatorname{L}_{\partial_{1}}, \operatorname{L}_{\partial_{2}}]$ is a continuous
derivation of $\Omega^{{\textstyle \cdot},\operatorname{sep}}_{A/k}$ commuting with $\mathrm{d}$,
and for all $a \in A$,
\[ [\operatorname{L}_{\partial_{1}}, \operatorname{L}_{\partial_{2}}](a)
=[\partial_{1},\partial_{2}](a)
= \operatorname{L}_{[\partial_{1},\partial_{2}]}(a)\ , \]
so
$[\operatorname{L}_{\partial_{1}}, \operatorname{L}_{\partial_{2}}] =
\operatorname{L}_{[\partial_{1},\partial_{2}]}$.
The functoriality of $\operatorname{L}$ follows from the same functoriality of
$\iota$ (and $\mathrm{d}$).
\end{pf}
\begin{lem} \label{lem4.1}
Suppose
$\Omega^{n+1,\operatorname{sep}}_{A/k}=0$ for some $n$. Then for any
$a \in A$, $\alpha \in \Omega^{n,\operatorname{sep}}_{A/k}$ and
$\partial \in \cal{T}(A)$, one has
$\operatorname{L}_{a \partial}(\alpha) = \operatorname{L}_{\partial}(a \alpha)$.
\end{lem}
\begin{pf}
First note that
\begin{equation} \label{eqn4.6}
\partial(a) \alpha - \mathrm{d}(a) \iota_{\partial}(\alpha) =
\iota_{\partial}(\mathrm{d}(a) \alpha) = 0 .
\end{equation}
Since
$\mathrm{d} \alpha = 0$,
$\iota_{a \partial(\alpha)} = a \iota_{\partial}(\alpha)$ and
$ = \partial(a) \alpha +
a \operatorname{L}_{\partial}(\alpha)$,
it follows that
$\operatorname{L}_{a \partial}(\alpha) = \operatorname{L}_{\partial}(a \alpha)$.
\end{pf}
Now assume $k$ is a perfect field.
\begin{prop} \label{prop4.3}
Let $A \in \mathsf{BCA}(k)$, and let $M$ be a finite type ST $A$-module.
Then for any $n \geq 0$,
$\cal{P}_{A/k}^{n,\operatorname{sep}}(M)$ is a finite type ST left $A$-module.
In particular, so is
$\cal{P}_{A/k}^{1,\operatorname{sep}} = A \oplus \Omega^{1,\operatorname{sep}}_{A/k}$, so $A$
is differentially of finite type over $k$ \rom{(}In the sense of
\cite{Ye1} Def.\ \rom{1.5.16)}.
\end{prop}
\begin{pf}
We may assume $A$ is a local ring. Choose a parametrization of $A$, i.e.\
a surjective morphism of BCAs
$F((\underline{s}))[\sqbr{\underline{t}}] \rightarrow A$.
Let $\underline{u} = (u_{1}, \ldots, u_{l})$ be a separating transcendency basis
for $F$ over $k$. By \cite{Ye1} Cor.\ 1.5.19,
$k \sqbr{\underline{u}, \underline{s}, \underline{t}} \rightarrow F((\underline{s}))[\sqbr{\underline{t}}]$
is topologically \'{e}tale (rel.\ to $k$). Therefore, using \cite{Ye1}
formula (1.4.2) and Theorem 1.5.11, it follows that
\[ \cal{P}_{F((\underline{s}))[\sqbr{\underline{t}}]/k}^{n,\operatorname{sep}} \cong
F((\underline{s}))[\sqbr{\underline{t}}] \otimes_{k\sqbr{\underline{u}, \underline{s}, \underline{t}}}
\cal{P}_{k \sqbr{\underline{u}, \underline{s}, \underline{t}} / k}^{n,\operatorname{sep}} \]
is a free ST left $F((\underline{s}))[\sqbr{\underline{t}}]$-module of finite rank.
Now in general, if $\phi : M \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} N$ is a strict surjection of ST
$k$-modules, then so is
$\phi \otimes \phi : M \otimes_{k} M \rightarrow N \otimes_{k} N$; and if
$M' \subset M$ and $N' \subset N$ and submodules such that
$\phi(M') \subset N'$, then
$\bar{\phi}: M / M' \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} N / N'$ is also strict.
This implies that
$\cal{P}_{F((\underline{s}))[\sqbr{\underline{t}}]/k}^{n,\operatorname{sep}} \rightarrow
\cal{P}_{A/k}^{n,\operatorname{sep}}$ is a strict surjection. Hence
$\cal{P}_{A/k}^{n,\operatorname{sep}}$ is a ST module of finite type over
$A$ (as a left module, via $a \mapsto a \otimes 1$).
Given a finite type ST $A$-module $M$ as above, use Lemma \ref{lem4.3}.
\end{pf}
\section{$\protect\cal{D}$-Modules over TLFs}
Henceforth $k$ is a fixed perfect field.
Let $K$ be a topological local field(TLF) over $k$. We need to understand
the structure of the ring $\cal{D}(K)$ of continuous differential operators.
First assume $k$ has
characteristic $p$. Let $M$ be a free ST $K$-module of finite rank. We know
from \cite{Ye1} Theorems 2.1.14 and 1.4.9
that $\cal{D}(K; M)$ admits the $p$-filtration
\[ \cal{D}(K ;M) = \operatorname{Diff}_{K/k}(M,M) =
\bigcup_{n=0}^{\infty} \operatorname{End}_{K^{(p^{n}/k)}}(M)\ . \]
Here $K^{(p^{n}/k)} = k \otimes_{k} K$, with
$1 \otimes \lambda = \lambda^{p^{n}} \otimes 1$ for $\lambda \in k$.
This filtration is cofinal with the order filtration - see \cite{Ye1} Lemma
1.4.8. According to ibid.\ Prop.\ 2.1.13, the relative Frobenius map
$K^{(p^{n}/k)} \rightarrow K$, $\lambda \otimes a \mapsto \lambda a^{p^{n}}$,
is a finite morphism in $\mathsf{TLF}(k)$.
In characteristic $0$, $\cal{D}(K)$ is a ``topologically \'{e}tale
localization'' of a Weyl algebra. Choose a parametrization
$K \cong F((\underline{s}))$ and a separating transcendency basis
$\underline{u}$ for $F$ over $k$. Let
$\underline{t} = (t_{1}, \ldots, t_{m}) := (\underline{u}, \underline{s}) =
(u_{1}, \ldots, s_{1} \ldots)$ be the concatenated sequence. Then
$k\sqbr{ \underline{t} } \rightarrow K$ is a flat, topologically \'{e}tale homomorphism
in $\mathsf{STComAlg}(k)$. The ring
$\cal{D}(k \sqbr{\underline{t}})$ is a Weyl algebra over $k$:
$\cal{D}(k \sqbr{\underline{t}}) \cong k \sqbr{\underline{t}} \otimes_{k}
k \sqbr{ \partial_{1},\ldots,\partial_{m} }$, where
$\partial_{i}:= \frac{\partial}{\partial t_{i}}$, and the multiplication
is determined by
$(1 \otimes \partial_{i})(t_{j} \otimes 1)=t_{j}
\otimes \partial_{i} + (\partial_{i} * t_{j}) \otimes 1$. By Prop.\
\ref{prop4.1}, we have
$\cal{D}(K) \cong K \otimes_{k \sqbr{\underline{t}}}
\cal{D}(k \sqbr{\underline{t}})$. Considering the
faithful action of $\cal{D}(K)$ on $K$, we get a presentation
\begin{equation} \label{eqn5.8}
\begin{array}{rcl}
\cal{D}(K) & \cong & K \otimes_{k} k \sqbr{\partial_{1},\ldots,\partial_{m}} \\
(1 \otimes \partial_{i})(a \otimes 1) & = &
a \otimes \partial_{i} + (\partial_{i} * a) \otimes 1
\end{array}
\end{equation}
for $i = 1, \ldots, m$ and $a \in K$
(i.e. $\cal{D}(K)$ is a smash product of $K$ and the universal enveloping
algebra of
the abelian $k$-Lie algebra spanned by the derivations $\partial_{i}$).
\begin{dfn}
Let $K$ be a TLF over $k$. Define
$\omega(K)$ to be the top degree component of
$\Omega^{{\textstyle \cdot}, \mathrm{sep}}_{K/k}$.
It is a free ST $K$-module of rank $1$.
\end{dfn}
At this point we can exhibit the canonical right $\cal{D}(K)$-module
structure on $\omega(K)$ (cf.\ \cite{Bo} Ch.\ VI \S 3.2).
\begin{prop} \label{prop5.1}
For any $K \in \mathsf{TLF}(k)$ there is a unique right
$\cal{D}(K)$-module structure on $\omega(K)$, written
$\alpha * D$, for $\alpha \in \omega(K)$ and $D \in \cal{D}(K)$,
such that:
\begin{enumerate}
\rmitem{i} If $D=a \in K$ then $\alpha * a = a \alpha$.
\rmitem{ii} If $D = \partial \in \cal{T}(K)$ then
$\alpha * \partial = - \operatorname{L}_{\partial}(\alpha)$, where
$\operatorname{L}_{\partial}$ is the Lie derivative \rom{(}see Prop.\
\rom{\ref{prop4.2})}.
\rmitem{iii} If $\operatorname{char} k=p$ and
$D \in \cal{D}^{p^{n}-1}(K)$ for some $n \geq 0$, then for
every $a \in K$,
\[ \langle D * a, \alpha \rangle_{K/K^{(p^{n}/k)}} =
\langle a, \alpha * D \rangle_{K/K^{(p^{n}/k)}}\ , \]
where $\langle - , - \rangle_{K/K^{(p^{n}/k)}}$ is the trace pairing of
\cite{Ye1} formula \rom{(2.3.8)}.
\end{enumerate}
\end{prop}
\begin{pf}
First assume $\operatorname{char} k=0$. Since
$[\operatorname{L}_{\partial_{i}},\operatorname{L}_{\partial_{j}}]=0$,
$\alpha * \partial_{i} := - \operatorname{L}_{\partial_{i}}(\alpha)$ is an action
of
$k \sqbr{\partial_{1}, \ldots, \partial_{m}}$ on $\omega(K)$. According to
the presentation (\ref{eqn5.8}), in order to extend this to a
right action of $\cal{D}(K)$ it suffices to show that
\[ -a \operatorname{L}_{\partial_{i}}(\alpha) = - \operatorname{L}_{\partial_{i}}(a
\alpha) + \partial_{i}(a) \alpha \]
which is true since $\operatorname{L}_{\partial_{i}}$ is an even derivation of
$\Omega^{{\textstyle \cdot},\operatorname{sep}}_{K/k}$ and
$\operatorname{L}_{\partial_{i}}(a)= \partial_{i}(a)$. By Lemma \ref{lem4.1},
condition (ii) holds for an arbitrary
derivation $\partial = \sum a_{i} \partial_{i}$, $a_{i} \in K$.
Next consider the case $\operatorname{char} k=p$. Let
$D \in \cal{D}^{n}(K)$. By \cite{Ye1} Lemma 1.4.8, $D$ is
$K^{(p^{n}/k)}$-linear. The trace
pairing $\langle -,- \rangle_{K/K^{(p^{n}/k)}}$ is perfect (\cite{Ye1}
Prop.\ 2.3.9), so by
adjunction $D$ acts on $\omega(K)$. The functoriality of the trace
guarantees that this action is independent of $n$. We thus get a right
action satisfying conditions (i) and (iii). In order to check (ii) it
suffices to look at $\partial=\partial_{i}$.
Let $\beta:= \mathrm{d} t_{1} \wedge \cdots \wedge \mathrm{d} t_{i-1}
\wedge \mathrm{d} t_{i+1} \wedge \cdots \wedge \mathrm{d} t_{m}$.
We can compute the difference:
\begin{eqnarray*}
\lefteqn{\langle \partial_{i}(a),b \mathrm{d} t_{i} \wedge \beta
\rangle_{K/K^{(p^{n}/k)}} -
\langle a,-\operatorname{L}_{\partial_{i}}(b \mathrm{d}
t_{i} \wedge \beta) \rangle_{K/K^{(p^{n}/k)}} } \blnk{30mm}\\
& & = \operatorname{Tr}_{K/K^{(p/k)}}(\partial_{i}(ab) \mathrm{d}
t_{i} \wedge \beta) \\
& & = \operatorname{Tr}_{K/K^{(p/k)}}(\mathrm{d}(ab \beta)) = 0
\end{eqnarray*}
since $\operatorname{Tr}_{K/K^{(p/k)}}$ commutes with $\mathrm{d}$ and vanishes
on $\Omega^{\leq m-1,\operatorname{sep}}_{K/k}$.
\end{pf}
Let $\cal{D}(K)^{\circ}$ denote the opposite ring of $\cal{D}(K)$.
\begin{prop} \label{prop5.11}
The right $\cal{D}(K)$ action on $\omega(K)$ of the previous proposition
induces a canonical isomorphism of filtered $k$-algebras
\[ T_{K} : \cal{D}(K)^{\circ} \stackrel{\simeq}{\rightarrow}
\cal{D}(K; \omega(K)). \]
\end{prop}
\begin{pf}
If $\operatorname{char} k = p$ we have, for every $n \geq 0$, an isomorphism (of
$K$-$K$-bimodules)
$T^{n}: \operatorname{End}_{K^{(p^{n}/k)}}(K)^{\circ}
\stackrel{\simeq}{\rightarrow} \operatorname{End}_{K^{(p^{n}/k)}}(\omega(K))$
induced by adjunction. In the limit we get $T_{K}$.
If $\operatorname{char} k = 0$, choose a topologically \'{e}tale homomorphism
$k \sqbr{\underline{t}} \rightarrow K$. Let $T_{\underline{t}} : \cal{D}(K) \rightarrow \cal{D}(K)$ be
the involution such that $T_{\underline{t}} |_{K}$ is the identity and
$T_{\underline{t}}(\partial_{i})=-\partial_{i}$ (cf.\ formula (\ref{eqn5.8})).
Let
$\phi_{\underline{t}}:K \stackrel{\simeq}{\rightarrow} \omega(K)$
be the $K$-linear isomorphism defined by
$\phi_{\underline{t}}(1)= \mathrm{d} t_{1} \wedge \cdots \wedge \mathrm{d} t_{m}$.
Then for any $D \in \cal{D}(K)$,
\[ T_{K}(D)= \phi_{\underline{t}} \circ T_{\underline{t}}(D) \circ \phi_{\underline{t}}^{-1}
\in \cal{D}(K; \omega(K)). \]
\end{pf}
\begin{cor} \label{cor5.10}
Let $K,\hat{K} \in \mathsf{BCA}(k)$ be fields and let $K \rightarrow \hat{K}$ be a
topologically \'{e}tale homomorphism in $\mathsf{STComAlg}(k)$.
Then $\omega(K) \rightarrow \omega(\hat{K})$ is a homomorphism of right
$\cal{D}(K)$-modules.
\end{cor}
\begin{pf}
In characteristic $0$ this follows from the covariance of the Lie derivative.
In positive characteristics it follows from the fact that the trace map
commutes with base change, cf.\ \cite{Ye1} Prop.\ 2.3.11.
\end{pf}
On the category $\mathsf{TLF}(k)$ there is a functorial residue map. To
each morphism $f : K \rightarrow L$ it assigns a homomorphism of differential
graded ST left $\Omega^{{\textstyle \cdot},\operatorname{sep}}_{K/k}$-modules,
$\operatorname{Res}_{L/K} = \operatorname{Res}_{f} : \Omega^{{\textstyle \cdot},\operatorname{sep}}_{L/k} \rightarrow
\Omega^{{\textstyle \cdot},\operatorname{sep}}_{K/k}$ (cf.\ \cite{Ye1} Thm.\ 2.4.3).
The residue pairing
\[ \begin{array}{c}
\langle - , - \rangle_{L/K} : L \times \omega(L) \rightarrow \omega(K) \\
\langle a , \alpha \rangle_{L/K} = \operatorname{Res}_{L/K}(a \alpha)
\end{array} \]
is a perfect pairing of ST $K$-modules, in the sense that the induced map
$\omega(L) \rightarrow \operatorname{Hom}_{K}^{\operatorname{cont}}(L, \omega(K))$ is bijective
(cf.\ \cite{Ye1} Thm.\ 2.4.22 - Topological Duality).
\begin{thm} \label{thm5.1}
Let $K \in \mathsf{TLF}(k)$ and assume that $k \rightarrow K$ is a morphism in
$\mathsf{TLF}(k)$. Given a DO $D \in \cal{D}(K)$, let
$D^{\vee} \in \operatorname{End}_{k}(\omega(K))$
be its adjoint relative to the residue pairing
$\langle -,- \rangle_{K/k}$.
Then for every $\alpha \in \omega(K)$,
\[ D^{\vee}(\alpha) = \alpha * D . \]
In other words, the adjoint action of $\cal{D}(K)$ on $\omega(K)$
coincides with the canonical right action.
\end{thm}
\begin{pf}
We must show that for all $a \in K$, $\alpha \in \omega(K)$ and $D
\in \cal{D}(K)$, $\langle D * a, \alpha \rangle_{K/k} = \langle
a, \alpha * D \rangle_{K/k}$. In characteristic $p$ this follows
immediately from condition (iii) of Prop.\ \ref{prop5.1} and the
functoriality of the residue maps (\cite{Ye1} Thm.\ 2.4.2).
in characteristic $0$ first choose a parametrization
$K \cong F((\underline{t}))= F((t_{1},\ldots,t_{n}))$. Then $k \rightarrow F$ is finite
separable
and any $k$-linear DO is also $F$-linear.
Given $\lambda \in F$,
$\underline{i} \in {\Bbb Z}^{n}$ and $1 \leq j \leq n$, write
$\operatorname{dlog}(\underline{t}) := \operatorname{dlog}(t_{1}) \wedge \cdots \wedge
\operatorname{dlog}(t_{n}) = t_{1}^{-1} \mathrm{d} t_{1} \wedge \cdots \wedge
t_{n}^{-1} \mathrm{d} t_{n}$
and
$\partial_{j} := \frac{\partial}{\partial t_{j}}$.
Then
\begin{eqnarray*}
\lefteqn{\operatorname{L}_{\partial_{j}}( \lambda \underline{t}^{\underline{i}}
\operatorname{dlog}(\underline{t}) ) =} \\
& & (-1)^{j-1} \mathrm{d} ( \lambda t_{j}^{-1} \underline{t}^{\underline{i}}
\operatorname{dlog}(t_{1}) \wedge \cdots \wedge \operatorname{dlog}(t_{j-1}) \wedge
\operatorname{dlog}(t_{j+1}) \wedge \cdots \wedge \operatorname{dlog}(t_{n}))
\end{eqnarray*}
so
$\operatorname{Res}_{K/k} ( \operatorname{L}_{\partial_{j}}( \lambda \underline{t}^{\underline{i}} )) =0$.
By continuity we conclude that
\begin{equation} \label{eqn5.10}
\operatorname{Res}_{K/k} ( \operatorname{L}_{\partial_{j}} (\alpha) ) =0
\end{equation}
for all $\alpha$.
To prove the theorem it suffices to consider either $D=b \in K$ or
$D=\partial_{j}$.
For $D=b$ we get
\[ \langle a, \alpha * b \rangle_{K/k} =
\langle a, b \alpha \rangle_{K/k} = \operatorname{Res}_{K/k}(ab \alpha) =
\langle ab , \alpha \rangle_{K/k} =
\langle b * a, \alpha \rangle_{K/k} . \]
For $D=\partial_{j}$ we use ``integration by parts'':
\begin{eqnarray*}
\lefteqn{ \langle \partial_{j} * a , \alpha \rangle_{K/k} -
\langle a , \alpha * \partial_{j} \rangle_{K/k} =} \\
& & \operatorname{Res}_{K/k}( \operatorname{L}_{\partial_{j}} (a) \alpha +
a \operatorname{L}_{\partial_{j}} (\alpha) ) =
\operatorname{Res}_{K/k}( \operatorname{L}_{\partial_{j}} (a \alpha)) = 0
\end{eqnarray*}
by (\ref{eqn5.10}).
\end{pf}
\section{Duals of Finite Type Modules}
The purpose of this subsection is to establish the existence of a canonical
dual module $\operatorname{Dual}_{A} M$ to every finite type ST $A$-module $M$.
If $k \rightarrow A$ is a morphism in $\mathsf{BCA}(k)$, then we set
$\operatorname{Dual}_{A} M := \operatorname{Hom}_{k}^{\mathrm{cont}}(M, k)$, endowed with the
fine $A$-module topology. Otherwise we define $\operatorname{Dual}_{A} M$ using
differential operators, and show this definition is independent of choices
made by a base change argument, which reduces things to the case when
$k \rightarrow A$ is a morphism.
Recall that $k$ is a fixed perfect field.
For a TLF $K$, $\omega(K)$ is the top degree component of
$\Omega^{{\textstyle \cdot}, \mathrm{sep}}_{K/k}$, a rank $1$ free ST $K$-module.
\begin{dfn} \label{def6.1}
Let $A,K \in \mathsf{BCA}(k)$ be a local ring and a field,
respectively, and let $\sigma: K \rightarrow A$ be a morphism in $\mathsf{BCA}(k)$.
For any finite type ST $A$-module $M$ define
\[ \operatorname{Dual}_{\sigma} M := \operatorname{Hom}_{K; \sigma}^{\operatorname{cont}}
(M, \omega(K)) , \]
the set of continuous $K$-linear homomorphisms, where $M$ is a $K$-module
via $\sigma$. Put on $\operatorname{Dual}_{\sigma} M$ the {\em fine $A$-module
topology}.
\end{dfn}
\begin{rem} \label{rem6.1}
The module $\operatorname{Hom}_{K; \sigma}^{\operatorname{cont}}(M, \omega(K))$, with the
(weak) $\operatorname{Hom}$ topology, is a ST $A$-module. Therefore the identity map
$\operatorname{Dual}_{\sigma} M \rightarrow \operatorname{Hom}_{K; \sigma}^{\operatorname{cont}}(M, \omega(K))$
is continuous. However, this will not be a homeomorphism unless
$\sigma :K \rightarrow A$ is a pseudo coefficient field and $M$ is a finite length
module.
\end{rem}
Let $A$ be a commutative noetherian local ring, with maximal ideal ${\frak m}$,
and let $I$ be an injective hull of $A / {\frak m}$. Then
$M \mapsto \operatorname{Hom}_{A}(M,I)$ is a duality between finite type (i.e.\
finitely generated) $A$-modules and cofinite type (i.e.\ artinian)
$A$-modules. The module $\operatorname{Hom}_{A}(M,I)$ is called a {\em Matlis dual}
of $M$ (cf.\ \cite{LC} \S 4).
\begin{lem} \label{lem6.1}
Let $\sigma : K \rightarrow A$ and $M$ be as in Def.\ \ref{def6.1}.
\begin{enumerate}
\rmitem{a} Suppose $\tau : L \rightarrow A$ and $f : K \rightarrow L$ are morphisms in
$\mathsf{BCA}(k)$, with $L$ a field, and $\sigma = \tau \circ f$. Then
the map
\[ \begin{array}{rcl}
\operatorname{Dual}_{\tau} M & \rightarrow & \operatorname{Dual}_{\sigma} M \\
\phi & \mapsto & \operatorname{Res}_{L/K} \circ \phi
\end{array} \]
is an isomorphism of ST $A$-modules.
\rmitem{b} The (untopologized) $A$-module $\operatorname{Dual}_{\sigma} M$ is a Matlis
dual of $M$. In particular, Taking $M = A$, it follows that
$\operatorname{Dual}_{\sigma} A$ is an injective hull of $A / {\frak m}$.
As a ST $A$-module, $\operatorname{Dual}_{\sigma} M$ is of cofinite type.
\end{enumerate}
\end{lem}
\begin{pf}
(a)\ First consider the case when $\tau: L \rightarrow A$ is finite; so $A$ has the
fine $L$-module topology (Prop.\ \ref{prop2.1} (a)).
Then $M$ is a free ST $L$-module of
finite rank. By Topological Duality
(\cite{Ye1} Thm.\ 2.4.22),
$\operatorname{Dual}_{\tau} M \rightarrow \operatorname{Dual}_{\sigma} M$
is bijective, and it is an isomorphism of ST $A$-modules since both modules
have the fine $A$-module topologies.
Next assume $\tau: L \rightarrow A$ is a pseudo coefficient field. Because
$\omega(K)$ (resp.\ $\omega(L)$) is a simple, separated ST $K$-module
(resp.\ $L$-module), and
$M \cong \lim_{\leftarrow n} M / {\frak m}^{n} M$,
we can use \cite{Ye1} Prop.\ 1.2.22 to conclude that
\begin{equation} \label{eqn6.5}
\operatorname{Dual}_{\sigma} M = \bigcup_{n = 1}^{\infty}
\operatorname{Hom}_{K; \sigma}^{\operatorname{cont}}(M / {\frak m}^{n} M, \omega(K))
\end{equation}
and similarly for $L$.
For any $n \geq 1$, $M / {\frak m}^{n} M$ is a finite type ST
$A / {\frak m}^{n}$-module, so we are back to the first step.
For the general situation, we may factor $\tau$ through some pseudo
coefficient field $\tau' : L' \rightarrow A$ (cf.\ Lemma \ref{lem2.2} (d)),
and use the functoriality of the residue maps.
\medskip \noindent (b)\
By part (a) we can assume that $\sigma : K \rightarrow A$ is a pseudo coefficient
field. Then in (\ref{eqn6.5}) we can drop the superscript ``$\operatorname{cont}$'',
in which case the statement is well known (cf.\ \cite{LC} p.\ 63 Example 1).
\end{pf}
Let $A, \hat{A}$ be local BCAs, with maximal ideals
${\frak m}, \hat{{\frak m}}$ respectively, and let $v : A \rightarrow \hat{A}$ be an
intensification homomorphism. Note that $v$, being a local
homomorphism, is faithfully flat. Let $\sigma : K \rightarrow A$ be a morphism
in $\mathsf{BCA}(k)$, with $K$ a field. Assume that there is an
intensification homomorphism $u : K \rightarrow \hat{K}$
and a morphism $\hat{\sigma} : \hat{K} \rightarrow \hat{A}$ s.t.\
$\hat{A} \cong A \otimes_{K}^{(\wedge)} \hat{K}$.
\begin{prop} \label{prop6.1}
Let $M$ be a finite type ST $A$-module, and set
$\hat{M} := \hat{A} \otimes_{A} M$. Then any
$\phi \in \operatorname{Dual}_{\sigma} M$ has a unique extension
$\hat{\phi} \in \operatorname{Dual}_{\hat{\sigma}} \hat{M}$. The resulting continuous
homomorphism
\[ q_{v; \sigma}^{M} : \operatorname{Dual}_{\sigma} M \rightarrow
\operatorname{Dual}_{\hat{\sigma}} \hat{M} \]
is injective, and induces an isomorphism of ST $\hat{A}$-modules
\[ 1 \otimes q_{v; \sigma}^{M}:
\hat{A} \otimes_{A} \operatorname{Dual}_{\sigma} M \stackrel{\simeq}{\rightarrow}
\operatorname{Dual}_{\hat{\sigma}} \hat{M} . \]
\end{prop}
\begin{pf}
Let $n := \operatorname{res.dim} \sigma$. Then we can extend $\sigma$ to a
a pseudo coefficient field
$K((\underline{s})) = K((s_{1}, \ldots, s_{n})) \rightarrow A$, and extend $\hat{\sigma}$
to $\hat{K}((\underline{s})) \rightarrow \hat{A}$. By replacing $K, \hat{K}$ with
$K((\underline{s})), \hat{K}((\underline{s}))$ we can then assume that
$\sigma, \hat{\sigma}$ are pseudo coefficient fields. This puts us in the
setup of Prop.\ \ref{prop3.2}. For $i \geq 0$ define
\[ H^{i} := \operatorname{Hom}_{K; \sigma}^{\operatorname{cont}}(M / {\frak m}^{i+1} M, \omega(K))
\subset \operatorname{Dual}_{\sigma} M\, \]
and similarly define $\hat{H}^{i}$.
Since $\operatorname{Dual}_{\sigma} M$ and $\operatorname{Dual}_{\hat{\sigma}} \hat{M}$ both have
the fine topologies, it suffices to exhibit an isomorphism
$\hat{A} \otimes_{A} H^{i} \stackrel{\simeq}{\rightarrow} \hat{H}^{i}$, with
$\hat{\phi} := 1 \otimes \phi$ extending $\phi$. By Prop.\ \ref{prop3.2} (a),
\[ \hat{K} \otimes_{K} (M / {\frak m}^{i+1} M) \cong
\hat{A} \otimes_{A} (M / {\frak m}^{i+1} M) \cong \hat{M} / {\frak m}^{i+1} \hat{M} . \]
Since $K \rightarrow \hat{K}$ is topologically \'{e}tale,
$\hat{K} \otimes_{K} \omega(K) \stackrel{\simeq}{\rightarrow} \omega(\hat{K})$. Therefore
$\hat{K} \otimes_{K} H^{i} \stackrel{\simeq}{\rightarrow} \hat{H}^{i}$; and again by Prop.\
\ref{prop3.2} (a), $\hat{A} \otimes_{A} H^{i} \stackrel{\simeq}{\rightarrow} \hat{H}^{i}$.
\end{pf}
Let $A$ be a local BCA with maximal ideal ${\frak m}$. Suppose
$\sigma, \sigma' : K \rightarrow A$ are pseudo coefficient fields, such that
$\sigma \equiv \sigma'\ (\operatorname{mod} {\frak m})$.
Let $M$ be a finite type ST $A$-module. Given a nonzero element
$x \in M$, its order with respect to ${\frak m}$ is
\[ \operatorname{ord}_{{\frak m}}(x) := \operatorname{max} \{ n\ |\ x \in {\frak m}^{n} M \} . \]
If $\operatorname{ord}_{{\frak m}}(x) =n$, then the symbol of $x$ is its image in
${\frak m}^{n} / {\frak m}^{n+1} \subset \operatorname{gr}_{{\frak m}} M$.
\begin{dfn} \label{def6.3}
An ${\frak m}$-filtered $K$-basis of $M$ is
a sequence $\underline{x} = (x_{0}, x_{1}, \ldots)$ of elements of $M$,
such that the symbols of $x_{0}, x_{1}, \ldots$ form a $K$-basis of
$\operatorname{gr}_{{\frak m}} M$, and such that
$\operatorname{ord}_{\frak{m}}(x_{i}) \leq \operatorname{ord}_{\frak{m}}(x_{i+1})$.
\end{dfn}
Choose such a basis $\underline{x}$.
Then any $x \in M$ is expressed uniquely as a convergent sum
\[ x = \sum_{i} \sigma'(\lambda_{i}) x_{i} =
\sum_{i} \sigma(\mu_{i}) x_{i} \]
with $\lambda_{i}, \mu_{i} \in K$. Define functions $D_{ij} : K \rightarrow K$
by the equation
\[ \sigma'(\lambda) x_{i} = \sum_{j} \sigma(D_{ij}(\lambda)) x_{j} . \]
\begin{lem} \label{lem6.2}
$D_{ij} \in \cal{D}(K)$, i.e.\ it is a continuous differential operator
over $K$ relative to $k$.
\end{lem}
\begin{pf}
Pick two indices $i_{0},i_{1}$, and let
$n:= \operatorname{max} \{ \operatorname{ord}_{{\frak m}}(x_{i_{0}}), \operatorname{ord}_{{\frak m}}(x_{i_{1}}) \}$.
We can compute the function $D_{i_{0} i_{1}}$ for the module
$M / {\frak m}^{n+1} M$ instead of $M$. Define
\[ A^{-} := \sigma(K) \oplus {\frak m} = \sigma'(K) \oplus {\frak m} \subset A . \]
This is a local BCA, with $A^{-} / {\frak m} \cong K$,
and $A^{-} \rightarrow A$ is a finite morphism. Let $l$ be the length of
$M / {\frak m}^{n+1} M$ over $A^{-}$, and let
$E,E' : K^{l} \stackrel{\simeq}{\rightarrow} M / {\frak m}^{n+1} M$ be the $K$-linear homeomorphisms
\begin{eqnarray*}
E(\lambda_{0}, \ldots, \lambda_{l-1}) & := &
\sum_{i=0}^{l-1} \sigma(\lambda_{i}) x_{i} \\
E'(\lambda_{0}, \ldots, \lambda_{l-1}) & := &
\sum_{i=0}^{l-1} \sigma'(\lambda_{i}) x_{i} \\
\end{eqnarray*}
($M / {\frak m}^{n+1} M$ is a free ST $K$-module via $\sigma$ and via $\sigma'$).
According to \cite{Ye1} Prop.\ 1.4.4, $E,E^{-1},E'$ and $(E')^{-1}$ are
DOs over $A^{-}$, relative to $k$. Set
\[ D:= E^{-1} \circ E' : K^{l} \stackrel{\simeq}{\rightarrow} K^{l} \ , \]
which is a DO over $A^{-}$, and hence over $K$. Expanding $D$ as an
$l \times l$ matrix with entries in $\cal{D}(K)$, one gets
$D = [D_{ij}]$.
\end{pf}
One can easily show that
\[ \operatorname{ord}_{K}(D_{ij}) \leq 2 \operatorname{max} \{ -1,
\operatorname{ord}_{\frak{m}}(x_{j}) - \operatorname{ord}_{\frak{m}}(x_{i}) \} \]
and
$D_{ii} = 1$. Thus the matrix of DOs looks like this:
\[ [ D_{ij} ] =
\left[ \begin{array}{cccc}
1 & * & * \\
0 & 1 & * & \cdot \ \cdot \\
0 & 0 & 1 \\
& : \\
\end{array} \right] \]
\begin{dfn} \label{dfn6.2}
In the situation described above, define a function
\[ \Psi_{\sigma, \sigma'}^{M} : \operatorname{Dual}_{\sigma} M \rightarrow \operatorname{Dual}_{\sigma'}
M \]
by the equation
\begin{equation} \label{eqn6.6}
\Psi_{\sigma, \sigma'}^{M}(\phi) (\sum_{i} \sigma'(\lambda_{i}) x_{i}) =
\sum_{i,j} \lambda_{i} (\phi(x_{j}) * D_{ij})
\end{equation}
for $\phi \in \operatorname{Dual}_{\sigma} M$ and $\lambda_{i} \in K$.
\end{dfn}
The second sum in (\ref{eqn6.6}) makes sense, since there are only finitely
many nonzero terms in it. At first glance this somewhat strange definition
seems to depend on the basis $\underline{x}$. We shall soon see that there is no
dependence on the basis, and that in fact $\Psi_{\sigma, \sigma'}^{M}$ is an
isomorphism of ST $A$-modules.
Immediately from the definition we get:
\begin{lem} \label{lem6.0}
$\Psi^{M}_{\sigma, \sigma'}$ is a $k$-linear bijection, with inverse
$\Psi^{M}_{\sigma', \sigma}$. Given a third pseudo coefficient field
$\sigma'': K \rightarrow A$ one has
\[ \Psi^{M}_{\sigma, \sigma''} =
\Psi^{M}_{\sigma'', \sigma'} \circ \Psi^{M}_{\sigma, \sigma'}. \]
\end{lem}
Further properties of $\Psi^{M}_{\sigma, \sigma'}$ are less obvious.
\begin{lem} \label{lem6.3}
Under the combined assumptions of Prop.\ \ref{prop6.1} and Def.\
\ref{dfn6.2}, one has
\[ \Psi_{\hat{\sigma}, \hat{\sigma}'}^{\hat{M}} \circ
q_{v; \sigma}^{M} = q_{v; \sigma'}^{M} \circ
\Psi_{\sigma, \sigma'}^{M}. \]
Here we are using the $\hat{{\frak m}}$-filtered basis
$(1 \otimes x_{0}, 1 \otimes x_{1}, \ldots)$ on $\hat{M}$ to define
$\Psi_{\hat{\sigma}, \hat{\sigma}'}^{\hat{M}}$.
\end{lem}
\begin{pf}
The DOs $\hat{D}_{ij} \in \cal{D}(\hat{K})$ which appear in the definition of
$\Psi_{\hat{\sigma}, \hat{\sigma}'}^{\hat{M}}$ are precisely the images
of the DOs
$D_{ij} \in \cal{D}(K)$ under the natural ring homomorphism
$\cal{D}(K) \rightarrow \cal{D}(\hat{K})$. By Cor.\ \ref{cor5.10},
$\omega(K) \rightarrow \omega(\hat{K})$ is a homomorphism of right
$\cal{D}(K)$-modules.
\end{pf}
\begin{lem} \label{lem6.4}
In the situation of Def.\ \ref{dfn6.2},
suppose in addition that $k \rightarrow K$ is a morphism in $\mathsf{BCA}(k)$. Then
for any $\phi \in \operatorname{Dual}_{\sigma} M$, one has
\[ \operatorname{Res}_{K/k} \circ \phi =
\operatorname{Res}_{K/k} \circ \Psi_{\sigma, \sigma'}^{M}(\phi). \]
\end{lem}
\begin{pf}
Say $\phi(x_{i}) = \alpha_{i} \in \omega(K)$. Given
$x = \sum_{i} \sigma'(\lambda_{i}) x_{i} \in M$, with
$\lambda_{i} \in K$, we have by definition
$x = \sum_{i,j} \sigma(D_{ij} * \lambda_{i}) x_{j}$. So
\[ \operatorname{Res}_{K/k} \circ \phi (x) =
\operatorname{Res}_{K/k} (\sum_{i,j} (D_{ij} * \lambda_{i}) \alpha_{j}) . \]
On the other hand, setting
$\phi' := \Psi_{\sigma, \sigma'}^{M}(\phi)$, one has
\[ \operatorname{Res}_{K/k} \circ \phi' (x) =
\operatorname{Res}_{K/k} (\sum_{i,j} \lambda_{i} (\alpha_{j} * D_{ij})) . \]
By linearity and continuity, it suffices to prove that for all
$i,j \geq 0$:
\[ \operatorname{Res}_{K/k} \left( (D_{ij} * \lambda_{i}) \alpha_{j} \right)
= \operatorname{Res}_{K/k} \left( \lambda_{i} (\alpha_{j} * D_{ij}) \right)\ ; \]
but this is done in Thm.\ \ref{thm5.1}.
\end{pf}
\begin{lem} \label{lem6.5}
Let $K \in \mathsf{BCA}(k)$ be a field. There exists a field
$\hat{K} \in \mathsf{BCA}(k)$ and a homomorphism $u : K \rightarrow \hat{K}$
in $\mathsf{STComAlg}(k)$ such that $k \rightarrow \hat{K}$ is a morphism
in $\mathsf{BCA}(k)$ and $u$ is an intensification. Moreover we can choose
$u$ to be dense.
\end{lem}
\begin{pf}
Choose a parametrization $K \cong F((\underline{t}))$. $F$ is a finitely generated
field extension of $k$; let $\underline{s} = (s_{1}, \ldots, s_{m})$ be a
transcendency basis for $F/k$. Then $k(\underline{s}) \rightarrow F$ is a finite morphism
in $\mathsf{BCA}(k)$. The map
$k(\underline{s}) \rightarrow k((\underline{s}))$ is certainly an intensification homomorphism.
Applying finitely ramified base change (Thm.\ \ref{thm3.1})
we get a dense intensification homomorphism
$K \rightarrow K \otimes_{k(\underline{s})}^{(\wedge)} k((\underline{s}))$. Thus the BCA
$K \otimes_{k(\underline{s})}^{(\wedge)} k((\underline{s}))$ is a reduced cluster of TLFs,
and we can take $\hat{K}$ to be any local factor of it.
\end{pf}
\begin{prop} \label{prop6.2}
Let $A$ be a local BCA with maximal ideal ${\frak m}$, and let
$\sigma, \sigma' : K \rightarrow A$ be two pseudo coefficient fields, such that
$\sigma \equiv \sigma'\ (\operatorname{mod} {\frak m})$. Let $M$ be a finite type ST
$A$-module. Then the map
$\Psi_{\sigma, \sigma'}^{M}$
is an isomorphism of ST $A$-modules, independent of the ${\frak m}$-filtered
$K$-basis
$\underline{x} = (x_{0}, x_{1}, \ldots)$.
\end{prop}
\begin{pf}
First we reduce the problem to the case when $K \stackrel{\simeq}{\rightarrow} A / {\frak m}$, i.e.\
when $\sigma, \sigma'$ are coefficient fields. Let $A^{-}$ be the algebra
$\sigma(K) \oplus {\frak m} \subset A$, cf.\ proof of Lemma \ref{lem6.2}.
The map $\Psi_{\sigma, \sigma'}^{M}$ is the
same when restricting $M$ to an $A^{-}$-module, so we may replace $A$ with
$A^{-}$.
Choose an intensification homomorphism $u : K \rightarrow \hat{K}$ as in Lemma
\ref{lem6.5}, and define $\hat{A} := A \otimes_{K}^{(\wedge)} \hat{K}$,
w.r.t.\ the morphism $\sigma : K \rightarrow A$. So the homomorphism
$v : A \rightarrow \hat{A}$ is also an intensification,
$\hat{A}$ is local with maximal ideal
$\hat{{\frak m}} = \hat{A} \cdot v({\frak m})$, and
$\hat{A} / \hat{{\frak m}} \cong \hat{K}$. Let
$R: \operatorname{Dual}_{\hat{\sigma}} \hat{M} \stackrel{\simeq}{\rightarrow}
\operatorname{Hom}_{k}^{\mathrm{cont}}(\hat{M}, k)$
be the $\hat{A}$-linear isomorphism
$\phi \mapsto \operatorname{Res}_{\hat{K} / k} \circ \phi$ of Lemma \ref{lem6.1},
and similarly define $R'$.
According to Lemmas \ref{lem6.3} and \ref{lem6.4}, the diagram
\medskip \noindent
\begin{equation} \label{eqn6.10}
\setlength{\unitlength}{0.30mm}
\begin{array}{ccccc}
\operatorname{Dual}_{\sigma} M & \lrar{q_{v; \sigma}^{M}}
& \operatorname{Dual}_{\hat{\sigma}} \hat{M} \\
\ldar{\Psi_{\sigma, \sigma'}^{M}} &
& \ldar{\Psi_{\hat{\sigma}, \hat{\sigma}'}^{\hat{M}}}
& \ldrar{R} \\
\operatorname{Dual}_{\sigma'} M & \lrar{q_{v; \sigma'}^{M}}
& \operatorname{Dual}_{\hat{\sigma}'} \hat{M} & \lrar{R'}
& \operatorname{Hom}_{k}^{\mathrm{cont}}(M, k)
\end{array}
\end{equation}
is commutative. Since $q_{v; \sigma}^{M}$ and $q_{v; \sigma'}^{M}$
are injections, we deduce
the independence of $\Psi_{\sigma, \sigma'}^{M}$ of the basis
$\underline{x}$, and that $\Psi_{\sigma, \sigma'}^{M}$
is an $A$-linear bijection. Since both
$\operatorname{Dual}_{\sigma} M$ and $\operatorname{Dual}_{\sigma'} M$ have the fine topologies,
$\Psi_{\sigma, \sigma'}^{M}$ is in fact a homeomorphism.
\end{pf}
\begin{prop} \label{prop6.3}
Under the hypothesis of Prop.\ \ref{prop6.2},
suppose $\tau, \tau' : L \rightarrow A$ are pseudo coefficient fields,
and $f : K \rightarrow L$ is a (finite) morphism in $\mathsf{BCA}(k)$, such that
$\tau \equiv \tau'\ (\operatorname{mod} {\frak m})$, $\sigma = \tau \circ f$ and
$\sigma' = \tau' \circ f$. Then for any $\phi \in \operatorname{Dual}_{\tau} M$ one has
\begin{equation} \label{eqn6.7}
\Psi_{\sigma, \sigma'}^{M} (\operatorname{Tr}_{f} \circ \phi) =
\operatorname{Tr}_{f} \circ \Psi_{\tau, \tau'}^{M}(\phi) .
\end{equation}
\end{prop}
\begin{pf}
After making a reduction as in prop.\ \ref{prop6.2}, we can assume that
$L \stackrel{\simeq}{\rightarrow} A / {\frak m}$. Now set
$A^{-} := \sigma(K) \oplus {\frak m} \subset A$. Choose a homomorphism
$u : K \rightarrow \hat{K}$ as in Lemma \ref{lem6.5}, and define BCAs
$\hat{A}^{-} := A^{-} \otimes_{K}^{(\wedge)} \hat{K}$ and
$\hat{A} := A \otimes_{K}^{(\wedge)} \hat{K}$, w.r.t.\ the morphisms
$\sigma : K \rightarrow A^{-} \rightarrow A$. Let $v : A \rightarrow \hat{A}$ be the resulting
intensification homomorphism. The algebra $\hat{A}^{-}$ is local, with
maximal ideal $\hat{A}^{-} \cdot v({\frak m})$. Denote by $\hat{\frak{r}}$ the
Jacobson radical of $\hat{A}$; so
$\hat{\frak{r}} = \hat{A} \cdot v({\frak m})$. Set
$\hat{L} := \hat{A} / \hat{\frak{r}} \cong L \otimes_{K} \hat{K}$.
For each $\hat{{\frak m}} \in \operatorname{Max} \hat{A}$ denote by
$f_{\hat{{\frak m}}} : K \rightarrow L_{\hat{{\frak m}}}$,
$v_{\hat{{\frak m}}} : A \rightarrow \hat{A}_{\hat{{\frak m}}}$ and
$u_{\hat{{\frak m}}} : L \rightarrow \hat{L}_{\hat{{\frak m}}}$ the localized homomorphisms.
We have
$\hat{A}_{\hat{{\frak m}}} \cong A \otimes_{L}^{(\wedge)} \hat{L}_{\hat{{\frak m}}}$,
and there are coefficient fields
$\hat{\tau}_{\hat{{\frak m}}}, \hat{\tau}'_{\hat{{\frak m}}} :
\hat{L}_{\hat{{\frak m}}} \rightarrow \hat{A}_{\hat{{\frak m}}}$
extending $\tau, \tau'$. All the claims above follow from Prop.\
\ref{prop3.2}.
Let $\hat{M} := \hat{A} \otimes_{A} M$. For every
$\hat{{\frak m}} \in \operatorname{Max} \hat{A}$ there is a homomorphism
\[ q_{v_{\hat{{\frak m}}}; \tau}^{M} : \operatorname{Dual}_{\tau} M \rightarrow
\operatorname{Dual}_{\hat{\tau}_{\hat{{\frak m}}}} \hat{M}_{\hat{{\frak m}}}\, \]
and a corresponding homomorphism $q_{v_{\hat{{\frak m}}}; \tau'}^{M}$,
which, by Lemma \ref{lem6.3}, intertwine
$\Psi_{\tau, \tau'}^{M}$ with
$\Psi_{\hat{\tau}_{\hat{{\frak m}}}, \hat{\tau}_{\hat{{\frak m}}}'}^{
\hat{M}_{\hat{{\frak m}}}}$.
There are also (injective) homomorphisms
$q_{v; \sigma}^{M}$ and $q_{v; \sigma'}^{M}$.
Since the trace maps satisfy
\[ u \circ \operatorname{Tr}_{f} = \sum_{\hat{{\frak m}}}
\operatorname{Tr}_{f_{\hat{{\frak m}}}} \circ u_{\hat{{\frak m}}} \]
we get
\[ q_{v; \sigma}^{M} (\operatorname{Tr}_{f} \circ \phi) =
\sum_{\hat{{\frak m}}} \operatorname{Tr}_{f_{\hat{{\frak m}}}} \circ q_{v_{\hat{{\frak m}}}; \tau}^{M}
(\phi) \]
and similarly with $\sigma', \tau'$,
so the problem is reduced to the case when $k \rightarrow K$ is a morphism.
In this case, using \ref{lem6.4} twice and the transitivity of residues,
we get
\[ \operatorname{Res}_{K/k} \circ \Psi_{\sigma, \sigma'}^{M} (\operatorname{Tr}_{f} \circ \phi)
= \operatorname{Res}_{K/k} \circ \operatorname{Tr}_{f} \circ \Psi_{\tau, \tau'}^{M}(\phi) \]
which, in virtue of Lemma \ref{lem6.1} (a), implies formula
(\ref{eqn6.7}).
\end{pf}
We are ready to prove the first main result of this article.
\begin{thm} \label{thm6.1} \rom{(Dual Modules)}\
Let $A$ be a local Beilinson completion algebra over $k$, and let $M$ be a
finite type semi topological $A$-module.
Then the following data exist:
\begin{enumerate}
\rmitem{a} A ST $A$-module
$\operatorname{Dual}_{A} M$, called the {\em dual module} of $M$.
\rmitem{b} For every morphism
$\sigma : K \rightarrow A$ in $\mathsf{BCA}(k)$, with $K$ a field, an
isomorphism of ST $A$-modules
\[ \Psi_{\sigma}^{M} : \operatorname{Dual}_{A} M \stackrel{\simeq}{\rightarrow} \operatorname{Dual}_{\sigma} M =
\operatorname{Hom}^{\mathrm{cont}}_{K; \sigma}(M, \omega(K)). \]
\end{enumerate}
These data satisfy, and are completely determined by the following
conditions:
\begin{enumerate}
\rmitem{i} Let $f : K \rightarrow L$ and $\tau : L \rightarrow A$ be
morphisms in $\mathsf{BCA}(k)$, with $K,L$ fields, and let
$\sigma := \tau \circ f$. Then for any $\phi \in \operatorname{Dual}_{A} M$,
\[ \Psi_{\sigma}^{M}(\phi) = \operatorname{Res}_{f} \circ \Psi_{\tau}^{M}(\phi). \]
Here
$\operatorname{Res}_{f}: \omega(L) \rightarrow \omega(K)$ is the residue map in
$\mathsf{TLF}(k)$, cf.\ \cite{Ye1} \S \rom{2.4}.
\rmitem{ii} Denote by ${\frak m}$ the maximal ideal of $A$.
If $\sigma,\sigma' : K \rightarrow A$ are pseudo coefficient fields such that
$\sigma \equiv \sigma'\ (\operatorname{mod} {\frak m})$, then
\[ \Psi_{\sigma'}^{M} = \Psi_{\sigma,\sigma'}^{M} \circ \Psi_{\sigma}^{M}, \]
where $\Psi_{\sigma,\sigma'}^{M}$ is the isomorphism defined in
Def.\ \ref{dfn6.2}.
\end{enumerate}
\end{thm}
Observe that if $A=K$ is a TLF, then there
is a canonical isomorphism
$\operatorname{Dual}_{K} M$ \linebreak
$\cong \operatorname{Hom}_{K}(M, \omega(K))$,
corresponding to the identity morphism $K \rightarrow K$, thought of as a
coefficient field.
\begin{pf}
The proof is divided into four steps.
\medskip \noindent (1)\
Fix a coefficient field
$\tau_{0} : L_{0} = A / {\frak m} \rightarrow A$, and set
$\operatorname{Dual} M := \operatorname{Dual}_{\tau_{0}} M$. Given another coefficient field
$\tau : L_{0} \rightarrow A$, we are forced by condition (ii) to define
$\Psi_{\tau}^{M} := \Psi_{\tau_{0}, \tau}^{M}$. For any other coefficient field
$\tau' : L_{0} \rightarrow A$ this condition is satisfied, on account of Lemma
\ref{lem6.0}; condition (i) is irrelevant.
\medskip \noindent (2)\
Now let $\sigma : K \rightarrow A$ be a pseudo coefficient field which factors
through some coefficient field
$\tau : L_{0} \rightarrow A$ (if $\sigma$ is a quasi coefficient field then there is
precisely one such $\tau$). Define
$\Psi_{\sigma}^{M} : \operatorname{Dual} M \stackrel{\simeq}{\rightarrow} \operatorname{Dual}_{\sigma}$ to be
$\phi \mapsto \operatorname{Tr}_{L_{0} / K} \circ \Psi_{\tau_{0}, \tau}^{M}(\phi)$,
as is forced by condition (i). According to Prop.\ \ref{prop6.3}, this
definition is independent of the coefficient field $\tau$.
\medskip \noindent (3)\
Let $\sigma : K \rightarrow A$ be any pseudo coefficient field. Choose some
pseudo coefficient field $\sigma' : K \rightarrow A$ such that
$\sigma \equiv \sigma'\ (\operatorname{mod} {\frak m})$ and such that $\sigma'$ factors
through some coefficient field. For example, take
$\sigma' := \tau_{0} \circ \pi \circ \sigma$, where $\pi : A \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} L_{0}$
is the natural projection. Define
$\Psi_{\sigma}^{M} := \Psi_{\sigma', \sigma}^{M} \circ
\Psi_{\sigma'}^{M}$. Prop.\ \ref{prop6.3} shows that this definition is
independent of the choice of $\sigma'$, and furthermore it shows that
conditions (i) and (ii) hold for all pseudo coefficient fields.
\medskip \noindent (4)\
Finally let $\sigma : K \rightarrow A$ be a morphism with
$\operatorname{res.dim} \sigma \geq 1$. Choose a
factorization $\sigma = \tau \circ f$, with
$\tau : L \rightarrow A$ a pseudo coefficient field and $f : K \rightarrow L$ a morphism.
Define
$\Psi_{\sigma}^{M}(\phi) := \operatorname{Res}_{f} \circ \Psi_{\tau}^{M}(\phi)$,
$\phi \in \operatorname{Dual} M$. Now condition (ii) is no longer relevant.
To verify condition (i) it suffices to prove the
independence of this definition on $\tau$. So suppose that $\sigma$ also
factors into $\sigma = \tau' \circ f'$.
First assume there exists some finite morphism $g : L \rightarrow L'$ such that
$\tau = \tau' \circ g$ and $f' = g \circ f$. Then applying condition (i) to
$\tau = \tau' \circ g$, we get
\begin{equation} \label{eqn6.9}
\operatorname{Res}_{f} \circ \Psi_{\tau}^{M}(\phi) =
\operatorname{Res}_{f} \circ \operatorname{Tr}_{g} \circ \Psi_{\tau'}^{M}(\phi) =
\operatorname{Res}_{f'} \circ \Psi_{\tau'}^{M}(\phi)
\end{equation}
for $\phi \in \operatorname{Dual} M$. By taking $L'$ to be the separable closure of
$L$ in $L_{0}$, and then using formula (\ref{eqn6.9}), we can assume that
$L \rightarrow L_{0}$ is purely inseparable.
It remains to consider the case when $L,L' \subset L_{0}$, and both
$L \rightarrow L_{0}$ and $L' \rightarrow L_{0}$ are purely inseparable. Choose $j >> 0$
such that
$L_{0}^{(p^{j}/k)} \subset L \cap L'$. Define
$L_{1} := K L_{0}^{(p^{j}/k)} \subset L_{0}$ and let
$\tau_{1}, \tau_{1}' : L_{1} \rightarrow A$
be the restrictions of $\tau, \tau'$. To finish the verification use formula
(\ref{eqn6.9}) twice more.
\end{pf}
\section{Traces on Dual Modules}
As before, $k$ is a fixed perfect field.
Suppose $A$ is a local BCA.
Then $\operatorname{Dual}_{A}: M \mapsto \operatorname{Dual}_{A} M$ is a functor on the category
of finite type ST $A$-modules.
Given a finite type ST $A$-module $M$ and an element $x \in M$, let
$\rho_{x} : A \rightarrow M$ be the function $a \mapsto ax$. As in \cite{LC} Lemma
4.1, and by our Lemma \ref{lem6.1} (b),
sending $\phi \in \operatorname{Dual}_{A} M$ to the homomorphism
$x \mapsto \operatorname{Dual}_{A}(\rho_{x})(\phi)$ gives an isomorphism
$\operatorname{Dual}_{A} M \rightarrow \operatorname{Hom}_{A}(M, \cal{K}(A))$.
Any BCA $A$ over $k$ decomposes into local factors:
$A = \prod_{{\frak m} \in \operatorname{Max} A} A_{{\frak m}}$, as ST $k$-algebras. Any
morphism in $\mathsf{BCA}(k)$ decomposes accordingly.
\begin{dfn} \label{dfn7.3}
Let $A$ be a BCA over $k$. Define
\[ \cal{K}(A) := \bigoplus_{{\frak m} \in \operatorname{Max} A} \operatorname{Dual}_{A_{{\frak m}}} A_{{\frak m}}.
\]
Given any ST $A$-module $M$, define
\[ \operatorname{Dual}_{A} M := \operatorname{Hom}_{A}^{\operatorname{cont}}(M, \cal{K}(A)) \]
with the (weak) $\operatorname{Hom}$ topology.
\end{dfn}
With this definition $\operatorname{Dual}_{A}$ is an additive functor
$\mathsf{STMod}(A)^{\circ} \rightarrow \mathsf{STMod}(A)$.
In view of the previous discussion and Prop.\ \ref{prop1.1} (2), there is
no conflict of definitions when
$A$ is local and $M$ is a ST $A$-module of finite type.
\begin{prop} \label{prop7.1} \rom{(Covariance of dual modules)}\
Let $v: A \rightarrow \hat{A}$ be an intensification homomorphism between two BCAs.
Given a ST $A$-module $M$, set $\hat{M} := \hat{A} \otimes_{A} M$. Then
there is a unique homomorphism in $\mathsf{STMod}(A)$,
\[ q_{v}^{M} : \operatorname{Dual}_{A} M \rightarrow \operatorname{Dual}_{\hat{A}} \hat{M}, \]
with the following properties:
\begin{enumerate}
\rmitem{i} If $\phi: M \rightarrow N$ is a homomorphism in $\mathsf{STMod}(A)$,
then
\[ q_{v}^{M} \circ \operatorname{Dual}_{A}(\phi) =
\operatorname{Dual}_{\hat{A}} (1 \otimes \phi) \circ q^{N}_{v}. \]
In other words,
$q_{v}: \operatorname{Dual}_{A} \rightarrow \operatorname{Dual}_{\hat{A}} (\hat{A} \otimes_{A} -)$
is a natural transformation of functors.
\rmitem{ii} If $M$ is a ST $A$-module of finite type then the induced
homomorphism
\[ 1 \otimes q^{M}_{v} : \hat{A} \otimes_{A} \operatorname{Dual}_{A} M
\rightarrow \operatorname{Dual}_{\hat{A}} \hat{M} \]
is an isomorphism.
\rmitem{iii} Let $\sigma : K \rightarrow A$ be a morphism in $\mathsf{BCA}(k)$.
Suppose $K$ is a field, $A$ is local, and there is an intensification
homomorphism $u : K \rightarrow \hat{K}$ s.t.\
$\hat{A} \cong A \otimes_{K}^{(\wedge)} \hat{K}$. Then for any ST $A$-module
of finite type $M$,
\[ q^{M}_{v} = (\Psi_{\hat{\sigma}}^{\hat{M}})^{-1} \circ
q^{M}_{v; \sigma} \circ \Psi_{\sigma}^{M} . \]
\rmitem{iv} If $w: \hat{A} \rightarrow \Hat{\Hat{A}}$ is another
flat, finitely ramified, radically unramified and topologically
\'{e}tale homomorphism, then
$q_{w \circ v} = q_{w} \circ q_{v}$.
\end{enumerate}
These properties characterize $q^{M}_{v}$.
\end{prop}
\begin{pf}
We may assume $A$ is local. Let us first check uniqueness. If $M$ is a
finite type ST $A$-module, this follows from condition (iii). If $M$ has
the fine topology then $M \cong \lim_{\alpha \rightarrow} M_{\alpha}$ with each
$M_{\alpha}$ a finite type module. By Lemma \ref{lem1.1} (4) we get
$\operatorname{Dual}_{A} M \cong \lim_{\leftarrow \alpha} \operatorname{Dual}_{A} M_{\alpha}$,
and we may use condition (i). Finally any ST $A$-module $M$ is a quotient of a
module $\tilde{M}$ which has the fine topology, and
$\operatorname{Dual}_{A} M \hookrightarrow \operatorname{Dual}_{A} \tilde{M}$.
To define $q^{M}_{v}$ for $M$ of finite type amounts, essentially,
to repeating the steps of the proof of Thm.\ \ref{thm6.1}, using Lemma
\ref{lem6.3} at every step. For a general ST $A$-module $M$, let
$q^{M}_{v}$ be the canonical continuous homomorphism
\[ \operatorname{Hom}_{A}^{\operatorname{cont}}(M, \cal{K}(A)) \rightarrow
\operatorname{Hom}_{\hat{A}}^{\operatorname{cont}}(\hat{A} \otimes_{A} M, \cal{K}(\hat{A})) \]
induced by
$q_{v} = q_{v}^{A} : \cal{K}(A) \rightarrow \cal{K}(\hat{A})$.
\end{pf}
\begin{dfn} \label{dfn7.2}
Let $K,A \in \mathsf{BCA}(k)$, with $K$ a field, and let
$\sigma : K \rightarrow A$ be a morphism. Define
\[ \operatorname{Res}_{A/K} = \operatorname{Res}_{\sigma} : \cal{K}(A) \rightarrow
\omega(K) \]
to be the function sending
\[ \phi = \sum_{\frak{m}} \phi_{\frak{m}} \in \cal{K}(A) =
\bigoplus_{\frak{m}} \operatorname{Dual}_{A_{\frak{m}}} A_{\frak{m}} \]
to
$\sum_{\frak{m}} \Psi_{\sigma}^{A_{\frak{m}}}(\phi_{\frak{m}})(1) \in
\omega(K)$. Here $\frak{m}$ runs through the maximal ideals of $A$.
\end{dfn}
The residue map $\operatorname{Res}_{A/K}$ is $K$-linear. It is also continuous:
this follows from the adjunction formula, Lemma \ref{lem1.2}
(cf.\ Remark \ref{rem6.1}).
Because of the transitivity of residues, if there is a factorization
$\sigma: K \exar{ f } L \exar{ \tau } A$, then
$\operatorname{Res}_{A/K}= \operatorname{Res}_{L/K} \circ \operatorname{Res}_{A/L}$.
Here is the second main result of this article:
\begin{thm} \label{thm7.2}
\rom{(Traces)}\ Let $f : A \rightarrow B$ be a morphism in $\mathsf{BCA}(k)$.
There is a unique continuous $A$-linear homomorphism
\[ \operatorname{Tr}_{B/A} = \operatorname{Tr}_{f} : \cal{K}(B) \rightarrow \cal{K}(A) \]
having the following properties:
\begin{enumerate}
\rmitem{i} (Transitivity)\ Given another morphism $g : B \rightarrow C$, one has
\[ \operatorname{Tr}_{C/A} = \operatorname{Tr}_{B/A} \circ \operatorname{Tr}_{C/B} . \]
\rmitem{ii} (Base Change )\ Suppose $u : A \rightarrow \hat{A}$ is an intensification
homomorphism. Let
$\hat{B} := B \otimes_{A}^{(\wedge)} \hat{A}$, $v : B \rightarrow \hat{B}$
and $\hat{f} : \hat{A} \rightarrow \hat{B}$ be the algebras and homomorphisms gotten
by intensification base change (cf.\ Thm.\ \ref{thm3.1}). Then
\[ q_{u} \circ \operatorname{Tr}_{B/A} =
\operatorname{Tr}_{\hat{B}/\hat{A}} \circ q_{v}, \]
where $q_{u}, q_{v}$ are the homomorphisms of Prop.\ \ref{prop7.1}.
\rmitem{iii} If $A$ is a field, then
$\operatorname{Tr}_{B/A} = \operatorname{Res}_{B/A} : \cal{K}(B) \rightarrow \cal{K}(A) = \omega(A)$.
\rmitem{iv} The map
\[ \cal{K}(B) \rightarrow \operatorname{Hom}_{A}^{\mathrm{cont}}(B, \cal{K}(A)) \]
induced by $\operatorname{Tr}_{B/A}$ is bijective.
\end{enumerate}
\end{thm}
\begin{pf}
We may assume both $A,B$ are local, with maximal ideals ${\frak m}, \frak{n}$.
Given any morphism $\sigma: K \rightarrow A$ with $K$ a field, define
$\operatorname{Tr}_{B/A; \sigma} : \cal{K}(B) \rightarrow \cal{K}(A)$
by
\begin{equation} \label{eqn7.2}
\operatorname{Tr}_{B/A; \sigma} := (\Psi_{\sigma}^{A})^{-1} \circ
\operatorname{Dual}_{\sigma}(f) \circ \Psi_{f \circ \sigma}^{B}
\end{equation}
where $\operatorname{Dual}_{\sigma}(f)(\phi) = \phi \circ f$ for
$\phi \in \operatorname{Dual}_{f \circ \sigma} B$.
The claim is that $\operatorname{Tr}_{B/A; \sigma}$ is independent of $\sigma$.
Let $\tau : L = A / {\frak m} \rightarrow A$ be any coefficient field. It suffices to
prove that
$\operatorname{Tr}_{B/A; \sigma} = \operatorname{Tr}_{B/A; \tau}$. To do so we choose an
intensification homomorphism $K \rightarrow \hat{K}$ s.t.\
$k \rightarrow \hat{K}$ is a morphism of BCAs, and set
$\hat{A} := A \otimes_{K}^{(\wedge)} \hat{K}$,
$\hat{B} := B \otimes_{K}^{(\wedge)} \hat{K}$ and
$\hat{L} := L \otimes_{K}^{(\wedge)} \hat{K}$.
Let $\hat{\tau} : \hat{L} \rightarrow \hat{A}$ be the unique extension of $\tau$.
Note that by Prop.\ \ref{prop3.2},
$\hat{A} \cong A \otimes_{L}^{(\wedge)} \hat{L}$.
According to Prop.\ \ref{prop7.1} (iii),
\[ q_{\hat{A} / A} \circ \operatorname{Tr}_{B / A; \sigma} =
\operatorname{Tr}_{\hat{B} / \hat{A}; \hat{\sigma}} \circ q_{\hat{B} / B} \]
and similarly for $\tau$. Since $\rho : k \rightarrow \hat{A}$ is a morphism,
we get (using Thm.\ \ref{thm6.1} (i))
\[ \operatorname{Tr}_{\hat{B} / \hat{A}; \hat{\sigma}} =
\operatorname{Tr}_{\hat{B} / \hat{A}; \rho} =
\operatorname{Tr}_{\hat{B} / \hat{A}; \hat{\tau}} . \]
But $q_{\hat{A} / A}$ is injective, so the claim is proved. Our arguments
also imply properties (i), (ii) and (iii).
Let us now prove that $\operatorname{Tr}_{B/A}$ is continuous. First assume that
$\operatorname{res.dim} f \leq 1$. Then
$\cal{K}(B)$, being a cofinite type ST $B$-module, actually has the fine
$A$-module topology. (cf.\ \cite{Ye1} Def.\ 3.3 or Def.\ 3.2.1 (b.ii)). Since
$\operatorname{Tr}_{B/A} : \cal{K}(B) \rightarrow \cal{K}(A)$ is $A$-linear, it is
continuous. Now assume $\operatorname{res.dim} f = n > 1$.
Consider the prime ideal
$\frak{p} := \operatorname{Ker}(A \rightarrow \kappa_{n-1}(B))$.
We can assume that $\frak{p} \neq {\frak m}$, by replacing (if necessary) $A$
with $A [\sqbr{ t }]$, and sending $t$ to a parameter of
$\cal{O}_{n}(B)$. Thus $A / \frak{p}$ is a DVR and
$C := \lim_{\leftarrow i} (A / \frak{p}^{i})_{\frak{p}}$
is a BCA. The morphism $A \rightarrow B$ factors into morphisms $A \rightarrow C \rightarrow B$,
both of $\operatorname{res.dim} < n$. By
induction $\operatorname{Tr}_{B/C}$ and $\operatorname{Tr}_{C/A}$ are continuous, and
$\operatorname{Tr}_{B/A} = \operatorname{Tr}_{B/C}\ \circ \operatorname{Tr}_{C/A}$.
Finally to prove (iv), take a coefficient field $\sigma: K \rightarrow A$. Then
\[ \Psi^{B}_{f \circ \sigma} : \cal{K}(B) \rightarrow
\operatorname{Dual}_{f \circ \sigma} B =
\operatorname{Hom}_{K}^{\mathrm{cont}}(B, \omega(K)) \]
is bijective. On the other hand, one easily sees that
\[ \operatorname{Hom}_{A}^{\mathrm{cont}}(B, \cal{K}(A)) \rightarrow
\operatorname{Hom}_{K}^{\mathrm{cont}}(B, \omega(K)) \]
is injective, so
$\cal{K}(B) \stackrel{\simeq}{\rightarrow} \operatorname{Hom}_{A}^{\mathrm{cont}}(B, \cal{K}(A))$.
\end{pf}
\begin{rem} \label{rem7.1}
Suppose $A, B$ are BCAs,
$f: A \rightarrow B$ is a continuous $k$-algebra homomorphism, and $M$ is a
torsion type ST $A$-module. For instance,
$A, B$ could be any complete local $k$-algebras which are residually
finitely generated over $k$, $f$ could be any
local homomorphism, and $M$ any $0$-dimensional $A$-module.
If $M$ has finite length, define
\[ f_{\#} M := \operatorname{Dual}_{B} (B \otimes_{A} \operatorname{Dual}_{A} M). \]
Otherwise $M = \lim_{\alpha \rightarrow} M_{\alpha}$ where each $M_{\alpha}$ has
finite length, and we set
$f_{\#} M := \lim_{\alpha \rightarrow} f_{\#} M_{\alpha}$.
This gives a functor
$f_{\#} : \mathsf{STMod}_{\mathrm{tors}}(A) \rightarrow
\mathsf{STMod}_{\mathrm{tors}}(B)$. Note that
$f_{\#} \cal{K}(A) = \cal{K}(B)$. If $f$ is a morphism in $\mathsf{BCA}(k)$,
the trace map $\operatorname{Tr}_{f}: \cal{K}(B) \rightarrow \cal{K}(A)$ defines a trace map
$\operatorname{Tr}_{f} : f_{\#} M \rightarrow M$ for any $M$.
The collection of data
$(\mathsf{STMod}_{\mathrm{tors}}(A), f_{\#})$ is a realization (and
generalization) of Lipman's pseudofunctor on $0$-dimensional modules;
cf.\ \cite{Hg}.
\end{rem}
\section{Duals of Continuous Differential Operators}
In this section we consider a continuous differential operator
$D : M \rightarrow N$, and construct a dual operator
$\operatorname{Dual}_{A}(D) : \operatorname{Dual}_{A} N \rightarrow \operatorname{Dual}_{A} M$.
The idea is to use the right $\cal{D}(K)$-module structure of
$\omega(K)$, for a TLF $K$.
Let $A$ be a local BCA with maximal ideal ${\frak m}$, and let
$\sigma : K \rightarrow A$ be a pseudo coefficient field. Given two finite type
ST $A$-modules $M,N$, choose ${\frak m}$-filtered $K$-bases
$\underline{x} = (x_{0}, x_{1}, \ldots)$ and
$\underline{y} = (y_{0}, y_{1}, \ldots)$ for $M$ and $N$, respectively (cf.\ Def.\
\ref{def6.3}).
Suppose $D : M \rightarrow N$ is a continuous DO over $A$ relative to $k$. For
$i,j \geq 0$ let
$D_{ij} : K \rightarrow K$ be the functions such that, for $\lambda \in K$,
\[ D(\sigma(\lambda) x_{i}) = \sum_{j} \sigma(D_{ij}(\lambda)) y_{j} . \]
Then, just like in Lemma \ref{lem6.2}, $D_{ij} \in \cal{D}(K)$.
\begin{dfn} \label{dfn8.1}
Let
$\operatorname{Dual}_{\sigma}(D) : \operatorname{Dual}_{\sigma} N \rightarrow \operatorname{Dual}_{\sigma} M$
be the function taking
$\phi \in \operatorname{Dual}_{\sigma} N$ to
\[ \operatorname{Dual}_{\sigma}(D) (\phi) :
\sum_{i} \sigma(\lambda_{i}) x_{i} \mapsto
\sum_{i,j} \lambda_{i} (\phi(y_{j}) * D_{ij}) . \]
\end{dfn}
There is no reference in the notation ``$\operatorname{Dual}_{\sigma}(D)$'' to the bases
$\underline{x}, \underline{y}$. This is not an oversight - as we shall see, this function
is independent of the bases. First, another definition:
\begin{dfn} \label{dfn8.0}
Let $M$ be a ST $A$-module (not necessarily of finite type).
Define the residue pairing to be
\begin{eqnarray*}
\langle -,- \rangle_{A/K}^{M} & : & M \times \operatorname{Dual}_{A} M \rightarrow \omega(K) \\
\langle x, \phi \rangle_{A/K}^{M} & = & \operatorname{Res}_{A/K}(\phi(x))
\end{eqnarray*}
where $\operatorname{Res}_{A/K}$ is as in Def.\ \ref{dfn7.2}.
\end{dfn}
\begin{rem} \label{rem8.1}
Suppose $K$ is discrete (i.e.\ $\operatorname{dim} K = 0$) and $M$ is a finite type
or a cofinite type ST
$A$-module. Then the topology on $M$ is $K$-linear (cf.\ \cite{Ye1} Prop.\
3.2.5). As a topological vector space over $K$, $M$ is strongly
reflexive, in the sense of \cite{Ko} \S 13.3. One can show that the strong
$\operatorname{Hom}_{K}$ topology on
$\operatorname{Dual}_{A} M \cong \operatorname{Hom}_{K}^{\mathrm{cont}}(M, \omega(K))$ coincides
with the fine $A$-module topology on it. Hence \linebreak
$\langle -,- \rangle_{A/K}^{M}$ is a perfect pairing also from the point
of view of \cite{Ko}.
\end{rem}
\begin{lem} \label{lem8.2} \mbox{ }
\begin{enumerate}
\rmitem{a} Suppose $\operatorname{ord}_{K}(D) = 0$, i.e.\ $D$ is $K$-linear. Then
$\operatorname{Dual}_{\sigma}(D) (\phi) = \phi \circ D$ for all
$\phi \in \operatorname{Dual}_{\sigma} N$.
\rmitem{b} Suppose $k \rightarrow K$ is a morphism in $\mathsf{BCA}(k)$. Then
for all $\phi \in \operatorname{Dual}_{\sigma} N$,
\[ \operatorname{Res}_{K/k} \circ \operatorname{Dual}_{\sigma}(D) (\phi) =
\operatorname{Res}_{K/k} \circ \phi \circ D . \]
In other words, $\operatorname{Dual}_{\sigma}(D)$ is adjoint to $D$ with respect to the
the residue pairings
$\langle -,- \rangle_{A/k}^{M}$ and
$\langle -,- \rangle_{A/k}^{N}$.
\end{enumerate}
\end{lem}
\begin{pf}
One has $D_{ij} = \mu_{ij} \in K \subset \cal{D}(K)$, where
$D(x_{i}) = \sum_{j} \sigma(\mu_{ij}) y_{j}$. Now simply plug this into the
definition of $\operatorname{Dual}_{\sigma}(D)$.
\medskip \noindent (b)\ Say
$\phi(y_{j}) = \alpha_{j} \in \omega(K)$. Given
$x = \sum_{i} \sigma(\lambda_{i}) x_{i} \in M$, with $\lambda_{i} \in K$,
we have by the definition of the DOs $D_{ij}$:
\[ D(x) = \sum_{i,j} \sigma(D_{ij} * \lambda_{i}) y_{j}\ , \]
so
\begin{eqnarray*}
\lefteqn{ \langle D(x), \phi \rangle_{A/k}^{N} =
\operatorname{Res}_{K/k} \circ \phi \circ D (x) }\\
& & = \operatorname{Res}_{K/k} (\sum_{i,j} (D_{ij} * \lambda_{i}) \alpha_{j}) =
\sum_{i,j} \langle D_{ij} * \lambda_{i}, \alpha_{j} \rangle_{K/k}.
\end{eqnarray*}
On the other hand, by the definition of $\operatorname{Dual}_{\sigma}(D)$,
\begin{eqnarray*}
\lefteqn{ \langle x, \operatorname{Dual}_{\sigma}(D)(\phi) \rangle_{A/k}^{M} =
\operatorname{Res}_{K/k} \circ (\operatorname{Dual}_{\sigma}(D) (\phi)) (x) } \\
& & = \operatorname{Res}_{K/k} (\sum_{i,j} \lambda_{i} (\alpha_{j} * D_{ij})) =
\sum_{i,j} \langle \lambda_{i}, \alpha_{j} * D_{ij} \rangle_{K/k}.
\end{eqnarray*}
Now use Thm.\ \ref{thm5.1}.
\end{pf}
\begin{lem} \label{lem8.3} \mbox{ }
\begin{enumerate}
\rmitem{a} $\operatorname{Dual}_{\sigma}(D) : \operatorname{Dual}_{\sigma} N \rightarrow
\operatorname{Dual}_{\sigma} M$
is a continuous DO over $A$, relative to $k$, of order $\leq \operatorname{ord}_{A}(D)$.
It is independent of the ${\frak m}$-filtered $K$-bases $\underline{x}, \underline{y}$.
\rmitem{b} Let ${\frak m} \subset A$ be the maximal ideal. Suppose
$\sigma' : K \rightarrow A$ is another pseudo coefficient field, s.t.\
$\sigma' \equiv \sigma\ (\operatorname{mod} {\frak m})$. Then
\[ \operatorname{Dual}_{\sigma'}(D) = \Psi_{\sigma, \sigma'}^{M} \circ
\operatorname{Dual}_{\sigma}(D) \circ \Psi_{\sigma', \sigma}^{N} . \]
\rmitem{c} Suppose $\tau : L \rightarrow A$ is another pseudo coefficient field, and
$f : K \rightarrow L$ is a (finite) morphism in $\mathsf{BCA}(k)$, s.t.\
$\sigma = \tau \circ f$. Then for each $\phi \in \operatorname{Dual}_{\tau} N$,
\[ \operatorname{Dual}_{\sigma}(D) (\operatorname{Tr}_{f} \circ \phi) = \operatorname{Tr}_{f} \circ
\operatorname{Dual}_{\tau}(D) (\phi) . \]
\end{enumerate}
\end{lem}
\begin{pf}
The proof resembles that of Prop.\ \ref{prop6.2}. Choose an intensification
homomorphism $u : K \rightarrow \hat{K}$
such that $k \rightarrow \hat{K}$ is a morphism. Let
$\hat{A} := A \otimes_{K}^{(\wedge)} \hat{K}$ and $v : A \rightarrow \hat{A}$.
Replacing $A$ with each of the localizations $\hat{A}_{\hat{{\frak m}}}$,
$\hat{{\frak m}} \in \operatorname{Max} \hat{A}$, allows us to assume that $k \rightarrow K$ is
itself a morphism in $\mathsf{BCA}(k)$. By Lemma \ref{lem8.2} (b) we see
that $\operatorname{Dual}_{\sigma}(D)$ is the adjoint of $D$ w.r.t.\
the residue pairings $\langle -,- \rangle_{A/k}^{M}$ and
$\langle -,- \rangle_{A/k}^{N}$, so in particular it is independent of the
${\frak m}$-filtered $K$-bases $\underline{x}, \underline{y}$. It also follows
that for any $a \in A$,
\[ [\operatorname{Dual}_{\sigma}(D), a] = - \operatorname{Dual}_{\sigma}([D,a]) :
\operatorname{Dual}_{\sigma} M \rightarrow \operatorname{Dual}_{\sigma} N, \]
bounding the order of the operator $\operatorname{Dual}_{\sigma}(D)$.
Here ``$[-,-]$'' denotes the commutator.
Parts (b),(c) of the present lemma are similarly proved, using Lemma
\ref{lem6.3}.
As for the continuity of $\operatorname{Dual}_{\sigma}(D)$, it can be deduced from the
fact that it is a linear combination of the continuous operators $D_{ij}$
appearing in its definition.
\end{pf}
The ST $A$-module $\cal{K}(A)$ is separated. Therefore for any ST $A$-module
$M$, the canonical surjection $M \mbox{$\rightarrow \! \! \! \! \! \rightarrow$} M^{\operatorname{sep}}$ induces an isomorphism
$\operatorname{Dual}_{A} M^{\operatorname{sep}} \stackrel{\simeq}{\rightarrow} \operatorname{Dual}_{A} M$.
Here is the third main result of the paper:
\begin{thm} \label{thm8.1} \rom{(Duals of Continuous DOs)}\
Let $A$ be a BCA over $k$. Let $M$ and $N$ be ST $A$-modules with the fine
topologies, and let $D : M \rightarrow N$ be a continuous DO over $A$ relative to
$k$. Then there is a unique function
\[ \operatorname{Dual}_{A}(D) : \operatorname{Dual}_{A} N \rightarrow \operatorname{Dual}_{A} M, \]
satisfying the conditions below:
\begin{enumerate}
\rmitem{i} $\operatorname{Dual}_{A}(D) : \operatorname{Dual}_{A} N \rightarrow \operatorname{Dual}_{A} M$ is a
continuous DO over $A$ relative to $k$, of order $\leq \operatorname{ord}_{A}(D)$.
\rmitem{ii} (Transitivity)\ if $E : N \rightarrow P$ is another such operator,
then $\operatorname{Dual}_{A}(E \circ D) = \operatorname{Dual}_{A}(D) \circ \operatorname{Dual}_{A}(E)$.
\rmitem{iii} (Linearity)\
if $D$ is $A$-linear, then $\operatorname{Dual}_{A}(D)$ is the homomorphism
$\phi \mapsto \phi \circ D$, for
$\phi \in \operatorname{Dual}_{A} N = \operatorname{Hom}^{\mathrm{cont}}_{A}(N, \cal{K}(A))$.
\rmitem{iv} (Base change)\ let $v : A \rightarrow \hat{A}$ be an intensification
homomorphism, and let
$\hat{D} : (\hat{A} \otimes_{A} M)^{\operatorname{sep}}
\rightarrow (\hat{A} \otimes_{A} N)^{\operatorname{sep}}$ be the unique extension of $D$. Then
\[ \operatorname{Dual}_{\hat{A}}(\hat{D}) \circ q^{N}_{v} =
q^{M}_{v} \circ \operatorname{Dual}_{A}(D), \]
where $q^{M}_{v}, q^{N}_{v}$ are the homomorphisms of Prop.\ \ref{prop7.1}.
\rmitem{v} Assume $\sigma: K \rightarrow A$ is a morphism in $\mathsf{BCA}(k)$ s.t.\
$D$ is $K$-linear. Then $\operatorname{Dual}_{A}(D)$ is the adjoint to $D$ w.r.t.\ the
residue pairings
$\langle - , - \rangle^{M}_{A/K}$ and
$\langle - , - \rangle^{N}_{A/K}$.
\rmitem{vi} Suppose $A$ is local and $M,N$ are finite type ST
$A$-modules.
Given a pseudo coefficient field $\sigma : K \rightarrow A$, one has
\[ \Psi_{\sigma}^{M} \circ \operatorname{Dual}_{A}(D) = \operatorname{Dual}_{\sigma} (D)
\circ \Psi_{\sigma}^{N} . \]
Here $\Psi_{\sigma}^{M}, \Psi_{\sigma}^{N}$ are the isomorphisms of Thm.\
\ref{thm6.1}, and $\operatorname{Dual}_{\sigma} (D)$ is the function defined in Def.\
\ref{dfn8.1}.
\end{enumerate}
\end{thm}
\begin{rem}
Trivially, the category $\mathsf{Mod}(A)$ of
$A$-modules and $A$-linear homomorphisms, and the category
$\mathsf{STMod}_{\operatorname{fine}}(A)$ of ST $A$-modules with fine topologies
and continuous
$A$-linear homomorphisms, are equivalent (under the functor
$\operatorname{untop} : \mathsf{STMod}(A) \rightarrow \mathsf{Mod}(A)$ which forgets
the topology). However, if we take the same classes of objects, but
enlarge the set of morphisms between two objects to be DOs and continuous
DOs, respectively, these new categories are no longer equivalent. This is
so at least when $\operatorname{char} k = 0$ and $\operatorname{res.dim} A \geq 1$.
Our results are valid only for continuous DOs.
\end{rem}
\begin{pf}
Let $M,N$ be finite type ST $A$-modules. Using Lemma \ref{lem8.3}, and
proceeding just like in the proofs of Theorems \ref{thm6.1} and
\ref{thm7.2}, we arrive at a function $\operatorname{Dual}_{A}(D)$ which satisfies
conditions (i)-(iv), (vi).
As for condition (v), after a base change $K \rightarrow \hat{K}$ we reduce to the
case when $k \rightarrow A$ is a morphism. Now we can use Lemma \ref{lem8.2} (b).
Now let $M,N$ be ST $A$-modules with fine topologies. After possibly
applying $(-)^{\operatorname{sep}}$ to these modules, we may assume they are separated.
Choose an isomorphism $M \cong \lim_{\alpha \rightarrow} M_{\alpha}$, with the
$M_{\alpha}$ modules of finite type. Let $N_{\alpha}$ be the $A$-module
$A \cdot D(M_{\alpha}) \subset N$, endowed with the fine topology.
Say $d = \operatorname{ord}_{A}(D)$. Because
$\cal{P}_{A/k}^{d,\operatorname{sep}}(M_{\alpha})$ is a finite type ST $A$-module
(by Prop.\ \ref{prop4.3}), $N_{\alpha}$ is of finite type, and
$D_{\alpha} := D|_{M_{\alpha}} : M_{\alpha} \rightarrow N_{\alpha}$ is continuous.
Let $\psi : \lim_{\alpha \rightarrow} N_{\alpha} \rightarrow N$ be the inclusion, and set
\[ \operatorname{Dual}_{A}(D) := (\lim_{\leftarrow \alpha}
\operatorname{Dual}_{A}(D_{\alpha}) ) \circ \operatorname{Dual}_{A}(\psi) . \]
Since the functor $\operatorname{Dual}_{A}$ sends $\lim_{\rightarrow}$ to $\lim_{\leftarrow}$
(cf.\ Lemma \ref{lem1.1} (4)),
this extended
definiton of $\operatorname{Dual}_{A}(D)$ satisfies all the conditions of the theorem.
\end{pf}
Occasionally we shall abbreviate $\operatorname{Dual}_{A} M$ to $M^{\vee}$, and
$\operatorname{Dual}_{A}(D)$ to $D^{\vee}$.
\begin{cor} \label{cor8.2} \mbox{ }
\begin{enumerate}
\rmitem{a} Let $M,N$ be each either finite type or cofinite type ST
$A$-modules, and let
$D \in \operatorname{Diff}_{A/k}^{\operatorname{cont}}(M,N)$. Then under the canonical
isomorphisms $M \stackrel{\simeq}{\rightarrow} M^{\vee \vee}$ and $N \stackrel{\simeq}{\rightarrow} N^{\vee \vee}$, one has
$D \mapsto D^{\vee \vee}$.
\rmitem{b} With $M,N$ as in \rom{(a)}, the map
$\operatorname{Diff}_{A/k}^{\operatorname{cont}}(M,N) \rightarrow
\operatorname{Diff}_{A/k}^{\operatorname{cont}}(M^{\vee},N^{\vee})$,
$D \mapsto D^{\vee}$, is an anti-isomorphism of filtered $A$-$A$-bimodules.
In particular,
$\cal{D}(A;$ \linebreak
$\cal{K}(A)) \cong \cal{D}(A)^{\circ}$
as filtered $k$-algebras.
\end{enumerate}
\end{cor}
\begin{pf}
(a)\ Using base change we can assume that $k \rightarrow A$ is a morphism in
$\mathsf{BCA}(k)$. Then both $D$ and $D^{\vee \vee}$ are adjoints to
$D^{\vee}$ w.r.t.\ the residue pairings
$\langle -,- \rangle_{A/k}^{M}$ and
$\langle -,- \rangle_{A/k}^{N}$.
\medskip \noindent (b)\
Immediate from part (a).
\end{pf}
Here are a couple of examples to illustrate the scope of our results:
\begin{exa} \label{exa8.2}
Suppose $A$ is a noetherian, local, residually finitely generated
$k$-algebra. Let $I$ be an injective hull of the residue field
$A / \frak{m}$. Then $I$ is (non-canonically) a right $\cal{D}(A)$-module,
and moreover
$\operatorname{Diff}_{A/k}(I,I) \cong \cal{D}(\hat{A})^{\circ}$, where
$\hat{A}$ is the $\frak{m}$-adic completion. This is because $\hat{A}$ is
a BCA, there exists
an isomorphism of $\hat{A}$-modules $I \cong \cal{K}(\hat{A})$, and any
DO $I \rightarrow I$ is automatically continuous for the $\frak{m}$-adic topology.
\end{exa}
\begin{exa} \label{exa8.3}
Let $A$ be a BCA.
Suppose $M^{{\textstyle \cdot}}$ is a bounded complex with each $M^{q}$ a finite type
ST $A$-module, and $D: M^{q} \rightarrow M^{q+1}$ a continuous DO (for instance,
$M^{{\textstyle \cdot}} = \Omega^{{\textstyle \cdot}, \mathrm{sep}}_{A/k}$). Then
$\operatorname{Dual}_{A} M^{{\textstyle \cdot}}$ is also a complex (of cofinite type modules), and
a standard spectral sequence argument shows that the homomorphism of
complexes
\[ M^{{\textstyle \cdot}} \rightarrow \operatorname{Dual}_{A} \operatorname{Dual}_{A} M^{{\textstyle \cdot}} \]
(in the abelian category of untopologized $k$-modules)
is a quasi-isomorphism.
\end{exa}
\begin{question} \label{que8.1}
In the example above, suppose the complex $M^{{\textstyle \cdot}}$ is acyclic. Is the
same true of the dual complex $\operatorname{Dual}_{A} M^{{\textstyle \cdot}}$? A slight variation
is: suppose
$\operatorname{rank}_{k} \mathrm{H}^{q} M^{{\textstyle \cdot}}$ \linebreak
$< \infty$ for all $q$. Is the same
true for $\operatorname{Dual}_{A} M^{{\textstyle \cdot}}$?
\end{question}
\begin{cor} \label{cor8.3}
Let $f: A \rightarrow B$ be a morphism in $\mathsf{BCA}(k)$, let $M$ (resp.\ $N$)
be a ST $A$-module (resp.\ $B$-module) with the fine topology, and let
$D \in \operatorname{Diff}^{\mathrm{cont}}_{A/k}(M,N)$. Then there is a DO
\[ \operatorname{Dual}_{B/A}(D) = \operatorname{Dual}_{f}(D) : \operatorname{Dual}_{B} N \rightarrow
\operatorname{Dual}_{A} M . \]
The asignment $D \mapsto \operatorname{Dual}_{f}(D)$ satisfies the obvious
generalizations of conditions (i)-(v) of Thm. \ref{thm8.1}. For instance
(iii): if $D$ is $A$-linear, then
$\operatorname{Dual}_{f}(D)(\phi) = \operatorname{Tr}_{B/A} \circ \phi \circ D$.
\end{cor}
\begin{pf}
We may assume that $M$ is a finite type ST $A$-module, and that $N$ is
separated. So $D$ factors into
$M \exar{ \mathrm{d}^{n}_{M} } \cal{P}^{n,\mathrm{sep}}_{A/k}(M)
\exar{ \phi } N$, with
$\phi \in \operatorname{Hom}_{A}^{\mathrm{cont}}(\cal{P}^{n,\mathrm{sep}}_{A/k}(M), N)$
and $n \geq \operatorname{ord}_{A}(D)$. Let
$\phi^{\vee} : \operatorname{Dual}_{B} N \rightarrow \operatorname{Dual}_{A}
\cal{P}^{n,\mathrm{sep}}_{A/k}(M)$
be the homomorphism
$\psi \mapsto \operatorname{Tr}_{B/A} \circ \psi \circ \phi$, for
$\psi \in \operatorname{Dual}_{B} N =\operatorname{Hom}_{B}^{\mathrm{cont}}(N, \cal{K}(B))$.
Define
$\operatorname{Dual}_{f}(D) := \operatorname{Dual}_{A}(\mathrm{d}^{n}_{M}) \circ \phi^{\vee}$.
The transitivity and uniqueness properties follow from base change
and the uniqueness of adjoints.
\end{pf}
\begin{exa} \label{exa8.1}
If $f : A \rightarrow B$ is a morphism of BCAs,
the trace map
$\operatorname{Tr}_{B/A}: \cal{K}(B) \rightarrow \cal{K}(A)$ and
the continuous DGA homomorphism
$\Omega^{{\textstyle \cdot}, \mathrm{sep}}_{A/k} \rightarrow \Omega^{{\textstyle \cdot}, \mathrm{sep}}_{B/k}$
induce a map
$\operatorname{Tr}_{B/A}: \operatorname{Dual}_{B} \Omega^{{\textstyle \cdot}, \mathrm{sep}}_{B/k} \rightarrow
\operatorname{Dual}_{A} \Omega^{{\textstyle \cdot}, \mathrm{sep}}_{A/k}$,
which by the corollary is a homomorphism of complexes. This fact is important
for the construction of the De Rham - residue double complex in \cite{Ye2}.
\end{exa}
|
1996-05-13T13:33:14 | 9605 | alg-geom/9605005 | en | https://arxiv.org/abs/alg-geom/9605005 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9605005 | null | A.Levin, M.Olshanetsky | Double coset construction of moduli space of holomorphic bundles and
Hitchin systems | 19 pages, Latex | Commun.Math.Phys. 188 (1997) 449-466 | 10.1007/s002200050173 | MPI-96-60; ITEP-TH-13/96 | null | We present a description of the moduli space of holomorphic vector bundles
over Riemann curves as a double coset space which is differ from the standard
loop group construction. Our approach is based on equivalent definitions of
holomorphic bundles, based on the transition maps or on the first order
differential operators. Using this approach we present two independent
derivations of the Hitchin integrable systems. We define a "superfree" upstairs
systems from which Hitchin systems are obtained by three step hamiltonian
reductions. A special attention is being given on the Schottky parameterization
of curves.
| [
{
"version": "v1",
"created": "Mon, 13 May 1996 11:30:45 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Levin",
"A.",
""
],
[
"Olshanetsky",
"M.",
""
]
] | alg-geom | \section{} command!!!
\newcommand{\sect}[1]{\setcounter{equation}{0}\section{#1}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newtheorem{predl}{Proposition}[section]
\newtheorem{defi}{Definition}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{theor}{Theorem}[section]
\def\mathbin{>\mkern -8mu\lhd}{\mathbin{>\mkern -8mu\lhd}}
\begin{titlepage}
\setcounter{footnote}0
\begin{center}
\hfill ITEP TH-7/96\\
\hfill MPI-96-60
\hfill hep-th/9605005\\
\vspace{0.3in}
{\LARGE\bf Double coset construction of moduli space of holomorphic
bundles and Hitchin systems.}\\
\vspace{0.15in}
\bigskip
\bigskip
{\Large A.Levin\footnote{E-mail address: [email protected]
}}$\phantom{hj}^{\dag}$,
{\Large M.Olshanetsky\footnote{E-mail address:
[email protected]}}$\phantom{hj}^{\ddag}$,
\\
\bigskip
{\sf Max-Planck-Institut f\"{u}r Mathematik, Bonn}\\
\bigskip
\bigskip
\begin{quotation}{
$\phantom{hj}^{\dag}$ --
{\it On leave from International Institute for Nonlinear Studies at
Landau Inst, Vorob'iovskoe sch. 2, Moscow, 117940, Russia}
$\phantom{hj}^{\ddag}$ --
{\it On leave from ITEP, Bol.Cheremushkinskaya, 25,
Moscow, 117 259, Russia}\\
}
\end{quotation}
\end{center}
\end{titlepage}
\title{}
\author{}
\maketitle
\date{}
\begin{abstract}
We present a description of the moduli space of holomorphic vector
bundles over Riemann curves as a double coset space which is differ
from the standard loop group construction. Our approach is based on
equivalent definitions of holomorphic bundles, based on the
transition maps
or on the first order differential operators. Using this approach we
present two independent derivations of the Hitchin integrable systems.
We define a "superfree" upstairs systems from which Hitchin systems are
obtained by three step hamiltonian reductions. A special attention is
being given on the Schottky parameterization of curves.
\end{abstract}
\section{Introduction}
The moduli space of holomorphic vector bundles over Riemann surfaces are
popular subject in algebraic geometry and number theory. In
mathematical physics they were investigated due to relations with the
Yang-Mills theory \cite{AB} and the Wess-Zumino-Witten theory
\cite{KZ,B}. The conformal blocks in the WZW theory satisfy the
Ward identities which take a form of differential equations on the
moduli space \cite{TUY,BSch}. In this approach the moduli space is
described
as a double coset space of a loop group defined on a small circle
on a Riemann surface \cite{PS}.
The main goal of the paper is an alternative description of the moduli
space and the Hitchin integrable systems \cite{H} based on this construction.
We start with a special group valued
field on a Riemann surface which is defined as a map from a holomorphic
basis in a
vector bundle to a $C^{\infty}$ basis.
This field is an analogous of the tetrade field in
the General Relativity and
we call it the Generalized Tetrade Field (GTF).
The holomorphic structures
can be extracted from GTF. They are
described via the holomorphic transition maps, or by means of
the operators $d''$ .
The former are invariant under the action of
the global $C^{\infty}$ transformations, while the later
under the action
of the local holomorphic transformations.
It allows to define the moduli space as a double
coset space of GTF with respect to the actions of the local holomorphic
transformations and the global $C^{\infty}$ transformations.
We introduce a cotangent
bundle to GTF and invariant symplectic structure on it. The
cotangent bundle to the moduli of holomorphic bundles can be obtained by
the symplectic
factorizations over the action of two types of
commuting gauge transformations.
This cotangent bundle is a phase space of the Hitchin integrable
systems \cite{H}.
The tetrade fields in their turns are sections of the principle bundle
over the Riemann
surface, which satisfy some constraints equations. We interpret them as
moment constraints in a big "superfree system"
with a special gauge symmetry.
This space is a cotangent bundle to the principle bundle.
Thus the Hitchin systems are obtained by the three step
symplectic reductions from this space.
We investigate specially our reductions in terms
of Schottky parameterization,
which is a particular case of the general construction.
This parameterization was
used to derived the Knizhnik-Zamolodchikov-Bernard equations on the higher
genus curves \cite{B,Lo,I}. On the other hand the
quantum second order Hitchin Hamiltonians coincide with them on the
critical level.
\section{Moduli of holomorphic vector bundles}
\setcounter{equation}{0}
Let $\Sigma=\Sigma_{g}$ be a nondegenerate Riemann curve of genus $g$ with
$g>1$ .
We will consider in this section a set of stable holomorphic structures on
complex vector bundles
over $\Sigma$ \cite{AB}. To define them we proceed in two ways
based on the \^{C}ech and the Dolbeault cohomologies.
Eventually, we come to the moduli space ${\cal L}$ of stable holomorphic
bundles over $\Sigma_{g}$ and represent them as a double coset space
(Proposition 2.3).
\bigskip
{\bf 1.} Consider a vector bundle $V$ over $\Sigma_{g}$ .
To be more concrete we assume
that the
structure group of $V$ is $GL(N,{\bf C})$.
Let ${\cal U}_a,~a=1,\ldots$ be a covering of $\Sigma_{g}$ by open subsets.
We consider two bases in
$V$ the holomorphic $\{e^{hol}\}$ basis and the smooth
$C^{\infty}$ $\{e^{C^{\infty}}\}$ one.
In local coordinates
$(z_a\in{\cal U}_a)$ $$e^{hol}_a=e^{hol}(z_a),~
e^{C^{\infty}}_a=e^{C^{\infty}}(z_a,\bar{z}_a).$$
Let $h$ be the transition map between them
$h_a=h(z_a,\bar{z}_a)$.
Then locally in ${\cal U}_a$ we have
\beq{a1}
h_ae^{C^{\infty}}_a=e_a^{hol}.
\end{equation}
We can consider $h_a$ as the sections $\Omega_{C^{\infty}}^0({\cal U}_a , P)$
of the adjoint bundle
$P=$Aut $V$. We call the field $h$ a
generalized tetrade field (GTF).
It follows from the definitions of the bases that there exists a global
section for $ e_{C^{\infty}}$
\beq{a3}
e^{C^{\infty}}_a(z_a,\bar{z}_a)= e^{C^{\infty}}_b(z_b(z_a),
\bar{z}_b(\bar{z}_a)),~
z_a\in{\cal U}_{ab}={\cal U}_a\cap{\cal U}_b\neq\emptyset,
\end{equation}
where $z_b=z_b(z_a)$ are holomorphic functions
defining a complex structure
on $\Sigma_g$. On the other hand the
transformations of $e^{hol}$ are holomorphic maps
\beq{a4}
e^{hol}_a(z_a)=g_{ab}(z_a)e^{hol}_b(z_b(z_a)),~~g_{ba}(z_b)=g_{ab}^{-1}(z_a(z_b))
\end{equation}
$$
g_{ab}
\in \Omega_{hol}^0({\cal U}_{ab},{\rm Aut}~ V),
~~(\bar{\partial} g_{ab}=0,~\bar{\partial}=\partial_{\bar z_a}).
$$
These matrix functions define the holomorphic structure
in the vector bundle $V$.
We can describe the same holomorphic structure working with the smooth
basis $e^{C^{\infty}}$ in $V$. Let
\beq{a5}
\bar{A}_a=h_a^{-1}\bar{\partial}h_a.
\end{equation}
Then the basis $ e^{C^{\infty}}$ is annihilated by the operator
$d''_A|_{{\cal U}_{a}}=\bar{\partial}+\bar A_a$
$$
(\bar{\partial}+\bar A_a)e^{C^{\infty}}_a=0.
$$
The GTF transformations $h$ in (\ref{a1}) by no means free.
Let ${\cal R}_\Sigma$ be the subset of sections in $P$
which satisfies the following
conditions
\beq{a6}
{\cal R}_\Sigma=\{h\in\Omega_{C^{\infty}}^0({\cal U}_a, P)~|
~h_a^{-1}\bar{\partial}h_a|_{{\cal U}_{ab}}=
h_b^{-1}\bar{\partial}h_b|_{{\cal U}_{ab}}, ~\forall~
{\cal U}_{ab}\neq\emptyset,~a,b=1,\ldots\},
\end{equation}
$$
(\bar{A}_a(z_a)=\bar{A}_b(z_b(z_a)),~z_a\in{\cal U}_{ab}).
$$
\bigskip
\begin{predl}
Conditions (\ref{a1}) and (\ref{a6}) are equivalent.
\end{predl}
{\it Proof.}
Since $e^{C^{\infty}}_b=e^{C^{\infty}}_a$ in ${\cal U}_{ab}$
(\ref{a1}) implies
\beq{a7}
g_{ab}=h_ah_b^{-1}.
\end{equation}
Then the holomorphicity of $g_{ab}$ implies (\ref{a6}) .
If $h\in{\cal R}_\Sigma$, then
(\ref{a7}) defines the transition map for some holomorphic basis $e^{hol}$.
The basis $e^{hol}h$ satisfies (\ref{a3}) and therefore can be taken as
$e^{C^{\infty}}$.
$\Box$
\bigskip
Consider the group
\beq{a8}
{\cal G}_{\Sigma}=\{\Omega_{C^{\infty}}^0({\cal U}_a,P),~a=1,\ldots\}.
\end{equation}
It transforms local basses of $V$ over ${\cal U}_a$.
The group acts on itself by the left
and right multiplications.
There are two subgroups of ${\cal G}_{\Sigma}$. Let
$x_a\in\Omega_{C^{\infty}}^0({\cal U}_a,P)$. Then
\beq{a10}
{\cal G}_{\Sigma}^{hol}=
\{x_a\rightarrow f_ax_a~|~f\in\Omega^0_{hol}(\Sigma,P)\},
\end{equation}
\beq{a11}
{\cal G}_{\Sigma}^{C^{\infty}}=\{x_a\rightarrow x_a\varphi_a~|~
\varphi\in \Omega_{C^{\infty}}^0(\Sigma,P),~
\varphi(z_b(z_a),\bar{z}_b(\bar{z}_a))= \varphi(z_a,\bar{z}_a)~
z_a\in{\cal U}_{ab}\}.
\end{equation}
We can consider the GTF (\ref{a6}) as a subset in ${\cal G}_{\Sigma}$.
We have the following evident statement
\bigskip
\begin{predl}
The left and right actions of ${\cal G}_{\Sigma}^{hol}$ and
${\cal G}_{\Sigma}^{C^{\infty}}$
leave invariant ${\cal R}_{\Sigma}$.
\end{predl}
\bigskip
In other words
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
{\cal G}_{\Sigma}:~&{\cal G}_{\Sigma}&\mapright{right~ mltpl.}&{\cal G}_{\Sigma}\cr
\cup &\cup & &\cup \cr
{\cal G}_{\Sigma}^{C^{\infty}}:~
&{\cal R}_{\Sigma}&\mapright{right ~mltpl.}&{\cal R}_{\Sigma}\cr}
$$
and
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
{\cal G}_{\Sigma}:~&{\cal G}_{\Sigma}&\mapright{left~ mltpl.}&{\cal G}_{\Sigma}\cr
\cup &\cup & &\cup \cr
{\cal G}_{\Sigma}^{hol}:~&{\cal R}_{\Sigma}&\mapright{left~ mltpl.}&{\cal R}_{\Sigma}\cr}
$$
{\bf 2}.
Consider the space of holomorphic structures on the bundles $V$ and $P$.
Since \\ $g>1$ there is an open subset of stable holomorphic
structures. The holomorphic structures can be defined in two ways.
In the first type of the construction, which we call the D-type,
the holomorphic structures are defined by
the covariant operators. For $V$ they are
$$
d''_A:\Omega_{C^{\infty}}^{0}(\Sigma,V)\rightarrow\Omega_{C^{\infty}}^{0,1}(\Sigma,V).
$$
It means that $\bar{A}$ satisfies (\ref{a6}).
The holomorphic structure is consistent with the complex structure on $\Sigma_{g}$
such that for any section
$s\in\Omega_{C^{\infty}}^{0}(\Sigma,V)$ and $f\in C^{\infty}(\Sigma)$
$d''_A(fs)=(\bar{\partial}f)s+fd''_As.$
The space of holomorphic structures ${\cal L}_{\Sigma}^D$ on $P$
is defined in the similar way
\beq{a12}
{\cal L}_{\Sigma}^D=
\{d''_A=\bar{\partial}+\bar{A}:\Omega_{C^{\infty}} ^0(\Sigma,P)\rightarrow
\Omega_{C^{\infty}}^{(0,1)}(\Sigma,P)\}.
\end{equation}
with the action in the adjoint representation. The stable holomorphic structures
${\cal L}_{\Sigma}^{D,st}$
are an open subset
in (\ref{a12}). The automorphisms of the holomorphic structures are given
by the action of the gauge group ${\cal G}_{\Sigma}^{C^{\infty}}$ (\ref{a11})
\beq{a12a}
d''_A\rightarrow \varphi^{-1}d''_A\varphi,~\varphi\in{\cal G}_{\Sigma}^{C^{\infty}}.
\end{equation}
They preserve the subset ${\cal L}_{\Sigma}^{D,st}$.
{\it The moduli space ${\cal L}$ of stable holomorphic structures} on $P$
is the quotient space
\beq{a13}
{\cal L}={\cal L}_{\Sigma}^{D,st}/{\cal G}_{\Sigma}^{C^{\infty}}.
\end{equation}
It is a smooth complex manifold with tangent space at $\bar{A}$
is isomorphic to \\
$H^{(0,1)}(\Sigma,{\rm Lie}({\rm GL}(N,{\bf C})))$.
Its dimension is given by the Riemann-Roch theorem
\beq{a14}
\dim{\cal L}=N^2(g-1)+1.
\end{equation}
The left action of the gauge transformations ${\cal G}_{\Sigma}^{hol}$ (\ref{a10})
does not change \\
$\bar{A}_a=h_a^{-1}\bar{\partial} h_a,~a=1,\ldots$. Therefore the space
${\cal L}_{\Sigma}^{D}$ (\ref{a12}) can be represented as the quotient space
${\cal L}_{\Sigma}^{D}={\cal G}_{\Sigma}^{hol}\backslash {\cal R}_{\Sigma}$.
There is an open subset in ${\cal R}_{\Sigma}^{st}$ such that
the subset of the stable holomorphic structures is the quotient space
$$
{\cal L}_{\Sigma}^{D,st}={\cal G}_{\Sigma}^{hol}\backslash{\cal R}_{\Sigma}^{st}.
$$
The main statement of this section follows immediately from (\ref{a13})
\bigskip
\begin{predl}
The moduli space ${\cal L}$ of stable holomorphic structures on $P$ can be
represented as the double
coset space
\beq{a15}
{\cal L}=
{\cal G}_{\Sigma}^{hol}\backslash {\cal R}_{\Sigma}^{st}/{\cal G}_{\Sigma}^{C^{\infty}}.
\end{equation}
\end{predl}
\bigskip
{\bf 3.} An alternative description of the holomorphic structures
in terms of the \^{C}ech cohomologies, which we call
the C-type construction is
based on the transition maps (\ref{a4}), (\ref{a7}).
The collection of transition maps
\beq{a16}
{\cal L}_{\Sigma}^{Ch}=\{g_{ab}(z_a)=h_a(z_a)h^{-1}_b(z_b(z_a)),~z_a\in
{\cal U}_{ab},~a,b=1,\ldots ,\}.
\end{equation}
defines the holomorphic structures on $V$ or $P$ depending on the choice
of the representations. Again we choose the open subset of stable holomorphic
structures ${\cal L}_{\Sigma}^{C,st}$ in ${\cal L}_{\Sigma}^{Ch}$.
The gauge group ${\cal G}_{\Sigma}^{hol}$ acts as the automorphisms of
${\cal L}_{\Sigma}^{C,st}$
\beq{a17}
g_{ab}\rightarrow f_ag_{ab}f^{-1}_b,~f_a=f(z_a),~
f_b=f_b(z_b(z_a)),~
f\in{\cal G}_{\Sigma}^{hol}.
\end{equation}
The space ${\cal L}_{\Sigma}^{Ch}$ has a transparent description in terms of graphs.
Consider the skeleton of the covering $\{{\cal U}_a,~a=1,\ldots\}$. It is an
oriented graph,
whose vertices are some fixed inner points in ${\cal U}_a$ and edges $L_{ab}$
connect those $V_a$ and $V_b$ for which $U_{ab}\neq\emptyset$.
We choose an orientation of the graph, saying that $a>b$ on the edge
$L_{ab}$ and put
the holomorphic function $z_b(z_a)$ which defines the holomorphic map
from ${\cal U}_a$ to ${\cal U}_b$.
Then the space ${\cal L}_{\Sigma}^{Ch}$ can
be defined by the following data. To each edge $L_{ab},~a>b$ we attach a matrix
valued function $g_{ab}\in{\rm GL}(N,{\bf C})$ along with $z_b(z_a)$. The gauge fields
$f_a$ are living on the
vertices $V_a$ and the gauge transformation is (\ref{a17}).
The moduli space of stable holomorphic bundles
is defined as the factor space under this action
\beq{a18}
{\cal L}={\cal G}_{\Sigma}^{hol}\backslash{\cal L}_{\Sigma}^{Ch,st}.
\end{equation}
The tangent space to the moduli space in this approach is
$H^1(\Sigma,{\rm Lie}({\rm GL}(N,{\bf C})))$ extracted from the \^{C}ech complex.
Though ${\cal L}_{\Sigma}^{Ch,st}$ differs from ${\cal L}_{\Sigma}^{D,st}$ we obtain
the same moduli space ${\cal L}$ of stable holomorphic structures on $P$ due
to the equivalence of the Dolbeault and the \^{C}ech cohomologies.
In this construction the right action of ${\cal G}_{\Sigma}^{C^{\infty}}$
(\ref{a11}) leaves the transition maps $g_{ab}$ invariant. Therefore
\beq{a19}
{\cal L}_{\ti{\Sigma}}^{Ch,st}={\cal R}_{\Sigma}^{st}/{\cal G}_{\Sigma}^{C^{\infty}}.
\end{equation}
Taking into account (\ref{a18}) we come to the same construction of the
moduli space as the double coset space (\ref{a15}).
\bigskip
{\bf 4.} We fit the components of our construction in the
exact bicomplex
\begin{center}
${\bf M}$
\end{center}
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
& &\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0&\mapright{}&\Omega_{hol}^0({\cal U}_{ab}, P)&\mapright{i}
&\Omega_{C^{\infty}}^0({\cal U}_{ab},P) & \mapright{} &
\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{ab},{\rm End}V)& \mapright{\bar{\partial}}\cr
& &\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0&\mapright{}&\Omega_{hol}^0({\cal U}_{a},P)
&\mapright{i}
&\Omega_{C^{\infty}}^0({\cal U}_{a},P) & \mapright{} &
\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{a},{\rm End}V)
& \mapright{\bar{\partial}}\cr
& &\mapup{i}& &\mapup{i}& &\mapup{i}&\cr
0&\mapright{}&\Omega_{hol}^0(\Sigma,P)&\mapright{i}
&\Omega_{C^{\infty}}^0(\Sigma,P) & \mapright{} &
\Omega_{C^{\infty}}^{(0,1)}(\Sigma,{\rm End}V)\}&
\mapright{\bar{\partial}}\cr
& &\mapup{}& &\mapup{}& &\mapup{}&\cr
& &0& &0& &0&\cr}
$$
Here $\partial^C$ are the \^{C}ech differentials, $i$ are
the augmentations. The right arrows from
$\Omega_{C^{\infty}}^0(*,P)$ to $\Omega_{C^{\infty}}^{(0,1)}(*,{\rm End}V)$
are of the type $h\rightarrow h^{-1}\bar{\partial} h$. We have
$$
g_{ab}\in\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{ab},{\rm End}V),~
h_a\in\Omega_{C^{\infty}}^0({\cal U}_{a},P),
$$
$$
\delta\bar{A}\in\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{a},{\rm End}V).
$$
If these fields satisfy the tetrade conditions (\ref{a1}),(\ref{a4}) or
(\ref{a6}) then they lie in the images of $i$. The Dolbeault cohomologies
$H^{(0,1)}(\Sigma,{\rm End}V)$ that define the tangent space to the moduli
space are living in $\Omega_{C^{\infty}}^{(0,1)}(\Sigma,{\rm End}V)$ and the
\^{C}ech cohomologies $H^1(\Sigma,{\rm End}V)$
in $\Omega_{hol}^0({\cal U}_{ab},{\rm End}V)$.
Their equivalence can be derived from the properties of the double spectral
sequence.
The gauge transformations also can be incorporated in the exact bicomplex
\begin{center}
${\bf G}$:
\end{center}
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
&&\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0&\mapright{}&\Omega_{hol}^0({\cal U}_{ab},{\rm End}P)&\mapright{i}
&\Omega_{C^{\infty}}^0({\cal U}_{ab},{\rm End}P) & \mapright{\bar{\partial}} &
\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{ab},{\rm End}P)
& \mapright{\bar{\partial}}\cr
&&\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0&\mapright{}&\Omega_{hol}^0({\cal U}_{a},{\rm End}P)&\mapright{i}
&\Omega_{C^{\infty}}^0({\cal U}_{a},{\rm End}P) & \mapright{\bar{\partial}} &
\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{a},{\rm End}P)
& \mapright{\bar{\partial}}\cr
&&\mapup{i}& &\mapup{i}& &\mapup{i}&\cr
0&\mapright{}&\Omega_{hol}^0(\Sigma,{\rm End}P)&\mapright{i}
&\Omega_{C^{\infty}}^0(\Sigma,{\rm End}P) & \mapright{\bar{\partial}} &
\Omega_{C^{\infty}}^{(0,1)}(\Sigma,{\rm End}P)
& \mapright{\bar{\partial}}\cr
& &\mapup{}& &\mapup{}& &\mapup{}&\cr
& &0& &0& &0&\cr}
$$
Let
$\epsilon^{hol}\in{\rm Lie}({\cal G}_{\Sigma}^{hol}),
~\epsilon^{C^{\infty}}\in{\rm Lie}({\cal G}_{\Sigma}^{C^{\infty}}).$ Then
$$
\epsilon^{hol}\in{\rm Image~of} (\Omega_{hol}^0({\cal U}_{a},{\rm End}P))~
{\rm in}~\Omega_{C^{\infty}}^0({\cal U}_{a},{\rm End}P),
$$
$$
\epsilon^{C^{\infty}}\in{\rm Image~of} (\Omega_{C^{\infty}}^0(\Sigma,{\rm End}P)~
{\rm in}~
\Omega_{C^{\infty}}^0({\cal U}_{a},{\rm End}P).
$$
The actions of the gauge group
(see (\ref{a12a}) and (\ref{a17}))
\beq{a20}
\delta^{C^{\infty}}\bar{A}_a=\bar{\partial}\epsilon^{C^{\infty}}_a+[\bar{A}_a,\epsilon^{C^{\infty}}_a],
\end{equation}
\beq{a21}
\delta^{hol}g_{ab}=\epsilon^{hol}_ag_{ab}-g_{ab}\epsilon^{hol}_b.
\end{equation}
More generally, ${\bf M}$ is the bigraded ${\bf G}$ module.
The action of ${\bf G}$ is consistent with the both differential $\partial^C$
and $\bar{\partial}$. The differentiations take into account the bigradings of
${\bf M}$ and ${\bf G}$.
The actions (\ref{a20}),(\ref{a21}) are particular cases of these actions .
\section{ The Schottky specialization.}
\setcounter{equation}{0}
We apply the general scheme to the particular covering of $\Sigma_{g}$
based on the Schottky parameterization.
Consider the Riemann sphere with $2g$ circles
${\cal A}_a,{\cal A}'_a,~
a=1,\ldots g$. Each circle lies in the
external part of others. Let $\gamma_a$ be $g$ projective
maps ${\cal A'}_a=\gamma_a {\cal A}_a,~\gamma_a\in PSL(2)$. The Schottky group
$\Gamma$ is a free group
generated by $\gamma_a,a=1\ldots g$.
The exterior part of all the circles
$$
\ti{\Sigma}={\bf P}^1/\cup_{b=1}^{2g}D_b
$$ is the fundamental domain of $\Gamma$.
The surface $\Sigma$ is obtained from $\ti{\Sigma}$ by the pairwise gluing
of the circles ${\cal A'}_a=\gamma_a {\cal A}_a$ and
${\cal A}_a$.
We have only one nonsimpliconnected 2d cell ${\cal U}_a\sim\ti{\Sigma}$
with selfintersections ${\cal U}_{aa'}={\rm vicinity}~{\cal A}_a
={\rm vicinity}~{\cal A'}_a$.
We choose $g$ local coordinates $z_a,~a=1,\ldots,g$, which define
the parameterizations of the internal disks of ${\cal A}_a$ circles.
In this case the holomorphic maps can be written as
$z_{a'}(z_a)=\gamma_a(z_a)$. The GTF ${\cal R}_\Sigma$ (\ref{a6}) is a twisted field $h$
on $\ti{\Sigma}$
$$~h^{-1}\bar{\partial}h(z_a,\bar{z}_a)=
h^{-1}\bar{\partial}h(\gamma_a(z_a),\overline{\gamma_a(z_a)}), ~a=1,\ldots,g.
$$
In the definition of ${\cal G}_{\Sigma}^{C^{\infty}}$ (\ref{a11}) "the
periodicity conditions" take the form
$$
\varphi(\gamma_a(z_a),\bar{\gamma}_a(\bar{z}_a))= \varphi(z_a,\bar{z}_a),~
z_a\in{\rm vicinity~of~}{\cal A}_{a}.
$$
The transition maps (\ref{a4}),(\ref{a7}) defining ${\cal L}_{\Sigma}^{Ch}$ are
\beq{a22}
g_a=g_{aa'}(z_a)=
h(z_a,\bar{z}_a)h^{-1}(\gamma_a(z_a),\bar{\gamma}_a(\bar{z}_a)),~
a=1,\ldots,g.
\end{equation}
The gauge group ${\cal G}_{\Sigma}^{hol}$ acts as a global holomorphic
transformation on $\ti{\Sigma}$. In the local coordinates we have
\beq{a23}
g_{a}\rightarrow f_ag_{a}f^{-1}_{\gamma_a},~f_a=f(z_a),~g_a=g_a(z_a),~
f_{\gamma_a}=f(\gamma_a(z_a)).
\end{equation}
In local coordinates $g_a$ have the form of Laurent polynomials.
$g_a(z_a)=\sum g_a^{(k)}z_a^k.$
Thus in
this parameterization the set of holomorphic structures on the vector bundles
${\cal L}_{\Sigma}^{Ch}$
can be
identified with the collection of the loop groups
$L(GL_a(N,{\bf C}))$. But in fact, taking into account the adjoint action of
the gauge group
(\ref{a23}), one concludes that the precise form of components
is the semidirect product
$ L({\rm GL}(N,{\bf C}))\mathbin{>\mkern -8mu\lhd} PSL(2)=\{g(z)\mathbin{>\mkern -8mu\lhd}\gamma(z)\}.$ Thus
\beq{a24}
{\cal L}_{\Sigma}^{Ch}=\oplus_{a=1}^g L_a({\rm GL}(N,{\bf C}))\mathbin{>\mkern -8mu\lhd} PSL(2)_a,
\end{equation}
where the subgroups $\{PSL(2)_a\},a=1\ldots g$ are responsible for
the complex structure on $\Sigma$.
To define the stable bundles one should choose an open subset in
$L_a(GL(N,{\bf C}))$.
Consider the bundles over genus $g=1$ curves. Though the bundles are unstable
this case can be completely described in the wellknown
terms. The Schottky parameterization means the realization
of elliptic curve as an annulus. Let
$~\gamma(z)=qz,~q=\exp(2\pi i\tau )$. The holomorphic bundles are defined by
the loop group extended by the shift operator
\beq{a25}
{\cal L}_{\Sigma}^{Ch}= L(GL(N,{\bf C}))\mathbin{>\mkern -8mu\lhd}\exp(2\pi i\tau z\partial) .
\end{equation}
The gauge action (\ref{a23})
$$g(z)\rightarrow f(z)g(z)f^{-1}(qz)$$
transforms $g(z)$ to a $z$ independent diagonal form, up to the action
of the complex affine Weyl group $\hat{W}$. Let $W$ be the $A_{N-1}$
Weyl group (the permutations of the Cartan elements). Then
$\hat{W}= ({\bf Z}R^{\vee}\tau+{\bf Z}R^{\vee})\mathbin{>\mkern -8mu\lhd} W$
($R^{\vee}$ is the dual root system). The moduli space ${\cal L}$
in this case is
the Weyl alcove. The comparison of two description of
holomorphic structures on elliptic curves (\ref{a25}) and (\ref{a12})
was carried out
in \cite{EF,EKh} in terms of two loop current algebras and invariants of
their coadjoint actions.
In general case $(g>1)$ the gauge transform (\ref{a23}) allows to choose
$g_a$ as constant matrices. They are defined up to the common conjugation by
a ${\rm GL}(N,{\bf C})$ matrix. Thus the moduli space of holomorphic bundles in the
(\ref{a24}) description are defined as the quotient
$$
{\cal L}\sim(\underbrace{{\rm GL}(N,{\bf C})\oplus\ldots\oplus{\rm GL}(N,{\bf C})}_{g})/{\rm GL}(N,{\bf C}).
$$
Since the center of ${\rm GL}(N,{\bf C})$ acts trivially we obtain $\dim{\cal L}=
N^2(g-1)+1$ (see (\ref{a14})).
\section{Symplectic geometry in the double coset picture}
\setcounter{equation}{0}
Here we consider the Hitchin integrable systems which are defined on
the cotangent bundle $T^*{\cal L}$
to the moduli of stable holomorphic bundles ${\cal L}$. As it was done in the
original work
\cite{H} this space is derived as a symplectic quotient of
$T^*{\cal L}_{\Sigma}^D$ under the gauge action of ${\cal G}_{\Sigma}^{C^{\infty}}$.
We will come to the same systems by the three step symplectic reductions from
some big upstairs space. The main object of this section is
the commutative diagram (\ref{b8}), which describes these reductions and
intermediate spaces.
{\bf 1}. First, as intermediate step, consider the Hitchin description of
$T^*{\cal L}$.
The upstairs phase space is the cotangent bundle $T^*{\cal L}_{\Sigma}^D$ to the
space ${\cal L}_{\Sigma}^D$ (\ref{a12}) of holomorphic structures
on the bundle $P$
in the Dolbeault picture.
It is the space of pairs
\beq{b1}
T^*{\cal L}_{\Sigma}^D=
\{\phi,~d''_A,~~~\phi\in\Omega_{C^{\infty}}^{(1,0)}(\Sigma,({\rm End}V)^*)\} .
\end{equation}
The field $\phi$ is called the Higgs field and the bundle $T^*{\cal L}_{\Sigma}^D$
is the Higgs bundle.
We can consider the Higgs field as a form
$$\phi\in\Omega_{C^{\infty}}^{0}(\Sigma,({\rm End}V)^*\otimes K),$$
where $K$ is the canonical bundle on $\Sigma$.
Locally on ${\cal U}_a$
$$
d''_a=\bar{\partial}+\bar A_a,~\bar A_a=h^{-1}_a\bar{\partial} h_a ,
~h_a\in\Omega_{C^{\infty}}^0({\cal U}_a,{\cal R}_\Sigma).
$$
The symplectic form on it is
\beq{b2}
\omega^D=\int_{\Sigma}{\rm tr}(D\phi,Dd''_A).
\end{equation}
The action of the gauge group ${\cal G}_{\Sigma}^{C^{\infty}}$ (\ref{a11}) on
$d_{ A}''$ (\ref{a12a}) with
$$
\phi\rightarrow\varphi^{-1}\phi\varphi
$$
is a symmetry of $T^*{\cal L}_{\Sigma}^D$. It defines the moment map
$$
\mu_{{\cal G}_{\Sigma}^{C^{\infty}}}(\phi,\bar A):~T^*{\cal L}_{\Sigma}^D\rightarrow
{\rm Lie}^*({\cal G}_{\Sigma}^{C^{\infty}}),
$$
$$\mu_{{\cal G}_{\Sigma}^{C^{\infty}}}(\phi,\bar A)=[d_{A}'',\phi].
$$
For the zero level moment map $[d_{A}'',\phi]=0$ the Higgs field becomes
holomorphic
$$\phi\in H^0(\Sigma,({\rm End}V)^*\otimes K) .$$
The symplectic quotient
$\mu^{-1}(0)/{\cal G}_{\Sigma}^{C^{\infty}}=
T^*{\cal L}_{\Sigma}^D//{\cal G}_{\Sigma}^{C^{\infty}}$
is identified with the cotangent
bundle to the moduli space $T^*{\cal L}$. The Hitchin
commuting integrals are constructed
by means of $(1-j,1)$ holomorphic differentials $\nu_{j,k},k=1,\ldots$:
\beq{b3}
I^D_{j,k}=\int_{\Sigma}\nu^D_{j,k}{\rm tr}\phi^j.
\end{equation}
Since the space of these differentials $H^0(\Sigma,K\otimes T^j)$ ($K$ is the
canonical class,\\ $T^j$ is $(-j,0)$ forms)
has dimension $(2j-1)(g-1)$ for $j>1$ and $g$ for $j=1$ we have $N^2(g-1)+1$
independent commuting integrals, providing the complete integrability
of the Hamiltonian systems (\ref{b2}),(\ref{b3}). The integrals (\ref{b3})
define the Hitchin map
$$H^0(\Sigma,({\rm End}V)^*\otimes K)\rightarrow H^0(\Sigma, K^j).$$
\bigskip
{\bf 2.} The same system can be derived starting from the cotangent bundle
$T^*{\cal L}_{\Sigma}^{Ch}$ to the holomorphic structures on $P$ defined
in the C-type description (\ref{a16}). Now
\beq{b4}
T^*{\cal L}_{\Sigma}^{Ch}=\{\eta_{ab},g_{ab}|
~\eta_{ab}\in\Omega_{hol}^{(1,0)}({\cal U}_{ab},({\rm End}V)^*),~
g_{ab}\in\Omega_{hol}^0({\cal U}_{ab},P)\},
\end{equation}
This bundle can be endowed with the symplectic structure by means of
the Cartan-Maurer one-forms on $\Omega_{hol}^0({\cal U}_{ab},P)$.
Let
$\Gamma_a^b(C,D)$ be a path in ${\cal U}_{ab}$ with the end points
in the triple intersections
$C\in {\cal U}_{abc}={\cal U}_{a}\cap{\cal U}_{b}\cap{\cal U}_{c}$,
$D\in {\cal U}_{abd}$.
We can put the data (\ref{b4}) on the fat graph corresponding to the covering
$\{{\cal U}_a\}$.
The edges of the graph are $\{\Gamma_a^b(CD)\}$
and $\{\Gamma_b^a(DC)\}$ with opposite orientation.
We assume that the covering
is such that the orientation of edges defines the oriented contours
around the faces ${\cal U}_a$. The fields $\eta_{ab},g_{ab}$ are attributed
to the edge $\Gamma_a^b(CD)$, while $\eta_{ba},g_{ba}$ to $\Gamma_b^a(DC)$.
The last pair is not independent - $(g_{ab}^{-1}=g_{ba})$ (see (\ref{a4})).
Its counterpart
in the dual space is
\beq{b4a}
\eta_{ab}(z_a)=g_{ab}(z_a)\eta_{ba}(z_b(z_a))g_{ab}^{-1}(z_a).
\end{equation}
The symplectic structure is defined by the form
\beq{b5}
\omega^{Ch}=\sum_{\rm edges}
\int_{\Gamma_a^b(CD)}D{\rm tr}(\eta_{ab}(z_a)(Dg_{ab}g_{ab}^{-1})(z_a)).
\end{equation}
Here the sum is taken over the edges of the oriented graph obtained
from the fat graph after the identifications of fields (\ref{b4a}).
In other words
we consider only the edge $\Gamma_a^b$ with the fields $g_{ab},\eta_{ab}$
and forget about the edge $\Gamma_b^a$.
Since $\eta_{ab}$ and $g_{ab}$ are holomorphic in ${\cal U}_{ab}$,
the definition is
independent on a choice of the path $\Gamma_a^b$ within ${\cal U}_{ab}$.
The symplectic form is invariant under the gauge transformations (\ref{a17})
supplemented by
\beq{b6}
\eta_{ab}\rightarrow f_a\eta_{ab}f_a^{-1}.
\end{equation}
The set of invariant
commuting integrals on $T^*{\cal L}_{\Sigma}^{Ch}$ is
\beq{b7}
I^{Ch}_{j,k}=\sum_{\rm edges}\int_{\Gamma_a^b(CD)}
\nu^{Ch}_{(j,k)}(z_a){\rm tr}(\eta_{ab}^j(z_a)),
\end{equation}
where $\nu^{Ch}_{j,k}$ are $(1-j,0)$ differentials, which are related locally
to the $(1-j,1)$ differentials as $\nu^D_{j,k}=\bar{\partial} \nu^{Ch}_{j,k}$.
We can consider the system on the defined above graph $L_{ab}$
which
is dual to $\Gamma_a^b(CD)$. The fields $g_{ab},\eta_{ab}, a,b=1\ldots$ are
defined on edges, while the gauge transformations $f_a$ live on
vertices.
The moment map is
$$
\mu_{{\cal G}_{\Sigma}^{hol}}(\eta_{ab},g_{ab}):~T^*{\cal L}_{\Sigma}^{Ch}\rightarrow
{\rm Lie}^*({\cal G}_{\Sigma}^{hol}).
$$
According to (\ref{a21}) the Hamiltonian generating the gauge
transformations is
$$
F_{\epsilon^{hol}}=
\sum_{\rm edges}
\int_{\Gamma_a^b(CD)}
{\rm tr}(\eta_{ab}(z_a)\epsilon_a^{hol}(z_a))-
{\rm tr}(\eta_{ab}(z_a)g_{ab}(z_a)\epsilon_b^{hol}((z_b(z_a))g_{ab}(z_a)^{-1})=
$$
$$
\sum_{\rm edges}
\int_{\Gamma_a^b(CD)}
{\rm tr}(\eta_{ab}(z_a)\epsilon_a^{hol}(z_a))-
{\rm tr}(\eta_{ba}(z_b(z_a))\epsilon_b^{hol}(z_b(z_a)))=
$$
$$
\sum_a\int_{\Gamma_a}\sum_b{\rm tr}(\eta_{ab}(z_a)\epsilon_a^{hol}(z_a)),
$$
where $\Gamma_a$ is an oriented contour around ${\cal U}_a$.
The moment equation $\mu_{{\cal G}_{\Sigma}^{hol}}=0$ can be read off
from $F_{\epsilon^{hol}}$. It means that $\eta_{ab}$
is a boundary value of some
holomorpfic form defined on ${\cal U}_a$
\beq{b8a}
\eta_{ab}(z_a)=H_a(z_a),~{\rm for}~z_a\in{\cal U}_{ab},~H_a\in
\Omega^{(1,0)}_{hol}({\cal U}_a,({\rm End}V)^*).
\end{equation}
The reduced system is again the cotangent bundle to the moduli space of
holomorphic bundles
$$
T^*{\cal L}={\cal G}_\Sigma^{hol}\backslash\backslash T^*{\cal L}_{\Sigma}^{Ch}=
{\cal G}_\Sigma^{hol}\backslash\mu_{{\cal G}_{\Sigma}^{hol}}^{-1}(0),
$$
which has dimension $2N^2(g-1)+2$.
\bigskip
{\bf 3}. To get the cotangent bundle $T^*{\cal L}$ via the symplectic reduction
we can start from $T^*{\cal R}_\Sigma$ using the double coset
representation (\ref{a15}). Then $T^*{\cal L}_{\Sigma}^D$
or $T^*{\cal L}_{\Sigma}^{Ch}$ are obtained on the intermediate stages of the two step
reduction under the actions of ${\cal G}_\Sigma^{hol}$ or ${\cal G}_\Sigma^{C^\infty}$.
Since these groups act from different sides on ${\cal R}_\Sigma$
their actions commute and the result
of the reduction procedure is independent on their order. But the space
${\cal R}_\Sigma$, as we already have remarked, is not free -
its elements satisfy (\ref{a6}).
We will represent the constraints (\ref{a6}) as moment constraints and
consider the "superfree" space - cotangent bundle to the group ${\cal G}_{\Sigma}$
(\ref{a8}). More exactly we will consider (Theorem 4.1) the three step
symplectic reductions which result in
the following commutative "tadpole" diagram
\bigskip
\beq{b8}
\begin{array}{ccccc}
& &\fbox{$T^*{\cal G}_{\Sigma}$}& &\\
& {\cal G}_{\Sigma}^{A}&\downarrow& &\\
& &\fbox{$T^*{\cal R}_{\Sigma}$}& &\\
&{\cal G}_\Sigma^{hol}\swarrow& &\searrow{\cal G}_{\Sigma}^{C^{\infty}}&\\
\fbox{$T^*{\cal L}_{\Sigma}^D$}&
& & &\fbox{$T^*{\cal L}_\Sigma^{Ch}$}\\
&{\cal G}_{\Sigma}^{C^{\infty}}\searrow&&\swarrow{\cal G}_\Sigma^{hol}&\\
&&\fbox{$T^*{\cal L}$}&&
\end{array}
\end{equation}
\bigskip
To begin with we define the initial data on $T^*{\cal G}_{\Sigma}$ and
the gauge group ${\cal G}_{\Sigma}^{A}$.
To construct $T^*{\cal G}_{\Sigma}$ we consider three dual elements
$$
\Psi_a\in\Omega_{C^{\infty}}^{(1,1)}({\cal U}_{a},({\rm End}V)^*),~
\eta_{ab},\eta_{ba}\in\Omega_{C^{\infty}}^{(1,0)}({\cal U}_{ab}, ({\rm End}V)^*),
$$
$$
\xi_{ab},\xi_{ba}\in\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{ab},({\rm End}V)^*).
$$
Cotangent bundle $T^*{\cal G}_{\Sigma}$ is the set of fields
$(\Psi,\eta,\xi,h)$. We endow it with the symplectic structure.
Consider the same fat graph with edges $\Gamma_a^b(CD)$ and $\Gamma_b^a(DC)$ as
in {\bf 4.2}.
Then
\beq{b9}
\omega_{\Sigma}=D\{\sum_a\int_{{\cal U}_a}{\rm tr}(\Psi_aDh_ah_a^{-1})+
\sum_{b}[\int_{\Gamma_a^b}{\rm tr}(\eta_{ab}Dh_ah_a^{-1})+
\int_{\Gamma_{a}^b}{\rm tr}(\xi_{ab}h_a^{-1}Dh_a)]\}.
\end{equation}
We assume as before that
paths $\Gamma_a^b,\Gamma_a^c,\dots$ can be unified in
a closed oriented contour $\Gamma_a\subset{\cal U}_a$ .
The integral over ${\cal U}_{a}$ means
in fact the integral over a part of ${\cal U}_{a}$ restricted by
the contour $\Gamma_a$. Thus the first sum can be replaced by
the integration over $\Sigma$. To maintain the independence of $\omega_{\Sigma}$
on the choice of the contours $\Gamma_a$ we introduce the following "gauge"
symmetry. Its action defines of variations of fields along with variations
of contours. Let $\Gamma'_a$ be another contour and $\delta{\cal U}_{a}$ be the
corresponding variation of the integration domain. There is the
integral relation between fields coming from the Stokes theorem,
providing the independence of $\omega_{\Sigma}$
\beq{b9a}
\int_{\delta{\cal U}_a}{\rm tr}(\Psi_aDh_ah_a^{-1})=
[\int_{\Gamma'_a}{\rm tr}(\eta_{ab}Dh_ah_a^{-1})+
\int_{\Gamma'_{a}}{\rm tr}(\xi_{ab}h_a^{-1}Dh_a)]\end{equation}
$$
-[\int_{\Gamma_a}{\rm tr}(\eta_{ab}Dh_ah_a^{-1})+
\int_{\Gamma_a}{\rm tr}(\xi_{ab}h_a^{-1}Dh_a)].
$$
In other words, the variation of contours is compensated by the variation of
the field $\Psi$.
The form $\omega_{\Sigma}$ (\ref{b9}) is invariant under the actions of
${\cal G}_\Sigma^{hol}$ :
\beq{b12}
h_a\rightarrow f_ah_a,~\Psi_a\rightarrow f_a\Psi_af_a^{-1},~
\eta_{ab}\rightarrow f_a\eta_{ab}f_a^{-1},~
\xi_{ab}\rightarrow\xi_{ab},
\end{equation}
and ${\cal G}_{\Sigma}^{C^{\infty}}$
\beq{b13}
h_a\rightarrow h_a\varphi_a,~\Psi_a\rightarrow\Psi_a,~\eta_{ab}\rightarrow\eta_{ab},~
\xi_{ab}\rightarrow\varphi_a^{-1}\xi_{ab}\varphi_a.
\end{equation}
We extend these group transformations by the following affine action of
the group
\beq{b10}
{\cal G}_{\Sigma}^{A}=
\{s_{ab}\in\Omega_{C^{\infty}}^0({\cal U}_{ab},P)|s_{ab}=s_{ba}\}
\end{equation}
on $\xi_{ab}$
$$
\xi_{ab}\rightarrow\xi_{ab}-s_{ab}^{-1}(\bar{\partial}+h_a^{-1}\bar{\partial} h_a)s_{ab}+h_a^{-1}\bar{\partial} h_a
$$
leaving the other fields untouched. This action commutes with
${\cal G}_{\Sigma}^{hol}$,
but does not commute with ${\cal G}_{\Sigma}^{C^{\infty}}$. ${\cal G}_{\Sigma}^{A}$
can be imbedded in the bicomplex ${\bf G}$ (see (\ref{b10})).
On the Lie algebra level
we have
\beq{b11}
\xi_{ab}\rightarrow\xi_{ab}-(\bar{\partial}\epsilon_{ab}^A +[h_a^{-1}\bar{\partial} h_a,\epsilon^A_{ab}])
\end{equation}
$$
(\epsilon_{ab}^A\in{\rm Lie}({\cal G}_{\Sigma}^{A})
=\{\Omega_{C^{\infty}}^0({\cal U}_{ab},{\rm End}V)|~\epsilon_{ab}^A=\epsilon_{ba}^A\}.
$$
\bigskip
\begin{predl}
The form $\omega_{\Sigma}$ (\ref{b9}) is invariant under the action of
${\cal G}_{\Sigma}^{A}$.
\end{predl}
{\sl Proof}. From (\ref{b11})
$$\delta_{\epsilon_{ab}^A}\xi_{ab}=-(\bar{\partial}\epsilon_{ab}^A+[h_a^{-1}\bar{\partial} h_a,\epsilon_{ab}^A]),$$
where $\epsilon^A\in{\rm Lie}({\cal G}_{\Sigma}^{A})$. Then
$$
-\delta_{\epsilon^A}\omega_{\ti{\Sigma}}=
\sum_{a}\sum_b
\int_{\Gamma_a^b}{\rm tr}\{D([h_a^{-1}\bar{\partial} h_a,\epsilon_{ab}^A])h_a^{-1}Dh_a
+\bar{\partial}\epsilon_a^AD(h_a^{-1}Dh_a)+[h_a^{-1}\bar{\partial} h_a,\epsilon_a^A]D(h_a^{-1}Dh_a)\}
$$
$$
=-\sum_{a}\sum_b
\int_{\Gamma_a^b}{\rm tr}\{[D(h_a^{-1}Dh_a),h_a^{-1}Dh_a]+
\bar{\partial} D(h_a^{-1}Dh_a)+[h_a^{-1}\bar{\partial} h_a,D(h_a^{-1}Dh_a)],\epsilon_{ab}^A\}.
$$
Then direct calculations show that the sum under the integral in front
of $\epsilon_{ab}^A$ vanishes.
Therefore $\omega_{\Sigma}$ is invariant under these
transformations. $\Box$
\bigskip
More generally, the dual fields
$(\Psi_a,\eta_{ab},\eta_{ba},\xi_{ab},\xi_{ba})$
can be incorporated in a general pattern of two exact
${\bf G}$ bimoduli:
\begin{center}
${\bf M'}^*$
\end{center}
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
&\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0\rightarrow&\Omega_{hol}^{(1,0)}({\cal U}_{ab}, ({\rm End}V)^*)&\mapright{i}
&\Omega_{C^{\infty}}^{(1,0)}({\cal U}_{ab},({\rm End}V)^*) & \mapright{\bar{\partial}} &
\Omega_{C^{\infty}}^{(1,1)}({\cal U}_{ab},({\rm End}V)^*)& \mapright{\bar{\partial}}\cr
&\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0\rightarrow&\Omega_{hol}^{(1,0)}({\cal U}_{a},({\rm End}V)^*)
&\mapright{i}
&\Omega_{C^{\infty}}^{(1,0)}({\cal U}_{a},({\rm End}V)^*)& \mapright{\bar{\partial}} &
\Omega_{C^{\infty}}^{(1,1)}({\cal U}_{a},({\rm End}V)^*)
& \mapright{\bar{\partial}}\cr
&\mapup{i}& &\mapup{i}& &\mapup{i}&\cr
0\rightarrow&\Omega_{hol}^{(1,0)}(\Sigma,({\rm End}V)^*)&\mapright{i}
&\Omega_{C^{\infty}}^{(1,0)}(\Sigma,({\rm End}V)^*)& \mapright{\bar{\partial}} &
\Omega_{C^{\infty}}^{(1,1)}(\Sigma,({\rm End}V))^*&
\mapright{\bar{\partial}}\cr
&\mapup{}& &\mapup{}& &\mapup{}&\cr
&0& &0& &0&\cr}
$$
\bigskip
\begin{center}
${\bf M''}^*$
\end{center}
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
&\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0\rightarrow&\Omega_{antihol}^{(0,1)}({\cal U}_{ab}, ({\rm End}V)^*)&\mapright{i}
&\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{ab},({\rm End}V)^*) & \mapright{\partial} &
\Omega_{C^{\infty}}^{(1,1)}({\cal U}_{ab},({\rm End}V)^*)& \mapright{\partial}\cr
&\mapup{\partial^C}& &\mapup{\partial^C}& &\mapup{\partial^C}&\cr
0\rightarrow&\Omega_{antihol}^{(0,1)}({\cal U}_{a},({\rm End}V)^*)
&\mapright{i}
&\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{a},({\rm End}V)^*)& \mapright{\partial} &
\Omega_{C^{\infty}}^{(1,1)}({\cal U}_{a},({\rm End}V)^*)
& \mapright{\partial}\cr
&\mapup{i}& &\mapup{i}& &\mapup{i}&\cr
0\rightarrow&\Omega_{antihol}^{(0,1)}(\Sigma,({\rm End}V)^*)&\mapright{i}
&\Omega_{C^{\infty}}^{(0,1)}(\Sigma,({\rm End}V)^*)& \mapright{\partial} &
\Omega_{C^{\infty}}^{(1,1)}(\Sigma,({\rm End}V))^*&
\mapright{\partial}\cr
&\mapup{}& &\mapup{}& &\mapup{}&\cr
&0& &0& &0&\cr}
$$
\bigskip
We remind that
$$\Psi_a\in\Omega_{C^{\infty}}^{(1,1)}({\cal U}_{a},({\rm End}V)^*),~
\eta_{ab},\eta_{ba}\in\Omega_{C^{\infty}}^{(1,0)}({\cal U}_{ab},({\rm End}V)^*),
$$
$$
\xi_{ab},\xi_{ba}\in\Omega_{C^{\infty}}^{(0,1)}({\cal U}_{ab},({\rm End}V)^*).
$$
We will see that after the symplectic reductions these fields will obey some
special constraints. Now we have all initial data
to start from the top of the diagram (\ref{b8}) -the fields, the symplectic form
$\omega_\Sigma$ (\ref{b9}) and the gauge groups actions
(\ref{b12}),(\ref{b13}),(\ref{b10}).
\bigskip
\begin{theor}
There exist two ways of symplectic reductions represented by the
commutative diagram (\ref{b8}) which leads from $T^*{\cal G}_{\Sigma}$
to the cotangent bundle to the moduli space $T^*{\cal L}$.
\end{theor}
To prove Theorem we shall go down along the diagram.
\bigskip
{\bf 4}.
Consider first the action of ${\cal G}_{\Sigma}^{A}$ (\ref{b11}).
Let $T^*{\cal R}_{\Sigma}=\{\Psi,\eta,h\}$ and $h$ is GTF
with symplectic form
\beq{b14}
\omega_{\Sigma}=D\{\sum_a[\int_{{\cal U}_a}{\rm tr}(\Psi_aDh_ah_a^{-1})+
\sum_{b}\int_{\Gamma_a^b(CD)}{\rm tr}(\eta_{ab}Dh_ah_a^{-1})]\}.
\end{equation}
\begin{lem}
$$T^*{\cal R}_{\Sigma}=T^*{\cal G}_{\Sigma}//{\cal G}_{\Sigma}^{A}=
\mu_{{\cal G}_{\Sigma}^{A}}^{-1}(0)/{\cal G}_{\Sigma}^{A}.
$$
\end{lem}
{\sl Proof}. It follows from (\ref{b9}),(\ref{b10}) that
the Hamiltonian of ${\cal G}_{\Sigma}^{A}$ action is
$$
F_{\epsilon^A}=\sum_{a>b}F_{ab},~F_{ab}=
\int_{\Gamma_a^b(CD)}{\rm tr}(\epsilon_{ab}^Ah_a^{-1}\bar{\partial} h_a)+
\int_{\Gamma_b^a(DC)}{\rm tr}(\epsilon_{ba}^Ah_b^{-1}\bar{\partial} h_b).
$$
In fact the one-form
$$
DF_{ab}=\int_{\Gamma_a^b(CD)}\{{\rm tr}(\bar{\partial}\epsilon_{ab}^Ah_a^{-1}D h_a)+
{\rm tr}(\epsilon_{ab}^A[h_a^{-1}\bar{\partial} h_a,h_a^{-1}D h_a])\}
$$
can be obtain from $\omega_\Sigma$ (\ref{b9}) by the action of the
vector field generated by ${\cal G}_{\Sigma}^{A}$ (\ref{b10}).
But
$\epsilon_{ab}^A=\epsilon_{ba}^A$ (\ref{b10}). Putting the moment equal to zero
$\mu_{{\cal G}_{\Sigma}^{A}}=0$ we come to the constraints
$h_a^{-1}\bar{\partial} h_a=h_b^{-1}\bar{\partial} h_b$, which are exactly (\ref{a6}).
Note that the gauge transform (\ref{b11}) allows to fix $\xi_{ab}=0$.
Thus the symplectic quotient
$T^*{\cal R}_\Sigma=T^*{\cal G}_\Sigma//{\cal G}_{\Sigma}^{A}$
has the field content
$(\Psi,\eta,h\in\Omega_{C^{\infty}}^0(\Sigma,{\cal P}))$ with $\omega_{\Sigma}$
(\ref{b14}). $\Box$
\bigskip
{\bf 5.}
Consider the action of ${\cal G}_\Sigma^{hol}$ (\ref{a21}),(\ref{b12}) on
$T^*{\cal R}_\Sigma$,
which corresponds to the left arrow in the diagram (\ref{b9}). We will prove
\bigskip
\begin{lem}
$$
T^*{\cal L}_{\Sigma}^D={\cal G}_\Sigma^{hol}\backslash\backslash T^*{\cal R}_\Sigma=
{\cal G}_\Sigma^{hol}\backslash\mu_{{\cal G}_\Sigma^{hol}}^{-1}(0),
$$
where $T^*{\cal L}_{\Sigma}^D$ is defined by (\ref{b1}) with the symplectic
structure (\ref{b2}).
\end{lem}
{\sl Proof}. From (\ref{b12}) and (\ref{b14}) we read off the hamiltonian
of the gauge fields
$$
F_{\epsilon^{hol}}=
\sum_{a}[
\int_{{\cal U}_{a}}{\rm tr}(\Psi_a\epsilon_a^{hol})+
\sum_{b}
\int_{\Gamma_a^b}{\rm tr}(\eta_{ab}\epsilon_a^{hol})].
$$
On ${\cal U}_a$ we can put $\Psi_a=\bar{\partial}(\ti{\Phi}_a+H_a)$, where
$\ti{\Phi}_a\in\Omega_{C^{\infty}}^{(1,0)}({\cal U}_a,({\rm End}V)^*$ and $H_a$
is an arbitrary element from
$\Omega_{hol}^{(1,0)}({\cal U}_{a},({\rm End}V)^*)$ (see
${\bf M'}^*$). Then
$$
F_{\epsilon^{hol}}=
\sum_{a}\sum_{b}
\int_{\Gamma_a^b}{\rm tr}((\ti{\Phi}_a+H_a+
\eta_{ab})\epsilon_a^{hol}).
$$
Resolving the moment constraint $\mu_{{\cal G}_\Sigma^{hol}}=0$ gives
\beq{b14d}
\eta_{ab}(z_a,\bar{z}_a)=
-\ti{\Phi}_a(z_a,\bar{z}_a)-H_a(z_a),~z_a\in{\cal U}_{ab}.
\end{equation}
By means of the Stokes theorem $\omega_{\Sigma}$ (\ref{b14}) can be transformed
to the form
$$
\omega_{\Sigma}=
D\{
\sum_{a}[\int_{{\cal U}_{a}}{\rm tr}(\bar{\partial}(\ti{\Phi}_a+H_a)Dh_ah_a^{-1})+
\sum_{b}\int_{\Gamma_a^b}{\rm tr}(\eta_{ab}Dh_ah_a^{-1})]\}=
$$
$$
-\sum_{a}D\int_{{\cal U}_{a}}{\rm tr}((\ti{\Phi}_a+H_a)\bar{\partial} (Dh_ah_a^{-1})).
$$
Let
\beq{b14c}
\phi_a=-h^{-1}_a(\ti{\Phi}_a+H_a)h_a .
\end{equation}
Remind that $H_a=H_a(z_a)$ is an
arbitrary holomorphic function on ${\cal U}_a$.
We will choose it in a such way that $\phi_a$ becomes a global section in
$ \Omega_{C^{\infty}}^{(1,0)}(\Sigma,({\rm End}V)^*) $. In other words
\beq{b14e}
(\phi_a-\phi_b)|_{\Gamma_a^b}=0.
\end{equation}
In fact, since $g_{ab}=h_ah_b^{-1}$,
\beq{b14b}
h_a(\phi_b-\phi_a)h^{-1}_a=(\ti{\Phi}_a-g_{ab}\ti{\Phi}_bg_{ab}^{-1})
+(H_a-g_{ab}H_bg_{ab}^{-1}),
\end{equation}
where the second term is holomorphic.
Consider the integral $I_a$ over the contour $\Gamma_a=\cup_b\Gamma_a^b$ around
${\cal U}_a$
$$
I_a=-\int_{\Gamma_a}\sum_b\frac
{(\ti{\Phi}_a-g_{ab}\ti{\Phi}_bg_{ab}^{-1})(y)}
{z-y}dy.
$$
Due to the Sokhotsky-Plejel theorem \cite{Mu} $I_a$ is holomorphic
inside and outside $\Gamma_a$. It has a jump
$\ti{\Phi}_a-g_{ab}\ti{\Phi}_bg_{ab}^{-1}$ on the contour.
Let
$$
H_a=I_a ~{\rm in}~{\cal U}_a,
$$
$$
H_b=g_{ab}^{-1}I_ag_{ab}~{\rm outside}~{\cal U}_a.
$$
Therefore the functions
$H_a$ and $g_{ab}H_bg_{ab}^{-1}$ defining by this integral provide the
vanishing of the left hand side (\ref{b14b}).
The symplectic form
$\omega_{\Sigma}$ in terms of $\phi$ and $\bar{A}$ can be rewritten as
$$
\omega_{\Sigma}= \sum_{a}D\int_{{\cal U}_{a}}{\rm tr}(\phi_a D\bar{A}_a)=
D\int_{\Sigma}{\rm tr}(\phi D\bar{A}).
$$
This form coincides with $\omega^D$ (\ref{b2}) for $T^*{\cal L}_\Sigma^D$.
The field $\phi$ is invariant under
the ${\cal G}_\Sigma^{hol}$ action (\ref{b12}).
Therefore the symplectic reduction by the gauging ${\cal G}_\Sigma^{hol}$
leaves us with the fields $\phi$ and $h$ and the symplectic structure
(\ref{b2}).
In other words $T^*{\cal L}_\Sigma//{\cal G}_\Sigma^{hol}=T^*{\cal L}_\Sigma^D$.
$\Box$
\bigskip
We can now move down along the left side of diagram (\ref{b9}) as it
was described in {\bf 1.} and obtain eventually $T^*{\cal L}$.
It will be instructive to look on relations between two type
of dual fields
$\eta$ (\ref{b14d}) and $\phi$ (\ref{b14c}) that arise after
these two consecutive reductions.
On the first step we found that $\eta$ are
boundary valued forms
\beq{b15a}
\eta_{ab}(z_a,\bar{z}_a)=h_a^{-1}\phi h_a|_{{\cal U}_{ab}}.
\end{equation}
Moreover, it follows from (\ref{b14e}) that
\beq{b15b}
\eta_{ab}(z_a,\bar{z}_a)=
g_{ab}((z_a,\bar{z}_a)\eta_{ba}(z_b(z_a),\bar{z}_b(\bar{z}_a))
g_{ab}((z_a,\bar{z}_a)^{-1}.
\end{equation}
The second reduction gives $\bar{\partial}\phi+[\bar{A},\phi]=0$ (see {\bf 1.}).
It is equivalent to $\bar{\partial}\eta=0$, due to (\ref{b15a}).
\bigskip
{\bf 6}.
Now look on the right side of the diagram.
\begin{lem}
$$
T^*{\cal L}_{\Sigma}^{Ch}=T^*{\cal R}_\Sigma//{\cal G}_{\Sigma}^{C^{\infty}}=
\mu_{{\cal G}_{\Sigma}^{C^{\infty}}}^{-1}(0)/{\cal G}_{\Sigma}^{C^{\infty}},
$$
where $T^*{\cal L}_{\Sigma}^{Ch}$ is the cotangent bundle (\ref{b4})
with $\omega^{Ch}$
(\ref{b5}).
\end{lem}
{\sl Proof.}
The gauge action of ${\cal G}_{\Sigma}^{C^{\infty}}$
(\ref{b13}) on $T^*{\cal R}_\Sigma$ defines
the Hamiltonian (see (\ref{b14}))
\beq{b14a}
F_{\epsilon^{C^{\infty}}}=
\sum_a\{\int_{{\cal U}_{a}}{\rm tr}(h_a^{-1}\Psi_ah_a\epsilon^{C^{\infty}}_a)+
\sum_{b}\int_{\Gamma_a^b}{\rm tr}(h_a^{-1}\eta_{ab}h_a\epsilon^{C^{\infty}}_a)\}.
\end{equation}
Consider the zero level of the moment map
$$
\mu_{{\cal G}_{\Sigma}^{C^{\infty}}}:~
T^*\ti{\cal L}_\Sigma\rightarrow {\rm Lie}^*({\cal G}_{\Sigma}^{C^{\infty}}).
$$
From the first terms in (\ref{b14a}) we obtain
$$
\Psi_a=0,~a=1,\ldots.
$$
This choice of $\Psi$ breaks the invariance with respect
to replacements of contours . But if $\bar{\partial}\eta_{ab}=0$ then the exact form
of the path $\Gamma_a^b(C,D)$ is nonessential. Note that this choice
is consistent with
the definition of $\eta$ (\ref{b14e}) $(\eta_{ab}=H_a$ in the
${\cal G}_\Sigma^{hol}$ reduction).
Picking up in the second sum in (\ref{b14a}) integrals over two
neighbor edges
we come to the condition
$$
\int_{\Gamma_a^b}{\rm tr}(h_a^{-1}\eta_{ab}h_a\epsilon^{C^{\infty}}_a)+
\int_{\Gamma_b^a}{\rm tr}(h_b^{-1}\eta_{ba}h_b\epsilon^{C^{\infty}}_b)=0.
$$
Since $\epsilon^{C^{\infty}}\in{\rm Lie}({\cal G}_{\Sigma}^{C^{\infty}})$ it is
"periodic" $\epsilon^{C^{\infty}}_a(z_a)=\epsilon^{C^{\infty}}_b(z_b(z_a))$.
It gives the following form of constraints
$$
(h_a^{-1}\eta_{ab}h_a)(z_a)=(h_b^{-1}\eta_{ba}h_b(z_b(z_a))),
~z_a\in{\cal U}_{ab},
$$
or
\beq{b15}
\eta_{ab}(z_a)=g_{ab}(z_a)\eta_{ba}(z_b(z_a))g^{-1}_{ab}(z_a), ~
(g_{ab}(z_a)=h_a(z_a,\bar{z}_a)h_b^{-1}(z_b(z_a),\overline{z_b(z_a)})),
\end{equation}
which is just the twisting property of $\eta$ (\ref{b4a}). Furthermore,
the symplectic form
$\omega_{\Sigma}$ (\ref{b14}) due to vanishing the field $\Psi$ now is
$$
\omega_{\Sigma}=\sum_{a>b}D[\int_{\Gamma_a^b(CD)}{\rm tr}(\eta_{ab}Dh_ah_a^{-1})+
\int_{\Gamma_b^a(DC)}{\rm tr}(\eta_{ba}Dh_bh_b^{-1})].
$$
Taking into account that
$$
(Dg_{ab}g_{ab}^{-1})(z_a)=
Dh_a(z_a)h_a^{-1}(z_a)-h_a(z_a)(h_b^{-1}Dh_b)(z_b(z_a))h_a(z_a)^{-1}
$$
and the moment constraint (\ref{b15}) we can rewrite $\omega_{\Sigma}$ as
$$
\omega_{\Sigma}=\sum_{\rm edges}
\int_{\Gamma_a^b(CD)}D{\rm tr}(\eta_{ab}(z_a)(Dg_{ab}g_{ab}^{-1})(z_a)).
$$
It is just
$\omega^C$ (\ref{b5}) in the
C-type description of holomorphic bundles. We have the same
field content and the same symplectic structure as in $T^*{\cal L}_{\Sigma}^{Ch}.$
Therefore
$T^*{\cal R}_\Sigma//{\cal G}_{\Sigma}^{C^{\infty}}=T^*{\cal L}_{\Sigma}^{Ch}.$ $\Box$
The last step on the right side of diagram was described in {\bf 2}.
Its completes the proof of Theorem.
\section {Schottky description of Hitchin systems}
\setcounter{equation}{0}
{\bf 1}. Now consider the last step in the diagram (\ref{b8})
in the Schottky parameterization. Since in this case we have only one topologically
nontrivial cell
$\ti{\Sigma}$
the symplectic reduction is differ from the described in {\bf 4.2} for
the standard covering.
In this case the holomorphic fields
$\eta_a,g_a=g_a(z_a),~a=1,\ldots,g$ live in vicinities
${\cal V}_a$ of
${\cal A}_a$-cycles, and $z_a$ are local parameters in the internal disks.
(see (\ref{a22})).
The phase space is
$$
T^*{\cal L}_{\Sigma}^{Ch}=\{\eta_{a},g_{a}|
~\eta_{a}\in\Omega_{hol}^{(1,0)}({\cal V}_{a},({\rm End}V)^*),~
g_{a}\in\Omega_{hol}^0({\cal V}_{a},P)\}.
$$
In other words in accordance with (\ref{a24})
$$
T^*{\cal L}_{\Sigma}^{Ch}=\oplus_{a=1}^g T^*L_a(GL(N,{\bf C})),
$$
and the loop groups $L_a(GL(N,{\bf C}))$ are extended by the
projective transformations of $z_a$ as in (\ref{a24}).
The symplectic form on this object is (see(\ref{b5})
\beq{c1}
\omega^{Ch}=\sum_{a=1}^gD\int_{{\cal A}_{a}}
{\rm tr}(\eta_{a}(z_a),Dg_{a}g^{-1}_{a}(z_a)).
\end{equation}
The gauge transformations (\ref{a17}),(\ref{b6}) act as the common conjugations
by global holomorphic
in $\ti{\Sigma}$ matrix functions
\beq{c2}
\eta_a(z_a)\rightarrow f(z_a)\eta_a f^{-1}(z_a),~~
g_a\rightarrow f(z_a)g_a(z_a)f^{-1}(\gamma_a(z_a)).
\end{equation}
The invariant commuting Hamiltonians (\ref{b7}) in this parameterization
are
\beq{c3}
I^C_{j,k}=\sum_{a}\int_{{\cal A}_{a}}\nu^C_{(j,k)}(z_a){\rm tr}(\eta_{a}^j(z_a)),
\end{equation}
The gauge transform (\ref{c2}) produces the moment map
$\mu_{{\cal G}_{\Sigma}^{hol}}$,
which takes the form
$$
\mu_{{\cal G}_{\Sigma}^{hol}}=
\eta_{a}(\gamma_a(z_a))-(g_{a}^{-1}\eta_{a}g_{a})(z_a),~a=1,\ldots g.
$$
Assume as above that $\mu_{{\cal G}_{\Sigma}^{hol}}=0$:
\beq{b17}
\eta_{a}(\gamma_a(z_a))-(g_{a}^{-1}\eta_{a}g_{a})(z_a)=0,~a=1,\ldots g,
\end{equation}
which is twisting property (\ref{b4a}) in the Schottky picture.
\bigskip
{\bf 2}.
The solutions of the moment equations are known in a few degenerate cases
\cite{N}. We will consider here as an example of the above construction
holomorphic bundles over elliptic curves with a marked point.
Define the elliptic curve as the quotient
$$\Sigma_{\tau}={\bf C}^*/q^{\bf Z},~q=\exp 2\pi i\tau.~$$
In this case
$$T^*{\cal L}_{\Sigma}^{Ch}\sim(\eta(z),g(z);p,s)$$
where $s\in{\rm GL}(N,{\bf C})$ is a group element in the marked point $z=1$ and
$p\in {\rm Lie}^*({\rm GL}(N,{\bf C}))$. In addition to (\ref{c2})
$$
p\rightarrow f(z)pf^{-1}(1),~s\rightarrow f(1)s.
$$
The one form $\eta(z)$ has a pole in the singular point $z=1$.
The symplectic form (\ref{c1}) on these objects is
\beq{c4}
\omega^{Ch}=D\int_{{\cal A}}
{\rm tr}(\eta(z)Dgg^{-1}(z))+D{\rm tr}(pDss^{-1}).
\end{equation}
The transition map $g(z)$ can be diagonalized by (\ref{c2}):
$$
g(z)=\exp 2\pi i \vec{u}=\exp\{{\rm diag} ~ 2\pi i(u_1,\ldots, u_N)\},
$$
where $u_j$ are $z$-independent.
We keep the same notation for the transformed $\eta(z)=
\Sigma_{n\in {\bf Z}}\eta_{j,k}^{(n)}z^n$.
The moment equation (\ref{c4}) takes the form
$$
\eta(qz)-(g^{-1}\eta g)(z)=p\delta(z),
$$
Rewrite it as
$$
q^n\eta_{j,k}^{(n)}-e^{2\pi i(x_k-x_j)}\eta_{j,k}^{(n)}=p_{j,k}^{(n)}.
$$
After the resolving the moment constraints we find
$$\eta_{j,j}(z)=w_j,~p_{jj}=0,$$
$$\eta_{j,k}=-\frac
{1}{2\pi i }
\frac
{\theta(u_j-u_k-\zeta)\theta'(0)}
{\theta(u_j-u_k)\theta(\zeta)},~z=\exp 2\pi i \zeta,$$
where $w_j$ are new free parameters and
$\theta(\zeta)=\sum_{n\in {\bf Z}}e^{\pi i(n^2\tau+2n\zeta)}$.
The symplectic form (\ref{c4}) on the reduced space takes the form
$$
\omega^{red}=D\vec{w}\cdot D\vec{u} +
{\rm tr} D(J s^{-1}Ds),
$$
and $J$ defines the coadjoint orbit $p=s^{-1}Js$.
Consider the quadratic Hamiltonian (\ref{c3}).
After the reduction $H$ takes
the form of the N-body elliptic Calogero Hamiltonian
with the spins \cite{Kr}:
$$
H= \frac{1}{2}(\vec{w}\cdot \vec{w}+\frac{1}{4\pi^2}
\sum_{j>k}^N[p_{j,k}p_{k,j}\wp(u_j-u_k|\tau)+E_2(\tau)]).
$$
Here $E_2(\tau)$ is the normalized Eisenstein series.
\bf Acknowledgments.
{\sl We would like to thank V.Fock, B.Khesin, N.Nekrasov and A.Rosly
for illuminating discussions.
We are grateful to the Max Planck Institute for Mathematik in Bonn
for the hospitality where this work was prepared. The work of A.L.
is supported in part by the grant INTAS 944720 and the grant ISF NSR-300.
The work of M.O. is supported in part by the grant
CEE-INTAS 932494 and the grant RFFI-96-02-18046 }
\small{
|
1996-05-08T06:23:10 | 9605 | alg-geom/9605001 | en | https://arxiv.org/abs/alg-geom/9605001 | [
"alg-geom",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9605001 | Parshin | A.N.Parshin | Vector Bundles and Arithmetical Groups I. The higher Bruhat-Tits tree | to appear in an english translation of the Proc. Steklov Math.
Institute, vol. 208 33 pages, LaTeX, emlines.sty | null | null | null | null | We define and study a simplicial complex which is a homogeneous space for the
group $PGL(2, K)$ over a two-dimensional local field $K$. The complex is a
generalization of the tree studied by F. Bruhat, J. Tits, J.-P. Serre and P.
Cartier in the 60's and early 70's. Such complex can be canonically attached to
the triples $x \in C \subset X$ where $X$ is an algebraic surface, $C$ is an
irreducible curve and $x$ is a smooth point on $C$ and $X$. This construction
can be used for a description of the isomorphism set of vector bundles on $X$.
| [
{
"version": "v1",
"created": "Tue, 7 May 1996 14:30:19 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Parshin",
"A. N.",
""
]
] | alg-geom | \section{Local Fields}
We begin with the main definition of the higher adelic theory.
\par\smallskip
{\sc Definition 1}.
Let $K$ and $k$ be fields. We say that $K$ is a $n$-{\em dimensional
local field} with $k$ as {\em last residue field} if the field $K$ has
the following structure. Either $n = 0$ or $K$ is the quotient field of a
(complete) discrete valuation ring ${\cal O}_{K}$ whose residue field is
a local field of dimension
$n-1$ with last residue field $k$. If $K', K''$, ... are the intermediate
residue fields from the definition then we will write $K/K'/K''.../k$ for
the structure.The {\em first residue field} will be denoted mostly as
$\bar K$.
\par\smallskip
In the sequel we restrict ourselves by the case of $n = 2$ ( the general
case was considered in \cite{P}).
A typical example (which is quite sufficient if we have in mind the
applications to algebraic surfaces) is the field of iterated power series
$$ K = k((u))((t)) $$
with an obvious inductive local structure on it
$$ {\cal O}_{K} = k((u))[[t]],\, \bar K = k((u)). $$
(see \cite[ch.2]{FP}) for other examples and classification theorem for
complete local fields). Let us mention that the choice of {\em local
parameters} $t, u $ in our example does not follow from the local
structure.
For technical reasons we do {\em not} assume as usual that the discrete
valuation rings which enter in our definition are the complete rings. We
have reduction map $p :{\cal O}_{K} \rightarrow {\bar K} $ and we denote
by $\wp$ and $m$ the maximal ideals of the local rings
${\cal O}_{K}$ and ${\cal O}_{\bar K}$ correspondingly. Also we denote
by $t, u$ the generators of these ideals.
Let
$$ {\cal O}'_{K} = p^{-1}({\cal O}_{\bar K}) $$
be a subring in $K$.
Then we have a tower of valuation rings for the valuations $\nu^{(i)}$~
of rank $i = 0, 1, 2$:
$$ {\cal O}_{(0)} \supset {\cal O}_{(1)} \supset {\cal O}_{(2)},$$
where $ {\cal O}_{(0)} = K,~{\cal O}_{(1)} = {\cal O}_{K},~{\cal O}_{(2)}
= {\cal O}'$.
For valuation groups
$$ \Gamma_{K}^{(i)} = K^{*}/({\cal O}_{(i)})^{*},~i = 1,2,
~\Gamma_{K} = \Gamma_{K}^{(2)} $$
there is a filtration
$$ \Gamma_{K}^{(2)} \rightarrow \Gamma_{K}^{(1)} , $$
which will be reduced to one homomorphism in our case. We denote it by
$\pi$.
This filtration defines on $\Gamma_{K}$ a structure of {\em ordered}
group. If we need to show the local structure we will write
$\Gamma_{K/.../k}$ instead of $\Gamma_{K}$.If we chose local parameters
$t, u$ of the field $K$ then the order becomes the lexicographical order.
Inside the group $\Gamma_{K}$ there is a subset
$\Gamma_{K}^{+}$ of non-negative elements.
In our situation we have two valuations (of ranks 1 and 2). They will be
denoted by $ \nu $ and $ \nu'$ correspondingly.
If $K \supset {\cal O}$ is a fraction field of a subring ${\cal O}$ we call
${\cal O}$-submodules $a \subset K$ fractional ${\cal O}$-ideals (or
simply fractional ideals).
\begin{theorem}.
The local rings ${\cal O}_{(i)} $ $i = 0, 1, 2$ have the following
properties:
\par\smallskip
i) $$ {\cal O}'/m = k,~K^{*} = \{ t \} \{ u \}({\cal O}')^{*},
~({\cal O}')^{*} = k^{*}(1 + m);$$
ii) every finitely generated fractional ${\cal O'}$-ideal $a$ is a
principal one and
$$a = m_{i, n} = (u^{i}t^{n}),~i,n \in \mbox{\bf Z}; $$
iii) every infinitely generated fractional ${\cal O'}$-ideal $a$ is equal to
$$a = \wp_{n} = (u^{i}t^{n} \mid \mbox{for all~}
i \in \mbox{\bf Z}),~n \in \mbox{\bf Z}; $$
iv) if $\wp_{(i,j)} = \wp \mbox{ for (i,j) = (2,1) and } \wp_{(i,j)} = (0)
\mbox{ for (i,j) = (i,0)},$
$$ {\cal O}_{(i)} \supset \wp_{(i,i-1)} \supset ... \supset \wp_{(i,1)}
\supset \wp_{(i,0)}, $$
then
$$ \mbox{Hom}_{{\cal O}'}({\cal O}_{(i)}, {\cal O}_{(j)}) = \left\{
\begin{array}{ll}
{\cal O}_{(j)}, & i \geq j, \\
\wp_{(j,i)}, & i < j.
\end{array} \right.
$$
\end{theorem}
{\sc Proof}.
The multiplicative structure of the field $K$ can be deduced immediately
from the corresponding results for the fields of dimension 1 (see, for
example,
\cite{S1}). We get also that $\nu':K^{*} \rightarrow
\Gamma_{K}$ is a valuation, if we introduce on $\Gamma_{K}$ the lexicographical
order. Thus for any $x, y \in K^{*}$ we have
\begin{equation}
\label{valuation}
x = ay ~\mbox{with} ~a \in {\cal O}' \Longleftrightarrow ~\nu'(x)
\geq \nu'(y).
\end{equation}
Let now $a = (x_{1},...,x_{n})$ be a finitely generated ${\cal
O}'$-module. If $x \in a$ then $x = \sum a_{i}x_{i},~a_{i}
\in {\cal O}'$.
From here we see that $min_{x \in a-(0)} \nu'(x)$ ~exists and it
can be achieved for some $x_{0} \in a$. This shows that $a = (x_{0})$.
If $b \subset K$ is an infinitely generated module then for some $i$ the
group $b \otimes {\cal O}_{(i)}$ will be a ${\cal O}_{(i)}$-module with
one generator. Let
$i$ be the largest index with this property. Then $\nu^{(i)}$ has it's
minimum on $b$ and $\min~\nu^{(j)}$ equals to infinity for $j > i$. This
gives our claim.
The last property must be checked only for $i < j$ ( otherwise it is
obvious). Explicitly it means that
$$ \mbox{Hom}_{{\cal O}'}(K, {\cal O}) = \mbox{Hom}_{{\cal O}'}
(K, {\cal O}') = (0)$$
and
$$ \mbox{Hom}_{{\cal O}'}({\cal O}, {\cal O}') = \wp .$$
Both the equalities followed from the results already proved on structure
of ideals in the ring ${\cal O}'$.
{\bf Remark 1}.
These non-noetherian rings play an important role in the whole theory.
Usually the higher local fields appear as fields attached to some chain
of the subschemes of decreasing codimension \cite[ch.4, 7]{FP} and many
structures related with them can be interpreted in terms of simplicial
stucture on the partially ordered set of such chains. It seems that the
rings ${\cal O}_{(i)}$ cannot be described in these terms. It would be
interesting to extend the simplicial language (as described in
\cite[ch.7]{FP}) to cover these rings also.
\section{BN-pairs}
Let $G = \mbox{SL}(n, K)$ where $K$ is a two-dimensional local field. We
put
$$
B = \left( \begin{array}{llll}
{\cal O}' & {\cal O}'& \dots & {\cal O}'\\
m & {\cal O}'& \dots & {\cal O}'\\
& & \dots & \\
m & m & \dots & {\cal O}'
\end{array} \right),
$$
We denote in such way the subgroup of $G$ consisting of the matrices
whose entries satisfy the written conditions. Also let $N$ be the
subgroup of monomial matrices.
\par\smallskip
{\sc Definition 2. }
Let
$$ T = B \bigcap N = \left( \begin{array}{lll}
({\cal O}')^{*} & \dots & 0 \\
& \ddots & \\
0 & \dots & ({\cal O}')^{*}
\end{array} \right)
$$
The group
$$W_{K/{\bar K}/k)} = N/T.$$
will be called the
{\em Weyl group}
We also introduce
$$
P = \left( \begin{array}{lll}
{\cal O}'& \dots & {\cal O}'\\
& \dots & \\
& \dots & \\
{\cal O}'& \dots & {\cal O}'
\end{array} \right) \bigcap G,
$$
$$
A = \left\{ \left( \begin{array} {lll}
t^{i_{1}} u^{j_{1}} & \dots & 0 \\
0 & \dots & 0 \\
& \ddots & \\
0 & \dots & t^{i_{n}} u^{j_{n}} \\
\end{array} \right),
\mbox{~for all } k~i_{k},j_{k} \in \mbox{\bf Z} \right\}.
$$
If the matrices in this definition satisfy the additional condition : the
integer vectors $(i_{1}, j_{1}), \dots, (i_{n}, j_{n})$ are
lexicographically ordered, then we get a subset $A_{+}$. We set also
$$
U = \left( \begin{array}{lll}
1 & \dots & K \\
& \ddots & \\
0 & \dots & 1
\end{array} \right)
$$
\begin{theorem}
We have the following decompositions in the group $G$:
\begin{quotation}
i) the Bruhat decomposition
$$ G = \bigcup_{w \in W} BwB $$
ii) the Cartan decomposition
$$ G = \bigcup_{a \in A_{+}} PaP $$
iii) the Iwasawa decomposition
$$ G = \bigcup_{a \in A} PaU $$
\end{quotation}
where all the unions are disjoint ones.
\end{theorem}
{\sc Proof} can be given as a generalization of the known proofs of these
facts for local fields of dimension 1 ( see, for example, \cite[ch.
VI]{G} for the Cartan and Iwasawa decompositions and \cite[theorem
3.15]{GI} for the Bruhat decomposition). We only outline the main steps
here.
{\sc Existence. } This can be done by standard application of elementary
trasformations to the rows and columns of matrices from the group $G$.
Let $e_{i, j}(\lambda)$ be an elementary matrix with $\lambda$ on
$(i, j)$-th place. Now let $g = (a_{k, l}) \in G$ and for some
$~k, i, j$~
$\nu'(a_{k, i}) \le (\mbox{or}<~) \nu'(a_{k, j}) $. Then after a
multiplication from the right by
$e_{i, j}(\lambda)$,~$\lambda = - a_{k, i}^{-1} a_{k, j}$ we get 0 on
the $(k, j)$-th place. By~(\ref{valuation}), $\lambda \in {\cal O}'
~(\mbox{or}~m)$. The same fact is true for the multiplication
by $e_{i, j}(\lambda)$ from the left.
Multiplying the given matrix from $G$ by $e_{i, j}(\lambda)$
~with~ $\lambda \in {\cal O}'$~for~$i < j$~and~$\lambda \in m$~for~$i > j$,
from the left and from the right we can get a monomial matrix after
several steps. This gives the Bruhat decomposition.
In other two cases we need also to multiply by permutation
matrices (after a multiplication by a suitable matrix from $T$ they
belong to $ SL(n, {\cal O}')$). Also we have to change $m$~on~${\cal
O}'$~ in the second restriction on matrices $e_{i, j}(\lambda)$ given
above.
{\sc Uniqueness.} Let $L = {\cal O}'e_{1} \oplus \dots \oplus {\cal
O}'e_{n}$ be a free ${\cal O}'$-submodule of the space $V$. If
$x \in V$~ ~$g \in \mbox{GL}(V)$, then we put
\begin{equation}
\label{val1}
\nu'(x) = \mbox{min}_{i} \nu'(x_{i}),~\nu'(g) = \mbox{min}_{i, j}
\nu'(a_{i, j}),
\end{equation}
where $x = x_{1}e_{1} + \dots + x_{n}e_{n}$.
\par\smallskip
\begin{lemma}. $\nu'(x) \in \Gamma_{K} \cup \infty $ ~and we have:
i) $\nu'(x) = \mbox{min}_{x \in \lambda L}~ \nu'(\lambda),$
ii) $\nu'(g) = \nu'(pgq), ~\mbox{if}~p, q \in
\mbox{Stab}(L) \cong \mbox{GL}(n, {\cal O}'),$
iii) $\nu'(g) = \mbox{min}_{\nu'(x) = 0}~ \nu'(g(x))
= \mbox{min}_{x \in L}~ \nu' (g(x)).$
\end{lemma}
\par\smallskip
These properties can be checked precisely as in the case of discrete
valuation rings. We denote $\nu'(x)$ by $\nu'_{L}(x)$ because it
depends only on the submodule $L$.
We show how to get the uniqueness for the Bruhat decomposition.
Let $L_{k} = me_{1} \oplus \dots \oplus me_{k} \oplus {\cal O}'e_{k
+ 1} \dots \oplus {\cal O}'e_{n},~k = 0, ..., n - 1$ --- free
${\cal O}'$-submodules in $V$. We put
\begin{equation}
\label{val2}
\delta_{rkl}(g) = \mbox{min}_{x \in \wedge^{r} L_{k}}
\nu'_{\wedge^{r}L_{l}}(\wedge^{r}g(x)),
\end{equation}
where $r = 1, ..., n$~and~$\wedge^{r}L_{k}$ is a $r$-th external
power of the module $L_{k}$~in~$\wedge^{r}V$. Then we can prove:
$$\delta_{rkl}(g') = \delta_{rkl}(g),~\mbox{if}~g' \in BgB,$$
(since ~$\forall k, B(L_{k}) = L_{k}$) and
$$\mbox{if}~w, w' \in N~\mbox{and}~\forall r, k, l
~\delta_{rkl}(w) = \delta_{rkl}(w'),~\mbox{then}~w'w^{-1} \in T.$$
It gives our claim. The uniqueness for the Cartan decomposition can
be proved along the same lines (with one module $L$ instead of all
$L_{k}$) . The uniqueness for the Iwasawa decomposition is a direct
computation.
The proof is finished.
\par\smallskip
{\bf Remark 2.} The same type decompositions also exist for the group
$\mbox{GL}(n, K)$.
We also conjecture that the decompositions of the theorem (and the
known decompositions for the parabolic(parahoric) subgroups in Tits
theory
\cite[ch. IV, \S 2.5]{B}) can be included in some general theorem
formulated in an appropriate simplicial language using the rings
${\cal O}_{(i)}$.
\par\smallskip
Let us study the Weyl group $W$ more carefully.
It contains the following elements of order two
$$ s_{i} = \left( \begin{array} {llllllll}
1 & \dots & 0 & & & 0 & \dots & 0 \\
& \ddots & & & & & & \\
0 & \dots & 1 & & & & & 0 \\
0 & \dots & & 0 & 1 & & \dots & 0 \\
0 & & & -1 & 0 & & \dots & 0 \\
0 & \dots & & & & 1 & \dots & 0 \\
& & & & & & \ddots & \\
0 & \dots & 0 & & & 0 & \dots & 1
\end{array} \right), i = 1, ..., n - 1;
$$
$$ w_{1} = \left( \begin{array} {rllll}
0 & 0 & \dots & 0 & t \\
0 & 1 & \dots & 0 & 0 \\
& & \dots & & \\
& & \dots & & \\
& & \dots & 1 & 0 \\
-t^{-1} & 0 & \dots & 0 & 0
\end{array} \right),~w_{2} = \left( \begin{array} {rllll}
0 & 0 & \dots & 0 & u \\
0 & 1 & \dots & 0 & 0 \\
& & \dots & & \\
& & \dots & & \\
& & \dots & 1 & 0 \\
-u^{-1} & 0 & \dots & 0 & 0
\end{array} \right).
$$
If $n = 2$ then we denote by $ w_{0}$ the element $s_{1}$ of the group
$G$. For general $n$ let $S$ be the constructed set of elements of the
Weyl group.Then $ \#S = n + 1 $ ( and $\mbox{rk}(G) + m$ for
$m$-dimensional field) and we have
\par\smallskip
\begin{theorem}.
The Weyl group $W$ has the following properties:
\begin{quotation}
i) $W$ is generated by the set $S$ of it's elements of order two,
ii) there exists an exact sequence
$$
0 \rightarrow E(= \mbox{Ker } \Sigma) \rightarrow W_{K/\bar K/k}
\rightarrow W_{K} \rightarrow 1, \\
$$
where
$$ \Sigma: \Gamma_{K} \oplus \stackrel{n} \dots \oplus \Gamma_{K}
\rightarrow \Gamma_{K} $$
is a summation map and $ W_{K}$ is isomorphic to the symmetric group
$\mbox{Symm}_{n}$ of $n$ elements,
iii) the elements $s_{i},~i = 1, \dots, n - 1$ define a splitting of
the exact sequence and the subgroup
$<s_{1}, \dots, s_{n - 1}>$
acts on $E$ by permutations,
iv) if $n = 2$ then the group $W$ has a presentation
$$W = <w_{0}, w_{1}, w_{2}/w_{0}^{2} = w_{1}^{2} = w_{2}^{2} = e,
(w_{0}w_{1}w_{2})^{2} = e>,$$
v) the Weyl groups of the group $G$ (for $n = 2$) over the local fields
$K,~K/{\bar K}, ~K/{\bar K}/k$ can be related by the following diagram
$$
\begin{array}{lllllllll}
& & 0 & & 0 & & & & \\
& & \uparrow & & \uparrow &
& & & \\
0 & \rightarrow & \Gamma_{K/\bar K} & \rightarrow & W_{K/\bar K} & \rightarrow
& W_{K} & \rightarrow & 0 \\
& & \uparrow & & \uparrow &
& \| & & \\
0 & \rightarrow & \Gamma_{K/\bar K/k} & \rightarrow & W_{K/\bar K/k} &
\rightarrow & W_{K} & \rightarrow & 0 \\
& & \uparrow & & \uparrow & & & & \\
& & \Gamma_{\bar K/k} & = & {\bf Z} & & & & \\
& & \uparrow & & \uparrow & & & & \\
& & 0 & & 0 & & & &
\end{array}
$$
\end{quotation}
\end{theorem}
\par\smallskip
{\sc Proof}. The claims i) - iii) and v) follow from a direct
computation. We have to use the multiplicative structure of the local
field $K$ and the structure of it's valuation rings (theorem 1 of
the previous section).
Let us deduce the presentation iv). It is easy to see that the
elements $w_{0}, w_{1}, w_{2}$ satisfy the conditions of the theorem. In
order to show that there are no other relations we observe that
according to the claim ii) of the theorem, the group $W$ can be
presented by some generators $a, b$ (free generators of the subgroup
$E$), $w_{0}$, and defining relations:
$$ w_{0}^{2} = e,~w_{0}aw_{0} = a^{-1},~w_{0}bw_{0} = b^{-1},~ab = ba. $$
We may assume that $a = w_{0}w_{1}$ and $b = w_{0}w_{2}$. It is
enough to show that these relations are equivalent to the relations of
the claim iv). Indeed, we have
$$ w_{0}aw_{0} = w_{0}w_{0}w_{1}w_{0} = (w_{0}w_{1})^{-1} $$
and similarly for $b$. Then
$$ e = (w_{0}w_{1}w_{2})^{2} = w_{0}w_{1}w_{2}w_{0}w_{1}w_{0}w_{0}w_{2} =
ab^{-1}a^{-1}b, \mbox{i. e.} ~ab = ba. $$
The same formulas will also give the equivalence between our
defining relations in the opposite direction also.
The theorem is proved.
{\sc Corollary }.
{\em The pair $(W,S)$ is not a Coxeter group
and furthermore there is no subset $S$ of involutions in $W$ such that
$(W,S)$ will be a Coxeter group}.
{\sc Proof.} We prove the second claim at once. Assume that the opposite
is true and consider the map of $S$ into the quotient-group $W_{K}$.
Choose an involution $s$ from the image of this set. Then the set $S'$ of
elements $S$ mapping to $s$ will generate the Coxeter group $W'$
\cite[ch. IV, \S 1.8]{B}.
By the theorem, it will be an extension of the free abelian group $ E$ of
rank $ > 1$ by a group of order 2. We see simultaneously that there is a
subgroup of $E$ which has rank $ > 1$ and on which the quotient-group
acts as a multiplication by $ - 1$.
We show that this is impossible for a Coxeter group. Let us consider
the Coxeter diagram of the pair $(W',S')$(for its definition and the
properties we need see
\cite[ch. IV, \S 1.9]{B}). It is clear that $\# S' > 1$ and the diagram
contains at least two vertices. If they are not connected by an edge then
the group has to contain a subgroup $ \mbox{\bf Z}/2 \oplus
\mbox{\bf Z}/2$ ,
which is obviously wrong. If the edge connecting the two vertices is
marked by some finite number $m > 2$, then our group has to contain
a
subgroup $\mbox{\bf Z}/m$ which is also impossible. It remains that the
diagram of our group is connected and all the edges are marked by the
symbol $\infty$. Now if we have only two vertices then
$W'$ cannot include a free abelian subgroup of rank
$> 1$. If the number of vertices $ > 2$ then $W'$ must
contain a free subgroup of rank $ > 1$ and this is also impossible.
\par\smallskip
{\bf Remark 3}.
If we consider the Weyl group for the $\mbox{SL}(2, K)$ defined over
$n$-dimensional local field then it will have $n+1$ generators
$w_{0}, \dots, w_{n}$ and the defining relations will be
$w_{i}^{2} = 1,~(w_{0}w_{i}w_{j})^{2} = 1$ for all $i,j$. It is not
a Coxeter group also.
\par\smallskip
{\bf Remark 4}.
Here we see the first basic difference between the Tits theory and ours.
The formalism of the
$BN$-pairs cannot be applied in our situation, at least, without some
substantial modifications. Nevertheless some corollaries of the Tits
axioms are valid, for example the Bruhat decomposition (see theorem 2
above)
In our situation there exists still some weaker form of the Tits axioms
(from \cite[ch. IV.2]{B}). More precisely they will be true only
partially and for $n = 2$
we can replace them by the following formula. Let
$$ w = \left( \begin{array}{ll}
0 & x^{-1} \\
- x & 0
\end{array} \right),~v = v'(x),~w(y) = \left( \begin{array}{ll}
0 & y^{-1} \\
- y & 0
\end{array} \right)
$$
If $s = w_{1}$ then there are three possibilities:
$$
\begin{array}{clll}
v \geq 0 & (BwB)(BsB) & = & BwsB \\
(0,-1) < v < 0 & (BwB)(BsB) & = & BwsB \bigcup_{(1,-1) + v \leq v'(y) <
(0,1) - v} Bw(y)B \\ v \leq (0,-1) & (BwB)(BsB) & = & BwsB
\bigcup_{(1,-1) + v \leq v'(y) < (0,1) + v} Bw(y)B
\end{array}
$$
We have the same expression for the diagonal elements $w \in W$. And if
$s = w_{0}, w_{2}$ then the Tits axiom T3
$$ BwBsB \subset BwB \cup BwsB $$
will be valid. These expressions can be deduced by
straightforward but rather lengthy computations using the elementary
transformations from the proof of theorem 2.
{\sc Problem 1.} To generalize the notion of $BN$-pair in order to
include both the Tits axioms and the infinite decompositions for
non-Coxeter groups which appear here.
For the BN-pairs attached to the algebraic groups in the Bruhat-Tits
theory we also know some finiteness property for the double classes
$BwB$. Namely, they are the finite unions of the cosets $Bg$.
This property is important for the definition of the Hecke rings (see
\cite{IM}). It is easy to see that this property is not preserved in
the higher dimensions. Thus the usual construction cannot be done in
our case.
{\sc Problem 2}. To define an analog of the Hecke ring for the
groups over $n$-dimensional local fields for $n > 1$.
\section{ Bruhat-Tits tree over local field of dimension 1}
{\sc The Complex} $\Delta(G, K)$.
First we assume that the field $K$ has no additional structure. Then the
spherical building of $G = PGL(V)$ over $K$ is a complex $\Delta(G, K)$
whose vertices are lines $l$ in the space $V$. All simplices of higher
dimension are degenerate and thus it's dimension equals zero. The group
$G(K)$ of rational points over $K$ acts on $\Delta(G, K)$ in a transitive
way.
Let $B$ be a Borel subgroup of $G$,
\begin{eqnarray*}
B
& = &
\left( \begin{array}{ll}
K^{*} & K \\
0 & K^{*}
\end{array} \right) ,
\end{eqnarray*}
Then $B$ is the stabilizer of a line in $V$ and
thus $B$ is a stabilizer of a vertex of $\Delta(G, K)$ . This gives us a
one to one correspondence between the Borel subgroups and the stabilizers
of the vertices.
The next important object inside $\Delta(G, K)$ is an {\em
appartment}~$\Sigma$. To specify it we need to choose a maximal torus $T$
of $G$
\begin{eqnarray*}
T
& = &
\left( \begin{array}{ll}
K^{*} & 0 \\
0 & K^{*}
\end{array} \right)
\end{eqnarray*}
or equivalently a splitting $ V = l_{1} \oplus l_{2} $. This means that
the torus $T$ will fix the pair of vertices corresponding to the lines
$l_{1}$ and $l_{2}$. And this pair is called an appartment. It's
stabilizer is the normalizer $N$ of the torus $T$,
\begin{displaymath}
N = \left( \begin{array}{ll}
K^{*} & 0 \\
0 & K^{*}
\end{array} \right)
\bigcup
\left( \begin{array}{ll}
0 & K^{*} \\
K^{*} & 0
\end{array} \right) .
\end{displaymath}
The group $ W = N/T $
is called the {\em Weyl group}. In our case it is of order two and has as
a generator an involution
\begin{eqnarray*}
w_{0}
& = &
\left( \begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array} \right)
\end{eqnarray*}
We see that the appartments are precisely the orbits of the Weyl group W.
{\sc The Complex} $\Delta(G, K/k)$.
Now we turn to
the case when the field $K$ is a local field (of dimension 1) with residue
field
$k$.
Denote by ${\cal O}$ the
valuation ring ${\cal O}_{K}$, by $u$ a generator of the maximal ideal $m$
and by
$\nu :K \rightarrow \Gamma_{K} \cong \mbox{\bf Z}$ the valuation homomorphism.
Again $G = PGL(V)$.
We define the euclidean building
$\Delta(G, K/k)$ as a one-dimensional complex constructed from classes of
lattices in $V$.
A lattice $L$ is an ${\cal O}_{K}$ -submodule in $V$ which is free and of
rank 2. A class $<L>$ of lattices is the set of all lattices $aL$ for $a
\in K^{*}$. We say that two classes $<L>$ and $<L'>$ are connected as the
vertices by an edge iff for some choice of $L$ and $L'$ we have an exact
sequence
$$ 0 \rightarrow L' \rightarrow L \rightarrow k \rightarrow 0. $$
This is equivalent to existence of a maximal totally ordered chain of
${\cal O}$-submodules in $V$ which is invariant under multiplication on
$K^{*}$ and contain $L$ and $L'$ \cite{BT2}.
Then from the combinatorial point of view $\Delta(G, K/k)$ is a homogeneous
tree
\cite{S2}.
We denote by $\Delta_{i}(G, K/k)$ the set of $i$-dimensional simplices.
From the construction we deduce easily the following property which we
will use in the sequel:
\begin{quotation}
{\sc Link property}
Let $P \in \Delta_{0}(K/k)$ be represented by a lattice $L$. Then the set
of edges going from $P$ is in one to one canonical correspondence with
the set of lines
${\bf P}(V_{P})$ in the vector space $V_{P} = L/mL$ of dimension 2 over k.
The last
set does not depend on the choice of $L$ (or better to say that there are
canonical isomorphisms between these ${\bf P}(V_{P})$ for different $L$'s
in the same class $<L>$).
\end{quotation}
In particularly, if $k = {\mbox{\bf F}}_{q}$ is a finite field then
$\Delta(G, K/k)$ is locally finite.
The group
$SL(V)$ acts on $\Delta(G, K/k)$ in the following way. It is transitive on the
edges $\in \Delta(G, K/k)$ and has two orbits on the vertices $\in \Delta(K/k)$.
There is a {\em type} of the
vertices $P$ which has two values. To understand this consider an exact
sequence
$$ 0 \rightarrow PGL^{+}(V) \rightarrow PGL(V) \rightarrow \mbox{\bf
Z}/2\mbox{\bf Z} \rightarrow 0 $$
where the last homomorphism is $\nu(det(.))$ {\em mod} 2. The group
$PGL(V)$ acts on $\Delta_{0}(G, K/k)$ in a transitive way and the group
$PGL^{+}(V)$ has the same two orbits as it's subgroup $SL(V)$.
Now let
\begin{eqnarray*}
B
& = &
\left( \begin{array}{ll}
{\cal O} & {\cal O} \\
m & {\cal O}
\end{array} \right)
\end{eqnarray*}
be the subgroup of $SL(V)$ consisting of the matrices whose entries
satisfy the written conditions. Then $B$ is a stabilizer of an edge of
the
$\Delta(G, K/k)$
and all the stabilizers look like this in an appropriate coordinate
system of $V$. The stabilizers of the boundary vertices of the edge are
$$P_{0} = \left( \begin{array}{rr}
{\cal O} & {\cal O} \\
{\cal O} & {\cal O}
\end{array} \right), P_{1} = \left( \begin{array}{ll}
{\cal O} & m^{-1} \\
m & {\cal O}
\end{array} \right) $$
We define the subgroup $N$ as above (so it does not reflect the local
structure on $K$). Instead of the maximal torus we take
\begin{eqnarray*}
T
& = &
\left( \begin{array}{ll}
{\cal O}^{*} & 0 \\
0 & {\cal O}^{*}
\end{array} \right)
\end{eqnarray*}
and the Weyl group $W = N/T$ is an extension
$$ 0 \rightarrow \mbox{\bf Z} \rightarrow W \rightarrow \mbox{\bf
Z}/2\mbox{\bf Z} \rightarrow 0 $$
Here we can identify the {\bf Z} with the valuation group $\Gamma_{K}$
and the {\bf Z}/2{\bf Z} with the previous Weyl group of $G$ over the
field $K$ without local structure.
We have a new involution
\begin{eqnarray*}
w_{1}
& = &
\left( \begin{array}{cl}
0 & u \\
-u^{-1} & 0
\end{array} \right)
\end{eqnarray*}
and the group $W$ is generated by $w_{0}$ and $w_{1}$.
The appartments $\Sigma$ are now the infinite lines:
\\
\\
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\put(70.00,2.00){\circle*{2.00}}
\put(50.00,2.00){\circle*{2.00}}
\put(50.00,6.00){\makebox(0,0)[cc]{$x_{n-1}$}}
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\put(90.00,6.00){\makebox(0,0)[cc]{$x_{n+1}$}}
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\put(14.00,2.00){\makebox(0,0)[cc]{$\dots$}}
\end{picture}
\\
The group $T$ acts trivially on $\Sigma$ and it's stabilizer coincides
with $N$. Thus appartments are the orbits of $W$. The vertices of
$\Sigma$ can be represented by lattices
\begin{equation}
\label{chain}
x_{n} = <L_{n}>, ~L_{n} = {\cal O} \oplus m^{n}, -\infty < n < \infty
\end{equation}
The action of $W$ on $\Sigma$ can now be easily described. If $w \in
\mbox{\bf Z}$ then $w$ acts by a translation of even length, and if $w
\not\in \mbox{\bf Z}$ then $w$ acts as an involution with a unique fixed
point $x_{n_{0}}$:
$$ w(x_{n_{0} + n}) = x_{n_{0} - n}$$
It can be proved that all the appartments look like this and thus they
are in one to one correspondence with the splittings $V = l_{1} \oplus
l_{2}$ of the space $V$.
{\sc Relations between } $\Delta(G, K)$ {\sc and} $\Delta(G, K/k)$. With
the local field $K$ of dimension 1 we can connect two local fields,
namely $K$ itself and $k$. They are local fields of dimension 0. Thus we
have three buildings attached to $G$ : $\Delta(G, K/k),
\Delta(G, K)$ and $\Delta(G, k)$.
The remark made above (the Link property) shows that the {\em link} of a
point $P \in \Delta(G, K/k)$ ( = the boundary of the $Star(P)$) is
isomorphic to $\Delta(G, k)$. The group
$G(k)$ acts on the last building, $P_{0}$ acts on the link of $P$ and the
isomorphism
between the buildings is an equivariant respective canonical homomorphism from
$P_{0}$ onto $G(k)$ (reduction map {\em mod} $m$).
To formalize the connection with $\Delta(G, K)$ we define a {\em boundary
point} of a tree as a class of half-lines such that intersection of any
two half-lines from the class is a half-line in both of them. We have now
an isomorphism of $G(K)$-sets between the set of boundary points and
$\Delta(G, K)$. If the half-line is represented by
$L_{n} = {\cal O} \oplus m^{n}, n > 0$ then the corresponding vertex in
$\Delta(K)$ is the line $K \oplus (0)$ in $V$.
It seems reasonable to consider the complexes $\Delta(G, K)$ and
$\Delta(G, K/k)$ together.
Denote by $\Delta_{.}[1](G, K/k)$ the complex of lattices introduced
above. We define the tree of $G$ as a union
$$ \Delta_{.}(G, K/k) = \Delta_{.}[1](G, K/k) \cup \Delta_{.}[0](G, K) $$
where $\Delta_{.}[1](G, K/k) = \Delta(G, K/k)$ and $\Delta_{.}[0](G, K/k)
$ is a complex of classes of ${\cal O}$-submodules in $V$ isomorphic
to
$K \oplus {\cal O}$
and we will call the subcomplex $\Delta_{.}[0](G, K/k)$ the {\em
boundary} of the tree. The definition of the boundary gives a topology on
$\Delta_{0}(G, K/k)$ which is discrete on both subsets $\Delta_{0}[1]$
and $\Delta_{0}[0]$.
Let $P_{n} = <L_{n}>$ and $L_{n} = {\cal O}e_{1} + m^{n}e_{2}$. If $P \in
\Delta[0]$ is represented by a line $l_{1} = Ke_{1}$ then $P_{n}
\rightarrow P$ since $\cap L_{n} = {\cal O}e_{1}$ belongs to a unique line,
namely to $l_{1}$ (see \cite[ch.II.1.1]{S2}). We can interpret the
points from $\Delta[0]$ as classes of ${\cal O}$-submodules which
are isomorphic to $K \oplus {\cal O}$ (see lemma 2 below). Then we
have $P = <L>,~L = Ke_{1} + {\cal O}e_{2}$ instead of $l_{1}$ and
the definition of the convergence can be given as $ \cup m^{-
n}L_{n} = L$.
It is easy to
extend it to 1-simplexes. In our case their limits at infinity will be
the degenerate simplexes. Thus we have a structure of a simplicial
topological space on the tree. It is simply a simplicial object in the
category of topological spaces. This topology is stronger then the
topology usually introduced to connect these complexes together (see
\cite{C1}).
Now the connections between the buildings over local fields of dimension
0 and 1 can be summarized as follows.
$$ \mbox{For any} P \in \Delta_{0}[1](PGL(V), K/k), \mbox{Link}(P) =
\Delta_{.}(PGL(V_{P}), k) $$
$$ \Delta_{.}[0](PGL(V), K/k) = \Delta_{.}(PGL(V), K) $$
The last subset can be called a star (or link) at infinity. It will be
interesting to define the last notion in purely simplicial terms (see
remark 6 below).
\section{Bruhat-Tits tree over local field of dimension 2}
As above let $G = \mbox{PGL}(V)$ be projective linear group of a vector
space $V$ of dimension 2 over a field $K$ and we assume now that $K$ is a
two-dimensional local field.
\par\smallskip
{\sc Definition 3.} {\em The vertices of the Bruhat-Tits tree}.
$$ \Delta_{0}[2](G, K/{\bar K}/k) = \{ \mbox{classes of}~{\cal O}'
\mbox{-submodules } L \subset
V : L \cong {\cal O}' \oplus {\cal O}' \}, $$
$$\Delta_{0}[1](G, K/{\bar K}/k) = \{ \mbox{classes of}~{\cal O}'
\mbox{-submodules } L \subset V
: L \cong {\cal O}' \oplus {\cal O} \}, $$
$$\Delta_{0}[0](G, K/{\bar K}/k) = \{ \mbox{classes of}~{\cal O}'
\mbox{-submodules } L \subset V
: L \cong {\cal O}' \oplus K \}. $$
The two submodules $L$ and $L'$ belong to one class $<L>
= <L'>$, iff $L = aL'$, with $a \in K^{*}$.
$$\Delta_{0}(G, K/{\bar K}/k) = \Delta_{0}[2](G, K/{\bar K}/k) \bigcup
\Delta_{0}[1](G, K/{\bar K}/k)
\bigcup \Delta_{0}[0](G, K/{\bar K}/k)$$
We say that the points from $\Delta_{0}[2]$ are the {\em inner} points,
the points from $\Delta_{0}[1]$ are the {\em inner boundary} points and
the points from $\Delta_{0}[0]$ are the {\em external boundary} points.
Sometimes we will delete $G$ and $K/\bar K/k$ from our notation if this
does not lead to a confusion.
We have defined the vertices only. For the simplices of higher dimension
we have the following
\begin{quotation}
{\sc Definition 4.}
Let $\{ L_{\alpha}, \alpha \in I \}$ be a set of ${\cal O}'$-submodules in
$V$. We say that $\{ L_{\alpha}, \alpha \in I \}$ is a {\em chain} iff:
i) for any $\alpha \in I$ and for any $a \in K^{\star}$ there exists an
$\alpha' \in I$ such that $aL_{\alpha} = L_{\alpha'}$,
ii) the set $\{ L_{\alpha}, \alpha \in I \}$ is totally ordered by the
inclusion.
$\{ L_{\alpha}, \alpha \in I \}$ is a {\em maximal chain} iff it cannot
be included in a strictly larger set satisfying the same conditions i) and ii).
We say that $<L_{0}>, <L_{1}>, ... , <L_{m}>$ belong to a {\em simplex} of
dimension
$m$ iff the $L_{i}, i = 0, 1, ..., m$ belong to a maximal chain of
${\cal O}'$-submodules in $V$. The faces and the degeneracies can be defined in
a standard way (as a deletion or a repetition of some vertex).
\end{quotation}
Thus the set $\Delta_{.}(G, K/\bar K/k)$ becomes a {\em simplicial set}.
The group $G = \mbox{PGL}(V)$ acts on ${\cal O}'$-modules. This gives a
simplicial action on $\Delta_{.}(G, K/\bar/k)$.
\par\smallskip
\begin{proposition}.
The set of all maximal chains of ${\cal O}'$ -submodules in the space
$V$ will be
exhausted by the following three possibilities:
\par\smallskip
i)$~\dots \supset m_{i, n}L \supset m_{i, n}L' \supset m_{i + 1, n}L
\supset m_{i + 1, n}L' \supset \dots \dots \supset m_{i, n + 1}L
\supset m_{i, n + 1}L' \supset m_{i + 1, n + 1}L \supset \dots,~i, n
\in \mbox{\bf Z}, $
where $<L>, <L'> \in \Delta_{0}(G, K/\bar K/k)[2] $ and
$L \cong {\cal O}' \oplus {\cal O}', ~L' \cong m \oplus {\cal O}'.$
\par\smallskip
ii)$~\dots \supset m_{i, n}L \supset m_{i + 1, n}L \supset m_{i + 2, n}
\supset \dots \dots \supset m_{i, n}L' \supset m_{i + 1, n}L' \supset
\dots \dots \supset m_{i, n + 1}L \supset m_{i + 1, n + 1}L \supset \dots ,
~i, n \in \mbox{\bf Z},$
where $<L>, <L'> \in \Delta_{0}(G, K/\bar K/k)[1] $ and
$L \cong {\cal O}' \oplus {\cal O}, ~L' \cong \wp \oplus {\cal O}'.$
\par\smallskip
iii) $~\dots \supset m_{i, n}L \supset m_{i + 1, n}L \supset \dots
\supset m_{i, n + 1}L \supset m_{i + 1, n + 1}L \supset \dots ,
i, n \in \mbox{\bf Z}$
where $<L>~\in \Delta_{0}(G, K/\bar K/k)[0] $ .
\end{proposition}
\par\smallskip
{\sc Proof.} The chains in the claim of our proposition can be
completed by the ${\cal O}'$-modules which are isomorphic to ${\cal
O}
\oplus {\cal O}$ and thus do not belong to the modules from
definition 3. Then a part (with $n = 0$) of the chain of first type
will look as follows:
\begin{equation}
\label{chain1}
\dots \supset {\cal O}L \supset \dots \supset L \supset L' \supset
mL \supset \dots \supset \wp L \supset \dots,
\end{equation}
where ${\cal O}L = {\cal O}L' \cong {\cal O} \oplus {\cal O}$~and
$\wp L = \wp L' \cong \wp \oplus \wp $. There is an isomorphism
${\cal O}L / \wp L \cong {\bar K} \oplus {\bar K}.$
For the same part of the chain of second type we have:
\begin{eqnarray}
\dots \supset {\cal O}L \supset \dots \supset L \supset mL \supset
\dots \supset {\cal O}L' \supset \dots \nonumber \\
\dots \supset L' \supset mL'\supset
\dots \supset \wp L \supset \dots, \label{chain2}
\end{eqnarray}
where ${\cal O}L \cong {\cal O} \oplus {\cal O},~{\cal O}L' \cong
\wp \oplus {\cal O}$~and $\wp L \cong \wp \oplus \wp $. Again there
exist isomorphisms
${\cal O}L / {\cal O}L' \cong {\bar K},~{\cal O}L' / \wp L
\cong {\bar K}.$
The last chain has the following structure:
\begin{equation}
\label{chain3}
\dots \supset {\cal O}L \supset \dots \supset L \supset
mL \supset \dots \supset \wp L \supset \dots ,
\end{equation}
where $L \cong K \oplus {\cal O}'$ ~and $ {\cal O}L / \wp L
\cong {\bar K}$.
Let us go to the proof of our proposition. It is easy to see that
the modules which we have inserted into our chains are the unions
(intesections)
of those $L_{\alpha}$ which are just to the right (left) of them.
Furthermore, if $L_{\alpha'}$ is the module, located after
$L_{\alpha}$,
then $L_{\alpha} / L_{\alpha'} \cong k $~(theorem 1). It follows
that all the chains from i) -- iii) are maximal ones. Now let
$L_{\alpha}$ be an arbitrary maximal chain satisfying to the
definition 3. We consider three cases:
1) For some $\alpha~L_{\alpha} \cong K \oplus {\cal O'}$. Then all
modules $~m_{i, n}L_{\alpha}$ enter into our chain, i.e. it
will coincide with the chain from iii).
2) For some $\alpha~L_{\alpha} \cong {\cal O}' \oplus {\cal O}' $.
Again all modules $m_{i, n}L_{\alpha}$ belong to the chain, but now
$L_{\alpha} / mL_{\alpha}$ has dimension 2 over $k$ and since our chain
is supposed to be a maximal one there exists a module $L'$ between these
two. All $m_{i, n}L'$ are in the chain, which should coincide with the
chain from i).
3) Now if $L_{\alpha} \cong {\cal O} \oplus {\cal O}'$, then the
chain contain subchains $\dots \supset m_{i, n}L_{\alpha} \supset
m_{i + 1, n}L_{\alpha} \supset \dots $, lying in between the modules
$\wp_{n}L_{\alpha}$ and $\wp_{n + 1}L_{\alpha}$. Choose some $n$.
The intersection of all $m_{i, n}L_{\alpha}$ for varying $i$ gives us a module
$L'' \supset \wp_{n + 1}L_{\alpha}$. An "empty" place which we have to the
right of $L''$ can be filled out if we
set $L' = \varphi^{-1}({\cal O}_{\bar K})$,
where $\varphi: L'' \rightarrow L''/\wp L \cong {\bar K}$. Then all
"multiples" $m_{i, n}L'$ should be presented in the chain because
of it's maximality. We see that the chain constructed in such way is
equal to the chain from ii).
The proposition is proved.
\par\smallskip
{\sc Corollary.} {\em The simplicial set $\Delta_{.}$ is a disconnected
union of it's subsets $\Delta_{.}[m], ~m = 0, 1, 2$. The dimension of the
subset $\Delta_{.}[m]$ equals to 0 for $m = 0$ and 1 for $m = 1, 2$} .
\par\smallskip
This is obvious. We need only note that all vertices of any simplex
can be represented by the modules of the same type and that in the
case of subset $\Delta_{.}[0]$ the chains of the type iii) contain only
{\em one} class of modules.
\par\smallskip
{\sc Definition 5.}{\em The projection map}.
For any ${\cal O}'$-module $L$ we have a ${\cal O}$-module
$M = L \otimes _{\cal O'} {\cal O}$ . This gives a map
$$ \pi : \Delta_{.}(K/{\bar K}/k) \rightarrow \Delta_{.}(K/{\bar K}) $$
in the tree of the same group $G$ over the field $K$, which we consider
as a local field of dimesnion 1 over ${\bar K}$.
\par\smallskip
\begin{proposition}.
The map $\pi$ has the following properties:
i) $\pi$ is a simplicial $G$-equivariant surjective map,
ii) $\pi$ induces a bijective map of the set $ \Delta_{.} (G, K/\bar
K/k)[0]$ onto the set $\Delta_{.}(G, K/\bar K)[0]$,
iii) if $\sigma = <L> \in \Delta_{0}(G, K/\bar K)[1]$, then there exist
simplicial and \\
$\mbox{Stab}(<L>)$-equivariant isomorphisms
$$ \pi^{-1}(\sigma) \bigcap \Delta_{.}(G, K/\bar K/k)[2] \cong
\Delta_{.}(\mbox{PGL}(L/\wp L),{\bar K}/k)[1],$$
$$ \pi^{-1}(\sigma) \bigcap \Delta_{.}(G, K/\bar K/k)[1] \cong
\Delta_{.}(\mbox{PGL}(L/\wp L),{\bar K}/k)[0],$$
where $L/\wp L$ is a vector space of dimension 2 over $\bar K$. Also
we have
$$ \pi^{-1}(\sigma) \bigcap \Delta_{.}(G, K/\bar K/k)[0]
= \emptyset,$$
iv) if two vertices from $\Delta_{0}(G, K/\bar K/k)[2]$ are connected by
an edge then they belong to the same fiber of the map $\pi,$
v) the image of any edge $\sigma \in \Delta_{1}(K/\bar K/k)[1] $ will be
(non-degenerate) edge in $\Delta_{.}(K/\bar K),$
vi) if $\sigma = (\dots \supset L \supset L' \supset \dots ) \in
\Delta_{1}(G, K/\bar K)[1]$, then $\pi^{-1}(\sigma)$ consists of
one edge, connecting vertices from $\pi^{-1}(<L>) \bigcap \Delta_{0}(G,
K/\bar K/k)[1]$ and $\pi^{-1}(<L'>) \bigcap \Delta_{0}(G, K/\bar
K/k)[1]$.
\end{proposition}
\par\smallskip
{\sc Proof.} The property i) is obvious.
Let $<L> \in \Delta_{0}(K/\bar K/k)[0]$~ and let $l \subset L$ be the
set of elements from $L$, divisible in $L$ by all $a \in K^{*}$.
\par\smallskip
\begin{lemma}.
The correspondence $<L> \mapsto l \subset V$~is a bijection between
$\Delta_{0}[0] $ and $\mbox{\bf P}(V)$.
\end{lemma}
\par\smallskip
{\sc Proof}. If $L = Ke_{1} \oplus {\cal O}'e_{2},$ then
$l = Ke_{1}$ and depends only on class $<L>$ . We can get all the lines
in such way. Now let $L = Ke_{1} \oplus {\cal O}'e_{2},~M = Ke_{1} \oplus
{\cal O}'e_{2}'$. If $e_{2}' = ae_{1} + be_{2},~a,b \in K$, then $M
= Ke_{1} + {\cal O}'be_{2} = bL$,~\\
i.e.~$<L>~=~<M>$.
This is also true for $\Delta_{.}(K/\bar K)[0]$. The claim ii)
follows since the projection commutes with the constructed
correspondence.
\par\smallskip
\begin{lemma}.
Let $L$ be a ${\cal O}'$-submodule in $V$ and
$L \cong {\cal O}' \oplus {\cal O}'$. Then for any point $P \in
\Delta_{0}[2]$~there exists a unique module $L'$ such that $<L'>~= P,\\
~L' \subset L$~
and one of the following equivalent conditions are true:
i) ~$L' \not \subset mL,$
ii)~$L/L' \cong {\cal O}'/a$, where $a$ is a principal ideal,
iii)~$L/L'$ is a module of rank 1.
\end{lemma}
\par\smallskip
{\sc Proof}. Take some module $L''$ in the class of the vertex
$P$. By the Cartan decomposition (theorem 2) there exists
a basis $e_{1, 2}$ in $V$ such that $L = {\cal O}'e_{1} \oplus {\cal
O}'e_{2},
~L'' = a_{1}e_{1} \oplus a_{2}e_{2},~a_{1, 2}$ are principal
ideals.
The standard arguments (see \cite[ch. II, \S1.1]{S2}) give the claim
of the lemma.
We now prove property iii). Fix a module $L_{0} \cong {\cal O}'
\oplus {\cal O}',~L_{0} \otimes_{{\cal O}'} {\cal O} = L$, i.e.
~$<L_{0}> \in
\pi^{-1}(\sigma) \cap \Delta_{0}[2]$. If $P \in \pi^{-1}(\sigma)$,
then according to lemma the vertex $P$ can be represented as $<L'>$ .
Then $L' = m_{i, n}e_{1} +
{\cal O}'e_{2},~L_{0} = {\cal O}'e_{1} + {\cal O}'e_{2}$~ and from
the equality
$\pi <L_{0}>~ = \pi <L'>~ =~ <L>$~ we get that $n = 0$. It follows
that $L' \supset \wp L_{0}$~and~ $L'$~defines a free
${\cal O}_{\bar K}$-module~$L'/\wp L_{0} \subset L_{0}/\wp L_{0}
\subset L/\wp L$~of rank 2 in space $L/\wp L$.
This correspondance gives the first bijection from iii). It is easy
to see that it preserves the simplicial structure of both sets and it is
equivariant under the stabilizer of the vertex $<L>$.
To construct the second bijection from ii) we take $P \in \pi^{-1}(\sigma)
\cap \Delta_{0}[1]$. If $P =~ <L'>$ then $<L' \otimes {\cal O}>~ =~ <L>$.
Changing the module $L'$ to an equivalent one we can assume that
$L' \otimes {\cal O} = L$ and $L' \subset L$. All such modules $L'$
can be transformed into one by a multipiliction by some
$a \in {\cal O}^{*}$. We have a map
$L' \rightarrow L/ \wp L$. The image $\mbox{Im}~L'$ will be
a ${\cal O}_{\bar K}$
-module in $L/ \wp L$ isomorphic to ${\bar K} \oplus {\cal O}_{\bar
K}$. As we saw the class $<\mbox{Im}~L'>$ will be defined in a
unique way. It defines a point in $\Delta_{0}(\mbox{PGL}(L/ \wp
L),~{\bar K}/k)[1]$. The constructed correspondence will be a
bijection with the properties
we need.
The last claim from iii) follows from the property ii) proved above.
To get iv) it is enough to apply lemma 2 and proposition 1, i).
The property v) can be seen from the description of the
chain~(\ref{chain2}) which represents the edge $\sigma$.
We need only to take it's quotient by the ideal $\wp$.
We check now the last property from the proposition. Let $P =~<L>,\\
~Q =~<L'>$
be two vertices of the tree $\Delta_{.}(K/\bar K)$ connected by an edge
$\sigma$. It is posible to choose a basis in $V$ such that $L = {\cal O}e_{1}
+ {\cal O}e_{2},~L' = \wp e_{1} + {\cal O}e_{2}$.
Then ${\cal O}'$-modules $M = {\cal O}'e_{1} + {\cal O}e_{2}$ and~
$M' = \wp e_{1} + {\cal O}'e_{2}$ will represent the boundary
points of the fibers $\pi^{-1}(P)$~and~$\pi^{-1}(Q)$ correspondingly. By
the
proposition 1, ii) they are connected by an edge which is mapped onto
an edge $\sigma$.
Thus the set $\pi^{-1}(\sigma)$~is not empty.
It consists of only one edge.
To make this clear we denote by $M, M'$ the modules which represent the
vertices of the edge lying over $\sigma$. By the proposition 1, ii) they
belong to a chain as in (\ref{chain2}). Now we remark that the set of
lines of the space $L/ \wp L$ is bijective to the following sets of
simplices of our trees:
\begin{itemize}
\item the set of the edges from $\Delta_{.}(K/\bar K)$ going out from the
vertex $P$~(link property, see section 3).
\item the set $\pi^{-1}(P) \cap \Delta_{1}[1]$~ of the boundary points
of the fiber $\pi^{-1}(Q)$~(the bijection constructed above).
\end{itemize}
From the definition of the bijection we conclude that the line
corresponding to the vertex $<M>$,
coincides with the line corresponding to the edge $\sigma$, i. e.
the vertex $<M>$ will be defined uniquely. As this is true also for the
vertex $<M'>$ we get that the edge connecting them will be also defined
in an unique way.
The proposition is proved.
\par\smallskip
{\sc Corollary 1}.{\em Any vertex $P \in \Delta_{0}[1]$~ belongs to
precisely one edge}.
\par\smallskip
{\sc Corollary 2}. {\em If $P \in \Delta_{0}(K/\bar K)[1]$ then the
stabilizer
$G_{P} \subset G$ of the vertex $P$ acts on the fiber $\pi^{-1}(P)$
by the reduction map}
$$ G_{P} \cong \mbox{SL}(2, {\cal O}_{K}) \rightarrow \mbox{SL}(2, {\bar K}).$$
Here we have fixed the modules $L$ with $<L> = P$ and $L_{0}$ with
$<L_{0}>~\in \pi^{-1}(P) \cap \Delta_{0}[2]$ .
\par\smallskip
We see that "inside" our construction there are five trees coming
from the dimensions $\leq 1$, namely
$$\Delta_{.}(K/{\bar K}),
\Delta_{.}(K), \Delta_{.}({\bar K}/k), \Delta_{.}({\bar K}), \Delta_{.}(k).$$
The first one is the tree which is the target of the projection map
$\pi$, the second one is the external boundary
and the three last trees will occur infinitely many times.
Thus the constructed simplicial set will be a disconnected union of it's
connected components. The $\Delta_{.}[2]$-piece of our tree is an
infinite disconnected union of the usual Bruhat-Tits trees = fibers of
the map $\pi$. The $~\Delta_{.}[1]$-piece will be an infinite
disconnected union of the edges.
In order to change this and to have a possibility
to pass from one fiber to another one has to use
some topology which will be a generalization of the topology we have
introduced in section 3.
\par\smallskip
{\sc Definition 6.}We say that a sequence $P_{n} \in \Delta_{0}[2]$
converges to $Q \in \Delta_{0}[1]$ iff there is a basis of $V$ and a
sequence $i(n)$ of integers such that $i(n) \rightarrow \infty$ as $n
\rightarrow \infty$ and for large $n$
$P_{n}$ and
$Q$ can be represented by the following modules
$$ P_{n} = <{\cal O}' \oplus m^{i(n)}>, ~Q = <{\cal O} \oplus {\cal O}'>$$
Also a sequence $Q_{n}$ from $\Delta_{0}[1]$ converges to a point $R$ from
$\Delta_{0}[0]$ iff in some basis and for some sequence $i(n)$ as above
$$ Q_{n} = <{\cal O}' \oplus {\wp}^{i(n)}>,~R = <K \oplus {\cal O}'>$$
Combining these two definitions we can get also a condition for a
sequence of points from $\Delta_{0}[2]$ to converge to a point from
$\Delta_{0}[0]$.
We introduce a topology on $\Delta_{0}(G, K/\bar K/k)$~as a discrete one
on any of the sets $\Delta_{0}[m]$ and for which the sequences introduced
above
are the only convergent sequences on the whole set.
$\Delta_{0}$. The convergence on the set of simplices
$\Delta_{1}$
can be defined as convergence of their vertices.
\par\smallskip
\begin{theorem}. $\Delta_{.}(G, K/\bar K/k)$ is a simplicial
topological space. Let $\mid \Delta_{.} \mid $ be it's geometrical
realization. Then
\begin{quotation}
i) $\mid \Delta_{.} \mid$ is a connected contractible topological space
of dimension 1 having a cell structure,
ii) if $x \in \mid \Delta_{.} \mid$ then $x$ has a neighbourhood
homeomorphic to an interval, if $x \notin \Delta_{0}[2]$, and to a
bouquet of (finite number if \\ $k = {\mbox{\bf F}}_{q}$ ) intervals
otherwise.
iii) the group $G$ acts on $\mid \Delta_{.} \mid$ by homeomorphisms
iv) $\mid \pi \mid$ is a continous map
v) if $\sigma = <L> \in \Delta_{0}(G, K/\bar K)[1]$ then the fiber
$\pi^{-1}(\sigma)$
is isomorphic to $\Delta_{.}(\mbox{PGL}(L/ \wp L), K/\bar K)$ as a
simplicial topological space.
\end{quotation}
\end{theorem}
We refer to \cite{D} for the notions of simplicial topological space and
it's geometrical realization.
{\sc Proof} can be given by a direct check
with an application of the proposition 2 and of the corresponding facts
for the trees $\Delta_{.}(K/\bar K)$
~and~$\Delta_{.}(\bar K/k)$ related to the local fields of dimension 1.
{\bf Remark 5}. $\mid \Delta_{.} \mid$ is not a CW-complex even if $n =
1$ and
$k = {\mbox{\bf F}}_{q}$
but it is a closure finite complex. Also we note that
$\mid \Delta_{.}(K/\bar K/k)
\mid $ is not a compact space just as in the case of local fields of
dimension 1.
We can make the results proved more transparent by drawing all that in
the following picture where the dots of different kinds belong to the
different
$\Delta_{.}[m]$-pieces of the tree:
\\
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\emline{93.00}{109.00}{39}{93.00}{109.00}{40}
\emline{93.00}{109.00}{41}{93.00}{109.00}{42}
\emline{52.00}{105.00}{43}{53.00}{109.00}{44}
\end{picture}
\\
\begin{center}
Pic. 1
\end{center}
Usually the buildings are defined as combinatorial complexes having a
system of subcomplexes called appartments (see, for example,
\cite{R, T}).
We show how to introduce them in our case.
\par\smallskip
{\sc Definition 7.} Let us fix a basis
$e_{1}, e_{2} \in V$. The {\em appartment},~defined by
this basis is the following set
$$ \Sigma_{.} = \bigcup_{0 \leq m \leq 2} \Sigma_{.}[m], $$
where
$$ \Sigma_{0}[m] = \left\{ \begin{array}{l}
<L> \mid L = a_{1}e_{1} \oplus a_{2}e_{2}, \\
\mbox{where }
a_{1}, a_{2}~\mbox{are } {\cal O}'\mbox{-submodules in } K \\
\mbox{and there exists a permutation }
~s , \\
\mbox{such that }
a_{s(1)} \cong {\cal O}_{(2)} = {\cal O}',~a_{s(2)} \cong {\cal O}_{(m)}
\end{array} \right\}.
$$
$ \Sigma_{.}[m] $ is the minimal subcomplex having $\Sigma_{0}[m]$ as vertices.
\par\smallskip
Let us denote the edge connecting the vertices $P$~and~ $Q$~by $\sigma(P, Q)$.
\begin{proposition}.
In some basis we have the following relations:
i) if
$$
\begin{array}{ccccccc}
x_{i,n} & = & <m_{i,n} \oplus {\cal O}'> & = & <{\cal O}' \oplus m_{-i,-n}>
,& & \\
y_{n} & = & <m_{i,n} \oplus {\cal O}> & = & <m_{j,n} \oplus {\cal O}>
& = & <{\cal O}' \oplus \wp^{-n} > , \\
z_{n}
& = & <{\cal O} \oplus m_{i,-n}> & = & <{\cal O} \oplus m_{j,-n}>
& = & <\wp^{n} \oplus {\cal O}' >,
\end{array}
$$
$$ x_{0} =~<K \oplus {\cal O}'>,~x_{\infty} =~<{\cal O}' \oplus K>, $$
then
$$ \Sigma_{0}[2] = \{ x_{i, n} \mid i, n \in \mbox{\bf Z} \},~
\Sigma_{1}[2] = \{ \sigma(x_{i, n},~x_{i + 1, n}) \mid i, n
\in \mbox{\bf Z} \},$$
$$ \Sigma_{0}[1] = \{ y_{n}, z_{n} \mid n \in \mbox{\bf Z} \},
~\Sigma_{1}[1] = \{ \sigma(y_{n},~z_{n}) \mid n \in \mbox{\bf Z} \}, $$
$$ \Sigma_{.}[0] = \{ x_{0}, x_{\infty} \}, $$
ii) let $\mbox{Stab}(\sigma)$ be a stabilizer of a simplex $\sigma$ in
the subgroup $SL(V)$.
Then
$$ \mbox{Stab}(x_{i,n}) = \left( \begin{array}{ll}
{\cal O}' & m_{i, n}\\
m_{-i, -n} & {\cal O}'
\end{array} \right),
\mbox{Stab}(\sigma(x_{i, n},~x_{i + 1, n})) = \left( \begin{array}{ll}
{\cal O}' & m_{i + 1, n}\\
m_{-i, -n} & {\cal O}'
\end{array} \right) $$
$$\mbox{Stab}(z_{n}) = \left( \begin{array}{ll}
{\cal O} & \wp^{n} \\
\wp^{-n + 1} & {\cal O}'
\end{array} \right), \mbox{Stab}(y_{n}) = \left(
\begin{array}{ll}
{\cal O}'& \wp^{n + 1} \\
\wp^{-n} & {\cal O}
\end{array} \right), $$
$$\mbox{Stab}(\sigma(y_{n - 1},~z_{n})) = \left( \begin{array}{ll}
{\cal O}'& \wp^{n} \\
\wp^{-n + 1} & {\cal O}'
\end{array} \right) $$
$$\mbox{Stab}(x_{0}) = \left( \begin{array}{ll}
K^{*} & K \\
0 & K^{*}
\end{array} \right),
\mbox{Stab}(x_{\infty}) = \left( \begin{array}{ll}
K^{*} & 0 \\
K & K^{*}
\end{array} \right).
$$
The stabilizers in the $PGL(V)$ are represented by the matrices from
$GL(V)$ satisfying the same conditions.
\end{proposition}
\par\smallskip
{\sc Proof }. It is obvious that all the vertices from i) belong to
$\Sigma$. It follows from the theorem 1 (section 1) that there are
no other
vertices. It is also clear that the simplicial complex described in i) is
a minimal complex containing it's vertices.
The formulas for the stabilizers (property ii) can be confirmed by
direct computations.
\par\smallskip
Thus the simplicial structure of an appartment can be presented as
the following triangulation of compactified line {\bf R }$ \cup
-\infty,~\infty$:
\\
\\
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\put(105.00,4.00){\makebox(0,0)[cc]{$\dots$}}
\put(80.00,4.00){\makebox(0,0)[cc]{$\dots$}}
\put(41.00,4.00){\makebox(0,0)[cc]{$\dots$}}
\put(16.00,4.00){\makebox(0,0)[cc]{$\dots$}}
\put(10.00,10.00){\makebox(0,0)[cc]{$x_{0}$}}
\put(20.00,10.00){\makebox(0,0)[cc]{$y_{n - 1}$}}
\put(35.00,10.00){\makebox(0,0)[cc]{$z_{n}$}}
\put(60.00,10.00){\makebox(0,0)[cc]{$x_{i,n}$}}
\put(70.00,10.00){\makebox(0,0)[cc]{$x_{i+1,n}$}}
\put(85.00,10.00){\makebox(0,0)[cc]{$y_{n}$}}
\put(100.00,10.00){\makebox(0,0)[cc]{$z_{n + 1}$}}
\put(115.00,10.00){\makebox(0,0)[cc]{$x_{i,n+1}$}}
\put(130.00,10.00){\makebox(0,0)[cc]{$x_{\infty}$}}
\end{picture}
\\
\begin{center}
Pic. 2
\end{center}
\begin{theorem}
The appartments $\Sigma{.}$ have the following properties:
\begin{quotation}
i) any two simplices are contained in an appartment,
ii) for any two apppartments
$\Sigma,\Sigma'$ there
exists an isomorphism $i:\Sigma \rightarrow \Sigma'$ such that
$i\mid_{\Sigma \cap \Sigma'} = \mbox{ identity}$,
iii) for any appartment $\bar \Sigma \subset \Delta_{.}(G, K/\bar K)$
there exists a unique appartment $\Sigma \subset
\Delta_{.}(G, K/\bar K/k)$ such that $ \pi(\Sigma) = \bar \Sigma, $
iv) a geometrical realization $\mid \Sigma_{.} \mid$ of an appartment
$\Sigma_{.}$ is homeomorphic to a closed interval,
v)~$\Sigma_{.} = \{ \sigma \in \Delta_{.} \mid \forall g \in T ~g(\sigma) =
\sigma \},~N(\Sigma_{.}) \subset \Sigma_{.}$
and the Weyl group $W$ acts on $\Sigma_{.}$.
If $w \in W$ is an involution then it has a fixed point
$x_{i_{0},n_{0}} \in \Sigma_{0}[2]$ and $w$ is a reflection:
$$w(x_{i,n}) = x_{2i-i_{0},2n-n_{0}},$$
$$w(y_{n_{0}+n}) = z_{n_{0}-n},
~w(z_{n_{0}+n}) = y_{n_{0}-n},~w(x_{0}) = x_{\infty}.$$
If $w \in \Gamma_{K} \cong \mbox{\bf Z} \oplus \mbox{\bf Z} \subset W$ then
$w = (0, 1)$ acts as a shift of the whole structure to the right
$$ w(x_{i,n}) = x_{i,n+2},$$
$$w(y_{n}) = y_{n+2},~w(z_{n}) = z_{n+2},
w(x_{0}) = x_{0},~w(x_{\infty}) = x_{\infty}. $$
The element $w = (1, 0)$ acts as a shift on the points $x_{i,n}$ but
leaves fixed the points in the inner boundary
$$ w(x_{i,n}) = x_{i+2,n},$$
$$w(y_{n}) = y_{n},~w(z_{n}) = z_{n},
~w(x_{0}) = x_{0},~w(x_{\infty}) = x_{\infty}. $$
Under the map $W_{K/{\bar K}/k} \rightarrow
W_{K/{\bar K}}$ this action goes to the action of Weyl group $W_{K/{\bar
K}}$ on an appartment of the tree $\Delta_{.}(K/{\bar K})$.
\end{quotation}
\end{theorem}
\par\smallskip
{\sc Proof.} If we compare the modules belonging to an appartment
according to proposition 3, ii) and the modules belonging to an
appartment of the tree over a local field of dimension 1 (see
(\ref{chain}) in section 3) we will see that they will go one to another
under the projection map. Thus we can find an appartment
$\Sigma$~in~$\Delta_{.}(K/\bar K/k)$,
projecting onto any given appartment of the tree
$\Delta_{.}(K/\bar K)$.
Note that an appartment always contains an edge from
$\Delta_{1}[1]$. Then from proposition 2, vi) we get that $\Sigma$
will be defined in an unique way. We have proved iii).
Let us prove the property i). For any two simplices there exists a
subcomplex in $\Delta_{.}(K/\bar K/k)$, having the same combinatorial and
topological structure as the line from picture 2 and containing our
simplices.
This is obvious from the picture 1. The image of this complex will be an
infinite chain, {\em i. e.} an appartment $\bar \Sigma$ in the tree
$\Delta_{.}(K/\bar K)$ (see section 3). By the result proved above, all
edges of our subcomplex which belong to
$\Delta_{.}[1]$ will also belong to an appartment
$\Sigma$ lying over $\bar \Sigma$. Next we consider the trees which are
the fibers of $\pi$. Looking at them we see that the other simplices of
our subcomplex also belong to $\Sigma$ (if two appartments in the usual
Bruhat-Tits tree have the same boundary points then they coincide).
To get ii) we remark that the intersection $\Sigma \cap \Sigma'$
will be an "interval", consisting of all the simplices
lying between two extreme points.
From picture 2 we see the existence of an isomorphism with the properties
which we need.
The property iv) is obvious. In v) we check only the first claim.
The other formulas can be deduced by direct computations. Let $\sigma
\notin \Sigma_{.}$ and let $\sigma$ be a vertex. Connect this
vertex with $\Sigma$ by a minimal "path" (= interval of an
appartment). This path will enter into the appartment $\Sigma$ at an
inner point
(corollary 1 of proposition 2).
Let $P$ be a vertex nearest to $\Sigma$ belonging to this path. Then $P$
belong to a usual Bruhat-Tits tree and there exists
$g \in G$ such that $g(P) \not = P$. Thus $g(\sigma)
\not = \sigma$. For the usual tree this property will follow from
the link property (section 3). Namely, if $P_{0}$ is a point of an
appartment then the group $T$ acts in a simply transitive way on all
edges coming out from
$P_{0}$ and not lying in the appartment.
The theorem is proved.
\par\smallskip
We note that the transformations from the Weyl group will be continous
but not necessarily smooth maps
of compactified line {\bf R }$ \cup -\infty,
\infty $ into itself.
If $P, Q \in \Sigma_{0}$,~then the subcomplex in $\Sigma_{.}$,
containing all the simplices lying between $P$ and $Q$ will be called
{\em a path} from $P$ to $Q$ (see picture 2).
\par\smallskip
{\sc Corollary}. {\em For any two vertices $P, Q \in \Delta_{0},~P \not =
Q$ there exists a unique path $PQ$ between them}.
\par\smallskip
As in the usual theory of the Bruhat-Tits tree we can introduce some
intrinsically defined metric over our tree.
If $<L>,<L'> \in \Delta_{0}(G)[2]$ then by Cartan decomposition (theorem
2 of section 2) there exists a basis
$e_{1}, e_{2}$ in $V$ such that
$$L = {\cal O}'e_{1} \oplus {\cal O}'e_{2},
~L' = a_{1}e_{1} \oplus a_{2}e_{2}, $$
where $a_{1}, a_{2}$ are some fractional ${\cal O}'$-ideals and
$\nu'(a_{1}) \geq \nu'(a_{2})$.
\par\smallskip
{\sc Definition 8}
~$d( <L>, <L'> ) = \nu'(a_{1}) - \nu'(a_{2}), $
where $<L>,<L'>$ are two vertices from $\Delta_{0}[2]$.
\par\smallskip
\begin{theorem}.
The function $ d(.,.)$ is a correctly defined metric on the set
$\Delta_{0}[2]$ having non-archimedean values in $\Gamma_{K}^{+}$. It
has the following properties
\begin{quotation}
i) $d(.,.)$ is invariant under the action of $G$.
ii)the projection map $\pi$ is a distance-decreasing map, precisely
$$ d(\pi(x), \pi(y)) = \pi (d(x, y))$$
iii)for any appartment $\Sigma$ there exists a simplicial map
$ \rho: \Delta_{.} \rightarrow \Sigma_{.} $
which is a retraction onto $\Sigma$ and which is a distance-decreasing
map on subset $\Delta_{.}[2]$.
iv) let $u, t$ be local parameters of the field $K$ and $P, Q \in
\Delta_{0}[2]$. Then $d(P, Q) = (m, n)$ and we have
$$ n = d(\pi P, \pi Q) \mbox{~in~} \Delta(K/\bar K)[1],$$
$$ m = d(Q, Q') \mbox{~in~}
\Delta(\bar K/k)[1] \cong
\pi^{-1}(Q) \cap \Delta(K/\bar K/k)[2], $$
where $Q' =
\left( \begin{array}{ll}
t^{n} & 0\\
0 & 1
\end{array} \right) P \in \pi^{-1}(Q) $
v) if $R \in PQ$, then $d(P, R) + d(R, Q) = d(P, Q)$,
vi) for $P, Q, P', Q' \in \Delta_{0}[2]$ there exist $g \in G$ such that
$gP = P', gQ = Q'$~if and only if~$d(P, Q) = d(P', Q')$.
\end{quotation}
\end{theorem}
\par\smallskip
{\sc Proof.} If we change a module inside it's class then the same number
will be added to the $\nu'(a_{1})$ and $\nu'(a_{2})$. Consequently, their
difference will be unchanged. The properties i) and ii) follows directly
from the definition. Let us show how to construct the retracting map.
Take an edge $\sigma$ of the appartment $\Sigma$. Then
$\Sigma_{.} - \sigma$ can be decomposed into two pieces
$\Sigma_{+}$ and $\Sigma_{-}$.
Let $0$ and $\infty$ be the points from the external boundary of the
appartment. We assume that $0~(\infty)$ are the limit points for
$\Sigma_{+}~(\Sigma_{-})$.
For any point $P \in \Delta_{0}[0]$ which does not belong to the
appartment there is a unique shortest path which connects $P$ with some
point
$Q(P)$ of the appartment.
Thus the whole external boundary $\Delta_{.}[0]$ can be divided into two pieces
$\Delta[0]_{+}$ and $\Delta[0]_{-}$.
The first piece $\Delta[0]_{+}$ will contain 0 and all the points which
are connected with $\Sigma_{+}$. All the other points will belong to
$\Delta[0]_{-}$. We start to construct $\rho$ from the external boundary:
$$ \rho (\Delta[0]_{+}) = 0, ~\rho (\Delta[0]_{-}) = \infty .$$
Then if $P \in \Delta[0]_{+}, ~P \not = 0$ there are two paths
connecting the point $Q(P)$ with external boundary: the path between
$P$~and~$Q(P)$, and the path between $0$~and~$Q(P)$ (a part of $\Sigma_{+}$).
There exists a unique simplicial bijection $s_{P}$ of one path onto
another one. Let us put
$$\rho (\sigma) = s_{P}(\sigma), \mbox{~if}~\sigma \mbox{~lies on the
path between }P~\mbox{and}~Q(P).$$
The same definition works for $\Delta[0]_{-}$. It is straightforward
that the constructed map is correctly defined on the whole tree and
satisfies all the conditions from iii).
Properties iv) and v) follows from ii) and direct computations
(compare with
theorem 5, v) ). To get vi) we first observe that we can assume
$P' = P$ (since $G$ is transitive on the tree) and $\pi(Q) = \pi(Q')$
(apply the same property for the tree $\Delta(K/\bar K)$). Now let
$n = d(P, Q) = d(P, Q')$~and we assume $n > 1$(otherwise we are done
by the property v) for the tree $\Delta(\bar K/k)$).Let $R$ be a common
point of the paths
$PQ$~and~$PQ'$ such that the intersection of the paths $RQ$ ~and~$RQ'$
is $R$. Let us denote by $R_{0}$ the inner boundary point of the
path $RP$ which is closest to $R$. Then $R, R_{0}$ belong to the same
fiber as $Q$~and~$Q'$ and the
equality $d(P, Q) = d(P, Q')$ is equivalent to $d(R, Q) = d(R, Q') $
in the tree $\Delta(\bar K/k)[1] \cong \Delta' =
\pi^{-1}(\pi (Q)) \cap \Delta(K/\bar K/k)[2]$. By proposition 3, ii) the
stabilizer $G'$ of the points $R_{0}$~and $P$ has the matrix form
$ \left( \begin{array}{ll}
{\cal O'} & {\cal O}\\
m_{i, n} & {\cal O'}
\end{array} \right) $~for some $i$. By the corollary 2
of proposition 2 $G'$ acts on $\Delta'$ as a group of upper
triangular matrices
$ \left( \begin{array}{ll}
{\cal O}^{*}_{\bar K} & {\bar K}\\
0 & {\cal O}^{*}_{\bar K}
\end{array} \right) $. This group acts
transitively on the boundary of $\Delta'$ outside $R_{0}$ and thus
under our distance condition it will move $Q$ to $Q'$.
The theorem is proved.
\par\smallskip
The last general notion which will be mentioned here is the type of the
vertices and also of the simplices. Let us consider an exact sequence
$$0 \rightarrow PGL^{+}(V) \rightarrow PGL(V) \rightarrow
\Gamma_{K}/2\Gamma_{K} \rightarrow 0$$
where the right hand map is $\nu'(det(.))$ {\em mod} 2. As we know
$$\Gamma_{K}/2\Gamma_{K} \cong \mbox{\bf Z}/2\mbox{\bf Z} \oplus
\mbox{\bf Z}/2\mbox{\bf Z}$$
It can be shown that the stabilizers of the vertices belong to the subgroup
$PGL^{+}(V)$ and thus we have a canonical map
$$ \Delta_{0}[2] \rightarrow \Gamma_{K}/2 \Gamma_{K}$$
which assign to the vertices {\em four} possible values, their type. The
type of a simplex will be then a subset of $\Gamma_{K}/2 \Gamma_{K}$. The
type is invariant under the action of $SL(V)$ and the fundamental domain
of this action is a disjoint union of two edges which are mapped by the
projection map on the adjacent vertices of an edge in the tree
$\Delta_{.}(K/{\bar K})$.
The integer points of the lattice $\Gamma_{K}$ can be located on an real
plane and it seems more reasonable to have a srtucture of dimension 2 on
our simplicial set.
In the case of local field $K$ of dimension $n$
the number of types equals to $2^{n}$
and the building of $G$ (see\cite{P}) could have a dimension depending on
$n$.
But if we would like to preserve one of most important features of the
Tits theory - the geometrical structure of reflections, walls, chambers
and so on then we are forced to introduce the simplicial structure as we
did above. The reason is that the involutions from the Weyl group have
very small fixed point set on the lattice $\Gamma_{K}$ (see theorem 3
above). In particularly, in dimension two they have the points as the
fixed points but not the lines as would be the case if the dimension of
our building were two.
{\bf Remark 6}.
Our use of the topology was rather artificial. It seems there should
exist a purely simplicial construction which binds the
$\Delta_{.}[m]$-pieces of the tree together. We can define $
\Delta_{.}(G) = \Delta_{.}[2] * \Delta_{.}[1]
* \Delta_{.}[0], $
where * is a join of the simplicial complexes. Then the group
$G \times G \times G $
will act on the whole $\Delta_{\mbox{max}}(G)$ in a transitive way and we
will have a one to one correspondence between subgroups of this larger
group and the simplexes of the new complex which has a dimension 4. The
same remark is true for the groups of higher rank over arbitrary local
fields \cite{P}.
We also add the following problem.
{\sc Problem 3.} It is well known that the buildings of the group
$\mbox{PGL}(V)$ (and in particularly the Bruhat-Tits tree) can be defined
as classes of norms on the space $V$ \cite[II, 1.1]{BT2, S2} . There is
no doubt that this approach can be developed also for the higher
buildings of this group also. But this should give directly a geometrical
realization of the simplicial set $\Delta_{.}(G)$ which was defined in
\cite{P}.
|
1997-02-17T08:45:51 | 9605 | alg-geom/9605002 | en | https://arxiv.org/abs/alg-geom/9605002 | [
"alg-geom",
"math.AG"
] | alg-geom/9605002 | Yuri G. Prokhorov | Yuri G. Prokhorov | On Extremal Contractions from Threefolds to Surfaces: the Case of One
non-Gorenstein Point | LaTeX2e, to appear in Contemporary Math. AMS | In Birational algebraic geometry (Baltimore, MD, 1996), Contemp.
Math., vol. 207, 119-141. Amer. Math. Soc., Providence, RI, 1997. | null | null | null | Let $f\colon X\to S$ be an extremal contraction from a threefold with only
terminal singularities to a surface. We study local analytic structure such
contractions near degenerate fiber $C$ in the case when $C$ is irreducible and
$X$ has on $C$ only one non-Gorenstein point.
| [
{
"version": "v1",
"created": "Sat, 11 May 1996 16:26:20 GMT"
},
{
"version": "v2",
"created": "Mon, 17 Feb 1997 07:45:53 GMT"
}
] | 2010-05-12T00:00:00 | [
[
"Prokhorov",
"Yuri G.",
""
]
] | alg-geom | \section*{Introduction}
Let $V$ be a non-singular algebraic projective threefold over ${\bb{C}}$
of Kodaira dimension
$\kappa(V)=-\infty$. It follows from the Minimal Model Program
that $V$ is birationally equivalent to an algebraic projective
threefold $X$
with only terminal ${\bb{Q}}$-factorial singularities and with
a fiber space structure
$f\colon X\to S$ over a lower dimensional variety $S$ such that $-K_X$
is $f$-ample
and relative Picard number of $X/S$ is one.
Thus, at least in principle, one can reduce the problem of studying
birational
structure of $V$ to studying structure of $S$ and the fiber space $X/S$.
In the present paper we investigate the local structure of
$X/S$ near singular fibers in the case $\dim(S)=2$.
In this situation it is easy to see that a general fiber is a conic
in ${\bb{P}}^2$.
We will work in the analytic situation.
\begin{definition}\label{def}
Let $(X,C)$ be a germ of a three-dimensional complex
space along a compact reduced curve $C$ and let
$(S,0)$ be a germ of a
two-dimensional normal complex space.
Suppose that $X$ has at worst terminal singularities.
Then we say that a proper morphism
$f\colon (X,C)\to (S,0)$ is a {\it Mori conic bundle}
if \par
\begin{enumerate}
\renewcommand\labelenumi{(\roman{enumi})}
\item $(f^{-1}(0))_{\mt{red}}=C$;
\item $f_*\cc{O}_X=\cc{O}_S$;
\item $-K_X$ is ample.
\end{enumerate}
\end{definition}
The following conjecture is interesting for applications of
Sarkisov program to the
rationality problem for conic bundles \cite{Iskovskikh1},
\cite{Iskovskikh}
or studying of ${\bb{Q}}$-Fano
threefolds with extremal contractions onto a surface \cite{Pro1}.
\begin{conjecture}\label{11}
Let $f\colon (X,C)\to (S,0)$ be a Mori conic bundle.
Then $(S,0)$ is a DuVal singularity of type $A_n$.
\end{conjecture}
It was proved in \cite{Cut} that if $X$ is Gorenstein, then
$(S,0)$ is non-singular and $f\colon (X,C)\to (S,0)$ is a
"usual" conic bundle,
i.~e. there exists an embedding
$(X,C)\subset\bb{P}^2\times (S,0)$ such that
all the fibers of $f$ are conics in ${\bb{P}}^2$.
Earlier \cite{Pro2} (see also \cite{Pro3}) we classified all
Mori conic bundles containing
only points of indices $\le 2$. In this case $(S,0)$ is
non-singular or
DuVal of type $A_1$.
In general,
Conjecture \ref{11} follows from the following special
case of Reid's general elephant conjecture
(see \cite{Pro}, \cite{Pro1}).
\begin{conjecture}\label{reidd}
Let $f\colon (X,C)\to (S,0)$ be a Mori conic bundle.
Then the linear system $|-K_X|$ contains a divisor having
only DuVal singularities.
\end{conjecture}
It can be generalized as follows.
\begin{conjecture}
\label{log}
Let $f\colon (X,C)\to (S,0)$ be a Mori conic bundle. Then for
any $n\in{\bb{N}}$
the following holds
\par\medskip\noindent
$(*_n)$\qquad
There exists a divisor $D\in |-nK_X|$ such that
$K_X+\frac{1}{n}D$ is log-terminal.
\par\medskip\noindent
\textup{(}Note that \textup{\ref{log}} $(*_1)$ is equivalent
to \textup{\ref{reidd}}\textup{)}.
\end{conjecture}
By \cite[Theorem 4.5]{Kawamata},
for every threefold $X$ with terminal singularities
there exists a projective bimeromorphic morphism
$q\colon X^{q}\to X$ called {\it ${\bb{Q}}$-factorialization} of $X$
such that $X^{q}$ has only terminal
${\bb{Q}}$-factorial along $q^{-1}(C)$ singularities and $q$ is an
isomorphism in
codimension 1. If $f\colon (X,C)\to (S,0)$ is a Mori conic bundle,
then applying the Minimal Model Program to $X^{q}$ over $(S,0)$
we obtain a new Mori conic bundle $f'\colon (X^{\prime},C')\to (S,0)$
over the same base surface $S$ and
$\rho (X^{\prime},C^{\prime})/(S,0)$=1.
In particular $C'$ is irreducible by
Corollary~\ref{vanishing}.
Therefore it is sufficient to prove Conjecture~\ref{11}
assuming that $C$ is irreducible.
\par
In the this paper we will study Mori conic bundles
$f\colon (X,C)\to (S,0)$ with irreducible
central fiber. In this situation $X$ contains at most three
singular points (Corollary~\ref{<=3}). Arguments similar to
\cite[0.4.13.3]{Mori-flip} show also that $X$ can contain at
most two
non-Gorenstein points
\cite{Pro1}. Our results concern only the case when $X$ contains
only one non-Gorenstein singular point.
It will be proved in this case (Theorem~\ref{main}) that
the point $(S,0)$ may be either
non-singular or DuVal of type $A_1$ or $A_3$. We also show that
either $(*_1)$ or $(*_2)$ hold and give a
classification if $(S,0)$ is singular.
Our main tool will be Mori's technique \cite{Mori-flip}.
\subsection*{Acknowledgments.}
I would like to acknowledge discussions that I had on this
subject with Professors V.I.~Iskovskikh,
V.V.~Shokurov and M.~Reid.
I have been working on this problem at
Max-Planck Institut f\"ur Mathematik and
the Johns Hopkins University. I am very grateful them
for hospitality and support.
\section{Preliminary results}
The following is a consequence of the Kawamata-Viehweg
vanishing theorem (see \cite[\S 4]{Nakayama},
\cite[1-2-5]{KMM}).
\begin{theorem}
Let $f\colon (X,C)\to (S,0)$ be a
Mori conic bundle. Then
$R^if_*\cc{O}_X=0$, $i>0$.
\begin{flushright
\end{theorem}
\begin{corollary1}
\textup{(}cf. \cite[(1.2)-(1.3)]{Mori-flip}\textup{)}
\label{vanishing}
\label{tree}
Let $f\colon (X,C)\to (S,0)$ be a
Mori conic bundle. Then
\begin{enumerate}
\renewcommand\labelenumi{(\roman{enumi})}
\item For an arbitrary ideal $\cc{I}$ such that
$\mt{Supp}(\cc{O}_X/\cc{I})\subset C$ we have,
$H^1(\cc{O}_X/\cc{I})=0$.
\item
The fiber $C$
is a tree of non-singular rational curves
(i.~e. $p_a(C')\le 0$ for any subcurve $C'\subset C$).
\item
If $C$ has $\rho$ irreducible components, then
$$
\mt{Pic}(X)\simeq H^2(C,\bb{Z})\simeq\bb{Z}^{\oplus\rho}.
$$
\begin{flushright
\end{enumerate}
\end{corollary1}
\subsection{Terminal singularities.}
\label{terminal}
Let $(X,P)$ be a terminal singularity of index $m\ge 1$
and let $\pi\colon (X\3,P\3 )\to (X,P)$ be the canonical cover.
Then $(X\3,P\3 )$ is a terminal singularity of index 1 and
$(X,P)\simeq (X\3,P\3)/\cyc{m}$.
It is known (see \cite{Pagoda}) that $(X\3,P\3)$ is a
hypersurface singularity, i.~e. there exist an
$\cyc{m}$-equivariant embedding
$(X\3,P\3)\subset ({\bb{C}}^4,0)$.
Fix a character $\chi$ that generates $\mt{Hom}(\cyc{m},{\bb{C}}^*)$.
For
$\cyc{m}$-semi-invariant $z$ define weight
$\mt{wt}(z)$ as an integer defined $\mod m$ such that
$$
\mt{wt}(z)\equiv a\mod m\qquad {\rm iff}
\qquad
\zeta(z)=\chi(\zeta)^a\cdot z\quad
\text{for all}\quad \zeta\in\cyc{m}.
$$
\begin{theorem1}
\textup{(}see \cite{Mori-term}, \cite{YPG}\textup{)}
\label{cl-term}
In notations of \textup{\ref{terminal}}
$(X\3,P\3)$ is analytically
$\cyc{m}$-isomorphic to
a hypersurface $\phi=0$ in
$(\bb{C}^4_{x_1,x_2,x_3,x_4},0)$ such that $x_1,x_2,x_3,x_4$
are semi-invariants and we have one of the following two series:
\begin{itemize}
\item[{\rm (i)}]
\textup{(}the main series\textup{)}
$\mt{wt}(x_1,x_2,x_3,x_4;\phi )\equiv (a,-a,b,0;0)\mod m$, or
\item[{\rm (ii)}]
\textup{(}the exceptional case\textup{)} ${m=4}$ and
$\mt{wt}(x_1,x_2,x_3,x_4;\phi )\equiv (a,-a,b,2;2)\mod 4$, \par\noindent
\end{itemize}
where $a$, $b$ are integers prime to $m$.
\end{theorem1}
Note that in case (i) the variety $X\3$ can be non-singular.
In this situation $(X,P)\simeq ({\bb{C}}^3,0)/\cyc{m}(a,-a,b)$.
Such singularities are called {\it terminal cyclic
quotient singularities} and
denoted also by $\frac{1}{m}(a,-a,b)$.
\begin{theorem1}
\textup{(}see \cite{YPG}\textup{)}
\label{g.e.}
Let $(X,P)$ be a germ of terminal singularity.
Then a general member $F\in |-K_X|$ has only DuVal
singularity (at $P$).
\end{theorem1}
\subsubsection{}
\label{term-cl}
Terminal singularities can be classified in terms of
a general member
$F\in |-K_X|$ and its preimage $F\3\colon ={\pi}^{-1}(F)$
under
the canonical cover $\pi\colon X\3\to X$ \cite{YPG}:
\par\bigskip
\begin{center}
{\large{
\begin{tabular}{|c|c|c|}
\hline
index&type&cover \quad $F\3\to F$\\
\hline
\multicolumn{3}{|c|}{{\rm the main series}}\\
\hline
$m$ & $cA/m $ & $A_{k-1}\stackrel{m:1}{\longrightarrow}A_{km-1} $\\
\hline
$2$ & $cAx/2$ & $A_{2k-1}\stackrel{2:1}{\longrightarrow}D_{k+2} $\\
\hline
$2$ & $cD/2 $ & $D_{k+1}\stackrel{2:1}{\longrightarrow}D_{2k} $\\
\hline
$3$ & $cD/3 $ & $D_{4}\stackrel{3:1}{\longrightarrow}E_{6} $\\
\hline
$2$ & $cE/2 $ & $E_{6}\stackrel{2:1}{\longrightarrow}E_{7} $\\
\hline
\multicolumn{3}{|c|}{{\rm the exceptional case}}\\
\hline
$4$ & $cAx/4$ & $A_{2k-2}\stackrel{4:1}{\longrightarrow}D_{2k+1}$\\
\hline
\end{tabular}
}}
\end{center}
\par\bigskip
Cyclic quotient singularities are included in type $cA/m$.
\begin{definition1}
Let $X$ be a normal variety and $\mt{Cl}(X)$ be its
Weil divisor class group. The subgroup of
$\mt{Cl}(X)$ consisting of Weil divisor classes
which are ${\bb{Q}}$-Cartier is called the {\it semi-Cartier
divisor class group}. We denote it by $\mt{Cl}^{sc}(X)$.
\end{definition1}
\begin{theorem1}
\textup{(}see \cite{Pagoda},\cite{Kawamata}\textup{)}
\label{generator}
Let $(X,P)$ be a germ of 3-dimensional singularity.
Then $\mt{Cl}^{sc}(X,P)\simeq\cyc{m}$ and it is
generated by the class of $K_{X}$.
Moreover the local fundamental group of
$(X,P)$ is cyclic:
$\pi_1(X-\{P\})\simeq\cyc{m}$.
\end{theorem1}
\begin{definition1}
Let $(S,0)$ be a two-dimensional log-terminal singularity.
It is well known (see \cite{Kawamata}) that
$(S,0)$ is a quotient singularity,
i.~e. $(S,0)\simeq ({\bb{C}}^2,0)/G$, where $G\subset GL(2,{\bb{C}})$ is a
finite subgroup without quasi-reflections.
The natural cover $({\bb{C}}^2,0)\to (S,0)$
we call the {\it topological cover} and the order of $G$ we call
the {\it topological index} of $(S,0)$.
\end{definition1}
The following is an easy consequence of \ref{terminal}.
\begin{lemma1}
\label{index-surf-threef}
Let $(X,P)$ be a germ of a terminal threefold
singularity of index $m>1$ and
$(F,P)\subset (X,P)$ be a germ of
irreducible surface. Assume that $F$ is ${\bb{Q}}$-Cartier and
$(F,P)$ is DuVal with topological index $n$.
\begin{itemize}
\item[{\rm (i)}]
Then $n$ is divisible by $m$.
\item[{\rm (ii)}]
Moreover if $n=m$, then $(X,P)$ is a cyclic
quotient singularity and $(F,P)$ is of type $A_{m-1}$.
\end{itemize}
\end{lemma1}
\begin{proof}
Let $F\3:=\pi^{-1}(F)$. The divisor $F\3$ is Cartier on $X\3$, because
point $(X\3,P\3)$ is terminal of index 1
(see \cite[Lemma 5.1]{Kawamata}).
On the other hand $F\3$ is non-singular outside $P\3$.
Therefore $F\3$ is irreducible and normal. The cover $F\3\to F$ is \'etale
outside $P$. Hence the topological cover of $(F,P)$ can be facorized through
$F\3\to F$. This proves (i). In conditions (ii) $F\3$ is non-singular
and so is $X\3$.
\end{proof}
\begin{proposition}
\textup{(}see \cite{Mori-flip}\textup{)}
\label{cd}
Let $(X,P)$ be a germ of terminal singularity of
index $m$, $(C,P)\subset (X,P)$ be a germ of smooth curve and
$\pi \colon (X\3 , P\3 )\to (X,P)$ be the canonical cover and
$C\3:=(\pi^{-1}(C))_{\mt{red}}$. Then
\begin{itemize}
\item[{\rm (i)}]
for an arbitrary $\xi\in\mt{Cl}^{sc}(X,P)$,
there exists an effective (Weil) divisor $D$
such that $[D]=\xi$ and $\mt{Supp}(D)\cap C=\{ P\}$.
\item[{\rm (ii)}]
$\xi\to (D\cdot C)_P$ induces a homomorphism
$$
\mt{cl}(C,P)\colon \mt{Cl}^{sc}(X,P)\to\frac{1}{m}\bb{Z}/\bb{Z}
\subset\bb{Q}/\bb{Z}.
$$
\end{itemize}
\begin{flushright
\end{proposition}
\begin{definition}
(see \cite{Mori-flip})
\label{shushu}
Let things be as in \ref{cd}. $X\supset C$ is called
{\it \textup{(}locally\textup{)} imprimitive} at $P$ of
{\it splitting degree} $d$
if $\mt{cl}(C,P)\colon \mt{Cl}^{sc}(X,P)\to\frac{1}{m}\bb{Z}/\bb{Z}$
is not an isomorphism and the order of
$\mt{Ker}(\mt{cl}(C,P))$ is equal $d$.
The order of $\mt{Coker}(\mt{cl}(C,P))$ is called the
{\it subindex} of $P$ and usually denoted by $\bar{m}$.
(Note that $\bar{m} d=m$).
If $\mt{cl}(C,P)\colon \mt{Cl}^{sc}(X,P)\to\frac{1}{m}\bb{Z}/\bb{Z}$
is an isomorphism, then $P$ is said to be
{\it \textup{(}locally\textup{)} primitive}.
In this case we put $d=1$, $\bar{m}=m$.
\end{definition}
In the situation above the preimage $C\3:=\pi^{-1}(C)$
under the canonical cover $\pi\colon X\3\to X$ has
exactly $d$ irreducible components.
\subsection{Construction, \cite{Mori-flip}.}
\label{Mori-covering}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a
Mori conic bundle with irreducible central fiber and let
$P\in X$ be a point of index $m>1$.
Assume that $(X,C)$ is imprimitive of
splitting degree $d$ at $P$.
Take an effective Cartier divisor $H$
such that $H\cap C$ is a smooth point of $X$
and $(H\cdot C)=1$ and take an effective Weil ${\bb{Q}}$-Cartier divisor
$D$ such that $D\cap C=\{ P\}$ and
$D$ is a generator of $\mt{Cl}^{sc}(X,P)$ (see \textup{\ref{cd}}). Then
the divisor
$$
D:=(m/d)D-((mD\cdot C)/d)H
$$
is a $d$-torsion element in $\mt{Cl}^{sc}(X,C)$. It defines a
finite Galois
$\cyc{d}$-morphism $g\6\colon X'\to X$ such that
$P':={g'}^{-1}(P)$
is one point, $g'$ is \'etale over $X-\{ P\}$
(hence $X'$ has only terminal singularities),
index of $(X',P')$ is equal to $m/d$, $C':=({g'}^{-1}(C))_{\mt{red}}$
is a union of $d$ $\bb{P}^1$'s meeting only at $P'$,
and each irreducible component of $C'$ is primitive at $P'$. \par
\section{Local invariants $w_P$ and $i_P$}
In this section we following Mori \cite{Mori-flip}
introduce numerical invariants $w_P$ and $i_P$.
There is nothing new in this section. The material is contained only
for convenience of the reader.
\subsection{}
Let $X$ be a normal three-dimensional
complex space with only terminal
singularities and let $C\subset X$ be a
reduced non-singular curve. Denote by
$\cc{I}_C$ the ideal sheaf of $C$ and $\omega_X:={\cc{O}}_X(K_X)$.
As in \cite{Mori-flip}, we consider the
following sheaves on $C$:
$$
\begin{array}{l}
{\mt{gr}}^0_C\omega:=\hbox{{\rm torsion-free part of}}\ \omega_X/(\cc{I}_C\omega_X),\\
\\
{\mt{gr}}^1_C{\cc{O}}:=\hbox{{\rm torsion-free part of}}\ \cc{I}_C/\cc{I}^2_C.\\
\end{array}
$$
Since $C$ is non-singular, we have
$$
\omega_X/(\cc{I}_C\omega_X)={\mt{gr}}^0_C\omega\oplus\mt{Tors},
\qquad
\cc{I}_C/\cc{I}^2_C={\mt{gr}}^1_C{\cc{O}}\oplus\mt{Tors}.
$$
\subsection{}
Let $m$ be the index of $X$.
The natural map
$$
(\omega_X\otimes\cc{O}_C)^{\otimes m}\to
\cc{O}_C(mK_X)
$$
induces an injection
$$
\beta\colon ({\mt{gr}}^0_C\omega)^{\otimes m}\to \cc{O}_C(mK_X).
$$
Denote
$$
w_P:=(\mt{length}_P\mt{Coker}\beta)/m.
$$
Note that if $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ is a Mori conic bundle,
then $\deg {\mt{gr}}^0_C\omega <0$, (because $\deg \cc{O}_C(mK_X)<0$).
\subsection{}
We also have the natural map
$$
\begin{CD}
{\mt{gr}}^1_C{\cc{O}}\times{\mt{gr}}^1_C{\cc{O}}\times\omega_C
@>>>
\omega_X\otimes\cc{O}_C
\to
{\mt{gr}}^0_C\omega,\\
x\times y\times zdt@>>> zdx\wedge dy\wedge dt\\
\end{CD}
$$
that induces a map
$$
\alpha\colon \wedge^2({\mt{gr}}^1_C{\cc{O}})\otimes
\omega_C\to{\mt{gr}}^0_C\omega.
$$
Let
$$
i_P:=\mt{length}_P\mt{Coker}(\alpha ).
$$
Note that $i_P=0$ if $X$ is smooth at $P$.
\begin{lemma1}
\textup{(}\cite[2.15]{Mori-flip}\textup{)}
\label{non}
If $(X,P)$ is singular, then $i_P\ge 1$.\begin{flushright
\end{lemma1}
\begin{example}
\label{example}
Let $(X,P)$ be a cyclic quotient of type $1/\bar{m}(1,-1,\bar{m}+1)$
($\bar{m}$ is even) and let $\pi\colon ({\bb{C}}^3_{x_1,x_2,x_3},0)\to (X,P)$ be
its canonical cover.
Consider the curve $C\3:=\{x_3=x_2^2-x_1^{2\bar{m}-2}=0\}\subset{\bb{C}}^3$.
This curve has exactly two components
$C\3(1), C\3(2)=\{x_3=x_2\pm x_1^{\bar{m}-1}=0\}$ permuted under the action
of $\cyc{2\bar{m}}$.
Thus the image $C:=\pi(C\3)$ is a smooth irreducible curve,
$X$ is locally imprimitive along $C$ with splitting degree $2$.
The curve $C$ is naturally isomorphic to $C\3(1)/\cyc{\bar{m}}$
and a local uniformizing parameter on $C$ is $x_1^{\bar{m}}$.
It is easy to see that
\begin{itemize}
\item[(i)]
${\cc{O}}_{C,P}\simeq{\bb{C}}\{x_1^{\bar{m}}\}$.
\item[(ii)]
${\cc{O}}_C(mK_X)\simeq
{\cc{O}}_C(d x_1\wedge d x_2\wedge d x_3)^m$,\qquad\qquad
${\mt{gr}}^0_C\omega \simeq {\cc{O}}_C(x_1^{\bar{m}-1}
x_1\wedge d x_2\wedge d x_3)^m$.
\item[(iii)]
${\mt{gr}}^1_C{\cc{O}}\simeq{\cc{O}}_C(x_1^{\bar{m}-1}x_3)\oplus
{\cc{O}}_C(x_1^2(x_2^2-x_1^{2\bar{m}-2}))$.
\item[(iv)]
$w_P=(\bar{m}-1)/\bar{m}$,\quad $i_p=2$.
\end{itemize}
\end{example}
From definitions we have
\begin{proposition}
\label{ocenki-1}
If $C\simeq\bb{P}^1$, then
$$
\deg{\mt{gr}}^1_C{\cc{O}}=2+\deg{\mt{gr}}^0_C\omega -\sum_Pi_P,
$$
$$
(K_X\cdot C)=\deg{\mt{gr}}^0_C\omega +\sum_Pw_P.\quad
$$
\begin{flushright
\end{proposition}
\begin{proposition}
\label{ocenki-2}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a Mori conic bundle. Then
$$
\deg{\mt{gr}}^1_C{\cc{O}}\ge-2.
$$
\end{proposition}
\begin{proof}
Consider the exact sequence
$$
0\to\cc{I}_C/\cc{I}_C^{2}\to\cc{O}_X/\cc{I}_C^{2}
\to \cc{O}_C\to 0.
$$
By Corollary \ref{vanishing} $H^1(\cc{O}_X/\cc{I}_C^{2})=0$
and since $H^0(\cc{O}_X/\cc{I}_C^{2})
\to H^0(\cc{O}_C)$ is onto, we have
$H^1(\cc{I}_C/\cc{I}_C^{2})=0$.
Hence $H^1({\mt{gr}}^1_C{\cc{O}})=0$.
It gives us our assertion.
\end{proof}
\begin{corollary1}
\label{grw}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a Mori conic bundle.
Then
$$
(-K_X\cdot C)+\sum w_P+\sum i_P\le 4.
$$
\end{corollary1}
\begin{remark3}
In the case of flipping contraction we always have
$\deg{\mt{gr}}^0_C\omega=-1$ (see \cite{Mori-flip}) but in our situation
of Mori conic bundles
there are examples (e.~g. \ref{99} (i)) with $\deg{\mt{gr}}^0_C\omega=-2$.
Nevertheless it is easy to see from \ref{ocenki-1} and \ref{ocenki-2}
that $-1\ge\deg{\mt{gr}}^0_C\omega\ge -3$. J. Koll\'ar observe also that from vanishing
$R^1f_*{\cc{O}}_X(K_X+H)=0$ for any ample Cartier divisor $H$, we have
$\deg{\mt{gr}}^0_C\omega =-1$ or $-2$.
\end{remark3}
\begin{corollary}
\label{<=3}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a Mori conic bundle.
Then $(X,C)$
contains at most three singular points.
\end{corollary}
\begin{proof}
It follows from Corollary \ref{grw} and Lemma \ref{non}.
\end{proof}
As in \cite{Mori-flip} we need the following construction to
compute invariants $w_P$ and $i_P$ locally.
\subsection{}
Let $(X,P)$ be a terminal point of index $m$ and let
$C\subset (X,P)$ be a smooth curve.
Let $\pi\colon (X\3,P\3)\to (X,P)$ be the canonical cover and
$C\3:=(\pi^{-1}(C))_{\mt{red}}$.
We suppose that $(X,P)$ has splitting degree $d$ along $C$ and subindex
$\bar{m}$
(i.~e. $C\3$ has exactly $d$ components and $m=\bar{m} d$).
By \ref{terminal}, there exists an $\cyc{m}$-equivariant embedding
$X\3\subset{\bb{C}}^4$. Let $\phi=0$ be an equation of $X\3$ in ${\bb{C}}^4$.
Let $C\3_1$ be an irreducible component of $C\3$.
For $z\in{\cc{O}}_{X\3}$ we define $\mt{ord}(z)$ as
the order of vanishing of $z$ on the normalization of $C\3_1$.
It is easy to see that $\mt{ord}(z)$ does not depend on our choice of
a component $C\3_1$,
if $z$ is a semi-invariant.
All the values $\mt{ord}(z)$ form a semigroup which is denoted by
$\mt{ord}(C\3)$. Obviously the "coordinate" values $\mt{ord}(x_1),\dots,\mt{ord}(x_4)$
generate $\mt{ord}(C\3)$.
\begin{proposition-definition1}
\label{notations}
\label{nor}
By \cite{Mori-flip}, one can
choose a coordinate system $x_1, x_2, x_3, x_4$ in
${\bb{C}}^4$ and a character $\chi\colon \cyc{m}\to{\bb{C}}^*$
such that the following conditions hold.
\begin{itemize}
\item[{\rm (i)}]
Coordinates $x_1, x_2, x_3, x_4$ are semi-invariants with
weights satisfying \textup{\ref{cl-term}}.
\item[{\rm (ii)}]
$\mt{ord}(x_i)\equiv\mt{wt}(x_i)\mod \bar{m}$ for all $i=1,2,3,4$.
\item[{\rm (iii)}]
Each component of $C\3$ is parameterized as
$$
x_i= \chi(g)^{\mt{wt}(x_i)}t^{\mt{ord}(x_i)},\quad i=1,2,3,4,\qquad
g\in\cyc{m}/\cyc{\bar{m}}.
$$
\item[{\rm (iv)}]\label{normal}
{\it Normalizedness property.}
For any $i=1,2,3,4$ there is no semi-invariants $y$
such that $\mt{wt}(y)\equiv\mt{wt}(x_i)\mod m$ and $\mt{ord}(y)<\mt{ord}(x_i)$.
In particular $\mt{ord}(x_i)<\infty$ for all $i$.
\item[{\rm (v)}]
There exists an invariant function $z$ on $X\3$ such that
$\mt{ord}(z)=\bar{m}$.
In particular in the case of the main series one has
$\mt{ord}(x_4)=\bar{m}$, $\mt{wt}(x_4)\equiv 0\mod m$.
\end{itemize}
Such a coordinate system is said to be {\it normalized}.
Note that we still may permute $x_1$, $x_2$ and may replace
a character $\chi$ with $\chi'$ if $\chi'\equiv \chi\mod \bar{m}$.\begin{flushright
\end{proposition-definition1}
\begin{proposition}
\textup{(}see \cite[2.10]{Mori-flip}\textup{)}
\label{computation-w}
Under the notations and conditions \textup{\ref{notations}}
one has
$$
w_P=\min \{ \mt{ord}(\psi)\quad |\quad \psi\in{\cc{O}}_{X\3,P\3},\quad
\mt{wt}(\psi)\equiv-\mt{wt}(x_3)\mod m\}.
$$
\begin{flushright
\end{proposition}
\begin{proposition}
\textup{(}cf. \cite[0.4.14.2]{Mori-flip}\textup{)}
\label{computation-k}
Let $F\in |-K_{(X,P)}|$ be a general member. Then in a normalized
coordinate system we have
$(F\cdot C)_P=\mt{ord}(x_3)/\bar{m}$.
\begin{flushright
\end{proposition}
\begin{proposition}
\textup{(}see \cite[2.12]{Mori-flip}\textup{)}
\label{computation-i}
Under the notations of \textup{\ref{notations}},
one has
$$
i_P\bar{m}=\bar{m}-\mt{ord}(x_4)-\bar{m} w_P+\min_{\phi_1,\phi_2\in
J\3_0}[\phi,\phi_1,\phi_2],
$$
where $J\3_0$ is the invariant part of the ideal sheaf
$J\3$ of $C\3$ in ${\bb{C}}^4$, $\phi$ is an equation of
$X\3$ in ${\bb{C}}^4$ and
$$
[\psi_1,\psi_2,\psi_3]:=
\mt{ord}\partial (\psi_1,\psi_2,\psi_3)/\partial (x_1,x_2,x_3).
$$
is the Jacobian determinant.\begin{flushright
\end{proposition}
By semi-additivity of $[\ ,\ ,\ ]$ and
because $x_4\phi\in J\3_0$, we have
\begin{corollary1}
\textup{(}see \cite[(2.15.1)]{Mori-flip}\textup{)}
\label{predv}
In notations above
$$
i_P\bar{m}\ge -\bar{m} w_P+\min_{\psi_1,\psi_2,\psi_3
\in J\3_0}[\phi_1,\phi_2,\phi_3]. \quad
$$
\begin{flushright
\end{corollary1}
\subsubsection{}
\label{choice}
It is clear that the ideal $J\3_0$ is generated by
invariants of the form
$\psi-x_4^n$ where $\psi$ is an invariant monomial with
$\mt{ord}(\psi)=\mt{ord}(x_4^n)=n\bar{m}$. Thus we can assume in \ref{predv}
that $\phi_i=\psi_i-x_4^{n_i}$, for $i=1,2,3$, where
each $\psi_i$ is an invariant monomial in $x_1,x_2,x_3$.
For any element
$\psi=x_1^{b_1}x_2^{b_2}x_3^{b_3}x_4^{b_4}-x_4^{n}$ define
the vector ${\mt{ex}}_4(\psi)\in{\bb{Z}}^3$ by ${\mt{ex}}_4(\psi)=(b_1, b_2, b_3)$.
\begin{lemma1}
\textup{(}see \cite[2.14]{Mori-flip}\textup{)}
Let $\psi_1,\psi_2,\psi_3$ be as in \textup{\ref{choice}}.
Then
$$
[\psi_1,\psi_2,\psi_3]=
\left\{
\begin{array}{ll}
\sum_{i=1}^3\mt{ord}(\psi_i)-\sum_{i=1}^3\mt{ord}(x_i)
& \hbox{{\rm if}}\quad {\mt{ex}}_4(\psi_i), i=1,2,3 \\
&\hbox{{\rm are independent,}}\\
&\\
\infty &\hbox{{\rm otherwise}}.\quad
\end{array}
\right.
$$
\begin{flushright
\end{lemma1}
An invariant monomial $\psi$ in $x_1, x_2, x_3$ is said
to be {\it simple}
if it cannot be presented as a product of two non-constant
invariant monomials.
\begin{corollary1}
\textup{(}see \cite[proof of 2.15]{Mori-flip}\textup{)}
\label{formula}
Under the notations and conditions of \textup{\ref{notations}}
for some simple different invariant monomials $\psi_1$,
$\psi_2$, $\psi_3$
in $x_1, x_2, x_3$ we have
$$
\begin{array}{l}
\bar{m} i_P\ge(\mt{ord}(\psi_1)-\mt{ord}(x_1))+(\mt{ord}(\psi_2)-\mt{ord}(x_2))+\\
\qquad (\mt{ord}(\psi_3)-\mt{ord}(x_3)-\bar{m} w_P).
\end{array}
\leqno(*)
$$
Moreover up to permutations $\psi_1$, $\psi_2$, $\psi_3$
we may assume that $\psi_i=x_i\nu_i$, where $\nu_i$, $i=1,2,3$ also
are monomials. In particular,
all three terms in the formula are non-negative and
$$
\bar{m} i_P\ge\mt{ord}(\nu_1)+\mt{ord}(\nu_2)+\mt{ord}(\nu_3)-\bar{m} w_P.
\leqno(**)
$$
\begin{flushright
\end{corollary1}
\section{Easy lemmas}
\subsection{Construction.}
\label{covering}
Let $f\colon (X,C)\to (S,0)$ be a
Mori conic bundle.
Assume that $(S,0)$ is singular.
Then $(S,0)$ is a log-terminal point
(see e.~g. \cite{KoMM} or \cite{Ishii}) and
the topological cover $h\colon (S\6,0)\simeq
(\bb{C}^2,0)\to (S,0)$ is non-trivial.
Let $X\6$ be the normalization of $X\times_{S}S\6$ and
$G=\mt{Gal}(S\6/S)$. Then we have the diagram
$$
\begin{CD}
X\6@>{g}>>X\\
@V{f\6}VV @V{f}VV\\
S\6@>{h}>>S\\
\end{CD}
$$
The group $G$ acts on $X\6$ and clearly
$X=X\6/G$. Since the action of $G$ on $S\6-\{ 0\}$ is free,
so is the action of $G$ on $X\6-C\6$, where
$C\6:=({f\6}^{-1}(0))_{\mt{red}}$.
Therefore $X\6$ has only terminal singularities
and the induced action of $G$ on $X\6$ is free
outside of a finite set
of points (see e.~g. \cite[6.7]{CKM}).
Since $K_{X\6}=g^*(K_X)$,
$f\6\colon (X\6,C\6)\to (S\6,0)$ is a Mori conic bundle
with non-singular base surface.
\par
As an easy consequence of the construction above and \ref{generator}
we have
\begin{proposition}
\textup{(}see \cite{Pro}, \cite{Pro1}, \cite{Kollar}\textup{)}
\label{ooo}
Let $f\colon (X,C)\to (S,0)$ be a Mori conic bundle.
Then $(S,0)$ is a cyclic quotient singularity.
\end{proposition}
\begin{corollary}
Let $f\colon (X,C)\to (S,0)$ be a Mori conic bundle
and let $T$ be the torsion part of the Weil divisor class group
$\mt{Cl}(X)$. Then
\begin{itemize}
\item[{\rm (i)}]
$T$ is a cyclic group,
\item[{\rm (ii)}]
the topological index of $(S,0)$ is equal to the order of $T$.
\end{itemize}
In particular, $(S,0)$ is non-singular iff $T=0$.
\end{corollary}
\begin{proof}
Consider the diagram from \ref{covering}. We claim that $\mt{Cl}(X\6)$
is torsion-free. Indeed in the opposite case there is
a finite \'etale in codimension 1 cover $X'\to X\6$ and then it must
be \'etale outside finite number
singular points on $C\6$. One can take the Stein factorization
$X'\to S'\to S\6$. Then $S'\to S\6$ will be \'etale outside $0$.
This contradicts smoothness of $S\6$. Therefore $\mt{Cl}(X\6)$
is torsion-free and $X\6\to X$ is the maximal abelian cover of
$X$. It gives us $T\simeq\cyc{n}$, where $n$ is the topological index of
$(S,0)$.
\end{proof}
\begin{proposition}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a
Mori conic bundle with irreducible central fiber.
Suppose that the point $(S,0)$ is singular of topological index $n>1$.
Then for $f\6 \colon (X\6,C\6)\to (S\6,0)$ of \textup{\ref{covering}}
we have the following two possibilities.
\begin{itemize}
\item[{\rm (i)}] \textup{(}primitive case\textup{)}
$C\6$ is
irreducible, $X$ has at least two non-Gorenstein points.
\item[{\rm (ii)}] \textup{(}imprimitive case\textup{)}
$C\6$ has $d>1$
irreducible components, they all pass through one point $P\6$ and
they do not intersect elsewhere. If in this case $m\6$ is the index
of $P\6$ and $n$ is the topological index of $(S,0)$, then the index
of the point $g(P)\in X$ is equal to $nm\6$. \end{itemize}
\end{proposition} \begin{proof} If $C\6$ is irreducible, then the
action of $G\simeq\cyc{n}$ on $C\6\simeq{\bb{P}}^1$ has exactly two fixed
points $P_1\6$ and $P_2\6$. Their images $g(P_1\6)$ and $g(P_2\6)$
are points of indices $>1$. Assume that $C\6$ is reducible. Then
$G\simeq\cyc{n}$ acts on components $\{C\6_i\}$ of $C\6$
transitively. Since $\cup C\6_i$ is a tree, $\cap C\6_i$ is a point
and this point must be fixed under $G$-action. The rest is obvious.
\end{proof}
\begin{corollary}
\label{odnatochka}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a
Mori conic bundle with irreducible central fiber.
Assume that $X$ contains only one non-Gorenstein point $P$.
Then the following conditions are equivalent
\begin{itemize}
\item[{\rm (i)}]
$(S,0)$ is singular of topological index $n$,
\item[{\rm (ii)}]
$C$ is locally imprimitive
\textup{(}see
\textup{\ref{shushu}}\textup{)}
at $P$ of splitting degree $n$.
\end{itemize}
\end{corollary}
\begin{lemma}
\label{s-even}
\label{even}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a minimal Mori conic bundle.
Assume that $f$ is imprimitive of splitting degree $d$. Then $d$ is
even.
\end{lemma}
\begin{proof}
Consider the base change \ref{covering}.
Then $f\6\colon (X\6,C\6)\to (S\6,0)$ is a Mori conic
bundle with $\rho (X\6)=d$.
Since $-K_{X\6}$ is $f\6$-ample, the Mori cone
$\overline{NE}((X\6,C\6)/(S\6,0))\subset \bb{R}^{\rho}$
is generated by classes of $C_1\6,\dots, C_d\6$. Thus each $C_i\6$
generates an extremal ray $R_i$ and
$\overline{NE}((X\6,C\6)/(S\6,0))\subset \bb{R}^{\rho}$
is simplicial. The contraction of each extremal face of
$\overline{NE}((X,C)/(S,0))$
gives us an extremal neighborhood in the sence of
\cite{Mori-flip} (not necessary flipping).
As in the introduction we
apply the Minimal Model Program to $(X\6,C\6)$ over $(S\6,0)$:
$$
\begin{array}{ccccc}
(X\6,C\6)& &\stackrel{p}{\mathop{\rm {- - }\to}\nolimits}& &(Y,L)\\
&\searrow& &\swarrow& \\
& &(S\6,0) & & \\
\end{array}
$$
Here $(Y,L)\to (S\6,0)$ is a Mori conic bundle with $\rho(Y,S\6)=1$.
Then the map $p$ contracts at least $d-1$ divisors.
Let $E$ be a proper transforms one of them on $X\6$.
Since $f\6$ is flat $\Gamma :=f\6(E)$ is a curve on $S\6$.
Moreover for a general point $s\in\Gamma$ the preimage
${f\6}^{-1}(s)$ is a reducible conic, so ${f\6}^{-1}(s)=\ell_1+\ell_2$.
Therefore ${f\6}^{-1}(\Gamma)$ has exactly two irreducible components
$E\supset\ell_1$
and $E'\supset\ell_2$.
Consider the orbit $\{E=E_1, E_2, \dots, E_k\}$ of $E$ under
the action of $\cyc{d}$.
Obviously, every $f\6(E_i)$ is a curve on $S\6$.
We have $\sum E_i\sim aK_{X\6}$ for some $a\in{\bb{Q}}$
because $\cyc{d}$-invariant part of
$\mt{Pic}(X\6)\otimes{\bb{Q}}$ is $\mt{Pic}(X)\otimes{\bb{Q}}\simeq{\bb{Q}}$.
But for a general fiber $\ell:={f\6}^{-1}(s)$, over $s\in S\6$ we have
$(\ell\cdot\sum E_i)=0$, $(\ell\cdot K_{X\6})<0$, so $a=0$,
and $\sum E_i\sim 0$. Further $(\ell_1\cdot E)<0$, $(\ell_1\cdot E')>0$.
It gives us $(\ell_1\cdot E_i)>0$ for some
$E_i\in\{E=E_1, E_2, \dots, E_k\}$. Then $E'=E_i$, i.~e.
there exists $\sigma\in\cyc{d}$ such that $\sigma(E)=E'$.
From the symmetry we get that the orbit $\{E=E_1, E_2, \dots, E_k\}$
may be divided into couples of divisors $E_j$, $E_j'$ such that
${f\6}(E_j)={f\6}(E_j')$ is a curve. Thus both $k$ and $d$ are even.
\end{proof}
\begin{lemma}
\label{warwick}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a Mori conic bundle
with irreducible central fiber and with a unique point $P$ of index
$m>1$. Assume that $P$ is an imprimitive of subindex $\bar{m}$
and splitting degree $d$ \textup{(}so $m=\bar{m}d$\textup{)}.
Then
\begin{itemize}
\item[{\rm (i)}]
$2\bar{m}\equiv 0\mt{mod} d$,
\item[{\rm (ii)}]
$(-K_X\cdot C)=1/\bar{m}$.
\end{itemize}
\end{lemma}
\begin{corollary1}
\label{warwick1}
In conditions \textup{\ref{warwick}}, $\bar{m}\ge d/2$.
In particular, if $d\ge 3$, then $\bar{m}>1$.
\end{corollary1}
\begin{proof}
\label{xaxa}
Again consider the base change \ref{covering}.
Terminal singularities are rational
and therefore Cohen-Macaulay \cite{Kempf}.
By \cite[Theorem 23.1]{Matsumura}, $f\6$ is flat.
Let $X\6_s$ be a general fiber of $f\6$.
Then $X\6_s\equiv nC\6$ for some $n\in {\bb{N}}$.
We have
$$
2=(-K_{X\6}\cdot X\6_s)=n(-K_{X\6}\cdot C\6)=n(-K_{X\6}\cdot \sum C\6(i))=
nd(-K_{X\6}\cdot C\6(1)).
$$
It is clear that
$(-K_{X\6}\cdot C\6(1))\in\frac{1}{\bar{m}}{\bb{Z}}$ because $-\bar{m} K_{X\6}$
is Cartier.
If $(-K_{X\6}\cdot C\6(1))=\delta /\bar{m}$ for some $\delta \in{\bb{N}}$,
then $2=nd\delta /\bar{m}$ or equivalently $2\bar{m}=nd\delta $.
This proves (i).
Since every $(X\6,C(i))$ is a primitive extremal neighborhood
and $-K_{X\6}$ is a generator of
$\mt{Cl}^{sc}(X\6,P\6)\simeq\cyc{\bar{m}}$, $(\delta ,\bar{m})=1$.
Thus $\delta=1$ by Lemma \ref{even}.
On the other hand,
$(-K_{X}\cdot C)=(-K_{X\6}\cdot C\6(1))=\delta /\bar{m}$,
because $K_{X\6}=g^*(-K_{X})$.
The rest is obvious.
\end{proof}
Using Lemma \ref{computation-k} we obtain.
\begin{corollary1}
\label{a3p}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a Mori conic bundle.
Assume that $(S,0)$ is singular and
$X$ contains the only one non-Gorenstein
point $P$. Then in notations of \textup{\ref{nor}} one has
$\mt{ord}(x_3)\equiv 1\mod \bar{m}$.\begin{flushright
\end{corollary1}
The following construction explains why Conjecture $(*_2)$ can
be useful for studying Mori conic bundles.
\subsection{Construction (double cover trick \cite{Kawamata}).}
\label{construction-II}
Let $f\colon (X,C)\to (S,0)$ be a Mori conic bundle and let
$D\in |-2K_X|$ be a general member.
Assume that $K_X+1/2D$ is log-terminal (i.~e. $(*_2)$ holds).
Then there exists a double cover $h\colon Y\to X$ with
ramification divisor
$D$ such that $Y$ has only canonical Gorenstein singularities
(see \cite[8.5]{Kawamata}).
Thus we have two morphisms
$$
Y\stackrel{h}{\longrightarrow}X
\stackrel{f}{\longrightarrow} S.
$$
By the Hurwitz formula, $K_Y=h^*(K_X+1/2D)=0$.
Therefore the composition map $g\colon Y\to S$ gives us
an elliptic fiber space structure on $Y$.
\section{The main result}
The main result of this section is the following
\begin{theorem}
\label{main}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a
Mori conic bundle with only one non-Gorenstein point $P$.
Then one of the following holds
\subsubsection{}
\label{main0}
$(S,0)$ is non-singular.
\subsubsection{}
\label{main1}
$(S,0)$ is a DuVal point of type $A_1$ and splitting degree
of $P$ is $2$ and $X$ contains at most one more singular
\textup{(}Gorenstein\textup{)} point.
There are the following subcases:
\begin{itemize}
\item[{\rm (i)}]
\textup{(}cf. \cite{Pro2}, \cite{Pro3}\textup{)}
\label{war2}\label{99}
$f\colon (X,C)\to (S,0)$ is a quotient of
a conic bundle $f\6\colon (X\6,C\6)\to ({\bb{C}}^2,0)$ by
$\cyc{2}$, $P$ is
the only singular point.
In some coordinate system $(x,y,z;u,v)$ in ${\bb{P}}^2\times{\bb{C}}^2$
the variety $X\6$ can be defined by the equation
$$
x^2+y^2+\psi(u,v)z^2=0, \qquad \psi(u,v)\in{\bb{C}}\{u,v\},
$$
where $\psi(u,v)$ has no multiple factors
and consists only of monomials of even degree.
Here
$X\6\to{\bb{C}}^2$ is the natural projection and the
action of $\cyc{2}$ has the form
$$
u\to -u,\qquad v\to -v,\qquad x\to -x,\qquad y\to y,\qquad z\to z.
$$
Moreover
\par
\mbox{{\rm (a)}}
if $\mt{mult}_{(0,0)}\psi(u,v)=2$, then $(X,P)$ is of type $cA/2$,
\par
\mbox{{\rm (b)}}
if $\mt{mult}_{(0,0)}\psi (u,v)\ge 4$, then $(X,P)$ is of type $cAx/2$.
\item[{\rm (ii)}] $(X,P)$ is of type $cAx/4$, more precisely
$$
(X,P)\simeq
(\{y_1^2-y_2^2+y_3\varphi_3+y_4\varphi_4=0\}/\cyc{4}(1,3,3,2),
$$
$$
C= \{y_1^2-y_2^2=y_3=y_4=0\})/\cyc{4},
$$
where $\varphi_3$, $\varphi_4$ are semi-invariants with weights
$$
\mt{wt}(\varphi_3)\equiv 3\mod 4,\qquad
\mt{wt}(\varphi_4)\equiv 0\mod 4
$$
\textup{(}this is a point of type $II^{\vee}$ in
Mori's classification \cite{Mori-flip}
\textup{)}.
\item[{\rm (iii)}]
$(X,P)$ is the cyclic quotient singularity
${\bb{C}}^3/\cyc{4}(1,3,1)$ and
$$
C=\{y_1^2-y_2^2=y_3=0\}/\cyc{4}.
$$
\item[{\rm (iv)}] $\bar{m}$ is even and $\ge 4$,
$(X,P)$ is the cyclic quotient singularity
${\bb{C}}^3/\cyc{2\bar{m}}(1,-1,\bar{m}+1)$
$$
C= \{ y_2^2-y_1^{2\bar{m}-2}=y_3=0\}\ni 0)/\cyc{2\bar{m}}
$$
\textup{(}this is a point of type $IC^{\vee}$ in
\cite{Mori-flip}, see also example
\textup{\ref{example}}\textup{)}.
\item[{\rm (v)}] $\bar{m}$ is even, $(X,P)$ is of type $cA/m$,
where $m=2\bar{m}$. More precisely
$$
(X,P)\simeq \{y_1y_2+y_2\varphi_1+y_3\varphi_2
+(y_4^2-y_1^{2\bar{m}})\varphi_3
=0\}/\cyc{2\bar{m}}(1,-1,\bar{m}+1,0)
$$
and
$$
C=\{y_2=y_3=y_4^2-y_1^{2\bar{m}}=0\}/\cyc{2\bar{m}},
$$
where $\varphi_i$, $i=1,2,3$ are semi-invariants with weights
$$
\mt{wt}(\varphi_1,\varphi_2,\varphi_3)\equiv (1,\bar{m} -1,0) \mod 2\bar{m}.
$$
\end{itemize}
\subsubsection{}
\label{main2}
$(S,0)$ is a DuVal point of type $A_3$,
$(X,P)$ is a cyclic quotient singularity of type
${\bb{C}}^3/\cyc{8}(a,-a,1)$,
splitting degree of $P$ is $4$
and $X$ has no other singular points. There are two subcases
\begin{itemize}
\item[{\rm (i)}]
$a=1,\qquad\qquad C=\{y_1^{4}-y_2^{4}=y_3=0\}/\cyc{8}$,
\item[{\rm (ii)}]
$a=3,\qquad\qquad C=\{y_2^2-y_3^2=y_2y_3-y_1^2=0\}/\cyc{8}$.
\end{itemize}
\end{theorem}
\begin{remark}
We will show that
in \ref{main1}, \ref{main2} a general member of
$|-K_X|$ does not contain $C$ except \ref{main1} (iv), (v).
In these two cases Conjecture $(*_2)$ holds.
Using the same arguments one can prove that in \ref{main0}
at least one of Conjectures $(*_1)$, $(*_2)$ or $(*_3)$ holds.
More precise results for the case of non-singular base surface
will be published elsewhere.
\end{remark}
Before starting the proof we consider examples of Mori conic bundles
such as in \ref{main1} (ii), (iii) and \ref{main2}.
Unfortunately I don't know examples of Mori conic bundles
as in \ref{main1} (iv) or (v).
\begin{example}
Let $Y\subset{\bb{P}}^3_{x,y,z,t}\times{\bb{C}}^2_{u,v}$ be a non-singular
subvariety given by the equations
$$
\left\{
\begin{array}{l}
xy=ut^2\\
(x+y+z)z=vt^2
\end{array}
\right.
$$
It is easy to check that $Y$ is non-singular.
The projection $g\colon Y\to{\bb{C}}^2$ gives us an elliptic fibration
whose fibers are intersections of two quadrics in ${\bb{P}}^3$. The
special fiber $g^{-1}(0)$ is the union of four lines meeting at one point
$Q:=\{(x,y,z,t;u,v)=(0,0,0,1;0,0)\}$.
Consider also the following action of $\cyc{8}$ on $Y$:
\begin{multline*}
x\to\varepsilon^{-3}z,\qquad y\to\varepsilon(x+y+z),
\qquad z\to-\varepsilon y,\qquad
t\to t,\\
u\to \varepsilon^{-2}v,\qquad v\to-\varepsilon^{2}u,
\end{multline*}
where $\varepsilon:=\exp(2\pi i/8)$.
Then a generator of $\cyc{8}$ has only one fixed point $Q$ and the quotient
$(Y,Q)$ is terminal of type $\frac{1}{8}(1,-1,3)$.
The locus of fixed point for $\cyc{2}\subset\cyc{8}$
consists of the point $Q$ and the divisor $B:=\{t=0\}\cap Y$.
Let $h\colon Y\to Y':=Y/\cyc{2}$ be the quotient morphism.
By the Hurwitz formula $0=K_Y=h^*K_{Y'}+B$, hence $h^*(-K_{Y'})=B$.
Therefore $-K_{Y'}$ is ample over ${\bb{C}}^2$ and $Y'\to{\bb{C}}^2$ is a Mori conic
bundle with singular point $h(Q)$ of type $\frac{1}{2}(1,1,1)$ and
reducible central fiber. Note that
$$
g\colon Y\stackrel{h}{\longrightarrow}X
\stackrel{f}{\longrightarrow} S.
$$
is the double cover trick \ref{construction-II}.
Further the induced action $\cyc{4}=\cyc{8}/\cyc{2}$ on $Y'$
is free outside point $h(Q)$ and permutes components of the central fiber.
Therefore the anticanonical divisor of $X:=Y'/\cyc{4}$ is ample over
${\bb{C}}^2/\cyc{4}$ and the central fiber of the projection
$X\to{\bb{C}}^2/\cyc{4}$ is irreducible.
Finally we obtain the Mori conic bundle $X:=Y/\cyc{8}\to{\bb{C}}^2/\cyc{4}$
such as in \ref{main2}.
The diagram
$$
\begin{CD}
Y'@>>>X\\
@V{f'}VV @V{f}VV\\
{\bb{C}}^2@>>>{\bb{C}}^2/\cyc{4}
\end{CD}
$$
is exactly as in the construction \ref{covering}.
\end{example}
\begin{example}
Let $Y\subset{\bb{P}}^3_{x,y,z,t}\times{\bb{C}}^2_{u,v}$
is given by the equations
$$
\left\{
\begin{array}{l}
xy=ut^2\\
z^2=u(x^2+y^2)+vt^2
\end{array}
\right.
$$
Consider the action of $\cyc{4}$ on $Y$:
$$
x\to y,\qquad y\to -x,
\qquad z\to iz,\qquad
t\to t,\qquad u\to-u,\qquad v\to-v.
$$
As above
$X:=Y/\cyc{4}\to{\bb{C}}^2/\cyc{2}$
is a Mori conic bundle such as in \ref{main1} (iii)
with unique singular point
of type
$\frac{1}{4}(1,1,3)$
\end{example}
\begin{example}
Let $Y\subset{\bb{P}}^3_{x,y,z,t}\times{\bb{C}}^2_{u,v}$
is given by the equations
$$
\left\{
\begin{array}{l}
xy=(u^{2k+1}+v)t^2,\qquad k\in{\bb{N}}\\
z^2=u(x^2-y^2)+vt^2
\end{array}
\right.
$$
Define the action of $\cyc{4}$ on $Y$ by
$$
x\to iy,\qquad y\to ix,
\qquad z\to iz,\qquad
t\to t,\qquad u\to-u,\qquad v\to-v.
$$
Then
$X:=Y/\cyc{4}\to{\bb{C}}^2/\cyc{2}$
is a Mori conic bundle such as in \ref{main1} (ii)
with unique singular point
of type
$cAx/4$.
\end{example}
\subsection{}
To begin the proof of Theorem \ref{main} we
suppose that $(S,0)$ is singular
(otherwise we have case \ref{main0}). Then by \ref{odnatochka}
the point $P$ is imprimitive.
Denote by $m$, $d$ and $\bar{m}$ its index, splitting degree and subindex,
respectively.
First we prove the existence of good divisors in $|-K_X|$
(with two exceptions \ref{main1} (iv), (v)).
As in \cite[\S 7]{Mori-flip} we will use local methods to find
good divisors in $|-K_X|$ or $|-2K_X|$.
\begin{lemma}
\textup{(}cf. \cite[ proof of 7.3]{Mori-flip}\textup{)}.
\label{assumptions}
Let $f\colon (X,C\simeq{\bb{P}}^1)\to (S,0)$ be a
Mori conic bundle with only one non-Gorenstein point $P$.
Let $\pi\colon (X\3,P\3)\to (X,P)$ be the canonical
cover and $X\3\subset{\bb{C}}^4$
be an embedding as in \textup{\ref{cl-term}}.
Then a general member $F\in |-K_X|$ does not contain $C$
and has only DuVal singularity at $P$ iff
we have $\mt{ord}(x_3)<\bar{m}$ in a normalized coordinate system
$(x_1,\dots,x_4)$ in ${\bb{C}}^4$. \textup{(}By \textup{\ref{a3p}},
$\mt{ord}(x_3)<\bar{m}$
implies $\mt{ord}(x_3)=1$, if $(S,0)$ is singular.\textup{)}
\end{lemma}
\begin{proof}
Assume that $\mt{ord}(x_3)<\bar{m}$.
From Lemma \ref{warwick} we have $(-K_X\cdot C)=1/\bar{m}$.
Take a general member $F\in|-K_{(X,P)}|$. Proposition
\ref{computation-k} yields $(F\cdot C)_P=\mt{ord}(x_3)/\bar{m}<1$.
Then $F+K_X$ can be considered as
a Cartier divisor on $X$ and in our case
$-1<((F+K_X)\cdot C)<1$. Therefore $(F\cdot C)=(-K_X\cdot C)$.
Since $\mt{Pic}(X)\simeq{\bb{Z}}$, $F\in|-K_X|$ and by Theorem
\ref{g.e.}, $F$ has only DuVal singularity at $P$.
The inverse implication is obvious.
\end{proof}
Thus it is sufficient to show only $\mt{ord}(x_3)<\bar{m}$.
First we consider the exceptional series of terminal points
\ref{cl-term}, (ii).
\begin{lemma}
\textup{(}cf. \cite[4.2]{Mori-flip}\textup{)}.
\label{except}
If $(X,P)\supset C$ is an imprimitive point of type $cAx/4$, then
$\bar{m}=2$ and
$$
\begin{array}{cccccc}
&x_1&x_2&x_3&x_4&\\
\mt{wt} &1 &3 &3 &2 &\mod 4 \\
\mt{ord} &1 &1 &1 &2 & \\
\end{array}
$$
\end{lemma}
\begin{proof} (cf. \cite[3.8]{Mori-flip})
From Corollary \ref{warwick1}, one has $\bar{m}=2$.
By \ref{notations}, $\mt{ord}(x_4)\equiv 0\mod \bar{m}$ and
there exists an invariant monomial $\psi$ such that $\mt{ord}(\psi)=\bar{m}=2$.
Since $\mt{wt}(x_i)\not\equiv 0\mod 4$ for $i=1,2,3,4$, we have $\psi=x_ix_j$.
Up to permutation $\{1, 2\}$, we may assume that
$\mt{wt}(x_2)\equiv\mt{wt}(x_3)$. Therefore $\mt{ord}(x_2)=\mt{ord}(x_3)$.
It gives us $\psi=x_1x_2$ or $x_1x_3$, whence
$\mt{ord}(x_1)=\mt{ord}(x_2)=\mt{ord}(x_3)=1$.
If necessary we can replace the character $\chi$ with
$-\chi$ to obtain $\mt{wt}(x_1)\equiv 1$.
Finally
$\mt{wt}(x_4)\equiv\mt{wt}(x_1^2)\equiv 2\mod m$ gives as $\mt{ord}(x_4)=2$.
\end{proof}
Since $\mt{ord}(x_3)=1$, we have a good divisor in $|-K_X|$ in this case.
By \ref{nor}, components $C\3(1), C\3(2)$ of curve $C\3$ are
images of
$$
t\longrightarrow (t,t,t,t^2),\qquad t\longrightarrow (it,-it,-it,-t^2).
$$
Therefore $C\3$ can be given by the equations
$x_4-x_1^2=x_2-x_3=x_2^2-x_1^2=0$. By changing coordinates as
$y_1=x_1$, $y_2=x_2$,
$y_3=x_3-x^2$,
$y_4=x_4-x_1^2$ we obtain case \ref{main1} (ii) of our theorem.
\par
From now we assume that $(X,P)$ is from the main series.
In particular $\mt{wt}(x_4)\equiv 0\mod m$, $\mt{ord}(x_4)=\bar{m}$.
\begin{lemma}
\label{matrix}
Let $f(X,C\simeq {\bb{P}}^1)\to (S,0)$ be a Mori conic bundle.
Assume that $P\in X$ is the only point of index $m>1$.
Assume further that $(X,C)$ is imprimitive at $P$ of splitting degree $d$.
Then one of the following holds
\subsubsection{}
\label{matrix1}
A general member of $|-K_X|$ does not contain $C$ and
has DuVal singularity at $P$.
\textup{(}By \textup{\ref{assumptions}}, this is equivalent to
$\mt{ord}(x_3)=1$\textup{)}.
\subsubsection{}
\label{matrix2}
$d=2$, $\bar{m}$ is even
a general member of $|-2K_X|$ does not contain $C$,
and up to permutation $x_1,\dots,x_4$ in some
normalized coordinate system we have
\begin{itemize}
\item[{\rm (i)}]
\quad
$
\begin{array}{llllll}
& x_1 & x_2 & x_3 & x_4& \\
\mt{ord} & 1 & \bar{m}-1 & \bar{m}+1 & \bar{m} & \\
\mt{wt} & 1 & -1 & \bar{m}+1 & 0 &\mod m\\
\end{array}
$
\qquad\qquad $\bar{m}\ge 4$\quad or
\item[{\rm (ii)}]
\quad
$
\begin{array}{llllll}
& x_1 & x_2 & x_3 & x_4 &\\
\mt{ord} & 1 & 2\bar{m}-1 & \bar{m}+1 & \bar{m} &\\
\mt{wt} & 1 & -1 & \bar{m}+1 & 0 &\mod m \\
\end{array}
$
\end{itemize}
\end{lemma}
\subsection{Proof.}
\label{assumptions1}
Denote $a_i:=\mt{ord}(x_i)$. Since $(\mt{wt}(x_i), m)=1$, for $i=1,2,3$,
one has $(a_i,\bar{m})=1$, $i=1,2,3$.
For \ref{matrix2} it is sufficient to prove that
$a_3<\bar{m}$. Let $a_3>\bar{m}$.
First suppose that $\mt{ord}(x_1^m)=2\bar{m}$.
We claim that in this situation \ref{matrix2} holds.
Indeed, then
$a_1m=a_1\bar{m} d=2\bar{m}$, hence $d=2$, $a_1=1$, $m=2\bar{m}$.
Thus $\mt{wt}(x_1)\equiv\mt{wt}(x_3)\equiv 1\mod\bar{m}$.
By the assumptions in \ref{normal} and because $a_3>\bar{m}$,
$\mt{wt}(x_1)\not\equiv
\mt{wt}(x_3)\mod m$. Since $(\mt{wt}(x_1), m)=(\mt{wt}(x_3), m)=1$,
$\bar{m}$ is even.
Changing a generator of $\cyc{m}$ if necessary we may achieve that
$\mt{wt}(x_1)\equiv 1\mod m$. The rest follows from normalizedness.
Taking into account Lemma \ref{even}, from now we may assume that
$\mt{ord}(x_1^m)>3\bar{m}$, and by symmetry,
$\mt{ord}(x_2^m)>3\bar{m}$. We will derive a contradiction with $a_3>\bar{m}$.
\subsection{}
\label{oc}
By Corollaries \ref{grw} and \ref{formula}, there exist different
simple
invariant monomials $\psi_1$, $\psi_2$, $\psi_3$
such that
$$
3\bar{m}\ge i_P\bar{m}\ge \mt{ord}(\psi_1)-a_1+\mt{ord}(\psi_2)-a_2
+\mt{ord}(\psi_3)-a_3-\bar{m}w_P.
$$
Remind that the last inequality can be rewritten as
$$
3\bar{m}\ge i_P\bar{m}\ge \mt{ord}(\nu_1)+\mt{ord}(\nu_2)
+\mt{ord}(\nu_3)-\bar{m}w_P,
$$
where $\nu_i x_i=\psi_i$, $i=1,2,3$.
In particular $\mt{ord}(\nu_1)+\mt{ord}(\nu_2)\le 3\bar{m}$.
Since $\mt{wt}(\nu_1)\equiv\mt{wt}(x_2)\mod m$, $\mt{wt}(\nu_2)\equiv\mt{wt}(x_1)\mod m$,
by \ref{normal}, we have $a_1+a_2\le 3\bar{m}$.
Consider the following cases.
\subsection{Case $a_1+a_2=3\bar{m}$.}
Then we have $\mt{ord}(\nu_1)+\mt{ord}(\nu_2)=3\bar{m}$.
Therefore $i_P=3$, $w_P<1$ (by the inequality in \ref{grw}) and
$\mt{ord}(\nu_3)=\bar{m} w_P<\bar{m}$ by \ref{oc}. Since $a_3>\bar{m}$,
the monomial $\nu_3$ does not depend on $x_3$.
Up to permutation of $x_1$, $x_2$ we may assume that $\nu_3=x_1^\alpha$
for some $\alpha\in{\bb{N}}$ such that $\alpha\mt{wt}(x_1)+
\mt{wt}(x_3)\equiv 0\mod m$ and
$\alpha a_1<\bar{m}$. Thus $\psi_3=x_1^\alpha x_3$ and
$a_1<\bar{m}$, $a_2>2\bar{m}$.
\par
Further from normalizedness and from $\mt{wt}\nu_1\equiv\mt{wt} x_2\mod m$,
we have $\mt{ord}(\nu_1)=a_2>2\bar{m}$, so $\mt{ord}(\nu_2)=a_1<\bar{m}$.
It gives us $\nu_2=x_1$, $\psi_2=x_1x_2$ and
$\mt{ord}(\psi_1)=3\bar{m}$. By the assumption in \ref{assumptions1},
$\psi_1$ is not a power of $x_1$. If $\nu_1$ depends on $x_2$,
then $\psi_1=x_1x_2=\psi_2$, a contradiction.
Hence $\psi_1=x_1^\beta x_3^\gamma$, where $\beta,\gamma>0$,
$a_1 \beta + a_3 \gamma=3\bar{m}$ and
$\mt{wt}(x_1)\beta +\mt{wt}(x_3)\gamma\equiv 0\mod m$.
Since $a_3>\bar{m}$, $\gamma\le 2$.
On the other hand if $\gamma=1$, then $x_1^{\beta -\alpha}=\psi_1/\psi_3$
is an invariant
of order $\le 2\bar{m}$,
a contradiction with assumptions in \ref{assumptions1}.
Thus $\gamma=2$. It gives us $a_1\beta +2a_3=3\bar{m}$ and $a_1\alpha+a_3=2\bar{m}$.
Then $(2\alpha-\beta )a_1=\bar{m}$, so $a_1=1$ and $2\alpha-\beta =\bar{m}$,
because $(a_1,\bar{m})=1$. On the other hand
$\beta \mt{wt}(x_1)+2\mt{wt}(x_3)\equiv 0\mod m$ and
$\alpha\mt{wt}(x_1)+\mt{wt}(x_3)\equiv 0\mod m$, so
$\bar{m}\mt{wt}(x_1)=(2\alpha-\beta)\mt{wt}(x_1)\equiv 0\mod m$, the contradiction with
$(m,\mt{wt}(x_1))=1$.
\subsection{}
Thus $a_1+a_2\le 2\bar{m}$.
Then there are no invariant monomials in $x_1, x_2, x_3$
of order $<\mt{ord}(x_1x_2)$.
Up to permutation of $\psi_1$ and $\psi_2$ we can take $\psi_1=x_1x_2$.
Since $\psi_2$ is simple, it depends only on $x_2, x_3$.
\subsection{Case $a_1+a_2=2\bar{m}$. }
Simple invariant monomials of order $\le 2\bar{m}$ may be only of the form
$x_1x_2$ or $x_i^\alpha x_3$, $i=1,2$\quad
(if $\alpha\mt{wt}(x_i)+\mt{wt}(x_3)\equiv 0\mod m$
and $\alpha a_i+a_3=2\bar{m}$).
On the other hand by the inequality in \ref{oc},
$$
\mt{ord}(\psi_2)\le 3\bar{m}+a_1+a_2-\mt{ord}(\psi_1)=3\bar{m}.
$$
There are two subcases.
\subsubsection{Subcase $\mt{ord}(\psi_2)=2\bar{m}$.}
\label{jjjj}
Then $\psi_2=x_2^\alpha x_3$, because $\psi_2$ depends on $x_2$.
Therefore $a_3<2\bar{m}$, $a_2\alpha<\bar{m}$ and $a_1>\bar{m}$.
Further, if we take in \ref{computation-w}
$\psi=x_2^\alpha$, we obtain $\bar{m} w_P=\mt{ord}(x_2^\alpha)=a_2\alpha<\bar{m}$.
It follows from inequality \ref{oc} that
$\mt{ord}(\psi_3)\le \bar{m} w_P+ a_3+\bar{m}\le 3\bar{m}$, hence $\mt{ord}(\psi_3)=3\bar{m}$.
Using $\partial\psi_3/\partial x_3\ne 0$ we get
$\psi_3\in\{ x_1x_3,\ x_2^\gamma x_3,\ x_2^\gamma x_3^2\}$.
But if $\psi_3=x_1x_3$, then $\mt{wt}(x_3)\equiv \mt{wt}(x_2)\mod m$,
and by normalizedness $a_3=a_2<\bar{m}$, a contradiction.
\par
If $\psi_3=x_2^\gamma x_3$, then
$\mt{ord}(x_2^{\gamma-\alpha})=\mt{ord}(\psi_3/\psi_2)=\bar{m}$.
We derive a contradiction with assumptions in \ref{assumptions1}.
\par
Thus $\psi_3=x_2^\gamma x_3^2$, where
$\gamma \mt{wt}(x_2)+2\mt{wt}(x_3)\equiv 0\mod m$
and $\gamma a_2+2a_3=3\bar{m}$. Consider the monomial
$\nu=x_2^{\alpha-\gamma}=x_2\psi_2/\psi_3$. We have $\mt{ord}(\nu)=a_3-\bar{m}<a_3$.
On the other hand
$\mt{wt}(\nu)\equiv\mt{wt}(x_3)\mod m$.
Hence $\mt{wt}(\nu)\equiv \mt{wt}(x_3)\mod m$ and $\mt{ord}(\nu)<\mt{ord}(x_3)$,
a contradiction with normalizedness.
\subsubsection{Subcase $\mt{ord}(\psi_2)=3\bar{m}$. }
By the inequality in \ref{oc}, $i_P=3$, $\mt{ord}(\psi_3)=a_3+\bar{m} w_P$ and by
Corollary \ref{grw}, $w_P<1$. Therefore $\mt{ord}(\psi_3)<a_3+\bar{m}$.
Thus one has $\psi_2\in \{x_2^\beta x_3^2,\quad x_2^\beta x_3\}$.
But if $\psi_2=x_2^\beta x_3^2$, then $a_2\beta+2a_3=3\bar{m}$.
Hence $a_3<2\bar{m}$, $a_2<\bar{m}$, $a_1>\bar{m}$ and $\mt{ord}(\psi_3)=2\bar{m}$.
So $\mt{ord}(\psi_3)=2\bar{m}$ and $\psi_3$ depends on $x_2, x_3$.
Then we can permute $\psi_2$, $\psi_3$ and get the subcase above.
\par
Therefore $\psi_2=x_2^\beta x_3$. Then $a_2\beta+a_3=3\bar{m}$, $a_3<3\bar{m}$.
We claim that $a_2<\bar{m}$, $a_1>\bar{m}$.
Indeed if otherwise, we have $\beta=1$, $\mt{wt}(x_3)\equiv\mt{wt}(x_1)$ and
by normalizedness $a_3=a_1<\bar{m}$, a contradiction.
Further $\mt{ord}(\psi_3)\le 3\bar{m}$ gives us
$$
\psi_3\in\{x_2^\gamma x_3, \quad x_2^\gamma x_3^2, \quad x_1x_3\}.
$$
If $\psi_3=x_2^\gamma x_3$, then
the monomial $\psi_2/\psi_3=x_2^{\beta-\gamma}$ is invariant and has
order $\le \bar{m}$, a contradiction with assumptions in \ref{assumptions1}.
But if $\psi_3=x_2^\gamma x_3^2$, then we can permute
$\psi_2$, $\psi_3$ and derive a contradiction as above.
Therefore $\psi_3$ depends on $x_1, x_3$, so
$\psi_3=x_1x_3$. Again from normalizedness we have $a_3=a_2<\bar{m}$,
a contradiction.
\subsection{Case $a_1+a_2=\bar{m}$.} Recall that $\psi_1=x_1x_2$.
As above by the inequality in \ref{oc}
$$
\mt{ord}(\psi_2)\le 3\bar{m},
$$
and if the equality holds, then $i_P=3$, $w_P<1$ and
$$
\mt{ord}(\psi_3)=a_3+\bar{m} w_P<a_3+\bar{m}.
$$
Since $\psi_1=x_1x_2$ and $\psi_2$ are different simple invariant monomials,
we have that $\psi_2$ depends on $x_3$.
Then obviously $\mt{ord}(\psi_2)\ge 2\bar{m}$.
Consider subcases.
\subsubsection{Subcase $\mt{ord}(\psi_2)=2\bar{m}$.}
Then it is easy to see $\psi_2=x_2^\beta x_3$.
It implies that
$$
a_2\beta+a_3=2\bar{m},\qquad \mt{wt}(x_1)\beta\equiv \mt{wt}(x_3)\mod m.
$$
Moreover we may assume that $\beta\ge 2$ (otherwise we derive
a contradiction
as in \ref{jjjj}).
If we take in \ref{computation-w}
$\psi=\psi_2/x_3=x_2^\beta$, we get
$w_P<1$. Thus $\mt{ord}(\psi_3)\le \bar{m}+\bar{m} w_P+a_3<3\bar{m}+\bar{m} w_P$ and
$\mt{ord}(\psi_3)\le 3\bar{m}$.
We claim that $\mt{ord}(\psi_3)=3\bar{m}$.
Indeed, if otherwise, we have $\psi_3=x_1^\gamma x_3$,
where $a_1\gamma +a_3=2\bar{m}$.
It gives us $a_1(\gamma -1)+a_2(\beta-1)+2a_3=3\bar{m}$.
Since $\beta\ge 2$, $\gamma =1$. But then $x_1x_3$ is an invariant
and $a_3=a_2<\bar{m}$, a contradiction.
\par
Thus $\mt{ord}(\psi_3)=3\bar{m}$.
Taking into account that $\psi_3$ and $\psi_3/\psi_2$ are
not powers of $x_i$, $i=1,\dots ,4$, we obtain
$$
\psi_3\in\{ x_1^\gamma x_3,\quad x_1^\gamma x_3^2,
\quad x_2^\gamma x_3^2\}.
$$
If $\psi_3= x_1^\gamma x_3$, then
$a_1\gamma +a_3=3\bar{m}$, so
$a_3=3\bar{m}-a_1\gamma $ and $a_1(\gamma +\beta)=\bar{m} (\beta+1)$.
Since $(a_1,\bar{m})=1$, $\gamma +\beta=\bar{m} k$, where $k$ is a
positive integer.
Then $\beta=a_1\bar{m} k-1\ge\bar{m}-1$.
The equality $a_2\beta+a_3=2\bar{m}$ is possible only if
$a_1=a_2=1$, $\bar{m}=2$, but then $\beta=1$, a contradiction.
\par
Assume that $\psi_3= x_1^\gamma x_3^2$.
Then $a_1\gamma +2a_3=3\bar{m}$, $a_1(\gamma -1)+a_2(\beta-1)+3a_3=4\bar{m}$.
Since $\beta\ge 2$, $\gamma =1$.
Whence
$$
3\bar{m}=a_1+2a_3=a_1+2(2\bar{m}-a_2\beta)=4\bar{m}+a_1-2a_2\beta=
(4-2\beta)\bar{m}+(1+2\beta)a_1.
$$
Thus $a_1(2\beta+1)=\bar{m}(2\beta-1)$.
Since $(a_1,\bar{m})=1$, it implies $\bar{m}=2\beta+1$, $a_1=2\beta-1$.
On the other hand $\mt{wt}(x_1)+2\mt{wt}(x_3)\equiv 0\mod m$ and
$\mt{wt}(x_1)\beta\equiv \mt{wt}(x_3)\mod m$ imply $\bar{m}=1+2\beta\equiv 0\mod m$.
So $\mt{wt}(x_1)(1+2\beta)\equiv 0\mod m$, a contradiction.
\par
Finally assume that $\psi_3= x_2^\gamma x_3^2$.
Then $a_2\gamma +2a_3=3\bar{m}$ and $a_2\beta+a_3=2\bar{m}$.
It gives us $\bar{m}=a_2(2\beta-\gamma )$, so $a_2=1$, $\bar{m}=2\beta-\gamma $.
On the other hand, $\mt{wt}(x_1)\gamma \equiv 2\mt{wt}(x_2)\mod m$,
because $\psi_3$ is an
invariant. Whence
$2\mt{wt}(x_1)\beta\equiv \mt{wt}(x_1)\gamma \mod m$
and $\bar{m}=2\beta-\gamma \equiv 0\mod m$,
a contradiction.
\subsubsection{Subcase $\mt{ord}(\psi_2)=3\bar{m}$.}
Since $\psi_2$ depends on $x_2$ and $x_3$,
$a_3<3\bar{m}$. We claim that $a_3<2\bar{m}$. Indeed, if otherwise, we have
$\psi_2= x_2^\beta x_3$
and $\mt{ord}(\psi_3)=3\bar{m}$, hence $\psi_3= x_1^\alpha x_3$.
Thus $a_1\alpha +a_3=3\bar{m}$ and $a_2\beta+a_3=3\bar{m}$ give us
$(\alpha-1)a_1+(\beta-1)a_2+2a_3=5\bar{m}$.
So $\alpha=1$ or $\beta=1$. But then $x_1x_3$ or $x_2x_3$ is an
invariant. By normalizedness $a_3<\bar{m}$, a
contradiction.
\par
Therefore
$a_3<2\bar{m}$, hence $\mt{ord}(\psi_3)=2\bar{m}$,
$$
\psi_3\in\{ x_1^\gamma x_3,\quad x_2^\gamma x_3
\},
\qquad
\psi_2\in\{x_2^\beta x_3,\quad x_2^\beta x_3^2\}.
$$
But then
we can permute $\{\psi_2,\ \psi_3\}$ and $\{x_1,\ x_2\}$ and get the
previous subcase.
This proves Lemma \ref{matrix}.
\begin{flushright
\subsection{Proof of Theorem \ref{main}.}
\label{lll}
First we treat \ref{matrix2} (cf. \cite[(4.4)]{Mori-flip}).
We claim that in case (i) of \ref{matrix2} $(X,P)$ is a
cyclic quotient singularity.
Let $C\3=C\3(1)\cup C\3(2)$ be irreducible decomposition of
$C\3=(\pi^{-1}(C))_{\mt{red}}$.
The component $C\3(1)$ is the image of
$t\longrightarrow (t, t^{\bar{m}-1}, t^{\bar{m}+1}, t^{\bar{m}})$. The curve
$C\3(1)$ is
a complete intersection:
$$
C\3(1)=\{ x_1x_2-x_4=x_3-x_1^{\bar{m}+1}=x_2-x_1^{\bar{m}-1}=0 \}
$$
and so is $C\3$ :
$$
C\3=\{ x_1x_2-x_4=x_3-x_1^{\bar{m}+1}=x_2^2-x_1^{2\bar{m}-2}=0 \}.
$$
Therefore the equation of $X\3$ can be written as
$$
\phi=(x_1x_2-x_4)\varphi_1+
(x_3-x_1^{\bar{m}+1})\varphi_2+
(x_2^2-x_1^{2\bar{m}-2})\varphi_3=0,
$$
where $\varphi_i=\varphi_i(x_1,x_2,x_3,x_4)$ are semi-invariants.
Since $\mt{wt}(x_3)\not\equiv \pm \mt{wt}(x_1),\mt{wt}(x_2)$ and because
$(X,P)$ is of type $cA/m$, the equation
of $\phi$ must contain either $x_1x_2$ or $x_4$ (see \cite{Mori-term}).
But this is possible only if $\varphi_1(0,0,0,0)\ne 0$.
Thus $\phi$ always contains $x_4$ and $X\3$ is non-singular.
We obtain case (iv) of \ref{main1}.
Case (ii) of \ref{matrix2} is treated by the similar way.
\subsection{}
\label{iiii}
From now we will suppose that $f$ is not from \ref{main1} (iv) or (v).
Consider the base change from \ref{covering}
$$
\begin{CD}
X\6 @>{g}>>& X\\
@V{f\6}VV @V{f}VV\\
S\6 @>{h}>> S
\end{CD}
$$
If $\bar{m}=1$, then $X\6$ is Gorenstein and $f\6$ is a "classical" conic bundle
\cite{Cut} (it means that $S\6$ is non-singular and
every fiber of $f\6$ is a conic in ${\bb{P}}^2$).
Since $(X,C)$ is imprimitive,
the central fiber $C\6$ has exactly two components.
The group $\cyc{2}$ permutes
these components. In this case it is not difficult to
write down an equation of $X\6$ explicitly (see \cite{Pro2} or \cite{Pro3}).
We obtain case \ref{main1}, (i).
\par
Assume now that $\bar{m}>1$.
By Lemma \ref{matrix},
the general member $F\in |-K_X|$ does not contain $P$
and has DuVal point at $P$.
The cover $f_F\colon (F,P)\to (S,0)$ is finite of degree 2.
Thus $(S,0)$ is a quotient of $(F,P)$ by an involution.
\begin{proposition1}
\textup{(} see \cite{Cat}\textup{)}
\label{catanese}
Let $(F,P)$ be a germ of DuVal singularity and
$\tau \colon (F,P)\to (F,P)$ be an (analytic) involution. Then
up to analytic isomorphism there are only the following
possibilities.
\label{catanese2}
\par\bigskip
$$
\begin{array}{|c|c|c||c|c|c|}
\hline
\mt{no.}&(F,P)&(F,P)/\tau&\mt{no.}&(F,P)&(F,P)/\tau\\
\hline
&\qquad&\qquad\qquad\qquad&&\qquad&\qquad\\
1) &\hbox{{\rm any}}&\hbox{{\rm smooth}} &
6) &A_{2k+1} &{\bb{C}}^2/\cyc{4k+4}(1,2k+1) \\
2) &A_{2k+1} &D_{k+3} &
7) &E_6 &A_2 \\
3) &E_6 &E_7 &
8) &A_{2k} &{\bb{C}}^2/\cyc{2k+1}(1,2k-1) \\
4) &D_k &D_{2k-2} &
9) &D_k &A_1 \\
5) &A_k &A_{2k+1} &
10)&A_{2k+1} &A_k \\
\hline
\end{array}
$$
\end{proposition1}
\par\bigskip
\begin{remark3}
In cases 6) and 8) the dual graphs of corresponding minimal
resolutions
of $(F,P)/\tau$ are
$$
\begin{array}{c}
{\scriptstyle -4}\\
\circ\\
\end{array}
\qquad
(\hbox{{\rm for}}
\quad k=0),
\leqno 6)
$$
$$
\underbrace{
\begin{array}{ccccccc}
{\scriptstyle -3}&&{\scriptstyle -2}&
&{\scriptstyle -2}&&{\scriptstyle -3}\\
\circ&\text{---}&\circ&\cdots&\circ&\text{---}&\circ\\
\end{array}}_{k+1}
\quad
(\hbox{{\rm for}}
\quad k>0)
\leqno 6)
$$
$$
\underbrace{
\begin{array}{ccccccccc}
{\scriptstyle -2}&&{\scriptstyle -2}&
&{\scriptstyle -2}&&{\scriptstyle -2}&
&{\scriptstyle -3}\\
\circ&\text{---}&\circ&\cdots&\circ&\text{---}&\circ&\text{---}&\circ\\
\end{array}}_{k}
\leqno 8)
$$
As noticed by Shokurov, these singularities are exactly all log-terminal
singularities with discrepancies $\ge -1/2$.
\end{remark3}
\subsubsection{Proof of Theorem \ref{main} (continue).}
Since $(S,0)=(F,P)/\tau$ is a cyclic quotient singularity by
Proposition \ref{ooo},
cases 1)-4) are impossible.
Further by assumptions in \ref{iiii} we have $\bar{m}>1$, hence $m>d$.
Remind that in our situation $d$ is the order of local fundamental
group $\pi_1(S-\{0\})$ (see \ref{odnatochka}).
In particular, $d=n+1$ if $(S,0)$ is a point of type $A_n$.
Thus in case 5) we have $m>d=2k+2\ge 4$.
But $(F,P)$ is a point of type $A_k$ and has the topological index $k+1$.
This contradicts to \ref{index-surf-threef}. The same arguments show that
the cases 6),7),8) are also impossible (if $\bar{m}>1$).
In case 9) $d=2$. From the list of \ref{term-cl} we see that
$m>2$ only if $(X,P)$ is of type $cAx/4$ and then $\bar{m}=2$, $m=4$.
This is case (ii) of \ref{main1} by Lemma \ref{except}.
\par
Finally consider case 10). Then $d=k+1$.
In the classification \ref{term-cl} $(X,P)$ is of type
$cA/n$. Point $(F,P)$ has topological index $2k+2$ and $d=k+1$, so by
Lemma \ref{index-surf-threef}
$(k+1)\bar{m}=d\bar{m}=m\le 2(k+1)$, whence $\bar{m}=2$.
Moreover $(X,P)$ is a cyclic quotient singularity in this case
(because an equality in \ref{index-surf-threef} holds).
Lemma \ref{warwick} gives us then $d=2$ or $4$.
\par
Suppose that $d=2$ (and $\bar{m}=2$).
From Lemma \ref{matrix}, $a_3=1$. By changing a character $\chi$
if necessary we get $\mt{wt}(x_3)\equiv 1\mod 4$. Up to permutation $\{1,\ 2\}$
we also have $\mt{wt}(x_1)\equiv 1\mod 4$ and $\mt{wt}(x_2)\equiv 3\mod 4$.
Then by normalizedness, $a_1=1$ and $a_2=1$ or $3$.
Obviously $\mt{wt}(x_4)\equiv 0\mod 4$ and $a_4=\bar{m}=2$.
Thus a component $C\3(1)$ of $C\3$ is the image of
$t\longrightarrow (t,t^{a_2},t,t^2)$,
where $a_2=1$ or $3$. But if $a_2=3$, then $C\3$ can be given by
$x_2-x_1^3=x_3-x_1=x_4^2-x_1^4=0$. Whence
an equation of $X\3$ in ${\bb{C}}^4$ in a normalized coordinate system
is
$\phi=(x_2-x_1^3)\varphi_1+(x_3-x_1)\varphi_2+(x_4^2-x_1^4)\varphi_3=0$.
Since $(X,P)$ is a cyclic quotient, $\phi$ must contain the term $x_4$,
a contradiction. Therefore $a_2=1$ and $C\3$ is given by
$x_2^2-x_1^2=x_3-x_1=x_4-x_1x_2=0$. By changing coordinates
we obtain \ref{main1}(iii).
\par
If $d=4$, then similarly one can see
$a_3=1$ and $\mt{wt}(x_3)\equiv 1\mod 8$. Again an equation of $X\3\subset{\bb{C}}^4$
in a normalized coordinate system
contains the term $x_4$, so it also contains a monomial $\phi_0\ne x_4$
such that $\mt{wt}(\phi_0)\equiv 0$ and $\mt{ord}(\phi_0)=\mt{ord}(x_4)=2$.
Then $\phi_0=x_1x_2$ and $a_1=a_2=1$. Denote $a:=\mt{wt}(x_1)$.
It is easy to see that we can take $a\in \{1,\ 3\}$.
If $a=1$, then $C\3=\{ x_2^4-x_1^4=x_3-x_1=x_4-x_1x_2=0\}$.
If $a=3$, then one can check that
$C\3=\{ x_2^2-x_3^2=x_2x_3-x_1^2=x_4-x_1x_2=0\}$.
After changing coordinates we obtain \ref{main2}.
\par
To complete the proof we have to show that $X$ can contain at most one
Gorenstein singular point if $(S,0)$ is of type $A_1$ and
contains no Gorenstein singular point if $(S,0)$ is of type $A_3$.
Indeed in diagram \ref{covering}
$f\6\colon (X\6,C\6)\to (S\6,0)$ is a Mori conic bundle with reducible
central fiber $C\6$. Therefore by \ref{vanishing}, $\rho(X\6/S\6)>1$ and
each component $C\6(i)$ of $C\6$ generates an extremal ray on $X\6$
over $(S\6,0)$. Thus each $(X\6,C\6(i))$ is an (not necessary flipping)
extremal neighborhood in the sense of \cite{Mori-flip}.
By \cite[Theorem (6.2)]{Mori-flip}, we see that $(X\6,C\6(i)$
contains at most one Gorenstein singular point and so does $(X,C)$
because $X\6\to X$ is \'etale outside $P$.
\par
In case \ref{main2} $X\6$ has index 2, whence
$(-K_{X\6}\cdot C\6(i))\in \frac{1}{2}{\bb{Z}}$.
On the other hand as in \ref{xaxa} $f\6\colon (X\6,C\6)\to (S\6,0)$ is flat.
Therefore for a general fiber $L$ of $f\6$ we have $L\equiv r\sum C\6(i)$,
where $r\in{\bb{N}}$. Thus $2=(-K_{X\6}\cdot L)=4r(-K_{X\6}\cdot C\6(i))$, so
$1/2r\in\frac{1}{2}{\bb{Z}}$. It gives us $r=1$, $L\equiv \sum C\6(i)$.
Whence the scheme-theoretical central fiber of $f\6$ over $0$ is
generically reduced. Then $C\6$ is a complete intersection of Cartier
divisors inside
$X\6$. Since $C\6$ has a unique singular point $P\6$, $X\6$ has exactly one
singular point (at $P\6$) along $C\6$ and so has $X$ (at $P$).
In case (i) of \ref{main1} one can use the same arguments to show
that the singular point $P$ is unique.
This proves Theorem \ref{main}.
\begin{flushright
\par
Finally we propose examples of Mori conic bundles over a
non-singular base surface (case \ref{main0} of Theorem).
\begin{example}
Let $Y\subset{\bb{P}}^3_{x,y,z,t}\times{\bb{C}}^2_{u,v}$
is given by the equations
$$
\left\{
\begin{array}{l}
xy-z^2=ut^2\\
x^2=uy^2+v(z^2+t^2)
\end{array}
\right.
$$
It is easy to check that $Y$ is non-singular.
The projection ${\bb{P}}^3\times{\bb{C}}^2\to{\bb{C}}^2$ gives us an elliptic fibration
$g\colon Y\to{\bb{C}}^2$. A general fiber of $g$ is an
intersection of two quadrics in
${\bb{P}}^3$ and the central fiber $g^{-1}(0)$ is multiple ${\bb{P}}^1$.
Let $\cyc{2}$ acts on $Y$ by
$$
x\to -x,\qquad
y\to -y,\qquad
z\to -z,\qquad
t\to t,\qquad
u\to u,\qquad
v\to v.
$$
The locus of fixed points consist of the divisor $B:=\{ t=0\}\cap Y$
and the isolated point $(x,y,z,t;u,v)=(0,0,0,1;0,0)$.
Consider the quotient mrphism $h\colon Y\to X:=Y/\cyc{2}$.
By the Hurwitz formula $0=K_Y=h^*(K_X)+D$, where $h^*(-K_X)=D$.
Whence $-K_X$ is ample over ${\bb{C}}^2$.
Therefore the quotient $f\colon X\to {\bb{C}}^2$ is a Mori conic bundle with
irreducible central fiber that contains
only one singular point of type $\frac{1}{2}(1,1,1)$.
\end{example}
The following example shows that the total space $X$
of a Mori conic bundle can contain two Gorenstein terminal
points (cf. \cite[0.4.13.1]{Mori-flip}).
\begin{example}
Let $Y\subset{\bb{P}}^3_{x,y,z,t}\times{\bb{C}}^2_{u,v}$
is given by the equations
$$
\left\{
\begin{array}{l}
x^2=uz^2+vt^2\\
y^2=ut^2+vz^2
\end{array}
\right.
$$
As above the quotient $X:=Y/\cyc{2}(1,1,1,0;0,0)\to {\bb{C}}^2$
is a Mori conic bundle with
irreducible central fiber.
Singular points of $X$ are one point of
type $\frac{1}{2}(1,1,1)$ and two ordinary double points.
\end{example}
\bibliographystyle{amsalpha}
|
1996-04-11T11:32:36 | 9604 | alg-geom/9604009 | en | https://arxiv.org/abs/alg-geom/9604009 | [
"alg-geom",
"math.AC",
"math.AG"
] | alg-geom/9604009 | Sinan Sertoz | Sinan Sertoz | Arf Rings and Characters | LaTeX 2.09, 15 pages, to appear in Note di Matematica | null | null | null | null | Algebraic curve branches can be classified according to their multiplicity
sequences. Arf's solution to this problem using Arf closures and possible
implementations of Henselization are discussed.
| [
{
"version": "v1",
"created": "Thu, 11 Apr 1996 08:34:26 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sertoz",
"Sinan",
""
]
] | alg-geom | \section{Introduction}
In 1949 an article by Cahit Arf \cite{arf} appears in the Proceedings of the
London Mathematical Society. In this article Arf solves
the classification problem of singular curve branches based
upon their multiplicity sequence. However, the geometric nature of the
problem is so hidden behind the algebraic ideas and subsequent
constructions
that immediately following Arf's article an article by
Du~Val \cite{duval2} appears, beginning with the
words: ``Cahit Arf's results being
severely algebraic in form,...''. There Du~Val provides the general
reader with the
necessary geometric ideas that lie behind the scene. This is most
appropriate since it was Du~Val who had formulated the final form of
the problem in his Istanbul article \cite{duval1}
to which Arf refers quite warmly
in his Proceedings article.
In this work I will attempt to describe both the problem
and the solution, using as few machinery as possible, and mostly in
today's terminology, and analyze the `Arf idea' for future prospects.
The problem and its solution form
one of those rare occasions where the solution supplied by an article
answers the question of another article, and does so
without altering the question
to suit the answer! I will describe this solution and then discuss some
future prospects. However, I must add that neither Arf nor Du~Val can
be held responsible for the yet unsubstantiated optimism that
surrounds the ideas I will express in the concluding remarks of
this article. I rely on the dexterity of my students to acquit me in
history for any hopes expressed in the final section.
In 1985, fresh out of the graduate school,
I was hired by Erdo\u{g}an \c{S}uhubi as a research assistant to
T\"{U}B{\.I}TAK's Gebze Research Center. The person to whom I would assist
and collaborate with was none other than Cahit Arf himself.
In this article while trying to convey the joy and excitement that
dominated our discussions on this topic I may inadvertently give away
some trade secrets that I learned from him, for which I apologize
from him beforehand. It is a privilege for me to dedicate this
work, albeit humble, to Professor Cahit Arf on the occasion of his
eighty fifth birthday, with gratitude and respect.
\section{The Setup and the Problem}
\subsection{Heuristic Arguments}
The main problem dominating the area is to
understand the behaviour of a curve at its
singularity. Before we attempt any definition however, we must agree to
choose and fix an underlying field to work with. Let us
call our base field \mbox{\bf k}.
For drawing pictures and extracting intuition \mbox{\bf k} =\mbox{\emre R} ~
is appropriate but for
most geometric applications \mbox{\bf k} =\mbox{\emre C} ~is used. Algebraic geometers
generally chose \mbox{\bf k} ~to be an algebraically closed field of any
characteristic. Arf's arguments however will work for any field \mbox{\bf k} ~of
any characteristic.
\subsection{The First Example: The Node}
Let us begin with an example. Consider the curve $C$ defined by the
equation ~$y^2=x^2(x+1)$
in the affine plane $\mbox{\emre A}_{\mbox{\bf k}}^2$. The curve $C$ is the nodal
curve which has a singularity at the origin where the curve intersects
itself. The usual way of `correcting' this singularity is by changing
the space in which the curve lies and hoping that the curve will
behave better given a more suitable environment.
For this we blow up \mbox{\emre A}$^2$ at the origin and
consider the monoidal transform of the curve in the blow up. This
involves first replacing \mbox{\emre A}$^2$ by the new space
\begin{eqnarray}
B_2=\{ ((x,y),[u_1:u_2])\in \mbox{\emre A}^2\times\mbox{\emre P}^1\; |\; u_1y=u_2x\;\}
\end{eqnarray}
which is easily seen to be smooth.
If $\pi :B_2\rightarrow\mbox{\emre A}^2$ is the projection on the $\mbox{\emre A}^2$ component
then $\pi^{-1}(0,0)$ is isomorphic to \mbox{\emre P}$^1$. It is denoted by $E$ and
is called the exceptional divisor. Note that
$\pi^{-1}(\mbox{\emre A}^2-(0,0))$ is isomorphic to $B_2-E$. In particular
$\pi^{-1}(C-(0,0))$ is isomorphic to $C-(0,0)$.
So we have carried the smooth part $C-(0,0)$ of the singular curve $C$
to a new space $B_2$. Now we look for a substitute for the missing point.
A natural way of doing this is by taking the closure
of $\pi^{-1}(C-(0,0))$ in $B_2$, which we denote by $\tilde{C}$.
The question
now is whether $\tilde{C}$ is smooth or not. For this we consider
local coordinates on $B_2$ and examine the equation of $\tilde{C}$ in these
coordinates. Let $U_i$ be the subset of $B_2$ consisting of the points
with $u_i\neq 0$, $i=1,2$. Note that $\{ U_1, U_2\}$ is an open cover
for $B_2$. Define the local
coordinates of $B_2$ as
\begin{eqnarray}
X=x,\;\; Y=u_2/u_1 \;\;\; {\rm in }\; U_1, \\
X=y, \;\; Y=u_1/u_2 \;\;\; {\rm in }\; U_2.
\end{eqnarray}
With this notation the exceptional divisor $E$ intersects $U_i$ along
the line $X=0$, for $i=1,2$. The equation of $\tilde{C}$ becomes
\begin{eqnarray}
Y^2=X+1 &~~& {\rm in }\;\; U_1 \\
1=XY^3+Y^2 &~~& {\rm in }\;\; U_2.
\end{eqnarray}
We see that $\tilde{C}$ is smooth in both of these coordinate
neighbourhoods. Moreover $\tilde{C}$ intersects $E$ at the points
$(0,\pm 1)$ in both charts. The numbers in the y-component
correspond to the slopes of
the tangent lines to $C$ at the origin in $\mbox{\emre A}^2$.
\subsection{The Second Example: The Cusp}
Next consider another example;
Define $C'$ to be the cusp in $\mbox{\emre A}^2$ given by the equation
$y^2=x^5$. Denote by $\tilde{C'}$ the closure of
$\pi^{-1}(C'-(0,0))$ in $B_2$. The equation of $\tilde{C'}$ becomes
\begin{eqnarray}
Y^2=X^3 &~~& {\rm in }\;\; U_1 \\
1=X^3Y^5 &~~& {\rm in }\;\; U_2.
\end{eqnarray}
We see that $\tilde{C'}$ intersects $E$ at
the point $(0,0)$ in $U_1$ and is singular there whereas it is smooth in
$U_2$ and does not intersect $E$ there. Judging from the way the
equation of $C'$ is transformed under the blow up operation we conclude
that if we apply another blow up operation to $\tilde{C'}$ in $U_1$ at
$(0,0)$ then the curve will be transformed to a smooth curve of the
form $Y^2=X$. The tangent line to the
curve $C'$ at the origin is horizontal and this is reflected in the
$Y$ component of the point where $\tilde{C'}$ intersects $E$ in $U_1$.
\subsection{The General Case}
The crucial information coded in the singularity seems to surface
at the intersection points of the transformed curve with the exceptional
divisor. If we could work with `one piece of information' at a time, then
after each blow up there would be only one intersection with the
exceptional curve and we would continue our analysis from there
on. For this purpose we restrict our attention to such pieces of
information at each singular point on the curve. When we later make
this concept precise we will call it a branch of the curve.
One significant information about the singular point
is the multiplicity of
that point. To find the multiplicity of a point we first count the
number of points a general line intersects the curve. This number is
also known as the degree of the curve. Then we consider a general line
passing through the singular point and count at how many other points
it intersects the curve. We subtract this number from the degree of
the curve and call it the multiplicity of the singular point. This is
reasonable since this difference must count the contribution of that
singular point. Algebraically speaking,
a plane curve is given by a polynomial of
degree $n$ and the number of intersection points of this curve with a
line corresponds to the number of roots of this polynomial after a
linear substitution is made. The number of roots is equal
to the degree of the polynomial when \mbox{\bf k} ~is algebraically closed.
The situation is similar in $n$-space.
We can thus associate to each singular point its multiplicity. The
multiplicity of a smooth point is 1 by the above definition.
Assume that $p_0$ is a singular point of a curve and that the blow up
of the curve at this point intersects the exceptional divisor at only
one point and further more assume that the same is true for the
subsequent transforms. This property ad hocly describes a singular curve
branch. We
then obtain a sequence $\{ (p_i, m_i)\}$, where $p_i$ is obtained from
$p_{i-1}$ by blowing up and $m_i$ is the multiplicity of $p_i$.
For ease of notation we can only consider the sequence $\{ m_i\}$
which is called the multiplicity sequence of the branch. We now agree
to call two singular branches equivalent if their multiplicity
sequences are the same. The problem is then to classify all singular
branches up to this equivalence class.
\subsection{Technical Formulation}\label{sec:technical}
We observed in the previous sections that the nodal curve $C$ had two parts
to its singularity at the origin whereas the cuspidal curve $C'$
had only one. How can we
recognize this phenomena by looking at their equations? Clearly we wish the
equation of $C$ to split up as the product of two parts and the
equation of $C'$ remain irreducible. The expression $y^2-x^2(x+1)$
is irreducible in the ring $\mbox{\bf k} [x,y]$. We may say that this ring is
unnecessarily small since it corresponds to global polynomial
functions on \mbox{\emre A}$^2$, whereas we are interested only in what happens
at the origin. Therefore we can look at $k[x,y]_{(x,y)}$, the
localization of $k[x,y]$ at its maximal ideal $(x,y)$. This ring
represents the regular functions at the origin and should fit to our
geometric purpose of focusing our attention to the origin.
However $y^2-x^2(x+1)$ is still irreducible in this ring.
This
hints to us that we are probably not working in the right rings.
Each irreducible component of $y^2-x^2(x+1)$
should be of the form $y=\pm x\sqrt{x+1}$. But $\sqrt{x+1}$ is not an element
of the rings $\mbox{\bf k} [x,y]$ and $k[x,y]_{(x,y)}$.
Therefore we must find a ring in which
$\sqrt{x+1}$ exists. Observe however that $x+1$ is the square of
1 when computed modulo the maximal ideal corresponding to the
origin. This suggests that we should look at the Henselization of the
local ring $k[x,y]_{(x,y)}$. (see \cite{milne, david}
for a discussion of Henselization.)
On the other hand the completion of $k[x,y]_{(x,y)}$ with respect to its
maximal ideal always satisfies Hensel's lemma and it can be used at
this stage. In fact in the formal power series ring $\mbox{\bf k}[[x,y]]$ we can write
$\sqrt{x+1}=\pm (1+x/2-x^2/8+\cdots )$. Hence the equation
$y^2-x^2(x+1)=0$ splits up as $(y-x(1+x/2-x^2/8+\cdots
))(y+x(1+x/2-x^2/8+\cdots )$.
The irreducibility of the expression $y^2-x^2(x+1)$ corresponds to the
fact that
$\mbox{\bf k}[x,y]/(y^2-x^2(x+1))$, the ring of polynomial functions on
$\tilde{C}$, is an integral domain. We are interested in what happens
at the origin so we localize with respect to the maximal ideal
corresponding to the origin. We can in fact first localize and then
consider the quotient to obtain the ring
$k[x,y]_{(x,y)}/(y^2-x^2(x+1))$.
This is an integral domain. Completing this ring with respect to
its maximal ideal we obtain a ring with zero divisors since
$y^2-x^2(x+1)$ which corresponds to zero can be split up as in the
above discussion. Note however that the equation $y^2-x^5$
continues to stay irreducible even after completing the relevant
ring.
In fact $x$ is never a square in the
ring $R[x]$ where $R$ is a commutative ring with unity. It can only be
a square if $R$ is a suitably chosen noncommutative ring. In our case
the coefficient rings are always fields so $x$ will never split.
(for a discussion of localizations, quotients and completions see
\cite{zariski, atiyah}.)
We can now give a technical definition for a curve branch. Consider
the prime ideal describing an irreducible curve in $n$ space
with a singularity at
the origin. The ideal it generates inside the formal power series
with $n$ indeterminates may split up into components. Each such
component is a branch of the curve passing through the origin. For a
geometric description see \cite{shafarevich}.
\subsection{Du Val's Formulation}\label{sec:duval}
In a much neglected article \cite{duval1} Du Val summarizes the stage
for the classification problem of singular curve branches and
formulates the question whose answer he claims will lead to a
complete understanding of the situation. It is left to the reader to check
that the description of the problem in this section agrees with the
one given in the previous sections.
Define a curve branch $C$ in $n$-space by the following formal
parameterization;
\begin{eqnarray}
x_1 & = & \phi_1(t) \nonumber \\
x_2 & = & \phi_2(t) \nonumber \\
\vdots & \vdots & \vdots \label{eq:main} \\
x_n & = & \phi_n(t) \nonumber
\end{eqnarray}
where each $\phi_i(t)$ is a formal power series in $t$
with coefficients from the field \mbox{\bf k}. We want the branch to pass through
the origin so we impose the condition that the constant
term of each $\phi_i(t)$ is
zero. Assume that the order of $\phi_1(t)$ is lowest among the
others. We want to blow up the branch at the origin and
write the parameterization of the transformed branch in the coordinate
chart where it intersects the exceptional divisor. The blow up of
\mbox{\emre A}$^n$ ~can be described as
\begin{eqnarray}
B_n=\{ ((x_1,...,x_n), [a_1:\cdots :a_n])\in \mbox{\emre A}^n\times\mbox{\emre P}^{n-1} \;
| \; x_ia_j=x_ja_i, \; 1\leq i,j \leq n\;\}.
\end{eqnarray}
In $U_1$ the local coordinates can be written as
\begin{eqnarray}
X_1=x_1,\; X_2:=a_2/a_1, ... , X_n=a_n/a_1.
\end{eqnarray}
Since $\phi_1(t)$ was chosen to have the smallest order, $U_1$ is the
chart in which the transformed branch intersects the exceptional
divisor. The parameterization of the transformed branch is now given by
\begin{eqnarray*}
X_1 & = & \phi_1(t) \\
X_2 & = & \phi_2(t)/ \phi_1(t) \\
\vdots & \vdots & \vdots \\
X_n & = & \phi_n(t)/ \phi_1(t).
\end{eqnarray*}
Observe that each $X_i$ is again expressed as a formal power series in
$t$. The multiplicity of a branch given by such a representation is
equal to the order of the lowest order series appearing in the
representation.
If the branch $C$ has its singularity $p_1$ located at
the origin then its blow up intersects the
exceptional divisor $E_1$ of the first blow up at a point $p_2$.
Semple in \cite{semple} defines $p_2$ to
be proximate to $p_1$. The second blow up is
centered at $p_2$ and intersects the exceptional divisor $E_2$ of the
second blow up at $p_3$. The transform of $E_1$ also intersects $E_2$
at $p_{12}$. If $p_3=p_{12}$ then we say that $p_3$ is proximate to
$p_1$ and $p_2$. Otherwise it is proximate only to $p_2$. In general
if a certain $p_{i+j}$, with $i,j>0$ lies
on $E_i$ or on any transform of $E_i$ under the subsequent blow ups
then we say that $p_{i+j}$ is proximate to $p_{i}$. Nowadays we use
the term ``infinitely close'' instead of ``proximate''. (see
\cite{semple,hartshorne}).
The sum of the
multiplicities of the points $p_1$, $p_2$,...,$p_m$ is called the m-th
multiplicity sum. Du Val in \cite{duval1} a point $p_i$ as
a leading point if the
number of points to which it is proximate is less than the number of
points to which $p_{i+1}$ is proximate. After this bombardment of
definitions comes the most relevant definition: the multiplicity sum
corresponding to a leading point is called a character of the
curve. In other words if $p_i$ is proximate to less points than
$p_{i+1}$ then the sum of the multiplicities of the points $p_1$,...,
$p_i$ is called a character of the branch. The set of all the
characters of the curve was later called the Arf characters. To the
best of my knowledge the first person who first observed the
significance of these numbers and called them ``characters''
was Du Val in \cite{duval1}
and Arf was the first person who could
explicitly calculate them, in \cite{arf}.
Du Val then defines an algorithm, which he calls the modified
Jacobian algorithm, to calculate the multiplicity sequence when these
characters are known. (The Jacobian algorithm, and also the modified
Jacobian algorithm for that matter, calculates the greatest common
divisor of a given set of positive integers.) Finally at the end of
\cite{duval1} he opens a section with the formidable title of
``Outstanding Questions'' and lists some natural questions related to
characters.
\begin{question}[Du Val]How do you find the characters of
a branch if only the local
parameterization with formal power series is available? \label{q1} \\
(see \ref{ans1} for a complete answer).
\end{question}
\begin{question}[Du Val]
Given a set of
positive integers how do you know that they are the characters of some
branch? \label{q2} \\
(see \ref{ans2} for a complete answer).
\end{question}
\begin{question}[Du Val]
Can you find the smallest dimensional space into which the curve branch
can be projected without changing its multiplicity sequence? \label{q3}
\\ (see \ref{ans3} for a complete answer).
\end{question}
Arf recalls that he objected to the amount of
geometric consideration that was clouding the problem,
when Du Val first gave a talk on this subject at
Istanbul University. It must have been 1945.
He claimed that
there was a very algebraic pattern in the problem which could be
solved if one could forget the great geometrical significance of the
problem. Naturally Du Val asked him to work on this. The next day Arf was
homebound with a severe cold so he decided he might as well
think about this problem. Next week when he returned to
work he had in his pocket, scribbled as usual on small pieces of
paper, his own
ticket to immortality...!
\section{The Solution}
In this section we will describe the tools that are developed by Arf
to solve the above problem. In the course of these descriptions it may
seem to the reader that we have strayed away from the problem. But
despite mounting evidence against we will be doing geometry and all
this will be justified at the end when we describe how these pieces
fall in to complete the jigsaw.
\subsection{Generalities and Some Notation}\label{sec:general}
We will be working in the formal power series ring $\mbox{\bf k}[[t]]$
of a single indeterminate $t$. If $H$ is a subring of $\mbox{\bf k}[[t]]$ then
we define
\begin{eqnarray}
W(H) & = & \{ ord \alpha \; |\; \alpha \in H\; \} \\
& = & \{ i_0=0<i_1<\cdots <i_r<\dots\;\}
\end{eqnarray}
The integers $i_0,i_1,...$ form a semigroup of the additive group of
nonnegative integers \mbox{\emre N}. We assume that $H$ is always so
chosen that the
semigroup $W(H)$ contains all integers large enough. In other words if
$\nu_l$ denotes the greatest common divisor of the integers
$i_1,i_2,...,i_l$, then for $\rho$ large enough we want
$\nu_{\rho}=1$. This is not a serious restriction since if $H$ does not
satisfy this condition then $H$ can be transformed by an automorphism
of $\mbox{\bf k}[[t]]$ into a subring $H'$ which satisfies this condition.
Arf does not carry out this transformation but chooses
and fixes a appropriate $T\in\mbox{\bf k}[[t]]$ whose order is $\nu=gcd W(H)$,
and assumes throughout that his ring $H$ can be considered as a subring
of the power series ring in the variable $T$ if necessary, see \cite[p
258, Remarque]{arf}.
For each $i_r$ in
$W(H)$ let $S_{i_r}$ be an element of $H$ with ord$S_{i_r}=i_r$.
We define an ideal $I_h$ by
\begin{eqnarray}
I_h&=&\{\alpha\in H\; |\; ord\alpha \geq h\;\}.
\end{eqnarray}
It can be shown that the inverse of any element in $H$ of order zero
is again an element of $H$. With this in mind we define the {\bf set}
$I_h/S_h$ as
\begin{eqnarray}
I_h/S_h &=& \{ \alpha S_h^{-1}\in H\; |\; \alpha\in I_h\;\}.\label{eq:arf1}
\end{eqnarray}
This set consists of certain elements of $H$ closed under addition
since $I_h$ is an ideal. But the product of any two elements from this
set need not be in the set. We want to consider the smallest subring of $H$
containing the set $I_h/S_h$. It turns out that this ring is
independent of the particular element $S_h$ we have chosen. So we
define the {\bf ring}
\begin{eqnarray}
[I_h]&=&\mbox{\rm the smallest subring of $H$ containing $I_h/S_h$}.
\label{eq:arf2}
\end{eqnarray}
Similarly for a semigroup $G=\{i_0=0<i_1<i_2<\cdots \}$
of nonnegative integers and for $h\in G$ define
\begin{eqnarray}
G_h &=& \{\alpha\in G\;|\; \alpha \geq h\;\} \\
G_h-h &=& \{ (\alpha-h)\in G\;|\; \alpha \geq h\;\} \label{eq:arf3}\\
{[} G_h{]} &=& \mbox{\rm the semigroup of nonnegative integers}
\nonumber \\
& &\mbox{generated by the set $G_h-h$} .\label{eq:arf4}
\end{eqnarray}
\subsection{Arf Rings and Arf Semigroups}
It is clear now that only for special rings $H$ the set $I_h/S_h$ will
already be a ring. We single out such rings and give them a name:\\
\begin{definition}[Arf Ring] {\rm \cite[p 260]{arf}} \\
{A subring $H$ of the formal power series ring
$k[[t]]$ is
called an Arf ring if the set $I_h/S_h$ is a ring for every nonzero
$S_h\in H$. (i.e. $I_h/S_h=[I_h$ for every $S_H\in H$. See
the equations \ref{eq:arf1} and \ref{eq:arf2}.)}
\end{definition}
Similarly we select out special semigroups: \\
\begin{definition}[Arf Semigroup] {\rm \cite[p 260]{arf} } \\
{ A semigroup $G$ of nonnegative integers is
called an Arf semigroup if the set $G_h-h$ is a semigroup for every
$h\in G$. (i.e. $G_h-h=[G_h]$ for every $h\in G$. See the equations
\ref{eq:arf3} and \ref{eq:arf4}.)
\ref{eq:arf2})}
\end{definition}
What if a ring $H$ does not satisfy this condition? We then associate
to it a ring which does:
\begin{definition}[Arf Closure of a Ring] {\rm \cite[p 263]{arf} }\\
{If $H$ is a subring of
$\mbox{\bf k}[[t]]$ then we define the Arf closure $\mbox{}^{\ast} H$ of $H$ to be the
smallest Arf ring in $\mbox{\bf k}[[t]]$ containing $H$.}
\end{definition}
We similarly define Arf closure for semigroups:
\begin{definition}[Arf Closure of a Semigroup] {\rm \cite[p 263]{arf}}\\
{If $G$ is a
subsemigroup of the nonnegative integers \mbox{\emre N}$=\{0,1,2,...\}$ then
we define the Arf closure $\mbox{}^{\ast} G$ of $G$ to be the smallest Arf
semigroup in \mbox{\emre N} ~containing $G$.}
\end{definition}
It remains to check that these definitions are not void. The
ring $\mbox{\bf k}[[t]]$ itself is obviously an Arf ring. So for any subring $H$
the collection of Arf rings in $\mbox{\bf k}[[t]]$ containing $H$ is not empty
and by Zorn's lemma must have a smallest element which we call $\mbox{}^{\ast}
H$. Here ordering is done with respect to inclusion. Similarly the
semigroup \mbox{\emre N} ~is an Arf semigroup and thus the definition of Arf
closure for semigroups is not void.
\subsection{Arf Characters}
In the previous section we saw that we are building parallel
constructions in algebra and arithmetic but it was not clear from
their definitions that they would interact in a meaningful way.
We are now ready to observe a crucial interaction.
If $H$ is a subring of the power series ring, as described in section
\ref{sec:general}, first take its Arf closure $\mbox{}^{\ast} H$ and then
look at $W(\mbox{}^{\ast} H)$, the semigroup of orders of the Arf closure. There
is a smallest semigroup $g_{\chi}$ in \mbox{\emre N} ~whose Arf closure is equal to
$W(\mbox{}^{\ast} H)$. This semigroup $g_{\chi}$ has a minimal generating set
$\chi_1,...\chi_n$ which generates it over \mbox{\emre N}. These are the Arf
characters of $H$:
\begin{definition}[Arf Characters] {\rm \cite[p 265]{arf}}\\
If $H$ is a subring of
$\mbox{\bf k}[[t]]$ as described in \ref{sec:general}, then the characteristic
semigroup of $H$ is defined to be the smallest semigroup $g_{\chi}$ in \mbox{\emre N}
~with $\mbox{}^{\ast} g_{\chi}=W(\mbox{}^{\ast} H)$. The semigroup $g_{\chi}$
can be generated over
\mbox{\emre N} ~with a minimal set of positive
integers $\chi_1,...,\chi_n$ which are
defined to be the characters of $H$.
\end{definition}
As is mentioned in section \ref{sec:duval} the concept of characters
seems to have originated with Du Val's article \cite{duval1}. However
the following ideas appear for the first time in Arf's article
\cite{arf} for the explicit purpose of solving the problem raised in
question \ref{q3}.
\begin{definition}[Bases, Base Characters, Dimension] {\rm \cite[pp
271-274]{arf} } \label{def:bases} \\
If $H$ is an Arf ring let $X_1$ be an
element in $H$ of smallest positive order. $X_1,...,X_{n-1}$
having been chosen let $X_n$ be an element of smallest order in $H$
not included in the Arf closure of the ring
$\mbox{\bf k}[X_1,...,X_{n-1}]$. (The ring $\mbox{\bf k}[X_1,...,X_r]$ is defined as
consisting of the elements of the form $\sum\alpha_{j_1j_2\cdot
j_r}X_1^{j_1}X_2^{j_2}\cdots X_n^{j_r}$ where
$\alpha_{j_1j_2\cdots j_r}\in\mbox{\bf k}$ and the summation is taken over all
$(j_1,j_2,...,j_r)\in\mbox{\emre N}^r$.)
Since $W(H)$ is finitely generated over \mbox{\emre N}
~this process terminates and we obtain a finite collection of elements $\{
X_1,...,X_m\}$. Such a collection is called a base of $H$. If we
denote their orders by $\chi_i=ord(X_i)$ for $i=1,...,m$, then the
numbers $ \chi_1,...,\chi_m$ are called the base characters of
$H$. The number $m$ is called the dimension of $H$.
\end{definition}
These definitions are backed up with concrete
constructions. If $G$ is a semigroup of \mbox{\emre N} ~then a method of
constructing its Arf closure is described in \cite[p 263]{arf}.
If $H$ is a subring of $\mbox{\bf k}[[t]]$ the a method of constructing its Arf
closure is given in \cite[p 267]{arf}. With these methods the
characters can be calculated in a finite number of steps. But not
content with this Arf gives a direct method of computing the
characters of any Arf semigroup, \cite[p 277]{arf}.
\subsection{Solving the Problems}
The setup being as in section \ref{sec:duval} we now explain the
procedure for answering the questions \ref{q1}, \ref{q2} and \ref{q3}.
At our disposal we only have some elements $\phi_1(t),..., \phi_n(t)$
of $\mbox{\bf k}[[t]]$ coming from the parameterization of the branch $C$ as in
the equation
\ref{eq:main}. We now answer the question \ref{q1}.
\begin{answer} Denote by $H$ the ring $\mbox{\bf k}[\phi_1(t),..., \phi_n(t)]$
generated by the $\phi_i(t)$'s in the formal power series ring
$k[[t]]$. (See the definition \ref{def:bases}
for a description of the ring
$\mbox{\bf k}[\phi_1(t),...,\phi_n(t)]$.) First construct its Arf closure $\mbox{}^{\ast} H$.
and then construct the smallest
semigroup $g_{\chi}$ whose Arf closure if $W(\mbox{}^{\ast} H)$. The minimal
generators of $g_{\chi}$ are the characters of the given branch $C$.
Or alternatively use the method described in \cite[p 277]{arf} to
find the characters directly from $W(\mbox{}^{\ast} H)$.
\label{ans1}
\end{answer}
The validity of this answer is proved in \cite[p 266, Th\'eor\`eme
3]{arf} where Arf shows that Du Val's Jacobian algorithm applied to
these characters gives the sought for multiplicity sequence.
Now we come to the question \ref{q2} of knowing
whether a given set of positive integers $ 0<\gamma_1<\cdots <\gamma_l$
are characters of an actual branch.
\begin{answer} If $G$ denotes the semigroup of \mbox{\emre N} ~generated by
$ 0<\gamma_1<\cdots <\gamma_l$, then the characters of $\mbox{}^{\ast} G$ form a
subset of $\{ \gamma_1,...,\gamma_l\}$, \cite[p 256, Th\'eor\`eme
2]{arf}. If any element from the
semigroup $\mbox{}^{\ast} G$ is added to this set of characters then the
resulting set of integers will give the same multiplicity sequence
when Du Val's modified Jacobian algorithm \cite[p 108]{duval1}
is applied to them, \cite[p 266, Th\'eor\`eme 3]{arf}.
Once $\mbox{}^{\ast} G$ is known one can construct
elements $\phi_1(t),..., \phi_r(t)$ of $\mbox{\bf k}[[t]]$ such that the ring
$\mbox{}^{\ast} H=\mbox{\bf k}[\phi_1(t),...,\phi_l(t)]$ is an Arf ring and that
$W(\mbox{}^{\ast} H)=\mbox{}^{\ast} G$, \cite[p 277 and p 282, Th\'eor\`eme 6]{arf}.
\label{ans2}
\end{answer}
And we finally come to the question \ref{q3} of finding the
smallest dimensional space into which the branch $C$ given by the
equations \ref{eq:main}
can be projected without changing its multiplicity sequence.
\begin{answer} If $H$ denotes the ring $\mbox{\bf k}[\phi_1(t),...,
\phi_n(t)]$, defined as in the definition \ref{def:bases}, then
the space of smallest dimension into which the
branch $C$ can be projected without changing its multiplicity
sequence has its dimension equal to the dimension of $\mbox{}^{\ast}
H$. (Recall the definition of the dimension of $\mbox{}^{\ast} H$ as
given in the definition
\ref{def:bases}.)
\label{ans3}
\end{answer}
To prove the validity of this answer Arf defines in \cite[p 273]{arf}
a generating system as a set of elements $Y_1(t),...,Y_{\nu}(t)$ in
$H$ such that the Arf closure of
$\mbox{\bf k}[Y_1(t)-Y_(0),...,Y_{\nu}(t)-Y_{\nu}(0)]$ is equal to $\mbox{}^{\ast}
H$. Then on \cite[p 274]{arf} shows that the smallest number of elements
required in a generating system is the dimension of $\mbox{}^{\ast} H$. And
finally in \cite[p 279, Th\'eor\`eme 5]{arf} he shows how to construct
a ring $\mbox{}^{\ast} H$ of a given set of characters such that $\mbox{}^{\ast} H$ has
the smallest possible dimension.
For the reader who wants to see only a geometric argument summarizing
all this we refer to the last section of Arf's article
\cite[pp 285-287]{arf} where he
relates all this ``algebraic interpretation'' to the problems raised
by Du Val in \cite{duval1}.
\subsection{Silent Heros: Base Characters}
In the statements of the above answers it was not necessary to quote
even the existence of base characters let alone their importance.
However they play a crucial
role in shedding light into the whole scene besides actually answering
question \ref{q3}. To understand the role they play in this set up
first observe that if $\mbox{}^{\ast} H$ is an Arf ring then $\mbox{}^{\ast}
H_h:=[I_{i_h}]$
is also an Arf ring. (See section \ref{sec:general} for the
notation. In particular note that $W(\mbox{}^{\ast} H)=\{ i_0<i_1<i_2<\cdots \}$
and hence the subscript $h$ counts the number of possible
constructions up to that stage.) Arf shows that the characters of
each $\mbox{}^{\ast} H_h$ are determined by characters of $\mbox{}^{\ast} H$ but the base
characters of each $\mbox{}^{\ast} H_h$ constitute a new set of characters. And
he then sets out to demonstrate how they can be constructed, \cite[pp
274-275]{arf}. Thus he obtains for each $\mbox{}^{\ast} H$ a set of invariants:
the characters of $\mbox{}^{\ast} H$ and the base characters of $\mbox{}^{\ast} H_h$ for
each $h\geq 0$.
Moreover in\cite[p 282, Th\'eor\`eme 6]{arf} he shows
that if $l_c$ denotes the number of characters of $\mbox{}^{\ast} H$ and if $l_b$
denotes the least possible number of base characters of $\mbox{}^{\ast} H$
(which he shows
how to construct in \cite[p 279, Th\'eor\`eme 5]{arf}) then for any
integer $n$ with $l_b\leq n\leq l_c$ there exists an Arf ring who has
the same characters and is of dimension $n$. To show that all this is
possible he calculates a concrete example in \cite[pp 283-284]{arf}
where he starts with an Arf ring $\mbox{}^{\ast} G$ and first calculates its
characters. Then he calculates all possible sets of invariants that
can be associated to $\mbox{}^{\ast} G$, as he explained how to do in \cite[pp
277-278]{arf}, and finishes the example by producing an actual $\mbox{}^{\ast}
H$ for each set of invariants having that set as its set of
invariants. So a careful rereading of Arf's article shows that the
algebraic structure of a branch is totally understood with the help of
base characters. Despite their importance they have never been given
the ``Arf'' adjective which they silently deserved...
\section{Concluding Remarks}
The most significant follow up of Arf rings came with Lipman's 1971
article \cite{lipman} in American Journal of Mathematics. In fact it
seems that Lipman was the first mathematician to coin the expression
``Arf Rings'' in the literature. He seems to have been motivated by
the similarity of ideas in Arf rings and in Zariski's theory of
saturations. In this article he studies the condition
that Arf singled out for his rings and relates them to Zariski's ideas in
analyzing the singularities.
The basic idea seems simple; if there is
a singularity then the local ring there `misses' something and the
idea is to fill these `gaps' in a controlled manner so as to
understand the nature of the singularity. A similar idea in a much
grand scale was utilized by Hironaka in \cite{hironaka} where he
measures how far his local rings are from being regular. Then Bennett
showed that in higher dimensions Hilbert functions can be used
effectively to measure these `gaps', see \cite{bennet}. Arf's idea,
being simple and fundamental surfaces every now and then in the study
of singularities. Recently the Italian and Spanish mathematicians are
working on ideas around Arf rings.
As already mentioned in section \ref{sec:technical} the completion of
the local ring at the singularity may be too large. Henselization may
be enough to find a complete set of invariants but the calculations
may be involved, to say the least. In higher dimensions instead of checking
the deviation from regularity by Hilbert functions I believe that a
direct description of the `gaps' may be much more useful. The recent
developments on Gr\"obner bases provides a margin of hope that this is
possible. This particular speculation
aims to provoke several definitions trying
to pinpoint a feature in the ring as a `gap'...
On the other hand the innocent looking structure of a branch involves
questions about the structure of complete rings which are for some
cases answered satisfactorily by Cohen in \cite{cohen}, however the
theory is far from being complete.
When the topic is on curves one can
speculate forever! But past the basic definitions comes a land of no
man... I remember what I heard years ago in a conference on curves:
``To learn some modesty one should study curve theory...''
{\small \bf Acknowledgments:}{\small ~I thank Professors G. Ikeda and
A. H. Bilge for inviting me to give a talk at the symposium for the
occasion of Professor Arf's eighty fifth birthday. The occasion has
rekindled my interest and awakened fond memories. Professor Lipman
has kindly communicated to me his informal thoughts and memories on the
topic. Professor Eisenbud has answered a crucial question about
complete rings, which saved a lot of time and worry for me in the
preparation process of this manuscript. Even though I freely used
the information I received from them the same cannot be said about their
wisdom which I no doubt failed to transfer to the manuscript due to
my own inability to recognize a wise word when I see one... Last but
not least my thanks are to Professor Arf for being a motivation and
a convincing example for us.}
|
1996-04-29T00:57:15 | 9604 | alg-geom/9604022 | en | https://arxiv.org/abs/alg-geom/9604022 | [
"alg-geom",
"math.AG"
] | alg-geom/9604022 | Rahul Pandharipande | R. Pandharipande | The Chow Ring of the Non-Linear Grassmannian | 17 pages, AMSLatex | null | null | null | null | Let M_{P^k}(P^r, d) be the moduli space of unparameterized maps \mu:P^k ->
P^r satisfying \mu^*(O(1))= O(d). M_{P^k}(P^r,d) is a quasi-projective variety,
and, in case k=1, M_{P^1}(P^r,d) is the fundamental open cell of Kontsevich's
space of stable maps \bar{M}_{0,0}(P^r,d). It is shown that the Q-coefficient
Chow ring of M_{P^k}(P^r,d) is canonically isomorphic to the Chow ring of the
Grassmannian Gr(P^k, P^r)= M_{P^k}(P^r,1).
| [
{
"version": "v1",
"created": "Sun, 28 Apr 1996 22:52:15 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{$\bold{Summary}$}
\label{intro}
Let $\Bbb{C}$ be the ground field of complex numbers.
Let $1\leq k \leq r$ be integers.
The Grassmannian $\bold G(\bold P^k,\bold P^r)$ of
projective $k$-planes in $\bold P^r$ can be viewed as
the moduli space of (unparameterized) regular maps from
$\bold P^k$ to $\bold P^r$ of degree $1$.
Let $M_{\bold P^k}(\bold P^r,d)$ be the
coarse moduli space of (unparameterized) regular
maps $\mu:\bold P^k \rightarrow \bold P^r$
satisfying $\mu^{*}({\cal{O}}_{\bold P^r}(1))\stackrel{\sim}{=} {\cal{O}}_{\bold P^k}(d)$.
Two maps
$$\mu:\bold P^k \rightarrow \bold P^r, \ \mu':\bold P^k \rightarrow \bold P^r$$
are equivalent for the moduli problem if
there exists an element
$\sigma\in \bold{PGL}_{k+1}$ satisfying
$\mu' \circ \sigma = \mu$.
If $\mu:\bold P^k \rightarrow \bold P^r$ is a non-constant
regular map, it is easy to
show that $dim(Im(\mu))=k$ and $\mu: \bold P^k \rightarrow Im(\mu)$
is a {\em finite} morphism. The space $M_{\bold P^k}(\bold P^r,d)$
is a natural non-linear generalization of the Grassmannian.
In section (\ref{construct}),
$M_{\bold P^k}(\bold P^r,d)$ will be constructed
via Geometric Invariant Theory. $M_{\bold P^k}(\bold P^r,d)$
is an irreducible, normal, quasi-projective variety
with finite quotient singularities.
Let $A_i\big(M_{\bold P^k}(\bold P^r,d)\big)
\otimes {\Bbb Q}$
be the Chow group (tensor ${\Bbb Q}$)
of $i$-cycles modulo linear equivalence.
Since the space $M_{\bold P^k}(\bold P^r,d) $
has finite quotient singularities, the Chow groups $\bigoplus
(A_i\otimes
{\Bbb Q})$
naturally form a graded ring via intersection. Since
${\Bbb Q}$-coefficients are required for the intersection theory,
all Chow groups considered here will be taken with
${\Bbb Q}$-coefficients.
Let $Ch(k,r,d)$ denote the Chow ring of $M_{\bold P^k}(\bold P^r,d) $.
The ring $Ch(k,r,1)$ is simply the Chow ring of the
linear Grassmannian $\bold G(\bold P^k,\bold P^r)$.
The main result of this paper is a determination of
$Ch(k,r,d)$.
\begin{tm}
\label{mainn}
There is a {\em canonical} isomorphism of graded rings
$$\lambda: Ch(k,r,d) \rightarrow Ch(k,r,1).$$
\end{tm}
Let $\overline{M}_{0,n}(\bold P^r,d)$ be the coarse
moduli space of $n$-pointed Kontsevich stable maps from
a genus $0$ curve to $\bold P^r$.
Let ${M}_{0,n}(\bold P^r,d) \subset \overline{M}_{0,n}(\bold P^r,d) $
denote the non-empty open set corresponding to
$n$-pointed, stable maps from $\bold P^1$ to $\bold P^r$.
The complement of ${M}_{0,n}(\bold P^r,d)$ in
$\overline{M}_{0,n}(\bold P^r,d)$ consists of the stable
maps with reducible domains.
A foundational treatment of these moduli spaces of pointed
stable maps of genus $0$ curves can be found in [K], [KM], and [FP].
The spaces $M_{0,0}(\bold P^r,d)$ and $M_{\bold P^1}(\bold P^r,d)$
are identical. The following corollary
is therefore a special case of Theorem (\ref{mainn}).
\begin{cor}
\label{corr}
The Chow ring (with ${\Bbb Q}$-coefficients) of
$M_{0,0}(\bold P^r,d)$ is canonically isomorphic to
the Chow ring of the Grassmannian $\bold G(\bold P^1, \bold P^r)$.
\end{cor}
Corollary
(\ref{corr}) is related by loose analogy to
results and conjectures on the Chow ring of $M_g$.
C. Faber has studied the subring of the Chow ring
of $M_g$ generated by certain geometric classes.
Faber has conjectured a presentation of this subring (which
may be the entire Chow ring of $M_g$). The conjectured
ring looks like the cohomology ring of a compact
manifold -- for example, it satisfies Poincar\'e duality.
$M_{0,0}(\bold P^r,d)\subset
\overline{M}_{0,0}(\bold P^r,d)$ is a zero-pointed open cell
analogous to $M_g\subset \overline{M}_g$. Corollary (\ref{corr}), then, is
analogous to Faber's conjectures.
In [GP], the Poincar\'e polynomial of
$\overline{M}_{0,n}(\bold P^r,d)$ is computed.
The {\em virtual} Poincar\'e polynomial of
${M}_{0,0}(\bold P^r,d)$ is needed as a preliminary
result. It was found the virtual Poincar\'e polynomial
of ${M}_{0,0}(\bold P^r,d)$ is essentially the Poincar\'e
polynomial of $\bold{G}(\bold P^1, \bold P^r)$. This observation
provided the starting point for Theorem (\ref{mainn}).
Thanks are especially due to E. Getzler for many discussions
about the geometry of the space $M_{0,0}(\bold P^r,d)$.
The theory of
equivariant Chow groups ([EG], [T]) plays an essential role
in the proof of Theorem (\ref{mainn}). The author wishes
to thank D. Edidin, W. Graham, and B. Totaro for
the long discussions in which this theory was explained.
The author has also benefitted from conversations with
P. Belorouski, W. Fulton, and H. Tamvakis.
\section{$M_{\bold P^k}(\bold P^r,d)$}
\label{construct}
A family of degree $d$ maps of $\bold P^k$ to $\bold P^r$ consists of the
data $(\pi:\cal{P}\rightarrow S, \ \mu: \cal{P}\rightarrow \bold P^r)$
where:
\begin{enumerate}
\item[(i.)]
$S$ is a noetherian scheme of finite type over $\Bbb{C}$.
\item[(ii.)] $\pi:\cal{P}\rightarrow S$ is a flat projective morphism
with geometric fibers isomorphic to $\bold P^k$.
\item[(iii.)] The restriction of $\mu^{*}({\cal{O}}_{\bold P^r}(1))$
to each geometric fiber of $\pi$ is isomorphic to
${\cal{O}}_{\bold P^k}(d)$.
\end{enumerate}
Two families of maps over $S$,
$$(\pi:\cal{P}\rightarrow S, \ \mu), \ (\pi':\cal{P}'\rightarrow S, \ \mu' )$$ are
isomorphic if there exists an isomorphism of
schemes $\sigma: \cal{P} \rightarrow \cal{P}'$
such that
$$\mu=\mu' \circ \sigma, \ \pi = \pi' \circ \sigma.$$
Let $\cal{M}_{\bold P^k}(\bold P^r,d)$ be the
contravariant functor from schemes to sets defined
as follows. $\cal{M}_{\bold P^k}(\bold P^r,d) \ (S)$ is the
set of isomorphism classes of families over $S$ of
degree $d$ maps from $\bold P^k$ to $\bold P^r$.
A coarse moduli space $M_{\bold P^k}(\bold P^r,d)$
is easily obtained via Geometric Invariant Theory.
Care is taken
here to exhibit $M_{\bold P^k}(\bold P^r,d)$ as a quotient
of a proper $\bold{GL}_{k+1}$-action with finite stabilizers.
In section
(\ref{genarg}),
the equivariant Chow groups of this $\bold{GL}_{k+1}$-action
will be analyzed.
Let
$$U(k,r,d) \subset \bigoplus_{0}^{r}H^0(\bold P^k, {\cal{O}}_{\bold P^k}
(d))$$
be the Zariski open locus of basepoint free
$(r+1)$-tuples of polynomials. There is
a natural $\bold{GL}_{k+1}$-action on
$\bigoplus_{0}^{r}H^0(\bold P^k, {\cal{O}}_{\bold P^k}
(d))$ obtained from the naturally linearized action
of $\bold{GL}_{k+1}$ on $\bold P^k$.
This $\bold{GL}_{k+1}$-action
leaves
$U(k,r,d)$ invariant. Note, since every
regular map $\mu: \bold P^k \rightarrow \bold P^r$ is
finite onto its image, $\bold{GL}_{k+1}$
acts with {\em finite stabilizers} on $U(k,r,d)$.
Let $\bold{1}\stackrel{\sim}{=} \Bbb{C}$ be a $1$ dimensional complex vector
space with the trivial $GL_{k+1}$-action.
Let $Det$ be the $1$ dimensional determinant representation
of $GL_{k+1}$. For convenience, let $Z$ denote
$\bigoplus_{0}^r
H^0(\bold P^k, {\cal{O}}_{\bold P^k}(d))$.
There is a $GL_{k+1}$-equivariant inclusion
$$U(k,r,d) \subset \bold P\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big)$$
obtained by the following equation:
\begin{equation}
\label{contort}
\xi \in U(k,r,d) \rightarrow [\ 1\otimes\
(1\oplus (\xi\otimes \cdots \otimes \xi) \oplus \xi)\ ].
\end{equation}
The representations $Det$ and $Sym^q(Z)$ in (\ref{contort})
occur to obtain the correct
G.I.T. linearization. The final $Z$ factor occurs
to insure (\ref{contort}) is an inclusion (consider
scaling $U(k,r,d)$ by a constant
$q^{th}$-root of unity).
\begin{lm}
\label{geintth}
Consider the naturally linearized
action of $GL_{k+1}$ on $$
\bold P\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big).$$
Then, for $q>k+1$,
$$U(k,r,d) \subset
\bold P\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big)
^{stable}.$$
\end{lm}
\begin{pf}
The Lemma is a consequence of the
Numerical Criterion of stability.
A development of Geometric Invariant Theory
can be found in [MFK] and [N]. Let
$V_{k+1}$ be a $(k+1)$-dimensional $\Bbb{C}$-vector
space such that $\bold P^k= \bold P (V_{k+1})$.
Let $\overline{v}=v_0, \ldots, v_{k}$ be a basis
of $V_{k+1}$ with
integer weights $w_0, \ldots, w_{k}$ (not all zero).
Let $\xi \in U(k,r,d)$ correspond to a basepoint
free map determined (in the basis $\overline{v}$)
by $[f_0, \ldots , f_r]$ where
each $f_l$ is an an element of $Sym^d(V_{k+1}^*)$.
The diagonal coordinates of
$$\xi \in
\bold P\big( Det \otimes\
(\bold{1} \oplus Sym^q( Z) \oplus Z ) \big)$$
with respect to the $\Bbb{C}^*$-action determined by the
weights and basis $\overline{v}$ are the following:
$$ 1\otimes 1 \in Det\otimes \bold{1},$$
\begin{equation}
\label{termmm}
1 \otimes (\xi \otimes \cdots \otimes
\xi) \in Det \otimes Sym^q\big(
\bigoplus_{0}^{r} Sym^d(V_{k+1}^*)\big),
\end{equation}
$$ 1 \otimes \xi \in Det \otimes \big(\bigoplus_{0}^r
Sym^d(V_{k+1}^*)\big).$$
The weight of $1\otimes 1 \in Det\otimes \bold{1}$ is
$\sum_{0}^{k} w_i$.
Since the polynomials $\{f_l\}$ do
not simultaneously vanish at $[1,0, \ldots, 0]\in \bold P^k$,
one of the coefficients of $v_0^{*d}$ among the
polynomials $\{ f_l\}$ must be non-zero.
Similarly non-zero coefficients of $v_1^{*d}, \ldots, v_{k}^{*d}$
can be found among the polynomials $\{f_l\}$.
Therefore, the terms
\begin{equation}
\label{jayy}
1\otimes (v_j^{*d} \otimes \cdots \otimes v_j^{*d})
\end{equation}
occur in (\ref{termmm}) and have
weight $-qd\cdot w_j + \sum_{0}^{k}
w_i$.
There are now two cases. First assume $\sum_{0}^{k} w_i >0$,
then $1\otimes 1$ has positive weight.
If $\sum_{0}^{k} w_i \leq 0$, there must exist $j$ such that
$w_j <0$. Let $w_j$ be the negative
weight of greatest absolute value.
Hence, for all $i$, if $w_i<0$, then $-w_j +w_i \geq 0$.
Finally, since $q>k+1$,
$$-qd \cdot w_j + \sum_{0}^{k} w_i >0.$$
The term (\ref{jayy}) therefore has positive weight.
The Numerical Criterion
implies $\xi$ is a stable point for the
$\bold{GL}_{k+1}$-action.
\end{pf}
As a consequence of Lemma (\ref{geintth}),
$U(k,r,d)/\bold{GL}_{k+1}$ exists as a quasi-projective
variety. Standard arguments show that the
space
$$M_{\bold P^k}(\bold P^r,d) \stackrel{\sim}{=} U(k,r,d)/\bold{GL}_{k+1}$$
has the desired functorial properties.
Note: the family of maps
\begin{equation}
\label{fammy}
(\pi:\cal{P} \rightarrow S, \ \mu:
\cal{P}\rightarrow \bold P^r)
\end{equation}
may not be a Zariski locally
trivial $\bold P^k$-bundle over $S$. A Galois cover
construction is required to obtain the canonical algebraic
morphism
\begin{equation}
\label{cann}
S \rightarrow M_{\bold P^k}(\bold P^r, d)
\end{equation} induced by the family
(\ref{fammy}). Alternatively, one can
define a map to $M_{\bold P^k}(\bold P^r,d)$ locally
in the \'etale topology on $S$. The morphism (\ref{cann})
is then obtained via {\em descente}.
Since $M_{\bold P^1}(\bold P^r,d)$ and $M_{0,0}(\bold P^r,d)$
coarsely represent the same functor, these spaces are
canonically isomorphic.
Since $U(k,r,d)$ is nonsingular, contained in the
stable locus, and the $GL_{k+1}$-action has finite stabilizers,
Luna's Etale Slice Theorem can be applied to conclude $M_{\bold P^k}
(\bold P^r,d)$ has finite quotient singularities (see [L]).
Luna's Theorem requires a characteristic zero hypothesis.
Finally, since $U(k,r,d)$ is equivariant and contained in a
G.I.T. stable locus, the group action
\begin{equation}
\label{proppp}
GL_{k+1} \times U(k,r,d) \rightarrow U(k,r,d)
\end{equation}
is a proper action. This is established in [MFK], Corollary (2.5).
The properness of the action (\ref{proppp}) is needed in
section (\ref{genarg}).
\section{\bf{The Homomorphism $\lambda:Ch(k,r,d) \rightarrow Ch(k,r,1)$}}
\label{homlam}
Let
$\nu: \bold P^r \rightarrow \bold P^r$ be a regular
map satisfying $\nu^{*}({\cal{O}}_{\bold P^r}(1)) \stackrel{\sim}{=} {\cal{O}}_{\bold P^r}(d)$.
The map $\nu$ induces a canonical morphism
$\tau_\nu: \bold G(\bold P^k, \bold P^r)\rightarrow M_{\bold P^k}(\bold P^r,d)$
by the following considerations.
Let $\pi: \cal{P} \rightarrow \bold{G}(\bold P^k, \bold P^r)$ be the
tautological $\bold P^k$-bundle over the Grassmannian.
Since
\begin{equation}
\label{famm1}
\cal{P} \subset \bold G(\bold P^k,\bold P^r) \times
\bold P^r,
\end{equation} there is
a canonical projection $\eta: \cal{P} \rightarrow \bold P^r$.
Let $\mu: \cal{P} \rightarrow \bold P^r$ be determined
by $\mu= \nu \circ \eta$. The family
\begin{equation}
\label{famm}
(\pi: \cal{P} \rightarrow \bold{G}(\bold P^k, \bold P^r),\
\mu: \cal{P} \rightarrow \bold P^r)
\end{equation}
is a family over $\bold G(\bold P^k, \bold P^r)$
of degree $d$ maps from $\bold P^k$ to
$\bold P^r$. Since $M_{\bold P^k}(\bold P^r,d)$ is a coarse moduli
space, the family (\ref{famm}) induces
a morphism from the base to moduli:
$$\tau_{\nu}:\bold{G}(\bold P^k, \bold P^r) \rightarrow
M_{\bold P^k}(\bold P^r,d).$$
Let $\tau_{\nu}^*$ be the ring homomorphism induced by pull-back:
$$\tau^*_{\nu}: Ch(k,r,d) \rightarrow Ch(k,r,1).$$
Since $M_{\bold P^k}(\bold P^r,d)$ has finite quotient singularities,
the pull-back map $\tau^*_{\nu}$ is well defined (see [V]).
\begin{pr}
\label{homomor}
The homomorphism
$\tau_{\nu}^*$ does not depend upon $\nu$ and is
a graded ring isomorphism.
\end{pr}
\noindent
Let $\lambda: Ch(k,r,d) \rightarrow Ch(k,r,1)$ be the
ring isomorphism $\tau_{\nu}^*$ for any regular map $\nu$.
Theorem (\ref{mainn}) is a
consequence of Proposition (\ref{homomor}).
The proof of Proposition (\ref{homomor}) will
be undertaken in several steps.
First the independence result will be established
in Lemma (\ref{indy5}).
A surjectivity Lemma will be also be proven
in this section. The injectivity
of $\tau_{\nu}^*$ will
be proven in section (\ref{genarg}).
\begin{lm}
\label{indy5}
The homomorphism
$\tau_{\nu}^*$ does not depend upon $\nu$.
\end{lm}
\begin{pf}
Let $U(r,r,d) \subset \bigoplus_{0}^{r} H^0(\bold P^r,
{\cal{O}}_{\bold P^r}(d))$ be the Zariski open locus of
basepoint free $(r+1)$-tuples of polynomials as defined
in section
(\ref{construct}). There is a tautological morphism
$$\nu_U: U(r,r,d) \times \bold P^r \rightarrow \bold P^r.$$
The tautological family (\ref{famm1}) over the
Grassmannian pulls-back to a tautological family $\cal{P}_U$ over
$$\bold G(\bold P^k, \bold P^r) \times U(r,r,d).$$
$\cal{P}_U$ is equipped with a canonical projection
$$\eta_U: \cal{P}_U \rightarrow U(r,r,d) \times \bold P^r.$$
Let $\mu_U= \nu_U \circ \eta_U$. The map $\mu_U$ defines
a family of degree $d$ maps from $\bold P^k$ to $\bold P^r$
over $\bold G(\bold P^k, \bold P^r) \times U(r,r,d)$. There
is an induced map
$$\tau_{U}:\bold G(\bold P^k, \bold P^r) \times U(r,r,d)
\rightarrow M_{\bold P^k}(\bold P^r,d).$$
The morphism $\tau_{\nu}$ is induced
by the composition of the inclusion
$$i_{\nu}:\bold G(\bold P^k, \bold P^r)
\rightarrow \bold G(\bold P^k, \bold P^r) \times [\nu]
\subset \bold G(\bold P^k, \bold P^r) \times U(r,r,d)$$
with $\tau_{U}$.
Hence, $\tau_{\nu}^*= i_{\nu}^* \circ \tau_U^*$
Since $U(r,r,d)$ is an open set in affine space,
$i_{\nu}^*= i_{\nu'}$ for any two maps
$[\nu], [\nu '] \in U(r,r,d)$.
\end{pf}
If $k=r$, then $\bold G(\bold P^r, \bold P^r)$ is a point
and $\tau_{\nu}^*$ is surjective. Assume $k<r$.
Let $1\leq j \leq r-k$. Let $H_j \subset \bold P^r$
be a linear subspace of codimension $k+j$.
Define an algebraic subvariety
$C(H_j)\subset M_{\bold P^k}(\bold P^r,d)$ by the following
condition. $C(H_j)$ is the set of maps that meet
$H_j$. $C(H_j)$ is easily seen to be an irreducible
subvariety of codimension $j$ in $M_{\bold P^k}(\bold P^r,d)$.
There is a natural $\bold{GL}_{r+1}$-action on $M_{\bold P^k}
(\bold P^r,d)$ obtained from the symmetries of $\bold P^r$.
Let $\xi \in \bold{GL}_{r+1}$. Certainly
$$\xi( \ C(H_j)\ )= C(\ \xi(H_j)\ ).$$
Since $\bold{GL}_{r+1}$ is a connected rational
group, the class $\sigma_j$ of $C(H_j)$ in the Chow ring
$Ch(k,r,d)$ is well-defined (independent of $H_j$).
\begin{lm}
\label{abcd} The pull-back of the class $\sigma_j$ for
$1\leq j \leq r-k$ is determined by:
$$\tau_{\nu}^*(\sigma_j)= d^{k+j}\cdot \sigma_j.$$
\end{lm}
\begin{pf}
Let $\nu: \bold P^r \rightarrow \bold P^r$ be a fixed
morphism satisfying $\nu^*({\cal{O}}_{\bold P^r}(1))\stackrel{\sim}{=}
{\cal{O}}_{\bold P^r}(d)$. Let $H_j\subset \bold P^r$ be
a general (with respect to $\nu$) linear space.
By Bertini's Theorem,
$\nu^{-1}(H_j)$ is a nonsingular complete
intersection of $k+j$ hypersurfaces of degree $d$
in $\bold P^r$. The set theoretic inverse image
$\tau_{\nu}^{-1}( C(H_j))$ is the set of $k$-planes of
$\bold P^r$ meeting $\nu^{-1}(H_j)$. A simple
tangent space argument shows that
the scheme theoretic inverse image
$\tau_{\nu}^{-1}(C(H_j))$ is generically reduced. Hence,
$$\tau_{\nu}^* (\sigma_j) = [\tau_{\nu}^{-1}(C(H_j))]\in Ch(k,r,1).$$
It remains to determine $[\tau_{\nu}^{-1}(C(H_j))]\in Ch(k,r,1).$
Recall $\pi:\cal{P}\rightarrow \bold G(\bold P^k,\bold P^r)$
is the tautological $\bold P^k$-bundle over the Grassmannian.
Let $L$ be the Chern class of the line bundle
$\eta^*({\cal{O}}_{\bold P^r}(1))$ on $\cal{P}$.
The following equations hold:
$$\pi_*(L^{k+j})= \sigma_j,$$
$$\pi_*( (d\cdot L)^{k+j})= [\tau_{\nu}^{-1}(C(H_j))].$$
These equations imply $\tau_{\nu}^*(\sigma_j)=d^{k+j}\cdot
\sigma_j.$
\end{pf}
Consider the $d=1$ case, $\bold G(\bold P^k, \bold P^r)\stackrel{\sim}{=}
M_{\bold P^k}(\bold P^r,1)$. There is a tautological
bundle sequence on $\bold G(\bold P^k, \bold P^r)$:
$$0 \rightarrow S \rightarrow \Bbb{C}^{r+1} \rightarrow Q \rightarrow 0.$$
$Q$ is a bundle of rank $r-k$. For
$1 \leq j \leq r-k$, let $c_j(Q) \in Ch(k,r,1)$
be the $j^{th}$ Chern class of $Q$. It is well known that
$$c_j(Q)=\sigma_j.$$
Also, the classes $c_j(Q)\in Ch(k,r,1)$ generate
$Ch(k,r,1)$ as ring. These facts can be found, for example,
in [F]. Therefore, the following Lemma
is a consequence of Lemma (\ref{abcd}).
\begin{lm}
The homomorphism
$\tau_{\nu}^*: Ch(k,r,d) \rightarrow Ch(k,r,1)$ is surjective.
\end{lm}
\noindent
In fact, the subring of $Ch(k,r,d)$ generated by
$\sigma_1, \ldots, \sigma_{r-k}$ surjects onto
$Ch(r,k,1)$ via $\tau_{\nu}^*$.
\section{\bf{Generation of $Ch(1,r,d)$}}
\label{genn}
In order to complete the proof of Proposition (\ref{homomor}),
results on the generation of $Ch(k,r,d)$ are needed.
In this section, a special argument in the $k=1$ case is
developed. In sections (\ref{eqchg})-(\ref{genarg}), a general
generation argument using the theory of equivariant Chow
groups is established. The general argument also
covers the $k=1$ case. The special method for the
$k=1$ case involves a natural stratification of
$M_{\bold P^1}(\bold P^r,d)$. Unfortunately, this stratification
does not easily generalize when $k>1$.
Let $0\leq j\leq r-1$. Let $\sigma_0\in Ch(1,r,d)$ be the
unit (the fundamental class). Let $\sigma_{j\neq 0}$ be the class
defined in section (\ref{homlam}).
\begin{pr}
\label{gen}
The elements $\sigma_i \cdot \sigma_j$ ($0\leq i \leq j \leq r-1$)
span a ${\Bbb Q}$-basis of $Ch(1,r,d)$.
\end{pr}
The proof of Proposition (\ref{gen}) uses the
3-pointed moduli space of maps $M_{0,3}(\bold P^r,d)$.
Let $1,2,\infty \in \bold P^1$ be three marked points.
There is a natural isomorphism:
\begin{equation}
\label{udef}
M_{0,3}(\bold P^r,d)\stackrel{\sim}{=} \bold P(U)=U(1,r,d)/\Bbb{C}^*
\subset \bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))
\end{equation}
where $U(1,r,d)$ is the basepoint free locus (see section
(\ref{construct})).
An element of $\bold P(U)$ corresponds to a degree $d$ map
from $\bold P^1$ to $\bold P^r$ with the
three markings $1,2,\infty\in \bold P^1$.
A map $[\mu]\in M_{0,3}(\bold P^r,d)$ corresponds to
a point in $\bold P(U)$ by identifying the three markings of $[\mu]$
with the points $1,2,\infty\in \bold P^1$. A tangent space
argument shows this identification is an isomorphism
of schemes (both are non-singular varieties).
The proof of Proposition (\ref{gen})
is a refinement of the methods that
appear in [P].
For $0\leq j \leq r-1$, let $H_j \subset \bold P^r$
be a linear space of codimension $1+j$. For
$0 \leq a,b \leq r-1$, let $C(H_a,H_b)\subset M_{0,0}
(\bold P^r,d)$ be the subvariety of maps meeting $H_a$ and $H_b$
(where $H_a$ and $H_b$ are in general position).
A simple argument shows the
equation $$[C(H_a,H_b)]=\sigma_a \cdot \sigma_b$$
holds in $Ch(1,r,d)$.
Note: intersection with the hyperplane $H_0$ imposes
no condition on the maps. In particular,
$C(H_0, H'_0)= M_{0,0}(\bold P^r,d)$.
\begin{lm}
\label{maxx}
Let $0\leq a,b \leq r-1$.
Assume $(a,b) \neq (r-1,r-1)$.
Let $H_a, H_b \subset \bold P^r$ be linear spaces of
codimension $1+a, 1+b$
in general position.
Let $H_{a+1}\subset H_a$, $H_{b+1}\subset H_b$ be
linear spaces of codimension $1$.
The natural map
\begin{equation}
\label{mlem}
C(H_{a+1}, H_b)\
\cup \ C(H_a, H_{b+1})\ \cup \
C(H_0,H_{a}\cap H_{b})
\rightarrow
C(H_a,H_b)
\end{equation}
yields a surjection on Chow groups
of proper codimension in $C(H_a, H_b)$.
If the linear spaces $H_{a+1}$, $H_{b+1}$, or $H_a \cap H_b$
are empty, the corresponding cycle on the left in (\ref{mlem})
is taken to
be empty. By the assumption $(a,b)\neq (r-1,r-1)$,
not all cycles are empty.
\end{lm}
\begin{pf}
Let $F$ be a hyperplane in general position
with respect to $H_a$ and $H_b$.
Let $\overline{N}=\overline{M}_{0,3}(\bold P^r,d)$ and
$\overline{M}=\overline{M}_{0,0}(\bold P^r,d)$.
Let $N$, $M$ be the unbarred moduli spaces.
Let
$e_i:\overline{N} \rightarrow \bold P^r$ be the natural
evaluation maps for the markings $1\leq i \leq 3$.
Let $$X= e_1^{-1}(F) \cap e_2^{-1}(H_a) \cap
e_3^{-1}(H_b).$$
$X$ is closed subvariety of $\overline{N}$.
The natural forgetful morphism
$\rho: X\rightarrow \overline{M}$ is proper. Also $\rho(X) \cap M=C(H_a,H_b)$.
Let $Z\subset C(H_a,H_b)$ be the open set of $\rho(X)$
corresponding to Kontsevich
stable maps satisfying the following conditions:
\begin{enumerate}
\item[(i.)] The domain curve is $\bold P^1$.
\item[(ii.)] The map meets $H_a$ and $H_b$.
\item[(iii.)] The map does not pass through $F\cap H_a$,
$F\cap H_b$, or $H_a \cap H_b$.
\end{enumerate}
Let $[\mu] \in Z$ be a element.
By condition (iii), the image $Im(\mu)\subset \bold P^r$ can not be
contained in $F$, $H_a$, or $H_b$.
Moreover, by (iii), $\rho^{-1}(Z) \subset N$.
Hence, the map $\rho^{-1}(Z)\rightarrow Z$
has finite fibers. Since $\rho^{-1}(Z)\rightarrow Z$ is
a proper morphism with finite fibers, it is a finite morphism.
Therefore, if $A_i(\rho^{-1}(Z)) \otimes {\Bbb Q}=0$, then
$A_i(Z) \otimes {\Bbb Q}=0$.
The set $\rho^{-1}(Z)\subset N \stackrel{\sim}{=} \bold P(U)$ (see (\ref{udef}) above)
is isomorphic to a quasi-projective variety
in $\bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))$.
The quasi-projective subvariety $$\rho^{-1}(Z)\subset
\bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))$$ can be identified as follows.
Let $L_1\subset \bold P(U)$ correspond
to maps $\mu:\bold P^1 \rightarrow \bold P^r$ satisfying $\mu(1)\in F$.
Let $L_2$, $L_{\infty}$ be the linear
subspaces in $\bold P(U)$ where $\mu(2)\in H_a$, $\mu(\infty)\in H_b$.
Let $L_1\cap L_2 \cap L_{\infty}= I\subset \bold P(U)$.
Let $D\subset I$ be the union of the
three hypersurfaces of maps meeting
the linear spaces $F\cap H_a$, $F\cap H_b$, and
$H_a\cap H_b$ respectively.
Since $(a,b) \neq (r-1,r-1)$,
$F\cap H_a$ or $F\cap H_b$ is non-empty. Therefore,
$D\subset I$ is a subvariety of codimension $1$.
Then
$$\rho^{-1}(Z)= I \setminus D.$$
$I$ is an open set of a linear subspace of
$\bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))$. Since $D$ is of codimension $1$ in $I$,
all the Chow groups of $\rho^{-1}(Z)$ of proper codimension vanish.
Hence all the Chow groups (tensor ${\Bbb Q}$) of $Z$ of proper
codimension also
vanish.
By definition, $Z\subset C(H_a, H_b)$.
The complement of $Z$ in $C(H_a, H_b)$ is the set of maps
meeting $F\cap H_a$, $F\cap H_b$, or $H_a\cap H_b$.
Therefore, the complement of $Z$ in $C(H_a,H_b)$ is the union
of three cycles:
\begin{equation}
\label{onion3}
C(F\cap H_a, H_b)\ \cup\ C(H_a,F\cap H_b)\ \cup\
C(H_0,H_a\cap H_b).
\end{equation}
Since the Chow groups of $Z$ vanish in proper
codimension, the Chow groups of the union (\ref{onion3})
surject onto the Chow groups of $C(H_a, H_b)$ (in proper
codimension).
\end{pf}
\noindent A vanishing result is also required.
\begin{lm}
\label{four4}
Chow groups in proper codimension of
$C(H_{r-1}, H'_{r-1})$ vanish.
\end{lm}
\begin{pf}
Let $F$ be a hyperplane in general position
with respect to two distinct points $p=H_{r-1}$ and
$q=H'_{r-1}$.
The notation $N\subset\overline{N}$, $M\subset \overline{M}$
of Lemma (\ref{maxx}) will be used.
Let $$X=e_1^{-1}(F) \cap e_2^{-1}(p)\cap e_3^{-1}(q).$$
Let $\rho: X \rightarrow \overline{M}$ be the proper forgetful morphism.
Again, $\rho(X) \cap M=C(p,q)$.
Let $Z\subset C(p,q)$ be the open set of $\rho(X)$
corresponding to Kontsevich
stable maps satisfying the following conditions:
\begin{enumerate}
\item[(i.)] The domain curve is $\bold P^1$.
\item[(ii.)] The map meets $p$ and $q$.
\end{enumerate}
Note $F\cap p$, $F\cap q$, and $p\cap q$ are empty.
By these conditions on $Z$, the map $\rho^{-1}(Z)\rightarrow Z$ is
finite and proper.
Therefore, if $A_i(\rho^{-1}(Z)) \otimes {\Bbb Q}=0$, then
$A_i(Z) \otimes {\Bbb Q}=0$. Also, $\rho^{-1}(Z) \subset N$.
The quasi-projective subvariety $$\rho^{-1}(Z)\subset
\bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))$$ can be identified as follows.
Let $L_1\subset \bold P(U)$ correspond
to maps $\mu:\bold P^1 \rightarrow \bold P^r$ satisfying $\mu(1)\in F$.
Let $L_2$, $L_{\infty}$ be the linear
subspaces in $\bold P(U)$ where $\mu(2)\in p$, $\mu(\infty)\in q$.
Let $L_1\cap L_2 \cap L_{\infty}= I\subset \bold P(U)$.
Then
$$\rho^{-1}(Z)= I $$
$I$ is an open set of a linear subspace of
$\bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))$.
Let $\overline{I}$ be the closure of $I$. It will be shown
that $\overline{I}\setminus I$ has codimension $1$ in $\overline{I}$.
The Chow groups of $\rho^{-1}(Z)$ of proper codimension therefore
vanish.
Hence all the Chow groups of $Z$ of proper
codimension also
vanish.
Let $[A_0, \ldots, A_r]$ be homogeneous coordinates on $\bold P^r$.
Let $$F=(A_0-A_r),\ p=[1,0, \ldots,0], \ q=[0,\ldots,0,1].$$
Let $[S,T]$ be homogeneous coordinates on
$\bold P^1$. Let $1,2,\infty\in \bold P^1$ be the points
$[1,1]$, $[1,0]$, $[0,1]$ respectively.
An element $[\mu]\in \bold P(\bigoplus_0^r H^0(\bold P^1,
{\cal{O}}_{\bold P^1}(d)))$ is given by
an $r$-tuple of degree $d$ homogeneous
polynomials in $S$ and $T$ : $[f_0, \ldots, f_r]$.
The element $[\mu]$ is in $I$ if and only if
\begin{enumerate}
\item[(i.)] $f_0,\ldots, f_r$ span a basepoint free
linear system on $\bold P^1$.
\item[(ii.)] $f_0(1,1)=f_r(1,1)$.
\item[(iii.)] $T$ divides $f_1, \ldots, f_r$.
\item[(iv.)] $S$ divides $f_0, \ldots, f_{r-1}$.
\end{enumerate}
The additional condition
$$S \ \ divides\ \ f_r$$ is a codimension
$1$ condition contained in the set $\overline{I}\setminus I$.
Hence $\overline{I}\setminus I$ has codimension $1$ in $I$.
\end{pf}
\noindent
Repeated application of Lemma (\ref{maxx}) shows the ring
$Ch(1,r,d)$ is generated (as a ${\Bbb Q}$-vector space) by the
classes $[C(H_a, H_b)]$ and the Chow groups of
$C(H_{r-1}, H'_{r-1})$. Lemma (\ref{four4}) shows
the Chow groups of $C(H_{r-1}, H'_{r-1})$ vanish in
proper codimension. Hence the classes
$[C(H_a,H_b)]=\sigma_a \cdot \sigma_b$ generate $Ch(1,r,d)$.
Via the classical
Schubert calculus,
the classes $\sigma_a \cdot \sigma_b$ for $0\leq a,b \leq r-1$
are easily seen to span a {\em basis} of the Chow ring of the
linear
Grassmannian $Ch(1,r,1)$. Consider the ring homomorphism
$$\tau^*_{\nu}: Ch(1,r,d) \rightarrow Ch(1,r,1)$$
defined in section (\ref{homlam}).
By Lemma (\ref{abcd}),
$$\tau^*_{\nu}(\sigma_0)=\sigma_0,$$
$$\forall a> 0, \ \ \tau^*_{\nu}(\sigma_a)= d^{1+a}\sigma_a,$$
$$\forall a,b> 0, \ \ \tau^*_{\nu}(\sigma_a
\cdot \sigma_b)= d^{2+a+b}\sigma_a\cdot \sigma_b,$$
Therefore, the elements $\sigma_a \cdot \sigma_b$ for
$0\leq a,b \leq r-1$ are independent in $Ch(1,r,d)$.
Since generation was established above, Proposition (\ref{gen})
is proven.
In case $k=1$, the injectivity of $\tau_{\nu}^*$ has been
proven.
\section{Equivariant Chow Groups}
\label{eqchg}
Let $G$ be a group.
Let $G\times X \rightarrow X$ be a left group action.
In topology, the $G$-equivariant cohomology of
$X$ is defined as follows. Let $EG$ be a contractable
topological space equipped with a free left $G$-action and
quotient $EG/G=BG$.
Consider the left action of $G$ on $X\times EG$ defined
by:
$$g(x,b)= (g(x), g(b)).$$
$G$ acts freely on $X\times EG$. Let
$X\times_{G} EG$ be the (topological) quotient.
The $G$-equivariant cohomology of
of $X$, $H_G^*(X)$, is defined by:
$$H_G^*(X) = H^*_{sing}(X\times_{G} EG).$$
If $X$ is a a locally trivial principal $G$-bundle,
then $X\times_{G} EG$ is a locally trivial
fibration of $EG$ over the quotient $X/G$.
In this case, $X\times_{G} EG$ is homotopy equivalent
to $X/G$ and
$$H_G^*(X) = H^*_{sing}(X\times_{G} EG) \stackrel{\sim}{=} H^*_{sing}(X/G).$$
For principal bundles, computing the
equivariant cohomology ring is equivalent
to computing the cohomology of the quotient.
There is an analogous equivariant theory of Chow groups
developed by D. Edidin, W. Graham, and B. Totaro in
[EG], [T]. Let $G$ be a linear algebraic group.
Let $G\times X \rightarrow X$ be a linearized, algebraic $G$-action.
The algebraic analogue of $EG$ is attained by
approximation. Let $V$ be a $\Bbb{C}$-vector space. Let
$G\times V \rightarrow V$ be an
algebraic representation of of $G$.
Let $W\subset V$ be a $G$-invariant open set satisfying:
\begin{enumerate}
\item[(i.)] The complement of $W$ in $V$ is of codimension greater than q.
\item[(ii.)] $G$ acts on $W$ with trivial stabilizers.
\item[(iii.)] There exists a geometric quotient $W\rightarrow W/G$.
\end{enumerate}
$W$ is an approximation of $EG$ up to codimension $q$.
By (iii) and the assumption of
linearization, a geometric quotient $X\times _{G} W$ exits as
an algebraic variety. Let $d=dim(X)$, $e=dim( X\times _{G} W)$.
The equivariant Chow groups are defined by:
\begin{equation}
\label{defff}
A^{G}_{d-j}(X)= A_{e-j}(X\times _{G} W)
\end{equation}
for $0\leq j \leq q.$
An argument is required to check these equivariant
Chow groups are well-defined (see [EG]).
The basic functorial properties of equivariant
Chow groups are
established in [EG]. In particular, if $X$ is
nonsingular, there is a natural intersection
ring structure on $A_i^{G}(X)$.
Let $Z$ be a variety of dimension $z$.
For notational convenience, a superscript will
denote the Chow group codimension:
$$A^{G}_{z-j}(Z) = A^j_G(Z), \ A_{z-j}(Z)=A^j(Z).$$
In particular, equation (\ref{defff}) becomes:
$$\forall\ 0\leq j \leq q, \ \
A_{G}^{j}(X)= A^j(X\times _{G} W).$$
The following result of [EG]
will be used in section (\ref{genarg}).
\begin{pr}
\label{dane}
Let $\Bbb{C}$ be the ground field of complex numbers.
Let $X$ be a quasi-projective variety. Let $G$ be a reductive
group.
Let $G\times X \rightarrow X$ be a linearized, proper, $G$-action
with finite stabilizers. Let $X\rightarrow X/G$ be a
quasi-projective
geometric quotient. Then, there are natural
isomorphisms for all $j$:
$$ A^{j}_G(X) \otimes {\Bbb Q} \stackrel{\sim}{=} A^{j}(X/G) \otimes {\Bbb Q}.$$
\end{pr}
\section{\bf{The Chow Ring of the Grassmannian
and} $A_*^{GL}(pt)$}
\label{grdwk}
Let $0 \rightarrow S \rightarrow \Bbb{C}^{r+1} \rightarrow Q \rightarrow 0$ be
the tautological sequence on $\bold G(\bold P^k, \bold P^r)$.
The following presentation of the Chow ring will be used
in section (\ref{genarg}). Let
$$c_1, \ldots, c_{k+1} \in Ch(k,r,1)$$ be the Chern
classes of the rank $k+1$ bundle $S$.
These classes generate $Ch(k,r,1)$.
Let $$c(S)= 1+ c_1 \ t+ c_2 \ t^2 + \cdots + c_{k+1}
\ t^{k+1},$$
$${1 \over c(S)} = 1+p_1(c_1) \ t+ p_2(c_1,c_2)
\ t^2+ p_3(c_1,c_2,c_3)\ t^3+
\cdots $$ where the latter is
the formal inverse in the formal power series ring
$\Bbb{C}[c_1, \ldots, c_{k+1}][[t]]$.
The ideal of relations among $c_1, \ldots, c_{k+1}$ in $Ch(k,r,1)$
is generated by
the polynomials $$\{p_j \ | \ j> r-k\}.$$ Geometrically,
these relations are obtained from the vanishing of the $j^{th}$
Chern class of the rank $r-k$ bundle $Q$ for $j>r-k$.
In section (\ref{genarg}), a basic result on push-forwards is needed.
\begin{lm}
\label{bozo}
Let $\pi:\bold P(S) \rightarrow \bold G(\bold P^k,\bold P^r)$
be the canonical projection. Let ${\cal{O}}_{\bold P(S)}(1)$
be the canonical line bundle on $\bold P(S)$. Then, for $l\geq k$,
\begin{equation}
\label{ecc}
\pi_*( \ c_1^l({\cal{O}}_{\bold P(S)}(1))\ )= p_{l-k} \in Ch(k,r,1).
\end{equation}
\end{lm}
\begin{pf}
Let $\xi= c_1({\cal{O}}_{\bold P(S)}(1))$.
Certainly, $\pi_*(\xi^k)= 1= p_0$.
The equation
$$ \xi^{k+1} + c_1\ \xi^k+ \cdots +c_k \ \xi+ c_{k+1}=0$$
recursively yields (\ref{ecc}).
\end{pf}
The equivariant Chow ring $A_*^{\bold{GL}_{k+1}}(pt)$ is computed
to motivate the construction in (\ref{genarg}).
The notation of section (\ref{construct}) will be used here.
Let $V_{k+1}$ be a fixed $k+1$- dimensional complex
vector space such that $\bold P(V_{k+1})=\bold P^k$,
Let $U(k,n,1) \subset \bigoplus_{0}^{n}V^*_{k+1}$
be the basepoint free locus.
The codimension of the complement of
$U(k,n,1)$ is easily found to be $n-k+1$.
$\bold{GL}(V_{k+1})$ acts on $U(k,n,1)$ with trivial stabilizers.
As determined in section (1), there is a geometric quotient
$$U(k,n,1)/\bold{GL}(V_{k+1}) \stackrel{\sim}{=} \bold G(\bold P^k,\bold P^n).$$
By the definition of the equivariant Chow ring,
$$A^{j}_{\bold{GL}_{k+1}} (pt)= A^{j}(\bold G(\bold P^k,\bold P^n))$$
for $0\leq j \leq n-k$.
By the presentation of the Chow ring of $\bold G(\bold P^k,\bold P^n)$
given above, the relations
among the generators $c_1, \ldots, c_{k+1}$
start in codimension $n-k+1$. Hence, $A^{*}_{\bold{GL}_{k+1}}(pt)$
is freely generated (as a ring) by $c_1, \ldots, c_{k+1}$ where
$c_j \in A^{j}_{\bold{GL}_{k+1}}(pt)$.
\section{\bf{The Generation Argument}}
\label{genarg}
Again, let $U(k,n,1) \subset \bigoplus_{0}^{n}V^*_{k+1}$
be the basepoint free open set. As $n\rightarrow \infty$,
$U(k,n,1)$ approximates
$E\bold{GL}_{k+1}$.
By the definitions,
$$A^{j}_{\bold{GL}_{k+1}}(U(k,r,d)) \stackrel{\sim}{=}
A^{j}\big( U(k,r,d) \times_{\bold{GL}_{k+1}} U(k,n,1)\big)$$
for $0 \leq j \leq n-k$.
Recall
$$U(k,r,d) \subset \bigoplus_{0}^{r} Sym^d(V_{k+1}^*)$$
is the basepoint free locus.
There is a natural $\bold{GL}(V_{k+1})$-equivariant
open inclusion,
$$U(k,r,d) \times U(k,n,1) \subset \bigoplus_{0}^{r} Sym^d(V_{k+1}^*)
\times U(k,n,1),$$
which yields an open inclusion
$$U(k,r,d) \times_{\bold{GL}_{k+1}}
U(k,n,1) \subset \bigoplus_{0}^{r} Sym^d(V_{k+1}^*)
\times_{\bold{GL}_{k+1}} U(k,n,1).$$
Let $0 \rightarrow S \rightarrow \Bbb{C}^{n+1} \rightarrow Q \rightarrow 0$
be the tautological sequence on $\bold G(\bold P^k,\bold P^n)$.
It is routine to verify
$$\bigoplus_{0}^{r} Sym^d(V_{k+1}^*)
\times_{\bold{GL}_{k+1}} U(k,n,1) = \bigoplus_{0}^{r} Sym^d(S^*)$$
where the latter is the total space of the bundle
$\bigoplus_{0}^{r} Sym^d(S^*)$ over $\bold G(\bold P^k,\bold P^n)$.
Let $D$ be the complement of $U(k,r,d) \times_{\bold{GL}_{k+1}}
U(k,n,1)$ in $\bigoplus_{0}^{r}Sym^d(S^*)$.
The Chow ring of $\bigoplus_{0}^{r} Sym^d(S^*)$ is isomorphic to
$Ch(k,n,1)$ via pull back.
Let $dim$ be the dimension of the variety
$ \bigoplus_{0}^{r} Sym^d(S^*)$.
Let
$$i_D: A_{dim-j}(D) \rightarrow A^j(\bigoplus_{0}^{r}
Sym^d(S^*))$$
be the map obtained by the inclusion
$D\subset \bigoplus_{0}^{r} Sym^d(S^*)$.
There are exact sequences
of Chow groups
$$ A_{dim-j}(D) \rightarrow A^{j}(\bigoplus_{0}^{r}Sym^d(S^*)) \rightarrow
A^{j}\big(U(k,r,d) \times_{\bold{GL}_{k+1}}
U(k,n,1)\big)\rightarrow 0.$$
Let $c_1, \ldots, c_{k+1}$ be the classes of
$Ch(k,n,1)$ defined in section (\ref{grdwk}).
\begin{lm}
\label{bilt}
$p_j(c_1, \ldots, c_{k+1})
\in Im(i_D) \subset A^{j}(\bigoplus_{0}^{r}Sym^d(S^*))$
for all $j>r-k$.
\end{lm}
\begin{pf}
The proof involves an auxiliary construction.
Let $\pi:\bold P(S) \rightarrow \bold G(\bold P^k,\bold P^n)$
be the projective bundle associated to $S$.
Let $T= \pi^{*}\big( \bigoplus_{0}^{r}
Sym^d(S^*) \big)$. Denote the total
space of the bundle $T$ also by $T$.
There is a commutative diagram.
\begin{equation*}
\begin{CD}
T@>>> \bold P(S) \\
@V{\pi}VV @V{\pi}VV \\
\bigoplus_{0}^{r}Sym^d(S^*) @>>> \bold G(\bold P^k,\bold P^n) \\
\end{CD}
\end{equation*}
There is a tautological rational evaluation map
$$\gamma: T \ - \ - \ \rightarrow \bold P^r.$$
A point $\tau \in T$ is a triple
$$\tau=(v, V\subset \Bbb{C}^{n+1}, (f_0, f_1, \ldots, f_r))$$
where $v$ is element of the $k$-dimensional projective
space $\bold P(V)$ and $f_i \in Sym^d(V^*)$.
The rational map $\gamma$ is obtained by
$$\gamma (\tau)= [f_0(v), \ldots, f_r(v)].$$
Let $\overline{D}$ be the set of elements $\tau\in T$
such that all the $f_i$ vanish at $v$. $\overline{D}$ is
the locus where $\gamma$ is undefined. The important
fact is
$$\pi(\overline{D})= D \subset \bigoplus_{0}^{r}Sym^d(S^*)$$
and $\pi: \overline{D} \rightarrow D$ is a projective, birational
morphism.
The Lemma will be proved by finding the
class of $\overline{D}$ in $A_*(T)$ and pushing forward.
Let $L$ be the line bundle ${\cal{O}}_{\bold P(S)}(d)$ on
$\bold P(S)$. Let $L$ also denote the pull-back of
${\cal{O}}_{\bold P(S)}(d)$ to $T$.
The rational map $\gamma$ is obtained from
$r+1$ tautological sections of $L$ on $T$.
There is a natural equivalence
$$ H^0(\bold G(\bold P^k,
\bold P^n), Sym^d(S^*)) \stackrel{\sim}{=} H^0(\bold P(S),L).$$
Also, there is a natural inclusion
$$ H^0(\bold G(\bold P^k,\bold P^n),\
Sym^d(S^*) \otimes \bigoplus_{0}^{r} Sym^d(S)\ ) \subset H^0(T,L).$$
Since the bundle $Sym^d(S^*) \otimes Sym^d(S)$ has a
canonical identity section,
the bundle
$Sym^d(S^*)\otimes \bigoplus _{0}^{r} Sym^d(S))$ has
$r+1$ canonical sections.
It is straightforward to verify these $r+1$ sections
$z_0, \ldots, z_r$ of $L$ on $T$
yield the rational map $\gamma$. The base locus
$\overline{D}$ is the common zero locus of the
sections $z_0, \ldots, z_{r}$. In fact, $\overline{D}$ is
a nonsingular variety of pure codimension $r+1$.
Explicit equations show $\overline{D}$ is
nonsingular complete intersection. Hence
$[\overline{D}]= c_1(L)^{r+1} \in A_*(T)$.
Certainly $\pi_*(\ c_1(L)^{r+1}\ ) \in Im (i_D)$.
Also, for all $\alpha \geq 0$,
$$\pi_*(\ c_1(L)^{r+1+\alpha}\ )=
\pi_*([\overline{D}] \cap c_1(L)^{\alpha})\in Im(i_D).$$
It remains to compute $\pi_*(c_1(L)^{r+1+\alpha})
\in A_*(\bigoplus_{0}^{r} Sym^d(S^*))$.
But since push-forward commutes with flat pull-back, it suffices
to consider $c_1(L)^{r+1+\alpha} \in A_*(\bold P(S))$ and compute
$\pi_*(c_1(L)^{r+1+\alpha})\in Ch(k,n,1)$. By Lemma (\ref{ecc}),
since $c_1(L)= d\cdot c_1({\cal{O}}_{\bold P(S)}(1))$,
$$ \pi_*(c_1(L)^{r+1+\alpha})= d^{r+1+\alpha} \cdot
p_{r-k+1+\alpha}.$$
Hence $p_{j} \in Im(i_D)$ for all $j>r-k$.
\end{pf}
By Lemma (\ref{bilt}) and the presentation of $Ch(k,r,1)$
given in section (\ref{grdwk}), the following inequality is obtained:
\begin{equation}
\label{donn}
dim_{{\Bbb Q}} \ A^{j}_{\bold{GL}_{k+1}}(U(k,r,d)) \leq dim_{{\Bbb Q}}\
A^{j}(\bold G(\bold P^k,\bold P^r)).
\end{equation}
Recall $\bold{GL}_{k+1} \times U(k,r,d) \rightarrow U(k,r,d)$
is a proper group action with finite stabilizers and
geometric quotient $M_{\bold P^k}(\bold P^r,d)$. Hence,
by Proposition (\ref{dane}) and inequality (\ref{donn}),
$$dim_{{\Bbb Q}} \ A^{j}(M_{\bold P^k}(\bold P^r,d) \leq dim_{{\Bbb Q}}
\ A^{j}(\bold G(\bold P^k,\bold P^r)).$$
The surjectivity of $\tau_{\nu}^*:Ch(k,r,d) \rightarrow Ch(k,r,1)$
obtained in section (\ref{homlam}) implies
$$dim_{{\Bbb Q}} \ A^{j}(M_{\bold P^k}(\bold P^r,d) \geq dim_{{\Bbb Q}}
\ A^{j}(\bold G(\bold P^k,\bold P^r)).$$
Therefore $\tau_{\nu}^*$ is injective.
The proofs of Proposition (\ref{homomor})
and Theorem (\ref{mainn}) are complete.
Since the subring of $Ch(r,k,d)$ generated by the
classes $\sigma_1, \ldots, \sigma_{r-k}$ surjects onto
$Ch(r,k,1)$ via $\tau_{\nu}^*$, $Ch(r,k,d)$ is generated
(as a ring) by the classes $\sigma_1, \ldots, \sigma_{r-k}$.
|
1996-04-24T10:11:35 | 9604 | alg-geom/9604014 | en | https://arxiv.org/abs/alg-geom/9604014 | [
"alg-geom",
"math.AG"
] | alg-geom/9604014 | E. Looijenga | Eduard Looijenga, Valery L. Lunts | A Lie algebra attached to a projective variety | AMSTeX v2.1, 46 pages | null | 10.1007/s002220050166 | null | null | Each choice of a K\"ahler class on a compact complex manifold defines an
action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty
set of such K\"ahler classes is given, then we prove that the corresponding
$\slt$-copies generate a semisimple Lie algebra. We investigate the formal
properties of the resulting representation and we work things out explicitly in
the case of complex tori, hyperk\"ahler manifolds and flag varieties. We pay
special attention to the cases where this leads to a Jordan algebra structure
or a graded Frobenius algebra.
| [
{
"version": "v1",
"created": "Wed, 24 Apr 1996 08:05:42 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Looijenga",
"Eduard",
""
],
[
"Lunts",
"Valery L.",
""
]
] | alg-geom | \section{\global\advance\headnumber
by1\global\labelnumber=0{{\the\headnumber}.\ }}
\define\label{(\global\advance\labelnumber by1 \the\headnumber
.\the\labelnumber )\enspace}
\NoBlackBoxes
\document
\topmatter
\title
A Lie algebra attached to a projective variety
\endtitle
\rightheadtext{Lie algebra attached to a projective variety}
\leftheadtext{Eduard Looijenga and Valery Lunts}
\author
Eduard Looijenga and Valery A.\ Lunts$^*$
\endauthor
\address
Faculteit Wiskunde en Informatica, Rijksuniversiteit Utrecht,
PO Box 80.010, 3508 TA Utrecht, The Netherlands
\endaddress
\email
looijeng\@math.ruu.nl
\endemail
\address
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
\endaddress
\email
vlunts\@ucs.indiana.edu
\endemail
\toc\nofrills
{Table of Contents}
{\eightpoint
\head \;\; Introduction\hfill 1\endhead
\head 1. Lefschetz modules\hfill 4\endhead
\head 2. Jordan--Lefschetz modules\hfill 12\endhead
\head 3. Geometric examples of Jordan type I: complex tori\hfill 18\endhead
\head 4. Geometric examples of Jordan type II: hyperk\"ahlerian manifolds\hfill
24\endhead
\head 5. Filtered Lefschetz modules\hfill 30\endhead
\head 6. Frobenius--Lefschetz modules\hfill 33\endhead
\head 7. Appendix: a property of the orthogonal and symplectic Lie
algebra's\hfill\; 41\endhead}
\endtoc
\thanks
$^*$Supported by the National Science Foundation.
\endthanks
\keywords
N\'eron--Severi group, Hodge structure, Jordan algebra, abelian variety,
hyperk\"ahler manifold
\endkeywords
\subjclass
Primary 17B20,14C30,17C50; Secondary 32J27,\break 14K10
\endsubjclass
\abstract
Each choice of a K\"ahler class on a compact complex manifold
defines an action of the Lie algebra $\slt$ on its total complex cohomology. If
a nonempty set of such K\"ahler classes is given, then we prove that the
corresponding $\slt$-copies generate a semisimple Lie algebra. We investigate
the formal properties of the resulting representation and we work things out
explicitly in the case of complex tori, hyperk\"ahler manifolds and flag
varieties. We pay special attention to the cases where this leads to a Jordan
algebra structure or a graded Frobenius algebra.
\endabstract
\endtopmatter
\head
Introduction
\endhead
\noindent
Let $X$ be a projective manifold of complex dimension $n$. If $\kappa\in
\Hm ^2(X)$ is an ample class, then cupping with it defines an operator
$e_{\kappa}$ in the total complex cohomology (denoted here by $\Hm (X)$) of
degree $2$ and the hard Lefschetz theorem asserts that for $s=0,\dots ,n$,
$e_{\kappa}^s$ maps $\Hm ^{n-s}(X)$ isomorphically onto $\Hm ^{n+s}(X)$. As is
well-known, this is equivalent to the exististence of a (unique)
operator $f_{\kappa}$ in $\Hm (X)$ of degree $-2$ such that the commutator
$[e_{\kappa},f_{\kappa}]$ is the operator $h$ which on $\Hm ^k(X)$ is
multiplication by $k-n$. The elements $e_{\kappa},h,f_{\kappa}$ make up a Lie
subalgebra $\g _{\kappa}$ of $\gl (\Hm (X))$ isomorphic to $\sli (2)$ and the
decomposition of $\Hm (X)$ as a $\g _{\kappa}$-module into isotypical summands
is just the primitive decomposition: the primitive cohomology in degree $n-s$
generates the isotypical summand associated to the irreducible representation
of
dimension $s+1$. As these operators respect the Hodge decomposition (in the
sense that $e_{\kappa}$ resp.\ $f_{\kappa}$ has bidegree $(1,1)$ resp.\
$(-1,-1)$), the Hodge structure on $\Hm (X)$ is entirely determined
by the Hodge structure on the primitive cohomology.
However, the primitive decomposition usually depends in a
nontrivial way on the choice of $\kappa$. This we regard as a fortunate fact,
as it often leads to finding an even smaller Hodge substructure of
$\Hm (X)$ that determines the one on $\Hm (X)$. To be explicit, let us define
the {\it
N\'eron--Severi Lie algebra} $\g _{NS}(X)$ as the Lie subalgebra of $\gl (\Hm
(X))$ generated by the $\g _{\kappa}$'s with $\kappa$ an ample class. This Lie
algebra is defined over $\Q$ and is evenly graded by the adjoint action of its
semisimple element $h$ (with its degree $2k$ summand acting as transformations
of bidegree $(k,k)$). We prove in this paper that it is also semisimple. So if
we regard $\Hm (X)$ as a representation of this Lie algebra, then the subspace
of $\Hm (X)$ annihilated by the negative degree part of $\g _{NS}(X)$ is a
Hodge substructure that determines the one on $\Hm (X)$. Notice that this Hodge
substructure is itself still invariant under the degree zero part of $\g
_{NS}(X)$ (which is a reductive Lie subalgebra). Despite its naturality, this
idea appears to be new (although a note by \cite{Verbitsky 1990}, of which we
were not aware of when we started this research, is suggestive in this repect.)
Whereas the $e_{\kappa}$'s commute, the corresponding $f_{\kappa}$'s don't
in general. This makes it difficult to compute the N\'eron--Severi Lie algebra
in practice. It is often helpful when we know of a morphism from $X$ to another
projective manifold $Y$ whose base and fibers are well-understood: for example,
the fact that the associated Leray spectral sequence degenerates yields (among
other things) the existence of a copy of $\g _{NS}(Y)$ in $\g _{NS}(X)$. This
is
an ingredient of our proof that the N\'eron--Severi Lie algebra of flag variety
of a simple complex Lie group is ``as big as possible'' (reflected by the fact
that its Hodge structure is as simple as possible): it is the Lie algebra of
infinitesimal automorphisms of a naturally defined bilinear form (which is
either symmetric or skew) on its cohomology.
But if the $f_{\kappa}$'s happen to commute, then we are in a very interesting
situation: the N\'eron--Severi Lie algebra has degrees $-2$, $0$ and $2$ only
and the (complexified) N\'eron--Severi group acquires the structure of a
Jordan
algebra without preferred unit element. For abelian varieties this is a
classical fact, although, as far as we know, it had not been seen from this
point of view. The N\'eron--Severi Lie algebra appears here as a natural
companion of the Mumford--Tate group: the latter helps us to find the Hodge
ring
as a ring of invariants, whereas the decomposition of the Hodge ring into
$\Q$-irreducible representations of the N\'eron--Severi Lie algebra helps us
to
say more about its structure. (For example, the subring generated by the
divisor
classes is one such irreducible summand.)
Here are some variants of this construction: instead of working with complex
projective manifolds, we could do this for compact complex manifolds that admit
a K\"ahler metric and replace the ample classes by K\"ahler classes. Or we
could
even take all cohomology classes of degree $2$ that have the Lefschetz
property;
clearly, the complex structure has now become irrelevant. The resulting Lie
algebra's are again semisimple and we call them the {\it K\"ahler Lie algebra}
and the {\it total Lie algebra} of the manifold respectively. Examples of
interest here are complex tori and hyperk\"ahler manifolds; in both cases we
get
Jordan algebra structures. In a different direction, we can take for $X$ a
projective variety and take instead its complex intersection homology, even
with
values in a variation of polarized Hodge structure.
These examples lead us to formalize the situation by means of what we have
called a {\it Lefschetz module}. This is essentially a graded vector space
equipped with a set of commuting degree two operators that have
the Lefschetz property, such that the Lie algebra generated by the
corresponding
$\slt$-triples is semisimple. So this vector space becomes a
representation of a semisimple Lie algebra, and it was one of our goals to
classify the representations that so arise. Although we found some rather
restrictive properties, we did not succeed in this.
\smallskip
We now briefly decribe the contents of the separate sections.
In section $1$
we introduce the notion that is central to this paper, that of a Lefschetz
module, and discuss its basic properties. If a Lefschetz module has a
compatible Hodge structure, as is the case for the cohomology of a projective
manifold, then there is also defined its Mumford--Tate group and we compare the
two notions. We next define and discuss the closely related notion of a
Lefschetz pair. This is followed by a partial classification of such pairs in
case the associated Lie algebra is of classical type.
In section $2$ we concentrate on the case when the $f_{\kappa}$'s commute. We
show that the resulting structure is essentially that of a Jordan algebra and
that is why a complete classification is available. We are also led to a
remarkable class of Frobenius algebra's associated to each Jordan algebra, some
of which we describe explicitly.
The next two sections are devoted to examples of K\"ahler manifolds that
give rise to Lefschetz modules of Jordan type.
First we compute the total Lie algebra and the K\"ahler Lie algebra of a
complex
torus. Then we turn our attention to the N\'eron--Severi Lie algebra of an
abelian variety $A$ and express it in terms of the endomorphism algebra of
$A$.
We find that that this N\'eron--Severi Lie algebra intersects $\End
(A)\otimes\C$ in a Lie ideal of $\End (A)\otimes\C$ and we describe the
complementary ideal.
Our treatment of hyperk\"ahler manifolds (in section 4) follows essentially
\cite{Verbitsky 1995}, a preprint that in turn is partly based on a
preliminary version of the present paper. As an application we show how the
Hodge structure on the cohomology algebra of a compact hyperk\"ahlerian
manifold
is expressed in terms of the Hodge structure on its degree two part. We also
give an alternative description of the Beauville-Bogomolov quadratic form on
the
N\'eron--Severi group.
Thus the abelian varieties and the hyperk\"ahler manifolds produce the
classical Jordan algebra's. The exceptional Jordan algebra can be realized
topologically and we ask whether it is realizable by a Calabi-Yau threefold.
Section $5$ is about filtered Lefschetz modules. The example to keep in mind
here
is the Leray filtration on $\Hm (X)$ defined by a surjective morphism $f:X\to
Y$
of projective manifolds. We apply this to the case where $f$ is a projective
space bundle. In combination with a theorem proved in the appendix we are then
able to determine the N\'eron--Severi Lie algebra of a flag variety. It would
be
interesting to do the same for the intersection homology of Schubert
varieties.
In section $6$ we investigate another interesting class of Lefschetz modules,
which we have called {\it Frobenius--Lefschetz} modules. These arise as the
Lefschetz submodule of the cohomology of a projective manifold generated by its
unit element. The Jordan--Lefschetz modules are among them and we suspect that
the remaining simple Frobenius--Lefschetz modules are ``tautological
representations'' of orthogonal or symplectic Lie algebra's. The main result
\refer{6.8} of this section supports this belief: it says that any other simple
Frobenius--Lefschetz module must be a representation of an exceptional Lie
algebra.
\smallskip
We began this work in the Fall of 1990, when both of us were at the
University of Michigan in Ann Arbor. We would like to thank its Mathematics
Department for providing so stimulating working conditions. One of us
(Looijenga) thanks in particular Igor Dolgachev for many
inspiring discussions (then and later) regarding the subject matter of this
paper as well as closely related questions. Our work continued during the
Spring of
1991, while Looijenga was at the University of Utah. He gratefully remembers
the friendly atmosphere he encountered there. After an interruption we
resumed work on this paper in the academic year 1994-95. We thank Tonny
Springer for some helpful references and Bertram Kostant, Misha
Verbitsky and Yuri Zahrin for useful conversations.
\head
\section Lefschetz modules
\endhead
\noindent\label
We fix a field $K$ of characteristic zero. Let $\Mb$ be a $\Z$-graded
$K$-vector space of finite dimension and denote by $h:M\to M$ the
transformation
that is multiplication
by $k$ in degree $k$. So a linear transformation $u:M\to M$ has degree
$k$ if and only if $[h,u]=ku$.
We say that a linear transformation $e:M\to M$ of degree $2$ has the {\it
Lefschetz property} if for all integers $k\ge 0$, $e^k$ maps
$M_{-k}$ isomorphically onto $M_k$. According to the Jacobson--Morozov lemma
this is equivalent to the existence of $K$-linear transformation $f$ in $M$ of
degree $-2$ such that $[e,f]=h$. This $f$ is then unique and $(e,h,f)$ is a
$\sli (2)$-triple: the assignment
$$
\pmatrix
0 & 1\\
0 & 0
\endpmatrix
\mapsto e,\quad
\pmatrix
1 & 0\\
0 & -1
\endpmatrix
\mapsto h,\quad
\pmatrix
0 & 0\\
1 & 0
\endpmatrix
\mapsto f
$$
defines a representation of $\sli (2)$. If $h$ and $e$ happen to be contained
in a semisimple Lie subalgebra $\g\subset \gl (M)$, then so is $f$.
Now let $\a$ be a finite dimensional $K$-vector space. We regard $\a$ as
a graded abelian Lie algebra which is homogeneous of degree two. We say that a
graded Lie homomorphism $e :\a\to \gl (M)$ has the Lefschetz property
if for some $a\in\a$, $e_a$ has that property. Notice that the set of $a\in\a$
with the Lefschetz property is always Zariski open in $\a$. For $a$ in this
open set, we have defined the operator $f_a$ such that
$(e_a,h,f_a)$ is $\sli (2)$-triple. This defines a rational map $f:\a\to\gl
(M)$ in
the sense of algebraic geometry. We let $\g (\a ,M)$ denote the Lie
subalgebra of $\gl (M)$ generated by the transformations $e_a,f_a$. If $\a$ is
merely an abelian group that acts on $M$ by operators of degree $2$, then the
linear extension $\a\otimes K\to\gl (M)$ is a Lie homomorphism and we then
often
write $\g (\a ,M)$ for $\g (\a\otimes K ,M)$. The following example shows that
this Lie algebra need not act reductively in $M$.
\smallskip
{\it Example.} Consider the graded $\slt$-representation $M=\slt \oplus K^2$,
where $\slt =Ke+Kh+Kf$ has the adjoint representation (with its usual
grading) and $K^2$ is the trivial representation in degree zero. Define an
operator $e'$ of degree $2$ in $M$ by $e'(xf+yh+zf,u,v)=(ve,z,0)$.
Then $ee'=e'e=0$, so that $e$ and $e'$ span an abelian Lie algebra $\a$. Since
$e$ has the Lefschetz property, the Lie algebra $\g=\g (\a ,M)$ is defined.
Now $\a$ kills $(0,1,0)$, but the line spanned by this vector has no
$\g$-invariant complement. This example was chosen as to make $\g$
infinitesimally preserve a nondegenerate quadratic form on $M$ (namely
$(xf+yh+zf,u,v)\mapsto -2xz+y^2+2uv$). A smaller example without that property
is the submodule $\slt\oplus K\oplus 0$.
\smallskip
Notice that $\g (\a ,M)$ is evenly graded and that the grading is induced
from the action of $\ad _h$. We say that $(\a ,M)$ is a {\it Lefschetz module}
if $\g (\a ,M)$ is semisimple. In case $M\not=0$, we call
greatest integer $n$ with $M_n\not= 0$ (or equivalently, $M_{-n}\not= 0$) the
{\it depth} of $M$.
The collection
of Lefschetz modules is closed under direct sums, tensor products and taking
duals. Also, a Lefschetz module $M$ has always the decomposition $M
=M_{\ev}\oplus M_{\odd}$, where $M_{\ev}$ (resp.\ $M_{\odd}$) is the direct
sum
of the $M_k$'s with $k$ even (resp.\ odd). Since any representation of a
semisimple Lie algebra is reductive, the category of Lefschetz modules of $\a$
is semisimple. A Lefschetz $\a$-module $M$ is irreducible as a
Lefschetz module if and only if it is irreducible as a $\g (\a ,M)$-module.
There is also an exterior direct sum and tensor product: if $(\a ',M')$
and $(\a '',M'')$ are Lefschetz modules, then we have defined Lefschetz
modules
$$
\align
(\a '\times\a '',M'\boxplus M''),&\quad
e_{(a',a'')}(m',m'')=(e_{a'}m',e_{a''}m'')\\
(\a '\times\a '',M'\boxtimes M''),&\quad
e_{(a',a'')}(m'\otimes
m'')=e_{a'}m'\otimes m'' + m'\otimes e_{a''}m''.
\endalign
$$
The associated Lie algebra is in the first case equal to $\g (\a ',M')\times\g
(\a '',M'')$. This is also true in the second case if both factors are
nonzero.
The preceding discussion showed that when studying Lefschetz modules we may
restrict ourselves to irreducible ones. The following lemma allows the further
reduction of having the associated Lie algebra {\it simple}.
\proclaim{\label Lemma}
Let $M$ be an irreducible Lefschetz $\a$-module and let $\g (\a ,M)=\g
'\times\g ''$ be a decomposition of Lie algebra's. Then this decomposition is
graded and there exist irreducible
Lefschetz $\a$-modules $M'$ and $M''$ such that $M\cong M'\otimes M''$
as Lefschetz $\a$-modules with $\g '$ resp.\ $\g ''$ corresponding to $\g (\a
,M')$ resp.\ $\g (\a ,M'')$.
\endproclaim
\demo{Proof}
Since the grading of $\g$ is the eigen space decomposition of $\ad
_h$ it is immediate that upon writing $h=(h',h'')\in \g'\times\g''$, $\g
^{(i)}$
gets a grading from $\ad _{h^{(i)}}$ making the decomposition a graded one.
Since $M$ is an irreducible module of the semisimple Lie algebra $\g (\a ,M)$,
it must have the form $M'\otimes M''$ with $M^{(i)}$ a $\g ^{(i)}$-module. This
is compatible with the gradings. If the rational
map $f:\a\to\g _{-2}=\g '_{-2}\oplus\g ''_{-2}$ is written $(f',f'')$, then
$[h,f]=-2f$ implies $[h' ,f']=-2f'$ and $[h'' ,f'']=-2f''$. So $M^{(i)}$ a
Lefschetz module of $\a$ with the stated property.
\enddemo
\label Given a Lefschetz module $M$ of $\a$, then an invariant
bilinear form on $M$ is a bilinear map $\phi :M\times M\to K$ that defines a
morphism of Lefschetz modules $M\otimes M\to K$ (where $\a$ acts trivially on
$K$): so $\phi$ is zero on $M_k\times M_l$ unless $k+l=0$ and $\a$ preserves
the form $\phi$ infinitesimally: $\phi (e_a m,m')+\phi (m,e_a m')=0$ for all
$m,m'\in M$ and $a\in\a$. If $a$ is a Lefschetz element, then the
Jacobson--Morozov lemma implies that $f_a$ also preserves $\phi$
infinitesimally. So $\g (\a ,M)$ is
then a subalgebra of $\aut (M,\phi)$. If $\phi$ is nondegenerate and symmetric
(resp.\ skew-symmetric), then we call $(M, \phi)$ an {\it orthogonal} (resp.\
{\it symplectic}) representation. Since a nonzero invariant bilinear form on an
irreducible representation is either orthogonal or symplectic, any
Lefschetz module with nondegenerate bilinear form is the perpendicular direct
sum of Lefschetz modules that are irreducible orthogonal, irreducible
symplectic, or the direct sum of an irreducible Lefschetz module with its
dual.
\smallskip
\label Many Lefschetz modules have the additional structure of an
algebra. Let $A=\oplus _{i=0}^{2n} A_i$ be a graded-commutative algebra with
$A_0=K$. We say that $A$ is a {\it Lefschetz algebra of depth $n$} if $A[n]$ is
a
Lefschetz module of depth $n$ over $A_2$. Such a Lefschetz module can be
endowed
with an invariant $(-)^n$-symmetric bilinear form: let $\int :A\to K$ be a
linear
form that is an isomorphism in degree $2n$ and zero in all other degrees and
define $\phi (a, b):=(-1)^q\int (ab)$ if $a$ is homogeneous of degree $n+2q$ or
$n+2q+1$. If this form is nondegenerate (which is for instance the case when
$A[n]$ is irreducible as a Lefschetz module), then the form
$(a,b)\mapsto\int (ab)$ is also nondegenerate and so $A$ becomes a Frobenius
algebra (in the graded sense).
\medskip\label
Let $M$ be a graded real vector space. A {\it Hodge structure of total weight
$d$} on $M$ consists of a bigrading on its
complexification: $M\otimes\C =\oplus _{p,q\in\Z} M^{p,q}$ such that
(i) $M_k\otimes\C=\oplus _{p+q=k+d} M^{p,q}$ for all $k$ and
(ii) complex conjugation interchanges $M^{p,q}$ and $M^{q,p}$.
These data are conveniently described in terms of an action of the
{\it Deligne torus} on $M$. We recall \cite{Deligne 1979} that this is
two-dimensional torus $\bS$ defined over $\Q$ that is obtained from $\GL (1)$
by
restricting scalars from $\C$ to $\R$. It comes with two characters $z$, $\bar
z$
that are each others complex conjugate and generate the character group. Their
product is for obvious reasons called the {\it norm character} and is denoted
$\Nm$. There is also a natural homomorphism $w:\GL (1)\to \bS$ which on the
real
points is given by the inclusion $\R ^{\times}\subset\C ^{\times}=\bS (\R)$. We
follow Deligne's convention by letting $\bS$ act on $M$ so that $M^{p,q}$
becomes the eigen space of $z^{-p}\bar z^{-q}$ (this action is defined over
$\R$). A
positive number $t>0$, viewed as an element of $\C ^{\times}=\bS (\R)$, acts on
$M_k$ as multiplication by $t^{-k-d}$. So $w(t)$ acts on $M$ as
$t^{-d-h}(=t^{-d}\exp(-\log (t)h))$.
The action of $\sqrt{-1}\in\C ^{\times}=\bS (\R)$ on $M_{\C}$ is on $M^{p,q}$
multiplication by $(\sqrt{-1})^{q-p}$; it is a real
operator, called the {\it Weil operator}; we denote it by $J$.
Suppose we are also given a nondegenerate $(-)^d$-symmetric form $\phi :M\times
M\to\R$ that is zero on $M^{p,q}\times M^{p',q'}$ unless $(p+p',q+q')=(d,d)$
(this
is equivalent to $\phi (gm,gm')=\Nm (g)^{-d}\phi (m,m')$ for all $g\in\bS (\C
)$).
Let $e:M\to M$ be a real operator of bidegree $(1,1)$ which preserves $\phi$
infinitesimally. Clearly, $e$ commutes with $J$
and it is easily checked that for $k\ge 0$,
the map $H^k_e :M_{-k}\otimes\C\times M_{-k}\otimes\C\to\C$ defined by
$$
H^k_e(m,m'):=\phi (e^k m,J\overline{m'}),
$$
is a Hermitian form. We say that $e$ is a {\it polarization} of $(M,\phi)$
if for all $k\ge 0$, $H^k_e$ is definite on $\Ker (e^{k+1}|M_{-k}\otimes \C)$.
\proclaim{\label Proposition}
Let $(M,\phi)$ be as above. Let $\a$ be a real abelian Lie algebra with a pure
weight two Hodge structure that acts morphically on $(M,\phi )$ (i.e., the
action is by mutually commuting transformations of degree $2$ that preserve
$\phi$
infinitesimally and with $\a\otimes M\to M$ a morphism of Hodge structures).
Assume that for some $a\in\a ^{1,1}$, $e_a$ polarizes $(M ,\phi)$.
Then $M$ is a Lefschetz module of $\a$ and $\g (\a, M)$ is a semisimple Lie
algebra defined over $\R$ that preserves $\phi$ infinitesimally.
\endproclaim
\demo{Proof}
The nondegenerateness of the Hermitian form $H^k_a$ on $\Ker
(e_a^{k+1}| M_{-k}\otimes\C )$ implies that $e_a^k$ is injective on this
subspace.
It is easily checked that this, together with the nondegenerateness of $\phi$,
implies that $e_a$ satisfies the Lefschetz property. So $\g (\a ,M)$ is
defined.
If we regard $\phi$ as an element of $M^*\otimes M^*$ of degree zero, then the
fact that $\phi$ is killed by $e_a$ implies that it is killed by $f_a$. So $\g
(\a, M)$ preserves $\phi$ infinitesimally.
We next show that $\g (\a ,M)$ is reductive; since $\g (\a ,M)$ is generated by
commutators, it then follows that $\g (\a ,M)$ semisimple. To this end we
observe
that the image of $\a$ in $\gl (M)$ is normalized by the Weil operator $J$. So
the same is true for $\g (\a ,M)$. As $J$ is semisimple,
it is therefore enough to show that any
subspace $N\subset M\otimes\C$ that is invariant under both $\g (\a ,M)$ and
$J$
is nondegenerate with respect to $\phi$: then its $\phi$-perpendicular space
will be an invariant complement. Consider the primitive decomposition of
$N$ with respect to $e_a$: $N=\oplus _{k\ge 0}\C[e_k]P_{-k}(N)$, where
$P_{-k}(N):= \Ker (e_a^{k+1}|N_{-k})$. This decomposition is
$\phi$-perpendicular
and so we need to show that $\phi$ is nondegenerate on each summand
$\C[e_k]P_{-k}(N)$. For this we observe that $P_{-k}(N)$ is $J$-invariant.
Since
$H^k_a$ is definite on $P_{-k}(N)$, it follows from the definition of $H^k_a$
that $\phi$ is nondegenerate on $P_{-k}(N)+ e^kP_{-k}(N)$. The fact that
$e_k$ leaves $\phi$ infinitesimally invariant then implies that $\phi$ is
nondegenerate on $\C[e_k]P_{-k}(N)$.
\enddemo
\medskip\label
We briefly explain the relation between $\g (\a ,M)$
and the Mumford--Tate group. For this we have to assume that $M$, its grading,
$\a$, and the action of $\a$ on $M$ are all defined over $\Q$. Then $\g (\a
,M)$
is as a Lie subalgebra of $\gl (M)$ also defined over $\Q$. We further assume
that $\a$ acts by transformations of bidegree $(1,1)$. Then $\g (\a
,M)^{2k}$ acts by transformations of bidegree $(k,k)$, in other words, for all
$g\in\bS$ and $x\in\g (\a ,M)$ we have $gxg^{-1}=\Nm (g)^{-h}$.
Consider the image of $\bS$ in $\GL (M)\times
\GL (1)$, where the second map is given by the norm. One defines the {\it
Mumford--Tate group} of $M$, $\operatorname{MT}(M)$, as the smallest
$\Q$-subgroup of $\GL (M)\times \GL (1)$ containing this image. It is clear
that
this is actually a subgroup of $(\times _k\GL (M_k))\times\GL (1)$. The
projection of $\MT (M)$
onto the last factor is still called the {\it norm character} and denoted
likewise. The identity
$$ gxg^{-1}=\Nm (g)^{-h}$$
is now valid for all
$g\in\MT (M)$ and $x\in\g (\a ,M)$. This shows in particular that the adjoint
action of $\MT (M)$ on $\gl (M)$ leaves $\g (\a ,M)$ invariant.
\medskip\label
This suggests to combine the Mumford--Tate group and the group associated to
the
Lie algebra $\g (\a ,M)$ into a single group: if $G (\a ,M)$ denotes the closed
subgroup of $\GL (M)$ with Lie algebra $\g (\a ,M)$, then $\MT (M)G (\a ,M)$ is
a
reductive algebraic group defined over $\Q$. Its Lie algebra has a natural
Hodge
structure; it is obtained by composing the homomorphism $\bS \to \MT (M)G (\a
,M)$ with the adjoint action. In this set up the the r\^ole of the Deligne
torus
is played by the semidirect product $\bS\ltimes SL(2)$, where $s\in \bS$ acts
on $SL(2)$ as conjugation by the diagonal matrix $\diag (z(s)^{-1},\bar
z(s))$.
So the $\Q$-homomorphism $w: GL(1)\to\bS\ltimes SL(2)$, which on the real
points
is given by $t\in\R ^{\times}\mapsto (t,\diag (t,t^{-1}))$, maps onto a
central
subgroup. A a polarized Hodge structure of weight $d$ on $(M ,\phi)$ can now
be
thought of as a certain representation of this group on $M$ with $w(t)$ acting
as
multiplication by $t^{-d}$.
The corresponding action of $\bS$ on $\slt$ is given by $s\mapsto (-z(s)-\bar
z(s))\ad _h$, so that for the resulting Hodge structure on $\slt$, the
bidegrees
of $e,h,f$ are $(1,1)$, $(0,0)$, $(-1,-1)$ respectively. In this spirit one can
also enhance the notion of a set of Shimura data, as defined by
\cite{Deligne}.
\medskip\label
It is high time to give the examples that motivated the preceding definitions.
Let $X$ be a compact K\"ahlerian manifold of dimension $n$. We take for $M$ its
shifted total complex cohomology $\Hm (X)[n]$ and we let $\phi$ be defined by
$$
\phi (\alpha ,\beta ):=(-1)^q\int _{X}\alpha\cup\beta
$$
if $\alpha$ is homogeneous of degree $n+2q$ or $n+2q+1$. (We shall
always suppose that $\Hm (X)$ is equipped with form and $\aut \Hm (X)$ will
stand
for the Lie algebra of endomorphisms of $\Hm (X)$ that preserve this form
infinitesimally.) The fundamental theorems of Hodge theory tell us that $M$
comes with a
Hodge structure of total weight $n$ and that $\phi$ together with cupping with
a
K\"ahler class defines a polarization of $M$. So by proposition \refer{1.6}
$\Hm (X)[n]$ is a Lefschetz module over $\Hm ^2(X)$. The
corresponding semisimple Lie subalgebra of $\aut \Hm (X)$ will be called the
{\it
total Lie algebra} of $M$ and be denoted $\g _{\tot}(X)$; it is defined over
$\Q$.
It is equivalent to say that the cohomology algebra of $X$ is a Lefschetz
algebra.
Clearly, $\g _{\tot}(X)$ is independent of the complex structure. For example,
if
$X$ is a product of an even number of circles, then $\g _{\tot}(X)$ is
defined.
If we want to take the complex structure into account, then it is more natural
to regard $\Hm (X)[n]$ as module over $\Hm ^{1,1}(X)$. This is by \refer{1.6}
also
a Lefschetz module structure. We shall refer to the associated
Lie algebra as the {\it K\"ahler Lie algebra} of $X$ and denote it by $\g
_K(X)$.
For a complex projective manifold we can restrict further and take for $\a$ the
N\'eron--Severi group $\NS (X)$. We call the corresponding semisimple Lie
algebra
the {\it N\'eron--Severi Lie algebra} of $X$ (denoted $\g _{NS}(X)$). It is
defined over $\Q$. Notice that the N\'eron--Severi Lie algebra and the
Mumford--Tate group behave in opposite ways under specialization: the former
gets
bigger, whereas the latter gets smaller.
If one of these Lie algebra's $\g _*(X)$ is defined over a subfield
$K\subset\C$, then we often write $\g _*(X;K)$ for the corresponding Lie
algebra
of $K$-points.
\smallskip\label
Here is another example. Let $V$ be a complex vector space and $W\subset
GL(V)$ a finite complex reflection group acting effectively (that is, $V^W=\{
0\}$). This group acts naturally in the symmetric algebra of $\Sym (V)$.
According to a theorem of Chevalley, the subalgebra of invariants, $\Sym (V)^W$
is a polynomial algebra on $\dim (V)$ homogeneous generators. Let $I$ be the
ideal generated by the invariants of positive degree. Then the quotient $\Sym
(V)/I$ is a graded complete intersection algebra. As a $W$-representation it is
isomorphic to the regular representation. In case $W$ is a Weyl group, then
after doubling the degrees, $\Sym (V)/I$ has the interpretation of the
cohomology algebra of a flag variety. From this we see that for a suitable
regrading, $\Sym (V)/I$ is a Lefschetz representation for the obvious action of
$V$. (We do not know whether this is true for an arbitrary reflection group.)
We shall determine the Lie algebra $\g (V,\Sym (V)/I)$ in \refer{5.8}.
\smallskip\label The N\'eron--Severi Lie algebra can also be defined when $X$
is
an irreducible projective variety: take for $M$ the total intersection
cohomology $\IH (X)$ with the same shift in the grading. There is in general
no such thing as a cup product on this graded vector space, but the cohomology
ring of $X$ acts on $M$ and according to \cite{Saito} polarizations have
the Hodge--Lefschetz property. This even extends to the case where we take
intersection cohomology with values in a local system defined on a Zariski
open-dense subset that underlies a polarized variation of Hodge structure.
There is an invariant form $\phi$ defined as in the previous example. The
natural setting here is that of polarizable Hodge modules.
\medskip
We return to general properties of Lefschetz modules.
\proclaim{\label Proposition}
The Lie algebra $\g (\a ,M)$ is a Lefschetz module of $\a$. If
$a\in\a$ is such that $f_a$ is defined, then the Lie subalgebra $\g (\a
,M)_{\ge 0}$ (resp.\ $\g (\a ,M)_{\le 0}$) is generated by $\g (\a ,M)_0$ and
$e_a$ (resp.\ $f_a$). We have
$$
U\g (\a ,M) =U(\g (\a ,M)_{>0}).U(\g (\a ,M)_{0}).U(\g (\a ,M)_{<0}).
$$
\endproclaim
\demo{Proof} The first statement is clear and the second follows from this. The
third is clear also.
\enddemo
We define the {\it primitive subspace} of $M$ as the set of vectors killed by
$\g (\a ,M)_{<0}$. It is a graded $\g (\a ,M)_0$-subrepresentation of $M$ that
we denote by $\Prim (M)$. Since $\g (\a ,M)_{<0}$ is nilpotent, $\Prim (M)\not=
0$. The previous proposition yields:
\proclaim{\label Corollary}
In the situation of the previous proposition, the primitive subspace $\Prim
(M)$ is the maximal $\g (\a ,M)_{0}$-invariant subspace contained in $\Ker
(f_a)$. Hence $M$ is irreducible as a $\g (\a ,M)$-representation if and
only if $\Prim (M)$ is irreducible as a $\g (\a ,M)_{0}$-representation (and if
$M\not=0$, then $\Prim (M)$ is the summand of $M$ of lowest degree).
\endproclaim
\smallskip The preceding suggests to shift, in Tannakian spirit, the emphasis
from modules to Lie algebra's. For suppose that conversely, we are given a
semisimple Lie algebra $\g$, a simple element $h\in\g$ (in the sense of
appearing
as the middle element of an $\slt$-triple) and an abelian subalgebra $\a$ of
$\g$
such that \roster
\item"{(i)}" the adjoint representation of $\g$ makes $\g$ a
Lefschetz module over $\a$, i.e., there is a rational map $f :\a \to \g_{-2}$
so
that for $e$ in the domain of $f$, we have an $\sli (2)$-triple $(e,h,f_e)$ and
\item"{(ii)}" $\g$ is as a Lie algebra generated by $\a$ and the image of $f$.
\endroster
If $M$ is a finite dimensional representation of $\g$, then $h$ determines a
grading
of $M$ and every $e\in\a$ in the domain of $f$ has the Lefschetz property in
$M$
with respect to this grading. So $M$ is then a Lefschetz module of $\a$. Since
$\g$
is generated by $\a$ and the image of $f$, it follows that $\g (\a ,M)$ is just
the
image of $\g$ in $\gl (M)$. This reduces the classification of Lefschetz
modules
to classifying triples $(\g, h,\a)$ as above. We shall call such a triple a
{\it
Lefschetz triple} and its first two items, $(\g ,h)$, a {\it
Lefschetz pair}. If we are given a Lefschetz pair $(\g ,h)$, then we say that
an
associated Lefschetz triple $(\g, h,\a)$ is {\it saturated} if $\a$ is maximal
for
this property. The stabilizer $G_h$ of $h$ in the adjoint group $G$ permutes
these, but we do not know whether this action is transitive or even whether it
has only finitely many orbits.
\smallskip
\label Let $(\g ,h)$ be a Lefschetz pair. Choose a Cartan subalgebra $\h$ of
$\g$
that contains $h$. It is clear that then $\h\subset\g _{0}$. Let $R\subset\h
^*$
denote the set of roots of $\h$ in $\g $ and let $R_k$ be the set of
$\alpha\in R$ such that $\alpha (h)=k$, or equivalently, $\g ^{\alpha}\subset\g
_k$
(remember that only even values of $k$ occur). The subset $R_0$ is a closed
root
subsystem of $R$; it is the set of roots of $\h$ in $\g _0$. Choose a root
basis $B\subset R$ such that $\alpha (h)\ge 0$ for all $\alpha\in B$ (we then
say
that $\h$ and $B$ are {\it adapted}). Since $h$ is the semisimple element of a
$\slt$-triple, the elements of $B$ take on $h$ values in $\{ 0,1,2\}$
\cite{Bourbaki}, Ch.~VIII, \S 11, Prop.~5. Since these are also even, we get a
decomposition $B=B_0\sqcup B_2$. Clearly $B_0$ will be a root basis of $R_0$.
According to \cite{Bourbaki}, Ch.~VIII, \S 11, Prop.~8 all $\slt$-triples with
$h$ as semisimple element are conjugate under the stabilizer $G_h$ of $h$ in
the
adjoint group $G$ of $\g $. So if $(e,h,f)$ is such a triple and
$M$ is any representation of $\g$, then the isomorphism class of $M$ as a
representation of this $\slt$-copy only depends on $(\g ,h)$. We call it the
{\it
$\slt$-type} of $M$. The following property narrows down the possible subsets
$B_2\subset B$. Let $V(k)$ denote the standard irreducible representation of
$\slt$ of dimension $k+1$ ($k=1,2,\dots$).
\proclaim{\label Proposition}
Let $(\g ,h)$ be a Lefschetz pair and let $M$ be an irreducible representation
of
$\g$ of depth $n$. Then the dimensions of the irreducible
$\slt$-representations
that occur in the $\slt$-type of $M$ make up an arithmetic progression with
increment $2$. In other words, there exists an integer $r$ with $0\le r\le
\lfloor {1\over 2}n\rfloor$ such that $\dim M_{-n}<\dim M_{-n+2}<\cdots <\dim
M_{-n+2r}=\dim M_{-n+2r+2}=\cdots =\dim M_{n-2r}<\dim M_{n-2r+2}<\cdots <\dim
M_n$. Moreover, $r>0$ unless (i) the image of $\g$ in $\gl (M)$ is reduced to
$\slt$ with $M\cong V(n)$ or (ii) the $\slt$-type of $M$ consists of a number
of
copies of $V(1)$.
\endproclaim
\demo{Proof} Denote this set of dimensions by $I$.
The irreducibility of $M$ implies that the elements of $I$ all have the
same parity. Suppose $I$ is not an arithmetic progression with increment $2$.
Then there exists an integer $k$ such that $M$ contains $V(k)$ (the standard
irreducible $\slt$-representation of dimension $k+1$) and $V(k+2l)$ for some
$l\ge 2$, but not $V(k+2)$. Let $(e,h,f)$ be an $\slt$-triple in $\g$
containing
$h$ and decompose $M$ as $M=M'\oplus M''$ with $M'$ resp.\ $M''$ the sum of
the
irreducible subrepresentations of $\slt$ of $\dim\le k+1$ resp.\ $\ge k+5$. Any
linear transformation in $M$ of degree two that commutes with $e$ must preserve
this decomposition, because any $K[e]$-linear homomorphism
$$
V(n)\cong K[e]/(e^{n+1})[n]\to K[e]/(e^{m+1})[m]\cong V(m)
$$
of degree two is zero if $|n-m|>2$. If $(\g ,h,\a )$ is a Lefschetz triple with
$e\in\a$, then this applies in particular to any $e'\in\a$. If $e'$ has the
Lefschetz property, then $f_{e'}$ will also preserve this decomposition (since
$f_{e'}$ is unique) and hence $\g$ will. This contradicts our assumption that
$M$ is irreducible.
The second statement is proved in a similar way: suppose $M\cong V(n)\otimes
P$
for some nonzero vector space $P$ with $n\ge 2$. If $e'$ is any element of
$\a$,
then the fact that $e'$ and $e$ commute, implies that $e'$ acts as $e\otimes
\sigma$ for some $\sigma\in\gl (P)$. In this way, $\a$ maps onto subspace
$\bar\a$ of $\gl (P)$ that contains the identity of $P$. The elements of
$e\otimes\bar\a$ commute in $V(n)\otimes P$ and hence the elements of $\bar\a$
commute in $P$ (here we use that $n\ge 2$).
If $\dim P =1$, then we see that the image of $\a$ in $\gl (M)$ consists of
multiples of $e$. This implies that the image of $\g$ in $\gl (M)$ is a copy of
$\slt$ and that $M\cong V(n)$. We now show that $\dim P\ge 2$ is impossible.
For
this we may assume that $K$ is algebraically closed. Then the commutative Lie
algebra $\bar\a$ leaves invariant a line $L\subset P$. So every $e'\in\a$
leaves
invariant $V(n)\otimes L$. If $e'$ has the Lefschetz property in $M$, then it
also has that property in $V(n)\otimes L$, and so the associated operator $f'$
leaves $V(n)\otimes L$ invariant. It follows that $\g$ leaves $V(n)\otimes L$
invariant. This again contradicts the irreducibility of $M$. \enddemo
The question which subset $B_2\subset B$ defines the semisimple element of an
$\slt$-triple is not difficult to answer for the classical Lie algebra's (see
\cite{Springer-St} and the discussion below) and for the exceptional Lie
algebra's this was tabulated by \cite{Dynkin 1952a}. (The weighted Dynkin
diagrams in these tables that matter here are only those that have all weights
$0$ or $2$.) The preceding proposition leads us to discard more possibilies,
but
we have not seriously studied the interesting question whether what is thus
left
actually occurs.
\smallskip
\label Let us carry out this procedure in case $\g$ is a classical Lie
algebra.
So we assume that $\g$ is simple and classical and we let $V$ be a standard
representation of $\g$ (of dimension $l+1, 2l+1,2l,2l$ in case $\g$ is of type
$A_l,B_l,C_l,D_l$ respectively). The element $h$ induces a grading on $V$. The
degrees that occur all have the same parity; we refer to this as the {\it
parity} of $V$. Let us recall that the $\slt$-invariant bilinear forms on
$V(k)$
are generated by a nonzero $(-)^k$-symmetric form. So an finite dimensional
$\slt$-representation of even parity always admits a nondegenerate invariant
symmetric form, whereas it admits a nondegenerate invariant
skew-symmetric form if and only if all multiplicities are even. In the case of
odd parity it is just the other way around.
The $\slt$-triple $(e,h,f)$ in $\g$ determines a primitive decomposition of
$V$. According to \refer{1.15} the set of positive integers $i\ge 0$ for which
$V(i)$ appears in the $\slt$-module $V$ is of the form $\{ n,n-2,\dots
,n-2r\}$
(with $n-2r\ge 0$). So if we put $k:=\lfloor n/2\rfloor$, then the dimensions
$d_t:=\dim V_{-n+2t}$ ($i=0,1,\dots ,k$) satisfy
$$
1\le d_0<d_1<\dots <d_r=d_{r+1}=\cdots =d_k.\tag{*}
$$
By the remark above, the $d_i$'s must all be even in the orthogonal cases with
odd
parity and in the symplectic cases with even parity.
We choose an $\h$-invariant basis of $V$ indexed as in
\cite{Bourbaki}: $(e_1,\dots ,e_{l+1})$ in case $A_l$, $(e_1,\dots
,e_l,e_0,e_{-l},\dots ,e_{-1})$ in case $B_l$ and in the cases $C_l$ and $D_l$,
$(e_1,\dots ,e_l,e _{-l}\dots,e_{-1},)$. The same shall apply to our
labeling of the simple roots $(\alpha _1,\alpha _2,\dots \alpha _l)$ (as
recalled below).
\smallskip
{\it Case $A_l$.} We let $\alpha _i(\diag (\lambda _1,\dots
,\lambda _{l+1}))=\lambda _i-\lambda _{i+1}$.
We find that the elements of $B_2$ are the simple roots with index
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_k, d_0+\cdots +d_{k-1}+2d_k,\dots
,d_0+2(d_1+\cdots +d_{k-1}+d_k)$ (then $l=2(d_0+\cdots +d_k)$) or
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_k, d_0+\cdots +2d_{k-1}+d_k,\dots
,d_0+2(d_1+\cdots +d_{k-1})+d_k$ (then $l=2(d_0+\cdots +d_{k-1})+d_k$).
So $B_2$ is symmetric with respect to the natural
involution of $B$. Notice that no two elements of $B_2$ will be adjacent in
the Dynkin diagram: if that would be the case, then $d_i=1$ for some $i>0$ and
\refer{*} shows that this is impossible.
\smallskip
{\it Case $B_l$.} Since $\dim V=2l+1$, its parity must be even. This
means that $n$ is even and $d_k$ is odd; we have
$l=d_0\cdots +d_{k-1}+{1\over 2}(d_k-1)$. We let
$$
\alpha _i(\diag (\lambda _1,\dots\lambda _l,0,-\lambda _{-l},\dots,-\lambda
_{-1}))= \cases
\lambda _i-\lambda _{i+1}&\text{if $i=1,\dots ,l-1$;}\\
\lambda _l&\text{if $i=l$.}
\endcases
$$
We find that the elements of $B_2$ are the simple roots with index
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_{k-1}$. As in the previous case we see that
no two elements of $B_2$ will be adjacent in the Dynkin diagram.
\smallskip
{\it Case $C_l$.}
We let
$$
\alpha _i(\diag (\lambda _1,\dots ,\lambda _l,-\lambda _{-l},\dots,-\lambda
_{-1}))= \cases
\lambda _i-\lambda _{i+1}&\text{if $i=1,\dots ,l-1$;}\\
2\lambda _l&\text{if $i=l$.}
\endcases
$$
In the case of odd parity, $n$ is odd and $l=d_0+\cdots +d_k$. The elements
of $B_2$ are the simple roots with index $d_0,d_0+d_1,\dots ,d_0+\cdots
+d_k=l$.
In the case of even parity $n$ and $d_0,\dots ,d_k$ are even. We have
$l=d_0+\cdots +d_{k-1}+{1\over 2}d_k$ and the elements of $B_2$ are
simple roots with index $d_0,d_0+d_1,\dots ,d_0+\cdots +d_{k-1}$.
In both cases no two elements of $B_2$ will be adjacent in the Dynkin diagram.
\smallskip
{\it Case $D_l$, $\l\ge 4$.}
We let
$$
\alpha _i(\diag (\lambda _1,\dots ,\lambda _l,-\lambda _{-l},\dots,-\lambda
_{-1}))= \cases
\lambda _i-\lambda _{i+1}&\text{if $i=1,\dots ,l-1$}\\
\lambda _{l-1}+\lambda _l&\text{if $i=l$;}\\
\endcases
$$
In the case of odd parity, $n$ is odd and all $d_i$'s are even. We have
$l= d_0+\cdots +d_{k-1}+d_k$ and the elements of
$B_2$ are simple roots with index $d_0,d_0+d_1,\dots ,d_0+\cdots +d_k$. We
claim that no two such elements are adjacent in the Dynkin diagram. For this
can
only happen when $d_k=2$. In view of \refer{*}, this implies that $d_i=2$ for
all
$i$, in other words, $V\cong V(n)\oplus V(n)$. But since $n>1$, this is
excluded
by \refer{1.15}.
In the case of even parity, $n$ and $d_k$ are even. We have $l=
d_0+\cdots +d_{k-1}+{1\over 2}d_k$. Suppose first that $d_k\ge 4$. Then the
elements of $B_2$ are the simple roots with index $d_0,d_0+d_1,\dots
,d_0+\cdots
+d_{k-1}$. In that case $\alpha _l,\alpha _{l-1}\in B_0$ and $\alpha _1\in B_2$
if and only if $d_0=1$. Clearly no two elements of $B_2$ are adjacent.
If $d_k=2$, then by \refer{*}, $d_0=1$ and $d_2=\cdots =d_k=2$, in other words,
$V\cong V(2k)\oplus V(2k-2)$. Then $l=2k$ is even and the elements of $B_2$ are
in position $1,3,5,\dots,2k-3,2k-1,2k$. No two elements of $B_2$ are
adjacent.
\smallskip
For future reference we record:
\proclaim{\label Corollary}
When $\g$ is simple and classical, no two elements of $B_2$ are connected.
\endproclaim
\head
\section Jordan--Lefschetz modules
\endhead
\noindent
In this section we discuss a particularly nice class of Lefschetz triples. We
introduce them via the following proposition.
\proclaim{\label Proposition}
Let $M$ be a Lefschetz $\a$-module. Then the following two properties are
equivalent
\roster
\item"{(i)}" The graded Lie algebra $\g (\a ,M)$ has degrees $-2$, $0$ and
$2$
only.
\item"{(ii)}" The operators $f_a$ with $a\in\a$ in the domain of $f$,
mutually commute. \endroster
\endproclaim
\demo{Proof} We only prove the nontrivial implication $(ii)\Rightarrow (i)$.
Write $\g$ for $\g (\a ,M)$.
If $a\in\a$ is such that $f_a$ is defined, then regard $\g$ as an $\slt$-module
via the $\slt$-triple $(e_a ,h,f_a )$ and let $V(a)\subset\g$ be sum of the
irreducible summands of dimension $1$ and $3$. Notice that $V(a)$ contains the
image of $\a$. So the intersection $V$ of the $V(a)$'s also contains the image
of $\a$. We have $V=V_{-2}\oplus V_0\oplus V_2$ where $V_{2k}$ can be
characterized as the subspace of $\g _{2k}$ that is
annihilated by all the operators $\ad ^{-k+2}(e_a)$, or alternatively, by all
the operators $\ad ^{k+2}(f_a)$. Since the operators $e_a$ resp.\ $f_a$
mutually commute it follows that $V$ is invariant
under these operators. Hence $V=\g$ and so $\g$ has degrees $-2$, $0$
and $2$ only. \enddemo
As we are interested in the graded Lie algebra's that so arise, we make the
following definition. Say that a Lefschetz pair $(\g ,h)$ is a {\it
Jordan--Lefschetz pair} if $(\g ,h,\g _2)$ is a Lefschetz triple. It is
clear that such a Lefschetz triple is saturated.
\proclaim{\label Proposition}
If $(\g ,h)$ is a Jordan--Lefschetz pair, then $\g =\g
_{-2}\oplus\g _0\oplus\g _2$ and $U\g =U\g _2.U\g_0.U\g _{-2}$.
\endproclaim
\demo{Proof}
If $e\in\g _2$ is a Lefschetz
element, then $[e,\g _2]=0$. The first assertion now follows from the primitive
decomposition under $\ad _e$. The second is a consequence of this.
\enddemo
\proclaim{\label Corollary}
Let $(\g ,h)$ be a Jordan--Lefschetz pair and let $M$ be a finite dimensional
representation of $\g$ and regard $M$ as a Lefschetz module of $\g _2$. Then
$M$
is generated as a $U\g _2$-module by $\Prim (M)$.
\endproclaim
Our terminology is easily explained: according to \cite{Springer}, \S 2.21,
a Jordan--Lefschetz pair defines a ``Jordan algebra without unit
element''. These have been classified. Let
us give a quick proof of this classification. It is based on two lemma's.
In what follows, $(\g ,h)$ is a Jordan--Lefschetz pair with $\g$ simple and
and $\h\subset\g$ and $B$ are adapted.
\proclaim{\label Lemma} The subset $B_2$ is a singleton.
\endproclaim
\demo{Proof}
If not, then there is chain $(\beta ,\alpha _1,\dots ,\alpha _N,\beta ')$ with
$\beta ,\beta '\in B_2$ be distinct and $\alpha _i \in
B_0$. But then is $\beta +\alpha _1+\cdots \alpha _N +\beta '$ a root and in
$R_4$, which is supposed to be empty.
\enddemo
It follows that $h$ is twice a fundamental coweight.
The following lemma concerns a general property of root systems.
\proclaim{\label Lemma}
Let $\beta\in B$ and let $R(\beta )$ be the set of positive roots that have
$\beta$-coefficient one. Then $R(\beta )+R(\beta )$ does not contain a
root if and only if $R(\beta )$ contains the highest root.
\endproclaim \demo{Proof}
It is clear that if $R(\beta )$ contains the highest root, then
$R(\beta )+R(\beta )$ cannot contain a root. If on the other hand
the highest root has $\beta$-coefficient $\ge 2$, then it follows from
\cite{Bourbaki}, Ch.~VI,\S 1, Prop.~19 that there exists a sequence of roots
$\alpha _1,\dots ,\alpha _r$ such that $\alpha _1\in B$, $\alpha _{i+1}-\alpha
_i\in B$ for $i=1,\dots ,r-1$ and $\alpha _r$ is the highest root. If $\alpha
_i$ is the last root in this sequence for which the coefficient of $\beta$ is
one, then $\alpha _i$ and $\beta$ are elements of $R(\beta)$ whose sum is a
root.
\enddemo
\proclaim{\label Corollary} Let $\beta$ be the unique element of $B_2$.
Then the pair $(B,B-\{\beta\})$ is of the following type:
\roster
\item"{}" $(A_{2m-1},A_{m-1}+A_{m-1})$ ($m\ge 1$),
\item"{}" $(B_m,B_{m-1})$ ($m\ge 2$),
\item"{}" $(C_m,A_{m-1})$ ($m\ge 2$),
\item"{}" $(D_m,D_{m-1})$ ($m\ge 5$),
\item"{}" $(D_{2m},A_{2m-1})$ ($m\ge 2$) or
\item"{}" $(E_7,E_6)$.
\endroster
Conversely, every item of this list determines an isomorphism class of
Jordan--Lefschetz pairs.
\endproclaim
\demo{Proof} The pairs $(B,\beta )$ with the property of the previous lemma are
those in this list plus the following: in case $A_l$ we can take $l$
and $\beta$ arbitrary, in case $D_{2l+1}$ any end of the Dynkin diagram,
and $B$ of type $E_6$ with $\beta$ the
end of the branch of length $3$. These addional possibilities disappear if we
want $h$ to be a simple element (that is, $h=[e,f]$ for certain $e\in\g _2$ and
$f\in \g _{-2}$): for the classical cases $A_l$ and $D_{2l+1}$ this follows
from the
discussion following \refer{1.16}) and for $E_6$ this follows from table 18 of
\cite{Dynkin}. On the other hand, case $3''$ of table 19 of this reference
shows
that in case $E_7$ this element $h$ is simple. The discussion below will show
that
all the other listed cases occur as well. \enddemo
\label We continue with the Jordan--Lefschetz pair $(\g ,h)$ and let $\h$, $B$
and $\beta\in B$ as above. Let $\varpi\in\h ^*$ be the fundamental weight
that is zero on $B^{\vee}-\{\beta ^{\vee}\}$ and $1$ on $\beta ^{\vee}$. We
say that an irreducible representation $M$ of $\g$ is a {\it Jordan--Lefschetz
module of level $k$} (where $k$ is a positive integer) if its lowest weight is
$-k\varpi$; for $k=1$, we shall also call it a {\it fundamental}
Jordan--Lefschetz
module.
Notice that $M$ then has depth $k\varpi (h)$. A more intrinsic
description of these modules is the following: given a Jordan--Lefschetz pair
$(\g ,h)$, then an irreducible representation $M$ of $\g$ is a
Jordan--Lefschetz
module if and only if it has a nonzero vector stabilized by $\g _{-2}+\g _0$.
\proclaim{\label Lemma}
Let $M$ be an irreducible representation $M$ of the Jordan--Lefschetz pair
$(\g ,h)$ and let $n$ be its depth as a Lefschetz module.
Then $M$ is a Jordan--Lefschetz module if and only if $\dim M_{-n}=1$.
If these equivalent conditions are fulfilled, then
the natural map $\g ^2\otimes M_{-n}\to M_{-n+2}$ is an isomorphism.
\endproclaim
\demo{Proof} Let $\lambda\in\h ^*$ be the lowest weight of $M$.
Then the lowest weight space $M^{-\lambda}$ is
contained in $M_{-n}$. Hence
$\dim M_{-n}=1$ is equivalent to $M^{-\lambda}=M_{-n}$. The latter is
equivalent to: $\g _{-2}+\g _0$ stabilizes $M^{-\lambda}$, which in turn is
equivalent to $\lambda$ vanishing on $B_0$.
Suppose the two conditions satisfied. The fact that $\g _{-2}+\g _0$ is the
$\g$-stabilizer of $M_{-n}$ implies that $\g ^2\otimes M_{-n}\to M_{-n+2}$
is injective. Since $M$ is as a $U\g _2$-module generated by $M_{-n}$, it is
also surjective.
\enddemo
\label Let us describe the fundamental Jordan--Lefschetz representations
in the classical cases (i.e., those that are not of type $(E_7,E_6)$). We only
do this over the complex numbers.
\smallskip
{\it Case $(A_{2m-1},A_{m-1}+A_{m-1})$}: $(\sli (2m),\sli (m)\times\sli (m))$.
Let $V$ be a vector space of dimension $2m$, $V=V_{-1}\oplus V_{1}$ a direct
sum
decomposition into subspaces of dimension $m$. We take $\g :=\sli (V)$ and let
$h\in \sli (V)$ be $\pm 1$ on $V_{\pm 1}$.
Then $\g _0$ maps isomorphically onto $\sli (V_{-1})\times\sli (V _1)\times\C
h$
and $\g _2\cong \Hom (V_{-1},V_1)$ resp.\ $\g _{-2}\cong \Hom (V_1,V_{-1})$.
The corresponding fundamental representation is $M:=\wedge ^m V$ with lowest
degree piece $M_{-m}=\wedge ^mV_{-1}$.
\smallskip
{\it Case $(B_m,B_{m-1})$ or $(D_m,D_{m-1})$}: $(\so (n),\so (n-2))$ with
$m=2n+1$ resp.\ $m=2n$ ($n\ge 2$).
Let $V$ be a vector
space of dimension $n$ equipped with a nondegenerate symmetric bilinear form
and let $V_{\pm 2}$ be isotropic lines in $V$ such that $V_{-2}\oplus V_2$ is
nondegenerate. Let $V_0$ be the orthogonal complement of $V_{-2}\oplus V_2$ in
$V$. We take $\g =\so (V)$ and let $h\in\so (V)$ be the element with the eigen
space decomposition $V_{-2}\oplus V_0\oplus V_2$. Then $\g _0=\so (V_0)\times
\gl (V_2)$ and $\g _{\pm 2}$ projects isomorphically to $\Hom (V_0,V_{\pm
2})$. We take $M=V$.
\smallskip
{\it Case $(C_m,A_{m-1})$}: $(\sy (2m),\sli (m))$ ($m\ge 2$). Let $V$ be a
vector
space of dimension
$2m$ equipped with a nondegenerate symplectic form and let $V=V_{-1}\oplus V_1$
be a decomposition of $V$ into totally isotropic subspaces of dimension $m$. We
take $\g =\sy (V)$ and let $h\in\sy (V)$ be the element with the eigen space
decomposition $V_{-1}\oplus V_1$. Then $\g _0$ maps isomorphically to $\gl
(V_1)$ and $\g _{\pm 2}$ is naturally isomorphic to the space of symmetric
elements in $(V_{\pm 1})^{\otimes 2}$. We take for $M$ the primitive quotient
of
$\wedge ^mV$ (i.e., the quotient by $\wedge ^{m-2} V\wedge\omega$, where
$\omega\in \wedge ^2V$ is the dual of the symplectic form).
\smallskip
{\it Case $(D_{2m},A_{2m-1})$}: $(\so (4m),\sli (2m))$ ($m\ge 2$).
Let $V$ be a vector space of
dimension $4m$ equipped with a nondegenerate symmetric bilinear form and let
$V=V_{-1}\oplus V_1$ be a decomposition of $V$ into totally isotropic subspaces
of dimension $2m$. We take $\g =\so (V)$ and let $h\in\so (V)$ be the element
with the eigen space decomposition $V_{-1}\oplus V_1$. Then $\g _0$ maps
isomorphically to $\gl (V_1)$ and $\g _{\pm 2}$ maps onto the skew elements in
$(V_{\pm 1})^{\otimes 2}$. Consider the spinor representation $\wedge
^{\bullet} V_1$. (We recall that this factors through the representation of the
Clifford algebra of $V$ on $\wedge ^{\bullet} V_1$ for which $v\in V_1$ acts as
wedging with $v$ and $v\in V_{-1}$ acts as the interior product under the
obvious isomorphism $V_{-1}\cong V_1^*$.) It splits into a direct sum of
subrepresentations $\wedge ^{\ev} V _1$ and $\wedge ^{\odd} V _1$. They are
irreducible and nonisomorphic and correspond to the case when $\beta$ is an end
of the Dynkin diagram connected with a branch point. For even $m$ both
are orthogonal and for odd $m$ both are symplectic. We take
$M=\wedge ^{\ev} V _1[m]$.
\smallskip
\label The Jordan--Lefschetz modules give rise to Frobenius
algebra's with remarkable properties. Let $(\g ,h)$ be a Jordan pair and let
$M$
be a Jordan--Lefschetz module of $(\g ,h)$ of depth $n$. Then $M$ is a monic
module
over the commutative algebra $U\g _2$. So if $I\subset U\g _2$ denote the
annihilator of $M$, then $I$ defines the origin in $\Spec (U\g _2)=\g _2^*$
with
local algebra $A:=U\g _2/I$. The latter is an evenly graded Lefschetz algebra
that
has $M$ as a free graded module of rank one. The next proposition shows that it
has a lot of automorphisms. Let $\g _0'$ be the Lie subalgebra of $\g _0$ that
kills $1\in A_0$ (or equivalently, kills $M_{-n}$); this Lie algebra is
complementary to the span of $h$ in $\g _0$.
\proclaim{\label Proposition}
The Lie algebra $\g '_0$ acts on $A$ as derivations and so the associated Lie
subgroup of $\GL (A)$ is a group of algebra automorphisms of $A$. Moreover, the
Lie subgroup $G_0\subset\GL (A)$ associated to $\g _0$ has a dense orbit in
$A_2$ consisting of Lefschetz elements.
\endproclaim
\demo{Proof}
If $u\in\g _0$ and $e\in\g _2$, then for all $x\in A$, $u(ex)=
[u,e]x+ e(ux)$. Since $A$ is generated by $\g _2$, it follows with induction
that $u$ acts as a derivation if (and only if) $u$ kills $1$. The last
assertion
follows from a well-known result \cite{Bourbaki}, Ch.~VIII, \S 11, Prop.\ 6.
\enddemo
\label
A decomposition of $\g$ into simple components corresponds to a decomposition
of
$A$ as a tensor product of algebra's, so little is lost in assuming that $\g$
is
simple. The form $\int$ defined in \refer{1.4} makes of $A$ a Frobenius
algebra
with soccle $A_{2n}$; at the same time $\int$ defines a generator of $A_{2n}$
that
serves as the identity element for another algebra structure defined by the
action of $U\g _{-2}$. Here is a description of the
algebra's associated to the fundamental Jordan--Lefschetz modules in all
cases.
\smallskip
{\it Case $(A_{2m-1},A_{m-1}+A_{m-1})$}:
Let $W$, $W'$ be vector spaces of dimension $m$ and let $A:= \oplus _{k=0}^m
\wedge ^k W\otimes\wedge ^ kW'$. This is just the subalgebra of the graded
algebra $\wedge ^{\bullet}W\otimes\wedge ^{\bullet}W'$ generated by $W\otimes
W'$. It is the fundamental Frobenius algebra associated to $(\sli (W\oplus
W'),h=(1_W,-1_{W'}))$. (To see the relation with the description given in
\refer{2.9}, take $V_{-1}=(W')^*$ and $V_1=W$ and observe that a choice of a
generator of $\wedge ^mV_{-1}$ identifies $\wedge ^{m-k}V_{-1}$ with
$\wedge ^k W'$ and hence $\wedge ^m(V_1\oplus V_{-1})$ with $A$.)
Here is a presentation of this algebra. Choose bases $(w_1,\dots
,w_m)$ of $W$ and $(w'_1,\dots ,w'_m)$ of $W'$. Then $x_{ij}:=w_i\otimes w'_j$,
($i,j=1,\dots ,m$) generate $A$ as an algebra and a set of defining relations
is
$x_{ij}x_{kl}+x_{il}x_{kj}=0$.
\smallskip
{\it Case $(D_{2m},A_{2m-1})$}: Let $W$ be a
vector space of dimension $2m$, then let $A$ be the subalgebra of $\wedge
^{\bullet}W$ generated by $\wedge ^2W$. This is the
fundamental Frobenius algebra associated to $\so (W\oplus W^*),
h=(1_W,-1_{W^*}))$.
A presentation of $A$ is a follows $(w_1,\dots ,w_m)$ is a basis of $W$, then
generators for $A$ are $\omega _{ij}:=w_i\wedge w_j$ ($1\le i,j\le m$), subject
to the linear relations $\omega _{ij}+\omega _{ji}=0$ and the quadratic
relations $\omega _{ij}\omega _{kl}+\omega _{il}\omega _{kj}=0$. So this is a
quotient of the algebra of the previous case.
\smallskip
{\it Case $(C_m,A_{m-1})$}: Let $W$ be a vector space of dimension $m$. Then
a model for the fundamental Frobenius algebra associated to $\sy (W\oplus
W^*), h=(1_W,-1_W))$ is the subalgebra $A$ of $\wedge
^{\bullet}W\otimes\wedge ^{\bullet}W$ generated by the symmetric elements
$w\otimes w$, $w\in W$. If $(w_1,\dots ,w_m)$ is a
basis of $W$, then generators for $A$ are $u _{ij}:=w_i\otimes w_j +w_j\otimes
w_i$ ($1\le i,j\le m$). A defining set of relations is $u_{ij}=u_{ji}$ and
$u_{ij}u_{jk}+u_{jj}u_{ik}=0$.
\smallskip
{\it Cases $(B_n,B_{n-1})$ and $(D_n,D_{n-1})$:} Let $(W,q:W\to K)$ be a
vector space of dimension $m$ endowed with a nondegenerate quadratic form.
Consider the graded vector space $A:=K\oplus W\oplus K\mu$, where $K$ has
degree zero, $W$ has degree $2$ and $\mu$ is a generator of a
one-dimensional $K$-vector space that has degree $4$. Equip $A$ with
the quadratic form $\tilde q(x+w+x'\mu)=q(w)-xx'$.
Make it also a graded algebra by letting $1\in K$ be the identity element and
letting for $w,w'\in W$, $w.w'=b(w,w')\mu$, where $b$ is the bilinear form
associated to $q$. Then $A$ is the fundamental Frobenius algebra associated to
$(\so (A,\tilde q), h=(-2,0_W,2))$.
If $(w_1,\dots ,w_m)$ is a $q$-orthogonal basis of $W$ then a presentation of
$A$
has generators $w_1,\dots ,w_m$ and relations $w_iw_j=0$ ($i\not=j$) and
$q(w_i)^{-1}w_iw_i=q(w_j)^{-1}w_jw_j$.
\smallskip
{\it Case $(E_7,E_6)$:}
Let $W$ be a finite dimensional $K$-vector space
and let be given a cubic form $c:\Sym ^3(W)\to K$ that does not factor through
proper linear quotient of $W$. The latter condition implies that the associated
homomorphism $\tilde c:\Sym ^2W\to W^*$ is surjective. Consider the graded
vector
space $A:=K\oplus W\oplus W^*\oplus K\mu$ with respective summands in degree
$0$, $2$, $4$, $6$ (here $\mu$ is a generator of a one dimensional vector
space)
and use $\tilde c$ and the obvious pairing $W\times W^*\to
K\mu$ to give $A$ the structure of a commutative graded $K$-algebra. We
require that multiplication makes $A[3]$ a Jordan--Lefschetz module over
$A_2$.
This forces $\dim W=27$ and after possibly passing to an algebraic closure of
$K$, $c$ will be unique up to a linear transformation. The group $G_c$ of
$g\in GL(W)$ that leave $c$ invariant is of type $E_6$ and the
associated Lie algebra $\g$ is of type $E_7$. (This Lie algebra can be
characterized as the Lie algebra of linear transformations of $A$ that leaves
invariant a certain quartic form on $A$.) If we regard $A$ as a quotient of the
symmetric algebra of $W$, then we checked by means of the program \cite{LiE}
that the relations are again quadratic: the ideal $I\subset\Sym (W)$ that
defines $A$ is generated by $\ker (\tilde c)$ (the latter is an irreducible
representation of $G_c$ whose highest weight is twice that of $W$).
\medskip
The Jordan--Lefschetz algebra's of higher level can be expressed
in terms of a fundamental one:
\proclaim{\label Proposition}
Let $A$ be a fundamental Jordan--Lefschetz algebra
for a Jordan--Lefschetz pair $(\g ,h)$. Let $k$ be a positive integer and give
$\Sym
^k(A)$ the structure of an algebra by identifying it with the algebra of
symmetric
invariants in $A^{\otimes k}$. Then the subalgebra $A(k)$ of $\Sym ^k(A)$
generated
by $A_2$ is a Jordan--Lefschetz algebra of level $k$ for $(\g ,h)$.
\endproclaim
\demo{Proof}
We regard $A$ as an fundamental representation of $\g$ with lowest weight space
spanned by its unit element $1\in A$. Then the irreducible representation of
$\g$
with lowest weight $k$ times the one of $A$ is contained in $A^{\otimes k}$ as
the $U\g$-submodule generated by the unit element $1\otimes\cdots\otimes 1$.
But this is also the $U\g_2$-submodule generated by this element. In other
words,
this is the subalgebra of $A^{\otimes k}$ generated by the elements
$a\otimes 1\otimes\cdots\otimes 1 + 1\otimes a\otimes\cdots\otimes 1 +\cdots +
1\otimes 1\otimes\cdots\otimes a$, with $a\in A_2$.
\enddemo
In section 4 we will be concerned with the algebra's of higher level in the
cases
$(B_n,B_{n-1})$ and $(D_n,D_{n-1})$, and that is why we want to describe them
here
in more explicit terms. Let $(W,q)$ and $(A,\tilde q)$ be as under the relevant
case above.
\proclaim{\label Proposition} Fix an integer $k\ge 1$ and let $I_k$ be the
ideal in $\Sym (W)$ generated by the $k+1$-st powers of $q$-isotropic vectors
(i.e., the $w^{k+1}$ for which $q(w)=0$). Then $A(k)=\Sym (W)/I_k$. Moreover,
the
subalgebra of $\Sym (W)/I_k$ of $\so (W)$-invariants is $K(u)/(u^{k+1})$,
where $u\in \Sym ^2(W)$ represents the inverse form of
$q$ on the dual of $W$. The soccle of $A(k)$ is spanned by
the image of $u^k$ and if we identify $W$ with its image in $A(k)$, then
$x^2u^{k-1}=q(x)u^k$.
\endproclaim
\demo{Proof} Consider the graded algebra obtained by dividing $\Sym (W)$ out
by the
ideal generated by $u$, $\Sym (W)/(u)$, and let $W(d)$ denote the image of
$\Sym
^d(W)$. Then $W(d)$ is an irreducible representation of $\so (W)$ that can be
identified with the subrepresentation of $\Sym ^d(W)$ which is linearly spanned
the
pure $d$th powers of $q$-isotropic vectors in $W$. In particular, $I_k$ is the
ideal
in $\Sym (W)$ generated by $W(k+1)$. It is easy to see that
$$
\Sym ^d(W)=\oplus _{i=0}^{\lfloor d/2\rfloor} u^iW(d-2i),
$$
as graded $\so (W)$-representations. A Clebsch--Gordan
rule asserts that for $p\ge q\ge 0$, the image of $W(p)\otimes W(q)\to \Sym
^{p+q}W$ under the multiplication map is $\oplus _{i=0}^q u^iW(p+q-2i)$. So the
cokernel can be identified with $u^{q+1}\Sym ^{p-q-2}(W)$. This remains so if
we replace $W(q)$ by $\Sym ^q(W)$. It follows that
$$\align
\Sym (W)/I_k =&\Sym ^0(W)\oplus\cdots \oplus\Sym ^{k-1}(W)\oplus\Sym ^k(W)\\
&\oplus u\Sym ^{k-1}(W)\oplus u^2\Sym ^{k-1}(W)\oplus \cdots\oplus u^k\Sym
^0(W).
\endalign
$$
If $\tilde u\in\Sym ^2(A)$ denotes the symmetric tensor dual to $\tilde q$,
then
$A(k)$ is the image of $\Sym ^k(A)$ in $\Sym (A)/(\tilde u)$. If we write
$A=Kt\oplus W\oplus K\mu$, with $\deg t=0$, then $\Sym (A)=K[t,\mu ]\Sym W$
with $\tilde u$
corresponding to $t\mu +u$. Hence we have the following identity of graded $\so
(W)$-representations:
$$
A(k)=\oplus _{i=0}^k t^{k-i}\Sym ^i(W)\,\oplus\,\oplus _{i=0}^k\mu
^{k-i}\Sym ^i(W).
$$
We know that there is a surjective graded algebra homomorphism
$\Sym (W)\to A(k)$. This homomorphism is $\so (W)$-equivariant and hence
contains $W(k+1)$ in its kernel. We therefore have a graded, $\so
(W)$-equivariant, algebra epimorphism $\Sym (W)/I_k\to A(k)$. The
$\so (W)$-decompositions that we found for both source and target show that
this must be an isomorphism.
The last two assertions are clear.
\enddemo
\head
\section Geometric examples of Jordan type I: complex tori
\endhead
\noindent
In this section we describe the total Lie algebra of a complex torus, the
K\"ahler Lie algebra of a complex torus, and the N\'eron--Severi Lie algebra of
an
abelian variety.
\medskip\label We begin with determining the total Lie algebra of a complex
torus.
As observed before this is independent of its complex structure. We first
recall
the construction of the spinor representations.
\smallskip
Let $V$ be a real finite dimensional vector space.
For $\alpha\in V^*$, left exterior product with
$\alpha$ defines a map $e_{\alpha}:\wedge
^{\bullet}V^*\to \wedge ^{\bullet +1}V^*$ of degree one. Dually, contraction
with $a\in V$ defines a map $i _a: \wedge^{\bullet}V^*\to \wedge
^{\bullet -1}V^*$ of degree minus one. Both are derivations of odd degree:
$i_a(\omega\wedge\omega ')= i_a(\omega )\wedge\omega '+(-)^{\deg\omega}\omega
\wedge i_a(\omega ')$ and similarly for $e_{\alpha}$. The anti-commutator
$i_a e_{\alpha}+e_{\alpha}i_a$ is simply multiplication by $\alpha (a)$.
Iterated use of this identity shows that for $a,b\in V$ and $\alpha,\beta\in
V^*$, we have
$$
\align
[i_a i_b,e_{\alpha} e_{\beta}] =&
-\alpha (b) e_{\beta}i_a +\alpha (a) e_{\beta}i_b
+\beta (b) e_{\alpha}i_a - \beta (a) e_{\alpha} i_b\\
&+(-\alpha (a)\beta (b)+\beta (a)\alpha (b))\bold{1}_{\wedge ^{\bullet}V^*}
.\tag{**}
\endalign
$$
We rewrite this as follows: if we denote by $\sigma(a\wedge b,\alpha\wedge\beta
)\in\gl (V^*)$ the transformation
$$
\xi\mapsto
-\alpha (b)\xi (a)\beta +\alpha (a)\xi (b)\beta
+\beta (b)\xi (a)\alpha - \beta (a)\xi (b)\alpha ,
$$
and write $\tilde\sigma(a\wedge b,\alpha\wedge\beta )$ for its extension as a
degree
zero derivation in $\wedge ^{\bullet}V^*$, then the righthand side of
eq.~\refer{**} is simply the sum of $\tilde\sigma(a\wedge b,\alpha\wedge\beta
)$
and the scalar operator that is multiplication by $-{1\over 2}\Tr
(\sigma(a\wedge
b,\alpha\wedge\beta ))$.
With the help of this formula we determine the Lie subalgebra $\g$ of
$\gl (\wedge ^{\bullet} V^*)$ generated by these operators. In fact,
by means of a Clifford construction we shall identify it with the Lie
subalgebra $\so (V\oplus V^*)$ of $\gl (V\oplus V^*)$ of infinitesimal
automorphisms of the quadratic form $q(x,\xi )=\xi (x)$.
The semi-simple element $u:=(-\bold{1}_V,+\bold{1}_{V^*})\in\so (V\oplus V^*)$
defines a grading of the latter with degrees $2$, $0$ and $-2$. We define a Lie
algebra isomorphism
$$
\psi _0: \gl (V ^*)\to \so (V\oplus V^*)_0;\quad
\psi _0(\sigma )(x,\xi )=(-\sigma ^*(x),\sigma (\xi )).
$$
We also have isomorphisms of abelian Lie algebra's
$$
\align
\psi _2: \wedge ^2V^*\to \so (V\oplus V^*)_2;&\quad
\psi _2(\alpha\wedge\beta )(x,\xi)=(0,\alpha (x)\beta -\beta (x)\alpha ),\\
\psi _{-2}: \wedge ^2V\to \so (V\oplus V^*)_{-2};&\quad
\psi _{-2}(a\wedge b)(x,\xi)=(\xi (a)b-\xi (b)a,0).
\endalign
$$
\proclaim{\label Proposition}
The maps $\psi _2(\alpha\wedge\beta )\mapsto e_{\alpha}e_{\beta}$,
$\psi _{-2}(a\wedge b)\mapsto i_ai_b$ extend to a graded Lie algebra
isomorphism of $\so (V\oplus V^*)$ onto $\g$ that maps $\psi _0(\sigma )$ to
$\tilde\sigma -{1\over 2}\Tr (\sigma )\bold{1}_{\wedge ^{\bullet}V^*}$, where
$\tilde\sigma$ denotes the extension of $\sigma$ as a degree zero derivation
of $\wedge ^{\bullet}V^*$.
\endproclaim
\demo{Proof} In view of formula \refer{**} it suffices to show that
$[\psi _{-2}(a\wedge b),\psi _2(\alpha\wedge\beta )]=
\psi _0(\sigma(a\wedge b,\alpha\wedge\beta ))$.
This is straightforward.
\enddemo
The pair $(\so (V\oplus V^*),u)$ is a Jordan--Lefschetz pair (it is a real
form of
case $(D_{2n},A_{2n-1})$).
Now let $X$ be a real torus of even dimension $2n$. We identify the
universal cover of $X$ with $\Hm _1(X;\R )$.
We will write $V$ for this real
vector space (of dimension $2n$) so that $\Hm (X;\R )=\wedge
^{\bullet}V^*$. The rational homology defines a rational structure
$V_{\Q}\subset V$. Let $\kappa\in \wedge ^2V^*$ be nondegenerate. If $\alpha
_{\pm 1},\dots ,\alpha _{\pm n}$ is a basis of $V^*$ such that $\kappa =\sum
_{k=1}^n \alpha _k\wedge\alpha _{-k}$ then $e_{\kappa}=\sum _{k=1}^n e_{\alpha
_k}e_{\alpha _{-k}}$. If $a_{\pm 1},\dots a_{\pm _n}$ is the dual basis, then
we
see from eq.~\refer{**} that
$$
[e_{\kappa},\sum _{k=1}^n i_{a_{-k}}i_{a_k}]=-n+\sum _{k=1}^n
(e_{\alpha _k}i_{a_k}+e_{\alpha _{-k}}i_{a_{-k}}).
$$
Since this element acts on $\wedge
^lV^*$ as multiplication by $-n+l$, it follows that $f_{\kappa}$ is defined
and equal to $\sum _{k=1}^n i_{a_{-k}}i_{a_k}$. The nondegenerate $2$-forms
make up a nonempty open subset of $\wedge ^2V^*$ and therefore span that
space.
The corresponding $2$-vectors form
an open subset of $\wedge ^2V$ and so $\g _{\tot}(X)$ is generated by $\g
_2\oplus\g _{-2}$ as a Lie algebra. Combining this with the above computation
gives:
\proclaim{\label Proposition}
There is a natural identification $(\g _{\tot}(X;\R ),h )\cong (\so (V^*\oplus
V),u)$; this is a real form of the case $(D_{2n},A_{2n-1})$. Furthermore, $\Hm
^{\ev}(X)[n]$ is a semispinorial representation of
$g _{\tot}(X;\R )$ and a fundamental Jordan--Lefschetz module of $\Hm ^2(X,\R
)$.
\endproclaim
\medskip
We now assume that $X$ comes with a complex structure. We shall
determine its K\"ahler Lie algebra. Let $V^*$ resp.\ $\bar V^*$ denote the
$\R$-dual of $V$ equipped with the complex structure
$J^*$ resp.\ $-J^*$. The quadratic form $q$ defined above is invariant under
the
complex structure $(J,-J^*)$ and so extends to a Hermitian form on $V\oplus\bar
V^*$. Let $\su (V\oplus V^*)$ be the Lie algebra of the corresponding special
unitary group. Since $u\in\su (V\oplus \bar V^*)$, it inherits a grading with
degrees $-2$, $0$ and $2$. In fact, $(\su (V\oplus \bar V^*),u)$ is a
Jordan--Lefschetz pair. It is a real form of case $(A_{2n-1},
A_{n-1}+A_{n-1})$.
The $J^*$-invariant elements of $\wedge _{\R}^2V^*$ are
precisely the real $(1,1)$-forms. So they make up the span of the K\"ahler
classes of $X$. Now a straightforward verification shows that $\psi$ maps
the $J$-invariants of $\wedge _{\R}^2V$ resp.~$J^*$-invariants of $\wedge _{\R}
^2V^*$ onto $\su (V\oplus \bar V^*)_{-2}$ resp.~$\su (V\oplus \bar V^*)_2$. We
have:
\proclaim{\label Proposition}
There is a natural identification $(\g _{K}(X;\R ),h )\cong (\su (V\oplus
\bar V^*),u)$; this is a real form of the case $(A_{2n-1},A_{n-1}+A_{n-1})$.
Furthermore, the subspace $\oplus _k\Hm ^{k,k}(X)[n]$ is a fundamental
Jordan--Lefschetz module of $\g _{K}(X)$.
\endproclaim
\demo{Proof}
The first assertion is clear from the preceding discussion.
The choice of a generator of $\wedge _{\C}^nV$
determines an isomorphism $\wedge^{n-k}_{\C}V\cong \wedge ^k_{\C} V^*$.
This yields a graded isomorphism
$$
\wedge ^n_{\C}(V\oplus \bar V^*)=\oplus _k \wedge ^{n-k}_{\C}V\otimes\wedge ^k
_{\C}\bar V^*\cong\oplus _k\wedge ^k _{\C}V^*\otimes\wedge ^k_{\C}\bar V^*[n]
\cong \oplus _k \Hm ^{k,k}(X)[n].
$$
The last assertion follows from this.
\enddemo
\medskip
We next determine the N\'eron--Severi Lie algebra (or rather the Lie algebra
of
its rational points) of an abelian variety $X$. We adhere to the convention to
denote the $\Q$-algebra $\End (X)\otimes\Q$ by $\End ^0(X)$ and
we write $V_{\Q}$ for $\Hm _1(X;\Q )$ and $V$ for $\Hm _1(X; \R )$. We think of
$\End ^0(X)$ as a subalgebra of $\End (V_{\Q })$;
if $J:V\to V$, $J^2=-\bold{1}_V$, is the
complex structure determined by the one of $X$, then $\End ^0(X)$ is
centralizer of $J$ in $\End (V)$ intersected with $\End
(V_{\Q})$. Likewise, $\NS (X)\otimes\Q$ may be identified with the
$J$-invariants in $\wedge ^2V^*$ intersected with $\wedge ^2V_{\Q}^*$.
If $\kappa\in \NS (X)$ is a polarization, then taking adjoints with respect to
this form defines an (anti-)involution ${}^{\dagger}$ in $\End (V)$:
$$
\kappa (\sigma v,w)=\kappa (v, \sigma ^{\dagger}w),
$$
which preserves $\End ^0(X)$; it is called the {\it Rosati involution}
defined by $\kappa $. A well-known fact can be stated as follows:
\proclaim{\label Proposition}
If $\lambda\in\NS (X)$ then the restriction of $[e_{\lambda},f_{\kappa}]\in\gl
(\wedge ^{\bullet} V^*)$ to $V^*$ is in $\End ^0(X)$ (acting on $V^*$
contragradiently) and is invariant under ${}^{\dagger}$; this defines an
isomorphism of $\NS (X)$ onto the ${}^{\dagger}$-invariants in $\End ^0(X)$.
\endproclaim
\demo{Proof} For any bilinear form $\lambda:V\times
V\to\R$ there is a unique $\sigma\in\End (V)$ such that $\lambda (a,b)=\kappa
(\sigma _{\lambda} a, b)$. The condition that $\lambda$ be anti-symmetric is
equivalent to that $\sigma _{\lambda}$ be ${}^{\dagger}$-invariant; the
condition that $\lambda$ be $J$-invariant to that $\sigma _{\lambda}$ be
$J$-equivariant. If $\lambda$ is skew-symmetric and is regarded as an element
of $\wedge ^2V^*$, then eq.~ \refer{**} shows that
$[e_{\lambda},f_{\kappa}]|V^*$ is equal to $\pm\sigma ^*_{\lambda}$ plus a
scalar operator. The proposition follows.
\enddemo
\label Let us write $\End ^0(X)^{\pm }$ for the $\pm 1$-eigen space of
${}^{\dagger}$ in
$\End ^0(X)$. Since ${}^{\dagger}$ is an anti-involution, $\End ^0(X)^-$ is a
Lie subalgebra of $\End ^0(X)$ and $\End ^0(X)^+$ is a module of this Lie
algebra.
The group of units $(\End (X)\otimes\R)^{\times}$ acts on $\NS (X)\otimes\R$
and it is well-known that the polarisations are contained in a single orbit. So
the Rosati involutions are all conjugate under
$(\End (X)\otimes\R)^{\times}$. Let $\uf (X)$ denote the set of
elements in $\End ^0(X)$ that are anti-invariant with respect to all
Rosati involutions. This is clearly a Lie ideal in $\End ^0(X)$.
\proclaim{\label Proposition}
The N\'eron--Severi Lie algebra of the abelian variety $X$ is of Jordan type.
Its degree $2$ summand is canonically isomorphic to $\NS (X)\otimes\Q$. Its
degree $0$ summand can be identified with the Lie ideal of $\End ^0(X)$
that is generated by $\End ^0(X)^+$ and this
isomorphism makes $h$ correspond to a scalar operator in $\End ^0 (X)$.
Moreover, $\End ^0(X) =\g _{NS}(X;\Q )_0\times \uf (X)$.
\endproclaim
\demo{Proof} We only prove the last two assertions, as the others just sum up
the preceding discussion. For a Jordan pair $(\g ,h)$, $[\g _2,\g
_{-2}]$ generates $\g _0$ and so \refer{3.5} implies that $\g _{NS}(X)_0$ is
the Lie algebra generated by the elements invariant under some Rosati
involution. As the Rosati involutions are dense in a single $(\End
(X)\otimes\R)^{\times}$-conjugacy class, a standard argument
shows that $\g _{NS}(X,\R )_0$ must be the Lie ideal in $\End (X)\otimes\R$
generated by $(\End (X)\otimes\R )^+$.
If $\sigma\in \End ^0(X)^-$ and $\tau\in \End ^0(X)^+$, then
$\Tr (\sigma\tau )=\Tr ((\sigma\tau )^{\dagger})=\Tr (-\tau\sigma )=-\Tr
(\sigma\tau )$
(here $\Tr$ denotes the $\Q$-trace). This shows that $\End ^0(X)^-$ is the
the orthoplement of $\End ^0(X)^+$ in $\End ^0(X)$ with respect to
the trace form. So $\uf (X)$ is the orthogonal complement of the span of the
elements fixed by some Rosati involution. This orthoplement is an ideal as
well and hence equal to $\g _{NS}(X)_0$.
\enddemo
\label
In order to convert this into a more explicit statement, we first make a
standard reduction.
The N\'eron--Severi Lie algebra of $X$ only depends on the isogeny type. So by
the
Poincar\'e's complete reducibility theorem we may without loss of generality
assume that the abelian variety is of the form
$$
X=X_1^{m_1}\times\cdots \times X_k^{m_k}
$$
with $X_1,\dots ,X_k$ simple, pairwise non-isogenous abelian varieties. Since
the N\'eron--Severi group of $X$ is just the direct sum of the
N\'eron--Severi groups of its isotypical factors, the same is true for the
N\'eron--Severi Lie algebra:
$$
\g _{NS}(X)=\g _{NS}(X_1^{m_1})\times\cdots \times\g _{NS}(X_k^{m_k}).
$$
We therefore assume that $X$ is a power $A^m$ of a simple abelian variety
$A$.
We first concentrate on $A$.
(For a discussion and the proofs of the properties that we are going to use
we refer to \cite{Lange-Birk}.) Since $A$ is simple, $\End ^0(A)$ is a skew
field over $\Q$. We shall write $F$ for it and denote the center of $F$ by $K$.
Then $\Hm _1(A;\Q )$ is in
a natural way a $K$-vector space. We fix a polarization $\kappa$ so that there
is defined a corresponding Rosati involution ${}^{\dagger}$. This involution is
positive
in the sense that for every nonzero $g\in F$, the action of $g^{\dagger}g$ on
the $\Hm _1(A;\Q)$ has positive trace over $\Q$.
The involution that $^{\dagger}$ induces in $K$ is independent of $^{\dagger}$:
for any embedding of $K$ in the complex field it is given by complex
conjugation. We therefore denote it by $\bar{}$.
The subfield $K_0\subset K$ fixed by $\bar{}$ is totally real; so if
$\bold{e}_0$ stands for the set of embeddings of $K_0$ in $\R$, then
$K_0\otimes _{\Q}\R $ is as a real vector space
canonically isomorphic to $\R ^{\bold{e}_0}$. The cardinality $e_0$ of
$\bold{e}_0$ is the degree of $K_0$ over $\Q$. The isomorphism between $\NS
(A)\otimes\Q$ and the ${}^{\dagger}$-invariants $F^+$ in $F$ gives $\NS
(A)\otimes\Q$ the
structure of a $K_0$-vector space as well. This is also independent of
$\dagger$. There are four cases:
\smallskip
{\it The case of totally real multiplication.} Then $F=K=K_0$, in particular,
the involution is trivial on $F$.
\smallskip
{\it The case of totally indefinite quaternion multiplication.} Here $K=K_0$
and $F$ is a $K_0$-form of $\End (2)$: there is an $\R$-algebra isomorphism
$F\otimes _{\Q}\R\cong\End (2,\R )^{\bold{e}_0}$ such that the involution
corresponds
to the transpose in every summand. In particular, $F^+$ is a $K_0$-form of the
space of binary quadratic forms.
\smallskip
{\it The case of totally definite quaternion multiplication.}
Here also $K=K_0$ and $F$
is a $K_0$-form of the quaternion algebra $\quat$ over $K_0$: there is an
$\R$-algebra isomorphism $F\otimes _{\Q}\R\to \quat ^{\bold{e}_0}$ such that
${}^{\dagger}$ corresponds to quaternion
conjugation in every summand. Since the self-conjugate elements in $\quat$ are
the reals, it follows that $F^+=K_0$.
\smallskip
{\it The case of totally complex multiplication.} The field $K$ has no real
embedding (so is a totally imaginary quadratic extension of $K_0$) and $F$ is
a
$K$-form of $\End (d)$: there is an $\R$-algebra isomorphism $F\otimes
_{\Q}\R\to\End (d,\C )^{\bold{e}_0}$ such that the involution corresponds to
the
conjugate transpose in every summand. So $F^+$ is a $K_0$-form of the space of
Hermitian $d\times d$-matrices.
\medskip According to Albert all these cases occur. Recall that an
isogeny type of a polarized abelian variety is given by rational vector space
$V_{\Q}$, a nondegenerate symplectic form $\kappa$ on $V_{\Q}$ and
a complex structure $J$ on the realification $V$ of $V_{\Q}$ such that
$\kappa$
is $J$-invariant and $\kappa (a,Ja)>0$ for all nonzero $a\in V$.
In order to realize the above cases, we fix a free finitely generated left
$F$-module $W$ and a nondegenerate skew-symmetric Hermitian form $\phi :
W\times
W\to F$ (i.e., $\phi (b,a)=\phi (a,b)^{\dagger}$ and $\phi$ $F$-linear in the
first
variable). Such a $\phi$ can be brought into a standard form: if the
involution
is trivial ($F=K=K_0$), then $\dim _FV$ must be even, say $2r$, and there
exists a basis $(e_{\pm 1},\dots ,e_{\pm r})$ such that $\phi (a,b)=\sum
_{i=1}^r a_ib_{-i}-a_{-i}b_i$; if it is not, then there exists a basis
$(e_1,\dots ,e_r)$ of $W$ and nonzero $u_i\in F$ with $u_i^{\dagger}=-u_i$
($i=1,\dots
,r)$ such that $\phi (a,b)=\sum _{i=1}^r a_iu_ib_i^{\dagger}$. We take for
$V_{\Q}$ the
$\Q$-vector space underlying $W$ and let $\kappa :=\Tr _{F/\Q}\phi
:V_{\Q}\times
V_{\Q}\to\Q$. Then one can find an complex structure $J$ on $V$ which commutes
with $F$, preserves $\kappa$ and is such that $\kappa (a,Ja) >0$ for all
nonzero
$a\in V$. Under some mild restrictions (given in \cite{Shimura}, \S 4, Thm.~5
ff.), one can also arrange that the centralizer of $J$ in $\End (V_{\Q})$ is no
more than $F$; this means that $F$ appears as the endomorphism algebra
tensorized with $\Q$ of any abelian variety associated to $(V_{\Q},J)$.
\medskip
Let us define a $K_0$-Lie subalgebra of $\End (2m,F)$ by:
$$
\sku (2m,F,\dagger )=\{
\pmatrix
A& B\\
C& -{}^t\! A^{\dagger}
\endmatrix\right) | A,B,C\in\End (m,F); B={}^{t}\! B^{\dagger}, C={}^{t}\!
C^{\dagger}\}.
$$
This is the Lie algebra of infinitesimal automorphisms of the skew-hermitian
form
$$
\sum _{k=1}^m(z_kw^{\dagger}_{-k}-z_{-k}w^{\dagger}_k).
$$
It is a reductive $K_0$-Lie algebra whose center is the space of
scalars $\lambda\in K$ with $\lambda ^{\dagger}=-\lambda$. So $\sku
(2m,F,\dagger )$ is
semisimple unless we are in the case of totally complex multiplication. We
grade
this Lie algebra by means of the semisimple element
$$
u_m:=
\pmatrix
-\bold{1}_m& 0\\
0&\bold{1}_m
\endmatrix\right) \in \sku (2m,F,\dagger )
$$
so that $A$, $B$ and $C$ parametrize the summands of degree $0$, $-2$ and $2$
respectively. Let $\g (2m,F,\dagger )$ denote the $K_0$-Lie subalgebra of
$\sku (2m,F,\dagger )$ generated by the summands of degree $2$ and $-2$; let
$\uf (m,F,\dagger )$
denote the union of $\GL (m,F)$-conjugacy classes in $\End (m,F)$ made up of
anti-invariants with respect to the involution $A\mapsto {}^tA^{\dagger}$ and
identify
$\uf (m,F,\dagger )$ with a subspace of $\sku (2m,F,\dagger )_0$ in an obvious
way.
The proof of the following lemma is left to the reader.
\proclaim{\label Lemma}
The pair $(\g (2m,F,\dagger ),u_m)$ is a Jordan pair.
The space $\uf (m,F,\dagger )$ is a Lie ideal in
$\sku (2m,F,\dagger )$ supplementary to $\g (2m,F,\dagger )$. It is
trivial except in the following cases:
\roster
\item $m=1$ and $F$ is totally definite quaternion: then $\g (2,F,\dagger
)\cong
\sli (2,K_0)$ and $\uf (1, F)$ can be identified with the
${}^{\dagger}$-anti-invariants in $F$ or
\item or $K$ is totally complex: then $\g (2m,F,\dagger )$ consists of
the matrices for which $A$ has its $K$-trace in $K_0$, whereas $\uf (m, F)$ can
be identified with the purely imaginary scalars in $K$ (i.e., the $\lambda\in
K$ with $\bar\lambda =-\lambda $).
\endroster
\endproclaim
Notice that in the exceptional cases the connected Lie subgroup of
$\GL(m,F\otimes _{\Q}\R )$ with Lie algebra $\uf (m,F,\dagger )\otimes _{\Q}\R
$ is a
product of $e_0$ copies of $U(1)$ resp.\ $SU(2)$ and hence compact.
\proclaim{\label Theorem}
The graded Lie algebra $\g _{NS}(A^m;\Q )\times\uf (A^m)$ is in a natural way a
product of graded $K_0$-Lie algebra's and as such it is isomorphic
(factor by factor) to $\g (2m,F,\dagger )\times \uf (m,F,\dagger )=\sku
(2m,F,\dagger )$.
\endproclaim
\demo{Proof} We make the identification $\Hm _1(A^m;\Q)=\Hm _1(A;\Q )^m$. This
identifies the algebra $\End (A^m)\otimes\Q$ with $\End (m, F)$. We polarize
each summand by means of $\kappa$. Then the sum of these polarizations is a
polarization of $A^m$ and the correponding Rosati involution in $\End (m, F)$
is given by $\sigma\mapsto {}^{t}\sigma ^{\dagger}$. Hence $\NS (A^m)\otimes\Q$
can be
identified with the space of $^{\dagger}$-hermitian matrices in $\End (m, F)$.
This
identifies $\g _{NS}(A^m;\Q)_2$ resp.~$\g _{NS} (A^m;\Q)_{-2}$ with $\g
(2m,F,\dagger )_2$ resp.~$\g (2m,F,\dagger )_{-2}$. Since the N\'eron--Severi
Lie algebra is
generated by these summands, it follows that this extends to an isomorphism
of $\g _{NS}(A^m;\Q )$ onto $\g (2m,F,\dagger )$. The rest is easy.
\enddemo
We describe the situation in each of the four cases:
\smallskip
{\it Totally real multiplication.} Then $\g _{NS}(A^m;\Q )$ is a
$K_0$-form of $\sy (2m)$ and we have $\g _{NS}(A^m;\Q )_0=\End
(A^m)\otimes\Q\cong\End (m,K_0)$.
This is a $K_0$-form of the case $(C_m,A_{m-1})$.
\smallskip
{\it Totally indefinite quaternion multiplication.} Then $\g
_{NS}(A^m;\Q )$ is a $K_0$-form of $\sy (4m)$ and $\g _{NS}(A^m;\Q )_0=\End
(A^m)\otimes\Q\cong\gl (2m;K_0)$.
This is a $K_0$-form of the case $(C_{2m},A_{2m-1})$.
\smallskip
{\it Totally definite quaternion multiplication.}
The Lie algebra $\g _{NS}(A;\Q )$ is isomorphic to $\sli (2,K_0)$ and so
$\g _{NS}(A;\Q )_0\cong K_0$, whereas $\End (A)$ is a quaternion
$K_0$-algebra. For $m\ge 2$, $\g _{NS}(A ^m;\Q )$ is a $K_0$-form of $\so
(4m)$ and $\g _{NS}(A;\Q )_0=\End (A^m)\otimes\Q\cong \End (2m,K_0)$.
This is a $K_0$-form of the case $(D_{2m},A_{2m-1})$.
\smallskip
{\it Totally complex multiplication.} The Lie algebra $\g _{NS}(A^m;\Q )$
is a $K_0$-form of $\sli (2md)$.
The inclusion $\g _{NS}(A^m;\Q )_0\subset \End (A^m)\otimes\Q$
corresponds to $\sli (md,K)\times K_0\bold{1}_m\subset\gl (md,K)$.
This is a $K_0$-form of the case $(A_{2md-1},A_{md-1}+A_{md-1})$.
\smallskip In all these cases, the subalgebra of $\Hm (A)$ generated by the
N\'eron--Severi group is a Jordan--Lefschetz algebra, or rather a tensor
product
of such: if $A(k)$ denotes the Jordan--Lefschetz $K_0$-algebra of level $k$
associated to $K_0$-Jordan pair $(\g _{NS}(A ^m;\Q ),h)$, then the subalgebra
of
$\Hm (A;\R )$ generated by the N\'eron--Severi group can be identified with
the tensor product of the $\R$-algebra's $\R\otimes _{\sigma}A(k)$, where
$\sigma$ runs over $\bold{e}_0$ and $k$ can be calculated from the equality
$ke_0.\depth A(1)=\dim _{\C}A^m$. In the case of complex multiplication, $k$
will be divisible by $d$.
\medskip\label
It is clear that the Hodge algebra $\Hdg (X)\subset \Hm (X)$ (i.e., the complex
span of the rational part of $\oplus _k\Hm ^{k,k}(X)$) is $\g
_{NS}(X)$-invariant and contains the subalgebra generated by the
N\'eron--Severi
classes as a $\g _{NS}(X)$-submodule. So the Hodge conjecture for $X$ is
basically concerned with the primitive subspace $\Prim (\Hdg (X))$ (in the
sense
of \refer{1.13}) in positive cohomological degree. In this way the
N\'eron--Severi Lie algebra neatly supplements the Mumford--Tate group in
helping us (at least in principle) to understand the Hodge ring: the latter
characterizes $\Hdg (X)$ as the ring of invariants of the Mumford--Tate group,
whereas the decomposition of $\Hdg (X)$ into $\Q$-irreducible representations
of
$\g _{NS}(X)$ tells us among other things which classes do not come from
divisors. To illustrate the point, let us observe that $\uf (X)$ kills the
N\'eron--Severi group, hence kills the subalgebra of $\Hdg (X)$
generated by this group. So as soon as $\uf (X)$ acts nontrivially
on $\Hdg (X)$, $X$ will have Hodge classes that are not in this subalgebra.
This often happens when $\uf (X)\not= 0$ \cite{Moonen-Zah}.
\head
\section Geometric examples of Jordan type II: hyperk\"ahlerian manifolds
\endhead
\noindent\label
In the previous section we saw that complex tori and abelian varieties furnish
examples of the Jordan--Lefschetz algebra's that are of type $(A
_{2m-1},A_{m-1}+A_{m-1})$, $(C_m,A_{m-1})$ and $(D_{2m},A_{2m-1})$. The
classical
Jordan--Lefschetz algebra's that remain are those of type $(B_m,B_{m-1})$ and
$(D_m,D_{m-1})$ and the purpose of this section is provide geometric examples
of
them. One way to get such examples is to take a compact K\"ahler surface $X$:
then
$\Hm ^{\ev}(X;\R )[2]$ is a fundamental Jordan--Lefschetz module of $\Hm
^2(X;\R )$
of the desired type. The corresponding Lie algebra is $\so (\phi )$, where
$\phi$
is the form defined in \refer{1.9} and its degree zero part is $\so (\Hm
^2(X;\R))\times\R h$. A more interesting class of examples was found by M.\
Verbitsky, and this is what we will discuss in what follows.
\medskip
Let $\quat$ be a
quaternion algebra over $\R$. We denote its trace by $\Tr :\quat\to\R$ so that
${1\over 4}\Tr$ is the projection onto the real subfield. Elements of the
kernel of $\Tr$ are called {\it pure quaternions}; they make up a Lie algebra
that we denote by $\quat _0$. The {\it pure part} of $a\in\quat$ is its
projection in $\quat _0$: $a_0:=a-{1\over 4}\Tr (a)$.
We have also defined the norm $\Nm :\quat\to \R$, $\Nm (a)=a\bar a$ (where
$\bar{}$ is the natural anti-involution that is $-1$ on $\quat _0$).
The set $\quat _1$ of elements of norm $1$ is a Lie subgroup of the group of
units $\quat ^{\times}$ of $\quat$ and has $\quat _0$ as its Lie algebra. It is
isomorphic to $SU(2)$. The unit sphere in
$\quat _0$, $\quat _0\cap \quat _1$, is precisely the set of square roots of
$-1$ in $\quat$ and so effectively parametrizes the field homomorphisms $\C\to
\quat$. It is a conjugacy class of $\quat ^{\times}$.
Let $T$ be a left $\quat$-module of finite rank $m\ge 1$, equipped with
positive
definite real inner product $\la\, ,\,\ra :T\times T\to\R$ that is
$\quat$-invariant (this gives rise to $\quat$-Hermitian form). We write $V$ for
its
$\R$-dual $\Hom (T,\R)$ and we let $\quat$ act on the latter on the right.
Every $J\in\quat _0\cap \quat _1$ gives $T$ the structure of a complex vector
space of dimension $2m$. Since the inner product is $\quat$-invariant,
$H_J(x,y):=\la x,y\ra -\sqrt{-1}\la Jx,y\ra$ is a $J$-Hermitian form on $V$.
Its imaginary part is antisymmetric and thus determines a 2-form
$\kappa _J\in\wedge ^2V$. Wedging with $\kappa _J$ defines an operator in
$\wedge V [2m]$ that we denote by $e_J$. It has the Lefschetz property: the
corresponding degree $-2$ operator $f_J$ is characterized by
$f_J=\star e_J\star ^{-1}$. This makes sense for any nonzero element
$a\in\quat _0$: $e_a$ has the Lefschetz property and $f_a=\Nm (a)^{-1}
\star e_a\star ^{-1}$. It is clear that the $f_a$'s commute.
We denote the Lie algebra generated by these elements by $\g (V)$.
This applies in particular to the left $\quat$-module that underlies
$\quat$ itself (with inner product given by the norm). So there is defined
a Lie algebra $\g (\quat )$.
This is actually the universal case, because an orthogonal splitting
of $T$ into $m$ $\quat$-lines allows us to identify $T$ with an orthogonal
direct sum $\quat\oplus\cdots\oplus\quat$. This induces a graded algebra
isomorphism $\wedge ^{\bullet}V\cong (\wedge ^{\bullet}_{\R}
\quat)\otimes\cdots\otimes (\wedge ^{\bullet}_{\R}\quat)$ that is
compatible with the actions of $e_a$ and $h$. The Lie algebra $\g (V)$ now
appears as $\g (\quat )$ acting on the $m$-fold tensor power of its defining
representation. In particular, we find an isomorphism of graded Lie algebra's
$\g (\quat)\cong\g (V)$ that extends the identifications between the actions
of $e_a$, $a\in\quat _0$.
The following lemma gives more information about the nature of this Lie
algebra.
\proclaim{\label Lemma}
\roster
\item"{(i)}" $(\g (\quat ), h)$ is a Jordan--Lefschetz pair with
$\g (\quat )_2$ canonically isomorphic to the vector space underlying
$\quat _0$.
\item"{(ii)}" We have a natural isomorphism $\g (\quat
)_0\cong\quat _0\times\R h$, where $\quat _0$ is regarded as the Lie algebra
of
$\quat _1$. The given action of $\quat _1$ on $\wedge ^{\bullet}V$ integrates
the action of this summand.
\item"{(iii)}" $(\g (\quat ),h)$ is isomorphic to the orthogonal Lie
algebra defined by the form $x_1x_5+x_2^2+x_3^2+x_4^2$ with $h$ corresponding
to $\text{diag}(-1,0,0,0,1)$.
\item"{(iv)}" The subalgebra $M\subset\wedge ^{\bullet}V$ generated by
the
$\kappa _J$'s is invariant under the star operator and $\g (\quat )$, and
$M[2m]$ becomes a Jordan--Lefschetz module of $(\g (\quat ), h)$ of level $m$.
\endroster
\endproclaim
\demo{Proof} In view of the preceding discussion, we may assume that $m=1$.
Besides the established fact that the $e_a$'s and the $f_a$'s commute among
each other, one verifies that
$$
[e _a, f_b]= -(ab^{-1})_0 +{1\over 4}\Tr (ab^{-1})h,
$$
where $h$ defines the grading and $(ab^{-1})_0\in\quat _0$ operates on $\wedge
^{\bullet}V$ on the right via the $\quat$-module structure on $V$. One further
checks that $e_a$ and $f_a$ commute with the action of $\quat$. The assertions
then follow in a straightforward manner, but let us
nevertheless make some remarks that make the verification rather simple and
give
a clearer picture as well. We keep assuming that $m=1$.
The star operator $\star$ in $\wedge ^{\bullet}V$ defines an involution
in $\wedge ^2V$ whose eigen spaces we denote $(\wedge ^2V)^{\pm}$.
There are two interesting symmetric bilinear forms on $\wedge ^{\bullet}V$.
One,
denoted $\phi$, is characterized by
$$
\phi (x,y):=\int x\wedge y \quad\text{ if } \deg (x)\ge 2,
$$
where $\int :\wedge ^{\bullet}V\to\wedge ^4V\cong\R$ is the obvious projection.
It has the
property that the $e_u$'s and $h$ leave this form infinitesimally
invariant, so that $\g\subset\so (\phi )$.
The other form is the natural extension of the inner product: $\la
x,y\ra =\int (x\wedge \star y)$. So on $(\wedge ^2V)^+$ both are equal, whereas
on $(\wedge ^2V)^-$ they are opposite, and these eigen spaces are perpendicular
for both forms. The $\kappa _J$'s make up the unit sphere in the space of
selfdual $2$-forms $(\wedge ^2V)^+$. Since the wedge product of a selfdual form
and an antiselfdual form is zero, we find that each $e_u$
annihilates $(\wedge ^2V)^-$, so that $\g (\quat )$ acts trivially on this
space. Hence $\g (\quat )$ acts on $\wedge ^{\ev}V$ via $M=\R\oplus(\wedge
^2V)^+\oplus\wedge ^4V$. One checks that the $e _J$'s and $f_J$'s
generate all of $\so (M, \phi )$, so that we have a surjective Lie homomorphism
$\g (\quat )\to \so (M, \phi )$ (a quick way to see this is to invoke
\refer{2.1}
and our classification of Jordan--Lefschetz pairs: the former implies that the
image is a Jordan--Lefschetz pair with $3$-dimensional degree two summand and
latter
shows that such a pair must be of type $B_2$). This homomorphism is in fact an
isomorphism. The action on $\wedge ^{\odd} V=V\oplus \star V$ can be
understood as follows:
$V$ is after complexification no longer irreducible as a $\quat$-module: we can
write $V\otimes\C=P\oplus \bar P$ where $P$ is a
$\quat$-invariant complex plane. Such a plane is totally isotropic with
respect to the complexified inner product. The group $\quat _1$ acts on $P$
faithfully with image $SU(P)$. Now $P\oplus\star\bar P$ is
$\g (\quat)\otimes\C$-invariant and comes with a natural symplectic form. This
symplectic form is infinitesimally preserved by the $e_a$'s so that we have a
Lie homomorphism $\g (\quat)\otimes\C\to\sy (P\oplus\star\bar P)$. This is an
isomorphism (defining a spinor representation of $\g (\quat)$).
\enddemo
\label Let $(X,g)$ be a compact connected Riemann manifold of dimension $4m$.
Assume that the holonomy group at $p\in M$, $G_p\subset GL(T_pM)$, is
isomorphic to $U(m,{\quat })\subset GL(4m)$. The group $U(m,{\quat })$ is
the group of quaternionic transformations that preserve a positive definite
sesquilinear inner product; it is a maximal compact subgroup of the complex
symplectic group of genus $2m$ (and is often denoted by $Sp (m)$, but we
avoid that notation since it may lead to confusion with the way we refer to the
symplectic groups). The centralizer of $U(m,{\quat })$ in $\End (4m)$ is the
algebra of quaternions. So the centralizer of $G_p$ in $\End (T_pM)$ is a
quaternion algebra $\quat (p)$ that gives $T_pM$ the structure of a (left)
vector
space over the skew field $\quat (p)$. The subalgebra's $\quat (p)$, $p\in M$,
make up a subbundle of $\End (TM)$ that is flat for the Levi-Civita connection
and has trivial monodromy. This allows us to identify each of them with the
algebra of flat sections of this bundle. Although that subalgebra
depends on $(M,g)$, we will somewhat ambiguously denote it by
$\quat$. So $TM$ is naturally endowed with an action of $\quat$. The metric is
an eigen tensor of this action with character the norm.
Any $J\in\quat _0\cap\quat _1$ (notation is as above) defines an
almost-complex structure on $X$. This almost-complex structure flat with
respect
to the Levi-Civita connection and (hence) integrable. It combines with the
given metric on $X$ to a K\"ahler structure on $X$; we denote the
K\"ahler form by $\kappa _J$. In particular $J$ preserves the harmonic forms;
if we identify $\Hm (X;\R )$ with the space of harmonic forms on $X$, then this
action is just the Weil operator (which on $\Hm ^{p,q}(X,J)$ is multiplication
by $\sqrt{-1}^{-p+q}$).
The assignment $J\mapsto \kappa _J$ extends linearly to $\quat _0$ and this
extension is an isomorphism of $\quat _0$ onto a space of harmonic
$2$-forms, of which the nonzero elements are K\"ahler classes. We denote the
image of this map by $\a$. We think of $\a$ as $3$-plane in $\Hm (X;\R )$ and
call it the {\it characteristic $3$-plane} of the metric. Its nonzero elements
have the Lefschetz property, so that is defined the Lie algebra $\g (\a ,\Hm
(X;\R ))$.
\proclaim{\label Proposition}
\roster
\item"{(i)}"
There is a unique isomorphism $\g (\quat)\cong \g (\a
,\Hm (X;\R ))$ (of graded Lie algebra's) that extends the identifications
between
the actions of $e_a$ $a\in\quat$ and the semisimple element $h$ defining the
grading.
\item"{(ii)}" Under the isomorphism of (i), the action of the Lie group
$\quat _1$ on the space of harmonic forms integrates the action of the
semisimple part $\g (\quat )'_0$ of $\g (\quat )_0$ on $\Hm (X)$. This action
preserves the algebra structure on $\Hm (X)$ so that $\g (\quat )'_0$ acts on
$\Hm (X)$ by derivations.
\item"{(iii)}" The subalgebra
$A_{\a}\subset \Hm (X;\R )$ generated by $\a$ is invariant under the star
operator and $\g (\quat )$ and $A_{\a}[2m]$ is a Jordan--Lefschetz module of
$\g
(\quat )$ of level $m$.
\endroster
\endproclaim
\demo{Proof} Part (i) follows from the observation that for every $x\in X$, the
harmonic forms define a subspace of the exterior algebra of the cotangent
space of $X$ at $x$ that is invariant under both $\star$ and cupping with
alefschetz operator $\kappa _a$ with $a\in\quat _0$ nonzero.
For (ii) we remember that every $J\in \quat _1\cap\quat _0$ acts as a Weil
operator. So $\cos\theta +J\sin\theta $ acts on $\Hm ^{p,q}(X,J)$ as
multiplication by $\exp{(-p+q)\theta}$, hence acts as an algebra automorphism.
This proves that $\quat _1$ acts by algebra automorphism.
As for (iii), note that $A_{\a}$ is additively spanned by the subalgebra's $\R
[a]$, with $a\in\a$ nonzero. As
such a subalgebra is invariant under $\star$, so is $A_{\a}$. Hence $A_{\a}$ is
also invariant under $f_a$. The rest of the assertion is clear.
\enddemo
Most of the preceding statement is due to Verbitsky (he proved a weaker form of
(ii)).
\cite{Beauville} shows that once $X$ admits one Riemann metric with
$U(m,\quat )$ as holonomy group, then it admits many of them. As we have seen
above any such metric defines a characteristic $3$-plane in $\Hm ^2(X;\R)$.
Using
a theorem of S.T.\ Yau, he proves among other things the following: \roster
\item"{(i)}" There is a nonempty open subset of the $3$-plane Grassmannian of
$\Hm ^2(X,\R )$ parametrizing characteristic planes.
\item"{(ii)}" There is a nonzero symmetric bilinear
form $q_0$ on $\Hm ^2(X,\R )$ with the property that for every characteristic
$3$-plane $H$ defined by the metric $g$, its $g$-orthogonal complement
$H^{\perp}$ coincides with its $q_0$-orthogonal complement and $q_0$ and $g$
define
proportional forms on $H$ (with positive ratio) and $H^{\perp}$
(with negative ratio).
\endroster
So $q_0$ is unique up to positive factor and is nondegenerate of signature
$(3,b_2(X)-3)$.
This will imply the following analogue of \refer{4.4}, which is also mostly
due
to \cite{Verbitsky 1995}.
\proclaim{\label Proposition} We then have:
\roster
\item"{(i)}" The pair $(\g _{\tot}(X;\R),h)$ is of Jordan--Lefschetz type of
type $(B,B)$ or $(D,D)$ with $\g _{\tot}(X;\R)$ isomorphic to $\so (4,
b_2(X)-2)$.
\item"{(ii)}" We have natural identifications $\g _{\tot}(X;\R)_2\cong \Hm
^2(X;\R
)$ and $\g _{\tot}(X;\R)_0\cong \so (q_0)\times\R h$.
The semisimple part of $\g _{\tot}(X;\R)_0$, $\g _{\tot}(X;\R)'_0\cong\so
(q_0)$, acts on $\Hm (X;\R )$ by derivations.
\item"{(iii)}" The subalgebra $A$ of $\Hm (X;\R )$ generated by $\Hm ^2(X;\R )$
is
invariant under $\g _{\tot}(X;\R)$ and $A[2m]$ is a Jordan--Lefschetz module of
$\g _{\tot}(X;\R)$ of level $m$.
\endroster
\endproclaim
\demo{Proof} According to the previous proposition, the operators $f_a$ and
$f_b$
commute when $a$ and $b$ are nonzero elements of a characteristic $3$-plane. By
(i) this is therefore the case for $a$ and $b$ in a nonempty open subset of
$\Hm ^2(X;\R )$. Since the expression $[f_a,f_b]$ is rationally dependent on
its
arguments, it follows that $f_a$ and $f_b$ commute whenever both are defined.
Let $\g _2$ resp. $\g _{-2}$ denote the abelian Lie subalgebra's of
$\gl (\Hm (X;\R ))$ spanned by the $e_a$'s resp. $f_a$'s and let
let $\g _0$ be the Lie subalgebra generated by $[\g _2,\g _{-2}]$.
\smallskip
{\it Claim 1.} $\g _0 =\g _0'\times\R h$ (with $\g _0 '=[\g _0 ,\g _0 ]$) and
$\g _0'$ consists of derivations of $\Hm (X;\R )$. Moreover, the image of $\g
_0'$
in $\gl (\Hm ^2(X,\R ))$ is $\so (q_0)$.\par
Proof. Notice that by the above (rationality) argument,
$\g _0$ is already generated by the brackets $[e_a,f _b]$ with $a,b$ nonzero
and contained in a characteristic $3$-plane. If we fix such a $3$-plane $\a$,
then the Lie subalgebra of $\g _0$ generated by the
$[e_a,f_b]$, $a,b$ nonzero elements of $\a$, is $\g (\a ,\Hm (X;\R ))'_0\times
\R h$ with $\g (\a ,\Hm (X;\R ))'_0$ acting as infinitesimal algebra
automorphisms of $\Hm (X)$, that is, as derivations. Moreover
$\g (\a ,\Hm (X;\R ))'_0$ leaves $\q _0$ invariant. So $\g '_0$ acts as
derivations and maps naturally to $\so (q_0)$. This last homomorphism is
surjective, because $\so (q_0)$ is generated by its elements that kill the
$q_0$-orthogonal complement of a characteristic $3$-plane.
\smallskip
{\it Claim 2.} $\ad _{\g _0}$ leaves $\g _2$ and $\g _{-2}$ invariant.\par
Proof. Let $u\in \g '_0$. Since $u$ is a derivation,
we have for every $a\in \Hm ^2(X;\R )$ and $z\in \Hm (X;\R )$, that
$[u,e_a](z)=u(a.z)-a.u(z)= u(a).z= e_{u(a)}(z)$. So $\ad _u$ leaves the space
of operators $e_a$ invariant. If $G_0'\subset GL(\Hm (X;\R ))$ denote the
connected
closed subgroup with Lie algebra $\g '_0$, then for every $g\in G_0'$ we have
$ge_ag^{-1}=e_{g(a)}$ and $ghg^{-1}=h$. Hence if $f_a$ is defined, then
$gf_ag^{-1}=f_{g(a)}$. It follows that $\Ad _{G '_0}$ leaves $\g _{-2}$
invariant. The same assertion then holds for $\ad _{\g '_0}$.
\smallskip
We conclude that $\g :=\g _{-2}+\g _0 +\g _{-2}$ is a Lie subalgebra of
$\gl (\Hm (X;\R ))$ and hence equal to $\g _{\tot}(X;\R)$.
Clearly, $(\g ,h)$ is a Jordan--Lefschetz pair.
\smallskip
{\it Claim 3.} $A$ is an irreducible $\g$-submodule that has $1$ as lowest
weight
vector and $\g$ acts faithfully on $A$. Moreover, $\g _0'$ maps isomorphically
onto $\so (q_0)$.\par
Proof. From $U\g =U\g _2.U\g _0.U\g _{-2}$ and and the fact that $\g _0+\g
_{-2}$ stabilizes the unit element it follows that $A$ is $\g$-invariant and
has
$1$ as lowest weight vector. It is clear that the kernel of the homomorphism
$\g\to\gl (M)$ is contained
in $\g '_0$. This kernel acts trivially on $\Hm ^2(X;\R )$, hence has zero Lie
bracket with any $e_a$ (here we use that $\g _0'$ acts by derivations).
Applying the Jacobson--Morozov theorem to $e_a$ acting on $\g$, we also find
that the kernel has zero Lie bracket with $f_a$, when defined. So the kernel
has
zero Lie bracket with $\g _2$, $\g _{-2}$ and hence also with $\g _0=[\g _2,\g
_{-2}]$. Since $\g$ is semisimple, this implies that the kernel is trivial.
We have already seen that the map $\g _0'\to\so (q_0)$ is surjective. Since
$\g _0'$ acts by derivations, its action on $A$ is completely determined by its
restriction to $\Hm ^2(M,\R )$. Hence it is also injective.
\smallskip
The theorem now follows easily.
\enddemo
\proclaim{\label Corollary}
The algebra structure on $\Hm (X;\Q)$ and the Hodge structure on $\Hm ^2(X)$
determine the Hodge structure on all of $\Hm (X)$.
\endproclaim
\demo{Proof} Let $\tilde G$ be the closed connected $\Q$-subgroup of $GL (\Hm
(X))$ with Lie algebra $\g _{tot}(X)'_0$. Then its image $G$ in $GL(\Hm ^2(X))$
is an orthogonal group of rank $\ge 3$ and the projection $\tilde G\to G$ has
finite kernel. The Hodge structure on $\Hm ^2(X)$ is given by a real
representation of the Deligne torus $\bS$ on $\Hm ^2(X)$. This defines an
$\R$-morphism of algebraic groups $\bS\to G$. Similarly, the Hodge structure on
$\Hm (X)$ is given by an $\R$-morphism of algebraic groups $\bS\to\tilde G$.
The
latter must be a lift of the former. But clearly such a lift is unique.
\enddemo
\label{\it Example.} Let $T$ be a complex torus of complex dimension $2$ and
let $m$ be an positive integer. As Beauville explains, the $(m+1)$-fold
symmetric product $S^{m+1}(T)$ of $T$ admits a natural nonsingular resolution
$T^{[m+1]}\to S^{m+1}(T)$ (the Hilbert scheme paramatrizing finite subschemes
of $T$ of length $m+1$). Composition of this resolution with the ``sum map''
$S^{m+1}(T)\to T$ gives a fibration $T^{[m+1]}\to T$ which is locally trivial
in the \'etale sense. Let $K_m$ be the fiber over the origin. This fiber is
simply connected and if $m\ge 2$, then $\Hm ^2(K_m)$ (with its Hodge structure)
is canonically isomorphic to the direct sum of $\Hm ^2(T)$ and the span of the
class of the exceptional divisor restricted to $K_m$. So in that case, $\dim
\Hm ^2(K_m)=7$. Beauville shows that for a generic choice of $T$, $K_m$ admits
a quaternion K\"ahler metric. Hence \refer{4.5} implies that
$\g _{\tot}(K_m;\R )\cong \so (4,5)$.
The cohomology of $K_2$ is computed in \cite{Salamon}. We can interpret his
result as saying that as a $\g _{\tot}(K_m;\R )$-module, $\Hm (X)$ is the
orthogonal direct sum of the subalgebra generated by $\Hm ^2(K_2)$, a trivial
representation of dimension $80$ and the spinor representation (of dimension
$16$). Their Hodge polynomials are
$1+(s^2+5st+t^2)+(s^4+5s^3t+16s^2t^2+5st^3+t^4)+(s^4t^2+5s^3t^3+s^2t^4)+s^4t^4$,
$80s^2t^2$ and $(s^2t+st^2)+(s^4t+s^3t^2)$ respectively.
\medskip
We have already looked at algebra's such as $A$ in \refer{2.14}. Invoking
\refer{2.14} we find that if $u\in A_4\subset\Hm ^4(X)$ represents the dual of
the
Beauville-Bogomolov form $q_0$, then $a\mapsto\int _Xa^2u ^{m-1}$ is
proportional to $q_0$. There is a natural choice for $u$:
\proclaim{\label Theorem}
Let $p\in \Hm ^4(X;\R )$ be the image of the first Pontryagin class of $X$
under
orthogonal projection of $\Hm (X;\R)$ onto $A$. Then the form $r$ on $\Hm
^2(X;\R )$ defined by $r(a)=\int _X a^2p^{m-1}$ is a nonzero multiple of
Beauville-Bogomolov form.
\endproclaim
We first show:
\proclaim{\label Proposition}
The Lie algebra $\g _{\tot}(X;\R)'_0$ acts trivially on the
subalgebra $P(X)$ generated by the Pontryagin classes.
\endproclaim
\demo{Proof} Let $z$ be an element in this subalgebra of degree $4k$.
Choose a metric $g$ with holonomy group $\cong U(m,\quat )$. This determines an
action of $\quat$ on $\Hm (X;\R )$ and a characteristic $3$-plane $\a\subset
\Hm ^2(X;\R )$. If $J\in\quat _0\cap\quat _1$, then with respect to the complex
structure defined by $J$, $P^{4k}(X)$ consists of classes of bidegree
$(2k,2k)$.
Equivalently: the Weil operator $J$ leaves $P(X)$ invariant. As this is true
for all $J\in \quat _0\cap\quat _1$, it follows that $P(X)$ is invariant under
all of $\quat _1$. Hence $P(X)$ is killed by the Lie algebra $\quat _0$. Since
these Lie algebra's generate the semisimple part of
$\g _{\tot}(X;\R )_0$, the proposition follows.
\enddemo
\demo{Proof of \refer{4.8}}
Denote by $\pi :\Hm (X;\R )\to M$ the $\phi$-orthogonal projection onto $M$. So
if
$a\in \Hm ^2(X;\R )$, then $\pi (a)\in M$ is characterized by the property
that
for all $b\in \Hm (X;\R )$ we have $\int _X ab=\int _X \pi (a)b$. This is a
$\g
_{\tot}(A;\R )$-equivariant projection. Since $p_1(X)$ is $\g _{\tot}(M;\R
)$-invariant, $p=\pi (p_1(X))$ must be a multiple of $u$ and so all we need to
see
is that $p\not=0$. But this follows from a theorem of \cite{Chen-Ogiue} which
states that for some K\"ahler class $\kappa $, $\int _X \kappa ^{2m-2}p_1(X)$
is
positive.
\enddemo
\medskip
\label We have now seen that all the classical Jordan--Lefschetz algebra's
arise
geometrically. The question comes up whether the same is true for the
exceptional
case of type $E_7$ (that corresponds to a Jordan algebra of dimension $27$).
In
the topological setting the answer is yes: if the $27$-dimensional vector space
$W$
in \refer{2.11} is equipped with an integral structure $W_{\Z}$ such that the
cubic
form only takes integral even values (this is indeed possible), then it follows
from theorems of Wall and Jupp that there is a simply connected closed oriented
$6$-manifold $X$ for which the integral cohomology ring is isomorphic to the
corresponding integral algebra $A_{\Z}$ (see \cite{Okon-vdVen} for a general
discussion). Now $H^{\ev}(X;\Z )$, does not change if take a connected sum of
$X$
with a number of copies of $S^3\times S^3$. We wonder whether such a manifold
admits a complex structure. Since all our interesting examples have trivial
canonical bundle we are inclined to make this question more specific
by asking:
\smallskip{\it Question.} Does there exist a Calabi-Yau $3$-fold with Picard
group of rank $27$ such that its N\'eron--Severi Lie algebra is of type $E_7$?
\smallskip We notice that the mirror dual family of such a Calabi-Yau manifold
will, if it exists, have its period mapping take values in a Hermitian domain
of type $E_7$.
\head
\section Filtered Lefschetz modules
\endhead
\noindent\label We recall a version of the
Jacobson--Morozov lemma. If $e$ is a nilpotent transformation in a vector space
$M$, then there is a unique
nonincreasing filtration $W^{\bullet}$ preserved by $e$ such that $e$ has
the Lefschetz property in $\Gr _W^{\bullet}(M)$. Any $\slt$-triple $(e,h,f)$
containing $e$ descends to an $\slt$-triple in $\Gr _W(M)$ and splits the
filtration (so the $k$-eigen space of $h$ is a supplement of $W^{k+1}$ in
$W^k$). We shall refer to $W^{\bullet}$ as the {\it Lefschetz filtration} of
$e$.
\proclaim{\label Lemma}
Let $\g$ be a reductive Lie algebra, $\s\subset\g$ a commutative subalgebra
consisting of semisimple elements and $\chi\in\s ^*$ a character of
$\s$ in $\g$. Then for every nilpotent $e\in\g ^{\chi}$ there exists a $f\in\g
^{-\chi}$ such that $(e, [e,f],f)$ is an $\sli (2)$-triple.
\endproclaim
\demo{Proof}
Choose an $\slt$-triple $(e,h',f')$ containing $e$. Let $f''$ be the
$\g ^{-\chi}$-component of $f'$. Then $h:=[e,f'']$ is the $\g ^0$-component of
$h'$ and so $[h,e]$ is the $\g ^{\chi}$-component of $[h',e]=2e$ and hence
equal
to $2e$. Since $h$ is in the image of $\ad (e)$, the pair $(e,h)$ is
by \cite{Bourbaki}, Ch.~VIII, \S 11, Lemme $6$, extendable to a $\slt$-triple
$(e,h,f)$. This remains an $\slt$-triple if we replace $f$ by its $\g
^{-\chi}$-component and so the lemma follows.
\enddemo
Let $(\a ,M)$ be a Lefschetz module and $\hor ^{\bullet}M$ a nonincreasing
filtration on the graded vector space underlying $M$ (so $\hor ^kM=\oplus
_l\hor ^kM_l$) which is preserved by $\a$. We shall refer to this filtration
as the {\it horizontal filtration}. Then the associated {\it
vertical filtration} is defined by $\ver _kM:=\sum _r \hor ^{r-k}M_r$. This
filtration is nondecreasing; we call the corresponding grading of $\Gr
_{\ver }M$ the {\it vertical grading}. The notations $\Gr _{\hor}M$ and $\Gr
^{\ver}M$ refer to the same vector space but with different gradings:
$\Gr _{\hor}^kM_r=\Gr ^{\ver}_{r-k}M_r$ has horizontal degree $k$ and vertical
degree $r-k$. The notation $\Gr M$ refers to their common bigraded structure.
Suppose now that some $a\in\a$ preserves the vertical
grading (i.e., $e_a (\hor ^kM)\subset \hor ^{k+2}M$ for all
$k$) and has the Lefschetz property in $\Gr _{\hor}M$:
$e_a ^k$ sends $\Gr _{\hor}^{-k}M$ isomorphically onto $\Gr _{\hor}^kM$.
It is then immediate that
the horizontal filtration is the Lefschetz filtration of the transformation
$e_a$ in $M$. If we apply \refer{5.2} to $e:=e_a$ and $\s:=\C h$, we find an
$\slt$-triple $(e_a,h_{\hor},f_a)$ in $\g (\a ,M)$ with $f_a$ of total degree
$-2$ and $h_{\hor}$ of total degree $0$. So $f_a$ will map $\hor ^kM_r$ to
$\hor
^{k-2}M_{r-2}$. This shows that $f_a$ preserves the vertical filtration. It
also
follows that $h_{\hor}=[e_a,f_a]$ has this property. It is clear that the eigen
spaces of $h_{\hor}$ split the horizontal filtration. This element commutes
with $h$, so if we put $h_{\ver}:=h-h_{\hor}$, then the eigen spaces of the
commuting pair $(h_{\hor},h_{\ver})$ define a bigrading of $M$ that identifies
$M$ with $\Gr M$. The eigen spaces of $(h_{\hor},h_{\ver})$ under the adjoint
representation also define a bigrading of $\g (\a ,M)$. The image of $\a$ in
$\g
(\a ,M)$ need not be bigraded.
\proclaim{\label Proposition} Let $\a _{\hor}\subset\a$ be the set of $a\in\a $
that preserve the vertical grading and suppose that $\Gr _{\hor}M$ is a
Lefschetz module of $\a _{\hor}$. Then we can write $h=h_{\hor}+h_{\ver}$
with $h_{\hor}$ and $h_{\ver}$ semisimple elements of $\g (\a, M)$ that have
integral eigen values and commute with each other (so for the resulting
bigrading of $M$, $M_{k,l}$ gets identified with $\Gr _{\hor}^kM_{k+l}$).
The span of the components of $\a _{\hor}$ of lowest horizontal degree $2$
make up an abelian subalgebra $\a _{2,0}$ of $\g (\a ,M)_{2,0}$ that has the
Lefschetz property in $M$ with respect to the horizontal grading. Moreover,
$\g (\a _{2,0},M_{\hor})$ is a subalgebra of $\g (\a ,M)_{\bullet ,0}$ that
maps isomorphically onto $\g (\a _{\hor},\Gr _{\hor}M)$.
If in addition, $\Gr ^{\ver}M$ is a Lefschetz module of $\a$, then
the span of the components of $\a$ of highest vertical degree $2$ make up an
abelian subalgebra $\a _{0,2}$ of $\g (\a ,M)_{0,2}$ that has the Lefschetz
property in $M$ with respect to the vertical grading. Moreover, $\g (\a
_{0,2},M_{\ver})$ is a subalgebra of $\g (\a ,M)_{0,\bullet}$ that maps
isomorphically onto $\g (\a ,\Gr ^{\ver}M)$. The obvious map $\g (\a
_{2,0},M_{\hor})\times \g (\a _{0,2},M_{\ver})\to\g (\a ,M)$ is then an
injective homomorphism of Lie algebra's.
\endproclaim
\demo{Proof}
Everything follows from the preceding or is obvious except the very last
statement.
We claim that any ``horizontal'' $\slt$-triple $(e',h',e')$ acting in $\Gr M$
commutes
with any ``vertical'' $\slt$-triple $(e'',h'',e'')$ acting in $\Gr M$. This
just
follows from the fact that $(e',h')$ commutes with $(e'',h'')$ and the fact
that
in either case the last member is a rational expression in the first two. It
follows that $\g (\a _{\hor},\Gr _{\hor}M)$ and $\g (\a ,\Gr ^{\ver}M)$
commute.
The same is therefore true for their bigraded lifts $\g (\a _{2,0},M)$ and
$g (\a _{0,2},M)$ in $\g (\a ,M)$. To see that $\g (\a _{2,0},M)\cap \g (\a
_{0,2},M)=0$, note that this intersection has bidegree $(0,0)$ and is normal
in either of them. If it were nonzero, then it would contain a simple factor of
$\g (\a _{2,0},M)$ of bidegree $(0,0)$. But this is impossible since $\g (\a
_{2,0},M)$ is (as a Lie algebra) generated by its degree $\pm 2$ summands. The
proposition follows.
\enddemo
Let $f:X\to Y$ be a fibration of projective manifolds which is topologically
locally trivial and let $n$ and $m$ be the complex dimensions of $X$ and $Y$
repectively, so that $d:=n-m$ is the complex fiber dimension. Following
\cite{Deligne 1968} the Leray spectral sequence of $f$ degenerates: if
$L^
{\bullet}$ denotes the Leray filtration of $\Hm ^{\bullet}(X)$, then $\Gr
^k_L\Hm ^r(X)\cong \Hm ^k(Y,R^{r-k}f_*\C )$.
\proclaim{\label Proposition}
If $h$ denotes the basic semisimple element of $\g _{NS}(X)$, then we can write
$h=h_{\hor}+h_{\ver}$ with $h_{\hor}$ and $h_{\ver}$
semisimple elements of $\g _{NS}(X)$ that have integral eigen values and
commute
with each other so that for the resulting bigradings of $\Hm (X)$ and $\g
_{NS}(X)$
have the following properties: $\Hm (X)_{k,l}$ gets identified with
$\Hm ^{k+m}(Y,R^{l+d}f_*\C
)$ and
$$
\g (\NS (Y),\Hm ^{\bullet}(Y,Rf_*\C )[m])\times\g (\NS
(X/Y),\Hm (Y,R^{\bullet}f_*\C [d]))
$$
lifts (uniquely) to a
bigraded Lie subalgebra of $\g _{NS}(X)$.
\endproclaim
\demo{Proof} We prove the proposition by verifying
the hypotheses of the previous proposition for $M:=\Hm ^{\bullet}(X)[n]$ with
as
horizontal filtration the Leray filtration appropriately shifted: $\hor
^kM_r:=L^{d+k}\Hm ^{n+r}(M)$.
First recall that $R^lf_*\C$ underlies a variation of Hodge structure of weight
$l$. If $\xi \in \NS (X)$ is
ample relative $f$, then its image in $\Hm ^0(Y,R^2f_*\C)$ has the
Lefschetz property in the graded local system $R^{\bullet}f_*\C$ and induces
a polarization in each summand. This implies that $\xi$ has the
Lefschetz property in $\Hm ^k(Y,R^{\bullet}f_*\C [d])$ and satisfies the
hypotheses of \refer{1.6}.
So $\Hm ^k(Y,R^{\bullet}f_*\C )[d]$ is a Lefschetz module over $\NS (X/Y)$.
If $\eta\in\NS (Y)$ is a polarization, then cupping with $\eta ^k$ defines an
isomorphism $\Hm ^{m-k}(Y,R^lf_*\C )\to \Hm ^{m+k}(Y,R^lf_*\C )$ \cite{Zucker},
\cite{Saito} and $\eta\cup$ has the Lefschetz property in $\Hm
^{\bullet}(Y,R^lf_*\C )[m]$. We apply \refer{1.6} again and find that all the
hypotheses of \refer{5.3} are fulfilled. \enddemo
{\it Remarks.} This splitting of the Leray filtration is certainly a splitting
that is invariant under the action of $\NS (Y)$. We do not know however whether
it can be chosen to be a splitting of $\Hm (Y)$-modules.
If the graded local system $R^{\bullet}f_*\C$ is trivial (which is the case
when $Y$ is simply connected), then by the universal coefficient theorem, $\Hm
^k(Y,R^{\bullet}f_*\C )\cong \Hm ^k(Y)\otimes \Hm ^{\bullet}(X_y)$, where $X_y$
is a fiber. If we take $y$ sufficiently general, then $\NS (X/Y)$ restricts
isomorphically to $\NS (X_y)$ and then $\g (\NS (X/Y),\Hm (Y,R^{\bullet}f_*\C
[d]))$ can be identified (via the preceding isomorphism) with $\g
_{NS}(X_y)$.
\medskip\label
The conditions $X$ and $Y$ nonsingular and $f$ topologically
locally trivial can all be eliminated in the context
of Hodge modules: if $f:X\to Y$ is a morphism of projective
varieties, and $E$ is a polarized Hodge module on $X$ of pure weight, then
according to \cite{Saito} the Leray spectral sequence for $f_*E$ degenerates
and the Leray filtration satisfies all the hypotheses of \refer{5.3} (here no
shifting is necessary). If the map from a space $Z$ to a fixed singleton is
denoted $a_Z$, then we find that $\g (\NS (X),a_{X *}(E))$ contains graded Lie
subalgebra's isomorphic to $\g (\NS (Y),a_{Y*}\Hm ^{\bullet}(f_*E))$ and $\g
(\NS
(X/Y),a_{Y*}\Hm ^{\bullet}(f_*E))$ that centralize each other. (Here the
cohomology
is taken in the sense of Hodge modules; for a representing complex of
constructible sheaves, this amounts to taking perverse cohomology.)
\proclaim{\label Proposition}
Let $f:X\to Y$ be a (topologically locally trivial) fibration of projective
manifolds with fiber $\PP ^d$. Then $\g _{NS}(X)$ contains a Lie subalgebra
isomorphic to $\g _{NS}(Y)\times\slt$. If this subalgebra is equal to
$\g _{NS}(X)$, then the characteristic classes of this bundle are trivial,
i.e.,
$\Hm (X)=\Hm (Y)\otimes \Hm (\PP ^n)$ as $\Hm (Y)$-algebra's.
\endproclaim
\demo{Proof} As an algebra, $\Hm (X)$ is a simple integral extension
of $\Hm (Y)$: $\Hm (X)=\Hm (Y)[\xi ]/(P)$ with $P$ a monic polynomial of degre
$d+1$:
$P=\xi ^{d+1} +c_1\xi ^d+\cdots +c_{d+1}$. We make $P$
unique by requiring that $c_1=0$; then $c_2,\dots ,c_{d+1}$ are the
characteristic classes of our bundle. Here $\xi $ is any element of $\NS
(X)\otimes\Q $ that spans a supplement of $\NS (Y)\otimes\Q$ in $\NS
(X)\otimes\Q $. Take a bigrading on $\Hm (X)$ as in the previous proposition so
that we get an embedding of $\g _{NS}(Y)\times\slt$ in
$\g _{NS}(X)$. If it is surjective, then take for $\xi $ a nonzero element
of $\NS (X)$ that projects in $\Hm ^2(X)_{(-\dim Y,-d+2)}$. Cupping with $\xi $
then corresponds to a nonzero element of the $\slt$ factor. So we have $\xi
^{d+1}=0$. Hence all the $c_i$'s are zero. \enddemo
\proclaim{\label Theorem}
Let $f:X\to Y$ be a $\PP ^d$-bundle involving projective manifolds. Assume that
the N\'eron--Severi Lie algebra of $Y$ is maximal: $\g _{NS}(Y)=\aut (\Hm
(Y))$,
and
that $\Hm (Y)$ is not an inner product space of dimension $4$. Then either the
characteristic classes of $f$ are trivial and $\g _{NS}(X)\cong \g
_{NS}(Y)\times\slt$ or the N\'eron--Severi Lie algebra of $X$ is maximal.
\endproclaim
\demo{Proof} Suppose not all the characteristic classes are
trivial and let $\xi \in\NS (X)\otimes\Q$ be such that
$\Hm (X)=\Hm (Y)[\xi ]/(P)$ with $P=\xi ^{d+1} +c_2\xi ^{d-1}+\cdots +c_{d+1}$
as in the proof of \refer{5.6}. Then
$\g _{NS}(X)$ contains a copy of $\g _{NS}(Y)\times\slt$, but $\xi $,
viewed as an element of $\g _{NS}(Y)$, commutes with neither factor. So the
simple component of $\g _{NS}(X)$ that contains $\xi $ contains $\g
_{NS}(Y)\times\slt$ as well. It remains to apply the theorem of the appendix.
\enddemo
\proclaim{\label Theorem}
Let $X$ be the flag space of a simple complex algebraic group. Then its
N\'eron--Severi Lie algebra is equal to $\aut (\Hm (X))$.
\endproclaim
\demo{Proof} In case the group is of rank one, then $X=\PP ^1$ and then the
assertion is clear. So assume that the rank is $\ge 2$. Then $X$ admits an
iterated fibered structure
$$
X=X_s\to X_{s-1}\to\cdots \to X_0
$$
with $X_0$ a singleton, $X_t\to X_{t-1}$ a projective space bundle with
positive fiber dimension, and $s\ge 2$. Iterated application of \refer{5.6}
yields an embedding of $(\slt )^s$ in $\g _{NS}(X)$. In view of the theorem of
the appendix it is therefore enough to show that $\g _{NS}(X)$ is simple. Were
that not the case then we could write nontrivially $\g _{NS}(X)=\s _1\times \s
_2$ with $\s _i$ nonzero semisimple. In that case, let $H_i$ be the $\s
_i$-submodule of $\Hm (X)$ generated by the unit element. As $\Hm (X)$ is
generated
by $\NS (X)$ it follows that $\Hm (X)=H_1\otimes H_2$ as algebra's. However,
such
a decomposition is precluded by Borel's description of $\Hm (X)$. According to
this theory, there is up to scalar a unique quadratic form on $\NS (X)\otimes\C
$
that becomes a relation for $\Hm ^4(X)$. This form is nondegenerate, and so
$\Hm (X)$ cannot be the tensor product of two graded subalgebra's (see also
\refer{1.2}).
\enddemo
\label The Borel description of $\Hm (X)$ actually shows that as an algebra,
$\Hm (X)$ is isomorphic to $\Sym (V)/I$, where $V$ is the complexified weight
lattice of $G$ and $I$ is the ideal generated by the Weyl group invariant
homogeneous forms
of positive degree. In other words, we are in the case of the example
discussed
in \refer{1.10} and the above theorem gives a complete description of $\g
(V,\Sym (V)/I)$ in case $W$ is an irreducible Weyl group.
Another interesting class of projective manifolds with the property that their
rational cohohomology is generated by the N\'eron--Severi group are the
Knudsen--Mumford moduli spaces $\Cal{M}_0^n$ of stable $n$-pointed curves of
genus zero. It it likely that here also the N\'eron--Severi Lie algebra of
$\Cal{M}_0^n$ equals $\aut (\Hm (\Cal{M}_0^n))$ (compare theorem \refer{6.8}
below).
\head
\section Frobenius--Lefschetz modules
\endhead
\noindent\label
We say that a Lefschetz module $(M,\a )$ of depth $n$ is {\it Frobenius} if it
satisfies the following three properties:
\roster
\item"{(1)}" $\Prim M_{-n}=1$ is of dimension one (and so $M$ is
irreducible),
\item"{(2)}" the map $\a\otimes M_{-n}\to M_{-n+2}$ is an
isomorphism,
\item"{(3)}" $M$ is generated as a $U\a$-module by $M_{-n}$.
\endroster
If only the first two conditions are satisfied we say that $M$ is a {\it
quasi-Frobenius} of depth $n$ and if instead of (3), we have
\roster
\item"{(3$'$)}" the $U\a$-module generated by $M_{-n}$ contains $M_{-n+2k}$ for
$k\le d$,
\endroster
then we say that $M$ is {\it Frobenius up to order $d$}.
\smallskip
Observe that if $A$ is Lefschetz algebra of depth $n$, then $A[n]$ is a
Frobenius--Lefschetz module of $A_2$ if and only if $A$ is generated by $A_2$.
Moreover, any Frobenius--Lefschetz module is of this form.
We also note that a Jordan--Lefschetz module is Frobenius. The property
of being quasi-Frobenius is a useful one, as it turns out to be a rather
strong approximation to being Frobenius, that (somewhat in contrast to the
latter) is generally easy to verify in practice. Another reason of our interest
in this notion is that it occurs naturally in geometric examples:
\proclaim{\label Proposition} Let $X$ be a connected compact K\"ahler
(resp. complex projective) manifold. Then the $\g _K(X)$-submodule
(resp. $\g _{NS}(X)$-submodule) of $\Hm (X)$ generated by $\Hm ^0(X)$ is
quasi-Frobenius
as a Lefschetz module of $\Hm ^{1,1}(X)$ (resp. $\NS (X)$).
\endproclaim
\demo{Proof}
In the K\"ahler case, it is clear that this submodule is contained in $\oplus
_k\Hm ^{k,k}(X)$. So its intersection with $\Hm ^2(X)$ is $\Hm ^{1,1}(X)$ and
the
proposition follows. The N\'eron--Severi case is proved similarly.
\enddemo
The following proposition explains our terminology for it shows that a
Lefschetz module $(M,\a )$ is Frobenius if and only if $M$ is
Frobenius ($=$ Gorenstein) as a $U\a$-module.
\proclaim{\label Proposition}
Let $(\a ,M)$ be a Frobenius--Lefschetz module of depth $n$.\break
Then there exists
a nondegenerate $\g (\a ,M)$-invariant $(-)^n$-symmetric bilinear form on $M$.
\endproclaim
\demo{Proof} Since $\dim M_{-n}=1$, we also have $\dim M_n=1$.
The choice of a generator $u\in M_{-n}$ identifies $M$ with a
graded quotient $R$ of the symmetric algebra $U\a$ (with a shift of degree).
Pick a nonzero linear form $\int :R_{2n}\to\C$, and define a graded bilinear
form
$\langle\, ,\,\rangle :M\times M\to\C$ by $\langle au, bu\rangle =(-1)^k\int
(abu)$ if $a\in R^{2k}$ $b\in R^{2n-2k}$. This form
is symmetric or skew according to whether $n$ is even or odd.
We claim that it is $\g (\a, M)$-invariant. For if we regard
$\langle\, ,\,\rangle$ as
an element of the Lefschetz module $(M\otimes M)^*$, then it is of degree
zero and annihilated by $\a$. So if $f_a$ ($a\in \a$) is
defined, then $\langle\, ,\,\rangle$ is primitive for the $\slt$-triple
$(e_a,h,f_a)$ and hence annihilated by $f_a$. This proves our claim.
Since $M$ is irreducible, $\langle\, ,\,\rangle$ must be nondegenerate.
\enddemo
\medskip
\label {\it Example.} The following example is not just instructive, it also
will be used in a proof (of theorem \refer{6.8}).
We give $V(2l)=K[e]/(e^{2l+1})[2l]$ the unique $\slt$-invariant quadratic form
for which the inner product of $1$ and $e^{2l}$ is $(-1)^l$, so the inner
product of $e^p$ and $e^q$ is $(-1)^{l+p}$ if $q=l-p\in\{ 0,\dots ,l\}$ and
zero otherwise. Let $V:=V(2k)\oplus V(2k-2)$, $k\ge 2$, regarded as the
orthogonal direct sum of $K[e]$-modules. Any endomorphism of $V$ that
commutes with $e$ is $K[e]$-linear and so representable by a $2\times 2$
matrix with coefficients in $K[e]$. We readily compute that the intersection
of the centralizer of $e$ with $\so (V)_2$ is the set of matrices of the form
$$
\pmatrix
ae & be^2\\
b & ce\\
\endpmatrix
$$
with $a,b,c$ scalars. These do not mutually commute, so any
subspace $\a\subset\so (V)_2$ that defines Lefschetz module structure on $V$
and contains $e$ is of $\dim \le 2$. Hence $\a$ will be spanned
by $e$ and an element $e'$ that can be represented
by a matrix of the above type with $c=-a$. If $a\not=0=b$, then it is easy to
see $\g (\a ,V)\cong\sli (2)\times\sli (2)$. Suppose therefore that $b\not=0$.
We normalize $e'$ to make $b=1$.
Then $e$ and $e'$ satisfy: $(e')^2= (a^2+1)e^2$, $e^{2k+1}=0$ and
$e^{2k-1}(e'-ae)=0$. This suggests to take as a new basis for $\a$
$a_{\pm}:=e'\pm\sqrt{a^2+1}e$. Then the relations become:
$a_+a_-=a_+^{2k+1}=a_-^{2k+1}=0$ plus a relation of degree $2k$. It is then
clear that as a $\C [a_-,a_+]$-module, $V$ is generated by $V_{-2k}$.
(One can now show that $\g (\a ,V)=\so (V)$, but we will not need this in what
follows.)
Now assume that $k\ge 3$ and consider the semispinorial representation $W$ of
$\so (V)$. Let us first recall how $W$ is obtained. The inner product is
nondegenerate on the plane $V_0$. Let $F_0$ and $F_0'$ be complementary
isotropic lines in $V_0$ and put $F:=F_0 +\sum _{i>0} V_{2i}$ and
$F':=F_0'+\sum
_{i<0} V_{2i}$. These are complementary isotropic subspaces of $V$. One of the
semispinorial representations of $\so (V)$ can be realized on $W:=\wedge^{\ev}
F[n]$, with $n=k^2$. Now $W$ has as its lowest degree summand a line in degree
$-n$ spanned by $1\in \wedge^0 F$. The action of $\a$ on $\wedge^{\ev} F$ is as
follows: if $f\in F_0$ and $f'\in F_0'$ are such that their inner product is
$1$, then we have $$ a\cdot (x_1\wedge\cdots \wedge x_{2l})=
-f\wedge a(f')\wedge x_1\wedge\cdots \wedge x_{2l}+
\sum _{i=1}^{2l} x_1\wedge\cdots\wedge a(x_i)\wedge\cdots \wedge x_{2l},
$$
where $a\in\a$ and $x_1,\dots ,x_{2l}\in F$.
A calculation shows that $a_{\pm}(f')=\pm ca_{\pm}(f)$, with $c$ a nonzero
constant. This enables us to determine the $\C [a_-,a_+]$-submodule of $W$
generated by $1\in W_{-n}$: we find that in degree
$\le -n+6$ it is the span of $f,\, f\wedge a_{\pm}(f),\, f\wedge
a_{\pm}^2(f),\, a_+(f)\wedge a_-(f),\, f\wedge a_{\pm}^3(f)+a_{\pm}(f)\wedge
a_{\pm}^2(f),\, a_{\pm}(f)\wedge a_{\mp}^2(f)$. This contains the summands of
degree $\le -n+4$ of $W$, but not $W_{-n+6}$: since $n=k^2\ge 9$, $f\wedge
a_{\pm}^3(f)$ is not in this span.
So $W$ is Frobenius up to order $2$, but not up to order $3$.
\bigskip\label
Let $(\g ,h,\a )$ be a Lefschetz triple, $\h$ a Cartan subalgebra of $\g$
containing $h$ and $B$ a root basis of $R=R(\g ,\h )$ whose members are $\ge
0$ on $h$. Let $M$ be an irreducible representation of $\g$ with
$-\lambda\in\h ^*$ as lowest weight (with respect to $B$). We give a simple
criterion for $M$ to be quasi-Frobenius--Lefschetz module.
The lowest weight subspace of $M$, $M^{-\lambda}$, is a line. The
$\g$-stabilizer of this line is the standard parabolic subalgebra $\p =\p _X$
of
$\g$, where $X$ is the set of $\alpha\in B$ with $\lambda (\alpha ^{\vee})=0$.
In other words, $\p $ is the direct sum of $\h$ and the root spaces of
roots $\beta$ with $\lambda (\beta ^{\vee})\le 0$.
We endow $M$ with the grading defined by $h$ and denote the grade by a
subscript. It is clear that the depth $n$ of $M$ is
equal to $\lambda (h)$. So $M^{-\lambda}\subset M_{-n}$. The map that assigns
to
$x\in\g$ the homomorphism $x|M^{-\lambda}\in\Hom (M^{-\lambda},M)$
induces an injective map $\g /\p\to \Hom (M^{-\lambda},M/M^{-\lambda})$. In
particular, if $\p _2$ denotes the degree two part of $\p$:
$$
\p _2:=\sum _{\alpha\in R_2, \lambda (\alpha ^{\vee})=0} \g ^{\alpha},
$$
then we have an injective map $\g _2/\p _2\to \Hom (M^{-\lambda},M_{-n+2})$.
We conclude:
\proclaim{\label Lemma}
Suppose $\dim M_n=1$. Then $M$ is quasi-Frobenius if and only if $\a$
is a supplement of $\p _2$ in $\g _2$.
\endproclaim
\label If $\g$ is a Lie algebra, then call a representation $M$ of $\g$ {\it
tautological} if $\g$ maps isomorphically onto the Lie algebra of
infinitesimal isometries of a nondegenerate ($\pm$)-symmetric form on $M$.
So $\g$ is then a symplectic or orthogonal Lie algebra and
if $\g\not\cong \so (4)$, then $M$ is a fundamental representation of $\g$
with highest weight at an end of the Dynkin diagram.
We have not found an example of a simple Frobenius--Lefschetz module that is
not of Jordan--Lefschetz type or tautological. The following theorem says that
such an example must involve an exceptional Lie algebra.
\medskip
\proclaim{\label Theorem}
A simple Frobenius--Lefschetz module that is not a Jordan-Lef-schetz module and
whose Lie algebra is simple and of classical type, is tautological.
\endproclaim
\label
Before we begin the proof, we derive some general properties of
quasi-Frobenius--Lefschetz modules. So from now on, $M$ is a
quasi-Frobenius--Lefschetz module and we retain the notation introduced above.
Since $\p$ contains $\g _0$, we have $X\supset B_0$. We denote $X\cap B_2$ by
$B_2^{\p}$ and we let $B_2^{\a }:=B_2\setminus X=B_2-B_2^{\p}$.
For $\beta\in B_2$, we denote by $R_2(\beta )$ the set of roots in $R_2$ that
have coefficient one on $\beta$ and put
$$
\g _2(\beta):=\sum _{\gamma\in R_2(\beta )}\g ^{\gamma}.
$$
Since $R_2$ is the disjoint union of the $R_2(\beta )$'s,
$\g _2$ is the direct sum of the $\g _2(\beta)$'s.
Each $\g _2(\beta)$ is a $\g _0$-invariant subspace of $\g$. Notice that
$\p _2=\sum _{\beta\in B_2^{\p }} \g _2(\beta )$ so that
$$
\bar\a :=\sum _{\beta\in B_2^{\a }} \g _2(\beta ).
$$
is a $\g _0$-invariant supplement of $\p _2$ in $\g _2$. By
\refer{6.6}, $\a$ is the graph of a linear map $\phi :\bar\a\to\p _2$.
\proclaim{\label Lemma}
For every $\beta '\in B_2^{\p}$ there exists a $\beta\in B_2^{\a}$ such that
the map $\g _2(\beta )\to\g _2(\beta ')$ induced by $\phi$ is nonzero.
\endproclaim
\demo{Proof}
Suppose not. Then $\a$ centralizes the fundamental coweight
$p _{\beta}$
corresponding to $\beta$. We show that then $[f(e),p_{\beta}]=0$, whenever
$f(e)$ is defined. This will imply that $\a\cup f(\a)$ is in the centralizer of
$p_{\beta}$ and thus contradict the fact that $\a\cup f(\a )$ generates $\g$ as
a Lie algebra.
Write $f(e)=\sum _k f_k$ with $[h_{\beta},f_k]=kf_k$. By homogeneity,
$(e,h,f_0)$ is then also an $\slt$-triple and by uniqueness of $f(e)$, we then
have $f(e)=f_0$.
\enddemo
Virtually all the properties that we shall derive about
quasi-Frobenius--Lefschetz modules come from the following lemma.
\proclaim
{\label Lemma}
The spaces $\bar\a$ and $\phi (\bar\a)$ are abelian subalgebras
and $[X,\phi (Y)]=[Y,\phi (X)]$ for all $X,Y\in\bar\a$.
(In particular, $[X,\phi (Y)]=0$ if $X\in\g _2(\beta )$, $Y\in\g _2(\beta ')$
with $\beta ,\beta '\in B_2^{\a}$ distinct.)
\endproclaim
\demo{Proof} Let $\gamma$ and $\gamma '$ be distinct roots of $R_2$ that have a
$B_2^{\a}$-coefficient equal to one and let $X\in \g^{\gamma}$, $Y\in\g
^{\gamma '}$. Since $\a$ is abelian, we have
$$\align
0&=[X+\phi (X),Y+\phi (Y)]\\
&=[X,Y]+([X,\phi (Y)]+[\phi (X),Y])+[\phi (X),\phi (Y)].
\endalign
$$
The three groups of terms belong to direct sums of root spaces that do not
intersect and so each of them must be zero.
\enddemo
\proclaim{\label Corollary}
Let $\beta\in B_2^{\a}$, $C$ its
connected component in $B-B_2^{\p}$. Then:
\roster
\item"{(i)}" $\beta$ is the unique element of
$B_2\cap C$ and the coefficient of $\beta$ in the highest root with support in
$C$ is $1$,
\item"{(ii)}" if $\beta '\in B_2^{\p}$ is connected with $C$ and $\beta _1\in
B_2^{\a}$, then $\phi $ induces a nonzero map $\g _2(\beta _1)\to \g
_2(\beta ')$ if and only if $\beta _1=\beta$,
\item"{(iii)}" if $B_2^{\p}\not=\emptyset$,
then $\beta$ is an end of $B$ or not connected with $B_0$.
\endroster
\endproclaim
\demo{Proof} (i) If $C\cap B_2$ contains an element distinct from $\beta$, then
let $\beta '$ be such an element that is not separated from $\beta$ by
another member $B_2$. Then $\beta '\in B_2^{\a}$. We then can find a $\gamma\in
R_2(\beta )$ such that $\beta '+\gamma$ is a root. But this contradicts the
fact
that $\g_2 (\beta )$ and $\g _2(\beta ')$ commute.
The fact that $\g _2(\beta )$ is abelian implies that the sum of no two
elements
of $R_2(\beta )$ is a root. So the coefficient of $\beta$ in the
highest root with support in $C$ is $1$ (see \refer{2.5}).
(ii) Let $\beta '\in B_2^{\p}$ and $\beta _1\in B_2^{\a}$ be as in the
statement. Choose $\gamma _1\in R_2(\beta _1)$ and $\gamma '\in R_2(\beta ')$
such that the map $\g ^{\gamma _1}\to \g ^{\gamma '}$ induced by $\phi$ is
nonzero and choose $\gamma\in R_2(\beta )$ such that $\gamma +\gamma '$ is a
root. So if $X^{\gamma}\in \g ^{\gamma}$ and $X^{\gamma _1}\in \g ^{\gamma _1}$
are generators, then the $\g ^{\gamma +\gamma '}$ component of
$[X_{\gamma},\phi (X_{\gamma _1})]$ is nonzero. By \refer{6.11}, this implies
that $\beta _1=\beta$.
(iii) For this assertion we prove that if $C-\{\beta\}$ is connected with
$\beta '\in B_2^{\p}$, then $C-\{\beta\}$ is connected and separates $\beta '$
from $\beta$. Since the Dynkin diagram is a tree this amounts to showing that
the union $Y$ of connected components of
$C-\{\beta\}$ that do not separate $\beta$ and $\beta '$ is empty.
Suppose this is not the case: $Y\not=\emptyset$. Since $\beta '$ is separated
from $Y$ by $\beta$, all the roots of $R_2(\beta ')$ will have zero coefficient
on $Y$. According to (ii) there exist
a $\gamma _1\in R_2(\beta )$ and a $\gamma '\in R_2(\beta ')$
such that $\phi (X^{\gamma _1})$ (with $X^{\gamma _1}\in\g ^{\gamma _1}$) has
nonzero $\g^{\gamma '}$-component. Choose $\gamma\in R_2(\beta )$ such that
$\gamma +\gamma '$ is a root and $\gamma-\gamma _1$ has a nonzero coefficient
on
an element of $Y$. Let $X^{\gamma}\in\g ^{\gamma}$ be a generator and consider
the identity $$
[X^{\gamma},\phi (X^{\gamma _1})]=[X^{\gamma _1},\phi (X^{\gamma})].
$$
The $\g^{\gamma +\gamma'}$-component of the lefthand side is clearly nonzero.
So this is also the case for the righthand side. This implies that
$-\gamma _1+\gamma +\gamma '\in R_2(\beta ')$. This is therefore a root whose
$Y$-coefficients are zero. However, these coefficients are those of
$-\gamma _1+\gamma$ and so we arrive at a contradiction.
\enddemo
\proclaim{\label Corollary}
If $B_2^{\p}=\emptyset$, then $B_2$ is a singleton, $\a =\bar\a =\g _2$ and $M$
is a Jordan--Lefschetz module.
\endproclaim
\demo{Proof} The first part of previous corollary implies that $B_2=B_2^{\a}$
is
a singleton, $\{\beta\}$, say. Then $\bar\a$ is the sum of the root spaces $\g
^{\gamma}$ with $\gamma\in R_2$. Since $\bar\a$ is abelian, the sum of two
elements of $R_2$ is never a root. Hence $R_2\cup R_0$ contains all the
positive
roots. This implies that $\bar\a=\g _2$. Since the lowest weight of $M$
is a negative multiple of the fundamental weight at the vertex labeled by
$\beta$, it is a Jordan--Lefschetz module.
\enddemo
Corollary \refer{6.12} does not exploit \refer{6.11} to the fullest. For
instance, we have:
\proclaim{\label Lemma}
Suppose that $C$ is connected with some $\beta '\in B_2^{\p}$ and assume that
$C\cup\{\beta '\}-\{\beta\}$ is of type $A_l$, $l\ge 2$. Then
$C$ is a string and $\beta$ has no greater root length then $\beta '$.
\endproclaim
\demo{Proof} Let us number the roots of $C\cup\{\beta '\}-\{\beta\}$ in
order: $\beta ',\alpha _1,\dots ,\alpha _{l-1}$ and let $\beta$ be connected
with $\alpha _k$, $1\le k\le l-1$.
According to \refer{6.12-ii}, $\phi$ induces a nonzero map
$\phi _{\beta ',\beta}:\g _2(\beta )\to \g _2(\beta ')$ such that
$[X,\phi _{\beta ,\beta '} (Y)]$ is symmetric in $(X,Y)\in \g (\beta )\times\g
(\beta )$. The simple roots $\{\alpha _1,\dots ,\alpha _{l-1}\}$ define a Lie
subalgebra $\s\subset\g$ isomorphic to $\sli (l)$. We denote its fundamental
weights by $\varpi _i\in (\h \cap\s)^*$ ($i=1,\dots ,l-1$).
Any root $\gamma '$ of $R_2(\beta ')$
will have the form $\gamma +\alpha _1\cdots +\alpha _i$ (including the case
$i=0$: $\gamma =\gamma '$), such that $\gamma \in R_2(\beta ')$ has all its
$C$-coefficients zero. We take $\gamma '\in R_2(\beta ')$ such that $\phi
_{\beta ',\beta}$ has nonzero component on $\g ^{\gamma '}$. Then
$$
V:=\g ^{\gamma}+\g ^{\gamma +\alpha _1}+\cdots + \g ^{\gamma +\alpha
_1+\cdots +\alpha _{l-1}}.
$$
is the $\s$-subrepresentation of $\g _2(\beta ')$ generated by $\g
^{\gamma '}$. It is irreducible with highest weight $\varpi _1$ and is
therefore
a standard representation of $\s$.
Suppose that $k\notin\{1, l-1\}$. In view of the classification of Dynkin
diagrams, $\beta$ has then the same root length as $\beta '$.
\smallskip {\it Claim 1:} The set of roots $R_2(\beta )$ is the orbit of
$\beta$
under the action of the Weyl subgroup $W$ defined by the subroot system
$\{\alpha
_1\dots ,\alpha _{l-1}\}$ of $R$.
\smallskip
Proof. Notice that $R_2(\beta )$ consists of positive roots that are linear
combinations of $\beta ,\alpha _1,\dots ,\alpha _{l-1}$ with the coefficient of
$\beta$ being $1$. Each $W$-orbit in $R_2(\beta )$
contains a root $\delta =\beta +r_1\alpha _1+\cdots +r_{l-1}\alpha _{l-1}$ in
the closed $W$-chamber opposite the fundamental one:
$\delta (\alpha _i^{\vee})\le 0$ for $i=1,\dots ,l-1$. This means
that $2r_i\le r_{i-1}+r_{i+1}$ for $i\not=k$ (where we put $r_0=r_l=0$) and
$2r_k\le r_{k-1}+r_{k+1}+1$. It is not difficult
to verify that this can only happen when all $r_i$'s are zero, i.e., when
$\delta =\beta$. So $R_2(\beta )=W\beta$.
\smallskip {\it Claim 2}: $\g (\beta)$ is as a $\s$-representation isomorphic
to $\wedge ^kV$.
\smallskip
Proof. According to the previous claim the weights of $\g (\beta)$ with
respect to $\h$ are in a single $W$-orbit. This implies that $\g (\beta)$
is irreducible as a $\s$-representation. Since
$\beta$ defines the minus the fundamental weight $-\varpi _k$ of the root
system generated by $\{\alpha _1,\dots ,\alpha _{l-1}\}$, this representation
must be equivalent to $\wedge ^kV$.
\smallskip
We identify $\g (\beta)$ with $\wedge ^kV$ so that $\phi$ induces a nonzero
linear map $\psi :\wedge ^kV\to V$. The Lie bracket defines a bilinear
$\s$-equivariant map from $V\times\wedge ^kV$ to a representation space
of $\s$ and so we may think of it as a projection onto $\s$-subrepresentation
of $V\otimes\wedge ^kV$. This subrepresentation is nonzero since for instance
the root spaces $\g^{\gamma +\alpha _1+\cdots +\alpha _k}$ and $\g^{\beta}$
have nonzero Lie bracket. Moreover, $[\psi (x),y ]$ is symmetric in $(x,y)$. We
show that this is impossible.
We observe that $\wedge
^kV\otimes V$ decomposes into two irreducible representations: one is $\wedge
^{k+1}V$ and the other is the space $U$ spanned by the elements $x_1\otimes
x_1\wedge\cdots \wedge x_k$.
Suppose first that the image of the Lie bracket
has a nonzero projection on $\wedge ^{k+1}V$. Then
$$
\psi (x_1\wedge\cdots \wedge x_k)\wedge y_1\wedge\cdots\wedge y_k=
\psi (y_1\wedge\cdots \wedge y_k)\wedge x_1\wedge\cdots\wedge x_k
$$
for all $x_i,y_i\in V$. This identity shows that if
$\psi (y_1\wedge\cdots \wedge y_k)\not= 0$, then
each $x_i$ must be in the span of $\psi (x_1\wedge\cdots \wedge x_k),
y_1,\cdots,
y_k$. Since $\psi$ is not identically zero, we deduce (by letting $y_1,\dots
,y_k$ vary) that each $x_i$ is proportional to $\psi (x_1\wedge\cdots
\wedge x_k)$ or that $k=l-1$. The last case is excluded and the first case
implies that $k=1$, which is excluded as well. So the bracket map has image in
$U$. The $\s$-equivariant projection $\pi _U: V\otimes \wedge ^kV\to U$ is
given by
$$
\align
\pi &_U(x_0\otimes (x_1\wedge\cdots \wedge x_k))=\\
&={k\over k+1} x_0\otimes (x_1\wedge\cdots \wedge x_k)-{1\over k+1}
\sum _{i=1}^k (-1)^ix_i\otimes (x_0\wedge x_1\wedge\cdots \wedge\widehat
x_i\wedge\cdots\wedge x_k).
\endalign
$$
The same reasoning as above shows that if $\pi _U(\psi (x)\otimes y)$ is
symmetric in $(x,y)$ arguments, then $\psi =0$. This proves the first assertion
of the lemma.
\smallskip
The proof of the second assertion uses a similar argument. Suppose
that the root length of $\beta$ is greater than that of $\beta '$. In view of
the classification this can only happen when $\beta$ is connected with $\alpha
_{l-1}$ so that $C\cup\{\beta '\}$ is of type $C_{l+1}$.
\smallskip {\it Claim 3}: $\g (\beta)$ is as a $\s$-representation isomorphic
to the space $\Sym ^2(V^*)$ of quadratic forms on $V$.
\smallskip
Proof. Note that $W$ has two orbits in $R _2(\beta )$: the orbit of the long
root $\beta$ and the orbit of the short root $\alpha _{l-1}+\beta$. The
$\s$-representation generated by $\g ^{\beta}$ has $\g ^{\beta}$ as lowest
weight space. The lowest weight
is $-2\varpi _{l-1}$ and the corresponding representation is therefore $\Sym
^2(V^*)$. The weights of this representation are just the elements of $R
_2(\beta )$ and so the claim follows.
\smallskip
We finish the argument as before. The contraction mapping $V\otimes \Sym
^2(V^*)\to V^*$ is equivariant and surjective and its kernel $U'$ is an
irreducible representation of $\s$.
We first show that the Lie bracket cannot induce a nonzero mapping $V\otimes
\Sym ^2V^*\to V^*$. For then
$\psi : \Sym ^2(V^*)\to V$ is a nonzero linear map such that the expression
$\xi (\psi (\eta ^2))\xi$ is symmetric in $\xi,\eta\in V^*$. Since $\Sym
^2(V^*)$ is spanned by the squares, there exists a $\eta\in V^*$ with $\psi
(\eta ^2)\not= 0$. It then follows that $\eta$ is proportional to $\xi$ for
almost all $\xi$, which is absurd since $\dim V\ge 2$.
To finish the argument we now suppose that the Lie bracket induces a nonzero
mapping $V\otimes \Sym ^2V^*\to U'$. The equivariant section of the above
contraction map assigns to $\xi\in V^*$ the symmetrization
$\sym (\bold{1}_V\otimes \xi)\in V\otimes \Sym ^2(V^*)$. So the equivariant
projection $V\otimes \Sym ^2V^*\to U'$ is given by
$v\otimes\xi ^2\mapsto v\otimes\xi ^2 -\sym (\xi (v)\bold{1}_V\otimes \xi)$.
This means that the expression $\xi (x)^2\psi (\eta ^2) -\xi (\psi (\eta
^2))\xi (x)x$
(with $\xi,\eta\in V^*$ and $x\in V$) is symmetric in $\xi$ and $\eta$. So if
$\eta (x)=0$, then $\xi (x)^2\psi (\eta ^2) -\xi (\psi (\eta ^2))\xi (x)x=0$.
By taking $\xi (x)\not= 0$, we see that $\psi (\eta ^2)$ and $x$ must be
proportional. As this is true for all $x\in V$ with $\eta (x)=0$, it follows
that $\eta (\psi (\eta ^2))=0$. If we substitute $\eta =t_1\eta _1+t_2\eta _2$,
and take the $t_1^2t_2$-coefficient, we find that $\eta _1\psi (\eta _1\eta
_2)=0$. This means that $\psi$ is identically zero, which contradicts our
assumption.
\enddemo
\proclaim{\label Proposition}
Let $(\a ,M)$ be an irreducible Lefschetz module of depth $n$ with
$\dim M_{-n}=1$. Suppose that an irreducible representation
of $\g (\a ,M)$ whose highest weight is $k\ge 1$ times that of $M$,
is Frobenius up to order $l$, with $1\le l\le k$. Then $(M ,\a )$ is
quasi-Frobenius and $M_{-n+2i}=\bar\a ^iM_{-n}$ for $i\le l$.
\endproclaim
\demo{Proof}
Let $u\in M_{-n}$ be nonzero and let $M(k)$ be the $\g (\a
,M)$-subrepresentation of $\Sym ^k(M)$ generated by $u^k$. Then
$M(k)$ is irreducible and has highest weight $k$ times that of $M$.
It is also a Lefschetz $\a$-module of depth $kn$ with $u^k$ spanning
$M(k)_{-kn}$. By assumption, $M(k)$ is Frobenius up to order $l$.
In particular, $\a (u^k)=\bar\a (u^k)=M(k)_{-kn +2}
$. In view of the fact that $M(k)_{-kn+2}=u^{k-1}M_{-n+2}$, it follows
that $\a M_{-n}=\bar\a M_{-n}=M_{-n+2}$, so that $M$ is quasi-Frobenius also.
Hence for $i\le l$, the map
$$
\Sym ^i(\bar\a )\otimes u^k\to \Sym ^i(M_{-n+2})u^{k-i},\quad
a_1a_2\cdots a_i\otimes u^k\mapsto a_1(u)\cdots a_i(u) u^{k-i}
$$
is an isomorphism. There is an obvious projection of $\Sym ^k(M)_{-kn+2i}$
onto
its subspace $\Sym ^i(M_{-n+2})u^{k-i}$ which makes the above map factor as
$$
\Sym ^i(\bar\a )\otimes u^k\to M(k)_{-kn+2i}\to \Sym ^i(M_{-n+2})u^{k-i},
$$
with the first map given by the $\bar\a$-action on $M(k)$.
The image of that first map is just $\bar\a ^i(u^k)$ and hence
$\dim\, \bar\a ^i (u^k)=\dim \Sym ^i(\bar\a )=\dim \Sym ^i(\a )$.
Since $M(k)$ is
quasi-Frobenius up to order $l\ge i$, we also have $\dim M(k)_{-kn+2i}\le \dim
\Sym ^i(\a )$. It follows that $M(k)_{-kn+2i}=\bar\a ^i(u^k)$. Taking the
projection in the summand $M_{-n+2i}u^{k-i}$, then gives that $\bar\a
^i(u)=M_{-n+2i}$.
\enddemo
\medskip\label {\it Example \refer{6.4} continued.} In this case
$\bar\a$ is the intersection of $\so (V)$ with $\Hom (V_{-2},F_0)+\Hom
(F_0',V_2)$. We may identify $\bar\a$ with $F_0\wedge V_2$ and if do so, then
its action on $W$ is given by the wedge product. From this it is immediate
that $\bar\a ^2$ acts trivially on $W$. So the previous proposition
implies that any irreducible representation of $\so (V)$ with lowest weight a
multiple of that of $W$ is not Frobenius.
\demo{Proof of \refer{6.8}}
In view of \refer{6.13}, the assumption that $(M,\a )$ is not a
Jordan--Lefschetz module implies that $B_2^{\p}$ is nonempty. According to
\refer{1.17}, $B_2$ is totally disconnected. So it follows from \refer{6.12}
that the elements of $B_2^{\a}$ are ends of the Dynkin diagram. Let the
numbers
$n$, $k$, $r$ and $1\le d_0<d_1<\cdots <d_r=d_{r+1}=\cdots =d_k$ have the same
meaning as in \refer{1.16}.
\smallskip
{\it Case $A_l$.} Since $B_2^{\a}$ consists of ends, we must have
$d_0=1$. Hence $M$ has lowest weight of the form $-p\varpi _1-q\varpi _l$,
with $p,q$ nonnegative integers. According to \refer{6.3}, $M$
is self-dual, so that $p=q$ and $B_2^{\a}=\{\alpha _1,\alpha _l\}$.
Notice that for $p=q=1$ we get the adjoint representation of $\g =\sli (V)$.
This representation is not quasi-Frobenius: its lowest degree summand is
$\sli (V)_{-2n}=\Hom (V_n,V_{-n})$ and so every element of $\a (\sli
(V)_{-2n})$
must be in $\Hom (V_{n-2},V_{-n})$, which is a proper subspace of $\sli
(V)_{-2n+2}=\Hom (V_n,V_{-n+2})\oplus \Hom (V_{n-2},V_{-n})$.
So by \refer{6.15}, $M$ cannot be quasi-Frobenius either and therefore this
case does not occur.
\smallskip
{\it Case $B_l$.} Then $n$ is even, $d_k$ is odd and $l=d_0+\cdots
+d_{k-1}+{1\over 2}(d_k-1)$. The elements of $B_2$ are in position
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_{k-1}$. So in this case $d_0=1$ and
$B_2^{\a}=\{\alpha _1\}$, the simple root at the tautological vertex of $B$.
\smallskip
{\it Case $C_l$, odd parity.}
Then $n$ is odd and $l=d_0+\cdots +d_k$. The elements of $B_2$ are in position
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_k=l$. The last element is the large simple
root, so by \refer{6.14} cannot belong to $B_2^{\a}$. Hence $d_0=1$ and
$B_2^{\a}=\{\alpha _1\}$, the simple root at the standard tautological vertex
of
$B$.
\smallskip
{\it Case $C_l$, even parity.}
Then $n$ and $d_0,\dots ,d_k$ are even and $l=d_0+\cdots
+d_{k-1}+{1\over 2}d_k$. The elements of $B_2$ are in position
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_{k-1}$. Neither the first nor the last root
are among them, so this case cannot occur.
\smallskip
{\it Case $D_l$, odd parity.} Then $n$ is odd, all $d_i$'s are even and
$l= d_0+\cdots +d_{k-1}+d_k$. The elements of $B_2$ are in position
$d_0,d_0+d_1,\dots ,d_0+\cdots +d_k$ and $d_k\ge 4$.
Now $\alpha _{l-d_k},\alpha _l\in B_2$, whereas $\alpha _i\in B_0$ for the
intermediate indices $i=l-d_k+1,\dots ,l-1$. Since $d_k\ge 4$, it follows from
\refer{6.14}, that we cannot have $\alpha _l\in B_2^{\a}$. So $d_0=1$ and
$B_2^{\a}=\{\alpha _1\}$, the simple root at the tautological vertex of $B$.
\smallskip
{\it Case $D_l$, even parity.} Then $n$ and $d_k$ are even and $l=
d_0+\cdots +d_{k-1}+{1\over 2}d_k$. If $d_k\ge 4$, then the elements of $B_2$
are in position $d_0,d_0+d_1,\dots ,d_0+\cdots +d_{k-1}$. So $\alpha _{l-1}$
and $\alpha _l$ do not belong to this set. It follows from \refer{6.12} that
$d_0=1$ and $B_2^{\a}=\{\alpha _1\}$, the simple root at the tautological
vertex of $B$.
Suppose now $d_k=2$. Then according to \refer{1.16}
$d_0=1$ and $d_1=\cdots =d_k=2$, in other words, $V\cong V(2k)\oplus V(2k-2)$
($k\ge 2)$. If $k=2$, then we are in the $D_4$-case with all ends belonging
to $B_2$ and the center in $B_0$. According to \refer{6.12} this can only be if
$B_2^{\a}$ is a singleton. So let us assume that $k\ge 3$; we are then in the
case of example \refer{6.4} and its continuation \refer{6.16}.
According to \refer{6.12},
$B_2^{\a}$ cannot contain both $\alpha _{l-1}$ and $\alpha _l$. Suppose it
contains one of them, say $\alpha _l$.
If $B_2^{\a}$ also contains $\alpha _1$, then
$\bar\a$ is the sum of the root spaces corresponding to the four roots $\alpha
_1,\alpha _1+\alpha _2, \alpha _l,\alpha _l+\alpha _{l-2}$. But this
contradicts
our finding in \refer{6.4} that $\dim\a\le 2$. We also cannot have
$B_2^{\a}=\{\alpha _l\}$: if that were the case,
then $M$ has lowest weight $-p\varpi _l$ ($p>0$). The representation with
lowest weight $-\varpi _l$ is a semispinorial representation $W$ as discussed
\refer{6.4} and so this is excluded by \refer{6.16}.
\smallskip
We conclude that we are in the orthogonal
or symplectic case and that $B_2^{\a}=\{\alpha _1\}$ always. So $M$ has
lowest weight $-p\varpi _1$, with $p$ a positive integer. To finish the
argument, we must show that $p=1$. For this we invoke \refer{6.15}, with $V$
taking the r\^ole of $M$. Notice that in all cases the assumption
$B_2^{\p}\not=\emptyset$ implies that $n>2$. The lowest degree part of $V$,
$V_{-n}$, is one dimensional and $\bar\a\subset\Hom (V_{-n},V_{-n+2})+\Hom
(V_{n-2},V_n)$. So $\bar\a V_n=V_{-n+2}$, but $\bar\a V_{-n+2}=0$, whereas
$V_{-n+4}\not= 0$ (here we use that $n>2$). So \refer{6.15} implies that
$p=1$.
\enddemo
\head
\section Appendix: a property of the orthogonal and symplectic Lie algebra's
\endhead
\noindent The purpose of this appendix is to prove the following theorem.
\proclaim{Theorem}
Let $U_i$ ($i=1,\dots ,k$) be finite dimensional complex vector spaces ($k\ge
2$) endowed with a nondegenerate form (symmetric or skew) and assume that
no $U_i$ is an inner product space of dimension $2$. If we give
$U_1\otimes\cdots\otimes U_k$ the product form
$$
\la u_1\otimes\cdots\otimes u_k,v_1\otimes\cdots\otimes v_k\ra :=
\la u_1,v_1\ra\cdots\la u_k,v_k\ra ,
$$
then every simple Lie subalgebra of $\aut (U_1\otimes\cdots\otimes U_k)$ that
contains $\aut (U_i)$ or contains a copy of $\sli (2)$ in $\aut (U_i)$ acting
irreducibly on $U_i$ ($i=1,\dots k$) is equal to $\aut (U_1\otimes\cdots\otimes
U_k)$.
\endproclaim
Before we begin the proof we make some preliminary observations and
recall two results of Dynkin.
As before, $V(d)$ denotes the standard irreducible $\slt$-module of dimension
$d+1$, which we regard as the $d$-fold symmetric product of the tautological
representation of $\slt$.
\proclaim{\label Lemma}
The decomposition of $\gl (V(d))$ into irreducible $\slt$-submodules is
$$
\gl (V(d))=\oplus _{i=0}^{d-1} \gl ^{(i)}(V(d)),
$$
where $\gl ^{(i)}(V(d))$ is the $\slt$-submodule generated by $e^i$.
Furthermore, $\gl ^{(0)}(V(d))$ consists of the scalars, $\gl ^{(1)}(V(d))$
can be identified with the image of $\slt$ in $\gl (V(d))$ and
$\gl ^{(\odd)}(V(d))=\aut (V(d))$.
\endproclaim
\demo{Proof}
Let $W$ be an irreducible $\slt$-submodule of $\gl (V(d))$ of dimension $m+1$.
If $T\in W$ is a highest weight vector, then $[e,T]=0$ and $[h,T]=mT$. Since
$V(d)$ is a monic $\C [e]$-module, it follows that $T$ is a polynomial in $e$
(of degree $\le d$, of course). Since $[h,e^i]=2i e^i$ it follows that
$m$ is even and that $T$ is proportional to $e^{{1\over 2}m}$. On the other
hand
is clear that for $i=0,\dots ,m$, $e^i$ is coprimitive of weight $2i$ and hence
generates an irreducible $\slt$-submodule of $\gl (V(d))$ of dimension $2i+1$.
This proves the first part of the lemma. The identity $\langle e^ix,y\rangle
=(-)^i \langle x,e^iy\rangle$ shows that $e^i\in\aut (V(d))$ if and only if $i$
is odd. \enddemo
\proclaim{\label Lemma}
In the situation of the previous lemma, we have for $i=0,\dots ,m$ that
$f^i\in \gl ^{(i)}(V(d))$. Furthermore, $u_i:=\ad _f^ie^i$ resp.\
$h_i:=[e^i,f^i]$ is a semisimple element in $\gl ^{(i)}(V))$ resp.\ $\aut
(V(d))$ which commutes with $h=[e,f]$. For $d\ge 3$ and $2\le i\le d$, we have
$h_i\notin\slt$. \endproclaim
\demo{Proof} There is an inner automorphism of $\slt$ that sends $e$ to $f$ and
so the first statement follows. For the second, regard $V(d)$ as the space of
homogeneous polynomials of degree $d$ in two variables $x,y$ and let $e$
resp.\ $f$ act as $x\partial /\partial y$ resp.\ $y\partial /\partial x$. The
calculation is then straightforward: $x^ky^l$ is an eigen vector of
$h_i$ with eigen value $c_{k,l}:=k(k-1)\cdots (k-1+d)(l+1)(l+2)\cdots (l+d)-
(k+1)(k+2)\cdots (k+d)l(l-1)\cdots (l-1+d)$. We have $c_{k,l}=-c_{k,l}$ and so
$h_i\in \aut (V(d))$. A simple verification shows that
under the given constraints, $h_i$ is not proportional to $h$ and hence not in
$\slt$. It is clear that $[h,u_i]=0$. Hence $u_i$ preserves each eigen space of
$h$ and so $u_i$ is semisimple.
\enddemo
We will also need two theorems due to Dynkin \cite{Dynkin 1952b}.
\proclaim{\label Theorem}
(Dynkin) Let $\s$ be a Lie subalgebra of $\aut (V(d))$
that contains $\sli (2)$ as a proper subalgebra. Assume $d\not= 6$. Then $\s
=\aut (V(d))$.
\endproclaim
\proclaim{\label Theorem}
(Dynkin) Let $U$ and $V$ be vector spaces with a nondegenerate
form (symmetric or skew), neither of which is an
inner product spaces of dimension $4$. Then every semisimple Lie
subalgebra $\g$ of $\aut (U\otimes V)$ that strictly contains $\aut (U)\times
\aut (V)$ is equal to $\aut (U\otimes V)$.
\endproclaim
The reason for excluding $4$-dimensional inner product spaces is that such a
space is of the form $V=W_1\otimes W_2$ with $W_i$ a
symplectic plane and $\aut (V)=\aut (W_1)\times \aut (W_2)$. In that case $\aut
(U\otimes W_1)\times\aut (W_2)$ is a semisimple Lie subalgebra of $\aut
(U\otimes V)$ that strictly contains $\aut (U)\times \aut (V)$ (at least, if
$\dim U\ge 3$). However this algebra is not simple and as we shall see the
exceptions disappear if $\g$ is assumed to be simple.
\medskip
Let $U$ be vector space of finite dimension $\ge 2$ with a nondegenerate
$\epsilon$-symmetric form and denote its Lie algebra of infinitesimal
automorphisms by $\aut (U)$. Let $\g _{\pm}(U)$ be the set of $x\in\sli (U)$
satisfying $\la xu,u'\ra =\pm\la u,xu'\ra$ and let $\g _0(U)$ denote the
scalar
operators in $\gl (U)$.
\proclaim{\label Lemma} Suppose that $U$ is not an inner product space of
dimension two. Then $\gl _-(U)=\aut (U)$ and $\gl (U)=\g _-(U)\oplus \g
_0(U) \oplus \g _+(U)$ is an $\aut (U)$-invariant decomposition. The summands
are irreducible, except when $U$ is an inner product space of dimension $4$. If
$\dim U >2$, then $[\g _+(U),\g _+(U)]=\g _ -(U)$ and for $\epsilon=\pm$,
there exists $Y\in\g _{\epsilon}(U)$ such that $Y^2\in\g _+(U)+\g _0(U)$ is not
a scalar and has nonzero trace.
\endproclaim
\demo{Proof}
The first statements are well-known. If $\dim U >2$, then
$[\g _+(U),\g _+(U)]$ is a nontrivial subspace of $\g _ -(U)$ and hence equal
to it (since $\g _ -(U)$ is irreducible). The last statement is an easy
exercise.
\enddemo
The following lemma describes the exceptional case of theorem \refer{7.3}. This
is (at least implicit) in the tables and in any case, the
proof is straightforward.
\proclaim{\label Lemma } Let $\s :=\gl ^{(1)}(V(6))+\gl ^{(5)}(V(6))$. This is
a
simple Lie subalgebra of $\gl (V(6))$ of type $G_2$ and any semisimple Lie
algebra of $\aut (V(6))$ that strictly contains $\slt$ contains $\s$.
The subspace $\g _+(V(6))$ is an irreducible representation of $\s$ (of
dimension $27$).
\endproclaim
We now treat the essential part of the case $k=2$ of the theorem. The proof is
however typical for the way we prove it in general.
\proclaim{\label Proposition}
Let $d,d'$ be positive integers and let $\g$ be a
semisimple Lie subalgebra of $\aut (V(d)\otimes V(d'))$ which contains $\slt
\times\slt$, but is not contained in $\aut (V(d))\times\aut (V(d'))$. Then $\g$
contains $\aut (V(d))\times\aut (V(d'))$.
\endproclaim
\demo{Proof} We abbreviate $V:=V(d)$ and $V':=V(d')$.
The irreducible $\slt\times\slt$-submodules of $\gl (V\otimes V')$ are
$\gl ^{(k)}(V)\otimes \gl ^{(l)}(V')$, where $0\le k\le d$ and
$0\le l\le d'$. The submodule $\aut (V\otimes V')$ is the sum of the $\gl
^{(k)}(V)\otimes \gl ^{(l)}(V')$ with $k+l$ odd. We are given that $\g$
contains
the summands indexed by the pairs $(1,0)$, $(0,1)$
and some $(i,j)$ with $i$ and $j$ both positive and with odd sum.
Suppose $i$ is even and $j$ is odd.
We first prove that $\g\supset \aut (V)\otimes\bold{1}$.
According to \refer{7.2}, $u_j\in \gl ^{(j)}(V')$ is nonzero semisimple with
integral eigen values and so $\Tr (u^2)\not=0$.
Consider $h_i\otimes u _j^2=[e^i\otimes u_j,f^i\otimes u_j]\in\g$. This
element must have a nonzero component in some $\gl ^{(k)}(V)\otimes\bold{1}$
with $k$ odd and $\not= 1$.
If $d\not=6$, then theorem \refer{7.3} implies that $\aut
(V)\otimes\bold{1}\subset\g$.
If $d=6$, then it follows that $\g$ contains $\s\otimes\bold{1}$, with $\s$ a
Lie algebra of type $G_2$ as described in \refer{7.6}.
Since $i$ is even and positive, it follows from \refer{7.6} that $\g$ contains
$\g _+(V)\otimes\gl ^{(j)}(V')$. If $X,Y\in \g _+(V)$, then
$[X,Y]\otimes h_j^2=[X\otimes h_j,Y\otimes h_j]\in\g$ and we find as
before that $[X,Y]\otimes\bold{1}\in\g$. The elements $[X,Y]$,
$X,Y\in\g _+(V)$, span $\g _-(V)$ by \refer{7.5} and so also in this case
$\g\supset \aut (V)\otimes\bold{1}$.
We next prove that $\bold{1}\otimes\aut (V')\subset\g$. If $d'=1$, there is
nothing to show, so suppose $d'\ge 2$.
In passing we have shown that $\g$ contains the summand $\gl
^{(k)}(V)\otimes u _j^2$ with $k$ odd. It is easily verified that for
$d'\ge 2$, $u_j^2$ is not a scalar and so $\g$ contains a summand $\gl
^{(k)}(V)\otimes \gl ^{(l)}(V')$ with $l>0$ (and necessarily even). So the same
argument as above (with $(i,j)$ replaced by $(k,l)$) shows that
$\bold{1}\otimes\aut (V')\subset\g$.
\enddemo
\proclaim{\label Proposition}
Let $d$ be a positive integer and $U$ a finite dimensional vector
space with a nondegenerate $\epsilon$-symmetric form that is not an inner
product space of dimension two. If $\g$ is a semisimple Lie subalgebra of $\aut
(U\otimes V(d))$ which contains $\aut (U)\times\slt$, but is not
contained in $\aut (U)\times\aut (V(d))$, then $\g$ contains
$\aut (U)\times\aut (V(d))$.
\endproclaim
The proof of this proposition is analogous to the proof of the
proposition preceding it (relying sometimes on \refer{7.5} instead of
\refer{7.2}) and so we omit it.
\demo{Proof of the theorem} As any $4$-dimensional inner product space is the
tensor product of two sumplectic planes, we may assume that no $U_i$ is of
that type. Then for $k=2$ the
assertion follows from the conjunction of \refer{7.7}, \refer{7.8} and theorem
\refer{7.4}. We proceed with induction on $k$ and assume $k\ge 3$.
In case $k=3$ and all factors
$U_1,U_2,U_3$ symplectic planes, then
$$
\aut (U_1\otimes U_2\otimes
U_3)=\sli (U_1)\times\sli (U_2)\times\sli (U_3)+\sli (U_1)\otimes\sli
(U_2)\otimes\sli (U_3).
$$
Since the last summand is irreducible as a
$\sli (U_1)\times\sli (U_2)\times\sli (U_3)$-module and contains a nonzero
element of $\g$, the theorem is then immediate.
Assume therefore that we are not in this situation.
First note that $\aut (U_1\otimes\cdots\otimes U_k)$ is the direct sum of the
summands $\g _{\epsilon _1}(U_1)\otimes\cdots\otimes\g _{\epsilon _k}(U_k)$
with $\epsilon _i\in \{ -,0,+\}$ with the value $-$ being taken an odd number
of
times (if $U_i$ is a symplectic plane, then $\epsilon _i$ cannot take the value
$+$). By assumption $\g$ contains a nonzero element in a summand
$\g _{\epsilon _1}(U_1)\otimes\cdots\otimes\g _{\epsilon _k}(U_k)$ with at
least
two nonzero $\epsilon _i$'s.
If $\epsilon _i=0$ for some $i$, then let $J$ denote
the set of indices $j$ with $\epsilon _j\not= 0$. We wish to apply our
induction hypothesis to $U:=\otimes _{j\in J}U_j$. For this, we need to know
that $\prod _{j\in J}\aut
(U_j)$ is contained in a simple component of $\g\cap\aut (U)$. The latter is
certainly reductive. The simple component of $\g\cap\aut (U)$ that contains
$\aut (U_j)$ ($j\in J)$ must also contain $\otimes _{l\in J}\g _{\epsilon
_l}(U_l)$ (since no nonzero element of $\g _{\epsilon _j}(U_j)$ commutes with
the elements of $\aut (U_j)$) and so the desired property holds. Our induction
hypothesis therfore applies and we conclude that $\g$ contains $\aut (U)$. Now
$U$ cannot be the tensor product of two symplectic planes $U_i\otimes U_j$, for
in that case $\aut (U)=\sli (U_i)\times \sli (U_j)$, so that $\epsilon _i =0$
or $\epsilon _j=0$. Hence $U$ will not be a
$4$-dimensional inner product space and we may therefore apply the induction
hypothesis once more to the tensor product of $U$ and the $U_i$ with $\epsilon
_i=0$ and conclude that $\g =\aut (U_1\otimes\cdots\otimes U_k)$.
We next deal with the case when $\epsilon _i\not= 0$ for all $i$.
Suppose first that not all factors $U_i$ are symplectic planes. Since at least
one $\epsilon _i$ is $-$, we can by renumbering assume that
$\epsilon _1=-$ and $U_2$ is not a symplectic plane. We then show that
$\epsilon _3,\dots ,\epsilon _k$ can be made zero while keeping $\epsilon _1,
\epsilon _2$ nonzero, so that this takes us to the case considered above.
With the help of \refer{7.2} and \refer{7.5} we can find $X_1,X_2\in\g _-(U_1)$
with $[X_1,X_2]\not= 0$ and
$Y_j\in\g _{\epsilon}(U_i)$ ($j=2,\dots k$) such that $Y_2^2$ is not scalar,
$Y_j^2$ is not traceless for $j\ge 3$, and $X_i\otimes Y_2\otimes\cdots\otimes
Y_k\in\g$ ($i=1,2$). Then $[X_1,X_2]\otimes Y_2^2\otimes\cdots\otimes Y_k^2$
is an element of $\g$ whose component in $\g _{-}(U_1)\otimes\g
_+(U_2)\otimes\bold{1}\otimes\cdots\otimes\bold{1}$ is nonzero.
It remains to do the case when all factors $U_i$ are symplectic planes
(and so all $\epsilon _i$'s are $-$). We then choose $X_i\in\sli (U_i)$ for
$i=1,2,3$ and $Y_j\in\sli (U_j)$ for $j=1,\dots ,k$ such that $\Tr
(Y_j^2)\not=0$ for $j\ge 4$ and $Z:=[X_1\otimes X_2\otimes X_3, Y_1\otimes
Y_2\otimes Y_3]\notin\sli (U_1)\times\sli (U_2)\times\sli (U_3)$. Then
$$
[X_1\otimes X_2\otimes X_3\otimes Y_4\otimes\cdots\otimes Y_k,
Y_1\otimes\cdots\otimes Y_k]=Z\otimes Y_4^2\cdots Y_k ^2
$$
is an element of $\g$ with a nonzero component in
$\sli (U_1)\otimes\sli (U_2)\otimes\sli
(U_3)\otimes\bold{1}\otimes\cdots\otimes\bold{1}$.
Therefore, $\g$ contains $\aut (U_1\otimes U_2\otimes U_3)$ and the induction
proceeds.
\enddemo
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\bye
|
1996-04-29T21:35:00 | 9604 | alg-geom/9604024 | en | https://arxiv.org/abs/alg-geom/9604024 | [
"alg-geom",
"math.AG"
] | alg-geom/9604024 | Dan Abramovich | Dan Abramovich | A high fibered power of a family of varieties of general type dominates
a variety of general type | Latex2e (in latex 2.09 compatibility mode). To get a fun-free version
change the `FUN' variable to `n' on the second line (option dedicated to my
friend Yuri Tschinkel). Postscript file with color illustration available on
http://math.bu.edu/INDIVIDUAL/abrmovic/fibered.ps | null | 10.1007/s002220050149 | null | null | We prove the following theorem:
Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive
dimensional varieties of general type, with $B$ irreducible. Then there exists
an integer $n>0$, a positive dimensional variety of general type $W_n$, and a
dominant rational map $X^n_B \das W_n$.
| [
{
"version": "v1",
"created": "Mon, 29 Apr 1996 19:27:01 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Abramovich",
"Dan",
""
]
] | alg-geom | \section{INTRODUCTION}\subtitle{We Are Introduced to Our Main
Theorem, and the Story Begins.}
\noindent %
We work over $\Bbb{C}$.
\subsection{Statement} We prove the following theorem:
\begin{th}[Fibered power theorem] \label{fibered-power}
Let $X\rightarrow} \newcommand{\dar}{\downarrow B$ be a smooth family of positive dimensional varieties of general
type, with $B$ irreducible. Then there exists an integer $n>0$, a positive
dimensional variety of
general type $W_n$, and a dominant rational map $X^n_B \das W_n$.
Specifically, let $m_n:X^n_B \das W_n$ be the $n$-pointed birational-moduli
map. Then for sufficiently large $n$, $W_n$ is a variety of general
type.
\end{th}
The latter statement will be explained in section \ref{kollars}. This solves
``Conjecture H'' of \chm, \S 6.1 as well as the question at the end of
remark 1.3 in \cite{av}.
Following Viehweg's suggestions in \cite{vletter}, the fibered
power theorem is proved by way of the following theorem:
\begin{th} \label{fiberedmax}
Let $X\rightarrow} \newcommand{\dar}{\downarrow B$ be a smooth family of positive dimensional varieties of general
type, with $B$ irreducible, and ${\operatorname{Var}}(X/B) = \dim B$. Then there exists an
integer $n>0$ such that the fibered power $X^n_B$ is of general type.
\end{th}
\subsection{Main ingredients} The starting point is a theorem of Koll\'ar,
which roughly speaking says that given $f:X\rightarrow} \newcommand{\dar}{\downarrow B$ a morphism of smooth
irreducible projective varieties whose generic fiber is a variety of general
type, and ${\operatorname{Var}}(X/B) = \dim B$, then
for large $m$ the saturation of $f_*(\omega^m_f)$ has many sections. A very
useful trick of Viehweg shows that this implies that for large $m$ the sheaf
$\omega_f^m$ itself has many sections, that is, $\omega_f$ is big.
Following
\chm, one would like to use
these sections pulled back to the fibered powers $f_n:X^n_B\rightarrow} \newcommand{\dar}{\downarrow B$ of $X$ over
$B$ to overcome
any possible negativity in $\omega_B$. Unfortunately, the fibered powers may
become increasingly singular, and it is not easy to tell who wins in the race
between the positivity of $\omega_{f_n}$ and the so called adjoint conditions
imposed by
the singularities of $X^n_B$. The fact that $\omega_f$ may have positivity
``by accident'', as shown by the example in \chm, \S 6.1 of plane quartics,
shows that
something more is needed - the fibration $X\rightarrow} \newcommand{\dar}{\downarrow B$ should be ``straightened
out'' before we can use sections coming from Koll\'ar's theorem.
Semistable reduction would be sufficient for this purpose, but
unfortunately semistable reduction for families of fiber
dimension $>2$ over a base of dimension $>1$ is yet unknown.
It is not known whether unipotent
monodromies would suffice. The case of
curves was done in \chm\ using the moduli
space of stable curves, and the case of surfaces was done in \cite{hassett}
using the moduli space of stable surfaces.
Lacking such constructions in higher
dimensions, we will use a variant of semistable reduction, introduced by de
Jong \cite{dj}. This variant involves, after a suitable base change and
birational modification, a Galois cover
$Y\rightarrow} \newcommand{\dar}{\downarrow X$, such that $Y\rightarrow} \newcommand{\dar}{\downarrow B$ is a composition of families of curves with at
most nodes. The fibers now are much better controlled, but we are left with
descending differential forms from $Y^n_B$ to $X^n_B$. This is done by studying
the behavior of the ramification ideals in the fibered powers.
\subsection{Arithmetic background and Applications} Results of this type are
motivated by Lang's conjecture. See, {\em e.g.}, \chm, \cite{hassett},
\cite{a}, \cite{av}, \cite{p}.
Let $K$ be a number field (or any field finitely generated over $\Bbb Q$), and
let $X$ be a variety of general type over $K$. According to a well known
conjecture of S. Lang (see \cite{langbul}, conjecture 5.7), the set of
$K$-rational points $X(K)$ is not Zariski dense in $K$. In \chm, it is shown
that this conjecture of Lang implies the existence of a uniform bound $B(K,g)$
on the number of $K$ rational points on curves of genus $g$ (a stronger
implication arises if one assumes a stronger version of Lang's
conjecture). This
result was later refined in \cite{a}, and the ultimate result of this type was
recently obtained by P. Pacelli in \cite{p}, to wit:
\begin{th}[Pacelli \cite{p}, Theorem 1.1] Assume that Lang's conjecture
is true. Let $g\geq 2$ and $d\geq 1$ be
integers, and let $K$ be a field as above. Then there exists an integer
$P_K(d,g)$, such that for any
extension $L$ of $K$ of degree $d$, and any curve $C$ of genus $g$
defined over $L$, one has $$\#C(L)\leq P_K(d,g).$$
\end{th}
The main geometric ingredient in the above mentioned results is the
``Correlation Theorem'' of \chm, which is Theorem \ref{fibered-power} for
curves. In \cite{chm}, \S 6, a version of Theorem \ref{fibered-power} was
conjectured (``Conjecture H''), with the suggestion that strong uniformity
results would follow from such a theorem. This was further investigated in
\cite{av}, where it is shown that Theorem \ref{fibered-power} gives an
alternative proof for Pacelli's
theorem, as well as other strong implication results for curves and higher
dimensional varieties. As an example of a result which does not follow from
Pacelli's theorem, we have (see \cite{av}, Corollary 3.4 and theorem 1.7):
\begin{cor} Assume that the weak Lang conjecture holds. Fix a number field $K$
and an integer $d$. Then there is a uniform bound $N$ for the number of points
of degree $d$ over $K$ on any curve $C$ of genus $g$ and gonality $>2d$ over
$K$. In fact, $N$ depends only on $g,d$ and the degree $[K:{\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}]$.
\end{cor}
Another conjecture of Lang states that if $X$ is a complex variety of general
type, the union of rational curves on $X$ is not Zariski dense. In
\cite{av} a few implications of this conjecture were investigated, see
\cite{av} \S 3. Using \cite{p}, proposition 2.8, P. Pacelli is able to
obtain the following remarkable result (\cite{p}, corollary 5.4): Lang's
conjecture about rational curves implies that there is a bound $P_{geom}(d,g)$
such
that for any complex curve $C$ of genus $g$, and curve $D$ of gonality $d$, the
number of nonconstant morphisms $D\rightarrow} \newcommand{\dar}{\downarrow C$ is bounded by $P_{geom}(d,g)$. This
result can again be deduced from theorem \ref{fibered-power}, but in an
unsatisfactory way - one has to reprove proposition 2.8 of \cite{p}, repeating
some of the steps, and therefore Pacelli's direct method is more appropriate.
Another direction where our theorem falls short of obtaining a definite result
is the logarithmic case. Here again Pacelli's methods should directly imply
results regarding uniform boundedness of stably integral points on elliptic
curves (see \cite{a-int} for the definition). One suspects that in the future a
fibered power theorem will be available for log-varieties. At the moment, the
main difficulties seem to lie in obtaining an appropriate generalization of the
theorems of Koll\'ar and Viehweg.
\subsection{Acknowledgements} I would like to thank F. Hajir and R. Gross,
whose question kept me thinking about the problem through a period when no
significant result seemed to be possible. Thanks to B. Hassett, J. de Jong,
J. Koll\'ar, P. Pacelli, R. Pandharipande, E. Viehweg and F. Voloch,
for helpful discussions. The realization that pluri-nodal fibration have
mild singularities was inspired by Hassett's results in \cite{hassett}, \S 4.
The understanding of the utility of such fibrations was reinforced by Pacelli's
results.
\section{PRELIMINARIES}\subtitle{We Set Up Some Terminology
about Families of Varieties and State a Lemma about Groups.}
\subsection{Definitions}
A variety is called an {\bf r-G variety} if it has only rational
Gorenstein (and hence canonical) singularities. For a Gorenstein variety $X$ to
be r-G, it is necessary and sufficient that for any resolution of singularities
$r:Y \rightarrow} \newcommand{\dar}{\downarrow X$ one has $r_*\omega_Y = \omega_X$ (see \cite{elkik}, II).
We say that a flat morphism of irreducible varieties $Y\rightarrow} \newcommand{\dar}{\downarrow B$ is {\bf mild},
if for any dominant $B_1 \rightarrow} \newcommand{\dar}{\downarrow B$ where $B_1$ is r-G, we have that ${Y_1} =
Y\times_B B_1$ is r-G. Note that, by induction, if $Y\rightarrow} \newcommand{\dar}{\downarrow B$ is mild then the
fibered powers $Y^n_B\rightarrow} \newcommand{\dar}{\downarrow B$ are mild as well.
An {\bf alteration} is a projective, surjective, generically finite morphism
of irreducible varieties. This is slightly different from the definition in
\cite{dj}, where propernes is assumed rather than projectivity. An
alteration $B_1\rightarrow} \newcommand{\dar}{\downarrow B$ is {\bf Galois} if there
exists a finite group $G\subset{\operatorname{Aut}}_B B_1$ such that $B_1/G\rightarrow} \newcommand{\dar}{\downarrow B$ is
birational.
A {\bf fibration} is a projective morphism of irreducible normal varieties
whose
general fibers
are irreducible and normal.
Given a fibration $X\rightarrow} \newcommand{\dar}{\downarrow B$ and an alteration $B_1\rightarrow} \newcommand{\dar}{\downarrow B$ we denote by
$X\tilde{\times}_B B_1$ the {\bf main component } of $X\times_B B_1$. Namely, if
$\eta_{B_1}$ is the generic point of $B_1$, then $X\tilde{\times}_B B_1 =
\overline{X\times_B \eta_{B_1}}$.
A {\bf family} is a flat fibration.
A family $Y\rightarrow} \newcommand{\dar}{\downarrow Y_1$ is called a {\bf nodal fibration} if every fiber is a
curve with at most ordinary
nodes. A family $Y\rightarrow} \newcommand{\dar}{\downarrow B$ is called a {\bf pluri-nodal fibration} if it is a
composition of nodal fibrations $Y\rightarrow} \newcommand{\dar}{\downarrow Y_1\rightarrow} \newcommand{\dar}{\downarrow \cdots \rightarrow} \newcommand{\dar}{\downarrow B$. Note that while
nodal fibrations are generically smooth, this is not the case with pluri-nodal
fibrations.
Given a line bundle $L$ and an ideal sheaf $\I$ on a variety $X$, we say that
$L\otimes \I$ is {\bf big} if for some $k>0$ the rational map associated to
$H^0(X, L^{\otimes k}\otimes \I^k)$ is birational to the image. It readily
follows that, if $L\otimes \I$ is big, and $J$ is an ideal sheaf, then for
sufficiently large $k$ we have that $L^{\otimes k}\otimes \I^kJ$ is big. The
definition differs
somewhat from Koll\'ar's definition in \cite{kollar}. In section \ref{kollars}
we will refer to sheaves which
are ``big'' in Koll\'ar's sense as {\em weakly big} \newcommand{\subtitle}[1]{ }: we say that a sheaf $\F$ is
{\bf weakly big} \newcommand{\subtitle}[1]{ }, if for any ample $L$ there is an integer $a$ such that
$Sym^a(\F)\otimes L^{\otimes (-1)} $ is weakly positive (see \cite{kollar},
p.367, (vii)).
\subsection{Group theory} For a finite group $G$ let $e(G) =
\operatorname{lcm}\{\operatorname{ord}(g)|g\in G\}$. We will use the following
obvious lemma:
\begin{lem}
Let $G$ be a finite group. Then for any $n$, we have $e(G^n) = e(G)$.
\end{lem}
\section{RAMIFICATION}\subtitle{We Encounter Ramification Ideals
Measuring Differences Between Pluricanonical Sheaves in a Quotient Situation,
and Show that These Ideals Can Be Controlled in Various Cases.}
Let $V$ be a quasi projective r-G variety, $G\subset {\operatorname{Aut}}(V)$ a finite
group. Let $W =
V/G$, and $q:V\rightarrow} \newcommand{\dar}{\downarrow W$ the quotient map. Let $r:W_1 \rightarrow} \newcommand{\dar}{\downarrow W$ be a
resolution of singularities. Note that $W$ is normal, therefore it is regular
in codimension 1. We can pull back sections of pluricanonical sheaves on
the nonsingular locus $W_{\mbox{\small ns}}$ and extend them into the
pluricanonical sheaf of $V$. Thus, for an
integer
$n>0$ we have an injective morphism $\phi_n: q^*r_*\omega^n_{W_1} \rightarrow} \newcommand{\dar}{\downarrow
\omega^n_V$.
Define the {\bf $n$-th ramification ideal} ${\cal J}} \newcommand{\I}{{\cal I}_n ={\cal J}} \newcommand{\I}{{\cal I}_n(G,V)=
\operatorname{Ann}\operatorname{Coker} \phi_n$.
\begin{lem}\begin{enumerate} \item We have ${\cal J}} \newcommand{\I}{{\cal I}_n\otimes \omega^n_V \cong
q^*r_*\omega^n_{W_1}$.
\item For any integer $k>0$ we have ${\cal J}} \newcommand{\I}{{\cal I}_n^k \subset {\cal J}} \newcommand{\I}{{\cal I}_{nk}$.
\item The ideals ${\cal J}} \newcommand{\I}{{\cal I}_n$ are locally defined: if
$V'\subset V$ is a $G$-invariant open subset, then
${\cal J}} \newcommand{\I}{{\cal I}_n(G,V')={\cal J}} \newcommand{\I}{{\cal I}_n(G,V)|_{V'}$.
\item The ideals ${\cal J}} \newcommand{\I}{{\cal I}_n$ are independent of the choice of resolution $r:W_1 \rightarrow} \newcommand{\dar}{\downarrow
W$.
\item The ideals ${\cal J}} \newcommand{\I}{{\cal I}_n$ can be also obtained if we use a partial resolution
$r:W_1 \rightarrow} \newcommand{\dar}{\downarrow W$ where $W_1$ is r-G.
\end{enumerate}
\end{lem}
{\bf Proof.}
Since $\omega_V$ is by assumption
invertible, we have (1).
For the same reason (2) follows: if
$\omega=\prod_{i=1}^k \omega_i$ where $\omega_i$ are local sections of
$\omega_V^n$ and if $f=\prod_{i=1}^k f_i$ where $f_i\in {\cal J}} \newcommand{\I}{{\cal I}_n$, then we can
write $f_i\cdot\omega_i = \sum g_{i,j}\cdot(q^*r_* \eta_{i,j})$ and expanding
we get that $f\omega$ is a local section of $q^*r_*\omega^{nk}_{W_1}$. It
would be nice to have an actual equality for high $n$. Part (3) follows by
definition. Parts (4) and (5) follow
by noticing that for a birational morphism $r':W_2\rightarrow} \newcommand{\dar}{\downarrow W_1$ with $W_2$
nonsingular, we have $r'_*\omega_{W_2}^n =\omega_{W_1}^n$ in both cases. \qed
Traditionally, one studies ramification by reducing to the case where both $V$
and $W$ are regular. Most of the results below follow that line, with the
exception of Proposition \ref{product}, where the author finds it liberating,
if not essential, to avoid unnecessary blowups.
The ideals ${\cal J}} \newcommand{\I}{{\cal I}_n$ give conditions for invariant differential forms
to descend to regular forms on the quotient:
\begin{prp}\label{invariants} Given an integer $n>0$ we have
$$ (q_* (\omega^{n}_V\otimes {\cal J}} \newcommand{\I}{{\cal I}_n))^G = r_* \omega^{n}_{W_1}.$$
\end{prp}
{\bf Proof:} A local section of $ (q_* (\omega^{n}_V\otimes {\cal J}} \newcommand{\I}{{\cal I}_n))^G$ can be
written
as $\sum q_*(f_i) r_*(s_i)$, where $f_i$ are $G$ invariant, therefore $f_i =
q^*g_i$. \qed
The ideals ${\cal J}} \newcommand{\I}{{\cal I}_n$ are bounded below in terms of multiplicities (here we first
use the assumption on rational singularities):
\begin{prp}[\chm\ \S 4.2, lemma 4.1]\label{multipl} Let $\Sigma_{G,V}=\Sigma
\subset V$ be the locus of fixed points:
$$ \Sigma = \{x\in V | \exists g\in G, g(x) = x\},$$ viewed as a closed reduced
subscheme, with ideal $\I_\Sigma$. Then $\I_{\Sigma}^{n\cdot (e(G)-1)}\subset
{\cal J}} \newcommand{\I}{{\cal I}_n$.
\end{prp}
{ {\bf Proof.} Let $V_1$ be the normalization of $W_1$ in ${\Bbb
C}(V)$. Let $W_1'\subset W_1$ be the open subset over which both $V_1$ and the
branch locus $B_{V_1/W_1}$ are nonsingular. The codimension of $W_1\setminus
W_1'$ is at least 2. Let $V_1'$ be the inverse image of $W_1'$. We have a
diagram
$$\begin{array}{lcl} V_1' & \stackrel{s}{\rightarrow} \newcommand{\dar}{\downarrow} & V \\
\dar q_1 & & \dar q \\
W_1' & \stackrel{r'}{\rightarrow} \newcommand{\dar}{\downarrow} & W
\end{array}
$$
Let $\omega$ be a $G$-invariant $n$-canonical form on $V$, vanishing to order
$n\cdot
(e(G)-1)$ on $\Sigma$. To show that $\omega$ descends to $W_1$ it suffices to
descend it to $W_1'$, since the codimension of the complement is at least
2. Since $V$ is r-G, $\omega' = s^*\omega$ is a regular $n$-canonical form on
$V_1'$, vanishing to
order $n\cdot (e(G)-1)$ on $B_{V_1/W_1}$. The
subgroup fixing a general point of a component of $B_{V_1/W_1}$ is cyclic, and
the action is given formally by $u_1\mapsto \zeta_k u_1, u_i\mapsto u_i$ for
some root of unity
$\zeta_k, k\leq e(G)$, where $u_i$ are local parameters, $u_1$ a uniformizer
for $B_{V_1/W_1}$. Formally at such a point, the quotient
map is given by $w_1=u_1^k, w_i=u_i, \, i>1$. By assumption, $\omega'$ can be
written
in terms of local parameters as $\omega'=f(u) (u^{k-1}du_1\wedge\cdots\wedge
du_m)^n=f(u) {q_1}^*(dw_1\wedge\cdots\wedge dw_m)^n$. The invariance implies
that $f(u) = {q_1}^*g(w)$ and therefore $\omega' = {q_1}^* g(w)
(dw_1\wedge\cdots\wedge dw_m)^n$.}\qed
{\bf Remark.} It is not difficult to obtain the following refinement of this
proposition (see analogous case in \cite{kollar}, lemma 3.2): let
$B=q(\Sigma)_{\mbox{\small red}}$, and let $I_B$ be the defining ideal. Then
$q^{-1}\I_B^{\lfloor n(1-{1\over{e(G)}})\rfloor}\subset J_n$.
Recall that if a group $G$ acts on a variety $V$, a line bundle $L$ and an
ideal $\I$ then the ring of invariant sections $\oplus_{k\geq 0}
H^0(Y,L^{\otimes k}\otimes
\I^k)^G$ has the same dimension as the ring of sections $\oplus_{k\geq 0}
H^0(Y,L^{\otimes k}\otimes \I^k)$. This allows us to have:
\begin{cor}[See more general statement in
\cite{p}, Lemma 4.2]\label{prod-quotient}
Let $X$ be a variety of general type and let
$G={\operatorname{Aut}}_\bfc(\bfc(X))$ be its birational automorphism group. Then for some
$n>0$
the quotient variety $X^n/G$, where $G$ acts diagonally, is of general type.
\end{cor}
{
{\bf Proof.} Applying Hironaka's equvariant resolution of singularities, we may
assume that $X$ is regular and $G={\operatorname{Aut}} X$. Let $p_i:X^n\rightarrow} \newcommand{\dar}{\downarrow X$ be
the projection onto the $i$-th factor. Choose $n$ large enough
so that $\omega^n_X\otimes \I_{\Sigma_{G,X}}$ is big. Therefore
$\omega^n_{X^n}\otimes (\sum p_i^{-1} \I_{\Sigma_{G,X}})^n$ is big. But $$(\sum
p_i^{-1}\I_{\Sigma_{G,X}})^n \subset \I_{\Sigma_{G,X^n}} \subset
J_n(G,X^n),$$ giving the result.
}\qed
Let $\Sigma \subset V$ be the locus of fixed points, and let
$\Sigma = \Sigma_1 \cup \Sigma_2 $ be a closed decomposition. Then ${\cal J}} \newcommand{\I}{{\cal I}_n$ is
supported along $\Sigma$, and can be written as
${\cal J}} \newcommand{\I}{{\cal I}_n={\cal J}} \newcommand{\I}{{\cal I}_{n,\Sigma_1}\cap {\cal J}} \newcommand{\I}{{\cal I}_{n,\Sigma_2}$.
Applying \ref{multipl} we obtain:
\begin{cor}\label{decompose-ramification} We have $
(\I_{\Sigma_2}^{e(G)})^n \cdot{\cal J}} \newcommand{\I}{{\cal I}_{n,\Sigma_1}\subset {\cal J}} \newcommand{\I}{{\cal I}_n$.
\end{cor}
Our goal is to apply our propositions to powers of mild families.
First, let $f:V\rightarrow} \newcommand{\dar}{\downarrow B$ be mild. Assume that $B$ is
r-G. As before, let
$G\subset
{\operatorname{Aut}}_B(V)$, $W = V/G$, and
$q:V\rightarrow} \newcommand{\dar}{\downarrow W$ the quotient map.
Let $p_i: V^m_B\rightarrow} \newcommand{\dar}{\downarrow V$ be the $i$-th projection.
We naturally have $G^m\subset {\operatorname{Aut}}_B(V^m_B)$ acting on all components.
We denote by $q_m:V^m_B \rightarrow} \newcommand{\dar}{\downarrow W^m_B$ the associated map. Let $r:W_1\rightarrow} \newcommand{\dar}{\downarrow W$ be a
resolution of singularities.
Define ${\cal J}} \newcommand{\I}{{\cal I}_{m,n} = \prod p_i^{-1} {\cal J}} \newcommand{\I}{{\cal I}_{n}$.
\begin{lem}\label{smoothproduct} Assume that $W_1\rightarrow} \newcommand{\dar}{\downarrow B$ is mild. Then
${\cal J}} \newcommand{\I}{{\cal I}_{m,n} \subset {\cal J}} \newcommand{\I}{{\cal I}_n(G^m,V^m_B)$.
\end{lem}
{\bf Proof.} Denote $r_m:W_m = (W_1)_B^m\rightarrow} \newcommand{\dar}{\downarrow W^m_B $ and $p_{i,W}:W_m\rightarrow} \newcommand{\dar}{\downarrow W_1$
the $i$-th projection. Since $V\rightarrow} \newcommand{\dar}{\downarrow B$ and
$W_1\rightarrow} \newcommand{\dar}{\downarrow B$ are mild, we have that
$$\omega^n_{V^m_B/B} =
\otimes_i \, p_i^* (\omega^n_{V/B}) \quad \mbox{ and }\quad \omega^n_{W_m/B}
= \otimes_i \, p_{i,W}^* (\omega^n_{W_1/B}).$$ Suppose a local section $w$ of
$\omega^n_{V^m_B/B}$ is
a monomial written as $w = \prod p_i^*w_i$,
and suppose $f\in {\cal J}} \newcommand{\I}{{\cal I}_{m,n}$ is a monomial written as $f=\prod p_i^*f_i$. Then
$fw = \prod p_i^*(f_i w_i)$ is a local section of
$q_m^*{r_m}_*\omega^n_{W_m/B}$. \qed
\begin{prp}\label{product} There exists a closed subset $F\subset B$ such that
$$(\I_F^{e(G)})^n\cdot{\cal J}} \newcommand{\I}{{\cal I}_{m,n} \subset {\cal J}} \newcommand{\I}{{\cal I}_n(G^m,V^m_B).$$
\end{prp}
{\bf Proof.} Let $F\subset B$ be the discriminant locus of $W_1\rightarrow} \newcommand{\dar}{\downarrow B$, and
$U=B\setminus F$. Now apply \ref{smoothproduct} and
\ref{decompose-ramification}. \qed
{\bf Remark.} It follows from the remark after \ref{multipl} that already
$$(\I_F^{\lfloor n(1-{1\over{e(G)}})\rfloor})\cdot{\cal J}} \newcommand{\I}{{\cal I}_{m,n} \subset
{\cal J}} \newcommand{\I}{{\cal I}_n(G^m,V^m_B).$$
We will often need to perform base
changes for fibrations. We need to find a condition on the base changed
fibration which guarantees that the original variety is of general type. This
is provided by the following proposition (which is probably well known):
\begin{prp}\label{base-change} Given an alteration $\rho:B_1 \rightarrow} \newcommand{\dar}{\downarrow B$ between
smooth projective
varieties,
there exists an ideal sheaf
$\I\subset {\cal{O}}} \newcommand{\calp}{{\cal{P}}_{{B_1}}$ with the following property: given a fibration $f:Y\rightarrow} \newcommand{\dar}{\downarrow
B$, with ${Y_1}\rightarrow} \newcommand{\dar}{\downarrow
Y\tilde{\times}_B B_1$ a resolution of singularities,
${f_1}: {Y_1} \rightarrow} \newcommand{\dar}{\downarrow {B_1}$ the induced projection,
such that
$\omega_{{f_1}} \otimes {f_1}^{-1}\I$ is big, then $Y$ is of general
type.
\end{prp}
First a lemma:
\begin{lem} \begin{enumerate}
\item Let $g:Y_1\rightarrow} \newcommand{\dar}{\downarrow Y$ be a generically finite morphism of smooth projective
varieties. Let $B\subset Y$ be the branch locus. Then
there exists an effective $g$-exceptional divisor $E$ on $Y_1$ and an injection
$\omega_{Y_1}(-g^*B) \rightarrow} \newcommand{\dar}{\downarrow g^*\omega_Y\otimes {\cal{O}}} \newcommand{\calp}{{\cal{P}}_{Y_1}(E)$.
\item If $\omega_{Y_1}(-g^*B)$ is big, then $\omega_Y$ is big as well.
\end{enumerate}
\end{lem}
{\bf Proof.} The pull-back morphism $g^*\omega_Y\rightarrow} \newcommand{\dar}{\downarrow \omega_{Y_1}$ gives
$g^*\omega_Y= \omega_{Y_1}(-R-E)$ where $E$ is an effective exceptional
divisor and $R$ is the ramification divisor. Clearly $R < g^*B$.
Assume that $\omega_{Y_1}(-g^*B)$ is big. Then $ g^*\omega_Y\otimes
{\cal{O}}} \newcommand{\calp}{{\cal{P}}_{Y_1}(E)$ is big. Let $ Y_1\stackrel{g_1}{\rightarrow} \newcommand{\dar}{\downarrow} Y'\stackrel{s}{\rightarrow} \newcommand{\dar}{\downarrow} Y$ be
the Stein factorization. Since $Y'$ is normal and $E$ is $g_1$-exceptional we
have that $s^*\omega_Y\otimes {g_1}_*{\cal{O}}} \newcommand{\calp}{{\cal{P}}_{Y_1}(E)=s^*\omega_Y$
therefore
$s^*\omega_Y$ is big. Since $s$ is finite we have that $\omega_Y$ is big.
{\bf Proof of \ref{base-change}.} Choose a nonzero ideal $\I_0\subset
{\cal{O}}} \newcommand{\calp}{{\cal{P}}_{B_1}$ with an injection $\I_0\subset \omega_{B_1}$, and an ideal
$\I_1\subset {\cal{O}}} \newcommand{\calp}{{\cal{P}}_{B}$ such that $\omega_{B_1}\otimes\rho^{-1}\I_1\subset
\rho^*\omega_B$. Given a fibration $f:Y\rightarrow} \newcommand{\dar}{\downarrow B$, with $g:Y_1\rightarrow} \newcommand{\dar}{\downarrow Y$ as above, we
have that the ideal $\I_1$ vanishes on the branch locus of $g$. Set $\I =
\I_1\rho^{-1}\I_2$. Assume that $\omega_{Y_1/B_1}\otimes g^{-1}\I$ is big,
then
$\omega_{Y_1}\otimes (\rho\circ g)^{-1}\I_2$ is big, therefore
$\omega_{Y_1}(-g^*B)$ is big, and by the lemma we have that $\omega_Y$ is
big. \qed
\section{MAXIMAL VARIATION AND KOLL\'AR'S
THEOREMS}\label{kollars}\subtitle{We Reduce Our Theorem to the Maximal
Variation Case, and Quote a Big Theorem Of Koll\'ar Producing Sections.}
\subsection{Pointed birational moduli} The following is an
immediate generalization of Koll\'ar's generic moduli theorem (\cite{kollar},
2.4):
\begin{th}[Pointed birational moduli theorem] Let $f:X\rightarrow} \newcommand{\dar}{\downarrow B$ be a family of
varieties of general type. There exist open sets
$U\subset B$ and $V\subset f^{-1}U$, and projective varieties $Z$ and
$W_n,\quad n\geq 1$, with a diagram:
$$\begin{array}{lclclcl}
V^n_B & \rightarrow} \newcommand{\dar}{\downarrow & V^{n-1}_B & \rightarrow} \newcommand{\dar}{\downarrow & \cdots & \rightarrow} \newcommand{\dar}{\downarrow & U \\
\dar m_n & & \dar m_{n-1} & & & &\dar m_0 \\
W_n & \rightarrow} \newcommand{\dar}{\downarrow & W_{n-1} & \rightarrow} \newcommand{\dar}{\downarrow & \cdots & \rightarrow} \newcommand{\dar}{\downarrow & Z
\end{array}
$$
satisfying the following requirements:
\begin{enumerate}
\item The morphisms $m_n$ are dominant.
\item If $P=(P_1,\ldots,P_n), P'=(P'_1,\ldots,P'_n)\in V^n_B, f_n(P)=b,
f_n(P')=b'\in U$, then
$m_n(P) = m_n(P')$ if and only if there exists a birational map $g:V_b\das
V_{b'}$ which is defined and invertible at $P_i$, such that $g(P_i) = P'_i$.
\item For general $b\in U$, let $G$ be the birational automorphism group of
$X_b$, then the fiber of $W_n$ over $m_0(b)$ is birational to $X_b^n/G$, where
$G$ acts diagonally.
\item There are canonical generically finite rational maps
$W_{nk}\das (W_n)^k_Z$.
\end{enumerate}
\end{th}
{\bf Sketch of proof:} Parts (3) and (4) follow from (2). The proof of (1) and
(2) is a
simple modification of \cite{kollar}, 2.4, where we let $PGL$ act on the
universal family over the Hilbert scheme and its fibered powers.\qed
\subsection{Reduction of theorem \ref{fibered-power} to theorem
\ref{fiberedmax}} Recall
by corollary \ref{prod-quotient} that for sufficiently large $n$ the
general fiber of $W_n\rightarrow} \newcommand{\dar}{\downarrow Z$ is of general type. Also, a simple lemma below
shows that for large $n$ the
family $W_n\rightarrow} \newcommand{\dar}{\downarrow Z$ is of maximal variation.
Assuming that theorem \ref{fiberedmax} holds true, we have that for large $k$
the variety $(W_n)^k_Z$ is of general type, therefore $W_{nk}$ is of general
type. For any $n'>nk$, applying the additivity theorem (Satz III of
\cite{viehweg1}) to
$W_{n'}\das W_{nk}$ we have that
the variety $W_{n'}$ is of general type. Therefore $X^{n'}_B$
dominates a variety of general type.
\begin{lem} Suppose $X\rightarrow} \newcommand{\dar}{\downarrow B$ is a one dimensional family of varieties of
general type, ${\operatorname{Var}}(X/B) =1 $, and $G\subset{\operatorname{Aut}}_BX$. Then for sufficiently
large
$n$, the quotient by the diagonal action $ W_n = X^n_B/G \rightarrow} \newcommand{\dar}{\downarrow B$ has
${\operatorname{Var}}(W_n/B)=1$.
\end{lem}
{\bf Proof.} This is immediate from the theorems of Kobayashi-Ochiai (see
\cite{d-m}) and Maehara (see \cite{moriwaki}). { Using Proposition
\ref{prod-quotient}, choose $n$ so that
the general fiber of $W_{n_0}$ over $B$ is of general type. We show that
$Var(W_{n_0+1}/B)=1$, and by induction this follows for any higher $n$. Assume
the opposite. We have the projection map $W_{n_0+1}\rightarrow} \newcommand{\dar}{\downarrow W_{n_0}$. The theorem
of Maehara implies that $Var(W_{n_0}/B)=0$: a family of varieties of genral
type dominated by a fixed variety is isotrivial. The theorem of
Kobayashi-Ochiai implies that the map $W_{n_0+1}\rightarrow} \newcommand{\dar}{\downarrow W_{n_0}$ is isotrivial: a
family of rational maps from a fixed variety to a fixed variety of general type
is isotrivial. But the general fiber of $(W_{n_0+1})_b\rightarrow} \newcommand{\dar}{\downarrow (W_{n_0})_b$ is
isomorphic to $X_b$ (one only needs to avoid the fixed point set of the
action) - implying that $X\rightarrow} \newcommand{\dar}{\downarrow B$ is isotrivial.}\qed
\subsection{Koll\'ar's big theorem} Here we introduce the main source
for global sections.
\begin{th}[Koll\'ar's big theorem, \cite{kollar}, I, p. 363]
Suppose that $\pi:X\rightarrow} \newcommand{\dar}{\downarrow B$ is a fibration of positive dimensional varieties of
general type, and ${\operatorname{Var}}(X/B) = \dim B$. Assume both $X$
and $B$ are smooth.
There is an integer $n>0$ such that
the sheaf $\pi_*\omega_\pi^n$ is {\em weakly big} \newcommand{\subtitle}[1]{ }. \qed
\end{th}
Koll\'ar's use of {\em weakly big} \newcommand{\subtitle}[1]{ } requires saturations, which means that the
sections obtained may have poles over exceptional divisors of the map $X\rightarrow} \newcommand{\dar}{\downarrow
B$. From this one first deduces:
\begin{cor}[\cite{viehweg}, Corollary 7.2] Suppose that $\pi:X\rightarrow} \newcommand{\dar}{\downarrow B$ is as
above. There is a divisor $D$ on $X$ such that ${\operatorname{codim}} (\pi({\operatorname{supp }}
B))>1$, and such that $\omega_\pi(D)$ is big. \qed
\end{cor}
We still have the annoying divisor $D$. Our method below will allow us to
ignore it, but actually a trick of Viehweg (\cite{viehweg}, lemma 7.3) makes it
easier. Viehweg simply applies the theorem above to $X'\rightarrow} \newcommand{\dar}{\downarrow B'$ where $X'$ is a
desingularization
of a flattening of $X$, where any exceptional divisor for $X'/B'$ is
exceptional for $X'/X$. Since $\omega_{B'/B}$ is effective, one immediately
obtains:
\begin{th}[Koll\'ar-Viehweg]\label{kv} Suppose that $\pi:X\rightarrow} \newcommand{\dar}{\downarrow B$ is as
above. Then $\omega_\pi$ is big. \qed
\end{th}
\section{PLURI-NODAL REDUCTION}\subtitle{We Prove a Pluri-Nodal Reduction
Lemma and Show That Pluri-Nodal Families Are Mild}
\subsection{Statement} Let $X_0\rightarrow} \newcommand{\dar}{\downarrow B_0$ be a fibration. We need to dominate
it by a pluri-nodal fibration, so that it becomes a quotient by the action of a
finite group.
To this end, we prove the following theorem, which is a variant of de Jong's
results in \cite{dj}, sections 6-7. The proof is based on that of de Jong.
\begin{lem}[{Galois pluri-nodal reduction lemma}]\label{pluri-nodal}
There exists a diagram
$$ \begin{array}{ccc} Y & \rightarrow} \newcommand{\dar}{\downarrow & X_0 \\
\dar & & \dar \\
B_1 & \rightarrow} \newcommand{\dar}{\downarrow & B_0\end{array} ,$$
and a finite group $G \subset {\operatorname{Aut}}_{X_0\tilde{\times}_{B_0} B_1}Y$ such that $B_1\rightarrow} \newcommand{\dar}{\downarrow
B_0$
is an
alteration, $Y/G \rightarrow} \newcommand{\dar}{\downarrow X_0\tilde{\times}_{B_0} B_1$ is birational and $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ is
a pluri-nodal fibration.
\end{lem}
{\bf Proof.}
We proceed by induction. The setup is as follows: suppose we have $X\rightarrow} \newcommand{\dar}{\downarrow Z
\rightarrow} \newcommand{\dar}{\downarrow B$ a pair of
fibrations, where $X\rightarrow} \newcommand{\dar}{\downarrow Z$ is pluri-nodal, and a finite
group $G_0\subset{\operatorname{Aut}}_B(X\rightarrow} \newcommand{\dar}{\downarrow Z)$. We also assume that we have a birational
morphism
$X/G_0\rightarrow} \newcommand{\dar}{\downarrow X_0\tilde{\times}_{B_0}B$. We will produce a diagram
$$\begin{array}{ccccccc} X' & \rightarrow} \newcommand{\dar}{\downarrow & Z'' & \rightarrow} \newcommand{\dar}{\downarrow & Z' & \rightarrow} \newcommand{\dar}{\downarrow & B' \\
\dar & & \dar & & & & \dar \\
X & \rightarrow} \newcommand{\dar}{\downarrow & Z & & \longrightarrow & & B
\end{array}
$$
with the following properties:
\begin{enumerate}
\item the vertical arrows are alterations,
\item the horizontal arrows are fibrations,
\item the morphism $Z''\rightarrow} \newcommand{\dar}{\downarrow Z'$ is a nodal fibration,
\item $X' = X\times_Z Z''$, and therefore $X'\rightarrow} \newcommand{\dar}{\downarrow Z'$ is pluri-nodal,
\item there is a finite group
$G' = G_0\times G'' \subset {\operatorname{Aut}}_{B'}(X'\rightarrow} \newcommand{\dar}{\downarrow Z''\rightarrow} \newcommand{\dar}{\downarrow Z')$, and
\item the morphism $X'/G'' \rightarrow} \newcommand{\dar}{\downarrow X\tilde{\times}_B B'$ is birational, and therefore
$X'/G'\rightarrow} \newcommand{\dar}{\downarrow X_0\tilde{\times}_{B_0}B'$ is birational.
\end{enumerate}
The basis of the induction is $X_0\rightarrow} \newcommand{\dar}{\downarrow X_0\rightarrow} \newcommand{\dar}{\downarrow B_0$ with $G_0$ trivial. The
induction ends with $Z'\rightarrow} \newcommand{\dar}{\downarrow B'$ being birational, in which case we set $Y :=
X', \quad B_1 := Z', \quad G := G'$ and the lemma will be proved.
Let
$G_Z\subset{\operatorname{Aut}}_BZ$ be the image of $G_0$, and denote $W=Z/G_Z$.
\begin{lem} There exists a dominant rational map
$Z/G_Z\das P\rightarrow} \newcommand{\dar}{\downarrow B$, where $P\rightarrow} \newcommand{\dar}{\downarrow B$ is a projective bundle, such
that $\dim(P) = \dim(Z)-1$, and such that the generic fiber of $Z$ over $P$ is
geometrically irreducible.
\end{lem}
{\bf Proof.} This is obvious in case $\operatorname{rel.dim }(Z/B)=1$, so assume
$\operatorname{rel.dim }(Z/B)>
1$. Denote this relative dimension by $r$. Since we are looking for a rational
map, we may replace $B$ by its
generic point $\eta$, and replace $Z$ by $Z_\eta$. Let $W=Z/G$, choose an
embedding $W\subset {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N$, and let $f:Z\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N$ be the
induced morphism. According to \cite{jou}, 6.3(4), for general hyperplane
$H\subset{\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N$ we have $f^{-1}H$ geometrically irreducible. Continuing by
induction, there is a linear series $({\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^{r-1})^*$ of dimension $r-1$ of
hyperplanes in ${\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N$ such that the general fiber of
$Z\das {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^{r-1}$ is a geometrically irreducible curve. \qed
The normalization of the closure of the graph of the
rational map $Z\das P$
gives a $G_Z$-equivariant resolution of
indeterminacies
$$ \begin{array}{ccc} Z_1 & \rightarrow} \newcommand{\dar}{\downarrow & P \\
\dar & & \\
Z & & \end{array}. $$
Let $X_1 = X\times_Z Z_1$. Then $X_1\rightarrow} \newcommand{\dar}{\downarrow Z_1$ is pluri-nodal, and the action
of $G_0$ on $X$ lifts to $X_1$ (if $x_1 = (x,z_1)\in X_1$ and $g\in G_0$ then
$(g(x),g(z_1))\in X_1$ as well).
We will now perform a canonical nodal reduction for $Z_1\rightarrow} \newcommand{\dar}{\downarrow P$ using the
Kontsevich space of stable maps.
The generic
fiber of $Z_1 \rightarrow} \newcommand{\dar}{\downarrow P$ is a normal curve, and therefore smooth. Choose a
projective embedding $Z_1\subset {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N$. Let $d$ be the degree of the generic
fiber of
$Z_1\rightarrow} \newcommand{\dar}{\downarrow P$ and let $g$ be its
genus. By
\cite{bm}, theorem 3.14, there exists a proper Deligne-Mumford
stack $\OM_{g,0}(Z_1,d)$ parametrizing stable maps $C\rightarrow} \newcommand{\dar}{\downarrow Z_1$ of
curves of genus $g$ and degree $d$. By \cite{pan} this stack admits a
projective coarse
moduli space. In particular, this implies that there is a finite cover $\rho:M
\rightarrow} \newcommand{\dar}{\downarrow
{\OM_{g,0}}(Z_1,d)$ where $M$ is a projective scheme
admitting a stable map $(C\rightarrow} \newcommand{\dar}{\downarrow M, f:C\rightarrow} \newcommand{\dar}{\downarrow Z_1)$ whose moduli map is $\rho$.
Let $\eta\in P$ be the generic point. The pair $((Z_1)_{\eta}\rightarrow} \newcommand{\dar}{\downarrow \eta,
(Z_1)_{\eta}\hookrightarrow Z_1)$ is a stable map of genus $g$ and degree $d$,
therefore we have a rational map $P\das
{\OM_{g,0}}(Z_1,d)$.
We can choose a normal resolution of indeterminacies
$$ \begin{array}{ccc} P_2 & {\rightarrow} \newcommand{\dar}{\downarrow} & M\\
\dar & & \\
P & & \end{array} $$
such that there is a finite group $G_1\subset{\operatorname{Aut}}_{P}P_2$ with $P_2/G_1\rightarrow} \newcommand{\dar}{\downarrow P$
birational. Let $Z_2 = C\times_MP_2$. We have an induced stable map $(Z_2\rightarrow} \newcommand{\dar}{\downarrow
P_2,f_2:Z_2\rightarrow} \newcommand{\dar}{\downarrow Z_1)$, in particular $Z_2\rightarrow} \newcommand{\dar}{\downarrow P_2$ is nodal. Over the generic
point of $P_2$ this coincides with $Z_1\times_{P} P_2$.
Since stable reduction
over a normal base is unique when it exists (see \cite{d-o}, 2.3), the action
of $G_1$ lifts to
$Z_2$, and it lifts
to $X_2 = X_1\times_{Z_1}Z_2$ as well by pulling back as before. Let $P_2\rightarrow} \newcommand{\dar}{\downarrow
B_2\rightarrow} \newcommand{\dar}{\downarrow B$ be the Stein
factorization. Since the Stein factorization is unique we have canonically an
action of $G_1$ on $B_2$. Let $G_2\subset G_1$ be the subgroup acting trivially
on
$B_2$. Then
$G=G_0\times G_2\subset{\operatorname{Aut}}_{B_2}(X_2\rightarrow} \newcommand{\dar}{\downarrow P_2)$. We have $X_2\rightarrow} \newcommand{\dar}{\downarrow P_2$
pluri-nodal,
and
$X_2/G_2\rightarrow} \newcommand{\dar}{\downarrow X\tilde{\times}_B B_2$ birational. If we denote $X' := X_2, \quad Z''
:=
Z_2, \quad Z' :=
P_2, \quad B' := B_2 $ and $G'' := G_2$ we have obtained the goal of the
induction step. \qed
\subsection{Mild Singularities}\label{mild}
We want to show that pluri-nodal fibrations are mild. This seems to be well
known (see \cite{hassett}, \S 4), but in our case we can give a
proof which is sufficiently short to include here. The following lemma is
well known (see \cite{viehweg}, lemma 3.6):
\begin{lem}
Let $Y\rightarrow} \newcommand{\dar}{\downarrow B$ be a nodal fibration such that $B$ is smooth and the discriminant
locus is a divisor of normal crossings. Then $Y$ is r-G.
\end{lem}
{
(The proof is by taking formal coordinates near a singular point of the form
$xy = t_1^{k_1}\cdots t_r^{k^r}$, and either resolving singularities
explicitly or noting that this is a toroidal singularity.)}
\begin{prp}
Let $Y\rightarrow} \newcommand{\dar}{\downarrow B$ be a nodal fibration such that $B$ is r-G. Then $Y$ is r-G.
\end{prp}
{\bf Proof.} Let $r:B_1 \rightarrow} \newcommand{\dar}{\downarrow B$ be a resolution of singularities, $Y_1\rightarrow} \newcommand{\dar}{\downarrow B_1$
the pullback, and assume that the discriminant locus of $Y_1\rightarrow} \newcommand{\dar}{\downarrow B_1$ is a
divisor of normal crossings. Let $f:Y_1\rightarrow} \newcommand{\dar}{\downarrow Y$ be the induced map. Then
$r_*\omega_{B_1} = \omega_B$ and $f^*\omega_{Y/B} = \omega_{Y_1/B_1}$, and by
the projection formula we obtain that $f_*\omega_{Y_1} = \omega_Y$.\qed
By induction we obtain:
\begin{cor}
If $\pi:Y\rightarrow} \newcommand{\dar}{\downarrow B$ is a pluri-nodal fibration where $B$ is r-G, then $Y$ is r-G.
In particular the $n$-th fibered power $Y^n_B$ is r-G.
\end{cor}
Thus pluri-nodal fibrations are mild.
\section{PROOF OF THE THEOREM}\subtitle{Our Main Theorem Arrives at an
Enchanted Place, and We Leave It There.}
Let $X_0\rightarrow} \newcommand{\dar}{\downarrow B_0$ be a smooth projective family
of varieties
of general type of maximal variation. Choose a model
${X}\rightarrow} \newcommand{\dar}{\downarrow {B}$ where both $X$ and $B$ are projective nonsingular. By
\ref{pluri-nodal} we may
assume, after an
alteration $B_1\rightarrow} \newcommand{\dar}{\downarrow {B}$, that we have a birational morphism
$g_0:Y/G=X_1 \rightarrow} \newcommand{\dar}{\downarrow X\tilde{\times}_{B}B_1$ where
$\pi_Y:Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ is a pluri-nodal
fibration and $G\subset{\operatorname{Aut}}_{B_1}Y_1$ a finite group. Choose a resolution of
singularities $r:X_2\rightarrow} \newcommand{\dar}{\downarrow X_1$ and denote by $\pi_2:X_2\rightarrow} \newcommand{\dar}{\downarrow B_1$ the projection.
We have a diagram:
\begin{equation}\label{diagram}
\begin{array}{lclcl}
& & Y & & \\
& & \dar q & & \\
X_2 & \stackrel{r}{\longrightarrow} \newcommand{\das}{\dashrightarrow}& X_1 & \stackrel{g_0}{\longrightarrow} \newcommand{\das}{\dashrightarrow} & X \\
& \searrow^{\pi_2} & \dar & & \dar \\
& & B_1 & \rightarrow} \newcommand{\dar}{\downarrow & B
\end{array}
\end{equation}
According to
\ref{product} (where we set $V=Y$ and $W=X_1$) there is an ideal $\I_F\subset
{\cal{O}}} \newcommand{\calp}{{\cal{P}}_{B_1}$
such that $(\I_F^{e(G)\cdot n})\cdot{\cal J}} \newcommand{\I}{{\cal I}_{m,n} \subset {\cal J}} \newcommand{\I}{{\cal I}_n(G^m,Y^m_{B_1})$.
For arbitrary integer $m>0$ let ${\cal{X}}} \newcommand{\cle}{{\cal{E}}_m\rightarrow} \newcommand{\dar}{\downarrow X^m_B$ be a resolution of
singularities of the main component, and let $\W_m\rightarrow} \newcommand{\dar}{\downarrow (X_1)^m_{B_1}$ be a
resolution of
singularities, dominating ${\cal{X}}} \newcommand{\cle}{{\cal{E}}_m$.
According to \ref{base-change} (applied to $B_1\rightarrow} \newcommand{\dar}{\downarrow B$) there is an ideal
$\I\subset {\cal{O}}} \newcommand{\calp}{{\cal{P}}_{B_1}$, such that for any $m$, if $\omega_{\W_m/B_1}\otimes \I$
is big then ${\cal{X}}} \newcommand{\cle}{{\cal{E}}_m$ (and therefore $(X_0)^m_{B_0}$) is of general type.
By the Koll\'ar - Viehweg theorem \ref{kv}, $\omega_{\pi_2}$ is big. Therefore
$q^*r_*\omega_{\pi_2}$ is big. We
have by definition that $\omega_{\pi_Y}\otimes {\cal J}} \newcommand{\I}{{\cal I}_1(G,Y)$ is big.
Therefore,
for sufficiently large $n$ we have that $\omega^{n}_{\pi_Y}\otimes
{{\cal J}} \newcommand{\I}{{\cal I}_n}\I\I_F^{e(G)}$ is big. Pulling back along all the projections
$p_i:Y^m_{B_1}\rightarrow} \newcommand{\dar}{\downarrow
Y$ we have that $\omega^n_{Y^m_{B_1}/B_1}\otimes {{\cal J}} \newcommand{\I}{{\cal I}_{m,n}}\I^m\I_F^{m\cdot
e(G)}$
is big. In particular, if $m>n$, we have that
$\omega^n_{Y^m_{B_1}/{B_1}}\otimes {{\cal J}} \newcommand{\I}{{\cal I}_{m,n}}\I^n\I_F^{n\cdot e(G)}$ is big. By
\ref{product} we have that $\omega^n_{Y^m_{B_1}/{B_1}}\otimes
{\cal J}} \newcommand{\I}{{\cal I}_n(G^m,Y^m_{B_1})\I^n$ is big.
Taking invariants and using \ref{invariants}, $\omega_{\W_m/B_1}\otimes \I$ is
big, and by \ref{base-change} we have that $(X_0)^m_{B_0}$
is of general type for large $m$.
\subsection{An alternative approach.}\label{alternative} The following argument
gives a
variation on the proof which is more in line with \cite{kollar} and
\cite{viehweg}. Having chosen the diagram (\ref{diagram}), we can alter it as
follows: using semistable reduction in codimension 1 (see \cite{te} II, and
\cite{kawamata}, theorem 17), we
can find a nonsingular alteration $B_1'\rightarrow} \newcommand{\dar}{\downarrow B_1$, a
variety $X_2'\rightarrow} \newcommand{\dar}{\downarrow B_1'$, and a birational morphism $X_2'\rightarrow} \newcommand{\dar}{\downarrow
X_2\tilde{\times}_{B_1}B_1'$ satisfying the following conditions
\begin{enumerate}
\item The discriminant locus $\Delta$ of $X_2'\rightarrow} \newcommand{\dar}{\downarrow B_1'$ is a divisor of normal
crossings. Set $F=Sing(\Delta)$ and $U=B_1'\setminus F
\stackrel{i}{\hookrightarrow} B_1'$.
\item The restriction $X_2'|_U\rightarrow} \newcommand{\dar}{\downarrow U$ is semistable, in particular it is mild
(see \cite{viehweg}, lemma 3.6).
\end{enumerate}
Let $X_1'= X_1\times_{B_1}B_1'$
and $Y'= Y\times_{B_1}B_1'$. We can replace $B_1,X_1,X_2, Y$ by
$B_1',X_1',X_2', Y'$ and assume that conditions (1) and (2) are satisfied.
Let $\pi_{X_{(m)}}:X_{(m)}\rightarrow} \newcommand{\dar}{\downarrow B_1$ be the main component of
$(X_1)^m_B$. Choose a resolution of singularities $W_m\rightarrow} \newcommand{\dar}{\downarrow X_{(m)} $, and
let
$\pi_{W_m}:W_m \rightarrow} \newcommand{\dar}{\downarrow B_1$ be the associated projection. Denote $\F_{m,n}=
{\pi_2}_*\omega_{\pi_{W_m}}$ and ${\cal G}} \newcommand{\K}{{\cal K}_{m,n}=(\F_{m,n})^{**}$. Since the
restriction of $W_m$ to $U$ is mild, we have that ${\cal G}} \newcommand{\K}{{\cal K}_{m,n} =
i_*i^*\F_{m,n}$. Applying \ref{smoothproduct}, we obtain:
\begin{enumerate}
\item We have natural morphisms
$$ {\cal G}} \newcommand{\K}{{\cal K}_{m_1,n}\otimes {\cal G}} \newcommand{\K}{{\cal K}_{m_2,n} \rightarrow} \newcommand{\dar}{\downarrow {\cal G}} \newcommand{\K}{{\cal K}_{m_1+m_2,n} $$
(by pulling back sections to $W_{m_1+m_2}$ over $U$, multiplying and
extending).
\item We have natural morphisms $$ {\cal G}} \newcommand{\K}{{\cal K}_{m,n_1}\otimes {\cal G}} \newcommand{\K}{{\cal K}_{m,n_2} \rightarrow} \newcommand{\dar}{\downarrow
{\cal G}} \newcommand{\K}{{\cal K}_{m,n_1+n_2} $$
(by multiplying sections).
\item We have $${\cal G}} \newcommand{\K}{{\cal K}_{m,n}\otimes \I_F^{n(e(G)-1)} \subset \F_{m,n}\subset
{\cal G}} \newcommand{\K}{{\cal K}_{m,n}$$ (by \ref{product}. Notice that the remark after \ref{product} shows
that $\I_F^n$ suffices).
\end{enumerate}
By Kollar's theorem $G_{1,n}$ is weakly big} \newcommand{\subtitle}[1]{ \ for sufficiently large $n$. We can
choose an ideal $\I$ as in \ref{base-change}. By (2) above, for sufficiently
large $n$ we have that $G_{1,n}\otimes \I\I_F^{e(G)}$ is big, and using (1)
above we have that for sufficiently large $m$, we have that $G_{m,n}\otimes
\I^m\I_F^{n\cdot e(G)}$ is big, therefore by (3) $\F_{m,n}\otimes\I^m$ is big,
which is what we need.
|
1996-04-24T17:07:18 | 9604 | alg-geom/9604015 | en | https://arxiv.org/abs/alg-geom/9604015 | [
"alg-geom",
"math.AG"
] | alg-geom/9604015 | Gerd Mueller | Gerd M\"uller | Symmetries of Surface Singularities | LaTeX, 24 pages, hard copies available | null | null | 952, Utrecht University | null | The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal
surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which
contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to
conjugation. In all cases except the cyclic quotient singularities the
connected component $G_1$ of the unit equals ${\Bbb C}^*$. The induced action
of $G$ on the minimal good resolution of $X$ embeds the finite group $G/G_1$
into the automorphism group of the central curve $E_0$ of the exceptional
divisor. We describe $G/G_1$ as a subgroup of ${\rm Aut}\: E_0$ in case $E_0$
is rational as well as for simple elliptic singularities. Moreover, sufficient
conditions for $G$ to be a direct product $G_1 \times G/G_1$ are presented.
Finally, it is shown that $G/G_1$ acts faithfully on the integral homology of
the link of $X$.
| [
{
"version": "v1",
"created": "Wed, 24 Apr 1996 15:59:36 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Müller",
"Gerd",
""
]
] | alg-geom | \section{Introduction}
The study of reductive group actions on a normal surface singularity $X$
is facilitated by the fact that the group ${\rm Aut}\: X$ of automorphisms of
$X$ has a maximal reductive algebraic subgroup $G$ which contains every
reductive algebraic subgroup of ${\rm Aut}\: X$ up to conjugation. If $X$ is
not weighted homogeneous then this maximal group $G$ is finite (Scheja,
Wiebe). It has been determined for cusp singularities by Wall. On the
other hand, if $X$ is weighted homogeneous but not a cyclic quotient
singularity then the connected component $G_1$ of the unit coincides
with the ${\Bbb C}^*$ defining the weighted homogeneous structure (Scheja,
Wiebe and Wahl). Thus the main interest lies in the finite group
$G/G_1$. Not much is known about $G/G_1$. Ganter has
given a bound on its order valid for Gorenstein singularities which are
not log-canonical. Aumann-K\"orber has determined $G/G_1$ for all
quotient singularities.
\\[1.5ex]
We propose to study $G/G_1$ through the action of $G$ on the minimal good
resolution $\tilde{X}$ of $X$. If $X$ is weighted homogeneous but not
a cyclic quotient singularity let $E_0$ be the central curve of the
exceptional divisor of $\tilde{X}$. We show that the natural homomorphism
$G\to{\rm Aut}\: E_0$ has kernel ${\Bbb C}^*$ and finite image. In particular, this
reproves the result of Scheja, Wiebe and Wahl mentioned above.
Moreover, it allows to view $G/G_1$ as a subgroup of
${\rm Aut}\: E_0$. For simple elliptic singularities it equals
$({\Bbb Z}_b\times{\Bbb Z}_b)\rtimes{\rm Aut}_0\, E_0$ where $-b$ is the self intersection
number of $E_0$, ${\Bbb Z}_b\times{\Bbb Z}_b$ is the group of $b$-torsion points of the
elliptic curve $E_0$ acting by translations,
and ${\rm Aut}_0\, E_0$ is the group of automorphisms
fixing the zero element of $E_0$. If $E_0$ is rational then $G/G_1$ is the
group of automorphisms of $E_0$ which permute the intersection points
with the branches of the exceptional divisor while preserving the Seifert
invariants of these branches. When there are exactly three branches we
conclude that $G/G_1$ is isomorphic to the group of automorphisms of the
weighted resolution graph. This applies to all non-cyclic quotient
singularities as well as to triangle singularities. We also
investigate whether the maximal reductive automorphism group is a direct
product $G\simeq G_1\times G/G_1$. This is the case, for instance, if
the central curve $E_0$ is rational of even self intersection number or if
$X$ is Gorenstein such that its nowhere zero $2$-form $\omega$ has degree
$\pm 1$. In the latter case there is a ``natural'' section
$G/G_1\hookrightarrow G$ of $G\to G/G_1$ given by the group of automorphisms
in $G$ which fix $\omega$. For a simple elliptic singularity one has
$G\simeq G_1\times G/G_1$ if and only if $-E_0\cdot E_0=1$.
\\[1.5ex]
In the weighted homogeneous case, we show that $G/G_1$ acts
faithfully on the homology $H_1(L,{\Bbb Z})$ of the link of $X$. For a hypersurface
in ${\Bbb C}^3$, not an $A_k$-singularity, defined by a weighted homogeneous
polynomial $f$ this will be rephrased in terms of the group $H$ of linear
right equivalences of $f$. Namely, with $F$ denoting the Milnor fibre of $f$,
the group $H$ acts faithfully on the Milnor lattice $H_2(F,{\Bbb Z})$ and
intersects the monodromy group in the cyclic group generated by the
monodromy operator.
\\[1.5ex]
The results of this paper were obtained during a stay at the Mathematical
Institute of Utrecht University. I thank its members for their hospitality
and the DFG for financial support. I profited from discussions with
Theo de Jong, Felix Leinen, Eduard Looijenga, Peter Slodowy, Tonny Springer,
Joseph Steenbrink and Jan Stevens.
Especially I wish to thank Dirk Siersma
for his suggestions and support.
\section{The action on the central curve}
Throughout this paper let $X=(X,0)$ denote a normal surface singularity
with local ring $(\O_X,m_X)$.
We are interested in algebraic subgroups $G$ of the group ${\rm Aut}\: X$ of
automorphisms of $X$. This means that $G\le{\rm Aut}\: X$ is an abstract
subgroup equipped with the structure of an algebraic group such that
the natural representations of $G$ on all higher cotangent spaces
$m_X^k/m_X^{k+1}$ of $X$ are rational.
There is an abundance of unipotent
algebraic subgroups of ${\rm Aut}\: X$, see \cite[Theorem 5]{MMonats}.
But the situation
becomes simpler if one restricts to reductive groups:
\begin{trivlist}
\item[] \bf Theorem 1. \it
There is a maximal reductive algebraic subgroup $G\le{\rm Aut}\: X$
containing every
reductive algebraic subgroup of ${\rm Aut}\: X$ up to conjugation.
\item[] Proof. \rm
This follows from \cite[Theorem 1]{HM} since a normal surface singularity
(just as any isolated singularity) can be defined by polynomials, see
\cite[Theorem 3.8]{A}. If one wants to avoid the use of
\cite[Theorem 1]{HM}, which depends on a deep result of Popescu and
Rotthaus, one can argue more elementary as follows: Let $\hat{X}$ be
the algebroid space corresponding to the formal completion $\widehat{\O_X}$
of $\O_X$.
By \cite[Satz 4]{MCrelle}
the group of automorphisms of $\hat{X}$
contains a maximal reductive subgroup $G$. It is enough to make $G$ act
on $X$ itself, with the same representation on the cotangent space,
\cite[Lemma]{HM}. If $G$ is finite then \cite[Remark on p.\ 184]{HM}
applies. Otherwise, $G$ contains some ${\Bbb C}^*$. It follows from a result of
Scheja and Wiebe \cite[3.1]{SW} that in this case the normal surface
$X$ in fact admits a good ${\Bbb C}^*$-action, i.\ e., $X$ is weighted
homogeneous. Then \cite[Satz 5]{MCrelle} applies.
\hfill$\Box$
\end{trivlist}
From now on, $G$ will always denote the maximal reductive automorphism
group of the normal surface singularity $X$. As mentioned in the preceding
proof, $G$ is finite if $X$ is not weighted homogeneous. Otherwise,
we may assume that the
connected component $G_1$ of the unit contains at least the ${\Bbb C}^*$
defining the weighted homogeneous structure. In fact,
Scheja and Wiebe \cite[section 3]{SW2} and
Wahl \cite[3.6.2]{WPSPM} showed
that $G_1={\Bbb C}^*$ unless $X$ is a cyclic quotient singularity.
We shall reprove this below with a different method which, at the same time,
will yield information on the finite group $G/G_1$.
\\[1.5ex]
Let $(\tilde{X},E)\to (X,0)$ be the minimal good resolution with exceptional
divisor $E$. There is a natural homomorphism ${\rm Aut}(X,0)\to{\rm Aut}(\tilde{X},E)$
obtained from the universal property of the minimal good resolution.
Here ${\rm Aut}(\tilde{X},E)$ denotes the group of germs of automorphisms
of $\tilde{X}$ along $E$.
In the weighted homogeneous case, the minimal good resolution was described
by Orlik and Wagreich \cite[Theorem 2.3.1]{OW}: If $X$ is not a cyclic
quotient singularity then $E$ has a component
$E_0$ which is uniquely determined by the property that $E_0$ has positive
genus $g$ or intersects at least three other components of $E$. The
exceptional divisor is star shaped with central curve $E_0$ and a certain
number $r$ of branches of rational curves. Here $r\ge3$ if $E_0$ is rational
and $r\ge0$ otherwise. In deriving this from \cite{OW} one has to be
aware of a result of Brieskorn \cite[Korollar 2.12]{B}: If $X$ can be
resolved by a
chain of rational curves then $X$ is a cyclic quotient singularity.
We obtain a natural homomorphism
${\rm Aut}\: X\to{\rm Aut}\: E_0$. Recall that ${\rm Aut}\: E_0$ is finite if $E_0$ has
genus $g\ge2$ and ${\rm Aut}\: E_0={\rm PSL}(2,{\Bbb C})$ if $E_0$ is rational. Finally, if
$E_0$ is an elliptic curve then
${\rm Aut}\: E_0=E_0\rtimes{\rm Aut}_0\, E_0$. Here the Abelian group $E_0$ acts on
itself by translations and the group ${\rm Aut}_0\, E_0$ of automorphisms
fixing the zero element is cyclic of
order 6, 4 or 2 if the $j$-invariant of $E_0$ is 0, 1 or else.
\begin{trivlist}
\item[] \bf Theorem 2. \it
Suppose that $X$ is weighted homogeneous but not a cyclic quotient
singularity. Then the restriction $\rho:G\to{\rm Aut}\: E_0$
of the natural map described above
has kernel ${\Bbb C}^*$ and finite image. Hence $G_1={\Bbb C}^*$ and $\rho$
embeds $G/G_1$ into ${\rm Aut}\: E_0$.
\item[] Proof. \rm
Let $X\subseteq({\Bbb C}^n,0)$ be a minimal embedding of the singularity $X$ in a
smooth germ.
By \cite[Satz 6]{MCrelle} the action of $G$ on $X$ can be extended to an
action on $({\Bbb C}^n,0)$ which is linear in suitable coordinates. Moreover,
we may assume that $X$ is defined by equations which are weighted
homogeneous polynomials in the chosen coordinates. Hence the action on
the analytic space germ $X$ is induced from a global rational action
on the affine algebraic variety $X$. We claim that the action on the
germ $(\tilde{X},E)$ is induced from a global rational
action $G\times \tilde{X}\to\tilde{X}$ on the non-singular surface
$\tilde{X}$. In fact, one obtains a $G$-equivariant good resolution
$X'\to X$ by succesively blowing up in $G$-invariant centres and
normalizing.
By the universal properties of blowing up and normalization, the
action $G\times X'\to X'$ is rational. The minimal good resolution is
obtained from $X'$ by blowing down $G$-invariant systems of exceptional
curves of the first kind. This gives the
claim. We conclude that $G\times E_0\to E_0$ is rational.
\item[]
The closed normal subgroup $K=\ker\:\rho\le G$ is reductive.
Consider the representation of $K$ on
the tangent space ${\rm T}_p\tilde{X}$ at some point $p\in E_0$. By Cartan's
Uniqueness Theorem \cite{K} it is faithful. Since $K$ is trivial on
the subspace ${\rm T}_p E_0$ and the representation is completely reducible
we see that $K$ has a faithful one dimensional representation. Thus $K$
is a subgroup of ${\Bbb C}^*$. It will follow that $K={\Bbb C}^*$ if we
show that ${\rm im}\:\rho$ is finite.
This is obvious if $E_0$ has genus $g\ge2$. But also for $g=1$ since
${\Bbb C}^*$ and hence every connected reductive group can only act trivially
on an elliptic curve. In the remaining
case $g=0$ consider the $r$ intersection
points of $E_0$ with other components of $E$. They have to be
permuted by any element of ${\rm im}\:\rho$. We obtain a homomorphism
${\rm im}\:\rho\to{\rm S}_r$ to the symmetric group which is injective since
$r\ge3$ and since automorphisms of the projective line have at most
two fixed points.
\hfill $\Box$
\end{trivlist}
\begin{trivlist}
\item[] {\it Remarks.} \rm
(i) Let $\Gamma$ denote the weighted dual graph defined by the minimal
good resolution and ${\rm Aut}\:\Gamma$ its group of automorphisms. Let $r$ be
the number of branches emanating from the central curve of genus $g$.
We have a natural homomorphism ${\rm im}\:\rho\to{\rm Aut}\:\Gamma$. A non-trivial
automorphism of $E_0$ has at most $2g+2$ fixed points, \cite[V.1.1]{FK}.
Thus, if
$r>2g+2$ then $G/G_1$ embeds into ${\rm Aut}\:\Gamma$. This is always the case
if $g=0$ since then $r\ge3$. The latter result was previously obtained
(with a different proof) by Aumann-K\"orber \cite[3.9]{A-K}.
\item[] (ii) If $G/G_1\to{\rm Aut}\:\Gamma$ is injective we obtain sufficient
conditions in terms of the weighted resolution graph for $G/G_1$ to be
trivial or cyclic. (Recall \cite[III.7.7]{FK}
that a finite group of automorphisms of a smooth
complete curve fixing a point is cyclic.)
\item[] (iii) The exact sequence
$1\to G_1\to G\to G/G_1\to 1$ splits if and only if
$G\simeq G_1\times G/G_1$ is a direct product over $G_1$. This follows
from the fact \cite[Proposition 3.10]{WPSPM} that $G_1={\Bbb C}^*$ is central
in $G$.
\item[] (iv) Let $X={\Bbb C}^2/H$ be a cyclic quotient singularity where the
group $H$ is generated by
$\left(\begin{array}{cc}
\zeta & 0 \\
0 & \zeta^q \end{array}\right)$
with $\zeta^n=1$ and $(q,n)=1$. Then $G_1={\rm GL}(2,{\Bbb C})/H$ if $q=1$ and
$G_1={\Bbb C}^*\times{\Bbb C}^*$ else, see \cite[3.6.2]{WPSPM}.
Also in this case, there is a natural homomorphism $G/G_1\to{\rm Aut}\:\Gamma$
which, in fact, is bijective, \cite[3.11, 3.12]{A-K}.
\item[] (v) Suppose that $X$ is not necessarily weighted homogeneous
but the exceptional
divisor $E$ has a component $E_1$ which is fixed by every automorphism
of $(\tilde{X},E)$. (This is, e.~g., the case for determinantal
rational singularities of multiplicity $m\ge3$ since their exceptional
divisor has a unique component $E_1$ of self intersection number $-m$,
\cite[Theorem 3.4]{Wrat}.)
Then there is again a natural homomorphism
$G\to{\rm Aut}\: E_1$. The proof of Theorem 2 shows that its kernel is
cyclic if $X$ is assumed to be not weighted homogeneous.
\item[] (vi) If $X$ is rational or minimally elliptic
but not weighted homogeneous and not a cusp singularity then the
kernel $K$ of the natural homomorphism $G\to{\rm Aut}\:\Gamma$ is cyclic. In
fact, by \cite[Lemma 1.3]{B} and \cite[Proposition 3.5]{Lauell}
all components of $E$ are rational but $\Gamma$ is not a chain nor a cycle.
(Note that, besides the cusp singularities, also simple elliptic and
cyclic quotient singularities are excluded because they are weighted
homogeneous.) Hence there is a component
$E_1$ intersecting at least three other components. As $E_1$ is rational
$K$ acts trivially on $E_1$. The proof of Theorem 2 shows that $K$ is
cyclic. The special case of non weighted homogeneous exceptional
unimodal singularities will be discussed in more detail in Example 3
of section 4.
For cusp singularities the kernel of $G\to{\rm Aut}\:\Gamma$ is Abelian
but not necessarily cyclic, \cite{Wcus}.
\item[] (vii) When we are considering the maximal reductive automorphism
group $G$ of some weighted homogeneous singularity $X$ we may choose
coordinates such that, at the same time, $G$ acts linearly on the ambient
space and $X$ is defined by weighted homogeneous polynomials, see the
beginning of the proof of Theorem 2. This does not mean that if we start
with a weighted homogeneous polynomial defining some $X$ then $G$ will
be linear in the given coordinates. For instance, let $X\subseteq({\Bbb C}^3,0)$
be defined by $f=x_1^3-2x_3x_2^2-x_3^2$ which is weighted homogeneous of
weights 4, 3, 6 and degree 12. The ${\Bbb C}^*$-action is given by
$t\cdot(x_1,x_2,x_3)=(t^4x_1,t^3x_2,t^6x_3)$. Suppose that
$G\le{\rm Aut}({\Bbb C}^3,0)$
contains this ${\Bbb C}^*$ and defines a maximal reductive subgroup of
${\rm Aut}\: X$. Since ${\Bbb C}^*$ is central in $G$, see Remark (iii) above,
it is easily seen that $G\cap{\rm GL}(3,{\Bbb C})={\Bbb C}^*$. But
$G$ is larger than ${\Bbb C}^*$ since $X$ is isomorphic to the singularity
$E_6$ defined by $g=x_1^3+x_2^4-x_3^2$ whose maximal reductive automorphism
group clearly contains ${\Bbb C}^*\times{\Bbb Z}_2$ where ${\Bbb Z}_2$ acts by
$(x_1,x_2,x_3)\mapsto(x_1,x_2,-x_3)$. It follows from Remark (i) that,
in fact, ${\Bbb C}^*\times{\Bbb Z}_2$ is the maximal reductive automorphism group of
$E_6$ and hence of $X$.
\end{trivlist}
\section{The finite group $G/G_1$}
In the weighted homogeneous case,
we are going to study the finite group $G/G_1$ viewed as a subgroup of
the automorphism group of the central curve via the natural homomorphism
$\rho$.
First consider singularities $X$ such that the exceptional divisor is
irreducible, $E=E_0$, of genus $g$. Let $N\to E_0$ be the normal bundle
of $E_0\subseteq\tilde{X}$ with zero-section $E_0\subseteq N$. It follows from work
of Grauert \cite[\S 4]{Grau} (see \cite[6.2]{Wag}) that the germs
$(\tilde{X},E_0)$ and $(N,E_0)$ are isomorphic if the self intersection number
satisfies $E_0\cdot E_0<4 - 4g$. We determine $G$ for these singularities:
\begin{trivlist}
\item[] \bf Theorem 3. \it
Let $\pi:N\to E_0$ be a negative line bundle on a smooth complete curve
$E_0$ of genus $g\ge1$, and let $X$ be the singularity obtained by
contracting the zero-section $E_0\subseteq N$.
\item[] {\rm (i)} Let $\tilde{G}$ be the group of automorphisms $g$ of the
manifold $N$ which restrict to an automorphism $g_0$ of the zero-section
such that $g_0\circ\pi=\pi\circ g$. Then $\tilde{G}$ is the maximal
reductive automorphism group of $X$.
\item[] {\rm (ii)} Let $g=1$ and $-b=E_0\cdot E_0$. Then the natural map
$\rho:G\to{\rm Aut}\: E_0$
has image $({\Bbb Z}_b\times{\Bbb Z}_b)\rtimes{\rm Aut}_0\, E_0$ where
${\Bbb Z}_b\times{\Bbb Z}_b$ is the group of $b$-torsion points of the Abelian group
$E_0$ and ${\rm Aut}_0\, E_0$ is the group of automorphisms fixing the zero
element. The exact sequence $1\to G_1\to G\to G/G_1\to 1$ splits
if and only if $b=1$.
\item[] Proof. \rm
Let $\tilde{\rho}:\tilde{G}\to{\rm Aut}\: E_0$ be the restriction map. It has
kernel ${\Bbb C}^*$. Let $g=1$. We may assume that $N=\O(-b\cdot p_0)$ for
some $p_0\in E_0$. Further we may assume that $p_0$ is the zero element $0$
of the group $E_0$. We claim that $\phi\in{\rm Aut}\: E_0$ belongs to
${\rm im}\:\tilde{\rho}$ if and only if $p=\phi(0)$ is a $b$-torsion point
of $E_0$. In fact, for $\psi=\phi^{-1}$ let
$\psi_*:\O(-b\cdot p)\to\O(-b\cdot 0)$ be the induced map satisfying
$\psi\circ\pi=\pi\circ\psi_*$ (where $\pi$ denotes both bundle
projections). If $\phi=g_0=\tilde{\rho}(g)$ for some $g\in\tilde{G}$
then $g\circ\psi_*:\O(-b\cdot p)\to\O(-b\cdot 0)$ is an isomorphism of
line bundles. Conversely, if $\Psi:\O(-b\cdot p)\to\O(-b\cdot 0)$ is a
line bundle isomorphism then $\Psi\circ \phi_*$ is an element of $\tilde{G}$
restricting to $\phi$. Now observe that the line bundles $\O(-b\cdot p)$
and $\O(-b\cdot 0)$ are isomorphic if and only if the divisors
$-b\cdot p$ and $-b\cdot 0$ are linearly equivalent if and only if
$bp=0$ in the group $E_0$. This proves the claim. It follows from
${\rm Aut}\: E_0= E_0\rtimes{\rm Aut}_0\,E_0$ that
${\rm im}\:\tilde{\rho}=({\Bbb Z}_b\times{\Bbb Z}_b)\rtimes{\rm Aut}_0\, E_0$. Since
$\tilde{\rho}$ has finite image also for $g\ge2$ we conclude that
$\tilde{G}$ is reductive in any case.
\item[]
Now $\tilde{G}$ induces a reductive algebraic subgroup of ${\rm Aut}\: X$.
Hence $\tilde{G}$ is contained in a maximal reductive subgroup $G$
viewed as a group of automorphisms of the germ $(N,E_0)$. As both
$\tilde{G}$ and $G$ are one dimensional they are equal if they have the
same image in ${\rm Aut}\: E_0$. But every $g\in G$ induces a
$\tilde{g}\in\tilde{G}$ with the same restriction to $E_0$ since
$N\to E_0$ is the normal bundle of $E_0\subseteq N$.
\item[]
Let us return to the case $g=1$. If $b=1$ then the divisor $-0$ defining
$N$ is invariant under ${\rm im}\:\rho={\rm Aut}_0\, E_0$. This yields a section
${\rm im}\:\rho\to\tilde{G}=G$.
To prove the non-existence of such a section for $b\ge2$
we write $E_0={\Bbb C}/\L$ where the lattice
$\L$ is generated by the primitive periods $\omega_1,\omega_2$ with
${\rm Im}\:\omega_2/\omega_1>0$. Let $\phi:E_0\to E_0$ be the translation
given by $z\mapsto z+\a$ with $\a=\omega_1/b$. Set $\a_0=-(b+1)/2b\cdot\omega_1$
and $\a_k=\a_0+k\a$ for
$k\in{\Bbb Z}$. Then the divisor $D_1=\sum_{k=1}^b\a_k$ on $E_0$ is
$\phi$-invariant and $\phi$ induces an automorphism of $\O(-D_1)$ over
$\phi$. It is given by
\[s_1(z)\mapsto s_1(z+\a)
\]
with $s_1$ denoting a rational section of $\O(-D_1)$. Because
$\a_1+\ldots+\a_b=0$ in ${\Bbb C}$ the divisors $D_1$ and
$D_0=b\cdot 0$ are linearly equivalent, say $D_1-D_0=(h_1)$ for some
rational function $h_1$ on $E_0$. Using a rational section $s_0$ of
$N=\O(-D_0)$ one defines a line bundle isomorphism $N\to\O(-D_1)$ by
\[s_0(z)\mapsto h_1(z)s_1(z).
\]
We obtain an automorphism $\tilde{\phi}\in\tilde{G}=G$ of $N$ over $\phi$
given by
\[s_0(z)\mapsto\frac{h_1(z)}{h_1(z+\a)}s_0(z+\a).
\]
Now let $\b=\omega_2/b$, $\psi$ the corresponding translation,
$\b_0=-(b+1)/2b\cdot\omega_2$, $\b_k=\b_0+k\b$,
$D_2=\sum_{k=1}^b\b_k$, $D_2-D_0=(h_2)$ and $\tilde{\psi}\in G$
the induced map. Then
$\tilde{\psi}^{-1}\tilde{\phi}^{-1}\tilde{\psi}\tilde{\phi}\in G$
is given by
\[s_0(z)\mapsto\frac{h(z)h(z+\a+\b)}{h(z+\a)h(z+\b)}s_0(z)
\]
where $h=h_1/h_2$. As this commutator is contained in
the kernel of $\rho$ the function
$h(z)h(z+\a+\b)h(z+\a)^{-1}h(z+\b)^{-1}$ is constant, say equal to
$\l\in{\Bbb C}^*$. If we assume that $\rho$ admits a section, hence
that $G={\Bbb C}^*\times B$ with a subgroup $B$, see Remark (iii) of section 2,
then the commutator must be contained in $B$ and $\l=1$. We
are going to show that $\l=\exp(2\pi i/b)$, hence $b=1$.
\item[]
Let $\zeta$ and $\sigma$ be Weierstrass' $\zeta$- and $\sigma$-function.
One has
\[\sigma(z+\omega_i)=-\exp(\eta_i(z+\omega_i/2))\cdot\sigma(z)
\]
with $\eta_i=2\zeta(\omega_i/2)$ for $i=1,2$, see
\cite[II.1.13, Satz 3]{HC}. The elliptic function $h$ has
$\a_1,\ldots,\a_b$ and $\b_1,\ldots,\b_b$ as complete systems of zeros
and poles. Since $\a_1+\ldots+\a_b=\b_1+\ldots+\b_b$ it follows from
\cite[II.1.14, Satz 1]{HC} that
\[h(z)=\prod_{k=1}^b\frac{\sigma(z-\a_k)}{\sigma(z-\b_k)}
\]
up to a constant. One calculates
\begin{eqnarray*}
\frac{h(z)h(z+\a+\b)}{h(z+\a)h(z+\b)} & = &
\frac{\sigma(z-\a_b)\sigma(z-\b_0)\sigma(z+\b-\a_0)\sigma(z+\a-\b_b)}
{\sigma(z-\a_0)\sigma(z-\b_b)\sigma(z+\b-\a_b)\sigma(z+\a-\b_0)} \\
& = & \exp(\eta_1\b-\eta_2\a) \\
& = & \exp((\eta_1\omega_2-\eta_2\omega_1)/b).
\end{eqnarray*}
Legendre's relation $\eta_1\omega_2-\eta_2\omega_1=2\pi i$, see
\cite[II.1.11]{HC}, gives the claim.
\hfill$\Box$
\end{trivlist}
Let $X$ be weighted homogeneous but not a cyclic quotient singularity.
For $i=1,\ldots,r$ let $E_{ij}$, $j=1,\ldots,l_i$, be the curves on
the $i$-th branch
of the exceptional divisor of the minimal good resolution. Assume that
$E_{i1}$ intersects the central curve $E_0$, say in the point $p_i$,
and that $E_{ij}$ intersects $E_{i,j+1}$. Let $-b=E_0\cdot E_0$ and
$-b_{ij}=E_{ij}\cdot E_{ij}$ denote the self intersection numbers. Finally,
consider the Hirzebruch-Jung continued fractions
\[\a_i/\b_i=b_{i1}-1/(b_{i2}-1/(\ldots-1/b_{i,l_i})\ldots)
\]
with coprime positive integers $\b_i<\a_i$.
\begin{trivlist}
\item[] \bf Theorem 4. \it
Let $X$ be weighted homogeneous but not a cyclic quotient singularity
and suppose that the central curve is rational.
\item[] {\rm (i)} Then the image of $\rho:G\to{\rm Aut}\: E_0$ equals
the group $A$ of automorphisms of $E_0$ which permute the points
$p_1,\ldots,p_r$ while preserving the pairs $(\a_i,\b_i)$.
\item[] {\rm (ii)} If $b$ is even or if $A$ is cyclic or if $A$ is
dihedral of order $2q$ with $q$ odd then the maximal reductive automorphism
group is a direct product $G\simeq{\Bbb C}^*\times A$.
\item[] Proof. \rm
Obviously, ${\rm im}\:\rho\subseteq A$. For the other inclusion
we use Pinkham's description \cite[Theorem 5.1]{P} of the affine
coordinate ring of a weighted homogeneous surface. Let the divisor $D_0$
correspond to the conormal bundle of $E_0\subseteq\tilde{X}$. Consider the
divisor
\[D=D_0-\sum_{i=1}^r\b_i/\a_i\cdot p_i
\]
with rational coefficients.
Then the coordinate ring of the affine algebraic variety $X$ is isomorphic,
as a graded ring,
to $\bigoplus_{k=0}^\infty L(kD)$ where $L(kD)$ denotes the space
of rational functions $f$ on $E_0$ with $(f)\ge-kD$. Now recall the
classification of the finite subgroups of ${\rm PSL}(2,{\Bbb C})$ and their orbits
on the projective line, \cite[Chapter I, \S 6]{L}. The cyclic groups
have a fixed point. The dihedral group of order $2q$ has orbits of
length 2 and $q$. And for the tetrahedral, the octahedral and the
icosahedral group the greatest common divisor of the lengths of the
orbits is 2. Thus, if $E_0$ is rational and one of the hypotheses of (ii)
is fulfilled then there exists on $E_0$ an $A$-invariant divisor $D_1$
of degree $b$. The sum in the definition of $D$ is $A$-invariant. Since
$E_0$ is rational the two divisors $D_0,D_1$ of the same degree $b$
are linearly equivalent. Hence we may replace $D_0$ by $D_1$ and assume
that $D$ itself is $A$-invariant.
Now there is an obvious action of $A$ on
$\bigoplus_{k=0}^\infty L(kD)$. We obtain a homomorphism
$A\to{\rm Aut}\: X:\phi\mapsto\phi'$ whose image may be assumed to lie in $G$.
One checks that $\rho(\phi')=\phi$ for all $\phi\in A$. Hence
$A={\rm im}\:\rho$ and the exact sequence
$1\to{\Bbb C}^*\to G\to A\to 1$ splits. Then $G\simeq {\Bbb C}^*\times A$, see
Remark (iii) of section 2.
To prove (i) in the cases not covered by
(ii) apply (ii) to the cyclic groups generated by the elements of $A$.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] \bf Corollary. \it
Let $X$ be weighted homogeneous such that the exceptional divisor
consists of exactly three branches emanating from a rational central curve.
Then $G\simeq{\Bbb C}^* \times {\rm Aut}\:\Gamma$ where $\Gamma$ denotes the weighted dual
resolution graph.
\item[] Proof. \rm
Recall from Remark (i) of section 2 the injection $A\to{\rm Aut}\:\Gamma$. In the
present case it is an isomorphism since three points on the projective line
have no moduli. Part (ii) of the Theorem applies as ${\rm Aut}\:\Gamma$ is
trivial or cyclic of order 2 or isomorphic to the symmetric group
${\rm S}_3$, i.\ e., dihedral of order 6.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] \it Remarks. \rm
(i) The Corollary applies to all non-cyclic quotient singularities.
In this special case the result was previously obtained by
Aumann-K\"orber \cite[3.12]{A-K}. She uses the fact \cite[3.6.3]{WPSPM}
that for a quotient singularity ${\Bbb C}^2/H$ the maximal reductive
automorphism group is ${\rm N}(H)/H$ where ${\rm N}(H)$ denotes the
normalizer of $H$ in ${\rm GL}(2,{\Bbb C})$. Then she explicitly computes this
normalizer for every $H$. The Corollary applies, as well, to triangle
singularities, see Example 2 below.
\item[] (ii) The proof of Theorem 4 shows $G\simeq{\Bbb C}^*\times A$ if $g$ is
arbitrary and the divisor $\sum_{i=1}^r p_i$ corresponds to the
conormal bundle. Such singularities (with $g=1$) have been considered
by Tomaru \cite{T}.
\end{trivlist}
We end this section by showing that every finite group appears
as $G/G_1$ for a suitable $X$.
\begin{trivlist}
\item[] \bf Theorem 5. \it
Let $E_0$ be an arbitrary smooth complete curve and let $A\le{\rm Aut}\: E_0$
be an arbitrary finite subgroup. Then there is a weighted homogeneous
normal surface singularity $X$ with central curve $E_0$ and maximal
reductive automorphism group $G\simeq{\Bbb C}^*\times A$.
\item[] Remark. \rm Hurwitz showed that every finite group
can be realized as a subgroup of ${\rm Aut}\: E_0$ for some smooth complete
curve $E_0$, see \cite{Green}.
\item[] {\it Proof of the Theorem.}
We first show that there exist finitely many $A$-orbits $B_1,\ldots,B_k
\subseteq E_0$ such that $A$ consists of exactly those
$\phi\in{\rm Aut}\: E_0$ with $\phi(B_j)=B_j$ for all $j=1,\ldots,k$.
Take some $A$-orbits $B_1,\ldots,B_l$ and let $A_1$ be the group of
$\phi\in{\rm Aut}\: E_0$ with $\phi(B_j)=B_j$ for $j=1,\ldots,l$. If the
number of elements of $B_1\cup\ldots\cup B_l$ is sufficiently big
(say, $\ge3$ if $E_0$ has genus $g=0$ and $\ge5$ if $g=1$) then $A_1$
is finite. We have $A\subseteq A_1$. If the inclusion is strict then a generic
$A_1$-orbit (not containing any of the finitely many points which are
fixed by some element of $A_1$) has more elements than $A$. Choose an
$A$-orbit $B_{l+1}$ contained in a generic $A_1$-orbit. Then the group
$A_2$ of elements of $A_1$ which moreover map $B_{l+1}$ onto itself
satisfies $A\subseteq A_2\subsetneq A_1$. After finitely many steps we arrive
at the claim. Now attach pairs $(\a_i,\b_i)$ of coprime positive integers
$\b_i<\a_i$ to the points $p_1,\ldots,p_r$ of
$B_1\cup\ldots\cup B_k$ in such a way that two points get the same label
if and only if they belong to the same orbit.
Then $A$ is the group of
automorphisms of $E_0$ which permute the points $p_1,\ldots,p_r$
while preserving the pairs $(\a_i,\b_i)$. Moreover, choose an $A$-invariant
divisor $D_0$ on $E_0$ of degree $b>\sum_{i=1}^r \b_i/\a_i$. Then
Pinkham's construction yields a weighted homogeneous $X$ with the given
data and such that its maximal reductive automorphism group is isomorphic
to ${\Bbb C}^*\times A$, see the proof of Theorem 4.
\hfill$\Box$
\end{trivlist}
\section{Examples}
Catanese \cite[section 2]{C} has studied involutions of rational double
points.
We extend this to all weighted homogeneous singularities with rational
central curve and three branches.
To determine the occuring quotients we need two Lemmas. As usual, the weight
$-2$ is omitted in the resolution graphs.
\begin{trivlist}
\item[] \bf Lemma 1. \it
Let $X=X_{b,1}$ be the cyclic quotient singularity with weighted graph
\[\Aone{-b}
\]
Let $\sigma$ be an involution of $X$ and consider its action on the minimal
resolution $\tilde{X}$ and on the exceptional curve $E$.
\item[] {\rm (i)} If $\sigma$ fixes $E$ pointwise then
$X/\sigma$ has weighted graph
\[\Aone{-2b}
\]
Otherwise there are the following cases:
\item[] {\rm (ii)} If $b$ is odd then one of the two fixed points of $\sigma$
on $E$ is an isolated fixed point on $\tilde{X}$ and the other is not.
Then $X/\sigma$ has weighted graph
\[\Atwo{-(b+1)/2}{}
\]
\item[] {\rm (iii)} If $b$ is even then either both fixed points of $\sigma$
on $E$ are isolated fixed points on $\tilde{X}$ or both are not. The
corresponding weighted graphs are
\[\Athree{}{-(b+2)/2}{}
\qquad\mbox{and}\qquad
\Aone{-b/2}
\]
\item[] Proof. \rm
Let $q$ be a fixed point of $\sigma$ lying on $E$. If $q$ is an isolated
fixed point it gives rise to an $A_1$-singularity in $\tilde{X}/\sigma$
which can be resolved by inserting a rational $(-2)$-curve. Otherwise,
$q$ is mapped onto a smooth point of $\tilde{X}/\sigma$. A resolution of
$\tilde{X}/\sigma$ clearly resolves $X/\sigma$. Thus, according to
whether there are two or one or no isolated fixed points lying on $E$
the quotient $X/\sigma$ has weighted graph
\[\Athree{}{-c}{}
\qquad\mbox{or}\qquad
\Atwo{-c}{}
\qquad\mbox{or}\qquad
\Aone{-c}
\]
with some $c$. To determine $c$ and to prove the other assertions
write $X=X_{b,1}={\Bbb C}^2/H$ where $H$ is generated by
$\left(\begin{array}{cc}
\zeta & 0 \\
0 & \zeta \end{array}\right)$
and $\zeta$ is a primitive $b$-th root of unity. As mentioned in Remark
(iii) of section 2 the maximal reductive automorphism group of $X$ is
${\rm GL}(2,{\Bbb C})/H$. Hence $X/\sigma\simeq{\Bbb C}^2/\Sigma$ with some group
$\Sigma\le{\rm GL}(2,{\Bbb C})$ containing $H$ as a subgroup of index two.
Clearly $\Sigma $ is Abelian. So we may assume that it consists of
diagonal matrices.
\item[]
Consider first the case that $\Sigma$ is cyclic, say generated by $\t$.
We may assume that $\t^2=
\left(\begin{array}{cc}
\zeta & 0 \\
0 & \zeta \end{array}\right)$.
If $\t=
\left(\begin{array}{cc}
\eta & 0 \\
0 & \eta \end{array}\right)$
with a $2b$-th root of unity $\eta$ then $X/\sigma\simeq{\Bbb C}^2/\t=X_{2b,1}$ with
graph as in (i). On
$\tilde{X}=\{(z_1,z_2,(w_1:w_2))\in {\Bbb C}^2\times\P_1,\; z_1w_2^b=z_2w_1^b\}$
we have $\sigma(z,w)=(-z,w)$ and $\sigma$ fixes $E=0\times\P_1$ pointwise. The
next possibility is $\t=
\left(\begin{array}{cc}
\eta & 0 \\
0 & -\eta \end{array}\right) =
\left(\begin{array}{cc}
\eta & 0 \\
0 & \eta^{b+1} \end{array}\right)$
where again $\eta$ is a $2b$-th root of unity. If $b$ is even then $2b$
and $b+1$ are coprime and $X/\sigma\simeq{\Bbb C}^2/\t=X_{2b,b+1}$. The continued
fraction expansion of $2b/(b+1)$ shows that the graph is the first one
in (iii). If $b$ is odd then $b$ and $(b+1)/2$ are coprime and
$\t^b=
\left(\begin{array}{cc}
-1 & 0 \\
0 & 1 \end{array}\right)$.
Since $\t$ acts on ${\Bbb C}^2/\t^b\simeq{\Bbb C}^2$ by
$\left(\begin{array}{cc}
\eta^2 & 0 \\
0 & \eta^{b+1} \end{array}\right) =
\left(\begin{array}{cc}
\zeta & 0 \\
0 & \zeta^{(b+1)/2} \end{array}\right)$
we see $X/\sigma\simeq X_{b,(b+1)/2}$ with graph as in (ii). In these two cases
the assertion on the fixed points is obvious from the graphs.
\item[]
Now consider the case that $\Sigma$ is not cyclic. Then
$\Sigma=H\times<\t>$ with some involution $\t\in{\rm GL}(2,{\Bbb C})$, and $b$ must
be even. As
$\left(\begin{array}{cc}
-1 & 0 \\
0 & -1 \end{array}\right)\in H$ we have
$\t=\left(\begin{array}{cc}
-1 & 0 \\
0 & 1 \end{array}\right)$ without loss of
generality and $\Sigma$ contains the reflection group $T$ generated by
$\t$ and $-\t$.
The generator of $H$ acts on ${\Bbb C}^2/T\simeq{\Bbb C}^2$ by
$\left(\begin{array}{cc}
\zeta^2 & 0 \\
0 & \zeta^2 \end{array}\right)$.
Hence $X/\sigma\simeq X_{b/2,1}$ and the graph is the second one in (iii).
Moreover, $\sigma(z,w)=(z,-w)$ on $\tilde{X}$.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] {\it Remark.} The statement of the Lemma is true also for $b=1$,
i.~e., if
$X=({\Bbb C}^2,0)$ and $\tilde{X}$ is the blow up of $0$. For $b=2$ it
appears in \cite[p.~80]{S}.
\end{trivlist}
\begin{trivlist}
\item[] \bf Lemma 2. \it
Let $X$ be weighted homogeneous but not a cyclic quotient singularity.
Let $\phi$ be an automorphism of $X$ of finite order and consider its
action on the minimal good resolution $\tilde{X}$. Suppose that $\phi$
fixes the intersection point $p$ of two components $E_1,E_2$ of the
exceptional divisor $E$ and that $p$ is not an isolated fixed point of
$\phi$ on $\tilde{X}$. Then near $p$ the fixed point locus coincides with
$E_1$ or $E_2$.
\item[] Proof. \rm
Near $p$ the automorphism $\phi$ can be linearized. In suitable local
coordinates $x,y$ it is given by $\phi(x,y)=(x,\l y)$ with some $\l\not=1$.
As $E$ is star shaped it is not possible that $\phi$ interchanges $E_1$
and $E_2$. Hence they are left invariant. It is easily seen that a
smooth $\phi$-invariant curve through $p$ different from the fixed point
locus $\{y=0\}$ must be tangent to $\{x=0\}$. Since $E_1$ and $E_2$
intersect transversely the Lemma is proven.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] \bf Proposition 1. \it
Let $X$ be weighted homogeneous such that the exceptional divisor consists
of exactly three branches emanating from a rational central curve.
\item[] {\rm (ii)} If ${\rm Aut}\:\Gamma$ is trivial then $X$ has (up to
conjugation) exactly one involution, namely the one contained in ${\Bbb C}^*$.
\item[] {\rm (ii)} If ${\rm Aut}\:\Gamma$ is not trivial then besides
$\sigma_1\in{\Bbb C}^*$ there are (up to conjugation) exactly two more involutions
$\sigma_2$ and $\sigma_3=\sigma_1\sigma_2$ in ${\rm Aut}\: X$. They can be distinguished by
the property that for one of them, say $\sigma_2$, the quotient $X/\sigma_2$
is a cyclic quotient singularity whereas $X/\sigma_3$ is not. Here we agree
that the smooth germ $({\Bbb C}^2,0)$ is called a cyclic quotient singularity,
too.
\item[] Proof. \rm
(i) is clear from the Corollary of Theorem 4. In (ii) we have
$G\simeq{\Bbb C}^*\times {\rm Aut}\:\Gamma$ with ${\rm Aut}\:\Gamma={\Bbb Z}_2$ or $\rm S_3$. Since
$\rm S_3$
has, up to conjugation, only one involution there are only three
involutions to consider in $G$. We may assume that $\sigma_2$ and $\sigma_3$
interchange the second and third branch which therefore get identified
in the quotient. Both involutions fix the intersection point of $E_0$
with the first branch and have a second fixed point $q$ on the
projective line $E_0$. In suitable local coordinates $x,y$ around $q$
with $E_0=\{x=0\}$ we have $\sigma_1(x,y)=(-x,y)$, $\sigma_2(x,y)=(x,-y)$ and
$\sigma_3(x,y)=(-x,-y)$. Hence $q$
is mapped onto a smooth point in $\tilde{X}/\sigma_2$, but in the resolution
of $\tilde{X}/\sigma_3$ there appears a new branch consisting of a single
$(-2)$-curve.
The only isolated fixed points possibly lying on the first branch are
intersection points of components plus, maybe, one point on the curve
at the end of the branch. Thus, after resolving the $A_1$-singularities
appearing in the quotient we are left with a chain of rational curves.
We conclude that $X/\sigma_2$ is a cyclic quotient singularity. And it will
follow that $X/\sigma_3$ has a star shaped graph with three branches (and
hence is not a cyclic quotient singularity) if we can show that the
chain of rational curves arising from the first branch of $X$ cannot be
blown down completely to a smooth point. From Lemma 1 one sees that there
is only one possible way for a $(-1)$-curve to appear. Namely, if the
branch contains a $(-2)$-curve $E_i$, not fixed pointwise, such that the
two fixed points on $E_i$ are not isolated fixed points of the involution
on $\tilde{X}$. But then Lemma 2 shows that the neighbouring components
have to be fixed pointwise. Using Lemma 1 again we see that the
configuration
\[\Athree{-b_{i-1}}{}{-b_{i+1}}
\]
produces
\[\Athree{-2b_{i-1}}{-1}{-2b_{i+1}}
\]
in the quotient, hence
\[
\unitlength0.8mm
\begin{picture}(30,9)
\multiput(5,6)(20,0){2}{\circle*{2}}
\put(0,0){\makebox(0,0)[b]{$-(2b_{i-1}-1)$}}
\put(30,0){\makebox(0,0)[b]{$-(2b_{i+1}-1)$}}
\put(5,6){\line(1,0){20}}
\end{picture}
\]
after blowing down the $(-1)$-curve. Consequently, the blowing down does
not create new $(-1)$-curves. This implies the claim.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] {\it Example 1.} Proposition 1 applies to each non-cyclic
quotient singularity $X$.
As can be seen from \cite{R} the exceptional divisor has a branch
consisting of a single $(-2)$-curve. It follows from Lemma 1 that this
branch disappears in the quotient with respect to the involution
$\sigma_1\in{\Bbb C}^*$. As in the proof of Proposition 1 one then shows that
$X/\sigma_1$ is a cyclic quotient singularity. Of course, it is a simple
task to determine it explicitly in each case. From \cite{R} one sees
that ${\rm Aut}\:\Gamma$ is non-trivial if and only if $X$ is dihedral or
$X=T_m$ is tetrahedral with $m\equiv1$ or 5 mod 6. In these cases let
$\sigma_3$ be the involution for which $X/\sigma_3$ is not a cyclic quotient
singularity. If $X$ is dihedral then $X/\sigma_3$ is dihedral again. And
for $X=T_m$, $m\equiv1$ or 5 mod 6, one obtains $T_m/\sigma_3\simeq O_m$, an
octahedral singularity. More precisely, if $m=6(b-2)+1$ then $T_m$ has
graph
\[\Esix{}{}{}{-b}{}{}
\]
Using Lemmas 1 and 2 one sees that $T_m/\sigma_3$ has graph
\[\Dfive{}{-4}{-(b+1)/2}{}{}
\]
if $b$ is odd, but
\[\Eseven{}{}{}{-(b+2)/2}{}{}{}
\]
if $b$ is even. For $m=6(b-2)+5$ the graphs of $T_m$ and
$O_m\simeq T_m/\sigma_3$
are
\[\Dfour{}{-3}{-b}{-3}
\]
and
\[\Dfour{}{-3}{-(b+1)/2}{-4}
\]
if $b$ is odd, but
\[\Dsix{}{-3}{-(b+2)/2}{}{}{}
\]
if $b$ is even.
\end{trivlist}
Before discussing the next examples we make a digression.
\begin{trivlist}
\item[] \bf Lemma 3. \it For a normal surface singularity $X$ let
\[V(X)=\Gamma(X-0,\Omega^2)/L^2(X-0)
\]
be the vector space of $2$-forms defined
on a deleted neighbourhood of the singular point modulo the subspace
of square integrable forms. Let $H\le{\rm Aut}\:X$ be a finite subgroup and
$\pi:X\to X/H$ the quotient map. Then the pullback of forms induces an
injection $V(X/H)\hookrightarrow V(X)^H$.
Here the upper index $H$ denotes the subspace of $H$-invariants. For
the geometric genus $p_g(X)=\dim\:V(X)$, see \cite[Theorem 3.4]{Laurat},
this yields
\[p_g(X/H)\le p_g(X)
\]
with strict inequality if the representation of $H$ on $V(X)$ is not
trivial. If $\pi$ is unramified outside the singular point then
$V(X/H)=V(X)^H$.
\item[] Proof. \rm
For $\a\in\Gamma(X/H-0,\Omega^2)$ the pullback $\pi^*\a\in\Gamma(X-0,\Omega^2)$
is $H$-invariant. And it is square integrable if and only if $\a$ is so.
If $\pi$ is unramified outside $0$ then every $H$-invariant form $\a'$ on
$X-0$ induces a form $\a$ on $X/H-0$ with $\a'=\pi^*\a$.
\hspace*{\fill}$\Box$
\end{trivlist}
Let $X$ be weighted
homogeneous but not a cyclic quotient singularity. Suppose that $X$ is
Gorenstein with nowhere zero $2$-form $\omega\in\Gamma(X-0,\Omega^2)$. We
may assume \cite[p.~56]{Ga} that $\omega$ is $G$-equivariant:
$g^*\omega=\chi(g)\cdot\omega$ for all $g\in G$ with some character
$\chi:G\to{\Bbb C}^*$.
Let $\bar{G}$ be the kernel of $\chi$.
Ganter \cite[Lemma 8.3]{Ga} has shown
that $X\to X/\bar{G}$ is unramified outside the singular point.
In particular, $\omega$ induces on $X/\bar{G}-0$ a nowhere zero $2$-form
and $X/\bar{G}$ is Gorenstein.
There is an integer $\varepsilon$ such that $\chi(t)=t^{-\varepsilon}$
for all $t\in{\Bbb C}^*=G_1$. It is known \cite[Corollary 3.3]{Wjac} that $\varepsilon<0$
if and only
if $\varepsilon=-1$ if and only if $X$ is a rational double point. And $\varepsilon=0$
if and only if $X$ is simple elliptic.
It may be mentioned at this place that Ganter \cite[Theorem 8.5]{Ga}
has obtained the bound
\[|G/G_1|\le 42\cdot(-P_X\cdot P_X)/\varepsilon
\]
if $X$ is not log-canonical. The invariant $-P_X\cdot P_X$ can be
calculated (using the notation of section 3) as
\[-P_X\cdot P_X=
\frac{(2g-2+r-\sum_{i=1}^r1/\a_i)^2}{b-\sum_{i=1}^r\b_i/\a_i},
\]
see \cite[Theorem 3.2]{WPP}.
If we now assume that $\varepsilon=\pm1$ then
$G={\Bbb C}^*\times\bar{G}$. Thus we obtain a ``natural'' section
$G/G_1\hookrightarrow G$ of $G\to G/G_1$.
\begin{trivlist}
\item[] {\it Example 2.} Consider the triangle singularities $X=D_{p,q,r}$
with graph
\[\Dfour{-r}{-p}{-1}{-q}
\]
It is known that they are minimally elliptic (i.~e., Gorenstein with
$p_g(X)=1$, see \cite{Lauell})
and have $\varepsilon=1$, \cite{Wjac}. Hence $G={\Bbb C}^*\times\bar{G}$.
As $V(X)$ is spanned by the nowhere zero form $\omega$ we see that
$X/H$ is minimally elliptic for $H\subseteq\bar{G}$ but $p_g(X/H)=0$
(i.~e., $X/H$ is rational) if $H\not\subseteq\bar{G}$. Let us determine
$X/\bar{G}$
in case $\bar{G}\simeq{\rm Aut}\:\Gamma$ is not trivial. First take $X=D_{p,q,q}$
with $p\not=q$. Then $\bar{G}$ is generated by the involution $\sigma_3$ of
Proposition 1. Hence $D_{p,q,q}/\bar{G}\simeq D_{2p,2,q}$ by an application
of Lemmas 1 and 2.
For $X=D_{p,p,p}$ we have $\bar{G}\simeq{\rm S}_3$. Let $H\le\bar{G}$ be
the cyclic subgroup of order three. As $H$ acts freely on $\tilde{X}-E$
the two fixed points lying on $E_0$ must be isolated. Lemma 4 below
shows $D_{p,p,p}/H\simeq D_{3,3,p}$. Then
$D_{p,p,p}/\bar{G}\simeq D_{3,3,p}/\sigma_3\simeq D_{2,3,2p}$.
\end{trivlist}
\begin{trivlist}
\item[] \bf Lemma 4. \it Let $X=X_{b,1}$ with weighted graph
\[\Aone{-b}
\]
Let $\sigma$ be an automorphism of $X$ of order three and consider its
action on $\tilde{X}$ and on $E$. If $\sigma$ fixes $E$ pointwise then
$X/\sigma$ has weighted graph
\[\Aone{-3b}
\]
Otherwise there are the following possibilities:
\[\Afour{}{}{-(b+3)/3}{-3}
\qquad\mbox{or}\qquad
\Aone{-b/3}
\]
if
$b\equiv0$ {\rm mod} $3$,
\[\Athree{-3}{-(b+2)/3}{-3}
\qquad\mbox{or}\qquad\qquad
\Athree{-(b+2)/3}{}{}
\]
if
$b\equiv1$ {\rm mod} $3$, and
\[\Afive{}{}{-(b+4)/3}{}{}
\qquad\mbox{or}\qquad\qquad
\Atwo{-(b+1)/3}{-3}
\]
if
$b\equiv2$ {\rm mod} $3$.
The number of isolated fixed points lying on $E$ can be seen from the
graphs.
\item[] Proof. \rm
Similar to the proof of Lemma 1.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] {\it Example 2 (continued).} Consider the fourteen triangle
singularities which can be embedded as a hypersurface in $({\Bbb C}^3,0)$,
see \cite[Theorem 3.13 and section V]{Lauell}. In eleven cases only
one of the
three weights of the ${\Bbb C}^*$-action is odd so that the involution
$\sigma_1\in{\Bbb C}^*$ is a reflection. By \cite[4.2]{MDiss} the quotient
$X/\sigma_1$ will be smooth or an isolated hypersurface singularity, hence
(as it is rational) a rational double point.
For $D_{3,3,6}=Q_{12}$ and $D_{3,4,5}=S_{12}$ the ${\Bbb C}^*$-action has
exactly two odd weights. The quotient $D_{3,3,6}/\sigma_1$ is a rational
triple point with graph
\[\Esix{-3}{}{}{}{}{}
\]
And $D_{3,4,5}/\sigma_1$ coincides with the icosahedral quotient singularity
$I_{13}$ of graph
\[\Esix{}{}{-3}{}{}{}
\]
This seems to be a quite interesting singularity: It plays an exceptional
role in Manetti's \cite{Ma} study of smooth curves on rational surface
singularities.
Finally, for $X=D_{3,3,5}=Z_{13}$ all three weights are odd. Hence
$X\to X/\sigma_1$ is unramified outside the singular point. In fact, it is
the canonical Gorenstein cover \cite[section 4]{Wrat} of the rational
quadruple point $X/\sigma_1$ with graph
\[
\unitlength0.8mm
\begin{picture}(80,49)
\multiput(0,6)(20,0){5}{\circle*{2}}
\multiput(40,26)(0,20){2}{\circle*{2}}
\put(43,26){\makebox(0,0)[l]{$-3$}}
\put(0,6){\line(1,0){80}}
\put(40,6){\line(0,1){40}}
\end{picture}
\]
\end{trivlist}
\begin{trivlist}
\item[] {\it Example 3.}
Consider the fourteen exceptional families $f_a=f_0+a\cdot g$, $a\in{\Bbb C}$,
of unimodal singularities, \cite[part II]{AGV}.
Here the $f_0$ are weighted homogeneous and
define exactly the fourteen triangle singularities $X_0$ which can be
embedded as a hypersurface in $({\Bbb C}^3,0)$.
For $a\not= 0$ the $f_a$ are semi-weighted homogeneous and
define a singularity $X_1$ whose analytic type is independent of $a$.
The family $f_a$ is topologically trivial. Hence $X_0$ and
$X_1$ have the same resolution graph $\Gamma$, \cite[Theorem 2]{N}. We claim
that the maximal reductive automorphism group of $X_1$ is
${\Bbb Z}_2\times{\rm Aut}\:\Gamma$. For $i=0,1$ let $H_i^*$ be the group of
$\phi\in{\rm GL}(3,{\Bbb C})$ with $\phi f_i=c\cdot f_i$ for some $c\in{\Bbb C}^*$. By
looking at the polynomial $f_0$ one sees
that there is a subgroup $B\subseteq{\rm GL}(3,{\Bbb C})$, isomorphic to ${\rm Aut}\:\Gamma$,
centralized by ${\Bbb C}^*$, intersecting ${\Bbb C}^*$ trivially and such that
${\Bbb C}^*\times B\subseteq H_0^*$. It follows from Proposition 2 in section 5
below that ${\Bbb C}^*\times B=H_0^*$.
Looking at $g$, which is in fact a monomial of (weighted) degree
$\deg\:g=\deg\:f_0+2$, one sees that the subgroup ${\Bbb Z}_2\times B$ is
contained in $H_1^*$. Now let $G$ be a maximal reductive automorphism
group of $X_1$ acting on $({\Bbb C}^3,0)$ by contact equivalences of $f_1$
and containing ${\Bbb Z}_2\times B$.
For each $\phi\in G$ there is a unit $u$, say with constant term $c$,
such that
\begin{equation}
f_1\circ\phi=u\cdot f_1=c\cdot f_0+c\cdot g+
\;\mbox{terms of degree}\; \ge d+3
\end{equation}
where $d=\deg\:f_0$. Here we have used that the weights $w_i$ are $\ge 3$.
Let $\phi^i$ be the components of $\phi$ and write
$\phi^i=\sum\phi_{ij}$ where $\deg\:\phi_{ij}=w_i+j$.
It follows from $f_1\circ\phi=u\cdot f_1$ via
\cite[Theorem 2.1]{GHP} that in the sum for $\phi^i$ only indices $j\ge0$
occur. Writing $\phi_j=(\phi_{1j},\phi_{2j},\phi_{3j})$ we obtain
\begin{equation}
f_1\circ\phi=f_0\circ\phi_0+(\partial_xf_0\circ\phi_0)\cdot\phi_1
+\;\mbox{terms of degree}\; \ge d+2
\end{equation}
and $f_0\circ\phi_0=c\cdot f_0$. In most of the fourteen cases (namely,
if ${\rm Aut}\:\Gamma=1$) the only monomial of degree $w_i$ is $x_i$. But also in
the remaining cases one easily sees that $f_0\circ\phi_0=c\cdot f_0$
forces $\phi_0$ to be linear. Therefore $\phi\mapsto\phi_0$ defines a
homomorphism $G\to H_0^*={\Bbb C}^*\times B$. It is not clear that $\phi_0$ is
the linear part of $\phi$. But the usual linearization trick produces an
analytic map germ $\psi$ with $\psi_0=1$ and
$\phi_0\circ\psi=\psi\circ\phi$ for all $\phi\in G$. It follows from
$\psi_0=1$ that the linear part of $\psi$ is triangular with trivial
diagonal (if the weights are suitably ordered) and hence that $\psi$
is invertible. Consequently $G$ is mapped isomorphically onto its image
in ${\Bbb C}^*\times B$. It remains to show that this image is contained in
${\Bbb Z}_2\times B$. As $G$ contains $B$ it is enough to consider
$\phi\in G$ with $\phi_0\in{\Bbb C}^*$, say
$\phi_0(x)=t\cdot x=(t^{w_1}x_1,t^{w_2}x_2,t^{w_3}x_3)$.
Comparing (1) and (2) once more we obtain
\[0=(\partial_xf_0(t\cdot x))\cdot\phi_1
=\sum_it^{d-w_i}\cdot\phi_{i1}\cdot\partial_{x_i}f_0.
\]
Since the partials of $f_0$ form a regular sequence in $\O_3$ we conclude
that the $\phi_{i1}$ are contained in the Jacobian ideal $j(f_0)$.
In each of the fourteen cases, the degree of every partial is larger than
$w_i+1$ for all $i$.
Hence $\phi_1=0$. Now we have a more precise
version of (2):
\[f_1\circ\phi=f_0(t\cdot x)+g(t\cdot x)
+(\partial_xf_0(t\cdot x))\cdot\phi_2+
\;\mbox{terms of degree}\;\ge d+3.
\]
We obtain
$c\cdot g=t^{d+2}\cdot g+(\partial_xf_0(t\cdot x))\cdot\phi_2$. The
second summand must vanish because otherwise $g\in j(f_0)$ which is
not the case. Then $c\cdot g=t^{d+2}\cdot g$ and
$c\cdot f_0=f_0(t\cdot x)=t^d\cdot f_0$ imply $t=\pm1$ proving the claim.
\end{trivlist}
\begin{trivlist}
\item[] {\it Example 4.}
Consider the quadrilateral singularities $X$ with graph
\[
\unitlength0.8mm
\begin{picture}(40,43)
\multiput(0,20)(20,0){3}{\circle*{2}}
\multiput(20,0)(0,40){2}{\circle*{2}}
\put(0,14){\makebox(0,0)[b]{$-p$}}
\put(40,14){\makebox(0,0)[b]{$-q$}}
\put(23,40){\makebox(0,0)[l]{$-r$}}
\put(23,0){\makebox(0,0)[l]{$-s$}}
\put(0,20){\line(1,0){40}}
\put(20,0){\line(0,1){40}}
\end{picture}
\]
Of course, the analytic type depends on the cross ratio of the four
intersection points on the central curve $E_0\simeq\P_1$.
The symmetric group ${\rm S}_4$ acts on
${\Bbb C}-\{0,1\}$ via the cross ratio. The subgroup ${\Bbb Z}_2\times{\Bbb Z}_2$
consisting of $1$ and the three products of two disjoint transpositions
acts trivially. The quotient map is
\[j:{\Bbb C}-\{0,1\}\to{\Bbb C}:\l\mapsto
\frac{4(\l^2-\l+1)^3}{27\l^2(\l-1)^2}.
\]
In particular, if the four intersection points on the central curve
have cross ratio $\l$ (with respect to some numbering) then
$j(X)=j(\l)$ is an invariant of the singularity (independent of the
numbering). The generic orbit of ${\rm S}_4$ on ${\Bbb C}-\{0,1\}$ has
cardinality six. There are two exceptional orbits: One of them,
corresponding to $j=0$, consists of the two solutions of
$\l^2-\l+1=0$, and the other, corresponding to $j=1$, consists of
$-1$, $2$ and $1/2$. These well known facts are enough to determine,
in each case, the group
$A\simeq G/G_1$ of automorphisms
of $\P_1$ which permute the intersection points while preserving the
self intersection numbers. If $p=q=r=s$ then $A$ coincides with the
alternating group ${\rm A}_4$ for $j(X)=0$. For $j(X)=1$ it is dihedral
of order
$8$ and for $j(X)\not=0,1$ it is ${\Bbb Z}_2\times{\Bbb Z}_2$.
If $p=q=r\not=s$ then $A\simeq{\Bbb Z}_3$ for $j(X)=0$, $A\simeq{\Bbb Z}_2$
generated by a transposition for
$j(X)=1$, and $A=1$ for $j(X)\not=0,1$.
For the remaining cases let $\l$ be the cross ratio of the four
intersection points $p_1,\ldots,p_4$ (in this order) and suppose that
the curves intersecting in $p_1$ and $p_2$ have equal self intersection
number $p=q$ whereas $r$ and $s$ are different from $p$. There is an
automorphism of $\P_1$ interchanging $p_1$ and $p_2$ while fixing $p_3$
and $p_4$ if and only if $\l=-1$. If $r\not=s$ we see $A\simeq{\Bbb Z}_2$
for $\l=-1$ and $A=1$ else.
If $r=s$
and $\l=-1$ then $A\simeq{\Bbb Z}_2\times{\Bbb Z}_2$ is generated by the two
transpositions interchanging $p_1$ and $p_2$, respectively $p_3$ and $p_4$.
Finally, if $r=s$ and $\l\not=-1$
then $A\simeq{\Bbb Z}_2$ is generated by the product of those two
transpositions.
The quadrilateral singularities
are minimally elliptic with $\varepsilon=1$, \cite{Lauell} and \cite{Wjac}.
Hence $G=G_1\times\bar{G}$ is a direct product, $X/H$ is
rational for finite subgroups $H\le G$ with $H\not\subseteq\bar{G}$ and
$X/H$ is minimally elliptic if $H\subseteq\bar{G}$.
The reader is invited to determine the quotient $X/\bar{G}$. In each
case, it is a triangle or a quadrilateral singularity.
\end{trivlist}
\section{The action on the homology of the link}
The link of a weighted homogeneous surface singularity $(X,0)$ is
a deformation retract of $X-0$ with $X$ denoting the corresponding affine
algebraic variety. The group $G$ acts on $X-0$.
Since the connected subgroup $G_1$ acts trivially on
integral homology there is an action of $G/G_1$ on
$H_1(X-0,{\Bbb Z})$.
\begin{trivlist}
\item[] \bf Theorem 6. \it
Let $X$ be weighted homogeneous.
Then the group $G/G_1$ acts faithfully on $H_1(X-0,{\Bbb Z})$.
\item[] Proof. \rm
In the sequel all homology and cohomology modules are with integral
coefficients. We need to recall how the homology of the link is expressed
in terms of resolution data, see e.~g.\ \cite[section 4]{LW}.
Let $s$ be the number of components of the exceptional
divisor $E$ of the minimal good resolution $\tilde{X}$ and let $g$ be the
genus of the central curve $E_0$. (We suppose that $X$ is not a cyclic
quotient singularity. For those the proof is left to the reader.)
Since $E$ is a deformation retract
of $\tilde{X}$ we have $H_2(\tilde{X})\simeq H_2(E)\simeq {\Bbb Z}^s$ and
$H_1(\tilde{X})\simeq H_1(E)\simeq H_1(E_0)\simeq {\Bbb Z}^{2g}$. Write
$L=\tilde{X}-E\simeq X-0$. Then Lefschetz Duality gives
$H_1(\tilde{X},L)\simeq H^3(E)=0$ and
$H_2(\tilde{X},L)\simeq H^2(E)\simeq H_2(E)'$ with the prime denoting the
dual ${\Bbb Z}$-module. Observe that all identifications are equivariant with
respect to $G/G_1$.
The exact homology sequence of the pair
$(\tilde{X},L)$ comes down to
\[H_2(E)\stackrel{j}{\to}H_2(E)'\to H_1(L)\to H_1(E_0)\to 0
\]
yielding the exact sequence
\[0\to{\rm coker}\:j\to H_1(L)\to H_1(E_0)\to 0.
\]
Here $j$ denotes the adjoint of the intersection product:
\[j(a):b\mapsto a\cdot b \quad\mbox{for}\quad a,b\in H_2(E).
\]
As the intersection matrix is negative definite the discriminant group
${\rm coker}\:j$ is torsion and hence equals the torsion subgroup $H_1(L)_t$
of $H_1(L)$. Recall from \cite[V.3.1]{FK} that ${\rm Aut}\: E_0$ acts
faithfully on $H_1(E_0)$ for $g\ge2$. This is not true for $g=1$ but then
it is easily seen that the group ${\rm Aut}_{p_0}\, E_0$ of automorphisms
fixing a point $p_0\in E_0$ acts faithfully on $H_1(E_0)$.
Now consider the natural homomorphism $G/G_1\to{\rm Aut}\:\Gamma$ and
let $\phi\in G/G_1$ be trivial on $H_1(L)$. We claim that
triviality on $H_1(L)_t$ forces $\phi$ to act trivially on $\Gamma$.
Accepting this for the moment, we are done in case
$g=0$ since then $G/G_1\to{\rm Aut}\:\Gamma$ is injective, see
Remark (i) of section 2. For $g\ge1$ we conclude that $\phi$, viewed as
an automorphism of the central curve $E_0$, has to fix the $r$ intersection
points of $E_0$ with other components of $E$. As $\phi$ is trivial on the
free part $H_1(E_0)$ of $H_1(L)$, too, we obtain $\phi=1$ in all cases
except for simple elliptic $X$. Then,
by Theorem 3 we have
$G/G_1\simeq({\Bbb Z}_b\times{\Bbb Z}_b)\rtimes{\rm Aut}_0\, E_0$
with $-b=E_0\cdot E_0$. It is known \cite[pp.~282 - 283]{LW}
that ${\Bbb Z}_b\times{\Bbb Z}_b$ is mapped isomorphically onto the group
${\rm Hom}({\Bbb Z}^2,{\Bbb Z}_b)$ of automorphisms of $H_1(L)$ which act trivially
on both $H_1(L)_t\simeq{\Bbb Z}_b$ and $H_1(E_0)\simeq{\Bbb Z}^2$. As ${\rm Aut}_0\, E_0$
acts faithfully on $H_1(E_0)$ we conclude that $G/G_1$ acts faithfully
on $H_1(L)$.
\item[] Let us prove the claim. For $i=1,\ldots,r$ let $E_{ij}$,
$j=1,\ldots,l_i$, be the curves on the $i$-th branch of the exceptional
divisor (counted beginning at the centre), and $-b_{ij}=E_{ij}\cdot E_{ij}$
as in section 3. Moreover, write $E_{00}=E_0$ and $-b=E_0\cdot E_0$.
Let $\l_{ij}$ be the basis of $H_2(E)'$ dual to the basis $E_{ij}$ of
$H_2(E)$. The action of $G/G_1$ on $H_2(E)$ is given by permutation
of the curves: $\phi E_{ij}=E_{\phi(i,j)}$. Thus we have
\[(\phi\l_{ij})(a)=a_{\phi(i,j)}\quad\mbox{for all}\quad
a=\sum a_{ij}E_{ij}\in H_2(E).
\]
Triviality of $\phi$ on $H_1(L)_t={\rm coker}\: j$ means
\[\phi\l-\l\in{\rm im}\: j\quad\mbox{for all}\quad\l\in H_2(E)'.
\]
Now assume that $\phi$ is not trivial on $\Gamma$. Then one may assume that
$\phi E_{11}=E_{21}$. There is $x=\sum x_{ij}E_{ij}\in H_2(E)$ such
that $\phi\l_{11}-\l_{11}=j(x)$, i.~e., $x\cdot a=a_{21}-a_{11}$ for all
$a\in H_2(E)$. This gives $x\cdot E_{11}=-1$, $x\cdot E_{21}=1$, and
$x\cdot E_{ij}=0$ for all other $(i,j)$. By looking at the curves on the
$i$-th branch one obtains $x_{i,l_i-1}=b_{i,l_i}\cdot x_{i,l_i}$, then
$x_{i,l_i-2}=b_{i,l_i-1}\cdot x_{i,l_i-1}-x_{i,l_i}
=(b_{i,l_i-1}-1/b_{i,l_i})\cdot x_{i,l_i-1}$ and so on up to
\[x_{00}=\a_i/\b_i\cdot x_{i1}+\gamma_i
\]
where $\a_i/\b_i$ is the Hirzebruch-Jung continued fraction of the $i$-th
branch and $\gamma_i=-1$, $1$ or $0$ according to $i=1$, $2$ or else.
Finally, $x\cdot E_0=0$ implies
\[0=-b\cdot x_{00}+\sum_{i=1}^r x_{i1}
=(\sum_{i=1}^r \b_i/\a_i -b)\cdot x_{00}
\]
because $(\a_1,\b_1)=(\a_2,\b_2)$. The intersection matrix being negative
definite we have $b>\sum_{i=1}^r\b_i/\a_i$ and $x_{00}=0$. Then
$1=\a_1/\b_1\cdot x_{11}$ with $\a_1/\b_1>1$ and $x_{11}\in{\Bbb Z}$ yields
a contradiction proving the claim.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] \it Remark. \rm
The proof shows that $G/G_1$ even acts faithfully on the torsion
subgroup of $H_1(X-0,{\Bbb Z})$ if $r>2g+2$.
\end{trivlist}
Consider a weighted homogeneous normal surface singularity $X$ which can be
embedded as a hypersurface in $({\Bbb C}^3,0)$ but which is not an
$A_k$-singularity. We may choose coordinates such that its maximal
reductive automorphism group $G$ acts linearly on ${\Bbb C}^3$ and such that
${\Bbb C}^*=G_1\le G$ acts diagonally. Then $X$ is defined by a weighted
homogeneous polynomial $f$, say of degree $d$.
\begin{trivlist}
\item[] \bf Proposition 2. \it
In this situation $G$ equals the group $H^*$ of $\phi\in{\rm GL}(3,{\Bbb C})$ with
$\phi f=c\cdot f$ for some $c\in{\Bbb C}^*$. The subgroup $H$ of
$\phi\in{\rm GL}(3,{\Bbb C})$ with $\phi f=f$ is finite. The intersection
$H\cap G_1=H\cap{\Bbb C}^*$ is the cyclic group ${\Bbb Z}_d$ of $d$-th roots of unity
and
$G/G_1\simeq H/{\Bbb Z}_d$.
\item[] Proof. \rm
First look at $H$. This is an algebraic subgroup of ${\rm GL}(3,{\Bbb C})$. So one has
to show that its Lie algebra $\bf h$ consisting of all derivations
$D=\sum \l_i\partial_{x_i}$ with linear forms $\l_i$ such that
$Df=0$ is reduced to $0$. For $f\in m^3$ this is shown by a standard
argument which can be found at several places in the literature, e.~g.\
in \cite{OS}. Now suppose that $f\notin m^3$. Since $f$ does not define
an $A_k$-singularity we may assume $f=x_1^2+g$ with $g\in m^3$. The argument
just mentioned shows that every $D\in\bf h$ is of form
$D=x_1\sum_{i>1}a_i\partial_{x_i}$ with $a_i\in{\Bbb C}$. But then $Df=0$
clearly implies $D=0$. Now turn to $H^*$. This is also an algebraic
subgroup being the image under the projection
${\rm GL}(3,{\Bbb C})\times{\Bbb C}^*\to{\rm GL}(3,{\Bbb C})$ of the algebraic group of
all pairs $(\phi,c)$ with $\phi f=c\cdot f$. Its Lie algebra consisting
of all
$D=\sum\l_i\partial_{x_i}$ such that $Df\in{\Bbb C}\cdot f$ is one dimensional
spanned by the Euler derivation. As ${\Bbb C}^*\subseteq H^*$ this implies that
$H^*$ is reductive. Because $G$ is a group of linear contact equivalences
of $f$ and because ${\Bbb C}^*$ is central in $G$ one has $G\subseteq H^*$. But
then $G=H^*$ by maximality. The remaining assertions are obvious.
\hfill$\Box$
\end{trivlist}
\begin{trivlist}
\item[] {\it Example 5.} Let $X$ be defined by $f=x_1^d+x_2^d+x_3^d$ with
$d\ge 3$ and let $H,H^*$ be as above. Since the three weights of the
${\Bbb C}^*$-action are equal to $1$ any automorphism of $({\Bbb C}^3,0)$
commuting with ${\Bbb C}^*$ is linear in the given coordinates. This shows
that $H^*$ is the maximal reductive automorphism group of $X$, i.\ e.,
the coordinates are well chosen, compare Remark (vii) of section 2. We
clearly have ${\Bbb Z}_d^3\rtimes{\rm S}_3\subseteq H$. To
prove equality take $\phi\in{\rm GL}(3,{\Bbb C})$ with $\phi f=f$. The ideal
$((x_1x_2x_3)^{d-2})$ generated by the Hessian of $f$ is $\phi$-stable.
Consequently $(x_1x_2x_3)$ is $\phi$-stable and, after permutation,
the ideals $(x_1)$, $(x_2)$ and $(x_3)$ must be $\phi$-stable. This
shows $H={\Bbb Z}_d^3\rtimes{\rm S}_3$. We conclude that
$G/G_1\simeq H/{\Bbb Z}_d$ has order $6d^2$. When $d$ is a multiple of $3$ we
obtain examples where $G\to G/G_1$ does not admit a section. In fact,
then the third root of unity $\zeta\in{\Bbb C}^*\le G$ is contained in the
commutator subgroup of $G$, namely
$\zeta=\phi\sigma\phi^{-1}\sigma^{-1}$ with
$\phi(x_1,x_2,x_3)=(\zeta^2x_1,\zeta x_2,x_3)$ and
$\sigma(x_1,x_2,x_3)=(x_2,x_3,x_1)$. Clearly this prevents $G$ from being a
direct product of ${\Bbb C}^*$ and some subgroup.
(Note that for $d=3$ we are discussing the simple elliptic singularity
obtained by contracting the zero-section of a line bundle of degree
$-3$ on the elliptic curve of $j$-invariant $0$.)
On the contrary, if $3$ does not divide $d$ then $G$ is a direct product
over ${\Bbb C}^*$. To prove this consider the normal subgroup
$N={\Bbb Z}_d^3\cap{\rm SL}(3,{\Bbb C})$ of ${\Bbb Z}_d^3$. It has trivial intersection
with the center ${\Bbb Z}_d$ of $H$ because $d$ is not a multiple of $3$.
Then $|N|=d^2$ implies ${\Bbb Z}_d^3={\Bbb Z}_d\times N$. As $N$ is
${\rm S}_3$-invariant we conclude $H={\Bbb Z}_d\times B$ for
$B=N\rtimes {\rm S}_3$ and then $G={\Bbb C}^*\times B$.
\end{trivlist}
Returning to the general situation as described in Proposition 2,
choose an $H$-invariant Hermitean inner product on ${\Bbb C}^3$ and let
$\bar{B}_\varepsilon$ be the corresponding closed ball of small radius $\varepsilon$. One
has the Milnor fibration
\[f^{-1}(\bar{D}_\delta-0)\cap\bar{B}_\varepsilon\to \bar{D}_\delta-0
\]
where $\bar{D}_\delta\subseteq{\Bbb C}$ is a small closed disc. Then clearly
the group $H$ of
right equivalences of $f$ acts on the Milnor fibre $F$. Observe that by
an equivariant version \cite[Lemma 4]{DEhr} of the Ehresmann Fibration
Theorem any two Milnor fibres are $H$-equivariantly diffeomorphic.
Moreover, let $M=H_2(F,{\Bbb Z})$ equipped with the intersection form be the
Milnor lattice, $O(M)$ its group of isometries, and $W\le O(M)$ the
monodromy group.
\begin{trivlist}
\item[] \bf Theorem 7. \it
The homomorphism $H\to O(M)$ is injective and $H\cap W\simeq{\Bbb Z}_d$,
generated by the monodromy operator.
\item[] Proof. \rm
Consider the Jacobian ideal $j(f)$ and $U=\O_3/j(f)$. Clearly $H$ acts
on $U$. Wall \cite{Wsym}, see also \cite{OS}, has constructed an
isomorphism $H_2(F,{\Bbb C})\simeq U'\otimes \Lambda^3{\Bbb C}^3$ of $H$-modules.
Let $\eta\in H$ be trivial on $M=H_2(F,{\Bbb Z})$. As
the basis element $1$ of $U$ is an eigenvector of eigenvalue $1$
we conclude
that $\det\:\eta=1$ and hence that $\eta$ must be trivial on $U$.
Consequently $\eta x_i\equiv x_i$ mod $j(f)$ for the coordinate functions
$x_i$. Because $X$ is not an $A_k$-singularity we may assume that
$x_1$ is the only linear form contained in $j(f)$. Hence $\det\:\eta=1$
implies
\[\eta=\left(
\begin{array}{ccc} 1 & * & * \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right)
\]
Thus $\eta$, having finite order, must be trivial.
\item[]
As $X$ is defined by a
weighted homogeneous polynomial of degree $d$ the $d$-th root of unity
in $H$ induces a monodromy diffeomorphism $F\to F$. Therefore the
monodromy operator has order $d$ and is contained in $H$ as acting on $M$.
To show $H\cap W\subseteq {\Bbb Z}_d$ consider the exact homology sequence of the
pair $(F,\partial F)$. By Lefschetz Duality it looks as follows:
\[H_2(F,{\Bbb Z})\stackrel{j}{\to}H_2(F,{\Bbb Z})'\to H_1(\partial F,{\Bbb Z})\to0
\]
where $j$ is the adjoint of the intersection form. The monodromy group
$W$ is generated by Picard-Lefschetz transformations. These are
reflections at hyperplanes orthogonal to the vanishing cycles. In
particular, $W$ acts trivially on $M/M'$. Therefore $H\cap W$ acts
trivially on $H_1(\partial F,{\Bbb Z})$. Because the fibre bundle
$f^{-1}(\bar{D}_\delta)\cap S_\varepsilon\to \bar{D}_\delta$ is trivial and the link
$L=X\cap S_\varepsilon$ is a deformation retract of $X-0$ there are $H$-equivariant
isomorphisms $H_1(\partial F,{\Bbb Z})\simeq H_1(L,{\Bbb Z})\simeq H_1(X-0,{\Bbb Z})$.
Then Theorem 6 implies that any $\eta\in H\cap W$ must be contained
in $H\cap G_1={\Bbb Z}_d$.
\hfill$\Box$
\end{trivlist}
|
1997-07-16T16:38:14 | 9604 | alg-geom/9604016 | en | https://arxiv.org/abs/alg-geom/9604016 | [
"alg-geom",
"math.AG"
] | alg-geom/9604016 | Pat Gilmer | Patrick M. Gilmer | Floppy Curves with Applications to Real Algebraic Curves | AmS-TeX- Version 2.1, 38 pages with 16 figures,needs epsf.tex The
estimate 9.6 has been improved with corresponding changes in 4.1. The
exposition in the proof of 3.4 has been improved. Other minor changes | Real Algebraic Geometry and Ordered Systems, ed. C. Delzell, J.
Madden, Cont. Math. 253, American Math Society (2000) 39-76 | 10.1090/conm/253 | null | null | We show how one may sometimes perform singular ambient surgery on the complex
locus of a real algebraic curve and obtain what we call a floppy curve. A
floppy curve is a certain kind of singular surface in CP(2), more general than
the complex locus of a real nodal curve. We derive analogs for floppy curves of
known restrictions on real nodal curves. In particular we derive analogs of
Fielder's congruence for certain nonsingular curves and Viro's inequalities for
nodal curves which generalize those of Arnold and Petrovskii for nonsingular
curves. We also obtain a determinant condition for certain curves which are
extremal with respect to some of these equalities. One may prohibit certain
schemes for real algebraic curves by prohibiting the floppy curves which result
from singular ambient surgery. In this way, we give a new proof of Shustin's
prohibition of the scheme $1<2\coprod 1<18>>$ for a real algebraic curve of
degree eight.
| [
{
"version": "v1",
"created": "Wed, 24 Apr 1996 20:31:10 GMT"
},
{
"version": "v2",
"created": "Wed, 16 Jul 1997 14:37:45 GMT"
}
] | 2015-12-22T00:00:00 | [
[
"Gilmer",
"Patrick M.",
""
]
] | alg-geom | |
1996-04-28T05:21:55 | 9604 | alg-geom/9604019 | en | https://arxiv.org/abs/alg-geom/9604019 | [
"alg-geom",
"math.AG"
] | alg-geom/9604019 | Lars Ernstrom | Lars Ernstr\"om and Gary Kennedy | Recursive Formulas for the Characteristic Numbers of Rational Plane
Curves | LaTeX2e with xy-pic v3.2 | null | null | null | null | We derive recursive equations for the characteristic numbers of rational
nodal plane curves with at most one cusp, subject to point conditions, tangent
conditions and flag conditions, developing techniques akin to quantum
cohomology on a moduli space of stable lifts.
| [
{
"version": "v1",
"created": "Sun, 28 Apr 1996 03:18:05 GMT"
}
] | 2016-08-15T00:00:00 | [
[
"Ernström",
"Lars",
""
],
[
"Kennedy",
"Gary",
""
]
] | alg-geom | \section{Introduction}
\par
Although of recent vintage, Kontsevich's recursive formula $$
N_d=\sum_{d_1+d_2=d}N_{d_1}N_{d_2}\left[d_1^2d_2^2\binom{3d-
4}{3d_1-2}- d_1^3d_2\binom{3d-4}{3d_1-1}\right]
$$
for the number $N_d$ of rational plane curves of degree $d$ through $3d-1$
general points is already a celebrated result. The formula is wonderfully
simple: it determines all such characteristic numbers, beginning with the
triviality $N_1 = 1$.
\par
In his proof, Kontsevich uses a compactification ${\overline M}_{0,n}(\pp,d)$ of the space of
$n$-pointed maps from $\pl$ to the projective plane, called the space of {\em
stable maps}. The key fact about these spaces is that (for $n \geq 4$)
they carry three linearly equivalent divisors, called the {\em special boundary
divisors}, which are the inverse images of the special points $0$, $1$, and
$\infty$, under a natural map from ${\overline M}_{0,n}(\pp,d)$ to the space of stable
rational curves with four distinguished points. (As is well-known, the latter
space is isomorphic to $\pl$.) This linear equivalence readily yields
Kontsevich's identity. Furthermore, each divisor is a union of components,
each of which is isomorphic to a fiber product
$$
{\overline M}_{0,n_1+1}(\pp,d_1)
\times_{\pp}
{\overline M}_{0,n_2+1}(\pp,d_2);
$$
as a consequence, one can interpret Kontsevich's formula as the assertion that
a certain new product on the cohomology group of $\pp$ (tensored with a
certain power series ring) is associative. The resulting associative ring is
called
the {\em quantum cohomology}.
\par
In this paper we show that Kontsevich's compactifications can also be used to
derive recursive formulas for these other classical characteristic numbers:
\begin{align*}
N_d(a,b,c)=&\text{ the number of rational plane curves of degree $d$
through $a$ general points, tangent to}\\ &\text{$b$ general lines, and
tangent to $c$ general lines at a specified general point on each line}\\
&\quad\text{(where $a+b+2c=3d-1$).}
\end{align*}
\par
In examining questions of tangency, it is natural to work with the incidence
correspondence $I$ of points and lines in $\pp$. However, the space of stable
maps to $I$ is too large; instead we need to consider only those stable maps
which can be lifted from maps to $\pp$, and degenerations of such maps.
Thus we will define a space ${\overline M}^1_{0,n}(\pp,d)$ of {\em stable lifts}. The superscript
$1$ indicates that $I$ parametrizes first-order jets of curves in $\pp$;
similar definitions for higher order jet bundles will be considered in another
paper. As a subspace of
the space of stable maps to the incidence correspondence, ${\overline M}^1_{0,n}(\pp,d)$ inherits
the special boundary divisors. A chief part of our project is to describe the
components of these divisors. We show, for example, that for many
components the general point represents a map from a curve with {\em
three} components, with the central component mapping to a fiber of $I$ over
$\pp$.
\par From the linear equivalence of the special boundary divisors, we
extract many recursive formulas. We show, in an admittedly {\it ad hoc}
fashion, that these formulas suffice to determine all the characteristic
numbers $N_d(a,b,c)$, starting with the four trivial cases in degree $1$.
At the same time we obtain recursive formulas which determine, in all cases,
three other types of characteristic numbers:
\begin{align*}
C_d(a,b,c;1)=&\text{ the number of rational plane curves of degree $d$, with
one cusp, through $a$ points,}\\
&\text{tangent to $b$ lines, and tangent to $c$ lines at $c$ specified
points}\\
&\quad\text{(where $a+b+2c=3d-2$);}\\ C_d(a,b,c;h)=&\text{ the number of
such curves having the cusp on a specified line}\\
&\quad\text{(where $a+b+2c=3d-3$);}\\
C_d(a,b,c;h^2)=&\text{ the number of such curves having the cusp at a
specified point}\\
&\quad\text{(where $a+b+2c=3d-4$).}
\end{align*}
In fact, it is easier to determine these numbers than to avoid them, since the
special boundary divisors on the space of stable lifts include, in addition to
the
components already described, other components corresponding to cuspidal
rational curves.
\par
Because of the occurrence of stable maps with automorphism, it is most
convenient to utilize the framework of algebraic stacks and the corresponding
intersection theory (with rational coefficients) as defined by Vistoli
\cite{Vistoli}. For automorphism-free stable maps the intersection theory of
the stack coincides with the usual intersection theory of the corresponding
moduli scheme. However, on the stack ${\overline M}^1_{0,n}(\pp,d)$ of
stable lifts there are components of the special boundary divisors for which
the general member has automorphism group of order two. Therefore we will
encounter the fractional intersection number $1/2$. Another possibility would
be to work with equivariant Chow groups as defined by Edidin and Graham
\cite{EdidinGraham}.
\par
Here is an outline of the paper.
In \S\ref{srmaps} we recall Kontsevich's notion of stable maps, the
definition of the Gromov-Witten invariants, and the basic linear equivalence
which leads to Kontsevich's recursive formula. As we have said, this linear
equivalence implies the associativity of the quantum product,
but---since in our project we do not use any analogous ``higher-order''
product---we refrain from discussing this point.
In \S\ref{sectionlifts} we introduce spaces of stable lifts for $\pp$, by
associating to a general stable map its lift to the projectivized tangent
bundle of $\pp$.
In \S\ref{sectionHOGW} we use these spaces to define first-order Gromov-Witten
invariants, and discuss their significance in enumerative geometry.
In \S\ref{sectiondivisors} we describe the structure of the special boundary
divisors on the space of stable lifts.
In \S\ref{sectionpotentials} we use the
physicists' method of quantum potentials to write down generating functions
for the Gromov-Witten invariants, and derive the basic identity
(\ref{ijklrel}), analogous to the associativity identity for the quantum
product.
In \S\ref{sectionrecursion} we derive, from selected cases of the basic
identity, recursive equations for the characteristic numbers;
we also present tables of characteristic numbers through degree $5$.
\par
We believe that these methods will work for a large class of surfaces. We also
believe that they will enable us to calculate the higher-order characteristic
numbers of curves, defined by conditions of higher-order tangency to lines or
other specified curves. Here the appropriate parameter spaces are, we believe,
those originally defined by Semple (later investigated in \cite{Collino} and
\cite{CKtrip}). We intend to pursue this matter with Susan Colley. A theory
for more general varieties and higher genus seems possible as a consequence
of the results of Behrend \cite{Behrend} and those of Li and Tian
\cite{LiTian2}.
However, the enumerative significance of the Gromov-Witten invariants is,
in this more general set-up, not obvious.
\par
In recent papers, Pandharipande has constructed an algorithm for computing the
numbers $N_d(a,b,0)$ \cite{Pandharipande2}, and Francesco and Itzykson have
found a recursive relation that determines the same numbers, with the
restriction $a\geq 3$, given characteristic numbers of lower degree
\cite{FrancescoItzykson}.
\par
This project was inspired by W.~Fulton's lecture series at the AMS summer
institute at Santa Cruz 1995.
We have benefited greatly from discussions with R.~Pandharipande,
and from his preprints. We also wish to thank P.~Aluffi, C.~Ban,
S.~Colley, B.~Crauder,
D.~Edidin, S.~Kleiman, D.~Laksov, L.~McEwan, and S.~Yokura for their help
during the preparation of this paper.
\par
\section{Stable maps and Gromov-Witten invariants}\label{srmaps}
As general references for the material in this section we suggest
\cite{BehrendManin}, \cite{Fulton2}, \cite{Kontsevich}, \cite{KontsevichManin},
\cite{LiTian} and \cite{Pandharipande}. In this section we will utilize the
intersection theory of the coarse moduli space of stable maps. Generic members
of the special boundary divisors of the space of stable
maps have no automorphisms. Therefore, for our purposes, the intersection
theory of the stack of stable maps is the same as that of the coarse moduli
space. Throughout this paper, we will be working over an algebraically closed
field of characteristic zero. Given a variety or stack $X$, homology $A_*(X)$
and cohomology $A^*(X)$ will denote the rational equivalence groups with
coefficients in $\q$.
\par
Suppose that $C$ is a connected reduced curve of arithmetic genus zero
whose singularities are at worst nodes. Then $C$ must be a tree of $\pl$'s.
We will call the nodes {\em points of attachment}. Suppose that on $C$ we
have $n$ distinct points $p_1,\dots,p_n$, none of which is a point of
attachment; we call these points {\em markings}. A {\em special point} is a
point of attachment or a marking.
\par
Now suppose that $X$ is a nonsingular projective variety. Following
Kontsevich, we define an {\em $n$-pointed stable (genus $0$) map to $X$} to be
a
map $\mu\colon C \to X$ from such a curve, subject to the following
condition: if the restriction of $\mu$ to a component is constant, then that
component contains at least three special points. (Since we never consider
stable maps from curves of higher arithmetic genus, we will generally omit the
parenthesized phrase.) A {\em family of stable maps} consists of a flat, proper
map $\pi\colon \mathcal C\to S$, a
map $\mu\colon\mathcal C\to X$, and $n$ sections
$\{p_t\}_{t=1,\dots,n}$ of $\pi$, such that, for each geometric fiber
$\mathcal C(s)$
of $\pi$, the restriction of $\mu$ to this fiber, with the markings $p_t(s)$,
is
an $n$-pointed stable map. A {\em morphism} over $S$ from one
such family $(\pi\colon \mathcal C \to S,\mu,\{p_t\})$ to another
$(\pi^\prime\colon \mathcal C^\prime \to
S,\mu^\prime,\{p^\prime_t\})$
is a morphism $\tau\colon \mathcal C\to \mathcal C^\prime$ over $S$
such that $\mu=\mu^\prime\circ \tau$ and $p_t\circ \tau=p_t^\prime$.
\par
Suppose that $\varphi$ is a specified class in $A_1(X)$. Kontsevich
showed that there is a coarse moduli space ${\overline M}_{0,n}(X,\varphi)$ for
isomorphism classes of stable maps whose image in $X$ represents
$\varphi$ \cite[2.4.1]{KontsevichManin}, \cite[\S1, p. 336]{Kontsevich},
\cite[Theorems 1 and 2]{Pandharipande}.
We call ${\overline M}_{0,n}(X,\varphi)$ the {\em space of stable (genus $0$) maps}. There are $n$
{\em evaluation maps} $e_t:{\overline M}_{0,n}(X,\varphi)\to X$ which associate to the point
representing a stable map $\mu$ the images $\mu(p_t)$ of the
markings. If $f:X \to Y$ is a morphism, then there is a morphism
$Mf:{\overline M}_{0,n}(X,\varphi) \to {\overline M}_{0,n}(Y,f_*(\varphi))$ which associates to the point representing
$\mu$ the point representing the stable map obtained from $f \circ \mu$ by
contracting any components which have become unstable; if $f_*(\varphi)=0$, we
must assume that $n \geq 3$. In particular, under this hypothesis
there is a ``forgetful'' morphism from ${\overline M}_{0,n}(X,\varphi)$ to ${\overline M}_{0,n}$, the
space of stable rational $n$-pointed curves. \par If $n \geq 4$ then we can
compose the forgetful morphism with another sort of forgetful morphism
${\overline M}_{0,n} \to {\overline M}_{0,4}$, this time forgetting about all markings except
the first four. The space ${\overline M}_{0,4}$ is isomorphic to $\pl$. It has a
distinguished point $P(12\mid 34)$ representing the two-component curve
having the first two markings on one component and the latter two on the
other; similarly there are two other distinguished points $P(13\mid 24)$ and
$P(14\mid 23)$. Their inverse images on ${\overline M}_{0,n}(X,\varphi)$ are three linearly
equivalent divisors $D(12\mid 34)$, $D(13\mid 24)$, and $D(14\mid 23)$,
called the {\em special boundary divisors}.
\par We allow the index set to be an arbitrary finite
set $A$ rather than just $\{1,\dots,n\}$; we then write
$\amoduli{A}{\varphi}$ for the moduli space. Suppose that $A_1 \cup A_2$
is a partition of $\{1,\dots,n\}$ and that $\varphi_1$, $\varphi_2$ are
classes in $A_1(X)$ whose sum is $\varphi$. Suppose that $\{\star\}$ is a
single-element set. Then the fiber product $$
D(A_1,A_2,\varphi_1,\varphi_2) =
\amoduli{A_1\cup\{\star\}}{\varphi_1} \times_X
\amoduli{A_2\cup\{\star\}}{\varphi_2} $$ is naturally a subspace of
$\amoduli{A}{\varphi}$; the typical point represents a map from a curve
with two components, with the point of attachment corresponding to the
point indexed by $\star$. In Figure 2.1 the attachment point and the image
thereof are marked $\bullet$.
$$
\xy
0;<0.5cm,0cm>:
(-1,-1);(1,1)**\dir{-};
(-1,1);(1,-1)**\dir{-};
(0,0)*{\bullet};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,-0.5);(6.5,0.75)**\dir{-};
(4,1);(6,1)**\crv{(5,-3)};
(4.41,-0.3)*{\bullet}
\endxy
$$
\begin{center}
{\bf Figure 2.1} Stable map.
\end{center}
\par
\bigskip
If $X$ is a product of homogeneous
spaces (for example, if $X$ is a projective space or a flag variety) then the
dimension of ${\overline M}_{0,n}(X,\varphi)$ (if it is nonempty) is
$\dim X + \int_\varphi c_1(T_X) + n - 3$ \cite{Kontsevich}, \cite[Proposition
1]{Fulton2}. Furthermore the divisor
$D(12\mid 34)$ is the sum
\begin{equation}\label{donetwo} D(12\mid
34)=\sum D(A_1,A_2,\varphi_1,\varphi_2) \end{equation}
over all
partitions in which $1$ and $2$ belong to $A_1$, and $3$ and $4$ belong to
$A_2$.
\par
Suppose that
$\gamma_1,\dots,\gamma_n$ are elements of $A^*X \otimes \q$. Then
the {\em Gromov-Witten invariant} associated to these classes and to
$\varphi$ is the number
$$
N_{\varphi}(\gamma_1\cdots\gamma_n)
=\int e_1^*(\gamma_1) \cup \dots \cup e_n^*(\gamma_n)
\cap [{\overline M}_{0,n}(X,\varphi)],
$$
the degree of the top-dimensional component. If the classes
$\gamma_1,\dots,\gamma_n$ are homogeneous, then the Gromov-Witten
invariant is nonzero only if the sum of their codimensions is the dimension
of the moduli space.
(If the moduli space is empty, we declare the Gromov-Witten invariant to be
zero.)
\par
We now restrict our attention to the case $X=\pp$.
The class $\varphi$ of a curve must be some multiple $dh$ of the class $h$ of
a line. We will write ${\overline M}_{0,n}(\pp,d)$ for the space of $n$-pointed stable maps; its
dimension is $3d-1+n$ (unless $d=0$ and $n \leq 2$, in which case
the space is empty). We define
$$
N_d=N_{dh}(h^2 \cdots h^2).
$$
By standard transversality arguments, one can show that $N_d$ is the
number of rational plane curves of degree $d$ through $3d-1$ general points.
\par
The linear equivalence of the special boundary divisors
$D(12\mid 34)$ and $D(13\mid 24)$ implies, for each
choice of $\varphi$ and of $\gamma_1,\dots,\gamma_n$,
the numerical equality
\begin{equation}
\label{numeq}
\int e_1^*(\gamma_1) \cup \dots \cup e_n^*(\gamma_n)
\cap [D(12\mid 34)] =
\int e_1^*(\gamma_1) \cup \dots \cup e_n^*(\gamma_n)
\cap [D(13\mid 24)].
\end{equation}
By (\ref{donetwo}) and the similar decomposition of the divisor $D(13\mid
24)$, this is an equation among various values of $N_d$. For example, if
$n=3d+1$, $\varphi=dh$, $\gamma_1=\gamma_2=h$, and
$\gamma_t=h^2$ for $t \geq 3$, then (\ref{numeq}) is Kontsevich's
recursive formula!
\par
Equation (\ref{numeq}) can also be interpreted as an equation among formal
power series. Let $T_0=1$, $T_1=h$, and $T_2=h^2$, so that an arbitrary class
$\gamma$ is a linear combination $y_0 T_0 + y_1 T_1 + y_2 T_2$. Define
the {\em potential} by
\begin{equation}
{\mathcal P}=\sum_{n \geq 0}\frac{1}{n!}
\sum_{\varphi}
N_{\varphi}(\gamma^n),
\end{equation}
which one can show is equal to
$$
\frac{1}{2}(y_0^2y_2+y_0y_1^2)
+ \sum_{d \geq 1} N_d e^{dy_1}\frac{y_2^{3d-1}}{(3d-1)!}.
$$
The first term, called the {\em classical potential}, encodes the intersection
product on $\pp$; the second term is called the {\em quantum potential}.
Using these potentials, we can translate the special case
\begin{align*}
\int e_1^*(T_i) & \cup e_2^*(T_j) \cup e_3^*(T_k) \cup e_4^*(T_l)
\cup e_5^*(\gamma) \cup \dots \cup e_n^*(\gamma)
\cap [D(12\mid 34)] \\
&=
\int e_1^*(T_i) \cup e_2^*(T_j) \cup e_3^*(T_k) \cup e_4^*(T_l)
\cup e_5^*(\gamma) \cup \dots \cup e_n^*(\gamma)
\cap [D(13\mid 24)]
\end{align*}
of (\ref{numeq}) into the partial differential equation
\begin{equation}
\sum_{s=0}^2 \frac{\partial^3{\mathcal P}}{\partial y_i\partial y_j\partial y_s}
\frac{\partial^3{\mathcal P}}{\partial y_k\partial y_l\partial y_{2-s}} =
\sum_{s=0}^2 \frac{\partial^3{\mathcal P}}{\partial y_i\partial y_k\partial y_s}
\frac{\partial^3{\mathcal P}}{\partial y_j\partial y_l\partial y_{2-s}}.
\end{equation}
The case $i=j=2$, $k=l=1$ is equivalent to Kontsevich's formula.
\section{Lifts of stable maps}\label{sectionlifts}
Denote by $I=\bp(T_{\pp})$ the projectivized tangent
bundle of the projective plane; it is the variety of point-line incidence in
$\pp\times\dpp$.
Suppose that $\mu \colon (\pl,p_1,\dots,p_n) \to \pp$ is a nonconstant
map with $n$ distinct marked points.
For a general point $x$ in $\pl$
define $\tilde\mu\colon\pl\to I$ by
$\tilde\mu(x)=(\mu(x),\mu^\prime(x))$, where $\mu^\prime(x)$ is
the tangent line to $\mu(\pl)$ at $\mu(x)$. The construction extends to all
of $\pl$: even at points where the map $\mu$ is singular there is a unique
tangent direction associated to the branch of $\mu(\pl)$ at $x$.
The map $\tilde\mu\colon (\pl,p_1,\dots,p_n)\to I$ is called the {\em
strict lift} of $\mu$. (By symmetry, we may also define the strict lift of a
nonconstant map to $\dpp$.)
$$
\xy
0;<0.5cm,0cm>:
(-10,0)*{\pl};
(-8.5,0);(-6.5,0)**\dir{-};
?>*\dir{>};
(-7.5,-0.5)*{\mu};
(-5,0)*{\pp};
(-8.5,1.5);(-6.5,3.5)**\dir{-};
?>*\dir{>};
(-5,5)*{I};
(-5,3.5);(-5,1.5)**\dir{-};
?>*\dir{>};
(-8,3)*{\tilde\mu};
(-1,1);(1,-1)**\dir{-};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,0.5);(5.5,0)**\crv{(8,-0.5)&(4,-1.5)};
(5.5,0);(6,0.5)**\dir{-};
(1.5,1.5);(3.5,3.5)**\dir{-};
?>*\dir{>};
(5,3.5);(5,1.5)**\dir{-};
?>*\dir{>};
(4,5.5);(5.4,4.9)**\crv{(8,4.5)&(4,3.5)};
(5.55,5.05);(6,5.5)**\dir{-};
\endxy
$$
\begin{center}
{\bf Figure 3.1} The strict lift of a map from $\pl$ to $\pp$.
\end{center}
\par
\bigskip
Strict lifts do not behave nicely in families of maps. For example if $\mu$ is
an immersion of degree $d$ and its image is a nodal curve, then the number
of nodes is $\delta=(d-1)(d-2)/2$ and the class of the curve is
$$\check{d}=d(d-1)-2\delta=2d-2.$$
Thus the homology class of $\tilde \mu(\pl)$
in $I$ is $d$ times the strict lift of a line plus $2d-2$ times the strict lift
of a
dual line.
We will say that it has {\em bidegree} $(d,2d-2)$. If instead $\mu(\pl)$ is a
rational curve with $(d-1)(d-2)/2-1$ nodes and one cusp then the bidegree of
$\tilde\mu(\pl)$ is
$(d,2d-3)$. Thus in a family degenerating a node to a cusp, the strict lifts of
the
members in the family do not piece together.
\par
Suppose that we have a family of stable maps $(\pi\colon \mathcal C \to
S,\mu\colon\mathcal C\to I,\{p_t\})$ whose general member is the strict
lift of an immersion $\pl\to \pp$. Then we say that the members of this
family are {\em stable lifts} of the corresponding members of the family
of maps to $\pp$. Note that a stable lift of a map of degree $d$ has bidegree
$(d,2d-2)$; for example (supposing there are no markings) the stable lift of a
$d$-fold branched cover of a line consists of the strict lift together with
maps to the fibers of $I$ over the $2d-2$ branch points. In Figure 3.2 we
illustrate the case $d=3$. Ramification and branch points are marked by
$\times$.
\par
$$
\xy
0;<0.5cm,0cm>:
(-1,0);(1,0)**\dir{-};
(-0.6,-1);(-0.6,1)**\dir{-};
(-0.6,0)*{\bullet};
(-0.6,0)*{\times};
(-0.2,-1);(-0.2,1)**\dir{-};
(-0.2,0)*{\bullet};
(-0.2,0)*{\times};
(0.2,-1);(0.2,1)**\dir{-};
(0.2,0)*{\bullet};
(0.2,0)*{\times};
(0.6,-1);(0.6,1)**\dir{-};
(0.6,0)*{\bullet};
(0.6,0)*{\times};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,0);(6,0)**\dir{-};
(4.4,-1);(4.4,1)**\dir{-};
(4.4,0)*{\bullet};
(4.8,-1);(4.8,1)**\dir{-};
(4.8,0)*{\bullet};
(5.2,-1);(5.2,1)**\dir{-};
(5.2,0)*{\bullet};
(5.6,-1);(5.6,1)**\dir{-};
(5.6,0)*{\bullet};
(4.4,0)*{\times};
(4.8,0)*{\times};
(5.2,0)*{\times};
(5.6,0)*{\times};
(0,-1.5);(0,-3.5)**\dir{-};
?>*\dir{>};
(5,-1.5);(5,-3.5)**\dir{-};
?>*\dir{>};
(-1,-5);(1,-5)**\dir{-};
(-0.6,-5)*{\times};
(-0.2,-5)*{\times};
(0.2,-5)*{\times};
(0.6,-5)*{\times};
(1.5,-5);(3.5,-5)**\dir{-};
?>*\dir{>};
(4,-5);(6,-5)**\dir{-};
(4.4,-5)*{\times};
(4.8,-5)*{\times};
(5.2,-5)*{\times};
(5.6,-5)*{\times};
\endxy
$$
\begin{center}
{\bf Figure 3.2} The stable lift of a three-cover of a line in $\pp$ with
ramification at four points.
\end{center}
\par
\bigskip
For simplicity of notation, we will write ${\overline M}_{0,n}(I,(d,2d-2))$ for the stack of stable
(genus $0$) maps representing the class of a curve of bidegree $(d,2d-2)$. Let
${\overline M}^1_{0,n}(\pp,d)$ be the closed substack of ${\overline M}_{0,n}(I,(d,2d-2))$ which represents stable lifts;
we
call it the {\em stack of stable lifts}. We continue to use the notations
$e_1,\dots,e_n$ for the evaluation maps ${\overline M}^1_{0,n}(\pp,d) \to I$.
Note that the map ${\overline M}^1_{0,n}(\pp,d) \to {\overline M}_{0,n}(\pp,d)$ (inclusion followed by
projection) is a birational morphism, whose inverse is the {\em lifting map}
$\lambda$ which associates to each immersion its strict lift.
\begin{prop}\label{stableliftd1d2}
Let $C$ be a curve consisting of two irreducible component, each isomorphic to
$\pl$ and attached at a single point $p$.
Let $\mu\colon C \to \pp$ be a map defined by mapping one component of $C$
to an irreducible curve of degree $d_1\geq 1$ and mapping the other
component to an irreducible curve of degree $d_2\geq 1$, so that $\mu$ is
represented by a point of ${\overline M}_{0,0}(\pp,d_1+d_2)$.
Assume that both maps are immersions and that they are transverse at $p$.
\par
Then the stable lift of $\mu$ is unique. It is a map $\tilde\mu$ from a curve
$\tilde C$ consisting of three components, each isomorphic to $\pl$. On the
cental component $\tilde\mu$ is a map of degree $2$ to the pencil of tangent
directions at $p$, with the points of attachment mapping to the tangent
directions of $C$; the map is ramified at these attachment points. On each
peripheral component $\tilde\mu$ is the strict lift of a component of $\mu$.
\end{prop}
\par
See Figure 3.3.
\par
$$
\xy
0;<0.5cm,0cm>:
(-1,-1);(1,1)**\dir{-};
(-1,1);(1,-1)**\dir{-};
(0,0)*{\bullet};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,-0.5);(6.5,0.75)**\dir{-};
(4,1);(6,1)**\crv{(5,-3)};
(4.41,-0.3)*{\bullet};
(0,6);(0,4)**\dir{-};
(0,5.333)*{\bullet};
(0,5.333)*{\times};
(0,4.667)*{\bullet};
(0,4.667)*{\times};
(-1,5.333);(1,5.333)**\dir{-};
(-1,4.667);(1,4.667)**\dir{-};
(1.5,5);(3.5,5)**\dir{-};
?>*\dir{>};
(4.41,6);(4.41,4)**\dir{-};
(4,4.667);(6,4.667)**\dir{-};
(4.41,4.667)*{\bullet};
(4.41,4.667)*{\times};
(4,6);(6,6)**\crv{(6,5)};
(4.41,5.78)*{\bullet};
(4.41,5.78)*{\times};
(0,3.5);(0,1.5)**\dir{-};
?>*\dir{>};
(5,3.5);(5,1.5)**\dir{-};
?>*\dir{>};
\endxy
$$
\begin{center}
{\bf Figure 3.3} Stable lift of Proposition \ref{stableliftd1d2}.
\end{center}
\par
\bigskip
\begin{proof}
The versal deformation theory of a plane node is given by the local equation
$xy=\epsilon$. Thus any family degenerating to $\mu$ will be the pullback of a
family which, in appropriate local coordinates, has total space isomorphic to a
neighborhood of the origin, with the fibers of the family being the curves
$xy=\epsilon$ and with the map being the identity. Locally around the origin,
$I$ is a trivial $\pl$-bundle; in coordinates the family of stable lifts is
given by
$$ (x,y)\mapsto \bigg((x,y),[x:-y]\bigg).$$
\par
According to \cite[section 3.2]{Pandharipande}, there is a unique extension of
the family of lifts, obtained by blowing up the points of indeterminacy,
possibly after a base change. We choose to make the base change which has the
effect of replacing our original family by the family $xy=\epsilon^2$. (In
fact, one can show that this base change is unavoidable.) Then we blow up the
origin on this singular surface. In the resulting family, the fiber over
$\epsilon=0$ is reduced and has a new component mapping to the pencil of
directions at the origin; the map, which is of degree 2, is ramified at the
points of attachment to the strict transforms of the two original components.
\end{proof}
But in general a stable map may have more than one stable lift.
Figure 3.4 illustrates a stable lift of a degree three cover of a line with
double ramification at two points. The two maps into the fibers over the branch
points are of degree two, ramified at the attachment point and at one other
point. The latter point may appear anywhere on the fiber. Thus the stable lift
in this situation is not unique; there is a two dimensional family of
lifts corresponding to a single stable map.
$$
\xy
0;<0.5cm,0cm>:
(-1,0.333);(1,0.333)**\dir{-};
(-0.333,-1);(-0.333,1)**\dir{-};
(-0.333,0.333)*{\bullet};
(-0.333,0.333)*{\times};
(-0.333,-0.333)*{\times};
(0.333,-1);(0.333,1)**\dir{-};
(0.333,0.333)*{\bullet};
(0.333,0.333)*{\times};
(0.333,-0.333)*{\times};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,0.333);(6,0.333)**\dir{-};
(4.667,-1);(4.667,1)**\dir{-};
(4.667,0.333)*{\bullet};
(4.667,0.333)*{\times};
(4.667,-0.333)*{\times};
(5.333,-1);(5.333,1)**\dir{-};
(5.333,0.333)*{\bullet};
(5.333,0.333)*{\times};
(5.333,-0.333)*{\times};
(0,-1.5);(0,-3.5)**\dir{-};
?>*\dir{>};
(5,-1.5);(5,-3.5)**\dir{-};
?>*\dir{>};
(-1,-5);(1,-5)**\dir{-};
(-0.333,-5)*{\times};
(0.333,-5)*{\times};
(1.5,-5);(3.5,-5)**\dir{-};
?>*\dir{>};
(4,-5);(6,-5)**\dir{-};
(4.667,-5)*{\times};
(5.333,-5)*{\times};
\endxy
$$
\begin{center}
{\bf Figure 3.4} A stable lift of a map with double ramification is not
unique.
\end{center}
\par
\bigskip
\begin{prop}\label{liftclassif} Let $\tilde\mu\colon \tilde C\to I$ be an
$n$-pointed stable lift. Let $\mu\colon C \to \pp$ be the stable map
obtained by composing $\mu$ with the projection $I\to \pp$, forgetting
about all markings, and (if necessary) contracting components which have
become unstable. Then, for each irreducible component $\tilde C_i$ of $\tilde
C$, the
restricted map $\tilde \mu\vert_{\tilde C_i}\colon \tilde C_i\to I$ is one of
the following:
\begin{trivlist}
\item[(1)] the strict lift of a map obtained by restricting $\mu$ to a
component of $C$,
\item[(2)] a map into a fiber of $I$ over some point $x$ in $\pp$, where $x$ is
either the image of an attachment point of $C$ or a singularity of the
restriction of $\mu$ to some component of $C$,
\item[(3)] a constant map.
\end{trivlist}
\end{prop}
\par
\begin{proof}
We may assume that $n=0$. It then suffices to prove that each component is of
type (1) or (2).
\par
By definition $\tilde\mu$ is a member of a family of stable maps from
$\tilde{\mathcal C}$ to $I$ over a base $S$, where the generic member of the family is
the strict lift of a map to $\pp$. We may and will assume that the base $S$ is
a nonsingular curve. Consider the family of maps ${\mathcal C}$ to $\pp$ over the same
base $S$ obtained by composing with the projection $I\to \pp$, forgetting about
all markings, and (if necessary) contracting components which have become
unstable.
We will next reconstruct the family of maps to $I$, and conclude
the facts about the special member $\tilde\mu$.
Consider the open set $U$ of ${\mathcal C}$ over $S$ that is the complement of the
singular points (the nodes or attachment points) of curves in the family.
Then there is a map
$$T_{U/S}\to T_\pp$$ from the relative tangent bundle of $U$
over $S$ to the tangent bundle of $\pp$.
Composing with the map
$$T_\pp - \text{zero section}\to \bp(T_\pp)=I$$
we get a rational map from $\bp(T_{U/S})=U$ and hence from ${\mathcal C}$ to $I$,
which is indeterminate exactly at the attachment points and singularities of
the map $\mu$ on components of the special member $C$.
For generic members of ${\mathcal C}$ the rational map is clearly the lifting map.
Finally, by \cite[Prop.3.3]{Pandharipande} the rational map may be extended by
blowing up the points of indeterminacy. Furthermore by
\cite[Prop.3.2]{Pandharipande} the extension is unique and thus we recover the
map $\tilde\mu$ as the special member after sufficient blowing-up.
The result now follows from what we have concluded about the points of
indeterminacy.
\end{proof}
\par
\section{Characteristic numbers}\label{sectionHOGW}
\par
The rational cohomology of $I$ is given by
$$
A^*(I)=\q[h,\check{h}]/(h^3,\check{h}^3,h^2+\check{h}^2-h\check{h}), $$
where $h$ is the first Chern class of the pullback of the line bundle
$\mathcal O_{\pp}(1)$ on $\pp$
and $\check{h}$ is the pullback of the line bundle $\mathcal O_{\dpp}(1)$ on $\dpp$.
Note that the fundamental class of the strict lift of a line is $\check{h}^2$,
and that the class of the strict lift of a dual line is $h^2$.
We fix the following basis for $A^*(I)$:
$$
\{T_0,T_1,T_2,T_3,T_4,T_5\}=\{1,h,h^2,\check{h},\check{h}^2,h^2\check{h}\}.
$$
With respect to this basis the fundamental class of the diagonal $\Delta$
in $I \times I$ has the simple decomposition
\begin{equation}
\label{diag}
[\Delta] = \sum_{s=0}^5 [T_s] \times [T_{5-s}].
\end{equation}
\par
Suppose that $d$ is a positive integer, and that
$\gamma_1,\dots,\gamma_n$ are elements of $A^*(I) \otimes \q$.
We define the {\em first-order Gromov-Witten invariant} by
$$
N_d(\gamma_1\cdots\gamma_n)=\int e_1^*(\gamma_1) \cup \dots \cup
e_n^*(\gamma_n)
\cap [{\overline M}^1_{0,n}(\pp,d)].
$$
Suppose that $a$ of the $\gamma_t$'s equal the class $h^2$, that $b$ of
them equal the class $\check{h}^2$, and that the remaining $c$ of them equal
$h^2\check{h}$, where $a+b+2c=3d-1$. In this case we denote the Gromov-Witten
invariant by $N_d(a,b,c)$.
\par
\begin{thm}\label{gwenumsign2} $N_d(a,b,c)$ is the number of rational plane
curves of degree $d$ through $a$ general points, tangent to $b$ general
lines, and tangent to $c$ general lines at a specified general point on
each line.
\end{thm}
\par
(We will say that such a curve is {\em incident to $c$ specified flags.})
\begin{proof}
Associated to each class $e_t^*h$ and $e_t^*\check{h}$ there is a complete
basepoint-free linear system parametrized by lines in $\pp$ or by lines in
$\dpp$. Applying the Kleiman-Bertini Theorem repeatedly
\cite[Corollary 5, p.291]{Kleiman4}, we find that the intersection of general
members of the linear systems is regular away from the singular locus of
${\overline M}^1_{0,n}(\pp,d)$. By a dimension count it is a set of reduced points of
${\overline M}^1_{0,n}(\pp,d)$. By the same dimension count, each point of the intersection
corresponds to the strict lift of an immersion $\pl \to \pp$. Thus no point
of the singular locus is in the intersection.
\end{proof}
The following Proposition is analogous to \cite[2.2.3 and 2.2.4]
{KontsevichManin} and \cite[p. 9]{Fulton2}.
\begin{prop}\label{GWprop}
\begin{trivlist}
\par
\item[(1)] For all $d\geq 1$ and all $\gamma_1,\dots,\gamma_{n-1}$,
$$
N_d(\gamma_1\cdots\gamma_n\cdot 1)=0. $$
\item[(2)] If $\gamma_n$ is a divisor class in
$A^1(I)$, then for all $d\geq 1$ and all $\gamma_1,\dots,\gamma_{n-1}$,
$$
N_d(\gamma_1\cdots\gamma_n)=
N_d(\gamma_1\cdots\gamma_{n-1})\int\gamma_n\cap [C], $$
where $[C]=d \check{h}^2 +(2d-2) h^2$ is the class of the strict lift of a
rational nodal plane curve of degree $d$.
\end{trivlist}
\end{prop}
\par
For $d\geq 3$ let $C_{0,n,\{\star\}}(\pp,d)$ be the open substack of
${\overline
M}_{0,n+1}(\pp,d)$ consisting of maps $\mu\colon \pl\rightarrow \pp$ for
which $\mu(\pl)$ has $(d-1)(d-2)/2-1$ nodes and exactly one cusp marked
$\star$, coinciding with the $(n+1)$st marking. The strict lift of any such map
has bidegree $(d,2d-3)$. Thus
there is a lifting map
$$\lambda_C\colon C_{0,n,\{\star\}}(\pp,d)\to {\overline
M}_{0,n+1}(I,(d,2d-3)).$$
The closure of
$C_{0,n,\{\star\}}(\pp,d)$ will be denoted by ${\overline C}_{0,n,\{\star\}}(\pp,d)$; its dimension is
$3d-2+n$.
We define the {\em stack of cuspidal stable lifts} ${\overline C}^1_{0,n,\{\star\}}(\pp,d)$ to be the
closure of
$\lambda_C(C_{0,n+1}(\pp,d))$ in ${\overline M}_{0,n}(\pp,(d,2d-3))$; its
dimension is likewise $3d-2+n$.
Let $e_1,\cdots,e_n$ and $e_C$ be the evaluation maps ${\overline C}^1_{0,n,\{\star\}}(\pp,d) \to I$.
For each $d \geq 3$ and each $\gamma_1,\cdots,\gamma_n,\gamma_C \in A^*(I)$
we define the {\em cuspidal first-order Gromov-Witten invariant} by
$$
C_d(\gamma_1\cdots\gamma_n;\gamma_C)=\int e_1^*(\gamma_1) \cup \dots \cup
e_n^*(\gamma_n) \cup e_C^*(\gamma_C)
\cap [{\overline C}^1_{0,n,\{\star\}}(\pp,d)].
$$
Suppose that among $\gamma_1,\dots,\gamma_n$ there are $a$ occurences of
the class $h^2$, also $b$ occurences of $\check{h}^2$, and $c$ occurences of
$h^2\check{h}$, where $a+b+2c=3d-1-\operatorname{codim}(\gamma_C)$. In this case we denote the
cuspidal Gromov-Witten invariant by $C_d(a,b,c;\gamma_C)$.
\par
\begin{thm}
\par
\begin{trivlist}
\item[(1)] $C_d(a,b,c;1)$ is the number of rational plane curves of degree
$d$, with one cusp, through $a$ general points, tangent to $b$ general
lines, and incident to $c$ general flags.
\par
\item[(2)] $C_d(a,b,c;h)$ is the number of such curves having the cusp on a
specified general line.
\par
\item[(3)] $C_d(a,b,c;h^2)$ is the number of such curves having the cusp at a
specified point.
\par
\item[(4)] $C_d(a,b,c;\check{h})$ is the number of such curves for which the cusp
tangent line passes through a specified point.
\par
\item[(5)] $C_d(a,b,c;\check{h}^2)$ is the number of such curves for which the
cusp tangent line is a specified line.
\par
\item[(6)] $C_d(a,b,c;h^2\check{h})$ is the number of such curves for which the
cusp is a specified point and the cusp tangent is a specified line through
that point.
\end{trivlist}
\end{thm}
\begin{proof}
The proof is almost identical to the proof of Theorem \ref{gwenumsign2}.
\end{proof}
\begin{prop}
\begin{trivlist}
\par
\item[(1)] For all $d\geq 3$ and all $\gamma_1,\dots,\gamma_{n-1}$,
$$
C_d(\gamma_1\cdots\gamma_{n-1}\cdot 1;\gamma_C)=0. $$
\item[(2)] If $\gamma_n$ is a divisor class in
$A^1(I)$, then for all $d\geq 3$ and all $\gamma_1,\dots,\gamma_{n-1}$,
$$
C_d(\gamma_1\cdots\gamma_n;\gamma_C)=
C_d(\gamma_1\cdots\gamma_{n-1};\gamma_C)\int (\gamma_n\cap [C]), $$
where
$[C]=d \check{h}^2 +(2d-3) h^2$ is the class of a the strict lift of a rational
plane curve of degree $d$ with one cusp and $(d-1)(d-2)/2-1$ nodes.
\end{trivlist}
\end{prop}
\par
\section{Special boundary divisors on the space of stable lifts}
\label{sectiondivisors}
\par
The components of the special boundary divisors of
${\overline M}_{0,n}(I,(d,2d-2))$ are indexed by the set of $4$-tuples
$\left(A_1,A_2,(d_1,c_1),(d_2,c_2)\right)$
where $d_1+d_2=d$, $c_1+c_2=2d-2$,
and $A_1\cup A_2=\{1,\dots,n\}$ is a partition in which two of the four numbers
$1,2,3,4$ belong to $A_1$ and the other two to $A_2$. The general member of the
corresponding divisor is a stable map with two components, one with
markings indexed by $A_1$ and of bidegree $(d_1,c_1)$, and the other with
markings indexed by $A_2$ and of bidegree $(d_2,c_2)$ \cite{Pandharipande}.
Since the general stable map of ${\overline M}^1_{0,n}(\pp,d)$ is not contained in this
divisor, its restriction to ${\overline M}^1_{0,n}(\pp,d)$ is a divisor on the space of stable
lifts, which we will denote by $D\left(A_1,A_2,(d_1,c_1),(d_2,c_2)\right)$.
This divisor may be empty; for example, $D\left(A_1,A_2,(d,0),(0,2d-2)\right)$
is empty for all $A_1$ and $A_2$.
\par
The linear equivalence of the special boundary divisors on
${\overline M}_{0,n}(I,(d,2d-2))$ passes to their restrictions:
$$
D(12\mid 34)\cong D(13\mid 24)\cong D(14\mid 23),
$$
where
\begin{equation}\label{sumpart}
D(12\mid 34)=\sum D\left(A_1,A_2,(d_1,c_1),(d_2,c_2)\right),
\end{equation}
the sum over all partitions in which $1,2\in A_1$ and
$3,4\in A_2$; the other two divisors have similar decompositions.
Thus for all choices of cohomology classes
$\gamma_1,\cdots,\gamma_n$
we have a numerical equality
\begin{equation}
\int e^*_1(\gamma_1)\cup \dots \cup e^*_n(\gamma_n)
\cap [D(12\mid 34)]
=\int e^*_1(\gamma_1)\cup \dots \cup e^*_n(\gamma_n)
\cap [D(14\mid 23)].
\end{equation}
\par
To obtain these equations in an explicit form, we must identify the
components of $D(12\mid 34)$ and $D(14\mid 23)$, at least to the extent
that they affect our calculations. We say that a divisor $D$ on ${\overline M}^1_{0,n}(\pp,d)$
is {\em irrelevant} if for any
$\gamma_1,\dots,\gamma_n$ in $A^*(I)$ we have
\begin{equation}
\label{vanish}
\int e^*_1(\gamma_1)\cup \dots \cup e^*_n(\gamma_n)\cap[D]=0.
\end{equation}
Other divisors are said to be {\em relevant}.
\par
Let $\tau\colon{\overline M}^1_{0,n}(\pp,d)\to {\overline
M}_{0,0}(\pp,d)$ be the map which composes stable maps $\mu\colon C\to I$
with $I\to \pp$, forgets all markings and contracts any components which
have become unstable.
\begin{prop}\label{unless}
A component $D$ of $D\left(A_1,A_2,(d_1,c_1),(d_2,c_2)\right)$ is irrelevant unless
$\operatorname{codim}(\tau(D))\leq 1$ in ${\overline M}_{0,0}(\pp,d)$.
\end{prop}
\begin{proof}
It suffices to establish (\ref{vanish}) when $\gamma_1,\dots,\gamma_n$ are
elements of the basis $\{T_0,T_1,T_2,T_3,T_4,T_5\}$.
Assume first that one of the classes, say $\gamma_n=1$ is the identity.
Then by the projection formula applied to the morphism
$\pi\colon{\overline M}^1_{0,n}(\pp,d)\to\lamoduli{n-1}{d}$ which forgets the $n$th
marking, it follows that the above degree is equal to $$
\int e^*_1(\gamma_1)\cup \dots \cup e^*_{n-1}(\gamma_{n-
1})\cap\pi_*[D]. $$
However, $\pi_*[D]$ vanishes unless
$\dim(\pi(D))=\dim(D)=\dim(\lamoduli{n-1}{d})$, so
$\pi(D)=\lamoduli{n-1}{d}$ and thus $\operatorname{codim}(\tau(D))=0$. \par
It remains to consider the case when none of the $\gamma_t$'s is the
identity. We will count dimensions. The dimension of ${\overline M}^1_{0,n}(\pp,d)$ is
$3d-1+n$, and
\begin{equation*}
\operatorname{codim}(e_i^*(h))=\operatorname{codim}(e_i^*(\check{h}))=1, \quad
\operatorname{codim}(e_i^*(h^2))=\operatorname{codim}(e_i^*(\check{h}^2))=2, \quad
\operatorname{codim}(e_i^*(h^2\check{h}))=3. \end{equation*}
Let $a$ be the number of pullbacks
of $h^2$, $b$ the number of pullbacks of $\check{h}^2$ and $c$ the number of
pullbacks of $h^2\check{h}$.
Then the class $e^*_1(\gamma_1)\cap \dots \cap
e^*_n(\gamma_n)$ has dimension 1 if $a+b+2c=3d-2$. The number of pullbacks
of the divisor
classes $h$ and $\check{h}$ does not influence the vanishing nor the dimension of
the class. The effect of a divisor class on the degree of the class,
evaluated on
$[D]$ is multiplication by $d$ in the case of $h$, and multiplication by $2d-2$
in the case of $\check{h}$, by Proposition \ref{GWprop}. \par
Assume $\operatorname{codim}(\tau(D))\geq 2$; then $\dim(\tau(D))\leq 3d-3$, and a dimension
count checks that there are no stable maps in $\tau(D)$ through $a$ general
points, tangent to $b$ lines and incident to $c$ flags with $a+b+2c=3d-2$.
Hence one must put at least one condition on a map to a fiber of
$I\to \pp$ contracted by $f$. However, because of Proposition
\ref{liftclassif} this will not bring down the count $3d-2$ of conditions
on the
maps in $\tau(D)$. Indeed, putting one tangency condition on a map to a
fiber will, in case of a fiber over the image of an attachment point, induce an
extra point condition on the maps in $\tau(D)$, and in the case of a fiber over
a singularity of maps in $\tau(D)$, induce the condition that a singularity
should be on a line; also a codimension 1 condition. Similarly, putting two
tangent conditions or one flag condition on a map to a fiber will induce two
extra point conditions or the condition that a singularity of the map should be
at a fixed point, respectively. These are both codimension 2 conditions. It is
clearly impossible to put point conditions on the fiber maps, so one concludes
that the evaluation is zero unless $\operatorname{codim}(\tau(D))\leq 1$. \end{proof}
\begin{cor}\label{irrel}
A component $D$ of $D\left(A_1,A_2,(d_1,c_1),(d_2,c_2)\right)$
is irrelevant unless the bidegrees $(d_1,c_1)$ and $(d_2,c_2)$ are either
\begin{trivlist}
\item[(1)] $(d_1,2d_1-2)$ and $(d_2,2d_2)$ with $d_1\geq 1$, $d_2\geq 0$
and $d_1+d_2=d$, or \item[(2)] $(d,2d-3)$ and $(0,1)$ with $d\geq 2$,
\end{trivlist}
or vice versa.
\end{cor}
\begin{proof}
First case: assume that $d_1=d$ and $d_2=0$ or vice versa.
The locus of curves in ${\overline M}_{0,0}(\pp,d)$ of degree $d$ and class
$c$ is dense if $c=2d-2$, has codimension 1 if $c=2d-3$ and codimension at
least 2 for all other values of $c$. (The case $d=2$ is special. Here $\tau(D)$
could parametrize degree two covers of lines in $\pp$.) \par
Second case: $d_1\geq 1$ and $d_2\geq 1$. We have that $\tau(D)$ is a subset of
the irreducible divisor $D(d_1,d_2)$ on ${\overline M}_{0,0}(\pp,d)$. Therefore
$D$ is irrelevant unless $\tau(D)=D(d_1,d_2)$. Now, by Proposition
\ref{stableliftd1d2} the stable lift of a generic member of $D(d_1,d_2)$ is a
map from a curve with three components and the bidegrees of the map, restricted
to the three components, are $(d_1,2d_1-2)$, $(0,2)$ and $(d_2,2d_2-2)$. Thus,
if $D$ is relevant
then the bidegrees must be $(d_1,2d_1-2)$ and $(d_2,2d_2)$, or vice versa.
\end{proof}
\par
We will next give a detailed description of the relevant divisors. To
accomplish this we need to introduce some more
special stacks of stable maps.
\par
The stack $M_{0,n+2}(I,(0,2))$ parametrizes maps of degree $2$ from $\pl$ to a
fiber of $I$ over $\pp$. Let $M_{0,n+2,\{\star,\diamond\}}(I,(0,2))$ be the
substack in which we demand that the last two markings $\star$ and $\diamond$
are the ramification points; let ${\overline
M}_{0,n,\{\star,\diamond\}}(I,(0,2))$
be its closure in ${\overline M}_{0,n+2}(I,(0,2))$.
In the same way, for maps of degree 2 to a fiber of $I$ over $\dpp$, we define
${\overline M}_{0,n,\{\star,\diamond\}}(I,(2,0))$.
\par
We define
\begin{equation}\label{emoduli}
{\overline M}^1_{0,n,\{\star\}}(\pp,(d,2d))=\bigcup_{A_1\cup A_2=\{1,\dots,n\}}
{\overline M}^1_{0,A_1\cup \{\diamond\}}(\pp,d)\times_I {\overline
M}_{0,A_2,\{\star,\diamond\}}(I,(0,2))
\end{equation}
where each fiber product is defined by using the evaluation maps corresponding
to $\diamond$; it is a closed substack of ${\overline M}_{0,n+1}(I,(d,2d))$.
The general member of each component of ${\overline
M}^1_{0,n,\{\star\}}(\pp,(d,2d))$
represents a map from a curve with two components. The restriction of the map
to one component is the strict lift of a map to a rational nodal curve; the
restriction to the other component is as described in the previous paragraph.
Similarly, we define a closed substack of ${\overline M}_{0,n+1}(I,(2,1))$ by
\begin{equation}\label{ctwomoduli}
{\overline C}^1_{0,n,\{\star\}}(\pp,2)=\bigcup_{A_1\cup A_2=\{1,\dots,n\}}
{\overline
M}_{0,A_1,\{\star,\diamond\}}(I,(2,0))\times_I {\overline
M}_{0,A_2\cup\{\diamond\}}(I,(0,1)).
\end{equation}
The general member of each component of ${\overline
C}^1_{0,n,\{\star\}}(\pp,2)$
represents a map from a curve with two components. The restriction of the map
to one component is a degree $2$ cover of a line of $\pp$ ; the restriction to
the other component is a map of degree $1$ to the pencil of all tangent
directions through the ramification point $\diamond$ of the first component.
The other ramification point is specially marked $\star$, and there are $n$
additional markings. The reason for the notation is that the
bidegree $(2,1)$ follows the pattern of bidegrees $(d,2d-3)$ for cuspidal
curves.
\par
\newpage
\begin{prop}\label{ldivisors}
Ignoring irrelevant components, we have the following isomorphisms of
divisors in ${\overline M}^1_{0,n}(\pp,d)$:
\begin{trivlist}
\item[(1)] $D\left(A_1,A_2,(d,2d-2),(0,0)\right)\cong
{\overline M}^1_{0,A_1\cup \{\star\}}(\pp,d)
\times_I {\overline M}_{0,A_2\cup\{\star\}}(I,(0,0))$ for
$d\geq 1$.
\item[(2)] $D\left(A_1,A_2,(d,2d-3),(0,1)\right)\cong {\overline
C}^1_{0,A_1,\{\star\}}(\pp,d)\times_I{\overline
M}_{0,A_2\cup\{\star\}}(I,(0,1))$ for
$d\geq 2$.
\item[(3)] $D\left(A_1,A_2,(d_1,2d_1-2),(d_2,2d_2)\right)\cong
{\overline M}^1_{0,A_1\cup\{\star\}}(\pp,d_1)\times_I {\overline
M}^1_{0,A_2,\{\star\}}(\pp,(d_2,2d_2))$ for $d_1,d_2>0$ with $d_1+d_2=d$.
\end{trivlist}
In particular each relevant component occurs with multiplicity one.
\end{prop}
\begin{proof}
\par
(1) Note that on the right side we have a fiber product of smooth
morphisms, hence an irreducible stack.
The forgetful morphism
$$
D\left(A_1,A_2,(d,2d-2),(0,0)\right)
\to {\overline M}^1_{0,A_1\cup \{\star\}}(\pp,d)
$$
is the restriction of the projection
$$
{\overline M}_{0,A_1\cup \{\star\}}(I,(d,2d-2))
\times_I {\overline M}_{0,A_2\cup\{\star\}}(I,(0,0))
$$
onto its first factor.
Comparing dimensions, we conclude that
we have the desired isomorphism of stacks.
\par
To see that the multiplicity is one, we remark that the lifting
map $\lambda$ extends to general points of our divisor.
Indeed, a general point represents the lift of a stable map from
a curve with two components, consisting of an immersion onto a rational
degree $d$ curve together with a constant map to some nonsingular point.
The unique stable lift consists of the strict lift of the immersion
together with the constant map to the unique tangent direction at the
point. (See Figure 5.1.)
Since $\lambda$ is a birational morphism, and since
$D(A_1,A_2,d,0)$ is a reduced divisor on ${\overline M}_{0,n}(\pp,d)$,
our divisor is likewise reduced.
\par
$$
\xy
0;<0.5cm,0cm>:
(-1,-1);(1,1)**\dir{-};
(-1,1);(1,-1)**\dir{-};
(0,0)*{\bullet};
(-0.5,0.5)*{\bullet};
(0.5,-0.5)*{\bullet};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,0.5);(5.5,0)**\crv{(8,-0.5)&(4,-1.5)};
(5.5,0);(6,0.5)**\dir{-};
(5,0.185)*{\bullet};
(1.5,1.5);(3.5,3.5)**\dir{-};
?>*\dir{>};
(5,3.5);(5,1.5)**\dir{-};
?>*\dir{>};
(4,5.5);(5.4,4.9)**\crv{(8,4.5)&(4,3.5)};
(5.55,5.05);(6,5.5)**\dir{-};
(5,5.185)*{\bullet};
\endxy
$$
\par
\bigskip
\begin{center}
{\bf Figure 5.1} Stable lift of $D\left(A_1,A_2,(d,2d-2),(0,0)\right)$,
with two markings in $A_2$ shown.
\end{center}
\par
(2) For $d \geq 3$ we again observe that on the right side we have a fiber
product of smooth morphisms, hence an irreducible stack. By Proposition
\ref{unless}, the image $\tau(D)$ of a relevant component $D$ of
$D\left(A_1,A_2,(d,2d-3),(0,1)\right)$ must have codimension at most $1$.
But a point of $\tau(D)$ cannot represent an immersion onto a nodal curve,
since the lift of such a curve has bidegree $(d,2d-2)$.
Nor can it represent a map to a reducible curve, since by Proposition
\ref{stableliftd1d2} the stable lift of a general map of this type has a
component of bidegree $(0,2)$.
Hence $\tau(D)$ must be the divisor on ${\overline M}_{0,0}(\pp,d)$
representing cuspidal curves. By Proposition \ref{liftclassif}, the stable lift
of a general cuspidal curve consists of its strict lift together with a map to
the pencil of tangent directions at the cusp.
Hence we have the
desired isomorphism of stacks (ignoring irrelevant components).
\par
To see that the multiplicity is one, we first consider the case when $A_2$
is empty. By a local calculation we will verify that the lifting map
$\lambda$ extends to general points of our divisor.
The deformation theory of a node degenerating into a cusp is given in local
coordinates by $x^3+\epsilon x^2=y^2$. We may parametrize this family by
$$
(\epsilon,t)\mapsto (t^2-\epsilon,t^3-t\epsilon).
$$
Then the family of strict lifts is given by
$$
(\epsilon,t)\mapsto
\bigg((t^2-\epsilon,t^3-t\epsilon),[2t:3t^2-\epsilon]\bigg)
$$
for $\epsilon\ne 0$.
This map is not determined at $\epsilon=0$, $t=0$. Blowing up the point of
indeterminacy by introducing projective coordinates $[\epsilon_1:t_1]$
satisfying $\epsilon t_1=t\epsilon_1$, we extend the map to
$$
\bigg((\epsilon,t),[\epsilon_1:t_1]\bigg)\mapsto
\bigg((t^2-\epsilon,t^3-t\epsilon),[2t_1:3tt_1-\epsilon_1]\bigg).
$$
The special map for $\epsilon=0$ is (locally) the stable lift of $D$. It
has two components: the strict lift of the cuspidal curve, with
bidegree$(d,2d-3)$, together with a map to the pencil of tangent directions
at the cusp, with bidegree $(0,1)$.
(See Figure 5.2.)
Thus, as claimed, the lifting map $\lambda$ is defined at the point of
$D\left(A_1,\emptyset,(d,2d-3),(0,1)\right)$ representing the cuspidal
curve. And thus, by the same argument as in part (1), this divisor is
reduced.
\par
$$
\xy
0;<0.5cm,0cm>:
(-1,-1);(1,1)**\dir{-};
(-1,1);(1,-1)**\dir{-};
(0,0)*{\bullet};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,-1);(4,1)**\dir{-};
(6,-1);(6,1)**\crv{(2,0)};
(4,0)*{\bullet};
(0,-1.5);(0,-3.5)**\dir{-};
?>*\dir{>};
(5,-1.5);(5,-3.5)**\dir{-};
?>*\dir{>};
(-1,-6);(1,-4)**\dir{-};
(0,-5)*{\bullet};
(1.5,-5);(3.5,-5)**\dir{-};
?>*\dir{>};
(6,-4);(4,-5)**\crv{(6,-5)};
(6,-6);(4,-5)**\crv{(6,-5)};
(4,-5)*{\bullet};
\endxy
$$
\par
\bigskip
\begin{center} {\bf Figure 5.2} Stable lift of
$D\left(A_1,A_2,(d,2d-3),(0,1)\right)$, $d\geq 3$.
\end{center}
\par
Interpreting this calculation in a different way, we have shown that, for
the unique stable lift of a stable map to a cuspidal curve, we can build a
one-parameter family for which the general member is the strict lift of an
immersion and for which the special member is the specified cuspidal stable
lift; since the special member appears in the family with multiplicity one,
the divisor $D\left(A_1,\emptyset,(d,2d-3),(0,1)\right)$ must be reduced.
Similarly, we can build such a family when $A_2$ is nonempty. We assume we
are at a general point of the divisor
$D\left(A_1,A_2,(d,2d-3),(0,1)\right)$, so that the corresponding map has
just two components, with the markings indexed by $A_2$ lying on the $\pl$
mapping to the pencil of directions at the cusp. Using the same family of
maps as above, we let $L_1, L_2, \dots$ be lines through the origin in the
$\epsilon$-$t$ plane whose directions correspond to these markings, and let
the markings on nearby members of the family be the points of intersection
with these lines. Then blowing up the origin, as above, creates a family
whose special member is the specified cuspidal stable lift, with the
specified markings. Since this member appears in the family with
multiplicity one, the divisor
$D\left(A_1,A_2,(d,2d-3),(0,1)\right)$ must be reduced.
\par
We now consider the case $d=2$. The image $\tau(D)$ of a relevant component
$D$ of $D\left(A_1,A_2,(2,1),(0,1)\right)$ must have codimension at most
$1$. But a point of $\tau(D)$ cannot represent a map to a nonsingular
conic, since the lift of such a curve has bidegree $(2,2)$, nor a map to a
pair of distinct lines, since by Proposition \ref{stableliftd1d2} the
stable lift of such a map has a component of bidegree $(0,2)$. Hence
$\tau(D)$ must be the stack of degree $2$ covers of lines in $\pp$. By
Proposition \ref{liftclassif}, we have the desired isomorphism of stacks
(again ignoring irrelevant components).
\par
To see that the multiplicity is one, we consider a general point of the
divisor. This point represents the stable lift of a degree $2$ cover of a
line. A deformation of such a map is given by
$$
(\epsilon,t)\mapsto (\epsilon t,t^2),
$$
and the family of lifts is given by
$$
(\epsilon,t)\mapsto \bigg((\epsilon t,t^2),[\epsilon:2t]\bigg).$$
Blowing up the point of indeterminacy $\epsilon=t=0$ by introducing
projective coordinates $[\epsilon_1:t_1]$ satisfying $\epsilon
t_1=t\epsilon_1$, we extend the family of lifts to
$$
\bigg((\epsilon,t),[\epsilon_1:t_1]\bigg)\mapsto \bigg((\epsilon
t,t^2),[\epsilon_1:2t_1]\bigg). $$
We do a similar blowup at the other ramification point
(located at $t = \infty$). Then the
special member $\epsilon=0$ has one component that is the strict lift of
the cover of the line and two components mapping into the pencils of
tangent directions at the ramification points.
(See Figure 5.3.)
As in the case $d \geq 3$, we can arrange for the markings on the special
member to appear in any specified configuration. Since the special member
appears in the family with multiplicity $1$, the divisor
$D\left(A_1,A_2,(2,1),(0,1)\right)$ must be reduced.
\par
$$
\xy
0;<0.5cm,0cm>:
(-1,0);(1,0)**\dir{-};
(-0.333,-1);(-0.333,1)**\dir{-};
(-0.333,0)*{\bullet};
(-0.333,0)*{\times};
(0.333,-1);(0.333,1)**\dir{-};
(0.333,0)*{\bullet};
(0.333,0)*{\times};
(1.5,0);(3.5,0)**\dir{-};
?>*\dir{>};
(4,0);(6,0)**\dir{-};
(4.667,-1);(4.667,1)**\dir{-};
(4.667,0)*{\bullet};
(4.667,0)*{\times};
(5.333,-1);(5.333,1)**\dir{-};
(5.333,0)*{\bullet};
(5.333,0)*{\times};
(0,-1.5);(0,-3.5)**\dir{-};
?>*\dir{>};
(5,-1.5);(5,-3.5)**\dir{-};
?>*\dir{>};
(-1,-5);(1,-5)**\dir{-};
(-0.333,-5)*{\times};
(0.333,-5)*{\times};
(1.5,-5);(3.5,-5)**\dir{-};
?>*\dir{>};
(4,-5);(6,-5)**\dir{-};
(4.667,-5)*{\times};
(5.333,-5)*{\times};
\endxy
$$
\par
\bigskip
\begin{center}
{\bf Figure 5.3} Stable lift of $D\left(A_1,A_2,(d,2d-3),(0,1)\right)$,
$d=2$.
\end{center}
\par
(3) The image $\tau(D)$ of a relevant component $D$ of
$D\left(A_1,A_2,(d_1,2d_1-2),(d_2,2d_2)\right)$ must have codimension at
most $1$. A point of $\tau(D)$ cannot represent a map to an irreducible
curve of degree $d$. (Nor, in case $d=2$, can it represent a double cover
of a line.) Hence $\tau(D)$ must be a special boundary divisor on
${\overline M}_{0,0}(\pp,d)$:
$$
\tau(D) =
{\overline M}_{0,\{\star\}}(\pp,d_1)
\times_\pp
{\overline M}_{0,\{\star\}}(\pp,d_2)
$$
for some $d_1$ and $d_2$.
A general point of $D$ represents a
stable lift of the sort of map described in Proposition \ref{stableliftd1d2},
which tells us that the unique stable lift has three components
of bidegrees
$(d_1,2d_1-2)$, $(0,2)$ and $(d_2,2d_2-2)$, and
thus shows that we have the desired isomorphism of stacks. (Refer again to
Figure 3.3.)
\par
To see that the divisor is reduced, we use the family of maps described in
the proof of Proposition \ref{stableliftd1d2}, with total space the surface
$xy=\epsilon^2$. The special member $\epsilon=0$ occurs with multiplicity
$1$; thus the divisor must be reduced. Furthermore we can specify markings
on the nearby members so as to obtain any configuration of specified
markings on the central component of the special member, as follows.
We introduce projective coordinates $[x_1:y_1:\epsilon_1]$, so that the
central component is the curve $x_1y_1=\epsilon_1^2$. Then to obtain
$[x_1:y_1:\epsilon_1]$ as a marking on the central component, we use
$(\frac{x_1}{\epsilon_1}\epsilon,\frac{y_1}{\epsilon_1}\epsilon,\epsilon)$
as a marking on the general member.
\end{proof}
\par
\section{Quantum potentials}\label{sectionpotentials} \par
We now introduce potential functions for each of the seven different kinds
of stable maps we are considering. We are not concerned with questions of
convergence, and all of these potentials should be interpreted as formal
power series in the indeterminates $y_0,\dots,y_5$, where
$$
\gamma=y_0T_0+\dots+y_5T_5
$$
is an arbitrary element of $A^*(I) \otimes \q$
(or in some cases as formal power series in two sets of indeterminates).
Our definitions are inspired by Proposition \ref{ldivisors} and the
auxiliary equations (\ref{emoduli}), (\ref{ctwomoduli}).
\par
We define ${\mathcal P}$ to be the classical potential of the incidence
correspondence:
$$
{\mathcal P}=\sum_{n \geq 3} \frac{1}{n!}
\int e^*_1(\gamma)\cup \dots \cup e^*_n(\gamma) \cap [{\overline M}_{0,n}(I,(0,0))].
$$
We begin the summation at $n=3$ since otherwise the moduli stacks are
empty. In fact the only nonzero term is the first one, which encodes the
intersection product:
$$
{\mathcal P}=
\frac{y_0^2y_5}{2}+y_0y_1y_4+y_0y_2y_3+\frac{y_1^2y_3}{2}+\frac{y
_1y_3^2}{2}.
$$
\par
Similarly, we define a quantum potential
$$
{\mathcal N}=\sum_{\substack{n \geq 0 \\ d\geq 1}} \frac{1}{n!}
\int e^*_1(\gamma)\cup \dots \cup e^*_n(\gamma) \cap [{\overline M}^1_{0,n}(\pp,d)].
$$
By Proposition \ref{GWprop} we have
$$
{\mathcal N}=\sum_{\substack{d\geq 1 \\a +b+2c=3d-1\\ a,b,c\geq 0}}
\frac{N_d(a,b,c)y_2^ay_4^by_5^c\exp(dy_1+(2d-2)y_3)}{a!b!c!}. $$
\par
To define the quantum potential for cuspidal stable lifts we need to
introduce a second arbitrary cohomology class
$$
\delta=z_0T_0+\dots+z_5T_5.
$$
The potential is
\begin{align*}
{\mathcal C}
&=\sum_{\substack{n \geq 0 \\ d\geq 2}} \frac{1}{n!}
\int e^*_1(\gamma)\cup \dots \cup e^*_n(\gamma)
\cup e_C^*(\delta)
\cap [{\overline C}^1_{0,n,\{\star\}}(\pp,d)] \\
&=\sum_{\substack{d\geq 2\\ a+b+2c=3d-2-\dim(T_s)\\a,b,c\geq
0\\s=0,\dots,5}} \frac{z_s C_d(a,b,c;T_s)y_2^ay_4^by_5^c\exp(dy_1+(2d-
3)y_3)}{a!b!c!}.
\end{align*}
\par
We will also need three potentials corresponding to maps to a
projective line.
The potential for stable maps of degree 1 to a fiber of $I$ over $\pp$ is
\begin{align*}
{\mathcal F}&=\sum_{n \geq 0} \frac{1}{n!}
\int e^*_1(\gamma)\cup \dots \cup e^*_n(\gamma) \cap
[{\overline M}_{0,n}(I,(0,1))] \\
&=\left(\frac{y_4^2}{2}+y_5\right)\exp(y_3).
\end{align*}
\par
The potential for stable maps of degree 2 to such a fiber, with the
ramification at two specially marked points, is
\begin{equation*}
{\mathcal R}=\sum_{n \geq 0} \frac{1}{2n!}
\int e^*_1(\gamma) \cup \dots \cup e^*_n(\gamma)
\cup e^*_{n+1}(\delta) \cup \cup e^*_{n+2}(\delta)
\cap
[{\overline M}_{0,n,\{\star,\diamond\}}(I,(0,2))].
\end{equation*}
Here it is important, in case $n=0$, that we use the {\em stack} of stable
maps of this type, since every such map has a nontrivial automorphism. To
write ${\mathcal R}$ in an explicit way, we use the Gromov-Witten invariants for
such maps:
\begin{equation*}
R(a,b,c;\delta_1\cdot\delta_2)=
\int e^*_1(\gamma_1) \cup \dots \cup e^*_n(\gamma_1)
\cup e^*_{n+1}(\delta_1) \cup e^*_{n+2}(\delta_2)
\cap
[{\overline M}_{0,n,\{\star,\diamond\}}(I,(0,2))],
\end{equation*}
where $a$ of the $\gamma_t$'s equal the class $h^2$, $b$ of them equal the
class $\check{h}^2$, and the remaining $c$ of them equal $h^2\check{h}$. Unless $n=0$
we may interpret these as the number of maps satisfying $a$ point
conditions, $b$ tangency conditions and $c$ flag conditions, plus the
conditions $\delta_1$ and $\delta_2$ at the two ramification points; if
$n=0$, however, the invariant is a fraction:
\begin{align*}
R(0,2,0;T_3\cdot T_3)=&2, & R(0,0,1;T_3\cdot T_3)=&1, & R(0,1,0;T_3\cdot
T_4)=&1,\\
R(0,0,0;T_4\cdot T_4)=&\frac{1}{2}, & R(0,0,0;T_3\cdot T_5)=&\frac{1}{2}.
\end{align*}
Thus
$$
{\mathcal R}=\left\{\frac{z_3^2}{2}(y_4^2+y_5)+
z_3z_4y_4+\frac{z_3z_5}{2}+\frac{z_4^2}{4}\right\}\exp(2y_3).
$$
\par
Similarly, the potential for stable maps of degree 2 to a fiber of $I$
over the dual projective plane $\dpp$, with the ramification at two
specially marked points, is
\begin{align*}
{\mathcal L} &= \sum_{n \geq 0} \frac{1}{2n!}
\int e^*_1(\gamma) \cup \dots \cup e^*_n(\gamma)
\cup e^*_{n+1}(\delta) \cup e^*_{n+2}(\delta)
\cap
[{\overline M}_{0,n,\{\star,\diamond\}}(I,(2,0))] \\
&=\left\{\frac{z_1^2}{2}(y_2^2+y_5)+ z_1z_2y_2+\frac{z_1z_5}{2}+
\frac{z_2^2}{4}\right\}\exp(2y_1).
\end{align*}
\par
Finally we have the potential ${\mathcal E}$ for maps represented by the stacks
${\overline M}^1_{0,n,\{\star\}}(\pp,(d,2d))$:
\begin{align*}
{\mathcal E}&=\sum_{\substack{n \geq 0 \\ d\geq 1}} \frac{1}{n!}
\int e^*_1(\gamma) \cup \dots \cup e^*_n(\gamma)
\cup e^*_{n+1}(\delta)
\cap
[{\overline M}^1_{0,n,\{\star\}}(\pp,(d,2d))] \\
&= \sum_{\substack{d\geq 1\\
a+b+2c=3d+1-\dim(T_s)\\a,b,c\geq 0\\s=0,\dots,5}}
\frac{z_sE_d(a,b,c;T_s)y_2^ay_4^by_5^c\exp(dy_1+(2d-3)y_3)}{a!b!c!},
\end{align*}
where $E_d(a,b,c;\delta)$ denotes the Gromov-Witten invariant for such maps
subject to $a$ point conditions, $b$~tangency conditions and $c$ flag
conditions, and subject to the condition
$\delta$ at the ramification point.
\par
Each of the divisors described in Proposition \ref{ldivisors} is a fiber
product
$M_1 \times_I M_2$ inside the moduli stack ${\overline M}_{0,n}(I,(d,2d-2))$, and each therefore
fits into a fiber square
$$
\xymatrix{
M_1 \times_I M_2 \ar[r]^\iota \ar[d]
& M_1 \times M_2 \ar[d]\\
I^{n+1}
\ar[r]^\Delta & I^{n+2}. }
$$
in which $\Delta$ is the diagonal inclusion that repeats the last factor.
The components of the vertical morphism on the right are the various
evaluation maps to $I$; the last two $e_{n+1}$ and $e_{n+2}$ become equal
when restricted to $M_1 \times_I M_2$.
By the decomposition (\ref{diag}) of the diagonal class, we have
\begin{equation}
\label{diageq}
\iota_*\left(e_1^*(\gamma_1)\cup\dots\cup e_n^*(\gamma_n)\right)
=\sum_{s=0}^5
e_1^*(\gamma_1)\cup\dots\cup e_n^*(\gamma_n)
\cup e_{n+1}^*(T_s)
\cup e_{n+2}^*(T_{5-s}).
\end{equation}
There is a partition $A_1 \cup A_2=\{1,\dots,n\}$, so that the evaluation
maps $e_t$ indexed by $t \in A_1$ factor through $M_1$ and those indexed by
$A_2$ factor through $M_2$. Taking degrees in (\ref{diageq}), we obtain the
following Proposition. (For details of this computation, see \cite[Lemma,
p.15]{Fulton2}.)
\begin{prop}\label{diagprop}
In this situation
\begin{align*}
\int e_1^*(\gamma_1)\cup\dots\cup e_n^*(\gamma_n)
& \cap [M_1 \times_I M_2] \\
& =\sum_{s=0}^5
\int
\bigcup_{t\in A_1}e_t^*(\gamma_t)
\cup e_{n+1}^*(T_s)
\cap [M_1]
\cdot
\int
\bigcup_{t\in A_2}e_t^*(\gamma_t)
\cup e_{n+2}^*(T_{5-s})
\cap [M_2].
\end{align*}
\end{prop}
\par
To use Proposition \ref{diagprop}, we need the following elementary
observation, which is a consequence of the chain rule.
\begin{prop}\label{potdiff}
Let $\gamma=y_0T_0+y_1T_1+\dots+y_5T_5$ and
$\delta=z_0T_0+z_1T_1+\dots+z_5T_5$. Suppose that ${\mathcal K}$
is a power series in $y_0,\dots,y_5,z_0,\dots,z_5$ which can be written in
the following way:
$$
{\mathcal K}=\sum_{\substack{n>0 \\m>0
}}\frac{K(\gamma^n;\gamma^{\prime m})}{n!m!}. $$
Let $k_1,\dots,k_r$ and $l_1,\dots,l_s$ have values in $\{0,\dots,5\}$.
Then
$$
\frac{\partial {\mathcal K}}{\partial y_{k_1}\dots \partial y_{k_r}\partial
z_{l_1}\dots \partial z_{l_s}}= \sum_{\substack{n>0 \\m>0
}}\frac{K(\gamma^nT_{k_0}\dots T_{k_r};\gamma^{\prime
m}T_{l_0}\dots T_{l_r})}{n!m!}. $$
\end{prop}
\par
\begin{prop}\label{Epot}
$\displaystyle {\mathcal E}=\sum_{s=0}^5
\frac{\partial {\mathcal N}}{\partial y_s}
\frac{\partial {\mathcal R}}{\partial z_{5-s}}.$
\end{prop}
\begin{proof}
Set
$\gamma_1=\dots=\gamma_n=\gamma=y_0T_0+y_1T_1+\dots+y_5T_5$.
Apply Proposition \ref{diagprop} to each component of
${\overline M}^1_{0,n,\{\star\}}(\pp,(d,2d))$, as listed in (\ref{emoduli}).
Sum
this identity over all components, over all $d\geq 1$, and over all $n\geq
1$. Then apply Proposition \ref{potdiff}.
\end{proof}
\par
Let ${\mathcal C}_2$ be the leading term ($d=2$) of the potential ${\mathcal C}$. Then a
similar argument, applied to (\ref{ctwomoduli}), yields the following
identity.
\par
\begin{prop}\label{Ctwopot}
$\displaystyle {\mathcal C}_2=\sum_{s=0}^5
\frac{\partial {\mathcal L}}{\partial z_s}
\frac{\partial {\mathcal F}}{\partial y_{5-s}}.$
\end{prop}
\par
The same sort of argument yields the basic identity from which we will
derive recursive equations for characteristic numbers. Let $i$, $j$, $k$, and
$l$ be integers between 1 and 5. Let
\begin{equation*}
{\mathcal G}(ij\mid kl)=
\sum_{n \geq 4} \int e_1^*(T_i) \cup e_2^*(T_j) \cup e_3^*(T_k) \cup
e_4^*(T_l) \cup e_5^*(\gamma) \cup \dots \cup e_n^*(\gamma) \cap [D(12\mid
34)].
\end{equation*}
Then, by (\ref{sumpart}), Corollary \ref{irrel} and Proposition
\ref{ldivisors},
\begin{align*}
{\mathcal G}(ij\mid kl)=
\sum_{s=0}^5 \bigg\{ &\frac{\partial^3
{\mathcal N}}{\partial y_i \partial y_j \partial y_s}\frac{\partial^3 {\mathcal P}}{\partial
y_k\partial y_l \partial y_{5-s}}
+\frac{\partial^3 {\mathcal N}}{\partial y_k
\partial y_l \partial y_s}\frac{\partial^3 {\mathcal P}}{\partial y_i\partial y_j
\partial y_{5-s}}\\
+&\frac{\partial^3 {\mathcal C}}{\partial y_i \partial y_j \partial
z_s}\frac{\partial^3 {\mathcal F}}{\partial y_k\partial y_l \partial y_{5-s}}
+\frac{\partial^3 {\mathcal C}}{\partial y_k \partial y_l \partial
z_s}\frac{\partial^3 {\mathcal F}}{\partial y_i\partial y_j \partial y_{5-s}}\\
+&\frac{\partial^3 {\mathcal N}}{\partial y_i \partial y_j
\partial y_s}\frac{\partial^3 {\mathcal E}}{\partial y_k\partial y_l \partial z_{5-s}}
+\frac{\partial^3 {\mathcal N}}{\partial y_k \partial y_l \partial
y_s}\frac{\partial^3 {\mathcal E}}{\partial y_i\partial y_j \partial z_{5-s}}
\bigg\}.
\end{align*}
The linear equivalence of the divisors $D(12\mid 34)$
and $D(13\mid 24)$ immediately implies the basic identity,
which we record as a Theorem.
\par
\begin{thm}\label{ijklrel}
For each $i,j,k$ and $l$ in $\{1,2,3,4,5\}$, there is an identity $$
{\mathcal G}(ij \mid kl)={\mathcal G}(il \mid jk).
$$
\end{thm}
\par
\section{Recursive formulas for characteristic
numbers}\label{sectionrecursion}
\par
In this section we will
derive recursive
formulas for the characteristic numbers $N_d(a,b,c)$,
$C_d(a,b,c;1)$, $C_d(a,b,c;h)$, and $C_d(a,b,c;h^2)$.
For characteristic numbers of the first type, the base cases are
$$
N_1(2,0,0)=N_1(0,0,1)=1 \quad \text{and} \quad N_1(0,2,0)=0.
$$
\par
The cuspidal characteristic numbers make sense only for $d \geq 2$.
In the case $d=2$ they are defined using the stack
${\overline C}^1_{0,n,1}(\pp,2)$ whose decomposition is shown in
(\ref{ctwomoduli}).
Proposition \ref{Ctwopot} tells us how to calculate the corresponding
potential ${\mathcal C}_2$ from the known potentials ${\mathcal L}$ and ${\mathcal F}$.
Equating coefficients, we find that the only nonzero cuspidal
characteristic numbers in degree $d=2$ are
\begin{align*}
C_2(2,1,0;T_1)=&2, & C_2(0,1,1;T_1)=&1, & C_2(1,2,0;T_1)=&1, &
C_2(1,0,1;T_1)=1,\\ C_2(1,1,0;T_2)=&1, & C_2(0,2,0;T_2)=&\frac{1}{2}, &
C_2(0,0,1;T_2)=&\frac{1}{2}, & C_2(0,1,0;T_5)=\frac{1}{2}.
\end{align*}
The fact that some of these characteristic numbers are fractions reflects
the fact that the generic map of this type has a nontrivial automorphism.
One could also calculate these characteristic numbers directly, again
taking into account the automorphisms.
\par
In addition to the desired characteristic numbers, we will also calculate
the characteristic numbers $E_d(a,b,c;T_s)$ for maps represented by the
stacks ${\overline M}^1_{0,n,1}(\pp,(d,2d))$, which appear as coefficients
in the potential ${\mathcal E}$. We are not particularly interested in these
numbers, but using them simplifies many of the formulas.
\par
\begin{prop}\label{ENrels}
\begin{align*}
&E_d(a,b,c;\check{h})=dbN_d(a,b-1,c)+b(b-1)N_d(a+1,b-2,c)+cN_d(a+1,b,c-1)
\text{ for $a+b+2c=3d$;}\\
&E_d(a,b,c;\check{h}^2)=\frac{d}{2}N_d(a,b,c)+bN_d(a+1,b-1,c)
\text{ for $a+b+2c=3d-1$;}\\
&E_d(a,b,c;h^2\check{h})=\frac{1}{2}N_d(a+1,b,c)\text{ for $a+b+2c=3d-2$;}\\
&E_d(a,b,c;T_s)=0 \text{ if $s = 0$, $1$ or $2$.}
\end{align*}
\end{prop}
\begin{proof}
Use the identity of Proposition \ref{Epot}; equate the coefficients.
\end{proof}
\par
\begin{thm} \label{algthm} There is a recursive algorithm, based on Theorem
\ref{ijklrel}, for
calculating all the characteristic numbers $N_d(a,b,c)$,
$C_d(a,b,c;1)$, $C_d(a,b,c;h)$, $C_d(a,b,c;h^2)$,
$E_d(a,b,c;\check{h})$, $E_d(a,b,c;\check{h}^2)$, and $E_d(a,b,c;h^2\check{h})$.
\end{thm}
Pandharipande \cite[Proposition 4]{Pandharipande2} proved that there is an
explicit algorithm for calculating all the numbers $N_d(a,b,0)$ and gave
explicit formulas for the numbers $C_d(3d-2,0,0;1)$ \cite[Proposition
5]{Pandharipande2}. \par
\begin{proof}
The proof will use induction on $d$.
Assume that all characteristic numbers have been
determined for degrees less than $d$. Note that the last three types of
characteristic numbers (the $E_d$'s) are determined by the other types;
hence we need only to worry about the $N_d$'s and $C_d$'s.
To simplify the argument we will denote
by $[d-1]$ any expression involving characteristic numbers for curves of
degree $d-1$ or less.
\par
First we list various cases of Theorem \ref{ijklrel} which immediately
determine a
characteristic number in degree $d$; each of these identities can be
written in the form
$$
N_d(a,b,c)=[d-1].
$$
\par
Equation $1122$ (meaning this case of the identity in Theorem \ref{ijklrel})
determines $N_d(a,b,c)$ for $a\geq 3$.
\par
Equation $1224$ determines $N_d(a,b,c)$ for $a\geq 2$ and $c\geq 1$.
\par
Equation $1155$ determines $N_d(a,b,c)$ for $a\geq 1$ and $c\geq 2$.
\par
Equation $1455$ determines $N_d(a,b,c)$ for $c\geq 3$.
\par
This leaves only six of the numbers $N_d(a,b,c)$ undetermined, namely
\begin{align*}
&N_d(2,3d-3,0), & &N_d(1,3d-4,1), & &N_d(0,3d-5,2),\\ &N_d(1,3d-2,0), &
&N_d(0,3d-3,1), & &N_d(0,3d-1,0). \end{align*}
\par
Similarily (for $d \geq 3$) there are equations of the form: $$
C_d(a,b,c;h^2)=[d-1].
$$
\par
Equation $2445$ determines $C_d(a,b,c;h^2)$ for $a\geq 1$ and $c\geq
1$.
\par
Equation $4455$ determines $C_d(a,b,c;h^2)$ for $a\geq 2$ and $c\geq
1$.
\par
Equation $2244$ determines $C_d(a,b,c;h^2)$ for $a\geq 2$.
\par
This leaves
\begin{align*}
&C_d(1,3d-5,0;h^2), & &C_d(0,3d-6,1;h^2), & & C_d(0,3d-4,0;h^2) \end{align*}
undetermined.
\par
Next, we exhibit in matrix form 8 independent equations involving the
undetermined characteristic numbers listed above, except $N_d(0,3d-1,0)$.
$$
\left(\begin{smallmatrix} 2(3d-3) & -d & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 3d-4 & -2d & 0 & 0 & 0 & 0 & 0\\
2(d-2) & 0 & 0 & -d(3d-2) & d^2 & 0 & 0 & 0\\ 0 & 0 & 2(3d-5) & 0 & 0 & 0 & -d
& 0\\
0 & 0 & 2(3d-5) & 0 & 0 & -(3d-5) & 0 & 0\\ 0 & 3d-3 & -4d+4 & 0 & (3d-4)(3d-3)
& 0 & 0 & -(3d-4)d\\ 0 & d-2 & 0 & 0 & -d(3d-3) & 0 & 0 & d^2\\ 0 & 0 & 0 & 0
& -(3d-4)d & d^2 & 0 & 0\\ \end{smallmatrix}\right) \left(\begin{smallmatrix}
N_d(2,3d-3,0)\\ N_d(1,3d-4,1)\\ N_d(0,3d-5,2)\\ N_d(1,3d-2,0)\\ N_d(0,3d-
3,1)\\ C_d(1,3d-5,0;h^2)\\ C_d(0,3d-6,1;h^2)\\ C_d(0,3d-4,0;h^2)
\end{smallmatrix}
\right) = [d-1]
$$
The rows of this equation correspond to the following equations:
1124, 1145, 1123, 1445, 2345, 1345, 1135, and 1135 again (for a different
choice of $a$, $b$, and $c$).
The determinant of the matrix is
$$
- 12 d^6(3 d - 5) (d - 1) (3 d - 4) (3 d - 2),
$$
which is nonzero for all integers
$d\geq 2$. Thus these equations determine the eight degree $d$
characteristic numbers listed in the column vector.
\par
Next, the equation $1134$ can be written
\begin{align*}
C_d(a,b,c;h)&=-C_d(a,b-1,c;h^2)b
+\frac{1}{d^2}\bigg\{N_d(a,b,c+1)d(2d-2)\\
&+N_d(a+1,b+1,c)(2-d) +N_d(a,b+2,c)d\bigg\}+[d-1].
\end{align*}
It determines all the numbers $C_d(a,b,c;h)$ except $C_d(0,3d-3,0;h)$.
\par
Equation $1133$ can be written
\begin{align*}
C_d(a,&b,c;1)=-C_d(a,b-1,c;h)b-C_d(a,b,c-1;h^2)c-C_d(a,b-
2,c;h^2)\binom{b}{2}\\ &+\frac{1}{d^2}\bigg\{4N_d(a+1,b,c)(d-
1)+N_d(a,b+1,c)(3d^2-4d)\bigg\}+[d-1]\end{align*}
It determines all the numbers $C_d(a,b,c;1)$ except $C_d(0,3d-2,0;1)$.
\par
It remains to determine the three numbers $N_d(0,3d-1,0)$, $C_d(0,3d-3,0;h)$
and $C_d(0,3d-2,0;1)$.
The two equations $1133$ and $1134$ just listed give two equations. Another
is given by $3344$. To show that they are linearly independent we write the
equations in matrix form and compute the determinant.
\begin{equation*}
\left(\matrix (3d-1)d(4-3d) & d^2 & 1\\
-(3d-2)(3d-1)d & d^2 & 0\\
(3d-3)(3d-2)(3d-1) & (3d-3)(7d+2) & (3d-3)(3d-2) \endmatrix\right)
\left(\matrix N_d(0,3d-1,0)\\ C_d(0,3d-3,0;h)\\ C_d(0,3d-2,0;1)
\endmatrix \right)=
\left(\matrix \text{known} \\ \text{known} \\ \text{known} \endmatrix
\right)
\end{equation*}
Here, ``known" means expressions involving already determined numbers of
degree $d$ or less. The determinant of the matrix is
$$
6 d (d - 1) (3 d - 1) (3 d - 2) (d^2 - 4 d - 1), $$
which is nonzero for all integers $d\geq 2$. This shows that we can find the
characteristic numbers in degree $d$ from those in characteristic $d-1$.
\end{proof}
We will next give an explicit recursive algorithm, developed and
implemented by the first author, which determines all the
characteristic numbers of Theorem \ref{algthm}. The algorithm will use 7
equations compared to the 17 used in the proof of Theorem \ref{algthm}.
A drawback is, however, that the recursion is not as transparent as the
recursion of the proof. Some degree $d+1$ characteristic numbers must be
computed before all numbers in degree $d$ may be computed. \par
Using equation $1122$ we derive
the following formula, which we call equation $1122a$. We must assume that
$a\geq3$. In all sums
$d_1,d_2>0$ and $a_1,a_2,b_1,b_2,c_1,c_2\geq 0$. \begin{align*}
&N_d(a,b,c)=\\
&\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a-
1\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2) \left[2d_1^2d_2\binom{a-
3}{a_1-1}
-d_1^3d_2\binom{a-3}{a_1}
-d_1d_2^2\binom{a-3}{a_1-1}
\right]\binom b{b_1}\binom c{c_1}\\
&+\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) \left[2d_1d_2\binom{a-
3}{a_1-2}
-d_1^2\binom{a-3}{a_1-1}
-d_2^2\binom{a-3}{a_1-2}
\right]\binom b{b_1}\binom c{c_1}\\
\end{align*}
If we put $a=3d-1$ and $b=c=0$ we recover Kontsevich's formula \cite[Claim
5.2.1]{KontsevichManin}. If we put $c=0$ we recover the recursive formula of
Francesco and C. Itzykson \cite[2.95, p.104]{FrancescoItzykson}.
\par
The next equation $1122b$ is a rewrite of $1122a$ in a very specific case.
We use
$1122a$ with $d$ replaced by $d+1$, $a=3$, $b=3d-1$ and $c=0$. Then we solve
for $N_d(0,b,0)$, which can only occur in the first sum of the right hand side
of $1122a$. There are two cases:
\begin{align*}
&(d_1,a_1,b_1,c_1)=(d,0,b,0), &(d_2,a_2,b_2,c_2)=(1,2,0,0), \end{align*}
and the same values with the indexes 1 and 2 interchanged. Using
Proposition \ref{ENrels} we find
\begin{equation*}
E_1(2,0,0;\check{h}^2)=\frac{1}{2},
\quad
E_d(0,b,0;\check{h}^2)=\frac{d}{2}N_d(0,b,0)+bN_d(1,b-1,0).
\end{equation*}
Then we have the following equation called 1122b:
\begin{align*}
&N_d(0,b,0)=\frac{1}{d^3}\bigg\{-N_{d+1}(3,b,0)-d^2bN_d(1,b-1,0)\\
&+\sum_{\substack{d_1+d_2=d+1\\ a_1+a_2=2\\b_1+b_2=b\\
(d_i,a_i,b_i)\neq (d,0,b)}}
N_{d_1}(a_1,b_1,0)E_{d_2}(a_2,b_2,0;\check{h}^2) \left[2d_1^2d_2\binom{0}{a_1-
1}
-d_1^3d_2\binom{0}{a_1}
-d_1d_2^2\binom{0}{a_1-1}
\right]\binom b{b_1}\\
&+\sum_{\substack{d_1+d_2=d+1\\ a_1+a_2=3\\b_1+b_2=b}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) \left[2d_1d_2\binom{0}{a_1-
2}
-d_1^2\binom{0}{a_1-1}
-d_2^2\binom{0}{a_1-2}
\right]\binom b{b_1}\\
\end{align*}
\par
Equation $1155$: If $a\geq1$ and $c\geq 2$ then \begin{align*}
&N_d(a,b,c)=\\
&\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a-1\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2) \left[2d_1^2d_2\binom{c-
2}{c_1-1}
-2d_1^2d_2\binom{c-2}{c_1-2}
-d_1^3\binom{c-2}{c_1}
\right]\binom {a-1}{a_1}\binom b{b_1}\\
&+\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) \bigg[2d_1d_2\binom {c-
2}{c_1-1}
-d_1^2\binom {c-2}{c_1}
-d_2^2\binom {c-2}{c_1-2}\bigg]\binom{a-1}{a_1-1}\binom b{b_1}
\end{align*}
\par
Equation $1123a$:
If $c\geq 1$ then
\begin{align*}
N_d(a,b&,c)=\frac{1}{d^2}\bigg\{N_d(a+1,b+1,c-1)-N_d(a+2,b,c-1)(d-2)\\
+&\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a+1\\b_1+b_2=b\\c_1+c_2=c-1}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2) \left[2d_1d_2^2\binom
{a}{a_1-1}
-2d_1^2d_2\binom {a}{a_1}\right]\binom{b}{b_1}\binom{c-1}{c_1}\\
&\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a+2\\b_1+b_2=b\\c_1+c_2=c-
1}} N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h})
\bigg[2d_2^2\binom{a}{a_1-2}-2d_1d_2\binom{a}{a_1-1}\bigg]
\binom{b}{b_1}\binom{c-1}{c_1} \bigg\}
\end{align*}
\par
Equation $1123b$: This is just a rewrite of 1123a. If $a\geq 1$ and $b\geq 1$
then \begin{align*}
N_d(a,b&,c)=(d-2)N_d(a+1,b-1,c)+d^2N_d(a-1,b-1,c+1)\\
+&\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a\\b_1+b_2=b-
1\\c_1+c_2=c}} N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2)
\left[2d_1^2d_2\binom {a-1}{a_1}-2d_1d_2^2\binom {a-1}{a_1-1}
\right]\binom{b-1}{b_1}\binom c{c_1}\\
&\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a+1\\b_1+b_2=b-
1\\c_1+c_2=c}} N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h})
\bigg[2d_1d_2\binom{a-1}{a_1-1}-2d_2^2\binom{a-1}{a_1-2}\bigg]
\binom{b-1}{b_1}\binom c{c_1} \end{align*}
\par
Equation $2245$: This is not an obvious recursion equation.
Given $a\geq 2,b\geq 1,c\geq 1$ and $d$ with $a+b+2c=3d-2$, we have that
\begin{align*}
0&=\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2)d_1 \bigg[\binom{a-2}{a_1-
2}\binom {b-1}{b_1}\binom{c-1}{c_1} +\binom{a-2}{a_1}\binom {b-1}{b_1-
1}\binom{c-1}{c_1-1}\\ &-\binom{a-2}{a_1-1}\binom {b-1}{b_1}\binom{c-
1}{c_1-1} -\binom{a-2}{a_1-1}\binom {b-1}{b_1-1}\binom{c-1}{c_1}\bigg]\\
&+\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a+1\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) \bigg[\binom{a-2}{a_1-
3}\binom {b-1}{b_1}\binom{c-1}{c_1} +\binom{a-2}{a_1-1}\binom{b-1}{b_1-
1}\binom{c-1}{c_1-1}\\ &-\binom{a-2}{a_1-2}\binom {b-1}{b_1}\binom{c-
1}{c_1} -\binom{a-2}{a_1}\binom{b-1}{b_1-1}\binom{c-1}{c_1}\bigg].\\
\end{align*}
To get a recursion equation, we rewrite Equation 2245 with $d$ replaced by
$d+1$, $a=2$
and $b+c=3d-1$. Then we solve for $N_d(0,b,c)$, which can only occur in the
first sum of the right hand side of $2245$. There are two cases:
\begin{align*}
&(d_1,a_1,b_1,c_1)=(d,0,b,c), &(d_2,a_2,b_2,c_2)=(1,2,0,0), \end{align*}
and the same values with the indexes 1 and 2 interchanged. Using
Proposition \ref{ENrels} we find
\begin{equation*}
E_1(2,0,0;\check{h}^2)=\frac{1}{2},
\quad
E_d(0,b,c;\check{h}^2)=\frac{d}{2}N_d(0,b,c)+bN_d(1,b-1,c).
\end{equation*}
We have, for $b\geq 1$, $c\geq 1$, equation 2245:
\begin{align*}
&N_d(0,b,c)=\frac{1}{d}\bigg\{-bN_d(1,b-1,c)-\\
&\sum_{\substack{d_1+d_2=d+1\\
a_1+a_2=2\\b_1+b_2=b\\c_1+c_2=c\\(d_i,a_i,b_i,c_i)\neq (d,0,b,c)}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2)d_1 \bigg[\binom{0}{a_1-
2}\binom {b-1}{b_1}\binom{c-1}{c_1} +\binom{0}{a_1}\binom {b-1}{b_1-
1}\binom{c-1}{c_1-1}\\ &-\binom{0}{a_1-1}\binom {b-1}{b_1}\binom{c-
1}{c_1-1} -\binom{0}{a_1-1}\binom {b-1}{b_1-1}\binom{c-1}{c_1}\bigg]\\
&+\sum_{\substack{d_1+d_2=d+1\\
a_1+a_2=3\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) \bigg[\binom{0}{a_1-
3}\binom {b-1}{b_1}\binom{c-1}{c_1} +\binom{0}{a_1-1}\binom{b-1}{b_1-
1}\binom{c-1}{c_1-1}\\ &-\binom{0}{a_1-2}\binom{b-1}{b_1}\binom{c-
1}{c_1} -\binom{0}{a_1}\binom{b-1}{b_1-1}\binom{c-1}{c_1}\bigg] \bigg\}
\end{align*}
\par
Now we are ready to present the algorithm for calculating the numbers
$N_d(a,b,c)$ with $a+b+2c=3d-1$. The inputs to the algorithm consist of the
trivial degree 1 numbers, which are zero except for $N_1(2,0,0)=1$ and
$N_1(0,0,1)=1$. It is also understood that the instruction ``use equation
$ijkl$'' means to use this equation and then to replace all occurences of
the $E_d$'s by means of Proposition \ref{ENrels}.
\par
\begin{tabbing}
$N_d(a,b,c):=$ \= {\bf if} $(d=1)$ {\bf then} \= {\bf if} $(a=2$ {\bf and $b=0$
and $c=0)$ then $1$} \\ \> \> {\bf else if $(a=0$ and $b=0$ and $c=1)$ then
$1$ }\\ \> \> {\bf else} 0 \\
\> {\bf else if $\big((a\geq 4)$ or $(a=3$ and $c\geq 1)\big)$ then} use
equation 1122a\\
\> {\bf else if $(a\geq 1$ and $c\geq 2)$ then} use equation 1155\\ \> {\bf
else if $(c\geq 3)$ then} use equation 1123a\\ \> {\bf else if $\big((a=3$ and
$c=0)$ or $(a=2$ and $c=1)$ or $(a=1$ and $c=1)$}\\ \> \> {\bf or $(a=2$ and
$c=0)$ or $(a=1$ and $c=0)\big)$ then} use equation 1123b\\
\> {\bf else if $\big((a=0$ and $c=2)$ or $(a=0$ and $c=1)\big)$ then} use
equation 2245\\ \> {\bf else} use equation 1122b.
\end{tabbing}
\par
We finally present tables of characteristic numbers for $d \leq 5$.
In the tables of characteristic numbers $N_d(a,b,c)$, columns have the same
$a$, rows have the same $c$, and $b$ is given by
$3d-1-a-2c$.
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|} \hline
$c\downarrow$ & \multicolumn{6}{|c|}{$N_2(a,b,c)$} \\ \hline 2 & 1 & 1
&\multicolumn{4}{c|}{} \\ \cline{1-5} 1 & 1 & 2 & 2 & 1 &
\multicolumn{2}{c|}{} \\ \hline 0 & 1 & 2 & 4 & 4 & 2 & 1 \\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{tabular}
\par
\smallskip
\par
{\bf Table 7.1} Characteristic numbers for plane conics.
\end{center}
\par
\bigskip
\begin{center}
\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{9}{|c|}{$N_3(a,b,c)$} \\ \hline 4 & 4 & \multicolumn{8}{c|}{}
\\ \cline{1-4} 3 & 16 & 12 & 6 & \multicolumn{6}{c|}{} \\ \cline{1-6} 2 & 56
& 56 & 40 & 20 & 8 & \multicolumn{4}{c|}{} \\ \cline{1-8} 1 & 148 & 200 &
196 & 136 & 68 & 28 & 10 & \multicolumn{2}{c|}{} \\ \hline 0 & 400 & 600 &
756 & 712 & 480 & 240 & 100 & 36 & 12 \\ \hline $a\to$ & 0 & 1 & 2 & 3 & 4
& 5 & 6 & 7 & 8 \\ \hline \end{tabular}
\par
\smallskip
\par
{\bf Table 7.2} Characteristic numbers for plane rational nodal cubics.
\end{center}
\bigskip
\par
The numbers for nodal cubics were calculated by Zeuthen \cite{Zeuthen},
Maillard \cite{Maillard}, Schubert \cite{Schubert}, Sacchiero \cite{Sacchiero},
Kleiman and Speiser \cite{KleimanSpeiser}, Aluffi \cite{Aluffi2} and
Pandharipande \cite{Pandharipande2}.
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{12}{|c|}{$N_4(a,b,c)$} \\ \hline 5 & 120 & 60 &
\multicolumn{10}{c|}{} \\ \cline{1-5} 4 & 816 & 528 & 264 & 108 &
\multicolumn{8}{c|}{} \\ \cline{1-7} 3 & 5040 & 3960 & 2472 & 1224 & 504 &
180 & \multicolumn{6}{c|}{} \\ \cline{1-9} 2 & 26408 & 25352 & 19424 &
11840 & 5816 & 2408 & 872 & 284 & \multicolumn{4}{c|}{} \\ \cline{1-11}
1 & 124592 & 140912 & 130824 & 97496 & 58208 & 28392 & 11792 & 4304 & 1416
& 428 & \multicolumn{2}{c|}{} \\ \hline 0 & 581904 & 728160 & 783584 &
699216 & 505320 & 295544 & 143040 & 59424 & 21776 & 7200 & 2184 & 620 \\
\hline $a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline
\end{tabular}
\par
\smallskip
{\bf Table 7.3} Characteristic numbers for plane rational nodal quartics.
\end{center}
\par
\bigskip
The numbers for nodal quartics with $c=0$ were calculated by Pandharipande
\cite{Pandharipande2}.
\par
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{7}{|c|}{$N_5(a,b,c)$} \\ \hline 7 & 840 &
\multicolumn{6}{c|}{} \\ \cline{1-4} 6 & 9120 & 4560 & 1920 &
\multicolumn{4}{c|}{} \\ \cline{1-6} 5 & 88560 & 52800 & 26160 & 11040 &
4080 & \multicolumn{2}{c|}{} \\ \hline 4 & 792432 & 548064 & 318432 &
155904 & 65712 & 24432 & 8184 \\ \hline 3 & 6347808 & 5092128 & 3442536 &
1961952 & 950880 & 400128 & 149448 \\ \hline
2 & 46200256 & 42546112 & 33296896 & 22024720 & 12343168 & 5928736 &
2489872 \\ \hline
1 & 317706976 & 327704960 & 292474400 & 222844672 & 144185600 & 79521536
& 37862920 \\ \hline
0 & 2150306368 & 2414524160 & 2397491872 & 2069215552 & 1532471744 &
969325888 & 526105120 \\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \end{tabular}
\par
\smallskip
\begin{tabular}{|r|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{8}{|c|}{$N_5(a,b,c)$} \\ \hline 3 & 50496 & 15672
& \multicolumn{6}{c|}{} \\ \cline{1-5} 2 & 932656 & 316960 & 99088 & 28816
& \multicolumn{4}{c|}{} \\ \cline{1-7} 1 & 15857120 & 5946992 & 2028160 &
636896 & 186080 & 51040 &
\multicolumn{2}{c|}{} \\ \hline
0 & 248204432 & 103544272 & 38816224 & 13258208 & 4173280 & 1222192 &
335792 & 87304 \\ \hline $a\to$ & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\
\hline \end{tabular}
\par
\smallskip
{\bf Table 7.4} Characteristic numbers for plane rational nodal quintics.
\end{center}
\par
\bigskip
The numbers with $c=0$ and $a\geq 3$ were calculated by Francesco and
Itzykson \cite[2.97, p.104]{FrancescoItzykson}. \par
We use the same kind of equations to
derive the characteristic numbers $C_d(a,b,c;1)$, $C_d(a,b,c;h)$ and
$C_d(a,b,c;h^2)$.
\par
Equation $1144$:
For any $a$, $b$ and $c$ with $a+b+2c=3d-4$ we have that
\begin{align*}
&C_d(a,b,c;h^2)=\frac{1}{d^2}\bigg\{2dN_d(a,b+1,c+1)-N_d(a+1,b+2,c)\\
&+\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a\\b_1+b_2=b+2\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2)
\left[2d_1^2d_2\binom{b}{b_1-1}-d_1d_2^2\binom{b}{b_1-2}
-d_1^3\binom{b}{b_1}\right]\binom {a}{a_1}\binom c{c_1}\\
&+\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a+1\\b_1+b_2=b+2\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h})
\bigg[2d_1d_2\binom{b}{b_1-1}
-d_1^2\binom{b}{b_1}-d_2^2\binom{b}{b_1-2} \bigg]\binom{a}{a_1-
1}\binom c{c_1}\bigg\} \end{align*}
\par
In the tables of the numbers $C_d(a,b,c;h^2)$ the number of tangent conditions
$b$ is given by $3d-4-a-2c$.
\par
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|} \hline
$c\downarrow$ & \multicolumn{6}{|c|}{$C_3(a,b,c;h^2)$} \\ \hline 2 & 4 &
2 & \multicolumn{4}{c|}{} \\ \cline{1-5} 1 & 14 & 12 & 6 & 2 &
\multicolumn{2}{c|}{} \\ \hline 0 & 32 & 44 & 38 & 20 & 8 & 2 \\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.5} Characteristic numbers for plane rational 1-cuspidal cubics;
cusp at a specified point.
\end{center}
\par
\bigskip
The numbers for cubics were calculated by Schubert \cite{Schubert}, by Kleiman
and Speiser \cite{KleimanSpeiser2}, and by Aluffi \cite{Aluffi2}.
\par
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{9}{|c|}{$C_4(a,b,c;h^2)$} \\ \hline 4 & 18 &
\multicolumn{8}{c|}{} \\ \cline{1-4} 3 & 132 & 72 & 30 &
\multicolumn{6}{c|}{} \\ \cline{1-6} 2 & 816 & 580 & 312 & 132 & 46 &
\multicolumn{4}{c|}{} \\ \cline{1-8} 1 & 4084 & 3760 & 2636 & 1420 & 616 &
224 & 70 & \multicolumn{2}{c|}{} \\ \hline 0 & 17444 & 19912 & 17904 &
12392 & 6700 & 2964 & 1112 & 364 & 102 \\ \hline $a\to$ & 0 & 1 & 2 & 3 & 4
& 5 & 6 & 7 & 8 \\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.6} Characteristic numbers for plane rational 1-cuspidal quartics;
cusp at a specified point.
\end{center}
\par
\bigskip
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|} \hline
$c\downarrow$ & \multicolumn{6}{|c|}{$C_5(a,b,c;h^2)$} \\ \hline 5 &
1080 & 480 & \multicolumn{4}{c|}{} \\ \cline{1-5} 4 & 10728 & 5808 & 2592 &
984 & \multicolumn{2}{c|}{} \\ \hline 3 & 95760 & 61872 & 32988 & 14736 &
5664 & 1920 \\ \hline 2 & 747952 & 575864 & 366256 & 193744 & 86864 &
33832 \\ \hline 1 & 5169728 & 4692096 & 3544608 & 2224088 & 1170192 &
526608 \\ \hline 0 & 33071072 & 34336864 & 30314016 & 22428704 & 13882384
& 7264872 \\ \hline $a\to$ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{tabular}
\par
\smallskip
\begin{tabular}{|r|c|c|c|c|c|c|} \hline
$c\downarrow$ & \multicolumn{6}{|c|}{$C_5(a,b,c;h^2)$} \\
\hline 2 & 11704 & 3656 & \multicolumn{4}{c|}{} \\ \cline{1-5} 1 & 207356 &
72888 & 23208 & 6752 & \multicolumn{2}{c|}{} \\ \hline 0 & 3276944 &
1300816 & 462744 & 149504 & 44272 & 12024 \\ \hline $a\to$ & 6 & 7 & 8 & 9
& 10 & 11 \\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.7} Characteristic numbers for plane rational 1-cuspidal quintics;
cusp at a specified point.
\end{center}
\par
\bigskip
Equation $1134$:
For any $a$, $b$ and $c$ with $a+b+2c=3d-3$ we have that
\begin{align*}
C_d(a,b&,c;h)=-bC_d(a,b-1,c;h^2)
+\frac{1}{d^2}\bigg\{d(2d-2)N_d(a,b,c+1)+(2-d)N_d(a+1,b+1,c)\\
+dN_d(a,b+2,c)&+\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a\\b_1+b_2=b+1\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2)
\bigg[2d_1d_2^2\binom{b}{b_1-1}
-2d_1^2d_2\binom{b}{b_1}\bigg]\binom{a}{a_1}\binom c{c_1}\\
&+\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a+1\\b_1+b_2=b+1\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) \bigg[2d_2\binom{b}{b_1-1}
-2d_1^2d_2\binom{b}{b_1}
\bigg]\binom{a}{a_1-1}\binom c{c_1}
\bigg\}
\end{align*}
\par
In the tables of the numbers $C_d(a,b,c;h)$ the number of tangent conditions
$b$ is given by $3d-3-a-2c$.
\par
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{7}{|c|}{$C_3(a,b,c;h)$} \\ \hline 3 & 6 &
\multicolumn{6}{c|}{} \\ \cline{1-4} 2 & 20 & 18 & 10 &
\multicolumn{4}{c|}{} \\ \cline{1-6} 1 & 42 & 60 & 54 & 30 & 12 &
\multicolumn{2}{c|}{} \\ \hline 0 & 72 & 132 & 186 & 168 & 96 & 42 & 12 \\
\hline $a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6\\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.8} Characteristic numbers for plane rational 1-cuspidal cubics;
cusp on a specified line.
\end{center}
\par
\bigskip
The numbers for cubics were calculated by Schubert \cite{Schubert}, and
Aluffi \cite{Aluffi2}.
\par
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{10}{|c|}{$C_4(a,b,c;h)$} \\ \hline 4 & 210 & 120 &
\multicolumn{8}{c|}{} \\ \cline{1-5} 3 & 1200 & 900 & 510 & 228 &
\multicolumn{6}{c|}{} \\ \cline{1-7} 2 & 5640 & 5404 & 3968 & 2232 & 1006 &
380 & \multicolumn{4}{c|}{} \\ \cline{1-9}
1 & 21844 & 26344 & 24812 & 17956 & 10084 & 4604 & 1774 & 592 &
\multicolumn{2}{c|}{} \\ \hline
0 & 81324 & 111776 & 128992 & 118296 & 84284 & 47284 & 21816 & 8552 & 2926
& 864 \\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.9} Characteristic numbers for plane rational 1-cuspidal quartics;
cusp on a specified line.
\end{center}
\par
\bigskip
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|} \hline
$c\downarrow$ & \multicolumn{6}{|c|}{$C_5(a,b,c;h)$} \\ \hline 6 & 1800
& \multicolumn{5}{c|}{} \\ \cline{1-4} 5 & 17160 & 9600 & 17160 &
\multicolumn{3}{c|}{} \\ \cline{1-6} 4 & 144024 & 96864 & 53520 & 24744 &
9840 & \multicolumn{1}{c|}{} \\ \hline 3 & 1068168 & 851832 & 561660 &
307056 & 141864 & 56808 \\ \hline 2 & 6944128 & 6542664 & 5123864 &
3327352 & 1805040 & 834488 \\ \hline 1 & 41321296 & 44794720 & 41150000 &
31591528 & 20223208 & 10897256 \\ \hline 0 & 239546016 & 286715392 &
299655536 & 267383808 & 200794048 & 126634616 \\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{tabular}
\par
\smallskip
\begin{tabular}{|r|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{7}{|c|}{$C_5(a,b,c;h)$} \\ \hline 3 & 20172 &
\multicolumn{6}{c|}{} \\ \cline{1-4} 2 & 336544 & 120864 & 39272 &
\multicolumn{4}{c|}{} \\ \cline{1-6} 1 & 5040940 & 2045080 & 741368 &
243568 & 73216 &
\multicolumn{2}{c|}{} \\ \hline
0 & 67751352 & 31325416 & 12761768 & 4659408 & 1544416 & 469112 & 130896
\\ \hline
$a\to$ & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.10} Characteristic numbers for plane rational 1-cuspidal quintics;
cusp on a specified line.
\end{center}
\par
\bigskip
Equation $1133$:
For any $a$, $b$ and $c$ with $a+b+2c=3d-2$ we have that
\begin{align*}
C_d(a,b&,c;1)=-bC_d(a,b-1,c;h)-cC_d(a,b,c-1;h^2)-\binom{b}{2}C_d(a,b-
2,c;h^2)\\ &+\frac{1}{d^2}\bigg\{4(d-1)N_d(a+1,b,c)+(3d^2-4d)N_d(a,b+1,c)
\\ &-4\cdot\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}^2)
d_1d_2\binom{a}{a_1}\binom{b}{b_1}\binom{c}{c_1}\\ &-
4\cdot\sum_{\substack{d_1+d_2=d\\
a_1+a_2=a+1\\b_1+b_2=b\\c_1+c_2=c}}
N_{d_1}(a_1,b_1,c_1)E_{d_2}(a_2,b_2,c_2;\check{h}) d_2^2\binom{a}{a_1-
1}\binom{b}{b_1}\binom{c}{c_1} \bigg\}
\end{align*}
\par
In the tables of the numbers $C_d(a,b,c;1)$ the number of tangent conditions
$b$ is given by $3d-2-a-2c$. \begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{8}{|c|}{$C_3(a,b,c;1)$} \\ \hline 3 & 6 & 6 &
\multicolumn{6}{c|}{} \\ \cline{1-5} 2 & 12 & 18 & 18 & 12 &
\multicolumn{4}{c|}{} \\ \cline{1-7} 1 & 18 & 36 & 54 & 54 & 36 & 18 &
\multicolumn{2}{c|}{} \\ \hline 0 & 24 & 60 & 114 & 168 & 168 & 114 & 60 &
24\\ \hline $a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.11} Characteristic numbers for plane rational 1-cuspidal cubics.
\end{center}
\par
\bigskip
The numbers for cubics were calculated by Schubert \cite{Schubert}, by Kleiman
and Speiser \cite{KleimanSpeiser2}, and by Aluffi \cite{Aluffi2}. Note how
the numbers reflect the fact that the dual of a cuspidal cubic is also a
cuspidal
cubic, with duality between point and tangent conditions. \par
Given a plane cuspidal cubic $C$, under the morphism $C\to
\check{C}$ the image of the cusp of $C$ is the point of flex tangency on
$\check{C}$ and vice versa. Therefore we have the following alternative
enumerative significance: $C_3(a,b,c;h^2)$ (listed in Table 7.5) is equal to
the number of cuspidal
cubics with given flex, through $b$ points, tangent to $a$ lines and
incident to
$c$ flags, and $C_3(a,b,c;h)$ (listed in Table 7.8) is equal to the number of
cuspidal cubics with
flex tangent through a given point, through $b$ points, tangent to $a$ lines
and incident to $c$ flags.
\par
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{11}{|c|}{$C_4(a,b,c;1)$} \\ \hline 5 & 90 &
\multicolumn{10}{c|}{} \\ \cline{1-4} 4 & 450 & 360 & 216 &
\multicolumn{8}{c|}{} \\ \cline{1-6} 3 & 2016 & 2016 & 1566 & 936 & 450 &
\multicolumn{6}{c|}{} \\ \cline{1-8} 2 & 7344 & 9228 & 9096 & 6948 & 4134 &
2004 & 828 & \multicolumn{4}{c|}{} \\ \cline{1-10}
1 & 24012 & 35568 & 43452 & 42012 & 31644 & 18792 & 9198 & 3852 & 1422 &
\multicolumn{2}{c|}{} \\ \hline
0 & 75924 & 126720 & 180288 & 212976 & 201132 & 149364 & 88560 & 43668 &
18486 & 6912 & 2304\\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.12} Characteristic numbers for plane rational 1-cuspidal
quartics.
\end{center}
\par
\bigskip
\begin{center}
\par
\begin{tabular}{|r|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{7}{|c|}{$C_5(a,b,c;1)$} \\ \hline 6 & 7560 & 4320 &
\multicolumn{5}{c|}{} \\ \cline{1-5} 5 & 58680 & 41040 & 23400 & 11160 &
\multicolumn{3}{c|}{} \\ \cline{1-7} 4 & 417528 & 343584 & 234432 & 132408
& 63216 & 26208 & \multicolumn{1}{c|}{} \\ \hline
3 & 2600136 & 2522808 & 2038572 & 1367856 & 766296 & 365976 & 152748 \\
\hline
2 & 14519232 & 16336416 & 15523656 & 12319632 & 8150736 & 4535952 &
2167776 \\ \hline
1 & 76377456 & 96562656 & 105583536 & 97988544 & 76305384 & 49814136 &
27552636 \\ \hline
0 & 392798880 & 543805632 & 664607952 & 704860128 & 637644672 & 486730080
& 313485192 \\ \hline
$a\to$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \end{tabular}
\end{center}
\par
\smallskip
\begin{center}
\begin{tabular}{|r|c|c|c|c|c|c|c|} \hline $c\downarrow$ &
\multicolumn{7}{|c|}{$C_5(a,b,c;1)$} \\ \hline 3 & 56952 &
\multicolumn{6}{c|}{} \\ \cline{1-4} 2 & 910056 & 342312 & 117216 &
\multicolumn{4}{c|}{} \\ \cline{1-6} 1 & 13170816 & 5553864 & 2103264 &
725616 & 230616 &
\multicolumn{2}{c|}{} \\ \hline
0 & 172272672 & 82282248 & 34794432 & 13240080 & 4592952 & 1467792 &
435168
\\ \hline
$a\to$ & 7 & 8 & 9 & 10 & 11 & 12 & 13\\ \hline \end{tabular}
\par
\smallskip
{\bf Table 7.13} Characteristic numbers for plane rational 1-cuspidal
quintics.
\end{center}
\par
\bigskip
The four series of characteristic numbers of Theorem \ref{algthm} have been
calculated up to degree 10 and are available upon request to the authors. The
Maple source code for the calculation may also be requested.
\newpage
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1999-07-14T05:59:10 | 9604 | alg-geom/9604021 | en | https://arxiv.org/abs/alg-geom/9604021 | [
"alg-geom",
"math.AG"
] | alg-geom/9604021 | Rahul Pandharipande | R. Pandharipande | The Symmetric Function h^0(\bar{M}_{0,n}, L_1^{x_1} \tensor...\tensor
L_n^{x_n}) | 11 pages, AMSLatex | null | null | null | null | Let \bar{M}_{0,n} be the moduli space of pointed, genus 0 curves. Let L_i
denote the line bundle on \bar{M}_{0,n} associated to the i-th marked point
(the fiber of L_i is the cotangent space of the pointed curve at the i-th
point). Y_n=h^0(\bar{M}_{0,n}, L_1^{x_1} \tensor... \tensor L_n^{x_n}) is a
symmetric function of the variables x_1,... x_n. Let R be the ring of symmetric
functions in infinitely many variables. An explicit linear transformation T:
R-> R is found such that Y_n= T^{n-3} (1).
| [
{
"version": "v1",
"created": "Sun, 28 Apr 1996 22:49:08 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{Introduction}
\subsection{Summary of Results}
Let $n\geq 3$ be an integer. Let $(C,\ p_1,\ldots, p_n)$ be a
reduced, connected, (at worst) nodal
curve of arithmetic genus $0$ with $n$ nonsingular
marked points. $(C, \ p_1, \ldots, p_n)$ is {\em Deligne-Mumford
stable} if $w_C(p_1+ \ldots +p_n)$ is ample (where $w_C$ is
the dualizing sheaf).
Let $\barr{M}_{0,n}$ denote the fine moduli space of Deligne-Mumford stable,
$n$-pointed, genus $0$ curves.
A foundational treatment of $\barr{M}_{0,n}$ can be found in [Kn].
Let $S_n=\{1,2, \ldots, n\}$ be
the marking set.
For each $i\in S_n$, a line bundle ${\cal{L}}_i$
on $\barr{M}_{0,n}$ is obtained by the following prescription: the
fiber of ${\cal{L}}_i$ at the moduli point
$[C, p_1, \ldots, p_n]\in \barr{M}_{0,n}$ is
the cotangent space $T^*_{C,p_i}$.
\begin{pr}
\label{vanish}
Let $x_1, x_2, \ldots, x_n$ be non-negative integers.
$$\forall k>0, \ \ H^k(\barr{M}_{0,n}, {\cal{L}}_1^{x_1}\otimes
{\cal{L}}_2^{x_2} \otimes \cdots \otimes {\cal{L}}_n^{x_n})=0.$$
\end{pr}
For non-negative $x_i$,
let $\gamma_n(x_1,x_2, \ldots, x_n)=h^0(\barr{M}_{0,n}, {\cal{L}}_1^{x_1}\otimes
{\cal{L}}_2^{x_2} \otimes \cdots \otimes {\cal{L}}_n^{x_n})$.
By Proposition (\ref{vanish}) and the Hirzebruch-Riemann-Roch
Theorem, $\gamma_n$ is a polynomial function: $\gamma_n \in {\Bbb Q}[x_1,x_2,
\ldots, x_n]$. The symmetric group $\Sigma_n$ acts
naturally on $\barr{M}_{0,n}$
by permuting the markings. The action of $\Sigma_n$
permutes the isomorphism classes of the line bundles ${\cal{L}}_i$
in the obvious manner. Therefore, $\gamma_n$ is a {\em symmetric}
polynomial in $x_1, x_2, \ldots, x_n$. Since $\barr{M}_{0,n}$ is
$n-3$ dimensional, $\gamma_n$ is of degree (at most) $n-3$.
Let $\sigma_1, \sigma_2, \ldots, \sigma_n$ denote the
elementary symmetric functions in $x_1, x_2, \ldots, x_n$.
\begin{eqnarray*}
\sigma_1 &=& x_1+ \ldots + x_n \\
\sigma_2 &=& x_1x_2 + x_1x_3 + \ldots +x_{n-1} x_n \\
& \vdots & \\
\sigma_n & =& x_1x_2 \cdots x_n. \\
\end{eqnarray*}
The following equivalence is well-known:
$$ {\Bbb Q}[\sigma_1, \ldots, \sigma_n] = \Big( {\Bbb Q}[x_1, \ldots, x_n]
\Big)^{\Sigma_n}.$$ Hence, $\gamma_n\in {\Bbb Q}[\sigma_1, \ldots, \sigma_n]$.
Let $R= {\Bbb Q}[\sigma_1, \sigma_2, \ldots]$ be the
infinite polynomial ring in the variables $\{\sigma_d\}|_{d\in {\Bbb N}^+}$.
Define a grading on $R$ by assigning to the
variable $\sigma_d$ the weight $d$.
For $f\in R$, the {\em degree} of $f$ is the highest weight of
the monomials that appear in $f$.
A ${\Bbb Q}$-linear transformation $T: R\rightarrow R$ is defined
as follows. Let
$f\in R$. Let $e=degree(f)$.
Therefore, $f\in {\Bbb Q}[\sigma_1, \sigma_2, \ldots, \sigma_e]$.
The element $f$ corresponds to a symmetric function in
the variables $x_1, \ldots, x_m$ provided $m\geq e$.
Let $g(x_1, \ldots, x_m)$ be defined
for non-negative integers $x_i$ by
\begin{equation}
\label{Tdef}
g(x_1, \ldots, x_m)= f(x_1, \ldots, x_m)+
\sum _{i=1}^{m} \sum_{j=0}^{x_i-1}
f(x_1, \ldots,x_{i-1}, j, x_{i+1},
\ldots, x_{m}).
\end{equation}
The function $g$ is manifestly a symmetric polynomial
in $x_1, \ldots, x_m$ of degree (at most) $e+1$. Therefore
$g\in {\Bbb Q}[\sigma_1, \sigma_2, \ldots, \sigma_e, \sigma_{e+1}]$.
The element $g$ is independent
of $m$ provided $m\geq e+1$. Define $T(f)=g \in R$ for $m$
in the stable range $m\geq e+1$.
\begin{pr}
\label{solution}
The
function
$\gamma_n \in {\Bbb Q}[\sigma_1, \ldots, \sigma_n]$ is determined by:
$$\gamma_n= T^{n-3}(1).$$
\end{pr}
\noindent
The author has benefited from conversations with W. Fulton,
M. Kapranov, and M. Thaddeus.
\subsection{Computing $T$}
Computing $T$ is tedious but straightforward.
If $f=1$, then $degree(f)=0$. $T(1)$ can be computed
in the variable $x_1$. By equation (\ref{Tdef}),
$$g(x_1)= f(x_1)+ \sum_{j=0}^{x_1-1} f(j) = 1+x_1 = 1+\sigma_1.$$
Hence $T(1)=1+\sigma_1 $.
Let $n=4$. $\overline{M}_{0,4} \stackrel{\sim}{=} \Bbb {P}^1$. Via this
isomorphism,
${\cal{L}}_i \stackrel{\sim}{=} {\cal{O}}_{\Bbb {P}^1}(1)$ for $1\leq i \leq 4$. The
function $\gamma_4$ is easily evaluated:
\begin{eqnarray*}
\gamma_4(x_1,x_2,x_3,x_4) & = & h^0(\Bbb {P}^1, {\cal{O}}_{\Bbb {P}^1}(x_1+x_2+x_3+x_4)) \\
& = &
1+x_1+x_2+x_3+x_4 \\ & = & 1+\sigma_1 \\ & = & T(1).
\end{eqnarray*}
Next, let $f=\sigma_1$.
$T(\sigma_1)$ can be computed in the variables $x_1, x_2$.
\begin{eqnarray*}
g(x_1,x_2)& =& f(x_1,x_2)+ \sum_{j=0}^{x_1-1} f(j,x_2)
+ \sum_{j=0}^{x_2-1} f(x_1,j) \\
& = & x_1+x_2 +{1\over 2} (x_1^2-x_1) + x_1x_2 +
{1\over 2} (x_2^2-x_2) + x_1x_2 \\
& = & {1\over 2}\sigma_1 +{1\over 2} \sigma_1^2 +\sigma_2 \\
\end{eqnarray*}
Therefore, $T(\sigma_1)={1\over 2}\sigma_1 +{1\over 2} \sigma_1^2 +\sigma_2$.
By Proposition (2),
$$\gamma_5=T^2(1)=T(\sigma_1+1)=T(\sigma_1)+T(1)=
1+{3\over 2}\sigma_1+ {1\over 2}\sigma_1^2+\sigma_2.$$
Proposition (2) leads to an easy calculation of $h^0(\overline{M}_{0,n}
, {\cal{L}}_1^{x_1})$. By relation (\ref{Tdef}), for $n\geq 4$,
$$h^0(\overline{M}_{0,n}
, {\cal{L}}_1^{x_1}) = \sum_{j=0}^{x_1} h^0(\overline{M}_{0,{n-1}},
{\cal{L}}_1^{x_1}).$$
It is easily seen $h^0(\overline{M}_{0,n}, {\cal{L}}_1^{x_1})={n-3+x_1\choose
x_1}$ uniquely satisfies the above recursion and the boundary
conditions at $n=3$. The global sections of ${\cal{L}}_1^{x_1}$
can also be computed by examining the
morphism $\overline{M}_{0,n}\rightarrow \Bbb {P}^{n-3}$
determined by $H^0(\overline{M}_{0,n}, {\cal{L}}_1)$ (see
section (\ref{express})).
The function $\gamma_n$ is tabulated below for small values of $n$:
\begin{eqnarray*}
\gamma_3 &= & 1\\
\gamma_4 &=& 1+\sigma_1\\
& & \\
\gamma_5 &=& 1+{3\over 2}\sigma_1+ {1\over 2}\sigma_1^2+\sigma_2\\
& & \\
\gamma_6 &=& 1+{11\over 6}\sigma_1+\sigma_1^2+\sigma_2+{1\over 6}\sigma_1^3+\sigma_1\sigma_2+2\sigma_3\\
& & \\
& & \\
\gamma_7 &=& 1+{25\over 12}\sigma_1+{35\over 24}\sigma_1^2 + {5\over 4} \sigma_2
+{5\over 12}\sigma_1^3+ {5\over 4} \sigma_1\sigma_2 -{1\over 4}\sigma_3 \\
& &\ \ \ +{1\over 24}\sigma_1^4+{1\over 2}\sigma_1^2\sigma_2 +{5\over 2}\sigma_1\sigma_3
+{1\over 4}\sigma_2^2 +{11\over 2}\sigma_4 \\
& & \\
& & \\
\gamma_8 &=& 1+{137\over 60}\sigma_1+ {15\over 8}\sigma_1^2+ {5\over 4}\sigma_2
+{17\over 24}\sigma_1^3+{11\over 6}\sigma_1\sigma_2 +{11\over 4}\sigma_3 \\
& & \ \ \ +{1\over 8}\sigma_1^4+{3\over 4}\sigma_1^2\sigma_2 -{3\over 4}\sigma_1\sigma_3
+{1\over 4}\sigma_2^2 -{21\over 2}\sigma_4 \\
& & \ \ \ +{1\over 120}\sigma_1^5 +{1\over 6}\sigma_1^3\sigma_2 +{3\over2}\sigma_1^2\sigma_3
+{1\over 4}\sigma_1\sigma_2^2 +{17\over 2}\sigma_1\sigma_4+\sigma_2\sigma_3 +19 \sigma_5 \\
\end{eqnarray*}
\section{The Line bundles ${\cal{L}}_i$}
\subsection{Expressions for ${\cal{L}}_i$}
\label{express}
Let $\rho:U_n\rightarrow \barr{M}_{0,n}$ be the universal family of
pointed, stable curves. The are $n$ sections
$s_1, \ldots, s_n$ of $\rho$ corresponding to the markings.
Consider the marking $i$. The projection $\rho$
takes $s_i$ isomorphically to $\barr{M}_{0,n}$.
Let $$x=[C, p_1, \ldots, p_n]\in \barr{M}_{0,n}.$$
Let $y\in s_i$ satisfy $\rho(y)=x$.
The normal bundle to $s_i$ in $U_n$ at $y$ is canonically the
the tangent space $T_{C,p_i}$. Therefore
\begin{equation}
\label{bay}
c_1({\cal{L}}_i)\stackrel{\sim}{=} \rho_*(-s_i^2).
\end{equation}
Note $U_n$ is canonically isomorphic to $\overline{M}_{0,n+1}$.
${\cal{L}}_{n+1}$ on $U_n$ can be expressed as follows:
\begin{equation}
\label{om}
{\cal{L}}_{n+1}\stackrel{\sim}{=} \omega_{\rho}(s_1+\ldots + s_n)
\end{equation}
where $\omega_{\rho}$ is the relative dualizing
sheaf of the morphism $\rho: U_n\stackrel{\sim}{=} \overline{M}_{0,n+1}
\rightarrow \barr{M}_{0,n}$.
The proof of (\ref{om}) requires a square diagram:
\begin{equation*}
\begin{CD}
\overline{M}_{0,n+1\#} @>{\pi_{n+1}}>> \overline{M}_{0,n+1}\\
@V{\nu}VV @V{\rho}VV \\
\overline{M}_{0,n\#} @>{\pi_{n}}>> \barr{M}_{0,n}
\end{CD}
\end{equation*}
$\barr{M}_{0,n+1\#}$ is the moduli
space of $n+2$-pointed genus 0 stable curves with marking
set $\{1, \ldots, n+1, \#\}$.
The morphisms $\nu$, $\rho$, $\pi_n$, and $\pi_{n+1}$ are
all contraction
maps.
Let $D_{n+1, \#}$
be the
boundary divisor of $\barr{M}_{0,n+1\#}$
corresponding to the partition $$\{n+1,\#\} \cup \{1, \ldots, n\}.$$
$D_{n+1, \#}$ is the section of ${\pi_{n+1}}$ corresponding to
the marking $n+1$.
A simple examination of the blow-ups involved in the
above square yields:
$$\pi_{n+1}^*(\omega_{\rho}(s_1+\ldots + s_n))|_{D_{n+1,\#}} \stackrel{\sim}{=}
\omega_{\nu}|_{D_{n+1, \#}}.$$
Since $\omega_{\nu}|_{D_{n+1, \#}} \stackrel{\sim}{=} -D_{n+1, \#}|_{D_{n+1, \#}}$,
(\ref{om}) is established by (\ref{bay}).
Another method of viewing ${\cal{L}}$ is as follows.
Fix $n$ marked points in general linear position in $\Bbb {P}^{n-2}$.
M. Kapranov has shown the closure in the Hilbert scheme
of the locus of rational normal curves passing through
the $n$ marked points is canonically $\barr{M}_{0,n}$ ([K]).
The universal curve over the Hilbert scheme restricts
to $U_n$ over $\barr{M}_{0,n}$. Since the universal
curve over the Hilbert scheme naturally maps
to $\Bbb {P}^{n-2}$, a morphism $$\mu:U_n \rightarrow \Bbb {P}^{n-2}$$ is
obtained.
$U_n$ is canonically isomorphic to $\overline{M}_{0,n+1}$.
There is an isomorphism:
\begin{equation}
\label{pullb}
{\cal{L}}_{n+1} \stackrel{\sim}{=} \mu^*({\cal{O}}_{\Bbb {P}^{n-2}}(1)).
\end{equation}
On each fiber of $\rho: U_n \rightarrow \barr{M}_{0,n}$,
$\mu^*({\cal{O}}_{\Bbb {P}^{n-2}}(1))$ is isomorphic to
$\omega_{\rho}(s_1+\ldots + s_n)$ (see [K]). Since
both line bundles
$\mu^*({\cal{O}}_{\Bbb {P}^{n-2}}(1))$ and $\omega_{\rho}(s_1+\ldots + s_n)$
are trivial on the sections $s_i$ of $\rho$,
$$\mu^*({\cal{O}}_{\Bbb {P}^{n-2}}(1)) \stackrel{\sim}{=} \omega_{\rho}(s_1+\ldots + s_n)$$
on $U_n$. Now (\ref{pullb}) follows from (\ref{om}).
Finally, it is useful to express ${\cal{L}}_{n+1}$ on $\barr{M}_{0,n+1}$
as a linear combination of boundary divisors.
Let $j,k\in S_{n}$ be distinct markings.
\begin{equation}
\label{lincom}
{\cal{L}}_{n+1} \stackrel{\sim}{=} \sum_{n+1 \in A\subset S_{n+1},\ j,k\notin A} D_A
\end{equation}
where the set $A$ is a subset of the marking set $S_{n+1}$
satisfying $|A|\geq 2$.
As above, $D_A$ is the boundary divisor
corresponding to the partition $S_{n+1}=A\cup A^c$.
Recall the map $$\mu:\barr{M}_{0,n+1} \stackrel{\sim}{=} U_n \rightarrow \Bbb {P}^{n-2}.$$
Consider the unique hyperplane $H\subset \Bbb {P}^{n-2}$
passing through
the $n-2$ marked points $S_n \setminus \{j,k\}$.
The divisor $\mu^*(H)$ is easily seen to be the
right side of (\ref{lincom}). By (\ref{pullb}),
the isomorphism of (\ref{lincom}) is established.
See [W] for another proof of (\ref{lincom}).
\label{lineb}
\subsection{Contraction}
It will be helpful in the sequel to denote
${\cal{L}}_i$ on $\barr{M}_{0,n}$ by ${\cal{L}}_{i,n}$.
Let $n\geq 3$.
Consider the contraction morphism
$$\rho: \barr{M}_{0, n+1} \rightarrow \barr{M}_{0,n}$$
obtained by omitting the marking $n+1$.
\begin{lm}
\label{comp}
$\rho^*({\cal{L}}_{i,n}) + D_{i,n+1} \stackrel{\sim}{=} {\cal{L}}_{i, n+1}. $
\end{lm}
\begin{pf}
Let $j,k\in S_n$ satisfy $j,k\neq i$.
By the results of section (\ref{lineb})
$$\rho^*({\cal{L}}_{i,n}) \stackrel{\sim}{=} \sum_{i \in A\subset S_n,\ j,k\notin A}
\rho^*(D_A).$$
A comparison with
$${\cal{L}}_{i,n+1} \stackrel{\sim}{=} \sum_{i \in A\subset S_{n+1},\ j,k\notin A} D_A$$
yields Lemma (\ref{comp}).
\end{pf}
\section{The Proof of Proposition (\ref{vanish})}
\subsection{The Induction Ladder}
Consider the following (infinite) commutative diagram:
\begin{equation}
\label{ladder}
\begin{CD}
\barr{M}_{0,n+1\#} @>{\nu_n}>>
\overline{M}_{0,n\#} @>{\nu_{n-1}}>> \overline{M}_{0,n-1 \#}\\
@V{\pi_{n+1}}VV @V{\pi_{n}}VV @V{\pi_{n-1}}VV \\
\barr{M}_{0,n+1} @>{\rho_{n}}>>
\overline{M}_{0,n} @>{\rho_{n-1}}>> \barr{M}_{0,n-1}
\end{CD}
\end{equation}
The maps are all contraction morphisms:
$\pi_n$ contracts $\#$, $\nu_{n-1}$ contracts $n$,
$\rho_{n-1}$ contracts $n$.
The diagram starts with $\barr{M}_{0,3}$ on the
lower right corner and extends left.
Denote the composition $\rho_n \circ \cdots \circ \rho_{m-1}$
by $\rho_{m,n}$ (similarly for $\nu$).
Recall ${\cal{L}}_{i,n}$ is the line bundle
corresponding to the $i^{th}$ marking on $\barr{M}_{0,n}$.
For $m>n$, the pull-back line bundle on $\barr{M}_{0,m}$,
$\rho_{m,n}^*({\cal{L}}_{i,n})$,
is also denoted by the same symbol ${\cal{L}}_{i,n}$.
Let ${\cal{L}}_{\#,n\#}$ be the line bundle corresponding
to the marking $\#$ on $\barr{M}_{0,n\#}$. Again
${\cal{L}}_{\#,n\#}$ will also denote the pull-back line
bundle, $\nu_{m,n}^*({\cal{L}}_{\#,n\#})$ on $\barr{M}_{0,m\#}$ for $m>n$.
Proposition (1) is established by an induction on the
ladder (\ref{ladder}). The following Lemmas are
needed in the induction.
\begin{lm}
\label{d} Let $n\geq 3$. The line bundle ${\cal{L}}_{\#,n\#}$ behaves well
under ladder base changes:
\begin{enumerate}
\item[(i.)]
For all $k\geq 0$ and $m\geq n$, $\ \pi_{m*}({\cal{L}}^k_{\#,n\#})\stackrel{\sim}{=}
\rho_{m,n}^*\pi_{n*}({\cal{L}}^k_{\#,n\#}).$
\item[(ii.)]
For all $k\geq 0$ and $m\geq n$,
$\ R^1\pi_{m*} ({\cal{L}}^k_{\#,n\#})=0.$
\end{enumerate}
\end{lm}
\begin{pf}
By (\ref{pullb}), ${\cal{L}}_{\#,n\#}$ is generated by global
sections on $\barr{M}_{0,n\#}$. Therefore ${\cal{L}}^k_{\#,n\#}$ is
generated globally on
$\barr{M}_{0,m\#}$. Any line bundle generated by global
sections on a genus 0, stable, pointed curve has
no higher cohomology. Hence, ${\cal{L}}^k_{\#,n\#}$ has no
higher cohomology on the fibers of $\pi_m$.
By the Cohomology and Base Change Theorems,
$$R^1\pi_{m*} ({\cal{L}}^k_{\#,n\#})=0.$$
Consider the fiber product:
$$\tau_m: \barr{M}_{0,m}\times_ {\barr{M}_{0,n}} \barr{M}_{0,n\#} \rightarrow \barr{M}_{0,m}.$$
Again by Base Change, $\tau_{m*}({\cal{L}}^k_{\#,n\#}) \stackrel{\sim}{=} \rho_{m,n}^*
\pi_{n*}({\cal{L}}^k_{\#,n\#})$. There is
natural map
\begin{equation}
\label{nat}
\barr{M}_{0,m\#} \rightarrow \barr{M}_{0,m}\times_ {\barr{M}_{0,n}} \barr{M}_{0,n\#}
\end{equation}
which commutes with $\pi_m$ and $\tau_m$.
It is easy to check that the natural map of
vector bundles $\tau_{m*}({\cal{L}}^k_{\#,n\#})
\rightarrow \pi_{m*}({\cal{L}}^k_{\#,n\#})$
induced by (\ref{nat}) is an isomorphism on fibers.
\end{pf}
\begin{lm}
\label{dd}
Let $n\geq 4$.
The following exact sequences of
vector bundles exist on $\barr{M}_{0,n}$ for
each pair of integers $a,b \geq 0$:
\begin{equation}
\label{w}
0 \rightarrow \pi_{n*}({\cal{L}}^{a+1}_{\#,n-1\#}\otimes {\cal{L}}^b_{\#,n\#})
\rightarrow \pi_{n*}({\cal{L}}^{a}_{\#,n-1\#}\otimes {\cal{L}}^{b+1}_{\#,n\#})
\rightarrow {\cal{L}}^a_{n,n} \rightarrow 0.
\end{equation}
\end{lm}
\begin{pf}
By Lemma (\ref{comp}),
there is a linear equivalence ${\cal{L}}_{\#,n-1\#} +
D_{\#,n} \stackrel{\sim}{=} {\cal{L}}_{\#, n\#}$.
Note: $${\cal{L}}_{\#, n\#}|_{D_{\#,n}}=0,$$
\begin{equation}
\label{jonny}
{\cal{L}}_{\#,n-1\#}|_{D_{\#,n}} \stackrel{\sim}{=} -D_{\#,n}|_{D_{\#,n}} .
\end{equation}
Tensoring the
sequence (\ref{ww}) with ${\cal{L}}^a_{\#, n-1\#}\otimes
{\cal{L}}^{b+1}_{\#,n\#}$
yields (\ref{www}).
\begin{equation}
\label{ww}
0 \rightarrow {\cal{O}}(-D_{\#,n}) \rightarrow {\cal{O}} \rightarrow {\cal{O}}_{D_{\#,n}} \rightarrow 0
\end{equation}
\begin{equation}
\label{www}
0 \rightarrow {\cal{L}}^{a+1}_{\#,n-1\#}\otimes {\cal{L}}^b_{\#,n\#}
\rightarrow {\cal{L}}^{a}_{\#,n-1\#}\otimes {\cal{L}}^{b+1}_{\#,n\#}
\rightarrow {\cal{L}}^a_{\#,n-1\#}|_{D_{\#,n}} \rightarrow 0.
\end{equation}
Equivalence (\ref{jonny}) implies
$\pi_{n*}({\cal{L}}^a_{\#,n-1\#}|_{D_{\#,n}}) \stackrel{\sim}{=} {\cal{L}}^a_{n,n}$.
By a global sections argument as in Lemma (\ref{d}),
the terms of sequence (\ref{www}) have vanishing
higher direct images under $\pi_n$.
Sequence
(\ref{w}) is obtained by pushing-forward sequence (\ref{www}).
\end{pf}
\begin{lm}
\label{ddd} Let $n\geq 3$.
Let ${\cal{N}}$ be a line bundle on $\barr{M}_{0,n}$ satisfying the
following condition
for all non-negative integers $z_3,\ldots, z_n$
and all $k>0$ :
\begin{equation}
\label{f}
H^k(\barr{M}_{0,n}, {\cal{N}}\otimes \bigotimes_{i=3}^n {\cal{L}}^{z_i}_{i,i})=0.
\end{equation}
Then for all $b\geq 0$ and $k>0$,
\begin{equation}
\label{ff}
H^k(\barr{M}_{0,n}, \pi_{n*}({\cal{L}}^b_{\#,n\#})\otimes {\cal{N}})=0.
\end{equation}
\end{lm}
\begin{pf}
If $b=0$, the vanishing (\ref{ff}) is a consequence
of (\ref{f}) since $\pi_{n*}({\cal{O}})\stackrel{\sim}{=} {\cal{O}}$. Assume $b>0$.
The proof is a simple consequence of Lemmas (\ref{d}) and (\ref{dd}).
Consider
the sequences for $0\leq j \leq b-1$ obtained from (\ref{w}) by tensoring
with ${\cal{N}}$:
$$0 \rightarrow \pi_{n*}({\cal{L}}^{j+1}_{\#,n-1\#}\otimes
{\cal{L}}^{b-j-1}_{\#,n\#})\otimes {\cal{N}}
\rightarrow \pi_{n*}({\cal{L}}^{j}_{\#,n-1\#}\otimes
{\cal{L}}^{b-j}_{\#,n\#})\otimes {\cal{N}}
\rightarrow {\cal{L}}^j_{n,n} \otimes {\cal{N}} \rightarrow 0.$$
The vanishing (\ref{ff}) is reduced by these sequences and
repeated application of (\ref{f}) to:
$$\forall b\geq 0, \ \forall k>0, \ \ \
H^k(\barr{M}_{0,n}, \pi_{n*}({\cal{L}}^b_{\#,n-1\#})\otimes {\cal{N}})=0.$$
But since, $\pi_{n*}({\cal{L}}^b_{\#,n-1\#})\stackrel{\sim}{=} \rho_{n-1}^*
\pi_{n-1*}({\cal{L}}^b_{\#,n-1\#})$ by Lemma (\ref{d}),
it suffices to show:
$$\forall b\geq 0, \ \forall k>0, \ \ \
H^k(\barr{M}_{0,n}, \rho^*_{n-1}\pi_{n-1*}({\cal{L}}^b_{\#,n-1\#})\otimes {\cal{N}})=0.$$
Pulling-back the sequences (\ref{w}) for $n-1$ to $\barr{M}_{0,n}$ via $\rho_{n-1}$,
tensoring with ${\cal{N}}$, and repeatedly applying (\ref{f}),
the vanishing (\ref{ff}) is further reduced to:
$$\forall b\geq 0, \ \forall k>0, \ \ \
H^k(\barr{M}_{0,n}, \rho^*_{n-1}\pi_{n-1*}({\cal{L}}^b_{\#,n-2\#})\otimes {\cal{N}})=0.$$
Applying Lemma (\ref{d}), it suffices to show
$$\forall b\geq 0, \ \forall k>0, \ \ \
H^k(\barr{M}_{0,n}, \rho^*_{n,n-2}\pi_{n-2*}({\cal{L}}^b_{\#,n-2\#})\otimes {\cal{N}})=0.$$
This process finally reduces the original claim (\ref{ff})
to:
$$\forall b\geq 0, \ \forall k>0, \ \ \
H^k(\barr{M}_{0,n}, \rho^*_{n,3}\pi_{3*}({\cal{L}}^b_{\#,3\#})\otimes {\cal{N}})=0.$$
Since $\pi_{3*}({\cal{L}}^b_{\#,3\#})$ is a trivial bundle over
the point $\barr{M}_{0,3}$, the
Lemma is proven.
\end{pf}
\subsection{The Induction}
A slightly stronger version of Proposition (\ref{vanish}) is
needed for the induction:
\noindent {$\bold {Proposition\ 1'}.$}
Let $n\geq 3$.
Let $x_{i,j}$ (for $1\leq i \leq j$, $3\leq j \leq n$ )
be non-negative integers.
$$\forall k>0, \ \ H^k(\barr{M}_{0,n},\ \bigotimes_{i,j} {\cal{L}}_{i,j}^{x_{i,j}})=0.$$
\begin{pf}
The Proposition certainly holds for $n=3$ since $dim(\barr{M}_{0,3})=0$.
Assume the Proposition holds for $n$.
Let $x_{i,j}$ (for $1\leq i \leq j$, $3\leq j \leq n$) be non-negative
integers. Let $y_1, \ldots, y_n, y_{\#}$ also be non-negative
integers. Let ${\cal{L}}_{1,n\#}, \ldots, {\cal{L}}_{n,n\#}, {\cal{L}}_{\#, n\#}$
denote the line bundles on $\barr{M}_{0,n\#}$ corresponding to the markings
$1,\ldots,n, \#$ .
Consider the following line bundle on $\barr{M}_{0,n\#}$:
$${\cal{L}}(y_1,\ldots, y_n, y_{\#})\otimes {\cal{N}} \stackrel{\sim}{=}
{\cal{L}}^{y_{\#}}_{\#,n\#} \otimes
\bigotimes_{t=1}^n {\cal{L}}^{y_t}_{t,n\#} \otimes
\bigotimes_{i,j} {\cal{L}}_{i,j}^{x_{i,j}}.$$
where ${\cal{N}}=\bigotimes_{i,j} {\cal{L}}_{i,j}^{x_{i,j}}$ and
$${\cal{L}}(y_1,\ldots, y_n, y_{\#})=
{\cal{L}}^{y_{\#}}_{\#,n\#} \otimes
\bigotimes_{t=1}^n {\cal{L}}^{y_t}_{t,n\#}.$$
It suffices to prove for all $k>0$,
$H^k(\barr{M}_{0,n\#}, {\cal{L}}(y_1, \ldots, y_n, y_{\#}) \otimes {\cal{N}})=0.$
As before, let $\pi_n$ denote the contraction
map $\pi_n: \barr{M}_{0,n\#} \rightarrow \barr{M}_{0,n}$.
For each $1\leq t \leq n$,
$$ {\cal{L}}_{t,n} + D_{t,\#} \stackrel{\sim}{=} {\cal{L}}_{t,n\#}.$$
The following exact sequences on $\barr{M}_{0,n\#}$
therefore exist for
each $1\leq t \leq n$ and non-negative integers $a,b$:
\begin{equation}
\label{redd}
0 \rightarrow {\cal{L}}^{a+1}_{t,n}\otimes {\cal{L}}^b_{t,n\#}
\rightarrow {\cal{L}}^{a}_{t,n}\otimes {\cal{L}}^{b+1}_{t,n\#}
\rightarrow {\cal{L}}^a_{t,n}|_{D_{t,\#}} \rightarrow 0.
\end{equation}
Via the natural identification $\pi_n:D_{t,\#} \stackrel{\sim}{=}
\barr{M}_{0,n}$, the following isomorphisms hold:
$${\cal{L}}_{\#, n\#} |_{D_{t,\#}} \stackrel{\sim}{=} {\cal{O}},$$
\begin{equation}
\label{rest}
{\cal{L}}_{t, n\#}|_{D_{t,\#}} \stackrel{\sim}{=} {\cal{O}},
\end{equation}
$$\forall s\neq t, \ {\cal{L}}_{s,n\#} |_{D_{t,\#}} \stackrel{\sim}{=} {\cal{L}}_{s,n},$$
$$\forall\ 1\leq i \leq j, \ 3\leq j \leq n, \ \
{\cal{L}}_{i,j} |_{D_{t,\#}} \stackrel{\sim}{=} {\cal{L}}_{i,j}.$$
By
repeated use of (\ref{redd}) along with the
inductive assumption of cohomology
vanishing on $\barr{M}_{0,n} \stackrel{\sim}{=} D_{t,\#}$,
it suffices to prove
$$\forall k>0, \ \ H^k(\barr{M}_{0,n\#}, {\cal{L}}_{\#,n\#}^{y_{\#}}\otimes {\cal{N}}')=0,$$
$${\cal{N}}' \stackrel{\sim}{=} \bigotimes_{i=1}^n {\cal{L}}^{y_i}_{i,n} \otimes {\cal{N}}.$$
In effect, the sequences (\ref{redd}) are used to
convert factors of ${\cal{L}}_{i,n\#}$ to ${\cal{L}}_{i,n}$
one at a time. The remaining factors of ${\cal{L}}_{\#,n\#}$
are handled by the following method.
Since $R^1\pi_{n*}( {\cal{L}}_{\#,n\#}^{y_{\#}}\otimes {\cal{N}}')=0$, for all
$k>0$,
$$ H^k(\barr{M}_{0,n\#}, {\cal{L}}_{\#,n\#}^{y_{\#}}\otimes {\cal{N}}')\stackrel{\sim}{=}
H^k(\barr{M}_{0,n},\pi_{n*}( {\cal{L}}_{\#,n\#}^{y_{\#}}\otimes {\cal{N}}')).$$
Since $\pi_{n*}( {\cal{L}}_{\#,n\#}^{y_{\#}}\otimes {\cal{N}}')\stackrel{\sim}{=}
\pi_{n*}( {\cal{L}}_{\#,n\#}^{y_{\#}})\otimes {\cal{N}}'$
and ${\cal{N}}'$ satisfies the vanishing (\ref{f}) by the
inductive assumption on $\barr{M}_{0,n}$, the proof of the
Proposition is completed by Lemma (\ref{ddd}).
\end{pf}
\section{The Proof of Proposition (2)}
The proof of Proposition ($1'$) gives a recursive (in $n$) method
to calculate $$h^0(\barr{M}_{0,n},\ \bigotimes_{i,j} {\cal{L}}_{i,j}^{x_{i,j}})$$
for non-negative $x_{i,j}$.
There are 3 reasons why this full recursion is not pursued here:
\begin{enumerate}
\item[(i.)]
The full recursion is complicated.
\item[(ii.)]
More data is required in the recursion
than is needed in Proposition (2).
\item[(iii.)] The full recursion
does not respect the symmetry of the variables in
Proposition (2).
\end{enumerate}
Fortunately there is simple way to sidestep these
problems.
Recall $\gamma_n(x_1,x_2, \ldots, x_n)=h^0(\barr{M}_{0,n}, {\cal{L}}_{1,n}^{x_1}\otimes
{\cal{L}}_{2,n}^{x_2} \otimes \cdots \otimes {\cal{L}}_{n,n}^{x_n})$
is a symmetric function of degree (at most) $n-3$ in
$x_1, \ldots, x_n$. The map
from symmetric functions in $x_1, \ldots, x_n$ to
to symmetric functions in $x_1, \ldots, x_{n-1}$ given by
setting $x_n=0$ is
bijective for symmetric functions of degree at most $n-1$.
Both spaces are spanned by monomials in
$\sigma_1, \ldots, \sigma_{n-1}$ of degree at most $n-1$.
Hence $\gamma_n$ is completely determined by
$\gamma_n(x_1, \ldots, x_{n-1}, 0)$.
Suppose the symmetric function $\gamma_n$ is known.
To determine $\gamma_{n+1}$, it suffices to
know $\gamma_{n+1}(x_1, \ldots, x_{n},0)$. The vanishing
of the last entry leads to a great simplification in the
recursion of Proposition ($1'$).
Let
$$\gamma_{n+1}(x_1, \ldots, x_n, 0)= h^0(\barr{M}_{0,n\#},
\bigotimes_{t=1}^{n}{\cal{L}}_{t,n\#}^{x_t}).$$
From the sequences (for $1\leq t \leq n$)
\begin{equation}
\label{seqq}
0 \rightarrow {\cal{L}}^{a+1}_{t,n}\otimes {\cal{L}}^b_{t,n\#}
\rightarrow {\cal{L}}^{a}_{t,n}\otimes {\cal{L}}^{b+1}_{t,n\#}
\rightarrow {\cal{L}}^a_{t,n}|_{D_{t,\#}} \rightarrow 0
\end{equation}
and the restriction equations (\ref{rest}),
the following relation is easily deduced:
$$\gamma_{n+1}(x_1, \ldots, x_n, 0)=
\gamma_n(x_1, \ldots, x_n) +
\sum _{i=1}^{n} \sum_{j=0}^{x_i-1}
\gamma_n(x_1, \ldots,x_{i-1}, j, x_{i+1},
\ldots, x_{n}).$$
The
sequences (\ref{seqq}), as before, are used
to convert factors of ${\cal{L}}_{t,n\#}$ to ${\cal{L}}_{t,n}$.
By Proposition ($1'$), all higher cohomology
vanishes.
It is the omission of the factor ${\cal{L}}_{\#,n\#}$ that
simplifies the recursion.
This concludes the proof of Proposition (2).
|
1996-04-04T12:21:40 | 9604 | alg-geom/9604005 | en | https://arxiv.org/abs/alg-geom/9604005 | [
"alg-geom",
"math.AG"
] | alg-geom/9604005 | Carlos Simpson | Carlos Simpson | The Hodge filtration on nonabelian cohomology | LaTeX | null | null | null | null | This is partly a survey article on nonabelian Hodge theory, but we also give
proofs of results that have only been announced elsewhere. In the introduction
we discuss a wide range of recent work on this subject and give some
references. In the body of the paper, we discuss Corlette's nonabelian Hodge
theorem, Hitchin's quaternionic structure on the moduli space of
representations of $\pi _1(X)$ (for a compact K\"ahler manifold $X$), and
Deligne's construction of the resulting twistor space. We then mention the
interpretation of Deligne's space of $\lambda$-connections as the Hodge
filtration of the nonabelian de Rham cohomology $M_{DR}(X,G)= H^1(X,G)$. We go
on to prove several properties of this Hodge filtration, such as Griffiths
transversality and regularity of the Gauss-Manin connection; a local triviality
property coming from Goldman and Millson's analysis of the singularities; and a
weight property coming from Langton's theory. We construct a compactification
of $M_{DR}$ using the Hodge filtration with its weight property. At the end, we
define a nonabelian version of the Noether-Lefschetz locus, and prove that it
is algebraic if the base of the fibration is compact. We pose the open problem
of studying degenerations of nonabelian Hodge structure sufficiently well to be
able to prove algebraicity of the Noether-Lefshetz locus even when the base is
quasiprojective.
| [
{
"version": "v1",
"created": "Thu, 4 Apr 1996 10:22:17 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Simpson",
"Carlos",
""
]
] | alg-geom | \section*{The Hodge filtration on nonabelian cohomology}
Carlos Simpson\newline
Laboratore Emile Picard\newline
UMR 5580, CNRS
Universit\'e Paul Sabatier\newline
31062 Toulouse CEDEX, France
\numero{Introduction}
Whereas usual Hodge theory
concerns mainly the usual or abelian cohomology of an algebraic variety---or
eventually the rational homotopy theory or nilpotent completion of $\pi _1$
which are in some sense obtained by extensions---nonabelian Hodge theory
concerns the cohomology of a variety with nonabelian coefficients. Because of
the basic fact that homotopy groups in higher dimensions are abelian, and since
cohomology theories can generally be interpreted as spaces of maps into
classifying (or Eilenberg-MacLane) spaces, nonabelian cohomology occurs
essentially only in degree $1$. There are certainly some degree $2$ aspects
which are as of yet totally untouched; and the same goes for the degree $1$
case with twisted coefficient systems. (See however \cite{kobe} for a
direction of development combining the nonabelian coefficients in degree $1$
with abelian coefficients in higher degrees). If we leave these aside, we are
left with the case of $H^1(X,G)$ for $G$ a nonabelian group. It is most
natural to interpret this cohomology as a groupoid, or, when $G$ is a
group-scheme, to interpret $H^1(X,G)$ as a stack. It is the stack of flat
principal $G$-bundles on $X$. Recall from the usual abelian
case that in order to obtain a Hodge structure, we must consider cohomology
with complex coefficients. The analogue in the nonabelian case is that we must
take as coefficient group a group-scheme $G$ over the complex numbers (and in
fact it should be affine too). This then is the domain of application of the
work that has been done in nonabelian Hodge theory: the study of
properties and additional structure on the moduli
stack ${\cal M} (X,G):= H^1(X, G)$ which are the analogues in an appropriate sense
of the main structures or properties of abelian cohomology.
By its nature, the first nonabelian cohomology is an invariant of the
fundamental group $\pi _1(X)$. The study of nonabelian Hodge theory may thus
be thought of as the study of fundamental groups of algebraic varieties or
compact K\"ahler manifolds. It is important to note, specially in light of
Toledo's examples of $\pi _1(X)$ not residually finite, that the study of $\pi
_1(X)$ via its nonabelian cohomology, i.e. via the spaces of homomorphisms
$\pi _1(X)\rightarrow G$, will only ``see'' a certain part of $\pi _1(X)$ and
in particular will not at all see the intersection of subgroups of finite
index. It is an interesting question to try to understand what Hodge-theoretic
methods could say about this more mysterious part of the fundamental group.
We start in \S 2 by reviewing Corlette's nonabelian Hodge theorem
\cite{Corlette} (cf also \cite{Donaldson} and \cite{DietrichOsawa}) which is
actually a generalization of the theorem of Eells and Sampson \cite{E-S}. This
theorem allows us to choose a prefered metric on any flat bundle. By a Bochner
technique on K\"ahler manifolds (\cite{Siu} \cite{Jost-Yau} \cite{Corlette}) a
harmonic metric is in fact pluriharmonic, and we recover in this way the
holomorphic data of a {\em Higgs bundle}. The correspondence in the other
direction characterizes exactly which Higgs bundles arise in this way
\cite{Hitchin} \cite{CVHS}.
We then mention (refering to \cite{NitsureModuli},
\cite{Moduli} for proofs) the
existence of moduli spaces for all of the objects in question. The space
$M_{DR}(X,G)$ is the moduli scheme for principal holomorphic $G$-bundles with
integrable connection. It is a coarse moduli space for the moduli stack ${\cal M}
_{DR}(X,G)$; it is this moduli stack which should be thought of as the
nonabelian de Rham cohomology, and the moduli space is a convenient
scheme-theoretic version. The space $M_{Dol}(X,G)$ is the moduli scheme for
semistable principal Higgs bundles with vanishing rational Chern classes; again
it is a coarse moduli space for ${\cal M} _{Dol}(X,G)$, the moduli
stack which is what should be thought of as the nonabelian
Dolbeault cohomology. The harmonic metric construction and the Bochner
technique (together with the inverse construction) give a homeomorphism
$M_{DR}(X,G) \cong M_{Dol}(X,G)$ which is ${\cal C} ^{\infty}$ on the smooth points.
The above work is the result of a long series of generalizations of the
original work of Narasimhan and Seshadri. Without being exhaustive, I should
at least mention the names of Mumford, Gieseker, Maruyama, Mehta and Ramanathan
for the constructions of moduli spaces; and Donaldson, Uhlenbeck, Yau, Deligne
and Beilinson for the inverse construction to the harmonic map construction.
See the introductions and references of \cite{HBLS}, \cite{Moduli} for more
detailed historical references.
\begin{center}
$\ast \hspace*{2cm} \ast \hspace*{2cm} \ast$
\end{center}
These constructions (or their predecessors in work of Eells and Sampson
\cite{E-S} \cite{Sampson} and Siu \cite{Siu}) are the starting point for
much of
the work which has been done in nonabelian Hodge theory in the past several
years. As many aspects are covered by other lectures (and their corresponding
articles), and in any case many of the papers on the subject contain
survey-like
introductions, I will not try to survey all of these topics in the main part
of the paper but will just mention some of them here in the introduction.
Hitchin was interested from the beginning in the completely integrable
holomorphic hamiltonian system given by the moduli space of Higgs bundles
\cite{HitchinDuke}. This direction of research has branched off toward the
Verlinde formula, quantization and so forth. I won't try to give references as
this gets away from our principal concern of Hodge theory.
One of the principal applications of Hodge theory has always been to give
restrictions on the topological type of varieties and their subvarieties. The
nonabelian version presents this same feature. The existence of all of
the various structures described above on the nonabelian cohomology $H^1(X, G)$
and various related considerations give restrictions on which groups can be
fundamental groups of compact K\"ahler manifolds, or more generally on which
homotopy types can arise. Some of these results such as
\cite{Carlson-Toledo} \cite{Jost-Yau} \cite{Sampson} \cite{Siu} pre-date the
general Hodge-theoretic point of view given above, being based on harmonic map
considerations \`a la Eells-Sampson and Siu. Others come directly from the
full correspondence between Higgs bundles and local systems and the subsidiary
fact that Higgs bundles invariant under the natural action of ${\bf G}_m = {\bf C}
^{\ast}$ correspond to variations of Hodge structure \cite{HBLS}.
The restrictions on higher homotopy types are generally of two sorts:
either one
uses information about a homotopy type to obtain additional information on the
harmonic map (such as its rank) and then concludes that such a harmonic map
cannot exist (\cite{Siu} \cite{Carlson-Toledo} \cite{CorletteRigidity}); or one
uses various notions about the cohomology of local systems to rule out higher
homotopy types (these are the restrictions coming from work of
Green-Lazarsfeld \cite{G-L}, Beauville \cite{Beauville}, Arapura
\cite{Arapura}, recently Hironaka \cite{Hironaka}, also \cite{ENS}---the idea
of using these results to get restrictions will be explored in \S 3
below since
many of these papers don't explicitly mention the aspect ``restrictions on
homotopy types'' which comes out of their results).
Gromov has an $L^2$-Hodge theoretic argument to rule out free (and certain
amalgamated) products of groups \cite{Gromov}. This is particularly
interesting in relation to the theory we sketch here, because it allows one to
``see'' the whole fundamental group (for example, the amalgamated product of
two groups with no subgroups of finite index is ruled out, which would
evidently
beimpossible to do by looking at representations into linear groups).
Gromov and
Schoen \cite{GromovSchoen} have also developed a generalization of the
harmonic
map theory to cover harmonic maps into negatively curved Euclidean buildings.
Coupled with a Lefschetz technique \cite{LefHolLeaves} this gives results about
fundamental groups and in particular seems to give an alternative proof of the
result about amalgamated products. This technique is generalized in
\cite{Jost-Zuo}
\cite{Katzarkov}.
With all of these restrictions on fundamental groups coming from Hodge theory,
one might well wonder if there are any interesting fundamental groups at all.
Toledo's example of a non residually finite fundamental group
(\cite{Toledo}---there have since been several other generalizations) shows
that the family of groups which can occur is, on the contrary, quite
complicated. We can still ask whether the part ``seen'' by nonabelian Hodge
theory with linear groups as coefficients can be nontrivial, for example, are
there nontrivial ways of obtaining positive dimensional moduli spaces (other
than easily known ways using curves and abelian varieties)? The answer here
is affirmative too, and in fact the Higgs bundle picture is
essential for calculating what happens to obtain examples \cite{Families}.
One of the main types of results has been the {\em factorization theorem}.
This type of result relates the fundamental group and the geometry of $X$.
The typical type of statement is that if $\rho : \pi _1(X)\rightarrow
\Gamma$ is
a certain type of representation then it must {\em factor:} there is a
morphism of varieties $X\rightarrow Y$ such that $\rho$ factors through $\pi
_1(Y)$. Perhaps the original result of this type is that of Siu \cite{Siu}
(cf also Beauville's appendix to \cite{Catanese}) stating that when $\Gamma$
is the fundamental group of a Riemann surface, then any $\rho$ must factor
through a Riemann surface $Y$. Gromov's result of \cite{Gromov} also passes
through a similar type of statement: certain $L^2$ cohomology classes on the
universal cover of $X$ must factor through maps to a Riemann surface. One of
the first statements involving factorization through a higher dimensonal
variety is that of Zuo \cite{Zuo}; and we now have a fairly complete picture
of this type of result (cf \cite{Jost-Zuo} \cite{Jost-Zuo2}
\cite{Katzarkov} \cite{KatzarkovPantev} \cite{Zuo} \cite{ZuoBook}): any
nonrigid
representation to a linear group $G$ must factor through a variety $Y$ of
dimension less than or equal to the rank of $G$. Note that the example of
\cite{Families} shows that we do not always get factorization through a
curve...but apart from this we do not know for sure if the bound rank of $G$ is
sharp. See also \cite{Mok} and various generalizations for
additional geometric information on the factorization variety $Y$.
Using the theory of harmonic maps to buildings mentioned above, one can
extend these factorization theorems to the case of representations into
linear groups over $p$-adic fields not going into a maximal compact subgroup
\cite{Jost-Zuo} \cite{Katzarkov} \cite{ZuoBook}.
These factorization techniques obviously have a certain application to the
Shafarevich conjecture \cite{Kollar}. This has been pursued by Katzarkov,
Lasell, Napier, Ramachandran (\cite{KatzaSha} \cite{KatzaRama}
\cite{LasellRama}
\cite{Napier} \cite{NapierRama}), see also Zuo \cite{ZuoBook}. The main problem
is that only the part of the fundamental group seen by a given linear
representation can be treated. They obtain a full proof of the Shafarevich
conjecture for surfaces whose fundamental group injects into a linear group.
Another recent development worth mentioning is Reznikov's proof of the Bloch
conjecture that the Chern-Simons classes of flat bundles on K\"ahler manifolds
are torsion \cite{Reznikov}.
The last principal area of work I would like to mention, one where there is
still a fair amount to be done, is the noncompact (quasiprojective) case. The
problem is to do the analogues of everything which we discuss in the compact
K\"ahler (or projective algebraic) case, in the case of a quasiprojective
variety. This problem becomes much more difficult in dimension $\geq 2$.
Aside from the dimension distinction, the problem can be divided into several
parts.
The
first part is to obtain the analogue of Corlette's theorem. The main
difficulty here is to get a starting point for a heat equation minimization
process, that is to say an equivariant map of finite energy. If the
eigenvalues of the monodromy at infinity are not of norm one, this becomes
impossible and the problem is more difficult. Modulo this difficulty, the
problem has been solved by Corlette \cite{CorletteToulouse} and
Jost and Zuo \cite{Jost-Zuo}.
The next problem is to obtain the analogues of
the Bochner results, yielding a Higgs bundle. This is discussed to some extent
in \cite{Jost-Zuo}. There may be a problem with Chern classes in general.
The other aspect of this problem is that the appropriate Higgs bundle notion
must include some data at infinity, namely a {\em parabolic structure}.
The problem of associating a parabolic Higgs bundle with nice properties, to a
harmonic bundle, is treated in Biquard \cite{Biquard} in the case when the
divisor at infinity is smooth.
Biquard also provides the converse
construction: given a parabolic Higgs bundle satisfying appropriate
conditions, he gets back a Yang-Mills connection and hence a representation.
This provides a relatively complete generalization in the case of smooth
divisor at infinity. What is left open for the moment is to treat the case
where
the divisor at infinity has normal crossings.
A fourth aspect of the problem is to construct moduli spaces. This is now
well understood for Higgs bundles and the like, due to work of Yokogawa and
Maruyama \cite{MaruYokogawa} \cite{Yokogawa}. I think there is still a little
work (probably not too hard) left to be done on the side of {\em filtered local
systems}, which are the general representation-like objects which correspond to
parabolic Higgs bundles.
It remains to be seen how all of these aspects fit together, and then to
proceed with the generalizations of all of the further structures inherited by
the moduli spaces in the compact case (i.e. the structures we will discuss
in the present paper).
\begin{center}
$\ast \hspace*{2cm} \ast \hspace*{2cm} \ast$
\end{center}
Rather than going into further detail on all of these
applications and developments, I would like in the body of the paper to
concentrate on a fundamental aspect---the nonabelian Hodge filtration
\cite{NAHT}. We will discuss many of the basic subjects surrounding the Hodge
filtration, such as the quaternionic structure and twistor spaces. And we give
proofs of the results announced in \cite{NAHT}, in particular the
compactification of $M_{DR}$ which is a consequence of (and practically
equivalent to) the Hodge filtration. The goal will be in the last section to
introduce an open problem, that of studying degenerations of nonabelian Hodge
structure coming from a degenerating family of varieties.
After our discussion of Corlette's theorem, its converse and the moduli spaces
in \S\S 1-2, we turn in \S 3 to a discussion of Hitchin's quaternionic
structure
on the moduli space for representations \cite{Hitchin} \cite{Fujiki}.
We give an application
to cohomology jump-loci retrieving the results of Green and Lazarsfeld via an
argument of Deligne and along the way see how these results give restrictions
on the higher homotopy type of non-simply connected K\"ahler manifolds.
In \S 4 we look at Deligne's
complex analytic construction of the twistor space corresponding to the
quaternionic structure \cite{DeligneLetter}.
He
obtains the twistor space by glueing two copies of a family $M_{Hod}$
deforming between the moduli space $M_{DR}$ of vector bundles with integrable
connection and the moduli space $M_{Dol}$ of Higgs bundles. This deformation
(parametrized by ${\bf A}^1$)
is the moduli space of {\em vector bundles with $\lambda$-connections};
over $\lambda = 0$ a $\lambda$-connection is just a Higgs field, whereas over
$\lambda \neq 0$ a $\lambda$-connection is $\lambda$ times an integrable
connection.
In \S 5 we explain the analogy with Rees modules which allows us to interpret
the space $M_{Hod}$ as the Hodge filtration on $M_{DR}$.
As justification we establish in \S 6 the relationship with the Morgan-Hain
Hodge filtration on the nilpotent completion of the fundamental group
\cite{Morgan} \cite{HainKth}.
We then proceed with certain results about the Hodge filtration in the
nonabelian case, notably Griffiths transversality for
its variation in a family,
and regularity of the Gauss-Manin connection at singular points of a family
(\S
8). In order to do this, we first introduce in \S 7 the notion of {\em formal
groupoid} \cite{Berthelot} \cite{Illusie}. This provides a general framework
for looking at connections, Higgs fields and so forth, and in particular allows
us to actions of these types of things on schemes rather than just vector
bundles.
In a detour \S 9 we investigate Goldman-Millson theory \cite{GoldmanMillson}
for the
local structure of $M_{Hod}$. The {\em isosingularity principle} which says
that the singularities of $M_{Dol}$ are the same as those of $M_{DR}$,
generalizes to give a trivalization of $M_{Hod}$ formally along prefered
sections. This allows us to conclude, for example, that $M_{Hod}$ is flat over
${\bf A}^1$.
Then we come to a properness property of $M_{Hod}$; the limits of
${\bf G}_m$-orbits always exist (\S 10). This is the analogue of the classical
property of the Hodge filtration, that $F^0$ is the whole space.
This weight property allows us to obtain a compactification of $M_{DR}$ in \S
11, by taking
the quotient of an open set in $M_{Hod}$ by the action of ${\bf G}_m$.
This compactification was announced without proofs in \cite{NAHT} so we take
this opportunity to provide a complete version of the argument.
(Drinfeld recently informed me that some of his students have obtained a
compactification of the moduli space of logarithmic connections on ${\bf P} ^1$
with singularities at a finite set of points, also using the method of
$\lambda$-connections but independently of \cite{NAHT}.)
At the end of the
section we revisit Griffiths transversality in terms of this compactification:
it says that the Gauss-Manin connection on $M_{DR}$ has poles of order $1$ at
infinity in the compactification. This gives a picture of a compact space
$\overline{M_{DR}}(X/S,G)$ with a lift of a frame vector field on $S$ to a
vector field having a simple pole at infinity. A similar interpretation holds
for the regularity of the Gauss-Manin connection.
In the penultimate section 12 we define the {\em nonabelian Noether-Lefschetz
locus} $NL(X/S, GL(n))$. If $X\rightarrow S$ is a family then this is
essentially the locus of $s\in S$ where $X_s$ supports an integral variation of
Hodge structure. It is the nonabelian analogue of the classical
Noether-Lefschetz locus of Hodge cycles. If $S$ is projective then we can see
that $NL(X/S,GL(n))$ is algebraic (as would be a consequence of the Hodge-type
conjecture that one could
formulate, that integral variations of Hodge structure are motivic). We
conjecture that this is true even for $S$ quasiprojective, which would be a
nonabelian version of the result of \cite{CDK}.
In the last section we present an open problem, the problem of understanding
the degeneration of all of our structures (Hodge filtration, Gauss-Manin
connection, quaternionic structure, etc.) near degenerations of a family
$X/S$. This problem is motivated by the problem of proving that $NL(X/S)$
is algebraic when $S$ is quasiprojective---the nonabelian analogue of the
work of Cattani, Deligne and Kaplan for the classical Noether-Lefschetz
locus of Hodge cycles \cite{CDK}.
I would like to thank P. Deligne for sharing with me his ideas on how to
construct Hitchin's twistor space. This construction provided the starting
point for everything (new) done below.
Everything is over the field ${\bf C}$ of complex numbers.
\numero{The nonabelian Hodge theorem}
Suppose $X$ is a Riemannian manifold with basepoint $x\in X$, and suppose
$G$ is a reductive algebraic group. A representation
$\rho : \pi _1(X,x)\rightarrow G$ corresponds to a flat principal left
$G$-bundle over $X$ (in other words a locally constant sheaf of
principal homogeneous spaces for $G$ over $X$), or equally well to a
${\cal C} ^{\infty}$ principal $G$-bundle $P$ with an integrable
connection $D$. We think of a connection on a principal bundle as a
$G$-invariant operator $\nabla$ from functions on $P$ to sections of
$T^{\ast}(X)|_P$ (satisfying a Leibniz rule with respect to functions pulled
back from the base). Such an operator then has a square $\nabla ^2$ from
functions on $P$ to sections of $\bigwedge ^2T^{\ast}(X)|_P$, and the
integrability condition is $\nabla ^2=0$. The flat principle bundle is the
sheaf of $\nabla$-horizontal sections of $P$.
Fix a maximal compact subgroup $K\subset G$. A {\em $K$-reduction} for a
principal bundle $P$ is a ${\cal C}^{\infty}$ principal $K$-subbundle
$P_K\subset P$ giving $P=P_K \times ^KG$.
A flat principal $G$-bundle $(P,\nabla )$ gives rise to a flat family of
homogeneous spaces over $X$ which we can write as $P\times ^G (G/K)$. If
$P_K\subset P$ is a $K$-reduction for $P$ then the image of $P_K \times (eK)$
in $P_K\times ^KG$ is a smooth section of the bundle $P\times ^G (G/K)$.
This smooth section can also be thought of as a $\rho$-equivariant map
$\phi :\tilde{X}\rightarrow G/K$. We define the {\em energy} of the
$K$-reduction or of its associated equivariant map by
$$
{\cal E} (\phi ) := \int _X | d\phi | ^2,
$$
where the integral is taken with respect to the volume form on $X$, and
the norm of the differential $d\phi$ is measured with respect to an invariant
metric on $G/K$ (note that one has to fix a $K$-invariant metric on the
complement ${\bf p} $ to ${\bf k} \subset {\bf g}$ when making this discussion---if
$G$ is semisimple then we can fix the Killing form as a canonical choice).
An equivariant map or $K$-reduction $P_K$ is called {\em harmonic} if it is a
critical point of ${\cal E} (\phi )$.
The Euler-Lagrange equation for for a harmonic equivariant map is $d^{\ast}
d\phi =0$. This is a nonlinear equation with the Laplacian as its principal
term.
The main theorem in this subject is the following generalisation of the
theorem of Eells and Sampson \cite{E-S}:
\begin{theorem} {\rm (Corlette \cite{Corlette})}
If $\rho$ is a representation such that $\rho (\pi
_1(X,x))$ is Zariski-dense in $G$ (or such that the Zariski closure is itself
reductive), then there exists a harmonic equivariant map $\phi$.
\end{theorem}
{\em Proof:}
We indicate here a variant of Corlette's proof which might be useful for
people with an algebraic geometry background. Assume that the Zariski
closure $G$ is semisimple (the general reductive case may then be obtained by
using the linear theory of harmonic forms for ${\bf C} ^{\ast}$ representations).
Eells and
Sampson \cite{E-S} prove the existence of harmonic maps from $X$ to a compact
negatively curved manifold $M$ by a heat-equation minimization technique. We
can start off with an equivariant map $\phi _0$ and apply the same heat
equation to obtain a family of maps $\phi _t$. We get the same local
estimates. In particular, for $x,y\in \tilde{X}$ the distance from $\phi _t
(x)$ to $\phi _t(y)$ is a bounded function of $t$. Furthermore, if for any one
point $x$ we can show the ``$C^0$-estimate'' that $\phi _t(x)$ stays in a
compact subset of $G/K$ then the estimates of \cite{E-S} will allow us to show
that the $\phi _t$ converge to a smooth harmonic map $\phi$. Note, for
example, that the $C^0$-estimate is not true if the Zariski closure of $\rho
(\pi _1(X,x))$ is not reductive (in fact, the existence of an equivariant
harmonic map implies reductivity of the Zariski closure). In the original case
of \cite{E-S} the target was a compact manifold so this problem was avoided.
Now apply a little bit of geometric invariant invariant theory to get the
$C^0$-estimate. Choose elements $g_t\in G$ bringing us
back to the basepoint: $g_t\phi _t(x) = eK$. Fix a set of generators $\gamma
_i$ for $\pi _1(X,x)$. From \cite{E-S} we know that the distance from
$\phi _t(x)$ to $\phi _t(\gamma _ix)$ remains bounded. As the distance on
$G/K$ is $G$-invariant, we have
$$
d(g_t\phi _t(x), g_t\phi _t(\gamma_i x))\leq C.
$$
On the other hand the equivariance of $\phi _t$ gives $\phi _t(\gamma
x)=\rho (\gamma_i )\phi _t(x)$ so (also plugging in $g_t\phi _t(x) = eK$) we
get
$$
d(eK, g_t\rho (\gamma_i )g_t^{-1}eK)\leq C.
$$
Since $K$ is compact the map $G\rightarrow G/K$ is proper so the $g_t\rho
(\gamma_i )g_t^{-1}$ remain in a compact subset of $G$.
The representation variety $R:= Hom (\pi _1(X,x), G)$ embedds in a product
$G\times \ldots \times G$ by $\rho \mapsto (\ldots, \rho (\gamma _i),
\ldots )$. The group $G$ acts on $R$
by the {\em adjoint action} $Ad(g)(\rho )(\gamma ):= g\rho (\gamma )g^{-1}$,
and this is compatible with the above embedding via the adjoint action in each
variable of $G\times \ldots \times G$. The previous paragraph tells us
that, in our situation, $Ad(g_t)\rho $ remain in a compact subset of $R$.
The basic information from the geometric invariant theory of spaces of
representations of finitely generated groups, is that the hypothesis that
$\rho
(\pi _1(X,x))$ is Zariski-dense in $G$ implies that the
$Ad(G)$-orbit of $\rho$ is closed in $R$. This is well known
\cite{LubotskyMagid} but we discuss it anyway in the next two paragraphs.
If $G=GL(n,{\bf C} )$ then $\rho$ corresponds to an irreducible $n$-dimensional
representation which we denote $V_{\rho }$. If $V_0$ is a representation in
the closure of the orbit of $\rho$ then we have a family of representations
$\{
V_t\}$ parametrized by $t$ in a smooth curve with
$V_0$ being the value at a point $0$ and $V_t\cong V_{\rho}$ for $t\neq 0$.
By
semicontinuity there is a nontrivial morphism of representations from $V_{\rho
}$ to $V_0$, but since $V_{\rho}$ is irreducible this must be an isomorphism
and
we get that $V_0$ is in the orbit of $V_{\rho}$.
To prove this for a semisimple group
$G$ note that $G$ admits a faithful irreducible representation $V$; this
gives a
composed representation $V_{\rho}$ of $\pi _1(X,x)$. Since $\rho$ is
Zariski-dense, $V_{\rho}$ is irreducible. Suppose we have a family $\rho_t$ of
representations (parametrized by an affine curve) with $\rho _t \sim \rho$ for
$t\neq 0$ and $\rho _0$ different from $\rho$. This gives a family of linear
representations $V_t$ of $\pi _1(X,x)$; as above, semicontinuity and
irreducibility of $V_{\rho}$ imply that $V_0\cong V_{\rho}$. This then
implies that $\rho$ and $\rho_0$ are conjugate by an automorphism of $G$ where
furthermore this automorphism is a limit of inner automorphisms. The group of
outer automorphisms being finite (hence discrete) we conclude that $\rho_0$ and
$\rho$ are conjugate by an inner automorphism, that is $\rho _0$ is in the
$Ad(G)$-orbit of $\rho$.
Now we complete the proof. The $Ad(g_t)\rho$
remain in a compact subset of the orbit $Ad(G)\rho$ because of
the fact that the orbit is closed.
Since $\rho
(\pi _1(X,x))$ is Zariski-dense in $G$, the stabilizer of $\rho$ is just the
center of $G$, which is finite (since we have assumed that $G$ is reductive).
In particular the map $G\rightarrow Ad(G)\rho$ given by the action on $\rho$
is proper, so the $g_t$ themselves remain in a compact subset of $G$. Finally
this implies that the $\phi (x)= g_t^{-1}eK$ remain in a compact subset of
$G/K$, which is the $C^0$-estimate we need.
\hfill $\Box$\vspace{.1in}
{\em Remark:} Donaldson proved this theorem for rank $2$ representations
independently of Corlette \cite{Donaldson}. It was also proved by Diederich
and Ohsawa for representations into $SL(2, {\bf R} )$ \cite{DietrichOsawa}.
On the other hand there have since been several generalizations to the
noncompact case, for example by Corlette \cite{CorletteToulouse} and Jost and
Zuo \cite{Jost-Zuo} \cite{Jost-Zuo2}.
\subnumero{The K\"ahler case}
Assume now that $X$ is a compact K\"ahler manifold.
Let $\omega$ denote the K\"ahler form (of a K\"ahler metric which we choose);
let $\Lambda$ denote the adjoint of wedging with $\omega$; and let $\partial$
and $\overline{\partial}$ denote the operators coming from the complex structure. A
holomorphic principal bundle $P$ may be considered as a ${\cal C} ^{\infty}$
principal bundle together with a $G$-invariant operator $\overline{\partial}$ from
functions on $P$ to sections of $\Omega ^{0,1}_X|_P$ satisfying the
appropriate Leibniz rule and $\overline{\partial} ^2 =0$.
We say that a section, map or whatever is {\em pluriharmonic} if it is
harmonic when restricted to any locally defined smooth complex subvariety.
This
condition is independant of the choice of metric. The classical Bochner formula
states that harmonic forms are pluriharmonic. The fundamental result about
equivariant harmonic maps on K\"ahler manifolds is just the analogue:
\begin{proposition}
If $\phi$ is a harmonic equivariant map from $\tilde{X}$ to $G/K$ then
$\phi$ is pluriharmonic.
\end{proposition}
{\em Proof:}
See \cite{Siu} \cite{Jost-Yau} and \cite{Corlette}.
\hfill $\Box$\vspace{.1in}
Suppose $P$ is a flat principal $G$-bundle with flat connection denoted by
$d = d' + d''$. Suppose $P_K$ is a $K$-reduction corresponding to equivariant
harmonic map $\phi$. We can decompose the connection into a component $d^+$
preserving $P_K$ and a component $a$ orthogonal to $P_K$. Then decompose
according to type, $d^+ = \partial + \overline{\partial}$ and $a = \theta ' + \theta
''$. We obtain the decompositions
$$
d' = \partial +
\theta '
$$
and
$$
d'' = \overline{\partial} + \theta ''.
$$
The components
orthogonal to $P_K$ operate on functions only via the restrictions of the
functions to the fiber, which is to say that they operate on functions via
the Lie algebra $ad (P)= P\times ^G{\bf g} $ of $G$-invariant vector fields on
$P$. Thus these components are sections $\theta '$ of $ad
(P)\otimes \Omega ^{1,0}_X$ and $\theta ''$ of $ad
(P)\otimes \Omega ^{0,1}_X$. The pluriharmonic map equations translate into:
$$
\overline{\partial} ^2 = 0;
$$
$$
\overline{\partial} \theta ' + \theta ' \overline{\partial} = 0 \;\; (\mbox{which we write}\;\;
\overline{\partial} (\theta ')=0);
$$
and
$$
[\theta ' , \theta ' ]=0.
$$
In the last equation the form coefficients are wedged and the Lie algebra
coefficients bracketed.
We also obtain of course the complex-conjugate equations for $\partial$ and
$\theta ''$. These are complex conjugates in view of the hermitian or
antihermitian properties of $\partial + \overline{\partial}$ and $\theta ' + \theta ''$
respectively.
The first equation says that $(P,\overline{\partial} )$ has a structure of holomorphic
principal bundle. This is in general different from the structure of
holomorphic principal bundle $(P,d'' )$ which comes from the flat structure.
The second equation says that $\theta '$ corresponds to a holomorphic section
which we now denote simply by $\theta \in H^0(X, ad(P)\otimes \Omega ^1_X)$;
and the third equation says $[\theta , \theta ]= 0$.
We define a {\em principal Higgs bundle} to be a holomorphic principal
$G$-bundle $P$ together with $\theta \in H^0(X, ad(P)\otimes \Omega ^1_X)$
such that $[\theta , \theta ]= 0$. From the previous results, a flat
principal
$G$-bundle $P$ with harmonic $K$-reduction $P_K$ gives a principal Higgs
bundle $(P, \theta )$.
If $(P,\theta )$ is a principal Higgs bundle and $V$ is a
linear representation of $G$ then we obtain an associated Higgs bundle
$(E,\theta _E)$ where $E=P\times ^GV$ and $\theta _E \in H^0(X, End(E)\otimes
\Omega ^1_X)$ is the associated form associated to $\theta$.
Recall that a Higgs bundle $(E,\theta )$ is {\em stable} if for any subsheaf
$F\subset E$ preserved by $\theta$ we have $deg (F)/r(F) < deg (E)/r(E)$ (the
notion of degree depends on choice of K\"ahler class). Say that $E$ is {\em
polystable} if it is a direct sum of stable Higgs bundles of the same slope
(degree over rank). Say that a principal Higgs bundle $P$ is {\em polystable}
if for every representation $V$, the associated Higgs bundle is polystable.
If the generalized first Chern classes of $P$ (corresponding to all degree one
invariant polynomials) vanish then it is enough to check this for one
faithful representation $V$ (cf \cite{HBLS} p. 86).
\begin{theorem}
\label{Corr}
Suppose $\rho : \pi _1(X)\rightarrow G$ with $G$ reductive, and suppose that
the Zariski closure of the image of $\rho$ is reductive. Let ${\cal P}$ be the
associated flat bundle and ${\cal P}_K$ be a pluriharmonic reduction. The structure
of principal Higgs bundle $(P,\theta )$ obtained above doesn't depend on choice
of pluriharmonic reduction $P_K$. The principal Higgs bundle has vanishing
rational Chern classes and is polystable. Furthermore, any polystable
principal Higgs bundle with vanishing rational Chern classes arises from a
unique representation $\rho$ in this way.
\end{theorem}
{\em Proof:}
See \cite{Hitchin} \cite{CVHS} \cite{HBLS}.
\hfill $\Box$\vspace{.1in}
\subnumero{Moduli spaces}
Fix a reductive complex algebraic group $G$ and a smooth projective variety
$X$.
Let $R_{DR}(X,x,G)$ denote the moduli scheme of principal $G$-bundles with
integrable connection and frame at $x\in X$ constructed in \cite{Moduli};
similarly let $R_{Dol}(X,x,G)$ denote the moduli scheme of semistable principal
Higgs bundles with vanishing rational Chern classes with a frame at $x\in X$;
and finally let $R_{B}(X,x,G):= Hom (\pi _1(X,x), G)$ denote the space of
representations of the fundamental group in $G$. In all three cases these
schemes represent the appropriate functors. We call these spaces the de Rham,
Dolbeault and Betti {\em representation spaces}.
The group $G$ acts on each of the representation spaces. In all three cases,
all points are semistable for an appropriate linearized line bundle, so by
\cite{GIT} the universal categorical quotients
$$
M_{DR}(X,G):=R_{DR}(X,x,G)//G
$$
$$
M_{Dol}(X,G):=R_{Dol}(X,x,G)//G
$$
$$
M_{B}(X,G):=R_{B}(X,x,G)//G
$$
exist
\cite{LubotskyMagid} \cite{Moduli} \cite{NitsureModuli}. They are independent
of the choice of basepoint. The points of these quotients parametrize the
closed orbits in the representation spaces. The closed orbit in the closure of
an orbit corresponding to a given representation is the
{\em semisimplification}
of the representation. Two points in a representation space map to the same
point in the moduli space if and only if their semisimplifications coincide.
For some purposes it is useful to think about the {\em moduli stacks} instead.
These are the stack-theoretic quotients
$$
{\cal M}_{DR}(X,G):=R_{DR}(X,x,G)/G
$$
$$
{\cal M}_{Dol}(X,G):=R_{Dol}(X,x,G)/G
$$
$$
{\cal M}_{B}(X,G):=R_{B}(X,x,G)/G .
$$
Properly speaking, it is the moduli stacks which should be thought of as the
first nonabelian cohomology stacks. The moduli spaces are the
hausdorffifications or associated coarse moduli spaces for the stacks (they
universally co-represent the functors $\pi _0$ of the stacks).
We have a complex analytic isomorphism
$R_{DR}(X,x,G)^{\rm an}\cong R_B(X,x,G)^{\rm
an}$ compatible with the action of $G$ coming from the Riemann-Hilbert
correspondence between holomorphic systems of ODE's and their monodromy
representations. This projects to the universal categorical quotients
(\cite{Moduli} \S 5) giving $M_{DR}(X,G)^{\rm an}\cong M_B(X,G)^{\rm
an}$ as well as to the stack quotients giving
${\cal M} _{DR}(X,G)^{\rm an}\cong {\cal M} _B(X,G)^{\rm
an}$.
The correspondence of Theorem \ref{Corr} gives an isomorphism between the
underlying sets of points of $M_{DR}(X,G)$ and $M_{Dol}(X,G)$, because the
points of these spaces correspond exactly to
representations which have reductive Zariski closure (really the
corresponding de Rham or Dolbeault analogues of this notion defined using the
Tannakian formalism). This isomorphism is a homeomorphism of underlying
topological spaces \cite{Moduli} which we thus write
$$
M_{DR}(X,G)^{\rm top} \cong M_{Dol}(X,G)^{\rm top}.
$$
Hitchin's original point of view \cite{Hitchin} was slightly different, in
that he constructed a single moduli space for all objects, and noted that it
had several different complex structures. This amounts to the same thing if
one ignores the algebraic structures (and in fact it is difficult to say
anything concrete about the relationship between the algebraic structures).
The
homeomorphism between the moduli spaces does not lift to a homeomorphism
between the representation spaces (cf the counterexample of
\cite{Moduli} II, pp
38-39). I don't know what happens when we look at the stacks, so I'll give that
as a question for future research.
{\em Question:} Does there exist a natural homeomorphism
${\cal M}_{DR}(X,G)^{\rm top} \cong {\cal M}_{Dol}(X,G)^{\rm top}$ inducing the previous
one on moduli spaces?
In order to attack this question one must first define the notion of the
''underlying topological space'' of a stack.
\numero{The quaternionic structure on the moduli space}
Let
$M^{\rm sm}_{DR}(X,G)$ (resp. $M^{\rm sm}_{Dol}(X,G)$,
$M^{\rm sm}_{B}(X,G)$) denote the
open subset of smooth points of $M^{\rm sm}_{DR}(X,G)$ (resp.
$M^{\rm sm}_{Dol}(X,G)$,
$M^{\rm sm}_{B}(X,G)$) parametrizing Zariski-dense representations
(the notion of
Zariski denseness makes sense for the de Rham or Dolbeault spaces using the
Tannakian point of view). Then the isomorphism
$$
M_{DR}^{\rm sm}(X,G)^{\rm top} \cong M_{Dol}^{\rm sm}(X,G)^{\rm top}
$$
is ${\cal C}^{\infty}$ (and even real analytic) \cite{Hitchin} \cite{Fujiki}.
Denoting by $M^{\rm sm}(X,G)$ the differentiable manifold underlying these
isomorphic spaces, we obtain two complex structures $I$ and $J$ on
$M^{\rm sm}(X,G)$ coming respectively from $M_{Dol}^{\rm sm}
(X,G)$ and $M_{DR}^{\rm sm}(X,G)$.
The tangent space to $M^{\rm sm}_{DR}(X,G)$ (resp. $M^{\rm sm}_{Dol}(X,G)$,
$M^{\rm sm}_{B}(X,G)$) at a point
corresponding to a principal bundle $P$ is $H^1_{DR}(X, ad(P))$
(resp. $H^1_{Dol}(X, ad(P))$, $H^1_{B}(X, ad(P))$). These tangent spaces have
natural $L^2$ metrics coming from the interpretation of classes as harmonic
forms.
\begin{theorem}
\label{quater}
{\rm (Hitchin \cite{Hitchin})} Put $K=IJ$. Then the triple
$(I,J,K)$ is a quaternionic structure for the manifold $M^{\rm sm}(X,G)$.
Furthermore if $g$ denotes the natural Riemannian metric on $M^{\rm sm}(X,G)$
obtained from the $L^2$ metric on the tangent space induced by the harmonic
metric (these are the same up to a constant for all structures) then $g$ is a
K\"ahler metric for each of the structures $(I,J,K)$, in other words $M^{\rm
sm}(X,G)$ becomes a hyperk\"ahler manifold.
\end{theorem}
{\em Proof:} See \cite{Hitchin} for the result when $X$ is a curve. The
theorem for any $X$ follows from
the corresponding theorem for a curve, using the embedding $M(X,G)\subset
M(C,G)$ for a curve $C$ which is a complete intersection of hyperplane sections
in $X$. On the other hand Fujiki proves this for a general K\"ahler manifold
$X$ \cite{Fujiki} (where the method of taking hyperplane sections is no longer
available).
We will
see how to calculate that $I,J,K$ form a quaternionic structure in the
proof of Theorem \ref{twistor} below.
\hfill $\Box$\vspace{.1in}
The embedding $M(X,G)\subset M(C,G)$ obtained from a hyperplane section is
complex analytic for all structures, along any naturally defined subvariety
such as the Whitney strata of the singular locus. Consequently the smooth
points of the underlying reduced scheme structure of these strata inherit
quaternionic (and even hyperk\"ahler) structures.
\subnumero{The twistor space}
The notion of {\em twistor space} of a quaternionic manifold $N$ is explained,
for example, in \cite{HitchinBourbaki}. Suppose $N$ is a manifold with three
integrable complex structures $(I,J,K)$ defining a quaternionic structure on
each tangent space. We identify ${\bf P} ^1$ with the sphere $x^2 + y^2 + z^2 =
1$ by the stereographic projection; this gives
$$
\lambda =u+ iv \leftrightarrow (x=\frac{1-|\lambda |^2}{1+|\lambda |^2},
y=\frac{2u}{1+|\lambda |^2},
z=\frac{2v}{1+|\lambda |^2}).
$$
The {\em twistor space} $TW(N)$ is a
complex manifold with a ${\cal C} ^{\infty}$ trivialisation
$$
TW(N) \cong N\times {\bf P} ^1
$$
such that the projection $TW(N)\rightarrow {\bf P} ^1$ is holomorphic; such that
for any $n\in N$ the section $\{ n\} \times {\bf P} ^1\subset TW(N)$ is
holomorphic; and such that for any $\lambda \in {\bf P} ^1$ the complex
structure on
$M\times \{ \lambda \}$ is $xI + yJ + zK$ where $(x,y,z)$ corresponds to $t$
via
the stereographic projection defined above (note that $(xI + yJ + zK)^2=-1$).
If we denote by $I_{u+iv}$ the complex structure corresponding to $\lambda =u+
iv\in {\bf A}^1$ then one can see using the above definitions that the formula
$$
I_{u+iv} = (1 -uK + vJ)^{-1} I (1 -uK + vJ)
$$
holds (we'll need this in the proof of Theorem \ref{twistor} below).
This definition serves to determine an almost complex structure on $TW(N)=
N\times {\bf P} ^1$. The almost complex structure is integrable
\cite{HitchinBourbaki} \cite{AHS} \cite{HKLR} \cite{Salamon}; in our case we
will indicate below an explicit construction which is integrable so we can
avoid using the general integrability result.
The twistor space has various other structures, notably an antilinear
involution $\sigma$ covering the antipodal involution $\sigma _{{\bf P} ^1}$
of ${\bf P} ^1$. Define $\sigma (n,t):= (n, \sigma _{{\bf P} ^1}(t))$. This is
antilinear because it is antilinear in the horizontal directions (along
prefered sections) and in vertical directions because $-xI-yJ-zK$ is the
complex conjugate complex structure to $xI+yJ+zK$.
The prefered sections are by definition $\sigma$-invariant.
If $N$ is a quaternionic vector space of quaternionic rank $r$ then the twistor
space may be constructed by hand. It the direct sum bundle ${\cal O} _{{\bf P}
^1}(1)^{2r}$ over ${\bf P} ^1$. In this case one can check that the prefered
sections are the only $\sigma$-invariant sections.
\subnumero{Application: subvarieties of $M(X,{\bf G}_m )$ defined by cohomological
conditions}
The quaternionic structure gives a nice way of looking at the results of
Green-Lazarsfeld \cite{G-L} on subvarieties defined by cohomological
conditions. In \cite{G-L} they look at the subvarietes $\Sigma
^i_k(Pic^0)\subset Pic ^0(X)$ of line bundles ${\cal L}$ with $h^i({\cal L} )\geq k$.
They show that these are unions of translates of subtori of $Pic ^0(X)$.
We
look at the subvarieties $\Sigma ^i_k(M)\subset M(X,GL(n) )$
consisting of those local systems $V$ such that $h^i(X,V)\geq k$.
We have not specified whether we look at $M_B$, $M_{DR}$ or $M_{Dol}$ because
the same locus is defined in all three cases, and they correspond under the
homeomorphisms $M_B\cong M_{DR}\cong M_{Dol}$. This is due to the fact that
if $\rho$ is a representation corresponding to vector bundle $V$ with
integrable connection and corresponding to Higgs bundle $E$ then the
interpretation of cohomology classes as harmonic forms and the K\"ahler
identities between the laplacians \cite{HBLS} gives isomorphisms
$$
H^i(X, \rho )\cong H^i_{DR}(X, V) \cong H^i_{Dol}(X, E)
$$
(see \cite{HBLS} Lemma 2.2). Now $\Sigma ^i_k(M_{Dol})$ is a
complex analytic subvariety of $M_{Dol}$ whereas $\Sigma ^i_k(M_{DR})$ is a
complex analytic subvariety of $M_{DR}$. At any smooth point of the reduced
subvariety, the tangent space of $\Sigma ^i_k(M)$ is preserved by both complex
structures $I$ and $J$ of the quaternionic structure; thus at smooth points
$\Sigma ^i_k$ is a quaternionic submanifold of $M^{\rm sm}$. This puts a big
restriction on the possibilities for $\Sigma ^i_k$ (for example, it must have
real dimension divisible by $4$ i.e. even complex dimension; the same is true
for any stratum in its Whitney stratification, for intersections of various
$\Sigma ^i_k$, etc.).
Deligne pointed out \cite{DeligneLetter} that we can use this compatibility
with
the quaternionic structure to recover the results of \cite{G-L}. This method
(which I described as an alternative in \cite{ENS}) adds to the many various
points of view on \cite{G-L} that are now available (\cite{Arapura}
\cite{Beauville} \cite{Catanese} \cite{ENS}). It seems worthwhile to mention
the quaternionic point of view here, since similar considerations may come
into
play for local systems of higher rank.
Deligne makes use of the following observation which is probably classical.
\begin{lemma}
Any locally defined smooth quaternionic subvariety of a quaternionic
vector space is flat (i.e. a linear subspace).
\end{lemma}
{\em Proof:}
If it were not flat, the second fundamental form would be a quaternionic
quadratic form, but an easy calculation shows that this cannot
exist.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\label{subtori}
If $G={\bf G}_m$ then the $\Sigma ^i_k (M)\subset M(X,{\bf G}_m )$ are unions of
translates of subtori.
\end{corollary}
{\em Proof:} The universal covering of $M(X,G)$ is just $H^1(X, {\bf C} )$ and
the quaternionic structure here is linear. Thus the previous lemma
(which is a local statement) applies to show that $\Sigma ^i_k(M)$
is flat at smooth points of reduced
irreducible components. A standard argument (such as in \cite{G-L}) gives the
conclusion.
\hfill $\Box$\vspace{.1in}
This property and its generalizations (one can show that the translates of
subtori are translations by torsion points \cite{Beauville}
\cite{Catanese} \cite{Arapura} \cite{ENS}) have implications for the
topology of $X$. One can easily fabricate examples of homotopy types such
that the corresponding jump loci are not translates of subtori.
We give a crude version here involving additions of $2$- and $3$-cells to a
torus (one can analyze in a similar way examples made by
adding cells of any dimensions). Put $\Gamma := {\bf Z} ^a$ with $a$
even, and put $U_1= K(\Gamma , 1)$ (which we can take as a real torus). Note
that $$ {\bf C} \Gamma \cong {\bf C} [t_1, t_1^{-1},\ldots , t_a, t_a^{-1}] $$
is the Laurent polynomial ring in $a$ variables, and
$$
M_B(U_1, {\bf G}_m )=Hom
(\Gamma , {\bf G}_m )\cong {\bf G}_m ^a.
$$
In fact one can canonically identify $M_B(U_1,{\bf G}_m )\cong Spec ({\bf C} \Gamma )$
(be careful that this reasoning only works well for $G={\bf G}_m$). Let $u\in U_1$
be the basepoint and let $U_2$ be obtained from $U_1$ by attaching $m$
$2$-spheres at $u$. Finally let $U_3$ be obtained by attaching $\ell$
$3$-cells to $U_2$ with attaching maps $\alpha _i\in \pi _2(U_2, u)$ for
$i=1,\ldots , \ell $. We calculate the cohomology jump loci $\Sigma
^i_k(U_3)\subset M_B(U_3,{\bf G}_m )$ for $i=2,3$. Note that $U_1\hookrightarrow U_3$
induces an isomorphism on $\pi _1$ so $M_B(U_3,{\bf G}_m ) = Spec ({\bf C} \Gamma )$ too.
If $L$ is a rank one local system corresponding to a nontrivial representation
$\rho : \Gamma \rightarrow {\bf G}_m$ then $H^2(U_2, L)\cong L_x^m$ and
Mayer-Vietoris gives an exact sequence
$$
0 \rightarrow H^2(U_3, L)\rightarrow L_x^m \stackrel{A(\rho )}{\rightarrow}
L_x^{\ell } \rightarrow H^3(U_3, L)\rightarrow 0.
$$
The matrix $A(\rho )$ comes from the attaching maps: we have $\pi _2(U_2)\cong
({\bf C} \Gamma )^m$ so the collection $\{ \alpha _i\}$ can be considered as an
$\ell \times m$ matrix $A$ with coefficients in ${\bf C} \Gamma$. The matrix
$A(\rho )$ is obtained by evaluating $A$ at the algebra homomorphism ${\bf C}
\Gamma \stackrel{\rho}{\rightarrow } {\bf C}$. In this case the jump loci $\Sigma
^2_k(U_3) = \Sigma ^3_{k+\ell - m}(U_3)$ are the sets of $\rho \in M_B(U_3,
\Gamma )$ where $A(\rho )$ has rank $\leq m-k$. In particular they are defined
by the ideals of $m-k$ by $m-k$ minors of $A$. Since our choice of $\alpha
_i$ and hence of $A$ is arbitrary (except that the matrix must actually have
coefficients in ${\bf Z} \Gamma $), we can get our jump loci to be any subscheme
of ${\bf G}_m ^a$ defined by equations in ${\bf Z} \Gamma$, that is any subscheme
defined over ${\bf Z}$.
We can, for example, get the jump loci to be subschemes which are not of even
complex dimension and in any case not unions of translates of subtori---this
gives constructions of many homotopy types which cannot be the homotopy types
of complex K\"ahler manifolds. We can arrange that the jump loci do not go
through the identity representation (or even any torsion point), in
particular this characteristic of the homotopy type will not be seen by
rational homotopy theory (we can insure that the cohomology with constant
coefficients and even the rational homotopy type are those of the real torus
e.g. an abelian variety of dimension $a/2$, or equally well those
of any complex subvariety with the same $\pi _1$).
Getting back to the result of Corollary \ref{subtori}, we can recover the
results of Green and Lazarsfeld by looking at the Dolbeault realization.
There
is a natural embedding $Pic ^0(X)\subset M_{Dol}(X,{\bf G}_m )$ sending a line
bundle
${\cal L}$ to the Higgs bundle $({\cal L} , 0)$. The Dolbeault cohomology of
$({\cal L} , 0)$
is just the direct sum of the $H^{i-k}(X, {\cal L} \otimes \Omega ^k_X)$. It is a
consequence of semicontinuity that the jump loci for a direct sum must contain
as irreducible components the jump loci for each of the factors. Thus the
irreducible components of $\Sigma ^i_k (Pic ^0)$ are among the irreducible
components of $\Sigma ^i_k(M_{Dol})$, and the conclusion of the corollary
implies the result of \cite{G-L}.
\numero{Deligne's construction of
$TW(M^{\rm sm})$}
In \cite{DeligneLetter}
Deligne indicated a complex analytic construction of the complex
manifold $TW(M^{\rm sm})$ (the idea is based on some properties that Hitchin
established). This is interesting because the construction given above of the
quaternionic structure on $M^{\rm sm}(X,G)$ comes from the homeomorphism
$M_{DR}(X,G) \cong M_{Dol}(X,G)$ which itself comes from the non -complex
analytic harmonic metric construction. Of course
the trivialization $TW(M^{\rm sm}(X,G))\cong M^{\rm sm}(X,G)\times {\bf P} ^1$
depends on the harmonic metric construction.
It turns out that Deligne's construction of $TW(M^{\rm sm}(X,G))$ is useful for
two other things that were not mentioned in \cite{DeligneLetter}: (1) it gives
an approach to defining the nonabelian analogue of the Hodge filtration on
$M_{DR}(X,G)$; and (2) it gives a way of compactifying $M_{DR}(X,G)$.
Both of these
were announced in \cite{NAHT} but with only brief sketches of proofs. In this
paper we will fill in the details about these two things and give some natural
extensions of these ideas. Before doing that, we review Deligne's
construction, since it has not otherwise appeared in print (to my knowledge).
An {\em antilinear morphism $T\rightarrow T'$} between two complex analytic
spaces is a morphism of ringed spaces such that the composition ${\bf C}
\rightarrow {\cal O} _{T'} \rightarrow {\cal O} _T$ is the complex conjugate of the
structural morphism ${\bf C} \rightarrow {\cal O} _T$. An {\em antilinear involution}
of $T$ is an antilinear morphism $\sigma : T\rightarrow T$ with $\sigma ^2=1$.
Let $\sigma _{{\bf P} ^1}$ denote the antilinear antipodal involution of ${\bf P} ^1$.
If $z$ is the standard linear coordinate on ${\bf A}^1$ then $\sigma (z)=
-\overline{z}^{-1}$. Let $\sigma _{{\bf G}_m}$ denote the restriction of $\sigma$
to ${\bf G}_m \subset {\bf P} ^1$.
The data of a morphism of complex analytic spaces $T\rightarrow {\bf P} ^1$
together with an antilinear involution $\sigma $ covering $\sigma _{{\bf P} ^1}$
is equivalent to the data of a morphism $T'\rightarrow {\bf A}^1$
and an antilinear involution $\sigma '$ of $T'_{{\bf G}_m}:=T'\times _{{\bf
A}^1} {\bf G}_m$. Given $T'$ and $\sigma '$,
let $\overline{T}'$ denote the complex conjugate analytic space (that is the
same ringed space but with structural morphism ${\bf C} \rightarrow {\cal O}
_{\overline{T}'}$ the complex conjugate of the structural morphism for $T'$).
The involution $\sigma '$ becomes a complex linear isomorphism
$$
T'_{{\bf G}_m} \cong \overline{T}'_{{\bf G}_m},
$$
which we can use to glue $T'$ to $\overline{T}'$ to obtain $T$. By
construction $T$ comes with an antilinear involution $\sigma$ (it comes from
the tautological antilinear morphism $T'\rightarrow \overline{T}'$ and its
inverse). One can see that ${\bf P} ^1$ is obtained from ${\bf A}^1$ and the
involution $\sigma _{{\bf G}_m}$ by the same construction, so we obtain our map
$T\rightarrow {\bf P} ^1$ compatible with involutions.
To give a holomorphic $\sigma$-invariant section $\eta :{\bf P} ^1\rightarrow T$
it
suffices to give a holomorphic section $\eta ' : {\bf A}^1 \rightarrow T'$
such that $\eta ' |_{{\bf G}_m}$ is $\sigma '$-invariant.
Hitchin noticed that the twistor space $TW(M^{\rm sm})$ comes equipped with
an action of ${\bf G}_m$ identifying the fibers over all different $\lambda \in {\bf G}_m
\subset {\bf P} ^1$ \cite{Hitchin} (note however that the twistor space of a
general hyperk\"ahler manifold doesn't
come equipped with such an action). Deligne's idea is to use this and the
remark of the previous paragraph to obtain a direct construction of
$TW(M^{\rm sm})$.
For simplicity we treat the case
$G=GL(n,{\bf C} )$ but the case of a general reductive group can be treated
directly by working with principal bundles, or indirectly using the Tannakian
formalism such as in \cite{Moduli} (note that for the constructions of moduli
spaces the indirect Tannakian method is the only one I know of).
Deligne makes the following definition.
Suppose $\lambda : S\rightarrow {\bf A}^1$ is a morphism. A {\em
$\lambda$-connection} on a vector bundle $E$ over $X\times S$ consists of an
operator
$$
\nabla : E\rightarrow E\otimes _{{\cal O}} \Omega ^1_{X\times S/S}
$$
such that $\nabla (ae) = \lambda e\otimes d(a) + a\nabla (e)$ (Leibniz
rule multiplied by $\lambda$) and such that $\nabla ^2 = 0$ as defined in the
usual way (integrability).
Note that if $\lambda = 1$ then this is the same as the usual notion of
connection, whereas if $\lambda = 0$ then this is the same as the notion of
Higgs field making $(E,\nabla )$ into a Higgs bundle \cite{Hitchin}
\cite{HBLS}.
\begin{proposition}
\label{ConstructMhod}
Fix $x\in X$.
The functor which to $\lambda : S\rightarrow {\bf A}^1$ associates the set of
triples $(E,\nabla , \beta )$ where $E$ is a vector bundle on $X\times S$,
$\nabla$ is a $\lambda$-connection on $E$ (such that the resulting Higgs
bundles over
$\lambda=0$ are semistable with vanishing rational Chern classes), and
$\beta : E|_{\{ x\} \times S}
\cong {\cal O} _S^n$ is a frame, is
representable by a scheme $R_{Hod}(X,x,GL(n)) \rightarrow {\bf A}^1$. The
group
$GL(n)$ acts on $R_{Hod}(X,x,GL(n))$ by change of frame and all points are
semistable for this action (with respect to the an appropriate linearized
bundle). The geometric-invariant theory quotient $M_{Hod}(X,GL(n))\rightarrow
{\bf A}^1$ is a universal categorical quotient. In particular the fibers of
$M_{Hod}$ over $\lambda = 0$ and $\lambda = 1$ are $M_{Dol}(X,GL(n))$ and
$M_{DR}(X,GL(n))$
respectively.
\end{proposition}
{\em Proof:}
This follows by applying the results of \cite{Moduli}
to the ring $\Lambda ^R$ defined in \cite{Moduli} p. 87.
\hfill $\Box$\vspace{.1in}
{\em Remark:}
We can define the notion of $\lambda$-connection on a principal
bundle, and obtain the corresponding statement for principal $G$-bundles.
We obtain schemes $R_{Hod}(X,x,G)$ and $M_{Hod} (X,G)$.
The construction is done by applying
the Tannakian considerations of \cite{Moduli} \S 9.
Concerning the terminology $R_{Hod}$ and $M_{Hod}$:
this reflects the fact that,
as we shall see below, these spaces incarnate the Hodge filtrations on
$R_{DR}$ and $M_{DR}$.
Let ${\cal M} _{Hod}(X,GL(n))$ (or $M_{Hod}(X,G)$) denote the stack-theoretic
quotient of $R_{Hod}(X,x,GL(n))$ by $GL(n)$ (or $R_{Hod}(X,x,G)$ by $G$).
The group ${\bf G}_m$ acts on the functor $\{ (E,\nabla , \beta )\}$ over its action
on ${\bf A}^1$: if $t\in {\bf G}_m (S)$ and $(E,\nabla , \beta )$ is a
$\lambda$-connection then $(E, t\nabla , \beta )$ is a $t\lambda$ connection.
Since $R_{Hod}(X,x,GL(n))$ represents the functor, we get an action of ${\bf G}_m$ on
$R_{Hod}(X,x,GL(n))$ covering its action on ${\bf A}^1$. Since $M_{Hod}
(X,GL(n))$ is a universal
categorical quotient, this descends to an action on $M_{Hod}(X,GL(n))$.
This action
serves to identify the fibers over any $\lambda , \lambda ' \neq 0$ in ${\bf
A}^1$---they are all isomorphic to $M_{DR}(X,GL(n))$.
The space $M_{Hod}(X,GL(n))$ will play the role of the space $T'$ in
constructing the
twistor space $T$. According to our general discussion, in order to obtain
$T$ by glueing, it suffices to have an antilinear involution $\sigma '$ of
$M_{Hod}(X,GL(n))|_{{\bf G}_m}$. As we have seen above, the action of ${\bf G}_m$ gives an
isomorphism
$$
M_{Hod}(X,GL(n))|_{{\bf G}_m} \cong M_{DR}(X,GL(n)) \times {\bf G}_m .
$$
On the other hand we have an antilinear involution $\tau $ of $M_B(X,GL(n))$
obtained
by setting $\tau (\rho )$ equal to the dual of the complex conjugate
representation (where complex conjugation is taken with respect to the real
structure $GL(n,{\bf R} )$; the dual of the complex conjugate is also the complex
conjugate with respect to the compact real form). To be totally explicit, for
$\gamma \in \pi _1(X,x)$ we set $\tau (\rho )(\gamma ):= \;\; ^t\overline{\rho
(\gamma )}^{-1}$. The complex analytic isomorphism $M_B(X,GL(n))^{\rm an}\cong
M_{DR}(X,GL(n))^{\rm an}$ given by the Riemann-Hilbert correspondence
allows us to
interpret $\tau$ as an antilinear involution of $M_{DR}(X,GL(n))$. Finally
we define
the involution $\sigma '$ of $M_{DR}(X,GL(n)) \times {\bf G}_m $ by the formula
$\sigma '
(u,\lambda )= (\tau (u), -\overline{\lambda}^{-1})$. Using $\sigma '$ and
$T'=M_{Hod}(X,GL(n))$ in the general recipe given above, we obtain a space $T$
which we
denote $M_{Del}(X,GL(n))\rightarrow {\bf P} ^1$.
The complex conjugate scheme $\overline{T'}= \overline{M_{Hod}(X,GL(n))}$
which appears above can be identified with $M_{Hod}(\overline{X}, GL(n))$.
{\em Exercise:} write down the glueing isomorphism between
$M_{Hod}(X, GL(n))$ and $M_{Hod}(\overline{X}, GL(n))$ over ${\bf G}_m
\subset {\bf A}^1$. Note that it will be analytic but not algebraic
(depending on the Riemann-Hilbert correspondence).
We note rapidly some properties of $M_{Del}(X,GL(n))$ which are immediate
consequences
of the construction. The fiber of $M_{Del}(X,GL(n))\rightarrow {\bf P} ^1$
over a point
$\lambda \in {\bf P} ^1$ (which is denoted using the coordinate system of the
first embedding of ${\bf A}^1$ which corresponds to the part concerning $X$)
is equal to $M_{Dol}(X,GL(n))^{\rm an}$ if $\lambda =0$; the fiber is
isomorphic to
$$
M_{DR}(X,GL(n))^{\rm an}\cong M_{B}(X,GL(n))^{\rm an}\cong
M_{DR}(\overline{X} ,GL(n)))^{\rm an} $$
if $\lambda \neq 0,\infty $; and the fiber is equal to
$M_{Dol}(\overline{X},GL(n))^{\rm an}$ if $\lambda = \infty$.
There is an analytic action of ${\bf G}_m$ covering the standard action on ${\bf P} ^1$
(this action is constructed by glueing the natural action over the first open
set $M_{Hod}(X,GL(n))^{\rm an}$ with the composition with $i$ of the natural
action on the second open set $M_{Hod}(\overline{X},GL(n))^{\rm an}$).
The antilinear involution $\sigma $ of $M_{Del}(X,GL(n))$ comes from the
first version
of the construction discussed above.
{\em Remark:} Let $R_{Hod}(X,x,GL(n))$ denote
the representation space constructed with reference to the basepoint $x\in X$.
We can construct,
exactly as above, an involution $\sigma$ which can be thought of as an
isomorphism between the inverse images of ${\bf G}_m \subset {\bf A}^1$ in
$R_{Hod}(X,x,GL(n))$ and $R_{Hod}(\overline{X},
\overline{x},GL(n))$.
We obtain $R_{Del}(X,x,GL(n))\rightarrow {\bf P} ^1$ by glueing
$R_{Hod}(X,x,GL(n))$ to
$R_{Hod}(\overline{X},
\overline{x},GL(n))$ using this isomorphism. The group $GL(n,{\bf C} )$ acts
analytically and on each open subset the associated moduli space
$M_{Hod}(X,GL(n))$ is a universal categorical quotient in the analytic category
(\cite{Moduli} \S 5). The glueing (along invariant open sets which are
pullbacks of open sets in the quotients) preserves this property, so
$$
R_{Del}(X,x,GL(n))\rightarrow M_{Del} (X,GL(n))
$$
is a universal categorical quotient by the action of $GL(n , {\bf C} )$ in the
analytical category. There is again an action of ${\bf G}_m$ on
$R_{Del}(X,x,GL(n))$ and the fibers are again respectively
$R_{Dol}(X,x,GL(n))^{\rm an}$, $R_B(X,x,GL(n))^{\rm an}$ and
$R_{Dol}(\overline{X},
\overline{x},GL(n))$ over $\lambda = 0$, $\lambda \neq 0,\infty$, and
$\lambda =
\infty$ in ${\bf P} ^1$.
Denote by ${\cal M} _{Del}(X,GL(n))$ the stack-theoretic quotient of
$R_{Del}(X,x,GL(n))$ by $GL(n)$; it is an analytic stack with a morphism to
${\bf P} ^1$.
We now show how a {\em harmonic bundle} defines a section
${\bf P} ^1\rightarrow M_{Del}(X,GL(n))$ which we refer to as a {\em prefered
section}. As mentioned before, it suffices to obtain a $\sigma '$-invariant
section ${\bf A}^1\rightarrow M_{Hod}(X,GL(n))$.
Suppose $P$ is a flat principal $GL(n)$-bundle. Choose a pluriharmonic
$K$-reduction $P_K$ and consider the decomposition defined previously
$$
d' = \partial + \theta ',
$$
$$
d'' = \overline{\partial} + \theta ''.
$$
For $\lambda \in {\bf A}^1$ we define a holomorphic structure
$$
\overline{\partial} _{\lambda} := \overline{\partial} + \lambda \theta '',
$$
and an operator
$$
\nabla _{\lambda} := \lambda \partial + \theta ' .
$$
We claim that $\overline{\partial} _{\lambda}$ is an integrable holomorphic structure and
$\nabla _{\lambda}$ an integrable holomorphic $\lambda$-connection on
$(P,\overline{\partial} _{\lambda})$. The equations $\overline{\partial} ^2=0$, $[\theta '' ,
\theta ''
]=0$ and $(d'')^2=0$ imply that $\overline{\partial} (\theta '' )=0$ and hence $\overline{\partial}
_{\lambda}^2=0$. Similarly, $\overline{\partial} (\theta ')=0$ and $\partial (\theta
'')=0$ and furthermore we get
$$
\overline{\partial} \partial + \partial \overline{\partial} + \theta ' \theta '' + \theta '' \theta
' = 0
$$
which gives $[\overline{\partial} _{\lambda} ,\nabla _{\lambda}]=0$; finally by the
same argument as previously $\partial (\theta ')=0$ so $\nabla _{\lambda
}^2=0$. This gives the claim. Note that at $\lambda = 0$ we recover the
Higgs bundle structure $(\overline{\partial} , \theta ')$ which we know to be
polystable with vanishing Chern classes. This construction thus gives a
section ${\bf A}^1\rightarrow M_{Hod}(X,GL(n))$. It is holomorphic in
$\lambda$
(since $\lambda$ appears linearly in the equations).
We have to check that our section is $\sigma '$-invariant over ${\bf G}_m \subset
{\bf A}^1$. This is a bit technical so feel free to skip it! A point of
$M_{Hod}(X,GL(n))$ can be represented as a quadruple $(E,\delta ' , \delta '',
\lambda )$ where $E$ is a ${\cal C} ^{\infty}$ bundle, $\lambda \in {\bf C}$, $\delta $
is an operator satisfying Leibniz' rule for $\lambda \partial$, and
$\delta ''$ is an operator satisfying Leibniz' rule for $\overline{\partial}$, such
that $(\delta ' )^2=0$, $(\delta '' )^2=0$, and $\delta ' \delta '' +
\delta '' \delta '=0$. If $\lambda \neq 0$ this corresponds to a flat
bundle $(E, \lambda ^{-1}\delta ' + \delta '' )$. The dual complex conjugate
flat bundle (corresponding to the dual of the complex conjugate representation
on $X$) is $(\overline{E}^{\ast},\overline{\delta ''}^{\ast} +\overline
{\lambda}^{-1}
\overline{\delta '}^{\ast})$ (the superscript $\; ^{\ast}$ on the operators
means the induced operators on the dual). If we take the point obtained by
multiplying this complex conjugate flat bundle by $-\overline{\lambda }^{-1}$,
we obtain the point
$$ \sigma (E,\delta ', \delta '', \lambda ) = (\overline{E}^{\ast},
-\overline{\lambda}^{-1} \overline{\delta ''} ^{\ast},
\overline{\lambda}^{-1} \overline{\delta '}^{\ast}, -\overline{\lambda}^{-1}).
$$
We have to check that this operation preserves our preserved section, which is
the collection of points of the form $(E, \lambda \partial + \theta ',
\overline{\partial} + \lambda \theta '', \lambda )$. We have an isomorphism
$\overline{E}^{\ast} \cong E$ given by the harmonic metric, and via
this isomorphism the dual complex conjugation operation has the
following effect on operators:
$$
\partial \leftrightarrow \overline{\partial} ,
$$
$$
\theta ' \leftrightarrow - \theta '
$$
(this is from the definition of $\overline{\partial} + \partial $ and $\theta '+\theta ''$
as the components parallel to and perpendicular to the unitary structure). In
view of these formulae, when we apply the operation $\sigma$ to such a point we
get
$$
\sigma (E, \lambda \partial + \theta ', \overline{\partial} + \lambda \theta '',
\lambda ) = (\overline{E}^{\ast}\cong E,
-\overline{\lambda}^{-1}\partial + \theta ', \overline{\partial}
-\overline{\lambda}^{-1} \theta '', -\overline{\lambda}^{-1} )
$$
which is indeed a point on our prefered section.
It is clear from the definition that through any point of $M_{Del}(X,GL(n))$
passes
exactly one prefered section.
The set of prefered sections gives a set-theoretic trivialization
$$
M_{Del}(X,GL(n))\cong M_B(X,GL(n))\times {\bf P} ^1.
$$
This trivialisation is in fact a homeomorphism, as can be seen by using the
techniques of \cite{Moduli} which are used in proving that
$M_{DR}(X,GL(n)) ^{\rm top} \cong M_{Dol}(X,GL(n))^{\rm top}$. This is also
verified in \cite{Fujiki}.
Let $M_{Del}^{\rm sm}(X,GL(n))$ denote the open subset of $M_{Del}(X,GL(n))$
where the
projection to ${\bf P} ^1$ is smooth. By the etale local triviality of $M_{Hod}$
(explained in \S 9 below) a point in $M_{Del}(X,GL(n))$ lies in
$M_{Del}^{\rm sm}(X,GL(n))$ if and only if it is a smooth point of the fiber
$M_{Dol}(X,GL(n))$, $M_{DR} (X,GL(n))$ or $M_{Dol}(\overline{X}, GL(n))$.
The trivialisation via
prefered sections gives
$$
M_{Del}^{\rm sm}(X,GL(n))^{\rm top} \cong
M^{\rm sm}(X,GL(n))^{\rm top} \times {\bf P} ^1.
$$
This is in fact a ${\cal C}^{\infty}$ isomorphism, as
follows from the construction of the prefered sections and the fact that
the harmonic maps or metrics vary smoothly with parameters (since they are
solutions of the appropriate kind of nonlinear elliptic equation).
\begin{theorem}
\label{twistor}
{\rm (Deligne)}
The space $M_{Del}^{\rm sm}(X,GL(n))$ with all of its structures is
analytically
isomorphic to the twistor space $TW(M^{\rm sm})$; via this isomorphism,
the prefered section trivialisations of $M_{Del}^{\rm sm}(X,GL(n))$ and
$TW(M^{\rm sm})$ coincide.
\end{theorem}
{\em Proof:}
This is actually a consequence of the properties obtained by Hitchin for his
twistor space in \cite{Hitchin}. For intrepid readers, we indicate a
self-contained calculation---partly because this also serves to show that
$(I,J,K)$ defined a quaternionic structure in the first place.
Both the twistor
space and $M_{Del}^{\rm sm}(X,GL(n))$ are ${\cal C} ^{\infty}$ isomorphic to the
product $M^{\rm sm}\times {\bf P} ^1$. This gives the isomorphism between the two.
Furthermore we know in both cases that the horizontal sections $\{ x\} \times
{\bf P} ^1$ are holomorphic, so the isomorphism is analytic in the horizontal
direction. We have to check that this isomorphism is compatible with the
complex structures in the vertical direction. Choose a tangent direction to
$M^{\rm sm}$ which we will look at first in the Dolbeault realization. The
tangent direction can be thought of as a change of operator $\overline{\partial} + \theta '
\mapsto \overline{\partial} + \theta ' + \alpha $ where $\alpha $ is an endomorphism-valued
form representing the cohomology class of the tangent vector. We may
(by gauging back if necessary) assume that the associated harmonic metric
remains fixed; the infinitesimal change $\alpha$ then induces a change of
operator $\partial + \theta '' \mapsto \partial + \theta '' + \beta $. Write
$\alpha = \alpha ' + \alpha ''$ and $\beta = \beta ' + \beta ''$ according to
type. In
terms of the isomorphism $E\cong \overline{E}^{\ast}$ the condition that the
fixed metric still relates our new operators is
$$
\beta '= \overline{\alpha ''}^{\ast}
$$
$$
\beta '' = -\overline{\alpha '}^{\ast}.
$$
One can see that if $\alpha$ is harmonic then the form $\beta$
defined by these formulas is also harmonic. If we denote by $B(\alpha )$ the
form $\beta$ defined by these formulas then $B$ becomes an endomorphism of the
space of harmonic forms. It is antilinear (that is $Bi=-iB$), and $B^2=-1$.
Thus $B$ is another complex structure which forms part of a quaternionic triple
with $i$.
The complex structure $I$ on $M_{Dol}(X,GL(n))$ corresponds to
multiplication of
$\alpha$ by $i$ (because $\alpha$ is the representative of our tangent vector
in the Dolbeault realization). The complex structure on $M_{DR}(X,GL(n))$ is
the operator on $\alpha$ which causes $\alpha + \beta = \alpha + B(\alpha )$
to
be multiplied by $i$. Thus we have the formula
$$
I(1+ B)\alpha = (1+B)J\alpha .
$$
From whence $J=IB$. This now shows that the pair $(I,J)$
form a part of a quaternionic triple, for which $B=-K$ (Theorem
\ref{quater}).
For $\lambda \in {\bf A}^1$ the change of associated
$\lambda$-connection is
$$
\lambda \partial + \theta ' +
\overline{\partial} + \lambda \theta ''\mapsto \lambda \partial + \theta ' +
\overline{\partial} + \lambda \theta ''+ \lambda \beta ' + \alpha ' + \alpha '' + \lambda
\beta '' .
$$
Thus if $I_{\lambda}$ denotes the complex structure on the fiber
of $M_{Del}(X,GL(n))$ over $\lambda \in {\bf A}^1$ then we get the formula
$$
I(1+ \lambda B) = (1+ \lambda B)I_{\lambda}.
$$
One has to be careful about what $\lambda B$ means: if $\lambda = u+ iv$ then
$\lambda B = uB + vIB$. We obtain (replacing $B$ by $-K$):
$$
I_{u+iv} = (1 -uK + vJ)^{-1} I (1 -uK + vJ).
$$
This coincides with the formula given in \S 3.
\hfill $\Box$\vspace{.1in}
{\bf Question:} Are the prefered sections the only sections which are
preserved by the involution $\sigma$?
This is certainly locally true, since the normal bundle to a prefered
section is a direct sum of ${\cal O} _{{\bf P} ^1}(1)$. In fact, locally the morphism
from the space of all sections to the product of any two distinct fibers is
an isomorphism. If we take two antipodal fibers then $\sigma$ gives an
antilinear involution of the product of the two fibers, and the prefered
sections correspond to the fixed points.
Hitchin's discussion in \cite{HitchinBourbaki} (Theorem 1) is actually a bit
unclear on this point: as written the converse in Theorem 1 would imply that
the answer is yes in general, but one can easily imagine that he meant only to
look at the real sections in the given family of sections. A glance at
\cite{HKLR} didn't resolve the problem, so I think that the answer to the
above question is not known.
An affirmative answer would mean that $(M_{Del}(X,GL(n)), \sigma )$ determines
the twistor space structure and in particular
the isomorphism $M_{DR} \cong M_{Dol}$, an interesting point since the
construction of $(M_{Del}(X,GL(n)), \sigma )$ is entirely complex analytic,
so we could bypass the nonlinear elliptic theory necessary to define the
harmonic metrics---conceptually speaking at least.
The answer to this question is `yes' for the twistor space of a
quaternionic vector space. As a consequence we obtain this property for the
moduli space of rank one representations:
\begin{theorem}
Suppose $G={\bf G}_m$. Then the prefered sections are the only $\sigma$-invariant
sections of $TW(M^{\rm sm})\rightarrow {\bf P} ^1$.
\end{theorem}
{\em Proof:}
The moduli space is a quotient $M = H^1(X, {\bf C} )/H^1(X, {\bf Z} )$ (as can be
seen by a flat version of the exponential exact sequence). The quaternionic
structure is the quotient by the lattice of a linear quaternionic structure on
$H^1(X, {\bf C} )$. Thus the twistor space is the quotient
$$
TW(M)= TW(H^1(X, {\bf C} )) /H^1(X, {\bf Z} ).
$$
Since ${\bf P} ^1$ is simply connected, the sections from ${\bf P} ^1$ to $TW(M)$ are
just projections of sections from ${\bf P} ^1$ to $TW(H^1(X, {\bf C} )$. The
involution $\sigma$ acts compatibly on everything. From the theory of the
twistor space for quaternionic vector spaces (which is just a bundle which is
a direct sum of ${\cal O} _{{\bf P} ^1}(1)$) we see that through any point of
$TW(H^1(X, {\bf C} ))$ there is a unique $\sigma$-invariant section; this gives
the same result on $TW(M)$ which implies the theorem.
\hfill $\Box$\vspace{.1in}
\numero{The Hodge filtration}
In \cite{Hitchin} Hitchin introduced an $S^1$ action on the moduli space of
representations. This was taken up again in \cite{HBLS} as a ${\bf C}
^{\ast}$-action. This action is defined via the isomorphism $M_B^{\rm top}
\cong M_{Dol}^{\rm top}$: $t\in {\bf C} ^{\ast}$ sends the Higgs bundle $(E,\theta
)$ to $(E, t\theta )$.
The ${\bf C} ^{\ast}$ or $S^1$ actions are the analogue in
nonabelian Hodge theory of the Hodge decomposition of cohomology coming from
harmonic forms. In the usual case, the Hodge decomposition does not vary
holomorphically with parameters, because it includes complex conjugate
information. Similarly, if the variety is defined over a small field, there is
no particular reason for the Hodge decomposition to be defined over a small
field. In order to obtain something which comes from algebraic geometry and
thus has the properties of holomorphic variation, and compatibility with fields
of definition, one looks at the {\em Hodge filtration} of the algebraic de Rham
cohomology. We will define and investigate the analogue for nonabelian
cohomology.
Begin with the following observation. Suppose $V$ is a vector space with
complete decreasing filtration $F^{\cdot}$ (complete means that the
filtration starts with $V$ and ends with $\{ 0\}$). Define a locally free
sheaf $\xi (V,F)$ over ${\bf A}^1$ with action of ${\bf G}_m$ as
follows. Let $j: {\bf G}_m \rightarrow {\bf A}^1$ denote the inclusion. Then $\xi
(V,F)$ is the subsheaf of $j_{\ast}(V\otimes {\cal O} _{{\bf G}_m})$ generated by the
sections of the form $z^{-p}v_p$ for $v_p \in F^pV$ (where $z$ denotes the
coordinate on ${\bf A}^1$). Conversely if $W$ is a locally free sheaf on ${\bf
A}^1$ with action of ${\bf G}_m$ then we obtain a decreasing filtration $F$ on the
fiber $W_1$ of $W$ over $1\in {\bf A}^1$ by looking at orders of poles of
${\bf G}_m$-invariant sections. These constructions are inverses.
The locally free sheaf $\xi (V,F)$ is the tilde of the {\em Rees module} of
$(V,F)$.
If $(V,F)$ is a filtered vector space then the fiber $\xi (V,F) _0$ over $0\in
{\bf A}^1$ is naturally identified with the associated-graded $\bigoplus
F^p/F^{p+1}$.
Let $\Sigma$ be a sheaf of sets on the big
etale site ${\cal X}$. We define a {\em filtration ${\cal F}$ of $\Sigma$} to be a sheaf
of sets with morphism $\Sigma _{{\cal F}}\rightarrow {\bf
A}^1$ together with action of ${\bf G}_m$ (here an action means
a morphism $\Sigma _{{\cal F}} \times {\bf G}_m \rightarrow \Sigma _{{\cal F}}$ satisfying
the usual
axioms) and an isomorphism $\Sigma _{{\cal F}} \times _{{\bf A}^1} \{ 1\} \cong
\Sigma$. Note that $\Sigma _{{\cal F}}$ may be interpreted as a sheaf on ${\cal X}
/{\bf A}^1$, and using this interpretation we can make a similar definition
for sheaves of objects of any appropriate category. We obtain a similar
definition for stacks (or homotopy-sheaves of spaces, or even $n$-stacks or
$\infty$-stacks once those are defined).
Normally we will be interested in the case where $\Sigma$ is represented by a
scheme or eventually an algebraic stack, in this case we expect $\Sigma
_{{\cal F}}$ to be a scheme or at least an algebraic stack.
{\em Caution:} As we will see in one of our main examples in the section on
formal categories, the notion of filtration of a sheaf of sets in the context
of stacks is different from the notion of filtration in the context of sets,
in other words we might have $\Sigma _{{\cal F}}$ a stack whereas $\Sigma$ is a set.
Now, getting back to our main discussion, in terms of this definition the
space $M_{Hod}\rightarrow {\bf A}^1$ with action of ${\bf G}_m$ is a filtration on
$M_{DR}$. We call this the {\em
Hodge filtration} on $M_{DR}$. Similarly $R_{Hod}$ is the Hodge filtration on
$R_{DR}$. And most properly speaking, it is ${\cal M} _{Hod}$ which
provides the Hodge filtration on the nonabelian cohomology stack ${\cal M} _{DR}$.
In the next section we will see how this filtration is compatible with the
usual Hodge filtration on the nilpotent completion of the fundamental group.
The idea of interpreting the Hodge filtration in this way is very closely
related to the interpretation of Deninger \cite{Deninger}. Essentially he
looks at a derivation expressing the infinitesimal action instead of the full
action of ${\bf G}_m$. In turn he refers to Fontaine \cite{Fontaine} (1979) for a
reworking of Hodge theory from this point of view (which is what led Fontaine
to all of his rings such as $B_+^{\rm cris}$\ldots I guess\ldots ).
A word about purity. If $(V,F,\overline{F})$ is a vector space with two
filtrations (which can be complex conjugates with respect to a real structure,
for example) then $\xi (V, F, \overline{F})$ is a vector bundle over ${\bf P}
^1$ obtained by glueing $\xi (V,F)$ to $\xi (V,\overline{F})$ much as in \S 4.
The two filtrations define a Hodge decomposition pure of weight $w$ if and only
if the vector bundle $\xi (V, F, \overline{F})$ is a direct sum of copies of
${\cal O} _{{\bf P} ^1}(w)$.
The construction $M_{Del}$
is in effect the nonabelian
analogue of the construction $\xi (V,F,\overline{F})$ where $F$ is replaced by
the ``filtration'' $M_{Hod}$. The fact that this construction gives the
twistor space for a quaternionic structure is equivalent to the statement that
the normal bundle along any prefered section is a direct sum of ${\cal O} _{{\bf P}
^1}(1)$ (cf \cite{HitchinBourbaki}, \cite{HKLR}). This can be interpreted as
purity of weight one. I don't know how far one can go toward making this
analogy more precise than it is.
\numero{The nilpotent completion of $\pi _1$ and representations near the
identity}
We will justify our definition of $M_{Hod}$ as the Hodge filtration by making
the connection with the usual Hodge filtration on the nilpotent completion of
the fundamental group \cite{Morgan} \cite{HainKth}. For simplicity we work
with
the group algebra ${\bf C} \pi _1^{\wedge }$ (completed at the augmentation
ideal). The mixed Hodge structure on $\pi _1$ is usually defined via the mixed
Hodge structure on
${\bf C} \pi
_1^{\wedge }$ and the inclusion of the Lie algebra corresponding to $\pi _1$
into this group algebra.
We first consider the relationship between the completed group algebra and the
completions of the spaces of representations at the identity.
Suppose $A$ is an augmented ${\bf C}$-algebra which is complete with respect to
the augmentation ideal $J_A$. Let $R(A,n)$ denote the functor of artinian
local
${\bf C}$-algebras $B$ defined by setting
$$
R(A,n)(B):= Hom ^{\rm aug}(A, M_n(B))
$$
where $Hom ^{\rm aug}$ denotes the set of algebra homomorphisms
sending $J_A$ to the ideal $M_n({\bf m} _B)$ (${\bf m} _B$ denotes the maximal
ideal of $B$).
If $A={\bf C} \Gamma ^{\wedge }$ is the completion of the group algebra of a
finitely presented group $\Gamma$ then $R(A,n)$ is pro-representable by the
completion at the identity representation $R(\Gamma , GL(n))^{\wedge}$ of the
space of representations of $\Gamma$ in $GL(n)$ (there are probably more
abstract conditions on $A$ which could be used to insure
representability but we don't need those here).
Conversely let ${\cal C}$ denote the category of algebras which are direct
produts of
algebras of the form $M_n({\bf C} )$. Suppose $\Upsilon : {\cal C} \rightarrow
\mbox{ForSch}$ is a functor from ${\cal C}$ to the category of formal schemes,
compatible with products (so we can think of $\Upsilon$ as a collection of
formal schemes $\Upsilon _n$ together with morphisms of functoriality
corresponding to morphisms of products of algebras in ${\cal C}$). Then we define
${\cal A} (\Upsilon )$ to be the algebra of natural transformations $\Upsilon
\rightarrow 1_{{\cal C} }$. The elements of ${\cal A}(\Upsilon )$ are functions $a$
which for each $n$ associate a section $a_n: \Upsilon _n\rightarrow M_n({\bf C} )$
with the $a_n$ compatible with morphisms of products of objects of ${\cal C} $.
\begin{lemma}
\label{Algs}
Suppose $A={\bf C} \Gamma ^{\wedge}$ is the completion of the group algebra of a
finitely presented group. Then the $R(A,n)$ give a functor $R(A,\cdot ):{\cal C}
\rightarrow \mbox{ForSch}$ and we can recover $A$ by the construction of the
previous paragraph:
$$
A={\cal A}(R(A,\cdot )).
$$
\end{lemma}
\hfill $\Box$\vspace{.1in}
{\em Remark:}
The morphisms of functoriality defining $R(A,\cdot )$ can be obtained from the
morphisms of functoriality of $R(\Gamma , GL(n))$ for morphisms between
products of groups $GL(n)$.
Now we investigate the Hodge filtrations. Suppose $X$ is a smooth projective
variety. We obtain a family of formal completions $R_{Hod}(X,x,GL(n))^{\wedge}
\rightarrow {\bf A}^1$, with an action of ${\bf G}_m$. The technique of Goldman
and Millson used in (\cite{Moduli} \S 10) to give an isomorphism
$R_{Dol}(X,x,GL(n))^{\wedge}\cong R_{DR}(X,x,GL(n))^{\wedge}$ actually gives a
trivialization $$
R_{Hod}(X,x,GL(n))^{\wedge}
\cong R_{Dol}(X,x,GL(n))^{\wedge}\times ^{\wedge} {\bf A}^1,
$$
with the action of ${\bf G}_m$ coming from the action defined in \cite{HBLS} on
$R_{Dol}(X,x,GL(n))^{\wedge}$ and the standard action on ${\bf A}^1$.
We discuss this further in \S 9 below.
We obtain a functor $R_{Hod}(X,x,\cdot ): {\cal C} \rightarrow \mbox{ForSch}/{\bf
A}^1$ which we think of as a family of functors parametrized by ${\bf A}^1$
(with action of ${\bf G}_m$). Because of the trivialization we can apply the
previous lemma. This family of functors gives rise to a completed algebra
${\cal A}$ over ${\bf A}^1$, by a relative version of the construction of
Lemma \ref{Algs} (which poses no problem since everything is a product).
Conversely starting from ${\cal A}$ we get back the
$R_{Hod}(X,x,GL(n))^{\wedge}$. Finally, the fiber ${\cal A}_1$ over $1\in {\bf
A}^1$ is isomorphic to the completed group algebra ${\bf C} \pi _1(X,x)^{\wedge}$
again by the above discussion. This family of algebras together with
${\bf G}_m$-action (which as we have seen is equivalent to the data of the $R_{Hod}$
functorially in $n$) corresponds to a filtration of ${\bf C} \pi
_1(X,x)^{\wedge}$. We claim that this filtration is the Hodge filtration of
Morgan-Hain \cite{Morgan} \cite{HainKth}.
To see this, note a consequence of the trivializations and ${\bf G}_m$-actions
in the above discussion, that the filtration on ${\bf C} \pi _1(X,x)^{\wedge}$
corresponding to ${\cal A}$ is just the filtration associated to the grading
given by the ${\bf G}_m$-action. This ${\bf G}_m$-action is that which was defined in
\cite{HBLS}. Finally, in \S\S 5-6 of \cite{HBLS} it was verified that this
${\bf G}_m$-action gives rise to the Hodge filtration of Morgan-Hain.
To sum up, starting with the completed group algebra
${\bf C} \pi _1(X,x)^{\wedge}$ and the Morgan-Hain Hodge filtration $F$ we can
form the family of algebras ${\cal A}=\xi ( {\bf C} \pi _1(X,x)^{\wedge},F)$ over
${\bf A}^1$ with ${\bf G}_m$-action; then the family of completed representation
spaces associated to this family of algebras is isomorphic (together with
${\bf G}_m$-action) to the completion $R_{Hod}(X,x,GL(n))^{\wedge}$ along the
identity
section. In the other direction, the data of the completions
$R_{Hod}(X,x,GL(n))^{\wedge}$ functorially in $GL(n)$ serve to define (via
Lemma \ref{Algs}) a family of completed algebras ${\cal A}$ over ${\bf A}^1$,
again with ${\bf G}_m$-action and isomorphism
${\cal A}_1\cong {\bf C} \pi _1(X,x)^{\wedge}$, and this family yields the
Morgan-Hain Hodge filtration on ${\bf C} \pi _1(X,x)^{\wedge}$ by reversing the
construction $\xi$.
Thus the completion of our Hodge filtration at the identity representation
corresponds to the Hodge filtration on the nilpotent completion of the
fundamental group. The whole Hodge filtration $R_{Hod}(X,x,G)$ or
$M_{Hod}(X,G)$ should be thought of as an analytic continuation of the Hodge
filtration on the nilpotent completion.
It might be interesting to try to express the existence of this analytic
continuation in terms of estimates on the mixed Hodge structure on
${\bf C} \pi _1(X,x)^{\wedge}$ in the spirit of Hadamard's technique
\cite{Hadamard}.
\subnumero{Writing formulas}
We can combine what we know so far to sketch a method which should allow,
in principle, to write down the local equations for the correspondence
$M_{Dol}\cong M_{DR}$ (and hence for the quaternionic structure) near a
complex variation of Hodge structure in $M^{\rm sm}$.
The method sketched above should work equally well along any
prefered section coming from a complex variation of Hodge structure $\rho$.
The Hodge filtration on the relative Malcev completion \cite{HainMalcev} should
give a Hodge filtration $F$ on the complete local ring $\widehat{{\cal O}} _{M_{DR},
\rho}$; then taking $\xi (\widehat{{\cal O}} _{M_{DR},
\rho}, F)$ we get a family of complete local rings over ${\bf A}^1$, and
taking a formal spectrum we get a formal scheme along ${\bf A}^1$. The real
structure or at least the invariant indefinite hermitian form underlying the
variation $\rho$ should give an involution allowing us to glue the formal
scheme with itself to get a formal scheme along ${\bf P} ^1$. We should get back in
this way the formal completion of $M_{Del}$ along the prefered section.
The
normal bundle of $M_{Del}$ along a prefered section is a direct sum of ${\cal O}
_{{\bf P} ^1}(1)$'s, so the space of sections near the given section (which has a
structure of formal scheme in this case) will map isomorphically to the product
of any two fibers. Taking two antipodal fibers (for example the fibers over
$0$ and $\infty$) we obtain explicitly the involution on the space of sections
and the prefered sections are those which are invariant. Finally looking at
the isomorphism from the space of sections to the product of the formal
completions of $M_{Dol}$ and $M_{DR}$, the space of invariant sections gives
the graph of the real analytic isomorphism $M_{Dol}\cong M_{DR}$.
One can imagine following out this entire construction explicitly to obtain
the Taylor series for the isomorphism $M_{Dol}\cong M_{DR}$ near the point
$\rho$. The only ingredients are the Hodge filtration and the real structure
(and of course an analysis of the space of sections of our formal scheme, but
this is an algebraic question).
One can see just from the existence of this method that the algebraically
closed field generated by the coefficients of the Hodge filtration on the
relative Malcev completion (and their complex conjugates) will contain the
coefficients of the power series for the isomorphism $M_{Dol}\cong M_{DR}$ and
hence for the power series of the quaternionic structure.
I have not checked the details of this construction any more than what is
written above.
\numero{Formal categories}
One of the main properties of the Hodge filtration on usual abelian cohomology
is Griffiths transversality. This is a property of the variation of the Hodge
filtration with respect to the Gauss-Manin connection which arises from a
smooth family of varieties. We would like to obtain a similar property for
our nonabelian Hodge filtration. Let's first look at how to interpret the
usual Griffiths transversality in terms of the construction $\xi$.
Suppose
$S$ is a smooth variety and $V$ is a vector bundle with integrable connection
$\nabla$. Suppose $F^{\cdot}$ is a decreasing filtration of $V$ by subbundles.
Then $F^{\cdot}$ satisfies the Griffiths transversality condition $\nabla F^p
\subset F^{p-1}\otimes \Omega ^1_S$ if and only if the action of
$T(S)$ on $V$ extends to a ${\bf G}_m$-invariant action of the sheaf
$T(S\times {\bf
A}^1/{\bf A}^1) (-S\times \{ 0\} )$ (of
relative tangent vector fields vanishing to order one along $S\times \{ 0\}$)
on $\xi (V, F^{\cdot})$. This can be seen by calculating directly with the
definition of $\xi (V, F)$ (cf Lemma \ref{VBonXHod} below).
In the nonabelian context suppose $X\rightarrow S$ is a smooth projective
morphism. We have a family $M_{DR}(X/S,G)$ of moduli spaces over $S$ and we
would like our ``Griffiths transversality'' to say that the lifting of vector
fields on $S$ to vector fields on $M_{DR}(X/S,G)$ given by the Gauss-Manin
connection, extends to a ${\bf G}_m$-invariant lifting of sections of $T(S\times {\bf
A}^1/{\bf A}^1) (-S\times \{ 0\} )$ to vector fields on $M_{Hod}(X/S, G)$.
The difficulty in making this precise is that the Gauss-Manin
connection can no longer be interpreted in terms of vector fields if
$M_{DR}(X/S,G)$ is not smooth---so the above interpretation makes sense only on
the smooth points. The calculations in terms of vector fields are also
difficult
to follow through. To remedy these problems we introduce the point of view of
formal categories.
Recall that the Gauss-Manin connection on $M_{DR}(X/S, G)$ is an isomorphism
$$
p_1^{\ast}M_{DR}(X/S, G)| _{(S\times S)^{\wedge}} \cong
p_2^{\ast}M_{DR}(X/S, G)| _{(S\times S)^{\wedge}}
$$
where $p_1, p_2: S\times S\rightarrow S$ are the projections and
$(S\times S)^{\wedge}$ is the formal completion of the diagonal. If we set
$N:=(S\times S)^{\wedge}$ then the pair $(S,N)$ has a structure of category in
the category of formal schemes. The notion of formal category is a
generalization of this example. It provides a general framework for operations
on families of things over $S$.
A {\em formal category} is a pair $(X,N)$ consisting of a scheme $X$ and a
formal scheme $M$ mapping to $X\times X$, together with a structure of
category, that is morphisms $N\times _XN\rightarrow N$ giving composition and
$X\rightarrow N$ giving the identity, subject to the usual axioms for a
category. A formal category gives in a natural way a presheaf of
categories on $Sch/{\bf C} $. A {\em formal groupoid} is a formal category such
that
the values of the associated presheaf are groupoids. We say that a formal
category is {\em of smooth type} if $X$ is smooth,
the underlying scheme of $N$ is the scheme $X$ (via the identity morphism), and
$N$ is formally smooth.
Let $X_N$ denote the stack over $Sch/{\bf C}$ associated to the presheaf of
groupoids given by $(X,N)$. We have a morphism $p:X\rightarrow X_N$.
Note that $N$ represents the functor $X\times _{X_N}X$, so we can recover
$(X,N)$ from the stack $X_N$ with its morphism $X\rightarrow X_N$. In
practice we confuse the notions (and notations) of formal groupoid $(X,N)$ and
associated stack $X_N$.
Suppose $(X,N)$ is a formal groupoid of smooth type. Note that the
structure sheaf ${\cal O}$ of $Scxh/{\bf C}$ restricts to a sheaf of
rings which we also denote by ${\cal O}$ on $Sch /X_N$. There is a complex of
${\cal O}$-modules over $X_N$ which we denote by $p_{\ast}\Omega ^{\cdot}_{X/X_N}$
with differential denoted $d$, giving a resolution $$ {\cal O} \rightarrow
p_{\ast}\Omega ^{0}_{X/X_N}\rightarrow p_{\ast}\Omega ^{1}_{X/X_N}\rightarrow
\ldots \rightarrow p_{\ast}\Omega ^{n}_{X/X_N}\rightarrow 0
$$
(here $n=dim(N/X)$ is the relative (formal) dimension of the formally smooth
scheme $N$ over $X$ via either of the projections).
The notation $p_{\ast}\Omega ^{\cdot}_{X/X_N}$ is justified by
the fact that each component of this resolution is actually the direct image
of a locally free sheaf $\Omega ^i_{X/X_N}$ on $X$. This locally free sheaf
comes from $\Omega ^i_{N/X}$ by descent from $N=X\times _{X_N}X$ to $X$. Note
that the differential is in general a differential operator (of first order)
between these component sheaves, so it becomes a morphism only over $X_N$.
A {\em local system $V$ on $X_N$} is a sheaf on $Sch /X_N$ which locally
isomorphic to ${\cal O} ^a$. This is equivalent to a vector bundle $V_X$ on $X$
together with {\em $N$-connection} $\varphi : V_X\rightarrow V_X \otimes _{{\cal O}
_X} \Omega ^1_{X/X_N}$ satisfying an integrability condition $\varphi ^2=0$.
We obtain
the resolution $p_{\ast}\Omega ^{\cdot}_{X/X_N}\otimes _{{\cal O}} V$ of $V$,
which we can use to calculate the cohomology of
$V$ over $X_N$. Note that the morphism $p$ is cohomologically trivial for
coherent sheaves on $X$. So the cohomology of $V$ may be calculated as the
hypercohomology of the complex $\Omega ^{i}_{X/X_N}\otimes _{\cal O} V$ of Zariski
(or etale) sheaves on $X$; in particular there is a spectral sequence
$$
H^i(X, \Omega ^j_{X/X_N}\otimes _{{\cal O}}V)\Rightarrow H^{i+j}(X_N, V).
$$
The cohomology groups are finite dimensional vector spaces if $X$ is
proper.
To a formal category $(X,N)$ of smooth type we can associate a split almost
polynomial sheaf of rings of differential operators $\Lambda _N$
\cite{Illusie} \cite{Moduli}. It is the sheaf of rings associated to the
differentials in the above complex. Note that $p_{\ast}({\cal O} _N )$ is naturally
a projective limit of locally free sheaves on $X$. We can construct $\Lambda
_N$ as the continuous dual, which is a union of locally free sheaves. The ring
structure is dual to the cogebra structure
$$
p_{\ast}({\cal O} _N )\rightarrow p_{\ast}({\cal O} _N )\otimes _{{\cal O} _X}p_{\ast}({\cal O} _N
)
$$
which itself comes from the composition morphism $N \times _XN\rightarrow N$.
A local
system $V$ on $X_N$ is the same thing as a $\Lambda _N$-module; the underlying
${\cal O}_X$-module is $V_X$ and the $\Lambda _N$-module structure is given by
$\varphi$.
A {\em principal $G$-bundle} on $X_N$ means a $G$-torsor on the stack $X_N$.
This is the same thing as a principal $G$-bundle $P$ on $X$ together with
an isomorphism $\varphi :s^{\ast}(P)\cong b^{\ast}(P)$ on $N$ (where
$s,b:N\rightarrow X$ are the two tautological morphisms), such that
$\varphi$ satisfies the appropriate cocycle condition on $N\times _XN$.
There is a notion of semistability for local systems over $X_N$ which is
analogous to the usual notion: we can define a notion of coherent sheaf over
$X_N$ (a coherent sheaf on $X$ with descent data to $X_N$) and a bundle over
$X_N$ is semistable if for every $X_N$-subsheaf, the normalized Hilbert
polynomial is less than or equal to that of the original object. Again as usual
we can define the notion of semistability of a principal $G$-bundle on $X_N$.
Lacking a direct proof of the conservation of semistability by tensor product
in the case of local systems over a general formal category (this is a good
question for further research), we put in the definition here that the local
systems associated to all representations of $G$ should be semistable.
When considering a relative situation $X_N\rightarrow S$, semistability means
semistability on each fiber $(X_N)_s$ (it is an open condition on the
base $S$).
Because, in all of the examples we interested in in this paper, it is
necessary to include a condition of vanishing rational Chern classes when
defining the moduli spaces, we put this directly into the definition in the
formal category setting. Of course, for formal groupoids different from our
examples, this condition may not necessarily be a sensible one; and even in our
examples, it may also be interesting to consider other components of the
moduli spaces. One could make the same definitions and obtain moduli spaces
without this condition, but we include the condition here for simplicity of
notation.
Suppose $(X,N)\rightarrow S$ is a morphism from a formal groupoid of smooth
type to a base scheme $S$, such that $X$ is smooth and proper over $S$ and $N$
is formally smooth over $S$. Suppose $x: S\rightarrow X$ is a section. Define
the functor ${\cal R} (X_N/S, x, G)$ which to an $S$-scheme $S'$ associates $\{
(P,\varphi , \beta )\}$ where $(P, \varphi )$ is a principal
$G$-bundle over $X_N\times _SS'$, semistable and with vanishing rational Chern
classes relative to $S$'; and $\beta : x^{\ast}(P) \cong G\times S'$ is a
framing along the section $x$.
\begin{theorem}
\label{ConstructMN}
The functor ${\cal R} (X_N/S, x,
G)$ is representable by a scheme which we denote by $R(X_N/S, x, G)\rightarrow
S$. Furthermore all points are semistable for the action of $G$ so a universal
categorical quotient $M(X_N/S,G)= R(X_N/S,x, G)// G$ exists.
\end{theorem}
This theorem follows from the interpretation of local systems over $X_N$ as
$\Lambda _N$-modules \cite{Moduli}; the construction of the moduli and
representation schemes for $\Lambda _N$-modules; and from the tannakian point
of view used in \cite{Moduli} \S 9.
\hfill $\Box$\vspace{.1in}
We denote the {\em stack quotient} by
$$
{\cal M} (X_N/S,G):= R(X_N/S,x, G)/ G.
$$ This is the first
relative nonabelian cohomology of $X_N/S$ with coefficients in $G$.
It is an algebraic stack.
\subnumero{The basic examples}
The main example (which we used to introduce this section and which you meet in
most treatments of the subject---cf \cite{Berthelot} for example) is the formal
groupoid obtained by setting $N:= (X\times X)^{\wedge}$ (completion along the
diagonal). We denote this formal groupoid (or the associated stack) by
$X_{DR}$. In this stack there is at most one morphism between any pair of
objects, so the stack is equivalent to a sheaf of sets which we also denote
$X_{DR}$. The sheaf of sets is the quotient of $X$ by the equivalence relation
$N$: heuristically we identify any two points which are infinitesimallly close
together, and it is this infinitesimal glue which makes it so that $X_{DR}$
actually reflects the topology of the underlying usual space. More precisely
if $S$ is any scheme over ${\bf C}$ then the $S$-valued points of $X_{DR}$ are
the $S$-valued points of $X$ modulo the relation that two
points are equivalent if their restrictions to the underlying reduced scheme
$S^{\rm red}$ are the same; except that we have to divide by this
equivalence relation and then sheafify. Since $X$ is smooth, any $S^{\rm
red}$-valued point extends, locally on $S$, to an $S$-valued point. Thus after
taking the quotient and sheafifying the result is simply that
$X_{DR}(S)=X(S^{\rm red})$.
The sheaf of rings of differential operators associated to the formal groupoid
$X_{DR}$ is just the full ring $\Lambda _{DR}$ of differential operators on
$X$ \cite{Moduli}. A principal bundle on $X_{DR}$ is
just a principal bundle on $X$ with integrable connection, and a vector bundle
or local system over $X_{DR}$ is just a vector bundle on $X$ with integrable
connection.
We recover
$$
M(X_{DR}, G)=M_{DR}(X,G)
$$
and similarly for the representation spaces and stacks
$$
R(X_{DR},x, G)=R_{DR}(X,x,G), \;\;\;\; {\cal M} (X_{DR}, G)={\cal M} _{DR}(X,G).
$$
The cohomology of $X_{DR}$ with coefficients in a local system is just the
algebraic de Rham cohomology of $X$ with coefficients in the corresponding
vector bundle with integrable connection.
We now define a formal groupoid $X_{Dol}$ which gives rise to the Dolbeault
theory in the same way as $X_{DR}$ gave rise to the de Rham theory.
In this formal groupoid
the object object is $X$ and the morphism object is the formal completion of
the
zero section in the tangent bundle of $X$, lying over the diagonal in $X\times
X$. A principal bundle on $X_{Dol}$ is just a principal Higgs bundle; a local
system is a Higgs bundle; the cohomology of a local system is the Dolbeault
cohomology; and the associated sheaf of rings of differential operators is the
ring $\Lambda _{Dol}$ defined in \cite{Moduli}.
We recover
$$
M(X_{Dol}, G)=M_{Dol}(X,G)
$$
and similarly for the representation spaces and stacks.
Now we define a formal groupoid $X_{Hod} \rightarrow {\bf A}^1$ which serves
as a deformation from $X_{DR}$ to $X_{Dol}$ and from which we can recover the
moduli spaces $M_{Hod} (X, G)$. It is the stack associated to the
realisation of the nerve of the presheaf of groupoids given by a formal
groupoid which we denote by $\widetilde{X}_{Hod} \rightarrow {\bf A}^1$. The
object object is $$ Ob(\widetilde{X}_{Hod}):=X\times {\bf A}^1.
$$
Let $Y$ be the complement of the strict transform of $X\times X\times \{ 0\}$
in
the blow-up of $X\times X\times {\bf A}^1$ along $\Delta (X) \times \{ 0\}$.
(Here $\Delta (X)$ is the diagonal.) There is a unique composition
$$
Y\times _{X\times {\bf A}^1}Y \rightarrow Y
$$
compatible with the trivial composition
$$
(X\times X\times {\bf A}^1)\times _{X\times {\bf A}^1}
(X\times X\times {\bf A}^1)
$$
via the morphism $Y\rightarrow X\times X\times {\bf A}^1$.
There is a morphism $\Delta ':X \times {\bf A}^1 \rightarrow Y$ covering the
inclusion $\Delta : X\rightarrow X\times X\times {\bf A}^1$. This section
provides an identity for the above composition---in particular, setting the
morphism object equal to $Y$ would define a category. As with the case of
$X_{DR}$ itself, we take the formal completion of this morphism object: the
morphism object $Mor( \widetilde{X}_{Hod })$ is defined to be the formal
completion of $Y$ along $\Delta '(X\times {\bf A}^1)$.
The formal groupoid defined in this way
is a groupoid;
it maps to ${\bf A}^1$;
it has fiber over $\{ 0\}$ equal to the formal groupoid defining $X_{Dol}$;
and its fiber over $\{ t\}$ for any $t\neq 0$ is equal to the formal groupoid
defining $X_{DR}$.
The general construction of Theorem \ref{ConstructMN} gives back for $X_{Hod}$
over ${\bf A}^1$ the result of Proposition \ref{ConstructMhod}:
$$
M(X_{Hod}/{\bf A}^1, G)= M_{Hod}(X,G)
$$
and similarly for the representation space and moduli stack
$$
R(X_{Hod}/{\bf A}^1,x, G)=R_{Hod}(X,x,G), \;\;\;\; {\cal M} (X_{Hod}/{\bf A}^1,
G)={\cal M} _{Hod}(X,G).
$$
\begin{lemma}
\label{VBonXHod}
In the case $G=GL(n)$ a section of the moduli stack
${\bf A}^1\rightarrow {\cal M} (X_{Hod}/{\bf A}^1,GL(n))$ preserved by ${\bf G}_m$ (or
more precisely with action of ${\bf G}_m$ specified) corresponds to a
vector bundle with filtration satisfying Griffiths transversality. The
relative cohomology of such a family of local systems is just the sheaf over
${\bf A}^1$ with action of ${\bf G}_m$ corresponding to the induced filtration on the
cohomology of the local system.
\end{lemma}
{\em Proof:}
A section of the moduli stack with action of ${\bf G}_m$ is just a vector bundle on
$X_{Hod}$ with action of ${\bf G}_m$. In particular we have a vector bundle on
the underlying scheme
$X\times {\bf A}^1$ together with action of ${\bf G}_m$; by a relative version of
the inverse of $\xi$ this corresponds to a bundle $V$
on $X$ with filtration by subbundles $F^p$ such that the associated-graded is a
bundle. The descent data to $X_{Hod}$ are determined by the descent data over
${\bf G}_m \subset {\bf A}^1$, and by the ${\bf G}_m$-invariance of our section, these
are determined by the descent data over $1\in {\bf A}^1$, which is to say an
integrable connection on $V$. The original bundle over $X\times {\bf A}^1$
is the locally free sheaf $\xi (V,F)=\sum t^{-p} F^p$. The statement that the
connection extends to descent data for this bundle down to $X_{Hod}$ is
equivalent to the condition
$$
t\nabla (\sum t^{-p} F^p)\subset (\sum t^{-p}
F^p)\otimes \Omega ^1_X,
$$
which translates to $\nabla F^p \subset
F^{p-1}\otimes \Omega ^1_X$---the Griffiths transversality condition.
\hfill $\Box$\vspace{.1in}
\numero{The Gauss-Manin connection}
Suppose $X\rightarrow S$ is a smooth projective morphism. Then we obtain
$$
M_{DR}(X/S, G)\rightarrow S,
$$
a family whose fiber over $s\in S$ is $M_{DR}(X_s,G)$. This family has an
algebraic integrable connection \cite{NAHT} \cite{Moduli}. We can interpret
the
connection in terms of formal categories in the following way (this is a
simple variant of the crystalline interpretation of \cite{Moduli}). We have a
morphism $X_{DR}\rightarrow S_{DR}$ and the fiber product
$X_{DR}\times _{S_{DR}}S$ has a structure of smooth formal groupoid over $S$,
which we call $X_{DR/S}$. The morphism space is the formal completion of
the diagonal in $X\times _SX$. The moduli stack ${\cal M} _{DR}(X/S)$ is just the
nonabelian cohomology of $X_{DR/S}$ relative to $S$ with coefficients in $G$,
or in our previous notations $$
{\cal M} _{DR}(X/S, G) = {\cal M} (X_{DR/S}/S, G).
$$
The same holds for the moduli spaces and representation spaces.
But we could equally well take the
nonabelian cohomology of $X_{DR}$ relative to $S_{DR}$. We obtain a stack over
$S_{DR}$ which, when pulled back to $S$, gives ${\cal M} _{DR}(X/S, G)$. To put
this
another way, we get descent data for ${\cal M} _{DR}(X/S, G)$ from $S$ down to
$S_{DR}$. This is exactly the data of an integrable connection which is
the nonabelian version of the Gauss-Manin connection.
One can see that the associated analytic connection on the analytic family is
the same as that induced by the fact that (locally over the base) all of the
fibers of ${\cal M} _{DR}(X/S)$ are of the form ${\cal M} _B(\Gamma )$ for $\Gamma $ the
fundamental group of the fiber (\cite{Moduli}, Theorem 8.6).
In abelian Hodge theory there are two principal results about the Gauss-Manin
connection: Griffiths transversality with respect to the Hodge filtration, and
regular singularities at the singular points of a family. We obtain their
analogues for nonabelian cohomology by using the theory of formal categories
and following the above description of the connection.
These properties are easily obtained by using a variant of our construction of
the connection. Suppose $X_N \rightarrow S_K$ is a morphism of formal
categories of smooth type such that the fiber product $X_N\times _{S_K} S$ is a
formal groupoid of smooth type on $X/S$. Then the schemes $M(X_N\times
_{S_K}S/S,G)$ and stacks ${\cal M} (X_N\times _{S_K}S/S,G)$ have descent data down
to $S_K$, that is they are pullbacks of sheaves or stacks on $Sch /S_K$. The
same is true for the $R(X_N\times _{S_K}S/S,x, G)$ if $x: S_K\rightarrow X_N$
is a section.
\subnumero{Griffiths transversality}
Suppose $X\rightarrow S$ is a smooth family of projective varieties (and we
now ask that the base be smooth, although we don't need it to be
projective). Then we obtain a morphism of formal categories
$X_{Hod} \rightarrow S_{Hod}$ over ${\bf A}^1$. Put
$$
X_{Hod /S} := X_{Hod} \times _{S_{Hod}} (S\times {\bf A}^1).
$$
It is given by a smooth formal groupoid on $X\times {\bf A}^1$
relative to $S\times {\bf A}^1$. The relative
nonabelian cohomology is
$$
{\cal M} _{Hod}(X/S, G):= {\cal M} (X_{Hod/S}/S, G)\rightarrow S\times {\bf A}^1.
$$
This morphism is provided with an action of the formal groupoid $S_{Hod}$
(i.e. ${\cal M} _{Hod}(X/S, G)$ is the pullback of a stack over $S_{Hod}$). In
particular over $\lambda \neq 0$ we get back the action of $S_{DR}$, that is to
say the Gauss-Manin connection. There is a ${\bf G}_m$-action
compatible with everything. This whole situation is the nonabelian
analogue of Griffiths transversality, an interpretation which, comparing with
the abelian case, is justified by Lemma {VBonXHod}.
After we discuss the compactification below we will give another
interpretation in terms of poles of the Gauss-Manin connection at infinity.
\subnumero{Regularity of the Gauss-Manin connection}
Suppose $S'\subset S$ is an open subset whose complement is a
divisor $D$ with normal crossings. Define $S_{DR}(\log D)$ to be the
formal groupoid which is associated to
the bundle of vector fields tangent to $D$. To construct it
explicitly we first treat the case where $S$ is a smooth curve and $D$ a point.
Then $S\times S$ is a smooth surface. Blow up at the point $(D,D)$,
and take the formal completion along
strict transform of the diagonal, as
morphism scheme. The maps of a smooth scheme into the blow up minus the
transform of $D\times S$ are the same as maps into $S\times
S$ which send the intersection with the divisor $D\times S$ to the
point $(D,D)$. From this description we obtain the composition. The formal
completion then becomes a formal groupoid. Now, for any $(S, D)$, glue this
construction in along each component of the divisor (or more precisely along
an $i$-fold intersection of the divisor in an $n$-dimensional $S$,
glue the product of $i$ copies of this construction with $n-i$ copies of
a smooth curve with the usual de Rham construction.
A vector bundle or local system over $S_{DR} (\log D)$ is just a vector bundle
on $S$ with integrable connection with logarithmic singularities along $D$. In
particular, the condition that a vector bundle on $S'_{DR}$
(corresponding to a vector bundle with integrable connection on $S'$) extends
to
a bundle on $S_{DR}(\log D)$ is equivalent to the condition that the
connection have regular singularities.
Now suppose that we have a projective morphism of smooth
varieties $f:S\rightarrow S$ and a divisor $D\subset
S$ which has normal crossings, such that $Y:=f^{-1}(D)$ has normal
crossings, and such that $f:X'\rightarrow S'$ is smooth where $S'=S-D$
and $X'=f^{-1}(S')=X-Y$.
There is a morphism of formal groupoids $S_{DR}(\log D)\rightarrow S_{DR}$. Let
$$
X_{DR} (\log D):= X_{DR}\times _{S_{DR}}S_{DR}(\log D).
$$
Again we have a morphism of formal groupoids
$$
X_{DR} (\log D)\rightarrow S_{DR} (\log D),
$$
and the fiber product
$$
X_{DR/S} (\log D):= X_{DR} (\log D)\times _{S_{DR} (\log D)}S
$$
is a formal groupoid on $X$ over $S$. Even though $X/S$ is not smooth, this
formal groupoid corresponds to a split almost polynomial sheaf of rings of
differential operators $\Lambda _{DR/S (\log D)}$ on $X$ over $S$. Thus the
moduli problems are solved by \cite{Moduli} so the first relative nonabelian
cohomology stack ${\cal M} (X_{DR/S} (\log D)/S, G)$ is an algebraic stack; the
associated representation functor for objects provided with a frame along a
fixed section (not passing through the singular points) is
representable by a scheme, and the universal categorical quotient scheme
$M(X_{DR/S} (\log D)/S,G)$ exists.
Finally, since
$X_{DR/S}(\log D)$ comes from fiber product as in our general situation, these
moduli spaces, moduli stacks and representation spaces descend to $S_{DR}(\log
D)$. This statement is the regularity of the Gauss-Manin connection.
If $G=GL(n)$ then we can do slightly better. Recall that
$M(X_{DR/S} (\log D)/S,GL(n))$
parametrizes bundles with descent data to the formal groupoid $X_{DR/S} (\log
D)$, which are semistable with vanishing rational Chern classes on the fibers
$X_s$. But over the singular fibers, where the extra structure is a
connection away from the singularities but simply a logarithmic connection near
the singularities, it is also natural to consider objects which are no longer
bundles but torsion-free sheaves. This moduli space, which we can denote by
$M^{\rm tf}(X_{DR/S} (\log D)/S,GL(n))$,
will have the advantage that, when combined with the techniques used below for
compactifying $M_{DR}(X_s, GL(n))$, will give a compact total space over
projective $S$.
{\em Caution:} One might be tempted to try the above argument with
$X_{DR}/S_{DR}$. In this case the fiber product $X_{DR}\times _{S_{DR}}S$
no longer has the required smoothness properties, and the relative moduli stack
is no longer an algebraic stack---otherwise we would have an extension of the
Gauss-Manin connection over the singularities! In the case $G= {\bf G}_a$ for
example this would give an extension of the Gauss-Manin connection for ordinary
first cohomology, easily seen as impossible in most examples.
{\em Exercise:} Explain what goes wrong in an example (such as a family of
curves aquiring a node) if we try to do the previous construction with
$X_{DR}/S_{DR}$. This can be done in the context of abelian cohomology.
\numero{Etale local triviality of $M_{Hod}$}
\subnumero{Goldman-Millson theory}
A deformation problem is often
controlled by a differential graded Lie algebra (dgla) $L^{\cdot}$ over the
base field which we are assuming is ${\bf C}$. As explained in
\cite{GoldmanMillson} this means that the deformations with values in an artin
local ring $A$ (with maximal ideal ${\bf m}$) correspond to elements $\eta \in
L^1\otimes _{{\bf C}} {\bf m}$ such that $$
d(\eta ) + \frac{1}{2}
[\eta , \eta ] = 0.
$$
The isomorphisms between deformations $\eta $ and $\eta '$ correspond to
elements $s\in L^0\otimes _{{\bf C}} {\bf m}$ with
$$
\eta ' = d(s) + e^{-s} \eta e^s .
$$
If $P$ is a principal $G$-bundle, let $A^{\cdot} (ad \, P)$ denote the graded
Lie algebra of ${\cal C} ^{\infty}$ forms on $X$ with coefficients in the adjoint
bundle $P\times ^G{\bf g}$ (with Lie bracket combining the wedge of forms and the
Lie bracket of ${\bf g}$). If $(P,\theta )$ is a principal Higgs bundle structure
then we obtain a dgla $(A^{\cdot}(ad\, P), \overline{\partial} + \theta )$ which
gives the deformation theory of $(P,\theta )$. If $(P,\nabla )$ is a principal
bundle with integrable connection then we obtain
a dgla $(A^{\cdot}(ad\, P), \nabla )$ which gives
the deformation theory of $(P,\nabla )$ \cite{GoldmanMillson}.
More generally, suppose $(X,N )$ is a formal groupoid structure for
$X$; then we have the algebra of differentials
$\Omega ^{\cdot}_{X/X_N}$ (cf \S 7 above). Let
$$
A^{i}_{N}:= \bigoplus _{p+q=i} A^{0,q} (\Omega ^p_{X/X_{N}}
$$
This is a differential graded algebra with differential equal to $\overline{\partial} +
\delta $ where $\delta$ is the first order differential operator corresponding
to the differential of $p_{\ast}\Omega ^{\cdot}_{X/X_{N}}$.
If $P$ is a principal $G$-bundle over $X_{N}$ then we obtain a dgla
$$
A^{\cdot}_{N}(ad\, P):= A^{\cdot}_{N} \otimes _{{\cal O}} (P\times ^G{\bf g} ).
$$
The differential comes from that of $A^{\cdot}_{N}$. This dgla controls the
deformation theory of $P$ as a principal bundle on $X_{N}$ (we leave the proof
as an exercise following \cite{GoldmanMillson} and \cite{Moduli} \S 10).
If
$X_{N} = X_{DR}$ then we get $A^{\cdot}_{DR}(ad\, P):=(A^{\cdot}(ad\, P),
\nabla )$, whereas if $X_{N} = X_{Dol}$ then we get $A^{\cdot}_{Dol}(ad\,
P):=(A^{\cdot}(ad\, P), \overline{\partial} + \theta )$.
If $L^{\cdot}$ is a dgla then let $H^{\cdot}(L^{\cdot})$ denote the dgla of
cohomology with differential equal to zero. Recall that we say that
$L^{\cdot}$ is {\em formal} if there is a quasiisomorphism between
$L^{\cdot}$ and $H^{\cdot}(L^{\cdot})$. According to the theory of Goldman and
Millson \cite{GoldmanMillson}, a quasiisomorphism induces an equivalence of
deformation theories, and on the other hand the deformation theory of a dgla
with zero differential is {\em quadratic}, in other words the universal
deformation space is the quadratic cone in $H^1$ which is defined as the zero
scheme of the map $H^1 \rightarrow H^2$, $\eta \mapsto [\eta , \eta ]$. It
follows that the deformation theory of a formal dgla is quadratic.
We need a relative version of this theory for $X\times {\bf A}^1/{\bf A}^1$
near a prefered section. It is actually an interesting question (which I
don't
think has yet been addressed) to develop a relative version of
Goldman-Millson theory in all generality. In our case we are helped by the
fact that the total space is a product. Also we will restrict to deformations
over artinian base (whereas in a better version one should consider
arbitrary base with nilpotent ideal).
Suppose $P$ is a flat principal $G$-bundle with harmonic $K$-reduction $P_K$
and associated operators $d'=\partial + \theta '$ and $d'' = \overline{\partial} + \theta
''$. We define the differential graded Lie algebra over ${\bf C} [\lambda ]$
$$
A^{\cdot}_{Hod}:=(A^{\cdot} (ad \, P) \otimes _{{\bf C}} {\bf C} [\lambda ],
\lambda \partial + \theta ' + \overline{\partial} + \lambda \theta '' ).
$$
Note that the differential has square zero and, as varying with parameters,
gives a deformation from the de Rham to the Dolbeault differentials.
Notice also that the components of $A^{\cdot}_{Hod}$ are flat ${\bf C} [\lambda
]$-modules. For any dgla $L^{\cdot}$ over ${\bf C} [\lambda ]$ where the
components are flat, we define a stack of groupoids over the category of
artinian local ${\bf C} [\lambda ]$-algebras $(B, {\bf m} )$ in the same way as
above (except that tensor products are taken over ${\bf C} [\lambda ]$): the
objects are elements
$\eta \in L^1\otimes _{{\bf C} [\lambda ]} {\bf m}$ such that
$$
d(\eta ) + \frac{1}{2}
[\eta , \eta ] = 0.
$$
The isomorphisms between deformations $\eta $ and $\eta '$ correspond to
elements $s\in L^0\otimes _{{\bf C} [\lambda ]} {\bf m}$ with
$$
\eta ' = d(s) + e^{-s} \eta e^s .
$$
Again we have the same theorem that quasiisomorphisms between flat dgla's
induce equivalences of deformation groupoids. And, on the other hand, the dgla
$A^{\cdot}_{Hod}$ gives the deformation theory of ${\cal M} _{Hod}$ near the
prefered section corresponding to $P$. More precisely the groupoid defined
above for $(B,{\bf m} )$ is equivalent to the groupoid of morphisms
$Spec (B)\rightarrow {\cal M} _{Hod}$ with isomorphism between the morphism $Spec
(B/{\bf m} )\rightarrow {\cal M} _{Hod}$ and the composed morphism
$Spec
(B/{\bf m} )\rightarrow {\bf A}^1 \rightarrow {\cal M} _{Hod}$ (where the first
morphism is the projection to ${\bf A}^1=Spec ({\bf C} [\lambda ])$ and the second
morphism is the prefered section corresponding to $P$). The proofs are
left as exercises.
Finally, the dgla $A^{\cdot}_{Hod}$ defined above is formal over ${\bf C} [\lambda
]$. Let $\ker(\partial + \theta '')[\lambda ]$ denote the subcomplex of
$A^{\cdot}_{Hod}$ consisting of forms $\alpha$ with $\partial \alpha + \theta
'' \alpha = 0$. Since this operator doesn't depend on $\lambda$, the subcomplex
is just a tensor product of the usual complex $\ker(\partial + \theta '')$ with
${\bf C} [\lambda ]$. Furthermore the subcomplex is a sub-dgla; and finally note
that the differential of the sub-complex is $\overline{\partial} + \theta '$ which also
doesn't depend on $\lambda$. The ``principle of two types''
(cf Lemma 2.2 of \cite{HBLS}) implies that
the morphisms
$$
H^{\cdot} (A^{\cdot}_{Hod}) \leftarrow \ker (\partial + \theta
'' )[\lambda ] \rightarrow A^{\cdot}_{Hod}
$$
are quasiisomorphisms.
Finally, let ${\cal H} ^{\cdot}$ denote the ${\bf C}$-vector space of harmonic forms in
$A^{\cdot}(ad \, P)$ (which is the same for the flat connection or the
Dolbeault operator $\overline{\partial} + \theta '$ or, for that matter, anything in
between). Then the morphism $$ {\cal H} ^{\cdot } \otimes _{{\bf C} } {\bf C} [\lambda ]
\rightarrow H^{\cdot} (A^{\cdot}_{Hod}) $$ is an isomorphism of graded vector
spaces (the space of harmonic forms does not {\em a priori} have a product
structure). This shows that $H^{\cdot} (A^{\cdot}_{Hod})$ is flat over ${\bf C}
[\lambda ]$ so we can apply the quasiisomorphism result to conclude that the
deformation theory of $A^{\cdot}_{Hod}$ is the same as that of the graded Lie
algebra $H^{\cdot} (A^{\cdot}_{Hod})$. Finally, in order to obtain
local triviality we have to show that there is a product structure on ${\cal H}
^{\cdot }$ such that the isomorphism
$H^{\cdot} (A^{\cdot}_{Hod})\cong {\cal H} ^{\cdot } \otimes _{{\bf C} } {\bf C} [\lambda ]
$ becomes an isomorphism with product structure. This can be seen by
identifying ${\cal H} ^{\cdot}$ with the cohomology of the complex
$\ker(\partial + \theta '')$ and noting that this latter has a product
structure. With this result, the deformation theory
along the prefered section becomes a product. We have shown this result for
deformations over artinian base, so we get the result on the level of formal
completions. Artin approximation then gives that locally in the etale topology
at any point of a prefered section, ${\cal M} _{Hod}$ is a product. Any point (even
non-semisimple) is isomorphic to a point in a neighborhood of a semisimple
point, so we obtain local triviality at any point in the union ${\cal M}
_{Hod}(X,G)$ of components corresponding to bundles with vanishing rational
Chern classes.
The formal trivialization along the prefered section is the total-space version
of the {\em isosingularity principle} stated in the introduction and
\S 10 of \cite{Moduli}.
We can sum up in the following theorem.
\begin{theorem}
\label{LocTriv}
Suppose $X$ is a smooth projective variety. Let
$M_{Hod}(X,G)\rightarrow {\bf A}^1$ denote the union of components
corresponding to objects with vanishing rational Chern classes. Then etale
locally (above) $M_{Hod}(X,G)$ is a product, in other words any point $P\in
M_{Hod}(X,G)$ over $\lambda \in {\bf A}^1$ admits an etale neighborhood $P\in
U\rightarrow M_{Hod}(X,G)$ with an etale morphism $U\rightarrow
M_{Hod}(X,G)_{\lambda }\times {\bf A}^1$. The same holds for $R_{Hod}
(X,x,G)$ and the moduli stack ${\cal M} _{Hod}(X,G)$. \end{theorem} \hfill $\Box$\vspace{.1in}
\begin{corollary}
The morphism $M_{Hod}(X, G, 0) \rightarrow {\bf A}^1$ is flat (and similarly
for $R_{Hod} (X,x,G)$ and the moduli stack ${\cal M} _{Hod}(X,G)$).
\end{corollary}
\hfill $\Box$\vspace{.1in}
On the other hand, if we have a family of varieties $X\rightarrow S$ then
the Gauss-Manin connection guarantees that the family $M_{DR}(X/S,G)\rightarrow
S$ is etale locally a product. It seems reasonable to guess that these two
results combine into the following.
\begin{conjecture}
\label{triv}
Suppose $X\rightarrow S$ is a smooth projective family. Let
$M_{Hod}(X/S,G)\rightarrow S\times {\bf A}^1$ denote the relative $M_{Hod}$
space. Then etale locally (above) $M_{Hod}(X/S,G)$ is a product, in other
words any
point $P\in M_{Hod}(X/S,G)$ over $(s,\lambda )\in S\times {\bf A}^1$ admits an
etale neighborhood $P\in U\rightarrow M_{Hod}(X/S,G)$ with an etale morphism
$U\rightarrow M_{Hod}(X/S,G)_{(s,\lambda )}\times S\times {\bf A}^1$.
The same holds for $R_{Hod} (X/S,x,G)$ and the moduli stack
${\cal M} _{Hod}(X/S,G)$.
\end{conjecture}
To prove this one would have to analyze much more closely the deformation
theory. In particular, the fact that the total space is a product, which
helped a lot in the previous argument, is no longer there to help us here.
We do give an argument showing this conjecture over smooth points (i.e.
showing that the morphism to $S$ is smooth at smooth points of the fibers) in
the subsection ``Griffiths transversality revisited'' of \S 11 below.
A consequence of this conjecture would be that the deformation class of
$M_{Hod}$ over a point $(s, 0)$ (we take $\lambda = 0$ because the theory over
$\lambda \neq 0$ is trivial due to the Gauss-Manin connection) is determined by
a class $\zeta \in H^1(M_{Dol}(X_s), \Theta_{M_{Dol}})\otimes (TS_s \oplus {\bf C}
)$, where $\Theta_{M_{Dol}}$ is the etale sheaf of infinitesimal automorphisms
of $M_{Dol}$.
\numero{A weight property for the Hodge filtration}
From its very definition, the usual Hodge filtration on cohomology has the
property that $F^0 V=V$. In terms of the Rees bundle this translates to the
statement that if $v\in V$ then the ${\bf G}_m$-orbit ${\bf G}_m v$ has a limit point,
i.e. it extends to a section ${\bf A}^1 \rightarrow \xi (V,F )$. We will
establish a similar property for $M_{Hod} $ and ${\cal M}_{Hod}$.
We don't explicitly review the notion of sheaf of rings of differential
operators $\Lambda$ on $X/S$, for $X\rightarrow S$ a projective morphism, from
\cite{Moduli}. Recall that for any formal groupoid of
smooth type we obtain an almost-polynomial sheaf of rings of differential
operators as described in \cite{Illusie} and in \S 7 above. The
$\Lambda$-modules are just the coherent sheaves on $X$ with descent data down
to $X_M$.
On the other hand, in the case which interests us (the formal groupoid
$X_{Hod}$ on $X\times {\bf A}^1$ over ${\bf A}^1$) we can explicitly describe
the sheaf of rings $\Lambda$ (it is the sheaf of rings denoted $\Lambda ^R$
in \cite{Moduli}). Note first of all that $\Lambda _{DR}$ (corresponding to
$X_{DR}$) is just the sheaf of rings of differential operators. It has a
filtration $\Lambda _{DR}^i$ being the differential operators of order $\leq
i$. Define a decreasing filtration by indexing negatively, $F^{-i}=\Lambda
^i_{DR}$. This filtration is compatible with the ring structure so the
construction $\xi$ gives a sheaf of rings on $X\times {\bf A}^1$ over ${\bf
A}^1$,
$$
\Lambda _{Hod} = \xi (\Lambda _{DR},F).
$$
It is the sheaf of rings associated to the formal groupoid $X_{Hod}$.
The relative moduli theory for $\Lambda _{Hod}$-modules on $X\times {\bf
A}^1/{\bf A}^1$ yields the moduli space $M_{Hod}$ and representation space
$R_{Hod}$. The stack theoretic quotient gives the moduli stack ${\cal M} _{Hod}$.
\subnumero{Langton theory}
We recall the notations and terminology of \cite{Moduli}. In particular,
$p$-semistablity and $p$-stability refer to Gieseker's definition involving
Hilbert polynomials. We will work with $G=GL(n)$ at the start.
The following theorem is the generalisation of Langton's theory
\cite{Langton} of properness of moduli spaces, in the case of sheaves of
$\Lambda$-modules (M. Maruyama pointed out to me that Langton's theory carries
over in this type of general context).
\begin{theorem}
\label{ThmA}
Suppose $S=Spec (A)$ where $A$ is a discrete valuation ring with fraction field
$K$ and residue field $A/{\bf m} = {\bf C}$. Let $\eta$ denote the generic
point and $s$ the closed point of $S$. Suppose $X\rightarrow S$ is a projective
morphism of schemes with relatively very ample ${\cal O} (1)$ on $X$. Suppose
$\Lambda$ is a split almost polynomial sheaf of rings of differential operators
on $X/S$ as in \cite{Moduli}. Suppose ${\cal F} $ is a sheaf of
$\Lambda$-modules on $X$ which is relatively of pure dimension $d$, flat over
$S$, and such that the generic fiber ${\cal F} _{\eta}$ is $p$-semistable. Then
there exists a sheaf of $\Lambda$-modules ${\cal F} '$ on $X$ which is relatively of
pure dimension $d$, flat over $S$, and such that ${\cal F} '_{\eta} \cong {\cal F}
_{\eta}$ and also ${\cal F} '_s$ is $p$-semistable.
\end{theorem}
{\em Proof:}
Langton's proof carries over into our situation. We give a brief sketch
for compatibility with our notations. Let $p_X(\cdot )$ denote the absolute
normalized Hilbert polynomial of a sheaf on $X$ with proper support, and let
$p_{X/S}(\cdot )$ denote the relative normalized Hilbert polynomial of
a sheaf flat over $S$.
Let ${\cal F} _n$ denote the sheaf of $\Lambda$-modules
$$
{\cal F} _n := {\cal F} \otimes _A A/{\bf m}^{n+1}.
$$
It is of pure dimension $d$ on $X$. Let
$$
{\cal F} _n \rightarrow {\cal G} _n
$$
denote the destabilizing quotient, that is the quotient with the minimal
normalized Hilbert polynomial. (Note that if ${\cal G} _0 = {\cal F} _0$ then ${\cal F} _0$
is $p$-semistable and we're done---so we assume that this is not the case). We
make the following claims:
\newline
1. Let $p_0=p_X({\cal G} _0)$. Then for all $n$, $p_X({\cal G} _n)=p_0$.
\newline
2. There are morphisms ${\cal G} _n \rightarrow {\cal G} _{n-1}$ compatible with the
morphisms ${\cal F} _n \rightarrow {\cal F} _{n-1}$.
\newline
3. There there is $q$ such that for $n\geq
q$ we have ${\cal G} _n \stackrel{\cong}{\rightarrow} {\cal G} _{q}$.
{\em Proof of 1:} Assume it is known for $n-1$. We have an exact sequence of
$\Lambda$-modules $$
0\rightarrow {\cal F} _0 \rightarrow {\cal F} _n \rightarrow {\cal F} _{n-1} \rightarrow 0.
$$
From this, we obtain a quotient ${\cal F} _n \rightarrow {\cal G} _{n-1}$, with
$p_X({\cal G} _{n-1})=p_0$. This shows that the normalized Hilbert
polynomial of the destabilizing quotient ${\cal G} _n$ is $\leq p_0$. On the other
hand,
$$
p_X({\rm im}({\cal F} _0 \rightarrow {\cal G} _n)) \leq p_X({\cal G} _n)
$$
(since ${\cal G} _n$ is $p$-semistable)---unless this morphism is zero in which
case ${\cal G} _n = {\cal G} _{n-1}$ and we're done anyway.
Therefore (by the definition of $p_0$) we have that $p_0 \leq p_X({\cal G} _n)$.
This proves claim (1).
{\em Proof of 2:} Since ${\cal G} _{n-1}$ is a quotient of ${\cal F} _n$ with
$p_X({\cal G} _{n-1})=p_0$, it factors through the destabilizing quotient giving the
morphism ${\cal G} _n \rightarrow {\cal G} _{n-1}$.
{\em Proof of 3:} Suppose not. We may assume that $A$ is complete.Let ${\cal G} :=
\lim _{\leftarrow} {\cal G} _n$. This gives a quotient of ${\cal F}$ destabilizing
${\cal F}$ over the generic point.
Now we proceed with the construction of ${\cal F} '$. Starting with ${\cal F}$ (and
assuming that ${\cal F} _0$ is not $p$-semistable), we construct the
quotient ${\cal G} _q$
as above. By statement (3), ${\cal G} _q$ is the maximal quotient of
${\cal F}$ which has normalized Hilbert polynomial $\leq p_0$ and which is supported
over some $Spec (A/{\bf m}^n$.
Let ${\cal F} ^{(1)}$ be the kernel of
the map ${\cal F} \rightarrow {\cal G} _q$. Let $p_1$ be the normalized Hilbert
polynomial of the destabilizing quotient ${\cal Q}$ of ${\cal F} ^{(1)}\otimes _A A/{\bf
m}$. We claim that $p_1 > p_0$. To see this, suppose to the contrary that
$p_1\leq p_0$. Let ${\cal K}$ be the kernel of the map ${\cal F} ^{(1)}\rightarrow {\cal Q}$,
and let ${\cal G} '= {\cal F} /{\cal K}$. Then ${\cal G} '$ is a quotient of ${\cal F}$ which is an
extension of ${\cal G} _q$ by ${\cal Q}$; in particular its normalized Hilbert polynomial
is $\leq p_0$. Furthermore ${\cal G} '$ is supported over $Spec (A/{\bf
m}^{q+2})$. This contradicts maximality of ${\cal G} _q$, showing that $p_1 > p_0$.
Now start with ${\cal F} ^{(1)}$ and repeat the same construction to obtain ${\cal F}
^{(2)}$ etc.; and for each $i$ let $p_i$ be the normalized Hilbert polynomial
of the destabilizing quotient of ${\cal F} ^{(i)}\otimes _A A/{\bf m}$ (we stop if
${\cal F} ^{(i)}\otimes _A A/{\bf m}$ is $p$-semistable).
We have $p_0 < p_1 < p_2 < \ldots $. Since all of these sheaves are
flat over $S$ (they are subsheaves of ${\cal F}$ and hence without $A$-torsion), the
Hilbert polynomials of ${\cal F} ^{(i)}\otimes _A A/{\bf m}$ are all equal to the
Hilbert polynomial of ${\cal F} \otimes _AK$. Finally, the set of $\Lambda$-modules
with a given Hilbert polynomial and with destabilizing quotient having
normalized
Hilbert polynomial $\geq p_0$, is bounded. Thus the set of possible
normalized Hilbert polynomials of the destabilizing quotients is finite. This
shows that the process must
stop. At the stopping point ${\cal F} ^{(i)}\otimes _A A/{\bf m}$ is
$p$-semistable, and we take ${\cal F} ' := {\cal F} ^{(i)}$.
\hfill $\Box$\vspace{.1in}
\subnumero{Application to $M_{Hod}$}
We apply Langton theory to limits of ${\bf G}_m$-orbits in $M_{Hod}$. In fact this
applies equally well to the moduli stack ${\cal M} _{Hod}$. Suppose $p\in {\cal M}
_{Hod}(X_s, GL(n))$. The ${\bf G}_m$-orbit of $p$ is a morphism ${\bf G}_m \rightarrow
{\cal M} _{Hod}$ which corresponds to a $\lambda$-connection $({\cal F} ', \nabla ')$ on
$X_s\times {\bf G}_m $ (where $\lambda : {\bf G}_m \rightarrow {\bf A}^1$ is the
projection of the orbit).
\begin{corollary}
\label{Extn}
With the notations of the above paragraph, there is an extension
$({\cal F} , \nabla )$ of $({\cal F} ', \nabla ')$ to a $\lambda$-connection on
$X_s\times {\bf A}^1$, such that ${\cal F} |_{X_s\times \{ 0\} }$ is a
bundle, is semistable and has vanishing rational Chern classes.
\end{corollary}
{\em Proof:}
First of all note that there exists an extension $({\cal F} _1,\nabla _1)$. For this
note that $p$ corresponds to a $\lambda (1)$-connection $({\cal E} , \varphi )$ on
$X_s$, and ${\cal F} '= p_1^{\ast} ({\cal E} )$ on $X\times {\bf G}_m$ with $\nabla = t\varphi
$ (here $t$ denotes the coordinate on ${\bf G}_m$). We can simply put ${\cal F}
_1=p_1^{\ast}({\cal E} )$ and $\nabla _1= t\varphi$ on $X\times {\bf A}^1$.
Theorem \ref{ThmA} now implies that there exists an extension $({\cal F} , \nabla )$
which is semistable over $X_s\times \{ 0\}$. Note that $\lambda (0)=0$ so the
restriction to $X_s\times \{ 0 \}$ is a Higgs sheaf. By flatness of ${\cal F}$ over
${\bf A}^1$, the restriction has vanishing rational Chern classes. By
(\cite{HBLS} Theorem 2 p. 39), our restriction is actually a bundle. \hfill $\Box$\vspace{.1in}
\begin{corollary}
\label{limits}
Suppose now that $G$ is any reductive group.
If $p\in M_{Hod} (X/S, G)$ then the limit $\lim _{t\rightarrow 0}t\cdot p
$ exists in $M_{Hod}(X/S, G)$.
\end{corollary}
{\em Proof:}
We can choose an injection $G\hookrightarrow GL(n)$. By a variant
of \cite{Moduli} Corollary 9.15 concerning $M_{Hod}$ (we can get this by using
the topological trivialization $M_{Hod} \cong M_{DR} \times {\bf A}^1$ which is
functorial in $G$) the induced map $M_{Hod} (X_s,G)\rightarrow M_{Hod}(X_s,
GL(n))$ is finite. Since the limit exists in $M_{Hod}(X_s, GL(n))$ by the
previous corollary, it exists in $M_{Hod}(X_s, G)$.
\hfill $\Box$\vspace{.1in}
{\bf Question:} What happens for non-reductive groups? If $G={\bf G}_a$ then
the limits again exist (this is exactly the weight property of the Hodge
filtration refered to at the start of the section), so it seems likely that
this will be true in general.
\begin{lemma}
\label{proper}
Suppose $G$ is a reductive group.
Let ${\bf V}\subset M_{Hod}(X, G)$ be the fixed point set of the
${\bf G}_m$-action (note that ${\bf V}$ is concentrated over the origin so in fact
$V\subset M_{Dol}(X, G)$). Then ${\bf V}$ is proper over $S$.
\end{lemma}
{\em Proof:}
The fixed point set lies over the origin in ${\bf A}^1$, so it is just the
fixed
point set of the ${\bf G}_m$-action on the moduli space of Higgs bundles. For
$G=GL(n)$ this fixed point set is proper by (\cite{Hitchin},
\cite{NitsureModuli}, \cite{Moduli} Theorem 6.11). For any $G$, argue
as in the previous corollary. Alternatively one can obtain properness using
Langton theory as above.
\hfill $\Box$\vspace{.1in}
\numero{Compactification of $M_{DR}$}
The space $M_{Hod}(X/S,G)\rightarrow {\bf A}^1$ together with the
action of ${\bf G}_m$ and the isomorphism between $M_{DR}(X/S,G)$ and the fiber over
$\lambda =1$, allow us to compactify $M_{DR}(X/S,G)$ relative to $S$. This
depends on the properness results of the previous section.
\subnumero{Structure theory for ${\bf G}_m$-orbits and construction of some
quotients}
Suppose $X\rightarrow S$ is a projective morphism with an action of ${\bf G}_m$
covering the trivial action on $S$. Choose a relatively very ample line
bundle ${\cal L}$ and a compatible action of ${\bf G}_m$. Let $V_i$ denote the connected
components of the fixed point set $V$. For each $i$ there is an integer
$\alpha _i$ such that $t\in {\bf G}_m$ acts by $t^{\alpha _i}$ on ${\cal L} |_{V_i}$.
Define a partial ordering $\leq$ on $V$, by saying that $u \leq v$ if there
is a sequence of points $x_1,\ldots , x_m\in X$ with
$$
\lim _{t\rightarrow 0} x_1 =u
$$
$$
\lim _{t\rightarrow 0} x_k = \lim _{t\rightarrow \infty} x_{k+1}
$$
$$
\lim _{t\rightarrow \infty} x_m =v .
$$
Notice that if $u\leq v$ and $u\in V_i$, $v\in V_j$ then $\alpha _i \geq
\alpha
_j$ (and if $\alpha _i=\alpha _j$ then $u=v$).
Suppose $V= V_{+} \cup V_{-}$ is a decomposition of
the fixed point set into two disjoint closed subsets (which are consequently
unions of connected components), with the properties that
$$
v\in V_{+}, \;\; u\in V, \;\; u\leq v \; \Rightarrow \; u\in V_{+}.
$$
and
$$
v\in V_{-}, \;\; u\in V, \;\; u\geq v \; \Rightarrow \; u\in V_{-}.
$$
Put
$$
Y_{+} = \{ y\in X \; : \;\; \lim _{\lambda \rightarrow \infty}
\lambda \cdot y \in V_{+}\}
$$
and
$$
Y_{-} = \{ y\in X \; : \;\; \lim _{\lambda \rightarrow 0}
\lambda \cdot y \in V_{-}\} .
$$
These are disjoint closed subsets. They are closed by an argument
similar to the proof of properness in Theorem \ref{ThmD} below. They
are disjoint because if there existed $y\in Y_{+}\cap Y_{-}$ then
$$
\lim _{\lambda \rightarrow 0}
\lambda \cdot y \; \leq \;
\lim _{\lambda \rightarrow \infty}
\lambda \cdot y,
$$
so we would obtain two points $u,v$ with $u\in V_{-}$ and $v\in V_{+}$ but
$u\leq v$; whence $u\in
V_{+}$ and $v\in V_{-}$ (by the conditions on $V_{-}$ and $V_{+}$)
contradicting the disjointness of $V_+$ and $V_-$. Finally, note that $Y_{+}$
and $Y_{-}$ are, by the nature of their definitions, ${\bf G}_m$-invariant.
Let $U:= X-Y_{+}-Y_{-}$. This is a ${\bf G}_m$-invariant open set in $X$.
\begin{theorem}
\label{ThmD}
With the above notations, a universal geometric quotient $U/{\bf G}_m$ exists. It
is separated and proper over $S$.
\end{theorem}
{\em Remark:} When the subsets $V_+$ and $V_-$ are defined by choosing $a\in
{\bf Q} - {\bf Z}$ and setting $V_+ = \bigcup _{\alpha _i > a}V_i$ and
$V_- = \bigcup _{\alpha _i < a}V_i$ then the quotient
defined above is just the geometric invariant theory quotient of the set of
semistable points (for the linearized action obtained when the
linearization is
translated by $a$). In particular, in this case the quotient is projective. I
don't know if the quotient given by this theorem will be projective in
general, nor if it is projective in our example (the compactification of
$M_{DR}$) below.
{\em Proof:}
Let $X^{(pre)}$ denote the set of pre-stable points
\cite{GIT}. In our case it is easy to see that $X^{(pre)}=X-V$ is just
the complement of the fixed point set. Mumford constructs a universal
geometric
quotient $\phi :X^{(pre)}\rightarrow X^{(pre)}/{\bf G}_m$. This morphism is
submersive
so $ \phi (U)$ is open, and by the universality we obtain a universal
geometic
quotient $\phi : U\rightarrow U/{\bf G}_m$. The only problem is to prove that
$U/{\bf G}_m$ is separated and proper over $S$.
Suppose $R$ the henselian local ring of ${\bf C} [z]$ at the origin $P$,
with maximal ideal ${\bf m}$ and residue field $R/{\bf m} = {\bf C}$.
Let $K$ be
the fraction field of $R$, and let $z\in R$ denote a uniformizing parameter.
Let $\tilde{K}$ denote the algebraic closure of $K$ and let $\tilde{R}$ denote
the normalization of $R$ in $\tilde{K}$. The extension $\tilde{K}$ is obtained
from $K$, as $\tilde{R}$ is obtained from $R$, by adjoining the elements
$z^{1/n}$. Let $\tilde{{\bf m}}$ denote the maximal ideal of $\tilde{R}$. Note
that $\tilde{R}$ is a valuation ring with ${\bf Q}$ as value group, $\tilde{{\bf
m}}$ is the valuation ideal, and $\tilde{R}/\tilde{m} = {\bf C}$.
Any finite extension of $K$ is isomorphic to $K$ (by changing the
parameter).
Suppose $\eta :Spec (K)\rightarrow U$ is a point. We have to show that
there is $\varphi \in {\bf G}_m (\tilde{K})$ such that $\varphi \eta $ extends to a
point $Spec (\tilde{R})\rightarrow U$, and furthermore that $\varphi$ is
unique up to ${\bf G}_m (\tilde{R})$.
The action of ${\bf G}_m$ on the point $\eta $ gives a morphism $Spec
(K)\times {\bf G}_m\rightarrow X$ which completes to $Spec
(K)\times {\bf P} ^1 \rightarrow X$.
Let
$$
\eta _0 := \lim _{t\rightarrow 0} t\cdot \eta
$$
$$
\eta _{\infty} := \lim _{t\rightarrow \infty} t\cdot \eta
$$
as points $Spec (K)\rightarrow X$. These complete to points $Spec
(R)\rightarrow X$. There is a scheme $W$ (the closure of the
graph of the previous morphism in $Spec (R) \times X$) with a diagram
$$
\begin{array}{ccccc}
Spec (K)\times {\bf P} ^1 & \hookrightarrow & W& \rightarrow &X \\
\downarrow && \downarrow && \downarrow \\
Spec (K)&\hookrightarrow & Spec (R) &\rightarrow &S
\end{array}
$$
where the vertical arrows are proper, and
where ${\bf G}_m$ acts compatibly on everything in the top row. The fiber of $W$
over the closed point of $Spec(R)$ decomposes as a string of ${\bf P} ^1$'s
meeting at fixed points for the action. Let $y_1,\ldots , y_r$ denote the
images in $X$ of the
fixed points in the string of ${\bf P} ^1$'s over the origin. Since $W$ was
taken as the closure of the graph, these points are distinct. We can order
them
so that $y_1= \eta _0(P)$, $y_r = \eta _{\infty}(P)$, and $y_i$ is joined to
$y_{i+1}$ by a ${\bf P}^1$ in the fiber. In this case for a general point $x$ on
the ${\bf P} ^1$ joining $y_i$ to $y_{i+1}$ we have $(\ast )$
$$
\lim _{t\rightarrow 0} t\cdot x = y_i,\;\;\;
\lim _{t\rightarrow \infty} t\cdot x = y_{i+1}.
$$
In particular, refering to our partial ordering above we have
$$
\eta _0(P) = y_1 < y_2 < \ldots < y_r = \eta _{\infty}(P).
$$
Note that since
$\eta \in U(K)$ we have $\eta _0 \in V_+(K)$ and $\eta _{\infty} \in V_-(K)$.
Thus $y_1\in V_+$ and $y_r\in V_-$ (as $V_+$ and $V_-$ are closed).
By the conditions on $V_+$ and $V_-$ there is $k$ such that
$y_1,\ldots , y_k \in V_+$ and $y_{k+1},\ldots , y_r \in V_-$.
From the definitions of $Y_+$ and $Y_-$ as well as the the property $(\ast )$
we find that the ${\bf P} ^1$ joining $y_i$ to $y_{i+1}$ lies in $Y_+$ if $i<k$ and
in $Y_-$ if $i>k$, whereas it meets $U$ if $i=k$. The uniqueness of the
${\bf G}_m$-orbit meeting $U$ in the closed fiber gives the separatedness. Choose
$\varphi \in {\bf G}_m (\tilde{K})$ so that $\varphi \eta : Spec (\tilde{K}
)\rightarrow W$ completes to a point $Spec (\tilde{R})\rightarrow W$ with $P$
mapping to a general point on the ${\bf P} ^1$ joining $y_k$ to $y_{k+1}$. This
gives the desired $\varphi$ for properness.
\hfill $\Box$\vspace{.1in}
We obtain the following theorem as a corollary.
\begin{theorem}
\label{ThmB}
Suppose $Z\rightarrow S$ is an $S$-scheme on which ${\bf G}_m$ acts (acting
trivially on $S$). Suppose that the fixed point set $W\subset Z$ is proper
over $S$, and that for any $z\in Z$ the limit $\lim _{t\rightarrow 0}t\cdot z
$ exists in $W$. Let $U\subset Z$ be the subset of points $z$ such that the
limit $\lim _{t\rightarrow \infty} t\cdot z$ does not exist in $Z$. Then $U$
is open and there exists a geometric quotient $Q=U/{\bf G}_m$ by the action of
${\bf G}_m$. This geometric quotient is separated and proper over $S$.
\end{theorem}
{\em Proof:} Chose a
${\bf G}_m$-linearized very ample line bundle ${\cal L}$ on $Z$ (this exists by
\cite{GIT}). Then ${\bf G}_m$ acts in a locally finite way on $H^0(Z, {\cal L} )$ so
we may choose a fixed subspace which gives a projective embedding of $Z$.
Thus we may assume that $Z\subset {\bf P} ^N$ as a locally closed subscheme, and
that ${\bf G}_m$ acts linearly on ${\bf P} ^N$ preserving $Z$ and inducing the given
action there. Taking the graph we can consider this as an embedding $Z\subset
{\bf P} ^N \times S$. Let $X$ be the
subscheme closure of $Z$ in ${\bf P} ^N \times S$ (that is, the subscheme
defined by the homogeneous ideal of forms which vanish on $Z$). Note that
$X$ is projective over $S$ and that ${\bf G}_m$ acts on $X$ preserving the open set
$Z$. Let $V$ denote the fixed point set in $X$, and let $V_{+}:=W$ be the fixed
point set in $Z$. Let $V_{-}:= V\cap (X-Z)$ denote the fixed point set in the
complement. Note that the complement $X-Z$ is closed, hence proper over $S$,
so $V_{-}$ is proper over $S$. By hypothesis $V_{+}$ is proper over $S$. We
obtain a decomposition $V= V_{+}\cup V_{-}$ as a disjoint union of two closed
subsets.
Suppose $u,v\in V$ with $v\leq u$. This means that there is a sequence of
points $v_0=v , \ldots , v_n = u$ such that $v_i$ is joined to $v_{i+1}$ by a
${\bf G}_m$-orbit (i.e. there is an orbit whose limits are $v_i$ at $\lambda
\rightarrow 0$ and $v_{i+1}$ at $\lambda \rightarrow \infty$). Suppose $v_i\in
V_{-}$. Then the orbit corresponds to a point $x\in X$ with $\lim _{\lambda
\rightarrow 0}\lambda \cdot x = v_i$. But if $x\in Z$ then our hypothesis
would give $v_i\in V_{+}$, so $x$ must be in $X-Z$. Since $X-Z$
is closed, the
other limit $v_{i+1}$ must be in $X-Z$ also. We thus show by
induction that if
$v=v_0$ is in $V_{-}$ then so is $u=v_n$. The contrapositive says that if $u$
is in $V_{+}$ then so is $v$.
We have shown on the one hand that if $v\in V_{-}$ and
$u\in V$ with $v\leq u$ then $u\in
V_{-}$; and on the other hand that if $u\in V_{+}$ and $v\in V$ with
$v\leq u$ then $v\in V_{+}$.
We are now ready to apply the general construction above. Define the subsets
$Y_{+}$ and $Y_{-}$ as before, and let $U'$ be the complement $U'=
X-Y_{+}-Y_{-}$. We claim that $Y_{+}$ is the set of points $x\in Z$ such
that $\lim _{\lambda \rightarrow \infty}\lambda \cdot x \in Z$.
Recall that $Y_{+} := \{ x\in X \; : \;\; \lim _{\lambda \rightarrow \infty}
\lambda \cdot x \in V_{+}\}$. But if $x\in Z$ with
$\lim _{\lambda \rightarrow \infty}\lambda \cdot x \in Z$ then this limit is in
$V\cap Z = V_{+}$. On the other hand, if $x\in X$ with $\lim _{\lambda
\rightarrow \infty} \lambda \cdot y \in V_{+}$ then $x\not \in (X-Z)$ because
$X-Z$ is closed and $V_{+}\cap (X-Z)=\emptyset$. This shows the claim. We next
show that $Y_{-}= X-Z$. To see this recall that
$Y_{-} := \{ x\in X \; : \;\; \lim _{\lambda \rightarrow 0}
\lambda \cdot x \in V_{-}\}$. If $x\in Z$ then by hypothesis
$\lim _{\lambda \rightarrow 0}
\lambda \cdot x \in Z$ and $V_{-}\cap Z= \emptyset$, so this shows that
$Y_{-}\subset X-Z$. On the other hand if $x\in X-Z$ then since $X-Z$ is
closed, $\lim _{\lambda \rightarrow 0}
\lambda \cdot x \in X-Z$ and hence this limit is in $Y_{-}$, which shows that
$Y_{-}=X-Z$.
With the two statements of the previous paragraph we obtain that the
complement $U'$ of $Y_{+}$ and $Y_{-}$ is equal to the subset of points
of $Z$ whose limits at $\lambda \rightarrow \infty$ do not exist in $Z$, that
is to say that $U'$ is the same as ths subset $U$ described
in the statement of the theorem. The result of Theorem \ref{ThmD}
now gives the
universal geometric quotient $U/{\bf G}_m$ which is separated and proper over $S$.
\hfill $\Box$\vspace{.1in}
\subnumero{Relative compactification of $M_{DR}(X/S, G)$}
Suppose $G$ is a reductive group and $X\rightarrow S$ a smooth projective
morphism. Apply Theorem \ref{ThmB} to $Z=M_{Hod} (X/S,G)$. The hypotheses
on $Z$
are given by Corollary \ref{limits} and Lemma \ref{proper}. Note that the open
set $U$ certainly contains the open set $$
M_{Hod} (X/S,G)\times _{\bf A}^1 {\bf G}_m \cong M_{DR}(X/S,G)\times {\bf G}_m
$$
as a ${\bf G}_m$-invariant open set. Since the quotient $U\rightarrow Q$ is a
geometric quotient, the image of the open set $M_{DR}(X/S,G)\times {\bf G}_m$ is a
geometric quotient of $M_{DR}(X/S,G)\times {\bf G}_m$, but we already know the
geometric quotient here, it is just $M_{DR}(X/S,G)$. Thus our quotient $Q$
contains $M_{DR}(X/S,G)$ as an open set, and $Q$ is proper over $S$.
We have proved the following theorem.
\begin{theorem}
\label{Th33}
If $G$ is a reductive group and $X\rightarrow S$ a smooth projective morphism,
then there exists a natural relative compactification $\overline{M}_{DR}(X/S,
G)$ proper over $S$ and containing $M_{DR}(X/S,G)$ as an open subset.
\end{theorem}
{\em Proof:}
Take $\overline{M}_{DR}(X/S,G)$ to be the quotient
$Q$ of the previous paragraph. \hfill $\Box$\vspace{.1in}
{\em Remark:}
There is
a natural stack-theoretic compactification $\overline{{\cal M} }_{DR}(X/S, GL(n))$
containing ${\cal M} _{DR}(X/S,GL(n))$ as an open subset and satisfying the
valuative criterion of properness over $S$. The valuative criterion comes
from Corollary \ref{Extn}. We state this only in the case $G=GL(n)$ because the
finiteness result (\cite{Moduli} Corollary 9.15) used to pass to any group $G$
in the proof of Corollary \ref{limits} is only available for the moduli spaces,
not for the moduli stacks.
There is a natural Cartier divisor
on ${\cal M} _{Hod}(X/S,GL(n))$ given as the pullback of the divisor $\{ 0\}
\subset {\bf A}^1$. This divisor is ${\bf G}_m$-invariant, so it projects to a
Cartier divisor in the stack-theoretic quotient $\overline{{\cal M}
}_{DR}(X/S,GL(n))$; and the open set ${\cal M} _{DR}(X/S, GL(n))$ is just the
complement of this divisor. Thus the ``divisor at infinity'' exists as a
natural Cartier divisor.
In the moduli space compactification $\overline{M}_{DR}(X/S, G)$ the divisor
at infinity is only a Weil divisor. We can define an intermediate {\em
orbifold compactification} by taking the quotient $M_{Hod}(X/S, G)/{\bf G}_m$ in the
sense of stacks. Here the divisor at infinity is again a Cartier divisor.
In the orbifold compactification there may be orbifold points corresponding to
fixed points of finite subgroups of ${\bf G}_m$. In the scheme-theoretic
compactification these project to certain quotient singularities. They
correspond to objects which are like systems of Hodge bundles (or variations
of Hodge structure) except that the Hodge bundles are only indexed by a cyclic
group instead of ${\bf Z}$ so the Kodaira-Spencer map $\theta$ can go ``around and
around'' to no longer be nilpotent.
The
orbifold structure of the divisor at infinity is that of the quotient $U\times
_{{\bf A}^1}\{ 0\} /{\bf G}_m$. But this is the quotient of the open subset of
$M_{Dol}(X/S, G)$ corresponding to Higgs bundles with non-nilpotent Higgs
field, by the action of ${\bf G}_m$.
\subnumero{Interpretation of the Hodge filtration in terms of the
compactification}
Let $\overline{M}'_{DR}$ denote the orbifold compactification described
above. There is a principal ${\bf G}_m$-bundle over this space, it is just the
total space from before taking the quotient. This principal bundle corresponds
to a line bundle ${\cal L}$. One can see that ${\cal L} =
{\cal O} _{\overline{M}'_{DR}}(-D)$
where $D$ is the divisor at infinity. Conversely, from the data of
$\overline{M}'_{DR}$ and the divisor at infinity $D$ (which has multiplicity
one) we obtain a line bundle and hence a principal ${\bf G}_m$-bundle with a section
defined over $M_{DR}$. There is only one function on this total space which
is constant on the multiples of our copy of $M_{DR}$ so this fixes the morphism
to ${\bf A}^1$. The total space is $M_{Hod}^{\ast}$, the open subset which is
the complement of the locus of Higgs bundles with nilpotent Higgs field. Thus
we recover most but not all of the Hodge filtration $M_{Hod}$ from our
compactification with its divisor at infinity.
\subnumero{Griffiths transversality revisited}
In the relative case we have obtained a family of compactifications
$$
\overline{M_{DR}}(X/S)\rightarrow S.
$$
On the other hand, recall that $M_{DR}(X/S)$ has the Gauss-Manin connection
which, analytically, translates the fact that $M_{DR}(X/S)^{\rm an}$ is
locally over $S^{\rm an}$ a product of the form $S^{\rm an}\times M_B$ where
$M_B$ is the moduli space of representations of the fundamental group of the
fiber $X_s$.
The Griffiths transversality condition basically says that the Gauss-Manin
connection has poles of order $1$ along the divisor at infinity.
In order to make this precise we restrict ourselves to a case where the moduli
space is smooth. Fix a family $X\rightarrow S$ over a base $S$, and suppose $S$
is a smooth curve. Suppose for simplicity that $G$ and Chern class data $c$ are
fixed so that the corresponding unions of components $M_{DR}(X_s, G)_c$ is
smooth and equidimensional (for example $X/S$ a family of curves,
$G=PGL(n)$ and $c$ means we look at bundles of degree $d$ prime to $n$).
We obtain
$$
M_{Hod} (X/S, G)_c \rightarrow S\times {\bf A}^1.
$$
We claim that this map is smooth. This is a special case of Conjecture
\ref{triv}, and requires some care. Apply the criterion of (\cite{Hartshorne}
Chapter III Lemma 10.3.A---for which Hartshorne refers to Bourbaki and Altman
and Kleiman) where $t$ is the coordinate on ${\bf A}^1$. We have to show that
$t$ is not a zero divisor upstairs, and that $M_{Dol}(X/S, G)_c\rightarrow S$ is
flat. Since all of the fibers of our map are smooth, the only way $t$ could
be a zero divisor is if there were an irreducible component lying over $0\in
{\bf A}^1$. But Theorem \ref{LocTriv} shows that this is not the case. To
show that $M_{Dol}(X/S, G)_c\rightarrow S$ is flat it suffices (again in
view of
the smoothness of the fibers) to show that no irreducible
component of $M_{Dol}(X/S,G)_c$ lies over a point in the curve $S$. Any
component of $M_{Dol}(X_s,G)_c$ is contained in the closure of
$M_{DR}(X_s,G)_c
\times {\bf G}_m$, so any component of $M_{Dol}(X/S,G)_c$ is contained in the
closure
of $M_{DR}(X/S,G)_c \times {\bf G}_m$. If $n$ denotes the dimension of any
components
of $M_{DR}(X_s,G)_c$ (by hypothesis these dimensions are all the same)
then the
dimension of any component of $M_{Hod}$ is at least $n+2$ and (since $M_{Dol}$
is defined by one equation $t=0$ inside $M_{Hod}$) the dimension of any
component of $M_{Dol}(X/S,G)_c$ must be at least $n+1$. But the fibers
$M_{Dol}(X_s, G)_c$ are all of dimension $n$, so they cannot contain
irreducible components of $M_{Dol}(X/S,G)_c$. This proves that our map is
flat. As the fibers are smooth, the map is smooth.
Now we can get back to the thread of our discussion.
Let $U\subset M_{Hod} (X/S, G)_c $ denote
the open set used in defining the compactification. Let
$\overline{M}'_{DR}(X/S,G)$ denote the orbifold compactification of
$M_{DR}(X/S,G)$ obtained by taking the quotient $U/{\bf G}_m$ in the sense of
stacks. This can introduce orbifold points at places where the stabilizer
is a nontrivial finite subgroup of ${\bf G}_m$ (it has to be the $m$-th roots of
unity). These orbifold points would be replaced by the corresponding cyclic
quotient singularities in the usual compactification defined previously. The
advantage here is that $\overline{M}'_{DR}(X/S,G)$ is smooth over $S$.
Let $D\subset \overline{M}'_{DR}(X/S,G)$ denote the divisor at infinity. It
is reduced (since $M_{Hod}$ is smooth over ${\bf A}^1$). The Gauss-Manin
connection can be interpreted as a lifting of vector fields on $S$ to vector
fields on $M_{DR}(X/S, G)$. If $p$ denotes the projection to $S$, we obtain a
vector field with coefficients in the line bundle $p^{\ast}(\Omega ^1_S )$
which we denote as
$$
\eta \in H^0(M_{DR}(X/S,G), T(M_{DR}(X/S,G))\otimes p^{\ast}\Omega ^1_S ).
$$
Note that it projects to the identity section of $T(S)\otimes \Omega ^1_S={\cal O}
_S$. This means that flowing along $\eta$ takes us from one fiber of $p$ to
another.
\begin{theorem}
The Griffiths transversality property says that the vector field $\eta$ has
simple poles along $D$, and the residue is tangent to $D$. More precisely
let
$$
{\cal F} := \frac{T(\overline{M}'_{DR}(X/S,G))\otimes p^{\ast}\Omega ^1_S \otimes
{\cal O} _D(D) }{T(D) \otimes p^{\ast}\Omega ^1_S \otimes
{\cal O} _D(D)}
$$
(which is supported on $D$),
then
$$
\ker (H^0(\overline{M}'_{DR}(X/S,G),
T(\overline{M}'_{DR}(X/S,G))\otimes p^{\ast}\Omega ^1_S \otimes
{\cal O} (D)) \rightarrow H^0(D, {\cal F} ).
$$
\end{theorem}
{\em Proof:}
We leave this to the reader.
\hfill $\Box$\vspace{.1in}
It should be interesting to study the behavior of the dynamical system given
by this vector field with poles. The transport between fibers
$M_{DR}(X_s,G)$ and $M_{DR}(X_t,G)$ has the effect of composing the
analytic isomorphisms
$$
M_{DR}(X_s, G)^{\rm an} \cong M_B(X_s, G)^{\rm an} = M_B(X_t,G)^{\rm an}
\cong
M_{DR}(X_t, G)^{\rm an}
$$
where the left and right isomorphisms are the Riemann-Hilbert correspondence
and the middle equality comes from the isomorphism of fundamental groups
(which depends on the path we choose from $s$ to $t$).
{\em Exercise:}
Interpret the regularity of the Gauss-Manin connection in way
similar to the above interpretation of Griffiths transversality.
On the moduli space $M(X_{DR/S}(\log D)/S, G)$ the lifts of vector fields
given by the Gauss-Manin connection will have simple poles along inverse image
of the singular set in $S$.
\subnumero{Compactifications of spaces of $\Lambda$-modules}
By a technique similar to our construction of the relative compactification of
$M_{DR}$ we have the following general theorem.
\begin{theorem}
\label{Thm3}
Suppose $X\rightarrow S$ is a projective flat morphism, and $\Lambda$ is a
split almost polynomial sheaf of rings of differential operators on $X/S$.
Let $M(\Lambda ,P)\rightarrow S$ denote the moduli space of semistable
$\Lambda$-modules with Hilbert polynomial $P$ on $X/S$. Then there exists a
relative compactification, a scheme $\overline{M(\Lambda )}\rightarrow
S$ containing $M(\Lambda )$ as an open set and which is proper over $S$.
\end{theorem}
{\em Proof:}
The proof is the same as the previous one, with the following changes.
We replace the ring $\Lambda _{Hod}$ by the ring $\xi (\Lambda , F)$ on
$X\times {\bf A}^1$
for the filtration $F$ of $\Lambda$ by degree. Even if not admitted in the
definition of $M(\Lambda ,P)$, we must now admit torsion-free objects in the
space $M(\xi (\Lambda , F), P)$ used to get the compactification.
\hfill $\Box$\vspace{.1in}
We could even obtain the same result with $\Lambda$-modules which are of pure
dimension $d < dim (X/S)$---this is an interpretation of the statement for
$deg (P)=d$. The proof is again exactly the same.
\subnumero{A total space compactification of $M_{DR}(X/S, GL(n))$}
Suppose $S$ is smooth, projective, with $S'=S-D$ the complement of a divisor
with normal crossings. Suppose $X\rightarrow S$ is smooth over $S'$ (we
denote $X':= X\times _SS'$) and has inverse image of $D$ being a divisor with
normal crossings. In this situation
We can get a compactification for the total space $M_{DR}(X'/S', GL(n))$.
Let $\Lambda = \Lambda _{DR/S (\log D)}$ be the split almost polynomial
sheaf of rings of differential operators corresponding to the formal
groupoid $X_{DR/S}(\log D)$ defined in \S 8. Apply the construction of
Theorem \ref{Thm3} to obtain a compactification. We can describe this more
precisely. There is a formal groupoid $X_{Hod/S}(\log D)$ combining all of
the constructions of \S 8, with underlying scheme $X\times {\bf A}^1$. Let
$M^{\rm tf} (X_{Hod/S}(\log D), GL(n))$ denote the moduli space for torsion
free semistable objects on $X_{Hod/S}(\log D)$ (with vanishing rational Chern
classes). Our {\em total space compactification} is
$$
\overline{M}^{\rm tf}(X_{DR/S}(\log D), GL(n)):= U/{\bf G}_m
$$
where $U\subset M^{\rm tf} (X_{Hod/S}(\log D), GL(n))$ is the open set of
points $p$ such that $\lim _{t\rightarrow \infty} tp$ does not exist. It is
proper over $S$ and since $S$ itself is proper, it is compact.
The necessity to
include torsion-free sheaves over the singular fibers is why we must assume
here that the structure group is $GL(n)$.
Combining the interpretations of Griffiths transversality and regularity of
Gauss-Manin (exercise), we get that the lifts of vector fields on $S$ given
by the Gauss-Manin connection on $M_{DR}(X/S, GL(n))$, have simple poles at all
components of infinity in the total space compactification of $M_{DR}(X/S,
GL(n))$.
\numero{The nonabelian Noether-Lefschetz locus}
A. Beilinson made a comment to the effect that one could get the moduli for
${\bf Z}$-variations of Hodge structure as an intersection between the integral
representations, and the filtered local systems. Of course for $X$ fixed the
moduli space is just a finite set of points, but this becomes interesting when
we let $X$ vary in a family.
For this section we suppose $G=GL(n)$.
Suppose $X\rightarrow S$ is a smooth projective morphism. Let $V\subset
M_{Dol}(X/S,GL(n))$ denote the fixed point set of the ${\bf G}_m$-action; it is also
the moduli space for systems of Hodge bundles or equivalently for complex
variations of Hodge structure \cite{CVHS}. Let
$V_{DR}$ denote the image in $M_{DR}(X/S,GL(n))$ (note that this is not a
complex
analytic subset). On the other hand, let
$M_B(X_s, GL(n,{\bf Z} ))\subset M_B(X_s, GL(n))$ denote the image of $Hom (\pi
_1(X_s,x), GL(n,{\bf Z} ))$ (it is the subset of integral representations), and let
$M_{DR}(X/S, GL(n,{\bf Z} ))$ denote the
subset of points of $M_{DR}(X/S,GL(n))$ which over each fiber $X_s$ correspond
to elements of $M_B(X_s,GL(n,{\bf Z} ))$. Note that $M_{DR}(X/S,GL(n,{\bf Z} ))$
is a complex
analytic subset of $M_{DR}(X/S,GL(n))$. Finally put
$$
NL(X/S,GL(n)):= V_{DR}\cap M_{DR}(X/S,GL(n,{\bf Z} ) ).
$$
There is a morphism $NL(X/S,GL(n))\rightarrow S$.
\begin{theorem}
For each $s\in S$, the fiber $NL(X/S,GL(n))_s$ is the set of
isomorphism classes
of integral representations $\rho$ such that $\rho \oplus \rho$ underlies an
integral variation of Hodge structure. The morphism $NL(X/S,GL(n))
\rightarrow S$ is
proper, and $NL(X/S,GL(n))$ has a unique structure of normal analytic
variety such
that the inclusions $NL(X/S,GL(n))\rightarrow M_{DR}(X/S,GL(n))$ and
$NL(X/S,GL(n))
\rightarrow M_{Dol}(X/S,GL(n))$ are complex analytic.
\end{theorem}
{\em Remark:} In this first treatment of the subject, we are ignoring
a possibly more natural non-reduced or non-normal structure of complex
analytic space on $NL(X/S,GL(n))$.
{\em Proof:} If $\rho \in NL(X/S,GL(n))_s$ then $\rho$ is a complex
variation of
Hodge structure on $X_s$, and $\rho$ is integral. It is easy to see that $\rho
\oplus \rho$ has a structure of integral variation of Hodge structure (see for
example the arguments in \cite{DeligneMostowVol}, \cite{Hodge1},
\cite{HBLS}). Conversely if $\rho \oplus \rho$ has a structure of variation of
Hodge structure then $\rho$ is fixed by the ${\bf G}_m$ action, so $\rho$ lies in
$V$.
To see that $NL(X/S,GL(n))\rightarrow S$ is proper, it suffices to work locally
near a point $s_0\in S$. By the main result of \cite{DeligneMostowVol}
(done with small variations in the parameters, which still works)
there is a finite subset of points of $M_B(X_{s_0}, GL(2n,{\bf Z} ))$ which
correspond to representations are doubles of $n$-dimensional
representations and which could possibly be ${\bf Z}$-variations of Hodge structure
on $X_s$ with $s$ in a given relatively compact neighborhood of $s_0$. Over
this neighborhood, $NL(X/S,GL(n))$ is the intersection of $V_{DR}$ (which is
proper over $S$ \cite{Moduli}) with this finite set of sections; thus
$NL(X/S,GL(n))$ is proper over $S$.
We show the complex analyticity of $NL(X/S,GL(n))$ inside
$M_{DR}(X/S,GL(n))$. The question is local over $S$, so we can fix a
neighborhood $U$ of $s\in S$ and a section $\sigma : U\rightarrow M_{DR}
(X/S,GL(n,{\bf Z} )
)$ corresponding to an integral representation of $\pi _1(X_s)$; we have to
show
that $\sigma ^{-1}(V_{DR})$ is an analytic subset of $U$.
Our representation $\rho$ corresponds to a local system $W$ on $X_s$.
Let $W_i$ denote the complex irreducible factors of $W$. Then $\sigma ^{-1}
(V_{DR})$ is the intersection of the subsets $N_i$ of points in $u$ where
$W_i$ admits a complex variation of Hodge structure. It suffices to show that
$N_i$ are complex analytic. If $W_i$ does not admit a flat hermitian form
then $N_i$ is empty, so we can assume that it does admit such a form. This
form $\langle \; ,\; \rangle$ is uniquely determined up to a
scalar. Fix an
integer $w$, and data of ranks $r_i$ and degrees $d_i$. Let
$$
HF (X/S, W_i, r_i, d_i ) \rightarrow U
$$
denote the parameter scheme for filtrations $F^{\cdot}$ of the local systems
$W_i(s)$ (considered as vector bundles with integrable connection on the
fibers $X_s$) with $r(F^i)=r_i$ and $deg (F^i)=d_i$, and satisfying the
Griffiths transversality condition. It is an analytic variety over $U$, in
fact
the pullback of a quasiprojective variety over $M_{DR}(X/S,GL(n))$ by the
section
$\sigma$ (the parameter variety is the closed subscheme of the Hilbert scheme
of filtrations defined by the Griffiths transversality condition). Let
$$
HF (X/S, W_i, \langle \; ,\; \rangle , w, r_i, d_i ) \subset
HF (X/S, W_i, r_i, d_i )
$$
denote the open subset of filtrations such that $F^{\cdot}$
and $F^{\cdot , \perp}$ together determine a Hodge structure of weight $w$ on
the fiber over every $x\in X$. The morphism
$$
HF (X/S, W_i, \langle \; ,\; \rangle , w, r_i, d_i )\rightarrow U
$$
is
injective (because there is at most one structure of complex variation of
Hodge structure on the irreducible representation $W_i$ up to translations,
but the translations are fixed by specifying the ranks $r_i$), so the image is
an analytic subset. There are only a finite number of possible sets of
degrees
and ranks which can occur (and it suffices to consider $w=0$ for example) so
the union of this finite number of analytic subsets is $N_i$. This shows
that $N_i$ is analytic, hence that the intersection is analytic. Thus
(taking the union over the finite number of sections we have to consider)
$NL(X/S,GL(n))$ is an analytic subset of $M_{DR}(X/S,GL(n))$.
From the above construction one gets that over any component of
$NL(X/S,GL(n))$ the
Hodge filtration of the resulting variation of Hodge structure varies
analytically. Since the associated Higgs bundle is the associated graded of
the Hodge filtration (with $\theta$ as
the projection of the connection, which
also varies analytically), the associated Higgs bundle varies analytically with
the point in $NL(X/S,GL(n))$, that is to say that $NL(X/S,GL(n))$ is an
analytic
subset of
$M_{Dol}(X/S,GL(n))$.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\label{NLalg}
IF $S$ is projective, then $NL(X/S,GL(n))$ has a structure of normal projective
variety such that the morphisms $NL(X/S,GL(n))\rightarrow M_{DR}(X/S, GL(n))$
and $NL(X/S,GL(n))\rightarrow M_{Dol}(X/S, GL(n))$ are algebraic morphisms.
\end{corollary}
{\em Proof:} We can think of $NL(X/S,GL(n))$ as the normalization of an
analytic
subvariety of $M_{DR}(X/S,GL(n))$ which is proper over $S$. In particular
it is a
closed subvariety of $\overline{M}_{DR}(X/S,GL(n))$ so we can apply GAGA
to say that
it is algebraic. Again by GAGA the morphism to $M_{Dol}(X/S,GL(n))$ is
algebraic.
\hfill $\Box$\vspace{.1in}
After the relatively surprising results of the theorem and this
corollary, the reader might well be asking if $NL(X/S,GL(n))$ isn't just a
finite
collection of points. In fact, $NL(X/S,GL(n))$ can
have positive dimensional components.
For example if $Z$ is a surface with a ${\bf Z}$-variation of Hodge structure and
then if $X/S$ is a pencil of curves on $Z$, the family of restrictions of the
variation on $Z$ gives a component of $NL(X/S,GL(n))$ dominating $S$. More
generally
if $X/S$ contains a pencil of curves on $Z$ as a subfamily, then we get a
component of $NL(X/S,GL(n))$ dominating this subfamily.
One might ask whether all positive dimensional components of $NL(X/S,GL(n))$
come
from such a construction.
We can also ask for an extension of the result of Corollary \ref{NLalg} to
the quasiprojective case.
\begin{conjecture}
\label{NLalgQPcase}
If $S$ is a quasiprojective variety then $NL(X/S, GL(n))$ is an algebraic
variety and the morphisms to $M_{DR}(X/S, GL(n))$ and
$M_{Dol}(X/S, GL(n))$ are algebraic.
\end{conjecture}
This conjecture would be the nonabelian analogue of the result of \cite{CDK}.
It is a globalized version of a problem Deligne posed to me some time ago, of
obtaining a generalization of the finiteness results of
\cite{DeligneMostowVol} uniformly near singularities. In
\cite{CDK} it is explained how their result would be a consequence of the
Hodge conjecture. In a similar way, Conjecture \ref{NLalgQPcase} would be a
consequence of the following nonabelian version of the Hodge conjecture.
\begin{conjecture}
\label{NAHC}
The points of $NL(X_s, GL(n))$ (i.e. the ${\bf Z}$-variations of Hodge structure)
are motivic representations on $X_s$.
\end{conjecture}
I don't know who first thought of this conjecture but that must have been a
long time ago. Of course there is little chance of making any progress on
this---I have presented it only for the light it sheds on Conjecture
\ref{NLalgQPcase}.
The main problem in proving Conjecture \ref{NLalgQPcase} is to make
a local analysis around the singularities of a good completion of the family
$X/S$ to $\overline{X}/\overline{S}$. We hope that the total
space compactification
constructed above will be useful for studying this algebraicity question
locally at the singularities of the family.
\numero{An open problem: degeneration of nonabelian Hodge structure}
I would like to end this paper by proposing an open problem for further
research. This is the problem (motivated at the end of the previous section)
of studying the degeneration of nonabelian Hodge structure which arises from a
degenerating family of varieties $X\rightarrow S$. It is already complicated
enough to study the case of a degenerating family of curves, so we can suppose
that $X\rightarrow S$ is a family of curves. Let $0\in S$ denote a point where
the fiber $X_0$ has a simple singularity (node). Let $U=S-\{ 0\}$ and suppose
$X_U$ is smooth over $U$. We assume $G=GL(n)$ for reasons we will see in a
minute. Then the relative moduli space $M_{DR}(X_U/U , GL(n))$ over $U$ is
provided with the Gauss-Manin connection; with a family of hyperk\"ahler
structures; with a family of Hodge filtrations satisfying Griffiths
transversality; and with a Noether-Lefschetz locus $NL(X_U/U, GL(n))\subset
M_{DR}(X_U/U, GL(n))$ for which
we would like to prove algebraicity. The problem, then, is to study the
degeneration of all of these structures as one approaches $0\in S$.
The first step in studying degenerations of variations of Hodge structure in
the abelian case was to have a canonical extension of the underlying
holomorphic bundle, and a regular singularity theorem for the flat connection.
We have obtained the analogues of these things for the nonabelian case. Note
that in defining a canonical extension, it is important to get something
which is proper over $S$. This was perhaps not apparent in the abelian case,
where nobody cares about the fact that a vector space is noncompact! But in
the nonabelian case, a noncompact extension would leave room for some
asymptotic behavior as one goes off to infinity, difficult to see.
To get a compactification of $M_{DR}(X_U/U, GL(n))$ we take the total space
compactification $\overline{M}^{\rm tf}(X_{DR/S}(\log 0)/S, GL(n))$ which comes
from looking at the
moduli of torsion free sheaves with $\lambda$-connection logarithmic at the
singularities, and applying the construction of \S\S 10-11. After doing
all of this we obtain a compactification of
$M_{DR}(X_U/U, GL(n))$ which is the analogue of the canonical extension
(together with the Hodge filtration which corresponds to the fiberwise
compactification). The regularity of the Gauss-Manin connection, coupled with
Griffiths transversality, give that the Gauss-Manin connection on
$M_{DR}(X_U/U, GL(n))$ over $U$ has poles of order one at infinity (both along
the fiberwise infinity and at the singular point $0\in S$).
We need a good local description of this compactified moduli space near all
points at infinity, specially torsion-free sheaves on the singular fibers but
also at infinity in the fiberwise direction.
One approach to the analysis of the degeneration might be to directly analyze
the lifted vector field giving the Gauss-Manin connection with its simple
poles, and try to deduce asymptotic properties of the transport along this
vector field.
Another option would be to try to analyze the degeneration of the
hyperk\"ahler structure, thinking of it as a family of quaternionic structures
on the fixed variety $M_B(X_s, GL(n))$.
Among the goals of this study should be: to prove algebraicity of the
Noether-Lefschetz locus, in a nonabelian version of the work of
Cattani-Deligne-Kaplan \cite{CDK}; to obtain a nonabelian version of the
Clemens-Schmid exact sequence; to obtain estimates and asymptotic
expansions for everything in the spirit of the $SL_2$ and nilpotent orbit
theorems; and finally to be able to apply all of this to obtain a devissage
principle for nonabelian cohomology (even with coefficients in higher homotopy
types \cite{kobe}), i.e. a version of the Leray spectral sequence.
|
1996-04-10T14:33:28 | 9604 | alg-geom/9604008 | en | https://arxiv.org/abs/alg-geom/9604008 | [
"alg-geom",
"math.AG"
] | alg-geom/9604008 | Bas Edixhoven | Bas Edixhoven (University of Rennes 1) | On a result of Imin Chen | latex, 5 pages, hard copies available | null | null | null | null | We give another proof of Imin Chen's result that the jacobian of the modular
curve X(p)_{non-split}, for p a prime number, is isogeneous to the new part of
the jacobian of X_0(p^2), using only the representation theory of the group
GL_2(Z/pZ). In fact, we prove a generalization of Chen's result for objects
with an action by GL_2(Z/pZ) in any pseudo-abelian Q-linear category.
| [
{
"version": "v1",
"created": "Wed, 10 Apr 1996 13:41:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Edixhoven",
"Bas",
"",
"University of Rennes 1"
]
] | alg-geom | \section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
-.2ex}{2.3ex plus .2ex}{\large\bf}}
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\begin{document}
\title{On a result of Imin Chen.}
\author{Bas Edixhoven}
\maketitle
\section{Introduction, notation and results.}\label{section1}
The aim of this text is to give another proof of a recent result of
Imin Chen, concerning certain identities among zeta functions
of modular curves, or, equivalently, isogenies between products of
jacobians of these curves. I want to thank Imin Chen for pointing
out a mistake in an earlier version of this text.
For $n\geq1$ an integer, let $X(n)_{\Bbb Q}$ be the modular
curve which is the compactified moduli space (coarse if $n<3$)
of pairs $(E/S/{\Bbb Q},\phi)$, where $S$ is a ${\Bbb Q}$-scheme, $E/S$ is
an elliptic curve and $\phi\colon ({\Bbb Z}/n{\Bbb Z})_S^2\to E[n]$ an
isomorphism of group schemes over~$S$. By construction, the group
${\rm GL}_2({\Bbb Z}/n{\Bbb Z})$ acts from the right on $X(n)_{\Bbb Q}$: an
element $g$ sends $(E/S/{\Bbb Q},\phi)$ to $(E/S/{\Bbb Q},\phi\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}} g)$.
This action induces a left action of the jacobian $J(n)_{\Bbb Q}$
of~$X(n)_{\Bbb Q}$.
Let $p$ be a prime number. Let $X$ denote $X(p)_{\Bbb Q}$ and $G$ the
group ${\rm GL}_2({\Bbb F}_p)$.
We will consider the following subgroups of $G$: the standard
``maximal torus'' $T$ consisting of diagonal matrices, a non-split
maximal torus $T'$ obtained by choosing an ${\Bbb F}_p$-basis of
a field ${\Bbb F}_{p^2}$ of $p^2$ elements, the normalizers $N$ of $T$
and $N'$ of~$T'$. Note that $N/T$ and $N'/T'$ are both of order~2.
Finally, let $B_+$ and $B_-$ denote the two Borel subgroups
containing $T$; $B_+$ is the subgroup of upper triangular matrices
and $B_-$ the one of lower triangular matrices.
The quotients of $X$ by some of these subgroups have the following
interpretations. The quotient $X/T'$ is usually denoted
$X(p)_{\mbox{\rm\scriptsize non-split}}$. The constructions
\begin{subeqn}\label{eqn1.0.1}
(E/S,\phi) \mapsto (E/S,\langle \phi(1,0)\rangle), \qquad
(E/S,\phi) \mapsto (E/S,\langle \phi(0,1)\rangle)
\end{subeqn}
induce isomorphisms
\begin{subeqn}\label{eqn1.0.2}
X/B_+\;\;\tilde{\longrightarrow}\;\; X_0(p)_{\Bbb Q}, \qquad
X/B_-\;\;\tilde{\longrightarrow}\;\; X_0(p)_{\Bbb Q}
\end{subeqn}
The construction
\begin{subeqn}\label{eqn1.0.3}
(E/S,\phi) \mapsto (E_1/S,\ker(\phi_2\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}\phi_1^*))
\end{subeqn}
where $\phi_1\colon E\to E_1$ (resp. $\phi_2\colon E\to E_2$) is the
isogeny whose kernel is the subgroup scheme generated by $\phi(1,0)$
(resp. $\phi(0,1)$), induces an isomorphism
\begin{subeqn}\label{eqn1.0.4}
X/T\;\;\tilde{\longrightarrow}\;\;X_0(p^2)_{\Bbb Q}
\end{subeqn}
Under this isomorphism the Atkin-Lehner
involution $w_{p^2}$ of $X_0(p^2)_{\Bbb Q}$ corresponds to the non-trivial
element of~$N/T$; the two maps $X/T\to X/B_+$ and $X/T\to X/B_-$
correspond to the two standard degeneracy maps from $X_0(p^2)_{\Bbb Q}$
to~$X_0(p)_{\Bbb Q}$.
The result of Chen is the following, see \cite[Theorem~1 and \S10]{Chen}.
\begin{theorem}\label{theorem1.1} {\bf (Chen)}
The jacobian of $X_0(p^2)_{\Bbb Q}$ is isogeneous to the product of the
jacobian of $X(p)_{\mbox{\rm\scriptsize non-split}}$ by the square of the jacobian of
$X_0(p)_{\Bbb Q}$. The jacobian of $X_0(p^2)_{\Bbb Q}/\langle w_{p^2}\rangle$
is isogeneous to the product of the jacobian of $X/N'$ by the
jacobian of~$X_0(p)_{\Bbb Q}$.
\end{theorem}
The proof given by Chen is to show that the traces of the Hecke
operators $T_n$ ($n$ prime to $p$) on the jacobians in the theorem
satisfy the identities required to conclude by the Eichler--Shimura
relations and Faltings's isogeny theorem that one has the desired
isogenies. We will prove a generalization of Theorem~\ref{theorem1.1}
using only the representation theory of~$G$ and some elementary properties
of abelian varieties.
For a field $k$, let ${\rm AV}(k)$ denote the category of abelian varieties
over~$k$. Let ${\QQ\otimes\AV}(k)$ denote the category of abelian varieties over $k$
``up to isogeny'', i.e., its objects are those of ${\rm AV}(k)$ and for two
objects $A$ and $B$ one has
${\rm Hom}_{{\QQ\otimes\AV}(k)}(A,B) = {\Bbb Q}\otimes{\rm Hom}_{{\rm AV}(k)}(A,B)$. For $A$ an abelian
variety over $k$ we denote by ${\Bbb Q}\otimes A$ the corresponding object
of~${\QQ\otimes\AV}(k)$. By construction, $A$ and $B$ are isogeneous if and only
if ${\Bbb Q}\otimes A$ and ${\Bbb Q}\otimes B$ are isomorphic. The categories
${\QQ\otimes\AV}(k)$ are ${\Bbb Q}$-linear, semi-simple and abelian.
Recall (e.g., see \cite[\S1]{Scholl1}), that an additive category ${\cal C}$
is called pseudoabelian if for every object $M$ of ${\cal C}$ every
idempotent $f$ in ${\rm End}(M)$ has a kernel (or, equivalently, an image).
If ${\cal C}$ is additive, pseudoabelian and $f$ in ${\rm End}(M)$ is an
idempotent in ${\cal C}$, then the natural morphism from ${\rm im}(f)\oplus\ker(f)$
to $M$ is an isomorphism. The categories ${\QQ\otimes\AV}(k)$ are clearly additive
and pseudoabelian.
For each subgroup $H$ of $G$ we define
\begin{subeqn}\label{eqn1.1.1}
{\rm pr}_H := \frac{1}{|H|}\sum_{h\in H} h \in {\Bbb Q}[G]
\end{subeqn}
Hence ${\rm pr}_H$ is the idempotent of ${\Bbb Q}[G]$ that projects on the
$H$-invariants. For two subgroups $H_1$ and $H_2$ of $G$ such that
$\langle H_1\cup H_2\rangle = H_1H_2$, one has
${\rm pr}_{H_1}{\rm pr}_{H_2}={\rm pr}_{\langle H_1\cup H_2\rangle}$. For $H$ a subgroup
and $g$ in $G$ one has $g{\rm pr}_Hg^{-1}={\rm pr}_{gHg^{-1}}$, hence ${\rm pr}_H$ is
a central idempotent if and only if $H$ is a normal subgroup.
For each irreducible representation $V$ of $G$ over ${\Bbb Q}$ let $e_V$
be the corresponding central idempotent in~${\Bbb Q}[G]$ which projects on
the $V$-isotypical part. If $V$ is absolutely irreducible, of dimension~$d$
and with character $\chi$, one has:
\begin{subeqn}\label{eqn1.1.2}
e_V := \frac{d}{|G|}\sum_{g\in G} \chi(g^{-1})g
\end{subeqn}
We will use only one idempotent of the form $e_V$, namely, with $V$ the
representation with character $\pi^-(1)$ (see Table~\ref{table2.1}). This
representation is the $p$-dimensional irreducible subrepresentation of
the induction of the trivial representation from $B_+$ to~$G$. It is
clearly absolutely irreducible and it exists over~${\Bbb Q}$.
Let us for the moment admit the following proposition, whose proof
will be given in the next section.
\begin{proposition}\label{proposition1.2}
Suppose that $p\neq2$. The elements
${\rm pr}_{T'}(1-{\rm pr}_G)$ and ${\rm pr}_T(1-e_{\pi^-(1)})(1-{\rm pr}_G)$
of the ring ${\Bbb Q}[G]$ are conjugate idempotents. Likewise, the elements
$({\rm pr}_{N'}+{\rm pr}_{B_+})(1-{\rm pr}_G)$ and ${\rm pr}_{N}(1-{\rm pr}_G)$ are conjugate
idempotents.
\end{proposition}
Our generalization of Chen's result is simply the following direct
consequence of Proposition~\ref{proposition1.2}.
\begin{theorem}\label{theorem1.3}
Suppose that $p\neq2$. Take elements $u$ and $v$ of ${\Bbb Q}[G]^*$ such that
\begin{eqnarray*}
u{\rm pr}_{T'}(1-{\rm pr}_G)u^{-1} & = & {\rm pr}_T(1-e_{\pi^-(1)})(1-{\rm pr}_G)\\
v({\rm pr}_{N'}+{\rm pr}_{B_+})(1-{\rm pr}_G)v^{-1} & = & {\rm pr}_{N}(1-{\rm pr}_G)
\end{eqnarray*}
Let ${\cal C}$ be a ${\Bbb Q}$-linear pseudoabelian additive category. Let $M$ be an
object of ${\cal C}$ with an action by the group~$G$; this gives a morphism
of rings ${\Bbb Q}[G]\to{\rm End}(M)$. Then $u$ induces an isomorphism
$$
{\rm pr}_{T'}(1-{\rm pr}_G)M \;\;\tilde{\longrightarrow}\;\; {\rm pr}_T(1-e_{\pi^-(1)})(1-{\rm pr}_G)M
$$
Likewise, $v$ induces an isomorphism
$$
{\rm pr}_{N'}(1-{\rm pr}_G)M\oplus{\rm pr}_{B_+}(1-{\rm pr}_G)M \;\;\tilde{\longrightarrow}\;\;
{\rm pr}_N(1-{\rm pr}_G)M
$$
\end{theorem}
To see that Theorem~\ref{theorem1.1} is a special case, apply
Theorem~\ref{theorem1.3} to ${\cal C}:={\QQ\otimes\AV}({\Bbb Q})$ and take
$M={\Bbb Q}\otimes{\rm jac}(X)$, with ${\rm jac}(X)$ the jacobian of~$X$. For any subgroup
$H$ of $G$ one then has ${\rm pr}_HM={\Bbb Q}\otimes{\rm jac}(X/H)$. In this case
${\rm pr}_G$ acts as zero on $M$, since $X/G$ has genus zero.
The idempotent $e_{\pi^-(1)}$, acting on
${\Bbb Q}\otimes{\rm jac}(X/T)={\Bbb Q}\otimes J_0(p^2)$, projects on the old part, which is
a product of two copies of ${\Bbb Q}\otimes J_0(p)$ (one way to see this is to
note that the space of $T$-invariants in the representation corresponding
to $\pi^-(1)$ is the direct sum of the two $1$-dimensional spaces of $B_+$
and $B_-$-invariants).
One also has to
use the interpretations of the $X/H$ as explained in
the beginning of this section. For the case $p=2$, note that $X(2)_{\Bbb Q}$ has
genus zero.
\section{The proof of Proposition~1.2.}\label{section2}
The notation is as in the previous section, in particular, $G={\rm GL}_2({\Bbb F}_p)$.
We suppose that $p\neq2$.
We will need to do some calculations involving the irreducible characters
of~$G$, so for convenience of the reader and to fix the notation, we
include its character table, taken from~\cite{Cartier}:
\begin{tabel}{The character table of $G$.}\label{table2.1}
$$
\renewcommand{\arraystretch}{1.5}
\begin{array}{|l||c|c|c|c|}
\hline
\mbox{\rm conjugacy class of} & \pi(\alpha,\beta),\;\;\alpha\neq\beta &
\pi(\Lambda),\;\;\Lambda^p\neq\Lambda & \alpha\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}\det & \pi^-(\alpha) \\
\hline \hline
\bigl({x\atop 0}{0\atop x}\bigr)\;\;x\in{\Bbb F}_p^* &
(p+1)\alpha(x)\beta(x) & (p-1)\Lambda(x) & \alpha(x)^2 & p\alpha(x)^2 \\
\hline
\bigl({x\atop 0}{0\atop y}\bigr)\;\;x,y\in{\Bbb F}_p^*\;\;x\neq y &
\alpha(x)\beta(y)+\alpha(y)\beta(x) & 0 & \alpha(x)\alpha(y) &
\alpha(x)\alpha(y) \\
\hline
\bigl({x\atop 0}{1\atop x}\bigr)\;\;x\in{\Bbb F}_p^* &
\alpha(x)\beta(x) & -\Lambda(x) & \alpha(x)^2 & 0 \\
\hline
\bigl({z\atop 0}{0\atop z^p}\bigr)\;\;z\in{\Bbb F}_{p^2}^*\;\;z^p\neq z &
0 & -\Lambda(z)-\Lambda(z^p) & \alpha(z^{p+1}) & -\alpha(z^{p+1}) \\
\hline
\end{array}
$$
\end{tabel}
In this table $\alpha$ and $\beta$ denote characters ${\Bbb F}_p^*\to{\overline{\QQ}}^*$ and
$\Lambda$ denotes a character ${\Bbb F}_{p^2}^*\to{\overline{\QQ}}^*$. For each effective
character $\chi$ of $G$ we denote by $V_\chi$ some ${\overline{\QQ}}[G]$-module with
character~$\chi$. For each irreducible $\chi$ and each of the subgroups
$H\subset G$ mentioned at the beginning of \S\ref{section1}, we will need to
know the dimension $\dim(V_\chi^H)$ of the set of $H$-invariants in~$V_\chi$.
These dimensions are given in the following table, in which
$\delta(x,y)$ denotes the Kronecker symbol, i.e., $\delta(x,y)=1$ if $x=y$
and $\delta(x,y)=0$ otherwise.
\begin{tabel}{The dimensions of the spaces $V_\chi^H$. }\label{tabel2.2}
$$
\renewcommand{\arraystretch}{1.5}
\begin{array}{|l||c|c|c|c|}
\hline
& \pi(\alpha,\beta) & \pi^-(\alpha) & \alpha\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}\det & \pi(\Lambda) \\
\hline \hline
T & \delta(\alpha\beta,1) & \delta(\alpha,1)+\delta(\alpha^2,1) &
\delta(\alpha,1) & \delta(\Lambda^{p+1},1) \\
\hline
N & \delta(\alpha(-1),1)\delta(\alpha\beta,1) &
\delta(\alpha(-1),1)\delta(\alpha^2,1) & \delta(\alpha,1) &
\delta(\Lambda^{p+1},1)-\delta(\Lambda^{(p+1)/2},1) \\
\hline
T' & \delta(\alpha\beta,1) & -\delta(\alpha,1)+\delta(\alpha^2,1) &
\delta(\alpha,1) & \delta(\Lambda^{p+1},1) \\
\hline
N' & \delta(\alpha(-1),1)\delta(\alpha\beta,1) &
-\delta(\alpha,1)+\delta(\alpha(-1),1)\delta(\alpha^2,1) &
\delta(\alpha,1) &
\delta(\Lambda^{p+1},1)-\delta(\Lambda^{(p+1)/2},1) \\
\hline
B & 0 & \delta(\alpha,1) & \delta(\alpha,1) & 0 \\
\hline
\end{array}
$$
\end{tabel}
We will not give the computation of this table in detail, since it is
a straightforward application of the theory of representations of finite
groups, see for example~\cite{Serre1}. As an example, let us do the
case $\chi=\pi(\Lambda)$ and $H=N'$ (the other computations are in fact
easier). The group $N'$ can be identified with the subgroup of
${\rm GL}_{{\Bbb F}_p}({\Bbb F}_{p^2})$ generated by ${\Bbb F}_{p^2}^*$ and $\sigma$, where
$\sigma$ is the automorphism of order two of~${\Bbb F}_{p^2}$. Then $N'$ is
the disjoint union of $T'={\Bbb F}_{p^2}^*$ and $T'\sigma$. The conjugacy
class in $G$ of $z\in T'$ is the conjugacy class
of~$({z\atop0}{0\atop z^p})$. The conjugacy class of $z\sigma$ is
the one of~$({z^{(p+1)/2}\atop 0}{0\atop -z^{(p+1)/2}})$.
One has:
\begin{subeqn}\label{eqn2.2.1}
\dim(V_\chi^H) = \dim {\rm Hom}_H({\rm Res}^G_H(V_\chi),{\overline{\QQ}}) =
\frac{1}{|H|}\sum_{g\in H}\chi(g)
\end{subeqn}
The sum over the elements of $T'$ can be written as:
\begin{subeqn}\label{eqn2.2.2}
-\sum_z\left(\Lambda(z)+\Lambda(z^p)\right) + (p+1)\sum_x\Lambda(x)
\end{subeqn}
In this sum, $z$ runs through ${\Bbb F}_{p^2}^*$ and $x$ through ${\Bbb F}_p^*$.
The first of the two terms of (\ref{eqn2.2.2}) gives zero, the second
contributes ${1\over2}\delta(\Lambda^{p+1},1)$ to~$\dim(V_\chi^H)$.
The sum over the elements of $T'\sigma$ can be written as
\begin{subeqn}\label{eqn2.2.3}
\sum_{z^{(p+1)/2}\in{\Bbb F}_p}
\left(\Lambda(z^{(p+1)/2})+\Lambda(-z^{(p+1)/2})\right) -
\sum_z\left(\Lambda(z^{(p+1)/2})+\Lambda(-z^{(p+1)/2})\right)
\end{subeqn}
The first of the two terms of (\ref{eqn2.2.3}) contributes
${1\over2}\delta(\Lambda^{p+1},1)$ to~$\dim(V_\chi^H)$ and the second
term contributes~$-\delta(\Lambda^{(p+1)/2},1)$. This completes the
computation of~$\dim(V_\chi^H)$.
As promised, we will now give a proof of Proposition~\ref{proposition1.2}.
In fact, that proposition is a direct consequence of the following one.
\begin{proposition}\label{proposition2.3}
Define $\overline{{\Bbb Q}[G]}:={\Bbb Q}[G]/({\rm pr}_G)$ and denote the projection
${\Bbb Q}[G]\to\overline{{\Bbb Q}[G]}$ by $u\mapsto\overline{u}$. Then the elements
$\overline{{\rm pr}_{T'}}$ and $\overline{{\rm pr}_T}(1-\overline{e_{\pi^-(1)}})$
of the ring $\overline{{\Bbb Q}[G]}$ are conjugate idempotents. Likewise, the elements
$\overline{{\rm pr}_{N'}}+\overline{{\rm pr}_{B_+}}$ and $\overline{{\rm pr}_{N}}$ are conjugate idempotents.
\end{proposition}
\begin{proof}
Consider the first statement. Both elements are clearly idempotents.
The ${\Bbb Q}$-algebra ${\Bbb Q}[G]$ is a product of matrix algebras over division
rings. Using Table~\ref{tabel2.2} one verifies that the two elements in
question generate, in each factor, two left ideals of the same dimension
over ${\Bbb Q}$ (actually, one verifies this after extension of scalars to
${\overline{\QQ}}$). Lemma~\ref{lemma2.4} then implies that the two elements are
conjugates.
The proof of the second statement is almost the same. The element
$\overline{{\rm pr}_{N'}}+\overline{{\rm pr}_{B_+}}$ is an idempotent because $T'B_+=G$. The rest
of the proof runs as before.
\end{proof}
\begin{lemma}\label{lemma2.4}
Let $\Delta$ be a division ring. Let $0\leq k\leq n$ be integers. Then
the group ${\rm GL}_n(\Delta)$ acts transitively (by conjugation) on the set
of idempotents of rank $k$ in ${\rm M}_n(\Delta)$.
\end{lemma}
\begin{proof}
Consider the right $\Delta$-module $\Delta^n$. Then ${\rm M}_n(\Delta)$
can be viewed as ${\rm End}_\Delta(\Delta^n)$. The map that associates to an
idempotent of rank $k$ its kernel and image is a bijection between the set
of such idempotents and the set of pairs of $\Delta$-submodules $(V_1,V_2)$
such that $\dim_\Delta(V_2)=k$ and $\Delta^n=V_1\oplus V_2$. One verifies
easily that ${\rm Aut}_\Delta(\Delta^n)$ acts transitively on the set of such
pairs.
\end{proof}
|
1994-02-15T23:01:09 | 9402 | alg-geom/9402011 | en | https://arxiv.org/abs/alg-geom/9402011 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9402011 | null | M.S.Ravi, J. Rosenthal and X.Wang | Degree of the generalized Pl\"ucker embedding of a Quot scheme and
Quantum cohomology | 18 pages, Latex document | null | 10.1007/s002080050173 | null | null | We compute the degree of the generalized Pl\"ucker embedding $\kappa$ of a
Quot scheme $X$ over $\PP^1$. The space $X$ can also be considered as a
compactification of the space of algebraic maps of a fixed degree from $\PP^1$
to the Grassmanian $\rm{Grass}(m,n)$. Then the degree of the embedded variety
$\kappa (X)$ can be interpreted as an intersection product of pullbacks of
cohomology classes from $\rm{Grass}(m,n)$ through the map $\psi$ that evaluates
a map from $\PP^1$ at a point $x\in \PP^1$. We show that our formula for the
degree verifies the formula for these intersection products predicted by
physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We
arrive at the degree by proving a version of the classical Pieri's formula on
the variety $X$, using a cell decomposition of a space that lies in between $X$
and $\kappa (X)$.
| [
{
"version": "v1",
"created": "Tue, 15 Feb 1994 21:59:42 GMT"
}
] | 2017-01-06T00:00:00 | [
[
"Ravi",
"M. S.",
""
],
[
"Rosenthal",
"J.",
""
],
[
"Wang",
"X.",
""
]
] | alg-geom | \section{The cellular decomposition of $A^q_{m,p}$}
\setcounter{equation}{0}
In this section we summarize those
results obtained in~\cite{wa94} which we will use
in this paper. The space $A^q_{m,p}$ consists of equivalence
classes of polynomial matrices
$M(s)=(M_{ij}(s))_{1\le i\le m,\ 1\le j\le n}$
such that the degree of any $m\times m$
minor of $M$ is at most $q$ and at least one of these minors is
a non-zero polynomial. Two such
matrices $M(s)$ and $M'(s)$ are considered equivalent, if there exists a
unimodular polynomial matrix $U(s)$ such that $M'(s)=U(s)M(s)$.
\begin{de}
Given any $m\times n$ polynomial matrix
$M(s)$, there exist
unique $\nu=(\nu_{1},\dots,\nu_{m})$ with $\nu_{1}\leq \cdots \leq
\nu_{m}$ and
$$\sum_{i=1}^{m}\nu_{i}=
\mbox{maximum degree of $m\times m$ minors of $M(s)$} $$
and an $m\times m$
unimodular matrix $U(s)$ such that the matrix $M'(s)=U(s)M(s)$
has row degrees $\nu_{1}\leq \cdots \leq
\nu_{m}$.
The numbers $\nu =(\nu_{1}, \ldots , \nu_{m})$ are
called the ordered Kronecker indices of the equivalence class of
$M(s)$.
\end{de}
A matrix $M(s)$ is called row reduced if
the ordered Kronecker indices are equal to the degrees of
the rows of $M(s)$.
$M(s)$ is row reduced if and only if the high order coefficient matrix
of $M(s)$ has full rank, where the high order coefficient matrix
of a polynomial matrix $M(s)$ is a matrix whose entries of the $i$-th row
are the coefficients of $s^{\nu_i}$ of the $i$-th row of $M(s)$ where
$\nu_i$ is the highest power of $s$ in the $i$-th row of $M(s)$.
\begin{de}(\cite{fo75}\cite{wa94})
Given an $m\times n$ row reduced polynomial matrix $M(s)$ with $M_h$
the high order coefficient matrix of $M(s)$,
the $i$-th pivot index $\mu_i'$ is the largest integer such
that the submatrix of $M_h$ formed from the intersection of columns
$\mu_1',\dots,\mu_i'$ with the rows corresponding to
indices $\leq \nu_i$ has rank $i$. The ordered pivot indices
$\mu=(\mu_1,\mu_2,\dots,\mu_m)$ of the equivalence class of
$M(s)$
are the indices obtained from
$(\mu_1',\mu_2',\dots,\mu_m')$ by reordering such that $\mu_i<\mu_{i+1}$
if $\nu_i=\nu_{i+1}$.
\end{de}
Let $\tilde{I}(m,p)$ be the set of $m$-tuple of integers defined by
$$
\tilde{I}(m,p)=\{\alpha=(\alpha_1,\dots,\alpha_m)|
1\le \alpha_1<\cdots<\alpha_m,\ \alpha_k\neq \alpha_l \bmod n\mbox{ for }
k\ne l\}.
$$
\begin{de} For each equivalence class $M\in A^q_{m,p}$ with Kronecker
index $\nu =(\nu_1,\dots ,\nu_m)$ and ordered pivot index
$\mu=(\mu_1,\dots ,\mu_m)$ we define a new index
$\alpha=(\alpha_1,\dots \alpha_m)\in \tilde{I}(m,p)$
by
$\alpha_l :=n\nu_l+\mu_l$.
\end{de}
Further for each index $\alpha\in \tilde{I}(m,p)$ we define
\begin{equation}\label{norm2}
|\alpha|=\sum_{l=1}^m (\alpha_l-l)-
\sum_{l=2}^m\sum_{k=1}^{l-1}\left[ {\frac{\alpha_l-\alpha_k}{n}}\right]
\end{equation}
where $[r]$ is the largest integer less than or equal to $r$.
We also define a partial order on $\tilde{I}(m,p)$ as follows:
first associate with each
$\alpha=(\alpha_1,\dots,\alpha_m)$ an infinite
sequence:
$$
f(\alpha)=(f_1(\alpha),f_2(\alpha),\cdots)
$$
where
$$
\{ f_l(\alpha) \}
=\{\alpha_j+k(n)\mid j=1,\dots,m\mbox{ and }k=0,1,2,\dots,\}
$$
and order the set such that
$$f_1(\alpha)<f_2(\alpha)<\cdots,$$
and then define the partial order on $\tilde{I}(m,p)$ by
\begin{equation}\label{partial2}
\alpha\leq \beta\mbox{ if and only if }
f_l(\alpha)\leq f_l(\beta)\mbox{ for all $l$}.
\end{equation}
Next we define a topology on $A^q_{m,p}$. Let ${\cal P}^q_{m,p}$
be the set of all $m\times n$ full rank polynomial
matrices of degree at most $q$ and denote with
${\cal P}^{q,r}_{m,p}$ the subset of ${\cal P}^q_{m,p}$ formed by all
matrices whose entries are polynomials of degree at most $r$.
Then
$${\cal P}^{q,0}_{m,p}\subset {\cal P}^{q,1}_{m,p}\subset
{\cal P}^{q,2}_{m,p}\subset \cdots$$
with the union $${\cal P}^q_{m,p}=
\bigcup_{r=0}^{\infty}{\cal P}^{q,r}_{m,p}.$$
The set of all $m\times n$ polynomial matrices whose entries
are polynomials of degree at most $r$ is an affine space
$${\bb C}^{mn(r+1)}$$
and the conditions that the degrees of the $m\times m$ minors are
at most $q$ are polynomial conditions on ${\bb C}^{mn(r+1)}$
which defines an algebraic set.
${\cal P}^{q,r}_{m,p}$ is a Zariski open set of this algebraic set.
Take the Zariski topology on ${\cal P}^{q,r}_{m,p}$.
The direct limit of the topologies on
${\cal P}^{q,r}_{m,p}\ \ r=0,1,\dots$ defines a topology
on ${\cal P}^q_{m,p}$. In other
words, a subset of ${\cal P}^q_{m,p}$ is open if and only if its
intersection with ${\cal P}^{q,r}_{m,p}$ is open as a subset of
${\cal P}^{q,r}_{m,p}$ for each $r$. The topology which
we take on $A^q_{m,p}$ is the quotient
topology under row equivalence,
i.e.
a subset $U$ of $A^q_{m,p}$ is open if, and only if
the subset $V$ of $P^q_{m,p}$ formed by all the polynomial
matrices in the equivalence classes of $U$ is open.
Since the minors of an $m\times n$ matrix are polynomials of its
entries,
any polynomial condition on $K^q_{m,p}$ induces a polynomial
condition on ${\cal P}^q_{m,p}$. So the map $\pi$ defined in\r{plu} is
continuous under the topology we defined on $A^q_{m,p}$.
Let $C_\alpha$ be the subset of $A^q_{m,p}$ consisting all the
elements with index $\alpha$. The main result given
in~\cite{wa94} can then be summarized in the following
proposition:
\begin{pr}~\cite{wa94} \label{c}
\begin{itemize}
\item[1)] $C_\alpha$ is an open cell of dimension $|\alpha|$.
\item[2)] $C_\alpha\cap C_\beta=\emptyset$ if $\alpha\neq
\beta$.
\item[3)] $\overline{C}_\alpha={\displaystyle
\bigcup_{\beta\in \tilde{I}(m,p) \atop \beta\leq\alpha}}C_\beta$.
\end{itemize}
\end{pr}
\section{Generalized Schubert Subvarieties of $K^q_{m,p}$}
\setcounter{equation}{0}
We fix the coordinates of $K^q_{m,p}$ first.
For each $i=(i_1,\dots,i_m)$, $0\le i_1<\cdots< i_m\le n$, let
$$
z_{(i;0)}t^q+z_{(i;1)}t^{q-1}s+\cdots+z_{(i;q)}s^q
$$
be the $m\times m$ minor of an $M(s,t)\in X$ consisting of the $i_1$th
through $i_m$th columns. Then
$$
z=(z_{(i;d)})_{0\le i_1<\cdots< i_m\le n,\ 0\le d \le q}.
$$
is the homogeneous coordinate of the image of $M$ in $K^q_{m,p}$.
Let $I(m)$ be the set of $m$-tuple of integers defined by
$$
I(m)=\{i=(i_1,\dots ,i_m)\,|\, 1\le i_1<\dots <i_m\}.
$$
Define
\begin{equation}\label{norm1}
|i|=\sum_{l=1}^m (i_l-l)
\end{equation}
and the partial order
\begin{equation} \label{partial1}
(i_1,\dots,i_m)\leq (j_1,\dots,j_m)\mbox{ if and only if }
i_l\leq j_l\mbox{ for all $l$}
\end{equation}
on $I(m)$.
Let $e_l$ be the vector whose $l$th component is $1$ and all the other
components are zero and
$$F_l={\rm sp}\ \{e_1,\dots,e_l\}.$$
For any $i=(i_1,\dots,i_m)\in I(m)$, $i\le (p+1,\dots,n)$, let $S_i$
be the Schubert variety $S(F_{i_1},\dots,F_{i_m})$ under the standard
flag $F_{i_1}\subset\cdots\subset F_{i_m}$; i.e.
\begin{equation}\label{schubert}
S_i=S(F_{i_1},\dots,F_{i_m})
=\{x\in{\rm Grass}(m,n)| \dim x\cap F_{i_l}\ge l\}.
\end{equation}
Then $S_i$ is a subvariety of dimension $|i|$ defined by
\begin{equation}\label{svariety}
S_i=\{x\in {\rm Grass}(m,n)
\mid x_j=0\mbox{ for all }j\not\leq i\}
\end{equation}
where $(x_i)$ are the Pl\"ucker coordinates of a point
$x\in{\rm Grass}(m,n)$.
We would like to generalize the Schubert varieties in
Grass$(m,n)=K^0_{m,p}$ to $K^q_{m,p}$.
For each $d\leq q$, the subset $\{z\in K^q_{m,p}\, |\, z_{(i:l)}=0\mbox{
for all }l>d\mbox{ and }i\}$ can be identified naturally with $K^d_{m,p}$.
By abuse of notation we will denote this subset as $K^d_{m,p}\subseteq
K^q_{m,p}$.
Let $\psi_d$ be the projection of
$z\in K^d_{m,p}\subseteq K^q_{m,p}$
on its components $(z_{(i;d)})$; i.e.
\begin{equation}\label{psidef}
\psi_d(z)=\{ z_{(i;d)}| 1\leq i_1<\cdots<i_m\leq n\}.
\end{equation}
Then $\psi_d$ is a rational map (it is in fact a central projection) and
$$
\psi_d(K^d_{m,p})={\rm Grass}(m,n).
$$
A first attempt to generalize Schubert varieties
could be to take $\psi_d^{-1}(S_i)$ in $K^d_{m,p}\subseteq K^q_{m,p}$
for each $d\le q$ and for each Schubert variety $S_i$
in ${\rm Grass}(m,n)$.
Unfortunately this is not a closed set.
A better definition comes from the cell decomposition of $A^q_{m,p}$.
But we first need the following definition.
\begin{de}
For each $(i;d)$, $i=(i_1,\dots,i_m)$, $1\leq i_1<\cdots<i_m\leq n$,
and $d=km+r$, $0\leq r<m$, let $\alpha =(i;d)$ where
$\alpha=(\alpha_1,\dots,\alpha_m)$
is defined through:
\begin{equation}\label{ialpha}
\alpha_l=\left\{\begin{array}{ll}
k(n)+i_{l+r} & \mbox{for } l=1,2,\dots,m-r\\
(k+1)(n)+i_{l-m+r} & \mbox{for } l=m-r+1,\dots,m.
\end{array}\right.
\end{equation}
\end{de}
Then $C_\alpha$ with $\alpha=(\alpha_1,\dots,\alpha_m)=(i;d)$ is the
``thickest'' cell among all the cells $C_\beta$ in $\pi^{-1}(K^d_{m,p})$
such that $$\psi_d(\pi(C_\beta))\subset S_i.$$
It can be proved that $\pi(\overline{C}_\alpha)$ is a
subvariety (see Lemma~\ref{lem37} and Proposition~\ref{irreducible}) and
$\psi_d(\pi(\overline{C}_\alpha))=S_i$. So $\pi(\overline{C}_\alpha)$ is
the variety we want.
In order to give a definition similar to (\ref{svariety}) we
re-index the coordinates of ${\bb P}\supset K^q_{m,p}$,
as
$z_{(i;d)}=z_\alpha.$
We set
$$
I(m,p)=\{\alpha=(\alpha_1,\dots,\alpha_m)\, |\, 1\le\alpha_1
<\cdots <\alpha_m,\ \alpha_m-\alpha_1<n\}.
$$
If $i\in I(m)$ and $i\leq (p+1,\dots,n)$ then $\alpha=(i;d)\in I(m,p)$
and for each $\alpha\in I(m,p)$ there exists unique $i\in I(m)$,
$i\le (p+1,\dots,n)$, and
$d$ such that $\alpha
= (i;d)$. Further, for $I(m,p)$
the notion of the partial order~(\ref{partial2}) agrees with the partial
order~(\ref{partial1}) and
the notion of $|\alpha|$ defined in~(\ref{norm2}) reduces
to~(\ref{norm1}). So $I(m,p)$ is a subset of $\tilde{I}(m,p)$ as well as
$I(m)$.
\begin{re}\label{remarki}
In a partially ordered set, an element $\alpha$ is said to
cover another element $\beta$ if $\beta<\alpha$ and there exists
no $\gamma$ such that $\beta<\gamma<\alpha$~\cite{kr86}.
{}From the definition of the
partial order one can see that for any
$\alpha=(i;d)$ and $\beta=(j;b)$ in $I(m,p)$,
$\alpha$ covers $\beta$ if and only if either
\begin{itemize}
\item[a.] $b=d$ and $i$ covers $j$ or
\item[b.] $b=d-1$, $i=(1,i_2,\dots,i_m)$ and $j=(i_2,\dots,i_m,n)$
for some $1<i_2<\cdots<i_m<n$.
\end{itemize}
\end{re}
\begin{de}\label{zdef}
For any $\alpha\in I(m,p)$, $\alpha=(i;d)\le (p+1,\dots n;q)$, let
$Z_\alpha$, or $Z_i^d$, be the closed subset of $K^q_{m,p}$
defined by $$
Z_\alpha=Z_i^d
=\{z\in K^q_{m,p}| z_\beta=0\mbox{ for all }\beta\not\leq\alpha \}
$$
and
$O_\alpha$, or $O_i^d$, be the open set of $Z_\alpha$ defined by
$$
O_\alpha=O_i^d
=\{z\in Z_\alpha| z_\alpha\neq 0 \}
$$
\end{de}
\begin{pr}\label{o}
$$
Z_\alpha=\bigcup_{\beta\in I(m,p), \atop \beta\leq\alpha} O_\beta.
$$
\end{pr}
\begin{proof}
Follows from the definition.
\end{proof}
\begin{lem}\label{lem37}
Let $\alpha=(i;d)$. Then
\begin{itemize}
\item[1)]
$
O_\alpha=\bigcup \pi(C_\beta),
$
where the union is over all cells in $A^q_{m,p}$
with Kronecker indices
$
\nu=(\nu_1,\dots,\nu_m),\ \sum \nu_l=d
$ and pivot index
$\mu=(\mu_1,\dots,\mu_m)$ such that
$\{\mu_1,\dots,\mu_m\}=\{i_1,\dots,i_m\}
$
as unordered sets.
\item[2)]
\begin{equation}\label{z}
Z_\alpha=\bigcup_{\beta\in \tilde{I}(m,p),
\atop \beta\leq \alpha}\pi(C_\beta)=\pi
(\overline{C}_\alpha). \end{equation}
\end{itemize}
\end{lem}
\begin{proof}
Using the echelon form of the elements in $C_\alpha$
(see~\cite{wa94}) one can see that
$
\pi (C_\alpha)\subset Z_\alpha.
$
Since
$C_\beta\subset \overline{C}_\alpha$
for all $\beta$ with
Kronecker indices
$\nu,\
\sum \nu_l =d$
and pivot indices as above
and since $\pi$ is continuous,
$
\bigcup \pi(C_\beta)\subset Z_\alpha.
$
Furthermore,
$z_\alpha\neq 0$
for all the points in $\pi(C_\beta)$.
So
$\bigcup \pi(C_\beta)\subset O_\alpha.
$
On the other hand, for any $z\in O_\alpha$ let $M(s)\in\pi^{-1} (z)$
be row reduced. Then by looking at the high order coefficient
matrix~\cite{fo75,wa94} of $M(s)$ one concludes immediately that
$M$ must have Kronecker index
$\nu,\
\sum \nu_l =d
$ and pivot index
$\mu$
such that $\{\mu_{1}, \ldots , \mu_{m}\}=\{i_1,\dots,i_m\}.
$
Therefore
$\bigcup \pi(C_\beta)=O_\alpha$
and (\ref{z}) follows because of Proposition~\ref{c} and
Proposition~\ref{o}.
\end{proof}
\begin{pr}\label{irreducible}
$Z_\alpha$ is an irreducible subvariety of $K^q_{m,p}$ of dimension
$|\alpha|$.
\end{pr}
\begin{proof}
Since $\overline{C}_\alpha$ is
irreducible and $\pi$ is one to one on the open set consisting
of all matrices whose maximal minors do not have any common factors,
$Z_\alpha=\pi(\overline{C}_\alpha)
$
is irreducible and
$\dim Z_\alpha=\dim
\overline{C}_\alpha=|\alpha|.
$ \end{proof}
\begin{re}
{}From Proposition~\ref{o} and Remark~\ref{remarki} one can see
immediately that
$$\psi_d (Z^d_i)=S_i$$
where $S_i$ is the Schubert variety of Grass$(m,n)$ defined
by~(\ref{svariety}). In the next section we shall show that
$Z_\alpha$ is birationally equivalent to the Schubert variety
$S_\alpha\subset {\rm Grass}(m,n(q+1))$
(see Proposition~\ref{pro41}). This is one of the
reasons why we use two kinds of indices to label the variety.
When $d=0$, $Z^0_i$ reduces to the Schubert variety~$S_i$.
\end{re}
\section{Degree of $Z_\alpha$}
\setcounter{equation}{0}
To prove our formula for the degree we shall apply B\'{e}zout's
theorem on the projective space ${\bb P}$. By \cite{fu84}
Prop. 8.4, if $Z\subset {\bb P}$
is a variety and $H$ is a hyperplane such that
$$Z\bigcap H = \bigcup Z_i
$$ where
$Z_i$ are irreducible subvarieties with $\dim Z_i=\dim Z-1$ then
the degree of $Z$ is given through the formula $\deg Z=\sum
m_i\deg Z_i$ where $m_i$ is the multiplicity of $Z$ and $H$ along
$Z_i$. Furthermore, by \cite{fu84} Remark 8.2, $m_i=1$
if $Z$ is generically non-singular along $Z_i$ and
generically meets $H$ transversally along $Z_i$.
We first construct a rational map from a
Schubert variety
of the Grassmannian
${\rm Grass}(m,n(q+1))$ into $Z_\alpha$.
Consider an $m\times n(q+1)$ full rank matrix
$$
Q\in {\rm Grass}(m,n(q+1)).
$$
Let the Pl\"{u}cker coordinate of
$Q\in{\rm Grass}(m,n(q+1))\subset\tilde{\bb P} $ be
$x=(x_i)$ where $i=(i_1,\dots,i_m)$
and $x_i$ is the $m\times m$ minor of $Q$ consisting of the
$i_1$th through $i_m$th columns. Let
$$
SC_i=\{x\in S_i|x_i\neq 0\}
$$
where $S_i$ is the Schubert variety defined by (\ref{svariety}) with
$n$ replaced by $n(q+1)$.
Then $SC_i$ is a cell and
$$SC_i\cap SC_j=\emptyset,\mbox{ if }\ i\neq j\mbox{ and }
$$
$$
S_i=\bigcup_{j\in I(m), \atop j\leq i} SC_j.
$$
Each $Q\in SC_i$ has a unique echelon form
\begin{equation}\label{echelon}
Q=\left[\begin{array}{ccccccccccccccc}
&&&i_1&&&&i_2&&\cdots&&i_m&&&\\
\ast &\cdots&\ast&1&0&\cdots&0&0&0&\cdots&0&0&0&\cdots&0\\
\vdots&&\vdots&0&\ast&\cdots&\ast&1&0&\cdots&0&0&0&\cdots&0\\
\vdots&&\vdots&\vdots&\vdots&&\vdots&0&\ast&\cdots&0&0&0&\cdots&0\\
\vdots&&\vdots&\vdots&\vdots&&\vdots&\vdots&\vdots&&
\vdots&\vdots&\vdots&&\vdots\\
\ast&\cdots&\ast&0&\ast&\cdots&\ast&0 &\ast&\cdots&\ast&1&0&\cdots&0
\end{array}\right]
\end{equation}
For
$$Q=[Q_0|Q_1|\cdots|Q_q]\in {\rm Grass}(m,n(q+1))
$$
where $\{Q_l\}$ are $m\times n$ matrices,
and $$
i=(i_1,\dots,i_m),\ 1\leq i_1<\cdots<i_m\leq
n,$$ let
$$
z_{(i;0)}+z_{(i;1)}s+\cdots+z_{(i;pq)}s^{pq}$$
be the $m\times m$ minor formed
by the $i_1$th through $i_m$th columns of the polynomial matrix
\begin{equation}\label{poly}
Q(s)=Q_0+Q_1s+\cdots+Q_qs^q.
\end{equation}
Define a rational map
$\tau : {\rm Grass}(m,n(q+1))\rightarrow {\bb P}$ by
$$
\tau (Q)=(z_{(i;d)})_{1\le i_1<\cdots<i_m\le n,\ 0\le d\le q}
$$
for all points $Q\in {\rm Grass}(m,n(q+1))$
for which the maximum degree of the minors of $Q(s)$ is at most $q$
and at least one minor is non-zero
(these points form a locally closed subset).
Since each $z_{(i;d)}$ is a linear combination of $m\times m$
minors of $Q$, which in turn are the coordinates of $\tilde{\bb P}$
there exists a linear subspace $\tilde{E}\subset\tilde{\bb P}$
such that
$\tau$ is the restriction to ${\rm Grass}(m,n(q+1))$ of the linear
projection $\tau :\tilde{\bb P}
-\tilde{E}\rightarrow {\bb P}$.
We have
\begin{equation}\label{subset}
\tau(S_\alpha)\subset Z_\alpha.
\end{equation}
For a fixed $\alpha=(\alpha_1,\dots,\alpha_m)\in I(m,p)$,
let
$$
\tilde{U}_\alpha=
\bigcup_{\beta=(\beta_1,\dots,\beta_m)\in I(m,p),\atop
\beta\leq\alpha\ {\rm and}\ \beta_1>\alpha_m-n} SC_\beta
\subset {\rm Grass}(m,n(q+1))
$$
and
$$
U_\alpha=
\bigcup_{\beta=(\beta_1,\dots,\beta_m)\in I(m,p),\atop
\beta\leq\alpha\ {\rm and}\ \beta_1>\alpha_m-n} C_\beta
\subset A^q_{m,p}.
$$
Then $\tilde{U}_\alpha$ and $U_\alpha$ are open sets of $S_\alpha$
and $\overline{C}_\alpha$, respectively.
Let $\phi$ be defined by
$$
\phi(Q)=Q(s)
$$
where $Q(s)$ is defined by (\ref{poly}).
Then the following diagram commutes:
$$
\begin{array}{c}
\tilde{U}_\alpha \ \stackrel{\textstyle \phi}{\rightarrow} \ U_\alpha \\
\tau \searrow\ \ \swarrow \pi\\
Z_\alpha
\end{array}.
$$
If $Q$ is in the echelon form of (\ref{echelon}), then $Q(s)=\phi (Q)$
is in the echelon form defined in~\cite[Proposition 3.5]{wa94}.
Furthermore, if $T$ is the elementary unimodular row operation
which add an $s^k$ multiple of the $l$th row of $Q(s)$ to the
$r$th row, then $\phi^{-1}(T(Q(s)))$ is in
$$SC_{(\beta_1,\dots,\beta_{r-1},\beta_{r+1},
\dots,\beta_l+kn)},
$$ i.e.
$$\phi^{-1}(T(Q(s))\not\in \tilde{U}_\alpha.
$$
Therefore $\phi:\tilde{U}_{\alpha} \rightarrow U_\alpha$
is one to one and onto.
\begin{pr}\label{pro41}
$S_\alpha$ and $Z_\alpha$ are birationally equivalent under $\tau$.
\end{pr}
\begin{proof}
Let the open set $U$ be defined by
\begin{equation}\label{open}
U=\{Q\in \tilde{U}_\alpha|
\mbox{the $m\times m$ minors of $\phi(Q)$ are relative prime}\}.
\end{equation}
Since
$\pi=\tau\circ \phi^{-1}$
and $\pi:\phi(U)\rightarrow Z_\alpha$ is one to one~\cite{fo75},
$\tau: U\rightarrow Z_\alpha$ is one to one, which means that
$S_\alpha$ is birationally equivalent to
$Z_\alpha$ (see~\cite{ha77}, Chapter I, Corollary 4.5).
\end{proof}
The following proposition generalizes the classical
Pieri formula and it is one of the main results of this paper.
\begin{pr} \label{zint}
Let $H_\alpha$ be the hyperplane of ${\bb P}$ defined by setting
$z_\alpha=0$. Then
\begin{equation}\label{intersection}
Z_{\alpha}\bigcap H_{\alpha}
=\bigcup_{\beta\in I(m,p),\atop
\beta<\alpha,\ |\beta|=|\alpha|-1}Z_{\beta}
\end{equation}
and the multiplicity of $Z_\alpha$ and $H_\alpha$ along $Z_\beta$
is one.\end{pr}
\begin{proof}
(\ref{intersection}) follows from Proposition~\ref{o}. So
the only thing we need to prove is that the multiplicity of
the intersection along each $Z_{\beta}$ is one.
Let
$$\tilde{H}_\alpha=\overline{\tau^{-1}(H_\alpha)}=
\tau^{-1}(H_\alpha)\cup \tilde{E}.$$
Then
$$\tau(\tilde{H}_\alpha)=H_\alpha.
$$
$\tilde{H}_\alpha$ is a hyperplane in $\tilde{\bb P}$ defined by
$$0=x_\alpha+\mbox{a linear combination of $x_i$'s
with $i\not\leq \alpha$}.
$$
So
$$S_\alpha\cap \tilde{H}_\alpha=\{x\in S_\alpha|x_\alpha=0\}
=\bigcup_{i\in I(m),\atop
i<\alpha,\ |i|=|\alpha|-1} S_i.
$$
$\tau$
restricted to the open set $U$ defined by~(\ref{open}) is an
isomorphism into an open subset of $Z_\alpha$ and
$U\bigcap S_\beta\neq \emptyset$. So $\tau$ is
a birational isomorphism between $S_\alpha$ and $Z_\alpha$ and
between $S_\beta$ and $Z_\beta$ respectively.
Now, by Pieri's formula~\cite{kl76} applied on the
Grassmannian, the intersection multiplicity of $S_\alpha$ and
$\tilde{H}_\alpha$ along $S_\beta$ is one. By \cite{fu84}~Remark~8.2
and~\ref{pro41},
the intersection multiplicity of $Z_\alpha$ and
$H_ \alpha$ along
$Z_\beta$ is also one.
\end{proof}
By B\'{e}zout's Theorem (see discussion following Prop.~8.4~\cite{fu84})
we have the following:
\begin{lem} \label{deglem}
$$\deg Z_\alpha=
\sum_{\beta\in I(m,p),\atop
\beta<\alpha,\ |\beta|=|\alpha|-1} \deg Z_\beta.
$$ \end{lem}
\begin{th} \label{mainth1}
The degree of $Z_\alpha$ is equal to the number of maximal totally
ordered subsets
of
\begin{equation}\label{indexset}
I_\alpha=\{\beta\in I(m,p)|\beta\leq \alpha\}.
\end{equation}
\end{th}
\begin{proof}
We use induction on the dimension $|\alpha|$ of the
variety $Z_\alpha$. When $|\alpha|=0$, $Z_\alpha=Z_{(1,\dots,p)}$
is a point and its degree is $1$. Assume that the degree of
$Z_\beta$ is equal to the number of maximal totally
ordered subsets of $I_\beta$
for any $|\beta|=|\alpha|-1$.
Notice that a set $I$ is a maximal totally ordered subset of
$I_\beta$ for some $\beta$ covered by $\alpha$ if and only if
$I\cup\{\alpha\}$ is a maximal totally ordered subset of $I_\alpha$.
By Lemma~\ref{deglem},
\begin{eqnarray*}
\deg Z_\alpha&=&
\displaystyle{\sum_{\beta\in I(m,p),\atop
\beta< \alpha,\ |\beta|=|\alpha|-1}}
\deg Z_\beta \\
&=&\displaystyle{\sum_{\beta\in I(m,p),\atop
\beta< \alpha,\ |\beta|=|\alpha|-1}}
\mbox{\# of totally ordered subsets of $I_\beta$}\\
&=&\mbox{\# of totally ordered subsets of $I_\alpha$}.
\end{eqnarray*}
\end{proof}
\begin{co}
The degree of $K^q_{m,p}$ is equal to the number of maximal totally
ordered subsets of
$$I_{(p+1,p+2,\dots,n;q)}.$$
\end{co}
In the next section we shall show that the number of maximal totally ordered
subsets is also measured by the formula conjectured in Physics. In order
to show this equivalence, we find it useful to give an abstract
characterization of a function $d(\alpha_1,\dots,\alpha_m)$ that measures the
degree of $Z_\alpha$ as follows:
\begin{co}
The function
\begin{equation}
d(\alpha_1,\dots,\alpha_m):=\deg Z_\alpha
\end{equation}
is the unique solution of the partial recurrence relation
\begin{equation} \label{recurr}
d(\alpha_1,\dots,\alpha_m)=\sum_{l=1}^m
d(\alpha_1,\dots,\alpha_l-1,\dots,\alpha_m)
\end{equation}
subject to the boundary conditions
\begin{eqnarray}
d(\dots,k,k,\dots)&=&0,\label{bound1}\\
d(k,\dots,k+n)&=&0.\label{bound2}
\end{eqnarray}
and subject to the initial conditions
\begin{eqnarray}
d(1,2,\dots,m)&=&1,\label{ini1}\\
d(0,\dots,\alpha_m)&=&0\hspace{2em}
\mbox{ for $\alpha_m<n$.} \label{ini2}
\end{eqnarray}
\end{co}
In~\cite{ra94b} we use Theorem~\ref{mainth1} and Remark~\ref{remarki}
to give explicit formulas for these degrees.
\section{The conjecture from Physics}
\setcounter{equation}{0}
In this section we shall first give a precise formulation of the
conjecture.
We then interpret the degrees of the subvarieties
$Z_i^d$ that we have just computed through this conjecture. Finally,
we prove that our formula for their degrees
agrees with the conjecture. We shall not address the physics
behind the conjecture at
all. We refer the reader to~\cite{wi94}~\cite{va92} for a
discussion of the physical aspects.
The cohomology ring of the Grassmannian is generated by the Chern classes
$X_1,\dots ,X_m$ of the canonical subbundle $S$ on ${\rm Grass}(m,n)$.
The complex codimension of the class $X_i$ is $i$ and the
cohomology class $X_m^p$ is Poincar\`e dual to the class of a point.
Let us fix a point $x\in{\bb P}^1$. Then there is an evaluation
map $\psi : X'\to
{\rm Grass}(m,n)$ that sends a map to its value at $x$.
Now any cohomology class on ${\rm Grass}(m,n)$ can be pulled back via
$\psi$ to $X$. The conjecture predicts
the intersection products of such classes. The first problem with this
is that $X'$ is not a compact space, so one has to interpret these
products on the compactification. But the introduction of the compactification
introduces other questions, namely, do these intersection products
depend on the compactification? In other words, does the boudary component
added to $X'$ change the intersection product.These
questions are somewhat subtle in the case of maps from Riemann
surfaces of higher genus to the Grassmanian
and have been dealt with in~\cite{be93}. In our particular case of maps
from ${\bb P}^1$ to ${\rm Grass}(m,n)$, the compactification chosen by us, namely
the Quot scheme $X$, is a smooth, irreducible variety of
dimension $mp+nq$ for all $m,\, n$ and $q$~\cite{st87}. Further, there is
a universal bundle $\tilde S$ over the Quot scheme $X$ that
extends the pullback $\psi^*(S)$ on $X'$ to all of $X$.
Thus, as in~(\cite{be93}, Section 5.1),
for any set of integers $a_i$ such that $ia_i=mp+nq$
we can define the
intersection products of $\psi^*(X_i)$ unambiguously through
the following definition:
\begin{equation}\label{intdef}
<\psi^*X_1^{a_1}\dots \psi^*X_m^{a_m}>:=
<c_1^{a_1}\cdots c_m^{a_m}>\end{equation}
where $c_i$ is the $i^{\rm th}$ Chern class of the bundle
$\tilde S$ on $X$.
Let $q_1\dots ,q_m$ be the Chern roots of the canonical subbundle $S$ above,
so $X_i$ is the $i^{\rm th}$ elementary symmetric function
in the $q_j$. Let
$$W=\sum_{i=1}^m(\frac{q_i^{n+1}}{n+1}+(-1)^m q_i).$$
This is called the Landau-Ginzburg potential by physicists. Since
$W$ is a symmetric polynomial in the $q_i$, it can
be expressed as a polynomial $W=W(X_1,\dots ,X_m)$
in the elementary symmetric functions.
Let
\begin{equation}
h(X_1,\dots ,X_m):=\det[\frac{\partial^2 W}{\partial X_i \partial X_j}]
\end{equation}
be the determinant
of the Hessian of $W(X_1,\dots ,X_m)$. If $a$ is a
multiindex as above
then the intersection product
$<\psi^*X_1^{a_1}\cdots \psi^*X_m^{a_m}>$ is an integer and the conjecture
says that this number is computed by the formula
\begin{equation}\label{conj}
<\psi^*X_1^{a_1}\cdots \psi^*X_m^{a_m}>=(-1)^{m(m-1)/2}
\sum_{dW=0} \frac{X_1^{a_1}\dots X_m^{a_m}}{h}.\end{equation}
The summation in the above formula is over the finite number
of critical points of the function $W(X_1,\dots ,X_m)$.
We will now interpret the degrees that we have computed in terms
of these intersection products. Recall
that $\kappa$ is the map from $X$ to ${\bb P}$ defined by the
line bundle $\wedge^m (\tilde S)$ on $X$ whose Chern class
is $c_1(\tilde S)$. Also
$\kappa(\overline{\psi^{-1}(S_i)})=Z^q_i$. Thus the degree of the
subvariety $Z^q_i\subset {\bb P}$ is given by
$<\psi^*(s_i)\cdot (\psi^*(X_1))^
{{\rm dim}Z^q_i}>$, where
$s_i(X_1,\dots ,X_m)$
is the Schubert cocycle Poincar\`e dual to $S_i$.
Further, for $d\le q$, by a
similar argument, the degree of $Z^d_i=<\psi^*(s_i)\cdot (\psi^*(X_1))^
{{\rm dim}Z^d_i}>$.
\begin{th}\label{phythm}
The degree of the subvariety $Z^d_i$ as given by Theorem~\ref{mainth1} also
equals:
\begin{equation}\label{voila}
\begin{array}{ccc}
\deg Z^d_i&=&<\psi^*(s_i)\cdot (\psi^*(X_1))^
{{\rm dim}Z^d_i}>\\
&=&(-1)^{m(m-1)/2}
\sum_{dW=0} \frac{s_i}{h}{X_1}^{\dim Z^d_i}\end{array}\end{equation}
where $h$ is the Hessian of the Landau-Ginzburg potential and
and $s_i$ is the Schubert cocycle of $S_i\subset{\rm Grass}(m,n)$.
In particular one has
\begin{equation}
\deg K^q_{m,p}=(-1)^{m(m-1)/2}
\sum_{dW=0} \frac{X_1^{mp+nq}}{h}
\end{equation}
\end{th}
In order to establish a proof we will first make
some simplifications in the formulas and we will
reformulate the theorem into an equivalent theorem
dealing with properties of the ring of symmetric
functions ${\cal Z}[q_{1},\ldots ,q_{m}]^{{\cal S}_{m}}$ only.
Some of these simplifications can also be found
in~\cite{wi94}, page 42.
Observe that the Jacobian
$$\det \left[ {\frac{\partial X_i}{\partial q_j} }\right]$$
for the change of variables from $q_i$ to
$X_j$ is given by the Vandermonde determinant
$\Delta =\prod_{j<k}(q_j-q_k)$
and on the critical points the transformation of the Hessian is
given through
$$\det \left.\left[ {\frac{\partial^2 W}{\partial q_i\partial q_j}}\right]
\right|_{dW=0}
=\det \left[ \frac{\partial^2 W}{\partial X_i\partial X_j}\right]
{\left( {\det \left[ {\frac{\partial X_i}{\partial q_j} }\right] }\right)}^2.$$
Thus the polynomial $h$ in terms of the Chern roots $q_j$ is given by
$$h(q_1,\ldots ,q_m)=\frac{n^m\cdot(q_1\dots q_m)^{n-1}}{\Delta^2}.$$
The Schubert cocycle $s_i$ can be identified with
a Schur symmetric function. For this consider the partition
$$\mu := (p+1-i_1,p+2-i_2,\ldots ,p+m-i_m).$$
Let
\begin{equation}
|\mu|:=\mu_1+\cdots+\mu_m=mp-|i|.
\end{equation}
It is well known that the Schubert cocycle $s_i$ can be identified
with the Schur symmetric function $s_\mu$ and $s_\mu$
has a classical representation due to Jacobi ($\sim$1835)
as a quotient of two alternating functions
resulting in a symmetric function:
\begin{equation} \label{jacobi}
s_{\mu}(q_1,\dots ,q_m) \; = \;
\frac{\det [q_{i}^{\mu_{j}+m-j}]}{\det[q_{i}^{m-j}]},
i,j=1,\ldots ,m.
\end{equation}
Also, $\Delta=0$ if $q_i=q_j$ so the summation in~(\ref{voila}) is over
all subsets $I$ consisting of $m$ distinct roots of the
the polynomial
$z^n+(-1)^{m}$. On $I$: $(q_1\cdots q_m)^n=1$, so
\begin{equation}
h(q_1,\ldots ,q_m)=\frac{n^m}{(q_1\cdots q_m)\Delta^2}
\end{equation}
Thus to prove Theorem~\ref{phythm} it suffices to
prove the following equivalent theorem:
\begin{th}
The degree of the subvariety $Z^d_i$ as given by Theorem~\ref{mainth1} also
equals:
\begin{equation} \label{degform}
\deg Z^d_i=\frac{(-1)^{m(m-1)/2}}{n^m}
\sum_{I} (q_1\cdots q_m)
(q_1+\cdots+q_m)^{mp-|\mu |+nq}
\Delta^2 s_\mu
\end{equation}
In particular one has
$$\deg K^q_{m,p}=\frac{(-1)^{m(m+1)/2}}{n^m}
\sum_{I} (q_1\cdots q_m)\left( {\prod_{i<j}(q_i-q_j)^2}\right)
{(q_1+\cdots+q_m)}^{mp+nq}.$$
\end{th}
The proof we will present is purely combinatorial,
and is just based
on Proposition~\ref{zint}. In order to establish
the proof we will show that\r{degform} satisfies
the recurrence relation\r{recurr}, the boundary
conditions\r{bound1} and\r{bound2} and
the initial conditions\r{ini1},\r{ini2}.
\begin{proof}
{}From Jacobi's identity\r{jacobi} and the fact that
$q_i^{n}=(-1)^{m+1}$ it is clear that
formula\r{degform} satisfies both boundary
conditions\r{bound1} and\r{bound2}.
Note that if we let
$$D(\alpha_1,\dots,\alpha_m):=\det[q_i^{n-\alpha_j}],$$
then
$$\det[q_i^{\mu_j+m-j}]=D(i_1,\dots,i_m)=D(\alpha_1,\dots,\alpha_m)$$
where $\alpha=(i;d)$ is defined by~(\ref{ialpha}). So the
recurrence property\r{recurr}
follows from the Pieri type formula:
$$(q_1 +\dots +q_m)\cdot D(\alpha_1,\dots,\alpha_m)=\sum_j
D(\alpha_1,\dots,\alpha_j-1,\dots,\alpha_m).$$
In order to complete the proof it is therefore enough
to show that the initial conditions\r{ini1} and\r{ini2}
are satisfied. Equivalently we have to show that
$$\frac{(-1)^{m(m-1)/2}}{n^m}
\sum_{I}
(q_1\ldots q_m) \Delta^2 s_\mu=\left\{\begin{array}{ll}
1&\mbox{if $\mu=(p^m)$}\\
0&\mbox{if $\mu_1> p$ and $|\mu|=mp$}\end{array}\right. .
$$
We will treat both cases simultaneously.
Note that
$$\Delta=\prod_{j<k} (q_j-q_k)=\det
\left[ \begin{array}{ccc}
1&\ldots &1 \\
\vdots & & \vdots \\
{q_m}^{m-1}&\ldots & {q_1}^{m-1}
\end{array} \right]$$
and
$$
s_{\mu}(q_1,\dots ,q_m)=
\frac{\det [q_{i}^{\mu_{j}+m-j}]}{\Delta}.$$
So we have
\begin{eqnarray}
&&
\frac{(-1)^{m(m-1)/2}}{n^m}
\sum_{I}
(q_1\cdots q_m)\Delta^2 s_\nu\nonumber \\
&=&
\frac{(-1)^{m(m-1)/2}}{n^m}
\sum_{I}
\Delta \det [q_i^{\mu_j+m+1-j}]\nonumber \\
&=& \label{product}
\frac{1}{n^m}
\sum_{I}\det
\left[ \begin{array}{ccc}
1&\ldots &1 \\
\vdots & & \vdots \\
{q_1}^{m-1}&\ldots & {q_m}^{m-1}
\end{array} \right]
\left[ \begin{array}{ccc}
{q_1}^{\mu_1+m}&\ldots &{q_1}^{\mu_m+1} \\
\vdots & & \vdots \\
{q_m}^{\mu_1+m}&\ldots & {q_m}^{\mu_m+1}
\end{array} \right]
\end{eqnarray}
Since the summation over the index set $I$ involves
all roots of the polynomial $z^n+(-1)^m$ we can view
the right hand side of the last expression as
a symmetric polynomial in
${\cal Z}[y_{1},\ldots ,y_n]^{{\cal S}_n}$,
where $\{ y_{1},\ldots ,y_n\} $ represent all roots
of $z^n+(-1)^m$. This fact is most conveniently
expressed by the Cauchy Binet formula, i.e. the
expression in\r{product} is equal to
\begin{eqnarray}
&=&
\frac{1}{n^m}
\det
\left[ \begin{array}{ccc}
1&\ldots &1 \\
\vdots & & \vdots \\
{y_1}^{m-1}&\ldots & {y_n}^{m-1}
\end{array} \right]
\left[ \begin{array}{ccc}
{y_1}^{\mu_1+m}&\ldots &{y_1}^{\mu_m+1} \\
\vdots & & \vdots \\
{y_n}^{\mu_1+m}&\ldots & {y_{n}}^{\mu_m+1}
\end{array} \right] \nonumber \\
&=&
\frac{1}{n^m}
\det
\left[ \begin{array}{cccc}
p(\mu_1 +m)&p(\mu_2 +m-1)&\ldots &p(\mu_m+1) \\
p(\mu_1 +m+1)&p(\mu_2 +m)&\ldots &p(\mu_m+2) \\
\vdots & & & \vdots \\
p(\mu_1 +2m-1)&p(\mu_2 +2m-2)&\ldots &p(\mu_m+m)
\end{array} \right]
\end{eqnarray}
where $p(k):=\sum_i y_i^k$ is the $k$th power symmetric
function in
${\cal Z}[y_{1},\ldots ,y_{n}]^{{\cal S}_n}$.
Note that
$$
p(k)=\left\{\begin{array}{ll}
n(-1)^{q(m-1)}&\mbox{if $k=qn$}\\
0&\mbox{otherwise}\end{array}\right.$$
Assume now that $\mu =(p^m)$. Then one verifies that
the last expression evaluates to
a diagonal matrix with all diagonal entries equal to
$(n)(-1)^{m-1}$.
But this just means that\r{product} correctly evaluates to 1.
On the other hand
if $\mu \neq (p^m)$ and $|\mu | =mp$ we have $\mu_m < p$.
But then the last column in the matrix
$\left[ { p(\mu_j+m+i-j)}\right] $ is zero what completes the proof.
\end{proof}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1994-02-08T00:00:00 | 9402 | alg-geom/9402004 | en | https://arxiv.org/abs/alg-geom/9402004 | [
"alg-geom",
"math.AG"
] | alg-geom/9402004 | Alexander A. Borisov | A. Borisov | Boundedness theorem for Fano log-threefolds | 16 pages, LaTeX v2.09, abstract corrected | null | null | null | null | The main purpose of this article is to prove that the family of all Fano
threefolds with log-terminal singularities with bounded index is bounded.
| [
{
"version": "v1",
"created": "Sun, 6 Feb 1994 23:50:41 GMT"
},
{
"version": "v2",
"created": "Mon, 7 Feb 1994 22:59:39 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Borisov",
"A.",
""
]
] | alg-geom | \section{Introduction}
First of all, let me recall necessary definitions and list some known results
and conjectures in direction of bondedness of Fano manifolds. All varieties in
this paper are over field of complex numbers.
\begin{Definition}
Normal variety $X$ is called the variety with log-terminal singularities
if $mK_X$ is a Cartier divisor for some integer $m$ and there exists a
resolution $\pi:Y\longrightarrow X$ of singularities of $X$ such that
exceptional divisors $F_i$ of $\pi$ have simple normal crossings and in
formula $K_Y=\pi^*K_X+\sum(a_iF_i)$ all $a_i>-1$
\end{Definition}
\begin{Definition}
Index (or Gorenstein index) of variety X is a minimal natural number m, s.t.
$mK_X$ is a Cartier divisor. Of course, index is defined for $Q$-Gorenstein
varieties only.
\end{Definition}
\begin{Definition}
Three-dimensional algebraic variety $X$ is called Fano log-threefold if the
following conditions hold.
$1)$ $X$ has log-terminal singularities,
$2)$ $X$ is Q-factorial,
$3)$ Picard number $\rho(X)=1$,
$4)$ $-K_X$ is ample.
\end{Definition}
\begin{Remark} This is only my terminology inspired by the term "Q-Fano
threefolds".
\end{Remark}
The following statement will be proven in this paper.
{\bf Main Theorem.
For an arbitrary natural $n$ all Fano log-threefolds of index $n$ lie in
finite number of families.}
\begin{Remark} Unfortunately, no effective bound on any invariant of $X$ will
be given because of Noetherian induction in the section $4.$
\end{Remark}
Here are some results in the direction of boundedness of Fano manifolds.
1) Boundedness theorem for smooth Fano manifolds of an arbitrary direction
is proven by Koll\'ar, Miyaoka and Mori in \cite{KoMiMo1}. Before this result
there were several proofs with extra condition $\rho(X)=1$. Three-dimensional
smooth case was also treated before by a long work of many authors beginning
with Fano itself. See \cite{Isk} for discussion.
2)Two-dimensional Fano varieties are traditionally called Del Pezzo surfaces.
Smooth (=terminal) case is fairly easy and the answer is the following.
$P^1\times{P^1}, P^2$ with $0$ to $8$ blown-up points in general position. (The
generality of position may be stated precisely.)
I should notice here that there are many difficult problems concerning Del
Pezzo surfaces if basic field is NOT algebraically closed.
Log-terminal case (with arbitrary Picard number) was studied by Alexeev and
Nikulin (see\cite{Ni}). One of the main results in this direction is
boundedness under the condition of bounded multiplicity of singularities. Let
me mention that by using methods of this paper one can obtain a new simple
proof of some intermediate result, namely boundedness under the condition of
bounded index.
3)The model case of toric varieties of arbitrary dimension is treated in
\cite{BB},
see also \cite{B}.
4) Boundedness of Fano threefolds with $\rho=1$ and terminal singularities
is proven by Kawamata in \cite{Ka1}.
5) Boundedness of Fano threefolds with terminal singularities with no extra
conditions is announced by Mori.
All these results justify the following conjecture.
\begin{Conjecture} The family of all Fano varieties of given dimension with
discrepancies of singularities greater (or greater or equal) $-1+\epsilon$,
where $\epsilon$ is an arbitrary given positive real number, is bounded.
\end{Conjecture}
\begin{Remark} This conjecture is so natural that probably many people
suspected it but I didn't see it published. Batyrev proposed the weaker variant
of this conjecture, where the condition on discrepancies is replaced by the
condition of boundedness of index. (\cite{Ba2}) Very recently Alexeev told me
that he also stated the above conjecture as a part of a general phenomenon
noticed by Shokurov that some geometric invariants (in this case minimal
discrepancies of Fano varieties) can accumulate only from above (below). See
\cite{Alex} for a discussion.
\end{Remark}
I am expressing my thanks to V. Iskovskih who encouraged me to work in this
direction. I am glad to thank V. Shokurov and V. Alexeev who invited me to the
geometry seminar at Johns Hopkins University and whose remarks simplified and
even corrected this paper. I also want to thank my brother Lev for helpful
discussions.
\section{Preliminary remarks and first lemmas}
In \cite{Ko} Koll\'ar proved that all three-dimensional normal varieties $X$
with an ample Cartier divisor $D$ lie in finite number of families if two
higher coefficients of Hilbert polynomial $P(m)=\chi(mD)$ are bounded. In our
case of three-dimensional Fano varieties of index $n$ it works as follows. Let
$D$
be equal to $-nK_X$. Then it is a Cartier divisor and it follows from general
theory of Riemann-Roch that
$\chi(O_X(-mnK_X))=\frac1{12}(-K_X)^3nm(nm+1)(2nm+1)+\alpha m+\beta$,
where $\alpha$ and $\beta$ are some constants depending on $X$.
Therefore in order to prove the Main Theorem we only need to prove that
$(-K_X)^3$ is bounded. The following lemma shows that in our case it is also
equivalent to the condition that $h^0(-2nK_X)$ is bounded.
\begin{Lemma}\label{L1}
For arbitrary Fano log-threefold $X$ of index $n$ (actually, only conditions
(1) and (3) are used) the following inequality holds.
$h^0(-2nK_X)\ge{(-K_X)^3 (\frac53 n^3+\frac12 n^2)-1}$
\end{Lemma}
{\bf Proof} By the Kawamata-Vieweg vanishing theorem $h^i(-mnK_X)=0$, $ i>0,
m\ge {0}$. Therefore $h^o(-mnK_X)=\frac1{12}(-K_X)^3nm(nm+1)(2nm+1)+\alpha
m+\beta$ for $m\ge{0}$. Let us consider "the second derivative at 1".
$h^o(-2nK_X)-2h^o(-nK_X)+h^0(O_X)=(-K_X)^3(\frac53 n^3+\frac12 n^2).$
Now the statement of lemma follows from the fact that $h^0(O_X)=1$ and
$h^0(-nK_X)\ge{0}$.
\begin{Lemma}\label{L2}
Suppose $v\in V$ - is a closed point of $k$-dimensional variety with
multiplicity of local ring $r$, $D$ is a semiample $Q$-Cartier divisor on $V$.
Suppose further that the general point $x$ of $V$ can be connected to $v$ by
some curve $\gamma _x$, such that $\gamma _x\cdot D\le d$
Then $D^k\le r\cdot d^k$.
\end{Lemma}
{\bf Proof} For sufficiently large $m$ such that $mD$ is a Cartier divisor one
have that $h^o(O_V(mD))=\frac{m^kD^k}{k!}+O(m^{k-1})$. Therefore if $D^k>r\cdot
d^k$ then for $m>>0$ one can find a non-zero global section $s\in H^0(O_V(mD))$
such that its image by trivialization map of $O_V(mD)$ in $v$ lies in
$(md+1)$th power of maximal ideal of point $v$. Then every curve $\gamma _x$
lies in $Supp(s)$, that is impossible.
\begin{Remark}
The above lemma is very general. In applications $V$ will be our Fano
log-threefold $X$ and $D$ will be $(-K_X)$.
\end{Remark}
\section{Covering family and first division into cases}
\begin{Remark} (about notations). We will often consider birational varieties.
Doing this we will usually identify curves on different varieties if they
coincide in their general points. Namely, let $X\leftarrow - \rightarrow
X^\prime$ and $L\subset X, L^\prime\subset X^\prime$ be curves. Then $L$ and
$L^\prime$ are identified if there are Zariski open subsets $U\subset X$ and
$U^\prime\subset X^\prime$, such that the above rational map is defined on them
and $U\cong U^\prime$, $L\cap U \cong L^\prime \cap U^\prime\ne\emptyset$ via
it. The identified curves will be usually denoted by the same symbol. The same
convention will be used for two-dimensional subvarieties. If it is necessary to
point out that, say, simple divisor $S$ is considered on variety $X$ it will be
denoted by $S_X$. Another convention is that $\{l\}$ will denote the family of
curves with general element $l$ and $\{H\}$ will denote the LINEAR system of
Weil divisors with general element $H$. It will be clear in every particular
case why these conventions agree with each other.
\end{Remark}
Now we start to prove our Main Theorem. Suppose $X$ is a Fano log-threefold,
$\pi^Y_X :Y\longrightarrow X$ is its $Q$-factorial terminal modification,
$\pi^{Y_1}_Y\longrightarrow Y$ is a resolution of isolated singularities of
$Y$. By the Miyaoka-Mori theorem (\cite{MM}, see also \cite{Ka2}) there exists
a covering family of rational curves $\{l\}$, such that $l\cdot (-K_X)\le 6$.
The family $\{l\}$ is free on $Y_1$ that is small full deformation of $l$
covers small neighborhood of $l$. ( See \cite{Na}.) We can and will denote by
$\{l_{Y_1}\}$ full family, that is (some Zariski open subset of) a component of
the scheme of morphisms from $P^1$ to $Y_1$. Consider the RC-fibration $\phi :
Y_1-{ - }\rightarrow Z$, associated with $\{l\}$. (See \cite{KoMiMo1},
\cite{KoMiMo2}.) The following cases are possible.
$(0)$ $dimZ=0$. In \cite{KoMiMo2} such $X$ are called primitive. It implies
that two general points of $Y_1$ can be joined by chain of no more than $3$
curves from $\{l\}$. It follows now from one of the "gluing lemmas"
(\cite{KoMiMo2}) that we can glue them together and obtain new family
$\{l^{\prime}\}$. Then we can apply to it lemma \ref{L2} and obtain that
$(-K_X)^3 \le (3\cdot l\cdot (-K_X))^3 \le (3\cdot 6)^3$
$(1)$ $dimZ=1$. In this case after some additional blowing-up $\tilde{Y}
\longrightarrow Y_1$ we obtain a morphism $\phi _{\tilde{Y}}:\tilde{Y}
\longrightarrow Z$. Here $Z\cong{P^1}$, because $X$ is rationally connected
(see \cite{KoMiMo2}).
$(2)$ $dimZ=2$. In this case general $l\in \{l\}$ is smooth and does not
intersect with another general $l$ on $Y_1$. And it is exactly the general
fiber of the RC-fibration.
We will proceed by the following way. First of all we will treat the case (1).
Doing this we will require $l\cdot (-K_X)$ to be bounded not by $6$ but only by
an arbitrary constant depending on $n$. After that we will reduce the case (2)
to the case (1) but for some new family $\{l^{\prime}\}$ where $l^{\prime}\cdot
(-K_X)$ will be bounded.
\section{The treatment of case (1)}
Let $S$ be a general fiber of our RC-fibration. As we already mentioned, the
image of RC-fibration is rational. This implies that S are linear equivalent on
$Y_2$ and therefore on $X$. Notice that it can happen that $\{l\}$ does not
connect two general points of $S$ immediately. But it will always be true if we
glue two examples of $\{l\}$. (See \cite{KoMiMo2}.) Therefore we will assume,
that $\{l\}$ is a connecting family on $S$. Evidently, $l^2\ge1$ on a smooth
surface $\tilde{S}=S_{\tilde{Y}}$. The condition that $X$ is Q-factorial with
Picard number $1$ implies that $S_X=\alpha H, \alpha >0$, where $H=(-2nK_X)$.
We will assume up to the end of this section that $l\cdot H\le \rho,$ where
$\rho$ is some constant depending on $n.$
\begin{Proposition}\label{S1}
If $h^0(H)>2(\rho +1)^2$ then $\alpha\le\frac12$.
\end{Proposition}
{\bf Proof} Let $S_1$ and $S_2$ be two general surfaces from $\{S_X\}$.
Let $l_1, l_2,...l_{\rho+1} \subset S_1$ and $l_{\rho+2}, l_{\rho+3},...
l_{2\rho+2} \subset S_2$ be general curves from $\{l\}$. We have that $H\cdot
l\le \rho$, therefore
$$dimH^0(O_X(H))-dimH^0(J_{l_i}\cdot O_X(H))\le dimH^0(O_{l_i}(H))\le\rho +1.$$
(Here $J_{l_i} $ is an ideal sheaf of the curve $l_i\subset X$.)
This implies that $$codim(\bigcap^{2\rho +2}_{i=1}H^0(J_{l_i}\cdot
O_X(H)))\le\sum^{2\rho +2}_{i=1}(\rho+1)=2(\rho+1)^2.$$
If $h^0(H)>2(\rho+1)^2$ then there exists a divisor $H^*\in |H|$, such that all
$l_i\subset H^*$. Suppose $\pi_1$ is a composition of birational morphism
$\tilde{S_1}\longrightarrow S_1$ and embedding $S_1\longrightarrow X$. If $H^*$
does not contain $S_1$ then $(\pi^{\tilde{Y}}_Y)^* (H^*)$ does not contain
$\tilde{S}_1$ but at the same time contains preimages of $l_i$. On
$\tilde{S}_1$ we have $(\pi^*_1H^*)\cdot l\ge\sum l_i\cdot l\ge\rho+1$. It
contradicts to the fact that $(\pi_1^*(H^*)\cdot l=H\cdot l\le\rho$. Therefore
$H^*$ contains $S_1$ and, by the same arguments, $S_2$. This implies that
$\alpha \le \frac12.$
We will always assume below that $\alpha \le \frac12.$
\begin{Proposition}\label{S2}
For arbitrary $S\in \{S\}$ on $X$, arbitrary positive integer $k$
$h^i(X,J_S\cdot O_X(kH))=0$ for every $i>O$.
\end{Proposition}
{\bf Proof} $S$ is a simple divisor, therefore $J_S=O_X(-S)$, where $O_X(-S)$
is a divisorial sheaf sheaf, associated with Weil divisor $(-S)$. After that,
$O_X(kH)$ is an invertible sheaf, therefore $J_S\cdot O_X(kH) =O_X(kH-S)$. Now
one can apply Kawamata-Vieweg vanishing theorem (see the reformulation of it in
\cite{Al}) because $kH-S-K_X=(k-\alpha+\frac1{2n})H$ is ample for $k \ge 1.$
\begin{Proposition}\label{S3}
For all $k>0$, $l>0$ $h^i(S,O_S(kH))=O.$
\end{Proposition}
{\bf Proof} It follows from exact sequence
$0 \longrightarrow J_S \cdot O_X(kH) \longrightarrow O_X(kH) \longrightarrow
O_S(kH) \longrightarrow 0$,
vanishing theorem and proposition \ref{S2}.
\begin{Proposition}\label{S4}
All surfaces $S_X$ for given $n$ and $\rho = c(n)$ lie in finite number of
families.
\end{Proposition}
{\bf Proof} By a result of Koll\'ar (\cite{Ko}) and proposition \ref{S3} it is
enough to prove the boundedness of coefficients of Hilbert polynomial
$P(k)=\chi(O_S(kH))=h^0(O_S(kH)), k\ge 1.$ For this purpose we will prove that
there exists some constant $c_1(n,\rho )$ such that for all $k\ge 1$ it is true
that $h^0(O_S(kH))\le k^2\cdot c_1(n,\rho )$. It implies the boundedness of
coefficients by the following arguments. Suppose $P(k)=a_2k^2+a_1k+a_0$.
Evidently, $0\le a_2\le c_1$. Therefore $|a_1|=|P(2)-P(1)-3a_2|\le 4c_1$. After
that, $|a_0|=|P(1)-a_2-a_1|\le 5c_1$.
In order to prove that $h^0(O_S(kH))\le k^2\cdot c_1(n,\rho )$ consider the
following construction. By applying several times gluing lemma to a free family
$\{l\}$ on $\tilde{S}$ (\cite{KoMiMo2}) we obtain families $\{l_k\}$ such that
$l_k=k\cdot l$ as divisors on $\tilde{S}$ and therefore on $S$. (Here "=" means
algebraic equivalence.) Notice that the natural map $\mu _k
:H^0(S,O_S(kH))\longrightarrow H^0(l_{k\rho +1},O_{l_{k\rho +1}}(kH))$ is
injective. Otherwise there should have been some $D\in |kH|$ containing
$l_{k\rho +1}$ but not containing $S$. As in the proof of proposition \ref{S1}
we obtain a contradiction by intersecting with general $l$. The fact that
$\mu_k$ is injective implies that $h^0(O_S(kH))\le h^0(l_{k\rho +1},O_{l_{k\rho
+1}}(kH))\le l_{k\rho +1}(kH)+1=(k\rho +1)k\rho +1$, that is what we need.
\begin{Proposition}\label{S5}
In the condition of the above proposition there is a constant $c_2(n,\rho )$,
such that on every general $S_X$ EVERY two points can be joined by some
irreducible curve $\gamma$, such that $\gamma\cdot (-K_X)\le c_2$.
\end{Proposition}
{\bf Proof} It is a straightforward consequence of boundedness of $S_X$ with
$H|_{S_X}$. Indeed, it is true for a general element of every one of families
in proposition \ref{S4} and Noetherian induction on base completes the proof.
\begin{Remark} Of course, two GENERAL points of $S_X$ are already connected by
$l$, but the above proposition gives much more.
\end{Remark}
Now we can complete the treatment of the case $(1)$. By the definition of Fano
log-threefold $\rho (X)=1$ therefore two general $S_X$ intersect with each
other. Moreover, they intersect along some curve $C$ because $X$ is
Q-factorial. We know that $\{S_X\}$ is a linear system, therefore all of them
contain $C$. It may happen that $C$ lies in $Sing(X)$, but the multiplicity of
$X$ in a general point $x_0\in C$ is bounded by $2n$, because the index of $X$
is bounded by $n$. (By canonical cover trick it is a factor of $CDV$
singularity that is analytically isomorphic to $(DV-point)\times (disk).$)
Therefore we can apply lemma \ref{L2} to $X, (-K_X), x_0$ to obtain a bound on
$(-K_X)^3.$
\section{Two lemmas}
In this section we will prove some adjunction lemma and a lemma about accurate
resolution that will be used in next section to treat the case $(2)$. However,
these lemmas themself are interesting enough to deserve a separate section.
\begin{Lemma}\label{LA}(adjunction)
Suppose $X$ is a three-dimensional variety and $S$ is simple Weil divisor on
it, such that $(K_X+S)$ is Q-Cartier. Suppose $\{L\}$ is a covering family of
curves on $S$, $\hat{S}$ is a minimal resolution of normalization of $S$. Then
$K_{\hat{S}}\cdot L\le (K_X+S)\cdot L.$
\end{Lemma}
{\bf Proof} Denote by $\pi$ the natural morphism $\hat{S}\longrightarrow X.$
Then by the proposition 3.2.2 of \cite{SH} $K_{\hat{S}}=\pi ^*(K_X+S)-D$, where
D is an effective divisor. The rest is trivial.
\begin{Remark}
The above lemma is due to Shokurov. In the first variant of this paper I
formulated and proved it only under condition that singularities of $X$ were
isolated which is enough for applications.
\end{Remark}
\begin{Lemma}\label{LR}(accurate resolution)
Suppose $X$ is a Q-factorial three-dimensional variety, $E\subset X$ is a
simple Weil divisor, $\{L\}$ is a covering family of curves on E. Suppose
further that there exists a covering family $\{l\}$ on $X$, such that $l\cdot
E\ge 1$ and a linear system $|H|$ on $X$, such that the following inequalities
hold true. ($c_i$ are some nonnegative constants.)
$1)$ $H\cdot l\le c_1$
$2)$ $H\cdot L\le c_2$
$3)$ $K_X\cdot L\le c_3$
$4)$ $-E\cdot L\le c_4$
Then $h^0(H)>1+(c_1+1)(c_2+c_1c_4+1)$ implies that there exists a resolution
$Y\longrightarrow X$, such that $\{L\}$ have no base points on $E_Y$ and
$K_Y\cdot L\le c_3+2(c_2+c_1c_4).$
\end{Lemma}
\begin{Remark}
The proof of this lemma will be pretty long. It will take the rest of the
section.
\end{Remark}
\begin{Remark}
In some sense this lemma is a very weak substitute for the following conjecture
for which I have a lot of evidence.
\end{Remark}
{\bf Accurate Resolution Conjecture} For an arbitrary Q-Gorenstein threefold
$X$ there exists a resolution of singularities $\pi:Y\longrightarrow X,$ such
that for EVERY Q-Cartier divisor $H$ on $X$ containing a curve $L_X$ not lying
in Sing(X) the following inequality holds true.
$(K_Y+D_Y)\cdot L_Y \le (K_X+D_X)\cdot L_X$
$ $
First of all we will introduce some convenient notations. Let $\{D\}$ be a
linear system of Weil divisors. We will denote by $H^0(\{D\})$ the
corresponding vector subspace in $H^0(O_X(D))$, where $O_X(D)$ is a divisorial
sheaf, associated with $D$. Reversely, for a linear subspace $V\subset
H^0(\{D\})$ let $|V|$ be the corresponding linear system. Divisor that
corresponds to $s\in H^0(O_X(D))$ will be denoted by $(s)$. Section that
determines divisor $D$ will be called "equation" of $D.$ Of course, it is
defined up to multiplicative constant. By definition
$h^0(\{D\})=dimH^0(\{D\})=dim\{D\}+1.$
For the purpose of convenience we introduce the concept of $L$-base of linear
system in the following way. Suppose $\{D\}$ is a linear system of Weil
divisors, $\{L\}$ is a family of curves parameterized by base $S.$ For every
nonempty Zariski open subset $U\subset S$ let $V(U,\{D\})$ be a linear subspace
in $H^0(\{D\})$, spanned by $s,$ such that $(s)$ contains $L_u$ for some $u\in
U.$ Evidently, $V(U^\prime\bigcap U^{\prime \prime},\{D\})\subset
V(U^\prime,\{D\})\bigcap V(U^{\prime \prime},\{D\})$ and $H^0(\{D\})$ is
finite-dimensional. Therefore there exists the minimal $V(U^*,\{D\}),$ such
that $V(U^*,\{D\})\subset V(U,\{D\})$ for every $U\subset S.$ Then
$|V(U^*,\{D\})|$ will be called $L$-base of $\{D\}$ and denoted by $\{D\}^L.$
\begin{Proposition}\label{S6}
$h^0(\{D\}^L)\ge h^0(\{D\})-L\cdot D-1$
\end{Proposition}
{\bf Proof} Suppose $\{D\}^L=|V(U^*,\{D\})|,$ $u\in U^*.$ We can also assume
that $L_u$ is not contained in $Sing(X).$ Choose on $L_u$ points $x_1,$ $x_2,$
. . . , $x_d,$ $L\cdot D<d\le L\cdot D+1$ lying in nonsingular part of $X.$ The
condition of vanishing in $x_1,$ $x_2,$ . . . , $x_d$ determines a subspace in
$H^0(\{D\})$ of codimension no greater than $d$ and $d\le L\cdot D+1.$ Now we
just notice that for every $s$ from this subspace $(s)$ contains $L_u$, because
otherwise we would have a contradiction by intersecting it with $L_u$.
$ $
Define a new linear system $\{H_*\}$ by the following procedure. Denote $|H|$
by $\{H_0\}$ and for every nonnegative integer $i$ let $\{H_{i+1}\}$ be a
movable part of $\{H_i\}^L.$ Evidently, $\{H_i\}$ will eventually stabilize.
This stabilized $\{H_i\}$ will be our $\{H_*\}.$ It is evident that $\{H_*\}$
is movable and $\{H_*\}=\{H_*\}^L.$ (Here we set as definition that trivial
linear systems $\emptyset$ and $|O_X|$ are movable.)
\begin{Proposition}\label{S6*}
If $h^0(H)>1+(c_1+1)(c_2+c_1c_4+1)$ then $\{H_*\}$ is not trivial
\end{Proposition}
{\bf Proof} First of all, let $\{H\}^L=a_iE+D_i+\{H_{i+1}\}$, where $a_i\ge 0$,
$D_i$ does not contain E. Notice that if $a_i=0$ then
$\{H_{i+1}\}^L=\{H_{i+1}\}$ and the procedure stabilizes. From the other hand,
$\sum a_i\le c_1$ because $E\cdot L\ge 1$ and $H\cdot l\le c_1.$ Therefore
$\{H_*\}=\{H_{[c_1]+1}\}.$ It is easy to see that for all $i$ $H_i\cdot L\le
H\cdot L+c_1(-E\cdot L)\le c_2+c_1c_4.$ Therefore by proposition \ref{S6} we
have that $h^0(\{H_*\})\ge h^0(H)-(c_1+1)(c_2+c_1c_4+1)>1$. This implies
$\{H_*\}$ is not trivial.
$ $
We also have from the above proof that $H_*\cdot L\le c_2+c_1c_4.$ Apply to
$K_X+2\{H_*\}$ Alexeev Minimal Model Program (\cite{Al}). Namely, let $\pi
:Y_1\longrightarrow X$ be a terminal modification of $K_X+2\{H_*\}$ in sense of
Alexeev.
\begin{Proposition}\label{S7}
Under the above notations the following is true.
$(1)$ $Y_1$ is Q-factorial and have at worst terminal singularities.
$(2)$ $\{\pi ^\prime H_*\}$ is free. Here $\{\pi ^\prime H_*\}$ is a inverse
image of linear system $\{H\}$ in sense of Alexeev, that is general element of
$\{\pi ^\prime H_*\}$ is $\pi ^\prime H_*$ for general $H_*\in \{H\}.$
$(3)$ $K_{Y_1}\cdot L\le c_3+2(c_2+c_1c_4)$
\end{Proposition}
{\bf Proof} Parts $(1)$ and $(2)$ are proved the same way as lemma $1.22$ in
\cite{Al}. Part $(3)$ is a corollary of the following chain of inequalities.
$K_{Y_1}\cdot L\le (K_{Y_1}+2(\pi ^\prime H_*))\cdot L \le (K_X+2H_*)\cdot L\le
c_3+2(c_2+c_1c_4)$
Here the middle inequality is due to the following argument. By definition of
terminal modification $K_{Y_1}+2(\pi ^\prime H_*)$ is $\pi -nef$ and therefore
in adjunction formula $K_{Y_1}+2(\pi ^\prime H_*)=\pi ^*(K_X+2H_*)+\sum
a_iD_i,$ where $D_i$ are exceptional divisors, all $a_i\le 0.$
$ $
For the rest of the section we will use the following notations. Suppose $D_i,$
$i=1,...,k$ are exceptional divisors of morphism $\pi .$ For an arbitrary Weil
divisor $F$ on $X$ we will say that discrepancy of $F$ is a $k$-tuple
$\{discr_{D_i}(F)\}$ of discrepancies of $F$ in $D_i$, that is numbers
$discr_{D_i}(F)$ from the formula $\pi ^*F=\pi ^\prime (F)+\sum
discr_{D_i}(F)D_i.$
In these notations we have the following lemma.
\begin{Lemma}\label{L5}
Suppose $F=(s), s\in H^0(O_X(F))$. Suppose $s=\sum \alpha _j s_j,$ where
$(s_j)=F_j.$ Then for all $D_i$ $discr_{D_i}(F)\ge \min{_j}discr_{D_i}(F_j)$
and for a general $\{\alpha _j\}$ for given $\{s_j\}$ this inequality becomes
an equality.
\end{Lemma}
{\bf Proof} Suppose $rF$ is a Cartier divisor. In a neighborhood of generic
point $\pi (D_i)$ the sheaf $O_X(rF)$ can be trivialized. With respect to this
trivialization the local equation $f$ of divisor $rF$ is, by Newton binomial
formula, a linear combination of local equations $f_{(\gamma)}$ of divisors
$\sum \gamma _jF_j$, where $\sum \gamma _j=r,$ $\gamma _j\in {\bf Z}_{\ge 0}.$
By the definition, $discr_{D_i}(F)=\frac1r discr_{D_i}(rF)$ and
$discr_{D_i}(rF)$ is just an image of $f$ by a valuation on function field $\bf
C(X)$ of variety $X$ corresponding to $D_i.$ Therefore for arbitrary $\{\alpha
_j\}$ $discr_{D_i}(rF)\ge \frac1r\min{_j}discr_{D_i}(\sum \gamma _jF_j)\ge
\min{_j}discr_{D_i}(F_j)$ and for general $\{\alpha_j\}$ it becomes an
equality.
$ $
Suppose now that $P_1\subset \{H_*\}$ is a set of all divisors $H_*$ containing
some $L\in \{L\}.$ Suppose a general element of $P_1$ has discrepancy
$\{d_i\}.$ Denote the set of all divisors from $P_1$ with such discrepancy by
$P.$
\begin{Proposition}\label{S9}
"Equations" of $H_*$, $H_*\in P,$ span $H^0(\{H_8\}).$
\end{Proposition}
{\bf Proof} For a general $L\in \{L\}$ divisors $H_*\in P,$ containing $L$
constitute a nonempty Zariski open subset in linear system of divisors from
$\{H\}$ containing $L.$ Therefore their "equations" span the corresponding
subspace in $H^0(\{H_*\}).$ By definition $\{H_*\}=\{H_*\}^L,$ so we are done.
\begin{Proposition}\label{S8}
$\{L\}$ have no base points on $E_{Y_1}.$
\end{Proposition}
{\bf Proof}
Proposition \ref{S9} and lemma \ref{L5} applied together imply that discrepancy
of general element of linear system $\{H_*\}$ equals $\{d_i\}.$ Therefore for
every $H_*\in P$ $\pi ^\prime H_*\in \{\pi ^\prime H_*\}.$ Moreover, the linear
equivalence between divisors $\pi ^\prime H_*$ is given by the same functions
from $\bf C(Y_1)=\bf C(X)$ as between corresponding divisors $H_*.$ Therefore
the proposition \ref{S9} implies that "equations" of $\pi ^\prime H_*,$ where
$H_*\in P,$ span $H^0(\{\pi ^\prime H_*\}).$.
Suppose all $L$ on $Y_1$ pass through some point $y.$ Then all $\pi ^\prime
H_*,$ where $H_*\in P,$ contain $y$. But it is in contradiction with
proposition \ref{S7}, $(2),$ so proposition \ref{S8} is proven.
$ $
To complete the proof of the whole Accurate Resolution Lemma it is enough to
choose an arbitrary resolution of singularities $Y\longrightarrow Y_1.$ Then
$Y\longrightarrow X$ will satisfy all the requirements of accurate resolution.
\section{Treatment of case (2)}
Now we are in situation and notations of case $(2).$ (See section $3$.)
\begin{Proposition}\label{S10}
On $Y_1$ there exists a divisor $E$ which is exceptional with respect to
morphism $\pi^{Y_1}_X:Y_1\longrightarrow X$ such that $E\cdot l\ge 1.$
\end{Proposition}
{\bf Proof} Suppose $C$ is some general enough curve on the image $Z$ of
RC-fibration $\phi .$ Suppose $D\subset X$ is an image by $\pi^{Y_1}_X$ of the
surface $[ {\phi^{-1}(C)}].$ (Here parenthesis means Zariski closure.) The
general $l_{Y_1}$ does not intersect $\phi^{-1}(C)$ and, therefore,
$[{\phi^{-1}(C)}].$ ($Y_1$ is smooth therefore $\{l\}$ is free, see \cite{Na}.)
So, if $l_{Y_1}$ does not intersect with exceptional divisors of $\pi^{Y_1}_X$
then $l_X\cdot D=0,$ that is impossible because $X$ is Q-factorial and $\rho
(X)=1.$ Q.E.D.
$ $
Notice that if $E\cdot l\ge 1$ then general $l_{Y_1}$ intersects with $E$ in
general points because $\{l_{Y_1}\}$ is free. Two cases are possible.
$(A)$ There exists such $E\subset Y_1$ that is exceptional with respect to the
morphism $\pi ^{Y_1}_Y.$
$(B)$ Family $\{l_Y\}$ is free. Then there exists $E\subset Y$ that is
exceptional with respect to $\pi ^Y_X.$
The proof is generally the same in both cases but some technical details are
different. We begin with the case $(A).$ By the relative version of the usual
Minimal Model Program morphism $\pi ^{Y_1}_Y$ can be decomposed into extremal
contractions and flips, relative over $Y.$ Suppose $\pi ^{Y_3}_{Y_2}$ is the
first that contract some divisor $E_{Y_3},$ for which $l\cdot E_{Y_3}\ge 1.$
Suppose $\hat{E_{Y_3}}$ is a minimal resolution of $E_{Y_3}.$
\begin{Proposition}\label{11a} {\bf (Case (A))}
There exists a covering family $\{L\}$ of rational curves on $E_{Y_3},$ such
that the following conditions hold true.
$(1)$ $L\cdot K_{Y_3}<0$
$(2)$ $-L\cdot E_{Y_3}<3$
$(3)$ $L$ does not admit a nontrivial 2-point deformation on $\hat E_{Y_3}$,
that is a deformation with two fixed points, whose image is not in $L.$
\end{Proposition}
{\bf Proof} Suppose $\pi ^{Y_3}_{Y_2}(E_{Y_3})$ is a curve. Then we can choose
$\{L\}$ to be the fibers of $\pi ^{Y_3}_{Y_2} |_{E_{Y_3}}.$ Then $(1)$ is true
by the definition of extremal contraction. Suppose $\tilde E_{Y_3}$ is a
normalization of $E_{Y_3}.$ Then $\{L\}$ does not have base points on $\tilde
E_{Y_3}$ and therefore $L$ does not pass through its singularieties. This
easily implies $(3).$ The condition $(2)$ follows from the fact that (by lemma
\ref{LA}) $$(K_{Y_3}+E_{Y_3})\cdot L\ge K_{\hat E_{Y_3}}\cdot L=-2>-3.$$
Suppose now that $\pi ^{Y_3}_{Y_2}(E_{Y_3})$ is a point. Consider a minimal
model $F$ of $\hat E_{Y_3}.$ The surface $\hat E_{Y_3}$ is birationally ruled
or rational therefore we have two possibilities for $F$:
$1)$ $F\cong P^2$
$2)$ $F$ is ruled, there is a morphism $\theta :F\longrightarrow C$
We let $\{L\}$ be the family of planes on $P^2$ in the first case and the
family of fibers of $\theta$ in the second one. It evidently satisfies the
condition $(3).$ The condition $(1)$ holds for arbitrary curve on $E_{Y_3}$.
The condition $(2)$ again follows from the fact that $$(K_{Y_3}+E_{Y_3})\cdot
L\ge K_{\hat E_{Y_3}}\cdot L\ge -3.$$ The proposition is proven.
$ $
Now we can apply the Accurate Resolution Lemma (lemma \ref{LR}.) Here $X$ means
$Y_3$, $H$ means $(\pi ^{Y_3}_X)^*(-2nK_X)$ and constants will be as follows.
$c_1=12n,$ $c_2=0,$ $c_3=0,$ $c_4=3.$
We see that if $h^0(-2nK_X)$ is big enough there exists a resolution
$Y_4\longrightarrow Y_3$ such that $K_{Y_4}\cdot L\le 2(3\cdot 12n)=72n$ and
$\{L\}$ have no base points on $E_{Y_4}.$
\begin{Proposition}\label{S12}
$L$ does not admit a nontrivial 2-point deformation on $Y_4.$
\end{Proposition}
{\bf Proof} If such deformation existed it would be a deformation on $E_{Y_4}$
by rigidity lemma. (About this lemma see \cite{CKM}, section 1. I must only
notice that it is not stated there correctly, one should add a condition of
flatness of morphism $f.$ It was noticed by several people, my attention was
brought to it by Iskovskikh.) The system $\{L\}$ has no base points on
$E_{Y_4}$ therefore $L$ does not pass through the singularities of
normalization $\tilde{E_{Y_4}}$ of the surface $E_{Y_4}.$ Resolution of
singularities $\hat{E_{Y_4}}$ is naturally mapped to $\hat {E_{Y_3}}$ therefore
2-point deformation of $L$ on $E_{Y_4}$ gives deformation on $\tilde{E_{Y_4}}$
and then on $\hat{E_{Y_4}},$ and then on $\hat {E_{Y_3}}.$ The last is
impossible by the choice of $L$, Q.E.D.
$ $
Now we can apply to $\{L\}$ and $\{l\}$ the gluing lemma on $Y_4$ (see
\cite{KoMiMo1}) to obtain a new covering family of rational curves $\{l^\prime
\}.$ But now the image of RC-fibration corresponding to $\{l^\prime \}$ has
dimension $1$ or $0.$ And $l^\prime \cdot (-K_X) \le (1+dimY_4+L\cdot
K_{Y_4})(l\cdot (-K_X))\le 6(4+72n).$ So we managed to reduce the case $(2A)$
to cases $(1)$ and $(0),$ as it was promised at the end of section 3.
$ $
Now we consider the case $(B).$ Similarly to the case $A,$ we have the
following statement.
\begin{Proposition}\label{11b} {\bf (Case (B))}
There exists a covering family $\{L\}$ of rational curves on $E_{Y},$ such that
the following conditions hold true.
$(1)$ $L\cdot K_{Y}<0$
$(2)$ $-L\cdot E_{Y}<3$
$(3)$ $L$ does not admit a 2-point nontrivial deformation on $\hat E_{Y_3}.$
$(4)$ $\pi ^Y_X(L)$ is a point
\end{Proposition}
{\bf Proof} If $\pi ^Y_X(E_Y)$ is a curve let $\{L\}$ be the family of fibers
of $\pi ^Y_X|_{E_Y}.$ If $\pi ^Y_X(E_Y)$ is a point then let it come from the
minimal model of $\hat E_Y$ as in the proof of proposition \ref{11a}. As in the
case $(A),$ $K_{\hat E_Y}\cdot L$ is $-2$ or $-3.$ Conditions $(3)$ and $(4)$
are evidently satisfied, we only need to prove $(1)$ and $(2).$ In order to do
it consider the adjunction formula for $\pi ^Y_X,$ multiplied by $L:$
$$K_Y\cdot L=\sum_{E_i\ne E_Y}a_iE_iL+aE_Y\cdot L
{}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(*) $$
Here $a_i$ and $a$ are discrepancies, they are of form $(-\frac mn),$ $m\in
\{0,1,...,n-1\},$ where $n$ is an index of $X.$ (Discrepancies are nonpositive
because $Y$ is a terminal modification of $X.$) We have the following chain of
inequalities.
$$-3\le K_{\hat E_Y}\cdot L\le (1+a)E_Y\cdot L+\sum_{E_i\ne E_Y}a_iE_iL\le
(1+a)E_Y\cdot L$$
Here the middle inequality follows from lemma \ref{LA} and formula $(*),$ and
the right from nonpositivity of $a_i.$ Therefore $1+a\ge \frac1n$ implies that
either $-E_Y\cdot L\le 0$ or $-E_Y\cdot L\le 3n.$ Therefore $-E_Y\cdot L\le
3n.$
Now the condition $(1)$ follows from the following chain of inequalities.
$$K_Y\cdot L=\sum_{E_i\ne E_Y}a_iE_iL+aE_Y\cdot L\le aE_YL\le 3n$$
Here the right inequality holds because of the following argument. We know that
$-1<a\le 0$ therefore $E_YL\ge 0$ implies $aE_YL\le 0$ and $E_YL< 0$ implies
$aE_YL\le -E_YL.$ Q.E.D.
$ $
Again, as in case $(A),$ we apply the Accurate Resolution Lemma (lemma
\ref{LR}.) The only difference is that now we have $Y$ instead of $Y_3$ and
constants are as follows.
$c_1=12n,$ $c_2=0,$ $c_3=3n,$ $c4=3n.$
Again if $h^O(-2nK_X)$ is big enough there exists an accurate resolution $Y_4.$
We have again that $L$ does not admit nontrivial 2-point deformation on $Y_4$.
(Arguments from the proof of proposition \ref{S12} work without any problems
because of condition $(4)$ of proposition \ref{11b}.) So we can apply gluing
lemma from \cite{KoMiMo1}. The bound on $l^\prime \cdot (-K_X)$ will be the
following.
$l^\prime \cdot (-K_X) \le (4+L\cdot K_{Y_4})(l\cdot (-K_X))\le
(4+3n+2(12n\cdot 3n))\cdot 12n=12n(4+3n+72n^2).$
So we completed the treatment of case $(2B)$. Our Main Theorem is finally
proven.
|
1994-06-22T20:44:12 | 9402 | alg-geom/9402003 | en | https://arxiv.org/abs/alg-geom/9402003 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9402003 | Andras Szenes | Andras Szenes | The combinatorics of the Verlinde formulas | 13 pages, amslatex file. (Some errors corrected) | null | null | null | null | A residue formula is given for the Verlinde formula, which allows one to
calculate its coefficients as a polynomial in the level and connects it to the
Riemann-Roch formula on the moduli space of vector bundles on a curve.
| [
{
"version": "v1",
"created": "Fri, 4 Feb 1994 03:10:29 GMT"
},
{
"version": "v2",
"created": "Fri, 4 Feb 1994 19:03:32 GMT"
},
{
"version": "v3",
"created": "Wed, 22 Jun 1994 18:43:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Szenes",
"Andras",
""
]
] | alg-geom | \section{Introduction}
In this short note we discuss the origin and properties of the Verlinde
formulas and their connection with the intersection numbers of moduli
spaces. Given a simple, simply connected Lie group $G$, the Verlinde
formula is an expression $V^G_k(g)$ associated to this group depending
on two integers $k$ and $g$. For $G=\SL_2$ the formula is \be
\label{evenintro} V_k^{\SL_2}(g) = \sum_{j=1}^{k-1}
\frwd{k}{2\sin^2\frac{j \pi}{k}}^{g-1}. \end{equation}
We describe $V^G_k$ for general groups in \secref{fusion}. These formulas
were first written down by E. Verlinde \cite{fusion} in the context of
Conformal Field theory. The interest towards them in Algebraic Geometry
stems form the fact that they give the Hilbert function of moduli spaces of
principle bundles over projective curves. More precisely, let $C$ be a
smooth projective curve of genus $g$, and let $\M_C^G$ be the moduli space
of principal $G$-bundles over $C$ (cf. e.g. \cite{kunara} and references
therein). Then there is an ample line bundle $\cL$ over $\M_C^G$ such that
\be \label{main}
\dim \cH^0(\M_C^G,\cL^k) = V_{k+h}^G(g),
\end{equation}
where $h$ is the dual Coxeter number of $G$. This statement requires
some modifications for a general simple $G$, but it holds for $\SL_n$
(\cite{BSz,faltings,bela,kunara}).
Proving \eqref{main} is important, but in this paper we will
address a different question: what can be said about the moduli spaces
knowing \eqref{main}? Accordingly, first we concentrate on
understanding the formula.
Two rather trivial aspects of \eqref{main} are that
\begin{itemize}
\item $V^G_k(g)$ is integer valued,
\item $V^G_k(g)$ is a polynomial in $k$.
\end{itemize}
Note that looking at the formula itself, none of this is
obvious. Our goal is to explain these properties and
connect them to the intersection theory of $\M^G_C$.
The paper is structured as follows: in \secref{fusion} we discuss some of
the ideas of Topological Field Theory, which explain the structure of the
formula for general $G$ and show its integrality (cf.
\cite{kac,phd,gepner,arnb}). In \secref{resi} we give our main result, a
residue formula for $V^G_k$ for $G=\SL_n$. Such a formula gives an explicit
way of calculating the coefficients of $V^G_k$ as a polynomial in $k$.
Finally, in
\secref{application} we give an application of our formulas: a ``one-line
proof of'' \eqref{main}.
This paper is intended as an announcement and overview. As a result, few
proofs will be given, and even most of those will be sketchy. A more
complete treatment will appear separately.
{\bf Acknowledgements.} I am grateful to Raoul Bott, my thesis advisor,
for suggesting to me this circle of problems and helping me with advice
and ideas along the way. I would like to thank Noam Elkies for useful
discussions.
I am thankful to the organizers of the Durham Symposium on Vector bundles,
Peter Newstead and Bill Oxbury, for their help and for the opportunity to
present my work.
\section{Topological Field Theory and Fusion Algebras} \label{fusion}
This section is independent from the rest of the paper. It contains a
quick and rather formal overview of the structure of Topological Field
Theories \cite{segal,atiyah} and Verlinde's calculus \cite{fusion}.
Consider a finite dimensional vector space $F$ (the space of fields)
with a marked element $1\in F$ (the vacuum). Assume that a number
$F(g)_{v_1,v_2,\dots v_n}$ (correlation functions) is associated to every
topological Riemann surface of genus $g$, with elements of the algebra
$v_1,v_2,\dots v_n\in F$ inserted at $n$ punctures, which satisfies the
following axioms: \begin{description} \item[Normalization]
$F(0)_{1,1,1} = 1$, \item[Invariance] $F(g)_{v_1,\dots} =
F(g)_{1,v_1,\dots}$, \item[Linearity] $F(g)_{v_1,\dots}$ is linear in
$v_i$. \end{description} Introduce the symmetric linear 3-form
$\om:F\tensor F\tensor F\arr \C$ by $\om(u,v,w) = F(0)_{u,v,w}$, the
bilinear form $(u,v)=F(0)_{u,v}$ and the trace $\int u = F(0)_{u}$.
Assume that $(,)$ is {\bf non-degenerate}, and fix a pair of bases
$\{u_i,u^i\}$ of $F$, dual with respect to this form, that is
$(u_i,u^j)=\delta_{ij}$.
\begin{description}
\item[Verlinde's fusion rule] $F(g)_{v_1,\dots} = \sum_{i}
F(g-1)_{u_i,u^i,v_1\dots}. $
\end{description}
One can extend $F$ to disconnected surfaces by the axiom:
\begin{description}
\item[Multiplicativity] $F$ is
multiplicative under disjoint union.
\end{description}
\begin{rem} These axioms serve as an algebraic model of certain
relations among the Hilbert functions of various moduli spaces. The number
$F(g)_{v_1,v_2,\dots v_n}$ represents the dimension of the space of
sections of a certain line bundle over a moduli space of parabolic bundles
with weights depending on the insertions $v_1,v_2,\dots v_n$. The fusion
axiom describes how the space of sections of a line bundle decomposes over
a family of curves degenerating to a nodal curve (cf.
\cite{faltings,tsueya}).
\end{rem}
\begin{lem} \label{twisted:moments} The axioms above define the
structure of an associative and commutative algebra on $F$, by the
formula $vw=\sum_i\om(v,w,u^i)u_i$, compatible with $(,)$ and $\int$.
Then if we denote the invariantly defined element $\sum_i u_i u^i\in F$
by $\al$, we have
\be \label{numberofCB} F(g)_{v_1,v_2,\dots,v_n} = \int
\al^g v_1v_2\dots v_n. \end{equation} \end{lem}
Now assume in addition that the algebra $F$ is {\bf semisimple}. Then it
has the form $ F\cong L^2(S,\mu),$ where $S=\Spec F$ is a finite set
and the complex measure $\mu$ can be given via a function $\mu:
S\arr\C$.
The elements of $F$ become functions on $S$ and the trace
$\int$ turns out to be the actual integral with respect to
$\mu$. Now take the following pair of dual bases:
$\{\del_s,\del_s/\mu(s)|\; s\in S\}$, where $\del_s(x)=\del_{sx}$. We
call this the spectral basis. Using this basis and \eqref{numberofCB},
we obtain the following formula:
\be \label{diagonal:formula}
F(g) = \sum_{s\in S} \mu(s)^{1-g}.
\end{equation}
This formula resembles \eqref{evenintro}, but what is the appropriate
algebra?
\subsection{Fusion algebras}
Here we construct the fusion algebras for arbitrary simple, simply
connected Lie groups. First we need to introduce some standard
notation.
\begin{notation} In this paragraph we will use the compact form of
simple Lie groups, still denoting them by the same letter. Thus let $G$ be
a compact, simply connected, simple Lie group, $\g$ its complexified Lie
algebra, $T$ a fixed maximal torus, and $\gt$ the complexified Lie algebra
of $T$. Denote by $\La$ the unit lattice in $\gt$ and by $\Wt\in\ts$ its
dual over $\Z$, the weight lattice. Let $\De\in\Wt$ be the set of roots
and $W$ the Weyl group of $G$. A fundamental domain for the natural action
of the Weyl group on $T$ is called an alcove; a fundamental domain for the
associated action on $\ts$ is called a chamber. We will use the
multiplicative notation for weights and roots, and think of them as
characters of $T$. The element of $\ts$ corresponding to a weight $\la$
under the exponential map will be denoted by $L_\la$.
Fix a dominant chamber $\goC$ in $\ts$ or a corresponding alcove $\alc$
in $T$. This choice induces a splitting of the roots into positive
($\De^+$) and negative ($\De^-$) ones. For a weight $\la$, denote its
Weyl antisymmetrization by $\A\la=\sum_{w\!\in\! W} \sigma(w)w\cdot\la$,
where $\sigma:W\arr\pm1$ is the standard character of $W$. According to
the Weyl character formula, for a dominant weight $\la$, the character
of the corresponding irreducible highest weight representation is
$\chi_\la=\A\la\rho/\A\rho$, where $\rho$ is the square root of the
product of the positive roots.
The ring $R(G)$, the representation ring of $G$, can be identified with
$R(T)^W$, the ring of Weyl invariant linear combinations of the weights.
Denote by $d\mu_T$ be the normalized Haar measure on $T$. If we endow
$T/W$ with the Weyl measure $$d\mw = \A\rho \A\bar\rho \;d\mu_T,$$
then $R(G)$ becomes a pre-Hilbert space with orthonormal basis
$\{\cl\}$, i.e. one has \newline
$\int_{T/W}\cl\chi_\mu\,d\mw = \delta_{\la,\bar{\mu}}.$
\qed \end{notation}
We need to introduce an integer parameter denoted by $k$ called
the {\em level}, which can be thought of as an element of
$\cH^3(G,\Z)\cong\cH^4(BG,\Z)\cong\Z$, and in turn can be identified
with a Weyl-invariant integral inner product on $\gt$.
The {\em basic} invariant inner product on $\gt$ corresponding to
$k=1$ is specified by the condition $(H_\th,H_\th)=2$, where $H_\th\in\gt$
is the coroot of the highest root $L_\th$. It has the following
properties (see \cite[\S 6]{kac},\cite[Ch.4]{loopgr}):
\begin{itemize}
\item For the induced inner product on $\ts$, we have $(L_\th,L_\th)=2$.
\item For $\la\!\in\!\Wt$,
the inner product $(L_\th,L_\la)$ is an integer, and (,) is the smallest
inner product with this property.
\item The Killing form is equal to $-2h (,)$, where $h = (L_\th,L_\rho)+1$ is
the dual Coxeter number of $G$.
\end{itemize}
The basic inner product also gives an identification $\nu:\ts\arr\gt$
between $\ts$ and $\gt$, by the formula $\beta(x)=(\nu(\beta),x)$.
\subsection{The simply-laced subgroup}
Let $\De_l\in\De$ be the set of long roots of $G$. Denote by $\Wt_r$ the
lattice in $\ts$ generated by $\De$, and by $\Wt_l$ the lattice generated
by $\De_l$. By definition $\La$ is the dual of $\Wt$ over $\Z$ with respect
to the canonical pairing $\<\, ,\?$ between $\ts$ and $\gt$. The dual
of $\Wt_r$ is the center lattice in $\gt$. Denote
the dual of $\Wt_l$ by $\La_l$.
The root system $\De_l$ corresponds to a subgroup $G_l$ of $G$ with
maximal torus $T$ and Weyl group $W_l\subset W$, which is generated by
reflections corresponding to the elements of $\De_l$. Denote the center
of $G_l$ by $Z_l$. Then $Z_l$ can be described as the set of elements of $T$
invariant under $W_l$, and we have $\exp^{-1}\La_l=Z_l$. It is important
to note that in view of the second property of $(,)$ above, $\nu:\Wt_l \maps
\La$ is an isomorphism.
Since $\Wt$ is paired to $\La$ and $\Wt_l$ is paired to
$\La_l$ over $\Z$, it follows that $\nu:\Wt\maps
\La_l$ is also an isomorphism. Then the map
$\exp\cdot\nu:\{\al\!\in\!\goC|\,(L_\th,\al)\leq 1\}\maps\alc$ is a
bijection.
Naturally, if $G$ is simply laced, then $G_l=G$. For the non-simply laced
groups one has the following subgroups:
\begin{itemize}
\item $\Spin_{2n}\subset\Spin_{2n+1} $
\item $\SU_2^n\subset\eSp_n $
\item $\SU_3\subset\Gtwo $
\item $\Spin_8\subset\Ffour $
\end{itemize}
\subsection{The definition of the fusion algebra}
We give a different definition from the standard one via co-invariants
of infinite dimensional Lie algebras \cite{tsueya}, but one which is
very natural from the point of view of representation theory.
To motivate the construction, recall the procedure of holomorphic
induction \cite{bott}: the flag variety $\F=G/T$ has a complex structure
and every character $\la$ of $T$ induces a holomorphic equivariant line
bundle $\cL_\la=G\times_\la \C$ over $\F$. Then one can define the
induction map $\I:R(T)\arr R(G)$ as a homomorphism of additive groups by
the formula $\la\mapsto \sum (-1)^{i}\cH^i(F,\cL_\la)$, where the
cohomology groups in the latter expression are thought of as
$G$-modules. The Borel-Weil-Bott theorem then says that
\begin{align}
&\text{for }\la\text{ dominant }\I(\la)=\chi_\la\\
\intertext{and}
&\I(\la') = \sigma(w)\I(\la), \text{ whenever } \la'\rho =
w(\la\rho)\text{ for some } w\in W. \label{indrel}
\end{align}
Note that $\I$ is not expected to be a ring homomorphism.
This procedure applies to the loop group $\lpg$ as well
(\cite{loopgr,kumar,mathi}). Once the action of the central elements is
fixed as $c\mapsto c^k$, where $k\in \N$ is the level, again, we have a map
$\bar{\I}:R(T)\mapsto R_k(\lpg)$. This last object $R_k(\lpg)$ has only
additive structure, since the tensor product of two level $k$
representations has level $2k$. The role of the Weyl group is played by the
affine Weyl group $W_k$ obtained by adjoining to $W$ the translation by
$(k+h) L_\th$. Again the Borel-Weil-Bott theorem applies, and
\eqref{indrel}, with $W$ replaced with $W_k$, gives a description of the
kernel of $\bar{\I}$.
Since $W\subset W_k$, the map $\bar{\I}$ factors through $\I$, and as a
result we have a map $\J:R(G) \mapsto R_k(\lpg)$. It is easy to see from
the \eqref{indrel} that the set of characters: $\Xi _k = \{\cl |\,
(L_\th,L_\la)\leq k\}$ forms a basis of $R_k(\lpg)$ if we identify $\J(\cl)$
with $\cl$.
\begin{lem} \label{quot}
The additive group $R_k(\lpg)$ can be endowed with a ring
structure $F_k^G$ so that the map $\J$ becomes a homomorphism of rings.
\end{lem}
The algebra $F_k^G$ is called the {\em fusion algebra} of $G$ of level $k$.
As noted above, we can consider $\Xi_k$ to be a basis of $F^G_k$. Endow
$F_k^G$ with the trace function $\int$ by the formula $\int\cl=0$ except
for the trivial character $\chi_1$, which has trace equal to 1. Also
note that since $\Spec(R(G))=T/W$ and $F_k^G$ is a quotient of $R(G)$, we
expect $\Spec(F_k^G)\subset T/W$.
\begin{lem} \label{measure}
$F_k^G$ can be identified with ``$L^2$'' of the finite normalized measure
space $Z_k=\{t\!\in\! T |\, t^{k+h}\!\in\! Z_l, \, t\;\text{\rm is
regular}\}/W$, with measure $d\mu_k$ given by the function
\be \label{lu}
\dfrac{\A\rho(t)\A\bar\rho(t)}{|Z_l|(k+h)^r}. \end{equation}
\end{lem}
Note the surprising fact that the discrete measure remains unchanged up
to a normalization factor as $k$ varies.
Now we can define the quantity $V^G_k(g)$ which appeared in
\eqref{main} as the number associated to a Riemann surface of
genus $g$ and the fusion algebra $F^G_{k-h}$. Combining
\eqref{diagonal:formula} and \eqref{lu} we obtain
$$ V^G_k(g) = \sum_{t\in T_r/W, t^k\in Z_l}
\frwd{|Z_l|k^r}{\A\rho(t)\A\bar\rho(t)}^{g-1},$$
where $T_r$ is the set of regular elements of $T$. This can be easily
seen to give \eqref{evenintro} for the case of $G=\SU_2$. Indeed,
embedding the maximal torus of $\SU_2$ into $\C$ as the unit circle,
we have: $\rho(z)=z$, $Z_l=\pm 1$, $h=2$ and $A(z) = 1/z$.
Finally, note that since the relations in the fusion algebras given in
\eqref{indrel} have integer coefficients, and the $\{\khi_\la\}$ form an
orthonormal basis of $F$, we see that $\int\al^g$ from \eqref{numberofCB}
has to be an integer. This proves that $V^G_k(g)$ is an integer for all
groups and values of $k$ and $g$.
\begin{rem} That this definition of the fusion algebras is equivalent
to the standard one via coinvariants of current algebras
\cite{tsueya} can be shown to be
equivalent to Verlinde's conjecture on the diagonalization of the fusion
rules. which gives a formula for the product in $F^G_k$ using the
$S$-matrix. The definition given above is simpler to use for calculations
and it gives the correct prescription for non simply connected groups (see
also \cite{gepner}).
\end{rem}
\section{Residue formulas} \label{resi}
In this section we study $V_k^G(g)$ as a function of $k$. We show that
$V_k^G(g)$ is a polynomial in $k$ for $G=\SL_3$, and give a simple
formula for the coefficients of this polynomial. The generalization of
these results to $\SL_n$ is straightforward.
Consider the case $G=\SL_2$ first. Again, as at the end of the previous
section embed the maximal torus of $\SL_2$ into $\C\subset\pone$.
Consider the differential form $$\mu=\frac{dz}{z}\frac{z+\zin}{z-\zin}
\text{ on }\Pro^1.$$ This form has simple poles: at $z=\pm 1$ with
residue 1, and at $z=0,\infty$ with residue $-1$. Thus if we pull back
$\mu$ by the $k$-th power map we obtain a differential form $\mu_k$
with poles at the $2k$-th roots of unity and residues $+1$, and simple
poles at $z=0,\infty$ with residue $-k$ . It is given by the following
formula: $$ \mu_k = k
\frac{dz}{z}\frac{ z^{k}+z^{-k}}{z^{k}-z^{-k}}.$$
Note that $\mu_k$ is invariant under multiplication by a $2k$th root
of unity and under the Weyl reflection $z\arr 1/z$.
Now suppose we have a function $f(z)$, with poles only at $z=\pm 1$,
vanishing at 0 and $\infty$, and invariant under the substitution
$z\arr z^{-1}$. Then by applying the Residue Theorem to
the differential form $f\mu_k $ and using the Weyl symmetry at hand,
we have
$$\sum_{j=1}^{k-1}f(\exp(\pi Ij/k)) =
-\Res_{z=1} \mu_k f(z),$$
where $I^2=-1$.
Applying this argument to the function
$$f(z)=\frwd{2k}{-(z-z^{-1})^2}^{g-1}$$ we obtain the formula
$$V^{\SL_2}_k(g) =(-1)^{g} (2k)^{g-1}
\Res_{z=1} \frac{k\, dz}{z} \frac{z^{k}+z^{-k}}{z^{k}-z^{-k}}
\frwd{1}{(z-z^{-1})^2}^{g-1}.$$
Now using the invariance of the residue under substitutions we can
obtain different formulas for $V^{\SL_2}_k(g)$. For example, the
polynomial nature of $V^{\SL_2}_k(g)$ becomes transparent if we
perform the substitution $z\arr \exp(Ix)$: $$V^{\SL_2}_k(g) =
-(2k)^{g-1}\Res_{x=0} \frac{k\cot(kx)\,dx}{(2\sin x)^{2(g-1)}}.$$ It
is easy to check using this formula that the degree of $V^{\SL_2}_k(g)$
as a polynomial in $k$ is $3(g-1)$, which, as expected, coincides with
the dimension of $\M^{\SL_2}_C$.
Before we proceed, we need an understanding of higher dimensional
residues. The notion that a top dimensional differential form has an
invariantly defined number assigned to it, does not carry over to the
higher dimensions. The correct object in $\C^n$ is
$\Res:\cH^n_{\loc}(\Om^n,\C^n) \mapsto \C$ mapping from the $n$th local
\v Cech cohomology group in a neighborhood of 0 with values in holomorphic
$n$-forms to
complex numbers. To define this map let $\om$ be a meromorphic $n$-form
defined in a neighborhood of 0 in $\C^n$. Then $\om$ can be represented in
the form $dz_1\,dz_2\,\dots dz_n h(z)/f(z)$ where $f$ and $h$ are
holomorphic functions. The additional data necessary to represent an
element of $\cH^n_{\loc}(\Om^n,\C^n)$ is a splitting of $f$ into the
product of $n$ functions $f=a_1a_2\dots a_n$. Such a splitting defines $n$
open sets in the a neighborhood of 0: $A_i = \{a_i\neq 0\}$. These define
a local \v Cech cocycle. A detailed explanation of this and an algorithm to
calculate the residue can be found in \cite{grha,harts}.
We will call a differential $n$-form with such a splitting a {\em
residue form}.
\begin{defn} A non-trivial residue form $\om$ is called flaglike if $a_i$ only
depends on $z_1,\dots z_i$. This notion depends on choice and the order of
the coordinates $z_1,\dots, z_n$. \end{defn}
\begin{lem} \label{flag}
Let $\om$ be a flaglike residue form. Then
$$ \Res(\om) = \Res_{z_n}\dots\Res_{z_1} \om.$$
Here $\Res_{z_i}$ is the ordinary 1-dimensional residue, taken assuming
all the other variables to be constants.
\end{lem}
The proof is straightforward. Note that the order of the variables is
important, while there is some freedom in the way the denominator is split
up.
For simplicity we restrict ourselves to the case of $\SL_3$. According
to \lemref{measure} and \eqref{diagonal:formula} the Verlinde formula
can be written as
\be \label{eseltri}
V_k^{\SL_3}(g) = (3k^2)^{g-1}\sum_{i,j,k-i-j > 0}
(8\sin(i\pi/k)\sin(j\pi/k)\sin((i+j)\pi/k))^{-2(g-1)}
\end{equation}
Now we can write down the main result of the paper:
\begin{thm} \label{result}
\begin{multline} \label{fores}
V_k^{\SL_3}(g) = (3k^2)^{g-1} \Res_{Y=1} \Res_{X=1}
\frac{X^k+X^{-k}}{X^k-X^{-k}}\;\frac{Y^k+Y^{-k}}{Y^k-Y^{-k}}\times\\
\times\frac{(-1)^{g-1}}{((X-X^{-1})(Y-Y^{-1})(XY-(XY)^{-1}))^{2(g-1)}}
\frac{k^2\,dX\,dY}{XY}.
\end{multline}
\end{thm}
The proof is analogous to the case of $\SL_2$. Denote the residue form in
\eqref{fores} by $\om_k(g)$.
Again at the points
$p_{ij} = (e^{iI\pi/k},e^{jI\pi/k})$, with $i,j,k-i-j\ge 1$ the
residues of $\om_k(g)$ reproduce the sum \eqref{eseltri}. However,
now it is not immediately obvious that the residue theorem can
localize this sum at the point $(1,1)$, since the residue form in
\eqref{fores} has non-trivial residues at other points as well. To
illustrate the situation consider the matrix $M_k$ whose $(i,j)$th entry
is the residue of $\om_k(g)$ taken at the point $p_{ij}$
instead of $(1,1)$, where $i,j=0,\dots,k-1$.
Example for $g=2$: $$ M_6 =
\left [\begin {array}{cccccc} 166&-45&-29&-18&-29&-45
\\\noalign{\medskip}-45&36&9&9&36&-45\\\noalign{\medskip}-29&9&4&9&-29
&36\\\noalign{\medskip}-18&9&9&-18&9&9\\\noalign{\medskip}-29&36&-29&9
&4&9\\\noalign{\medskip}-45&-45&36&9&9&36\end {array}\right ]
. $$
We can apply the Residue Theorem to ``each column'' by fixing a value
of $X$. By degree count, one can see that $\om(g)$ has trivial residues at
$Y=0,\infty$ and this implies
that the sum of the entries in each column of $M_k(g)$ is 0. Next, note that
$M_k(0,i)=M_k(i,0)$, since these residues are {\em split}, i.e. they have
the form $dX\,dY\, X^{-m}Y^{-n} f(X,Y)$, where $f$ is holomorphic at the
point where the residue is taken.
Now to prove the Theorem it is sufficient to show that
$M_k(j,0)=M_k(j,k-j)$ for every $j>0$. To see this, note that both residues
are simple (first order) in $X$ at $\al=\exp(j\pi/k)$. This means that
after taking the $X$-residue, we are left with the form $$ \om_\al = \const
\frac{dY}{Y}
\frac{Y^k+Y^{-k}}{Y^k-Y^{-k}} \frac{1}{((Y-Y^{-1})(\al
Y-\al^{-1}Y^{-1}))^{2(g-1)}}.$$ The two numbers we need to compare are the
residues of this form at $0$ and $\al$ respectively. But these two
residues clearly coincide since $\om_\al$ is invariant under the
substitution $Y\arr \al^{-1}Y^{-1}$. \qed
The formula for $G=\SL_n$ reads as follows:
\be \label{slnverl}
V^G_k(g) = (-1)^{n-1+(g-1)|\Delta^+|} (nk^{n-1})^{g-1} \Res_{X_{n-1}=1}\dots
\Res_{X_1=1}
W^{-2(g-1)}
\prod_{i=1}^{n-1} \frac{X_i^k+1}{X_i^k-1} \cdot \frac{k\,dX_i}{2X_i}
\end{equation}
where $X_i,\; i=1,\dots,n-1$ are the simple (multiplicative) roots and $W =
\prod_{\al\in\Delta^+} (\al^{\fel}-\al^{-\fel})$ is the Weyl
measure.
As we pointed out after \lemref{flag}, the ordering of the variables
matters when taking the subsequent residues. In the special case of
$G=\SL_3$ this ordering does not matter (i.e. $M_k(i,j)=M_k(j,i)$), but for
higher rank groups a finer argument is necessary.
Finally, note that similarly to the case of $\SL_2$, \eqref{slnverl} gives
a simple prescription for calculating the coefficients of $V^G_k(g)$ as a
polynomial in $k$, via the exponential substitution. For example, for
$\SL_3$ we obtain
\be \label{verlslt} V^{\SL_3}_k(g)= (3k^2)^{g-1} \Res_{y=0}\Res_{x=0}
\frac{k^2cot(kx)\cot(ky)\, dx\,dy}{(8\sin(x)\sin(y)\sin(x+y))^{2(g-1)}}.
\end{equation}
A different generating function was obtained for the case of $G=\SL_3$ by
Zagier \cite{zagverl}.
\section{Multiple $\zeta$-values and intersection numbers of moduli spaces}
\label{application}
In this final section we show how \eqref{main} and \eqref{fores} can be
related via the Riemann-Roch formula to Witten's conjectures on the
intersection numbers of moduli spaces. Our argument below gives a quick
proof of \eqref{main} for $\SL_n$ assuming Witten's formulas. This is a
generalization of the work of Thaddeus who considered the case of $\SL_2$
\cite{thad}.
\subsection{Multiple $\zeta$-values and intersection numbers of moduli
spaces}
Consider the case of $\SL_2$ first. If we want to find the asymptotic
behavior of $V^{\SL_2}_k$ for large $k$, the best way to think about the
formula is that it is a discrete approximation to the (divergent)
integral $\int_0^1 \sin(\pi x)^{-2(g-1)}\, dx$. To find the leading
asymptotics, we can replace $\sin(x)$ by $x$, and taking the large $k$
limit we obtain: $ (k/2)^{g-1}\sum_{j=1}^\infty (k/(j\pi))^{2(g-1)} =
k^{3(g-1)}\zeta(2(g-1))/(2^{g-1}\pi^{2(g-1)})$. This can be easily
proven, and in fact, a generalization of this formula for arbitrary
groups appeared in Witten's work \cite{wittzeta}.
Below we will concentrate on the case of $\SL_3$, however the formulas can
be extended to $\SL_n$ as well.
If we perform the trick above for $\SL_3$, up to a constant, the leading
behavior of the Verlinde formula appears to be $$ V^{\SL_3}_g(k) \sim
\const k^{8(g-1)}/\pi^{6(g-1)} \sum_{i,j=1}^\infty (ij(i+j))^{-2(g-1)}.
$$
One can write down more general sums, e.g.:
$$S(a,b,c) = \sum_{i,j=1}^\infty i^{-a} j^{-b} (i+j)^{-c},$$
closely related to the so-called {\em multiple zeta values} \cite{zagzet}.
It was discovered by Witten that all intersection numbers of moduli
spaces are given by combinations of multiple $\zeta$-values
\cite{wittvol,wittzeta}. Below we give a couple of useful formulas for them.
We restrict ourselves to the case $S(2g,2g,2g)$ for simplicity. Similar
formulas exist in greater generality.
\begin{lem} \mbox{}
\be \label{bern}
S(2g,2g,2g) = \fel\int_0^1 \bar{B}_{2g}(x)^3 \, dx,
\end{equation}
where $\bar{B}_n(x)$ is a modified {\rm n}th Bernoulli polynomial,
$\bar{B}_n(x)= -(2\pi I)^n B_n(x)/n!$.
\be \label{zetares}
S(2g,2g,2g) =
\frac{1}{3}\Res_{(0,0)} \cot(x)\cot(y) (xy(x+y))^{-2g}.
\end{equation}
\end{lem}
Sketch of Proof: The first formula follows from the definition of the
Bernoulli polynomials:
$$ \bar{B}_n(x) = \sum_{j\neq 0} e^{2Ij\pi x}/j^n. $$
Indeed, substituting this into $\fel\int_0^1 B_{2g}(x)^3 \, dx,$ one obtains
$S(2g,2g,2g)$ on the nose; the coefficient $1/2$ is a combinatorial
factor.
The proof of the second formula is similar to the proof of
\thmref{result}. On has to apply the Residue Theorem in two steps. That
the residue at infinity vanishes follows from the expansion $\cot(x)$:
$$\pi\cot(\pi x) = \sum_{n\in\Z} (x-n)^{-1}.\qed $$
\subsection{Intersection numbers of the moduli spaces} First we recall
some facts about the cohomology of the moduli spaces. We will ignore that
the moduli spaces are not smooth in general, and accordingly, we will
assume the existence of a universal bundle, Riemann-Roch formula, etc.
However, formulas analogous to \eqref{result} exist for the smooth moduli
spaces as well (e.g. when the degree and rank are coprime for $\SL_n$), and
all of our statements are rigorous for these cases. Some of the singular
moduli spaces (e.g. vector bundles) can be handled using the methods of
\cite{BSz}. We will also ignore certain difficulties which arise for
$\Spin_{n}$, $n>6$, and the exceptional groups, where the ample line
bundle exists only for $k=0\mod l$, for some $l$, depending on the type of
the group. In these cases the Verlinde formula is a polynomial only when
restricted to these values. Thus what follows should perceived as a
scheme of a proof, which works as it is in some cases, but requires
modification and more work in greater generality.
There is a universal principal $G$-bundle $U$ over the space
$C\times\M^G_C$, which induces a map $\M^G_C\arr BG$, and consequently
a map $s:\cH^*(BG)\arr \cH^*(\M^G_C)\tensor\cH^*(C)$.
Recall that $\cH^*(BG) \cong\Sym(\g^*)^G$, the space of $G$-invariant
polynomial functions on $\g$. This is a polynomial ring itself in
$\rank(G)$ generators and it is isomorphic by restriction to
$S^G=\Sym(\gt^*)^W$. For every $\al\in\cH_i(C)$ and $P\in S^G$ we obtain a
cohomology class of $\al\cap s(P)\in\cH^{2i-j}(\M^G_C)$, the
$\al$-component of $s(P)$. In fact, $s$ induces a map $\bs:\cH_*(C)\cap
S^G\arr\cH^*(\M^G_C)$, where $\cH^*(C)\cap S^G$ is the free commutative
differential algebra generated by the ring $S^G$ and the differentials
of negative degree modeled on $\cH_*(C)$. For the case of $\SL_n$ and
coprime degree and rank it is known that $\bs$ is surjective \cite{AB,kirwan}.
In
particular, denoting the fundamental class of $C$ by $\eta_C$, and the
basic invariant scalar product from \secref{fusion} by $P_2$ we obtain a
class $\om = \eta_C\cap s(P_2)\in \cH^2(\M^G_C),$ which turns out to be the
first Chern class of the line bundle from \eqref{main}. To simplify the
notation, below we omit the map $s$ and also $\al$ if $\al=1$, when writing
down the classes $\cH^*(\M^G_C)$. Thus $1\cap s(P)$ will be denoted simply
by $P$.
Any power series in the variables $\al\cap P$ can be integrated over
$\M^G_C$ and these numbers are called the intersection numbers of the
moduli space. Naturally, only the terms of degree $\dim \M^G_C=
\dim(G)(\on{genus}(C)-1)$ will contribute.
Witten, using non-rigorous methods, gave a complete description of these
intersection numbers in the most general case \cite{wittzeta}. His
formulas are combinations of multiple $\zeta$-values, and are rather difficult
to calculate. In this paper, we will focus only on a subset of these
intersection numbers, which are of the form $\int_{\M} \om^l P$, where
$l\in\N$ and $P$ is a not necessarily homogeneous Weyl-symmetric function
on $\gt$.
\begin{conj} \label{conjecture}
For every group $G$, there exists a residue form $\Omega^G$ depending on
$g$, defined in a neighborhood of $0\in\gt$, the Cartan subalgebra of $G$,
such that
\be \label{conj}
\int_{\M} e^{\om} P = \Res_{\on{at}\; 0\in\gt} \Omega^G P,
\end{equation}
\end{conj}
For $G=\SL_n$,
\be
\label{intersect}
\Omega = n^{g-1} \Res_{x_{n-1}=0}\dots\Res_{x_1=0} \prod_{\al\in
\Delta^+} L_\al^{2(g-1)} \prod_{i=1}^{n-1} \cot(x_i) \, dx_i,
\end{equation}
where the $x_i$-s are halves of the simple (additive) roots of $\SL_n$,
ordered according to the Dynkin diagram.
Let us write down the formula for $\SL_3$ more explicitly and inserting the
``grading'':
\be \label{special}
\int_{\M} e^{k\om} P = (3k^2)^{g-1}
\Res_{y=0} \Res_{x=0}
\frac{k^2\cot(kx)\cot(ky)\,dx\,dy}{(8xy(x+y))^{2(g-1)}}P
\end{equation}
\begin{rem}
It can be shown that \eqref{intersect} is consistent with Witten's
formulas. We will not give the proof here, but note that the link
between the two types of formulas is given by equalities like
\eqref{zetares}.
At the moment we do
not know $\Omega^G$ for general $G$.
Our formulas seem to be related to those given in the
works of Jeffrey and Kirwan \cite{jeffkir,jeffkirnew}. \qed
\end{rem}
Finally, we present another evidence for \eqref{intersect}:
the consistency with the Verlinde formula. First we need a few facts
about the moduli spaces. Fix a curve $C$ of genus $g$ and a group $G$.
They will be omitted from the notation.
\begin{lem} \label{properties}\mbox{}
\begin{enumerate}
\item $c_1(T_{\M}) = h\om$,
\item $p(T_{\M}) = c(\on{Ad} U_z)^{2(g-1)}=\prod_{\al\in\De} (1+\al)^{2(g-1)}$,
\item $\ahat(T_{\M}) =
\prod_{\al\in\De^+}\frwd{\al/2}{\sinh(\al/2)}^{2(g-1)}$.
\end{enumerate}
Here $h$ is the dual Coxeter number of $G$, $p$ denotes the total
Pontryagin class, $c$ the total Chern class, $U_z$ is the bundle over
$\M$ obtained by restricting the universal principal bundle $U$ to a
slice $z\times\M$ for some $z\in C$ and $\on{Ad} U_z$ is the vector bundle
associated to $U_z$ via the adjoint representation of $G$.
\end{lem}
For the proof of the first two statements in some partial cases see
\cite{AB}. The second statement
statement follows from the Kodaira-Spencer construction. From the second
statement we find that the Pontryagin roots of $T_\M$ are the roots of the
Lie algebra $\g$, and this in turn implies the third statement.
Finally, we can put everything together. We will calculate $\dim
\cH^0(\M_C^G, \cL^k)$. Again, consider $G=\SL_3$ for simplicity.
First, the Kodaira vanishing theorem applies to
$\cL^k$, because the canonical bundle of $\M$ is negative, (this follows
from the \lemref{properties}(1), see also
\cite{BSz}). Thus we can replace the dimension of $\cH^0$ by the
Euler characteristic, and apply the Riemann-Roch theorem:
$$\dim\cH^0(\M_C^G, \cL^k)=
\chi(\M_C^G, \cL^k) = \int_{\M} e^{k\om} \Todd(\M). $$ According to
\lemref{properties}(1), and using the standard shifting trick we can rewrite
this integral as $$\int_{\M} e^{(k+h)\om} \ahat(\M). $$
We can calculate this integral using \eqref{special} and
\lemref{properties}(3), and the result is exactly $V^G_k(g)$ according to
\eqref{verlslt}. This proves \eqref{main}.\qed
\newpage
|
1994-02-16T03:33:45 | 9402 | alg-geom/9402012 | en | https://arxiv.org/abs/alg-geom/9402012 | [
"alg-geom",
"math.AG"
] | alg-geom/9402012 | null | Ludmil Katzarkov | Factorization theorems for the representations of the fundamental groups
of quasiprojective varieties and some applications | 50 pages, LaTeX | null | null | null | null | In this paper, using Gromov-Jost-Korevaar-Schoen technique of harmonic maps
to nonpositively curved targets, we study the representations of the
fundamental groups of quasiprojective varieties. As an application of the above
considerations we give a proof of a weak version of the Shafarevich Conjecture.
| [
{
"version": "v1",
"created": "Wed, 16 Feb 1994 02:33:20 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Katzarkov",
"Ludmil",
""
]
] | alg-geom | \section{Lefschetz theorem}
In this section we generalize a result of Simpson \cite{3}. The
Clemens-Lefschetz-Simpson theorem we prove is of independent
interest but it is also one of the key arguments in the next sections.
Let $X$ be a smooth projective variety and let $\alpha _{1},
\alpha _{2}, \ldots ,\alpha _{n}$ be linearly independent holomorphic one-forms
in $H^{0}(X, \Omega^{1}_{X})$. Denote by $\widetilde{X}$ the universal
covering of $X$. Then by integrating the pullbacks of the forms $\alpha
_{1}, \alpha _{2}, \ldots ,\alpha _{n}$ over $\widetilde{X}$ we can define a
map
\[ g: \widetilde{X} \longrightarrow {\Bbb C}^{n}.\]
Let us first consider the case $n=2$.
\begin{theo} Let $\alpha _{1}, \alpha _{2}$ be two linearly
independent holomorphic one forms. Then one of the following cases holds:
\begin{list}{\alph{bean})}{\usecounter{bean}}
\item The map $g$ has connected fibers.
\item There is a holomorphic map with connected fibers from $X$ to a normal
projective variety $Y$ such that $1 \leq \dim Y \leq 2$ and the forms $\alpha
_{1}, \alpha _{2}$ are pullbacks from $Y$.
\item The third case is a combination of the previous two. Namely, up to a
linear change of the coordinates in ${\Bbb C}^{2}$ the first form $\alpha
_{1}$ is a pullback by a map $X \longrightarrow Y$, where Y is an algebraic
curve. The second form $\alpha _{2}$ gives us a map $ \widetilde{X}
\longrightarrow {\Bbb C}$ with connected fibers.
\end{list}
\end{theo}
{\bf Proof.} We borrow heavily from Simpson's arguments from \cite{3},
where he proves the case $n=1$.
Let $Alb (X)$ be the Albanese variety of $X$. The forms
$\alpha_{1}, \alpha_{2}$ are pullbacks of forms on $Alb (X)$, which we will
also call $\alpha _{1}, \alpha_{2}$.
Define $B$ as the biggest abelian subvariety over which both forms $\alpha
_{1}$ and $\alpha_{2}$ considered as a sections of
$\Omega^{1}_{Alb (X)}$ are zero. Let
\[ A=Alb(X)/B.\]
We have the natural map
\[a : X \longrightarrow A.\]
Let $l : \widetilde{X} \longrightarrow X$ be the universal covering map.
Suppose first that the dimension of $A$ is zero. But this means that the forms
$\alpha _{1}, \alpha_{2}$ are the zero forms. Therefore $\dim_{\Bbb{C}}A \geq
1$.
Consider the map $g$ . Let $Q$ be the subset of ${\Bbb C^{2}}$ over which the
map $g$ has multiple fibers. From the construction of $g$ it follows that $Q$
is a constructible set in ${\Bbb C^{2}}$. We show now that if $\dim_{\Bbb C}Q
> 0$ then either case b) or c) of the theorem holds.
Assume that $\dim_{\Bbb C}Q > 0$. Then we have that $Q$ is an analytic
subvariety of ${\Bbb C}^{2}$ and $\dim_{\Bbb C}Q = 1$.
We consider now the composition of the maps
\[g: \widetilde{X} \longrightarrow {\Bbb C}^{2} \longrightarrow {\Bbb C}^{1},\]
where the map
\[p : {\Bbb C}^{2} \longrightarrow {\Bbb C}^{1} = {\Bbb C}^{2} / l \]
is just a quotient by some line $l$ in ${\Bbb C^{2}}$.
Let us choose coordinates $t_{1}$ and $t_{2}$ in ${\Bbb C}^{2}$ so that
$\alpha_{1}=g^{*}(dt_{1})$ and $\alpha_{2}=g^{*}(dt_{2})$.
There are two main cases to be dealt with.
Case $\alpha$) There is a line
\[l: c_{1}t_{1} +c_{2}t_{2}=0 ,\]
contained in $Q$. Here $c_{1}$ and $c_{2}$ are some complex numbers.
In this case the map
\[g: \widetilde{X} \longrightarrow {\Bbb C}^{2} \longrightarrow {\Bbb C}^{1}
\]
is given by integration of a holomorphic one form. Applying Simpson's version
from \cite{3} we obtain a map $X \longrightarrow Y$, where $Y$ is an algebraic
curve. This places us in cases b) or c) of the theorem. We will be in case
b) if there exists another line in $Q$, which is transversal to $l$.
Otherwise we get case c) of the theorem.
\medskip
Case $\beta$) $Q$ does not contain a line.
Let $x$ be a smooth point in $Q$. Then we need to consider two possibilities.
\begin{enumerate}
\item The differential $g_{*y}$ is the zero map on $T\widetilde{X}_{y}$ for
every $y$ from $g^{-1}(x)$.
We show that in this situation we have a map $X \longrightarrow Y$, where $Y$
is an algebraic surface. Consider the map
\[a : X \longrightarrow A.\]
Let the map $f : X \longrightarrow Y$ be the Stein factorization of $a$.
Pulling back the map $f : X \longrightarrow Y$ to the
universal coverings of $X$ and $Y$, $\widetilde{X}$ and $\widetilde{Y}$, we
get that the map
\[ g : \widetilde{X} \longrightarrow {\Bbb C}^{2} \]
factors through the map
\[\tilde{f} : \widetilde{X} \longrightarrow \widetilde{Y}.\]
Suppose now that $\dim_{\Bbb{C}}Y>2$. Obviously a connected component of the
fiber of $\tilde{f}$ maps to a component of $v$, where $v$ is the map
\[v:\widetilde{Y} \longrightarrow {\Bbb C^{2}}, \]
with the property that $g=\tilde{f}\circ v$. Note that under the assumption
$\dim_{\Bbb{C}}Y > 2$ we have that dimension of the fiber of $v$ is at least
one.
But the fact that $T\widetilde{X}_{y}$ for every $y$ from $g^{-1}(x)$ goes to
a point under $g_{*}$ means that the restrictions and projections of the forms
$\alpha_{1}$ and $ \alpha_{2}$ on a connected component of $g^{-1}(x)$ are
equal to zero. This implies that $g^{-1}(x)$ goes to a discrete set of points
under $ \widetilde{f}$, which contradicts the fact that the dimension of the
fiber of $v$ is at least one. Therefore there exists a map $X \longrightarrow
Y$, where $Y$ is an algebraic surface, so we are in case b) of the theorem.
\item For every $y$ from $g^{-1}(x)$, $g_{*}$ sends $T\widetilde{X}_{y}$ to
the same line in
$T \times {\Bbb C^{2}}$. Let $N$ be the irreducible component of $Q$ passing
through $x$.
Since $Q$ does not contain a line we can assume that the two form
\[ dt_{1} \wedge dt_{2}\]
is zero when restricted to $N$ and $ dt_{1}, dt_{2}$ are linearly independent
at every point $x$ in $N$ .
In this case the Castelnuovo- De Francis theorem gives us a holomorphic map
$s : l(g^{-1}(N)) \longrightarrow Y$, where $Y$ is an algebraic curve. We can
see that the map $g : g^{-1}(N) \longrightarrow \widetilde{Y}$ factors through
the map $\widetilde{s} : g^{-1}(N) \longrightarrow \widetilde{Y}$.
Consider now the map
\[a:X \longrightarrow A.\]
Let $J=a(X)$. We prove next that $\dim_{\Bbb C} J \leq2$. Suppose not. From the
construction of $a$ we see that the map
\[g: \widetilde{X} \longrightarrow {\Bbb C}^{2} \]
factors through the maps
\[\widetilde{a}: \widetilde{X} \longrightarrow \widetilde{J},\]
\[v: \widetilde{J} \longrightarrow {\Bbb C}^{2}.\]
To get a contradiction we need to show that there exists a fiber $V$ of the map
$v$ such that $\dim_{\Bbb C}V=0$.
We know that $\widetilde{Y} $ is contained in $\widetilde{J}$. But
by the universality of the Albanese map for $l(g^{-1}(N))$ we can see that
the map
\[s: l(g^{-1}(N)) \longrightarrow Y \]
is nothing else but the map
\[a:X \longrightarrow A,\]
restricted on $ l(g^{-1}(N)$. Recall that the map
\[g: \widetilde{X} \longrightarrow {\Bbb C}^{2} \]
was the pullback of $a : X \longrightarrow A$ to the universal covers of $X$
and $A$. Consider now a point $x$ in $N$. It follows from the construction that
$g^{-1}(x)$ is contained $ g^{-1}(N)$. We conclude that $l(g^{-1}(x))$ has
discrete image under $s$ and therefore it has a discrete image under the map
\[\widetilde{a}:\widetilde{X} \longrightarrow \widetilde{J}.\]
In this way we obtain a fiber $V=\widetilde{a}(g^{-1}(x))$ of the map $v$ such
that $\dim_{\Bbb C}V=0$. Therefore $\dim_{\Bbb C} J \leq 2$ and we are in case
b) of the theorem.
\end{enumerate}
So far we have shown that the map
\[ g: \widetilde{X} \longrightarrow \Bbb C^{2}\]
has no multiple fibers over an open set whose complement is of codimension
greater than two. Now we show that in this case part a) of the theorem holds.
To finish the argument we use the discussion of Gromov and Schoen \cite{1}
paragraph 9 about the Stein factorization for nonproper varieties. The only
thing we need to check is if the coresponding leave space is Hausdorff. This
follows easily from Lemma 9.3 in \cite{1}. After that the version of the
Stein factorization theorem from (see Theorem 3 \cite{33}) implies that the
map $g$ exists, namely we have $g$ factoring as
\[ g': \widetilde{X} \longrightarrow \Bbb C^{2}\]
and a finite covering
\[ m: {\Bbb C}^{2} \longrightarrow {\Bbb C}^{2}.\]
To show that the map $g$ has connected fibers we need to show that $m$ is an
isomorphism. But since the branch locus of $m$ produces a multiple fibers of
$g$ we conclude that $m$ is etale outside some subset in $\Bbb{C}^{2}$ of
codimension two. Then due to the purity theorem ( SGA) we conclude that $m$
is etale. But $\pi_{1}({\Bbb{C}}^{2} ) = 0$ and hence $m$ is biholomorphic.
Another way to see that $m$ is biholomorphic is to observe that since $Q$ has
real codimension at least 4 then ${\Bbb C}^{2} \setminus Q$ is simply connected
and therefore it has one, namely the trivial nonramified covering. Therefore
$g$ has connected fibers in codimension two. We finish the proof by observing
that semicontinuity ensures all fibers of $g$ are connected.
\hfill $\Box$
Now we state the Clemens-Lefschetz-Simpson theorem for arbitrary $n$. We
sketch a proof emphasizing the details where it differs from the case $n=1$.
Recall that $\alpha _{1}, \alpha _{2}, \ldots ,\alpha _{n}$ are $n$ are
holomorphic one forms on $X$ and the map
\[g: \widetilde{X} \rightarrow \Bbb C^{n} \]
was defined on $\widetilde{X}$ by integrating them on $ \widetilde{X}$.
\begin{theo}[Clemens-Lefschetz-Simpson] Let $\alpha _{1}, \alpha _{2},\ldots
,\alpha _{n}$ be linearly independent holomorphic one forms on $X$. Then one
of the following cases holds:
\begin{list}{\alph{inner})}{\usecounter{inner}}
\item The map $g$ has connected fibers.
\item There is a holomorphic map with connected fibers from $X$ to the
projective normal variety $Y$ such that $ 1 {\leq }\dim Y {\leq n} $ and the
forms $\alpha _{1}, \alpha _{2}, \ldots ,\alpha _{n}$ are pullbacks from $Y$.
\item The third case is a combination of the previous two. Namely after a
linear change of the coordinates in ${\Bbb C}^{n}$ some of the forms $\alpha
_{1}, \ldots ,\alpha _{k}$ come as a pullbacks of a map $X \longrightarrow Y$,
where $Y$ is an algebraic variety $1 \leq \dim Y \leq k < n$. The rest of
the forms give us a map
$g': \widetilde{X} \rightarrow {\Bbb C}^{n-k} $, with connected fibers.
\end{list}
\end{theo}
{\bf Proof.} The proof splits again into two cases.
\begin{enumerate}
\item $Q$ contains a hyperplane - then we reduce to the Simpsons's version of
the theorem.
\item $Q$ does not contain a hyperplane. We argue in the same way as in the
case of $n=2$ . Instead of applying the Castelnuovo De Francis theorem we apply
a generalization of it due to Z. Ran \cite{R}.
\begin{theo}[Z. Ran] Let $\alpha _{1}, \alpha _{2},\ldots ,\alpha _{n}$ be
linearly independent holomorphic one forms on $X$ and such that
\[ \alpha _{1} \wedge \alpha _{2} \wedge \ldots \wedge \alpha _{n} =0.\]
Then there exists a complex torus $A$ and an analytic map
\[f : X \longrightarrow A ,\]
such that $f(X)$ is a proper annalytic subvariety of $A$ , $\dim f(X)\leq n$,
and the forms \linebreak $\alpha _{1}, \alpha _{2}, \ldots ,\alpha _{n}$ are
pullbacks from $A$.
\end{theo}
\end{enumerate}
\begin{flushright}
$\Box$
\end{flushright}
In section 5 we give a version of the Clemens-Lefschetz-Simpson
theorem for quasiprojective varieties.
\section{Harmonic maps to buildings-basic facts}
\subsection{}
Following Gromov-Schoen and Korevaar-Schoen-Jost we briefly describe some
facts of the theory of the harmonic maps to buildings, which are essential for
our discussion. For a more detailed treatment one can look at their original
papers \cite{1}, \cite{16}. Also in this section we define a new object: a
spectral covering, which we assign to every nonconstant harmonic map to a
building.
Suppose from now on that $X$ is a smooth projective variety and $\widetilde{X}$
is the universal cover of $X$. Let $G$ be a simple Lie group of rank $r$ over
${\Bbb Q}_{p}$ and let $B$ be the corresponding Euclidean building. (For
definitions and more detailed account on buildings see \cite{6}.) $G$ acts
on $B$ by automorphisms and if there exists a representation of $\pi_{1}(X)$
in $G$, so does $G$.
Following \cite{1} one can define an energy functional for every
equivariant continuous nonconstant map $U:\widetilde{X}{\longrightarrow}B$ and
we can ask if there exists a minimum for this functional - the harmonic map.
The following remarkable result gives the answer to this question.
\begin {theo}[Gromov-Schoen] If the action of $\pi_{1}(X)$ on the building
has no fixed point, then there exists an equivariant harmonic map $U :
\widetilde{X} \longrightarrow B$.
\end{theo}
{\bf Proof.} See \cite{1}. \hfill $\Box$
Another important result from \cite{1} which we will need is the
following:
\begin {theo}[Gromov-Schoen] The singular set of the equivariant harmonic map
$U : \widetilde{X} \longrightarrow B$ is of codimension at least two.
\end {theo}
All these results hold in the case when the building is locally compact. For
some of our considerations we need the same results in the nonlocally compact
situation. In this case we use the following:
\begin {theo}[Korevaar -Schoen] The previous two theorems hold in the case of
length spaces of non-positive curvature, including nonlocally compact
buildings.
\end{theo}
{\bf Proof.} The theorem follows from proposition 2.6.5 in \cite{16}. The
actual proof was given by R. Schoen in his course at IAS Princeton 1992-1993
and will appear elsewhere. R. Schoen informed me that the same result was
proven J. Jost. \hfill $\Box$
Using this map $U$ we construct a new object - a spectral covering. As we will
see later this construction is not unique.
In every apartment $Ap$ of the building we have locally a canonical choice of
coordinates modulo similarity maps and actions of the affine Weyl group
$\widetilde{W}$. This means that locally we can choose one coordinate $z_{1}$
and consider also $e_{1}(z_{1}) , \ldots ,e_{w}(z_{1})$, where $e_{1}\ldots
e_{w}$ are the elements of the usual Weyl group, and $w$= number of the
elements in $W$.
Let $U^{*} T^{*}Ap $ be the pullback of the cotangent bundle of $Ap$ and let
$d$ denote the corresponding exterior derivative in $T^{\star}Ap $. Then we can
define, modulo the action of $W$, the differential forms $de_{1}(z_{1}),
\ldots ,de_{w}(z_{1})$ globally on the whole building. Observe that after
taking differentials we can make everything invariant under the translations
of the affine Weyl group. Consider the complexified differentials
$U^{\star}de_{1}(z_{1}) , \ldots ,U^{\star}de_{w}(z_{1})$. Due to the
harmonicity of $U$ the (1,0) part of each of the forms $U^{\star}de_{1}(z_{1})
, \ldots, U^{\star}de_{w}(z_{1})$ is
holomorphic. (This follows essentially from the fact that due to the Corlette
vanishing theorem (see \cite{1}), $U$ is actually pluriharmonic outside the
singular set). We will denote these new holomorphic one forms again by
$U^{\star}de_{1}(z_{1}) ,\ldots ,U^{\star}de_{w}(z_{1})$. Since $U$ is an
equivariant map these holomorphic one forms descend to forms on $X$, defined
modulo a $W$-action. Let $h_{1}\ldots h_{r}$ be a basis for all invariant
polynomials of $G$. We apply them to $U^{\star}de_{1}(z_{1}),\ldots ,
U^{\star}de_{w}(z_{1})$. According to \cite{1} the map $U$ is Lipschitz and
therefore we get that the holomorphic differentials $h_{1}, \ldots ,
h_{r}$ are bounded. Using Theorem 3.2 we conclude that $h_{1}, \ldots ,
h_{r}$ extend to a holomorphic differentials on the whole $X$, namely $h_{1},
\ldots , h_{r}$ extend to elements of $H^{0}(X,{\rm
Symm}^{d_{1}}\Omega^{1}_{X}), \ldots, H^{0}(X,{\rm
Symm}^{d_{r}}\Omega^{1}_{X})$ respectively.
\begin{rem} {\rm If the group $G$ is not simply connected one needs include in
$W$ the fundamental group of $G$ as well.}
\end{rem}
\subsection{}
Here we give one way of defining a spectral covering:
We define a spectral covering corresponding to the elements $h_{1},\ldots
,h_{r}$ from $H^{0}(X,{\rm Symm}^{d_{1}}{\Omega^{1}_{X}}),$ $\ldots,
H^{0}(X,{\rm Symm}^{d_{r}}{\Omega^{1}_{X}})$ respectively to be the zero
scheme ${\frak S}$ of the section $\det{(\rho(\theta) - \lambda\cdot id_{\bf
V})}$ , where $id_{\bf V}$ is the identity in a vector space $V$ such that
$\dim {\bf V} = w$. Here by $\det{(\rho(\theta) - \lambda\cdot id_{\bf V})}$ we
mean the following :
\begin{enumerate}
\item For every element of $G$ the coefficients of the characteristic
polynomial $\det{(\rho(\theta) - \lambda\cdot id_{\bf V})}$ (here $\lambda $
is a number ) are combinations of the invariant polynomials $h_{1},\ldots
,h_{r}$.
\item Now if we interpret $h_{1},\ldots ,h_{r}$ as elements from
$H^{0}(X,{\rm Symm}^{d_{1}}{\Omega^{1}_{X}}), \ldots , H^{0}(X,{\rm
Symm}^{d_{r}}{\Omega^{1}_{X}})$ and $ \lambda$ as the tautological section of
$T^{*}X$ we obtain $\det{(\rho(\theta) - \lambda\cdot id_{\bf V})}$ as an
element from $H^{0}(T^{*}X,{\rm Symm}^{w}{\pi^{*}\Omega^{1}_{X}})$.
\end{enumerate}
For more detailed account on spectral coverings see \cite{8}.
Some of these coverings may be nonreduced and reducible. In such cases we
work with $S' = {\frak S}_{\rm red}$. Finally we take $S$ - the equivariant
desingularization of $S'$. (Note that the existence of this desingularisation
follows from the equivariant version of the famous Hironaka's theorem.)
\begin{defi} $S$ is called the factorizing spectral covering corresponding to
$U$.
\end{defi}
For the proof of the main theorem we need only the following three properties
of $S$:
\begin{list}{\arabic{inner})}{\usecounter{inner}}
\item The forms $U^{\star}de_{1}(z_{1}), \ldots , U^{\star}de_{w}(z_{1})$ are
well defined holomorphic one forms on $S$;
\item There is a $W$ action on at least an open set of $S$;
\item $S$ is smooth.
\end{list}
Therefore we can define $S$ as a finite covering of a blow-up of $X$ with the
above three properties. An alternative way of constructing $S$ is given in
\cite{8} and \cite{9}. We describe this construction now .
First choose $r$ generically linearly independent holomorphic one forms out of
$U^{\star}de_{1}(z_{1}),\ldots ,$ $U^{\star}de_{w}(z_{1})$. The multivalued
form $U^{\star}de_{1}(z_{1}),\ldots , U^{\star}de_{r}(z_{1})$ defines
an $r$-fold covering $S'$ in $T^{*}X$. Now we define $S$ to be the Galois
closure of the function field extension ${\Bbb C}(S')/{\Bbb C}(X)$.
It is clear that from the above definitions that the spectral covering is
not unique. In the proof of the main theorem we show that the factorizing
properties of a given representation do not depend on the spectral covering
we have chosen.
The following fact is going to play an essential role in the proof of the
Shafarevich-Koll\'{a}r Conjecture.
\begin{lemma} The singular set $Q$ of the map $U^{'} : S \longrightarrow B$ is
contained in the union of the zeros of $U^{\star}de_{1}(z_{1}),\ldots ,
U^{\star}de_{w}(z_{1})$.
\end{lemma}
{\bf Proof.} The proof follows from the fact that according to Gromov and
Schoen ${\rm codim}_{S}Q \leq 2$ and that $U^{'}(Q)$ is contained in the faces
of the chambers of $B$. The existence of the intrinsic derivative, proved in
\cite{1} and \cite{16}, implies existence of a kernel $T^{*}S$ and therefore
$U^{\star}de_{1}(z_{1}),\ldots , U^{\star}de_{w}(z_{1})$ are equal to zero on
$Q$. \hfill $\Box$
\begin{rem} Observe that over some open set in $S$ , namely outside $Q$ we can
use the Castelnouvo-de Francis theorem to get a factorization (see \cite{2}).
\end{rem}
\subsection{Example}
Let us consider the case of the group $SL(3,{\Bbb Q_{p}})$. The Weyl group in
this case is going to be $S_{3}$. Then the
coordinates $e_{1}(z_{1})\ldots e_{6}(z_{1})$ can be thought as
six roots of unity on the face of $B$, which in this case is nothing else but
an equilaterial triangle.
The complexified differentials $U^{\star}de_{1}(z_{1}),\ldots
,U^{\star}de_{6}(z_{1})$ are
$U^{\star}de_{1}(z_{1}), \ldots, U^{\star}de_{3}(z_{1}) ,
-U^{\star}de_{1}(z_{1}), \ldots, -U^{\star}de_{3}(z_{1})$.
In this case we have
$h_{1}=-(U^{\star}de_{1}(z_{1})^{2}+U^{\star}de_{2}(z_{2})^{2}+U^{\star}de_{3}(z_{3})^{2})$
$h_{2}=U^{\star}de_{1}(z_{1})^{2}.U^{\star}de_{2}(z_{2})^{2}.U^{\star}de_{3}(z_{3})^{2}$.
The spectral covering $\widetilde{S}$ in this case is given by the following
section in $H^{0}(X,{\rm Symm}^{6}\Omega^{1}_{X})$:
$\lambda^{6}-h_{1}\lambda^{4}+h_{1}^{2}/2{\lambda^{2}}-h_{2}^{2}$,
where $\lambda$ is the tautological one form.
If, for example,
$U^{\star}de_{1}(z_{1})=U^{\star}de_{2}(z_{2})\neq 0$, then the factorizing
spectral covering $S$ is the desingularization of the zeros of the section
$\lambda^{2}-U^{\star}de_{1}(z_{1})^{2}$, which belongs to $H^{0}(X,{\rm
Symm}^{2}\Omega^{1}_{X})$.
\section { The main theorem and its proof}
\subsection{}
The main idea is to use the abundance of holomorphic one forms over the
factorizing spectral covering and to apply to it the Clemens-Lefschetz-Simpson
theorem we have proven above. This gives the factorization over an open set of
$X$. To finish the proof we use a new technique developed by J\'anos Koll\'ar
in \cite{30} and \cite{31}.
Recall some notation from section 2. Let $X$ be a smooth projective variety
and let $\widetilde{X}$ be the universal cover of $X$. Let $G$ be a simple Lie
group of rank $r$ over an arbitrary local field of characteristic zero, and
let $B$ be the corresponding Euclidean building ($B$ could be non locally
finite). $G$ acts on $B$ by isometries and if we have a representation
$\varrho:\pi_{1}(X) \longrightarrow G $, so does $\pi_{1}(X)$. If $\pi_{1}(X)$
acts on $B$ without a fixed point then it follows from the theory of
Gromov-Korevaar-Schoen-Jost that there exists a nonconstant equivariant
harmonic map $U:\widetilde{X}\longrightarrow B$.
\begin {theo} Let $U:\widetilde{X}\longrightarrow B$ be an equivariant
harmonic map which corresponds to the Zariski dense representation $\varrho:
\pi_{1}(X) \longrightarrow G$. Then there exists a holomorphic map $x:S
\longrightarrow Y $ where $Y$ is a normal projective variety such that $1 \leq
dim(Y) \leq r$. Moreover there exist $S^{0}$ and $Y^{0}$ - Zariski open
sets in $S$ and $Y$ respectively such that $\varrho: \pi_{1}(S^{0} )
\longrightarrow G$ factors through a representation $\varrho{\prime}:
\pi_{1}(Y^{0}))\longrightarrow G$.
\end {theo}
The fact that $\pi_{1}(X)$ acts on $B$ without fixed points allows us to
define spectral covering $\widetilde{S}$ the corresponding to $\varrho:
\pi_{1}(X) \longrightarrow G$. Take the corresponding $S$. Since the map
$U:\widetilde{X}\longrightarrow B$ is a nonconstant map we have at least one
holomorphic one form $\alpha _{1} $ on $S$.
If we have the forms $\alpha _{1},\ldots ,\alpha _{q}$, $1\leq q \leq r$,
then we are in position to apply the Clemens-Lefschetz -Simpson theorem.
Suppose that part a) or a) of this theorem holds in our situation.
Let $\bar{S}$ be the universal cover of $S$. Consider the map
$g: \bar{S} \longrightarrow {\Bbb C}^{q}$ defined as in section 2 by
integration of $\alpha _{1}, \ldots ,\alpha _{q}$ over
$\bar{S}$. Let $re(g)$ be the real part of this map. Obviously
$U : \widetilde{X}\longrightarrow B$ provides us with the
$\pi_{1}(S)$-equivariant harmonic map $U' :\bar{S}\longrightarrow B$.
\begin{lemma} The fiber of $re(g) :\bar{S} \longrightarrow {\Bbb R}^{q}$ maps
to a point under $U'$.
\end {lemma}
{\bf Proof.} We use an argument of \cite{1}, namely the local version of the
Stein factorization theorem, which says that in a small neighborhood $\Omega$
in $\bar {S}$ the map $g$ decomposes into a composition of a holomorphic map
$\Omega\longrightarrow D$, where $D$ is an open ball in ${\Bbb C}^{q}$,
followed by a harmonic map $u : D \longrightarrow B$. So we get that each
intersection of the fiber of $re(g)$ with a small neighborhood $\Omega$ goes
to a point in $B$. But now using the fact that the fiber is connected and
all the maps are continuous we see that the lemma holds. Observe that this
argument also works in the neighborhoods around the critical points of this
map. What one does is first approximate the singular fibers by nonsingular
ones, and then use the fact that $g$ is a continuous map. \hfill $\Box$
Now using the above Lemma we construct an isometry ${\Bbb R}^{q}
\longrightarrow B$. Before we do this we show how we are going to use this
isometry.
Recall that we are in case a) or c) of the
Clemens-Lefschetz-Simpson theorem with $l=q$. The image of
$U$ is all of ${\Bbb R}^{q}$. But this means that the action of $\pi_{1}(X)$
on the building $B$ preserves ${\Bbb R}^{q}$, so the action on the building
at infinity preserves a flat. Hence it is contained in some parabolic
subgroup in $G$. This contradicts the Zariski density of $\varrho$. In fact
one can prove this without going to the building at infinity. Fixing ${\Bbb
R}^{q}$ is equivalent to fixing ${\Bbb R}^{q}$ pointwise up to the action of
some finite group and this is again a contradiction.
This rules out the cases a) and c) of the
Clemens-Lefschetz-Simpson theorem.
Now we prove the existence of the isometry ${\Bbb R}^{q}\longrightarrow B$.
\begin{lemma} In the cases when part a) or c) of the
Clemens-Lefschetz-Simpson theorem holds we always have an isometry ${\Bbb
R}^{q} \longrightarrow B$.
\end {lemma}
{\bf Proof.} The case $q=1$ was proven by Simpson \cite{4}. Following our
proof of the
Clemens -Lefschetz -Simpson theorem we are going to work for simplicity with
the case $q=2$.
Start with the map $f:\bar{S}\longrightarrow{\Bbb C}$ defined by integration of
$\alpha _{1}$ over $\bar{S}$. Since the singular sets of the map $f$ are
compact sets in $\bar{S}$ and the fibers $F$ of $f$ are connected (recall we
are in case a) or c) of the
Clemens-Lefschetz-Simpson theorem), we can find a real line ${\Bbb R}$ in
${\Bbb C}$ over which the map $f$ has only smooth points. Now through every
point of this real line we can define another real line in the fiber of the
map $t : F \longrightarrow {\Bbb C}$. Here we again use the fact that the
singular sets of the map $t$ are compact sets in $F$ and the fibers $F'$ of
$t$ are connected. This means that we can not only find such a line
but also require that it passes through the point of our initial line ${\Bbb
R}$ in ${\Bbb C}$. We can do this in a continuous way using the fact that the
singular sets of the maps above are in codimension at least two. Since every
${\Bbb R}$-bundle over ${\Bbb R}$ is trivial we have a smooth ${\Bbb R}^{2}$
in $\bar{S}$.
Now using Lemma 4.1 we find smooth map $\Bbb R^{2}\longrightarrow B$. To show
that this is an isometry we apply Simpson's argument, which uses that the
forms $\alpha _{1}$ and $\alpha _{2}$ are defined by $U$. Thus the
differentials of $U$ and $re(g)$ are the same and therefore the differential
of $U$ is equal to the identity.
$\Box$
From now on we are going to work only with case B) of the Clemens -Lefschetz
-Simpson theorem. We need first to construct the factorization map. Part b)
of the Clemens-Lefschetz-Simpson theorem gives us a map $S\longrightarrow Y$.
So we have a $\pi_{1}(S)$ - equivariant harmonic map
$U\prime:\bar{S}\longrightarrow B$. Let $\widetilde{Y}$ be the universal cover
of $Y$.
\begin{lemma} The map $U\prime:\bar{S}\longrightarrow B$ factors through a
map
$u_{0}:\widetilde{Y}\longrightarrow B$, namely $U\prime=u_{0}.a$, where $a $ is
the map $a:\bar{S}\longrightarrow\widetilde{Y}$.
\end{lemma}
{\bf Proof.}
For simplicity we again consider the case $dim_{\Bbb R} B=2$ and two
holomorphic one-forms $\alpha _{1}$, $\alpha _{2}$ such that $\alpha
_{1}\wedge \alpha _{2}\neq0$ generically on $\bar{S}$. The forms $\alpha
_{1}$, $\alpha _{2}$ come from holomorphic one-forms $\beta _{1}$, $\beta
_{2}$ on $\widetilde{Y}$. This way we get that the map
$r_{1}:\bar{S}\longrightarrow \Bbb R^{2}$, given by integration of $\alpha
_{1}$, $\alpha _{2}$ over $\bar{S}$, factors through the map $r_{2}:Y
\longrightarrow \Bbb R^{2}$, given by integration of $\beta _{1}$, $\beta _{2}$
on $\widetilde{Y}$. Namely we have $r_{1}=r_{2}.a$. But then the fiber of the
map $a:\bar{S}\longrightarrow\widetilde{Y}$ is contained in the fiber $J$ of
the map $r_{1}:\bar{S}\longrightarrow \Bbb R^{2}$. By Lemma 4.1 $J$ is mapped
to a point under $U$ and so is the fiber of the map $a$. This proves the
factorization.
$\Box$
Define $X^{0}$ to be the Zariski dense set in $X$ obtained by throwing away
the branch locus of the map $S \longrightarrow X $ and the images in $X$ of
the exceptional sets in $S$. Let $S^{0}$ be the preimage of $X^{0}$ in $S$.
The fundamental group $\pi_{1}(S^{0})$ maps to a group with finite index in
$\pi_{1}(X^{0})$. The representation $\varrho(\pi_{1}(X))$ is Zariski dense in
$G$ as is $\varrho (\pi_{1}(X^{0}))$ since $\pi_{1}(X^{0})$ surjects to
$\pi_{1}(X)$. But since we know that $\pi_{1}(S^{0})$ maps to a group with
finite index in $\pi_{1}(X^{0})$, we can conclude that
$\varrho(\pi_{1}(S^{0}))$ is Zariski dense in $G$.
From the previous lemma we have that the harmonic map
$\bar{U}:\widetilde{S^{0}}\longrightarrow B$ factors through the map
$\bar{u_{0}}:\widetilde{Y^{0}} \longrightarrow B$, namely
$\bar{U}=\bar{u_{0}}.a$. (Here $\widetilde{S^{0}}$ and $\widetilde{Y^{0}}$,
are the universal covers of $S^{0}$ and $Y^{0}$ respectively.) Using the
properties of the pullback map for the inclusion $ i: \widetilde{S^{0}}
\longrightarrow \bar{S }$ and the fact that $\pi_{1}(S^{0})$ surjects onto
$\pi_{1}(S)$ we conclude that the map $\bar{U}:\widetilde{S^{0}}\longrightarrow
B$ is equivariant with respect to $\pi_{1}(S^{0})$.
We show that the action of $\varrho:\pi_{1}(S^{0})$ factors through an
action
of $\varrho{\prime}:(\pi_{1}(Y^{0}))$. Take $\gamma$ an element of
$\pi_{1}(S^{0})$ such that
$a_{\star}(\gamma)=1$. Then for any $x$ in $\widetilde{S^{0}}$ we have
$a_{\star}(\gamma.x)=
a_{\star}(x)$. We use the same notation: $a$ for both maps
$a:S\longrightarrow Y$ and $a:\bar{S}\longrightarrow\widetilde{Y}$ .
Then using the equivariance of $U$ we have
\[\varrho(\gamma)\bar{U}(x)=\bar{U}(\gamma.x)=\bar{u_{0}}.a(\gamma.x)=\bar{u_{0}}.a(x)=\bar{U}(x).\]
In the same way one can see that $\bar{u_{0}}$ is equivariant for the action
of $\varrho{\prime}(\pi_{1}(Y^{0}))$ on $B$.
\[\bar{u_{0}}(a_{\star}(\gamma)a(x))=\bar{u_{0}}a(\gamma x)=\bar{U}(\gamma
x)=\varrho(\gamma)\bar{U}( x)=
\varrho(a_{\star}\gamma)\bar{U}(ax).\]
We need to show that $\varrho(\gamma)=1$ if $\gamma$ is in $Ker (a
_{\star})$. Recall that $a$ has connected fibers, so the map
$a_{\star}:\pi_{1}(S^{0})\longrightarrow \pi_{1}(Y^{0})$ is a surjective map.
But $Ker (a _{\star})$ is a normal subgroup in $\pi_{1}(S^{0})$ thus
$\varrho(Ker (a _{\star}))$ is normal in $G$. Since $G$ is a simple Lie group
over $K$ there are two possibilities:
1) $\varrho(Ker (a _{\star}))=G$;
2) $\varrho(Ker (a _{\star}))$ is contained in the center $Z(G)$.
Suppose that we are in case 1). But from the computation above we know that
$\varrho(Ker (a _{\star}))$ fixes $\bar{U}(X^{0})$, an open set in $U(X)$,
which is a contradiction since $G$ does not fix any point in $B$.
In case 2) we obtain that $\varrho(\gamma)=1$ since we are working only with
centerless representations.
$\Box$
Let $\varrho:\pi_{1}(X) \longrightarrow G$ be a Zariski dense nonrigid
representation of the fundamental group of $X$ to some complex simple Lie
group $G$. Then there exists a curve $I$ in the moduli space of
representations passing through
$\varrho:\pi_{1}(X) \longrightarrow G$ in direction of which $\rho$ deforms
nontrivialy. Since the moduli space of representations is an affine variety the
curve $I$ is affine as well and let $\bar{I}$ be the compactification of $I$.
Consider now the representation $\bar{\rho}$ to $\bar{G}$, where $ \bar{G} $
is defined over the field of fractions of $I$. Let $O_{p}$ be the
localization at some point $p \in \bar{I} \ I$. Let $Z[T]$ be an extension of
the ring of integers of $O_{p}$ which contains all coefficients of
$\bar{\rho}$ localized at the point $p$. Consider now $I$ as a curve over $Spec
Z[T]$. The fact that $\varrho:\pi_{1}(X) \longrightarrow G$ is a Zariski
dense nonrigid representation implies that $\bar{\rho}$ is a Zariski
dense nonrigid representation. Observe that there are still going to be a
point $q$ in the curve $\bar{I}$ which is not in $Spec Z[T]$. Let
$\chi(\bar{\rho})$ is a character of $\bar{\rho}$. Moving over $\bar{I}
\subset Spec Z[T]$ we see that $\chi(\bar{\rho})$ is unbounded at $q$.
Therefore we conclude that the representation $\bar{\rho}$ to $\bar{G}$ is
not contained in any bounded subgroup in $\bar{G}$.
The following corollary is an easy consequence of the previous result:
\begin{corr} Let $U$ be an equivariant harmonic map
$U:\widetilde{X}\longrightarrow B$ and $\varrho:\pi_{1}(X) \longrightarrow G$
be a Zariski dense nonrigid representation, and let $X$ be a projective
variety. Then one of the following possibilities holds:
A) The $Im(\varrho(\pi_{1}(X))$ is contained in a maximal compact subgroup
in $G$ or
B) There exists a holomorphic map $S \longrightarrow Y$ and a map
$u_{0}:S^{0} \longrightarrow Y^{0} $ with $S^{0}$ and $Y^{0} $ Zariski open
sets in $S$ and $Y$ respectively, such that $\varrho( \pi_{1}(S^{0})) $
factors through a representation $\varrho{\prime}: \pi_{1}(Y^{0})
\longrightarrow G$, where $Y$ is a normal projective variety such that $1 \leq
dim(Y) \leq r$.
\end{corr}
Using this corollary we give a proof of Theorem 1.2. It follows from remark 3.1
and \cite{2} , paragraph 4 that using the properties of the spectral covering
we can mod out $S^{0}$ by $W$ and get a map
\[h:X^{0}\longrightarrow Y^{0} ,\]
, which comes from the morphism $h:X \longrightarrow Y/W $, where $X^{0}$ is
an open set in $X$ and $Y^{0}$ is an open set in $Y/W$. From now on we
denote all modifications of $Y$, namely blow ups and finite nonramified covers,
by $Y$ if not stated otherwise.
Denote the generic fiber of $h:X^{0} \longrightarrow Y^{0}$ by $Z^{1}$.
Let us first resolve the singularities of $Y$. This might change the
fundamental group of $Y$ but we need only that the fundamental group of $X$
does not change, which follows from the fact that $X$ is a smooth projective
variety.
We finish the proof of the main theorem by generalizing 4.8.1. in \cite{30}.
According to corollary 4.1 we have the following sequence:
\[ \pi_{1}(X^{0}) \longrightarrow \pi_{1}(Y^{0}) \longrightarrow G.\]
What we need to show is that $\pi_{1}(Z^{1})$ belongs to the kernel of the map
\[ \pi_{1}(X^{0}) \longrightarrow \pi_{1}(Y^{0}).\]
We cannot do that directly on $X$ but by generalizing 4.8.1. in \cite{30} we
show that this is possible on some finite nonramified covering of $X$, $X(T)$.
Observe that
\[ im[\pi_{1}(X^{0}) \longrightarrow \pi_{1}(Y^{0})]\subset Ker \rho.\]
\begin{defi} Let $P$ be a subgroup of $ \pi_{1}(X)$ and let $X(P)$ be a
covering of $X$ with fundamental group $P$. Consider now the Stein
factorization of the map $X(P) \longrightarrow Y$ and
define $X(P) \longrightarrow Y(P) $ to be the map with connected fibers in this
factorization.
\end{defi}
Note that $Y(P)$ will be an analytic variety even when the covering $
Y(P)\longrightarrow Y$ is infinite. This follows from the most general
version of the Stein factorization theorem (see Theorem 3 \cite{33}).
\begin{defi} Define $ \Omega$ to be the intersection of all subgroups $P$ in $
\pi_{1}(X)$ such that $H\subseteq P $, where
$H=im[\pi_{1}(Z^{1})\longrightarrow \pi_{1}(X)]$ and the covering $
Y(P)\longrightarrow Y$ has finite ramification indexes.
\end{defi}
Such an $\Omega$ is well defined due to the fact that there exists at least
one such a $P$ - $\pi_{1}(X)$.
Define $K=Ker (\varrho) $.
\begin{lemma} The covering $Y(K) \longrightarrow Y$ has finite ramification
indexes.
\end{lemma}
{\bf Proof.}
After intersecting $X$ with sufficiently many generic hyperplanes we get a
finite ramified covering $X \cap H \longrightarrow Y$. This covering obviously
has finite ramification index. But since the covering $X(K)\longrightarrow X$
is nonramified we know the indeces of the covering $Y(K) \longrightarrow Y$
are also finite.
$\Box$
\begin{lemma} There exists a finite index subgroup $T$ in $\pi_{1}(X)$ such
that:
\[H \subseteq R \subseteq \Omega \subseteq \pi_{1}(X ),\]
where $R$ is the kernel of the map $\pi_{1}(X(T))\longrightarrow
\pi_{1}(Y(T))$.
\end{lemma}
{\bf Proof.} The proof is the same as in 4.8.1 \cite{30}.
$\Box$
To finish the proof of 1.2 we need only to observe that
\[ R \subseteq \Omega \subseteq K \]
and this gives us the complete factorization
\[X(T) \longrightarrow Y(T).\]
As an almost immediate corollary we have:
\begin{corr}
Let $\varrho:\pi_{1}(X) \longrightarrow G$ be a Zariski dense nonrigid
representation of the fundamental group of $X$ to some complex simple Lie
group $G$. Then there exist:
1) a finite etale cover $X'$ of a blow up of $X$;
2) a smooth projective variety $Y$ of positive dimension $l$ less then or
equal to the rank $r$ of $G$ over $\Bbb{C}$;
3) and a holomorphic map $h:X' \longrightarrow Y $
such that $\varrho:\pi_{1}(X') \longrightarrow G$ factors through a
representation of $\pi_{1}(Y) $.
\end{corr}
{\bf Proof.} The fact that $\varrho:\pi_{1}(X) \longrightarrow G$ is a
Zariski dense nonrigid representation of the fundamental group of $X$ to
some complex simple Lie group $G$ implies that there exists a curve in the
moduli space of representations passing through
$\varrho: \pi_{1}(X) \longrightarrow G$, which intersects infinity in the
moduli space of representations in the point $p$. Let $O_{p}$ be the local ring
of this point as a point of the curve described above. This way we obtain a
representation $\bar{\rho}$ to $\bar{G}$, where $ \bar{G} $ is defined over
the field of fractions of the completion of $O_{p}$, which is also Zariski
dense and is not contained in any bounded subgroup in $\bar{G}$. We finish the
proof by applying Theorem 1.2 .
$\Box$
\begin{rem} In the last section we give a different proof of the
last part of Theorem 1.2. There we use the theory of the Shafarevich maps.
\end{rem}
\begin{rem} Observe that if we work with group of finite index in
$\pi_{1}(X)$ we get again the conclusions of the main theorem.
\end{rem}
\section{The quasiprojective case.}
In this section we are going to work with $X=X_{1}\setminus D$ - a
quasiprojective variety , where $D$ is the divisor at infinity and $X_{1}$ is
the compactification of $X$. Our goal is to prove theorem similar to that of
the previous section. Of course to start the whole procedure we need to show
that there exists an equivariant continuous map of finite energy
$U:\widetilde{X}\longrightarrow B$. To be able to show this we require that
our divisor at infinity is a divisor with normal crossing with unipotent
monodromy around it. We can cover $X$ by finitely many open sets since it is
compactly embedded. Using the fact that we have a unipotent monodromy at
infinity we see that the fundamental groups of the open sets which cover the
divisor at infinity are contained in maximal compact subgroups. A simple
computation shows that by simultaneous conjugation by some elements we can
make all of the finitely many generators of these groups have integer
coefficients
and hence they are contained in the same maximal
compact subgroup. Using the fact that the maximal compact subgroups fix a point
in $B$ we get an equivariant continuous map of finite energy from every open
set which covers the divisor at infinity to $B$, namely the map to the fixed
point. Using the standard center of mass construction we obtain the
equivariant continuous map of finite energy $U:\widetilde{X}\longrightarrow B$.
We are going to make sense of spectral covering in the quasiprojective case.
First we formulate and give a sketch of the proof of Corlette vanishing
theorem in the quasiprojective case.
Let $X$ be a quasiprojective variety with universal covering $\widetilde{X}$.
Consider now a representation of the fundamental group of $X$ to some Lie group
defined over an arbitrary nonarchimedian field. Let $B$ be the corresponding
Euclidean building. Let us assume also that the divisor at infinity in $X$ is a
divisor with normal crossings and the monodromy around it is unipotent. We have
the following theorem, the proof of which is similar to the proof of the
original theorem of Gromov and Schoen.
It is clear that in the situation above, provided that
$\pi_{1}(X)$ acts on $B$ without fixed points, to every Zariski dense
representaion we assign a harmonic map $U:\widetilde{X}{\longrightarrow}B$
for which the whole theory of Gromov, Korevaar, Schoen and Jost works.
Now take a regular point $x_{0}$ of $\widetilde{X}$ (for the definition of
regular point see \cite{1}). Then the image of the ball $B-\sigma(x_{0})$ is
contained in at least one flat $F$ in $B$. Let $\nabla $ denote the pullback
connection and let $d_{\nabla}$ denote the corresponding exterior derivative
operator on p-forms with values in $U^{*}TF$. Let ${\delta} _{\nabla}$ denote
its formal adjoint. The differential $dU$ then defines a 1-form with values in
$U^{*}TF$, and we have the following:
\begin {theo}(Corlette ) Let $X$ be a quasiprojective variety, let $ \omega $
be a parallel p-form on $\widetilde{X}$, and let $U$ be a harmonic map
$U:\widetilde{X}{\longrightarrow}B$.
Then in the neighborhood of $x_{0}$, a regular point for $U$, the form $w
{\wedge}dU$
satisfies ${\delta} _{\nabla}(w {\wedge} dU){\equiv}0$.
\end {theo}
{\bf Proof.} The statement of this theorem is local so the proof is almost the
same as in \cite{1}. First we exhaust $X$ by compact sets $X_{i}$. After that
one needs to choose the right cutfunction and apply to each $X_{i}$ theorem
7.2 from \cite{1}. But since the statement is local we just choose the same
functions as in \cite{1}.
$\Box$
\begin{corr} In the situation above $U$ is pluriharmonic.
\end{corr}
{\bf Proof.} Proof is as in \cite{1}.
$\Box$
First we define the spectral covering . As in the compact case we have the one
forms \linebreak $U^{\star}de_{1}(z_{1}),\ldots , U^{\star}de_{w}(z_{1})$
defined on $X=X_{1}\setminus D$ up to action of the Weyl group. (Here $z_{1}$
is a an arbitrary choice of a coordinate on a given chamber of $B$ , which we
extend using the action of the group.) Again we apply to them the basis of the
$G$-invariant for $G$ polynomials and obtain the forms $h_{1}, \ldots ,
h_{r}$. Due to the fact that $U$ is a finite energy map it follows from
\cite{1} that the forms $h_{1}, \ldots , h_{r}$ are $L_{2}$ bounded and
therefore have at most log poles. The unipotency of the loops around infinity
gives us that the residues of $h_{1}, \ldots , h_{r}$ are rational. Therefore
over some finite covering of $X_{1}$ they are holomorphic. Using the procedure
of section 3 we build now the spectral covering $S$ over $X_{1}$.
The theorem above gives us the forms $U^{\star}de_{1}(z_{1}),\ldots,
U^{\star}de_{w}(z_{1})$ defined on $X=X_{1}\setminus D$ up to an action of
the Weyl group. Therefore over the spectral covering $S$ we obtain the forms
$\alpha _{1}, \alpha _{2},...,\alpha _{r}$, where $r$ is the rank of the group
$G$. Observe that the forms $\alpha _{1}, \alpha _{2},...,\alpha _{r}$ are
$L_{2}$ bounded and therefore have at most log poles. This can be seen in the
following way: Using \cite{1}(Theorem 6.3) we see that a loop around $D$
fixes a point $s$ in $B$. Therefore the map $U:D \longrightarrow B$ gives us
a weakly subharmonic map $d^{2}(U(x),s)$ and then the same arguments as in
\cite{1}( Theorem 6.3) imply the above statement.
Now we state the quasiprojective version of the Clemens -Lefschetz -Simpson
theorem.
Observe that if $l:S \longrightarrow X$ is the spectral coveing then
$S=S^{1} \setminus l^{*}(D)$ .
Following Iitaka \cite{I} we introduce the Albanese map $alb:S \longrightarrow
Alb(S)$ for quasiprojective varieties. As in the case of projective varieties
it is defined by integrals of holomorphic one forms on $S$. Here $Alb(S)$ is a
semiabelian variety- a group extension of $Alb(S^{1})$ by
$(\Bbb{C}^{*})^{\times l}$.
Define also
\[ A=Alb(S)/B.\]
We have the natural map
\[a:S \longrightarrow A.\]
Here $B$ is again the maximal abelian subvariety over which the forms
$\alpha _{1}$, $\alpha _{2}$,..., $\alpha _{n}$ are zero as a forms on
$Alb(S^{1})$.
Let us also define the map
\[g:\widetilde {S}\longrightarrow \Bbb{C}^{n}\]
as a pullback of $a$. In this situation we have the following :
\begin{theo}(Clemens - Lefschetz - Simpson) Let $S$ be a smooth
quasiprojective variety. Let $\alpha _{1}, \alpha _{2},...,\alpha _{r}$ be
holomorphic one forms on $X$ with at most log poles at $l^{*}(D)$. Then one
of the following cases holds:
A) The map $g$ has connected fibers.
B) There is a holomorphic map with connected fibers from $S$ to the projective
normal variety $Y$ such that $ 1 {\leq }dimY {\leq n} $ and the forms $\alpha
_{1}, \alpha _{2},...,\alpha _{n}$ are pullbacks from $Y$.
C)The third case is a combination of the previous two. Namely after a linear
change of the coordinates in $\Bbb {C}^{n}$ some of the forms $\alpha
_{1},...,\alpha _{k}$ come as a pullbacks of a map $X \longrightarrow Y$,
where $Y$ is an algebraic variety $1 {\leq }\dim Y {\leq k}<n $. The rest of
the forms give us a map
$g^{'}: \widetilde{X} \rightarrow \Bbb C^{l} $, where $l \leq n-k $, with
connected fibers.
\end{theo}
The proof is similar to the proof in the projective case.
Now we formulate the version of our main theorem for a quasiprojective variety
$X$.
\begin{theo} Let $X=X^{1}\setminus D$ be a smooth quasiprojective variety and
$\varrho:\pi_{1}(X) \longrightarrow G$ be a
Zariski dense representation of the fundamental group of $X$ with unipotent
monodromy around $D$, where $D$ is a divisor with a normal crossing. Let $G$ be
a simple Lie group over $K$. Then
A. either the image of $\varrho$ is in a maximal compact subgroup of $G$ or,
B. there exist:
1) a blow up $ X' $ of a finite etale cover of $X^{1}$;
2) a smooth projective variety $Y$ of positive dimension $l$ less than or
equal to the rank $r$ of $G$ over $K$;
3) A holomorphic map $h:X' \longrightarrow Y $ such that $\varrho:\pi_{1}(X')
\longrightarrow G$ factors through a representation of $\pi_{1}(Y) $ and such
that the pullback of $D$ in $X'$ is a pullback from a divisor on $Y$.
\end{theo}
We give an application of the above theorem, which was suggested by J.
Koll\'ar.
According to N.Mok (see \cite{21}) every real Zariski dense discrete
representation in $SL(n,\Bbb{C})$ of noncompact type of the fundamental group
of any compact K\"{a}hler manifold after some blow up and finite nonramified
covering factors though the representation of the fundamental group of
projective algebraic variety of general type.
Let $X=X^{1}\setminus D$ be a quasiprojective variety such that $X^{1
}$ has Kodaira dimension zero and let $\varrho:\pi_{1}(X) \longrightarrow
SL(n,\Bbb{C})$ be a real Zariski dense discrete representation of the
fundamental group of
$X$. The hypothesis of the Mok's theorem send us in case B) of the above
theorem. Since $ X' $ is a finite etale cover of blow up of $X^{1}$ it has
also Kodaira dimension equal to zero . Mok's theorem also tells us that $Y$
from part B) of the previous theorem is an algebraic variety of general type.
Therefore we obtain a holomorphic map $h:X' \longrightarrow Y $ from a variety
with Kodaira dimension zero to a variety of general type and this impossible,
due to a theorem of Kawamata \cite{27}. We have obtained the following:
\begin{corr}Let $X=X^{1}\setminus D$ be a quasiprojective variety such that
$X^{1}$ has Kodaira characteristic zero. Then:
\[\varrho:\pi_{1}(X) \longrightarrow SL(n,\Bbb{R})\]
is a finite group.
\end{corr}
\section{Factorization theorems for complexes of groups.}
In \cite{1} Gromov and Schoen proved that if the fundamental group of a
quasiprojective variety $X$ admits a decomposition as an amalgamated product
of groups, then $X$ admits a surjective holomorphic map to an algebraic curve.
In this section we extend this result to higher dimensional complexes of
groups. This gives a partial answer to a question Gromov stated in \cite{22}
(section 7). In what follows we are working with the negatively curved
complexes of groups defined by Benakli in \cite{32}.
The key idea in \cite{1} is that the Baas-Serre theory assigns to an
amalgamated product of groups a tree to which one applies the theory of
harmonic maps. According to \cite{17} and \cite{18}(Theorem 6.4) we can assign
to every negatively curved two dimesional complex of groups a connected
negatively curved simplicial cell complex of dimension 2 with a finite set of
isometry types of cells $T$. Let $G(T)$ be the universal covering of $T$.
The fundamental group $\pi_{1}(T)$ (see \cite{18}) acts on $G(T)$ by
simplicial isometries . Let $X$ be a projective variety such that there
exists a surjective homomorphism $\theta:\pi_{1}(X)\longrightarrow
\pi_{1}(T)$. Therefore $\pi_{1}(X)$ acts on $G(T)$ without fixed points. Using
the fact that $G(T)$ is contractible (\cite{17} and \cite{18}) we apply the
Korevaar-Schoen theorem to get a harmonic map
$U:\widetilde{X}\longrightarrow G(T)$. In this section we work only with $G(T)$
of the type defined by Benakli in \cite{32} paragraph 7. Namely they are
constructed by using the baricentric subdivision of a regular hyperbolic
polyhedra.
The following theorem is a consequence of the previous sections:
\begin{theo} Every representation of the fundamental group of projective
varieties onto the fundamental group of some negatively curved 2-dimensional
polyhedra, comes from the representation of the fundamental group of an
orbicurve.
\end{theo}
{\bf Proof.} The only thing we need to show is how to construct the spectral
covering.
To do this we first choose local coordinates on $G(T)$ . Using the
symmetries, defined on $G(T)$ (see \cite{17} and \cite{18}) we can extend
these coordinates to the all of $G(T)$. To obtain holomorphic forms, and
consequently a spectral covering, we need vanishing theorems. But since in
the case of $G(T)$ we do not have a statement about the codimension of the
singularities of the corresponding harmonic map, we need to do some extra work,
which is the essence of the theorem. First we use the following theorem
(\cite{32}):
\begin{theo} (Benakli) $G(T)$ can be embedded isometrically in a hyperbolic
space $\Bbb{H}^{3}$
in such a way that $G(T)$ is a "deformation retract " of $\Bbb{H}^{3}_{1}$.
\end{theo}
Here we denote by $\Bbb{H}^{3}_{1}$ the 3-hyperbolic space $\Bbb{H}^{3}$
with the vertexes of every ideal polytop thrown away. Observe that
$\Bbb{H}^{3}_{1}$ is contractible.
We prove a version of the vanishing theorem for $G(T)$ closely following
\cite{1}. We omit the details of the argument in \cite{1}, emphasizing only
the differences.
Let us first mention that Benakli's construction can be made $\pi_{1}(T)$ -
equivariant. For our argument we need a special kind of retraction. Namely to
show that the harmonic maps $U_{t}:\widetilde{X}\longrightarrow
\Bbb{H}^{3}_{t}$ exist for every $t$, we need to make sure that the curvature
of $\Bbb{H}^{3}_{t}$ is nonpositive for every $t$.
We do that by retracting $\Bbb{H}^{3}_{1}$ to $G(T)$ equidistantly. Observe
that the $\Bbb{H}^{3}_{t}$ are going to be singular for every $t$. The crucial
fact, which helps us avoid these difficulties is that $\Bbb{H}^{3}_{1}$ has
dimension greater than $G(T)$.
\begin{lemma} $\Bbb{H}^{3}_{t}$ is negatively curved for every $t$.
\end{lemma}
The proof of this statement is a standard application of the Gauss formula for
the curvature of a hypersurface.
\begin{theo} The map $U_{t}: \widetilde{X} \longrightarrow \Bbb{H}^{3}_{t}$ is
pluriharmonic for every $t>0$.
\end{theo}
{\bf Proof.} Observe that the singular set of $G(T)$ consists of the centers of
the ideal hyperbolic polyhedra and the edges, which come out of them.
Therefore we need to worry only about ball $b$ around a regular points $x_{0}
$ in $\widetilde{X}$ which maps to the ideal polyhedra on the boundary of
$\Bbb{H}^{3}_{t}$ which can be retracted to $G(T)$. We consider 3 different
cases:
A) The map $U_{t}: \widetilde{X} \longrightarrow \Bbb{H}^{3}_{t}$ maps a ball
$b$ around a regular point $x_{0}
$ in $\widetilde{X}$ to a face in the boundary of $\Bbb{H}^{3}_{t}$ but far
from the edges . In this case the proof of the theorem follows from Theorem (
7.3) \cite{1}.
B) The image of a ball $b$ around a regular point $x_{0}
$ in $\widetilde{X}$ under the map $U_{t}: \widetilde{X} \longrightarrow {\Bbb
H}^{3}_{t}$ contains an edge comming out from the center of an ideal
hyperbolic polyhedra. In this situation
$\Bbb{H}^{3}_{t}$ looks localy like a product of a tree and $\Bbb{R}^{2}$.
Therefore Theorem ( 7.2) \cite{1} applies and we can find for every $t$ a
sequence of functions $ \psi_{i,t}$ which:
1) vanish in a neighborhood of the set $S_{t}$ of $U_{t}$;
2) tend to 1 on $X \backslash S_{t}$ and;
3) such that:
\[ \lim_{t\rightarrow \infty}{} \lim_{i\rightarrow 0}{\int_{X}^{}{\parallel
\bigtriangledown \bigtriangledown U_{t}}\parallel \psi_{i,t} d\mu}=0 .\]
Following Theorem ( 7.2) \cite{1} we get:
\begin{theo} Let $X$ be a smooth projective variety, $ \omega $ a parallel
p-form on $ \widetilde{X}$ and let the image of a ball $b$ around a regular
point $x_{0}
$ in $\widetilde{X}$ under the map $U_{t}: \widetilde{X} \longrightarrow
\Bbb{H}^{3}_{t}$ contain an edge of an ideal hyperbolic polyhedra but be far
from a vertex. Then in some small neighborhood around $x_{0}$ the form $\omega
\wedge dU$ satiesfies
\[ \delta_{\bigtriangledown}(\omega \wedge dU)\equiv 0.\]
\end{theo}
Here the notations are the same as in theorem 5.2.
Now in the same way as in Theorem ( 7.3) \cite{1} we obtain:
\begin{theo} The map $U_{t}: \widetilde{X} \longrightarrow \Bbb{H}^{3}_{t}$
is pluriharmonic.
\end{theo}
C) The image of a ball $b$ around a regular point $x_{0}
$ in $\widetilde{X}$ under the map $U_{t}: \widetilde{X} \longrightarrow
\Bbb{H}^{3}_{t}$ contains a translated center of an ideal hyperbolic
polyhedra. This case cannot be done as the previous two . $\Bbb{H}^{3}_{t}$
looks localy like a product $P$ of an open polyhedra (namely we have taken the
botom face away) and an open interval. We can approximate $P$ by products
$C_{t}$ of intervals and cones. Following the last example in the
Introduction of \cite{1}, we can also approximate the singular metric $g_{t}$
on $C_{t}$ by regular metrics $g_{ts}$ with $K\leq 0$. This construction is
similar to the polyhedral immersion of a 2-disk into $\Bbb{H}^{4}$ descibed in
\cite{L}. To show that the maps $U_{ts} : \widetilde{X} \longrightarrow b$
are pluriharmonic for every regular metric $g_{ts}$ on $C_{t}$, we use the
same argument as in B). Namely we use Theorem ( 7.2) \cite{1} to find for
every $t$ a sequence of functions $ \psi_{i,t}$.
According to \cite{15} the limit of pluriharmonic maps is a pluriharmonic map
so we obtain a pluriharmonic map to $C_{t}$ with singular metric $g_{t}$ on it.
Now we approximate $P$ by the $C_{t}$, and using again that limit of
pluriharmonic maps is a pluriharmonic map, we get that the map $U_{t}:
\widetilde{X} \longrightarrow \Bbb{H}^{3}_{t}$ is pluriharmonic.
$\Box$
\begin{rem} The above argument works in the same way for a truncated two
dimensional polyhedra (\cite{32}). Part C) of the argument does not generalize
in higher dimensions.
\end{rem}
An easy corollary of the above theorem is the following:
\begin{corr} The map $U_{t}: \widetilde{X} \longrightarrow G(T)$ is
pluriharmonic.
\end{corr}
We are now ready to finish the proof of Theorem 6.1. After we get the
pluriharmonicity of the map $U:\widetilde{X}\longrightarrow G(T) $ we can
construct the spectral covering. As before the pluriharmonicity implies that if
we take the (0,1) part of the complexified differentials $du_{1}, du_{2}$ we
obtain holomorphic differentials over some spectral covering.
The same argument as in the proof of Theorem 1.2 gives a complete
factorization $ h:X \longrightarrow Y$. Let say that we are in case A) of
the Clemens-Lefschetz -Simpson theorem. Then we have a local isometry $l:
\Bbb{R}^{2}\longrightarrow G(T)$ as in Lemma 4.2. We obtain $\Bbb{R}^{2}$ as
the real part of the map defined by the integration of the holomorphic forms
over the spectral covering $S$. The fact that $ G(T)$ is strictly negatively
curved make the existence of such a local isometry impossible. Following
Lemma 4.2 we obtain in the same way a contradiction in the case when $dim_{\Bbb
{C}} Y=2$.
According to the Clemens-Lefschetz -Simpson theorem there is a possibility of
an isometry $l: \Bbb{R}^{1}\longrightarrow G(T)$. But then if the image is
isometric to $ \Bbb{R}^{1}$ we will have subcomplexes fixed under the action
of $\pi_{1}(X) $ and this contradicts the surjectivity of
$\theta:\pi_{1}(X)\longrightarrow \pi_{1}(T)$.
Therefore the Clemens-Lefschetz -Simpson theorem that there is a
factorization through the fundamental group of an orbicurve.
$\Box$
A conjecture stated by J. Carlson and D. Toledo \cite{31} says that if
$\Gamma$ is a K\"{a}hler group then $H^{2}(\Gamma)$ is nontrivial. As a simple
corollary of the above theorem 6.1 we have:
\begin{corr} If there exists a surjective homomorphism $\theta:\pi_{1}(X)
\longrightarrow \pi_{1}(T)$ then the above conjecture is true for $\pi_{1}(X)$.
\end{corr}
Inspired by \cite{22} we describe a construction which we use in the next
section. According to the previous theorem, if we work with negatively curved
length spaces we get a nice description of the representations, namely they are
coming from the representations of the fundamental group of an orbicurve. It is
natural to try to hyperbolize the objects we are working with. This can be done
in the case of a building $B$. (Following \cite{23} we can replace every
chamber in every apartment of the building by a negatively curved complex with
boundary the boundary of the chamber. )
According to the result of R. Charney and M. Davis \cite{23} we obtain the
hyperbolized building $HB$, which is strictly negatively curved. The
functoriality of the construction of Charney and Davis provides us with an
action of $G$, the corresponding group of the initial building $B$, over $HB$.
Unfortunately there is one disadvantage of the whole construction - $HB$ has
a nontrivial topology. So we can use the technique developed above only on
representations $\varrho: \pi_{1}(X) \longrightarrow G$, for which we know
there exists a $G$- equivariant continuous map
$U:\widetilde{X}{\longrightarrow}HB$. Along the lines of the previous
corollary we have:
\begin{corr} Every Zariski dense representation $\varrho: \pi_{1}(X)
\longrightarrow G$
for which there exists an $G$- equivariant continuous map
$U:\widetilde{X}{\longrightarrow}HB$, such that $G$ acts on $HB$ without a
fixed point, factors through the representation of the fundamental group of an
orbicurve.
\end {corr}
Let us give some sufficient conditions for the existence of a $G$-
equivariant continious map $U:\widetilde{X}{\longrightarrow}HB$.
\begin{theo} There exists a $G$- equivariant continuous map
$U:\widetilde{X}{\longrightarrow}HB$ if the group homomorphism
$\pi_{1}(X)\to Out (PI)$ induced by $\varrho: \pi_{1}(X) \longrightarrow G$
can be lifted to a group
homomorphism $\pi_{1}(X)\to Aut(PI)$. Here $PI$ is the fundamental group of
$HB$ and $Out(PI)$ and $Aut(PI)$ denote the groups of
outer automorphisms and automorphisms of $PI$ respectively.
\end{theo}
{\bf Proof.}
It is easy to see the necessity of this condition directly. It has
a natural interpretation in terms of gerbes (see \cite{34} and \cite{35}). Let
us explain the construction of the gerbes we use in this situation. For this
we need to choose a universal covering $\widetilde{HB}\to HB $ of
$HB=K(PI,1)$.
Then we have an exact sequence of groups:
\[ 1\longrightarrow PI \longrightarrow Homeo(\widetilde{HB}, HB)
\longrightarrow Homeo(HB) \longrightarrow 1 ,\]
where $Homeo(\widetilde{HB}, HB)$ is the group of homeomorphisms of
$\widetilde{HB}$ , which cover some homeomorphism of $HB$.
Since we have a principal bundle over $HB$ with group $Homeo(HB)$, there is
a gerbe whose objects are local liftings of the structure group to
$Homeo(\widetilde{HB})$, and whose arrows are isomorphisms of
$Homeo(\widetilde{HB})$ - bundles, which
induce the identity on the $Homeo(HB)$-bundle. The band of this gerbe is
locally isomorphic to the band associated to the constant sheaf of
$PI$ , but there is an outer twisting represented by a class in
$H^{1}(X,Out(PI))$, namely the class of the
homomorphism $\varrho: \pi_{1}(X) \longrightarrow G$. The obstruction to
realizing the given band as arising from some sheaf of groups over $X$ is
exactly the obstruction
to lifting the above class to a class in $H^{1}(X , Aut(PI))$.
$\Box$
Let us see how all this applies to the situation, where we let $HB$ be
the hyperbolized $SL(3,\Bbb(Q)_{p})$ building. It easy to see that in this case
$PI=F_{SL(3,\Bbb(Q_{p}))}$, where $F_{SL(3,\Bbb(Q_{p}))}$ is a free group
with $SL(3,\Bbb(Q_{p}))$ many generators. $Out(F_{SL(3,\Bbb(Q_{p}))})$
contains $SL(3,\Bbb(Q_{p}))$. What we need now, to claim the existence of a
$G$- equivariant continuous map $U:\widetilde{X}{\longrightarrow}HB$, is the
splitting of the following exact sequence:
\[1\longrightarrow F_{SL(3,\Bbb(Q_{p}))} \longrightarrow
Aut(F_{SL(3,\Bbb(Q_{p}))} \longrightarrow SL(3,\Bbb(Q_{p})\longrightarrow 1,\]
over $T$, where $T$ is the set theoretic image of $SL(3,\Bbb(Q_{p}))$ in
$Out(F_{SL(3,\Bbb(Q_{p}))})$.
To have this splitting we need to make $SL(3,\Bbb(Q_{p}))$ act on the graph we
have assigned to $F_{SL(3,\Bbb(Q_{p}))}$ (the graph in this case looks like
rose with one vertex and $SL(3,\Bbb(Q_{p})$ edges) while fixing the vertex.
For this we need to make a canonical choice of an initial point for $PI$ in
$HB$. A sufficient condition for this is, for example, if the image of
$\widetilde{X}$ under $U$ is homotopy equivalent to a tree which is fixed
under the action of the image of $\varrho: \pi_{1}(X) \longrightarrow G$. If
this condition is satisfied, we change in a canonical way the generators of
$F_{SL(3,\Bbb(Q_{p}))}$ when we are change the initial point for
$F_{SL(3,\Bbb(Q_{p}))}$ in $HB$. One case when the above condition is
satisfied is when the whole image of $U$ is a tree which is fixed under the
action of the image of $\varrho: \pi_{1}(X) \longrightarrow G$. This is an
equivariant harmonic map $U$ having dimension 1. Unfortunately in general this
condition is hard to check.
Using the technique of F. Paulin \cite{40} we prove:
\begin{theo} Let X be a smooth projective variety, and let $\Gamma $ be a
word hyperbolic group acting on the corresponding $\Bbb R$-tree without fixing
a vertex . Let us assume that $Out(\Gamma)$ is an infinite group and let
$\rho:\pi_{1}(X) \longrightarrow \Gamma$ be a surjective homomorphism. Then
$\rho:\pi_{1}(X) \longrightarrow \Gamma$ factors through a representation of
the fundamental group of an orbicurve.
\end{theo}
{\bf Proof.} According to Paulin \cite{40} $\Gamma$ has an fixed point free
action on $T$, an $\Bbb R$ tree. Using a theorem of Reeps \cite{40} we
conclude that this action is simplicial. We obtain $T$ by deforming the Cayley
graph of $\Gamma$ , using that $Out(\Gamma)$ is an infinite group. From the
Schoen- Korevaar - Jost theorem we know that there exists a harmonic $
\pi_{1}(X)$-equivariant map to $T$. In the same way as in the proof of theorem
6.1 we show that $\rho: \pi_{1}(X) \longrightarrow \Gamma$ factors through a
representation of the fundamental group of an orbicurve. We rule out the
possibility of part A) of the Clemens-Lefschetz -Simpson theorem by using the
fact that in the situation above we do not have parabolic action of $\Gamma$ on
infinity of $T$, namely $\Gamma$ does not have fixed point of the infinity
of $T$.
Another way to show that $\rho(\pi_{1}(X)): \longrightarrow \Gamma$ factors
through a representation of the fundamental group of an orbicurve, is to
follow the proof of theorem 8.1 \cite{15}.
$\Box$
\begin{rem} We think that one can use the harmonic map technique to prove the
Morgan - Shalen conjecture for K\"{a}hler groups .The conjecture states
\cite{40}, that every group with free action on an $\Bbb R $ tree is a free
product of an abelian and a surface group.
\end{rem}
\section{Some results about the integrality of representations}
In this section we discuss a partial verification of the following two
conjectures of Simpson \cite{11}. We are going to consider representations
$\varrho: \pi_{1}(X) \longrightarrow G$ , where $G$ is simple Lie group
defined over $k$, an algebraically closed field of characteristic zero. In
this section we work only with $G=SL(n,k)$.
\begin{con} Let $\varrho: \pi_{1}(X) \longrightarrow G$ be a rigid semisimple
representation. Then $\varrho \otimes k$ is a direct summand over $k$ in the
monodromy representation of a motive (i.e. comes from geometry ) over $X$.
\end{con}
\begin{con} Let $\varrho: \pi_{1}(X) \longrightarrow G$ be a rigid semisimple
representation. Then it is integral, in other words it is conjugate to a
representation whose matrix coefficients are algebraic integers.
\end{con}
(Note that the first Conjecture would imply the second.)
Here rigid representation means that every representation which is nearby in
the
affine variety of representations is conjugate to it. In the case of an
irreducible representation (the case we are working with), rigid is equivalent
to the fact that the corresponding point in the moduli space of
representations is isolated.
The goal of this section is to verify the second conjecture in some cases.
We proceed following Simpson \cite{3} . We assign to every Zariski dense
rigid representation a new Zariski dense rigid representation $\varrho:
\pi_{1}(X) \longrightarrow G^{1}$ over a local nonarchimedean field. The
procedure goes as follows. Observe that the moduli space of representations is
defined over $\Bbb Q$, and since we are working with a rigid representation we
can find an isomorphic representation defined over $\bar{\Bbb Q}$.
Let $E$ be a finite extension of $\Bbb Q$ defined to be the extension
which contains
all coefficients of our representation, and let $O$ denote the ring of
integers in $E$.
Let $E_{p}$ denote the field of fractions of the completion of $O$ in $p$ ,
for some prime $p$. Let $G^{1}$ be the new group
over $E_{p}$ and use $\varrho$ again for the representation $\varrho:
\pi_{1}(X) \longrightarrow G^{1}$. Since $E_{p}$ is a local field then we are
in the situation of the Theorem 4.2. Namely,
using the Bruhat-Tits theory we can attach to $G^{1}$ a building $B$.
Following the Gromov and Schoen theory we can attach a harmonic map to
$\varrho$. Therefore we get:
\begin{corr} Let $\varrho: \pi_{1}(X) \longrightarrow G$ be a Zariski dense
rigid representation. Then one of the following holds:
A) For every prime $p$ the image of $\varrho: \pi_{1}(X) \longrightarrow G^{1}$
is contained in some maximal compact subgroup in $G^{1}$.
B)For some prime $p$ the image of $\varrho: \pi_{1}(X) \longrightarrow G^{1}$
is not contained in some maximal compact subgroup in $G^{1}$. Then there
exist:
1) a finite etale cover $X'$ of a blow up of $X$;
2) a smooth projective variety $Y$ of positive dimension $l$ less or equal
to the rank $r$ of $G$ over $\Bbb{C}$;
3) and a holomorphic map $h:X' \longrightarrow Y $
such that $\varrho:\pi_{1}(X') \longrightarrow G$ factors through a
representation of $\pi_{1}(Y) $.
\end{corr}
In case A) we apply theorem os Simpson's \cite{C} to obtain that
$ \rho$ is an integer representation, namely a representation $\varrho:
\pi_{1}(X) \longrightarrow GL(r,{\Bbb {Z}})$, $r \geq n$.
Now we define special type of representations for which case B) of the above
theorem does not occur, namely conjecture 7.2 is satisfied. The idea is that
since curves do not have many rigid representations then if something factors
through a curve should not have either.
\begin{defi} Take $\varrho: \pi_{1}(X) \longrightarrow G^{1}$ to be a Zariski
dense representation in $G^{1}$ which is assigned to $\varrho: \pi_{1}(X)
\longrightarrow G$. Define the corresponding forms
$de_{1}(z_{1}), \ldots , de_{1}(z_{1})$, and the holomorphic one forms
$\alpha _{1}, \alpha _{2},..., \alpha _{n} \neq 0$ on $\widetilde{S}$ . If all
of these forms span a one dimensional subbundle of the cotangent bundle of
$\widetilde{S}$ over a Zariski open set we say that $\varrho: \pi_{1}(X)
\longrightarrow G$ is of dimension one.
\end{defi}
\begin{theo} Let $\varrho: \pi_{1}(X) \longrightarrow G$ be a rigid Zariski
dense representation of dimension one. Then it is integral, in other words it
is conjugate to a representation whose matrix coefficients are algebraic
integers.
\end{theo}
{\bf Proof.}
The case of $G=SL(2,\Bbb{C})$ - representations was proven by Simpson
\cite{3} for any rigid representation. Note that every Zariski dense
representation to $SL(2,\Bbb{C})$ is of rank one.
For groups of higher ranks we use the fact that our representation is of
dimension one. In this situation we can use Theorem 1.1, namely, using the
Bruhat-Tits theory we can attach to $G^{1}$ a building $B$ and then attach,
following Gromov-Schoen theory, a harmonic map to $B$. Since our representation
is of dimension one, Theorem 1.2 gives us that there exists a finite
nonramified covering $X^{1} \longrightarrow X $ such that the fundamental group
of $X^{1}$ factors though the fundamental group of some smooth algebraic curve
$Y^{1}$. Now following \cite{3} we can actually factor the original
representation $\varrho: \pi_{1}(X) \longrightarrow G^{1}$ working with the
fundamental group of an orbicurve, $\pi_{1}(Y,O)$, instead of with the
fundamental group of $\pi_{1}(Y^{1})$.
We proceed by applying a theorem of N. Katz.
\begin{theo} (Katz) If $\varrho:\pi_{1}(Y,O)\longrightarrow G$ is
cohomologicaly rigid then it is motivic.
\end{theo}
To be able to apply the above theorem we need to show that rigidity in terms
of moduli spaces and the cohomological rigidity define the same objects. This
is the subject of the next two lemmas.
Let us first give the precise definitions .
\begin{defi} Let $Y \cong \Bbb{P}^{1}$. We say that
$\varrho:\pi_{1}(Y,O)\longrightarrow G$ is a physically rigid representation
if and only if the conjugacy class $\varrho:\pi_{1}(Y,O)\longrightarrow G$ is
uniquely determined by its local monodronomies.
\end{defi}
Let us introduce a new notion-the notion of cohomological rigidity.
\begin{defi} Let $Y \cong \Bbb{P}^{1}$ and let
$\varrho:\pi_{1}(Y,O)\longrightarrow G$ be a representation. Let $F$ be the
bundle over $Y \cong \Bbb{P}^{1}$ corresponding to
$\varrho:\pi_{1}(Y,O)\longrightarrow G$. We say that
$\varrho:\pi_{1}(Y,O)\longrightarrow G$ is cohomologically rigid if and only if
\[\chi (X, j_{*}(End F))=2.\]
\end{defi}
For more detailed treatment of the notions of physical and cohomological
rigidity see \cite{36}.
Now we show that the notions of rigidity in terms of moduli spaces and the
cohomological rigidity coincide.
\begin{lemma} Let $\varrho:\pi_{1}((Y,O)) \longrightarrow G$ be a Zariski
dense representation which is rigid in the moduli sense. Then $Y \cong
\Bbb{P}^{1}$ with orbistructure in finitely many points.
\end{lemma}
{\bf Proof.} Let $a_{1},\ldots ,a_{g},b_{1},\ldots ,b_{g}, r_{1},\ldots ,r_{t}$
be the generators of
$\pi_{1}((Y,O))$ .(Here $g$ is the genus of $Y$, and $t$ is the number of the
points with an orbistructure ). Consider the map:
\[ s:G_{1}\times\ldots \times G_{2g} \times C_{1} \times\ldots \times C_{t}
\longrightarrow G. \]
where $G_{1},\ldots , G_{2g}$ are $2g$ copies of $G$ and $C_{1},\ldots ,
C_{t}$ are the conjugacy classes $SL(n,\Bbb{C})/Z(r_{1}),$ $\ldots ,$
$SL(n,\Bbb{C})/Z(r_{t})$.
Here the $Z(r_{i})$ are the centralizers of the $\rho(r_{i})$ in
$SL(n,\Bbb{C})$. This map is defined to be just the multiplication of the
corresponding elements. The dimension of the maximal component of the preimage
of 1 under this map gives an estimate for the dimension of the tangent space
to the moduli space of representations $\varrho:\pi_{1}(Y,O)\longrightarrow G$.
This dimension is equal to:
\[2g \dim G+\sum \dim SL(n,\Bbb{C})/Z(r_{i})- \dim G.\]
Since $\varrho:\pi_{1}(Y,O)\longrightarrow G$ is a rigid representation, then
this dimension should be equal to zero. But the number we have computed above
is always positive ( $n\geq 3$ ) unless $Y \cong \Bbb{P}^{1}$.
$\Box$
The following was shown in \cite{36}:
\begin{theo} (Katz) Let $Y \cong \Bbb{P}^{1}$. Then the notions of physical
rigidity and cohomological rigidity over $Y$ are the same.
\end{theo}
The second lemma we need is the following:
\begin{lemma} Let $Y \cong \Bbb{P}^{1}$. Then the notions of physical rigidity
and rigidity in terms of the moduli space of representations for an orbigroup
over $Y$ are the same.
\end{lemma}
{\bf Proof.} Let us first show that physical rigidity implies moduli space
rigidity. Observe that $ r_{1},....,r_{t}$ are semisimple elements in $G$. We
need to show that if we have two representations of $\pi_{1}(Y,O)$ that we
can deform one to another, then these two representations are conjugate,
providing we know they are physically rigid. But since $ r_{1},....,r_{t}$
are semisimple, if we deform them they need to stay in the same conjugacy
classes. Therefore they have the same local monodromies, and the physical
rigidity implies that they are actually conjugate.
Now we show that moduli space rigidity implies physical rigidity. Let us start
with a rigid, in terms of moduli spaces, representation
$\varrho:\pi_{1}(Y,O)\longrightarrow G$.
We have already estimated the dimension of the component of the moduli space
of representations which contains $\varrho$. Namely this dimension is less than
or equal to:
\[ \sum \dim SL(n,\Bbb{C})/Z(r_{i})- \dim G.\]
Observe that:
\[\sum_{i=1} ^{t} \dim Z(r_{i})=\left( \sum_{i=1} ^{t} \sum_{ j=1} ^{k_{i}}
n_{ij}^{2}\right)-1.\]
(Here $ k_{i}$ is the number of the different eigenvalues of $ \rho(r_{i})$ and
$n_{ij}$ are the sizes of the corresponding Jordan blocks.)
We also have:
\[\dim SL(n,\Bbb{C})/Z(r_{i})=(n^{2}-1)-(\left( \sum
n_{ij}^{2}\right)-1)=n^{2}- \sum n_{ij}^{2}.\]
Since $\varrho:\pi_{1}(Y,O)\longrightarrow G$ is rigid in terms of the moduli
spaces we have that:
\[\dim SL(n,\Bbb{C})\geq n^{2}-1=\sum_{i=1}^{t}\left\{ (n^{2}-1)-(\sum_{ j=1}
^{n} n_{ij}^{2})-1\right\}-(n^{2}-1).\]
This is equivalent to:
\begin{equation}
2(n^{2}-1) \geq \sum _{i=1}^{t} (n^{2}-\sum_{ j=1} ^{n} n_{ij}^{2})
\end{equation}
But from the Grothendieck -Ogg-Shafarevich formula (see \cite{36} ) we know
that:
\begin{equation}
\chi(X, j_{*}(End F)) = (2 - t) (n^{2}) + \sum_{i=1, j=1} ^{t,n} n_{ij}^{2}.
\end{equation}
( Here $\chi(X, j_{*}(End F))$ is the Euler characteristic of $ j_{*}(EndF)$
and $j$ is the embedding $ j: \Bbb{P}^{1} -\; finite \; set \; of \; points
\hookrightarrow \Bbb{P}^{1}$.
Combining (1) and (2) we obtain:
\[\chi(X, j_{*}(End F)=\geq 2.\]
On the other hand we have started with an irreducible representation
$\varrho:\pi_{1}(Y,O)\longrightarrow G$ we have:
\[\dim H^{0}(X, j_{*}(End F))= \dim H^{2}(X ,j_{*}(End F))=1\]
Then $\dim H^{1}(X, j_{*}(End F))=0$ , which means that the representation
$\varrho:\pi_{1}(Y,O)\longrightarrow G$ is cohomologicaly rigid. Now we are in
a position to apply theorem 7.2 and we conclude that
$\varrho:\pi_{1}(Y,O)\longrightarrow G$ is motivic.
But if $\varrho:\pi_{1}(Y,O)\longrightarrow G$ is motivic then the image of
$\varrho:\pi_{1}(Y,O)\longrightarrow G^{1}$ is in a bounded subgroup in
$G^{1}$. This can be seen as follows:
The fact $\varrho:\pi_{1}(Y,O)\longrightarrow G$ means that as representation
$\varrho$ extends from a discrete representation of the fundamental group of a
complex variety extends to a representation of the profinite completion of the
fundamental group i.e. it extends to a representation of $X$ as an algebraic
variety. But under continuous representation the compact profinite completion
of the fundamental group goes to a compact image . So we get that the image of
$\varrho:\pi_{1}(Y,O)\longrightarrow G^{1}$ is in a bounded subgroup in
$G^{1}$.
Therefore the corresponding harmonic map is the constant map so the forms
$\alpha _{1}, \alpha _{2},..., \alpha _{n} \neq 0$ do not exists. This
contradicts our assumption that we have a factorization through an orbicurve.
So the second case does not occur.
$\Box$
In the process of the proof of this theorem we have obtained the following
corollary:
\begin{corr} The preimage of 1 under the map defined above
\[s:C_{1}\times... \times C_{t}\longrightarrow G,\]
is connected.
\end{corr}
\begin{rem} The argument above depends a lot on the fact that we are working
with
$SL(n,\Bbb{C})$ (the dimensions of $SL(n,\Bbb{C})/Z(r_{i})$, a duality argument
which is hidden in the computation of $\dim H^{2}(X ,j_{*}(End F))=1$), but
we believe that it is true for all simple complex Lie groups.
\end{rem}
Now we are going to extend the field of application of the above theorem to
the quasiprojective case.
\begin{theo} Let $X$ be a quasiprojective variety and let $\varrho: \pi_{1}(X)
\longrightarrow G$ be a rigid Zariski dense representation of dimension one
and type B) with unipotent monodromy at infinity.
Then $\varrho: \pi_{1}(X) \longrightarrow G$ is integral.
\end{theo}
{\bf Proof.} From section 5 we know that we can define a spectral covering
for quasiprojective variety. The proof of theorem 7.4 just repeats the
arguments in the proof of theorem 7.2.
$\Box$
Now we are going to formulate 3 different sufficient conditions for which we
get a factorization through an orbicurve:
1)It follows from the work of Arapura, Bressler and Ramachandrachan \cite{19}
that if $H^{1}EL_{2}(\pi_{1}(X))$ \linebreak $\neq 0$ then every Zariski dense
representation $\varrho: \pi_{1}(X) \longrightarrow G$ factors through a
representation of the fundamental group of an orbicurve.
2)It follows from \cite{20} that if $H^{1}EL_{2}(S) \neq 0$, where $S$ is the
factorizing spectral covering, then every Zariski dense representation
$\varrho: \pi_{1}(X) \longrightarrow G$ factors through a representation of
the fundamental group of an orbicurve.
3)It follows from \cite{2} that if $Prym_{\sigma}(S,X)$ is nontrivial then
every Zariski dense representation $\varrho: \pi_{1}(X) \longrightarrow G$
factors through a representation of the fundamental group of an orbicurve.
Therefore we arrive at:
\begin{corr} If $\varrho: \pi_{1}(X) \longrightarrow G$ is a rigid Zariski
dense representation and one of the above conditions is satisfied, then this
representation is integral.
\end{corr}
There is one more case for which we were able to prove the fact that every
rigid representation is integral.
\begin{corr} Let X be a smooth projective variety such that $\pi_{1}(X) $ is
a word hyperbolic group, acting on the corresponding $\Bbb R$-tree. Assume
that $Out(\Gamma)$ is an infinite group. Let $\rho:\pi_{1}(X) \longrightarrow
SL(n,\Bbb C)$ be a rigid Zariski dense representation such that
$Ker(\rho)\supseteq Ker(R)$. Then $\rho: \pi_{1}(X) \longrightarrow SL(n,\Bbb
C)$ is integral.
Here $Ker(R)$ is the kernel of the map
\[R:\pi_{1}(X) \longrightarrow Isom(T),\]
where $Isom(T)$ is the group of isometries of the tree $T$, defined in the
proof of theorem 6.7.
\end{corr}
{\bf Proof.} It follows from theorems 6.1 , 7.1 and 6.7.
$\Box$
In some cases it is possible to prove that every rigid representation is
motivic without assuming that the harmonic map to the building $B$ is of
dimension one. It follows from Corrolary 6.2 that:
\begin {corr} Every rigid Zariski dense representation $\varrho: \pi_{1}(X)
\longrightarrow G$ for which there exists an $G$- equivariant nonconstant
continuous map $U:\widetilde{X}{\longrightarrow}HB$, is integral.
\end{corr}
{\bf Proof.} If the action of $G^{1}$ fixes a point in $HB$ then the image of
$\varrho: \pi_{1}(X) \longrightarrow G$ is contained in a maximal compact
subgroup of $G$. Using again the theorem of Baas we get that our
representation is integral. Now if we have an action of $G$ without a fixed
point we are in a position to apply Corollary 6.2.
In this case it is very easy to prove (see \cite{16}) the pluriharmonicity of
$U$ since we keep almost the same metric on the boundary of every chamber.
Then we get a spectral covering as in Theorem 7.1. Also the spectral
covering, which we obtain has an involution on it. Following \cite{3} and
\cite{4} we obtain an involution which acts on the curve $Y$ as well. We mod
out the spectral covering $S$ and the curve $Y$ by $\Bbb{Z}_{2}$ and we have
a holomorphic map between the quotients. The rest of the proof is the same as
in the main theorem. We finish it by applying theorem 7.3.
$\Box$
The above corollary actually implies that if there exists an $\rho$-
equivariant nonconstant continuous map $U:\widetilde{X}{\longrightarrow}HB$
and $ \rho$ is a rigid representation then the action of $\rho$ on $B$ should
have a fixed point.
We think of the above corollary as a generalization of the tree case, since the
tree is negatively curved. Unfortunately as we saw in Theorem 6.3 it is in
general impossible to make $G$ act on simply connected negatively curved
spaces. So one should try another approach for the general Simpson
conjecture. It seems one way of doing that would be to generalize the result
of Katz to higher dimensions. While at the moment the Conjecture 7.1 and
Conjecture 7.2 look out of reach in their complete generality the following
conjecture looks more doable at this moment.
\begin{con} Every rigid Zariski dense representation $\varrho: \pi_{1}(X)
\longrightarrow G$ of a group $ \pi_{1}(X)$ with the property $T$ is
integrable.
\end{con}
\section{The Shafarevich conjecture}
Inspired by the results in \cite{30} we make in this section a connection
between the
Shafarevich map and the results of this paper. We recall the definition of
the Shafarevich map \cite{30} , and describe some of its properties.
\begin{defi} Let $X$ be a smooth projective variety. Then we call a rational
map
\[Sh:X ----\rightarrow Sh(X),\]
the Shafarevich map of $X$ iff:
1) $Sh(X)$ is a normal projective variety;
2)The rational map $Sh:X \longrightarrow Sh(X)$ has connected fibers and;
3) there are countable many closed subvarieties $D_{i}$ in $X$ ($D_{i}\neq X$)
such that for every irreducible subvariety $Z$ in $X$, not in the union of
the $D_{i}$
we have:
\[Sh(Z)= {\rm point \; iff} im[\pi_{1}(Z')\longrightarrow \pi_{1}(X)] {\rm is
\; finite}.\]
\end{defi}
It is easy to see that if the rational map $Sh:X \longrightarrow Sh(X)$
exists, it is unique up to birational equivalence.
One can give a relative version of this definition with respect to a normal
subgroup $H$ of $\pi_{1}(X)$ .To do that one needs to require in part 3) of
the definition above that $im[\pi_{1}(Z')\longrightarrow \pi_{1}(X)] \cap H $
has a finite index in $im[\pi_{1}(Z')\longrightarrow \pi_{1}(X)]$. In this
case we write $Sh^{H}:X \longrightarrow Sh^{H}(X)$ for the corresponding
rational map.
The following theorem is proven in \cite{30}.
\begin{theo}( J\'anos Koll\'ar) The rational maps $Sh:X \longrightarrow Sh(X)$
and $Sh^{H}:X ---\longrightarrow Sh^{H}(X)$ exist.
\end{theo}
The properties of the Shafarevich map we have described above provide us with
enough information to prove the following theorem:
\begin{theo} Let X be a smooth projective variety which has type B (from
Theorem 1.1) Zariski dense representation of its fundamental group to a Lie
group $G$ defined over a field with discrete valuation. Then:
\[dim_{\Bbb{C}}Sh^{H}(X)=dim_{\Bbb{C}}(Y) \leq rank_{\Bbb{C}}(G),\]
where Y is the variety defined in Theorem 1.1 and $H=Ker (\varrho) $. Moreover
we have that some finite nonramified coverings of $Y$ and $ Sh(X(\Gamma)$
(defined below) are birationally isomorphic.
\end{theo}
{\bf Proof.} From the construction of the holomorphic map $X \longrightarrow
Y$ in the proof of Theorem 1.2 we know that if $Z$ is the general fiber of
this map we have that the intersection of the subgroups
$R=im[\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ and $H=Ker (\varrho) $ is a
subgroup of finite index in $R$. This implies that $Sh^{H}:X \longrightarrow
Sh^{H}(X)$ factors through $X \longrightarrow Y$ and we get:
\[ rank_{\Bbb{C}}(G) \geq dim_{\Bbb{C}} Y \geq dim _{\Bbb{C}} Sh^{H}(X).\]
To get the inverse inequality we argue as follows:
Let us first do two reductions.
Denote by $L$ the intersection of all subgoups of finite index in $\pi_{1}(X)$.
According to remark 4.4 the conclusions of the main theorem are still true if
we work with $\pi_{1}(X) \not L$. Therefore we can assume that
\[ H=\bar{H}. \]
Define now $R=im[\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$.
The second reduction is to show that we can always assume that $R$ is
contained in $H$. If $R$ is not contained in $H$ then
$R=im[\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ intersects $H=Ker (\varrho) $
in a subgroup of finite index in $im[\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$.
Then let the image of this finite subgroup be the finite subgroup $ \Pi$ in $
\pi_{1}(X)/H $. Now we do the following modification:
Observe that $ \pi_{1}(X)/H $ is a residually finite group due to the fact
that $G$ is an affine algebraic group. Therefore since we have mod out $
\pi_{1}(X)$ by $L$ and $H=\bar{H} $ we can find a subgroup of finite index
$\Theta $ in $ H=\bar{H}/H $ which does not intersect $ \Pi$. We conclude that
for the covering , which corresponds to the group $\Theta $, $R$ is containe
in $H$.
So from now on we assume that $R$ is contained in $H$.
Applying 4.8 \cite{30} we see that there exists some subgroup $ \Gamma$ of
finite index in $ \pi_{1}(X)$ such that for the finite nonramified covering
$X(\Gamma)$ we have a complete factorization through $Sh^{H}(X(\Gamma))$.
The construction goes as follows. We have already constructed $X(\Gamma)$ in
section 4. According to 4.8 there exists a map:
\[f_{*}: \pi_{1}(X(\Gamma)) \longrightarrow \pi_{1}(Sh^{H}(X(\Gamma))) \]
such that
\[Ker f_{*} \subseteq \bar{R} \subseteq \bar{H} =H.\]
But $Y$ was defined as the variety of minimal dimension, through which we have
a factorization. Therefore:
\[ dim_{\Bbb{C}}Sh^{H}(X) \geq dim_{\Bbb{C}} Y.\]
Combining this with the previous inequality we get:
\[dim_{\Bbb{C}}Sh^{H}(X)=dim_{\Bbb{C}}(Y) \leq rank_{\Bbb{C}}(G).\]
$\Box$
We prove now a weak version of the Shafarevich conjecture.
The Shafarevich conjecture says that for every smooth projective variety $X$
there exists a there exists a Stein manifold ${\bf Sh} (\widetilde{X} )$ and a
proper map with connected fibers $ {\bf Sh}: \widetilde{X}\longrightarrow {\bf
Sh}(\widetilde{X})$.
An easy consequence of previous theorem is the following theorem:
\begin{theo}Let X be a smooth projective variety which has a type B (from
Theorem 1.2) Zariski dense representation $\rho$ of its fundamental group to a
Lie group $G$ defined over a local field $K$. Define $H=Ker(\rho)$. Then $Y$
is isomorphic to $Sh^{H}(X')$ and the map
\[{\bf Sh}^{H}:X'\longrightarrow {\bf Sh}^{H}(X'),\]
is a morphism. Here $ X'$ and $Y$ are the same as in Theorem 1.2.
\end{theo}
{\bf Proof.}Without lost of generality we can work with the case when
$H=Ker(\rho)$ is finite.
It is clear that if $Z$ is a subvariety in $X$ and the fundamental group of $Z$
is finite then
$Z$ goes to a point in $B$, where $B$ is the corresponding building. This
follows from a very general theorem of M. Bridson saying that every finite
group, which acts on a CAT space has a fixed point. Therefore the corresponding
harmonic maps are just constants. We need to show that if $Z$ is a subvariety
of $X$ with a finite fundamental group then then it goes to a point in $Y$.
According to Gromov and Schoen the map $U:\widetilde{X}\longrightarrow B$ is
essentially regular, namely the intrinsic derivative exists everywhere (see
\cite{1}). We claim that $U^{*}\alpha_{i}$ is zero on $Z$. On the intersection
of the smooth locus of $U$ and $Z$ this is obviously true. For the singular
points we use the same argument as in the proof of Lemma 3.1, namely the
existence of the intrinsic derivative. This existence implies existence of a
kernel of $U^{*}$ in $T^{*}X'$ and we conclude $U^{*}\alpha_{i}$ is zero on
$ Z$. This implies that $Z$ will go to a point in $Y$ even if $Z$ it is
contained in the singular set of $U$.
Another way to show that $Z$, a subvariety of $X$ with a finite fundamental
group goes to a point in $Y$ is the following. Using theorem 6.3 \cite{1} we
obtain the following estimate, which follows from the existence of the
intrinsic derivative
\[\lim_{x\rightarrow S}{\|dU(x)\|}=0,\]
where $S$ is the singular set of $U$.
According to \cite{1} $U(S)$ is contained in the closed faces of the
simplices of the highest dimension. Now using the above estimate and
approximating
$U(S)$ in the normal directions of the faces containing it
we get that $U^{*}\alpha_{i}$ is zero on $ Z$.
\hfill $\Box$
Therefore we obtain the following :
\begin{corr} If the conditions of the above theorem we have $H=ker \rho$ is a
finite group then the Shafarevich-Koll\'ar conjecture is true.
\end{corr}
{\bf Proof.} According to the previous theorem the map
\[{\bf Sh}:X'\longrightarrow {\bf Sh}(X'),\]
is a morphism. Consider the morphism
\[h:X \longrightarrow Y \]
from the proof of the main theorem in section 4.
Assume that there is a subvariety $Z$ in $X$ with the property that
$im[\pi_{1}(Z)\longrightarrow \pi_{1}(X)]$ is finite and it does not go to a
point in $Y$. But then pulling it back to $X' $ we get a subvariety $Z^{'}$ in
$ X'$ such that $ \pi_{1}(Z^{'})$ is a finite group due to a theorem of
Campana \cite{31}. Therefore
$im[\pi_{1}(Z^{'})\longrightarrow \pi_{1}(X')]$ is a finite group and $Z^{'}$
goes to a point in $ {\bf Sh}(X')$. Now using the map $ {\bf Sh}(X')
\longrightarrow Y $ we obtain a contradiction since the image of $Z^{'}$ in
$Y$ is the same as the image of $Z$ in $Y$. Therefore $Y={\bf Sh}(X)$.
\hfill $\Box$
The Shafarevich-Koll\'ar conjecture in general is connected with another
question of Gromov:
Can we find a faithful discrete cocompact action of every K\"{a}hler word
hyperbolic group to a space with $K \leq 0$ ?
We believe that the Shafarevich-Koll\'ar conjecture follows for every
K\"{a}hler group for which this could be done.
Now we show two examples for which the above condition is satisfied and we can
actually prove the Shafarevich conjecture.
\begin{corr} Let $X$ is a compact K\"{a}hler manifold and $\pi_{1}(X)$ is an
amalgamated product of two groups and $\pi_{1}(X)$ is word hyperbolic. Then the
Shafarevich-Koll\'ar conjecture is true for $X$.
\end{corr}
{\bf Proof.} According to \cite{1} we have a faithful discrete cocompact
action of $\pi_{1}(X)$ on a tree. The rest is just repeating the argument from
the proof of Theorem 8.3.
\hfill $\Box$
In the same way Theorem 6.1 implies :
\begin{corr} Let $X$ be a compact K\"{a}hler manifold and $\pi_{1}(X)$ has
an a faithful discrete cocompact action on a two dimensional complexes defined
in section 6. Then the Shafarevich-Koll\'ar conjecture is true for $X$.
\end{corr}
Now we discuss some possible applications of the Shafarevich-Koll\'ar
conjecture.
One possible application of the above statement is to try to answer the
following question of M. Ramachandran \cite{M}.
\begin{con}( M. Ramachandran) Let X be a smooth projective variety which has a
type B (from Theorem 1.2) Zariski dense representation $\rho$ of its
fundamental group to some Lie group $G$ defined over a local field $K$.
Define $H=Ker(\rho)$ and assume $H$ is a finite group.
Then one of the following holds:
1) The universal cover of $X$ satisfies the Bochner - Hartogs property, namely
for every $\alpha \in A^{0,1}_{c}(\widetilde{X})$ satisfying
\[\bar{\partial} \alpha =0 ,\]
there exists $u \in C^{\infty}_{c}(\widetilde{X})$ such that
\[ \bar{\partial} u = \alpha.\]
2) $\pi_{1}(X)$ is comesurable to the fundamental group of a compact Riemann
surface.
\end{con}
Here $ A^{0,1}_{c}(\widetilde{X})$ is the space of compactly supported smooth
(0,1) forms on $\widetilde{X}$ and $ C^{\infty}_{c}(\widetilde{X})$ is the
space of all compactly supported smooth functions on $\widetilde{X}$ with
complex values.
\begin{rem} M. Ramachandran actually stated the conjecture in much bigger
generality namely for every smooth projective variety.
\end{rem}
\bigskip
\noindent
We hope to answer this question by using the maps
\[U:\widetilde{X} \longrightarrow B\]
and
\[{\bf Sh}^{H}:X'\longrightarrow {\bf Sh}^{H}(X').\]
to produce enough plurisubharmonic functions on $\widetilde{X}$.
We study now some questions connected with the topological nature of the map
\[{\bf Sh}^{H}:X'\longrightarrow {\bf Sh}^{H}(X').\]
We give a partial answer to some questions posed by J\'anos Koll\'ar
at the end of \cite{27} . These questions are about how much the Shafarevich
variety ${\bf Sh}(X)$ depends on the original variety $X$ e.g. does a
deformation of $X$ give a deformation of ${\bf Sh}(X)$ an so on.
\begin{corr} Let $X$ be a smooth projective variety which has nonrigid
finite kernel Zariski dense representation in $SL(2,\Bbb{C})$. Let
$X_{t}$ , $t \in $ to the unit disc $\Delta$ , be a holomorphic deformation of
$X$. Then ${\bf Sh}(X_{t})$ is a holomorphic deformation of ${\bf Sh}(X)$.
\end{corr}
{\bf Proof.} The moduli space of $\lambda$ - connections (see \cite{35})
deforms together with $X$ . The moduli space of $\lambda$-connections
itself is a deformation of the moduli space of $SL(2,\Bbb{C})$ representations
to the moduli space of $SL(2,\Bbb{C})$ Higgs bundles. Since we have fixed the
representation we get a constant holomorphic section over $\Delta\times
\Bbb{P}_{1}$, which specializes at $0$ of $\Bbb{P}_{1}$ to an Higgs bundle. As
we said the moduli space of $SL(2,\Bbb{C})$ Higgs bundles is a deformation of
the moduli space $\lambda$ - connections so we get a holomorphic section to
the family of moduli spaces of $SL(2,\Bbb{C})$ Higgs bundles. over $ \Delta$.
Also we have a holomorphic map from the moduli space of $SL(2,\Bbb{C})$ Higgs
bundles to the $Hilb (T^{*}(X))$, which corresponds to every Higgs bundle a
spectral covering (see \cite{3}). We get this way a holomorphic deformation
of the corresponding spectral coverings and as a consequence a family of
curves $C_{t}$ through which according to our main theorem the representation
factors.
But since this is an finite kernel Zariski dense representation we
have ${\bf Sh}(X_{t}(\Gamma))=C_{t}(\Gamma)$, where $C_{t}(\Gamma)$ is the
spectral covering of $X_{t}(\Gamma)$. Observe that the map ${\bf
Sh}(X_{t}(\Gamma))\longrightarrow {\bf Sh}(X_{t})$ has the same Galois group as
$X_{t}(\Gamma) \longrightarrow X_{t}$. Therefore we can mod out by this
group and get that ${\bf Sh}(X_{t})$ deforms itself. \hfill $\Box$
Even more is true:
\begin{corr} Let $X$ be a smooth projective varietiy which has nonrigid
finite kernel Zariski dense representation $\rho:\pi_{1}(X) \longrightarrow
SL(2,\Bbb{C})$. Let $Y$ be another smooth projective varieties which is
homeomorphic to $X$. Then ${\bf Sh}(X)$ is homeomorphic to $Sh(Y)$.
\end{corr}
{\bf Proof.} Due to the fact that we have a finite kernel Zariski dense
representation we conclude that the $\pi_{1}(X)$ is infinite and therefore
${\bf Sh}(X(\Gamma))$ is a curve. Therefore $\pi_{1}(X)= \pi_{1}(Y)$ and let
say that $C_{X(\Gamma)}={\bf Sh}(X(\Gamma))$ and $C_{Y(\Gamma)}={\bf
Sh}(Y(\Gamma))$
are the curves through which our representation factors. The only thind we need
to show is that $g(C_{X})=g( C_{Y})$, where $g(C_{X})$ is the genus of
$C_{X}$. According to a theorem of Siu ( see for example \cite{5} ) if we have
a map from $\pi_{1}(X)$ to $\pi_{1}(C_{Y})$ we have a holomorphic map
$h:X(\Gamma)\longrightarrow C^{'}_{Y}$ to some other curve $ C^{'}_{Y}$. Due to
the fact that $C_{X}={\bf Sh}(X(\Gamma))$ we get that $g(
C_{Y})=g(C^{'}_{Y})\geq g(C_{X})$. Now to finish the proof we change the places
of $X$ and $Y$ in the above argument.
\hfill $\Box$
\begin{rem} The above corollary works for $X_{t}$ homotopy equivalent to $X$
as well as for ${\bf Sh}_{K}(X)$ instead of ${\bf Sh}(X)$
, where $K$ is the kernel of $ \rho$ and $K$ is an infinite group.
\end{rem}
The following two corollaries are straightforward consequences from section 7.
\begin{corr} The corollary above is true if instead of
a Zariski dense representation in $SL(2,\Bbb{C})$ we require the existence of a
finite kernel surjective homomorphism of the fundamental group of $X$ to the
fundamental group of a 2 dimensional negatively curved complex of groups (see
paragraph 6).
\end{corr}
\begin{corr} The corollary above is true if instead of
a Zariski dense representation in $SL(2,\Bbb{C})$ we require the existence of a
nonrigid Zariski dense representation of dimension 1 (see paragraph 7) to any
simple complex Lie group.
\end{corr}
In the same way we treat the case when $Y$ is a surface.
\begin{theo} Let $X$ be a smooth projective variety which has a finite
kernel Zariski dense representation $\rho:\pi_{1}(X) \longrightarrow G $ of
type B in $G$, where $G$ is a simple Lie group such that $rank_{\Bbb{C}} G
=2$. Let
$X_{t}$ be a holomorphic deformation of $X$. Then ${\bf Sh}(X_{t}(\Gamma))$ is
a holomorphic deformation of ${\bf Sh}(X(\Gamma))$.
\end{theo}
{\bf Proof.} The assumptions of the theorem imply that we have a factorization
through the representation of the fundamental group of $Y$, where $Y$ is either
an algebraic curve or an algebraic surface. We have considered the first case
in the previous corollary.
Let us consider now the case $dim{\bf Sh}(X(\Gamma))=2 $.
Observe that ${\bf Sh}(X_{t}(\Gamma))$ is defined only up to birational
isomorphism. Therefore when we say that ${\bf Sh}(X_{t}(\Gamma))$ is a
holomorphic deformation of ${\bf Sh}(X(\Gamma))$, we mean that the minimal
model of ${\bf Sh}(X_{t}(\Gamma))$ is a holomorphic deformation of the minimal
model of ${\bf Sh}(X(\Gamma))$. Since $X_{t}$ is a holomorphic deformation of
$X$ and we work with the same representation in the same way as in the
previous corollary, we can show that the corresponding spectral coverings of
$X_{t}$ and $X$ are holomorphic deformations of each other. What we need to
show is that if $Y$ is a holomorphic deformation of $Y_{t}$ then the minimal
model of $Y$ is a holomorphic deformation of $Y_{t}$. But in case of surfaces
with $ \kappa(Y) \geq 0 $ (this is exactly our case since $ {\bf
Sh}(X(\Gamma))$ has a large fundamental group) this follows from a theorem of
Iitaka. The fact that $Y$ and ${\bf Sh}(X(\Gamma))$ are isomorphic proves the
theorem.\hfill $\Box$
The following nonvanishing theorem is a consequence of (3).
\begin{theo}
Let $X $ be a smooth fourfold of general type with a type B (from Theorem
1.1) Zariski dense representation of its fundamental group $\rho: \pi_{1}(X)
\longrightarrow G $. Here $G$ is a rank 2 or 3 simple Lie group over field
with a discrete valuation. Then either:
1) $\rho: \pi_{1}(X) \longrightarrow G $ factors through a representation of
the fundamental group of an orbicurve or;
2) $P_{n}$=$H^{0} (X, nK_{X}) $ for $n\geq4$ is not zero.
\end{theo}
{\bf Proof.} According to Theorem 1.2 our representation factors through a
representation of the fundamental group of some variety $Y$ of dimension less
than or equal to 3. If this $dimY= 1 $ then we are in part 1) of the theorem.
Consider the case where the dimension of $Y$ is at least 2. The previous
theorem tells us that $Y$ is birational to $Sh^{H}(X)$. Following J\'{a}nos
Koll\'{a}r \cite{30}
(4.5 and 5.8) we can work with a morphism $X^{1} \longrightarrow S $, where
$X^{1}$ is a nonramified finite covering of $X$ and $S$, a smooth variety
with generically large fundamental group.
It follows from \cite{30} that in this case nonvanishing theorems for $X^{1}$
imply nonvanishing theorems for $X$. We need to consider the following cases:
1) $dimS=3$ and $S$ is of general type. By \cite{30} (10.1 ) we have that
$P_{m}(S)\geq1$ for $m\geq2$. Observe also that the fibers $X_{s}$ of the map
$X^{1}\longrightarrow S$ are curves of general type and therefore we have
$P_{1}(X_{s})\geq 1$. Using \cite{30} (10.4) we obtain:
\[ P_{n}(X) \geq P_{n-2}(S) \geq 1, \; {\rm for} \; n\geq 4.\]
2) $dimS=3$ and $S$ is abelian variety. Then we can apply \cite{30} (8.10
), which gives a strong nonvanishing. Strong nonvanishing is equivalent to:
\[ h^{0}(X^{1},K_{X^{1}} \otimes D)\neq 0 \; {\rm for} \; D \; {\rm big \;
divisor \; on} \; X^{1} \; {\rm and} \; X^{1} {\rm big \; birational \; to }
X. \]
In our situation we can make $3K_{X} = D $ since $X$ is of general type and
therefore:
\[P_{n}(X) \geq 1 \; {\rm for} \; n\geq 4. \]
This argument actually applies in the previous case too. We need only that
the fundamental group of $S$ is generically large.
3) $dimS=2$ and $S$ is abelian. We know that the fiber $X_{s}$ of
$X^{1}\longrightarrow S$ is a surface of general type. If we have a complete
factorization $X^{1}\longrightarrow S$ of the representation $\varrho$ we get a
contradiction, since $\varrho$ is a Zariski dense representation and we cannot
have a Zariski dense representation of a free abelian group in a simple Lie
group $G$. Therefore we need to work with a finite ramified cover of $S$ - a
surface of general type with generically large fundamental group, which we
will also denote by the letter $S$. We are in a position to apply \cite{30}
(10.4).
We have:
\[P_{n}(X_{s}) \geq 1 \; {\rm for} \; n\geq 2,\]
and also
\[ P_{n}(S) \geq 1 \; {\rm for} \; n\geq 2.\]
Therefore we obtain from \cite{30} (10.4):
\[ P_{n}(X) \geq P_{n-2}(S)\geq 1 \; {\rm for} \; n\geq 4.\]
4) $dimS=2$ and $S$ is of general type. Then the fiber $X_{s}$ of
$X^{1}\longrightarrow S$ is also a surface of general type. In this case we
argue as before, again using \cite{30} (10.4).
\begin{rem} Obviously the parts 1) and 2) are not mutually exclusive.
\end{rem}
$\Box$
In the last application we are going to be concerned with the nonexistence of
some representations for a special class of algebraic varieties. If we think
of $Hom (\pi_{1}(X), G)$ as the first nonabelian cohomology group, then the
nonexistence of certain representations is a sort of vanishing theorems.
\begin{defi} We say that an algebraic variety $X$ has a generically large
fundamental group iff for every subvariety $Z$ of $X$ outside some union of
divisors $ \bigcup{D_{i}}$ we have that:
\[im[\pi_{1}(Z')\longrightarrow \pi_{1}(X)]\]
is an infinite group.
\end{defi}
\begin{corr} Let $X$ be a variety with a generically large fundamental
group. Then:
1)A finite nonramified covering of $X$ is birationaly isomorphic to $Sh(X)$
and
2) $X$ does not have Zariski dense, finite kernel representations in a
complex Lie group $G$ if $dim_{\Bbb{C}}(X)>rank _{\Bbb{C}}(G)$.
\end{corr}
{\bf Proof.} Part 1) of this corollary follows immediately from the definition
of the Shafarevich variety. Part 2) of this corollary follows from theorem
8.3 since:
\[dim_{\Bbb{C}}X=dim_{\Bbb{C}}Sh(X)=dim_{\Bbb{C}}(Y) \leq rank_{\Bbb{C}}(G),\]
and this contradicts our assumption that:
\[dim_{\Bbb{C}}(X)>rank _{\Bbb{C}}(G).\]
$\Box$
\section{Final remarks}
By definition all of our spectral coverings are zero shemes of sections in the
cotangent bundle of $X$. Then obviously they can be deformed to the zero
section. The zero section taken with some multiplicity (in the case of
buildings this multiplicity was equal to $w$) corresponds to a harmonic map
to a point in the Euclidean building $B$ corresponding to $G$. Therefore we
have an action of $\pi_{1}(X)$ on $B$ which fixes a point. In all of our
arguments before we have excluded the case where $\pi_{1}(X)$ fixes a point of
$B$ b. But if the fixed point $P$ is not a point at infinity of $B$ we
assign to the action of $\pi_{1}(X)$ the constant equivariant harmonic map
$\widetilde{X }\longrightarrow P$. Obviously all holomorphic differentials in
this case are equal to zero and the spectral covering $S$ is just $X$, the
zero section of the cotangent bundle of $X$. Therefore every nonrigid subgroup
$G(B)$ (the group of isometries of $B$) which cannot be deformed to
another subgroup of $G(B)$ fixing a point of $B$, cannot be the fundamental
group of a smooth projective variety. Of course one needs to find (if
possible) the proper notion of deformation. It is clear that such a statement
might then be true in much more general situations, namely for nonrigid
subgroups of the group of isometries of quite general nonpositively curved
length spaces.
Another direction one can try to apply the techniques developed in this paper
is to study the p-adic uniformization defined in \cite{13}, \cite{14},
\cite{15}.
Using the techniques described above one can try to study the discrete
subgroups of the infinite dimensional Lie groups $SDiff (D)$, where $D$ is a
Riemannian domain.
It follows from \cite{27}, \cite{28} and \cite{29} that these groups are
nonpositively curved in some sense if $D=T^{2} $, $D=T^{n} $ or $D=S^{2} $.
Then using the harmonic map technique one obtains some finiteness results.
These remarks are going to be an object of future considerations.
|
1994-02-03T21:12:43 | 9402 | alg-geom/9402002 | en | https://arxiv.org/abs/alg-geom/9402002 | [
"alg-geom",
"math.AG"
] | alg-geom/9402002 | Victor Batyrev | Victor V. Batyrev and Lev A. Borisov | Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds | 17 pages, Latex | null | null | null | null | We introduce a special class of convex rational polyhedral cones which allows
to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$,
where $r$ is a positive integer and d is the dimension of critical string vacua
with central chatge $c = 3d$. It is conjectured that the natural combinatorial
duality satisfies by these cones corresponds to the mirror involution. Using
the theory of toric varieties, we show that our conjecture includes as special
cases all already known examples of mirror pairs proposed by physicists and
agrees with previous conjectures of the authors concerning explicit
constructions of mirror manifolds. In particular we obtain a mathematical
framework which explains the construction of mirrors of rigid Calabi-Yau
manifolds.
| [
{
"version": "v1",
"created": "Thu, 3 Feb 1994 19:59:39 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Borisov",
"Lev A.",
""
]
] | alg-geom | \section{Introduction}
\noindent
This paper is devoted to the problem of finding an appropriate
mathematical framework which explains explicit constructions
of mirror pairs of Calabi-Yau manifolds. In this context, the existence
of rigid Calabi-Yau manifolds furnish a fundamental obstruction to
a mathematical formulation of mirror symmetry motivated
by a natural involution in $N=2$ superconformal field theories.
In recent works \cite{schimm1,schimm2}, Schimmrigk came
to the conclusion that the class of Calabi-Yau manifolds
{\em is not appropriate setting for mirror symmetry}.
For every positive integer $r$, Schimmrigk has proposed
a new class of K\"ahler manifolds of dimension $(d + 2(r-1))$
which generalizes the class of $d$-dimensional Calabi-Yau varieties.
Although the first Chern class of
{\em generalized Calabi-Yau manifolds} does not vanish for $r >1$, it is
possible to derive from these manifold the massless spectrum of
critical string vacua. Typical examples of the generalized Calabi-Yau
manifolds are obtained from quasi-smooth hypersurfaces
of degree $w$ in weighted projective spaces
\[ {\bf P}(w_1, \ldots, w_{d + 2r}) \]
where the degree $w$ and the weights $w_1, \ldots, w_{d + 2r}$ are
related by the condition:
\[ w_1 + \cdots + w_{d + 2r} = rw. \]
Schimmrigk has shown that some classes of usual $d$-dimensional
Calabi-Yau manifolds $V$ which can be described
as complete intersections of $r$ hypersurfaces in products of
$r$ copies of weighted projective spaces naturally give rise to
generalized Calabi-Yau manifolds $M$ of dimension
$(d- 2(r-1))$ embedded as hypersurface
in some higher-dimensional weighted projective space. Moreover,
the correspondence $V \mapsto M$ induces the canonical
inculison of the Hodge $(p,q)$-spaces
\[ H^{p,q}(V) \subset H^{p +r -1, q + r -1}(M). \]
It is important to remark that not every generalized Calabi-Yau manifold
of dimension $(d + 2(r-1))$
can be obtained by this method from some usual $d$-dimensional
Calabi-Yau manifold.
\medskip
In \cite{bat.dual,bat.straten,borisov} there were proposed
a combinatorial approach to the construction
of mirror pairs using theory of toric varieties \cite{danilov,oda}.
The main idea of
this approach is the interpretation of the mirror duality in terms of the
classical duality for convex sets. It is a natural ask whether the same
approach can be applied to rigid and generalized Calabi-Yau manifolds?
The main purpose of this paper is to show that the answer is positive.
\bigskip
In Section 2, we define a special class of convex rational polyhedral
cones which we call {\em reflexive Gorenstein cones}.
Every reflexive Gorenstein cone $\sigma$ canonically defines the
projective toric Fano variety
${\bf P}_{\sigma}$ together with the ample invertible sheaf
${\cal O}_{{\bf P}_{\sigma}}(1)$ on it such that
some $r$-tensor power of ${\cal O}_{{\bf P}_{\sigma}}(1)$ is
isomorphic to the anticanonical sheaf of ${\bf P}_{\sigma}$ (in particular
${\bf P}_{\sigma}$ has only Gorenstein singularities).
The zeros of global sections of ${\cal O}_{{\bf P}_{\sigma}}(1)$ are
generalized Calabi-Yau manifolds. For special simplicial
reflexive Gorenstein cones, we obtain generalized Calabi-Yau
manifolds considered by Schimmrigk.
The class of reflexive Gorenstein cones of fixed dimension admits a
natural involution $\sigma \rightarrow \check{\sigma}$ which we conjecture
to correspond the mirror involution in $N =2$ superconformal theories.
The rest of the paper is devoted to arguments which confirm this
conjecture.
In Section 3, we give a general overview for
the method of reduction of complete
intersections in toric varieties to hypersurfaces in higher dimensional
toric varieties. This method allows to construct reflexive
Gorenstein cones and
generalized Calabi-Yau manifolds from usual Calabi-Yau complete
intersections in Gorenstein toric Fano varieties. The relation
between Hodge structures of Calabi-Yau complete intersections
and the corresponding generalized Calabi-Yau manifolds enable
to use results from \cite{bat.var}
to determine variations of Hodge structure for Calabi-Yau complete
intersections in toric varieties.
In Section 4, we check that the duality between nef-partitions
defining Calabi-Yau complete intersections in Gorenstein toric
Fano varieties \cite{borisov} agree with the duality between
reflexive Gorenstein cones.
In Section 5, we show that the duality for reflexive Gorenstein cones
agree also with the explicit construction of mirrors for rigid Calabi-Yau
manifolds.
\medskip
{\bf Acknowledgements.} We are grateful to I.V. Dolgachev, H. Esnault,
S. Katz, Yu.I. Manin, D. van Straten, and R. Schimmrigk for helpful
discussions.
\bigskip
\section{Reflexive Gorenstein cones}
\noindent
Let $\overline{M}$ and $\overline{N} =
{\rm Hom}(\overline{M}, {\bf Z})$ be dual free abelian groups
of rank $\overline{d}$, $\overline{M}_{\bf R}$ and
$\overline{N}_{\bf R}$ the real scalar extensions
of $\overline{M}$ and $\overline{N}$,
$ \langle *, * \rangle\; : \; \overline{M}_{\bf R}
\times \overline{N}_{\bf R} \rightarrow
{\bf R} $
the natural pairing.
We consider $\overline{M}$ (resp. $\overline{N}$) as
a maximal lattice in $\overline{M}_{\bf R}$
(resp. in $\overline{N}_{\bf R}$).
\begin{dfn} {\rm
Let $e_1, \ldots, e_k$ be elements of $M$.
By a finite rational polyhedral cone
\[\sigma = {\bf R}_{\geq 0} \langle e_1, \ldots, e_k \rangle
\subset \overline{M}_{\bf R} \]
generated by $\{ e_1, \ldots, e_k \}$ we mean the set of all $x =
\lambda_1 e_1 + \cdots + \lambda_k e_k \in \overline{M}_{\bf R}$ such that
$\lambda_i \geq 0$, $\lambda_i \in {\bf R}$ $(i =1, \ldots, k)$.
}
\end{dfn}
\begin{dfn}
{\rm Let $\sigma$ be a
finite rational polyhedral cone in $\overline{M}_{\bf R}$.
Then the {\em dual cone} is the set
\[ \check{\sigma} = \{ y \in \overline{N}_{\bf R} \mid
\langle x, y \rangle \geq 0 \mbox{ for all } x \in \sigma \}. \] }
\end{dfn}
\begin{dfn} {\rm
If a $\overline{d}$-dimensional cone
$\sigma \subset \overline{M}_{\bf R}$ satisfies
the condition $\sigma \cap -\sigma
=0$, then we can uniquely choose
the {\em minimal set of integral generators} of $\sigma$ which
is defined as the set of all primitive $\overline{M}$-lattice vectors
on $1$-dimensional faces of $\sigma$. By a {\em generator} of
$\sigma$ we will always mean a primitive $\overline{M}$-lattice
vector on a $1$-dimensional face one of $\sigma$. }
\end{dfn}
\begin{dfn}
{\rm Let $\sigma$ be a finite rational polyhedral cone in
$\overline{M}_{\bf R}$ satisfying the condition
$\sigma \cap -\sigma =0$.
Then the cone $\sigma$ is called {\em Gorenstein} if
there exists an element $n_{\sigma} \in N$ such that
$\langle e, n_{\sigma} \rangle =1$ for each generator $e$ of $\sigma$.}
\end{dfn}
\begin{rem}
{\rm If $\sigma \subset \overline{M}_{\bf R}$ is a Gorenstein cone
whose dimension equals the dimension of $\overline{M}_{\bf R}$,
then the element $n_{\sigma}$ is uniquely defined. }
\end{rem}
\begin{dfn}
{\rm A Gorenstein cone $\sigma \subset \overline{M}_{\bf R}$
is called {\em reflexive} if the
dual cone $\check{\sigma} \subset \overline{N}_{\bf R}$
is also Gorenstein. In this case,
the positive integer
\[ r_{\sigma} = \langle m_{\check{\sigma}}, n_{\sigma} \rangle \]
will be called the {\em index} of $\sigma$ (or of $\check{\sigma})$.
Since the notion of reflexive Gorenstein cone strongly depends on
the choice of the maximal sublattice $\overline{M}$ in the
$\overline{d}$-dimensional real vector space, we will say
that reflexive Gorenstein cones are defined by pairs
$(\sigma, \overline{M})$. }
\end{dfn}
\medskip
The notions of Gorenstein cones and reflexive Gorenstein cones
can be interpreted via toric geometry.
\begin{prop}
Let ${\bf A}_{\sigma}= {\rm Spec}\,{\bf C} \lbrack \check{\sigma} \cap
\overline{N} \rbrack$ be the affine $\overline{d}$-dimensional
toric variety associated with a rational polyhedral
cone $\sigma$. Then $\sigma$ is a Gorenstein cone if and only if
${\bf A}_{\sigma}$ has only Gorenstein singularities.
\label{gorenstein}
\end{prop}
{\em Proof. } The statement follows from the characterization of
M. Reid for Gorenstein toric singularities \cite{reid}. \hfill $\Box$
\medskip
By a lattice polyhedron in a finite dimensional real vector space
${\cal U}$
over ${\bf R}$ we always mean a convex polyhedron whose vertices
belong to some fixed maximal sublattice in ${\cal U}$.
\begin{dfn}
{\rm Let $\sigma$ be a Gorenstein cone. Then
\[ \Delta_{\sigma} = \{ x \in \sigma \mid \langle x, n_{\sigma}
\rangle =1 \} \]
is a $(\overline{d} -1)$-dimensional convex lattice polyhedron
which we call the {\em support} of $\sigma$. }
\end{dfn}
\begin{rem} {\rm
Every lattice polyhedron $\Delta$ in a $d$-dimensional
space $M_{\bf R}$ can be considered as a support of
$(d+1)$-dimensional Gorenstein cone $\sigma_{\Delta} \subset
\overline{M}_{\bf R} = {\bf R} \oplus M_{\bf R}$
defined as
\[ \sigma_{\Delta} = \{
( \lambda, \lambda x) \in \overline{M}_{\bf R}
\mid \lambda \in {\bf R}_{\geq 0}, \; x \in \Delta \}. \]
}
\end{rem}
\begin{dfn}
{\rm \cite{bat.dual} A lattice polyhedron is called {\em reflexive} if
$\sigma_{\Delta}$ is a reflexive Gorenstein cone of
index $1$. If $\Delta$ is a reflexive polyhedron, then
the support of the dual reflexive Gorenstein cone $\check{\sigma}_{\Delta}$
is another reflexive polyhedron $\Delta^*$ which is called {\em dual} to
$\Delta$. }
\label{def.ref}
\end{dfn}
\begin{prop}
A Gorenstein cone $\sigma$ is reflexive cone of index $r$
if and only $r\Delta_{\sigma}$ is a reflexive polyhedron.
\label{reflex.cone}
\end{prop}
{\em Proof. } Let $\sigma \subset \overline{M}_{\bf R}$ be a
Gorenstein cone and $r$ a positive
integer. Define new lattices $M' \subset \overline{M}_{\bf R}$ and
$N' \subset \overline{N}_{\bf R}$ as follows
\[ M' = \{ x \in \overline{M} \mid
\langle x, n_{\sigma} \rangle = 0\,(\mbox{\rm mod
$r$ }) \}, \]
\[ N' = \overline{N} + {\bf Z} n_{\sigma}'\; \mbox{\rm where }\; n_{\sigma}'
= \frac{1}{r}n_{\sigma}. \]
Then the lattice $N'$ is dual to the lattice
$M'$ and the pair $(\sigma, M')$ defines again a Gorenstein cone
whose support is $r\Delta_{\sigma}$.
Assume now that the pair $(\check{\sigma}, \overline{N})$ defines a reflexive
Gorenstein cone of index $r$. Since $m_{\check{\sigma}} \in M'$ and
$\langle m_{\check{\sigma}}, e \rangle = 1$ for every $\overline{N}$-integral
generator $e$ of ${\check{\sigma}}$, we obtain that $e$ is also
a $N'$-integral generator of ${\check{\sigma}}$.
Thus the pair $(\check{\sigma}, N')$ defines a reflexive
Gorenstein cone of index $1= \langle m_{\check{\sigma}}, n_{\sigma}'\rangle$.
By \ref{def.ref}, $r\Delta_{\sigma}$ is reflexive.
Analogous arguments show the part "if".
\hfill $\Box$
The next statement follows immediately from the
equivalent characterizations of reflexive polyhedra in \cite{bat.dual}:
\begin{prop}
A lattice polyhedron $\Delta$ is reflexive if and only if
$\Delta$ is the support of global sections of the ample anticanonical
sheaf on a Gorenstein toric Fano variety.
\label{reflex}
\end{prop}
Using \ref{reflex.cone} and \ref{reflex}, we obtain:
\begin{coro} Let $\sigma$ be a Gorenstein cone.
We define the degree of an element
$m \in \sigma \cap \overline{M}$ as
${\rm deg}\, m = \langle m, n_{\sigma} \rangle$.
Let ${\bf P}_{\sigma}= {\rm Proj}\,{\bf C} \lbrack {\sigma} \cap
\overline{M} \rbrack$ be the corresponding
projective $(\overline{d}-1)$-dimensional
toric variety. Then $\sigma$ is reflexive Gorenstein cone
of index $r$ if and only if
${\bf P}_{\sigma}$ is a Gorenstein toric Fano variety such that
the anticanonical sheaf on ${\bf P}_{\sigma}$ is isomorphic to
${\cal O}_{{\bf P}_{\sigma}}(r) =
{\cal O}_{{\bf P}_{\sigma}}(1)^{\otimes r}$.
\label{characteriz}
\end{coro}
By the adjunction formula, one has:
\begin{coro}
Let $\sigma$ be a reflexive Gorenstein cone. Then the zeros of
a general global section of ${\cal O}_{{\bf P}_{\sigma}}(1)$ define
a $(\overline{d} -2)$-dimensional algebraic variety $Z$ with
only Gorenstein toroidal singularities such that
the anticanonical sheaf on $Z$ is ${\cal O}_Z(r-1)$.
\label{zeros}
\end{coro}
\begin{dfn}
{\rm We call the $(\overline{d} -2)$-dimensional algebraic varieties
$Z$ obtained as zeros of global sections of ${\cal O}_{{\bf P}_{\sigma}}(1)$
(\ref{zeros}) {\em generalized Calabi-Yau manifolds associated
with the reflexive Gorenstein cone $\sigma$.}}
\end{dfn}
\begin{exam}
{\rm Let ${\bf P}(w_1, \ldots, w_{\overline{d}})$ be a
$(\overline{d}-1)$-dimensional weighted projective spaces whose
weights satisfy the condition $w_1 + \cdots + w_{\overline{d}} = r w_0$
for some positive integers $r$ and $w_0$ such that $w_i$ divides
$w_0$ for all $i =1, \ldots, \overline{d}$.
Let $\sigma = {\bf R}^d_{\geq 0} \subset {\bf R}^{\overline{d}}$
be the positive octant.
Define the maximal lattice $\overline{M} \subset {\bf R}^{\overline{d}}$
as
\[ \overline{M} = \{ (x_1, \ldots, x_{\overline{d}}) \in
{\bf Z}^{\overline{d}} \mid w_1 x_1 + \cdots + w_{\overline{d}}
x_{\overline{d}} = 0\, (\mbox{\rm mod } w_0) \}. \]
Then the pair $(\sigma, \overline{M})$ defines an example
of a reflexive Gorenstein cone of index $r$.
In this case the associated with $\sigma$
generalized Calabi-Yau manifolds
are hypersurfaces of degree $w_0$ in
${\bf P}(w_1, \ldots, w_{\overline{d}})$. }
\label{weighted}
\end{exam}
Now we formulate the main conjecture:
\begin{conj}
Every pair of $\overline{d}$-dimensional
dual refelxive Gorenstein cones $\sigma$ and $\check{\sigma}$
of index $r$ give rises to a $N=2$ superconformal theory with central
charge $c = 3( \overline{d} - 2(r-1))$. Moreover, the superpotentials
of the corresponding Landau-Ginzburg theories define two
families of generalized Calabi-Yau manifolds
associated with $\sigma$ and $\check{\sigma}$ which are exchanged by
the mirror involution.
\label{conjecture}
\end{conj}
\begin{rem}
{\rm By \ref{def.ref}, the duality between reflexive Gorenstein
cones is equivalent to the duality between the supporting reflexive polyhedra.
This shows that Conjecture \ref{conjecture}
includes the one of \cite{bat.dual}. }
\end{rem}
\bigskip
\section{From complete intersections to hypersurfaces}
\noindent
In this section we discuss the general procedure which ascribes to a complete
intersection in toric variety a hypersurface in another toric variety.
This method is essentially due to Danilov and Khovanski\v{i} \cite{dan.khov}
(see also \cite{esnault,terasoma}).
Then we show that the hypersurfaces that arise in this way give rise
to reflexive Gorenstein cones if and only if the original complete intersection
is Calabi-Yau variety (Prop. \ref{reflexivecone}).
We also discuss when the the reflexive Gorenstein cones come from
some Calabi-Yau complete intersections.
\medskip
Let $M$ be a free abelian group of rank $d$, $M_{\bf R}$ the real scalar
extension of $M$.
Let $\Delta_1, \ldots, \Delta_r \subset M_{\bf R}$ be
lattice polyhedra supporting global sections of
semi-ample invertible sheaves
${\cal O}_X(D_1), \ldots, {\cal O}_X(D_r)$
on a $d$-dimensional toric variety $X$.
Without loss of generality we will always assume that ${\rm dim}\,
\Delta_1 + \cdots + \Delta_r = d$.
Let ${\bf Z}^r$ be the standard $r$-dimensional lattice,
${\bf R}^r$ its real scalar
extension. We put $\overline{M} = {\bf Z}^r \oplus
M$, $\overline{d} = d + r$, and define
the cone $\sigma \subset \overline{M}_{\bf R}$
as
\[ \sigma = \{ ( \lambda_1, \ldots, \lambda_r, \lambda_1 x_1 + \cdots
+ \lambda_r x_r ) \in \overline{M}_{\bf R}
\mid \lambda_i \in {\bf R}_{\geq 0}, \; x_i \in \Delta_i, \;
i =1, \ldots r \}. \]
Then $\sigma$ is a $\overline{d}$-dimensional
Gorenstein cone, where $n_{\sigma}$ is an element of the dual lattice
$\overline{N}$ defined uniquely by the conditions
\[ \langle x, n_{\sigma} \rangle = 0 \; \mbox{ for } x \in M_{\bf R} \subset
\overline{M}_{\bf R}; \]
\[ \langle e_i, n_{\sigma} \rangle = 1\; \mbox{ for } i = 1, \ldots, r, \]
where $\{e_1, \ldots, e_r \} $ is the standard basis of ${\bf Z}^r \subset
\overline{M}$.
\begin{rem}
{\rm The supporting polyhedron $\Delta_{\sigma}$ coincides
with the $(\overline{d} -1)$-dimensional polyhedron
$\Delta_1 * \cdots * \Delta_r$ considered by
Danilov and Khovanski\v{i} (\cite{dan.khov} \S 6). }
\end{rem}
\begin{prop}
Denote by $Y$ the $(\overline{d} -1)$-dimensional toric variety which is
the toric ${\bf P}^{r-1}$-bundle over $X:$
\[ Y = {\bf P}({\cal O}_X(D_1) \oplus \cdots \oplus {\cal O}_X(D_r)). \]
Let ${\cal O}_Y(-1)$ be the Grothendieck tautological sheaf on $Y$.
Then ${\cal O}_Y(1)$ is semi-ample, and
${\bf P}_{\sigma}$ is the birational image of the toric morphism
\[ \alpha \; :\; Y \rightarrow {\bf P}_{\sigma} \]
defined by global sections of ${\cal O}_Y(1)$.
In particular the polyhedron $\Delta_{\sigma}$ supports the
global sections of ${\cal O}_Y(1)$.
\label{reduction}
\end{prop}
{\em Proof.} Let $\pi\, :\, Y \rightarrow X$ be the
canonical projection. Since $\pi$ agrees with the
torus actions on $X$ and $Y$, we obtain the natural torus action
on ${\cal O}_Y(1)$. Since
\[ \pi_* {\cal O}_Y(1) = {\cal O}_X(D_1) \oplus \cdots
\oplus {\cal O}_X(D_r) \]
is the direct sum of sheaves generated by global sections,
${\cal O}_Y(1)$ is also generated by global sections; i.e.,
${\cal O}_Y(1)$ is semi-ample.
In order to determine the polyhedron supporting the global sections
of ${\cal O}_Y(1)$, it suffices to compute the $(d+r-1)$-dimensional
torus action on
\[ H^0({\cal O}_Y(1)) \cong H^0({\cal O}_X(D_1) \oplus \cdots
\oplus {\cal O}_X(D_r)). \]
The latter is trivial, since we know the $(r-1)$-dimensional
torus action on homogeneous coordinates in ${\bf P}^{r-1}$
and the $d$-dimensional
torus action on $H^0({\cal O}_X(D_i))$ $(i =1, \ldots, r)$
defined by the lattice points in the polyhedron $\Delta_i$.
\hfill $\Box$
\begin{coro}
Every global section $s$ of ${\cal O}_Y(1)$ defines uniquely
global sections $s_i$ of ${\cal O}_X(D_i)$ such that
\[ \pi_*( s) = (s_1, \ldots , s_r) \in
H^0({\cal O}_X(D_1)) \oplus \cdots \oplus H^0({\cal O}_X(D_r)), \]
and vise versa, every $r$-tuple of sections $s_i \in H^0({\cal O}_X(D_i))$
$( i =1, \ldots, r)$ defines a global section $s \in
H^0({\cal O}_Y(1))$.
\label{sections}
\end{coro}
\begin{coro}
Let $s \in H^0({\cal O}_Y(1))$ and
$s_i \in H^0({\cal O}_X(D_i))$
$( i =1, \ldots, r)$ be global sections as in \ref{sections}.
Denote by $V_s$ the hypersurface in $Y$ defined by $s = 0$, and by
$V_{s_i}$ $( i =1, \ldots, r)$ the hypersurfaces in $Y$ defined by
$s_i = 0$. Then $Y \setminus V_s$ is locally trivial in Zariski topology
${\bf C}^{r-1}$-bundle over
\[ X \setminus \bigcap_{i=1}^r V_{f_i}. \]
\label{bundle}
\end{coro}
\begin{rem}
{\rm The statement in \ref{bundle} implies the isomorphism
\[ H^{i}_c( X \setminus \bigcap_{i=1}^r V_{f_i}) \cong
H^{i + 2r -2}_c (Y \setminus V_s) \]
which sends the $(p,q)$-component in
cohomology with compact supports of
$X \setminus \bigcap_{i=1}^r V_{f_i}$
to $(p + r -1,q+ r-1)$-component in the cohomology with
compact supports of $Y \setminus V_s$.
The relation between Hodge structures of the complements to the
higher dimensional hypersurfaces and to the complete
intersections was used for
estimations of the Hodge type of algebraic subvarieties \cite{esnault}
and in the proof of weak global Torelli theorem \cite{terasoma}.
One can consider \ref{bundle} also as a version of the Lagrange method
proposed by Danilov and Khovanski\v{i} (\cite{dan.khov}, \S 6) for
affine hypersurfaces}.
\end{rem}
\begin{prop}
The cone $\sigma \subset \overline{M}_{\bf R}$ is a
reflexive Gorenstein cone of index $r$ if and only if
${\cal O}_X(D_1 + \cdots + D_r)$
is isomorphic to the anticanonical sheaf
on $X$.
\label{reflexivecone}
\end{prop}
{\em Proof.} Standard calculations show that the
canonical sheaf ${\cal K}_Y$ on $Y$ is isomorphic to the tensor product
\[ {\cal O}_Y(-r) \otimes \pi^* {\cal K}_X \otimes
\pi^* \Lambda^r({\cal O}_X(D_1) \oplus \cdots
\oplus {\cal O}_X(D_r)). \]
Since
\[ \Lambda^r({\cal O}_X(D_1) \oplus \cdots
\oplus {\cal O}_X(D_r)) \cong {\cal O}_X(D_1 + \cdots + D_r), \]
we have
\[ {\cal K}_Y^{-1} \cong {\cal O}_Y(r) \otimes \pi^* {\cal K}_X^{-1}
\otimes \pi^* {\cal O}_X(-D_1 - \cdots - D_r). \]
Assume that ${\cal O}_X(D_1 + \cdots + D_r)$ is isomorphic to
the anticanonical sheaf ${\cal K}^{-1}_X$
on $X$. Then ${\cal O}_Y(r)$ is isomorphic to the anticanonical sheaf on
$Y$. Therefore, by \ref{characteriz} and \ref{reduction},
$\sigma$ is reflexive.
The "only if" part is left to reader. \hfill $\Box$
\medskip
The following example of the reduction of Calabi-Yau
complete intersections to a higher-dimensional generalized
Calabi-Yau manifolds is due to Schimmrigk \cite{schimm2}.
\begin{exam}
{\rm Let $k, l$ be two positive integers. Define $V$ as a
Calabi-Yau complete intersection of two hypersurfaces in ${\bf P}^k
\times {\bf P}^l$ having bidegrees $(k+1, 1)$ and $(0,l+1)$. Then
the corresponding generalized Calabi-Yau manifolds associated
with the reflexive Gorenstein cone of index $2$ are hypersurfaces
of degree $(k+1)l$ in the $(k+l-1)$-dimensional weighted projective space
\[ {\bf P} (\underbrace{(l-1),\ldots,(l-1)}_{k+1},
\underbrace{(k+1), \ldots,(k+1)}_{l+1}).\]
}
\end{exam}
\begin{dfn}
{\rm A $\overline{d}$-dimensional reflexive Gorenstein cone
$\sigma \subset \overline{M}_{\bf R}$ is
called {\em split} if there exist two lattice polyhedra
$\Delta_1, \Delta_2 \subset M_{\bf R}$ of
dimension $d = \overline{d} - 2$
such that
\[ \sigma \cong \{ ( \lambda_1, \lambda_2, \lambda_1 x_1 +
\lambda_2 x_2 ) \in \overline{M}_{\bf R}
\mid \lambda_i \in {\bf R}_{\geq 0}, \; x_i \in \Delta_i, \;
i =1, 2 \}. \]}
\label{split1}
\end{dfn}
\begin{dfn}
{\rm A $\overline{d}$-dimensional reflexive Gorenstein cone
$\sigma \subset \overline{M}_{\bf R}$ of index $r$ is
called {\em completely split} if there exist $r$ lattice polyhedra
$\Delta_1, \ldots, \Delta_r \subset M_{\bf R}$ of
dimension $d = \overline{d} - r$
such that
\[ \sigma \cong \{ ( \lambda_1, \ldots, \lambda_r, \lambda_1 x_1 + \cdots
+ \lambda_r x_r ) \in \overline{M}_{\bf R}
\mid \lambda_i \in {\bf R}_{\geq 0}, \; x_i \in \Delta_i, \;
i =1, \ldots r \}. \]}
\label{split2}
\end{dfn}
\begin{rem}
{\rm Cones that come from Calabi-Yau
complete intersections are completely split. On
the other hand, every splitting of the reflexive Gorenstein cone gives rise to
the family of complete intersections.
However it's not clear whether there exist reflexive Gorenstein cones that
could
be split in several essentially different ways, so that they come from
different Calabi-Yau varieties.}
\label{diffCYsamecones}
\end{rem}
There exist simple examples of reflexive Gorenstein cones of
index $r >1$ which are not split (and hence are not completely split):
\begin{exam}
{\rm Let $\sigma = {\bf R}_{\geq 0}^{\overline{d}} \subset
{\bf R}^{\overline{d}}$ be the positive octant. Assume that
$\overline{d} = kr$ where $k, r$ are positive integers and
$k,r >1$. Define the lattice $\overline{M}$ as
\[ \overline{M} = \{ (x_1, \ldots, x_{\overline{d}}) \in
{\bf Z}^{\overline{d}} \mid x_1 + \cdots + x_{\overline{d}} = 0\,
\mbox{\rm mod $k$} \}. \]
Then the pair $(\sigma, \overline{M})$ defines a reflexive
Gorenstein cone of index $r$ which is not split. Indeed,
if there were two $(\overline{d} - 2)$-dimensional polyhedra
$\Delta_1, \Delta_2$ having the property described in
\ref{split1}, then for any two vertices $v_1 \in \Delta$,
$v_2 \in \Delta_2$ we could find two generators
$e_1 = (1,0,v_1)$, $e_2 = (0,1,v_2)$ of the cone $\sigma$ such
that the segment $\lbrack e_1, e_2 \rbrack$ would have no
interior $\overline{M}$-lattice points. On the other hand,
it is clear that for any two generators of $\sigma$
the segment $\lbrack e_1, e_2 \rbrack$ always
contains $k-1$ interior $\overline{M}$-lattice points. }
\end{exam}
\begin{rem}
{\rm In Section 5 we consider an example of a $3d$-dimensional
cone reflexive Gorenstein cone
$\sigma$ which is completely split, but the dual cone
$\check{\sigma}$ is not split.}
\end{rem}
\section{Complete intersections and nef-partitions}
\noindent
There is an important class of reflexive Gorenstein cones $\sigma$
such that both $\sigma$ and $\check{\sigma}$ are completely split.
These cones correspond to so called {\em nef-partitions} introduced
in \cite{borisov}.
We use notations from the
previous section and assume that $X$ is a Gorenstein
Fano toric variety. Let $T \subset X$ be the dense open torus
orbit, $E_1, \ldots, E_k$ irreducible components of
$X \setminus T$. Then ${\cal O}(E_1 + \cdots + E_k)$ is
is naturally isomorphic to the anticanonical sheaf ${\cal K}_X^{-1}$
on $X$. Denote by $I$ the set $\{ 1, \ldots, k \}$. Put
$E=\sum_{i \in I} E_i.$
\begin{dfn}
{\rm The decomposition of the index set $I$ into a disjoint union of $r$ sets
$I_j,\,j=1, \ldots, r$ is called a {\em nef-partition} if all
$D_j=\sum_{i\in I_j}E_i$ are
semi-ample Cartier divisors on $X$. By abuse of notations, we
will call by nef-partition also the set of convex lattice polyhedra
$\Pi = \{ \Delta_1, \ldots, \Delta_r \}$ such that $\Delta_i$ is
the support of global sections of ${\cal O}(D_i)$ $i =1, \ldots, r$. }
\label{defnef}
\end{dfn}
\begin{dfn}
{\rm Let $\Pi = \{ \Delta_1, \ldots, \Delta_r \}$ be a nef-partition,
$\Sigma$ is the fan in $N_{\bf R}$ defining the Gorenstein toric
variety $X$. For every $E_i$ ($ i =1, \ldots, k$), we denote by $e_i$
the primitive $N$-lattice generator of the $1$-dimensional cone of
$\Sigma$ corresponding to the divisor $E_i$. Define the lattice
polyhedron $\nabla_j$ $(j =1, \ldots, r)$ as the convex hull
\[ \nabla_i = {\rm Conv}(\{0\} \cup \bigcup_{j \in J_i} \{ e_j \}). \]
}
\label{defdual}
\end{dfn}
\begin{rem}
{\rm Since $D_i = \sum_{j \in J_i} E_i$ defines a convex piecewise linear
function $\psi_i$ such that $\psi_i(e_j) = 1$ if $j \in J_i$ and
$\psi_i(e_j) = 0$ otherwise, we can define $\Delta_1,
\ldots, \Delta_r$ as
\[ \Delta_i = \{ x \in M_{\bf R} \mid \langle x, y \rangle \geq -
\psi_i(y) \}, \; i=1, \ldots, r. \]
In the sequel, we will always assume this definition which
immediately implies that all polyedra
$\Delta_1, \ldots, \Delta_r$ contain $0 \in M_{\bf R}$. }
\end{rem}
The main result of \cite{borisov} is the following:
\begin{theo}
The set $\Pi^* = \{ \nabla_1, \ldots, \nabla_r \}$ is also a nef-partition.
In other words, there exist another $d$-dimensional Gorenstein toric Fano
variety $X^*$ which compactifies the dual torus $T^*$ and the index set
$J^* = \{ 1, \ldots , l \}$ for the irreducible components
$E_1^*, \ldots, E_l^*$ of $X^* \setminus T^*$
such that the $\nabla_i$ is the support of global sections
of the semi-ample sheaf ${\cal O}(D_i^*) =
{\cal O} (\sum_{j \in J^*_i} E_i^*)$ $( i =1, \ldots, r)$ where
$J_1^* \cup \cdots \cup J_r^*$ is a splitting of $I^*$ into a
disjoint union.
\end{theo}
The combinatorial involution on $\Pi \mapsto \Pi^*$ is conjectured
to give rise to the mirror symmetry for the families of Calabi-Yau complete
intersections in $X$ and $X^*$. Some results which confirm this conjecture
are obtained in \cite{bat.bor}.
On the other hand, once we have
\[ {\cal K}^{-1}_X = {\cal O}_X(D_1 + \cdots + D_r) \]
and
\[ {\cal K}^{-1}_{X^*} = {\cal O}_{X^*}(D_1^* + \cdots + D_r^*) \]
we can follow the pro\-ce\-du\-re of the pre\-vious sec\-tion
and get two ref\-le\-xive $\overline{d}$-di\-men\-sio\-nal
Gorenstein cone $\sigma \subset \overline{M}_{\bf R}$ and
$\sigma^* \subset \overline{N}_{\bf R}$.
The purpose of this section is to relate the involution for
nef-partitions \cite{borisov} to the involution
for reflexive Gorenstein cones.
First we recall one property
of dual nef-partitions:
\begin{prop} {\rm \cite{borisov}}
Let $x \in \Delta_i$, $y \in \nabla_j$. Then
\[ \langle x , y \rangle \geq -1\;\; \mbox{ if $ i = j$}, \]
\[ \langle x , y \rangle \geq 0\;\; \mbox{ if $ i \neq j$}. \]
\label{property}
\end{prop}
The main statement is contained in the following theorem.
\begin{theo}
Assume that the canonical pairing $\langle \cdot , \cdot \rangle \, :
M \times N \rightarrow {\bf Z}$ is extended to the pairing
between $\overline{M} = {\bf Z}^r \oplus M$ and $\overline{N} = {\bf Z}^r
\oplus N$ as
\[ \langle (a_1, \ldots,a_r, m), (b_1, \ldots, b_r, n) \rangle =
\sum_{i =1}^r a_i b_i + \langle m, n \rangle. \]
Then the cone $\sigma$ is dual to $\sigma^*$; i.e., the
dual nef-partitions in the sense of {\rm \cite{borisov}} give rise to
the dual reflexive Gorenstein cones.
\label{nefpart-refcone}
\end{theo}
{\em Proof.} By \ref{property}, if $(a_1, \ldots,a_r, m) \in \sigma$ and
$(b_1, \ldots, b_r, n) \in \sigma^*$, then
\[ \langle (a_1, \ldots,a_r, m), (b_1, \ldots, b_r, n) \rangle \geq 0. \]
Therefore $\sigma^* \subset \check{\sigma}$.
Let $(b_1, \ldots, b_r, n) \in \check{\sigma}$. Then, for any
$x \in \Delta_i$, one has $\langle x, n \rangle \geq - b_i$
$(i = 1, \ldots, r)$. Using the same arguments as in the proof
of Prop. 3.2 in \cite{borisov}, we obtain that
$ n \in b_1 \nabla_1 + \cdots + b_r \nabla_r$. Therefore $\check{\sigma}
\subset \sigma^*$.
\hfill $\Box$
\begin{rem}
{\rm Several different nef-partitions can give rise to the same reflexive
Gorenstein cone, so the duality for nef-partitions carries more
information than that for the cones. From the geometrical point of view,
the nef-partition corresponds to the family of Calabi-Yau
complete intersections
together with some special degeneration of the
family into the union of the strata of dim $d-r$. It is not yet clear
if the dual family depends upon the degeneration or not, which is related to
the
question in Remark \ref{diffCYsamecones}. Of course, there are no such
problems in the case of hypersurfaces, or, equivalently, when the index of the
reflexive Gorenstein cones is $1$.}
\end{rem}
\section{Mirrors of rigid Calabi-Yau manifolds}
\noindent
All already known constructions of the mirror correspondence
for rigid Calabi-Yau manifolds were originated from
the explicit identification of
the minimal superconformal models of Gepner \cite{gepner} with
the special Landau-Ginzburg superpotentials \cite{greene.vafa.warner}. This
indentification allows to apply the orbifolding modulo some finite
symmetry group \cite{greene.plesser,roan}.
Our purpose is to show that the orbifold-construction of mirrors
of rigid Calabi-Yau varieties agrees with the duality for reflexive
Gorenstein cones.
\bigskip
We consider in details the example
of the rigid $d$-dimensional Calabi-Yau manifold associated with
the superconformal theory $1^{3d}$ which is the tensor
product of $3d$ copies of the level-1 theories.
\medskip
Let $E_0 ={\bf C} / {\bf Z} \langle 1, \tau \rangle$
be the unique elliptic curve having an authomorphism of
order $3$; i.e., $J(E_0) = 0$, $\tau = e^{\pi i/3}$
and $E_0$ is isomorphic to the Fermat cubic in ${\bf P}^2$.
Notice that
the action of the group ${\bf Z}/3{\bf Z}$ on $E_0$ has exactly $3$
fixed points which we denote by $p_0, p_1, p_2$. Let
\[ G = \{ (g_1,\ldots, g_d) \in ({\bf Z}/3{\bf Z})^d \mid
g_1 + \cdots g_d = 0\, (\mbox{mod } d) \}. \]
Then $G$ is the maximal subgroup in $({\bf Z}/3{\bf Z})^d$
whose action on the product $X = (E_0)^d$
leaves invariant the holomorphic $d$-form
$z_1 \wedge \cdots \wedge z_d$. We denote by
$Z$ the geometric quotient $X/G$ considered as a $d$-dimensional
orbifold.
Then the mirror involution in the $1^{3d}$ superconformal
theory shows that $Z$ is mirror symmetric to the $(3d-2)$-dimensional
Fermat cubic $Y \subset {\bf P}^{3d-1}$. One sees a
geometric confirmation of this duality from
the following statement:
\begin{prop}
Let $\hat{Z}$ be a maximal projective crepant partial resolution of
quotient singularities of $Z$ {\rm \cite{bat.dual}}. Then
\[ h^{1,1} (\hat{Z}) = h^{3d-3,1}(Y). \]
\end{prop}
{\em Proof.} Using the standard technique based on the consideration
of the Jacobian ring associated with the homogeneous equation of
$Y$, we obtain that a basis of $H^{3d-3,1}(Y)$ can be identified with
all square-free monomials of degree $3$ in $3d$ variables. Therefore
\[ h^{3d-3,1}(Y) = { 3d \choose 3 } = \frac{ d (3d-1)(3d-2) }{2}. \]
On the other hand, there exists exactly $d$ linearly independent
$G$-invariant $(1,1)$-forms on $X$: $dz_1\wedge d \overline{z}_1,
\ldots, dz_d \wedge d\overline{z}_d$. Thus, we have $h^{1,1}(Z) = d$. It
remains to compute the number of exceptional divisors on $\hat{Z}$.
Let $\gamma_1 \,: \, \hat{Z} \rightarrow Z$ be the maximal projective
crepant partial resolution, $\gamma_2\,: \, X \rightarrow Z$ the
quotient by $G$.
One easily sees that the $\gamma_1$-image of an exceptional
divisor $D \subset \hat{Z}$
in $Z$ is either $(d-2)$-dimensional, or $(d-3)$-dimensional.
In the first case, $\gamma_1(D)$ is the $\gamma_2$-image of
a $G$-invariant codimension-2 subvariety defined by
the conditions $z_i, z_j \in \{ p_0, p_1, p_2\}$
$(i \neq j)$, and the $\gamma_1$-fiber
over general point of $\gamma_1(D)$ is the exceptional locus
of the crepant resolution of the $2$-dimensional
Hirzebruch-Jung singularity of type
$A_2$, i.e., it consists of two irreducible components. In the second
case, $\gamma_1(D)$ is the $\gamma_2$-image of
a $G$-invariant codimension-3 subvariety defined by
conditions $z_i, z_j, z_k \in \{ p_0, p_1, p_2\}$,
and the $\gamma_1$-fiber
over general point of $\gamma_1(D)$
is an irreducible surface.
Therefore the number of the exceptional divisors equals
\[ 2 \cdot 3^2 \cdot { d \choose 2} + 3^3 \cdot { d \choose 3 }. \]
This immediately implies
\[ h^{1,1}(\hat{Z}) = d + 2 \cdot 3^2 \cdot
{ d \choose 2} + 3^3 \cdot { d \choose 3 } = \frac{ d (3d-1)(3d-2) }{2}. \]
\hfill $\Box$
\medskip
The combinatorial interpretation of the mirror duality between
$Y$ and $Z$ is based on the representation of $X$ and $Z$ as
complete intersections in toric varieties. Let ${\bf P}_{\Delta}$
be the $2$-dimesnional toric variety associated with
the reflexive polygon $\Delta = {\rm Conv}\{ (1,0),(0,1),(-1,-1) \}$.
We can also define ${\bf P}_{\Delta}$ in ${\bf P}^3$
by the equation $u_0^3 = u_1 u_2 u_3$.
\begin{prop}
Let $C \subset {\bf P}_{\Delta}$ be a curve defined by an
equation $\lambda_1 u_1 + \lambda_2 u_2 + \lambda_3 u_3 = 0$.
Then $C$ is isomorphic to $E_0$.
\end{prop}
{\em Proof. } It is sufficient to notice that the mapping
\[ q \;: \; {\bf P}_{\Delta} \rightarrow {\bf P}_{\Delta} \]
\[ (u_0, u_1,u_2,u_3) \mapsto (e^{2\pi i/3}u_0, u_1,u_2,u_3) \]
induces an authomorphism of order $3$ of $C$ with three fixed points. \hfill
$\Box$
\begin{coro}
The $d$-dimensional variety $X$ is a complete intersection
of $d$ nef-divisors in the $2d$-dimensional
toric variety $({\bf P}_{\Delta})^d$.
\end{coro}
\begin{coro}
The mapping $q$ induces the action of $({\bf Z}/3{\bf Z})^d$ on
$({\bf P}_{\Delta})^d$ such that $Z$ becomes a
complete intersection in the geometric quotient
$({\bf P}_{\Delta})^d/G$.
\end{coro}
\begin{prop}
Let $\sigma \subset {\bf R}^{3d}$ be positive octant. Define
\[ \overline{M} = {\bf Z}^{3d} + {\bf Z}(\frac{1}{3}, \ldots,
\frac{1}{3}). \]
Then the pair $(\sigma, M)$ defines a
$3d$-dimensional Gorenstein reflexive cone associated with
$Z$ as a complete intersection in $({\bf P}_{\Delta})^d/G$.
\end{prop}
{\em Proof. } We notice that the $3$-dimensional
reflexive cone $\sigma_{\Delta}$ can be described
as the positive octant in ${\bf R}^3$ with respect to
the lattice
\[ M = {\bf Z}^3 + {\bf Z}(\frac{1}{3},\frac{1}{3},\frac{1}{3}). \]
Thus the pair $((\sigma_{\Delta})^d, M^d)$ is the
$3d$-dimensional Gorenstein reflexive cone associated with
$X$ as a complete intersection in $({\bf P}_{\Delta})^d$.
It remains to compute the sublattice $\overline{M} \subset M^d$
corresponding to modding out of $({\bf P}_{\Delta})^d$ by $G$.
It is clear that ${\bf Z}^{3d} \subset \overline{M}$ and
$M^d/\overline{M}$ must be isomorphic to
$G$. On the other hand, by construction,
$\overline{M}$ must be invariant under
$d$-element permutations in $M^d$. These conditions define
$\overline{M}$ uniquely as
${\bf Z}^{3d} + {\bf Z}({1}/{3}, \ldots, {1}/{3})$. \hfill
$\Box$.
Using \ref{weighted}, we have:
\begin{coro}
Let ${\sigma} \subset {\bf R}^{3d}$
be positive octant. Define
\[ \overline{N} = \{ (x_1, \ldots, x_{3d}) \in
{\bf Z}^{3d} \mid x_1 + \cdots + x_{3d} = 0( \mbox{\rm mod 3}). \]
Then the pair $(\sigma, N)$ defines the dual to $(\sigma, M)$
reflexive Gorenstein cone with respect to the standard scalar
product on ${\bf R}^{3d}$. In particular,
the reflexive Gorenstein pair $( \sigma, \overline{N})$
corresponds to cubic hypersurfaces in ${\bf P}^{3d-1}$.
\end{coro}
\begin{rem}
{\rm If we choose an intermediate lattice $M'\, : \;
\overline{M} \subset M' \subset M^d$ and the corresponding dual intermediate
lattice $N'\, : \; N^d \subset N' \subset \overline{N}$, then we obtain
another pair of dual reflexive Gorenstein cones. In particular,
if $d =3$, then one can obtain a rigid Calabi-Yau 3-fold $Z'$ by modding out
$E_0 \times E_0 \times E_0$ by the diagonal action of ${\bf Z}/3{\bf Z}
\subset G \cong ({\bf Z}/3{\bf Z})^2$. The
mirror symmetric generalized Calabi-Yau manifolds are then obtained
as quotients of $7$-dimensional cubics by ${\bf Z}/3{\bf Z}$. This particular
case was considered in \cite{cand.derr.parkes}.}
\end{rem}
|
1994-04-13T17:38:39 | 9402 | alg-geom/9402007 | en | https://arxiv.org/abs/alg-geom/9402007 | [
"alg-geom",
"math.AG"
] | alg-geom/9402007 | Valery Alexeev | Valery Alexeev | Boundedness and $K^2$ for log surfaces | This version: a TeX fix only. The old TeX version did not work with
pdflatex, producing strange characters | Internat. J. Math. 5 (1994), no.6, 779-810 | null | null | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ | Let $\epsilon, C$ be two positive real numbers, and $\mathcal C \subset
\mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j
B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then
(1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such
that $(X,B)$ is $\epsilon$-log terminal and $-(K_X + B)$ is nef (excluding only
those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du
Val singularities), is bounded.
(2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs
$(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, is a DCC set.
(3) The class $\{(X,B)\}$ of pairs such that $(X, B)$ is semi log canonical,
$K_X + B$ is ample, $(K_X + B)^2 = C$ and $b_j \in \mathcal C$, is bounded.
| [
{
"version": "v1",
"created": "Wed, 9 Feb 1994 00:58:27 GMT"
},
{
"version": "v2",
"created": "Wed, 13 Apr 1994 15:38:19 GMT"
},
{
"version": "v3",
"created": "Thu, 16 Feb 2017 21:24:14 GMT"
}
] | 2017-02-20T00:00:00 | [
[
"Alexeev",
"Valery",
""
]
] | alg-geom | \section{Introduction}
\label{Introduction}
\begin{say}
The aim of this paper is to generalize to the singular case the
following statements which in the nonsingular case are easy and well-known:
\begin{enumerate}
\item The class of smooth
surfaces with ample anticanonical divisor $-K$, also known as Del
Pezzo surfaces, is bounded.
\item The class of smooth
surfaces with ample canonical divisor $K$ with $K^{2}\le
C$ is bounded.
\item The class of smooth
surfaces with ample canonical divisor $K$ with $K^{2}=
C$ is bounded.
\item The set $\{K_{X}^{2}\}$, where $X$ is a smooth surface with
ample $K_{X}$, is the set $\Bbb N$ of positive integer numbers.
\end{enumerate}
\end{say}
\begin{say}
Of course, the last of these statements is trivial, and the second and the
third ones are equivalent. In the singular case, however, the
set $\{K_{X}^{2}\}$ {\em a priori\/} is just a certain subset in the
set of positive
rationals. We shall prove that under natural and rather weak
conditions this set has a rigid structure: it satisfies the descending
chain condition, abbreviated below to
D.C.C. for short. The boundedness for surfaces with
constant $K^{2}$ will be proved under much weaker conditions on
singularities than for surfaces with $K^{2}$ only bounded.
\end{say}
\begin{say}
In order to be useful, the generalizations should come from interesting
real-life examples. The examples should also suggest what conditions on
singularities are most natural.
The section~\ref{Applications} contains several applications, due to
J.Koll\'ar,
G.Xiao and others, that provide the motivation for our boundedness
theorems.
The most important application is the
projectiveness of the coarse moduli space of stable surfaces.
Two others are a formula bounding the automorphism group of a
(possibly, singular) surface of general type by $cK^{2}$, and
a theorem on the uniform plurigenera of elliptic threefolds.
\end{say}
\begin{say}
\label{proved in this paper}
These are the precise statements of the results proved in this paper:
\begin{enumerate}
\label{proved here}
\item Fix $\varepsilon>0$.
Consider all projective surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR $\varepsilon$-log terminal and $-(K_{X}+B)$ is
nef excluding only those for which at the same time $K_{X}$
is numerically trivial, $B$ is zero and $X$ has at worst Du Val
singularities.
Then the class $\{X\}$ is bounded
(theorem~\ref{bound for general when -K nef}).
\item Fix $\varepsilon>0$, a constant $C$ and a D.C.C. set $\cal C$.
Consider all surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR $\varepsilon$-log terminal, $K_{X}+B$ is
big and nef, $b_{j}\in \cal C$ and $(K+B)^{2}\le C$.
Then the class $\{(X,\operatorname{supp} B)\}$ is bounded
(theorem~\ref{bound for varepsilon-terminal, K big, nef}).
\item Fix a constant $C$ and a D.C.C. set $\cal C$.
Consider all surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR semi-log canonical, $K_{X}+B$ is
ample, $b_{j}\in \cal C$ and $(K+B)^{2}= C$.
Then the class $\{(X,\sum b_{j}B_{j})\}$ is bounded
(theorem~\ref{K2=const for semi-log canonical}).
\item Fix a D.C.C. set $\cal C$.
Consider all surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR semi-log canonical, $K_{X}+B$ is
ample and $b_{j}\in \cal C$.
Then the set $\{ (K_{X}+B)^{2} \}$ is a D.C.C. set
(theorem~\ref{DCC for semi-log canonical}).
\end{enumerate}
\end{say}
\begin{say}
The necessary definitions will be given in the section
{}~\ref{Standard definitions}. As the
first approximation, the reader can
look at the results obtained by dropping $B$. It is
interesting to note that none of
the conditions above can be weakened, see the examples in section
{}~\ref{Applications}.
(Of course, it is always possible to choose an entirely different way
of generalizing the nonsingular case and pick a set of conditions
which is orthogonal to ours.)
\end{say}
\begin{say}
I became interested in the subject reading
\cite{Kollar92} by Koll\'ar, and that is where most of the motivation
for the present paper comes from. The statements (3) and (4) above solve
and generalize a Koll\'ar's conjecture formulated there. A hope for
proving this conjecture came to me when I looked at the preprint
\cite{Xiao91} by Xiao, and especially at his proof of Proposition 5. I
am most indebted to G.Xiao for answering many of my questions
concerning his preprint. I am also thankful to V.V.Shokurov and
K.Matsuki for useful discussions.
\end{say}
\begin{say}
Throughout, we work with projective algebraic schemes over an
algebraically closed field of arbitrary
characteristic.
\end{say}
\section{Standard definitions}
\label{Standard definitions}
\begin{say}
For a normal variety $X$, $K_{X}$ or simply $K$ will always denote the
class of linear equivalence of the canonical Weil divisor.
\end{say}
\begin{defn}
An {\em $\Bbb R$-divisor\/} $D=\sum d_{i}D_{i}$
is a linear combination
of prime Weil divisors with real coefficients,
i.e.\ an element of $N^{1}\otimes \Bbb R$.
An $\Bbb R$-divisor is said to be
$\Bbb R$-Cartier if it is a combination
of Cartier divisors with real coefficients, i.e.\ if it belongs to
$Div(X)\otimes\Bbb R \subset N^{1}(X)\otimes\Bbb R$.
The $\Bbb Q$-divisors and $\Bbb Q$-Cartier
divisors are defined in a similar fashion.
\end{defn}
\begin{defn}
Consider an $\Bbb R$-divisor
$K+B=K_{X}+\sum b_{j}B_{j}$ and assume that
\begin{enumerate}
\item $K+B$ is $\Bbb R$-Cartier
\item $0\le b_{j}\le1$
\end{enumerate}
For any resolution $f:Y\to X$ look at the natural formula
\begin{eqnarray}
\label{definition of codiscrepancies}
K_{Y}+B^{Y}=
f^{*}(K_{X}+\sum b_{j}B_{j})= K_{Y}+\sum b_{j}f^{-1}B_{j} + \sum b_{i}F_{i}
\end{eqnarray}
or, equivalently,
\begin{eqnarray}
\label{definition of log discrepancies}
K_{Y}+\sum b_{j}f^{-1}B_{j} + \sum F_{i}=
f^{*}(K_{X}+\sum b_{j}B_{j}) + \sum f_{i}F_{i}
\end{eqnarray}
Here $f^{-1}B_{j}$ are the proper
preimages of $B_{j}$ and $F_{i}$ are the exceptional divisors of
$f:Y\to X$.
The coefficients $b_{i},b_{j}$ are called codiscrepancies, the
coefficients $f_{i}=1-b_{i},f_{j}=1-b_{j}$ -- log discrepancies.
\end{defn}
\begin{rem}
In fact, $K+B$ is not a usual $\Bbb R$-divisor but rather a special
gadget consisting of a linear class of a Weil divisor $K$ (or a
corresponding reflexive sheaf) and an honest $\Bbb R$-divisor $B$.
This, however, does not cause any confusion.
\end{rem}
\begin{defn}
An $\Bbb R$-Cartier divisor $K+B$ (or a pair $(X,B)$) is called
\begin{enumerate}
\item log canonical, if the log discrepancies $f_{k}\ge0$
\item Kawamata log terminal, if $f_{k}>0$
\item canonical, if $f_{k}\ge0$
\item terminal, if $f_{k}>0$
\item $\varepsilon$-log canonical, if $f_{k}\ge\varepsilon$
\item $\varepsilon$-log terminal, if $f_{k}>\varepsilon$
\end{enumerate}
for every resolution $f:Y\to X$.
\end{defn}
\begin{say}
The name $\varepsilon$-log terminal was suggested to me by V.V.Shokurov.
\end{say}
\begin{defn}
In the two-dimensional case
we shall say that $K+B$ is MR log canonical, MR Kawamata log terminal
etc. if we require the previous inequalities to hold not for all
resolutions $f:Y\to X$ but only for a distinguished one,
the minimal desingularization.
\end{defn}
\begin{say}
In the surface case
all the standard theorems of the log Minimal Model Program are valid under
very weak assumptions, see the section~\ref{MMP in dimension 2}. In
particular, assuming $K+B$ to be MR Kawamata log terminal or MR log
canonical is sufficient for all applications.
\end{say}
\begin{defn}
Let $X$ be a reduced scheme satisfying the Serre's condition $S_{2}$ and
$B=\sum b_{j}B_{j}$, $0\le b_{j} \le 1$ be an $\Bbb R$-divisor.
An $\Bbb R$-Cartier divisor $K_{X}+B$ (or a pair $(X,B)$) is called
semi-log canonical if $f_{k}\ge0$ in the above formula
{}~\ref{definition of log discrepancies} for every {\em semiresolution}
$f:Y\to X$ (see \cite{FAAT} 12.2.1).
\end{defn}
\begin{say}
In dimension 2 and characteristic 0 the semi-log canonical
singularities are classified in \cite{KollarShepherdBarron88}. This is
the complete list (modulo analytic isomorphism): nonsingular points,
cones over nonsingular elliptic curves, double normal crossing points
$xy=0$, pinch points $x^{2}=y^{2}z$; and all finite quotients of above.
In codimension one the semi-log canonical schemes have only normal
crossing points.
\end{say}
\begin{say}
\label{reduce semi-log canonical to log canonical}
Let $\nu: X^{\nu}\to X$ be a normalization of $X$,
$X^{\nu}=\cup X_{m}$ be a decomposition into irreducible components
and define $B_{m}$ on $X_{m}$ to be $\nu^{-1}(B)$ plus the double
intersection locus. Then
\begin{displaymath}
K_{X^{\nu}}+\sum B_{m}= \nu^{*}(K_{X}+B)
\end{displaymath}
and $K_{X_{m}}+B_{m}$ are log canonical ( \cite{FAAT} 12.2.2,4).
\end{say}
\begin{defn}
We shall say that $K_{X}+B$ is MR semi-log canonical if $K_{X}$ is
semi-log canonical and all $K_{X_{m}}+B_{m}$ are MR log canonical.
\end{defn}
\section{Examples}
\label{Applications}
\begin{say}
I include the following three examples
from \cite{Kollar92}, in a generalized form.
They help to understand
what kind of boundedness results for singular surfaces are desirable.
After the theorems of this paper are proved, these applications are no
longer conjectural.
\end{say}
\begin{exmp}[Moduli of stable surfaces of general type]
\label{general type}
It is well known that the G.I.T. construction of a complete and projective
moduli space $\overline {\cal M}_{g}$ of stable curves does not work for
surfaces of general type.
As a bare minimum, the complete moduli space should
parameterize surfaces appearing as semistable degenerations of
nonsingular ones, and these have semi-log canonical singularities. In
particular, they can have singularities of arbitrarily large
multiplicities. On the other hand, by \cite{Mumford77} 3.19 a normal
surface with a singularity of multiplicity $\ge7$ is not
asymptotically Chow-(Hilbert-)stable,
so the usual G.I.T. construction does not go through.
The following is a rather
general theorem which allows to circumvent this difficulty.
\end{exmp}
\begin{thm}[Koll\'ar, \cite{Kollar90}, 2.6]
Let $\cal C$ be an open class of $\Bbb Q$-polarized varieties with
Hilbert function $H(t)$. Assume that the corresponding moduli functor
${\cal M \cal C}$
is separated, functorially polarized, semipositive, bounded, complete
and has tame automorphisms. Then $\cal M \cal C$ is coarsely
represented by a projective scheme $\bold M \bold C$.
\end{thm}
\begin{say}
Applying this theorem to surfaces of general type one considers
stable surfaces, introduced for this purpose by Koll\'ar and
Shepperd-Barron in \cite{KollarShepherdBarron88}.
These are defined as surfaces with semi-log
canonical singularities (in particular, they are reduced but not
necessarily irreducible) such that $(\omega_{X}^{N})^{**}$ is ample
for some $N>0$.
Some of the conditions in the theorem are
easy to check. Separateness follows directly from the uniqueness of the
canonical model (in any dimension). Stable surfaces have finite
automorphism groups by Iitaka \cite{Iitaka82} 11.12
(and this is also true in higher
dimensions). Completeness follows from the semi-stable reduction
theorem and the log Minimal Model Theory (in dimension $\dim X+1$, or,
as some say, $\dim X+1/2$). Semipositiveness is harder but it is proved in
\cite{Kollar90}. The boundedness is exactly what we are concerned with in
this paper. So if we want to include this application, the boundedness
theorem for surfaces with positive $K$ and constant $K^{2}$ should be
powerful enough as to include the case of semi-log canonical
singularities. And it is indeed, see~\ref{K2=const for semi-log
canonical} with empty $\cal C$.
\end{say}
\begin{say}
In fact,~\ref{K2=const for semi-log canonical} is strong enough so that
we are able to prove the projectiveness of the moduli space
$\overline {\cal M}_{(K+B)^{2}}$
of
surfaces with semi-log canonical and ample $K+B$ where $B$ is a reduced
divisor (or, more generally, an $\Bbb R$-divisor with coefficients in
a D.C.C. set $\cal C$) and $(K+B)^{2}=C$. This is a direct
generalization of the moduli space $\overline {\cal M}_{g,k}$ of pointed
stable curves (see \cite{Knudsen83}). This includes, for example, a
projective moduli space, an open subset of which parameterizes smooth K3
surfaces $X$ with ample and reduced divisors $B$ with normal crossings.
It is quite
interesting to see what is the relation between this moduli space and the
usual moduli space of polarized K3 surfaces. This
will be carried out in detail elsewhere.
\end{say}
\begin{say}
The following easy trick allows to reduce the boundedness of semi-log
canonical surfaces that are {\em a priori \/} not irreducible to the
boundedness of irreducible normal log canonical surfaces (see \cite{Kollar92}).
Let $X^{\nu}=\cup X_{m}$
be a decomposition into irreducible components as in
{}~\ref{reduce semi-log canonical to log canonical}.
Then
\begin{displaymath}
K_{X}^{2}= \sum (K_{X_{m}}+ B_{m})^{2}
\end{displaymath}
So if we have enough information about possible values of
$(K+B)^{2}$ and know the
boundedness for normal surfaces with log canonical singularities, this
should help us in the general situation.
This shows that it is natural to consider not
only the canonical divisor $K$ but also the canonical divisor with a
``boundary'' $K+B$.
The following example explains the importance of coefficients $b_{j}$ in
$B=\sum b_{j}B_{j}$ other than 1.
\end{say}
\begin{exmp}[Automorphisms of log surfaces of general type]
\label{cor auts of surfaces of general type}
The characteristic of the base field is assumed to be zero for this
application.
Consider a nonsingular surface with ample $K_{X}$. A general fact is
that the group of biregular automorphisms
of $X$ $\quad G=\operatorname{Aut} (X)$ is finite. Let $\pi:X\to Y$ denote a quotient
morphism. Then by the Hurwitz formula one has
\begin{displaymath}
K_{X}= \pi^{*} (K_{Y}+ \sum (1-1/n_{j})B_{j})
\end{displaymath}
where $B_{j}$ are the branching divisors. It follows that
$K_{Y}+B=K_{Y}+ \sum (1-1/n_{j})B_{j}$ is log terminal and that
\begin{displaymath}
|\operatorname{Aut}(X)|= \frac{K_{X}^{2}}{(K_{Y}+B)^{2}},
\end{displaymath}
so if $(K_{Y}+B)^{2})^{2} \ge 1/c$ then $|\operatorname{Aut}(X)|\le cK_{X}^{2}$. The
reader will certainly recognize that for curves this is the original
construction of Hurwitz which gives $\operatorname{Aut}(X)\le 42(2g_{X}-2)$ because
$2g_{Y}-2+\sum (1-1/n_{j})\ge 1/42$ if it is $>0$. It was Xiao's idea
to use the same construction for surfaces of general type in
\cite{Xiao91}. Using this and other methods he proves that for
nonsingular surfaces with ample canonical class one has
$|\operatorname{Aut}(X)|\le42^{2}K_{X}^{2}$. As an application of~\ref{DCC for
semi-log canonical} we have the following theorem.
\end{exmp}
\begin{thm}
\label{my thm on auts}
Fix a D.C.C. set $\cal C$. Then for every surface $X$ with $K+B$ ample,
semi-log canonical, and with $b_{j}\in\cal C$ one has the following
bound for the
automorphism group interchanging components of $B$ with the same
coefficients
\begin{displaymath}
|\operatorname{Aut}(X,B)|\le c(\cal C)(K+B)^{2}
\end{displaymath}
where the constant $c(\cal C)$ depends only on the set $\cal C$.
\end{thm}
\begin{pf}
The group
$G=\operatorname{Aut}(X,B)$ is known to be finite, cf.~\cite{Iitaka82}, 11.12.
Consider $\pi:X\to Y=X/G$.
Now use the same formulas
\begin{eqnarray*}
K+B=\pi^{*}(K_{Y}+D) \text{ and }
|\operatorname{Aut}(X,B)|= \frac{(K+B)^{2}}{(K_{Y}+D)^{2}}
\end{eqnarray*}
It easily follows that $K_{Y}+D$ is also semi-log canonical. The
coefficients of $D$ belong
to the set
\begin{displaymath}
\{ 0\le 1-(1-\sum n_{j }b_{j})/m_{i}\le1 \,|\, b_{j}\in\cal C,
\text{ } m_{i},n_{j}\in \Bbb N \}
\end{displaymath}
which is obviously also a D.C.C. set. Now we only need to see that a
D.C.C. set of positive numbers is bounded from below by a positive
constant.
\end{pf}
\begin{say}
Here is one more application of~\ref{DCC for semi-log canonical}
due to J.Koll\'ar \cite{Kollar92}.
\end{say}
\begin{cor}[Uniform plurigenera of elliptic 3-folds]
For a smooth 3-fold of Kodaira dimension 2 in characteristic 0, there
exists an absolute constant $N$ such that $h^{0}(NK)\ne0$.
\end{cor}
\begin{say}
This was previously known for Kodaira dimension 0 (Kawamata,
\cite{Kawamata86}),
dimension 1 (Mori, \cite{Mori92})
and for dimension 3 and $\chi(\cal O_{X})\le M$ (Fletcher, \cite{Fletcher87}).
The D.C.C. set used in this application is
\begin{displaymath}
\cal C=\{ \frac{1}{12},\ldots{},\frac{11}{12},1-\frac{1}{k} \,|\,
k\in\Bbb N \}
\end{displaymath}
\end{say}
\begin{say}
The following series of examples show that the restrictions of
{}~\ref{proved here} are
in a sense the weakest possible, and that none of them can be weakened
further.
\end{say}
\begin{exmp}
Consider the cone over an elliptic curve embedded
by a complete linear system of arbitrary degree in some projective
space.
The family of these cones is, evidently, not bounded. The
singularities are simple elliptic and they are log canonical, $-K$ is ample.
\end{exmp}
\begin{exmp}
There are infinitely many, I would say, hopelessly many
types of log Del Pezzo surfaces, i.e.\
surfaces with ample $-K$ and log terminal (=quotient in dimension 2
and characteristic 0) singularities. These include all surfaces $\Bbb
P^{2}/G$, $G$ a finite subgroup in $PGL(3)$, for example. So, for the
ample $-K$ the condition on singularities should be stronger than just
log canonical or log terminal. The $\varepsilon$-log terminal
condition seems to be the best substitute.
\end{exmp}
\begin{exmp}
The next example shows that
the condition $b_{j}\le 1-\varepsilon$ is necessary even for
smooth surfaces.
Consider $\Bbb P^{2}$ with two lines
$B_{1}$ and $B_{2}$ and the surface obtained by
blowing up the point of the
intersection of these lines, then several times the point of the
intersection of the first line and the exceptional divisor. Choosing
$b_{1}$ very close to 1 and taking for $K+B$ the full preimage
$f^{*}(K_{P^{2}}+b_{1}B_{1}+b_{2}B_{2})$
changed a little bit in the exceptional
divisors, one easily obtains an infinite sequence
of smooth surfaces with ample $-(K+B)$ and a strictly increasing Picard number.
The same construction works for surfaces with ample $K+B$ as well.
\end{exmp}
\begin{exmp}
This example shows that the set $\cal C$ in
{}~\ref{proved in this paper} has to satisfy the D.C.C.
To
see this, take a rational ruled surface $\Bbb F_{e}$, $e\ge 2$ and
$B=(2+a)/4 (B_{1}+B_{2}+B_{3}+B_{4})$, where $B_{i}$ are general
elements in the linear system $|s_{e}+ef|$, $s_{e}$ is the exceptional
section, $f$ is a fiber. If $a$ is the positive root of the quadratic
equation $ea^{2}+(e-2)a=1$ then $K+B$ is ample and $(K+B)^{2}=1$. Note
that as $e\to \infty$, $(2+a)/4$ approaches its limit $ 1/2$ from
above.
\end{exmp}
\begin{rem}
It was conjectured by V.V. Shokurov in \cite{Shokurov88} that all
sets naturally appearing in conjunction with log Minimal Model Program
have to satisfy either
the descending or the ascending chain conditions. The first
example of this general phenomenon was given in \cite{Alexeev89} for
the set of Fano indices of log Del Pezzo surfaces. See
\cite{Alexeev91a}, \cite{Alexeev93a}, \cite{GrassiKollar92}
for further examples.
\end{rem}
\section{Some methods for proving boundedness}
\label{Some methods for proving boundedness}
\begin{defn}
One says that a certain class of schemes $\cal B$ is bounded
if there exists a morphism $f:\cal X \to \cal S$ between two schemes
of finite type such that every scheme in $\cal B$ appears as one of
the geometric fibers of $f$, not necessarily in a one-to-one way. We do not
require that every geometric fiber of $f$ belongs to $\cal B$.
Usually, though, the class is defined by a combination of
algebraic conditions
some of which are open and others are closed. In this case one can
find a constructible subset in $S$, points of which parameterize
exactly elements of the class $\cal B$.
\end{defn}
\begin{defn}
In the same way, we say that a class $\cal B$ of schemes with closed
subschemes $\{(X,Z)\}$ is bounded if there exist three schemes of
finite type -- $\cal X$, a closed subscheme $\cal Z \subset \cal X$,
and $\cal S$, and a morphism $f:\cal X\to \cal S$ such that every
element of $B$ appears as a fiber of $f$.
\end{defn}
\begin{defn}
Finally, we can assign certain coefficients to subschemes of $X$
and then by boundedness of $\{(X,\sum b_{j}B_{j})\}$ we mean that
all $\{(X,B_{j})\}$ are bounded in the previous sense and, in
addition, that there are
only finitely many
possibilities for the sets of coefficients $\{b_{j}\}$.
\end{defn}
\begin{defn}
A polarization on a scheme $H$ is a class of an ample Cartier divisor.
A $\Bbb
Q$-polarization on a normal variety is a $\Bbb Q-$Cartier divisor a,
positive multiple of which is a polarization.
It is possible to define an $\Bbb R$-polarization on some nonnormal
varieties too
when there is a suitable notion of an $\Bbb R$-divisor. A semi-log
canonical scheme with an ample $\Bbb R$-Cartier divisor $K+B$ would be
an example.
\end{defn}
\begin{say}
Consider a class of polarized reduced schemes over an algebraically
closed field $k$ $(X,H)$ with a fixed
Hilbert function $\cal H(t)=\chi(tH)$. It is known that this class is
bounded provided any of the following conditions are satisfied:
\begin{enumerate}
\item $\dim X=2$ (Matsusaka \cite{Matsusaka86} for normal
surfaces, Koll\'ar \cite{Kollar85} for the general case).
\item $\dim X=3$, $X$ are normal and $\operatorname{char} k=0$ ( \cite{Kollar85}).
\item $X$ are nonsingular and $\operatorname{char} k=0$ (Matsusaka's Big
Theorem, see \cite{Matsusaka86}).
\end{enumerate}
\end{say}
\begin{say}
Moreover, if $\operatorname{char} k=0$ and $X$ are normal, then only the first two
coefficients
of $\cal H(t)$, i.e.\ , up to constants, $H^{\dim X}$ and $H^{\dim
X-1}K_{X}$, are important, by the Riemann-Roch inequalities of
Koll\'ar-Matsusaka
\cite{KollarMatsusaka83}.
In dimension two this is also true in arbitrary characteristic, see
Lemma 2.5.2 \cite{Kollar85}.
\end{say}
\begin{lem}\label{main method for proving boundedness}
\begin{enumerate}
\item Fix $C>0$ and $N\in \Bbb N$, and
let $\cal B= \{ (X,H) \}$ be a class of normal $\Bbb Q$-polarized
surfaces such that $NH$ is Cartier, $H^{2}\le C$.
Then
the class $\cal B$ is bounded.
\item Fix $C,C'>0$ and $N\in \Bbb N$. Let
$\cal B= \{ (X,H,B) \}$ be a class of normal
$\Bbb Q$-polarized surfaces with
subschemes, where $B$ is either a divisor with $HB\le C'$ or
a $0$-dimensional subscheme of length $\le$ $C'$. Then
the class $B$ is bounded.
\end{enumerate}
\end{lem}
\begin{pf}
The first part is just a reformulation of the above remarks. In $
(2)$, since the class $\{(X,H)\}$ is bounded, all the surfaces can be
embedded by a uniform multiple of $H$ in the same projective space
$\Bbb P$. Then the subschemes $B$ satisfying the conditions above are
parameterized by a certain Hilbert scheme $\bold H$, and the pairs
$(B\subset X)$ are parameterized by a closed subscheme of $\Bbb P\times \bold
H$.
\end{pf}
\begin{lem}
\label{boundedness by blowingup}
Let $\{X\}$ be a certain class of schemes and assume that every scheme
$X$ is isomorphic to a blowup $Bl_{Z}Y$ of a scheme $Y$ at a subscheme
$Z$. If the class $\{(Y,Z)\}$ is bounded then the class $\{X\}$ is
bounded.
\end{lem}
\begin{pf}
After subdividing $\cal S$ we can assume that both $\cal Y$ and $\cal
Z$ are flat over $\cal S$. Then
$Bl_{\cal Z}\cal Y$ will give a required family.
\end{pf}
\begin{say}
I shall use one more method for proving boundedness given below, and a
similar method for proving the descending chain condition.
They sound almost
trivial but I still would like to formulate them explicitly.
\end{say}
\begin{thm}
\label{kinky method for proving boundedness}
Let $\cal B$ be a certain class of schemes. Assume that for every
infinite sequence $\{X_{s}\in \cal B\}$ there exists an infinite
subsequence $\{ X_{s_{k}} \}$ which is bounded. Then the class $\cal
B$ is bounded.
\end{thm}
\begin{thm}
\label{kinky method for proving DCC}
Let $\cal C$ be an ordered set. Assume that for every
infinite sequence $\{ x_{s}\in \cal C\}$ there exists an infinite
nondecreasing
subsequence $\{ x_{s_{k}} \}$. Then the set $\cal
C$ satisfies the descending chain condition.
\end{thm}
\section{Additional definitions and easy technical results}
\label{Additional definitions and easy technical results}
\begin{defn}
For two $\Bbb R$-divisors we write $D_{1}\ge D_{2}$
(resp.~$D_{1}> D_{2}$) if
$D_{1}-D_{2}$ is effective, i.e.\ the coefficients for all prime
components are nonnegative
(resp. is effective and nonzero).
\end{defn}
\begin{defn}
For two $\Bbb R$-divisors on possibly different normal varieties
of the same dimension we write
$D_{1}\underset{c}{\ge} D_{2}$ (resp.
$D_{1}\underset{c}{>} D_{2}$) if
$$\underset{n\to\infty}{\varliminf}
\frac{h^{0}(nD_{1})-h^{0}(nD_{2})}{n^{\dim}}\ge0$$
(resp. is strictly positive). Here $n$ is assumed to be divisible
enough.
\end{defn}
\begin{defn}
One says that an $\Bbb R$-Cartier divisor $D$ is {\em nef\/} if
$DC\ge0$ for any effective curve $C$.
\end{defn}
\begin{defn}
The {\em Iitaka dimension} or the {\em $D$-dimension} of a Weil divisor
$D$ on a normal variety,
denoted
by $\nu(D)$, is
the number in the formula $h^{0}(nD) \sim n^{\nu(D)}$ as $n\to\infty$,
unless $h^{0}(nD)=0$ for all $n>0$, in which case $\nu(D)$
is defined to be equal to $-\infty$. A divisor with $\nu(D)=\dim X$ is
called {\em big}. The {\em Kodaira dimension} of a variety $X$ is
defined as $\nu(K_{Y})$ where $Y$ is a resolution of singularities of $X$.
\end{defn}
\begin{lem}
\label{easy inequalities}
\begin{enumerate}
\item $D_{1}\ge D_{2}$ implies $D_{1}\underset{c}{\ge} D_{2}$,
\item if $H^{i}(nD_{1})=H^{i}(nD_{2})=0$ for $i>0$, $n\gg0$,
then $D_{1}\underset{c}{\ge} D_{2}$ (resp.
$D_{1}\underset{c}{>} D_{2}$) is equivalent to
$D_{1}^{\dim}\ge D_{2}^{\dim}$ (resp.
$D_{1}^{\dim}>D_{2}^{\dim}$),
\item in particular, (2) is applicable when $D_{1}$ and
$D_{2}$ are ample.
\end{enumerate}
\end{lem}
\begin{pf}
(1) is clear, (2) and (3) trivially follow from the Riemann--Roch
formula and the Serre vanishing theorem.
\end{pf}
\begin{lem}
\label{inequalities}
Let $D_{1}$ be an ample $\Bbb R$-divisor on a normal variety $X$, $f:Y\to
X$ be a birational projective morphism with $Y$ normal,
and write
$$D_{2}=f^{*}(D_{1})+\sum e_{j}E_{j}+\sum f_{i}F_{i}$$
where the divisors $F_{i}$ are exceptional for $f$ and $E_{j}$ are
not.
Then
\begin{enumerate}
\item if $e_{j}\le0$ then $D_{2}\underset{c}{\le}D_{1}$
\item if, moreover, for some index $e_{j_{0}}<0$ or $f_{i_{0}}<0$ then
$D_{2}\underset{c}{<}D_{1}$
\item assume that
$e_{j}\le0$,
$D_{2}$ is {\em nef\/};
then all $f_j\le0$ i.e.\ $D_{2}\le f^{*}(D_{1})$
\item assume that
$e_{j}\le0$,
$D_{2}$ is {\em nef\/}
and that $D_{1}\underset{c}{=}f^{*}(D_{2})$; then
$D_{2}=D_{1}$ i.e.\ all $e_{j}=f_{i}=0$
\end{enumerate}
\end{lem}
\begin{pf}
Most of the statements are elementary, others follow from the
Negativity of Contractions Lemma, see \cite{Shokurov91}, 1.1 or
\cite{FAAT}, 2.19.
\end{pf}
\begin{lem}
\label{only surfaces}
Let $D_{1}$ be a divisor on a nonsingular surface $X$ such that
$H^{i}(nD_{1})=0$ for $i>0,$ $n\gg0$. Let $D_{2}$ be a nonzero
divisor such that $-D_{2}$ is not quasieffective
(i.e.\ there exists an ample divisor $H$ such
that $-D_{2}H<0$). Assume that
$D_{2}\underset{c}{\le}D_{1}$. Then $D_{2}^{2}\le D_{1}^{2}$.
\end{lem}
\begin{pf}
Use the Riemann--Roch theorem for $nD_{1},nD_{2}$
and the fact that $h^{2}(nD_{2})=h^{0}(K-nD_{2})=0$
for $n\gg0$.
\end{pf}
\section{The diagram method}
\label{The diagram method}
\begin{say}
In proving the boundedness results for surfaces with nef $-(K+B)$ the
main part will be to bound rank of the Picard group. A way for
doing this, called the diagram method,
comes from the theory of reflection groups in hyperbolic
spaces. Given a convex polyhedron in a hyperbolic space it is possible
to bound the dimension of this space purely combinatorially, provided
subpolyhedra of the main polyhedron
satisfy certain arithmetic properties. The diagram method was
successfully applied to surfaces in the works of V.V.Nikulin
\cite{Nikulin90c}, \cite{Nikulin89b}, \cite{Nikulin90a},
\cite{Nikulin89a}
and of the author \cite{Alexeev89}, and also to Fano 3-folds in
\cite{Nikulin90b}, \cite{Nikulin93a}.
The vector space in question is $\operatorname{Pic}(X)\otimes \Bbb R$ which is
hyperbolic by the Hodge Index theorem or some linear subspace of it,
the polyhedron is generated by exceptional
curves.
In the next section I shall give one more application of the diagram method.
But first we need a few definitions and facts.
\end{say}
\begin{defn}
An {\em exceptional curve\/} on a surface is an irreducible curve $F$
with $F^{2}<0$. The set of all exceptional curves on the surface $X$
will be denoted by $\operatorname{Exc}(X)$. A {\em $(-n)$-curve} is a smooth rational
curve $F$ with $F^{2}=-n$.
\end{defn}
\begin{say}
To a set of exceptional curves we associate a {\em
weighted graph}. Each curve $F$ corresponds to a vertex of weight
$-F^{2}$ and two vertices $F_{1}$ and $F_{2}$ are connected by an
edge of weight $F_{1}F_{2}$. An edge of weight 1 will be called {\em
simple}. We also assign to every vertex a nonnegative
number,
the arithmetical genus of the corresponding curve. However, in the
situation of interest to us all exceptional curves will have genus zero, as
{}~\ref{genera only zero} shows.
The (possibly infinite) graph corresponding to all exceptional curves
on a surface $X$ will be denoted by $\Gamma(\operatorname{Exc}(X))$.
\end{say}
\begin{defn}
A finite set of exceptional curves $\{ F_{i} \,|\, i=1\ldots{}r \}$
(and the corresponding weighted
graph) is called {\em elliptic\/} (resp.~ {\em parabolic, hyperbolic\/}) if the
matrix $(F_{i_{1}}F_{i_{2}})$ has signature $(0,r)$
(resp.~$(0,r-1)$, $(1,r-1))$.
\end{defn}
\begin{defn}
A finite set of exceptional curves
(and the corresponding weighted
graph) is called {\em Lanner\/} if it is hyperbolic but any proper subset
of it is not.
\end{defn}
\begin{say}
The following theorem will be the most important for our purposes, see
\cite{Nikulin89a}, 3.4--6 for the proof.
\end{say}
\begin{thm}
\label{Nikulin's main theorem}
Let $X$ be a nonsingular surface such that the Iitaka dimension of the
anticanonical divisor is nonnegative. Let us assume that for certain
constants $d$,$c_{1}$,$c_{2}$ the following conditions hold:
\begin{enumerate}
\item the diameter of any Lanner subgraph $\cal L\subset\Gamma(\operatorname{Exc}(X))$
does not exceed $d$;
\item if $\nu(-K)=2$ then for any connected elliptic subgraph
$\cal E \subset \Gamma(\operatorname{Exc}(X))$ with $n$ vertices, the number
of (unordered) pairs of its vertices on distance $\rho$,
where $1\le \rho \le d-1$, does not exceed $c_{1}n$, and the
number of pairs on distance $\rho$, where $d\le \rho \le
2d-1$, does not exceed $c_{2}n$;
\item if $\nu(-K)=1$ then for any $(-1)$-curve $E$ for which
$EP>0$ ($P$ is the positive part of the Zariski
decomposition for $-K$ and won't be used later) and
for any connected elliptic subgraph
$\cal E \subset \Gamma(\operatorname{Exc}(X))$ with $n+1$ vertices
containing $E$, the number
of pairs
of its vertices
different from $E$ and on distance $\rho$,
where $1\le \rho \le d-1$, does not exceed $c_{1}n$, and the
number of pairs on distance $\rho$, where $d\le \rho \le
2d-1$, does not exceed $c_{2}n$;
\item if $\nu(-K)=0$, $-K=\sum_{b_{j}>0}b_{j}B_{j}$
then for any connected elliptic subgraph
$\cal E \subset \Gamma(\operatorname{Exc}(X))$ with $n+m$ vertices,
$m$ of which $E_{1}\ldots{}E_{m}$
correspond to $(-1)$- or $(-2)$-curves different from
$B_{j}$, the number
of pairs of its vertices different from
$E_{1}\ldots{}E_{m}$ and on distance $\rho$,
where $1\le \rho \le d-1$, does not exceed $c_{1}n$, and the
number of pairs on distance $\rho$, where $d\le \rho \le
2d-1$, does not exceed $c_{2}n$.
\end{enumerate}
Then
\begin{enumerate}
\item if $\nu(-K)=2$ then $\operatorname{rk}\operatorname{Pic}(X)\le96(c_{1}+c_{2}/3)+69$;
\item if $\nu(-K)=1$ then $\operatorname{rk}\operatorname{Pic}(X)\le96(c_{1}+c_{2}/3)+70$;
\item if $\nu(-K)=0$ then $\#(B_{j})\le96(c_{1}+c_{2}/3)+68$.
\end{enumerate}
\end{thm}
\begin{say}
We need a few definitions for blowing up and down weighted graphs. They
are merely reformulations on the language of graphs of
usual operations of blowing up points on a nonsingular surface.
\end{say}
\begin{defn}
A weighted graph is said to be {\em minimal} if it does
not contain vertices of weight 1 and arithmetical genus 0.
\end{defn}
\begin{defn}
{\em Blowing up} a vertex $F$ is the operation on a weighted
graph consisting
of adding a new vertex $E$ of weight 1
and of arithmetical genus 0,
connected only with the vertex
$F$ by a simple edge and increasing weight of $F$ by 1.
\end{defn}
\begin{defn}
{\em Blowing up} a simple edge $F_{1}F_{2}$ is the operation on a
weighted graph consisting
of adding a new vertex $E$ of weight 1
and of arithmetical genus 0,
connected only with the vertices
$F_{1}$ and $F_{2}$ by simple edges, removing the edge between them
and increasing weights of $F_{1}$ and $F_{2}$ by 1.
\end{defn}
\begin{say}
Blowing up
can be easily defined in a more general situation but we won't need
it.
\end{say}
\begin{defn}
{\em Blowing down} is the inverse operation to blowing up.
\end{defn}
\begin{defn}
{\em The canonical class} $K=K(\cal \Gamma)$ of a weighted graph $\cal
\Gamma$ is the function on vertices defined by the formula
\begin{displaymath}
KF_{i}=-F_{i}^{2}-2+2p_{a}(F_{i})
\end{displaymath}
\end{defn}
\begin{defn}
{\em The log discrepancies} $f_{i}$ for a finite weighted graph $\cal
\Gamma$ are defined as solutions of the following system of linear
equations (if exist):
\begin{displaymath}
(K+\sum (1-f_{i})F_{i})F_{j}=0 \text{\quad for all }j
\end{displaymath}
Note that this system has a unique solution if the matrix
$(F_{i}F_{j})$ is invertible (for example if the graph $\Gamma$
is elliptic or hyperbolic).
\end{defn}
\begin{defn}
A weighted graph $\Gamma$ is said to be {\em log terminal} if for every
elliptic subgraph $\Gamma' \subset \Gamma$ all log discrepancies of
$\Gamma'$ are positive.
\end{defn}
\begin{defn}
\label{def of condition star for graphs}
We say that a finite graph $\cal \Gamma=\{F_{i}\}$ satisfies {\em the
condition $*(\varepsilon)$} if there exist constants $0\le b_{i}\le
1-\varepsilon<1$ such that
$(K+\sum b_{i}F_{i})F_{j}\le0$ for all vertices $F_{j}$.
\end{defn}
\begin{lem}
\label{condition star}
If $\cal \Gamma$ satisfies $*(\varepsilon)$ then every subgraph $\cal
\Gamma_{1} \subset \Gamma$ and every graph $\cal \Gamma_{2}$ obtained
from $\cal \Gamma$ by blowing down several vertices of weight 1 also
satisfy $*(\varepsilon)$.
\end{lem}
\begin{pf}
Evident.
\end{pf}
\begin{lem}
If an elliptic graph
$\cal \Gamma$ satisfies $*(\varepsilon)$ then for all the log
discrepancies $f_{i}\ge 1-b_{i} \ge \varepsilon$.
\end{lem}
\begin{pf}
This is well known and follows easily from the negative definiteness
of $(F_{i}F_{j})$, see for example \cite{Alexeev92}, 3.1.3.
\end{pf}
\begin{lem}
\label{genera only zero}
If an elliptic graph
$\cal \Gamma$ satisfies $*(\varepsilon)$ then
every vertex $F$ has arithmetical genus 0 and
its
weight does not exceed $2/\varepsilon$.
\end{lem}
\begin{pf} Follows from
\begin{eqnarray*}
-2 \le 2p_{a}(F)-2 = (K+F)F = (K+ (1-\varepsilon)F)F + \varepsilon
F^{2} \le \\
(K+\sum b_{j}B_{j})F + \varepsilon F^{2} \le \varepsilon F^{2} <0
\end{eqnarray*}
\renewcommand{\qed}{}
\qed
\end{pf}
\begin{thm}
\label{minimal elliptic graph}
Every minimal elliptic log terminal
graph is a
tree with at most one fork and of type $A_{n}$, $D_{n}$ or $E_{6,7
\text{ or }8}$. There exists a constant $S_{1}(\varepsilon)$ depending
only on $\varepsilon$ such that $\sum (-F^{2}_{i}-2) \le
S_{1}(\varepsilon)$
if such a graph satisfies $*(\varepsilon)$.
\end{thm}
\begin{pf}
The first part of the statement is well known,
see for example \cite{Alexeev92}. The second part follows
from the explicit description of elliptic graphs with all log
discrepancies $\ge\varepsilon>0$ given in \cite{Alexeev93a}, 3.3.
\end{pf}
\begin{thm}
\label{minimal Lanner graph}
There exist $\le 14(2/\varepsilon)+29$
Lanner graphs with simple edges
such that every Lanner graph
with more than 5 vertices satisfying $*(\varepsilon)$ can be
obtained from one of them by blowing up several vertices and edges.
Each of these graphs is a tree or a cycle or a cycle and one more
vertex.
Each of these graphs has only simple edges and
every vertex has at most 3 neighbors.
\end{thm}
\begin{pf}
It follows from the theorems of V.V.Nikulin, \cite{Nikulin89a}, 4.4.18,19,21
which are valid for arbitrary log terminal Lanner graphs.
Some of the graphs in \cite{Nikulin89a} have a vertex of arbitrary
positive weight $b$ but in our situation $b\le 2/\varepsilon$ by
{}~\ref{condition star} and~\ref{genera only zero}.
\end{pf}
\begin{say}
A typical example of a Lanner graph in the above statement is the
chain containing three vertices of weights 1, 1 and $b\ge 1$. We won't
need explicit description of these graphs, just knowing that for every
$\varepsilon$ there are finitely many of them will be sufficient.
\end{say}
\section[Boundedness for surfaces with nef $-(K+B)$
]{Boundedness for surfaces with nef $\bold -(K+B)$}
\label{Boundedness for surfaces with nef -(K+B)}
\begin{lem}
\label{graphs on good surfaces satisfy the condition star}
Let $X$ be a nonsingular surface and assume that $-(K+\sum
b_{j}B_{j})$ is nef, where $0\le b_{j} \le 1-\varepsilon <1$.
Then
every finite subgraph
$\Gamma\subset \Gamma(\operatorname{Exc}(X))$ satisfies the condition $*(\varepsilon)$.
\end{lem}
\begin{pf}
Evident.
\end{pf}
\begin{lem}
\label{structure when -K ge0}
Let $X$ be a nonsingular surface and assume that $-(K+\sum
b_{j}B_{j})$ is nef, where $0\le b_{j} \le 1-\varepsilon <1$.
Then one of the following is true:
\begin{enumerate}
\item all $b_{j}=0$ and $K$ is numerically trivial
\item $X$ is a rational surface, obtained by blowing up several points
from $\Bbb P^{2}$ or $\Bbb F_{n}$ with $n\le 2/\varepsilon$
\item $X$ is an elliptic ruled surface without exceptional
curves, i.e.\
a projectivization of a rank 2 locally free
sheaf $\cal E$ on an elliptic curve $C$ and $\cal E$ is isomorphic
to $\cal O\oplus \cal F$, $\cal F\in \operatorname{Pic}^{0}(C)$
or to one of the only two, up to
tensoring with an invertible sheaf,
nonsplilttable rank 2 bundles on $C$
\end{enumerate}
\end{lem}
\begin{pf}
This lemma is practically proved in \cite{Nikulin89a}, 4.2.1
under weaker assumptions on $b_{j}$ and in arbitrary
characteristic. We only need to see that in the case (2) one has $n\le
2/\varepsilon$ by~\ref{genera only zero}.
\end{pf}
\begin{thm}
\label{thm A1}
Let $X$ be a nonsingular surface and assume that $-(K+\sum
b_{j}B_{j})$ is nef, and $0\le b_{j} \le 1-\varepsilon <1$. Then
there exists a constant $A_{1}(\varepsilon)$ which depends only on
$\varepsilon$ such that
\begin{displaymath}
\operatorname{rk}\operatorname{Pic}(X)\le A_{1}(\varepsilon)
\end{displaymath}
\end{thm}
\begin{pf}
In order to prove this theorem we have to check the conditions (1) and
(2--4) for graphs satisfying $*(\varepsilon)$. Among them, (1) is the
hardest.
Given an arbitrary graph satisfying $*(\varepsilon)$ and containing a
vertex of weight 1, we can blow this vertex down and the new graph will
satisfy the same condition by~\ref{condition star}. After blowing down
several vertices of an elliptic graph we get a minimal graph
described in~\ref{minimal elliptic graph}.
It is well known that for an elliptic graph
blowing down vertices of weights 1 in any order yields the same graph,
so we can
backtrack the situation by blowing up edges and vertices on the minimal
elliptic graph {\em in arbitrary order}. All intermediate graphs
again should satisfy $*(\varepsilon)$.
For a Lanner graph, removing any two vertices gives an elliptic graph.
So, basically, we can do the same thing. Contracting several vertices
we obtain one of the finitely many graphs of
{}~\ref{minimal Lanner graph}, and we can backtrack
the situation blowing up,
in any order,
edges and vertices that do not
affect two
arbitrarily chosen vertices.
Again, all intermediate graphs should satisfy
$*(\varepsilon)$, moreover, they all should be Lanner.
Using these considerations, the proof easily follows from the lemmas
{}~\ref{graphs: blow up vertices}, \ref{graphs: few neighbors},
\ref{graphs: blow up edges}, \ref{graphs: blow up till get E9} below.
\begin{lem}
\label{graphs: blow up vertices}
Fix a weighted graph $\Gamma$ and pick one of its vertices $F$.
Blow
it up to get the vertex $E_{1}$, then blow up $E_{1}$ to get $E_{2}$,
and so on. Call the intermediate graphs
$\cal \Gamma_{1},\cal \Gamma_{2}\ldots{}$
Then for $k\gg0$ the graph $\cal \Gamma_{k}$ is not Lanner.
\end{lem}
\begin{pf}
It is easy to see, using only elementary linear algebra, that there
exists a fractional linear function $f(k)$
(which is an appropriately normalized determinant)
so that the condition for
the graph $\cal \Gamma-E_{k}$ not to be hyperbolic
is equivalent to $f(k)\ge0$. But then
the condition for $\Gamma$ to be hyperbolic is equivalent to
$f(\infty)<0$ and hence $f(k)<0$ for $k\gg0$.
\end{pf}
\begin{lem}
\label{graphs: few neighbors}
In any satisfying $*(\varepsilon)$ graph
which is elliptic or is Lanner with more than 5 vertices,
each
vertex has at most $2/\varepsilon-2$ neighbors.
\end{lem}
\begin{pf}
Indeed, in each of the initial graphs in~\ref{minimal elliptic graph}
and~\ref{minimal Lanner graph} every vertex
has at most 3 neighbors,
so to produce a vertex with $d$ neighbors one has to
blow up some vertex $\ge d-3$ times, which means that the weight of
this vertex will be $\ge d-2$. Now use~\ref{genera only zero} .
\end{pf}
\begin{lem}
\label{graphs: blow up edges}
Fix an elliptic weighted graph $\Gamma$ consisting of two vertices
connected by a
simple edge.
Then among all graphs $\Gamma'$
obtained from $\Gamma$ by blowing up only edges,
there exist $\le S_{2}(\varepsilon)$ graphs satisfying $*(\varepsilon)$,
where the last function $S_{2}(\varepsilon)$ depends only on $\varepsilon$.
\end{lem}
\begin{pf}
For any vertex $F$ in any such graph $\Gamma'$ let us introduce {\em
the height} $h(F)$ as the minimal number of blowups needed to obtain
this vertex from $\Gamma$. Consider this minimal sequence of blowups.
Note that it is unique.
On each step we get two chains of vertices of weight $\ge2$ and a
single vertex of weight 1 between them. After every blowup the sum
$\sum (-F^{2}_{i}-2)$ increases by 1. Therefore by~\ref{minimal
elliptic graph}
$h(F)\le S_{1}(\varepsilon)/2$ and there are $\le 2^{S_{1}/2-1}$ such
vertices. Finally,
there exist only finitely many graphs
$\Gamma'$ that contain only vertices of bounded height.
\end{pf}
\begin{lem}
\label{graphs: blow up till get E9}
Fix an elliptic weighted graph $\Gamma$ consisting of two vertices
connected by a
simple edge. Blow up the edge to get
the vertex $E_{1}$, then blow up $E_{1}$ to get $E_{2}$,
and so on. Call the intermediate graphs
$\cal \Gamma_{1},\cal \Gamma_{2}\ldots{}$.
Then for $k>5$ the graph $\cal \Gamma_{k}$ is not log terminal, i.e.\ it
does not satisfy
$*(\varepsilon)$ for any $\varepsilon>0$.
\end{lem}
\begin{pf}
Indeed, $\cal \Gamma_{6}-E_{6}$ has type $E_{9}$ or worse.
\end{pf}
{\em End of the proof of~\ref{thm A1}.} The conditions (2--4) of
{}~\ref{Nikulin's main theorem} follow
easily from~\ref{graphs: few neighbors}.
Indeed, the number of pairs in (2--4) on distance
$\rho$ is bounded by $n/2(2/\varepsilon-2)^{\rho}$.
To prove that the condition (1) of
{}~\ref{Nikulin's main theorem} is satisfied, consider any of
the finitely many Lanner graphs of
{}~\ref{minimal Lanner graph}. We can assume that this graph has at
least 6 vertices. We shall prove that
there is a bound, in terms of $\varepsilon$, on the number of possible
blowups that can be done preserving the condition $*(\varepsilon)$ and
the property of the whole graph to remain Lanner.
As was mentioned before, fixing any two vertices, the order of
blowups not affecting these vertices is unimportant.
First, by
{}~\ref{graphs: blow up vertices},
vertices can be blown up only finitely many times
(again, there is a bound in terms of $\varepsilon$). Then an edge
between some two vertices
should be blown up. By~\ref{graphs: blow up edges},
different edges between these two vertices
can be blown up only finitely many times. Then some of the vertices
may acquire more neighbors, but again their number is limited by
{}~\ref{graphs: few neighbors}.
After that one of the new branches may grow longer, but its length is
bounded by~\ref{graphs: blow up till get E9}.
Then, again there may be a few edges blown up,
and then a few vertices may acquire some new neighbors. Finally, on
this stage if some of the branches grow longer, a minimal elliptic
intermediate
subgraph should appear which has more than one fork. But this is
impossible by~\ref{minimal elliptic graph}.
Therefore, all conditions (1) and (2--4) of
{}~\ref{Nikulin's main theorem} are satisfied with
constants
$d(\varepsilon)$, $c_{1}(\varepsilon)$, $c_{2}(\varepsilon)$ that depend only
on $\varepsilon$. In the cases $\nu(-K)=2\text{ or }1$ we immediately get
the upper bound on the Picard number. In the case $\nu(-K)=0$,
$-K=\sum_{b_{j}>0} b_{j}B_{j}$ one has
\begin{displaymath}
K^{2}\ge -(\#B_{j})(2/\varepsilon)
\end{displaymath}
by~\ref{genera only zero}
and if $X$ is rational
we have $\operatorname{rk}\operatorname{Pic}(X)=10-K^{2}$ by Noether's formula. In the remaining two
cases of
{}~\ref{structure when -K ge0} one certainly has $\operatorname{rk}\operatorname{Pic}(X)\le20$.
\end{pf}
\begin{thm}
\label{bound for nonsingular when -K nef}
Fix $\varepsilon>0$. Consider all nonsingular surfaces
$X$ with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$0\le b_{j}\le 1-\varepsilon <1$ and $-(K+B)$ nef excluding only those
for which at the same time $K_{X}$ is numerically trivial and $B$ is
zero.
Then the class $\{X\}$ is bounded.
\end{thm}
\begin{pf}
Indeed, in the case (3) of
{}~\ref{structure when -K ge0}
there are only three deformation types.
All other surfaces, except $\Bbb P^{2}$,
are obtained from $\Bbb F_{n}$, $n\le 2/\varepsilon$
by $\le A_{1}(\varepsilon)-2$ blowups.
Now use~\ref{boundedness by blowingup}.
\end{pf}
\begin{thm}
\label{bound for general when -K nef}
Fix $\varepsilon>0$.
Consider all projective surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR $\varepsilon$-log terminal and $-(K_{X}+B)$ is
nef excluding only those for which at the same time $K_{X}$
is numerically trivial, $B$ is zero and $X$ has at worst Du Val
singularities.
Then the class $\{X\}$ is bounded.
\end{thm}
\begin{pf}
By~\ref{bound for nonsingular when -K nef} we already know that the
class of minimal desingularizations $\{Y\}$ of surfaces $\{X\}$ is
bounded. This, however, does not yet guarantee the boundedness of the
class $\{X\}$. To prove it we shall use a ``sandwich'' argument: we
shall prove that there exist two birational morphisms $Y\to X\to Z$
with $\{Z\}$ also bounded.
By~\ref{structure when -K ge0}
we can assume that $X$ is rational, moreover, by
{}~\ref{thm A1} the rank
of the Picard group of the minimal resolution of $X$ is effectively bounded.
We consider the following cases:
\begin{case}$B$ is nonempty or $K$ is not numerically trivial.
We want to show that we can find a contraction $X\to Z$ so that
$-K_{Z}$ is ample or is relatively ample with respect to some $\Bbb
P^{1}$-fibration.
First, assume that there exists an $\Bbb R$-divisor $D$ such that
$K+B+D$ is
numerically trivial
and satisfies the same condition
$*(\varepsilon)$. According to our conditions, $B+D$ is nonempty.
If we decrease one of the coefficients in $K+B+D$, the result will be
not nef. Therefore, according to the Minimal Model Program there exists an
extremal ray and the corresponding contraction.
If $E$ is a component of $B+D$ with $E^{2}\le 0$
then $K+B+D-eE$ for $0<e\ll1$ has a nonnegative
intersection with $E$. Hence, there always exists an extremal contraction
which does not contract $E$,
so the image of $B+D$ will be again nonempty.
Performing several such contractions we
arrive either at a surface $Z$ with $\operatorname{rk}\operatorname{Pic}(Z)=1$ and ample $-K_{Z}$
or at a surface $Z$ with $\operatorname{rk}\operatorname{Pic}(Z)=2$ which has a $\Bbb
P^{1}$-fibration. Because the rank of the Picard group of the minimal
resolution is bounded and the self-intersection numbers of exceptional
curves are bounded from below,
there are only finitely many types of singularities that $Z$ can have and
all of them can be effectively described in terms of $\varepsilon$.
Therefore, in the case of $\operatorname{rk}\operatorname{Pic}(Z)=1$
a fixed multiple of $-K_{Z}$ is an ample Cartier divisor and
$K_{Z}^{2}$ is effectively bounded from above.
In the case of a $\Bbb P^{1}$-fibration,
$-K_{Z}+(\lceil 2/\varepsilon \rceil -1)\times fiber$ is ample, and
the same argument applies. The class $\{Z\}$ is bounded by
{}~\ref{main method for proving boundedness}.
Now, instead of proving that such a divisor $D$ as above exists, consider an
arbitrary ample divisor $A$ on $X$ and observe that we can as
well use $D'/N$ in the place of $D$, where $D'$ is a general element
of a very ample linear system $|N(\delta A-K-B)|$ for $N\gg0$,
$0<\delta\ll \varepsilon$.
The surface $Y$ is obtained from $Z$ by blowing up a certain closed
subscheme.
Again, from the boundedness of the Picard number of the resolution of
$Y$ and the squares of exceptional curves, we see that the length of
this subscheme is effectively bounded in terms of $\varepsilon$.
Now use~\ref{boundedness by blowingup}.
\end{case}
\begin{case}$B$ is empty, $K$ is numerically trivial but $X$ has
worse than Du Val singularities.
Consider
a partial resolution $f:Y\to X$ dominated by
the minimal resolution of singularities of $X$ such that,
letting $K_{Y}+B^{Y}=f^{*}(K_{X})$, $B^{Y}$ has positive coefficients
in all exceptional divisors of $f$.
By the previous case,
there exists a polarization $H$ on $Y$ with $H^{2}$ and $K_{Y}H$ bounded.
Also, there exists
a lower bound, in terms of $\varepsilon$,
for $b_{j}^{Y}$. Since for any ample divisor $H$,
\begin{displaymath}
\sum b_{j}^{Y}B_{j}H = -K_{Y}H
\end{displaymath}
we can bound all $B_{j}H$ and, therefore,
for $H'=f_{*}(H)$
we can bound $H^{\prime 2}$, $H'K_{X}$ and an integer $N$ such that $NH$ is
Cartier. Then apply~\ref{main method for proving boundedness} again.
\end{case}
\renewcommand{\qed}{}
\end{pf}
\begin{cor}
\label{bound for log terminal when -K nef}
The class of projective complex surfaces
$X$ with $K$ numerically trivial and log terminal and
singularities worse than only Du Val, is bounded.
\end{cor}
\begin{pf}
Indeed, by \cite{Blache92}, Theorem C,
$GK_{X}$ is Cartier for some $G\in\{ 1,2\ldots{}21 \}$,
so every such surface is $1/21$-log terminal.
\end{pf}
\begin{thm}
\label{thm A2 for general}
Fix $\varepsilon>0$. Then there exists a constant $A_{2}(\varepsilon)$
such that for any projective surface
$X$ with $-(K+B)$ nef and $K+B$
MR $\varepsilon$-log terminal,
\begin{displaymath}
\sum b_{j}\le A_{2}(\varepsilon)
\end{displaymath}
\end{thm}
\begin{pf}
Indeed, for an ample Cartier divisor $H$
\begin{displaymath}
\sum b_{j} \le \sum b_{j}B_{j}H \le -KH
\end{displaymath}
and on each surface $X$ as above we can find $H$ with a bounded $-KH$.
\end{pf}
\begin{say}
The following theorem, which applies only to the case when $B$ is empty
and $-K$ is ample, but which in this case is stronger than ours,
belongs to V.V.Nikulin.
\end{say}
\begin{thm}[Nikulin \cite{Nikulin89a} 4.7.2]
Fix $N>0$. Then the class of surfaces with ample $-K$,
log terminal $K$ and singularities of multiplicities $\le N$ is
bounded.
\end{thm}
\section[Boundedness for surfaces with big and nef ${K+B}$
]{Boundedness for surfaces with big and nef $K+B$}
\label{Boundedness for surfaces with big and nef K+B}
\begin{lem}
\label{decrease one coefficient}
Assume that on a surface $X$, $K+B=K+b_{0}B_{0}+\sum_{j>0} b_{j}B_{j}$
is big and MR log canonical.
Then one of the following is true:
\begin{enumerate}
\item $K+\sum_{j>0}b_{j}B_{j}$ is big;
\item there exists $0\le b'_{0}<b_{0}$ such that
$K+xB_{0}+\sum_{j>0}b_{j}B_{j}$ is big iff $x>b'_{0}$,
and there exists a morphism
$f:X\to X'$ such that
the $\Bbb R$-divisor
$D=f_{*}(K+b'_{0}B_{0}+\sum_{j>0}b_{j}B_{j})$
on $X'$ is nef but not numerically trivial, and $D^{2}=0$.
Moreover, if $b_{0}'>0$ then the
linear system $|ND|$, $N\gg0$ and divisible, defines a
$\Bbb P^{1}$-fibration $\pi:X'\to C$ to a nonsingular
curve $C$;
\item there exists $0\le b'_{0}<b_{0}$ such that
$K+xB_{0}+\sum_{j>0}b_{j}B_{j}$ is big iff $x>b'_{0}$,
and there exists a morphism
$f:X\to X'$ such that the $\Bbb R$-divisor
$D=f_{*}(K+b'_{0}B_{0}+\sum_{j>0}b_{j}B_{j})$
is numerically trivial.
\end{enumerate}
\end{lem}
\begin{pf}
It is well known that a divisor on a projective variety is
big if and only if it is a
sum of an ample and an effective
divisors. Hence, the property of being big is an
open property. Similarly,
the property of $(X,K+B)$ not to have a minimal model with the image
of $K+B$ nef is also an open property,
modulo the Cone and Contraction theorems of
Minimal Model Program. Indeed, it is equivalent to existence of a
covering family $\{C_{t}\}$ with $(K+B)C_{t}<0$.
Therefore, in the cases (2) and (3)
$K+b'_{0}B_{0}+\sum_{j>0}b_{j}B_{j}$ has a minimal model $X'$
with nef, but not big $D$.
If $f_{*}(B_{0})D<0$ then the Minimal Model Program applied to
$K+D-\delta B_{0}$, $0<\delta\ll1$, gives a $\Bbb P^{1}$-fibration.
In the case (2) with $b_{0}'=0$,
if the characteristic of the ground field is 0,
then some multiple of $D$ is base point free and defines an elliptic
fibration. We won't need this fact, however. In positive characteristic the
same is true but the proof requires using the classification theory,
which we would like to avoid.
\end{pf}
\begin{say}
The constants $A_{1}(\varepsilon)$, $A_{2}(\varepsilon)$ in the
following theorem were defined in
{}~\ref{thm A1},~\ref{thm A2 for general}.
\end{say}
\begin{thm}
\label{throw out almost all coefficients}
Assume that on a surface $X$, $K+B=K+\sum_{j\in J} b_{j}B_{j}$
is big and
MR $\varepsilon$-log terminal.
Then there exists a subset
of indices $J'=J'_{1}\cup J'_{2} \subset J$
such that $K+B=K+\sum_{j\in J'} b_{j}B_{j}$
is big and
\begin{displaymath}
|J'_{1}| \le A_{1}(\varepsilon) +1 \text{, }
\sum_{j\in J'_{2}}b_{j} \le A_{2}(\varepsilon)
\end{displaymath}
\end{thm}
\begin{pf}
Decrease the coefficient $b_{0}$. We have 3
cases as in~\ref{decrease one coefficient}.
\setcounter{case}{0}
\begin{case}
Pick another coefficient and continue.
\end{case}
\setcounter{case}{2}
\begin{case}
Find the maximal resolution $X''$ of $X'$ dominated by $X$:
\begin{displaymath}
X \overset g\to X'' \overset h \to X'
\end{displaymath}
such that
\begin{displaymath}
K_{X''}+B''= h^{*}(f_{*}(K+b'_{0}B_{0}+\sum_{j>0}b_{j}B_{j}))
\end{displaymath}
has all
nonnegative coefficients. Let $J'\subset J$ be the set of indices for
the nonnegative $b_{j}''$ in the latter divisor, including zeros. Then
$K_{X}+\sum_{j\in J'}b_{j}B_{j}$ is big since it contains
\begin{displaymath}
c(K+B)+(1-c)f^{*}(f_{*}(K+b'_{0}B_{0}+\sum_{j>0}b_{j}B_{j}))\equiv
\text{ big + nef}
\end{displaymath}
for $0<c\ll1$. Part of the divisors $B_{j}$, $j\in J'$, lie in the
exceptional set of $X'' \to X'$ so their number is bounded by
$\operatorname{rk}\operatorname{Pic}(X'')-1$,
which is less than or equal to $A_{1}(\varepsilon)-1$ by~\ref{thm A1}.
The sum of the coefficients for the others is bounded by
$A_{2}(\varepsilon)$ by~\ref{thm A2 for general}. Also, we should not
forget the divisor
$B_{0}$ itself.
\end{case}
\setcounter{case}{1}
\begin{case}
Let $b'_{0}$ be as in
{}~\ref{decrease one coefficient}, (2).
To begin the proof, let us first assume that
the linear system $|ND|$, $N\gg0$ is base point free and defines a morphism
with connected fibers
$\pi\circ f:X\to C$ to a nonsingular curve $C$.
For every fiber
$F_{k}=\sum F_{kl}$ of $\pi\circ f$ consider the maximal rational number
$f_{k}$ such that
\begin{displaymath}
K+b'_{0}B_{0}- \sum f_{k}F_{k}
\end{displaymath}
has at least one nonnegative coefficient in $F_{kl}$, say, $F_{k0}$.
To present the idea of the proof more clearly, assume than we can contract
for every $k$ all components of $F_{k}$ other than $F_{k0}$
to obtain a morphism
\begin{displaymath}
f''':X\to X'''
\end{displaymath}
Note that the coefficients of components of
\begin{displaymath}
K_{X'''}+B'''= f'''_{*}(K+b'_{0}B_{0} -\sum f_{k}F_{k})
\end{displaymath}
are all nonnegative.
\begin{subcase}$K_{X'''}+B'''$ is nef but not numerically trivial.
Then $K_{X}+B'''+ (b_{0}-b'_{0})B_{0}$ is big,
so we can use $J'$ corresponding to $B'''$
plus $B_{0}$ itself, i.e.\ to all $B_{j}$ not
in fibers of $\pi\circ f$. Evidently,
\begin{displaymath}
\sum_{j\in J'-0}b_{j} \le 2 < A_{2}(\varepsilon)
\end{displaymath}
\end{subcase}
\begin{subcase} The opposite to the subcase 1.
At the same time $K_{X'''}+B'''+\sum b_{k0}B_{k0}$ is nef but not big.
Hence, there
exist several $F_{k0}$, say, $F_{00},F_{10}\ldots{}F_{s0}$ such that
\begin{displaymath}
K_{X'''}+B'''+b'_{00}F_{00}+ \sum_{i=1}^{s} b_{i0}B_{i0}
\end{displaymath}
is numerically trivial.
Find the maximal partial resolution $X''$ of $X'''$ dominated by $X$:
\begin{displaymath}
X \overset g\to X'' \overset h \to X'''
\end{displaymath}
such that
\begin{displaymath}
K_{X''}+B''= h^{*}(K_{X'''}+B''' +b'_{00}F_{00}+\sum_{i=1}^{s}b_{i0}B_{i0}
\end{displaymath}
has all nonnegative coefficients.
Then choose $J'$ as in the case 3. Again, we can divide $J'$ into
two parts, $J'_{1}$ and $J'_{2}$ and count indices as above. We
shouldn't forget one
more divisor, $F_{00}$, to arrive at the final estimate.
\end{subcase}
In this proof we assumed the existence of a fibration $\pi\circ f:X\to
C$. If there is a component $B_{k}$ of $b_{0}'B_{0}+\sum_{j>0}b_{j}B_{j}$
that intersects $D$ positively (i.e.\ ``horizontal'')
then applying the Cone and the
Contraction theorems to $K+b_{0}'B_{0}+\sum_{j>0}b_{j}B_{j}-\epsilon
B_{k}$ for $0<\epsilon\ll1$ several times we obtain such a $(\Bbb
P^{1})$-fibration.
The case when no such component $B_{k}$ exists corresponds to an elliptic
fibration. We could still prove that a multiple of $D$ is
base point free, i.e.\ the abundance theorem,
(cf.~\cite{TsunodaMiyanishi83}), but this would involve some
classification theory. Instead, we can use the same argument as above
with not actual fibers but fibers in the numerical sense. Consider
a connected component of $\operatorname{supp}(b_{0}'B_{0}+\sum_{j>0}b_{j}B_{j})$.
Then by the Hodge Index theorem the corresponding graph is elliptic or
parabolic.
A parabolic graph corresponds to a fiber, an elliptic graph --
to a part of a fiber, the
case when several curves of the fiber have coefficients 0.
Then argue as above.
As for the existence of a contraction of components of $F_{k}$ other
than $F_{k0}$,
note that all we need in this
proof is the contraction $g:X\to X''$ and it can be
constructed without obtaining $X'''$ first. Now, $g:X\to X''$ exists
because the corresponding
graph is log
terminal, cf.~ section~\ref{MMP in dimension 2}.
\end{case}
\renewcommand{\qed}{}
\end{pf}
\begin{cor}
\label{cor to throw out almost all coefficients}
\begin{displaymath}
|J'| \le A_{1}(\varepsilon)+A_{2}(\varepsilon)/\min(b_{j}) +1
\end{displaymath}
\end{cor}
\begin{thm}
\label{can decrease all by delta}
Assume that on a surface $X$, $K+B=K+\sum b_{j}B_{j}$
is big and MR log canonical.
Further assume that $b_{j}$ belong to a D.C.C. set $\cal C$. Then there
exists a constant
$\delta(\cal C)>0$ depending only on the set $\cal C$ such that
$K+\sum (b_{j}-\delta)B_{j}$ is big.
\end{thm}
\begin{pf}
Assume the opposite. Then, similarly to the proof of
{}~\ref{decrease one coefficient}, there exists an
infinite sequence $\delta_{n}\to0$ and a sequence of surfaces $X'_{n}$
such that the direct image of
$K+\sum (b_{j}-\delta_{n})B_{j}$,
which we will denote by $K+B'_{n}$,
is numerically trivial on a
general fiber of a $\Bbb P^{1}$-fibration or simply is numerically
trivial. In the former case for some nonnegative integers $k_{j}$ we get
\begin{displaymath}
\sum k_{j}(b_{j}-\delta_{n})=2
\end{displaymath}
which is trivially impossible. In the latter case we get a
contradiction as follows. If $B'_{n}$ is
not empty then $K+B'_{n}-\alpha B_{1}$ is not nef and the corresponding
extremal ray does not contain $B_{1}$. Therefore, after several
contractions we can assume that $\operatorname{rk}\operatorname{Pic} X'_{n}=1$. Then the number
of irreducible components in $B'_{n}$ is less than $3/\min(\cal C)+1$
and the coefficients of $B'_{n}$ give an infinite
strictly increasing sequence of chains $\{b_{n,k}\}$. This is impossible by
\cite{Alexeev93a}, 5.3. Analyzing the proof of 5.3 shows that the
condition for $K+B'_{n}$ to be log canonical can be weakened to MR log
canonical.
\end{pf}
\begin{thm}
\label{main diagram}
Fix $C>0$ and a D.C.C. set $\cal C$.
Then there exist a bounded class of
surfaces with divisors $(Z,D)$ such that for every surface $X$ with
$K+B$ nef, big, MR log canonical, with $b_{j}\in \cal C$ and
$(K+B)^{2}\le C$
there exists a diagram
\begin{displaymath}
\begin{CD}
Y @>g>> Z\\
@VfVV\\
X
\end{CD}
\end{displaymath}
in which
\begin{enumerate}
\item $Y$ is the minimal desingularization of $X$,
\item defining $K_{Y}+B^{Y}=f^{*}(K+B)$,
$D=g(\operatorname{supp} B^{Y}\cup \operatorname{Exc}(f))$ where $\operatorname{Exc}(f)$ is the union of
exceptional divisors of $f$.
\end{enumerate}
\end{thm}
\begin{pf}
Start with a divisor $K_{Y}+f^{-1}B+\sum F_{i}$ on $Y$.
By~\ref{can decrease all by delta} we can decrease the coefficients of
$B^{Y}$ to obtain a divisor $B'$ with the following properties:
\begin{enumerate}
\item $K_{Y}+B'$ is big,
\item coefficients of $B'$ belong to a {\em finite} set
$\cal C'$ of {\em rational} numbers that depends only on
the original set $\cal C$,
\item all the coefficients of $B'$ are less than
$1-\varepsilon<1$, i.e.\ $K_{Y}+B'$ is MR $\varepsilon$-log
terminal. Here $\varepsilon$ depends again only on $\cal C$.
\end{enumerate}
Next use~\ref{throw out almost all coefficients} to obtain a big
divisor $K_{Y}+B''$. The coefficients of $B''$ belong to the same set
$\cal C'$ and, moreover, the number of components of $B''$ is bounded
in terms of $\cal C$.
Apply the log Minimal Model Program in a slightly more general than
usual form
described in section~\ref{MMP in dimension 2} (because $K_{Y}+B''$ is
not necessarily log canonical)
to obtain a log canonical model $g:Y\to Z$
of $K_{Y}+B''$. Then $H=g(K_{Y}+B'')$ is ample and gives a $\Bbb
Q$-polarization on $Z$.
{}From the classification of surface log terminal singularities
(cf. \cite{Alexeev93a} 3.3)
it follows
that $Z$ can have only finitely many types of singularities.
Indeed, for each of these singularities number of exceptional curves
on the minimal
resolution is bounded, and all log discrepancies are
$\ge\varepsilon>0$,
so the corresponding weights are less than $2/\varepsilon$.
Therefore
there exists an integer $N(\cal C)$ such that $NH$ is a polarization
of $Z$. Note that by construction $H\underset{c}\le K+B$,
hence
\begin{displaymath}
H^{2}\le (K+B)^{2} \le C
\end{displaymath}
by~\ref{easy inequalities}.
Also, $HK\le H^{2}$ and
\begin{displaymath}
p_{a}(NH)=1/2(N^{2}H^{2}+NHK)\ge0.
\end{displaymath}
Hence, there are only finitely many possible values for $H^{2}$ and $HK$.
Now we use~\ref{main method for proving boundedness} to
conclude that the class of surfaces $\{Z\}$ is bounded. To prove that
the class $\{(Z,D)\}$ is bounded we also
have to show that $DH=g(\operatorname{supp} B\cup \operatorname{Exc}(f))H$ is bounded.
This is clear for the components of $g(\operatorname{supp} B'')$. So let us
consider the rest of $g(\operatorname{supp} B\cup \operatorname{Exc}(f))$.
First, consider divisors $D_{k}$ on $Y$
with $g(D_{k})\ne\text{point}$ which are not
$f$-exceptional $(-2)$-curves.
Their coefficients $d_{k}$ in $K_{Y}+B^{Y}$ are bounded from below by
$\min(1/3,\cal C)$. We have $g^{*}(H)\le K_{Y}+{B^{Y}}$ by
{}~\ref{inequalities}, so we
get the desired bound from
\begin{eqnarray*}
(K_{Y}+B^{Y}-g^{*}(H))g^{*}(H) \le
(K_{Y}+B^{Y})g^{*}(H) \le
(K_{Y}+B^{Y})^{2} \le C
\end{eqnarray*}
For $f$-exceptional (-2)-curves one has
\begin{displaymath}
E=g^{*}(H)+1/2\sum(Hg(D_{k}))D_{k}\underset{c}{\le}K+B^{Y}
\end{displaymath}
by~\ref{inequalities} because we add to $g^{*}(H)$
only $f$-exceptional divisors and
\begin{displaymath}
E^{2}\ge H^{2}+1/2\sum(Hg(D_{k}))^{2}.
\end{displaymath}
On the other hand,
by lemma~\ref{only surfaces}
\begin{displaymath}
E^{2}\le (K_{Y}+B^{Y})^{2}\le C.
\end{displaymath}
This concludes the proof.
\end{pf}
\begin{thm}
\label{bound for varepsilon-terminal, K big, nef}
Fix $\varepsilon>0$, a constant $C$ and a D.C.C. set $\cal C$.
Consider all surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR $\varepsilon$-log terminal, $K_{X}+B$ is
big and nef, $b_{j}\in \cal C$ and $(K+B)^{2}\le C$.
Then the class $\{(X,\operatorname{supp} B)\}$ is bounded
\end{thm}
\begin{pf}
Consider a diagram as in~\ref{main diagram} above.
Because the family $(Z,D)$ is bounded,
changing $Z$ and $D$ we can assume
that $Z$ is nonsingular and that the only singularities of $D$
in the exceptional set for $Y\to Z$ are nodes.
But then $Y$ is obtained from $Z$ by several
blowups at these nodes of $D$
and their number depends only on $\varepsilon$. Indeed,
since $K_{Y}+B^{Y}$ is nef, when
blowing up the point of intersection of two curves with coefficients
$b_{1}$ and $b_{2}$, the new coefficient should be $\le
b_{1}+b_{2}-1$. On the other hand, all coefficients in $K_{Y}+B^{Y}$
are nonnegative. Since every $b_{k}\le1-\varepsilon<1$, only finitely
many such blowups can be done.
By~\ref{boundedness by blowingup}
we conclude that the family $(Y,\operatorname{supp} B\cup \operatorname{Exc}(f))$ is bounded.
The surfaces $Y$ come with a polarization, call it $H_{Y}$ with
bounded $H_{Y}^{2}$, $H_{Y}K_{Y}$, $H_{Y}(\operatorname{supp} B\cup \operatorname{Exc}(f))$.
Since there are only finitely many configurations
for the exceptional curves $\operatorname{Exc}(f)$ and they are all log terminal,
hence rational, we can construct a polarization $H_{X}$ on $X$ with
bounded $H_{Z}^{2}$, $H_{Z}K_{Z}$, $H_{Z}(\operatorname{supp} B)$.
At this point, we can use~\ref{main method for proving boundedness}
one more time.
\end{pf}
\section[Descending Chain Condition]{Descending Chain Condition}
\begin{say}
The aim of this section is to prove the following
\end{say}
\begin{thm}
\label{DCC for semi-log canonical}
Fix a D.C.C. set $\cal C$.
Consider all surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR semi-log canonical, $K_{X}+B$ is
ample and $b_{j}\in \cal C$.
Then the set $\{ (K_{X}+B)^{2} \}$ is a D.C.C. set.
\end{thm}
\begin{say}
The theorem will be proved in several steps, under weaker and weaker
conditions.
\end{say}
\begin{thm}
\label{DCC for Kawamata log terminal}
Theorem~\ref{DCC for semi-log canonical} holds for $K+B$
MR Kawamata log terminal.
\end{thm}
\begin{pf}
We can certainly assume that there exists a
constant $C$ such that $(K+B)^{2}\le C$. By
{}~\ref{main diagram}, we know
that there exists a diagram
\begin{displaymath}
\begin{CD}
Y @>g>> Z\\
@VfVV\\
X
\end{CD}
\end{displaymath}
with $Z$ and $D=\operatorname{supp} B\cup \operatorname{Exc}(f)$ bounded.
Assume first that $Z$ and $D$ are actually fixed. Then the proof immediately
follows from~\ref{kinky method for proving DCC} and the following
theorem, since for surfaces $H^{2} \le (g^{*}g_{*}H)^{2}$.
\begin{thm}
\label{thm main trick with sequences}
Let $\{Y^{(s)} \,|\, s\in \Bbb N \}$ be a sequence of surfaces as above.
Then changing $(Z,D)$ and picking a subsequence one has for every $s<t$
\begin{equation}
\label{votono}
g^{(t)*}(g^{(s)}_{*}(K+B^{Y^{(s)}})){\le}
K+B^{Y^{(t)}}
\end{equation}
\end{thm}
\begin{pf}
\setcounter{case}{0}
Let me outline the strategy first.
After
canceling $K$ on both sides of the inequality~\ref{votono}, we have to
compare the coefficients for all prime divisors.
We are going to look at
coefficients in $B^{Y^{(t)}}$.
Some of them come from $B_{X^{(t)}}$ and hence
belong to the set
$\cal C$. Since this set satisfies a descending chain condition,
we can hope that after passing to a subsequence all the
required inequalities will be satisfied.
Then, the other part of coefficients in $B^{Y^{(t)}}$ comes from the
exceptional divisors of $f^{(t)}$. {\em A priori}, there is no
information about these coefficients except that they are nonnegative
and are less than 1. But, by
lemma~\ref{inequalities} as soon as we prove the inequalities for the
first part of $B^{Y^{(t)}}$, the inequalities for the exceptional
divisors will follow automatically. Finally, the fact that
$K+B^{Y^{(s)}}$ is MR Kawamata
log terminal will mean that for every fixed $s$, after modifying $Z$
and $D$, there are only {\em finitely many} inequalities we have to
check.
Consider $g^{(s)}:Y^{(s)}\to Z$ and the divisors
$g_{*}(K_{Y^{(s)}}+B^{Y^{(s)}})$ on $Z$. We shall denote
various prime divisors by $D_{\ldots{}}$ and the corresponding
sequences of coefficients in $K+B^{Y^{(s)}}$ by $x^{(s)}_{\ldots{}}$,
where $\ldots{}$ stands for a certain index.
For any fixed divisor $D_{\ldots{}}$, after
picking a subsequence, we can
assume that the sequence $x^{(s)}_{\ldots{}}$ is either increasing or
decreasing (not necessarily strictly) and has the limit
$c_{\ldots{}}$. Below we shall talk, for example, about a divisor
$D_{\ldots{}}$ that we get by blowing up a particular point $P$ on $Z$.
Picking a subsequence, we can assume that this point is blown up for
all the surfaces $Y^{(s)}$ or is not blown up at all
and then the analysis of the coefficients $x^{(s)}_{\ldots{}}$
is superfluous. The important fact that we shall use is that all the
coefficients $x_{\ldots{}}$ are strictly less than 1.
\begin{step}
Changing $(Z,D)$ we can assume that the points being blown up are
nonsingular on $Z$ and are at
worst the nodes of $D$. Moreover, if we blow up a point $P$ which is better
than a node then for any divisor $D_{k}$ over $P$ in $B^{Y^{(t)}}$ the
corresponding coefficient in
$g^{(t){*}}(g_{*}^{(s)}(K+B^{Y^{(s)}}))$
is negative, so we have the inequality~\ref{votono}.
Therefore, below we consider only the situation of two normally
crossing divisors $D_{1}$ and $D_{2}$ and the corresponding numbers
$c_{1},c_{2}\le1.$
\end{step}
\begin{step}
Points $P$ with $c_{1}=c_{2}=1$.
Blow up $P$ to get $D_{3}$. If $c_{3}<1$, change $Z$ to this new blown
up surface and go on to the next step. If $c_{3}=1$ then for any divisor
$D_{k}$ over $P$ we have $c_{k}=1$ by a very simple computation taking
into account that all $K+B^{Y^{(s)}}$ are nef.
Since the coefficients $x^{(s)}_{1}$ and
$x^{(s)}_{2}$ are both less than 1, for the inequality~\ref{votono} to be
true for $s=1$
over $P$ we need to check only finitely many divisors and the
inequalities are satisfied after picking a subsequence
because $\underset{s\to\infty}{\lim} x^{(s)}_{k}=c_{k}=1$. Then do the same for
$s=2$ and so on. For the new sequence the inequalities~\ref{votono} are
satisfied for all divisors over $P$.
\end{step}
\begin{step}
Points $P$ with $c_{1}=1$, $c_{2}<1.$
Let us change notation slightly in this step. Denote $D_{2;1}=D_{2}$,
and then
$D_{2;k+1}$ will be the exceptional divisor of a blowup at $D_{1}\cap
D_{2;k}$, $k=1,2,3\ldots{}$ Then denote by $D_{3;k}$ the exceptional
divisor of a blowup at the point $D_{2;k}\cap D_{2;k+1}$, by $D_{4;k}$
the exceptional divisor of a blowup at $D_{2;k}\cap D_{3;k}$ (or at
$D_{3;k}\cap D_{2;k+1}$) and so on.
The claims below are elementary and follow immediately from the fact
that $K+B^{Y^{(s)}}$ are nef.
\begin{claim}
In this way we obtain {\em finitely\/}
many sequences $\{ x^{(s)}_{p;k} \,|\,
k\in \Bbb N \}$ that are the only possible nonnegative
coefficients in $B^{Y^{(s)}}$.
\end{claim}
Note that $c_{2;1}\ge c_{2;2}\ge \ldots{}$ and denote by
$c=\underset{k\to\infty}{\lim}c_{2;k}$.
\begin{claim}
There exist naturally defined positive integers $n(p)$ such that
\begin{displaymath}
\underset{k\to\infty}{\lim}c_{p;k}= 1-n(p)(1-c)
\end{displaymath}
\end{claim}
\begin{claim}
\label{convex}
If for some $k,s$ $x^{(s)}_{2;k+2}< x^{(s)}_{2;k+1}>d$ then for the same $k,s$
one also has $x^{(s)}_{p;k}>1-n(p)(1-d)$
\end{claim}
\bigskip
Now consider two cases.
\begin{case}
There exists $k_{0}$ such that for infinitely many $s$
\begin{displaymath}
x^{(s)}_{2;k_{0}}\le c
\end{displaymath}
Pick a subsequence of $s$ with this property. Change $Z$ by its
$(k_{0}-1)$-blowup. Then again for the inequality~\ref{votono} to be
true for $s=1$ we have to check only finitely many divisors and we get the
corresponding inequalities because $c_{p;k}\ge 1-n(p)(1-c)$
by~\ref{convex} with $d=c$. Then proceed
with $s=2$ and so on.
\end{case}
\begin{case}
For any $k$ there exist only finitely many $s$ such that
\begin{displaymath}
x^{(s)}_{2;k}\le c
\end{displaymath}
Then~\ref{convex} with $d=c$ implies that
for any fixed $p,k$ there exist only finitely many $s$ such that
\begin{displaymath}
x^{(s)}_{p;k}\le 1-n(p)(1-c)
\end{displaymath}
Define a set
\begin{displaymath}
\cal A_{2}= \{ x^{(s)}_{2;k} \,|\, x^{(s)}_{2;k}>c; \, k,s\in \Bbb N \}
\end{displaymath}
Fix $k_{0}$ such that $c_{2;k_{0}}< \min\cal A_{2}\cap \cal C$, which
is possible because $\cal C$ is a D.C.C. set. Then pick a subsequence
of $s$ with the following property:
\begin{equation}
x\in \cal A_{2}, \, x< x^{(s)}_{2;k_{0}} \text{ implies } x\notin \cal C
\end{equation}
Introducing
\begin{displaymath}
\cal A_{p}= \{ x^{(s)}_{p;k} \,|\, x^{(s)}_{p;k}>1-n(p)(1-c); \, k,s\in \Bbb N \}
\end{displaymath}
we can also assume that
\begin{equation}
x\in \cal A_{p}, \, x< 1-n(p)(1-x^{(s)}_{2;k_{0}}) \text{ implies } x\notin \cal C
\end{equation}
Now, again, as above, we change $Z$ by its $(k_{0}-1)$-blowup and arrange
the inequality~\ref{votono} to be true for $s=1$ picking a
subsequence. The important thing here is that the relevant inequalities
may fail only for $x\notin \cal C$ which is OK
by lemma~\ref{inequalities}.
Then proceed with the same procedure for $s=2$ and so on.
\end{case}
Checking for a vicious circle, I emphasize that in this step we change
$Z$ for each point $P$ only once.
\end{step}
\begin{step}
Points with $c_{1}<1$, $c_{2}<1$.
Change $Z$ by blowing it up several times so that for any future
blowup all divisors should have negative coefficients in $B^{Y^{(s)}}$.
\end{step}
At this point we achieved that the inequality~\ref{votono} is satisfied
for any divisor which is exceptional for $g^{(s)}$. Finally, let us
settle it for the divisors $D_{k}$ on $Z$ itself.
\begin{step}
$D_{1}$ is a divisor on $Z$.
If $\{ x^{(s)}_{1} \}$ is an increasing sequence, we are done. Otherwise,
omitting finitely many $s$, we have $x^{(s)}_{1}< \min \{x^{(s)}_{1}\} \cap \cal C$, so
the coefficients $x^{(s)}_{1}$ correspond to exceptional divisors and
we are OK by lemma~\ref{inequalities}.
\end{step}
So far, we assumed that $(Z,D)$ is fixed.
To prove the general case we use~\ref{kinky method for proving DCC}. Pick
an arbitrary sequence of surfaces satisfying the initial conditions.
Since by~\ref{main diagram}
the class $(Z,D)$ is bounded, there are only
finitely many numerical possibilities for $Z$, $D$ and the subsequent
blowups so, after picking a subsequence,
we can argue in the same way as we did for the fixed $(Z,D)$.
\renewcommand{\qed}{}
\qed
\end{pf}
\begin{rem}
Several important ideas of
the proof given here are taken directly
from the Xiao's proof of
Proposition~5, \cite{Xiao91}.
Unfortunately, I do not understand the arguments of
\cite{Xiao91} completely and therefore cannot say precisely
how close I follow them. However, it is
clear that
the Xiao's proof may apply only to a set $\cal C$
with a single limit point at 1.
\end{rem}
\begin{thm}
\label{DCC for log canonical}
Theorem~\ref{DCC for semi-log canonical} holds for $K+B$ MR log
canonical.
\end{thm}
\begin{pf}
Let $\{X^{(s)},K+B^{(s)}\}$ be a sequence with strictly decreasing
$(K+B)^{2}$. For each $X^{(s)}$ consider the maximal log crepant
(i.e.\ all new log codiscrepancies equal~1)
partial resolution $f^{(s)}$ which is dominated by the minimal
desingularization. Let $E_{i}$ be the divisors with coefficients~1 in
$f^{*}(K+B)$. Then for the appropriate choice of
$\epsilon_{i}\to0$ the
divisors $f^{*}(K+B)-\sum \epsilon_{i}E_{i}$ will be ample and
MR Kawamata log
terminal, the coefficients will belong to a {\em new}, but again a
D.C.C., set,
and the sequence of the squares will still be strictly decreasing.
So we get a contradiction by~\ref{kinky method for proving DCC}
and \ref{DCC for Kawamata log terminal}.
\end{pf}
{\em End of the proof of theorem~\ref{DCC for semi-log canonical}.}
By~\ref{reduce semi-log canonical to log canonical},
\begin{displaymath}
(K+B)^{2}= \sum (K_{X_{m}}+B_{m})^{2}
\end{displaymath}
and all $K_{X_{m}}+B_{m}$ are log canonical. By
{}~\ref{DCC for log canonical} every summand in this formula belongs
to a D.C.C. set, and so does the sum.
\end{pf}
\section[Boundedness for the constant $(K+B)^{2}$
]{Boundedness for the constant $(K+B)^{2}$}
\begin{say}
In this section we shall prove the following theorem. Again, it will
be done first for $K+B$ MR Kawamata log terminal and MR log canonical.
\end{say}
\begin{thm}
\label{K2=const for semi-log canonical}
Fix a constant $C$ and a D.C.C. set $\cal C$.
Consider all surfaces $X$
with an $\Bbb R$-divisor $B=\sum b_{j}B_{j}$ such that
$K_{X}+B$ is MR semi-log canonical, $K_{X}+B$ is
ample, $b_{j}\in \cal C$ and $(K+B)^{2}= C$.
Then the class $\{(X,\sum b_{j}B_{j})\}$ is bounded.
\end{thm}
\begin{thm}
Theorem~\ref{K2=const for semi-log canonical} holds for $K+B$ MR
Kawamata log terminal. Moreover, in this case $K+B$ may be taken to
be big and nef instead of ample.
\end{thm}
\begin{pf}
Again, first assume that in the diagram~\ref{main diagram} $(Z,D)$ is
fixed. Then, for the general case we use~\ref{kinky method for proving
boundedness} since $(Z,D)$ moves in a bounded family.
By lemma~\ref{inequalities} the inequalities~\ref{votono} become
equalities only when
\begin{displaymath}
g^{(s)*}(K+\overline B)=K+B^{Y^{(s)}} \, \text{ for every }\, s \,
\text{ with a fixed }\, \overline B
\end{displaymath}
Since the log canonical model of $K+B^{Y^{(s)}}$
is the image of the linear system $|N(K+B^{Y^{(s)}})|$, $N\gg0$,
all surfaces in the sequence have the same log canonical model.
This means that the class $\{(Z,\overline{ B})\}$ is bounded and since
all coefficients in $B^{Y^{(s)}}$ are less than~1, that $\{(Y,B)\}$ is
bounded.
Since
\begin{displaymath}
K_{Y^{(s)}}+B^{Y^{(s)}}= f^{*}(K_{X^{(s)}}+B_{X^{(s)}}),
\end{displaymath}
we get a diagram $Y^{(s)}\to X^{(s)} \to Z$
and the class $\{(X,B)\}$ is bounded by the ``sandwich'' principle.
\end{pf}
\begin{thm}
\label{K2=const for log canonical}
Theorem~\ref{K2=const for semi-log canonical} holds for $K+B$ MR log
canonical.
\end{thm}
\begin{pf}
As in the previous proof,
by~\ref{main diagram} we first assume existence of the diagram
\begin{displaymath}
\begin{CD}
Y @>g>> Z\\
@VfVV\\
X
\end{CD}
\end{displaymath}
with $Z$ and $D=\operatorname{supp} B\cup \operatorname{Exc}(f)$ fixed.
Pick an infinite sequence of surfaces $\{ X^{(s)} \}$ and for
$K_{Y^{(s)}}+B^{Y^{(s)}}=f^{(s)*}(K_{X^{(s)}}+B^{(s)})$
write
\begin{displaymath}
(K_{Y^{(s)}}+B^{Y^{(s)}})^{2}= Base^{(s)} + Defect^{(s)}
\end{displaymath}
where
\begin{eqnarray*}
Base^{(s)}= (g^{(s)}_{*}(K_{Y^{(s)}}+B^{Y^{(s)}}))^{2}
\\Defect^{(s)}= (K_{Y^{(s)}}+B^{Y^{(s)}})^{2}-
(g^{(s)}_{*}(K_{Y^{(s)}}+B^{Y^{(s)}}))^{2}
\end{eqnarray*}
Note that $Defect^{(s)}\le0$ for all $s$.
Now if $K+B$ were MR Kawamata log terminal then from
{}~\ref{thm main trick with sequences} we could
conclude that, after picking a subsequence, $\{ Base^{(s)} \}$ is a
nondecreasing sequence and
that $Defect^{(s)}\to0$. But because every MR log
canonical $K+B$ can be approximated by MR Kawamata log terminal
divisors as in the proof of
{}~\ref{DCC for log canonical}, we can conclude just the same in the present
situation. But then $Base^{(s)} + Defect^{(s)}=C$ implies that
$Defect^{(s)}=0$ and $Base^{(s)}=C$,
i.e.\
\begin{displaymath}
g^{(s)*}(K+\overline B)=K+B^{Y^{(s)}} \, \text{ for every }\, s \,
\text{ with a fixed }\, \overline B
\end{displaymath}
Since $K+B^{X^{(s)}}$
is the image of the linear system $|N(K+B^{Y^{(s)}})|$, $N\gg0$,
we get $(X^{(s)},B)=(Z,\overline{ B})$, so
all surfaces in the sequence are actually isomorphic to each other.
In the general situation $(Z,D)$ is not fixed but the class $(Z,D)$ is
bounded, so we get that the class $(X,B)$ is bounded.
\end{pf}
{\em End of the proof of~\ref{K2=const for semi-log canonical}. }
\begin{pf}
Let $X^{\nu}=\cup X_{m}$ be the normalization of $X$. Then as in
{}~\ref{reduce semi-log canonical to log canonical}
\begin{displaymath}
(K+B)^{2}= \sum_{m} (K_{X_{m}}+B_{m})^{2},
\end{displaymath}
the coefficients of $B_{m}$ belong to $\cal C\cup \{1\}$ and
$K_{X_{m}}+B_{m}$ are log canonical.
Applying~\ref{DCC for log canonical} and~\ref{K2=const for log
canonical} we see that there are only finitely many
possibilities for $(K_{B_{m}})^{2}$ and that all
$(X,\operatorname{supp} B_{k})$ belong to a bounded class. Now the class $\{X\}$ is
bounded by the Conductor Principle \cite{Kollar85} 2.3.5. It
immediately follows that in fact $\{(X,\operatorname{supp} B)\}$ is bounded.
\end{pf}
\section[On the log Minimal Model Program for surfaces,
in arbitrary characteristic
]{On log MMP for surfaces}
\label{MMP in dimension 2}
\begin{say}
The Log Minimal Model Program in dimension two is undoubtedly much
easier than in higher dimensions. There are two main circumstances that
distinguish the surface case.
The first difference is that
the Log Minimal
Model Program in dimension two is characteristic free, see
\cite{Mori82} for nonsingular surfaces and \cite{TsunodaMiyanishi83}
for open surfaces. The Cone Theorem follows from the nonsingular case
by a simple argument as in \cite{TsunodaMiyanishi83}. The proofs of
the Contraction Theorem and of the Log Abundance Theorem in higher
dimension use the Kodaira vanishing theorem which fails in positive
characteristic. However, in dimension two they can be proved
using the contractibility conditions of Artin \cite{Artin62} because
the corresponding configurations of curves are rational. In fact, in
characteristic $p>0$ contractibility conditions are even weaker than
in characteristic 0.
The second difference is the existence of a unique minimal resolution of
singularities.
The statements in this section and their proofs are elementary.
I provide
them here because I need them for this paper and they are slightly
more general than those appearing elsewhere.
\end{say}
\begin{thm}[The Cone Theorem]
Let $X$ be a normal surface and $K+B=K_{X}+\sum b_{j}B_{j}$ be an
$\Bbb R$-Cartier divisor with $b_{j}\ge0$. Let $A$ be an ample divisor on
$X$. Then for any $\epsilon>0$ the Mori-Kleiman cone of effective
curves $\overline{NE}(X)$ in $NS(X)\otimes\Bbb R$ can be written as
\begin{displaymath}
\overline{NE}(X)= \overline{NE}_{K+B+\epsilon A}(X) +\sum R_{k}
\end{displaymath}
where, as usual, the first part consists of cycles that have positive
intersection with $K+B+\epsilon A$ and $R_{k}$ are finitely many
extremal rays. Each of the extremal rays is generated by an effective curve.
\end{thm}
\begin{pf}
It suffices to prove this theorem for the minimal desingularization
$Y$ of
$X$ with $K_{Y}+B^{Y}=f^{*}(K+B)$ instead of $K+B$ since every effective
curve
on $X$ is an image of an effective curve on $Y$,
and so $\overline{NE}(X)$ is the image of $\overline{NE}(Y)$ under a
linear map $f_{*}:N_{1}(Y)\to N_{1}(X)$.
Then for a
nonsingular surface the statement easily follows from
the usual Cone Theorem
\cite{Mori82} by the same argument as in \cite{TsunodaMiyanishi83}, 2.5.
The reason for this is, certainly, that for any curve $C\ne B_{j}$
one has $C(K+B)\ge CK$.
The set of
extremal rays for $K+B+\epsilon A$ is a subset of extremal rays for
$ K+\epsilon A$, possibly enlarged by some of the curves $B_{j}$.
\end{pf}
\begin{thm}[Contraction Theorem]
Let $X$ be a projective surface with MR log canonical
$K+B=K+ \sum b_j B_j$. Let $R$ be an
extremal ray for $K+B$. Then there exists a nontrivial projective
morphism $\phi_{R}:X\to Z$ such that $\phi_{R*}(\cal O_{X})=\cal
O_{Z}$ and $\phi(C)=pt$ iff the class of $C$ belongs to $R$, and
$K_{Z}$ is also log canonical. If $K_{X}$ is Kawamata log terminal
then so is $K_{Z}$.
\end{thm}
\begin{pf}
By the previous theorem $R$ is generated by an irreducible curve $C$.
If $C^{2}>0$ then as in \cite{Mori82} 2.5, $f^{*}(C)$ on the minimal
resolution (defined, according to Mumford, for every Weil divisor
$C$) is in the interior of $\overline{NE}(Y)$, so $C$ is in the
interior of $\overline{NE}(X)$. This is possible only if $\rho(X)=1$
and then
$\phi_{R}$ maps the whole $X$ to a point.
If $C^{2}=0$ then
the graph corresponding to $\operatorname{supp}(f^{*}(C))$ is parabolic and it
is rational because $KC<0$.
By the Riemann--Roch formula $h^{0}(nf^{*}(C))$ grows linearly in $n$,
so for some $N\gg0$ the linear system $|Nf^{*}(C)|$ is base point free and
defines a projective morphism to a curve. Similarly to \cite{Mori82},
2.5.1, this is possible only if $\rho(X)=2$.
Now let us assume that $C^{2}<0$. Consider the minimal resolution
$f:Y\to X$ and
let $F_{i}$ be the maximal number of exceptional curves of $f$ such
that $C\cup_{i} F_{i}$ is connected. Then, since $(K+B)C<0$ and
the quadratic form $|F_{i}F_{i'}|$ is negative definite, all the
log discrepancies of the graph $C\cup_{i} F_{i}$ are strictly
positive, so this graph is log terminal. Looking at the list of the
minimal log
terminal graphs (for example, in \cite{Alexeev92}) one easily observes that
they
are all rational in the sense of Artin \cite{Artin62}.
So are all the nonminimal
log terminal elliptic graphs obtained from them by simple
blowups.
Hence, by
\cite{Artin62} the configuration of curves $C\cup_{i}F_{i}$ can be
contracted to a normal point on a projective surface $Z$. By normality
of $X$, this defines a morphism $\phi_{R}:X\to Z$ satisfying the
required properties.
\end{pf}
\begin{thm}[Easy Log Abundance Theorem]
Let $X$ be a projective surface with MR Kawamata log terminal
$K+B=K+\sum b_{j}B_{j}$.
Assume
that $K+B$ is big and nef. Then for some $N\gg0$ the linear system
$|N(K+B)|$ is base point free and defines a birational projective
morphism $f:X\to Z$ with $N(K+B)=f^{*}(H)$ for some ample divisor
$H$ on $Z$. $K_{Z}$ is also Kawamata log terminal.
\end{thm}
\begin{pf}
We have $(K+B)^{2}>0$.
If $K+B$ is not ample then by the Nakai-Moishezon criterion of ampleness
there exists a curve $C$ such that $(K+B)C=0$.
By the Hodge Index theorem, $C^{2}<0$.
Then, as in the
previous theorem, $C$ can be contracted to a normal point on a
projective surface $Z_{1}$, $K+B=g^{*}(K_{Z_{1}}+g_{*}(B))$ and
the latter divisor is again MR Kawamata log terminal.
Now the statement follows
by induction on rank of the Picard group of $Z_{1}$.
\end{pf}
\begin{rem}
It is elementary to generalize the above three theorems to a relative
situation of an arbitrary
projective morphism $X\to S$ to a variety $S$.
\end{rem}
\section{Concluding remarks, generalizations, open questions}
\label{Concluding remarks, generalizations, open questions}
\begin{say}
Following the logics of the theorems presented here one should expect
that there exist infinitely many types of surfaces with ample
canonical divisor $K$, $K^{2}\le C$
and {\em empty\/} $B$ if we allow singularities worse
than $\varepsilon$-log terminal,
for example arbitrary quotient singularities. It would be interesting
to see the examples.
\end{say}
\begin{say}[Effectiveness]
Note that it is quite straightforward to obtain effective formulas for
$A_{1}$ in~\ref{thm A1} and $S_{1}$ in~\ref{minimal elliptic graph}.
However, methods of this paper do not provide
effective estimates for functions $A_{2}$ in~\ref{thm A2 for
general} and $c(\cal C)$ in~\ref{my thm on auts}.
\end{say}
\begin{say}
I would go as far as to conjecture that the conditions for the
boundedness formulated in~\ref{proved here}
are the most natural ones and that direct
analogs of all four our main theorems
should be true in any dimension.
\end{say}
\begin{say}
In the case of $K$ negative, i.e.\ Fano varieties, there are several
results supporting this conjecture. Most importantly, it is true for
toric log Fano varieties with empty $B$ by Borisovs
\cite{BorisovBorisov92}.
Fano 3-folds with terminal
singularities (i.e.\ 1-log terminal) are bounded (see Kawamata
\cite{Kawamata89} for $\Bbb Q$-factorial case, general case is due to
Mori, unpublished). Log Fano 3-folds of fixed index $N$
and Picard number 1
(in particular, they are $(1/N)$-log terminal) are bounded by
\cite{BorisovA93}.
\end{say}
\begin{say}
In the case of positive $K$ nothing is known in dimension greater than
two.
\end{say}
\begin{say}
Let me list the main points where dimension two is essential in the
proofs: diagram method, lemma~\ref{only surfaces}, lemma~\ref{can
decrease all by delta}. It could also be argued that all proofs
involving $D^{2}$ do not have a chance to be generalized to higher
dimensions because only in dimension $n=2$ $D^{n}$ behaves well under
birational transformations. However, this obstacle can be avoided by
using systematically $\underset{c}\le$ instead of $\le$ and then
applying~\ref{easy inequalities}.
\end{say}
\begin{say}
Theorem~\ref{bound for nonsingular when -K nef} has applications to
singularities of curves on surfaces with positive $-K$ such as the
projective plane or Del Pezzo surfaces.
For example, it implies that an irreducible curve of
degree $d$ for $d\gg0$ cannot have tangents of high multiplicity as
compared to $d$. It would be interesting to spell this connection out
and compare with known results.
\end{say}
\begin{say}
Most of the results of this paper remain valid if one works in the
category of algebraic spaces of dimension two
instead of the category of algebraic surfaces.
\end{say}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1994-02-09T14:47:39 | 9402 | alg-geom/9402005 | en | https://arxiv.org/abs/alg-geom/9402005 | [
"alg-geom",
"math.AG"
] | alg-geom/9402005 | null | Giorgio Ottaviani and G\"unther Trautmann | The Tangent Space at a Special Symplectic Instanton Bundle on $P_{2n+1}$ | 13 pages, LaTex (Reason for resubmission: correction of a Tex
misprint in the title page) | null | null | null | null | Mathematical instanton bundles on $ P_3$ have their analogues in rank--$2n$
instanton bundles on odd dimensional projective spaces $ P_{2n+1}$. The
families of special instanton bundles on these spaces generalize the special
'tHooft bundles on $ P_3$. We prove that for a special symplectic instanton
bundle $ E$ on $ P_{2n+1}$ with $c_2=k$ $h^1End( E) = 4(3n-1) k +
(2n-5)(2n-1)$. Therefore the dimension of the moduli space of instanton bundles
grows linearly in $k$. The main difference with the well known case of $ P_3$
is that $h^2End( E)$ is nonzero, in fact we prove that it grows quadratically
in $k$. Special symplectic instanton bundles turn out to be singular points of
the moduli space. Such bundles $ E$ are $SL(2)$--invariant and the result is
obtained regarding the cohomology groups of $ E$ as $SL(2)$--representations.
| [
{
"version": "v1",
"created": "Mon, 7 Feb 1994 18:53:00 GMT"
},
{
"version": "v2",
"created": "Wed, 9 Feb 1994 14:48:00 GMT"
}
] | 2016-08-14T00:00:00 | [
[
"Ottaviani",
"Giorgio",
""
],
[
"Trautmann",
"Günther",
""
]
] | alg-geom | \section*{Introduction}\addcontentsline{toc}{section}{Introduction}
Mathematical instanton bundles on $\P_3$ have their analogues in rank--$2n$
instanton bundles on odd dimensional projective spaces $\P_{2n+1}$. The
families of special instanton bundles on these spaces, which generalize the
special 'tHooft bundles on $\P_3$, were constructed and described in \cite{OS}
and \cite{ST}. More general instanton bundles have recently been constructed
in \cite{AO2}. Let $MI_{2n+1}(k)$ denote the moduli space of all instanton
bundles on $\P_{2n+1}$ with second Chern class $c_2=k$. In order to obtain a
first impression of this space it is important to know its tangent dimension
$h^1End({\cal E})$ at a stable bundle ${\cal E}$ and the dimension $h^2End({\cal E})$ of the
space of obstructions to smoothness.
In this paper we prove that for a special symplectic
bundle ${\cal E} \in MI_{2n+1}(k)$
\[
h^2 End({\cal E}) = (k-2)^2 {2n-1\choose 2}.
\]
Such bundles are stable by \cite{AO1}.
So for $n \ge 2$ the situation is quite different to that of $\P_3$, where
this number becomes zero, which was shown in \cite{HN}. Since $H^iEnd({\cal E}) =
0$ for $i \ge 3$, our result and the Hirzebruch--Riemann--Roch formula, see
Remark \ref{2.4},
\[
h^1End({\cal E}) - h^2End({\cal E}) = -k^2{2n-1\choose 2} + k(8n^2) + 1-4n^2
\]
give
\[
h^1End({\cal E}) = 4(3n-1) k + (2n-5)(2n-1).
\]
Therefore the dimension of $MI_{2n+1}(k)$ grows linearly in $k$, whereas the
difference $h^1End({\cal E}) - h^2End({\cal E})$ becomes negative for $n \ge 2$ and
grows quadratically in $k$. A more important consequence, however, is that
in general $MI_{2n+1}(k)$ cannot be smooth at special symplectic bundles, see
section 4 and \cite{AO2}.
In order to derive our result we fix a 2--dimensional vector space $U$ and
consider the natural action of $SL(2)$ on $\P_{2n+1} = \P(U \otimes S^nU)$ as
in \cite{ST}. The special instanton bundles are related to the
$SL(2)$--homomorphisms $\beta$, see \ref{1.4}, and are $SL(2)$--invariant. We
prove that there is an isomorphism of $SL(2)$--representations
\[
H^2(End\,{\cal E}) \cong S^{k-3}(U) \otimes S^{k-3}(U) \otimes S^2(U \otimes
S^{n-2}U).
\]
\vfill
{\bf Acknowledgement}\quad This work was supported by the
{\bf Deutsche Forschungs
Gemeinschaft}. The first author wishes to thank the Fachbereich Mathematik of
the University of Kaiserslautern, where this work was begun, for its
hospitality.
\newpage
\section*{Notation}
\begin{itemize}
\item Throughout the paper $K$ denotes an algebraically closed ground field of
characteristic 0.
\item $U$ denotes a 2--dimensional $K$--vector space, $S_n = S^nU$ its $n$th
symmetric power and $V_n = U \otimes S_n$.
\item There is the natural exact squence of $GL(U)$--equivariant maps for any
$k,\, n \ge 1$
\[
0 \to \Lambda^2 U \otimes S_{k-1} \otimes S_{n-1} \buildrel
\beta\over\to S_k
\otimes S_n \buildrel \mu\over\to S_{k+n} \to 0
\]
where $\mu$ is the multiplication map and $\beta$ is defined by $(s\wedge t)
\otimes f \otimes g \mapsto sf \otimes tg - tf \otimes sg$. This sequence
splits and leads to the Clebsch--Gordan decomposition of $S_k \otimes S_n$ by
induction. When we tensorize the sequence with $U$ we obtain the exact
sequence
\[
0 \to \Lambda^2 U \otimes S_{k-1} \otimes V_{n-1}
\buildrel\beta\over\to S_k
\otimes V_n \buildrel\mu\over\to V_{k+n} \to 0.
\]
\item $\P = \P_{2n+1} = \P V_n$ is the projective space of one dimensional
subspaces of $V_n$.
\item The terms vector bundle and locally free sheaf are used synonymously.
\item ${\cal O}(d)$ denotes the invertible sheaf of degree $d$ on $\P$, $\Omega^p$
the locally free sheaf of differential $p$--forms on $\P$, such that
$\Omega^p(p) = \Lambda^p {\cal Q}^\vee$ where ${\cal Q} = {\cal T}(-1)$ is the canonical
quotient bundle on $\P$.
\item We use the abbreviations ${\cal F}(d) = {\cal F} \otimes_{{\cal O}} {\cal O}(d)$ for any
sheaf ${\cal F}$ of ${\cal O}$--modules on $\P,\; H^i{\cal F} = H^i({\cal F}) = H^i(\P, {\cal F}),\;
h^i{\cal F} = \dim\, H^i{\cal F}$. If $E$ is a finite dimensional $K$--vector space,
$E \otimes {\cal O}$ denotes the sheaf of sections of the trivial bundle $\P \times
E$, and $E \otimes {\cal F} = (E \otimes {\cal O}) \otimes_{\cal O} {\cal F}$. We also write
$m{\cal F} = K^m \otimes {\cal F}$.
\item We use the Euler sequence $0\to \Omega^1(1) \to V^\vee \otimes {\cal O} \to
{\cal O} (1) \to 0$ and the derived sequences in its Koszul complex $0 \to
\Omega^p(p) \to \Lambda^p V^\vee \otimes {\cal O} \to \Omega^{p-1} (p) \to 0$
without extra mentioning.
\item $Ext^i({\cal F}, {\cal G}) = Ext^i_{\cal O} (\P, {\cal F}, {\cal G})$ for any two ${\cal O}$--modules
${\cal F}$ and ${\cal G}$.
\end{itemize}
\newpage
\section{Instanton bundles}
\begin{sub}\label{1.1}
{\rm An instanton bundle on $\P = \P_{2n+1}$ with
instanton number $k$ or a $k$--instanton is an algebraic vector bundle ${\cal E}$
on $\P$ satisfying:
\begin{itemize}
\item[(i)] ${\cal E}$ has rank $2n$ and Chern polynomial $c({\cal E}) =
(1-h^2)^{-k} = 1 + k h^2 + \ldots$.
\item[(ii)] ${\cal E}$ has natural cohomology in the range $-2n-1 \le d \le
0$, that is for any $d$ in that range $h^i{\cal E}(d) \not= 0$ for at most one
$i$.
\end{itemize}
A $k$--instanton bundle ${\cal E}$ is called {\bf symplectic} if there is an
isomorphism ${\cal E} \buildrel \varphi\over\to {\cal E}^\vee$ satisfying $\varphi^\vee =
-\varphi$. In this case the spaces $A$ and $B$ below are Serre--duals of
each other, since
$H^{2n}({\cal E}(-2n-1))^\vee \cong H^1{\cal E}^\vee(-1) \cong H^1{\cal E}(-1)$.
{\bf Remark}: In the original definition in \cite{OS} the additional
conditions
\begin{itemize}
\item[(iii)] ${\cal E}$ is simple, that is $Hom({\cal E},{\cal E}) = K$,
\item[(iv)] the restriction of ${\cal E}$ to a general line is trivial
\end{itemize}
are imposed. It was shown in \cite{AO1} that (iii) is already a consequence
of (i) and (ii). Condition (iv) seems to be independent but we do not need
it in this paper. By \cite{ST} special instantons satisfy (iv).}
\end{sub}
\begin{sub}\label{1.2}
{\rm Let now $A, B, C$ be vector spaces of dimensions $k, k,
2n(k-1)$ respectively. A pair of linear maps
\[
A \buildrel a \over \longrightarrow B \otimes \Lambda^2 V^\vee_n,\quad B
\otimes V^\vee_n \buildrel b\over\longrightarrow C
\]
corresponds to a pair of sheaf homomorphisms
\[
A \otimes {\cal O}(-1) \buildrel\tilde{a}\over\longrightarrow B \otimes \Omega^1
(1), \quad B \otimes \Omega^1(1) \buildrel\tilde{b}\over\to C \otimes {\cal O}.
\]
Here $\tilde{a}$ is the composition of the induced homomorphisms $A \otimes
{\cal O}(-1) \to B \otimes \Lambda^2 V^\vee_n \otimes {\cal O}(-1) \longrightarrow B
\otimes \Omega^1(1)$ and $\tilde{b}$ is the composition of the induced
homomorphismus $B \otimes \Omega^1(1) \longrightarrow B \otimes V^\vee_n
\otimes
{\cal O} \to C \otimes {\cal O}$. Conversely, $a$ and $b$ are determined by
$\tilde{a}$ and $\tilde{b}$ respectively as $H^0 (\tilde{a}(1))$ and
$H^0(\tilde{b}^\vee)^\vee$. Moreover, the sequence
\begin{equation}
A \otimes {\cal O}(-1) \buildrel \tilde{a}\over\longrightarrow B \otimes
\Omega^1(1) \buildrel \tilde{b}\over\longrightarrow C \otimes {\cal O}
\end{equation}
is a complex if and only if the induced sequence
\[
A \longrightarrow B \otimes \Lambda^2 V^\vee_n \longrightarrow C \otimes
V^\vee_n
\]
is a complex. We say that (1) is a {\bf monad} if it is a complex and if in
addition $\tilde{a}$ is a subbundle and $\tilde{b}$ is surjective.}
\end{sub}
\begin{proposition}\label{1.3}
The cohomology sheaf ${\cal E} = Ker\,
\tilde{b}/Im\, \tilde{a}$ of a monad (1) is a $k$--instanton and conversely
any $k$--instanton is the cohomology of a monad (1). Moreover, the spaces
$A, B, C$ of such a monad can be identified with \mbox{$H^{2n}{\cal E}(-2n-1)$,}
$H^1{\cal E}(-1),\; H^1{\cal E}$ respectively.
\end{proposition}
Sketch of a proof: if a monad (1) is given it is easy to derive the
properties of the definition. Conversely using Beilinson's spectral
sequence, Riemann--Roch and in particular (ii), one obtains a monad with the
identification of the vector spaces as in \cite{OS}. The map $b$ is then
nothing but the
natural map $H^1{\cal E}(-1) \otimes V^\vee_n \to H^1{\cal E}$ and the map $a$ is given
as the composition of the cup product
\[
H^{2n}{\cal E}(-2n-1) \otimes \Lambda^2V_n \to H^{2n}{\cal E} \otimes \Omega^{2n-1}
(-1)
\]
and the natural isomorphisms
\[
H^{2n}{\cal E}\otimes\Omega^{2n-1}(-1) \cong H^{2n-1}{\cal E} \otimes
\Omega^{2n-2}(-1)
\cong \ldots \cong H^1{\cal E}(-1)
\]
arising from the Koszul sequences and condition (ii), see \cite{V} in case of
$\P_3$.\\
\begin{sub}\label{1.4}{\rm{\bf Existence and special instanton bundles}:
Using the special
structure $V_n = U \otimes S^n U$ and the Clebsch--Gordan type exact sequence
\[
0 \longrightarrow \Lambda^2 U \otimes S_{k-2} \otimes V_{n-1}
\buildrel\beta\over\longrightarrow S_{k-1} \otimes V_n
\buildrel\mu\over\longrightarrow V_{k+n+1} \longrightarrow 0,
\]
see notation, we define the special homomorphism
\[
S^\vee_{k-1} \otimes \Omega^1(1) \buildrel\tilde{b}\over\longrightarrow
\Lambda^2 U^\vee \otimes S^\vee_{k-2} \otimes V^\vee_{n-1} \otimes {\cal O}
\]
by $b = \beta^\vee$. We denote ${\cal K} = Ker\;(\tilde{b})$. It was shown in
\cite{ST} that $\tilde{b}$ is surjective and that
\[
H^0 {\cal K}(1) \subset S^\vee_{k-1} \otimes H^0 \Omega^1(2)
\]
can be identified with a canonical injective $GL(U)$--homomorphism
\[
S^\vee_{2n+k-1} \otimes \Lambda^2 U^\vee \buildrel\kappa\over\to
S^\vee_{k-1} \otimes
\Lambda^2 V^\vee_n,
\]
dual to the map
\[
S_{k-1} \otimes \Lambda^2V_n \to S_{2n+k-1} \otimes \Lambda^2U
\]
which is defined by $f \otimes (s \otimes g) \wedge (t \otimes h) \mapsto
(fgh) \otimes
(s \wedge t)$.
In order to construct instanton bundles we have to find $k$--dimensional
subspaces
\[
A \subset S^\vee_{2n+k-1} \otimes \Lambda^2 U^\vee \subset
S^\vee_{k-1}
\otimes \Lambda^2 V^\vee_n
\]
such that the induced homomorphism $\tilde{a}$ is a subbundle. By \cite{ST},
Lemma 3.7.1, this is the case exactly when $\P A \subset \P (S^\vee_{2n+k-1})$
does not meet the secant variety $Sec_n(C_{2n+k-1})$ of $(n-1)$--dimensional
secant planes of the canonical rational curve $C_{2n+k-1}$ of $\P
S^\vee_{2n+k-1}$, given by $u \mapsto u^{2n+k-1}$. By dimension reasons such
subspaces exist, \cite{ST}, 3.7, and hence instanton bundles exist.
A $k$--instanton bundle ${\cal E}$ is called {\bf special} if the map $b$ of its
monad is isomorphic to the $GL(U)$--homomorphism $\beta^\vee$, that is if
there are isomorphisms $\varphi$ and $\psi$ and $g \in GL(V_n)$ with the
commutative diagram
\unitlength1cm
\begin{picture}(9,4)
\put(3.75,1.5){$S^\vee_{k-1} \otimes V^\vee_n$}
\put(6.25,1.6){\vector(1,0){1}}
\put(6.75,1.7){$\beta^\vee$}
\put(7.75,1.5){$\Lambda^2 U^\vee \otimes S^\vee_{k-2} \otimes V^\vee_{n-1}.$}
\put(3.45,2.5){$\varphi \otimes g^\vee$}
\put(4.75,3.0){\vector(0,-1){1}}
\put(4.95,2.5){$\wr\wr$}
\put(8.65,2.5){$\psi$}
\put(8.95,3.0){\vector(0,-1){1}}
\put(9.05,2.5){$\wr\wr$}
\put(3.05,3.3){$H^1{\cal E}(-1) \otimes V^\vee_n$}
\put(6.25,3.4){\vector(1,0){1}}
\put(6.75,3.5){$b$}
\put(8.45,3.3){$H^1{\cal E}$}
\end{picture}
Whereas in \cite{ST} the family of all special $k$--instanton bundles was
described, examples of different types of general instanton bundles were found
in \cite{AO2}.}
\end{sub}
\begin{remark}\label{1.5}
{\rm If ${\cal E}$ is special and symplectic then, in addition to
the special $GL(U)$--homomorphism $b = \beta^\vee$ of its monad, the map $a$
is given by an element $\alpha \in S^\vee_{2n+2k-2}$ as $a = \kappa \circ
\tilde{\alpha}$ where $S_{k-1} \buildrel \tilde{\alpha}\over\to
S^\vee_{2n+k-1}$ is defined by $\tilde{\alpha}(f)(g) = \alpha(fg)$ and
$S^\vee_{2n+k-1} \buildrel \kappa\over\to S^\vee_{k-1} \otimes \Lambda^2
V^\vee_n$ is as above, \cite{ST}, 4.3 and 5.8. In particular $a$ is a
$GL(U)$--homomorphism, too, and can be represented by a persymmetric matrix.}
\end{remark}
\begin{remark}\label{1.6}
{\rm It is shown in \cite{AO1} that special symplectic
instanton bundles are stable in the sense of Mumford--Takemoto.}
\end{remark}
\newpage
\section{Representing $\bf Ext^2({\cal E},{\cal E})$}
\begin{proposition}\label{2.1}
Let ${\cal E}$ be a symplectic $k$--instanton and let ${\cal N}$ be the kernel of the
monad (1). Then $Ext^2({\cal E},{\cal E}) \cong H^2({\cal N} \otimes {\cal N})$.
\end{proposition}
Proof:\quad The monad (1) gives rise to the exact sequences
\begin{equation
0 \longrightarrow {\cal N} \longrightarrow B \otimes \Omega^1(1) \buildrel
\tilde{b}\over\longrightarrow C \otimes {\cal O} \longrightarrow 0
\end{equation}
and
\begin{equation
0 \longrightarrow A \otimes {\cal O} (-1) \longrightarrow {\cal N} \longrightarrow
{\cal E}
\longrightarrow 0.
\end{equation}
After tensoring we have the exact sequences
\begin{equation
0 \longrightarrow A \otimes {\cal N}(-1) \longrightarrow {\cal N} \otimes {\cal N}
\longrightarrow {\cal E} \otimes {\cal N} \longrightarrow 0
\end{equation}
and
\begin{equation
0 \longrightarrow A \otimes {\cal E} (-1) \longrightarrow {\cal N} \otimes {\cal E}
\longrightarrow {\cal E} \otimes {\cal E} \longrightarrow 0.
\end{equation}
Since ${\cal E} \cong {\cal E}^\vee$ we obtain $Ext^2({\cal E},{\cal E}) \cong H^2({\cal E} \otimes
{\cal E})$. Sequence (2) implies $h^2{\cal N}(-1) = h^3{\cal N}(-1) = 0$ and from this and
(3) also $h^2{\cal E}(-1) = h^3{\cal E}(-1) = 0$. Now sequences (4) and (5) yield
isomorphisms $H^2({\cal E} \otimes {\cal E}) \cong H^2({\cal N} \otimes {\cal E}) \cong H^2({\cal N}
\otimes {\cal N})$.\hfill$\Box$
\begin{sub}\label{2.2}
{\rm In order to represent $H^2({\cal N} \otimes {\cal N})$ we note that the sequence
(2) is part of the exact diagram
\begin{equation}
\begin{array}{ccccccccc}
& & 0 & & 0 & & & &\\
& & \downarrow & & \downarrow & & & &\\
0 & \longrightarrow & {\cal N} & \longrightarrow & B\otimes\Omega^1(1) &
\buildrel\tilde{b}\over\longrightarrow
& C \otimes {\cal O} & \longrightarrow & 0\\
& & \downarrow & & \downarrow & & \| & &\\
0 & \longrightarrow & H \otimes{\cal O}& \longrightarrow &B\otimes
V^\vee\otimes {\cal O} & \buildrel b\over\longrightarrow
& C \otimes {\cal O} & \longrightarrow & 0\\
& & \downarrow & & \downarrow & & & & \\
& & B \otimes {\cal O}(1) & = & B \otimes {\cal O}(1) & & & & \\
& & \downarrow & & \downarrow & & & &\\
& & 0 & & 0 & & & &
\end{array}
\end{equation}
where $H$ is the kernel of the operator $b$, which is surjective because
$\tilde{b}$ is surjective. The
left--hand column of (6) gives us after tensoring by $\Omega^1(1)$
\begin{equation
B \otimes H^0\Omega^1(2) \buildrel\delta\over\cong H^1({\cal N} \otimes
\Omega^1(1)) \mbox{ and } H^2({\cal N} \otimes \Omega^1(1)) = 0.
\end{equation}
Since $\tilde{b}$ is the Beilinson representation of ${\cal N}$, we have the
commutative diagram
\begin{equation
\begin{array}{ccc}
H^1{\cal N}(-1) \otimes H^0 {\cal O}(1) & \buildrel cup\over\longrightarrow &
H^1{\cal N}\\
\|\wr & & \|\wr\\
B \otimes V^\vee & \buildrel b\over\longrightarrow & C.
\end{array}
\end{equation}
Moreover, $\delta$ in (7) coincides also with cup:
\begin{equation
\begin{array}{cl}
B \otimes H^0 \Omega^1(2) &
\buildrel\delta\over{\buildrel\longrightarrow\over\approx} H^1({\cal N} \otimes
\Omega^1(1))\\
\|\wr & \nearrow_{\mbox{cup}}\\
H^1{\cal N}(-1)\otimes H^0\Omega^1(2) &
\end{array}
\end{equation}
Tensoring the top row of (6) with ${\cal N}$ and using (7) we obtain the following
diagram with exact row:
\begin{equation
\begin{array}{ccccc}
0 \to H^1({\cal N}\otimes{\cal N})\to
& B \otimes H^1({\cal N} \otimes \Omega^1(1))
& \to
& C \otimes H^1({\cal N})
& \to H^2({\cal N} \otimes{\cal N}) \to 0\\
& \|\wr & & \|\wr & \\
& B \otimes B \otimes \Lambda^2 V^\vee_n & \buildrel\Phi\over\longrightarrow
& C \otimes C. &
\end{array}
\end{equation}
It follows that
\begin{equation
H^2({\cal N} \otimes {\cal N}) = Coker(\Phi) = Ker(\Phi^\vee)^\vee.
\end{equation}
}\end{sub}
\begin{lemma}\label{2.3}
The induced operator $\Phi$ is the composition $B \otimes B \otimes \Lambda^2
V^\vee_n \buildrel id \otimes \sigma\over\longrightarrow B \otimes B \otimes
V^\vee_n \otimes V^\vee_n \buildrel b \otimes b\over\longrightarrow C \otimes
C$, where $\sigma$ denotes the canonical desymmetrization.
\end{lemma}
Proof:\quad The computation of $\Phi$ is achieved by the diagram
\unitlength1cm
\begin{picture}(15,9)
\put(1,8){$B\otimes B\otimes \wedge^2 V^\vee_n$}
\put(0,6){$B\otimes H^1 {\cal N} (-1) \otimes H^0\Omega^1(2)$}
\put(0.50,4){$B\otimes H^1({\cal N}\otimes \Omega^1(1))$}
\put(1.2,2){$C\otimes H^1{\cal N}$}
\put(1.5,0){$C\otimes C$}
\put(2.1,7.6){\vector(0,-1){1}}
\put(2.1,5.6){\vector(0,-1){1}}
\put(2.1,3.6){\vector(0,-1){1}}
\put(2.1,1.6){\vector(0,-1){1}}
\put(2.2,7.1){$\approx$}
\put(2.2,5.1){$id_B \otimes cup$}
\put(2.2,3.1){$H^1 (id_{\cal N} \otimes \tilde{b})$}
\put(2.2,1.1){$\approx$}
\put(5.2,8.1){\vector(1,0){4}}
\put(5.2,6.1){\vector(1,0){4}}
\put(5.2,4.1){\vector(1,0){4}}
\put(6.2,8.3){$id_{B\otimes B}\otimes \sigma$}
\put(6.2,6.3){$id\otimes H^0(\iota (1))$}
\put(5.7,4.3){$id_B\otimes H^1(id_{\cal N}\otimes \iota)$}
\put(10.7,8){$B\otimes B\otimes V^\vee_n \otimes V^\vee_n$}
\put(9.7,6){$B\otimes H^1 {\cal N}(-1)\otimes V^\vee_n\otimes H^0 {\cal O}(1)$}
\put(10.7,4){$B\otimes V^\vee_n \otimes H^1 {\cal N}$}
\put(10.9,2){$B\otimes V^\vee_n \otimes C$}
\put(12.4,7.6){\vector(0,-1){1}}
\put(12.4,5.6){\vector(0,-1){1}}
\put(12.4,3.6){\vector(0,-1){1}}
\put(12.6,7.1){$\approx$}
\put(12.6,5.1){$id_{B\otimes V^\vee_n} \otimes cup$}
\put(12.6,3.1){$\approx$}
\put(9.1,1.6){\vector(-4,-1){5}}
\put(9.1,3.6){\vector(-4,-1){5}}
\put(6.7,2.5){$b\otimes id_{H^1{\cal N}}$}
\put(6.7,0.5){$b\otimes id_C$}
\put(7.3,7.2){I}
\put(7.3,5.2){II}
\put(7.3,3.5){III}
\put(7.3,1.8){IV}
\end{picture}
In this diagram $\iota$ denotes the canonical inclusion $\Omega^1(1)
\hookrightarrow V^\vee_n \otimes {\cal O}$, and up to $\Lambda^2 V^\vee_n \cong
H^0\Omega^1(2)$ and $V^\vee_n \cong H^0 {\cal O}(1)$ the map $\sigma$ can be
identified with $H^0(\iota(1))$. Therefore, the square I is commutative.
Square II is a canonically induced diagram of cup--operations and commutative
using $B \cong H^1 {\cal N}(-1)$. The triangle III is induced by the commutative
triangle
\[
\begin{array}{cl}
B\otimes {\cal N} \otimes \Omega^1(1) & \buildrel id\otimes
\iota\over\longrightarrow B \otimes V^\vee \otimes {\cal N}\\
\quad\downarrow \tilde{b} \otimes id & \swarrow_{b \otimes id} \\
C \otimes {\cal N} &
\end{array}
\]
and hence commutative, and the commutativity of IV results just from the
identification $H^1 {\cal N} \cong C$. Now by definition the composition of the
left--hand column is $\Phi$ and the composition of the right--hand column is
$id_B \otimes id_{V^\vee_n} \otimes b$ since $b$ is defined by (8).
It follows that $\Phi = (b \otimes id_C) \circ (id_B \otimes id_{V^\vee_n}
\otimes b) \circ (id_{B \otimes B} \otimes \sigma) = (b \otimes b) \circ (id
\otimes \sigma)$.
\begin{remark}\label{2.4}
{\rm If ${\cal E}$ is a $k$--instanton bundle it is easily checked
that $h^i{\cal E}(d) = h^i{\cal E}^\vee(d) = 0$ for $i \ge 2$ and $d \ge -1$. Using
${\cal E}^\vee \otimes {\cal N}$ again it follows that $Ext^i({\cal E},{\cal E})
= H^i({\cal E}^\vee\otimes {\cal E}) = H^i({\cal E}^\vee \otimes {\cal N}) = 0$ for $i \ge 3$.
This and the Riemann--Roch formula, which can also ad hoc be derived from the
monad representation, give
\[
h^1({\cal E}^\vee \otimes {\cal E}) - h^2({\cal E}^\vee\otimes {\cal E}) = -k^2{2n-1\choose 2} +
8kn^2 - 4n^2 + 1.
\]}
\end{remark}
\newpage
\section{Determination of $\bf Ext^2({\cal E},{\cal E})$}
We are now able to determine $Ext^2({\cal E},{\cal E})$ as a $GL(2)$--representation
space in case of a special instanton bundle. In that case $b$ is the dual of
the operator $\beta : \Lambda^2 U \otimes S_{k - 2} \otimes V_{n-1} \to
S_{k - 1} \otimes V_n$, see notation or \ref{1.4}. Then $\Phi^\vee$ is
the composition of $\beta \otimes \beta$ and the multiplication map $V_n
\otimes V_n \to \Lambda^2V_n$. In order to simplify we choose a fixed basis
$s,t \in U$ and the isomorphism $\Lambda^2 U \cong k$ given by $s \wedge t$.
Then
\[
S_{k - 2} \otimes S_{k - 2} \otimes V_{n-1} \otimes V_{n-1}
\buildrel \Phi^\vee\over\to S_{k - 1} \otimes S_{k - 1} \otimes
\Lambda^2 V_n
\]
is explicitly given by
\begin{eqnarray*}
\Phi^\vee(g \otimes g' \otimes v \otimes v') & = & sg \otimes sg'
\otimes (tv
\wedge tv') - sg \otimes tg' \otimes (tv \wedge sv')\\
& - & tg \otimes sg' \otimes (sv \wedge tv') + tg \otimes tg' \otimes (sv
\wedge sv').
\end{eqnarray*}
In order to determine the kernel of $\Phi^\vee$ we consider the
$GL(U)$--homomorphism
\[
S_{k-3} \otimes S_{k-3} \otimes V_{n-2} \otimes V_{n-2} \buildrel
\epsilon'\over\to S_{k-2} \otimes S_{k-2} \otimes V_{n-1} \otimes V_{n-1}
\]
defined similarly by
\begin{eqnarray*}
\epsilon' (f \otimes f' \otimes u \otimes u') & = & sf \otimes sf' \otimes
tu \otimes tu' - sf \otimes tf' \otimes su \otimes tu'\\
& - & tf \otimes sf' \otimes tu \otimes su' + tf \otimes tf' \otimes su
\otimes su'.
\end{eqnarray*}
Up to the order of factors the map $\epsilon'$ is the tensor product $\beta'
\otimes \beta'$ where $\beta' : S_{k-3} \otimes V_{n-2} \to S_{k-2} \otimes
V_{n-1}$ is defined as $\beta$. Hence, $\epsilon'$ is injective. Finally,
we define $\epsilon$ as the composition
\[
S_{k-3} \otimes S_{k-3} \otimes S^2V_{n-2} \stackrel{id\otimes\iota}{
\longrightarrow} S_{k-3} \otimes S_{k-3} \otimes V_{n-2} \otimes
V_{n-2} \buildrel\epsilon'\over\to S_{k-2} \otimes S_{k-2}
\otimes V_{n-1} \otimes V_{n-1}
\]
where $\iota$ is the canonical desymmetrization. Then also $\epsilon$ is
injective.
\begin{proposition}\label{3.1}
$(S_{k-3} \otimes S_{k-3} \otimes S^2V_{n-2}, \epsilon)$ is the kernel of
$\Phi^\vee$.
\end{proposition}
Proof: A straightforward computation shows that $Im(\epsilon) \subset
Ker(\Phi^\vee)$. In order to show equality we reduce $Ker(\Phi^\vee)$ modulo
$Im(\epsilon)$ using canonical bases of the vector spaces. A more elegant
proof using Clebsch--Gordan decompositions seems much harder to achieve. Let
us denote the bases as follows:
\begin{tabular}{lll}
basis of $S_{k-3}:$ & $e_\alpha = s^{k-3-\alpha} t^\alpha$ &$ 0 \le \alpha
\le k-3$\\[0.5ex]
basis of $S_{k-2}:$ & $f_\alpha = s^{k-2-\alpha}t^\alpha $&$ 0 \le \alpha
\le k-2$\\[0.5ex]
basis of $S_{k-1}:$ &$ g_\alpha = s^{k-1-\alpha}t^\alpha $&$ 0 \le
\alpha \le k-1$\\[1.0ex]
basis of $V_{n-2}: $&$ u_\mu = s \otimes s^{n-2-\mu}t^\mu$ &$ 0 \le \mu
\le n-2$\\
& $\bar{u}_\mu = t \otimes s^{n-2-\mu} t^\mu$ & \\[0.5ex]
basis of $V_{n-1}: $&$ x_\mu = s \otimes s^{n-1-\mu}t^\mu$ &$ 0 \le \mu
\le n-1$\\
& $\bar{x}_\mu = t \otimes s^{n-1-\mu} t^\mu $& \\[0.5ex]
basis of $V_n: $&$ y_\mu = s \otimes s^{n-\mu} t^\mu$ &$ 0 \le \mu
\le n$\\
& $\bar{y}_\mu = t \otimes s^{n-\mu}t^\mu.$ &
\end{tabular}
For the basis $f_\alpha \otimes f_\beta \otimes x_\mu \otimes x_\nu,\;
f_\alpha \otimes f_\beta \otimes x_\mu \otimes \bar{x}_\nu,\; f_\alpha \otimes
f_\beta \otimes \bar{x}_\mu \otimes x_\nu,\, f_\alpha \otimes f_\beta \otimes
\bar{x}_\mu \otimes \bar{x}_\nu$ we use the index tuplets $(\alpha, \beta,
\mu, \nu),\; (\alpha, \beta, \mu,\bar{\nu}),\; (\alpha, \beta, \bar{\mu},\nu),
\; (\alpha, \beta, \bar{\mu}, \bar{\nu})$ respectively. The set
of these indices will be ordered {\bf lexicographically} with the additional
assumption that always $\mu < \bar{\nu}$. Then, for example, $(\alpha,
\beta, \mu, \bar{\nu}) < (\alpha, \beta, \bar{\lambda}, \delta)$.
Accordingly, the coefficients of an element $\xi \in S_{k-2} \otimes S_{k-2}
\otimes V_{n-1} \otimes V_{n-1}$ will be denoted by $c(\alpha, \beta, \mu,
\nu),\; c(\alpha, \beta, \mu, \bar{\nu}),\; c(\alpha, \beta, \bar{\mu},
\nu),\; c(\alpha, \beta, \bar{\mu}, \bar{\nu})$.
By the formula for $\Phi^\vee$ we obtain the
\begin{lemma}\label{3.2}
Let $\xi \in S_{k-2} \otimes S_{k-2} \otimes V_{n-1} \otimes V_{n-1}$.
\begin{itemize}
\item[(i)] The coefficient of $\Phi^\vee(\xi)$ at the basis element $g_\alpha
\otimes g_\beta \otimes y_\mu \wedge \bar{y}_\nu$ in $S_{k-1} \otimes S_{k-1}
\otimes \Lambda^2V_n$ is
$\phantom{-}c(\alpha, \beta, \mu-1, \overline{\nu-1}) - c(\alpha, \beta,
\overline{\nu-1}, \mu-1)$\\[0.5ex]
$-c(\alpha, \beta-1, \mu-1, \bar{\nu}) + c(\alpha, \beta-1, \overline{\nu-1},
\mu)$\\[0.5ex]
$-c(\alpha-1, \beta, \mu, \overline{\nu-1}) + c(\alpha-1, \beta, \bar{\nu},
\mu)$\\[0.5ex]
$+c(\alpha-1, \beta-1, \mu, \bar{\nu}) - c(\alpha-1, \beta-1, \bar{\nu},
\bar{\mu}).$
Here we agree that each of these coefficients is 0 if one of $\alpha,
\alpha-1, \beta,
\beta-1 \not\in [0,k-2]$ or if one of $\mu, \mu-1, \nu, \nu-1 \not\in [0,
n-1]$.
\item[(ii)] Analogous statements hold for the coefficient of $\Phi^\vee(\xi)$
at $g_\alpha \otimes g_\beta \otimes y_\mu \wedge y_\nu$ for $\mu < \nu$
(without bars) and at $g_\alpha \otimes g_\beta \otimes \bar{y}_\mu \wedge
\bar{y}_\nu$ for $\mu < \nu$ (with two bars).
\end{itemize}
\end{lemma}
\begin{lemma}\label{3.3} Let the notation be as above. If $\Phi^\vee(\xi) =
0$ then:
\begin{itemize}
\item[(i)] If $c(\alpha, \beta, \mu, \nu)$ is the first non--zero coefficient
of $\xi$ (in the lexicographical order), then $0 < \mu \le \nu$.
\item[(ii)] If $c(\alpha, \beta, \mu, \bar{\nu})$ is the first non--zero
coefficient of $\xi$, then $\mu \not= 0,\; \nu \not= 0$.
\item[(iii)] $c(\alpha, \beta, \bar{\mu}, \nu)$ is never a first non--zero
coefficient of $\xi$.
\item[(iv)] If $c(\alpha, \beta, \bar{\mu}, \bar{\nu})$ is the first non--zero
coefficient of $\xi$, then $0 < \mu \le \nu$.
\end{itemize}
\end{lemma}
Proof: (i)\quad Let $c(\alpha, \beta, \mu, \nu)$ be the first coefficient of
$\xi$. Then, by Lemma \ref{3.2} the coefficient of $0 = \Phi^\vee(\xi)$ at
$g_\alpha \otimes g_\beta \otimes y_{\mu+1} \wedge y_{\nu+1}$ is
\begin{eqnarray*}
0 & = & c(\alpha, \beta, \mu, \nu) - c(\alpha, \beta, \nu, \mu) - c(\alpha,
\beta-1, \mu, \nu + 1) + c(\alpha, \beta-1, \nu, \mu + 1)\\
& - & c(\alpha-1, \beta, \mu + 1, \nu) +
c(\alpha-1, \beta, \nu + 1, \mu) -
\ldots
\end{eqnarray*}
Since $c(\alpha, \beta, \mu, \nu)$ is the first coefficient, only the first two
in this formula could be non--zero because the others have smaller index in
the lexicographical order. Hence
\[
c(\alpha, \beta, \mu, \nu) = c(\alpha, \beta, \nu, \mu).
\]
If $\mu > \nu$ then $c(\alpha, \beta, \nu, \mu)$ would be earlier and
non--zero. Hence, $\mu \le \nu$. Assume now that $\mu = 0$. The
coefficient of $\phi^\vee(\xi)$ of $g_\alpha \otimes g_{\beta + 1} \otimes y_0
\wedge y_{\nu+1}$ is
\begin{eqnarray*}
0 & = & c(\alpha, \beta + 1, -1, \nu) - c(\alpha, \beta+1, \nu, -1)\\
& - & c(\alpha, \beta, -1, \nu+1) + c(\alpha, \beta, \nu, 0) \mp \ldots
\end{eqnarray*}
In this sum all but $c(\alpha, \beta, \nu, 0)$ are automatically zero because
$(\alpha-1, \beta, \ldots) \le (\alpha, \beta, 0, \nu)$ and $-1$ occurs.
Hence, $c(\alpha, \beta, 0, \nu) = c(\alpha, \beta, \nu, 0) = 0$,
contradiction.
The statements (ii), (iii), (iv) are proved analogously. \hfill $\Box$\\
Now we continue the proof of Proposition \ref{3.1}. We reduce an element
$\xi \in Ker(\Phi^\vee)$ to $0\, mod\, Im(\epsilon)$ using Lemma \ref{3.3}.
\begin{itemize}
\item[a)] Assume that the first non--zero coefficient of $\xi$ is
\[
c(\alpha, \beta, \mu, \nu).
\]
Then by Lemma \ref{3.3}\quad $0 < \mu \le \nu$.
Then the element
\[
\xi' = \xi - c(\alpha, \beta, \mu, \nu) \epsilon (e_\alpha \otimes e_\beta
\otimes u_{\mu-1} \cdot u_{\nu-1})
\]
belongs to $Ker(\Phi^\vee)$. We have
$\epsilon(e_\alpha \otimes e_\beta \otimes u_{\mu-1} \cdot u_{\nu-1})$\\
$ = f_\alpha \otimes f_\beta \otimes (x_\mu \otimes x_\nu + x_\nu \otimes
x_\mu)$\\
$- f_\alpha \otimes f_{\beta+1} \otimes (x_{\mu-1} \otimes x_\nu + x_{\nu-1}
\otimes x_\mu)$\\
$- f_{\alpha + 1} \otimes f_\beta \otimes (x_\mu \otimes x_{\nu-1} + x_\nu
\otimes x_{\mu-1})$\\
$+ f_{\alpha +1} \otimes f_{\beta + 1} \otimes (x_{\mu-1} \otimes x_{\nu-1} +
x_{\nu-1} \otimes x_{\mu-1})$
and therefore $\xi'$ is a sum of monomials of index $> (\alpha, \beta, \mu,
\nu)$. Hence, we can assume that $\xi\, mod\, Im(\epsilon)$ has no
coefficient with index $(\alpha, \beta, \mu, \nu)$.
\item[b)] By Lemma \ref{3.3} we can assume that the first non--zero
coefficient of $\xi$ has index $(\alpha, \beta, \mu, \bar{\nu})$ or $(\alpha,
\beta, \bar{\mu}, \bar{\nu})$. In the first case we know by Lemma \ref{3.3}
that $0 < \mu, \nu$. When we consider again
\[
\xi' = \xi - c(\alpha, \beta, \mu, \bar{\nu}) \epsilon (e_\alpha \otimes
e_\beta \otimes u_{\mu-1} \cdot \bar{u}_{\nu-1})
\]
we have $\phi^\vee(\xi') = 0$ and $\xi'$ is a sum of monomials of index
$>(\alpha, \beta, \mu, \bar{\nu})$. Hence, we may assume that $\xi\, mod\,
Im(\epsilon)$ has $c(\alpha, \beta, \bar{\mu}, \bar{\nu})$ as first non--zero
coefficient. Again by Lemma \ref{3.3}\quad $0 < \mu, \nu$ and
\[
\xi' = \xi - c(\alpha, \beta, \bar{\mu}, \bar{\nu}) \epsilon (e_\alpha
\otimes e_\beta \otimes \bar{u}_{\mu-1} \cdot \bar{u}_{\nu-1})
\]
is a sum of monomials of index $>(\alpha, \beta, \bar{\mu}, \bar{\nu})$.
This finally shows that $\xi = 0\, mod\, Im(\epsilon)$.
\end{itemize}
This completes the proof of Proposition \ref{3.1}.
\newpage
\section{Conclusions}
By Proposition \ref{2.1}, Proposition \ref{3.1}, (11) and Lemma \ref{2.3}
we have determined the space
$Ext^2({\cal E}, {\cal E})$. Together with Remark \ref{2.4} we obtain
\begin{theorem}\label{4.1} For any special symplectic $k$--instanton bundle
${\cal E}$ on $\P_{2n+1}$
\begin{itemize}
\item[(1)] $Ext^2({\cal E},{\cal E}) \cong S^\vee_{k-3} \otimes S^\vee_{k-3} \otimes
S^2V^\vee_{n-2}$
\item[(2)] $\dim\, Ext^2({\cal E},{\cal E}) = (k-2)^2 {2n-1\choose 2}$
\item[(3)] $\dim\, Ext^1({\cal E},{\cal E}) = 4k(3n-1) + (2n-5)(2n-1)$.
\end{itemize}
\end{theorem}
Let $MI_{2n+1}(k)$ denote the open part of the Maruyama scheme of
semi--stable coherent sheaves on $\P_{2n+1}$ with Chern polynomial
$(1-h^2)^{-k}$
consisting of instanton bundles. By \cite{AO1} any special symplectic
instanton bundle ${\cal E}$ is stable. Therefore, $Ext^1({\cal E},{\cal E})$ can be
identified with the tangent space of $MI_{2n+1}(k)$ at ${\cal E}$. In \cite{AO2}
deformations ${\cal E}'$ of special symplectic instanton bundles in $MI_{2n+1}(k)$
have been found for $n = 2$ and $k = 3,4$ which satisfy $Ext^2({\cal E}', {\cal E}') =
0$.
This shows that in these cases there
are components $MI'_{2n+1}(k)$ of $MI_{2n+1}(k)$ of the expected dimension
$4(3n-1) k+(2n-5)(2n-1)$ containing the set of special instanton bundles. In
particular, see \cite{AO2}:
{\it for $k = 3,4$ the moduli space
$MI_5(k)$ is singular at least in special symplectic bundles.}
However, in case $2n+1 = 3$ we obtain the vanishing result of \cite{HN}:
{\it
any special $k$--instanton bundle ${\cal E}$ on $\P_3$ satisfies $Ext^2({\cal E},{\cal E}) =
0$ and is a smooth point of $MI_3(k)$,}
since any rank--2 instanton bundle is
symplectic.
\newpage
\addcontentsline{toc}{section}{References}
|
1992-04-02T13:33:21 | 9204 | alg-geom/9204001 | en | https://arxiv.org/abs/alg-geom/9204001 | [
"alg-geom",
"math.AG"
] | alg-geom/9204001 | null | Marc Coppens and Takao Kato | Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane
Curves | 13 pages, LaTeX 2.09 | null | null | null | null | Let $\Gamma$ be a plane curve of degree $d$ with $\delta$ ordinary nodes and
no other singularities. If $P$ is a smooth point on $\Gamma$ then the
Weierstrass gap sequence at $P$ is considered as that at the corresponding
point on the normalization of $\Gamma$. A smooth point $P\in\Gamma$ is called a
total inflection point if $i(\Gamma ,T;P)=d$ where $T$ is the tangent line to
$\Gamma$ at $P$. There are many possible Weierstrass gap sequences at total
inflection points. Our main results are: Among them (1) There exists a pair
$(P,\Gamma )$ such that the gap sequence at $P$ is the minimal (in the sense of
weight). (2) There exists a pair $(P,\Gamma )$ such that the gap sequence at
$P$ is the maximal (resp. up to 1 maximal). And we characterize these cases in
the sense of location of nodes.
| [
{
"version": "v1",
"created": "Thu, 2 Apr 1992 17:28:22 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Coppens",
"Marc",
""
],
[
"Kato",
"Takao",
""
]
] | alg-geom | \section{Introduction}
Let ${\mit \Gamma}$ be a plane curve of degree $d$ with $\delta$ ordinary nodes and no
other singularities. Let $C$ be the normalization of ${\mit \Gamma}$. Let
$g=\fracd{(d-1)(d-2)}{2}-\delta$; the genus of $C$. We identify smooth
points of ${\mit \Gamma}$ with the corresponding points on $C$. In particular, if
$P$ is a smooth point on ${\mit \Gamma}$ then the Weierstrass gap sequence at $P$
is considered with respect to $C$. A smooth point $P\in{\mit \Gamma}$ is called an
$(e-2)$-inflection point if $i({\mit \Gamma} ,T;P)=e\ge 3$ where $T$ is the
tangent line to ${\mit \Gamma}$ at $P$
(cf. Brieskorn--Kn\"{o}rrer\cite[p.~372]{B}). Of course, $e\le d$ and a
1-inflection point is an ordinary flex. In particular, a
$(d-2)$-inflection point is called a total inflection point.
Let $N$ be the semigroup consisting of the non-gaps of $P$, so
${\bbb N}-N=\{\alpha_1<\alpha_2<\cdots <\alpha_g\}$ is the Weierstrass gap
sequence of $P$. Clearly $\{ d-1,d\}\subset N$, so
$N_d:=\{ a(d-1)+bd|a,b\in{\bbb N}\}\subset N$ (see also Lemma \ref{lem:1}).
Let $k=\min\{\ell\in{\bbb N}|\delta\le\fracd{\ell (\ell +3)}{2}\}$ and let
$$
N_{d,\delta}^{(1)}=N_d\cup\{ n\in{\bbb N}|n\ge (d-k-3)d+\fracd{k(k+3)}{2}
-\delta +2\}.
$$
Let ${\bbb N}-N_{d,\delta}^{(1)}=\{\alpha_1^{(1)}<\alpha_2^{(1)}<\cdots <
\alpha_g^{(1)}\}$. One has $\alpha_i\ge\alpha_i^{(1)}$ for $1\le i\le g$. So
$N_{d,\delta}^{(1)}$ is the minimal (in the sense of weight) possible
semigroup of non-gaps.
For $\delta\in\{ 0,1\}$, one has $N=N_{d,\delta}^{(1)}$. For $\delta\ge 2$ there
exist pairs of $({\mit \Gamma} ;P)$ as above with $N\ne N_{d,\delta}^{(1)}$.
We give a list of all possible values for $N$ in case $2\le\delta\le 5$.
(see end of \S 1).
Define $N_{d,1}^{({\rm max})}=N_{d,1}^{({\rm max2})}=N_{d,1}$ and, by
means of induction, for $\delta\ge 2$,
\begin{eqnarray*}
N_{d,\delta}^{({\rm max})} & = & N_{d,\delta -1}^{({\rm max})}\cup
\{ (d-\delta -2)d+1\} \\
N_{d,\delta}^{({\rm max2})} & = & N_{d,\delta -1}^{({\rm max2})}\cup
\{ (d-\delta -2)d+\delta\}.
\end{eqnarray*}
$N_{d,\delta}^{({\rm max})}$ (resp. $N_{d,\delta}^{({\rm max2})}$) is a
semigroup if and only if $d\ge 2\delta +1$ (resp. $2\delta$). Let
\begin{eqnarray*}
{\bbb N}-N_{d,\delta}^{({\rm max})} & = &
\{\alpha_1^{({\rm max})}<\alpha_2^{({\rm max})}<\cdots <\alpha_g^{({\rm max})}\} \\
{\bbb N}-N_{d,\delta}^{({\rm max2})} & = &
\{\alpha_1^{({\rm max2})}<\alpha_2^{({\rm max2})}<\cdots <\alpha_g^{({\rm max2})}\}.
\end{eqnarray*}
We prove that $\alpha_i\le\alpha_i^{({\rm max})}$ for $1\le i\le g$ and if
$N\ne N_{d,\delta}^{({\rm max})}$, then $\alpha_i\le\alpha_i^{({\rm max2})}$ for
$1\le i\le g$ (Lemma \ref{lem:max}). So $N_{d,\delta}^{({\rm max})}$ (resp.
$N_{d,\delta}^{({\rm max2})}$) is the maximal (resp. up to 1 maximal)
semigroup of non-gaps.
Our main results are the following:
\begin{enumerate}
\item There exist pairs $({\mit \Gamma} ;P)$ such that $N=N_{d,\delta}^{(1)}$
(\ref{thm:general}),
\item If $d\ge 2\delta +1$ (resp. $d\ge 2\delta$) then there exist pairs
$({\mit \Gamma} ;P)$ such that $N=N_{d,\delta}^{(max)}$ (resp. $N=N_{d,\delta}^{(max2)}$)
(\ref{prop:max}).
\end{enumerate}
The existence of Weierstrass points with gap sequence
${\bbb N}-N_{d,\delta}^{(1)}$ is already proved in \cite{K} for the case
$\delta =\fracd{d^2-7d+12}{2}$. The method used in that paper is completely
different from ours. It has the advantage of not using plane models but
the proof looks more complicated. It might be possible to prove our
existence result in this way completely, but it might become very
complicated. We didn't try it. Also, it gives an affirmative answer to
Question 1 in \cite{C} for the case $s=n+1$. It is not clear to us at the
moment how to generalize the proof for the cases with $s\ge n+2$.
\section{Generalities and low values for $\delta$}
To start, we deal with the case $\delta =0$.
\vs 2
\begin{lem}
Let ${\mit \Gamma}$ be a smooth plane curve of degree $d$ and let $P$ be a
total inflection point of ${\mit \Gamma}$. Then $N_d=N_{d,0}^{(1)}$ is the
semigroup of non-gaps of $P$.
\label{lem:0}
\end{lem}
\vs 1
{\it Proof\/}. Let $T$ be the tangent line at $P$, $L_1$ be a general
line passing through $P$ and let $L_2$ be a general line not passing
through $P$. Then the curve $C(a,b)=aT+bL_1+(d-3-a-b)L_2$ is canonical
adjoint, if $0\le a\le d-3, 0\le b\le d-3-a$. Then we have
$i({\mit \Gamma} .C(a,b);P)=ad+b$. Hence,
$\{ ad+b+1:0\le a\le d-3, 0\le b\le d-3-a\}$ is the gap sequence at $P$.
This completes the proof.
\vs 2
In order to study the case $\delta >0$, we prove some lemmas. For the rest
of this section, ${\mit \Gamma}$ is a plane curve of degree $d$ with $\delta (>0)$
ordinary nodes $s_1,\dots ,s_{\delta}$ as its only singularities. Also
$P\in{\mit \Gamma}$ is a total inflection point.
\vs 2
\begin{lem}
The set of nongaps at $P$ contains $N_{d,0}$.
\label{lem:1}
\end{lem}
\vs 2
{\it Proof\/}. Assume that $n\in N_{d,0}$. Let $\alpha =\left[\fracd{n-1}{d}
\right] +1$, $\ell$ be the equation of $T$ (the tangent line at $P$),
$\ell_0$ be the equation of a general line passing through $P$ and let
$\ell_1$ be the equation of a general line. Considering
$$
\frac{\ell_0^{\alpha d-n}\ell_1^{\alpha +n-\alpha d}}{\ell^{\alpha}},
$$
we obtain that $n$ is a nongap at $P$.
\vs 2
\begin{lem}
Let $\gamma$ be a curve of degree less than $d$ so that
$i(\gamma ,{\mit \Gamma} ;P)=k\ge d$. Then, $T$ is a component of $\gamma$, i.~e. there
is a curve $\gamma'$ of degree $\deg\gamma -1$ such that $\gamma =\gamma'T$.
\label{lem:2}
\end{lem}
\vs 2
{\it Proof\/}. Since $i(T,{\mit \Gamma} ;P)=d$ and $i(\gamma ,{\mit \Gamma} ;P)\ge d$, by
Namba's lemma \cite[Lemma 2.3.2]{namba} (cf. Coppens and Kato
\cite[Lemma 1.1]{ck1} for a generalization), we have
$i(T,\gamma ;P)\ge d>\deg\gamma$.
Hence we have the desired result by Bezout's theorem.
\vs 2
By a successive use of this lemma we have:
\vs 2
\begin{lem}
Let $\gamma$ be a canonical adjoint curve such that
$i(\gamma ,{\mit \Gamma} ;P)=\alpha d+\beta$ $(0\le\alpha\le d-3, 0\le\beta\le d-3-\alpha )$.
Then, there is an adjoint curve $\gamma'$ of degree $d-3-\alpha$ such that
$\gamma =T^{\alpha}\gamma'$ and $i(\gamma',{\mit \Gamma} ;P)=\beta$.
\label{lem:3}
\end{lem}
\vs 2
Using Lemma \ref{lem:3} we have the following corollaries:
\vs 2
\begin{cor}
If $\delta\ge 1$, then $i(\gamma ,{\mit \Gamma} ;P)<(d-3)d$ for every canonical adjoint
curve $\gamma$, hence $(d-3)d+1$ is a nongap at $P$.
\label{cor:1}
\end{cor}
\vs 2
\begin{cor}
Assume that $\delta\ge 2$. Then, $(d-4)d+\beta +1$ $(\beta =0$ or $1)$ is a gap
if and only if there is a line $L_0$ such that
$s_1,\dots ,s_{\delta}\in L_0$. Moreover, in this case, the following three
conditions are equivalent:
\begin{enumerate}
\item $P\notin L_0$, {\rm (resp.} $P\in L_0)$,
\item $(d-4)d+1$ {\rm (resp.} $(d-4)d+2)$ is a gap,
\item $(d-3-\alpha )d+1+\alpha$ $(\alpha =1,\dots ,\delta -1)$ {\rm (resp.}
$(d-3-\alpha )d+1$ $(\alpha =1,\dots ,\delta -1))$ are nongaps.
\end{enumerate}
\label{cor:2}
\end{cor}
\vs 2
{\it Proof\/}. The existence of the line $L_0$ and the equivalence
between (i) and (ii) follows immediately from Lemma \ref{lem:3}.
Assume that $(d-4)d+1$ is a gap. If $(d-3-\alpha )d+\alpha +1$
$(1\le\alpha\le\delta -1)$ is a gap then Lemma \ref{lem:3} provides an adjoint
curve $\gamma'$ of degree $\alpha$ with $i(\gamma' ,{\mit \Gamma} ;P)=\alpha$. So $\gamma'$ has
$s_1,\dots ,s_{\delta}$ as common points with $L_0$. Bezout's theorem
implies that $\gamma'=\gamma''L_0$ where $\gamma''$ is a curve of degree $\alpha -1$
with $i(\gamma'',{\mit \Gamma} ;P)=\alpha$ (since $P\notin L_0$). Namba's lemma implies
$\gamma''=\gamma'''T$, but then $i(\gamma'',{\mit \Gamma} ;P)\ge d$, so
$\delta\ge\alpha +1\ge d+1$. A contradiction since $s_1,\dots ,s_{\delta}$ are
collinear.
Assume that $(d-4)d+2$ is a gap. If $(d-3-\alpha )d+1$ $(1\le\alpha\le\delta -1)$
is a gap then Lemma \ref{lem:3} provides an adjoint curve $\gamma'$ of
degree $\alpha$ with $i(\gamma' ,{\mit \Gamma} ;P)=0$. But $\gamma'=\gamma''L_0$ and
$P\in L_0$, hence a contradiction.
Assuming (iii), we obtain (ii) because the number of gaps has to be $g$.
\vs 2
Using Lemma \ref{lem:1} and Corollary \ref{cor:2}, we are able to
determine the gap sequence in case that $s_1,\dots ,s_{\delta}$ are
collinear.
Checking case by case by use of Lemmas \ref{lem:1} and \ref{lem:3}, we
show a table of possible nongaps $N_{d,\delta}$ for $1\le\delta\le 5$.
\newpage
{
\parindent -4.5em
\begin{tabular}[t]{|p{13em}|p{22.5em}|}
\hline
$N_{d,1}=N_{d,0}\cup\{ (d-3)d+1\}$ &
general\\
\hline\hline
$N_{d,2}^{(1)}=N_{d,1}\cup\{ (d-4)d+2\}$ & general\\
\hline
$N_{d,2}^{(2)}=N_{d,1}\cup\{ (d-4)d+1\}$ & $s_1, s_2, P$ are collinear\\
\hline\hline
$N_{d,3}^{(1)}=N_{d,2}^{(1)}\cup\{ (d-4)d+1\}$ & general\\
\hline
$N_{d,3}^{(2)}=N_{d,2}^{(1)}\cup\{ (d-5)d+3\}$ & $s_1, s_2, s_3$ are
collinear but not $P$\\
\hline
$N_{d,3}^{(3)}=N_{d,2}^{(2)}\cup\{ (d-5)d+1\}$ & $s_1, s_2, s_3, P$ are
collinear\\
\hline\hline
$N_{d,4}^{(1)}=N_{d,3}^{(1)}\cup\{ (d-5)d+3\}$ & general\\
\hline
$N_{d,4}^{(2)}=N_{d,3}^{(1)}\cup\{ (d-5)d+2\}$ & $s_1,\dots ,s_4$ general
but $i(\gamma ,{\mit \Gamma} ;P)=2$ where $\gamma$ is the conic passing through
$s_1,\dots ,s_4,P$\\
\hline
$N_{d,4}^{(3)}=N_{d,3}^{(1)}\cup\{ (d-5)d+1\}$ & $s_1,s_2,s_3,P$ are
collinear but not $s_4$\\
\hline
$N_{d,4}^{(4)}=N_{d,3}^{(2)}\cup\{ (d-6)d+4\}$ & $s_1,s_2,s_3,s_4$ are
collinear but not $P$\\
\hline
$N_{d,4}^{(5)}=N_{d,3}^{(3)}\cup\{ (d-6)d+1\}$ & $s_1,s_2,s_3,s_4,P$ are
collinear\\
\hline\hline
$N_{d,5}^{(1)}=N_{d,4}^{(1)}\cup\{ (d-5)d+2\}$ & general\\
\hline
$N_{d,5}^{(2)}=N_{d,4}^{(1)}\cup\{ (d-5)d+1\}$ & $s_1,\dots ,s_5$ general
but $\exists$ conic $\gamma$ passing through $s_1,\dots ,s_5, P$ and
$i(\gamma ,{\mit \Gamma} ;P)=1$\\
\hline
$N_{d,5}^{(3)}=N_{d,4}^{(2)}\cup\{ (d-5)d+1\}$ & $s_1,\dots ,s_5$ general
but $\exists$ conic $\gamma$ passing through $s_1,\dots ,s_5, P$ and
$i(\gamma ,{\mit \Gamma} ;P)=2$\\
\hline
$N_{d,5}^{(4)}=N_{d,4}^{(1)}\cup\{ (d-6)d+4\}$ & $s_1,\dots ,s_4$ are
collinear but not $s_5, P$\\
\hline
$N_{d,5}^{(5)}=N_{d,4}^{(3)}\cup\{ (d-6)d+1\}$ & $s_1,\dots ,s_4, P$ are
collinear but not $s_5$\\
\hline
$N_{d,5}^{(6)}=N_{d,4}^{(4)}\cup\{ (d-7)d+5\}$ & $s_1,\dots ,s_5$ are
collinear but not $P$\\
\hline
$N_{d,5}^{(7)}=N_{d,4}^{(5)}\cup\{ (d-7)d+1\}$ & $s_1,\dots ,s_5, P$ are
collinear\\
\hline\hline
\end{tabular}
}
\newpage
\section{General Case ($\delta\ge 2$)}
Remember the definition of $N_{d,\delta}^{(1)}$, let
$k=\min\{\ell\in{\bbb N}|\delta\le\fracd{\ell (\ell +3)}{2}\}$.
Then
$$
N_{d,\delta}^{(1)}=N_d\cup\{ n\in{\bbb N}|n\ge (d-k-3)d+\fracd{k(k+3)}{2}
-\delta +2\}.
$$
In this section, we prove that for $({\mit \Gamma} ;P)$ general, the semigroup of
non-gaps of $P$ is equal to $N_{d,\delta}^{(1)}$.
\vs 1
Let ${\bbb P}_{\ell}\cong{\bbb P}^{\ell (\ell +3)/2}$ be the linear system of
divisors of degree $\ell$ on ${\bbb P}^2$. Let
$$
{\bbb P}_{\ell}(s_1,\dots ,s_{\delta})=\{\gamma\in{\bbb P}_{\ell}|s_1,\dots ,
s_{\delta}\in\gamma\},
$$
and let
$$
{\bbb P}_k(s_1,\dots ,s_{\delta};m)=\{\gamma\in{\bbb P}_k(s_1,\dots ,
s_{\delta})|i({\mit \Gamma} ,\gamma ;P)\ge m\}.
$$
\vs 1
\begin{lem}
Assume that
$$
(*)\qquad\left\{
\begin{array}{ll}
{\bbb P}_{\ell}(s_1,\dots ,s_{\delta})=\emptyset\quad & {\rm if\ }\ell <k,\\
{\bbb P}_k(s_1,\dots ,s_{\delta};m)=\emptyset\quad & {\rm if\ }m>
\fracd{k(k+3)}{2}-\delta .
\end{array}\right.
$$
Then the Weierstrass gap sequence of ${\mit \Gamma}$ at $P$ is given by
${\bbb N}^+-N^{(1)}_{d,\delta}$.
\label{lem:general}
\end{lem}
\vs 2
{\it Proof\/}. By Lemma \ref{lem:1}, every element of $N_{d,0}$ is a
nongap. For $0\le n\le d-3$ the natural number not belonging to
$N_{d,0}$ are $nd+1,\dots ,nd+(d-n-2)$. Assume such a number $nd+\beta$
(hence $0\le n\le d-3$, $1\le\beta\le d-n-2$) is a gap. Then there exists a
canonical adjoint curve $\gamma$ of ${\mit \Gamma}$ with
$$
i(\gamma ,{\mit \Gamma} ;P)=nd+\beta -1.
$$
Lemma \ref{lem:3} gives us that there exists
$\gamma'\in{\bbb P}_{d-3-n}(s_1,\dots ,s_{\delta})$ with
$i(\gamma',{\mit \Gamma} ;P)=\beta -1$. But the hypothesis $(*)$ implies that this is
impossible for $d-3-n<k$, i.e. $n>d-3-k$ or for $n=d-3-k$ and
$\beta -1>\fracd{k(k+3)}{2}-\delta$. So, the only possible gaps are
$$
\begin{array}{llcl}
1, \ \ \ 2, & \dots & \dots ,& d-2\\
d+1, \ d+2, & \dots & \dots ,& 2d-3\\
2d+1, 3d+2, & \dots & \dots ,& 3d-4\\
& \dots & \dots & \\
(d-4-k)d+1, & \dots & \dots ,& (d-3-k)d-(d-2-k)\\
(d-3-k)d+1, & \dots & \dots ,& (d-3-k)d+\fracd{k(k+3)}{2}-\delta +1.
\end{array}
$$
Since these are $g$ mumbers, we obtain the gaps of $C$ at $P$. It is
clear that this set is ${\bbb N}^+-N^{(1)}_{d,\delta}$.
\vs 2
\begin{thm}
The hypothesis $(*)$ in Lemma {\rm \ref{lem:general}} occurs.
\label{thm:general}
\end{thm}
\vs 1
{\it Proof\/}.
(Inspired by the proof of Proposition 3.1 in \cite{T}).
Take a union of $d$ general lines in ${\bbb P}^2$: ${\mit \Gamma}_0=L_1\cup L_2\cup
\cdots\cup L_d$.
Let $P_1=L_1\cap L_2$, $\{ P_2,P_3\} =L_3\cap (L_1\cup L_2)$ and so on.
Take $0\le\delta\le\fracd{(d-1)(d-2)}{2}$. The statement $(*)$ holds for
${\mit \Gamma}_0$ instead of ${\mit \Gamma}$ and $s_1=P_1,\dots ,s_{\delta}=P_{\delta}$ and $P_0$
suitably chosen on $L_d$.
Indeed, let $k=\min\{\ell\in{\bbb N}|\delta\le\fracd{\ell (\ell +3)}{2}\}$.
Take $\ell <k$ and assume that
$\gamma\in{\bbb P}_{\ell}(P_1,\dots ,P_{\delta})$. Since
$$
\{P_{\frac{(\ell +1)\ell}{2}+1},\dots ,P_{\frac{(\ell +2)(\ell +1)}{2}}
\} = L_{\ell +2}\cap (L_1\cup\cdots\cup L_{\ell +1})\subset\gamma
$$
one has $\gamma =\gamma_{\ell -1}\cup L_{\ell +2}$ with
$\gamma_{\ell -1}\in{\bbb P}_{\ell -1}(P_1,\dots ,
P_{\frac{(\ell +1)\ell}{2}})$. Continuing this way one finds
$$
\gamma =L_{\ell +2}\cup\gamma_{\ell -1}=L_{\ell +2}\cup L_{\ell +1}\cup
\gamma_{\ell -2}=\cdots =L_{\ell +2}\cup\cdots\cup L_4\cup\gamma_1,
$$
where $\gamma_j\in{\bbb P}_j(P_1,\dots ,P_{\frac{(j+2)(j+1)}{2}}),
(j=1,\dots ,\ell -1)$. Since $P_1, P_2, P_3$ are not collinear, this is
impossible.
This already proves that ${\bbb P}_{\ell}(P_1,\dots ,P_{\delta})=\emptyset$
for $\ell <k$. In particular
${\bbb P}_k(P_1,\dots ,P_{\frac{(k+1)(k+2)}{2}})=\emptyset$. This implies
$\dim ({\bbb P}_k(P_1,\dots ,P_{\delta}))=\fracd{k(k+3)}{2}-\delta$. Because
$\delta\le\fracd{(d-1)(d-2)}{2}$, $\{ P_1,\dots ,P_{\delta}\}\cap L_d=
\emptyset$. So if some element of ${\bbb P}_k(P_1,\dots ,P_{\delta})$ would
contain $L_d$ then ${\bbb P}_{k-1}(P_1,\dots ,P_{\delta})\ne\emptyset$, a
contradiction.
Hence, ${\bbb P}_k(P_1,\dots ,P_{\delta})$ induces a linear system of
dimension $\fracd{k(k+3)}{2}-\delta$ on $L_d$. For $P_0$ general on $L_d$
and $\gamma\in{\bbb P}_k(P_1,\dots ,P_{\delta})$, this implies
$i(\gamma, L_d;P_0)\le\fracd{k(k+3)}{2}-\delta$, hence
$$
{\bbb P}_k(P_1,\dots ,P_{\delta};m)=\emptyset\quad {\rm if\ }m>
\fracd{k(k+3)}{2}-\delta .
$$
\vs 1
{\raggedright{\sc Claim}}: There exists a smooth (affine) curve $T$ and
$0\in T$ and a family of plane curves of degree $d$
\setlength{\unitlength}{1mm}
\begin{picture}(70,40)(-40,0)
\put(2,6.5){$T$}
\put(0,20){$p$}
\put(26,20){$p_T$}
\put(2,32){${\cal C}$}
\put(40.5,31.5){$T\times{\bbb P}^2$}
\put(3,31){\vector(0,-1){19}}
\put(9.5,33.5){\oval(3,3)[l]}
\put(9.5,32){\vector(1,0){29}}
\put(9.5,35){\line(1,0){10}}
\put(39,29){\vector(-2,-1){34}}
\end{picture}
\newline
with $\delta$ sections $S_1,\dots ,S_{\delta}:T\to{\cal C}$ satisfying the
following properties:
\begin{enumerate}
\item $p^{-1}(0)={\mit \Gamma}_0=L_1\cup\cdots\cup L_d$:
\item $S_i(0)=P_i$ for $1\le i\le\delta$:
\item for $r\in T-\{ 0\}$, $p^{-1}(r)$ is an irreducible curve,
$S_i(r)$ is an ordinary node for $p^{-1}(r)$ and $p^{-1}(r)$ has no other
singularities, $P_0$ is a total inflection point on $p^{-1}(r)$.
\end{enumerate}
(For short, we call this a suited family of curves on ${\bbb P}^2$
containing ${\mit \Gamma}_0$ preserving the first $\delta$ nodes and the total
inflection point $P_0$.)
Because of semi-continuity reasons it follows that for a general $r\in T$
the curve $p^{-1}(r)$ satisfies the statement $(*)$. So it is
sufficient to prove the claim.
\vs 2
In order to prove the claim we start as follows. Let
$\pi_1:X_1\to{\bbb P}^2$ be the blowing-up of ${\bbb P}^2$ at $P_0$. Let
$E_1$ be the exceptional divisor and let $L_{d,1}$ be the proper
transform of $L_d$. Let $P^{(1)}=L_{d,1}\cap E_1$. Blow-up $X_1$ at
$P^{(1)}$ obtaining $\pi_2:X_2\to X_1$ with the exceptional divisor $E_2$
and let $L_{d,2}$ be the proper transform of $L_{d,1}$. Let
$P^{(2)}=L_{d,2}\cap E_2$ and continue until one obtains
$$
\pi:X=X_d\stackrel{\pi_d}{\rightarrow}X_{d-1}
\stackrel{\pi_{d-1}}{\rightarrow}\cdots\stackrel{\pi_2}{\rightarrow}X_1
\stackrel{\pi_1}{\rightarrow}{\bbb P}^2.
$$
Write $L_i$ for $\pi^{-1}(L_i)$ for $1\le i\le d-1$ and let
$$
{\mit \Gamma}'_0=L_1+\dots +L_{d-1}+L_{d,d}.
$$
For $1\le i\le d-1$, let $\mu_i=\pi_{i+1}\circ\cdot\circ\pi_d$ and let
$L$ be a general line on ${\bbb P}^2$. Then
$$
{\mit \Gamma}'_0\in{\bbb P}:=|d\pi^*(L)-\left(\sum_{i=1}^{d-1}\mu^*_i(E_i)\right)
-E_d|
$$
We are going to use a theorem of Tannenbaum \cite[Theorem 2.13]{tannen}.
Since $L_{d,d}.K_X\ge 0$, we are not allowed to take $Y={\mit \Gamma}'_0$ on $X$ in
Tannenbaum's Theorem. Therefore we first prove the existence of an
irreducible curve ${\mit \Gamma}'_1$ in ${\bbb P}$ with enough nodes.
{}From Tannenbaum's Theorem it follows that there is a quasi-projective
family ${\bbb P}_d((d-1)(d-2)/2)\subset{\bbb P}_d$ of dimension
$\fracd{d(d+3)}{2}-\fracd{(d-1)(d-2)}{2}$ such that a general element
belongs to a suited family of curves on ${\bbb P}^2$ containing ${\mit \Gamma}_0$
and preserving the first $\fracd{(d-1)(d-2)}{2}$ nodes.
The condition $i(\gamma ,L_d;P_0)\ge d$ for $\gamma\in{\bbb P}_d((d-1)(d-2)/2)$
are at most $d$ linear condition. Let
$$
{\bbb P}_d((d-1)(d-2)/2;d)=\{\gamma\in{\bbb P}_d((d-1)(d-2)/2)|
i(\gamma ,L_d;P_0)\ge d\} .
$$
One has ${\mit \Gamma}_0\in{\bbb P}_d((d-1)(d-2)/2;d)$ and
$$
\dim ({\bbb P}_d((d-1)(d-2)/2;d))\ge\frac{d(d+3)}{2}-\frac{(d-1)(d-2)}{2}
-d=2d-1.
$$
Let $\tilde{\bbb P}$ be an irreducible component of
${\bbb P}_d((d-1)(d-2)/2;d)$ containing ${\mit \Gamma}_0$. Since ${\mit \Gamma}_0$ is smooth
at $P_0$, a general element of $\tilde{\bbb P}$ is smooth at $P_0$. Let
${\mit \Gamma}_1$ be a general element of $\tilde{\bbb P}$. If ${\mit \Gamma}_1$ is not
irreducible then $i({\mit \Gamma}_1,L_d;P_0)=d$ implies that $L_d$ is an
irreducible component of ${\mit \Gamma}_1$. Since
$\{ P_1,\dots ,P_{\frac{(d-1)(d-2)}{2}}\}\cap L_d=\emptyset$ also ${\mit \Gamma}_1$
possesses $\fracd{(d-1)(d-2)}{2}$ nodes none of them belonging to $L_d$.
This implies ${\mit \Gamma}_1=L_d\cup{\mit \Gamma}_2$, where ${\mit \Gamma}_2$ belongs to a family of
plane curves of degree $d-1$ on ${\bbb P}^2$ containing
$L_2\cup\cdots\cup L_d$ and preserving the $\fracd{(d-1)(d-2)}{2}$ nodes.
Clearly, if a union of at least two of the lines $L_2,\dots , L_d$
become irreducible in this deformation, some nodes have to disappear.
Since this is not allowed, ${\mit \Gamma}_2$ is the union of $d-1$ lines. But this
would imply $\dim (\tilde{\bbb P})=2d-2$, a contradiction. This proves
that ${\mit \Gamma}_1$ is irreducible.
Moreover ${\mit \Gamma}_1$ belongs to a suited family of curves on ${\bbb P}^2$
containing ${\mit \Gamma}_0$ preserving the first $\fracd{(d-1)(d-2)}{2}$ nodes and
the total inflection point $P_0$. Because of semi-continuity, we can
assume that $(*)$ holds for the first $\delta$ nodes of ${\mit \Gamma}_1$.
Let ${\mit \Gamma}'_1$ be the proper transform of ${\mit \Gamma}_1$ on $X$. Then
${\mit \Gamma}'_1\in{\bbb P}$ and we can apply Tannenbaum's Theorem to obtain a
suited family of curves on $X$ belonging to ${\bbb P}$ containig ${\mit \Gamma}'_1$
and preserving the first $\delta$ nodes of ${\mit \Gamma}'_1$. Projecting on
${\bbb P}^2$ we obtain a suited family of curves on ${\bbb P}^2$
containing ${\mit \Gamma}_1$, preserving the first $\delta$ nodes of ${\mit \Gamma}_1$ and the
total inflection point $P_0$. This completes the proof of the claim.
\vs 2
Let
$$
{\bbb P}_d(d,\delta )=\left\{
\begin{array}{ll}
\gamma\in{\bbb P}_d: & \gamma{\rm\ is\ irreducible;}\\
& \gamma{\rm\ has\ a\ total\ inflection\ point\ and}\\
& \gamma{\rm\ has\ }\delta{\rm\ ordinary\ nodes\ and\ no\ other\ singularities}
\end{array}\right\} .
$$
Then Ran \cite[The irreducibility Theorem (bis)]{ran} proves that
${\bbb P}_d(d;\delta)$ is irreducible. This implies:
\vs 1
\begin{thm}
The normalization of a general nodal irreducible plane curve of degree
$d$ with $\delta$ nodes and possessing a total inflection point $P$ has in
general Weierstrass gap sequence given by $N^{(1)}_{d,\delta}$ at $P$.
\end{thm}
\section{Case: Maximal Weight}
Assume that $\delta\le d-2$ and remember the definition for
$N_{d,\delta}^{({\rm max})}$ and $N_{d,\delta}^{({\rm max2})}$ in the
introduction.
Let $P$ be a total inflection point on the nodal plane curve ${\mit \Gamma}$ of
degree $d$ with $\delta$ nodes, let $\alpha_1 <\cdots <\alpha_g$ be the
Weierstrass gap sequence of $P$ and let
$N={\bbb N}-\{\alpha_1,\dots ,\alpha_g\}$ be the semigroup of non-gaps of $P$.
\vs 1
\begin{lem}
For $1\le i\le g$ one has $\alpha_i\le\alpha_i^{({\rm max})}$. Moreover if
$N\ne N_{d,\delta}^{({\rm max})}$, then $\alpha_i\le\alpha_i^{({\rm max2})}$
for $1\le i\le g$.
\label{lem:max}
\end{lem}
\vs 1
{\it Proof\/}. For $\delta\le 2$ see \S 1, so assume that $\delta\ge 3$.
Let $\alpha_{i,j}=(d-i-2)d+j$, $1\le j\le i\le d-2$. They are just the
members of ${\bbb N}-N_d$.
Since $N_d\subset N$, by Lemma \ref{lem:1}, $N$ is the union of $N_d$ and
$\delta$ values of $\alpha_{i,j}$. Moreover, if $\alpha\in N$ then
$\{\alpha +d-1,\alpha + d\}\subset N$. So, if the number of values $\alpha_{i',j}$
belonging to $N$ with $i'<i$ is less than $\delta$, then $\alpha_{i,j_0}\in N$
for some $1\le j_0\le i$. Each of $N_{d,\delta}^{({\rm max})}$ and
$N_{d,\delta}^{({\rm max2})}$ does not possess two values
$\alpha_{i,j_1}\ne\alpha_{i,j_2}$ for each $i$. Hence, if
$\{\alpha_{2,1},\alpha_{2,2}\}\subset N$, then
$$
^{\#}\{\alpha_{i',j'}\in N|i'<i,j'\ge j\}\ge\ ^{\#}\{
\alpha_{i',j'}\in N_{d,\delta}^{({\rm max2})}|i'<i,j'\ge j\}\quad{\rm for}\
\forall i,j.
$$
So, we have $\alpha_k\le\alpha_k^{({\rm max2})}$ for $1\le k\le g$. In
particular, $\alpha_k\le\alpha_k^{({\rm max})}$.
But if $\{\alpha_{2,1},\alpha_{2,2}\}\not\subset N$, then
$N\in\{ N_{d,\delta}^{({\rm max})}, N_{d,\delta}^{({\rm max2})}\}$ because
of Corollary \ref{cor:2}.
This completes the proof of the lemma.
\vs 2
\begin{prop}
If $d\ge 2\delta+1$, then $N_{d,\delta}^{({\rm max})}$ occurs as the semigroup
of the non-gaps of a total inflection point and if $d\ge 2\delta$,
then so does $N_{d,\delta}^{({\rm max2})}$.
\label{prop:max}
\end{prop}
\vs 1
{\it Proof\/}. Fix $\delta +1$ points $P, P_1,\dots ,P_{\delta}$ on an
arbitrary line $L$. For $i=1,\dots ,\delta$, take general lines $L_i$ and
$L'_i$ passing through $P_i$. Let $T$ be a general line passing through
$P$ and let $C$ be a curve of degree $d-2\delta -1$ which does not pass
through any one of $P, P_1,\dots ,P_{\delta}$ and the common point of each
pair of the above curves. Let
\begin{eqnarray*}
C_1 & = & dL\\
C_2 & = & T+C+L_1+L'_1+\cdots +L_{\delta}+L'_{\delta}.
\end{eqnarray*}
Let ${\bbb P}$ be the pencil generated by $C_1$ and $C_2$. By Bertini's
theorem, a general element ${\mit \Gamma}$ of ${\bbb P}$ is a curve of degree $d$
with $\delta$ ordinary nodes at $P_1,\dots ,P_{\delta}$ as its only
singularities and $P$ is a total inflection point of ${\mit \Gamma}$ with tangent
line $T$. In particular, if ${\mit \Gamma}$ would not be irreducible then
${\mit \Gamma} =T+{\mit \Gamma}'$. But then $T$ would be a fixed component of ${\bbb P}$,
which is not true. Hence ${\mit \Gamma}$ is irreducible. Because of Corollary
\ref{cor:2}, the semigroup of nongaps of $P$ is
$N_{d,\delta}^{({\rm max})}$.
Next, we prove the latter part. Fix $\delta$ points $P_1,\dots ,P_{\delta}$ on
an arbitrary line $L$ and a point $P$ not on $L$. For $i=1,\dots ,\delta$,
let $L_i$ be the line joining $P$ and $P_i$ and let $L'_i$ be general
lines passing through $P_i$. Let $T$ and $T'$ be general lines passing
through $P$ but not any of $P_i$ and let $C$ be a curve of degree
$d-\delta -2$ which does not pass through any one of $P, P_1,\dots ,P_{\delta}$
and the common point of each pair of the above curves. Let
\begin{eqnarray*}
C_1 & = & 2(L_1+\cdots +L_{\delta})+(d-2\delta )T'\\
C_2 & = & L+T+C+L'_1+\cdots +L'_{\delta}.
\end{eqnarray*}
Let ${\bbb P}$ be the pencil generated by $C_1$ and $C_2$. Again, by
Bertini's theorem, a general element ${\mit \Gamma}$ of ${\bbb P}$ is a curve of
degree $d$ with $\delta$ ordinary nodes at $P_1,\dots ,P_{\delta}$ as its only
singularities and $P$ is a total inflection point of ${\mit \Gamma}$ with tangent
line $T$. Also ${\mit \Gamma}$ is irreducible, by Corollary \ref{cor:2}, the
semigroup of nongaps of $P$ is $N_{d,\delta}^{({\rm max2})}$.
\vs 2
{\raggedright {\sc Remark 3.3.}} Define
$N_{d,3}^{({\rm max3})}=N_{d,3}^{({\rm max4})}=N_{d,3}^{(1)}$ and for
$\delta >3$ we define inductively $N_{d,\delta}^{({\rm max3})}=
N_{d,\delta -1}^{({\rm max3})}\cup\{(d-\delta -1)d+1\}$ and
$N_{d,\delta}^{({\rm max4})}=N_{d,\delta -1}^{({\rm max4})}\cup\{(d-\delta -1)d
+\delta -1\}$. As above one can check that, for $\delta\ge 3$ and
$N\not\in\{ N_{d,\delta}^{({\rm max})}, N_{d,\delta}^{({\rm max2})}\}$ one has
$\alpha_k\le\alpha_k^{({\rm max3})}$ for $1\le k\le g$ and, for $\delta\ge 5$ and
$N\not\in\{ N_{d,\delta}^{({\rm max})}, N_{d,\delta}^{({\rm max2})},
N_{d,\delta}^{({\rm max3})}\}$ one has $\alpha_k\le\alpha_k^{({\rm max4})}$ for
$1\le k\le g$. Moreover $N_{d,\delta}^{({\rm max3})}$ (resp.
$N_{d,\delta}^{({\rm max4})}$) occurs if and only if exactly $\delta -1$ nodes
are on a line $L_0$ and $P\in L_0$ (resp. $P\not\in L_0$). As above one
can also discuss the existence.
If one wants to continue, then one has to start making an analysis of the
case where the nodes are on a conic. Another direction of further
investigation could be: let $3\le\delta'\le\fracd{d}{2}$, what is the
general situation for $N$ if $\delta'$ nodes are on a line ? Probably
reasoning as in \S 2, one obtains an answer.
\newpage
|
1994-03-17T15:35:54 | 9403 | alg-geom/9403013 | fr | https://arxiv.org/abs/alg-geom/9403013 | [
"alg-geom",
"math.AG"
] | alg-geom/9403013 | Jean Valles | Valles Jean | Complexes inattendus de droites de saut (Unexpected complex of jumping
lines) | 7 pages, Latex | null | null | null | null | We prove here the following results: \begin{th} Let $E$ a rank 2 vector
bundle over ${\bf P}_3$, if $C$ is a reduced irreducible curve of ${\bf
P}_3^{\vee}$ such that $E_H$ is unstable for all $H\in C$ then $C$ is a line.
\end{th} We define now the set $W(E)$ as the set of planes $H$ such that the
restricted bundle $E_H$ is unstable (that means non semi-stable).
\begin{th} Let $E$ a rank 2 vector bundle over ${\bf P}_3$, with first chern
class $c_1=c_1(E)$, $L$ a line and an integer $n\ge 0$. The following
conditions are equivalent:
\begin{description}
\item[(i)] $L^{\vee}\subset W(E)$ and $H^0(E_H(-n+[-c_1/2]))\neq 0$ for a
general point $H\in L^{\vee}$. \item[(ii)] There exist $m>0$ and a section
$t\in H^0(E(m+[-c_1/2]))$ such that the zero variety of $t$ contains the
infinitesimal neighbourhood of order $(m+n-1)$ of $L$.
\end{description}
| [
{
"version": "v1",
"created": "Thu, 17 Mar 1994 14:34:40 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Jean",
"Valles",
""
]
] | alg-geom | \section{Introduction.} Soit $E$ un fibr\'e stable de rang 2 de
premi\`ere classe de chern $c_1\le -3$. Les droites $L$ telles que
$H^0(E_L)\neq 0$ forment un sous sch\'ema de la grassmanienne des
droites de ${\bf P}_3$ dont la codimension attendue est au moins 2 (cf.
\cite{G-P.2}). Pourtant on connait de nombreux exemples de fibr\'es pour
lesquels ce sous sch\'ema est un complexe (voir exemple 1.2).\\ Dans
\cite{G-P} Gruson et Peskine annoncent le r\'esultat suivant :\\ \\{\bf
Lemme B.5 : }{\it Soit $E$ un fibr\'e de rang 2 sur ${\bf P}_3$ tel que
$c_1(E)\le -3$ et $H^0(E)=0$. Si $K$ est un complexe r\'eduit
irr\'eductible de droites tel que $H^0(E_L)\neq 0$ pour $L\in K$, il
existe une courbe r\'eduite irr\'eductible $\Gamma$ de ${\bf P}_3$
telle que les droites de $K$ sont les droites rencontrant $\Gamma$. De
plus il existe un entier $l$ et une section $s$ de $E(l)$ dont la
vari\'et\'e des z\'eros contient le $(l-1)$-{i\`eme} voisinage
infinit\'esimal de $\Gamma$.}\\ \\ \indent Les auteurs me signalent que
la preuve est incompl\`ete. Plus pr\'ecis\'ement la d\'emonstration de
l'irr\'eductibilit\'e g\'eom\'etrique de $K$ au dessus de ${\bf P}_3$
sur laquelle cette preuve repose comporte une erreur.\\ \indent Nous
donnons ici une d\'emonstration de cet \'enonc\'e dans le cas o\`u le
complexe $K$ est form\'e des droites d'une famille de dimension 1 de
plans. Cela peut paraitre surprenant car une telle famille est toujours
g\'eom\'etriquement r\'eductible except\'e dans le cas o\`u elle est
form\'ee des plans contenant une droite. C'est bien entendu le seul cas
possible. En effet, consid\'erons la courbe $C$ de ${\bf P}_3^{\vee}$
naturellement associ\'ee au complexe $K$ (voir lemme 1), on montre :
\begin{theo} Soit $E$ un fibr\'e stable de rang 2 sur ${\bf P}_3$, si
$C$ est une courbe r\'eduite irr\'eductible de ${\bf P}_3^{\vee}$ telle
que $E_H$ est instable pour tout $H\in C$ alors $C$ est une droite.
\end{theo} {\it Remarque.} On d\'eduit imm\'ediatement de ce
th\'eor\`eme que l'ensemble des hyperplans $H$ tels que $E_H$ est
instable (c'est \`a dire non semi-stable) not\'e $W(E)$ est de dimension
au plus 1 dans ${\bf P}_3^{\vee}$ (ce r\'esultat est d\'ej\`a prouv\'e
par Coanda \cite{Coanda}).\\ \\ De plus, on a l'\'enonc\'e suivant,
directement inspir\'e du lemme B.5 : \begin{theo}
Soit $E$ un fibr\'e stable de rang 2 sur ${\bf P}_3$ de premi\`ere
classe de chern $c_1=c_1(E)$, $L$ une droite et $n$ un entier $\ge 0$.
Les conditions suivantes sont \'equivalentes: \begin{description}
\item[(i)]
$L^{\vee}\subset W(E)$ et $H^0(E_H(-n+[-c_1/2]))\neq 0$ pour $H$ un
point g\'en\'eral de $L^{\vee}$. \item[(ii)] Il existe $m>0$ et une
section $t\in H^0(E(m+[-c_1/2]))$ dont la vari\'et\'e des z\'eros
contient le $(m+n-1)$-{i\`eme} voisinage infinit\'esimal de $L$.
\end{description} \end{theo} \subsection{Remarques pr\'eliminaires.}
Rappelons qu'un fibr\'e $E$ de rang deux sur ${\bf P}_3$
est stable si et seulement si $H^0(E([-c_1/2]))=0$, o\`u $[.]$
d\'esigne la partie enti\`ere et $c_1$ la premi\`ere classe de chern de
$E$.\\ On supposera dans les d\'emonstrations des th\'eor\`emes 1 et 2
que $c_1=0 \,\,\mbox{ou} \, -1$.\\ \\ {\it Remarque 1.} Les
th\'eor\`emes 1 et 2 impliquent le lemme B.5 lorsque le complexe $K$ est
form\'e des droites d'une famille de dimension 1 de plans (voir lemme
1).\\ {\it Remarque 2.} Les th\'eor\`emes 1 et 2 sont vrais sur ${\bf
P}_n,\, n\ge 3$.\\ {\it Remarque 3.} Le lemme B.5 est \'evident lorsque
$E$ n'est pas stable.\\ \\ D\'emontrons la remarque 3 pour $c_1(N)=-3$.
Dans ce cas $H^0(N(1))\neq 0$ et l'unique section non nulle de $N(1)$
s'annule le long d'une courbe $Z$ : $$ 0 \rightarrow O_{{\bf
P}_3}(-1)\rightarrow N\rightarrow {\cal I}_Z(-2)\rightarrow 0.$$ On en
d\'eduit que $L\in K$ si et seulement si $L\cap Z \neq \emptyset$. Le
complexe $K$ \'etant r\'eduit irr\'eductible, ceci prouve l'existence
d'une courbe r\'eduite irr\'eductible $\Gamma$ telle que ${\cal I}_Z
\subset {\cal I}_{\Gamma}.$\\ \\ Le lemme suivant \'etablit la
correspondance entre le complexe $K$ de droites de saut d'un fibr\'e
$E$, complexe form\'e des droites d'une famille de dimension 1 de plans,
et la courbe $C$ de $W(E)$. \begin{lem} Soit $N$ un fibr\'e de rang 2
stable tel que $c_1(N)\le -2$, on a l'\'equivalence : $$ H^0(N_L)\neq 0,
\,\,{\rm\mbox{pour tout}}\,\,L\subset H \Longleftrightarrow H^0(N_H)\neq
0.$$ \end{lem} {\it d\'emonstration :} L'implication ($\Leftarrow$)
r\'esulte de la suite exacte suivante : $$0\longrightarrow
N_H(-1)\longrightarrow N_H\longrightarrow N_L\longrightarrow 0.$$
Inversement, d'apr\`es Grauert-Mulich le fibr\'e $N_H$ est instable. Si
$H^0(N_H)=0$, on peut donc supposer que $H^0(N_H(1))\neq 0$, soit,
$$0\longrightarrow O_H(-1)\longrightarrow N_H\longrightarrow {\cal
I}_{Z_H}(c_1+1)\longrightarrow 0,$$ o\`u $Z_H$ est un groupe de points
de $H$. On en d\'eduit que pour une droite g\'en\'erale $L$ de $H$,
$H^0(N_L)=0$, ce qui contredit l'hypoth\`ese.\\ \\ {\small {\bf
Remerciements :} Je remercie Christian Peskine qui a dirig\'e ce
travail et Iustin Coanda pour ses conseils et les nombreuses et utiles
discussions que nous avons eues.} \subsection{Exemple.} Avant de
d\'emontrer les th\'eor\`emes 1 et 2 donnons un exemple de fibr\'e
${\cal E}$ poss\`edant une famille de dimension 1 de plans instables et
d\'ecrivons une section de ce fibr\'e ayant les propri\'et\'es
annonc\'ees dans le th\'eor\`eme 2.\\ \\ {\it Exemple.} On reprend
l'exemple de Gruson-Peskine d\'ecrit par Mei-Chu Chang \cite{M.C.C}:
Soient $(r+1)$ plans $H_1,...,H_{r+1}\, (r\geq 2)$
articul\'es autour d'une droite $D$, et dans chaque plan $H_i$ une
courbe $X_i$ de degr\'e $(2r-1)$; les $(r+1)$ courbes sont choisies 2
\`a 2 disjointes. On note:
$$ X=\bigcup_{i=1}^{(r+1)} X_i \quad \mbox{o\`u}\quad X_i\subset H_i .$$
Il existe un fibr\'e ${\cal E}$ stable de rang 2 tel que $c_1({\cal
E})=0$ et une extension \begin{equation}
0\longrightarrow O_{{\bf P}_3}\longrightarrow {\cal E}(r)
\longrightarrow {\cal I}_X(2r)\longrightarrow 0. \end{equation} En
effet comme $w_X=O_X(2r- 4)$, on a $\mbox{Ext}^1({\cal I}_X,w_{{}_{{\bf
P}_3}}(4-2r))\neq 0$, et ${\cal E}$ est stable car $H^0({\cal
I}_X(r))=0$.\\
Soit $H$ un plan g\'en\'eral contenant $D$, alors $H$ coupe proprement
$X$, et $(1)$ reste exacte apr\`es tensorisation par $O_H(1-2r)$ \[
0\longrightarrow O_H(1-2r)\longrightarrow {\cal
E}_H(1-r)\longrightarrow {\cal I}_{X\cap H}(1)\longrightarrow 0. \]
Comme $X\cap H\subset D$ on a $H^0({\cal I}_{X\cap H}(1))\neq 0$, donc
$H^0({\cal E}_H(1-r))\neq 0$ (on remarque de plus que $H^0({\cal
E}_H(-r))=0$ pour tout plan $H$).
Nous avons montr\'e que $D^{\vee}$ est une droite de ${\bf
P}_3^{\vee}$ dont les points correspondent \`a des plans instables du
fibr\'e ${\cal E}$. \\ \\ \indent D\'ecrivons maintenant une section de
ce fibr\'e ayant les propri\'et\'es annonc\'ees. La suite exacte (1)
montre que
$\, H^0({\cal I}_X(r+1))\simeq H^0({\cal E}(1)).$
La courbe $X$ est contenue dans la r\'eunion de $(r+1)$ plans. Cette
surface induit une section non nulle $t$ de ${\cal E}(1)$. On note
$\Gamma$ le lieu des z\'eros de la section $t$ (c'est une courbe car
$H^0({\cal E})=0$).\\ Montrons que $\Gamma$ contient le $(r-1)$-{i\`eme}
voisinage infinit\'esimal de $D$. Sinon soit
$H$ un plan contenant $D$ et coupant proprement $\Gamma$. Dans ce cas
la section $t_H\in H^0({\cal E}_H(1))$ s'annule en codimension 2 le long
du groupe de points $\Gamma\cap H$ \begin{equation} 0 \longrightarrow
O_H\longrightarrow {\cal E}_H(1)\longrightarrow {\cal I}_{\Gamma\cap H}
(2)\longrightarrow 0. \end{equation} Il en r\'esulte que $H^0({\cal
E}_H(-1))=0$ , ce qui contredit $H\in W({\cal E})$. Donc la section
$t_H$ s'annule le long d'une courbe (d'\'equation $f_H=0$) contenant $D$
et \'eventuellement d'un groupe de points. Si le degr\'e de cette
courbe est $k\le (r-1)$, la section $t_H/f_H\in H^0({\cal E}_H(1-k))$
s'annule en codimension 2, et on trouve $H^0({\cal E}_H(-k))=0$.
C'est impossible car $H^0({\cal E}_H(1-r))\neq 0$.\\
Par cons\'equent la restriction de $\Gamma$ \`a un plan g\'en\'eral
contenant $D$ est une courbe plane de degr\'e $r$. On en d\'eduit que
$\Gamma$ contient le $(r-1)$-{i\`eme} voisinage infinit\'esimal de $D$.\\
\\ Montrons de plus que le support de $\Gamma$ est contenu dans $D$.
En effet supposons que $\Gamma$ poss\`ede une autre composante
irr\'eductible $\Gamma_1$.
Soit $H$ le plan contenant $\Gamma_1$ et $L$. Le lieu des z\'eros de la
section $t_H \in H^0({\cal E}_H(1))$ contient une courbe de degr\'e $\ge
(r+1)$.
Alors $H^0({\cal E}_H(-r))\neq 0$, ce qui est impossible.\\ \\ Le lieu
des z\'eros de la section $t\in H^0({\cal E}(1))$ est une courbe
$\Gamma$ telle que : \begin{description} \item{-} $\deg (\Gamma)=r^2+r$
et $p_a(\Gamma)=1-r^2-r$ (o\`u $p_a$ est le genre arithm\'etique).
\item{-} $\Gamma$ contient le $(r-1)$-i\`eme voisinage infinit\'esimal
de $D$. \item{-} Le support de $\Gamma$ est inclus dans $D$.
\end{description} \section{D\'emonstration du th\'eor\`eme 1}
La d\'emonstration est divis\'ee en deux \'etapes.
On v\'erifie tout d'abord que $C$ est plane. On d\'emontre ensuite que
c'est une droite en interpr\'etant la singularit\'e apparaissant au
point correspondant au plan de ${\bf P}_3^{\vee}$ contenant la courbe.\\
D'apr\`es le th\'eor\`eme de semi continuit\'e la fonction
$h^0(E_H(-k))$ est minimale sur un ouvert non vide $U$ de $C$. Soit
$n\ge 0$ tel que $h^0(E_H(-n))=1$ pour un point $H\in U$
(remarquons que $n\ge 1$ lorsque $c_1(E)=0$). \paragraph*{Etape 1 : } La
d\'emonstration de cette premi\`ere partie repose essentiellement sur la
remarque bien connue suivante que nous ne red\'emontrons pas.\\ \\ {\it
Remarque 1.} Soit $H$ un plan g\'en\'eral de $C$ et $s_H$ l'unique
section non nulle de $H^0(E_H(-n))$. On note $Z(s_H)$ le sch\'ema des
z\'eros de $s_H$, si $L$ est une droite de $H$ on a : $$\deg (O_{L\cap
Z(s_H)})\ge r \Longleftrightarrow H^0(E_L(-n-r))\neq 0.$$ \indent On
suppose que $C$ n'est pas plane, i.e. la famille de plans correspondant
aux points de $C$ ne poss\`ede pas de point fixe. Soit $H_i$ et $H_j$
deux points g\'en\'eraux de $U$. Comme $Z(s_{H_i})\cap
Z(s_{H_j})=\emptyset$, on a d'apr\`es la remarque 1, $h^0(E_{H_i\cap
H_j}(-n))=1$. Par cons\'equent les sections $s_{H_i}\in
H^0(E_{H_i}(-n))$ et $s_{H_j}\in H^0(E_{H_j}(-n))$ uniques \`a une
constante pr\`es coincident sur $H_i\cap H_j$ i.e. $s_{H_i}(x)=\lambda
s_{H_j}(x)$ pour $x\in H_i\cap H_j$ et $\lambda \in {\bf C}$.\\ \\ Soit
${\bf F} \subset {\bf P}_3 \times {\bf P}_3^{\vee}$ la vari\'et\'e
d'incidence
points-plans de ${\bf P}_3$. Posons $X:= {\bf F}\cap({\bf P}_3 \times
C) $ et consid\'erons
les projections : $ X\stackrel{p}\rightarrow{\bf P}_3 $ et $
X\stackrel{q}\rightarrow C $. Les fibres de $X$ au dessus de $C $ sont
des ${\bf P}_2$, et au dessus de ${\bf P}_3$ elles correspondent aux
sections hyperplanes de $C $. On a : $$h^0(p^*E(-n)_{\mid
q^{-1}(H)})=h^0(E_H(-n))=1 \quad\mbox{pour tout}\,\,H\in U.$$ Le
faisceau $q_*p^*E(-n)$ est coh\'erent de rang 1 sur $C$ sans torsion
donc inversible. Le morphisme canonique $q^*q_*p^*E(-n)\rightarrow
p^*E(-n)$ s'annule le long d'un sous sch\'ema ferm\'e $Z$ de $X$ de
codimension $\ge 2$. Il induit un morphisme compos\'e $\phi $ au dessus
de ${\bf P}_3$ : $$ X\setminus Z \stackrel{\phi}\longrightarrow {\bf
P}_{{}_{{\bf P}_3}} (E) $$ $$ p\searrow \hspace{0.5cm}\swarrow \pi
\hspace{0.2cm}$$ $${\bf P}_3 $$ L'image de $\phi $ est un ouvert dans un
diviseur r\'eduit irr\'eductible $D$ de ${\bf P}_{{}_{{\bf P}_3}}(E) $.
Le diviseur $D$ correspond \`a une section $t\in H^0(S_kE(m))$ (o\`u
$S_kE(m)$ est la $k$-i\`eme puissance sym\'etrique de $E$). La fibre
g\'en\'erale $p^{-1}(x)$ de $ X\setminus Z$ au dessus de ${\bf P}_3$
consiste en $d=\deg C$ points $(x,H_1),...,(x,H_d)$. L'image de
$(x,H_i)$ par $\phi$ est donn\'ee par le point $s_{H_i}(x)$ de ${\bf
P}_{{}_{{\bf P}_3}}(E_x) $ o\`u $s_{H_i}$ est l'unique section non nulle
de $E_{H_i}$. Il r\'esulte de la remarque 1 que les points
$(x,H_1),...,(x,H_d)$ s'envoient sur le m\^eme point de ${\bf
P}_{{}_{{\bf P}_3}}(E_x) $. \\ \\On en d\'eduit que la fibre
g\'en\'erale de $D$ au dessus de ${\bf P}_3$ est irr\'eductible et
qu'elle consiste en un point simple par lissit\'e g\'en\'erique. Le
diviseur $D$ est donc birationnel \`a ${\bf P}_3$. Il en r\'esulte que
$t$ est une section de $E(m)$ avec $m>0$ car $E$ est stable. Quitte \`a
enlever la composante de codimension 1,
on peut supposer que $t$ s'annule en codimension 2 le long d'une courbe
$\Gamma$.\\ \\Les sections $s_H\in H^0(E_H(-n))$ et $t_H \in
H^0(E_H(m))$ sont proportionnelles. Par hypoth\`ese $
H^0(E_H(-n-1))=0$, la section $s_H$ s'annule alors en codimension $\ge
2$. Il existe $f_H\in H^0(O_H(m+1))$ tel que $t_H=f_Hs_H$. Mais pour un
plan g\'en\'eral H, $t_H$ s'annule aussi en codimension 2 (car la
courbe $C$ ne poss\`ede pas de point fixe), ce qui est impossible car
$m>0$. \paragraph*{Etape 2:} Soit $X$ le plan de ${\bf P}_3^\vee$
contenant la courbe $C$, et $x\in {\bf P}_3$ le point correspondant \`a
$X$. On consid\`ere l'\'eclatement ${\bf \widetilde{P}}_3 $ de ${\bf
P}_3$ en $x$ et les morphismes naturels $p:{\bf \widetilde{P}}_3
\rightarrow {\bf P}_3 $ et $q:{\bf \widetilde{P}}_3 \rightarrow {\bf
P}_2 $. On a $ {\bf \widetilde{P}}_3=\mbox{Proj}_{{}_{{\bf
P}_3}}(\bigoplus_i {\cal M}_x^i) $ et $
{\bf \widetilde{P}}_3={\bf P}_{{}_{{\bf P}_2}}(O_{{\bf P}_2}(1) \oplus
O_{{\bf P}_2} ) $.\\
On appelle $\Delta$ le diviseur exceptionnel, le morphisme $q_{\mid
\Delta}: \Delta \rightarrow {\bf P}_2 $ est un isomorphisme. De plus si
$l$ est un point de ${\bf P}_2$ on a : $$ (p^*E)_{\mid q^{-1}(l)} \simeq
E_L. $$
Le morphisme $q$ \'etant plat, le th\'eor\`eme de semi-continuit\'e
implique que $ h^0((p^*E)_{\mid q^{-1}(l)})$ atteint son minimum sur
un ouvert non vide $U$ de $ {\bf P}_2$. \noindent Alors il existe $
a\geq (n+1)$ tel que $ h^0((p^*E)_{\mid q^{-1}(l)}(-a))=h^0(E_L(-a))=1$
pour $l\in U$. L'entier $a$ est $\ge (n+1)$ d'apr\`es la remarque 1.
\noindent Le faisceau $q_*p^*E(-a) $ est un faisceau coh\'erent de rang
1, il est r\'eflexif donc inversible sur $ {\bf P}_2$. Soit $k$ tel
que $q_*p^*E(-a)=O_{{\bf P}_2}(-k)$.
On consid\`ere le morphisme canonique:
$$\widetilde{s}:q^*O_{{\bf P}_2}(-k)=q^*q_*p^*E(-a)\longrightarrow
p^*E(-a).$$ Il est injectif. En effet pour $l\in U$, $ p^*E(-a)_{\mid
q^{-1}(l)} \simeq O_L\bigoplus O_L(-a+c_1)$. D'apr\`es le th\'eor\`eme
de changement de base, la restriction de $\widetilde{s}$ \`a la fibre de
$l$ est en fait le morphisme d'\'evaluation (de l'unique section non
nulle de $E_L(-a)$) qui ne s'annule pas,
$$\widetilde{s}_{\mid q^{-1}(l)}:H^0(E_L(-a))\otimes
O_L\longrightarrow E_L(-a).$$ Le lieu des z\'eros du morphisme
$\widetilde{s}$ est alors contenu dans la r\'eunion des fibres
$q^{-1}(l)$ o\`u $L$ est une droite dont l'ordre de saut est $>a$. On
sait (cf lemme 9, \cite{Barth}) que ce lieu des z\'eros ne contient pas
d'hypersurface, c'est \`a dire qu'il ne contient au plus qu'un nombre
fini de fibres. \\ \\ Le morphisme $\widetilde{s}$ induit donc une
section non nulle $\widetilde{t}$ de $ p^*E(-a)\otimes q^*O_{{\bf
P}_2}(k)$ qui s'annule le long d'une courbe de $ {\bf \widetilde{P}}_3$.
On remarque que $H^0(E(-a))= 0$ implique $k>0$.\\
Comme $p_*q^*O_{{\bf P}_2}(k) \simeq {\cal M}_x^k(k)$, l'image directe
sur ${\bf P}_3$ de la section $\widetilde{t}$ est
une section non nulle $t\in H^0(E(-a)\otimes {\cal M}_x^k(k))$, ce qui
d\'emontre $k>a$. La section $t$ induit une suite exacte: $$
0\longrightarrow O_{{\bf P}_3}\stackrel{t} \longrightarrow E(k-a)
\longrightarrow {\cal
I}_{\Gamma}(2k-2a+c_1)\longrightarrow 0 $$ o\`u $\Gamma$ est une courbe
qui contient le $(k-1)$-i\`eme voisinage
infinit\'esimal de $x$, i.e ${\cal I}_{\Gamma}\subset {\cal M}_x^k$.\\
Soient $H$ un point g\'en\'eral de $C$, $t_H$ la restriction de $t$ au
plan $H$ et $s_H$ l'unique section non nulle de $E_H(-n)$. On
consid\`ere le morphisme de fibr\'es sur $H$ suivant: $$ O_H(n)\oplus
O_H(a-k)\stackrel{(s_H,t_H)} \longrightarrow E_H. $$
Le d\'eterminant de ce morphisme est une section de ${\cal
M}_{x,H}^k(c_1+k-a-n)$. Etant donn\'e que $(c_1 +k-a-n)<k$ cette section
est nulle i.e $s_H \wedge t_H \equiv 0$ \\ Le lieu des z\'eros de la
section $s_H$ \'etant de codimension deux, on en d\'eduit qu'il existe
$$ f_H\in H^0(O_H(k-a+n))\ /\ t_H=f_H s_H.$$ Ceci montre que la courbe
du plan $H$ d'equation $ f_H=0 $ est contenue dans $\Gamma$. En
cons\'equence tout plan de $C$ contient une composante irr\'eductible
de $\Gamma$. Il est clair que $\Gamma$ contient une droite $L$ telle
que $C=L^{\vee}$. \section{D\'emonstration du th\'eor\`eme 2} {\bf
(ii)} $\Rightarrow$ {\bf (i)}
On note $t_H\in H^0(E_H(m))$ la restriction de $t$ \`a un plan
g\'en\'eral $H$ contenant $L$. Par hypoth\`ese $\,{\cal I}_{\Gamma}.O_H
\subset {\cal I}_{L,H}^{(m+n)}\simeq O_H(-m-n) $. En d'autres termes,
la vari\'et\'e des z\'eros de $t_H$ contient une hypersurface
(d'\'equation $f_H=0$) de degr\'e $(m+n)$. Alors $t_H/f_H$ est une
section non nulle de $H^0(E_H(-n))$.\\ \\ {\bf (i)} $\Rightarrow$ {\bf
(ii)}
Dans le cas pr\'esent, le complexe de droites associ\'e \`a $L^{\vee}$
(droites rencontrant $L$) est g\'eom\'etriquement irr\'eductible. Nous
reprenons donc les id\'ees de la d\'emonstration du lemme B.5 pour
\'etablir le r\'esultat. \\ \\ On reprend la construction de l'\'etape
1 (th\'eor\`eme 1). Consid\'erons la vari\'et\'e d'incidence
points-plans ${\bf F}$ de ${\bf P}_3$. Posons $X:= {\bf F}\cap({\bf
P}_3 \times L^{\vee}) $ et consid\'erons
les projections : $ X\stackrel{p}\rightarrow{\bf P}_3 $ et $
X\stackrel{q}\rightarrow L^{\vee} $. Comme pr\'ec\'edemment on a un
morphisme compos\'e $\phi $ au dessus de ${\bf P}_3$ : $$ X\setminus Z
\stackrel{\phi}\longrightarrow {\bf P}_{{}_{{\bf P}_3}} (E),$$ et
l'image de $ X\setminus Z$ est un diviseur $D$ de ${\bf P}_{{}_{{\bf
P}_3}} (E) $.
Le complexe \'etant g\'eom\'etriquement irr\'eductible $D$ est
birationnel \`a ${\bf P}_3$.
Il en r\'esulte que $D$ correspond \`a une section $t$ de $E(m)$ avec
$m>0$ car $E$ est stable. La section $t$ s'annule le long d'une courbe
$\Gamma$.\\ \\ Soit $H\supset L$, les sections $s_H\in H^0(E_H(-n))$ et
$t_H \in H^0(E_H(m))$ sont proportionnelles. Par hypoth\`ese $
H^0(E_H(-n-1))=0$, la section $s_H$ s'annule alors en codimension $\ge
2$. On en d\'eduit qu'il existe $f_H\in H^0(O_H(m+n))$ telle que
$t_H=f_Hs_H$.\\ \\ Ceci montre que la courbe du plan $H$ d'\'equation
$f_H=0$ est contenue dans $\Gamma$. Par cons\'equent tout plan $H$ de
$L^{\vee}$ contient une composante irr\'eductible de $\Gamma$ de degr\'e
$(m+n)$. Il est clair que la courbe $\Gamma$ contient le
$(m+n-1)$-i\`eme voisinage infinit\'esimal de $L$.
|
1994-03-11T06:02:06 | 9403 | alg-geom/9403008 | en | https://arxiv.org/abs/alg-geom/9403008 | [
"alg-geom",
"math.AG"
] | alg-geom/9403008 | Ishida | Masa-Nori Ishida | Torus embeddings and algebraic intersection complexes | 59 pages, latex | null | null | null | null | We describe the intersection complex of any perversity of a toric variety
completely in terms of the associated fan. It is described by a finite complex
of finite dimensional graded vector spaces which we call graded exterior
modules. The intersection complexes of the middle perversity are treated in the
second part following this article.
| [
{
"version": "v1",
"created": "Fri, 11 Mar 1994 04:53:47 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ishida",
"Masa-Nori",
""
]
] | alg-geom | \section*{Introduction}
In \cite{GM2}, Goreskey and MacPherson defined and constructed
intersection complexes for topological pseudomanifolds.
The complexes are defined in the derived category of sheaves
of modules over a constant ring sheaf.
Since analytic spaces are of this category, any algebraic variety
defined over ${\bf C}$ has an intersection complex for each perversity.
The purpose of this paper is to give an algebraic description of the
intersection complex of a toric variety.
Namely, we describe it as a finite complex of coherent sheaves
whose coboudary map is a differential operator of order one.
Let $Z_{\rm h}$ be the complete toric variety associated to a complete
fan $\Delta$.
For each $\sigma\in\Delta$, let $X(\sigma)_{\rm h}$ be the associated
closed subvariety of $Z_{\rm h}$.
For each perversity ${\bf p}$, we construct a bicomplex
$(\mathop{\rm ic}\nolimits_{\bf p}(Z_{\rm h})^{\bullet,\bullet}, d_1, d_2)$ with the following
properties.
(1) $\mathop{\rm ic}\nolimits_{\bf p}(Z_{\rm h})^{i,j} = \{0\}$ for $(i,j)\not\in[0,r]\times[-r,0]$.
(2) Each $\mathop{\rm ic}\nolimits_{\bf p}(Z_{\rm h})^{i,j}$ is a direct sum for $\sigma\in\Delta$
of free ${\cal O}_{X(\sigma)_{\rm h}}$-modules of finite rank.
(3) $d_1$ is an ${\cal O}_{Z_{\rm h}}$-homomorphism and $d_2$ is a differential
operator of order one.
(4) The associated single complex $\mathop{\rm ic}\nolimits_{\bf p}(Z_{\rm h})^\bullet$ is
quasi-isomorphic to the $r$-times dimension shifts to the right of
the intersection complex defined in \cite{GM2}.
In other words, our complex belongs to
``Beilinson-Bernstein-Deligne-Gabber scheme'' (cf. \cite[2.3,(d)]{GM2}
and \cite[2.1]{BBD}).
In \S1, we introduce an abelian category $\mathop{\rm GM}\nolimits(A(\sigma))$ of
finitely generated graded $A(\sigma)$-modules, where $A(\sigma)$
is the exterior algebra of the ${\bf Q}$-vector space $N(\sigma)_{\bf Q}$
defined by a cone $\sigma$.
In \S2, we define an additive category $\mathop{\rm GEM}\nolimits(\Delta)$ for
a finite fan $\Delta$.
Each object $L$ of this category is a collection
$(L(\sigma)\mathrel{;}\sigma\in\Delta)$ of $L(\sigma)\in\mathop{\rm GM}\nolimits(A(\sigma))$.
We define a dualizing functor ${\bf D}$ on the category
$\mathop{\rm CGEM}\nolimits(\Delta)$ of finite complexes in $\mathop{\rm GEM}\nolimits(\Delta)$.
A perversity ${\bf p}$ on $\Delta$ is defined to be a
${\bf Z}$-valued map on $\Delta\setminus\{{\bf 0}\}$.
The intersection complex $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ as
an object of $\mathop{\rm CGEM}\nolimits(\Delta)$ is defined and constructed
in \S2.
In \S3, we work on the toric variety $Z(\Delta)$ associated
to the fan $\Delta$.
For each $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$, we define a finite
bicomplex $\Lambda_{Z(\Delta)}(L)^{\bullet,\bullet}$ of
coherent ${\cal O}_{Z(\Delta)}$-modules whose second
coboundary map is a differential operator of order one.
We consider the normal analytic space $Z_{\rm h} := Z(\Delta)_{\rm h}$
associated to the toric variety $Z(\Delta)$ in \S4.
The bicomplex $\mathop{\rm ic}\nolimits_{\bf p}(Z_{\rm h})^{\bullet,\bullet}$
stated above is defined to be the bicomplex
$\Lambda_{Z(\Delta)_{\rm h}}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^{\bullet,\bullet}$
on this analytic space.
When $\Delta$ is a complete fan, we show that the
intersection cohomologies are described in termes of the
complex $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$.
The middle perversity ${\bf m}$ is defined by ${\bf m}(\sigma) := 0$
for all $\sigma\in\Delta\setminus\{{\bf 0}\}$.
In \cite{Ishida3}, we will discuss on $\mathop{\rm ic}\nolimits_{\bf m}(\Delta)^\bullet$
and prove the decomposition theorem for a barycentric
subdivision of the fan.
\section*{Notation}
We denote by ${\bf Z}$ the ring of rational integers and by ${\bf Q}$, ${\bf R}$
and ${\bf C}$ the fields of rational numbers, real numbers and
complex numbers, respectively.
For a free ${\bf Z}$-module $F$ of finite rank, we denote
$F_{\bf Q} := F\otimes{\bf Q}$ and $F_{\bf R} := F\otimes{\bf R}$.
We denote a complex $E^\bullet$ of modules or of sheaves of
modules simply by $E$ when the substitution of the dot by
an integer is not suitable.
In particular, we prefer to write $G(E)^\bullet$ rather than
$G(E^\bullet)$, if $G$ is a functor between categories of complexes.
However, we left the dot in representing the cohomologies
$\H^i(E^\bullet)$.
For a complex $(E^\bullet, d_E)$ and a integer $n$, the complex
$(E[n]^\bullet, d_{E[n]})$ is defined by $E[n]^i := E^{i+n}$ and
$d_{E[n]}^i := (-1)^nd_E^{i+n}$ for $i\in{\bf Z}$.
By a bicomplex $(E^{\bullet,\bullet}, d_1, d_2)$, we mean a
naive double complex \cite[0.4]{Deligne}, i.e., a double complex
satisfying $d_1\cdot d_2 = d_2\cdot d_1$.
The associated single complex $(E^\bullet, d)$ is defined by
$E^k :=\bigoplus_{i+j=k}E^{i,j}$ for every $k\in{\bf Z}$ and
$d(x) := d_1^{i,j}(x) + (-1)^id_2^{i,j}(x)$ for
$(i, j)\in{\bf Z}\times{\bf Z}$ and $x\in E^{i,j}$.
In general, we follow \cite{Deligne} for the sign
configuration of complexes.
By a $d$-complex and a $\partial$-complex, we mean complexes in
an additive category whose coboundary
maps are denoted by $d$ and $\partial$, respectively.
Some impotant bicomplexes in this paper has $d_1 = d$ and
$d_2 =\partial$.
\section{The exterior algebras and modules}
\setcounter{equation}{0}
Let $r$ be a non-negative integer, and let $N$ and $M$ be
mutually dual free ${\bf Z}$-modules of rank $r$ with the pairing
$\langle\;,\;\rangle\mathrel{:} M\times N\longrightarrow{\bf Z}$.
This pairing is extended ${\bf R}$-bilinearly to
$\langle\;,\;\rangle\mathrel{:} M_{\bf R}\times N_{\bf R}\longrightarrow{\bf R}$.
By a {\em cone} in $N_{\bf R}$, we mean a strongly convex rational
polyhedral cone \cite[Chap.1,1.1]{Oda1},
i.e., a cone $\sigma$ is equal to
${\bf R}_0n_1 +\cdots +{\bf R}_0n_s$ for a finite subset
$\{n_1,\cdots, n_s\}$ of $N$ and satisfies
$\sigma\cap(-\sigma) =\{0\}$, where
${\bf R}_0 :=\{c\in{\bf R}\mathrel{;} c\geq 0\}$.
The cone $\{0\}$ is denoted by ${\bf 0}$.
For each cone $\sigma$ we set $r_\sigma :=\dim\sigma$.
For a cone $\sigma$ in $N_{\bf R}$, we define
$N(\sigma) := N\cap(\sigma + (-\sigma))\simeq{\bf Z}^{r_\sigma}$ and
$N[\sigma] := N/N(\sigma)\simeq{\bf Z}^{r-r_\sigma}$.
Hence, $N(\sigma)_{\bf R}$ is the real subspace $\sigma + (-\sigma)$
of $N_{\bf R}$.
On the other hand, we set
$\sigma^\bot :=\{x\in M_{\bf R}\mathrel{;}\langle x, a\rangle = 0,\forall a\in\sigma\}$,
$M[\sigma] := M\cap\sigma^\bot$ and $M(\sigma) := M/M[\sigma]$.
Then $N(\sigma)_{\bf R}\subset N_{\bf R}$ and $M[\sigma]_{\bf R}\subset M_{\bf R}$
are orthogonal complement of each other with respect to the pairing.
These notations are defined so that $N(\sigma)$ and $M(\sigma)$
as well as $N[\sigma]$ and $M[\sigma]$ are mutually dual,
respectively.
In this article, we often treat finite dimensional
graded ${\bf Q}$-vector spaces.
Let $V$ be such a ${\bf Q}$-vector space.
Then, we denote by $V_j$ the vector subspace consisting of zero
and the homogeneous elements of degree $j$ for each $j\in{\bf Z}$.
For two finite dimensional ${\bf Q}$-vector spaces $V$ and
$W$, we define the grading of $V\otimes_{\bf Q} W$ by
\begin{equation}
(V\otimes_{\bf Q} W)_k :=\bigoplus_{i+j=k}V_i\otimes_{\bf Q} W_j
\end{equation}
for each $k\in{\bf Z}$.
We identify $V\otimes_{\bf Q} W$ with $W\otimes_{\bf Q} V$ by the
identifications
\begin{equation}
\label{left and right op}
a_i\otimes b_j = (-1)^{ij}b_j\otimes a_i
\end{equation}
for $a_i\in V_i, b_j\in W_j$ for all $i, j\in{\bf Z}$.
We denote by $A(M_{\bf Q})$ the exterior algebra
$\bigwedge^\bullet M_{\bf Q}$ over the rational number field ${\bf Q}$.
Then $A(M_{\bf Q})$ is a graded ${\bf Q}$-algebra with
$A(M_{\bf Q})_i :=\bigwedge^i M_{\bf Q}$ for each $i\in{\bf Z}$.
The algebra $A(N_{\bf Q}) :=\bigwedge^\bullet N_{\bf Q}$ is more important
in our theory.
The grading of $A(N_{\bf Q})$ is defined negatively, i.e.,
$A(N_{\bf Q})_i :=\bigwedge^{-i} N_{\bf Q}$ for each $i$.
For a cone $\sigma$ in $N_{\bf R}$,
$A(N(\sigma)_{\bf Q}) :=\bigwedge^\bullet N(\sigma)_{\bf Q}$ is a graded
subalgebra of $A(N_{\bf Q})$.
In order to simplify the notation, we set $A := A(N_{\bf Q})$ and
$A^* := A(M_{\bf Q})$.
For a cone $\sigma$ in $N_{\bf R}$, we set
$A(\sigma) := A(N(\sigma)_{\bf Q})\subset A$
and $A^*[\sigma] := A(M[\sigma]_{\bf Q})\subset A^*$.
Let $C =\bigoplus_{p\in{\bf Z}}C_p$ be a graded ${\bf Q}$-subalgebra
of $A$ or $A^*$.
For a graded left $C$-module $V =\bigoplus_{q\in{\bf Z}}V_q$, we define
the {\em associated} graded right $C$-module structure on $V$ by
\begin{equation}
\label{leftright}
xa := \sum_{q\in{\bf Z}}(-1)^{pq}ax_q
\end{equation}
for $a\in C_p$ and $x =\sum_{q\in{\bf Z}}x_q\in V$.
Conversely, if $V$ is a graded right $C$-module, then the asssociated
graded left $C$-module structure is defined similarly.
In both cases, we see easily that $a(xb) = (ax)b$ for $a, b\in C$
and $x\in V$.
Hence $V$ is a two-sided $C$-module and we call it simply a
$C$-module.
The following lemma is checked easily.
\begin{Lem}
\label{Lem 1.1}
Let $V, V'$ be graded $C$-modules for $C$ as above.
If $W$ is a homogeneous left or right $C$-submodule of $V$,
then it is a two-sided $C$-submodule of $V$.
If $f\mathrel{:} V\rightarrow V'$ is a homogeneous homomorphism as left or
right $C$-modules of degree zero, then it is a homomorphism
of two-sided $C$-modules.
\end{Lem}
Let $\mathop{\rm GM}\nolimits(C)$ be the abelian category of finitely generated
graded $C$-modules where the morphisms are defined to be
homogeneous homomorphisms of degree zero.
By definition, every object in $\mathop{\rm GM}\nolimits(C)$ is
finite-dimensional as a ${\bf Q}$-vector space.
Since $M_{\bf Q}$ is the dual ${\bf Q}$-vector space of $N_{\bf Q}$, each
homogeneous element $a\in A^*_p$ for any $p\in{\bf Z}$ induces a
homogeneous ${\bf Q}$-linear map $i(a)\mathrel{:} A\rightarrow A$ (cf.\ \cite[5.14]{Greub})
which is usually called the right interior product.
Here note that the degree of the interior product $i(a)$ is $p$,
since the indices of $A$ are given negatively.
It is known that this operation induces a graded right
$A^*$-module structure on $A$ (cf.\ \cite[(5.50)]{Greub}).
With the associated left $A^*$-module structure, we regard
$A$ as a two-sided $A^*$-module.
\begin{Lem}
\label{Lem 1.2}
Let $\sigma$ be a cone in $N_{\bf R}$.
Then the left operations of $A(\sigma)$ and
$A^*[\sigma]$ on $A$ commute with each other.
This is also true for the right operations.
\end{Lem}
{\sl Proof.}\quad
For $a\in A^*[\sigma]_p$, $b\in A(\sigma)_q$ and
$x\in A_s$, we prove the equalities
\begin{eqnarray}
b(ax) & = & a(bx)\;, \label{Lem 1.2 1} \\
(xa)b & = & (xb)a\;, \label{Lem 1.2 2} \\
(bx)a & = & (-1)^{pq}b(xa)\;. \label{Lem 1.2 3}
\end{eqnarray}
Since $N(\sigma)_{\bf Q}$ is the orthogonal complement of
$M[\sigma]_{\bf Q}$ and $(bx)a = i(a)(b\wedge x)$, the equality
(\ref{Lem 1.2 3}) is equal to \cite[(5.58)]{Greub}.
By this equality, we have
$b(ax) = (-1)^{ps}b(xa) = (-1)^{ps+pq}(bx)a = a(bx)$ and
$(xa)b = (-1)^{pq+qs}b(xa) = (-1)^{qs}(bx)a =(xb)a$.
These are the equalities (\ref{Lem 1.2 1}) and (\ref{Lem 1.2 2}).
\QED
Let $\sigma$ be a cone in $N_{\bf R}$ and let $V$ be a graded
$A(\sigma)$-module.
Then the above lemma implies that $V_A := V\otimes_{A(\sigma)}A$
has a structure of $A^*[\sigma]$-module such that
$a(u\otimes x) = u\otimes ax$ for $u\in V$, $x\in A$ and
$a\in A^*[\sigma]$.
\begin{Lem}
\label{Lem 1.3}
Let $V$ be a finitely generated graded $A(\sigma)$-module
with a homogeneous ${\bf Q}$-basis $\{u_1,\cdots, u_s\}$ and let
$\{x_1,\cdots, x_r\}$ be a ${\bf Q}$-basis of $N_{\bf Q}$ such that
$N(\sigma)_{\bf Q}$ is generated by $\{x_1,\cdots, x_k\}$ for
$k :=r_\sigma$.
Then $V_A$ is a free $A^*[\sigma]$-module with the
basis $\{u'_1,\cdots, u'_s\}$, where
$u'_i :=u_i\otimes(x_{k+1}\wedge\cdots\wedge x_r)$ for each $i$.
\end{Lem}
{\sl Proof.}\quad
Let $E$ be the subspace ${\bf Q} x_{k+1} +\cdots +{\bf Q} x_r$ of $N_{\bf Q}$.
Since the operation of $A^*[\sigma]$ on $A$ is defined
by the interior products, we have
$A(E) = A^*[\sigma](x_{k+1}\wedge\cdots\wedge x_r)$ in $A$.
Hence $A = A(\sigma)\otimes_{\bf Q} A(E)$ is equal to
$(A(\sigma)\otimes_{\bf Q} A^*[\sigma])(x_{k+1}\wedge\cdots\wedge x_r)$.
Hence
\begin{equation}
V_A = V\otimes_{A(\sigma)}A\simeq
V\otimes_{\bf Q} A^*[\sigma](x_{k+1}\wedge\cdots\wedge x_r)
\end{equation}
as $A^*[\sigma]$-modules and we get the lemma.
\QED
Let $\sigma$ be a cone of $N_{\bf R}$.
We set
\begin{equation}
\det(\sigma) := \bigwedge^{r_\sigma}N(\sigma)\simeq{\bf Z}
\end{equation}
and $\det(\sigma)_{\bf Q} :=\det(\sigma)\otimes{\bf Q}$.
We denote by $\mathop{\rm Det}\nolimits(\sigma)_{\bf Q}$ the graded ${\bf Q}$-vector space
defined by $(\mathop{\rm Det}\nolimits(\sigma)_{\bf Q})_{-r_\sigma} :=\det(\sigma)_{\bf Q}$
and $(\mathop{\rm Det}\nolimits(\sigma)_{\bf Q})_j :=\{0\}$ for $j\not = -r_\sigma$.
For a finitely generated graded $A(\sigma)$-module $V$,
we define a graded $A(\sigma)$-module $\d_\sigma(V)$ as follows.
We define graded ${\bf Q}$-vector spaces $\d_\sigma^{\rm left}(V)$ and
$\d_\sigma^{\rm right}(V)$ by
\begin{equation}
\d_\sigma^{\rm left}(V) =\d_\sigma^{\rm right}(V) :=
\mathop{\rm Hom}\nolimits_{\bf Q}(V,\mathop{\rm Det}\nolimits(\sigma)_{\bf Q}) =
\bigoplus_{i\in{\bf Z}}\mathop{\rm Hom}\nolimits_{\bf Q}(V_{-r_\sigma-i},\det(\sigma)_{\bf Q})\;.
\end{equation}
For $x\in V$ and $y\in\d_\sigma^{\rm left}(V)$
(resp.\ $z\in\d_\sigma^{\rm right}(V)$) we denote the operation by
$(y, x)$ (resp.\ (x, z)).
The right $A(\sigma)$-module structure of $\d_\sigma^{\rm left}(V)$
(resp.\ the left $A(\sigma)$-module structure of
$\d_\sigma^{\rm right}(V)$) is defined by
\begin{equation}
(ya, x) := (y, ax)\;\;\;(\hbox{ resp.\ }(x, az) := (xa, z))
\end{equation}
for $a\in A(\sigma)$.
There exists a unique homogeneous isomorphism
$\varphi\mathrel{:}\d_\sigma^{\rm left}(V)\rightarrow\d_\sigma^{\rm right}(V)$
degree zero such that $(y, x) = (-1)^{pq}(x, \varphi(y))$
for homogeneous elements $x\in V_p$ and $y\in\d_\sigma^{\rm left}(V)_q$
for all $p, q\in{\bf Z}$.
We define $\d_\sigma(V)$ to be the identification of
$\d_\sigma^{\rm left}(V)$ and $\d_\sigma^{\rm right}(V)$ by the
isomorphism $\varphi$.
Namely, we have $(y, x) = (-1)^{pq}(x, y)$ for $x\in V_p$ and
$y\in\d_\sigma(V)_q$.
Here note that $(y, x) = 0$ if $p + q\not= -r_\sigma$.
It is easy to see that the induced left and right
$A(\sigma)$-module structures on $\d_\sigma(V)$ have the
compatibility (\ref{leftright}).
By definition, we have
\begin{equation}
\label{eq of d-dual}
\dim_{\bf Q}\d_\sigma(V)_j =\dim_{\bf Q} V_{-r_\sigma-j}
\end{equation}
for every $j\in{\bf Z}$.
For $V\in\mathop{\rm GM}\nolimits(A(\sigma))$, we define an $A(\sigma)$-homomorphism
$\iota\mathrel{:} V\rightarrow\d_\sigma(\d_\sigma(V))$ by
$(y,\iota(x)) := (y, x)$ for $x\in V$ and $y\in\d_\sigma(V)$.
It is easy to see that the symmetric equality
$(\iota(x), y) = (x, y)$ holds.
Since $V$ is a finite dimensional ${\bf Q}$-vector space, the
pairings are perfect and $\iota$ is an isomorphism.
We call $\iota$ the {\em canoniacal isomorphism}.
For a homomorphism $f\mathrel{:} V\rightarrow W$ in $\mathop{\rm GM}\nolimits(A(\sigma))$,
we define $\d_\sigma(f)$ to be the natural induced homomorphism
$\d_\sigma(W)\rightarrow \d_\sigma(V)$ of $A(\sigma)$-modules,
i.e., the equality $(y, f(x)) = (\d_\sigma(f)(y), x)$ for
$x\in V$ and $y\in\d_\sigma(W)$.
It is clear by definition that the correspondence $V\mapsto\d_\sigma(V)$
is a contravariant exact functor from $\mathop{\rm GM}\nolimits(A(\sigma))$ to itself.
If $\pi$ is of dimension $r$, then $A(\pi) = A$ and
$\d_\pi$ is a functor from $\mathop{\rm GM}\nolimits(A)$ to itself.
We denote this functor by $\d_N$ which does not depend
on the choice of $\pi$.
Let $\sigma$ and $\rho$ be cones in $N_{\bf R}$ with $\sigma\prec\rho$.
For $V$ in $\mathop{\rm GM}\nolimits(A(\sigma))$, we denote by $V_{A(\rho)}$ the
graded $A(\rho)$-module
$V\otimes_{A(\sigma)}A(\rho) = A(\rho)\otimes_{A(\sigma)}V$,
where we identify $x\otimes a$ with $(-1)^{pq}a\otimes x$ for
$x\in V_p$ and $a\in A(\rho)_q$.
For a morphism $f\mathrel{:} V\rightarrow V'$ in $\mathop{\rm GM}\nolimits(A(\sigma))$, we denote
$f_{A(\rho)} := f\otimes1_{A(\rho)}\mathrel{:} V_{A(\rho)}\rightarrow V'_{A(\rho)}$.
The correspondence $V\mapsto V_{A(\rho)}$ is a covariant functor from
$\mathop{\rm GM}\nolimits(A(\sigma))$ to $\mathop{\rm GM}\nolimits(A(\rho))$.
Let $H$ be a linear subspace of $N(\rho)_{\bf Q}$ such that
$N(\rho)_{\bf Q} = N(\sigma)_{\bf Q}\oplus H$.
Then $A(\rho) = A(\sigma)\otimes_{\bf Q} A(H)$ and
$V_{A(\rho)} = V\otimes_{\bf Q} A(H)$ for any $V$ in $\mathop{\rm GM}\nolimits(A(\sigma))$.
This implies that the functor is exact.
Similarly, the functor from $\mathop{\rm GM}\nolimits(A(\sigma))$ to $\mathop{\rm GM}\nolimits(A)$ defined
by $V\mapsto V_A$ is exact.
For $V\in\mathop{\rm GM}\nolimits(A(\sigma))$, we define an $A(\sigma)$-homomorphism
\begin{equation}
\label{d_ sigma rho}
\varphi\mathrel{:}\d_\sigma(V)\rightarrow\d_\rho(V_{A(\rho)})
\end{equation}
by $(\varphi(y), xa) := \phi_\rho((y, x)a)$ for $x\in V$,
$y\in\d_\sigma(V)$ and $a\in A(\rho)$, where $\phi_\rho$ is the
homogeneous projection $A(\rho)\rightarrow A(\rho)_{-r_\rho} =\mathop{\rm Det}\nolimits(\rho)_{\bf Q}$.
\begin{Lem}
\label{Lem 1.4}
Let $\sigma,\rho$ be cones in $N_{\bf R}$ with $\sigma\prec\rho$.
Then the homomorphism (\ref{d_ sigma rho}) induces $A(\rho)$-isomorphism
$\d_\rho(V_{A(\rho)})\simeq\d_\sigma(V)_{A(\rho)}$ for every $V$ in
$\mathop{\rm GM}\nolimits(A(\sigma))$.
\end{Lem}
{\sl Proof.}\quad
Consider the case $V = A(\sigma)$.
Let $\phi_\sigma\mathrel{:} A(\sigma)\rightarrow\mathop{\rm Det}\nolimits(\sigma)_{\bf Q} = A(\sigma)_{-r_\sigma}$
be the homogeneous projection.
For $u\in A(\sigma)$, the corresponding element $\phi_\sigma u$ of
$\d_\sigma(A(\sigma))$ is given by
$(\phi_\sigma u, x) :=\phi_\sigma(u\wedge x)$ for $x\in A(\sigma)$.
Hence $\d_\sigma(A(\sigma))$ is a free $A(\sigma)$-module
generated by $\phi_\sigma$.
Similarly, $\d_\rho(A(\rho))$ is equal to $\phi_\rho A(\rho)$.
By the definiton of $\varphi$ in (\ref{d_ sigma rho}), we have
$\varphi(\phi_\sigma) = \phi_\rho$.
Hence $\varphi$ induces an isomorphism
$\d_\sigma(A(\sigma))_{A(\rho)}\simeq\d_\rho(A(\rho))$.
For general $V$, we take an exact sequence
\begin{equation}
\bigoplus_{i=1}^m A(\sigma)x_i\mathop{\longrightarrow}\limits^{f}
\bigoplus_{j=1}^n A(\sigma)y_j\longrightarrow V\lra0
\end{equation}
of graded left $A(\sigma)$-modules, where $\{x_1,\cdots, x_m\}$
and $\{y_1,\cdots, y_n\}$ are homogeneous bases.
Then by the exactness of the functors, we get exact sequences
\begin{equation}
0\longrightarrow\d_\sigma(V)_{A(\rho)}\longrightarrow\bigoplus_{j=1}^n y_j^*A(\rho)
\mathop{\longrightarrow}\limits^{{}^{\rm t}f}\bigoplus_{i=1}^mx_i^*A(\rho)
\end{equation}
and
\begin{equation}
0\longrightarrow\d_\rho(V_{A(\rho)})\longrightarrow\bigoplus_{j=1}^n y_j^*A(\rho)
\mathop{\longrightarrow}\limits^{{}^{\rm t}f}\bigoplus_{i=1}^mx_i^*A(\rho)
\end{equation}
of graded right $A(\rho)$-modules.
Hence we get the required isomorphism.
\QED
Let $V^\bullet$ be a finite $d$-complex of graded $A(\sigma)$-modules
in $\mathop{\rm GM}\nolimits(A(\sigma))$ with $d = (d_V^i\mathrel{:} i\in{\bf Z})$.
We define the complex $\d_\sigma(V)^\bullet$ by
$\d_\sigma(V)^i := \d_\sigma(V^{-i})$ for $i\in{\bf Z}$.
The coboudary map $d = (d_{\d_\sigma(V)}^i)$ is defined by
\begin{equation}
d_{\d_\sigma(V)}^i := (-1)^{i+1}\d_\sigma(d_V^{-i-1})
\mathrel{:}\d_\sigma(V)^i\longrightarrow\d_\sigma(V)^{i+1}
\end{equation}
for each $i\in{\bf Z}$ (cf. \cite[1.1.5]{Deligne}).
Note that we have the principle to put the dot sign of complexes
at the right end.
Since $\d_\sigma$ is an exact functor, the cohomology group
$\H^p(\d_\sigma(V)^\bullet)$ is isomorphic to
$\d_\sigma(\H^{-p}(V^\bullet))$ as a graded ${\bf Q}$-vector space.
By the equality (\ref{eq of d-dual}), we get the following lemma.
\begin{Lem}
\label{Lem 1.5}
Let $\sigma\subset N_{\bf R}$ be a cone and let
$V^\bullet$ be a finite $d$-complex in $\mathop{\rm GM}\nolimits(A(\sigma))$.
Then
\begin{equation}
\dim_{\bf Q}\H^p(\d_\sigma(V)^\bullet)_q =
\dim_{\bf Q}\H^{-p}(V^\bullet)_{-r_\sigma-q}
\end{equation}
for any integers $p,q$.
In particular, we have
\begin{equation}
\dim_{\bf Q}\H^p(\d_N(V)^\bullet)_q =\dim_{\bf Q}\H^{-p}(V^\bullet)_{-r-q}
\end{equation}
if $V^\bullet$ is a finite $d$-complex in $\mathop{\rm GM}\nolimits(A)$.
\end{Lem}
For a homomorphism $f\mathrel{:} V^\bullet\rightarrow W^\bullet$ in $\mathop{\rm CGM}\nolimits(A(\sigma))$,
the homomorphism
\begin{equation}
\d_\sigma(f)\mathrel{:}\d_\sigma(W)^\bullet\rightarrow\d_\sigma(V)^\bullet
\end{equation}
is defined as the collection
\begin{equation}
\{\d_\sigma(f)^i =\d_\sigma(f^{-i})\mathrel{;} i\in{\bf Z}\}\;.
\end{equation}
For a homogeneous ${\bf Q}$-subalgebra $C$ of $A$,
we denote by $\mathop{\rm CGM}\nolimits(C)$ the category of finite $d$-complexes
in $\mathop{\rm GM}\nolimits(C)$.
It is easy to see that $\d_\sigma$ is a contravariant exact
functor of the abelian category $\mathop{\rm CGM}\nolimits(A(\sigma))$ to itself.
Let $V^\bullet$ be an object of $\mathop{\rm CGM}\nolimits(C)$.
Hence each $V^i =\bigoplus_{j\in{\bf Z}}V_j^i$ is in $\mathop{\rm GM}\nolimits(C)$.
For each $i, j\in{\bf Z}$, we denote by
$d^{i,j}\mathrel{:} V_j^i\rightarrow V_j^{i+1}$ the homogeneous component of
$d^i$ of degree $j$.
For each integer $k$, the {\em gradual truncation} below
$(\mathop{\rm gt}\nolimits_{\leq k}V)^\bullet$ is the homogeneous subcomplex
of $V^\bullet$ defined by
\begin{equation}
(\mathop{\rm gt}\nolimits_{\leq k}V)_j^i :=\left\{
\begin{array}{lll}
V_j^i & \hbox{ if } & i+j < k \\
\mathop{\rm Ker}\nolimits d^{i,j} & \hbox{ if } & i+j = k \\
\{0\} & \hbox{ if } & i+j > k
\end{array}
\right.
\end{equation}
and the gradual truncation above
$(\mathop{\rm gt}\nolimits^{\geq k}V)^\bullet$ is the homogeneous quotient
complex of $V^\bullet$ defined by
\begin{equation}
(\mathop{\rm gt}\nolimits^{\geq k}V)_j^i :=\left\{
\begin{array}{lll}
\{0\} & \hbox{ if } & i+j < k \\
\mathop{\rm Coker}\nolimits d^{i-1,j} & \hbox{ if } & i+j = k \\
V_j^i & \hbox{ if } & i+j > k\;.
\end{array}
\right.
\end{equation}
Since $C$ is graded negatively, $(\mathop{\rm gt}\nolimits_{\leq k}V)^\bullet$
and $(\mathop{\rm gt}\nolimits^{\geq k}V)^\bullet$ are $d$-complexes in $\mathop{\rm GM}\nolimits(C)$.
Hence, these are covariant functors from $\mathop{\rm CGM}\nolimits(C)$
to itself.
The variant gradual truncations $(\widetilde{\mathop{\rm gt}\nolimits}_{\leq k}V)^\bullet$
and $(\widetilde{\mathop{\rm gt}\nolimits}^{\geq k}V)^\bullet$ are defined by
\begin{equation}
(\widetilde{\mathop{\rm gt}\nolimits}_{\leq k}V)_j^i :=\left\{
\begin{array}{lll}
V_j^i & \hbox{ if } & i+j\leq k \\
\Im d^{i-1,j} & \hbox{ if } & i+j = k+1 \\
\{0\} & \hbox{ if } & i+j > k+1
\end{array}
\right.
\end{equation}
and
\begin{equation}
(\widetilde{\mathop{\rm gt}\nolimits}^{\geq k}V)_j^i :=\left\{
\begin{array}{lll}
\{0\} & \hbox{ if } & i+j < k-1 \\
\Im d^{i,j} & \hbox{ if } & i+j = k-1 \\
V_j^i & \hbox{ if } & i+j\geq k\;,
\end{array}
\right.
\end{equation}
respectively.
It is easy to see that $(\widetilde{\mathop{\rm gt}\nolimits}_{\leq k}V)^\bullet$ is
quasi-isomorphic to $(\mathop{\rm gt}\nolimits_{\leq k}V)^\bullet$, while
$(\widetilde{\mathop{\rm gt}\nolimits}^{\geq k}V)^\bullet$ is quasi-isomorphic to
$(\mathop{\rm gt}\nolimits^{\geq k}V)^\bullet$ (cf.\cite[p.93]{GM2}).
It is clear that
\begin{equation}
\label{H of gt leq}
\H^p((\mathop{\rm gt}\nolimits_{\leq k}V)^\bullet)_q =\left\{\begin{array}{ll}
\H^p(V^\bullet)_q & \hbox{ for } p + q\leq k \\
\{0\} & \hbox{ for } p + q > k
\end{array}\right.
\end{equation}
and
\begin{equation}
\H^p((\mathop{\rm gt}\nolimits^{\geq k}V)^\bullet)_q =\left\{\begin{array}{ll}
\{0\} & \hbox{ for } p + q < k \\
\H^p(V^\bullet)_q & \hbox{ for } p + q\geq k\;.
\end{array}\right.
\end{equation}
An object $V^\bullet$ in $\mathop{\rm CGM}\nolimits(C)$ is said to be
{\em acyclic} if $\H^p(V^\bullet) = \{0\}$ for all integers $p$.
\begin{Lem}
\label{Lem 1.6}
Let $\sigma$ be a cone of $N_{\bf R}$.
Let $V^\bullet$ be in $\mathop{\rm CGM}\nolimits(A(\sigma))$ and $k$ be an integer.
Then the $d$-complex $(\mathop{\rm gt}\nolimits_{\leq k}\d_\sigma(V))^\bullet$ is acyclic
if and only if $(\mathop{\rm gt}\nolimits^{\geq -r_\sigma -k}V)^\bullet$ is.
Similarly, the $d$-complex $(\mathop{\rm gt}\nolimits^{\geq k}\d_\sigma(V))^\bullet$ is
acyclic if and only if $(\mathop{\rm gt}\nolimits_{\leq -r_\sigma -k}V)^\bullet$ is.
If $V^\bullet$ is in $\mathop{\rm CGM}\nolimits(A)$, then these assertions with
$\d_\sigma$ replaced by $\d_N$ and $r_\sigma$ replaced by
$r$ hold.
\end{Lem}
{\sl Proof.}\quad
By (\ref{H of gt leq}),
$(\mathop{\rm gt}\nolimits_{\leq k}\d_\sigma(V))^\bullet$ is acyclic if and only if
$\H^p(\d_\sigma(V)^\bullet)_q =\{0\}$ for $p + q\leq k$.
By Lemma~\ref{Lem 1.5}, this is equivalent to the condition
\begin{equation}
\dim_{\bf Q}\H^p(V^\bullet)_q =
\dim_{\bf Q}\H^{-p}(\d_\sigma(V)^\bullet)_{-r_\sigma-q} = 0
\end{equation}
for $p + q\geq -r_\sigma - k$.
This condition means that $(\mathop{\rm gt}\nolimits^{\geq-r_\sigma-k}V)^\bullet$ is acyclic.
The second assertion is similarly proved.
The last assertion is obtained by taking $\sigma$ with
$r_\sigma = r$.
\QED
\section{The graded exterior modules on a fan}
\setcounter{equation}{0}
Let $\Delta$ be a finite fan of $N_{\bf R}$ \cite[1.1]{Oda1}.
We introduce an additive category $\mathop{\rm GEM}\nolimits(\Delta)$ which contains
$\mathop{\rm GM}\nolimits(A(\sigma))$ as full subcategories for all $\sigma\in\Delta$.
Let $V$ be in $\mathop{\rm GM}\nolimits(A(\sigma))$ and $W$ in $\mathop{\rm GM}\nolimits(A(\rho))$.
If $\sigma\prec\rho$, then $A(\sigma)\subset A(\rho)$ and
$W$ has an induced structure of graded $A(\sigma)$-module.
A morphism $f\mathrel{:} V\rightarrow W$ in $\mathop{\rm GEM}\nolimits(\Delta)$
is defined to be a homogeneous $A(\sigma)$-homomorphism
of degree zero.
If $\sigma$ is not a face of $\rho$, we allow only the zero
map as a morphism even if $A(\sigma)$ happens to be contained
in $A(\rho)$.
Consequently, the additive category $\mathop{\rm GEM}\nolimits(\Delta)$ of
{\em graded exterior modules} on $\Delta$ is defined as follows.
A graded exterior module $L$ on $\Delta$ is a collection
$(L(\sigma)\mathrel{;}\sigma\in\Delta)$ of objects $L(\sigma)$
in $\mathop{\rm GM}\nolimits(A(\sigma))$ for $\sigma\in\Delta$.
A {\em homomorphism} $f\mathrel{:} L\rightarrow K$ of graded exterior modules
on $\Delta$ is a collection $f = (f(\sigma/\rho))$ of morphisms
\begin{equation}
f(\sigma/\rho)\mathrel{:} L(\sigma)\longrightarrow K(\rho)
\end{equation}
in $\mathop{\rm GM}\nolimits(A(\sigma))$ for all pairs $(\sigma,\rho)$ of cones
in $\Delta$ with $\sigma\prec\rho$.
For $f\mathrel{:} L\rightarrow K$ and $g\mathrel{:} K\rightarrow J$, the composite
$(g\cdot f)\mathrel{:} L\rightarrow J$ is defined by
\begin{equation}
(g\cdot f)(\sigma/\rho) :=
\sum_{\tau\in F[\sigma,\rho]}g(\tau/\rho)\cdot f(\sigma/\tau)
\end{equation}
for $\sigma,\rho$ with $\sigma\prec\rho$, where
$F[\sigma,\rho]$ is the set of the faces $\tau$ of $\rho$
with $\sigma\prec\tau$.
The direct sum of finite objects in $\mathop{\rm GEM}\nolimits(\Delta)$ is defined
naturally.
An object $V$ of $\mathop{\rm GM}\nolimits(A(\sigma))$ is also regarded as an object of
$\mathop{\rm GEM}\nolimits(\Delta)$ by defining $V(\sigma) := V$ and $V(\rho) := \{0\}$
for $\rho\not= \sigma$.
In this sense, we may write
$L =\bigoplus_{\sigma\in\Delta}L(\sigma)$.
A homomorphism $f\mathrel{:} L\rightarrow K$ is said to be {\em unmixed} if
$f(\sigma/\rho) = 0$ for any $\sigma,\rho$ with $\sigma\not= \rho$.
If $f$ is unmixed, $\mathop{\rm Ker}\nolimits f$, $\mathop{\rm Coker}\nolimits f$ and $\Im f$ are defined
naturally as an object of $\mathop{\rm GEM}\nolimits(\Delta)$.
We denote by $\mathop{\rm UGEM}\nolimits(\Delta)$ the category of the objects
of $\mathop{\rm GEM}\nolimits(\Delta)$ with the class of homomorphisms restricted
to unmixed ones.
It is easy to see that $\mathop{\rm UGEM}\nolimits(\Delta)$ is an abelian category.
Let $f\mathrel{:} L\rightarrow K$ be a homomorphism in $\mathop{\rm GEM}\nolimits(\Delta)$.
We say that $L$ is a submodule or a subobject of $K$,
if $f$ is unmixed and $f(\sigma/\sigma)\mathrel{:} L(\sigma)\rightarrow K(\sigma)$
is an inclusion map for every $\sigma\in\Delta$.
If $L$ is a submodule of $K$, then we define an object $K/L$ in
$\mathop{\rm GEM}\nolimits(\Delta)$ by $(K/L)(\sigma) := K(\sigma)/L(\sigma)$ for
$\sigma\in\Delta$.
Namely, we have a short exact sequence
\begin{equation}
0\longrightarrow L\longrightarrow K\longrightarrow K/L\longrightarrow 0
\end{equation}
in $\mathop{\rm UGEM}\nolimits(\Delta)$.
We denote $\hat\Delta :=\Delta\cup\{\alpha\}$ and call it an
{\em augmented fan} where $\alpha$ is an imaginary cone.
We define $A(\alpha) := A$.
The category $\mathop{\rm GEM}\nolimits(\hat\Delta )$ is defined similarly by supposing
$\sigma\prec\alpha$ for all $\sigma\in\hat\Delta$.
An object $L$ of $\mathop{\rm GEM}\nolimits(\Delta)$ is also regarded as that of
$\mathop{\rm GEM}\nolimits(\hat\Delta )$ by setting $L(\alpha) := \{0\}$.
For each $\rho\in\hat\Delta$, an additive covariant functor
\begin{equation}
\i_\rho^*\mathrel{:}\mathop{\rm GEM}\nolimits(\hat\Delta)\longrightarrow\mathop{\rm GM}\nolimits(A(\rho))
\end{equation}
is defined by
\begin{equation}
\i_\rho^*(L) :=\bigoplus_{\sigma\in F(\rho)}%
L(\sigma)_{A(\rho)}\;,
\end{equation}
where $F(\rho)$ is the set of faces of $\rho$ and
we suppose $F(\rho) =\hat\Delta$ if $\rho =\alpha$.
Recall that $L(\sigma)_{A(\rho)}$ is the graded
$A(\rho)$-module $L(\sigma)\otimes_{A(\sigma)}A(\rho)$
for each $\sigma$.
We usually denote by $\Gamma$ the functor $\i_\alpha^*$
For a homomorphism $f\mathrel{:} L\rightarrow K$ in $\mathop{\rm GEM}\nolimits(\hat\Delta)$, the
$(\sigma,\tau)$-component of the homomorphism
\begin{equation}
\begin{array}{ccccc}
\i_\rho^*(f) & \mathrel{:} & \i_\rho^*(L) & \longrightarrow & \i_\rho^*(K) \\
& & \| & & \| \\
& & \displaystyle\bigoplus_{\sigma\in F(\rho)}L(\sigma)_{A(\rho)}
& & \displaystyle\bigoplus_{\tau\in F(\rho)}K(\tau)_{A(\rho)}
\end{array}
\end{equation}
is defined to be $f(\sigma/\tau)_{A(\rho)}$ if
$\sigma\prec\tau$ and zero otherwise.
For each $\rho\in\Delta$, the additive covariant functor
\begin{equation}
\i_\rho^!\mathrel{:}\mathop{\rm GEM}\nolimits(\hat\Delta)\longrightarrow\mathop{\rm GM}\nolimits(A(\rho))
\end{equation}
is defined by $\i_\rho^!(L) := L(\rho)$.
For $f\mathrel{:} L\rightarrow K$, the homomorphism $\i_\rho^!(f)$ is defined
to be $f(\rho/\rho)$.
For cones $\sigma,\tau$ with $\sigma\prec\tau$ and
$r_\tau = r_\sigma + 1$, we define the {\em incidence isomorphism}
$q'_{\sigma/\tau}\mathrel{:}\det(\sigma)\rightarrow\det(\tau)$ of free
${\bf Z}$-modules of rank one as follows.
By the condition, $N(\tau)/N(\sigma)$ is a free ${\bf Z}$-module
of rank one.
We take $a\in N(\tau)\cap\tau$ such that the class of
$a$ in $N(\tau)/N(\sigma)$ is a generator.
Then we define $q'_{\sigma/\tau}(w) := a\wedge w$ for
$w\in\det(\sigma)$.
For cones $\sigma,\rho$ with $\sigma\prec\rho$ and
$r_\rho = r_\sigma + 2$, there exists exactly two cones
$\tau$ with $\sigma\prec\tau\prec\rho$ and
$r_\tau = r_\sigma + 1$.
Let these cones be $\tau_1,\tau_2$.
Then the equality
\begin{equation}
\label{codim 2 eq}
q'_{\sigma/\tau_1}\cdot q'_{\tau_1/\rho} +
q'_{\sigma/\tau_2}\cdot q'_{\tau_2/\rho} = 0
\end{equation}
holds (cf. \cite[Lem.1.4]{Ishida1}).
For a subset $\Phi\subset\Delta$ and an integer $i$, we set
\begin{equation}
\Phi(i) :=\{\sigma\in\Phi\mathrel{;} r_\sigma = i\}\;.
\end{equation}
A subset $\Phi$ of $\Delta$ is said to be {\em locally star closed}
if $\sigma,\rho\in\Phi$, $\tau\in\Delta$ and $\sigma\prec\tau\prec\rho$
imply $\tau\in\Phi$.
For a locally star closed subset $\Phi$ of $\Delta$, we define
a complex $E(\Phi,{\bf Z})^\bullet$ of free ${\bf Z}$-modules as follows.
For each integer $i$, we set
\begin{equation}
E(\Phi,{\bf Z})^i :=\bigoplus_{\sigma\in\Phi(i)}\det(\sigma)\;.
\end{equation}
For $\sigma\in\Phi(i)$ and $\tau\in\Phi(i+1)$, the
$(\sigma,\tau)$-component of the coboundary map
\begin{equation}
d^i\mathrel{:} E(\Phi,{\bf Z})^i\rightarrow E(\Phi,{\bf Z})^{i+1}
\end{equation}
is defined to be $q'_{\sigma/\tau}$.
The equality $d^{i+1}\cdot d^i = 0$ follows from
(\ref{codim 2 eq}) for every $i$.
We say that a locally star closed subset $\Phi\subset\Delta$
is {\em $1$-complete} if, for each $\sigma\in\Phi(r-1)$, there exist
exactly two $\tau$'s in $\Phi(r)$ with $\sigma\prec\tau$.
If the finite fan $\Delta$ is complete \cite[Thm.1.11]{Oda1}, then
$\Delta(\sigma{\prec}) :=\{\rho\in\Delta\mathrel{;}\sigma\prec\rho\}$
is $1$-complete for every $\sigma\in\Delta$.
If $\Phi$ is $1$-complete, then we can define an augmented complex
$E(\hat\Phi,{\bf Z})^\bullet$ for $\hat\Phi :=\Phi\cup\{\alpha\}$ by
defining $r_\alpha := r+1$, $\det(\alpha) :=\bigwedge^r N$ and
$q'_{\tau/\alpha} := {\rm id}$ for every $\tau\in\Phi(r)$
with respect to the identification $\det(\tau) =\det(\alpha)$.
In particular, $E(\hat\Phi,{\bf Z})^i = E(\Phi,{\bf Z})^i$ for $i\not= r+1$
and $E(\hat\Phi,{\bf Z})^{r+1} =\det(\alpha)$.
When $\Delta$ is complete, $E(\hat\Delta(\sigma{\prec}),{\bf Z})^\bullet$
is acyclic for every $\sigma\in\Delta$.
Actually, $\H^i(E(\Delta(\sigma{\prec}),{\bf Z})^\bullet)$ is equal
to the $(i{-}r_\sigma{-}1)$-th reduced cohomology group
of an $(r{-}r_\sigma{-}1)$-dimensional sphere, and hence it
vanishes if $i\not= r$.
The $r$-th cohomology is killed by
$E(\hat\Phi,{\bf Z})^{r+1} =\det(\alpha)$.
We denote by $\mathop{\rm CGEM}\nolimits(\Delta)$ and $\mathop{\rm CGEM}\nolimits(\hat\Delta)$ the category
of finite $d$-complexes in $\mathop{\rm GEM}\nolimits(\Delta)$ and $\mathop{\rm GEM}\nolimits(\hat\Delta)$,
respectively.
Let $(L^\bullet, d_L)$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then, for each $\rho\in\Delta$, we get an object
$(L(\rho)^\bullet, d_L(\rho/\rho))$ of $\mathop{\rm CGM}\nolimits(A(\rho))$
which we denote simply $L(\rho)^\bullet$.
For $\rho,\mu\in\Delta$ with $\rho\prec\mu$, we set
$F[\rho,\mu] :=\{\sigma\in F(\mu)\mathrel{;}\rho\prec\sigma\}$.
Then the equality $d_L\cdot d_L = 0$ implies that
\begin{equation}
\label{CGEM1}
\sum_{\sigma\in F[\rho,\mu]}%
d_L^{i+1}(\sigma/\mu)\cdot d_L^i(\rho/\sigma) = 0
\end{equation}
for each integer $i$.
In particular, if $r_\mu =r_\rho + 1$, then the collection
$(d_L^i(\rho/\mu)\mathrel{;} i\in{\bf Z})$ defines a homomorphism of
complexes $d_L(\rho/\mu)\mathrel{:} L(\rho)^\bullet\rightarrow L(\mu)[1]^\bullet$
since then $F[\rho,\mu] =\{\rho,\mu\}$ and the equality (\ref{CGEM1})
imply the commutativity of the diagram
\begin{equation}
\begin{array}{ccc}
\makebox[40pt]{}L(\rho)^i
& \mathop{\longrightarrow}\limits^{\textstyle d_L^i(\rho/\rho)}
& L^{i+1}(\rho)\makebox[70pt]{} \\
d_L^i(\rho/\mu)\downarrow &
& \downarrow d_L^{i+1}(\rho/\mu)\\
\makebox[40pt]{}L(\mu)[1]^i
&\mathop{\longrightarrow}\limits^{\textstyle d_{L[1]}^i(\mu/\mu)}
& L(\mu)[1]^{i+1}\makebox[70pt]{}
\end{array}
\;,
\end{equation}
where $d_{L[1]}^i(\mu/\mu) = - d_{L}^{i+1}(\mu/\mu)$.
Conversely, assume that complexes
$L(\rho)^\bullet\in\mathop{\rm CGM}\nolimits(A(\rho))$ for $\rho\in\Delta$
and homomorphisms
\begin{equation}
d_L^i(\sigma/\tau)\mathrel{:} L(\sigma)^i\longrightarrow L(\tau)^{i+1}
\end{equation}
for $\sigma,\tau\in\Delta$ with $\sigma\prec\tau$ and
$i\in{\bf Z}$ are given.
If they satisfy (\ref{CGEM1}) for all $(\rho,\mu)$ and
$i\in{\bf Z}$, then we get a complex $(L^\bullet, d_L)$ in
$\mathop{\rm CGEM}\nolimits(\Delta)$.
An object $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ is said to be
{\em shallow} if $d_L(\sigma/\rho) = 0$ for any $\sigma,\rho$
with $r_\rho - r_\sigma\geq 2$.
In order to define a shallow object $L^\bullet$ of
$\mathop{\rm CGEM}\nolimits(\Delta)$, it is sufficient to give the following data
(1), (2) and check the condition (3).
(1) A complex $L(\sigma)^\bullet\in\mathop{\rm CGM}\nolimits(A(\sigma))$ for
each $\sigma\in\Delta$.
(2) A homomorphism
$d(\sigma/\tau)\mathrel{:} L(\sigma)^\bullet\rightarrow L(\tau)[1]^\bullet$
for each pair $(\sigma,\tau)$ of cones in $\Delta$ with
$\sigma\prec\tau$ and $r_\tau = r_\sigma + 1$.
(3) The equality
\begin{equation}
d(\tau_1/\rho)^{i+1}\cdot d(\sigma/\tau_1)^i +
d(\tau_2/\rho)^{i+1}\cdot d(\sigma/\tau_2)^i = 0
\end{equation}
holds for all $i\in{\bf Z}$ and all pairs $(\sigma,\rho)$ with
$\sigma\prec\rho$ and $r_\rho = r_\sigma + 2$, where
$\tau_1,\tau_2$ are the dual cones with
$\sigma\prec\tau_i\prec\rho$ and $r_{\tau_i} = r_\sigma + 1$.
For an object $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, we define
a shallow object $\tilde L^\bullet$ as follows.
For each $\rho\in\Delta$ and $i\in{\bf Z}$, we set
\begin{equation}
\label{def of tilde L}
\tilde L(\rho)^i :=
\bigoplus_{\sigma\in F(\rho)}\;\bigoplus_{\eta\in F(\sigma)}
\det(\rho)\otimes\det(\sigma)^*\otimes
L(\eta)_{A(\rho)}^{-r_\rho+r_\sigma+i}\;,
\end{equation}
where $\det(\sigma)^* :=\mathop{\rm Hom}\nolimits_{\bf Z}(\det(\sigma),{\bf Z})$.
For
\begin{equation}
\label{def of tilde L +1}
\tilde L(\rho)^{i+1} :=
\bigoplus_{\tau\in F(\rho)}\;\bigoplus_{\zeta\in F(\tau)}
\det(\rho)\otimes\det(\tau)^*\otimes
L(\zeta)_{A(\rho)}^{-r_\rho+r_\tau+i+1}\;,
\end{equation}
the $((\sigma,\eta),(\tau,\zeta))$-component of the
coboundary map
$d(\rho/\rho)^i\mathrel{:} \tilde L(\rho)^i\rightarrow \tilde L(\rho)^{i+1}$
is defined to be zero map except for the cases (a) $\sigma =\tau$ and
$\eta\prec\zeta$, or (b) $\tau\prec\sigma$, $r_\sigma = r_\tau + 1$
and $\eta =\zeta$. In case (a), the component is defined to be
$(-1)^{r_\rho - r_\sigma}{\rm id}
\otimes d(\eta/\zeta)^{-r_\rho+r_\sigma+i}$, and
in case (b), it is defined to be $(-1)^{r_\rho - r_\sigma-1}
1_{\det(\rho)}\otimes(q'_{\tau/\sigma})^*\otimes{\rm id}$, where
${\rm id}$'s are identity maps of the corresponding parts.
For $\rho\prec\mu$ with $r_\mu = r_\rho + 1$, the homomorphism
$d(\rho/\mu)\mathrel{:}\tilde L(\rho)^\bullet\rightarrow\tilde L(\mu)[1]^\bullet$
is defined to be the tensor product of $q'_{\rho/\mu}$ and the
natural inclusion map.
It is easy to check that $\tilde L(\rho)^\bullet$ is actually
a complex and $d(\rho/\mu)$ is a homomorphism of complexes.
The condition (3) in the definition of shallow complexes is
satisfied by the equality (\ref{codim 2 eq}).
We define a homomorphism $f_L\mathrel{:} L^\bullet\rightarrow\tilde L^\bullet$
as follows.
For $\tau,\rho\in\Delta$ with $\tau\prec\rho$, the
$A(\tau)$-homomorphism
$f_L(\tau/\rho)^i\mathrel{:} L(\tau)^i\rightarrow\tilde L(\rho)^i$ is the
inclusion map to the component
$\det(\rho)\otimes\det(\sigma)^*\otimes
L(\eta)_{A(\rho)}^{-r_\rho+r_\sigma+i}$ for $\sigma =\rho$
and $\eta =\tau$ in the description (\ref{def of tilde L}).
The compatibility with the coboundary maps is checked easily.
A homomorphism $f\mathrel{:} L^\bullet\rightarrow K^\bullet$ of $d$-complexes of
graded exterior modules on $\Delta$ is said to be a
{\em quasi-isomorphism} if
$f(\sigma/\sigma)\mathrel{:} L(\sigma)^\bullet\rightarrow K(\sigma)^\bullet$ is a
quasi-isomorphism of $d$-complexes of $A(\sigma)$-modules
for every $\sigma\in\Delta$.
The following proposition shows that any object in $\mathop{\rm CGEM}\nolimits(\Delta)$
is quasi-isomorphic to a shallow one.
\begin{Prop}
\label{Prop 2.1}
For any $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, the homomorphism
$f_L\mathrel{:} L^\bullet\rightarrow\tilde L^\bullet$ is quasi-isomorphic.
\end{Prop}
{\sl Proof.}\quad
We have to check that
$f_L(\rho/\rho)\mathrel{:} L(\rho)^\bullet\rightarrow\tilde L(\rho)^\bullet$
is a quasi-isomorphism for each $\rho\in\Delta$.
We define a decreasing filtration $\{F^k\}$ on the complex
$V^\bullet :=\tilde L(\rho)^\bullet$ as follows.
For each integer $k$, $F^k(V)^i$ is defined to be the direct
sum of the components
\begin{equation}
\det(\rho)\otimes\det(\sigma)^*\otimes
L(\eta)_{A(\rho)}^{-r_\rho+r_\sigma+i}
\end{equation}
in the description (\ref{def of tilde L}) for $\sigma,\eta$
with $r_\eta\geq k$.
Then $F^k(V)^0 = V^\bullet$ and $F^k(V)^\bullet$ is a
subcomplex of $V^\bullet$ for each $k\geq 0$.
We can decompose the quotient complex
$F^k(V)^\bullet/F^{k+1}(V)^\bullet$ to a direct sum
\begin{equation}
\bigoplus_{\eta\in F(\rho)(k)}V_{k,\eta}^\bullet\;,
\end{equation}
where $V_{k,\eta}^\bullet$ is the part consisting of the
components related to $\eta$.
Then we see that $V_{k,\eta}^\bullet$ is isomorphic to the
associated single complex of the bicomplex
\begin{equation}
\det(\rho)\otimes\mathop{\rm Hom}\nolimits(E(F[\eta,\rho],{\bf Z}),{\bf Z})[-r_\rho]^\bullet
\otimes L(\eta)_{A(\rho)}^\bullet\;,
\end{equation}
where we denote by $\mathop{\rm Hom}\nolimits(E,{\bf Z})^\bullet$ the complex defined by
\begin{equation}
\mathop{\rm Hom}\nolimits(E,{\bf Z})^i :=\mathop{\rm Hom}\nolimits(E^{-i},{\bf Z})
\end{equation}
and $d^i := (-1)^{i+1}(d_E^{-i-1})^*$ for $i\in{\bf Z}$.
If $k < r_\rho$, then $\eta\not=\rho$ and $E(F[\eta,\rho],{\bf Z})^\bullet$
is acyclic, and hence so is $V_{k,\eta}^\bullet$.
Hence $V^\bullet$ is quasi-isomorphic to the subcomplex
$F^{r_\rho}(V)^\bullet$.
Since $f_L(\rho/\rho)$ is an isomorphism from $L(\rho)^\bullet$
onto $F^{r_\rho}(V)^\bullet$, it is a quasi-isomorphism to
$V^\bullet =\tilde L(\rho)^\bullet$.
\QED
For a complex $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, we define a
shallow complex ${\bf D}(L)^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ as follows.
For each $\rho\in\Delta$, we set
\begin{equation}
{\bf D}(L)(\rho)^\bullet :=
\det(\rho)\otimes\d_\rho(\i_\rho^*L)[-r_\rho]^\bullet\;.
\end{equation}
Let $\rho,\mu\in\Delta$ satisfy $\rho\prec\mu$ and
$r_\mu = r_\rho + 1$.
By the definition of $\i_\mu^*$ and $\i_\rho^*$, we see that
$(\i_\rho^*L^i)_{A(\mu)}$ is a direct summand of
$\i_\mu^*L^i$.
Let $p(\mu/\rho)^i\mathrel{:}\i_\mu^*L^i\rightarrow(\i_\rho^*L^i)_{A(\mu)}$ be
the natural projection map.
We see that $\{p(\mu/\rho)^i\mathrel{;} i\in{\bf Z}\}$ defines a homomorphism
$p(\mu/\rho)\mathrel{:}\i_\mu^*L^\bullet\rightarrow(\i_\rho^*L)_{A(\mu)}^\bullet$.
Hence we get a homomorphism
\begin{equation}
\label{dual rho mu}
\d_\mu(p(\mu/\rho))\mathrel{:}\d_\mu((\i_\rho^*L)_{A(\mu)})^\bullet\longrightarrow
\d_\mu(\i_\mu^*L)^\bullet\;.
\end{equation}
Since
\begin{equation}
\d_\mu((\i_\rho^*L)_{A(\mu)})^\bullet =
\d_\rho(\i_\rho^*L)_{A(\mu)}^\bullet
\end{equation}
by Lemma~\ref{Lem 1.4}, we get a homomorphism
\begin{equation}
i(\rho/\mu)\mathrel{:}\d_\rho(\i_\rho^*L)^\bullet\longrightarrow
\d_\mu(\i_\mu^*L)^\bullet
\end{equation}
as the composite of the inclusion
$\d_\rho(\i_\rho^*L)^\bullet\rightarrow\d_\rho(\i_\rho^*L)_{A(\mu)}^\bullet$
and (\ref{dual rho mu}).
We define
\begin{equation}
d_{{\bf D}(L)}(\rho/\mu)\mathrel{:}{\bf D}(L)(\rho)^\bullet\longrightarrow
{\bf D}(L)(\mu)[1]^\bullet
\end{equation}
to be $q_{\rho/\mu}\otimes i(\rho/\mu)[-r_\rho]$.
By the equality (\ref{codim 2 eq}), the condition (3)
of the construction of shallow complexes
is satisfied and we get a complex ${\bf D}(L)^\bullet$.
\begin{Lem}
\label{Lem 2.2}
Let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then, there exists a quasi-isomorphism
$\varphi\mathrel{:} L^\bullet\rightarrow {\bf D}({\bf D}(L))^\bullet$.
\end{Lem}
{\sl Proof.}\quad
We prove that ${\bf D}({\bf D}(L))^\bullet$ is isomorphic to
$\tilde L^\bullet$.
Then the lemma follows from Proposition~\ref{Prop 2.1}.
Let $\rho$ be an element of $\Delta$.
For each integer $i$,
\begin{eqnarray*}
& & {\bf D}({\bf D}(L))(\rho)^i \\
& = & \bigoplus_{\sigma\in F(\rho)}%
\det(\rho)\otimes\d_\rho({\bf D}(L)(\sigma)_{A(\rho)}^{r_\rho-i}) \\
& = & \bigoplus_{\sigma\in F(\rho)}\bigoplus_{\eta\in F(\sigma)}%
\det(\rho)\otimes\d_\rho(\det(\sigma)\otimes
\d_\sigma(L(\eta)_{A(\sigma)}^{-r_\rho+r_\sigma+i})_{A(\rho)}) \\
& = & \bigoplus_{\sigma\in F(\rho)}\bigoplus_{\eta\in F(\sigma)}%
\det(\rho)\otimes\det(\sigma)^*\otimes\d_\rho(
\d_\sigma(L(\eta)_{A(\sigma)}^{-r_\rho+r_\sigma+i})_{A(\rho)})\;.
\end{eqnarray*}
By Lemma~\ref{Lem 1.4}, we have a natural isomorphism
\begin{equation}
\d_\rho(\d_\sigma(L(\eta)_{A(\sigma)}^{-r_\rho+r_\sigma+i})_{A(\rho)})
\simeq\d_\rho(\d_\rho(L(\eta)_{A(\rho)}^{-r_\rho+r_\sigma+i}))\;.
\end{equation}
We identify the last $A(\rho)$-module with
$L(\eta)_{A(\rho)}^{-r_\rho+r_\sigma+i}$ by the canonical
isomorphism for all $(\sigma,\eta)$.
we know ${\bf D}({\bf D}(L))(\rho)^i =\tilde L(\rho)^i$ for every
$\rho\in\Delta$ and $i\in{\bf Z}$, however the coboundary map is not
equal to that of $\tilde L^\bullet$.
We consider the descriptions (\ref{def of tilde L}) and
(\ref{def of tilde L +1}) with replacing $\tilde L(\rho)^i$ and
$\tilde L(\rho)^{i+1}$ by ${\bf D}({\bf D}(L))(\rho)^i$ and
${\bf D}({\bf D}(L))(\rho)^{i+1}$, respectively.
We can check that the $((\sigma,\eta),(\tau,\zeta))$-component of the
coboundary map
$d(\rho/\rho)^i\mathrel{:} \tilde {\bf D}({\bf D}(L))(\rho)^i\rightarrow{\bf D}({\bf D}(L))(\rho)^{i+1}$
is the zero map except for the cases (a) $\sigma =\tau$ and
$\eta\prec\zeta$, or (b) $\tau\prec\sigma$, $r_\sigma = r_\tau + 1$
and $\eta =\zeta$. In case (a), the component is calculated to be
$(-1)^{r_\rho + 1}{\rm id}\otimes d(\eta/\zeta)^{-r_\sigma+r_\sigma+i}$,
and in case (b), it is
$(-1)^{i+1}1_{\det(\rho)}\otimes(q'_{\tau/\sigma})^*\otimes{\rm id}$.
For each $\rho$ and $i$, we define an isomorphism
\begin{equation}
\varphi(\rho)^i\mathrel{:}\tilde L(\rho)^i\longrightarrow{\bf D}({\bf D}(L))(\rho)^i
\end{equation}
by defining its restriction to the component
\begin{equation}
\det(\rho)\otimes\det(\sigma)^*\otimes
L(\eta)_{A(\rho)}^{-r_\rho+r_\sigma+i}
\end{equation}
to be $(-1)^{(r_\rho+i)(r_\sigma+1)}$ times the
identity map to the same component of ${\bf D}({\bf D}(L))(\rho)^i$.
Then we see that the collection $\{\varphi(\rho)^i\}$ defines
an unmixed isomorphism $\tilde L^\bullet\rightarrow{\bf D}({\bf D}(L))^\bullet$.
\QED
We can define a similar functor
\begin{equation}
\hat{\bf D}\mathrel{:}\mathop{\rm CGEM}\nolimits(\hat\Delta)\longrightarrow\mathop{\rm CGEM}\nolimits(\hat\Delta)
\end{equation}
for an augmented 1-complete fan $\hat\Delta$ by the convention
that $r_\alpha = r+1$, $F(\alpha) =\hat\Delta$ and
$q_{\tau/\alpha} := 1_{\det N}$ for $\tau\in\Delta(r)$.
When $L^\bullet$ is an object of $\mathop{\rm CGEM}\nolimits(\Delta)$,
$\hat{\bf D}(L)^\bullet$ is in $\mathop{\rm CGEM}\nolimits(\hat\Delta)$.
Since $\det(\alpha) =\det N$, we get an exact sequence
\begin{equation}
0\rightarrow\hat{\bf D}(L)(\alpha)^\bullet\longrightarrow
\hat{\bf D}(L)^\bullet\longrightarrow{\bf D}(L)^\bullet\rightarrow 0
\end{equation}
in $\mathop{\rm CGEM}\nolimits(\hat\Delta)$ as well as an exact sequence
\begin{equation}
\label{augmented exact}
0\rightarrow\det N\otimes\d_N(\Gamma(L))[-r-1]^\bullet\longrightarrow
\Gamma(\hat{\bf D}(L))^\bullet\longrightarrow\Gamma({\bf D}(L))^\bullet\rightarrow 0
\end{equation}
in $\mathop{\rm CGM}\nolimits(A)$ by applying $\Gamma$.
\begin{Lem}
\label{Lem 2.3}
Assume that $\Delta$ is a complete fan.
Then, for any $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, the complex
of $A$-modules $\Gamma((\hat{\bf D}(L)))^\bullet$ is acyclic.
\end{Lem}
{\sl Proof.}\quad
For each $i\in{\bf Z}$, we have
\begin{equation}
\label{hat dual}
\Gamma(\hat{\bf D}(L))^i =
\bigoplus_{\rho\in\hat\Delta}\;\bigoplus_{\sigma\in F(\rho)}
\det(\rho)\otimes\d_\rho(L(\sigma)_{A(\rho)}^{-r_\rho-i})_A\;.
\end{equation}
For
\begin{equation}
\Gamma(\hat{\bf D}(L))^{i+1} =
\bigoplus_{\mu\in\hat\Delta}\;\bigoplus_{\tau\in F(\mu)}
\det(\mu)\otimes\d_\mu(L(\tau)_{A(\mu)}^{-r_\mu-i-1})_A\;,
\end{equation}
the $((\rho,\sigma),(\mu,\tau))$-component of the coboundary
map is nonzero only for (a) $\rho =\mu$ and $\tau\prec\sigma$,
or (b) $\rho\prec\mu$, $r_\mu = r_\rho + 1$ and $\sigma =\tau$.
In case (a), the component is
$(-1)^{i+1}{\rm id}\otimes\d_\rho(d(\tau/\sigma)_{A(\rho)}^{-r_\rho-i-1})_A$,
and in case (b), it is $q'_{\rho/\mu}\otimes{\rm id}$, where
${\rm id}$'s are the identities of the corresponding parts,
respectively.
For $V^\bullet :=\Gamma(\hat{\bf D}(L))^\bullet$, we introduce a
deceasing filtration $\{F^k\}$ as follow.
For each integer $i$, we define $F^k(V)^i$ to be the direct
sum of the components
$\det(\rho)\otimes\d_\rho(L(\sigma)_{A(\rho)}^{-r_\rho-i})_A$ of
$\Gamma(\hat{\bf D}(L))^i$ in the description (\ref{hat dual})
for all pairs $(\rho,\sigma)$ with $r_\sigma\geq k$.
Then $F^k(V)^\bullet$ is a subcomplex of $V^\bullet$ for
every $k\in{\bf Z}$, and $F^k(V)^\bullet/F^{k+1}(V)^\bullet$ is
a direct sum of complexes
$\bigoplus_{\sigma\in\Delta(k)}V_\sigma^\bullet$, where
$V_\sigma^\bullet$ is the part related to each $\sigma$.
We see that $V_\sigma^\bullet$ is isomorphic to the
associated single complex of the bicomplex
\begin{equation}
E(F[\sigma,\alpha],{\bf Z})^\bullet\otimes
\d_\sigma(L(\sigma))_A^\bullet\;.
\end{equation}
Since $\Delta$ is complete, $E(F[\sigma,\alpha],{\bf Z})^\bullet$
is acyclic for every $\sigma\in\Delta$.
Hence $F^k(V)^\bullet/F^{k+1}(V)^\bullet$ is also acyclic
for every $k$.
Since $F^0(V)^\bullet = V^\bullet$ and $F^{r+1}(V)^\bullet =\{0\}$,
$V^\bullet$ is acyclic.
\QED
By the long exact sequence obtained from the exact sequence
(\ref{augmented exact}), we get the following corollary.
\begin{Cor}
\label{Cor 2.4}
If $\Delta$ is complete, then there exists an isomorphism
\begin{equation}
\H^{-p}(\d_N(\Gamma(L))^\bullet)\simeq
\H^{r-p}(\Gamma({\bf D}(L))^\bullet)
\end{equation}
in $\mathop{\rm GM}\nolimits(A)$ for each integer $p$.
\end{Cor}
The following proposition is a consequence of Lemma~\ref{Lem 1.5}
and Corollary~\ref{Cor 2.4}.
\begin{Prop}
\label{Prop 2.5}
Assume that $\Delta$ is complete.
For any integer $p, q$, the equality
\begin{equation}
\dim_{\bf Q}\H^p(\Gamma(L)^\bullet)_q =
\dim_{\bf Q}\H^{r-p}(\Gamma({\bf D}(L))^\bullet)_{-r-q}
\end{equation}
holds.
\end{Prop}
A finite fan $\Delta$ of $N_{\bf R}$ is said to be {\em lifted complete}
if there exists a rational line $\ell$ of $N_{\bf R}$ going through
the origin with the following property.
Let $\bar\sigma$ be the image of $\sigma$ in the quotient $N_{\bf R}/\ell$
for each $\sigma\in\Delta$.
Then (1) $\dim\bar\sigma =r_\sigma$ for every $\sigma\in\Delta$,
(2) $\bar\sigma\not=\bar\tau$ for any distinct $\sigma,\tau\in\Delta$
and (3) $\Delta_\ell :=\{\bar\sigma\mathrel{;}\sigma\in\Delta\}$ is a
complete fan of $N_{\bf R}/\ell$.
For a lifted complete fan, the associated toric variety has
an action of the multiplicative algebraic group ${\bf G}_{\rm m}$, and it
has a complete toric variety of dimension $r-1$ as the
geometric quotient in the sense of Mumford's geometric invariant
theory.
Let $\Delta$ be a lifted complete fan with respect to $\ell$ and
let $\ell^+$ be one of the one-dimensional cones contained
in $\ell$.
By the property (1), $\dim(\tau +\ell^+) =r_\tau + 1$ for every
$\tau\in\Delta$.
The {\em oriented} lifted complete fan $\tilde\Delta$ is defined
to be $\Delta\cup\{\beta\}$ where $\beta$ is an imaginary
cone of dimension $r$.
We suppose $\sigma\prec\beta$ for every $\sigma\in\Delta$.
We define $\det(\beta) :=\det N$ and
$q'_{\tau/\beta} := q'_{\tau/\tau'}$ for
$\tau\in\Delta(r{-}1)$, where $\tau' :=\tau +\ell^+$
For any star closed subset $\Phi\subset\tilde\Delta$, the
complex $E(\Phi,{\bf Z})^\bullet$ is defined similarly as in the
previous case.
The complex $E(\tilde\Delta(\sigma{\prec}),{\bf Z})^\bullet$ is
acyclic for every $\sigma\in\Delta$ since it is isomorphic to
the augmented complex
$E(\hat\Delta_\ell,{\bf Z})^\bullet$.
Each lifted complete fan has two orientations according to the
choice of $\ell^+$, but it does not depend on the choice
of the line $\ell$.
Here we give three typical examples of oriented lifted complete fans.
(1)
Let $C\subset N_{\bf R}$ be a closed convex cone of dimension $r$
which may not be strongly convex and which is not equal to $N_{\bf R}$.
Let $\partial C$ be the boundary set of $C$.
Then a finite fan $\Delta$ with the support $\partial C$ is
lifted complete.
Any $\ell$ which intersects the interior of $C$ satisfies the
condition.
As the natural orientation, we take $\ell^+ :=\ell\cap C$.
(2)
Let $\Phi$ be a simplicial complete fan of $N_{\bf R}$ and let $\gamma$
be a one-dimensional cone in $\Phi$.
Set
\begin{equation}
\Delta :=
\{\sigma\in\Phi\mathrel{;}\gamma\not\prec\sigma,\sigma+\gamma\in\Phi\}\;.
\end{equation}
Then $\Delta$ is a lifted complete fan and $\ell^+ :=\gamma$
defines an orientation (cf.\cite{Oda2}).
(3)
Let $N'$ be a free ${\bf Z}$-module of rank $r - 1$, $\Phi$ a
finite complete fan of $N'_{\bf R}$ and $h$ a real-valued
continuous function on $N'_{\bf R}$ which is linear on each cone
$\sigma\in\Phi$ and has rational values on $N'_{\bf Q}$.
Then the fan $\Delta =\{\sigma'\mathrel{;}\sigma\in\Phi\}$ of
$N_{\bf R} := N'_{\bf R}\oplus {\bf R}$ is lifted complete, where
$\sigma' :=\{(x,h(x))\mathrel{;} x\in\sigma\}$ for each $\sigma\in\Phi$.
We take $\ell^+ :=\{0\}\times{\bf R}_0$ as the orientation.
This type of fan is treated in \cite{Oda3}.
Let $\tilde\Delta =\Delta\cup\{\beta\}$ be an oriented
lifted complete fan.
The category $\mathop{\rm GEM}\nolimits(\tilde\Delta)$ is defined by setting
$A(\beta) := A$.
Then the functors
\begin{equation}
\i_\beta^*\mathrel{:}\mathop{\rm GEM}\nolimits(\tilde\Delta)\longrightarrow\mathop{\rm GM}\nolimits(A)
\end{equation}
and
\begin{equation}
\tilde{\bf D}\mathrel{:}\mathop{\rm GEM}\nolimits(\tilde\Delta)\longrightarrow\mathop{\rm GEM}\nolimits(\tilde\Delta)
\end{equation}
are defined similarly as $\i_\alpha^*$ and $\hat{\bf D}$ in
the case of augmented 1-complete fans, respectively.
We denote also by $\Gamma$ the functor $\i_\beta^*$.
We omit the proofs of the following results, since they are
similar to those of the corresponding results for an
augmented complete fan.
\begin{Lem}
\label{Lem 2.6}
Let $\tilde\Delta =\Delta\cup\{\beta\}$ be an oriented
lifted complete fan.
Then, for any $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, the complex
of $A$-modules $\Gamma(\tilde{\bf D}(L))^\bullet$ is acyclic, and
there exists an exact sequence
\begin{equation}
0\longrightarrow\det N\otimes\d(\Gamma(L))^\bullet[-r]\longrightarrow
\Gamma(\tilde{\bf D}(L))^\bullet\longrightarrow
\Gamma({\bf D}(L))^\bullet\longrightarrow 0\;.
\end{equation}
\end{Lem}
\begin{Cor}
\label{Cor 2.7}
Let $\Delta\cup\{\beta\}$ be an oriented lifted complete fan.
Then for any $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$, there exists an isomorphism
\begin{equation}
\H^{-p}(\d_N(\Gamma(L))^\bullet)\simeq
\H^{r-p-1}(\Gamma({\bf D}(L))^\bullet)
\end{equation}
in $\mathop{\rm GM}\nolimits(A)$ for each integer $p$.
\end{Cor}
\begin{Prop}
\label{Prop 2.8}
Let $\Delta\cup\{\beta\}$ be an oriented lifted complete fan.
For any integer $p, q$, the equality
\begin{equation}
\dim\H^p(\Gamma(L)^\bullet)_q =
\dim\H^{r-p-1}(\Gamma({\bf D}(L))^\bullet)_{-r-q}
\end{equation}
holds.
\end{Prop}
Let $\Delta$ be a finite fan of $N_{\bf R}$ and let $\Phi$ be
a subfan of $\Delta$.
For $K^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Phi)$, we denote by the same symbol
$K^\bullet$ the trivial extension to $\Delta$, i.e., we define
$K(\sigma)^\bullet :=\{0\}$ for $\sigma\in\Delta\setminus\Phi$.
For $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ and $\sigma\in\Delta$, we
set
\begin{equation}
\i_\sigma^\circ(L)^\bullet :=
\i_\sigma^*(L|F(\sigma){\setminus}\{\sigma\})^\bullet\;,
\end{equation}
where $(L|F(\sigma){\setminus}\{\sigma\})^\bullet$ is the
restriction of $L^\bullet$ to $F(\sigma)\setminus\{\sigma\}$.
We see the case $\Phi =\Delta\setminus\{\pi\}$ for a maximal element
$\pi\in\Delta$.
Let $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$.
Since
\begin{equation}
\i_\pi^\circ(L)^i =
\bigoplus_{\sigma\in F(\pi)\setminus\{\pi\}}L(\sigma)_{A(\pi)}^i
\end{equation}
and $\i_\pi^*(L)^i = L(\pi)^i\oplus\i_\pi^\circ(L)^i$ for
each $i\in{\bf Z}$, there exists an exact sequence
\begin{equation}
0\rightarrow L(\pi)^\bullet\longrightarrow\i_\pi^*(L)^\bullet\longrightarrow
\i_\pi^\circ(L)^\bullet\rightarrow 0
\end{equation}
in $\mathop{\rm CGM}\nolimits(A(\pi))$.
In other words, $\i_\pi^*(L)^\bullet$ is equal to the
mapping cone of the homomorphism
\begin{equation}
\label{hom of ext}
\phi\mathrel{:}\i_\pi^\circ(L)[-1]^\bullet\longrightarrow L(\pi)^\bullet
\end{equation}
whose component for each $\sigma\in F(\pi)\setminus\{\pi\}$ is
$d(\sigma/\pi)_{A(\pi)}$.
The extension of $(L|\Phi)^\bullet$ to $L^\bullet$ is determined
by the above homomorphism $\phi$.
Actually, if $(L|\Phi)^\bullet\in\mathop{\rm CGEM}\nolimits(\Phi)$,
$L(\pi)^\bullet$ and the homomorphism (\ref{hom of ext}) is
given, then we get the extension $L^\bullet$.
We define a functor $\j_!^\Phi\mathrel{:}\mathop{\rm CGEM}\nolimits(\Phi)\rightarrow\mathop{\rm CGEM}\nolimits(\Delta)$
as follows.
For $K^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Phi)$, we define $\j_!^\Phi(K)^\bullet$
by
\begin{equation}
\j_!^\Phi(K)(\sigma)^\bullet := K(\sigma)^\bullet
\end{equation}
for $\sigma\in\Phi$ and
\begin{equation}
\j_!^\Phi(K)(\pi)^\bullet := \i_\pi^\circ(K)[-1]^\bullet
= \i_\pi^*(K)[-1]^\bullet\;.
\end{equation}
In particular,
\begin{equation}
\j_!^\Phi(K)(\pi)^{i+1} =
\bigoplus_{\sigma\in F(\pi)\setminus\{\pi\}}K(\sigma)_{A(\pi)}^i\;.
\end{equation}
We define the extension $\j_!^\Phi(K)^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$
of $K^\bullet$ by the identity map
\begin{equation}
\i_\pi^\circ(K)[-1]^\bullet\longrightarrow\j_!^\Phi(K)(\pi)^\bullet\;.
\end{equation}
Let $(L^\bullet, d_L)$ and $(K^\bullet, d_K)$ be $d$-complexes
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
By an {\em unmixed} homomorphsm $f\mathrel{:} L^\bullet\rightarrow K^\bullet$, we means a
collection $\{f^i\mathrel{;} i\in{\bf Z}\}$ of unmixed homomorphisms $f^i\mathrel{:} L^i\rightarrow K^i$
such that $d_K^i\cdot f^i = f^{i+1}\cdot d_L^i$ for every $i\in{\bf Z}$.
Note that $d_L^i$ and $d_K^i$ are not necessary unmixed.
Let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then there exists a unique unmixed homomorphism
$\j_!^\Phi(L|\Phi)^\bullet\rightarrow L^\bullet$ which is the
identity map on $\Phi$.
The homomorphism
\begin{equation}
\j_!^\Phi(L|\Phi)(\pi)^\bullet =\i_\pi^\circ(L)[-1]^\bullet
\rightarrow L(\pi)^\bullet
\end{equation}
is defined to be (\ref{hom of ext}).
We can define a functor $\j_!^\Phi\mathrel{:}\mathop{\rm CGEM}\nolimits(\Phi)\rightarrow\mathop{\rm CGEM}\nolimits(\Delta)$
for general subfan $\Phi\subset\Delta$ as the composite of
the above functors.
However, we will not use the general case.
Let $\pi$ be a maximal element of $\Delta$.
For $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ and an integer $k$, we define
$\mathop{\rm gt}\nolimits_\pi^{\geq k}L^\bullet$ by
\begin{equation}
(\mathop{\rm gt}\nolimits_\pi^{\geq k}L)(\sigma)^\bullet =
\left\{
\begin{array}{ll}
L(\sigma)^\bullet & \hbox{if } \sigma\not=\pi \\
\mathop{\rm gt}\nolimits^{\geq k}L(\pi)^\bullet & \hbox{if } \sigma=\pi\;.
\end{array}
\right.
\end{equation}
There exists a natural unmixed homomorphism
$L^\bullet\longrightarrow\mathop{\rm gt}\nolimits_\pi^{\geq k}L^\bullet$
such that the components for $\sigma\not=\pi$ are the identity
maps and the component for $\pi$ is the natural surjection
$L^\bullet(\pi)\rightarrow\mathop{\rm gt}\nolimits^{\geq k}(L(\pi))^\bullet$.
This homomorphism is a quasi-isomorphism if and only if
$\mathop{\rm gt}\nolimits_{\leq k-1}(L^\bullet(\pi))$ is acyclic.
Two objects $L^\bullet, K^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ are
said to be {\em quasi-isomorphic} if there exist a finite
sequence $L_0^\bullet, L_1^\bullet,\cdots, L_{2k}^\bullet$
of objects in $\mathop{\rm CGEM}\nolimits(\Delta)$ with $L^\bullet = L_0^\bullet$
and $L_{2k}^\bullet = K^\bullet$ and quasi-isomorphisms
$L_{2i-2}^\bullet\rightarrow L_{2i-1}^\bullet$ and
$L_{2i}^\bullet\rightarrow L_{2i-1}^\bullet$ for $i = 1,\cdots, k$.
Some lemmas on this definition are given at the end this section.
A map ${\bf p}\mathrel{:}\Delta\setminus\{{\bf 0}\}\rightarrow{\bf Z}$ is called
a {\em perversity} on a finite fan $\Delta$.
We prove the following theorem.
\begin{Thm}
\label{Thm 2.9}
Let ${\bf p}$ be a perversity on $\Delta$.
Then there exists a finite $d$-complex $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$
of graded exterior modules on $\Delta$ satisfying
the following conditions.
{\rm (1)} $\H^0(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})^\bullet) ={\bf Q}$ and
$\H^i(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})^\bullet) =\{0\}$ for $i\not= 0$.
{\rm (2)} For $\sigma\in\Delta\setminus\{{\bf 0}\}$ and
$i, j\in{\bf Z}$ with $i + j\leq{\bf p}(\sigma)$, we have
$\H^i(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^\bullet)_j =\{0\}$.
{\rm (3)} For $\sigma\in\Delta\setminus\{{\bf 0}\}$ and
$i, j\in{\bf Z}$ with $i + j\geq{\bf p}(\sigma)$, we have
$\H^i(\i_\sigma^*(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet)_j =\{0\}$.
Futhermore, if $L^\bullet$ is another finite $d$-complex
satisfying the above conditions, then $L^\bullet$ is
quasi-isomorphic to $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$.
\end{Thm}
{\sl Proof.}\quad
We prove the theorem by induction on the number of
cones in $\Delta$.
If $\Delta =\{{\bf 0}\}$, then we set
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})^0 ={\bf Q}$ and
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})^i =\{0\}$ for $i\not= 0$.
Then (1) and the last assertion are clearly satisfied.
Assume that $\Delta\not=\{{\bf 0}\}$.
Let $\pi\in\Delta$ be a cone of maximal dimension.
We assume that the $d$-complex $\mathop{\rm ic}\nolimits_{\bf p}(\Phi)^\bullet$ exists
for $\Phi =\Delta\setminus\{\pi\}$
Let $L^\bullet :=\j_!^\Phi\mathop{\rm ic}\nolimits_{\bf p}(\Phi)^\bullet$.
We define
\begin{equation}
\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet :=\mathop{\rm gt}\nolimits_\pi^{\geq{\bf p}(\pi)+1}L^\bullet\;.
\end{equation}
Since
\begin{equation}
\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\pi)^\bullet =
\mathop{\rm gt}\nolimits^{\geq{\bf p}(\pi)+1}L(\pi)^\bullet\;,
\end{equation}
the truncation $\mathop{\rm gt}\nolimits_{\leq{\bf p}(\pi)}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\pi))^\bullet$
is the zero complex.
On the other hand, $\i_\pi^*\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ is equal to
the mapping cone
\begin{equation}
(L(\pi)[1]\oplus\mathop{\rm gt}\nolimits^{\geq{\bf p}(\pi)+1}L(\pi))^\bullet
\end{equation}
of the natural surjection
\begin{equation}
L(\pi)^\bullet\longrightarrow\mathop{\rm gt}\nolimits^{\geq{\bf p}(\pi)+1}L(\pi)^\bullet\;.
\end{equation}
There exists an exact sequence
\begin{equation}
0\longrightarrow\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\pi)}L(\pi)^\bullet\longrightarrow L(\pi)^\bullet
\longrightarrow\mathop{\rm gt}\nolimits^{\geq{\bf p}(\pi)+1}L(\pi)^\bullet\longrightarrow 0
\end{equation}
of $d$-complexes in $\mathop{\rm GM}\nolimits(A(\pi))$.
Hence $\i_\pi^*\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ is quasi-isomorphic to
\begin{equation}
(\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\pi)}L(\pi))[1]^\bullet =
\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\pi)-1}(L(\pi)[1])^\bullet
\end{equation}
and to $\mathop{\rm gt}\nolimits_{\leq{\bf p}(\pi)-1}(L(\pi)[1])^\bullet$.
Hence $\mathop{\rm gt}\nolimits^{\geq{\bf p}(\pi)}\i_\pi^*\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ is acyclic.
This is equivalent to the condition (3).
For the last assertion, it is sufficient to prove the following
lemma.
\begin{Lem}
\label{Lem 2.10}
Let $\pi\not={\bf 0}$ be a maximal element of $\Delta$ and let
$\Phi :=\Delta\setminus\{\pi\}$.
Let $L^\bullet, K^\bullet$ be objects of $\mathop{\rm CGEM}\nolimits(\Delta)$ which
satisfy the conditions of {\rm Theorem~\ref{Thm 2.9}}, and
assume that there exists a quasi-isomorphism
$(L|\Phi)^\bullet\rightarrow(K|\Phi)^\bullet$.
Then, $L^\bullet$ and $K^\bullet$ is connected by a finite
sequence of unmixed quasi-isomorphisms.
\end{Lem}
{\sl Proof.}\quad
By the condition (2), the natural homomorphism
\begin{equation}
L^\bullet\rightarrow\mathop{\rm gt}\nolimits_\pi^{\geq {\bf p}(\sigma)+1}L^\bullet
\end{equation}
is a quasi-isomorphism.
Since $\i_\pi^*(L)^\bullet$ is the mapping cone of the
homomorphism
$\j_!^\Phi(L|\Phi)^\bullet\rightarrow L(\pi)^\bullet$,
the condition (3) implies the homomorphism
\begin{equation}
\mathop{\rm gt}\nolimits_\pi^{\geq {\bf p}(\sigma)+1}(\j_!^\Phi(L|\Phi))^\bullet\rightarrow
\mathop{\rm gt}\nolimits_\pi^{\geq {\bf p}(\sigma)+1}L^\bullet
\end{equation}
is also a quasi-isomorphism.
There are similar quasi-isomorphisms for $K^\bullet$.
We are done since the quasi-isomorphism
$(L|\Phi)^\bullet\rightarrow(K|\Phi)^\bullet$ induces a quasi-isomorphism
\begin{equation}
\mathop{\rm gt}\nolimits_\pi^{\geq {\bf p}(\sigma)+1}(\j_!^\Phi(L|\Phi))^\bullet\rightarrow
\mathop{\rm gt}\nolimits_\pi^{\geq {\bf p}(\sigma)+1}(\j_!^\Phi(K|\Phi))^\bullet\;.
\end{equation}
\QED
Thus we complete the proof of Theorem~\ref{Thm 2.9}
In the rest of this paper, we denote by $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$
the $d$-complex constructed in the proof of the above theorem,
and we call it the intersection complex of the fan $\Delta$
with the perversity ${\bf p}$.
It does not depend on the choice of the order of the induction,
and is uniquely determined by $\Delta$ and ${\bf p}$.
If $\Phi$ is a subfan of $\Delta$, then the construction
implies that $\mathop{\rm ic}\nolimits_{\bf p}(\Phi)^\bullet$ is isomorphic to the
restriction of $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ to $\Phi$.
In particular,
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^\bullet =\mathop{\rm ic}\nolimits_{\bf p}(F(\sigma))(\sigma)^\bullet$.
Hence the complex $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^\bullet$ of
$A(\sigma)$-modules depends only on ${\bf p}$ and $\sigma$.
\begin{Prop}
\label{Prop 2.11}
Let $\sigma$ be a nonzero cone of $\Delta$ and let ${\bf p}$
be a perversity on $\Delta$.
Then (1) $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)_j^i =\{0\}$ unless
$(i, j)\in [1, r_\sigma]\times[-r_\sigma, 0]$,
(2) $\i_\sigma^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))_j^i =\{0\}$ unless
$(i, j)\in [0, r_\sigma-1]\times[-r_\sigma, 0]$ and
(3) $\i_\sigma^*(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))_j^i =\{0\}$ unless
$(i, j)\in [0, r_\sigma]\times[-r_\sigma, 0]$.
Furthermore, $\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))_j^i =\{0\}$ unless
$(i, j)\in [0, r]\times[-r, 0]$.
\end{Prop}
{\sl Proof.}\quad
We prove the proposition by induction on $r_\sigma$.
Note that $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})_j^i =\{0\}$ unless
$(i, j) = (0, 0)$ by the construction of $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$.
Let $\eta,\sigma$ be cones in $\Delta$ with $\eta\prec\sigma$.
Recall that, if we take an $(r_\sigma - r_\eta)$-dimensional
linear subspace $H$ of $N(\sigma)_{\bf Q}$ such that
$N(\sigma)_{\bf Q} = N(\eta)_{\bf Q}\oplus H$, then
$V_{A(\sigma)} = V\otimes_{\bf Q} A(H)$ for $V\in\mathop{\rm GM}\nolimits(A(\eta))$.
Hence, if $V_j =\{0\}$ unless $j\in [a, b]$ for integers
$a, b$ with $a\leq b$, then $(V_{A(\sigma)})_j =\{0\}$
unless $j\in [a-(r_\sigma - r_\eta), b]$.
Since
\begin{equation}
\i_\sigma^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^i =
\bigoplus_{\eta\in F(\sigma)\setminus\{\sigma\}}
\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\eta)_{A(\sigma)}^i\;,
\end{equation}
(2) is a consequence of (1) for
$\eta\in F(\sigma)\setminus\{\sigma\}$ which are true by
the assumption of the induction.
Since
\begin{equation}
\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^i =
\mathop{\rm gt}\nolimits^{\geq{\bf p}(\sigma)+1}(\i_\sigma^\circ\mathop{\rm ic}\nolimits_{\bf p}(\Delta)[-1])^i\;,
\end{equation}
(1) follows from (2).
Since
\begin{equation}
\i_\sigma^*(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^i =
\i_\sigma^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^i\oplus\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^i
\end{equation}
for every $i\in{\bf Z}$, (3) follows from (1) and (2).
Since
\begin{equation}
\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^i =
\bigoplus_{\sigma\in\Delta}\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)_A^i
\end{equation}
for every $i$, the last assertion is a consequence of
(1) for all $\sigma\in\Delta\setminus\{{\bf 0}\}$.
\QED
\begin{Cor}
\label{Cor 2.12}
For any perversity ${\bf p}$ on $\Delta$, ${\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$ is
quasi-isomoprhic to $\mathop{\rm ic}\nolimits_{-{\bf p}}(\Delta)^\bullet$.
\end{Cor}
{\sl Proof.}\quad
It is sufficient to show that ${\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$
satisfies the conditions of the theorem for the perversity $-{\bf p}$.
(1) is satisfied since ${\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))({\bf 0})^0 ={\bf Q}$ and
${\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))({\bf 0})^i =\{0\}$ for $i\not= 0$ by the
definition of ${\bf D}$.
We check the conditions (2) and (3) for each $\sigma$ in
$\Delta\setminus\{{\bf 0}\}$.
Since
${\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))(\sigma)^\bullet =
\det(\sigma)\otimes\d_\sigma(\i_\sigma^*\mathop{\rm ic}\nolimits_{\bf p}(\Delta))[-r_\sigma]^\bullet$,
the condition (3) of the theorem and Lemma~\ref{Lem 1.6} imply
that $\mathop{\rm gt}\nolimits_{\leq-{\bf p}(\sigma)}({\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))(\sigma))^\bullet$
is acyclic, i.e., (2) for $-{\bf p}$.
By Lemma~\ref{Lem 2.2}, $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^\bullet$ is
quasi-isomorphic to ${\bf D}({\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))(\sigma)^\bullet$.
Hence $\mathop{\rm gt}\nolimits_{\leq{\bf p}(\sigma)}({\bf D}({\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))(\sigma))^\bullet$
is acyclic by (2) for $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\sigma)^\bullet$.
Since
\begin{equation}
\det(\sigma)\otimes\d_\sigma(\i_\sigma^*{\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))^\bullet =
{\bf D}({\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))(\sigma)[r_\sigma]^\bullet\;,
\end{equation}
Lemma~\ref{Lem 1.6} implies that
$\mathop{\rm gt}\nolimits^{\geq-{\bf p}(\sigma)}(\i_\sigma^*{\bf D}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))^\bullet$
is acyclic, i.e., (3) for $-{\bf p}$.
\QED
Two homomorphisms $f, g\mathrel{:} L^\bullet\rightarrow K^\bullet$ in
$\mathop{\rm CGEM}\nolimits(\Delta)$ are said to be homotopic if there exists a
collection of homomorphisms
$\{u^i\mathrel{:} L^i\rightarrow K^{i-1}\mathrel{;} i\in{\bf Z}\}$ in $\mathop{\rm GEM}\nolimits(\Delta)$
such that
\begin{equation}
f^i - g^i = d_K^{i-1}\cdot u^i + u^{i+1}\cdot d_L^i
\end{equation}
for every $i\in{\bf Z}$.
If $f$ and $g$ are homotopic, then
$f(\sigma),g(\sigma)\mathrel{:} L(\sigma)^\bullet\rightarrow K(\sigma)^\bullet$
and
$\i_\sigma^*(f),\i_\sigma^*(g)\mathrel{:}\i_\sigma^*L^\bullet\rightarrow\i_\sigma^*K^\bullet$
for $\sigma\in\Delta$ as well as
$\Gamma(f),\Gamma(\gamma)\mathrel{:}\Gamma(L)^\bullet\rightarrow\Gamma(K)^\bullet$
are homotopic as complexes in abelian categories.
Actually, it is sufficient to take
$\{u^i(\sigma/\sigma)\}$, $\{\i_\sigma^*(u^i)\}$ and $\{\Gamma(u^i)\}$,
respectively.
In particular, the maps of the cohomologies induced by the
two homomorphisms of the complexes are respectively equal.
We give here some elementary lemmas on the quasi-isomorphism
property in $\mathop{\rm CGEM}\nolimits(\Delta)$.
\begin{Lem}
\label{Lem 2.13}
Let $f_1\mathrel{:} L_1^\bullet\rightarrow L_2^\bullet$ be a quasi-isomorphism
and $f_2\mathrel{:} L_1^\bullet\rightarrow L_3^\bullet$ a homomorphism
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then there exist $L_4^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$,
a quasi-isomorphism $g_1\mathrel{:} L_3^\bullet\rightarrow L_4^\bullet$ and
a homomorphism $g_2\mathrel{:} L_2^\bullet\rightarrow L_4^\bullet$ such that
the homomorphisms $g_2\cdot f_1$ and $g_1\cdot f_2$ are homotopic.
If $f_2$ is a quasi-isomorphism, then so is $g_2$.
\end{Lem}
\begin{Lem}
\label{Lem 2.14}
Let $g_1\mathrel{:} L_3^\bullet\rightarrow L_4^\bullet$ be a quasi-isomorphism
and $g_2\mathrel{:} L_2^\bullet\rightarrow L_4^\bullet$ a homomorphism
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then there exist $L_1^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$,
a quasi-isomorphism $f_1\mathrel{:} L_1^\bullet\rightarrow L_2^\bullet$ and
a homomorphism $f_2\mathrel{:} L_1^\bullet\rightarrow L_3^\bullet$ such that
the homomorphisms $g_2\cdot f_1$ and $g_1\cdot f_2$ are homotopic.
If $g_2$ is a quasi-isomorphism, then so is $f_2$.
\end{Lem}
We get these lemmas by setting $L_4^\bullet$ and
$L_1^\bullet$ the mapping cones of the homomorphisms
$L_1^\bullet\rightarrow L_2^\bullet\oplus L_3^\bullet$ and
$L_2^\bullet\oplus L_3^\bullet\rightarrow L_4^\bullet$, respectively.
By applying these lemmas, we get the following equivalent
conditions.
\begin{Lem}
\label{Lem 2.15}
For $L^\bullet, K^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, the following
conditions are equivalent.
(1) $L^\bullet$ is quasi-isomorphic to $K^\bullet$.
(2) There exists $J^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ and
quasi-isomorphisms $L^\bullet\rightarrow J^\bullet$ and
$K^\bullet\rightarrow J^\bullet$.
(3) There exists $I^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ and
quasi-isomorphisms $I^\bullet\rightarrow L^\bullet$ and
$I^\bullet\rightarrow K^\bullet$.
\end{Lem}
\begin{Lem}
\label{Lem 2.16}
Let $f_1\mathrel{:} L_1^\bullet\rightarrow K_1^\bullet$ be a homomorphism
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
Assume that $L_2^\bullet, K_2^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$ are
quasi-isomorphic to $L_1^\bullet$ and $K_1^\bullet$, respectively.
Then there exist $K_3^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, a quasi-isomorphism
$K_2^\bullet\rightarrow K_3^\bullet$ and a homomorphism
$f_2\mathrel{:} L_2^\bullet\rightarrow K_3^\bullet$ such that the diagrams
\begin{equation}
\begin{array}{ccc}
\H^i(\Gamma(L_1)^\bullet)
& \longrightarrow & H^i(\Gamma(K_1)^\bullet) \\
\wr | & & \wr | \\
\H^i(\Gamma(L_2)^\bullet)
& \longrightarrow & H^i(\Gamma(K_3)^\bullet) \\
\end{array}
\end{equation}
of the cohomologies are commutative for all $i\in{\bf Z}$.
There exist also $L_3^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$, a quasi-isomorphism
$L_3^\bullet\rightarrow L_2^\bullet$ and a homomorphism
$f_3\mathrel{:} L_3^\bullet\rightarrow K_2^\bullet$ with the similar compatibility
with $f_1$.
\end{Lem}
{\sl Proof.}\quad
By Lemma~\ref{Lem 2.15}, there exists $L_0^\bullet$ and
quasi-isomorphisms $g_1\mathrel{:} L_0^\bullet\rightarrow L_1^\bullet$ and
$g_2\mathrel{:} L_0^\bullet\rightarrow L_2$.
By applying Lemma~\ref{Lem 2.14} for $f_1\cdot g_1$ and
$g_2$, we get $K_0^\bullet$ with a quasi-isomorphism
$K_1^\bullet\rightarrow K_0^\bullet$ and a homomorphism
$h_0\mathrel{:} L_2^\bullet\rightarrow K_0^\bullet$ with the compatibility
condition.
Since $K_0^\bullet$ is quasi-isomorphic to $K_2$,
there exists $K_3^\bullet$ and quasi-isomorphisms
$h_1\mathrel{:} K_0^\bullet\rightarrow K_3^\bullet$ and
$K_2^\bullet\rightarrow K_3^\bullet$ by Lemma~\ref{Lem 2.15}.
It is sufficient to set $f_3 := h_1\cdot h_0$.
The second assertion is proved similarly.
\QED
\section{The algebraic theory on toric varieties}
\setcounter{equation}{0}
In this section, we construct a functor from the category of
graded exterior modules to that of complexes on the toric
variety associated to the fan.
For a finite fan $\Delta$, we denote by $Z(\Delta)$ the associated
toric variety defined over ${\bf Q}$ (cf.\cite{Oda1}).
We start with the case of an affine toric variety.
Assume that $\Delta$ is $F(\pi)$, i.e. the set of all faces of
a cone $\pi$ in $N_{\bf R}$.
We denote by ${\bf Q}[M]$ the group ring $\bigoplus_{m\in M}{\bf Q}{\bf e}(m)$
defined by ${\bf e}(m){\bf e}(m') ={\bf e}(m + m')$ for $m, m'\in M$ and ${\bf e}(0) = 1$.
This ${\bf Q}$-algebra has a grading in the free ${\bf Z}$-module $M$.
For a subset $U$ of $M$, we denote
${\bf Q}[U] :=\bigoplus_{m\in U}{\bf Q}{\bf e}(m)$.
Note that $1\not\in{\bf Q}[U]$ if $0\not\in U$.
For the subsemigroup $M\cap\pi^\vee\subset M$,
we denote by $S(\pi)$ the $M$-graded ${\bf Q}$-subalgebra
${\bf Q}[M\cap\pi^\vee]$ of ${\bf Q}[M]$.
Then the affine toric variety $Z(F(\pi))$ is equal to
$\mathop{\rm Spec}\nolimits S(\pi)$.
The algebraic torus $T_N$ is equal to $\mathop{\rm Spec}\nolimits{\bf Q}[M]$ and the reduced
complement $Z(F(\pi))\setminus T_N$ is defined by the ideal
$J(\pi) :={\bf Q}[M\cap\mathop{\rm int}\nolimits\pi^\vee]$.
The {\em logarithmic de Rham complex}
$\Omega_{S(\pi)}(\log J(\pi))^\bullet$ is defined as follows.
We set
\begin{equation}
\Omega_{S(\pi)}(\log J(\pi))^1 := S(\pi)\otimes M
\end{equation}
and
\begin{equation}
\Omega_{S(\pi)}(\log J(\pi))^i :=
\bigwedge^i\Omega_{S(\pi)}(\log J(\pi))^1 =
S(\pi)\otimes\bigwedge^iM
\end{equation}
for $0\leq i\leq r$, where the exterior powers are taken as
an $S(\pi)$-module and as a ${\bf Z}$-module, respectively.
These are clearly free $S(\pi)$-modules.
By the notation $A^* = A(M_{\bf Q}) =\bigwedge^\bullet M_{\bf Q}$,
the direct sum $\bigoplus_{i=0}^r\Omega_{S(\pi)}(\log J(\pi))^i$
is equal to the $M$-graded free $S(\pi)$-module
\begin{equation}
S(\pi)\otimes_{\bf Q} A^*
=\bigoplus_{m\in M\cap\pi^\vee}{\bf Q}{\bf e}(m)\otimes_{\bf Q} A^*\;.
\end{equation}
The ${\bf Q}$-endomorphism $\partial$ of this $S(\pi)$-module
is defined to be the $M$-homogeneous morphism such
that the the restriction to the component ${\bf Q}{\bf e}(m)\otimes_{\bf Q} A^*$
is $1\otimes d_m$ for each $m\in M\cap\pi^\vee$, where $d_m$
is the left multiplication of $m$.
Since
$\partial(\Omega_{S(\pi)}(\log J(\pi))^i)\subset\Omega_{S(\pi)}(\log
J(\pi))^{i+1}$
for each $i$, $\Omega_{S(\pi)}(\log J(\pi))^\bullet$ is a
$\partial$-complex of $M$-graded ${\bf Q}$-vector spaces.
For each face $\sigma$ of $\pi$, $\pi^\vee\cap\sigma^\bot$ is a
face of the dual cone $\pi^\vee\subset M_{\bf R}$.
Furthermore, it is known that the correspondence
$\sigma\mapsto\pi^\vee\cap\sigma^\bot$ defines a bijection from
$F(\pi)$ to $F(\pi^\vee)$ \cite[Prop.A.6]{Oda1}.
For each $\sigma\in F(\pi)$, we denote by $P(\pi;\sigma)$ the
$M$-homogeneous prime ideal
\begin{equation}
{\bf Q}[M\cap(\pi^\vee\setminus\sigma^\bot)] =
\bigoplus_{m\in M\cap(\pi^\vee\setminus\sigma^\bot)}{\bf Q}{\bf e}(m)
\end{equation}
of $S(\pi)$.
The $M$-homogeneous quotient ring $S(\pi)/P(\pi;\sigma)$ is denoted
by $S(\pi;\sigma)$. We denote the image of ${\bf e}(m)$ in $S(\pi;\sigma)$
for $m\in M\cap\pi^\vee\cap\sigma^\bot$ also by ${\bf e}(m)$.
Since $M[\sigma] = M\cap\sigma^\bot$, we have a description
$S(\pi;\sigma) =\bigoplus_{m\in M[\sigma]\cap\pi^\vee}{\bf Q}{\bf e}(m)$.
Let $J(\pi;\sigma)$ be the ideal
${\bf Q}[M[\sigma]\cap\mathop{\rm rel.\,int}\nolimits(\pi^\vee\cap\sigma^\bot)]$ of $S(\pi;\sigma)$.
The $\partial$-complex $\Omega_{S(\pi;\sigma)}(\log J(\pi;\sigma))^\bullet$
is defined to be $S(\pi;\sigma)\otimes_{\bf Q} A^*[\sigma]$ with the
$M$-homogeneous ${\bf Q}$-homomorphism $\partial$ defined similarly as above,
where $A^*[\sigma] =\bigwedge^\bullet M[\sigma]_{\bf Q}$.
Note that $\Omega_{S(\pi;{\bf 0})}(\log J(\pi;{\bf 0}))^\bullet$ is equal to
$\Omega_{S(\pi)}(\log J(\pi))^\bullet$.
We denote by $\mathop{\rm Coh}\nolimits(S(\pi))$ the category:
\begin{description}
\item[object:] A finitely generated $M$-graded $S(\pi)$-module.
\item[morphism:] An $M$-homogeneous $S(\pi)$-homomorphism
of $M$-degree zero.
\end{description}
A ${\bf Q}$-homomorphism $\delta\mathrel{:} F\rightarrow G$ of $S(\pi)$-modules is said to be
a {\em differential operator of order one} if the map
$(\delta\cdot f - f\cdot\delta)\mathrel{:} F\rightarrow G$ defined by
$(\delta\cdot f - f\cdot\delta)(x) :=\delta(fx) - f(\delta(x))$
is an $S(\pi)$-homomorphism for every $f\in S(\pi)$.
We denote by $\mathop{\rm CCohDiff}\nolimits(S(\pi))$ the category:
\begin{description}
\item[object:] A finite $\partial$-complex $F^\bullet$ such that
$F^i$'s are in $\mathop{\rm Coh}\nolimits(S(\pi))$ and $\partial$ is $M$-homogeneous
of $M$-degree zero and is a differential operator of order one.
\item[morphism:] An $M$-homogeneous $S(\pi)$-homomorphism
of $M$-degree zero.
\end{description}
We construct a functor $\Lambda_{S(\pi)}$ from the category
$\mathop{\rm GEM}\nolimits(F(\pi))$ of graded exterior modules on $F(\pi)$ to
this category $\mathop{\rm CCohDiff}\nolimits(S(\pi))$.
Let $\rho$ be a cone in $F(\pi)$.
Recall that an object $V$ of $\mathop{\rm GM}\nolimits(A(\rho))$ is a
finitely generated graded $A(\rho)$-module and the $A$-module
$V_A$ has a structure of a free $A^*[\rho]$-module
(cf. Lemma~\ref{Lem 1.3}).
Each $m\in M[\rho] = M\cap\rho^\bot$ is a homogeneous element of
$A^*[\rho]$ of degree one.
We denote by $d_m$ the left operation of $m$ on $V_A$.
Then $d_m^2 = 0$ since $m\wedge m = 0$.
For each $m\in M[\rho]$, we denote by $V_A(m)^\bullet$
the $\partial$-complex defined by $V_A(m)^i :=(V_A)_i$
for each $i\in{\bf Z}$ and $\partial := d_m$.
We set $\Lambda_{S(\pi)}^\rho(V)^i := S(\pi;\rho)\otimes_{\bf Q}(V_A)_i$
for each integer $i$.
We define the $\partial$-complex $\Lambda_{S(\pi)}^\rho(V)^\bullet$
by
\begin{equation}
\Lambda_{S(\pi)}^\rho(V)^\bullet :=S(\pi;\rho)\otimes_{\bf Q} V_A =
\bigoplus_{m\in M[\rho]\cap\pi^\vee}{\bf Q}{\bf e}(m)\otimes_{\bf Q} V_A(m)^\bullet\;,
\end{equation}
i.e., the its $m$-component of $\partial$ is $1_{{\bf Q}{\bf e}(m)}\otimes d_m$
for every $m\in M[\rho]\cap\pi^\vee$.
In order to check that $\partial$ is a differential operator of
order one, it is sufficient to show the $S(\pi;\rho)$-linearlity of
$\partial\cdot {\bf e}(m_0) -{\bf e}(m_0)\cdot \partial$ for $m_0\in M\cap\pi^\vee$.
Since $P(\pi;\rho)$ is the annihilator of $S(\pi;\rho)$,
$\partial\cdot {\bf e}(m_0) -{\bf e}(m_0)\cdot \partial = 0$ if $m_0\not\in M[\rho]$.
Assume $m_0, m_1\in M[\rho]\cap\pi^\vee$ and $x\in V_A$.
For any $m\in M[\rho]\cap\pi^\vee$, we have
\begin{eqnarray*}
& & (\partial\cdot {\bf e}(m_0) -{\bf e}(m_0)\cdot \partial)({\bf e}(m){\bf e}(m_1)\otimes x) \\
& = & \partial({\bf e}(m + m_0 + m_1)\otimes x) -{\bf e}(m_0)\partial({\bf e}(m + m_1)\otimes
x) \\
& = & {\bf e}(m + m_0 + m_1)\otimes(m + m_0 + m_1)x
-{\bf e}(m + m_0 + m_1)\otimes(m + m_1)x \\
& = & {\bf e}(m + m_0 + m_1)\otimes m_0x \\
& = & {\bf e}(m + m_0 + m_1)\otimes(m_0 + m_1)x
-{\bf e}(m + m_0 + m_1)\otimes m_1x \\
& = & {\bf e}(m)(\partial\cdot {\bf e}(m_0) -{\bf e}(m_0)\cdot \partial)({\bf e}(m_1)\otimes x)\;.
\end{eqnarray*}
Hence $\partial\cdot {\bf e}(m_0) -{\bf e}(m_0)\cdot \partial$ is an
$S(\pi;\rho)$-homomorphism.
\begin{Prop}
\label{Prop 3.1}
Let $\rho$ be in $F(\pi)$ and $V$ in $\mathop{\rm GM}\nolimits(A(\rho))$.
Then $\Lambda_{S(\pi)}^\rho(V)^\bullet$ is isomorphic to a finite
direct sum of subcomplexes which are isomorphic to dimension shifts
of $\Omega_{S(\pi;\rho)}(\log J(\rho))^\bullet$.
\end{Prop}
{\sl Proof.}\quad
By Lemma~\ref{Lem 1.3}, $V_A$ is a free $A^*[\rho]$-module.
Let $\{x_1,\cdots, x_s\}$ be a homogeneous basis.
By the definition of $\partial$, the decomposition
\begin{equation}
\Lambda_{S(\pi)}^\rho(V)^\bullet =
\bigoplus_{i=1}^sS(\pi;\rho)\otimes_{\bf Q} A^*[\rho]x_i
\end{equation}
is a direct sum of subcomplexes.
Furthermore, there exists an isomorphism
\begin{equation}
\Omega_{S(\pi;\rho)}(\log J(\rho))[-\deg x_i]^\bullet\longrightarrow
(S(\pi;\rho)\otimes_{\bf Q} A^*[\rho]x_i)^\bullet
\end{equation}
for each $i$.
\QED
For a homomorphism $f\mathrel{:} V\rightarrow W$ in $\mathop{\rm GM}\nolimits(A(\rho))$, the
$S(\pi)$-homomorphism
$\Lambda_{S(\pi)}^\rho(f)\mathrel{:}\Lambda_{S(\pi)}^\rho(V)^\bullet\rightarrow%
\Lambda_{S(\pi)}^\rho(W)^\bullet$ of $\partial$-complexes is defined
by $\Lambda_{S(\pi)}^\rho(f) = 1_{S(\pi)}\otimes f_A$.
Since $f_A$ is an $A^*[\rho]$-homomorphism, $\Lambda_{S(\pi)}^\rho(f)$
commutes with the coboudary maps of $\Lambda_{S(\pi)}^\rho(V)^\bullet$
and $\Lambda_{S(\pi)}^\rho(W)^\bullet$.
\begin{Prop}
\label{Prop 3.2}
Let $\sigma,\rho$ be cones in $F(\pi)$ with $\sigma\prec\rho$.
For $V$ in $\mathop{\rm GM}\nolimits(A(\sigma))$, there exists a natural isomorphism
\begin{equation}
\Lambda_{S(\pi)}^\sigma(V)^\bullet\otimes_{S(\pi;\sigma)}S(\pi;\rho)\simeq
\Lambda_{S(\pi)}^\rho(V_{A(\rho)})^\bullet
\end{equation}
of $M$-graded $\partial$-complexes.
\end{Prop}
{\sl Proof.}\quad
$\Lambda_{S(\pi)}^\sigma(V)^\bullet\otimes_{S(\pi;\sigma)}S(\pi;\rho)$
is actually an $M$-graded $\partial$-complex since it is the quotient
of $\Lambda_{S(\pi)}^\sigma(V)^\bullet$
by the $M$-homogeneous subcomplex
$P(\pi;\rho)\Lambda_{S(\pi)}^\sigma(V)^\bullet$.
Both sides are equal to $S(\pi;\rho)\otimes_{\bf Q} V_A$ by the
identification $(V_{A(\rho)})_A = V_A$.
The differential operators commute with the identification
since the $m$-components of them for $m\in M[\rho]\cap\pi^\vee$
are both the operation $d_m$.
\QED
For $L$ in $\mathop{\rm GEM}\nolimits(F(\pi))$, we define the $\partial$-complex
$\Lambda_{S(\pi)}(L)^\bullet$ in $\mathop{\rm CCohDiff}\nolimits(S(\pi))$ by
\begin{equation}
\Lambda_{S(\pi)}(L)^\bullet :=
\bigoplus_{\sigma\in F(\pi)}\Lambda_{S(\pi)}^\rho(L(\rho))^\bullet\;,
\end{equation}
where the coboundary map $\partial$ is also defined as the direct sum.
Let $f\mathrel{:} L\rightarrow K$ be a morphism in $\mathop{\rm GEM}\nolimits(F(\pi))$.
We define an $M$-homogeneous $S(\pi)$-homomorphism
$\Lambda_{S(\pi)}(f)\mathrel{:}\Lambda_{S(\pi)}(L)^\bullet\rightarrow%
\Lambda_{S(\pi)}(K)^\bullet$ of $\partial$-complexes as follows.
For $\sigma,\rho\in F(\pi)$, the $(\sigma,\rho)$-component of
the morphism
\begin{equation}
\begin{array}{ccccc}
\Lambda_{S(\pi)}(f) & \mathrel{:} & \Lambda_{S(\pi)}(L)^\bullet
& \longrightarrow & \Lambda_{S(\pi)}(K)^\bullet \\
& & \| & & \| \\
& &
\bigoplus_{\sigma\in F(\pi)}\Lambda_{S(\pi)}^\sigma(L(\sigma))^\bullet
& &
\bigoplus_{\rho\in F(\pi)}\Lambda_{S(\pi)}^\rho(K(\rho))^\bullet
\end{array}
\end{equation}
is defined to be the composite of the natural surjection
\begin{equation}
\Lambda_{S(\pi)}^\sigma(L(\sigma))^\bullet\rightarrow
\Lambda_{S(\pi)}^\sigma(L(\sigma))^\bullet%
\otimes_{S(\pi;\sigma)}S(\pi;\rho)
= \Lambda_{S(\pi)}^\rho(L(\sigma)_{A(\rho)})^\bullet
\end{equation}
and
\begin{equation}
1_{S(\pi;\rho)}\otimes f(\sigma/\rho)_A\mathrel{:}
\Lambda_{S(\pi)}^\rho(L(\sigma)_{A(\rho)})^\bullet\longrightarrow
\Lambda_{S(\pi)}^\rho(K(\rho))^\bullet
\end{equation}
if $\sigma\prec\rho$ and is defined to be the zero map otherwise.
Then $\Lambda_{S(\pi)}$ is a covariant functor from $\mathop{\rm GEM}\nolimits(\Delta)$
to $\mathop{\rm CCohDiff}\nolimits(S(\pi))$.
Let $k$ be a field.
For a topological space $X$, we denote by $k_X$ the sheaf of
rings with the constant stalk $k$.
A $k_X$-module on $X$ is called a $k$-sheaf and
a $k_X$-homomorphism of $k_X$-modules is called simply a
$k$-homomorphism.
In this paper, we treat only the cases $k ={\bf Q}$ and $k ={\bf C}$.
Let ${\cal O}_X$ be a sheaf of commutative $k$-algebras.
A $k$-homomoprhism $f\mathrel{:} F\rightarrow G$ of ${\cal O}_X$-modules $F, G$
is said to be a differential operator of order one if the
$k$-homomorphism $f(U)\mathrel{:} F(U)\rightarrow G(U)$ is a differential
operator of order one of ${\cal O}_X(U)$-modules
for every open subset $U\subset X$.
For a finitely generated $S(\pi)$-module $E$, we denote by ${\cal F}(E)$
the associated coherent sheaf on the affine scheme $\mathop{\rm Spec}\nolimits S(\pi)$.
If $d\mathrel{:} E\rightarrow G$ is a differential operator of order one of
finitely generated $S(\pi)$-modules, then it defines a differential
operator ${\cal F}(d)\mathrel{:}{\cal F}(E)\rightarrow{\cal F}(G)$ of order one of
${\cal O}_{Z(F(\pi))}$-modules on the affine toric variety $Z(F(\pi))$.
If $(E^\bullet, \partial = (\partial_E^i))$ is a $\partial$-complex
such that each $E^i$ is a finitely generated $S(\pi)$-module and
$\partial$ is a differential operator of order one, then we
denote by ${\cal F}(E)^\bullet$ the $\partial$-complex on
$Z(F(\pi))$ with the coboudary map
$\partial = ({\cal F}(\partial_E^i))$.
Let $\mu$ be an element of $F(\pi)$.
Take an element $m\in M\cap\mathop{\rm rel.\,int}\nolimits(\pi^\vee\cap\mu^\bot)$.
Since $\mu^\vee =\pi^\vee +{\bf R}(-m)$ \cite[Cor.A.7]{Oda1},
$S(\mu) :={\bf Q}[M\cap\mu^\vee]$ is equal to the localization
$S(\pi)[{\bf e}(m)^{-1}]$.
For any $\rho\in F(\pi)$, we see easily that
$S(\pi;\rho)\otimes_{S(\pi)}S(\mu)$ is equal to $S(\mu;\rho)$
if $\rho\prec\mu$ and is $\{0\}$ otherwise.
\begin{Lem}
\label{Lem 3.3}
Let $\mu$ be an element of $F(\pi)$.
For $\rho\in F(\mu)$ and $V$ in $\mathop{\rm GM}\nolimits(A(\rho))$, the localization
$\Lambda_{S(\pi)}^\rho(V)^\bullet\otimes_{S(\pi)}S(\mu)$
is equal to $\Lambda_{S(\mu)}^\rho(V)^\bullet$.
For $L$ in $\mathop{\rm GEM}\nolimits(F(\pi))$, the localization
$\Lambda_{S(\pi)}(L)^\bullet\otimes_{S(\pi)}S(\mu)$
of $\partial$-complex is equal to
$\Lambda_{S(\mu)}(L|{F(\mu)})^\bullet$, where $L|{F(\mu)}$ is the
restriction of $L$ to $F(\mu)$.
\end{Lem}
{\sl Proof.}\quad
The first equality is clear as $S(\mu)$-modules.
The coboundary maps $\partial$ are compatible with the
inclusion
$\Lambda_{S(\pi)}^\rho(V)^\bullet\subset\Lambda_{S(\mu)}^\rho(V)^\bullet$
since they are defined as the direct sums of
${\bf Q}{\bf e}(m)\otimes_{\bf Q} V_A(m)^\bullet$'s.
Since differential operators are extended to localizations uniquely,
they are equal as $\partial$-complexes.
Since $\Lambda_{S(\pi)}(L)^\bullet$ and $\Lambda_{S(\mu)}(L)^\bullet$
are defined as the direct sums of
$\Lambda_{S(\pi)}^\rho(L(\rho))^\bullet$ for $\rho\in F(\pi)$ and
$\Lambda_{S(\mu)}^\rho(L(\rho))^\bullet$ for $\rho\in F(\mu)$,
respectively, the second assertion follows from the first.
\QED
Let $\Delta$ be a finite fan of $N_{\bf R}$.
The toric variety $Z(\Delta)$ has the affine open covering
$\bigcup_{\pi\in\Delta}Z(F(\pi))$.
The reduced divisor $D(\Delta) := Z(\Delta)\setminus T_N$
is defined by the ideal $J(\pi)\subset S(\pi)$ on each affine
open subscheme $Z(F(\pi))$.
The logarithmic de Rham complex
$\Omega_{Z(\Delta)}(\log D(\Delta))^\bullet$ is defined by
\begin{equation}
\Omega_{Z(\Delta)}(\log D(\Delta))^\bullet :=
{\cal O}_{Z(\Delta)}\otimes_{\bf Q} A^*\;.
\end{equation}
We define the coboudary map $\partial = (\partial^i\mathrel{:} i\in{\bf Z})$
\begin{equation}
\partial^i\mathrel{:}\Omega_{Z(\Delta)}(\log D(\Delta))^i\longrightarrow
\Omega_{Z(\Delta)}(\log D(\Delta))^{i+1}
\end{equation}
so that the restriction to $Z(F(\pi))$ is equal to $\partial$
of ${\cal F}(\Omega_{S(\pi)}(\log J(\pi)))^\bullet$ for each
$\pi\in\Delta$.
Although it is common to write a logarithmic de Rham complex
as $\Omega_X^\bullet(\log D)$, we put the dot at the right end
as $\Omega_X(\log D)^\bullet$ for the compatibility with the other
notation in this paper.
For each $\sigma\in F(\rho)$, the subscheme $\mathop{\rm Spec}\nolimits S(\rho;\sigma)$
of $Z(F(\rho))$ is denoted by $X(\rho;\sigma)$.
In particular, $X(\rho;\rho)$ is the algebraic torus
$T_{N[\rho]} :=\mathop{\rm Spec}\nolimits{\bf Q}[M[\rho]]$ of dimension $r - r_\rho$.
In order to simplify the notation, we set $T := T_N$ and
$T[\rho] := T_{N[\rho]}$ for each $\rho\in\Delta$.
Then the toric variety $Z(\Delta)$ is decomposed as the
disjoint union
\begin{equation}
\bigcup_{\rho\in\Delta}T[\rho]
\end{equation}
of $T$-orbits \cite[Prop.1.6]{Oda1}.
For each $\sigma\in\Delta$, we denote by $X(\Delta;\sigma)$ or
simply $X(\sigma)$ the union of $X(\rho;\sigma)$ for $\rho\in\Delta$
with $\sigma\prec\rho$. $X(\sigma)$ is a $T$-invariant irreducible
closed subvariety of $Z(\Delta)$.
For each $\sigma\in\Delta$, let $N[\sigma] := N/N(\sigma)$.
For each $\rho\in\Delta$ with $\sigma\prec\rho$, we denote by
$\rho[\sigma]$ the image of $\rho$ in $N[\sigma]_{\bf R} = N_{\bf R}/N(\sigma)_{\bf R}$.
Then $\Delta[\sigma] :=\{\rho[\sigma]\mathrel{:}\rho\in\Delta,\sigma\prec\rho\}$
is a fan of $N[\sigma]_{\bf R}$.
It is known that $X(\sigma)\subset Z(\Delta)$ is equal to
the toric variety $Z(\Delta[\sigma])$ with the torus $T[\sigma]$
\cite[Cor.1.7]{Oda1}.
The reduced complement $X(\sigma)\setminus T[\sigma]$ is denoted by
$D(\Delta;\sigma)$ or $D(\sigma)$.
We define
\begin{equation}
\Omega_{X(\sigma)}(\log D(\sigma))^\bullet :=
{\cal O}_{X(\sigma)}\otimes_{\bf Q} A^*[\sigma]\;
\end{equation}
and $\partial$ of it is defined so that the restriction to the
affine open subscheme $Z(F(\rho))$ is equal to that of
${\cal F}(\Omega_{S(\rho;\sigma)}(\log J(\rho;\sigma)))^\bullet$ for
every $\rho\in\Delta$ with $\sigma\prec\rho$.
For each $V\in\mathop{\rm GM}\nolimits(A(\sigma))$, we define the
$\partial$-complex
$\Lambda_{Z(\Delta)}^\sigma(V)^\bullet$ on $Z(\Delta)$ by
\begin{equation}
\Lambda_{Z(\Delta)}^\sigma(V)^\bullet :=
{\cal O}_{X(\sigma)}\otimes_{\bf Q} V_A\;.
\end{equation}
The coboudary map $\partial$ is defined so that the restriction to
each open set $Z(F(\rho))$ is equal to that of
${\cal F}(\Lambda_{S(\rho)}^\sigma(V))^\bullet$ for every
$\rho\in\Delta$ with $\sigma\prec\rho$.
Note that $M$-gradings of $\Lambda_{S(\rho)}^\sigma(V)^\bullet$
for $\rho\in\Delta$ induce a natural $T$-action on the $\partial$-complex
$\Lambda_{Z(\Delta)}^\sigma(V)^\bullet$.
The following propositions follow from Propositions~\ref{Prop 3.1}
and \ref{Prop 3.2}, respectively.
\begin{Prop}
\label{Prop 3.4}
Let $\rho$ be in $\Delta$ and $V$ in $\mathop{\rm GM}\nolimits(A(\rho))$.
Then $\Lambda_{Z(\Delta)}^\rho(V)^\bullet$ is isomorphic to a
finite direct sum of subcomplexes which are isomorphic to dimension
shifts of $\Omega_{X(\rho)}(\log D(\rho))^\bullet$.
\end{Prop}
\begin{Prop}
\label{Prop 3.5}
Let $\sigma,\rho$ be cones in $\Delta$ with $\sigma\prec\rho$.
For $V$ in $\mathop{\rm GM}\nolimits(A(\sigma))$, there exists a natural
$T$-equivariant isomorphism
\begin{equation}
\Lambda_{Z(\Delta)}^\sigma(V)^\bullet\otimes_{{\cal O}_{X(\sigma)}}
{\cal O}_{X(\rho)}\simeq
\Lambda_{Z(\Delta)}^\rho(V_{A(\rho)})^\bullet
\end{equation}
of $\partial$-complexes.
\end{Prop}
Let $V^\bullet$ be in $\mathop{\rm CGM}\nolimits(A(\sigma))$.
Then the bicomplex
$\Lambda_{Z(\Delta)}^\sigma(V)^{\bullet,\bullet}$ is defined by
\begin{equation}
\Lambda_{Z(\Delta)}^\sigma(V)^{i,j} :=
\Lambda_{Z(\Delta)}^\sigma(V^i)^j
\end{equation}
for $i, j\in{\bf Z}$ and $d_1 :=d$ and $d_2 :=\partial$.
We denote by $\Lambda_{Z(\Delta)}^\sigma(V)^\bullet$ the
associated single complex and by $\delta$ the
coboundary map.
For each object $L$ of $\mathop{\rm GEM}\nolimits(\Delta)$, we set
\begin{equation}
\Lambda_{Z(\Delta)}(L)^\bullet :=
\bigoplus_{\sigma\in\Delta}%
\Lambda_{Z(\Delta)}^\sigma(L(\sigma))^\bullet\;.
\end{equation}
For a morphism $f\mathrel{:} L\rightarrow K$ in $\mathop{\rm GEM}\nolimits(\Delta)$, the
$T$-equivariant homomorphism
\begin{equation}
\Lambda_{Z(\Delta)}(f)\mathrel{:}\Lambda_{Z(\Delta)}(L)^\bullet%
\longrightarrow\Lambda_{Z(\Delta)}(K)^\bullet
\end{equation}
is defined naturally.
Then $\Lambda_{Z(\Delta)}$ is a covariant functor from
$\mathop{\rm GEM}\nolimits(\Delta)$ to the category $\mathop{\rm CCohDiff}\nolimits(Z(\Delta))$ which
is defined naturally as the globalization of
$\mathop{\rm CCohDiff}\nolimits(S(\pi))$.
Let $L^\bullet$ be a $d$-complex in $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then the bicomplex $\Lambda_{Z(\Delta)}(L)^{\bullet,\bullet}$
is defined by
\begin{equation}
\Lambda_{Z(\Delta)}(L)^{i,j} :=\Lambda_{Z(\Delta)}(L^i)^j
\end{equation}
for $i,j\in{\bf Z}$.
Note that $d_1 := d$ of this bicomplex is a ${\cal O}_{Z(\Delta)}$-homomorphism
and $d_2 :=\partial$ is a differential operator of order one.
If there is no danger of confusion, we denote by
$\Lambda_{Z(\Delta)}(L)^\bullet$ the associated single complex.
The coboudary map, which we denote by $\delta$, is a differential
operator of order one.
For each integer $j$, we denote by $\Lambda_{Z(\Delta)}(L)_j^\bullet$
the $d$-complex $\Lambda_{Z(\Delta)}(L)^{\bullet,j}$.
Then $\Lambda_{Z(\Delta)}(L)_j^\bullet$ is a finite
$d$-complex in the category of coherent ${\cal O}_Z$-modules.
If the fan $\Delta$ is complete, then the toric variety $Z(\Delta)$
is complete.
For the functor $\Gamma\mathrel{:}\mathop{\rm GEM}\nolimits(\Delta)\rightarrow\mathop{\rm GM}\nolimits(A)$ defined in
Section~1, we get the following lemma.
\begin{Lem}
\label{Lem 3.6}
Assume that $\Delta$ is a complete fan.
Let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
For any integers $p, q$, we have an isomorphism
\begin{equation}
\H^p(Z(\Delta),\Lambda_Z(L)_q^\bullet)\simeq
\H^p(\Gamma(L)^\bullet)_q
\end{equation}
of finite dimensional ${\bf Q}$-vector spaces, where
the lefthand side is the hepercohomology group of the
complex of coherent sheaves on $Z(\Delta)$.
\end{Lem}
{\sl Proof.}\quad
For each $\sigma\in\Delta$, we have
$\H^0(X(\sigma),{\cal O}_{X(\sigma)}) ={\bf Q}$ and
$\H^p(X(\sigma),{\cal O}_{X(\sigma)}) = \{0\}$ for $p > 0$, since
$X(\sigma)$ is a complete toric variety \cite[Cor.2.8]{Oda1}.
Since $\Lambda_Z(L)_q^p =\Lambda_Z(L)^{p,q}$ is a direct sum for
$\sigma\in\Delta$ of free ${\cal O}_{X(\sigma)}$-modules for every
$p\in{\bf Z}$, the hypercohomology $\H^p(Z(\Delta),\Lambda_Z(L)_q^\bullet)$
is equal to the $p$-th cohomology of the complex
$\Gamma(Z,\Lambda_Z(L)_q)^\bullet$ of ${\bf Q}$-vector spaces.
For any $\sigma\in\Delta$ and $p, q\in{\bf Z}$, we have
\begin{equation}
\Gamma(Z,\Lambda_Z^\sigma(L(\sigma))_q)^p
= (L(\sigma)_A^p)_q\;.
\end{equation}
Hence $\Gamma(Z,\Lambda_Z(L)_q)^\bullet$ is isomorphic
to $\Gamma(L)_q^\bullet$ as a complex of ${\bf Q}$-vector spaces.
\QED
\section{The analytic theory on toric varieties}
\setcounter{equation}{0}
For any ${\bf Q}$-algebra $B$ and any ${\bf Q}$-scheme $X$, we denote by
$B_{\bf C}$ and $X_{\bf C}$ the scalar extensions $B\otimes_{\bf Q}{\bf C}$ and
$X\times_{\mathop{\rm Spec}\nolimits{\bf Q}}\mathop{\rm Spec}\nolimits{\bf C}$, respectively.
When $X$ is of finite type over ${\bf Q}$, we denote by $X_{\rm h}$ the
analytic space associated to the algebraic ${\bf C}$-scheme $X_{\bf C}$
\cite[Chap.1,\S6]{Hartshorne}.
For a coherent sheaf $F$ on $X$, the pulled-back coherent sheaf
on $X_{\bf C}$ and the associated analytic coherent sheaf on $X_{\rm h}$
are denoted by $F_{\bf C}$ and $F_{\rm h}$, respectively.
Let $f\mathrel{:} F\rightarrow G$ be a differential operator of order one over
${\bf Q}$.
Then it is easy to see that the ${\bf C}$-homomorphism
$f_{\bf C}\mathrel{:} F_{\bf C}\rightarrow G_{\bf C}$ on $X_{\bf C}$ obtained by scalar extension
is a differential operator of order one over ${\bf C}$.
By \cite[16.8]{EGA4}, the differential operator $f_{\bf C}\mathrel{:} F_{\bf C}\rightarrow G_{\bf C}$
is decomposed uniquly to $u\cdot d_{X_{\bf C}}^1$ where
$d_{X_{\bf C}}^1\mathrel{:} F\rightarrow{\cal P}_{X_{\bf C}}^1(F)$ is a canonical
${\bf C}$-homomorphism \cite[16.7.5]{EGA4} and $u\mathrel{:}{\cal P}_{X_{\bf C}}^1(F)\rightarrow G$
is a ${\cal O}_{X_{\bf C}}$-homomorphism.
For the definition of ${\cal P}_{X_{\bf C}}^1(F)$, see
\cite{EGA4}.
Since these homomorphisms are canonically pulled-back to
$X_{\rm h}$, we get a differential operator $f_{\rm h}\mathrel{:} F_{\rm h}\rightarrow G_{\rm h}$
of order one of the analytic coherent sheaves on $X_{\rm h}$.
Let $\rho$ be a cone in $N_{\bf R}$.
By the notation of the scalar extensions,
\begin{equation}
S(\rho)_{\bf C} = S(\rho)\otimes_{\bf Q}{\bf C} ={\bf C}[M\cap\rho^\vee]
\end{equation}
and
\begin{equation}
S(\rho;\sigma)_{\bf C} = S(\rho;\sigma)\otimes_{\bf Q}{\bf C} =
{\bf C}[M[\sigma]\cap\rho^\vee]
\end{equation}
for each $\sigma\in F(\rho)$.
Since $S(\rho;\sigma) = S(\rho)/P(\rho;\sigma)$, $S(\rho;\sigma)_{\bf C}$
is the quotient of $S(\rho)_{\bf C}$ by the prime ideal
$P(\rho;\sigma)_{\bf C} ={\bf C}[M\cap(\rho^\vee\setminus\sigma^\bot)]$.
We fix a finite fan $\Delta$ of $N_{\bf R}$ in this section.
We denote simply by $Z$ the toric variety $Z(\Delta)$.
Since $Z$ is normal, so are the toric variety $Z_{\bf C} = Z(\Delta)_{\bf C}$
over ${\bf C}$ and the analytic space $Z_{\rm h} = Z(\Delta)_{\rm h}$.
For each $\sigma\in\Delta$, a $T$-invariant irreducible closed
subvariety $X(\sigma)$ of $Z$ was defined in Section~3.
Hence $X(\sigma)_{\bf C}$ and $X(\sigma)_{\rm h}$ are irreducible
closed subvarieties of $Z_{\bf C}$ and $Z_{\rm h}$, respectively.
Let $n$ be an element of $N$.
The group homomorphism $M\rightarrow{\bf Z}$ defined by
$m\mapsto\langle m, n\rangle$ induces a homomorphism of group rings
${\bf C}[M]\rightarrow{\bf C}[t, t^{-1}]$ and the associated morphism
$\lambda_n\mathrel{:}\mathop{\rm Spec}\nolimits{\bf C}[t, t^{-1}]\rightarrow T_{\bf C} =\mathop{\rm Spec}\nolimits{\bf C}[M]$, where
$t$ is the monomial corresponding to $1\in{\bf Z}$.
We call $\lambda_n$ the {\em one-parameter subgroup} associated
to $n$ \cite[1.2]{Oda1}.
We denote also by $\lambda_n$ the associated map
${\bf C}^*\rightarrow T_{\rm h}$ of complex Lie groups.
If $n\in N\cap\rho$ for a cone $\rho\in\Delta$, then
${\bf C}[M\cap\rho^\vee]$ is mapped to ${\bf C}[t]$.
Hence the one-parameter subgroup is extended to a regular
map $\lambda_n\mathrel{:}{\bf C}\rightarrow Z(F(\rho))_{\bf C}\subset Z_{\bf C}$, uniquely.
For each $m\in M$, the monomial ${\bf e}(m)\in{\bf C}[M]$ is regarded as a
character $T_{\rm h}\rightarrow{\bf C}^*$.
The composite ${\bf e}(m)\cdot\lambda_n\mathrel{:}{\bf C}^*\rightarrow{\bf C}^*$
of $\lambda_n$ and ${\bf e}(m)$ is equal to the map
$t\mapsto t^{\langle m, n\rangle}$.
Since $Z_{\bf C}$ is a toric variety, the group $T_{\rm h}$ acts on
$Z_{\rm h}$ analytically.
For $a\in T_{\rm h}$ and $x\in Z_{\rm h}$, we denote by $ax$ the
corresponding point of $Z_{\rm h}$ by the action.
Let $f$ be a complex analytic function on an open subset $U$ of $Z_{\rm h}$.
For each $n\in N$, the derivation $\partial_nf$ of $f$ is defined
by
\begin{equation}
\partial_nf(x) :=\left.\frac{d}{dt}\right|_{t=1}f(\lambda_n(t)x) =
\lim_{t\rightarrow 1}\frac{f(x) - f(\lambda_n(t)x)}{1 - t}
\end{equation}
for $x\in U$.
Since $f(\lambda_n(t)x)$ is analytic in the variables $x, t$, the
function $\partial_nf$ is anlytic on $U$.
Hence we get a ${\bf C}$-derivation
$\partial_n\mathrel{:}{\cal O}_{Z_{\rm h}}\rightarrow{\cal O}_{Z_{\rm h}}$ of the structure sheaf.
For each $\sigma\in\Delta$, we denote by ${\cal P}(\sigma)_{\rm h}$
the ideal sheaf of ${\cal O}_{Z_{\rm h}}$ defining $X(\sigma)_{\rm h}\subset Z_{\rm h}$.
If $f$ is in ${\cal P}(\sigma)_{\rm h}(U)$, then so is $\partial_nf$
since $f$ is zero on $X(\sigma)\cap U$ and $X(\sigma)_{\rm h}$ is
closed by the action of $\lambda_n$.
\begin{Lem}
\label{Lem 4.1}
Let $\sigma$ be an element of $\Delta$ and let $y$
be a point in $X(\sigma)_{\rm h}$.
If $n$ is in $N\cap\mathop{\rm rel.\,int}\nolimits\sigma$, then the endomorphism
$(\partial_n)_y$ of the stalk $({\cal P}(\sigma)_{\rm h})_y$
is an automorphism as a ${\bf C}$-vector space.
\end{Lem}
{\sl Proof.}\quad
Let $U\subset Z_{\rm h}$ be an open neighborhood of $y$ such that
$\lambda_n(t)U\subset U$ for every $t$ with $|t|\leq 1$.
For $g\in{\cal P}(\sigma)_{\rm h}(U)$, we define
\begin{equation}
f(x) :=\int_0^1\frac{g(\lambda_n(s)x)}{s}ds\;,
\end{equation}
where the integration is taken on the real interval $[0,1]$.
Since $\lambda_n(0)x$ is in $X(\sigma)_{\rm h}$, the analytic fuction
$g(\lambda_n(s)x)$ on $U\times\{s\mathrel{;}|s|\leq 1\}$ has zero at the
divisor $(s = 0)$.
Hence $g(\lambda_n(s)x)/s$ is an analytic function.
Hence the integral $f$ is an analytic function on $U$.
By the definition, we have
\begin{equation}
f(\lambda_n(t)x) =\int_0^1\frac{g(\lambda_n(st)x)}{s}ds
=\int_0^t\frac{g(\lambda_n(u)x)}{u}du\;.
\end{equation}
Hence $\partial_nf = g$.
This implies that $\partial_n\mathrel{:}{\cal P}(\sigma)_{\rm h}(U)\rightarrow{\cal P}(\sigma)_{\rm h}(U)$
is surjective.
For $f\in{\cal P}(\sigma)_{\rm h}(U)$, suppose that $\partial_nf = 0$.
Then $(d/dt)f(\lambda_n(t)x) = 0$ for $t\in[0, 1]$ and
we have $f(x) = f(\lambda_n(0)x) = 0$.
Since $\partial_n$ is ${\bf C}$-linear, it is also injective.
Since $U$'s with this property form a fundamental system of
neighborhood of $y$, $\partial_n\mathrel{:}{\cal P}(\sigma)_{\rm h}\rightarrow{\cal P}(\sigma)_{\rm h}$
is isomorphic at the stalk of $y$.
\QED
Let $D = D(\Delta)$ be the complementary reduced divisor
of $T$ in $Z$.
We set
\begin{equation}
\Omega_{Z_{\rm h}}(\log D_{\rm h})^1 :=
\Omega_Z(\log D)_{\rm h}^1 ={\cal O}_{Z_{\rm h}}\otimes_{\bf Z} M\;.
\end{equation}
We define the ${\bf C}$-derivation
$\partial\mathrel{:}{\cal O}_{Z_{\rm h}}\rightarrow\Omega_{Z_{\rm h}}(\log D_{\rm h})^1$ as follows.
Let $n, n'$ be elements of $N$.
The equality $\lambda_{n + n'}(t) =\lambda_n(t)\lambda_{n'}(t)$
holds for $t\in{\bf C}^*$.
For an analytic function $f$ on an open subset $U$ of $Z_{\rm h}$,
we have
\begin{eqnarray*}
\lefteqn{\partial_{n + n'}f(x)} \\
& = &\lim_{t\rightarrow 1}\frac{f(x) - f(\lambda_n(t)\lambda_{n'}(t)x)}{1 - t} \\
& = &\lim_{t\rightarrow 1}\frac%
{f(x) - f(\lambda_n(t)x)}{1 - t} + \lim_{t\rightarrow 1}\frac%
{f(\lambda_n(t)x) - f(\lambda_{n'}(t)\lambda_n(t)x)}{1 - t} \\
& = &\partial_nf(x) +\partial_{n'}f(x)
\end{eqnarray*}
for any $x\in U$.
Hence the map $n\mapsto\partial_nf\in{\cal O}_{Z_{\rm h}}(U)$ is a
homomorphism, i.e., an element of $\mathop{\rm Hom}\nolimits_{\bf Z}(N,{\cal O}_{Z_{\rm h}}(U))$.
We define $\partial f$ to be the corresponding element of
$\Omega_{Z_{\rm h}}(\log D_h)(U)^1 ={\cal O}_{Z_{\rm h}}(U)\otimes_{\bf Z} M$.
Let $m$ and $n$ be elements of $M$ and $N$, respectively.
Since the character ${\bf e}(m)\mathrel{:} T_{\rm h}\rightarrow{\bf C}^*$ is a homomorphism,
\begin{equation}
{\bf e}(m)(\lambda_n(t)x)
= {\bf e}(m)(\lambda_n(t))\cdot {\bf e}(m)(x)
= t^{\langle m, n\rangle}\cdot {\bf e}(m)(x)\;.
\end{equation}
Hence $\partial_n{\bf e}(m) =\langle m, n\rangle {\bf e}(m)$ for every $n\in N$.
This implies $\partial{\bf e}(m) ={\bf e}(m)\otimes m$.
Hence we denote the global section $1\otimes m$ of
$\Omega_{Z_{\rm h}}(\log D_h)^1$ by $\partial{\bf e}(m)/{\bf e}(m)$ for
every $m\in M$, however it is common to write it by
$d{\bf e}(m)/{\bf e}(m)$.
Let $\{m_1,\cdots, m_r\}$ be a ${\bf Z}$-basis of $M$ and let
$\{n_1,\cdots, n_r\}$ be the dual basis of $N$.
If we write $\partial f =\sum a_i\otimes m_i$, we have
$a_i =\langle\partial f, n_i\rangle =\partial_{n_i}f$ for each $i$.
Hence
\begin{equation}
\label{eq df}
\partial f =\sum_{i=1}^r\partial_{n_i}f\,\frac{\partial{\bf e}(m_i)}{{\bf e}(m_i)}\;.
\end{equation}
In particular $\partial$ is a ${\bf C}$-derivation.
Since ${\bf e}(m)$'s form a ${\bf C}$-basis of the coordinate ring of
each affine toric variety, we know that this ${\bf C}$-derivation
$\partial$ is compatible with the algebraic ${\bf C}$-derivation
$\partial\mathrel{:}{\cal O}_{Z_{\bf C}}\rightarrow\Omega_{Z_{\bf C}}(\log D_{\bf C})^1$.
For each $0\leq i\leq r$ we set
\begin{equation}
\Omega_{Z_{\rm h}}(\log D_{\rm h})^i :=
\bigwedge^i\Omega_{Z_{\rm h}}(\log D_{\rm h})^1 =
{\cal O}_{Z_{\rm h}}\otimes_{\bf Z}\bigwedge^i M\;.
\end{equation}
Since $\Omega_{Z_{\rm h}}(\log D_{\rm h})^1$ is a free ${\cal O}_{Z_{\rm h}}$-module
of rank $r$, $\Omega_{Z_{\rm h}}(\log D_{\rm h})^i$ is free of rank ${}_rC_i$.
For $0 < i < r$, we define a pairing
\begin{equation}
{\cal O}_{Z_{\rm h}}(U)\times\bigwedge^i M\longrightarrow
\Omega_{Z_{\rm h}}(\log D_{\rm h})^{i+1}(U)
\end{equation}
by $(f, w)\mapsto\partial f\wedge w$ which induces a ${\bf C}$-homomorphism
\begin{equation}
\partial\mathrel{:}\Omega_{Z_{\rm h}}(\log D_{\rm h})^i\rightarrow\Omega_{Z_{\rm h}}(\log D_{\rm h})^{i+1}
\end{equation}
of sheaves which we denote also by $\partial$.
We check easily that $\partial\cdot \partial = 0$, and we get a
$\partial$-complex
$\Omega_{Z_{\rm h}}(\log D_{\rm h})^\bullet$ which we call the
{\em logarithmic de Rham complex} on $Z_{\rm h}$.
For any $\sigma\in\Delta$, we see easily by the description
(\ref{eq df}) of $\partial f$ that
${\cal P}(\sigma)_{\rm h}\Omega_{Z_{\rm h}}(\log D_{\rm h})^\bullet$
is a subcomplex of $\Omega_{Z_{\rm h}}(\log D_{\rm h})^\bullet$.
For each $\sigma\in\Delta$, we denote by $\bar I_\sigma$ the closed
immersion $X(\sigma)_{\rm h}\rightarrow Z_{\rm h}$ and by $I_\sigma$ the immersion
$T[\sigma]_{\rm h}\rightarrow Z_{\rm h}$.
For a ${\bf C}$-sheaf $F$ on $Z_{\rm h}$, we denote by $\bar I_\sigma^*F$
and $I_\sigma^*F$ the pull-back of $F$ to $X(\sigma)_{\rm h}$ and
$T[\sigma]_{\rm h}$, respectively.
Note that, even if $F$ is an ${\cal O}_{Z_{\rm h}}$-module, the pull-back
is taken as a ${\bf C}$-sheaf.
\begin{Lem}
\label{Lem 4.2}
For every $\sigma\in\Delta$, the $\partial$-complex
$\bar I_\sigma^*({\cal P}(\sigma)_{\rm h}\Omega_{Z_{\rm h}}(\log D_{\rm h}))^\bullet$
on $X(\sigma)_{\rm h}$ is homotopically equivalent to the zero complex.
\end{Lem}
{\sl Proof.}\quad
We have to show that the identity map of this $\partial$-complex is
homotopic to the zero map.
Take an element $n_0$ in $N\cap\mathop{\rm rel.\,int}\nolimits\sigma$.
Let $h$ be the ${\cal O}_{Z_{\rm h}}$-homomorphism of degree $-1$ of the
graded ${\cal O}_{Z_{\rm h}}$-module $\Omega_{Z_{\rm h}}(\log D_{\rm h})^\bullet$
induced by the right interior product
$i(n_0)\mathrel{:}\bigwedge^\bullet M\rightarrow\bigwedge^\bullet M$.
Then the ${\bf C}$-homomorphism $h\cdot \partial + \partial\cdot h$ on
$\Omega_{Z_{\rm h}}(\log D_{\rm h})^\bullet = {\cal O}_{Z_{\rm h}}\otimes_{\bf Z}\bigwedge^\bullet M$
is equal to $\partial_{n_0}\otimes 1$ (cf. \cite[13.4]{Danilov}).
By Lemma~\ref{Lem 4.1}, this induces an automorphism of
$\bar I_\sigma^*({\cal P}(\sigma)_{\rm h}\Omega_{Z_{\rm h}}(\log D_{\rm h}))^\bullet$
as a ${\bf C}$-sheaf.
Let $u$ be the inverse isomorphism of
$\bar I_\sigma^*(h\cdot \partial + \partial\cdot h)$.
Since
\begin{equation}
(h\cdot \partial + \partial\cdot h)\cdot\partial =
\partial\cdot(h\cdot \partial + \partial\cdot h) =
\partial\cdot h\cdot \partial\;,
\end{equation}
we have
$\bar I_\sigma^*\partial\cdot u = u\cdot\bar I_\sigma^*\partial
=u\cdot I_\sigma^*(\partial\cdot h\cdot \partial)\cdot u$.
Set $h' := u\cdot\bar I_\sigma^*h$.
Then we have
\begin{equation}
h'\cdot\bar I_\sigma^*\partial +\bar I_\sigma^*\partial\cdot h' =
u\cdot I_\sigma^*(h\cdot \partial + \partial\cdot h) = 1\;.
\end{equation}
\QED
Let $\sigma$ be an element of $\Delta$.
We set $N[\sigma] := N/(N\cap(\sigma+(-\sigma)))$.
This is naturally the dual ${\bf Z}$-module of
$M[\sigma] = M\cap\sigma^\bot$.
It is known that the closed subscheme $X(\sigma)$ of $Z$ is
naturally identified with the toric vatiety $Z(\Delta[\sigma])$
with the torus $T[\sigma]$ \cite[Cor.1.7]{Oda1}.
Furthermore, the action of $T[\sigma]$ on $Z(\Delta[\sigma])$
is equivariant with that of $T$ with respect to the natural
surjection $T\rightarrow T[\sigma]$.
We set $D(\sigma)_{\rm h} := X(\sigma)_{\rm h}\setminus T[\sigma]_{\rm h}$
and
\begin{equation}
\Omega_{X(\sigma)_{\rm h}}(\log D(\sigma)_{\rm h})^\bullet :=
{\cal O}_{X(\sigma)_{\rm h}}\otimes_{\bf Z}{\bigwedge}\!{}^\bullet M[\sigma]\;.
\end{equation}
Then $\Omega_{X(\sigma)_{\rm h}}(\log D(\sigma)_{\rm h})^\bullet$ has the
$\partial$-complex structure so that it is identified with
the logarithmic de Rham complex of $Z(\Delta[\sigma])_{\rm h}$
similarly as $\Omega_{Z_{\rm h}}(\log D_{\rm h})^\bullet$ for
$Z_{\rm h} =Z(\Delta)_{\rm h}$.
\begin{Lem}
\label{Lem 4.3}
Let $\sigma$ be an elemet of $\Delta$.
For $\rho\in\Delta$ with $\sigma\prec\rho$, the $\partial$-complex
$\bar I_\rho^*({\cal P}(\rho)_{\rm h}\Omega_{X(\sigma)_{\rm h}}(\log D(\sigma)_{\rm h}))^\bullet$
of ${\bf C}$-sheaves on $X(\rho)_{\rm h}$ is homotopically equivalent
to the zero complex.
\end{Lem}
{\sl Proof.}\quad
For the toric variety $X(\sigma)_{\rm h} = Z(\Delta[\sigma])_{\rm h}$, $X(\rho)_{\rm h}$
is the closed subvariety associated to $\rho[\sigma]\in\Delta[\sigma]$.
The closed subvariety $X(\rho)_{\rm h}$ of $X(\sigma)_{\rm h}$ is defined by the
image of ${\cal P}(\rho)_{\rm h}$ in ${\cal O}_{X(\sigma)_{\rm h}}$.
Hence, this is a consequence of Lemma~\ref{Lem 4.2} applied
to the toric variety $Z(\Delta[\sigma])_{\rm h}$.
\QED
Let $\rho$ be an element of $\Delta$ and $V$ an object of $\mathop{\rm GM}\nolimits(A(\rho))$.
Then the $\partial$-complex $\Lambda_Z^\rho(V)^\bullet$ defined in
Section~3 induces a $\partial$-complex $\Lambda_{Z_{\bf C}}^\rho(V)^\bullet$
and its analytic version $\Lambda_{Z_{\rm h}}^\rho(V)^\bullet$.
By Proposition~\ref{Prop 3.1}, $\Lambda_{Z_{\rm h}}^\rho(V)^\bullet$ is
isomorphic to a finite direct sum of dimension shifts of
$\Omega_{X(\rho)_{\rm h}}(\log D(\rho)_{\rm h}))^\bullet$.
In particular, we get the following corollary by Lemma~\ref{Lem 4.3}.
\begin{Cor}
\label{Cor 4.4}
Let $\sigma$ be an element of $\Delta$ and $V$ in $\mathop{\rm GM}\nolimits(A(\sigma))$.
For $\rho\in\Delta$ with $\sigma\prec\rho$, the $\partial$-complex
$\bar I_\rho^*({\cal P}(\rho)_{\rm h}\Lambda_{Z_h}^\sigma(V))^\bullet$
on $X(\rho)_{\rm h}$ is homotopically equivalent to the zero complex.
\end{Cor}
We recall some general notation of derived categories
(cf.\cite{Verdier}).
Let $X$ be a locally compact topological space and let
$A({\bf C}_X)$ be the abelian category of the ${\bf C}$-sheaves on $X$.
We denote by $C^+({\bf C}_X)$ and $D^+({\bf C}_X)$ the category of the
complexes bounded below in $A({\bf C}_X)$ and the derived category
of it, respectively.
For a continuous map $f\mathrel{:} X\rightarrow Y$, let
$f^*\mathrel{:} A({\bf C}_Y)\rightarrow A({\bf C}_X)$ be the the pull-back functor
which is exact.
We denote also by $f^*$ the induced functors
$C^+({\bf C}_Y)\rightarrow C^+({\bf C}_X)$ and $D^+({\bf C}_Y)\rightarrow D^+({\bf C}_X)$.
If the direct image functor with proper support
$f_!\mathrel{:} A({\bf C}_X)\rightarrow A({\bf C}_Y)$ is of finite cohomological dimension,
the functor $f^!\mathrel{:} D^+({\bf C}_Y)\rightarrow D^+({\bf C}_X)$ is defined
\cite[2.2]{Verdier}.
Note that this condition is satisfied for any regular morphisms
of finite dimensional analytic spaces.
For $V^\bullet\in\mathop{\rm CGM}\nolimits(A(\sigma))$, the bicomplex
$\Lambda_{Z_{\rm h}}^\sigma(V)^{\bullet,\bullet}$ is defined
as the analytic version of $\Lambda_Z^\sigma(V)^{\bullet,\bullet}$.
The associated single complex is denoted by
$\Lambda_{Z_{\rm h}}^\sigma(V)^\bullet$.
For $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$, we get the bicomplex
$\Lambda_{Z_{\rm h}}(L)^{\bullet,\bullet}$ and its associated
single complex $\Lambda_{Z_{\rm h}}(L)^\bullet$, similarly.
We denote by $\delta$ the coboundary map of
$\Lambda_{Z_{\rm h}}(L)^\bullet$.
\begin{Prop}
\label{Prop 4.5}
For $\rho\in\Delta$ and a $d$-complex $L^\bullet$ in
$\mathop{\rm CGEM}\nolimits(\Delta)$, there exist a quasi-isomorphism
\begin{equation}
I_\rho^*\Lambda_{Z_{\rm h}}(L)^\bullet\simeq
I_\rho^*\Lambda_{Z_{\rm h}}^\rho(\i_\rho^*L)^\bullet
\end{equation}
as $\delta$-complexes of ${\bf C}$-sheaves on $T[\rho]_{\rm h}$ and an
isomorphism
\begin{equation}
I_\rho^!\Lambda_{Z_{\rm h}}(L)^\bullet\simeq
I_\rho^*\Lambda_{Z_{\rm h}}^\rho(\i_\rho^!L)^\bullet
\end{equation}
in the derived category $D^+({\bf C}_{T[\rho]_{\rm h}})$.
\end{Prop}
{\sl Proof.}\quad
Let $\sigma$ be an element of $F(\pi)$ with $\sigma\prec\rho$.
For $V$ in $\mathop{\rm GM}\nolimits(A(\sigma))$, there exists an exact sequence
\begin{equation}
0\longrightarrow{\cal P}(\rho)_{\rm h}\Lambda_{Z_{\rm h}}^\sigma(V)^\bullet\longrightarrow
\Lambda_{Z_{\rm h}}^\sigma(V)^\bullet\mathop{\longrightarrow}\limits^{\lambda(V)}
\Lambda_{Z_{\rm h}}^\rho(V_{A(\rho)})^\bullet\longrightarrow 0\;.
\end{equation}
In particular, the homomorphism $I_\rho^*\lambda(V)$ of
$\partial$-complexes is a quasi-isomorphism by Corollary~\ref{Cor 4.4}.
For each $L^i$, we get a quasi-isomorphism
\begin{equation}
I_\rho^*\lambda(L^i)\mathrel{:} I_\rho^*\Lambda_{Z_{\rm h}}(L^i)^\bullet\longrightarrow
I_\rho^*\Lambda_{Z_{\rm h}}^\rho(\i_\rho^*L^i)^\bullet
\end{equation}
as a collection of quasi-isomorphisms $I_\rho^*\lambda(L^i(\sigma))$
for $\sigma\in F(\rho)$.
We get the first quasi-isomorphism as the collection of
$I_\rho^*\lambda(L^i)$'s for $i\in{\bf Z}$.
Since $\i_\rho^!L^\bullet = L(\rho)^\bullet$, it is enough to
show that $\bar I_\rho^!\Lambda_{Z_{\rm h}}^\sigma(L^i(\sigma))$'s are
quasi-isomorphic to the zero complex for all
$\sigma\in F(\rho)\setminus\{\rho\}$ and $i\in{\bf Z}$.
By proposition~\ref{Prop 3.1}, $\Lambda_{Z_{\rm h}}^\sigma(L^i(\sigma))$
is isomorphic to the finite direct sum of dimension shifts
of the logarithmic de Rham complex
$\Omega_{X(\sigma)_{\rm h}}(\log D(\sigma)_{\rm h})^\bullet$ on $X(\sigma)_{\rm h}$.
By \cite[Prop.1.2]{Oda2}, the logarithmic de Rham complex
is quasi-isomorphic to the direct image $Rj_*{\bf C}_{T[\sigma]}$ for
the open immersion $j\mathrel{:} T[\sigma]_{\rm h}\rightarrow X(\sigma)_{\rm h}$.
Since $T[\sigma]_{\rm h}\cap X(\rho)_{\rm h} =\emptyset$,
$\bar I_\rho^!Rj_*{\bf C}_{T[\sigma]_{\rm h}}$ is equivalent to zero.
Hence $\bar I_\rho^!\Lambda_{Z_{\rm h}}^\sigma(L^i(\sigma))$ is also zero
in the derived category.
\QED
For any ${\bf Q}$-vector space $W$, we denote by $\bar W$ the scalar
extension $W\otimes_{\bf Q}{\bf C}$.
If $W$ is graded, so is $\bar W$.
Let $\rho$ be an element of $\Delta$.
For $V$ in $\mathop{\rm GM}\nolimits(A(\rho))$, we denote by $\bar V_{T[\rho]_{\rm h}}$ the
constant sheaf on $T[\rho]_{\rm h}$ with the stalk $\bar V$.
We regard it as a $\partial$-complex of ${\bf C}$-sheaves by setting
$\bar V_{T[\rho]_{\rm h}}^i := (\bar V_i)_{T[\rho]_{\rm h}}$ and $\partial = 0$.
For $N[\rho] := N/N(\rho)$, we set $\det[\rho]_{\bf Q} :=\det N[\rho]_{\bf Q}$.
We denote by $\mathop{\rm Det}\nolimits[\rho]$ the graded ${\bf Z}$-module defined by
$(\mathop{\rm Det}\nolimits[\rho])_{-r+r_\rho} :=\det[\rho]$ and
$(\mathop{\rm Det}\nolimits[\rho])_i :=\{0\}$ for $i\not= -r+r_\rho$.
Here note that $\mathop{\rm rank}\nolimits N[\rho] = r-r_\rho$.
We define a homomorphism
$(\bar V\otimes\mathop{\rm Det}\nolimits[\rho])_{T[\rho]_{\rm h}}^\bullet\rightarrow
I_\rho^*(\Lambda_{Z_{\rm h}}^\rho(V))^\bullet$ as follows.
We take a ${\bf Q}$-linear subspace $H$ of $N_{\bf Q}$ such that
$N_{\bf Q} = N(\rho)_{\bf Q}\oplus H$.
Then we have a natural isomorphisms
$N[\rho]_{\bf Q} = N_{\bf Q}/N(\rho)_{\bf Q}\simeq H$ and
$\det[\rho]_{\bf Q} =\det N[\rho]_{\bf Q}\simeq\det H\subset A_{-r+r_\rho}$.
We denote by $\mathop{\rm Det}\nolimits H$ the graded ${\bf Q}$-vector space defined by
$(\mathop{\rm Det}\nolimits H)_{-r+r_\rho} :=\det H$ and $(\mathop{\rm Det}\nolimits H)_i :=\{0\}$ for
$i\not= -r+r_\rho$.
By Lemma~\ref{Lem 1.3}, $V_A = V\otimes_{A(\rho)}A$ is equal
to the free $A^*[\rho]$-module
$(V\otimes_{\bf Q}\mathop{\rm Det}\nolimits H)\otimes_{\bf Q} A^*[\rho]$.
The restriction of $\partial$ of
$\Lambda_Z^\rho(V)^\bullet ={\cal O}_Z\otimes_{\bf Q} V_A$ to the
constant subsheaf ${\bf Q}{\bf e}(0)\otimes_{\bf Q} V_A$ is zero.
Hence the composite of the natural homomorphisms
\begin{equation}
\bar V\otimes\mathop{\rm Det}\nolimits[\rho]\longrightarrow\bar V\otimes_{\bf Q}\mathop{\rm Det}\nolimits H\longrightarrow
\bar V_A\longrightarrow {\bf C}{\bf e}(0)\otimes_{\bf C}\bar V_A
\end{equation}
defines a homomorphism
$(\bar V\otimes\mathop{\rm Det}\nolimits[\rho])_{T[\rho]_{\bf C}}\rightarrow%
\Lambda_{Z_{\bf C}}^\rho(V)|_{T[\rho]_{\bf C}}$
of $\partial$-complexes of ${\bf C}$-sheaves on the algebraic torus
$T[\rho]_{\bf C}$
We denote by $\phi_V^H$ the associated homomorphism
\begin{equation}
(\bar V\otimes\mathop{\rm Det}\nolimits[\rho])_{T[\rho]_{\rm h}}^\bullet\longrightarrow
I_\rho^*(\Lambda_{Z_{\rm h}}^\rho(V))^\bullet
\end{equation}
on the smooth analytic space $T[\rho]_{\rm h}$.
\begin{Prop}
\label{Prop 4.6}
Let $\rho\in\Delta$ and let $V$ be an object of $\mathop{\rm GM}\nolimits(A(\rho))$.
Then the homomorphism $\phi_V^H$ is a quasi-isomorphism for any
$H\subset N_{\bf Q}$ with $N_{\bf Q} = N(\rho)_{\bf Q}\oplus H$.
For another $K\subset N_{\bf Q}$ with $N_{\bf Q} = N(\rho)_{\bf Q}\oplus K$,
the homomorphisms $\phi_V^H$ and $\phi_V^K$ are locally homotopic.
\end{Prop}
{\sl Proof.}\quad
Since $V$ is of finite dimension, we prove the proposition
by induction on the dimension of $V$.
If $V =\{0\}$, then the $\partial$-complexes are trivial and the
assertion is clear.
Assume $\dim V\geq 1$.
Let $k$ be the maxiaml integer with $V_k\not=\{0\}$.
We take a vector subspace $V'_k\subset V_k$ of codimension one.
By setting $V'_i := V_i$ for $i\not= k$, we get a homogeneous
subspace $V'\subset V$ of codimension one.
Since $A(\rho)$ is graded negatively, $V'$ is an object of $\mathop{\rm GM}\nolimits(A(\rho))$.
We get an exact sequence
\begin{equation}
0\longrightarrow V'\longrightarrow V\longrightarrow{\bf Q}(-k)\longrightarrow 0
\end{equation}
of graded $A(\rho)$-modules, where $Q(a)$ is the graded
${\bf Q}$-vector space defined by $Q(a)_{-a} :={\bf Q}$ and
$Q(a)_i :=\{0\}$ for $i\not= -a$.
Set $c := r - r_\rho$.
Then we get a commutative diagram
\begin{equation}
\begin{array}{ccccccccc}
0 &\rightarrow &(\bar V'\otimes\mathop{\rm Det}\nolimits[\rho])_{T[\rho]_{\rm h}}^\bullet &\rightarrow
&(\bar V\otimes\mathop{\rm Det}\nolimits[\rho])_{T[\rho]_{\rm h}}^\bullet &\rightarrow
&({\bf C}({-}k)\otimes\mathop{\rm Det}\nolimits[\rho])_{T[\rho]_{\rm h}}^\bullet &\rightarrow & 0 \\
& &\hbox{ }\downarrow\phi_{V'}^H & &\hbox{ }\downarrow\phi_V^H & &
\hbox{ }\downarrow\phi_{{\bf Q}(-k)}^H & & \\
0 &\rightarrow & I_\rho^*(\Lambda_{Z_{\rm h}}^\rho(V'))^\bullet &\rightarrow
&I_\rho^*(\Lambda_{Z_{\rm h}}^\rho(V))^\bullet &\rightarrow
&I_\rho^*(\Lambda_{Z_{\rm h}}^\rho({\bf Q}(-k)))^\bullet &\rightarrow & 0
\end{array}
\end{equation}
By the induction assumption, $\phi_{V'}^H$ is a quasi-isomorphism
of $\partial$-complexes of ${\bf C}$-sheaves.
On the other hand, $I_\rho^*(\Lambda_{Z_{\rm h}}^\rho({\bf Q}(-c)))^\bullet$ is equal
to the analytic de Rham complex on the complex manifold $T[\rho]_{\rm h}$.
Since $\phi_{{\bf Q}(-k)}^H$ is the dimension shift of the natural
homomorphism ${\bf C}_{T[\rho]_{\rm h}}\longrightarrow\Omega_{T[\rho]_{\rm h}}^\bullet$, this is a
quasi-isomorphism, by the complex analytic version of the
Poincar\'e Lemma.
Hence $\phi_V^H$ is also a quasi-isomorphism.
Clearly, $\phi_{{\bf Q}(-k)}^H$ does not depend on the choice of $H$.
Hence $\phi_{{\bf Q}(-k)}^H -\phi_{{\bf Q}(-k)}^K = 0$.
By assumption, $\phi_{V'}^H -\phi_{V'}^K$ is locally homotopic
to zero.
Hence, we know that $\phi_V^H -\phi_V^K$ gives zero maps on the
cohomology sheaves.
Since $\bar V_{T[\rho]_{\rm h}}$ is a locally free ${\bf C}$-sheaf,
$\phi_V^H -\phi_V^K$ is locally homotopic to zero.
\QED
Recall that $Z_{\rm h}$ has the docomposition
$\bigcup_{\sigma\in\Delta}T[\sigma]_{\rm h}$ into $T_{\rm h}$-orbits.
For each integer $0\leq i\leq r$, we define
\begin{equation}
Z_{\rm h}^{2i} = Z_{\rm h}^{2i+1} =:
\bigcup_{\scriptstyle\sigma\in\Delta\atop\scriptstyle r_\sigma\geq r-i}
T[\sigma]_{\rm h}\;.
\end{equation}
Then we have a filtration
\begin{equation}
Z_{\rm h} = Z_{\rm h}^{2r}\supset Z_{\rm h}^{2r-1}\supset\cdots
\supset Z_{\rm h}^2\supset Z_{\rm h}^1\supset Z_{\rm h}^0
\end{equation}
of $Z_{\rm h}$ satisfying the conditions of the topological
stratification in \cite[1.1]{GM2}.
The intersection complex of a stratified space is defined for a
sequence of integers $({\bf p}(2),{\bf p}(3),{\bf p}(4),\cdots)$ which
is called a {\em perversity} \cite[2.0]{GM2}.
Since $Z_{\rm h}$ is a complex analytic space of dimension $r$, only
${\bf p}(i)$'s for even $i$ less than or equal to $2r$ are relevant
for the intersection complex.
Let ${\bf p} = ({\bf p}(2),{\bf p}(4),{\bf p}(6),\cdots,{\bf p}(2r))$ be a sequence of
integers with ${\bf p}(2) = 0$ and ${\bf p}(2i)\leq{\bf p}(2i+2)\leq{\bf p}(2i)+2$ for
$i=1,\cdots, r-1$ as a perversity for $Z_{\rm h}$.
We denote by the same symbol ${\bf p}$ the perversity on $\Delta$
defined by ${\bf p}(\sigma) :={\bf p}(2r_\sigma) -r_\sigma + 1$
for $\sigma\in\Delta\setminus\{{\bf 0}\}$.
Note that, for the middle perversity ${\bf m} := (0, 1, 2,\cdots, r-1)$,
we have ${\bf m}(\sigma) = 0$ for all $\sigma\in\Delta\setminus\{{\bf 0}\}$.
We consider the $\delta$-complex $\Lambda_{Z_{\bf C}}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$
which is the associated single complex of the bicomplex
$\Lambda_Z(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^{\bullet,\bullet}$.
By \cite[3.3, AX1]{GM2}, the intersection complex $\mathop{\rm IC}\nolimits_{\bf p}(Z_{\rm h})^\bullet$
in the derived category $D^{\rm b}({\bf C}_{Z_{\rm h}})$ of bounded
complexes of ${\bf C}$-sheaves is characterized by the following
properties.
For the convenience of our use, we set
$F :=\mathop{\rm IC}\nolimits_{\bf p}(Z_{\rm h})[-r]^\bullet$.
Note that $n$ in \cite[3.3]{GM2} is $2r$ in our case.
(a) The restriction of $F$ to $T_{\rm h}$ is quasi-isomorphic
to ${\bf C}_{T_{\rm h}}[r]$.
(b) ${\cal H}^i(F)$ is a trivial sheaf for $i < -r$.
(c) For any $\sigma\in\Delta\setminus\{{\bf 0}\}$,
$\H^i(F_x) =\{0\}$ for $i \geq{\bf p}(2r_\sigma) - r + 1$.
(d) For any $\sigma\in\Delta\setminus\{{\bf 0}\}$,
$\H^i((I_\sigma^!F)_x) =\{0\}$ for $i \leq{\bf p}(2r_\sigma) - r + 1$.
\begin{Thm}
\label{Thm 4.7}
Let ${\bf p} = ({\bf p}(2),{\bf p}(4),{\bf p}(6),\cdots,{\bf p}(2r))$ be a perversity.
The complex of ${\bf C}$-sheaves $\Lambda_{Z_{\rm h}}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$
is isomorphic to $\mathop{\rm IC}\nolimits_{\bf p}(Z_{\rm h})[-r]^\bullet$.
\end{Thm}
{\sl Proof.}\quad
We set $L^\bullet :=\Lambda_{Z_{\rm h}}(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$.
It is sufficient to show that $L^\bullet$ satisfies the above
properties.
Let $\sigma$ be in $\Delta\setminus\{{\bf 0}\}$.
Then the stratum $T[\sigma]_{\rm h}$ is of dimension $2r - 2r_\sigma$
in the real dimension.
By Propositions~\ref{Prop 4.5} and \ref{Prop 4.6}, we have an
isomorphism
\begin{equation}
\H^i(L^\bullet_x)\simeq
\H^{i+r-r_\sigma}(\i_\sigma^*\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet)\otimes_{\bf Q}{\bf C}
\end{equation}
for every point $x\in T[\sigma]_{\rm h}$.
By the condition (3) of Theorem~\ref{Thm 2.9}, the cohomology
$\H^i(\i_\sigma^*\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet)$ vanishes for
$i\geq{\bf p}(\sigma) ={\bf p}(2r_\sigma) - r_\sigma + 1$.
Hence $\H^i(L_x^\bullet) = 0$ for
$i\leq{\bf p}(2r_\sigma) - r + 1$.
By Propositions~\ref{Prop 4.5} and \ref{Prop 4.6}, we also have
\begin{equation}
\H^i((I_\sigma^!L)_x^\bullet)\simeq
\H^{i+r-r_\sigma}(\i_\sigma^!\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet)\otimes_{\bf Q}{\bf C}
\end{equation}
for all $x\in T[\sigma]_{\rm h}$.
The condition (2) of Theorem~\ref{Thm 2.9} implies
$\H^i(\i_\sigma^!\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet) = 0$ for
$i\leq{\bf p}(\sigma) ={\bf p}(2r_\sigma) - (r_\sigma) + 1$.
Hence $\H^i((I_\sigma^!L)_x^\bullet) = 0$ for $i\leq{\bf p}(2r_\sigma) - r + 1$.
Hence, by the Axioms [AX1] of \cite[3.3]{GM2}, $L^\bullet$ is the
intersection complex with the perversity ${\bf p}$.
\QED
\begin{Prop}
\label{Prop 4.8}
Assume that $\Delta$ is a complete fan.
Let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then there exists a natural isomorphism
\begin{equation}
\H^k(\Lambda_{Z_{\rm h}}(L)^\bullet)\simeq
\bigoplus_{p+q=k}\H^p(\Gamma(L)^\bullet)_q\otimes_{\bf Q}{\bf C}
\end{equation}
for every integer $k$.
\end{Prop}
{\sl Proof.}\quad
Since any $X(\sigma)$ is a complete toric variety,
$\H^i(X(\sigma),{\cal O}_{X(\sigma)}) = 0$
for every $i > 0$ \cite[Cor.2.8]{Oda1}.
By \cite{GAGA}, we have
\begin{equation}
\H^i(X(\sigma)_{\rm h},{\cal O}_{X(\sigma)_{\rm h}}) =\left\{
\begin{array}{cc}
{\bf C} & \;\;\;\hbox{ if }\;\; i = 0 \\
\{0\} & \;\;\;\hbox{ if }\;\; i > 0
\end{array}
\right.
\end{equation}
Since $\Lambda_{Z_{\rm h}}^\sigma(L^i(\sigma))$ is a free
$O_{X(\sigma)_{\rm h}}$-module for any $\sigma$ and $i$,
we have an isomorphism
\begin{equation}
\H^k(\Lambda_{Z_{\rm h}}(L)^\bullet) =
\H^k(\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))^\bullet)\;,
\end{equation}
where $\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))^\bullet$ is the single
complex of ${\bf C}$-vector spaces associated to the bicomplex
$\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))^{\bullet,\bullet}$.
Since the global sections are locally $M$-homogeneous of
degree $0\in M$, $d_2 =\partial$ of the last bicomplex is zero.
Hence the spectral sequence \begin{equation}
E_1^{p,q} :=\H^p(\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))_q^\bullet)
\Longrightarrow
\H^{p+q}(\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))^\bullet)
\end{equation}
degenerates at $E_1$-terms.
Consequently, the cohomology group
$\H^k(\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))^\bullet)$ is equal to the
direct sum
\begin{equation}
\bigoplus_{q\in{\bf Z}}\H^{k-q}(\Gamma(Z_{\rm h},\Lambda_{Z_{\rm h}}(L))_q^\bullet)\\
= \bigoplus_{q\in{\bf Z}}\H^{k-q}(\Gamma(L)^\bullet)_q\otimes_{\bf Q}{\bf C}\;.
\end{equation}
\QED
\section{The Serre duality}
\setcounter{equation}{0}
In this section, we again fix a finite fan $\Delta$ of $N_{\bf R}$
and let $Z := Z(\Delta)$ be the associated toric variety defined
over ${\bf Q}$.
Let ${\bf p}$ be a perversity on $\Delta$.
Then $\Lambda_Z(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^{\bullet,\bullet}$ is a
bicomplex of coherent ${\cal O}_Z$-modules such that $d_1 = d$
is a ${\cal O}_Z$-homomorphism and $d_2 =\partial$ is a
differential operator of order one.
Note that $\Lambda_Z(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^{i,j} =\{0\}$
unless $0\leq i\leq r$ and $-r\leq j\leq 0$ by
Proposition~\ref{Prop 2.11}.
For each integer $j$, we set
\begin{equation}
\Omega_j({\bf p}; Z) :=\Lambda_Z(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))_{-j}^\bullet\;.
\end{equation}
Clearly, $\Omega_j({\bf p}; Z)$ is the zero complex unless $0\leq j\leq r$.
Since $d$ is an ${\cal O}_Z$-homomorphism, $\Omega_j({\bf p}; Z)$ is a
finite complex in the category of coherent ${\cal O}_Z$-modules.
For the top perversity $\t$, we have
\begin{equation}
\Omega_j(\t; Z)^i =
\bigoplus_{\sigma\in\Delta(i)}\det(\sigma)\otimes
\bigwedge^{-j}N[\sigma]\otimes{\cal O}_{X(\sigma)}\;,
\end{equation}
where $N[\sigma] := N/N(\sigma)$.
For $\sigma\in\Delta(i)$ and $\tau\in\Delta(i+1)$, the
$(\sigma,\tau)$-component of the coboundary map is the
tensor product of $q'_{\sigma/\tau}\mathrel{:}\det(\sigma)\rightarrow\det(\tau)$
and the natural surjection
\begin{equation}
\bigwedge^{-j}N[\sigma]\otimes{\cal O}_{X(\sigma)}\rightarrow
\bigwedge^{-j}N[\tau]\otimes{\cal O}_{X(\tau)}
\end{equation}
if $\sigma\prec\tau$ and is the zero map otherwise.
For finite complexes $B^\bullet, C^\bullet$ of coherent
sheaves on $Z$, we denote by ${\cal H} om_{{\cal O}_Z}(C, B)^{\bullet,\bullet}$
the bicomplex with the component
\begin{equation}
{\cal H} om_{{\cal O}_Z}(C, B)^{i,j} :=
{\cal H} om_{{\cal O}_Z}(C_j, B_i)
\end{equation}
for each pair $(i, j)$ of integers.
We denote by ${\cal H} om_{{\cal O}_Z}(C, B)^\bullet$ the associated
single complex.
The coboundary map
\begin{equation}
d^k\mathrel{:}{\cal H} om_{{\cal O}_Z}(C, B)^k\longrightarrow{\cal H} om_{{\cal O}_Z}(C, B)^{k+1}
\end{equation}
is determined as follows.
For an integer $i$, the restriction of $d^k$ to the component
${\cal H} om_{{\cal O}_Z}(C^i, B^{i+k})$ is the sum of
\begin{equation}
(d_B^{i+k})_*\mathrel{:}
{\cal H} om_{{\cal O}_Z}(C^i, B^{i+k})\longrightarrow{\cal H} om_{{\cal O}_Z}(C^i, B^{i+k+1})
\end{equation}
and
\begin{equation}
(-1)^{k+1}(d_C^{i-1})^*
{\cal H} om_{{\cal O}_Z}(C^i, B^{i+k})\longrightarrow{\cal H} om_{{\cal O}_Z}(C^{i-1}, B^{i+k})
\end{equation}
(cf.\cite[1.1.10,(ii)]{Deligne})
\begin{Thm}
\label{Thm 5.1}
Let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
For each integer $q$, there exists a natural isomorphism
\begin{equation}
{\cal H} om_{{\cal O}_Z}(\Lambda_Z(L)_q,\Omega_0(\t; Z)\otimes\det N)^\bullet
\simeq\Lambda_Z({\bf D}(L))_{-r-q}^\bullet\;.
\end{equation}
\end{Thm}
{\sl Proof.}\quad
For each pair $(i, j)$ of integers, we have
\begin{eqnarray}
\lefteqn{
{\cal H} om_{{\cal O}_Z}(\Lambda_Z(L)_q,
\Omega_0(\t; Z)\otimes\det N)^{i,j}} \\
& = &
\bigoplus_{(\sigma,\rho)}\mathop{\rm Hom}\nolimits_{\bf Q}((L(\sigma)_A^{-j})_q,
\det(\rho)\otimes\det N)\otimes_{\bf Q}{\cal O}_{X(\rho)}\;,
\end{eqnarray}
where the sum is taken over all pairs $(\sigma,\rho)$ with
$\sigma\in\Delta$, $\rho\in\Delta(i)$ and $\sigma\prec\rho$.
We identify $\mathop{\rm Hom}\nolimits_{\bf Q}((L(\sigma)_A^{-j})_q,\det N_{\bf Q})$ with
$\d_N(L(\sigma)_A^{-j})_{-r-q}$ through the set of right
operations $\d_N^{\rm right}(L(\sigma)_A^{-j})_{-r-q}$.
Hence
\begin{eqnarray}
\lefteqn{
{\cal H} om_{{\cal O}_Z}(\Lambda_Z(L)_q,
\Omega_0(\t; Z)\otimes\det N)^k} \\
\label{Groth dual}
& = &
\bigoplus_{\rho\in\Delta}
\det(\rho)\otimes
\d_N((\i_\rho^*L^{-r_\rho+k})_A)_{-r-q}\otimes_{\bf Q}{\cal O}_{X(\rho)} \\
& = &
\bigoplus_{\rho\in\Delta}
\det(\rho)\otimes
(\d_\rho(\i_\rho^*L^{-r_\rho+k})_A)_{-r-q}\otimes_{\bf Q}{\cal O}_{X(\rho)} \\
& = &
\Lambda_Z({\bf D}(L))_{-r-q}^k\;.
\end{eqnarray}
The equality of the coboundary maps is also checked.
Namely, for $\rho,\mu\in\Delta$, the $(\rho,\mu)$-component
of $d^k$'s are nonzero only for (a)
$\rho\prec\mu$ and $r_\mu = r_\rho + 1$, or (b) $\rho =\mu$.
In case (a), they are both equal to the tensor product of
$q'_{\rho/\mu}$, inclusion map
\begin{equation}
\d_N((\i_\rho^*L^{-r_\rho+k})_A)_{-r-q}\rightarrow
\d_N((\i_\mu^*L^{-r_\rho+k})_A)_{-r-q}
\end{equation}
and the natural surjection ${\cal O}_{X(\rho)}\rightarrow{\cal O}_{X(\mu)}$
in the description (\ref{Groth dual}).
In case (b), they are both equal to
$(-1)^{k+1}{\rm id}\otimes\d_N(\i_\rho^*(d_L)^{k-1})\otimes{\rm id}$.
\QED
Let $S$ be a scheme of finite type over a field and let
$D_{\rm coh}^+(S)$ be the derived category of complexes
bounded below of ${\cal O}_S$-modules with coherent cohomologies.
The Grothendieck theory of residues and duality \cite{RD} says that
there exists an object of $D_{\rm coh}^+(S)$ which is called the
dualizing complex, and the Serre duality theorem for nonsingular
projective varieties is generalized to $S$ by using the the
dualizing complex in place of the canonical invertible sheaf.
For the toric variety $Z$, the dualizing complex which is
denoted by ${\cal C}^\bullet(Z,\Omega_0^\vee)$ in
\cite[\S5]{Ishida2} is described as follows.
For the compatibility with the notation of this paper,
we write it by ${\cal C}(Z,\Omega_0^\vee)^\bullet$.
For each integer $-r\leq i\leq 0$, we set
\begin{equation}
{\cal C}(Z,\Omega_0^\vee)^i :=
\bigoplus_{\sigma\in\Delta(r+i)}{\cal O}_{X(\sigma)}\otimes\det M[\sigma] =
\bigoplus_{\sigma\in\Delta(r+i)}\Omega_{X(\sigma)}(\log D(\sigma))^{-i}\;.
\end{equation}
For $\sigma,\tau\in\Delta$, the subvariety $X(\tau)$ of $Z$
is contained in $X(\sigma)$ if and only if $\sigma\prec\tau$.
When $\sigma\prec\tau$, let
$\varphi_{\sigma/\tau}\mathrel{:}{\cal O}_{X(\sigma)}\rightarrow{\cal O}_{X(\tau)}$ be the
natural surjection.
The component of the coboudary map
\begin{equation}
\begin{array}{ccc}
{\cal C}^i(Z,\Omega_0^\vee)
& \mathop{\longrightarrow}\limits^{\textstyle d}
& {\cal C}^{i+1}(Z,\Omega_0^\vee) \\
\| & & \| \\
\bigoplus_{\sigma\in\Delta(r+i)}{\cal O}_{X(\sigma)}\otimes\det M[\sigma]
& &
\bigoplus_{\tau\in\Delta(r+i+1)}{\cal O}_{X(\tau)}\otimes\det M[\tau]
\end{array}
\end{equation}
for $\sigma\in\Delta(r{+}i)$ and $\tau\in\Delta(r{+}i{+}1)$
is defined to be $\varphi_{\sigma/\tau}\otimes q_{\sigma/\tau}$
if $\sigma\prec\tau$ and the zero map otherwise, where
$\varphi_{\sigma/\tau}$ is the natural surjection
${\cal O}_{X(\sigma)}\rightarrow{\cal O}_{X(\tau)}$.
For the definition of $q_{\sigma/\tau}$, see \cite[\S1]{Ishida2}.
With respect to the identifications
$\det M[\sigma]\otimes\det N =\det(\sigma)$ and
$\det M[\tau]\otimes\det N =\det(\tau)$, the isomorphism
$q'_{\sigma/\tau}$ defined in \S2 is equal to
$q_{\sigma/\tau}\otimes 1_{\det N}$.
As a special case of \cite[Thm.5.4]{Ishida2}, the $d$-complex
${\cal C}(Z,\Omega_0^\vee)^\bullet$ is quasi-isomorphic to
the residual complex $f_Z^\Delta{\bf Q}$ \cite[VI,\S3]{RD} for the
structure morphism $f_Z\mathrel{:} Z\rightarrow\mathop{\rm Spec}\nolimits{\bf Q}$, i.e., it is a global
dualizing complex of the ${\bf Q}$-scheme $Z$.
For a finite complex $B^\bullet$ of ${\cal O}_Z$-modules with
coherent cohomology sheaves, the object
$R{\cal H} om(B, f_Z^\Delta{\bf Q})^\bullet$ in the derived category
$D_{\rm coh}^+(Z)$ is called the Grothendieck dual of
$B^\bullet$.
It is easy to see that $\Lambda_Z(\mathop{\rm ic}\nolimits_\t(\Delta))_0^\bullet$ is
isomorphic to $({\cal C}(Z,\Omega_0^\vee))\otimes(\det N))[-r]^\bullet$.
\begin{Cor}
\label{Cor 5.2}
For each integer $0\leq j\leq r$, the Grothendieck dual of the
$d$-complex $\Lambda_Z(L)_{-j}^\bullet$ is quasi-isomorphic to
$\Lambda_Z({\bf D}(L))_{j-r}[r]^\bullet$.
\end{Cor}
{\sl Proof.}\quad
Since $\Lambda_Z(L)_{-j}^\bullet$ is a direct sum of free
${\cal O}_{X(\sigma)}$-modules for $\sigma\in\Delta$, the dual
$R{\cal H} om(\Lambda_Z(L)_{-j},{\cal C}(Z,\Omega_0^\vee))^\bullet$
in the derived category $D_{\rm coh}^+(Z)$ is represented by the complex
${\cal H} om(\Lambda_Z(L)_{-j},{\cal C}(Z,\Omega_0^\vee))^\bullet$
by \cite[Lem.3.6]{Ishida2}.
Hence we get the corollary by Theorem~\ref{Thm 5.1}.
The following corollary follows from Corollaries~\ref{Cor 2.12} and
\ref{Cor 5.2}.
\begin{Cor}
\label{Cor 5.3}
For each integer $0\leq j\leq r$, the Grothendieck dual of the
$d$-complex $\Omega_j({\bf p}; Z)$ is quasi-isomorphic to
$\Omega_{r-j}(-{\bf p}; Z)[r]$.
\end{Cor}
We assume that $\Delta$ is a complete fan.
Then $Z$ is a complete variety.
Let $F^\bullet$ be a finite complex of coherent ${\cal O}_Z$-modules.
By the Grothendieck duality theorem \cite[VI,Thm.3.3]{RD} applied for the
proper morphism $f_Z\mathrel{:} Z\rightarrow\mathop{\rm Spec}\nolimits{\bf Q}$, we have a natural isomorphism
\begin{equation}
\H^i(Z, R{\cal H} om(F,{\cal C}(Z,\Omega_0^\vee))^\bullet)
\simeq\mathop{\rm Hom}\nolimits(\H^{-i}(Z,F^\bullet),{\bf Q})\;.
\end{equation}
\begin{Thm}
\label{Thm 5.4}
Assume that $\Delta$ is a complete fan.
Then the equality
\begin{equation}
\dim_{\bf Q}\H^i(Z,\Omega_j({\bf p}; Z)) =\dim_{\bf Q}\H^{r-i}(Z,\Omega_{r-j}(-{\bf p}; Z))
\end{equation}
holds for any integers $i, j$.
\end{Thm}
{\sl Proof.}\quad
By Corollary~\ref{Cor 5.2}, this is a consequence of the
Grothendieck-Serre duality theorem.
We can also prove the equality directly as follows.
We have equalities
\begin{equation}
\dim_{\bf Q}\H^i(Z,\Omega_j({\bf p}; Z)^\bullet) =
\dim_{\bf Q}\H^i(\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet)_{-j}
\end{equation}
and
\begin{equation}
\dim_{\bf Q}\H^{r-i}(Z,\Omega_{r-j}({\bf p}; Z)^\bullet) =
\dim_{\bf Q}\H^{r-i}(\Gamma(\mathop{\rm ic}\nolimits_{-{\bf p}}(\Delta)^\bullet)_{-r+j}
\end{equation}
by Lemma~\ref{Lem 3.6}.
Since $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ is quasi-isomorphic to
${\bf D}(\mathop{\rm ic}\nolimits_{-{\bf p}}(\Delta))^\bullet$ by Corollary~\ref{Cor 2.12},
we get the equality by Proposition~\ref{Prop 2.5}.
\QED
|
1994-03-11T06:06:53 | 9403 | alg-geom/9403009 | en | https://arxiv.org/abs/alg-geom/9403009 | [
"alg-geom",
"math.AG"
] | alg-geom/9403009 | Ishida | Masa-Nori Ishida | Torus embeddings and algebraic intersection complexes, II | 47 pages, latex | null | null | null | null | In the previous paper, we describe the intersection complexes of a toric
variety as a finite complex of graded exterior modules on the associated fan.
In this second part, we rewrite it explicitly by the barycentric subdivision of
the fan. We get the decomposition theorem of the intersection homologies for a
barycentric subdivision of a fan in the case of middle perversity. We get also
the diagonal theorems I and II. These theorems give a new proof of the
g-comnjecture on a simplicial polytope which was proved by R. Stanley.
| [
{
"version": "v1",
"created": "Fri, 11 Mar 1994 04:57:51 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ishida",
"Masa-Nori",
""
]
] | alg-geom | \section*{Introduction}
\setcounter{equation}{0}
The theory of toric varieties is based on the fact that
each toric variety of dimension $r$ has an associated fan,
i.e., a finite set of rational polyhedral cones in the
$r$-dimensional real space.
The theory of intersection homologies was introduced by Goresky
and MacPherson in \cite{GM1} and \cite{GM2}.
The intersection homologies are obtained by special complexes
of sheaves on the variety, which are called the intersection
complexes.
The intersection complex of a given variety has a variation
depend on a sequence of integers which is called a perversity
\cite[\S2]{GM2}.
It is known that the complex with the middle perversity is most
important for normal complex varieties.
The decomposition theorem and strong Lefschetz theorem
for the intersection homologies with the middle perversity
were proved in \cite{BBD}.
Since these theories applied for toric varieties are used in
the combinatorial problems (cf.\cite{Stanley2}), it was
expected that these theory restricted to toric varieties are
described and proved by an combinatorial method in terms of the
associated fans (cf.\cite{Oda3}).
In \cite{Ishida3}, we introduced the additive category
$\mathop{\rm GEM}\nolimits(\Delta)$ of graded exterior modules on a finite fan
$\Delta$.
We defined and constructed the intersection complex
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ as a finite complex in this category
for each perversity ${\bf p}$.
We defined a natural functor from the category of finite
complexes in $\mathop{\rm GEM}\nolimits(\Delta)$ to that of complexes of sheaves
on the toric variety associated to the fan.
It was shown that the intersection complex of the toric
variety is obtained by applying this functor to the
intersection complex on the fan with the corresponding
perversity.
The intersection homologies of the toric varieties
are also calculated by the complex $\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$,
where $\Gamma$ is an additive functor from $\mathop{\rm GEM}\nolimits(\Delta)$
to an abelian category of graded modules over an exterior
algebra.
In this article, we work only on the fan and the category
$\mathop{\rm CGEM}\nolimits(\Delta)$ of finite complexes in $\mathop{\rm GEM}\nolimits(\Delta)$.
By considering the barycentric subdivision algebra of
a finte fan $\Delta$, we define an object $\mathop{\rm SdP}\nolimits(\Delta)^\bullet$
of $\mathop{\rm CGEM}\nolimits(\Delta)$, and we show that the intersection complex of
the fan is obtained as a quotient of $\mathop{\rm SdP}\nolimits(\Delta)^\bullet$
for every perversity.
For a barycentric subdivision of a finite fan, we prove in
\S2 a decomposition theorem of the intersection complex with
the middle perversity.
In \S3, we study the case of simplicial fans.
We show that the intersection complexes of a simplicial fan
defined for any perversities between the top and the bottom
perversities are mutually quasi-isomorphic.
Some known vanishing theorems on the cohomologies of toric
varieties associated to a simplicial fan are proved in terms
of the category $\mathop{\rm CGEM}\nolimits(\Delta)$.
In \S4, we get the first and the second diagonal theorems as a
consequence of the decomposition theorem in \S2
and the results for simplicial fans in \S3.
Let $\H^i(\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet)_j$ be the
homogeneous degree $j$ part of the $i$-th cohomology of
$\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet$.
Then this is a finite dimensional ${\bf Q}$-vector space
and $\H^i(\Gamma(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet)_j =\{0\}$ for
$(i, j)\not\in [0, r]\times[-r, 0]$.
The first diagonal theorem says that, if $\Delta$ is a
complete fan and ${\bf p}$ is the middle perversity, then
this vector space is zero unless $i = j + r$.
The second diagonal theorem is for a cone of dimension $r$.
The strong Lefschets theorem used by Stanley \cite{Stanley1}
in the proof of $g$-conjecture on the number of faces of a
simplicial convex polytope can be replaced by this diagonal
theorem (cf.\cite[Cor.4.5]{Oda3}).
\section*{Notation}
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In the first part \cite{Ishida3} of this series of papers, the
coboundary maps of complexes are denoted by $d$ or $\partial$, and
called ``$d$-complexes'' and ``$\partial$-complexes'', respectively.
Where $\partial$ was used to represent the exterior derivatives
of logarithmic de Rham complexes and their direct sums.
In this paper, we need not use $\partial$-complexes
since we avoid to work on varieties.
Hence the $d$-complexes are simply called complexes in
this article.
The notation $E^\bullet$ means that $E$ is a complex and
its component of degree $i$ is $E^i$.
When we apply to it a functor $F$ to a category of complexes,
we usually denote by $F(E)^\bullet$ the resulting complex.
When $E^\bullet$ is a complex in an abelian category, its
$i$-the cohomology is denoted by $\H^i(E^\bullet)$.
Neither derived categories nor derived functors appear in
this article.
\section{Barycentric subdivision algebras and resolutions}
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Let $N$ be a free ${\bf Z}$-module of a fixed finite rank $r\geq 0$.
By {\em cones} in $N_{\bf R} := N\otimes_{\bf Z}{\bf R}$, we always
mean strongly convex rational polyhedral cones
(cf.\cite[Chap.1,1.1]{Oda1}).
We denote by ${\bf 0}$ the trivial cone $\{0\}$.
For a cone $\sigma$, we set $r_\sigma :=\dim\sigma$
and $N(\sigma) := N\cap(\sigma +(-\sigma))$.
Since $\sigma$ is a rational cone, $N(\sigma)$ is a
free ${\bf Z}$-module of rank $r_\sigma$.
For each cone $\sigma$ in $N_{\bf R}$, we set
\begin{equation}
\det(\sigma) :=\bigwedge^{r_\sigma}N(\sigma)\simeq{\bf Z}\;.
\end{equation}
For cones $\sigma,\tau$ in $N_{\bf R}$ with $\sigma\prec\tau$
and $r_\tau - r_\sigma = 1$, we define the {\em incidence
isomorphism}
$q'_{\sigma/\tau}\mathrel{:}\det(\sigma)\rightarrow\det(\tau)$
as follows.
By the short exact sequence
\begin{equation}
0\longrightarrow N(\sigma)\longrightarrow N(\tau)\longrightarrow N(\tau)/N(\sigma)\longrightarrow 0
\end{equation}
of free ${\bf Z}$-modules, we get an isomorphism
\begin{equation}
\label{zzzq1}
(N(\tau)/N(\sigma))\otimes\det(\sigma)\simeq\det(\tau)\;.
\end{equation}
Since $\sigma$ is a codimension one face of $\tau$,
the image of $\tau$ in $N(\tau)_{\bf R}/N(\sigma)_{\bf R}$ is a
half line with the edge $0$.
We define the isomorphism
\begin{equation}
\label{zzzq2}
\det(\sigma)\simeq(N(\tau)/N(\sigma))\otimes\det(\sigma)
\end{equation}
by the isomorphism ${\bf Z}\rightarrow N(\tau)/N(\sigma)$ such that $1$
is mapped into the image of $\tau$ in $N(\tau)_{\bf R}/N(\sigma)_{\bf R}$.
The isomorphism $q'_{\sigma/\tau}$ is defined to be the
composite of (\ref{zzzq2}) and (\ref{zzzq1}).
Namely, if the class $\bar a$ of $a\in N\cap\tau$ in
$N(\tau)/N(\sigma)$ is a generator, then
$q'_{\sigma/\tau}(w) = a\wedge w$ for $w\in\det(\sigma)$.
We get the following lemma (cf.\cite[Lem.1.4]{Ishida1}).
\begin{Lem}
\label{Lem 1.1}
Let $\sigma,\rho$ be cones in $N_{\bf R}$ with $\sigma\prec\rho$
and $r_\rho - r_\sigma = 2$.
Then there exist exactly two cones $\tau$ with
$\sigma\prec\tau\prec\rho$ and $r_\tau = r_\sigma + 1$.
Let $\tau_1,\tau_2$ be the two cones.
Then the equality
\begin{equation}
q'_{\tau_1/\rho}\cdot q'_{\sigma/\tau_1} +
q'_{\tau_2/\rho}\cdot q'_{\sigma/\tau_2} = 0
\end{equation}
holds.
\end{Lem}
{\sl Proof.}\quad
The first assertion is a consequence of the fact that
any two-dimensional cone has exactly two edges
(cf.\cite[Prop.1.3]{Ishida1}).
By definiton, both $q'_{\tau_1/\rho}\cdot q'_{\sigma/\tau_1}$
and $q'_{\tau_2/\rho}\cdot q'_{\sigma/\tau_2}$ are
isomorphisms $\det(\sigma)\simeq\det(\rho)$ of the free
${\bf Z}$-modules of rank one.
Hence it is sufficient to show that these two isomorphisms
have mutually opposite signs.
Take elements $a\in(\tau_1\setminus\sigma)\cap N$,
$b\in(\tau_2\setminus\sigma)\cap N$ and
$x\in\det(\sigma)\setminus\{0\}$.
Then $(q'_{\tau_1/\rho}\cdot q'_{\sigma/\tau_1})(x)$
has the same sign with $b\wedge a\wedge x$, while
$(q'_{\tau_2/\rho}\cdot q'_{\sigma/\tau_2})(x)$
has the same sign with $a\wedge b\wedge x$.
Hence they have distinct signs.
\QED
Let $\sigma$ and $\tau$ be nonsingular cones generated by
$\{n_1,\cdots, n_s\}$ and $\{n_1,\cdots, n_{s+1}\}$, respectively,
for a ${\bf Z}$-basis $\{n_1,\cdots, n_r\}$ of $N$ and for an integr
$0\leq s < r$ (cf.\cite[Thm.1.10]{Oda1}).
Then, it is easy to see that
\begin{equation}
q'_{\sigma/\tau}(n_1\wedge\cdots\wedge n_s) =
n_{s+1}\wedge n_1\wedge\cdots\wedge n_s =
(-1)^sn_1\wedge\cdots\wedge n_s\wedge n_{s+1}\;.
\end{equation}
If $\sigma$ and $\tau$ are simplicial cones generated
by $\{n_1,\cdots, n_s\}$ and $\{n_1,\cdots, n_{s+1}\}$,
respectively, for a ${\bf Q}$-basis $\{n_1,\cdots, n_r\}$ of $N_{\bf Q}$,
then
\begin{equation}
q'_{\sigma/\tau}(n_1\wedge\cdots\wedge n_s) =
a\cdot n_{s+1}\wedge n_1\wedge\cdots\wedge n_s
\end{equation}
for a positive rational number $a$, where we denote by
the same symbol $q'_{\sigma/\tau}$ its extension to
$\det(\sigma)\otimes{\bf Q}\rightarrow\det(\tau)\otimes{\bf Q}$.
For a cone $\pi$ in $N_{\bf R}$, we denote by $F(\pi)$ the set
of faces of $\pi$.
The zero cone ${\bf 0}$ and $\pi$ itself are elements of
$F(\pi)$.
For cones $\eta,\pi$ of $N_{\bf R}$ with $\eta\prec\pi$, we define
the ``closed interval''
\begin{equation}
F[\eta,\pi] :=\{\sigma\in F(\pi)\mathrel{;}\eta\prec\sigma\}
\end{equation}
and the ``open interval''
\begin{equation}
F(\eta,\pi) := F[\eta,\pi]\setminus\{\eta,\pi\}\;.
\end{equation}
We also use the notation
\begin{equation}
F[\eta,\pi) := F[\eta,\pi]\setminus\{\pi\}\;.
\end{equation}
Let $\Delta$ be a finite fan \cite[1.1]{Oda1}.
We call a subset $\Phi\subset\Delta$ {\em star closed} if
$\sigma\in\Phi$ and $\sigma\prec\rho\in\Delta$ imply
$\rho\in\Phi$.
We call $\Phi$ {\em locally star closed} if
$\sigma,\rho\in\Phi$ and $\sigma\prec\rho$
imply $F[\sigma,\rho]\subset\Phi$.
Let $\Phi$ be a locally star closed subset of a finite
fan $\Delta$.
We define a finite complex $E(\Phi,{\bf Z})^\bullet$ of free
${\bf Z}$-modules as follows.
For each integer $i$, we set
\begin{equation}
E(\Phi,{\bf Z})^i :=\bigoplus_{\sigma\in\Phi(i)}\det(\sigma)\;,
\end{equation}
where $\Phi(i) :=\{\sigma\in\Phi\mathrel{;} r_\sigma = i\}$.
For $\sigma\in\Phi(i)$ and $\tau\in\Phi(i+1)$, the
$(\sigma,\tau)$-component of the coboundary map
\begin{equation}
d^i\mathrel{:} E(\Phi,{\bf Z})^i\longrightarrow E(\Phi,{\bf Z})^{i+1}
\end{equation}
is defined to be $q'_{\sigma/\tau}$ if $\sigma\prec\tau$
and the zero map otherwise.
We have $d^{i+1}\cdot d^i = 0$ for every $i$ by
Lemma~\ref{Lem 1.1}.
Here we explain the relation of this complex
with the similar complex $C^\bullet(\Phi,{\bf Z})$ in
\cite{Ishida1}.
Let $M$ be the dual ${\bf Z}$-module of $N$ and let
$M_{\bf R} := M\otimes_{\bf Z}{\bf R}$.
For a cone $\sigma$ in $N_{\bf R}$, we set
$\sigma^\perp :=\{x\in M_{\bf R}\mathrel{;}\langle x, a\rangle = 0,\forall a\in\sigma\}$,
where $\langle\; ,\;\rangle\mathrel{:} M_{\bf R}\times N_{\bf R}\rightarrow{\bf R}$ is the natural
pairing.
For a cone $\sigma$ in $N_{\bf R}$, we set
$M[\sigma] := M\cap\sigma^\perp$, which is a free ${\bf Z}$-module
of rank $r - r_\sigma$.
We set ${\bf Z}(\sigma) :=\bigwedge^{r - r_\sigma}M[\sigma]$.
For cones $\sigma,\tau$ with $\sigma\prec\tau$ and
$r_\tau = r_\sigma + 1$, the isomorphism
\begin{equation}
q_{\sigma/\tau}\mathrel{:}{\bf Z}(\sigma)\longrightarrow{\bf Z}(\tau)
\end{equation}
is defined as follows.
Let $p := r - r_\sigma$.
Then $M[\sigma]$ and $M[\tau]$ are free ${\bf Z}$-modules of
rank $p$ and $p-1$, respectively.
We take an element $n_1$ in $N$ such that the homomorphism
$\langle\;, n_1\rangle\mathrel{:} M[\sigma]\rightarrow{\bf Z}$ is zero on the submodule
$M[\tau]$ and maps $M[\sigma]\cap\tau^\vee$ onto
$\{c\in{\bf Z}\mathrel{;} c\geq 0\}$.
Then we define
\begin{equation}
q_{\sigma/\tau}(m_1\wedge\cdots\wedge m_p) :=
\langle m_1, n_1\rangle(m_2\wedge\cdots\wedge m_p)
\end{equation}
for $m_1\in M[\sigma]$ and $m_2,\cdots, m_p\in M[\tau]$.
This definition does not depend on the choice of $n_1$.
$C^i(\Phi,{\bf Z})$ was defined to be the direct sum
$\bigoplus_{\sigma\in\Phi(i)}{\bf Z}(\sigma)$ and the
$(\sigma,\tau)$-component of
$d^i\mathrel{:} C^i(\Phi,{\bf Z})\rightarrow C^{i+1}(\Phi,{\bf Z})$ was defined
to be $q_{\sigma/\tau}$ for each $(\sigma,\tau)$ with
$\sigma\prec\tau$.
For each cone $\sigma$, we define an identification
$\det(\sigma) ={\bf Z}(\sigma)\otimes\bigwedge^r N$ as follows.
Let $\{n_1,\cdots, n_r\}$ and $\{m_1,\cdots, m_r\}$ be
mutually dual ${\bf Z}$-basis of $N$ and $M$ such that
$M[\sigma]$ is generated by $\{m_1,\cdots, m_p\}$
and $N(\sigma)$ is generated by $\{n_{p+1},\cdots, n_r\}$.
Then we identify the generator
$(m_p\wedge\cdots\wedge m_1)\otimes(n_1\wedge\cdots\wedge n_r)$
of ${\bf Z}(\sigma)\otimes\bigwedge^rN$ with the generator
$n_{p+1}\wedge\cdots\wedge n_r$ of $\det(\sigma)$.
It is easy to see that $q'_{\sigma/\tau}$ and $q_{\sigma/\tau}$
are compatible with respect to this identification.
In particular, we have
$E(\Phi,{\bf Z})^\bullet = C^\bullet(\Phi,{\bf Z})\otimes\bigwedge^rN$.
By this observation, we interpret some elementary
vanishing lemmas in \cite{Ishida1} and \cite{Ishida2}
to the following lemma.
Topologically, these are due to the contractability of convex sets.
\begin{Lem}
\label{Lem 1.2}
{\rm (1)} Let $\rho$ be a cone of $N_{\bf R}$.
If $\rho\not={\bf 0}$, then all the cohomologies of the
complex $E(F(\rho),{\bf Z})^\bullet$ are zero.
{\rm (2)} Let $\Delta$ be a finite fan such that the
support $\rho :=|\Delta|$ is a cone.
Let $\eta$ be an element of $\Delta$ and let
$\Phi :=\{\sigma\in\Delta\mathrel{;}
\eta\prec\sigma,\sigma\cap\mathop{\rm rel.\,int}\nolimits\rho\not=\emptyset\}$,
where $\mathop{\rm rel.\,int}\nolimits\rho$ is the interior of $\rho$ in $N(\rho)_{\bf R}$.
Then
\begin{equation}
\H^i(E(\Phi,{\bf Z})^\bullet) =\left\{
\begin{array}{ll}
\{0\} & \hbox{ for } i\not= r_\rho \\
\det(\rho) & \hbox{ for } i = r_\rho
\end{array}
\right.\;.
\end{equation}
\end{Lem}
{\sl Proof.}\quad
(1) is a special case of \cite[Prop.2.3]{Ishida1}
as it is mensioned in the article after the proposition.
For (2), let $N' := N/N(\eta)$ and let $C$ be the image of
$\rho$ in $N'_{\bf R}$.
Then $C$ is a rational polyhedral cone which is
not necessary strongly convex.
Set $\Delta(\eta{\prec}) :=\{\sigma\mathrel{;}\sigma\in\Delta,\eta\prec\sigma\}$.
For each $\sigma\in\Delta(\eta{\prec})$, let $\sigma[\eta]$
be the image of $\sigma$ in $N'_{\bf R}$.
Then $\Delta[\eta] :=\{\sigma[\eta]\mathrel{;}\sigma\in\Delta(\eta{\prec})\}$
is a fan of $N'_{\bf R}$ with the support $C$.
We set $\Psi :=\{\sigma[\eta]\in\Delta[\eta]\mathrel{;}
\sigma[\eta]\cap\mathop{\rm rel.\,int}\nolimits C\not=\emptyset\}$.
Since
$\sigma[\eta]\cap\mathop{\rm rel.\,int}\nolimits C\not=\emptyset$ if and only if
$\sigma\in\Phi$, the complex $E(\Phi,{\bf Z})^\bullet$ is naturally
isomorphic to $E(\Psi,{\bf Z})[-r_\eta]^\bullet\otimes\det(\eta)$.
Then the lemma is a consequence of \cite[Lem.1.6]{Ishida2}
applied for $\pi = C$ and $\Sigma =\Delta[\eta]$.
Note that $C^i(\Phi,{\bf Z}_{1,0})$ in \cite{Ishida2} is equal to
$C^{i+r}(\Phi,{\bf Z})$ for every $i$ and the coboundary
map is equal to that of $C^\bullet(\Phi,{\bf Z})$.
Hence $\H^i(C^\bullet(\Phi,{\bf Z}_{1,0}))$ is equal to
$\H^{i+r}(C^\bullet(\Phi,{\bf Z}))$.
(In the statement of \cite[Lem.1.6]{Ishida2},
``$\pi\subset|\Sigma|\pi+(-\pi)$'' is a misprint of
``$\pi\subset|\Sigma|\subset\pi+(-\pi)$''.)
\QED
As in \cite[\S1]{Ishida3}, we denote by $A$ the exterior algebra
$\bigwedge^\bullet N_{\bf Q}$ and define the grading of $A$ by
$A_i :=\bigwedge^{-i}N_{\bf Q}$ for $i\in{\bf Z}$, where the indexes are
written as subscripts.
For a cone $\rho$, the subalgebra $\bigwedge^\bullet N(\rho)_{\bf Q}$
of $A$ is denoted by $A(\rho)$.
For a graded ${\bf Q}$-subalgebra $C\subset A$, we denote by
$\mathop{\rm GM}\nolimits(C)$ the abelian category of finitely generated
left $A$-modules.
We denote by $\mathop{\rm CGM}\nolimits(C)$ the abelian category of finite
complexes in $\mathop{\rm GM}\nolimits(C)$.
We do not take the quotient by homotopy equivalences in the
definition of morphisms in $\mathop{\rm CGM}\nolimits(C)$ in order to keep the
explicitness of the theory.
We write the complex degree by a superscript and the
graded module degree by a subscript.
Let $\Delta$ be a finite fan as before.
We consider the additive categories $\mathop{\rm GEM}\nolimits(\Delta)$ and
$\mathop{\rm CGEM}\nolimits(\Delta)$ as in \cite{Ishida3}.
Namely, an object $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ has the following
structure.
(1) $L$ is a finite dimensional ${\bf Q}$-vector space with the
decomposition
\begin{equation}
L =\bigoplus_{\sigma\in\Delta}\;\bigoplus_{i,j\in{\bf Z}}%
L(\sigma)_j^i\;.
\end{equation}
(2) For each $\sigma\in\Delta$ and for each $i\in{\bf Z}$,
$L(\sigma)^i :=\bigoplus_{j\in{\bf Z}}L(\sigma)_j^i$ is in the
category $\mathop{\rm GM}\nolimits(A(\sigma))$.
(3) For each $\sigma\in\Delta$,
$L(\sigma)^\bullet :=\bigoplus_{i\in{\bf Z}}L(\sigma)^i$ is in the
category $\mathop{\rm CGM}\nolimits(A(\sigma))$.
The coboundary map is denoted by $d_L(\sigma/\sigma)$.
(4) For each pair $(\sigma,\tau)$ of distinct cones in $\Delta$
with $\sigma\prec\tau$ and for each $i\in{\bf Z}$, a homomorphism
$d_L^i(\sigma/\tau)\mathrel{:} L(\sigma)^i\rightarrow L(\tau)[1]^i = L(\tau)^{i+1}$
in $\mathop{\rm GM}\nolimits(A(\sigma))$ is given, where we consider
$L(\tau)[1]^i\in\mathop{\rm GM}\nolimits(A(\sigma))$ by the inclusion
$A(\sigma)\subset A(\tau)$.
(5) For each pair $(\sigma,\rho)$ of cones in $\Delta$ with
$\sigma\prec\rho$ and for each integer $i$, the equality
\begin{equation}
\label{eq in (5)}
\sum_{\tau\in F[\sigma,\rho]}
d_L^{i+1}(\tau/\rho)\cdot d_L^i(\sigma/\tau) = 0
\end{equation}
holds.
We call $\dim_{\bf Q} L$ the {\em total dimension} of $L^\bullet$.
Note that the homomoprhism
$d_L(\sigma/\tau)\mathrel{:} L(\sigma)^\bullet\rightarrow L(\tau)[1]^\bullet$
in (4) is not a homomorhism of complexes in general, i.e.,
it may not commute with the coboundary maps.
However, the condition (5) implies that this is a homomorphism
of complexes if $r_\tau - r_\sigma = 1$, since then
$F[\sigma,\rho] =\{\sigma,\rho\}$.
A homomorphism $f\mathrel{:} L^\bullet\rightarrow K^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$
consists of $\{f^i\mathrel{:} L^i\rightarrow K^i\mathrel{;} i\in{\bf Z}\}$ which is
compatible with $d_L$ and $d_K$.
A homomorphism $f$ is said to be {\em unmixed} if
$f(\sigma/\tau)\mathrel{:} L(\sigma)^\bullet\rightarrow K(\tau)^\bullet$
is a zero map whenever $\sigma\not=\tau$.
When $f$ is unmixed, the homomorphism of complexes
$f(\sigma/\sigma)\mathrel{:} L(\sigma)^\bullet\rightarrow K(\sigma)^\bullet$
is denoted simply by $f(\sigma)$ for each $\sigma\in\Delta$.
An ummixed homomorphism has the kernel and the cokernel
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
For a finite fan $\Delta$, an objects
$\P(\Delta)^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ is defined as follows.
For each $\sigma\in\Delta$, we define
$\P(\Delta)(\sigma)^\bullet :=
(\det(\sigma)\otimes A(\sigma))[-r_\sigma]^\bullet$, i.e.,
\begin{equation}
\P(\Delta)(\sigma)^i :=\left\{
\begin{array}{ll}
\det(\sigma)\otimes A(\sigma) & \hbox{ if } i = r_\sigma \\
\{0\} & \hbox{ if } i\not= r_\sigma
\end{array}
\right.\;.
\end{equation}
For $\sigma\in\Delta(i)$ and $\rho\in\Delta(i+1)$ with
$\sigma\prec\rho$, the $A(\sigma)$-homomorphism
$d^i(\sigma/\rho)\mathrel{:}\P(\Delta)(\sigma)^i\rightarrow\P(\Delta)(\rho)^{i+1}$
is defined by the homomorphism
\begin{equation}
q'_{\sigma/\rho}\otimes\lambda_{\sigma/\rho}\mathrel{:}
\det(\sigma)\otimes A(\sigma)\longrightarrow \det(\rho)\otimes A(\rho)\;,
\end{equation}
where $\lambda_{\sigma/\rho}\mathrel{:} A(\sigma)\rightarrow A(\rho)$
is the natural inclusion map.
The equality (\ref{eq in (5)}) follows from
Lemma~\ref{Lem 1.1}.
As we see later, there exists an unmixed surjection
$\P(\Delta)^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$ to the
intersection complex \cite[Thm.2.9]{Ishida3}
for the top perversity $\t$.
However, for a general perversity, such a description of the
intersection complex in terms of $\P(\Delta)^\bullet$ is not
possible.
We make it possible by replacing $\P(\Delta)^\bullet$ by its
{\em barycentric resolution} $\mathop{\rm SdP}\nolimits(\Delta)^\bullet$.
For a finite set $\Phi$ of nontrivial cones of $N_{\bf R}$,
we define the {\em barycentric subdivision algebra} $B(\Phi)$
as follows.
We take an indeterminate $y(\sigma)$ for each $\sigma\in\Phi$,
and denote by $W(\Phi)$ the ${\bf Q}$-vector space with the basis
$\{y(\sigma)\mathrel{;}\sigma\in\Phi\}$.
The graded ${\bf Q}$-algebra $B(\Phi)$ is defined to be the quotient
of the exterior ${\bf Q}$-algebra $\bigwedge^\bullet W(\Phi)$ by
the two-sided ideal generated by
\begin{equation}
\{y(\sigma)\wedge y(\rho)\mathrel{;}\sigma,\rho\in\Phi\hbox{ such that }
\sigma\not\prec\rho\hbox{ and }\rho\not\prec\sigma\}\;.
\end{equation}
The grading $B(\Phi) =\bigoplus_{i=0}^\infty B(\Phi)^i$ is induced
by that of $\bigwedge^\bullet W(\Phi)$.
Note that the indexes of $B(\Phi)$ are given as superscripts.
We denote also by $y(\sigma)$ its image in $B(\Phi)^1$.
We denote by $u\cdot v$ or $uv$ the multiplicaiton
of $u, v$ in $B(\Phi)$.
We denote by $\mathop{\rm Sd}\nolimits(\Phi)$ the set of sequences
$(\sigma_1,\cdots,\sigma_k)$ of distinct cones
$\sigma_1,\cdots,\sigma_k$ with
$\sigma_1\prec\cdots\prec\sigma_k$ including the case of
length zero.
For $\alpha = (\sigma_1,\cdots,\sigma_k)$ in
$\mathop{\rm Sd}\nolimits(\Phi)$, we define $\max(\alpha) :=\sigma_k$
if $k > 0$ and $\max(\alpha) :={\bf 0}$ if $k = 0$.
For each nonnegative integer $k$, we denote by $\mathop{\rm Sd}\nolimits_k(\Phi)$
the subset of $\mathop{\rm Sd}\nolimits(\Phi)$ consisting of the sequences of length $k$.
By the assumption ${\bf 0}\not\in\Phi$, we have
$\mathop{\rm Sd}\nolimits_k(\Phi) =\emptyset$ for $k > r$.
For each $\alpha = (\sigma_1,\cdots,\sigma_k)$, we set
$z(\alpha) :=
y(\sigma_1)\cdots y(\sigma_k)\in B(\Phi)^k$,
where we understand $z(\alpha) = 1$ if $k = 0$.
The following lemma is clear by the definition of $B(\Phi)$.
\begin{Lem}
\label{Lem 1.3}
For each integer $0\leq i\leq r$, the ${\bf Q}$-vector space
$B(\Phi)^i$ has the basis
\begin{equation}
\{z(\alpha)\mathrel{;}\alpha\in\mathop{\rm Sd}\nolimits_i(\Phi)\}\;,
\end{equation}
while $B(\Phi)^i =\{0\}$ for $i > r$.
\end{Lem}
Set $Y(\Phi) :=\sum_{\sigma\in\Phi}y(\sigma)\in B(\Phi)^1$.
By defining the coboundary map $d^i\mathrel{:} B(\Phi)^i\rightarrow B(\Phi)^{i+1}$
to be the multiplication of $Y(\Phi)$ to the left,
we regard $B(\Phi)^\bullet$ as a complex of ${\bf Q}$-vector spaces.
Let $\Delta$ be a finite fan of $N_{\bf R}$.
We consider the barycentric subdivision algebra
$B(\Delta{\setminus}\{{\bf 0}\})$.
For each $\rho\in\Delta\setminus\{{\bf 0}\}$,
$B(F({\bf 0},\rho))$ is a graded subalgebra of
$B(\Delta{\setminus}\{{\bf 0}\})$.
We set
$B(\rho) := B(F({\bf 0},\rho))y(\rho)\subset B(\Delta{\setminus}\{{\bf 0}\})$
for each $\rho\in\Delta{\setminus}\{{\bf 0}\}$.
By Lemma~\ref{Lem 1.3}, $B(\rho)$ is a ${\bf Q}$-vector space
with the basis
$\{z(\alpha)y(\rho)\mathrel{;}\alpha\in\mathop{\rm Sd}\nolimits(F({\bf 0},\rho))\}$.
We regard $B(\rho)^\bullet$ a complex by defining
$B(\rho)^i := B(F({\bf 0},\rho))^{i-1}y(\rho)$ for $i\in{\bf Z}$
and defining the coboundary map to be the multiplication of
$Y(F({\bf 0},\rho))$ to the left.
Note that $B(\rho)^i\not=\{0\}$ only for $1\leq i\leq r_\rho$,
where $r_\rho :=\dim\rho$.
For the zero cone ${\bf 0}$, let $B({\bf 0})^\bullet$ be the
complex defined by $B({\bf 0})^0 :={\bf Q}$ and $B({\bf 0})^i :=\{0\}$
for $i\not= 0$.
For each $\beta\in\mathop{\rm Sd}\nolimits(\Delta{\setminus}\{{\bf 0}\})$,
$z(\beta)$ is in $B(\rho)$ if
and only if $\max(\beta) =\rho$.
If $\rho\not={\bf 0}$, then $\max(\beta) =\rho$ means
that $z(\beta) = z(\alpha)y(\rho)$ for some
$\alpha\in\mathop{\rm Sd}\nolimits(F({\bf 0},\rho))$.
Hence we get the decomposition
\begin{equation}
\label{zzdecomp}
B(\Delta{\setminus}\{{\bf 0}\}) =
\bigoplus_{\rho\in\Delta}B(\rho)
\end{equation}
as a ${\bf Q}$-vector space by Lemma~\ref{Lem 1.3}.
However this is not a direct sum of the complexes.
We introduce a decreasing filtration $\{F^k(B(\Delta{\setminus}\{{\bf 0}\}))\}$ by
\begin{equation}
F^k(B(\Delta{\setminus}\{{\bf 0}\})) :=
\bigoplus_{i=k}^r\bigoplus_{\rho\in\Delta(i)}B(\rho)\;.
\end{equation}
Then it is easy to see that each $F^k(B(\Delta{\setminus}\{{\bf 0}\}))$ is a
subcomplex of $B(\Delta{\setminus}\{{\bf 0}\})^\bullet$ for each $k$, and
\begin{equation}
F^k(B(\Delta{\setminus}\{{\bf 0}\}))^\bullet/%
F^{k+1}(B(\Delta{\setminus}\{{\bf 0}\}))^\bullet =
\bigoplus_{\rho\in\Delta(k)}B(\rho)^\bullet
\end{equation}
as complexes.
For $\rho,\mu\in\Delta$, with $\rho\prec\mu$ and $\rho\not=\mu$,
we define a ${\bf Q}$-linear map
\begin{equation}
\label{zzzmult}
\varphi_{\rho/\mu}\mathrel{:} B(\rho)\longrightarrow B(\mu)
\end{equation}
to be the multiplication of $y(\mu)$ to the left, i.e.,
$w\mapsto y(\mu)w$.
If $r_\mu - r_\rho = 1$, then $\varphi_{\rho/\mu}$ is
a homomorphism $B(\rho)^\bullet\rightarrow B(\mu)[1]^\bullet$
of complexes, however it is not the case if $r_\mu - r_\rho > 1$.
We consider the complex of $A(\rho)$-modules
\begin{equation}
(B(\rho)\otimes_{\bf Q} A(\rho))^\bullet :=
B(\rho)^\bullet\otimes_{\bf Q} A(\rho)\;.
\end{equation}
By definition, $B(\rho)^i\otimes_{\bf Q} A(\rho)_j\not=\{0\}$ only for
$1\leq i\leq r_\rho$ and $-r_\rho\leq j\leq 0$, if $\rho\not={\bf 0}$.
The barycentric resolution $\mathop{\rm SdP}\nolimits(\Delta)^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$
of $\P(\Delta)^\bullet$ is defined as follows.
For each $\rho\in\Delta$, we set
\begin{equation}
\mathop{\rm SdP}\nolimits(\Delta)(\rho)^\bullet :=
(B(\rho)\otimes_{\bf Q} A(\rho))^\bullet\;.
\end{equation}
In particular,
\begin{equation}
\mathop{\rm SdP}\nolimits(\Delta)(\rho)^i =
\bigoplus_{\alpha\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\rho))}%
({\bf Z} z(\alpha)y(\rho))\otimes A(\rho)
\end{equation}
for each $1\leq i\leq r_\rho$ if $\rho\not={\bf 0}$, while
$\mathop{\rm SdP}\nolimits(\Delta)({\bf 0})^0 ={\bf Q}$ and $\mathop{\rm SdP}\nolimits(\Delta)({\bf 0})^i =\{0\}$
for $i\not= 0$.
For $\rho,\mu\in\Delta$ with $\rho\prec\mu$ and $\rho\not=\mu$,
the $(\rho,\mu)$-component
\begin{equation}
d_{\mathop{\rm SdP}\nolimits(\Delta)}(\rho/\mu)\mathrel{:}
\mathop{\rm SdP}\nolimits(\Delta)(\rho)^\bullet\longrightarrow\mathop{\rm SdP}\nolimits(\Delta)(\mu)[1]^\bullet
\end{equation}
of the coboundary map of $\mathop{\rm SdP}\nolimits(\Delta)^\bullet$ is defined to
be the tensor product of $\varphi_{\rho/\mu}\mathrel{:} B(\rho)\rightarrow B(\mu)$
defined at (\ref{zzzmult}) and the natural inclusion map
$A(\rho)\rightarrow A(\mu)$.
Note that this is not a homomorphism of complexes, if
$r_\mu - r_\rho > 1$.
In order to check the equality (\ref{eq in (5)}) for
$\mathop{\rm SdP}\nolimits(\Delta)^\bullet$, it is sufficient to show the
equality
\begin{equation}
\sum_{\tau\in F[\rho,\mu]}\varphi_{\tau/\mu}\cdot\varphi_{\rho/\tau}
= 0\;,
\end{equation}
where $\varphi_{\rho/\rho}$ and $\varphi_{\mu/\mu}$ are the
multiplication of $Y(F({\bf 0},\rho))$ and $Y(F({\bf 0},\mu))$
to the left, respectively.
This equality follows from the equality
\begin{equation}
(\;y(\mu)Y(F({\bf 0},\rho)) + Y(F({\bf 0},\mu))y(\mu) +
\sum_{\tau\in F(\rho,\mu)}y(\mu)y(\tau)\;)\;y(\rho) = 0
\end{equation}
in $B(\Delta{\setminus}\{{\bf 0}\})$ which is checked easily.
For each $\rho\in\Delta$, we define a covariant functor
\begin{equation}
\i_\rho^\circ\mathrel{:}\mathop{\rm CGEM}\nolimits(\Delta)\longrightarrow\mathop{\rm CGM}\nolimits(A(\rho))
\end{equation}
as follows.
For $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ and for each $i\in{\bf Z}$,
we set
\begin{equation}
\i_\rho^\circ(L)^i :=
\bigoplus_{\sigma\in F[{\bf 0},\rho)}L(\sigma)_{A(\rho)}^i\;.
\end{equation}
For $\sigma,\tau\in F[{\bf 0},\rho)$ with $\sigma\prec\tau$,
the $(\sigma,\tau)$-component of
$d^i\mathrel{:}\i_\rho^\circ(L)^i\rightarrow\i_\rho^\circ(L)^{i+1}$ is defined
to be the $A(\rho)$-homomorphism induced by $d_L^i(\sigma/\tau)$.
Recall that a similar functor $\i_\rho^*$ was defined in
\cite[\S2]{Ishida3}.
We get the definition of $\i_\rho^*$ by replacing
$F[{\bf 0},\rho)$ in the definition of $\i_\rho^\circ$
by $F[{\bf 0},\rho] = F(\rho)$.
For $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ and $\rho\in\Delta$, we get
an exact sequence
\begin{equation}
0\longrightarrow L(\rho)^\bullet\longrightarrow\i_\rho^*(L)^\bullet
\longrightarrow\i_\rho^\circ(L)^\bullet\longrightarrow 0
\end{equation}
in $\mathop{\rm CGM}\nolimits(A(\rho))$.
Since $\i_\rho^*(L)^i = L(\rho)^i\oplus\i_\rho^\circ(L)^i$
for each integer $i$, $\i_\rho^*(L)^\bullet$ is equal to
the mapping cone of a homomoprhism
\begin{equation}
\phi(L,\rho)\mathrel{:}\i_\rho^\circ(L)^\bullet\longrightarrow L(\rho)[1]^\bullet
\end{equation}
of complexes.
The restriction of $\phi(L,\rho)^i$ to the component
$L(\sigma)_{A(\rho)}^i$ is the $A(\rho)$-homomorphism
induced by $d_L^i(\sigma/\rho)$ for each $\sigma$.
Let $f\mathrel{:} L^\bullet\rightarrow K^\bullet$ be a homomorphism
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
For $\rho\in\Delta$, consider the following diagram
\begin{equation}
\begin{array}{ccc}
\label{circle diagram}
\makebox[20pt]{}\i_\rho^\circ(L)^\bullet &
\mathop{\longrightarrow}\limits^{\i_\rho^\circ(f)} &
\i_\rho^\circ(K)^\bullet\makebox[10pt]{} \\
{\phi(L,\rho)}\downarrow &
&\downarrow{\phi(K,\rho)} \\
\makebox[20pt]{}L(\rho)[1]^\bullet &
\mathop{\longrightarrow}\limits^{f(\rho)[1]} &
K(\rho)[1]^\bullet\makebox[10pt]{}
\end{array}
..
\end{equation}
Let $\{u^i\mathrel{;} i\in{\bf Z}\}$ be the collection of
$A(\rho)$-homomorphisms
$u^i\mathrel{:}\i_\rho^\circ(L)^i\rightarrow K(\rho)^i = K(\rho)[1]^{i-1}$
induced by $\{f^i(\sigma/\rho)\mathrel{;}\sigma\in F[{\bf 0},\rho)\}$.
Then restriction of the equality $d_K\cdot f = f\cdot d_L$
to $\i_\rho^\circ(L)^i$ implies
\begin{equation}
\phi(K,\rho)^i\cdot\i_\rho^\circ(f)^i + d_{K(\rho)[1]}^i\cdot u^i =
f(\rho)[1]^i\cdot\phi(L,\rho)^i + u^{i+1}\cdot d_{\i_\rho^\circ(L)}^i\;.
\end{equation}
Hence the difference
$\phi(K,\rho)^i\cdot\i_\rho^\circ(f)^i - f(\rho)[1]^i\cdot\phi(L,\rho)^i$
is homotopy equivalent to the zero map.
In particlar, the diagram (\ref{circle diagram}) induces a
commutative diagram of cohomologies.
If $f$ is unmixed, then all $u^i$'s are zero and the
diagram (\ref{circle diagram}) is commutative.
Let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta)$.
$K^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ with a homomorphism
$f\mathrel{:} K^\bullet\rightarrow L^\bullet$ is said to be a {\em subcomplex}
if $f$ is unmixed and
$f(\sigma)\mathrel{:} K(\sigma)^\bullet\rightarrow L(\sigma)^\bullet$
is an inclusion map of complexes for every $\sigma\in\Delta$.
By a {\em homogeneous element} of $L^\bullet$, we mean
an element of $L(\sigma)_j^i$ for some $\sigma\in\Delta$ and
$i, j\in{\bf Z}$.
For a set $S$ of homogeneous elements of $L^\bullet$, there
exists a unique subcomplex $\langle S\rangle^\bullet$ of $L^\bullet$
generated by $S$.
The complex $\langle S\rangle^\bullet$ is described inductively as follows.
For $\sigma\in\Delta$ and $i, j\in{\bf Z}$, let
$S(\sigma, i, j) := S\cap L(\sigma)_j^i$.
We denote by $\langle S(\sigma)\rangle^\bullet$ the
$A(\sigma)$-subcomplex of $L(\sigma)^\bullet$ generated by
$S(\sigma) :=\bigcup_{i,j\in{\bf Z}}S(\sigma, i, j)$ for
each $\sigma\in\Delta$.
We set $\langle S\rangle({\bf 0})^\bullet :=\langle S({\bf 0})\rangle^\bullet$.
Let $\rho$ be in $\Delta\setminus\{{\bf 0}\}$, and assume that
we already know $\langle S\rangle(\sigma)^\bullet$ for
$\sigma\in F[{\bf 0},\rho)$.
Then $\langle S\rangle(\rho)^\bullet$ is the $A(\rho)$-subcomplex
of $L(\rho)^\bullet$ given by
\begin{equation}
\label{eq in S}
\langle S\rangle(\rho)^\bullet =
\phi(L,\rho)(\i_\rho^\circ(\langle S\rangle))^\bullet +
\langle S(\rho)\rangle^\bullet\;.
\end{equation}
Let ${\bf p}$ be a perversity of $\Delta$, i.e., ${\bf p}$ is a map
$\Delta\setminus\{{\bf 0}\}\rightarrow{\bf Z}$.
We denote by $k_{\bf p}(\Delta)^\bullet$ the subcomplex of
$\mathop{\rm SdP}\nolimits(\Delta)^\bullet$ generated by
\begin{equation}
\bigcup_{\sigma\in\Delta\setminus\{{\bf 0}\}}\;%
\bigcup_{i+j\leq{\bf p}(\sigma)}\mathop{\rm SdP}\nolimits(\Delta)(\sigma)_j^i\;.
\end{equation}
Note that if we set this set $S$, then $\langle S(\sigma)\rangle^\bullet$
is equal to the gradual truncation
$\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\sigma)}\mathop{\rm SdP}\nolimits(\Delta)(\sigma)^\bullet$
(cf.\cite[\S1]{Ishida3}).
\begin{Lem}
\label{Lem 1.4}
Let $\Delta$ be a finite fan.
For each $\rho\in\Delta\setminus\{{\bf 0}\}$, the homomorphism
\begin{equation}
\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)\mathrel{:}\i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta))^\bullet
\longrightarrow \mathop{\rm SdP}\nolimits(\Delta)(\rho)[1]^\bullet
\end{equation}
is an isomorphism.
\end{Lem}
{\sl Proof.}\quad
Since $\mathop{\rm SdP}\nolimits(\Delta)^i =\{0\}$ for $i < 0$ and
$\mathop{\rm SdP}\nolimits(\Delta)(\rho)^0 =\{0\}$ for $\rho\not={\bf 0}$,
it is sufficient to show that $\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)^i$
is an isomorphism for $i\geq 0$.
For $i = 0$, we have
$i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta))^0 ={\bf Z}\otimes A(\rho)$ and
$\mathop{\rm SdP}\nolimits(\Delta)(\rho)^1 ={\bf Z} y(\rho)\otimes A(\rho)$, and
$\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)^0$ is the isomorphism given by
$1\otimes 1\mapsto y(\rho)\otimes 1$.
Assume $i > 0$.
The free $A(\rho)$-module
$\mathop{\rm SdP}\nolimits(\Delta)(\rho)^{i+1} = B(\rho)^{i+1}\otimes_{\bf Q} A(\rho)$,
has the basis $\{z(\alpha)y(\rho)\mathrel{;}\alpha\in\mathop{\rm Sd}\nolimits_i(F({\bf 0},\rho))\}$
which is decomposed to the disjoint union
\begin{equation}
\bigcup_{\sigma\in F({\bf 0},\rho)}
\{z(\beta)y(\sigma)y(\rho)\mathrel{;}\beta\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\sigma))\}\;.
\end{equation}
On the other hand, the component $\mathop{\rm SdP}\nolimits(\Delta)(\sigma)_{A(\rho)}^i$
of $\i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta))^i$ has the basis
\begin{equation}
\{z(\beta)y(\sigma)\mathrel{;}\beta\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\sigma))\}
\end{equation}
for each $\sigma\in F({\bf 0},\rho)$.
Since
$\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)^i(z(\beta)y(\sigma)) =
(-1)^iz(\beta)y(\sigma)y(\rho)$ for each $z(\beta)y(\sigma)$,
$\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)^i$ induces an isomoprhism from
$\mathop{\rm SdP}\nolimits(\Delta)(\sigma)_{A(\rho)}^i$ to the submodule of
$\mathop{\rm SdP}\nolimits(\Delta)(\rho)^{i+1}$ generated by
$\{z(\beta)y(\sigma)y(\rho)\mathrel{;}\beta\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\sigma))\}$.
We are done, since $\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)^i$ is the direct sum of
these isomorphisms for $\sigma\in F({\bf 0},\rho)$.
\QED
We define an unmixed homomorphism
\begin{equation}
\psi_\Delta\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)^\bullet\longrightarrow\P(\Delta)^\bullet
\end{equation}
of complexes in $\mathop{\rm CGEM}\nolimits(\Delta)$ as follows.
We define a homomorphism
\begin{equation}
\label{sdp to p}
\psi_\Delta(\rho)\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)(\rho)^\bullet
\longrightarrow\P(\Delta)(\rho)^\bullet
\end{equation}
in $\mathop{\rm CGM}\nolimits(A(\rho))$ for each $\rho\in\Delta$.
Let $k := r_\rho$.
Then $\psi_\Delta(\rho)^i := 0$ for $i\not= k$ since
$\P(\Delta)(\rho)^i =\{0\}$.
For $\alpha\in\mathop{\rm Sd}\nolimits_{k-1}(F({\bf 0},\rho))$, we define
$a :=\psi_\Delta(\rho)^k(z(\alpha)y(\rho))$ to be the
generator of $\det(\rho)\subset\P(\Delta)(\rho)^k$ which
is determined by the orientation of the sequence
$\alpha = (\sigma_1,\cdots,\sigma_{k-1})$, i.e.,
if we take $a_i\in N\cap\mathop{\rm rel.\,int}\nolimits\sigma_i$ for $i = 1,\cdots, k-1$
and $a_k\in N\cap\mathop{\rm rel.\,int}\nolimits\rho$, then $a$ is the generator of
$\det(\rho)$ which has the same sign with
$a_1\wedge\cdots\wedge a_k\in\det(\rho)$.
The commutativity of $\psi_\Delta$ and the coboundary
maps is checked easily.
The only one nontrivial commutativity is of
the component for $\mathop{\rm SdP}\nolimits(\Delta)(\rho)^{k-1}$ and
$\P(\Delta)(\mu)^k$ with $\mu =\rho$.
For each $\alpha\in\mathop{\rm Sd}\nolimits_{k-2}(F({\bf 0},\rho))$, there exist
exactly two $\beta_1,\beta_2\in\mathop{\rm Sd}\nolimits_{k-1}(F({\bf 0},\rho))$
which contains $\alpha$ as a subsequence.
We get the commutativity, since $\beta_1$ and $\beta_2$
have mutually opposite orientations and hence the equality
\begin{equation}
\psi_\Delta(\rho)^k(z(\beta_1)y(\rho)) +
\psi_\Delta(\rho)^k(z(\beta_2)y(\rho)) = 0
\end{equation}
holds.
By the definition, $\psi_\Delta(\rho)$ is surjective.
The following lemma implies that the kernel is generated by
\begin{equation}
\label{ker of sdp to p}
\bigcup_{i=1}^{r_\rho-1}\mathop{\rm SdP}\nolimits(\Delta)(\rho)^i
\end{equation}
as a subcomplex for each $\rho\in\Delta\setminus\{{\bf 0}\}$.
\begin{Lem}
\label{Lem 1.5}
The above unmixed homomorphism $\psi_\Delta$ is a
quasi-isomorphism in $\mathop{\rm CGEM}\nolimits(\Delta)$.
\end{Lem}
{\sl Proof.}\quad
Since $\mathop{\rm SdP}\nolimits(\Delta)({\bf 0}) =\P(\Delta)({\bf 0}) ={\bf Q}$,
$\psi_\Delta({\bf 0})$ is a quasi-isomorphism.
Let $\Phi$ be a maximal subfan of $\Delta$ such that
the restriction of $\psi_\Delta$ to $\Phi$ is a quasi-isomorphism.
Suppose $\Phi\not=\Delta$, and let $\rho$ be a minimal
element of $\Delta\setminus\Phi$.
Since $\psi_\Delta$ is unmixed, we get a commutative diagram
\begin{equation}
\begin{array}{ccc}
\makebox[70pt]{}\i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta))^\bullet &
\mathop{\longrightarrow}\limits^{\i_\rho^\circ\psi_\Delta} &
\i_\rho^\circ(\P(\Delta))^\bullet\makebox[50pt]{} \\
{\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)}\downarrow &
&\downarrow{\phi(\P(\Delta),\rho)} \\
\makebox[70pt]{}\mathop{\rm SdP}\nolimits(\Delta)(\rho)[1]^\bullet &
\mathop{\longrightarrow}\limits^{\psi_\Delta(\rho)[1]} &
\P(\Delta)(\rho)[1]^\bullet\makebox[50pt]{}
\end{array}
..
\end{equation}
By Lemma~\ref{Lem 1.4}, $\phi(\mathop{\rm SdP}\nolimits(\Delta),\rho)$ is an
isomorphism, while $\i_\rho^\circ\psi_\Delta$ is a quasi-isomorphism
since it depends only on the restriction of $\psi_\Delta$ to $\Phi$.
Since the mapping cone $\i_\rho^*(\P(\Delta))^\bullet$ of
$\phi(\P(\Delta),\rho)$ is equal to
$E(F(\rho),{\bf Z})^\bullet\otimes A(\rho)$, it has trivial
cohomologies by Lemma~\ref{Lem 1.2},(1).
Hence $\phi(\P(\Delta),\rho)$ is also a quasi-isomorphism.
By the commutative diagram, $\psi_\Delta(\rho)$ is also a
quasi-isomorphism.
This contradicts the maximality of $\Phi$.
\QED
\begin{Lem}
\label{Lem 1.6}
For $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ and a homomorphism
$f_0\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)({\bf 0})^\bullet\rightarrow L({\bf 0})^\bullet$,
there exists a unique unmixed homomorphism
$f\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)^\bullet\rightarrow L^\bullet$ with
$f({\bf 0}) = f_0$.
\end{Lem}
{\sl Proof.}\quad
We prove the lemma by induction on the number of
cones in $\Delta$.
If $\Delta =\{{\bf 0}\}$, then the assertion is clear.
Assume that $\Delta\not=\{{\bf 0}\}$ and $\pi$ is a
maximal element of $\Delta$.
Set $\Delta' :=\Delta\setminus\{\pi\}$ and assume that
$f'\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta')^\bullet\rightarrow (L|\Delta')^\bullet$
is the unique extension of $f_0$.
Let $f_1\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)(\pi)^\bullet\rightarrow L(\pi)^\bullet$
be a homomorphism in $\mathop{\rm CGM}\nolimits(A(\pi))$.
Then $f'$ is extended to an unmixed homomorphism
$f\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)^\bullet\rightarrow L^\bullet$ by $f(\pi) := f_1$
if and only if the diagram
\begin{equation}
\begin{array}{ccc}
\makebox[70pt]{}\i_\pi^\circ(\mathop{\rm SdP}\nolimits(\Delta))^\bullet &
\mathop{\longrightarrow}\limits^{\i_\pi^\circ f'} &
\i_\pi^\circ(L)^\bullet\makebox[50pt]{} \\
{\phi(\mathop{\rm SdP}\nolimits(\Delta),\pi)}\downarrow &
&\downarrow{\phi(L,\pi)} \\
\makebox[70pt]{}\mathop{\rm SdP}\nolimits(\Delta)(\pi)[1]^\bullet &
\mathop{\longrightarrow}\limits^{f_1[1]} & L(\pi)[1]^\bullet\makebox[50pt]{}
\end{array}
\end{equation}
is commutative.
Since $\phi(\mathop{\rm SdP}\nolimits(\Delta),\pi)$ is an isomorphism by Lemma~\ref{Lem 1.4},
such a $f_1$ exists uniquely.
\QED
\begin{Thm}
\label{Thm 1.7}
Let ${\bf p}$ be a perversity of $\Delta$.
Let
\begin{equation}
\varphi(\Delta,{\bf p})\mathrel{:}
\mathop{\rm SdP}\nolimits(\Delta)^\bullet\longrightarrow\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet
\end{equation}
be the unmixed homomorphism obtained by extending the
identity $\mathop{\rm SdP}\nolimits(\Delta)({\bf 0}) =\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0}) ={\bf Q}$.
Then $\varphi(\Delta,{\bf p})$ is surjective and the kernel
is equal to $k_{\bf p}(\Delta)^\bullet$.
\end{Thm}
{\sl Proof.}\quad
We prove the theorem by induction.
Namely, let $\Phi$ be the maximal subfan of $\Delta$
such that $\varphi(\Delta,{\bf p})(\sigma)$ is surjective
and the kernel is equal to $k_{\bf p}(\Delta)(\sigma)^\bullet$
for $\sigma\in\Phi$.
Since $k_{\bf p}(\Delta)({\bf 0}) =\{0\}$, we have ${\bf 0}\in\Phi$.
Suppose $\Phi\not=\Delta$ and let $\rho$ be a minimal
element of $\Delta\setminus\Phi$.
Since $\varphi(\Delta,{\bf p})$ is unmixed,
we get the following commutative diagram.
\begin{equation}
\begin{array}{ccccccccc}
0 & \rightarrow & \i_\rho^\circ(k_{\bf p}(\Delta))^\bullet &
\longrightarrow & \i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta))^\bullet &
\mathop{\longrightarrow}\limits^{w} & \i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet & \rightarrow & 0 \\
& & \hbox{ }\downarrow{\phi_1} & & \hbox{ }\downarrow{\phi_2}
& & \hbox{ }\downarrow{\phi_3} & & \\
0 & \rightarrow & k_{\bf p}(\Delta)(\rho)[1]^\bullet &
\mathop{\longrightarrow}\limits^{u} & \mathop{\rm SdP}\nolimits(\Delta)(\rho)[1]^\bullet &
\mathop{\longrightarrow}\limits^{v} & \mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\rho)[1]^\bullet & \rightarrow & 0
\end{array}
..
\end{equation}
The upper line of this diagram is exact by the assumption.
Among the vertical homomorphisms, $\phi_2$ is an isomorhism
by Lemma~\ref{Lem 1.4} and $\phi_3$ is the natural surjection
\begin{equation}
\i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet\longrightarrow
\mathop{\rm gt}\nolimits^{\geq{\bf p}(\rho)}(\i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))^\bullet
= \mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\rho)[1]^\bullet
\end{equation}
by the construction of $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ \cite[Thm 2.9]{Ishida3}.
Hence $v$ is surjective, while $u$ is an inclusion map.
Since there exists an exact sequence
\begin{equation}
0\rightarrow
\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)-1}(\i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))^\bullet
\rightarrow\i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet\rightarrow
\mathop{\rm gt}\nolimits^{\geq{\bf p}(\rho)}(\i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))^\bullet\rightarrow 0\;,
\end{equation}
$\mathop{\rm Ker}\nolimits\phi_3$ is equal to
$\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)-1}(\i_\rho^\circ(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)))^\bullet$.
Since $w$ is surjective, this is equal to the image of
$\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)-1}(\i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta)))^\bullet$
by $w$.
Hence
\begin{equation}
w^{-1}(\mathop{\rm Ker}\nolimits\phi_3) =
\i_\rho^\circ(k_{\bf p}(\Delta))^\bullet +
\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)-1}(\i_\rho^\circ(\mathop{\rm SdP}\nolimits(\Delta)))^\bullet\;.
\end{equation}
Since
\begin{equation}
\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)-1}(\mathop{\rm SdP}\nolimits(\Delta)(\rho))[1])^\bullet =
(\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)}(\mathop{\rm SdP}\nolimits(\Delta)(\rho)))[1]^\bullet\;,
\end{equation}
$\mathop{\rm Ker}\nolimits v =\phi_2(w^{-1}(\mathop{\rm Ker}\nolimits\phi_3))$ is equal to the sum of $\Im\phi_1$
and $(\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)}(\mathop{\rm SdP}\nolimits(\Delta)(\rho)))[1]^\bullet$
in $\mathop{\rm SdP}\nolimits(\Delta)(\rho)[1]^\bullet$.
On the other hand, $k_{\bf p}(\Delta)(\rho)^\bullet$ is the
subcomplex of $\mathop{\rm SdP}\nolimits(\Delta)(\rho)^\bullet$ generated by
$\Im\phi_1$ and
$\bigcup_{i+j\leq{\bf p}(\rho)}\mathop{\rm SdP}\nolimits(\Delta)(\rho)_j^i$ (cf.(\ref{eq in S})).
Since the subcomplex of $\mathop{\rm SdP}\nolimits(\Delta)(\rho)^\bullet$
generated by the last set is
$\widetilde{\mathop{\rm gt}\nolimits}_{\leq{\bf p}(\rho)}(\mathop{\rm SdP}\nolimits(\Delta)(\rho))^\bullet$,
the lower line is also exact.
This contradicts the maximality of $\Phi$, and we conclude
$\Phi =\Delta$.
\QED
For a finite fan $\Delta$, the covariant additive
functor $\Gamma\mathrel{:}\mathop{\rm GEM}\nolimits(\Delta)\rightarrow\mathop{\rm GM}\nolimits(A)$ is defined
as follows.
For $L\in\mathop{\rm GEM}\nolimits(\Delta)$, we set
\begin{equation}
\Gamma(L) :=\bigoplus_{\sigma\in\Delta}L(\sigma)_A\;.
\end{equation}
Let $f\mathrel{:} L\rightarrow K$ be a homomorphism in $\mathop{\rm GEM}\nolimits(\Delta)$.
For $\sigma,\tau\in\Delta$, the $(\sigma,\tau)$-component
of the homomorphism $\Gamma(f)\mathrel{:}\Gamma(L)\rightarrow\Gamma(K)$
is defined to be the $A$-homomorphism
$L(\sigma)_A\rightarrow K(\tau)_A$
induced by $f(\sigma/\tau)$ and is the zero map otherwise.
Let $L^\bullet$ be in $\mathop{\rm CGEM}\nolimits(\Delta)$.
Then $\Gamma(L)^\bullet$ is in $\mathop{\rm CGM}\nolimits(A)$, i.e.,
is a finite complex of graded $A$-modules.
For each integer $q$, the homogeneous component
$\Gamma(L)_q^\bullet$ is a complex of ${\bf Q}$-vector spaces.
Since $A(\rho)_A = A$ for all $\rho\in\Delta$, we have
\begin{equation}
\Gamma(\mathop{\rm SdP}\nolimits(\Delta)^i) =
B(\Delta{\setminus}\{{\bf 0}\})^i\otimes_{\bf Q} A
\end{equation}
for each $i\in{\bf Z}$.
By comparing the definitions of the coboundary maps,
we get the following
lemma.
\begin{Lem}
\label{Lem 1.8}
For any finite fan $\Delta$,
$\Gamma(\mathop{\rm SdP}\nolimits(\Delta))^\bullet$ is canonically isomorphic
to $B(\Delta{\setminus}\{{\bf 0}\})^\bullet\otimes_{\bf Q} A$ as a
complex of $A$-modules.
\end{Lem}
The {\em top perversity} $\t$ is defined by $\t(\sigma) := r_\sigma - 1$
for every nontrivial cone $\sigma$, while the
{\em bottom perversity} $\b$ is defined by $\b :=-\t$.
We use this notation for all finite fans.
\begin{Lem}
\label{Lem 1.9}
Let $\Delta$ be a finite fan.
Then the unmixed surjection $\varphi(\Delta,\t)\mathrel{:}
\mathop{\rm SdP}\nolimits(\Delta)^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$
induces an unmixed surjection
$\varphi\mathrel{:}\P(\Delta)^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$ such that
\begin{equation}
\mathop{\rm Ker}\nolimits\varphi(\sigma)^{r_\sigma} =
\det(\sigma)\otimes(N(\sigma)A(\sigma))
\end{equation}
for every $\sigma\in\Delta$.
\end{Lem}
Note that $N(\sigma)A(\sigma)$ is a two-sided maximal ideal
of $A(\sigma)$.
{\sl Proof.}\quad
The kernel of the homomorphism $\varphi(\Delta,\t)$ is
equal to $k_\t(\Delta)^\bullet$ by Theorem~\ref{Thm 1.7}.
By definition, this subcomplex is generated by the union of
\begin{equation}
\bigcup_{i+j<r_\sigma}\mathop{\rm SdP}\nolimits(\Delta)(\sigma)_j^i
\end{equation}
for all $\sigma\in\Delta\setminus\{{\bf 0}\}$.
Since $\mathop{\rm SdP}\nolimits(\Delta)(\sigma)_j^i$ is nonzero only for
$1\leq i\leq r_\sigma,\; -r_\sigma\leq j\leq 0$,
the condition $i+j<r_\sigma$ means all
$(i, j)$ except for $(r_\sigma, 0)$.
On the other hand, the kernel of
$\psi_\Delta(\sigma)\mathrel{:}
\mathop{\rm SdP}\nolimits(\Delta)(\sigma)^\bullet\rightarrow\P(\Delta)(\sigma)^\bullet$
is generated as a subcomplex by
\begin{equation}
\bigcup_{i=1}^{r_\sigma-1}\mathop{\rm SdP}\nolimits(\Delta)(\sigma)^i
\end{equation}
for each $\sigma\in\Delta\setminus\{{\bf 0}\}$ as we mensioned
before Lemma~\ref{Lem 1.5}.
Hence we get the surjection $\varphi$.
We get the lemma since
\begin{equation}
\mathop{\rm Ker}\nolimits\varphi(\sigma)^{r_\sigma}\simeq
\mathop{\rm Ker}\nolimits\varphi(\Delta,\t)(\sigma)^{r_\sigma}/
\mathop{\rm Ker}\nolimits\psi_\Delta(\sigma)^{r_\sigma}
\end{equation}
and this is equal to
\begin{equation}
\bigoplus_{j=-r_\sigma}^{-1}
\psi_\Delta(\sigma)(\mathop{\rm SdP}\nolimits(\Delta)(\sigma)_j^{r_\sigma})
= \bigoplus_{j=-r_\sigma}^{-1}\P(\Delta)(\sigma)_j^{r_\sigma}
\end{equation}
as a graded ${\bf Q}$-vector space.
This is equal to $\det(\sigma)\otimes(N(\sigma)A(\sigma))$.
\QED
We denote by $\bar A(\sigma)$ the $A(\sigma)$-module
$A(\sigma)/N(\sigma)A(\sigma)$ of lengh one.
By this lemma, we identify $\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$ with the
quotient complex of $\P(\Delta)^\bullet$ by $\mathop{\rm Ker}\nolimits\varphi$.
Namely we have
\begin{equation}
\label{top}
\mathop{\rm ic}\nolimits_\t(\Delta)(\sigma)^i =\left\{
\begin{array}{ll}
\det(\sigma)\otimes\bar A(\sigma) &
\hbox{ if }i = r_\sigma \\
\{0\} & \hbox{ if }i\not= r_\sigma
\end{array}
\right.
\end{equation}
for every $\sigma\in\Delta$.
\begin{Lem}
\label{Lem 1.10}
Let $\Delta$ be a finite fan.
Then $\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q =\{0\}$ for
$p, q\in{\bf Z}$ with $p > q + r$.
\end{Lem}
{\sl Proof.}\quad
Since
$\bar A(\sigma)_A =\bigwedge^\bullet(N_{\bf Q}/N(\sigma)_{\bf Q})$
and $\dim_{\bf Q} N_{\bf Q}/N(\sigma)_{\bf Q} = r - r_\sigma$,
\begin{equation}
\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))_q^p =
\bigoplus_{\sigma\in\Delta(p)}
\det(\sigma)\otimes(\bar A(\sigma)_A)_q =\{0\}
\end{equation}
for $q < -(r - p)$, i.e, for $p > q + r$.
Hence, for the component of degree $q$ of the complex
$\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet$, the cohomologies vanish for
$p > q + r$.
\QED
\section{The intersection complex of the middle perversity}
\setcounter{equation}{0}
Let $\Delta$ be a finite fan of $N_{\bf R}$.
The middle perversity ${\bf m}\mathrel{:}\Delta\setminus\{{\bf 0}\}\rightarrow{\bf Z}$
is defined by ${\bf m}(\sigma) := 0$ for every $\sigma$.
We denote the intersection complex $\mathop{\rm ic}\nolimits_{\bf m}(\Delta)^\bullet$
simply by $\mathop{\rm ic}\nolimits(\Delta)^\bullet$.
Since $-{\bf m} ={\bf m}$, the dual ${\bf D}(\mathop{\rm ic}\nolimits(\Delta))^\bullet$ is
quasi-isomorphic to $\mathop{\rm ic}\nolimits(\Delta)^\bullet$ by
\cite[Cor.2.12]{Ishida3}.
A finite fan $\Delta$ of $N_{\bf R}$ is said to be a
{\em lifted complete fan}, if it is a lifting of a complete
fan of an $(r-1)$-dimensional space (cf.\cite[\S2]{Ishida3}).
In particular, $\Delta$ is a lifted complete fan if
the support $|\Delta|$ is equal to the boundary of an
$r$-dimensional cone.
\begin{Thm}
\label{Thm 2.1}
Let $\Delta$ be a lifted complete fan of $N_{\bf R}$.
Then for any integers $p, q\in{\bf Z}$, the equality
\begin{equation}
\label{zzliftedcomplete}
\dim_{\bf Q}\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_{q} =
\dim_{\bf Q}\H^{r-1-p}(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_{-r-q}
\end{equation}
holds.
\end{Thm}
{\sl Proof.}\quad
Since ${\bf D}(\mathop{\rm ic}\nolimits(\Delta))^\bullet$ is quasi-isomorphic to
$\mathop{\rm ic}\nolimits(\Delta)^\bullet$ by \cite[Cor.2.12]{Ishida3},
this is a consequence of \cite[Prop.2.8]{Ishida3}.
\QED
\begin{Cor}
\label{Cor 2.2}
Let $\pi$ be an $r$-dimensional cone.
Then, for $\Delta := F(\pi)\setminus\{\pi\}$, the equality
(\ref{zzliftedcomplete}) folds for any integers $p, q\in{\bf Z}$.
\end{Cor}
Let $\Delta$ and $\Delta'$ be finite fans of $N_{\bf R}$.
Then $\Delta'$ is said to be a {\em subdivision} of $\Delta$
if $|\Delta| = |\Delta'|$ and, for every $\sigma\in\Delta'$
there exists $\rho\in\Delta$ with $\sigma\subset\rho$
(cf.\cite[Cor.1.16]{Oda1}).
If $\Delta'$ is a subdivision of $\Delta$, then there exists
a unique map $f\mathrel{:}\Delta'\rightarrow\Delta$ such that
$\sigma\cap\mathop{\rm rel.\,int}\nolimits f(\sigma)\not=\emptyset$ for each
$\sigma\in\Delta'$.
Actually, $f(\sigma)$ is defined to be the minimal cone in
$\Delta$ which contains $\sigma$.
The map $f$ is also called a subdivision.
Let $f\mathrel{:}\Delta'\rightarrow\Delta$ be a subdivision.
For $\rho\in\Delta$, we denote by $f^{-1}(\rho)$ the
subset $f^{-1}(\{\rho\})$ of $\Delta'$.
Clearly, $f^{-1}(\rho)$ is a locally star closed subset
of $\Delta'$.
For $L\in\mathop{\rm GEM}\nolimits(\Delta')$, we define the {\em direct image}
$f_*L\in\mathop{\rm GEM}\nolimits(\Delta)$ by
\begin{equation}
(f_*L)(\rho) :=
\bigoplus_{\sigma\in f^{-1}(\rho)}L(\sigma)_{A(\rho)}
\end{equation}
for each $\rho\in\Delta$.
For a homomorphism $g\mathrel{:} L\rightarrow K$ in $\mathop{\rm GEM}\nolimits(\Delta')$,
the homomorphism $f_*(g)\mathrel{:} f_*L\rightarrow f_*K$ is defined
as follows.
For $\rho,\mu\in\Delta$ with $\rho\prec\mu$ and
$\sigma\in f^{-1}(\rho),\;\tau\in f^{-1}(\mu)$,
the $(\sigma,\tau)$-component of
$f_*(g)(\rho/\mu)$ is the $A(\rho)$-homomorphism
$L(\sigma)_{A(\rho)}\rightarrow K(\tau)_{A(\mu)}$
induced by the $A(\sigma)$-homomoprhism
$g(\sigma/\tau)\mathrel{:} L(\sigma)\rightarrow K(\tau)$
if $\sigma\prec\tau$, otherwise it is defined to be the
zero map.
It is easy to see that $f_*$ is a covariant additive
functor from $\mathop{\rm GEM}\nolimits(\Delta')$ to $\mathop{\rm GEM}\nolimits(\Delta)$.
We denote also by $f_*$ the induced functor from
$\mathop{\rm CGEM}\nolimits(\Delta')$ to $\mathop{\rm CGEM}\nolimits(\Delta)$.
A sequence of homomorphisms $0\rightarrow L\rightarrow K\rightarrow J\rightarrow 0$ in
$\mathop{\rm GEM}\nolimits(\Delta)$ is said to be a short exact sequence if
$0\rightarrow L(\sigma)\rightarrow K(\sigma)\rightarrow J(\sigma)\rightarrow 0$ is exact
for every $\sigma\in\Delta$.
\begin{Lem}
\label{Lem 2.3}
Let $f\mathrel{:}\Delta'\rightarrow\Delta$ be a subdivision of a finite fan.
(1)
If $g\mathrel{:} L^\bullet\rightarrow K^\bullet$ is a quasi-isomoprhism in
$\mathop{\rm CGEM}\nolimits(\Delta')$, then direct image
$f_*(g)\mathrel{:} f_*L^\bullet\rightarrow f_*K^\bullet$ is also a
quasi-isomorphism.
(2)
If $0\rightarrow L\lraw{g}K\lraw{h}J\rightarrow 0$ is a short exact
sequence in $\mathop{\rm GEM}\nolimits(\Delta')$, then
$0\rightarrow f_*L\lraw{f_*(g)}f_*K\lraw{f_*(h)}f_*J\rightarrow 0$
is a short exact sequence in $\mathop{\rm GEM}\nolimits(\Delta)$.
\end{Lem}
{\sl Proof.}\quad
(1)
Let $\Phi$ be a maximal subfan of $\Delta'$ such that
$f_*(g|\Phi)\mathrel{:} f_*(L|\Phi)^\bullet\rightarrow f_*(K|\Phi)^\bullet$
is a quasi-isomorphism.
Suppose $\Phi\not=\Delta'$ and let $\tau$ be a minimal
element of $\Delta'\setminus\Phi$.
Let $\Phi' :=\Phi\cup\{\tau\}$ and $\rho := f(\tau)$.
We get a commutative diagram
\begin{equation}
\begin{array}{ccccccccc}
0 & \rightarrow & L(\tau)_{A(\rho)}^\bullet &
\longrightarrow & f_*(L|\Phi')(\rho)^\bullet &
\longrightarrow & f_*(L|\Phi)(\rho)^\bullet & \rightarrow & 0 \\
& & \hbox{ }\downarrow{\phi_1} & & \hbox{ }\downarrow{\phi_2}
& & \hbox{ }\downarrow{\phi_3} & & \\
0 & \rightarrow & K(\tau)_{A(\rho)}^\bullet &
\longrightarrow & f_*(K|\Phi')(\rho)^\bullet &
\longrightarrow & f_*(K|\Phi)(\rho)^\bullet & \rightarrow & 0 \\
\end{array}
..
\end{equation}
Among the vertical homomorphisms, $\phi_1$ is a
quasi-isomorphism since
$g(\tau/\tau)\mathrel{:} L(\tau)^\bullet\rightarrow K(\tau)^\bullet$ is
quasi-isomorphic and $A(\rho)$ is a free $A(\tau)$-module,
while $\phi_3$ is quasi-isomorphic by the assumption.
Hence $\phi_2 = f_*(g|\Phi')(\rho)$ is also quasi-isomorhic.
Since $f_*(L|\Phi')(\mu) = f_*(L|\Phi)(\mu)$ for $\mu\not=\rho$,
$f_*(g|\Phi')$ is a quasi-isomorphism in $\mathop{\rm CGEM}\nolimits(\Delta)$.
This contradicts the maximality of $\Phi$.
Hence $\Phi =\Delta'$.
(2)
Let $\Phi$ be a maximal subfan of $\Delta'$ such that
\begin{equation}
0\rightarrow f_*(L|\Phi)\lraw{f_*(g)}f_*(K|\Phi)\lraw{f_*(h)}f_*(J|\Phi)\rightarrow 0
\end{equation}
is a short exact sequence.
Suppose $\Phi\not=\Delta'$, and let $\tau$ be a minimal
element of $\Delta'\setminus\Phi$ and $\rho := f(\tau)$.
Let $\Phi' :=\Phi\cup\{\tau\}$.
It is sufficient to show that
\begin{equation}
0\rightarrow f_*(L|\Phi')\lraw{f_*(g)}f_*(K|\Phi')\lraw{f_*(h)}f_*(J|\Phi')\rightarrow 0
\end{equation}
is a short exact sequence, since it contradicts the maximality
of $\Phi$.
It is enough to check it for $\rho$.
Since
$0\rightarrow L(\tau)_{A(\rho)}\longrightarrow K(\tau)_{A(\rho)}\longrightarrow J(\tau)_{A(\rho)}\rightarrow 0$
is exact by the assumption, the exactness of
\begin{equation}
0\rightarrow f_*(L|\Phi')(\rho)\lraw{f_*(g)}f_*(K|\Phi')(\rho)
\lraw{f_*(h)}f_*(J|\Phi')(\rho)\rightarrow 0
\end{equation}
follows from the nine lemma of the homology algebra.
\QED
Although $\mathop{\rm GEM}\nolimits(\Delta)$ and $\mathop{\rm CGEM}\nolimits(\Delta)$ are not in
general abelian categories, we say that the functor $f_*$ is
exact in the sense that the property (2) of the above lemma holds.
The dualizing functor ${\bf D}$ defined in \cite[\S2]{Ishida3}
is an exact contravariant functor from $\mathop{\rm CGEM}\nolimits(\Delta)$ to
itself in this sense.
Actually, for $L^\bullet\in\mathop{\rm CGEM}\nolimits(\Delta)$ and $\rho\in\Delta$,
${\bf D}(L)(\rho)^\bullet$ is defined by the combination of
exact functors $\i_\rho^*$ and $\d_\rho$ (cf.\cite[\S2]{Ishida3}).
Let $\Delta$ be a finite fan.
For each $\rho\in\Delta\setminus\{{\bf 0}\}$, we take a
rational point $a(\rho)$ in the relative interior of $\rho$.
For each
$\alpha = (\rho_1,\cdots,\rho_k)\in\mathop{\rm Sd}\nolimits(\Delta{\setminus}\{{\bf 0}\})$,
let $c(\alpha)$ be the simplicial cone
${\bf R}_0a(\rho_1) +\cdots +{\bf R}_0a(\rho_k)$ if $k > 0$, and
let $c(\alpha) :={\bf 0}$ if $k = 0$.
Then
\begin{equation}
\Sigma :=\{c(\alpha)\mathrel{;}\alpha\in\mathop{\rm Sd}\nolimits(\Delta{\setminus}\{{\bf 0}\})\}
\end{equation}
is a simplicial subdivision of $\Delta$.
We call $\Sigma$ a {\em barycentric subdivision} of $\Delta$.
Clearly, it is not unique for $\Delta$ since it depends on
the choice of the set
$\{a(\rho)\mathrel{;}\rho\in\Delta\setminus\{{\bf 0}\}\}$.
Let $f\mathrel{:}\Sigma\rightarrow\Delta$ be a barycentric subdivision of
$\Delta$.
For $\alpha\in\mathop{\rm Sd}\nolimits(\Delta{\setminus}\{{\bf 0}\})$,
$f(c(\alpha)) =\max(\alpha)$ by the definition.
We define an unmixed homomorphism
$\lambda_\Delta\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)^\bullet\rightarrow f_*\P(\Sigma)^\bullet$
as follows.
For each $\alpha\in\mathop{\rm Sd}\nolimits(\Delta\setminus\{{\bf 0}\})$, let
$c(\alpha)$ be the corresponding cone in $\Sigma$ as above.
For each $\rho\in\Delta$ and integer $i$, we have
\begin{equation}
\mathop{\rm SdP}\nolimits(\Delta)(\rho)^i =
\bigoplus_{\beta\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\rho))}%
{\bf Z} z(\beta)y(\rho)\otimes A(\rho)
\end{equation}
and
\begin{equation}
\begin{array}{lll}
f_*\P(\Sigma)(\rho)^i & = &
\bigoplus_{\sigma\in f^{-1}(\rho)(i)}%
\det(\sigma)\otimes A(\rho) \\
& = &\bigoplus_{\beta\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\rho))}%
\det(c(\beta)+c(\rho))\otimes A(\rho)\;,
\end{array}
\end{equation}
where $c(\rho)$ is the one-dimensional cone generated
by $a(\rho)$.
For each
$\beta = (\sigma_1,\cdots,\sigma_{i-1})\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\rho))$,
let $z'(\beta,\rho)$ be the generator of
$\det(c(\beta)+c(\rho))\simeq{\bf Z}$ which has the same sign with
$a(\sigma_1)\wedge\cdots\wedge a(\sigma_{i-1})\wedge a(\rho)$
in $\det(c(\beta)+c(\rho))\otimes{\bf Q}$.
We define $\lambda_\Delta(\rho)^i$ to be the isomoprhism given by
$z(\beta)y(\rho)\otimes 1\mapsto z'(\beta,\rho)\otimes 1$
for all $\beta\in\mathop{\rm Sd}\nolimits_{i-1}(F({\bf 0},\rho))$.
It is easy to check the compatibility with the coboundary
maps.
Since $\lambda_\Delta(\rho)^i$'s are isomorphic, we get the
following lemma.
\begin{Lem}
\label{Lem 2.4}
Let $f\mathrel{:}\Sigma\rightarrow\Delta$ be a barycentric subdivision of
$\Delta$.
Then the above unmixed homomorphism
$\lambda_\Delta\mathrel{:}\mathop{\rm SdP}\nolimits(\Delta)^\bullet\rightarrow f_*\P(\Sigma)^\bullet$
is an isomorphism.
\end{Lem}
Let $\psi_\Sigma\mathrel{:}\mathop{\rm SdP}\nolimits(\Sigma)^\bullet\rightarrow\P(\Sigma)^\bullet$
be the natural unmixed quasi-isomorphism.
We define an unmixed homomorphism
\begin{equation}
\phi_{\Sigma/\Delta}\mathrel{:}
f_*\mathop{\rm SdP}\nolimits(\Sigma)^\bullet\longrightarrow\mathop{\rm SdP}\nolimits(\Delta)^\bullet
\end{equation}
by $\phi_{\Sigma/\Delta} :=
\lambda_\Delta^{-1}\cdot f_*(\psi_\Sigma)$.
\begin{Lem}
\label{Lem 2.5}
Let $f\mathrel{:}\Sigma\rightarrow\Delta$ be a barycentric subdivision and
let ${\bf p}$ and ${\bf q}$ be perversities of $\Delta$ and $\Sigma$,
respectively.
Then $\phi_{\Sigma/\Delta}(f_*k_{\bf q}(\Sigma))$ is contained
in $k_{\bf p}(\Delta)$ if ${\bf q}(\sigma)\leq{\bf p}(f(\sigma))$ for
every $\sigma\in\Sigma\setminus\{{\bf 0}\}$.
In this case, $\phi_{\Sigma/\Delta}$ induces a natural
homomorphism
\begin{equation}
f_*\mathop{\rm ic}\nolimits_{\bf q}(\Sigma)^\bullet\longrightarrow\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet\;.
\end{equation}
\end{Lem}
{\sl Proof.}\quad
By definition, $k_{\bf q}(\Sigma)$ is the subcomplex of
$\mathop{\rm SdP}\nolimits(\Sigma)^\bullet$ generated by
\begin{equation}
S :=\bigcup_{\sigma\in\Sigma\setminus\{{\bf 0}\}}\;%
\bigcup_{i+j\leq{\bf q}(\sigma)}\mathop{\rm SdP}\nolimits(\Sigma)(\sigma)_j^i\;.
\end{equation}
Hence it is sufficient to show that
\begin{equation}
\phi_{\Sigma/\Delta}(f(\sigma))(\mathop{\rm SdP}\nolimits(\Sigma)(\sigma)_j^i)
\subset k_{\bf p}(\Delta)(f(\sigma))
\end{equation}
for all $\sigma\in\Sigma\setminus\{{\bf 0}\}$ and $i, j$
with $i+j\leq{\bf q}(\sigma)$.
If $i < r_\sigma$, then $\mathop{\rm SdP}\nolimits(\Sigma)(\sigma)_j^i$ is
mapped to zero.
Hence the inclusion is obvious in this case.
We consider the case $i = r_\sigma$.
Let $\sigma\in\Sigma$ and let
$\alpha = (\sigma_1,\cdots,\sigma_{r_\sigma})$
be the corresponding element of
$\mathop{\rm Sd}\nolimits_{r_\sigma}(\Delta\setminus\{{\bf 0}\})$.
We set $\rho := f(\sigma) =\sigma_{r_\sigma}$.
Recall that $\mathop{\rm SdP}\nolimits(\Sigma)(\sigma)^{r_\sigma}$ is the free
$A(\sigma)$-module with the basis
$\{z(\beta)y(\sigma)\mathrel{;}\beta\in\mathop{\rm Sd}\nolimits_{r_\sigma-1}(F({\bf 0},\sigma))\}$.
Since $\phi_{\Sigma/\Delta}(\rho)(z(\beta)y(\sigma)\otimes 1)
=\pm z(\alpha)\otimes 1$
for every $\beta\in\mathop{\rm Sd}\nolimits_{r_\sigma-1}(F({\bf 0},\sigma))$,
\begin{equation}
\phi_{\Sigma/\Delta}(\rho)({\bf Z} z(\beta)y(\sigma)\otimes A(\sigma)_j)
= {\bf Z} z(\alpha)\otimes A(\sigma)_j
\subset{\bf Z} z(\alpha)\otimes A(\rho)_j
\end{equation}
for every $\beta\in\mathop{\rm Sd}\nolimits_{r_\sigma-1}(F({\bf 0},\sigma))$
and $j\leq{\bf q}(\sigma) - r_\sigma$.
Hence the image $\phi_{\Sigma/\Delta}(\rho)(S(\sigma))$ is equal to
$\bigcup_{j\leq{\bf q}(\sigma) - r_\sigma}{\bf Z} z(\alpha)\otimes A(\sigma)_j$
and is contained in $k_{\bf p}(\Delta)(\rho)^{r_\sigma}$ if
${\bf q}(\sigma)\leq{\bf p}(\rho)$.
\QED
Clearly, the condition of the above lemma is satisfied if
${\bf p}$ and ${\bf q}$ are the middle perversities.
In particular we get a natural unmixed homomoprhism
\begin{equation}
\label{zzichom}
\delta_{\Sigma/\Delta}\mathrel{:} f_*\mathop{\rm ic}\nolimits(\Sigma)^\bullet\longrightarrow\mathop{\rm ic}\nolimits(\Delta)^\bullet
\end{equation}
for a barycentric subdivision $f\mathrel{:}\Sigma\rightarrow\Delta$.
Let $f\mathrel{:}\Delta'\rightarrow\Delta$ be a subdivision and
let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta')$.
We define an unmixed homomorphism
\begin{equation}
\kappa(f, L)\mathrel{:} f_*{\bf D}(L)^\bullet\longrightarrow{\bf D}(f_*L)^\bullet
\end{equation}
as follows.
For $\rho\in\Delta$ and $i\in{\bf Z}$, we have
\begin{eqnarray}
\label{dual's image}
f_*{\bf D}(L)(\rho)^i & = &
\bigoplus_{\tau\in f^{-1}(\rho)}%
{\bf D}(L)(\tau)_{A(\rho)}^i \\
\label{dual's image2}
& = &
\bigoplus_{\tau\in f^{-1}(\rho)}\;\bigoplus_{\sigma\in F(\tau)}%
\det(\tau)\otimes\d_\sigma(L(\sigma)^{r_\tau-i})_{A(\rho)}\;,
\end{eqnarray}
while
\begin{eqnarray}
\label{image's dual}
{\bf D}(f_*L)(\rho)^i & = &
\bigoplus_{\eta\in F(\rho)}%
\det(\rho)\otimes\d_\eta(f_*L(\eta)^{r_\rho-i})_{A(\rho)} \\
& = &
\bigoplus_{\eta\in F(\rho)}\;\bigoplus_{\sigma\in f^{-1}(\eta)}%
\det(\rho)\otimes\d_\sigma(L(\sigma)^{r_\rho-i})_{A(\rho)} \\
& = &
\bigoplus_{\sigma\in f^{-1}(F(\rho))}%
\det(\rho)\otimes\d_\sigma(L(\sigma)^{r_\rho-i})_{A(\rho)}\;.
\end{eqnarray}
For $\tau\in f^{-1}(\rho)$ and $\sigma\in F(\tau)$, the
restriction of
\begin{equation}
\kappa(f, L)(\rho)^i\mathrel{:} f_*{\bf D}(L)(\rho)^i\longrightarrow{\bf D}(f_*L)(\rho)^i
\end{equation}
to the component
$\det(\tau)\otimes\d_\sigma(L(\sigma)^{r_\tau-i})_{A(\rho)}$
is defined to be the zero map if $r_\tau < r_\rho$.
If $r_\tau = r_\rho$, then $N(\tau) = N(\rho)$ and
$\det(\tau) =\det(\rho)$.
In this case, the component is defined to be the identity map to
$\det(\rho)\otimes\d_\sigma(L(\sigma)^{r_\rho-i})_{A(\rho)}$.
The commutativity of the diagram
\begin{equation}
\begin{array}{ccc}
\makebox[70pt]{}f_*{\bf D}(L)(\rho)^i &
\mathop{\longrightarrow}\limits^{d^i(\rho/\rho')} &
f_*{\bf D}(L)(\rho')^{i+1}\makebox[50pt]{} \\
{\kappa(f, L)(\rho)^i}\downarrow &
&\downarrow{\kappa(f, L)(\rho')^{i+1}} \\
\makebox[70pt]{}{\bf D}(f_*L)(\rho)^i &
\mathop{\longrightarrow}\limits^{d^i(\rho/\rho')} &
{\bf D}(f_*L)(\rho')^{i+1}\makebox[50pt]{}
\end{array}
\end{equation}
is checked by the definitions.
The only one nontrivial case is the commutativity for the component
$\det(\tau)\otimes\d_\sigma(L(\sigma)^{r_\tau-i})_{A(\rho)}$
of $f_*{\bf D}(L)(\rho)^i$ with $r_\tau = r_\rho - 1$.
Since $\kappa(f, L)(\rho)^i$ is a zero map on this
component, we have to show that the composite
$\kappa(f, L)(\rho')^{i+1}\cdot d_{f_*{\bf D}(L)}^i(\rho/\rho')$
is zero on it.
Since $r_\tau = r_\rho - 1$, there exist exactly two cones
$\tau_1,\tau_2$ in $f^{-1}(\rho)$ with $\tau\prec\tau_1,\tau_2$
and $r_{\tau_1} = r_{\tau_2} = r_\rho$.
We have $q'_{\tau/\tau_1} + q'_{\tau/\tau_2} = 0$ under the
identification $\det(\tau_1) =\det(\tau_2) =\det(\rho)$.
Hence the composite is zero on this component.
Thus we know that $\kappa(f, L)$ is an unmixed homomorphism
in $\mathop{\rm CGEM}\nolimits(\Delta)$.
\begin{Prop}
\label{Prop 2.6}
Let $f\mathrel{:}\Delta'\rightarrow\Delta$ be a subdivision and
let $L^\bullet$ be an object of $\mathop{\rm CGEM}\nolimits(\Delta')$.
Then the unmixed homomorphism
\begin{equation}
\kappa(f, L)\mathrel{:} f_*{\bf D}(L)^\bullet\longrightarrow{\bf D}(f_*L)^\bullet
\end{equation}
is quasi-isomorphic.
\end{Prop}
{\sl Proof.}\quad
We prove the proposition by induction on the total dimension
of $L^\bullet$.
The assertion is trivially true if $\dim_{\bf Q} L = 0$.
We assume that $\dim_{\bf Q} L > 0$.
Let $\sigma$ be a maximal element of $\Delta'$ such that
$L(\sigma)^\bullet$ is nontrivial.
Take the maximal integer $p$ such that $L(\sigma)^p\not=\{0\}$
and the minimal integer $q$ such that $L(\sigma)_q^p\not=\{0\}$.
We define an object $E_{\sigma,p,q}^\bullet$ of $\mathop{\rm CGEM}\nolimits(\Delta')$
as follows.
We define $E_{\sigma,p,q}(\tau)^\bullet :=\{0\}$ for
$\tau\in\Delta'$ with $\tau\not=\sigma$ and
$E_{\sigma,p,q}(\sigma)^i :=\{0\}$ for $i\not= p$.
The graded $A(\sigma)$-module $E_{\sigma,p,q}(\sigma)^p$
is defined to be $\bar A(\sigma)(-q)$, i.e.,
\begin{equation}
E_{\sigma,p,q}(\sigma)_j^p :=\left\{
\begin{array}{ll}
{\bf Q} & \hbox{ if } j = q \\
\{0\} & \hbox{ if } j\not= q
\end{array}
\right.\;.
\end{equation}
By taking a nontrivial $A(\sigma)$-homomoprhism
$g_0\mathrel{:} E_{\sigma,p,q}(\sigma)^p\rightarrow L(\sigma)^p$, we get an unmixed
homomorphism $g\mathrel{:} E_{\sigma,p,q}^\bullet\rightarrow L^\bullet$
such that $g(\sigma)^p = g_0$.
Let $K^\bullet$ be the cokernel of $g$.
Then we get a commutative diagram
\begin{equation}
\begin{array}{ccccccccc}
0 & \rightarrow & f_*{\bf D}(K)^\bullet &
\longrightarrow & f_*{\bf D}(L)^\bullet &
\longrightarrow & f_*{\bf D}(E_{\sigma,p,q})^\bullet & \rightarrow & 0 \\
& & \hbox{ }\downarrow{\kappa(f, K)}
& & \hbox{ }\downarrow{\kappa(f, L)}
& & \hbox{ }\downarrow{\kappa(f, E_{\sigma,p,q})} & & \\
0 & \rightarrow & {\bf D}(f_*K)^\bullet &
\longrightarrow & {\bf D}(f_*L)^\bullet &
\longrightarrow & {\bf D}(f_*E_{\sigma,p,q})^\bullet & \rightarrow & 0 \\
\end{array}
\;.
\end{equation}
Since the functors $f_*$ and ${\bf D}$ are exact, the two horizontal
lines in the diagram are short exact sequences.
Since $\dim_{\bf Q} K =\dim_{\bf Q} L - 1$, $\kappa(f, K)$ is a
quasi-isomorphism by the induction assumption.
Hence it is sufficient to show that $\kappa(f, E_{\sigma,p,q})$
is quasi-isomorphic.
Let $\rho$ be an element of $\Delta$.
If $f(\sigma)$ is not in $F(\rho)$, then both
$f_*{\bf D}(E_{\sigma,p,q})(\rho)^\bullet$ and
${\bf D}(f_*E_{\sigma,p,q})(\rho)^\bullet$ are zero.
We assume $\eta := f(\sigma)\in F(\rho)$.
Set
\begin{equation}
\Phi :=\{\tau\in\Delta'\mathrel{;}\sigma\prec\tau\in f^{-1}(\rho)\}\;.
\end{equation}
By (\ref{dual's image2}), we have
\begin{equation}
f_*{\bf D}(E_{\sigma,p,q})(\rho)^i =
\bigoplus_{\tau\in\Phi(p+i)}\det(\tau)\otimes\bar A(\sigma)_{A(\rho)}\;.
\end{equation}
Hence we know
\begin{equation}
\H^i(f_*{\bf D}(E_{\sigma,p,q})(\rho)^\bullet)\simeq
\H^{p+i}(E(\Phi,{\bf Z})^\bullet)\otimes\bar A(\sigma)_{A(\rho)}
\end{equation}
for $i\in{\bf Z}$.
On the other hand, ${\bf D}(f_*E_{\sigma,p,q})(\rho)^i$ is zero
if $i\not= r_\rho - p$, while
\begin{equation}
{\bf D}(f_*E_{\sigma,p,q})(\rho)^{r_\rho - p} =
\det(\rho)\otimes\bar A(\sigma)_{A(\rho)}
\end{equation}
by (\ref{image's dual}).
The cohomologies $\H^i(E(\Phi,{\bf Z})^\bullet)$ are zero for
$i\not= r_\rho$ and
$\H^{r_\rho}(E(\Phi,{\bf Z})^\bullet) =\det(\rho)$ by
Lemma~\ref{Lem 1.2},(2).
We know $\H^i(f_*{\bf D}(E_{\sigma,p,q})(\rho)^\bullet)$ is
zero for $i\not= r_\rho - p$ and isomorphic to
$\det(\rho)\otimes\bar A(\sigma)_{A(\rho)}$ for $i = r_\rho - p$.
Since $\kappa(f, E_{\sigma,p,q})(\rho)^{r_\rho-p}$ is surjective,
$\kappa(f, E_{\sigma,p,q})(\rho)$ is quasi-isomorphic.
\QED
The intersection complex has the following irreducibility.
\begin{Lem}
\label{Lem 2.7}
Let $\Delta$ be a finite fan, and ${\bf p}$ a perversity
of it.
Let $ L_1^\bullet$ and $L_2^\bullet$ be objects of
$\mathop{\rm CGEM}\nolimits(\Delta)$ which are quasi-isomorphic to
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)$.
If $h\mathrel{:} L_1^\bullet\rightarrow L_2^\bullet$ is a homomoprhism
such that
$h({\bf 0}/{\bf 0})\mathrel{:} L_1({\bf 0})^\bullet\rightarrow L_2({\bf 0})^\bullet$
is a quasi-isomorphism, then $h$ is a quasi-isomorphism.
\end{Lem}
{\sl Proof.}\quad
Suppose that $h$ is not quasi-isomorphic.
Let $\rho$ be a minimal element of $\Delta$ such that
$h(\rho/\rho)\mathrel{:} L_1(\rho)^\bullet\rightarrow L_2(\rho)^\bullet$
is not quasi-isomorphic.
By the assumption, $\rho$ is not equal to ${\bf 0}$.
By the minimality of $\rho$, the homomorphism
\begin{equation}
\i_\rho^\circ(h)\mathrel{:}
\i_\rho^\circ(L_1)^\bullet\rightarrow\i_\rho^\circ(L_2)^\bullet
\end{equation}
in $\mathop{\rm CGM}\nolimits(A(\rho))$ is a quasi-isomorphism.
For each of integer $i$, we have a
commutative diagram
\begin{equation}
\begin{array}{ccc}
\makebox[10pt]{}\H^i(\i_\rho^\circ L_1^\bullet) &
\mathop{\longrightarrow}\limits^{u^i} &
\H^i(\i_\rho^\circ L_2^\bullet)\makebox[10pt]{} \\
{\phi_1}\downarrow &
&\downarrow{\phi_2} \\
\makebox[10pt]{}\H^i(L_1(\rho)[1]^\bullet) &
\mathop{\longrightarrow}\limits^{v^i} &
\H^i(L_2(\rho)[1]^\bullet)\makebox[10pt]{}
\end{array}
,
\end{equation}
where $u^i$ and $v^i$ are the $A(\rho)$-homomorphism induced
by $\i_\rho^\circ(h)$ and $h(\rho/\rho)$, respectively.
Since $\i_\rho^\circ(h)$ is quasi-isomorphic, $u^i$ is an
isomorphism.
Since $L_2^\bullet$ is quasi-isomorphic to $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$,
$\phi_2$ is surjective by the construction of intersection
complexes \cite[Thm.2.9]{Ishida3}.
Hence $v^i$ is also surjective.
Since $\H^i(L_1(\rho)[1]^\bullet)$ and
$\H^i(L_2(\rho)[1]^\bullet)$ are finite dimensional
${\bf Q}$-vector spaces of same dimension, $v^i$ is an
isomorphism for each $i$, i.e., $h(\rho/\rho)$ is a
quasi-isomorphism.
This contradicts the assumption.
\QED
Since $\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})^0 ={\bf Q}$ and
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)({\bf 0})^i =\{0\}$ for $i\not= 0$,
$h({\bf 0}/{\bf 0})$ in the above lemma is quasi-isomorphic
if and only if the induced homomorphism
$\H^0(L_1({\bf 0})^\bullet)\rightarrow\H^0(L_2({\bf 0})^\bullet)$
is a nonzero map.
Let $f\mathrel{:}\Delta'\rightarrow\Delta$ be a subdivision and let
$L^\bullet$ be in $\mathop{\rm CGEM}\nolimits(\Delta')$.
Then, by the definition of the functor $f_*$,
the complex $\Gamma(f_*L)^\bullet$ is canonically
isomorphic to $\Gamma(L)^\bullet$.
In particular, if $f\mathrel{:}\Sigma\rightarrow\Delta$ is a barycentric
subdivision, then $\delta_{\Sigma/\Delta}$ in
(\ref{zzichom}) induces a homomorphism
\begin{equation}
\Gamma(\delta_{\Sigma/\Delta})\mathrel{:}\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet\longrightarrow
\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet
\end{equation}
in $\mathop{\rm CGM}\nolimits(A)$.
\begin{Thm}[Decomposition theorem]
\label{Thm 2.8}
Let $\Delta$ be a finite fan and $\Sigma$ a barycentric
subdivision of $\Delta$.
Then, for each integer $p$, the homomorphism of $A$-modules
\begin{equation}
\label{zzhomThm 2.7}
\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)\longrightarrow
\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)
\end{equation}
induced by $\Gamma(\delta_{\Sigma/\Delta})$ is a split surjection, i.e.,
is surjective and the kernel is a direct summand as an
$A$-module.
In particular,
\begin{equation}
\dim_{\bf Q}\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_{q}\leq
\dim_{\bf Q}\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)_{q}
\end{equation}
for any integers $p$, $q$.
\end{Thm}
{\sl Proof.}\quad
By applying the contravariant functor ${\bf D}$ to the
homomorphism (\ref{zzichom}), we get a homomorphism
\begin{equation}
\label{zzrevichom}
{\bf D}(\delta_{\Sigma/\Delta})\mathrel{:}{\bf D}(\mathop{\rm ic}\nolimits(\Delta))^\bullet
\longrightarrow{\bf D}(f_*\mathop{\rm ic}\nolimits(\Sigma))^\bullet\;.
\end{equation}
Since $\delta_{\Sigma/\Delta}({\bf 0})$ is an isomorphism, so is
${\bf D}(\delta_{\Sigma/\Delta})({\bf 0})$.
Since ${\bf D}(\mathop{\rm ic}\nolimits(\Sigma))^\bullet$ is quasi-isomorphic to
$\mathop{\rm ic}\nolimits(\Sigma)^\bullet$ by \cite[Cor.2.12]{Ishida3},
$f_*{\bf D}(\mathop{\rm ic}\nolimits(\Sigma))^\bullet$ and $f_*\mathop{\rm ic}\nolimits(\Sigma)^\bullet$
are also quasi-isomorphic by Lemma~\ref{Lem 2.3},(1).
By Proposition~\ref{Prop 2.6}, we get a quasi-isomorphism
\begin{equation}
\kappa(f,\mathop{\rm ic}\nolimits(\Sigma))\mathrel{:}
f_*{\bf D}(\mathop{\rm ic}\nolimits(\Sigma))^\bullet\longrightarrow{\bf D}(f_*\mathop{\rm ic}\nolimits(\Sigma))^\bullet\;.
\end{equation}
Hence ${\bf D}(f_*\mathop{\rm ic}\nolimits(\Sigma))^\bullet$ is quasi-isomorphic to
$f_*\mathop{\rm ic}\nolimits(\Sigma)^\bullet$.
On the other hand, ${\bf D}(\mathop{\rm ic}\nolimits(\Delta))^\bullet$ is quasi-isomorphic
to $\mathop{\rm ic}\nolimits(\Delta)^\bullet$ by \cite[Cor.2.12]{Ishida3}.
By applying \cite[Lem 2.16]{Ishida3} for the homomorphism
${\bf D}(\delta_{\Sigma/\Delta})$, we get $L^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta)$,
a quasi-isomorphism $g\mathrel{:} L^\bullet\rightarrow\mathop{\rm ic}\nolimits(\Delta)^\bullet$
and a homomorphism
$h\mathrel{:} L^\bullet\rightarrow f_*\mathop{\rm ic}\nolimits(\Sigma)^\bullet$ such that the
homomorphisms of cohomologies induced by $h$ is compatible
with those induced by ${\bf D}(\delta_{\Sigma/\Delta})$.
Note that all homomorphisms in $\mathop{\rm CGEM}\nolimits(\Delta)$ appeared
here are quasi-isomorphic on ${\bf 0}\in\Delta$.
Since $L^\bullet$ and $\mathop{\rm ic}\nolimits(\Delta)^\bullet$ are quasi-isomorphic,
the composite $\delta_{\Sigma/\Delta}\cdot h\mathrel{:}
L^\bullet\rightarrow\mathop{\rm ic}\nolimits(\Delta)^\bullet$
is a quasi-isomorphism by Lemma~\ref{Lem 2.7}.
Hence, for each integer $i$, the homomorphism (\ref{zzhomThm 2.7})
is surjective and $\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)$ is the
direct sum of the kernel and the image of the homomorphism
$\H^p(L^\bullet)\rightarrow\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)$ induced
by $h$.
\QED
\section{Intersection complexes for simplicial fans}
\setcounter{equation}{0}
\begin{Lem}
\label{Lem 3.1}
Let $\pi$ be a simplicial cone.
Then
\begin{equation}
\H^i(\mathop{\rm ic}\nolimits_\t(F(\pi))(\pi)^\bullet)_j\simeq\left\{
\begin{array}{ll}
{\bf Q} & \hbox{ if } (i, j) = (r_\pi, 0) \\
\{0\} & \hbox{ if } (i, j)\not= (r_\pi, 0)
\end{array}
\right.\;.
\end{equation}
On the other hand,
\begin{equation}
\H^i(\i_\pi^*\mathop{\rm ic}\nolimits_\t(F(\pi))^\bullet)_j\simeq\left\{
\begin{array}{ll}
{\bf Q} & \hbox{ if } (i, j) = (0, -r_\pi) \\
\{0\} & \hbox{ if } (i, j)\not= (0, -r_\pi)
\end{array}
\right.\;.
\end{equation}
\end{Lem}
{\sl Proof.}\quad
The first assertion follows from the description (\ref{top}).
Let $s := r_\pi$ and $\{x_1,\cdots, x_s\}\subset N_{\bf Q}$ be a
minimal generator of the simplicial cone $\pi$.
For each element $\sigma\in F(\pi)$, there exists a unique
subset $\{i_1,\cdots, i_p\}$ of $\{1,\cdots, s\}$ with
$i_1 <\cdots < i_p$ such that $\{x_{i_1},\cdots, x_{i_s}\}$
generates $\sigma$.
We denote $x(\sigma) := x_{i_1}\wedge\cdots\wedge x_{i_s}$.
For $\rho\in F(\pi)$, we have a description
\begin{equation}
\mathop{\rm ic}\nolimits_\t(F(\pi))(\rho)^i_{A(\pi)} =
\det(\rho)\otimes{\bigwedge}\!^\bullet(N(\pi)_{\bf Q}/N(\rho)_{\bf Q}) =
\bigoplus_{\sigma\in F(\rho')}\det(\rho)\otimes{\bf Q} x(\sigma)\;,
\end{equation}
where $\rho'$ is the complementary cone of $\rho$ in $\pi$,
i.e., the unique cone in $F(\pi)$ with $\rho\cap\rho' ={\bf 0}$
and $\rho +\rho' =\pi$.
Hence $\mathop{\rm ic}\nolimits_\t(F(\pi))(\rho)_{A(\pi)}^i$ has the
component $\det(\rho)\otimes{\bf Q} x(\sigma)$ if and only if
$\rho\in F(\sigma')$ for the complementary cone $\sigma'$
of $\sigma$.
Note that $x(\sigma)$ is a homogeneous element of degree
$-r_\sigma$ and $\sigma'$ is of dimension $r_\pi - r_\sigma$.
By this observation, we have
\begin{equation}
(\i_\pi^*\mathop{\rm ic}\nolimits_\t(F(\pi))^\bullet)_j\simeq
\bigoplus_{\sigma\in F(\pi)(-j)}E(F(\sigma'),{\bf Z})^\bullet\otimes{\bf Q} x(\sigma)
\end{equation}
for each integer $j$.
By Lemma~\ref{Lem 1.2},(1), this complex of ${\bf Q}$-vector spaces
has trivial cohomologies if $r_\pi + j > 0$ since
$\dim\sigma' = r_\pi + j$ for $\sigma\in F(\pi)(-j)$.
For $j = - r_\pi$, $F(\pi)(-j) =\{\pi\}$ and $\pi' ={\bf 0}$.
Hence $\H^0(\i_\pi^*\mathop{\rm ic}\nolimits_\t(F(\pi))^\bullet)_{-r_\pi}\simeq{\bf Q}$
and $\H^i(\i_\pi^*\mathop{\rm ic}\nolimits_\t(F(\pi))^\bullet)_{-r_\pi} =\{0\}$
for $i\not= 0$.
We get the lemma since the complexes are nontrivial only
for $-r_\pi\leq j\leq 0$.
\QED
\begin{Thm}
\label{Thm 3.2}
Let $\Delta$ be a simplicial finite fan.
Then, for any perversity ${\bf p}$ with $\b\leq{\bf p}\leq\t$,
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ is quasi-isomorphic to
$\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$.
\end{Thm}
{\sl Proof.}\quad
By \cite[Thm.2.9]{Ishida3}, it sufficient to show that
$\H^i(\mathop{\rm ic}\nolimits_{\bf p}(\Delta)(\rho)^\bullet)_j =\{0\}$
if $i +j\leq{\bf p}(\rho)$ and
$\H^i(\i_\rho^*(\mathop{\rm ic}\nolimits_{\bf p}(\Delta))^\bullet)_j =\{0\}$
if $i +j\geq{\bf p}(\rho)$ for $\rho\in\Delta\setminus\{{\bf 0}\}$.
Since $\mathop{\rm ic}\nolimits_{\bf p}(F(\rho))^\bullet$ is the restiction of
$\mathop{\rm ic}\nolimits_{\bf p}(\Delta)^\bullet$ to $F(\rho)$ and
$-r_\rho + 1\leq{\bf p}(\rho)\leq r_\rho - 1$ by the assumption, this
is a consequence of Lemma~\ref{Lem 3.1}.
The following theorem is equivalent to \cite[Prop.4.1]{Oda2}.
\begin{Thm}
\label{Thm 3.3}
Let $\Delta$ be a simplicial complete fan.
Then $\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q =\{0\}$ for
any integers $p, q\in{\bf Z}$ with $p\not= q + r$.
\end{Thm}
{\sl Proof.}\quad
By Lemma~\ref{Lem 1.10}, we have
$\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q =\{0\}$ for
$p > q + r$.
By the duality for complete fans \cite[Prop.2.5]{Ishida3},
we have $\H^p(\Gamma({\bf D}(\mathop{\rm ic}\nolimits_\t(\Delta)))^\bullet)_q =\{0\}$
for $p, q\in{\bf Z}$ with $(r-p) > (-r-q) + r$, i.e.,
with $p < q + r$.
${\bf D}(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet$ is quasi-isomorphic to
$\mathop{\rm ic}\nolimits_\b(\Delta)^\bullet$ by \cite[Cor.2.12]{Ishida3}.
We get the theorem, since $\mathop{\rm ic}\nolimits_\b(\Delta)^\bullet$ is
quasi-isomorphic to $\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$ by
Theorem~\ref{Thm 3.2}.
\QED
Let $f\mathrel{:}\Delta'\rightarrow\Delta$ be an arbitrary subdivision of
a finite fan.
We define an unmixed homomorphism
\begin{equation}
\label{image of top}
\bar\delta_{\Delta'/\Delta}\mathrel{:}
f_*\mathop{\rm ic}\nolimits_\t(\Delta')^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet
\end{equation}
in $\mathop{\rm CGEM}\nolimits(\Delta)$ as follows.
Note that
\begin{equation}
f_*\mathop{\rm ic}\nolimits_\t(\Delta')(\rho)^i :=
\bigoplus_{\sigma\in f^{-1}(\rho)(i)}
\det(\sigma)\otimes(\bar A(\sigma))_{A(\rho)}
\end{equation}
for each integer $i$, while
\begin{equation}
\mathop{\rm ic}\nolimits_\t(\Delta)(\rho)^i = \left\{
\begin{array}{ll}
\det(\rho)\otimes\bar A(\rho) &
\hbox{ if }i = r_\rho\\
\{0\} & \hbox{ if }i\not= r_\rho
\end{array}
\right.\;.
\end{equation}
For each $\sigma\in f^{-1}(\rho)$, the restriction to the
component $\det(\sigma)\otimes(\bar A(\sigma))_{A(\rho)}$
of the homomorphism
\begin{equation}
\bar\delta_{\Delta'/\Delta}(\rho)^i\mathrel{:}
f_*\mathop{\rm ic}\nolimits_\t(\Delta')(\rho)^i\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)(\rho)^i
\end{equation}
is defined to be zero if $i = r_\sigma < r_\rho$.
If $r_\sigma = r_\rho$, then $\det(\sigma) =\det(\rho)$
and $A(\sigma) = A(\rho)$, and the component is defined to
be the identity map to $\det(\rho)\otimes\bar A(\rho)$.
Let $N'$ be a free ${\bf Z}$-module of rank $r'$ and let
$f_0\mathrel{:} N\rightarrow N'$ be a surjective homomorphism.
Fans $\Delta$ of $N_{\bf R}$ and $\Phi$ of $N'_{\bf R}$ are said to
be compatible with $f_0$, if $f_0(\sigma)$ is contained in
a cone of $\Phi$ for every $\sigma\in\Delta$, where we
denote also by $f_0$ the linear map $N_{\bf R}\rightarrow N'_{\bf R}$.
We say a map $f\mathrel{:}\Delta\rightarrow\Phi$ a {\em morphism of fans}
if such $f_0$ is given and $f(\sigma)$ is the minimal cone
of $\Phi$ which contains $f_0(\sigma)$ for every
$\sigma\in\Delta$.
The direct image functor
\begin{equation}
f_*\mathrel{:}\mathop{\rm GEM}\nolimits(\Delta)\longrightarrow\mathop{\rm GEM}\nolimits(\Phi)
\end{equation}
is defined as follows.
Let $L$ be in $\mathop{\rm GEM}\nolimits(\Delta)$.
For each $\rho\in\Phi$, we set $f^{-1}(\rho) := f^{-1}(\{\rho\})$
similarly as in the case of subdivisions.
If $\sigma\in f^{-1}(\rho)$, then $f_0$ induces a homomorphism
$N(\sigma)\rightarrow N'(\rho)$ and a ring homomorphism
$A(\sigma)\rightarrow A'(\rho)$.
We set
\begin{equation}
(f_*L)(\rho) :=\bigoplus_{\sigma\in f^{-1}(\rho)}
L(\sigma)_{A'(\rho)}\;,
\end{equation}
where $L(\sigma)_{A'(\rho)} := L(\sigma)\otimes_{A(\sigma)}A'(\rho)$.
Let $g\mathrel{:} L\rightarrow K$ be a homomorphism in $\mathop{\rm GEM}\nolimits(\Delta)$.
For $\rho,\mu\in\Phi$ with $\rho\prec\mu$ and for
$\sigma\in f^{-1}(\rho)$ and $\tau\in f^{-1}(\mu)$, the
$(\sigma,\tau)$-component of the homomorphism
$f_*(g)(\rho/\mu)\mathrel{:}(f_*L)(\rho)\rightarrow(f_*K)(\mu)$
is the map induced by $g(\sigma/\tau)$ if $\sigma\prec\tau$
and is the zero map otherwise.
Let $\Delta$ and $\Phi$ be complete fans of $N_{\bf R}$ and
$N'_{\bf R}$, respectively, and let $f\mathrel{:}\Delta\rightarrow\Phi$ be a
morphism of fans associated with $f_0\mathrel{:} N\rightarrow N'$.
We take a generator $w$ of $\bigwedge^{r-r'}\ker f_0\simeq{\bf Z}$.
We define an unmixed homomorphism
\begin{equation}
\label{from direct image}
\bar\delta_{\Delta/\Phi}\mathrel{:}
f_*\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet\longrightarrow\mathop{\rm ic}\nolimits_\t(\Phi)[-r+r']^\bullet
\end{equation}
as follows.
Let $\rho$ be in $\Phi$.
For each $\sigma\in f^{-1}(\rho)$, the
$\sigma$-component of the homomorphism
\begin{equation}
\bar\delta_{\Delta/\Phi}(\rho)^i\mathrel{:} f_*\mathop{\rm ic}\nolimits_\t(\Delta)(\rho)^i
\longrightarrow\mathop{\rm ic}\nolimits_\t(\Phi)[-r+r'](\rho)^i
\end{equation}
is defined to be zero if $i\not= r_\rho + r -r'$ since
then
\begin{equation}
\mathop{\rm ic}\nolimits_\t(\Phi)[-r+r'](\rho)^i =
\mathop{\rm ic}\nolimits_\t(\Phi)(\rho)^{i-r+r'} =\{0\}\;.
\end{equation}
If $i = r_\rho + r -r'$, then
\begin{equation}
f_*\mathop{\rm ic}\nolimits_\t(\Delta)(\rho)^i =
\bigoplus_{\sigma\in f^{-1}(\rho)(r_\rho + r -r')}
\det(\sigma)\otimes\bar A(\sigma)_{A'(\rho)}
\end{equation}
and
\begin{equation}
\mathop{\rm ic}\nolimits_\t(\Phi)[-r+r'](\rho)^i =
\det(\rho)\otimes\bar A'(\rho)\;.
\end{equation}
If $\sigma\in f^{-1}(\rho)(r_\rho + r -r')$, then
$r_\sigma - r_\rho = r - r'$ and $\mathop{\rm Ker}\nolimits f_0\subset N(\sigma)$.
Hence $w\in\bigwedge^{r-r'}N(\sigma)$.
The $\sigma$-component of the homomorphism is defined to
be the tensor product of the isomorphism
$q_{\sigma/\rho}^w\mathrel{:}\det(\sigma)\rightarrow\det(\rho)$ which sends
$w\wedge a$ to $(\bigwedge^{r_\rho}f_0)(a)$ for
$a\in\bigwedge^{r_\rho}N(\sigma)$ and the natural isomorphism
$\bar A(\sigma)_{A'(\rho)}\simeq\bar A'(\rho)$, where
$\bigwedge^{r_\rho}f_0\mathrel{:}\bigwedge^{r_\rho}N(\sigma)\rightarrow\det(\rho)$
is the natural map induced by $f_0$.
Let $\eta$ be an element of a finite fan $\Delta$ of $N_{\bf R}$
and let $N'$ be the quotient free ${\bf Z}$-module $N/N(\eta)$ of
rank $r' := r - r_\eta$.
We set $\Delta(\eta{\prec}) :=
\{\sigma\in\Delta\mathrel{;}\eta\prec\sigma\}$ as before.
For each $\sigma\in\Delta(\eta{\prec})$, let $\sigma[\eta]$ be
the image of $\sigma$ in the quotient space
$N'_{\bf R} = N_{\bf R}/N(\eta)_{\bf R}$.
Then
\begin{equation}
\Delta[\eta] :=\{\sigma[\eta]\mathrel{;}\sigma\in\Delta(\eta{\prec})\}
\end{equation}
is a finite fan of $N'_{\bf R}$.
For $\tau\in\Delta[\eta]$, the notations $N'(\tau)\subset N'$ and
$A'(\tau)\subset A' :=\bigwedge^{r - r_\eta}N'_{\bf Q}$ are defined
similarly for $N'$ as we defined for $N$.
For $\sigma\in\Delta(\eta{\prec})$, $A'(\sigma[\eta])$
is a quotient ring of $A(\sigma)$ by the two-sided ideal
generated by $N(\eta)$.
Hence the category $\mathop{\rm GM}\nolimits(A'(\sigma[\eta]))$ of fintely generated
graded $A'(\sigma[\eta])$-modules is a full subcategory of
$\mathop{\rm GM}\nolimits(A(\sigma))$.
We define the categories $\mathop{\rm GEM}\nolimits(\Delta[\eta])$ and
$\mathop{\rm CGEM}\nolimits(\Delta[\eta])$, similarly.
We denote by $\epsilon_\eta$ the natural functor
\begin{equation}
\epsilon_\eta\mathrel{:}\mathop{\rm GEM}\nolimits(\Delta[\eta])\longrightarrow\mathop{\rm GEM}\nolimits(\Delta)
\end{equation}
defined by $\epsilon_\eta(L)(\sigma) := L(\sigma[\eta])$
if $\sigma\in\Delta(\eta{\prec})$ and
$\epsilon_\eta(L)(\sigma) :=\{0\}$ otherwise.
For $f\mathrel{:} L\rightarrow K$ in $\mathop{\rm GEM}\nolimits(\Delta[\eta])$,
$\epsilon_\eta(f)\mathrel{:}
\epsilon_\eta(L)\rightarrow\epsilon_\eta(K)$ is defined by
$\epsilon_\eta(f)(\sigma/\tau) := f(\sigma[\eta]/\tau[\eta])$
if $\sigma\in\Delta(\eta{\prec})$ and
$\epsilon_\eta(f)(\sigma/\tau) := 0$ otherwise.
We take a generator $w\in\det(\eta) =\mathop{\rm Ker}\nolimits f_0\simeq{\bf Z}$.
We define an isomorphism
\begin{equation}
\label{quotient iso}
h(\Delta,\eta, w)\mathrel{:}
\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))^\bullet\simeq
\epsilon_\eta(\mathop{\rm ic}\nolimits_\t(\Delta[\eta]))[-r_\eta]^\bullet
\end{equation}
as follows.
For each $\sigma\in\Delta(\eta{\prec})$, we have
\begin{equation}
\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))(\sigma)^{r_\sigma} =
\det(\sigma)\otimes\bar A(\sigma)\;,
\end{equation}
and
\begin{equation}
\epsilon_\eta(\mathop{\rm ic}\nolimits_\t(\Delta[\eta]))[-r_\eta](\sigma)^{r_\sigma}
= \det(\sigma[\eta])\otimes\bar A'(\sigma[\eta])
\end{equation}
by (\ref{top}), while
\begin{equation}
\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))(\sigma)^i =
\epsilon_\eta(\mathop{\rm ic}\nolimits_\t(\Delta[\eta]))[-r_\eta](\sigma)^i =\{0\}
\end{equation}
for $i\not = r_\sigma$.
Since $\bar A(\sigma) = A(\sigma)/N(\sigma)A(\sigma)$ and
$\bar A'(\sigma[\eta]) =
A'(\sigma[\eta])/N'(\sigma[\eta])A'(\sigma[\eta])$
and since the kernel of the surjection
$A(\sigma)\rightarrow A'(\sigma[\eta])$ is
$N(\eta)A(\sigma)\subset N(\sigma)A(\sigma)$,
$\bar A'(\sigma[\eta])$ is naturally isomorphic to
$\bar A(\sigma)$ as an $A(\sigma)$-module.
we define an isomorphism
\begin{equation}
h(\Delta,\eta, w)(\sigma)^{r_\sigma}\mathrel{:}
\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))(\sigma)^{r_\sigma}\longrightarrow
\epsilon_\eta(\mathop{\rm ic}\nolimits_\t(\Delta[\eta]))[-1](\sigma)^{r_\sigma}
\end{equation}
to be the tensor product of $q_{\sigma/\sigma[\eta]}^w$ and
this isomorphism.
The commutativity with the coboundary maps is checked
easily.
When $r_\eta = 1$, we take the primitive element of
$\eta\cap N$ as $w$ and denote the ismorphism simply by
$h(\Delta,\eta)$.
\begin{Lem}
\label{Lem 3.4}
Let $\eta$ be a cone of a complete fan $\Delta$.
We define $N'$, $f_0\mathrel{:} N_{\bf R}\rightarrow N'_{\bf R}$ and $\Delta[\eta]$
as above.
If $f_0$ induces a morphism $f\mathrel{:}\Delta\rightarrow\Delta[\eta]$ of
fans, then the homomorhism
\begin{equation}
\H^i(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec})))^\bullet)\longrightarrow
\H^i(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)
\end{equation}
induced by the inclusion map
$\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$
is injective for every $i\in{\bf Z}$.
\end{Lem}
{\sl Proof.}\quad
We take a generator $w$ of $\det(\eta)$.
By definitions, the composite of homomorphisms
\begin{equation}
\label{composite 3.4}
\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec})))^\bullet
\mathop{\longrightarrow}\limits^{\phi}
\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet\rightarrow
\Gamma(f_*\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet\rightarrow
\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta[\eta])[-r_\eta])^\bullet
\end{equation}
induced by the homomorphisms
$\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))^\bullet\rightarrow
\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$
and
$\bar\delta_{\Delta/\Delta[\eta]}\mathrel{:} f_*\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet\rightarrow
\mathop{\rm ic}\nolimits_\t(\Delta[\eta])[-r_\eta]^\bullet$
is equal to the homomoprhism
\begin{equation}
\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec})))^\bullet\longrightarrow
\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta[\eta])[-r_\eta])^\bullet
\end{equation}
induced by $h(\Delta,\eta, w)$.
Since $h(\Delta,\eta, w)$ is an isomorphism, the
homomorphisms of cohomologies induced by $\phi$ in
(\ref{composite 3.4}) are injective.
\QED
\begin{Thm}
\label{Thm 3.5}
Let $\Delta$ be a simplicial complete fan and let
$\eta$ be an element of $\Delta$.
Then the homomorphism
\begin{equation}
\H^i(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec})))^\bullet)\longrightarrow
\H^i(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)
\end{equation}
induced by the inclusion map
$\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$
is injective for every $i\in{\bf Z}$.
\end{Thm}
{\sl Proof.}\quad
Let $g\mathrel{:} L^\bullet\rightarrow K^\bullet$ be a quasi-isomorphism in
$\mathop{\rm CGEM}\nolimits(\Delta)$.
Then we get a commutative diagram
\begin{equation}
\begin{array}{ccc}
\makebox[20pt]{}\H^i(\Gamma(L|\Delta(\eta{\prec}))^\bullet) &
\mathop{\longrightarrow}\limits^{\phi_1} &
\makebox[10pt]{}\H^i(\Gamma(L)^\bullet) \\
{g_1}\downarrow &
&\downarrow{g_2} \\
\makebox[20pt]{}\H^i(\Gamma(K|\Delta(\eta{\prec}))^\bullet) &
\mathop{\longrightarrow}\limits^{\phi_2} &
\makebox[10pt]{}\H^i(\Gamma(K)^\bullet)
\end{array}
,
\end{equation}
for every $i\in{\bf Z}$.
Since $g$ is a quasi-isomorphism, $g_1$ and $g_2$ are isomorphisms.
Hence $\phi_1$ is injective if and only if $\phi_2$ is
injective.
This implies that, for the proof of the theorem, it is sufficient
to show the injectivity of the homomorphisms
\begin{equation}
\H^i(\Gamma(L|\Delta(\eta{\prec}))^\bullet)\longrightarrow
\H^i(\Gamma(L)^\bullet)
\end{equation}
for $i\in{\bf Z}$ for an $L^\bullet$ which is quasi-isomorphic to
$\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$.
Let $N' := N/N(\eta)$ and $f_0\mathrel{:} N_{\bf R}\rightarrow N'_{\bf R}$ be the
natural surjection.
We take a simplicial subdivision $u\mathrel{:}\Delta'\rightarrow\Delta$ such
that $\eta\in\Delta'$,
$\Delta'(\eta{\prec}) =\Delta(\eta{\prec})$ and
$f_0$ induces a morphism $f\mathrel{:}\Delta'\rightarrow\Delta'[\eta]$.
Then $u_*\mathop{\rm ic}\nolimits_\t(\Delta'(\eta{\prec}))^\bullet$ is
equal to $\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec}))^\bullet$.
Since
$\Gamma(u_*\mathop{\rm ic}\nolimits_\t(\Delta'))^\bullet =\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta'))^\bullet$,
the homomorphism
\begin{equation}
\label{inj 3.5}
\H^i(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta(\eta{\prec})))^\bullet) =
\H^i(\Gamma(u_*\mathop{\rm ic}\nolimits_\t(\Delta'(\eta{\prec})))^\bullet)\longrightarrow
\H^i(\Gamma(u_*\mathop{\rm ic}\nolimits_\t(\Delta'))^\bullet)
\end{equation}
is injective for every $i\in{\bf Z}$ by Lemma~\ref{Lem 3.4}.
By applying the dualizing functor to the homomorphism
\begin{equation}
\bar\delta_{\Delta'/\Delta}\mathrel{:}
u_*\mathop{\rm ic}\nolimits_\t(\Delta')^\bullet\longrightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet\;,
\end{equation}
we get a homomorphism
\begin{equation}
{\bf D}(\bar\delta_{\Delta'/\Delta})\mathrel{:}
{\bf D}(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet\longrightarrow{\bf D}(u_*\mathop{\rm ic}\nolimits_\t(\Delta'))^\bullet\;.
\end{equation}
By \cite[Cor.2.12]{Ishida3} and Theorem~\ref{Thm 3.2},
${\bf D}(\mathop{\rm ic}\nolimits_\t(\Delta'))^\bullet$ is quasi-isomorphic to
$\mathop{\rm ic}\nolimits_\t(\Delta')^\bullet$ in $\mathop{\rm CGEM}\nolimits(\Delta')$.
Hence, by Lemma~\ref{Lem 2.3}, (1) and Proposition~\ref{Prop 2.6},
$K^\bullet :={\bf D}(u_*\mathop{\rm ic}\nolimits_\t(\Delta'))^\bullet$ is quasi-isomorphic
to $u_*\mathop{\rm ic}\nolimits_\t(\Delta')^\bullet$.
By the injectivity of (\ref{inj 3.5}), the homomorphism
\begin{equation}
\label{inj K}
\H^i(\Gamma(K|\Delta(\eta{\prec}))^\bullet)\longrightarrow
\H^i(\Gamma(K)^\bullet)
\end{equation}
is injective for every $i\in{\bf Z}$.
On the other hand, $L^\bullet :={\bf D}(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet$ is
quasi-isomorphic to $\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$ by
\cite[Cor.2.12]{Ishida3} and Theorem~\ref{Thm 3.2}.
Hence the theorem is equivalent to the injectivity of the
homomorphism
\begin{equation}
\label{inj L}
\H^i(\Gamma(L|\Delta(\eta{\prec}))^\bullet)\longrightarrow
\H^i(\Gamma(L)^\bullet)
\end{equation}
for $i\in{\bf Z}$.
Let $\Phi :=\bigcup_{\sigma\in\Delta(\eta{\prec})}F(\sigma)$.
Then $\Phi$ is a common subfan of $\Delta$ and $\Delta'$
which contains $\Delta(\eta{\prec})$.
In particular,
$(u_*\mathop{\rm ic}\nolimits_\t(\Delta')|\Phi)^\bullet = (\mathop{\rm ic}\nolimits_\t(\Delta)|\Phi)^\bullet$.
Hence, by the definition of ${\bf D}$, we have
$(L|\Phi)^\bullet = (K|\Phi)^\bullet$ and hence
$(L|\Delta(\eta{\prec}))^\bullet =
(K|\Delta(\eta{\prec}))^\bullet$
We get a commutative diagram
\begin{equation}
\begin{array}{ccc}
\makebox[20pt]{}\H^i(\Gamma(L|\Delta(\eta{\prec}))^\bullet) &
\longrightarrow &
\makebox[10pt]{}\H^i(\Gamma(L)^\bullet) \\
\downarrow &
&\downarrow \\
\makebox[20pt]{}\H^i(\Gamma(K|\Delta(\eta{\prec}))^\bullet) &
\longrightarrow &
\makebox[10pt]{}\H^i(\Gamma(K)^\bullet)
\end{array}
,
\end{equation}
Since ${\bf D}(\bar\delta_{\Delta'/\Delta})$ induces the identity
map of $\H^i(\Gamma(L|\Delta(\eta{\prec}))^\bullet)$ and
$\H^i(\Gamma(K|\Delta(\eta{\prec}))^\bullet)$,
the injectivity of (\ref{inj L}) follows from that of
(\ref{inj K}).
\QED
The following theorem is essentially equal to
\cite[Thm.4.2]{Oda2}.
We write here the proof in our notation in order to show
that we need not use the corresponding toric varieties.
\begin{Thm}
\label{Thm 3.6}
Let $\tilde\Delta$ be a complete simplicial fan of
$N_{\bf R}$ and $\gamma$ a one-dimensional cone in $\tilde\Delta$.
Let $\Delta$ be the subfan
$\tilde\Delta\setminus\tilde\Delta(\gamma{\prec})$ of
$\tilde\Delta$.
Then
\begin{equation}
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q =\{0\}
\end{equation}
for all $p\not= q + r$.
\end{Thm}
{\sl Proof.}\quad
Since $\Delta(\gamma{\prec})$ is star closed in $\tilde\Delta$,
there exists a short sequence
\begin{equation}
\label{ic short exact}
0\rightarrow\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet
\longrightarrow\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta))^\bullet
\longrightarrow\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet\rightarrow 0\;.
\end{equation}
We consider the homogeneous degree $q$-part
\begin{equation}
\label{long exact}
\begin{array}{cccccc}
& & \rightarrow &
\H^{p-1}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta))^\bullet)_q &
\rightarrow &
\H^{p-1}(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q \\
\rightarrow &
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet)_q &
\rightarrow &
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta))^\bullet)_q &
\rightarrow &
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q \\
\rightarrow &
\H^{p+1}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet)_q &
\rightarrow
\end{array}
\end{equation}
of the long exact sequence obtained by (\ref{ic short exact})
for each integer $q$.
Since $\tilde\Delta$ is a simplicial complete fan, so is
the fan $\tilde\Delta[\gamma]$ of the $(r-1)$-dimensional
space $N'_{\bf R}$.
By the isomorphism
\begin{equation}
h(\tilde\Delta,\gamma)\mathrel{:}
\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec}))^\bullet\simeq
\epsilon_\gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta[\gamma]))[-1]^\bullet\;,
\end{equation}
we have an isomorphism
\begin{equation}
\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet\simeq
\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta[\gamma]))[-1]^\bullet\;,
\end{equation}
of graded $A$-modules.
By Theorem~\ref{Thm 3.3} applied for $\tilde\Delta[\gamma]$,
we have
\begin{equation}
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta[\gamma]))[-1]^\bullet)_q =
\H^{p-1}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta[\gamma]))^\bullet)_q =
\{0\}
\end{equation}
for $p - 1\not= q + r - 1$, i.e., for $p\not= q + r$.
Hence
$\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet)_q =\{0\}$
for $p\not= q + r$.
Since $\tilde\Delta$ is a simplicial complete fan of $N_{\bf R}$,
we have
\begin{equation}
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta))^\bullet)_q =\{0\}
\end{equation}
for $p\not= q + r$ by Theorem~\ref{Thm 3.3}.
If $p\not= q + r - 1, q + r$, then we have
$\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q =\{0\}$ by the
long exact sequence.
We have
$\H^{q+r-1}(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q =\{0\}$
for every $q$, since the homomorphism
\begin{equation}
\label{iso to the shift}
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet)\rightarrow
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta))^\bullet)
\end{equation}
is injective by Theorem~\ref{Thm 3.5}
\QED
\begin{Lem}
\label{Lem 3.7}
Under the same assumption as the above theorem,
the homomorphism
\begin{equation}
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta))^\bullet)_q\longrightarrow
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits_\t(\Delta))^\bullet)_q
\end{equation}
induced by the natural homomorphism
$\mathop{\rm ic}\nolimits_\t(\tilde\Delta)^\bullet\rightarrow\mathop{\rm ic}\nolimits_\t(\Delta)^\bullet$
is surjective.
\end{Lem}
{\sl Proof.}\quad
This lemma follows from the long exact sequence
(\ref{long exact}) in the proof of the theorem, since
$\H^{q+r+1}(\Gamma(\mathop{\rm ic}\nolimits_\t(\tilde\Delta(\gamma{\prec})))^\bullet)_q =\{0\}$.
\QED
\section{The diagonal theorems for a complete fan and a cone}
\setcounter{equation}{0}
In this section, we prove two diagonal theorems.
\begin{Thm}[The first diagonal theorem]
\label{Thm 4.1}
Let $\Delta$ be a complete fan of $N_{\bf R}$.
Then
\begin{equation}
\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_q =\{0\}
\end{equation}
for $p, q\in{\bf Z}$ with $p\not= q+r$.
\end{Thm}
{\sl Proof.}\quad
Let $\Sigma\rightarrow\Delta$ be a barycentric subdivision.
Since $\Sigma$ is a simplicial fan, $\mathop{\rm ic}\nolimits(\Sigma)^\bullet$
is quasi-isomorphic to $\mathop{\rm ic}\nolimits_\t(\Sigma)^\bullet$ by
Theorem~\ref{Thm 3.2}.
Since $\Delta$ is complete, so is $\Sigma$.
Hence $\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)_{q} =\{0\}$ for
$p\not= q + r$ by Theorem~\ref{Thm 3.3}.
We get the theorem, since
\begin{equation}
\dim_{\bf Q}\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_{q}\leq
\dim_{\bf Q}\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)_{q}
\end{equation}
for any $p, q$ by Theorem~\ref{Thm 2.8}.
\QED
\begin{Thm}
\label{Thm 4.2}
Let $\Delta$ be a finite fan which may not be complete.
Assume that there exists a complete fan $\tilde\Sigma$ and
a one-dimensional cone $\gamma\in\tilde\Sigma$ such that
$\Sigma :=\tilde\Sigma\setminus\tilde\Sigma(\gamma{\prec})$ is a
barycentric subdivision of $\Delta$.
Then
\begin{equation}
\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_q = 0
\end{equation}
for $p\not= q+r$.
\end{Thm}
{\sl Proof.}\quad
By Theorem~\ref{Thm 3.6}, we have
$\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)_q = 0$ for $p\not= q+r$.
Then the lemma is a consequence of Theorem~\ref{Thm 2.8}.
\QED
\begin{Thm}[The second diagonal theorem]
\label{Thm 4.3}
Let $\pi\subset N_{\bf R}$ be a cone of dimension $r$.
Then
\begin{equation}
\label{vanishing}
\H^p(\Gamma(\mathop{\rm ic}\nolimits(F(\pi){\setminus}\{\pi\}))^\bullet)_q =\{0\}
\end{equation}
unless $p + q\geq 0, p\not= q + r - 1$ or
$p + q\leq -1, p\not= q + r$.
\end{Thm}
{\sl Proof.}\quad
Since $0\leq r_\sigma\leq r-1$ for $\sigma\in F(\pi){\setminus}\{\pi\}$,
$\Gamma(\mathop{\rm ic}\nolimits(F(\pi){\setminus}\{\pi\}))_q^p =\{0\}$
unless $0\leq p\leq r-1$ and $-r\leq q\leq 0$
by \cite[Prop.2.11]{Ishida3}.
In order to prove the theorem, it is sufficient to show
the vanishing (\ref{vanishing}) for $p, q$ with $p\not= q + r - 1$
and $p + q\geq 0$, since then Theorem~\ref{Thm 2.1} implies the
vanishing (\ref{vanishing}) for $p, q$ with $p\not= q + r$
and $p + q\leq -1$.
We denote simply by $\mathop{\rm ic}\nolimits(\pi)^\bullet$ the complex
$\mathop{\rm ic}\nolimits(F(\pi))(\pi)^\bullet$ in $\mathop{\rm CGM}\nolimits(A)$.
Since
\begin{eqnarray}
\mathop{\rm ic}\nolimits(\pi)^\bullet & = &
\mathop{\rm gt}\nolimits^{\geq1}(\i_\pi^\circ(\mathop{\rm ic}\nolimits(F(\pi)))[-1])^\bullet \\
& = &
\mathop{\rm gt}\nolimits^{\geq1}(\Gamma(\mathop{\rm ic}\nolimits(F(\pi){\setminus}\{\pi\}))[-1])^\bullet\\
& = &
(\mathop{\rm gt}\nolimits^{\geq0}\Gamma(\mathop{\rm ic}\nolimits(F(\pi){\setminus}\{\pi\})))[-1]^\bullet\;,
\end{eqnarray}
we have
\begin{equation}
\H^p(\Gamma(\mathop{\rm ic}\nolimits(F(\pi){\setminus}\{\pi\}))^\bullet)_q =
\H^{p+1}(\mathop{\rm ic}\nolimits(\pi)^\bullet)_q
\end{equation}
for $p + q\geq 0$.
Hence it is sufficient to show that
$\H^p(\mathop{\rm ic}\nolimits(\pi)^\bullet)_q = 0$ for $p\not= q + r$.
We take a one-dimensional cone $\gamma$ of $N_{\bf R}$ such that
$-\gamma$ intersects the interior of $\pi$.
We set
\begin{equation}
\Delta := (F(\pi)\setminus\{\pi\})\cup
\{\gamma+\sigma\mathrel{;}\sigma\in F(\pi)\setminus\{\pi\}\}
\end{equation}
and
\begin{equation}
\tilde\Delta :=\Delta\cup\{\pi\}\;.
\end{equation}
Then $\tilde\Delta$ is a complete fan.
Let $\tilde\Sigma$ be a barycentric subdivision of
$\tilde\Delta$ and let $\eta\in\tilde\Sigma$ be the
one-dimensional cone which intersects the interior of $\pi$.
Then $\Sigma :=\tilde\Sigma\setminus\tilde\Sigma(\eta{\prec})$
is a barycentric subdivision of $\Delta$.
By Theorems~\ref{Thm 4.1} and \ref{Thm 4.2},
$\H^p(\Gamma(\mathop{\rm ic}\nolimits(\tilde\Delta))^\bullet)_q$ and
$\H^p(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_q$ are zero unless
$p = q + r$.
By the exact sequence
\begin{equation}
0\rightarrow\mathop{\rm ic}\nolimits^\bullet(\pi)\rightarrow\Gamma(\mathop{\rm ic}\nolimits(\tilde\Delta))^\bullet
\rightarrow\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet\rightarrow 0\;
\end{equation}
we get the exact sequence
\begin{equation}
\begin{array}{ccccccc}
0 & \rightarrow & \H^{q+r}(\mathop{\rm ic}\nolimits^\bullet(\pi))_q & \rightarrow &
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits(\tilde\Delta))^\bullet)_q &
\mathop{\rightarrow}\limits^{\varphi} & \\
& & \H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_q & \rightarrow &
\H^{q+r+1}(\mathop{\rm ic}\nolimits^\bullet(\pi))_q & \rightarrow & 0
\end{array}
\;,
\end{equation}
while we have $\H^p(\mathop{\rm ic}\nolimits^\bullet(\pi))_q =\{0\}$
for $p\not= q + r, q + r + 1$.
Hence it is sufficient to show the surjectivity of $\varphi$
for each $q\in{\bf Z}$.
Then we get a commutative diagram of canonical homomorphisms
\begin{equation}
\begin{array}{ccc}
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits(\tilde\Sigma))^\bullet)_q &
\mathop{\longrightarrow}\limits^{\varphi'} &
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits(\Sigma))^\bullet)_q \\
\hbox{ }\downarrow{\scriptstyle\psi'} & &\hbox{ }\downarrow{\scriptstyle\psi}
\\
\H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits(\tilde\Delta))^\bullet)_q &
\mathop{\longrightarrow}\limits^{\varphi} & \H^{q+r}(\Gamma(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_q
\end{array}
\;.
\end{equation}
Since $\tilde\Sigma$ is simplicial, $\mathop{\rm ic}\nolimits(\tilde\Sigma)^\bullet$
is quasi-isomorphic to $\mathop{\rm ic}\nolimits_\t(\tilde\Sigma)^\bullet$.
Hence $\varphi'$ in the diagram is surjective
by Lemma~\ref{Lem 3.7}.
Since $\Sigma$ is a barycentric subdivision of $\Delta$, $\psi$
is also surjcetive by Theorem~\ref{Thm 2.8}
Hence $\varphi$ is surjective for every $q$.
\QED
Note that the theorem says that the cohomologies may not
zero only for $(p, q) = (i-1, i-r)$ for $r/2 < i\leq r$
or $(p, q) = (i, i-r)$ for $0\leq i < r/2$.
Here we comment the relation with the results of \cite{Oda3}.
If the cone $\pi$ is of dimension $r$ and $F(\pi)\setminus\{\pi\}$
is a simplicial fan, then $\mathop{\rm ic}\nolimits(F(\pi){\setminus}\{\pi\})^\bullet$
is quasi-isomorphic to $\mathop{\rm ic}\nolimits_\t(F(\pi){\setminus}\{\pi\})^\bullet$
by Theorem~\ref{Thm 3.2}.
Hence the second diagonal theorem implies that
\begin{equation}
\label{II for simp}
\H^p(\Gamma(\mathop{\rm ic}\nolimits_\t(F(\pi){\setminus}\{\pi\}))^\bullet)_q =\{0\}
\end{equation}
unless $p + q\geq 0, p\not= q + r - 1$ or
$p + q\leq -1, p\not= q + r$.
Let $\Pi$ be a simplicial complete fan of $N_{\bf R}$.
Suppose a continuous map $h\mathrel{:} N_{\bf R}\rightarrow{\bf R}$ satisfies the
following conditions.
(1) $h$ is linear on each cone $\sigma\in\Pi$.
(2) $h$ has rational values on $N_{\bf Q}$.
(3) $h$ is strictly convex with respect to the fan $\Pi$,
i.e., $h(x) + h(y)\geq h(x + y)$ and the equality holds
only if $x, y$ are in a common cone of $\Pi$.
We set $N' := N\oplus{\bf Z}$ and
$\tilde\sigma :=\{(x,h(x))\mathrel{;} s\in\sigma\}\subset N'_{\bf R}$ for
each $\sigma\in\Pi$, and define
\begin{equation}
\tilde\Pi :=\{\tilde\sigma\mathrel{;}\sigma\in\Pi\}\;.
\end{equation}
Then $\tilde\Pi$ is a simplicial fan of the $(r+1)$-dimensional
space $N'_{\bf R}$ and $\tilde\Pi = F(\pi)\setminus\{\pi\}$ for
the $(r+1)$-dimensional cone
$\pi := \{(x, y)\mathrel{;} x\in N_{\bf R}, y\geq h(x)\}\subset N'_{\bf R}$.
We see that the complex $C^\bullet(\tilde\Pi,\tilde{\cal G}_\ell)$ defined
in \cite{Oda3} is isomorphic to $\mathop{\rm ic}\nolimits_\t(\tilde\Pi)_{\ell-r-1}^\bullet$
for each $0\leq\ell\leq r + 1$.
Hence by the second diagonal theorem, we have
\begin{equation}
\H^p(\tilde\Pi,\tilde{\cal G}_\ell) :=
\H^p(C^\bullet(\tilde\Pi,\tilde{\cal G}_\ell)) =\{0\}
\end{equation}
except for $(p,\ell) = (i-1, i)$ for $(r+1)/2 < i\leq r + 1$ or
$(p,\ell) = (i, i)$ for $0\leq i < (r+1)/2$.
In particular, it is zero for $(p,\ell) = (i, i)$ with
$r/2 < i$ and $(p,\ell) = (i-1, i)$ with $i < r/2 + 1$
since $r$ and $i$ are integers.
Hence we get the equivalent conditions of \cite[Cor.4.5]{Oda3}
for a (rational) fan $\Pi$ and a rational strictly convex
function $h$.
Note that these conditions for the rational case is a consequence
of the strong Lefschetz theorem (cf.\cite{Oda3}).
However irrrational fans are also treated in \cite{Oda3},
our theory can not be applied for irrational case.
\begin{Cor}
\label{Cor 4.4}
Let $\rho$ be a nontrivial cone in $N_{\bf R}$.
If $i + j\geq 0$ and $i\not= j + r_\rho - 1$ or if
$i + j\leq -1$ and $i\not= j + r_\rho$, then
$\H^i(\i_\rho^*(\mathop{\rm ic}\nolimits(F(\rho){\setminus}\{\rho\}))^\bullet)_j =\{0\}$.
\end{Cor}
{\sl Proof.}\quad
The cone $\rho$ is of maximal dimension in the real space
$N(\rho)_{\bf R}$ of dimension $r_\rho$.
Since the functor $\i_\rho^*$ for the fan $F(\rho)$ is equal
to $\Gamma$ of this space, this is a consequence of
Theorem~\ref{Thm 4.3}.
\QED
\begin{Cor}
\label{Cor 4.5}
Let $\rho$ be in a finite fan $\Delta$.
Then we have $\H^i(\mathop{\rm ic}\nolimits(\Delta)(\rho)^\bullet)_j =\{0\}$
unless $i + j\geq 1,\; i = j + r_\rho$, while
$\H^i(\i_\rho^*(\mathop{\rm ic}\nolimits(\Delta))^\bullet)_j =\{0\}$ unless
$i + j\leq -1,\; i = j + r_\rho$.
\end{Cor}
{\sl Proof.}\quad
By the construction of $\mathop{\rm ic}\nolimits(\Delta)^\bullet$ in
\cite[Thm.2.9]{Ishida3},
$\mathop{\rm ic}\nolimits(\Delta)(\rho)^\bullet$ is isomorphic to
$\mathop{\rm gt}\nolimits^{\geq 1}(\i_\rho^*(\mathop{\rm ic}\nolimits(F(\rho){\setminus}\{\rho\}))[-1])^\bullet$
in $\mathop{\rm CGM}\nolimits(A(\rho))$, and $\i_\rho^*(\mathop{\rm ic}\nolimits(\Delta))^\bullet$ is
quasi-isomorphic to
$\mathop{\rm gt}\nolimits_{\leq -1}(\i_\rho^*(\mathop{\rm ic}\nolimits(F(\rho){\setminus}\{\rho\})))^\bullet$.
Hence this is a consequence of Corollary~\ref{Cor 4.4}.
\QED
|
1995-08-01T03:37:09 | 9403 | alg-geom/9403015 | en | https://arxiv.org/abs/alg-geom/9403015 | [
"alg-geom",
"math.AG"
] | alg-geom/9403015 | Richard Hain | Richard M. Hain | Torelli Groups and Geometry of Moduli Spaces of Curves | 38 pages, AMS-LaTeX. Abstract of revised version: This is a
significanly revised and expanded version of the original paper. Several
significant typos in the section on heights have been corrected and four new
sections have been added. Among the new results are proofs of several
generalizations of the classical Franchetta Conjecture to curves with a level
structure and n marked points. Another section contains a computation of the
monodromy group of nth roots of the canonical bundle. The two | null | null | null | null | In this paper we give an exposition of Dennis Johnson's work on the first
homology of the Torelli groups and show how it can be applied, alone and in
concert with Saito's theory of Hodge modules, to study the geometry of moduli
spaces of curves. For example, we show that the picard groups of moduli spaces
of curves with a fixed level structure are finitely generated, classify all
"natural" normal functions defined over moduli spaces of curves with a fixed
level, and also "compute" the height paring between cycles over moduli spaces
of curves which are homologically trivial and disjoint over the generic point.
Several new sections have been added. These apply the results on normal
functions to prove generalizations of the classical Franchetta conjecture for
curves and abelian varieties. In one section, the monodromy group of nth roots
of the canonical bundle is computed.
| [
{
"version": "v1",
"created": "Sun, 27 Mar 1994 21:19:39 GMT"
},
{
"version": "v2",
"created": "Tue, 2 Aug 1994 22:30:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hain",
"Richard M.",
""
]
] | alg-geom | \section{Introduction}\label{intro}
The Torelli group $T_g$ is the kernel of the natural homomorphism
$\Gamma_g \to Sp_g({\Bbb Z})$ from the mapping class group in genus $g$ to
the group of $2g \times 2g$ integral symplectic matrices. It accounts
for the difference between the topology of ${\cal A}_g$, the moduli space of
principally polarized abelian varieties of dimension $g$, and ${\cal M}_g$,
the moduli space of smooth projective curves of genus $g$, and
therefore should account for some of the difference between their
geometeries. For this
reason, it is an important problem to understand its structure and its
cohomology. To date, little is known about $T_g$ apart from Dennis
Johnson's few fundamental results --- he has proved that $T_g$ is
finitely generated when $g\ge 3$ and has computed $H_1(T_g,{\Bbb Z})$. It
is this second result which will concern us in this paper. Crudely
stated, it says that there is an $Sp_g({\Bbb Z})$-equivariant isomorphism
$$
H^1(T_g,{\Bbb Q}) \approx PH^3(\Jac C,{\Bbb Q})
$$
where $C$ is a smooth projective curve of genus $g$, and $P$ denotes
primitive part. My aim in this paper is to give a detailed exposition
of Johnson's homomorphism
$$
PH^3(\Jac C,{\Bbb Q}) \to H^1(T_g,{\Bbb Q})
$$
and to explain how Johnson's computation, alone and in concert
with M.~Saito's theory of Hodge modules \cite{saito}, has some
remarkable consequences
for the geometry of ${\cal M}_g$. It implies quite directly, for example, that
for each $l$, the Picard group of the moduli space ${\cal M}_g(l)$ of genus
$g\ge 3$ curves with a level $l$ structure is finitely generated. Combined
with Saito's work, it enables one to completely write down all ``natural''
generically defined normal functions over ${\cal M}_g(l)$ when $g\ge 3$.
The result is that modulo torsion, all are half integer multiples of the
normal function of the cycle $C-C^-$. This is applied to give a new
proof of the Harris-Pulte Theorem \cite{harris,pulte}, which relates
the mixed Hodge structure on the fundamental group of a curve $C$ to
the algebraic cycle $C-C^-$ in its jacobian. We are also able to
``compute'' the archimedean height pairing between any two ``natural''
cycles in a smooth projective variety defined over the moduli space of
curves, provided they are homologically trivial over each curve, disjoint
over the generic curve, and satisfy the usual dimension restrictions.
The precise statement can be found in Section \ref{heights}.
The classical Franchetta conjecture asserts that the Picard group
of the generic curve is isomorphic to ${\Bbb Z}$ and is generated by
the canonical divisor. Beauville (unpublished), and later Arbarello
and Cornalba \cite{arb-corn}, deduced this from Harer's computation of
$H^2(\Gamma_g)$. As another application of the classification of
normal functions over ${\cal M}_g(l)$, we prove a ``Franchetta Conjecture''
for the generic curve with a level $l$ structure. The statement is that
the Picard group of the generic curve of genus $g$ with a level $l$
structure is finitely generated of rank 1 --- the torsion subgroup is
isomorphic to $({\Bbb Z}/l{\Bbb Z})^{2g}$; mod torsion, it is generated by the
canonical bundle if $l$ is odd, and by a square root of the canonical
bundle if $l$ is even. Our proof is only valid when $g\ge 3$;
it does not use the computation of $\Pic {\cal M}_g(l)$, which is
not known at this time. We also compute the Picard group of the generic
genus $g$ curve with a level $l$ structure and $n$ marked points.
Our results on normal functions are inspired by
those in the last section of Nori's remarkable paper \cite{nori} where
normal functions on finite covers of Zariski open subsets of
the moduli space of principally polarized abelian varieties are studied.
There are analogues of our main results for ${\cal A}_g(l)$, the moduli
space of principally polarized abelian varieties of dimension $g$
with a level $l$ structure. These results are similar to Nori's,
but differ. The detailed statements, as well as a discussion of the
relation between the results, are in Section \ref{abelian}. Our
results on abelian varieties seem to be related to some results of
Silverberg \cite{silverberg}.
Sections \ref{homom} and \ref{original} contain an exposition of the
three constructions of the Johnson homomorphism that are given in
\cite{johnson:survey}. Since no proof of their equivalence appears
in the literature, I have given a detailed exposition, especially
since the equivalence of two of these constructions is essential in
one of the applications to normal functions.
In Section \ref{root} Johnson's results is used to give an explicit
description of the quotient of $\Gamma_g$ by the kernel of its action
on all $n$th roots of the canonical bundle is given. A consequence
of this computation is the ``well known fact'' that the only roots of
the canonical bundle defined over Torelli space are the canonical bundle
itself and all
theta characteristics.
\medskip
\noindent{\it Acknowledgements.}\ First and foremost, I would like to
thank Eduard Looijenga for his hospitality and for stimulating discussions
during a visit to the University of Utrecht in the spring of 1992 during
which some of the work in this paper was done. I would also like to thank
the University of Utrecht and the Dutch NWO for their generous support
during that visit. Thanks also to Pietro Pirola and Enrico Arbarello for
pointing out to me that the non-existence of sections of the universal
jacobian implies the classical Franchetta conjecture. From this it was a
short step to the generalizations in Section \ref{franchetta}.
\section{Mapping Class Groups and Moduli}
At this time there is no argument within algebraic geometry to
compute the Picard groups of all ${\cal M}_g$, and one has to resort to
topology to do this computation. Let $S$ be a compact orientable
surface of genus $g$ with $r$ boundary components and let $P$ be an
ordered set of $n$ distinct marked points of $S -\partial S$. Denote
the group of orientation preserving diffeomorphisms of $S$ that fix
$P\cup \partial S$ pointwise by $\Diff ^+ (S,P \cup \partial S)$.
Endowed with the compact open topology, this is a topological group. The
mapping class group $\Gamma_{g,r}^n$ is defined to be its group of
path components:
$$
\Gamma_{g,r}^n = \pi_0 \Diff^+(S,P\cup\partial S).
$$
Equivalently, it is the group of isotopy classes of orientation
preserving diffeomorphisms of $S$ that fix $P\cup \partial S$
pointwise.
It is conventional to omit the decorations $n$ and $r$ when they
are zero. So, for example, $\Gamma_g^n = \Gamma_{g,0}^n$.
The link between moduli spaces and mapping class groups is
provided by Teichm\"uller theory. Denote the moduli space of
smooth genus $g$ curves with $n$ marked points by ${\cal M}_g^n$.
Teichm\"uller theory provides a contractible complex manifold
${\cal X}_g^n$ on which $\Gamma_g^n$ acts properly discontinuously---it
is the space of all complete hyperbolic metrics on $S-P$ equivalent
under diffeomorphisms isotopic to the identity. The
quotient $\Gamma_g^n\backslash {\cal X}_g^n$ is analytically
isomorphic to ${\cal M}_g^n$. It is useful to think of $\Gamma_g^n$ as
the orbifold fundamental group of ${\cal M}_g^n$.
One can compactify $S$ by filling in the $r$ boundary components of
$S$ by attaching disks. Denote the resulting genus $g$ surface by
$\overline{S}$. Elements of $\Gamma_{g,r}^n$ extend canonically to $\overline{S}$ to
give a homomorphism $\Gamma_{g,r}^n \to \Gamma_g$. Denote the
composite
$$
\Gamma_{g,r}^n \to \Gamma_g \to \Aut H_1(\overline{S},{\Bbb Z})
$$
by $\rho$. Since elements of $\Gamma_{g,r}^n$ are represented by
orientation preserving diffeomorphisms, each element of
$\Gamma_{g,r}^n$ preserves the intersection pairing
$$
q : \Lambda^2 H_1(\overline{S},{\Bbb Z}) \to {\Bbb Z}.
$$
Consequently, we obtain a homomorphism
$$
\rho : \Gamma_{g,r}^n \to \Aut (H_1(\overline{S},{\Bbb Z}),q) \approx Sp_g({\Bbb Z}).
$$
This homomorphism is well known to be surjective.
Denote the moduli space of principally polarized abelian varieties
of dimension $g$ by ${\cal A}_g$. Since this is the quotient of the Siegel
upper half plane by $Sp_g({\Bbb Z})$, it is an orbifold with orbifold
fundamental group $Sp_g({\Bbb Z})$. The period map
$$
{\cal M}_g^n \to {\cal A}_g
$$
is a map of orbifolds and induces $\rho$ on fundamental groups.
The Torelli group $T_{g,r}^n$ is the kernel of the homomorphism
$$
\rho : \Gamma_{g,r}^n \to Sp_g({\Bbb Z}).
$$
Since $\rho$ is surjective, we have an extension
$$
1 \to T_{g,r}^n \to \Gamma_{g,r}^n \to Sp_g({\Bbb Z}) \to 1.
$$
The Torelli group $T_g$ encodes the differences between the topology of
${\cal M}_g$ and ${\cal A}_g$ --- between curves and abelian varieties. More formally,
we have the Hochschild-Serre spectral sequence
$$
H^s(Sp_g({\Bbb Z}),H^t(T_{g,r}^n)) \implies H^{s+t}(\Gamma_{g,r}^n).
$$
Much more (although not enough) is known about the topology of the
${\cal A}_g$ than about that of the ${\cal M}_g$. For example, the rational
cohomology groups of the ${\cal A}_g$ stabilize as $g \to \infty$, and this
stable cohomology is known by Borel's work \cite{borel:triv}; it is
a polynomial ring generated by classes $c_1, c_3, c_5, \dots$, where
$c_k$ has degree $2k$. As with ${\cal A}_g$, the rational cohomology of
the ${\cal M}_g$ is known to stabilize, as was proved by Harer
\cite{harer:stab}, but the stable cohomology of the ${\cal M}_g$ is known
only up to dimension 4; the computations are due to Harer
\cite{harer:h3,harer:h4}.
Torelli space ${\cal T}_g^n$ is the quotient $T_g^n\backslash{\cal X}_g^n$ of
Teichm\"uller space. When $g\ge 3$, it is the moduli space of smooth
projective curves $C$, together with $n$ ordered distinct points and a
symplectic basis of $H_1(C,{\Bbb Z})$.
The Torelli group is torsion free. Perhaps the simplest way to see this
is to note that, by standard topology, since ${\cal X}_g^n$ is contractible,
each element of $\Gamma_g^n$
of prime order must fix a point of ${\cal X}_g^n$. If $\phi\in \Gamma_g^n$
fixes the point corresponding to the marked curve $C$, then there is
an automorphism of $C$ that lies in the mapping class $\phi$. Since
the automorphism group of a compact Riemann surface injects into
$\Aut H^0(C,\Omega_C^1)$, and therefore into $H_1(C)$, it follows that
$T_g^n$ is torsion free. Because of this, the Torelli space
${\cal T}_g^n$ is the classifying space of $T_g^n$.
One can view Siegel space $\goth h_g$ as the classifying space of
principally polarized abelian varieties of dimension $g$ together with
a symplectic basis of $H^1$. The period map therefore induces a
map
$$
{\cal T}_g^n \to \goth h_g
$$
which is 2:1 when $g\ge 2$, and ramified along the hyperelliptic
locus when $g\ge3$.
For a finite index subgroup $L$ of $Sp_g({\Bbb Z})$, let $\Gamma_{g,r}^n(L)$ be
the inverse image of $L$ in $\Gamma_{g,r}^n$ under the canonical
homomorphism $\Gamma_{g,r}^n \to Sp_g({\Bbb Z})$. It may be expressed
as an extension
$$
1 \to T_{g,r}^n \to \Gamma_{g,r}^n(L) \to L \to 1.
$$
Set ${\cal M}_{g}^n(L) = \Gamma_{g}^n(L)\backslash {\cal X}_g^n$.
We will call $\Gamma_{g,r}^n(L)$ the {\it level $L$} subgroup of
$\Gamma_{g,r}^n$, and we will say that points in ${\cal M}_{g}^n(L)$
are curves with a level $L$ structure and $n$ marked points.
The traditional moduli space of curves with a level $l$ structure,
where $l\in {\Bbb N}^+$, is obtained by taking $L$ to be the elements of
$Sp_g({\Bbb Z})$ that are congruent to the identity mod $l$.
Since the Torelli groups are torsion free, $\Gamma_{g,r}^n(L)$ is
torsion free when $L$ is. Note, however, that by the Lefschetz fixed
point formula, $\Gamma_{g,r}^n$ is torsion free when $n+2r > 2g+2$,
so that $\Gamma_{g,r}^n(L)$ may be torsion free even when $L$ is not.
\begin{proposition}
For all $g,n \ge 0$ and for each finite index subgroup $L$ of
$Sp_g({\Bbb Z})$, there is a natural homomorphism
$$
H^\bullet({\cal M}_g^n(L),{\Bbb Z}) \to H^\bullet(\Gamma_g^n(L),{\Bbb Z})
$$
which is an isomorphism when $\Gamma_g^n(L)$ is torsion free, and
is an isomorphism after tensoring with ${\Bbb Q}$ for all $l$.
\end{proposition}
\begin{pf}
Set $\Gamma=\Gamma_g^n$, $\Gamma(L) = \Gamma_g^n(L)$,
${\cal M}(L)={\cal M}_g^n(L)$ and ${\cal X}={\cal X}_g^n$. Let $E\Gamma$ be any
space on which $\Gamma$ acts freely and properly discontinuously%
---so $E\Gamma$ is the universal covering space of some model of
the classifying space of $\Gamma$. Since ${\cal X}$ is contractible, the
quotient $E\Gamma\times_{\Gamma(L)}{\cal X}$ of $E\Gamma \times
{\cal X}$ by the diagonal action of $\Gamma(L)$ is a model
$B\Gamma(L)$ of the classifying space of $\Gamma(L)$. The
projection $E\Gamma \times {\cal X} \to {\cal X}$ induces a map
$f: B\Gamma(L) \to {\cal M}(L)$ which
induces the map of the theorem. If $\Gamma(L)$ is torsion free, $f$ is a
homotopy equivalence. Otherwise, choose a finite index, torsion
free normal subgroup $L'$ of $L$. Then $\Gamma(L)$ is torsion free.
Set
$$
G = \Gamma(L)/\Gamma(L')\approx L/L'
$$
This is a finite group. We have the commutative diagram of Galois
$G$-coverings
$$
\begin{matrix}
B\Gamma(L') & \to & {\cal M}(L') \cr
\downarrow & & \downarrow \cr
B\Gamma(L) & \to & {\cal M}(L)
\end{matrix}
$$
where the top map is a homotopy equivalence. It therefore induces
a $G$-equivariant isomorphism
$$
H^\bullet({\cal M}(L')) \to H^\bullet(B\Gamma(L')).
$$
The result follows as the vertical projections induce isomorphisms
$$
H^\bullet({\cal M}(L),{\Bbb Q}) \stackrel{\sim}{\to} H^\bullet({\cal M}(L'),{\Bbb Q})^G
\text{ and }
H^\bullet(\Gamma(L),{\Bbb Q}) \stackrel{\sim}{\to} H^\bullet(\Gamma(L'),{\Bbb Q})^G.
$$
\end{pf}
The group $\Gamma_{g,r}^n(L)$ also admits a moduli interpretation
when $r > 0$, even though algebraic curves have no boundary
components. The idea is that a topological boundary component of a
compact orientable surface should correspond to a first order local
holomorphic coordinate about a cusp of a smooth algebraic curve.
Denote by ${\cal M}_{g,r}^n(L)$ the moduli space of smooth curves of
genus $g$ with a level $L$ structure and with $n$ distinct marked
points and $r$ distinct, non-zero cotangent vectors, where the
cotangent vectors do not lie over any of the marked points, and
where no two of the cotangent vectors are anchored at the same
point. This is a $\left({\Bbb C}^\ast\right)^r$ bundle over ${\cal M}_g^{r+n}(L)$.
\begin{proposition}
For all finite index subgroups $L$ of $Sp_g({\Bbb Z})$ and
for all $g,n,r \ge 0$, there is a natural homomorphism
$$
H^\bullet({\cal M}_{g,r}^n(L),{\Bbb Z}) \to H^\bullet(\Gamma_{g,r}^n(L),{\Bbb Z})
$$
which is an isomorphism when $\Gamma_{g,r}^n(L)$ is torsion free,
and is an isomorphism after tensoring with ${\Bbb Q}$ for all $L$.\qed
\end{proposition}
\section{The Johnson Homomorphism}
\label{homom}
Dennis Johnson, in a sequence of pioneering papers
\cite{johnson_1,johnson_2,johnson_3}, began a systematic
study of the Torelli groups. From the point of view of
computing the cohomology of the ${\cal M}_g$, the most important of
his results is his computation of $H_1(T_g^1)$ \cite{johnson_3}.
Let $S$ be a compact oriented surface of genus $g$ with a
distinguished base point $x_0$.
\begin{theorem}\label{john:h1}
There is an $Sp_g({\Bbb Z})$-equivariant homomorphism
$$
\tau_g^1 : H_1(T_g^1,{\Bbb Z}) \to \Lambda^3 H_1(S)
$$
which is an isomorphism mod 2-torsion.
\end{theorem}
Johnson has also computed $H_1(T_g^1,{\Bbb Z}/2{\Bbb Z})$. It is related to
theta characteristics. Bert van~Geemen has interesting ideas regarding
its relation to the geometry of curves.
A proof of Johnson's theorem is beyond the scope of this paper.
However, we will give three constructions of the homomorphism
$\tau_g^1$ and establish their equality.
We begin by sketching the first of these constructions: Since the Torelli
group is torsion free, there is a universal curve
$$
{\cal C} \to {\cal T}_g^1
$$
over Torelli space. This has a tautological section
$\sigma: {\cal T}_g^1 \to {\cal C}$.
There is also the jacobian
$$
{\cal J} \to {\cal T}_g^1
$$
of the universal curve. The universal curve can be imbedded in its
jacobian using the section $\sigma$---the restriction of this mapping
to the fiber over the point of Torelli space corresponding to $(C,x)$
is the Abel-Jacobi mapping
$$
\nu_x : (C,x) \to (\Jac C,0)
$$
associated to $(C,x)$. Since $T_g^1$ acts trivially on the first
homology of the curve, the local system associated to $H_1(C)$ is
framed. There is a corresponding topological trivialization of the
jacobian bundle:
$$
{\cal J} \stackrel{\sim}{\to} {\cal T}_g^1 \times \Jac C.
$$
Let $p : {\cal J} \to \Jac C$ be the corresponding projection onto the fiber.
Each element $\phi$ of $H_1(T_g^1,{\Bbb Z})$ can be represented by an
imbedded circle $\phi :S^1 \to {\cal T}_g^1$. Regard the
universal curve
${\cal C}$ as subvariety of ${\cal J}$ via the Abel-Jacobi mapping. Then the part
of the universal curve $M(\phi)$ lying over the circle $\phi$ is a 3-cycle
in ${\cal J}$. The Johnson homomorphism is defined by
$$
\tau_g^1(\phi) = p_\ast[M(\phi)] \in H_3(\Jac C,{\Bbb Z}) \approx
\Lambda^3 H_1(C,{\Bbb Z}).
$$
This definition is nice and conceptual, but is not so easy to work with.
In the remainder of this section, we remake this definition without
appealing to Torelli space. In the next section, we will give two more
constructions of it, both due to Johnson, and prove all three constructions
agree.
Recall that the {\it mapping torus} of a diffeomorphism $\phi$ of a
manifold $S$ is the quotient $M(\phi)$ of $S\times [0,1]$ obtained
by identifying $(x,1)$ with $(\phi(x),0)$:
$$
M(\phi) = S \times [0,1]\left\{(x,1) \sim (\phi(x),0)\right\}.
$$
The projection $S \times [0,1] \to [0,1]$ induces a bundle projection
$$
M(\phi) \to [0,1]/\left\{0\sim 1\right\} = S^1
$$
whose fiber is $S$ and whose geometric monodromy is $\phi$.
Now suppose that $\phi : (S,x_0) \to (S,x_0)$ is a diffeomorphism of
$S$ that represents an element of $T_g^1$. The mapping torus bundle
$$
M(\phi) \to S^1
$$
has a canonical section $\sigma : S^1 \to M(\phi)$ which takes
$t \in S^1$ to $(x_0,t)\in M(\phi)$.
Denote the ``jacobian'' of $S$, $H_\bullet(S,{\Bbb R}/{\Bbb Z})$,
by $\Jac S$. The next task is to imbed $M(\phi)$ into $\Jac S$ using
the section $\sigma$ of base points.
To this end, choose a basis $\omega_1,\dots, \omega_{2g}$ of
$H^1(S,{\Bbb Z})$. This gives an identification of $\Jac S$ with
$\left({\Bbb R}/{\Bbb Z}\right)^{2g}$. Choose closed,
real-valued 1-forms $w_1,\dots,w_{2g}$
that represent $\omega_1,\dots, \omega_{2g}$. These have integral periods.
Since $\phi$ acts trivially on $H^1(S)$, there are smooth functions
$f_j : S \to {\Bbb R}$ such that
$$
\phi^\ast w_j = w_j + df_j.
$$
These functions are uniquely determined if we insist, as we shall, that
$f_j(x_0)=0$ for each $j$. Set
$$
\vec{w} = (w_1,\dots,w_g)\text{ and } \vec{f} = (f_1,\dots, f_g).
$$
The map
$$
S \times [0,1] \to \Jac S
$$
defined by
$$
(x,t) \mapsto t\vec{f}(x) + \int_{x_0}^x \vec{w}
$$
preserves the equivalence relations of the mapping torus $M(\phi)$,
and therefore induces a map
$$
\nu(\phi) : (M(\phi),\sigma(S^1)) \to (\Jac S,0).
$$
Define $\tilde{\tau}(\phi)$ to be the homology class of $M(\phi)$ in
$H_3(\Jac S,{\Bbb Z})$:
$$
\tilde{\tau}(\phi) = \nu(\phi)_\ast[M(\phi)]\in \Lambda^3 H_1(S,{\Bbb Z}).
$$
\begin{proposition} If $\phi$, $\psi$ are diffeomorphisms of $S$ that
act trivially on $H_1(S)$, then
\begin{enumerate}
\item[(a)] $\tilde{\tau}(\phi)$ is independent of the choice of representatives
$w_1,\dots, w_g$ of the basis $\omega_1,\dots, \omega_{2g}$ of $H^1(S,{\Bbb Z})$;
\item[(b)] $\tilde{\tau}(\phi)$ is independent of the choice of basis
$\omega_1,\dots \omega_{2g}$ of $H^1(S,{\Bbb Z})$;
\item[(c)] $\tilde{\tau}(\phi)$ depends only on the isotopy class of $\phi$;
\item[(d)] $\tilde{\tau}(\phi\psi) = \tilde{\tau}(\phi) + \tilde{\tau}(\psi)$;
\item[(e)] $\tilde{\tau}(g \psi g^{-1}) = g_\ast \tilde{\tau}(\phi)$ for
all diffeomorphisms
$g$ of $S$, where $g_\ast$ is the automorphism of $\Lambda^3 H_1(S)$
induced by $g$.
\end{enumerate}
\end{proposition}
\begin{pf}
If $w'_1,\dots, w'_{2g}$ is another set of representatives of the $\omega_j$,
then there are functions $g_j : S \to {\Bbb R}$ such that
$w'_j = w_j + dg_j \text{ and } g_j(x_0) = 0$.
For each $s \in [0,1]$, the 1-form $w_j(s) = w_j + sdg_j$
is closed on $S$ and represents $\omega_j$. The map
$$
\nu_s : M(\phi) \to \Jac S
$$
defined using the representatives $w_j(s)$ takes $(x,t)$ to
$$
\left(t\Bigl(f_j(x) + s\bigl(g_j(\phi(x))-g_j(x)\bigr)\Bigr) + sg_j(x) +
\int_{x_0}^x w_j\right).
$$
Since this depends continuously on $s$, it follows that $\nu_0$ is
homotopic to $\nu_1$. The first assertion follows.
The second assertion follows from linear algebra. The proof of the
third assertion is similar to that of the first.
To prove the fourth assertion, observe that the quotient of
$M(\phi\psi)$ obtained by identifying $(x,1)$ with $(\psi(x),1/2)$ is
the union of $M(\phi)$ and $M(\psi)$. The map $\nu(\phi\psi)$ factors
through the quotient $M(\phi)\cup M(\psi)$ of $M(\phi\psi)$,
and its restrictions to $M(\phi)$ and $M(\psi)$ are
$\nu(\phi)$ and $\nu(\psi)$, respectively. Additivity follows.
Suppose that $g:(S,x_0) \to (S,x_0)$ is a diffeomorphism. The map
$(g,\text{id}) : S\times [0,1] \to S\times [0,1]$ induces a diffeomorphism
$$
F(g) : M(\phi) \to M(g \phi g^{-1}).
$$
To prove the last assertion, it suffices to prove that the
diagram
$$
\begin{CD}
M(\phi) @>{F(g)}>> M(g \phi g^{-1}) \cr
@V\nu(\phi)VV @VV\nu(g\phi g^{-1})V \cr
\Jac S @>g_\ast>> \Jac S \cr
\end{CD}
$$
commutes up to homotopy. In the proof of the first assertion, we
saw that the homotopy class of $\nu$ depends only on the basis
of $H^1(S,{\Bbb Z})$ and not on the choice of de~Rham representatives.
Set $w'_j = g^\ast w_j$. Since the diagram
$$
\begin{CD}
H_1(S) @>g_\ast>> H_1(S) \cr
@V\int w'_jVV @VV\int w_jV\cr
{\Bbb R} @= {\Bbb R}\cr
\end{CD}
$$
commutes, it suffices to prove that the diagram
$$
\begin{CD}
M(\phi) @>{F(g)}>> M(g \phi g^{-1}) \cr
@V{\nu'}VV @VV{\nu}V \cr
\left({\Bbb R}/{\Bbb Z}\right)^{2g} @>\text{id}>>
\left({\Bbb R}/{\Bbb Z}\right)^{2g} \cr
\end{CD}
$$
commutes, where $\nu$ is defined using $w_1,\dots w_{2g}$, and
$\nu'$ is defined using the representatives $w'_1,\dots w'_{2g}$.
This last assertion is easily verified.
\end{pf}
Recall that the homology groups of $T_g^1$ are $Sp_g({\Bbb Z})$ modules;
the action on $H_1(T_g)$ is given by
$$
g : [\phi] \mapsto [\tilde{g}\phi\tilde{g}^{-1}],
$$
where $g\in Sp_g({\Bbb Z})$ and $\tilde{g}$ is any element of
$\Gamma_g^1$ that projects to $g$ under the canonical
homomorphism.
\begin{corollary}
The map $\tilde{\tau}$ induces an $Sp_g({\Bbb Z})$-equivariant homomorphism
$$
\tau_g^1 : H_1(T_g^1,{\Bbb Z}) \to \Lambda^3 H_1(S,{\Bbb Z}).
$$
\end{corollary}
{}From $\tau_g^1$, we can construct a representation $\tau_g$ of
$H_1(T_g)$. The kernel of the natural surjection $T_g^1 \to T_g$
is isomorphic to $\pi_1(S,x_0)$. The composition of the induced map
$H_1(S,{\Bbb Z}) \to H_1(T_g^1,{\Bbb Z})$ with $\tau_g^1$ is easily seen to be the
canonical inclusion
$$
\underline{\phantom{x}}\times [S] : H_1(S,{\Bbb Z}) \hookrightarrow H_3(\Jac S,{\Bbb Z})
$$
induced by taking Pontrjagin product with $\nu_\ast[S]$. We therefore
have an induced $Sp_g({\Bbb Z})$-equivariant map
$$
\tau_g : H_1(T_g,{\Bbb Z}) \to \Lambda^3 H_1(S,{\Bbb Z}) /H_1(S,{\Bbb Z}).
$$
The following result of Johnson is an immediate corollary of
Theorem \ref{john:h1}.
\begin{theorem}\label{tau_g}
The homomorphism $\tau_g$ is an isomorphism modulo 2-torsion.
\end{theorem}
It is not difficult to boot strap up from Johnson's basic computation
to prove the following result.
\begin{theorem}\label{h1_alg}
There is a natural $Sp_g({\Bbb Z})$-equivariant isomorphism
$$
\tau_{g,r}^n : H_1(T_{g,r}^n,{\Bbb Q}) \to H_1(S,{\Bbb Q})^{\oplus(n+r)} \oplus
\Lambda^3 H_1(S,{\Bbb Q}) /H_1(S,{\Bbb Q}).
$$
\end{theorem}
An important consequence of Johnson's theorem is that the action
of $Sp_g({\Bbb Z})$ on $H_1(T_{g,r}^n,{\Bbb Q})$ factors through a rational
representation of the ${\Bbb Q}$-algebraic group $Sp_g$. Let $\lambda_1,
\dots, \lambda_g$ be a fundamental set of dominant integral weights
of $Sp_g$. Denote the irreducible $Sp_g$-module with highest weight
$\lambda$ by $V(\lambda)$. The fundamental representation of $Sp_g$ is
$H_1(S)$. It is well known (and easily verified) that
$$
\Lambda^3 H_1(S) \approx V(\lambda_1) \oplus V(\lambda_3).
$$
The previous result can be restated by saying that
$$
H_1(T_{g,r}^n,{\Bbb Q}) \approx
V(\lambda_3) \oplus V(\lambda_1)^{\oplus(n+r)}
$$
as $Sp_g$ modules.
\section{A Second Definition of the Johnson Homomorphism}
\label{original}
In this section we relate the definition of $\tau_g^1$ given in the
previous section to Johnson's original definition, which is defined
using the action of $T_g^1$ on the lower central series of
$\pi_1(S,x_0)$. It is better suited to computations. In order to
relate this definition to the one given in the previous section,
we need to study the cohomology ring of the mapping torus associated
to an element of the Torelli group.
Suppose that the diffeomorphism $\phi : (S,x_0) \to (S,x_0)$
represents an element of $T_g^1$. As explained in the previous
section, the associated mapping torus $M=M(\phi)$ fibers over $S^1$
and has a canonical section $\sigma$. This data guarantees that there
is a canonical decomposition of the cohomology of $M$.
Since $\phi$ acts trivially on the homology of $S$, the $E_2$-term of
the Leray-Serre spectral sequence of the fibration $\pi : M \to S^1$
satisfies
$$
E_2^{r,s} = H^r(S^1)\otimes H^s(S).
$$
This spectral sequence degenerates for trivial reasons. Consequently,
there is a short exact sequence
$$
0 \to H^1(S^1,{\Bbb Z}) \stackrel{\pi^\ast}{\to} H^1(M,{\Bbb Z}) \stackrel{i^\ast}
{\to} H^1(S,{\Bbb Z}) \to 0,
$$
where $\pi$ is the projection to $S^1$ and $i : S \hookrightarrow M$
is the inclusion of the fiber over the base point $t=0$ of $S^1$.
The section $\sigma$ induces a splitting of this sequence. Denote
$\pi^\ast$ of the positive generator of $H^1(S^1,{\Bbb Z})$ by $\theta$. Then
we have the decomposition
\begin{equation}\label{h1}
H^1(M,{\Bbb Z}) = H^1(S,{\Bbb Z}) \oplus {\Bbb Z}\theta.
\end{equation}
{}From the spectral sequence, it follows that we have an exact sequence
$$
0 \to \theta\wedge H^1(S,{\Bbb Z}) \to H^2(M,{\Bbb Z})
\stackrel{i^\ast}{\to} H^2(S,{\Bbb Z}) \to 0.
$$
Denote the Poincar\'e dual of a homology class $u$ in $M$ by $PD(u)$.
Since
$$
\int_S PD(\sigma) = \sigma\cdot S = 1
$$
it follows that the previous sequence can be split by taking the
positive generator of $H^2(S,{\Bbb Z})$ to $PD(\sigma)$. We therefore
have a canonical splitting
\begin{equation}\label{h2}
H^2(M,{\Bbb Z}) = {\Bbb Z} PD(\sigma) \oplus \theta\wedge H^1(S,{\Bbb Z}).
\end{equation}
The cup product pairing
$$
c : H^1(M)\otimes H^2(M) \to H^3(M) \approx {\Bbb Z}
$$
induces pairings between the summands of the decompositions
(\ref{h1}) and (\ref{h2}).
\begin{proposition}\label{prop}
The cup product $c$ satisfies:
\begin{enumerate}
\item[(a)] $c(\theta \otimes PD(\sigma)) = 1$;
\item[(b)] the restriction of $c$ to $H^1(S) \otimes PD(\sigma)$
vanishes;
\item[(c)] the restriction of $c$ to
$\theta \otimes \bigl(\theta\wedge H^1(S)\bigr)$ vanishes;
\item[(d)] the restriction of $c$ to
$H^1(S) \otimes \bigl( \theta\wedge H^1(S)\bigr)$ takes
$u \otimes (\theta\wedge v)$ to $-\int_S u\wedge v$.
\end{enumerate}
\end{proposition}
\begin{pf}
Since $\theta$ is the Poincar\'e dual of the fiber $S$, we have
$$
\int_M \theta \wedge PD(\sigma) = \int_M PD(S) \wedge PD(\sigma)
= S\cdot \sigma = 1.
$$
In the decomposition (\ref{h1}), $H^1(S)$ is identified with the kernel
of $\sigma^\ast: H^1(M) \to H^1(S^1)$; that is, with those
$u\in H^1(M)$ such that
$$
\int_\sigma u = 0.
$$
The second assertion now follows as
$$
\int_M u \wedge PD(\sigma) = \int_\sigma u
$$
for all $u \in H^1(M)$. The third and fourth assertions are easily
verified.
\end{pf}
To complete our understanding of the cohomology ring of $M$, we
consider the cup product
$$
\Lambda^2 H^1(M) \to H^2(M).
$$
Since $\theta\wedge \theta =0$, there is only one interesting part of
this mapping; namely, the component
$$
\Lambda^2 H^1(S) \to {\Bbb Z} PD(\sigma) \oplus \theta\wedge H^1(S).
$$
There is a unique function
$$
f_\phi : \Lambda^2 H^1(S,{\Bbb Z}) \to H^1(S,{\Bbb Z})
$$
such that
$$
u\wedge v \mapsto
\left(\int_S u\wedge v,-\theta\wedge f_\phi(u\wedge v)\right)
\in H^2(M,{\Bbb Z})
$$
with respect to the decomposition (\ref{h2}).
We can view $f_\phi$ as an element of
$$
H_1(S,{\Bbb Z}) \otimes \Lambda^2 H^1(S,{\Bbb Z}).
$$
Using Poincar\'e duality on the last two factors, $f_\phi$ can be
regarded as an element $F(\phi)$ of
$$
H_1(S,{\Bbb Z})\otimes \Lambda^2 H_1(S,{\Bbb Z}).
$$
There is a canonical imbedding of $\Lambda^3 H_1(S,{\Bbb Z})$ into
this group. It is defined by
$$
a\wedge b \wedge c \mapsto a\otimes (b\wedge c) +
b\otimes (c\wedge a) + c\otimes (a\wedge b).
$$
\begin{theorem}\label{image}
The invariant $F(\phi)$ of the cohomology ring of $M(\phi)$ is the
image of $\tilde{\tau}(\phi)$ under the canonical imbedding
$$
\Lambda^3 H_1(S,{\Bbb Z}) \hookrightarrow
H_1(S,{\Bbb Z})\otimes \Lambda^2 H_1(S,{\Bbb Z}).
$$
\end{theorem}
\begin{pf}
The dual of $\tau_g^1(\phi)$ is the map
$$
\Lambda^3 H^1(S) \to {\Bbb Z}
$$
defined by
$$
u \wedge v \wedge w \mapsto \int_{M(\phi)} u \wedge v \wedge w.
$$
Here we have identified $H^1(S)$ with $H^1(\Jac S)$ using the
canonical isomorphism
$$
\nu^\ast : H^1(\Jac S) \stackrel{\sim}{\to} H^1(S).
$$
The map $\nu(\phi) : M \to \Jac S$ collapses $\sigma$ to the point 0.
It follows that the image
of
$$
\nu(\phi)^\ast : H^1(\Jac S) \to H^1(M)
$$
lies in the subspace we are identifying with $H^1(S)$ in the
decomposition (\ref{h1}). Since the restriction of $\nu(\phi)$ to
the fiber over the base point $t=0$ of $S^1$ is the isomorphism
$\nu^\ast$, it follows that the diagram
$$
\begin{CD}
H^1(\Jac S) @>{\nu(\phi)^\ast}>> H^1(M) \cr
@V{\nu^\ast}VV @| \cr
H^1(S) @>{i}>> H^1(M) \cr
\end{CD}
$$
commutes, where $i$ is the inclusion given by the splitting
(\ref{h1}). That is, all the identifications we have made with $H^1(S)$
are compatible.
We will compute the dual of $\tau_g^1(\phi)$ using $F(\phi)$,
which we regard as a homomorphism
$$
F(\phi) : H^1(S) \otimes \Lambda^2 H^1(S) \to {\Bbb Z}
$$
It follows from (\ref{prop}) that this map takes
$u\otimes (v\wedge w)$ to
$$
\int_S u\wedge f_\phi(v\wedge w).
$$
The assertion that $F(\phi)$ lie in $\Lambda^3H_1(S)$ is equivalent
to the assertion that
$$
F(\phi)(u\otimes (v\wedge w)) = F(\phi)(v\otimes (w\wedge u))
=F(\phi)(w\otimes (u\wedge v)),
$$
which is easily verified using (\ref{prop}). The equality of
$F(\phi)$ and $\tau_g^1(\phi)$ follows as
$$
\tau_g^1(\phi)(u\wedge v\wedge w)
= \int_{M} u\wedge v\wedge w
= -\int_{M} u \wedge \theta \wedge f_\phi(v\wedge w)
= F(\phi) (u\otimes (v\wedge w)).
$$
\end{pf}
We are now ready to give Johnson's original definition of $\tau_g^1$.
Denote the lower central series of a group $\pi$ by
$$
\pi = \pi^{(1)} \supseteq \pi^{(2)} \supseteq \pi^{(3)}
\supseteq \cdots
$$
We regard the cup product
$$
\Lambda^2 H^1(S,{\Bbb Z}) \to H^2(S,{\Bbb Z}) \approx {\Bbb Z}
$$
as an element $q$ of $\Lambda^2 H^1(S,{\Bbb Z})$.
\begin{proposition}\label{lcs}
The commutator mapping
$$
[\phantom{x},\phantom{x}] : \pi_1(S,x_0) \times \pi_1(S,x_0) \to \pi_1(S,x_0)
$$
induces an isomorphism
$\Lambda^2 H_1(S,{\Bbb Z})/q \to \pi_1(S,x_0)^{(2)} /
\pi_1(S,x_0)^{(3)}$.
\end{proposition}
\begin{pf}
This follows directly from the standard fact (see \cite{serre}
or \cite{mks}) that if $F$ is a free group, the commutator
induces an isomorphism
$$
\Lambda^2 H_1(F) \stackrel{\sim}{\to} F^{(2)}/F^{(3)}
$$
and from the standard presentation of $\pi_1(S,x_0)$.
\end{pf}
An element $\phi$ of $T_g^1$ induces an automorphism of
$\pi_1(S,x_0)$. Since it acts trivially on $H_1(S)$,
$$
\phi(\gamma)\gamma^{-1} \in \pi_1(S,x_0)^{(2)}
$$
for all $\gamma\in \pi_1(S,x_0)$. From (\ref{lcs}), it follows
that $\phi$ induces a well defined map
$$
\hat{\tau}(\phi) : H_1(S,{\Bbb Z}) \to \Lambda^2 H_1(S,{\Bbb Z})/q
$$
Using Poincar\'e duality, we may view this as an element
$L(\phi)$ of
$$
H_1(S,{\Bbb Z}) \otimes \left(\Lambda^2 H_1(S,{\Bbb Z})/q\right).
$$
\begin{theorem}
The image of $F(\phi)$ in
$H_1(S,{\Bbb Z}) \otimes \left(\Lambda^2 H_1(S,{\Bbb Z})/q\right)$
is $L(\phi)$.
\end{theorem}
\begin{pf}
Since $H_1(S,{\Bbb Z}) \otimes \left(\Lambda^2 H_1(S,{\Bbb Z})/q\right)$
is torsion free, it suffices to show that the image of $F(\phi)$
in
$$
H_1(S,{\Bbb Q}) \otimes \left(\Lambda^2 H_1(S,{\Bbb Q})/q\right)
$$
is $L(\phi)$. For the rest of this proof, all (co)homology groups have
${\Bbb Q}$ coefficients.
For all groups $\pi$ with finite dimensional $H_1(\phantom{x},{\Bbb Q})$, the
sequence
\begin{equation}\label{ex-seq}
0 @>>> H^1(\pi) @>h^\ast>>
\left(\pi/\pi^{(3)}\right)^\ast @>{[\phantom{x},\phantom{x}]^\ast}>>
\Lambda^2 H^1(\pi) @>{\wedge}>> H^2(\pi)
\end{equation}
of ${\Bbb Q}$ vector spaces is exact. Here $(\phantom{x})^\ast$ denotes
the dual vector space, $h^\ast$ the dual Hurewicz homomorphism, and
$[\phantom{x},\phantom{x}]$ the map induced by the commutator. This can be proved
using results in either \cite[\S 2.1]{chen} or
\cite[\S 8]{sullivan:inf}.
We apply this sequence to the fundamental group
of the mapping torus. Choose $m_0=(x_0,0)$ as the base point of
$M$. Since $M$ fibers over the circle with fiber $S$,
we have an exact sequence
$$
1 \to \pi_1(S,x_0) \to \pi_1(M,m_0) \to {\Bbb Z} \to 0.
$$
The section $\sigma$ induces a splitting ${\Bbb Z} \to \pi_1(M,m_0)$.
Denote the image of $1$ by $\sigma$. Observe that if
$\gamma\in \pi_1(S,x_0)$, then
$$
\sigma \gamma \sigma^{-1} = \phi(\gamma).
$$
It follows that the inclusion $\pi_1(S,x_0) \hookrightarrow
\pi_1(M,m_0)$
induces isomorphisms
$$
\pi_1(S,x_0)^{(k)} \approx \pi_1(M,m_0)^{(k)}
$$
for all $k> 1$ and, as above, that $\sigma$ induces an
isomorphism
$$
H_1(M) = H_1(S) \oplus {\Bbb Q}\Sigma,
$$
where $\Sigma$ denotes the homology class of $\sigma$.
It also follows that for all $a\in H_1(S)$
$$
\tilde{\tau}(\phi)(a) = [\Sigma,a] \in \pi_1(M)^{(2)}/\pi_1(M)^{(3)}
\approx \pi_1(S)^{(2)}/\pi_1(S)^{(3)}.
$$
Using (\ref{prop}) and the exact sequence (\ref{ex-seq}), we
see that for all $u\in H_1(S)$, the image of $f_\phi^\ast (u)$ in
$$
\Lambda^2 H_1(S)/q
$$
is $[\Sigma,u]$, which is $\hat{\tau}(\phi)(u)$ as we have seen. The
result follows.
\end{pf}
The composite of the inclusion
$$
\Lambda^3 H_1(S,{\Bbb Z}) \hookrightarrow H_1(S,{\Bbb Z})\otimes \Lambda^2 H_1(S,{\Bbb Z})
$$
with the quotient mapping
$$
H_1(S,{\Bbb Z})\otimes \Lambda^2 H_1(S,{\Bbb Z}) \to
H_1(S,{\Bbb Z})\otimes \left(\Lambda^2 H_1(S,{\Bbb Z})/q\right)
$$
is injective. One way to see this is to tensor with ${\Bbb Q}$ and
note that both of these maps are maps of $Sp_g$ modules. One can
then use the fact that $\Lambda^3 H_1(S)$ is the sum of the first
and third fundamental representations of $Sp_g$ to check the result.
The following result is therefore a restatement of (\ref{image}).
\begin{corollary}\label{equality}
$L(\phi)$ lies in the image of the canonical injection
$$
\Lambda^3 H_1(S,{\Bbb Z}) \hookrightarrow
H_1(S,{\Bbb Z})\otimes \left(\Lambda^2 H_1(S,{\Bbb Z})/q\right)
$$
and the corresponding point of $\Lambda^3 H_1(S)$ is
$\tau_g^1(\phi)$.
\end{corollary}
In his fundamental papers, Johnson defines $\tau_g^1(\phi)$ to
be $L(\phi)$. The other two definitions we have given were stated
in \cite{johnson:survey}.
\section{Picard Groups}
In \cite{mumford}, Mumford showed that
$$
c_1 : \Pic {\cal M}_g \otimes {\Bbb Q}\to H^2({\cal M}_g,{\Bbb Q})
$$
is an isomorphism. Using Johnson's computation of $H_1(T_g,{\Bbb Q})$ and
the well known result (\ref{pic}), we will prove the analogous
statement for all ${\cal M}_{g,r}^n(L)$ when $g\ge 3$. The novelty
lies in the variation of the level, and not in the variation
of the decorations $r$ and $n$. The first, and principal, step is to
establish the vanishing of the $H^1({\cal M}_{g,r}^n(L))$.
\begin{proposition}\label{h1-level}
Suppose that $L$ is a finite index subgroup of $Sp_g({\Bbb Z})$.
If $g \ge 3$, then $H^1({\cal M}_{g,r}^n(L),{\Bbb Z})=0$.
\end{proposition}
Since $H^1(\phantom{X},{\Bbb Z})$ is always torsion free, it suffices to
prove that $H^1({\cal M}_g(L),{\Bbb Q})$ vanishes. We will prove a stronger
result.
\begin{proposition}\label{calc}
Suppose that $L$ is a finite index subgroup of $Sp_g({\Bbb Z})$ and that
$g\ge 3$. If $V(\lambda)$ is an irreducible representation of $Sp_g$
with highest weight $\lambda$, then
$$
H^1(\Gamma_{g,r}^n(L),V(\lambda)) =
\begin{cases}
{\Bbb Q}^{r+n} & \text{if } \lambda = \lambda_1; \cr
{\Bbb Q} & \text{if }\lambda = \lambda_3;\cr
0 & otherwise.
\end{cases}
$$
Consequently, $H^1({\cal M}_{g,r}^n(L),{\Bbb Z})$ vanishes for all $r,n$ when
$g\ge 3$.
\end{proposition}
\begin{pf}
It follows from the Hochschild-Serre spectral sequence
$$
H^r(L,H^s(T_{g,r}^n\otimes V(\lambda))) \implies
H^{r+s}(\Gamma_{g,r}^n(L),V(\lambda))
$$
that there is an exact sequence
\begin{multline*}
0 \to H^1(L,V(\lambda)) \to
H^1(\Gamma_{g,r}^n(L),V(\lambda)) \\
\to H^0(L,H^1(T_{g,r}^n\otimes V(\lambda)))
@>{d_2}>> H^2(L,V(\lambda)).
\end{multline*}
By a result of Ragunathan
\cite{ragunathan}, the first term vanishes when $g\ge 2$. By
(\ref{h1_alg}), the third term vanishes except when $\lambda$ is
either $\lambda_1$ or $\lambda_3$. This proves the result except
in the cases when $\lambda$ is either $\lambda_1$ or $\lambda_3$.
In these exceptional cases, the third term has
rank $r+n$ or 1, respectively. To complete the proof, we need to show
that the differential $d_2$ is zero.
There are several ways to do this. Perhaps the most straightforward is
to use the result, due to Borel \cite{borel:twisted}, which asserts
that the last group vanishes when $g\ge 8$. This establishes the result
when $g\ge 8$. When $r\ge 1$, the vanishing of $d_r$ for all $g\ge 3$
follows as the diagram
$$
\begin{CD}
H^0(L,H^1(T_{g,r}^n\otimes V(\lambda))) @>d_2>> H^2(L,V(\lambda)) \cr
@AAA @AAA \cr
H^0(L,H^1(T_{g+8,r}^n\otimes V(\lambda))) @>d_2>> H^2(L_{g+8},V(\lambda))\cr
\end{CD}
$$
commutes. Here $L_{g+8}$ is any finite index subgroup of $Sp_{g+8}({\Bbb Z})$
such that
$$
L_{g+8}\cap Sp_g({\Bbb Z}) \subseteq L
$$
and the vertical maps are induced by the ``stabilization map''
$$
\Gamma_{g,r}^n(L) \to \Gamma_{g+8,r}^n(L_{g+8}).
$$
When $r=0$ and $\lambda=\lambda_1$, there is nothing to prove. This
leaves only the case $r=0$ and $\lambda = \lambda_3$ which follows
as the diagram
$$
\begin{CD}
H^0(L,H^1(T_g^n\otimes V(\lambda))) @>d_2>> H^2(L,V(\lambda)) \cr
@AAA @AAA \cr
H^0(L,H^1(T_{g,1}^n\otimes V(\lambda))) @>d_2>> H^2(L,V(\lambda))\cr
\end{CD}
$$
which arises from the homomorphism $\Gamma_{g,1}^n \to \Gamma_g^n$,
commutes.
\end{pf}
Denote the category of ${\Bbb Z}$ mixed Hodge structures by ${\cal H}$.
We shall denote the group of ``integral $(0,0)$ elements''
$\Hom_{\cal H}({\Bbb Z},H)$ of a mixed Hodge structure $H$ by $\Gamma H$.
Suppose that $X$ is a smooth variety. Since $H^1(X,{\Bbb Z})$ is torsion
free, we can define
$$
W_1H^1(X,{\Bbb Z}) = W_1H^1(X,{\Bbb Q}) \cap H^1(X,{\Bbb Z}).
$$
This is a polarized, torsion free Hodge structure of weight 1. Set
$$
JH^1(X) = {W_1H^1(X,{\Bbb C}) \over W_1H^1(X,{\Bbb Z}) + F^1W_1H^1(X,{\Bbb C})}.
$$
This is a polarized Abelian variety.
\begin{theorem}\label{pic}
If $X$ is a smooth variety, then there is a natural exact sequence
$$
0 \to JH^1(X) \to \Pic X \to \Gamma H^2(X,{\Bbb Z}(1)) \to 0.
$$
\end{theorem}
Alternatively, this theorem may be stated as saying that the
cycle map
$$
\Pic X \to H_{\cal H}^2(X,{\Bbb Z}(1))
$$
is an isomorphism, where $H_{\cal H}^\bullet$ denotes Beilinson's absolute
Hodge cohomology, the refined version of Deligne cohomology
defined in \cite{beilinson:ahc}.
\begin{pf}
Choose a smooth completion $\overline{X}$ of $X$ for which $\overline{X}-X$
is a normal crossings divisor $D$ in $\overline{X}$ with smooth components.
Denote the dimension of $X$ by $d$. From the usual exponential
sequence, we have a short exact sequence
$$
0 \to JH^1(\overline{X}) \to \Pic \overline{X} \to \Gamma H^2(X,{\Bbb Z}) \to 0.
$$
{}From \cite[(1.8)]{fulton}, we have an exact sequence
$$
CH^0(D) \to \Pic \overline{X} \to \Pic X \to 0.
$$
The Gysin sequence
$$
0 \to H^1(\overline{X}) \to H^1(X) \to H_{2d-2}(D)(-2d) \to H^2(\overline{X}) \to H^2(X)
\to H_{2d-3}(D)(-2d)
$$
is an exact sequence of ${\Bbb Z}$ Hodge structures. Since $H_{2d-2}(D)(-2d)$ is
torsion free and of weight 2, it follows that
$$
W_1H^1(X,{\Bbb Z}) = H^1(\overline{X},{\Bbb Z}),
$$
and therefore that $JH^1(X) = JH^1(\overline{X})$.
Next, since each component $D_i$ of $D$ is smooth, it follows that
$$
H_{2d-3}(D)(-2d) = \bigoplus_i H^1(D_i,{\Bbb Z})(-1),
$$
and is therefore torsion free and of weight 3. It follows that the
sequence
$$
H_{2d-2}(D)(-2d) \to \Gamma H^2(\overline{X}) \to \Gamma H^2(X) \to 0
$$
is exact. Since the cycle map
$$
CH^0(D) \to H_{2d-2}(D)
$$
is an isomorphism \cite[(1.5)]{fulton}, the result follows.
\end{pf}
It is now an easy matter to show that the Picard groups of the
${\cal M}_{g,r}^n(L)$ are finitely generated.
\begin{theorem}
Suppose that $L$ is a finite index subgroup of $Sp_g({\Bbb Z})$.
If $g\ge 3$, then for all $r,n$, the Chern class map
$$
c_1 : \Pic {\cal M}_{g,r}^n(L) \to \Gamma H^2({\cal M}_{g,r}^n(L),{\Bbb Z})
$$
is an isomorphism when $\Gamma_{g,r}^n(L)$ is torsion free, and is
an isomorphism after tensoring with ${\Bbb Q}$ in general.
\end{theorem}
\begin{pf}
The case when $\Gamma_{g,r}^n(L)$ is torsion free follows directly
from (\ref{h1-level}) and (\ref{pic}). To prove the assertion in
general, choose a finite index normal subgroup $L'$ of $L$ such that
$\Gamma_{g,r}^n(L')$ is torsion free. Let
$$
G = \Gamma_{g,r}^n(L) / \Gamma_{g,r}^n(L') \approx L/L'.
$$
Then it follows from the Teichm\"uller description of moduli spaces
that the projection
$$
\pi : {\cal M}_{g,r}^n(L') \to {\cal M}_{g,r}^n(L)
$$
is a Galois covering with Galois group $G$. It follows from the first
case that
$$
c_1 :\Pic {\cal M}_{g,r}^n(L') \to \Gamma H^2({\cal M}_{g,r}^n(L'),{\Bbb Z})
$$
is a $G$-equivariant isomorphism. The result now follows as the
projection $\pi$ induces isomorphisms
$$
\Pic {\cal M}_{g,r}^n(L)\otimes{\Bbb Q} \approx
H^0(G,\Pic {\cal M}_{g,r}^n(L')\otimes {\Bbb Q})
$$
and
$$
\Gamma H^2({\cal M}_{g,r}^n(L),{\Bbb Q}) \approx \Gamma H^0(G,H^2({\cal M}_{g,r}^n(L'),{\Bbb Q})).
$$
\end{pf}
If we knew that $H^2(T_g,{\Bbb Q})$ were finite dimensional and a rational
representation of $Sp_g$, we would know from Borel's work
\cite{borel:triv} that $H^2({\cal M}_{g,r}^n(L),{\Bbb Q})$ would be independent
of the level $L$, once $g$ is sufficiently large; $g\ge 8 $ should
do it --- cf.\ \cite{borel:twisted}. It would then follow, for
sufficiently large $g$, that the Picard number
of ${\cal M}_{g,r}^n(L)$ is $n+r+1$. At present it is not even known whether
$H^2(T_g,{\Bbb Q})$ is finite dimensional. The computation
of this group, and the related problem of finding a presentation of
$T_g$ appear to be deep and difficult. It should be mentioned that
the only evidence for the belief that the Picard number of each
${\cal M}_g(L)$ is one comes from Harer's computation \cite{harer:spin}
of the Picard numbers of the moduli spaces of curves with a
distinguished theta characteristic.
\section{Normal Functions}
In this section, we define abstract normal functions which generalize
the normal functions of Poincar\'e and Griffiths.
We begin by reviewing how a family of homologically trivial
algebraic cycles in a family of smooth projective varieties gives
rise to a normal function.
Suppose that $X$ is a smooth variety. A homologically trivial
algebraic $d$-cycle in $X$ canonically determines an element of
$$
\Ext^1_{\cal H}({\Bbb Z},H_{2d+1}(X,{\Bbb Z}(-d))).
$$
This extension is obtained by pulling back the exact sequence
$$
0 \to H_{2d+1}(X,{\Bbb Z}(-d)) \to H_{2d+1}(X,Z,{\Bbb Z}(-d)) \to H_{2d}(Z,{\Bbb Z}(-d))
\to \cdots
$$
of mixed Hodge structures along the inclusion
$$
{\Bbb Z} \to H_{2d}(|Z|,{\Bbb Z}(-d))
$$
that takes 1 to the class of $Z$.
When $H$ is a mixed Hodge structure all of whose weights are
non-positive, there is a natural isomorphism
$$
J H \approx \Ext^1_{\cal H}({\Bbb Z},H),
$$
where
$$
JH = {H_{\Bbb C} \over F^0H_{\Bbb C} + H_{\Bbb Z}}.
$$
(This is well known---see for example \cite{carlson}. Our
conventions will be taken from \cite[(2.2)]{hain:heights}.)
When $X$ is projective,
Poincar\'e duality provides an isomorphism of the complex torus
$JH_{2d+1}(X,{\Bbb Z}(-d))$ with the Griffiths intermediate jacobian
$$
\Hom_{\Bbb C}(F^dH^{d+1}(X),{\Bbb C})/H_{2d+1}(X,{\Bbb Z}).
$$
The point in $JH_{2d+1}(X,Z(d))$ corresponding to the cycle $Z$
under this isomorphism is $\int_\Gamma$, where $\Gamma$ is a
real $2d+1$ chain that satisfies $\partial \Gamma = Z$.
Now suppose that ${\cal X} \to T$ is a family of smooth projective varieties
over a smooth base $T$. Suppose that ${\cal Z}$ is an algebraic cycle in
${\cal X}$ which is proper over $T$ of relative dimension $d$. Denote the
fibers of ${\cal X}$ and ${\cal Z}$ over $t\in T$ by $X_t$ and $Z_t$, respectively.
The set of $H_{2d+1}(X_t,{\Bbb Z}(-d))$ form a variation of Hodge structure
${\Bbb V}$ over $T$ of weight $-1$. We can form the relative intermediate
jacobian
$$
{\cal J}_d \to T;
$$
this has fiber $JH_{2d+1}(X_t,{\Bbb Z}(-d))$ over $t\in T$. The family of
cycles ${\cal Z}$ defines a section this bundle. Such a section is what
Griffiths calls {\it the normal function of the cycle ${\cal Z}$}
\cite{griffiths}. Griffiths' normal functions generalize those of
Poincar\'e.
We will generalize this notion further. Before we do, note that the
elements of $\Ext^1_{\cal H}({\Bbb Z},H_{2d+1}(X_t,{\Bbb Z}(-d)))$ defined by the
cycles $Z_t$ fit
together to form a variation of mixed Hodge structure over $T$. It
follows from the main result of \cite{g-n-p} that this variation is
good in the sense of \cite{steenbrink-zucker} along each curve in $T$,
and is therefore good in the sense of Saito \cite{saito}.
Suppose that $T$ is a smooth variety and that ${\Bbb V} \to T$ is a
variation of Hodge structure over $T$ of negative weight. Denote the
bundle over $T$ whose fiber over $t\in T$ is
$$
JV_t \approx \Ext^1_{\cal H}({\Bbb Z},V_t)
$$
by $J{\cal V}$.
\begin{definition}
A holomorphic section $s: T \to J{\cal V}$ of $J{\cal V} \to T$ is a {\it normal
function} if it defines an extension
$$
0 \to {\Bbb V} \to {\Bbb E} \to {\Bbb Z}_T \to 0
$$
in the category ${\cal H}(T)$ of good variations of mixed Hodge structure
over $T$.
\end{definition}
\begin{remark}\label{norm}
We know from the preceding discussion that families of
homologically trivial cycles in a family ${\cal X} \to T$ define normal
functions in this sense.
\end{remark}
The asymptotic properties of good variations of mixed Hodge structure
guarantee that these normal functions have nice properties.
\begin{lemma}[Rigidity]\label{rigid}
If ${\Bbb V} \to T$ and ${\Bbb V}' \to T$
are two good variations of mixed Hodge structure over $T$ with the
same fiber $V_{t_0}$ (viewed as a mixed Hodge structure) over some
point $t_0$ of $T$ and with the same monodromy representations
$$
\pi_1(T,t_0) \to \Aut V_{t_0},
$$
then ${\Bbb V}_1$ and ${\Bbb V}_2$ are isomorphic as variations.
\end{lemma}
\begin{pf}
The proof is a standard application of the theorem of the fixed part.
The local system $\Hom_{\Bbb Z}({\Bbb V},{\Bbb V}')$
underlies a good variation of mixed Hodge structure. From Saito's
work \cite{saito}, we know that each cohomology group of a variety
with coefficients in a good variation of mixed Hodge structure has a
natural mixed Hodge structure. So, in particular,
$$
H^0(T,\Hom_{\Bbb Z}({\Bbb V},{\Bbb V}'))
$$
has a mixed Hodge structure, and the restriction map
$$
H^0(T,\Hom_{\Bbb Z}({\Bbb V},{\Bbb V}')) \to \Hom_{\Bbb Z}(V_{t_0},V_{t_0}')
$$
is a morphism. The result now follows as there are natural
isomorphisms
$$
H^0(T,\Hom_{\Bbb Z}({\Bbb V},{\Bbb V}')) \approx
\Hom_{{\Bbb Z}\pi_1(T,t_0)}(V_{t_0},V_{t_0}')
$$
and
$$
\Gamma H^0(T,\Hom_{\Bbb Z}({\Bbb V},{\Bbb V}')) \approx \Hom_{{\cal H}(T)}({\Bbb V},{\Bbb V}'),
$$
where ${\cal H}(T)$ denotes the category of good variations of mixed
Hodge structure over $T$.
\end{pf}
\begin{corollary}\label{rigidity}
Two normal functions $s_1, s_2 : T \to J{\cal V}$ are equal if and only
if there is a point $t_0 \in T$ such that $s_1(t_0) = s_2(t_0)$ and
such that the two induced homomorphisms
$$
(s_j)_\ast : \pi_1(T,t_0) \to \pi_1(J{\cal V},s_1(t_0))
$$
are equal. \qed
\end{corollary}
\section{Extending Normal Functions}
\label{extending}
The strong asymptotic properties of variations of mixed Hodge
structure imply that almost all normal functions extend across
subvarieties where the original variation of Hodge structure is
non-singular. Suppose that $X$ is a smooth variety and that
${\Bbb V}$ is a variation of Hodge structure over $X$ of negative
weight. Denote the associated intermediate jacobian bundle by
${\cal J} \to X$.
\begin{theorem}\label{extend}
Suppose that $U$ is a Zariski open subset of $X$ and that
$s: U \to {\cal J}|_U$ is a normal function defined on $U$. If
the weight of ${\Bbb V}$ is not $-2$, then $s$ extends to a normal
function $\tilde{s} : X \to {\cal J}$.
\end{theorem}
\begin{pf}
Write $U=X-Z$. By Hartog's Theorem, it suffices to show that $s$
extends to a normal function on the complement of the union of the
singular locus of $Z$ and the union of the components of $Z$ of
codimension $\ge 2$ in $X$. That is, we may assume that $Z$ is a
smooth divisor.
The problem of extending $s$ is local. By taking
a transverse slice, we can reduce to the case where $X$ is the
unit disk $\Delta$ and $Z$ is the origin.
In this case, we have a variation of Hodge structure over $\Delta$.
The normal function $s: \Delta^\ast \to {\cal J}$ corresponds to a good
variation of mixed Hodge structure ${\Bbb E}$ over the punctured disk
$\Delta^\ast$ which is an extension
$$
0 \to {\Bbb V}|_{\Delta^\ast} \to {\Bbb E} \to {\Bbb Z}_{\Delta^\ast}.
$$
To prove that the normal function extends, it suffices to show
that the monodromy of ${\Bbb E}$ is trivial, for then the local system
${\Bbb E}$ extends uniquely as a flat bundle to $\Delta$ and the Hodge
filtration extends across the origin as ${\Bbb E}$ is a good
variation.
Since ${\Bbb V}$ is defined on the whole disk, it has trivial monodromy.
It follows that the local monodromy operator $T$ of ${\Bbb E}$ satisfies
$$
(T-I)^2 = 0
$$
and that the local monodromy logarithm $N$ is $T-I$. Since
$E$ is a good variation, it has a relative weight filtration $M_\bullet$
(cf.\ \cite{steenbrink-zucker}), which is defined over ${\Bbb Q}$ and
satisfies $NM_l \subseteq M_{l-2}$. From the defining properties of
$M_\bullet$ (\cite[(2.5)]{steenbrink-zucker}), we have
$$
M_0 = {\Bbb E},\, M_{m} = {\Bbb V}, \text{ and } M_{m-1} = 0,
$$
where $m$ is the weight of ${\Bbb V}$.
In the case where $m=-1$, the proof that $N=0$ is simpler. Since this
case is the most important (as it is the one that applies to normal
functions of cycles), we prove it first. The condition $m=-1$ implies
that $M_{-2}=0$. Since $NM_0 \subseteq M_{-2}$, it follows that $N=0$
and consequently, that the normal function extends.
In general, we use defining property
\cite[(3.13)iii]{steenbrink-zucker} of good variations of mixed Hodge
structure which says that
$$
(E_t, M_\bullet, F^\bullet_{\lim})
$$
is a mixed Hodge structure and $N$ is an morphism of mixed Hodge
structures of type $(-1,-1)$, where $F^\bullet_{\lim}$ denotes the
limit Hodge filtration. In this case, $N$ induces a morphism
$$
{\Bbb Z} \approx Gr^M_0 \to Gr^M_{-2},
$$
which is zero if $m\neq -2$. Since $N$ is a morphism of mixed Hodge
structures, the vanishing of this map implies the vanishing of $N$.
\end{pf}
When $m=-2$, there are normal functions that don't extend. For example,
if we take ${\Bbb V} = {\Bbb Z}(1)$, then the bundle of intermediate jacobians is
the bundle $X\times {\Bbb C}^\ast$ and the normal functions are precisely
the invertible regular functions $f: X \to {\Bbb C}$ --- for details see,
for example, \cite[(9.3)]{hain:poly}.
\section{Normal Functions over ${\cal M}_{g,r}^n(L)$}
\label{normal_functions}
Throughout this section, we will assume that $g\ge 3$ and $L$ is a finite
index subgroup of $Sp_g({\Bbb Z})$ such that $\Gamma_{g,r}^n(L)$ is torsion free.
With this condition on $L$, ${\cal M}_{g,r}^n(L)$ is smooth. Each irreducible
representation of $Sp_g$ defines a polarized ${\Bbb Q}$ variation of Hodge
structure over ${\cal M}_{g,r}^n(L)$ which is unique up to Tate twist --- cf.\
(\ref{exist-unique}). It follows that every rational representation of
$Sp_g$ underlies a polarized ${\Bbb Z}$ variation of Hodge structure over
${\cal M}_g(L)$.
\begin{lemma}\label{fin_gen}
If ${\Bbb V} \to {\cal M}_{g,r}^n(L)$ is a good variation of Hodge
structure of negative weight whose monodromy representation
$$
\Gamma_{g,r}^n(L) \to \Aut V_\circ\otimes {\Bbb Q}
$$
factors through a rational representation of $Sp_g$ and contains no
copies of the trivial representation, then the group of
normal functions $s : {\cal M}_{g,r}^n(L) \to J{\cal V}$ is finitely generated
of rank bounded by
$$
\dim H^1(\Gamma_{g,r}^n(L),V_{\Bbb Z}).
$$
\end{lemma}
\begin{pf}
A normal function corresponds to a variation of mixed Hodge
structure whose underlying local system is an extension
$$
0 \to {\Bbb V} \to {\Bbb E} \to {\Bbb Z} \to 0
$$
of the trivial local system by ${\Bbb V}$.
One can form the semidirect product $\Gamma_{g,r}^n(L)\ltimes V_{\Bbb Z}$,
where the mapping class group acts on $V_{\Bbb Z}$ via a representation
$L\to \Aut V$. The monodromy representation of the local system
${\Bbb E}$ gives a splitting
$$
\rho : \Gamma_{g,r}^n(L) \to \Gamma_{g,r}^n(L)\ltimes V_{\Bbb Z}
$$
of the natural projection
\begin{equation}\label{mono}
\Gamma_{g,r}^n(L)\ltimes V_{\Bbb Z} \to \Gamma_{g,r}^n(L).
\end{equation}
The splitting is well defined up to conjugation by an element of $V_{\Bbb Z}$.
The first step in the proof is to show that an extension of ${\Bbb Q}$ by ${\Bbb V}$
in the category of ${\Bbb Q}$ variations of mixed Hodge structure is determined
by its monodromy representation. Two such variations can
be regarded as elements of the group
\begin{equation}\label{ext}
\Ext^1_{{\cal H}({\cal M}_{g,r}^n(L))}({\Bbb Q},{\Bbb V}).
\end{equation}
It is easily seen that their difference is an extension whose monodromy
representation factors through the homomorphism
$\Gamma_{g,r}^n \to Sp_g({\Bbb Q})$. It now follow from (\ref{decomp}) and the
assumption that ${\Bbb V}$ contain no copies of the trivial representation
that this difference is the trivial element of (\ref{ext}). The
assertion follows.
{}From \cite[p.~106]{maclane} it follows that the set of splittings of
(\ref{mono}), modulo conjugation by elements of $V_{\Bbb Z}$, is isomorphic
to
$$
H^1(\Gamma_{g,r}^n(L),V_{\Bbb Z}).
$$
It follows from (\ref{calc}) that this group is finitely generated
provided $V\otimes {\Bbb Q}$ does not contain the trivial representation.
Since normal functions are determined by their monodromy, the result
follows.
\end{pf}
If ${\Bbb V}$ contains the trivial representation, the group of normal
functions is an uncountably generated divisible group. For example,
if ${\Bbb V}$ has trivial monodromy, then all such extensions are pulled
back from a point. The set of normal functions is then
$$
\Ext^1_{\cal H}({\Bbb Z},V_o) \approx JV_o,
$$
where $V_o$ denotes the fiber over the base point.
\begin{theorem}\label{comp}
If, in addition, the fiber over the base point is an irreducible $Sp_g$
module with highest weight $\lambda$ and Hodge weight $m$, then the group
of normal functions $s : {\cal M}_{g,r}^n(L) \to J{\cal V}$ is finitely generated
of rank
$$
\dim H^1(\Gamma_{g,r}^n(L), V(\lambda))
=\begin{cases}
1 & \text{if }\lambda = \lambda_3\text{ and } m=-1; \cr
r+n & \text{if }\lambda = \lambda_1\text{ and } m=-1; \cr
0 & \text{ otherwise}.
\end{cases}
$$
\end{theorem}
The upper bounds for the rank of the group of normal functions follow
from (\ref{fin_gen}), (\ref{calc}), and the fact that the monodromy
representation associated
to a normal function has to be a morphism of variations of mixed Hodge
structure (\ref{morph}). It remains to show that these upper bounds are
achieved. We do this by explicitly constructing normal functions.
Multiples of the generators mod torsion of the normal functions associated
to $V(\lambda_1)$ can be pulled back from ${\cal M}_g^1(L)$ along the $n+r$
forgetful maps ${\cal M}_{g,r}^n(L) \to {\cal M}_g^1(L)$. There the normal function
can be taken to be the one that takes $(C,x)$ to the point
$(2g-2)x -\kappa_C$ of $\Pic^0 C$, where $\kappa_C$ denotes the
canonical class of $C$.
A multiple of the normal function associated to $\lambda_3$ can be pulled
back from ${\cal M}_g(L)$ along the forgetful map ${\cal M}_{g,r}^n(L) \to {\cal M}_g(L)$.
We will describe how this normal function over ${\cal M}_g(L)$ arises
geometrically. If $C$ is a smooth projective curve of genus $g$ and
$x\in C$, we have the Abel-Jacobi mapping
$$
\nu_x : C \to \Jac C.
$$
Denote the algebraic 1-cycle ${\nu_x}_\ast C$ in $\Jac C$ by $C_x$.
Denote the cycle $i_\ast C_x$ by $C_x^-$, where $i : \Jac C \to \Jac C$
takes $u$ to $-u$. The cycle $C_x - C^-$ is homologous to zero, and
therefore defines a point $\tilde{e}(C,x)$ in $JH_3(\Jac C,{\Bbb Z}(-1))$.
Pontrjagin product with the class of $C$ induces a homomorphism
$$
A:\Jac C \to JH_3(\Jac C,{\Bbb Z}(-1)).
$$
Denote the cokernel of $A$ by $JQ(\Jac C)$. It is not difficult to show
that
$$
\tilde{e}(C,x) - \tilde{e}(C,y) = A(x-y).
$$
It follows that the image of $\tilde{e}(C,x)$ in $JQ(\Jac C)$ is
independent of $x$. The image will be denoted by $e(C)$.
The primitive decomposition
$$
H_3(\Jac C,{\Bbb Q}) = H_1(\Jac C,{\Bbb Q}) \oplus PH_3(\Jac C,{\Bbb Q})
$$
is the decomposition of $H_3(\Jac C)$ into irreducible $Sp_g$
modules; the highest weights of the pieces being $\lambda_1$
and $\lambda_3$, respectively.
Fix a level $L$ so that ${\cal M}_g(L)$ is smooth. The union of the
$JQ(\Jac C)$ form the bundle ${\cal J}_{\lambda_3}$ of intermediate jacobians
over ${\cal M}_g(L)$ associated to the variation of Hodge structure of weight
$-1$ and monodromy the third fundamental representation $V(\lambda_3)$
of $Sp_g$.
\begin{theorem}
The section $e$ of ${\cal J}_{\lambda_3}$ is a normal function. Every other
normal function associated to this bundle is, up to torsion, a half
integer multiple of $e$.
\end{theorem}
\begin{pf}
This result is essentially proved in \cite{hain:completions}. We give
a brief sketch.
To see that $e$ is a normal function, consider the
bundle of intermediate jacobians $JH_3(\Jac C,{\Bbb Z}(-1))$ over
${\cal M}_g^1(L)$. It follows from (\ref{norm}) that $(C,x) \mapsto
\tilde{e}(C,x)$ is a normal function. The argument on page 97 of
\cite{hain:completions}, shows that there is a canonical quotient
of the variation corresponding to $\tilde{e}$. (It is the extension
$E$ in display 10 of \cite{hain:completions}.) This variation does
not depend on the base
point $x$, and is therefore constant along the fibers of
${\cal M}_g^1(L) \to {\cal M}_g(L)$. It follows that this quotient variation is the
pullback of a variation on ${\cal M}_g(L)$. This quotient variation is
classified by $e$. It follows that $e$ is a normal function.
Each normal function $f$ associated to this bundle of intermediate
jacobians induces an $L$ equivariant homomorphism
$$
f_\ast : H_1(T_g,{\Bbb Z}) \to H_1(JQ,{\Bbb Z}) \approx \Lambda^3 H_1(C,{\Bbb Z})/H_1(C,{\Bbb Z}).
$$
It follows from monodromy computation in
\cite[(4.3.5)]{hain:heights} (see also \cite[(6.3)]{hain:completions})
that $e_\ast$ is twice the Johnson homomorphism.
$$
\tau_g : H_1(T_g,{\Bbb Z}) \to \Lambda^3 H_1(C,{\Bbb Z})/H_1(C,{\Bbb Z}).
$$
Since this homomorphism is primitive --- i.e., not a non-trivial
integral multiple of another such normal function, all other normal
functions associated to $\lambda_3$ must have monodromy representations
which are half integer multiples of that of $e$. As we have seen in
the proof of (\ref{fin_gen}), such normal functions are determined, up
to torsion, by their monodromy representation. The result follows.
\end{pf}
I don't know how to realize $e/2$ as a normal function in this sense.
But I do know to construct a more general kind of normal function
associated to the 1-cycle $C$ in $\Jac C$ that does realize
$e/2$. It is a section of a bundle whose fiber over $C$ is a principal
$JQ(\Jac C)$ bundle. The details may be found in
\cite[p.~92]{hain:completions}.
\begin{remark}
Using the results in Section \ref{tech} and Theorem \ref{comp}, one
can easily show that the rank of the group of normal
functions in the theorem above is
$$
\dim \Gamma\Hom_{Sp_g({\Bbb Q})}(H_1(T_{g,r}^n,{\Bbb Q}),V_{{\Bbb Q},C}),
$$
where $H_1(T_{g,r}^n)$ is given the Hodge structure of weight $-1$
described in \S \ref{tech}.
\end{remark}
\section{Technical Results on Variations over ${\cal M}_g$}
\label{tech}
In this section, we prove several technical facts about variations of
mixed Hodge structure over moduli spaces of curves that were used
in Section \ref{normal_functions}. Throughout we will assume that $L$ has
been chosen so that $\Gamma_{g,r}^n(L)$ is torsion free.
\begin{proposition}\label{exist-unique}
The local system ${\Bbb V}(\lambda)$ over ${\cal M}_{g,r}^n(L)$ associated
to the irreducible representation of $Sp_g$ with highest weight
$\lambda$ underlies a good ${\Bbb Q}$ variation of (mixed) Hodge structure,
and this variation is unique up to Tate twist.
\end{proposition}
\begin{pf}
First observe that the local system ${\Bbb H}$ corresponding to the
fundamental representation $V(\lambda_1)$ occurs as a variation of Hodge
structure over ${\cal M}_{g,r}^n(L)$ of weight 1; it is simply the local system
$R^1\pi_\ast{\Bbb Q}$ associated to the universal curve ${\cal C} \to {\cal M}_{g,r}^n(L)$.
The existence of the structure of a good variation of Hodge structure
on the local system corresponding the the $Sp_g$ module with
highest weight $\lambda$ now follows using Weyl's construction of the
irreducible representations of $Sp_g$---see, for example,
\cite[\S 17.3]{fulton-harris}.
To prove uniqueness, suppose that ${\Bbb V}$ and ${\Bbb V}'$ are both good variations
of mixed Hodge structure corresponding to the same irreducible $Sp_g$ module.
{}From Saito \cite{saito}, we know that
$$
\Hom_{\Gamma_{g,r}^n(L)}({\Bbb V},{\Bbb V}')
$$
has a mixed Hodge structure. By Schur's lemma, this group is
one dimensional. It follows that this group is isomorphic to ${\Bbb Q}(n)$
for some $n$. It follows that ${\Bbb V}' = {\Bbb V}(n)$.
\end{pf}
\begin{proposition}\label{decomp}
If ${\Bbb E}$ is a good variation of ${\Bbb Q}$ mixed Hodge structure over
${\cal M}_g(L)$ whose monodromy representation factors through a rational
representation of the algebraic group $Sp_g$, then for each dominant
integral weight $\lambda$ of $Sp_g$, the $\lambda$-isotypical part
${\Bbb E}_\lambda$ of ${\Bbb E}$ is a good variation of mixed Hodge structure.
Consequently,
$$
{\Bbb E} = \bigoplus_\lambda {\Bbb E}_\lambda
$$
in the category of good variations of ${\Bbb Q}$ mixed Hodge structure
over ${\cal M}_g(L)$. Moreover, for each $\lambda$, there is a mixed Hodge
structure $A_\lambda$ such that
${\Bbb E}_\lambda = A_\lambda \otimes {\Bbb V}(\lambda)$.
\end{proposition}
\begin{pf}
Fix $\lambda$, and let ${\Bbb V}(\lambda) \to {\cal M}_g(L)$ be a variation of
Hodge structure whose fiber over some fixed base point is the
irreducible $Sp_g$ module with highest weight $\lambda$.
It follows from Saito's work \cite{saito} that
$$
A_\lambda := \Hom_{\Gamma_g(L)}\big({\Bbb V}(\lambda), {\Bbb E}\big) =
H^0\big({\cal M}_g(L),\Hom_{\Bbb Q}({\Bbb V}(\lambda),{\Bbb E})\big)
$$
is a mixed Hodge structure. Let
$$
{\Bbb E}' = \bigoplus_\lambda A_\lambda \otimes {\Bbb V}(\lambda).
$$
This is a good variation of mixed Hodge structure which is isomorphic
to ${\Bbb E}$ as a ${\Bbb Q}$ local system. Now
$$
\Hom_{\Gamma_g(L)}({\Bbb E}',{\Bbb E}) = \bigoplus_\lambda A_\lambda^\ast
\otimes \Hom_{\Gamma_g(L)}({\Bbb V}(\lambda),{\Bbb E}) = \bigoplus_\lambda
\Hom_{\Bbb Q}(A_\lambda,A_\lambda).
$$
The element of this group which corresponds to
$\text{id} : A_\lambda \to A_\lambda$ in each factor is an isomorphism
of local systems and an element of
$$
\Gamma\Hom_{\Gamma_g(L)}({\Bbb E}',{\Bbb E}).
$$
It is therefore an isomorphism of variations of mixed Hodge structure.
\end{pf}
The local system
$$
\left\{H_1(T_{g,r}^n)\right\}
$$
over ${\cal M}_{g,r}^n(L)$ naturally underlies a variation of mixed Hodge
structure of weight $-1$. The $\lambda_1$ isotypical component
is simply $r+n$ copies of the variation ${\Bbb V}(\lambda_1)$. We shall
denote this variation by ${\Bbb H}_1(T_{g,r}^n)$.
\begin{proposition}\label{morph}
Suppose that ${\Bbb V}$ is a variation of mixed Hodge structure over
${\cal M}_{g,r}^n(L)$ whose monodromy representation factors through a
rational representation of $Sp_g$. If ${\Bbb E}$ is an extension of
${\Bbb Q}$ by ${\Bbb V}$ in the category of variations of mixed Hodge structure
over ${\cal M}_{g,r}^n(L)$, then the restriction of the monodromy
representation to $H_1(T_{g,r}^n)$,
$$
{\Bbb H}_1(T_{g,r}^n) \to {\Bbb V},
$$
is a morphism of variations of mixed Hodge structure.
\end{proposition}
\begin{pf}
It suffices to prove the assertion for ${\Bbb Q}$ variations of mixed
Hodge structure. We will prove the case where $n=r=0$; the proofs
of the other cases being similar.
If the monodromy representation of ${\Bbb E}$ is trivial, the
result is trivially true. So we shall assume that the monodromy
representation is non-trivial.
Using the previous result, we can write
$$
{\Bbb V} = \bigoplus_\lambda {\Bbb V}_\lambda
$$
as variations of mixed Hodge structure over ${\Bbb V}$. By pushing
out the extension
$$
0 \to {\Bbb V} \to {\Bbb E} \to {\Bbb Q} \to 0
$$
along the projection ${\Bbb V} \to {\Bbb V}_{\lambda_3}$ onto the $\lambda_3$
isotypical component, we obtain an extension
$$
0 \to {\Bbb V}_{\lambda_3} \to {\Bbb E}' \to {\Bbb Q} \to 0.
$$
It follows from Johnson's computation that the restricted monodromy
representation of ${\Bbb E}$ factors through that of ${\Bbb E}'$:
$$
{\Bbb H}_1(T_g) \to {\Bbb V}_{\lambda_3} \to {\Bbb V}.
$$
We have therefore reduced to the case where ${\Bbb V}={\Bbb V}_{\lambda_3}$.
Let ${\Bbb V}(\lambda_3)$ be the unique variation of Hodge structure of
weight $-1$ over ${\cal M}_g(L)$ with monodromy representation given by
$\lambda_3$. Let ${\Bbb S}$ be the variation of mixed Hodge structure over
${\cal M}_g(L)$ given by the cycle $C - C^-$ that was constructed in Section
\ref{normal_functions}. It is an extension of ${\Bbb Q}$ by ${\Bbb V}(\lambda_3)$.
By \cite{saito}, the exact sequence
$$
0 \to \Hom_{\Gamma_g(L)}({\Bbb S},{\Bbb V}_{\lambda_3}) \to
\Hom_{\Gamma_g(L)}({\Bbb S},{\Bbb E}') \to \Hom_{\Gamma_g(L)}({\Bbb S},{\Bbb Q})
$$
is a sequence of mixed Hodge structures. The most right hand group is
easily seen to be isomorphic to ${\Bbb Q}(0)$; it is generated by the
projection ${\Bbb S} \to {\Bbb Q}$. The left hand group is easily seen to be
zero. It follows that
$$
\Hom_{\Gamma_g(L)}({\Bbb S},{\Bbb E}') \approx {\Bbb Q}(0).
$$
Since the monodromy representation of ${\Bbb S}$ is a morphism, it follows
that the monodromy representations of ${\Bbb E}'$ and ${\Bbb E}$ are too.
\end{pf}
\section{The Harris-Pulte Theorem}
As an application of the classification of normal functions above, we
give a new proof of the Harris-Pulte theorem which relates the mixed
Hodge structure on $\pi_1(C,x)$ to the normal function of the cycle
$C_x-C_x^-$ when $g\ge 3$. The result we obtain is slightly stronger.
Fix a level so that $\Gamma_g^1(L)$ is torsion free.
Denote by ${\Bbb L}$ the ${\Bbb Z}$ variation of Hodge structure of weight
$-1$ over ${\cal M}_g^1(L)$ whose fiber over the pointed curve $(C,x)$
is $H_1(C)$. Denote the corresponding holomorphic vector bundle
by ${\cal L}$. The cycle $C_x - C_x^-$ defines a normal function $\zeta$
which is a section of
$$
J\Lambda^3{\cal L} \to {\cal M}_g^1(L).
$$
Denote the integral group ring of $\pi_1(C,x)$ by ${\Bbb Z}\pi_1(C,x)$,
and its augmentation ideal by $I(C,x)$, or $I$ when there is no
possibility of confusion. There is a canonical mixed
Hodge structure on the truncated augmentation ideal
$$
I(C,x)/I^3.
$$
(See, for example, \cite{hain:geom}.)
It is an extension
$$
0 \to H_1(C)^{\otimes 2}/q \to I(C,x)/I^3 \to H_1(C) \to 0,
$$
where $q$ denotes the symplectic form. Tensoring with $H_1(C)$
and pulling back the resulting extension along the map
${\Bbb Z} \to H_1(C)^{\otimes 2}$, we obtain an extension
$$
0 \to H_1(C)\otimes \left(H_1(C)^{\otimes 2}/q\right)
\to E(C,x) \to {\Bbb Z} \to 0.
$$
Since the set of $I(C,x)$ form a good variation of mixed Hodge
structure over ${\cal M}_g^1(L)$ (\cite{hain:dht}), the set of $E(C,x)$
form a good variation of mixed Hodge structure ${\Bbb E}$ over ${\cal M}_g^1(L)$.
It therefore determines a normal function $\rho$ which is a section of
$$
J{\cal L}\otimes\left({\cal L}^{\otimes 2}/q\right) \to {\cal M}_g^1(L).
$$
Define the map
$$
\Phi: J\Lambda^3 {\cal L} \to J{\cal L}\otimes({\cal L}^{\otimes 2}/q)
$$
to be the one induced by the map
$$
\Lambda^3 {\Bbb L} \to {\Bbb L}^{\otimes 3} \to
{\Bbb L} \to {\Bbb L}\otimes({\Bbb L}^{\otimes 2}/q);
$$
the first map is defined by
$$
x_1 \wedge x_2 \wedge x_3 \mapsto \sum_\sigma
\text{sgn}(\sigma) x_{\sigma(1)} \otimes x_{\sigma(2)} \otimes x_{\sigma(3)}
$$
where $\sigma$ ranges over all permutations of $\{1,2,3\}$.
Our version of the Harris-Pulte Theorem is:
\begin{theorem}
The image of $\zeta$ under $\Phi$ is $2\rho$.
\end{theorem}
\begin{pf}
The proof uses (\ref{rigidity}). It is a straightforward consequence
of (\ref{equality}) that the monodromy representations of $\Phi(\zeta)$
and $2\rho$ are equal. It is also a straightforward matter
to use functoriality to show that both $\Phi(\zeta)$ and $2\rho$ vanish
at $(C,x)$ when $C$ is hyperelliptic and $x$ is a Weierstrass point
(cf.\ \cite[(7.5)]{hain:geom}.)
\end{pf}
\section{The Franchetta Conjecture for Curves with a Level}
\label{franchetta}
Suppose that $L$ is a finite index subgroup of $Sp_g({\Bbb Z})$, not necessarily
torsion free. Denote the generic point of ${\cal M}_g(L)$ by $\eta$. There is a
universal curve defined generically over ${\cal M}_g(L)$. Denote its fiber over
$\eta$ by ${\cal C}_g(L)_\eta$. In the statement below, $S$ denotes a
compact oriented surface of genus $g$.
\begin{theorem}\label{franch_conj}
For all $g\ge 3$ and all finite index subgroups $L$ of $Sp_g({\Bbb Z})$, the
group $\Pic {\cal C}_g(L)_\eta$ is finitely generated of rank 1. The
torsion subgroup is isomorphic to $H^0(L,H_1(S,{\Bbb Q}/{\Bbb Z}))$. Modulo torsion,
either it is generated by the canonical bundle, or by a divisor of degree
$g-1$.
\end{theorem}
This has a concrete statement when $L=Sp_g({\Bbb Z})(l)$, the congruence
subgroup of level $l$ of $Sp_g({\Bbb Z})$. It is not difficult to show
that the only torsion points of $\Jac S$ invariant under $L$ are
the points of order $l$. That is,
$$
H^0(L,H_1(S,{\Bbb Q}/{\Bbb Z})) \approx H_1(S,{\Bbb Z}/l{\Bbb Z}).
$$
In this case we shall denote ${\cal C}_g(L)_\eta$ by ${\cal C}_g(l)_\eta$.
During the proof of the theorem, we will show that, mod torsion,
$\Pic {\cal C}_g(l)_\eta$ is generated by a theta characteristic when $l$
is even. Combining this with the theorem, we have:
\begin{corollary}
If $g\ge 3$, then for all $l\ge 0$, $\Pic {\cal C}_g(l)_\eta$ is a
finitely generated group of rank one with torsion subgroup
isomorphic to $H_1(S,{\Bbb Z}/l{\Bbb Z})$. Modulo torsion, $\Pic {\cal C}_g(l)_\eta$
is generated by a theta characteristic when $l$ is even, and by the
canonical bundle when $l$ is odd. \qed
\end{corollary}
The case $g=2$, if true, should follow from Mess's computation of
$H_1(T_2)$ \cite{mess}. One should note that Mess proved that $T_2$
is a countably generated free group.
\begin{pf*}{Sketch of proof of Theorem \ref{franch_conj}}
We first suppose that $L$ is torsion free. In this case, the universal
curve is defined over all of ${\cal M}_g(L)$. Denote the restriction of
it to a Zariski open subset $U$ of ${\cal M}_g(L)$ by ${\cal C}_g(L)_U$. Set
$$
\Pic_{\rel U} {\cal C}_g(L) = \coker\{\Pic U \to \Pic {\cal C}_g(U)\}.
$$
Then
$$
\Pic {\cal C}_g(L)_\eta =
\lim_{\stackrel{\longrightarrow}{U}}
\Pic_{\rel U} {\cal C}_g(L),
$$
where $U$ ranges over all Zariski open subsets of ${\cal M}_g(L)$.
There is a natural homomorphism
$$
\mathrm{deg}: \Pic_{\rel U}{\cal C}_g \to {\Bbb Z}
$$
given by taking the degree on a fiber. Denote $\mathrm{deg}^{-1}(d)$
by $\Pic_{\rel U}^d {\cal C}_g(L)$.
We first compute $\Pic^0 {\cal C}_g(L)_\eta$. Each element of this group
can be represented by a line bundle over ${\cal C}_g(L)_U$ whose
restriction to each fiber of $\pi : {\cal C}_g(L)_U \to U$ is topologically
trivial. This line bundle has a section. By tensoring it with the
pullback of a line bundle on $U$, if necessary, we may assume that the
divisor of this section intersects each fiber of $\pi$ in only
a finite number of points. We therefore obtain a normal function
$$
s : U \to \Pic_{\rel U}^0 {\cal C}_g(L).
$$
Since the associated variation of Hodge structure is the unique one
of weight $-1$ associated to $V(\lambda_1)$, it follows from
(\ref{comp}) and (\ref{extend}) that this normal function is torsion.
It follows that
$$
\Pic^0 {\cal C}_g(L)_\eta = \Pic_{\rel U}^0 {\cal C}_g(L)
= H^0(L,H_1(S,{\Bbb Q}/{\Bbb Z})).
$$
Since this group is isomorphic to $H_1(S,{\Bbb Z}/l{\Bbb Z})$ when $L$ is the
congruence $l$ subgroup of $Sp_g({\Bbb Z})$, and since every finite index
subgroup of $Sp_g({\Bbb Z})$ contains a congruence subgroup by \cite{b-m-s},
it follows that $\Pic^0 {\cal C}_g(L)_\eta$ is finite for all $L$.
The relative dualizing sheaf $\omega$ of ${\cal C}_g(L)_U$ gives an
element of $\Pic^{2g-2} {\cal C}_g(L)_\eta$. Denote the greatest common
divisor of the degrees of elements of $\Pic {\cal C}_g(L)_\eta$ by $d$.
Observe that $d$ divides $2g-2$. Let $m=(2g-2)/d$. We will show that
$m=1$ or 2.
Choose an element $\delta$ of $\Pic^d {\cal C}_g(L)_\eta$. Then
$$
\omega - m\delta \in \Pic^0 {\cal C}_g(L)_\eta
$$
and is therefore torsion of order $k$, say. Replace $L$ by
$$
L' = L \cap Sp_g({\Bbb Z})(km).
$$
Observe that the natural map
$$
\Pic^0 {\cal C}_g(L)_\eta \to \Pic^0 {\cal C}_g(L')_\eta
$$
is injective. We can find
$$
\mu \in \Pic^0 {\cal C}_g(L')_\eta
$$
such that $m\mu = \omega - m\delta$. Then $\delta + \mu$ is an $m$th
root of the canonical bundle $\omega$. It appears to be well known
that the only non-trivial roots of the canonical bundle that can be
defined over ${\cal M}_g(L)$ are square roots. In any case, this follows from
the result (\ref{action}) in the next section. This implies that
$m$ divides 2, as claimed.
It follows from (\ref{action})) that square roots of the canonical
bundle are defined over ${\cal M}_g(l)$ if and only if $l$ is even. Combined
with the argument above, this shows that, mod torsion,
$\Pic^0 {\cal C}_g(l)_\eta$ is generated by $\omega$ if $l$ is odd, and by
a square root of $\omega$ if $l$ is even.
Our final task is to reduce the general case to that where $L$ is
torsion free. For arbitrary $L$, we have
$$
\Pic {\cal C}_g(L)_\eta =
\lim_{\stackrel{\longrightarrow}{U}}
\Pic_{\rel U} {\cal C}_g(L),
$$
where $U$ ranges over all smooth Zariski open subsets of ${\cal M}_g(L)$.
Choose a torsion free finite index normal subgroup $L'$ of $L$ and a
smooth Zariski open subset $U$ of ${\cal M}_g(L)$. Denote the inverse image
of $U$ in ${\cal M}_g(L')$ by $U'$. Then the projection $U' \to U$ is a
Galois cover with Galois group $G=L/L'$. It follows that
$$
\Pic_{\rel U} {\cal C}_g(L) = \Pic_{\rel U'} {\cal C}_g(L')^G.
$$
Since $\pi_1(U)$ surjects onto $\Gamma_g(L)$, and therefore onto $L$,
the result follows.
\end{pf*}
Denote the universal curve over the generic point $\eta$ of
${\cal M}_{g,r}^n(l)$ by ${\cal C}_{g,r}^n(l)_\eta$. The proof of the following
more general result is similar to that of Theorem \ref{franch_conj}.
\begin{theorem}
If $g\ge 3$, then for all $l\ge 0$, $\Pic{\cal C}_{g,r}^n(l)_\eta$ is a
finitely generated group of rank $r+n+1$ whose torsion subgroup
isomorphic to $H_1(S,{\Bbb Z}/l{\Bbb Z})$. Each of the $n$ marked points and the
anchor point of each of the $r$ marked cotangent vectors gives an element of
$\Pic^1{\cal C}_{g,r}^n(l)_\eta$. The pairwise differences of these points
generate a subgroup of $\Pic^0{\cal C}_{g,r}^n(l)_\eta$ of rank $r+n-1$.
Moreover, $\Pic^0{\cal C}_{g,r}^n(l)_\eta$ is generated by these differences
modulo torsion. Modulo $\Pic^0{\cal C}_{g,r}^n(l)_\eta$,
$\Pic{\cal C}_{g,r}^n(l)_\eta$ is generated by the class of one of the
distinguished points together with a theta characteristic when $l$ is
even, and by the canonical divisor when $l$ is odd. \qed
\end{theorem}
Note that the independence of the pairwise difference of the points
follows from the discussion following Theorem \ref{comp}.
\section{The Monodromy of Roots of the Canonical Bundle}
\label{root}
In this section we compute the action of $\Gamma_g$ on the set of
$n$th roots of the canonical bundle of a curve of genus $g$.
This action has also been computed by P.~Sipe \cite{sipe}, but
in quite a different form.
If $L$ is an $n$th root of the tangent bundle of a smooth projective
curve $C$, then its dual is an $n$th root of the canonical bundle.
That is, there is a one-one correspondence between $n$th roots of
the canonical bundle and $n$th roots of the tangent bundle of a curve.
As it is more convenient, we shall work with roots of the tangent bundle.
The first point is that roots of the tangent bundle are determined
topologically (cf.\ \cite[\S 3]{atiyah} and \cite{sipe}): denote the
${\Bbb C}^\ast$ bundle associated to the holomorphic tangent bundle $TC$ of
$C$ by $T^\ast$. Indeed, an $n$th root of $TC$ is a cyclic covering of
$T^\ast$ of degree $n$ which has degree $n$ on each fiber. The complex
structure on such a covering is uniquely determined by that on $T^\ast$.
The first Chern class of $TC$ is $2-2g$. So if $R$ is an $n$th root of
$K$, we have that $n$ divides $2g-2$. Since the Euler class of $T^\ast$
is $2-2g$, it follows from the Gysin sequence that there is a short exact
sequence
\begin{equation}\label{extension}
0 \to {\Bbb Z}/n{\Bbb Z} \to H_1(T^\ast,{\Bbb Z}/n{\Bbb Z}) \to H_1(C,{\Bbb Z}/n{\Bbb Z}) \to 0
\end{equation}
By covering space theory, an $n$th root of $TC$ is determined by
a homomorphism
$$
H_1(T^\ast,{\Bbb Z}/n{\Bbb Z}) \to {\Bbb Z}/n{\Bbb Z}
$$
whose composition with the inclusion
${\Bbb Z}/n{\Bbb Z} \hookrightarrow H_1(T^\ast,{\Bbb Z}/n{\Bbb Z})$ is the identity. That is,
we have the following result:
\begin{proposition}
There is a natural one-to-one correspondence between $n$th roots of the
the canonical bundle of $C$ and splittings of the sequence
(\ref{extension}). \qed
\end{proposition}
Throughout this section, we will assume $g \ge 3$.
Denote the set of $n$th roots of $TC$ by $\Theta_n$. This is a
principal $H_1(C,{\Bbb Z}/n{\Bbb Z})$ space. The automorphisms of this affine
space is an extension
$$
0 \to H_1(C,{\Bbb Z}/n{\Bbb Z}) \to \Aut \Theta_n
\stackrel{\pi}{\to} GL_{2g}({\Bbb Z}/n{\Bbb Z}) \to 1;
$$
the kernel being the group of translations by elements of $H_1(C,{\Bbb Z}/n{\Bbb Z})$.
The mapping class group acts on $\Theta_n$, so we have a homomorphism
$$
\Gamma_g \to \Aut \Theta_n.
$$
The composite of this homomorphism with $\pi$ is the
reduction mod $n$
$$
\rho_n : \Gamma_g \to Sp_g({\Bbb Z}/n{\Bbb Z})
$$
of the natural homomorphism. Denote the subgroup
$\pi^{-1}(Sp_g({\Bbb Z}/n{\Bbb Z}))$ of $\Aut \Theta_n$ by ${\cal K}_n$. It follows
that the action of $\Gamma_g$ on $\Theta_n$ factors through a
homomorphism
$$
\theta_n : \Gamma_g \to {\cal K}_n
$$
whose composition with the natural projection
${\cal K}_n \to Sp_g({\Bbb Z}/n{\Bbb Z})$ is $\rho_n$. In order to determine
$\theta_n$, we will need to compute its restriction
$$
\theta_n : H_1(T_g) \to H_1(C,{\Bbb Z}/n{\Bbb Z})
$$
to the Torelli group. First some algebra.
\begin{proposition}
There is a natural homomorphism
$$
\psi_g : H_1(T_g,{\Bbb Z}) \to H_1(C,{\Bbb Z}/(g-1){\Bbb Z}).
$$
\end{proposition}
\begin{pf}
By (\ref{tau_g}), there is a natural homomorphism
$$
\tau_g : H_1(T_g,{\Bbb Z}) \to
\Lambda^3 H_1(C,{\Bbb Z})/\big([C]\times H_1(C,{\Bbb Z})\big).
$$
Here we view $\Lambda^\bullet H_1(C)$ as the homology of $\Jac C$ and
$[C]$ denotes the homology class of the image of $C$ under
the Abel-Jacobi map. There is also a natural homomorphism
$$
p : \Lambda^3 H_1(C,{\Bbb Z}) \to H_1(C,{\Bbb Z})
$$
defined by
$$
p : x\wedge y \wedge z \mapsto
(x\cdot y)\, z + (y\cdot z)\, x + (z\cdot x)\, y.
$$
It is easy to see that the composite
$$
H_1(C,{\Bbb Z}) \stackrel{[C]\times}{\longrightarrow}
\Lambda^3 H_1(C,{\Bbb Z}) \stackrel{p}{\to} H_1(C,{\Bbb Z})
$$
is multiplication by $g-1$. It follows that $p$ induces a homomorphism
$$
\overline{p} : \Lambda^3 H_1(C,{\Bbb Z}) {\to} H_1(C,{\Bbb Z}/(g-1){\Bbb Z}).
$$
The homomorphism $\psi_g$ is the composite $\overline{p}\circ\tau_g$.
\end{pf}
Call a translation of $\Theta_n$ {\it even\/} if it is translation
by an element of $2H^1(C,{\Bbb Z}/n{\Bbb Z})$. If $n$ is odd, this is the set
of all translations. If $n=2m$, this is the proper subgroup of
$H^1(C,{\Bbb Z}/n{\Bbb Z})$ isomorphic to $H^1(C,{\Bbb Z}/m{\Bbb Z})$. It is not difficult to
see that there is a unique subgroup of ${\cal K}_n$ that is an extension
of $Sp_g({\Bbb Z}/n{\Bbb Z})$ by the even translations. We shall denote it by
${\cal K}_n^{(2)}$.
\begin{theorem}\label{action}
The image of the natural homomorphism $ \theta_n : \Gamma_g \to {\cal K}_n$
is ${\cal K}_n^{(2)}$. The restriction of $\theta_n$ to $T_g$ is the
composite of $\psi_g$ with the homomorphism
$$
H_1(C,{\Bbb Z}/(g-1){\Bbb Z}) @>r>> H_1(C,{\Bbb Z}/n{\Bbb Z}) @>{PD}>> H^1(C,{\Bbb Z}/n{\Bbb Z}),
$$
where $r(k)$ equals $2k$ mod $n$ and $PD$ denotes Poincar\'e duality.
In particular, the Torelli group acts trivially on $\Theta_n$ if and
only if $n$ divides 2.
\end{theorem}
\begin{pf}
First, it was proven by Johnson in \cite{johnson_2} that the kernel
of the composite
$$
T_g \to H_1(T_g) \stackrel{\tau_g}{\to}
\Lambda^3 H_1(C,{\Bbb Z})/\big([C]\times H_1(C,{\Bbb Z})\big)
$$
is generated by Dehn twists on separating simple closed curves. Using
this, it is easy to check that the restriction of $\theta_n$ to $T_g$
factors through $\tau_g$. In \cite{johnson_3}, Johnson shows that
$T_g$ is generated by Dehn twists on a bounding pair of disjoint
simple closed curves.\footnote{Actually, all we need is that
$\Lambda^3 H_1(C,{\Bbb Z})/\big([C]\times H_1(C,{\Bbb Z})\big)$ be generated
by the images under $\tau_g$ by such bounding pair maps.
This is easily checked directly.}
Now suppose that $\varphi$ is such a bounding pair map. There are
two disjoint imbedded circles $A$ and $B$ such that $\varphi$ equals
a positive Dehn twist about $A$ and a negative one about $B$.
When we cut $C$ along $A\cup B$, we obtain two surfaces, of
genera $g'$ and $g''$, say. Choose one of these components, and
let $a$ be the cycle obtained by orienting $A$ so that it is a
boundary component of this component. It is not difficult to show
that the image of $\varphi$ under $\psi_g$ equals
$$
-g'\, [a] \in H_1(C,{\Bbb Z}/(g-1){\Bbb Z}).
$$
where $g'$ is the genus of the chosen component. Since $g' + g'' = g-1$,
this is well defined. Next, one can use Morse theory to show that the
image of this same bounding pair map in $H^1(C,{\Bbb Z}/n{\Bbb Z})$ is $-2 g'PD(a)$,
from which the result follows. Full details of these computation will
appear elsewhere.
\end{pf}
\begin{corollary}
The only roots of the canonical bundle defined over Torelli space
are the canonical bundle itself and its $2^{2g}$ square roots. \qed
\end{corollary}
\begin{remark}
The homomorphism $\theta_{2g-2} :\Gamma_g \to {\cal K}_{2g-2}$ appears
in Morita's work --- cf. \cite[\S 4.A]{morita}.
\end{remark}
\section{Heights of Cycles defined over ${\cal M}_g(L)$}
\label{heights}
Suppose that $X$ is a compact K\"ahler manifold of dimension $n$
and that $Z$ and $W$ are two homologically trivial algebraic cycles
in $X$ of dimensions $d$ and $e$, respectively. Suppose that
$d+e = n-1$ and that $Z$ and $W$ have disjoint supports. Denote the
current associated to $W$ by $\delta_W$. It follows from the
$\partial\overline{\del}$-Lemma that there is a current $\eta_W$ of type
$(d,d)$ that is smooth away from the support of $Z$ and
satisfies
$$
\partial\overline{\del}\eta_W = \pi i \delta_W.
$$
The (archimedean) height pairing between $Z$ and $W$ is defined by
$$
\langle Z,W \rangle = - \int_Z \eta_W.
$$
This is a real-valued, symmetric bilinear pairing on such disjoint
homologically trivial cycles. It is important in number theory
(cf.\ \cite{beilinson:height}).
Now suppose that
$$
X \to {\cal M}_g(L)
$$
is a family of smooth projective varieties of relative dimension $n$.
Suppose that $Z\to {\cal M}_g(L)$ and $W\to {\cal M}_g(L)$ are families of
algebraic cycles in $X$ of relative dimensions $d$ and $e$, respectively,
where $d+e = n-1$. Denote the fiber of $X$, $Z$ and $W$ over $C \in {\cal M}_g(L)$
by $X_C$, $Z_C$ and $W_C$, respectively. Suppose that $Z_C$ and $W_C$ are
homologically trivial in $X_C$ and that they have disjoint supports for
generic $C\in {\cal M}_g(L)$.
We shall suppose that $L$ has been chosen so that every curve has
two distinguished theta characteristics $\alpha$ and $\alpha+\delta$,
where $\delta$ is a non-zero point of order 2 in $\Jac C$. We shall
also suppose that $g$ is odd and $\ge 3$. Write $g$ in the form $g=2d+1$.
Denote the difference divisor
$$
\left\{x_1+\dots + x_d -y_1 -\dots - y_d : x_j,y_j\in C\right\}
$$
in $\Jac C$ by $\Delta$, and the theta divisor
$$
\left\{x_1+\dots +x_{2d} - \alpha : x_j \in C\right\}
$$
in $\Jac C$ by $\Theta_\alpha$. By \cite[(4,1.2)]{hain:heights}, there
is a rational function $f_C$ on $\Jac C$ whose divisor is
$$
\Delta - {2d \choose d} \Theta_\alpha.
$$
Denote the unique invariant measure of total mass one on $\Jac C$
by $\mu$.
\begin{theorem}\label{height}
Suppose that $g$ is odd and $\ge 3$. Suppose that $Z$ and $W$ are
families of homologically trivial cycles over ${\cal M}_g(L)$ in a family of
smooth projective varieties $p:X\to {\cal M}_g(L)$, as above. If the
monodromy of the local system $R^{2d+1}p_\ast {\Bbb Q}_X$ factors through
a rational representation of $Sp_g$, then there is a rational function
$h$ on ${\cal M}_g(L)$, and rational numbers $a$ and $b$ such that
$$
\langle Z_C,W_C\rangle = a \left(
\log |h(C)| + 2b\left(\log |f_C(\delta)|
- \int_{\Jac C}\log |f_C(x)| d\mu(x)\right)\right).
$$
\end{theorem}
The numbers $a$ and $b$ are topologically determined, as will become
apparent in the proof. The divisor of $h$ is computable when one has
a good understanding of how the cycles $Z$ and $W$ intersect.
One should be able to derive a similar formula for even $g$ using Bost's
general computation of the height in \cite{bost} and results from
\cite{hain:completions}.
The proof of Theorem \ref{height} occupies the remainder of this
section. We only give a sketch. We commence by defining two
algebraic cycles in $\Pic^d C$. For $D\in \Jac C$, let $C^{(d)}_D$ be
the $d$-cycle in $\Pic^d C$ obtained by pushing forward the
fundamental class of the $d$th symmetric power of $C$ along the
map
$$
\{x_1,\dots,x_d\} \mapsto x_1 + \dots + x_d + D.
$$
Let $i$ be the automorphism of $\Pic^d C$ defined by
$i : x \mapsto \alpha - x$.
Define
$$
Z_D = C^{(d)}_D - i_\ast C^{(d)}_D.
$$
This is a homologically trivial $d$-cycle in $\Pic^d C$.
{}From \cite{bost} and \cite{hain:heights}, we know that
$$
\langle Z_0 , Z_\delta \rangle =
2\log |f_C(\delta)| - 2\int_{\Jac C}\log |f_C(x)| d\mu(x).
$$
So the content of the theorem is that there is a rational function
$h$ on ${\cal M}_g(L)$ and rational numbers $a$ and $b$ such that
$$
\langle Z,W \rangle =
a\left(\log |h(C)| + b\, \langle Z_0 , Z_\delta \rangle\right).
$$
The basic point, as we shall see, is that, up to torsion, all normal
functions over ${\cal M}_g(L)$ are half integer multiples of that of $C-C^-$,
as was proved in Section \ref{normal_functions}.
We will henceforth assume that the reader is familiar with the content
of \cite[\S 3]{hain:heights}. We will briefly review the most relevant
points of that section.
A {\it biextension} is a mixed Hodge structure $B$ with only three
non-trivial weight graded quotients: ${\Bbb Z}$, $H$, and ${\Bbb Z}(1)$, where
$H$ is a Hodge structure of weight $-1$. The isomorphisms
$$
Gr^W_{-2}B\approx {\Bbb Z}(1)\text{ and }Gr^W_0B\approx {\Bbb Z}
$$
are considered to be part of the data of the biextension. If one replaces
${\Bbb Z}$ by ${\Bbb R}$ in this definition, one obtains the definition of a real
biextension. To a biextension $B$ one can associate a real number
$\nu(B)$, called the {\it height} of $B$. It depends only on the
associated real biextension $B\otimes{\Bbb R}$.
To a pair of disjoint homologically trivial cycles in a
smooth projective variety $X$ satisfying
$$
\dim Z + \dim W + 1 = \dim X,
$$
there is a canonical biextension $B_{\Bbb Z}(Z,W)$, whose
weight graded quotients are
$$
{\Bbb Z},\quad H_{2d+1}(X,{\Bbb Z}(-d)),\quad {\Bbb Z}(1),
$$
where $d$ is the dimension of $Z$.
The extensions
$$
0 \to H_{2d+1}(X,{\Bbb Z}(-d)) \to B_{\Bbb Z}(Z,W)/{\Bbb Z}(1) \to {\Bbb Z} \to 0
$$
and
$$
0 \to {\Bbb Z}(1) \to W_{-1}B_{\Bbb Z}(Z,W) \to H_{2d+1}(X,{\Bbb Z}(-d)) \to 0
$$
are the those determined by $Z$ (directly), and $W$
(via duality) \cite[(3.3.2)]{hain:heights}. We have
$$
\nu(B_{\Bbb Z}(Z,W)) = \langle Z,W \rangle.
$$
The first step in the proof is to reduce the size of the biextension.
Suppose that $\Lambda={\Bbb Z}$ or ${\Bbb R}$, and that $B$ is a
$\Lambda$-biextension with weight $-1$ graded quotient $H$.
Suppose that there is an inclusion $i: A\hookrightarrow H$ of
$\Lambda$ mixed Hodge structures. Pulling back the extension
$$
0 \to \Lambda(1) \to W_1B \to H \to 0
$$
along $i$, we obtain an extension
$$
0 \to \Lambda(1) \to E \to C \to 0.
$$
If this extension splits, there is a canonical lift
$\tilde{\imath} : C \to B$ of $i$. The quotient $B/C$ is also a $\Lambda$
biextension.
\begin{proposition}\label{prune}
The biextensions $B_\Lambda(Z,W)$ and $B_\Lambda(Z,W)/C$ have the
same height.
\end{proposition}
\begin{pf}
This is a special case of \cite[(5.3.8)]{lear}. It
follows directly from \cite[(3.2.11)]{hain:heights}.
\end{pf}
We will combine this with (\ref{comp}) to prune the biextension
$B_{\Bbb Z}(Z_C,W_C)$ until its weight $-1$ graded quotient is either
trivial or else one copy of $V(\lambda_3)$.
First observe that if $B$ is a biextension and $B'$ a mixed Hodge
substructure of $B$ of finite index, then $B'$ is a biextension and
there is a non-zero integer $m$ such that $\nu(B') = m\nu(B)$. This can be
proved using \cite[(3.2.11)]{hain:heights}.
To prune the biextension ${\Bbb B}(Z,W)$ over ${\cal M}_g(L)$, we consider the portion
of the monodromy representation
$$
H_1(T_g) \to \Hom_{\Bbb Z}(Gr^W_{-1}B(Z_C,W_C),{\Bbb Z}(1))
$$
associated to the variation $W_{-1}{\Bbb B}(Z,W)$ over ${\cal M}_g(L)$. This map is
$Sp_g$ equivariant. Denote $Gr^W_{-1}{\Bbb B}(Z,W)$ by ${\Bbb H}$. This monodromy
representation corresponds to the map
$$
{\Bbb H} \to \left\{H^1(T_g,{\Bbb Z}(1))\right\}
$$
of local systems over ${\cal M}_g(L)$ which takes $h\in H_C$ to the
functional $\{\phi \mapsto \phi(h)\}$ on $H_1(T_g)$. For each
$C\in{\cal M}_g(L)$ this is a morphism of Hodge structures by (\ref{morph}).
Denote its kernel by $K_C$. These form a variation of Hodge structure
${\Bbb K}$ over ${\cal M}_g(L)$. If the monodromy representation is trivial on
$T_g$, then ${\Bbb K} = {\Bbb H}$. Otherwise, Schur's lemma implies that ${\Bbb H}/{\Bbb K}$
is isomorphic ${\Bbb V}(\lambda_3)$ placed in weight $-1$.
We can pull back the extension
$$
0 \to {\Bbb Q}(1) \to W_{-1}{\Bbb B}(Z,W) \to {\Bbb H} \to 0
$$
along the inclusion ${\Bbb K}\hookrightarrow {\Bbb H}$ to obtain an extension
\begin{equation}\label{putative}
0 \to {\Bbb Q}(1) \to {\Bbb E} \to {\Bbb K} \to 0.
\end{equation}
If this extension splits over ${\Bbb Q}$, then, by replacing the lattice in
${\Bbb B}_{\Bbb Z}(Z,W)$ by a commensurable one, we may assume that the
splitting is defined over ${\Bbb Z}$. This has the effect of multiplying the
height by a non-zero rational number. Once we have done this, the
inclusion ${\Bbb K} \hookrightarrow Gr^W_{-1}{\Bbb B}(Z,W)$ lifts to an inclusion
${\Bbb K} \hookrightarrow {\Bbb B}(Z,W)$. Using (\ref{prune}), we can replace
$B_{\Bbb Z}(Z_C,W_C)$ by ${\Bbb B}' = {\Bbb B}(Z,W)/{\Bbb K}$ without changing height of
the biextension. For the time being, we shall assume that (\ref{putative})
splits over ${\Bbb Q}$. This is the case, for example, when ${\Bbb H}$
contains no copies of the trivial representation, as follows from
(\ref{decomp}) since ${\Bbb K}$ is a trivial $T_g$ module by construction.
The weight $-1$ graded quotient of ${\Bbb B}'$
is either trivial or isomorphic to ${\Bbb V}(\lambda_3)$. This
biextension is defined over the open subset $U$ of ${\cal M}_g(L)$ where
$Z_C$ and $W_C$ are disjoint. The related variations $W_{-1}{\Bbb B}'$
and ${\Bbb B}'/{\Bbb Z}(1)$ are defined over all of ${\cal M}_g(L)$.
If ${\Bbb K}={\Bbb H}$, then ${\Bbb B}'$ is an extension of ${\Bbb Z}$ by ${\Bbb Z}(1)$. It therefore
corresponds to a rational function $h$ on ${\cal M}_g(L)$ which is defined
on $U$ (cf. \cite[(9.3)]{hain:poly}). It follows from
\cite[(3.2.11)]{hain:heights} that the height of this biextension ${\Bbb B}'$
is $C\mapsto \log|h(C)|$. This completes the proof of the theorem in
this case.
Dually, when the extension
$$
0 \to Gr^W_{-1}{\Bbb B}' \to {\Bbb B}'/{\Bbb Z}(1) \to {\Bbb Z} \to 0
$$
has finite monodromy, there is a rational function $h$ on ${\cal M}_g(L)$
such that the height of $B'$, and therefore $B(Z_C,W_C)$, is rational
multiple of $\log|h(C)|$.
We have therefore reduced to the case where ${\Bbb B}'$ has weight graded
$-1$ quotient ${\Bbb V}(\lambda_3)$ and where neither of the extensions
$$
0 \to {\Bbb V}(\lambda_3) \to {\Bbb B}'/{\Bbb Z}(1) \to {\Bbb Z} \to 0
$$
or
$$
0 \to {\Bbb Z}(1) \to W_{-1}{\Bbb B}' \to {\Bbb V}(\lambda_3) \to 0
$$
is torsion. We also have the biextension ${\Bbb B}''$ associated to the cycles
$Z_0$ and $Z_\delta$. It has these same properties. After replacing the
lattices in each by lattices of finite index, we may assume that the
extensions of variations $W_{-1}{\Bbb B}'$ and $W_{-1}{\Bbb B}''$ are isomorphic,
and that the ${\Bbb B}'/{\Bbb Z}(1)$ and ${\Bbb B}''/{\Bbb Z}(1)$ are isomorphic. As in
\cite[(3.4)]{hain:heights}, the
biextensions ${\Bbb B}'$ and ${\Bbb B}''$ each determine a canonically metrized
holomorphic line bundle over ${\cal M}_g(L)$. These metrized line bundles
depend only on the variations ${\Bbb B}/{\Bbb Z}(1)$ and $W_{-1}{\Bbb B}$, and are
therefore isomorphic. Denote this common line bundle by
${\cal B} \to {\cal M}_g(L)$. The biextensions ${\Bbb B}'$ and ${\Bbb B}''$ determine
(and are determined by) meromorphic sections $s'$ and $s''$ of ${\cal B}$,
respectively. There is therefore a meromorphic function $h$ on ${\cal M}_g(L)$
such that $s'' = hs'$. It follows from the main result of \cite{lear}
that this function is a rational function. (The philosophy is that
period maps of variations of mixed Hodge structure behave well at the
boundary.) The result follows as
$$
\nu(B''_C) = \log||s''(C)|| = \log|h(C)| + \log||s'(C)|| = \nu(B'_C)
+ \log |h(C)|.
$$
To conclude the proof, we now explain how to proceed when the
extension (\ref{putative}) is not split as a ${\Bbb Q}$ variation. Write
${\Bbb K} = {\Bbb T} \oplus {\Bbb T}'$, where ${\Bbb T}$ is the trivial submodule of ${\Bbb K}$
and ${\Bbb T}'$ is its orthogonal complement. This is a splitting in the
category of ${\Bbb Q}$ variations by (\ref{decomp}). It also follows from
(\ref{decomp}) that the restriction of (\ref{putative}) to ${\Bbb T}'$ is
split. Consequently, there is an inclusion of mixed Hodge structures
${\Bbb T}' \hookrightarrow {\Bbb B}(Z,W)$. As above, we may replace
${\Bbb B}(Z,W)$ by the biextension ${\Bbb B}'={\Bbb B}(Z,W)/{\Bbb T}'$ after rescaling
lattices. This only changes the height by a non-zero rational number.
The weight graded $-1$ quotient of ${\Bbb B}'$ is the sum of at most one
copy of ${\Bbb V}(\lambda_3)$ and a trivial variation of weight $-1$.
Now suppose that $B_1$ and $B_2$ are two biextensions. We can
construct a new biextension $B_1 + B_2$ from them as follows:
Begin by taking their direct sum. Pull this back along the diagonal
inclusion
$$
{\Bbb Z} \hookrightarrow {\Bbb Z}\oplus {\Bbb Z} = Gr^W_0 \left(B_1\oplus B_2\right)
$$
to obtain a mixed Hodge structure $B$ whose weight $-2$ graded
quotient is
$$
Gr^W_{-2}\left(B_1 \oplus B_2\right) = {\Bbb Z}(1) \oplus {\Bbb Z}(1).
$$
Push this out along the addition map
$$
{\Bbb Z}(1) \oplus {\Bbb Z}(1) \to {\Bbb Z}(1)
$$
to obtain the sought after biextension $B_1\boxplus B_2$. The following
result follows directly from \cite[(3.2.11)]{hain:heights}.
\begin{proposition}
The height of $B_1\boxplus B_2$ is the sum of the heights of $B_1$ and
$B_2$.
\end{proposition}
The biextension ${\Bbb B}'$ is easily seen to be the sum, in this sense, of
two biextensions. The first is constant with weight $-1$ quotient
equal to the trivial variation ${\Bbb T}$ and the second is a variation with
weight $-1$ quotient equal to ${\Bbb H}/{\Bbb K}$, which is either zero or one
copy of ${\Bbb V}(\lambda_3)$. Since the height of a constant biextension
is a constant, the result follows from the computation of the height
of a biextension with weight $-1$ quotient ${\Bbb V}(\lambda_3)$ above.
\section{Results for Abelian Varieties}
\label{abelian}
Denote the quotient of Siegel space $\goth h_g$ of rank $g$ by a finite index
subgroup $L$ of $Sp_g({\Bbb Z})$ by ${\cal A}_g(L)$. This is the moduli space of
abelian varieties with a level $L$ structure. In this section we state
results for ${\cal A}_g(L)$ analogous to those in Sections \ref{normal_functions}
and \ref{heights}. The proofs are similar, but much simpler, and are
left to the reader.
We call a representation of $Sp_g$ {\it even} if it has a symmetric
$Sp_g$-invariant inner product, and {\it odd} if it has a
skew symmetric $Sp_g$-invariant inner product. It follows from Schur's
Lemma that every irreducible representation of $Sp_g$ is either
even or odd. The even ones occur as polarized variations of Hodge
structure of even weight over each ${\cal A}_g(L)$, while the odd ones occur
as polarized
variations of Hodge structure only over ${\cal A}_g(L)$ of odd weight provided
$-I \notin L$. These facts are easily proved by adapting the arguments
in Section \ref{tech}.
The first theorem is the analogue of (\ref{fin_gen}) for abelian
varieties. It is similar to the result \cite{silverberg} of Silverberg.
The point in our approach is that $H^1(L,V)$ vanishes for all non-trivial
irreducible representations of $Sp_g$ by \cite{ragunathan}.
\begin{theorem}\label{norm_ab}
Suppose that $g\ge 2$ and that $L/\pm I$ is torsion free. If ${\Bbb V} \to {\cal A}_g(L)$
is a variation of Hodge structure of negative weight whose monodromy
representation is the restriction to $L$ of a rational representation of
$Sp_g$, then the group of generically defined normal functions associated
to this variation is finite.
\qed
\end{theorem}
Since there are no normal functions of infinite order over ${\cal A}_g(L)$,
we have the following
analogue of (\ref{height}). Suppose that $Z$ and $W$ are families of
homologically trivial cycles over ${\cal A}_g(L)$ in a family of smooth projective
varieties $p:X\to {\cal A}_g(L)$. Suppose that they are disjoint over the generic
point. Suppose further that $d + e = n-1$, where $d$, $e$ and $n$ are the
relative dimensions over ${\cal A}_g(L)$ of $Z$, $W$ and $X$,
respectively. Denote the fiber of $Z$ over $A\in {\cal A}_g(L)$ by $Z_A$, etc.
\begin{theorem}
If $g\ge 2$ and the monodromy of the local system $R^{2d+1}p_\ast {\Bbb Q}_X$
is the restriction to $L$ of a rational representation of $Sp_g$, then
there is a rational function $h$ on ${\cal A}_g(L)$ such that
$$
\langle Z_A, W_A \rangle = \log|h(A)|
$$
for all $A\in {\cal A}_g(L)$. \qed
\end{theorem}
One can formulate and prove analogues of these results for the moduli
spaces ${\cal A}_g^n(L)$ of abelian varieties of dimension $g$, $n$ marked
points, and a level $L$ structure.
We conclude this section with a discussion of Nori's results and their
relation to Theorems \ref{comp} and \ref{norm_ab}. We first recall
the main result of the last section of Nori's paper \cite{nori}.
\begin{theorem}[Nori]\label{nori_thm}
Suppose that $X$ is a variety that is an unbranched covering of a
Zariski open subset $U$ of ${\cal A}_g(L)$, where $L$ is torsion free.
Suppose that ${\Bbb V}$ is a variation of Hodge structure of negative weight
over $X$ that is pulled back from the canonical variation over ${\cal A}_g(L)$
of the same weight whose monodromy representation is irreducible and
has highest weight $\lambda$. Then the group of normal functions
defined on $X$ associated to this variation is finite unless
$$
\lambda = \begin{cases}
0 & \text{and $g \ge 2$ or;}\cr
\lambda_1 & \text{and $g\ge 3$ or;} \cr
\lambda_3 & \text{and $g=3$ or; }\cr
m_1\lambda_1 + m_2\lambda_2 & \text{$g=2$ and $m_1\ge 2$.}
\qed
\end{cases}
$$
\end{theorem}
This result may seem to contradict Theorem \ref{norm_ab}. The difference
can be accounted for by noting that Theorem \ref{norm_ab} only applies
to open subsets of the ${\cal A}_g(L)$, whereas Nori's theorem applies to
a much more general class of varieties which contains unramified
coverings of open subsets of the ${\cal A}_g(L)$. One instructive example is
${\cal M}_3(l)$, where $l$ is odd and $\ge 3$. The map ${\cal M}_3(l) \to {\cal A}_3(l)$ is
branched along the hyperelliptic locus. Theorem \ref{norm_ab} does not
apply. However, Nori's Theorem \ref{nori_thm} does apply --- remember,
normal functions in weight $-1$ extend by (\ref{extend}). In this
way we realize the normal function associated to $\lambda_3$ in
Nori's result. Also,
by standard arguments, for each $n$, there is an open subset $U$
of $M_3(l)$ and an unbranched finite cover $V$ of $U$ over which the
natural projection ${\cal M}_3^n(l) \to {\cal M}_3(l)$ has a section. From this
one can construct $n$ linearly independent normal sections of the
jacobian bundle defined over $V$. Note that Nori's result does apply
to $V$, whereas (\ref{norm_ab}) does not.
|
1994-03-18T16:22:50 | 9403 | alg-geom/9403014 | en | https://arxiv.org/abs/alg-geom/9403014 | [
"alg-geom",
"math.AG"
] | alg-geom/9403014 | Charles Walter | A. D. King and Charles H. Walter | On Chow Rings of Fine Moduli Spaces of Modules | 8 pages, LATeX | J. reine angew. Math. 461 (1995), 179-187 | null | null | null | Let $M$ be a complete nonsingular fine moduli space of modules over an
algebra $S$. A set of conditions is given for the Chow ring of $M$ to be
generated by the Chern classes of certain universal bundles occurring in a
projective resolution of the universal $S$-module on $M$. This result is then
applied to the varieties $G_T$ parametrizing homogeneous ideals of $k[x,y]$ of
Hilbert function $T$, to moduli spaces of representations of quivers, and
finally to moduli spaces of sheaves on ${\Bbb P}^2$, reinterpreting a result of
Ellingsrud and Str\o mme.
| [
{
"version": "v1",
"created": "Fri, 18 Mar 1994 15:23:41 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"King",
"A. D.",
""
],
[
"Walter",
"Charles H.",
""
]
] | alg-geom | \section{\@startsection{section}{1}{\z@}{-3.25ex plus
-1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf}}
\def\subsection{\@startsection
{subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}}
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\begin{document}
\maketitle
\begin{abstract}
Let $M$ be a complete nonsingular fine moduli space of modules over an
algebra $S$. A set of conditions is given for the Chow ring of $M$ to
be generated by the Chern classes of certain universal bundles
occurring in a projective resolution of the universal $S$-module on
$M$. This result is then applied to the varieties $G_T$ parametrizing
homogeneous ideals of $k[x,y]$ of Hilbert function $T$, to moduli
spaces of representations of quivers, and finally to moduli spaces of
sheaves on ${\Bbb P}^2$, reinterpreting a result of Ellingsrud and
Str\o mme.
\end{abstract}
In a recent paper \cite{ES} Ellingsrud and Str\o mme identified a set
of generators of the Chow ring of the moduli space of stable sheaves
of given rank and Chern classes on ${\Bbb P}^2$ (in the case where the
moduli space is smooth and projective). In this paper we formulate a
part of their argument as a general theorem about fine moduli spaces
of modules over an associative algebra. This provides a more widely
applicable method for showing that the Chow ring of a fine moduli
space is generated by the Chern classes of appropriate universal
sheaves. In particular we apply the method to verify a conjecture of
Iarrobino and Yam\'eogo concerning the Chow rings of the varieties
$G_T$ parametrizing homogeneous ideals in $k[x,y]$ with a given
Hilbert function. We also verify a conjecture of the first author
concerning Chow rings of moduli spaces of representations of quivers.
Let $S$ be an associative algebra over an algebraically closed field
$k$. By convention we will consider only left $S$-modules in this
paper. A flat family of $S$-modules over a $k$-scheme $X$ is a sheaf
$\cal F$ of $S\otimes {\cal O}_X$-modules on $X$, quasi-coherent and
flat over ${\cal O}_X$. At a (closed) point $x\in X$ the fiber ${\cal
F}(x)$ is an $S$-module. If $\cal C$ is a class of $S$-modules, then
a fine moduli space for $\cal C$ is a scheme $M$ equipped with a flat
family $\cal U$ all of whose fibers are in $\cal C$ and with the usual
universal property. Our general theorem is the following:
\begin{theorem}
\label{main}Let ${\cal C}$ be a class of $S$-modules, and $M$ a fine
moduli space for ${\cal C}$ which is a complete nonsingular
variety. Suppose further that
(i)\quad If $E\in {\cal C}$, then ${\rm Hom}_S(E,E)\cong k$, ${\rm Ext}
_S^1(E,E)\cong T_{[E]}M$, and ${\rm Ext}_S^p(E,E)=0$ for $p\geq 2$;
(ii)\quad If $E\ncong F$ are in ${\cal C}$, then ${\rm Hom}_S(E,F)=0$
and ${\rm Ext}_S^p(E,F)=0$ for $p\geq 2.$
(iii)\quad If ${\cal U}$ is the universal $S\otimes {\cal O}_M$-module
on $M$, then ${\cal U}$ has a universal projective resolution of
finite length $$ 0\rightarrow \bigoplus_jP_{rj}\otimes {\cal
E}_{rj}\rightarrow \cdots
\rightarrow \bigoplus_jP_{1j}\otimes {\cal E}_{1j}\rightarrow
\bigoplus_jP_{0j}\otimes {\cal E}_{0j}\rightarrow {\cal U}\rightarrow 0
$$
with the $P_{ij}$ projective $S$-modules such that $\dim _k{\rm Hom}
_S(P_{ij},P_{i^{\prime }j^{\prime }})$ is always finite, and the ${\cal E}
_{ij}$ are all locally free ${\cal O}_M$-modules of finite rank.
Then
(a)\quad The Chern classes of the ${\cal E}_{ij}$ generate the Chow ring $
A^{*}(M)$ as a ${\Bbb Z}$-algebra.
(b)\quad Numerical and rational equivalence coincide on $M$. In particular, $
A^{*}(M)$ is a free ${\Bbb Z}$-module.
(c)\quad If $k={\Bbb C}$, then the cycle map $A^{*}(M)\rightarrow H^{*}(M,{
\Bbb Z})$ is an isomorphism. In particular, there is no odd-dimensional
cohomology.
\end{theorem}
If $S$ is a graded algebra, then one may formulate a graded version of
the theorem by replacing ``module'' by ``graded module'' throughout
and using the degree-zero parts of ${\rm Hom}$ and ${\rm Ext}$. More
generally, one should be able to formulate a version of Theorem
\ref{main} for moduli spaces of objects in an abelian category of
$k$-vector spaces in any situation where one has a suitable notion of
a family of objects of the category.
Our first application of Theorem \ref{main} is to the Iarrobino
varieties $G_T$ which parametrize homogeneous ideals $I\subset k[x,y]$
of Hilbert function $T$. Here $T=(t_0,t_1,\ldots )$ is a sequence of
nonnegative integers such that $t_n=0$ for $n\gg 0$, and the points of
$G_T$ correspond to those $I$ such that $\dim_k(k[x,y]/I)_n = t_n$ for
all $n$. These smooth projective varieties were originally constructed
in \cite{I} in order to study the Hilbert-Samuel function
stratification of the punctual Hilbert scheme of a surface. They have
since been studied in several papers including \cite{Goe} \cite{IY}
and \cite{Y}. The fact that these $G_T$ are fine moduli spaces was
addressed formally in \cite{G} Kap.\ 2, Lemma 4.
Having fixed the Hilbert function $T$, the degree-$n$ graded pieces of
the quotient rings form a family of quotients of dimension $ t_n$ of
the space of binary forms of degree $n$. This induces a natural
morphism from $G_T$ to the Grassmannian of quotient spaces ${\rm Gr}
(t_n,n+1) $. Let ${\cal A}_n$ denote the pullback to $G_T$ of the
universal quotient bundle on ${\rm Gr}(t_n,n+1)$. Our result is
\begin{theorem}
\label{GT}Let $G_T$ be the Iarrobino variety parametrizing homogeneous
ideals $I\subset k[x,y]$ of Hilbert function $T$, and let the ${\cal A}_n$
be the universal bundles defined above. Then
(a)\quad The Chern classes of the ${\cal A}_n$ generate the Chow ring $
A^{*}(G_T)$ as a ${\Bbb Z}$-algebra.
(b)\quad Numerical and rational equivalence coincide on $G_T$. In
particular, $A^{*}(G_T)$ is a free ${\Bbb Z}$-module.
(c)\quad If $k={\Bbb C}$, then the cycle map $A^{*}(G_T)\rightarrow
H^{*}(G_T,{\Bbb Z})$ is an isomorphism. In particular, there is no
odd-dimensional cohomology.
\end{theorem}
Parts (b) and (c) were already known because $G_T$ has a cell decomposition
corresponding to the initial ideals with Hilbert function $T$ (cf.\
\cite{Goe} or \cite{IY}). Nevertheless, our methods give a new proof.
Our second application is to fine moduli spaces of representations of
a quiver without oriented cycles. These moduli spaces were
constructed in \cite{K}.
To fix notation, we recall that a quiver $Q$ is a directed graph,
specified by a finite set of vertices $Q_0$ and a finite set of arrows
$Q_1$ between the vertices together with two maps
$h,t{:}~Q_1\rightarrow Q_0$ specifying the head and tail of each
arrow. A representation of $Q$ consists of vector spaces $W_i$ for
each $i\in Q_0$ and $k$-linear maps $\phi_a:W_{ta}\to W_{ha}$ for each
$a\in Q_1$. A subrepresentation is a collection of subspaces
$W_i'\subset W_i$ such that $\phi_a(W_{ta}')\subset W_{ha}'$ for all $a$.
The dimension vector $\alpha \in {\Bbb N}^{Q_0}$ of a representation
$(W_i,\phi_a)$ is given by $\alpha_i = \dim_k(W_i)$.
To obtain a moduli space of representations of $Q$ of dimension vector
$\alpha$ one needs to introduce a notion of stability. Having chosen
$\theta = (\theta_i)\in {\Bbb R}^{Q_0}$ such that $\sum_i
\theta_i\alpha_i = 0$, we say that a representation $(W_i,\phi_a)$ is
$\theta$-stable if all (proper) subrepresentations $(W_i')$ satisfy
$\sum_i \theta_i \dim(W_i) > 0$. When $\alpha $ is an indivisible
dimension vector and $\theta $ is generic, there is a smooth fine
moduli space $M_Q(\alpha,\theta)$ of $\theta $-stable representations
of $Q$ of dimension vector $\alpha $ (\cite{K} Proposition 5.3). If
the quiver $Q$ has no oriented cycles, then this fine moduli space is
projective (\cite{K} Proposition 4.3). Note that this moduli
space may actually be empty. The conditions on $\alpha$ and $\theta$
which make it non-empty are more subtle (cf.\ \cite{K} Remark 4.5).
The universal representation over $M_Q(\alpha ,\theta )$ consists of
vector bundles ${\cal U}_i$ of rank $\alpha_i$ together with the
universal morphisms. We use Theorem \ref{main} to prove the
following, confirming the conjecture made in Remark 5.4 of \cite{K}.
\begin{theorem}\label{quiver}
Let $Q$ be a quiver without oriented cycles, and
$M=M_Q(\alpha,\theta)$ be a smooth projective fine moduli space of
$\theta$-stable representations of $Q$ of dimension vector $\alpha$.
Let ${\cal U}_i$ be the universal bundles on $M$ described above. Then
(a)\quad The Chern classes of the ${\cal U}_i$ generate the Chow ring
$A^{*}(M)$ as a ${\Bbb Z}$-algebra.
(b)\quad Numerical and rational equivalence coincide on $M$.
In particular, $A^{*}(M)$ is a free ${\Bbb Z}$-module.
(c)\quad If $k={\Bbb C}$, then the cycle map $A^{*}(M)\to H^{*}(M,{\Bbb Z})$
is an isomorphism. In particular, there is no odd-dimensional cohomology.
\end{theorem}
The outline of the paper is as follows. In the first section we prove
Theorem \ref{main} by adapting the method of Ellingsrud and Str\o mme. In
the second and third sections we apply Theorem \ref{main} to prove Theorems
\ref{GT} and \ref{quiver}. In the fourth section we explain how
Theorem \ref{main} may be used to prove Ellingsrud and Str\o mme's
original result for sheaves on ${\Bbb P}^2$.
\section{Proof of the Main Theorem}
In this section we prove Theorem \ref{main} by adapting a method of
Ellingsrud and Str\o mme.
Let $\delta $ be the class of the diagonal in $A^{*}(M\times M)$, and let $
p_1$ and $p_2$ denote the projections from $M\times M$ onto its two factors.
We will adapt the methods of \cite{ES} \S 2 to show that $\delta $ can be
written as a polynomial in the Chern classes of the $p_1^{*}({\cal E}_{ij})$
and the $p_2^{*}({\cal E}_{ij})$. The theorem will then follow from the
following result, which Ellingsrud and Str\o mme describe as ``a well-known
observation on varieties with decomposable diagonal class'':
\begin{theorem}
{\rm (\cite{ES} Theorem 2.1)} Let $X$ be a nonsingular complete variety.
Assume that the rational equivalence class $\delta $ of the diagonal $\Delta
\subseteq X\times X$ decomposes in the form
\begin{equation}
\label{diag}\delta =\sum_{i\in J}p_1^{*}\alpha _i\ p_2^{*}\beta _i
\end{equation}
where $p_1$ and $p_2$ are the projection of $X\times X$ onto its factors,
and $\alpha _i$, $\beta _i\in A^{*}(X)$. Then
(a)\quad The $\alpha _i$ generate $A^{*}(X)$ as a ${\Bbb Z}$-module.
(b)\quad Numerical and rational equivalence coincide on $X$. In particular, $
A^{*}(X)$ is a free ${\Bbb Z}$-module.
(c)\quad If $k={\Bbb C}$, then the cycle map $A^{*}(X)\rightarrow H^{*}(X,{
\Bbb Z})$ is an isomorphism. In particular, there is no odd-dimensional
cohomology.
(d)\quad Suppose the set $\{\alpha _i\}$ in (\ref{diag}) is minimal. Then $
\{\alpha _i\}$ and $\{\beta _i\}$ are dual bases with respect to the
intersection form on $A^{*}(X)$.
\end{theorem}
So we show how to write $\delta $ as a polynomial in the Chern classes
of the $p_1^{*}({\cal E}_{ij})$ and the $p_2^{*}({\cal E}_{ij})$ using
a method similar to \cite{ES} \S 2. First we write ${\cal P}_{\bullet
}$ for the projective resolution of the universal family of modules
${\cal U}$ in unaugmented form
$$
0\rightarrow \bigoplus_jP_{rj}\otimes {\cal E}
_{rj}\rightarrow \cdots \rightarrow \bigoplus_jP_{0j}\otimes {\cal E}
_{0j}\rightarrow 0.
$$
Then let ${\cal L}^{\bullet}={\cal H}om_{S\otimes {\cal O}_{M\times
M}}^{\bullet}(p_1^{*}{\cal P}_{\bullet},p_2^{*}{\cal P}_{\bullet})$.
Since
\begin{equation}
\label{Lp}{\cal L}^p=\bigoplus_{i^{\prime }-i=p}\left(
\bigoplus_{j,j^{\prime }}{\rm Hom}_S(P_{ij},P_{i^{\prime }j^{\prime
}})\otimes p_1^{*}{\cal E}_{ij}^{\vee }\otimes p_2^{*}{\cal
E}_{i^{\prime }j^{\prime }}\right) ,\
\end{equation}
${\cal L}^{\bullet}$ is a finite complex of locally free modules of
finite rank with the universal property that for any morphism of
$k$-schemes of the form $\phi {:}\ X\rightarrow M\times M$, we have
$H^p(\phi ^{*}{\cal L} ^{\bullet})={\cal E}xt_{S\otimes {\cal
O}_X}^p(\phi ^{*}p_1^{*}{\cal U},\phi ^{*}p_2^{*}{\cal U})$ for all
$p$. In particular ${\cal L}^{\bullet}$ is exact except in degrees 0
and 1. Indeed, if $d^p{:}\ {\cal L}^p\rightarrow {\cal L}^{p+1}$ is
the differential of ${\cal L}^{\bullet}$, then ${\cal L} ^{\bullet}$
is quasi-isomorphic to the short complex
$$
0\rightarrow {\rm cok}(d^{-1})\stackrel{\phi }{\longrightarrow }{\rm
\ker } (d^1)\rightarrow 0
$$
where $\phi $ is a map between locally free sheaves whose degeneracy
locus is exactly the diagonal. Our complex ${\cal L}^{\bullet}$ now
has all the essential properties of the complex ${\cal C}^{\bullet}$
of \cite{ES} Lemma 2.4. Hence by the same argument we have
$$
\delta =c_{\dim M}([{\rm \ker }(d^1)]-[{\rm cok}(d^{-1})])=c_{\dim
M}(\sum (-1)^{p+1}[{\cal L}^p]). $$ The formula (\ref{Lp}) and
standard formulas for Chern classes now permit us to write $\delta $
as a polynomial in the Chern classes of the $p_1^{*}{\cal E}_{ij}$ and
the $p_2^{*}{\cal E}_{i^{\prime }j^{\prime }}$. This completes the
proof of Theorem \ref{main}. {\hfill\mbox{$\Box$}}
\section{Iarrobino Varieties\label{iarro}}
In this section we apply the main theorem to the study of the
Iarrobino varieties $G_T$ which parametrize homogeneous ideals in
$k[x,y]$ with Hilbert function $T$. We prove Theorem \ref{GT}.
\paragraph{Proof of Theorem \ref{GT}.}
Let $S=k[x,y]$ and ${\cal S}=S\otimes _k{\cal O}_{G_T}$. The fact that
$G_T$ is a fine moduli space means that there is a universal sheaf of
homogeneous ${\cal S}$-ideals ${\cal I}$ and a universal quotient
sheaf of graded ${\cal S}$-modules ${\cal A}={\cal S}/{\cal
I}$. According to our construction, we have ${\cal A}=\bigoplus_n{\cal
A}_n$.
We now wish to apply Theorem \ref{main} using ${\cal A}$ as the
universal module. To do so we need to verify the various hypotheses.
First the cohomological ones. Because we are working with graded
modules, we must examine the degree 0 graded pieces of the internal
Hom and Ext. If $ A=S/I$ and $B=S/J$ are two graded quotients of $S$
with the same Hilbert function, then there are no nonzero morphisms of
degree 0 between $A$ and $B$ unless $A=B$, in which case ${\rm
Hom}_S(A,A)_0\cong k$, the homotheties. Some standard exact sequences
show that ${\rm Ext}_S^1(A,A)_0\cong {\rm Hom} _S(I,A)_0$ which is the
tangent space to $G_T$ at $[I]$ by the graded analog of Grothendieck's
formula for the tangent space of the Hilbert scheme (cf.\ \cite{PS} \S
4). Since $S$ is of global dimension $2$, the functor ${\rm Ext}^2_S$
is right exact, so the surjection $S \to B$ induces a surjection ${\rm
Ext}^2_S(A,S)_0 \to {\rm Ext}^2_S(A,B)_0$. But by local duality
(\cite{S} n$^{\circ}$ 72, Th\'eor\`eme 1) we have ${\rm
Ext}^2_S(A,S)_0^* \cong H^0_{\frak m}(A)_{-2} = A_{-2}$, which
vanishes. So ${\rm Ext}^2_S(A,B)_0 = 0$. Finally ${\rm
Ext}_S^p(A,B)=0$ for all $p\geq 3$ because $S$ has global dimension
$2$. Thus the cohomological hypotheses of Theorem \ref {main} are
fulfilled.
Now we exhibit a universal projective resolution of ${\cal A}$. Since
${\cal A}=\bigoplus {\cal A}_n$ is a $k[x,y]\otimes {\cal
O}_{G_T}$-module, multiplication by $x$ and $y$ define morphisms $\xi
$, $\eta {:}\ {\cal A} _n\rightarrow {\cal A}_{n+1}$. Then the
universal projective resolutions is
$$
0\rightarrow \bigoplus_nS(-n-2)\otimes _k{\cal A}_n\stackrel{\alpha }{
\longrightarrow }\bigoplus_nS(-n-1)^2\otimes _k{\cal
A}_n\stackrel{\beta }{\longrightarrow }\bigoplus_nS(-n)\otimes _k{\cal
A}_n\rightarrow {\cal A}\rightarrow 0
$$
where the morphisms are the standard ones
$$
\alpha =\left[
\begin{array}{c}
-y\otimes 1+1\otimes \eta \\
x\otimes 1-1\otimes \xi
\end{array}
\right] ,\qquad \beta =\left[
\begin{array}{cc}
x\otimes 1-1\otimes \xi & y\otimes 1-1\otimes \eta
\end{array}
\right] .
$$
Note that the sums are finite because each ${\cal A}_n$ is of rank $t_n$
which vanishes for $n\gg 0$. Finally the ${\rm Hom}_S(S(-i),S(-j))_0$ are
all finite-dimensional. So the remaining hypotheses of Theorem \ref{main}
are fulfilled.
The theorem now follows directly from Theorem \ref{main}. {\hfill\mbox{$\Box$}}
\section{Representations of Quivers\label{reps}}
In this section we prove Theorem \ref{quiver} by applying the main
theorem to fine moduli spaces of representations of quivers without
oriented cycles. To do this, we start with the well-established
observation (cf.\ for example \cite{Ben}) that representations of a
quiver $Q$ are the same as modules for the path algebra
$kQ$. This algebra is generated over $k$ by a set of orthogonal
idempotents $\{e_i\mid i\in Q_0\}$ and a further set of generators
$\{x_a\mid a\in Q_1\}$ such that $x_a=e_{ha}x_{a}e_{ta}$. A left
$kQ$-module $E$ corresponds to the representation of $Q$ consisting of
the vector spaces $W_i=e_iE$ for each $i\in Q_0$, and the $k$-linear
maps $\phi_a:W_{ta}\to W_{ha}$ giving multiplication by $x_a$ for each
$a\in Q_1$.
The algebra $kQ$ is always hereditary, i.e.\ of global
dimension $\leq1$, and is finite-dimensional if and only if the quiver
$Q$ has no oriented cycles.
\paragraph{Proof of Theorem \ref{quiver}.}
Let $S=kQ$. Then the indecomposable projective $S$-modules are $Se_i$
for $i\in Q_0$, and $S$ has the following minimal projective
resolution as an $S,S$-bimodule (or $S\otimes S^{op}$-module)
\[
0\rightarrow \bigoplus_{a\in
Q_1}Se_{ha}\otimes e_{ta}S\;{\frac{\buildrel d}{
\longrightarrow }}\;\bigoplus_{i\in Q_0}Se_i\otimes e_iS\;
{\frac{\buildrel \mu }{\longrightarrow }}\;S\rightarrow 0
\]
where $\mu $ is multiplication and $d(e_{ha}\otimes e_{ta})=x_a\otimes
e_{ta}-e_{ha}\otimes x_a$. We can use this resolution to calculate
the derived functors of ${\rm Hom}_S$, because ${\rm Hom}_S(E,F)={\rm
Hom}_{S,S}(S,{\rm Hom}_k(E,F))$ for any $E$ and $F$. We see
immediately see that ${\rm Ext}_S^i(E,F)=0$, for $i\geq 2$.
To check the other cohomological conditions, we first note that, by a
standard Schur's Lemma style argument, the stability condition implies
that for any two $\theta$-stable modules $E$ and $F$
$$
{\rm Hom}_S(E,F)=\cases{ k & if $E\cong F$, \cr 0 & otherwise.}
$$
Now $M$ is constructed as a GIT quotient of the
representation space
$$
{\cal R}(Q,\alpha )=\bigoplus_{a\in Q_1}{\rm Hom}(W_{ta},W_{ha})
$$
by the reductive group $GL(\alpha )=\prod_{i\in Q_0}GL(W_i)$, where
$W_i$ is a fixed vector space of dimension $\alpha _i$. If
$E=(W_i,\phi_a)$, then the tangent space $T_{[E]}(M)$ is
isomorphic to normal space at $\phi\in {\cal R}(Q,\alpha)$ to the
$GL(\alpha)$-orbit. This is the cokernel of
$$
d_{\phi}: \quad \bigoplus_{i\in Q_0}{\rm Hom}(W_i,W_i) \longrightarrow
\bigoplus_{a\in Q_1}{\rm Hom}(W_{ta},W_{ha})
$$
where $(d_\phi \gamma )_a=\phi _a\gamma _{ta}-\gamma _{ha}\phi _a$.
But this is exactly the complex which calculates ${\rm
Ext}^{*}_S(E,E)$, using the projective resolution of $S$ above. Thus
$T_{[E]}M\cong {\rm Ext}^1_S(E,E)$.
Finally, we obtain the necessary universal projective resolution
by tensoring ${\cal U}$ with the projective bimodule resolution of $S$,
giving
$$
0\rightarrow \bigoplus_{a\in Q_1}Se_{ha}\otimes {\cal
U}_{ta}\rightarrow \bigoplus_{i\in Q_0}Se_i\otimes {\cal
U}_i\rightarrow {\cal U}\rightarrow 0
$$
This completes the verification of the hypotheses for Theorem
\ref{main} and hence the proof of Theorem \ref{quiver}. {\hfill\mbox{$\Box$}}
\section{Sheaves on ${\Bbb P}^2$}
In \cite{ES} Ellingsrud and Str\o mme proved that the Chow ring of a
moduli space $M$ of stable sheaves on ${\Bbb P}^2$ of fixed rank and
Chern classes is generated by the Chern classes of three bundles on
$M$ in those cases where $M$ is smooth and projective. We show how
their result can be viewed as an application of our Theorem
\ref{main}.
We use notation derived from a recent paper of Le Potier \cite{L}.
Let $r$, $c_1$, $\chi $, and $m$ be integers such that $-r<c_1\leq
0$, $\chi \leq 0$, $\chi \leq r+2c_1$, and $m\gg 0$. Write $ n=-\chi
+r+c_1$. We consider representations of the quiver with triple edges
labeled $x_1,y_1,z_1$ and $x_2,y_2,z_2$
\[
\begin{array}{ccccccc}
\alpha : & & n+c_1 & & n & & n-(r+c_1) \\
& & \bullet & \vcenter{\lrabox\nointerlineskip\lrabox\nointerlineskip\lrabox} & \bullet & \vcenter{\lrabox\nointerlineskip\lrabox\nointerlineskip\lrabox} & \bullet \\
\theta : & & -(r+c_1)m+n & & (r+2c_1)m-2n+r & & -c_1m+n
\end{array}
\]
with dimension vector $\alpha$ as marked. Those representations
satisfying the symmetric relations $x_1y_2=y_1x_2$, $x_1z_2=z_1x_2$,
and $y_1z_2=z_1x_2$ form a closed subvariety ${\cal R}^{\rm sym}(Q,\alpha)$ of the
representation space ${\cal R}(Q,\alpha)$. Let $I$ be the two-sided
ideal of $kQ$ generated by the symmetric relations above, and let $S=
kQ/I$. For any $\alpha$ (resp.\ $\theta$) we say that an $S$-module
is of dimension vector $\alpha$ (resp.\ is $\theta$-stable) if it is
so as a $kQ$-module. Then for any $\theta$ the image of ${\cal R}^{\rm sym}(Q,\alpha)$ in
the moduli space $M_Q(\alpha,\theta)$ is a fine moduli space $M_{S}(\alpha,\theta)$
of $\theta$-stable $S$-modules of dimension vector $\alpha$.
Le Potier has established a result (\cite{L} Th\'eor\`eme 3.1) which
may be interpreted in this language as saying that for $\alpha$ and
$\theta$ as marked in the diagram above, $M_{S}(\alpha,\theta)$ is isomorphic to the
moduli space $M_{{\Bbb P} ^2}(r,c_1,\chi )$ of Gieseker-Maruyama
stable sheaves on ${\Bbb P}^2$ of rank $r$, determinant $c_1$ and
Euler characteristic $\chi $. The restrictions to $M_{S}(\alpha,\theta)$ of the
universal bundles ${\cal U}_i$ on $M_Q(\alpha,\theta)$ may be
identified with the universal bundles $R^1\pi _{*}( {\cal E}(-i))$ on
$M_{{\Bbb P}^2}(r,c_1,\chi )$. (Note that with these conventions, the
vertices of the quiver are labeled $2,1,0$ from left to right.) Lemma
2.2 of \cite{ES} may be interpreted as saying that the Ext groups for
$\theta$-stable $S$-modules are isomorphic to the Ext groups for the
corresponding stable sheaves on ${\Bbb P}^2$. Hence the cohomological
hypotheses in our Theorem \ref{main} can be verified for
$\theta$-stable $S$-modules by using properties of stable sheaves on
${\Bbb P}^2$.
Let $\overline{e}_i$ be the images in $S$ of the idempotents $e_i$ of
$kQ$ corresponding to the three vertices $2,1,0\in Q_0$ (cf.\
\S\ref{reps}). Then the indecomposable projective $S$-modules are
$S\overline{e}_i$. As in \S\ref{reps} the minimal projective
resolution of $S$ as an $S,S$-bimodule yields a projective resolution
of the universal $S$-module $\cal U$ on $M_{S}(\alpha,\theta)$ which is now of the
form
\[
0\rightarrow \bigoplus_{\mathop{3\
relations}}S\overline{e}_0\otimes{\cal U}_2 \rightarrow
\bigoplus_{a\in Q_1}S\overline {e}_{ha}\otimes {\cal
U}_{ta}\rightarrow \bigoplus_{i\in Q_0}S\overline {e}_i\otimes {\cal
U}_i\rightarrow {\cal U}\rightarrow 0.
\]
We could therefore apply Theorem \ref{main} to retrieve \cite{ES}
Theorem 1.1.
\paragraph{Remark.}
It is also possible to prove Theorem \ref{GT} by regarding the
Iarrobino varieties $G_T$ as moduli spaces for representations of a
quiver ``with relations'' and thus as moduli spaces for
modules over a finite-dimensional non-commutative algebra. If
$T=(t_0,t_1,\ldots ,t_q,0,0,\ldots )$, then the algebra will be of the
form $R=kQ/I$ where $Q$ is the quiver
\[
\begin{array}{ccccccccccccc}
\alpha : & & t_0 & & t_1 & & t_2 & & & & t_{q-1} & & t_q \\
& & \bullet & \vcenter{\lrabox\nointerlineskip\lrabox} & \bullet & \vcenter{\lrabox\nointerlineskip\lrabox} & \bullet &
\vcenter{\lrabox\nointerlineskip\lrabox} & \cdots & \vcenter{\lrabox\nointerlineskip\lrabox} & \bullet & \vcenter{\lrabox\nointerlineskip\lrabox} &
\bullet \\
\theta : & & - & & + & & + & & & & + & & +
\end{array}
\]
with $2q$ edges $x_1,y_1,x_2,y_2,\ldots ,x_q,y_q$, and $I$ is
generated by the relations $x_iy_{i+1} - y_ix_{i+1}$. There is a
clear correspondence between $R$-modules of dimension vector $\alpha$
(as marked) and graded $k[x,y]$-modules of Hilbert function $T$. When
$t_0=1$ and the coefficients of $\theta $ have the signs indicated
above, then the $\theta $-stable $R$-modules correspond exactly to
$k[x,y]$-modules which are generated in degree $0$, i.e.\ to modules
isomorphic to $k[x,y]/J$ for some homogeneous ideal $J$ of $k[x,y]$.
Hence $M_R(\alpha,\theta)\cong G_T$. The universal bundles ${\cal
U}_i$ on $M_R(\alpha,\theta)$ are exactly the universal bundles ${\cal
A}_i$ on $G_T$.
|
1994-03-06T05:46:12 | 9403 | alg-geom/9403006 | en | https://arxiv.org/abs/alg-geom/9403006 | [
"alg-geom",
"math.AG"
] | alg-geom/9403006 | Misha Verbitsky | Misha Verbitsky | Hyperkaehler Embeddings II | 14 pp, LaTeX 2.09 | GAFA vol. 5 no. 1 (1995) pp. 92-104 | null | null | null | In the first part, Hyperkaehler Embeddings and Holomorphic symplectic
Geometry I, we prove the following. Let $N$ be a closed analytic subvariety of
a generic deformation of a holomorphically symplectic compact manifold $M$.
Then the restriction of a holomorphic symplectic form is non-degenerate on $N$.
In particular, $N$ is even-dimensional. In present paper, we prove that there
exist a hyperkaehler metric on $M$, such that the embedding of $N$ to $M$ is
hyperkaehler.
| [
{
"version": "v1",
"created": "Sun, 6 Mar 1994 04:46:27 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Verbitsky",
"Misha",
""
]
] | alg-geom | \section{Hyperk\"ahler manifolds.}
\definition \label{_hyperkaehler_manifold_Definition_}
(\cite{_Beauville_},
\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a
Riemannian manifold $M$ endowed with three complex structures $I$, $J$
and $K$, such that the following holds.
\hspace{5mm} (i) $M$ is K\"ahler with respect to these structures and
\hspace{5mm} (ii) $I$, $J$ and $K$, considered as endomorphisms
of a real tangent bundle, satisfy the relation
$I\circ J=-J\circ I = K$.
\hfill
This means that the hyperk\"ahler manifold has the natural action of
quaternions ${\Bbb H}$ in its real tangent bundle.
Therefore its complex dimension is even.
Let $\mbox{ad}I$, $\mbox{ad}J$ and $\mbox{ad}K$ be the operators on the
bundles of differential forms over a hyperk\"ahler manifold
$M$ which are defined as follows. Define $\mbox{ad}I$.
Let this operator act as a complex structure operator
$I$ on the bundle of differential 1-forms. We
extend it on $i$-forms for arbitrary $i$ using Leibnitz
formula: $\mbox{ad}I(\alpha\wedge\beta)=\mbox{ad}I(\alpha)\wedge\beta+
\alpha\wedge \mbox{ad}I(\beta)$. Since Leibnitz
formula is true for a commutator in a Lie algebras, one can immediately
obtain the following identities, which are implied by the same
identities in ${\Bbb H}$:
\[
[\mbox{ad}I,\mbox{ad}J]=2\mbox{ad}K;\;
[\mbox{ad}J,\mbox{ad}K]=2\mbox{ad}I;\;
\]
\[
[\mbox{ad}K,\mbox{ad}I]=2\mbox{ad}J
\]
Therefore, the operators $\mbox{ad}I,\mbox{ad}J,\mbox{ad}K$
generate a Lie algebra $\goth{su}(2)$ acting on the
bundle of differential forms. We can integrate this
Lie algebra action to the action of a Lie group
$G_M=SU(2)$. In particular, operators $I$, $J$
and $K$, which act on differential forms by the formula
$I(\alpha\wedge\beta)=I(\alpha)\wedge I(\beta)$,
belong to this group.
{\bf Proposition 1.1:} There is an action of the Lie group $SU(2)$
and Lie algebra $\goth{su}(2)$ on the bundle of differential
forms over a hyperk\"ahler manifold. This action is
parallel, and therefore it commutes with Laplace operator.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
If $M$ is compact, this implies that there is
a canonical $SU(2)$-action on $H^i(M,{\Bbb R})$ (see
\cite{_so5_on_cohomo_}).
\hfill
Let $M$ be a hyperk\"ahler manifold with a Riemannian form $<\cdot,\cdot>$.
Let the form $\omega_I := <I(\cdot),\cdot>$ be the usual K\"ahler
form which is closed and parallel
(with respect to the connection). Analogously defined
forms $\omega_J$ and $\omega_K$ are
also closed and parallel.
The simple linear algebraic
consideration (\cite{_Besse:Einst_Manifo_}) shows that \hfill
$\omega_J+\sqrt{-1}\omega_K$ is of
type $(2,0)$ and, being closed, this form is also holomorphic.
It is called {\bf the canonical holomorphic symplectic form
of a manifold M}. Conversely, if there is a parallel
holomorphic symplectic form on a K\"ahler manifold $M$,
then this manifold has a hyperk\"ahler structure (\cite{_Besse:Einst_Manifo_}).
If some $compact$ K\"ahler manifold $M$ admits non-degenerate
holomorphic symplectic form $\Omega$, the Calabi-Yau
(\cite{_Yau:Calabi-Yau_}) theorem
implies that $M$ is hyperk\"ahler (Proposition 2.1).
This follows from the existence of a K\"ahler
metric on $M$ such that $\Omega$ is parallel for the Levi-Civitta
connection associated with this metric.
\hfill
Let $M$ be a hyperk\"ahler manifold with complex structures
$I$, $J$ and $K$. For any real numbers $a$, $b$, $c$
such that $a^2+b^2+c^2=1$ the operator $L:=aI+bJ+cK$ is also
an almost complex structure: $L^2=-1$.
Clearly, $L$ is parallel with respect to connection.
This implies that $L$ is a complex structure, and
that $M$ is K\"ahler with respect to $L$.
\hfill
\definition \label{_induced_structures_Definion_}
If $M$ is a hyperk\"ahler manifold,
the complex structure $L$ is called {\bf induced
by a hyperk\"ahler structure}, if $L=aI+bJ+cK$ for some
real numbers $a,b,c\:|\:a^2+b^2+c^2=1$.
\hfill
\hfill
If $M$ is a hyperk\"ahler manifold and $L$ is induced complex structure,
we will denote $M$, considered as a complex manifold with respect to
$L$, by $(M,L)$ or, sometimes, by $M_L$.
\hfill
Consider the Lie algebra $\goth{g}_M$ generated by ${ad}L$ for all $L$
induced by a hyperk\"ahler structure on $M$. One can easily see
that $\goth{g}_M=\goth{su}(2)$.
The Lie algebra $\goth{g}_M$ is called {\bf isotropy algebra} of $M$, and
corresponding Lie group $G_M$ is called an {\bf isotropy group}
of $M$. By Proposition 1.1, the action of the group is parallel,
and therefore it commutes with Laplace operator in differential
forms. In particular, this implies that the action of the isotropy
group $G_M$ preserves harmonic forms, and therefore this
group canonically acts on cohomology of $M$.
\hfill
\proposition \label{_G_M_invariant_forms_Proposition_}
Let $\omega$ be a differential form over
a hyperk\"ahler manifold $M$. The form $\omega$ is $G_M$-invariant
if and only if it is of Hodge type $(p,p)$ with respect to all
induced complex structures on $M$.
{\bf Proof:} Assume that $\omega$ is $G_M$-invariant.
This implies that all elements of ${\goth g}_M$ act trivially on
$\omega$ and, in particular, that $\mbox{ad}L(\omega)=0$
for any induced complex structure $L$. On the other hand,
$\mbox{ad}L(\omega)=(p-q)\sqrt{-1}\:$ if $\omega$ is of Hodge type $(p,q)$.
Therefore $\omega$ is of Hodge type $(p,p)$ with respect to any
induced complex structure $L$.
Conversely, assume that $\omega$ is of type $(p,p)$ with respect
to all induced $L$. Then $\mbox{ad}L(\omega)=0$ for any induced $L$.
By definition, ${\goth g}_M$ is generated by such $\mbox{ad}L(\omega)=0$,
and therefore ${\goth g}_M$ and $G_M$ act trivially on $\omega$. $\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{Holomorphic symplectic geometry.}
\definition \label{_holomorphi_symple_Definition_}
The compact K\"ahler manifold $M$ is called
holomorphically symplectic if there is a holomorphic 2-form $\Omega$
over $M$ such that $\Omega^n=\Omega\wedge\Omega\wedge...$ is
a nowhere degenerate section of a canonical class of $M$.
There, $2n=dim_{\Bbb C}(M)$.
Note that we assumed compactness of $M$.%
\footnote{If one wants to define a holomorphic symplectic
structure in a situation when $M$ is not compact,
one should require also the equation $\nabla'\Omega$ to held.
The operator $\nabla':\;\Lambda^{p,0}(M)\longrightarrow\Lambda^{p+1,0}(M)$
is a holomorphic differential defined on differential $(p,0)$-forms
(\cite{_Griffiths_Harris_}).}
One observes that the holomorphically symplectic
manifold has a trivial canonical bundle.
A hyperk\"ahler manifold is holomorphically symplectic
(see Section 1). There is a converse proposition:
\proposition \label{_symplectic_=>_hyperkaehler_Proposition_
(\cite{_Beauville_}, \cite{_Besse:Einst_Manifo_})
Let $M$ be a holomorphically
symplectic K\"ahler manifold with the holomorphic symplectic form
$\Omega$, a K\"ahler class
$[\omega]\in H^{1,1}(M)$ and a complex structure $I$.
There is a unique hyperk\"ahler structure $(I,J,K,(\cdot,\cdot))$
over $M$ such that the cohomology class of the symplectic form
$\omega_I=(\cdot,I\cdot)$ is equal to $[\omega]$ and the
canonical symplectic form $\omega_J+\sqrt{-1}\:\omega_K$ is
equal to $\Omega$.
\ref{_symplectic_=>_hyperkaehler_Proposition_} immediately
follows from the Calabi-Yau theorem (\cite{_Yau:Calabi-Yau_}).
$\:\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
For each complex analytic variety $X$ and a point $x\in X$,
we denote the Zariski tangent space to $X$ in $x$ by $T_xX$.
\definition \label{_non-dege_symple_Definition_
Let $M$ be a holomorphically symplectic
manifold and $S\subset M$ be its closed complex analytic subvariety.
Assume that $S$ is closed in $M$ and reduced. It is called
{\bf non-degenerately symplectic} if for each point $s\in S$
outside of the singularities of $S$ the restriction
of the holomorphic symplectic form $\Omega$ to $T_sM$
is nondegenerate on $T_s S\subset T_s M$, and the set
$Sing(S)$ of the singular points of $S$ is nondegenerately
symplectic. This definition refers to itself, but
since $dim\:Sing(S)<dim\:S$, it is consistent.
Of course, the complex dimension of a non-degenerately symplectic
variety is even.
\hfill
Let $M$ be a holomorphically symplectic K\"ahler manifold.
By \ref{_symplectic_=>_hyperkaehler_Proposition_},
$M$ has a unique hyperk\"ahler metric with
the same K\"ahler class and holomorphic symplectic form.
Therefore one can without ambiguity speak
about the action of $G_M$ on $H^*(M,{\Bbb R})$ (see Proposition 1.1).
Of course, this action essentially depends on the choice
of K\"ahler class.
\hfill
\definition \label{_generic_manifolds_Definition_}
Let $\omega\in H^{1,1}(M)$ be the K\"ahler of a K\"ahler matric
defined on a holomorphically symplectic
manifold $M$. We say that $\omega$ {\bf induces the $SU(2)$-action
of general type} when all elements of the group
\[ H^{pp}(M)\cap H^{2p}(M,{\Bbb Z})\]
are $G_M$-invariant. The
action of $SU(2)\cong G_M$ is defined by
\ref{_symplectic_=>_hyperkaehler_Proposition_}.
The holomorphically symplectic manifold $M$ is
called {\bf of general type} if there
exists a K\"ahler class on $M$ which induces
an $SU(2)$-action of general type.
\hfill
As \ref{_subvarieties_of_generic_mfold_are_nondege_symple_Theorem_}
implies, the holomorhically symplectic manifold
of general type has no Weil divisors. Therefore these manifolds
have connected Picard group. In particular, such manifolds
are never algebraic.
\hfill
\proposition \label{_generic_are_dense_Proposition_}
Let $M$ be a hyperk\"ahler manifold. Let $S$
be the set of induced complex structures over $M$. Let $S_0\subset S$
be the set of $R\in S$ such that the natural K\"ahler metric
on $(M,R)$ induces the $SU(2)$ action of general type.
Then $S_0$ is dense in $S$.
{\bf Proof:}
Let $A$ be the set of all $\alpha\in H^{2p}(M,{\Bbb Z})$ such that $\alpha$
is not $G_M$-invariant. The set $A$ is countable. For each
$\alpha\in A$, let $S_\alpha$ be the set of all $R\in S$
such that $\alpha$ is of type $(p,p)$ with respect to $R$.
The set $S_0$ of all induced complex structures of general type
is equal to $\{S\backslash\bigcup_{\alpha\in A}S_\alpha\}$.
Now, to prove \ref{_generic_are_dense_Proposition_}
it is sufficient to show that
$S_\alpha$ is a finite set for each $\alpha\in A$.
This would imply that $S_0$ is a complement of a
countable set to a 2-sphere $S$, and therefore dense in $S$.
As it follows from Section 1,
$\alpha$ is of type $(p,p)$ with respect to $R$ if and
only if $ad \:R(\alpha)=0$. Now, let $V$ be a representation
of $\goth{su}(2)$, and $v\in V$ be a non-invariant vector.
It is easy to see that the element $a\in \goth{su}(2)$
such that $a(v)=0$ is unique up to a constant, if it exists.
This implies that if $\alpha$ is not $G_M$-invariant
there are no more than two $R\in S$ such that
$ad R(\alpha)=0$.
Of course, these two elements
of $S$ are opposite to each other. $\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
One can easily deduce from the results in
\cite{_Todorov:Moduli_I_II_} and
from \ref{_generic_are_dense_Proposition_} that the set of points
associated with holomorphically symplectic
manifolds of general type is dense in the classifying space
of holomorphically symplectic manifolds.
\hfill
For a K\"ahler manifold $M$ , $m=dim_{\Bbb C} M$
and a form $\alpha\in H^{2p}(M,{\Bbb C})$, define
\[deg(\alpha):=\int_M L^{m-p}(\alpha)\]
where $L$ is a Hodge operator of exterior
multiplication by the K\"ahler form $\omega$
(see \cite{_Griffiths_Harris_}).
Of course, the degree of forms of Hodge type
$(p,q)$ with $p\neq q$ is equal zero,
so only $(p,p)$-form can possibly have
non-zero degree.
\hfill
We recall that the real dimension of a
holomorphically symplectic manifold is divisible by 4.
\theorem \label{_G_M_invariant_cycles_over_Theorem_}
(Theorem 2.1 of \cite{_part_one_}).
Let $M$ be a holomorhically symplectic K\"ahler manifold with
a holomorphic symplectic form $\Omega$.
Let $\alpha$ be a $G_M$-invariant form of non-zero degree.
Then the dimension of $\alpha$ is divisible by 4. Moreover,
\[ \int_M \Omega^n\wedge\bar\Omega^n\wedge\alpha=2^n deg(\alpha),\]
where $n=\frac{1}{4}(dim_{\Bbb R} M-dim\:\alpha)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\proposition \label{_subvarieties_of_gene_are_even-dimen_} %
(Theorem 2.2 of \cite{_part_one_}). Let $M$ be a holomorphic symplectic
manifold of general type. All closed analytic subvarieties
of $M$ have even complex dimension.
{\bf Proof:} \ref{_subvarieties_of_gene_are_even-dimen_} immediately
follows from \ref{_G_M_invariant_cycles_over_Theorem_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\theorem \label{_subvarieties_of_generic_mfold_are_nondege_symple_Theorem_} %
(Theorem 2.3 of \cite{_part_one_}).
Let $M$ be a holomorphic symplectic
manifold of general type. All reduced closed analytic
subvarieties of $M$ are non-degenerately symplectic.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Combining this with \ref{_symplectic_=>_hyperkaehler_Proposition_},
one obtains
\theorem
Let $M$ be a holomorphically symplectic
manifold of general type, and $S\subset M$ be its smooth differentiable
submanifold. If $S$ is analytic in $M$, it is a
hyperk\"ahler manifold. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
%
\section{When analytic implies tri-analytic} %
%
\label{_analytic_implies_trianalytic_Section}
Let $M$ be a compact hyperk\"ahler manifold, $dim_{\Bbb R} M =2m$.
Let $I$ be an induced complex structure. As usually, $(M,I)$
denotes $M$ considered as a K\"ahler manifold with the
complex structure defined by $I$.
\hfill
\definition\label{_tri-analytic_Definition_}
Let $N\subset M$ be a closed subset of $M$. Then $N$ is
called {\bf tri-analytic} if $N$ is an analytic subset
of $(M,I)$ for any induced complex structure $I$.
\hfill
Let $N\subset(M,I)$ be
a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$.
Let $[N]\in H_{2n}(M)$ denote the homology class
represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote
the Poincare dual cohomology class. Recall that
the hyperk\"ahler structure induces the action of
the group $G_M=SU(2)$ on the space $H^{2m-2n}(M)$.
The main result of this section is following:
\hfill
\theorem\label{_G_M_invariant_implies_trianalytic_Theorem_}
Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect
to the action of $G_M$ on $H^{2m-2n}(M)$. Then $N$ is tri-analytic%
\footnote{The number $n=dim_{\Bbb C} N$ is even by
\ref{_G_M_invariant_cycles_over_Theorem_}.}%
.%
\hfill
\ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following
important corollary:
\corollary \label{_hyperkae_embeddings_Corollary_}
Let $M$ be a holomorphically symplectic
manifold of general type, and $S\subset M$ be its smooth complex
submanifold. Let $\omega$ be the K\"ahler class
which induces an $SU(2)$-action of general type.
Pick a hyperk\"ahler metric $s$ associated with
$\omega$ by \ref{_symplectic_=>_hyperkaehler_Proposition_}.
Then the restriction of $s$ to $S$ is a hyperk\"ahler
metric.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
The rest of this section is dedicated to the proof
of \ref{_G_M_invariant_implies_trianalytic_Theorem_}.
\hfill
{\bf Proof of \ref{_G_M_invariant_implies_trianalytic_Theorem_}:}
We use the following proposition of linear algebra.
Let $V$ be a complex space and $V_{\Bbb R}$ be its underlying ${\Bbb R}$-linear
space. Let $W\subset V_{\Bbb R}$ be an ${\Bbb R}$-linear subspace of $V_{\Bbb R}$,
$dim\; W=2n$. Denote the Hermitian form on $V$ by
$\inbfpare {\cdot,\cdot}$. The ${\Bbb R}$-linear space $V_{\Bbb R}$ is
equipped with the positively defined symmetric scalar product
$(u,v):=Re\:\inbfpare {u,v}$. and the non-degenerated
symplectic form $\inangles{u,v}:= I\!m\,\inbfpare {u,v}$.
Let $(\cdot,\cdot)_W$ and $\omega=\inangles{\cdot,\cdot}_W$
be the restrictions of these forms to $W\subset V_{\Bbb R}$.
Clearly, $(\cdot,\cdot)_W$ is a positively defined
scalar product on $W$; therefore, $(\cdot,\cdot)_W$
in non-degenerate. By a scalar product on an $R$-vector
space, we define a {\bf volume form} as follows.
\hfill
\definition
Let $H$ be an ${\Bbb R}$-linear space equipped with a
positively defined scalar product. Let $h=dim\; H$. The exterrior
form $V\!ol\in \Lambda^h(H)$ is called {\bf a volume form}
if the the standard hypercube with the side 1
has the volume 1 in the measure defined by $V\!ol$.
\hfill
The proof of correctness of this definition can be found
in any linear algebra textbook.
Clearly, the volume form is defined up to a sign.
This sign is determined by the choice of orientation on $H$.
In the same manner
we define the top degree differential form called
{\bf a volume form} $V\!ol$ on any oriented
Riemannian manifold.
\hfill
Let $V\!ol$ be the volume form on $W$ defined by the scalar product.
Then $V\!ol$ is a non-zero element of the 1-dimensional linear
space
\[
\mbox{\bf Vol}=\Lambda^{2n}(W).
\]
The form $\omega^n=\inangles{\cdot,\cdot}_W^n$ is another
element of $\mbox{\bf Vol}$. The number $\frac{\omega^n}{Vol}\in {\Bbb R}$
is defined up to a sign, because $V\!ol$ is defined up to a sign.
Let $\eta_W:= |\frac{\omega^n}{Vol}|$. This number is an
invariant of $W$, $V$ and $\inbfpare{\cdot,\cdot}$.
\hfill
Consider the complex structure operator on $V$ as the
real endomorphism of $V_{\Bbb R}$:
\[ I:\;V_{\Bbb R}\longrightarrow V_{\Bbb R},\;\;\;
I^2=-1.
\]
\hfill
\proposition\label{_Wirtinger_Proposition_}
(Wirtinger's inequality)
Let $2n=dim_{\Bbb R} W$. Then $\eta_W\leq 2^n.$
Moreover, if $\eta_W=2^n$, then $I(W)=W$. In
other words, if $\eta_W=2^n$, then $W$ is a
complex subspace of $V$.
{\bf Proof:} \cite{_Stolzenberg_} page 7. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We return to the situanion when $M$ is a hyperk\"ahler manifold
and $N\subset (M,I)$ is a closed analytic subvariety of $(M,I)$.
Let $J$ be an induced complex structure, and
$N^0 \subset M$ be the set of non-singular
points of $N$. Denote the standard embedding
$N^0 \hookrightarrow M$ by $\phi$.
Let $\omega_J\in \Lambda^{1,1}(M)$ be the K\"ahler
form induced by $J$. Denote the standard coupling of homology
and cohomology by
\[ \inangles{\cdot,\cdot}:\;
H_i(M)\times H^i(M)\longrightarrow {\Bbb C}.
\]
\hfill
\proposition\label{_restriction_is_coupling_Proposition_}
\[ \inangles{[N],\omega_J^n}=
\int\limits_{N^0}\phi^*(\omega_J^n).
\]
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $\inbfpare{\cdot,\cdot}_J$ denote the Hermitian
form on $M$ induced by the Riemannian metric and the
complex structure $J$. Let $\omega_J=\inangles{\cdot,\cdot}_J$
and $(\cdot,\cdot)$ denote its imaginary and real parts respectively.
It is clear that the scalar product $(\cdot,\cdot)$ is the original
Riemannian form on $M$. Thus, the real part of
$\inbfpare{\cdot,\cdot}_J$ does not depend on
the complex structure $J$.
Let $(\cdot,\cdot)_N:=\phi^*((\cdot,\cdot))$ be the
Riemannian form on $N^0$. Since $N$ is oriented,
$(\cdot,\cdot)_N$ defines a volume form $V\!ol\in \Lambda^{2n}(N^0)$.
By definition, $V\!ol$ is a nowhere degenerate section of the
1-dimensional\footnote {over ${\Bbb R}$}
vector bundle $\Lambda^{2n}(N^0)$.
Since $N$ is analytic in $(M,I)$, we have
$V\!ol=1/2^n\phi^*(\omega_I)^n\in\Lambda^{2n}(N^0).$
Let $x\in N^0$. Consider $V=T_xM$ as a complex
space with the complex structure induced by $J$.
Let $W=T_xN^0$. Then $W\subset V_{\Bbb R}$ defines a number
$\eta_W$ as in \ref{_Wirtinger_Proposition_}. This
number depends on $x$ and $J$. For any induced complex
structure $J$, we define a function
$\eta_J:\; N^0\longrightarrow {\Bbb R}^{\geq 0}$ which supplies the number
$\eta_W\in{\Bbb R}^{\geq 0}$ by the point $x\in N^0$.
\hfill
\proposition\label{_N_is_analytic_if_eta_is_constant_Proposition_}
Let $J$ be an induced complex structure.
The closed set $N\subset M$ is analytic with respect to $J$
if and only if
\[ \forall x\in N^0 \;\;\;\; \eta_J(x)=2^n. \]
{\bf Proof:} The implication
\hfill
\centerline{($N$ is analytic w. r. to $J$)
$\;\;\;\Rightarrow\;\;\;$ ($\eta_J(x)\equiv 2^n$)}
\hfill
\hspace{-1.5em}%
is clear because if $N$ is analytic with respect to $J$, then
$J(T_xN)=T_xN$ and $\eta_J(x)\equiv 2^n$ by
\ref{_Wirtinger_Proposition_}. We proceed proving
the converse implication. Assume that
$\forall x\in N^0$ we have $\eta_J(x)=2^n$. Then
$J(T_xN)=T_xN$ by \ref{_Wirtinger_Proposition_}.
Using Newlander-Nierenberg theorem, we see that
$N^0$ is an analytic subset of $(M,J)$.
Clearly, $N$ is a closure of $N^0$.
Since the closure of an analytic set is
also analytic, the set $N\subset M$ is also an
analytic subset of $(M,J)$. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
By definition of $\eta_J(x)$, we have
\begin{equation}\label{_eta_in_terms_of_omega_Equation_}
\int\limits_{N^0}\phi^*(\omega_J^{2n})=
\int\limits_{N^0} V\!ol\cdot \eta_J(x).
\end{equation}
By \ref{_Wirtinger_Proposition_}, we have $\eta_J(x)\leq 2^n$.
Clearly, the function
\[ \eta_J(x):\; N^0 \longrightarrow {\Bbb R}^{\geq 0}\]
is continous. Therefore
\[ \int\limits_{N^0} V\!ol\cdot \eta_J(x)
= 2^n \int\limits_{N^0} V\!ol
\]
if and only if $\eta_J(x)=2^n$ for every $x\in N^0$.
Combining this with
\eqref{_eta_in_terms_of_omega_Equation_}
and
\ref{_N_is_analytic_if_eta_is_constant_Proposition_},
we obtain the following statement:
\hfill
\proposition\label{_analiticity_in_terms_of_integrals_Proposition_}
\nopagebreak
\vspace{\baselineskip}
\hspace{15mm}$\displaystyle\int\limits_{N^0}\phi^*(\omega_J^{2n})=
2^n \int\limits_{N^0} V\!ol$
\hfill
\hspace{-1.9em}
if and only if $N$ is analytic with respect to $J$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\hfill
Since $N$ is analytic with respect to $J$, we have
\[ 2^n \int\limits_{N^0} V\!ol = \int\limits_{N^0}\phi^*(\omega_J^{2n}). \]
Combining \ref{_analiticity_in_terms_of_integrals_Proposition_} with
\ref{_restriction_is_coupling_Proposition_}, we see that
\ref{_G_M_invariant_implies_trianalytic_Theorem_} is implied
by the following statement.
\hfill
\proposition\label{_volumes_of_G_M_invariant_cycles_Proposition_}
Let $M$ be a compact hyperk\"ahler manifold, $dim_{\Bbb C} M=m$.
Let $[N]\in H_{2n}(M)$ be a homology class of $M$ such that
its Poincare dual cocycle $\inangles{N}\in H^{2m-2n}$ is
$G_M$-invariant. Let $I$ and $J$ be two induced complex
structures on $M$. Then $\inangles{[N], \omega_I^{2n}}=
\inangles{[N], \omega_J^{2n}}$.
{\bf Proof:} By definition,
\[ \inangles{[N], \omega_J^{2n}} =
\int_M\inangles{N} \wedge \omega_J^{2n}.
\]
Therefore all we need to show is that
the number
\[ deg_J\alpha:= \int_M\alpha\wedge \omega_J^2n\]
is independent on the choice of $J$ once the cohomology
class $\alpha$ is $G_M$-invariant.
Let $L_J:\; \Lambda^i(M)\longrightarrow \Lambda^{i+2}(M)$
denote the Hodge operator acting on differential forms
over $M$, $L_J(\eta)=\omega_J\wedge \eta$.
Let $\Lambda_J:\; \Lambda^i(M)\longrightarrow \Lambda^{i-2}(M)$
denote the adjoint operator. It is well known that
$L_J$, $\Lambda_J$ map harmonic form to harmonic ones.
Therefore one can consider $L_J$, $\Lambda_J$
as operators on the cohomology space $H^*(M)$.
Let $\goth a_M\subset End(H^*(M))$ be the Lie algebra
generated by $L_J$, $\Lambda_J$ for all induced complex
structures $J$. Let ${\goth g}_M\cong \goth{so}(3)$ be
the Lie algebra of $G_M$. Since $G_M$ non-trivially
acts on $H^*(M)$, we may consider ${\goth g}_M$ as a subalgebra
of $End(H^*(M))$. It is known that
$\goth a_M\cong \goth{so}(5)$ and that
${\goth g}_M$ considered as a Lie subalgebra of $End(H^*(M))$
lies in $\goth a_M\subset End(H^*(M))$
(see \cite{_so5_on_cohomo_}).
\hfill
Let $H^*(M)=\oplus_{l\in\Pi}H_l$ be the isotypic
decomposition of a $\goth a_M$-module
$H^*(M)$. We recall that {\bf isotypic decomposition}
of a representation of an arbitrary reductive Lie algebra
is defined as follows. For each $l\in \Pi$, where
$\Pi$ is a weight lattice of $\goth{a}_M$,
the module $H_l$ is a union of all simple $\goth{a}_M$-submodules
of $H^*(M)$ with a highest weight $l$.
One can easily see that the isotypic decomposition does not depend
on a choice of a Cartan subalgebra of $\goth{a}_M$. This follows,
for example, from Schuhr's lemma.
\hfill
Let $H_o$ be the $\goth a_M$-submodule of $H^*(M)$ generated by
$H^0(M)\cong {\Bbb C}$. Clearly, $H_o$ is an isotypic component of
$H^*(M)$ (see \cite{_part_one_} for details). Let $\alpha_o$ be the component
of
$\alpha$ which corresponds to the summand
$H_o\subset \oplus_{l\in\Pi}H_l$. Since ${\goth g}_M\subset \goth a_M$,
the isotypic component $\alpha_o$ of $\alpha$ is ${\goth g}_M$-invariant.
It was proven that $deg_J(\alpha_o)=deg_J(\alpha)$ for all
induced complex structures $J$ (see the paragraph right
after the proof of Lemma 3.1 in \cite{_part_one_}).
Let $I$, $J$, $K$ be a triple of induced complex structures
on $M$, such that
\[ I\circ J=-J\circ I= K. \]
Lemma 3.2 of \cite{_part_one_} implies that
\[ \alpha_o=c(L_I^2+ L_J^2+L_K^2)^{\frac{m-n}{2}} {\Bbb I} \]
where $ {\Bbb I} $ is a generator of $H^0(M)$ and
$c$ is a constant. In this notation, the equality
\[ deg_I(\alpha_o)= deg_J(\alpha_o)= deg_K(\alpha_o) \]
is obvious. We proved the following lemma
\hfill
\lemma \label{_deg_is_independent_on_I_J_K_Lemma_}
If $I$, $J$, $K$ are induced complex structures
on $M$, such that
\[ I\circ J=-J\circ I= K, \]
then
\[ deg_I(\alpha_o)= deg_J(\alpha_o)= deg_K(\alpha_o). \]
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We deduce \ref{_volumes_of_G_M_invariant_cycles_Proposition_}
from \ref{_deg_is_independent_on_I_J_K_Lemma_} as follows.
Let $V$ be the space of purely imaginary quaternions.
By definition, the space of purely imaginary quaternions
$V\subset \Bbb H$ is a three-dimensional vector space over ${\Bbb R}$':
\[ V= \{ t\in {\Bbb H} \; | \; \bar t =-t \}, \]
\[ V= \{ aI+bJ+cK,\; a,b,c \in {\Bbb R}\}. \]
The set of induced complex structures can be considered
as a sphere $S$ of radius 1 in $V$. For $I$, $J$ in $S$,
we have $I\circ J=-J\circ I$ if and only if the vector
$\stackrel\longrightarrow {(0,I)}\in V$ is perpendicular to
$\stackrel\longrightarrow {(0,J)}\in V$. On the other hand,
if $\stackrel\longrightarrow {(0,I)}\bot\stackrel\longrightarrow {(0,J)}$,
then $K:= I\circ J$ belongs to $S$ and the vectors
$\stackrel\longrightarrow {(0,I)}$, $\stackrel\longrightarrow {(0,J)}$,
$\stackrel\longrightarrow {(0,K)}$ are pairwise orthogonal.
We obtain that the triples
$I$, $J$, $K$ of induced complex structures which satisfy
\[ I\circ J=-J\circ I= K \]
are in one-to-one correspondence with the orthonormal repers in
$V$. Therefore \ref{_deg_is_independent_on_I_J_K_Lemma_}
implies the following statement:
\hfill
\lemma \label{_orthogonal_quaternions_and_degree_Lemma_}
Let $I$, $J$ be the induced complex structures such
that the vectors $\stackrel\longrightarrow {(0,I)}\in V$ and
$\stackrel\longrightarrow {(0,J)}\in V$ are orthogonal. Then
$deg_I(\alpha)=deg_J(\alpha)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_orthogonal_quaternions_and_degree_Lemma_} trivially
implies \ref{_volumes_of_G_M_invariant_cycles_Proposition_}.
\ref{_G_M_invariant_implies_trianalytic_Theorem_} is proven.
\hfill
{\bf Acknowledgements:}
I am very grateful to my advisor David Kazhdan for a warm support
and encouragement. Professor Y.-T. Siu was extremely helpful
kindly answering my questions. He provided me with the
reference to the Stolzenberg's book (\cite{_Stolzenberg_}).
F. A. Bogomolov cleared some of the misconceptions about
the holomorphic symplectic geometry that I had.
I am grateful to Roman Bezrukavnikov for interesting
discussions and insightful remarks.
I am also grateful to MIT math department
for allowing me the use of their computing
facilities.
\hfill
|
1994-03-02T17:47:29 | 9403 | alg-geom/9403003 | en | https://arxiv.org/abs/alg-geom/9403003 | [
"alg-geom",
"math.AG"
] | alg-geom/9403003 | Klaus Altmann | Klaus Altmann | Toric Q-Gorenstein Singularities | 17 pages, LaTeX, Preprint 9/93 Humboldt-Universitaet Berlin | null | null | null | null | For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q)
the vector space T^1 of infinitesimal deformations is related to the
complexified vector spaces of rational Minkowski summands of faces of Q.
Moreover, assuming Y to be an isolated, at least 3-dimensional singularity, Y
will be rigid unless it is even Gorenstein and dim Y=3 (dim Q=2). For this
particular case, so-called toric deformations of Y correspond to Minkowski
decompositions of Q into a sum of lattice polygons. Their Kodaira-Spencer-map
can be interpreted in a very natural way. We regard the projective variety P(Y)
defined by the lattice polygon Q. Data concerning the deformation theory of Y
can be interpreted as data concerning the Picard group of P(Y). Finally, we
provide some examples (the cones over the toric Del Pezzo surrfaces). There is
one such variety yielding Spec C[e]/e^2 as the base space of the semi-universal
deformation.
| [
{
"version": "v1",
"created": "Wed, 2 Mar 1994 16:47:03 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Altmann",
"Klaus",
""
]
] | alg-geom | \section{#1}
\protect\setcounter{secnum}{\value{section}}
\protect\setcounter{equation}{0}
\protect\renewcommand{\theequation}{\mbox{\arabic{secnum}.\arabic{equation}}}}
\setcounter{tocdepth}{1}
\begin{document}
\title{Toric $\,I\!\!\!\!Q$-Gorenstein Singularities}
\author{Klaus Altmann\\
\small Dept. of Mathematics, M.I.T., Cambridge, MA 02139, U.S.A.
\vspace{-0.7ex}\\ \small E-mail: [email protected]}
\date{}
\maketitle
\begin{abstract}
For an affine, toric $\,I\!\!\!\!Q$-Gorenstein variety $Y$ (given by
a lattice polytope $Q$) the vector space $T^1$ of infinitesimal deformations
is related to the complexified vector spaces
of rational Minkowski summands of faces of $Q$.\\
Moreover,
assuming $Y$ to be an isolated, at least 3-dimensional singularity, $Y$ will
be rigid unless it is even Gorenstein and $\mbox{dim}\,Y=3\;
(\mbox{dim}\,Q=2)$.\\
For this particular case,
so-called toric deformations of $Y$ correspond to Min\-kow\-ski decompositions
of $Q$ into a sum of lattice polygons. Their Kodaira-Spencer-map can be
interpreted in a very natural way.\\
We regard the projective variety $I\!\!P (Y)$ defined by the lattice
polygon $Q$. Data concerning the deformation theory of $Y$ can be interpreted
as data concerning the Picard group of $I\!\!P (Y)$.\\
Finally, we provide some examples (the cones over the toric Del Pezzo
surfaces). There is one such variety yielding $\mbox{Spec$\;$} I\!\!\!\!C[\varepsilon]/_
{\displaystyle \varepsilon^2}$ as the base space of the semi-universal
deformation.
\end{abstract}
\tableofcontents
\par
\vspace{2ex}
\sect{Introduction}\label{s1}
\neu{11}
In \cite{T1} and \cite{Mink} we investigated the deformation theory of
affine toric varieties $Y=\mbox{Spec$\;$} I\!\!\!\!C [\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M]$
(cf. \zitat{2}{1} for an explanation of the notations):\\
\par
It is always the first step to look at the vector space $T^1_Y$ of
infinitesimal deformations - it equals (if $Y$ admits isolated singularities
only) the tangent space on the base space $S$ of the semi-universal
deformation of $Y$. For toric $Y$ the space $T^1_Y$ is $M$-graded, and the
homogeneous pieces were computed in \cite{T1} (cf. \zitat{2}{2} of the present
paper).\\
\par
In \cite{Mink} we were interested in describing toric deformations of $Y$.
They are defined as those deformations (i.e. flat maps $f:X\rightarrow S$
endowed with an isomorphism $f^{-1}(0\in S) \cong Y$) such that the total
space $X$ together with the embedding of the special fiber $Y \hookrightarrow
X$
are toric.\\
\par
Toric deformations are ``really existing'' deformations -
in the sense that they admit reduced (even smooth) base spaces. Moreover,
we conjecture that (in case of isolated singularities $Y$)
the semi-universal deformation of $Y$ is toric over each irreducible component
of the reduced base space. (This is true for $\mbox{dim}\, Y = 2$ and also for
examples of higher dimension.)\\
\par
Toric deformations always arise as relative deformations
of $Y$ inside a greater affine toric variety $X$ containing $Y$ as a relatively
complete intersection. More strictly speaking, $Y\subseteq X$ is defined by a
so-called toric regular sequence $x^{r_0^1}-x^{r_1^1},\dots,
x^{r_0^m}-x^{r_1^m} \in \Gamma(X, {\cal O}_X)$.\\
\par
On the other hand, each toric regular sequence can be regarded as a flat map
$X\rightarrow I\!\!\!\!C^m$
by itself. This $m$-parameter deformation of $Y$ is called the standard
toric deformation induced by the given sequence.\\
\par
It is possible to compute the Kodaira-Spencer-map $\varrho: I\!\!\!\!C^m \rightarrow
T^1_Y$ corresponding to these standard toric deformations. Then, the first
observation is that $\varrho$ maps
the $i$-th canonical basis vector $e^i\in I\!\!\!\!C^m$ into the homogeneous piece
$T^1_Y(-\bar{r}^i)$, and $\bar{r}^i \in M$ is defined as the exponent of the
common image of $x^{r^i_0}$ and $x^{r^i_1}$ via the surjection
$\Gamma(X, {\cal O}_X)\longrightarrow\hspace{-1.5em}\longrightarrow\Gamma(Y, {\cal O}_Y)$.\\
\par
{\bf Definition:}
A toric regular sequence is called strongly homogeneous if
only $m+1$ different elements occur in the set $\{r^1_0,\,r^1_1,\dots,
\,r^m_0,\,r^m_1\}$. Then, the corresponding images $\bar{r}^1,\dots,\bar{r}^m$
coincide, and this element will be denoted by $\bar{r}\in
\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M$.
It equals the
negative degree of the Kodaira-Spencer-map.\\
(Toric regular sequences of length one are always strongly homogeneous.)\\
\par
The main result of \cite{Mink} is a complete combinatorial description of
those standard toric deformations that are induced by strongly homogeneous
toric regular sequences. They arise from certain Minkowski decompositions
of affine slices (induced by $\bar{r}\in M$) of the cone $\sigma$ (cf.
\zitat{3}{1} of the present paper).\\
\par
\neu{12}
The aim of this paper is to apply the previous results to the special case
of toric $\,I\!\!\!\!Q$-Gorenstein singularities. On the one hand, this notion is the
next one if we are looking for a wider class than that of complete
intersections (which yields no interesting deformation theory). On the other
hand, the property ``$\,I\!\!\!\!Q$-Gorenstein'' admits a very clear description
in the language of toric varieties and convex cones:\\
\par
In general, the dualizing sheaf $\omega$ on a Cohen-Macaulay variety
is defined as
\begin{itemize}
\item[(i)]
$\omega_P := \Omega_P^{\,\mbox{\scriptsize dim}\,P}$ (sheaf of the highest
differential forms) if $P$ is smooth, and
\item[(ii)]
$\pi_{\ast}\omega_Y:= \mbox{\underline{Hom}}_{{\cal O}_P}(\pi_{\ast}{\cal
O}_Y,\,
\omega_P)$ for flat and finite maps $\pi: Y \rightarrow P$.
\vspace{1ex}
\end{itemize}
\par
{\bf Definition:} A variety $Y$ is called ($\,I\!\!\!\!Q$-) Gorenstein if (the
reflexive hull of some tensor power of) $\omega_Y$ is an invertible sheaf
on $Y$.\\
\par
Since toric varieties are normal, the dualizing sheaf can be obtained as the
push forward of the canonical sheaf on its smooth part. Hence, in our
special situation, $\omega_Y$ equals the $T\mbox{(orus)}$-invariant complete
fractional ideal that is given by the order function mapping each fundamental
generator onto $1\in Z\!\!\!Z$ (cf. Theorem I/9 in \cite{Ke}).\\
\par
In particular, we obtain the following\\
\par
{\bf Fact:} Let $Y=\mbox{Spec$\;$} I\!\!\!\!C [\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M]$ be an affine
toric variety given by a cone $\sigma= \langle a^1,\dots, a^N \rangle$. (The
fundamental generators $a^i$ are assumed to be primitive elements of the
lattice that is dual to $M$.)\\
Then, $Y$ is $\,I\!\!\!\!Q$-Gorenstein, if and only if
there is a primitive element $R^{\ast}\in M$ and a natural number
$g\in I\!\!N$ such that
\[
\langle a^i,\, R^{\ast}\rangle = g \quad \mbox{for each } i=1,\dots,N.
\]
$Y$ is Gorenstein if and only if $g=1$, in addition.\\
\par
\neu{13}
Affine toric varieties of dimension two are always $\,I\!\!\!\!Q$-Gorenstein. The
deformation theory of these varieties (the two-dimensional cyclic quotient
singularities) is well known. For instance, $T^1_Y$ and the
number and dimension of the components of the reduced base space of the
semi-universal deformation have been computed (cf. \cite{Riem}, \cite{Arndt},
\cite{Ch}, \cite{St}).\\
Therefore, our investigation concerns the case of $Y$ being smooth in
codimension 2.\\
\par
Assume that $Y=\mbox{Spec$\;$} I\!\!\!\!C [\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M]$ is a
$\,I\!\!\!\!Q$-Gorenstein
variety (i.e. $\langle a^i,\, R^{\ast}\rangle = g$ for each $i=1,\dots,N$),
which is smooth in codimension 2. Denote by $Q$ the lattice polyhedron
$Q:= \mbox{Conv}(a^1,\dots,a^N)$.
Then, we obtain the following results:
\begin{itemize}
\item[(1)]
The graded pieces of $T^1_Y$ equal the vector spaces of Minkowski summands
of certain faces of $Q$ (cf. Theorem \zitat{2}{7}).
\item[(2)]
For the special case of an isolated singularity $Y$ this implies:
\begin{itemize}
\item
If $g\geq 2$, then $Y$ will be rigid, i.e. $T^1_Y=0$.
\item
If $Y$ is Gorenstein (i.e. $g=1$) and at least 4-dimensional, then $Y$ will
be rigid.
\item
Let $Y$ be a 3-dimensional Gorenstein singularity given by a plane, convex
$N$-gon $Q$. Then, $T^1_Y$ is concentrated in the single degree $-R^\ast$,
and it equals to the $(N-3)$-dimensional vector space of Minkowski summands of
$Q$.
\end{itemize}
(Cf. \zitat{2}{8} and \zitat{2}{9}.)
\item[(3)]
Keep assuming that $Y$ is a 3-dimensional, isolated, toric, Gorenstein
singularity. Then,
toric $m$-parameter deformations of $Y$
correspond to Minkowski decompositions of $Q$ into a sum of $m+1$ lattice
polygons. Using the previous description of $T^1_Y$, the Kodaira-Spencer-map
is the natural one.
\vspace{2ex}
\end{itemize}
\par
\neu{14}
Let us assume, for a moment, that the semi-universal deformation is, indeed,
toric over each component of the reduced base space $S_{\mbox{\scriptsize
red}}$.
Then, the absence of proper lattice summands of $Q$
would mean that $S$ is only 0-dimensional.\\
\par
On the other hand, this is possible even for $Y$ admitting a non-trivial
$T^1_Y$ (which equals the tangent space of $S$).
In (\ref{s4}.\ref{t44}.3) we will give a (three-dimensional) example of this
phenomenon:
The base space $S$ is a point with non-reduced structure, i.e. each deformation
is obstructed.\\
\par
\neu{15}
A toric Gorenstein variety $Y=\mbox{Spec$\;$} I\!\!\!\!C [\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M]$
is given by a lattice polytope $Q$ ($\sigma = \mbox{Cone}(Q)$).
On the other hand, each lattice polytope $Q$ induces a projective toric
variety (defined by the inner normal fan) endowed with an ample line
bundle ${\cal O}(1)$. We call this the polar variety of $Y$ - denoted by
${I\!\!P}(Y)$.\\
Then, the cone $P(Y)$ over the embedded projective variety
${I\!\!P}(Y)$ equals the affine toric variety which is given by the cone dual to
that of $Y$.\\
\par
In \S \ref{s4}, we interprete data of the deformation theory of
3-dimensional $Y$ as
data concerning divisors on the varieties ${I\!\!P}(Y)$ or $P(Y)\setminus
\{0\}$. Assuming
$Y$ having only an isolated singularity
we obtain the following relations:
\begin{itemize}
\item[(4)]
$T^1_Y \,=\; \mbox{Pic}(P(Y)\setminus \{0\}) \otimes_{Z\!\!\!Z} I\!\!\!\!C \,=\;
\mbox{Pic}\,{I\!\!P}(Y) \left/_{\displaystyle \!\!{\cal O}(1)}\right.
\otimes_{Z\!\!\!Z} I\!\!\!\!C$.
\item[(5)]
Toric $m$-parameter deformations of $Y$ correspond to splittings of
${\cal O}_{{I\!\!P}(Y)}(1)$ into a tensor product
of $m+1$ invertible sheaves that are nef.
\vspace{2ex}
\end{itemize}
\neu{16}
{\bf Acknowledgement:}
I am grateful to Duco van Straten for computing several semi-universal
deformations on the computer (using Macaulay).\\
Moreover, I want to thank Bernd Sturmfels for his special lecture
concerning the cone of Minkowski summands and the Gale transform of a given
polytope.\\
\par
\sect{The $T^1$ of toric $\,I\!\!\!\!Q$-Gorenstein singularities}\label{s2}
\neu{21}
Let us start with introducing some basic notations and recalling the
$T^1$-formulas of \cite{T1}:\\
\par
Let $M, N$ be free $Z\!\!\!Z$-modules of finite rank - endowed with a perfect
pairing $\langle .,.\rangle : N \times M \rightarrow Z\!\!\!Z$. Denote by
$M_{I\!\!R}$ and $N_{I\!\!R}$ the corresponding vector spaces (dual to each other)
obtained via base change with $I\!\!R$.\\
\par
Let $\sigma= \langle a^1,\dots, a^N \rangle \subseteq N_{I\!\!R}$ be a
top-dimensional, rational, polyhedral cone with apex.
The fundamental generators $a^i \in N$ are assumed
to be primitive elements of the lattice $N$.\\
\par
The dual cone $\stackrel{\scriptscriptstyle\vee}{\sigma}\subseteq M_{I\!\!R}$ of $\sigma$ is
defined as $\stackrel{\scriptscriptstyle\vee}{\sigma}:= \{r\in M_{I\!\!R} |\; \langle a,\, r\rangle
\geq 0 \;\mbox{ for each } a\in \sigma\}$. Denote by $E \subseteq
\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M$ the minimal (finite) set that
generates the semigroup $\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M$.
In particular, $Y:= \mbox{Spec}\,I\!\!\!\!C [\stackrel{\scriptscriptstyle\vee}{\sigma}\cap M] \subseteq
I\!\!\!\!C^E$.\\
\par
\neu{22} {\bf Theorem:} (cf. (2.3) and (4.4) of \cite{T1})
\begin{itemize}
\item[(1)]
The vector space $T^1_Y$ of infinitesimal deformations of $Y$ is $M$-graded.
For a fixed element $R\in M$ the homogeneous piece of degree $-R$ can be
computed as
\[
T_Y^1(-R)=\left(^{\displaystyle
L(E')}\!\left/{\displaystyle\sum\limits^N_{i=1}L(E_i)}
^{\makebox[0mm]{}}\right.\right)^{\ast}\otimes_{I\!\!R} I\!\!\!\!C
\]
$(E_i := \{s\in E\;|\;\;
(0\le)\langle a^i,s\rangle<\langle a^i,R\rangle\}
\quad (i=1,\ldots,N)\,;\quad
E' :=\; \bigcup\limits^N_{i=1} E_i\,;$\\
$L(\mbox{set}) := \mbox{$I\!\!R$-vector space of all linear dependences between
its elements}).$
\item[(2)]
If $Y$ is smooth in codimension 2 (i.e. if all 2-dimensional faces
$\langle a^i, a^j \rangle < \sigma$ are spanned by a part of a $Z\!\!\!Z$-basis of
the lattice $N$), then
with
\[
V_i := {\mbox{span}}_{I\!\!R}(E_i)=\left\{
\begin{array}{l@{\quad\mbox{for}\;\:}l}
0 &\langle a^i,R\rangle\le 0\\
\left[ a^i=0\right] \subseteq M_{I\!\!R} & \langle a^i,R\rangle =1\\
M_{I\!\!R} & \langle a^i,R\rangle\ge2
\end{array}
\right.
\]
we obtain the second formula
\[
T_Y^1(-R)=\mbox{Ker}\left[^{\displaystyle V_1\oplus\ldots\oplus V_N}\left/
_{\!\!\!\displaystyle\sum_{\langle a^i,a^j\rangle <\sigma}V_i\cap V_j}\right.
\longrightarrow\hspace{-1.5em}\longrightarrow
V_1+\ldots+V_N\right]^{\ast}.
\]
\end{itemize}
\par
\neu{23} {\bf Lemma:}
Let $Y$ be a $\,I\!\!\!\!Q$-Gorenstein variety, which is smooth in codimension 2.
If $R\in M$
is a degree such that $\langle a^i,\,R \rangle \geq 2$ for some $i \in
\{1,\dots,N\}$, then $T^1_Y(-R)=0$.\\
\par
{\bf Proof:}
Let $R^{\ast}\in M$ be as in \zitat{1}{2}, i.e. $\langle a^i,\,R^{\ast}
\rangle = g$ for $i=1,\dots,N$. Then,
\[
H:= \{a\in N_{I\!\!R}|\; \langle a,\, g\,R-R^{\ast}\rangle = 0\}
\]
is a hyperplane in $N_{I\!\!R}$ that subdivides the set of fundamental generators
of $\sigma$. $H^-, H$, and $H^+$ contain the elements $a^i$ meeting the
properties $\langle a^i,\,R\rangle \leq 0$, $\langle a^i,\,R\rangle =1$, and
$\langle a^i,\,R\rangle \geq 2$, respectively. Let us assume that the latter
class of generators is not empty.\\
\par
To use the second $T^1$-formula of the previous theorem, we fix a map
\[
\varphi: \{i\,|\; \langle a^i,R\rangle =1\} \rightarrow \{1,\dots,N\}
\]
such
that for each $a^i\in H$ the element $a^{\varphi(i)}$ is contained in $H^+$
and adjacent to $a^i$ (i.e. $\{a^i,\, a^{\varphi(i)}\}\subseteq\sigma$ spans a
2-dimensional face
of $\sigma$).\\
\par
Now, assume that we are given an element $v=(v_1,\dots,v_N)\in V_1\oplus
\dots\oplus V_N$ such that $v_1+\dots +v_N=0$. Adding the terms
$[-v_i\cdot e^i+v_i\cdot e^{\varphi(i)}]$ (for $\langle a^i,R\rangle =1$)
does not change the equivalence class of $v$ in
$^{\displaystyle V_1\oplus\ldots\oplus V_N}\!\!\left/
_{\!\!\!\!\!\displaystyle\sum_{\langle a^i,a^j\rangle <\sigma}
V_i\cap V_j}\right.$.
However,
non-trivial components survive for $\langle a^i,R\rangle \geq 2$ (corresponding
to $V_i=M_{I\!\!R}$) only.\\
\par
The set of these special generators $a^i$ is connected by 2-dimensional
faces of $\sigma$. Moreover, by slightly disturbing $R$ inside $M_{I\!\!R}$, we
can find a unique $a^{\ast}$ among these edges on which $R$ is maximal. Then,
each $a^i\in H^+$ is connected with $a^{\ast}$ by an $R$-monotone path
(consisting of 2-faces of $\sigma$) inside $H^+$.\\
Now, we can use the previous method of cleaning the components of $v$ once
more - the steps from $a^i$ to $a^{\varphi(i)}$ are replaced by the steps on
the path from $a^i$ to $a^{\ast}$. It remains an $N$-tuple $v$ which is
non-trivial at most at the $a^{\ast}$-place. On the other hand, the components
of $v$ sum up to $0$, but this yields $v=0$.
\hfill$\Box$\\
\par
\neu{24}
If $\langle a^i,\, R\rangle \leq 1$ for every $i \in \{1,\dots,N\}$, then
equality holds on some face $\tau < \sigma$. Now, $\tau$ is a
top-dimensional cone in the linear subspace $\tau - \tau \subseteq N_{I\!\!R}$,
and it defines a variety $Y_{\tau} = \mbox{Spec} \;I\!\!\!\!C[
^{\displaystyle\tau^{\scriptscriptstyle\vee}\cap M}\!/\!_{
\displaystyle\tau^{\bot}\cap M} ]$, which is even Gorenstein.
The corresponding element $R^{\ast}_{\tau} \in
^{\displaystyle M}\!/\!_{\displaystyle \tau^{\bot}\cap M}$
can be obtained as the image of $R$ as well as of $\frac{1}{g}R^{\ast}$
using the canonical projection $M_{I\!\!R} \longrightarrow\hspace{-1.5em}\longrightarrow
^{\displaystyle M_{I\!\!R}}\!/\!_{\displaystyle \tau^{\bot}}$.
\\
\par
{\bf Lemma:}
In general (even the $\,I\!\!\!\!Q$-Gorenstein assumption can be dropped)
let $\tau < \sigma$ be a face such that $\langle a^i, R\rangle
\geq 1$ for $a^i\in\tau$ and $\langle a^i, R\rangle \leq 0$ otherwise. Then,
$T^1_Y(-R) = T^1_{Y_{\tau}}(-[\mbox{image of }R])$.\\
In the special situation discussed previously this means
$T^1_Y(-R) = T^1_{Y_{\tau}}(-R^{\ast}_{\tau})$.\\
\par
{\bf Proof:}
The formula of Theorem \zitat{2}{2}(1) remains true if $E$ is replaced by
an arbitrary (not necessarily minimal) generating subset of $\sigma^{\scriptscriptstyle\vee}
\cap M$ - even a multiset could be allowed. Hence, for computing
$T^1_{Y_{\tau}}(-R^{\ast}_{\tau})$ we can use the image $\bar{E}$ of $E$
under the projection
\[
\sigma^{\scriptscriptstyle\vee}\cap M \longrightarrow\hspace{-1.5em}\longrightarrow ^{
\displaystyle (\sigma^{\scriptscriptstyle\vee}+\tau^{\bot})\cap M}\!\! \left/
\!\! _{\displaystyle \tau^{\bot}\cap M} \right. .
\]
For $a^i\in \tau$, the corresponding sets $\bar{E}_i \subseteq \bar{E}$
coincide with the images of the subsets $E_i$. For $a^i \notin \tau$, the
notion $ \bar{E}_i$ does not make sense, and the $E_i$ are empty, anyway. It
remains to show that the canonical map
\[
\left. ^{\displaystyle L(E')} \!\! \right/ \!\!
_{\displaystyle \sum_{a^i\in \tau} L(E_i)} \longrightarrow
\left. ^{\displaystyle L(\bar{E}')} \!\! \right/ \!\!
_{\displaystyle \sum_{a^i\in \tau} L(\bar{E}_i)}
\]
is an isomorphism.\\
\par
The vector space $\tau^{\bot}$ is generated by
$\tau^\bot\cap (\bigcap_{a^i\in\tau} E_i)$.
Hence, by choosing a basis among these elements, we can embed $\tau^{\bot}$
into $I\!\!R^{\tau^\bot\cap (\cap_\tau E_i)}$ to obtain a section of
\begin{eqnarray*}
I\!\!R^{\tau^\bot\cap (\cap_\tau E_i)} \subseteq
\bigcap_{a^i\in\tau} L(\bar{E}_i) \subseteq
L(\bar{E}') &\longrightarrow\hspace{-1.5em}\longrightarrow& \tau^\bot \\
(\dots,\lambda_r,\dots)_{r\in E'} & \mapsto & \sum_{r\in E'} \lambda_r \cdot
r \in \tau^\bot \subseteq M_{I\!\!R}.
\end{eqnarray*}
In particular, we obtain
\begin{eqnarray*}
L(\bar{E}_i) &=& L(E_i) \oplus \tau^\bot \;\; (a^i \in \tau)
\qquad\mbox{ and}\\
L(\bar{E}') &=& L(E') \oplus \tau^\bot.
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box
\end{eqnarray*}
\par
\neu{25}
Let $Q=\mbox{Conv}(a^1,\dots,a^N)$ be a $K$-polytope contained in some
$K$-vector space $A\!\!\!\!\!A$ ($K= I\!\!\!\!Q\mbox{ or } I\!\!R$). It can be described by
inequalities
\[
\langle \bullet,\, -c^v\rangle \leq \eta^v \quad (c^v\in A\!\!\!\!\!A^{\ast},\,
\eta^v \in K)
\]
corresponding to the facets of $Q$.
(The $c^v$ are the inner normal vectors of $Q$.)\\
\par
Let us denote by $\sigma \subset A\!\!\!\!\!A\times K$ the cone over $Q$ (embedded
as $Q\times\{1\}$). Then, the
pairs $[c^v,\eta^v]\in A\!\!\!\!\!A^{\ast}\times K$ are exactly the fundamental
generators of the dual cone $\stackrel{\scriptscriptstyle\vee}{\sigma}$.
For $i=1,\dots,N$ we define
\begin{eqnarray*}
F_i &:=& \{[c^v,\eta^v]\in A\!\!\!\!\!A^{\ast}\times K \,|\; \langle a^i,\,-c^v
\rangle = \eta^v\}\\
&=& \{\mbox{fundamental generators of the face } (a^i)^{\bot}\cap
\stackrel{\scriptscriptstyle\vee}{\sigma} \,<\; \stackrel{\scriptscriptstyle\vee}{\sigma}\}.
\end{eqnarray*}
Finally, denoting the set of all
fundamental
generators of $\stackrel{\scriptscriptstyle\vee}{\sigma}$
by $F':=\; \bigcup\limits^N_{i=1} F_i$,
we can construct the
$K$-vector space
\[
\tilde{T}^1(Q):=
\left(^{\displaystyle L(F')}\!\left/{\displaystyle\sum\limits^
N_{i=1}L(F_i)}^{\makebox[0mm]{}}\right.\right)^{\ast}.
\]
Now, each Minkowski summand $Q'$ of a scalar multiple of $Q$ is given by
inequalities $\langle \bullet,\, -c^v\rangle \leq {\eta '\,}^v$. We can define
its class $\varrho(Q')\in \tilde{T}^1(Q)$ as
\[
\varrho(Q')\, (q\in L(F')):= \sum_v q_v {\eta '\,}^v \in K.
\]
This definition is correct and depends on the translation class of $Q'$
only. Moreover, scalar multiples of $Q$ yield the zero class.\\
\par
On the other hand, the translation classes of Minkowski summands of scalar
multiples of $Q$ form a convex polyhedral cone which contains ``$Q$'' as an
interior point. Dividing by the relation ``$Q=0$'' yields a $K$-vector space
which we will call the vector space of Minkowski summands of scalar multiples
of $Q$. It has one dimension less than the cone of Minkowski summands.\\
\par
{\bf Theorem:} (cf. \cite{Sm})\quad
The map $\varrho$ induces an isomorphism between the vector space of
Minkowski summands of scalar multiples of $Q$ and the vector space
$\tilde{T}^1(Q)$.\\
\par
{\bf Remark:} The constructions made in \zitat{2}{5} do not depend on the
linear, but on the affine structure of $A\!\!\!\!\!A$.\\
\par
\neu{26}
{\bf Lemma:}
Let $Y$ be an affine
toric Gorenstein variety induced
from a lattice
polytope $Q$. Then, the vector space $T^1_Y(-R^\ast)$
equals the complexified vector space $\tilde{T}^1(Q)\otimes I\!\!\!\!C$ of
(rational or real) Min\-kow\-ski summands of scalar
multiples of $Q$ (modulo translations and scalar multiples of $Q$ itself).\\
\par
{\bf Proof:}
We will use the first $T^1$-formula of Theorem \zitat{2}{2}. For the special
degree $-R^{\ast}$ the sets $E_i$ equal
\[
E_i= \{s\in E\,|\; \langle a^i,\, s\rangle =0\} = E\cap (a^i)^{\bot}.
\]
In particular, they contain the sets $F_i$ constructed above. We obtain a
natural linear map
\[
\theta: T^1_Y(-R^\ast) = \left(^{\displaystyle L(E')}\!\left/
{\displaystyle\sum\limits^
N_{i=1}L(E_i)}^{\makebox[0mm]{}}\right.\right)^{\ast}\otimes_{I\!\!R} I\!\!\!\!C
\longrightarrow
\left(^{\displaystyle L(F')}\!\left/{\displaystyle\sum\limits^
N_{i=1}L(F_i)}^{\makebox[0mm]{}}\right.\right)^{\ast}\otimes_{I\!\!R} I\!\!\!\!C
= \tilde{T}^1(Q)\otimes I\!\!\!\!C,
\]
and it remains to prove that $\theta$ is an isomorphism.\\
\par
Let $s\in E'\subseteq \partial\sigma^{\scriptscriptstyle\vee}$ be an element that
is not a fundamental generator of $\stackrel{\scriptscriptstyle\vee}{\sigma}$. Then, there
is a minimal face $\alpha < \sigma^{\scriptscriptstyle\vee}$ containing $s$ (as a
relatively interior point), and we can choose some fundamental generators
$s^1,\dots, s^k \in \alpha$ such that $s=\sum_{j=1}^k {\lambda}_j\,s^j\;
({\lambda}_j \in I\!\!\!\!Q_{\geq 0})$.\\
\par
Now, for each $E_i$ containing $s$ (equivalent to $\alpha < (a^i)^{\bot}
\cap \stackrel{\scriptscriptstyle\vee}{\sigma}$) we have a decomposition
\[
L(E_i) = L(E_i \setminus \{s\}) \;\oplus\; I\!\!R\cdot [\mbox{relation }
s=\sum_{j=1}^k {\lambda}_j\,s^j].
\]
In particular, the second summand can be reduced in the expression for
$T^1_Y$. Since the map $\theta$ consists of such steps only, we are done.
\hfill$\Box$\\
\par
\neu{27}
We collect the results of \zitat{2}{3} - \zitat{2}{6}: Let $Y$ be an affine
toric $\,I\!\!\!\!Q$-Gorenstein variety given by a lattice polytope $Q =
\mbox{Conv}(a^1,\dots,a^N)$ contained in an affine hyperplane $[\langle
\bullet , R^\ast\rangle = g] \subseteq N_{I\!\!R}$ of lattice-distance $g$ from
$0\in N_{I\!\!R}$. Moreover, assume that $Y$ is smooth in codimension 2.\\
Then, the graded pieces of $T^1_Y$ are related to the vector spaces of
Minkowski summands of faces of $Q$. Using the notations of \zitat{2}{5} (and
$\tilde{T}^1(\emptyset):= 0$) we obtain the following two equivalent
descriptions of $T^1_Y$:\\
\par
{\bf Theorem:}
\begin{itemize}
\item[(1)]
Let $R\in M$, then
\begin{eqnarray*}
T^1_Y(-R) &=&
\left\{ \begin{array}{ll}
\tilde{T}^1(Q\cap[\frac{1}{g}R^\ast-R]^\bot) &=
\tilde{T}^1(\mbox{Conv}\{a^i\,|\,\langle a^i,R\rangle =1\})\\
& \mbox{ for } \frac{1}{g}R^\ast \geq R \mbox{ on } \sigma \;(\mbox{i.e.}
\langle a^i,R\rangle\leq 1 \;\forall i)\\
0 & \mbox{ otherwise}.
\end{array} \right.
\end{eqnarray*}
\item[(2)]
Let $\tau < \sigma$ be a face of $\sigma$, then
$T^1_Y\left( [-\frac{1}{g}R^\ast +
\mbox{int}(\sigma^{\scriptscriptstyle\vee}\cap\tau^\bot)]\cap M \right) = \tilde{T}^1
(Q\cap \tau)$. $T^1_Y$ vanishes in the remaining degrees.
\vspace{2ex}
\end{itemize}
\par
\neu{28}
With the same assumptions as in \zitat{2}{7} we obtain the following
applications of the previous theorem:
\pagebreak
\\
\par
{\bf Corollary:}
\begin{itemize}
\item[(1)]
If every 2-face of $Q$ is a triangle (for instance, if $Y$ is smooth in
codimension 3), then $Y$ is rigid, i.e. $T^1_Y=0$.
\item[(2)]
If $Y$ is Gorenstein ($g=1$) of dimension at least 4 ($\mbox{dim}\,Q\geq 3$),
then the existence of a 2-face of $Q$ that is not a triangle implies
$\mbox{dim}\, T^1_Y = \infty .$
\item[(3)]
Let $Y$ be not Gorenstein, i.e. $g\geq 2$. Then, $\mbox{dim}\, T^1_Y <
\infty$ implies $T^1_Y=0$.
\vspace{2ex}
\end{itemize}
\par
{\bf Proof:}
(1) If $Q$ is shaped that every 2-face is a triangle, then every (at least
2-dimensional) face of $Q$ will have this property, too. On the other hand,
Smilanski has shown that polytopes with only triangular 2-faces admit at
most trivial Minkowski decompositions (cf. \cite{Sm}, Corollary (5.2)), i.e.
$\tilde{T}^1=0$.\\
Moreover, faces of dimension smaller or equal than 1 of $Q$ cannot be
non-trivially decomposed, anyway.\\
\par
(2) Two-dimensional polygons with at least 4 vertices have a non-trivial
$\tilde{T}^1$. Hence, a non-triangular 2-face of $Q$ yields a proper face
$\tau < \sigma$ with $\tilde{T}^1(Q\cap \tau) \neq 0$.\\ On the other hand,
``proper'' means that $[-R^\ast + \mbox{int}(\sigma^{\scriptscriptstyle\vee}\cap\tau^\bot)]
\cap M$ contains infinitely many elements, and $T^1_Y$ is non-trivial in all
those degrees.\\
\par
(3) Assume that $T^1_Y \neq 0$, then there must be a face $\tau < \sigma$
and an element $-R\in [-\frac{1}{g}R^\ast +
\mbox{int}(\sigma^{\scriptscriptstyle\vee}\cap\tau^\bot)]\cap M$ such that $T^1_Y(-R) =
\tilde{T}(Q\cap\tau) \neq 0$.\\
For $\tau = \sigma$ we would obtain
$[-\frac{1}{g}R^\ast +
\mbox{int}(\sigma^{\scriptscriptstyle\vee}\cap\tau^\bot)]\cap M = \{-\frac{1}{g}R^\ast\}
\cap M = \emptyset$. Hence, $\tau<\sigma$ must be a proper face, and we can
argue as in (2).
\hfill$\Box$\\
\par
\neu{29}
Finally, we want to mention the case $\mbox{dim}\,Y = 3$. Then, $Y$ is an
isolated singularity, and it is given by a lattice $N$-gon $Q$.
\\
\par
{\em Case1: $Y$ is not Gorenstein (i.e. $g\geq 2$).}\\
Then, $Y$ is rigid. (This follows from (3) of the corollary in
\zitat{2}{8}.)\\
\par
{\em Case 2: $Y$ is Gorenstein (i.e. $g=1$).}\\
Then, $T^1_Y$ is concentrated in degree $-R^\ast$, and $T^1_Y=T^1_Y(-R^\ast)
= \tilde{T}^1(Q)$ has dimension $N-3$.\\
(The proper faces of $Q$ are Minkowski indecomposable. Hence, to produce a
non-trivial contribution to $T^1_Y$, the face $\tau$ in Theorem
\zitat{2}{7}(2) has to equal $\sigma$. That means, $T^1_Y$ is concentrated
in degree $-R^\ast$ only.\\
On the other hand, for computing the dimension of $T^1_Y(-R^\ast)
= \tilde{T}^1(Q)$ in our special case, use the second formula of Theorem
\zitat{2}{2}.)\\
\par
\sect{Really existing deformations of toric Gorenstein singularities}\label{s3}
\neu{31}
As in the previous chapter, we start with recalling the general result
concerning arbitrary affine toric varieties. We use the notations of
\zitat{2}{1}.\\
\par
Let $\bar{r}\in \stackrel{\scriptscriptstyle\vee}{\sigma}\cap M$ be a primitive element.
Then, each (strongly homogeneous)
toric regular sequence of degree $-\bar{r}$ and its corresponding standard
deformation of $Y$ arise in the following way:
\begin{itemize}
\item[(i)]
Define $(A\!\!\!\!\!A,\,I\!\!L)$ as the affine space (with lattice) induced by $\bar{r}\in
M$
\[
(A\!\!\!\!\!A,\,I\!\!L) := (N_{I\!\!R},\,N) \cap \{a\in N_{I\!\!R}|\; \langle a,\,\bar{r} \rangle
= 1 \}.
\]
By choosing an arbitrary base point $0\in I\!\!L$ the pair $(A\!\!\!\!\!A,\,I\!\!L)$ can be
regarded as a vector space with lattice. Moreover, we obtain an isomorphism
of lattices $I\!\!L\timesZ\!\!\!Z \stackrel{\sim}{\rightarrow} N$ via
$(a,\,g) \mapsto (a-0)+g\cdot 0$.
\item[(ii)]
$Q:= \sigma \cap A\!\!\!\!\!A$ is a (not necessarily compact) rational polyhedron
in $A\!\!\!\!\!A$.
Fix a Minkowski decomposition $Q=R_0 + \dots + R_m$ such that for each vertex
of $Q$ at least $m$ of its $m+1$ $R_i$-summands (which are uniquely determined
vertices of the polyhedra $R_i$) are contained in the lattice $I\!\!L$.
\item[(iii)]
Define $P\subseteq A\!\!\!\!\!A\times I\!\!R^{m+1}$ as the convex polyhedron
\[
P:= \mbox{conv$\;$} \left( \bigcup_{i=0}^m R_i\times \{e^i\} \right)
\]
and $\tilde{\sigma}:= \overline{I\!\!R_{\geq 0} \cdot P}$ as the closure of its
cone in $A\!\!\!\!\!A\timesI\!\!R^{m+1}$.\\
Moreover, if $r^i$ denotes the projection of $I\!\!L\timesZ\!\!\!Z^{m+1}$ onto the $i$-th
component of $Z\!\!\!Z^{m+1}$, we have found elements $r^0,\dots,r^m\in
\tilde{\sigma}^{\scriptscriptstyle\vee} \cap (I\!\!L\timesZ\!\!\!Z^{m+1})^{\ast}$.
\item[(iv)]
$\sigma\subseteq N_{I\!\!R}$ is the cone over $Q\subseteq A\!\!\!\!\!A$. Hence, the affine
embedding $A\!\!\!\!\!A \hookrightarrow A\!\!\!\!\!A\timesI\!\!R^{m+1}\; (a\mapsto (a;\,1,\dots,1))$
induces an embedding of lattices $N \hookrightarrow I\!\!L\timesZ\!\!\!Z^{m+1}$ such
that $N= (I\!\!L\timesZ\!\!\!Z^{m+1}) \cap \bigcap_{i=1}^m (r^i-r^0)^{\bot}$ and
$\sigma = \tilde{\sigma}\cap N_{I\!\!R}$.
\item[(v)]
$\{x^{r^1}-x^{r^0},\dots,x^{r^m}-x^{r^0}\}$ is a toric regular sequence in
$X:= \mbox{Spec$\;$} I\!\!\!\!C[\tilde{\sigma}^{\scriptscriptstyle\vee}\cap (I\!\!L\timesZ\!\!\!Z^{m+1})^{\ast}]$,
and $Y$ is equal to the special fiber of the corresponding
(flat) map $X\rightarrow
I\!\!\!\!C^m$.
\end{itemize}
(The proof can be found in \S 4 of \cite{Mink}.)\\
\par
{\bf Remark:} (1)
The assumption that the degree $-\bar{r}$ has to be a primitive element of the
lattice $M$ is not essential. However, the description of the corresponding
toric regular sequences becomes slightly more complicated in the genaral case
(cf. \S 3 of \cite{Mink}) - and we do not need it in the present paper.\\
(2) The previous method yields deformations of degrees contained in
$-(\sigma^{\scriptscriptstyle\vee}\cap M)$ only. Nevertheless, $T^1$ can be non-trivial in
other degrees, too.\\
\par
\neu{32}
As a direct consequence we obtain\\
\par
{\bf Theorem:}
Let $Y$ be an affine toric Gorenstein variety induced
from a lattice polytope $Q$. Then, toric $m$-parameter
deformations of degree $-R^{\ast}$ correspond to Minkowski
decompositions of
$Q$ into a sum $Q=R_0+\dots +R_m$ of $m+1$ lattice polytopes.\\
The Kodaira-Spencer-map maps the parameter space $I\!\!\!\!C^m$ onto the linear
subspace
$\mbox{span} (\varrho(R_0),\dots ,\varrho(R_m)) \subseteq \tilde{T}^1(Q) =
T^1_Y(-R^{\ast})\subseteq T^1_Y$.\\
\par
{\bf Proof:}
$(A\!\!\!\!\!A,I\!\!L)$ defined in (i) of the previous theorem is exactly
that affine space containing our polytope $Q$. Moreover, $Q$ coincides with
the polyhedron $Q:= \sigma \cap A\!\!\!\!\!A$ defined in (ii).
Since $Q$ is a lattice polytope, the conditions for the summands $R_i$
(cf. (ii) of the previous theorem) are equivalent to the property of being
lattice polytopes, too.\\
Finally, the claim concerning the Kodaira-Spencer-map follows from the
definitions of the maps $\varrho$ and $\theta$ in \zitat{2}{5} and
\zitat{2}{6}, respectively, and from Theorem (5.2) of \cite{Mink}.
\hfill$\Box$\\
\par
\neu{33}
Let us focus on the special case of $\mbox{dim}\, Y=3$. Let $Y$ be given by
a 2-dimensional lattice polygon $Q=\mbox{Conv}(a^1,\dots,a^N)$ with
primitive edges
$\vec{v}_i:=a^{i+1}-a^i\;
(i\in Z\!\!\!Z\left/_{\displaystyle \!\!N\,Z\!\!\!Z}\right)$,
i.e. $Y$ has an isolated singularity in $0 \in Y$. Then, we obtain\\
\par
{\bf Theorem:} Non-trivial toric $m$-parameter deformations of $Y$
correspond to non-trivial Minkowski decompositions of $Q$ into a sum of
$m+1$ lattice polygons, i.e. to decompositions of the set of edges of $Q$
into a disjoint union of $m+1$ subsets each suming up to 0.\\
\par
{\bf Proof:} This is an immediate consequence of Theorem \zitat{3}{2} and
the fact that $T^1_Y$ is concentrated in degree $-R^\ast$ (cf.
\zitat{2}{9}). Nevertheless, beeing a little more carefully, for this
conclusion the following fact has to be used: Toric, regular sequences
inducing a trivial Kodaira-Spencer-map always yield trivial (standard)
deformations. This is proved in \S 6 of \cite{Mink}.
\hfill$\Box$\\
\par
\sect{Polarity and Examples}\label{s4}
\neu{41}
We start with recalling some general facts concerning the relation between
lattice polytopes and projective toric varieties (cf. Chapter 2 of
\cite{Oda}).\\
\par
Let $Q\subseteq (A\!\!\!\!\!A,I\!\!L)$ be a lattice polytope. Then, the inner normal fan
$\Sigma$ induces a projective toric variety ${I\!\!P}(Q)$, and $Q$ itself
corresponds to an ample line bundle on it.\\
\par
Equivariant Weil divisors on
$I\!\!P(Q)$ are described by maps $\Sigma^{(1)} \stackrel{h}{\rightarrow}Z\!\!\!Z$.
Modulo principal divisors, they generate the whole divisor class group
$\mbox{Div}(I\!\!P(Q))$. Let $D_h$ be a Weil divisor on $I\!\!P(Q)$.
\begin{itemize}
\item[(i)]
$D_h$ is Cartier if and only if, on each top dimensional cone
$\alpha = \langle c^1,\dots,c^k\rangle \in\Sigma$, the map $h$ can be
represented as
\[
h(c^j) = \langle a_{\alpha},c^j\rangle \;(j=1,\dots,k) \quad \mbox{with }
a_{\alpha}\inI\!\!L.
\]
\item[(ii)]
A Cartier divisor $D_h$ is nef if and only if, moreover, $h\leq \langle
a_{\alpha},\bullet\rangle$ holds (for each top dimensional $\alpha\in\Sigma$)
on the whole 1-skeleton $\Sigma^{(1)}$. Then, the elements $a_{\alpha}$
form the vertex set of a polytope with inner normal cones containing the
corresponding cones of $\Sigma$.
\end{itemize}
On the other hand, if $Q'$ is a lattice polytope such that $\Sigma$ is a
subdivision of its inner normal fan (i.e. $Q'$ is a Minkowski summand of a
scalar multiple of $Q$), then we can use its vertices to define a map
$h(Q'):\Sigma^{(1)}\rightarrowZ\!\!\!Z$ via (i). We obtain a nef Cartier divisor
$D_{Q'}$ on $I\!\!P(Q)$ again.\\
The divisor $D_{Q'}$ is even ample if and only if $Q'$ and $Q$ induce the
same inner normal fan $\Sigma$. Equivalently, the elements $a_{\alpha}$ yield
different lattice points for different cones $\alpha$.\\
\par
Finally, we remark that variations by principal divisors correspond to
translations of the polytopes by lattice vectors only.\\
\par
\neu{42}
Let $Y$ be a 3-dimensional affine toric Gorenstein variety induced
by a lattice polygon $Q$, let
$Y$ having an isolated singularity in $0\in Y$. Then, we call
$I\!\!P(Q)$ the polar variety of $Y$, it will be denoted by $I\!\!P (Y)$.
\begin{itemize}
\item[(I)]
$\mbox{Pic}\,I\!\!P (Y)$ equals the group generated by the lattice Minkowski
summands of scalar multiples of $Q$.
If we denote by
${\cal O}_{I\!\!P (Y)}(1)$
the ample line bundle corresponding to $Q$ itself, then
\zitat{2}{9} tells us that
\[
T^1_Y \,=\; \mbox{Pic}\,{I\!\!P}(Y) \left/_{\displaystyle \!\!{\cal O}(1)}
\right. \otimes_{Z\!\!\!Z} I\!\!\!\!C.
\]
Let $P(Y)$ be the cone over $(I\!\!P (Y),\,{\cal O}(1))$. The pull back of
${\cal O}(1)$ is a principal divisor on $P(Y)\setminus \{0\}$. Hence,
\[
T^1_Y \,=\; \mbox{Pic}(P(Y)\setminus \{0\}) \otimes_{Z\!\!\!Z} I\!\!\!\!C.
\]
\item[(II)]
Theorem \zitat{3}{3} deals with Minkowski decompositions of $Q$ into a sum
of lattice polygons. In the language of the polar variety we obtain that
non-trivial toric $m$-parameter deformations of $Y$ correspond to non-trivial
decompositions of ${\cal O}_{I\!\!P (Y)}(1)$ into a tensor product
\[
{\cal O}_{{I\!\!P}(Y)}(1) = {\cal L}_0 \otimes \dots \otimes {\cal L}_m
\]
of $m+1$ nef invertible sheaves on $I\!\!P (Y)$. The tangent plane to this
deformation inside the semi-universal base space $S$ is spanned by the
classes $[{\cal L}_0], \dots, [{\cal L}_m] \in \mbox{Pic}\,{I\!\!P}(Y)
\left/_{\displaystyle \!\!{\cal O}(1)}\right. = \mbox{Pic}(P(Y)\setminus\{0\})
\subseteq T^1_Y$.
\vspace{2ex}
\end{itemize}
\par
{\bf Remark:}
The condition ``$0\in Y$ is an isolated singularity'' can be translated
into the
$I\!\!P (Y)$-language, too: Each closed equivariant subvariety of
$I\!\!P (Y)$
equals a linearly embedded projective space.\\
\par
\neu{43}
{\bf Conjecture:}
Take an arbitrary Minkowski decomposition of $Q$ into a sum of lattice
polytopes (equivalently: a decomposition of ${\cal O}_{I\!\!P (Y)}(1)$ into a
tensor
product of nef line bundles), project the summands into $T^1_Y = \tilde{T}^1
(Q)\otimes_{I\!\!R}I\!\!\!\!C = \mbox{Pic}(P(Y)\setminus\{0\})\otimes_{Z\!\!\!Z}I\!\!\!\!C$, and form
their linear
hull. Then, the union of all linear subspaces obtained in this way equals
the reduced base space of the semi-universal deformation of $Y$.\\
\par
The {\em cone} of Minkowski summands (i.e. the nef sheaves in $\mbox{Pic}\,
I\!\!P (Y)$) contains much more information than the so-called {\em space}
of Minkowski summands of $Q$ (i.e.
$\mbox{Pic}(P(Y)\setminus\{0\})\otimes_{Z\!\!\!Z}I\!\!R$).
Apart from the toric context, is there any such cone in deformation theory?
Projecting a certain interior point onto $0$ has to yield $T^1_Y$, then.\\
\par
Does the cone of Minkowski summands contain any information about the
non-reduced structure of the base space?\\
\par
\neu{44}
Finally, we want to present a special class of examples. We are looking for
those three-dimensional $Y$ that are, additionally to the usual assumptions,
cones over
projective toric varieties. (Do not mistake this property for $P(Y)$ being the
cone over $I\!\!P (Y)$.)\\
In \S 4 of \cite{Bat} it is shown that these $Y$ can be characterized as the
cones over two-dimensional toric Fano varieties with Gorenstein singularities.
(Then, $P(Y)$ admits the same property.)\\
\par
The corresponding $Q$ are exactly
those lattice polygons containing one and only one interior lattice point
(``reflexive polygons''). Choosing this point as the origin, the polar
polygon $Q^{\scriptscriptstyle\vee}:= \{r\in A\!\!\!\!\!A^{\ast}|\,\langle Q,\,r\rangle \leq 1\}$ is a
lattice polygon, too. Then, $\stackrel{\scriptscriptstyle\vee}{\sigma}=\mbox{Cone}(Q^{\scriptscriptstyle\vee})$,
and
$Y$ is the cone over the projective variety corresponding to $Q^{\scriptscriptstyle\vee}$.\\
\par
Reflexive polygons were classified in (4.2) of \cite{Rob}. Our additional
assumption of
$Y$ having only an isolated singularity
causes that only five polygons
$Q$ survive from the original list (containing 16 items). Including the
polar polygons $Q^{\scriptscriptstyle\vee}$, we will see nine ones, however.\\
\par
{\bf (\ref{s4}.\ref{t44}.1)}\\
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\\
$Y_1$ is the cone over $I\!\!P^1\timesI\!\!P^1$ embedded by ${\cal O}(2,2)$.\\
\par
$Q_1$ is a quadrangle, hence $T^1$ is one-dimensional. Moreover, $Q_1$ is the
Minkowski sum of two line segments, i.e. there is a really existing toric
1-parameter deformation. The total space is an isolated 4-dimensional cyclic
quotient singularity.\\
\par
In particular, the base space $S_1$ of the semi-universal
deformation of $Y_1$ equals $I\!\!\!\!C^1$.\\
\par
{\bf (\ref{s4}.\ref{t44}.2)}\\
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\\
$Y_2$ is the cone over $I\!\!P^2$ embedded by ${\cal O}(3)$. Since $Q_2$ is a
triangle, $Y_2$ is rigid.\\
\par
{\bf (\ref{s4}.\ref{t44}.3)}\\
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\\
$Y_3$ is the cone over the Del Pezzo surface of degree 8 (the blowing up of
$(I\!\!P^2,{\cal O}(3))$ in one point).\\
\par
The vector space $T^1$ is one-dimensional. The two-dimensional cone of the
rational Minkowski summands of scalar multiples of $Q_3$ is generated by two
triangles.\\
\par
However, there are no lattice polygons that are non-trivial Minkowski summands
of $Q_3$.
That means, $Y_3$ does not admit any toric deformation at all.\\
Indeed, as Duco van Straten has computed with Macaulay, the semi-universal
base space $S_3$ of $Y_3$ equals
$\mbox{Spec$\;$} I\!\!\!\!C[\varepsilon]/_{\displaystyle \varepsilon^2}$.\\
\par
{\bf (\ref{s4}.\ref{t44}.4)}\\
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\\
$Y_4$ is the cone over the Del Pezzo surface of degree 7 (obtained from
$(I\!\!P^2,{\cal O}(3))$ by blowing up two points, or from $(I\!\!P^1\timesI\!\!P^1,
{\cal O}(2,2))$ by blowing up one point).\\
\par
$T^1$ is two-dimensional, but $Q_4$ admits one decomposition into a
Minkowski sum of two lattice polygons only. $Q_4$ equals the sum of a line
segment and a triangle - this yields a 1-parameter deformation of $Y_4$,
the total space is the cone over $I\!\!P ({\cal O}_{I\!\!P^2}\oplus {\cal O}
_{I\!\!P^2}(1))$.\\
\par
The semi-universal base space $S_4$ is a complex line with one embedded
component (computed by Duco van Straten using Macaulay).\\
\par
{\bf (\ref{s4}.\ref{t44}.5)}\\
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\\
$Y_5$ is the cone over the Del Pezzo surface of degree 6 (obtained by blowing
up the projective variety of (\ref{s4}.\ref{t44}.4) in one more point).\\
$T^1$ is three-dimensional, and $Q_5$ admits two different extremal Minkowski
decompositions:
\begin{itemize}
\item[(i)]
$Q_5$ equals the sum of two triangles, the corresponding 1-parameter family
admits the cone over $I\!\!P^1\timesI\!\!P^1\timesI\!\!P^1$ as its total space.
\item[(ii)]
$Q_5$ also equals the sum of three line segments. This corresponds to a
two-parameter family with the cone over $I\!\!P^2\timesI\!\!P^2$ as its total space.
\end{itemize}
Again, Duco van Straten has computed the semi-universal base space - it is
reduced and equals the
transversal union of a complex plane with a complex line. These components
correspond to the toric deformations we have already seen.
|
1994-03-12T01:02:41 | 9403 | alg-geom/9403010 | en | https://arxiv.org/abs/alg-geom/9403010 | [
"alg-geom",
"math.AG"
] | alg-geom/9403010 | Bernd Siebert | Bernd Siebert and Gang Tian | On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and
Intriligator | 21 pages, latex with amstex-fonts | null | null | null | null | We observe a general structure theorem for quantum cohomology rings, a
non-homogeneous version of the usual cohomology ring encoding information about
(almost holomorphic) rational curves. An application is the rigorous
computation of the quantum cohomology of Grassmannians. As purely algebraic
consequence we prove a beautiful formula of Vafa and Intriligator for
intersection numbers of certain compactifications of moduli spaces of maps from
a Riemann surface (any genus) to G(k,n) which recently has excited many
mathematicians. The formula generalizes to any Fano manifold whose cohomology
ring can be presented as complete intersection.
| [
{
"version": "v1",
"created": "Sat, 12 Mar 1994 00:01:45 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Siebert",
"Bernd",
""
],
[
"Tian",
"Gang",
""
]
] | alg-geom | \section{Definition of quantum multiplications}
In this section, we recall the definition of quantum multiplications
given in \cite{ruan/tian}. The definition uses the
GRW-invariants as defined in \cite{ruan}.
A symplectic manifold $(M,\omega)$ of dimension $2n$ is called {\em
(semi)-positive} if for any $R=f_*[S^2]\in H_2(M,{\Bbb Z})$,
$f:S^2\rightarrow M$, with $[\omega](R)>0$ either $c_1(M)(R)>0$
($c_1(M)(R)\ge0$). Any Fano manifold is positive w.r.t.\ any K\"ahler
form. Let $J$ be a generic almost complex structure on $M$ tamed by
$\omega$ (i.e.\ $\omega(X,JX)>0$ $\forall\, X\in TM\setminus\{0\}$). A
$J$-holomorphic curve is a smooth map $f:\Sigma\rightarrow M$
satisfying $J\circ Df = Df\circ j$, where $\Sigma$ is a Riemann surface
(of genus $g$, say) and $j$ is its standard complex structure. This
last equation is a Cauchy-Riemann equation ${\bar\di} f=0$,
${\bar\di}=\frac{1}{2}(D-J\circ D\circ j)$. For a technical reason it is
convenient to look at the inhomogeneous equation ${\bar\di} f=\gamma$ with
$\gamma$ (the pull-back to $\Sigma$ of) a section of an appropriate
bundle over $\Sigma\times M$. Solutions of this equation are called
perturbed or $(J,\gamma)$-holomorphic curves. The GRW-invariant can be
defined as follows:
Let $R\in H_2(M,{\Bbb Z})$ and $B_1,B_2,\ldots,B_s$ be integral
homology classes in $H_*(M,{\Bbb Z})$ satisfying:
\[
\hspace{3cm}\sum_{i=1}^s (2n-\deg B_i)= 2c_1(M)(R)+2n(1-g).
\hspace{3cm}\mbox{(dim)}
\]
Every integral homology class can be represented by a so-called
pseudo-manifold (a certain simplicial complex, cf.\ \cite{ruan/tian}).
For simplicity, we shall also use $B_i$ to denote the pseudo-manifolds
representing these homology classes. Then if $(J,\gamma)$ is generic
and $B_i$ are in sufficiently general position (transversal w.r.t.\ the
evaluation map $\Sigma\times\{f\} \rightarrow M$), for $s$ generic
points $t_1,\ldots,t_s \in\Sigma$ there are only finitely many
$(J,\gamma)$-holomorphic curves $f$ satisfying: $f(t_i)\in B_i$,
$i=1,\ldots,s$ and $f_*[\Sigma]= R$. We define $\tilde\Phi
_{(R,\omega)}(B_1,\ldots,B_s)$ to be the algebraic sum of such $f$ with
appropriate sign according to the orientation. One can prove that
$\tilde\Phi_{(R,\omega)} (B_1,\ldots,B_s)$ is independent of the
choices of $J$, $j$, $\gamma$, and pseudo-manifolds representing
$B_1,\ldots,B_s$ provided the $B_i$ are transversal to the Gromov
boundary of the compactified moduli space of $(J,\gamma)$-holomorphic
curves. The Gromov boundary here consists of curves $C=f(\Sigma)\cup
g_1(S^2)\cup\ldots\cup g_k(S^2)$ with: $C$ connected; $f$ a
$(J,\gamma)$-holomorphic curve; $g_\nu$ a $(J,0)$-holomorphic rational
curve (a ``bubble''); $R=f_*[\Sigma]+\sum_\nu a_\nu {g_\nu}_*[S^2]$ for
some $a_\nu\in{\Bbb N}$. Transversality means that there are no such
curves intersecting each $B_i$. Furthermore, $\tilde\Phi_{(R,\omega)}
(B_1,\ldots,B_s)$ depends only on the deformation class $\{\omega\}$ of
$\omega$. Therefore we obtain the GRW-invariant
$\tilde\Phi_{(R,\{\omega\})}(B_1,\ldots,B_s)$. As a matter of notation
we define $\tilde\Phi_{(R,\omega)}(B_1,\ldots,B_s)$ to be zero unless
the dimensions match (dim).
We now make a simple but important remark: Let $J$ be an almost
complex structure such that any $J$-holomorphic curve with
$f_*[\Sigma] = R$ is regular, i.e.\ the linearization of the
Cauchy-Riemann operator at $f$ has trivial cokernel. Then $(J,0)$ is
generic and we can use exact $J$-holomorphic curves to compute the
invariant $\tilde\Phi_{(R, \{\omega\})}(B_1,\ldots,B_s)$. In case $J$ is
integrable and $f$ is an immersion, regularity at $f$ is equivalent to
the vanishing of $H^1(C,N_{C|M})$, $C=f(\Sigma)$ \cite[2.1.B.]{gromov}. In
particular the invariant is easily computable in certain cases as
follows:
\begin{lemma}\label{genericcrit}
Let $(M,\omega)$ be a K\"ahler manifold and $R\in H_2(M,{\Bbb Z})$ s.th.\
any holomorphic curve $C\subset M$ homologous to $R$ is non-singular
and has $H^1(C,N_{C|X})=0$. Let $B_1,\ldots,B_s\subset M$ be complex
submanifolds transversal to the evaluation map and the Gromov boundary.
Then
\[
\tilde\Phi_{(R,\{\omega\})}(B_1,\ldots,B_s)=\sum_C\,
\sharp(B_1\cap C)\cdot\sharp(B_2\cap C)\cdot\ldots\cdot\sharp(B_s\cap C),
\]
where the sum is over all holomorphic curves $C$ homologous to $R$.
\end{lemma}
Note that $\sharp(B_1\cap C)\cdot\ldots\cdot\sharp(B_s\cap C)$ is
precisely the number of ways to parametrize $C$ by $f:\Sigma\rightarrow
C$ with $t_i$ mapping to $B_i$, $i=1,\ldots,s$. Moreover, there are no
signs occurring in the formula since all spaces involved are
canonically oriented by their complex structure.
\begin{rem}\label{algmod}\rm
(This will only be used to comment on the relationship of our work with
\cite{bertram.etal}.)
More generally, if the moduli space ${\cal M}_{R,\Sigma}$ (for shortness
${\cal M}$ in the sequel) of holomorphic maps
$f:\Sigma\rightarrow M$ with $f_*[\Sigma]=R$ has the expected dimension
$c_1(M)(R)+n(1-g)$ (equivalently, $H^1(\Sigma,f^*T_M)=0$ for almost
all $f$) the GRW-invariants $\tilde\Phi_R$ have an interpretation as
intersection products of certain compactifications of ${\cal M}$: $M$ being
projective algebraic
there clearly exist projective compactifications of ${\cal M}$. Choose one,
say $\overline{\cal M}$. Then the evaluation map
\[
{\rm ev}:\Sigma\times{\cal M}\longrightarrow M,\ \ \
(t,f)\longmapsto f(t)
\]
extends rationally to $\overline{\cal M}$. Hence there is a modification
$\pi:\Gamma\rightarrow\Sigma\times\overline{\cal M}$, biregular over
$\Sigma\times{\cal M}$ such that ${\rm ev}$ has an extension $\tilde{\rm ev}:
\Gamma\rightarrow M$. For generic $t\in\Sigma$, $\Gamma_t:=\pi^{-1}(\{t\}\times
{\cal M})$ is a modification of $\overline{\cal M}$. We choose a common
desingularization
$\widehat{\cal M}$ of all the $\Gamma_t$, $t\in\Sigma$ generic (blow-up the sum
of ideals ${\cal I}_t$ whose blow-up desingularizes $\Gamma_t$, then
desingularize).
Let $\iota_t:\widehat{\cal M}\rightarrow\Gamma_t \hookrightarrow\Gamma$. Then
for $\beta_i\in H^*(M,{\Bbb Z})$ Poincar\'e-dual to $B_i$ and generic choices of
$t_1,\ldots,t_s\in\Sigma$
\[
\tilde\Phi_R(B_1,\ldots,B_s)
=(\iota_{t_1}^*\Phi^*\beta_1\wedge\ldots\wedge\iota_{t_s}^*\Phi^*\beta_s)
[\widehat{\cal M}].
\]
In fact, choosing $B_1,\ldots,B_s$ as generic pseudo-manifolds then
$(\Phi\circ \iota_{t_i})^{-1}(B_i)$ is Poincar\'e-dual to
$\iota_{t_i}^*\Phi^*\beta_i$ and the bordism argument of \cite{ruan}
should generalize to show
\[
\tilde\Phi_R(B_1,\ldots,B_s)=\sharp(\Phi\circ\iota_{t_1})^{-1}(B_1)\cap
\ldots\cap(\Phi\circ\iota_{t_s})^{-1}(B_s),
\]
($\sharp$ means algebraic sum with appropriate signs keeping track of
the orientations). A proper proof has to provide a bordism to a generic
situation carefully dealing with the singularities of ${\cal M}$ as well
as with transversality and orientations. Details will be given in
\cite{ruan/tian}.
{\hfill$\Diamond$}\vspace{1.5ex}
\end{rem}
To define a quantum multiplication on $H^*(M,{\Bbb Z})$, we introduce
the real-valued invariant
\[
\tilde{\Phi}_{[\omega]}(B_1,\ldots,B_s)=\sum_{R\in H_2(M,{\Bbb Z})}
\tilde{\Phi}_{(R,\{\omega\})}(B_1,\ldots,B_s)\,e^{-[\omega](R)}.
\]
In general there might be infinitely many terms contributing to the sum
(e.g.\ in the important and interesting case of Calabi-Yau manifolds)
and one faces a non-trivial convergence problem. In the positive case,
however, this sum is actually finite due to the dimension condition
(dim). Let us assume $M$ positive for the following (but see
Remark~\ref{cy-rem}). Note also that letting $\omega$ vary one gets
back all the invariants $\tilde{\Phi}_{(R,\{\omega\})}(B_1,\ldots,B_s)$
by solving some system of linear equations. The quantum multiplication
$\wedge_{\mbox{\scriptsize Q}}$ on $H^*(M,{\rm I \! R})$ is then characterized by the equation
\[
(\alpha\wedge_{\mbox{\scriptsize Q}}\beta)[A]=\tilde{\Phi}_{[\omega]}(\alpha^\vee,
\beta^\vee,A),
\]
where $\alpha$, $\beta\in H^*(M,{\Bbb Z})$ and $\ ^\vee$ means Poincar\'e-dual.
In terms of a basis $\{A_i\}$ for the torsion free part of
$H_*(M,{\Bbb Z})$ and $\{\alpha_i\}$ the Poincar\'e-dual basis of
$H^*(M,{\Bbb Z})$ we may state this more explicitely as
\[
\alpha_i\wedge_{\mbox{\scriptsize Q}}\alpha_j=\sum_{k,l}\eta^{lk}\hspace{1pt}\tilde{\Phi}_{[\omega]}(A_i,
A_j,A_k)\,\alpha_l,
\]
with $(\eta^{lk})_{kl}$ inverse to the intersection matrix
$(\eta_{ij})_{ij}=(A_i\cdot A_j)_{ij}$ ($\sum_k
\eta_{ij}\eta^{kj}=\delta_{ik}$). The associativity of the quantum
multiplication is highly non-trivial and shown by a careful analysis
of the degeneration of rational curves (following an idea of Witten)
in \cite{ruan/tian}.
A trivial, but decisive feature of this definition, that we are going
to exploit, is that its homogeneous part reduces to the cup product
\[
\alpha_i\wedge\alpha_j=\sum_{k,l}\eta^{lk}\cdot(A_i\cdot A_j\cdot A_k)
\,\alpha_l.
\]
\section{A presentation for quantum cohomology}
Denote by $\cz\langle X_1,\ldots,X_n\rangle$ the graded anticommutative
$\cz$-algebra with generators $X_i$ of degree $d_i$, i.e.\ with $X_i
X_j=(-1)^{d_i\cdot d_j}X_j X_i$. If $m$ of the $X_i$ have odd degree
this is isomorphic to $\left(\Lambda^*\cz^m\right) \otimes\left(
\mbox{Sym}^*\cz^{n-m}\right)$. We will call elements of this algebra
ordered polynomials (since in addition to the coefficients one has to
select an order among the factors in a monomial to determine its
sign).
Let $(M,\omega)$ be a positive symplectic manifold and
\[
H^*(M,\cz)=\cz\langle X_1,\ldots,X_n\rangle/(f_1,\ldots,f_k)
\]
be a presentation of the cohomology ring, $f_i=\sum_{|J|=\deg f_i}
a_{iJ}X^J$. (We use multiindex notation $J=(j_1,\ldots,j_n)$,
$X^J=X_1^{j_1}\wedge\ldots\wedge X_n^{j_n}$, $|J|=\sum_{i=1}^n j_i d_i$
etc.) Denote by $\wedge_{\mbox{\scriptsize Q}}$ the product in the quantum cohomology $H^*_{[\omega]}(M)$.
To distinguish clearly between calculations in $H^*(M,\cz)$ and in
$H^*_{[\omega]}(M)$, we use a $\widehat{\ }$ to mark elements of the quantum
cohomology ($\widehat{\ }$ might be thought of as a $\cz$-linear map
$\cz\langle X_1,\ldots,X_n\rangle\rightarrowH^*_{[\omega]}(M)$). An ordered polynomial
$f$ may be evaluated on generators of the quantum cohomology provided
the degrees match. We write $f(\hat X_1,\ldots,\hat X_n)$ as in the
commutative case. Especially, $\hat X^J:=\hat
X_1\wedge_{\mbox{\scriptsize Q}}\ldots\wedge_{\mbox{\scriptsize Q}}\hat X_1\wedge_{\mbox{\scriptsize Q}}\ldots\wedge_{\mbox{\scriptsize Q}}\hat X_n
\wedge_{\mbox{\scriptsize Q}}\ldots\wedge_{\mbox{\scriptsize Q}}\hat X_n$ with $\hat X_\nu$ occurring $j_\nu$-often
(note the difference from \mbox{$(X^J)\hat{\,}$ ).}
\begin{lemma}\label{generate}
$\hat X_1,\ldots,\hat X_n$ generate $H^*_{[\omega]}(M)$.
\end{lemma}
{\em Proof. }
By induction on the degree. So assume $\hat X_1,\ldots,\hat X_n$
generate $H^*_{[\omega]}(M)$ up to degree $d-1$. We want to show that the
monomials $(X^J)\hat{\,}$, $|J|=d$, can be written as linear combination
of (quantum) products of the $\hat X_\nu$. But by definition of the
multiplication in $H^*_{[\omega]}(M)$
\[
\hat X^J= (X^J)\hat{\,}+\sum_{|I|<d}a_I (X^I)\hat{\,}\, ,
\]
and by induction hypothesis $(X^I)\hat{\,}=\sum_{K\le I}b_{IK}\hat X^K$, so
\[
(X^J)\hat{\,}=\hat X^J-\sum_{K\le I\atop |I|<d}a_I b_{IK}\hat X^K.
\]
\vspace{-1cm}
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{2ex}
Next we are calculating $f_i(\hat X_1,\ldots,\hat X_n)$ (i.e. in the
quantum cohomology). Since $f_i$ is a relation in the cohomology ring, the
($\deg f_i$)-term vanishes. By the lemma we find an ordered polynomial
$g_i^{[\omega]}$ (depending on $[\omega]$) in $n$ variables
(say $T_1,\ldots,T_n$, $\deg T_i=\deg X_i$) of {\em lower degree} with
\[
f_i(\hat X_1,\ldots,\hat X_n)=g_i^{[\omega]}(\hat X_1,\ldots,\hat X_n)
\hspace{1cm}\mbox{in }H^*_{[\omega]}(M).
\]
Thus
$f_i^{[\omega]}(T_1,\ldots,T_n):=f_i(T_1,\ldots,T_n)-
g_i^{[\omega]}(T_1,\ldots,T_n) \in\cz\langle T_1,\ldots,T_n\rangle$ is
a non-trivial relation between $\hat X_1,\ldots,\hat X_n$.
\begin{theorem}\label{presentation}
$H^*_{[\omega]}(M)=\cz\langle T_1,\ldots,T_n\rangle/(f_1^{[\omega]},
\ldots,f_k^{[\omega]})$.
\end{theorem}
{\em Proof. }
Let ${\cal J}\subset\cz\langle T_1,\ldots,T_n\rangle$ be the ideal of
relations between the $\hat X_1,\ldots,\hat X_n$. Then
$(f_1^{[\omega]},\ldots,f_k^{[\omega]})\subset {\cal J}$ and $H^*_{[\omega]}(M)=\cz\langle
T_1,\ldots,T_n\rangle/{\cal J}$ by the lemma. Let $F\in {\cal J}\setminus\{0\}$.
Expand $F=F_d+F'$ with $F_d\neq0$ (weighted) homogeneous of degree $d$,
$d>0$, and $\deg F'<d$. Then
\[
F_d(\hat X_1,\ldots,\hat X_n)=-F'(\hat X_1,\ldots,\hat X_n)
\]
in the quantum cohomology for $F$ is a relation. But the highest degree
($\deg=d$) contribution to $F_d(\hat X_1,\ldots,\hat X_n)$ is just
$(F_d(X_1,\ldots,X_n))\hat{\,}\,$. Its vanishing implies
$F_d\in(f_1,\ldots,f_k)$, i.e.\ $F_d=\varphi(f_1,\ldots,f_k)$, $\varphi$ a
polynomial in $k$ variables. Thus
\[
\varphi(f_1^{[\omega]},\ldots,f_k^{[\omega]})=F_d+F''
\hspace{1cm}\mbox{in }\cz\langle T_1,\ldots,T_n\rangle
\]
with $\deg F''<d$, and we may write
$F=\varphi(f_1^{[\omega]},\ldots,f_k^{[\omega]}) +F'-F''$, $\deg F'-F''<
d=\deg F$. Proceeding by induction on the degree we finally see
$F\in(f_1^{[\omega]},\ldots,f_k^{[\omega]})$,
i.e.\ ${\cal J}=(f_1^{[\omega]},\ldots,f_k^{[\omega]})$ as claimed.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{2ex}
\begin{rem}\label{cy-rem}\rm
In the positive case $c_1(M)>0$ the contributions with $R\neq0$ give rise
to terms of lower degree by the dimension condition, so we were able to fix
$[\omega]$ and argue by considerations on the degree. Modulo
convergence problems in the semi-positive case mentioned at the end
of the previous chapter, Theorem~\ref{presentation} however remains valid.
Arguments involving the degree can be replaced by the linear
independence of terms $e^{-\lambda\cdot t}$ for various $\lambda$
in the algebra $\cz[[t]]$.
Another approach, which works without any positivity condition, is to
use formal power series over $H_2(M,{\Bbb Z})$ (cf.\ also \cite{piuni}): Let $N$
be the (obvious) completion of the group ring over $H_2(M,{\Bbb Z})$. Then
$H^*(M,{\Bbb Z})\otimes_{\Bbb Z} N$ appears as natural domain of definition of
quantum products for more general manifolds. Our arguments provide a
presentation of the quantum cohomology ring as quotient of ${\Bbb Z}\langle
T_1,\ldots,T_n\rangle\otimes_{\Bbb Z} N$ by ordered polynomials with coefficients
in $N$. Note also that in the positive case $H^*_{[\omega]}(M)$ may be viewed as
homeomorphic image of this formal quantum cohomology ring by sending
$R\in H_2(M,{\Bbb Z})$ to $e^{-[\omega](R)}$.
{\hfill$\Diamond$}\vspace{1.5ex}
\end{rem}
\section{Grassmann manifolds}
As is well-known (e.g.\ \cite[Ex.~14.6.6]{fulton}) the cohomology ring
(in fact even the Chow ring) of the Grassmann variety ${\rm G}(k,n)$ of
$k$-planes in $\cz^n$ has a presentation
\[
H^*({\rm G}(k,n),\cz)=\cz[c_1,\ldots,c_k]/(s_{n-k+1},\ldots,s_n),
\]
with $c_i$ corresponding to the Chern classes of the tautological
$k$-bundle $S$ and $s_j$ to its Segre classes viewed as polynomials in
$c_1,\ldots,c_k$ via
\begin{eqnarray*}
&(1+c_1+\ldots+c_k)(1+s_1+s_2+\ldots)=1,&\\
&\mbox{i.e. } s_j=-s_{j-1}c_1-\ldots-s_1c_{j-1}-c_j.&\hspace{3cm}(*)
\end{eqnarray*}
Note that $s_j$ is also the $j$-th Chern class of the universal
quotient bundle $Q$, which has rank $n-k$, so $s_j=0$ in
$H^*({\rm G}(k,n),\cz)$ for $j>n-k$. In fact, as a polynomial in $c_i$, $s_j$
lies in the relation ideal for $j>n$ by the recursion formula ($*$).
Under the canonical isomorphism $\Phi:{\rm G}(k,n)\simeq{\rm G}(n-k,n),\
\Lambda\mapsto(\cz^n/\Lambda)^*$, $S$ corresponds to the dual of the
universal quotient bundle $Q'$ on ${\rm G}(n-k,n)$ and $Q$ to the dual of the
tautological $(n-k)$-bundle $S'$. Thus $c_i$ and $s_j$ exchange (up to sign)
their roles and one might as well write
\[
H^*({\rm G}(k,n),\cz)=\cz[s_1,\ldots,s_{n-k}]/(c_{k+1},\ldots,c_n),
\]
this time with $c_i$ polynomials in $s_1,\ldots,s_{n-k}$ (this
presentation is actually better adapted to Schubert calculus,
i.e.\ geometry, see below). These remarks are made to emphasize the symmetry
between $c_i$ and $s_j$. Note also that the generators all have
even degree, so we need not worry about questions of sign.
Schubert calculus (e.g.\ \cite[\S~14.7]{fulton}) provides a basis of
$H^*({\rm G}(k,n),\cz)$ as a $\cz$-vector space (indeed a basis of integral
homology/cohomology as ${\Bbb Z}$-module), indexed by tuples
$(\lambda_1,\ldots,\lambda_k)$,
$n-k\ge\lambda_1\ge\ldots\ge\lambda_k\ge0$, via
\[
\{\lambda_1,\ldots,\lambda_k\}:=\det(s_{\lambda_i+j-i})_{1\le i,j\le k},
\]
i.e.\ by evaluating the Schur polynomial (S-function) associated to
$(\lambda_1,\ldots,\lambda_k)$ on the Segre classes of the tautological
$k$-bundle ($s_j=0$ for $j\not\in\{0,\ldots,n-k\}$).
$\{\lambda_1,\ldots,\lambda_k\}$ is (weighted) homogeneous of degree
$2\sum\lambda_i$. The Chern respectively Segre classes are given by
\begin{eqnarray*}
c_i&=&(-1)^i\{1,\ldots,1,0,\ldots,0\}\hspace{3ex}\mbox{(``1'' $i$-times)},\\
s_j&=&\{j,0,\ldots,0\}
\end{eqnarray*}
(for $c_i$ see \cite[Lemma~14.5.1]{fulton} or do an easy induction).
The connection to classical Schubert calculus is given by
Poincar\'e-duality. In fact, $\{\lambda_1,\ldots,\lambda_k\}\cap[{\rm G}(k,n)]$
may be represented by the Schubert varieties
\[
\Omega_{\underline{V}}(n-k+1-\lambda_1,\ldots,n-k+i-\lambda_i,\ldots,n-\lambda_k).
\]
To define the latter one has to fix a flag ${\underline{V}}=(V_1,\ldots,V_n)$,
$V_1\subset\ldots\subset V_n=\cz^n$ of linear subspaces of
$\cz^n$, $\dim V_i=i$. Then for $(a_1,\ldots,a_k)$,
$0\le a_1<\ldots<a_k\le n$
\[
\Omega_{\underline{V}}(a_1,\ldots,a_k):=\{\Lambda\in {\rm G}(k,n)\ |
\dim\Lambda\cap V_{a_i}\ge i, 1\le i\le k\}.
\]
The homology {\em class}
$(a_1,\ldots,a_n):=[\Omega_{\underline{V}}(a_1,\ldots,a_k)]$ is independent of
the choice of flag. We write $\{{\underline{\lambda}}\}^\vee:=(n-k+1-\lambda_1,\ldots,
n-k+i-\lambda_i,\ldots,n-\lambda_k)\in H_*({\rm G}(k,n),\cz)$. What is
important for us is that $[{\rm G}(k,n)]$ is Poincar\'e-dual to
$\{0,\ldots,0\}=1$ (trivial) and that the class of a point $[*]$ is
Poincar\'e-dual to $\{n-k,\ldots,n-k\}=s_{n-k}^k=(-1)^{n-k}c_k^{n-k}$
(which makes sense as $\dim {\rm G}(k,n)=k(n-k)$, but is less trivial: What
is the Poincar\'e-dual of $c_1^{k(n-k)}$?). The intersection of classes of
complementary dimension is especially easy: $|{\underline{\lambda}}|+|{\underline{\mu}}|=k(n-k)$,
then (``duality theorem''):
\[
\left(\{{\underline{\lambda}}\}\wedge\{{\underline{\mu}}\}\right)[G]
=\{{\underline{\lambda}}\}\cap\{{\underline{\mu}}\}^\vee
=\{{\underline{\lambda}}\}^\vee\cdot\{{\underline{\mu}}\}^\vee
=\left\{\begin{array}{lcl}1&,&\mbox{ if }{\underline{\mu}}={\underline{\lambda}}^*.\\
0&,&\mbox{ otherwise}.\end{array}\right.
\]
with ${\underline{\lambda}}^*:=(n-k-\lambda_k,\ldots,n-k-\lambda_1)$. In particular, we
get $\eta_{{\underline{\lambda}}{\underline{\mu}}}=\delta_{{\underline{\lambda}}{\underline{\mu}}}$ for the intersection matrix.
This ends our collection of facts concerning Grassmannians.
\vspace{2ex}
The product in $H^*_{[\omega]}(\grass)$ now reads
\[
\widehat{\{{\underline{\lambda}}\}}\wedge_{\mbox{\scriptsize Q}}\widehat{\{{\underline{\mu}}\}}=\sum_{R\in H_2({\rm G}(k,n),
{\Bbb Z})}\sum_{{\underline{\nu}}}\tilde\Phi_R\left(\{{\underline{\lambda}}\}^\vee,
\{{\underline{\mu}}\}^\vee,\{{\underline{\nu}}\}^\vee\right)\widehat{\{{\underline{\nu}}^*\}}
e^{-[\omega](R)}.\hspace{1cm}{(**)}
\]
Next we observe that $H^{1,1}({\rm G}(k,n))$ is spanned by the single
class $\{1,0,\ldots,0\}=s_1=-c_1$ and that $\det(Q)=(\det
S)^*$ is actually very ample, the corresponding embedding being the
Pl\"ucker embedding $\iota:{\rm G}(k,n)\hookrightarrow {\rm I \! P}(\Lambda^k\cz^n)$
(so $s_1=\iota^*c_1\left({\cal O}_{{\rm I \! P}(\Lambda^k{\,\rm I \!\!\! C}^n)}(1)\right)$).
Dually, $H_2({\rm G}(k,n),{\Bbb Z})$ is spanned by the single class
$\{n-k,\ldots,n-k,n-k-1\}^\vee=(1,\ldots,k-1,k+1)=:[L]$, and the sum
over $H_2({\rm G}(k,n),{\Bbb Z})$ is just running through the set
$\{d\cdot[L]\mid d\in {{\Bbb N}}_0\}$. Since
$T_{{\rm G}(k,n)}\simeq{\mbox{\rm Hom}\skp}(S,Q)\simeq S^*\otimes Q$, as one easily verifies
by using the standard local coordinates on ${\rm G}(k,n)$,
$c_1({\rm G}(k,n))={\mbox{\rm rk}}(Q)\cdot c_1(S^*)+{\mbox{\rm rk}}(S^*)\cdot c_1(Q)=n\cdot s_1$.
Recall from Chapter~1 that $\tilde\Phi_{d\cdot[L]}\left(\{{\underline{\lambda}}\}^\vee,
\{{\underline{\mu}}\}^\vee,\{{\underline{\nu}}\}^\vee\right)=0$ unless
$|{\underline{\lambda}}|+|{\underline{\mu}}|+|{\underline{\nu}}|=k(n-k)+c_1({\rm G}(k,n))
\left(d\cdot[L]\right)=k(n-k)+d\cdot n$. In calculating
$s_{n-k+i}(\hat{c}_1,\ldots,\hat{c}_k)$ (i.e.\ in $H^*_{[\omega]}(\grass)$,
$\hat{c}_i$ the generators corresponding to $c_i$ as in Chapter~2) we
have $|{\underline{\lambda}}|+|{\underline{\mu}}|\le n$, and ``='' only in the case of $s_n$. But
$|{\underline{\nu}}|\le k(n-k)=\dim {\rm G}(k,n)$, so we get no quantum contributions
besides in case of $s_n$ with $|{\underline{\nu}}|=k(n-k)$ and $d=1$. Thus
letting $\underline{\hat c}=(\hat{c}_1,\ldots,\hat{c}_k)$,
$s_{n-k+1}(\underline{\hat c})=\ldots=s_{n-1}(\underline{\hat c})=0$ in
$H^*_{[\omega]}(\grass)$ and
\begin{eqnarray*}
s_n(\underline{\hat c})
&=&-\hat{c}_1 s_{n-1}(\underline{\hat c})
-\ldots-\hat{c}_{k-1}s_{n-k+1}(\underline{\hat c})
-\hat{c}_k s_{n-k}(\underline{\hat c})\\
&=&-\hat{c}_k s_{n-k}(\underline{\hat c})\ =\
-\tilde\Phi_{[L]}\left(c_k^\vee,s_{n-k}^\vee,[*]\right)
\cdot e^{-[\omega](L)},
\end{eqnarray*}
where for the last equality we have used $(**)$.
Taking into account Theorem~\ref{presentation} to prove
Theorem~\ref{qrGrass} we are left with
\begin{prop}\label{Phi_L}
$\tilde\Phi_{[L]}\left(c_k^\vee,s_{n-k}^\vee,[*]\right)=(-1)^k$.
\end{prop}
For the proposition we are first classifying holomorphic curves homologous
to $[L]=(1,\ldots, k-1,k+1)$.
\begin{lemma}\label{minratcurve}
Let $C\subset {\rm G}(k,n)$ be a (rational) curve homologous to
$(1,\ldots,k-1,k+1)$. Then $C$ is a Schubert variety
$\Omega_{{\underline{V}}}(1,\ldots,k-1,k+1)$, i.e.\ there are linear subspaces
$U\subset W\subset\cz^n$, $\dim U=k-1$, $\dim W=k+1$, with
$C=\{\Lambda\in {\rm G}(k,n)\mid U\subset\Lambda\subset W\}$.
\end{lemma}
{\em Proof. }
Since $s_1[C]=1$ by the duality theorem, $\deg\iota(C)=1$, so the image
of $C$ under the Pl\"ucker embedding
$\iota:{\rm G}(k,n)\rightarrow{\rm I \! P}(\Lambda^k\cz^n)$ is a linear ${\rm I \! P}^1$. The
image of ${\rm G}(k,n)$ consists precisely of (rays of) {\em decomposable}
vectors $v_1\wedge\ldots\wedge v_k=\iota(\langle v_1,\ldots,v_k\rangle)
\in\Lambda^k\cz^n$. Locally the vectors $v_1,\ldots,v_k$ may be chosen
to vary smoothly with $\Lambda\in {\rm G}(k,n)$: In fact, the standard
(affine) coordinate neighbourhood of $\Lambda\in {\rm G}(k,n)$ is
${\mbox{\rm Hom}\skp}(\Lambda,\cz^n/\Lambda)$ with
$\Psi:{\mbox{\rm Hom}\skp}(\Lambda,\cz^n/\Lambda)\rightarrow {\rm G}(k,n)$,
$\varphi\mapsto\langle v+\varphi(v) \mid v\in\Lambda\rangle$ (in particular
$0\in{\mbox{\rm Hom}\skp}(\Lambda,\cz^n/\Lambda)$ corresponds to $\Lambda$). Now fixing
a basis $v_1,\ldots,v_k$ of $\Lambda$, $\iota\circ\Psi$ may be
represented (lifted to $\Lambda^k\cz^n$) by
\[
\varphi\mapsto\left(v_1+\varphi(v_1))\wedge\ldots\wedge(v_k+\varphi(v_k)\right).
\]
Thus choosing $\Lambda'\in C$ sufficiently close to $\Lambda\in C$ and letting
$e_1,\ldots,e_l$ be a basis of $\Lambda\cap\Lambda'$, completed by $e_{l+1},
\ldots,e_k$ and $e'_{l+1},\ldots,e'_k$ to a basis of $\Lambda$ and
$\Lambda'$ respectively, we have for $t\in\cz$ small
\[
e_1\wedge\ldots\wedge e_l\wedge e_{l+1}\wedge\ldots\wedge e_k
+t\cdot e_1\wedge\ldots\wedge e_l\wedge e'_{l+1}\wedge\ldots\wedge e'_k
=v_1(t)\wedge\ldots\wedge v_k(t)
\]
with $v_i(0)=e_i$ by construction. Taking
$\displaystyle\left.\frac{d}{dt}\right|_{
t=0}$ yields
\[
e_1\wedge\ldots\wedge e_l\wedge e'_{l+1}\wedge\ldots\wedge e'_k
=\dot{v}_1(0)\wedge e_2\wedge\ldots\wedge e_k+\ldots+
e_1\wedge\ldots\wedge e_{k-1}\wedge\dot{v}_k(0).
\]
As one sees by expanding $\dot{v}_i(0)$ in terms of a basis of $\cz^n$
containing $\{e_1,\ldots,e_k,$ $e'_{l+1},\ldots,e'_k\}$ we may gather the
linearly independent terms with $e_k$ and without $e_k$ to form two
equations. The left-hand side of the equation above belongs
to the latter ($\Lambda\neq\Lambda'$ $\Rightarrow$ $l<k$), so we get
\[
e_1\wedge\ldots\wedge e_l\wedge e'_{l+1}\wedge\ldots\wedge e'_k
=e_1\wedge\ldots\wedge e_{k-1}\wedge\left(\dot{v}_k(0)-\lambda\cdot
e_k\right)
\]
for some $\lambda\in\cz$ (s.th.\ $\dot{v}_k(0)-\lambda\cdot e_k$ lies
in the span of the basis vectors different from $e_k$). By linear
independence of wedge products of a basis of $\cz^n$ this shows
$l=k-1$. In view of the linearity of $\iota(C))$ we conclude
\[
\iota(C)=\left\{ [t\cdot e_1\wedge\ldots\wedge
e_{k-1}\wedge e'_k+ u\cdot e_1\wedge\ldots\wedge e_{k-1}\wedge
e_k]\ \Big|\ [t:u]\in{\rm I \! P}^1\right\},
\]
so $C$ is the Schubert variety
$\Omega_{{\underline{V}}}(1,\ldots,k-1,k+1)$ belonging to a flag ${\underline{V}}$ with
$V_{k-1}=\langle e_1,\ldots,e_{k-1}\rangle=:U$ and $V_{k+1}=\langle
e_1,\ldots,e_{k-1},e_k,e'_k\rangle=:W$.
{\hfill$\Diamond$}\vspace{1.5ex}
\begin{lemma}\label{holinvariant}
Let $A_1=\Omega_{\underline{V}^1}(n-k,n-k+1,\ldots,n-1)=\{1,\ldots,1\}^\vee
=(-1)^k c_k^\vee$, $A_2=\Omega_{\underline{V}^2}(1,n-k+2,\ldots,n)=
\{n-k,0,\ldots,0\}^\vee=s_{n-k}^\vee$,
$A_3=\{*\}=\Omega_{\underline{V}^3}(1,\ldots,k)
=\{n-k,\ldots,n-k\}^\vee$, where $\underline{V}^1$,
$\underline{V}^2$, $\underline{V}^3$ are three transversal flags
(i.e.\ $\dim V^1_i\cap V^2_j\cap V^3_k=\max\{0,i+j+k-2n\}$).
Then there is one and only one rational curve $C$ homologous to $[L]$
and with $C\cap A_i\neq\emptyset$, $i=1,2,3$. Moreover,
$C\cdot A_1=C\cdot A_2=C\cdot A_3=1$.
\end{lemma}
The lemma can be proved by doing intersection theory on the flag
manifold ${\rm F}(k-1,k+1;n)$, which parametrizes rational curves of minimal
degree by the preceding lemma, and using the two obvious maps
$\pi:{\rm F}(k-1,k+1;n)\rightarrow {\rm G}(k,n)$ and $p:{\rm F}(k-1,k+1;n)\rightarrow
{\rm G}(k,n)$ (calculate $(p_*\pi^*[A_1])\cdot(p_*\pi^*[A_2])
\cdot(p_*\pi^*[A_3])$). This method might be appropriate for more
general $A_1,A_2,A_3$, yet in our case an explicit linear algebra
argument is simpler and even more enlightening.
\vspace{2ex}
\noindent{\em Proof. }
We nee to find three $k$-planes $\Lambda^1$, $\Lambda^2$, $\Lambda^3$
and subspaces $U,W\subset\cz^n$, $\dim U=k-1$, $\dim W=k+1$ with
\begin{enumerate}
\item
$U\subset\Lambda^1\cap\Lambda^2\cap\Lambda^3\subset
\Lambda^1+\Lambda^2+\Lambda^3\subset W$.
\item
$\dim\Lambda^1\cap V^1_{n-k+(i-1)}\ge i$, $i=1,\ldots,k$.
\item
$V_1^2\subset\Lambda^2$.
\item
$\Lambda^3=V^3_k$,
\end{enumerate}
(1) says that $\Lambda^1,\Lambda^2,\Lambda^3$ lie on the rational curve
$C$ defined by $U$ and $W$ according to Lemma~\ref{minratcurve},
whereas (2)--(4) rephrase the conditions $\Lambda_i=A_i\cap C$,
$i=1,2,3$ respectively. We are now using transversality of the flags
$\underline{V}^1$, $\underline{V}^2$, $\underline{V}^3$. From (3), (4)
and (1) we readily deduce $W=V_k^3+V_1^2$, and (2) with $i=k$ shows
$\Lambda^1\subset V^1_{n-1}$, so by (4) and (1) we get $U=V_k^3\cap
V^1_{n-1}$. This choice of $U,W$ implies $\Lambda^1 =W\cap V^1_{n-1}$,
$\Lambda^2=U+V^2_1$, $\Lambda^3=V^3_k$. Conversely these
$\Lambda^1,\Lambda^2,\Lambda^3$ fulfill (1)--(4). {\hfill$\Diamond$}\vspace{1.5ex}
\vspace{2ex}
\noindent
{\em Proof of Proposition~\ref{Phi_L}.}
In view of the preceding lemma and the criterion for multiplicity one
(Lemma~\ref{genericcrit} with $R=[L]$) the only thing remaining to be
checked is transversality of $A_i$ w.r.t.\ the evaluation map
\[
\Sigma\times\{f:\Sigma\rightarrow{\rm G}(k,n)\}\rightarrow{\rm G}(k,n).
\]
The moduli space $\{f\}$ of holomorphic curves in question is in our
case isomorphic to the flag manifold ${\rm F}(k-1,k+1;n)
=\{(U,W)\}$ by Lemma~\ref{minratcurve} and \ref{holinvariant}. But
$\Lambda_i=A_i\cap f(\Sigma)$ clearly varies when $f$ (and hence $(U,W)$)
varies as seen explicitely in the proof of Lemma~\ref{holinvariant}.
Note also that the Gromov boundary is empty in this case. This follows either
from the explicit description of the moduli space as flag manifold
or from the fact that $L\in H_2({\rm G}(k,n),{\Bbb Z})$ is primitive and thus can not
be represented by any reducible holomorphic curve.
{\hfill$\Diamond$}\vspace{1.5ex}
This finishes the proof of Theorem~\ref{qrGrass}.
\section{Higher invariants}
The main results of \cite{ruan/tian} show how to compute higher
GRW-invariants (i.e.\ with more than three entries or for higher genus
Riemann surfaces) from the genus $0$ three-point functions inductively.
Namely, for $g>0$
\[
{\tilde\Phi}^g_{[\omega]}(B_1,\ldots,B_s)=\sum_{i,j}\eta^{ij}
{\tilde\Phi}^{g-1}_{[\omega]}(B_1,\ldots,B_s,A_i,A_j),
\]
$(\eta_{ij})$ the intersection matrix with respect to a basis $\{A_i\}$
of $H^*(M,\cz)$. More invariantly, the right-hand side of course is the
trace with respect to $\eta$ of the bilinear form
\[
H^*(M,{\rm I \! R})\times H^*(M,{\rm I \! R})\longrightarrow{\rm I \! R},\ \ \ (B',B'')\longmapsto
{\tilde\Phi}^{g-1}_{[\omega]}(B_1,\ldots,B_s,B',B'').
\]
Secondly, for $g=0$ and $1<r<s-1$ (otherwise trivial) we have the
{\em composition law}
\[
{\tilde\Phi}^0_{[\omega]}(B_1,\ldots,B_s)=\sum_{i,j}\eta^{ij}
{\tilde\Phi}^0_{[\omega]}(B_1,\ldots,B_r,A_i)
{\tilde\Phi}^0_{[\omega]}(A_j,B_{r+1},\ldots,B_s),
\]
a trace with respect to $\eta$ as well (for $s=4$ and $r=2$ this
equation states the associativity of quantum products.) Our goal in
this section is to give a closed formula for higher invariants in terms
of the relations $f_1^{[\omega]},\ldots,f_k^{\omega]}$ of the quantum
cohomology ring. Putting all ${\tilde\Phi}^g_{[\omega]}$ together for
different $s$ we get a $\cz$-linear map
\[
\langle\ \ \rangle_g:\cz\langle X_1,\ldots X_n\rangle\longrightarrow\cz,
\ \ X_1^{\nu_1}\ldots X_n^{\nu_n}\longmapsto{\tilde\Phi}^g_{[\omega]}
(X_1^\vee,\ldots,X_1^\vee,\ldots,X_n^\vee,\ldots,X_n^\vee),
\]
the Poincar\'e-dual $X_i^\vee$ of $X_i$ occurring $\nu_i$-times. But
from the composition law denoting by $\alpha_i$ the Poincar\'e-dual of
$A_i$
\[
\hat X_1^{\nu_1}\wedge_{\mbox{\scriptsize Q}}\ldots\wedge_{\mbox{\scriptsize Q}}\hat X_n^{\nu_n}=
\sum_{ij}\eta^{ij}\,{\tilde\Phi}^0_{[\omega]}(X_1^\vee,\ldots,
X_1^\vee,\ldots,X_n^\vee,A_i)\,\hat\alpha_j,
\]
so $\langle X_1^{\nu_1}\ldots X_n^{\nu_n}\rangle_0={\tilde\Phi}^0_{[\omega]}
(X_1^\vee,\ldots,X_1^\vee,\ldots,X_n^\vee,[M])$ is nothing but the coefficient
of the class $[\Omega]$ of the normalized volume form in $\hat X_1^{\nu_1}
\wedge_{\mbox{\scriptsize Q}}\ldots\wedge_{\mbox{\scriptsize Q}}\hat X_n^{\nu_n}$. That is, $\langle\ \ \rangle_0$
decomposes
\[
\cz\langle X_1,\ldots,X_n\rangle\longrightarrow
\cz\langle X_1,\ldots,X_n\rangle/(f_1^{[\omega]},\ldots,f_k^{[\omega]})
\simeq H^*_{[\omega]}(M)\stackrel{{\rm top}}{\longrightarrow}
H^{2n}(M,\cz),
\]
and $H^{2n}(M,\cz)$ is identified with $\cz$ by sending $[\Omega]$ to
$1$. In case $n=k$ and the generators have even degree, i.e.
$H^*(M,\cz)$ a (commutative) complete intersection ring, one can use
higher dimensional residues to express this map more explicitely (we
refer the reader to \cite{griffharr} and \cite{tsikh} for
the general facts on residues to be used):
\sloppy
Recall that the residue of $F\in\cz[X_1,\ldots,X_k]$ with respect to
a polynomial mapping $g=(g_1,\ldots,g_k):\cz^k\rightarrow\cz^k$ with
$g^{-1}(0)$ finite (or equivalently, $\cz[X_1,\ldots,X_k]/(g_1,\ldots,g_k)$
is artinian, i.e.\ finite dimensional as vector space over $\cz$) in
$a\in g^{-1}(0)$ is defined by
\[
{\rm res}_g(a;F):=\frac{1}{(2\pi i)^k}\int_{\Gamma_a^\varepsilon}
\frac{F}{g_1\cdots g_k}dX_1\ldots dX_k,
\]
with $\Gamma_a^\varepsilon=\{x\in U(a)\mid |g_i(x)|=\varepsilon\}$, $U(a)$ a
neighbourhood of $a$ with $g^{-1}(0)\cap U(a)=\{a\}$ and $\varepsilon$ so
small that $\Gamma_a^\varepsilon$ lies relatively compact in $U(a)$.
$\Gamma_a^\varepsilon$ is smooth for almost all $\varepsilon$ by Sard's Theorem and has a
canonical orientation by the $k$-form $d(\arg g_1)\wedge\ldots\wedge
d(\arg g_k)|\Gamma_a^\varepsilon$. (This local residue of course makes sense
for holomorphic $g$ and $F\in{\cal O}_a$, but the polynomial case, to
which the general case may easily be reduced, is sufficient for our
purposes). We define the total residue
\[
{\rm Res}_g(F):=\sum_{a\in g^{-1}(0)}{\rm res}_g(a;F),
\]
which is also known as Grothendieck residue symbol
${F\choose g_1,\ldots,g_k}$ in the context of duality
theory in algebraic geometry \cite{hartshorne}. Let
$J=\det\left(\frac{\partial g_i}{ \partial X_j}\right)$ be the
Jacobian of $g$. Then for regular values $y$ of $g$
\[
{\rm Res}_{g-y}(F)=\sum_{x\in g^{-1}(y)}\left(\frac{F}{J}\right)(x)=
{\rm tr\skp}\left(\frac{F}{J}\right)(y).
\]
Therefore ${\rm tr\skp}(F/J)$ extends holomorphically (surprise!) to a neighbourhood
of $0$ (the extension will be denoted ${\rm tr\skp}(F/J)$ as well) and
\[
{\rm Res}_g(F)={\rm tr\skp}\left(\frac{F}{J}\right)(0).
\]
One abstract feature in our setting is that
we have weights $d_i$ associated to $X_i$ and that our relations
$f_i^{[\omega]}$ form a standard basis of the relation ideal with respect to
these weights, i.e.\
\[
({\rm In}\hspace{1pt} f_1^{[\omega]},\ldots,{\rm In}\hspace{1pt} f_k^{[\omega]})=
(f_1,\ldots,f_k)={\rm In}\hspace{1pt} (f_1^{[\omega]},\ldots,f_k^{[\omega]}),
\]
where ``${\rm In}\hspace{1pt} $'' means taking initial forms. This is trivial in our
case since we started with the homogeneous generators $f_1,\ldots,f_k$
of the relation ideal in a presentation of $H^*(M,\cz)$. In this
situation one can describe the residue map algebraically as follows:
\begin{prop}\label{residues}\sloppy
Let $R=\cz[X_1,\ldots,X_k]/(g_1,\ldots,g_k)$ be artinian such that $\{g_i\}$
is a standard basis of the relation ideal
with respect to weights $d_i$ of $X_i$. Put
$N:=\sum_i\deg g_i-\sum_i d_i$, $R_{<N}:=\left\{F\in\cz[X_1,\ldots,X_k]\
|\ \deg F<N\right\}/(g_1,\ldots,g_k)$
and $J=\det\left(\frac{\partial g_i}{\partial X_j}\right)$. Then
\[
R=R_{<N}\oplus\cz\cdot J,
\]
and the total residue map ${\rm Res}_g:\cz[X_1,\ldots,X_k]
\rightarrow\cz$ factorizes via the projection onto the second factor
as follows
\[
\cz[X_1,\ldots,X_k]\stackrel{{\rm can}}{\longrightarrow
\hspace{-4.1ex}\longrightarrow}R
\stackrel{{\rm pr}_2}{\longrightarrow}\cz\cdot J\longrightarrow\cz,
\]
where the last map sends $J$ to $\dim_{\,\rm I \!\!\! C} R$.
\end{prop}
{\em Proof. }
To check the normalization we observe ${\rm
Res}_g(J)={\rm tr\skp}_g\Big(1\Big)(0)=$ degree of $g$ over $0=\dim_{\,\rm I \!\!\! C} R$ (the
latter equality is generally true by flatness if the covering space is
Cohen-Macaulay \cite{fischer}). Next it is well known that the
residue vanishes on elements of $(g_1,\ldots,g_k)$. The claim thus
reduces to $\ker({\rm Res}_g)/(g_1,\ldots,g_k)=R_{<N}$.
If $\deg F<N$ then $F dX_1\ldots dX_k/g_1\cdots g_k$ extends to a
rational differential form $\varphi$ on weighted projective space
$V={\rm I \! P}_{(1,d_1,\ldots,d_k)}$ with polar divisor $D_1+\ldots+D_k\in{\rm
Div}(V)$, $D_i$ the natural extension of the divisor $(g_i)$ to $V$.
The point of course is that the divisor at infinity $V\setminus\cz^k$
is not a polar divisor of $\varphi$. We claim $|D_1|\cap
\ldots\cap|D_k|\subset\cz^k$. In fact, the restriction of the
homogenization of $g_i$ to $V\setminus\cz^k\simeq
{\rm I \! P}_{(d_1,\ldots,d_k)}$ is just ${\rm In}\hspace{1pt} g_i$ and $V({\rm In}\hspace{1pt} g_1,
\ldots,{\rm In}\hspace{1pt} g_k) =\{0\}\in\cz^k$. The latter follows because
otherwise $\dim{\rm Spec}\hspace{1pt} \cz[X_1,\ldots,X_k]/ ({\rm
In}\hspace{1pt} g_1,\ldots,{\rm In}\hspace{1pt} g_k)>0$ by homogeneity. But from the standard
basis property we have $\dim_{\,\rm I \!\!\! C}\cz[X_1,\ldots,X_k]/({\rm In}\hspace{1pt} g_1,\ldots,
{\rm In}\hspace{1pt} g_k)=\dim_{\,\rm I \!\!\! C}\cz[X_1,\ldots,X_k]/(g_1,\ldots,g_k)<\infty$. --- We
may thus desingularize $V$ (at infinity) without violating the
discreteness of $|D_1|\cap\ldots\cap|D_k|$ (we use the same notations
for the pulled-back objects). Now the global residue theorem tells that
on the compact manifold $V$ the sum of the local residues of $\varphi$ with
respect to $D_1,\ldots,D_k$ equals zero (the local residue ${\rm
res}_g(a;F)$ is coordinate free in so far that it depends only on
the associated rational differential form
$FdX_1\ldots dX_k/{g_1\ldots g_k}$ and the divisors $(g_1),
\ldots,(g_k)$). This proves ${\rm Res}_g(F)=0$ in case $\deg
F<N$.
The second case is $F$ homogeneous of degree $>N$. We show $F\in({\rm
In}\hspace{1pt} g_1, \ldots,{\rm In}\hspace{1pt} g_k)$. This is an easy generalization
of a theorem of Macaulay (cf.\ e.g.\ \cite{tsikh}) to the weighted
situation. Namely, for any $G$ (wlog.\ homogeneous), setting $P=F\cdot
G$, $Q={\rm In}\hspace{1pt} (g_1) \cdots{\rm In}\hspace{1pt} (g_k)$, we have
$\frac{P}{Q}dX_1\ldots dX_k =(\deg P-\deg Q+\sum_i
d_i)^{-1}d\sigma=(\deg P-N)^{-1}d\sigma$ with
\[
\sigma=\frac{P}{Q}\sum_{j=1}^k(-1)^{j-1}d_j\cdot X_j\,
dX_1\ldots\widehat{dX_j}\ldots dX_k,
\]
where $\widehat{\hspace{3ex}}$ means that this entry is to be left out.
This is a simple check using the weighted Euler formula $\displaystyle
\sum_j d_j X_j\frac{\partial H}{\partial X_j}=\deg(H)\cdot P$, $H$ weighted
homogeneous (same proof as usual). Thus
\[
{\rm Res}_{{\rm In}\hspace{1pt} g}(F\cdot G)=\sum_a\int_{\Gamma_a^\varepsilon}d\sigma=0
\]
for all (homogeneous) $G\in\cz[X_1,\ldots,X_k]$. But this implies $F\in
({\rm In}\hspace{1pt} g_1,\ldots,{\rm In}\hspace{1pt} g_k)$ (``duality theorem'', cf.\
\cite{tsikh}
--- this is Poincar\'e duality in our case!). Modulo $(g_1,\ldots,g_k)$
this means that we may reduce $F$ to lower degree. So proceeding by
induction $N$ turns out to be ``top-degree'' in $R$ in that all elements
of $R$ can be represented by polynomials of degree $\le N$.
What remains to be checked is that for $F$ homogeneous of degree $N$,
${\rm Res}_g(F)\neq0$ or $F/(g_1,\ldots,g_k)\in R_{<N}$. But in the
first instance ${\rm Res}_{{\rm In}\hspace{1pt} (g)}(F\cdot
G)=0\ \forall\ G\in\cz[X_1,\ldots,X_k]$ as shown above, and again
$F\in({\rm In}\hspace{1pt} g_1,\ldots,{\rm In}\hspace{1pt} g_k)$, i.e.\ modulo
$(g_1,\ldots,g_k)$, $F$ may be represented by a polynomial of degree
$<N$.
{\hfill$\Diamond$}\vspace{1.5ex}
\begin{rem}\rm
The decomposition $R=R_{<N}\oplus\langle J\rangle$ is $\em not$ canonical but
rather depends on the particular presentation of $R$. This may be seen either
in elementary terms from the trannsformation formula for residues or as
manifestation of the choice of an isomorphism ${\rm Ext}^k_V({\cal O}_Z,
\Omega_V^k)\simeq H^0(V,{\cal O}_Z)$ in the duality morphism
\[
{\rm Ext}^k_V({\cal O}_Z,\Omega_V^k)\times H^0(V,{\cal O}_Z)
\longrightarrow\cz
\]
induced by the global residue. For quantum cohomology rings and
weightings coming from cohomology, however, $R_{<N}=\oplus_{d<N}H^d(M,\cz)$,
$N=\dim_\cz M$ and $\langle Y\rangle=H^{2N}(M,\cz)$, so the decomposition
{\em has} an invariant meaning in this case.
{\hfill$\Diamond$}\vspace{1.5ex}
\end{rem}
In the quantum cohomology ring the top-degree class $\cz\cdot J$ is thus
spanned by the class $[\Omega]$ of the volume form. Let $F_{[\Omega]}$
be a polynomial of degree $N$ representing $[\Omega]$ (modulo
$(f_i^{[\omega]})$ or modulo $(f_i)$, this will yield the same result),
and set $c= 1/{\rm Res}\hspace{1pt} _{f^{[\omega]}}(F_{[\Omega]})$. By the
interpretation of $\langle F\rangle_0$ as coefficient of $[\Omega]$ of
$F(\hat X_1,\ldots, \hat X_k)\in H^*_{[\omega]}(M)$ we conclude (for
$H^*(M,\cz)$ a complete intersection)
\[
\langle F\rangle_0=c\cdot{\rm Res}_{f^{[\omega]}}(F)=
c\cdot{\rm tr\skp}_{f^{[\omega]}}\left(\frac{F}{J}\right)(0).
\]
To incorporate the higher genus case we prove
\begin{lemma}
Notations as in the proposition and $F\in\cz[X_1,\ldots,X_k]$ let
$B_F$ be the bilinear form $R\times R\rightarrow\cz$, $(\alpha,\beta)
\mapsto{\rm Res}_g(F\cdot G_\alpha\cdot G_\beta)$ (with $G_\alpha$,
$G_\beta\in\cz[X_1,\ldots,X_k]$ representing $\alpha$,
$\beta$ --- this is well-defined), $\eta=B_1$ (``intersection form'').
Then
\[
{\rm tr\skp}_\eta B_F={\rm Res}_g(F\cdot J).
\]
\end{lemma}
{\em Proof. }
We give a basis-free, algebraic proof (some might find the brute force
method by adapting a basis to $\eta$ more enlightening).
By definition ${\rm tr\skp}_\eta B_F$ is the trace of the endomorphism
$\mu_F:R\rightarrow R$ of multiplication by (the class of) $F$.
Putting $Z={\rm Spec}\hspace{1pt} R$ we have $R=\oplus_{z\in g^{-1}(0)}{\cal O}_{Z,z}$.
For $z\in g^{-1}(0)$, $F-F(z)$ is nilpotent in ${\cal O}_{Z,z}$
(for it has value $0$ in $z$), so $\mu_{F-F(z)}$ has trace $0$ and
\[
{\rm tr\skp}(\mu_F|{\cal O}_{Z,z})={\rm tr\skp}(\mu_{F-F(z)}|{\cal O}_{Z,z})
+{\rm tr\skp}(\mu_{F(z)}|{\cal O}_{Z,z})=F(z)\cdot\dim_{\,\rm I \!\!\! C}{\cal O}_{Z,z}.
\]
Furthermore, $\dim_{\,\rm I \!\!\! C}{\cal O}_{Z,z}=\deg_z(g)$, the local mapping degree
of $g$ at $z$, so summing up we get
\[
{\rm tr\skp}_\eta B_F=\sum_{z\in g^{-1}(0)}\deg_z(g)\cdot F(z)={\rm tr\skp}_g(F),
\]
which is nothing but ${\rm Res}_g(F\cdot J)$ as claimed.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{3ex}
In view of the reduction formula to lower genus stated above we conclude
a rigorous version of the ``handle gluing formula'', previously
established in QFT by Witten \cite{witten0}.
\begin{prop}
Let $H^*(M,\cz)=\cz[X_1,\ldots,X_k]/(f_1,\ldots,f_k)$ (the commutative
complete intersection case) and $f_i^{[\omega]}$ the induced
relations in $H^*_{[\omega]}(M)$ as in Theorem~\ref{presentation}. Put
$J=\det\Big(\frac{\partial f_i^{[\omega]}}{\partial X_j}\Big)$. Then
\[
\langle F\rangle_g=\langle J\cdot F\rangle_{g-1}
\]
holds for all $F\in\cz[X_1,\ldots,X_k]$.
{\hfill$\Diamond$}\vspace{1.5ex}
\end{prop}
In other words, multiplication by $J$ acts as ``attaching a handle''
to our Riemann surface. Our considerations so far yield the main
result of the present chapter:
\begin{theorem}\label{main4}
Assumptions as in the preceding proposition then for all $F\in\cz[X_1,\ldots,
X_k]$ the following holds
\[
\langle F\rangle_g=c\cdot{\rm Res}_{f^{[\omega]}}(J^g\cdot F)
=c\cdot \lim_{\stackrel{\scriptstyle y\rightarrow0}{
\stackrel{\scriptstyle y\mbox{\scriptsize\ regular}}{\scriptstyle
\mbox{\scriptsize value of }f^{[\omega]} } } }
{\rm tr\skp}_{f^{[\omega]}}\left(J^{g-1}\cdot F\right)(y),
\]
with $c=1/{\rm Res}_{f^{[\omega]}}(F_{[\Omega]})$, $F_{[\Omega]}$ a
polynomial representing the class $[\Omega]\inH^*_{[\omega]}(M)$ of the normalized
volume form.
{\hfill$\Diamond$}\vspace{1.5ex}
\end{theorem}
We emphasize that for $g>0$ or $0\in\cz^k$ a regular value of $f^{[\omega]}$
the right-hand side has the form $\sum_\nu a_\nu F(y_\nu)$ with
$g^{-1}(0)=\{y_\nu\}$ and constants $a_\nu$ {\em independent} of $F$!
Note also that $0$ is a regular value of $f^{[\omega]}$ iff $\dim_{\,\rm I \!\!\! C}
{\cal O}_{Z,z}=1$ for all $z\in(f^{[\omega]})^{-1}(0)$, $Z={\rm Spec}\hspace{1pt}
\cz[X_1,\ldots,X_k]/(f_1^{[\omega]},\ldots,f_k^{[\omega]})$, which is
if and only if
\[
\dim_{\,\rm I \!\!\! C} H^*(M,\cz)=\sharp\left(f^{[\omega]}\right)^{-1}(0)
\]
(``$\ge$'' always).
\vspace{3ex}
For the case of ${\rm G}(k,n)$ we have a basis of Schubert classes $\{\lambda_1,
\ldots,\lambda_k\}$ parametrized by sequences $n-k\ge\lambda_1\ge\ldots
\ge\lambda_k\ge0$. The latter are in \mbox{1--1} correspondence with subsets
$\{\lambda_k+1,\lambda_{k-1}+2,\ldots,\lambda_1+k\}$ of $\{1,\ldots,n\}$,
so $\dim_{\,\rm I \!\!\! C} H^*({\rm G}(k,n),\cz)={n\choose k}$.
To find $n \choose k$ distinct elements in
$\left(f^{[\omega]}\right)^{-1}(0)$ we use a description of the
cohomology ring coming from the study of the corresponding
Landau-Ginzburg model in physics \cite{vafa0}: There is a function $W$
(``Landau-Ginzburg potential'') s.th.\ in the notations of
Theorem~\ref{qrGrass}
\[
Y_{n+1-i}(X_1,\ldots,X_k)=\frac{\partial W}{\partial X_i},
\]
and then $\displaystyle f_i^{[\omega]}=\frac{\partial W^{[\omega]}}{\partial
X_i}$ with $W^{[\omega]}:=W+(-1)^ke^{-[\omega](L)}\cdot X_1$. $W$ has a
simple description in terms of Chern roots, i.e.\ after
composition with $\Sigma:\cz^k\rightarrow\cz^k$,
$(\underline{\lambda})=(\lambda_1,\ldots,\lambda_k)
\mapsto\Sigma(\underline{\lambda})=
(-\sigma_1(\underline{\lambda}),\sigma_2(\underline{\lambda})
\ldots,(-1)^k\sigma_k(\underline{\lambda}))$,
$\sigma_i$ the elementary symmetric polynomials:
$W\circ\Sigma=-\frac{1}{n+1}\sum_i \lambda_i^{n+1}$. Thus
\[
W^{[\omega]}\circ\Sigma=-\sum_{i=1}^k\left(\frac{\lambda_i^{n+1}}{n+1}
-(-1)^k e^{-[\omega](L)}\cdot\lambda_i\right).
\]
This all is an easy formal consequence of the algebraic relations
between Chern and Segre classes on one side and Chern roots on the
other, cf.\ \cite{bertram.etal} for a mathematical account. Now
$\displaystyle\frac{\partial W^{[\omega]}}{\partial X_i}\Big(\Sigma
(\underline\lambda)\Big)=0$, $i=1,\ldots,k$, if $\displaystyle\frac{\partial
W^{[\omega]}\circ\Sigma}{\partial\lambda_i}(\underline{\lambda})=0$ and if
$\Sigma$ is non-degenerate in $(\underline{\lambda})$. The latter
obviously is equivalent to $\lambda_i\neq\lambda_j$ for $i\neq j$.
Solving the second equation means
\[
\lambda_i^n=(-1)^k e^{-[\omega](L)},\ \ \ i=1,\ldots,k
\]
which has $n\cdot(n-1)\cdots(n-k+1)$ solutions with distinct
$\lambda_i$. The fiber of $\Sigma$ over the regular value
$\Sigma(\underline\lambda)$ consists of the $k!$ permutations of
$\{\lambda_1,\ldots,\lambda_k\}$, so we get precisely $n\choose k$
distinct elements of $(f^{[\omega]})^{-1}(0)$, as wanted.
\begin{theorem}\label{VafIntForm}
{\rm\bf (Formula of Vafa and Intriligator)}
(Notations as in Theorem~\ref{qrGrass}.) For any $[\omega]\in
H^{1,1}({\rm G}(k,n))$ the set $C^{[\omega]}$ of critical points of
$W^{[\omega]}$ is finite and all of these are non-degenerate.
Moreover, for any $F\in\cz[X_1,\ldots,X_k]$ the following formula
for the genus $g$ GRW-invariants of ${\rm G}(k,n)$ holds:
\[
\langle F\rangle^{[\omega]}_g=
(-1)^{n+k(k-1)/2}\sum_{\underline{x}\in C^{[\omega]}}
\det\left(\frac{\partial^2 W^{[\omega]}}{\partial X_i\partial X_j}\right)^{g-1}
\hspace{-3ex}(\underline{x})\cdot F(\underline{x}).
\]
\vspace{-5ex}
{\hfill$\Diamond$}\vspace{1.5ex}
\end{theorem}
{\em Proof. }
What is left is a check of the normalization. This amounts to calculate
the residue of $[\Omega]=(-1)^{n-k} X_k^{n-k}$ with respect to $f^{[\omega]}$.
We set $a=(-1)^k e^{[\omega](L)}$. Using standard properties of residues
one gets
\[
{\rm Res}_{f^{[\omega]}}\left((-1)^{n-k} X_k^{n-k}\right)
=\frac{(-1)^n}{k!}{\rm Res}_{{\underline\lambda}^n-\underline a}
\left(\prod_{i=1}^k\lambda_i^{n-k}\prod_{i<j}(\lambda_i-\lambda_j)^2\right),
\]
where ${\underline\lambda}^n-\underline
a=(\lambda_1^n-a,\ldots,\lambda_k^n-a)$.
Note also that $\prod_{i<j}(\lambda_i-\lambda_j)^2$ is the squared Jacobian of
$\Sigma$ and that $k!$ is the degree of $\Sigma$. Since terms of degree less
than $k(n-1)$ modulo $(\lambda_i^n-a)$ have vanishing residue
(Proposition~\ref{residues}) only the term $\prod_{i=1}^k\lambda_i^{k-1}$ from
the expansion of $\prod_{i<j}(\lambda_i-\lambda_j)^2$ contributes. The
coefficient
of this term is $(-1)^{k(k-1)/2}\cdot k!$ as one sees by writing $\prod_{i<j}
(\lambda_i-\lambda_j)^2$ as determinant of a product of a Vandermonde-matrix
with its transposed. What is left is a multiple of the Jacobian $n^k\prod
\lambda_i^{n-1}$ of ${\underline\lambda}^n-\underline a$, the residue of
which is known to be the degree $n^k$ of ${\underline\lambda}^n-\underline a$.
Putting everything together we get the claimed normalization.
{\hfill$\Diamond$}\vspace{1.5ex}
\vspace{3ex}
In concluding let us comment on the connection with the mathematical
formulation
of this formula given by Bertram, Daskalopoulos and Wentworth
\cite{bertram.etal}.
They showed that for any Riemann surface $\Sigma$ the moduli space
${\cal M}(d,\Sigma)$ (denoted ${\cal M}$ in the sequel) of holomorphic
maps $f:\Sigma\rightarrow{\rm G}(k,n)$, $f_*[\Sigma]=d\cdot[L]$ has the expected
dimension provided the degree $d$ is sufficiently large. Moreover, the
compactification of ${\cal M}$ as Grothendieck quot scheme $\overline
{\cal M}_Q(d,\Sigma)$ (denoted $\overline{\cal M}$ in the sequel), which parametrizes
sheaf quotients ${\cal O}_\Sigma^n\rightarrow{\cal F}\rightarrow0$ with fixed
Hilbert-polynomial, is generically reduced with irreducible reduction; the
universal quotient sheaf $\tilde{\cal F}$ on $\Sigma\times{\cal M}$
has a {\em locally free} kernel $\tilde{\cal E}$ extending ${\rm ev}^*
S$ as subsheaf of ${\cal O}^n_{\Sigma\times\overline{\cal M}}$, $S$ the
tautological bundle on ${\rm G}(k,n)$. In particular it makes sense to talk about
the Chern classes $c_i(\tilde{\cal E})$. Let $\iota:\{t\}\times\overline{\cal M}
\rightarrow\Sigma\times\overline{\cal M}$ be the inclusion for some $t\in\Sigma$.
Then for $i_1,\ldots,i_k$ with $\sum_\nu \nu i_\nu=\dim{\cal M}=k(n-k)(1-g)+dn$
they declared
\[
\langle X_1^{i_1}\cdots X_k^{i_k}\rangle_\Sigma
:=\left(\iota^*c_1(\tilde{\cal E})^{\wedge i_1}\wedge\ldots\wedge
\iota^*c_k(\tilde{\cal E})^{\wedge i_k}\right)[\,\overline{\cal M}\,].
\]
Presumably this invariant does not in general coincide with the
GRW-invariant: As we tried to explain in Remark~\ref{algmod} the
GRW-invariants are computed from the Chern classes of $\tilde{\rm
ev}^*(S)$, where $\tilde {\rm ev}:\Gamma\rightarrow{\rm G}(k,n)$ is an
extension of ${\rm ev}:\Sigma\times {\cal M}\rightarrow{\rm G}(k,n)$ to some
blow-up $\pi:\Gamma\rightarrow\Sigma\times\overline{\cal M}$. The trouble
is that $\pi^*\tilde{\cal E}$, though a locally free subsheaf of
$\tilde{\rm ev}^*S$ of the same rank $k$, it need not coincide with the
latter. For instance this always happens if $\tilde{\cal F}$ has torsion
since then ${\cal O}_\Gamma^n/\pi^*\tilde {\cal E}\simeq\pi^*\tilde{\cal F}$ has
torsion as well. Additional contributions thus come precisely from the
first $k$ Chern classes of the torsion sheaf $\tilde{\rm
ev}^*{\cal O}(S)/\pi^*\tilde{\cal E}$ (one should desingularize $\Gamma$ to
make sense of these Chern classes). A general vanishing theorem for
these seems to be unlikely.
|
1994-07-15T18:34:23 | 9403 | alg-geom/9403012 | en | https://arxiv.org/abs/alg-geom/9403012 | [
"alg-geom",
"math.AG"
] | alg-geom/9403012 | Alexander A. Borisov | A. Borisov | Minimal discrepancies of toric singularities | 11 pages, Latex v2.09 | null | null | null | null | This is the major revision. The main purpose of this paper is to prove that
minimal discrepancies of $n$-dimensional toric singularities can accumulate
only from above and only to minimal discrepancies of toric singularities of
dimension less than $n$. I also prove that some lower-dimensional minimal
discrepancies do appear as such limit.
| [
{
"version": "v1",
"created": "Tue, 15 Mar 1994 15:13:30 GMT"
},
{
"version": "v2",
"created": "Fri, 15 Jul 1994 16:31:50 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Borisov",
"A.",
""
]
] | alg-geom | \section{Introduction}
First of all let me stress that minimal discrepancy of a variety $X$ I am
talking about is defined only for singular varieties and is the minimum of
discrepancies of exceptional divisors in all resolutions of singularities of
$X$. I don't allow to blow up anything in a smooth part of $X$ otherwise the
minimal discrepancy would always be no more than $1$ and this means loss of
information about $X.$
Now let me recall the following conjecture proposed by V. Shokurov.(\cite{Sh})
\begin{Conjecture}
For every natural $n$ minimal discrepancies of $n$-dimensional log-terminal
singularities can accumulate only from above.
\end{Conjecture}
In particular this conjecture implies that for every $n$ there exists a
positive constant $\epsilon (n),$ such that if all discrepancies of
$n$-dimensional variety $X$ are greater than $-\epsilon (n)$ then they are in
fact nonnegative, that is $X$ has at most canonical singularities.
In this paper the above conjecture will be proven for a particular case of
toric singularities. By the simple observation (see Lemma $2.1$) the problem
can be reduced to the case of cyclic quotient singularities for which some
results are obtained. They are in some sense best possible for nonterminal
singularities and also very informative for terminal case. What to do in case
of more general singularities is discussed in section $3.$ The main result of
the paper is the following. (Corollary 2.1)
{\bf Main Result. For every natural $n$ minimal discrepancies of
$n$-dimensional toric singularities can accumulate only from above and only
to minimal discrepancies of toric singularities of dimension less than $n.$}
\begin{Remark}
It can happen that infinitely many different toric singularities have the same
minimal discrepancy. I do not consider this as an accumulation of minimal
discrepancies. In this case the minimal discrepancy in question does not
necessarily come from lower dimension. It may be of the form (1+(minimal
discrepancy of lower-dimensional toric singularity.) However the only example
of that kind I know is rather trivial case of minimal discrepancy $n-1$ for
$2n-$dimensional singularities.
\end{Remark}
There is also the result in the opposite direction (see Theorem $2.2$) which
implies in particular that every minimal discrepancy of a toric singularity of
dimension $k$ is a limit of minimal discrepancies of $n-$dimensional toric
singularities for $n$ big enough. And it also implies that if this discrepancy
is non-positive then the only restriction on $n$ is that $n>k.$
I am glad to thank here V. Shokurov for his constant interest in this study and
helpful remarks about it.
\section{Proofs}
First of all, let me recall some basic facts about toric varieties. (See, for
example, \cite {BB}, \cite {D}, \cite {R}.) Every $n$-dimensional affine toric
variety $X$ is just a $Spec(R),$ where R is a ring generated by monomials
$x_1^{\alpha _1} x_2^{\alpha _2}...x_n^{\alpha _n},$ where $\{\alpha _1,\alpha
_2,...,\alpha _n\}$ is an integral point of some finitely generated convex cone
of full dimension $C(X)$ in $R^n.$ For several reasons it is more useful to
consider the dual cone $C^*(X)$ in the dual space $V=R^n.$ It does not
necessarily have dimension $n,$ but we will always assume that it will.
Otherwise $X$ would be isomorphic (not canonically) to a product of another
toric variety and an algebraic torus of positive dimension.
Now various conditions on singularities of $X$ have simple combinatorial
formulation in terms of $C^*(X).$ Namely, let us consider one-dimensional
extremal rays $l_1,l_2,...,l_k$ that generate this cone. They are rational
therefore we can pick on every $l_i$ an integral point $P_i$ which is the
closest one to zero. Then all $P_i$ lie in one hyper-plane if and only if $X$
is Q-Gorensteinian. $X$ is Q-factorial if and only if $k=n,$ that is $C^*$ (or
$C$) is simplicial. $X$ is regular if and only if $C^*(X)$ is regular which
means that it is simplicial and $P_i$ form a basis for the lattice. Moreover,
the Gorenstein index and minimal discrepancy of $X$ also have simple
description. Namely, let us consider the linear function $F$ on $V,$ such that
$F(P_i)=1.$ (This is possible exactly when $X$ is Q-Gorensteinian.) Then the
least common denominator of values of $F$ on non-zero points of $C^*$ is the
index. The minimal log-discrepancy which is by definition (1+(minimal
discrepanc!
y)) is the minimum of the above values among points in the interior of
non-regular sub-cones of $C^*(X).$ (If all sub-cones including $C^*(X)$ itself
are regular then $X$ is regular and its minimal discrepancy is undefined.) We
will pass freely from discrepancies to log-discrepancies mostly using the
latter in proofs and the former in statements.
There is one type of toric singularities which is particularly interesting for
our purposes. Namely, quotients of an affine plane $A^n$ by cyclic groups. This
corresponds to the case when $C^*$ is simplicial and the lattice $N$ of all
integral points in $V$ is generated by $P_i$ and only one extra element $x.$ We
can always assume that $x$ lies in the interior of $C^*$ otherwise the
singularity splits into the lower-dimensional singularity and torus. I want to
mention that the cyclic group in question is just a factor ($N/<P_i>$) and its
action can be reconstructed from the coordinates of $x$ in the basis $\{P_i\}.$
The following lemma reduces everything to this special case.
\begin{Lemma} The set of minimal discrepancies of toric singularities of
dimension $n$ coincides with that of cyclic quotients of dimension no greater
than $n.$
\end{Lemma}
{\bf Proof} Let $X$, $C^*\subset V$, $P_i$, $F$ be as above. Let $\epsilon$ be
the minimal log-discrepancy of $X.$ By the above combinatorial description
there exists an integral point $x\in C^*,$ such that $F(x)=\epsilon .$ Consider
the ray generated by $x.$ It intersects the polygon $P_1P_2...P_k$ in some
point $P.$ Standard combinatorial arguments show that there exists a simplex
$P_{i_1}P_{i_2}...P_{i_r}\subset P_1P_2...P_k,$ such that $P$ lies in its
interior. Of course, this simplex has dimension no greater than $n-1$ and its
interior means interior with respect to its own geometry.
Now let us stick to the subspace $W$ of $V$ generated by $P_{i_j}.$ Evidently,
$C^*\bigcap W$ will be a convex cone corresponding to some new toric variety
$X^\prime$ with the same minimal log-discrepancy $\epsilon .$ This $X^\prime$
is already Q-factorial but it is not a cyclic quotient yet. To produce out of
it a cyclic quotient let me do the following. Consider lattice $N^\prime
\subset N$ generated by $x,$ change coordinates in such a way that $N^\prime$
become a lattice of integral points and forget about $N.$ What we have now is a
cyclic quotient $X^{\prime \prime},$ which again has the same minimal
log-discrepancy $\epsilon,$ so the lemma is proven.
{\bf Remark} As one can easily see from the proof of the above lemma we can
assume that factor-group ($N/<P_i>$) is not only cyclic but also generated by
the element $x,$ which has "minimal log-discrepancy" (that is $F(x)=\epsilon
.$)
In the rest of the paper this element will be often called generating element.
The fact that it is not uniquely determined for a given singularity does not
cause any difficulties.
{}From now on we stick to this particular case of quotient singularities and
whenever we have a toric variety it is a cyclic quotient singularity. The above
lemma allows us to do it. Now let me notice that the results we are going to
prove are of two types. Most of them are negative in a sense that they restrict
where and how minimal discrepancies can accumulate. And there are some positive
results based on procedures that allow us to construct cyclic quotients with
prescribed minimal discrepancies starting with the given one. We begin with
negative results which all deal with the following situation.
Suppose we have a sequence of cyclic quotient singularities $\{X^\nu\}, \nu
=1,2,...,$ such that their log-discrepancies $\epsilon ^\nu$ are getting closer
and closer to some real number $\epsilon. $ Consider the standard simplex
$\Delta$ in $R^n$ defined by the inequalities $\alpha_i \ge 0$, $\sum
{\alpha_i} \le 1$ inside a standard hypercube $H$, defined by the inequalities
$0\le \alpha_i \le 1.$ By identifying simplexes $P_1^\nu P_2^\nu ...P_k^\nu $
with this standard one we obtain a sequence of points $\alpha^\nu \in H$ that
correspond to $x^\nu .$ By the compactness of $H$ there exists a subsequence
with a limit point $\alpha.$ We replace our sequence by this subsequence. By
the above remark $\epsilon ^\nu = \sum {\alpha_i^\nu}.$ Therefore $\epsilon =
\sum {\alpha_i}.$
Now all our negative results can be formulated as the sequence of statements
which will be proven in a row. Let me state this as a theorem.
\begin{Theorem} In the above notations the following is true.
Content-Length: 17452
$1)$ If $\epsilon ^\nu$ are not the same for big $\nu$ then $\alpha$ is on the
boundary of $H.$
$2)$ If $\epsilon ^\nu$ are the same for big $\nu$ then one can choose $\alpha$
on the boundary of $H$ which has the same $\epsilon$ and is also a limit of
some sequence of the same type.
$3)$ $\epsilon$ is rational.
$4)$ If $\epsilon^\nu$ are not the same for big $\nu$ they accumulate to
$\epsilon$ only from above.
$5)$ We can choose $\alpha$ as in $2)$ on some face of $H$ to be a generating
point of a cyclic quotient singularity if considered on this face. As a
corollary, $\epsilon$ is a minimal log-discrepancy for some lower-dimensional
toric singularity plus some nonnegative integer.
$6)$ Every face of $H$ is characterized by restricting some coordinates to be
$0$ and some coordinates to be $1.$ Under this remark statement 5 can be
strengthen by the restriction that for the face of $\alpha$ the number of $1-$s
is not greater than the number of $0-$s. Moreover if $\epsilon ^\nu$ are not
the same for big $\nu$ then the number of $1-$s is strictly less than the
number of $0-$s
\end{Theorem}
In order to prove this theorem let me introduce the notion of multiple of the
point in $H.$ It will be used a lot in the rest of the paper so it deserves to
be stated formally.
\begin{Definition}
For every point $\alpha =(\alpha_i)\in H$ and integer $m$ let $m-$th multiple
of $\alpha$ be the point $\alpha^{(m)}$ whose $i-th$ coordinate is $1$ if
$\alpha_i=1$ and $\{m\alpha_i\}$ otherwise. Note that for positive $m$ this
construction is continuous at the neighborhood of the boundary of $H.$
\end{Definition}
Now we begin the proof.
{\bf Statement 1).} Suppose $\alpha$ is in the interior of $H.$ Consider
$\alpha^{(m)}$ for all integer $m.$ Then the compactness of $H$ tells us that
there are two numbers $m_1 < m_2,$ such that $\alpha^{(m_i)}$ are very close,
for example closer than $\frac1{100} \times$ (distance from $\alpha$ to the
boundary of H). They may also coincide, we don't care. Then
$\alpha^{(m_1-m_2+1)}$ and $\alpha^{(m_2-m_1+1)}$ are evidently very close to
$\alpha^{(1)} = \alpha.$ We have several cases.
First of all suppose that sum of coordinates of one of the above two points is
less than $\epsilon$ (that means that sums of coordinates of
$\alpha^{(m_1-m_2+1)}$ and $\alpha^{(m_2-m_1+1)}$ are different.) Let it be
$\alpha^{(m_1-m_2+1)}.$ Then for $\nu$ big enough $\alpha^\nu$ is close enough
to $\alpha$ and $\alpha^{\nu,(m_1-m_2+1)}$ is close enough to
$\alpha^{(m_1-m_2+1)}$ and therefore the sum of coordinates of
$\alpha^{\nu,(m_1-m_2+1)}$ is less than $\epsilon^\nu,$ which is impossible.
Now suppose that sums of coordinates are the same. Then if there is a
subsequence of $\alpha^\nu$ for which $\epsilon^\nu$ accumulate to $\epsilon$
from above consider $(m_1-m_2+1)-$th multiples. Then for $\nu$ big enough from
this subsequence sum of the coordinates of $\alpha^{\nu,(m_1-m_2+1)}$ is less
then sum of the coordinates of $\alpha^{(m_1-m_2+1)},$ because $(m_1-m_2+1)
<0.$ Therefore it is less then $\epsilon^\nu,$ which is impossible. Similar
arguments work for the case when $\epsilon^\nu$ accumulate to $\epsilon$ from
below. We should just consider $(m_2-m_1+1)-$th multiples instead of
$(m_1-m_2+1)-$th ones and notice that $(m_2-m_1+1) \ge 2.$
\begin{Remark}
We did not prove that point inside $H$ cannot be a limit of generating points
of cyclic quotients with the {\bf same} discrepancy. And this indeed can
happen. The easiest example is given by two-dimensional canonical toric
singularities.
\end{Remark}
{\bf Statement 2).} Suppose we have a sequence of points $\alpha^\nu$ with the
same sum of coordinates $\epsilon.$ Consider those multiples of all these
points that have the same sum of coordinates $\epsilon.$ We will see very soon
that there are plenty of them. We have two cases.
First of all, suppose $\alpha$ has finite order in $H$ that is $\alpha^{(k)} =
0$ for some $k.$ Then for $\alpha^\nu$ that are close enough to $\alpha$
$(mk+1)-$multiples have sum of the coordinates $\epsilon$ Moreover when we make
$m$ run from zero to some number depending on $\nu$ they run following some
straight line with small intervals until they hit the boundary of $H.$ The
length of these intervals goes to zero when $\alpha^\nu$ go to $\alpha.$
Therefore we have infinitely many points in every neighborhood of the boundary
of $H,$ intersected with a hyper-plane $\sum{x_i} =\epsilon.$ Therefore there
exists a point on this boundary which is a limit of some sequence of these
points. To complete the argument it is enough to mention that each one of these
points is also a generating point for some quotient singularity with the same
discrepancy $\epsilon.$
Now suppose $\alpha$ has infinite order in $H.$ Nevertheless $\epsilon$ is
rational because $\epsilon =\epsilon^\nu .$ So we have infinitely many
multiples of $\alpha$ with the same sum of coordinates $\epsilon.$ Then
arguments similar to that of the above case show that whenever two of such
multiples are close to each other there is some other multiple that is close to
the boundary of $H.$ Again as before, there is a point on the boundary which is
a limit of a sequence of these multiples. Now we can just notice that every
multiple of $\alpha$ is a limit of multiples of $\alpha^\nu$ and the rest is
the same as above.
{\bf Statement 3).} Suppose $\epsilon$ is irrational. By previous statements we
can assume that $\alpha$ is on some face of $H.$ Then all its multiples are by
the definition on the same face. Now we want to prove that for some $m>0$
$\alpha ^{(m)}$ is close enough to $\alpha$ and sum of the coordinates of
$\alpha ^{(m)}$ is less than $\epsilon .$ This is not completely trivial,
because we require $m$ to be positive. Here is the proof. First of all, we can
stick to the face of $H$ $\alpha$ belongs to. Then we notice that all sums of
coordinates of $\alpha ^{(m)}$ are different because $\epsilon$ is irrational.
By the compactness argument there exist some positive integers $m_1<m_2$ such
that $\alpha ^{(m_1)}$ and $\alpha ^{(m_2)}$ are close enough. If the sum of
coordinates of $\alpha ^{(m_1)}$ is greater than the sum of coordinates of
$\alpha ^{(m_2)}$ it is enough to choose $m$ to be equal to $1+m_2-m_1.$
Otherwise we need one more step. Denote $m_3=1+m_1-m_2.$ Then everyth!
ing would have been OK, but $m_3$ is not positive. But we can find $m_4 < m_5$
of form $l(m_3+1),$ such that $\alpha ^{(m_4)}$ and $\alpha ^{(m_5)}$ are so
close that $\alpha ^{(m_3+m_5-m_4)}$ is still close enough to $\alpha$ and the
sum of its coordinates is still less than the sum of coordinates of $\alpha.$
Now for $m$ as above and $\nu$ big enough $\alpha ^\nu$ is close enough to
$\alpha$ therefore $\alpha ^{\nu ,(m)}$ is close enough to $\alpha ^{(m)}.$
(Here we really need that $m$ is positive because $\alpha$ lies on the boundary
of $H.$) But this means that for $\nu$ big enough sum of the coordinates of
$\alpha ^{\nu ,(m)}$ is less than sum of the coordinates of $\alpha ^\nu,$
which is impossible.
{\bf Statement 4).} Now $\epsilon$ is rational. The arguments similar to the
above allow us to find an integer $m>1$ such that $\alpha ^{(m)}$ is close to
$\alpha $ and has the same sum of coordinates. Namely, the compactness argument
tells that there are $m_1,m_2$ such that $\alpha ^{\nu ,(m_1)}$ and $\alpha
^{\nu ,(m_2)}$ are arbitrary close. (They may even coincide, we don't care.)
Then $m=1+m_2-m_1$ will satisfy all requirements.
Now if discrepancies $\epsilon ^\nu$ accumulate to $\epsilon$ from below then
for sufficiently large $\nu$ the sum of coordinates of $\alpha ^{\nu ,(m)}$ is
less than the sum of coordinates of $\alpha ^\nu.$ (By definition $m$ is
greater than 1 and $(\alpha ^{\nu ,(m)}-\alpha ^{(m)})=m(\alpha ^\nu -\alpha
).$) This completes the proof of the statement.
{\bf Statement 5).} We have $\alpha$ on some face of $H.$ Let us consider this
face and multiples of $\alpha$ on it. If there are infinitely many of them that
are in fact different then there are infinitely many of them with the same sum
of the coordinates $\epsilon,$ because $\epsilon$ is rational. Then arguments
of the proof of statement $2$ allow us to replace $\alpha$ so that it lie on a
face of lower dimension. We can do this until we come to $\alpha$ that has
finite order in the appropriate face. Now on this face $\alpha$ is a generating
point for a quotient singularity, because if some multiple of it has smaller
sum of coordinates in the face it has smaller sum of all coordinates and usual
arguments show that it is impossible.
{\bf Statement 6).} Suppose $\alpha$ has order $N$ in its face. Suppose this
face is determined by $k$ equalities of type $x_i =0$ and $l$ equalities of
type $x_i =1$ Consider $(1-N)-$th multiples of $\alpha ^\nu .$ Then the
corresponding $\epsilon-$s go to $\epsilon +k-l$ when $\alpha ^\nu$ go to
$\alpha.$ Therefore $k \ge l.$ Moreover, if $\epsilon ^\nu$ accumulate to
$\epsilon$ from above then $\epsilon-$s for $(1-N)-th$ multiples accumulate
from below. So case $k=l$ is also impossible.
This completes the proof of the theorem. The following corollary is formally
also restrictive but in fact as you can see from its proof it is a positive
result (or, more precisely, simple observation.)
\begin{Corollary}
Under the notations of the above theorem if $\epsilon ^\nu$ are not the sam for
big $\nu$ then $\epsilon$ is not just sum of lower-dimensional log-discrepancy
and integer but is a lower-dimensional log-discrepancy itself. If $\epsilon
^\nu$ are the same for big $\nu$ then $\epsilon$ is either a lower-dimensional
log-discrepancy or (1+(minimal log-discrepancy of dimension $\le (n-2)$)).
\end{Corollary}
{\bf Proof} This is a straightforward consequence of statement $6$ and the
following fact.
{\bf Fact} For arbitrary $m-$dimensional cyclic quotient one can construct
$(m+2)-$dimensional cyclic quotient whose minimal log-discrepancy is greater
than given exactly by $1.$
Construction that proves the above fact is as follows. Suppose the generating
point $\alpha$ has order $N.$ Then we construct new $(m+2)-$dimensional
singularity defined by the generating point $(\alpha ,\frac1N, 1-\frac1N ) $
which means that first $m$ coordinates remain the same and last two are as
specified.
\begin{Remark}
The main idea of the proof of the above theorem (namely use of multiples,
compactness and some sort of continuity) is very similar to that of the
boundedness theorem for toric Fano varieties with bounded discrepancies. (\cite
{BB}) However there everything is written using less geometrical language that
maybe hide this idea in formulas. Probably it can also be written using the
language of this paper, I just don't know anybody who really tried. I can do it
for canonical singularities, but not in general case.
\end{Remark}
Now let me state the most general positive result I know about what
lower-dimensional discrepancies can indeed appear as a limit. But before doing
this I would like to notice that by the evident reason of symmetry every
minimal log-discrepancy of cyclic quotient of dimension $n$ is no greater than
$\frac n2 .$
\begin{Theorem} Suppose we have an $m-$dimensional cyclic quotient generated by
$\alpha$ with minimal discrepancy $\epsilon.$ Denote $r=-[-\epsilon]$ so that
$r$ is the smallest real number which is greater or equal than $\epsilon .$
Then for all nonnegative integers $l$, $(\epsilon +l)$ is a limit of
$n-$dimensional log-discrepancies for all $n \ge m+r+2l.$
\end{Theorem}
{\bf Proof} There are in fact several ways of doing this for nonzero $l.$ The
freedom we have is basically due to the Fact in the proof of the above
corollary. I will show you just one way.
First of all let me consider the standard $n-$dimensional hypercube $H$ and
divide the set of coordinates $\{x_1, x_2 ... x_n\}$ into three parts as
follows. First $m$ of them will correspond to the coordinates of our given
$m-$dimensional singularity and will be still called $x_i$. Those with indexes
from $m+1$ to $m+l$ will be called $y_i, i=1,...,l.$ And those with indexes
from $m+1+l$ to $n$ will be called $z_i, i=1,...,n-m-l.$ Now we place our
$m-$dimensional singularity on the face of $H$ defined by equalities
$y_i=1,z_i=0.$ Let us denote by $T$ the point $(\alpha;1,...,1;0,...,0)$ that
corresponds to the generating point $\alpha.$ Suppose its order is $q$ that is
its $q-$th multiple is a vertex of $H.$ Consider vertex
$P=(0,...,0;0,...,0;1,..,1).$ (Here ";" divides $x_i$, $y_i$ and $z_i.$) Now
the generating points of $n-$dimensional singularities we are looking for lie
on the segment $PT$ close to $T.$ More precisely, they are given by formula
$\frac1NP+(1-\frac1N)T)!
$ where $q|(N-1).$ In order to complete the proof it is enough to show that for
all such points every multiple is either trivial or has greater or equal sum of
coordinates than the point itself.
So, consider the point $A=A_N$ defined as above. It is a straightforward
observation that it has order $N.$ Now arguments similar to those from the
proof of the statement $2$ of Theorem $2.1$ show that for every positive
integer $k<N$ $k$-th multiple of $A$ lies in the set $S_k$ defined by the
following procedure.
{\bf Procedure.}
Suppose $T_k$ is a $k-th$ multiple of $T.$ Draw a raw starting from $T_k$ and
parallel to the raw $[TP).$ When it hits the boundary of $H$ change the
corresponding $1-$s to $0-$s and $0-$s to $1-$s. Do it until the sum of lengths
of all segments drawn equals the length of $PT.$
So it is enough to prove that no point from this set can have greater sum of
coordinates than that of $A.$ In order to do it let me make several simple
observations. First of all let me notice that in fact when we draw our segments
we $\bf never$ change $1$ to $0$, we only change $O-$s to $1-$s. The reason is
that in all $x_i$ we draw in the negative direction and we cannot hit the
boundary on $y$ or $z$ for $k<N.$ Another observation is that "locally", that
is when we don't hit boundary, the sum of coordinates does not decrease because
sum of coordinates of $P$ is greater or equal than that of $T$ by the condition
$n \ge m+r+2l.$ Combined together these two observations evidently take care of
$k$ which are not divisible by $q.$ For $k$ divisible by $q$ we only need to
notice that we begin our procedure from the point $(0,...,0;0,...,0;0,...,0)$
but the first nontrivial segment starts from the point
$(1,...,1;0,...,0;0,...,0)$ whose sum of coordinates is greater than that !
of $A.$
\section{Some open questions}
There are several natural questions concerning the obtained results.
{\bf Question 1.} Is it true that EVERY minimal log-discrepancy of
$n-$dimensional cyclic quotient is a limit of minimal log-discrepancies of
$(n+1)-$dimensional cyclic quotients? The theorem above together with the
classification of $3-$dimensional terminal toric singularities implies that
this is true for $n\le 3.$ It is natural to try, maybe with computer, the case
$n=4.$ As far as I know the classification of $4-$dimensional toric terminal
singularities is not yet completed but there are some conjectures and a lot of
work is already done. See \cite{MMM} for details.
Other questions naturally arose when I tried to extend these results to more
general singularities.
{\bf Question 2.} Is it true that the set of minimal discrepancies of quotient
singularities with respect to arbitrary groups coincides with that of cyclic
quotients of the same dimension?
{\bf Question 3.} Is there an example of log-terminal singularity whose minimal
discrepancy is not a minimal discrepancy for any cyclic quotient of the same
dimension?
{\bf Question 4.} Is it true for arbitrary log-terminal singularities which are
not terminal that every (or at least one) divisorial valuation that corresponds
to the minimal discrepancy is given by a divisor on the Q-factorial terminal
modification in sense of Clemens-Koll\'ar-Mori? (Of course it is not true for
all valuations with negative discrepancy, but the question is about minimal
discrepancy.) It is true for toric singularities and in dimension $2$ and I
have no counterexamples in general case.
While stating these questions it would be unfair not to express my opinion
about them. I suspect that the answer to Question $1$ is "Yes" for many
singularities but not for all of them. The answer to Question $2$ is probably
"Yes". Example to Question $3$ probably also exists maybe even $3-$dimensional.
And I have been unable so far to find any evidence pro or against to Question
$4.$
|
1994-03-28T22:50:16 | 9403 | alg-geom/9403016 | fr | https://arxiv.org/abs/alg-geom/9403016 | [
"alg-geom",
"math.AG"
] | alg-geom/9403016 | Christoph Sorger | Christoph Sorger | Groupe de Picard de la vari\'et\e de modules des
th\^eta-caracteristiques des courbes planes | 48 pages, amslatex v1.1, french | null | null | null | null | Nous calculons le groupe de Picard de l'espace des modules
$\Theta_{\planp}(d)$ des th\^eta-caract\'eristiques des courbes planes (non
forc\'ement lisses) de degr\'e $d$. Nous montrons que pour $d\ge 6$, le groupe
de Picard de la composante paire est engendr\'e par le fibr\'e d\'eterminant
${\cal{D}}$ et l'image r\'eciproque ${\cal{L}}$, sous le morphisme support
sch\'ematique, de ${\cal{O}}_{\proj}(1)$ o\`u $\proj$ est l'espace des courbes
de degr\'e $d$ sur $\planp$. Pour $1\le d \le 4$, ce groupe est engendr\'e
uniquement par ${\cal{L}}$, le fibr\'e d\'eterminant \'etant trivial dans ces
cas. Dans le groupe $\Cl$ des diviseurs de Weil, le fibr\'e d\'eterminant admet
une racine: le fibr\'e pfaffien ${\cal{P}}$. De plus, pour $d\geq 4$ pair,
${\cal{L}}$ aussi admet une racine dans $\Cl$. Si $d=5$, le fibr\'e pfaffien
s'\'etend en un fibr\'e inversible et le groupe de Picard est engendr\'e par
${\cal{L}}$ et ${\cal{P}}$. On verra ainsi que la composante paire de
$\Theta_{\planp}(d)$ est localement factoriel si et seulement si $d=1,2,3$ ou
$d=5$. On obtient un r\'esultat analogue pour la composante impaire. Ensuite,
nous \'etudions la question d'existence d'une famille universelle sur un ouvert
de l'ouvert $U$ des th\^eta-caract\'eristiques dont le faisceau sous-jacent est
stable. Nous montrons que pour $d$ pair une telle famille ne peut exister,
tandis que pour $d$ impair une telle famille existe, pas sur $U$ entier, mais
localement dans la topologie de Zariski.
| [
{
"version": "v1",
"created": "Mon, 28 Mar 1994 20:49:01 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sorger",
"Christoph",
""
]
] | alg-geom | \section{Introduction}
Le pr\'esent travail poursuit l'\'etude de l'espace des modules de \cite{11}
des
th\^eta-caract\'eristiques des courbes (non forc\'ement lisses) trac\'ees sur
une surface
dans le cas du plan projectif. Nous calculons le groupe de Picard
des composantes irr\'eductibles et \'etudions les questions de
factorialit\'e locale et de l'existence d'une famille universelle.
Rappelons qu'on appelle th\^eta-caract\'eristique (g\'en\'eralis\'ee) d'une
courbe plane un ${\cal{O}}_{\planp}$-module ${\cal{F}}$ de
dimension $1$ muni d'une identification sym\'etrique $\sigma$
avec son
$\omega_{_{\planp}}$-dual. La courbe associ\'ee \`a $({\cal{F}},\sigma)$
est d\'efinie par le support sch\'ematique de ${\cal{F}}$, \ie le
$0$-\`eme id\'eal de Fitting de ${\cal{F}}$. Le degr\'e de cette courbe
s'appelle multiplicit\'e de ${\cal{F}}$. D'apr\`es $\cite{11}$,
il existe un espace de modules grossier $\Theta_{\planp}(d)$ pour
les th\^eta-caract\'eristiques semi-stables (\cf \ref{theta-def} pour
la d\'efinition) de multiplicit\'e $d$.
C'est une vari\'et\'e normale et projective de dimension
$\frac{d(d+3)}{2}$, muni d'un morphisme
$$\sigma_{_{\Theta}}:\Theta_{\planp}(d)\lra\proj^{M}$$ avec
$M=\frac{d(d+3)}{2}$, d\'efini
en associant \`a la classe d'une th\^eta-caract\'eristique semi-stable
son support sch\'ematique. Au-dessus de l'ouvert des
courbes lisses, $\sigma_{_{\Theta}}$ est \'etale de degr\'e $2^{2g}$,
la fibre au dessus d'une courbe lisse $C$ s'identifiant \`a l'ensemble
de th\^eta-caract\'eristiques classiques sur $C$.
Par contre, ce morphisme n'est pas quasi-fini: si $C$ est int\`egre
la fibre est finie si et seulement si $C$ n'a que des singularit\'es
simples (\cf Cook dans le m\^eme volume); si $C$ est \`a structure
multiple d'une courbe lisse (\ie $C=rC'$ en tant que diviseur de
Weil avec $C'$ lisse), la fibre contient l'espace des modules de
fibr\'es $\omega_{C'}$-quadratiques de rang $r$ sur la courbe $C'$.
Si $({\cal{F}},\sigma)$ est une famille de th\^eta-caract\'eristiques
param\'etr\'ee par une vari\'et\'e connexe, la dimension de l'espace des
sections de ${\cal{F}}_{s}$ est invariant modulo 2 pour $s\in S$.
(\cite{11}, 0.3 ou \cite{12}). Il s'ensuit que
$\Theta_{\planp}(d)$ a au moins deux composantes (qui seront non
vide pour $d\ge 4$) correspondant aux th\^eta-caract\'eristiques
$({\cal{F}},\sigma)$ telles que
$\h^{0}({\cal{F}})=0
\bmod 2$ et aux th\^eta-caract\'eristiques telles que
$\h^{0}({\cal{F}})=1 \bmod 2$. Si $d$ est impair on a une troisi\`eme
composante, qu'on note
$\Theta_{can}(d)$, correspondant aux th\^eta-caract\'eristiques {\em
canoniques},
\ie de la forme ${\cal{O}}_{C}(\frac{d-3}{2})$, o\`u $C$ est une
courbe de degr\'e $d$. Cette composante est toujours isomorphe \`a
l'espace des courbes de degr\'e
$d$ sur $\planp$. Dans ce qui suit, on appellera {\em paire} une
th\^eta-caract\'eristique non canonique telle que $\h^{0}({\cal{F}})=0
\bmod 2$. Une th\^eta-caract\'eristique non canonique telle que
$\h^{0}({\cal{F}})=1
\bmod 2$ sera appel\'e {\em impaire}. On notera $\Theta_{p}(d)$ (resp.
$\Theta_{i}(d)$) la composante des th\^eta-caract\'eristiques paires
(resp. impaires). D'apr\`es (\cite{11}, 0.5; \cite{1}
pour l'ouvert des courbes lisses) ces composantes sont
irr\'eductibles. La composante paire contient l'ouvert not\'e $I(d)$
des th\^eta-caract\'eristiques dites {\em ineffectives} (\ie
$\h^{0}({\cal{F}})=0$), la composante impaire contient l'ouvert
not\'e $SI(d)$ des th\^eta-caract\'eristiques dites {\em
semi-ineffectives} (\ie
$\h^{0}({\cal{F}})=1$).
Si $d=1$, toute th\^eta-caract\'eristique est canonique; si $d=2$,
$\Theta_{p}(d)$ est isomorphe \`a
$\proj_{5}$ et $\Theta_{i}(d)$ est vide. Enfin, si
$d=3$, $\Theta_{i}(d)$ est encore vide, toute th\^eta-caract\'eristique
telle que $\h^{0}({\cal{F}})$ soit impaire \'etant
canonique.
Par normalit\'e de $\Theta_{p}(d)$, le morphisme naturel
$$\rho:\Pic(\Theta_{p}(d))\lra\Cl(\Theta_{p}(d))$$
est une injection.
\begin{th}\label{PicPair} Si $d=3,4$, le
groupe de Picard de $\Theta_{p}(d)$ est un groupe ab\'elien libre \`a
un g\'en\'erateur. De plus, $\rho$ est
un isomorphisme pour $d=3$; si $d=4$, il est d'indice $2$.
Si $d\geq 5$, le groupe de Picard de $\Theta_{p}(d)$ est isomorphe \`a un
groupe ab\'elien libre \`a deux g\'en\'erateurs. De plus, $\rho$
est un isomorphisme pour $d=5$, d'indice $4$ si $d\geq 6$ est pair et
d'indice $2$ si $d\geq 7$ est impair.\par
En particulier, la vari\'et\'e
$\Theta_{p}(d)$ est localement factorielle si et seulement si $d=1,2,3$ ou
$d=5$.
\end{th}
Pour l'identification des g\'en\'erateurs on consid\'era l'espace de
modules $N_{\planp}(d,\chi)$ des ${\cal{O}}_{\planp}$-modules
semi-stables de dimension $1$, de degr\'e $d$ et de caract\'eristique
d'Euler-Poincar\'e $\chi$. Consid\'erons le morphisme d'oubli
$$\beta:\Theta_{\planp}(d)\lra N_{\planp}(d,0)$$ qui associe \`a la
classe d'une th\^eta-caract\'eristique semi-stables la classe du
faisceau semi-stable sous-jacent.
D'apr\`es \cite{9}, $N_{\planp}(d,\chi)$ est une vari\'et\'e
irr\'eductible, projective et normale. En outre,
$N_{\planp}(d,\chi)$ est localement factorielle et son groupe de
Picard est un groupe ab\'elien libre \`a deux g\'en\'erateurs, not\'es
${\cal{L}}_{N}$ et
${\cal{D}}_{N}$, d\'efinies fonctoriellement. Le fibr\'e inversible
${\cal{L}}_{N}$ s'identifie \`a l'image r\'eciproque de
${\cal{O}}_{\proj^{M}}(1)$ sous le morphisme support sch\'ematique
$\sigma_{_{M}}:N_{\planp}(d,0)\lra\proj^{M}$; le fibr\'e
${\cal{D}}_{N}$ est le fibr\'e d\'eterminant (\cf \ref{LePotierspace} pour la
d\'efinition) Notons ${\cal{L}}$ et ${\cal{D}}$ les images r\'eciproques de
${\cal{L}}_{N}$ et ${\cal{D}}_{N}$ sous $\beta$. Pour $d=5$,
le fibr\'e inversible ${\cal{D}}$ admet une racine:
le fibr\'e pfaffien, not\'e ${\cal{P}}$ (\cf \ref{defpfaff} pour la
d\'efinition). Alors on a pour le groupe de Picard
$$\Pic(\Theta_{p}(d))=
\begin{cases}
<{\cal{L}}>& \text{si $d=3,4$}\\
<{\cal{L}},{\cal{P}}>& \text{si $d=5$}\\
<{\cal{L}},{\cal{D}}>& \text{si $d\ge6$}\\
\end{cases}
$$
Soit $\Theta^{os}_{p}\subset\Theta_{p}$ l'ouvert des
th\^eta-caract\'eristiques $({\cal{F}},\sigma)$ tels que
${\cal{F}}$ soit stable en tant que faisceau. D'apr\`es \cite{11},
cet ouvert est lisse et l'on a
$$\Cl(\Theta_{p})=\Pic(\Theta_{p}^{os}(d)).$$
On verra aussi que si $d\geq 4$ est pair, l'image de
${\cal{L}}$ sous $\rho$ admet une racine dans $\Pic(\Theta_{p}^{os}(d))$,
not\'e ${\cal{R}}$. Si $d\geq 5$, l'image de ${\cal{D}}$ sous $\rho$
admet une racine dans $\Pic(\Theta_{p}^{os}(d))$,
le fibr\'e pfaffien ${\cal{P}}$. On verra qu'en fait
$$\Pic(\Theta_{p}^{os}(d))=
\begin{cases}
<{\cal{L}}>& \text{ si $d=3$}\\
<{\cal{R}}>& \text{ si $d=4$}\\
<{\cal{L}},{\cal{P}}>& \text{ si $d\geq 5$ est impair}\\
<{\cal{R}},{\cal{P}}>& \text{ si $d\geq 6$ est pair}\\
\end{cases}
$$
Ceci pr\'ecise l'assertion concernant la factorialit\'e locale.
Pour la composante impaire, on obtient
\begin{th}\label{PicImpair}\par Si $d\ge4$, avec $d\not=6$, le
groupe de Picard de
$\Theta_{i}(d)$ s'identifie \`a un groupe ab\'elien libre \`a deux
g\'en\'erateurs: les images r\'eciproques de ${\cal{L}}_{N}$ et
${\cal{D}}_{N}$. De plus, le morphisme naturel
$\Pic(\Theta_{i}(d))\lra\Cl(\Theta_{i}(d))$ est d'indice $2$.
Si $d=6$, le groupe de
Picard de $\Theta_{i}(d)$ a un g\'en\'erateur de plus provenant du
ferm\'e des th\^eta-caract\'eristiques telles que
$h^{0}({\cal{F}})\ge3$, qui est un diviseur irr\'eductible pour cette
valeur de
$d$.\par En particulier, pour $d\geq 4$, la vari\'et\'e $\Theta_{i}(d)$ n'est
pas localement factorielle.
\end{th}
Nous d\'emontrerons les th\'eor\`emes ci-dessus en \'etudiant
\begin{list}{-}{}
\item le groupe de Picard des ouverts $I(d)\subset\Theta_{p}(d)$
et $SI(d)\subset\Theta_{i}(d)$
\item les sous-sch\'emas correspondant aux th\^eta-caract\'eristiques
sp\'eciales (\ie ni ineffectives, ni semi-ineffectives).
\end{list}
Le groupe de Picard de $I(d)$ et $SI(d)$ se calculera en utilisant
le type particulier de la r\'esolution minimale des
th\^eta-caract\'eristique ineffectives et semi-ineffectives. Si le
support sch\'ematique est lisse, ce type particulier
est connu de Dixon \cite{3}, Catanese \cite{2} et
Laszlo \cite{7}. La d\'emonstration de Laszlo se g\'en\'eralise
facilement au cas des th\^eta-caract\'eristiques \`a support sch\'ematique
quelconque. Cependant, nous en donnons encore une autre
d\'emonstration permettant d'obtenir un r\'esultat davantage fonctoriel
et une description de la fl\`eche en termes de la cohomologie d'une
th\^eta-caract\'eristique donn\'ee, n\'ecessaire pour notre propos.
Pour l'\'etude des th\^eta-caract\'eristiques sp\'eciales nous d\'efinissons
les ensembles
$$\Theta^{r}(d)=\{[{\cal{F}},\sigma]\in\Theta(d)/\h^{0}({\cal{F}})\ge
r+1,
\h^{0}({\cal{F}})=(r+1)\bmod 2\}$$ qu'on munit de leur structure
sch\'ematique de vari\'et\'e d\'eterminantielle naturelle
d\'efinie dans la section \ref{BNlocus}. On y re-d\'emontre aussi que
$$\text{codim}_{\Theta}(\Theta^{r}(d))\le\frac{r(r+1)}{2},$$
si $\Theta^{r}(d)$ est non vide. En particulier $\Theta^{1}(d)$ est un
diviseur, si
non vide, puisque ce n'est pas une composante. L'\'enonc\'e de type
Brill-Noether
duquel on a besoin est le suivant:
\begin{th}\label{BN} Le diviseur $\Theta^{1}(d)$ est irr\'eductible et le
ferm\'e $\Theta^{3}(d)$ est de codimension au moins deux pour tout
$d$. Le ferm\'e $\Theta^{2}(d)$ est de codimension au moins deux sauf
si $d=6$ o\`u il est irr\'eductible de codimension un.
\end{th}
Nous d\'emontrons ce th\'eor\`eme en \'etudiant les ouverts $U_{a}(d)$ de
$\Theta(d)$ des th\^eta-caract\'eristiques de passant pas par un point donn\'e
$a\in\planp$. Le point essentiel ici est le fait que ces ouverts
ont une description particuli\`erement simple en
termes de fibr\'es de Higgs quadratiques sur la droite projective
Enfin, nous \'etudierons la question de l'existence d'une famille
universelle: Soit $U$ un ouvert non-vide de l'ouvert des
th\^eta-caract\'eristiques
${\cal{O}}$-stables paires (resp. impairs). On appelle famille
universelle sur $U$ une famille $({\cal{F}},\sigma)$ param\'etr\'ee par
$U$ telle que le morphisme modulaire induit $U\lra\Theta^{os}(d)$ soit
l'inclusion.
\begin{th}\label{UniFam} Soit $d\geq 3$. Si $d$ est pair il n'existe
pas de famille universelle sur $U$. Si $d$ est impair il existe une
famille universelle localement dans la topologie de Zariski, mais il
n'existe pas de famille universelle globalement sur
$\Theta^{os}(d)$.
\end{th}
{\em Notations:} Par vari\'et\'e alg\'ebrique on entend sch\'ema de type fini,
s\'epar\'e, sur un corps $k$
suppos\'e alg\'ebriquement clos de caract\'eristique 0.
Si
$f:X\lra S$ est un morphisme de vari\'et\'es alg\'ebriques, on note $X_{s}$ la
fibre de $f$ au-dessus du point $s\in S$ et si ${\cal{F}}$ est un faisceau
sur $X$, on note ${\cal{F}}_{s}$ la restriction de ${\cal{F}}$ \`a $X_{s}$.
Si $X$ est une vari\'et\'e de Cohen-Macaulay on d\'esigne par
$\omega_{_{X}}$ le faisceau dualisant. Si ${\cal{F}}$ est un faisceau de
Cohen-Macaulay de dimension $d$ sur une vari\'et\'e de Cohen-Macaulay de
dimension $n$, on note ${\cal{F}}^{\vee}$ le {\em $\omega$-dual}
de ${\cal{F}}$ \ie le faisceau
$\ul{\Ext}^{n-d}_{{\cal{O}}_{X}}({\cal{F}},\omega_{_{X}})$.
Si $(L^{\cdot},d_{L})$ est un complexe de
${\cal{O}}_{S}$-modules, on d\'esigne par $L^{\cdot}[m]$ le complexe
translat\'e de $m$ places \`a gauche, de diff\'erentielle $(-1)^{m}d_{L}$ et
par $\tau_{\leq m}(L^{\cdot})$ le sous-complexe de $L$ d\'efini par
$\dots\lra L^{m-2}\lra L^{m-1}\lra \Ker(d^{m})\lra 0$.
Si $(K^{\cdot},d_{K})$ est un autre complexe de
${\cal{O}}_{S}$-modules, le complexe de ${\cal{O}}_{S}$-modules
$\ul{\Hom}^{\cdot}(L^{\cdot},K^{\cdot})$ est d\'efini en degr\'e $n$ par
$\displaystyle \prod_{-p+q=n}\ul{\Hom}(L^{p},K^{q})$, de diff\'erentielle
$d(f)^{p}=d_{K}\circ f^{p}-(-1)^{n}f^{p+1}d_{L}$. Finalement, on note
$L^{\cdot*}$ le complexe $\ul{\Hom}(L^{\cdot},{\cal{O}}_{S})$.
Par $D(S)$ on d\'esigne la cat\'egorie d\'eriv\'ee de la cat\'egorie des
${\cal{O}}_{S}$-modules, par
$D_{c}(S)$ la sous-cat\'egorie pleine des complexes
\`a cohomologie coh\'erente, par $D^{b}(S)$ la sous-cat\'egorie pleine des
complexes born\'es \`a gauche et \`a droite et enfin, par $D^{b}_{c}(S)$
l'intersection dans $D(S)$ de $D_{c}(S)$ avec $D^{b}(S)$.
\bigskip\medskip
{\em Remerciements:} Je tiens \`a remercier Joseph Le Potier pour son
aide pendant la pr\'eparation du pr\'esent travail.
\np
\tableofcontents
\np
\section{Le module $\Theta_{\planp}(d)$ des th\^eta-caract\'eristiques
des courbes planes}
Nous allons rappeler les d\'efinitions
et r\'esultats de \cite{11} sur l'espace de modules des
th\^eta-caract\'eristiques semi-stables et celles
de \cite{9} sur le module des faisceaux semi-stables de dimension 1 sur le plan
projectif dont on a besoin dans la suite.
\subsection{Th\^eta-caract\'eristiques des courbes planes }
Soit ${\cal{F}}$ un faisceau pur de dimension $1$ sur le plan
projectif,
\ie tel que tout sous-faisceau non nul de ${\cal{F}}$ soit de
dimension $1$. C'est un faisceau de Cohen-Macaulay; il admet donc
une r\'esolution $$0\lra A\hfl{\alpha}{}B\lra{\cal{F}}\lra 0$$ par
des faisceaux localement libres sur $\planp$. On appelle {\em
support sch\'ematique} et on note
$\supps({\cal{F}})$ la courbe d\'efinie par $\det(\alpha)$. C'est une
courbe de Gorenstein dont le degr\'e est \'egale \`a la {\em
multiplicit\'e} de
${\cal{F}}$, not\'ee $\mult({\cal{F}})$, \ie au coefficient directeur
du polyn\^ome de Hilbert de
${\cal{F}}$. Posons
${\cal{F}}^{\vee}=\ul{\Ext}_{{\cal{O}}_{\planp}}^{1}
({\cal{F}},\omega_{_{\planp}}).$
Si l'on consid\`ere
${\cal{F}}$ comme faisceau de modules sur son support sch\'ematique,
on a ${\cal{F}}^{\vee}\simeq
{\cal{F}}^{*}\otimes_{{\cal O}_{C}}\omega_{_{C}}$, o\`u
${\cal{F}}^{\vee}=
\ul{\Hom}_{{\cal{O}}_{C}}({\cal{F}},\omega_{_{C}})$.
De plus, on a $\chi({\cal{F}}^{\vee})=-\chi({\cal{F}})$ et le
morphisme canonique d'\'evaluation
${\cal{F}}\lra{\cal{F}}^{\vee\vee}$ est un isomorphisme (\cite{11}, 1.1).
\begin{defi}\label{theta-def} On appelle th\^eta-caract\'eristique sur
$\planp$ la donn\'ee d'un couple $({\cal{F}},\sigma)$ form\'e d'un
${\cal{O}}_{\planp}$-module coh\'erent de dimension
$1$ et d'un isomorphisme sym\'etrique
$\sigma:{\cal{F}}\lra{\cal{F}}^{\vee}.$
\end{defi}
Si $({\cal{F}},\sigma)$ est une th\^eta-caract\'eristique, alors
$\chi({\cal{F}})=0$. Un {\em morphisme} de th\^eta-caract\'eristiques
$\varphi:({\cal{E}},\tau)\lra({\cal{F}},\sigma)$ est la donn\'ee d'un
morphisme de ${\cal{O}}_{\planp}$-modules
$\varphi:{\cal{E}}\lra{\cal{F}}$ tel que
$\tau=^{\vee}\varphi\circ\sigma\circ\varphi$. Soit
${\cal{E}}\subset{\cal{F}}$ un sous-faisceau coh\'erent de
${\cal{F}}$. On appelle {\em $\sigma$-orthogonal} de ${\cal{E}}$ et
on note ${\cal{E}}^{\ort}$ le noyau du morphisme compos\'e
${\cal{F}}\lra{\cal{F}}^{\vee}\lra{\cal{E}}^{\vee}.$ Un
sous-faisceau coh\'erent ${\cal{E}}\subset{\cal{F}}$ est dit {\em
$\sigma$-isotrope} si ${\cal{E}}\cap{\cal{E}}^{\ort}\not=(0)$. Si
l'on a ${\cal{E}}\subset{\cal{E}}^{\ort}$ on dit que ${\cal{E}}$
est {\em totalement $\sigma$-isotrope}.
Une th\^eta-caract\'eristique
$({\cal{F}},\sigma)$ est dite {\em semi-stable (resp. stable)} si
pour tout sous-faisceau totalement $\sigma$-isotrope non nul
${\cal{E}}\subset{\cal{F}}$ on a
$$\chi({\cal{E}})\le 0\text{(resp. $<$)}.$$ Une th\^eta-caract\'eristique
$({\cal{F}},\sigma)$ est semi-stable si et seulement si son faisceau
sous-jacent ${\cal{F}}$ est semi-stable. Une th\^eta-caract\'eristique stable
est somme directe orthogonale de th\^eta-caract\'eristiques dont les faisceaux
sous-jacents sont stables et 2 \`a 2 non-isomorphes (\cite{11},
Prop. 1.4). On dit qu'une th\^eta-caract\'eristique est ${\cal{O}}$-{\em
stable} si son faisceau sous-jacent est stable.
La cat\'egorie des th\^eta-caract\'eristiques semi-stables
n'est pas ab\'elienne.
On
peut n\'eanmoins d\'efinir une notion de
${\cal{S}}$-\'equivalence de deux faisceaux quadratiques: Soit
$({\cal{F}},\sigma)$ une th\^eta-caract\'eristique semi-stable. Alors
il existe une filtration par des sous-faisceaux coh\'erents totalement
isotropes
$ 0\subset{\cal{F}}_{1}
\subset\dots\subset{\cal{F}}_{\ell}\subset{\cal{F}}
$ telle que les faisceaux
${\cal{E}}_{i}={\cal{F}}_{i}/{\cal{F}}_{i-1}$, avec
${\cal{F}}_{0}=0$, soient stables de caract\'eristique
d'Euler-Poincar\'e nulle pour $i=1,\dots,\ell$ et telle que
$({\cal{F}}_{\ell}^{\ort}/{\cal{F}}_{\ell},\sigma)$ soit une
th\^eta-caract\'eristique stable. Une th\^eta-caract\'eristique
$({\cal{H}},\sigma)$ est dite {\em hyperbolique} si elle est
isomorphe \`a une th\^eta-caract\'eristique
$({\cal{G}}\osum{\cal{G}}^{\vee},\tau)$, o\`u ${\cal{G}}$ est stable
en tant que faisceau et o\`u $\tau$ est donn\'e par la matrice
suivante:
$$
\bordermatrix{%
& {\cal{G}} & {\cal{G}}^{\vee} \cr
{\cal{G}}^{\vee}& 0 & 1 \cr
{\cal{G}}& 1 & 0 \cr }
$$ Munissons ${\cal{H}}_{i}={\cal{E}}_{i}\osum{\cal{E}}_{i}^{\vee}$
de la structure hyperbolique $\sigma_{i}=\tau$. On d\'efinit la
th\^eta-caract\'eristique gradu\'e associ\'ee \`a $({\cal{F}},\sigma)$ comme
la somme directe orthogonale suivante:
$$
\gr({\cal{F}},\sigma)=
\left({\cal{F}}_{\ell}^{\ort}/{\cal{F}}_{\ell},\sigma\right)\osum
\left(\osum_{i=1}^{\ell}({\cal{H}}_{i},\sigma_{i})\right).
$$ La filtration n'est pas unique. Le gradu\'e par contre, ne d\'epend
pas, \`a isomorphisme pr\`es, de la filtration choisie. Par cons\'equent,
on dira que deux th\^eta-caract\'eristiques
$({\cal{E}},\tau)$ et $({\cal{F}},\sigma)$ sont {\em
${\cal{S}}$-\'equivalentes} si leurs gradu\'es associ\'es respectifs sont
isomorphes. On dira que $({\cal{F}},\sigma)$ est {\em poly-stable} si
$$({\cal{F}},\sigma)=\left(\osum_{i}{\cal{H}}_{i}^{m_{i}}\right)
\osum\left(\osum_{j}{\cal{F}}_{j}^{n_{j}}\right),$$
la somme directe \'etant orthogonale, les ${\cal{H}}_{i}$ (et
de m\^eme les ${\cal{F}}_{j}$) \'etant deux \`a deux non-isomorphes et
${\cal{H}}_{i}={\cal{E}}_{i}\osum{\cal{E}}_{i}^{\vee}$ \'etant hyperbolique
{\em stricte} \ie ${\cal{E}}_{i}\not\simeq{\cal{E}}_{i}^{\vee}$.
D'apr\`es ce qui pr\'ec\`ede le gradu\'e d'une th\^eta-caract\'eristique est
poly-stable
et une th\^eta-caract\'eristique poly-stable est isomorphe \`a son gradu\'e.
Les points ferm\'es du module des th\^eta-caract\'eristiques qui sera d\'efini
dans
la section suivante seront exactement les th\^eta-caract\'eristiques
poly-stables.
\subsection{La construction du module des th\^eta-caract\'eristiques sur
$\planp$.}\label{thetaconstruction}
Soit $S$ une vari\'et\'e alg\'ebrique, ${\cal{F}}$ un
${\cal{O}}_{S\times \planp}$-module, $S$-plat. On pose
$${\cal{F}}^{\vee}=\ul{\Ext}_{{\cal{O}}_{S\times \planp}}^{1}
({\cal{F}},pr_{2}^{*}\omega_{_{\planp}}).$$ Si ${\cal{F}}$
est une famille $S$-plate de faisceaux purs de dimension $1$, alors
${\cal{F}}^{\vee}$ est $S$-plat, le morphisme canonique
${\cal{F}}\lra{\cal{F}}^{\vee\vee}$ est un isomorphisme et l'on a
$({\cal{F}}^{\vee})_{s}\simeq({\cal{F}}_{s})^{\vee}$ (\cite{11}, 3.1). Une {\em
famille de th\^eta-caract\'eristiques}
para\-m\'etr\'ee par la vari\'et\'e
alg\'ebrique $S$ est la donn\'ee d'un couple
$({\cal{F}},\sigma)$, form\'e d'un ${\cal{O}}_{S\times X}$-module
coh\'erent
${\cal{F}}$, $S$-plat, et d'un isomorphisme sym\'etrique
$\sigma:{\cal{F}}\lra{\cal{F}}^{\vee}$.
Consid\'erons le foncteur
$\underline{\Theta}_{\planp}(d)$ associant \`a la vari\'et\'e alg\'ebrique
$S$ l'ensemble des classes d'iso\-morphisme de familles de
th\^eta-caract\'eristiques $({\cal{F}},\sigma)$, param\'etr\'ees par $S$,
telles que, pour tout point ferm\'e $s$ de $S$, la
th\^eta-caract\'eristique $({\cal{F}}_{s},\sigma_{s})$ soit semi-stable
et de polyn\^ome de Hilbert $P(m)=md$. Pour le foncteur
$\underline{\Theta}_{\planp}(d)$ il existe un espace de modules
grossier, not\'e $\Theta_{\planp}(d)$. C'est une vari\'et\'e projective
dont l'ensemble des points ferm\'es est l'ensemble des classes de
${\cal S}$- \'equivalence de th\^eta-caract\'eristiques semi-stables de
degr\'e
$d$ sur $\planp$. De plus, il existe un ouvert non-vide
$\Theta_{\planp}^{s}(d)\subset\Theta_{\planp}(d)$ dont les points
repr\'esentent les classes d'isomorphisme des th\^eta-caract\'eristiques
stables (\cite{11}, 0.2).
Dans ce qui suit on a besoin d'expliciter la construction de
$\Theta_{\planp}(d)$:
soit $N$ un entier tel que pour toute th\^eta-caract\'eristique semi-stable
$({\cal{F}},\sigma)$ le faisceau ${\cal{F}}(N)$ soit engendr\'e par
ses sections globales et de cohomologie sup\'erieure nulle.
Ceci est possible, la famille des th\^eta-caract\'eristiques semi-stables de
degr\'e
fix\'e \'etant limit\'ee (\cite{11}, Prop. 3.3]). Soit
$H=k^{dN}$ et
${\tr{H}}=H\otimes_{k}{\cal{O}}_{\planp}(-N)$.
Consid\'erons, pour toute vari\'et\'e alg\'ebrique
$S$, les triplets $({\cal{F}},\alpha,\sigma)$ form\'es d'un quotient
coh\'erent
${\tr{H}}_{S}\hfl{\alpha}{}{\cal{F}}$, $S$-plat, et d'un
isomorphisme sym\'etrique $\sigma:{\cal{F}}\lra{\cal{F}}^{\vee}$
satisfaisant aux conditions suivantes: pour tout point ferm\'e $s$ de
$S$, $({\cal{F}}_{s},\sigma_{s})$ est une th\^eta-caract\'eristique
semi-stable de multiplicit\'e $d$ et $\alpha$ induit un isomorphisme
$H\otimes{\cal{O}}_{S}\simeq pr_{1*}({\cal{F}}(N))$. Deux triplets
$({\cal{F}},\alpha,\sigma)$ et
$({\cal{F}}',\alpha',\sigma')$ sont dits \'equivalents s'il existe un
isomorphisme $\varphi:{\cal{F}}\lra{\cal{F}}'$ tel que
$\alpha'=\varphi\circ \alpha$ et tel que
$\sigma'\varphi=(^{\vee}\varphi)^{-1}\circ\sigma$. Notons par
$[{\cal{F}},\alpha,\sigma]$ la classe d'\'equivalence du triplet
$({\cal{F}},\alpha,\sigma)$ et par
$\underline{T}^{ss}(d,N)(S)$ l'ensemble des classes d'\'equivalence
de tels triplets. On obtient un foncteur
$$
\underline{T}^{ss}(d,N):\text{Vari\'et\'es alg\'ebriques}
\lra\text{Ensembles},
$$ dont on montre la repr\'esentabilit\'e (\cite{11}, 7.3).
Notons $T^{ss}(d,N)$ le sch\'ema qui le repr\'esente. Le groupe
$GL(H)$ op\`ere sur $T^{ss}(d,N)$ en associant \`a $g\in GL(H)$ et au
triplet
$[{\cal{F}},\alpha,\sigma]$ le triplet $[{\cal{F}},\alpha\circ
g^{-1},\sigma]$. D'apr\`es \cite{11}, l'espace de modules
$\Theta_{\planp}(d)$ s'identifie au quotient de Mumford de
$T^{ss}(d,N)/GL(H)$.
Le stabilisateur d'un point $[{\cal{F}},\alpha,\sigma]$ s'identifie
au groupe des automorphismes de la th\^eta-caract\'eristiques
$({\cal{F}},\sigma)$. En effet, soit
$\chi\in\Aut({\cal{F}},\sigma)$. De $\chi$ on d\'eduit un
isomorphisme $\H^{0}({\cal{F}}(N))\lra\H^{0}({\cal{F}}(N))$
induisant un isomorphisme $\overline\chi\in GL(H)$:
$$
\begin{diagram}
\H&\hfl{\alpha}{}&\H^{0}({\cal{F}}(N))\\
\vfl{\overline\chi}{}&&\vfl{\chi}{}\\
\H&\hfl{\alpha}{}&\H^{0}({\cal{F}}(N))
\end{diagram}
$$ qui est dans le stabilisateur. R\'eciproquement, un \'el\'ement
$g\in GL(H)$ du stabilisateur induit, comme
$[{\cal{F}},\alpha,\sigma] =[{\cal{F}},\alpha\circ g^{-1},\sigma]$,
un isomorphisme $\phi:{\cal{F}}\lra{\cal{F}}$ respectant $\sigma$.
Ces deux op\'erations sont inverses l'une de l'autre, d'o\`u
l'identification voulue.
La vari\'et\'e $T^{ss}(d,N)$ est une vari\'et\'e lisse (\cite{11},
Prop. 7.4) et
$GL(H)/\{\pm Id\}$ op\`ere librement sur l'ouvert des
th\^eta-caract\'eristiques
${\cal{O}}$-stables. Il en d\'ecoule que $\Theta_{\planp}(d)$ est
normale et n'a que des singularit\'es rationnelles. Il en d\'ecoule
aussi que l'ouvert
$\Theta_{\planp}^{os}(d)$ des th\^eta-caract\'eristiques
${\cal{O}}$-stables est lisse. La codimension du ferm\'e des
th\^eta-caract\'eristiques semi-stables, non ${\cal{O}}$-stables est au
moins
$2$ (\cite{11}, Prop. 8.6). On utilisera ce r\'esultat plus
tard sans le rappeler explicitement.
\subsection{La structure sch\'ematique de $\Theta^{r}(d)$.}
\label{BNlocus}
Soit $({\cal{F}},\sigma)$ une famille de th\^eta-caract\'eristiques
param\'etr\'ee
par la vari\'et\'e alg\'ebrique $S$. D\'efinissons les ensembles
$$S^{r}=\{s\in S/\h^{0}({\cal{F}}_{s})\geq r+1,
\h^{0}({\cal{F}}_{s})= r+1\bmod 2\}.$$
D'apr\`es le th\'eor\`eme de semi-continuit\'e et le th\'eor\`eme d'invariance
mod
$2$ les $S^{r}$ sont ferm\'es.
Pour la d\'efinition de la structure sch\'ematique des $S^{r}$, rappelons qu'on
appelle approximation de le cohomologie de ${\cal{F}}$
un complexe
$$E^{\cdot}\text{: }0\lra E^{0}\efl{d}{} E^{1}\lra 0$$, form\'e de
${\cal{O}}_{S}$-modules coh\'erents
localement libres, tel que
pour tout changement de base $S'\hfl{g}{}S$,
$$
\begin{diagram}
S'\times\planp&\hfl{g'}{}&S\times\planp\\
\vfl{f'}{}&&\vfl{}{f}\\
S'&\hfl{g}{}&S
\end{diagram}
$$
on ait $\H^{i}(g^{*}E^{\cdot})=\R^{i}f'_{*}g^{\prime *}{\cal{F}}$. En
particulier, pour $s\in S$, on a
$\H^{i}(E^{\cdot}_{s})=\H^{i}(X_{s},{\cal{F}}_{s})$.
Une telle approximation est dite anti-sym\'etrique si
$E^{1}=E^{0*}$ et si $d$ est anti-sym\'etrique.
Une approximation de la cohomologie globale de ${\cal{F}}$ existe
toujours; par contre pour une
approximation anti-sym\'etrique on n'a que l'existence, pour tout point
$s\in S$, d'un voisinage ouvert $U_{s}$ de $s$ telle que sur $U_{s}$ il
existe une approximation anti-sym\'etrique
$$0\lra Z^{0}\efl{d}{}Z^{*}\lra 0$$
de ${\cal{F}}$ avec $Z$ de rang $z=\h^{0}({\cal{F}}_{s})$ et $d(s)=0$.
Localement sur $U$, la structure sch\'ematique de $S^{r}$ est
d\'efinie par les mineurs d'ordre $(z-r)$ de $\alpha$. L'id\'eal engendr\'e par
ces mineurs ne d\'epend pas de l'approximation anti-sym\'etrique particuli\`ere
choisie. De ce fait, cette construction
d\'efinit donc un faisceau d'id\'eaux sur $S$ entier d\'efinissant ainsi la
structure sch\'ematique de $S^{r}$.
En particulier, si $s\in S^{r}\backslash S^{r+1}$, le sous-sch\'ema $S^{r}$ est
d\'efini au voisinage de $s$ par des $1$-mineurs d'une matrice
anti-sym\'etrique,
donc par $\frac{r(r-1)}{2}$ \'equations. Ceci d\'emontre, en utilisant le
lemme de
Krull, que
$$\codim_{S}(S^{r})\leq\frac{r(r-1)}{2},$$
si $S^{r}\backslash S^{r+1}$ est non-vide.
Consid\'erons maintenant la vari\'et\'e $T^{ss}(d)$ de la section
\ref{thetaconstruction} et le bon quotient
$\varphi:T^{ss}(d)\lra\Theta(d)$. Sur $T^{ss}(d)$ on a une famille
universelle $({\cal{F}},\sigma,\alpha)$. Soient les sous-sch\'emas
$T^{ss,r}(d)\subset T(d)$ d\'efinis comme ci-dessus. Ces sous-sch\'emas sont
invariants
sous l'action de $GL(H)$. On d\'efinit
$\Theta^{r}(d)$ comme \'etant l'image de $T^{ss,r}(d)$. D'apr\`es les
propri\'et\'es
de bons quotients, $\Theta^{r}(d)$ est un sous-sch\'ema ferm\'e de $\Theta(d)$.
L'ensemble sous-jacent \`a $\Theta^{r}(d)$ s'identifie \`a
$$\Theta^{r}(d)=\{[{\cal{F}},\sigma]\in\Theta(d)/\h^{0}({\cal{F}})\ge r+1,
\h^{0}({\cal{F}})=(r+1)\bmod 2\}.$$
\subsection{L'espace de modules $N_{\planp}(d,0)$ de Le
Potier}\label{LePotierspace}
Soit $\ul{N}_{\planp}(d,\chi)$ le foncteur qui associe \`a la vari\'et\'e
alg\'ebrique $S$ l'ensemble des classes d'isomorphismes de familles
$S$-plates de faisceaux semi-stables de dimension $1$ de degr\'e
$d$ et de caract\'eristique d'Euler-Poincar\'e $\chi$. D'apr\`es
\cite{10}, il existe un espace de modules grossier projectif, not\'e
$N_{\planp}(d,\chi)$, pour ce foncteur
et l'ensemble de ses points ferm\'es s'identifie aux classes de
${\cal{S}}$-\'equivalence de faisceaux semi-stables. De plus, on a un
morphisme, appel\'e morphisme support sch\'ematique,
$$\sigma_{N}:N(d,\chi)_{\planp}\lra\proj^{M},$$ avec $M=d(d+3)/2$, qui
associe \`a un faisceau semi-stable de dimension $1$ son support
sch\'ematique.
D'apr\`es \cite{9},
la vari\'et\'e
$N_{\planp}(d,\chi)$ est irr\'eductible, normale et localement
factorielle. De plus, son groupe de Picard est isomorphe \`a
\`a un groupe ab\'elien \`a deux g\'en\'erateurs. Pour la description de ces
g\'en\'erateurs dont
on a besoin dans ce qui suit, on va rappeler la construction de cet
espace de modules.
Elle se fait de la mani\`ere suivante: d'abord on montre
que la famille des faisceaux semi-stables de polyn\^ome de Hilbert
fix\'e est limit\'ee. Ensuite, on choisit un entier $N$ de fa\c con que
pour tout faisceau semi-stable ${\cal{F}}$ de dimension $1$, de
multiplicit\'e
$d$ et de caract\'eristique d'Euler-Poincar\'e $\chi$, le faisceau
${\cal{F}}(N)$ soit engendr\'e par ses sections, et que
$\H^1({\cal{F}}(N))=0.$ Un tel entier \'etant choisi, on consid\`ere
un espace vectoriel $\H$ de dimension
$n=dN+\chi$ et le fibr\'e vectoriel ${\tr{H}}= H_{k}\otimes{\cal
O}_{\planp}(-N)$. Le groupe $SL(H)$ op\`ere de mani\`ere naturelle sur
le sch\'ema de Hilbert-Grothendieck
$\Groth^{d,\chi}({\tr{H}})$ des faisceaux quotients de
${\tr{H}}$ de multiplicit\'e
$d$
et de caract\'eristique d'Euler-Poincar\'e $\chi$. Quitte \`a choisir un
entier $m$ suffisamment grand, ce sch\'ema se plonge dans une
grassmannienne par le plongement de Grothendieck: au quotient
${\tr{H}}\lra{\cal{F}}$ on associe le quotient
$H\otimes_{k}\H^{0}({\cal{O}}_{\planp}(m-N))\lra{\cal{F}}(m)$. De
plus, l'action de $SL(H)$ se rel\`eve en une action lin\'eaire sur la
grassmannienne. Maintenant, quitte \`a choisir $N$ et ensuite $m$
suffisamment grand, l'ouvert $\Omega^{ss}$ des points semi-stables pour
l'action de $SL(H)$ correspond aux faisceaux quotients de
${\tr{H}}$ qui sont semi-stables et tels que le morphisme
d'\'evaluation
$\H\ra\H^0({\cal{F}}(N))$ soit un isomorphisme \cite{10}. Alors
$N_{\planp}(d,\chi)$ s'identifie au quotient de Mumford
$$N_{\planp}(d,\chi)=\Omega^{ss}/SL(\H).$$
Venons-en \`a la description des g\'en\'erateurs de
$\Pic(N_{\planp}(d,0))$: ce sont les fibr\'e inversible
${\cal{L}}_{N}$, image r\'eciproque de
${\cal{O}}_{\proj^{M}}(1)$ sous le morphisme
$\sigma_{N}$, et le fibr\'e inversible ${\cal{D}}_{N}$, appel\'e fibr\'e
d\'eterminant, associ\'e au diviseur ``th\^eta'' des faisceaux
semi-stables ${\cal{F}}$ ayant au moins une section (\cite{9}, 1.1). On a une
autre
description, davantage fonctorielle, de
ces g\'en\'erateurs: consid\'erons pour cela l'alg\`ebre de Grothendieck
$K(\planp)$ de $\planp$. Elle est muni d'une forme quadratique \`a
valeurs enti\`ere, d\'efinie en associant \`a $u\in K(\planp)$ l'entier
relatif $\chi(u^{2})$. Cette forme provient d'une forme
quadratique sur
$K_{top}(\planp)$. On note $<\ ,\ >$ la forme bilin\'eaire associ\'e.
Si $x$ (resp. $y$) est de rang $r$, de degr\'e $d$ et de
caract\'eristique d'Euler-Poincar\'e $\chi$ (resp. $r',d',\chi'$), on
a:
$$<x,y>=r\chi'+r'\chi+dd'-rr'.$$ Soit ${\cal{F}}$ une famille
$S$-plate de faisceaux semi-stables de dimension 1 sur $\planp$,
param\'etr\'ee par la vari\'et\'e alg\'ebrique $S.$ Supposons de plus que le
groupe alg\'ebrique $G$ op\`ere sur $S$ et ${\cal{F}}$.
Consid\'erons le diagramme
$$\begin{diagram}
S\times X&\hfl{p}{}&X\\
\sfl{q}{}&&\cr S&&\\
\end{diagram}
$$ o\`u $p$ et $q$ sont les projections
canoniques. Alors \`a $u\in K(\planp)$, on peut associer l'\'el\'ement
$${\cal{L}}_{{\cal{F}}}(u)=\det(q_{!}({\cal F}.p^*(u))$$ du groupe
$\Pic^{G}(S)$ des fibr\'e inversibles sur $S$ muni d'une action de
$G$. On peut appliquer cette construction en particulier au
faisceau quotient universel sur $\Omega^{ss}\times X$. Le fibr\'e
inversible ${\cal{L}}_{{\cal{F}}}(u)$ est alors muni d'une action
de $GL(H)$. D\'esignons par $Z(d,\chi)$ l'orthogonal de
$(d,\chi)$ dans $K(\planp)$ par rapport \`a la forme quadratique
d\'efinie ci-dessus. Le r\'esultat de \cite{8} affirme
maintenant que pour $u\in Z(d,\chi)$ il existe dans
$\Pic(N_{\planp}(d,\chi))$ un \'el\'ement ${\cal{L}}(u)$ caract\'eris\'e
par la propri\'et\'e universelle suivante: pour toute famille plate
${\cal{F}}$ de faisceaux semi-stables sur $X$ de degr\'e $d$ et de
caract\'eristique d'Euler-Poincar\'e $\chi$ param\'etr\'ee par la vari\'et\'e
alg\'ebrique $S$ on a $${\cal{L}}_{{\cal F}}(u)
=f_{{\cal{F}}}^*({\cal{L}}(u))$$ o\`u $f_{{\cal F}}:S\lra
N_{\planp}(d,\chi)$ est le morphisme modulaire associ\'e \`a la famille
${\cal{F}}$.
Pour les g\'en\'erateurs ${\cal{L}}_{N}$ et ${\cal{D}}_{N}$ on a alors
${\cal{L}}_{N}={\cal{L}}(u)$ avec $u$ la classe d'un point,
et
${\cal{D}}_{N}={\cal{L}}(u)$ avec $u=-[{\cal{O}}_{\planp}]$.
\np
\section{L'ouvert des th\^eta-caract\'eristiques ne
passant pas par un point.}\label{Ua}
Soit $U_{a}$ l'ouvert correspondant aux th\^eta-caract\'eristiques
semi-stables ne passant pas par le point $a\in\planp$.
On note $U_{a,p}(d)$ (resp. $U_{a,i}(d)$) le ferm\'e de
$U_{a}(d)$ correspondant aux th\^eta-caract\'eristiques paires (resp.
impaires).
Soit de plus
$$U^{r}_{a}(d):=
\{[{\cal{F}},\sigma]\in U_{a}/\h^{0}({\cal{F}})\ge r+1,
\h^{0}({\cal{F}})=(r+1)\bmod 2\},$$
qu'on muni de sa structure de vari\'et\'e d\'eterminantielle naturelle.
L'objet de cette
section est de d\'emontrer le th\'eor\`eme suivant:
\begin{th}\label{codimU} L'ouvert $U_{a}(d)$ est, si $d$ est pair, r\'eunion de
deux composantes irr\'eductibles correspondant aux th\^eta-caract\'eristiques
paires et impaires; si $d$ est impair, il est r\'eunion de trois composantes
irr\'eductibles correspondant aux
th\^eta-caract\'eristiques paires, impaires et
canoniques. De plus, le ferm\'e $U^{1}_{a}(d)$ est irr\'eductible de
codimension $1$
pour $d\ge 5$, vide sinon. Le ferm\'e $U^{2}_{a}(d)$
est de codimension au moins deux sauf si $d=6$ o\`u il est
irr\'eductible de codimension $1$.
\end{th}
Ceci donnera,
en corollaire, apr\`es l'\'etude du compl\'ementaire $F_{a}(d)$ de
$U_{a}(d)$ le th\'eor\`eme \ref{BN} et aussi une nouvelle preuve de
l'irr\'eductibilit\'e des composantes paires et impaires
de \cite{11}.
Soit $\proj_{1}$ une droite du plan projectif ne passant
pas par ce point. Consid\'erons la projection de centre a sur la droite:
$$\pi:\planp-\{a\}\lra\proj_{1}$$
Les images directes sous $\pi$ des th\^eta-caract\'eristiques ne
passant pas par
$a$ se d\'ecrivent en termes de fibr\'es
$\omega_{_{\proj_{1}}}$-quadratiques sur la droite projective muni d'un
morphisme $\varphi:G\lra G(1)$ compatible avec la structure quadratique:
\subsection{Fibr\'es de Higgs quadratiques}
Rappelons bri\`evement quelques g\'en\'eralit\'es sur les fibr\'es de Higgs:
soit $C$ une courbe projective lisse et soit $L$ un fibr\'e inversible sur
$C$. On appelle {\em faisceau de Higgs} la donn\'ee d'un
couple $(G,\varphi)$ form\'e d'un faisceau coh\'erent $G$ et d'un
morphisme $\varphi:G\lra G\otimes L$. Si $G$ est localement libre on dira
{\em fibr\'e de Higgs}. Un morphisme de faisceaux de Higgs de
$F'=(G',\varphi')$ dans $F''=(G'',\varphi'')$ est la donn\'ee d'un morphisme
de faisceau $f:G'\lra G''$ tel que le diagramme suivant commute:
$$\begin{diagram} G'&\efl{f}{}&G''\\
\sfl{\varphi'}{}&&\sfl{\varphi''}{}\\
G'\otimes L&\efl{f\otimes id}{}&G''\otimes L\\
\end{diagram}
$$
Ainsi un sous-faisceau de Higgs est la donn\'ee d'un sous-faisceau $G'\subset
G$ tel que $\varphi(G')\subset G'\otimes L$.
Un fibr\'e de Higgs
est dit {\em semi-stable} (resp. {\em stable}) si pour tout
sous-faisceau propre de Higgs non nul $G'$ de
$F=(G,\varphi)$ on a
$$\mu(G')\le\mu(G) \text{ (resp. $<$)}.$$
Comme dans le cas des fibr\'es vectoriels il suffit de v\'erifier
cette condition pour les {\em sous-fibr\'es} de Higgs. Si
$F=(G,\varphi)$ est un faisceau de Higgs on note $H^{i}(C,F)$ la
cohomologie de
$F$,
\ie l'hypercohomologie du complexe
$$0\lra G\efl{\varphi}{}G\otimes L\lra 0.$$ Si
$F'=(G',\varphi')$ et $F''=(G'',\varphi'')$ sont deux faisceaux de Higgs le
faisceaux des homomorphismes de $F'$ dans $F''$ est d\'efini par
$(\ul{\Hom}(G',G''),\varphi)$ avec
$\varphi(s)=\varphi''s-s\varphi'$ o\`u $s$ est une section locale de
$\ul{\Hom}(G',G'')$. On note $\Ext^{q}(F',F'')$ la cohomologie
de $\ul{\Hom}(F',F'')$. Bien s\^ur, on a
$\Ext^{0}(F',F'')=\Hom(F',F'')$. Le $\omega_{_{C}}$-dual d'un faisceau de
Higgs $F=(G,\varphi)$ est d\'efini par la paire
$(G^{\vee},\psi)$ o\`u $\psi:G^{\vee}\lra G^{\vee}\otimes L$ est d\'efini par
la transpos\'ee $^{\vee}\varphi:G^{\vee}\otimes L^{*}\lra G^{\vee}$ de
$\varphi$. On le note $F^{\vee}$. Si $F$ est un fibr\'e de Higgs, le morphisme
canonique d'\'evaluation est un isomorphisme.
\begin{defi}
On appelle fibr\'e de Higgs quadratique la donn\'ee d'un fibr\'e de Higgs $F$
et d'un isomorphisme de Higgs sym\'etrique $\sigma:F\lra F^{\vee}$.
\end{defi}
Soit $(F,\sigma)$ un fibr\'e de Higgs quadratique, $F'\subset F$ un
sous-faisceau (de Higgs). On d\'efinit {\em l'orthogonal} de $F'$ et
l'on note
$F^{\prime\ort}$ le noyau du morphisme compos\'e $$F\lra F^{\vee}\lra
F^{\prime\vee}.$$ Un sous-faisceau $F'$ est dit isotrope si $F'\cap
F^{\prime\ort}\not=0$ et totalement isotrope si $F'\subset F^{\prime\ort}$.
Un fibr\'e de Higgs quadratique est dit {\em semi-stable} (resp. {\em
stable}) si pour tout sous-faisceau totalement isotrope
$F'=(G',\varphi)$ de $F=(G,\varphi)$ on a
$$\mu(G')\le\mu(G) \text{ (resp. $<$}).$$
\begin{lemme}\label{ss-ss}
Un fibr\'e de Higgs quadratique est semi-stable si et seulement si
son fibr\'e de Higgs sous-jacent est semi-stable.
\end{lemme}
\begin{proof}
Analogue \`a la proposition 1.4 de \cite{11}.
\end{proof}
Cette \'equivalence n'est pas vraie pour la notion de stabilit\'e. On
dira qu'un fibr\'e de Higgs quadratique est {\em ${\cal{O}}$-stable}
si son fibr\'e de Higgs sous-jacent est stable.
Soit $(G,\sigma)$ un fibr\'e vectoriel $\omega_{_{C}}$-quadratique.
L'isomorphisme $\sigma$ d\'efinit une involution $\imath$ sur
l'espace vectoriel $\Ext^{1}(G,G^{\vee})$. Le sous-espace vectoriel
des \'el\'ements sym\'etriques (resp. antisym\'etriques) est not\'e
$\Ext^{1}_{sym}(G,G^{\vee})$ (resp. $\Ext^{1}_{asym}(G,G^{\vee})$).
En notant $\Ext^{1}_{sym}(G,G)$ (resp.
$\Ext^{1}_{asym}(G,G)$) on suppose que l'on a identifi\'e $G$ et
$G^{\vee}$ via $\sigma$. \begin{defi}
Un fibr\'e vectoriel $\omega_{_{C}}$-quadratique $(G,\sigma)$ est dit rigide si
$$\Ext^{1}_{asym}(G,G)=0.$$
\end{defi}
Un fibr\'e de Higgs quadratique dont le fibr\'e vectoriel quadratique
sous-jacent est rigide est dit {\em quasi-rigide}.
\subsubsection{Le cas de la droite projective.}
Soit maintenant $C=\proj_{1}$ et $L={\cal{O}}_{\droitep}(1)$.
\'Etant donn\'e un
fibr\'e vectoriel $G$ sur $\droitep$ on \'ecrit
$G=\osum_{\ell\in\reln}r_{\ell}{\cal{O}}_{\proj_{1}}(\ell)$. La
suite des entiers $r_{\ell}\not=0$ s'appelle {\em spectre} de $G$,
les entiers $r_{\ell}$ s'appellent multiplicit\'es.
\begin{lemme} Soit $(G,\varphi,\sigma)$ un fibr\'e de Higgs quadratique
semi-stable. Alors le spectre de $G$ est connexe.
\end{lemme}
\begin{proof} En raison du lemme \ref{ss-ss} on peut supposer que
$(G,\varphi)$ est semi-stable, puis on applique le lemme $3.12$ de
\cite{9}.
\end{proof}
Soit $(G,\sigma)$ un fibr\'e quadratique, de spectre $r_{\ell}$.
Alors
$\Ext^{1}_{asym}(G,G)$ s'identifie \`a
$H^{1}(\Lambda^{2}G^{*}\otimes\omega_{_{\proj_{1}}})$. On obtient
$$\begin{diagram}
\ext^{1}_{asym}(G,G)&=&\displaystyle
h^{1}(\Lambda^{2}G^{*}\otimes\omega_{_{\proj_{1}}})\hfill\\
&=&\displaystyle
h^{1}\left(\osum_{\ell}\frac{r_{\ell}(r_{\ell}-1)}{2}
{\cal{O}}_{\proj_{1}}(-2\ell-2)
\osum
(\osum_{\ell<m}r_{\ell}r_{m}{\cal{O}}_{\proj_{1}}(-\ell-m-2))\right)\hfill\\
&=&\displaystyle
\sum_{\ell\ge0}\frac{r_{\ell}(r_{\ell}-1)}{2}(2\ell+1)
+\sum\begin{Sb} \ell<m\\ \ell+m\ge0\end{Sb} r_{\ell}r_{m}(\ell+m+1)\hfill\\
\end{diagram}
$$
On d\'eduit de ce calcul, en raison de la connexit\'e du spectre d'un
fibr\'e de Higgs quadratique semi-stable, le lemme suivant:
\begin{lemme}\label{rigide}
Le fibr\'e vectoriel $G$ sous-jacent \`a un fibr\'e de Higgs quadratique de
rang
$d$ \`a la fois semi-stable et quasi-rigide $(G,\varphi,\tau)$
s'identifie ou bien \`a $R_{0}=d{\cal{O}}_{\proj_{1}}(-1)$ ou bien \`a
$R_{1}=
{\cal{O}}_{\proj_{1}}\osum (d-2){\cal{O}}_{\proj_{1}}(-1) \osum
{\cal{O}}_{\proj_{1}}(-2)$ (si $r\ge 3$).
\end{lemme}
\subsection{Description de l'ouvert $U_{a}(d)$.}
Soit $a\in\planp$ et $\proj_{1}$ une droite du plan projectif ne passant
pas par ce point. Consid\'erons la projection de centre a sur la droite:
$$\pi:\planp-\{a\}\lra\proj_{1}$$
L'espace total du fibr\'e normal $N={\cal{O}}_{\droitep}(1)$
s'identifie \`a l'ouvert $\planp-\{a\}$.
On va d\'ecrire l'ouvert $U_{a}(d)$ des th\^eta-caract\'eristiques semi-stables
de degr\'e $d$ dont le support ne passe pas par $a$ en termes de fibr\'es de
Higgs quadratiques sur $\droitep$.
\begin{prop}\label{cat-equiv}
La cat\'egorie des
th\^eta-caract\'eristiques dont le support sch\'ematique ne passe pas par le
point $a$ est \'equivalente,
par image directe, \`a la cat\'egorie des $N$-fibr\'es
de Higgs quadratiques sur $\droitep$.
\end{prop}
\begin{proof} Rappelons d'abord l'\'equivalence des cat\'egories entre
les faisceaux purs de dimension $1$ dont le support ne passe pas
par le point
$a$ et les fibr\'es de Higgs sur $\droitep$
(\cf la proposition 3.10 de \cite{9}). La correspondance
se fait par image directe: si ${\cal{F}}$ est un faisceau pur de
dimension 1 dont le support $C=\supps({\cal{F}})$ ne passe pas par $a$ alors
l'image directe $G=\pi_{*}({\cal{F}})$ est localement libre (par puret\'e) de
rang $\mult({\cal{F}})$ et munie d'une structure de
$\pi_{*}({\cal{O}}_{N})$-module. La donn\'ee d'une telle structure \'equivaut
maintenant \`a se donner un morphisme $\varphi:G\lra G\otimes N$ d'o\`u la
structure de Higgs sur $G$. R\'eciproquement la donn\'ee d'un
fibr\'e vectoriel sur $\droitep$ muni d'une structure de
$\pi_{*}({\cal{O}}_{N})$-module d\'efinit un faisceau coh\'erent pur
${\cal{F}}$ sur N dont le support est fini au-dessus de $\droitep$: Soit
$\lambda$ la section canonique de $\pi^{*}(N)$ et consid\'erons le morphisme
$$\varphi-\lambda id_{G}:\pi^{*}(G)\lra\pi^{*}(G)\otimes\pi^{*}(N).$$
Le support sch\'ematique de ${\cal{F}}$ est alors d\'efini par le d\'eterminant
de ce morphisme (c'est la courbe spectrale associ\'e \`a $\varphi$) et
${\cal{F}}$ s'identifie au conoyau du morphisme
$(\varphi-\lambda id_{G})\otimes id_{N^{-1}}$.
Maintenant, le $\omega_{_{\planp}}$-dual de ${\cal{F}}$ s'identifie au
$\omega_{_{\droitep}}$-dual du fibr\'e de Higgs associ\'e:
\begin{lemme} Soit ${\cal{F}}$ un faisceau pur dont
le support ne passe pas par $a$ et $(G,\varphi)$ le fibr\'e de Higgs associ\'e
par la correspondance de la proposition \ref{cat-equiv}. Alors le fibr\'e de
Higgs associ\'e \`a ${\cal{F}}^{\vee}$ s'identifie au
$\omega_{_{\droitep}}$-dual $(G^{\vee},^{\vee}\varphi)$ de $(G,\varphi)$
\end{lemme}
\begin{proof} Soit $C$ le support sch\'ematique de ${\cal{F}}$.
Consid\'er\'e sur $C$, le faisceau ${\cal{F}}^{\vee}$ s'identifie \`a
$\ul{\Hom}_{{\cal{O}}_{C}}({\cal{F}},\omega_{C})$. On va consid\'erer le
morphisme fini $C\lra\droitep$ d\'efini par $\pi$. La dualit\'e de
Serre-Grothendieck fournit un isomorphisme
$$R\pi_{*}R\ul{\Hom}_{{\cal{O}}_{C}}({\cal{F}},\omega_{C})\simeq
R\ul{\Hom}_{{\cal{O}}_{\droitep}}(R\pi_{*}{\cal{F}},\omega_{\droitep})$$ dans
la cat\'egorie d\'eriv\'ee. Maintenant, comme ${\cal{F}}$ est un
${\cal{O}}_{C}$-module de Cohen-Macaulay, on a
$\ul{\Ext}^{i}({\cal{F}},\omega_{C})=0$ pour $i\ge 1$ \cite{12}. Par
cons\'equent,
les images
directes sup\'erieures \'etant nulles et $\pi_{*}{\cal{F}}$ \'etant localement
libre, l'isomorphisme ci-dessus se lit
$\pi_{*}\ul{\Hom}_{{\cal{O}}_{C}}({\cal{F}},\omega_{C})\simeq
\ul{\Hom}_{{\cal{O}}_{\droitep}}(\pi_{*}{\cal{F}},\omega_{\droitep})$.
Et puisque un isomorphisme dans la cat\'egorie d\'eriv\'ee entre deux
complexes de longueur $1$ d\'efinit un isomorphisme entre les objets
on a l'isomorphisme cherch\'e.
\end{proof}
La correspondance de la proposition est donc obtenu en associant \`a la
th\^eta-caract\'eristique $({\cal{F}},\sigma)$ le fibr\'e de Higgs quadratique
$(G,\varphi,\tau)$ o\`u $G=\pi_{*}({\cal{F}})$ et $\tau=\pi_{*}(\sigma)$.
\end{proof}
Remarquons que cette correspondance pr\'eserve la multiplicit\'e et la
caract\'eristique d'Euler-Poincar\'e (suite spectrale de Leray). En
particulier
${\cal{F}}$ est semi-stable (resp. stable) si et seulement si le
fibr\'e de Higgs associ\'e
$(G,\varphi)$ est semi-stable (resp. stable). De plus, si
$({\cal{F}},\sigma)$ est une th\^eta-caract\'eristique et $(G,\varphi,\tau)$
le fibr\'e de Higgs quadratique associ\'e alors si ${\cal{F}}'\subset{\cal{F}}$
est un sous-faisceau, le sous-faisceau de Higgs $\pi_{*}({\cal{F}}')$ de
$(G,\varphi)$ est d'orthogonal $\pi_{*}({\cal{F}}^{\prime\ort})$. En
particulier
$({\cal{F}},\sigma)$ est semi-stable (resp. stable) si et seulement si
$(G,\varphi,\tau)$ est semi-stable (resp. stable).
\subsubsection{Le foncteur $\protect\underline{HQ}(d,N)$}
Soient $d$ un entier et soit $N$ choisi de fa\c con \`a
ce que pour tout fibr\'e
de Higgs quadratique semi-stable $(G,\varphi,\tau)$ de rang $d$, $G(N)$
n'ait pas de cohomologie sup\'erieure et soit engendr\'e par ses sections
globales.
Ceci est possible, la familles des fibr\'es de Higgs quadratiques
semi-stables de rang fix\'e \'etant limit\'ee en raison de la
correspondance ci-dessus et le fait que la famille des
th\^eta-caract\'eristiques de multiplicit\'e fix\'ee est limit\'ee. Soit
$n=dN$,
$H=k^{n}$ et ${\tr{H}}=H\otimes_{k}{\cal{O}}_{\droitep}(-N)$. Consid\'erons,
pour toute vari\'et\'e alg\'ebrique $S$, les quadruplets
$(G,\alpha,\varphi,\tau)$ form\'es d'un quotient coh\'erent
${\tr{H}}_{S}\hfl{\alpha}{}G$, $S$-plat, d'une structure de Higgs
$\varphi$ sur $G$ et d'un isomorphisme de Higgs sym\'etrique
$\tau:G\lra G^{\vee}$ satisfaisant aux conditions suivantes:
pour tout point ferm\'e $s$ de $S$, $(G_{s},\varphi_{s},\tau_{s})$ est un
fibr\'e de Higgs quadratique de rang $d$ et $\alpha$ induit
un isomorphisme $H\otimes{\cal{O}}_{S}\simeq pr_{1*}(G(N))$. Deux
quadruplets $(G,\alpha,\varphi,\tau)$ et
$(G',\alpha',\varphi',\tau')$ sont dits \'equivalents s'il existe un
isomorphisme de fibr\'es de Higgs
$f:(G,\varphi)\lra (G',\varphi')$ tel que
$\alpha'=f\circ \alpha$ et tel que
$\tau'\circ f=(^{\vee}f)^{-1}\circ\tau$. Notons
par $[G,\alpha,\varphi,\tau]$ la classe d'\'equivalence du
quadruplet $(G,\alpha,\varphi,\tau)$ et par
$\ul{HQ}(d,N)(S)$ l'ensemble des classes d'\'equivalence de
tels quadruplets. Ceci d\'efinit un foncteur:
$$ \ul{HQ}(d,N):\text{Vari\'et\'es alg\'ebriques}
\lra\text{Ensembles}.
$$
Le foncteur qu'on obtient en supposant de plus que pour tout point ferm\'e
$s$ on a $[G_{s},\varphi_{s},\tau_{s}]$ semi-stable sera not\'e
$\ul{HQ}^{ss}(d,N)$, celui qu'on obtient en consid\'erant
au lieu des quadruplets seulement les triplets $[G,\alpha,\tau]$ (sans
structure de Higgs) sera not\'e $\ul{Q}(d,N)$.
On va repr\'esenter $\ul{HQ}^{ss}(d,N)$ de la mani\`ere suivante:
Soit $\Groth(d,N)$ le sch\'ema de
Hilbert-Grothendieck des quotients de $H\otimes{\cal{O}}_{\droitep}(-N)$ de
polyn\^ome de Hilbert $md$. Notons $\Groth_{0}$ l'ouvert
correspondant aux quotients $G$ tels que $H\simeq\H^{0}(G(N))$.
Cet ouvert param\`etre les classes d'\'equivalence des
fibr\'es vectoriels $G$ de rang $d$ et de caract\'eristique
d'Euler-Poincar\'e nulle muni d'un isomorphisme $\alpha:H\simeq G(N)$.
Ici, $(G,\alpha)$ et $(G',\alpha')$ sont \'equivalents s'il existe un
isomorphisme $f:G\lra G'$ tel que $\alpha'=f\circ\alpha$.
Soit $\underline{G}$ le quotient universel sur
$\Groth_{0}\times\droitep$ et consid\'erons le faisceau coh\'erent
$pr_{1*}(\underline{G}^{\vee}(N))$. Par hypoth\`ese ce faisceau est
localement libre de fibre $H^{0}(G^{\vee}(N))$ au
dessus du point repr\'esent\'e par $[G,\alpha]$.
Soit ${\cal{R}}$ son fibr\'e de
rep\`eres.
L'espace total de ${\cal{R}}$ param\`etre les classes d'\'equivalence
des fibr\'es vectoriels $G$ de rang $d$ et de caract\'eristique
d'Euler-Poincar\'e nulle muni d'un isomorphisme $\alpha:H\simeq G(N)$
et d'un isomorphisme $\beta:H\simeq G^{\vee}(N)$. Deux triplets sont
\'equivalents si s'il existe un isomorphisme $f:G\lra G'$ tel que
$\alpha'=f\circ\alpha$ et $\beta'=^{\vee}f^{-1}\circ\beta$.
On a une involution sur ${\cal{R}}$ en associant au triplet
$[G,\alpha,\beta]$ le triplet $[G^{\vee},\beta,\alpha]$. Soit
$Q$ le sch\'ema
des points fixes de ${\cal{R}}$ sous cette involution.
\begin{lemme}
Le sch\'ema $Q$ repr\'esente le foncteur $\ul{Q}$.
\end{lemme}
\begin{proof} Analogue \`a la d\'emonstration de la proposition 7.2 de
\cite{11}.
\end{proof}
Consid\'erons le triplet
universel $[\ul{G},\ul{\alpha},\ul{\tau}]$ sur
$$Q\times \droitep.$$
On peut identifier via $\ul\tau$ les faisceaux des
${\cal{O}}_{Q\times \droitep}$-modules
$\ul{\Hom}(\ul{G},\ul{G}(-3))$ et $\ul{\Hom}(\ul{G},\ul{G}^{\vee}(-3))$.
Soit $\ul{\Hom}_{sym}(\ul{G},\ul{G}^{\vee}(-3))$ le
sous-faisceau des homomorphismes sym\'etriques. Ce sous-faisceau d\'efinit via
l'identification ci-dessus un sous-faisceau de
$\ul{\Hom}(\ul{G},\ul{G}(-3))$ qu'on note
$\ul{\Hom}_{sym}(\ul{G},\ul{G}(-3))$.
Soit $HQ(d,N)$ le fibr\'e
vectoriel (au sens de Grothendieck) associ\'e au faisceau coh\'erent
$R^{1}pr_{1*}\ul{\Hom}_{sym}(\ul{G},\ul{G}(-3))$.
Sa fibre au-dessus de $[G,\alpha,\tau]$ s'identifie, par dualit\'e
de Serre, aux morphismes $\varphi:G\lra G(1)$ $\tau$-sym\'etriques,
\ie aux morphismes $\varphi$ tels que $(G,\varphi,\tau)$ soit un
fibr\'e de Higgs quadratique.
Le sch\'ema $HQ(d,N)$ repr\'esente
$\ul{HQ}(d,N)$. On note $HQ^{ss}(d,N)$ l'ouvert de $HQ(d,N)$
repr\'esentant
$\ul{HQ}^{ss}(d,N)$.
Soit $\ul{T}_{a}^{ss}(d,N)$ le sous-foncteur ouvert de
$\ul{T}^{ss}(d,N)$ d\'efini par les triplets
$[{\cal{F}},\alpha,\sigma]$ tels que
$a\not\in\supps({\cal{F}})$.
D'apr\`es le proposition \ref{cat-equiv}, on a
\begin{prop}\label{isom} L'image directe induit un
isomorphisme de foncteurs entre les foncteurs $\ul{T}_{a}^{ss}(d,N)$ et
$\ul{HQ}^{ss}(d,N)$.
\end{prop}
Le groupe $GL(H)$ op\`ere sur $HQ^{ss}(d,N)$. D'apr\`es la
proposition \ref{isom}, le quotient de Mumford s'identifie \`a
l'ouvert
$U_{a}(d)$. On note $HQ^{os}(d,N)$ (resp. $U_{a}^{os}(d)$)
l'ouvert correspondant aux th\^eta-caract\'eristiques
${\cal{O}}$-stables. L'op\'eration de $GL(H)/\{\pm 1\}$ est libre sur
$HQ^{os}(d,N)$ de quotient
$U_{a}^{os}(d)$. Ainsi, le morphisme
$$HQ^{os}(d,N)\lra U_{a}^{os}(d)$$ est lisse.
On note $Q^{os}(d,N)$ l'ouvert de $Q(d,N)$ d\'efini par l'image de
$HQ^{os}(d,N)$ sous la projection.
\subsection{D\'emonstration du th\'eor\`eme \protect\ref{codimU}}
L'ouvert $U_{a}(d)$ \'etant \'equidimensionnel (car $\Theta(d)$ l'est)
et le ferm\'e des th\^eta-caract\'eristiques semi-stables, non
${\cal{O}}$-stables \'etant de codimension au moins $2$, il suffit de
consid\'erer l'ouvert
$U_{a}^{os}(d)$. Dans ce qui suit on notera simplement
$HQ^{os}$ et $Q^{os}$ les sch\'emas $HQ^{os}(d,N)$ et
$Q^{os}(d,N)$.
Consid\'erons la projection
$$p:HQ^{os}\lra Q^{os}$$
et un triplet $[G,\alpha,\tau]$ correspondant \`a un point ferm\'e de
$Q^{os}$. Soit $O(G,\alpha,\tau)$ l'orbite dans $Q^{os}$ de
$[G,\alpha,\tau]$ sous l'action de $GL(H)$ . Remarquons que $\tau$ est
n\'ecessairement donn\'e par une constante non nulle \ie $\tau\in k^{*}$.
En effet, $Q^{os}$ est par d\'efinition l'image de $HQ^{os}$, qui repr\'esente
les th\^eta-caract\'eristique ${\cal{O}}$-stables ne passant par par $a$. Or
si $({\cal{F}},\sigma)$ est ${\cal{O}}$-stable $\sigma\in k^{*}$.
Il s'ensuit que l'orbite de d\'epend que de $G$.
La codimension de l'orbite de $[G,\alpha,\tau]$ est
donn\'ee par $\ext^{1}_{asym}(G,G)$.
Soit $Z(G)$ l'image
r\'eciproque de $O(G,\alpha,\tau)$ sous $p$ et
calculons sa codimension: au-dessus de l'orbite la dimension de la fibre
est donn\'ee par $\hom_{sym}(G,G(1))$. D'apr\`es la proposition
\ref{isom} et \cite{11}, la vari\'et\'e
$HQ^{os}$ est lisse de dimension $\frac{d(d+3)}{2}+n^{2}$. La
vari\'et\'e
$Q^{os}$ est lisse, elle aussi: c'est un ouvert d'un sch\'ema des points
fixes sous une involution d'une vari\'et\'e lisse. Sa dimension se calcule
suivant la suite exacte suivante: (\cite{11} corollaire 7.6)
$$0\ra\Hom_{sym}(G,G)\ra
T_{[G,\alpha,\tau]}Q\ra
T_{[G,\alpha]}\Groth\ra
\Ext^{1}_{sym}(G,G)\ra 0$$
On en d\'eduit que $\dim Q^{os}=n^{2}-d^{2}+\chi_{sym}(G,G)$.
Par la formule de Riemann-Roch $\chi_{sym}(G,G)=\frac{d(d+1)}{2}$ d'o\`u
$\dim Q^{os}=n^{2}-\frac{d(d-1)}{2}$. De ces calculs on obtient
que la codimension de l'image r\'eciproque est donn\'ee par
$$\text{ext}^{1}_{asym}(G,G)-\text{ext}^{1}_{sym}(G,G(1)).$$
Soit $r_{\ell}$ le spectre de $G$. Ce spectre est connexe par
semi-stabilit\'e. L'espace vectoriel $\Ext^{1}_{asym}(G,G)$
s'identifie \`a $H^{1}(\Lambda^{2}G^{*}\otimes\omega_{_{\proj_{1}}})$ et
l'espace vectoriel $\Ext^{1}_{sym}(G,G(1))$ \`a
$H^{1}(S^{2}G^{*}\otimes\omega_{_{\proj_{1}}}(1))$. Par cons\'equent:
$$
\text{ext}^{1}_{asym}(G,G)-\text{ext}^{1}_{sym}(G,G(1))=
\displaystyle
\sum_{\ell\ge 0}\left(\frac{r_{\ell}(r_{\ell}-1)}{2}-2\ell r_{\ell}\right)+
\sum\begin{Sb}\ell+m\ge 0\\ \ell<m\end{Sb}r_{\ell}r_{m}.
$$
Cette expression est toujours positive,
nulle dans exactement $3$ cas:
celui o\`u $G$ est \'egale \`a $R_{0}$ ou $R_{1}$
du lemme \ref{rigide} ou si
$$G={\cal{O}}(\ell)\osum{\cal{O}}(\ell-1)\osum\dots
\osum{\cal{O}}(-\ell-1)\osum{\cal{O}}(-\ell-2).$$
Les images r\'eciproques $Z(G)$ sont, dans ces trois cas, $GL(H)/\{\pm
id\}$-invariants et irr\'eductibles. Et comme ils d\'efinissent respectivement
les th\^eta-caract\'eristiques ineffectives, semi-ineffectives et canoniques on
d\'eduit la premi\`ere partie du th\'eor\`eme, la codimension du
compl\'ementaire de
$U_{a}^{2,os}(d)$ \'etant, sauf dans le cas des
th\^eta-caract\'eristiques canoniques,
sup\'erieure ou \'egale \`a $1$ d'apr\`es ce qui
pr\'ec\`ede. Maintenant l'expression ci-dessus vaut $1$ exactement,
toujours par connexit\'e du spectre, pour
\begin{list}{-}{}
\item $G=R_{2}=
2{\cal{O}}\osum(d-4){\cal{O}}(-1)\osum 2{\cal{O}}(-2)
\text{ ($d\ge 5$ pour avoir la connexit\'e})$\par et aussi pour
\item $G=R_{3}=
{\cal{O}}(1)\osum{\cal{O}}\osum2{\cal{O}}(-1)\osum{\cal{O}}(-2)
\osum{\cal{O}}(-3)
\text{ (ce qui impose $d=6$})$
\end{list}
Les images r\'eciproques $Z(G)$ sont, dans ces deux
cas, $GL(H)/\{\pm id\}$-invariants et irr\'eductibles.
Ils d\'efinissent, respectivement, les sous-sch\'emas localement ferm\'es
$U_{a}^{1,os}(d)\backslash U_{a}^{2,os}(d)$ et
$U_{a}^{2,os}(6)\backslash U_{a}^{3,os}(6)$ d'o\`u le th\'eor\`eme.
\cqfd
La d\'emonstration montre que le nombre maximal de sections
possibles d'une th\^eta-caract\'eristique semi-stable de degr\'e $d$ est
donn\'ee, si $d$ est impair par la dimension de l'espace des sections
de (avec $\ell=\frac{d-3}{2}$)
$$G={\cal{O}}(\ell)\osum{\cal{O}}(\ell-1)\osum\dots
\osum{\cal{O}}\osum{\cal{O}}(-1)\osum{\cal{O}}(-2)\dots
\osum{\cal{O}}(-\ell-1)\osum{\cal{O}}(-\ell-2),$$
si $d$ est pair par la dimension de l'espace des sections de
$$G={\cal{O}}(\ell)\osum{\cal{O}}(\ell-1)\osum\dots
\osum{\cal{O}}\osum2{\cal{O}}(-1)\osum{\cal{O}}(-2)\dots
\osum{\cal{O}}(-\ell-1)\osum{\cal{O}}(-\ell-2),$$
On en d\'eduit le corollaire suivant:
\begin{cor}
Soit $({\cal{F}},\sigma)$ une th\^eta-caract\'eristique semi-stable de degr\'e
$d$.
\begin{list}{-}{}
\item Si $d$ est impair on a
$\h^{0}({\cal{F}})\le\frac{(d-3)^2}{8}+\frac{3(d-3)}{4}+1$.
\item Si $d$ est pair on a
$\h^{0}({\cal{F}})\le\frac{(d-4)^2}{8}+\frac{3(d-4)}{4}+1$.
\end{list}
\end{cor}
En particulier, $\Theta^{1}(d)$ est vide si $d\leq 4$,
$\Theta^{2}(d)$ est vide si
$d\leq 5$ et $\Theta^{3}(d)$ est vide si $d\leq 7$.
\begin{lemme}
Si $d\ge 5$ il existe une th\^eta-caract\'eristique
$({\cal{F}},\sigma)$ semi-stable de degr\'e
$d$ telle que $\h^{0}({\cal{F}})=2$.
\end{lemme}
\begin{proof}
Si $d\ge 6$, on peut consid\'erer par exemple la somme directe orthogonale
${\cal{O}}_{C}\osum{\cal{O}}_{C}\osum(d-6){\cal{O}}_{\ell}(-1)$ avec
$C$ une courbe de degr\'e $3$ et $\ell$ une droite de $\planp$. Si $d=5$,
consid\'erons le fibr\'e vectoriel $G=2{\cal{O}}\osum{\cal{O}}(-1)\osum
2{\cal{O}}(-2)$ sur $\droitep$ et l'isomorphisme
$\sigma:G\lra G^{\vee}$ d\'efini par $1$ sur l'anti-diagonale, $0$
sinon. Soit de plus $\varphi:G\lra G(1)$ d\'efini par la matrice
suivante:
$$\left(\begin{matrix}
\alpha&0&0&0&0\\
0&\beta&0&0&0\\
\lambda&\mu&0&0&0\\
0&0&\mu&0&0\\
0&0&\lambda&0&0
\end{matrix}\right)
$$
avec $\lambda$ et $\mu$ des constantes non nuls, $\alpha$ et
$\beta$ non proportionnels. On v\'erifie cas par cas qu'aucun
sous-fibr\'e de $G$ contenant ${\cal{O}}$ n'est de Higgs. Ainsi
$(G,\varphi,\sigma)$ est un fibr\'e de Higgs quadratique semi-stable,
d\'efinissant, pour
$a\in\planp$ fix\'e, par le proc\'ed\'e ci-dessus une
th\^eta-caract\'eristique semi-stable de degr\'e
$5$ ne passant pas par $a$.
\end{proof}
\subsection{Le compl\'ementaire de $U_{a}(d)$ dans $\Theta(d)$.}
Consid\'erons le compl\'ementaire $F_{a,p}(d)$ de $U_{a}(d)$
dans $\Theta_{p}(d)$ (resp.
$F_{a,i}(d)$ de $U^{a}_{1}(d)$ dans $\Theta_{i}(d)$).
\begin{prop} Pour les ferm\'es $F_{a,p}(d)$ et $F_{a,i}(d)$ on a:
\begin{list}{\arabic{lc}.}{\usecounter{lc}}
\item le ferm\'e $F_{a,p}(d)$ est une hypersurface irr\'eductible de
$\Theta_{p}(d)$.
\item le ferm\'e $F_{a,i}(d)$ est une hypersurface irr\'eductible de
$\Theta_{i}(d)$.
\end{list}
\end{prop}
\begin{proof}
$1)$ Soient $a,b\in\planp$. Quitte \`a \'echanger $a$ et $b$ il s'agit de
montrer l'\'enonc\'e ci-dessus pour les th\^eta-caract\'eristiques
ne passant pas par $b$. D'abord on a besoin du lemme suivant:
\begin{lemme} On a, pour $[R_{0},\alpha,\tau]\in Q^{os}$,
l'isomorphisme suivant, avec $K=\Aut(R_{0},\tau)$:
$$Sym^{os}(R_{0},R_{0}(1))/K\simeq I_{a}^{os}(d)$$
o\`u $I_{a}^{os}(d)$ d\'esigne l'ouvert de $U_{a,p}(d)$ correspondant
aux th\^eta-caract\'eristiques ineffectives.
\end{lemme}
\begin{proof}
Le groupe $K$ s'identifie au stabilisateur de $[R_{0},\alpha,\tau]$
sous l'action de $GL(H)$ sur $Q^{os}$. Ce groupe agit sur
$Sym^{os}(R_{0},R_{0}(1))$ par conjugaison et l'on a le diagramme commutatif
$$
\begin{diagram}
Sym^{os}(R_{0},R_{0}(1))&\efl{i}{}&Z(G)\\
&\sefl{\pi'}{}&\sfl{\pi}{}\\
&&I_{a}^{os}(d)\\
\end{diagram}
$$
L'inclusion $i$ est compatible aux actions de $K$ et $GL(H)$ et $\pi$ est un
quotient g\'eom\'etrique.
Par suite, $\pi'$ est un quotient g\'eom\'etrique aussi,
d'o\`u le lemme.
\end{proof}
Si $(R,\varphi,\tau)$ est le fibr\'e de Higgs quadratique associ\'e \`a la
th\^eta-caract\'eristique $({\cal{F}},\sigma)$, le support du faisceau
${\cal{F}}$ est d\'efini par les couples $(x,\lambda)$ tels que
$$\det(\varphi(x)-\lambda\id_{R(x)})=0.$$ Maintenant
le fibr\'e
$\ul{Sym}(R_{0},R_{0}(1))$ est engendr\'e par ses sections globales.
Par cons\'equent l'image r\'eciproque sous le morphisme \'evaluation
$$Sym(R_{0},R_{0}(1))\lra \ul{Sym}(R_{0},R_{0}(1))(b)$$
de l'hypersurface des applications lin\'eaires sym\'etriques de $R_{0}\lra
R_{0}(1)$ de d\'eterminant nul est une hypersurface irr\'eductible de
$Sym^{os}(R_{0},R_{0}(1))$. Cette hypersurface est $K$-invariante et
d\'efinit, par passage au quotient, l'hypersurface des
th\^eta-caract\'eristiques de
$I_{a}^{os}(d)$ dont le support passe par $b$. Cette
hypersurface est donc irr\'eductible et comme le ferm\'e des
th\^eta-caract\'eristiques non
${\cal{O}}$-stables est de codimension au moins deux
ceci est encore vrai sans supposer la ${\cal{O}}$-stabilit\'e.
Pour voir que l'hypersurface
$F_{b}(d)$ est encore irr\'eductible dans $\Theta_{p}$, il suffit de
montrer qu'il existe des th\^eta-caract\'eristiques ayant au moins deux
sections lin\'eairement ind\'ependantes ne passant ni par $a$ ni par $b$. Pour
cela, consid\'erons $Sym(R_{2},R_{2}(1))$. Ce fibr\'e n'est pas
engendr\'e par ses sections. Cependant l'image de
$$Sym(R_{2},R_{2}(1))\lra \ul{Sym}(R_{2},R_{2}(1))(b)$$
est un espace lin\'eaire de codimension $3$ qui n'est pas inclus dans
l'hypersurface des applications lin\'eaires sym\'etriques de $R_{2}\lra
R_{2}(1)$ de d\'eterminant nul. Ceci d\'emontre la derni\`ere assertion
et par cons\'equent $(i)$.
$2)$ Un argument analogue \`a celui ci-dessus montre qu'on a, pour
$[R_{1},\alpha,\tau]\in Q^{os}$, l'isomorphisme suivant, avec
$K=\Aut(R_{1},\tau)$: $$Sym^{os}(R_{1},R_{1}(1))/K\simeq
SI_{a}^{os}(d),$$
o\`u $SI_{a}^{os}(d)$ d\'esigne l'ouvert de $U_{a,i}(d)$ correspondant
aux th\^eta-caract\'eristiques semi-ineffectives.
Maintenant le fibre
$\ul{Sym}(R_{1},R_{1}(1))$ n'est plus engendr\'e par ses sections.
Cependant l'intersection de l'hyperplan image de
$Sym(R_{1},R_{1}(1))$ dans $\ul{Sym}(R_{1},R_{1}(1))(b)$
avec
l'hypersurface
des application lin\'eaires sym\'etriques de $R_{1}\lra R_{1}(1)$ de
d\'eterminant nul est irr\'eductible de codimension $1$ dans l'image.
L'image r\'eciproque de cette intersection d\'efinit une hypersurface
de
$Sym^{os}(R_{1},R_{1}(1))$.
Cette hypersurface est $K$-invariante et
d\'efinit l'hypersurface des th\^eta-caract\'eristiques
de $SI_{a}^{os}(d)$ passant par $b$. Cette hypersurface est donc
irr\'eductible et ceci est encore vrai sans supposer la
${\cal{O}}$-stabilit\'e. Si $d\not=6$ ceci d\'emontre
$(ii)$, si
$d=6$ il faut encore voir qu'il existe des
th\^eta-caract\'eristiques ayant $3$ sections lin. ind\'ependantes ne
passant ni par $a$ ni par $b$.
Pour cela, consid\'erons l'\'evaluation
$$Sym(R_{3},R_{3}(1))\lra \ul{Sym}(R_{3},R_{3}(1))(b)$$
L'image est un espace lin\'eaire de codimension $5$ n'\'etant pas
inclus dans l'hypersurface des application lin\'eaires sym\'etriques
de $R_{2}\lra R_{2}(1)$ de d\'eterminant nul. Par cons\'equent il
existe de telles th\^eta-caract\'eristiques dans $U_{a}(6)$ qui
sont dans
$U_{b}(6)$, d'o\`u $(ii)$.
\end{proof}
\begin{cor} On a
\begin{list}{\arabic{lc})}{\usecounter{lc}}
\item L'hypersurface $\Theta^{1}(d)$ de $\Theta_{p}(d)$
est irr\'eductible, le ferm\'e $\Theta^{3}(d)$ est de codimension au
moins deux.
\item Le ferm\'e $\Theta^{2}(d)$ est de codimension au moins deux, sauf si
$d=6$ o\`u la codimension est $1$.
\end{list}
\end{cor}
\begin{proof}
L'\'enonc\'e est vrai pour l'ouvert $U_{a,p}(d)$. Il suffit donc de
montrer que
$\Theta^{1}(d)$ coupe $F_{a,p}(d)$ suivant un ferm\'e de codimension au
moins $2$. Soit $\ell$ une droite de $\planp$ passant par $a$,
$({\cal{F}},\sigma)$ la th\^eta-caract\'eristique somme directe orthogonale
$d{\cal{O}}(-1)$. Alors $({\cal{F}},\sigma)\in F_{a,p}(d)$. Par
semi-continuit\'e $F_{a}^{1}(d)$ est un ferm\'e de $F_{a,p}(d)$, par
irr\'eductibilit\'e il est de codimension au moins $1$ dans $F_{a,p}(d)$.
L'assertion sur $\Theta^{3}(d)$ se d\'eduit de l'assertion analogue pour
l'ouvert $U_{a,p}(d)$ et de ce qui pr\'ec\`ede.
La d\'emonstration de $2)$ est analogue \`a celle de $1)$.
\end{proof}
\np
\section{La
r\'esolution minimale d'une th\^eta-caract\'eristique semi-stable}
\label{resolution}
Soit $({\cal{F}},\sigma)$ une th\^eta-caract\'eristique de degr\'e $d$.
Consid\'erons la $k$-alg\`ebre gradu\'e $S=k[X_{0},X_{1},X_{2}]$ et le
$S$-module
gradu\'e $M=\osum_{n\in\reln}\H^{0}({\cal{F}}(n))$. La r\'esolution minimale
gradu\'e de $M$ fournit une r\'esolution $(\star)$:
$$
0\lra E \hfl{\phi}{}
F\lra {\cal{F}}\lra 0
$$
avec $E=\osum_{i=1}^{n}{\cal{O}}_{\planp}(-a_{i})$ et
$F=\osum_{i=1}^{n+1}{\cal{O}}_{\planp}(-b_{i})$. Le morphisme $\phi$
s'identifie \`a une $n\times n$-matrice $\phi_{ij}$ avec
$\phi_{ij}\in\Hom({\cal{O}}(-a_{i}),{\cal{O}}(-b_{i}))$. Par
minimalit\'e de la r\'esolution, il ne peut exister de $i,j$ tel que
$\phi_{ij}$ soit une constante non nulle. On utilisera ce fait un peu
plus loin.
En appliquant le
foncteur $\ul{\Hom}(\cdot,\omega_{_{\planp}})$ \`a la
r\'esolution $(\star)$ on voit,
comme deux r\'esolution minimales sont isomorphes, que $F\simeq E^{\vee}$, \ie
que
$-b_{i}=a_{i}-3$ apr\`es r\'enumerotation. Posons
$b=h^{1}({\cal{F}})=h^{0}({\cal{F}})$,
$c=h^{1}({\cal{F}}(1))=h^{0}({\cal{F}}(-1))$.
On a $b>c$.
\begin{prop} Supposons $({\cal{F}},\sigma)$ semi-stable.
Alors la r\'esolution minimale $(\star)$ est de la forme suivante:
\begin{list}{-}{}
\item si $b=0$ on a $n=d$ et $(a_{i})=(2,\dots,2)$
\item si $b=1$ on a $n=d-2$ et $(a_{i})=(2,\dots,2,3)$
\end{list}
De plus, on peut supposer $\phi$ sym\'etrique dans le deux cas.
\end{prop}
\begin{proof}
Cette proposition est due \`a Dixon, si $b=0$ \cite{3}, \`a
Catanese \cite{2} sinon (\cf aussi Laszlo
\cite{7}) dans le cas o\`u le support sch\'ematique de
$({\cal{F}},\sigma)$ est lisse. L'argument de Laszlo se g\'en\'eralise
\`a notre situation, quitte \`a remplacer
l'argument d'irr\'eductibilit\'e du support qu'il invoque par un
argument utilisant la semi-stabilit\'e de $({\cal{F}},\sigma)$.
Nous omettons les d\'etails ici en raison de la proposition suivante.
\end{proof}
\begin{prop}\label{Beilinson}
Soit $({\cal{F}},\sigma)$ une th\^eta-caract\'eristique semi-stable.
Si $({\cal{F}},\sigma)$ est ineffective, ${\cal{F}}$ est conoyau du
morphisme $d_{2}$ sym\'etrique et g\'en\'eriquement injectif,
donn\'e par la suite spectrale de Beilinson et $\sigma$:
$$\H^{1}({\cal{F}}(-1))\otimes{\cal{O}}_{\planp}(-2)
\hfl{d_{2}}{}\H^{1}({\cal{F}}(-1))^{*}\otimes {\cal{O}}_{\planp}(-1)$$\par
Si $({\cal{F}},\sigma)$ est semi-ineffective, alors ${\cal{F}}$ est
conoyau du morphisme $d_{2}$ sym\'etrique et g\'en\'eriquement injectif,
donn\'e par la suite spectrale de Beilinson et $\sigma$:
$$K\otimes{\cal{O}}_{\planp}(-2)\osum
H^{1}({\cal{F}})\otimes{\cal{O}}_{\planp}(-3)
\hfl{d_{2}}{}
K^{*}\otimes{\cal{O}}_{\planp}(-1)\osum
H^{1}({\cal{F}})^{*}\otimes{\cal{O}}_{\planp},
$$ o\`u
$K$ est le noyau de
l'application canonique
$\H^{1}({\cal{F}}(-1))\lra\H^{1}({\cal{F}})\otimes V$ avec
$V=H^{0}(Q)$, $Q$ \'etant le quotient tautologique sur $\planp$.
\end{prop}
\begin{proof}
Soit ${\cal{E}}$ un ${\cal{O}}_{\planp}$-module
coh\'erent. La suite spectrale de Beilinson est donn\'ee, en degr\'e $1$,
par:
$$E_{1}^{p,q}=\H^{q}(\planp,{\cal{E}}(p))\otimes\Lambda^{-p}Q^{*}.$$
Elle est d'aboutissement
${\cal{E}}$ en degr\'e $0$, nul sinon.
Soit $({\cal{F}},\sigma)$ une th\^eta-caract\'eristique de degr\'e $d$.
Supposons que
$h^{0}(\planp,{\cal{F}})\leq1$. Par d\'ecroissance de la suite des
$h^{1}({\cal{F}}(i))$, on a
$h^{1}(\planp,{\cal{F}}(1))=0$. Nous allons appliquer la suite
spectrale de Beilinson \`a ${\cal{F}}(1)$, puis tordre par
${\cal{O}}_{\planp}(-1)$.
$$
\begin{array}{|c|c|c|}
\hline
0 & 0 & 0\\
\hline
\H^{1}({\cal{F}}(-1))\otimes\Lambda^{2}Q^{*}(-1)&
\H^{1}({\cal{F}})\otimes Q^{*}(-1)&0\\
\hline
0&\H^{0}({\cal{F}})\otimes Q^{*}(-1)&
\H^{0}({\cal{F}}(1))\otimes {\cal O}(-1)\\
\hline
\end{array}
$$
Soient $H_{1}$ et $H_{2}$ les
${\cal{O}}_{\planp}$-modules de cohomologie du complexe
$$
0\lra \H^{1}({\cal{F}}(-1))\otimes\Lambda^{2}Q^{*}(-1)\hfl{d_{1}^{-2,1}}{}
\H^{1}({\cal{F}})\otimes Q^{*}(-1)\lra 0,
$$ et $K_{2}$ et $K_{3}$ les ${\cal{O}}_{\planp}$-modules de
cohomologie du complexe
$$
0\lra \H^{0}({\cal{F}})\otimes Q^{*}(-1)\hfl{d_{1}^{-1,0}}{}
\H^{0}({\cal{F}}(1))\otimes {\cal{O}}_{\planp}(-1) \lra 0.
$$ Le terme $E_{2}^{p,q}$ s'\'ecrit par cons\'equent:
$$
\begin{array}{|c|c|c|}
\hline
0 & 0 & 0\\
\hline
H_{1} & H_{2} & 0\\
\hline
0 & K_{2} & K_{3}\\
\hline
\end{array}
$$
Soient $L_{1}$ et $L_{3}$ les
${\cal{O}}_{\planp}$-modules de cohomologie du complexe
$$0 \lra H_{1}\hfl{d_{2}}{} K_{3} \lra 0$$ Par suite, on a pour le
terme $E_{3}$:
$$
\begin{array}{|c|c|c|}
\hline
0 & 0 & 0\\
\hline
L_{1}& H_{2} & 0\\
\hline
0 & K_{2} & L_{3}\\
\hline
\end{array}
$$
Du fait que $E_{3}=E_{\infty}$, on d\'eduit que
$K_{2}=L_{1}=0$ et qu'on a la suite exacte
$$0\lra L_{3}\lra{\cal{F}}\lra H_{2}\lra 0.$$
D\'eterminons $K_{3}:$ Si $b=0$, on a bien s\^ur
$K_{3}=\H^{0}({\cal{F}}(1))\otimes{\cal{O}}_{\planp}(-1)$. Si $b=1$,
consid\'erons le diagramme commutatif avec lignes et colonnes exactes, la
premi\`ere ligne \'etant la dualis\'e tordue par
$H^{0}({\cal{F}})\otimes{\cal{O}}_{\planp}(-1)$ de la suite exacte
d'Euler:
$$
\begin{diagram}
& & & & 0& & 0\\
& & & & \vfl{}{}& & \vfl{}{}\\
& & & & M_{1}&\ra& M_{2}\\
& & & & \vfl{}{}& & \vfl{}{}\\
0&\ra &\H^{0}({\cal{F}})\otimes Q^{*}(-1)
&\ra&\H^{0}({\cal{F}})\otimes
V^{*}\otimes {\cal{O}}(-1)&\ra&
\H^{0}({\cal{F}})\otimes {\cal{O}} &\ra&0\\
& & \Vert &&\vfl{}{}& &\vfl{}{}\\
0&\ra &\H^{0}({\cal{F}})\otimes
Q^{*}(-1)&\ra&\H^{0}({\cal{F}}(1))\otimes{\cal{O}}(-1)&
\ra&K_{3}&\ra& 0\\
& & & & \vfl{}{}& & \vfl{}{}\\
& & & & N_{1}&\ra& N_{2}\\
& & & & \vfl{}{}& & \vfl{}{}\\
& & & & 0& & 0\\
\end{diagram}
$$
Par le lemme du serpent, $M_{1}\simeq M_{2}$ et
$N_{1}\simeq N_{2}$.
Supposons d'abord que $M_{1}$ (et donc aussi $N_{1}$) nul. A ce moment-l\`a,
$N_{1}$ (et aussi $N_{2}$) est isomorphe \`a
$H\otimes{\cal{O}}_{\planp}(-1)$, o\`u $H$ est le conoyau de l'application
canonique $\H^{0}({\cal{F}})\otimes V^{*}\lra \H^{0}({\cal{F}}(1)).$
Par cons\'equent,
$K_{3}$ s'identifie \`a une extension de
$\H^{0}({\cal{F}})\otimes{\cal{O}}_{\planp}$ et
$H\otimes{\cal{O}}_{\planp}(-1)$. L'espace de telles extensions \'etant
nul, on a, si $b$=1,
$$K_{3}=\H^{0}({\cal{F}})\otimes{\cal{O}}_{\planp}\osum
H\otimes{\cal{O}}_{\planp}(-1).$$
Montrons que $M_{1}$ non nul est impossible. Sinon, ce faisceau est
n\'ecessairement isomorphe \`a
${\cal{O}}_{\planp}(-1)$. Mais ceci signifie que $K_{3}$ s'identifie
\`a ${\cal{O}}_{\ell}\osum{\cal{O}}(-1)^{d-2}$, $\ell$
\'etant une droite. Puisque $H_{1}$ est localement libre, ce
${\cal{O}}_{\ell}$ est contenu dans $L_{3}$ et par suite dans
${\cal{F}}$. Mais ceci est en contradiction avec la semi-stabilit\'e de
${\cal{F}}$, la caract\'eristique d'Euler-Poincar\'e de
${\cal{O}}_{\ell}$ \'etant strictement positive.
\begin{lemme}\label{Bdual}
La suite spectrale de Beilinson appliqu\'ee \`a ${\cal{F}}^{\vee}(1)$ puis
tordue
par ${\cal{O}}_{\planp}(-1)$ est naturellement isomorphe
\`a la suivante, dont le terme $E_{1}$ est donn\'e par
\smallskip
$$
\begin{diagram}
\H^{0}({\cal{F}}(1))^{*}
\otimes\Lambda^{2}Q^{*}(-1)&\efl{^{\vee}d_{1,{\cal{F}}}^{-1,0}}{}&
\H^{0}({\cal{F}})^{*}\otimes Q^{*}(-1)&&0\\
0&&\H^{1}({\cal{F}})^{*}
\otimes Q^{*}(-1)&\efl{^{\vee}d_{1,{\cal{F}}}^{-2,1}}{}&
\H^{1}({\cal{F}}(1))^{*}\otimes {\cal O}(-1)\\
\end{diagram}
$$
et le terme $E_{2}$ par
$$\Ker(^{\vee}d_{1,{\cal{F}}}^{-1,0})
\efl{^{\vee}d_{2}^{-2,1}}{}\Coker(^{\vee}d_{1,{\cal{F}}}^{-2,1}).$$
\end{lemme}
Admettons pour un instant ce lemme est terminons la d\'emonstration de la
proposition.
Montrons d'abord que $\sigma$ d\'efinit un isomorphisme
de $K_{3}^{\vee}$ avec $H_{1}$:
par functorialit\'e, $\sigma$ fournit un isomorphisme au niveau des suites
spectrales de Beilinson associ\'es \`a ${\cal{F}}$ et ${\cal{F}}^{\vee}$ ce qui
donne, en utilisant le lemme pr\'ec\'edent, le diagramme commutatif
$$
\begin{diagram}
\H^{1}({\cal{F}}(-1))\otimes \Lambda^{2}Q^{*}(-1)&\efl{d^{-2,1}_{1}}{}&
\H^{1}({\cal{F}})\otimes Q^{*}(-1)\\
\sfl{}{}&&\sfl{}{}\\
\H^{0}({\cal{F}}(-1))^{*}\otimes \Lambda^{2}Q^{*}(-1)&
\efl{^{\vee}d_{1,{\cal{F}}}^{-1,0}}{}&
\H^{0}({\cal{F}})^{*}\otimes Q^{*}(-1)\\
\end{diagram}
$$
Le morphisme induit au niveau des noyaux, donne l'isomorphisme
$\tau:H_{1}\lra K_{3}^{\vee}$ cherch\'e. On en d\'eduit, en consid\'erant la
suite
exacte qui d\'efinit $H_{1}$ et $H_{2}$ que $H_{2}$ est de degr\'e nul et de
caract\'eristique d'Euler-Poincar\'e nulle. Comme $H_{2}$ est de torsion en
tant que conoyau de ${\cal{F}}$, il est n\'ecessairement nul.
Par cons\'equent, ${\cal{F}}$ s'identifie au conoyau du morphisme
$d_{2}^{-2,1}:H_{1}\lra K_{3}$. En utilisant la functorialit\'e des suites
spectrales de Beilinson et la sym\'etrie de $\sigma$, on obtient le diagramme
commutatif suivant:
$$
\begin{diagram}
H_{1}&\efl{d_{2}^{-2,1}}{}&K_{3}\\
\sfl{\tau}{}&\sefl{d_{2}}{}&\sfl{^{\vee}\tau}{}\\
K_{3}^{\vee}&\efl{}{^{\vee}d_{2}^{-2,1}}&H_{1}^{\vee}\\
\end{diagram}
$$
\bigskip
On en d\'eduit que ${\cal{F}}$ s'identifie au conoyau du morphisme sym\'etrique
$d_{2}$
\end{proof}
{\em D\'emonstration du lemme \ref{Bdual}:} Pour d\'emontrer ce lemme, revenons
\`a
la d\'efinition de la suite spectrale de Beilinson: soit $D^{\cdot}$
le complexe
$$0\lra\Lambda^{2}Q^{*}\etimes{\cal{O}}_{\planp}(-2)\lra
Q^{*}\etimes{\cal{O}}_{\planp}(-1)\lra{\cal{O}}_{\planp}
\etimes{\cal{O}}_{\planp}\lra 0$$
donn\'e par la r\'esolution de la diagonale $\Delta\subset\planp\times\planp$
de Beilinson.
Alors on a $R_{p_{1*}}(D^{\cdot}\etimes{\cal{F}})={\cal{F}}$ dans
${\cal{D}}^{b}_{c}(\planp)$ et la suite spectrale de Beilinson est donn\'ee
par la suite spectrale des foncteurs hyperderiv\'es. Dans notre cas, la suite
spectrale consid\'er\'ee est celle obtenue en regardant le complexe
$$L^{\cdot}=D^{\cdot}\otimes_{{\cal{O}}_{\planp\times\planp}}
({\cal{O}}(-1)\etimes{\cal{O}}(1))\etimes{\cal{F}}.$$ Ceci s'applique de
m\^eme au faisceau
${\cal{F}}^{\vee}$. Consid\'erons maintenant le complexe
$M^{\cdot}=R\ul{\Hom}(L^{\cdot},p_{1}^{!}(\Lambda^{2}Q^{*}(-2)))$
dans $D^{b}_{c}(\planp\times\planp)$ o\`u
$p_{1}^{!}(G)=p_{1}^{*}(G)\otimes p_{2}^{*}(\omega_{_{\planp}})[2]$ pour un
${\cal{O}}_{\planp}$-module coh\'erent $G$. Par dualit\'e de
Grothendieck-Serre,
on a, dans
$D^{b}_{c}(\planp)$,
$$Rp_{1*}(M^{\cdot})=R\ul{\Hom}(Rp_{1*}(L^{\cdot}))={\cal{F}}^{\vee}$$
De plus, dans
$D^{b}_{c}(\planp\times\planp)$, on a
$$M^{\cdot}\simeq\ul{\Hom}(D^{\cdot},\Lambda^{2}Q^{*}\etimes
{\cal{O}}_{\planp}(-2))\otimes_{{\cal{O}}_{\planp\times\planp}}
({\cal{O}}(-1)\etimes{\cal{O}}(1))\etimes
R\ul{\Hom}({\cal{F}},p_{2}^{*}(\omega_{_{\planp}}))[2].$$ La suite spectrale
des foncteurs hyperd\'eriv\'es du dernier complexe est celle
cherch\'ee et l'isomorphisme de suites spectrales du lemme se d\'eduit de
l'isomorphisme naturel
$$D^{\cdot}
\simeq \ul{\Hom}(D^{\cdot},\Lambda^{2}Q^{*}\etimes
{\cal{O}}_{\planp}(-2))[2],$$
d\'eduit de l'isomorphisme $Q^{*}\etimes{\cal{O}}(-1)\simeq
Q\etimes{\cal{O}}(+1)\otimes_{{\cal{O}}_{\planp\times\planp}}
\Lambda^{2}Q^{*}\etimes{\cal{O}}(-2).$
\cqfd
Dans
la suite, on aura besoin d'une description de la r\'esolution
minimale d'une th\^eta-caract\'eristique $({\cal{F}},\sigma)$
semi-stable dans le cas o\`u
$b=2$. A ce moment l\`a,
$h^{1}({\cal{F}}(2))=h^{0}({\cal{F}}(-2))=0$ et, a priori, deux cas se
pr\'esentent suivant si $c=0$ ou $c=1$.
Consid\'erons d'abord le cas o\`u $c=1$.
Remarquons qu'on a $-a_{i}+3\ge-1$, \ie $a_{i}\le4$ ici,
$h^{1}({\cal{F}}(2))$ \'etant nul. De plus, on a forc\'ement $a_{i}\ge0$. En
effet, s'il existait un
$i$ tel que $a_{i}\le-1$ on aurait $a_{i}+a_{j}-3\le a_{j}-4\le0$
pour tout $j$ ce qui est en contradiction \`a la minimalit\'e de la
r\'esolution. Le fibr\'e $E$ est donc de la forme
$\displaystyle \osum_{i=0}^{4}n_{i}{\cal{O}}_{\planp}(-i)$. En
tordant la r\'esolution par ${\cal{O}}_{\planp}(-1)$, on
voit que $n_{4}=1$ car $c=1$. De plus, comme $\h^{0}({\cal{F}})=2$ ceci
impose $n_{0}\ge 1$ et $n_{0}-n_{3}=1$. Encore par minimalit\'e de la
r\'esolution $n_{0}>1$ est impossible, d'o\`u n\'ecessairement $n_{0}=1$ et
$n_{3}=0$. On
a donc la factorisation suivante:
$$ \begin{diagram} && 0 &
& 0 & &\\ && \vfl{}{} & & \vfl{}{} & &\\
0&\lra& {\cal{O}}_{\planp}&\hfl{}{}&{\cal{O}}_{\planp}(1)&\hfl{}{}
& {\cal{O}}_{\ell}(1)&\lra&0\\
&& \vfl{}{} & & \vfl{}{} & & \vfl{}{}\\
0&\lra& E &\hfl{}{}&E^{\vee}&\hfl{}{}& {\cal{F}}&\lra&0\\
\end{diagram}
$$
Le noyau de ${\cal{O}}_{\ell}(1)\lra{\cal{F}}$ est, par le lemme
du serpent, dans le conoyau de ${\cal{O}}_{\planp}\lra E$. Par cons\'equent,
puisque ce conoyau est localement libre, ${\cal{O}}_{\ell}(1)$
s'injecte dans ${\cal{F}}$ ce qui est en contradiction avec la
semi-stabilit\'e de $({\cal{F}},\sigma)$.
Par suite, $c$ est n\'ecessairement \'egale \`a $0$. Dans ce
cas l\`a, on a $1\le a_{i}\le3$,
$\displaystyle E=\osum_{i=1}^{3}n_{i}{\cal{O}}_{\planp}(-i)$ avec
$n_{3}=2$ et la factorisation suivante: $$
\begin{diagram}
&& 0 & & 0 & &\\
&& \vfl{}{} & & \vfl{}{} & &\\
0&\lra&
n_{1}{\cal{O}}_{\planp}(-1)&\hfl{}{}&2{\cal{O}}_{\planp}&\hfl{}{} &
K&\lra&0\\
&& \vfl{}{} & & \vfl{}{} & & \vfl{}{}\\
0&\lra& E &\hfl{}{}&E^{\vee}&\hfl{}{}& {\cal{F}}&\lra&0\\
\end{diagram}
$$
Si $n_{1}=2$, alors on a $K\subset{\cal{F}}$ mais $\chi(K)>0$ ce qui
contredit la semi-stabilit\'e de ${\cal{F}}$. Il reste finalement le cas o\`u
$n_{1}=1$ ou $n_{1}=0$. On a donc d\'emontr\'e la proposition suivante:
\begin{prop}
Soit $({\cal{F}},\sigma)$ une th\^eta-caract\'eristique semi-stable telle que
$\h^{0}({\cal{F}})=2$. Alors la r\'esolution minimale de
${\cal{F}}$ qui est soit de type $(I)$
$$
0\ra (\osum_{i=1}^{d-6}{\cal{O}}_{\planp}(-2))
\osum2{\cal{O}}_{\planp}(-3)\hfl{\phi}{}
(\osum_{i=1}^{d-6}{\cal{O}}_{\planp}(-1))
\osum2{\cal{O}}_{\planp}\ra{\cal{F}}\ra 0
$$
soit de type $(II)$
$$
0\ra{\cal{O}}(-1)\osum(\osum_{i=1}^{d-5}{\cal{O}}(-2))
\osum2{\cal{O}}_{}(-3)\hfl{\phi}{}
{\cal{O}}(-2)\osum(\osum_{i=1}^{d-5}{\cal{O}}(-1))
\osum2{\cal{O}}\ra{\cal{F}}\ra 0.
$$
\end{prop}
La premi\`ere r\'esolution impose $d\ge 6$. Donc si $d=5$, la r\'esolution est
n\'ecessairement du deuxi\`eme type.
\begin{prop} Si $({\cal{F}},\sigma)$ est une th\^eta-caract\'eristique de
$\Theta^{1}(d)\backslash\Theta^{2}(d)$ localement libre sur son support
alors la r\'esolution minimale de
${\cal{F}}$ est de type $(I)$.
\end{prop}
\begin{proof} Il s'agit de montrer que $n_{1}=1$ est impossible dans ce
cas-l\`a. Le morphisme ${\cal{O}}_{\planp}(-1)\lra 2{\cal{O}}_{\planp}$ du
diagramme est donn\'e par deux sections $\alpha$ et $\beta$. Supposons
$\alpha$ et $\beta$ proportionnel. A ce moment l\`a, $K$ a ${\cal{O}}_{\ell}$
comme sous-faisceau,
$\ell$ la droite d\'efinie par $\alpha$. Or ce sous-faisceau s'injecte dans
${\cal{F}}$ ce qui donne une contradiction \`a la semi-stabilit\'e de
${\cal{F}}$. Sinon, les sections $\alpha$ et $\beta$ d\'efinissent deux droites
distinctes qui se coupent en un point $a\in\planp$. Soit $F$ le sous-fibr\'e
${\cal{O}}_{\planp}(-1)$ de $E$. Son $\varphi$-orthogonal est donn\'e par la
somme directe de ${\cal{O}}_{\planp}(-1)$ avec $n_{2}{\cal{O}}_{\planp}(-2)$
et le noyau de la fl\`eche $2{\cal{O}}_{\planp}(-3)\lra{\cal{O}}_{\planp}(-2)$
d\'efini par
$\alpha$ et
$\beta$. Ce noyau est isomorphe \`a ${\cal{O}}_{\planp}(-4)$. Le quotient
$E/{\cal{F}}^{\ort}$ est isomorphe \`a ${\cal{I}}_{a}(-2)$ o\`u
${\cal{I}}_{a}$ est le faisceau d'id\'eaux du point $a$. Consid\'erons le
diagramme suivant avec suites verticales des $0$-suites, suites
horizontales exactes:
$$
\begin{diagram} &&&&0&&0\\ &&&&\sfl{}{}&&\sfl{}{}\\
0&\efl{}{}&F&\efl{}{}&E&\efl{}{}&E/F^{\ort}&\efl{}{}&0\\
&&\Vert&&\sfl{}{}&&\sfl{}{}\\
0&\efl{}{}&F&\efl{}{}&E^{\vee}&\efl{}{}&{\cal{F}}^{\vee}&\efl{}{}&0\\
&&&&\sfl{}{}&&\sfl{}{}\\ &&&&{\cal{F}}&\efl{}{}&\comp_{a}\\
&&&&\sfl{}{}&&\sfl{}{}\\ &&&&0&&0\\
\end{diagram}
$$ Chaque $0$-suite verticale n'a de la cohomologie uniquement en degr\'e $0$.
Soient ${\cal{H}}$, ${\cal{L}}$ et ${\cal{F}}'$ les faisceaux de cohomologie
respectivement. On a donc une suite exacte
$$0\lra{\cal{H}}\efl{}{}{\cal{L}}\efl{}{}{\cal{F}}'\efl{}{}0.$$ Soit
$K^{\cdot}$ le complexe
$0\ra F\ra E\ra E/F^{\ort}\ra 0$. Les suites spectrales
$'E_{2}^{p,q}=H^{p}(\ul{Ext}^{q}(K^{\cdot},\omega))$ et
$''E_{2}^{p,q}=\ul{Ext}^{p}(H^{q}(K^{\cdot},\omega))$ sont d'aboutissement
$\ul{{\tr{E}}xt}^{p+q}(K^{\cdot},{\cal{O}}_{\planp})$. On en d\'eduit la suite
exacte
$$0\efl{}{}{\cal{L}}\efl{}{}\ul{{\tr{E}}xt}^{0}(K^{\cdot},{\cal{O}}_{\planp})
\efl{}{}\comp_{a}\efl{}{}0$$ et l'identification
$\ul{{\tr{E}}xt}^{0}(K^{\cdot},{\cal{O}}_{\planp})={\cal{H}}^{\vee}.$ On
obtient un diagramme
$$
\begin{diagram} &&&&0&&0\\ &&&&\sfl{}{}&&\sfl{}{}\\
0&\efl{}{}&{\cal{H}}&\efl{}{}&{\cal{L}}&\efl{}{}&{\cal{F}}'&\efl{}{}&0\\
&&\Vert&&\sfl{}{}&&\sfl{}{}&&\\
0&\efl{}{}&{\cal{H}}&\efl{}{}&{\cal{H}}^{\vee}&\efl{}{}&
{{\cal{E}}}&\efl{}{}&0\\ &&&&\sfl{}{}&&\sfl{}{}\\ &&&&\comp_{a}&=&\comp_{a}\\
&&&&\sfl{}{}&&\sfl{}{}\\ &&&&0&&0\\
\end{diagram}
$$ Les th\^eta-caract\'eristiques ${\cal{F}}$ et ${{\cal{E}}}$ sont donc toutes
les deux extensions de $\comp_{a}$ avec ${\cal{F}}'$. La premi\`ere a deux, la
deuxi\`eme trois sections. Mais si ${\cal{F}}$ est localement libre sur en
$a$, l'espace de telles extensions est de dimension $1$. En particulier,
toutes ces extensions sont munies d'une forme quadratique. On obtient ainsi
une famille connexe de th\^eta-caract\'eristiques. Mais pour une telle famille
la dimension de l'espace des sections est invariante modulo
$2$, d'o\`u la contradiction cherch\'ee.
\end{proof}
\begin{cor}
La th\^eta-caract\'eristique g\'en\'erale de $\Theta^{1}\backslash\Theta^{2}$
admet
une r\'esolution de type $(I)$
\end{cor}
\begin{proof}
D'apr\`es (\cite{11} Prop. 8.3), le ferm\'e de $\Theta(d)$ des
th\^eta-caract\'eristiques non localement libres sur leur support
sch\'ematique,
est de codimension au moins $1$. Ce ferm\'e coupe, par irr\'eductibilit\'e de
$\Theta^{1}$, le ferm\'e
$\Theta^{1}$ suivant un ferm\'e de codimension au moins $2$, d'o\`u le
corollaire par la proposition pr\'ec\'edente.
\end{proof}
\np
\section{Groupe de Picard de la
composante paire}\label{GdPpair}
Consid\'erons la
composante $\Theta_{p}(d)$ des
th\^eta-caract\'eristiques semi-stables paires. On suppose, dans ce qui suit,
que $d\ge 3$.
\subsection{Le fibr\'e pfaffien}\label{defpfaff}
Soit $({\cal{F}},\sigma)$ une famille de th\^eta-caract\'eristiques
param\'etr\'ee
par la vari\'et\'e alg\'ebrique $S$. On suppose que $S$ est lisse et connexe,
que le sous-sch\'ema $S^{1}$ de $S$ (voir section \ref{BNlocus}) est de
codimension $1$ et que $S^{1}\backslash S^{2}$ est non vide. Ce sous-sch\'ema
est donc un diviseur de Weil et puisque
$S$ est lisse un diviseur de Cartier. D\'eterminons l'\'equation en un point
$s\in S^{1}\backslash S^{2}$: pour cela, on choisit une approximation
anti-sym\'etrique de la cohomologie de
${\cal{F}}$ d\'efinie au voisinage
$U_{s}$ de
$s$, qu'on peut supposer de la forme
$$\H^{0}({\cal{F}}_{s})\times U_{s}\efl{a}{}
\H^{0}({\cal{F}}_{s})^{*}\times U_{s},$$
avec $a:U_{s}\lra
\Lambda^{2}\H^{0}({\cal{F}}_{s})^{*}$.
Identifions via $\sigma$ et dualit\'e de Serre les espaces vectoriels
$\H^{0}({\cal{F}}_{s})^{*}$ avec $\H^{1}({\cal{F}}_{s})$ et consid\'erons le
morphisme
$$\Pf(a):U_{s}\lra\Lambda^{max}\H^{1}({\cal{F}}_{s})\simeq k$$
associant \`a $t\in U_{s}$ le pfaffien de la matrice anti-sym\'etrique
$a(t)$. Par d\'efinition, le sous-sch\'ema
$S^{1}$ est d\'efini au voisinage de $s\in S^{1}\backslash S^{2}$ par cette
\'equation.
Le fibr\'e inversible, unique \`a isomorphisme pr\`es, associ\'e \`a
$S^{1}$ sera appel\'e {\em fibr\'e pfaffien} et not\'e
${\cal{P}}_{({\cal{F}},\sigma)}$.
Si ${\cal{D}}_{{\cal{F}}}$ est le fibr\'e d\'eterminant associ\'e \`a la
famille
${\cal{F}}$, on a, sur l'ouvert $S^{1}\backslash S^{2}$ de $S^{1}$, que
$${\cal{P}}_{({\cal{F}},\sigma)}^{2}={\cal{D}}_{{\cal{F}}},$$ le
d\'eterminant
d'une matrice anti-sym\'etrique \'etant le carr\'e du pfaffien de celle-ci.
Par irr\'eductibilit\'e de $S^{1}$, le ferm\'e $S^{2}$ est de codimension $2$
dans
$S$. Par lissit\'e de $S$, le carr\'e du fibr\'e pfaffien est le fibr\'e
d\'eterminant
sur $S$ entier.
Consid\'erons maintenant le bon quotient
$T^{ss}_{p}(d)\lra\Theta_{p}(d)$ de la section \ref{thetaconstruction}. Soit
$({\cal{F}},\sigma,\alpha)$ la famille universelle sur $T^{ss}_{p}(d)$ et
$T^{ss,1}(d)$ le diviseur de Cartier comme ci-dessus. On verra dans la
section suivant que $\Theta^{1}(d)$ est int\`egre. Ainsi, $T^{os,1}_{p}(d)$
est irr\'eductible et comme le ferm\'e de
$T^{ss}_{p}(d)\backslash T^{os}_{p}(d)$
est de codimension $2$ dans $T^{ss}_{p}(d)$, l'ouvert
$T^{os,1}_{p}(d)\subset T^{ss,1}_{p}(d)$ est partout dense dans
$T^{ss,1}_{p}(d)$ ce qui montre que $T^{ss,1}_{p}(d)$ est irr\'eductible lui
aussi. Ce diviseur est
$GL(H)$-invariant et non-vide pour
$d\ge 5$. Soit de plus ${\cal{P}}_{({\cal{F}},\sigma)}$ le
fibr\'e pfaffien d\'efini comme ci-dessus,
${\cal{D}}_{{\cal{F}}}$ le fibr\'e d\'eterminant.
Ces fibr\'es sont des $GL(H)$-fibr\'es vectoriels inversibles.
\begin{lemme}\label{descentePD} Soit $d\ge 5$. Alors on a
\begin{list}{\arabic{lc})}{\usecounter{lc}}
\item Le fibr\'e pfaffien ${\cal{P}}_{({\cal{F}},\sigma)}$ descend \`a
$\Theta^{os}_{p}(d)$ pour
$d\geq 5$. De plus,
${\cal{P}}_{({\cal{F}},\sigma)}$ descend \`a
$\Theta_{p}(d)$ si et seulement si $d=5$.
\item Le fibr\'e d\'eterminant ${\cal{D}}_{{\cal{F}}}$ descend \`a
$\Theta_{p}(d)$.
\end{list}
\end{lemme}
On note par ${\cal{P}}$ le fibr\'e inversible obtenu par descente de
${\cal{P}}_{{\cal{F}}}$ et par
${\cal{D}}$ le fibr\'e inversible obtenu par descente de
${\cal{D}}_{{\cal{F}}}$.
\begin{proof} On utilisera le lemme de descente de
Kempf-Drezet-Narasimhan \cite{5} affirmant que si le groupe
r\'eductif $G$ op\`ere sur la vari\'et\'e alg\'ebrique
$Y$ et si $\pi:Y\lra Z$ est un bon quotient de $Y$ par $Z$, alors un
$G$-fibr\'e inversible
$\widetilde{{\cal{L}}}$ est de la forme $\pi^{*}({\cal{L}})$ pour
${\cal{L}}\in\Pic(Z)$ (\ie
$\widetilde{{\cal{L}}}$ {\em descend} \`a Z) si et seulement si l'action du
stabilisateur sur $\widetilde{{\cal{L}}}_{y}$ est triviale en tout point
$y\in Y$ d'orbite ferm\'ee.
Montrons la premi\`ere assertion: en un point
$[{\cal{F}}_{t},\sigma_{t},\alpha_{t}]\in T^{os}_{p}$ l'action du
stabilisateur $\{\pm 1\}$ sur
${\cal{P}}_{({\cal{F}}_{t},\sigma_{t})}$ est donn\'ee par
$$g\mapsto g^{\h^{1}({\cal{F}}_{t})}$$
L'action est donc triviale, $\h^{1}({\cal{F}}_{t})$ \'etant pair et, par le
lemme de Kempf, ${\cal{P}}_{({\cal{F}},\sigma)}$ descend \`a
$\Theta^{os}_{p}(d)$.
Maintenant, si $d=5$, toute th\^eta-caract\'eristique paire
$({\cal{F}},\sigma)$ de degr\'e $d$ ayant des sections est stable,
puisqu'il n'existe pas de th\^eta-caract\'eristique paire de degr\'e compris
entre $1$ et $4$ ayant des sections. L'action du stabilisateur est donc
triviale en tout point de $T^{ss}_{d}(d)$ et, par le lemme de Kempf,
${\cal{P}}_{({\cal{F}},\sigma)}$ descend \`a $\Theta_{p}(d)$. Si $d\geq 6$, on
consid\`ere la th\^eta-caract\'eristique somme directe orthogonale
$$({\cal{G}},\tau)={\cal{O}}_{C}\osum{\cal{O}}_{C'}\osum
(d-6){\cal{O}}_{\ell}$$ avec $\ell$ une droite, $C$ et $C'$ deux courbes
elliptiques non proportionnelles du plan projectif. Choisissons une
identification $\alpha:\H^{0}({\cal{G}}(N))\simeq k^{dN}$ et consid\'erons le
point $({\cal{G}},\tau,\alpha)$ de $T^{ss}_{p}(d)$. Le stabilisateur en
ce point s'identifie \`a $\{\pm 1\}\times\{\pm 1\}\times O(d-6)$ et son action
sur
$\Lambda^{2}(H^{1}({\cal{G}}))$ est donn\'ee par
$$(g_{1},g_{2},g_{3})\mapsto g_{1}^{\h^{1}({\cal{O}}_{C})}
g_{2}^{\h^{1}({\cal{O}}_{C'})}.$$
On voit alors que $(-1,1)$ et $(1,-1)$ n'op\`erent pas trivialement et que
${\cal{P}}_{({\cal{F}},\sigma)}$ ne peut descendre \'a
$\Theta^{os}_{p}(d)$ entier.
La deuxi\`eme assertion r\'esulte de \cite{9}
\end{proof}
\subsection{Le diviseur $\Theta^{1}(d)$}
Si $d\geq 5$, le diviseur $\Theta^{1}(d)$ est non-vide. Ce qui est
important pour notre propos est qu'il est int\`egre, comme le montre la
proposition suivante:
\begin{prop}
Le ferm\'e $\Theta^{1}(d)\subset\Theta_{p}(d)$ est
irr\'eductible et r\'eduit, lisse aux points ${\cal{O}}$-stables de
$\Theta^{1}(d)\backslash\Theta^{2}(d)$. \end{prop}
\begin{proof}
D\'emontrons d'abord l'\'enonc\'e concernant la lissit\'e. Si $d\leq4$,
l'\'enonc\'e est vide, supposons donc $d\geq5$. Soient
$({\cal{G}},\tau)\in\Theta^{1}(d)\backslash\Theta^{2}(d)$ une
th\^eta-caract\'eristique
${\cal{O}}$-stable, $S$ un voisinage \'etale de
$({\cal{G}},\tau)$ et $({\cal{F}},\sigma)$ une
famille universelle sur $S\times\Theta_{p}(d)$, \ie telle que le
morphisme modulaire d\'eduit de $({\cal{F}},\sigma)$ et $S$ soit le
rev\^etement
\'etale $S\lra \Theta_{p}(d)$ donn\'e. Il s'agit donc de montrer l'assertion
pour un point $s\in S$ dont l'image est
$({\cal{G}},\tau)$,
\ie tel que $({\cal{F}}_{s},\sigma_{s})\simeq({\cal{G}},\tau)$.
Choisissons, au voisinage de $s$, une approximation $E\efl{a}{}
E^{*}$ antisym\'etrique de la cohomologie de ${\cal{F}}$.
En g\'en\'eral, la vari\'et\'e d\'eterminantielle $S^{1}$ est lisse de
codimension
$1$ en $s$ si et
seulement si le morphisme canonique
$$\phi:T_{s}S\lra \Lambda^{2}(\Ker a_{s})^{*}$$
est surjectif. Comme dans \cite{6} on montre qu'on a un
diagramme anti-commutatif
$$\begin{diagram}
T_{s}S&\hfl{\phi}{}&\Lambda^{2}\H^{0}({\cal{F}}_{s})^{*}\\
\vfl{\psi}{}&\nefl{}{\chi}\\
\Ext^{1}_{asym}({\cal{F}}_{s},{\cal{F}}_{s})\\
\end{diagram}
$$
o\`u $\psi$ d\'esigne le morphisme de d\'eformation de Kodaira-Spencer et
$\chi$ le co-morphisme de Petri, \ie le morphisme d\'efini par l'accouplement
de Yoneda:
$$\H^{0}({\cal{F}}_{s})\times\Ext^{1}({\cal{F}}_{s},{\cal{F}}_{s})\lra
\H^{1}({\cal{F}}_{s})$$
Comme $\psi$
est surjectif en $s$ il suffira de montrer que $\chi$ est surjectif en $s$.
Pour montrer que $\chi$ est surjectif il suffira de montrer que le morphisme
naturel
$$\zeta:\Ext^{1}({\cal{G}},{\cal{G}})\lra
L(\H^{0}({\cal{G}}),\H^{1}({\cal{G}}))$$ est surjectif. En effet, si $f\in
\Lambda^{2}\H^{0}({\cal{G}})^{*}$, il existerait
$g\in\Ext^{1}({\cal{G}},{\cal{G}})$ tel $\zeta(g)=f$. Or
$g'=1/2(g-^{\vee}g)\in
\Ext^{1}_{asym}({\cal{G}},{\cal{G}})$ et $\chi(g')=f$.
Pour montrer la surjectivit\'e de $\zeta$
on utilise la r\'esolution de la proposition \ref{Beilinson} de
${\cal{G}}$:
$$
0\lra F\hfl{\varphi}{} F^{\vee}\lra{\cal{G}}\lra 0.
$$
avec $F=(\osum_{i=1}^{d-6}{\cal{O}}_{\planp}(-2))
\osum2{\cal{O}}_{\planp}(-3)$ ou
$F={\cal{O}}_{\planp}(-1)\osum(\osum_{i=1}^{d-6}{\cal{O}}_{\planp}(-2))
\osum2{\cal{O}}_{\planp}(-3).$
Consid\'erons maintenant l'accouplement
$$\H^{0}(F^{\vee})\times\Ext^{2}(F^{\vee},F)\lra\H^{2}(F)$$
On obtient un diagramme commutatif
$$\begin{diagram}
\Ext^{1}({\cal{G}},{\cal{G}})&\hfl{f}{}&\Ext^{2}(F^{\vee},F)\\
\vfl{\varphi}{}&&\vfl{\vartheta}{}\\
L(\H^{0}({\cal{G}}),\H^{1}({\cal{G}}))&\hfl{g}{}&
L(\H^{0}(2{\cal{O}}_{\planp}),\H^{2}(2{\cal{O}}_{\planp}(-3))\\
\end{diagram}
$$
o\`u les fl\`eches verticales sont donn\'es par
les accouplements et o\`u $f$ est
donn\'ee par la suite spectrale qui calcule les
$\Ext^{i}({\cal{G}},{\cal{G}})$ \`a partir de la r\'esolution de ${\cal{G}}$.
Le morphisme $f$ est surjectif et $g$ est un isomorphisme. De plus, $\vartheta$
est un isomorphisme par dualit\'e de Serre. Par suite, $\varphi$ est surjectif
aussi et $S^{1}\backslash S^{2}$ est lisse aux points ferm\'es correspondant
\`a
des th\^eta-caract\'eristiques ${\cal{O}}$-stables. Ceci d\'emontre l'assertion
de
la proposition concernant la lissit\'e.
On sait d\'ej\`a que la sous-vari\'et\'e $\Theta^{1}(d)$ est irr\'eductible de
codimension $1$. D'apr\`es ce que pr\'ec\`ede ce ferm\'e est g\'en\'eriquement
lisse.
Par suite il est r\'eduit, ce qui termine la d\'emonstration de la
proposition.
\end{proof}
\subsection{L'ouvert $I(d)\subset \Theta_{p}(d)$.}
Consid\'erons le fibr\'e vectoriel
$$E=A\otimes_{k}{\cal{O}}_{\planp}(-2),$$ avec
$A=k^{d}$. Soit
$\widetilde{X}$ l'espace vectoriel des morphismes sym\'etriques
$q:E\lra E^{\vee}$. Si $\ol{q}$ est
injectif comme morphisme de faisceaux, son conoyau ${\cal{F}}$ est de
dimension $1$, muni d'une structure de th\^eta-caract\'eristique $\sigma$
provenant de la sym\'etrie de $q$. Le faisceau ${\cal{F}}$ est forc\'ement
semi-stable. En effet, aucun sous-faisceau coh\'erent ${\cal{E}}$ de
${\cal{F}}$ ne peut avoir des sections non nulles, ce qui implique
$\chi({\cal{E}})\le 0$. On note
$X\subset\widetilde{X}$ l'ouvert des morphismes $q$ tels que $\ol{q}$ soit
injectif.
La groupe $G=GL(A)$ op\`ere sur $X$ en associant \`a au morphisme
sym\'etrique $q$ le morphisme sym\'etrique $g.q=^{t}g^{-1}\circ
q\circ g^{-1}$.
\begin{prop} Le morphisme modulaire $f:X\lra I(d)$ fait
de $I(d)$ un bon quotient de $X$ par $G$
\end{prop}
\begin{proof}
On aura besoin du lemme de transitivit\'e de bons quotients de
Seshadri:
\begin{lemme}
Soit $G$ un groupe alg\'ebrique, $N$ un sous-groupes distingu\'e ferm\'e
de $G$. Supposons que $G$ op\`ere sur les vari\'et\'es alg\'ebriques $X$ et
$Z$ et que la vari\'et\'e alg\'e\-brique $Y$ est un bon quotient de $X$ par
$N$. Soit de plus $g:X\lra Z$ un morphisme
$G$-\'equivariant
qui se factorise suivant un morphisme $G/N$-\'equivariant
$h:Y\lra Z$. Alors $g$ est un bon quotient si et seulement si $h$
est un bon quotient.
\end{lemme}
Consid\'erons le triplet universel
$[{\cal{F}},\sigma,\alpha]$ sur
$T^{ss}_{ineff}(d)\times\planp$, o\`u $T^{ss}_{ineff}(d)$ est l'ouvert de
$T^{ss}(d)$ correspondant aux triplets dont la th\^eta-caract\'eristique
sous-jacente est ineffective. Consid\'erons de plus le fibr\'e de rep\`eres $R$
au dessus de $T^{ss}_{ineff}(d)$ dont la fibre au-dessus de
$[{\cal{F}}_{s},\sigma_{s},\alpha_{s}]$
param\`etre les isomorphismes $A\simeq\H^{1}({\cal{F}}_{s}(-1))$. Le groupe
$GL(H)\times G$ op\`ere sur
$R$ et $R\lra T^{ss}_{ineff}(d)$ est un bon quotient de $R$ par
$GL(H)$, puisque localement triviale dans la topologie de
Zariski de groupe structural $GL(H)$.
La r\'esolution \ref{Beilinson} fournit un morphisme de
$R$ dans $X$. De plus, ce morphisme fait de $R$ un fibr\'e localement
trivial dans la topologie de Zariski de groupe structural
$G$. Ces morphismes forment un diagramme commutatif
$$
\begin{diagram}
R&\lra&X\\
\vfl{}{}&&\vfl{}{}\\
T^{ss}_{ineff}(d)&\lra& I(d)
\end{diagram}
$$
En appliquant maintenant le lemme de Seshadri deux fois, obtient la
proposition.
\end{proof}
\begin{prop}\label{codim} Soit $E$ un fibr\'e vectoriel tel que
$S^{2}E^{*}(-3)$ soit engendr\'e par ses sections globales. Soit $\Gamma$
l'espace des sections globales de $S^{2}E^{*}(-3)$ et $\Gamma^{i}$ l'ouvert
des sections induisant un morphisme injectif en tant que morphisme de
faisceau $E\efl{}{}E^{\vee}$. Alors le compl\'ementaire de $\Gamma^{i}$ dans
$\Gamma$ est de codimension au moins deux.
\end{prop}
\begin{proof}
Soit $r=\rang(E)$, $\gamma=\dim \Gamma$. Consid\'erons le diagramme suivant:
$$
\begin{diagram}
\Gamma\times\planp&\efl{ev}{}&S^{2}E^{*}(-3)\\
\sfl{\pi}{}\\
\Gamma\\
\end{diagram}
$$
Le morphisme d'\'evaluation, not\'e $ev$, est un morphisme de vari\'et\'es
alg\'ebriques surjectif. Soit $D\subset S^{2}E^{*}(-3)$ le sous-fibr\'e dont la
fibre au dessus de $x\in \planp$ consiste en les $q_{x}$, tels que
$q_{x}:E_{x}\efl{}{}E^{*}_{x}$ soit de rang inf\'erieur ou \'egal \`a $r-1$.
Remarquons que $D$ est de rang $r(r+1)/2-1$. Consid\'erons l'image r\'eciproque
$\widetilde{D}$ de $D$ sous le morphisme d'\'evaluation. On a
$\widetilde{D}=\{(q,x)/\rang q(x)\le r-1\}$ et
la vari\'et\'e $\widetilde{D}$ est irr\'eductible de dimension $\gamma+1$.
Consid\'erons la projection
$\widetilde{\pi}:\widetilde{D}\lra \Gamma$
obtenue par restriction de $\pi$ \`a $\widetilde{D}$.
Il s'agit de
voir que le ferm\'e $\Gamma_{1}$ de $\Gamma$ des points o\`u la fibre est de
dimension $2$ est de codimension au moins $2$.
Remarquons d'abord que $\Gamma_{1}$ est un ferm\'e strict, la fibre
g\'en\'erale de $\widetilde{\pi}$ \'etant une courbe lisse. Maintenant, la
codimension de
$\Gamma_{1}$ dans $\Gamma$ ne peut \^etre $2$. Sinon,
$\widetilde{\pi}^{-1}(\Gamma_{1})$ serait un ferm\'e de $\widetilde{D}$ de
dimension $\gamma+1$ et serait donc, par irr\'eductibilit\'e de
$\widetilde{D}$,
\'egale \`a $\widetilde{D}$. Contradiction.
\end{proof}
Pour les estimations de codimension de cette section et de la suivante, on a
besoin de la proposition suivante, variante de la proposition 1.5 de
\cite{4}.
\begin{prop}\label{etudediff}
Soit $({\cal{F}},\sigma)$ une famille de th\^eta-caract\'eristiques
param\'etr\'ee par la vari\'et\'e alg\'ebrique $S$. Soit
$${\cal{H}}=\Groth_{isotr}({\cal{F}}/S,P)\lra S$$
le sch\'ema de Hilbert relatif des paires $(s,{\cal{E}})$ d'un point $s$
de $S$ et d'un quotient coh\'erent ${\cal{F}}_{s}/{\cal{E}}$ de polyn\^ome de
Hilbert $P$ tel que
${\cal{E}}\subset{\cal{F}}$ soit totalement isotrope pour $\sigma_{s}$.
Alors pour tout point $t=(s,{\cal{E}})$ de ${\cal{H}}$ on a la suite exacte
$$T_{t}{\cal{H}}\lra T_{s}S\efl{\omega_{+}}{}
\Ext^{1}_{asym}({\cal{E}},{\cal{E}}^{\vee}),$$
o\`u $\omega_{+}$ est le compos\'e du morphisme de Kodaira-Spencer
$\kappa:T_{s}S\lra\Ext^{1}_{asym}({\cal{F}}_{s},{\cal{F}}_{s})$ et du
morphisme canonique $\Ext^{1}_{asym}({\cal{F}}_{s},{\cal{F}}_{s})\lra
\Ext^{1}_{asym}({\cal{E}},{\cal{E}})$.
\end{prop}
\begin{lemme}\label{codim2part1}
Soit $X^{os}\subset X$ l'ouvert des $q$ induisant des
th\^eta-caract\'eristiques ${\cal{O}}$-stables. Alors la codimension
du ferm\'e compl\'ementaire \`a $X^{os}$ dans $X$ est au moins deux.
\end{lemme}
\begin{proof}
On montre le lemme en deux \'etapes: d'abord $(i)$ on montre qu'on a
$\codim_{_{X}}(X\backslash X^{s})\geq 2$, o\`u $X^{s}$ d\'esigne
l'ouvert de
$X$ correspondant aux th\^eta-caract\'eristiques stables, puis $(ii)$ on
montre que $\codim_{_{X^{s}}}(X^{s}\backslash X^{os})\geq 2$, o\`u
$X^{s}$ d\'esigne l'ouvert correspondant aux th\^eta-caract\'eristique
${\cal{O}}$-stables.
Pour $(i)$ on utilise la proposition \ref{etudediff},
appliqu\'ee \`a la famille des th\^eta-caract\'eristiques param\'etr\'ee par
$X$ et
d\'efinie par le conoyau du morphisme universel:
$$q:A\otimes{\cal{O}}_{X^{os}\times\planp}(-2)\lra
A^{*}\otimes{\cal{O}}_{X^{os}\times\planp}(-1).$$
Le morphisme $\omega_{+}$ est surjectif, la famille en question \'etant
compl\`ete et le morphisme canonique
$\Ext^{1}_{asym}({\cal{F}}_{s},{\cal{F}}_{s}^{\vee})\lra
\Ext^{1}_{asym}({\cal{E}},{\cal{E}}^{\vee})$ \'etant surjectif.
On est donc ramen\'e \'a prouver que si $0\not={\cal{E}}\subset{\cal{F}}$
est un sous-faisceau totalement isotrope de ${\cal{F}}$ et de caract\'eristique
d'Euler-Poincar\'e $0$, la dimension de
$\Ext^{1}_{asym}({\cal{E}},{\cal{E}}^{\vee})$ est au moins $2$.
Or, d'apr\`es la proposition 8.5 de $\cite{11}$, on a
$$\dim\Hom^{1}_{asym}({\cal{E}},{\cal{E}}^{\vee})-
\dim\Ext^{1}_{asym}({\cal{E}},{\cal{E}}^{\vee})=
-\frac{d({\cal{E}})(d({\cal{E}})+3)}{2}$$
d'o\`u $\dim\Ext^{1}_{asym}({\cal{E}},{\cal{E}}^{\vee})\geq 2$ pour
$d({\cal{E}})\geq 1$.
Pour $(ii)$,
consid\'erons deux sous-espaces vectoriels non nuls $A'$ et $A''$ de
$A$ de dimension respectivement $a'$ et $a''$ et tels que
$A=A'\osum A''$. On a un morphisme
$$\phi:S^{2}A^{'*}\otimes V\times S^{2}A^{''*}\otimes V
\lra S^{2}A^{*}\otimes V$$
en associant \`a $(q',q'')$ le morphisme sym\'etrique
$$\left(
\begin{matrix}
q'&0\\
0&q''\\
\end{matrix}
\right)
$$
L'image de $\phi$ est un ferm\'e de $X$ de codimension $3a'a''$. Soit
$Y\subset X$ le $G$-satur\'e de cet image. Comme l'action de $GL(A)$ sur
$X^{s}$ est propre, l'image r\'eciproque de l'image de $Y$ est \'egal \`a $Y$.
Maintenant, le sous-groupe de
$G$ des matrices du type
$$\left(
\begin{matrix} g'&0\\ 0&g''\\
\end{matrix}
\right)
$$
avec $g'\in GL(A')$ et $g''\in GL(A'')$ stabilise l'image
de $\phi$ la codimension de $Y$ est au moins
$3a'a''-a^{2}+a^{'2}+a^{''2}=a'a''.$
Pour $a=d\geq 3$ ce nombre est au moins $2$.
Il reste, \cf (\cite{11},
1.4), \`a consid\'erer le cas des th\^eta-caract\'eristiques hyperboliques.
A ce moment-l\`a, $a$ est pair et l'on consid\`ere une d\'ecomposition
$A=A'\osum A'$. Un argument analogue \`a celui ci-dessus montre que
la codimension est aussi au moins deux dans ce cas, d'o\`u $(ii)$, ce qui
termine la d\'emonstration du lemme.
\end{proof}
Soit
$\ol{G}=G/\{\pm\id\}$.
D'apr\`es ce que pr\'ec\`ede, les seules fonctions
inversibles sur $X$ sont les constantes non nulles et le
groupe de Picard de $X$ est r\'eduit au fibr\'e en droites trivial. Il
en d\'ecoule que le groupe
$\Pic^{\ol{G}}(X)$ des $\ol{G}$-fibr\'es inversibles sur $X$ est
isomorphe au groupe
$\chi(\ol{G})$ des caract\`eres de $\ol{G}$. Ce groupe est isomorphe \`a
$\reln$. Nous choisissons pour g\'en\'erateur le caract\`ere
$$
\chi:f\mapsto
\left\{
\begin{diagram}\det^{-1}(f)& \text{si }d\text{ est pair}\hfill\null\\
\det^{-2}(f)& \text{si }d\text{ est impair}\hfill\null\\
\end{diagram}
\right.
$$
Soit ${\cal{L}}_{\chi}$ le $\ol{G}$-fibr\'e inversible associ\'e \`a
$\chi$. Puisque $\ol{G}$ op\`ere librement sur l'ouvert $X^{os}$ des
points correspondant aux th\^eta-caract\'eristiques
${\cal{O}}$-stables, tout $\ol{G}$-fibr\'e inversible descend \`a $
I^{os}(d)$. On d\'eduit que
$\Pic( I^{os}(d))$ est isomorphe \`a $\reln$, nous choisissons
pour g\'en\'erateur le fibr\'e inversible ${\cal{L}}$, obtenu
par descente de ${\cal{L}}_{\chi}$.
\subsection{Fin de la d\'emonstration du th\'eor\`eme \protect\ref{PicPair}.}
Consid\'erons le morphisme d'oubli
$\gamma:I^{os}(d)\lra N^{s}(d,0).$
\begin{lemme}\label{PullbackL}
$\gamma^{*}({\cal{L}}_{N})=
\begin{cases}{\cal{L}}^{\otimes 2}&\text{si d est pair}\\
{\cal{L}}&\text{si est impair}\\
\end{cases}
$
\end{lemme}
\begin{proof} On utilise la propri\'et\'e universelle qui d\'efinit
${\cal{L}}_{N}.$
Consid\'erons sur $X^{os}\times\planp$ le morphisme universel
$$q:A\otimes{\cal{O}}_{X^{os}\times\planp}(-2)\lra
A^{*}\otimes{\cal{O}}_{X^{os}\times\planp}(-1)$$
dont on note ${\cal{F}}$ le conoyau. Le
${\cal{O}}_{X^{os}\times\planp}$-module
${\cal{F}}$ d\'efinit une famille $X$-plate de faisceaux ${\cal{O}}$-stables de
dimension $1$ sur $\planp$. Soit $f_{{\cal{F}}}$ le morphisme modulaire
associ\'e \`a cette famille.
Par la propri\'et\'e universelle qui d\'efinit ${\cal{L}}_{N}$, on a
$$f^{*}_{{\cal{F}}}({\cal{L}}_{N})=\lambda_{{\cal{F}}}(u)\text{ avec $u$
la classe d'un point.}$$
Or la r\'esolution de ${\cal{F}}$ montre
que $\lambda_{{\cal{F}}}(u)$ s'identifie au $G$-fibr\'e inversible
$$(\det(A))^{-2}\otimes{\cal{O}}_{X^{os}}.$$ L'image r\'eciproque de
${\cal{L}}_{N}$ sous $f_{{\cal{F}}}$ s'identifie donc suivant si $d$ est pair
ou impair \`a ${\cal{L}}_{\chi}^{\otimes 2}$ ou ${\cal{L}}_{\chi}$.
Par construction on a un diagramme commutatif:
$$
\begin{diagram}
X^{os}\\
\sfl{\pi}{}&\sefl{}{f_{{\cal{F}}}}\\
I^{os}(d)&\efl{\gamma}{}&N^{s}(d,0)\\
\end{diagram}
$$
Il en d\'ecoule que, suivant si $d$ est pair ou impair $\gamma^{*}$ envoie
${\cal{L}}_{N}$ sur ${\cal{L}}^{\otimes 2}$ ou ${\cal{L}}$
ce qui d\'emontre le lemme.
\end{proof}
Consid\'erons le morphisme d'oubli $\beta:\Theta^{os}(d)\lra
N^{s}(d,0)$.
\begin{lemme}\label{PullbackD} On a
$\beta^{*}({\cal{D}}_{N})=
\left\{
\begin{diagram}{\cal{O}}&
\text{si $d\le 4$}\hfill\null\\
{\cal{P}}^{2}&
\text{si $d\ge 5$}\hfill\null\\
\end{diagram}
\right.
$
\end{lemme}
\begin{proof}
C'est \'evident d'apr\`es la propri\'et\'e universelle qui d\'efinit le
fibr\'e d\'eterminant et le fait que ${\cal{P}}^{2}={\cal{D}}$.
\end{proof}
Consid\'erons maintenant la restriction
$$i^{*}:\Pic(\Theta^{os}_{p}(d))\lra \Pic(I^{os}(d))$$
Puisque $\Theta^{os}_{p}(d)$
est lisse, le noyau de $i^{*}$ est donn\'e par $\reln{\cal{P}}$.
Le groupe de Picard de
$\Theta_{p}^{os}(d)$ est donc librement engendr\'e par ${\cal{L}}$ et
${\cal{P}}$.
Consid\'erons maintenant le diagramme commutatif de morphismes naturels
$$\begin{diagram}
\Pic(\Theta_{p}(d))&\efl{\rho}{}&\Pic(\Theta_{p}^{os}(d))\\
\nfl{}{}& &\nfl{}{}\\
\Pic(N_{\planp}(d,0))&\efl{\rho_{N}}{}&\Pic(N_{\planp}^{s}(d,0))\\
\end{diagram}
$$
Le morphisme
$\rho_{N}$ est un
isomorphisme, $N_{\planp}(d,0)$ \'etant localement factorielle; le
morphisme $\rho$ est injectif, $\Theta_{p}(d)$ \'etant
normale. Il d\'ecoule du lemme \ref{PullbackL} et du diagramme que, si $d\ge 3$
est impair, ${\cal{L}}$ est dans l'image de $\rho$. Si $d\ge 4$ est pair
${\cal{L}}^{\otimes 2}$ est de m\^eme dans
l'image de $\rho$, mais ${\cal{L}}$ ne l'est pas puisque
${\cal{L}}_{\chi}$ ne descend pas \`a $I(d)$.
Il d\'ecoule du lemme \ref{PullbackD} et du diagramme que ${\cal{P}}^{2}$ est
dans l'image de
$\rho$ pour $d\geq 5$, mais ${\cal{P}}$ ne l'est que pour $d=5$ d'apr\`es le
lemme \ref{descentePD}. Les fibr\'e inversibles
${\cal{P}}^{2}$ et suivant si $d$ est impair ${\cal{L}}$ ou si $d$ est
pair ${\cal{L}}^{\otimes 2}$, s'\'etendent donc, et ceci de fa\c con
unique, \`a $\Theta_{p}(d)$ entier.
Comme l'image r\'eciproque de
${\cal{O}}_{\proj^{N}}(1)$ sous le morphisme sch\'ematique
$$\sigma_{_{N}}:N(d,0)\lra\proj^{M},$$ avec $M=d(d+3)$,
s'identifie \`a ${\cal{L}}_{N}$ on obtient aussi le lemme suivant:
\begin{lemme}
Si $\sigma_{\Theta}:\Theta_{p}(d)\lra\proj^{N}$, avec $N=d(d+3)/2$, est le
morphisme qui associe \`a une th\^eta-caract\'eristique son support
sch\'ematique,
l'image r\'eciproque de ${\cal{O}}_{\proj^{N}}(1)$ s'identifie, si $d$ est
pair, \`a l'extension unique de ${\cal{L}}^{2}$ \`a $\Theta_{p}(d)$, et, si $d$
est impair, \`a l'extension
unique de ${\cal{L}}$ \`a $\Theta_{p}(d)$.
\end{lemme}
\np
\section{Groupe de Picard de la composante impaire}
Consid\'erons la
composante $\Theta_{i}(d)$ des
th\^eta-caract\'eristiques semi-stables impaires. On suppose, dans ce qui suit,
que $d\ge 4$.
\subsection{L'ouvert $ SI(d)\subset \Theta_{i}(d)$.}
Consid\'erons le fibr\'e vectoriel
$$E=H'\otimes_{k}{\cal{O}}_{\planp}(-2)\osum
H''\otimes_{k}{\cal{O}}_{\planp}(-3),$$ avec
$H'=k^{d-3}$ et $H''=k$.
Soit de plus
$\widetilde{X}$ l'espace vectoriel des morphisme
sym\'etriques $q:E\lra E^{\vee}$.
Si $\ol{q}$ est g\'en\'eriquement injectif
son conoyau ${\cal{F}}$ est de dimension $1$,
muni d'une structure de th\^eta-caract\'eristique $\sigma$ provenant de la
sym\'etrie de $q$. On note $X'\subset\widetilde{X}$ l'ouvert des morphismes $q$
tels que $\ol{q}$ soit g\'en\'eriquement injectif.
D'apr\`es la proposition \ref{codim} la codimension du compl\'ementaire de $X'$
dans $\widetilde{X}$ est au moins $2$.
Soit $X\subset X'$ l'ouvert de $X'$ des $q$ d\'efinissant une
th\^eta-caract\'eristique semi-stable.
\begin{prop}
La codimension du ferm\'e compl\'ementaire de $X$ dans $X'$ est au moins deux.
\end{prop}
\begin{proof}
On utilise la proposition \ref{etudediff} appliqu\'ee \`a la famille de
th\^eta-caract\'eristiques
$X'$-plate d\'efinie par le quotient du morphisme universel sur
$X'\times\planp$. On v\'erifie que le morphisme $\omega_{+}$ est surjectif et
l'on est donc ramen\'e \`a prouver que si ${\cal{E}}\subset{\cal{F}}$ est un
sous-faisceau totalement isotrope non nul et propre de ${\cal{F}}$ tel que
$\chi({\cal{E}})=1$, alors $\Ext^{1}({\cal{E}},{\cal{E}}^{\vee})$ est de
dimension au moins $2$. Mais ceci se montre facilement en utilisant
une variante de la proposition 8.5 de \cite{11}.
\end{proof}
Le groupe alg\'ebrique $G=\Aut(E)$ op\`ere sur $X$ en associant \`a $g\in G$ et
$q\in X$ l'\'el\'ement $g.q=^{\vee}(g^{-1})qg^{-1}$.
\begin{prop} Le morphisme modulaire $X\lra SI(d)$ est un bon
quotient
\end{prop}
\begin{proof}
Identifions d'abord les \'el\'ements de $X$ aux matrices
$$q=\left(\begin{matrix}
u&v\\
^{\vee}v&w\\
\end{matrix}
\right)
$$
avec $u\in L(S^{2}H^{'*},V^{*})$,
$v\in \Hom(H''\otimes {\cal{O}}(-3),H^{'*}\otimes {\cal{O}}(-1))$
et $w\in L(S^{2}H^{''*},W^{*})$ avec
$V=H^{0}({\cal{O}}(1)$ et $W^{*}=\H^{0}({\cal{O}}(3))$.
Soit $N\subset G$ le
sous-groupe distingu\'e et ferm\'e de $G$ form\'e des \'el\'ements
$$g=\left(\begin{matrix}
1&b\\
0&1\\
\end{matrix}\right)
$$
avec $b\in\Hom(H''\otimes{\cal{O}}(-3),H^{\prime *}\otimes{\cal{O}}(-1))$.
Le quotient $G/N$ s'identifie \`a $GL(H')\times GL(H'')$.
Le groupe alg\'ebrique $N$ op\`ere (comme sous-groupe de $G$) sur $X$.
Cette action est donn\'ee en associant \`a $g\in N$ et $q\in X$
$$g.q=\left(\begin{matrix}
u&-ub+v
\\ -^{\vee}bu+^{\vee}v&^{\vee}bub-(^{\vee}bv+^{\vee}vb)+w\\
\end{matrix}\right)
$$
Le morphisme $u$ est g\'en\'eriquement injectif. En effet, sinon, soit $N$
son noyau. On obtiendrait un diagramme commutatif
$$\begin{diagram}
&&0&&0\\
&&\sfl{}{}&&\sfl{}{}\\
&&N&=&N\\
&&\sfl{(i,0)}{}&&\sfl{(0,^{\vee}vi)}{}\\
0&\ra&H^{'}\otimes{\cal{O}}(-2)\osum H^{''}\otimes{\cal{O}}(-3)
&\ra&H^{'*}\otimes{\cal{O}}(-1)\osum H^{''*}\otimes{\cal{O}} &\ra&
{\cal{F}}&\ra&0\\
&&\sfl{}{}&&\sfl{}{}&&\Vert\\
0&\ra&\Im(u)\osum H^{''}\otimes{\cal{O}}(-3)
&\ra&H^{'*}\otimes{\cal{O}}(-1)\osum
\frac{H^{''}\otimes{\cal{O}}}{N} &\ra&
{\cal{F}}&\ra&0\\
&&\sfl{}{}&&\sfl{}{}\\
&&0&&0\\
\end{diagram}
$$
Mais $\frac{H^{''}\otimes{\cal{O}}}{N}$ est de dimension $1$ et, comme
$\Im(u)$ est sans torsion, s'injecterait dans ${\cal{F}}$, ce qui est
impossible en raison de la semi-stabilit\'e de ${\cal{F}}$. Par cons\'equent,
si le morphisme $ub=0$, on a $b=0$. On en d\'eduit que l'action de $N$ sur
$X$ est libre, puisque l'application donn\'ee par l'op\'eration
$$\Hom(H''\otimes{\cal{O}}(-3),H^{\prime *}\otimes{\cal{O}}(-1))\times X\lra
X\times X$$
est donc une immersion ferm\'ee.
Il existe un quotient g\'eom\'etrique $Y$ de $X$ par l'action de $N$.
Consid\'erons maintenant le fibr\'e de rep\`eres $R$ au
dessus de $T^{ss}_{s-ineff}(d)$ dont la fibre au-dessus de
$[{\cal{F}}_{s},\sigma_{s},\alpha_{s}]$ param\`etre les isomorphismes
$$\beta:k^{d-3}\simeq
Ker(\H^{1}({\cal{F}}_{s}(-1))\efl{d_{1}^{-2,1}}{}
\H^{1}({\cal{F}}_{s})) \text{ et }\gamma:k\simeq\H^{1}({\cal{F}}_{s}).$$
On a un diagramme commutatif
$$\begin{diagram}
R&\efl{a}{}&X/N\\
\sfl{p}{}&&\sfl{f}{}\\
T^{ss}_{s-ineff}(d)&\efl{}{}&SI(d)
\end{diagram}
$$
o\`u le morphisme $a$ est donn\'e par la proposition \ref{Beilinson} et la
propri\'et\'e universelle \'evidente de $X/N$. Maintenant $p$ est un bon
quotient. Par le lemme de Seshadri, $R\lra SI(d)$ l'est aussi.
Puisque $a$ est un bon quotient, $f$ l'est aussi, encore par le lemme de
Seshadri. Ce lemme applique \`a nouveau au morphisme modulaire
$X\lra SI(d)$ montre que ce dernier est un bon quotient, d'o\`u la
proposition.
\end{proof}
\begin{lemme} Soit $X^{os}\subset X$ l'ouvert des $q$ induisant des
th\^eta-caract\'eristiques ${\cal{O}}$-stables. Alors la codimension du ferm\'e
compl\'ementaire est au moins deux.
\end{lemme}
\begin{proof}
Analogue \`a la d\'emonstration du lemme \ref{codim2part1}.
\end{proof}
Soit $\ol{G}=\Aut(E)/\{\pm Id\}$. On d\'eduit des propositions
\ref{codim} et du lemme pr\'ec\'edent
que les seules fonctions inversibles sur $X$ sont les
constantes non nulles et que le groupe de Picard de $X$ est r\'eduit au
fibr\'e
en droites trivial. Il en d\'ecoule que le groupe
$\Pic^{\ol{G}}(X)$ des $\ol{G}$-fibr\'es inversibles sur $X$ est
isomorphe au groupe
$\chi(\ol{G})$ des caract\`eres de $\ol{G}$. Le groupe des
automorphismes de $E$ est form\'e de matrices $f$ de la forme
$$f=\left(\begin{matrix}
\alpha&\gamma \\
0 &\beta\\
\end{matrix}\right).$$ Le groupe des
caract\`eres de $\ol{G}$ est donn\'e par
$f\mapsto (\det \alpha)^k(\det \beta)^{\ell}$ o\`u les entiers $k$ et
$\ell$ satisfont \`a l'\'equation $(d-3)k+\ell=0\bmod 2$.
Si $d$ est impair, ce sont les couples $(k,\ell)\in\reln\times\reln$
avec $\ell$ pair. Si $d$ est pair, ce sont les couples
$(k,\ell)\in\reln\times\reln$ avec $k+\ell$ pair. Nous choisissons
pour g\'en\'erateurs les caract\`eres $$
\chi_{_{1}}:f\mapsto
\text{det}^{-1}(\alpha)\text{det}^{-(d-3)}(\beta)
$$
$$
\chi_{_{2}}:f\mapsto\text{det}^{-2}(\beta)
$$
Soient ${\cal{L}}_{1}$ et ${\cal{L}}_{2}$ les $\ol{G}$-fibr\'es
inversibles associ\'es respectivement aux caract\`eres $\chi_{_{1}}$ et
$\chi_{_{2}}$. Puisque $\ol{G}$ op\`ere librement sur l'ouvert
$X^{os}$ des points correspondant aux th\^eta-caract\'eristiques
${\cal{O}}$-stables, tout $\ol{G}$-fibr\'e inversible descend \`a
$SI^{os}(d)$. On d\'eduit que $\Pic(SI^{os}(d))$ est isomorphe \`a un
groupe ab\'elien libre \`a deux g\'en\'erateurs. Pour g\'en\'erateurs nous
choisissons le fibr\'e inversible
${\cal{L}}$, obtenu par descente de ${\cal{L}}_{1}$ et le fibr\'e
inversible ${\cal{D}}$, obtenu par descente de ${\cal{L}}_{2}$.
\subsection{Fin de la d\'emonstration du th\'eor\`eme \ref{PicImpair}.}
Soit $d\not=6$. Alors la codimension du ferm\'e des th\^eta-caract\'eristiques
telles que $h^{0}({\cal{F}})\ge3$ est au moins deux dans $\Theta_{i}(d)$,
d'apr\`es les r\'esultats de la section $2$. Si $d=6$, ce ferm\'e est de
codimension $1$.
\begin{question} Si $d=6$, ce ferm\'e est irr\'eductible. Est-il r\'eduit?
\end{question}
On en d\'eduit que, si $d\not=6$, alors on a
$\Pic(SI^{os}(d))\simeq\Pic(\Theta_{i}^{os}(d))$,
si $d=6$, alors on a $\Pic(\Theta_{i}^{os}(d))
\simeq\Pic(SI^{os}(d))\osum\reln$.
Consid\'erons le morphisme $\gamma:\Theta^{os}_{1}(d)\lra N_{\planp}(d,0)$.
\begin{lemme}
On a $\gamma^{*}({\cal{D}}_{N})={\cal{D}}$ et
$\gamma^{*}({\cal{L}}_{N})={\cal{L}}^{\otimes 2}
\otimes{\cal{D}}^{\otimes(d-4)}$
\end{lemme}
\begin{proof}
On utilise encore la propri\'et\'e universelle qui d\'efinit ${\cal{L}}_{N}$ et
${\cal{D}}_{N}$, cette fois-ci appliqu\'ee \`a la famille des faisceaux
semi-stables d\'efinie comme conoyau du morphisme universel
sur $X\times\planp$:
$$q:H'\otimes{\cal{O}}_{X\times\planp}(-2)\osum
H^{\prime\prime}\otimes{\cal{O}}_{X\times\planp}(-3)\lra
H^{\prime*}\otimes{\cal{O}}_{X\times\planp}(-1)\osum
H^{\prime\prime*}\otimes{\cal{O}}_{X\times\planp}.$$
Si $u$ est la classe d'un point,
${\cal{L}}_{{\cal{F}}}(u)$ s'identifie \`a
$(\det(H'))^{-2}\otimes(\det(H^{\prime\prime}))^{-2}\otimes{\cal{O}}_{X}$,
si
$u={\cal{O}}_{\planp}$, ${\cal{L}}_{{\cal{F}}}(u)$ s'identifie \`a
$(\det(H^{\prime\prime}))^{-2}\otimes{\cal{O}}_{X}$. De l\`a se d\'eduit le
lemme, comme dans
le cas de l'\'enonc\'e analogue pour la composante paire.
\end{proof}
Consid\'erons le diagramme commutatif de morphismes naturels
$$\begin{diagram}
\Pic(\Theta_{i}(d))&\efl{\rho}{}&\Pic(\Theta_{i}^{os}(d))\\
\nfl{}{}& &\nfl{}{}\\
\Pic(N(d,0))&\efl{\rho_{_{N}}}{}&\Pic(N^{s}(d,0))\\
\end{diagram}
$$
Le morphisme
$\rho_{_{N}}:\Pic(N(d,0))\lra \Pic(N^{s}(d,0))$ est un
isomorphisme, $N(d,0)$ \'etant localement factorielle; le
morphisme $\rho$ est injectif, $\Theta(d)$ \'etant normale.
Il d\'ecoule du lemme et du diagramme que ${\cal{D}}$ est dans l'image de
$\rho_{\theta}$. De plus ${\cal{L}}^{\otimes 2}$ est dans l'image
de $\rho_{\theta}$, mais ${\cal{L}}$ ne l'est pas, puisque
${\cal{L}}_{\chi_{1}}$ ne descend pas \`a $SI(d)$ entier.
Ceci termine la d\'emonstration de th\'eor\`eme
\ref{PicImpair}.
\np
\section{D\'emonstration du th\'eor\`eme \protect\ref{UniFam}}
Soit $d\geq4$ un entier pair. Supposons l'existence d'une famille universelle
sur un ouvert $U\subset\Theta^{os}_{p}(d)$
On obtient, par image r\'eciproque, une famille universelle $({\cal{G}},\tau)$
param\'etr\'ee par l'ouvert $U'$, image r\'eciproque de $U$ sous le
morphisme
$T_{p}^{ss}(d,N)\lra\Theta_{p}(d)$. Soit
$[{\cal{F}},\sigma,\alpha]$ le triplet universel sur
$T_{p}^{ss}(d,N)$ et consid\'erons le
faisceau ${\cal{L}}=\ul{\Hom}({\cal{G}},{\cal{F}})$. Ce faisceau est
un fibr\'e en droites sur $U'$, muni d'une action de $GL(H)$. Pour
$\alpha\in\{\pm id\}$ cette action est donn\'ee par
$$\alpha\lra\alpha\id_{{\cal{L}}}.$$ Par lissit\'e de $T_{p}^{ss}(d,N)$,
${\cal{L}}$ s'\'etend \`a
$T_{p}^{ss}(d,N)$ entier. Que l'action s'\'etende
est cons\'equence du lemme suivant, d\^u \`a Le Potier \cite{9}.
\begin{lemme}
Soit $X$ une vari\'et\'e affine, lisse et irr\'eductible sur
laquelle op\`ere le
groupe r\'eductif et connexe $G$, $f\in{\cal{O}}(X)$ un \'el\'ement non-nul,
invariant sous l'action de $G$. Soit $U_{f}$ l'ouvert d\'efini par $f\not=0$.
Alors si $L$ est un fibr\'e inversible sur $X$ toute action lin\'eaire de $G$
sur $L\restriction{U_{f}}$ s'\'etend \`a $X$ entier.
\end{lemme}
Consid\'erons maintenant la th\^eta-caract\'eristique suivante: Soit $\ell$ une
droite de $\planp$ et $({\cal{M}},\rho)$ d\'efinie par la somme directe
orthogonale
$$({\cal{M}},\rho)=\osum_{i=1}^{i=d}({\cal{O}}_{\ell}(-1),1)$$
Choisissons une identification $\beta$ de $H^{0}({\cal{M}}(N))$ avec $H$. Le
stabilisateur de $[{\cal{F}},\alpha,\sigma]$ sous l'action de $GL(H)$
s'identifie au groupe orthogonal $O(d)$. Ce stabilisateur op\`ere sur
${\cal{L}}_{[{\cal{M}},\beta,\sigma]}$. Une telle op\'eration est donn\'ee par
un caract\`ere de $O(d)$, donc de la forme
$$g\mapsto \det(g)^{n} \text{
avec } n=0,1.$$ Par densit\'e de $U'$ dans
$T_{p}^{ss}(d,N)$, l'op\'eration de $(-1)$
sur ${\cal{L}}_{[{\cal{M}},\rho,\beta]}$ est donn\'ee par $v\mapsto -v$
pour
$v\in{\cal{L}}_{[{\cal{M}},\rho,\beta]}$. Mais ceci n'est pas possible, vu
que $\det(-\id)=1$ pour $d$ pair, ce qui donne la contradiction cherch\'ee.
Soit $d\geq3$ un entier impair.
Consid\'erons la famille
universelle $[{\cal{F}},\sigma,\alpha]$ sur
$T^{os}(d,N)\times\planp$. Soient $p$ et $q$ les
projections canoniques dans le diagramme
$$
\begin{diagram}
T^{os}(d,N)\times\planp&\hfl{p_{2}}{}&\planp\\
\vfl{p_{1}}{}\\
T^{os}(d,N)\\
\end{diagram}
$$
Pour $u\in K(\planp)$, consid\'erons l'\'el\'ement suivant dans
$Pic^{GL(H)}(T^{os}(d,N)):$
$${\cal{L}}_{{\cal{F}}}(u)=\det(p_{1!}({\cal{F}}\otimes p_{2}^{*}(u)).$$
L'action de $\alpha=\pm 1$ est donn\'ee par
$$\alpha\mapsto\alpha^{<c,u>}$$
o\`u $c$ est la classe de $K_{top}(\planp)$ d\'efinie par $(0,d,0)$.
Choisissons $u=(0,d',0)$ avec $d'$ impair, ce qui donne $<c,u>=1\bmod 2$, et
posons
${\cal{L}}={\cal{L}}_{{\cal{F}}}(u)$.
On a une suite exacte
$$0\lra{\cal{A}}\lra{\cal{B}}\lra{\cal{F}}\lra 0$$
sur $T^{os}(d,N)\times\planp$ avec
${\cal{B}}=H\otimes_{_{k}}p^{*}({\cal{O}}_{\planp}(-N))$ et ${\cal{A}}$
localement libre. Consid\'erons sur $T^{os}(d,N)\times\planp$ l'injection
${\cal{A}}\otimes q^{*}({\cal{L}})\inject
{\cal{B}}\otimes q^{*}({\cal{L}})$.
Ces $Gl(H)$-fibr\'es descendent. Le conoyau sera une famille universelle
de th\^eta-caract\'eristiques sur $\Theta^{os}(d)$,
si ${\cal{L}}$ satisfait \'a
${\cal{L}}\otimes{\cal{L}}={\cal{O}}$. Maintenant pour tout point
de $T^{os}(d,N)$ il existe
un voisinage de Zariski $U'$ tel que
${\cal{L}}\otimes{\cal{L}}$ soit trivial sur $U'$. On peut supposer cet ouvert
$GL(H)$-invariant (car ${\cal{L}}\otimes{\cal{L}}$ descend),
d'o\`u l'existence d'une famille universelle
sur l'ouvert $U$, image de l'ouvert $U'$. Ainsi, il existe une famille
universelle de th\^eta-caract\'eristiques localement dans la topologie
de Zariski. Globalement sur $\Theta^{os}(d)$, il ne peut exister une telle
famille. Il suffit en effet de consid\'erer par exemple l'ouvert des
th\^eta-caract\'eristiques ineffectives. D'apr\`es ce que pr\'ec\`ede
l'existence d'une telle famille supposerait l'existence d'un
$GL(A)$-fibr\'e inversible d'ordre $2$ dans $\Pic^{GL(A)}(X)$
(avec les notations de la section \ref{GdPpair}). Mais on vu que ce
groupe est sans torsion.
\cqfd
\begin{question} Peut-on trouver un rev\^etement \'etale de degr\'e $2$ de
$\Theta(d)$ sur lequel on a une famille universelle?
\end{question}
\np
|
1994-05-17T17:51:00 | 9403 | alg-geom/9403004 | en | https://arxiv.org/abs/alg-geom/9403004 | [
"alg-geom",
"math.AG"
] | alg-geom/9403004 | Klaus Altmann | Klaus Altmann | The versal Deformation of an isolated toric Gorenstein Singularity | 42 pages, LaTeX | null | null | null | null | Given a lattice polytope Q in R^n, we define an affine scheme M(Q) that
reflects the possibilities of splitting Q into a Minkowski sum. On the other
hand, Q induces a toric Gorenstein singularity Y, and we construct a flat
family over M(Q) with Y as special fiber. In case Y has an isolated singularity
only, this family is versal. (This revised version contains the proof now.)
| [
{
"version": "v1",
"created": "Wed, 2 Mar 1994 16:47:26 GMT"
},
{
"version": "v2",
"created": "Tue, 17 May 1994 15:50:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Altmann",
"Klaus",
""
]
] | alg-geom | \section{#1}
\protect\setcounter{secnum}{\value{section}}
\protect\setcounter{equation}{0}
\protect\renewcommand{\theequation}{\mbox{\arabic{secnum}.\arabic{equation}}}}
\setcounter{tocdepth}{1}
\begin{document}
\title{The versal Deformation of an isolated toric Gorenstein Singularity}
\author{Klaus Altmann\footnotemark[1]\\
\small Dept. of Mathematics, M.I.T., Cambridge, MA 02139, U.S.A.
\vspace{-0.7ex}\\ \small E-mail: [email protected]}
\footnotetext[1]{Die Arbeit wurde mit einem Stipendium des DAAD unterst\"utzt.}
\date{}
\maketitle
\begin{abstract}
Given a lattice polytope $Q\subseteq I\!\!R^n$, we define an affine scheme
$\bar{{\cal M}}$ that
reflects the possibilities of splitting $Q$ into a Minkowski sum.\\
\par
On the other hand, $Q$ induces a toric Gorenstein singularity $Y$, and we
construct a flat family
over $\bar{{\cal M}}$ with $Y$ as special fiber. In case $Y$ has an isolated
singularity only, this
family is versal.
\end{abstract}
\tableofcontents
\par
\vspace{2ex}
\sect{Introduction}\label{s1}
\neu{11}
The whole deformation theory of an isolated singularity is encoded in its
so-called
versal deformation.\\
For complete intersection singularities this is a family over a
smooth base space - obtained by certain disturbations of the defining
equations.\\
As
soon as we are leaving this class of singularities, the structure of the family
or
even the base space will be more complicated.
It is well known that the base space might consist of several components or
might be non-reduced.\\
In \zitat{9}{2} we will present a (three-dimensional) example
of a singularity
admitting a fat point as base space of its versal deformation.\\
\par
\neu{12}
For two-dimensional cyclic quotient singularities (coinciding with the
two-dimensional affine toric varieties),
the computations of Arndt, Christophersen,
Koll\'{a}r/ Shephard-Barron,
Riemenschneider, and Stevens
provide a description of the versal family - in particular, number and
dimension
of the components of the reduced base (they are smooth) are computed.\\
Christophersen observed that
the total spaces over these components
are toric varieties again (cf.\ \cite{Ch}).
This suggests the conjecture that the entire deformation theory of affine
toric varieties keeps inside this category. It should be a
challenge to find the versal deformation, its base space, or the total spaces
over the components by purely combinatorial methods.\\
\par
\neu{13}
In the present paper we investigate the case of affine, toric
Gorenstein singularities $Y$ given by some lattice polytope $Q$.\\
\par
In \S \ref{s2} and \S \ref{s3} we start with describing an affine scheme
$\bar{{\cal M}}$ which
seems to be interesting independently from the toric or deformation stuff.
It describes the possibilities of spliting
$Q$ into Minkowski
summands. The underlying reduced space is an arrangment of planes corresponding
to
those Minkowski decompositions involving
summands, that are lattice polytopes themselfs.\\
\par
In \S \ref{s5} we
construct a flat family over $\bar{{\cal M}}$
with
the toric Gorenstein singularity $Y$ induced by $Q$ as special fiber.
Computing the Kodaira-Spencer as well as the obstruction map
shows that, in case that
the singularity is isolated, the family is versal (nevertheless trivial
for $\mbox{dim}\,Q\geq 3$).\\
In the general case, the Kodaira-Spencer map is an isomorphism onto the
homogeneous part of $T^1_Y$ with the most interesting multidegree
(cf.\ Theorem \zitat{6}{2}), and the obstruction map is still injective
(cf.\ Theorem \zitat{7}{2}).\\
\par
On the other hand, this family is embedded in a larger (non-flat) family that
equals
a morphism of affine toric varieties: The base space is given by the cone
$C(Q)$ of
Minkowski summands of positive multiples of $Q$, and the total space comes from
the
tautological cone over $C(Q)$ (cf.\ \S \ref{s4} and \S \ref{s5}). In
particular,
for affine, toric, isolated Gorenstein singularities, Christophersen's
observation
(cf.\ \zitat{1}{2})
keeps true (cf.\ \S \ref{s8}).\\
\par
Through the whole paper, an example accompanies the general theory. Further
examples can be
found in \S \ref{s9}.\\
\par
\neu{14}
{\em Acknowledgements:} I am very grateful to Duco van Straten
and Theo de Jong for constant
encouragement and valuable hints.\\
This paper was written during a one-year-stay at MIT.
I want to thank Richard Stanley and all the other people who made it possible
for me to work at this very interesting and stimulating place.\\
\par
\sect{The Minkowski scheme of a lattice polytope}\label{s2}
\neu{21}
Let $Q \subseteq I\!\!R^n$ be a lattice polytope, i.e.\ the vertices are contained
in
$Z\!\!\!Z^n$. We will always
assume that the edges do not contain any interior lattice points
(cf.\ \zitat{3}{6}), hence, after choosing orientations they are given by
primitive vectors $d^1,\dots, d^N \in Z\!\!\!Z^n$.\\
\par
{\bf Definition:}
{\em For every 2-face $\varepsilon <Q$ we define its sign vector
$\underline{\varepsilon}=(\varepsilon_1,\dots, \varepsilon_N) \in \{0,\pm 1\}^N$ by
\[
\varepsilon_i := \left\{
\begin{array}{cl}
\pm 1 & \mbox{if $d^i$ is an edge of $\varepsilon$}\\
0 & \mbox{otherwise}
\end{array} \right.
\]
such that the oriented edges $\varepsilon_i\cdot d^i$ fit to a cycle along the
boundary
of $\varepsilon$.
This determines $\underline{\varepsilon}$ up to sign, and we choose one of both
possibilities.
In particular, $\sum_i \varepsilon_i d^i =0$.}\\
\par
{\bf Example:}
Let us introduce the following example, which will be continued through the
paper:\\
For $Q$ we take the hexagon
\[
Q_6:= \mbox{Conv}\, \{ (0,0), (1,0), (2,1), (2,2), (1,2), (0,1) \} \subseteq
I\!\!R^2.
\vspace{2ex}
\]
\begin{center}
\unitlength=0.5mm
\linethickness{0.4pt}
\begin{picture}(100.00,64.00)
\put(20.00,20.00){\line(1,0){20.00}}
\put(40.00,20.00){\line(1,1){20.00}}
\put(60.00,40.00){\line(0,1){20.00}}
\put(60.00,60.00){\line(-1,0){20.00}}
\put(40.00,60.00){\line(-1,-1){20.00}}
\put(20.00,40.00){\line(0,-1){20.00}}
\put(16.00,10.00){\makebox(0,0)[cc]{$(0,0)$}}
\put(44.00,10.00){\makebox(0,0)[cc]{$(1,0)$}}
\put(70.00,36.00){\makebox(0,0)[cc]{$(2,1)$}}
\put(70.00,64.00){\makebox(0,0)[cc]{$(2,2)$}}
\put(34.00,64.00){\makebox(0,0)[cc]{$(1,2)$}}
\put(12.00,46.00){\makebox(0,0)[cc]{$(0,1)$}}
\put(30.00,22.00){\makebox(0,0)[cb]{$d^1$}}
\put(100.00,40.00){\makebox(0,0)[lc]{hexagon $Q_6$}}
\end{picture}
\end{center}
Starting with $d^1:= \overline{(0,0) (1,0)}$, the anticlockwise oriented edges
are denoted by
$d^1,\dots, d^6$. As vectors they equal
\[
\begin{array}{ccc}
d^1=(1,0); & d^2=(1,1); & d^3=(0,1);\\
d^4=(-1,0); & d^5=(-1,-1); & d^6=(0,-1).
\end{array}
\]
$Q_6$ is 2-dimensional, hence, it is its own unique 2-face $\varepsilon=Q$. For
$\underline{Q}$ we take
$\underline{Q}=(1,\dots,1)$.\\
\par
\neu{22}
We define the vector space $V \subseteq I\!\!R^N$ by
\[
V:= V(Q):= \{ (t_1,\dots,t_N)\, |\; \sum_i t_i \,\varepsilon_i \,d^i =0\;
\mbox{ for every 2-face } \varepsilon <Q\}.
\]
Then, $C(Q):= V \cap I\!\!R^N_{\geq 0}$ is a rational, polyhedral cone in $V$, and
its
points correspond to the Minkowski summands of positive multiples of $Q$:
Given a point $(t_1,\dots,t_N)\in C(Q)$, the corresponding polytope
$Q_{\underline{t}}$ is built by the
edges $t_i\cdot d^i$ instead of the plain $d^i$ used in $Q$ (cf.\
\zitat{4}{1}).\\
For a Minkowski summand $Q'$ of a positive multiple of $Q$ we will denote its
point
in the cone by $\varrho (Q')\in C(Q)$.\\
\par
({\em Example:} $\varrho(t\cdot Q) = (t,\dots,t) \in C(Q) \subseteq V \subseteq
I\!\!R^N.$)\\
\par
\neu{23}
For each 2-face $\varepsilon <Q$ and for each integer $k\geq 1$
we define the (vector valued)
polynomial
\[
g_{\varepsilon, k}(\underline{t}):= \sum_{i=1}^N t_i^k \, \varepsilon_i \,d^i\,.
\]
Using coordinates of $I\!\!R^n$ the $g_{\varepsilon, k}(\underline{t})$ turn into regular
polynomials - for each pair $(\varepsilon,k)$ we will get two linearly independent
ones.\\
\par
We obtain an ideal
\[
{\cal J} := \left( g_{\varepsilon,k} \, | \; \varepsilon < Q, \; k\geq 1 \right)
\subseteq \,I\!\!\!\!C [t_1,\dots,t_N],
\]
which defines an affine closed subscheme
\[
{\cal M} := \mbox{Spec}\, ^{\displaystyle \,I\!\!\!\!C [\underline{t}]} \!\! \left/ \!\!
_{\displaystyle {\cal J} }\right. \subseteq V_{\,I\!\!\!\!C} \subseteq \,I\!\!\!\!C^N.
\]
\par
{\bf Example:}
For our hexagon $Q_6$ introduced in \zitat{2}{1} we obtain
\[
{\cal J} = \left( t_1^k + t_2^k - t_4^k - t_5^k,\; t_2^k + t_3^k - t_5^k -
t_6^k\,|\; k\geq 1 \right).
\]
\\
Of course, finally many equations are sufficient to generate the ideal ${\cal J} $ -
but we
can even give an effective criterion to see which equations can be dropped:\\
\par
{\bf Proposition:}
{\em Let $\varepsilon <Q$ be a 2-face. Then, $\varepsilon$ is contained in a
two-dimensional
subspace of $I\!\!R^n$, and this vector space comes with a natural lattice
(the restriction of
the big lattice $Z\!\!\!Z^n$).\\
If $\varepsilon$ is contained in two different strips defined by pairs of
parallel lines of
lattice-distance $\leq k_0$ each, then the equations $g_{\varepsilon,k}\; (k>k_0)$
are
contained
in the ideal generated by $g_{\varepsilon,1}, \dots, g_{\varepsilon,k_0}$.}\\
\par
{\bf Proof:} cf.\ \zitat{3}{3}.\\
\par
{\bf Corollary:}
{\em If $Q$ is contained in $n$ linearly independent strips (defined by pairs
of
parallel hyperplanes) of lattice-thickness $\leq k_0$, then all polynomials
$g_{\varepsilon,k}$ with $k>k_0$ are superfluous.}\\
\par
{\bf Example:}
Obviously, $Q_6$ is contained in at least three strips of thickness 2. Hence,
${\cal J} $ is
generated in degree $\leq 2$:
\[
{\cal J} = \left( t_1 + t_2 - t_4 - t_5,\quad t_2 + t_3 - t_5 - t_6,\quad
t_1^2 + t_2^2 - t_4^2 - t_5^2,\quad t_2^2 + t_3^2 - t_5^2 - t_6^2 \right) .
\]
\vspace{-1ex}
\\
\neu{24}
Denote by $\ell$ the canonical projection
\[
\ell : \,I\!\!\!\!C^N \longrightarrow\hspace{-1.5em}\longrightarrow \;^{\displaystyle \,I\!\!\!\!C^N} \!\!\! \left/
\!_{\displaystyle \,I\!\!\!\!C\cdot (1,\dots,1)} \right.
= \;^{\displaystyle \,I\!\!\!\!C^N} \!\!\! \left/
\!_{\displaystyle \,I\!\!\!\!C\cdot \varrho(Q)} \right. .
\]
On the level of regular functions this corresponds to the inclusion\\
$\,I\!\!\!\!C [t_i-t_j\, | \; 1 \leq i,j \leq N ] \subseteq \,I\!\!\!\!C[t_1,\dots,t_N]$.\\
\par
{\bf Theorem:}
{\em \begin{itemize}
\item[(1)]
${\cal J} $ is generated by polynomials from $\,\,I\!\!\!\!C [t_i-t_j]$, i.e.\ ${\cal M} =
\ell^{-1}(\bar{{\cal M}})$ for some affine closed subscheme
$\bar{{\cal M}} \subseteq
\,^{\displaystyle V_{\,I\!\!\!\!C}} \!\!\! \left/
\!_{\displaystyle \,I\!\!\!\!C\cdot \varrho(Q)} \right.
\subseteq
\;^{\displaystyle \,I\!\!\!\!C^N} \!\!\! \left/
\!_{\displaystyle \,I\!\!\!\!C\cdot \varrho(Q)} \right. .$\\
($\bar{{\cal M}}$ is defined by the ideal ${\cal J} \cap \,I\!\!\!\!C [t_i-t_j]$.)
\item[(2)]
${\cal J} \subseteq \,I\!\!\!\!C[t_1,\dots,t_N]$ is the smallest ideal (i.e.\ ${\cal M}$ is
the largest closed subscheme of $\,I\!\!\!\!C^N$) with
\begin{itemize}
\item[(i)]
property (1) and
\item[(ii)]
containing the ``toric'' equations
\[
\prod_{i=1}^N t_i^{d_i^+} - \prod_{i=1}^N t_i^{d_i^-}\quad \mbox{ with}
\]
$\underline{d} \in Z\!\!\!Z^N \cap \mbox{span} \left\{
[\langle\varepsilon _1 d^1,c\rangle, \dots, \langle \varepsilon_N d^N,c \rangle ]\, | \;
\varepsilon <Q \mbox{ 2-face},\, c\in I\!\!R^n \right\}$. \\
(For an integer $h$ we denote
\[
h^+ := \left\{ \begin{array}{cl}
h & \mbox{ if } h \geq 0\\
0 & \mbox{ otherwise}
\end{array} \right.
\quad ; \qquad
h^- := \left\{ \begin{array}{ll}
0 & \mbox{ if } h \geq 0\\
-h & \mbox{ otherwise}
\end{array} \right. .)
\]
\end{itemize}
\end{itemize} }
{\bf Proof:} cf.\ \zitat{3}{4}.\\
\par
{\bf Example:}
Toric equations for $Q_6$ are for instance
$t_1\,t_2 - t_4\,t_5, \; t_2\,t_3 - t_5\,t_6,\,$ and $t_1\,t_6 -
t_3\,t_4$.\\
\par
\neu{25}
We want to describe the structure of the underlying reduced spaces of ${\cal M}$ or
$\bar{{\cal M}}$.\\
\par
First, we mention the following trivial observations concerning the cone
$C(Q)$:
\begin{itemize}
\item[(i)]
Minkowski summands $Q'$ of $Q$ (instead of a positive multiple of $Q$) are
characterized by the property $\varrho(Q) - \varrho(Q') \in I\!\!R^N_{\geq 0}$,
i.e.\ all
components of $\varrho(Q')$ have to be contained in the interval $[0,1]$.
\item[(ii)]
For a Minkowski summand $Q'$ (of some positive multiple of $Q$) the property of
being
a lattice polytope is equivalent to the fact that $\varrho(Q')\in Z\!\!\!Z^N$.
\end{itemize}
Now, let $Q=R_0+\dots +R_m$ be a decomposition of $Q$ into a Minkowski sum of
$m+1$
lattice polytopes. Then, the $N$-tuples $\varrho(R_0),\dots,\varrho(R_m)$
consist
of numbers 0 and 1 only, and they sum up to $(1,\dots,1)$. In particular, the
$(m+1)$-plane
$\;\,I\!\!\!\!C\cdot\varrho(R_0) + \dots + \,I\!\!\!\!C\cdot\varrho(R_m) \subseteq \,I\!\!\!\!C^N$
(or its $m$-dimensional image via $\ell$)
is contained in
${\cal M}$ (in $\bar{{\cal M}}$, respectively).\\
\par
{\bf Remark:}
\begin{itemize}
\item[(1)]
Those $(m+1)$-plane (or its image via $\ell$) is given by the linear equations
$t_i-t_j=0$ (if $d^i, d^j$ belong to a common summand $R_v$).
\item[(2)]
Refinements of Minkowski decompositions (they form a partial ordered set)
correspond to inclusions of the associated planes.
\vspace{2ex}
\end{itemize}
\par
{\bf Theorem:}
{\em ${\cal M}_{\mbox{\footnotesize red}}$ and $\bar{{\cal M}}_{\mbox{\footnotesize
red}}$
equal the union of those flats
corresponding to maximal Minkowski decompositions of $Q$ into lattice
summands.}\\
\par
{\bf Proof:} cf.\ \zitat{3}{5}.\\
\par
{\bf Example:}
${\cal M}(Q_6)$ and $\bar{{\cal M}}(Q_6)$ are already reduced schemes - for non-reduced
examples
cf.\ \S \ref{s9}. Let us study them directly:
\begin{itemize}
\item
The linear equations allow the following substitution:
\[
\begin{array}{rcl}
t &:=& t_1 \\
s_1 &:=& t_1 - t_3 \\
s_2 &:=& t_4 - t_2 \\
s_3 &:=& t_1 - t_4
\end{array}
\qquad
\begin{array}{rcl}
t_1 &=& t \\
t_2 &=& t-s_2-s_3 \\
t_3 &=& t-s_1 \\
t_4 &=& t-s_3 \\
t_5 &=& t-s_2 \\
t_6 &=& t-s_1-s_3 \, .
\end{array}
\]
\item
The two quadratic equations transform into $s_1\,s_3 = s_2\,s_3 = 0$.
\end{itemize}
In particular, $\bar{{\cal M}}$ is the union of a line and a 2-plane - corresponding
to the Minkowski
decompositions
\[
\begin{array}{rcl}
Q_6 &=& \mbox{Conv}\,\{(0,0), (1,0), (1,1)\} + \mbox{Conv}\,\{(0,0), (0,1),
(1,1)\} \; \mbox{ and}\\
Q_6 &=& \mbox{Conv}\,\{(0,0), (1,0)\} +
\mbox{Conv}\,\{(0,0), (0,1)\} + \mbox{Conv}\,\{(0,0), (1,1)\} .
\end{array}
\vspace{-2ex}
\]
\\
\par
\neu{26}
$\bar{{\cal M}}$ (or ${\cal M}=\ell^{-1}(\bar{{\cal M}})$) reflect the possibilities of
Minkowski decompositions of $Q$:
\begin{itemize}
\item
The underlying reduced space encodes the decompositions of $Q$ into lattice
summands.
\item
Extremal decompositions into rational summands are hidden in the scheme
structure
of $\bar{{\cal M}}$.\\
Its tangent space in 0 (the smallest affine space containing
$\bar{{\cal M}}$) equals $^{\displaystyle V_{\,I\!\!\!\!C}}\!\!\!\left/ \!
_{\displaystyle \,I\!\!\!\!C \cdot \varrho(Q)} \right.$ - it is the vector space arising
from
the cone $C(Q)$ of Minkowski summands by
killing the summands homothetic to $Q$.
\end{itemize}
Therefore, we will call $\bar{{\cal M}}$ the (affine) {\em Minkowski scheme} of
$Q$.\\
\par
{\bf Remark:}
The ideals defining ${\cal M}$ and $\bar{{\cal M}}$ are homogeneous. Hence, there are
projective versions of these schemes, too.\\
\par
\sect{Proof of the statements of \S 2}\label{s3}
\neu{31}
Using vectors $c \in Z\!\!\!Z^N$ (or selected $c\in I\!\!R^N$)
we can evaluate the edges $d^1,\dots,d^N$ to get integers
\[
d_1:= \langle \varepsilon_1 d^1, c \rangle , \dots, d_N:= \langle \varepsilon_N d^N, c
\rangle
\]
for every given 2-face $\varepsilon < Q$.
Doing so, the statements of \S \ref{s2} can be
reduced to much simpler lemmas, which we will present here.\\
\par
Then, all those lemmas are proved using the following recipe:
\begin{itemize}
\item[(i)]
Assume $d_i=\pm 1$ - then the lemmas reduce to well known facts concerning
symmetric functions.
\item[(ii)]
Move to the general case by specialization of variables.
\vspace{2ex}
\end{itemize}
\par
\neu{32}
For the whole \S \ref{s3} we use the following notations:
\vspace{1ex}\\
Let $d_1,\dots,d_N \in Z\!\!\!Z$ such that
$\, d_1,\dots, d_M \geq 0, \, d_{M+1},\dots, d_N \leq 0, \,$ and $\sum_{i=1}^N
d_i =0\,$.
\[
\begin{array}{rcccl}
g_k(\underline{t}) &:=& g_{\underline{d},k}(\underline{t}) &:=&
\sum_{i=1}^N d_i \, t_i^k\; , \\
p(\underline{t}) &:=& p_{\underline{d}}(\underline{t}) &:=&
t_1^{d_1}\cdot\dots\cdot t_M^{d_M} - t_{M+1}^{d_{M+1}}\cdot\dots\cdot
t_N^{d_N}.
\end{array}
\]
Denote by $\sigma_k$ and $s_k$ the $k$-th elementary symmetric polynomial and
the sum of the
$k$-th powers of a given set of variables, repectively.
\\
\par
{\bf Remark:}
For $1\leq i,j \leq M$ or $M+1 \leq i,j \leq N$,
identifying the two variables $t_i$ and $t_j$ (i.e.\ switching from
$\,I\!\!\!\!C[\underline{t}]$ to
$^{\displaystyle \,I\!\!\!\!C[\underline{t}]} \! / _{\displaystyle t_i-t_j}$) yields the
following
situation:
\begin{itemize}
\item
$t_i, t_j$ are replaced by a common new variable $\tilde{t}$ (i.e.\ $N$ is
replaced by $N-1$),
\item
$d_i, d_j$ are replaced by $\tilde{d}:= d_i + d_j$, but
\item
$g_k(\underline{t}), p(\underline{t})$ keep their shapes in the new set up.
\end{itemize}
In particular, the general situation can always be obtained via factorization
from the special
case $d_1=\dots = d_M=1;\; d_{M+1}=\dots = d_N=-1$ (and $N=2M$).
Renaming $t_i=x_i, \, t_{M+i}=y_i \,(i\leq M)$ it looks like
\begin{eqnarray*}
g_k(\underline{x}, \underline{y}) &= &
\left( \sum_{i=1}^M x_i^k \right) - \left( \sum_{i=1}^M y_i^k \right) =
s_k(\underline{x}) -
s_k(\underline{y})\, , \\
p(\underline{x}, \underline{y}) &= &
(x_1\cdot\dots\cdot x_M) - (y_1\cdot\dots\cdot y_M) = \sigma_M(\underline{x}) -
\sigma_M(\underline{y}).
\end{eqnarray*}
\\
\par
\neu{33}
{\bf Lemma:} {\em
If $k_0:= \sum_{i=1}^M d_i = -\sum_{i=M+1}^N d_i$, then the
polynomials $g_k \, (k>k_0)$ are
$\,I\!\!\!\!C[\underline{t}]$-linear combinations of the $g_1,\dots,g_{k_0}$. }
(This implies Proposition \zitat{2}{3}.)\\
\par
{\bf Proof:}
As previously discussed, we may regard the special case $d_i=\pm 1$.
In particular, this implies
$k_0=M$.\\
Now, for an arbitrary $k\,(>M)$, the
expression $s_k(\underline{x})$ is a polynomial in either the
$\sigma_1(\underline{x}),\dots, \sigma_M(\underline{x})$ or the
$s_1(\underline{x}), \dots,
s_M(\underline{x})$, say
\[
s_k(\underline{x}) = P_k\left( s_1(\underline{x}),\dots, s_M(\underline{x})
\right) .
\]
Then,
\[
g_k(\underline{x}, \underline{y}) = s_k(\underline{x}) - s_k(\underline{y}) =
P_k\left( s_1(\underline{x}), \dots,s_M(\underline{x})\right) -
P_k\left( s_1(\underline{y}), \dots,s_M(\underline{y})\right),
\]
but for each monomial $s_1^{e_1}\, s_2^{e_2} \dots s_M^{e_M}$ ocuring in $P_k$,
we have
\[
\begin{array}{l}
s_1(\underline{x})^{e_1}\cdot\dots\cdot s_M(\underline{x})^{e_M} -
s_1(\underline{y})^{e_1}\cdot\dots\cdot s_M(\underline{y})^{e_M} =
\vspace{1ex}\\
\qquad \begin{array}{r}
= \sum_{v=1}^M \sum_{i=1}^{e_v} [s_v(\underline{x}) - s_v(\underline{y})]\cdot
s_1(\underline{x})^{e_1}\dots s_{v-1}(\underline{x})^{e_{v-1}}\,
s_v(\underline{x})^{i-1} \cdot
\qquad\qquad \\
\cdot s_v(\underline{y})^{e_v-i}\, s_{v+1}(\underline{y})^{e_{v+1}} \dots
s_M(\underline{y})^{e_M}
\end{array}
\vspace{1ex}\\
\qquad \begin{array}{r} =
\sum_{v=1}^M g_v(\underline{x}, \underline{y})\cdot \sum_{i=1}^{e_v}
s_1(\underline{x})^{e_1}\dots s_{v-1}(\underline{x})^{e_{v-1}}\,
s_v(\underline{x})^{i-1} \cdot
\qquad\qquad\qquad \\
\cdot s_v(\underline{y})^{e_v-i}\, s_{v+1}(\underline{y})^{e_{v+1}} \dots
s_M(\underline{y})^{e_M},
\end{array}
\end{array}
\]
which proves the lemma.
\hfill$\Box$\\
\par
\neu{34}
{\bf Lemma:}
{\em \begin{itemize}
\item[(1)]
The ideal ${\cal J} := (g_k\, | \; k\geq 1) \subseteq \,I\!\!\!\!C[t_1,\dots, t_N]$ is
generated by polynomials in
$t_i - t_1\; (i=2,\dots,N)$ only.
\item[(2)]
${\cal J} $ is the smallest ideal generated by polynomials in $t_i-t_1$, which
additionally contains $p$.
\end{itemize} }
(This implies Theorem \zitat{2}{4}.)\\
\par
{\bf Proof:}
(1) Replacing $t_i$ by $t_i-t_1$ as arguments in $g_k$ yields
\begin{eqnarray*}
g_k(t_1-t_1,\dots,t_N-t_1) &=&
\sum _{i=1}^N d_i\, (t_i-t_1)^k = \sum_{i=1}^N d_i\cdot \left(\sum_{v=0}^k
(-1)^v \, t_1^v \,
t_i^{k-v}\right) \\
&=& \sum_{v=0}^k (-1)^v\, t_1^v \cdot \left( \sum_{i=1}^N d_i\, t_i^{k-v}
\right) =
\sum_{v=0}^k (-1)^v\, t_1^v \, g_{k-v} (\underline{t}).
\end{eqnarray*}
In particular, $\left( g_k(\underline{t})\, |\; k\geq 1 \right)$ and
$\left( g_k(\underline{t}- t_1)\, |\; k\geq 1 \right)$ are the same ideals in
$\,I\!\!\!\!C[\underline{t}]$.\\
\par
(2) The polynomial rings $\,I\!\!\!\!C[\underline{t}]$ and $\,I\!\!\!\!C[t_1,\,\underline{t}-t_1]$
are equal, i.e.
each polynomial $q(\underline{t})$ can uniquely be written as
\[
q(\underline{t}) = \sum_{v\geq 0} q_v (t_2-t_1, \dots , t_N-t_1) \cdot t_1^v.
\]
Moreover, if $J\subseteq \,I\!\!\!\!C[\underline{t}]$ is an ideal generated by
polynomials in
$\underline{t}-t_1$ only, then for each $q(\underline{t})\in J$ the components
$q_v$ are
automatically contained in $J$, too.\\
\par
Let us determine the components of the polynomial $p$ - we will start with our
special case
again:
\[
p(T+\underline{X}, T+\underline{Y}) = (T+X_1)\cdot \dots \cdot (T+X_M) -
(T+Y_1)\cdot \dots \cdot (T+Y_M)
\]
has $\sigma_k(\underline{X}) - \sigma_k(\underline{Y})$ as coefficient of
$T^{M-k} \;
(k=1,\dots, M)$. Now, there are a polynomial $P_k$ and a non-vanishing rational
number
$c_k$ (not depending on $M$) such that
\[
\sigma_k(\underline{X}) = P_k(s_1(\underline{X}),\dots,s_{k-1}(\underline{X}))
+
c_k\cdot s_k(\underline{X}).
\]
As in the proof of the previous lemma we obtain
\begin{eqnarray*}
\sigma_k(\underline{X}) - \sigma_k(\underline{Y}) &=&
\begin{array}[t]{r}
P_k(s_1(\underline{X}),\dots,s_{k-1}(\underline{X})) -
P_k(s_1(\underline{Y}),\dots,s_{k-1}(\underline{Y})) + \qquad \\
+ c_k\cdot s_k(\underline{X}) - c_k\cdot s_k(\underline{Y})
\end{array}\\
&=& \sum_{v=1}^{k-1} q_v(\underline{X}, \underline{Y})\cdot g_v(\underline{X},
\underline{Y})
+ c_k \cdot g_k(\underline{X}, \underline{Y})
\end{eqnarray*}
for some coefficients $q_v$. Specialization - first by $T\mapsto x_1,\,
X_i\mapsto x_i-x_1,\,
Y_i\mapsto y_i-x_1$, then followed by the usual one - shows that the ideal
generated by the
components $p_v(\underline{t}-t_1)$ of $p$ equals ${\cal J} $.
\hfill$\Box$\\
\par
\neu{35}
{\bf Lemma:} {\em
Let $\underline{c}=(c_1,\dots,c_N) \in \,I\!\!\!\!C^N$ be a point such that
$g_k(\underline{c})=0$ for each
$k\geq 1$. Then, for every fixed $c\in \,I\!\!\!\!C$, we have $\sum_{c_i=c} d_i =0$.}
(This implies Theorem \zitat{2}{5}.)\\
\par
{\bf Proof:}
The equations $\sum_{i=1}^N d_i\, c_i^k =0$ present 0 as a linear combination
of the vectors
$(c_i, c_i^2, c_i^3,\dots)$. On the other hand, it is the Vandermonte that
tells us that this linear
combination has to be a trivial one, i.e.\ the sum of the coefficients $d_i$
belonging to equal
variables vanishes.
\hfill$\Box$\\
\par
\neu{36}
The polytope $Q$ was assumed to have primitive edges only. Actually, we never
needed this fact
neither in the previous lemmata nor in their proofs. It is only important to
translate these results into
the language of Minkowski summands used in \S \ref{s2}.\\
\par
Droping this condition, similar constructions are possible. However, by
declaring some or all lattice
points contained in edges of $Q$ to be additional, artificial vertices of $Q$,
several possibilities
arise with equal rights. The two extremal cases (add
either no or all possible generalized vertices) seem to
be the most interesting ones.\\
\par
{\bf Remark:}
\begin{itemize}
\item[(1)]
For a natural number $g\inI\!\!N$, the polytopes $Q$ (with some fixed set of
possibly artificial vertices)
and $g\cdot Q$ (with the correponding set of vertices) induce the same
Minkowski scheme
$\bar{{\cal M}}$.
\item[(2)]
Let $Q_1\subseteq Q_2$ be the same polytopes with different sets of generalized
vertices.
Then, $\bar{{\cal M}}_1$ is a closed subscheme of $\bar{{\cal M}}_2$. It is defined by
identifying
the variables associated to those generalized edges of $Q_2$, that are
contained in the same
generalized edge of $Q_1$.
\vspace{1ex}
\end{itemize}
{\bf Conjecture:}
Let $Q$ be a lattice polytope such that each extremal Minkowski summand of $Q$
is a lattice
polytope, too. Then, using all generalized vertices of $Q$, the affine schemes
${\cal M}$ and
$\bar{{\cal M}}$ are reduced.\\
\par
In particular, if $Q$ is an arbitrary lattice polytope (with primitive edges),
then $\bar{{\cal M}}_Q$ would
be embedded in some reduced $\bar{{\cal M}}_{g\cdot Q}$. The non-reduced structure
of
$\bar{{\cal M}}_Q$ would arise as a germ of components visible in $\bar{{\cal M}}_{g\cdot
Q}$ only.\\
\par
\sect{The tautological cone over $C(Q)$}\label{s4}
\neu{41}
In \zitat{2}{2} we have introduced the cone $C(Q)$ of Minkowski summands of
$I\!\!R_{\geq 0} \cdot Q$. For an element $(t_1,\dots,t_N)\in C(Q)$ the
corresponding summand
$Q_{\underline{t}}$ was built by the edges $t_i\cdot d^i\; (i=1,\dots,N)$.
However, defining
$Q_{\underline{t}}$ as a particular polytope inside its translation class
requires a closer look:\\
\par
Assume that $0\in I\!\!R^n$ coincides with some vertex of the lattice polytope $Q$.
Then, each
vertex $a$ of $Q$ can be reached from there by some walk along the edges of $Q$
- we
obtain
\[
a= \sum_{i=1}^N \lambda_i\, d^i \; \mbox{ for some }
\underline{\lambda}= (\lambda_1,\dots, \lambda_N)\, , \; \lambda_i \in Z\!\!\!Z.
\]
Now, given an element $\underline{t}\in C(Q)$, we can define the corresponding
vertex
$a_{\underline{t}}$ (and finally the polytope $Q_{\underline{t}}$ as the convex
hull of all of them)
by
\[
a_{\underline{t}} := \sum_{i=1}^N t_i\, \lambda_i\, d^i.
\]
(The linear equations defining $V= \mbox{span}\,C(Q)$ ensure that this
definition does not depend
on the particular path from $0$ to $a$ through the 1-skeleton of $Q$.)\\
\par
\neu{42}
{\bf Definition:} {\em
The tautological cone $\tilde{C}(Q) \subseteq I\!\!R^n \times V \subseteq I\!\!R^{n+N}$
is defined as
\[
\tilde{C}(Q) := \{ (a,\,\underline{t})\, | \; \underline{t}\in C(Q); \, a\in
Q_{\underline{t}} \} .
\] }
\\
{\bf Remark:}
$\tilde{C}(Q)$ is (as $C(Q)$) a rational, polyhedral cone. It is generated by
the pairs
$(a^i_{\underline{t}^j}, \, \underline{t}^j)$ with
\begin{itemize}
\item[$\bullet$]
$a^i$ is a vertex of $Q$ and
\item[$\bullet$]
$\underline{t}^j$ is a fundamental generator of $C(Q)$.
\end{itemize}
(This follows from the simple rule
$(a_{\underline{t}+\underline{t}'},\,\underline{t}+\underline{t}')
= (a_{\underline{t}},\, \underline{t}) + (a_{\underline{t}'},\, \underline{t}')
$ for a vertex $a \in Q$
and $\underline{t}, \underline{t}' \in C(Q)$.)
\vspace{1ex}\\
\par
Defining $\sigma:= \mbox{Cone}(Q) \subseteq I\!\!R^{n+1}$ by puting $Q$ into the
hyperplane $(t=1)$,
we obtain a fiber product diagram of rational polyhedral cones:
\[
\begin{array}{ccc}
[\sigma \subseteq I\!\!R^{n+1}] & \stackrel{i}{\hookrightarrow} &
[\tilde{C}(Q) \subseteq I\!\!R^n \times V ]
\vspace{0.5ex}\\
\downarrow\!\!\!\!\!\makebox[.7pt]{ \mbox{\footnotesize pr}_{n+1} && \downarrow\!\!\!\!\!\makebox[.7pt]{ \mbox{\footnotesize pr}_V
\vspace{-0.5ex}\\
I\!\!R_{\geq 0} \quad& \stackrel{\cdot \varrho (Q)}{\hookrightarrow} & [C(Q)
\subseteq V]
\end{array}
\]
(The vertical maps are projections onto the $(n+1)$-th and the $V$-component,
respectively.
The inclusion $i$ is given by $(t\cdot a;\,t) \mapsto (t\cdot a;\, t,\dots,
t)$.)\\
\par
\neu{43}
The three cones $\sigma= \mbox{Cone}(Q) \subseteq I\!\!R^{n+1},\, \tilde{C}(Q)
\subseteq
I\!\!R^n\times V,\,$ and $C(Q)\subseteq V$ define affine toric varieties called $Y,
X,$ and $S$,
respectively. The corresponding rings of regular functions are
$A(Y)=\,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^{n+1}],\;
A(X)=\,I\!\!\!\!C[\tilde{C}(Q)^{\scriptscriptstyle\vee}\cap(Z\!\!\!Z^n\times V^\ast_{Z\!\!\!Z})],$
and $A(S)=\,I\!\!\!\!C[C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}]$.
These varieties come with the following maps:
\begin{itemize}
\item[(i)]
The diagram of \zitat{4}{2} induces a fiber product diagram
\[
\begin{array}{ccc}
Y & \stackrel{i}{\hookrightarrow} & X \\
\downarrow && \downarrow \!\!\pi \\
\,I\!\!\!\!C & \hookrightarrow & S\, .
\end{array}
\]
Both horizontal maps are closed embeddings.
(These claims will be checked in \ \zitat{4}{5} and \zitat{4}{8}(1).)
\item[(ii)]
$C(Q) = V \cap I\!\!R^N_{\geq 0}$ is contained in $I\!\!R^N_{\geq 0}$, and the
inclusion provides a
morphism $p: S \rightarrow \,I\!\!\!\!C^N$ defining functions $t_1,\dots, t_N$ on $S$.
The composition $\,I\!\!\!\!C \hookrightarrow S \stackrel{p}{\rightarrow} \,I\!\!\!\!C^N$ sends $t$
to
$(t,\dots,t)$.
\vspace{2ex}
\end{itemize}
\par
{\bf Remark:}
$Y$ is the affine toric Gorenstein singularity corresponding to the lattice
polytope $Q$. We will
use the map $\pi: X \rightarrow S$ to construct the versal deformation of
$Y$.\\
\par
\neu{44}
To study the toric varieties $Y, X,$ and $S$ it is important to understand the
dual cones of
$\sigma, \tilde{C}(Q),$ and $C(Q)$, respectively.
Let us start with the dual cone of $\sigma$:\\
\par
To each non-trivial $c\inZ\!\!\!Z^n$ we associate a vertex $a(c)$ of $Q$ and an
integer $\eta_0(c)$
meeting the properties
\begin{eqnarray*}
\langle Q,\, -c \rangle &\leq& \eta_0(c) \qquad \mbox{ and}\\
\langle a(c),\, -c \rangle &=& \eta_0(c).
\end{eqnarray*}
For $c=0$ we define $a(0):=0 \in I\!\!R^n$ and $\eta_0(0):=0 \in Z\!\!\!Z$.
\pagebreak[3]
\\
\par
{\bf Remark:}
\begin{itemize}
\item[(1)]
With respect to $Q$, $c\neq 0$ is the inner normal vector of the affine
supporting hyperplane
$[\langle \bullet,-c\rangle = \eta_0(c)]$ through $a(c)$. In particular,
$\eta_0(c)$ is uniquely determined,
while $a(c)$ is not.
\item[(2)]
Since $0\in Q$, the integers $\eta_0(c)$ are non-negative.
\vspace{2ex}
\end{itemize}
The dual cone of $\sigma$ is defined as
\[
\sigma^{\scriptscriptstyle\vee}:= \{ r\in I\!\!R^{n+1}\,|\; \langle\sigma,r\rangle \geq 0\}.
\]
By the definition of $\eta_0$, we have
\[
\partial\sigma^{\scriptscriptstyle\vee} \cap Z\!\!\!Z^{n+1} = \{[c,\eta_0(c)]\,|\; c\in Z\!\!\!Z^n\}\,.
\]
Moreover, if $c^1,\dots,
c^w \in Z\!\!\!Z^n\setminus 0$ are those elements producing irreducible pairs
$[c,\eta_0(c)]$ (i.e.\ not
allowing any non-trivial
lattice decomposition $[c,\eta_0(c)]=[c',\eta_0(c')] + [c'',\eta_0(c'')]$),
then the
elements
\[
[c^1,\eta_0(c^1)],\dots, [c^w,\eta_0(c^w)] , \, [\underline{0},1]
\]
form the minimal generator set for $\sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^{n+1}$ as a semigroup.
Among them are
all pairs $[c,\eta_0(c)]$ corresponding to facets (i.e.\ top dimensional faces)
of $Q$.\\
\par
We obtain a closed embedding $Y\hookrightarrow \,I\!\!\!\!C^{w+1}$. The coordinate
functions of
$\,I\!\!\!\!C^{w+1}$ will be denoted by $z_1,\dots,z_w,\,t$ corresponding to
$[c^1,\eta_0(c^1)],\dots, [c^w,\eta_0(c^w)], \, [\underline{0},1]$,
respectively.\\
\par
{\bf Example:}
We continue our example $Q_6$ from \S \ref{s2}. Here, the facets of $Q_6$ equal
its edges
$d^1,\dots,d^6$, and they are sufficient for producing
all irreducible pairs\\
$[c^1,\eta_0(c^1)],\dots, [c^6,\eta_0(c^6)]$. We have
\[
\begin{array}{ccccccccc}
c^1 &=& [0,1], & c^2 &=& [-1,1], & c^3 &=& [-1,0], \\
c^4 &=& [0,-1], & c^5 &=& [1,-1], & c^6 &=& [1,0]\, .
\end{array}
\]
The corresponding vertices are (for instance)
\[
a(c^6) = a(c^1) = (0,0), \quad
a(c^2) = a(c^3) = (2,1), \quad
a(c^4) = a(c^5) = (1,2),
\]
and we obtain
\[
\eta_0(c^1) = 0,\; \eta_0(c^2) = 1,\; \eta_0(c^3) = 2,\;
\eta_0(c^4) = 2,\; \eta_0(c^5) = 1,\; \eta_0(c^6) = 0\, .
\]
\\
\par
\neu{45}
Thinking of $C(Q)$ as a cone in $I\!\!R^N$ instead of $V$ allows dualizing
the equation $C(Q)=I\!\!R^N_{\geq 0} \cap V$ to get
$C(Q)^{\scriptscriptstyle\vee}= I\!\!R^N_{\geq 0} + V^\bot$. Hence, for $C(Q)$ as a cone in $V$
we obtain
\[
C(Q)^{\scriptscriptstyle\vee}= \;^{\displaystyle I\!\!R^N_{\geq 0} + V^\bot}\!\!\left/
_{\displaystyle V^\bot} \right.
= \mbox{Im}\, [I\!\!R^N_{\geq 0} \longrightarrow V^\ast].
\]
{\em As already happend with $I\!\!R^n$, we do not use different notations
for $I\!\!R^N$ and its dual space. However, writing down vectors we try to use
paranthesis and brackets for primal and dual ones, respectively.}\\
\par
The surjection $I\!\!R^N_{\geq 0}\longrightarrow\hspace{-1.5em}\longrightarrow C(Q)^{\scriptscriptstyle\vee}$ induces a map
$I\!\!N^N \longrightarrow C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}$, which does not
need to be
surjective at all. This leads to the following definition:\\
\par
{\bf Definition:}
{\em On $V^\ast_{Z\!\!\!Z}$ we introduce a partial ordering ``$\succeq$'' by
\[
\underline{\eta}\succeq \underline{\eta}' \quad \Longleftrightarrow
\quad \underline{\eta}-\underline{\eta}' \in \mbox{Im}\, [I\!\!N^N\rightarrow
V_{Z\!\!\!Z}^\ast] \subseteq C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}.
\] }
\\
On the geometric level, the non-saturated semigroup
$\mbox{Im}\, [I\!\!N^N\rightarrow
V_{Z\!\!\!Z}^\ast] \subseteq C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}$ corresponds to the
scheme theoretical
image ${\bar{S}}$ of $p:S\rightarrow \,I\!\!\!\!C^N$, and $S\rightarrow {\bar{S}}$ is its
normalization (cf.\ \zitat{5}{2}).\\
The equations of ${\bar{S}} \subseteq \,I\!\!\!\!C^N$ are collected in the kernel of
\[
\,I\!\!\!\!C[t_1,\dots,t_N]=\,I\!\!\!\!C[I\!\!N^N] \stackrel{\varphi}{\longrightarrow}
\,I\!\!\!\!C[ C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}] \subseteq \,I\!\!\!\!C[V^\ast_{Z\!\!\!Z}],
\]
and it is easy to see that
\begin{eqnarray*}
\mbox{Ker}\,\varphi &=& \left( \left.
\prod_{i=1}^N t_i^{d_i^+} - \prod_{i=1}^N t_i^{d_i^-}\, \right| \;
\underline{d}\in Z\!\!\!Z^N \cap V^\bot \right)\qquad \mbox{ with}\\
V^\bot &=& \mbox{span}
\left\{ \left.
[\langle\varepsilon _1 d^1,c\rangle, \dots, \langle \varepsilon_N d^N,c \rangle ]\, \right|
\;
\varepsilon <Q \mbox{ is a 2-face},\, c\in I\!\!R^n \right\}.
\end{eqnarray*}
{\bf Remark:}
Using our new notations, we can reformulate Theorem \zitat{2}{4} now:\\
${\cal M} \subseteq \,I\!\!\!\!C^N$ is the largest closed subscheme that is contained in
${\bar{S}}$ and, additionally, comes from
$^{\displaystyle \,I\!\!\!\!C^N} \!\!\! \left/
\!_{\displaystyle \,I\!\!\!\!C\cdot \varrho(Q)} \right.$ via $\ell$.\\
\par
On the other hand, dualizing the embedding $I\!\!R_{\geq 0} \hookrightarrow C(Q)$
yields
\[
\begin{array}{ccc}
C(Q)^{\scriptscriptstyle\vee}\cap V_{Z\!\!\!Z}^\ast & \longrightarrow\hspace{-1.5em}\longrightarrow & I\!\!N\\
\underline{\eta} & \mapsto & \sum_i \eta_i
\end{array}
\]
at the level of semigroups. This map
is surjective, even after restricting to the subset
$\mbox{Im}\, [I\!\!N^N\rightarrow V_{Z\!\!\!Z}^\ast] $: All vectors $e_i$ corresponding
to the functions $t_i$
map onto $1\in I\!\!N$.\\
\par
Geometrically this means that both maps $\,I\!\!\!\!C \rightarrow S$ and $\,I\!\!\!\!C \rightarrow
{\bar{S}}$ are closed
embeddings,
and the corresponding ideals are
$\left(x^{\underline{\eta}} - x^{\underline{\eta}'}\, \left|\; \underline{\eta}, \underline{\eta}'\in
C(Q)\cap V^\ast_{Z\!\!\!Z}
\mbox{ with } \sum_i \eta_i = \sum_i \eta_i' \right. \right)$ and
$(t_i-t_j\,|\; 1\leq i,j \leq N)$,
respectively.
In particular, we got a first contribution to proof the claims made in
\zitat{4}{3}(i).\\
\par
\neu{46}
In the next two sections we take a closer look at the dualized cone
$\tilde{C}(Q)^{\scriptscriptstyle\vee}$.\\
\par
{\bf Definition:} {\em
For $c \in Z\!\!\!Z^n$ let $\underline{\lambda}^c = (\lambda_1^c,\dots,
\lambda_N^c)\in Z\!\!\!Z^N$
describe some path from $0\in Q$ to $a(c)\in Q$ through the 1-skeleton of $Q$
(cf.\ \zitat{4}{1}).
Then,
\[
\underline{\eta}(c) := \left[ -\lambda^c_1\langle d^1,c\rangle, \dots,
-\lambda^c_N\langle d^N,c\rangle \right] \in Z\!\!\!Z^N
\]
defines an element $\underline{\eta}(c) \in V_{Z\!\!\!Z}^\ast$ not depending on the
choice of the
particular path $\underline{\lambda}^c$.\\ }
\par
(Let $\underline{\tilde{\lambda}}^c$ be a different path from $0$ to $a(c)$ -
it will differ from
$\underline{\lambda}^c$ by some linear combination
$\sum_{\varepsilon < Q} g_{\varepsilon}\, \underline{\varepsilon}$ ($g_{\varepsilon}\in Z\!\!\!Z \mbox{ for 2-faces
} \varepsilon < Q$) only.
In particular,
\[
\tilde{\lambda}_i^c \langle d^i,c\rangle - \lambda^c_i \langle d^i,c \rangle =
\sum_{\varepsilon < Q} g_{\varepsilon} \langle \varepsilon_i\,d^i, \, c\rangle,
\]
and we obtain $\underline{\eta}(c)_{\tilde{\lambda}} -
\underline{\eta}(c)_{\lambda} \in V^\bot$.)\\
\par
{\bf Lemma:}
{\em \begin{itemize}
\item[(i)]
$\underline{\eta}(0) = 0 \in V^\ast_{Z\!\!\!Z}$.
\item[(ii)]
For all $c\in Z\!\!\!Z^n$ we have $\underline{\eta}(c) \succeq 0$ (in the sense of
Definition \zitat{4}{5}).
\item[(iii)]
$\underline{\eta}$ is convex: $\sum_vg_v\,\underline{\eta}(c^v) \succeq
\underline{\eta}
(\sum_v g_v\, c^v)$ for natural numbers $g_v\inI\!\!N$.
\item[(iv)]
$\sum_{i=1}^N \eta_i(c) = \eta_0(c)$ for arbitrary $c\in Z\!\!\!Z^n$.
\vspace{2ex}
\end{itemize} }
\par
{\bf Proof:}
(ii) $a(c)$ is a vertex of $Q$ providing minimal value of the linear function
$\langle \bullet,c \rangle$.
In particular, we can choose a path $\underline{\lambda}^c$ from $0\in Q$ to
$a(c)$ such that this
function decreases in each step, i.e.\ $\lambda_i^c \langle d^i, c \rangle \leq
0 \;
(i=1,\dots,N)$.\\
\par
(iii) We define the following paths through the 1-skeleton of $Q$:
\begin{itemize}
\item
$\underline{\lambda}:=$ path from $0\in Q$ to $a(\sum_v g_v\, c^v)\in Q$,
\item
$\underline{\mu}^v:=$ path from $a(\sum_v g_v\, c^v)\in Q$ to $a(c^v)\in Q$
such that
$\mu_i^v \langle d^i, c^v \rangle \leq 0$ for each $i=1,\dots,N$.
\end{itemize}
Then, $\underline{\lambda}^v := \underline{\lambda} + \underline{\mu}^v$ is a
path from $0\in Q$
to $a(c^v)$, and for $i=1,\dots,N$ we obtain
\begin{eqnarray*}
\sum_v g_v \, \eta_i (c^v) - \eta_i \left( \sum_v g_v\, c^v \right) &=&
-\sum_v g_v\, (\lambda_i + \mu^v_i)\, \langle d^i,c^v \rangle
+ \lambda_i \left\langle d^i,\, \sum_v g_v\, c^v \right\rangle\\
&=& -\sum_v g_v\, \mu^v_i \, \langle d^i, c^v \rangle \geq 0\, .
\end{eqnarray*}
(iv) By definition of $\underline{\lambda}^c$ we have $\sum_{i=1}^N
\lambda_i^c\, d^i = a(c)$.
In particular,
\[
\sum_{i=1}^N \eta_i(c) = - \sum_{i=1}^N \langle \lambda^c_i\, d^i,\,c\rangle =
- \langle a(c),\, c \rangle = \eta_0(c).
\nopagebreak
\vspace{-2ex}
\]
\nopagebreak
\hfill$\Box$
\pagebreak[3]\\
\par
{\bf Example:}
In our hexagon $Q_6$ we choose the following paths from $(0,0)$ to the vertices
$a(c^1),\dots,a(c^6)$, respectively:
\[
\underline{\lambda}^6 = \underline{\lambda}^1 := \underline{0},\quad
\underline{\lambda}^2 = \underline{\lambda}^3 := [1,1,0,0,0,0],\quad
\underline{\lambda}^4 = \underline{\lambda}^5 := [1,1,1,1,0,0]\, .
\]
They provide
\[
\begin{array}{ccccccccc}
\underline{\eta}(c^1) &=& [0,0,0,0,0,0]\,, & \underline{\eta}(c^2) &=& [1,0,0,0,0,0]\,, &
\underline{\eta}(c^3) &=& [1,1,0,0,0,0]\,, \\
\underline{\eta}(c^4) &=& [0,1,1,0,0,0]\,, & \underline{\eta}(c^5) &=& [-1,0,1,1,0,0]\,, &
\underline{\eta}(c^6) &=& [0,0,0,0,0,0]\,.
\end{array}
\]
Since $[1,0,-1,-1,0,1] = [\langle d^1,[1,-1] \rangle,\dots,\langle
d^6,[1,-1]\rangle] \in V^\bot$,
the vector $\underline{\eta}(c^5)$ can be transformed into $[0,0,0,0,0,1]$.\\
\par
{\bf Remark:} The definitions of $a(c), \eta_0(c),$ and $\underline{\eta}(c)$
also make sense for general $c\in I\!\!R^n$. Then, $\eta_0(c)\in I\!\!R$ and
$\underline{\eta}(c)\in V^\ast$ do not need to be contained in the lattices
anymore.
The previous lemma will keep valid (even for $g_v\in I\!\!R_{\geq 0}$ in (iii)),
if the relation ``$\succeq 0$'' is replaced by the weaker version
``$\in C(Q)^{\scriptscriptstyle\vee}$''.\\
\par
\neu{47}{\bf Proposition:}
{\em \begin{itemize}
\item[(1)]
$\tilde{C}(Q)^{\scriptscriptstyle\vee} =
\left\{ \left. [c,\underline{\eta}]\in I\!\!R^n \times V^\ast \, \right| \;
\underline{\eta} - \underline{\eta}(c) \in C(Q)^{\scriptscriptstyle\vee} \right\}$
\item[(2)]
In particular, $[c, \underline{\eta}(c)] \in \tilde{C}(Q)^{\scriptscriptstyle\vee}$, and
moreover, it is the only preimage
of $[c, \eta_0(c)] \in \sigma^{\scriptscriptstyle\vee}$ via the surjection $i^{\scriptscriptstyle\vee}:
\tilde{C}(Q)^{\scriptscriptstyle\vee} \longrightarrow\hspace{-1.5em}\longrightarrow
\sigma^{\scriptscriptstyle\vee}$.
\item[(3)]
$[c^1,\underline{\eta}(c^1)],\dots, [c^w,\underline{\eta}(c^w)]$ and
$C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}$
(embedded as $[0,C(Q)^{\scriptscriptstyle\vee}]$) generate the semigroup $\tilde{C}(Q)^{\scriptscriptstyle\vee}
\cap \left(Z\!\!\!Z^n \times V_{Z\!\!\!Z}^\ast \right)$. (For recalling the definition of
the $c^1,\dots,c^w$,
cf.\ \zitat{4}{4}.)
\vspace{2ex}
\end{itemize} }
\par
{\bf Proof:}
(1) Let $[c,\underline{\eta}]\in I\!\!R^n \times V^\ast$ be given; if some
representative of
$\underline{\eta}$ in $I\!\!R^N$ is needed,
then it will be denoted by the same name. We have
the following equivalences:
\begin{eqnarray*}
[c,\underline{\eta}] \in \tilde{C}(Q)^{\scriptscriptstyle\vee} &\Longleftrightarrow&
\langle (Q_{\underline{t}}, \, \underline{t}),\, [c,\underline{\eta}] \rangle
\geq 0 \quad
\mbox{ for each } \underline{t} \in C(Q)\\
&\Longleftrightarrow&
\langle Q_{\underline{t}},\, c\rangle + \langle
\underline{t},\,\underline{\eta} \rangle \geq 0
\quad \mbox{ for each } \underline{t} \in C(Q)\\
&\Longleftrightarrow&
\langle a(c)_{\underline{t}},\, c\rangle + \langle
\underline{t},\,\underline{\eta} \rangle \geq 0
\quad \mbox{ for each } \underline{t} \in C(Q).
\end{eqnarray*}
Using some path $\underline{\lambda}^c$ we obtain:
\begin{eqnarray*}
[c,\underline{\eta}] \in \tilde{C}(Q)^{\scriptscriptstyle\vee} &\Longleftrightarrow&
\sum_{i=1}^N t_i\, \lambda_i^c \langle d^i,\,c\rangle + \langle \underline{t},\,
\underline{\eta} \rangle \geq 0
\quad \mbox{ for each } \underline{t} \in C(Q)\\
&\Longleftrightarrow&
\sum_{i=1}^N t_i\cdot \left(\lambda_i^c\, \langle d^i,\,c\rangle + \eta_i
\right) \geq 0
\quad \mbox{ for each } \underline{t} \in C(Q)\\
&\Longleftrightarrow&
\left[ \lambda_1^c\, \langle d^1,\,c\rangle + \eta_1,\dots,
\lambda_N^c\, \langle d^N,\, c \rangle + \eta_N \right] \in C(Q)^{\scriptscriptstyle\vee}.
\end{eqnarray*}
(2) By part (1) we know that for a $[c,\underline{\eta}]\in\tilde{C}(Q)^{\scriptscriptstyle\vee}$ it is
possible to choose
$I\!\!R^N$-representatives for
$\underline{\eta}, \underline{\eta}(c)$ such that $\eta_i\geq \eta_i(c)$ for $i=1,\dots,N$.\\
On the other hand, the two equalities $\sum_i \eta_i(c) = \eta_0(c)$ (cf.\ (iv)
of the previous lemma)
and $\sum_i \eta_i = \eta_0(c)$ (corresponding to the fact
$[c,\underline{\eta}] \mapsto [c,\eta_0(c)]$) imply $\underline{\eta} = \underline{\eta}(c)$ then.\\
\par
(3) Let $[c,\underline{\eta}] \in \tilde{C}(Q)^{\scriptscriptstyle\vee}$.
Then, $[c,\eta_0(c)]$ is representable as a non-negative
linear combination $[c,\eta_0(c)]=\sum_{v=1}^w p_v\, [c^v, \eta_0(c^v)]$
($p_v\in I\!\!N$ if $c\in Z\!\!\!Z^n$).
Since both elements $[c, \underline{\eta}(c)]$ and $\sum_vp_v [c^v,\underline{\eta}(c^v)]$
are
preimages of $[c, \eta_0(c)]$ via $i^v$, they must be equal by (2),
and we obtain
\[
\begin{array}{rcl}
[c,\, \underline{\eta}] &=& [c,\, \underline{\eta}(c)] + [0, \underline{\eta} - \underline{\eta}(c)]
\vspace{0.5ex}\\
&=&
\sum_v p_v\,[c^v,\underline{\eta}(c^v)] + [0, \underline{\eta} - \underline{\eta}(c)]\,.
\end{array}
\vspace{-3ex}
\]
\hfill$\Box$\\
\par
\neu{48}
Finally, we will take a short look at the geometrical situation reached at this
point.
\begin{itemize}
\item[(1)]
The linear map
\[
\begin{array}{ccc}
\tilde{C}(Q)^{\scriptscriptstyle\vee}\cap \left( Z\!\!\!Z^n\times V^\ast_{Z\!\!\!Z}\right) & \longrightarrow\hspace{-1.5em}\longrightarrow
&\sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^{n+1}\\
\,[c,\, \underline{\eta}] & \mapsto & [c,\,\sum_i\eta_i]
\end{array}
\]
is surjective ($[c,\underline{\eta}(c)]\mapsto [c,\eta_0(c)];\; [0,e_i]\mapsto
[0,1]$). Since
\[
x^{[c,\underline{\eta}]} - x^{[c,\underline{\eta}']} = x^{[c,\underline{\eta}(c)]}\cdot
(x^{[0,\underline{\eta}-\underline{\eta}(c)]} - x^{[0,\underline{\eta}'-\underline{\eta}(c)]}),
\]
the kernel of the corresponding
homomorphism between the semigroup algebras equals the ideal
\[
\left(x^{[0,\underline{\eta}]}-x^{[0,\underline{\eta}']}\, \left| \; \sum_i\eta_i = \sum_i
\eta_i'\right.\right) .
\]
In particular, the map $Y\hookrightarrow X$ is a closed embedding. Moreover,
comparing with the similar statement concerning $C(Q)^{\scriptscriptstyle\vee}$ and $I\!\!N$ at the
end of
\zitat{4}{5}, we obtain that the diagram of \zitat{4}{3}(i) is a fiber product
diagram, indeed.
\item[(2)]
The elements $[c^1,\underline{\eta}(c^1)], \dots, [c^w,\underline{\eta}(c^w)] \in
\tilde{C}(Q)$ induce some regular
functions $Z_1,\dots,Z_w$ on $X$. They definine a closed embedding
$X \hookrightarrow \,I\!\!\!\!C^w \times S$ lifting the ebedding $Y \hookrightarrow
\,I\!\!\!\!C^{w+1}$ of
\zitat{4}{4}. Moreover, for $i=1,\dots,N$, $Z_i$ is the only monomial function
lifting $z_i$ from
$Y$ to $X$.
\end{itemize}
We have obtained the following commutative diagram:
\[
\begin{array}{ccccccc}
Y & \hookrightarrow & \,I\!\!\!\!C^w\times\,I\!\!\!\!C & = & \,I\!\!\!\!C^w\times\,I\!\!\!\!C\\
\downarrow & \otimes & \downarrow && \downarrow {\scriptstyle \Delta}\\
X & \hookrightarrow & \,I\!\!\!\!C^w\times S & \stackrel{p}{\longrightarrow} & \,I\!\!\!\!C^w\times
\,I\!\!\!\!C^N\\
&& \downarrow & & \downarrow\\
&& S & \stackrel{p}{\longrightarrow} & \,I\!\!\!\!C^N & \stackrel{\ell}{\longrightarrow}
& \,I\!\!\!\!C^{N-1}\, .
\end{array}
\]
\\
\par
\sect{A flat family over $\bar{{\cal M}}$}\label{s5}
\neu{51}
{\bf Theorem:} {\em Denote by $\bar{X}$ and $\bar{S}$ the scheme theoretical
images of $X$ and
$S$ in $\,I\!\!\!\!C^w\times\,I\!\!\!\!C^N$ and $\,I\!\!\!\!C^N$, respectively. Then,
\begin{itemize}
\item[(1)]
$X\rightarrow \bar{X}$ and $S\rightarrow\bar{S}$ are the normalization maps.
\item[(2)]
$\pi:X\rightarrow S$ induces a map $\bar{\pi}:\bar{X}\rightarrow \bar{S}$, and
$\pi$ can
be recovered from $\bar{\pi}$ via base change $S\rightarrow \bar{S}$.
\item[(3)]
Restricting to ${\cal M}\subseteq \bar{S}$ and composing with $\ell$ turns
$\bar{\pi}$ into a
family
\[
\bar{X} \times_{\bar{S}}{\cal M} \stackrel{\bar{\pi}}{\longrightarrow} {\cal M}
\stackrel{\ell}{\longrightarrow}
\bar{{\cal M}}\,.
\]
It is flat in $0\in \bar{{\cal M}}\subseteq \,I\!\!\!\!C^{N-1}$, and the special fiber equals
$Y$.
\vspace{2ex}
\end{itemize} }
The proof of this theorem will fill \S \ref{s5}.\\
\par
\neu{52}
The ring of regular functions $A(\bar{S})$ is given as the image of the map
$\,I\!\!\!\!C[t_1,\dots,t_N] \rightarrow A(S)$. Since $Z\!\!\!Z^N\longrightarrow\hspace{-1.5em}\longrightarrow V_{Z\!\!\!Z}^\ast$ is
surjective, the rings
$A(\bar{S})\subseteq A(S) \subseteq \,I\!\!\!\!C[V^\ast_{Z\!\!\!Z}]$ have the same field of
fractions.\\
On the other hand, while $t$-monomials with negative exponents are involved in
$A(S)$, the
surjectivity of $I\!\!R^N_{\geq 0}
\longrightarrow\hspace{-1.5em}\longrightarrow C(Q)^{\scriptscriptstyle\vee}$ tells us that sufficiently high powers of those monomials
always come from $A(\bar{S})$. In particular, $A(S)$ is normal over
$A(\bar{S})$.\\
\par
$A(\bar{X})$ is given as the image
$A(\bar{X})=\mbox{Im}\, (\,I\!\!\!\!C[Z_1,\dots,Z_w,t_1,\dots,t_N]\rightarrow A(X))$.
Since $A(X)$ is generated by $Z_1,\dots,Z_w$ over its subring $A(S)$
(cf.\ Proposition \zitat{4}{7}(3)), the same arguments as for $S$ and $\bar{S}$
apply.
Hence, Part (1) of the previous theorem is proved.\\
\par
\neu{53}
Recalling that $z_1,\dots,z_w,\,t \in A(Y)$ stand for the monomials with
exponents
$[c^1,\eta_0(c^1)],\dots,[c^w,\eta_0(c^w)], [0,1] \in C(Q)^{\vee}\cap
V^\ast_{Z\!\!\!Z}$, respectively,
we obtain the following equations defining $Y \subseteq \,I\!\!\!\!C^{w+1}$:
\begin{eqnarray*}
f_{(a,b,\alpha,\beta)}(\underline{z},t) &:=& t^\alpha \,\prod_{v=1}^w z_v^{a_v} -
t^\beta \, \prod_{v=1}^w z_v^{b_v} \\
&& \hspace{-1cm} \mbox{with }
\begin{array}[t]{l}
a,b\in I\!\!N^w: \; \sum_v a_v\, c^v = \sum_v b_v\, c^v \quad \mbox{ and}\\
\alpha, \beta \in I\!\!N: \; \sum_v a_v\,\eta_0(c^v) + \alpha = \sum_v
b_v\,\eta_0(c^v) + \beta\, .
\vspace{2ex}
\end{array}
\end{eqnarray*}
{\bf Example:}
The singularity $Y_6$ induced by the hexagon $Q_6$ equals the cone over the Del
Pezzo
surface of degree 6 (obtained by blowing up three points of $(I\!\!P^2, {\cal
O}(3))$).
As a closed subset of $\,I\!\!\!\!C^7$, it is given by the following 9 equations:
\[
\begin{array}{ccc}
f_{(e_1,e_6+e_2,1,0)} = z_1\,t - z_6\,z_2, \; &
f_{(e_2,e_1+e_3,1,0)} = z_2\,t - z_1\,z_3, \; &
f_{(e_3,e_2+e_4,1,0)} = z_3\,t - z_2\,z_4 , \\
f_{(e_4,e_3+e_5,1,0)} = z_4\,t - z_3\,z_5, \; &
f_{(e_5,e_4+e_6,1,0)} = z_5\,t - z_4\,z_6, \; &
f_{(e_6,e_5+e_1,1,0)} = z_6\,t - z_5\,z_1 , \\
f_{(\underline{0},e_1+e_4,2,0)} = t^2 - z_1\,z_4, \; &
f_{(\underline{0},e_2+e_5,2,0)} = t^2 - z_2\,z_5, \; &
f_{(\underline{0},e_3+e_6,2,0)} = t^2 - z_3\,z_6 \, .
\end{array}
\]
\\
\par
\neu{54}
Defining $c:=\sum_v a_v\, c^v = \sum_v b_v\, c^v$ we can lift the equations of
$Y$ to the following
elements of $\,\,I\!\!\!\!C[Z_1,\dots,Z_w,\,t_1,\dots,t_N] \longrightarrow\hspace{-1.5em}\longrightarrow
A(\bar{S})[Z_1,\dots,Z_w]$:
\[
F_{(a,b,\alpha,\beta)}(\underline{Z},\underline{t}) := f_{(a,b,\alpha,\beta)}(\underline{Z},t_1) -
\underline{Z}^{[c,\underline{\eta}(c)]} \cdot
\left( \underline{t}^{\alpha e_1 +\sum_v a_v \underline{\eta}(c^v)} -
\underline{t}^{\beta e_1 +\sum_v b_v \underline{\eta}(c^v)} \right) \cdot
\underline{t}^{-\underline{\eta}(c)}\, .
\]
{\bf Remark:}
\begin{itemize}
\item[(1)]
The symbol $\underline{Z}^{[c,\underline{\eta}(c)]}$ means $\prod_{v=1}^w Z_v^{p_v}$ with
natural numbers
$p_v\in I\!\!N$ such that
$[c,\underline{\eta}(c)] = \sum_v p_v\, [c^v, \underline{\eta}(c^v)]$ or equivalently
$[c,\eta_0(c)] = \sum_v p_v\, [c^v, \eta_0(c^v)]$. This condition does not
determine the
coefficients $p_v$ uniquely - choose one of the possibilities.
\item[(2)]
By part (iii) of Lemma \zitat{4}{6}, we have $\sum_v a_v\underline{\eta}(c^v),\,
\sum_v b_v\underline{\eta}(c^v)
\succeq \underline{\eta}(c)$. In particular, representatives of the $\underline{\eta}$'s can be
chosen such that
all $t$-exponents occuring in monomials of $F$ are non-negative, i.e.
$F \in A(\bar{S})[Z_1,\dots,Z_w]$.
\item[(3)]
Mapping $F$ to $A(X)= \oplus_{[c,\underline{\eta}]} \,\,I\!\!\!\!C x^{[c,\underline{\eta}]}$
($[c,\underline{\eta}]$ runs through
all elements of $\tilde{C}(Q)^{\scriptscriptstyle\vee} \cap (Z\!\!\!Z^n\times V_{Z\!\!\!Z}^\ast)$;
$Z_v \mapsto x^{[c^v\!, \underline{\eta}(c^v)]},\, t_i \mapsto x^{[0,e_i]}$) yields
\begin{eqnarray*}
F_{(a,b,\alpha,\beta)} &=&
\begin{array}[t]{r}
\left( t_1^\alpha \, \prod_v Z_v^{a_v} -
\underline{Z}^{[c,\underline{\eta}(c)]} \, \underline{t}^{\alpha e_1 + \sum_v a_v\underline{\eta}(c^v) -
\underline{\eta}(c)} \right) - \qquad\\
- \left( t_1^\beta \, \prod_v Z_v^{b_v} -
\underline{Z}^{[c,\underline{\eta}(c)]} \, \underline{t}^{\beta e_1 + \sum_v b_v\underline{\eta}(c^v) -
\underline{\eta}(c)} \right)
\end{array}\\
& \mapsto &
\begin{array}[t]{r}
\left( x^{\alpha [0,e_1] + \sum_v a_v [c^v\!,\underline{\eta}(c^v)]} -
x^{[c,\underline{\eta}(c)] + \alpha [0,e_1] + \sum_v a_v [0,\underline{\eta}(c^v)] -
[0,\underline{\eta}(c)]} \right) - \quad\\
- \left( x^{\beta [0,e_1] + \sum_v b_v [c^v\!,\underline{\eta}(c^v)]} -
x^{[c,\underline{\eta}(c)] + \beta [0,e_1] + \sum_v b_v [0,\underline{\eta}(c^v)] -
[0,\underline{\eta}(c)]} \right)
\end{array}\\
& = & \; 0 \; - \; 0\, = \,0 \,,
\end{eqnarray*}
i.e.\ the polynomials $F_{(a,b,\alpha,\beta)}$ are equations
for $\bar{X} \subseteq \,I\!\!\!\!C^w \times \bar{S}$.
\vspace{2ex}
\end{itemize}
\par
{\bf Example:}
In the hexagon example, we obtain the following liftings:
\[
\begin{array}{rclcl}
F_{(e_1,e_6+e_2,1,0)} &=& (Z_1\,t_1 - Z_6\,Z_2 )- Z_1(t_1-t_1) &=&
Z_1\,t_1 - Z_6\,Z_2, \\
F_{(e_2,e_1+e_3,1,0)} &=&( Z_2\,t_1 - Z_1\,Z_3) -Z_2(t_1^2-t_1\,t_2)\,t_1^{-1}
&=&
Z_2\,t_2 - Z_1\,Z_3, \\
F_{(e_3,e_2+e_4,1,0)} &=& (Z_3\,t_1 - Z_2\,Z_4) - Z_3(t_1^2t_2-t_1\,t_2\,t_3)\,
t_1^{-1}t_2^{-1} &=&
Z_3\,t_3 - Z_2\,Z_4, \\
F_{(e_4,e_3+e_5,1,0)} &=& (Z_4\,t_1 - Z_3\,Z_5 )- Z_4(t_1\,t_2\,t_3 -
t_2\,t_3\,t_4)\, t_2^{-1}t_3^{-1}
&=& Z_4\,t_4 - Z_3\,Z_5, \\
F_{(e_5,e_4+e_6,1,0)} &=& (Z_5\,t_1 - Z_4\,Z_6) -
Z_5(t_1\,t_6-t_2\,t_3)\,t_6^{-1} &=&
Z_5\,t_5 - Z_4\,Z_6 ,\\
F_{(e_6,e_5+e_1,1,0)} &=& (Z_6\,t_1 - Z_5\,Z_1) - Z_6(t_1-t_6) &=&
Z_6\,t_6 - Z_5\,Z_1 , \\
F_{(\underline{0},e_1+e_4,2,0)} &=&( t_1^2 - Z_1\,Z_4) - (t_1^2-t_2\,t_3) = t_2\,t_3 -
Z_1\,Z_4 &=&
t_5\,t_6 - Z_1\,Z_4, \\
F_{(\underline{0},e_2+e_5,2,0)} &=& (t_1^2 - Z_2\,Z_5) - (t_1^2 - t_3\,t_4) &=&
t_3\,t_4 - Z_2\,Z_5, \\
F_{(\underline{0},e_3+e_6,2,0)} &=& (t_1^2 - Z_3\,Z_6) - (t_1^2-t_1\,t_2) &=&
t_1\,t_2 - Z_3\,Z_6 \, .
\end{array}
\]
\\
\par
\neu{55}
To obtain a complete list of equations defining $\bar{X} \subseteq \,I\!\!\!\!C^w \times
\bar{S}$,
we have to regard the kernel of the homomorphism
$A(\bar{S})[Z_1,\dots,Z_w]\longrightarrow\hspace{-1.5em}\longrightarrow A(\bar{X}) \subseteq A(X)$. It is generated by
the binomials
\[
\begin{array}[t]{r}
\underline{t}^{\underline{\eta}}\, Z_1^{a_1}\cdot\dots\cdot Z_w^{a_w} -
\underline{t}^{\underline{\mu}}\, Z_1^{b_1}\cdot\dots\cdot Z_w^{b_w} \quad \mbox{ such
that}\hspace{4cm}
\vspace{1ex}\\
\begin{array}[t]{l}
\sum_v a_v [c^v,\underline{\eta}(c^v)] + [0,\underline{\eta}] =
\sum_v b_v [c^v,\underline{\eta}(c^v)] + [0,\underline{\mu}] \, , \vspace{1ex}\\
\mbox{i.e.\ }
\begin{array}[t]{ll}
\bullet & c:= \sum_v a_v \, c^v = \sum_v b_v\, c^v\\
\bullet & \sum_v a_v\, \underline{\eta}(c^v) + \underline{\eta} = \sum_v b_v\, \underline{\eta}(c^v)
+ \underline{\mu}\, .
\end{array}
\end{array}
\end{array}
\]
However,
\begin{eqnarray*}
\underline{t}^{\underline{\eta}}\,\underline{Z}^{a} -
\underline{t}^{\underline{\mu}}\,\underline{Z}^{b} &=&
\begin{array}[t]{r}
\underline{t}^{\underline{\eta}}\cdot
\left( \prod_v Z_v^{a_v} - \underline{Z}^{[c ,\underline{\eta}(c)]} \, \underline{t}^{\sum_v a_v
\underline{\eta}(c^v) -
\underline{\eta}(c)} \right) - \qquad\\
- \underline{t}^{\underline{\mu}}\cdot
\left( \prod_v Z_v^{b_v} - \underline{Z}^{[c ,\underline{\eta}(c)]} \, \underline{t}^{\sum_v b_v
\underline{\eta}(c^v) -
\underline{\eta}(c)} \right)
\end{array}\\
& = &
\underline{t}^{\underline{\eta}}\cdot F_{(a,p,0,\alpha)} -
\underline{t}^{\underline{\mu}}\cdot F_{(b,p,0,\beta)}
\end{eqnarray*}
with $p\in I\!\!N^w$ such that $\sum_v p_v [c^v, \underline{\eta}(c^v)] = [c,
\underline{\eta}(c)]$,
$\alpha = \sum_v a_v \eta_0(c^v) - \eta_0(c)$, and
$\beta = \sum_v b_v \eta_0(c^v) - \eta_0(c)$.\\
In particular, $\mbox{Ker}\,(A(\bar{S})[\underline{Z}] \rightarrow A(X))$ is generated
by the polynomials
$F_{(a,b,\alpha,\beta)}$ introduced in \zitat{5}{4}.\\
\par
{\bf Remark:}
\begin{itemize}
\item[(1)]
The inaccuracy caused by writing $\underline{Z}^{[c,\underline{\eta}(c)]}$ for some
undetermined
$Z_1^{p_1}\cdot\dots\cdot Z_w^{p_w}$ (with $\sum_v p_v [c^v, \underline{\eta}(c^v)] =
[c, \underline{\eta}(c)]$)
does not matter: Choosing other coefficients $q_v$ with the same property
yields
\[
Z_1^{p_1}\cdot\dots\cdot Z_w^{p_w} - Z_1^{q_1}\cdot\dots\cdot Z_w^{q_w} =
F_{(p,q,0,0)}(\underline{Z},\underline{t}) = f_{(p,q,0,0)}(\underline{Z},t)\, .
\]
\item[(2)]
Using exponents $\underline{\eta}, \underline{\mu} \in Z\!\!\!Z^N$ (instead of $I\!\!N^N$), the
binomials
$\underline{t}^{\underline{\eta}}\,\underline{Z}^{a} -
\underline{t}^{\underline{\mu}}\,\underline{Z}^{b}$ generate the kernel of the map
\[
A(S)[\underline{Z}] = A(\bar{S})[\underline{Z}] \otimes_{A(\bar{S})} A(S) \longrightarrow\hspace{-1.5em}\longrightarrow
A(\bar{X}) \otimes_{A(\bar{S})} A(S) \longrightarrow\hspace{-1.5em}\longrightarrow A(X)\, .
\]
Since $\underline{Z}^a \otimes \underline{t}^{\underline{\eta}} -
\underline{Z}^b \otimes \underline{t}^{\underline{\mu}} =
\underline{Z}^{[c,\underline{\eta}(c)]} \otimes \left(
\underline{t}^{\sum_v a_v \underline{\eta}(c^v) - \underline{\eta}(c) + \underline{\eta}} -
\underline{t}^{\sum_v b_v \underline{\eta}(c^v) - \underline{\eta}(c) + \underline{\mu}} \right) = 0$
in $A(\bar{X}) \otimes_{A(\bar{S})} A(S)$, this implies that the
surjection $A(\bar{X}) \otimes_{A(\bar{S})} A(S) \longrightarrow\hspace{-1.5em}\longrightarrow A(X)$ is
injective, too. In particular, part (2) of our theorem is proved.
\vspace{2ex}
\end{itemize}
\par
\neu{56}
Both the point $0\in \bar{{\cal M}}$ and $\,I\!\!\!\!C\subseteq \bar{S}$ are given by
the equations $t_i-t_j=0 \; (1\leq i,j,\leq N)$. Modulo these relations, the
equations $F_{(a,b,\alpha,\beta)}$ of $\bar{X}\subset \,I\!\!\!\!C^w\times\bar{S}$
specialize to the equations $f_{(a,b,\alpha,\beta)}$ of $Y\subset
\,I\!\!\!\!C^{w+1}$.
In particular, $Y$ is the special fiber of the family $\bar{X}\times_{\bar{S}}
{\cal M} \rightarrow \bar{{\cal M}}$.\\
To show the flatness of this family we have to determine all relations
between the $f_{(a,b,\alpha,\beta)}$'s and lift them to relations between
the $F_{(a,b,\alpha,\beta)}$'s.\\
\par
There are three types of relations between the $f_{(a,b,\alpha,\beta)}$'s:
\begin{itemize}
\item[(i)]
$f_{(a,r,\alpha,\gamma)} + f_{(r,b,\gamma,\beta)} = f_{(a,b,\alpha,\beta)}$\\
with
$\begin{array}[t]{ll}
\bullet & \sum_va_vc^v = \sum_v r_v c^v = \sum b_v c^v\; \mbox{ and}\\
\bullet & \sum_v a_v\eta_0(c^v) +\alpha = \sum_vr_v\eta_0(c^v) +\gamma =
\sum_vb_v\eta_0(c^v) + \beta\, .
\end{array}$\\
For this relation, the same equation between the $F$'s is true.
\item[(ii)]
$t\cdot f_{(a,b,\alpha,\beta)} = f_{(a,b,\alpha+1,\beta+1)}\;$ lifts to
$\;t_1\cdot F_{(a,b,\alpha,\beta)} = F_{(a,b,\alpha+1,\beta+1)}$.
\item[(iii)]
$\underline{z}^r\cdot f_{(a,b,\alpha,\beta)} = f_{(a+r,b+r,\alpha,\beta)}$.
\vspace{1ex}\\
With $c:= \sum_va_vc^v = \sum_vb_vc^v,\; \tilde{c}:= c+ \sum_vr_vc^v$ we
obtain
\[
\begin{array}{l}
\underline{Z}^r\cdot F_{(a,b,\alpha,\beta)} - F_{(a+r,b+r,\alpha,\beta)} =
\vspace{1ex}\\
\qquad\begin{array}[t]{r}
=\underline{Z}^{[\tilde{c},\underline{\eta}(\tilde{c})]}
\cdot
\left( \underline{t}^{ \alpha e_1 + \sum_va_v\underline{\eta}(c^v) +
\sum_vr_v\underline{\eta}(c^v)} - \underline{t}^{ \beta e_1 + \sum_vb_v\underline{\eta}(c^v) +
\sum_vr_v\underline{\eta}(c^v)} \right)
\cdot \underline{t}^{-\underline{\eta}(\tilde{c})} - \\
- \underline{Z}^{[c,\underline{\eta}(c)]}\, \underline{Z}^r\cdot
\left( \underline{t}^{ \alpha e_1 + \sum_va_v\underline{\eta}(c^v)} -
\underline{t}^{ \beta e_1 + \sum_vb_v\underline{\eta}(c^v)} \right)\cdot
\underline{t}^{-\underline{\eta}(c)}
\end{array}
\vspace{1ex}\\
\qquad \begin{array}[t]{r} =
\left( \underline{t}^{ \alpha e_1 + \sum_va_v\underline{\eta}(c^v)-\underline{\eta}(c)} -
\underline{t}^{ \beta e_1 + \sum_vb_v\underline{\eta}(c^v)-\underline{\eta}(c)} \right)\cdot
\hspace{4cm}\\
\left( \underline{t}^{\underline{\eta}(c)+\sum_vr_v\underline{\eta}(c^v) - \underline{\eta}(\tilde{c})}
\underline{Z}^{[\tilde{c},\underline{\eta}(\tilde{c})]} - \underline{Z}^{[c,\underline{\eta}(c)]}
\underline{Z}^r \right)\,.
\end{array}
\end{array}
\]
Now, the inequalities
\[
\sum_va_v\underline{\eta}(c^v),\,\sum_vb_v\underline{\eta}(c^v) \succeq \underline{\eta}(c)
\;\mbox{ and }\;
\underline{\eta}(c)+\sum_vr_v\underline{\eta}(c^v) - \underline{\eta}(\tilde{c}) \succeq 0
\]
imply that
\begin{itemize}
\item
the first factor is contained in the ideal defining
$0\in \bar{{\cal M}}$, and
\item
the second factor is an equation of $\bar{X}\subseteq
\,I\!\!\!\!C^w\times\bar{S}$ (called $F_{(q,p+r,\xi,0)}$ in \zitat{7}{4}).
\end{itemize}
In particular, we have found a lift for the third relation, too.
\end{itemize}
The proof of Theorem \zitat{5}{1} is complete.\\
\par
\neu{57}
{\bf Example:}
For $Y_6$, the previously constructed family
is contained in $\,I\!\!\!\!C^6\times\,I\!\!\!\!C^6\stackrel{\mbox{pr}_2}{\longrightarrow}\,
\,I\!\!\!\!C^6/_{\displaystyle \,I\!\!\!\!C\cdot (1,\dots,1)}$. Its base space is defined by the 4
equations mentioned at the end of \zitat{2}{3}, and for the total space, the 9
equations of \zitat{5}{4}
have to be added.\\
\par
\sect{The Kodaira-Spencer map}\label{s6}
\neu{61}
Denote by $E\subseteq \sigma^{\scriptscriptstyle\vee}\cap Z\!\!\!Z^{n+1}$ the minimal generating
set
\[
E:= \{[c^1,\eta_0(c^1)],\dots,[c^w,\eta_0(c^w)],[\underline{0},1]\}
\]
mentioned in \zitat{4}{4}.
To each vertex $a^j\in Q$
(or equally named fundamental generator $a^j:=(a^j,1)\in \sigma$)
and each element
$R\in Z\!\!\!Z^{n+1}$ we associate the subset
\[
E^R_j:= E_{a^j}^R:= \{r\in E\, |\; \langle a^j,r\rangle < \langle
a^j,R\rangle\} \,.
\vspace{2ex}
\]
{\bf Theorem:} (cf. \cite{T1})
{\em
The vector space $T^1_Y$ of infinitesimal deformations of $Y$ is
$Z\!\!\!Z^{n+1}$-graded, and
in degree $-R$ it equals
\begin{eqnarray*}
T^1_Y(-R) &=&
\left( \left.^{\displaystyle L_{\,I\!\!\!\!C}\left( \cup_j
E_j^R\right)} \! \right/
\!\! \sum_j
L_{\,I\!\!\!\!C}(E_j^R) \right)^\ast
\end{eqnarray*}
($L(\dots)$ denotes the vector space of linear relations).
}
\\
\par
\neu{62}
There is a special degree $R^\ast=[\underline{0},1]\in Z\!\!\!Z^{n+1}$ corresponding to the
affine
hyperplane containing $Q$. The associated subsets of $E$ equal
\begin{eqnarray*}
E^{R^\ast}_j &=& E\cap (a^j)^\bot\\
&=& \{[c^v,\eta_0(c^v)]\, |\; \langle a^j,-c^v\rangle = \eta_0(c^v)\}\,.
\vspace{1ex}
\end{eqnarray*}
In \zitat{4}{6}, for each $c\in Z\!\!\!Z^n$, we have defined the linear form
$\underline{\eta}(c)\in V_{Z\!\!\!Z}^\ast$. Restricted to the cone $C(Q)$, it
maps $\underline{t}$ to
$\mbox{Max}\langle Q_{\underline{t}},-c\rangle = \langle a(c)_{\underline{t}},-c\rangle$.
This induces the following bilinear map:
\[
\begin{array}{cccccl}
\Phi: & ^{\displaystyle V_{Z\!\!\!Z}}\!/\!_{\displaystyle (1,\dots,1)}
& \times & L_{Z\!\!\!Z}(E\cap \partial\sigma^{\scriptscriptstyle\vee})
& \longrightarrow & Z\!\!\!Z\\
&\underline{t} & , & q & \mapsto & \sum_{v,i} t_i\, q_v\, \eta_i(c^v)\,.
\end{array}
\]
(Indeed, for $\underline{t}:=\underline{1}$ we obtain
$\sum_{v,i} q_v\, \eta_i(c^v) = \sum_v q_v\, \eta_0(c^v) = 0$ since $q\in
L_{Z\!\!\!Z}(E\cap \partial\sigma^{\scriptscriptstyle\vee})$.)\\
Moreover, if $q$ comes from one of the submodules
$L_{Z\!\!\!Z}(E_j^{R^\ast})\subseteq L_{Z\!\!\!Z}(E\cap\partial\sigma^{\scriptscriptstyle\vee})$,
we obtain
\begin{eqnarray*}
\Phi(\underline{t},q) &=&
\sum_v q_v\cdot \mbox{Max}\langle Q_{\underline{t}},-c^v\rangle =
\sum_v q_v\cdot \langle a_{\underline{t}}^j,-c^v\rangle\\
& = &
\langle a_{\underline{t}}^j,-\sum_v q_vc^v\rangle = 0\,.
\vspace{2ex}
\end{eqnarray*}
\par
{\bf Theorem:}
{\em
The Kodaira-Spencer map of the family $\bar{X}\times_{\bar{S}}{\cal M}
\rightarrow
\bar{{\cal M}}$ of \S \ref{s5} equals the map
\vspace{-1ex}
\[
T_0\bar{{\cal M}} = ^{\displaystyle V_{\,I\!\!\!\!C}}\!\!/\!_{\displaystyle (1,\dots,1)}\longrightarrow
\left( \left. ^{\displaystyle L_{\,I\!\!\!\!C}(E\cap\partial\sigma^{\scriptscriptstyle\vee})} \! \right/
\! \sum_j
L_{\,I\!\!\!\!C}(E_j^{R^\ast}) \right)^\ast
=T^1_Y(-R^\ast)
\]
induced by the previous pairing. Moreover, this map is an isomorphism.
}
\\
\par
{\bf Proof:}
Using the same symbol ${\cal J}$ for the ideal ${\cal J}\subseteq \,I\!\!\!\!C[t_1,\dots,t_N]$
and the intersection ${\cal J}\cap \,I\!\!\!\!C[t_i-t_j\,|\; 1\leq i,j\leq N]$
(cf. \zitat{2}{4}), our family
corresponds to the flat $\,I\!\!\!\!C[t_i-t_j]/_{\displaystyle {\cal J}} $-module
$\,I\!\!\!\!C[\underline{Z},\underline{t}]/_{\displaystyle ({\cal J}, F_\bullet (\underline{Z}, \underline{t}))}$.\\
\par
Now, we fix a non-trivial tangent vector $\underline{t}^0\in V_{\,I\!\!\!\!C}$. Via
$t_i\mapsto t+t_i^0\,\varepsilon$ it induces the infinitesimal family given by
the flat $\,I\!\!\!\!C[\varepsilon]/\!_{\displaystyle \varepsilon^2}$-module
\[
A_{\underline{t}^0} := \;\left.^{\displaystyle \,I\!\!\!\!C[\underline{z}, t, \varepsilon]}\! \right/ \!
_{\displaystyle (\varepsilon^2, F_\bullet(\underline{z}, t+\underline{t}^0\,\varepsilon))}\,.
\]
To obtain the associated $A(Y)$-linear map $I/_{\displaystyle I^2} \rightarrow A(Y)$
($I:=(f_\bullet (\underline{z},t))$ denotes the ideal of $Y$ in $\,I\!\!\!\!C^{w+1}$),
we have to compute the images of $f_\bullet (\underline{z},t)$ in
$\varepsilon \,A(Y) \subseteq
A_{\underline{t}^0}$ and divide them by $\varepsilon$:\\
\par
Using the notations of \zitat{5}{3} and \zitat{5}{4}, in
$A_{\underline{t}^0}$ it holds
\begin{eqnarray*}
0 &=& F_{(a,b,\alpha,\beta)}(\underline{z}, t+\underline{t}^0\,\varepsilon)\\
&=& \!\begin{array}[t]{l}
f_{(a,b,\alpha,\beta)}(\underline{z}, t+t^0_1\,\varepsilon) -\\
\quad - \underline{z}^{[c,\underline{\eta}(c)]}\cdot
\left( (t+\underline{t}^0\,\varepsilon)^{\alpha e_1 + \sum_v a_v
\underline{\eta}(c^v)-\underline{\eta}(c)}
- (t+\underline{t}^0\,\varepsilon)^{\beta e_1 + \sum_v b_v \underline{\eta}(c^v)-\underline{\eta}(c)}
\right)\,.
\end{array}
\end{eqnarray*}
The relation $\varepsilon^2=0$ yields
\[
f_{(a,b,\alpha,\beta)}(\underline{z}, t+t^0_1\varepsilon) =
f_{(a,b,\alpha,\beta)}(\underline{z}, t) + \varepsilon\cdot
(\alpha \,t^{\alpha-1} \,t_1^0 \,\underline{z}^a -
\beta \,t^{\beta-1}\, t_1^0 \,\underline{z}^b)\,,
\]
and similarly we can expand the other terms.
Eventually, we obtain
\begin{eqnarray*}
f_{(a,b,\alpha,\beta)}(\underline{z},t) &=&
\begin{array}[t]{l}
-\varepsilon\,t_1^0\, (\alpha\, t^{\alpha -1}\,\underline{z}^a -
\beta\, t^{\beta -1}\,\underline{z}^b )\,+\,
\varepsilon \, \underline{z}^{[c,\underline{\eta}(c)]}\,
t^{\alpha+\sum_v a_v \eta_0(c^v)-\eta_0(c)-1}\cdot
\vspace{1ex}\\
\qquad\qquad\qquad\qquad\cdot
\left[
t_1^0\, (\alpha-\beta) +
+\sum_i t_i^0\, \left(\sum_v (a_v-b_v)\eta_i(c^v) \right) \right]
\end{array}
\vspace{1ex}\\
&=&
\, \varepsilon\cdot x^{\sum_v a_v [c^v,\eta_0(c^v)] +\alpha-1} \cdot
\left( \sum_i t_i^0\, \left(\sum_v (a_v-b_v)\eta_i(c^v) \right) \right)\,.
\end{eqnarray*}
(In $\varepsilon \, A(Y)$ we were able to replace the variables $t$ and $z_i$
by $x^{[\underline{0},1]}$ and $x^{[c^v,\eta_0(c^v)]}$, respectively.)\\
\par
On the other hand, we use Theorem \zitat{3}{4} of \cite{T2}: Fixing
$R^\ast\in Z\!\!\!Z^{n+1}$, it is the element of $L_{\,I\!\!\!\!C}(E\cap\partial\sigma
^{\scriptscriptstyle\vee})^\ast$ given by $q\mapsto \sum_{i,v} t_i^0\,q_v\,\eta_i(c^v)$
that corresponds to the infinitesimal deformation of $T^1_Y(-R^\ast)$
defined by the map
\begin{eqnarray*}
^{\displaystyle I}\! \left/ _{\displaystyle I^2} \right. & \longrightarrow & A(Y)\\
t^\alpha\,\underline{z}^a - t^\beta\,\underline{z}^b & \mapsto &
\left( \sum_{i,v} t_i^0\, (a_v-b_v)\eta_i(c^v) \right)
\cdot x^{\sum_v a_v [c^v,\eta_0(c^v)] +\alpha-1}\,.
\vspace{-3ex}
\end{eqnarray*}
\hfill$\Box$\\
\par
\neu{63}
To dicuss the meaning of the homogeneous part $T^1_Y(-R^\ast)$ inside the
whole vector space $T^1_Y$, we have to look at the results of
\cite{Gor}:\\
\par
If $\mbox{dim}\, T^1_Y< \infty$ (for instance, if $Y$ has an isolated
singularity), then
\begin{itemize}
\item[(1)]
$T^1_Y=T^1_Y(-R^\ast)$, but
\item[(2)]
$T^1_Y=0$ for $\mbox{dim}\, Y\geq 4$.
\end{itemize}
In particular, the interesting cases arise from 2-dimensional
lattice polygons $Q$ with primitive edges
only. The corresponding 3-dimensional toric varieties $Y$ have an isolated
singularity, and
the Kodaira-Spencer map $T_0\bar{{\cal M}}\rightarrow T^1_Y$ is an
isomorphism.\\
\par
If $T^1_Y$ has infinite dimension, then this comes from the existence of
infinitly many non-trivial homogeneous pieces $T^1_Y(-R)$. Whenever
$\langle a^j, R\rangle \leq 1$ holds for all vertices $a^j\in Q$, we
have
\[
T^1_Y(-R) = V_{\,I\!\!\!\!C}(\mbox{conv}\{a^j\,|\; \langle a^j,R\rangle =1\})\,,
\]
i.e. $T^1_Y(-R)$ equals the vector space of Minkowski summands of some
face of $Q$. ($T^1_Y(-R)=0$ for all other $R\in Z\!\!\!Z^{n+1}$.)\\
In particular, $T^1_Y(-R^\ast)$ is a typical, but nevertheless extremal
and perhaps the most interesting part of $T^1_Y$.\\
\par
\sect{The obstruction map}\label{s7}
\neu{71}
Dealing with obstructions in the deformation theory of $Y$ involves the
$A(Y)$-module $T^2_Y$. Usually, it is defined in the following way:\\
\par
Let
$
m:= \{ ([a,\alpha],[b,\beta])\in I\!\!N^{w+1}\times I\!\!N^{w+1}\,|\;
\begin{array}[t]{l}
\sum_v a_v\,c^v=\sum_v b_v\,c^v;\\
\sum_v a_v\,\eta_0(c^v) +\alpha = \sum_v b_v \,\eta_0(c^v) +\beta\}
\end{array}
$\vspace{1ex}\\
denote the set
parametrizing the equations $f_{(a,b,\alpha,\beta)}$ generating the ideal
$I\subseteq \,I\!\!\!\!C[\underline{z},t]$ of $Y$.
Then,
\[
{\cal R}:=\mbox{Ker}\left( \,I\!\!\!\!C[\underline{z},t]^m \longrightarrow\hspace{-1.5em}\longrightarrow I\right)
\]
is the module of linear relations between these equations; it contains the
submodule ${\cal R}_0$ of the so-called Koszul relations.\\
\par
{\bf Definition:}\quad
$T^2_Y:=
\;^{\displaystyle \mbox{Hom}\,(^{\displaystyle {\cal R}}\! / \!
_{\displaystyle {\cal R}_0}, A(Y))} \! \left/ \!
_{\displaystyle \mbox{Hom}\,(\,I\!\!\!\!C[\underline{z},t]^m,A(Y))} \right.$\,.\\
\par
Now, we have a similar theorem for $T^2_Y$ as we had in \zitat{6}{1} for
$T^1_Y$;
in particular, we use the
notations introduced there.\\
\par
{\bf Theorem:} (cf. \cite{T2})
{\em
The vector space $T^2_Y$ is $Z\!\!\!Z^{n+1}$-graded, and
in degree $-R$ it equals
\[
T^2_Y(-R)=
\left( \frac{\displaystyle
\mbox{Ker}\,\left( \oplus_j L_{\,I\!\!\!\!C}(E_j^R) \longrightarrow
L_{\,I\!\!\!\!C}(E)\right)}{\displaystyle
\mbox{Im}\, \left( \oplus_{\langle a^i,a^j\rangle<Q} L_{\,I\!\!\!\!C}(E_i^R\cap E_j^R)
\rightarrow \oplus_i L_{\,I\!\!\!\!C}(E_i^R)\right)} \right)^\ast\,.
\]
}
\par
\neu{72}
In this section we build up the so-called obstruction map. It
detects all possibile infinitesimal extensions of our family over
$\bar{{\cal M}}$ to a flat family over some larger base space.
We follow the explanation given in \S 4 of \cite{RedFund}.\\
\par
As before,
\[
{\cal J}=(g_{\varepsilon,k}(\underline{t}-t_1)\,|\; \varepsilon<Q,\, k\geq 1) =
(g_{\underline{d},k}(\underline{t}-t_1)\,|\; \underline{d}\in V^\bot\capZ\!\!\!Z^N, \, k\geq 1)
\subseteq \,I\!\!\!\!C[t_i-t_j]
\]
denotes the homogeneous ideal
of the base space $\bar{{\cal M}}$. Let
\[
\tilde{\kI} := (t_i-t_j)_{i,j}\cdot {\cal J} +{\cal J}_1\cdot \,I\!\!\!\!C[t_i-t_j]
\subseteq
\,I\!\!\!\!C[t_i-t_j\,|\; 1\leq i,j\leq N]\,.
\]
Then, $W:= \,^{\displaystyle {\cal J}}\!\!\! \left/ \! _{\displaystyle \tilde{\kI}} \right.$ is a finitely
dimensional, $Z\!\!\!Z$-graded vector space ($W=\oplus_{k\geq 2} W_k$, and
$W_k$ is generated by the polynomials
$g_{\underline{d},k}(\underline{t}-t_1)$). It comes as the kernel in the exact
sequence
\[
0\rightarrow W \longrightarrow
\,^{\displaystyle \,I\!\!\!\!C[t_i-t_j]}\!\!\left/ \! _{\displaystyle \tilde{\kI}} \right.
\longrightarrow
\,^{\displaystyle \,I\!\!\!\!C[t_i-t_j]}\!\!\left/ \! _{\displaystyle {\cal J}} \right.
\rightarrow 0\,.
\]
Identifying $t$ with $t_1$ and $\underline{z}$ with $\underline{Z}$,
the tensor product with $\,I\!\!\!\!C[\underline{z},t]$ (over $\,I\!\!\!\!C$) yields the important,
exact sequence
\[
0\rightarrow W \otimes_{\,I\!\!\!\!C} \,I\!\!\!\!C[\underline{z},t]
\longrightarrow
\,^{\displaystyle \,I\!\!\!\!C[\underline{Z},\underline{t}]}\!\!\left/ \! _{\displaystyle \tilde{\kI}\cdot \,I\!\!\!\!C[\underline{Z},\underline{t}]}
\right.
\longrightarrow
\,^{\displaystyle \,I\!\!\!\!C[\underline{Z},\underline{t}]}\!\!\left/ \! _{\displaystyle {\cal J}\cdot \,I\!\!\!\!C[\underline{Z},\underline{t}]}
\right.
\rightarrow 0\,.
\vspace{2ex}
\]
Now, let $s$ be any relation with coefficients in $\,I\!\!\!\!C[\underline{z},t]$
between the equations $f_{(a,b,\alpha,\beta)}$,
i.e.
\[
\sum s_{(a,b,\alpha,\beta)} \, f_{(a,b,\alpha,\beta)} =0 \quad
\mbox{in } \,I\!\!\!\!C[\underline{z},t]\,.
\]
By flatness of our family (cf. \zitat{5}{6}), the components of $s$
can be lifted to $\,I\!\!\!\!C[\underline{Z},\underline{t}]$ obtaining an $\tilde{s}$ such that
\[
\lambda(s):=
\sum \tilde{s}_{(a,b,\alpha,\beta)} \, F_{(a,b,\alpha,\beta)} \mapsto 0 \quad
\mbox{in }
\,^{\displaystyle \,I\!\!\!\!C[\underline{Z},\underline{t}]}\!\!\left/ \! _{\displaystyle {\cal J}\cdot \,I\!\!\!\!C[\underline{Z},\underline{t}]}
\right.\,.
\]
In particular, each relation $s\in {\cal R}$ induces some element
$\lambda(s)\in
W\otimes_{\,I\!\!\!\!C} \,I\!\!\!\!C[\underline{z},t]$, which is well defined after the additional
projection to $W\otimes_{\,I\!\!\!\!C} A(Y)$.
This procedure describes a certain
element $\lambda\in T^2_Y\otimes_{\,I\!\!\!\!C}W= \mbox{Hom}(W^\ast, T^2_Y)$
called the obstruction map.\\
\par
{\bf Theorem:}
{\em
The obstruction map $\lambda: W^\ast\rightarrow T^2_Y$ is injective.
}\\
\par
{\bf Corollary:}
{\em
If $\mbox{dim}\, T^1_Y < \infty$,
our family equals the versal deformation of $Y$.
In general, we could say that it is ``versal in degree $-R^\ast$''.
}\\
\par
{\bf Proof:}
In \zitat{6}{2} we have proved that the Kodaira-Spencer map is an isomorphism
(at least onto the homogeneous piece $T^1_Y(-R^\ast)$). By a criterion also
described
in \cite{RedFund}, this fact combined with injectivity of the obstruction map
implies
versality.
\hfill$\Box$\\
\par
The remaining part of \S \ref{s7} contains the proof of the previous theorem.\\
\par
\neu{73}
We have to improve the notations of \S \ref{s4} and \S \ref{s5}.
Since $\bar{{\cal M}}
\subseteq \bar{S} \subseteq \,I\!\!\!\!C^N$, we were able to use the toric
equations (cf. \zitat{2}{4}) during computations modulo ${\cal J}$. In
particular, the exponents $\underline{\eta}\in Z\!\!\!Z^N$ of $\underline{t}$ needed be known
modulo $V^\bot$ only; it was enough to define $\underline{\eta}(c)$ as elements
of $V_{Z\!\!\!Z}^\ast$.\\
However, to compute the obstruction map, we have to deal with the smaller
ideal $\tilde{\kI}\subseteq {\cal J}$. Let us start with refining the definitions
of \zitat{4}{6}:
\begin{itemize}
\item[(i)]
For each vertex $a\in Q$ we
choose the following paths through the
1-skeleton of $Q$:
\begin{itemize}
\item[$\bullet$]
$\underline{\lambda}(a):=$ path from $0\in Q$ to $a\in Q$ .
\item[$\bullet$]
$\underline{\mu}^v(a):=$ path from $a\in Q$ to $a(c^v) \in Q$ such that
$\mu_i^v(a) \langle d^i, c^v \rangle \leq 0$ for each $i=1,\dots,N$.
\item[$\bullet$]
$\underline{\lambda}^v(a):= \underline{\lambda}(a) + \underline{\mu}^v(a)$ is then a path
from $0\in Q$ to $a(c^v)$, which depends on $a$.
\end{itemize}
\item[(ii)]
For each $c\in Z\!\!\!Z^n$ we use the vertex $a(c)$ to define
\[
\underline{\eta}^c(c):= \left[ -\lambda_1(a(c))\langle d^1,c\rangle,
\dots, -\lambda_N(a(c)) \langle d^N,c\rangle \right] \in Z\!\!\!Z^N
\]
and
\[
\underline{\eta}^c(c^v):= \left[ -\lambda_1^v(a(c))\langle d^1,c^v\rangle,
\dots, -\lambda_N^v(a(c)) \langle d^N,c^v\rangle \right] \in Z\!\!\!Z^N\,.
\]
\item[(iii)]
For each $c\in Z\!\!\!Z^n$ we fix a representation $c=\sum_v p_v^c\, c^v$
($p_v^c\in I\!\!N$) such that $\eta_0(c)=\sum_v p_v^c \,\eta_0(c^v)$. (That
means, $c$ is represented only by those generators $c^v$ that define
faces of $Q$ containing the face defined by $c$ itself.)
\end{itemize}
{\bf Remark:}
Let $a\in I\!\!N^w$. Denoting $c:=\sum_v a_vc^v$
we obtain
$\sum_v a_v\,\underline{\eta}^c(c^v) - \underline{\eta}^c(c) \in I\!\!N^N$
by arguments as in
Lemma \zitat{4}{6}.\\
Moreover, for the special representation $c=\sum_v p_v^c c^v$,
the equation
$\sum_v p_v^c\,\underline{\eta}^c(c^v) = \underline{\eta}^c(c)$ is true.\\
\par
Now, we improve the definition of the polynomials $F_\bullet(\underline{Z},\underline{t})$
given in \zitat{5}{4}. Let $a,b\in I\!\!N^w, \alpha,\beta\in I\!\!N$ such that
\[
c:= \sum_v a_v\, c^v = \sum_v b_v\, c^v\quad \mbox{and}\quad
\sum_v a_v\, \eta_0(c^v) + \alpha = \sum_v b_v \, \eta_0(c^v) + \beta\,.
\vspace{-0.5ex}
\]
Then,
\[
F_{(a,b,\alpha,\beta)}(\underline{Z},\underline{t}) := f_{(a,b,\alpha,\beta)}(\underline{Z},t_1) -
\underline{Z}^{\underline{p}^c} \cdot
\left( \underline{t}^{\alpha e_1 +\sum_v a_v \underline{\eta}^c(c^v)-\underline{\eta}^c(c)} -
\underline{t}^{\beta e_1 +\sum_v b_v \underline{\eta}^c(c^v)-\underline{\eta}^c(c)} \right)\, .
\vspace{2ex}
\]
\par
\neu{74}
We have to discuss the same three types of relations as we did in
\zitat{5}{6}. Since there is only one single element $c\in Z\!\!\!Z^n$ involved
in the relations (i) and (ii), computing modulo $\tilde{\kI}$ instead of ${\cal J}$
makes no difference in these cases -
we always obtain $\lambda(s)=0$.\\
\par
Let us regard the relation $s:=\left[\underline{z}^r\cdot f_{(a,b,\alpha,\beta)} -
f_{(a+r,b+r,\alpha,\beta)} =0\right]$ $\;(r\in I\!\!N^w$).
We will use the following notations:
\begin{itemize}
\item
$c:=\sum_v a_v \, c^v = \sum_v b_v \, c^v;\quad
\underline{p}:=\underline{p}^c;\quad
\underline{\eta}:= \underline{\eta}^c;$
\item
$\tilde{c}:=\sum_v (a_v+r_v) \, c^v = \sum_v (b_v+r_v) \, c^v =
\sum_v (p_v+r_v) \, c^v; \quad
\underline{q}:=\underline{p}^{\tilde{c}};\quad
\tilde{\underline{\eta}}:= \underline{\eta}^{\tilde{c}};$
\item
$\xi:= \sum_i\left(\left( \sum_v (p_v+r_v)\tilde{\eta}_i(c^v)\right)
-\tilde{\eta}_i(\tilde{c})\right) = \sum_v (p_v+r_v)\eta_0(c^v) -
\eta_0(\tilde{c})\,.$
\end{itemize}
Using the same lifting of $s$ to $\tilde{s}$ as in \zitat{5}{6} yields
\[
\begin{array}{rcl}
\lambda(s)&=&\!\begin{array}[t]{l}
\underline{Z}^r\cdot F_{(a,b,\alpha,\beta)}- F_{(a+r,b+r,\alpha,\beta)} \,-
\vspace{1ex}\\
\qquad\qquad -\,
\left( \underline{t}^{ \alpha e_1 + \sum_va_v\underline{\eta}(c^v)-\underline{\eta}(c)} -
\underline{t}^{ \beta e_1 + \sum_vb_v\underline{\eta}(c^v)-\underline{\eta}(c)} \right)\cdot
F_{(q,p+r,\xi,0)}
\end{array}
\vspace{2ex}\\
&=&\!
\begin{array}[t]{l}
-\underline{Z}^{p+r}\cdot \left( \underline{t}^{\alpha e_1 + \sum_v(a_v-p_v)\underline{\eta}(c^v)}
- \underline{t}^{\beta e_1 + \sum_v(b_v-p_v)\underline{\eta}(c^v)} \right)\, +
\vspace{1ex}\\
+\,
\underline{Z}^q\cdot \left( \underline{t}^{\alpha e_1 +
\sum_v(a_v+r_v-q_v)\underline{\tilde{\eta}}(c^v)}
- \underline{t}^{\beta e_1 + \sum_v(b_v+r_v-q_v)\underline{\tilde{\eta}}(c^v)} \right)\, -
\vspace{1ex}\\
-\,
\left( \underline{t}^{\alpha e_1 + \sum_v(a_v-p_v)\underline{\eta}(c^v)}
- \underline{t}^{\beta e_1 + \sum_v(b_v-p_v)\underline{\eta}(c^v)} \right) \cdot
\left(\underline{Z}^q\, \underline{t}^{\sum_v(p_v+r_v-q_v)\underline{\tilde{\eta}}(c^v)}
-\underline{Z}^{p+r}\right)
\end{array}
\vspace{2ex}\\
&=&\!
\begin{array}[t]{l}
\underline{Z}^q\cdot
\left( \underline{t}^{\alpha e_1 + \sum_v(a_v+r_v-q_v)\underline{\tilde{\eta}}(c^v)}
-\underline{t}^{\alpha e_1 + \sum_v(p_v+r_v-q_v)\underline{\tilde{\eta}}(c^v) +
\sum_v (a_v-p_v)\underline{\eta}(c^v)}
\right) -
\vspace{1ex}\\
\qquad - \underline{Z}^q\cdot\left(
\underline{t}^{\beta e_1 + \sum_v(b_v+r_v-q_v)\underline{\tilde{\eta}}(c^v)}
- \underline{t}^{\beta e_1 + \sum_v(p_v+r_v-q_v)\underline{\tilde{\eta}}(c^v) +
\sum_v (b_v-p_v)\underline{\eta}(c^v)} \right)\,.
\end{array}
\end{array}
\]
As in \zitat{5}{6}(iii), we can see that $\lambda(s)$ vanishes modulo ${\cal J}$ (or
even in
$A(\bar{S})$) - just identify $\underline{\eta}$ and $\underline{\tilde{\eta}}$.\\
\par
\neu{75}
In \zitat{7}{2} we already mentioned the isomorphism
\[
W\otimes_{\,I\!\!\!\!C} \,I\!\!\!\!C[\underline{z},t] \stackrel{\sim}{\longrightarrow}\,
^{\displaystyle {\cal J}\cdot \,I\!\!\!\!C[\underline{Z},\underline{t}]} \!\!\left/ \! _{\displaystyle \tilde{\kI}\cdot
\,I\!\!\!\!C[\underline{Z},\underline{t}]} \right.
\]
obtained by identifying $t$ with $t_1$ and $\underline{z}$ with $\underline{Z}$. Now, with
$\lambda(s)$,
we have obtained an element of the right hand side, which has to be interpreted
as an element of
$W\otimes_{\,I\!\!\!\!C} \,I\!\!\!\!C[\underline{z},t]$.\\
\par
{\bf Lemma:}
{\em
Let $A,B\in I\!\!N^N$ such that $\underline{d}:=A-B\in V^\bot$ (i.e. $\underline{t}^A-\underline{t}^B
\in
{\cal J}\cdot \,I\!\!\!\!C[\underline{Z},\underline{t}]$). Then, via the previously mentioned isomorphism,
$\underline{t}^A-\underline{t}^B$ corresponds to the element
\[
\sum_{k\geq 1} c_k\cdot g_{\underline{d},k}(\underline{t}-t_1)\cdot t^{k_0-k}
\in W\otimes_{\,I\!\!\!\!C}
\,I\!\!\!\!C[\underline{z},t]\,
\]
($k_0:=\sum_iA_i$; $c_k$ are the constants occured in \zitat{3}{4}).
In particular, the coefficients from $W_k$ vanish for $k> k_0$.
}\\
\par
{\bf Proof:}
First, we remark that it is allowed to assume that $A=\underline{d}^+$, $B=\underline{d}^-$,
i.e.
$\underline{t}^A-\underline{t}^B = p_{\underline{d}}(\underline{t})$ (cf. \zitat{3}{2}).
(Otherwise we could write this binomial as
\[
\underline{t}^A - \underline{t}^B = \underline{t}^C\cdot \left( \underline{t}^{\underline{d}^+} -
\underline{t}^{\underline{d}^-}\right)\;(C\in I\!\!N^N),
\]
and since
\[
\underline{t}^C = (t_1+[\underline{t}-t_1])^C \equiv t_1^{\sum_i C_i} \pmod{(t_i-t_j)}\,,
\vspace{0.5ex}
\]
we would obtain
\vspace{-0.5ex}
\[
\underline{t}^A-\underline{t}^B \equiv t_1^{\sum_iC_i}\cdot \left( \underline{t}^{\underline{d}^+} -
\underline{t}^{\underline{d}^-}\right) \pmod{\tilde{\kI}}\,. )
\vspace{2ex}
\]
In \zitat{3}{4} we have seen that
\[
p_{\underline{d}}(\underline{t}) = \sum_{k=1}^{k_0} t_1^{k_0-k}\cdot
\left( \sum_{v=1}^{k-1} q_{v,k}(\underline{t}-t_1)\cdot g_{\underline{d},v}(\underline{t}-t_1) +
c_k\cdot g_{\underline{d},k}(\underline{t}-t_1)\right)
\]
(with $k_0:= \sum_id_i^+$). Since $q_{v,k}(\underline{t}-t_1)\in (t_i-t_j)\cdot
\,I\!\!\!\!C[t_i-t_j]$, this implies
\[
p_{\underline{d}}(\underline{t}) \equiv \sum_{k=1}^{k_0} t_1^{k_0-k} \cdot c_k\cdot
g_{\underline{d},k}(\underline{t}-t_1) \pmod{\tilde{\kI}}\,.
\]
On the other hand, for $k>k_0$, Lemma \zitat{3}{3} tells us that
$g_{\underline{d},k}(\underline{t}-t_1)$
is a $\,I\!\!\!\!C[t_i-t_j]$-linear combination of the elements
$g_{\underline{d},1}(\underline{t}-t_1),\dots, g_{\underline{d},k_0}(\underline{t}-t_1)$.
Then, the degree $k$ part of the corresponding equation shows
$g_{\underline{d},k}(\underline{t}-t_1)\in \tilde{\kI}$.
\hfill$\Box$\\
\par
{\bf Corollary:}
{\em
Transfered to $W\otimes_{\,I\!\!\!\!C}\,I\!\!\!\!C[\underline{z},t]$, the element $\lambda(s)$ equals
\[
\sum_{k\geq 1} c_k \cdot g_{\underline{d},k}(\underline{t}-t_1)\cdot \underline{z}^q \cdot
t^{k_0-k}\quad
\mbox{ with} \begin{array}[t]{rcl}
\underline{d} &:=& \sum_v (a_v-b_v)\cdot \left(\underline{\tilde{\eta}}(c^v)-\underline{\eta}(c^v)
\right)\,,\\
k_0 &:=& \alpha + \sum_v (a_v+r_v)\, \eta_0(c^v) - \eta_0(\tilde{c})\,.
\end{array}
\]
The coefficients vanish for $k>k_0$.
}
\\
\par
{\bf Proof:}
We apply the previous lemma to both summands of the $\lambda(s)$-formula of
\zitat{7}{4}. For the first one we obtain
\begin{eqnarray*}
\underline{d}^a &=& \!
\begin{array}[t]{l}
[ \alpha e_1 + \sum_v(a_v +r_v -q_v)\, \tilde{\underline{\eta}}(c^v)] \, -
\vspace{0.5ex}\\
\qquad\qquad - \,[ \alpha e_1 + \sum_v(p_v+r_v-q_v) \, \tilde{\underline{\eta}}(c^v)
+\sum_v (a_v-p_v)\, \underline{\eta}(c^v) ]
\end{array}\\
&=&
\sum_v (a_v-p_v)\cdot \left(\underline{\tilde{\eta}}(c^v)-\underline{\eta}(c^v) \right)
\qquad \mbox{and}
\vspace{2ex}\\
k_0 &=&
\sum_i \left( \alpha e_1 +
\sum_v(a_v+r_v-q_v)\,\underline{\tilde{\eta}}(c^v)\right)_i\\
&=& \alpha + \sum_v(a_v+r_v-q_v) \,\eta_0(c^v)\,
=\, \alpha + \sum_v (a_v+r_v)\, \eta_0(c^v) - \eta_0(\tilde{c})\,.
\end{eqnarray*}
$k_0$ has the same value for both the $a$- and $b$-summand,
and
\[
\begin{array}{rcl}
\underline{d}=\underline{d}^a-\underline{d}^b &=&
\sum_v (a_v-p_v)\cdot \left(\underline{\tilde{\eta}}(c^v)-\underline{\eta}(c^v) \right) -
\sum_v (b_v-p_v)\cdot \left(\underline{\tilde{\eta}}(c^v)-\underline{\eta}(c^v) \right) \\
&=&
\sum_v (a_v-b_v)\cdot \left(\underline{\tilde{\eta}}(c^v)-\underline{\eta}(c^v) \right)\,.
\end{array}
\vspace{-3ex}
\]
\hfill$\Box$\\
\par
\neu{76}
Now, we try to approach the obstruction map $\lambda$ from the opposite
direction.
Using the description of $T^2_Y$ given in \zitat{7}{1} we construct an element
of $T^2_Y\otimes_{\,I\!\!\!\!C}W$, that afterwards turns out to equal $\lambda$.\\
\par
For a path $\varrho\in Z\!\!\!Z^N$ along the edges of $Q$, we will denote by
\[
\underline{d}(\varrho,c):= [\langle \varrho_1\,d^1,\,c\rangle, \dots,
\langle \varrho_N\,d^N,\,c\rangle]\in Z\!\!\!Z^N
\]
the vector showing the behaviour of $c\inZ\!\!\!Z^n$ passing each particular edge.
If, moreover, $\varrho$ comes from a closed path, $\underline{d}(\varrho,c)$ is also
contained in
$V^\bot$.\\
On the other hand, for each $k\geq 1$, we can use the $\underline{d}$'s from $V^\bot$
to get
elements $g_{\underline{d},k}(\underline{t}-t_1)\in W_k$ generating this vector space.
Composing both procedures we obtain, for each closed path $\varrho\in Z\!\!\!Z^N$, a
map
\[
\begin{array}{cccccl}
g^{(k)}(\varrho,\bullet):& I\!\!R^n&\longrightarrow &V^\bot &\longrightarrow &W_k\\
&c&&\mapsto && g_{\underline{d}(\varrho,c), k}(\underline{t}-t_1)\,.
\end{array}
\]
\par
{\bf Remark:}
\begin{itemize}
\item[(1)]
Taking the sum over all 2-faces we get a surjective map
\[
\sum_{\varepsilon<Q} g^{(k)}(\underline{\varepsilon},\bullet): \oplus_{\varepsilon<Q} \,I\!\!\!\!C^n \longrightarrow\hspace{-1.5em}\longrightarrow W_k\,.
\vspace{-2ex}
\]
\item[(2)]
Let $c\inZ\!\!\!Z^n$ (having integer coordinates is very important here).
If $\varrho^1, \varrho^2\inZ\!\!\!Z^N$ are two paths each connecting vertices
$a,b\in Q$ such that
\begin{itemize}
\item[$\bullet$]
$|\langle a,c\rangle - \langle b,c \rangle | \leq k-1\;$ and
\item[$\bullet$]
$c$ is monoton along both pathes (i.e.
$\langle \varrho^1_i \,d^i, c \rangle;\;
\langle \varrho^2_i \,d^i, c \rangle
\geq 0$
for $i=1,\dots,N$),
\end{itemize}
then $\varrho^1-\varrho^2\in Z\!\!\!Z^N$ will be a closed path yielding
$g^{(k)}(\varrho^1-\varrho^2, \,c)=0$ in $W_k$.
\vspace{2ex}
\end{itemize}
\par
{\bf Proof:}
The reason for (1) is the fact that the elements $\underline{d}(\varepsilon,c)$ ($\varepsilon
<Q\,$ 2-face; $c\in Z\!\!\!Z^n$) generate $V^\bot$ as a vector space.\\
For the proof of (2), we look at $\underline{d}:=\underline{d}(\varrho^1-\varrho^2,\,c)$.
Since
$d_i=\langle \varrho^1_i\,d^i,\,c\rangle - \langle \varrho^2_i\,d^i,\,c\rangle$
is the difference of two non-negative integers, we obtain
$d_i^+ \leq \langle \varrho_i^1\,d^i,\, c\rangle$.\\
Hence,
\[
\sum_i d_i^+ \leq \sum_i \langle \varrho^1_i\,d^i,\,c\rangle =
\langle b,c\rangle - \langle a,c\rangle \leq k-1\,,
\]
and as in \zitat{7}{5} we obtain $g_{\underline{d},k}(\underline{t}-t_1)
\in \tilde{\kI}$ by Lemma \zitat{3}{3}.
\hfill$\Box$
\vspace{2ex}\\
\par
Using the notations introduced in \zitat{6}{1} we obtain for $R:=k\, R^\ast,\;
k\geq 2$
\[
E_j^{kR^\ast}=
\{[c^v,\eta_0(c^v)]\,|\; \langle a^j,c^v\rangle + \eta_0(c^v) \leq k-1\}
\cup \{R^\ast\} \subseteq \sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^{n+1}\,.
\]
Then, we can define the following linear maps :
\[
\begin{array}{cccl}
\psi_j^{(k)}:& L(E_j^{kR^\ast}) & \longrightarrow & W_k\\
& q & \mapsto & \sum_v q_v\cdot
g^{(k)}\left(\underline{\lambda}(a^j)+\underline{\mu}^v(a^j) -
\underline{\lambda}(a(c^v)),\,c^v\right)\,.
\end{array}
\]
(The $q$-coordinate corresponding to $R^\ast\in E_j^{kR^\ast}$ is not used in
the
definition of $\psi_j^{(k)}$.)
\vspace{1ex}\\
\par
{\bf Lemma:}
{\em
Let $\langle a^i,a^j \rangle <Q$ be an edge of the polyhedron $Q$. Then, on
$L(E_i^{kR^\ast} \cap E_j^{kR^\ast}) = L(E_i^{kR^\ast}) \cap L(E_j^{kR^\ast})$,
the
maps $\psi_i^{(k)}$ and $\psi_j^{(k)}$ coincide.\\
In particular (cf. Theorem \zitat{7}{1}), the $\psi_j^{(k)}$'s induce a linear
map
$\psi^{(k)}: T^2_Y(-kR^\ast)^{\ast}\rightarrow W_k$.
}\\
\par
{\bf Proof:}
Let $q\in L(E_i^{kR^\ast} \cap E_j^{kR^\ast})$. Moreover, we denote by
$\varrho^{ij}\inZ\!\!\!Z^N$ the path consisting of the single edge running from
$a^i$ to $a^j$.\\
Then,
\[
\begin{array}{rcl}
\psi_i^{(k)}(q) - \psi_j^{(k)}(q) &=&
\sum_v q_v\cdot g^{(k)}\left(
\underline{\lambda}(a^i) + \underline{\mu}^v(a^i) -
\underline{\lambda}(a^j) + \underline{\mu}^v(a^j) ,\,c^v\right)
\vspace{1ex}\\
&=&
\! \begin{array}[t]{l}
g^{(k)}\left( \underline{\lambda}(a^i) - \underline{\lambda}(a^j) + \varrho^{ij},\,
\sum_v q_v \, c^v\right) \,+
\vspace{0.5ex}\\
\qquad\qquad +\,
\sum_v q_v\cdot g^{(k)}\left(
\underline{\mu}^v(a^i) - \underline{\mu}^v(a^j) - \varrho^{ij},\,c^v\right)\,,
\end{array}
\end{array}
\]
and both summands vanish for several reasons.
The first one is killed simply by the equality $\sum_v q_v\,c^v=0$. For the
second one we can use (2) of the previous remark:\\
If $q_v\neq 0$, then the assumption about $q$ implies the inequalities
\[
0\leq \langle a^i,c^v\rangle - \langle a(c^v),c^v \rangle\, ; \;
\langle a^j,c^v\rangle - \langle a(c^v),c^v \rangle \leq k-1\,.
\]
Hence, assuming w.l.o.g.
$\langle a^i,c^v\rangle \geq \langle a^j,c^v\rangle$, we
can take
$\varrho^1:= -\underline{\mu}^v(a^j)-\varrho^{ij}$
and
$\varrho^2:= -\underline{\mu}^v(a^i)$
to see that
$g^{(k)}\left(\underline{\mu}^v(a^i) - \underline{\mu}^v(a^j) - \varrho^{ij},
\,c^v\right) =0$.
\hfill$\Box$\\
\par
\neu{77}
{\bf Proposition:}
{\em
$\;\sum_{k\geq 1} c_k\,\psi^{(k)}$ equals $\lambda^\ast$,
the adjoint of the obstruction map.
}\\
\par
{\bf Proof:}
In Theorem (3.5) of \cite{T2} we gave a dictionary between the two
$T^2$-formulas mentioned in \zitat{7}{1}. Using this result we can
find an element of $\mbox{Hom}(^{\displaystyle \cal R}\!/\!_{\displaystyle {\cal R}_0},\,
W_k\otimes A(Y))$
representing $\psi^{(k)}\in T^2_Y\otimes W_k$ -
it sends relations of type (i) (cf. \zitat{5}{6}) to 0 and deals with relations
of type (ii) and (iii) in the following way:
\[
[\underline{z}^r\, t^{\gamma}\cdot f_{(a,b,\alpha,\beta)} -
f_{(a+r,b+r,\alpha+\gamma,\beta+\gamma)}=0] \mapsto
\psi_j^{(k)}(a-b)\cdot x^{\sum_v
(a_v+r_v)[c^v,\eta_0(c^v)]+(\alpha+\gamma-k)R^{\ast}}\,,
\]
if
\[
\langle (Q,1),\, \sum_v (a_v+r_v)\, [c^v,\eta_0(c^v)] + (\alpha + \gamma -k)
R^\ast\rangle \geq 0\,,
\]
and $j$ is such that
\[
\langle (a^j,1),\, \sum_v a_v \,[c^v,\eta_0(c^v)] +
(\alpha -k)R^\ast\rangle <0\,;
\]
otherwise the relation is sent to 0 (in particular, if there is not any $j$
meeting the desired property).
\vspace{0.5ex}\\
\par
On $Q$, the linear forms $c:=\sum_va_v\,c^v$ and
$\tilde{c}=\sum_v(a_v+r_v)c^v$ admit their minimal values at the vertices
$a(c)$
and $a(\tilde{c})$, respectively. Hence,
we can transform the previous formula into
\[
\begin{array}{l}
[\underline{z}^r\, t^{\gamma}\cdot f_{(a,b,\alpha,\beta)} -
f_{(a+r,b+r,\alpha+\gamma,\beta+\gamma)}=0] \mapsto
\psi_{a(c)}^{(k)}(a-b)\cdot x^{\sum_v (a_v+r_v)[c^v,\eta_0(c^v)]+
(\alpha+\gamma-k)R^{\ast}}
\vspace{1ex}\\
\begin{array}[t]{ll}
\mbox{if } &
\begin{array}[t]{l}
\sum_v(a_v+r_v)\eta_0(c^v)-\eta_0(\tilde{c})+(\alpha+\gamma-k) =\\
\qquad =
\langle (a(\tilde{c}),1),\, \sum_v (a_v+r_v)\, [c^v,\eta_0(c^v)] +
(\alpha + \gamma -k)
R^\ast\rangle \geq 0\,,
\end{array}
\vspace{0.5ex}
\\ &
\begin{array}[t]{l}
\sum_va_v\,\eta_0(c^v)-\eta_0(c)+(\alpha-k) =\\
\qquad =
\langle (a(c),1),\, \sum_v a_v \,[c^v,\eta_0(c^v)] +
(\alpha -k)R^\ast\rangle <0
\end{array}
\end{array}
\end{array}
\]
(and mapping to 0 otherwise).\\
\par
Adding the coboundary $h\in \mbox{Hom}\,(\,I\!\!\!\!C[\underline{z},t]^m,\, W_k\otimes A(Y))$
\[
h_{(a,\alpha), (b,\beta)}:=
\left\{ \begin{array}{ll}
\psi^{(k)}_{a(c)}(a-b)\cdot x^{\sum_v a_v [c^v,\eta_0(c^v)] +(\alpha -k)R^\ast}
&
\mbox{for } \sum_v a_v\,\eta_0(c^v)-\eta_0(c)+\alpha\geq k\,,\\
0 & \mbox{otherwise}
\end{array} \right.
\]
does not
change the class in $T^2_Y(-kR^\ast)$ (still representing $\psi^{(k)}$), but
improves the represantative from
$\mbox{Hom}(^{\displaystyle \cal R}\!/\!_{\displaystyle {\cal R}_0},\, W_k\otimes A(Y))$.
It still maps type-(i)-relations to 0, and moreover
\[
\begin{array}{l}
[\underline{z}^r\, t^{\gamma}\cdot f_{(a,b,\alpha,\beta)} -
f_{(a+r,b+r,\alpha+\gamma,\beta+\gamma)}=0] \mapsto
\vspace{1ex}\\
\quad\mapsto \left\{ \begin{array}{ll}
\left(\psi_{a(c)}^{(k)}(a-b)-
\psi_{a(\tilde{c})}^{(k)}(a-b)\right)
\cdot x^{\sum_v (a_v+r_v)[c^v,\eta_0(c^v)]+
(\alpha+\gamma-k)R^{\ast}} &
\mbox{for }
k_0+\gamma\geq k\\
0 & \mbox{otherwise}\,
\end{array}
\right.
\end{array}
\]
(with $k_0=\alpha + \sum_v(a_v+r_v)\, \eta_0(c^v)-\eta_0(\tilde{c})$).\\
\par
By definition of $\psi^{(k)}_j$ and $g^{(k)}$ we obtain
\[
\begin{array}{l}
\psi_{a(c)}^{(k)}(a-b)- \psi_{a(\tilde{c})}^{(k)}(a-b)\, =
\vspace{0.5ex}\\
\qquad=\,
\sum_v (a_v-b_v)\cdot g^{(k)}\left(
\underline{\lambda}(a(c)) + \underline{\mu}^v(a(c)) -
\underline{\lambda}(a(\tilde{c})) - \underline{\mu}^v(a(\tilde{c})),\;c^v\right)
\vspace{0.5ex}\\
\qquad=\,
\sum_v (a_v-b_v)\cdot g^{(k)}\left(
\underline{\lambda}^v(a(c)) -
\underline{\lambda}^v(a(\tilde{c})) ,\,c^v\right)
\vspace{0.5ex}\\
\qquad=\,
g_{\underline{d},\,k}(\underline{t}-t_1) \quad
\mbox{ with }
\begin{array}[t]{rcl}
\underline{d} &=&
\sum_v(a_v-b_v)\cdot \underline{d} \left(
\underline{\lambda}^v(a(c)) -
\underline{\lambda}^v(a(\tilde{c})) ,\,c^v\right)\\
&=& \sum_v (a_v-b_v)\cdot \left( \tilde{\underline{\eta}}(c^v)-\underline{\eta}(c^v)
\right)\,,
\end{array}
\end{array}
\]
and this completes our proof. Indeed,
\begin{itemize}
\item
for relations of type (ii)
(i.e. $r=0$; $\gamma=1$) we know $c=\tilde{c}$, hence, those relations map
onto 0;
\item
for relations of type (iii) (i.e. $\gamma=0$) we compare the previous
formula with the result obtained in Corollary \zitat{7}{5} -
the coefficients coincide, and
the monomial
$\underline{z}^q\,t^{k_0-k}\in \,I\!\!\!\!C[\underline{z},t]$ maps onto
$x^{\sum_v (a_v+r_v)[c^v,\eta_0(c^v)]+
(\alpha+\gamma-k)R^{\ast}}\in A(Y)$.
\vspace{-3ex}
\end{itemize}
\hfill$\Box$\\
\par
\neu{78}
It remains to show that the summands $\psi^{(k)}$ of $\lambda^\ast$ are
indeed surjective maps from $T^2_Y(-kR^\ast)^\ast$ to $W_k$. We will do
so by composing them with auxiliary surjective maps
$p^k: \oplus_{\varepsilon<Q} \,I\!\!\!\!C^n \longrightarrow\hspace{-1.5em}\longrightarrow T^2_Y(-kR^\ast)^\ast$
yielding
$\psi^{(k)}\circ p^k = \sum_{\varepsilon<Q} g^{(k)}(\underline{\varepsilon},\bullet)$. Then, the
result
follows from the first part of Remark \zitat{7}{6}.\\
\par
In \S 6 of \cite{T2} we used a short exact sequence of complexes called
\[
0\rightarrow L_{\,I\!\!\!\!C}(E^R)_{\bullet} \longrightarrow (\,I\!\!\!\!C^{E^R})_{\bullet}
\longrightarrow \mbox{span}_{\,I\!\!\!\!C}(E^R)_{\bullet}\rightarrow 0
\]
to obtain from Theorem \zitat{7}{1} an isomorphism
\[
T^2_Y(-R) \cong
\left(
\frac{\displaystyle \mbox{Im}\, [\oplus_{\varepsilon<Q} \,I\!\!\!\!C^{n+1}\rightarrow
\oplus_{\langle a^i,a^j\rangle < Q} \,I\!\!\!\!C^{n+1}] }
{\displaystyle \mbox{Im}\, [\oplus_{\varepsilon <Q} \mbox{span}_{\,I\!\!\!\!C}\,
(\cap_{a^j \in \varepsilon} E_j^R) \rightarrow
\oplus_{\langle a^i,a^j\rangle < Q} \,I\!\!\!\!C^{n+1}] }
\right)^\ast\,.
\]
Since $R^\ast=[\underline{0},1]\in E_j^{kR^\ast}$ for $k\geq 2$, the induced
surjective
map $\oplus_{\varepsilon<Q}\,I\!\!\!\!C^{n+1}\longrightarrow\hspace{-1.5em}\longrightarrow T^2_Y(-kR^\ast)^\ast$ factorizes through
$\oplus_{\varepsilon<Q} \,I\!\!\!\!C^{n+1}\!/\!_{\displaystyle\,I\!\!\!\!C\cdot R^\ast} =
\oplus_{\varepsilon<Q}\,I\!\!\!\!C^n$ yielding the auxiliary map $p^k$ just mentioned.
Taking a closer look at the construction of \cite{T2} \S 6,
we can give an explicit
description of $p^k$; eventually we will be able to compute $\psi^{(k)}
\circ p^k$.\\
\par
Let us fix some 2-face $\varepsilon<Q$. Assume that $d^1,\dots,d^M$ are
its counterclockwise orientated edges, i.e.\ the sign vector $\underline{\varepsilon}$ looks
like $\varepsilon_i=1$ for $i=1,\dots,M$ and $\varepsilon_j=0$ otherwise.
Moreover,
we denote the vertices of $\varepsilon<Q$ by $a^1,\dots,a^M$ such that $d^i$
runs from $a^i$ to $a^{i+1}$ ($M+1:=1$).\\
\par
Starting with a $[c,\eta_0]\in \,I\!\!\!\!C^{n+1}$ (and, as just mentioned, only the
$c\in\,I\!\!\!\!C^n$ is essential) we have to proceed as follows:
\begin{itemize}
\item[(i)]
For $i=1,\dots,M$ we represent $[c,\eta_0]$ as a linear combination of
elements of $E_i^{kR^\ast}\cap E_{i+1}^{kR^\ast}$. (This corresponds to
the lifting from $\mbox{span}_{\,I\!\!\!\!C}(E^R)_{\bullet}$ to
$(\,I\!\!\!\!C^{E^R})_{\bullet}$.)
\[
[c,\eta_0] =
\sum_v q_{iv}\, [c^v,\eta_0(c^v)] + q_i\, [\underline{0},1]\,,
\]
and $q_{iv}\neq 0$ implies $[c^v,\eta_0(c^v)]\in E_i^{kR\ast}\cap
E_{i+1}^{kR^\ast}$, i.e.
\[
\langle a^i, c^v\rangle + \eta_0(c^v) \leq k-1\,;\quad
\langle a^{i+1}, c^v\rangle + \eta_0(c^v) \leq k-1\,.
\]
\item[(ii)]
We map the result to $\oplus_{i=1}^M \,I\!\!\!\!C^{E_i^{kR^\ast}}$ by taking
succesive differences (corresponding to the application of the
differential in
the complex $(\,I\!\!\!\!C^{E^R})_{\bullet}$). The result is automatically
contained in $\mbox{Ker}\left( \oplus_i L(E_i^{kR^\ast})\rightarrow
L(E)\right)$, and its $i$-th summand is the linear relation
\[
\sum_v (q_{i,v}-q_{i-1,v})\cdot [c^v,\eta_0(c^v)] +
(q_i-q_{i-1})\cdot [\underline{0},1]=0\,.
\]
\item[(iii)]
Finally, we apply $\psi^{(k)}$ to obtain
\[
\begin{array}{rcl}
\psi^{(k)}(p^k(c)) &=&
\sum_{i=1}^M \sum_v (q_{i,v}-q_{i-1,v})\cdot
g^{(k)} \left(
\underline{\lambda}(a^i)-\underline{\lambda}(a(c^v)) + \underline{\mu}^v(a^i),\;c^v\right)
\vspace{1ex}\\
&=& \! \begin{array}[t]{l}
\sum_{i,v} g^{(k)} \left(
\underline{\lambda}(a^i)-\underline{\lambda}(a(c^v)) + \underline{\mu}^v(a^i),\;
q_{i,v}\,c^v\right)\,-
\vspace{0.5ex}\\
\qquad -\,
\sum_{i,v} g^{(k)} \left(
\underline{\lambda}(a^{i+1})-\underline{\lambda}(a(c^v)) + \underline{\mu}^v(a^{i+1}),\;
q_{i,v}\,c^v\right)
\vspace{1ex}
\end{array}\\
&=&
\sum_{i,v} g^{(k)} \left(
\underline{\lambda}(a^i) -\underline{\lambda}(a^{i+1})
+ \underline{\mu}^v(a^i)- \underline{\mu}^v(a^{i+1}),\;
q_{i,v}\,c^v\right)\,.
\end{array}
\]
\end{itemize}
Similar to the proof of Lemma \zitat{7}{6} we introduce the path $\varrho^i$
consisting of the single edge $d^i$ only. Then, if $q_{iv}\neq 0$
and w.l.o.g. $\langle a^i,c^v\rangle \geq \langle a^{i+1},c^v\rangle$,
the pair of paths $\underline{\mu}^v(a^i)$ and $\underline{\mu}^v(a^{i+1})+\varrho^i$
meets the assumption of Remark \zitat{7}{6}(2) (cf.\ (i)). Hence,
we can proceed as follows:
\[
\begin{array}{rcl}
\psi^{(k)}(p^k(c)) &=& \!
\begin{array}[t]{l}
\sum_{i,v} g^{(k)} \left(
\underline{\lambda}(a^i) -\underline{\lambda}(a^{i+1}) + \varrho^i,\,q_{iv}\,c^v \right)
\,+
\vspace{0.5ex}\\
\qquad\qquad\qquad +\,\sum_{i,v} g^{(k)} \left(
\underline{\mu}^v(a^i)- \underline{\mu}^v(a^{i+1}) - \varrho^i ,\,q_{iv}\,c^v \right)
\end{array}
\vspace{1ex}\\
&=&
\sum_{i=1}^M g^{(k)}\left(
\underline{\lambda}(a^i) -\underline{\lambda}(a^{i+1}) + \varrho^i,\,
\sum_vq_{iv}\,c^v \right)
\vspace{1ex}\\
&=&
\sum_{i=1}^M g^{(k)}\left(
\underline{\lambda}(a^i) -\underline{\lambda}(a^{i+1}) + \varrho^i,\,
c \right)
\vspace{1ex}\\
&=&
g^{(k)}\left( \sum_{i=1}^M \varrho^i,\,c \right)
\vspace{1ex}\\
&=&
g^{(k)}(\underline{\varepsilon},\,c) \,.
\vspace{0.1ex}
\end{array}
\]
\par
Hence, Theorem \zitat{7}{2} is proven.\\
\par
\sect{The components of the reduced versal family}\label{s8}
\neu{81}
The components of the reduced base space $\bar{{\cal M}}_{red}$ correspond to
maximal
decompositions of $Q$
into a Minkowski sum $Q=R_0+\dots+R_m$ with lattice polytopes $R_k\subseteq
I\!\!R^n$ as
summands.
Intersections of components are obtained by the finest Minkowski decompositions
of $Q$,
that are coarser than all the involved maximal ones.\\
\par
{\bf Theorem:}
{\em
Fix such a Minkowski decomposition. Then, the corresponding component (or
intersection of
components)
$\bar{{\cal M}}_0$
is isomorphic to
$^{\displaystyle \,I\!\!\!\!C^{m+1}}\!\!/\!{\displaystyle \,I\!\!\!\!C\cdot (1,\dots,1)}$, and the
restriction $X_0\rightarrow \,I\!\!\!\!C^m$
of the versal family can be described as follows:
\begin{itemize}
\item[(i)]
Define the cone
\vspace{-0.5ex}
\[
\tilde{\sigma}:= \mbox{Cone}\,\left( \bigcup_{k=0}^m (R_k\times \{e^k\})
\right)
\subseteq I\!\!R^{n+m+1}\, ,
\]
it contains $\sigma = \mbox{Cone}\,(Q\times\{1\})\subseteq I\!\!R^{n+1}$ via the
diagonal embedding
$I\!\!R^{n+1}\hookrightarrow I\!\!R^{n+m+1}\; ((a,1)\mapsto (a;1,\dots,1))$. The
inclusion
$\sigma \subseteq \tilde{\sigma}$ induces a closed embedding of the
affine toric varities defined by these cones - this gives $Y\hookrightarrow
X_0$.
\item[(ii)]
The projection $I\!\!R^{n+m+1} \longrightarrow\hspace{-1.5em}\longrightarrow I\!\!R^{m+1}$ provides $m+1$ regular functions on
$X_0$, i.e.\ we
obtain a map $X_0\rightarrow \,I\!\!\!\!C^{m+1}$. Composing this map with\\
$\ell: \,I\!\!\!\!C^{m+1}\longrightarrow\hspace{-1.5em}\longrightarrow ^{\displaystyle \,I\!\!\!\!C^{m+1}}\!\!/\!{\displaystyle \,I\!\!\!\!C\cdot
(1,\dots,1)}$ yields the
family.
\vspace{2ex}
\end{itemize} }
We will
use the \zitat{8}{2} and \zitat{8}{3} to prove the theorem.\\
\par
\neu{82}
We already know (cf.\ \zitat{2}{5}) that both the space
\[
\bar{{\cal M}}_0 =\; ^{\displaystyle \,I\!\!\!\!C^{m+1}}\!\!\!\!/\!_{\displaystyle \,I\!\!\!\!C\cdot
(1,\dots,1)}
\;\subseteq \;^{\displaystyle \,I\!\!\!\!C^N}\!\!\!\!/\!_{\displaystyle \,I\!\!\!\!C\cdot
(1,\dots,1)}
\]
and its pullback ${\cal M}_0\subseteq\,I\!\!\!\!C^N$ are
given by the equations $t_i-t_j=0$ (if $d^i,d^j$ belong to a common summand
$R_k$ of $Q$).\\
\par
There is a chain of inclusions
${\cal M}_0 \subseteq {\cal M} \subseteq \bar{S} \subseteq \,I\!\!\!\!C^N$, and
the map ${\cal M}_0 \hookrightarrow \bar{S}$ factorizes through an embedding
${\cal M}_0 \hookrightarrow S$. It is given by the surjection of $\,I\!\!\!\!C$-algebras
\[
\begin{array}{ccc}
\,I\!\!\!\!C[C(Q)^{\scriptscriptstyle\vee}\cap V_{Z\!\!\!Z}^\ast] &\longrightarrow\hspace{-1.5em}\longrightarrow& \,I\!\!\!\!C[T_0,\dots,T_m]\\
t_i & \mapsto & T_k \mbox{ with } d^i\in R_k
\end{array}
\]
coming from the linear map
\[
\begin{array}{ccccc}
I\!\!R^{m+1} & \hookrightarrow & V & \hookrightarrow & I\!\!R^N\\
e^k & \mapsto & \sum_{d^i\in R_k} e^i\, .
\end{array}
\]
The matrix of $I\!\!R^{m+1}\hookrightarrow I\!\!R^N$ equals
the incidence matrix between edges and Minkowski summands of $Q$, and the
space ${\cal M}_0=\,I\!\!\!\!C^{m+1}$ corresponds to the cone $I\!\!R^{m+1}_{\geq0}=I\!\!R^{m+1}\cap
C(Q)$.\\
\par
\neu{83}
The family $X_0\rightarrow\bar{{\cal M}}_0$ arises from $\bar{X}\times_{\bar{S}}{\cal M}
\rightarrow \bar{{\cal M}}$ via base change $\bar{{\cal M}}_0 \hookrightarrow \bar{{\cal M}}$.
We
obtain
\[
\begin{array}{ccccccl}
X_0 &= &
(\bar{X}\times_{\bar{S}}{\cal M}) \times _{\bar{{\cal M}}} \bar{{\cal M}}_0
&=&
\bar{X}\times_{\bar{S}} ({\cal M} \times _{\bar{{\cal M}}} \bar{{\cal M}}_0 )
&=&
\bar{X}\times_{\bar{S}} {\cal M}_0 \\
&=&
\bar{X}\times_{\bar{S}} (S\times_S {\cal M}_0)
&=&
(\bar{X}\times_{\bar{S}} S) \times_S {\cal M}_0
&=&
X\times_S {\cal M}_0\, .
\end{array}
\]
Hence, with $\tilde{\sigma}$ as defined is the theorem, it remains to show that
\[
\begin{array}{ccc}
\,I\!\!\!\!C[\tilde{C}(Q)^{\scriptscriptstyle\vee} \cap (Z\!\!\!Z^n \times V^\ast_{Z\!\!\!Z})]
&\stackrel{\psi}{\longrightarrow}&
\,I\!\!\!\!C[\tilde{\sigma}^{\scriptscriptstyle\vee}\cap Z\!\!\!Z^{n+m+1}]\\
\bigcup \!| && \bigcup \!|\\
\,I\!\!\!\!C[C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}] &\longrightarrow\hspace{-1.5em}\longrightarrow&
\,I\!\!\!\!C[I\!\!N^{m+1}] = \,I\!\!\!\!C[T_0,\dots,T_m]
\end{array}
\]
is a tensor product diagram:
\begin{itemize}
\item[(i)]
$\tilde{\sigma}$ is the preimage of $I\!\!R^{m+1}_{\geq 0}\subseteq C(Q)$ via the
projection
$\tilde{C}(Q)\longrightarrow\hspace{-1.5em}\longrightarrow C(Q)$. In particular, $\tilde{\sigma}\subseteq \tilde{C}(Q)$
causes a surjective
map $\psi_{I\!\!R}: \tilde{C}(Q)^{\scriptscriptstyle\vee} \longrightarrow\hspace{-1.5em}\longrightarrow \tilde{\sigma}^{\scriptscriptstyle\vee}$.
\item[(ii)]
To show surjectivity at the level of lattices (i.e.\ surjectivity of $\psi$) we
start with an element
$[c,\underline{\eta}]\in \tilde{C}(Q)^{\scriptscriptstyle\vee}$ and suppose its image
$\psi_{I\!\!R}([c,\underline{\eta}])$ to be contained in
$\tilde{\sigma}^{\scriptscriptstyle\vee}\cap Z\!\!\!Z^{n+m+1}$.\\
In particular, $c\in Z\!\!\!Z^n$, and we obtain
$[c,\underline{\eta}(c)]\in \tilde{C}(Q)^{\scriptscriptstyle\vee}\cap(Z\!\!\!Z^n\times V_{Z\!\!\!Z}^\ast)$
implying that
\[
[0,\underline{\eta}-\underline{\eta}(c)] = [c,\underline{\eta}] - [c,\underline{\eta}(c)] \in
[0,C(Q)^{\scriptscriptstyle\vee}] \subseteq
\tilde{C}(Q)^{\scriptscriptstyle\vee}
\]
maps to an element of $I\!\!N^{m+1}\subseteq \tilde{\sigma}^{\scriptscriptstyle\vee}\cap
Z\!\!\!Z^{n+m+1}$.\\
On the other hand, surjectivity of $C(Q)^{\scriptscriptstyle\vee}\cap V_{Z\!\!\!Z}^\ast \longrightarrow\hspace{-1.5em}\longrightarrow
I\!\!N^{m+1}$ causes that this
element can be reached by some $[0,\underline{\mu}]\in [0,\, C(Q)^{\scriptscriptstyle\vee}\cap
V_{Z\!\!\!Z}^\ast]$,
too.\\
Hence, $[c,\underline{\eta}(c) + \underline{\mu}] \in
\tilde{C}(Q)^{\scriptscriptstyle\vee}\cap(Z\!\!\!Z^n\times V_{Z\!\!\!Z}^\ast)$
is a lattice-preimage of $\psi_{I\!\!R}([c,\underline{\eta}])$.
\item[(iii)]
The same methods applies for showing that $\mbox{Ker}\,\psi$ is generated by
the same elements
as $\mbox{Ker}\,(\,I\!\!\!\!C[C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}] \rightarrow \,I\!\!\!\!C[I\!\!N^{m+1}])$:\\
If $[c^1,\underline{\eta}^1],\, [c^2,\underline{\eta}^2] \in
\tilde{C}(Q)^{\scriptscriptstyle\vee}\cap(Z\!\!\!Z^n\times V_{Z\!\!\!Z}^\ast)$ have the
same image in $\,I\!\!\!\!C[\tilde{\sigma}^{\scriptscriptstyle\vee}\capZ\!\!\!Z^{n+m+1}]$ (i.e.\
$x^{[c^1,\underline{\eta}^1]} -
x^{[c^2,\underline{\eta}^2]} \in \mbox{Ker}\,\psi$), then $c^1=c^2$, and the elements
\[
\underline{\mu}^1:=\underline{\eta}^1 - \underline{\eta}(c^1),\; \underline{\mu}^2:=\underline{\eta}^2 -
\underline{\eta}(c^2) \in
C(Q)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}
\]
have the same image in $\,I\!\!\!\!C[I\!\!N^{m+1}]$.\\
In particular,
$x^{\underline{\mu}^1} - x^{\underline{\mu}^2} \in \mbox{Ker}\,(\,I\!\!\!\!C[C(Q)^{\scriptscriptstyle\vee}\cap
V^\ast_{Z\!\!\!Z}] \rightarrow
\,I\!\!\!\!C[I\!\!N^{m+1}])$, and
\[
x^{[c^1,\underline{\eta}^1]} - x^{[c^2,\underline{\eta}^2]} = x^{[c^1,\underline{\eta}(c^1)]} \cdot
\left( x^{\underline{\mu}^1} - x^{\underline{\mu}^2} \right)\, .
\vspace{2ex}
\]
\end{itemize}
\par
\neu{84}
{\bf Example:}
At the end of \zitat{2}{5} we presented two decompositions of $Q_6$ into a
Minkowski sum of
lattice summands. Let us describe now the restrictions of the versal family to
the associated
components of $\bar{{\cal M}}$:
\begin{itemize}
\item[(i)]
Putting the two triangles $R_0,R_1$ into two parallel planes contained in
$I\!\!R^3$ yields an
octahedron as the convex hull of the whole configuration. Then,
$\tilde{\sigma}$ is the
(4-dimensional) cone over this octahedron
\[
\tilde{\sigma}=\langle (0,0;1,0),\, (1,0;1,0),\,(1,1;1,0),\,(0,0;0,1),\,
(0,1;0,1),\,(1,1;0,1) \rangle\, .
\]
\item[(ii)]
Looking at the second decomposition, we have to put three line segments
$R_0,R_1,R_2$ on
three parallel 2-planes in general position inside the affine space $I\!\!R^4$.
Taking the convex
hull of this configuration yields a 4-dimensional polytope that is dual
to (triangle)$\times$(triangle).\\
Again, $\tilde{\sigma}$ is the (5-dimensional) cone over this polytope
\[
\begin{array}{r}
\tilde{\sigma}=\langle (0,0;1,0,0),\, (1,0;1,0,0),\,(0,0;0,1,0),\,(0,1;0,1,0),
\qquad\\
(0,0;0,0,1),\,(1,1;0,0,1) \rangle\, .
\end{array}
\]
\end{itemize}
The total spaces over the components arise as the toric varieties defined by
$\tilde{\sigma}$. In our
example, they equal the cones over $I\!\!P^1\timesI\!\!P^1\timesI\!\!P^1$ and
$I\!\!P^2\timesI\!\!P^2$,
respectively.\\
\par
\sect{Further examples}\label{s9}
\neu{91}
Three examples of toric Gorenstein singularities arise as cones over the Del
Pezzo surfaces
obtained by blowing up $(I\!\!P^2,{\cal O}(3))$ in one, two, or three points,
respectively. They
correspond to the following polygons:
\vspace{2ex}\\
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\vspace{2ex}\\
Let us discuss these three examples:
\begin{itemize}
\item[(iv)]
The edges equal
\[
d^1=(1,0),\; d^2=(1,2),\; d^3=(-2,-1),\; d^4=(0,-1)\,,
\]
and they imply the following equations of the versal base space as closed
subscheme of
$^{\displaystyle \,I\!\!\!\!C^4}\!\!/\!_{\displaystyle \,I\!\!\!\!C\cdot (1,1,1,1)}$:
\[
t_1+t_2=2t_3,\quad t_3+t_4=2t_2,\quad t_1^2+t_2^2=2t_3^2,\quad
t_3^2+t_4^2=2t_2^2\,.
\]
Using the two linear equations, only two coordinates $t:=t_1,\, \varepsilon:=t_1-t_3$
are sufficient.
(We get the $t_i$'s back by
$t_1=t,\,t_2=t-2\varepsilon,\,t_3=t-\varepsilon,\,t_4=t-3\varepsilon$). Then, the two
quadratic equations transform into $2\varepsilon^2=0$, i.e.\ the versal base space is
a fat point.\\
On the other hand, $Q_4$ does not allow any splitting into a Minkowski sum
involving lattice
summands only. This reflects the triviality of the underlying reduced space.
(Cf.\ \zitat{9}{2}.)
\item[(v)]
The polygon $Q_5$ allows the decomposition into the sum of a triangle and a
line segment. In
particular, the reduced base space of the versal deformation of $Y_5$ has to be
a line.\\
We compute the true base space:
$d^1=(1,1),\,d^2=(-1,1),\,d^3=(-1,0),\\
d^4=(0,-1),\,d^5=(1,-1)$ yield the equations
\[
t_1-t_3=t_2-t_5=t_4-t_1\quad \mbox{and} \quad
t_1^2-t_3^2=t_2^2-t_5^2=t_4^2-t_1^2\,.
\]
With $t:=t_1,\, s_1:= t_1-t_3,\, s_2:=t_1-t_2$ and $t_1=t,\,
t_2=t-s_2,\,t_3=t-s_1,\\
t_4=t+s_1,\, t_5=t-s_1-s_2$, they turn into
\[
s_1^2=2s_1s_2=0\,.
\]
\item[(vi)]
This example was spread in the paper.
\vspace{2ex}
\end{itemize}
\par
\neu{92}
We will use the polygon $Q_4:= \mbox{Conv}\,\{(0,0),\,(1,0),\,(2,2),\,(0,1)\}$
of \zitat{9}{1}(iv) for a more detailed demonstration how the theory
works. In particular, we will describe the versal family of $Y_4$ over
$\mbox{Spec}\,^{\displaystyle \,I\!\!\!\!C[\varepsilon]}\!/\!_{\displaystyle \varepsilon^2}$:
\begin{itemize}
\item[(1)]
The $(t,\varepsilon)$-coordinates of $V$ correspond to the linear map
\[
\left( \begin{array}{cc}
1&0\\1&-2\\1&-1\\1&-3
\end{array} \right) :
I\!\!R^2 \stackrel{\sim}{\longrightarrow} V \hookrightarrow I\!\!R^4\, .
\]
We obtain
\begin{eqnarray*}
C(Q_4) &=& \{(a,b)\in I\!\!R^2\,|\; a\geq 0,\; a-2b\geq 0,\; a-b\geq0,\; a-3b\geq
0\}\\
&=& \{(a,b)\in I\!\!R^2\,|\; a\geq 0,\; a-3b\geq 0\}\\
&=& \langle [1,0],\, [1,-3]\rangle ^{\scriptscriptstyle\vee} = \langle (0,-1),\,(3,1)\rangle
\subseteq I\!\!R^2\, ,
\end{eqnarray*}
and the map $I\!\!N^4 \rightarrow C(Q_4)^{\scriptscriptstyle\vee}\cap V^\ast_{Z\!\!\!Z}$ sends $e_1, e_2,
e_3, e_4$
to $[1,0]$, $[1,-2]$, $[1,-1]$, $[1,-3]$, respectively.
In particular, this map is surjective, i.e.\ $S_4=\bar{S}_4$ and
$X_4=\bar{X}_4$.
\item[(2)]
To compute the tautological cone $\tilde{C}(Q_4)$,
we need the Minkowski summands associated to the two fundamental generators of
$C(Q_4)$:
\[
(Q_4)_{(0,-1)} = \mbox{Conv}\{(0,0),\,(2,4),\,(0,3)\} ,\;
(Q_4)_{(3,1)} = \mbox{Conv}\{(0,0),\,(3,0),\,(4,2)\}\, .
\]
Hence,
\[
\begin{array}{r}
\tilde{C}(Q_4) = \langle (0,0;0,-1);\, (2,4;0,-1);\, (0,3;0,-1);\, (0,0;3,1);\,
\qquad\\
(3,0;3,1);\, (4,2;3,1) \rangle \,.
\end{array}
\]
\item[(3)]
Now, we have all information to obtain the versal family of \\
$Y_4=\mbox{Spec}\, \,I\!\!\!\!C[\mbox{Cone}(Q_4)^{\scriptscriptstyle\vee}\cap Z\!\!\!Z^3]$:
\begin{itemize}
\item
Restrict the family $\;\mbox{Spec}\,\,I\!\!\!\!C[\tilde{C}(Q_4)^{\scriptscriptstyle\vee}\cap
Z\!\!\!Z^4]\rightarrow
\mbox{Spec}\,\,I\!\!\!\!C[C(Q_4)^{\scriptscriptstyle\vee}\cap Z\!\!\!Z^2]\subseteq \,I\!\!\!\!C^4\;$ to the subspace
$\,I\!\!\!\!C^2 \simeq V_{\,I\!\!\!\!C}\subseteq \,I\!\!\!\!C^4$, i.e.\ use the $(t,\varepsilon)$-coordinates
instead of
$(t_1,t_2,t_3,t_4)$.
\item
Compose the result with the projection $\,I\!\!\!\!C^2 \longrightarrow\hspace{-1.5em}\longrightarrow \,I\!\!\!\!C^1\; ((t,\varepsilon)\mapsto
\varepsilon)$. That means,
we do no longer regard $t$ as a coordinate of the base space.
\item
Finally, we restrict our family to the closed subscheme defined by the equation
$\varepsilon^2=0$.
\end{itemize}
\item[(4)]
To obtain equations, we could either take a closer look to the family
constructed so far, or we can
proceed more directly as described in \zitat{4}{6}, \zitat{5}{3}, and
\zitat{5}{4}:
\begin{itemize}
\item
Computing the minimal generator set of the semigroup
$\mbox{Cone}\,(Q_4)^{\scriptscriptstyle\vee}\capZ\!\!\!Z^3$,
we get the elements $[c^v;\eta_0(c^v)]$:
\[
\begin{array}{llll}
[c^1;\eta_0^1] = [0,1;0] ,&
[c^2;\eta_0^2] = [-1,1;1] ,&
[c^3;\eta_0^3] = [-2,1;2], \\
\,\![c^4;\eta_0^4] = [-1,0;2] ,&
[c^5;\eta_0^5] = [0,-1;2], &
[c^6;\eta_0^6] = [1,-2;2] ,\\
\,\![c^7;\eta_0^7] = [1,-1;1], &
[c^8;\eta_0^8] = [1,0;0] .
\end{array}
\vspace{1ex}
\]
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\put(17.00,53.00){\makebox(0,0)[rb]{$z_3$}}
\put(27.00,37.00){\makebox(0,0)[rt]{$z_4$}}
\put(37.00,27.00){\makebox(0,0)[rt]{$z_5$}}
\put(53.00,17.00){\makebox(0,0)[lt]{$z_6$}}
\put(53.00,30.00){\makebox(0,0)[lc]{$z_7$}}
\put(53.00,43.00){\makebox(0,0)[lb]{$z_8$}}
\put(43.00,53.00){\makebox(0,0)[lb]{$z_1$}}
\put(30.00,53.00){\makebox(0,0)[cb]{$z_2$}}
\put(38.00,42.00){\makebox(0,0)[rb]{$t$}}
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\vspace{2ex}
\\
Together with $[0,0;1]$, they induce coordinates $z_1,\dots,z_8,t$ on $Y_4$,
i.e.\ we have
obtained an embedding $Y_4 \hookrightarrow \,I\!\!\!\!C^9$.\\
(The sum of the three components of the vectors are always 1. In the figure we
have
drawn the first two coordinates.)
\item
$Y_4\subseteq \,I\!\!\!\!C^9$ is defined by the following 20 equations:
\[
\begin{array}{llll}
t^2-z_4z_8,&
t^2-z_1z_5,&
t^2-z_2z_7,&
z_1t-z_2z_8,\\
z_2t-z_3z_8,&
z_2t-z_1z_4,&
z_3t-z_2z_4,&
z_4t-z_3z_7,\\
z_4t-z_2z_5,&
z_5t-z_4z_7,&
z_5t-z_2z_6,&
z_6t-z_5z_7,\\
z_7t-z_5z_8,&
z_7t-z_1z_6,&
z_8t-z_1z_7,&
z_1z_3-z_2^2,\\
z_3z_5-z_4^2,&
z_4z_6-z_5^2,&
z_6z_8-z_7^2,&
z_3z_6-z_4z_5.
\end{array}
\]
\item
Choosing paths from $(0,0)\in Q_4$ to the other vertices, we obtain the list
\[
\underline{\eta}^1=[0,0,0,0], \;
\underline{\eta}^2=[1,0,0,0], \;
\underline{\eta}^3=[2,0,0,0],
\]\[
\underline{\eta}^4=[1,1,0,0]=[0,0,2,0], \;
\underline{\eta}^5=[0,2,0,0]=[0,0,1,1],
\]\[
\underline{\eta}^6=[0,0,0,2], \;
\underline{\eta}^7=[0,0,0,1], \;
\underline{\eta}^8=[0,0,0,0].
\]
\item
Now, we can lift our 20 equations to the ring
$\,I\!\!\!\!C[Z_1,\dots,Z_8,t_1,\dots,t_4]$:
\[
\begin{array}{llll}
t_1t_2-Z_4Z_8,&
t_2^2-Z_1Z_5,&
t_1t_4-Z_2Z_7,&
Z_1t_1-Z_2Z_8,\\
Z_2t_1-Z_3Z_8,&
Z_2t_2-Z_1Z_4,&
Z_3t_2-Z_2Z_4,&
Z_4t_3-Z_3Z_7,\\
Z_4t_2-Z_2Z_5,&
Z_5t_3-Z_4Z_7,&
Z_5t_2-Z_2Z_6,&
Z_6t_3-Z_5Z_7,\\
Z_7t_3-Z_5Z_8,&
Z_7t_4-Z_1Z_6,&
Z_8t_4-Z_1Z_7,&
Z_1Z_3-Z_2^2,\\
Z_3Z_5-Z_4^2,&
Z_4Z_6-Z_5^2,&
Z_6Z_8-Z_7^2,&
Z_3Z_6-Z_4Z_5.
\end{array}
\]
\item
Finally, we restrict the family to the versal base space
by switching to the $(t,\varepsilon)$-coordinates
and obeying the equation $\varepsilon^2=0$. Moreover, $t$ is no longer a coordinate
of the base space:
\[
\begin{array}{llll}
t(t-2\varepsilon)-z_4z_8,&
t(t-4\varepsilon)-z_1z_5,&
t(t-3\varepsilon)-z_2z_7,\\
z_1t-z_2z_8,&
z_2t-z_3z_8,&
z_2(t-2\varepsilon)-z_1z_4,\\
z_3(t-2\varepsilon)-z_2z_4,&
z_4(t-\varepsilon)-z_3z_7,&
z_4(t-2\varepsilon)-z_2z_5,\\
z_5(t-\varepsilon)-z_4z_7,&
z_5(t-2\varepsilon)-z_2z_6,&
z_6(t-\varepsilon)-z_5z_7,\\
z_7(t-\varepsilon)-z_5z_8,&
z_7(t-3\varepsilon)-z_1z_6,&
z_8(t-3\varepsilon)-z_1z_7,\\
z_1z_3-z_2^2,&
z_3z_5-z_4^2,&
z_4z_6-z_5^2,\\
z_6z_8-z_7^2,&
z_3z_6-z_4z_5.
\end{array}
\]
\end{itemize}
\end{itemize}
\neu{93}
At last we want to present an example involving more than only quadratic
equations for the versal
base space. Let $Q_8$ be the ``regular'' lattice 8-gon, it is contained in two
strips of lattice
thickness 3.
\vspace{2ex}\\
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\vspace{2ex}\\
$Q_8$ admits three maximal Minkowski decompositions into a sum of lattice
summands:
\vspace{-6ex}
\begin{itemize}
\item[(i)]
$Q_8\;=$
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\item[(ii)]
$Q_8\;=$
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\item[(iii)]
$Q_8\;=$
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\vspace{6ex}
\pagebreak
\end{itemize}
The decompositions (i), (ii) and (i), (iii) are refinements of the coarser
decompositions
\vspace{-5ex}
\\
$Q_8\;=$
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and
$\quad Q_8\;=$
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+
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\vspace{5ex}
\\
respectively.\\
\par
These facts translate directly into the geometry of the reduced base space of
the versal deformation
of $Q_8$:
\begin{itemize}
\item
It is embedded in some affine space $\,I\!\!\!\!C^5$ and equals the union of a 3-plane
with two
2-planes (through $0\in \,I\!\!\!\!C^5$).
\item
The two 2-planes each have a common line with the 3-dimensional component.
However, they
intersect each other in $0\in \,I\!\!\!\!C^5$ only.
\vspace{2ex}
\end{itemize}
On the other hand, we can write down the equations of the true versal base
space (as a closed
subscheme of $^{\displaystyle \,I\!\!\!\!C^8}\!\!/\!_{\displaystyle \,I\!\!\!\!C\cdot
(1,\dots,1)}$):
\[
t_1^k +t_2^k+t_8^k=t_4^k+t_5^k+t_6^k,\quad
t_2^k+t_3^k+t_4^k=t_6^k+t_7^k+t_8^k\quad
(k=1,2,3)\,.
\]
\\
\par
|
1996-03-11T06:20:28 | 9603 | alg-geom/9603008 | en | https://arxiv.org/abs/alg-geom/9603008 | [
"alg-geom",
"math.AG"
] | alg-geom/9603008 | Dan Edidin | Dan Edidin, William Graham | Equivariant intersection theory | LaTex. 49pages. This paper subsumes our previous preprint
"Equivariant Chow groups and the Bott residue formula", alg-geom/9508001.
Email for William Graham is [email protected]. This revision corrects
some minor errors and adds some references | null | null | null | null | In this paper we develop an equivariant intersection theory for actions of
algebraic groups on algebraic schemes. The theory is based on our construction
of equivariant Chow groups. They are algebraic analogues of equivariant
cohomology groups which satsify the formal properties of ordinary Chow groups.
In addition, they enjoy many of the properties of equivariant cohomology. The
principal results are: (1) We prove the existence of canonical intersection
products on the Chow groups of geometric quotients of smooth varieties- even
when the stabilizers of geometric points are non-reduced. (2) We construct a
Todd class map from equivariant $K$-theory of coherent sheaves to a completion
of equivariant Chow groups, and prove that a completion of equivariant
$K$-theory is isomorphic to the completion of equivariant Chow groups. (3) We
prove a localization theorem for torus actions and use it to give a
characteristic free proof of the Bott residue formula for actions of tori on
complete smooth varieties.
| [
{
"version": "v1",
"created": "Thu, 7 Mar 1996 17:38:59 GMT"
},
{
"version": "v2",
"created": "Thu, 7 Mar 1996 18:31:12 GMT"
},
{
"version": "v3",
"created": "Mon, 11 Mar 1996 02:51:51 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Edidin",
"Dan",
""
],
[
"Graham",
"William",
""
]
] | alg-geom | \section{Introduction}
The purpose of this paper is to develop an
equivariant intersection theory for actions
of linear algebraic groups on algebraic schemes. The
theory is based on our construction of equivariant Chow groups. They are
algebraic analogues of equivariant cohomology groups which satisfy
all the functorial properties of ordinary Chow groups. In addition,
they enjoy many of the properties of
equivariant cohomology.
The principal results of this paper are:\\
(1) If a group $G$ acts with finite stabilizers on a
scheme $X$, then rational equivariant Chow groups can be identified
with the rational Chow groups of a quotient. As a result, we show
that the rational Chow groups of quotients of smooth varieties by
group actions have a canonical ring structure. This extends and
simplifies previous work of Mumford (\cite{Mu}), Gillet (\cite{Gi})
and Vistoli (\cite{Vi}). In addition the integral Chow groups are an
invariant of the quotient stack $[X/G]$, so we can associate an {\it
integral} Chow ring to smooth quotient stacks.\\
(2) There is a Riemann-Roch isomorphism between a completion
of equivariant $K$-theory of coherent sheaves and a completion
of equivariant Chow groups. This extends the Riemann-Roch theorem
of Baum, Fulton and MacPherson to the equivariant case.\\
(3) There is a localization for torus actions relating the the
equivariant Chow groups of a scheme to the Chow groups of the fixed
locus. Such a theorem is a hallmark of other equivariant theories
such as cohomology and $K$-theory. The localization theorems in
equivariant cohomology and $K$-theory imply residue formulas such as
Bott's ( \cite{B-V}, \cite{A-B}, \cite{I-N}),
which can now be proved using intersection theory.
\medskip
Previous work on equivariant intersection theory (\cite{Br},
\cite{Gi}, \cite{Vi}) defined equivariant Chow groups using invariant
cycles on $X$. The definition we give of equivariant Chow groups, in
contrast, is modeled on Borel's definition of equivariant cohomology.
Borel's insight was to replace the original topological space $X$ by a
homotopic space $X \times EG$, where $EG$ is a contractible space on
which $G$ acts freely. Since $G$ acts freely, there is a nice
quotient $X_G$ of $X \times EG$ by $G$ and equivariant cohomology is
defined as the cohomology of $X_G$. To define equivariant Chow groups
one needs an appropriate algebraic replacement for $EG$. This was
supplied by Totaro \cite{To}, who used finite dimensional
representations to approximate the infinite dimensional space $EG$. In
particular if $V$ is a representation of $G$, let $U$ denote an open
set on which $G$ acts freely and has a
a quotient $U \rightarrow U/G$ which is a principal bundle
For any linear algebraic group, the
representation can be chosen so that $V-U$ has arbitrarily large
codimension. If $X$ is a $G$-scheme then, under mild hypotheses on
$G$ or $X$ (see below), $X \times U$ has a quotient $X \times^G U$ so
that $X \times U\rightarrow X \times^G U$ is a principal $G$-bundle.
The group $A_{\dim V +i - \dim G}(X \times^G U)$ is independent of $V$
as long as the codimension of $V-U$ is sufficiently large. This
defines the $i$-th equivariant Chow group $A_i^G(X)$.
Because $X \times U \rightarrow X \times^G U$ is
a principal $G$-bundle, cycles on $X \times^G U$ exactly
correspond to $G$-invariant cycles on $X \times U$. Since
we only consider cycles of codimension smaller
than the dimension of $X \times (V-U)$, we may in fact
view these as $G$-invariant cycles on $X \times V$.
In other words, instead of considering only $G$-invariant cycles on
$X$ we consider $G$-invariant cycles on $X \times V$ for any
sufficiently big representation of $G$. By enlarging the class
of cycles we allow, we obtain a theory with many good properties.
By construction, the equivariant Chow groups $A_*^G(X)$ inherit most
of the properties of ordinary Chow groups. In particular if $X$ is
smooth, then there is an intersection product on the equivariant Chow
groups $A_*^G(X)$, no matter how badly $G$ acts on $X$. When $G$ acts
properly and a quotient $X/G$ exists we prove (Theorem \ref{quotient})
that $A_*^G(X)_{{\bf Q}}$ = $A_*(X/G)_{{\bf Q}}$. As a result, this proves that
if $G$ acts properly on a smooth variety $X$, then the rational Chow
groups of a quotient $X/G$ have a canonical intersection product
(Corollary \ref{moduli}). This extends the results of Vistoli, who
proved such a theorem when $G$ acts with finite, reduced stabilizers.
This theorem should be useful for doing intersection theory on moduli
spaces of objects which posess infinitesimal automorphisms. It can
also be used to do intersection theory on toric varieties in arbitrary
characteristic. Furthermore, by avoiding the use of algebraic stacks,
our proof is much simpler.
Another interesting aspect of the theory is that the groups
$A_*^G(X)$ are actually an invariant of the {\em quotient stack}
$[X/G]$ (Proposition \ref{qstacks}).
Thus if $X$ is smooth, then there is an integral intersection
ring associated to the quotient stack $[X/G]$. When $[X/G]$
is Deligne-Mumford (i.e. $G$ acts with finite, reduced stabilizers)
then our ring tensored with ${\bf Q}$ agrees with the rings of Gillet
and Vistoli. It would be interesting to compute the torsion
in the equivariant Chow ring in examples of moduli stacks
such as curves of low genus.
Our results on quotient stacks also suggest that there should
be integral Chow rings associated to arbitrary smooth stacks.
Motivated by the equivariant Chow ring, we expect that this ring
would have torsion in arbitrarily high degree. However, we do not
know how to construct such a ring in general.
The connection between equivariant $K$-theory and equivariant Chow groups
is given by our
equivariant Riemann-Roch theorem (Theorem \ref{rockandroll}).
We prove that there is an isomorphism $\tau_X: \widehat{K_0^{'G}(X)}_{{\bf Q}}
\rightarrow \widehat{A_*^G(X)}_{{\bf Q}}$ between the completion of
$K$-theory along the augmentation ideal of the representation ring $R(G)$,
and the completion of the $A_*^G(X)$ along the augmentation
ideal of the equivariant Chow ring of a point.
Along the way we prove a theorem (Theorem \ref{completions})
which shows that the completion of $K_i^{'G}(X)$ along
the augmentation ideal of $R(G)$ is the same as the completion
along the augmentation ideal of $K^0_G(X)$. This result
is related to results of \cite{CEPT} and answers a special
case of a conjecture of \cite{kock}.
In the last part of the paper we prove a localization theorem
for torus actions. If a torus $T$ acts on $X$ with fixed
locus $X^T \subset X$, then $A_*^T(X) \otimes_{R_T} (R_T^+)^{-1} =
A_*(X^T) \otimes (R_T^+)^{-1}R_T$, where $R_T^+$ denotes the multiplicative
set of homogeneous elements of positive degree in $R_T =
Sym(\hat{T})$.
Finally, using the localization theorem, we prove the Bott residue
formula for actions of (split) tori on smooth complete varieties. This
formula has been recently applied in enumerative geometry (cf. \cite{E-S})
so we include an intersection theoretic proof.
Our line of argument follows that of \cite{A-B} using equivariant
intersection theory in place of equivariant cohomology.
(Note that Iversen and Nielsen \cite{I-N} gave an algebraic proof of
this formula - for smooth projective varieties - using equivariant $K$-theory.
Also, using techniques of algebraic deRahm homology,
H\"ubl and Yekutieli \cite{H-Y}
proved a version - in characteristic 0 - for the action
of an algebraic vector field with isolated fixed points.)
In Section \ref{mixed} we discuss extensions of the theory to
group schemes over a regular base scheme, and
in the appendix we prove a number of technical results about
group actions and quotients in arbitrary characteristic.
{\bf Acknowledgements:} We thank William Fulton,
Rahul Pandharipande and Angelo Vistoli for advice and encouragement.
We also benefitted from discussions with Burt Totaro and Amnon Yekutieli.
Thanks also to Holger Kley for suggesting the inclusion
of the cycle map to equivariant cohomology.
\section{Definitions and basic properties}
\subsection{Conventions and Notation}
Except in Section
\ref{mixed}, all schemes are assumed to be of finite type over a
field of arbitrary characteristic.
A variety is a reduced and irreducible scheme. An algebraic
group is always assumed to be linear.
If an algebraic group $G$ acts on a scheme $X$ then the action is said to
be {\it closed} if the orbits of geometric points are closed in $X$.
It is {\it proper} if the action map $G \times X \rightarrow X \times
X$ is proper. Finally, we say that it is {\it free} if the action map
is a closed embedding. By (\cite[Prop. 0.9]{GIT}) if the action is
free and a geometric quotient scheme $X/G$ exists, then $X$ is
a principal $G$ bundle over $X/G$.
Throughout the paper we will assume that at least one of the following
hypotheses on $X$ or $G$ is satisfied.\\
(1) $X_{red}$ is quasi-projective and the action is linearized
with respect to some projective embedding.\\
(2) $G$ is connected and $X_{red}$ equivariantly embeds as
a closed subscheme in a normal variety.\\
(3) $G$ is special in the sense of \cite{Sem-Chev}; i.e. all
principal $G$-bundles are locally trivial in the Zariski topology.
(Examples of special groups are tori, solvable and unipotent
groups as well as $GL(n)$, $SL(n)$, and $Sp(2n)$. Finite groups
are not special, nor or the orthogonal groups $SO(2n)$ and $SO(2n+1)$.)
\medskip
For simplicity of exposition, we will usually assume that $X$ is
equidimensional.
\subsection{Equivariant Chow groups} Let $X$ be an $n$-dimensional
scheme.
We will denote the $i$-th equivariant Chow
of $X$ group by $A^G_i(X)$. It is defined as follows.
Let $G$ be a $g$-dimensional algebraic group.
Choose an $l$-dimensional representation $V$ of $G$ such that $V$
has an open set $U$ on which $G$ acts freely and whose complement has
codimension more than $n-i$. Assume that a quotient
$U \rightarrow U/G$ (necessarily a principal bundle) exists.
(Such representations exist for any group; see Lemma \ref{q.exist}
of the Appendix.)
The principal bundle $U \rightarrow U/G$ is Totaro's
finite
dimensional approximation of the classifying bundle $EG \rightarrow BG$
(see \cite{To} and \cite{E-G}).
The diagonal
action on $X \times U$ is also free, and since one hypothesis (1)-(3)
holds, there is a quotient
$X_{red} \times U \rightarrow (X_{red} \times U)/G$
which is a principal $G$ bundle\footnote{Without any hypothesis on $X$
or $G$, we only know that the quotient exists as an algebraic space.}
(Prop \ref{inap}).
We will usually denote this quotient
by $(X _{red}\times^G U)$ or $X_G$.
\begin{def-prop} \label{keydef}
Set $A_i^G(X)$ (the $i$-th equivariant Chow group) to be
$A_{i+l-g}(X_G)$, where $A_*$ is the usual Chow group.
This group is independent of the representation
as long as $V- U$ has sufficiently high codimension.
\end{def-prop}
{\bf Remark.} In the sequel, the notation $U \subset V$ will
refer to an open set in a representation on which the action is
free. Because we are working with Chow groups, we
will, when no confusion can arise, abuse notation and act as if all
schemes are reduced.
\medskip
Proof of Definition-Proposition \ref{keydef}.
We will use Bogomolov's double fibration
argument.
Let $V_1$ be another representation of dimension $k$ such
that there is an open $U_1$ with a principal bundle quotient
$U_1 \rightarrow U_1/G$ and whose complement has
codimension at least $n-i$.
Let $G$ act diagonally on $V \oplus V_1$.
Then $V \oplus V_1$ contains an open set
$W$ which has
a principal bundle quotient $W/G$ and contains both
$U \oplus V_1$ and $V \oplus U_1$.
Thus, $A_{i+k+l-g}(X \times^G W) =
A_{i+k+l-g}(X \times^G (U \oplus V_1))$ since
$(X \times^G W)-(X \times^G (U \oplus V_1)$
has dimension smaller than $i+k+l-g$. On the
other hand, the projection $V \oplus V_1 \rightarrow V$
makes $X \times^G (U \oplus V_1)$ a vector
bundle over $X \times^G U$ with fiber $V_1$ and
structure group $G$. Thus, $A_{i+k+l-g}(X \times^G (U \oplus V_1))
= A_{i+l-g}(X \times^G U)$. Likewise,
$A_{i+k+l-g}(X \times^G W) =A_{i+k-g}(X \times^G U_1)$,
as desired.
$\Box$
\paragraph{Example} For the classical groups, the representations
and subsets can be constructed explicitly.
In the simplest case, if $G= {\bf G}_m$
then we can take $V$ to an $l$-dimensional
representation with weights one,
$U = V - \{0\}$, and $U/G = {\bf P}^{l-1}$. If $G = GL_n$,
take $V$ to be the vector space of $n \times p$
matrices ($p>n$), with $GL_n$ acting by left multiplication, and let
$U$ be the subset of matrices of maximal rank. Then $U/G$ is the
Grassmannian $Gr(n,p)$.
\paragraph{Remarks}
Now that we have defined equivariant Chow groups,
we will use the notation $X_G$ to mean a mixed quotient $X \times^G U$
for any representation $V$ of $G$. If we
write $A_{i+l-g}(X_G)$ then $V-U$ is assumed to have codimension
more than $n-i$ in $V$. (As above $n=\mbox{dim }X$, $l=\mbox{dim }V$
and $g =\mbox{dim }G$.)
If $Y \subset X$ is an $m$-dimensional $G$-invariant subvariety, then
it has a $G$-equivariant fundamental class $[Y]_G \in A_m^G(X)$.
However, unlike ordinary Chow groups, $A_i^G(X)$
can be non-zero for any $i \leq n$, including negative $i$.
The projection $X \times U \rightarrow U$ induces
a map $X_G \rightarrow U$ with fiber $X$. Restriction
to a fiber gives a map $i^*:A_*^G(X) \rightarrow A_*(X)$
from equivariant Chow groups to ordinary Chow groups. The
map is independent of the choice of fiber because any two
points of $U/G$ are rationally equivalent.
For any $G$-invariant subvariety $Y \subset X$,
$i^*([Y]_G) =[Y]$.
\subsection{Functorial properties}
In this section all maps $f: X \rightarrow Y$ are assumed to
be $G$-equivariant.
If $f: X \rightarrow Y$ is proper, then by descent,
the induced map $f_G: X_G \rightarrow Y_G$
is also proper. Likewise, if $f:X \rightarrow Y$ is flat of
relative dimension $k$ then $f_G:X_G \rightarrow Y_G$ is flat
of dimension $k$.
\begin{defn}
Define proper pushforward $f_*:A_i^G(X) \rightarrow A_i^G(Y)$,
and flat pullback $f^{*}:A_i^G(Y) \rightarrow A_{i-k}^G(X)$
as $f_{G*}:A_{i+l-g}(X_G) \rightarrow A_{i+l-g}(Y_G)$ and
$f_G^*:A_{i+l-g}(Y_G) \rightarrow A_{i+l-g-k}(X_G)$ respectively.
\end{defn}
\medskip
If $f: X \rightarrow Y$ is smooth, then
$f: X_G \rightarrow Y_G$ is also
smooth. Furthermore, if $f: X \rightarrow
Y$ is a regular embedding, then
$f \times id: X \times U \rightarrow Y \times U$
is a regular embedding. In the cartesian diagram
$$\begin{array}{ccc}
X \times U & \rightarrow & Y \times U\\
\downarrow & & \downarrow \\
X_G & \rightarrow & Y_G
\end{array}$$
the vertical arrows are flat and surjective so
by \cite[Prop. IV 3.5]{F-L} the map
$X_G \rightarrow Y_G$ is also a regular embedding.
In particular, if $f: X \rightarrow Y$ is
an l.c.i morphism in
the sense of \cite[Section 6.6]{Fulton},
then $f_G:X_G \rightarrow Y_G$ is also
l.c.i.
\begin{defn} \label{lci}
If $f: X \rightarrow Y$ is l.c.i. of codimension $d$
then define $f^{*}:A_i^G(Y) \rightarrow A_{i-d}^G(X)$
as $f^*_G: A_{i+l-g}(Y_G) \rightarrow A_{i+l-g-d}(X_G)$.
\end{defn}
\begin{prop}
The maps $f_*$ and $f^{*}$ above
are well defined.
\end{prop}
Proof: We will use the double fibration argument.
Let $V_1$ be another representation. Then we have
a cartesian diagram
$$\begin{array}{ccc}
X \times^G (U \oplus V_1) & \rightarrow &Y \times^G (U \oplus V_1)\\
\downarrow & & \downarrow\\
X \times^G U & \rightarrow & Y \times^G U
\end{array}$$
The vertical maps are flat, and their pullbacks
are the isomorphisms which allowed us to define
$A_i^G$. Since flat pullback is compatible with proper
pushfoward, the equivariant pushforward $f_*$ is well defined. Likewise
the flat pullback is compatible with flat and l.c.i
pullback, so $f^{*}$ is also well defined.
$\Box$
\medskip
\subsection{Chern classes}
Let $X$ be a scheme with a $G$ action, and let
$E$ be an equivariant vector bundle. For each $i$, $j$
define a map $c_j^G(E):A_i^G(X) \rightarrow A_{i-j}^G(X)$
as follows. Let $V$ be an $l$-dimensional representation such that
$V-U$ has high codimension. By hypothesis, there is
a principal bundle $X \times U \rightarrow X_G$. Thus
by \cite[Prop. 7.1]{GIT} there is a quotient $E_G$ of $E \times U$.
\begin{lemma}
$E_G \rightarrow X_G$ is a vector bundle.
\end{lemma}
Proof.
The bundle $E_G \rightarrow X_G$ is an affine bundle which locally
trivial in the \'etale topology since it becomes locally trivial after the
smooth
base change $X \times U \rightarrow X_G$. Also, the transition
functions are affine since they are affine when pulled back to $X \times U$.
Hence, by descent, $E_G \rightarrow
X_G$
is locally trivial in the Zariski topology and has affine transition
functions; i.e., $E_G$ is a vector bundle over $X_G$.
$\Box$
\medskip
Identify $A_i^G(X)$ and $A_{i-j}^G(X)$
with $A_{i+l-g}(X_G)$ and $A_{i-j+l-g}(X_G)$ respectively.
\begin{def-prop}
Define equivariant Chern classes $c_j^G(E):A_i^G(X) \rightarrow A_{i-j}^G(X)$
by $c_j^G(E)\cap \alpha= c_j(E_G) \cap \alpha \in A_{i-j+l-g}(X_G)$.
This definition does not depend on the choice of representation.
\end{def-prop}
Proof: Let $V_1$ be another representation. Then
the pullback of $E \times^G U$ to $X \times^G (U \oplus V_1)$
is isomorphic to the quotient $E \times^G (U \oplus V_1)$.
$\Box$
\medskip
Given the above propositions, equivariant Chow groups
satisfy all the formal properties of ordinary Chow groups
(\cite[Chapters 1-6]{Fulton}). In particular, if $X$ is
smooth, there is an intersection product on the
the equivariant Chow groups $A_*^G(X)$ which makes $\oplus A_*^G(X)$
into a graded ring.
\subsection{Operational Chow groups}
Define equivariant operational Chow groups
$A^i_G(X)$ as operations $c(Y \rightarrow X): A_*^G(Y) \rightarrow
A_{*-i}^G(Y)$
for every $G$-map $Y \rightarrow X$.
As for ordinary operational Chow groups (\cite[Chapter 17]{Fulton}),
these operations should be
compatible with the operations on equivariant Chow groups
defined above (pullback
for l.c.i. morphisms, proper pushforward, etc.) From this definition
it is clear that for any $X$, $A^*_G(X)$ has a ring structure.
The ring $A^*_G(X)$ is positively graded, and $A^i_G(X)$ can
be non-zero for any $i \geq 0$.
Note
that by construction, the equivariant Chern classes
defined above are elements of the equivariant operational
Chow ring.
\begin{prop} \label{opsmooth}
If $X$ is smooth of dimension $n$,
then $A^i_G(X) \simeq A_{n-i}^G(X)$.
\end{prop}
Proof. Define a map $A^i_G(X) \rightarrow A_{n-i}^G(X)$
by the formula $c \mapsto c \cap [X]_G$.
Define a map
$A_{n-i}^G(X)\rightarrow A^i_G(X)$, $\alpha \mapsto c_{\alpha}$
as follows. If $Y \stackrel{f} \rightarrow X$ is a
$G$-map, then since $X$ is smooth, the graph $\gamma_f: Y \rightarrow
Y \times X$ is a $G$-map which is a regular embedding. If $\beta \in
A^G_*(Y)$ set $c_\alpha \cap \beta= \gamma_f^*(\beta \times
\alpha)$ (note that the cartesian product of equivariant classes is
well defined).
\medskip
Claim (cf. \cite[Proposition 17.3.1]{Fulton}): $\beta \times
(c \cap [X]_G)
= c \cap (\beta \times [X_G])$.
\medskip
Given the claim, the formal arguments of
\cite[Proposition 17.4.2]{Fulton}
show that the two maps are inverses.
Proof of Claim: The equivariant class $\beta$ is represented
by a cycle on some $Y_G$, which we can assume to be the fundamental
class of a subvariety $W \subset Y_G$.
Let $\tilde{W}$ be the
inverse image of $W$ in $Y \times U$.
Then $\beta$ pulls back to $[\tilde{W}]_G$
by the equivariant projection map $Y \times U \rightarrow Y$.
By requiring $V - U$ to have sufficiently high codimension, we may
assume that the pullback on Chow groups is an
isomorphism
in the appropriate degrees. Replacing $Y$ by $Y_G$, we
may assume $\beta = [\tilde W]_G$. Since $\tilde W$ is $G$-invariant,
the projection $p: \tilde{W} \times X \rightarrow X$ is equivariant.
Thus,
$$(c \cap [X]_G) \times [\tilde{W}]_G = p^*(c \cap [X]_G)=
c \cap p^*([X]_G) = c \cap ([X_G] \times [\tilde{W}]_G).$$
$\Box$
\medskip
Let $V$ be a representation such that
$V- U$ has codimension more than $k$, and set $X_G =
X \times^G U$. In the remainder of the subsection we will
discuss the relation between $A^k_G(X)$ and $A^k(X_G)$ (ordinary
operational Chow groups).
Recall \cite[Definition 18.3]{Fulton} that an envelope
$\pi:\tilde{X} \rightarrow X$ is a proper map such that for any
subvariety $W \subset X$ there is a subvariety $\tilde{W}$ mapping
birationally to $W$ via $\pi$. In the case of group actions, we will
say that $\pi: \tilde{X} \rightarrow X$ is an {\it equivariant} Chow
envelope, if $\pi$ is $G$-equivariant, and if we can take $\tilde{V}$
to be $G$-invariant for $G$-invariant $V$. If there is an open set $X^0
\subset X$ over which $\pi$ is an isomorphism, then we say $\pi:
\tilde{X} \rightarrow X$ is a {\it birational} envelope.
\begin{lemma} If $\pi: \tilde{X} \rightarrow X$ is an
equivariant (birational) envelope, then
$p: \tilde{X}_G \rightarrow X_G$ is a (birational)
envelope ($\tilde{X}_G$ and $X_G$ are constructed
with respect to a fixed representation $V$). Furthermore,
if $X^0$ is the open set over which $\pi$ is an isomorphism
(necessarily $G$-invariant), then $p$ is an isomorphism
over $X^0_G = X^0 \times^G U$.
\end{lemma}
Proof: Fulton \cite[Lemma 18.3]{Fulton} proves that
the base extension of an envelope is an envelope.
Thus $\tilde{X} \times U \stackrel{\pi \times id}\rightarrow X \times U$
is an envelope. Since the projection $X \times U \rightarrow X$
is equivariant, this envelope is also equivariant.
If $W \subset X_G$ is a subvariety, let $W'$ be its inverse image
(via the quotient map) in $X \times U$. Let $\tilde{W'}$ be
an invariant subvariety of $\tilde{X} \times U$ mapping
birationally to $W'$. Since $G$ acts freely on $\tilde{X} \times U$
it acts freely on $\tilde{W'}$, and $\tilde{W} = \tilde{W'}/G$
is a subvariety of $\tilde{X}_G$ mapping birationally to $W$.
This shows that $\tilde{X}_G \rightarrow X_G$ is an envelope;
it is clear that the induced map $\tilde{X}_G \rightarrow
\tilde{X}$ is an isomorphism over $X_0^G$. $\Box$
\medskip
Suppose $\tilde{X} \stackrel{\pi}\rightarrow X$
is an equivariant envelope which is
an isomorphism over $U$. Let $\{S_i\}$ be the irreducible components
of $S= X -X^0$, and let $E_i = \pi^{-1}(S_i)$. Then $\{S_{i G}\}$
are the irreducible components of $X_G - X^0_G$ and
$E_{i G} = p^{-1}(S_{i G})$.
\begin{thm}
If $X$ has an equivariant non-singular envelope
$\pi: \tilde{X} \rightarrow X$ such that there is an
open $X^0 \subset X$ over which $\pi$ is an isomorphism, then
$A^k_G(X) = A^k(X_G)$.
\end{thm}
Proof: If $\pi: \tilde{X} \rightarrow X$ is an
equivariant non-singular envelope, then
$p: \tilde{X}_G \rightarrow X_G$
is also an envelope and $\tilde{X}_G$ is non-singular.
Thus, by \cite[Lemma 1.2]{Kimura}
$p^*:A^*(X_G) \rightarrow A^*(\tilde{X}_G)$ is injective.
The image of $p^*$ is described inductively
in \cite[Theorem 3.1]{Kimura}. A class
$\tilde{c} \in A^*(\tilde{X}_G)$ equals
$p^*c$ if and only if for each
$E_{i G}$ , $\tilde{c}_{| E_{i G}} = p^*c_i$
where $c_i \in A^*(E_i)$.
This description follows from formal properties of operational
Chow groups, and the exact sequence \cite[Theorem 2.3]{Kimura}
$$A^*(X_G) \stackrel{p}\rightarrow A^*(\tilde{X}_G)
\stackrel{p_1^* - p_2^*} \rightarrow A^*(\tilde{X}_G \times_{X_G}
\tilde{X}_G)$$ where $p_1$ and $p_2$ are the two projections
from $\tilde{X}_G \times_{X_G} \tilde{X}_G$.
By Proposition \ref{opsmooth} above, we know that
$A^k_G(\tilde{X}) = A^k(\tilde{X}_G)$.
We will show that $A^k_G(X)$ and $A^k(X_G)$ have the same image
in $A^k(\tilde{X}_G)$.
By Noetherian induction we may assume that
$A^k(S_i) = A^k((S_{i})_G)$. To prove the theorem, it suffices
to show that there is also an exact sequence of equivariant
operational Chow groups
$$0 \rightarrow A^*_G(X) \stackrel{\pi^*}\rightarrow A^*_G(\tilde{X})
\stackrel{p_{1}^* -p_{2}^*}\rightarrow A^*(\tilde{X} \times_X
\tilde{X})$$
This can be checked by working with the action of $A^*_G(X)$
on a fixed Chow group $A_{i}(X_G)$ and arguing as in Kimura's
paper.
$\Box$
\begin{cor}
If equivariant resolution of singularities holds (in particular
if the characteristic is 0), and $V-U$ has codimension more than $k$,
then
$A^k_G(X) = A^k(X_G).$
\end{cor}
Proof (c.f. \cite[Remark 3.2]{Kimura}).
We must show the existence of an equivariant envelope
$\pi:\tilde{X} \rightarrow X$. By equivariant
resolution of singularities, there is a resolution
$\pi_1:\tilde{X_1} \rightarrow X$ such
that $\pi_1$ is an isomorphism outside
some invariant subscheme $S \subset X$. By Noetherian
induction, we may assume that we have constructed an
equivariant envelope $\tilde{S} \rightarrow S$. Now
set $\tilde{X} = \tilde{X_1} \cup \tilde{S}$.
$\Box$
\subsection{Equivariant higher Chow groups}
In this section assume that $X$ is quasi-projective.
Bloch (\cite{Bl}) defined
higher Chow groups $A^i(X,m)$ as $H_m(Z^i(X,\cdot))$
where $Z^i(X,\cdot)$ is a complex whose $k$-th term
is the group of cycles of codimension $i$ in $X \times \Delta^k$
which intersect the faces properly. Since we prefer
to think in terms of dimension rather than codimension
we will define $A_p(X,m)$ as $H_m(Z_p(X,\cdot))$,
where $Z_p(X,k)$ is
the group of cycles of dimension $p+k$ in $X \times \Delta^k$
intersecting the faces properly. When $X$ is equidimensional
of dimension $n$, then $A_p(X,m) = A^{n-p}(X,m)$.
If $Y \subset X$ is closed, there is a localization long exact sequence.
The advantage of indexing by dimension rather than codimension is that
the sequence exists without assuming that $Y$ is equidimensional.
\begin{lemma}
Let $X$ be equidimensional, and let $Y \subset X$ be closed,
then there is a long exact sequence of higher Chow groups
$$\ldots \rightarrow A_p(Y,k) \rightarrow A_p(X,k) \rightarrow
A_p(X-Y,k) \rightarrow \\
\ldots \rightarrow A_p(Y) \rightarrow A_p(X) \rightarrow A_p(X-Y)
\rightarrow 0$$
(there is no requirement that $Y$ be equidimensional).
\end{lemma}
Proof. This is a simple consequence of
the localization theorem of \cite{Bl}.
By induction it suffices to prove the lemma when $Y$
is the union of two irreducible components $Y_1$, $Y_2$.
In particular, we will prove that the complexes
$Z_p(X-(Y_1 \cup Y_2),\cdot)$ and $\frac{Z_p(X,\cdot)}{Z_p(Y_1 \cup
Y_2,\cdot)}$ are quasi-isomorphic.
By the original localization theorem, $Z_p(X-(Y_1 \cup Y_2),\cdot)
\simeq \frac{Z_p(X-Y_1,\cdot)}{Z_p(Y_2-(Y_1 \cap Y_2),\cdot)}$ and
$Z_p(X-Y_1, \cdot) \simeq \frac{Z_p(X,\cdot)}{Z_p(Y_1)}$.
By induction on dimension, we can assume that the lemma holds for
schemes of smaller dimension, so
$Z_p((Y_2 - (Y_1 \cap Y_2),\cdot) \simeq \frac{Z_p(Y_2,\cdot)}{Z_p(Y_1 \cap
Y_2)}$. Finally
note that
$\frac{Z_p(Y_2),\cdot}{Z_p(Y_1 \cap Y_2)}=\frac{Z_p(Y_1 \cup
Y_2,\cdot)}{Z_p(Y_1,\cdot)}$.
Combining all our quasi-isomorphisms we have
$$Z_p(X-(Y_1 \cup Y_2), \cdot) \simeq \frac{\frac{Z_p(X,\cdot)}
{Z_p(Y_1,\cdot)}}{\frac{Z_p(Y_1 \cup Y_2),\cdot}{Z_p(Y_1,\cdot)}}
\simeq \frac{Z_p(X,\cdot)}{Z_p(Y_1 \cup Y_2,\cdot)}$$
as desired.
$\Box$
\medskip
If $X$ is quasi-projective with a $G$-action, we
can define equivariant higher Chow groups
$A_{i}^G(X,m)$ as $A_{i+l-g}(X_G,m)$, where
$X_G$ is formed from an $l$-dimensional representation
$V$ such that $V-U$ has high codimension.
The homotopy lemma for higher Chow groups shows
that $A_{i}^G(X,m)$ is well defined.
Our reason for constructing equivariant higher Chow groups
is to obtain a long exact sequence for a
$G$-invariant subscheme $Y$ of a quasi-projective scheme $X$
with a $G$-action.
\begin{prop} Let $X$ be an equidimensional $G$-scheme, and
let $Y \subset X$ be an invariant subscheme.
There is a long exact sequence of higher equivariant Chow groups
$$\ldots \rightarrow A_p^G(Y,k) \rightarrow A_p^G(X,k) \rightarrow
A_p^G(X-Y,k) \rightarrow \\
\ldots \rightarrow A_p^G(Y) \rightarrow A_p^G(X) \rightarrow A_p^G(X-Y)
\rightarrow 0. $$
$\Box$
\end{prop}
\subsection{Cycle Maps}
If $X$ is a complex algebraic variety with the action of
a complex algebraic group, then we can define
equivariant Borel-Moore homology $H_{BM, i}^G(X)$
as $H_{BM,i+2l-2g}(X_G)$ for $X_G = X \times^G U$.
As for Chow groups, the definition is independent
of the representation, as long as $V -U$ has sufficiently
high codimension, and we obtain a cycle map
$$cl:A^G_i(X) \rightarrow H_{BM,2i}^G(X)$$
compatible with the usual operations on equivariant
Chow groups (\cite[cf. Chapter 19]{Fulton}).
Note that $H_{BM,i}^G(X)$ is not the same as $H_i(X \times^G EG)$,
where $EG \rightarrow BG$ is the topological classifying bundle.
However, if $X$ is smooth, then $X_G$ is also smooth, and $H_{BM,i}(X_G)$
is dual to $H^{2n-i}(X_G)=H^{2n-i}(X \times^G EG)=H^{2n-i}_G(X)$,
where $n$ is the complex dimension of $X$. In this
case we can interpret the cycle
map as giving a map
$$cl: A^i_G(X) \rightarrow H^{2i}_G(X)$$
If $X$ is compact, and the open sets $U \subset V$ can be chosen so
that $U/G$ is projective, then
Borel-Moore homology of $X_G$ coincides with ordinary
homology, so $H^G_{BM*}(X)$ can be calculated with a compact model.
However
In general, however, $U/G$ is only quasiprojective.
If $G$ is finite, then $U/G$ is
never projective. If $G$ is a torus, then $U/G$ can be taken to be a
product of projective spaces. If $G = GL_n$, then $U/G$ can be taken
to be a Grassmannian (see the example in Section 2.1)
If $G$ is semisimple, then $U/G$ cannot be chosen
projective, for then the hyperplane class would be a nontorsion
element in $A^1_G$, but by \cite{E-G}
$A^*_G \otimes {\bf Q} \cong S(\hat{T})^W \otimes
{\bf Q}$, which has no elements of degree 1. Nevertheless for
semisimple (or reductive) groups we can obtain a cycle map
$$cl: A_*^G(X)_{{\bf Q}} \rightarrow H_{BM*}^T(X;{\bf Q})^W$$ by identifying
$A_*^G(X) \otimes {\bf Q}$ with $A_*^T(X)^W \otimes {\bf Q}$ and
$H_{BM*}^G(X;{\bf Q})$ with $H_{BM*}^T(X;{\bf Q})^W$; if $X$ is compact then
the last group can be calculated with a compact model.
\section{Intersection theory on quotients}
One of the uses of equivariant intersection theory is
to study intersection theory on quotient stacks
and their moduli. In particular, we show below that the rational
Chow groups of moduli spaces
which are group quotients of a smooth variety
have an intersection product -- even when there are
infinitesimal automorphisms.
\subsection{Chow groups of quotients}
Let $G$ act on a scheme $X$, and assume
that a geometric quotient\footnote
{In characteristic $p$, the definition of geometric quotient used
here is slightly weaker
than the one given in \cite{GIT}. See the appendix.}
$X \rightarrow X/G$ exists.
\begin{prop} \label{p.quotient}
If $G$ acts freely, then $A_*^G(X,m) = A_*(X/G,m)$ (the isomorphism
of higher Chow groups requires $X$ to be quasi-projective).
\end{prop}
Proof. If the action is free, then $(V \times X)/G$ is a vector bundle
over $X/G$. Thus $X_G$ is an open set in this bundle
with arbitrarily high codimension, and the proposition follows from homotopy
properties of (higher) Chow groups. $\Box$
\medskip
\begin{thm} \label{quotient}
If $G$ acts properly
on a quasi-projective variety
$X$, so that $X/G$ is quasi-projective,
then\\
(1)$A_*^G(X,m) \otimes {\bf Q} = A_*(X/G,m) \otimes {\bf Q}$ for all $m \geq 0$.\\
(2) There is an isomorphism of operational Chow rings
$$p^*:A^*(X/G)_{{\bf Q}} \stackrel{\simeq} \rightarrow A^*_G(X)_{{\bf Q}}.$$
\end{thm}
{\bf Remarks.} (1) If the action is proper,
then the stabilizers are complete.
Since $G$ is affine they must in fact be finite.
We will
sometimes mention that the stabilizers are finite for emphasis.
(2) When $X$ is the set of stable points for some
linearized action of $G$ and $X/G$ is quasi-projective, then the
action is proper.
(3) The condition that $X/G$ is quasi-projective is only
required because the localization long exact sequence for higher Chow
groups has only been proved for quasi-projective schemes.
\medskip
In practice, many interesting varieties arise as quotients
of a smooth variety by a connected
algebraic group
which acts with finite stabilizers.
Examples include simplicial toric varieties
and various moduli spaces such as curves, vector bundles, stable maps, etc.
There is a long history
of work on the problem of constructing an intersection product
on the rational Chow group of quotients of smooth varieties.
In characteristic 0,
Mumford (\cite{Mu}) proved the existence
of an intersection product on the rational Chow groups
of $\overline{{\cal M}}_g$, the moduli space of stable curves.
Gillet (\cite{Gi}) and Vistoli (\cite {Vi}) constructed intersection
products on quotients in arbitrary characteristic -- provided
that the stabilizers of geometric points are reduced. In characteristic 0,
Gillet (\cite[Thm 9.3]{Gi}) showed that his product on $\overline{{\cal M}}_g$ agreed
with Mumford's, and in \cite[Lemma 1.1]{Edidin} it was shown
that Vistoli's product also agreed with Mumford's.
As a corollary to Theorem \ref{quotient} we
obtain a simple proof of the existence of intersection products on the
rational Chow groups of quotients for a group acting with
finite but possibly non-reduced stabilizers. Furthermore, when the
stabilizers are reduced,
our
product agrees with Gillet's and Vistoli's (Proposition \ref{triprod}).
In particular, this
answers \cite[Conjecture 6.6]{Vi}
affirmatively for moduli spaces of quotient stacks.
\begin{cor} \label{moduli}
Let $Y$ be a quasi-projective variety
which is isomorphic to a geometric quotient
$X/G$, where $X$ is smooth and $G$ acts properly
(hence with finite stabilizers) on $X$. Then the rational Chow groups
$A_*(Y)_{{\bf Q}}$ have an intersection product. This product is
independent of the presentation of $Y$ as a quotient.
\end{cor}
Proof of Corollary \ref{moduli}. Since $X$ is smooth, the equivariant
Chow groups $A_*^G(X)$ have an intersection product induced by the
isomorphism $A^*_G(X) \rightarrow A_*^G(X)$ (with ${\bf Z}$ coefficients).
By Theorem \ref{quotient} $A_*^G(X)_{{\bf Q}} = A_*(Y)_{{\bf Q}}$ so the groups
$A_*(Y)_{{\bf Q}}$ inherit a ring structure. Since the intersection
product on $A_*(Y)$ is induced by the multiplication in $A^*(Y)$ it
depends only on $Y$.
$\Box$
\medskip
Proof of part (1) of Theorem \ref{quotient}.
For simplicity of exposition we give the proof
assuming that the group $G$ is connected of
dimension $g$. (This way we can assume that the set-theoretic
inverse image in $X$ of a subvariety of $X/G$ is a single variety rather than
a possible disjoint union of varieties.)
All coefficients -- including those of cycle groups -- are assumed to
be rational.
If $G$ acts properly on $X$, then $G$ acts
properly on $X \times \Delta^m$
by acting trivially on the second factor. In this case,
the boundary map of the higher
Chow group complex preserves invariant cycles, so there is
a subcomplex of invariant cycles $Z_*(X,\cdot)^G$.
Set $$A_*([X/G],m) = H_m(Z_*(X, \cdot)^G,\partial).$$
Now if $X \rightarrow X/G$ is a geometric quotient, then so
is $X \times \Delta^m \stackrel{\pi} \rightarrow X \times \Delta^m$.
Define a map
$\pi^*: Z_k(X,m) \otimes {\bf Q} \rightarrow Z_{k+g}(X,m)^G
\otimes {\bf Q}$ for all
$m$ as follows. Let $F \subset X/G \times \Delta^m$ be a $k+m$-dimensional
subvariety
intersecting the faces properly,
then $H = (\pi^{-1}F)_{red}$ is a $G$-invariant
$(k+m+g)$-dimensional subvariety of $X \times \Delta^m$ which intersects the
faces properly. Thus, $[H] \in Z_{k+g}(X,m)^G $.
Let $e_H$ be the order of the stabilizer
at a general point of $H$, and let $i_H$ be the degree of the purely
inseparable
extension $K(F) \subset K(H)^G$. Set $\pi^*[F] = \alpha_H[
H] \in Z_{k+g}^G(X,m)$, where $\alpha_H = \frac{e_H}{i_H}$.
Since $G$-invariant subvarieties of $X \times \Delta^m$
exactly correspond to subvarieties of $X/G \times \Delta^m$,
$\pi^*$ is an isomorphism of cycles for all $m$.
\begin{prop} \label{proper}
Let
$$\begin{array}{ccc}
Z & \stackrel{g} \rightarrow & X \\
\small{p} \downarrow & & \small{\pi} \downarrow\\
Q & \stackrel{f} \rightarrow & Y
\end{array}$$
be a commutative diagram of quotients with $f$ and $g$ proper.
Then $p^* \circ g_* = f_* \circ \pi^*$ as maps $Z_*(Q) \rightarrow
Z_*(Z,m)^G$.
\end{prop}
Proof of Proposition \ref{proper}. The proposition is an
immediate consequence of the following lemma.
\begin{lemma} Suppose $G$ acts properly (hence with finite
stabilizers) on varieties $Z$ and $X$. Let
$$\begin{array}{ccc}
Z & \stackrel{g} \rightarrow &X \\
\small{p} \downarrow & & \small{\pi} \downarrow\\
Q & \stackrel{f} \rightarrow & Y
\end{array}$$
be a commutative diagram of geometric quotients with
$f$ and $g$ finite and surjective. Then
$$\frac{e_Z}{i_Z} [K(Q):K(Y)] = \frac{e_X}{i_X} [K(Z):K(Z)].$$
\end{lemma}
Proof. Since we are checking degrees, we may replace $Y$ and $Q$
by $K(Y)$ and $K(Q)$, and $X$ and $Z$ by their generic fibers over
$Y$ and $Q$ respectively.
Then we have a commutative diagram of varieties.
$$\begin{array}{ccc}
Z & \rightarrow & X \\
\downarrow & & \downarrow\\
\mbox{spec}(K(Z)^G) & \rightarrow & \mbox{spec}(K(X)^G)\\
\downarrow & & \downarrow\\
\mbox{spec}(K(Q)) & \rightarrow & \mbox{spec}(K(Y))
\end{array}$$
Since $i_Z := [K(Z)^G:K(Q)]$ and $i_X :=[K(X)^G:K(Y)]$, it suffices
to prove
$$e_Z[K(Z)^G:K(X)^G] = e_X[K(Z):K(X)].$$
By \cite[Paragraph 6.5]{Borel} the
extensions $K(Z)^G \subset K(Z)$ and $K(X)^G \subset
K(X)$ are separable (transcendental).
Thus, after finite separable base extensions,
we may assume that there are sections $s: \mbox{spec}(K(X)^G) \rightarrow
U$ and $t: \mbox{spec}(K(Z)^G) \rightarrow
W$. In this case the stabilizer group schemes over $W$ and $U$
are
isomorphic to $K(Z)^G \times G$ and $K(X)^G \times G$ respectively
(\cite[Proof of Prop 0.7]{GIT}). Thus, $e_Z = [K(Z)^G \times K(G):
K(Z)]$ and $e_X = [K(X)^G \times K(G): K(X)]$. The lemma follows.
$\Box$
\begin{prop} \label{squiggy}
The map $\pi^*$ commutes with the boundary operator of the higher Chow groups.
In particular, there is an induced isomorphism of Chow groups
$$A_k(X/G,m) \simeq A_{k+g}([X/G],m).$$
\end{prop}
Proof of Proposition \ref{squiggy}.
If
$$\begin{array}{ccc}
Z & \stackrel{g} \rightarrow & X \\
\small{p} \downarrow & & \small{\pi} \downarrow\\
Q & \stackrel{f} \rightarrow & X/G
\end{array}$$
is a commutative diagram of quotients with $f$ and $g$ finite and
surjective, then $f_*$ and $g_*$ are surjective as
maps of cycles. Thus, by Proposition \ref{proper} it suffices
to prove $p^*:Z_*(Q) \rightarrow Z_*(X)^G$ commutes with
$\partial$.
By Proposition \ref{whizzbang} of the appendix, there is
a commutative diagram of quotients such that $p:Z \rightarrow Q$
is a principal bundle. Since $p$ is flat, $p^*$ commutes
with $\partial$ and Proposition \ref{squiggy} follows.
$\Box$
\medskip
Suppose $T \subset X$ is a $G$-invariant subvariety. Let $S \subset X/G$
be its image under the quotient map. Set $U = X -T$ and $V=X/G -U$.
Then we have two commuative diagrams of geometric quotients
$\begin{array}{ccc} T & \stackrel{i} \rightarrow & X\\
\small{\pi} \downarrow & & \small{\pi} \downarrow \\
S & \stackrel{i} \rightarrow & X/G
\end{array}$ \hspace{2.0in}
$\begin{array}{ccc} U & \stackrel{j} \rightarrow & X\\
\small{\pi} \downarrow & & \small{\pi} \downarrow \\
V & \stackrel{j} \rightarrow & X/G
\end{array}$
\begin{lemma} \label{iggypop}
Let $\alpha \in Z_k(X/G,m)$ and $\beta \in Z_k(S,m)$.
(1) $\pi^*j^* \alpha = j^* \pi^* \alpha$ in $Z_{k+g}^G(U,m)$.
(2) $\pi^*i_* \beta = i_*\pi^*\beta$ in $Z_{k+g}^G(X,m)$.
\end{lemma}
Proof of Lemma \ref{iggypop}. If $\alpha = [F]$ and $H= \pi^{-1}(F)_{red}$,
then $\pi^*j^*\alpha$ and $j^*\pi^*\alpha$ are both multiples of
$[H \cap U]$. Since $e_{[H \cap U]} = e_{[H]}$, and $i_{[H \cap U]} = i_{[H]}$,
the multiplicities are the same. This proves (1).
Part (2) was proved in Proposition \ref{proper}. $\Box$
As a consequence of Proposition \ref{squiggy} and Lemma \ref{iggypop},
we obtain
the following proposition.
\begin{prop} \label{snodgrass}
Let $T \subset X$ be an invariant subvariety. If $S$, $U$, and $V$
are as above, then there is a commutative diagram of isomorphisms
$$\begin{array}{ccccccc} \ldots \rightarrow & A_*([T/G],m) & \rightarrow &
A_*([X/G],m) & \rightarrow & A_*([U/G],m) & \rightarrow \ldots \\
& \simeq \uparrow & & \simeq \uparrow & & \simeq \uparrow & \\
\ldots \rightarrow & A_*(S,m) & \rightarrow &
A_*(X/G,m) & \rightarrow & A_*(V,m) & \rightarrow \ldots
\end{array}$$
$\Box$
\end{prop}
Next, note that there is a map
$$\alpha:A_*([X/G],m) \rightarrow A_*^G(X,m)$$
defined by the formula
$$[F] \in Z^G_*(X,m) \mapsto [F]_G$$ which commutes with equivariant
proper pushforward and equivariant flat pullback.
\begin{prop} \label{warhol}
If $G$ acts properly
on $X$, and a quasi-projective
geometric quotient $X \rightarrow X/G$ exists,
then $\alpha$ is an isomorphism.
\end{prop}
Proof of Proposition \ref{warhol}.
By the Nullstellensatz there is a point of
$X$ which is finite over the generic point of $X/G$.
Thus, by generic flatness, there is a locally closed
subvariety $Z \subset X$, and an open set $W \subset X/G$ such
that the projection $Z \rightarrow W$ is finite and flat. Let $U =
\pi^{-1}(W)$.
Since $G$ acts properly, the map
$G \times Z \rightarrow U$ is finite. Shrinking $Z$ (and thus $U$)
we may assume that $G \times Z \rightarrow U$ is also flat.
By Noetherian induction and the localization long exact sequence
(which exists by Proposition \ref{snodgrass})
it suffices to prove that $\alpha:A_*^G([U/G],m) \rightarrow A_*^G(U,m)$
is an isomorphism.
Taking Chow groups, we obtain a commutative diagram where all maps commute.
$$\begin{array}{ccc}
A_*^G(G \times Z,m) & \stackrel{\leftarrow} \rightarrow & A_*^G(U,m)\\
\small{\alpha} \downarrow & & \small{\alpha} \downarrow \\
A_*([G \times Z/G],m) & \stackrel{\leftarrow} \rightarrow & A_*([U/G])
\end{array}$$
(The right horizontal arrows are proper pushforward divided by the degree,
and the left horizontal arrows are flat pullback.)
Chasing the diagram shows that if the left vertical arrow is an isomorphism
then so is the right vertical arrow. Since $G$ acts freely,
$\alpha: A_*([G \times Z/G],m) \rightarrow A_*(G \times Z, m)^G$ is
an isomorphism by Proposition \ref{p.quotient}.
$\Box$.
We have now proved part (1) of Theorem \ref{quotient}.
\medskip
Proof of part (2) of Theorem \ref{quotient} (cf. \cite[Proposition 6.1]{Vi}).
Suppose $c \in A^*(X/G)_{{\bf Q}}$,
$Z \rightarrow X$ is a $G$-equivariant morphism, and $\alpha \in A_*^G(Z)$.
For any representation $V$, there are maps
$Z_G \rightarrow X_G \rightarrow X/G$. If $V$ is chosen so that
$\alpha$ is represented by a class $\alpha_V \in A_*(Z_G)$ we can define
$$p^*c \cap \alpha = c \cap \alpha_V. $$
As usual, this definition is independent of the representation,
so $p^*c \cap \alpha \in A_*^G(Z)$.
(1) $p^*$ is injective.
Proof of (1). Suppose $p^* \cap \alpha =0$ for all $G$-maps $Z \rightarrow
X$ and all $\alpha \in A_*^G(Z)$. By base change, it suffices to show
$c \cap x = 0$ for all $x \in A_*(X/G)_{{\bf Q}}$. By Proposition \ref{whizzbang}
there is a finite map $Y \rightarrow X/G$, and a principal bundle
$Z \rightarrow Y$ together with a finite $G$-map $Z \rightarrow X$.
Thus we obtain a commutative diagram
$$\begin{array}{ccc} Z_G & \stackrel{g} \rightarrow & X_G\\
\small{q}\downarrow & & \small{p} \downarrow \\
Y & \stackrel{f} \rightarrow & X/G
\end{array}$$
where the horizontal maps are proper and surjective.
Choose $y \in A_*(Y)$ so that $f_*(y) = x$.
Since $q$ is flat
$$0= c \cap q^*y =q^*(c \cap y).$$
Since $q^*$ is an isomorphism in the appropriate degrees,
$c \cap y = 0$. Thus
$$0 = f_*(c \cap y) = c \cap f_*y = c \cap x$$
as desired.
(2) $p^*$ is surjective.
Proof of (2). Suppose $d \in A^*_G(X)$. Define $c \in A^*(X/G)$
as follows: If $Y \rightarrow X/G$ and $y \in A_*(Y)$, set
$c \cap y = d \cap \pi^*y$ where $\pi: X\times_{X/G} Y \rightarrow
Y$ is the quotient map, and $\pi^*y \in A_*([X\times_{X/G} Y/G])_{{\bf Q}}
= A_*^G(X \times_{X/G} Y)_{{\bf Q}}$ is defined as above.
Then $p^*c = d \in A^*_G(X)$.
$\Box$.
\subsection{Intersection products on moduli}
Equivariant intersection theory gives a nice way of working with
cycles on a singular moduli space ${\cal M}$ which is a quotient $X/G$ of
a smooth variety by a group acting with finite stabilizers.
Given a subvariety $W \subset {\cal M}$ and
a family $Y \stackrel{p}\rightarrow B$ of schemes
parametrized by ${\cal M}$ there is a map $B \stackrel{f} \rightarrow
{\cal M}$. We wish to define
a class $f^*([W]) \in A_*B$ corresponding to how the image
of $B$ intersects $W$. This can be done (after tensoring with ${\bf Q}$) using
equivariant theory.
By Theorem \ref{quotient}, there is an isomorphism
$A_*({\cal M})_{{\bf Q}}= A_*^G(X)$ which takes
$[W]$ to the equivariant class $w= \frac{e_W}{i_W}[f^{-1}W]_G$.
Let $B_G \rightarrow B$ be the principal $G$-bundle $B \times_{[X/G]} X$,
(The fiber product is a scheme, although the product is taken over
the quotient stack $[X/G]$.
Typically, $B_G$ is the structure bundle of
a projective bundle ${\bf P}(p_*L)$ for a relatively very ample line bundle
$L$ on $Y$).
Since $X$ is smooth, there is an equivariant pullback
$f^*G: A_*^G(X) \rightarrow A_*^G(B_G)$ of the induced map $B_G \stackrel{f_G}
\rightarrow X$, so we
can define a class $f_G^*(w) \in A_*^G(B_G)$. Identifying
$A_*^G(B)$ with $A_*(B)$ we obtain our class $f^*(W)$.
\paragraph{Example (Moduli of stable curves)} Let $\overline{{\cal M}}_g$ be the moduli space
of curves. Let $\Delta_{i} \subset
\overline{{\cal M}}_g$ be the Weil divisor corresponding
to (stable) nodal curves which are formed by identifying a curve of genus $i$
to
a curve of genus $g-i$ at a point. Given a family of curves
$Y \stackrel{p} \rightarrow B$ there is a map
$B \rightarrow \overline{{\cal M}}_g$. We wish to define a cycle $\delta_B$
corresponding to the intersection of the image of $B$ with $\Delta_i$.
Such a class can be defined using Vistoli's intersection theory on the
Deligne-Mumford stack $F_{\overline{{\cal M}}_g}$ since there is a Gysin pullback
$A_*(B) \otimes {\bf Q} \stackrel{i^*}\rightarrow A_*B \otimes {\bf Q}$
corresponding to the inclusion $\delta_i \stackrel{i} \hookrightarrow
F_{\overline{{\cal M}}_g}$.
Let $\delta_i$
be the inverse image of $\Delta_i$ in $F_{\overline{{\cal M}}_g}$.
If the image of $B$ is completely contained
in $\Delta_i$, then, to calculate $\delta_B$ we need to use an
excess intersection
formula
$$\delta_B = i^*(\delta_i) = c_1({\cal O}_{\delta_i}(\delta_i)) \cap [B].$$
Similarly, if $\Theta \subset \overline{{\cal M}}_g$ is a subvariety of codimension
$d$ corresponding
a smooth substack $\theta \subset F_{\overline{{\cal M}}_g}$, then we
would like to assert that
$$\theta_B = c_d(N_{\theta}F_{\overline{{\cal M}}_g}) \cap [B]$$
where $N_{\theta}F_{\overline{{\cal M}}_g}$ is the normal bundle to
$\theta$ in $F_{\overline{{\cal M}}_g}$.
Unfortunately, such formulas were not fully developed in \cite{Vi},
so their use can not be completely justified.
Consider the equivariant situation.
The moduli space $\overline{{\cal M}}_g$ is a geometric quotient $H_g/G$
where $H_g$ is the (smooth) Hilbert scheme of pluricanonically
embedded stable curves and $G=PGL(N)$ for some $N$.
Given an abstract family $Y \rightarrow B$
let $Y_H \rightarrow B_H$ be the corresponding family of pluricanonically
embedded curves. Let $\Delta_i^H \stackrel{i_H} \hookrightarrow
H_g$ be the corresponding $G$ invariant divisor in $A_*^G(H_g)$.
Since $H_g$ is smooth, we obtain an equivariant line bundle
${\cal O}(\Delta_i^H)$.
Let $B_H \rightarrow H_g$ be the corresponding
equivariant map. Then by equivariant excess intersection (which follows
from ordinary excess intersection on schemes $(H_g \times U)/G$)
$$i_H^*(\Delta_i^H) = c_1({\cal O}_{\Delta_i^H}(\Delta_i^H)) \cap [B].$$
The corresponding formula in $A_*B \otimes {\bf Q}$ follows from
the identification of $A_*(B)$ with $A_*^G(B_H)$.
Similar formulas
(involving Chern classes of the equivariant normal bundle) hold
for cycles of higher codimension.
In \cite[Section 3]{Edidin} explicit excess intersection formulas were given
in codimension 1 and 2, for various nodal loci. The approach
there is similar to the discussion above, although
equivariant Chow groups were not used. Instead, a result of Vistoli (proved
in characteristic 0, although true in arbitrary characteristic)
was used to identify $A^1(B)$ (codimension-one cycles) with
$A^1(B_H)$ because $B_H \rightarrow B$ is a principal ${\bf P} GL(N)$ bundle.
The statements of \cite[Section 3]{Edidin} can proved in
in arbitrary characteristic using the methods outlined above.
\subsection{Chow groups of quotient stacks}
If $G$ acts on $X$ we let $[X/G]$ denote the quotient stack. This is
a stack in the sense of Artin, and exists without any assumptions on
the $G$-action. By the next proposition, the equivariant Chow groups
do not depend on the presentation as a quotient, so they are an invariant
of the stack.
\begin{prop} \label{qstacks}
Suppose that $[X/G] \simeq [Y/H]$ as quotient stacks. Then
$A_i^G(X) \simeq A_i^H(Y)$ for all $i$.
\end{prop}
Proof: Suppose $\mbox{dim } G =g$ and $\mbox{dim } H = h$. Let
$V_1$ be an $l$-dimensional representation of $G$, and $V_2$ an
$M$ dimensional representation of $H$. Let $X_G = X \times^G U_1$
and $Y_H = X \times^H U_2$, where $U_1$ (resp. $U_2$) is an open set
on which $G$ (resp. $H$) acts freely.
Since the diagonal of a quotient stack is representable,
the fiber product $Z=X_G \times_{[X/G]} Y_H$ is a scheme. This
scheme is a bundle over $X_G$ and $Y_H$ with fiber $U_2$ and $U_1$
respectively.
Thus, $A_{i+l-g}(X_G) = A_{i+l+m-g}(Z) = A_{i+m-h}(Y_H)$
and the proposition follows.
$\Box$
\medskip
\noindent
{\bf Remark.} Proposition \ref{qstacks} suggests that there should be a
notion of
Chow groups of an arbitrary algebraic stack which can have non-zero
torsion in arbitrarily high degree. This situation would be analogous to
the cohomology of quasi-coherent sheaves on the \'etale (or flat) site
(cf. \cite[p. 101]{D-M}).
\medskip
If $G$ acts properly with finite, reduced stabilizers, then $[X/G]$ is
a separated
Deligne-Mumford stack. The rational Chow groups $A_*([X/G]) \otimes
{\bf Q}$ were first defined by Gillet \cite{Gi} and coincide with the groups
$A_*([X/G]) \otimes {\bf Q}$ defined above.
More generally, if $G$ acts with finite stabilizers which are not reduced, then
then Gillet's definition
can be extended and we can define the ``naive'' Chow groups
$A_k([X/G])_{{\bf Q}}$ as the group generated by $k$-dimensional integral
substacks modulo rational equivalences. With this definition we
expect that $A_*^G(X)_{{\bf Q}} = A_*([X/G])_{{\bf Q}}$. To prove such an
isomorphism in general requires that the naive Chow groups of the stack satisfy
the homotopy property (i.e. if $F \rightarrow G$ is a vector bundle in
the category of stacks, then $A_*(F)_{{\bf Q}} = A_*(G)_{{\bf Q}})$.
However, if a quasi-projective
quotient exists, then Proposition \ref{warhol} can be restated in
the language of stacks as
\begin{prop}
Let $G$ be a $g$-dimensional group which acts
properly on a scheme $X$
(so the quotient $[X/G]$ is a separated Artin stack). Assume that
a quasi-projective moduli scheme $X/G$ exists for $[X/G]$.
Then $A_i^G(X) \otimes {\bf Q} = A_{i-g}([X/G]) \otimes {\bf Q}$.
$\Box$
\end{prop}
{\bf Remarks.} (1) Although $A_*^G(X) \otimes {\bf Q} = A_*([X/G]) \otimes {\bf Q}$,
the integral Chow groups may have non-zero torsion for all $i < \mbox{dim }
X$. It would be interesting to compute this torsion in examples
such as moduli spaces of curves of low genus.
(2) In general, a separated quotient stack should always
have an algebraic space as coarse moduli space. Thus, an extension
of the present theory to algebraic spaces would eliminate the need
for any assumptions in the proposition.
\medskip
With the identification of $A_*^G(X) \otimes {\bf Q}$ and $A_*([X/G]) \otimes {\bf Q}$
there are three intersection products on the rational Chow groups
of a smooth Deligne-Mumford quotient stack with a moduli
space -- the equivariant product, Vistoli's product
defined using a via a gysin pullback for regular embeddings of stacks,
and Gillet's product defined using the product in higher $K-$theory.
The next proposition shows that they are identical.
\begin{prop} \label{triprod}
If $X$ is smooth and $[X/G]$ is a separated
Deligne-Mumford stack
(so $G$ acts with finite, reduced stabilizers)
with a quasi-projective moduli space $X/G$, then
the intersection products on $A_*([X/G])_{{\bf Q}}$ defined by Vistoli and
Gillet are the same as the equivariant product on $A_*^G(X)_{{\bf Q}}$.
\end{prop}
Proof. If $V$ is an $l$-dimensional representation, then all three products
agree on the smooth quotient scheme (\cite{Vi}, \cite{Grayson})
$(X \times U)/G$. Since
the flat pullback of stacks $f:A^*([X/G])_{{\bf Q}} \rightarrow
A^*((X \times U)/G)_{{\bf Q}}$ commutes with all 3 products, and is an isomorphism
to arbitrarily high codimension, the proposition follows.
$\Box$
\medskip
\section{Equivariant Riemann-Roch}
In this section we construct an equivariant Todd class map and prove
an equivariant Riemann-Roch theorem for $G$-schemes.
The theorem involves
completions of equivariant $K$-groups and Chow groups
because the groups $A_*^G(X)$ (resp. $A^*_G(X)$) can have terms of
arbitrarily large negative (resp. positive) degree. Thus,
the Todd class and Chern character map to completions of
these groups. The map $\tau^G_X: K^{'G}_0(X)
\rightarrow \widehat{A_*^G(X)_{{\bf Q}}}$ factors through a completion
of $K_{0}^{'G}(X)$ and we obtain an isomorphism
$\tau^G_X: \widehat{K^{'G}_0(X)} \rightarrow \widehat{A_*^G(X)_{{\bf Q}}}$.
This section has two parts. In the first, we define
$\widehat{A_*^G(X,i)}$ and $\widehat{K^{'G}_i(X)}$ as completions of
$A_*^G(X,i)$ and $K^{'G}_i(X)$ along certain ideals. We then prove an
analogue of a theorem of Atiyah and Segal which gives a more geometric
description of these completions. In the second part we construct
the Todd class map $\tau_X^G:K_0^{'G}(X) \rightarrow \widehat{A_*^G(X)_{{\bf Q}}}$
and show that it induces an isomorphism
$\tau_X^G: \widehat{K^{'G}_i(X)} \stackrel{\simeq} \rightarrow
\widehat{A_*^G(X)_{{\bf Q}}}$.
The construction is an easy consequence of the
nonequivariant Riemann-Roch theorem and our geometric
description of the completions. Finally, we discuss a
conjecture of Vistoli.
\subsection{Completions of equivariant K-groups and Chow groups} Let
$R(G)$ denote the representation ring of $G$.
Let $K_0^G(X)$ denote the Grothendieck group of
$G$-equivariant vector bundles on $X$, and let $K_i^{'G}(X)$ denote
the $i$-th higher $K$-group of the category of $G$-equivariant
coherent sheaves (\cite{Thomason}). As in the
non-equivariant case, $K_0^G(X)$ is a ring under tensor product, and
$K_i^{'G}(X)$ is a module for that ring. Also, $K_0^G(X)$ and
$K_i^{'G}(X)$ are $R(G)$ modules via the isomorphism $R(G) \simeq
K_0^G(pt)=K_0^G$.
Let $P \subset K_0^G = R(G)$ denote the ideal of virtual
representations of dimension 0, and let
$\widehat{K_i'^G(X)}$ be the completion of $K_i'^G(X)$ along $P$.
Let $Q = A^+_G
\subset A^*_G$ be the augmentation ideal, and let
$\widehat{A_*^G(X,i)}$ be the completion of $A_*^G(X,i)$
along $Q$.
Let $\tilde{Q} = A^+_G(X) \subset A^*_G(X)$ denote the augmentation
ideal, then $Q A^*_G(X) \subset \tilde{Q}$. Let
$\widetilde{A_*^G(X)}$ denote the completion of $A^*_G(X)$ along
$\tilde{Q}$. Likewise, let $\tilde{P} \subset K_0^G(X)$ denote the ideal of
virtual bundles of rank 0 (the kernel of the rank map), then $P
K_0^G(X) \subset \tilde{P}$. Let $\widetilde{K_i'^G(X)}$ denote the
completion of $K_i'^G(X)$ along $\tilde{P}$.
We will show below that there are isomorphisms
$$
\widetilde{K_i'^G(X)} \cong \widehat{K_i'^G(X)}
$$
$$
\widetilde{A_*^G(X,i)} \cong \widehat{A_*^G(X,i)}.
$$
To show this, we will compare these completions with more geometrically
defined ones.
Partially order the set ${\cal V}$ of representations of $G$ by the rule
$W < V$ if $W$ is a summand in $V$. For each representation,
let $V^f$ be the open set of points whose orbits
are closed in $V$ and which have trivial stabilizer.
The collections
$\{K_i^{'G}(X \times V^f)\} _{ V \in {\cal V} }$ and
$\{A_*^G((X \times V^f),i)\}_{ V \in {\cal V} }$
are inverse systems since the inclusion $V^f \oplus W \hookrightarrow
(V\oplus W)^f)$ induces restriction maps
$$
K_i^{'G}(X \times (V\oplus W)^f) \rightarrow K_i^{'G}(X \times V^f \oplus W)
\simeq K_i^{'G}(X \times V^f)
$$
$$
A_*^{G}((X \times (V\oplus W)^f),i) \rightarrow
A_*^{G}((X \times V^f \oplus W),i)
\simeq A_*^{G}((X \times V^f),i).
$$
By identifying $K_i^{'G}(X \times V)$ with $K_i^{'G}(X)$ and
$A_*^{G}((X \times V),i)$ with $A_*^G(X,i)$ we obtain
restriction maps
$$r'_V:K_i^{'G}(X) \rightarrow K_i^{'G}(X \times V^f)$$
$$r_V:A_*^G(X,i) \rightarrow A_*^G((X \times V^f),i)$$
and thus inverse systems $\{r'_V(K_i^{'G} \}_{ V \in {\cal V} }$
and $\{r_V(A_*^{'G}(X,i))\}_{ V \in {\cal V} }$.
\begin{thm} \label{completions}
There are isomorphisms of completions
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \simeq \widehat{K_i^{'G}(X)}$$
and if $X$ is quasi-projective we also have
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \simeq \widetilde{K_i^{'G}(X)}$$
$$\lim_{\leftarrow V} r_V(A_*^{G}(X,i)) \simeq \widehat{A_*^{G}(X,i)}
\simeq \widetilde{A_*^{G}(X,i)}.$$
\end{thm}
Remark: A similar equality of completions was proved (for $K_0^{'G}$)
in \cite{CEPT} for actions of finite groups of
projective varieties defined over rings of integers of number fields.
\medskip
As a result of this identification of completions, we can prove
a particular case of a
conjecture of K\"ock (\cite{kock}) for arbitrary reductive groups acting
on regular schemes of finite type over a field. Set
$K(X,G) = \oplus K_i^G(X) = \oplus K_i^{'G}(X)$, and let
$\widetilde K(X,G)$ be the completion along the augmentation
ideal $\tilde{P}$ of $K_0^G(X)$.
\begin{cor} Let $X \stackrel{f} \rightarrow Y$ be a proper equivariant
morphism of quasi-projective, regular schemes. Then
$f_*:K(X,G) \rightarrow K(Y,G)$
is continuous with respect to the $\tilde{P}$-adic topologies.
\end{cor}
Proof. The pushforward
$f_*$ induces a map of inverse systems
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \rightarrow \lim_{\leftarrow V} R'_V(K_i^{'G}(Y))$$
The corollary follows from the identification of completions in
Theorem \ref{completions}. $\Box$
\medskip
Proof of Theorem \ref{completions}.
The two statements have essentially identical proofs, so we
will only prove the isomorphisms in $K$-theory. Furthermore,
the proof that
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \simeq \widetilde{K_i^{'G}(X)}$$
is virually identical to the proof that
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \simeq \widehat{K_i^{'G}(X)}$$
so we will only prove the latter. (The only difference in the proof
of the former is that we need to assume $X$ is quasi-projective
so we can compare the $\gamma$ filtration and the topological filtration
on $K_0^G(X)$.)
This statement is the analogue of
\cite[Theorem 2.1]{A-S}, except that we do not need the hypothesis
that $\widehat{K_i^{'G}(X)}$ is finite over $R(G)$. As in \cite{A-S}
we first prove the result for a torus and from this deduce the general
case. Our proof of the torus case is somewhat different from that of
\cite{A-S}, but the passage to the general case uses their arguments,
which we have repeated for completeness.
\medskip
{\em Step 1.} We first prove the result if $G=T$ is a torus.
We have filtrations of $K_i^{'G}(X)$ by the ideals $\mbox{ker }r'_V$
and by powers of the ideal $P = P_T$. It suffices to show that the
filtrations have bounded difference. This is a consequence of the
next two lemmas.
\begin{lemma}\label{koszul} Let $V$ be a representation of $T$ and $W
\subset V$ a subrepresentation of codimension $l$. Let $i: X \times
W \rightarrow X \times V$ be the inclusion. Then $i_*(K_i^{'T}({X
\times W})) \in P^l K_i^{'T}(X \times V)$.
\end{lemma}
Proof of Lemma \ref{koszul}: We can find a chain of $T$-invariant
subspaces $W=W_l \subset W_{l-1} \subset \ldots \subset W_0 = V$ where
the codimension of $W_j$ in $V$ is $j$. By induction on codimension,
it suffices to consider the case where the codimension of $W$ is $1$.
By the projection formula for equivariant $K$-theory, $i_*(K_i^{'T}(X
\times W))= i_*([{\cal O}_{X \times W}])K_i^{'T}(X)$. However,
$$
i_*([{\cal O}_{X \times W}]) = [{\cal O}_{X \times V}] - [(V/W)
\otimes_{k} {\cal O}_{X \times V}],
$$
which is in $P K_i^{'T}(X \times V)$. $\Box$
\begin{lemma}\label{bound}
$$P^{s}K_0^{'T} \subset \mbox{ker } r'_V \subset P^l K_0^{'T}(X)$$
for any $s > d + \mbox{dim }X - \mbox{dim }T$.
\end{lemma}
Proof of Lemma \ref{bound}. If $V$ is a representation of a torus
then $V^u=V - V^f$ is a finite union of linear subspaces
(Appendix, Proposition \ref{complement})
which by assumption have codimension at least $l$ in $V$. From the localization
long exact sequence
$$\ldots \rightarrow K_i^{'T}(X \times V^u) \stackrel{i_*} \rightarrow
K_i^{'T}(X \times V)
\stackrel{r'_V}\rightarrow K_i^{'T}(X \times V^f) \rightarrow \ldots$$
we know
that $\mbox{ker }r'_V\; = i_*(K_i^{'T}(X \times V^u))$. The image of
$K_i^{'T}(X \times V^u)$ is generated by the images of $K_i^{'T}(X \times W)$
for each linear space $W \subset V^u$. By Lemma \ref{koszul}, these
images are contained in $P^l K_i^{'T}(X \times V) = K_i^{'T}(X)$.
Hence $\mbox{ker }r'_V \subset P^l K_i^{'T}(X)$.
For the other inclusion, note that $K_i^{'T}(X \times V^f) =
K_i'(X_T)$, and $P^s K_0^{'T}(X \times V^f) \subset F^s K_i'(X_T)$,
where $F^{\cdot}$ denotes the $\gamma$-filtration on $R(G)$.
Since a point is projective,
$F^sK_i'(X_T) \subset F^s_{top}K_i^{'T}(X_T)$. Thus, if
$s > \mbox{dim }X_T$, then $F^s K_i'(X_T) = 0$. Hence
$P^{s}K_i^{'T} \subset \mbox{ker } r'_V$, as desired. $\Box$
This lemma implies the desired equality of completions for the case of
a torus.
\medskip
{\em Step 2.} We prove the result for $G=GL_n$. Let $j: T
\hookrightarrow G$ be the inclusion and let $j^*: K_i^{'G}(X)
\rightarrow K_i^{'T}(X)$ be the induced restriction map.
\begin{lemma} \label{summand}
There is a functorial map $j_*: K_i^{'T}(X) \rightarrow K_i^{'G}(X)$
such that $j_*j^*$ is the identity. Hence $K_i^{'G}(X)$ is a direct
summand in $K_i^{'T}(X)$.
\end{lemma}
Proof of Lemma \ref{summand}.
This is proved in \cite[Prop. 4.9]{Atiyah} for topological K-theory.
The same proof works in the algebraic setting: the main ingredient is the
projective bundle theorem, which was proved in this setting by Thomason
\cite[Theorem 3.1]{Thomason}. $\Box$
{}From the proof of Step 1, in computing $\lim_{\leftarrow V} K_i^{'T}(X \times
V^f)$ we need not consider all representations of $T$: it suffices to
consider the subsystem of representations of $T$ which are
restrictions of representations of $G$,
then $\mbox{ker }r^{'G}_V = \mbox{ker }r^{'T}_V \cap K_i^{'G}(X)$.
The submodule $K_i^{'G}(X)$ of $K_i^{'T}(X)$ inherits two topologies
from $K_i^{'T}(X)$: the topology induced by the ideals $\mbox{ker
}r^{'T}_V \cap K_i^{'G}(X) = \mbox{ker }r^{'G}_V$, and the topology
induced by powers of the ideal $P_T$. Because $K_i^{'G}(X)$ is a
direct summand in $K_i^{'T}(X)$, by Lemma \ref{bound} these topologies
coincide. On the other hand, as noted in \cite{A-S}, the ideals $P_T$
and $P_G R(T)$ have bounded difference,
so they induce the
same topology on $K_i^{'T}(X)$. The restriction of this topology to
$K_i^{'G}(X)$ is the topology induced by powers of the ideal $P_G$.
Putting these facts together, we conclude that $\lim_{\leftarrow V} K_i^{'G}(X
\times V^f) \simeq \widehat{K_i^{'G}(X)}$ for $G=GL_n$, as desired.
\medskip
{\em Step 3.} We now deduce the result for general $G$. Embed $G$
into $H = GL_n$. Then $K_i^{'G}(X) = K_i^{'H}(X \times^G H)$\footnote{Note
that $X \times^G H$ is a
scheme, because of our hypothesis on $X$ or $G$.}
(\cite[Proposition 6.2]{Thomason}).
As above, we may restrict our attention
to representations of $G$ which are restrictions of representations of
$H$, then
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \simeq \lim_{\leftarrow V} r'_V(K_i^{'H}(( X \times^G H)).$$
As noted in \cite{A-S}, the $P_H$-adic and $P_G$-adic topologies
coincide on any $R(G)$-module, and hence, by the result for $H=GL_n$,
we have
$$\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \simeq \widehat{K_i^{'G}(X)},$$
as desired. $\Box$
If $X$ is any $G$-scheme, let $K_i'^G(X)_{{\bf Q}} =
K_i'^G(X) \otimes {\bf Q}$, and $A_*^G(X,i)_{{\bf Q}} = A_*^G(X,i) \otimes {\bf Q}$.
\begin{cor} There are isomorphisms of completions
$$\lim_{\leftarrow V} \{ r'_V(K_i^{'G}(X))_{{\bf Q}} \} \simeq \widehat{K_i^{'G}(X)_{{\bf Q}}}
\simeq \widetilde{K_i^{'G}(X)_{{\bf Q}}}$$
and if $X$ is quasi-projective
$$\lim_{\leftarrow V} \{ r_V(A_*^{G}(X,i))\} \simeq
\widehat{A_*^{G}(X,i)_{{\bf Q}} }.$$
\end{cor}
Proof: The proof is the same as above, except we do not need $X$
to be quasi-projective to show that
$\tilde{P}^sK_i^{'G}(X)_{{\bf Q}} \subset \mbox{ker }r'_V$ for $s >> 0$. Instead,
we can apply the non-equivariant Riemann-Roch isomorphism of
\cite[Chapter 18]{Fulton} $\Box$
\medskip
\noindent{\bf Example.}
Since inverse limits do not commute with tensoring with ${\bf Q}$,
$\lim_{\leftarrow V} \{ r'_V(K_i^{'G}(X))_{{\bf Q}}
\}$ need not be equal to $ (\lim_{\leftarrow V} r'_V(K_i^{'G}(X))) \otimes
{\bf Q}$. For example if $G = {\bf Z} / 2{\bf Z}$ and $X = pt$
then
$$K_0^{'G}(X) = K_0^G(X) = R(G) = {\bf Z}[u]/(u^2-1).$$
In this case $\lim_{\leftarrow V} r'_V(K_0^{'G}(X)) \simeq {\bf Z}_{(2)}$ (the
2-adic integers). Thus, $(\lim_{\leftarrow V} r'_V(K_0^{'G}(X))) \otimes {\bf Q} = {\bf Q}_{2}$.
On the other hand, $\lim_{\leftarrow V}(r'_V(K_0^{'G}(X)_{{\bf Q}})= {\bf Q}$.
However, there is a map
$$
\lim_{\leftarrow V} r'_V(K_i^{'G}(X)) \rightarrow \lim_{\leftarrow V} \{ r'_V(K_i^{'G}(X))_{{\bf Q}} \}
$$
which induces a map
$$
( \lim_{\leftarrow V} r'_V(K_i^{'G}(X))) \otimes {\bf Q} \rightarrow \lim_{\leftarrow V} \{
r'_V(K_i^{'G}(X))_{{\bf Q}} \}.
$$
Likewise there is a map
$$
( \widehat{K_i^{'G}(X)} ) \otimes {\bf Q} \rightarrow\widehat{K_i^{'G}(X)_{{\bf Q}}}
$$
which need not be an isomorphism.
\subsection{The equivariant Riemann-Roch isomorphism} \label{eqrr}
Before stating the Riemann-Roch theorem we need to define the equivariant
Chern character. For this we need a suitable completion of
$A^*_G(X)$. Elements of this completion should operate on
$\widehat{A_*^G(X)}$. There are three candidates for this completion:
(1) $\lim_{\leftarrow V} A^*_G(X \times V^f)$;
(2) $\widetilde{A^*_G(X)}$, the completion with respect to $\tilde{Q}$;
(3) $\widehat{A^*_G(X)}$, the completion with respect to $Q$. \\
If $X$ is smooth, then all three completions are equal. In general,
we do
not know whether they are equal because of the lack
of a suitable exact sequence for operational Chow groups. However,
all of these operate on $\widehat{A_*^G(X)}$ by virtue of the
isomorphisms proved in the last subsection. There are maps
$$
\widehat{A^*_G(X)} \rightarrow \widetilde{A^*_G(X)} \rightarrow
\lim_{\leftarrow V} A^*_G(X \times V^f).
$$
The first map is because $Q \subset \tilde{Q}$. The second is because
the map $A^*_G(X) \rightarrow A^*_G(X \times V^f)$ induces a map
$\widetilde{A^*_G(X)} \rightarrow A^*_G(X \times V^f)$ (because high
powers of $\tilde{Q}$ map to zero in $A^*_G(X \times V^f)$). These
maps are compatible with restrictions and hence induce a map to the
inverse limit. We will define a Chern character map with image in
$\widetilde{A^*_G(X)}$.
\begin{defn}
Define the equivariant Chern character
$$
ch_G:K_0^G(X) \rightarrow \widetilde{A^*_G(X)}_{{\bf Q}}$$
by the formula
$$ch_G(E) = r + c_1^G(E) + \frac{1}{2}(c_1^G(E)^2 - 2c_2^G(E))
+ \ldots.$$
\end{defn}
Let $V$ be a representation of $G$ such that $V -V^f$ has codimension
more than $i$. If $E \rightarrow X$ is an equivariant vector bundle,
then $c_i^G(E)$ restricts to $c_i(E \times^G V^f)$
under the restriction map $A_G^i(X) \rightarrow A_G^i(X \times^G V^f)$.
Thus, the equivariant Chern character $ch_G:K_0^G(X) \rightarrow \widetilde{
A_G^*(X)}$ restricts to the ordinary Chern character
$ch_{X \times^G V^f}:K_0(X \times^G V^f) \rightarrow A^*(X \times^G V^f)$.
\begin{prop}
There is a factorization
$$ch_G:K_0^G(X) \rightarrow \widehat{K_0^G(X)} \rightarrow
\widetilde{K_0^G(X)} \widetilde{A^*_G(X)}_{{\bf Q}}.$$
\end{prop}
Proof. The proof follows from the fact that $ch(P^n)$ and
$ch(\tilde{P}^n)$ are contained in $\tilde{Q}^n$ for any $n>0$.
$\Box$
We will also denote the map $\widehat{K_0^G(X)} \rightarrow
\widetilde{A^*_G(X)}_{{\bf Q}}$ by $ch_G$.
\begin{thm} \label{rockandroll} (Equivariant Riemann-Roch)\\
There are maps
$$\tau^G_X: \widehat{K_{0}^{'G}(X)}
\rightarrow \widehat{A_*^G(X)_{{\bf Q}}}$$
with the following properties (cf. \cite[Chapter 18]{Fulton}):
(1) $\tau^G_X$ is covariant for equivariant proper morphisms.
(2) If $\epsilon \in K_0^G(X)$ and $\alpha \in \widehat{K_0^{'G}(X)}$
then $\tau^G_X(\epsilon \alpha) = ch^G_X(\epsilon) \cap \tau_X^G(\alpha)$.
(Recall that $\widetilde{A_G^*(X)}$ operates on $\widehat{A_*^G(X)}$
because of the isomorphisms of completions.)
(3) If $f:X \rightarrow Y$ is a $G$-equivariant l.c.i. morphism, then
$$ch^G f_*(\epsilon) = f_*(((Td^G(T_f) ch^G(\epsilon))$$
and
$$\tau_X^G(f^*\alpha) = Td^G(T_f)f^*\tau_X^G(\alpha).$$
(4) If $V \subset X$ is a $G$-invariant subvariety of dimension $k$, then
$$\tau_X^G({\cal O}_V - [V]_G) \in F_{k-1}(\widehat{A_*^G(X)_{{\bf Q}}})$$
where $F_{k-1}(\widehat{A_*^G(X)_{{\bf Q}}})$ denotes the subgroup of cycles
of ``dimension'' strictly less than $k$.
(5) $\tau_X^G$ factors through the map $\widehat{K_0^{'G}(X)}
\rightarrow\widehat{K_0^{'G}(X)_{{\bf Q}}}$ and induces an isomorphism
between $\widehat{K_0^{'G}(X)_{\bf Q}}$ and $\widehat{A_*^G(X)_{\bf Q}}$.
(6) If $X$ is quasi-projective, and $i > 0$, then there is an
isomorphism
$$\tau_X^G: \lim_{\leftarrow V}{K_i^{'G}(X \times V^f)_{\bf Q}} \stackrel{\simeq} \rightarrow
\lim_{\leftarrow V}{A_*^G((X \times V^f),i)_{\bf Q}}.$$
\end{thm}
Proof. When $i = 0$, the restriction maps $r_V$ and $r'_V$ are surjective.
Thus
by Theorem \ref{completions},
$$\widehat{K_0^{'G}(X)} = \lim_{\leftarrow V} K_0{'G}(X \times V^f)$$
and
$$\widehat{A_*^G(X)_{{\bf Q}}} = \lim_{\leftarrow V} (A_*^G(X \times V^f) \otimes {\bf Q}).$$
By non-equivariant Riemann-Roch (\cite[Chapter 18]{Fulton}),
for each representation $V$ of $G$ there is a
map
$$\tau_{X \times V_f}: K_0^{'G}(X \times V^f) \rightarrow A_*^G(X
\times V^f) \otimes {\bf Q}$$
satisfying the analogues of (1) - (5). To prove the theorem it
suffices to show that maps $\{\tau_{X \times V^f}\}$ are compatible
with the inverse system maps. This is quite straightforward. Let $V$
and $W$ be representations. Let $\pi:(X \times V^f \oplus W)/G
\rightarrow (X \times V^f)$. Then $\pi$ is smooth and $Td_{T_{\pi}} =
1$. Thus $\pi^* \cdot (\tau_{X \times V_f}) = \tau_{X \times V^f
\oplus W} \cdot \pi^*$. Likewise, if $i:(X \times^G V^f \oplus W)
\rightarrow (X \times^G (V \oplus W)^f)$ is the inclusion map, then
$i^* \cdot \tau_{X \times (V \oplus W)^f} =\tau_{X \times V^f \oplus
W} \cdot i^*$.
To prove (6) we argue as above, using Bloch's Riemann-Roch isomorphism
$$K_i(X \times^G V^f) \cong A_*((X \times^G V^f),i).$$
$\Box$
\paragraph{Vistoli's conjecture}
By composing the map above with the natural map $K_0^{'G}(X)
\rightarrow \widehat{K_0^{'G}(X)}$ we get a map
$\tau^G_X: K_{0}^{'G}(X)
\rightarrow \widehat{A_*^G(X)_{{\bf Q}}}.$
When $G$ acts on $X$ with finite reduced stabilizers then Vistoli
\cite{Vi3} stated a theorem which asserted the existence of
a map
$$\tau_X: K_0^{'G}(X) \otimes {\bf Q} \rightarrow A_*([X/G] \otimes {\bf Q})$$
satisfying properties (1)-(4) above (here $[X/G]$ is the
Deligne-Mumford quotient stack). By Theorem \ref{moduli}, $A_*([X/G])
\otimes {\bf Q}= A_*^G(X) \otimes {\bf Q}$. Thus $I^d(A_*^G(X) \otimes {\bf Q})=0$
for $d >>0$, so $\widehat{A_*^G(X)_{{\bf Q}}} = A_*^G(X) \otimes {\bf Q}$. Thus
Vistoli's map is a special case of our map $\tau_X^G$, since it is
uniquely determined by properties (1)-(4). Vistoli noted that this
map need not be an isomorphism and made the following conjecture
about its kernel.
\begin{conj} (\cite[Conjecture 2.4]{Vi3})
Suppose that $G$ acts on $X$ with finite reduced stabilizers.
If $\alpha \in ker(\tau^G_X: K_0^{'G}(X) \rightarrow A_*([X/G])_{{\bf Q}})$
then there exists an element $\epsilon \in K_0^G(X)$
with every non-zero rank (meaning $\epsilon$ is represented by
a complex of locally free sheaves whose homology is non-zero
at the generic point of every subvariety) such that $\epsilon \alpha = 0$.
\end{conj}
The results of this section identify
the kernel: it is exactly the kernel of the completion map
$K_0^{'G}(X)\otimes {\bf Q} \rightarrow \widehat{K_0^{'G}(X)_{\bf Q}}$.
\begin{prop}
Suppose $K_0^G(X)$ is Noetherian and $K_0^{'G}(X)$
is finitely generated over $K_0^{G}(X)$. Then
$\alpha \in ker\; \tau_X^G$ if and only if $(1+\delta) \alpha = 0$
for some $\delta \in K_0^G(X)$ of (virtual) rank 0.
\end{prop}
Proof.
The proof follows immediately from Krull's theorem. $\Box$
\section{Localization}
In this section we discuss properties of equivariant Chow groups
that are similar to properties of equivariant cohomology.
In the first part, we give the relationship between
$A_*^G(X)$ and $A_*^T(X)$ when $G$ is a connected reductive
group with maximal torus $T$. The remainder of the section
is devoted to actions of (split) tori. In particular, we prove
two localization theorems (Theorems \ref{lcztn}, \ref{xxx}).
Following ideas of \cite{A-B} they
yield a characteristic free proof of the Bott residue
formula for split torus actions on complete varieties over a field
of arbitrary characteristic (Theorem \ref{bott}).
\subsection{Connected reductive groups}
Denote by $A^*_G$ or $R_G$ the equivariant Chow ring of a point (the
equivariant Chow groups of a point have a ring structure since a point
is smooth). If $G$ is a connected reductive group then by
\cite{E-G}, $R_G \otimes {\bf Q}=
Sym(\hat{T})^W \otimes {\bf Q}$, where $\hat{T}$ is the group of characters
of the maximal torus and $W$ is the Weyl group. When $G$ is special
in the sense of \cite{Sem-Chev} then $R_G = Sym(\hat{T})^W$ exactly
(\cite{E-G}). Under this identification we will write
$R_G^d = A^d_G = A^G_{-d}$. Via pullback
from a point, $A_*^G(X)$ has the structure of an $R_G$-module.
If $G = T$ is a split torus, then $W$ is trivial, and the identification
$R_T = Sym(\hat{T})$ is given explicitly as follows. If $\lambda \in \hat{T}$,
let $k_{\lambda}$ denote the corresponding 1-dimensional
representation of $T$, and let $L_{\lambda}$ denote the line bundle
$U \times^T k_{\lambda} \rightarrow U /T$. The map $\hat{T}
\rightarrow A^1_T$ given by $\lambda \mapsto c_1(L_{\lambda})$ extends
to a ring isomorphism $Sym(\hat{T}) \rightarrow R_T$. If $f:T
\rightarrow S$ is a homomorphism of tori, then there is a pullback map
$f^*:\hat{S} \rightarrow \hat{T}$. This extends to a ring homomorphism
$f^*: Sym(\hat{S}) \rightarrow Sym(\hat{T})$, or in other words, a map
$f^*: R_S \rightarrow R_T$.
\begin{prop} Let $G$ be a connected reductive group with split maximal
torus $T$ and Weyl group $W$. Then
$A_*^G(X) \otimes {\bf Q} = A_*^T(X)^W \otimes {\bf Q}$. If $G$ is special
the isomorphism holds with integer coefficients.
\end{prop}
Proof: If $G$ acts freely on $U$, then so does
$T$. Thus for a sufficiently large representation $V$,
$A_{i}^T(X) = A_{i+l -t}((X \times U)/T)$ and
$A_i^G(X) = A_{i+l-g}((X \times U)/G)$ (here
$l$ is the dimension of $V$, $t$ the dimension of $T$
and $g$ the dimension of $G$). On the other hand,
$(X \times U)/T$ is $G/T$ bundle over $(X \times U)/G$.
Thus
$A_{k}((X \times U/T)) \otimes {\bf Q} =A_{k+g-t}((X \times U)/G)^W \otimes {\bf Q}$
and if $G$ is special, then the equality holds integrally (\cite{E-G})
and the proposition follows.
$\Box$
\medskip
Thus, for connected reductive groups,
to compute equivariant Chow groups (at least
with rational coefficients), it suffices to understand
equivariant Chow groups for tori. We begin with the following
proposition.
\begin{prop}
If $T$ acts trivially on $X$, then $A_*^T(X) = A_*(X) \otimes
R_T$.
\end{prop}
Proof. If the action is trivial then $(U \times X)/T= U/T \times X$.
The spaces $U/T$ can be taken to be products
of projective spaces, so $A_*(U/T \times X) = A_*(X) \otimes A_*(U/T)$.
$\Box$
\medskip
{\bf Remark.} If the action is trivial,
the pullbacks $A^*X_T \rightarrow A^*X$ and $A^*(U/T) \rightarrow
A^*X_T$ induce an inclusion of $A^*X \otimes R_T
\subset A^*_T(X)$ as a subring. If $X$ is smooth, then
the inclusion is an isomorphism by Proposition \ref{opsmooth}.
\subsection{Fixed loci and the localization theorem}
For the remainder of this section,
all Chow groups have rational coefficients, and
for simplicity of exposition, we assume that tori are split.
If $X$ is a scheme with a $T$-action,
we may put a closed subscheme structure
on the locus $X^T$ of points fixed by
$T$.
Now $R_T= Sym(\hat{T})$ is a polynomial ring.
Set ${\cal Q}= (R_T^+)^{-1} \cdot R_T$, where $R_T^+$
is the multiplicative system of homogeneous elements of positive degree.
\begin{thm} \label{lcztn}(localization)
The map $i^T_*:A_*(X^T) \otimes {\cal Q} \rightarrow A_*^T(X) \otimes
{\cal Q}$ is
surjective, and if $X$ is quasi-projective it is an isomorphism.
\end{thm}
{\bf Remark.} The quasi-projectivity assumption is needed to apply
the long exact sequence for higher Chow groups. The strategy of the
proof is similar to \cite[Theorem 5.3]{Th2}.
\medskip
Proof. Applying the localization exact sequence for higher equivariant
Chow groups
$$\ldots \rightarrow A_*^T(X^T) \rightarrow A_*^T(X) \rightarrow
A_*^T(X-X^T) \rightarrow 0$$
the theorem follows from the following proposition.
\begin{prop} \label{fix}
If $T$ acts on $X$
without fixed points, then there exists $r \in R_T^+$ such that
$r \cdot A_*^T(X,m)= 0$. (Recall that $A_*^T(X,m)$ refers to
$T$-equivariant higher Chow groups.)
\end{prop}
Suppose $f: T \rightarrow S$ is a homomorphism of tori. As discussed
above, there is a pullback map $f^*: A^*_S \rightarrow A^*_T$.
\begin{lemma} \label{t-map}(cf. \cite{A-B})
Suppose there is a $T$-map
$X \stackrel{\phi} \rightarrow S$.
Then $t\cdot A_*^T(X)= 0$ for any $t=f^*s$ with $s \in R_S^+$.
\end{lemma}
Proof of Lemma \ref{t-map}.
Since $A^*_S$ is generated in degree 1, we may
assume that $s$ has degree 1. After clearing denominators
we may assume that $s = c_1(L_s)$ for some line bundle
on a space $U/S$. The action of $t=f^*s$ on $A_*(X_T)$ is just
given by $c_1(\pi_T^*f^*L_s)$ where $\pi_T$ is the
map $U \times^T X \rightarrow U/T$.
To prove the lemma we will show that this bundle is trivial.
First note that $L_s = U \times^S k$ for some action of $S$ on
the one-dimensional vector space $k$.
The pullback bundle on $X_T$ is the line bundle
$$U \times^T(X \times k) \rightarrow X_T$$
where $T$ acts on $k$ by the composition of $f:T \rightarrow S$
with the original $S$-action.
Now define a map
$$\Phi: X_T \times k \rightarrow U \times^T(X \times k)$$
by the formula
$$\Phi(e,x,v) = (e,x,\phi(x)\cdot v)$$
(where $\phi(x) \cdot v$ indicates the original $S$ action).
This map is well defined since
\begin{eqnarray*}
\Phi(et,t^{-1}x,v) & = &(et, t^{-1}x, \phi(t^{-1}x) \cdot v)\\
& = & (et,t^{-1}x,t^{-1} \cdot(\phi(x) \cdot v))
\end{eqnarray*}
as required. This map is easily seen to be an isomorphism
with inverse $(e,x,v) \mapsto (e,x,\phi(x)^{-1} \cdot v)$.
$\Box$
\medskip
Proof of Proposition \ref{fix}. Since $A_*^G(X) = A_*^G(X_{red})$
we may assume $X$ is reduced. Working with each component
separately, we may assume $X$ is a variety. Let $X^0 \subset X$
be the ($G$-invariant) locus of smooth points.
By Sumhiro's theorem \cite{Sumihiro}, the action of
a torus on a normal variety is locally
linearizable (i.e. every point has an affine invariant
neighborhood). Using this theorem it is easy to see
that the set
$X(T_1) \subset X^0$ of points with stabilizer
$T_1$ can be given the structure of a locally closed
subscheme of $X$.
Furthermore, only finitely many subgroups can occur as stabilizers
(Appendix, Lemma \ref{porb}), so there is
some $T_1$ such that $U= X(T_1)$ is open in $X^0$, and thus in $X$.
The torus $T'=T/T_1$ acts without stabilizers,
but the action of $T'$ on $U$ is not a priori proper. However,
by \cite[Proposition 4.10]{Th1}, we can replace $U$
by a sufficiently small open set so that $T'$ acts freely
on $U$ and a principal bundle quotient $U \rightarrow U/T$
exists. Shrinking $U$ further, we can assume that this
bundle is trivial, so there is a $T$ map $U \rightarrow T'$.
Hence, by the lemma, $t \cdot A^T_*(U) = 0$ for any $t \in A_T^*$
which is pulled back
from $A^*_{T'}$.
Let $Z = X -U$. By induction on dimension, we may assume $p \cdot
A^T_*Z = 0$ for some homogeneous polynomial $p \in R_T$. From the
long exact sequence of higher Chow groups,
$$\ldots A_*^T(Z,m) \rightarrow A_*^T(X,m) \rightarrow A_*^T(U,m) \rightarrow
\ldots$$
it follows that $tp$ annihilates $A_*^T(X)$ where
$t$ is the pullback of a homogeneous element of degree $1$
in $R_S$.
\medskip
Remark: Using only the short exact localization sequence for ordinary
equivariant Chow groups (which does not require an assumption
of quasi-projectivity) shows that $i_*$ is surjective.
$\Box$
\subsection{Explicit localization and the integration formula}
The localization theorem in equivariant cohomology has a more explicit
version for manifolds. This yields an integration formula from which
the Bott residue formula is easily deduced (\cite{A-B}, \cite{B-V}).
In this section we prove the analogous results for equivariant Chow
groups of smooth varieties. Because equivariant Chow theory has
formal properties similar to equivariant cohomology, the arguments are
almost the same as in \cite{A-B}. As before we assume that all tori
are split.
Let $F$ be a scheme with a trivial $T$-action.
If $E \rightarrow F$ is a $T$-equivariant vector bundle on
$F$, then $E$ splits canonically into a direct sum of vector subbundles
$\oplus_{\lambda \in \hat{T}} E_{\lambda}$, where $E_{\lambda}$
consists of the subbundle of vectors in $E$ on which $T$ acts by the
character $\lambda$. The equivariant Chern classes of an
eigenbundle $E_{\lambda}$ are given by the following lemma.
\begin{lemma} \label{l.trivchern}
Let $F$ be a scheme with a trivial $T$-action, and let
$E_{\lambda} \rightarrow F$ be a $T$-equivariant vector bundle of rank
$r$ such
that the action of $T$ on each vector in $E_{\lambda}$ is given by the
character $\lambda$. Then for any $i$,
$$
c^T_i(E_{\lambda}) = \sum_{j \leq i}
\left( \begin{array}{c} r-j \\
i-j \end{array}
\right)
c_j(E_{\lambda}) \lambda^{i-j}.
$$
In particular the component of $c_r^T(E_{\lambda})$ in $R^r_T$ is
given by $\lambda^r$. $\Box$
\end{lemma}
As noted above, $A^*_T(F) \supset A^*F \otimes R_T$. The lemma
implies that $c^T_i(E)$ lies in the subring $A^*F \otimes R_T$.
Because $A^N F = 0$ for $N > \mbox{dim }F$, elements of $A^i F$, for
$i>0$, are nilpotent elements in the ring $A^*_T(F)$. Hence an
element $\alpha \in A^d F \otimes R_T$ is invertible in $A^*_T(F)$ if
its component in $A^0 F \otimes R^d_T \cong R^d_T$ is nonzero.
For the remainder of this section $X$ will denote a smooth variety
with a $T$ action. If $X$ is smooth then by \cite{Iv} the fixed locus
$X^T$ is also smooth. For each component $F$ of the fixed locus $X^T$
the normal bundle $N_FX$ is a $T$-equivariant vector bundle over $F$.
Note that the action of $T$ on $N_FX$ is non-trivial.
\begin{prop}
If $F$ is a component of $X^T$
with codimension $d$ then $c_d^T(N_FX)$ is invertible
in $A^*_T(F) \otimes {\cal Q}$.
\end{prop}
Proof: By (\cite[Proof of Proposition 1.3]
{Iv}), for each closed point $f \in F$, the tangent space
$T_fF$ is equal to $(T_fX)^T$, so $T$ acts with non-zero weights on the
normal space $N_f = T_fX/T_fF$. Hence the characters $\lambda_i$
occurring in the
eigenbundle decomposition of $N_FX$ are all non-zero. By the
preceding lemma, the component of $c_d^T(N_FX)$ in $R^d_T$ is nonzero.
Hence $c_d^T(N_FX)$ is invertible
in $A^*_T(F) \otimes {\cal Q}$, as desired. $\Box$
\medskip
Using this result we can get, for $X$ smooth,
the following more explicit version
of the localization theorem.
\begin{thm} \label{xxx}(Explicit localization)
Let $X$ be a smooth (not necessarily quasi-projective) variety with a torus
action.
Let $\alpha \in A_*^T(X) \otimes {\cal Q}$.
Then $$\alpha = \sum_F
i_{F*}\frac{i^*_F\alpha}{c_{d_F}^T(N_FX)},$$ where the sum is over the
components $F$ of $X^T$ and $d_F$ is the codimension of $F$ in $X$.
\end{thm}
Proof: By the surjectivity part of the localization theorem,
we can write $\alpha = \sum_F
i_{F*}(\beta_F)$. Therefore, $i^*_F\alpha = i^*_Fi_{F*}(\beta_F)$
(the other components of $X^T$ do not contribute); by the
self-intersection formula, this is equal to $ c_{d_F}^T(N_FX) \cdot
\beta_F$. Hence $\beta_F = \frac{i^*_F\alpha}{c_{d_F}^T(N_FX)}$ as
desired. $\Box$
\medskip
If $X$ is complete, then the projection $\pi_X: X \rightarrow pt$
induces push-forward maps $\pi^T_{X*}: A^T_* X \rightarrow R_T$ and
$\pi^T_{X*}: A^T_* X \otimes {\cal Q} \rightarrow {\cal Q}$. There
are similar maps with $X$ replaced with any component $F$ of $X^T$.
Applying $\pi^T_{X*}$ to both sides of the explicit localization
theorem, and noting that $\pi^T_{X*} i_{F*} = \pi^T_{F*} $, we deduce
the ``integration formula'' (cf. \cite[Equation (3.8)]{A-B}).
\begin{cor}
(Integration formula) Let $X$ be smooth and complete, and
let $\alpha \in A_*^T(X) \otimes {\cal Q}$. Then
$$\pi_{X*}(\alpha) = \sum_{F \subset X^T}
\pi_{F*}\{\frac{i^*_F\alpha}{c_{d_F}^T
(N_FX)}\}$$
as elements of ${\cal Q}$. $\Box$
\end{cor}
\medskip
{\bf Remark.} If $\alpha$ is in the image of the natural map $A_*^T(X)
\rightarrow A_*^T(X) \otimes {\cal Q}$ (which need not be injective),
then the equation above holds in the subring $R_T$ of ${\cal Q}$. The
reason is that the left side actually
lies in the subring $R_T$; hence so does the right side. In the
results that follow, we will have expressions of the form $z = \sum
z_j$, where the $z_j$ are degree zero elements of ${\cal Q}$ whose sum
$z$ lies in the subring $R_T$. The pullback map from equivariant to
ordinary Chow groups gives a map $i^*: R_T = A^T_* (pt) \rightarrow {\bf Q}
= A_* (pt)$, which identifies the degree 0 part of $R_T$ with ${\bf Q}$.
Since $\sum z_j$ is a degree 0 element of $R_T$, it is identified via
$i^*$ with a rational number. Note that $i^*$ cannot be applied to
each $z_j$ separately, but only to their sum. In the integration and
residue formulas below we will identify the degree 0 part of $R_T$
with ${\bf Q}$ and suppress the map $i^*$. \medskip
The preceding corollary yields an integration formula for
an element $a$ of the ordinary Chow group $A_0 X$, provided that $a$ is
the pullback of an element $\alpha \in A^T_0 X$.
\begin{prop}
Let $a \in A_0 X$, and suppose that $a = i^* \alpha$ for $\alpha \in
A^T_0 X$. Then
$$
\mbox{deg }(a) = \sum_F \pi^T_{F*}\{\frac{i^*_F\alpha}{c_{d_F}^T
(N_FX)} \}
$$
\end{prop}
Proof: Consider the commutative diagram
$$\begin{array}{ccc}
X & \stackrel{i} \hookrightarrow & X_T\\
\downarrow\scriptsize{\pi_X} & & \downarrow\scriptsize{\pi^T_X}\\
\mbox{pt} & \stackrel{i} \rightarrow & U/T .
\end{array}$$
We have $\pi_{X*}(a) = \pi_{X*} i^*(\alpha) = i^* \pi^T_{X*}(\alpha)$.
Applying the integration formula gives the result. $\Box$
\subsection{The residue formula}
Let $E \rightarrow X$ be a $T$-equivariant vector bundle of rank $r$
on a complete, smooth $n$-dimensional variety. Let $p(x_1, x_2, \ldots
, x_r)$ be a polynomial of weighted degree $n$, where the degree of
$x_i$ is $i$. The integration formula above will allow us to compute
$\mbox{deg }(p(c_1(E), \ldots, c_r(E)) \cap [X])$
in terms of the
restriction of $E$ to $X^T$.
As a notational shorthand, write $p(E)$
for $p(c_1(E), \ldots, c_r(E))$ and $p^T(E)$ for $p(c^T_1(E), \ldots,
c^T_r(E))$.
Write $p^T(E|_{F})$ for $p(c^T_1(E|_{F}), \ldots,
c^T_r(E|_{F})) = i^*_F p(c^T_1(E), \ldots, c^T_r(E))$. Notice that
$p(E) \cap [X] = i^* (p^T(E) \cap [X_T])$. We can therefore apply the
preceding proposition to get the Bott residue formula.
\begin{thm} \label{bott}
(Bott residue formula) Let $E \rightarrow X$ be a $T$-equivariant
vector bundle of rank $r$ on a complete, smooth $n$-dimensional
variety, and let $p(x_1, x_2, \ldots , x_r)$ be a polynomial of weighted
degree $n$. Then
$$
\mbox{deg }(p(E) \cap [X]) = \sum_{F \subset X^T}
\pi^T_{F*}\{\frac{p^T(E|_{F}) \cap
[F]_T}{c_{d_F}^T (N_FX)} \} .
$$
$\Box$
\end{thm}
By Lemma \ref{l.trivchern} the
equivariant Chern classes $c^T_i(E|_{F})$ and $c_{d_F}^T (N_FX)$ can
be computed in terms of the characters of the
torus occurring in the eigenbundle
decompositions of $E|_{F}$ and $N_FX$ and the Chern classes of the
eigenbundles.
The above formula can then be readily converted (cf. \cite{A-B}) to more
familiar forms of the Bott residue formula not involving equivariant
cohomology. We omit the details. If the torus $T$ is
1-dimensional, then degree zero elements of ${\cal Q}$ are rational
numbers, and the right hand side of the formula is just a sum of
rational numbers. This is the form of the Bott residue formula which
is most familiar in practice.
\section{Actions of group schemes over an arbitrary base} \label{mixed}
Let $S$ be a regular scheme, and let $X/S$ be an $S$-scheme
with an action of a connected reductive group scheme $G/S$. With appropriate
assumptions on $G/S$ (see below), it is possible to use
Seshadri's results on geometric reductivity over an arbitrary
base to extend much of our theory.
If $S = Spec({\bf Z})$ and $G/S$ is reductive, then the
theory goes through more or less intact. In particular, if
$X/{\bf Z}$ is a smooth scheme acted on by a reductive group scheme
$G/{\bf Z}$, then there is an equivariant Chow ring $A^*_G(X)$. Such a ring
should be useful for studying intersection theory on moduli in
mixed characteristic.
\subsection{Definitions}
\begin{defn} Let $G/S$ be a smooth group scheme. Let $E/S$ be a vector
bundle (i.e., $Spec(\mbox{Sym}({\cal E^{.}})$
where ${\cal E}/S$ is a locally free
$G$-module). The bundle $E/S$ is said to be a representation of
$G/S$ if there is an action $G \times E \rightarrow E$ which
is linear on each fiber.
\end{defn}
We assume the following condition on $G/S$:
(*) There exist representations $E/S$ with a non-empty open
set $U/S$ such that $G/S$ acts freely on $U$.
\begin{prop}
If $G/S$ is a smooth group scheme, then condition (*) above is satisfied
if either
(1) $G/S$ is the pullback of
a group scheme $G_R/Spec(R)$ where $R$ is a Dedekind domain.
(2) The geometric fibers of $G/S$ are all semisimple with trivial
center.
\end{prop}
Proof. By \cite[Lemma 1, Proposition 3]{Seshadri}, $R(G_R)$ is
a projective $R$-module which contains a finitely generated $G$-invariant
$R$-module. Since $R$ is a Dedekind domain, this module is projective.
Pulling back to $S$ gives the desired representation.
By \cite[Expose II]{SGA3}, the Lie algebra $Lie(G/S)$ is a vector
bundle over $S$. Since $G$ has trivial center, the adjoint action of
$G$ on the vector bundle $Lie(G/S) \times_S Lie(G/S)$ is generically
free.
$\Box$
\medskip
Henceforth we will assume that $G/S$ is reductive.
If condition (*) holds, we can find representations $E/S$ and
open set $U/S$ so that $E-U$ has arbitrarily high codimension.
By Seshadri's theorem (\cite{Seshadri}) there is a principal
bundle quotient $U \rightarrow U/G$.
The arguments of Proposition \ref{inap} yield
\begin{prop} \label{qbase}
Assume one of the following:
(1) $X/S$ equivariantly embeds in a projective bundle over $S$.
(2) $X/S$ is normal.
\noindent
Then a principal bundle quotient $X \times U \rightarrow X \times^G U$
exists.
$\Box$ \end{prop}
As a consequence of Proposition \ref{qbase} we can define
equivariant Chow groups.
\begin{defn} Assume that condition (*) on $G/S$ holds, as well
as one of the hypotheses (1) or (2) of Proposition \ref{qbase}.
Define the $i$-th equivariant Chow group as $A_{i+l-g}(X \times^G U)$
where $l = \mbox{dim }(U/S)$ and $g=\mbox{dim }(G/S)$. As for
algebraic schemes, the definition is independent of the representation.
\end{defn}
\subsection{Results over an arbitrary base}
Since most of the results of intersection theory hold for schemes
over a regular base (\cite[Chapter 19]{Fulton}), most of the results
on equivariant Chow groups also hold.\\
In particular, the functorial properties with respect to
proper, flat and l.c.i maps hold.\\
If $S = Spec(R)$ where $R$ is a Dedekind domain, then there
is an intersection product on $A_*^G(X)$ for $X/S$ smooth.\\
If $G$ acts freely on $X$ and a quotient $X/G$ exists,
then $A_*^G(X) = A_*(X/G)$.\\
If the stabilizer group scheme for the action of $G$ on
$X$ is finite over $S$, and a quotient $X/G$ exists,
then we expect that $A_*^G(X)_{{\bf Q}} = A_*(X/G)_{{\bf Q}}$. However,
to prove such a statement using the techniques of this paper
would require a localization long exact sequence for higher
Chow groups over an arbitrary base.\\
If $S$ is regular and $T/S$ is a split torus, then the equality of
completions of $K_i^{'T}(X)$ with respect to either the augmentation
ideal of $K_0^T(S)$ or the augmentation ideal of $K_0^T(X)$ holds (cf.
Theorem \ref{completions}). The analogous equality of completions for
(higher) Chow groups also holds. From the torus case, we can deduce
the corresponding equality of completions for the totally split (i.e.
pulled back from the split groups over $Spec({\bf Z})$) classical groups $G
= Sl(n,S)$, $Sp(2n,S)$. The argument is the same as in
Section \ref{eqrr}. The key point is that for
these groups $K_i^{'G}(X)$ is a direct summand in $K_i^{'T}(X)$; this
can be proved (as in Lemma \ref{summand}) by realizing $G/B$ as a
sequence of projective bundles and applying Thomason's projective
bundle theorem. (Note that for group schemes $G$, as for groups,
there is a scheme $G/B$.) For $G = SO(n,S)$, $G/B$ can be realized as
a sequence of quadric bundles, and the analysis of \cite{E-G1} applied
to deduce the result with rational
coefficients. Once the analogue of Theorem \ref{completions} is
proved, the corresponding Riemann-Roch statements follow.\\
A form of the localization theorem for split torus
actions also holds over an arbitrary base. However, we can only
prove a localization isomorphism if the fixed locus is regularly
embedded in $X$. Again, the obstruction is the lack of
a long exact sequence for higher Chow groups over an arbitrary base.\\
Finally, if $G/S$ is smooth but not reductive and $G$ embeds
as a closed subgroup of $GL(n,S)$, then
a quotient $X \times^G U$ exists as an algebraic space. To develop
an equivariant intersection theory in this case would require further
facts about Chow groups of algebraic spaces.
\section{Appendix}
Here, we collect some useful results about actions
of algebraic groups acting on algebraic schemes in arbitrary characteristic.
\subsection{Torus actions}
\begin{lemma} \label{porb}
If $X$ is a variety with an action of a torus $T$, then there is
an open $U \subset X$ so that the stabilizer is constant for
all points of $U$.
\end{lemma}
Proof: It suffices to prove the lemma
after finite base change, so we may assume that
$T$ is split. Let $\tilde X \rightarrow X$ be the normalization
map. This map is $T$-equivariant and is an isomorphism
over an open set. Thus we may assume $X$ is normal. By Sumihiro's
theorem, the $T$ action on $X$ is locally linearizable, so it
suffices to prove the lemma when $X = V$ is a vector space and
the action is diagonal.
If $V = k^n$, then let $U = (k^*)^n$. The $n$-dimensional
torus ${\bf G}_m^n$ acts transitively on $U$ in the obvious way.
This action commutes with the given action of $T$. Thus the stabilizer
at each closed point of $U$ is the same.
$\Box$
\begin{prop} \label{complement}
Let $V$ be a vector space with a linear action of a torus $T$. Let
$V^f$ be the set of points with closed orbits and trivial
stabilizers.
Then the set $V^u = V-V^f$ is a finite union of linear subspaces.
\end{prop}
Proof: Again, after finite base change we may assume that $T$ is
split. Let $V^c$ denote the set of points in $V$ whose $T$-orbits are
closed in $V$. We first prove that $V-V^c$ is a finite union of
linear subspaces. Choose a basis $\{ v_i \}$ on which $T$ acts
diagonally. If $\mbox{dim }T=1$, then the $T$-orbit of $v = \sum a_i
v_i$ is not closed if and only if the weights of the non-zero coordinates
are either all non-negative or all non-positive. Thus, $V-V^c$ is
defined by the vanishing of various subsets of coordinate hyperplanes,
hence is a finite union of linear subspaces. For $T$ of arbitrary
dimension, $T \cdot v$ is closed iff for all 1-dimensional subtori $S
\subset T$, $S \cdot v$ is closed (this follows from \cite[Prop.
2.4]{GIT}). This in turn holds if $S \cdot v$ is closed for a
sufficiently general $S \subset T$, so the result follows from the
case $\mbox{dim }T=1$.
To complete the proof we must show that the complement of the set
of points with trivial stabilizer is a union of linear subspaces.
This follows from two facts.\\
(1) If $G \subset T$ is a subgroup, then $L_G = \{{\bf v} \in V |
G \subset Stab({\bf v}) \}$ is a linear subspace.
(2) $V$ can be covered by a finite number of $L_G$'s by Lemma
\ref{porb}.
$\Box$
\subsection{Principal bundles}
\begin{lemma} \label{q.exist} (\cite{E-G})
Let $G$ be an algebraic group. For
any $i > 0$, there is a representation $V$ of $G$ and an open set
$U \subset V$ such that $V-U$ has codimension more than $i$
and such that a principal bundle quotient $U \rightarrow U/G$
exists.
\end{lemma}
Proof. Embed $G$ into $GL(n)$ for some $n$. Assume that
$V$ is a representation
of $GL(n)$ and $U \subset V$ is an open set such that a principal
bundle quotient $U \rightarrow U/GL(n)$ exists. Since $GL(n)$
is special, this principal bundle is locally trivial in the Zariski
topology. Thus $U$ is locally isomorphic to $W \times GL(n)$ for
some open $W \subset U/GL(n)$. A quotient $U/G$ can be constructed
by patching the quotients $W \times GL(n) \rightarrow W \times (GL(n)/G)$.
We have thus reduced to the case $G=GL(n)$. Since the action of
an affine group is locally finite, there as an equivariant
closed embedding of $G \hookrightarrow V$ into a sufficiently large
vector space $V/k$. Consider the open set $U \subset V$ of points with trivial
stabilizers
which are stable for the
$G$ action on $V$. Since $G$ acts freely on itself,
$G \subset U$; hence $U$ is non-empty.
Since the stabilizers are trivial, the action on
$U$ is free, and the GIT quotient $U \rightarrow U/G$ is a principal
bundle. Now if $V_1 = V \oplus V$, then
(\cite[Proposition 1.18]{GIT}) $U_1 = (U \oplus V) \cup (V \oplus U)
\subset V_1^s$. Thus a principal bundle quotient $U_1 \rightarrow U_1/G$
exists, and the codimension of $V_1 - U_1$ is strictly smaller
than the codimension of $V-U$. Thus, by taking the direct sum of
a sufficiently large number of copies $V$, we may assume that $V - U$
has arbitrarily high codimension. $\Box$
\medskip
Let $G$ be an algebraic group, let $U$ be a scheme on which $G$ acts
freely, and suppose that a principal bundle quotient $U \rightarrow
U/G$ exists.
\begin{prop} \label{inap}
Let $X$ be an algebraic scheme with a $G$ action.
Assume that at least one of the following hypotheses holds.\\
(1) $X$ is quasi-projective with a linearized $G$-action.\\
(2) $G$ is connected and $X$ is equivariantly embedded as a closed
subscheme of a normal variety.\\
(3) $G$ is special.\\
Then a principal bundle quotient
$X \times U \rightarrow (X \times^G U)$ exists.
\end{prop}
Proof. If $X$ is quasi-projective with a linearized action, then there
is an equivariant line bundle on $X \times U$ which is relatively
ample for the projection $X \times U \rightarrow U$. By \cite[Prop 7.1]{GIT}
a principal bundle quotient $X \times^G U$ exists.
Now suppose that $X$ is normal and $G$ is connected.
By Sumhiro's theorem \cite{Sumihiro}, $X$ can be covered
by invariant quasi-projective open sets which have a linearized
$G$ action. Thus, by \cite[Prop 7.1]{GIT} we can construct
a quotient $X_G = X \times^{G} U$ by patching the quotients of
the quasi-projective open sets in the cover.
If $X$ equivariantly embeds in a normal variety $Y$, then by the above
paragraph a principal bundle quotient $Y \times U \rightarrow Y
\times^G U$ exists. Since $G$ is affine, the quotient map is affine,
and $Y \times U$ can be covered by affine invariant open sets. Since
$X \times U$ is an invariant closed subscheme of $Y \times U$, $X
\times U$ can also be covered by invariant affines. A quotient $X
\times^G U$ can then be constructed by patching the quotients of the
invariant affines.
Finally, if $G$ is special, then $U \rightarrow U/G$ is a
locally trivial bundle in the Zariski topology. Thus
$U = \bigcup\{U_\alpha\}$ where
$\phi_{\alpha}:U_\alpha \simeq G \times W_\alpha$
for some open $W_\alpha \subset U/G$. Then $\psi_{\alpha}:
X \times U_{\alpha} \rightarrow X \times W_{\alpha}$ is a quotient,
where $\psi_{\alpha}$ is defined by the formula
$(x,w,g) \mapsto (g^{-1}x,w)$
(Here we assume that $G$ acts on the left on both factors
of $X \times G$).
\subsection{Quotients}
Following Vistoli, we define a geometric quotient $X \stackrel{\pi}
\rightarrow Y$
to be a map which satisfies properties i)-iii) of \cite[Definition
0.6]{GIT}. In particular, we do not require that ${\cal O}_Y = \pi_*({\cal O}
_X)^G$. The advantage of this definition is that is preserved under base
change. In characteristic 0 there are no inseparable extensions,
so our definition agrees with Mumford's
(\cite[Prop 0.2]{GIT}).
The following proposition is an analogue of \cite[Prop 2.6]{Vi}. The proof
is similar.
\begin{prop} \label{whizzbang}
Let $G$ act properly on a variety $X$ (hence with finite, but possibly
non-reduced stabilizers), so that a geometric quotient $X
\rightarrow Y$ exists. Then there is a commutative diagram of
quotients
$$\begin{array}{ccc} Z & \rightarrow & X\\
\downarrow & & \downarrow\\
Q & \rightarrow & Y
\end{array}$$
where $Z \rightarrow Q$ is a principal $G$-bundle and the horizontal
maps are finite and surjective.
\end{prop}
Proof.
By \cite[Lemma p. 14]{GIT}, there is a finite map $Q \rightarrow
Y$, with $Q$ normal,
so that the pullback $X_1 \stackrel{\pi}\rightarrow Q$ has a section
in the neighborhood of every point. Cover $Q$
by a finite number of open sets $\{U_\alpha\}$ so that
$X_1 \rightarrow Q$ has a section $U_\alpha \stackrel{s_{\alpha}}
\rightarrow
V_{\alpha}$ where $V_{\alpha} = \pi^{-1}(U_{\alpha})$.
Define a $G$-map
$$\phi_{\alpha}: G \times U_\alpha \rightarrow V_\alpha$$
by the formula
$$(g,y) \mapsto gs_\alpha(y).$$
The action is proper so
each $\phi_\alpha$ is proper. Since the stabilizers
are finite, $\phi_{\alpha}$ is in fact finite.
To construct a principal bundle $Z \rightarrow Q$ we must glue
the $G \times U_{\alpha}$'s along
their intersection. To do this we will find isomorphisms
$\phi_{\alpha\beta}: s_\alpha(U_{\alpha\beta}) \rightarrow
s_{\beta}(U_{\alpha\beta})$ which satisfy the cocycle
condition.
For each $\alpha, \beta$, let $I_{\alpha\beta}$ be the scheme
which parametrizes isomorphisms of $s_\alpha$ and $s_\beta$
over $U_{\alpha\beta}$ (i.e. a section $U_{\alpha\beta}
\rightarrow I_{\alpha\beta}$ corresponds to a global isomorphism
$s_\alpha(U_{\alpha\beta}) \rightarrow s_\beta(U_{\alpha\beta})$).
The scheme $I_{\alpha\beta}$ is finite
over $U_{\alpha\beta}$ (but possibly totally ramified
in characteristic
$p$) since it is defined by the cartesian diagram
$$\begin{array}{ccc}
I_{\alpha\beta} & \rightarrow & U_{\alpha\beta} \\
\downarrow & & \small{1 \times s_\beta} \downarrow\\
G\times U_{\alpha\beta} & \stackrel{1 \times \phi_\alpha} \rightarrow &
U_{\alpha\beta} \times V_{\alpha\beta}
\end{array}$$
(Note that $I_{\alpha\alpha}$ is the stabilizer of $s_\alpha(U_\alpha)$.)
Over $U_{\alpha\beta\gamma}$ there is a composition giving
multiplication morphisms which are surjective when $\gamma = \beta$.
$$I_{\alpha\beta} \times_{U_{\alpha\beta\gamma}} I_{\beta\gamma}
\rightarrow I_{\alpha\gamma}$$ which gives
multiplication morphisms which are surjective when $\gamma = \beta$.
After a suitable finite (but possibly inseparable) base change, we may
assume that there is a section $U_{\alpha\beta} \rightarrow
I_{\alpha\beta}$ for every irreducible component of $I_{\alpha\beta}$.
(Note that $I_{\alpha\beta}$ need not be reduced.) Fix an open set
$U_{\alpha}$. For $\beta \neq \alpha$ choose a section
$\phi_{\alpha\beta}: U_{\alpha\beta} \rightarrow I_{\alpha\beta}$.
Since the $I_{\alpha\beta}$'s split completely and $I_{\alpha\alpha}$
is a group scheme, there are sections
$\phi_{\beta\alpha}:U_{\alpha\beta} \rightarrow I_{\beta\alpha}$ so
that $\phi_{\alpha\beta} \cdot \phi_{\beta\alpha}$ is the identity
section of $U_{\alpha\alpha}$. For any $\beta, \gamma$ we can define a
section of $I_{\beta\gamma}$ over $U_{\alpha\beta\gamma}$ as the
composition $\phi_{\beta\alpha} \cdot \phi_{\alpha\gamma}$. Because
$I_{\beta\gamma}$ splits, the $\phi_{\beta\alpha}$'s extend to
sections
over $U_{\beta\gamma}$.
By construction, the $\phi_{\beta\gamma}$'s satisfy the cocycle condition.
We can now define $Z$ by gluing the sets $G \times U_{\beta}$
along the $\phi_{\beta\gamma}$'s.
$\Box$
|
1996-04-05T20:21:28 | 9603 | alg-geom/9603006 | en | https://arxiv.org/abs/alg-geom/9603006 | [
"alg-geom",
"math.AG"
] | alg-geom/9603006 | Yuri Zarhin | Yu. G. Zarhin | p-adic abelian integrals and commutative Lie groups | LaTeX2e | null | null | null | null | The aim of this paper is to propose an ``elementary" approach to Coleman's
theory of p-adic abelian integrals. Our main tool is a theory of commutative
p-adic Lie groups (the logarithm map); we use neither dagger analysis nor
Monsky-Washnitzer cohomology theory. Notice that we also treat the case of bad
reduction. A preliminary version of this paper appeared as Expos\'e 9 dans
``Problemes Diophantiens 88-89" (D. Bertrand, M. Waldschmidt), Publ. Math.
Univ. Paris VI 90(1990), 15 pp.
| [
{
"version": "v1",
"created": "Tue, 5 Mar 1996 21:35:55 GMT"
},
{
"version": "v2",
"created": "Thu, 7 Mar 1996 16:19:18 GMT"
},
{
"version": "v3",
"created": "Tue, 12 Mar 1996 21:24:52 GMT"
},
{
"version": "v4",
"created": "Fri, 5 Apr 1996 18:12:31 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Zarhin",
"Yu. G.",
""
]
] | alg-geom | \section{Logarithm maps}
Let $p$ be a prime, ${\bold Q}_p$ the field of $p-$adic numbers, ${\bold C}_p$ the completion
of its algebraic closure.
Let $K$ be a complete subfield of ${\bold C}_p$. Clearly, $K$ contains ${\bold Q}_p$. We will always deal with the valuation map
$$v:K^*\to{\bold Q}$$
normalized by the condition $v(p)=1$. We will view $v$ as a homomorphism of the (multiplicative) $K-$Lie group $K^*$ into the discrete (additive) $K-$Lie group ${\bold Q}$ of rational numbers.
Let $G$ be a commutative $K-$algebraic group, $\mbox{Lie}(G)$ its Lie algebra,
$\Omega^1_{\mbox{inv}}(G)$ the $K-$vector space of invariant differential forms of degree 1 on $G$. One may identify naturally $\Omega^1_{\mbox{inv}}(G)$ with the dual of $\mbox{Lie}(G)$, i.e.,
$$\Omega^1_{\mbox{inv}}(G) =\mbox{Hom}_K(\mbox{Lie}(G),K).$$
Let us consider the $K-$analytic Lie group $G(K)$. Its Lie algebra coincides
with $\mbox{Lie}(G)$ (\cite{SH}, Ch. 2, Sect. 2.3 or \cite{Weil1}, Appendix III).
If $u: G\to H$ is a homomorphism of $K-$algebraic groups, then it induces
a homomorphism $G(K)\to H(K)$ of the $K-$analytic Lie groups and the corresponding algebraic and analytic tangent maps $du:\mbox{Lie}(G)\to\mbox{Lie}(H)$ coincide.
Recall (\cite{Bourbaki}, Chapitre III, 7.6) the properties of the logarithm map
$$\mbox{log}_{G(K)}: G(K)_f\to \mbox{Lie}(G).$$
Here $G(K)_f$ is the smallest open subgroup of $G(K)$ such that the quotient
$G(K)/G(K)_f$ does not contain non-zero torsion elements,
$\mbox{log}_{G(K)}$ is a $K-$analytic homomorphism, whose tangent map
$$d\mbox{log}_{G(K)}:\mbox{Lie}(G)\to \mbox{Lie}(\mbox{Lie}(G))=\mbox{Lie}(G)$$
is the identity map. These properties determine $G(K)_f$ and $\mbox{log}_{G(K)}$ uniquely.
Here are some examples and remarks.
\begin{enumerate}
\item
Let $G=\mbox{Spec} K[t,t^{-1}]$ be the multiplicative group ${\mathbf G}_m$. Then
$G(K)=K^*, \mbox{Lie}(G)=K\cdot t \frac{d}{dt}\quad =K$ and $G(K)_f$ coincides with
the group of units $U_K=\{x\in K^*\mid v(x)=0\}$
(\cite{Bourbaki}, Ch. 3, Sect. 7, Ex. 3). The logarithm map
$\mbox{log}_{K^*}: U_K\to K$ coincides with the usual logarithm $\mbox{log}$
(defined via the usual convergent power series) on the subgroup of
``principal" units $U_K^1=\{x\in K^*\mid v(1-x)>0\}$.
\item
Let $G=\mbox{Spec}\ K[t]$ be the additive group ${\mathbf G}_a$ then $G(K)=G_f(K)=K$, $\mbox{Lie}(G)=K\cdot\frac{d}{dt}\quad =K$ and $\mbox{log}_K : K\to K$ is the identity map.
\item
Let $G$ be an abelian variety. Then $G(K)=G(K)_f$, because if $U$ is an open
subgroup of $G(K)$ then the quotient $G(K)/U$ is a torsion group
(see Coleman(\cite{C2}; the proof is based on results of Raynaud \cite{R}). If
$K$ is a finite algebraic extension of ${\bold Q}_p$, i.e., is locally compact, the equality follows easily from the compactness of $G(K)$.
\item
Let $G,H$ be commutative $K-$algebraic groups. Then
$(G\times H)(K)=G(K)\times H(K)$, $\mbox{Lie}(G\times H)=\mbox{Lie}(G)\times\mbox{Lie}(H),
(G\times H)(K)_f=G(K)_f\times H(K)_f$ and in obvious notation
$\mbox{log}_{(G\times H)(K)}=(\mbox{log}_{G(K)},\mbox{log}_{H(K)})$.
\item
If $u: G\to H$ is a homomorphism of $K-$algebraic groups, then $u(G(K)_f)\subset H(K)_f$ and
$$du\ \mbox{log}_{G(K)}=\mbox{log}_{H(K)} u.$$
\item
If $L$ is a closed algebraic $K-$subgroup of $G$ then
$$L(K)_f=G(K)_f\bigcap L(K)$$
and the restriction of $\mbox{log}_{G(K)}$ to $L(K)_f$ coincides with
$$\mbox{log}_{L(K)}:L(K)_f\to \mbox{Lie}(L)\subset \mbox{Lie}(G).$$
\item
Let $K' \subset {\bold C}_p$ be a complete overfield of $K$ and $G(K')$ be the $K'-$analytic Lie group of all $K'-$points of $G$. Then
$$G(K)_f=G(K')_f\bigcap G(K)$$
and the restriction of $\mbox{log}_{G(K')}$ to $G(K)_f$ coincides with
$$\mbox{log}_{G(K)}:G(K)_f\to \mbox{Lie}(G)\subset \mbox{Lie}(G)\otimes_K K'=\mbox{Lie}(G(K')).$$
\end{enumerate}
Since $G(K)/G(K)_f$ is torsion-free, one may easily extend $\mbox{log}_{G(K)}$ to a homomomorphism
$G(K)\to\mbox{Lie}(G)$, which automatically is analytic and whose tangent map is the identity map. Now we explain how this extension can be chosen canonically for all $G$ , in order to keep the functoriality properties.
\vskip 1cm
{\bf Step 1} We start with the case of $G={\mathbf G}_m$. Notice that $v$ defines an isomorphism between $C^*/U_K$ and the additive group ${\bold Q}$. So, in order to extend $\mbox{log}$ to a homomorphism ${\bold C}_p^*\to{\bold C}_p$, which sends $K^*$ into $K$, one has only to
choose $c \in K$ and fix a branch of $p-$adic logarithm
$$\mbox{log}=\mbox{log}^{(c)}:{\bold C}_p^*\to {\bold C}_p,$$
by putting
$$ \mbox{log}(p)=\mbox{log}^{(c)}(p):=c$$
\cite{C2}, \cite{Z1}.
(If $\mbox{Log}:K^*\to K$ is another branch of the logarithm then
$$\mbox{Log}=\mbox{log}+(\mbox{Log}(p)-\mbox{log}(p))v .)$$
If $K' \subset {\bold C}_p$ is a complete overfield of $K$ then $\mbox{log}({K'}^*)\subset K'$. Notice, that for any automorphism $\sigma$ of $K'/K$ which preserves the absolute value (i.e., $v$) ,
$$\mbox{log}(\sigma(x))=\sigma((\mbox{log}(x)) \quad\mbox{\rm for all } x \in {K'}^*.$$
\vskip 1cm
{\bf Step 2} Let us extend the logarithm map for a split torus $G={\mathbf G}_m^r$. We
have $G(K)=(K^*)^r, \mbox{Lie}(G(K))=K^r$ and extend the logarithm map as follows.
$$\mbox{log}^{(c)}_{(K^*)^r}(x_1, \ldots , x_r)=(\mbox{log}(x_1), \ldots \mbox{log}(x_r) ).$$
One may easily check that if $u:G={\mathbf G}_m^r\to H={\mathbf G}_m^n$ is a homomorphism of
algebraic $K-$tori then
$$\mbox{log}^{(c)}_{ H(K)_f}(ux)=du(\mbox{log}^{(c)}_{G(K)}(x)).$$
Indeed, there exists an integral $r\times n$-matrix $(a_{ij})$ such that $u$ acts via
$$((x_1,\ldots ,x_r)\mapsto (y_1,\ldots ,y_n), \quad y_j=\prod_{i=1}^r x_i^{a_{ij} }\quad (j=1,\ldots n)$$
and the matrix of the tangent linear map
$$du: K^r=\mbox{Lie}(G)\to\mbox{Lie}(H)=K^r$$
coincides with $(a_{ij})$. Now, one has only to notice that
$$\mbox{log}^{(c)}(\prod_{i=1}^r x_i^{a_{ij}})=\sum_{i=1}^r a_{ij} \mbox{log}^{(c)}(x_i ).$$
Let $K' \subset {\bold C}_p$ be a complete overfield of $K$. Then we may consider the
$K'-$analytic Lie group $G(K')=({K'}^*)^r$ with the $K'-$Lie algebra $\mbox{Lie}(G)_{K'}
={K'}^r$ and define the extended logarithm map
$$\mbox{log}^{(c)}_{({K'}^*)^r}: G(K')=({K'}^*)^r\to\mbox{Lie}(G)_{K'}={K'}^r, $$
$$ (x_1, \ldots , x_r)\mapsto(\mbox{log}(x_1), \ldots \mbox{log}(x_r) ).$$
Clearly, $\mbox{log}^{(c)}_{({K'}^*)^r}$ coincides with $\mbox{log}^{(c)}_{(K^*)^r}$ on
$G(K)=(K^*)^r\subset ({K'}^*)^r=G(K')$.
Notice that for any automorphism $\sigma$ of $K'/K$ which preserves the absolute value (i.e., $v$),
$$\mbox{log}^{(c)}_{({K'}^*)^r}(\sigma(x))=\sigma(\mbox{log}^{(c)}_{({K'}^*)^r}(x))
\quad\mbox{\rm for all } x \in ({K'}^*)^r=G(K').$$
\vskip 1cm
{\bf Step 3} Now, we do the case of arbitrary algebraic $K-$torus $T$. Let us
choose a fiinite Galois extension $K'/K$ , which sits in ${\bold C}_p$ and such that
$T$ splits over $K'$, i.e., there exists an isomorphism of $K'-$algebraic tori
$\phi:T_{K'}\cong {\mathbf G}_m^r$. This allows us to identify $T(K')$ with $({K'}^*)^r$ and
$\mbox{Lie}(T)_{K'}=\mbox{Lie}(T)\otimes_K K'$ with ${K'}^r$. Then there exists a
homomorphism (cocycle) of the Galois group $\Gamma$ of $K'/K$
$$\Gamma\to \mbox{Aut}({\mathbf G}_m^r)=\mbox{GL}(r,{\bold Z}),\quad \sigma\mapsto \psi_{\sigma},$$
such that the natural Galois action on $T(K')$ is defined by the formula
$$\psi_T(\sigma):x\mapsto \psi_{\sigma}(\sigma(x)) \quad
\mbox{for all } \sigma \in \Gamma, x \in ({K'}^*)^r = T(K').$$
(Here $\sigma(x)=\sigma(x_1,\ldots , x_r)=(\sigma x_1, \ldots , \sigma x_r)$
for $x=(x_1,\ldots , x_r) \in ({K'}^*)^r$.)
Clearly,
$$T(K)=\{x\in ({K'}^*)^r\mid x=\psi_{\sigma}(\sigma(x))
\forall \sigma \in \Gamma\},$$
$$\mbox{Lie}(T)=\{b \in {K'}^r\mid b=\psi_{\sigma}(\sigma(b))
\forall \sigma \in \Gamma\}.$$
It follows easily that
$$\mbox{log}^{(c)}_{({K'}^*)^r}(T(K)) \subset \mbox{Lie}(T)$$
and therefore one may define
$$\mbox{log}^{(c)}_{T(K)}:T(K)\to\mbox{Lie}(T)$$
as the restriction of $\mbox{log}^{(c)}_{({K'}^*)^r}$ to $T(K)$. One may easily
check, using the results of the previous step, that the definition of
$\mbox{log}^{(c)}_{T(K)}$ does not depend on the choice of isomorphism
between $T_{K'}$ and ${\mathbf G}_m^r$.
\vskip 1cm
{\bf Step 4} Now, assume that $G$ is a connected commutative linear algebraic group. Then $G$ is isomorphic to the product $T\times {\mathbf G}_a^n$ where $T$ is algebraic $K-$torus. Then
$$G(K)=T(K)\times K^n, \quad \mbox{Lie}(G)=\mbox{Lie}(T)\times K^n$$
and we define
$$\mbox{log}^{(c)}_{T(K)\times K^n}:T(K)\times K^n\to \mbox{Lie}(T)\times K^n$$
by the formula
$$\mbox{log}^{(c)}_{T(K)\times K^n}(x, a_1, \ldots , a_n)=(\mbox{log}^{(c)}_{T(K)}(x), a_1, \ldots , a_n).$$
\vskip 1cm
{\bf Step 5} Let $G$ be a connected commutative algebraic $K-$group. Then, by a theorem of Chevalley, $G$ sits in a short exact sequence
$$0\to L\to G\to A\to 0$$
where $L$ is a connected linear algebraic group and $A$ is an abelian variety. This gives rise to the short exact sequence of the $K-$Lie algebras
$$0\to\mbox{Lie}(L)\to\mbox{Lie}(G)\to \mbox{Lie}(A)\to 0$$
and to the exact sequence of the $K-$Lie groups
$$0\to L(K)\to G(K)\to A(K).$$
It follows easily that the image $U$ of $G(K)_f$ in $A(K)$ is an open subgroup of $A(K)$. Recall that the quotient $A(K)/U$ is a torsion group. This implies that the quotient $G(K)/(L(K)\ G(K)_f)$ is also a torsion group, because it is isomorphic to a subgroup of $A(K)/U$.
It follows that for each $x\in G(K)$ there exist a positive integer $m$ and an element $x_f\in G(K)_f$ such that
$$y=x^m (x_f)^{-1} \in L(K).$$
In other words,
$$x^m =x_f y, \quad x_f \in G_f(K), y\in L(K).$$
Now, let us put
$$\mbox{log}^{(c)}_{G(K)}(x)=\frac{1}{m}(\mbox{log}_{G(K)}(x_f) +\mbox{log}^{(c)}_{L(K)}(y) )\in \mbox{Lie}(G).$$
If
$$x^n =x'_f y', \quad x_f \in G_f(K), y\in L(K)$$
then $x_f^n y^n=(x'_f)^m (y')^m$, i.e.,
$$z=x_f^n (x'_f)^{-m} =(y')^m y^{-n} \in G(K)_f \bigcap L(K)=L(K)_f$$
and therefore
$$n\cdot \mbox{log}_{G(K)}(x_f)-m\cdot \mbox{log}_{G(K)}(x'_f)=\mbox{log}_{G(K)}(z)=\mbox{log}_{L(K)}(z)=\mbox{log}^{(c)}_{L(K)}(z).$$
This implies that
$$\frac{1}{m}(\mbox{log}_{G(K)}(x_f) +\mbox{log}^{(c)}_{L(K)}(y) )=\frac{1}{n}(\mbox{log}_{G(K)}(x'_f) +\mbox{log}^{(c)}_{L(K)}(y') )\in \mbox{Lie}(G),$$
i.e., the definition of $\mbox{log}^{(c)}_{G(K)}(x)$ does not depend on the choice
of $m$ and $x_f$. It also follows easily that $\mbox{log}^{(c)}_{G(K)}$ is a
homomorphism from $G(K)$ to $\mbox{Lie}(G)$, which coincides with $\mbox{log}_{G(K)}$ on $G(K)_f$.
\vskip 1cm
{\bf Step 6} Assume that $G$ is not necessarily connected commutative algebraic group. Let $G^0$ be its connected identity component. We write $n$ for the index of $G^0$ in $G$. Then
$\mbox{Lie}(G)=\mbox{Lie}(G^0)$ and $x^n\in G^0(K) \forall x\in G(K)$. We put
$$\mbox{log}^{(c)}_{G(K)}(x)=\frac{1}{n} \mbox{log}^{(c)}_{G^0(K)}(x^n).$$
\vskip 1cm
The following statement summarizes the results of the steps 1-6.
\vskip 1 cm
{\bf Theorem.} {\sl To each commutative} $K-${\sl algebraic group} $G$ {\sl one may attach a} $K-${\sl analytic homomomorphism}
$$\mbox{log}^{(c)}_{G(K)}\to \mbox{Lie}(G), $$
{\sl enjoying the following properties}:
\begin{enumerate}
\item
{\bf (Choice of branch)} {\sl If} $K={\mathbf G}_m$ {\sl then} $G(K)=K^*$ {\sl and}
$$\mbox{log}^{(c)}_{G(K)}=\mbox{log}^{(c)}:K^*\to K=K\cdot t \frac{d}{dt}.$$
\item
{\bf (Logarithm property)} {\sl The restriction of} $\mbox{log}^{(c)}_{G(K)}$ {\sl to }$G(K)_f$ {\sl coincides with} $\mbox{log}_{G(K)}$. {\sl In particular, its tangent map coincides with the identity map} $\mbox{Lie}(G)\to\mbox{Lie}(G)$.
\item
{\bf (Functoriality)} {\sl If} $G\to H$ {\sl is a homomorphism of commutative} $K-${\sl algebraic groups then}
$$du \ \mbox{log}^{(c)}_{G(K)}=\mbox{log}^{(c)}_{H(K)} u. $$
\item
{\bf (Product formula)} {\sl If} $G, H$ {\sl are commutative} $K-${\sl algebraic groups then} $(G\times H)(K)=G(K)\times H(K), \mbox{Lie}(G\times H)=\mbox{Lie}(G)\times\mbox{Lie}(H)$ {\sl and in obvious notation}
$$\mbox{log}^{(c)}_{(G\times H)(K)}=(\mbox{log}^{(c)}_{G(K)},\mbox{log}^{(c)}_{H(K)}).$$
\item
{\bf (Ground field extension)} {\sl Let} $K' \subset {\bold C}_p$ {\sl be a complete overfield of} $K$ {\sl and} $G'=G\times K'$ {\sl be the corresponding commutative} $K'-${\sl algebraic group. Then} $G'(K')=G(K)$ is a commutative $K'-${\sl analytic Lie group}, {\sl the} $K'-${\sl Lie algebra} $\mbox{Lie}(G')=\mbox{Lie}(G)\otimes_K K'$ {\sl and the restriction of} $\mbox{log}^{(c)}_{G'(K')}$ to $G(K)\subset G(K')=G'(K')$ {\sl coincides with}
$$\mbox{log}^{(c)}_{G(K)}:G(K)\to \mbox{Lie}(G)\subset \mbox{Lie}(G)\otimes_K K'=\mbox{Lie}(G'). $$
{\sl In addition, if} $\sigma$ {\sl is an automorphism of the field extension} $K'/K$, {\sl which preserves the absolute value, then}
$$\mbox{log}^{(c)}_{G'(K')}(\sigma(x))=\sigma(\mbox{log}^{(c)}_{G'(K')}(x)) \quad\mbox{\rm for all } x \in G(K')=G'(K').$$
\end{enumerate}
{\sl These properties determine the homomorphisms}
$\mbox{log}^{(c)}_{G(K)}$ {\sl uniquely}.
\section{N\'eron pairings as periods of logarithm maps}
Let us consider an extension $G$ of an abelian variety $A$ by the multiplicative group ${\mathbf G}_m$, i.e., assume that $G$ sits in a short exact sequence
$$0\to{\mathbf G}_m\to G\to A\to 0.$$
The Hilbert's theorem 90 implies the exactness of the corresponding sequence
$$0\to K^*\to G(K)\to A(K)\to 0.$$
Recall that the corresponding exact sequence of the $K-$Lie algebras
$$0\to K\ \to \mbox{Lie}(G) \to \mbox{Lie}(A)\to 0$$
is also exact. Let us consider the difference of the two branches of the logarithm map
$$\mbox{per}_G:\mbox{log}^{(c+1)}_{G(K)}-\mbox{log}^{(c)}_{G(K)}: G(K)\to \mbox{Lie}(G)$$
attached to $c+1$ and $c$ respectively. Clearly, $\mbox{per}_G$ is a locally constant homomorphism, which kills $G(K)_f$ and coincides with
$$v:K^*\to {\bold Q}\subset K\subset\mbox{Lie}(G)$$
on $K^*\subset G(K)$. Since $G(K)/(K^*\ G(K)_f)$ is a torsion group (see step 5 of the previous section),
$$\mbox{per}_G(G(K))\subset {\bold Q}\subset K\subset \mbox{Lie}(G)$$
and does not depend on the choice of $c$. Indeed, if $x\in G(K)$ and a positive integer $m$, $a\in K^*$ and $x_f\in G(K)_f$ satisfy $x^m=a x_f \in G(K)$ then
$$m\mbox{per}_G(x)=\mbox{per}_G(x_f)+\mbox{per}(a)=v(a)\in {\bold Q}$$
and therefore
$$\mbox{per}_G(x)= v(a)/m \in {\bold Q}$$
does not depend on the choice of $c$.
Since $\mbox{per}_G$ is locally constant, it may be viewed as a continuous homomorphism
$$\mbox{per}_G: G(K)\to {\bold Q} \subset {\bold R},$$
whose restriction to $K^*\subset G(K)$ coincides with
$$v:K^*\to{\bold Q}\subset {\bold R}.$$
These properties determine $\mbox{per}_G$ uniquely. Indeed, if $\mbox{per}':G(K)\to{\bold R}$ is
a a locally constant homomorphism, coinciding with $v$ on $K^*$ then the
difference $\mbox{per}_G-\mbox{per}':G(K)\to{\bold R}$ kills $K^*$ and a certain open subgroup
$U'$ of $G(K)$. The image $U$ of $U'$ in $A(K)$ is an open subgroup and
therefore $A(K)/U$ is a torsion group. The quotient $G(K)/(K^*\ U')$ is
isomorphic to a subgroup of $A(K)/U$ and therefore is also a torsion group. Since $\mbox{per}_G-\mbox{per}'$ kills $K^*\ U'$, one has only to notice that ${\bold R}$ is uniquely divisible and therefore $(\mbox{per}_G-\mbox{per}')(G(K))=0.$
Recall \cite{S3}(see also \cite{L}) that there exists a linear bundle $L$ over
$A$, which is algebraically equivalent to zero and such that $G$, viewed as a
principal ${\mathbf G}_m-$bundle over $A$,
is isomorphic to the principal ${\mathbf G}_m-$bundle
$$L^*=L\setminus \{\mbox{\rm the image of zero section}\}\to A.$$
Further, we will identify $G$ with $L^*$.
Let us consider the absolute value on $K$ defined by the formula
$$\mid a\mid _p = \mbox{exp} (-v(a)) \forall x \in K^*.$$
There is a {\sl canonical local height} (\cite{Z3}, Sect. 3; see also \cite{Z4})
$$\hat h_{-v,L}:L^*(K)=G(K)\to {\bold R},$$
which is a continuous homomorphism, coinciding with $-v$ on $K^*$. Since each neighborhood of the identity element of $G(K)$ contains an open subgroup,
$\hat h_{-v,L}$ must kill a certain open subgroup of $G(K)$, i.e., $\hat h_{-v,L}$ is locally constant. This implies that
$$\hat h_{-v,L}=-\mbox{per}_G.$$
Warning: $v$ as defined in \cite{Z3} coincides with our $-v$!
Recall \cite{Z1} (see also \cite{L}, \cite{Z3} ) a construction of N\'eron
pairings $\langle D,\ {\frak a}\rangle_{-v}$ \cite{N} in terms of $\hat h_{-v,L}$.
Here $D$ is a divisor on $X$, algebraically equivalent to zero, whose rational equivalence class coincides with (isomorphism class of)
$L$, and ${\frak a} = \sum a_xx$ is a $0-$dimensional cycle of degree zero on $X$, whose support lies in $X(K)$ and does not meet $\mbox{Supp}(D)$. Let $s_D$ be a non-zero rational section of
$L$, whose divisor coincides with $D$. Recall that $s_D$ is determined uniquely up to multiplication by an element of $K^*$. Then
$$\langle D,\ {\frak a}\rangle_{-v}= -\hat h_{-v,L}\bigg(\sum a_x s_D(x)\bigg)=-\sum
a_x \hat h_{-v,L}(s_D(x)).$$
It follows that
$$\langle D,\ {\frak a}\rangle_{-v}=\sum
a_x \mbox{per}_G(s_D(x)) \in {\bold Q}.$$
The rationality of $\langle D,\ {\frak a}\rangle_{-v}$ is well-known \cite{N} (see also \cite{L}) when $K$ is a discrete valuation field.
\section{Differentials of the third kind}
Let $V$ be an absolutely irreducible projective variety over $K$. We assume
that the set $V(K)$ of $K-$rational points is non-empty. Clearly, $V(K)$ is
dense in $V$ in the Zariski topology. Let $\mbox{Div}(V)$ be the group of divisors on $V$.
Recall that each divisor $D\subset \mbox{Div}(V)$ can be uniquely presented as a formal
linear combination $\sum c_Z Z$ where $Z$ runs through the set $V^{(1)}$ of
$K-$irreducible
closed subvarieties of codimension 1, $c_Z$ are integers vanishing for all but
finitely many $Z$. We will mostly deal with the tensor product
$\mbox{Div}_A(V)\otimes K$, whose elements are formal linear combinations
$\sum c_Z Z$ where $c_Z$ are elements of $K$, vanishing for all but finitely
many $Z\in V^{(1)}$. To each non-zero
$D=\sum_{Z\in V^{(1)}}c_Z Z\in\mbox{Div} (V)\otimes K$
we attach the finite set
$\mbox{supp} (D)=\{Z\in V^{(1)}\mid c_Z\ne 0\}\subset V^{(1)}$
and the codimension 1 subvariety $\mbox{Supp}(D)=\bigcup_{Z\in\mbox{supp}(D)} D \subset V$.
Recall \cite{S2}, \cite{FW} that a K\"ahler
differential $\omega$ on $V$ is a {\sl differential of the third kind} if locally
in the Zariski topology one may represent $\omega$ as a sum
$$\omega=\omega_{\mbox{reg}} + \sum c_i df_i/f_i $$
of regular differential $\omega_{\mbox{reg}}$ and a finite linear combination of
logarithmic differentials $df_i/f_i$ of non-zero rational functions
$f_i\in K(V)$ with coefficients $c_i\in K$. We write $\mbox{res}(\omega)$ for the
residue of $\omega$ \cite{S1,S2,FW}; by definition
$$\mbox{res}(\omega) \in \mbox{Div}_a(V) \otimes K \subset \mbox{Div}(V)\otimes K$$
where $\mbox{Div}_a(V)$ is the group of divisors on $V$, algebraically equivalent to
zero. The differential $\omega$ is regular outside $\mbox{Supp}(\mbox{res}(\omega))$. For
example, if $\omega=df/f$ is the logarithmic differential of (non-constant)
rational function $f$, then $\mbox{res}(\omega)=\mbox{div}(f)$ is the divisor of $f$ and
$\omega$ is regular outside $\mbox{Supp}(\mbox{div}(f))$. As usual, we agree that empty set is the support
of zero divisor and zero differential.
Further, we assume that the set $V(K)$ is non-empty. Let $S\subset V^{(1)}$ be a finite set
of irreducible codimension 1 subvarieties in $V$. We write $K[S]$ for the finite-dimensional
$K-$vector subspace of $\mbox{Div}_A(V)\otimes K$, generated by elements of $S$, i.e., the subspace,
consisting of elements, whose $\mbox{supp}$ is contained in $S$. We write $\Omega_{3,S}(V)$ for the
finite-dimensional $K-$vector space of differentials $\omega$ of the third kind on $V$ with
$\mbox{res}(\omega)\in K[S]$. We write $K[S]_a$ for the intersection of $k[S]$ and
$\mbox{Div}_a(V)\otimes K$ in $\mbox{Div}(V)\otimes K$. Clearly, $\Omega_{3,S}(V)$ contains the space
$\Omega^1(V)$ of differentials of the first kind, i.e., everywhere regular differentials on
$V$ and $\mbox{res}(\Omega_{3,S}(V))\subset K[S]_a$.
It follows from \cite{S1}, Lemma 6 and its proof (see also \cite{S2} and
\cite{FW}, especially, pp. 197--201) that there exists an extension
$$ 0 \to T \to G_S \to \mbox{Alb} (V) \to 0 $$
$G_S$ of the Albanese variety $\mbox{Alb} (V)$ of $V$ by a $K-$torus $T$ and a rational map $f: V \to G_S$, which is regular on the complement of union of $D \in S$ and satisfies the following properties. If $\Omega^1_{\mbox{inv}}(G_S)$ is the $K-$vector space of invariant differential forms of degree 1 on $G_S$ then $f^*\Omega^1_{\mbox{inv}}(G_S)=\Omega_{3,S}(V)$ and the map $\omega\mapsto f^*\omega$ is an isomorphism between $\Omega^1_{\mbox{inv}}(G_S)$ and $\Omega_{3,S}(V)$. In other words, for each $\omega\in \Omega_{3,S}(V)$ there exists exactly one invariant form $\omega_{\mbox{inv}}\in\Omega^1_{\mbox{inv}}(G_S)$ with $f^*\omega_{\mbox{inv}}=\omega$. The rational map $f$ is unique, up to a translation by an element of $G_S(K)$. The algebraic group $G_S$ is unique, up to an isomorphism.
So, in order to integrate $\omega\in \Omega_{3,S}(V)$ it suffices to know how to integrate $\omega_{\mbox{inv}}$ on $G_S$ and just put
$$\int_P^Q \omega =\int_{f(P)}^{f(Q)}\omega_{\mbox{inv}} \in K.$$
Now, let us use the duality between $\mbox{Lie}(G_S)$ and $\Omega^1_{\mbox{inv}}(G_S)$. Let us choose a branch $\mbox{log}^{(c)}$ of the $p-$adic logarithm and put
$$\int_x^y \omega_{\mbox{inv}} := \omega_{\mbox{inv}}(\mbox{log}^{(c)}_{G_S(K)}(y)-\mbox{log}^{(c)}_{G_S(K)}(x)) \in K.$$
The $K-$valued function(s) $u(P,Q)=u(P, Q,\omega) =\int_{f(P)}^{f(Q)}\omega_{\mbox{inv}}$ obviously enjoys the following natural properties:
\begin{itemize}
\item (Newton--Leibnitz rule) $u(P,Q)$ is a $K-$analytic function in $Q$ and its differential coincides with $\omega$;
\item ($K-$linearity) $u(P,Q)=u(P,Q,\omega)$ is a $K-$linear function in $\omega$;
\item (Additivity) $u(P,Q)+u(Q,R)=u(P,R)$ for all $P,Q,R \in V(K)\setminus \mbox{Supp}(\mbox{res}(\omega))$.
\item (Change of variables) Let $u: V\to W$ be a regular dominant map of absolutely irreducible projective
algebraic $K-$varieties, $\omega'$ a differential of the third kind on $W$ such that $u^*\omega'=\omega$.
Then
$$\int_P^Q\omega=\int_{u(P)}^{u(Q)}\omega' $$
whenever both integrals are defined.
\item
(Choice of branch) If $\omega=dz/z$ is a logarithmic differential of a rational function $z$ on $V$ then
$$\int_P^Q dz/z= \mbox{log}^{(c)}(z(Q))-\mbox{log}^{(c)}(z(P)).$$
\end{itemize}
\vskip 1cm {\bf Remark.} Using Lie theory over the fields of real and complex numbers (maximal compact subgroups), Bertrand and Philippon have constructed differentials of the third kind with prescribed residues and purely imaginary periods (\cite{BP}, Sect. 4, especially Remarque 5).
\section{Closed forms}
Let $X$ be an absolutely irreducible smooth (not necessarily projective)
variety over $K$. We assume that the set $X(K)$ is non-empty. Let $\omega$ be
a {\sl closed} regular differential form of degree 1 on $X$. In this section we define
$\int_P^Q \omega$ for $P,Q\in X(K)$. According to Satz 5 of
\cite{FW}, p. 197, there exist a commutative algebraic $K-$group $G$,
an invariant form $\omega_{\mbox{inv}}\in \Omega_{\mbox{inv}}^1(G)$ and a regular map $f: X\to G$ such that
$$f^* \omega_{\mbox{inv}}=\omega.$$
Now, we put
$$\int_P^Q \omega=\omega_{\mbox{inv}}(\mbox{log}^{(c)}_{G(K)}(f(Q))-\mbox{log}^{(c)}_{G(K)}(f(P))) \in K.$$
One may easily deduce from Satz 6 of \cite{FW}, p. 197 that
$\int_P^Q \omega$ does not depend on the choice of $G, f$ and $\omega_{\mbox{inv}}$. It
also follows easily that the integral enjoys the Newton--Leibnitz, $K-$linearity, additivity, choice of branch and change of variables properties.
|
1996-03-08T06:20:23 | 9603 | alg-geom/9603007 | en | https://arxiv.org/abs/alg-geom/9603007 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9603007 | Harald Skarke | Harald Skarke | Weight systems for toric Calabi-Yau varieties and reflexivity of Newton
polyhedra | Latex, 14 pages | Mod.Phys.Lett. A11 (1996) 1637-1652 | 10.1142/S0217732396001636 | TU Wien report TUW-96/04 | null | According to a recently proposed scheme for the classification of reflexive
polyhedra, weight systems of a certain type play a prominent role. These weight
systems are classified for the cases $n=3$ and $n=4$, corresponding to toric
varieties with K3 and Calabi--Yau hypersurfaces, respectively. For $n=3$ we
find the well known 95 weight systems corresponding to weighted $\IP^3$'s that
allow transverse polynomials, whereas for $n=4$ there are 184026 weight
systems, including the 7555 weight systems for weighted $\IP^4$'s. It is proven
(without computer) that the Newton polyhedra corresponding to all of these
weight systems are reflexive.
| [
{
"version": "v1",
"created": "Thu, 7 Mar 1996 11:45:23 GMT"
}
] | 2009-10-28T00:00:00 | [
[
"Skarke",
"Harald",
""
]
] | alg-geom | \section{Introduction}
A large class of Calabi--Yau manifolds can be described as hypersurfaces
defined by transverse polynomials in weighted projective 4 spaces $\IP^4$.
These spaces could be classified \cite{nms,kl94}, and the resulting list
showed a remarkable property, known as mirror symmetry: To nearly each such
variety there is a partner in the list such that the Hodge numbers
$h^{1,1}$ and $h^{2,1}$ of the two varieties are interchanged.
There are strong hints that mirror symmetry is not only a property
concerning spectra, but a full symmetry of the corresponding conformal
field theories.
Batyrev \cite{ba94} has suggested a far more powerful technique for the
construction of Calabi--Yau varieties: In his framework of reflexive
polyhedra, mirror symmetry (at the level of Hodge numbers) is manifest,
and it was checked by computer \cite{ca95,klun} that the Newton
polyhedra of all 7555 weight systems for $\IP^4$'s are reflexive, i.e.
that the older class of models is contained in this newer approach.
This makes the following goals look desirable:\\
(i) The classification of reflexive polyehedra, and\\
(ii) an explanation of the reflexivity of Newton polyhedra corresponding
to $\IP^4$'s.\\
A big step towards the solution of problem (i) was taken in ref. \cite{crp}:
There it was shown that all reflexive polyhedra are bounded by
polyhedra that can be described with the help of
certain weight systems or combinations of weight systems, and an
algorithm for the classification of these weight systems was proposed.
In the present work, I present a far more efficient algorithm and use
it to find all of the weight systems involved in the construction of
reflexive polyhedra in $n\le 4$ dimensions.
Then I show that the Newton polyhedra corresponding to any of these
weight systems (which contain the 7555 old ones as a small subset) are
reflexive, thus solving problem (ii).
There is another recent development that should be mentioned here:
It was found that, through black hole condensation, there seem to be physical
transitions between string theories compactified on different Calabi--Yau
manifolds \cite{st95,gms}, implying that there are connections between
the various moduli spaces.
In terms of toric geometry, such a transition may take place if
a reflexive polyhedron describing some Calabi--Yau hypersurface is
contained in the polyhedron describing some other Calabi--Yau hypersurface.
In this way it was shown that the moduli spaces of all Calabi--Yau
hypersurfaces of weighted $\IP^4$'s are connected \cite{cggk,acjm}.
The present work also provides a big step towards showing the connectedness
of the moduli spaces of all toric Calabi--Yau varieties: Using the fact
that the maximal Newton polyhedra corresponding to any of the weight systems
found here are reflexive and that any reflexive polyhedron is contained
in one of them, all that is left to do is to show the connectedness
of the maximal Newton polyhedra, which should be a straightforward
application of the tools developed in \cite{cggk,acjm}.
In section 2 I give a definition of reflexivity and a description of
some of the main ideas of ref. \cite{crp} used here. Then I describe
the new algorithm for the construction of weight systems required for
the classification of reflexive polyhedra and report the results
of the implementation of this algorithm on a computer.
In section 3 I give a proof that the Newton polyhedra corresponding to
any such weight system (or combination of weight systems) is
reflexive. I also give an explicit proof that the 7555 weight systems
for weighted $\IP^4$'s fall into the new set of weight systems.
\section{The classification of weight systems}
Consider a dual pair of lattices $\G\simeq \IZ^n$ and $\G^*$ and their rational
extensions $\G_\IQ$ and $\G_\IQ^*$; we denote the duality pairing
$\G_\IQ^*\times\G_\IQ\to\IQ$ by $\<\;,\;\>$.
A reflexive polytope $\D$ is an integer polytope in $\G_\IQ$
(i.e., a polytope in $\G_\IQ$ with vertices in $\G$) with exactly
one integer interior point (which we may choose to be the origin) such that
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
\D^*:=\{y\in \G^*_\IQ:\<y,x\>\ge-1\:\forall x\in\D\}
\eeq
is an integer polytope in $\G^*_\IQ$.
This is equivalent to the statement that all facets of $\D$ lie on
hyperplanes of integer distance 1 to the interior point
(a lattice hyperplane $H$ has integer distance $k$ to a lattice point $P$
if there are $k-1$ lattice hyperplanes parallel to $H$ between $H$ and $P$).
Let me briefly outline the basic ideas of the algorithm proposed in \cite{crp}
for the classification of reflexive polyhedra:
Given a reflexive polytope $\D$, we look for a set of hyperplanes
$H_i$, $i=1,\cdots ,k$ carrying facets of $\D$ such that the $H_i$
define a (generically non--integer) bounded polyhedron $Q\subset\G_\IQ$.
We also assume that $Q$ is minimal in the sense that there is no
polytope with the same properties but with a smaller number of facets.
Each $H_i$ corresponds to some vertex $\5V^i$ of $\D^*$,
and $Q^*$ (the convex hull of the $\5V^i$) is an integer polytope with
the interior point of $\D^*$ in its interior.
In ref. \cite{crp} we have defined a redundant coordinate system where
the $i$'th coordinate $P^i$ of some point $P\in\G$ is given by its
integer distance to the hyperplane $H_i$ (positive on the side of $\D$).
In this way we get a natural embedding of $\G$ in $\G^k\simeq\IZ^k$.
Whenever we use this type of coordinate system, we label the interior point,
which has coordinates $(1,\cdots,1)^T$, by $\ip$.
Making use of duality and the fact that the $H_i$ have distance 1 to $\ip$,
we see that $P^i=\<\5V^i,P\>+1$.
This coordinate system has several disadvantages: We require more
coordinates than with a $\IZ^n$ description, there is an ambiguity
about the choice of $Q$, and even the lattice is not always completely
determined.
The advantages, however, seem to be greater: Our description
naturally leads to pairing matrices between vertices of $\D^*$ and $\D$
which characterise dual pairs uniquely up to the choice of some sublattice,
and even this finite ambiguity vanishes when we consider pairing
matrices for all integer points of $\D^*$ and $\D$.
In this way we avoid all the cumbersome considerations about
equivalences of polyhedra that are mapped to each other by $GL(n,\IZ)$
transformations.
Then we have shown that $Q^*$ is composed of simplices (perhaps of lower
dimension) that have $\ip$ in their interiors.
To each of these simplices there corresponds a weight system {\bf q} in the
following way:
We define the weights $q_i$ to
be the barycentric coordinates of $\5{\hbox{\bf 1}}} \def\ip{\hbox{\bf 1}} \def\Z{M} \def\Q{Q$ (the interior point of $\D^*$)
w.r.t. the vertices $\5V^i$ of the simplex,
i.e. $\5{\hbox{\bf 1}}} \def\ip{\hbox{\bf 1}} \def\Z{M} \def\Q{Q=\sum q_i\5V^i$ with $\sum q_i=1$.
The $q_i$ are positive because $\5{\hbox{\bf 1}}} \def\ip{\hbox{\bf 1}} \def\Z{M} \def\Q{Q$ is in the interior of the simplex
defining them.
Then $\<\5{\hbox{\bf 1}}} \def\ip{\hbox{\bf 1}} \def\Z{M} \def\Q{Q,P\>=0$ for $P\in\G_\IQ$ implies $\sum q_i\<\5V^i,P\>=0$,
i.e. $\sum q_i(P^i-1)=0$ and $\sum q_iP^i=1$.
The number of independent equations of this type (i.e., the number of
weight systems involved in the construction), is $k-n$.
They define an n dimensional lattice $\G^n$ with $\G\subseteq\G^n\subset\G^k$.
Now it is easy to see that $\ip$ is the only integer point in the interior
of $Q$:
For any point in the interior of $Q$ all coordinates have to be positive,
i.e. for any integer point they have to be $\ge1$.
Comparing this with $\sum q_i(P^i-1)=0$ it is clear that this can be
fulfilled only by $P^i=1$ $\forall i$.
For the construction of reflexive polyhedra we certainly need weight
systems where $\ip$ is in the interior not only of $Q$, but also in
the interior of the maximal Newton polyhedron $\D_{\rm max}=Q\cap\G^n$
defined by a weight system.
The classification algorithm proposed in \cite{crp} involved the
consideration of minimal polytopes both in $\G$ and in $\G^*$ and
the construction of pairing matrices between these polytopes.
There is, however, a far simpler way of constructing all allowed weight
systems.
The new algorithm is based on the following observation:
Assume that a weight system $q_1,\cdots,q_l$
\footnote{In this paper I always denote the dimension of $\G$ by $n$
and the number of weights in a weight system by $l$; if $\G^n$ is defined
by a single weight system, then $l=n+1$}
allows a collection of points with
coordinates $x^i$, including the interior point with $x^i=1\;\forall i$.
If these points fulfill an equation of the type $\sum_{i=1}^la_ix^i=1$
with {\bf a}$\ne${\bf q}, then the weight system must also allow at least one
point with $\sum_{i=1}^la_ix^i>1$ and at least one point with
$\sum_{i=1}^la_ix^i<1$ to ensure that {\bf 1} is really in the interior
of the maximal Newton polyhedron defined by {\bf q}.
The latter inequality is the one that we actually use for the algorithm:
Starting with the point $\ip=(1,\cdots,1)^T$, we see that unless our
weight system is $\hbox{\bf q}=(1/l,\cdots,1/l)$, there must be at least one point
with $\sum_{i=1}^lx^i<l$. For $l\le5$ there are only a few possibilities,
and after choosing some point {\bf x}${}_1$, we can look for some simple
equation fulfilled by $\ip$ and {\bf x}${}_1$ and proceed in the same way.
For $l=3$ the classification is easily carried out by hand:
Unless $\hbox{\bf q}=(1/3,1/3,1/3)$, we need at least one point with $x^1+x^2+x^3<3$.
As points where no coordinate is greater than 1 would be in conflict with
the positivity of the weight system,
we need the point $(2,0,0)^T$ (up to a permutation of indices).
Now we note that $\ip$ and $(2,0,0)^T$ both fulfill $2x^1+x^2+x^3=4$,
so $\hbox{\bf q}=(1/2,1/4,1/4)$ or we need a point with $2x^1+x^2+x^3<4$.
The only point allowed by this inequality which leads to a sensible
weight system is $(0,3,0)^T$, leading to $\hbox{\bf q}=(1/2,1/3,1/6)$.
One should note how easily we have reproduced all weight systems in
comparison with the rather lengthy analysis in ref. \cite{crp}.
For $l=4$ we can either get $\hbox{\bf q}=(1/4,1/4,1/4,1/4)$ or we need a point
with $x^1+x^2+x^3+x^4<4$. Up to permutations, all possibilities are
exhausted by $(3,0,0,0)^T$, $(2,1,0,0)^T$ and $(2,0,0,0)^T$.
For the rest of the task the computer program
requires only a few seconds. The result is a list of 99 weight systems
which still have to be checked with respect to the property that the
maximal Newton polyhedra defined by them must have $\ip$ in their interiors.
For $l=5$ similar considerations show that, unless $q_i=1/5$ for
$i=1,\cdots,5$, the weight system must allow at least one of the
points $(4,0,0,0,0)^T$, $(3,1,0,0,0)^T$, $(2,2,0,0,0)^T$, $(2,1,1,0,0)^T$,
$(3,0,0,0,0)^T$, $(2,1,0,0,0)^T$ and $(2,0,0,0,0)^T$.
Given these starting points, a C program running on an HP 735/125 required two
days of system time to produce 200653 candidates for weight systems.
The next task is to find out whether the maximal Newton polyhedra defined by
the weight systems really have $\ip$ in their interiors.
It is straightforward to construct all points allowed by some {\bf q}.
Then one could in principle construct all facets and check that
$\ip$ does not lie on one of them or on the wrong side of one of them.
I have used a different approach: Starting with $l$ points of the
maximal Newton polyhedron which are independent in $\IQ^l$, it is easy to
calculate the barycentric coordinates of $\ip$ w.r.t these points.
If all of the barycentric coordinates are positive (in this case we can
identify them with the $\5{\hbox{\bf q}}$ system introduced in \cite{crp}),
$\ip$ is in the interior of the simplex defined by the $l$ points.
If some of the barycentric coordinates
are negative, one can substitute the point corresponding to the smallest
coordinate by a point on the other side of the hyperplane defined by the
remaining points and try the same procedure again.
The same strategy can be used in cases where only $l-1$ of the starting
points are independent.
If one of the barycentric coordinates is 0 while all others are
positive, the points corresponding to the positive coordinates
define a codimension one hyperplane with $\ip$ in its interior,
so one has to check whether there is at least one point on either
side of this hyperplane.
Depending on the starting points, this strategy produced results
more or less quickly.
The best version turned out to be the one where the starting points were
determined in a fashion very similar to the algorithm that produced
the original candidates for weight systems.
Another good strategy is to use points with maximal exponents as starting
points.
It turns out that exactly 95 of the 99 weight systems for $l=4$ have the
property that $\ip$ is in the interior of the corresponding
maximal Newton polyhedron.
These are precisely the well known 95 weight systems for weighted $\IP^4$'s
that have K3 hypersurfaces \cite{reid,fl89}.
For $l=5$ the situation is completely different:
The 7555 weight systems corresponding to weighted $\IP^4$'s that allow
transverse polynomials are just a small subset of the 184026 different
weight systems whose maximal Newton polyhedra contain $\ip$.
Later I will give a proof that in arbitrary dimensions weight systems
corresponding to weighted $\IP^n$'s have the property that their
maximal Newton polyhedra contain $\ip$.
The simplest weight systems that do not correspond to weighted $\IP^4$'s
are $(1,1,1,3,4)/10$ and $(1,1,1,4,5)/12$. Note that in the latter system
the first four weights are of Fermat type, whereas the last weight
is such that no monomial of the type $X^i(X^5)^\l$, which would be necessary
for transversality, is allowed.
The corresponding maximal Newton polyhedron has vertices $V_1=(12,0,0,0,0)^T$,
$V_2=(0,12,0,0,0)^T$, $V_3=(0,0,12,0,0)^T$, $V_4=(0,0,0,3,0)^T$,
$V_5=(2,0,0,0,2)^T$, $V_6=(0,2,0,0,2)^T$ and $V_7=(0,0,2,0,2)^T$.
The facets correspond to the hyperplanes $H_i: x^i=0$ for $i=1,\cdots,5$
and $H_6: 2x^4+3x^5=6$ (spanned by the $V_j$ with $j\ge 4$).
As $\ip$ fulfils $2x^4+3x^5=5$, it has integer distance 1 to
$H_6$ and the maximal Newton polyhedron is reflexive.
Its vertex pairing matrix, i.e. the matrix ${A^i}_j=\<\5V^i,V_j\>+1$
(with $\5V^i$ corresponding to $H_i$ for $i=1,\cdots,6$) is
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \pmatrix{12 & 0 & 0 & 0 & 2 & 0 & 0 \cr
0 & 12 & 0 & 0 & 0 & 2 & 0 \cr
0 & 0 & 12 & 0 & 0 & 0 & 2 \cr
0 & 0 & 0 & 3 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 2 & 2 & 2 \cr
6 & 6 & 6 & 0 & 0 & 0 & 0 \cr}. \eeq
The zeroes in the matrix indicate incidence relations: ${A^i}_j=0$
means that $V_j$ lies on $H_i$.
With the help of a program that can test any weight system with respect to
the property of having $\ip$ in the interior of the maximal Newton polyhedron,
there is an easy way to check the program for the construction of all such
weight systems:
Given a positive integer $d$, one can consider all weight systems of the type
$q_i=n_i/d$ with $n_i\in\IN$ and $\sum_{i=1}^ln_i=d$ and apply the interior
point check.
I have done this up to $d=230$ for $l=5$, resulting in approximately
50000 allowed weight systems which are identical with the ones the
original program produced.
All of the weight systems found in this way might be interesting
for theoretical reasons, in particular for a better understanding of the
connection between older approaches to weighted $\IP^n$'s and the toric
framework and for dealing with the question of whether the moduli space
of all Calabi--Yau varieties allowing a description in terms of reflexive
polyhedra is connected.
For our classification program, however, we need only those weight systems
where every coordinate hyperplane is spanned by points of the maximal Newton
polyhedron.
It is easy to write a program that checks for this property.
58 of the 95 weight systems for $l=4$ pass the test (see table II in the
appendix).
For $l=5$ only approximately one fifth of the weight systems is such
that each coordinate hyperplane is spanned by points of the maximal Newton
polyhedron.
Among the 7555 weights corresponding to weighted $\IP^4$'s slightly more
than half have this property.
As an illustration for the fact that weight systems without the above
mentioned property are redundant in the classification scheme, consider
the system $(40,41,486,1134,1701)/3402$.
Its maximal Newton polyhedron $\D_{\rm max}$ is the simplex whose vertices
are the columns of the matrix
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \pmatrix{83 & 1 & 0 & 0 & 0 \cr
2 & 82 & 0 & 0 & 0 \cr
0 & 0 & 7 & 0 & 0 \cr
0 & 0 & 0 & 3 & 0 \cr
0 & 0 & 0 & 0 & 2 \cr}. \eeq
The lines of this matrix correspond to the points in $\5\G$ dual to
the coordinate hyperplanes. Obviously the first two lines cannot correspond
to vertices of $\D_{\rm max}^*$. The lacking vertices of $\D_{\rm max}^*$ are
easily found to be $(84,0,0,0,0)$ and $(0,84,0,0,0)$.
Thus the vertex pairing matrix for $\D_{\rm max}^*$ and $\D_{\rm max}$ is
given by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \pmatrix{84 & 0 & 0 & 0 & 0 \cr
0 & 84 & 0 & 0 & 0 \cr
0 & 0 & 7 & 0 & 0 \cr
0 & 0 & 0 & 3 & 0 \cr
0 & 0 & 0 & 0 & 2 \cr} \eeq
which obviously corresponds to the Fermat type weight system
$(1/84,1/84,1/7,1/3,1/2)$.
We finish this section with a table of the numbers of various types
of weight systems for $l=5$.
In this table, ``span'' means the weight systems where each coordinate
hyperplane is spanned by points of the maximal Newton polyhedron, $\IP^4$ means
that the weights correspond to weighted $\IP^4$'s that allow transverse
polynomials, and in addition I have given the numbers of weight systems
containing and not containing a weight of $1/2$.
\noindent
\begin{tabular}{||l|c|c|c|c|c|c||} \hline\hline
& $\IP^4,\;q_5=1/2$ & $\IP^4,\;q_5<1/2$ & $\IP^4$ & $q_5=1/2$ & $q_5<1/2$
& total\\
\hline
span & 1309 & 2860 & 4169 & 14872 & 23858 & 38730 \\ \hline
total& 2390 & 5165 & 7555 & 97036 & 86990 &184026 \\
\hline\hline \end{tabular}\hfill\\[3mm]
\hfil Table I: Numbers of various types of weight systems with 5 weights
Tables III and IV in the appendix contain small sublists of the complete
list of weight systems (with small and large $d$).
\section{Some results derived without a computer}
The main aim of this section is to show that the maximal Newton polyhedra
corresponding to weights or combinations of weights constructed in the way
reported in the previous section are all reflexive.
As a prerequisite, we first need a technical lemma.
\noindent
{\bf Lemma 1:} Consider an integer pyramid $Pyr$ of height $h\ge 2$ in a
lattice
$\G\simeq\IZ^4$ and the pyramid $Pyr_{\rm double}$, which has the same peak
and the same shape as $Pyr$, but double height $2h$.
Then $Pyr_{\rm double}$ contains integer lattice points which are neither in
$Pyr$ nor in the base of $Pyr_{\rm double}$.\\[4pt]
{\it Proof:} Obviously it is sufficient to consider the case where
the base of $Pyr$ is a simplex in $\IZ^3$ (otherwise triangulate the
base and pick any simplex). Then the base of $Pyr_{\rm double}$ is a
simplex in $(2\IZ)^3$ and we may choose its vertices to be
$\1 0, 2\1e_1,2\1e_2$ and $2\1e_3$.
The peak has coordinates $(2x,2y,2z,2h)^T$.
The points in $Pyr_{\rm double}$ can be parameterized as
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
\l\pmatrix{2\cr 0\cr 0\cr 0\cr}+\m\pmatrix{0\cr 2\cr 0\cr 0\cr}
+\n\pmatrix{0\cr 0\cr 2\cr 0\cr}+\r\pmatrix{2x\cr 2y\cr 2z\cr 2h\cr}
\eeq
with $\l\ge0,\cdots,\r\ge 0$ and $\l+\m+\n+\r\le 1$.
With $x'=x\mao{mod}h$, $y'=y\mao{mod}h$, $z'=z\mao{mod}h$, (with
$0\le x'<h$ etc.) it is easily checked that the points
with $(\l_1,\m_1,\n_1,\r_1)=(h-x',h-y',h-z',1)/2h$
and $(\l_{h-1},\m_{h-1},\n_{h-1},\r_{h-1})=(x',y',z',h-1)/2h$
are integer points at heights 1 and $h-1$, respectively.
Clearly all parameters $\l_1, \cdots$ are positive.
With $\l_1+\m_1+\n_1+\r_1+\l_{h-1}+\m_{h-1}+\n_{h-1}+\r_{h-1}=2$,
at least one of the inequalities $\l_1+\m_1+\n_1+\r_1\le 1$
and $\l_{h-1}+\m_{h-1}+\n_{h-1}+\r_{h-1}\le 1$ must be fulfilled.
This means that at least one of the two points is a point of
$Pyr_{\rm double}$, and because of the height it is neither in $Pyr$ nor
in the base.
\hfill$\Box$\\[4pt]
{\bf Remarks:} The same proof works for lattices $\IZ^n$
with $n<4$.
For $n=5$ there is the following counterexample:
Let the base again be given by $\1 0$ and $2\1e_i$ and the peak by
$(2,2,2,2,4)^T$. With $h=2$, a point fulfilling the criterion of the
lemma would have to be at height 1. With the same ansatz as in the proof,
we would have $\r=1/4$ and $\l\ge 0, \m\ge 0, \cdots$ would have to be
at least $1/4$, resulting in $\l+\cdots+\r\ge5/4$ and a point outside
$Pyr_{\rm double}$.
Thus there is no integer point between the bases of $Pyr$ and
$Pyr_{\rm double}$.
\noindent
{\bf Theorem:} Four or lower dimensional maximal Newton polyhedra with $\ip$
in their interior are reflexive.\\[4pt]
{\it Proof:} Consider a collection of points in a maximal Newton polyhedron
$\D_{\rm max}$ spanning a hyperplane at distance $h\ge 2$ from $\ip$.
We take these points to define the base of the pyramid $Pyr$ of lemma 1 and
$\ip$ as the peak.
Then $Pyr_{\rm double}$ lies in
$(\D_{\rm max})_{\rm double}\subseteq \{x\in\G^n:x^i\ge -1\}$.
Only the base of $Pyr_{\rm double}$ can intersect with the boundary of
$(\D_{\rm max})_{\rm double}$, so the integer points of $Pyr_{\rm double}$
that are not in the base must have nonnegative coordinates, i.e. they must
be in $\D_{\rm max}$.
Thus the lemma ensures that there are points in the cone defined by $Pyr$
which are outside of $Pyr$, but within $\D_{\rm max}$. This means that the
hyperplane
defined by the base of $Pyr$ is not a bounding hyperplane of $\D_{\rm max}$.
Therefore every bounding hyperplane of $\D_{\rm max}$ must be at distance 1,
i.e. $\D_{\rm max}$ is reflexive.\hfill$\Box$\\[4pt]
{\bf Remark:} The theorem holds not only for maximal Newton polyhedra defined
by a single weight system with $l=n+1$, as mainly considered in this paper,
but also for maximal Newton polyhedra defined by several weight systems with
$l<n+1$ involved in the classification scheme of \cite{crp}.\\[4pt]
{\bf Examples:} The weight
system $q_i=1/5$, $i=1,\cdots,5$ contains the points $(2,0,0,0,3)^T$,
$(0,2,0,0,3)^T$, $(0,0,2,0,3)^T$, $(0,0,0,2,3)^T$ defining the
hyperplane $x^5=3$ (at distance 2 to $\ip$). The base of $Pyr_{\rm double}$
is the convex hull of the points $(3,-1,-1,-1,5)^T$, $(-1,3,-1,-1,5)^T$,
$(-1,-1,3,-1,5)^T$, $(-1,-1,-1,3,5)^T$, and the ``height one'' points
$(1,0,0,0,4)^T$, $(0,1,0,0,4)^T$, $(0,0,1,0,4)^T$, $(0,0,0,1,4)^T$
ensure that $x^5=3$ is not a bounding hyperplane. In the same way the
hyperplane $x^5=4$ (at distance 3 to $\ip$), with the points
$(1,0,0,0,4)^T$, $(0,1,0,0,4)^T$, $(0,0,1,0,4)^T$, $(0,0,0,1,4)^T$,
gives rise to a pyramid of height 6 with base points
$(1,-1,-1,-1,7)^T$, $(-1,1,-1,-1,7)^T$, $(-1,-1,1,-1,7)^T$, $(-1,-1,-1,1,7)^T$.
This time the integer point we are looking for is $(0,0,0,0,5)^T$.
The fact that maximal Newton polyhedra of weighted $\IP^4$'s are reflexive
has been
known for some time due to explicit computer calculations \cite{ca95,klun}.
In order to rederive this result without the help of a computer, we still
have to show that the list of 7555 weights for $\IP^4$'s is contained
in our complete list of weights whose maximal Newton polyhedra have $\ip$
in the interior.
The analogous statement holds in any dimension and also for
abelian orbifolds (with the transversality condition applied to
polynomials that are invariant under the twist group); although it looks
quite obvious to anyone who has worked with weighted $\IP^n$'s for
some time, the proof turns out to be rather technical.
\noindent
{\bf Lemma 2:} Maximal Newton polyhedra corresponding to weighted $\IP^n$'s
or abelian orbifolds of weighted $\IP^n$'s that allow transverse polynomials
have $\ip$ in their interiors.\\[4pt]
{\it Proof:} A transverse polynomial contains monomials of the type
$M^i=(X^i)^{a_i}$ or $M^i=(X^i)^{a_i}X^j$ (with $a_i\ge 2$) for each $i$.
These monomials define points which can be arranged in the matrix
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq A=\pmatrix{a_1&x&x&x&\cdots\cr
x&a_2&x&x&\cdots\cr
x&x&a_3&x&\cdots\cr
\cdots&&&&\cr}, \eeq
where in each column at most one of the $x$'s can be 1, whereas all others
are zero.
Let us first see that $A$ is regular: Assuming it were singular,
we could find a nontrivial vanishing linear combination of its lines, i.e.
we would have $\1\l\ne0$ with $\sum_i \l_iA^i{}_j=0$.
The specific form of our matrix implies that $\l_i=0$ if $M^i=(X^i)^{a_i}$
and $a_i\l_i+\l_{p(i)}=0$ if $M^i=(X^i)^{a_i}X^{p(i)}$.
Iterating this, we get $a_ia_{p(i)}\l_i-\l_{p(p(i))}=0$ etc.
At some point either $\l_{p(\cdots(i)\cdots)}=0$ or $p(\cdots(i)\cdots)=i$,
showing that indeed $\l_i=0$.
Thus $A$ is regular and we can solve $\sum_i A^i{}_j\5q^j=1$ for $\5q^j$.
Then $\sum_j\5q^j=\sum_{i,j}(A^{-1})^j{}_i=\sum_iq_i=1$, showing that the
$\5q^j$ are the barycentric coordinates of $\ip$ with respect to the
columns of $A$.
If all of the $\5q^j$ are positive, $\ip$ is in the interior of the simplex
defined by the columns of $A$.
Let us now assume that not all of the $\5q^j$ are positive:
Let $\5q^j\le 0$ for $j\in I_-$ and $\5q^j>0$ for $j\in I_+$, with
$I_-\cup I_+=\{1,\cdots,n\}$.
Now sum the equations defining the $\5q^j$ over $i\in I_-$ to get
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \sum_{i\in I_-}\sum_j A^i{}_j\5q^j=|I_-| \eeql{grausgl}
and split $\sum_j$ in $\sum_{j\in I_-}+\sum_{j\in I_+}$.
Then
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \sum_{i\in I_-}\sum_{j\in I_-} A^i{}_j\5q^j\le2\sum_{j\in I_-}\5q^j\eeq
because the diagonal elements are involved and
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \sum_{i\in I_-}\sum_{j\in I_+} A^i{}_j\5q^j\le\sum_{j\in I_+}\5q^j \eeq
because each column contains at most a single 1.
Thus the l.h.s. of eq. (\ref{grausgl}) fulfils
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \sum_{i\in I_-}\sum_j A^i{}_j\5q^j\le
2\sum_{j\in I_-}\5q^j+\sum_{j\in I_+}\5q^j=1+\sum_{j\in I_-}\5q^j\le 1
\eeq
with equality iff
$\sum_{j\in I_-}\5q^j=0$ and $\sum_{i\in I_-}\sum_{j\in I_+} A^i{}_j\5q^j=1$.
Therefore $I_-$ can have at most 1 element corresponding to some $\5q^j=0$, and
the corresponding line may contain no 0, i.e. up to permutations our matrix is
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq A=\pmatrix{a_1&1&1&1&\cdots\cr
0&a_2&0&0&\cdots\cr
0&0&a_3&0&\cdots\cr
\cdots&&&&\cr}. \eeq
$\ip$ is in the interior of the $n-1$ dimensional simplex defined by
all columns except the first.
This simplex lies on the hyperplane $x^1=1$.
The first point is above this hyperplane, and the transversality condition
ensures that there is also a point with $x^1=0$, i.e. a point below
this hyperplane.
Therefore $\ip$ is again in the interior of the maximal Newton polyhedron.
\hfill$\Box$\\
\bigskip
{\it Acknowledgements.} I would like to thank Max Kreuzer for a long
collaboration to which I owe my interest in reflexive polyhedra, and also
for helpful discussions in the context of the present work.
This work is supported
by the {\it Austrian National Bank} under grant number 5674.
\vfill\eject
\noindent
{\bf\Large Appendix: Various Tables}
In table II I list the 95 weight systems for $l=4$.
The last column indicates whether a weight system has the property that
coordinate hyperplanes are spanned by points of the maximal Newton polyhedron.
Table III contains all weight systems for $l=5$ with $d\le20$, whereas
table IV contains the weight systems with the largest values of $d$ for
the cases $q_5=1/2$ and $q_5<1/2$.
The last columns indicate whether a weight system corresponds to a weighted
$\IP^4$ that allows transverse polynomials.
All weights in table III have the property that
coordinate hyperplanes are spanned by points of the maximal Newton polyhedron,
whereas none of the weight systems in table IV fulfil this criterion.
\bigskip
\noindent
\begin{tabular}{||rrrr|r|r|} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & d & span \\ \hline
1 & 1 & 1 & 1 & 4 & y\\
1 & 1 & 1 & 2 & 5 & y\\
1 & 1 & 2 & 2 & 6 & y\\
1 & 1 & 1 & 3 & 6 & y\\
1 & 1 & 2 & 3 & 7 & y\\
1 & 2 & 2 & 3 & 8 & y\\
1 & 1 & 2 & 4 & 8 & y\\
1 & 2 & 3 & 3 & 9 & y\\
1 & 1 & 3 & 4 & 9 & y\\
1 & 2 & 3 & 4 & 10 & y\\
1 & 2 & 2 & 5 & 10 & y\\
1 & 1 & 3 & 5 & 10 & y\\
1 & 2 & 3 & 5 & 11 & y\\
2 & 3 & 3 & 4 & 12 & y\\
1 & 3 & 4 & 4 & 12 & y\\
2 & 2 & 3 & 5 & 12 & y\\
1 & 2 & 4 & 5 & 12 & y\\
1 & 2 & 3 & 6 & 12 & y\\
1 & 1 & 4 & 6 & 12 & y\\
1 & 3 & 4 & 5 & 13 & y\\
2 & 3 & 4 & 5 & 14 & y\\
2 & 2 & 3 & 7 & 14 & y\\
1 & 2 & 4 & 7 & 14 & y\\
3 & 3 & 4 & 5 & 15 & y\\
2 & 3 & 5 & 5 & 15 & y\\
1 & 3 & 5 & 6 & 15 & y\\
1 & 3 & 4 & 7 & 15 & y\\
1 & 2 & 5 & 7 & 15 & y\\
1 & 4 & 5 & 6 & 16 & y\\
2 & 3 & 4 & 7 & 16 & y\\
1 & 3 & 4 & 8 & 16 & y\\
1 & 2 & 5 & 8 & 16 & y\\
\hline\hline \end{tabular}\hfil
\hbox{\begin{tabular}{|rrrr|r|r|} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & d & span \\ \hline
2 & 3 & 5 & 7 & 17 & y\\
3 & 4 & 5 & 6 & 18 & y\\
1 & 4 & 6 & 7 & 18 & y\\
2 & 3 & 5 & 8 & 18 & y\\
2 & 3 & 4 & 9 & 18 & y\\
1 & 3 & 5 & 9 & 18 & y\\
1 & 2 & 6 & 9 & 18 & y\\
3 & 4 & 5 & 7 & 19 & y\\
2 & 5 & 6 & 7 & 20 & n\\
3 & 4 & 5 & 8 & 20 & y\\
2 & 4 & 5 & 9 & 20 & n\\
2 & 3 & 5 & 10 & 20 & y\\
1 & 4 & 5 & 10 & 20 & y\\
3 & 5 & 6 & 7 & 21 & n\\
1 & 5 & 7 & 8 & 21 & n\\
2 & 3 & 7 & 9 & 21 & y\\
1 & 3 & 7 & 10 & 21 & n\\
2 & 4 & 5 & 11 & 22 & n\\
1 & 4 & 6 & 11 & 22 & y\\
1 & 3 & 7 & 11 & 22 & n\\
3 & 6 & 7 & 8 & 24 & n\\
4 & 5 & 6 & 9 & 24 & y\\
1 & 6 & 8 & 9 & 24 & y\\
3 & 4 & 7 & 10 & 24 & y\\
2 & 3 & 8 & 11 & 24 & n\\
3 & 4 & 5 & 12 & 24 & y\\
2 & 3 & 7 & 12 & 24 & y\\
1 & 3 & 8 & 12 & 24 & y\\
4 & 5 & 7 & 9 & 25 & n\\
2 & 5 & 6 & 13 & 26 & n\\
1 & 5 & 7 & 13 & 26 & n\\
2 & 3 & 8 & 13 & 26 & n\\
\hline\hline \end{tabular}}\hfil
\hbox{\begin{tabular}{|rrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & d & span \\ \hline
5 & 6 & 7 & 9 & 27 & n\\
2 & 5 & 9 & 11 & 27 & n\\
4 & 6 & 7 & 11 & 28 & n\\
3 & 4 & 7 & 14 & 28 & y\\
1 & 4 & 9 & 14 & 28 & n\\
5 & 6 & 8 & 11 & 30 & n\\
3 & 4 & 10 & 13 & 30 & n\\
4 & 5 & 6 & 15 & 30 & y\\
2 & 6 & 7 & 15 & 30 & n\\
1 & 6 & 8 & 15 & 30 & y\\
2 & 3 & 10 & 15 & 30 & y\\
1 & 4 & 10 & 15 & 30 & y\\
4 & 5 & 7 & 16 & 32 & n\\
2 & 5 & 9 & 16 & 32 & n\\
3 & 5 & 11 & 14 & 33 & n\\
4 & 6 & 7 & 17 & 34 & n\\
3 & 4 & 10 & 17 & 34 & n\\
7 & 8 & 9 & 12 & 36 & n\\
3 & 4 & 11 & 18 & 36 & n\\
1 & 5 & 12 & 18 & 36 & n\\
5 & 6 & 8 & 19 & 38 & n\\
3 & 5 & 11 & 19 & 38 & n\\
5 & 7 & 8 & 20 & 40 & n\\
3 & 4 & 14 & 21 & 42 & y\\
2 & 5 & 14 & 21 & 42 & n\\
1 & 6 & 14 & 21 & 42 & y\\
4 & 5 & 13 & 22 & 44 & n\\
3 & 5 & 16 & 24 & 48 & n\\
7 & 8 & 10 & 25 & 50 & n\\
4 & 5 & 18 & 27 & 54 & n\\
5 & 6 & 22 & 33 & 66 & n\\
& & & & & \\
\hline\hline \end{tabular}}\hfill\\[3mm]
\hfil Table II: Weights for $l=4$
\noindent
\begin{tabular}{||rrrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & d & $\IP^4$ \\ \hline
1 & 1 & 1 & 1 & 1 & 5 & y\\
1 & 1 & 1 & 1 & 2 & 6 & y\\
1 & 1 & 1 & 2 & 2 & 7 & y\\
1 & 1 & 1 & 1 & 3 & 7 & y\\
1 & 1 & 2 & 2 & 2 & 8 & y\\
1 & 1 & 1 & 2 & 3 & 8 & y\\
1 & 1 & 1 & 1 & 4 & 8 & y\\
1 & 1 & 2 & 2 & 3 & 9 & y\\
1 & 1 & 1 & 3 & 3 & 9 & y\\
1 & 1 & 1 & 2 & 4 & 9 & y\\
1 & 2 & 2 & 2 & 3 & 10 & y\\
1 & 1 & 2 & 3 & 3 & 10 & y\\
1 & 1 & 2 & 2 & 4 & 10 & y\\
1 & 1 & 1 & 3 & 4 & 10 & n\\
1 & 1 & 1 & 2 & 5 & 10 & y\\
1 & 2 & 2 & 3 & 3 & 11 & y\\
1 & 1 & 2 & 3 & 4 & 11 & y\\
1 & 1 & 2 & 2 & 5 & 11 & y\\
1 & 1 & 1 & 3 & 5 & 11 & y\\
2 & 2 & 2 & 3 & 3 & 12 & y\\
1 & 2 & 3 & 3 & 3 & 12 & y\\
1 & 2 & 2 & 3 & 4 & 12 & y\\
1 & 1 & 3 & 3 & 4 & 12 & y\\
1 & 1 & 2 & 4 & 4 & 12 & y\\
1 & 2 & 2 & 2 & 5 & 12 & y\\
1 & 1 & 2 & 3 & 5 & 12 & y\\
1 & 1 & 1 & 4 & 5 & 12 & n\\
1 & 1 & 2 & 2 & 6 & 12 & y\\
1 & 1 & 1 & 3 & 6 & 12 & y\\
1 & 2 & 3 & 3 & 4 & 13 & y\\
1 & 1 & 3 & 4 & 4 & 13 & y\\
1 & 2 & 2 & 3 & 5 & 13 & y\\
1 & 1 & 3 & 3 & 5 & 13 & y\\
1 & 1 & 2 & 4 & 5 & 13 & n\\
1 & 1 & 2 & 3 & 6 & 13 & y\\
1 & 1 & 1 & 4 & 6 & 13 & y\\
2 & 2 & 3 & 3 & 4 & 14 & y\\
1 & 2 & 3 & 4 & 4 & 14 & n\\
2 & 2 & 2 & 3 & 5 & 14 & n\\
1 & 2 & 3 & 3 & 5 & 14 & n\\
1 & 2 & 2 & 4 & 5 & 14 & y\\
1 & 1 & 3 & 4 & 5 & 14 & n\\
1 & 2 & 2 & 3 & 6 & 14 & y\\
1 & 1 & 2 & 4 & 6 & 14 & y\\
1 & 2 & 2 & 2 & 7 & 14 & y\\
\hline\hline \end{tabular}\hfil
\hbox{\begin{tabular}{||rrrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & d & $\IP^4$ \\ \hline
1 & 1 & 2 & 3 & 7 & 14 & y\\
1 & 1 & 1 & 4 & 7 & 14 & n\\
2 & 3 & 3 & 3 & 4 & 15 & y\\
1 & 3 & 3 & 4 & 4 & 15 & y\\
2 & 2 & 3 & 3 & 5 & 15 & y\\
1 & 3 & 3 & 3 & 5 & 15 & y\\
1 & 2 & 3 & 4 & 5 & 15 & y\\
1 & 2 & 2 & 5 & 5 & 15 & y\\
1 & 1 & 3 & 5 & 5 & 15 & y\\
1 & 2 & 3 & 3 & 6 & 15 & y\\
1 & 1 & 3 & 4 & 6 & 15 & y\\
1 & 1 & 2 & 5 & 6 & 15 & n\\
1 & 2 & 2 & 3 & 7 & 15 & y\\
1 & 1 & 3 & 3 & 7 & 15 & y\\
1 & 1 & 2 & 4 & 7 & 15 & y\\
1 & 1 & 1 & 5 & 7 & 15 & y\\
2 & 3 & 3 & 4 & 4 & 16 & y\\
1 & 3 & 4 & 4 & 4 & 16 & y\\
2 & 2 & 3 & 4 & 5 & 16 & n\\
1 & 3 & 3 & 4 & 5 & 16 & y\\
1 & 2 & 4 & 4 & 5 & 16 & y\\
1 & 2 & 3 & 5 & 5 & 16 & n\\
1 & 1 & 4 & 5 & 5 & 16 & y\\
1 & 2 & 3 & 4 & 6 & 16 & y\\
1 & 1 & 4 & 4 & 6 & 16 & y\\
1 & 2 & 2 & 5 & 6 & 16 & n\\
1 & 1 & 3 & 5 & 6 & 16 & n\\
2 & 2 & 2 & 3 & 7 & 16 & y\\
1 & 2 & 3 & 3 & 7 & 16 & y\\
1 & 2 & 2 & 4 & 7 & 16 & y\\
1 & 1 & 3 & 4 & 7 & 16 & n\\
1 & 1 & 2 & 5 & 7 & 16 & y\\
1 & 2 & 2 & 3 & 8 & 16 & y\\
1 & 1 & 3 & 3 & 8 & 16 & y\\
1 & 1 & 2 & 4 & 8 & 16 & y\\
1 & 1 & 1 & 5 & 8 & 16 & y\\
2 & 3 & 3 & 4 & 5 & 17 & y\\
1 & 3 & 4 & 4 & 5 & 17 & n\\
2 & 2 & 3 & 5 & 5 & 17 & y\\
1 & 2 & 4 & 5 & 5 & 17 & n\\
1 & 2 & 3 & 5 & 6 & 17 & y\\
1 & 1 & 4 & 5 & 6 & 17 & n\\
2 & 2 & 3 & 3 & 7 & 17 & y\\
1 & 2 & 3 & 4 & 7 & 17 & y\\
1 & 2 & 2 & 5 & 7 & 17 & n\\
\hline\hline \end{tabular}}\hfil\\
\hbox{\begin{tabular}{||rrrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & d & $\IP^4$ \\ \hline
1 & 1 & 3 & 5 & 7 & 17 & y\\
1 & 2 & 3 & 3 & 8 & 17 & y\\
1 & 1 & 3 & 4 & 8 & 17 & y\\
1 & 1 & 2 & 5 & 8 & 17 & y\\
3 & 3 & 3 & 4 & 5 & 18 & n\\
2 & 3 & 4 & 4 & 5 & 18 & n\\
2 & 3 & 3 & 5 & 5 & 18 & y\\
1 & 3 & 4 & 5 & 5 & 18 & n\\
2 & 3 & 3 & 4 & 6 & 18 & y\\
1 & 3 & 4 & 4 & 6 & 18 & n\\
2 & 2 & 3 & 5 & 6 & 18 & y\\
1 & 3 & 3 & 5 & 6 & 18 & y\\
1 & 2 & 4 & 5 & 6 & 18 & n\\
1 & 2 & 3 & 6 & 6 & 18 & y\\
1 & 1 & 4 & 6 & 6 & 18 & y\\
2 & 2 & 3 & 4 & 7 & 18 & y\\
1 & 3 & 3 & 4 & 7 & 18 & n\\
1 & 2 & 4 & 4 & 7 & 18 & n\\
1 & 2 & 3 & 5 & 7 & 18 & n\\
1 & 1 & 4 & 5 & 7 & 18 & n\\
1 & 2 & 2 & 6 & 7 & 18 & n\\
1 & 1 & 3 & 6 & 7 & 18 & n\\
2 & 2 & 3 & 3 & 8 & 18 & y\\
1 & 2 & 3 & 4 & 8 & 18 & n\\
1 & 2 & 2 & 5 & 8 & 18 & y\\
1 & 1 & 3 & 5 & 8 & 18 & n\\
1 & 1 & 2 & 6 & 8 & 18 & y\\
2 & 2 & 2 & 3 & 9 & 18 & y\\
1 & 2 & 3 & 3 & 9 & 18 & y\\
1 & 2 & 2 & 4 & 9 & 18 & y\\
1 & 1 & 3 & 4 & 9 & 18 & n\\
1 & 1 & 2 & 5 & 9 & 18 & n\\
1 & 1 & 1 & 6 & 9 & 18 & y\\
3 & 3 & 4 & 4 & 5 & 19 & y\\
2 & 3 & 4 & 5 & 5 & 19 & n\\
1 & 3 & 4 & 5 & 6 & 19 & y\\
2 & 3 & 3 & 4 & 7 & 19 & n\\
1 & 3 & 4 & 4 & 7 & 19 & n\\
2 & 2 & 3 & 5 & 7 & 19 & n\\
1 & 3 & 3 & 5 & 7 & 19 & n\\
1 & 2 & 4 & 5 & 7 & 19 & y\\
1 & 2 & 3 & 6 & 7 & 19 & n\\
1 & 1 & 4 & 6 & 7 & 19 & n\\
1 & 3 & 3 & 4 & 8 & 19 & y\\
\hline\hline \end{tabular}}\hfil
\hbox{\begin{tabular}{||rrrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & d & $\IP^4$ \\ \hline
1 & 2 & 3 & 5 & 8 & 19 & n\\
1 & 1 & 3 & 6 & 8 & 19 & y\\
1 & 2 & 3 & 4 & 9 & 19 & y\\
1 & 2 & 2 & 5 & 9 & 19 & y\\
1 & 1 & 3 & 5 & 9 & 19 & y\\
1 & 1 & 2 & 6 & 9 & 19 & y\\
3 & 4 & 4 & 4 & 5 & 20 & y\\
3 & 3 & 4 & 5 & 5 & 20 & y\\
2 & 4 & 4 & 5 & 5 & 20 & y\\
2 & 3 & 5 & 5 & 5 & 20 & y\\
1 & 4 & 5 & 5 & 5 & 20 & y\\
2 & 3 & 4 & 5 & 6 & 20 & y\\
1 & 4 & 4 & 5 & 6 & 20 & n\\
2 & 2 & 5 & 5 & 6 & 20 & y\\
1 & 3 & 5 & 5 & 6 & 20 & n\\
1 & 2 & 5 & 6 & 6 & 20 & n\\
2 & 3 & 4 & 4 & 7 & 20 & n\\
2 & 3 & 3 & 5 & 7 & 20 & n\\
2 & 2 & 4 & 5 & 7 & 20 & n\\
1 & 3 & 4 & 5 & 7 & 20 & n\\
1 & 2 & 5 & 5 & 7 & 20 & n\\
2 & 2 & 3 & 6 & 7 & 20 & y\\
1 & 2 & 4 & 6 & 7 & 20 & y\\
1 & 1 & 5 & 6 & 7 & 20 & n\\
2 & 3 & 3 & 4 & 8 & 20 & y\\
1 & 3 & 4 & 4 & 8 & 20 & y\\
2 & 2 & 3 & 5 & 8 & 20 & n\\
1 & 3 & 3 & 5 & 8 & 20 & n\\
1 & 2 & 4 & 5 & 8 & 20 & y\\
1 & 2 & 3 & 6 & 8 & 20 & n\\
1 & 1 & 4 & 6 & 8 & 20 & y\\
2 & 2 & 3 & 4 & 9 & 20 & y\\
1 & 2 & 4 & 4 & 9 & 20 & y\\
2 & 2 & 2 & 5 & 9 & 20 & y\\
1 & 2 & 3 & 5 & 9 & 20 & y\\
1 & 1 & 4 & 5 & 9 & 20 & n\\
1 & 2 & 2 & 6 & 9 & 20 & y\\
2 & 2 & 3 & 3 & 10 & 20 & y\\
1 & 2 & 3 & 4 & 10 & 20 & y\\
1 & 1 & 4 & 4 & 10 & 20 & y\\
1 & 2 & 2 & 5 & 10 & 20 & y\\
1 & 1 & 3 & 5 & 10 & 20 & y\\
1 & 1 & 2 & 6 & 10 & 20 & y\\
& & & & & & \\
\hline\hline \end{tabular}}\hfill\\[3mm]
\hfil Table III: Weights for $l=5$ and $d\le 20$
\noindent
\hbox{\begin{tabular}{||rrrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & d & $\IP^4$ \\ \hline
\del
35 & 38 & 400 & 946 & 1419 & 2838 & n \\
36 & 37 & 401 & 948 & 1422 & 2844 & n \\
33 & 35 & 408 & 952 & 1428 & 2856 & n \\
31 & 37 & 408 & 952 & 1428 & 2856 & n \\
29 & 39 & 408 & 952 & 1428 & 2856 & n \\
27 & 41 & 408 & 952 & 1428 & 2856 & n \\
33 & 41 & 403 & 954 & 1431 & 2862 & n \\
\enddel
35 & 39 & 405 & 958 & 1437 & 2874 & n \\
34 & 35 & 414 & 966 & 1449 & 2898 & n \\
32 & 37 & 414 & 966 & 1449 & 2898 & n \\
31 & 38 & 414 & 966 & 1449 & 2898 & n \\
29 & 40 & 414 & 966 & 1449 & 2898 & n \\
28 & 41 & 414 & 966 & 1449 & 2898 & y \\
34 & 41 & 409 & 968 & 1452 & 2904 & n \\
33 & 37 & 420 & 980 & 1470 & 2940 & n \\
31 & 39 & 420 & 980 & 1470 & 2940 & n \\
29 & 41 & 420 & 980 & 1470 & 2940 & n \\
35 & 41 & 415 & 982 & 1473 & 2946 & n \\
35 & 36 & 426 & 994 & 1491 & 2982 & n \\
34 & 37 & 426 & 994 & 1491 & 2982 & n \\
33 & 38 & 426 & 994 & 1491 & 2982 & n \\
32 & 39 & 426 & 994 & 1491 & 2982 & n \\
31 & 40 & 426 & 994 & 1491 & 2982 & n \\
30 & 41 & 426 & 994 & 1491 & 2982 & n \\
29 & 42 & 426 & 994 & 1491 & 2982 & n \\
36 & 41 & 421 & 996 & 1494 & 2988 & y \\
35 & 37 & 432 & 1008 & 1512 & 3024 & n \\
31 & 41 & 432 & 1008 & 1512 & 3024 & n \\
36 & 37 & 438 & 1022 & 1533 & 3066 & n \\
35 & 38 & 438 & 1022 & 1533 & 3066 & n \\
34 & 39 & 438 & 1022 & 1533 & 3066 & n \\
33 & 40 & 438 & 1022 & 1533 & 3066 & n \\
32 & 41 & 438 & 1022 & 1533 & 3066 & n \\
31 & 42 & 438 & 1022 & 1533 & 3066 & n \\
35 & 39 & 444 & 1036 & 1554 & 3108 & n \\
33 & 41 & 444 & 1036 & 1554 & 3108 & n \\
37 & 38 & 450 & 1050 & 1575 & 3150 & n \\
34 & 41 & 450 & 1050 & 1575 & 3150 & n \\
37 & 39 & 456 & 1064 & 1596 & 3192 & n \\
35 & 41 & 456 & 1064 & 1596 & 3192 & n \\
38 & 39 & 462 & 1078 & 1617 & 3234 & n \\
37 & 40 & 462 & 1078 & 1617 & 3234 & n \\
36 & 41 & 462 & 1078 & 1617 & 3234 & y \\
37 & 41 & 468 & 1092 & 1638 & 3276 & n \\
39 & 40 & 474 & 1106 & 1659 & 3318 & n \\
38 & 41 & 474 & 1106 & 1659 & 3318 & n \\
37 & 42 & 474 & 1106 & 1659 & 3318 & n \\
39 & 41 & 480 & 1120 & 1680 & 3360 & n \\
40 & 41 & 486 & 1134 & 1701 & 3402 & n \\
41 & 42 & 498 & 1162 & 1743 & 3486 & y \\
\hline\hline \end{tabular}}\hfill\
\hbox{\begin{tabular}{||rrrrr|r|r||} \hline\hline
$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & d & $\IP^4$ \\ \hline
\del
20 & 33 & 199 & 471 & 690 & 1413 & n \\
19 & 35 & 200 & 473 & 692 & 1419 & n \\
13 & 42 & 204 & 476 & 693 & 1428 & n \\
15 & 38 & 204 & 476 & 695 & 1428 & n \\
21 & 32 & 201 & 476 & 698 & 1428 & n \\
19 & 30 & 204 & 476 & 699 & 1428 & n \\
21 & 26 & 204 & 476 & 701 & 1428 & n \\
\enddel
19 & 36 & 203 & 480 & 702 & 1440 & n \\
14 & 41 & 207 & 483 & 704 & 1449 & y \\
16 & 37 & 207 & 483 & 706 & 1449 & n \\
17 & 35 & 207 & 483 & 707 & 1449 & n \\
19 & 31 & 207 & 483 & 709 & 1449 & n \\
20 & 29 & 207 & 483 & 710 & 1449 & n \\
17 & 36 & 210 & 490 & 717 & 1470 & n \\
21 & 34 & 207 & 490 & 718 & 1470 & n \\
19 & 32 & 210 & 490 & 719 & 1470 & n \\
15 & 41 & 213 & 497 & 725 & 1491 & n \\
16 & 39 & 213 & 497 & 726 & 1491 & n \\
17 & 37 & 213 & 497 & 727 & 1491 & n \\
18 & 35 & 213 & 497 & 728 & 1491 & n \\
19 & 33 & 213 & 497 & 729 & 1491 & n \\
20 & 31 & 213 & 497 & 730 & 1491 & n \\
21 & 29 & 213 & 497 & 731 & 1491 & n \\
17 & 38 & 216 & 504 & 737 & 1512 & n \\
19 & 34 & 216 & 504 & 739 & 1512 & n \\
16 & 41 & 219 & 511 & 746 & 1533 & n \\
17 & 39 & 219 & 511 & 747 & 1533 & n \\
18 & 37 & 219 & 511 & 748 & 1533 & n \\
19 & 35 & 219 & 511 & 749 & 1533 & n \\
20 & 33 & 219 & 511 & 750 & 1533 & n \\
21 & 31 & 219 & 511 & 751 & 1533 & n \\
17 & 40 & 222 & 518 & 757 & 1554 & n \\
19 & 36 & 222 & 518 & 759 & 1554 & n \\
21 & 32 & 222 & 518 & 761 & 1554 & n \\
17 & 41 & 225 & 525 & 767 & 1575 & n \\
19 & 37 & 225 & 525 & 769 & 1575 & n \\
17 & 42 & 228 & 532 & 777 & 1596 & n \\
21 & 34 & 228 & 532 & 781 & 1596 & n \\
18 & 41 & 231 & 539 & 788 & 1617 & y \\
19 & 39 & 231 & 539 & 789 & 1617 & n \\
20 & 37 & 231 & 539 & 790 & 1617 & n \\
19 & 40 & 234 & 546 & 799 & 1638 & n \\
19 & 41 & 237 & 553 & 809 & 1659 & n \\
20 & 39 & 237 & 553 & 810 & 1659 & n \\
21 & 37 & 237 & 553 & 811 & 1659 & n \\
19 & 42 & 240 & 560 & 819 & 1680 & n \\
21 & 38 & 240 & 560 & 821 & 1680 & n \\
20 & 41 & 243 & 567 & 830 & 1701 & n \\
21 & 40 & 246 & 574 & 841 & 1722 & n \\
21 & 41 & 249 & 581 & 851 & 1743 & y \\
\hline\hline \end{tabular}}\hfill\\[3mm]
\hfil Table IV: Weights for $l=5$ and large $d$ (for $q_5=1/2$ and $q_5<1/2$)
\newpage
\def\LLab#1{\BP(0,0)\unitlength=1mm\put(-15,0){\makebox(0,0)[br]{\small#1}}\EP}
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\ifundefined{draftmode}
|
1996-03-22T06:20:27 | 9603 | alg-geom/9603015 | en | https://arxiv.org/abs/alg-geom/9603015 | [
"alg-geom",
"math.AG"
] | alg-geom/9603015 | Stein Arild Stromme | Geir Ellingsrud and Stein Arild Str{\o}mme | An intersection number for the punctual Hilbert scheme of a surface | AMSLaTeX v 1.2, 7 pages | null | null | null | null | Let S be a smooth projective surface, and consider the following two
subvarieties of the Hilbert scheme parameterizing closed subschemes of S of
length n:
A = {subschemes with support in a fixed point of S}
B = {subschemes with support in one (variable) point of S}
A and B have complementary dimensions in the Hilbert scheme. We prove that
the intersection number [A].[B] = n(-1)^(n-1), answering a question by H.
Nakajima.
| [
{
"version": "v1",
"created": "Thu, 21 Mar 1996 08:21:21 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ellingsrud",
"Geir",
""
],
[
"Strømme",
"Stein Arild",
""
]
] | alg-geom | \section{Introduction}
Let $S$ be a smooth projective surface over an algebraically closed
field $k$. For any natural number $n$ let $H_n$ denote the Hilbert
scheme parameterizing finite subschemes of $S$ of length $n$. It is
smooth and projective of dimension $2n$.
Let $P\in S$ be a point and let $M_n(P)\subseteq H_n$ be the closed
reduced subvariety consisting of points which correspond to subschemes
with support at $P$. Brian\c con proved that $M_n(P)$ is an irreducible
variety of dimension $n-1$, see \cite{Bria}.
Denote by $M_n=\cup_{P\in S}M_n(P)\subseteq H_n$ the subvariety whose
points correspond to subschemes with support in just one point. We may
map $M_n$ to $S$ by sending a point of $M_n$ to the point where the
corresponding subscheme is supported. The fiber of this map over a
point $P$ being the variety $M_n(P)$, we see that $M_n$ is irreducible
of dimension $n+1$.
The subvarieties $M_n$ and $M_n(P)$ are of complementary codimensions,
and hence the product of their rational equivalence classes (or dual
cohomology classes, if $k$ is the field of complex numbers) defines an
intersection number $\int_{H_n}^{}[M_n]\cdot [M_n(P)]$. The main content
of this note is the computation of that number. The result is:
\begin{thm}\label{main}
$\int_{H_n}^{}[M_n]\cdot[M_n(P)]=(-1)^{n-1}n.$
\end{thm}
One reason, pointed out to us by H.~Nakajima, to be interested in these
intersection numbers is the following. In case $k={\mathbf C}$ is the field
of complex numbers, G\"ottsche \cite{Goet1} computed the generating
series $ \sum_{m,n=0}^{\infty}\dim H^m(H_n,{\mathbf Q})t^n u^m $ and showed
that it may be expressed in terms of classical modular forms. These
forms are closely related to the trace of some standard
representations of respectively the infinite Heisenberg algebra and
the infinite Clifford algebra. In \cite{Naka} Nakajima defined a
representation of a product of these algebras, indexed over
$H^{*}(S,{\mathbf Q})$, on the space $\oplus_{m,n=0}^{\infty}H^{m}(H_n,{\mathbf Q})$.
He completely described this representation up to the determination of
a series of universal constants $c_n$ for $n=1,2,\ldots$, universal in
the sense that they do not depend on the surface. He also proved that
$c_n=\int_{H_n}^{}[M_n]\cdot[M_n(P)]$. Hence we have
\begin{thm}\label{NakaTheo}
The Nakajima constants are given by $c_n=(-1)^{n-1}n$.
\end{thm}
\section{Proof of the main theorem}
The proof will be an inductive argument comparing the number $c_n$
with $c_{n+1}$. To make this comparison, we shall make use of the
``incidence variety'', i.e., the closed, reduced subscheme of
$H_n\times H_{n+1}$ given by $H_{n,n+1}=\{(\xi,\eta)\mid \xi\subseteq \eta \}$. It
is known that $H_{n,n+1}$ is smooth and irreducible of dimension $2n+2$
(there are several proofs of this, see for example \cite{Chea-1} or
\cite{Tikh-2}).
There are obvious maps $f\:H_{n,n+1}\to H_n$ and $g\:H_{n,n+1}\to H_{n+1}$ induced
by the projections. There is also a natural map $q\:H_{n,n+1}\to S$
sending a pair $(\xi,\eta)$ to the unique point where $\xi$ and $\eta$
differ. Let $Z_n\subseteq H_n\times S$ be the universal subscheme. It is
finite and flat over $H_n$ of rank $n$. Let $\pi_n\:Z_n\to H_n $
denote the restriction of the projection. Furthermore, let
$H_n'\subH_n$ denote the open dense subset parameterizing local
complete intersection subschemes, and put $Z_n'=\pi_n^{-1} H_n'$.
In the next section we will prove the following results, which
shed light on the maps $g$ and $f$.
\begin{prop}\label{structure1}
The map $g\:H_{n,n+1}\toH_{n+1}$ factors naturally as
$g=\pi_{n+1}\circ\psi$, where $\psi\:H_{n,n+1}\to Z_{n+1}$ is canonically
isomorphic to ${\mathbf P}(\omega_{Z_{n+1}})$. In
particular, $\psi$ is birational and an isomorphism over $Z_{n+1}'$,
and $g$ is generically finite of degree $n+1$.
\end{prop}
\begin{prop}\label{structure2}
The map $\phi=(f,q)\:H_{n,n+1}\toH_n\times S$ is canonically isomorphic to
the blowing up of $H_n\times S$ along $Z_n$. In particular, over $Z_n'$,
the map $\phi$ is a
${\mathbf P}^1$-bundle.
\end{prop}
It follows that the fibers of $f$ over local complete intersection
subschemes $\xi\inH_n'$ are given as $f^{-1}(\xi)=\widetilde S(\xi)$, the
surface $S$ blown up along $\xi$. (This is also easy to see directly.)
The locus of pairs $(\xi,\eta)\in H_{n,n+1}$ where $\xi$ and $\eta$ have
the same support is a divisor in $H_{n,n+1}$ which we denote by $E$. This
is nothing but the exceptional divisor of the blowup morphism $\phi$.
On the fiber of $f$ over a local complete intersection $\xi$, it
restricts to the exceptional divisor of $\widetilde S(\xi)$.
Let $M_{n,n+1}=(g^{-1} M_{n+1})_{\text{red}}$. We need the following
strengthening of Brian\c con's result, also to be proved in the next
section.
\begin{prop}\label{structure3}
$M_{n,n+1}$ is irreducible, and $g$ maps it birationally to
$M_{n+1}$. In particular, all the $M_n$ are irreducible, and the
complete intersection subschemes form a dense open subset of $M_n$.
\end{prop}
Using the three propositions above, we have sufficient information to
carry out the intersection computation. Let us summarize the situation
in the following commutative diagram, where $B_n=\rho(M_{n,n+1})$:
\[
\begin{CD}
M_{n,n+1} @>\subseteq >> E @>j>> H_{n,n+1} @>\gamma=(g,q)>> H_{n+1}\times S \\
@VVV @V\rho VV @VV\phi=(f,q) V @VV pr_2V \\
B_n @>\subseteq >> Z_n @>i>> H_n\times S @> pr_2 >> S \\
@VVV @V\pi_n VV @VV pr_1 V \\
M_n@>\subseteq>> H_n @= H_n
\end{CD}
\]
\begin{lemma}\label{lemmaa}
$ g^*[M_{n+1}] = (n+1)\,[M_{n,n+1}]$ in $A^n(H_{n,n+1})$.
\end{lemma}
\begin{proof}
Since $g^{-1} M_{n+1}$ is a multiple structure on $M_{n,n+1}$ by
definition, and the codimensions of $M_{n+1}$ and $M_{n,n+1}$ are
the same, $g^*[M_{n+1}] = \ell\, [M_{n,n+1}]$ for some integer
$\ell$. Now use that $g_*[M_{n,n+1}]=[M_{n+1}]$ (by
\propref{structure3}) and the projection formula to get
\[
(n+1)[M_{n+1}] = g_* g^* [M_{n+1}] = g_* (\ell\, [M_{n,n+1}])=
\ell\,[M_{n+1}],
\]
proving that $\ell=n+1$.
\end{proof}
\begin{lemma}\label{lemmab}
$ [E]\cdot f^*[M_n] = n\,[M_{n,n+1}]$ in $A^n(H_{n,n+1})$.
\end{lemma}
\begin{proof}
Consider first $[M_{n,n+1}]_E\in A_{n+2}(E)$. Let $h=\pi_n\circ\rho\:
E\toH_n$. Since $M_{n,n+1}$ is the support of $h^{-1} M_n$ and its
codimension in $E$ equals $\operatorname{codim}(M_n,H_n)$, we have that
$h^*[M_n]=\ell\,[M_{n,n+1}]_E$ where $\ell$ is the multiplicity of
$h^{-1} M_n$ at the generic point $\eta$ of $M_{n,n+1}$. By
\propref{structure2}, $\rho$ is a smooth at $\eta$, so $\ell$ equals
also the multiplicity of $\pi_n^{-1} M_n$ at the generic point
$\rho(\eta)$ of $B_n$. But observing that $B_n$ maps isomorphically
to $M_n$, a similar argument as in the proof of \lemref{lemmaa}
shows that $\pi_n^*[M_n]=n\,[B_n]$, hence $\ell=n$.
We have shown that $h^*[M_n] = n\,[M_{n,n+1}]_E$ in $A_{n+2}(E)$. Apply
$j_*$ and the projection formula to get
\[
n\,[M_{n,n+1}] = j_*h^*[M_n]=j_*j^*f^*[M_n] = [E]\cdot f^*[M_n].
\]
\end{proof}
Combining the two lemmas above, we get
\begin{equation}
\label{vari}
\frac1{n+1}\,g^*[M_{n+1}] = \frac1n\,[E]\cdot f^*[M_n] \in A^n(H_{n,n+1}),
\end{equation}
and exactly parallel reasoning shows that also
\begin{equation}
\label{fast}
\frac1{n+1}\,g^*[M_{n+1}(P)] = \frac1n\,[E]\cdot f^*[M_n(P)] \in
A^{n+2} (H_{n,n+1}).
\end{equation}
We are now ready to prove \thmref{main}. Let $F$ be a general
fiber of $f$, for example corresponding to a reduced subscheme
$\xi$. Clearly, $f^*[M_n]\cdot f^*[M_n(P)] = c_n\,[F]$.
It is easy to see that $\int_F [E]^2 = -n$, and we get the
following computation:
\begin{alignat*}{3}
\frac{c_{n+1}}{n+1} &= \frac{1}{n+1}\int_{H_{n+1}}
[M_{n+1}][M_{n+1}(P)]&&\\[2mm]
& =\int_{H_{n,n+1}}
\frac1{n+1}\,{g^*[M_{n+1}]}\cdot\frac1{n+1}\,{g^*[M_{n+1}(P)]}&\quad
&\text{(proj.\ formula)}\\[2mm]
& = \int_{H_{n,n+1}} \frac1n\,{[E]f^*[M_n]}
\cdot \frac1n\,{[E]f^*[M_n(P)]} &\quad&\text{(\eqref{vari} and
\eqref{fast})}\\[2mm]
&=c_n \int_F \frac1{n^2} \,[E]^2 =\frac{-c_n}n.
\end{alignat*}
Now since trivially $c_1=1$, \thmref{main} follows by induction.
\section{The geometry of the incidence variety}
The aim of this section is to prove propositions \ref{structure1},
\ref{structure2}, and \ref{structure3} above. Some of the content
of this section may be found in \cite{Ell}, but for the benefit of
the reader we reproduce it here.
Consider a nested pair of subschemes $(\xi,\eta)\inH_{n,n+1}$, and let
$P=q(\xi,\eta)\in S$ be the point where they differ. There are
natural short exact sequence on $S$:
\begin{align}\label{basic0}
&0 \to {\mathcal I}_{\eta} \to {\mathcal I}_{\xi} \to k(P) \to 0\\
\label{basic1}
&0 \to k(P) \to {\mathcal O}_{\eta} \to {\mathcal O}_{\xi} \to 0.
\end{align}
The first of these shows that the fiber $\phi^{-1}(\xi,P)$ is
naturally identified with
the projective space ${\mathbf P}({\mathcal I}_{\xi}(P))$.
Dualizing \eqref{basic1} we arrive at another exact
sequence
\begin{equation}
\label{basic2}
0 \to \omega_{\xi} \to \omega_{\eta} \to k(P) \to 0,
\end{equation}
and this shows that the fiber $\gamma^{-1}(\eta,P)$ maps naturally to
${\mathbf P}(\omega_{\eta}(P))$. Dualizing again we see that \eqref{basic1}
and hence $\xi$ can be reconstructed from the right half of
\eqref{basic2}, so the map is an isomorphism.
It follows from \eqref{basic0} that
\begin{equation}
\label{basic3}
|\dim_k{\mathcal I}_{\xi}(P) - \dim_k{\mathcal I}_{\eta}(P)| \le 1.
\end{equation}
(If ${\mathcal F}$ is a coherent sheaf, ${\mathcal F}(P)$ means ${\mathcal F}\*k(P)$.)
Note also that for any pair $(\xi,P)\in
H_n\times S$, we have
\begin{equation}
\label{basic4}
\dim_k {\mathcal I}_{\xi}(P) = 1+ \dim_k \omega_{\xi}(P),
\end{equation}
most easily seen using a minimal free resolution of the local ring
${\mathcal O}_{\xi,P}$ over ${\mathcal O}_{S,P}$.
The sequences \eqref{basic0} and \eqref{basic2} can be naturally
globalized to the relative case of families of subschemes and points.
This way one easily proves \propref{structure1}, as well as the
following lemma:
\begin{lemma}\label{philemma}
Let ${\mathcal I}_n$ denote the sheaf of ideals of $Z_n$ in ${\mathcal O}_{H_n\times
S}$. Then there is an isomorphism $H_{n,n+1}\simeq{\mathbf P}({\mathcal I}_n)$ such that
$\phi$ corresponds to the tautological mapping ${\mathbf P}({\mathcal I}_n)\to
H_n\times S$.
\end{lemma}
We shall prove that the map $\phi\:H_{n,n+1}\to H_n \times S$ is the blow
up of $H_n \times S$ along the universal subscheme $Z_n$, by proving
a general proposition on blowing up codimension two subschemes, and
later verify its hypotheses in the case at hand.
Let $W$ be any irreducible algebraic scheme and $Z\subseteq W$ a subscheme
of codimension $2$ whose ideal we denote by ${\mathcal I}_{W}$. We assume that
${\mathcal O}_{Z}$ is of local projective dimension 2 over ${\mathcal O}_{W}$. For any
integer $i$ let $W_{i}=\{w\in W \mid \dim_k{\mathcal I}_{W}(w) = i\}$. Let
$\widetilde W$ be the blow up of $W$ along $Z$.
There is an obvious map from $\widetilde W$ to ${\mathbf P}({\mathcal I})$ due to the fact
that ${\mathcal I}{\mathcal O}_{\widetilde W}$ is invertible. We shall see that under
certain conditions this map is an isomorphism. (See also
\cite[Prop.~9]{Avramov}.)
\begin{prop}\label{blowup}
With the above hypothesis
\begin{enumerate}
\item[(a)] Suppose that $\operatorname{codim} W_{i}\ge i$ for all $i\ge2$. Then
${\mathbf P}({\mathcal I})$ is irreducible and isomorphic to $\widetilde W$.
\item[(b)] If furthermore $Z$ is irreducible and $\operatorname{codim}
W_{i}\ge i+1$ for $i\ge3$, then the exceptional divisor $E$ is
irreducible.
\end{enumerate}
\end{prop}
\begin{proof}
The assumption on the local projective dimension gives an exact
sequence
\begin{equation}\label{hilbertbirch}
0\to {\mathcal A} \til{M} {\mathcal B} \to {\mathcal I} \to 0
\end{equation}
where ${\mathcal A}$ and ${\mathcal B}$ are vector bundles on $W$ whose ranks
are $p$ and $p+1$ respectively, for some integer $p$.
Locally the map $M$ is given
by a $(p+1)\times p$-matrix of functions on $W$, and the
ideal ${\mathcal I}$ is locally generated by its maximal minors.
The sequence \eqref{hilbertbirch} induces a natural inclusion
${\mathbf P}({\mathcal I})\subseteq{\mathbf P}({\mathcal B})$. In fact, letting $\varepsilon\:{\mathbf P}({\mathcal B})\to W$ be
the structure map and $\tau\:\varepsilon^{*}{\mathcal B}\to{\mathcal O}_{{\mathbf P}({\mathcal B})}(1)$
the tautological quotient, ${\mathbf P}({\mathcal I}) $ is defined in
${\mathbf P}({\mathcal B})$ as the vanishing locus of the map $\tau\circ M$.
Consequently, any irreducible component of ${\mathbf P}({\mathcal I})$ has dimension at
least equal to $\dim( W)$.
Let $\phi\:{\mathbf P}({\mathcal I})\to W$ be the structure map. Over $W_i$, the map
$\phi$ is a ${\mathbf P}^{i-1}$-bundle. Hence by the assumption in $(a)$, we
have
\[ \dim \phi^{-1} W_i \le (\dim W - i) + (i-1) < \dim W
\]
for all
$i\ge 2$. It follows that $\phi^{-1}(W-Z)$ is dense in ${\mathbf P}({\mathcal I})$,
which is therefore irreducible.
To see that ${\mathbf P}({\mathcal I})$ is isomorphic to the blow up, we remark that it
follows from the resolution \eqref{hilbertbirch} that
${\mathcal I}{\mathcal O}_{{\mathbf P}({\mathcal I})}$ is an invertible ideal. This gives a map from
${\mathbf P}({\mathcal I})$ to $\widetilde W$, which over $\phi^{-1}(W-Z)$ is an inverse to
the map from $\widetilde W$ to ${\mathbf P}({\mathcal I})$ given above. As both spaces are
irreducible, the two maps are inverses to each other.
Under the assumption in $(b)$, it follows similarly that $\phi^{-1}
W_2$ is dense in the exceptional locus $\phi^{-1} Z$, as all $\phi^{-1} W_i$
are of strictly lower dimension if $i\ge 3$.
\end{proof}
To finish the proof of \propref{structure2},
we shall verify the conditions in proposition \ref{blowup}
for $Z=Z_{n}$ and $W=H_n\times S$.
Let $W_{i,n}$ be the set of points $(\xi,P)\in H_n\times S$ such that
the ideal ${\mathcal I}_{Z_{n}}$ needs exactly $i$ generators at $(\xi,P)$.
Equivalently,
\begin{equation}
W_{i,n}=\{(\xi,P)\in H_n\times S \mid \dim_k{\mathcal I}_{\xi}(P) = i\}.
\end{equation}
We shall show by induction on $n$
that $\operatorname{codim}(W_{i,n},H_n\times S) \ge 2i-2$, or
equivalently, that
\[
\dim W_{i,n}\le 2n+4-2i
\]
for all $i,n\ge1$. For $n=1$ this is evidently satisfied for all $i$.
Assume that the inequality holds for a given $n$, and all $i\ge1$.
Then it follows that
\[\dim\phi^{-1} W_{j,n}\le (2n+4-2j) + (j-1) = 2n+3-j \le 2n+4-i\]
for all $j\ge i-1$. By \eqref{basic3},
\[\gamma^{-1} W_{i,n+1}\subseteq
\phi^{-1} W_{i-1,n}\cup\phi^{-1} W_{i,n} \cup\phi^{-1} W_{i+1,n}.
\]
The fibers of $\gamma$ over $W_{i,n+1}$ are $(i-2)$-dimensional. Hence
\[
\dim W_{i,n+1}+(i-2) = \dim \gamma^{-1} W_{i,n+1} \le 2n+4-i
\]
and hence $\dim W_{i,n+1}\le 2(n+1) +4 - 2i$, as was to be shown.
Since $2i-2\ge i$ for $i\ge2$ and $2i-2\ge i+1$ for $i\ge 3$, the
proof of \propref{structure2} is now complete. Note that we have also
proved that the exceptional divisor $E$ is irreducible.
\subsubsection*{Proof of \propref{structure3}} The only hard part is
to show that $M_{n,n+1}$ is irreducible. We will apply
\propref{blowup} in the case where $W=M_{n}(P)\times S$ and $Z=Z_{n}\cap
W$. As this intersection is proper, the condition on local projective
dimension still holds in this case. Put $W_{i,n}'=W_{i,n}\cap W$.
Similar reasoning as in the last proof gives the inequality
$\operatorname{codim}(W_{i,n}',W)\ge i+1$ for all $n\ge 1$ and $i\ge 3$.
Now note that the exceptional divisor $\phi^{-1} Z$ is nothing but
$M_{n,n+1}$, which is therefore irreducible by \propref{blowup}.
|
1996-04-19T20:26:05 | 9603 | alg-geom/9603003 | en | https://arxiv.org/abs/alg-geom/9603003 | [
"alg-geom",
"dg-ga",
"hep-th",
"math.AG",
"math.DG"
] | alg-geom/9603003 | Teleman | Christian Okonek and Andrei Teleman | Seiberg-Witten invariants for manifolds with $b_+=1$, and the universal
wall crossing formula | LaTeX, 27 pages. To appear in Int. J. Math | null | null | null | null | In this paper we describe the Seiberg-Witten invariants, which have been
introduced by Witten, for manifolds with $b_+=1$. In this case the invariants
depend on a chamber structure, and there exists a universal wall crossing
formula. For every K\"ahler surface with $p_g=0$ and $q$=0, these invariants
are non-trivial for all $Spin^c(4)$-structures of non-negative index.
| [
{
"version": "v1",
"created": "Mon, 4 Mar 1996 17:33:23 GMT"
},
{
"version": "v2",
"created": "Sat, 9 Mar 1996 15:05:00 GMT"
},
{
"version": "v3",
"created": "Fri, 15 Mar 1996 16:52:46 GMT"
},
{
"version": "v4",
"created": "Fri, 19 Apr 1996 18:22:22 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Okonek",
"Christian",
""
],
[
"Teleman",
"Andrei",
""
]
] | alg-geom | \section{Introduction}
The purpose of this paper is to give a systematic description of the
Seiberg-Witten
invariants, which were introduced in [W], for manifolds with $b_+=1$. In
this situation,
compared to the general case $b_+>1$, several new features arise.
The Seiberg-Witten invariants for manifolds with $b_+>1$ are (non -
homogeneous) forms
$SW_{X,{\scriptstyle{\cal O}}}({\germ c})\in \Lambda^* H^1(X,{\Bbb Z})$, associated with an orientation
parameter
${\scriptstyle{\cal O}}$ and a class of a $Spin^c(4)$-structures ${\germ c}$ on $X$.
The invariants for manifolds with $b_+=1$ depend on a chamber structure;
they are
associated with data $({\scriptstyle{\cal O}}_1,{\bf H}_0,{\germ c})$, where $({\scriptstyle{\cal O}}_1,{\bf H}_0)$ are
orientation parameters and ${\germ c}$ is again the class of a
$Spin^c(4)$-structure on $X$. In
this case, the invariants are functions $SW_{X,({\scriptstyle{\cal O}}_1,{\bf
H}_0)}({\germ c}):\{\pm\}\longrightarrow
\Lambda^* H^1(X,{\Bbb Z})$.
One of the main results of this paper is the proof of a universal wall
crossing formula.
This formula, which generalizes previous results of [W], [KM] and [LL]
describes the difference
$SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(+)-SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(-)$ as an
abelian
$Spin^c(4)$-form. More precisely, on elements $\lambda\in
\Lambda^r\left(\qmod{H_1(X,{\Bbb Z})}{\rm Tors}\right)$ with $0\leq r \leq\min(
b_1,w_c)$,
we have:
$$\left[SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(+)-SW_{X,({\scriptstyle{\cal O}}_1,{\bf
H}_0)}({\germ c})(-)\right](\lambda)=\langle
\lambda\wedge\exp(-u_c),l_{{\scriptscriptstyle{\cal O}}_1}\rangle\ ,
$$
where $u_c\in \Lambda^2\left(\qmod{H_1(X,{\Bbb Z})}{\rm Tors}\right)$ is given by
$u_c(a\wedge b)=\frac{1}{2}\langle a\cup b\cup c,[X]\rangle$, $a,b\in
H^1(X,{\Bbb Z})$,
and $l_{{\scriptscriptstyle{\cal O}}_1}\in \Lambda^{b_1} H^1(X,{\Bbb Z})$ represents the orientation
${\scriptstyle{\cal O}}_1$ of
$H^1(X,{\Bbb R})$. Here $c$ is the Chern class of ${\germ c}$ and
$w_c:=\frac{1}{4}(c^2-3\sigma(X)-2e(X))$ is the index of ${\germ c}$.
This formula has some important consequences, e.g. it shows that Seiberg-Witten
invariants of manifolds with positive scalar curvature metrics are essentially
topological invariants. According to Witten's vanishing theorem [W], one has
$SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(\pm)=0$ for at least one element of
$\{\pm\}$, and the other value is determined be the wall crossing formula.
In the final part of the paper we show how to calculate $SW_{X,({\scriptstyle{\cal O}}_1,{\bf
H}_0)}({\germ c})(\pm)$ for K\"ahlerian surfaces. The relevant Seiberg-Witten
moduli spaces
have in this case a purely complex analytic description as Douady spaces
of curves
representing given homology classes: this description is essentially the
Kobaya\-shi-Hitchin correspondence obtained in [OT1].
Witten has shown that non-trivial invariants of K\"ahlerian surfaces with
$b_+>1$
must necessarily have index 0. This is not the case for surfaces with
$b_+=1$. We show
that a K\"ahlerian surface with $b_+=1$ and $b_1=0$ has $SW_{X,({\scriptstyle{\cal O}}_1,{\bf
H}_0)}({\germ c})(\{\pm\})=\{0,1\}$ or $SW_{X,({\scriptstyle{\cal O}}_1,{\bf
H}_0)}({\germ c})(\{\pm\})=\{0,-1\}$ as soon as the index of ${\germ c}$ is
non-negative. For these
surfaces the invariants are therefore completely determined by their
reductions modulo
2.
There exist examples of 4-manifolds with $b_+=1$ which possess - for every
prescribed
non-negative index - infinitely many classes ${\germ c}$ of
$Spin^c(4)$-structures with
$SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})\not\equiv 0$.
\section{ The twisted Seiberg-Witten equations}
Let $X$ be a closed
connected oriented 4-manifold, and let $c\in H^2(X,{\Bbb Z})$ be a class with
$c\equiv
w_2(X)$ (mod 2). A
\underbar{compatible} $Spin^c(4)$-bundle is a $Spin^c(4)$-bundle $\hat
P$ over
$X$ with
$c_1(\hat P\times_{\det}{\Bbb C})=c$ such that its $GL_+(4,{\Bbb R})$-extension $\hat
P\times_{\tilde\pi}GL_+(4,{\Bbb R})$ is isomorphic to the bundle of oriented frames in
$\Lambda^1_X$; here $\tilde\pi$ denotes the composition of the canonical
representation $\pi:Spin^c(4)\longrightarrow SO(4)$ with the inclusion
$SO(4)\subset GL_+(4,{\Bbb R})$. Let $\Sigma^{\pm}:=\hat P\times_{\sigma_{\pm}}{\Bbb C}^2$
be the associated spinor bundles with $\det\Sigma^{\pm}=\hat P\times_{\det}{\Bbb C}$
[OT1].
\begin{dt} A \underbar{Clifford} \underbar{map} of type $\hat P$ is
an orientation-preserving isomorphism \hbox{ $\gamma:\Lambda^1_X\longrightarrow
\hat P\times_{\pi}{\Bbb R}^4$}.
\end{dt}
The $SO(4)$-vector bundle $\hat P\times_\pi{\Bbb R}^4$ can be identified with the
bundle
${\Bbb R} SU(\Sigma^+,\Sigma^-)$ of real multiples of ${\Bbb C}$-linear isometries of
determinant 1 from $\Sigma^+$ to $\Sigma^-$.
A Clifford map
$\gamma$ defines a metric
$g_\gamma $ on $X$, a lift $\hat P\longrightarrow P_{g_\gamma}$ of the associated
frame bundle, and it induces isomorphisms
$\Gamma:\Lambda^2_{\pm}\longrightarrow su(\Sigma^{\pm})$ [OT1]. We denote by ${\cal
C}={\cal C}(\hat P)$ the space of all Clifford maps of type $\hat P$. The
quotient
$\qmod{{\cal C}}{{\rm im}[{\rm Aut}(\hat P)\longrightarrow {\rm Aut}(\hat P\times_\pi SO(4))]}$
para\-metrizes the set of all $Spin^c(4)$-structures of Chern class $c$,
whereas $\qmod{{\cal C}}{{\rm Aut}({\hat P\times_\pi SO(4))}}$ can be identified
with
the space ${\cal M}et_X$ of Riemannian metrics on $X$. In fact,
since ${\cal M}et_X$ is contractible, we have a natural isomorphism
$\qmod{{\cal C}}{ {\rm Aut}( \hat P)}
\textmap{\simeq}{\cal M}et_X\times
\pi_0\left(\qmod{{\cal C}} { {\rm Aut}( \hat P)} \right)
$, where the second factor is a ${\rm Tors}_2H^2(X,{\Bbb Z})$-torsor; it
para\-metrizes the set of equivalence classes of $Spin^c(4)$ -structures with
Chern class $c$ on $(X,g)$, for an arbitrary metric $g$. The latter
assertion follows
from the fact that the map $H^1(X, {{\Bbb Z}}_{2})\longrightarrow
H^1(X,\underline{Spin^c}(4))$ is
trivial for any 4-manifold $X$.
We use the symbol ${\germ c}$ to denote elements in $\pi_0\left(\qmod{{\cal C}}{ {\rm Aut}(
\hat P)}\right)$, and we denote by ${\germ c}_\gamma$ the connected component defined
by
$[\gamma]\in\qmod{{\cal C}} { {\rm Aut}( \hat P)}$.
A fixed Clifford map $\gamma$ defines a bijection
between unitary connections in
$\hat P\times_{\det}{\Bbb C}$ and $Spin^c(4)$-connections in $\hat P$ which lift (via
$\gamma$) the Levi-Civita connection in
$P_{g_\gamma}$, and allows to associate a Dirac operator ${\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A$ to a
connection
$A\in{\cal A}(\hat P\times_{\det}{\Bbb C})$.
\begin{dt} Let $\gamma$ be a Clifford map, and let $\beta\in Z^2_{\rm
DR}(X)$ be a
closed 2-form. The
$\beta$-twisted Seiberg-Witten equations are
$$\left\{\begin{array}{lll}
{\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{A}\Psi&=&0\\
\Gamma\left((F_A+{2\pi i}\beta)^+\right)&=&2(\Psi\bar\Psi)_0\ .
\end{array}\right. \eqno{(SW^{\gamma}_\beta)}
$$
\end{dt}
These twisted Seiberg-Witten equations arise
naturally in
connection with certain non-abelian monopoles [OT2], [T]. They should
\underbar{not} be regarded as perturbation of $(SW^\gamma_0)$, since later the
cohomology class of $\beta$ will be fixed.
Let ${\cal W}_{X,\beta}^{\gamma} $ be the moduli space of
solutions $(A,\Psi)\in{\cal A}(\det \Sigma^+)\times A^0(\Sigma^+)$ of
$(SW^{\gamma}_\beta)$ modulo the natural action
$((A,\Psi),f)\longmapsto
(A^{f^2},f^{-1}\Psi)$ of the gauge group
${\cal G}={\cal C}^{\infty}(X,S^1)$.
Since two Clifford maps lifting the same pair $(g,{\germ c})$ are equivalent modulo
${\rm Aut}(\hat P)$, the moduli space ${\cal W}_{X,\beta}^{\gamma}$ depends up to
\underbar{canonical} isomorphism only on $(g_\gamma,{\germ c}_\gamma)$ and $\beta$.
Now fix a class $b\in H^2_{\rm DR}(X)$, consider
$(SW^{\gamma}_\beta)$ as equation for a triple $(A,\Psi,\beta)\in {\cal
A}(\det \Sigma^+)\times A^0(\Sigma^+)\times b$, and let
${\cal W}_{X,b}^{\gamma}\subset\qmod{ {\cal A}(\det \Sigma^+)\times
A^0(\Sigma^+) \times b }{{\cal G}}$
be the (infinite dimensional) moduli space of solutions. Finally we need the
universal moduli space
${\cal W}_X \subset\qmod{ {\cal A}(\det \Sigma^+)\times
A^0\ (\Sigma^+)\times Z^2_{DR}(X)\times {\cal C} }
{{\cal G}}$
of solutions of $(SW^{\gamma}_\beta)$ regarded as equations for tuples
$(A,\Psi,\beta,\gamma)\in{\cal A}(\det \Sigma^+)\times
A^0(\Sigma^+)\times Z^2_{DR}(X)\times {\cal C} $.
We complete the spaces ${\cal A}(\det\Sigma^+)$, $A^0(\Sigma^{\pm})$ and $A^2$
with respect to the Sobolev norms $L^2_q$, $L^2_q$ and $L^2_{q-1}$, and
the gauge group ${\cal G}$ with respect to $L^2_{q+1}$,
but we suppress the Sobolev subscripts in our notations. As usual we denote
by the
superscript ${\ }^*$ the open subspace of a moduli space where the spinor
component is non-zero.
\begin{dt} Let $c\in H^2(X,{\Bbb Z})$ be characteristic. A pair
$(g,b)\in{\cal M}et_X\times H^2_{\rm DR}(X)$
is $c$-good if the g-harmonic
representant of $(c-b)$ is not $g$-anti-selfdual.
\end{dt}
A pair $(g,b)$ is $c$-good for every metric $g$ if
$(c-b)\ne\ 0$ and $(c-b)^2\geq 0$.
\begin{pr} Let $X$ be a closed oriented 4-manifold, and let $c\in H^2(X,{\Bbb Z})$ be
characteristic. Choose a compatible $Spin^c(4)$-bundle $\hat P$ and an
element ${\germ c}\in \pi_0\left(\qmod{ {\cal C}}{{\rm Aut}(
\hat P)} \right)$. \\
i) The projections
$p:{\cal W}_X \longrightarrow Z^2_{DR}(X)\times{\cal C}$ and $p_{\gamma,b}:{\cal
W}_{X,b}^{\gamma } \longrightarrow b$ are proper for all choices of $\gamma$ and $b$.\\
ii) ${\cal W}_X^* $ and ${{\cal W}_{X,b}^{\gamma}}^* $ are smooth
manifolds for all $\gamma$ and $b$ .\\
iii) ${{\cal W}_{X,b}^{\gamma}}^* ={\cal W}_{X,b}^{\gamma} $ if
$(g_\gamma,b)$ is
$c$-good.\\
iv) If $(g_\gamma,b)$ is $c$-good, then every pair $(\beta_0,\beta_1)$ of
regular
values of $p_{\gamma,b}$ can be joined by a smooth path
$\beta:[0,1]\longrightarrow b$ such that the fiber product
$[0,1]\times_{(\beta, p_{\gamma,b})}{\cal W}_{X,b}^{\gamma} $ defines a
smooth bordism between ${\cal W}_{X,\beta_0}^{\gamma} $ and ${\cal
W}_{X,\beta_1}^{\gamma} $. \\
v) If $(g_0,b_0)$, $(g_1,b_1)$ are $c$-good pairs which can be joined by a
smooth path of $c$-good pairs, then there is a
smooth path $(\beta,\gamma):[0,1]\longrightarrow Z^2_{\rm DR}(X)\times{\cal
C}$ with the following properties:\\
\hspace*{0.8cm}1. $[\beta_i]=b_i$ and $g_{\gamma_i}=g_i$ for $i=0, 1$. \\
\hspace*{0.8cm}2. $\gamma_t$ lifts $(g_{\gamma_t},{\germ c})$ and
$(g_{\gamma_t},[\beta_t])$ is $c$-good for every $t\in[0,1]$.\\
\hspace*{0.8cm}3. $[0,1]\times_{((\beta,\gamma),p)}{\cal
W}_X ^*$ is a smooth bordism between
${\cal W}_{X,\beta_0}^{\gamma_0} $ and
${\cal W}_{X,\beta_1}^{\gamma_1} $.\\
vi) If $b_+>1$, then any two $c$-good pairs $(g_0,b_0)$,
$(g_1,b_1)$ can be joined by a smooth path of $c$-good pairs .
\end{pr}
{\bf Proof: } \\
{\sl i)} See [KM], Corollary 3.\\
{\sl ii)} It suffices to show that, for a fixed class $b\in H^2_{\rm
DR}(X)$, the map
$$F:{\cal A}(\det\Sigma^+)\times A^0(\Sigma^+)\times b\longrightarrow
A^0(\Sigma^-)\times iA^0(su(\Sigma^+))
$$
given by
$F(A,\Psi,\beta)=\left({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A\Psi,\Gamma\left((F_A+2\pi
i\beta)^+\right)-2(\Psi\bar\Psi)_0\right)
$
is a submersion in every point $\tau=(A,\Psi,\beta)$ with $\Psi\ne 0$ and
${\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{A}\Psi=0$. To see this write $F=(F^1,F^2)$ for the components of
$F$, and consider a pair
$(\Psi^-,S)\in A^0(\Sigma^-)\times iA^0(su(\Sigma^+)$ which is
$L^2$-orthogonal to the image of $d_\tau(F)$. Using variations of $\beta$ by
exact forms, we see that $\Gamma^{-1}(S)\in i A^2_+$ is orthogonal to
$id^+(A^1)$, hence must be a harmonic selfdual form. This implies
$\left\langle d_\tau(F^2)(a,0,0),S\right\rangle=0$ for any variation
$a\in iA^1$, and therefore
$$Re\langle \gamma(a)(\Psi),\Psi^-\rangle= Re\left\langle
d_\tau(F^1)(a,0),(\Psi^-)\right\rangle=\left\langle
d_\tau(F)(a,0,0),(\Psi^-,S)\right\rangle=0
$$
for all $a\in iA^1$. Since $\Psi$ is
Dirac-harmonic and non-trivial, it cannot vanish on non-empty open sets.
Therefore, the multiplication map $\gamma(\cdot)\Psi:iA^1\longrightarrow
A^0(\Sigma^-)$ has
$L^2$-dense image, so that we must have $\Psi^-=0$. Using now the vanishing of
$\Psi^-=0$, we get
$$\langle d_\tau F^2(0,\psi),S\rangle=\langle
d_\tau F(0,\psi,0),(\Psi^-,S)\rangle=0
$$
for all variations $\psi\in A^0(\Sigma^+)$. Since $d_\tau
F^2(0,\cdot):A^0(\Sigma^+)\longrightarrow i A^0(su(\Sigma^+))$ has $L^2$-dense image,
we must also have $S=0$.\\
{\sl iii)} This is clear since the form $F_A+2\pi i\beta$ represents the
cohomology class
$-2\pi i(c-b)$: If $(A,0)$ was a solution of $(SW^{\gamma}_\beta)$, then
the $g_\gamma$-anti-selfdual form $\frac{i}{2\pi}F_A-\beta$ would be the
$g_\gamma$-harmonic representant of $c-b$. \\
{\sl iv)} This follows from {\sl ii)}, {\sl iii)} and Proposition
(4.3.10) of [DK].\\
{\sl v)} This is a consequence of {\sl ii)}, {\sl iii)}, Proposition
(4.3.10) of [DK], and
the fact that the condition "$c$-good" is open for fixed $c$.\\
{\sl vi)} For fixed $c$, the closed subspace of $Z^2_{\rm
DR}(X)\times{\cal C}$
consisting of pairs $(\beta, {\gamma})$ with $(g_\gamma,[\beta])$ not $c$-good
has codimension
$b_+$. Its complement is therefore connected when $b_+\geq 2$. This fact
was noticed by C. Taubes [Ta].
\hfill\vrule height6pt width6pt depth0pt \bigskip
\\
{\bf Remark:} Suppose $(g,b)$ is a $c$-good pair. Then the bordism type of
a smooth
moduli space ${\cal W}_{X,\beta}^{\gamma}$ with $g_\gamma=g$ depends only
on the pair $(g,{\germ c})$ and the class $b$ of $\beta$. Furthermore, this
bordism type
does not change as long as one varies $(g,b)$ in a smooth 1-parameter family of
$c$-good pairs. Note that the statement makes sense since the set
$\pi_0(\qmod{{\cal C}}{{\rm Aut}(\hat P)})$ to which ${\germ c}$ belongs was
defined independently of the metric.
\\
\section{ Seiberg-Witten invariants for 4-manifolds with $b_+=1$}
Let $X$
be a closed connected oriented 4-manifold , $c$ a characteristic element, and
$\hat P$ a compatible $Spin^c(4)$-bundle. We put
$${\cal B}(c)^*:= \qmod{{\cal A}(\det\Sigma^+)
\times(A^0(\Sigma^+)\setminus\{0\})}{{\cal G}}\ .$$
\begin{lm}
${\cal B}(c)^*$ has the weak homotopy type of $K({\Bbb Z},2)\times
K(H^1(X,{\Bbb Z}),1)$. There is a natural isomorphism
$$ \nu:{\Bbb Z}[u]\otimes \Lambda^*(\qmod{H_1(X,{\Bbb Z})}{\rm Tors})\longrightarrow H^*({\cal
B}(c)^*,{\Bbb Z})
.$$
\end{lm}
{\bf Proof: } The inclusion $S^1\subset{\cal G}$ defines a canonical exact sequence
$$1\longrightarrow S^1\longrightarrow {\cal G}\longrightarrow\overline{{\cal G}}\longrightarrow 1
$$
with $\overline{{\cal G}}:=\qmod{{\cal G}}{ S^1}$, and the exponential map
yields
a natural identification of $\overline{{\cal G}}$ with the product
$\qmod{{\cal C}^{\infty}(X,{\Bbb R})}{{\Bbb R}}\times H^1(X,{\Bbb Z})$. The choice of a base
point
$x_0\in X$ induces a splitting $ev_{x_0}:{\cal G}\longrightarrow S^1$ of the exact
sequence,
and therefore a homotopy equivalence of classifying spaces $B{\cal G}\longrightarrow
BS^1\times B\overline{{\cal G}}$; the homotopy class of this map is
independent of
$x_0$ when $X$ is connected. Since ${\cal
A}(\det \Sigma^+)\times(A^0(\Sigma^+)\setminus\{0\})$ is weakly contractible,
${\cal B}(c)^*$ has the weak homotopy type of $B{\cal G}$. We fix weak homotopy
equivalences ${\cal B}(c)^*\simeq B{\cal G}$,
$BS^1\simeq K({\Bbb Z},2)$ and $B\overline{{\cal G}}\simeq K(H^1(X,{\Bbb Z}),1)$ in the
natural homotopy classes. Since the homotopy class of the induced weak
homotopy equivalence ${\cal B}(c)^*\longrightarrow K({\Bbb Z},2)\times K(H^1(X,{\Bbb Z}),1)$ is canonical,
we obtain a natural isomorphism
%
$$H^*({\cal
B}(c)^*,{\Bbb Z})\simeq H^*( K({\Bbb Z},2))\otimes
H^*(K(H^1(X,{\Bbb Z}),1))\simeq {\Bbb Z}[u]\otimes\Lambda^*(\qmod{H_1(X,{\Bbb Z})}{\rm Tors})\ .$$
\hfill\vrule height6pt width6pt depth0pt \bigskip
{\bf Remark:} Let ${\cal G}_0$ be the kernel of the evaluation map
$ev_{x_0}:{\cal
G}\longrightarrow S^1$, and set ${\cal B}_0(c)^*:=\qmod{{\cal A}(\det\Sigma^+)
\times(A^0(\Sigma^+)\setminus\{0\})}{{\cal G}_0}$. The group $\qmod{{\cal
G}}{{\cal G}_0}\simeq S^1$ acts freely on ${\cal B}_0(c)^*$ and defines a
principal
$S^1$-bundle over ${\cal B}(c)^*$. The first Chern class of this bundle is
the class
$u$ defined above.\\
Suppose now that $(g,b)$ is a c-good pair, and fix ${\germ c}\in\pi_0(\qmod{{\cal
C}}{{\rm Aut}(\hat P)})$ . The moduli space
${\cal W}^{\gamma}_{X,\beta} $ is a compact manifold of dimension
$w_c:=\frac{1}{4}(c^2-2e(X)-3\sigma(X))$ for every lift $\gamma$ of $(g,{\germ c})$
and every regular value
$\beta$ of $p_{\gamma,b}$. It can be oriented by using
the canonical complex orientation of the line bundle
$\det{}_{\Bbb R}({\rm index} ({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D})) $
over
${\cal B}(c)^*$ together with a chosen orientation ${\scriptstyle{\cal O}}$ of the line
$\det(H^1(X,{\Bbb R}))\otimes\det({\Bbb H}^2_{+,g}(X)^{\vee})$. Let $[{\cal
W}^{\gamma}_{X,\beta} ]_{{\raisebox{-.5ex}{${\scriptstyle{\cal O}}$}}}\in H_{w_c}({\cal
B}(c)^*,{\Bbb Z})$ be the fundamental class associated with the choice of ${\scriptstyle{\cal O}}$.
\begin{dt} Let $X$ be a closed connected oriented 4-manifold, $c\in
H^2(X,{\Bbb Z})$ a
characteristic element, $(g,b)$ a $c$-good pair, ${\germ c}\in\pi_0(\qmod{{\cal
C}}{{\rm Aut}(\hat P)})$, and ${\scriptstyle{\cal O}}$ an orientation of the line
$\det(H^1(X,{\Bbb R}))\otimes\det({\Bbb H}^2_{+,g}(X)^{\vee})$. The corresponding
Seiberg-Witten form is the element
$$SW_{X,{\scriptstyle{\cal O}}}^{(g,b)}({\germ c})\in \Lambda^* H^1(X,{\Bbb Z})
$$
defined by
$$SW_{X,{\scriptstyle{\cal O}}}^{(g,b)}({\germ c})(l_1\wedge\dots\wedge l_r)=\left\langle\nu(l_1)
\cup\dots\cup\nu(l_r)\cup u^{\frac{w_c-r}{2}}, [{\cal W}^{\gamma}_{X,\beta}
]_{{\raisebox{-.5ex}{${\scriptstyle{\cal O}}$}}} \right\rangle\
$$
for decomposable elements
$l_1\wedge\dots\wedge l_r$ with $r\equiv w_c$ (mod 2) .
Here $\gamma$ lifts the pair $(g,{\germ c})$ and $\beta\in b$ is a regular value of
$p_{\gamma,b}$.
\end{dt}
{\bf Remark:} The form $SW_{X,{\scriptstyle{\cal O}}}^{(g,b)}({\germ c})$ is well-defined, since the
cohomology classes $u$, $\nu(l_i)$, as well as the trivialization of the
orientation line bundle extend to the quotient $\qmod{{\cal A}(\det
\Sigma^+)\times (A^0(\Sigma^+)\setminus\{0\})\times b}{{\cal G}}$, and
since, by
Proposition 4 {\sl iv)}, the homology class defined by $[{\cal
W}^{\gamma}_{X,\beta} ]_{{\raisebox{-.5ex}{${\scriptstyle{\cal O}}$}}}$ in this
quotient depends only on $(g_\gamma,{\germ c}_\gamma)$ and $b$.\\
Now there are two cases:
If $b_+>1$, then, by Proposition 4 {\sl v)}, {\sl vi)}, the
form $SW_{X,{\scriptstyle{\cal O}}}^{(g,b)}({\germ c})$ is also independent of $(g,b)$, since the
cohomology classes and the trivialization of the orientation line bundle
extend to
${\rm Aut}(\hat P)$-invariant objects on the universal quotient $\qmod{{\cal
A}(\det\Sigma^+)\times (A^0(\Sigma^+)\setminus\{0\})\times Z^2_{\rm
DR}(X)\times{\cal C}}{{\cal G}}$. Thus we may simply write
$SW_{X,{\scriptstyle{\cal O}}}({\germ c})\in
\Lambda^* H^1(X,{\Bbb Z})$. If $b_1=0$, then we obtain numbers which we denote by
$n_{{\germ c}}^{{\scriptscriptstyle{\cal O}}}$; these numbers can be considered as refinements of the numbers
$n_c^{{\scriptscriptstyle{\cal O}}}$ which were defined in [W]. Indeed, $n_c^{{\scriptscriptstyle{\cal O}}}=\sum_{{\germ c}}
n^{{\scriptscriptstyle{\cal O}}}_{{\germ c}}$, the summation being over all ${\germ c}\in\pi_0(\qmod{{\cal
C}}{{\rm Aut}(\hat P)})$.
Suppose now that $b_+=1$. There is a natural map ${\cal M}et_X\longrightarrow
{\Bbb P}(H^2_{\rm DR}(X))$ which sends a metric $g$ to the line ${\Bbb R}[\omega_+]\subset
H^2_{\rm DR}(X)$, where $\omega_+$ is any non-trivial $g$-selfdual harmonic
form. Let ${\bf H}$ be the hyperbolic space
$${\bf H}:=\{\omega\in H^2_{\rm DR}(X)|\ \omega^2=1\}\ .
$$
${\bf H}$ has two connected components, and the choice of one of them
orients the
lines ${\Bbb H}^2_{+,g}(X)$ for all metrics $g$. Furthermore, having fixed a component
${\bf H}_0$ of ${\bf H}$, every metric defines a unique $g$-self-dual form
$\omega_g$ with $[\omega_g{]\in\bf H}_0$.
\begin{dt} Let $X$ be a manifold with $b_+=1$, and let $c\in H^2(X,{\Bbb Z})$ be
characteristic. The \underbar{wall} associated with $c$ is the hypersurface
$c^{\bot}:=\{(\omega,b)\in{\bf H}\times H^2_{\rm DR}(X)|\
(c-b)\cdot\omega=0\}$.
The connected components of ${\bf H}\setminus c^{\bot}$ are called
\underbar{chambers} of type $c$.
\end{dt}
Notice that the walls are non-linear! Every characteristic element $c$ defines
precisely four chambers of type $c$, namely
$$C_{{\bf H}_0,\pm}:=\{(\omega,b)\in{\bf H}_0\times H^2_{\rm DR}(X)|\
\pm(c-b)\cdot\omega<0\}\ ,
$$
where ${\bf H}_0$ is one of the components of ${\bf H}$. Each of these four
chambers contains pairs of the form $([\omega_g],b)$. Let ${\scriptstyle{\cal O}}_1$ be an
orientation of $H^1(X,{\Bbb R})$.
\begin{dt} The Seiberg-Witten invariant associated with the data $({\scriptstyle{\cal O}}_1,{\bf
H}_0,{\germ c})$ is the function
$$
SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c}):\{\pm\}\longrightarrow \Lambda^* H^1(X,{\Bbb Z})
$$
given by $SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(\pm):=SW^{(g,b)}_{X,{\scriptstyle{\cal O}}}({\germ c})$, where
${\scriptstyle{\cal O}}$ is the orientation defined by $({\scriptstyle{\cal O}}_1,{\bf H}_0)$, and $(g,b)$ is a
pair such
that $([\omega_g],b)$ belongs to the chamber $C_{{\bf H}_0,\pm}$.
\end{dt}
{\bf Remark: } The intersection $c^{\bot}\cap {\bf H}\times\{0\}$ defines a
non-trivial wall in ${\bf H}\times\{0\}$ if and only if $c^2<0$. This means that
invariants of index $w_c<\frac{b_2-10}{4}+b_1$ could also be defined for the
chambers $C^{\pm}_{{\bf H}_0}:=\{\omega\in{\bf H}_0|\ \pm c\cdot \omega<0\}$.
However, when $c$ is rationally non-zero and $c^2\geq 0$, then ${\bf
H}_0\times\{0\}$ is entirely contained in one of the chambers $C_{{\bf
H}_0,\pm}$.\\
Note that, changing the orientation ${\scriptstyle{\cal O}}_1$ changes the invariant by a factor
$-1$, and that $SW_{X,({\scriptstyle{\cal O}}_1,-{\bf H}_0)}({\germ c})(\pm)=-SW_{X,({\scriptstyle{\cal O}}_1, {\bf
H}_0)}({\germ c})(\mp)$.\\ \\
{\bf Remark: } A different approach - adapting ideas from intersection theory to
construct "Seiberg-Witten multiplicities" - has been proposed by R. Brussee.\\
\section{The wall crossing formula}
In this section we prove a wall crossing formula for the complete
Seiberg-Witten
invariant $SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})$. This formula generalizes previous
results of [W], [KM] and [LL]. Our proof is based on ideas similar to the
ones in [LL],
but our method - using the real bow up
of the (singular) locus of reducible points as in [OT2] - has some advantages:
It allows us to construct explicitely a smooth bordism to which the cohomology
classes $u$, $\nu(l_i)$ extend in a natural way, and it enables us a to
give a simple description of that part of its boundary which lies on the
wall.
Our main result can be formulated by saying that the the difference
$SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(+)-SW_{X,({\scriptstyle{\cal O}}_1, {\bf H}_0)}({\germ c})(-)$ is an
abelian $Spin^c$-form. These abelian $Spin^c$-forms are topological invariants
which will be introduced in the first subsection. Using a remark of [LL],
we give an
explicit formula for these invariants in the case of manifolds with $b_+=1$.
\subsection{Abelian $Spin^c(4)$-forms }
Let $X$ be an oriented 4-manifold, $c\in H^2(X,{\Bbb Z})$ a characteristic
element, and
$\hat P$ a compatible
$Spin^c(4)$-bundle. Let $\Sigma^{\pm}$ be the associated spinor bundles, define
$L:=\det\Sigma^{\pm}$, and put:
%
$${\cal B}(L):=\qmod{{\cal A}(L)}{{\cal G}_0}\ ,\ \ {\cal
B}'(L):=\qmod{{\cal A}(L)}{{\cal G}_0^2}\ .$$
There is an obvious covering projection
$$s:{\cal B}'(L)\longrightarrow {\cal B}(L)$$
with fiber \ $\qmod{{\cal G}_0}{{\cal G}_0^2}=\qmod{H^1(X,{\Bbb Z})}{2H^1(X,{\Bbb Z})}$.
The homotopy equivalence ${\cal G}_0\textmap{\simeq} H^1(X,{\Bbb Z})$ induces
canonical
isomorphisms
$$\mu:\Lambda^*\left(\qmod{H_1(X,{\Bbb Z})}{\rm
Tors})\right)\textmap{\simeq} H^*({\cal B}(L),{\Bbb Z})\ ,$$
$$\nu':\Lambda^*\left(\qmod{H_1(X,{\Bbb Z})}{\rm
Tors})\right)\textmap{\simeq} H^*({\cal B}'(L),{\Bbb Z})\ ,$$
such that $s^*\circ\mu=2\nu'$.
Now fix a metric $g$ and let $h_c$ be the harmonic representant of the de Rham
class $c_{\rm DR}\in H^2_{\rm DR}(X)$. The spaces
${\cal B}(L)$, ${\cal B}'(L)$ are trivial fibre bundles over the affine
subspace
$h_c^+ +d^+(A^1)=h_c^++({\Bbb H}^2_{g,+}(X))^{\bot}\subset A^2_+$ via the maps induced by
$A \longmapsto \frac{i}{2\pi}F_A^+$.
For a given 2-form $\beta\in h_c^+ +d^+(A^1)$ let
$${\cal T}_\beta(L)\subset {\cal B}(L)\ ,\ \ {\cal
T}_{\beta}'(L)\subset{\cal B}'(L)$$
be the fibers of these maps over $\beta$. These fibers are tori, consisting of
equivalence classes of solutions of of the equation
$$F_A^++2\pi i\beta=0
$$
modulo ${\cal G}_0$ respectively ${\cal G}_0^2$. Indeed, the choice of a
solution $A_0\in{\cal A}(L)$ yields identifications
$${\cal T}_\beta(L)=\qmod{H^1(X,{\Bbb R})}{H^1(X,{\Bbb Z})}\ ,\ \ {\cal
T}_{\beta}'(L)=\qmod{H^1(X,{\Bbb R})}{2H^1(X,{\Bbb Z})}\ .$$
Now fix a Clifford map $\gamma\in{\cal C}$ of type $\hat P$ with $g_\gamma=g$,
and a 2-form $\beta\in h_c^+ +d^+(A^1)$. Let $S(A^0(\Sigma^+))$ be the
unit sphere in $A^0(\Sigma^+)$ with respect to the $L^2$-norm.
\begin{dt} The $\beta$-twisted $Spin^c(4)$-equations for a pair
$(A,\Phi)\in {\cal
A}(L)\times S(A^0(\Sigma^+))$ are the equations
$$\left\{\begin{array}{ccl}{\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A \Phi&=&0\\
F_A^++2\pi i\beta&=&0 \ .
\end{array}\right. \eqno{(S^\gamma_\beta)}
$$
\end{dt}
The gauge group ${\cal G}$ acts on ${\cal A}(L)\times S(A^0(\Sigma^+))$ by
$(A,\Phi)\cdot f= (A^{f^2},f^{-1}\Phi)$, letting invariant the space of
solutions of $(S^\gamma_\beta)$. We denote by $\hat {\cal T}_{\beta}'^\gamma(L)$
the moduli space of solutions; it is a
projective fiber space over the "Brill-Noether locus" in ${\cal
T}_{\beta}'(L)$.
Let
$$q:\qmod{{\cal A}(L)\times S(A^0(\Sigma^+))}{{\cal G}}\longrightarrow {\cal B}'(L)
$$
be the natural projection.
\begin{lm} There is a natural isomorphism
$${\Bbb Z}[u]\otimes
\Lambda^* \left(\qmod{H_1(X,{\Bbb Z})}{\rm Tors}\right)\longrightarrow H^*\left(\qmod{{\cal
A}(L)\times S(A^0(\Sigma^+))}{{\cal G}},{\Bbb Z}\right)
$$
whose restriction to $\Lambda^* \left(\qmod{H_1(X,{\Bbb Z})}{\rm Tors}\right)$ factors
as $q^*\circ\nu'$. The class $u$ restricts to the positive generator of the
second cohomology group ${\Bbb P}(A^0(\Sigma^+))$ of the fibers of $q$.
\end{lm}
This lemma, as well as the following proposition can be proved using the same
methods as in the proofs of Lamma 5 and Proposition 4.
Set $\delta_c:=\frac{1}{8}(c^2-\sigma(X))$.
\begin{pr} For generic elements $\beta\in h_c^+ + d^+(A^1)$, the moduli
space $\hat {\cal
T}_{\beta}'^\gamma(L)$ is a closed smooth manifold of dimension
$b_1+2\delta_c-2$. It can be oriented by choosing an orientation
${\scriptstyle{\cal O}}_1$ of $H^1(X,{\Bbb R})$. The fundamental class
$$[\hat {\cal
T}_{\beta}'^\gamma(L)]_{{\scriptscriptstyle{\cal O}}_1}\in
H_{b_1+2\delta_c-2}\left( \qmod{{\cal A}(L)\times S(A^0(\Sigma^+))}{{\cal
G}} ,{\Bbb Z}\right)$$
depends only on the component ${\germ c}_\gamma\in\pi_0\left(\qmod{{\cal
C}}{{\rm Aut}(\hat P)}\right)$.
\end{pr}
We can now define the abelian $Spin^c(4)$-forms.
\begin{dt} Let $X$ be a closed connected oriented 4-manifold, and ${\scriptstyle{\cal O}}_1$ an
orientation of $H^1(X,{\Bbb R})$. Let $c\in H^2(X,{\Bbb Z})$ be a characteristic element and
${\germ c}\in\pi_0\left(\qmod{{\cal C}}{{\rm Aut}(\hat P)}\right)$. The corresponding
$Spin^c(4)$-form is the element $\sigma_{X,{{\scriptscriptstyle{\cal O}}_1}}({\germ c})\in
\Lambda^*(H^1(X,{\Bbb Z}))$ defined by the formula
$$\sigma_{X,{\scriptscriptstyle{\cal O}}_1}({\germ c})(l_1\wedge\dots\wedge
l_r):=\langle\nu'(l_1)\cup\dots\cup\nu'(l_r)\cup
u^{\frac{b_1+2\delta_c-2-r}{2}},[\hat {\cal
T}_{\beta}'^\gamma(L)]_{{\scriptscriptstyle{\cal O}}_1}\rangle
$$
for decomposable elements $l_1\wedge\dots\wedge l_r$ with $r\equiv b_1$
(mod 2).
Here $\gamma$ induces ${\germ c}={\germ c}_\gamma$ and $\beta$ is generic.
\end{dt}
Note that the expected dimension $b_1+2\delta_c-2$ of the moduli space
$\hat {\cal T}_{\beta}'^\gamma(L)$ coincides with $w_c$ iff
$b_+=1$. The $Spin^c$-forms $\sigma_{X,{\scriptscriptstyle{\cal O}}_1}({\germ c})$ are topological invariants,
which can be explicitely computed. This is our next aim.\\
Let $pr_2:{\cal A}(L)\times X\longrightarrow X$ be the projection onto the second
factor, and put
$${\Bbb P}:=\qmod{pr_2^*(\hat P)}{{\cal G}_0}\ ,
$$
where ${\cal G}_0$ acts on $pr_2^*(\hat P)={\cal A}(L)\times\hat P$ by
$(A,\hat p)\cdot f =(A^{f^2}, f(\hat p))$. The universal
$Spin^c(4)$-bundle ${\Bbb P}$ over ${\cal B'}(L)\times X$ comes with a tautological
connection ${\Bbb A}$ in the
$X$-direction. Let ${\Bbb L}:=\qmod{pr_2^*(L)}{{\cal G}_0}$ be the universal line
bundle over ${\cal B}(L)\times X$; its pull back $(s\times{\rm id})^*({\Bbb L})$ to ${\cal
B}'(L)\times X$ is the universal determinant bundle ${\Bbb P}\times_{\det}{\Bbb C}$. The
Chern class $c_1({\Bbb L})$ has a K\"unneth decomposition $c_1({\Bbb L})=1\otimes c+
c_1({\Bbb L})^{1,1}$ whose $(1,1)$-component $c_1({\Bbb L})^{1,1}\in H^1({\cal
B}(L),{\Bbb Z})\otimes H^1(X,{\Bbb Z})$, considered as homomorphism
$\mu_1:\qmod{H^1(X,{\Bbb Z})}{\rm Tors}\longrightarrow H^1({\cal B}(L),{\Bbb Z})$, is given by the
restriction of the isomorphism $\mu$.
We fix a basis $(l_i)_{1\leq i\leq b_1}$ of
$\qmod{H_1(X,{\Bbb Z})}{\rm Tors}$ and let $l^i$ be the elements of the dual
basis. Then
%
$$c_1({\Bbb L})=1\otimes c+\sum\limits_{i=1}^{b_1} \mu(l_i)\otimes l^i\ ,$$
$$c_1({\Bbb P}\times_{\det}{\Bbb C})=1\otimes c+2\sum\limits_{i=1}^{b_1}\nu'(l_i)\otimes
l^i \ .$$
Let $\gamma:\Lambda^1\longrightarrow \hat P\times_{\pi}{\Bbb R}^4$ be a Clifford map and
$\raisebox{-.18ex}{$\gamma$}\hskip-5.5pt{\gamma}:pr_2^*(\Lambda^1)\longrightarrow
{\Bbb P}\times_{\pi}{\Bbb R}^4$ the induced isomorphism.
We restrict the universal objects
$({\Bbb P},\raisebox{-.18ex}{$\gamma$}\hskip-5.5pt{\gamma},{\Bbb A})$ to the subspace
${\cal T}'_{\beta}(L)\times X$ and denote the restrictions by the same
symbols. The Chern character of the virtual bundle
$index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{\Bbb A})$ over the torus ${\cal T}'_{\beta}(L)$ is
$$ch(index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{\Bbb A}))=\left[\left(e^{\frac{c}{2}+\sum \nu'(l_i)\otimes l^i
}\right)
\cup \left(1-\frac{1}{24}p_1(X)\right)\right]/[X]=
$$
$$=\left(e^{\frac{c}{2}+\sum \nu'(l_i)\otimes
l^i}\right)/[X]-\frac{1}{8}\sigma(X)=
$$
$$=-\frac{1}{8}\sigma(X)+\frac{1}{8}c^2+\frac{1}{3!}(\frac{c}{2}+ \sum
\nu'(l_i)\otimes l^i)^3/[X]+
\frac{1}{4!}(\frac{c}{2}+\sum \nu'(l_i)\otimes l^i)^4/[X] \ .
$$
To simplify this expression, put $c_{ij}:=\langle c\cup l^i\cup l^j,[X]\rangle$,
$l_{hijk}:=\langle l^h\cup l^i\cup l^j\cup l^k,[X]\rangle$. The numbers
$c_{ij}$ are
even since $c$ is characteristic and $(l^i\cup l^j)^2=0$. Substituting into the
formula above we find:
$$\begin{array}{cl}
ch(index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{\Bbb A}))=&-\frac{1}{8}\sigma(X)+\frac{1}{8}c^2+\frac{1}{2}
\sum\limits_{i<j}c_{ij}\ [\nu'(l_i)\cup
\nu'(l_j)]+\\ \\
&+\sum\limits_{h<i<j<k} l_{hijk}[\nu'(l_h)\cup\nu'(l_i)\cup \nu'(l_j)\cup
\nu'(l_k)]\ .
\end{array}$$
The cohomology class
$$
u_c:=\frac{1}{2} \sum_{i<j}c_{ij}\
[\nu'(l_i)\cup \nu'(l_j)]\in H^2({\cal T}'_{\beta}(L),{\Bbb Z}) $$
has the following invariant description: The assignment
$$
(a,b)\longmapsto \frac{1}{2}\langle c\cup a\cup b,[X]\rangle
$$
defines a ${\Bbb Z}$-valued skew-symmetric bilinear form on
$H^1(X,{\Bbb Z})$; using the isomorphism $H^1(X,{\Bbb Z}) {\simeq} 2H^1(X,{\Bbb Z})$, we get a
cohomology class in $H^2({\cal
T}'_{\beta}(L),{\Bbb Z})$ $=\Lambda^2(2H^1(X,{\Bbb Z})^{\vee})$, and this coincides with
$u_c$. Clearly $u_c=c_1(index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{{\Bbb A}}))$.
\begin{lm} { } [LL] Let $X$ be a 4-manifold with with $b_+=1$. Then
$$c_k(index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{{\Bbb A}}))=\frac{1}{k!}u_c^k \ .
$$
\end{lm}
{\bf Proof: } In the case $b_+=1$, all coefficients $l_{hijk} $ vanish, so
$ch_k(index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{{\Bbb A}}))=0$ for $k\geq 2$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Regard now $u_c\in H^2({\cal T}'_{\beta}(L),{\Bbb Z})$ as an element of
$\Lambda^2\left(\qmod{H_1(X,{\Bbb Z})}{\rm Tors}\right)$.
\begin{pr} Let $X$ be a closed connected oriented 4-manifold with $b_+=1$, and
${\scriptstyle{\cal O}}_1$ an orientation of $H^1(X,{\Bbb R})$. Let $c\in H^2(X,{\Bbb Z})$ be a characteristic
element, $\hat P$ a compatible $Spin^c(4)$-bundle and
${\germ c}\in\pi_0\left(\qmod{{\cal C}}{{\rm Aut}(\hat P)}\right)$. Choose the generator
$l_{{\scriptscriptstyle{\cal O}}_1}\in\Lambda^{b_1}(H^1(X,{\Bbb Z}))$ which defines the orientation ${\scriptstyle{\cal O}}_1$.
For every $\lambda\in\Lambda^r\left(\qmod{H_1(X,{\Bbb Z})}{\rm Tors}\right)$ with
$r\equiv b_1$ (mod 2) and $0\leq r\leq\min(b_1,w_c)$, we have:
$$\sigma_{X,{\scriptscriptstyle{\cal O}}_1}({\germ c})(\lambda)= \left\langle\lambda\wedge\exp(-u_c)
,l_{{\scriptscriptstyle{\cal O}}_1}\right\rangle \ .
$$
In all other cases
$\sigma_{X,{\scriptscriptstyle{\cal O}}_1}({\germ c})(\lambda)=0$.
\end{pr}
The proof follows from a more general result which we will now explain. \\
Let $T$ be a closed connected oriented manifold. Consider Hilbert vector
bundles $E$ and $F$ over $T$, and a smooth family of Fredholm operators
$q_t:E_t\longrightarrow F_t$ of constant index $\delta$. Suppose the map
$\tilde q: E\setminus\{0\} \longrightarrow F$ is a submersion, so that its zero locus
$\tilde
T:=Z(\tilde q)$ is a (finite dimensional) manifold which fibers over the
possibly
singular "Brill-Noether locus"
$BN_q:=\{ t\in T|\ \ker q_t\ne \{0\}\}$ of the family $q$. Put $\hat
T:=\qmod{\tilde
T}{{\Bbb C}^*}$. The projection $\hat p:\hat T\longrightarrow T$ induces a projective
fibration over
$BN_q$. Note that
$\hat T$ comes with a canonical cohomology class $u\in H^2(\hat T,{\Bbb Z})$
induced by the
dual of the ${\Bbb C}^*$-bundle $\tilde T\longrightarrow \hat T$; the restriction of $u$
to any fiber
$\hat p^{-1}(t)={\Bbb P}(\ker q_t)$ is the positive generator of the second
cohomology group
of this projective space. We wish to calculate the direct images
$\hat p_*(u^k)$ for all $k\in{\Bbb N}$, $k\geq\delta$ in terms of the Chern
classes of the (virtual)
index bundle $index(q)$ of the family $q$. In the particular case of Dirac
operators on a
4-manifold with $b_+=1$, similar computations have been carried out in
[LL] .
\begin{pr}Let $c_i=c_i(index(q))$ be the Chern classes of $index(q)$, and
define polynomials
$(p_k)_{k\geq \delta-1}$ by the recursive relations:
$$p_{\delta-1}=1,\ p_k=-\sum_{i=1}^{k-\delta+1} c_i p_{k-i} \ .
$$
For every non-negative integer $k\geq \delta$ we have
$$\hat p_*(u^k)= p_k(c_1,c_2,\dots) \ ,
$$
hence
$\hat p_*(u^{\delta-1})= 1\in H^0(T,{\Bbb Z})$ when $\delta>0$.
\end{pr}
{\bf Proof: } One can find a smooth family of Fredholm operators $(Q_t)_{t\in T}$,
$Q_t:E_t\oplus
{\Bbb C}^n\longrightarrow F_t $ with $Q_t|_{E_t\times\{0\}}=q_t$, such that $Q_t$ is
surjective of
positive index $n+\delta$ for every $t\in T$. The associated map
$\tilde Q : (E\oplus{\Bbb C}^n)\setminus\{0\} \longrightarrow F$ is a submersion
and $Z(\tilde Q)$ is a locally trivial fiber bundle over $T$ with standard
fiber ${\Bbb C}^{n+\delta}\setminus\{0\}$. Indeed, $Z(\tilde Q)$ is the
complement of the
zero section of the vector bundle
$$V:=\mathop{\bigcup}\limits_{t\in T}\ker Q_t\ .$$
The space $\tilde T$ can be identified with the
zero locus $Z(\zeta)$, of the map
$$\zeta:Z(\tilde Q) \longrightarrow {\Bbb C}^n$$
given by $\zeta(e,z)=z$. The map $\zeta$ is a submersion in all points of
$\tilde T$, since $\tilde q$ was such.
Let $p_V:{\Bbb P}(V)\longrightarrow T$ be the obvious projection, and denote by $U\in
H^2({\Bbb P}(V),{\Bbb Z})$ the Chern
class of the dual of the tautological bundle. The map $\hat p:\hat T\longrightarrow
T$ factors through
the inclusion $j:\hat T\hookrightarrow{\Bbb P}(V)$, and the fundamental class
$j_*[\hat T]$
is Poincar\' e dual to $U^n$. Therefore we have
$$\hat p_*(u^k)=[PD_T^{-1}\circ p_{V*}\circ j_*\circ PD_{\hat
T}](u^k)=[PD_T^{-1}\circ p_{V*}\circ PD_{{\Bbb P}(V)}](U^{k+n})= p_{V*}(U^{k+n}) .$$
Since $index(q)=[V]-[{\Bbb C}^n]\in K(T)$, we have $c_i(V)=c_i(index(q))=c_i$,
and therefore
$$U^{\delta+n}= -\sum_{i=1}^{\delta+n}
p_{V}^*(c_i)U^{\delta+n-i} \ .
$$
Multiplying with $U^{k-\delta}$ and using $ p_{V*}(U^{\delta+n-1})=1$, we
get the recursion relations
$$ p_{V*}(U^{k+n})=-\sum\limits_{i=1}^{\delta+n}p_{V*}(U^{k+n-i}) c_i\ ,$$
hence $\hat p_{V*}(U^{k+n})=p_k$ for $k\geq \delta-1$ by induction.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Now we can prove Proposition 14 by applying the result above to the map
$\hat p\hat{\cal T}_\beta'^\gamma(L)\longrightarrow{\cal T}'_\beta(L)$ and the family
${\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{{\Bbb A}}$
of Dirac operators over
${\cal T}'_\beta(L)$.
Since $c_k(index({\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_{{\Bbb A}}))=\frac{1}{k!} u_c^k$, we get $ p_*(u^{\delta-1+k})=
p_{\delta-1+k}=\frac{(-1)^k}{k!} u_c^k$, hence
$$\hat p _*(u^{\frac{w_c-r}{2}})=
\frac{ (-1) ^{\left[\frac{b_1-r}{2}\right]} }{\ \
\left[\frac{b_1-r}{2}\right]!\ }\ u_c\
^{\left[\frac{b_1-r}{2}\right]}
$$
for any non-negative integer $r$ with $ r\leq
\min(b_1,w_c)$ and $r\equiv b_1$ (mod 2).
Therefore, for every $\lambda\in \Lambda^r\left(\qmod{H_1(X,{\Bbb Z})}{\rm
Tors}\right)$, we
find
$$\begin{array}{cl}\sigma_{X,{\scriptscriptstyle{\cal O}}_1}({\germ c})(\lambda)&=\langle \hat
p^*\nu'(\lambda)\cup
u^{\frac{w_c-r}{2}},[\hat {\cal T}_\beta'^\gamma(L)]_{{\scriptscriptstyle{\cal O}}_1}\rangle=
\langle \nu'(\lambda)\cup \hat p_*(u^{\frac{w_c-r}{2}}),[ {\cal
T}_\beta' (L)]_{{\scriptscriptstyle{\cal O}}_1}\rangle\\ &= \frac{ (-1)^{\left[\frac{b_1-r}{2}\right]} }{\ \
\left[\frac{b_1-r}{2}\right]!\ }\left\langle \lambda\wedge u_c\
^{\left[\frac{b_1-r}{2}\right]},l_{{\scriptscriptstyle{\cal O}}_1}\right\rangle \ ,
\end{array}$$
which proves the proposition.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\subsection{Wall crossing}
The following theorem generalizes results of [W], [KM] and [LL].
\begin{thry} Let $X$ be a closed connected oriented 4-manifold with $b_+=1$, and
${\scriptstyle{\cal O}}_1$ an orientation of $H^1(X,{\Bbb R})$. For every class ${\germ c}$ of
$Spin^c(4)$-structures of Chern class $c$ and every component ${\bf H}_0$
of ${\bf
H}$, the following holds:
$$ SW_{X,({\scriptscriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(+) - SW_{X,({\scriptscriptstyle{\cal O}}_1,{\bf
H}_0)}({\germ c})(-) =\sigma_{X,{\scriptscriptstyle{\cal O}}_1}({\germ c})\ .
$$
\end{thry}
We need some preparations before we can prove the theorem.
Fix a compatible $Spin^c(4)$-bundle $\hat P$ with spinor bundles $\Sigma^{\pm}$
and determinant $L:=\det \Sigma^{\pm}$. Choose a Clifford map
$\gamma$ with ${\germ c}_\gamma={\germ c}$, set $g=g_\gamma$, and let
$\omega_g$ be the generator of ${\Bbb H}_{+,g}^2(X)$ whose class belongs to ${\bf
H}_0$. Put $s_c:= c\cdot [\omega_g]$, so that the $g$-harmonic
representant of $c-s_c[\omega_g]$ is $g$-anti-selfdual.
Consider first a cohomology class $b_0\in H^2_{\rm DR}(X)$ with
$(c- b_0)\cdot[\omega_g]=0$, and let $\beta_0 \in b_0$. The moduli space
${\cal W}_{\beta_0}:={\cal W}_{X,\beta_0}^\gamma$ contains the closed subset
of reducible solutions of the form
$(A,0)$, where $A$ solves the equation
$$F_A^++2\pi i\beta_0^+=0 \ .
$$
This closed subspace can be identified with ${\cal
T}_{ \beta_0^+}'(L)$.
Consider the equations
$$\left\{\begin{array}{lll}{\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A\Psi&=&0\\
\Gamma(F_A^++2\pi i\beta^+)&=&2(\Psi\bar\Psi)_0
\end{array}\right. \eqno{( {SW}^\gamma)}
$$
for a triple $(A,\Psi,\beta)\in {\cal A}(L)\times
A^0(\Sigma^+)\times Z^2_{\rm DR}(X)$, denote by ${\cal W}$ the
corresponding moduli space of solutions, and by $p:{\cal W}\longrightarrow Z^2_{\rm
DR}(X)$
the natural projection.
${\cal W}$ is singular in the points of the form
$[A,0,\beta]$. If such a triple is a solution, then
$(c-[\beta])\cdot[\omega_g] =0$,
and the singular part of ${\cal W}$ is
$${\cal S}=\mathop{\bigcup}\limits_{(c-[\beta])\cdot[\omega_g]}{\cal T}_{ \beta^+}'(L) \ .
$$
Now perform a "real blow up" of the singular locus ${\cal S}\subset{\cal
W}$ in
the $\Psi$-direction. This means, consider the equations
$$\left\{\begin{array}{lll}{\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A\Phi&=&0\\
\Gamma(F_A^++2\pi i\beta^+)&=&2t(\Phi\bar\Phi)_0
\end{array}\right. \eqno{(\hat{SW}^\gamma)}
$$
for a tuple $(A,\Phi,t,\beta)\in {\cal A}(L)\times
S(A^0(\Sigma^+))\times{\Bbb R}\times Z^2_{\rm DR}(X)$, where $S(A^0(\Sigma^+))$ is
the unit sphere in $A^0(\Sigma^+)$ with respect to the $L^2$-norm. Denote by
$\hat{\cal W}$ the moduli space of solutions of
$(\hat{SW}^\gamma)$ and by
$q_{\Bbb R}$, $q$ the natural projections on ${\Bbb R}$ and $Z^2_{\rm DR}(X)$
respectively.
Let
$\hat {\cal W}^{\geq 0}:=q_{\Bbb R}^{-1}({\Bbb R}_{\geq 0})$ be the closed subset defined by
the inequality $t\geq 0$, and, for a form $\beta\in Z^2_{\rm DR}(X)$, put
$\hat{\cal W}_{\beta}:=q^{-1}(\beta)$ and $\hat{\cal W}_{\beta}^{\geq
0}=\hat{\cal
W}_{\beta}\cap \hat {\cal W}^{\geq 0}$.
There is a natural map $\hat {\cal W}^{\geq 0}\textmap{\rho}{\cal W}$,
given by
$(A,\Phi,t,\beta)\longmapsto (A,t^{\frac{1}{2}}\Phi,\beta)$, which contracts the
locus $\hat{\cal S}:=\{t=0\}$ to ${\cal S}$ and defines a real analytic
isomorphism
$\hat {\cal W}^{\geq 0}\setminus \hat{\cal S}\longrightarrow {\cal W} \setminus{\cal S}$.
Note that
$$\hat{\cal S}=\mathop{\bigcup}\limits_{(c-[\beta])\cdot[\omega_g]}\hat{\cal T}_{\beta
^+}'(L)\ ,
$$
where $\hat{\cal T}_{\beta ^+}'(L)$ is a projective fibration over the
Brill-Noether locus in ${\cal T}_{\beta ^+}'(L)$.
\begin{lm} If $(g,[\beta])$ is $c$-good, then $q_{\Bbb R}|_{\hat{\cal
W}_{\beta}^{\geq 0}}$ is bounded below by a
positive number. The space $\hat{\cal
W}_{\beta}^{\geq 0}$ is open and closed in $\hat{\cal W}_{\beta}$, and it is
isomorphic to ${\cal W}_\beta$ via the map $\rho$.
\end{lm}
{\bf Proof: } If $\hat{\cal W}_{\beta}$ would contain a
sequence $[(A_n,\Phi_n,t_n,\beta)]_{n\in{\Bbb N}}$ with
$t_n\searrow 0$ , then, by Proposition 1 $i)$, we could find a subsequence
$(m_n)_{n\in{\Bbb N}}\subset{\Bbb N}$ such that $(A_n,t_n^{\frac{1}{2}}\Phi_n)_{n\in{\Bbb N}}$
converges to a point in
${\cal W}_\beta$. This point must be reducible.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{lm} $\hat{\cal W}$ is a smooth manifold .
\end{lm}
{\bf Proof: } Let $(\Psi^-,S)$ be orthogonal to ${\rm im} d_\tau(\hat F)$, where $\hat
F=(\hat F_1,\hat F_2)$ is the map given by the left-hand side of $\hat
{(SW^\gamma)}$, and $\hat\tau=(A,\Phi,t,\beta)$ is a point in ${\cal
A}(L)\times S(A^0(\Sigma^+))\times{\Bbb R}\times Z^2_{\rm DR}(X)$. Using
variations of
$\beta$, we get $i\Gamma^{-1}(S)\in d^*(A^3)\cap A^2_+$, hence $S=0$.
Using now variations of $A$ and the non-triviality of $\Phi$, we get
$\Psi^-=0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{lm}
The linear map $h:A^1\times{\Bbb R}\longrightarrow Z^2_{\rm DR}(X)$, given by
$h(\alpha,s)=s\omega_g+d\alpha$, is transverse to $q$.
\end{lm}
{\bf Proof: } The image of $dh$ is the subspace $d(A^1)\oplus{\Bbb H}^2_{+,g}(X)$ of $Z^2_{\rm
DR}(X)$ which coincides with the orthogonal complement of ${\Bbb H}^2_{-,g}(X)$ in
$Z^2_{\rm DR}(X)$. It suffices to show that ${\Bbb H}^2_{-,g}(X)$ is contained in the
image of $d_{\hat\tau}q$, for every $\hat\tau\in\hat{\cal W}$. But if
$\hat\tau=(A,\Phi,t,\beta)$ solves the equations
$(\hat {SW}^\gamma)$, then also
$\hat\tau_t:=(A,\Phi,t,\beta+r\omega_-)$ is a solution for every
$\omega_-\in{\Bbb H}_{-,g}(X)$ and every $r\in{\Bbb R}$. Therefore ${\Bbb H}_{-,g}(X)$ is
contained
in the image of $d_{\hat\tau} q$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Since $h$ is transverse to $q$, the fibre product ${\cal
V}:=(A^1\times{\Bbb R})\times_{(h,q)}
\hat{\cal W}$ is a smooth manifold, and, putting
$h_\alpha:=h(\alpha,\cdot)$ , we see that for any
$\alpha$ in a second category subset of $A^1$, the fibre product ${\cal
V}_\alpha:={\Bbb R}\times_{(h_\alpha,p)}\hat{\cal W}$ is a smooth submanifold of
${\cal V}$.
\begin{lm} The map $\theta:{\cal V}\longrightarrow {\Bbb R}$ , projecting
$(\alpha,s,A,\Phi,t,s\omega_g+d\alpha)$ to $t$, is a submersion. For any
$\alpha$ in a second category subset of $A^1$, the restricted map
$\theta|_{{\cal V}_\alpha}$ is a submersion in all points of $Z(\theta)\cap{\cal
V}_{\alpha}$.
\end{lm}
{\bf Proof: } ${\cal V}$ can be identified with the moduli space of tuples
$(\alpha,s,A,\Phi,t)\in A^1\times{\Bbb R}\times {\cal
A}(L)\times S(A^0(\Sigma^+))\times{\Bbb R}$ satisfying the equations
$$\left\{\begin{array}{ccl} {\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A\Phi&=&0\\
\Gamma(F_A^++2\pi i( s\omega_g+d^+\alpha))-2t(\Phi\bar\Phi)_0&=&0 \ .\\
\end{array}\right.
$$
Since the map defined by the left hand side of this system is a submersion
in every
tuple solving the equations, it suffices to show that the map
$$T:A^1\times{\Bbb R}\times {\cal
A}(L)\times S(A^0(\Sigma^+))\times{\Bbb R}\longrightarrow A^0(\Sigma^-)\times
A^0(su(\Sigma^+))\times{\Bbb R}\ ,$$
defined by
$$T(\alpha,s,A,\Phi,t)=\left(\begin{array}{c}{\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A\Phi\\
\Gamma(F_A^+ + 2\pi i( s\omega_g+d^+\alpha)) -2t(\Phi\bar\Phi)_0 \\ t
\end{array}\right)\ ,
$$
is a submersion in every point $v=(\alpha,s,A,\Phi,t)$ with
$[(\alpha,s,A,\Phi,t)]\in{\cal V}$. If
$(\Psi^-,S,r)$ is orthogonal to
${\rm im}(d_v T)$, use first variations of $s$ and $\alpha$ to get $S=0$, then
variations of $A$ to get
$\Psi^-=0$, and then variations of $t$ to get $r=0$.
The second assertion follows by applying Sard's Theorem to the projection
$Z(\theta)\longrightarrow A^1$ onto the first factor.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{lm} For any $\alpha$ in a second category subset of $A^1$, the
moduli space
$\hat {\cal T}_{(s_c\omega+d^+\alpha)}'(L)$ is a smooth manifold.
\end{lm}
{\bf Proof: } If $(\alpha,s,A,\Phi,0)\in Z(\theta)$, then the pair
$(g,[s\omega_g+d\alpha])$
cannot be $c$-good, hence the $s$-component of every point in $Z(\theta)$ must
be $s_c$. Thus there is a natural identification
$\hat {\cal T}_{(s_c\omega+d^+\alpha)}'(L)=Z(\theta)\cap{\cal
V}_{\alpha}$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Since a pair $([\omega_g],[h(\alpha,s)])$ belongs to the wall $c^{\bot}$
iff $s=s_c$, every point $(\alpha,s,A,\Phi,t,s\omega_g+d\alpha)\in {\cal
V}$ with $s\ne s_c$ must have a non-vanishing $t$-component.
\begin{lm} The map $\chi:{\cal V}\longrightarrow {\Bbb R}$, projecting
$(\alpha,s,A,\Phi,t,s\omega_g+d\alpha)$ to $ s$, is a submersion in every
point of ${\cal V}\setminus Z(\theta)$, in particular in all points
$(\alpha,s,A,\Phi,t,s\omega_g+d\alpha)$ with $s\ne s_c$.
\end{lm}
{\bf Proof: } It suffices to show that the map
$$U:A^1\times{\Bbb R}\times {\cal
A}(L)\times S(A^0(\Sigma^+))\times{\Bbb R}\longrightarrow A^0(\Sigma^-)\times
A^0(su(\Sigma^+))\times{\Bbb R}\ , $$
defined by
$$U(\alpha,s,A,\Phi,t)=\left(\begin{array}{c} {\raisebox{.16ex}{$\not$}}\hskip -0.35mm{D}_A\Phi\\
\Gamma(F_A^++2\pi i( s\omega_g+d^+\alpha)) -2t(\Phi\bar\Phi)_0 \\ s
\end{array}\right)\ ,
$$
is a submersion in every point $v=(\alpha,s,A,\Phi,t)$ with $t\ne 0$ . If
$(\Psi^-,S,r)$ is orthogonal to
${\rm im}(d_v U)$, we first use first variations of $\alpha$ to see that $S$ is
orthogonal to $d^+(A^1)$, then variations of $A$ to get
$\Psi^-=0$, and then
variations of $\Phi$ and $t$ ( $t\ne 0$ !) to get $S=0$. Finally, using
variations of
$s$ one finds $r=0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
%
Applying Sard's theorem again, we have
\begin{lm} Let $S_0$ be a countable subset of ${\Bbb R}\setminus\{s_c\}$. Then,
for every
$\alpha$ in a second category subset of
$A^1$, the restricted map
$\chi|_{{\cal V}_\alpha}$ is a
submersion in every point of $\mathop{\bigcup}\limits_{s\in S_0}Z(\chi|_{{\cal
V}_\alpha}-s)$
\end{lm}
\vspace{3mm}
Now we can prove the theorem. \\
{\bf Proof: } Fix $\alpha\in A^1$ such that $\theta|_{{\cal V}_\alpha}$ is a
submersion in every
point of $Z(\theta|_{{\cal V}_\alpha})$ and $\chi|_{{\cal V}_\alpha}$ is a
submersion in
every point of $Z(\chi|_{{\cal V}_\alpha}-(s_c\pm 1))$. Set
$\beta_0=h_\alpha(s_c)=s_c\omega_g+d\alpha$,
$b_0:=[\beta_0]=[s_c\omega_g]$, and $\beta_\pm:=h(\alpha, s_c\pm 1)$,
$b_\pm:=[\beta_\pm]$. Then
$([\omega_g],b_0)$ belongs to the wall $c^{\bot}$, and the intersections $\hat
{\cal W}_{\beta_\pm}:=\hat{\cal W}\cap\{\beta=\beta_\pm\}$ are smooth. Note
that $([\omega_g],b_{\pm})\in C_{{\bf H}_0,\pm}$.
The space
$$\bar{\cal V}^{\geq 0}:={\cal V}_\alpha \cap\{s_c-1\leq
s\leq s_c+1\}\cap\{t\geq 0 \} = [s_c-1,s_c+1]\times_{(h_\alpha,p)}\hat {\cal
W}^{\geq 0}$$
is a
smooth manifold with boundary $\hat {\cal W}^{\geq 0}_{\beta_+}\mathop{\bigcup}
\hat {\cal W}^{\geq 0}_{\beta_-}\mathop{\bigcup} \hat{\cal T}_{\beta_0^+}'(L)$ which
is isomorphic to ${\cal W}_{\beta_+}\mathop{\bigcup} {\cal W} _{\beta_-}\mathop{\bigcup} \hat{\cal
T}_{\beta_0^+}'(L)$ according to Lemma 17.
\dfigure 181mm by 177mm (bordy scaled 500 offset 0mm:)
Indeed, by the choice of $\alpha$, ${\cal V}_\alpha$ is smooth,
the projection on the
$t$ component is a submersion in all points of $\hat{\cal
T}_{\beta_0^+}'(L)$ (Lemma 20), and the projection on the $s$-component
is a submersion in the points of
$\hat {\cal W}^{\geq 0}_{\beta_\pm}$ (Lemma 22) .
Now use the orientation ${\scriptstyle{\cal O}}_1$ of $H^1(X,{\Bbb R})$ and the component ${\bf
H}_0$ of ${\bf H}$ to endow the smooth moduli spaces
$\hat {\cal W}^{\geq 0}_{\beta_\pm}={\cal W} _{\beta_\pm}$
with the corresponding orientations. The manifold
with boundary $\bar{\cal V}^{>0}$ can be oriented by ${\scriptstyle{\cal O}}_1$, ${\bf
H}_0$, and by choosing the natural orientation of the $s$-coordinate.
Then the
oriented boundary of $\bar{\cal V}^{>0}$ is
$$\partial\bar{\cal V}^{>0}=\hat {\cal W}^{\geq 0}_{\beta_+}\mathop{\bigcup}
\left(-\hat {\cal W}^{\geq 0}_{\beta_-}\right)\ .$$
Recall that the moduli space $\hat{\cal T}_{\beta_0^+}'(L)$ could be
oriented using \underbar{only} the orientation ${\scriptstyle{\cal O}}_1$ of $H^1(X,{\Bbb R})$.
To determine the sign of the part
$\hat{\cal T}_{\beta_0^+}'(L)$ of the oriented boundary $\partial\bar{\cal
V}^{\geq 0}$ is a technical problem, which can be solved by a careful
examination
of the Kuranishi model for ${\cal V}_\alpha$ in a point of $\hat{\cal
T}_{\beta_0^+}'(L)$. The final result is
$$\partial\bar{\cal V}^{\geq 0}=\hat {\cal W}^{\geq 0}_{\beta_+}\mathop{\bigcup}
\left(-\hat {\cal W}^{\geq 0}_{\beta_-}\right)-\hat{\cal
T}_{\beta_0^+}'(L)\ .$$
Since the cohomology classes $u$, $\nu(l_i)$ extend to $\hat
{\cal W}$ and ${\cal V}$, and their restrictions to the moduli space
$\hat{\cal
T}_{\beta_0^+}'(L )$ coincide with the corresponding classes defined in the
section above, the theorem follows from the relation
$$[\hat {\cal W}^{\geq 0}_{\beta_+}]_{{\scriptscriptstyle{\cal O}}}-[\hat {\cal W}^{\geq
0}_{\beta_-}]_{{\scriptscriptstyle{\cal O}}}-[\hat{\cal T}_{\beta_0^+}'(L)]_{{\scriptscriptstyle{\cal O}}_1}=0
$$
between the fundamental classes of the three moduli spaces. Here ${\scriptstyle{\cal O}}$ is the
orientation determined by ${\scriptstyle{\cal O}}_1$ and ${\bf H}_0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip\\
{\bf Remark:} Let $(X,g)$ be a manifold of positive scalar curvature,
and let $c\in H^2(X,{\Bbb Z})$ be a characteristic element such that $(g,0)$ is
$c$-good.
Then
$SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(\cdot)=0$ for at least one element in
$\{\pm\}$.
This element is determined by the sign in the inequality $\pm
c\cdot[\omega_g]<0$,
and the other value of
$SW_{X,({\scriptstyle{\cal O}}_1,{\bf H}_0)}({\germ c})(\cdot)$ is determined by the wall crossing
formula.
\section { Seiberg-Witten invariants of K\"ahler surfaces}
Let $(X,g)$ be a
K\"ahler surface with K\"ahler form
$\omega_g$, and let ${\germ c}_0$ be the class of the canonical
$Spin^c(4)$-structure of
determinant
$K_X^{\vee}$ on $(X,g)$. The corresponding spinor bundles are
$\Sigma^{+}=\Lambda^{00}\oplus\Lambda^{02}$, $\Sigma^{-}=\Lambda^{01}$ [OT1].
There is a natural bijection between classes of $Spin^c(4)$-structures ${\germ c}$ of
Chern class $c$ and isomorphism classes of line bundles
$M$ with $2c_1(M)-c_1(K_X)=c$. We denote by ${\germ c}_M$ the class defined by a
line bundle $M$. The spinor bundles of ${\germ c}_M$ are the tensor products
$\Sigma^{\pm}\otimes M$, and the map $\gamma_M:\Lambda^1_X\longrightarrow {\Bbb R}
SU(\Sigma^+\otimes M,\Sigma^-\otimes M)$ given by
$\gamma_M(\cdot)=\gamma_0(\cdot)\otimes{\rm id}_M$ is a Clifford map representing
${\germ c}_M$.
Let $C_0$ be the Chern connection in the anti-canonical bundle $K_X^{\vee}$. We
use the variable substitutions $A:=C_0\otimes B^{\otimes 2}$ with $B\in{\cal
A}(M)$ and $\Psi=:\varphi+\alpha\in A^0(M)\oplus A^{02}(M)$ to rewrite the
Seiberg-Witten equations for $(A,\Psi)$ in terms of $(B,\varphi+\alpha)\in{\cal
A}(M)\times[A^0(M)\oplus A^{02}(M)]$.
\begin{pr} Let $(X,g)$ be a K\"ahler surface, and $\beta\in A^{1,1}_{\Bbb R}$ a
closed
real $(1,1)$-form in the class $b$. Let $M$ be a Hermitian line bundle such
that $(2c_1(M)-c_1(K_X)-b)\cdot[\omega_g]<0$. A pair
$(B,\varphi+\alpha)\in {\cal A}(M)\times\left[A^0(M)\oplus A^{02}(M)\right]$
solves the $\beta$-twisted Seiberg-Witten equations
$(SW^{\gamma_M}_\beta )$ iff:
$$
\left\{\begin{array}{l}F_B^{20}=F_B^{02}=0\ \\
\alpha=0\ ,\ \ \bar\partial_B(\varphi)=0 \ \\
i\Lambda_gF_B+\frac{1}{2}\varphi\bar\varphi+(\frac{s}{2}-\pi\Lambda_g\beta)=0
\ .\end{array}\
\right. $$
\end{pr}
{\bf Proof: } The pair $(B,\varphi+\alpha)$ solves $(SW^{\gamma_M}_\beta )$ iff
$$\begin{array}{ll}F_A^{20}&=-\varphi\otimes\bar\alpha\\
F_A^{02}&=\ \alpha\otimes\bar\varphi\\
\bar\partial_B(\varphi)&=\ i\Lambda\partial_B(\alpha) \\
i\Lambda_g(F_A+2\pi i\beta)&
=- \left(\varphi\bar\varphi-*(\alpha\wedge\bar\alpha)\right).\end{array}\
$$
Using Witten's transformation $(B,\varphi+\alpha)\longmapsto(B,\varphi-\alpha)$,
we find
$\varphi\otimes\bar\alpha=\alpha\otimes\bar\varphi=0$, hence
$F_A^{20}=F_A^{02}=0$, so that
$\varphi$ or $\alpha$ must vanish. Putting $c:=2c_1(M)-c_1(K_X)$ and integrating
the last equation over $X$ we get
$$\frac{1}{2\pi}\int\limits_X(|\alpha|^2-|\varphi|^2)\frac{\omega_g^2}{2} =
\int\limits_X(\frac{i}{2\pi}F_A-\beta)\wedge\omega_g=(c-b)\cup[\omega_g]<0
\ ,$$
hence $\alpha=0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
Let ${\cal D}ou(m)$ be the Douady space of all effective
divisors $D$ on $X$ with $c_1({\cal O}_X(D))=m$.
\begin{thry}
Let $(X,g)$ be a connected K\"ahler surface, and let ${\germ c}_M$ be the class of
the $Spin^c(4)$-structure associated to a Hermitian line bundle
$M$ with $c_1(M)=m$. Let $\beta\in A^{1,1}_{{\Bbb R}}$ be a closed form
representing
the class $b$ such that
$ (2m-c_1(K_X)-b)\cup[\omega_g]<0$ ($>0$).
\hfill{\break}
i) \ If $c\ \not\in\ NS(X)$, then ${\cal W}_{X,\beta}^{\gamma_M}
=\emptyset$. If $c\in NS(X)$, then there is a natural real analytic
isomorphism
${\cal W}_{X,\beta}^{\gamma_M}\simeq {\cal
D}ou(m)$ $({\cal D}ou(c_1(K_X)-m))$.
\hfill{\break}
ii) ${\cal W}_{X,\beta}^{\gamma_M} $ is smooth at a point corresponding to
$D\in{\cal D}ou(m)$ iff
$h^0({\cal O}_D(D))=\dim_D{\cal D}ou(m)\ .$
This condition is always satisfied when $h^1({\cal O}_X)=0$. \hfill{\break}
iii) If ${\cal W}_{X,\beta}^{\gamma_M}$ is smooth at a point
corresponding to $D$,
then it has the expected dimension in this point iff $h^1({\cal O}_D(D))=0$.
\end{thry}
{\bf Proof: } Clearly $c\in NS(X)$ is a necessary condition for ${\cal
W}_{X,\beta}^{\gamma_M}\not =\emptyset$. Putting again $c:=2m-c_1(K_X)$, we
may assume that we are in the case $(c-b)\cdot[\omega_g]<0$, since the other one
can be reduced to it by Serre duality. Under this assumption ${\cal
W}^{\gamma_M}_{X,\beta}$ can be identified with the moduli space of holomorphic
pairs $(\bar\partial,\varphi)\in{\cal H}(M)\times A^0(M)$ for which the
generalized
vortex equation
$$i\Lambda_g F_h+\frac{1}{2}\varphi\bar\varphi^h
+(\frac{s}{2}-\pi\Lambda_g\beta)=0
$$
is solvable. The latter space is naturally isomorphic with the Douady space
${\cal
D}ou(m)$ [OT1]. The remaining assertions follow from the long exact cohomology
sequence of the structure sequence
$$0\longrightarrow {\cal O}_X\textmap{\cdot D}{\cal O}_X(D)\longrightarrow {\cal O}_D(D)\longrightarrow 0\ .
$$
\hfill\vrule height6pt width6pt depth0pt \bigskip
{\bf Remark:} We use the complex structure of the surface to orient
$H^1(X,{\Bbb R})$ and
${\Bbb H}^2_{+,g}(X)$. With this convention, the following holds:
The natural isomorphism
${\cal W}^{\gamma_M}_{X,\beta}\simeq {\cal D}ou(m)$ respects the orientation
when
$(2m-c_1(K_X)-b)\cup[\omega_g]<0$.
If $(2m-c_1(K_X)-b)\cup[\omega_g]>0$, then the isomorphism ${\cal
W}^{\gamma_M}_{X,\beta}\simeq {\cal D}ou(c_1(K_X)-m)$ multiplies the
orientation by the factor $(-1)^{\chi(M)}$.
The pull-back of the hyperplane class of ${\cal D}ou(m)$ is precisely $u$
when
$(2m-c_1(K_X)-b)\cup[\omega_g]<0$.
If
$(2m-c_1(K_X)-b)\cup[\omega_g]>0$, then the pull-back of the hyperplane class
of ${\cal D}ou(c_1(K_X)-m)$ is $-u$.
\\
Recall that an effective divisor $D$ on a connected complex surface $X$ is
$k$-connected iff $D_1\cdot D_2\geq k$ for every effective decomposition
$D=D_1+D_2$ [BPV]. Canonical divisors of minimal surfaces of Kodaira dimension
$\kappa=1, 2$ are $(\kappa-1)$-connected.
\begin{lm} Every connected complex surface with $p_g>0$ is oriented
diffeomorphic to a surface which possesses a 0-connected canonical divisor.
\end{lm}
{\bf Proof: } Let $X$ be a connected complex surface with $p_g>0$ and minimal model
$X_{\rm min}$. Let $K_{\rm min}$ be a 0-connected canonical divisor of $X_{\rm
min}$. Choose $b_2(X)-b_2(X_{\rm min})$ distinct points $x_i\in X_{\rm
min}\setminus{\rm supp}(K_{\rm min})$, and note that $X$ is diffeomorphic to the
blow up $\hat X_{\rm min}$ of $X_{\rm min}$ in these points. Denote by
$\sigma$
the projection $\sigma:\hat X_{\rm min}\longrightarrow X_{\rm min}$ and by $E$ the
exceptional divisor. Then $\hat K:=\sigma^*(K_{\rm min})+E$ is a canonical
divisor
on $\hat X_{\rm min}$. If $\hat K$ decomposes as $\hat K=D_1+ D_2$, then every
component of $E$ is contained in precisely one of the summands, and $K_{\rm
min}=\sigma_*(\hat K)=\sigma_* D_1+\sigma_* D_2$ is a decomposition of
$K_{\rm min}$. This implies $D_1\cdot D_2=\sigma_* D_1\cdot \sigma_* D_2\geq
0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{co} ([W]) All non-trivial Seiberg-Witten invariants of K\"ahler surfaces
with $p_g>0$ have index 0.
\end{co}
{\bf Proof: } Let $X$ be a K\"ahler surface with $p_g>0$. We may suppose that $X$
possesses a 0-connected canonical divisor $K$, defined by a holomorphic 2-form
$\eta$. Using a moduli space ${\cal W}^\gamma_{X,\eta}$ to calculate the
invariant as in [W], we find an effective decomposition $K=D_1+D_2$. This
implies
$w_c=-D_1\cdot D_2\leq 0$.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{co} Let $X$ be a K\"ahlerian surface with $p_g=0$ and $q=0$. Endow
$H^1(X,{\Bbb R})=0$ with the standard orientation and let ${\bf H}_0$ be the
component of
${\bf H}$ containing K\"ahler forms . If $m(m-c_1(K_X))\geq 0$, then we have
$$SW_{X,{\bf H}_0}({\germ c}_M)(+)=\left\{\begin{array}{ccc} 1&{\rm if} &{\cal
D}ou(m)\ne\emptyset\\
0 &{\rm if} &{\cal D}ou(m)=\emptyset \ ,
\end{array}\right.
$$
and
$$SW_{X,{\bf H}_0}({\germ c}_M)(-)= \left\{\begin{array}{ccc} 0&{\rm if} &{\cal
D}ou(m)\ne\emptyset\\
-1 &{\rm if} &{\cal D}ou(m)=\emptyset \ .
\end{array}\right.
$$
\end{co}
{\bf Proof: } Suppose ${\cal D}ou(m)\ne\emptyset$. Since $p_g=0$ and $q=0$, we
must have
${\cal D}ou(c_1(K_X)-m)=\emptyset$, hence $SW_{X,{\bf H}_0}({\germ c}_M)(-)=0$
by Theorem 10.
Now the wall crossing formula implies $SW_{X,{\bf H}_0}({\germ c}_M)(+)=1$. If
${\cal D}ou(m)=\emptyset$, then $SW_{X,{\bf H}_0}({\germ c}_M)(+)=0$ by Theorem
10, and
$SW_{X,{\bf H}_0}({\germ c}_M)(-)=-1$ again by the wall crossing formula.
\hfill\vrule height6pt width6pt depth0pt \bigskip
An interesting formulation is obtained under the additional assumption
${\rm Tors}_2
H^2(X,{\Bbb Z})=0$. Then a class
${\germ c}$ of $Spin^c(4)$-structures is determined by its Chern class $c$ and
$\frac{c_1(K_X)\pm c}{2}$ makes sense. If $c^2\geq c_1(K_X)^2$, then
$$SW_{X,{\bf H}_0}(c)(+)=\left\{\begin{array}{ccc} 1&{\rm if} &{\cal
D}ou(\frac{c_1(K_X)+c}{2})\ne\emptyset\\ \\
0 &{\rm if} &{\cal D}ou(\frac{c_1(K_X)+c}{2})=\emptyset \ ,
\end{array}\right.
$$
and
$$SW_{X,{\bf H}_0}(c)(-)=\left\{\begin{array}{ccc} -1 &{\rm if} &{\cal D}ou
(\frac{c_1(K_X)-c}{2})\ne\emptyset
\\ \\ 0&{\rm if} &{\cal
D}ou(\frac{c_1(K_X)-c}{2})=\emptyset\ . \end{array} \right.
$$
\\
{\bf Example:} Let $X={\Bbb P}^2$, let $h\in H^2({\Bbb P}^2,{\Bbb Z})$ be the class of the ample
generator, and let ${\bf H}_0$ be the component of ${\bf H}=\{\pm h\}$
which contains $h$. The classes of $Spin^c(4)$-structures are labelled by odd
integers $c$, and the index corresponding to $c$ is
$w_c=\frac{1}{4}(c^2-9)$. The
chambers of type $c$ contained in ${\bf H}_0\times H^2_{\rm DR}(X)$ are the
half-lines $C_{{\bf H}_0,\pm}=\{ b\in H^2_{\rm DR}({\Bbb P}^2)|\ \pm(c-b)\cdot h<0\}$.
Using Theorem 10 to
calculate the moduli spaces ${\cal W}_{{\Bbb P}^2,\beta}^{\gamma_{\frac{c-3}{2}}}$ we
find
$${\cal W}_{{\Bbb P}^2,\beta}^{\gamma_{\frac{c-3}{2}}}\simeq\left\{\begin{array}{lll}
|{\cal O}_{{\Bbb P}^2}(\frac{c-3}{2})|&{\rm if}&[\beta]>c\\
|{\cal O}_{{\Bbb P}^2}(\frac{-c-3}{2})|&{\rm if}&[\beta]<c \ .
\end{array}\right.
$$
Taking into account the orientation-conventions, we get by direct verification:
$$SW_{{\Bbb P}^2,{\bf H}_0}(c)(+)=\left\{\begin{array}{lll}
1&{\rm if}& c\geq 3\\
0&{\rm if}&c<3\ ,
\end{array}\right.\ \
SW_{{\Bbb P}^2,{\bf H}_0}(c)(-)=\left\{\begin{array}{lll}
-1&{\rm if}& c\leq-3\\
0&{\rm if}&c>-3\ .
\end{array}\right.$$
\dfigure 187mm by 186mm (plane scaled 350 offset 0mm:)
For every $c$, the subspace ${\bf H}_0\times\{0\}$ is contained in
the chamber on which $SW_{{\Bbb P}^2,{\bf H}_0}(c)$ vanishes.\\ \\
{\bf Remark:} Let $X=\hat{\Bbb P}^2$ be the blow-up of ${\Bbb P}^2$ in $r\geq
3$ points, and fix a non-negative even integer $w$. There exist infinitely many
solutions $(d;m_1,\dots,m_r)\in{\Bbb N}^{\oplus(r+1)}$ of the equation
%
$\frac{1}{2}w=\frac{1}{2}d(d+3)-\sum\limits_{i=1}^r\frac{m_i(m_i+1)}{2}\
.$
%
For every solution $(d;m_1,\dots,m_r)$ let $M$ be the
underlying line bundle of the linear system $|dL-\sum\limits_{i=1}^r m_i
E_i|$, and set $c:=2 c_1(M)-c_1(K_X)$. Then $w_c=w$ and we have $SW_{X,{\bf
H}_0}(c)(+)=1$, $SW_{X,{\bf H}_0}(c)(-)=0$ for the component ${\bf H}_0$
containing K\"ahler classes. Hence there exist infinitely many
characteristic classes $c$ with non-trivial Seiberg-Witten invariants and
prescribed index $w_c=w$.
\parindent0cm
\vspace{0.7cm}
{{\bf References}}\vskip 10pt
{\small
[BPV] Barth, W., Peters, C., Van de Ven, A.: {\it Compact complex surfaces},
Springer Verlag, 1984.
[DK] Donaldson, S.; Kronheimer, P..: {\it The Geometry of
four-manifolds}, Oxford Science Publications 1990.
[KM] Kronheimer, P.; Mrowka, T.: {\it The genus of embedded surfaces in
the projective plane}, Math. Res. Lett. 1, 1994, pp. 797-808.
[LL] Li, T.; Liu, A.: {\it General wall crossing formula}, Math. Res. Lett.
2, 1995,
pp. 797-810.
[OT1] Okonek, Ch.; Teleman A.: {\it The Coupled Seiberg-Witten Equations,
Vortices, and Moduli Spaces of Stable Pairs}, Int. J. Math. Vol. 6, No. 6,
1995,
pp. 893-910.
[OT2] Okonek, Ch.; Teleman A.: {\it Quaternionic monopoles}, Comm. Math. Phys. (to
appear).
[OT3] Okonek, Ch.; Teleman A.: {\it Seiberg-Witten invariants for manifolds with
$b_+=1$}, Comptes Rendus Acad. Sci. Paris (to appear).
[Ta] Taubes, C.: {\it The Seiberg-Witten and Gromov invariants} Math.
Res. Lett. 2, 1995, pp. 221-238.
[T] Teleman, A.: {\it Moduli spaces of monopoles}, Habilitationsschrift,
Z\"urich 1996 (in preparation).
[W] Witten, E.: {\it Monopoles and four-manifolds}, Math.
Res. Lett. 1, 1994, pp. 769-796.
\vspace{0.2cm}\\
%
Ch. Okonek: Mathematisches Institut, Universit\"at Z\"urich,
Winterthurerstr. 190,
\hspace*{2.2cm} CH-8057 Z\"urich \\
A. Teleman: \ Mathematisches Institut, Universit\"at Z\"urich,
Winterthurerstr. 190,
\hspace*{2.4cm}CH-8057 Z\"urich and\\
\hspace*{2.4cm}Faculty of Mathematics, University of Bucharest\\
\hspace*{2.4cm}e-mail: [email protected] ; [email protected]}
\end{document}
|
1996-03-04T06:20:20 | 9603 | alg-geom/9603001 | en | https://arxiv.org/abs/alg-geom/9603001 | [
"alg-geom",
"math.AG"
] | alg-geom/9603001 | Alice Silverberg | A. Silverberg and Yu. G. Zarhin | Images of $\ell$-adic representations and automorphisms of abelian
varieties | LaTeX2e | null | null | null | null | Suppose $F$ is either a global field or a finitely generated extension of
${\mathbf Q}$, $A$ is an abelian variety over $F$, and $\ell$ is a prime not
equal to the characteristic of $F$. Let $Z$ denote the center of the
endomorphism algebra of $A$. Let $G$ denote the group of ${\mathbf
Q}_\ell$-points of the identity connected component of the Zariski closure of
the image of the $\ell$-adic representation associated to $A$. We prove the
$\ell$-independence of the intersection of $G$ with the torsion subgroup of
$Z$. Our results provide evidence in the direction of the Mumford-Tate
Conjecture.
| [
{
"version": "v1",
"created": "Fri, 1 Mar 1996 16:58:50 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Silverberg",
"A.",
""
],
[
"Zarhin",
"Yu. G.",
""
]
] | alg-geom | \section{Introduction}
Suppose that $F$ is either a global field or a
finitely generated extension of ${\mathbf Q}$, ${A}$ is an abelian variety
over $F$, $\ell$ is a prime number, and $\ell \neq \mathrm{char}(F)$.
Let ${\mathfrak G}_\ell(F,{A})$ denote the algebraic envelope of the image
of the absolute Galois group of $F$ under the $\ell$-adic
representation associated to ${A}$, and
let ${\mathfrak G}_\ell(F,{A})^0$ denote its identity connected component.
In \S\ref{exclasses} we prove that
the intersection of
${\mathfrak G}_\ell(F,{A})^0({\mathbf Q}_\ell)$ with the torsion subgroup of the center of
$\mathrm{End}({A}) \otimes {\mathbf Q}$ is independent of $\ell$.
In the case where $F$ is a finitely generated extension of ${\mathbf Q}$,
this would follow from the Mumford-Tate Conjecture.
Our results do not assume the Mumford-Tate Conjecture,
and apply even in the positive
characteristic case, where there is no analogue of the
Mumford-Tate Conjecture.
The result in the characteristic zero case can therefore be viewed
as providing evidence in the direction of the Mumford-Tate conjecture.
Let $F_{\Phi,\ell}({A})$ be the smallest extension $F'$ of $F$ such that
${\mathfrak G}_\ell(F',{A})$ is connected. We call this extension the
$\ell$-connectedness extension, or connectedness extension.
The algebraic group ${\mathfrak G}_\ell(F,{A})$
and the field $F_{\Phi,\ell}({A})$ were
introduced by Serre (\cite{serre},
\cite{resume}, \cite{serremotives}), who
proved that if $F$ is a global field or a finitely
generated extension of ${\mathbf Q}$,
then $F_{\Phi,\ell}({A})$ is independent of $\ell$
(see \cite{serre},
\cite{resume}, Proposition 1.1 of \cite{LPmathann}, and
Proposition 6.14 of \cite{LPinv}). In such cases,
we will denote the field $F_{\Phi,\ell}({A})$ by $F_\Phi({A})$.
Our results rely heavily on the $\ell$-independence results
of Serre and their generalizations \cite{LPinv}.
Let $F(\mathrm{End}({A}))$
denote the smallest extension of $F$ over which all the endomorphisms
of ${A}$ are defined. Then
(see Proposition 2.10 of \cite{conn}),
$$F(\mathrm{End}({A})) \subseteq F_{\Phi,\ell}({A}).$$
By enlarging the ground field, we may assume that
$F = F(\mathrm{End}({A})) = F_{\Phi,\ell}({A})$. We then consider twists of
the $\ell$-adic representations associated to ${A}$.
The results of this paper follow from the
$\ell$-independence of the connectedness extensions associated
to these twists.
See \cite{connweil} for a study of the connectedness extensions
attached to twists of abelian varieties. See \cite{conn} for
conditions for the connectedness of ${\mathfrak G}_\ell(F,{A})$.
\section{Definitions, notation, and lemmas}
\label{notation}
Let ${\mathbf Z}$, ${\mathbf Q}$, and ${\mathbf C}$ denote respectively the integers, rational
numbers, and complex numbers.
If $G$ is an algebraic
group, let $G^0$ denote the identity connected component.
If $F$ is a field, let $F^s$ denote a separable closure of $F$ and let
${\bar F}$ denote an algebraic closure of $F$. If ${A}$ is an abelian variety
over a field $F$, write $\mathrm{End}_F({A})$
for the endomorphisms of ${A}$ which are defined over $F$,
write $\mathrm{Aut}_F({A})$ for the automorphisms of ${A}$ defined over $F$, let
$\mathrm{End}({A}) = \mathrm{End}_{F^s}({A})$, let $\mathrm{End}^0({A}) = \mathrm{End}({A}) \otimes_{\mathbf Z} {\mathbf Q}$,
and let $\mathrm{End}^0_F({A}) = \mathrm{End}_F({A}) \otimes_{\mathbf Z} {\mathbf Q}$.
If $\ell$ is a prime number and $\ell \neq \mathrm{char}(F)$, let
$T_\ell({A}) = {\displaystyle \lim_\leftarrow {A}_{\ell^r}}$
(the Tate module), let
$V_\ell({A}) = T_\ell({A}) \otimes_{{\mathbf Z}_\ell}{\mathbf Q}_\ell$, and let
$\rho_{{A},\ell}$
denote the $\ell$-adic representation
$$\rho_{{A},\ell} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(T_\ell({A})) \subseteq
\mathrm{Aut}(V_\ell({A})).$$
Let ${\mathfrak G}_\ell(F,{A})$
denote the algebraic envelope of the image of $\rho_{{A},\ell}$,
i.e., the Zariski closure in $\mathrm{Aut}(V_\ell({A}))$ of the image of
$\mathrm{Gal}(F^s/L)$ under $\rho_{{A},\ell}$.
Let $F_{\Phi,\ell}({A})$ be the smallest extension $F'$ of $F$
in $F^s$ such that ${\mathfrak G}_\ell(F',{A})$ is connected.
\begin{lem}[Lemma 2.7 of \cite{conn}]
\label{conncomp}
If ${A}$ is an abelian variety over a field $F$, $L$ is a finite
extension of $F$ in $F^s$, and $\ell$ is a prime number, then
$${\mathfrak G}_\ell(L,{A}) \subseteq {\mathfrak G}_\ell(F,{A})
\text{ and } {\mathfrak G}_\ell(L,{A})^0 = {\mathfrak G}_\ell(F,{A})^0.$$
In particular, if ${\mathfrak G}_\ell(F,{A})$
is connected, then ${\mathfrak G}_\ell(F,{A}) = {\mathfrak G}_\ell(L,{A})$.
\end{lem}
\begin{lem}
\label{conncomplem}
Suppose ${A}$ and ${B}$ are abelian varieties over a field $F$, $L$ is a finite
extension of $F$ in $F^s$, $\ell$ is a prime number,
$\ell \neq \mathrm{char}(F)$, ${\mathfrak G}_\ell(F,{A})$ is
connected, and ${A}$ and ${B}$ are isomorphic over $L$. Then:
\begin{enumerate}
\item[{(i)}] ${\mathfrak G}_\ell(F,{B})^0 = {\mathfrak G}_\ell(F,{A})$, and
\item[{(ii)}]${\mathfrak G}_\ell(L,{B})$ is connected, i.e.,
$F_{\Phi,\ell}({B}) \subseteq L$.
\end{enumerate}
\end{lem}
\begin{proof}
Since ${A}$ and ${B}$ are isomorphic over $L$, and ${\mathfrak G}_\ell(F,{A})$ is
connected, we have
$${\mathfrak G}_\ell(L,{B}) = {\mathfrak G}_\ell(L,{A}) = {\mathfrak G}_\ell(F,{A}) = {\mathfrak G}_\ell(F,{A})^0$$
$$= {\mathfrak G}_\ell(L,{A})^0 = {\mathfrak G}_\ell(L,{B})^0 = {\mathfrak G}_\ell(F,{B})^0,$$
using Lemma \ref{conncomp}. The result follows.
\end{proof}
\begin{lem}
\label{strcompat}
Suppose ${A}$ is an abelian variety over a global field $F$ and
$$c : \mathrm{Gal}(F^s/F) \to \mathrm{End}^0_F({A})^\times$$ is a continuous
homomorphism of finite order. For each
prime $\ell \ne \mathrm{char}(F)$ let
$$\rho_{\ell,c}: \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(V_{\ell}({A})),\quad
\sigma \mapsto c(\sigma) \rho_{{A},\ell}(\sigma)$$
be the twist of $\rho_{{A},\ell}$. Then $\{\rho_{\ell,c}\}$
constitutes a strictly compatible system of integral
$\ell$-adic representations of $\mathrm{Gal}(F^s/F)$.
More precisely, suppose $M$ is a finite Galois extension of
$F$ such that $c$ factors through $\mathrm{Gal}(M/F)$.
Let $S_\ell$ be the set of finite places $v$ of $F$ such that
either ${A}$ has bad reduction at $v$, $v$ is ramified in
$M/F$, or the residue characteristic of $v$ is $\ell$.
Let $v$ be a finite place of $F$,
$w$ a place of $F^s$ lying over $v$, and $\kappa_v$ and
$\kappa_w$ the residue fields at $v$ and $w$, respectively.
Let $\tau \in \mathrm{Gal}(F^s/F)$ be an element that acts as
the Frobenius automorphism of $\kappa_w/\kappa_v$.
Suppose that $v \notin S_\ell$, and let
$$\varphi_w = \rho_{\ell,c}(\tau) \in \mathrm{Aut}(V_{\ell}({A})).$$
Then $\rho_{\ell,c}$ is unramified at $v$,
the characteristic polynomial
$P_v(t)$ of $\varphi_w$ lies in ${\mathbf Z}[t]$ and does not depend on
the choice of $w$ and $\ell$,
and the roots of $P_v(t)$ all have complex absolute value
$\sqrt{\#\kappa_v}$.
\end{lem}
\begin{proof}
We use that $\{\rho_{{A},\ell}\}$ is a strictly compatible system
of integral $\ell$-adic representations of $\mathrm{Gal}(F^s/F)$
(see \cite{Weil} and I.2 of \cite{abreps}).
Let ${A}_v$ be the abelian variety over $\kappa_v$
which is the reduction of ${A}$ at $v$.
The choice of $w$ allows us to identify
the Tate modules $V_\ell({A})$ and $V_\ell({A}_v)$, and this
identification is compatible with the natural embedding
$\mathrm{End}^0({A}) \hookrightarrow \mathrm{End}^0({A}_v)$.
Let
$$Fr_w = \rho_{{A},\ell}(\tau) \in \mathrm{Aut}(V_{\ell}({A})).$$
Then
$$\varphi_w = c(\tau)Fr_w \in \mathrm{Aut}(V_\ell({A}))
\subseteq \mathrm{Aut}(V_\ell({A}_v)),$$
and the identification of $\mathrm{Aut}(V_\ell({A}))$ with $\mathrm{Aut}(V_\ell({A}_v))$
identifies $Fr_w$ with the Frobenius endomorphism of
${A}_v$ inside $\mathrm{Aut}(V_\ell({A}_v))$.
It follows from Weil's results on endomorphisms of abelian varieties
that $P_v(t)$ has rational coefficients
which do not depend on the choice of $w$ and $\ell$.
If $m = [M:F]$ then
$(c(\tau)Fr_w)^m = (Fr_w)^m \in \mathrm{End}({A}_v)$
and therefore all roots of $P_v(t)$ are algebraic
integers. Therefore, $P_v(t) \in {\mathbf Z}[t]$. Further,
Weil's results imply that the eigenvalues of $\varphi_w$
have absolute value $\sqrt{\#\kappa_v}$.
\end{proof}
\begin{thm}
\label{galois}
If $F$ is either a finitely generated extension of ${\mathbf Q}$ or
a function field in one variable over a finite field,
then every finite abelian group occurs as
a Galois group over $F$.
\end{thm}
\begin{proof}
See Theorem 3.12c of \cite{Saltman}, and
IV.2.1 and IV.1.2 of \cite{Matzat}.
\end{proof}
Next we define the Mumford-Tate group of a complex abelian variety ${A}$
(see \S2 of \cite{Ribet} or \S6 of \cite{Izv}). If ${A}$ is a complex abelian
variety, let $V = H_1({A}({\mathbf C}),{\mathbf Q})$ and consider the Hodge decomposition
$V \otimes {\mathbf C} = H_1({A}({\mathbf C}),{\mathbf C}) = H^{-1,0} \oplus H^{0,-1}$.
Define a homomorphism $\mu : {\mathbf G}_m \to GL(V)$ as follows. For
$z \in {\mathbf C}$, let $\mu(z)$ be the automorphism of $V \otimes {\mathbf C}$ which is
multiplication by $z$ on $H^{-1,0}$ and is the identity on $H^{0,-1}$.
\begin{defn}
The {\em Mumford-Tate group} $MT_{A}$ of ${A}$ is the smallest
algebraic subgroup of $GL(V)$, defined over ${\mathbf Q}$, which after extension of
scalars to ${\mathbf C}$ contains the image of $\mu$.
\end{defn}
If ${A}$ is an abelian variety over a subfield $F$ of ${\mathbf C}$, we fix an
embedding of ${\bar F}$ in ${\mathbf C}$. This gives an identification of
$V_\ell({A})$ with $H_1({A},{\mathbf Q})\otimes{\mathbf Q}_\ell$, and allows us to view
$MT_{A} \times {\mathbf Q}_\ell$ as a linear ${\mathbf Q}_\ell$-algebraic subgroup of
$GL(V_\ell({A}))$. Let
$$MT_{{A},\ell} = MT_{A} \times_{\mathbf Q} {\mathbf Q}_\ell.$$
Then $MT_{{A}}({\mathbf Q}_\ell) = MT_{{A},\ell}({\mathbf Q}_\ell)$.
The Mumford-Tate conjecture for abelian varieties (see \cite{serrereps})
may be reformulated as the equality of ${\mathbf Q}_\ell$-algebraic groups,
${\mathfrak G}_\ell(F,{A})^0 = MT_{{A},\ell}$.
\begin{conj}[Mumford-Tate Conjecture]
\label{mtconj}
If ${A}$ is an abelian variety over a finitely generated extension $F$ of
${\mathbf Q}$, then ${\mathfrak G}_\ell(F,{A})^0 = MT_{{A},\ell}$.
\end{conj}
The inclusion ${\mathfrak G}_\ell(F,{A})^0 \subseteq MT_{{A},\ell}$ was
proved by Piatetski-Shapiro \cite{ps}, Deligne \cite{sln900},
and Borovoi \cite{Borovoisb}.
It is well-known that $MT_A$ contains the homotheties
${\mathbf G}_m$ and that the centralizer of
$MT_A$ in $\mathrm{End}(V)$ is $\mathrm{End}^0(A)$. Therefore, the center of
$MT_A({\mathbf Q})$ contains $-1$ and is contained in the center of $\mathrm{End}^0(A)$.
\section{$\ell$-independence}
\label{exclasses}
Suppose that $F$ is either a
finitely generated extension of ${\mathbf Q}$ or a global field.
Suppose $F = F_\Phi({A})$, so that
${\mathfrak G}_\ell(F,{A}) = {\mathfrak G}_\ell(F,{A})^0 = {\mathfrak G}_\ell(L,{A})$
for all finite extensions $L$ of $F$.
It follows from \cite{faltings}, \cite{fw}, \cite{z0}, \cite{z1},
and VI.5 and XII.2 of \cite{Mori2}
that ${\mathfrak G}_\ell(F,{A})$ is a reductive
${\mathbf Q}_\ell$-algebraic group, whose centralizer in $\mathrm{End}(V_\ell({A}))$
is $\mathrm{End}({A}) \otimes {\mathbf Q}_\ell$. This implies that the center
of ${\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$ is contained in
$(Z \otimes {\mathbf Q}_\ell)^\times$, where $Z$ is the center
of $\mathrm{End}^0({A})$.
Let $\mu_\X$ denote the group of elements of finite order in
the center of $\mathrm{End}^0({A})$.
In the case where $F \subset {\mathbf C}$,
the Mumford-Tate Conjecture (Conjecture \ref{mtconj})
would imply that
$\mu_\X \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$
is the torsion subgroup of the center of $MT_{A}({\mathbf Q})$,
and therefore is independent of $\ell$.
In the following two results we prove that
$\mu_\X \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$
is independent of $\ell$ (without assuming the
Mumford-Tate Conjecture).
It follows from Weil's results on abelian varieties \cite{Weil}
(as was pointed out by Deligne; see 2.3 of \cite{serrereps}) that
${\mathfrak G}_\ell(F,{A})$ contains the homotheties ${\mathbf G}_m$.
In particular,
$$-1 \in {\mathfrak G}_\ell(F,{A})({\mathbf Q}_{\ell}).$$
\begin{thm}
\label{discondcor}
Suppose ${A}$ is an abelian variety over a finitely generated
extension $F$ of ${\mathbf Q}$,
and $F = F_{\Phi}({A})$. Let $\mu_\X$ denote the
group of elements of finite order in
the center of $\mathrm{End}^0({A})$.
Then $\mu_\X \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$ is independent of
the prime $\ell$.
\end{thm}
\begin{proof}
Over ${\mathbf C}$, we can view $A$ as ${\mathbf C}^d/L$ with $L$ a lattice in
${\mathbf C}^d$. Then $L'=\sum_{\gamma \in \mu_\X}\gamma(L)$ is a
$\mu_\X$-invariant lattice in ${\mathbf C}^d$ that contains $L$ as a
subgroup of finite index.
The complex abelian variety ${\mathbf C}^d/L'$
has a model $A'$ defined over a finite extension
$F'$ of $F$ such that ${A}$ and $A'$ are $F'$-isogenous and
$\mu_\X$ coincides with the set of elements of finite order in
the center of $\mathrm{End}(A')$. Since
${\mathfrak G}_\ell(F',A') = {\mathfrak G}_\ell(F',{A}) = {\mathfrak G}_\ell(F,{A})$,
we may assume without loss of generality that $\mu_\X$ coincides
with the set of elements of finite order in
the center of $\mathrm{End}({A})$.
By Theorem \ref{galois},
we can choose an abelian extension $M$ of $F$ such that
$\mathrm{Gal}(M/F)$ is isomorphic to $\mu_\X$. Let
$$\chi : \mathrm{Gal}(M/F) \to \mu_\X$$
be an isomorphism, let $c : \mathrm{Gal}(F^s/F) \to \mu_\X$ be the
composition of $\chi$ with the projection
$\mathrm{Gal}(F^s/F) \to \mathrm{Gal}(M/F)$, and
let ${B}$ denote the twist of ${A}$ by the cocycle
induced by $c$. By Lemma \ref{conncomplem}i,
${\mathfrak G}_\ell(F,{B})^0 = {\mathfrak G}_\ell(F,{A})$.
The character $c$ induces an isomorphism
$$\mathrm{Gal}(M/F_{\Phi}({B})) \cong \mu_\X \cap {\mathfrak G}_\ell(F,{B})^0({\mathbf Q}_\ell) =
\mu_\X \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell).$$
Since $\mathrm{Gal}(M/F_{\Phi}({B}))$ is independent of $\ell$, we are done.
\end{proof}
\begin{thm}
\label{discondcor2}
Suppose $F$ is a function field in one variable over
a finite field, ${A}$ is an abelian variety over $F$, and
$\ell$ is a prime number not equal to $\mathrm{char}(F)$.
Suppose $F = F_{\Phi}({A})$, and let $\mu_\X$ denote the
group of elements of finite order in the center of $\mathrm{End}^0({A})$.
Then $\mu_\X \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$ is independent of $\ell$.
\end{thm}
\begin{proof}
By Theorem \ref{galois} we
can choose an abelian extension $M$ of $F$ such that
$\mathrm{Gal}(M/F)$ is isomorphic to $\mu_\X$. Let
$$\chi : \mathrm{Gal}(M/F) \to \mu_\X$$
be an isomorphism, let $c : \mathrm{Gal}(F^s/F) \to \mu_\X$ be the
composition of $\chi$ with the projection
$\mathrm{Gal}(F^s/F) \to \mathrm{Gal}(M/F)$, and
define $\rho_{\ell,c} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(V_\ell(A))$
by $\rho_{\ell,c}(\sigma) = c(\sigma)\rho_{A,\ell}(\sigma)$.
For $F \subseteq F' \subseteq F^s$, let ${\mathfrak G}_{\ell,c}(F')$ denote
the Zariski closure of
$\rho_{\ell,c}(\mathrm{Gal}(F^s/F'))$. Let $F_{\Phi,c}$
denote the smallest extension $F'$ of $F$ in $F^s$ such that
${\mathfrak G}_{\ell,c}(F')$ is connected. By Lemma \ref{strcompat},
$\{\rho_{\ell,c}\}$ is a strictly compatible system of
integral $\ell$-adic representations.
Therefore by Proposition 6.14 of \cite{LPinv},
$F_{\Phi,c}$ is independent of $\ell$.
By definition,
$$\mathrm{Gal}(M/F_{\Phi,c}) \cong
\mu_\X \cap {\mathfrak G}_{\ell,c}(F)^0({\mathbf Q}_\ell) \quad {\text{ and }}
\quad {\mathfrak G}_{\ell,c}(M) = {\mathfrak G}_\ell(M,A).$$
Lemma \ref{conncomp} is valid with ${\mathfrak G}_{\ell,c}(F')$ in place of
${\mathfrak G}_\ell(F',A)$; the proof remains unchanged. Therefore,
$${\mathfrak G}_{\ell,c}(F)^0 = {\mathfrak G}_{\ell,c}(M)^0 = {\mathfrak G}_\ell(M,A)^0 = {\mathfrak G}_\ell(F,A).$$
Since $\mathrm{Gal}(M/F_{\Phi,c})$ is independent of $\ell$, we are done.
\end{proof}
|
1996-07-26T17:29:42 | 9603 | alg-geom/9603019 | en | https://arxiv.org/abs/alg-geom/9603019 | [
"alg-geom",
"math.AG"
] | alg-geom/9603019 | Tony Pantev | Fedor Bogomolov and Tony Pantev | Weak Hironaka theorem | 11 pages, minor corrections, version revised for publication LATEX 2e | null | null | null | null | The purpose of this note is to give a simple proof of the following theorem:
Let $X$ be a normal projective variety over an algebraically closed field $k$,
$\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$.
Then there exist a smooth projective variety $M$, a strict normal crossings
divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D =
R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also
use a multidimensional version of G.Belyi argument which allows us to simplify
the shape of a ramification divisor. By induction on the dimension of $X$ the
problem is reduced to resolving toroidal singularities. This process however is
too crude and does not permit any control over the structure of the birational
map $f$. A different proof of the same theorem was found independently by D.
Abramovich and A.J. de Jong. The approach is similar in both proofs but they
seem to be rather different in details.
| [
{
"version": "v1",
"created": "Sun, 24 Mar 1996 03:12:24 GMT"
},
{
"version": "v2",
"created": "Fri, 26 Jul 1996 14:04:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bogomolov",
"Fedor",
""
],
[
"Pantev",
"Tony",
""
]
] | alg-geom | \section{Introduction} \label{s1}
The existence of a smooth projective model for any proper algebraic
variety over an algebraically closed field of characteristic zero is one
of the most important results in algebraic geometry.
H.Hironaka \cite{h} proved a very strong version of the above statement.
For any such variety he established the existence of a sequence
of blow ups which resolve the singularities of the variety. Moreover at every
step the blow ups occur along smooth subvarieties of the singular locus.
In particular the process does not modify the nonsingular part of the initial
variety. This result proved to be even more useful than the existence
of a smooth model, but its proof was rather complicated.
Later on, E. Bierstone and P. Milman \cite{m}
and M. Spivakovsky \cite{s} found different versions of the resolution process
that are considerably simpler and canonical but somehow in principle their
proofs follow the same track.
In this note we want to give a proof
of the existence of a smooth projective model
for any variety in characteristic zero using an idea similar to A.J. de Jong's
approach in \cite{dj}.
We present a straightforward way to obtain a nonsingular model which
however does not permit any control over the intermediate steps.
The main result of this note is the following theorem
\begin{thm} \label{thm1} Let $X$ be a normal projective variety
over an algebraically closed field $k$, $\op{char} k = 0$ and let
$D \subset X$ be a
proper subvariety of $X$. Then there exist a smooth projective variety
$M$, a strict normal crossings divisor $R \subset M$ and a birational
morphism $f : M \to X$ with $f^{-1} D = R$
\end{thm}
Our proof uses induction on the dimension of $X$. It is organized as
follows. In section~2 we construct a finite
morphism $(\widehat{X},\widehat{D}) \to (P,S_{0})$ from a suitable blow up
$(\widehat{X},\widehat{D})$ of the pair $(X,D)$ to a pair $(P,S_{0})$ where
$P$ is a compactification of the total space of a line bundle over
${\mathbb P}^{n-1}$, $S_{0}$
is the image of a section of this line bundle and $\widehat{X} \to P$
is branched over a divisor $B = \sum S_{i}$ which is a sum of distinct
sections of $P \to {\mathbb P}^{n-1}$.
After this is achieved we proceed in section~3 by applying the induction
hypothesis to the pair $({\mathbb P}^{n-1},Z)$. Here $Z \subset {\mathbb P}^{n-1}$ is the
discriminant locus of the set of sections $\{ S_{i} \}$, i.e. $Z$ is the
unique closed reduced subscheme in ${\mathbb P}^{n-1}$ whose preimage in $P$ contains
all possible intersections of the $S_{i}$'s.
By induction we can find a birational morphism $\widetilde{{\mathbb P}}^{n-1}
\to {\mathbb P}^{n-1}$ which transforms $Z$
into a strict normal crossings divisor $\widetilde{Z}$. Put $\widetilde{P}$ for
the variety obtained from $P$ via a base change with $\widetilde{{\mathbb P}}^{
n-1}\to {\mathbb P}^{n-1}$. Then $(\widetilde{P} \to \widetilde{{\mathbb P}}^{n-1},
\{S_{i}\})$ is a family of pointed ${\mathbb P}^{1}$'s and an application of
F. Knudsen's stabilization theorem yields a smooth variety
$Q$ equipped with a contraction map onto $\widetilde{P}$ and such that the
complete preimage of $B$ in $Q$ is a normal crossings divisor. Now the fiber
product of $Q$ and $\widehat{X}$ has only abelian quotient singularities
which can be resolved by a toroidal blow-up. The resulting pair (M,R) will be
the resolution of $(X,D)$.
\begin{rem} \label{rem11} Recently A.J. de Jong \cite{dj} invented a
remarkable new approach which allowed him to
resolve in any characteristic at least some variety which dominates
the initial variety by a finite map.
The idea to fiber $X$ by curves
and use induction on the dimension came from de Jong's insight in
\cite{dj}. The important reduction in section~2 imitates the first part of
the proof of the famous theorem of Belyi \cite{b}.
\end{rem}
\begin{rem} \label{rem12} The fact that $\op{char} k = 0$ is used only
at last step of the argument in section~3 since in the case of a positive
characteristic it is hard to control the unramified coverings of the torus.
\end{rem}
\begin{rem} \label{rem13} A different proof of the same theorem was found
independently by {\em D. Abramovich and A.J. de Jong} \cite{A-dJ}. It uses
similar ingredients but is quite different in detail. In particular their
proof has the advantage of being better adapted to handling equivariant
resolutions.
\end{rem}
\bigskip
\noindent
{\bf Acknowledgments} We would like to thank A.J. de Jong for his inspiring
Santa Cruz lectures and D. Abramovich for suggesting that we use semistable
reduction for pointed curves of genus zero which allowed us to substantially
simplify our original application of the induction. We would also like to
thank the referee for the careful reading of the manuscript and for the many
valuable comments.
\section{The map to a ${\mathbb P}^1$ bundle}
Let $X$ be an irreducible variety of dimension $n$ over $k$ and let $D \subset
X$ be a closed subvariety.
The aim of this section is to prove the following theorem.
\begin{theo} \label{theo21}
There exists a finite
morphism $(\widehat{X},\widehat{D}) \to (P,S_{0})$ from a suitable blow up
$(\widehat{X},\widehat{D})$ of the pair $(X,D)$ to a pair $(P,S_{0})$ of
varieties satisfying:
\begin{list}{{\em (\roman{dummy})}}{\usecounter{dummy}}
\item $P = {\mathbb P}({\mathcal O}\oplus L)$ for some line bundle $L \to {\mathbb P}^{n-1}$.
\item $S_{0} \subset P$ is the image of a section of $L$.
\item The finite morphism $\widehat{X} \to P$ is branched over a divisor
$B$ of the form
\[
B = \sum_{i=1}^{k} S_{i} + S_{k+1},
\]
where the $S_{i}$'s, $i= 1,\ldots, k$ are images of distinct sections of
$L$ and $S_{k+1} := {\mathbb P}_{\infty}$ is the infinity section of
$P \to {\mathbb P}_{n-1}$.
\end{list}
\end{theo}
{\bf Proof.} Passing to a blow up of $(X,D)$ if necessary we may
assume without loss of generality that $X$ is normal and that
$D = \sum D_i$ with $D_{i}$ reduced and irreducible divisors.
We start with the following relative version of the Noether normalization
lemma.
\begin{lem}There exists a finite morphism $f : X \to {\mathbb P}^n$ which maps
$\widetilde{D}$ onto a hyperplane ${\mathbb P}^{n-1}$ of ${\mathbb P}^n$. \label{lem21}
\end{lem}
{\bf Proof.} In order to find such a map it is enough to consider the
embedding of $X$ into projective space via a very ample line bundle
which has a section vanishing on every $D_i$. This can be easily
achieved by taking a sufficiently big line bundle.
Let $X\subset {\mathbb P}^{m}$ be such an embedding and let $H \subset {\mathbb P}^{m}$
be the hyperplane containing all $D_i$. A straightforward dimension count
shows that the generic
subspace ${\mathbb P}^{N-n-1}\subset H$ does not intersect $X\cap H$ and therefore
we can take for $f$ the projection of ${\mathbb P}^m$ centered at such a subspace.
\hfill $\Box$
\begin{lem} The map $f$ can be chosen so that
there exists a point $o \in {\mathbb P}^n$ satisfying the following properties:
\begin{list}{{\em (\alph{dummy})}}{\usecounter{dummy}}
\item the preimage of $o$ consists
of a finite number of smooth points $x_i$ with $df_{x_i}$
being an isomorphism for every $x_{i}$.
\item the preimage of every line in ${\mathbb P}^n$ containing $o$ is a connected
and generically reduced curve.
\item the preimage of the generic line trough $o$ is irreducible.
\end{list} \label{lem22}
\end{lem}
{\bf Proof.} The subset of points satisfying (a) is open
in ${\mathbb P}^n$ due to the generic smoothness lemma. In order
to satisfy (b) it suffices to chose the projective embedding from
lemma~\ref{lem21} in such a way that all non-reduced curves in $X$
obtained as hyperplane sections constitute a subvariety of big codimension
(greater than $n-1$) in the corresponding Grassmanian. In other words we
want to choose the embedding $X \subset {\mathbb P}^{m}$ so that there exists a
projective subspace $V \subset {\mathbb P}^m$ of dimension $m-n$ with the property
\medskip
\noindent
\begin{description}
\item[(*)] For every projective subspace $V \subset M \subset {\mathbb P}^m$ of
dimension $m-n+1$ the intersection $M\cap X$ is a generically reduced
connected curve.
\end{description}
\medskip
\noindent
Due to the connectedness theorem \cite[III
Corollary 7.9]{ha}, \cite{fh} the preimage in $X$ of every line trough $o$
will be connected. Let $Y$ be the union of the singular locus of $X$ the
divisor $D$ and the ramification divisor of $f$. Furthermore
Bertini's theorem implies that the part of a general
hyperplane curve contained in $X\setminus Y$ will be reduced and irreducible
and therefore it suffices to show that for a suitable embedding of $X$ there
exists a $m-n$ dimensional projective subspace $V \subset {\mathbb P}^{m}$
satisfying
\medskip
\noindent
\begin{description}
\item[(**)] For every projective subspace $V \subset M \subset {\mathbb P}^m$
of dimension $m-n+1$ the intersection $M\cap Y$ is of dimension zero.
\end{description}
\medskip
\noindent
Consider the dimension jump locus
\[
\Gamma = \left\{ M \left| \dim M = m-n+1, \; \dim(M\cap Y) \geq
1 \right. \right\} \subset Gr(m-n+2,m+1),
\]
and let $\widetilde{\Gamma} = \{ (M,y) | y \in M \} \subset \Gamma \times Y$
be the corresponding incidence variety. The projection $\pi_{Y} :
\widetilde{\Gamma} \to Y$ surjects on $Y$ and its fiber over a general
point $y \in Y$ consists of all $M \subset {\mathbb P}^m$ of dimension $m-n+1$
such that $y \in M$ and $\dim(M\cap Y) \geq 1$. In particular $\dim
\widetilde{\Gamma} \leq 2n - 3$ and $\dim \Gamma \leq 2n -4$. Furthermore
the flag variety $F$ parametrizing flags $V \subset M \subset {\mathbb P}^m$ as
above has the usual double fibration structure
\[
{\divide\dgARROWLENGTH by 5
\begin{diagram}
\node[2]{F} \arrow{sw,l}{p} \arrow{se,l}{q} \\
\node{Gr(m-n+1,m+1)} \node[2]{Gr(m-n+2,m+1)}
\end{diagram}}
\]
where the fibers of $p$ are projective spaces of dimension $n-1$ and the fibers
of $q$ are projective spaces of dimension $m-n$. The locus in $Gr(m-n+1,m+1)$
consisting of all $V$'s that do not satisfy (**) is contained in
$q(p^{-1}(\Gamma))$ and hence has codimension that is $\geq (m-n+1)n - (m-n +
2n -4) = m(n+1) - n^2 + 4$. Thus composing our original embedding $X \subset
{\mathbb P}^m$ with a Veronese embedding of sufficiently high degree we can make this
codimension strictly positive which proves the lemma. \hfill $\Box$
\bigskip
Denote by $P_{1}$ the blow up of ${\mathbb P}^{n}$ at $o$ and by $\widehat{X}$ the
blow up of $X$ at the points $x_i$. Since the blow up at $o \in {\mathbb P}^n$
resolves the projection centered at $o$ we get a realization of $P_{1}$ as a
${\mathbb P}^1$ bundle over ${\mathbb P}^{n-1}$. There is a canonical isomorphism
$P_{1} \cong {\mathbb P}({\mathcal O}_{{\mathbb P}^{n-1}}\oplus {\mathcal O}_{{\mathbb P}^{n-1}}(1))$ which
gives a natural tautological line bundle ${\mathcal O}_{P_{1}}(1) \to P_{1}$ and the
preimage ${\mathbb P}_{\infty}$ of $o$ in $P_1$ is a section of ${\mathcal O}_{P_{1}}(1)$.
Alternatively, ${\mathbb P}_{\infty}$ is the divisor in ${\mathbb P}({\mathcal O}_{{\mathbb P}^{n-1}}
\oplus {\mathcal O}_{{\mathbb P}^{n-1}}(1))$ corresponding to the line subbundle
${\mathcal O}_{{\mathbb P}^{n-1}} \subset {\mathcal O}_{{\mathbb P}^{n-1}}\oplus {\mathcal O}_{{\mathbb P}^{n-1}}(1)$
and $P_1$ can be thought of as a compactification of the total space of
${\mathcal O}_{{\mathbb P}^{n-1}}(1)$ with the divisor ${\mathbb P}_{\infty}$ at infinity.
By construction $\widehat{X}$ is fibered by connected and generically reduced
curves $C_t, t\in {\mathbb P}^{n-1}$ and
maps fiberwise via $f$ to the ${\mathbb P}^1$-fibration $P_1$.
Due to the smoothness of $P_{1}$ the Zariski-Nagata purity theorem
\cite[X 3.1]{sga} implies that the branch locus $B_{1}$ of the
finite map $f : \widehat{X} \to P_1$ is a divisor. Moreover, since $C_{t}$ is
generically reduced for every $t$ we conclude that $B_{1}$ is a horizontal
divisor in $\pi : P_{1} \to {\mathbb P}^{n-1}$,
i.e. every component of $B_{1}$ is mapped finitelyonto ${\mathbb P}^{n-1}$ by $\pi$.
Denote by $N$ the degree of $B_{1}$ on the fibers of $\pi$.
Now we can apply Belyi's construction to simplify the branch divisor.
We will need the following technical result:
\begin{lem} Let $L$ be a line bundle on a variety $M$ and $B$ be a horizontal
divisor in the total space $\op{tot}(L)$ of $L$, which maps properly on $M$
under a natural projection $\pi : \op{tot}(L) \to M$.
Denote by $d$ the degree of the finite map $\pi : B \to M$.
There exists a canonical fiberwise morphism $p_B : \op{tot}(L) \to
\op{tot}(L^{\otimes d})$ which is polynomial of degree $d$ on any fiber $L_x,
x\in M$ and such that $B$ is the preimage of the zero section of $L^{\otimes
d}$ under $p_B$. The map $p_B$ is defined uniquely modulo multiplication by
invertible functions on $M$.
\end{lem}
{\bf Proof.} Consider the compactification $P_{L} := {\mathbb P}({\mathcal O}_M \oplus L) =
\op{Proj}(\op{Symm}^{\bullet}({\mathcal O}_{M}\oplus L^{-1}))$.
Let $M_{\infty}$ be the infinity section of $P_L$ and $y \in H^{0}(M,
\mathcal{O}_{M}) \subset H^{0}(P_{L}, {\mathcal O}_{P_{L}}(1))$ be a non-zero
element. Then $\op{div}(y) = M_{\infty}$.
Denote by $\lambda \in H^{0}(\op{tot}(L), \pi^*L)$ the tautological section
and let $x \in H^{0}(P_{L}, {\mathcal O}_{P_{L}}(1)\otimes \pi^{*}L)$ be the
section whose divisor is the zero section of $L$ normalized so that
$\lambda = x/y$. The fact that the intersection of $B$ with the generic
fiber of $\pi : P_L \to M$ is $d$ implies that ${\mathcal O}_{P_L}(B) =
{\mathcal O}_{P_{L}}(d)\otimes\pi^*A$ for some line bundle $A$ on $M$.
Moreover since $B \subset \op{tot}(L)$
it follows that ${\mathcal O}(B)_{|M_{\infty}} = {\mathcal O}$ and hence $\pi^{*}
A_{|M_{\infty}} \cong {\mathcal O}_{P_L}(-d)_{|M_{\infty}}$. On the
other hand $P_L = {\mathbb P}({\mathcal O}_M \oplus L)$ and by using that
$\pi_{|M_{\infty}}$ is an isomorphism we get that $A = L^{\otimes d}$.
Let $\varphi
\in H^{0}(P_{L},{\mathcal O}_{P_{L}}(d)\otimes\pi^*L^{\otimes d})$ be a section
with divisor $B$. Applying the projection formula to the pushforward by $\pi$
we get isomorphisms
\[
\begin{aligned}
H^{0}(P_{L}, & {\mathcal O}_{P_{L}}(d)\otimes\pi^*L^{\otimes d}) =
H^{0}(M,(\pi_{*}{\mathcal O}_{P_{L}}(d))\otimes L^{\otimes d}) = \\
& = H^{0}(M,S^d({\mathcal O}_M\oplus L^{-1})\otimes L^{\otimes d}) = \\
& = H^{0}(M,{\mathcal O}_M)\oplus H^{0}(M,L) \oplus \ldots \oplus
H^{0}(M,L^{\otimes d}).
\end{aligned}
\]
The image of $\varphi$ under this isomorphism can be decomposed as
$\varphi = (\varphi_{0}, \ldots, \linebreak \varphi_{d})$ with $\varphi_{i}
\in H^{0}(M,L^{\otimes i})$ and hence
\begin{equation} \label{eq21}
\varphi = (\pi^*\varphi_{0})x^d +(\pi^*\varphi_{1})x^{d-1}y + \ldots
+ (\pi^*\varphi_{d})y^d,
\end{equation}
and
\begin{equation} \label{eq22}
\varphi_{|\op{tot}(L)} = (\pi^*\varphi_{0})\lambda^d +(\pi^*\varphi_{1})
\lambda^{d-1} + \ldots
+ (\pi^*\varphi_{d}).
\end{equation}
Now the map
\[
\begin{array}{llcl}
p_{B} : & \op{tot}(L) & \longrightarrow & \op{tot}(L^{\otimes d}) \\
& \eta & \longrightarrow & (\pi^*\varphi_{0})\eta^d +(\pi^*\varphi_{1})
\eta^{d-1} + \ldots
+ (\pi^*\varphi_{d})
\end{array}
\]
has the desired properties. \hfill $\Box$
\begin{cor} The above map extends to a proper map of the compactifications
$p_B : P_L \to P_{L^{\otimes d}}$. The preimage of infinite section
$M_{\infty, L^{\otimes d}}$ is a $d$-multiple of $M_{\infty, L}$ \label{cor21}
\end{cor}
{\bf Proof.} The extension of $p_B$ is given by the right hand side of
(\ref{eq21}). \hfill $\Box$
\begin{cor} The map $p_B : \op{tot}(L) \to \op{tot}(L^{\otimes d})$ is
branched over a
divisor $B' \subset \op{tot}(L^{\otimes d})$ which is horizontal and of degree
$d-1$ along the fibers. \label{cor22}
\end{cor}
{\bf Proof.} From the definition of $p_{B}$ it is clear that the divisor
$B'$ is just the image under $p_{B}$ of the divisor of the section
$p_{B}'(\lambda) := d(\pi^*\varphi_{0})\lambda^{d-1} + \linebreak
(d-1)(\pi^*\varphi_{1})
\lambda^{d-2} + \ldots + (\pi^*\varphi_{d-1}) \in H^{0}(\op{tot}(L),
\pi^*L^{\otimes (d-1)})$. \hfill $\Box$
\begin{rem} Corollary~\ref{cor22} is a complete analogue of the crucial
observation of Belyi in \cite{b}.
\end{rem}
Now we can finish the proof of theorem~\ref{theo21}. Indeed let $P_{2} =
{\mathbb P}({\mathcal O}_{{\mathbb P}^{n-1}}\oplus {\mathcal O}_{{\mathbb P}^{n-1}}(N))$ and let $p_{B_{1}} :
P_{1} \to P_{2}$ be as in corollary~\ref{cor21}. Then according to
corollary~\ref{cor22} the branch divisor of $p_{B_{1}}\circ f$ is the union
of the zero section of ${\mathcal O}_{{\mathbb P}^{n-1}}(N)$ and a horizontal divisor $B_2$
of degree $N-1$ along the fibers. Now we can compose with $p_{B_{2}} + s$
where $s$ is a generic section of ${\mathcal O}_{{\mathbb P}^{n-1}}(N(N-1))$ and continue
until we get a map to a ${\mathbb P}^1$ bundle over ${\mathbb P}^{n-1}$ branched over a
union of distinct sections. \hfill $\Box$
\section{The induction step} In this section we use the induction hypothesis
to modify birationally the pair $(P,B\cup S_{0})$ obtained in
Theorem~\ref{theo21} so that $B\cup S_{0}$ becomes a divisor with strict
normal crossings. First we need the following proposition.
\begin{prop} \label{prop31} There exists a diagram
\[
{\divide\dgARROWLENGTH by 5
\begin{diagram}
\node{Q} \arrow{e,t}{\varepsilon} \arrow{se,b}{\pi} \node{\widetilde{P}}
\arrow{e} \arrow{s,l}{\tilde{p}} \node{P} \arrow{s,r}{p} \\
\node[2]{\widetilde{{\mathbb P}}^{n-1}} \arrow{e} \node{{\mathbb P}^{n-1}} \\
\node[2]{\widetilde{Z}} \arrow{n,J} \arrow{e} \node{Z} \arrow{n,J}
\end{diagram}}
\]
where
\begin{list}{{\em (\roman{dummy})}}{\usecounter{dummy}}
\item $\widetilde{{\mathbb P}}^{n-1} \to {\mathbb P}^{n-1}$ is a birational morphism.
$\widetilde{P} = \widetilde{{\mathbb P}}^{n-1}\times_{{\mathbb P}^{n-1}} P$ and
$\widetilde{Z}=\widetilde{{\mathbb P}}^{n-1} \times_{{\mathbb P}^{n-1}} Z$;
\item $\pi$ is a flat and proper morphism whose geometric fibers are
reduced and connected curves with at most ordinary double points.
Furthermore $\varepsilon : Q\setminus \pi^{-1}(\widetilde{Z}) \to \widetilde{P}
\setminus\tilde{p}^{-1}(\widetilde{Z})$ is an isomorphism;
\item $\varepsilon$ is a birational morphism which restricted
on a geometric fiber $Q_{x}$ of $\pi$ falls under one of the following
possibilities:
\begin{list}{{\em \alph{dummy})}}{\usecounter{dummy}}
\item $\varepsilon_{x} : Q_{x} \to \widetilde{P}_{x}$ is an isomorphism;
\item There is a (possibly disconnected) reduced rational curve $E \subset
Q_{x}$ having at most ordinary double points so that $\varepsilon_{x}(E) =
\Sigma$ is a finite set of closed points in $\widetilde{P}_{x}$ and
\[
\varepsilon_{x} : Q_{x}\setminus E \to \widetilde{P}_{x}\setminus \Sigma
\]
is an isomorphism;
\end{list}
\item $\widetilde{Z}$ and the preimage of $B\cup S_{0}$ in $Q$
are strict normal crossings divisors;
\item $Q$ is non-singular.
\end{list}
\end{prop}
{\bf Proof.} Denote by $Z\subset {\mathbb P}^{n-1}$ the discriminant locus of
the set of sections $S_{0}, S_{1}, \ldots, S_{k+1}$. That is $Z$ is the
smallest closed reduced subscheme in ${\mathbb P}^{n-1}$ whose preimage in $P$
contains all possible intersections of the $S_{i}$'s. By induction there
exists a birational morphism $\mu: \widetilde{{\mathbb P}}^{n-1} \to {\mathbb P}^{n-1}$
such that the reduced preimage $\widetilde{Z}$ of $Z$ is a divisor with strict
normal crossings. Denote by $\widetilde{P}$ the fiber product of $P$ with
$\widetilde{{\mathbb P}}^{n-1}$. Clearly $\widetilde{P} = {\mathbb P}(\mu^{*}L\oplus
\mathcal{O})$. Denote by $\tilde{s}_{i}$, $i = 0,\ldots, k+1$ the sections of
the projective bundle $\tilde{p} : \widetilde{P} \to \widetilde{{\mathbb P}}^{n-1}$
whose images are the divisors $\widetilde{S}_{i} := \mu^{*}S_{i}$, $i = 0,
\ldots ,k+1$.
The image $\widetilde{S}_{k+1}$ of the infinity section of $\widetilde{p}$
does not intersect any of the divisors $\widetilde{S}_{i}$, $i = 0,\ldots, k$
and thus will not cause any trouble. To deal with the bad intersections of
the remaining $k+1$ sections we appeal to F. Knudsen's stabilization theorem
\cite[Theorem~2.4]{k}. According to this theorem
given a flat family $C \to S$
of connected reduced curves with at most ordinary double points and $a$
distinct sections $\{s_{i} : S \to C\}_{i=1}^{a}$ plus an arbitrary extra
section $\Delta : S \to C$ there exist a canonical blow-up $q: C' \to C$ and
unique liftings of the sections $s_{1}, \ldots, s_{a}$ and $\Delta$ to sections
$s_{1}', \ldots, s_{a}',s_{a+1}'$ of $C' \to S$ so that for the induced
morphism $q_{s} : C'_{s} \to C_{s}$ on any geometric fiber $C'_{s}$ of
$C' \to S$ we have one of the following two cases
\begin{list}{\alph{dummy})}{\usecounter{dummy}}
\item $q_{s} : C'_{s} \to C_{s}$ is an isomorphism;
\item There is a rational component $E \subset C'_{s}$ such that $s_{a+1}(s)
\in E$, $q_{s}(E) = x$ is a closed point of $C_{s}$ and
\[
q_{s} : C'_{s}\setminus E \longrightarrow C_{s}\setminus \{ x\}
\]
is an isomorphism.
\end{list}
\medskip
\noindent
In particular, $C' \to S$ is again a flat family of reduced curves with
at most ordinary double points. Explicitly $q: C' \to C$ is given as
follows. Denote by $\mathcal{J}$ the $\mathcal{O}_{C}$ ideal defining
$\Delta$ and by $\mathcal{C}$ the cokernel of the diagonal embedding
\[
\mathcal{O}_{C} \longrightarrow \mathcal{J}^{\vee}\oplus \mathcal{O}_{C}(s_{1}
+\ldots + s_{a}).
\]
Then
\[
C' := \op{Proj}(\oplus_{i \geq 0} \mathcal{C}^{i}),
\]
and $q : C' \to C$ is the natural structure morphism.
The total space of $C'$ can be made smooth by a sequence of blow-ups with
smooth centers as long as $C$
and $S$ are smooth and the morphism $C \to S$ is smooth in the complement
of a divisor with strict normal crossings. This can be seen directly by
analyzing the local picture around the separated sections but instead of
doing that we prefer to invoke a much stronger general result of A.J. de
Jong. The \cite[Proposition~5.6]{dj}
guarantees that the total space of any split flat family of semistable
curves over a smooth base which is smooth over the complement of a strict
normal crossings divisor can be blown-up so that the resulting family is of the
same type and has a smooth total space. Thus the only thing that requires
checking is that the family $C'$ is split but this is clear since we can
label the components of every fiber of $C'$ by their distance from the
proper transform of the fiber in $C$.
\begin{rem}\label{rem31} The stabilization theorem is stated in \cite{k}
for families of stable curves since Knudsen is interested in obtaining a
morphism between the universal curve over the stack of $n$-pointed curves and
the stack of $n+1$ pointed curves. However the proof he gives in \cite{k}
is carefully designed to work for general flat families of reduced nodal
curves. In particular, the stability assumption is never used in his argument.
\end{rem}
\medskip
\noindent
Applying the stabilization theorem to each of the sections $\widetilde{S}_{1},
\ldots \widetilde{S}_{k}$ (or rather to the corresponding proper transforms)
in turn, we obtain a variety $Q$ together with a birational morphism
$\varepsilon : Q \to \widetilde{P}$ and $k+2$ distinct sections $t_{i} :
\widetilde{{\mathbb P}}^{n-1} \to Q$, $i = 0, \ldots,k+1$ lifting the sections
$\tilde{s}_{i}$. Since at every step we are blowing-up
only ideal sheaves whose support is contained in the preimage of
$\widetilde{Z}$ the construction of $Q$ guarantees the validity of items
(ii) and (iii) of the statement of the proposition. Furthermore due to part
b) of the stabilization theorem the images of the sections $t_{i}$ intersect
every fiber of $\pi$ at a smooth point. Thus the preimage of
$B\cup S_{0}$ in $Q$ is the same as the union of the images of the $t_{i}$'s
and $\pi^{-1}(\widetilde{Z})$. Since $\pi$ is flat and has singular fibers that
are trees of rational curves we conclude that $\pi^{-1}(\widetilde{Z})$ is
also a divisor with strict normal crossings which finishes the proof of the
proposition. \hfill $\Box$
\bigskip
To finish the proof of Theorem~A consider the
normalization $Y$ of the fiber product $X\times_{P} Q$. Denote by $Y^{\circ}$
the preimage of $Q\setminus \{ B
\cup S_{0} \}$ in $Y$. The normal variety $Y$ is a finite cover of the
smooth variety $Q$ branched along a divisor with strict normal crossings.
Since we are in characteristic zero this implies that $Y$ has only abelian
quotient singularities and is toroidal without self-intersections, i.e.
locally in the \'{e}tale topology (or formally) the embedding $Y^{\circ}
\subset Y$ is isomorphic to the embedding of an affine algebraic torus
$U$ in its toric variety $\overline{U}$ so that every component of the
complement $\overline{U}\setminus U$ is normal. Finally we invoke the
fundamental theorem \cite[Ch. II, \S 2, Theorem~11$\ast$]{kkms} about toric
resolutions according to which there exists a canonical sheaf of ideals on $I
\subset \mathcal{O}_{Y}$ such that the blow-up $Bl_{I}(Y)$ is non-singular.
|
1996-03-28T06:20:56 | 9603 | alg-geom/9603021 | en | https://arxiv.org/abs/alg-geom/9603021 | [
"alg-geom",
"math.AG"
] | alg-geom/9603021 | Alexander Givental | Alexander B. Givental | Equivariant Gromov - Witten Invariants | 48 pages, LaTeX v 2.09 | null | null | null | null | We develop general theory of equivariant quantum cohomology for ample Kahler
manifolds and prove the mirror conjecture for projective complete
intersections.
| [
{
"version": "v1",
"created": "Wed, 27 Mar 1996 03:56:46 GMT"
},
{
"version": "v2",
"created": "Wed, 27 Mar 1996 19:52:20 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Givental",
"Alexander B.",
""
]
] | alg-geom | \section{Moduli spaces of stable maps}
\label{sec1}
\setcounter{equation}{0}
It was M. Gromov \cite{Gr} who first
suggested to construct (and constructed some) topological invariants of
a symplectic manifold $X$
as bordism classes of spaces of pseudo-holomorphic curves in $X$. Recently
M. Kontsevich \cite{Kn} suggested the concept of {\em stable maps} which gives
rise to an adequate compactification of these spaces. We recall here some
basic facts from \cite{Kn} about these compactifications.
Let $(C,p)$ be a compact connected complex curve with only double singular
points
and with $n$ ordered non-singular {\em marked points} $(p_1,...,p_n)$.
Two holomorphic maps
$(C, p) \to X, \ (C', p') \to X$ to an almost-Kahler manifold $X$ are called
{\em equivalent} if they can be identified by a holomorphic isomorphism
$(C, p) \to (C', p')$. A holomorphic map $(C,p)\to X$ is called {\em stable}
if it does not have infinitesimal automorphisms. In other words, a map is
unstable if either it is constant on a genus $0$ irreducible component of
$C$ with $< 3$ {\em special} ($=$ marked or singular) points or
if $C$ is a torus, carries no marked points and the map is constant.
A stable map may have a non-trivial finite automorphism group.
According to Gromov's compactness theorem \cite{Gr}, any sequence of
holomorphic maps
$C\to X$ of a nonsingular compact curve $C$ has a subsequence
Hausdorff-convergent to a holomorphic map $\hat C\to X$ of (may be reducible)
curve $\hat C$ of the same genus $g$ and representing the same total homology
class $d\in H_2(X,\Bbb Z)$. A refinement of this theorem from \cite{Kn} says
that equivalence classes of stable maps $C\to X$ with given $g,n,d$ form a
single
compact Hausdorff space --- the moduli space of stable maps --- which we
denote $X_{g,n,d}$. Here $g=\operatorname{dim} H^1(C,{\cal O})=1-
\chi (C \backslash C^{\mbox{sing}})$.
In the case $X=pt$ the moduli spaces coincide with Deligne-Mumford
compactifications ${\cal M} _{g,n}$ of moduli spaces of genus $g$ Riemannian
surfaces with $n$ marked points. They are compact nonsingular orbifolds
(i.e. local
quotients of nonsingular manifolds by finite groups) and thus bear the rational
fundamental cycle which allows one to build up intersection theory.
In general, the moduli spaces $X_{g,n,d}$ are singular and may have ``wrong"
dimension, and the idea of the program started in \cite{KM, Kn} is to provide
$X_{g,n,d}$ with virtual fundamental cycles insensitive to perturbations of the
almost-Kahler structure on $X$.
In some nice cases however the spaces $X_{g,n,d}$ are already nonsingular
orbifolds of the ``right" dimension.
A compact complex manifold is called {\em
ample} if it is a homogeneous space of its Lie algebra of holomorphic
vector fields.
\begin{thm}[\cite{Kn, BM}]
If $X$ is ample then all non-empty moduli spaces
$X_{n,d}$ of genus $0$ stable maps are compact nonsingular complex
orbifolds of ``right" dimension $\langle c_1(T_x),d\rangle +\operatorname{dim}_{\Bbb C}X+n-3$.
\end{thm}
Additionally, there are canonical morphisms $X_{n,d}\to X_{n-1,d}, \
X_{n,d}\to{\cal M} _{0,n},\ X_{n,d}\to X^n$ between the moduli spaces $X_{n,d}$
called {\em forgetful, contraction} and {\em evaluation} (and defined
by forgetting one of the marked points, forgetting the map and evaluating
the map at marked points respectively). We refer to \cite{Kn, BM} for details
of their construction.
In the rest of this paper we will stick to ample manifolds; we
comment however on which results are expected to hold in greater generality.
A number of recent preprints by B. Behrend -- B. Fantechi, J.Li -- G. Tian,
T. Fukaya -- K. Ono shows that Kontsevich's ``virtual fundamental cycle''
program is being realized successfully and leaves no doubts that these
generalizations are correct. Still some verifications are necessary in order
make them precise theorems.
\section{Equivariant correlators}
\label{sec2}
\setcounter{equation}{0}
The Gromov-Witten theory borrows from quantum field theory the name {\em
(quantum) correlators} for numerical topological characteristics of the moduli
spaces $X_{n,d}$ (characteristic numbers) and borrows from bordism theory
the construction of such correlators as integrals
of suitable wedge-products of various universal cohomology classes
(characteristic classes of the GW theory) over the fundamental cycle.
We list here some such characteristic classes.
\begin{enumerate}
\item Pull-backs of cohomology classes from $X^n$ by the evaluation
maps $e_1\times\dots\times e_n:X_{g,n,d}\to X^n$ at the marked points.
\item Any polynomial of the first Chern classes $c^{(1)},\dots ,c^{(n)}$ of the
line bundles over $X_{g,n,d}$ consisting of tangent lines to the mapped curves
at the marked points. One defines these line bundles (by identifying the
Cartesian product of the forgetful and evaluation maps $X_{n+1,d}\to
X_{n,d}\times X$ with the {\em universal stable map} over $X_{n,d}$) as normal
line bundles to the $n$ embeddings $X_{n,d}\to X_{n+1,d}$ defined by the
$n$ marked points of the universal stable map. We will call these line
bundles {\em the universal tangent lines} at the marked points.
\item Pull-backs of cohomology classes of the Deligne - Mumford spaces
by contraction maps $\pi: X_{n,d}\to {\cal M} _{0,n}$. We will make use of the
classes $A_I:= A_{i_1,...,i_k} $ Poincare-dual to fundamental cycles of fibers
of forgetful maps ${\cal M} _{0,n}\to {\cal M} _{0,k}$.
\end{enumerate}
We define the {\em GW-invariant}
\[ A_I\langle \phi _1,...,\phi _n \rangle _{n,d} :=
\int _{X_{n,d}} \ \pi ^*A_I \wedge e_1^*\phi _1 \wedge ... \wedge e_n^*\phi _n .\]
It has the following meaning in enumerative geometry: it counts the number
of pairs ``a degree-$d$ holomorphic map ${\Bbb C} P^1 \to X$
with given $k$ points mapped to given $k$ cycles, a configuration of $n-k$
marked points mapped to the $n-k$ given cycles''.
\bigskip
Suppose now that the ample manifold $X$ is provided with a Killing action
of a compact Lie group $G$. Then $G$ act also on the moduli spaces of stable
maps. The evaluation, forgetful and contraction maps are $G$-equivariant,
and one can define correlators
$A_I\langle\phi_1,\dots \phi_n\rangle_{n,d}$ of {\em equivariant}
cohomology classes of $X$.
The equivariant cohomology $H^*_G(M)$ of a $G$-space $M$ is defined as the
ordinary cohomology $H^*(M_G)$ of the homotopic quotient
$M_G=EG\times_G M$ --- the total space of the $M$-bundle
$p:M_G\to BG$ associated with the universal principal $G$-bundle
$EG\to BG$. The characteristic class algebra $H^*(BG)=H^*_G$(pt) plays the role
of the coefficient ring of the equivariant theory (so that $H^*_G(M)$ is a
$H^*_G$(pt)-module). If $M$ is a compact manifold with smooth $G$-action, the
push-forward $p_*:H^*_G(M)\to H^*_G$(pt) (``fiberwise integration") provides
the equivariant cohomology of $M$ with intersection theory with values in
$H^*_G$(pt).
We introduce the {\em equivariant GW-invariants},
$A_I(\langle\phi_1,\dots ,\phi_n\rangle_{n,d}$, with
values in $H^*(BG)$, where $\phi_1,\dots ,\phi_n\in H^*_G(X)$. Values of such
invariants on fundamental cycles of maps $B\to BG$ are accountable for
enumeration of rational holomorphic curves in families of complex manifolds
with
the fiber $X$ associated with the principal $G$-bundles over a
finite-dimensional
manifold $B$.
\section{The WDVV equation}
\label{sec3}
\setcounter{equation}{0}
One of the main structural results about Gromov-Witten invariants ---
the composition rule \cite{RT},\cite{MS} --- expresses all genus-0
correlators via the 3-fixed point ones, (we denote them $\langle \phi_1, ...,
\phi _n \rangle _{n,d} $ since the corresponding $A_I=1$)
satisfying additionally the
so-called {\em Witten-Dijkgraaf-Verlinde-Verlinde} equation. We will see here
that the same result holds true for equivariant Gromov-Witten invariants
(at least in the ample case).
Following \cite{WD}, introduce the {\em potential}
\begin{equation}
\label{eq3.1}
F =\sum^\infty_{n=0} \ \frac{1}{n!} \ \sum_d q^d \
\langle t,\dots ,t\rangle_{n,d} \ .
\end{equation}
It is a formal function of $t\in H^*_G(X)$ with values in the coefficient ring
$\L=H^*_G(pt,{\Bbb C}[[q]]$). Here ${\Bbb C}[[q]]$ stands for some completion of
the group algebra ${\Bbb C}[H_2(X,{\Bbb Z})]$ so that the symbol
$q^d=q^{d_1}_1\dots q^{d_k}_k$ represents the class $(d_1,\dots ,d_k)$ in the
lattice ${\Bbb Z}^k=H_2(X,{\Bbb Z})$ of 2-cycles. Fundamental classes of
holomorphic curves in $X$ have non-negative coordinates with respect to a basis
of Kahler forms so that the formal power series algebra ${\Bbb C}[[q]]$ can be
taken on the role of the completion. Strictly speaking, the formula \ref{eq3.1}
defines $F$ up to a quadratic polynomial of $t$ since the spaces $X_{n,0}$ are
defined only for $n\geq 3$.
Denote $\nabla$ the gradient operator with respect to the equivariant
intersection pairing $\langle \ , \ \rangle$ on $H^*_G(X)$. It is defined over the
field of fractions of $H^*_G(pt)$. The WDVV equation is an identity between
third directional derivatives of $F$. It says that
\begin{equation}
\label{eq3.2}
\langle\nabla F_{\a,\b},\nabla F_{\gamma,\d}\rangle
\end{equation}
{\em is totally symmetric with respect to permutations of the four directions}
$\a,\b,\gamma,\d\in H^*_G(X)$.
\begin{thm} The WDVV equation holds for ample $X$.
\end{thm}
Notice that
\begin{equation}
\label{eq3.3}
\langle \nabla\int_X a\wedge t,\nabla\int_X b\wedge t\rangle=
\langle a,b\rangle
\end{equation}
has geometrical meaning of integration $\int_{\Delta\subset X\times X}
a\otimes b$ over the diagonal in $X\times X$.
In order to prove the non-equivariant version of the WDVV equation one
interprets the 4-point correlators $A_{1234}\langle\a,\b,\gamma,\d \rangle_{4,d}$
which are totally symmetric in $\a,\b,\gamma,\d$
as integrals over the fibers $\Gamma_\l$ of the contraction map
$\pi: X_{4,d}\to {\cal M} _4 ={\Bbb C} P^1$ and specializes the cross-ratio $\l$ to
$0, 1$ or $\infty$.
Stable maps corresponding to generic points of, say, $\Gamma_0$ are glued from
a pair
of maps $f_1: ({\Bbb C} P^1,p_1,p_2,a_1)\to X$,
$f_2:({\Bbb C} P^1,p_3,p_4,a_2)\to X$ of degrees $d_1+d_2=d$ with
three marked points each, satisfying the diagonal condition $f_1(a_1)=
f_2(a_2)$. One can treat such a pair as a point in $X_{3,d_1}\times
X_{d_2,3}$ situated on the inverse image $\Gamma_{d_1,d_2}$ of the diagonal
$\Delta \subset X\times X$ under the evaluation map $e_3\times e_3$.
The {\em glueing map} $\displaystyle{\sqcup_{d_1+d_2=d}} \Gamma_{d_1,d_2}\to\Gamma$ is
an isomorphism at generic points and therefore it identifies the analytic
fundamental cycles. This means that
\[
A_{1234}\langle\a,\b,\gamma\,d\rangle_{4,d} =\sum_{d_1+d_2=d}
\langle\nabla\langle\a,\b,t \rangle_{3,d_1},
\nabla\langle\gamma,\d,t \rangle_{3,d_2}\rangle \ .
\]
The above argument applies to the correlators
$A_{1234}\langle\a,\b,\gamma\,d, t,\dots ,t\rangle_{n+4,d}$ with additional marked points
and gives rise to
\begin{equation}
\label{eq3.4}
\langle\nabla F_{\a,\b},\nabla F_{\gamma,\d}\rangle =\sum^\infty_{n=0} \
\frac{1}{n!} \ \sum_d \ q^d
A_{1234}\langle\a,\b,\gamma,\d,t\dots t\rangle_{4+k,d}
\end{equation}
which is totally symmetric in $\a,\b,\gamma,\d$.
Ampleness of $X$ is used here only in order to make sure that the moduli
spaces
have fundamental cycles and that the diagonal in $X\times X$ consists of
regular values of the evaluation map $e_3\times e_3$.
In order to justify the above argument in the equivariant situation, it is
convenient to reduce the problem to the case of tori actions (using maximal
torus of $G$) and use the De Rham version of equivariant cohomology theory.
For a torus $G=(S^1)^r$ acting on a manifold $M$ the equivariant De Rham
complex
\cite{AB} consists of $G$-invariant differential forms on $M$ with
coefficients in
${\Bbb C} [u_1,\dots ,u_r]=H^*_G(pt)$, provided with the coboundary operator
$d_G=d+\sum^r_{s=1} u_si_s$ where $i_s$ are the operators of contraction
by the vector fields generating the action. Applying the ordinary Stokes
formula to $G$-invariant forms and $G$-{\em invariant} chains
we obtain well-defined functionals $H^*_G(M)\to{\Bbb C} [u]$ of
{\em integration over invariant cycles}. The identity \ref{eq3.4} follows
now from the obvious $G$-invariance of the analytic varieties $\Gamma_{\l }$,
$\Gamma$, $\Gamma_{d_1,d_2}$.
A similar argument proves a composition rule that reduces computation of all
equivariant correlators $A_I\langle ... \rangle$ to that of $\langle ... \rangle $.
\section{Ample vector bundles}
\label{sec4}
The following construction was designed by M. Kontsevich in order to extend the
domain of applications of WDVV theory to complete intersections in ample
Kahler manifolds.
Let $E\to X$ be an {\em ample} bundle, that is, a holomorphic vector
bundle spanned by its holomorphic sections. For stable
$f: (C,p)\to X$ (of degree $d$, with $n$ marked points), the spaces
$H^0(C,f^*E)$ form a holomorphic vector bundle $E_{n,d}$ over the moduli
space $X_{n,d}$.
If $f$ is glued from $f_1$ and $f_2$ as in the proof of
(\ref{eq3.4}), then $H^0(C,f^*E)= \operatorname{ker} (H^0(C_1,f^*_1E)
\oplus H^0(C_2,f^*_2E)
\stackrel{e_1-e_2}{\longrightarrow} e_1^*E=e_2^*E)$ where
$e_i: H^0(C_i,f^*_iE)\to e^*_iE$ is defined by evaluation of
sections at the marked point $a_i$.
This allows one to construct a solution $F$ to the WDVV equation
starting with an ample $G$-equivariant bundle $E$ and any invertible
$G$-equivariant multiplicative characteristic class $c$
(the total Chern class would be a good example).
Redefine
\[ \langle a,b\rangle \ := \ \int_X a\wedge b\wedge c(E) \ , \]
\[ \langle t,\dots ,t\rangle_{n,d} \ := \ \int_{X_{n,d}} e^*_1t \wedge
\dots e^*_n t \wedge c(E_{n,d}) \ , \]
\[ F(t) \ = \ \sum^\infty_{n=0} \frac{1}{k!} \sum_d q^d
\langle t,\dots ,t\rangle_{n,d} \ .\]
Then $\langle\nabla F_{\a,\b},\nabla F_{\gamma\d}\rangle$
{\em is totally symmetric in} $\a,\b,\gamma,\d$.
This construction bears a limit procedure from the total Chern class to the
(equivariant) Euler class, and the limit of $F$ corresponds to the GW-theory
on the submanifold $X'\subset X$ defined by an (equivariant) holomorphic
section $s$ of the bundle $E$. Namely, the section $s$ induces a
holomorphic section $s_{n,d}$ of $E_{n,d}$, and the (equivariant) Euler class
$Euler \ (E_{n,d})$ becomes represented by some cycle $[X'_{n,d}]$ situated
in the zero locus $X'_{n,d}:=s^{-1}_{n,d}(0)$ of the induced section. The
variety $X'_{n,d}$ consists of stable maps to $X'$, the Euler cycle
$[X'_{n,d}]$
plays the role of the virtual fundamental cycle in $X'_{n,d}$, and the
correlators
\[ \langle t, ... ,t \rangle _{n,d} := \int _{X_{n,d}} \ e_1^*t ... e_n^*t \ Euler \
(E_{n,d}) = \int _{[X'_{n,d}]} \ e_1^*t ... e_n^*t \]
are correlators of GW-theory on $X'$ between the classes $t$ which come from
the ambient space $X$.
\bigskip
Another solution of the $WDVV$-equation can be obtained from the bundles
$E'_{d,k}:= H^1(C,f^*E^*)$: one should put
$\langle a,b\rangle := \int_X a\wedge b\wedge c^{-1}(E^*)$,
$\langle t,\dots ,t\rangle_{n,d} = \int_{X_{n,d}}
e^*_1t\wedge\dots\wedge e^*_n t\wedge c(E'_{n,d})$ for $d\neq 0$
and $\langle t,\dots ,t\rangle_{n,0}=\int_{X_{n,0}} e^*_1t\wedge
\dots\wedge e^*_n t\wedge c(E^*)$.
\section{Quantum cohomology}
\label{sec5}
One interprets the WDVV equation as the associativity identity for the
{\em quantum cup-product} on $H^*_G(X)$ defined by
\[
\langle \a*\b,\gamma\rangle = F_{\a,\b,\gamma} \ .
\]
It is a deformation of the ordinary cup-product
(with $t$ and $q$ in the role of parameters) in the category
of (skew)-commutative algebras {\em with unity}:
\begin{equation}
\label{eq5.1}
\langle \a*1,\gamma\rangle = \langle\a,\gamma\rangle \ .
\end{equation}
Indeed, the push-forward by the forgetful map
$\pi : X_{n,d}\to X_{n-1,d}$ (with $n\geq 3$) sends
$1\in H^*_G(X_{n,d})$ to 0 unless $d=0$ and $k=3$ in which case
$X_{n,d}=X$ and $X_{n-1,d}$ is not defined.
The structure usually referred in the literature as the
{\em quantum cohomology algebra} corresponds to the restriction of the
deformation $*_{t,q}$ to $t=0$. As it is shown in \cite{KM}, in many
cases the function $F$ can be recovered on the basis of WDVV-equation from
the structural constants
$F_{\a,\b,\gamma}|_{t=0}(q)$ of the quantum cohomology algebra due to the following
symmetry of the potential $F$. Let $u \in H^2_G(X)$ and $(u_1,...,u_k)$ be
its coordinates with respect to the basis of the lattice
$({\Bbb Z} ^k)^*=H^2(X)=H^2_G(X)/H^2_G(pt)$ (so that $u_i \in H^*_G(pt)$).
Then
\begin{equation}
\label{eq5.2}
(F_{\a , \b ,\gamma })_u=\sum_{i=1}^k u_iq_i \partial F_{\a , \b ,\gamma }/\partial q_i
\quad \forall \a,\b,\gamma \in H^*_G(X) \ .
\end{equation}
The identity (\ref{eq5.2}) follows from the obvious push-forward formula
$\pi_*u=d_i u_i$.
The symmetry $(6)$ can be interpreted in the way that the quantum deformation
of the cup-product restricted to $t=0$ is equivalent to the deformation with
$q=1$ and $t$ restricted to the $2$-nd cohomology of $X$ (in the equivariant
setting it is better however to keep both parameters in place ---
see Sections $7, 8$).
\bigskip
In this paper, we will use the term {\em quantum cup-product} for the entire
$(q,t)$-deformation and {\em reserve the name {\em quantum cohomology algebra}
for the restriction of the quantum cup-product to $t=0$}.
I have heard some complaints about such terminology because it allows many
authors to compute quantum cohomology algebras without even mentioning the
deformation in $t$-directions. There are some indications however that (despite
the equivalence $(6)$) the $q$-deformation has a somewhat different nature than
the $t$-deformation. The loop space approach \cite{HG1} and our computations
in Sections $9$ -- $11$ seem to emphasize this distinction.
\bigskip
Quantum cohomology algebras of the classical flag manifolds have been
computed in
\cite{GK}, \cite{K} on the basis of several
conjectures about properties of $U_n$-{\em equivariant}
quantum cohomology (see also \cite{AS} where a slightly different formalism
was applied). The answer (in terms of generators and relations)
for complete flag manifolds $U_n/T^n$ is strikingly related to
conservation laws of Toda lattices. The conjectures named
in \cite{GK} the {\em product, induction} and {\em restriction}
properties and describing behavior of equivariant quantum cohomology under
some natural constructions, were motivated by interpretation of the
quantum cohomology in terms of Floer theory on the loop space $LX$.
Although a construction of the equivariant counterpart of the Floer - Morse
theory on $LX$ remains an open problem, the three conjectured axioms can
be justified within the Gromov-Witten theory. This was done by B.Kim \cite{K2}.
The induction and
restriction properties follow directly from definitions given in this paper
and hold for the entire quantum deformation (not only at $t = 0$), while the
``product'' axiom that the $G_1 \times G_2$-equivariant quantum cohomology
algebra of $X_1\times X_2$ is the tensor product of the $G_i$-equivariant
quantum cohomology algebras of the factors $X_i$ has been verified in
\cite{K2} for ample manifolds.
Behavior of the quantum cup-product at
$t\neq 0$ under the Cartesian product operation on the target manifolds is
much more complicated than the operation of the tensor product.
\section{Floer theory and $D$-modules}
\label{sec6}
Structural constants $\langle \a * \b,\gamma\rangle$ of the quantum cup-product
are derivatives $\partial_{\b}F_{\a,\gamma}$ of the same function. This allows to
interpret the WDVV-equation as integrability condition of some connections
$\nabla_\hbar $ on the tangent bundle $T_H$ of the space $H = H^*(X,{\Bbb C})$.
Namely, put $t = \sum t_{\a}p_{\a}$ where $p_1 = 1,p_2,\dots,p_N$ is a basis
in $H$ and define
\[
\nabla_\hbar = \hbar d - \sum(p_{\a}*)dt_{\a} \wedge:
\Omega^0(T_H) \to \Omega^1(T_H)
\]
where $p_{\a}*$ are operators of quantum multiplication by $p_{\a}$. Then
$\nabla_\hbar \circ \nabla_\hbar = 0$ {\em for each value of the parameter}
$\hbar$.
Notice that the integrability condition that reads ``the system of differential
equations $\hbar \partial_{\a}I = p_{\a} * I$ has solutions $I \in
\Omega^0(TH)$'' is actually obtained as a somewhat combinatorial statement
(the WDVV-equation) about coefficients of the series $F$.
In \cite{HG1}, \cite{HG} we attempted to improve this unsatisfactory
explanation of the integrability property by
describing a direct geometrical meaning of the solutions $I$ in terms of
$S^1$-equivariant Floer theory on the loop space $LX$. Briefly, the universal
covering $\widetilde{LX}$ carries the action of the covering transformation
lattice $\pi_2(X)$ with generators $q_1,\dots,q_k$ and the $S^1$-action by
rotation of loops which preserves natural symplectic forms $\o_1,\dots,\o_k$ on
$LX$ and thus defines corresponding Hamiltonians $H_1,\dots,H_k$ on
$\widetilde{LX}$ (the action functionals). The Duistermaat--Heckman forms
$w_i+ \hbar H_i$ (here $\hbar $ is the generator of $H_{S^1}^*(pt)$)
are equivariantly closed,
and operators $p_i$ of exterior multiplication by these forms have the
following Heisenberg commutation relations with the covering transformations:
\[
p_iq_j - q_jp_i = \hbar q_j\d_{ij}.
\]
Conjecturally, this provides $S^1$-equivariant Floer cohomology of
$\widetilde{LX}$ with a ${{\cal D}}$-module structure which is equivalent to the
above system of differential equations (restricted to $t = 0$, $q \ne 0$) and
reduces to the quantum cohomology algebra in the quasi-classical limit
$\hbar = 0$ (see \cite{HG1, GK}).
In this section we describe solutions to $\nabla_\hbar I = 0$ by imitating the
$S^1$-equivariant Floer theory (which is still to be constructed) within the
framework of Gromov--Witten theory. This construction turns out to be crucial
in our proof in Section 11 of the mirror conjecture for Calabi-Yau projective
complete intersections.
\bigskip
One may think of the graph of an algebraic
loop ${\Bbb C}P^1{\backslash}\{0,\infty\} \to X$ of degree $d$ as of a stable
map ${\Bbb C}P^1 \to X \times {\Bbb C}P^1$ of bidegree $(d,1)$. Our starting
point consists in interpretation of the moduli space $L_d(X)$ of such stable
maps as a degree-$d$ approximation to $\widetilde{LX}$ and application of
equivariant Gromov--Witten theory to the action of $S^1$ on the second factor
${\Bbb C}P^1$ with the fixed points $\{0,\infty\}$.
In the theorem below we assume $X$ to be ample. It is natural to expect
however that the theorem holds true whenever the non-equivariant Gromov--Witten
theory works for $X$ since the $S^1$-action is non-trivial only on the factor
${\Bbb C}P^1$ which is ample on its own.
Let $\langle \ ,\ \rangle$ be the Poincare pairing on $H = H^*(X,{\Bbb C})$.
The equivariant cohomology algebra $H_{S^1}^*(X \times {\Bbb C}P^1)$ is
isomorphic to $H \otimes_{\Bbb C} {\Bbb C}[p,\hbar ]/(p(p-\hbar ))$ with the
$S^1$-equivariant pairing
\[
(\varphi,\psi) = \frac {1}{2\pi i} \oint \frac {\langle \varphi,\psi\rangle
dp}{p(p-\hbar )}.
\]
Localization in $\hbar $ allows to introduce coordinates $\varphi = tp/\hbar +
\tau(\hbar -p)/\hbar $, $\tau,t \in H$, diagonalizing the equivariant pairing:
\[
((\tau,t),(\tau ',t')) = \frac {\langle t,t' \rangle - \langle
\tau,\tau '\rangle }{\hbar }.
\]
The potential
${{\cal F}}(t,\tau,h,q,q_0)$ satisfying the
equivariant WDVV-equation for $X \times {\Bbb C}P^1$ expands as
\[
{{\cal F}} = {{\cal F}}^{(0)} + q_0{{\cal F}}^{(1)} + q_0^2{{\cal F}}^{(2)}\dots
\]
according to contributions of stable maps of degree $0,1,2,\dots$ with respect
to the second factor. Denote $F = F(t,q)$ the potential (\ref{eq3.1}) of
the $GW$-theory for $X$.
\begin{thm}
{\em (a)} ${{\cal F}}^{(0)} = (F(t,q)-F(\tau,q))/\hbar $.
{\em (b)} The matrix $(\Phi_{\a\b}) :=
(\partial^2{{\cal F}}^{(1)}/\partial\tau_{\a}\partial t_{\b})$ is a fundamental
solution of $\nabla_{\pm \hbar }I = 0$:
\[
-\hbar \frac {\partial}{\partial \tau_{\gamma}} \Phi = p_{\gamma}(t)\Phi \ ,\]
\[ \hbar \frac {\partial}{\partial t_{\gamma}} \Phi^* = p_{\gamma}(\tau)\Phi^* \ ,\]
where $p_{\gamma} = (p_{\a}^{\b})_{\gamma}$, $\gamma = 1,\dots,N$, are matrices of quantum
multiplication by $p_1 = 1,\dots,p_N$, and $\Phi^*$ is transposed to $\Phi$.
\end{thm}
\noindent{\em Proof.}\ Moduli spaces of bidegree-$(d,0)$ stable maps to $X \times {\Bbb C}P^1$
coincide with $X_{n,d} \times {\Bbb C}P^1$. This implies (a) and shows that
the WDVV-equation for ${\cal F}$ {\em modulo} $q_0$ follows from the WDVV-equation
for $F$. Part (b) follows now directly from the WDVV-equation for ${\cal F}$
{\em modulo} $q_0^2$ and from
\[
\hbar \frac{\partial }{\partial t_1} \Phi _{ \a \b } = \Phi _{\a \b } =
-\hbar \frac{\partial }{\partial \tau _1 } \Phi _{\a \b }
\]
due to (\ref{eq5.1}) and (\ref{eq5.2}). Here $\partial /\partial t_1, \partial/\partial \t _1$ are
derivatives in the direction $1\in H^*(X)$ of the identity components of
$t$ and $\t$ respectively.
\bigskip
The following corollary is obtained by expressing equivariant correlators
$\Phi_{\a\b}$ via localization of equivariant cohomology classes of moduli
spaces $L_d(X)$ to fixed points of the $S^1$ action.
Define
\begin{equation}
\label{eq6.1}
\psi_{\a\b} = \sum_{n=0}^{\infty} \frac {1}{n!} \sum_d q^d\langle \frac
{p_{\b}}{\hbar +c} ,t,\dots,t,p_{\a}\rangle_{n+2,d}
\end{equation}
where $c$ is the first Chern class of the line bundle over $X_{k,d}$ introduced
in Section $1$ as ``the universal tangent line at the first marked
point'', and $\langle \frac {p_{\b}}{\hbar +c}, p_{\a}\rangle_{2,0} :=
\langle p_{\a}, p_{\b}\rangle $.
\begin{cor}
$\hbar \partial \psi /\partial t_{\gamma } = p_{\gamma}(t)\psi$,
i.e., the matrix $\psi$ is (another) fundamental solution of
$\nabla_{\hbar } I = 0$.
\end{cor}
\noindent{\em Proof.}\ A fixed point in $L_d(X)$ is represented by a stable map $C_0 \cup {\Bbb
C}P^1 \cup C_{\infty} \to X \times {\Bbb C}P^1$ where $\varphi_i: C_l \to X
\times \{i\}$ are stable maps of degrees $d_1 + d_2 = d$ connected by a
``constant loop'' ${\Bbb C}P^1 \stackrel{\simeq}{\rightarrow} \{x\} \times
{\Bbb C}P^1$. Thus components of $L_d(X)^{S^1}$ can be identified with
submanifolds in $X_{d_1,k_1+1}^{(0)} \times X_{d_2,k_2+1}^{(\infty)}$ defined
by the diagonal constraint $e_1(\varphi_0) = e_1(\varphi_{\infty})$, with
$\hbar^2(\hbar + c(0))(\hbar - c(\infty))$ to be the
equivariant Euler class of the normal bundle. This gives rise to
\begin{equation}
\label{eq6.2}
\hbar^2\Phi_{\a\b} = \sum_{\varepsilon,\varepsilon '}
\psi_{\a\varepsilon}(\tau ,-\hbar )\ \eta^{\varepsilon\e'} \psi_{\varepsilon '\b}(t,h)
\end{equation}
where $\sum \eta^{\varepsilon\e'}p_{\varepsilon} \otimes p_{\varepsilon'}$ is the coordinate expression
of the diagonal cohomology class of $X\subset X \times X$.
\bigskip
We give here several reformulations which will be convenient for computation
of quantum cohomology algebras in Sections $9$ -- $10$.
Consider the specialization of the connection $\nabla _{\hbar }$ to the
parameter subspace corresponding to the deformation of the
quantum cup-product along the $2$-nd cohomology (this is accomplished by
putting first $t=0$ and then replacing $q^d$ by $\exp (d,t)$ where
$t=(t_1,...,t_k)$
represents coordinates on $H^2(X)$ with respect to the basis
$p^{(1)},..., p^{k)} \in H^2(X)$ . In this new setting put
\[ s_{\a, \b} := \sum _d e^{dt} \langle p_{\b } \frac{e^{pt/\hbar }}{\hbar + c},
p_{\a }\rangle \]
where $pt:= \sum p^{(i)}t_i$.
{\bf Corollary 6.3.} {\em The matrix $(s_{\a ,\b }(t))$ is a fundamental
solution to
\[ \nabla _{\hbar } \ s = 0 : \ \hbar \frac{\partial}{\partial t_i} \ s = p^{(i)} * s .\] }
{\em Proof.} One should combine Corollary $6.2$ with iterative applications
of the following generalized symmetries $(5),(6)$:
\[ \langle f(c), ..., 1 \rangle _{n+1, d} = \langle \frac{f(0)-f(c)}{c},... \rangle _{n,d}
, \]
\[ \langle f(c), ..., p^{(i)} \rangle _{n+1, d} = d_i \langle f(c),... \rangle _{n,d} +
\langle p^{(i)} \frac{f(0)-f(c)}{c} , ... \rangle _{n,d} .\]
Here $f(c)$ is a function of $c$ with values in $H^*(X)$.
The symmetries are easily verified on the basis of the following geometrical
properties of universal tangent lines:
(i) Consider the push-forward along the map $\pi : X_{n+1,d}\to
X_{n,d} $ (forgetting the last marked point). It is easy to see that
the difference $\pi ^*(c)-c$ between the Chern class of the universal
tangent line at the $1$-st marked point and the pull-back of its counterpart
from $X_{n,d}$ is represented by the fundamental cycle of the section
$i: X_{n,d}\to X_{n+1,d}$ defined by the first marked point.
(ii) $i^* (c) = c$.
In particular $\pi _* (1/(\hbar +c)) = 1/[\hbar (\hbar +c)] $.
{\bf Corollary 6.4.} {\em Consider the functions
\[ s_{\b }:= \sum _d e^{dt }
\langle p_{\b } \frac{e^{pt/\hbar }}{\hbar + c}, 1\rangle _{2,d} \ .\]
Let $P(\hbar \partial/\partial t, \exp t, \hbar)$ be a differential operator
annihilating simultaneously all the functions $s_{\a }$. Then
the relation $P(p^{(1)},..., p^{(k)} ,q_1,...,q_k , 0)=0$ holds
in the quantum cohomology algebra of $X$ (we assume here that $P$
depends only on non-negative powers of $\hbar $).}
The functions $s_{\b }$ --- ``the first components of the vector-solutions
$s_{\a, \b}$'' --- generate a left $\cal{D}$-module with the {\em solution}
locally constant sheaf described by the flat connection $\nabla _{\hbar }$ and
with the characteristic variety isomorphic to the spectrum of the quantum
cohomology algebra.
\bigskip
All results of this section extend literally to the equivariant setting and/or
to the generalization to ample vector bundles described in Section $4$. We will
apply them in this extended form in Sections $9$--$11$.
\bigskip
{\em Remarks.}
$ 1$) The universal formula (\ref{eq6.1}) for solutions of $\nabla_\hbar I = 0$
was perhaps discovered independently by several authors. I first learned this
formula from R. Dijkgraaf. It can also be found in \cite{Db} in the
{\em axiomatic} context of conformal topological field theory. One can prove
it directly from a recursion relation (in the spirit of WDVV-equation) for
so-called {\em gravitational descendents} --- correlators involving the
first Chern classes of the universal tangent lines (or, in a slightly
different manner, by describing explicitly the divisor in $X_{n,d}$
representing $c$).
Our approach provides an interpretation of (\ref{eq6.1}) in
terms of fixed point localization in equivariant cohomology.
$ 2$) One can generalize our theorem to bundles over ${\Bbb C}P^1$ with the
fiber $X$. This seems to indicate that a straightforward ``open-string''
approach to $S^1$-equivariant Floer theory on $\widetilde{LX}$ would be more
powerful and flexible than the approximation by Gromov--Witten theory on $X
\times {\Bbb C}P^1$ described above.
$3$) Although the theorem provides a geometrical interpretation of
solutions to $\nabla_\hbar I = 0$,
it does not eliminate the combinatorial
nature of the integrability condition. Indeed, the theorem is deduced from an
equivariant WDVV-equation which in its turn can be interpreted as an
integrability condition. Of course one can explain it using the $S^1 \times
S^1$-equivariant WDVV-equation on $(X \times {\Bbb C}P^1) \times {\Bbb C}P^1$,
etc. It would be interesting to find out whether this process converges.
\section{Frobenius structures}
\label{sec7}
In \cite{Db}, B. Dubrovin studied geometrical structures defined by
solutions of WDVV-equations on the parameter space and reduced classification
of
generic solutions to the classification of trajectories of some Euler-like
non-autonomous Hamiltonian systems on $so_N^*$. We show here how this approach
to equivariant Gromov--Witten theory yields analogous Hamiltonian systems on
the
affine Lie coalgebras $\widehat{so}_N^*$.
The quantum cup-product on $H = H_G^*(X)$ considered as an $N$-dimensional
vector space over the field of fractions $K$ of the algebra $H_G^*(pt)$ defines
a formal {\em Frobenius structure} on $H$. The structure consists of the
following ingredients.
\begin{enumerate}
\item A symmetric $K$-bilinear inner product $\langle\ ,\ \rangle$,
\item a (formal) function $F: H \to K$ whose third directional derivatives
$\langle
a*b,c\rangle := F_{a,b,c}$ provide tangent spaces $T_tH$ with the Frobenius
algebra structure (i.e. associative commutative multiplication $*$
satisfying $\langle a*b, c\rangle = \langle a, b*c \rangle $).
\item The constant vector field $1\!\!1$ of unities of the algebras
$(T_tH, *)$ whose flow
preserves the multiplication $*$ (i.e. $L_{1\!\!1} (*)=0$).
\item Grading: In the non-equivariant case axiomatically studied
by B. Dubrovin it can be
described by the {\em Euler} vector field $E$, such that
the tensor fields $1\!\!1$, $*$ and $\langle \ , \ \rangle$ are homogeneous (i.e.
are eigen-vectors of the Lie derivative $L_E$) of degrees $-1, 1$ and $D$
respectively (where $D = \operatorname{dim} _{\Bbb C} X$ in the models arising from
the GW-theory). In the equivariant GW-theory this grading axiom should be
slightly modified since the grading of the
structural ring $H_G^*(pt)$ is non-trivial and thus the natural Euler
operator $L_E$ is ${\Bbb C}$-linear but not $K$-linear.
\end{enumerate}
\bigskip
The fact that the multiplication $*$ is defined on tangent vectors to $H$
means that the algebra $(\Omega ^0(T_H), *)$ can be naturally considered as
the algebra $K[L]$ of regular functions on some subvariety $L\subset T^*H$
in the cotangent bundle.
A point $t\in H$ is called {\em
semi-simple} if the algebra $(T_{t}H,*)$ is semi-simple, that is if $L \cap
T_{t}^*H$ consists of $N$ linearly independent points.
Flatness of the connection (defined on $T_H$)
\begin{equation}
\label{eq7.1}
\nabla_\hbar = \hbar d - \sum_{\a} p_{\a} * dt_{\a}
\end{equation}
implies \cite{GK} that $L$ is a Lagrangian submanifold in $T^*H$ near a
semisimple $t$. Following \cite{Db}, introduce local {\em canonical
coordinates} $(u_1,\dots,u_N)$ such that the sections $(du_1,\dots,du_N)$ of
$T^*H$ are the $N$ branches of $L$ near $t$, and transform the connections
$\nabla_\hbar$ to these local coordinates and to a (suitably normalized)
basis $f_1,\dots,f_N$ of vector field on $H$ diagonalizing the $*$-product.
The result of this transformation can be described as follows.
(a) The basis $\{f_i\}$ can be normalized in a way that in the transformed
form
\begin{equation}
\label{eq7.2}
\nabla _{\hbar } = \hbar d - \hbar A^1 \wedge - D^1\wedge
\end{equation}
of the connection $\nabla _{\hbar} $ with
$D^1 = \operatorname{diag}(du_1,\dots,du_N)$, and $A_{ij} =
V_{ij}(u) d(u_i-u_j)/(u_i-u_j)$ for all $i \ne j$,
we will have additionally $A_{ii} =0 \ \forall i$.
(b) The vector field $1\!\!1$ in the canonical coordinates assumes the form
$\sum_k \partial_k$ where $\partial_k := \partial/\partial u_k$ are
the canonical idempotents of the $*$-product:
\begin{equation}
\label{eq7.3}
\partial_i * \partial_j = \d_{ij}\partial_j.
\end{equation}
(c) The (remaining part of the) integrability condition $\nabla_{\hbar} ^2 =0$
reads
$d(A^1) = A^1 \wedge A^1$ or
\begin{equation}
\label{eq7.4}
\partial_i\phi_{\a}^j = \phi_{\a}^i V_{ij}/(u_i-u_j),\ i \ne j,
\end{equation}
where $(\phi_{\a}^j)$ is the transition matrix, $\partial/\partial t_{\a} =
\sum_i \phi_{\a}^i f_i$; it can be reformulated as compatibility of the
PDE system (\ref{eq7.4}) for $(\phi_{\a}^j)$ completed by
\begin{equation}
\label{eq7.5}
\sum_k \partial_k\phi_{\a}^j = 0.
\end{equation}
(d) The Frobenius property $\langle a*b,c\rangle = \langle a,b*c\rangle$ of the
$*$-product shows that the diagonalizing basis $\{f_i\}$ is orthogonal, that
its normalization by $\langle f_i,f_j \rangle = \d_{ij}$ obeys $A_{ii} = 0$
and, additionally, implies anti-symmetricity $A_{ij} = -A_{ji}$, or
\begin{equation}
\label{eq7.6}
V_{ij} = -V_{ji}.
\end{equation}
The presence of the grading axiom (4) of Frobenius structures over $K = {\Bbb
C}$ allows B.Dubrovin to describe anti-symmetric matrices
$V = (V_{ij}) \in so_N^*$
satisfying the integrability conditions (\ref{eq7.4}) and (\ref{eq7.5}) in
{\em quasi-homogeneous} canonical coordinates (i.e. $L_E u_i = u_i$ so that
$E = \sum u_k\partial_k$)
as trajectories of $N$ commuting non-autonomous Hamiltonian
systems (see \cite{Db}):
\[
\partial_iV = \{H_i,V\}
\]
where the Poisson-commuting non-autonomous quadratic Hamiltonians $H_i$
on $so_N^*$ are given by
\[
H_i = \frac {1}{2} \sum_{j \ne i} \frac {V_{ij}V_{ji}}{u_i-u_j} .
\]
Consider now the following model modification of the grading axiom:
$K = {\Bbb C}[[\lambda^{\pm 1}]]$, $\operatorname{deg} \lambda = 1$. In
quasi-homogeneous canonical coordinates $(u_1,\dots,u_N,\lambda)$ the Euler
vector field takes then on
\begin{equation}
\label{eq7.7}
L_E = \sum_k u_k\partial_k + \lambda\partial_{\lambda}.
\end{equation}
Introduce the connection operator
\[
{\Bbb V} = \lambda\partial_{\lambda} - V \in \widehat{so}_N^*
\]
and the qudratic Hamiltonians on the Poisson manifold $\widehat{so}_N^*$
\begin{equation}
\label{eq7.8}
{{\cal H}}_i({\Bbb V}) = \oint H_i(V) \frac {d\lambda}{\lambda} .
\end{equation}
\begin{prop}
The Hamiltonians ${{\cal H}}_1,\dots,{{\cal H}}_N$ are in involution. The operator
${\Bbb V}$ of a Frobenius manifold over $K$ satisfies the non-autonomous system
of Hamiltonian equations
\begin{equation}
\label{eq7.9}
\partial_i{\Bbb V} = \{{{\cal H}}_i,{\Bbb V}\},\ i = 1,\dots,N.
\end{equation}
The columns $\phi_{\a} = (\phi_{\a}^i)$ of the transition matrix are
eigen-functions of the connection operator ${\Bbb V}$:
\begin{equation}
\label{eq7.10}
{\Bbb V}\phi_{\a} := (\lambda\partial_{\lambda} - V)\phi_{\a} = \left( \frac
{n}{2} - \operatorname{deg} t_{\a} + 1\right) \phi_{\a}.
\end{equation}
\end{prop}
\noindent{\em Proof.}\ It can be obtained by a straightforward calculation quite analogous
to that in \cite{Db}.
\bigskip
In our real life the model equations (\ref{eq7.7}--\ref{eq7.10}) describe the
structure of Frobenius manifolds {\em over each semi-simple orbit of the
grading Euler field in the ground parameter space. This parameter space is the
spectrum of the coefficient algebra} $H_G^*(pt,{\Bbb C}) \otimes {\Bbb
C}[q^{\pm 1}_1,\dots,q^{\pm 1}_k]$ (its field of fractions can be taken on the
role of the ground field $K$). An orbit of the Euler vector field in this
parameter space is
semi-simple if the corresponding ${\Bbb C}$-Frobenius algebras are
semi-simple.
The equations (\ref{eq7.7}--\ref{eq7.10}) over semi-simple Euler orbits should
be complemented by the additional symmetries (\ref{eq5.2}).
In the next section we will show how the canonical coordinates of the axiomatic
theory of Frobenius structures emerge from localization formulas in equivariant
Gromov--Witten theory.
\section{Fixed point localization}
\label{sec8}
We consider here the case of a circle $T^1$ acting by Killing transformations
on a compact Kahler manifold $X$ with {\em isolated} fixed points only.
The case of tori actions with isolated fixed points requires only slight
modification of notations which we leave to the reader. Our
results are rigorous for ample $X$ (which includes homogeneous Kahher spaces of
compact Lie group and their maximal tori) while applications to general toric
manifolds (which are typically not ample) yet to be justified.
It is the Borel localization theorem that reduces computations in
torus-equivariant
cohomology to computations near fixed points. Let $\{ p_{\a}\} ,
\a=1,...,N$, be the fixed points of the action. We will denote with the same
symbols $p_{\a}$ the equivariant cohomology class of $X$ which restricts
to $1\in H^*_T(p_{\a})$ at $p_{\a}$ and to $0$ at all the other fixed points.
These classes
are well-defined over the field of fractions ${\Bbb C} (\l )$
of the coefficient ring $H^*_T(pt)={\Bbb C} [\l ]$ and form the basis
of canonical idempotents in the semi-simple algebra $H^*_T(X,{\Bbb C} (\l )$.
The equivariant Poincare pairing reduces to $\langle p_{\a},p_{\b}\rangle =
\d _{\a,\b}/e_{\b}$ where $e_{\a}\in {\Bbb C} [\l]$ is the equivariant Euler class
of the normal ``bundle'' $T_{p_{\a}}X\to p_{\a}$ to the fixed point.
The results described below apply to the setting of Section $4$ of a manifold
$X$ provided with an ample vector bundle in which case $e_{\a}$'s should be
modified accordingly.
\bigskip
The same localization theorem reduces computation of GW-invariants to that
near the fixed point set (orbifold) in the moduli spaces $X_{n,d}$.
A fixed point in
the moduli space is represented by a stable map to $X$ of a (typically
reducible) curve $C$ such that each component of $C$ is mapped to (the
closure of) an orbit of the complexified action $T_{{\Bbb C} }:X$. Any such an
orbit is either one of the fixed points $p_{\a}$ or isomorphic to $({\Bbb C}-0)$
connecting two distinct fixed points $p_{\a}$ and $p_{\b}$ corresponding to
$0$ and $\infty $. Respectively, there are two types of components of $C$:
\noindent (i) Each component of $C$ which carries $3$ or more special points
must be mapped to one of the fixed points $p_{\a}$.
\noindent (ii) All other components are multiple covers $z\mapsto z^d$ of the
non-constant orbits, and their special points may correspond only to $z=0$ or
$\infty$.
The {\em combinatorial structure}
of such a stable map can be described by a tree
whose edges correspond to {\em chains} of components of type (ii) and should
be labeled by the total degree of this chain as a curve in $X$, and vertices
correspond to the ends of the chains. The ends may carry $0$ or $1$ marked
point, or correspond to a (tree of) type-(i) components with $1$ or more
marked points and should be labeled by the indices of these marked points
and by the target point $p_{\a}$.
{\em The fixed stable maps with different combinatorial structure belong to
different connected components of the fixed point orbifold in $X_{n,d}$.}
The results below are based on the observation that a stable map with the
first $k\geq 3$ marked points in a {\em given} generic configuration
(i.e. with the given generic value of the contraction map $X_{n.d}\to
{\cal M} _{0,k}$) must have in an irreducible component $C_0$ in the underlying
curve $C$ which contains this given configuration of $k$ {\em special} points,
(so that the corresponding first $k$ marked points are located on the branches
outgoing these special points of $C_0$). The cause is hidden in the definition
of the contraction map (see \cite{Kn, BM}).
We will call the component $C_0$ {\em special}.
The observation applied to a fixed stable map of the circle action allows to
subdivide all fixed point components in $X_{3+n,d}$ into $N$ {\em types}
$p_i$ according
to the fixed points $p_i$ where the special component is mapped to. We
introduce the superscript notation $(...)^i$ for the contribution (via
Borel's localization formulas) of type-$p_i$ components into various
equivariant correlators. For example,
\[ F^i_{\a \b \gamma } =\sum _{n}\frac{1}{n!} \sum_d q^d
\langle p_{\a },p_{\b },p_{\gamma },t,...,t\rangle _{3+n,d}^i \]
where $t=\sum _{\a=1}^N t_{\a}p_{\a}$ is the general class in
$H^*_T(X,{\Bbb C}(\l))$, so that $F_{\a\b\gamma}=\sum_i F_{\a\b\gamma}^i$.
We introduce also the notations
\begin{itemize}
\item $\Psi _{\a\b}^i$ --- for contributions to $e_iF_{\a\b i}^i$ of those
fixed points which have the third marked point situated directly on the
special component $C_0$ (it is convenient here to introduce the normalizing
factor $e_i\in H^*_T(pt)$, the Euler class of the normal ``bundle'' to the
fixed point $p_i$ in $X$);
\item $\Psi _{\a}^i := \Psi _{\a 1\!\!1}^i=\sum_\b \Psi _{\a\b}^i$;
\item $D_{\a}^i$ --- for contributions to $e_iF_{\a i i}$ of those
fixed points which have the second and third marked points situated directly
on the special component;
\item $\Delta ^i$ --- for contributions to $e_iF_{i i i}$ of those fixed
points which have the first three marked points situated directly on the
special component;
\item $u_i=t_i+$ contributions to $e_iF_{i i}$ of all those fixed point
components in $X_{2+n,d}$ for which the first two marked points belong to
the same vertex of the tree describing the combinatorial structure.
\end{itemize}
The correlators $u_i$ can be also interpreted as contributions to the
{\em genus-$1$} equivariant correlators
\[ \sum _n\frac{1}{n!}\sum_d q^d (t,...,t)_{n,d} \]
with {\em given} complex structure of the elliptic curve of those $T$-invariant
classes which map the (only) genus-$1$ component of the curve $C$ to the
fixed point $p_i$.
\bigskip
{\bf Theorem 8.1.} {\em (a) The functions $u_1(t),...,u_N(t)$ are the canonical
coordinates of the Frobenius structure on $H^*_G(X,{\Bbb C} (\l))$.
(b) The functions $D_{\a}^i(t)$ are eigen-values of the quantum multiplication
by $p_{\a}$: $du_i=\sum _{\a} D_{\a }^i dt_{\a} $.
(c) The transition matrix $(\Psi_{\a}^i)$ provides simultaneous diagonalization
of the quantum cup product: $F_{\a\b\gamma}^i=\Psi_{\a}^i D_{\b}^i \Psi_{\gamma}^i$
and obeys the following orthogonality relations:
\[ \sum_i \Psi_{\a}^i\Psi_{\b}^i=\d _{\a\b}/e_{\b},
\sum _{\a} \Psi_{\a}^i\Psi_{\a}^j =\d _{ij} \ .\]
(d) The Euclidean structure on the cotangent bundle of the Frobenius
manifold (defined by the equivariant intersection pairing in $H^*_T(X)$)
in the canonical coordinates $u_i$ takes on $\langle du_i,du_j \rangle =
(\Delta^i)^2\d_{ij}e_j$ and additionally
\[ (\Delta ^i)^{-1}=\sum_{\a}\Psi_{\a}^i, \ \Psi_{\a}^i=\frac{D_{\a}^i}{\Delta^i},
\ \Psi_{\a\b}^i=\frac{D_{\a}^i D_{\b}^i}{\Delta^i} \ .\]
}
\bigskip
{\em Proof.} We first apply the localization
formula
\[ A_{1234}\langle ... \rangle _{n,d}=\sum _i A_{1234}\langle ...\rangle _{n,d}^i \]
to the $4$-point equivariant correlators with the fixed cross-ratio $z$
of the $4$ marked points and only after this specialize the cross-ratio
to $0$,$1$ or $\infty$. This gives rise to the {\em local} WDVV-identities
\[ \Psi_{\a\b}^i\Psi_{\gamma\d}^i \ \text{\em is totally symmetric in} \ \
{\a,\b,\gamma,\d} \]
which is independent of the global WDVV-equation.
When combined with the global identities
\[ A_{1234}\langle 1\!\!1,p_{\a},p_{\b},p_{\gamma} \rangle _{n,d}
=\langle p_{\a},p_{\b},p_{\gamma} \rangle _{n,d} \]
they yield the orthogonality relation $\sum_i \Psi _{\a}^i\Psi_{\b}^i=
\d_{\a\b}/e_{\b}$ and localization formulas
\[ F_{\a\b\gamma}=\sum_i\Psi_{\a\b}^i\Psi_{\gamma}^i \]
for the structural constants of the quantum cup-product.
A similar argument with $>4$-point correlators $A_{12345...}\langle ...\rangle ^i$
proves the diagonalization
\[ \langle p_{\a}*p_{\b},p_{\gamma}\rangle =
\sum_i \Psi_{\a}^iD_{\b}^i\Psi_{\gamma}^i/e_i \ ,\]
\[ \langle p_{\a}*p_{\b}*p_{\gamma},p_{\d}\rangle =
\sum _i \Psi_{\a}^i D_{\b}^i D_{\gamma}^i \Psi_{\d}^i/e_i \ \]
and the identities
\[ \Psi_{\a\b}^i=\Psi_{\a}^iD_{\b}^i, \
(\Delta ^i)^{-1}=\sum_{\a } \Psi_{\a}^i \ .\]
Finally, the identity $du_i=\sum_{\a} D_{\a}^idt_{\a} $ follows
directly from the definition of $u_i$ and implies that $u_1,...,u_N$
are the canonical coordinates of the Frobenius structure.
\section{ Projective complete intersections}
\label{sec9}
We are going to describe explicitly solutions of the
differential equations arising from quantum cohomology of projective complete
intersections. Lex $X$ be such a non-singular complete intersection in
$Y:={\Bbb C} P^n $ given by $r$ equations of the degrees $(l_1,...,l_r)$. If
$l_1+...+l_r=n+1$ then $X$ is a Calabi-Yau manifold and its quantum cohomology
is described by the mirror conjecture. In this and the next sections we study
respectively the cases $l_1+...+l_r < n$ and $l_1+...+l_r=n$
when the $1$-st Chern class of $X$ is still positive. In the case $l_1+...+l_r
> n+1$ (which from the point of view of enumerative geometry can be considered
as ``less interesting'' for rational curves generically occur only in
finitely many degrees) the ``mirror symmetry'' problem of hypergeometric
interpretation of quantum cohomology differential equations remains open.
Let $E_d$ be the Euler class of the vector bundle over the moduli space
$Y_{2,d}$ of genus $0$ degree $d$ stable maps $\phi : (C, x_0,x_1) \to {\Bbb C} P^n$
with two marked points, with the fiber $H^0(C, \phi ^* H^{l_1}\oplus ...
\oplus \phi^* H^{l_r})$ where $H^l$ is the $l$-th tensor power
of the hyperplane line bundle over ${\Bbb C} P^n$.
Consider the class
$$S_d(\hbar ):=\frac{1}{\hbar + c_1^{(0)}}\ E_d \ \in \ H^*(Y_{2,d})$$
where $c_1^{(0)}$ is the $1$-st Chern class of
the ``universal tangent line at the marked point $x_0$ '',
and $e_0, e_1$ are the evaluation maps. Due to the factor $E_d$ this
class represents the push forward along $X_{2,d} \to Y_{2,d}$ of the class
$1 /(\hbar + c_1^{(0)}) \ \in H^*(X_{2,d})$
(by the very construction of $X_{2,d}$ in Section $4$).
In the cohomology algebra ${\Bbb C} [P]/(P^{n+1})$ of ${\Bbb C} P^n$, consider the class
\[ S(t,\hbar):= e^{Pt/\hbar }\sum _{d=0}^{\infty } e^{dt}
(e_0)_* (S_d(\hbar )) \]
where $(e_0)_*$ represents the push-forward along the evaluation map
(and for $d=0$, when $Y_{2,d}$ is not defined, we take
$Euler \ (\oplus _j H^{\otimes l_j}) $ on the role of $(e_0)_* S_0$).
Considered as a function of $t$, $S$ is a curve in $H^*({\Bbb C} P^n)$ whose
components are solutions of the differential equation we are concerned
about. Indeed, according to Section $6$ a similar sum represents
the solutions of the quantum cohomology differential equation for $X$, and
$S$ is just the push-forward of that sum from $H^*(X)$ to $H^*(Y)$.
(Strictly speaking $S$ carries information only about correlators between
those classes which come from the ambient projective space; also if
$X$ is a surface $\operatorname{rk} H_2(X)$ can be greater than $1$ and $S$ mixes information
about the curves of different degrees in $X$ when they have the same degree
in $Y$.)
\bigskip
{\bf Theorem 9.1.} {\em Suppose that $l_1+...+l_r < n$. Then}
\[ S = e^{Pt/\hbar } \sum _{d=0}^{\infty } e^{dt}
\frac{ \Pi _{m=0}^{dl_1} (l_1P + m\hbar) ... \Pi _{m=0}^{dl_r} (l_rP+m\hbar)}
{\Pi _{m=1}^d (P+m\hbar )^{n+1} } .\]
The formula coincides with those in \cite{HG1, HG} (found by analysis
of toric compactifications of spaces of maps ${\Bbb C} P^1 \to {\Bbb C} P^n$) for
solutions of differential equations in $S^1$-equivariant cohomology of the
loop space.
\bigskip
{\bf Corollary 9.2.} (see \cite{HG1, HG})
{\em The components $s:=\langle P^i, S\rangle , i=0,...,n-r, $
of $S$ form a basis of solutions to the linear differential equation}
\[ (\hbar \frac{d}{dt})^{n+1-r}\ s \ = \
e^t \Pi _{j=1}^r \ l_j\ \Pi _{m=1}^{l_j-1}\
\hbar (l_j\frac{d}{dt} + 1) \ s \ .\]
This implies (combine \cite{HG} with \cite{BVS}) that the solutions
have an integral representation of the form
\[ \int _{\gamma ^{n-r}\subset X'_t} \ e^{(u_0+...+u_n)/\hbar } \ \
\frac{du_0\wedge ... \wedge du_n}{dF_0\wedge dF_1\wedge ... \wedge dF_r} \]
where
\[ F_0=u_0...u_n,\ F_1=u_1+...u_{l_1},\ F_2=u_{l_1+1}+...+u_{l_1+l_2},
... , \ F_r=u_{l_1+...+l_{r-1}+1}+...+u_{l_1+...+u_{l_r}} \]
and the ``mirror manifolds'' $X'_t$ are described by the equations
\[ X'_t=\{ (u_o,...,u_n)\ |\ F_0(u)=e^t,\ F_1(u)=1, ... ,\ F_r(u)=1 \} .\]
This proves for $X$ the mirror conjecture in the form suggested in
\cite{HG}.
\bigskip
{\bf Corollary 9.3.} {\em If $\operatorname{dim} _{{\Bbb C} } X \neq 2$ the cohomology class
$p$ of hyperplane section
satisfies in the quantum cohomology of $X$ the relation}
\[ p^{n+1-r}=l_1^{l_1}...l_r^{l_r} q p^{l_1+...+l_r-r} .\]
When $X$ is a surface the same relation holds true in the quotient of the
quantum cohomology algebra which takes in account only degrees of curves in
the ambient ${\Bbb C} P^n$ (we leave to figure out a precise description of this
quotient to the reader; quadrics ${\Bbb C} P^1\times {\Bbb C} P^1$ in ${\Bbb C} P^3$ provide
a good example:
$(p_1+p_2)^3 = 4q(p_1+p_2) \ \mod p_1^2=q=p_2^2 $.)
This corollary is consistent with the result of A. Beauville \cite{Bea}
describing quantum cohomology of complete intersections with
$\sum l_j \leq n+1 -\sum (l_j-1) $ and with results of M. Jinzenji \cite{J} on
quantum cohomology of projective hypersurfaces ($r=1$) with $l_1<n$.
\bigskip
{\bf Corollary 9.4.} {\em The number of degree $d$ holomorphic maps
${\Bbb C} P^1 \to X^{n-r} \subset {\Bbb C} P^n$,
which send $0$ and $\infty $ to two given
cycles and send $n+1-r$ given points in ${\Bbb C} P^1$ to
$n+1-r$ given generic hyperplane sections, is equal to
$l_1^{l_1}...l_r^{l_r}$ times
the number of degree $d-1$ maps which send $0$ and $\infty $ to the same cycles
and $l_1+...+l_r-r$ given points --- to $l_1+...+l_r-r$ given
hyperplane sections.}
This is the enumerative meaning of Corollary $9.3$; of course in this
formulation numerous general position reservations are assumed.
{\em Control examples.} $1.$ $l_1=...=l_r=1$: The above formulas for quantum
cohomology
and for solutions of the differential equations in the case of a hyperplane
section give rise to the same formulas with $n:=n-1$.
$2$. $n=5, r=1, l=2$: $X$ is the Plucker embedding of the grassmanian
$Gr_{4,2}$. Its quantum cohomology algebra is described by the relations
$c_1^3=2c_1 c_2,\ c_2^2-c_2c_1^2+q=0$ between the Chern classes of the
tautological plane bundle. For the $1$-st Chern class $p=-c_1$ of the
determinant line bundle we deduce the relation $p^5=4pq$ prescribed by
Corollary $9.3$.
\bigskip
We will deduce Theorem $9.1$ from its equivariant generalization.
Consider the space ${\Bbb C} ^{n+1}$ provided with the standard action of the
$(n+1)$-dimensional torus $T$. The equivariant cohomology algebra
of ${\Bbb C} ^{n+1}$ coincides with the algebra of characteristic classes
$H^*(BT^{n+1}) = {\Bbb C} [\l _0,...,\l _n]$. The equivariant cohomology algebra
of the projective space $({\Bbb C} ^{n+1} - 0)/{\Bbb C} ^{\times} $ in these notations
is identified with ${\Bbb C} [p,\l ]/ ((p-\l_0)...(p-\l_n))$ and the push-forward
$H_T^*({\Bbb C} P^n)\to H_T^*(pt)$ is given by the residue formula
\[ f(p,\l ) \mapsto \frac{1}{2\pi i} \oint
\ \frac{f(p,\l )dp}{(p-\l_0)...(p-\l_n)} \ .\]
Here $-p$ can be considered as the equivariant $1$-st Chern class of the
Hopf line bundle provided with the natural lifting of the torus action.
We will use $\phi _i:=\Pi _{j\neq i} (p-\l _j), \ i=0,...,n, $ as a basis in
$H_T^*({\Bbb C} P^n)$.
Consider the $T$-equivariant vector bundle $\oplus _{j=1}^r H^{\otimes l_j}$
and provide it with the fiberwise action of the additional $r$-dimensional
torus $T'$. The equivariant Euler class of this bundle is equal to
$ (l_1p-\l '_1)...(l_rp-\l '_r)$ where ${\Bbb C} [\l ']=H^*(BT')$.
Introduce the equivariant counterpart $S'$ of the class $S$ in the
$T\times T'$-equivariant cohomology of ${\Bbb C} P^n$. This means that we use
the equivariant class $p$ instead of $P$ and replace the Euler classes
$E_d$ and $c_1^{(0)}$ by their equivariant partners.
\bigskip
{\bf Theorem 9.5.} {Let $l_1+...+l_r < n$. Then}
\[ S'= e^{pt/\hbar} \sum _{d=0}^{\infty }
e^{dt} \frac{ \Pi _0^{dl_1} (l_1p-\l '_1 +m\hbar) ...\Pi _0^{dl_r}
(l_rp-\l '_r +m\hbar)}{\Pi _1^d (p-\l _0+m\hbar) ... \Pi _1^d (p-\l _n+m\hbar)}
\ .\]
Theorem $9.1$ follows from Theorem $9.5$ by putting $\l =0, \l '=0$ which
corresponds to passing from equivariant to non-equivariant cohomology.
The vector-function $S'$ satisfies the differential equation
\[ \Pi _{i=0}^r (\hbar \frac{d}{dt} -\l _i) \ S' =
e^t \Pi _{m=1}^{l_1} (l_1\hbar \frac{d}{dt} - \l '_1 + m\hbar ) ...
\Pi _{m=1}^{l_r} (l_r\hbar \frac{d}{dt} - \l '_r + m\hbar ) \ S' .\]
\bigskip
We intend to prove Theorem $9.5$ by means of localization of $S'$ to the
fixed point set of the torus $T$ action on the moduli spaces $Y_{2,d}$.
As it is shown in \cite{Kn}, all correlators of the equivariant theory on
${\Bbb C} P^n$ are computable at least in principle, and in practice the
computation reduces to a recursive procedure which can be understood as a
summation over trees and can be also formulated as a non-linear fixed point
(or critical point) problem. We will see below that in the case of correlators
$\langle \phi _i, S'\rangle $ certain reasons of a somewhat geometrical character
cause numerous cancellations between trees so that the recursive procedure
reduces to a ``summation over chains'' and respectively to a {\em linear}
recurrence equation. The formula of Theorem $9.5$ is simply the solution to
this equation.
\bigskip
In the proof of Theorem $9.5$ below we write down all formulas for for $r=1$
(it serves the case when $X$ is a hypersurface in ${\Bbb C} P^n$ of degree $l < n$).
The proof for $r > 1$ differs only by longer product formulas.
\bigskip
Let us abbreviate $c_1^{(0)}$ as $c$,
denote $E'_d$ the equivariant Euler class of the vector bundle over $Y_{2,d}$
whose fiber over the point $\psi : (C, x_0, x_1) \to Y={\Bbb C} P^n$ consists of
holomorphic sections of the bundle $ \psi ^* (H^l)$
{\em vanishing at} $x_0$, and introduce the following equivariant correlator:
\[ Z_i :=\sum _{d=0}^{\infty } q^d
\int _{Y_{2,d}} e_0^* (\phi _i) \frac{1}{\hbar + c} E'_d .\]
We have
\[ \langle \phi _i, S' \rangle = e^{\l _i t/\hbar } (l\l _i - \l ') (Z_i | _{q=e^t})
\]
\medskip
{\bf Proposition 9.6.}
\[ Z_i = 1 + \sum _{d>0} (\frac{q}{\hbar ^{n+1-l}})^d
\int _{Y_{2,d}} \frac{(-c)^{(n+1-l)d-1}}{1+c/\hbar} \ E'_d \
e_0^*(\phi _i) \ . \]
{\em Proof.} We have just dropped first several terms in the geometrical
series $1/(\hbar + c)$ since their degree added with the degrees of other
factors in the integral over $Y_{2,d}$ is still less than the dimension
of $Y_{2,d}$. It is important here that all the equivariant classes involved
including $\phi _i$ are defined in the equivariant cohomology over
${\Bbb C} [\l ,\l ']$ without any localization.
\medskip
It is a half of the geometrical argument mentioned above. The other half
comes from the description of the fixed point set in $Y_{2,d}$ given
in \cite{K}.
Consider a fixed point of the torus $T$ action on $Y_{2,d}$. It is represented
by a holomorphic map of a possibly reducible curve with complicated
combinatorial structure and with two marked points on some components. Each
component carrying $3$ or more special points is mapped to one of the $n+1$
fixed points of $T$ on ${\Bbb C} P^n$, and the other components are mapped
(with some multiplicity) onto the lines joining the fixed points and
connect the point-mapped components in a tree-like manner.
In the Borel localization formula for $\int e_0^*(\phi _i) ...$ the fixed
point will have zero contribution unless the marked point $x_0$ is mapped
to the $i$-th fixed point in ${\Bbb C} P^n$ (since $\phi _i$ has zero localizations
at all other fixed points.
Consider a fixed point curve $C$ whose marked point $x_0$ is indeed mapped to
the $i$-th fixed point in ${\Bbb C} P^n$. There are two options
(i) the marked point $x_0$ is situated on an irreducible component of $C$
mapped with some degree $d'$ onto the line joining the $i$-th
fixed point with the $j$-th fixed point in ${\Bbb C} P^n$ with $i\neq j$;
(ii) the marked point $x_0$ is situated on a component of $C$ mapped to
the $i$-th fixed point and carrying two or more other special points.
Consider first the option (ii) and the contribution of such a connected
component of the fixed point set in $Y_{2,d}$ to the Borel localization
formula for $\int c^{(n+1-l)d-1} ...$. The connected component itself
is the (product of the) Deligne-Mumford configuration space of, say, $s+1$
special points: the marked point $x_0$, may be the marked point $x_1$, and
respectively $s-1$ or $s$ endpoints of other components of $C$ mapped
onto the lines outgoing the $i$-th fixed point in ${\Bbb C} P^n$.
{\bf Lemma 9.7.} {\em The type (ii) fixed point component
in $Y_{2,d}$ has zero contribution to the Borel localization formula for}
$\int _{Y_{2,d}} c^{(n+1-l)d-1} ... $
{\em Proof.} Restriction of the class $c$ from $Y_{2,d}$ to the type (ii)
fixed point component coincides with the $1$-st Chern class of the line
bundle on the Deligne-Mumford factor ${\cal M} _{0, s+1}$ of the component
defined as ``the universal tangent line as the marked point $x_0$''
and is thus
nilpotent in the cohomology of the component. Since the number of straight
lines in a curve of degree $d$ does not exceed $d$ we find that the dimension
$s-2$ of the factor ${\cal M} _{0,s+1}$ is less than $d$ which in its turn does
not exceed $(n+1-l)d-1$ for $d>0$ (because we assumed that $n+1-l\geq 2$).
\medskip
Consider now the option (i). The irreducible component $C'$ of the curve
$C=C'\cup C''$
carrying the marked point $x_0$ is mapped with the multiplicity $d' \leq d$
onto the line joining $i$-th fixed point in ${\Bbb C} P^n$ with the $j$-th one
{\em while the remaining part $C''\to {\Bbb C} P^n$ of the map represents a fixed
point in $Y_{2,d-d'}$}. Moreover, the normal space to the fixed point
component at the type (i) point (the equivariant Euler class of the normal
bundle occurs in the denominator of the Borel localization formula) is the
sum of (a) such a space $N''$ for $C'' \to {\Bbb C} P^n$, (b) the space $N'$
of holomorphic vector fields along the map $C'\to {\Bbb C} P^n$ vanishing at
the fixed point $j$ factorized by infinitesimal reparametrizations of $C'$,
(c) the tensor product $L$ of the tangent lines to
$C'$ and $C''$ at their intersection point. Since the space $V$ of holomorphic
sections of $H^l$ restricted to $C$ (and vanishing at $x_0$) admits a similar
decomposition $V'\oplus V''$, we arrive to the following linear recursion
relation for $Z_i$.
{\bf Proposition 9.8.} {\em Put $z_i(Q,\hbar ):=
Z_i(\hbar ^{(n+1-l)}Q, \hbar )$. Then}
\[ z_i(Q, \hbar) = 1 + \sum _{j\neq i} \sum _{d'>0} Q^{d'} Coeff \ _i^j (d')
\ z_j(Q, (\l _j-\l _i)/d' ) \]
{\em where}
\[ Coeff \ _i^j(d')=
\frac{[(\l_j -\l_i)/d]^{(n+1-l)d'-1}}{1+(\l _i-\l _j)/d' \hbar}
\frac{Euler \ (V')}{Euler \ (N')} \ \phi _i |_{p=\l _i} \ . \]
{\em Proof.} Here $(\l _i-\l_j)/d' $ is the localization of $c$, and the key
point is that the equivariant Chern class of the line bundle $L$ over
$Y_{2,d-d'}$ is what we would denote $\hbar + c$ {\em for the moduli space}
$Y_{2,d-d'}$ but with $\hbar = (\l _j -\l _i)/d'$. This is how the recursion
for the correlators $z_i$ becomes possible. The rest is straightforward.
{\em Remark.} Our reduction to the linear
recursion relation can be interpreted in the following more geometrical way:
contributions
of all non-isolated fixed points cancel out with some explicit part of the
contribution from {\em isolated} fixed points; the latter are represented by
chains of multiple covers of straight lines connecting the two marked points.
\bigskip
Let us write down explicitly the factor $Coeff \ _i^j(d)$
from Proposition $9.8$
(compare with \cite{Kn}). $Coeff \ _i^j (d) =$
\[ \frac{
\Pi _{m=1}^{ld} (l\l _i -\l' + m(\l _j-\l_i)/d) [(\l_j-\l_i)/d]^{(n+1-l)d-1}
}{
d (1+(\l_i -\l_j)/\hbar d)
\Pi _{\a =0}^n \ _{m=1}^d \ _{(\a ,m)\neq (j, d)} (\l_i-\l_{\a}+m(\l_j-\l_i)/d)
} = \]
(here the product in the numerator is $Euler (V')$, the
denominator --- it is essentially $Euler (N')$ where however the cancellation
with $\phi _i |_{p=\l _i}$ is taken care of --- has been computed using the
exact sequence $0 \to {\Bbb C} \to {\Bbb C} ^{n+1} \otimes H \to T_Y \to 0$ of vector
bundles over $Y= {\Bbb C} P^n$, and the ``hard-to-explain'' extra-factor $d$
is due to the orbifold structure of the moduli spaces (the $d$-multiple map
of $C'$ onto the $(ij)$-line in ${\Bbb C} P^n$ has a discrete symmetry of order
$d$)
\[ = \frac{1}{[(\l_i-\l_j)/\hbar + d]}
\frac{\Pi _{m=1}^{ld} (\frac{(\l_i-\l')d}{\l_j-\l_i} +m)}
{\Pi _{\a =0}^n\ _{m=1}^d \ _{(\a ,m)\neq (j,d)}
(\frac{(\l_i-\l_{\a })d}{\l_j-\l_i} + m) } \ . \]
Now it is easy to check
{\bf Proposition 9.9.} {\em The correlators $z_i(Q, 1/\o )$ are power series
$\sum C_i(d) Q^d $ in $Q$ with coefficients $C_i(d)$ which are {\em reduced}
rational functions of $\o $ with poles of the order $\leq 1$ at
$\o = d'/(\l_j-\l_i)$ with $d'=1,...,d$. The correlators $z_i$ are uniquely
determined by these properties, the recursion relations of Proposition $9.8$
and the initial condition $C_i(0)=1$.}
\bigskip
The proof of Theorem $9.5$ is completed by the following
{\bf Proposition 9.10.} {\em The series}
\[ z_i=\sum _{d=0}^{\infty } Q^d
\frac{\Pi _{m=1}^{ld} ((l\l_i -\l')\o +m)}
{d! \Pi _{\a \neq i} \Pi _{m=1}^d ((\l_i -\l_{\a })\o +m)} \]
{\em satisfy all the conditions of Proposition $9.10$.}
{\em Proof.} The recursion relation is deduced by the decomposition of
the rational functions of $\o $ into the sum of simple fractions (or,
equivalently, from the Lagrange interpolation formula for each numerator
through its values at the roots of the corresponding denominator).
\section{Complete intersections with $l_1+...+l_r=n$}
\label{sec10}
Let $X\subset Y={\Bbb C} P^n$ be a non-singular complete intersection given by
equations of degrees $(l_1,...,l_r)$ with $l_1+...+l_r=n$. There are only
two points where our proof of Theorem $9.1$ would fail for such $X$. One of
them is the Lagrange interpolation formula in the proof of Proposition $9.10$.
Namely, the rational functions of $\o $ there are not reduced ---
the degree $dl$ of the numerator {\em is equal} to the degree $dn$ of the
corresponding denominator. The other one is Lemma $9.7$. Namely, we have the
following lemma instead.
{\bf Lemma 10.1.}
{\em The type (ii) fixed point component in $Y_{2,d}$ makes zero
contribution via Borel localization formulas to $\int _{Y_{2,d}} c^{d-1}...$
{\em unless} it consists of maps $(C'\cup C'', x_0,x_1)\to Y$ where $C'$ is
mapped to a fixed point in ${\Bbb C} P^n$ and carries both marked points, and
$C''$ is a disjoint union of $d$ irreducible components (intersecting $C'$ at
$d$ special points) mapped (each with multiplicity $1$) onto straight lines
outgoing the fixed point. All type (ii) components make zero contribution
to $\int _{Y_{2,d}} c^d ...$.}
Let us modify the results of Section $9$ accordingly. As we will see,
the LHS in Theorem $9.5$ is now only {\em proportional} to the RHS, and we will
compute the proportionality coefficient (a series in $q$) directly.
{\bf Proposition 10.2.} {\em Put $z_i(Q,\hbar ):= Z_i(\hbar Q,\hbar )$. Then}
\[ z_i(Q,\hbar) = 1 + \sum _{d>0} Q^d Coeff\ _i(d) +
\sum _{j\neq i} \sum _{d'>0} Q^{d'}
Coeff \ _i^j(d')\ z_j(Q,(\l_j-\l_i)/d') \]
{\em where $Coeff \ _i(d)$ is equal to the contribution of type (ii) fixed
point components to \newline
$\int _{Y_{2,d}} (-c)^{d-1} E_d' \ e_0^*(\phi _i) $, and}
\[ Coeff \ _i^j(d) = \frac{1}{[(\l_i-\l_j)/\hbar +d]}\
\frac{\Pi _{a=1}^r \Pi _{m=1}^{dl_a} (\frac{(l_a\l_i-\l_a')d}{\l_j-\l_i}+m)}
{\Pi _{\a =0}^n \ _{m=1}^d \ _{(\a ,m)\neq (j,d)}
(\frac{(\l_i-\l_{\a })d}{\l_j-\l_i} +m)} \ .\]
{\bf Corollary 10.3.} {\em The correlators $z_i(Q,1/\o )$ are power series
$\sum _d C_i(d) Q^d$ with coefficients
\[ C_i(d)=P_d(\o ,\l ,\l ')/\Pi _{\a }\Pi _{m=1}^d ((\l_i-\l_{\a })\o +m) \]
where $P_d = P_d^0 \o ^{nd} + ... $ is a polynomial in $\o $ of degree $nd$.
The correlators $z_i$ are uniquely determined by these properties, the
recursion relations of Proposition $10.2$ and the initial conditions
\[ \sum _d Coeff \ _i(d) Q^d = \sum _d Q^d \frac{P_d^0}
{d!\Pi _{\a \neq _i} (\l _i-\l_{a })^d} \ .\]}
{\bf Proposition 10.4.} {\em The series
\[ z_i' =\sum _{d=0}^{\infty } Q^d
\frac{\Pi _{a=1}^r \Pi _{m=1}^{l_a d} ((l_a\l_i-\l_a')\o +m)}
{d!\Pi _{\a \neq i}\Pi _{m=1}^d ((\l_i-\l_{\a })\o +m)} \ \]
satisfy the requirements of Corollary $10.3$ with the initial condition
\[ \sum _d Q^d \frac{\Pi _{a=1}^r (l_a\l_i-\l_a')^{l_a d} }
{ d!\Pi _{\a \neq i}
(\l_i-\l_{\a })^d} \ = \
\exp \{ Q \frac{\Pi _a (l_a\l_i -\l_a')^{l_a}}{\Pi _{\a \neq i}
(\l_i-\l_{\a })} \} . \]}
\bigskip
Now let us compute $Coeff \ _i(d)$ using the description of type (ii) fixed
point components given in Lemma $10.1$.
{\bf Proposition 10.5.} {\em Contribution of the type (ii) fixed point
components to \newline
$\sum _d Q^d \int _{Y_{2,d}} (-c)^{d-1} E'_d \ \phi _i $ is}
\[ \exp \{ Q \frac{\Pi _a (l_a\l_i-\l_a')^{l_a}}
{\Pi _{\a \neq i} (\l_i-\l_{\a })} \} \ \exp \{ - Q \ l_1!...l_r! \} \ .\]
{\em Proof.} Each fixed point component described in Lemma $10.1$ is isomorphic
to the Deligne - Mumford configuration space ${\cal M} _{0,d+2}$. Our computation
is based on the following known formula (see for instance \cite{Kn} ) for
correlators between Chern classes of universal tangent lines at the marked
points:
\[ \int _{{\cal M} _{0, k}} \frac{1}{(w_1+c_1^{(1)}) ... (w_k+c_1^{(k)})} =
\frac{(1/w_1+...1/w_k)^{k-3}}{w_1 ... w_k} \ .\]
Consider the type (ii) fixed point component specified by
the following combinatorial structure of stable maps: $d$ degree $1$
irreducible components join the $i$-th fixed point with the fixed points
with indices $j_1,...,j_d$. Using the above formula and describing explicitly
the normal bundle to this component in $Y_{2,d}$ and localization of the Euler
class $E_d'$ we arrive to the following expression for the contribution of this
component to $\int _{Y_{2,d}} (-c)^{d-1} \phi _i E_d'$:
\[ \Pi _{s=1}^d
\frac{\Pi _{a=1}^r \Pi _{m=1}^{l_a}(l_a\l_i-\l_a' +m(\l_{j_s}-\l_i))}
{(\l_i-\l_{j_s}) \Pi _{\a \neq j_s,i} (\l_{j_s}-\l_{\a })} \ . \]
Summation over all type (ii) components in all $Y_{2,d}$ with weights $Q^d$
gives rise to
\[ \exp \{ - Q \sum _{j\neq i}
\frac{\Pi _a \Pi _{m=1}^{l_a} (l_a\l_i-\l_a' +m(p-\l_i))}
{\Pi _{\a \neq j} (p-\l_{\a})} \ |_{p=\l_j} \} \ .\]
The exponent can be understood as a sum of residues at $p\neq \l_i, \infty $
and is thus opposite to the sum
\[ l_1!...l_r! -
\frac{\Pi _a (l_a\l_i-\l_a')^{l_a}}{\Pi _{\a \neq i} (\l_i-\l_{\a })}
\]
of residues at $\infty $ and $\l_i $.
{\bf Corollary 10.6.} $z_i(Q,1/\o )= z_i'(Q,\o )\ \exp (- l_1! ... l_r! Q)$.
{\em Proof.} Multiplication by a function of $Q$ does not destroy the
recursion relation of Proposition $10.2$ but changes the initial condition.
\bigskip
We have proved the following
{\bf Theorem 10.7.} {\em Suppose $l_1+...+l_r=n$. Then}
\[ S' = e^{ (pt - l_1!...l_r! e^t)/\hbar } \
\sum _{d=0}^{\infty } \ e^{dt} \
\frac{\Pi _0^{dl_1} (l_1p-\l_1'+m\hbar) ... \Pi _0^{dl_r} (l_rp-\l_r'+m\hbar)}
{\Pi _1^d (p-\l_0 +m\hbar ) ... \Pi _i^d (p-\l_n +m\hbar )} \ .\]
\[ S = S' |_{\l =0, \l' =0} =
e^{ (Pt - l_1! ... l_r! e^t)/\hbar }
\frac{ \Pi _{j=1}^r \Pi _{m=0}^{dl_j} (l_j P+m\hbar)}
{\Pi _{m=1}^d (P+m\hbar)^{n+1}} \ \ (\text{mod} \ P^{n+1}) \ .\]
{\bf Corollary 10.8.} {\em Let $D=\hbar d/dt + l_1!...l_r! e^t$. Then}
\[ D^{n+1-r} S = l_1...l_r e^t \Pi _{j=1}^r
(l_j D+\hbar)...(l_j D +(l_j-1)\hbar ) \ S .\]
{\bf Corollary 10.9.} {\em In the quantum cohomology algebra of $X$ the class
$p$ of hyperplane sections satisfies the following relation
(with the same reservation in the case $\operatorname{dim} X \leq 2$ as in Corollary $9.3$):}
\[ (p + l_1!...l_r! q)^{n+1-r}=
l_1^{l_1} ... l_r^{l_r} q (p+l_1!...l_r!q)^{n-r} \ .\]
{\em Control examples.} $1$.
$X=pt$ in ${\Bbb C} P^1$ ($n=1, r=1, l=1$). The above relation
takes on $p+q=q$, or $p=0$. Since $P^2=0$, we also find from Theorem 10.7 that
$ S=P \exp (-e^t) \sum _d e^{dt}/d! = P $, or $\langle 1, S\rangle =1 $ as it should
be for the solution of the differential equation $\hbar d/dt \ s = 0$ that
arises from quantum cohomology of the point.
$2$. $X={\Bbb C} P^1$ embedded as a quadric into ${\Bbb C} P^2$ ($n=2, r=1, l=2$).
We get $(p+2q)^2=4q(p+2q)$, or $p^2=4q^2$. Taking into account that $p$ is
twice the generator in $H^2({\Bbb C} P^1)$ and the line in ${\Bbb C} P^1$ has the degree
$2$ in ${\Bbb C} P^2$ we conclude that this is the correct relation in the quantum
cohomology of ${\Bbb C} P^1$. This example was the most confusing for the author:
predictions of the loop space analysis \cite{HG1} appeared totally unreliable
because they gave a wrong answer for the quadric in ${\Bbb C} P^2$. As we see now,
the loop space approach gives correct results if $l_1+...+l_r<n$ and require
``minor'' modification (by the factor $\exp (-l_1!...l_r! q /\hbar )$ )
in the boundary
cases $l_1+...l_r=n$; the quadric on the plane happens to be one of such cases.
$3$. $n=3, r=1, l=3$. We have $(p+6q)^3 = 27q (p+6q)^2 $, or $p^3=9qp^2+6^3q^2p
+ 27\cdot 28 q^3$. In particular, $\langle p * p, p\rangle = 9 q \langle p, p \rangle +
6^3q^2\langle p, 1\rangle + 27\cdot 28 q^3 \langle 1, 1\rangle = 27 q +0+0$ which indicates
that there should exist $27$ discrete lines on a generic cubical surface in
${\Bbb C} P^3$.
\section{Calabi-Yau projective complete intersections}
\label{sec11}
Let $L_d(Y)$ denote, as in Section $6$, the moduli space of stable maps
$\psi: {\Bbb C} P^1 \to {\Bbb C} P^n \times {\Bbb C} P^1$ of bidegree $(d,1)$ with $2$ marked
points mapped to ${\Bbb C} P^n \times \{ 0\} $ and ${\Bbb C} P^n \times \{ \infty \} $
respectively. Let ${\cal E} _d$ denote the equivariant Euler class of the
vector bundle over $L_d(Y)$ with the fiber
$H^0({\Bbb C} P^1, \psi ^*(E))$ where $E$ is the bundle on ${\Bbb C} P^n \times {\Bbb C} P^1$
induced from our ample bundle $\oplus _a H^{l_a}$ by the projection to the
first factor.
Consider the equivariant correlator
\[ \Phi = \int _Y \ Euler^{-1}(E) \ S'(t,\hbar ) \ S'(\t ,-\hbar) \ = \]
\[ = \sum _{d,d'} e^{dt} e^{d'\t} \sum _i \frac{\Pi _a (l_a \l_i -\l'_a)}
{\Pi _{j\neq i} (\l_i-\l_j)}
\int _{Y_{2,d}} E'_d \frac{e^{pt/\hbar } e_0^*(\phi _i)}{\hbar + c} \
\int _{Y_{2,d'}} E'_{d'} \frac{e^{-p\t /\hbar} e_0^*(\phi _i)}{-\hbar +c} .\]
In the case $l_1+...+l_r < n$ it is easy to check using the explicit formula
for $S'$ from Theorem $9.5$ that
\[ \Phi = \frac{1}{2\pi i} \oint e^{p(t-\t )/\hbar} [ \sum _d e^{d\t }
\frac{\Pi _a \Pi _{m=0}^{l_a d} (l_a p-\l'_a -m\hbar)}
{\Pi _{j=0}^n \Pi _{m=0}^d (p-\l_j - m\hbar)}] \ dp. \]
This is an equivariant version of a formula found in \cite{HG1} in the context
of loop spaces and toric compactifications of spaces of rational maps. Namely,
consider the projective space $L'_d$ of $(n+1)$-tuples of polynomials in
one variable of degree $\leq d$ each, up to a scalar factor (notice that
$L'_d$ has the same dimension $d(n+1)+n$ as $L_d$). It inherits the
component-wise action of the torus $T^{n+1}$ and the action of
$S^1$ by the rotation
of the variable (``rotation of loops''). Integration over the equivariant
fundamental cycle in $L'_d$ is given by the residue formula
\[ f(p,\l , \hbar) \mapsto \frac{1}{2\pi i}\oint \frac{ f dp }
{\Pi _{j=0}^n \Pi _{m=0}^d (p -\l_j - m\hbar ) } .\]
Consider the equivariant vector bundle over $L'_d$ such that substitution
of the $(n+1)$ polynomials into $r$ (invariant) homogeneous equations
in ${\Bbb C} P^n$ of degrees $l_1, ..., l_r$ produces a section of this bundle.
The equivariant Euler class of the bundle is
\[ {\cal E} '_d = \Pi _{a=1}^r \Pi _{m=0}^{l_a d} (l_a p -\l'_a - m\hbar ) .\]
The formula for $\Phi $ indicates that there should exist a close relation
between the spaces $L_d$ and $L'_d$. This relation is described in the
following lemma whose proof will be given in the end of this Section.
{\bf The Main Lemma.} {\em There exists a natural
$S^1\times T^{n+1}$-equivariant map
$\mu : L_d\to L'_d$. Denote $-p$ the equivariant $1$-st Chern class of the Hopf
bundle over $L'_d$ induced by $\mu $ to $L_d$. Then}
\[ \Phi (t,\t) =
\sum _d e^{d\t } \int _{L_d} e^{p(t-\t )/\hbar} {\cal E} _d .\]
Define $\Phi ' (q, z, \hbar ) := \Phi | t=\t +z\hbar, q=e^{\t}$. (By the way
the limit of the series $\Phi '$ at $\hbar =0$ has the topological meaning
of what is called in \cite{GK} the {\em generating volume function}, and the
meaning of this limit procedure in terms of differential equations satisfied
by $\Phi $ is the {\em adiabatic approximation}.)
{\bf Corollary 11.1.} $\Phi '(q,z):= \sum_d q^d \int _{L_d} e^{pz} {\cal E} _d
=$
\[ = \frac{1}{2\pi i} \oint e^{pz} \sum _d \frac{ E_d(p,\l, \l',\hbar )}
{\Pi _{j=0}^n \Pi _{m=0}^d (p-\l_j -m\hbar)} dp \]
{\em where $E_d = \mu _* ({\cal E} _d)$ is a {\em polynomial} (of degree
$<(n+1)d$) of all its variables.}
{\em Proof.} The integrals $E^{(k)} = \int _{L_d} p^k {\cal E} _d, \ k=0,...,
\operatorname{dim} L'_d$, which determine the push-forward $\mu _* ({\cal E})$
are polynomials in $(\l, \l', \hbar )$. The matrix
$\int _{L'_d} p^{i+\operatorname{dim} L'_d - j}$ is triangular with all eigenvalues equal
to $1$. This means that there exists a unique polynomial in $p$ with
coefficients {\em polynomial in} $(\l, \l', \hbar)$ which represents
the push-forward with any given polynomials $E^{(k)}(\l ,\l', \hbar)$.
The last argument also proves
{\bf Proposition 11.2.}{ \em Suppose that a series
\[ s= \sum _d q^d \frac{ P_d(p,\l,\l',\hbar)}
{ \Pi _j \Pi _{m =0}^d (p-\l_j -m\hbar)} \]
with coefficients $P_d$ which are polynomials of $p$ of degree $\leq \operatorname{dim} L_d$
has the property that for every $k=0,1,2,...$ the $q$-series $\oint s p^k dp $
has polynomial coefficients in $(\l, \l', \hbar)$. Then the coefficients of
all $P_d$ are polynomials of $(\l,\l',\hbar)$, and vice versa.}
\bigskip
The coefficient $E_d(p,\l,\l',\hbar)$ in the series $\Phi '$ has the total
degree $(l_1+...+l_r)d+r$ according to the dimension of the vector bundle
whose Euler class it represents. Consider the following operations with the
series $\Phi $:
(i) multiplication by a series of $e^t$ and / or $e^{\t }$;
(ii) simultaneous change of variables $t\mapsto t+f(e^t),
\t \mapsto \t +f(e^{\t} )$.
(iii) multiplication by $\exp [C (f(e^t)-f(e^{\t }))/\hbar ]$
(here the factor $C$ should be a linear function of $(\l, \l')$ in order
to obey homogeneity).
{\bf Proposition 11.3.} {\em The property of the series $\Psi $ to generate
polynomial coefficients $E_d(p,\l,\l' \hbar)$ is invariant with respect
to the operations (i),(ii),(iii).}
{\em Proof.} The polynomiality property of coefficients in $\Phi '$ is
equivalent, due to Proposition $11.2$, to the fact that for all $k$ the
$q$-series $(\partial /\partial z)^k |_{z=0} \Phi ' $ has polynomial coefficients.
Multiplication by a series of $q$ does not change this property, which proves
the invariance with respect to multiplication by functions of $e^{\t }$.
The roles of $t$ and $\t $ can be interchanged by the substitutions
$p \mapsto p+\hbar d, \hbar \mapsto -\hbar $ in each summand of $\Phi $.
This proves the invariance with respect to multiplication by functions
of $e^t$.
The operation (ii) transforms $\Phi '$ to
\[ \frac{1}{2\pi i} \sum _d q^d e^{d f(q)} \oint \
\exp \{ p\frac{z\hbar + f(qe^{z\hbar})-f(q)}{\hbar } \} \
\frac{E_d(p)}{\Pi _j \Pi _m (p-\l_j -m\hbar)} \ dp .\]
Since the exponent is in fact divisible by $\hbar $, the derivatives in $z$
at $z=0$ still have polynomial coefficients. This proves the invariance
with respect to (ii). The case of the operation (iii) is analogous.
\bigskip
We are going to use the above polynomiality and invariance
properties of the correlator $\Phi $ in order to describe quantum cohomology
of {\em Calabi-Yau} complete intersections in ${\Bbb C} P^n$ (in which case
$l_1+...+l_r = n+1$). We will use this polynomiality in conjunction
with recursion relations based on
the fixed point analysis of Sections $9, 10$. The result can be roughly
formulated in the following way: the hypergeometric functions of Theorem $9.5$
in the case $l_1+...+l_r =n+1$ can be transformed to the correlators $S'$ by
the operations (i),(ii),(iii). Notice that in the Calabi -- Yau case
all our formulas are homogeneous with the grading $\deg q = 0,
\deg p =\deg \hbar =\deg \l =\deg \l' =1, \deg z =-1$. In particular the
transformations (i)--(iii) also preserve the degree $\operatorname{dim} L_d$ of the
coefficients $E_d$ in $\Phi '$. In the ``positive'' case $l_1+...+l_r\leq n$
where $\deg q = n+1 -\sum l_a > 0$ the transformations (i)--(iii) in fact
increase degrees of the coefficients $E_d$ and are ``not allowed''. The only
exception is the operation (iii) with $f(q)=\text{const} \ q$ in the case
$l_1+...+l_r=n$ when $\deg q =1$. In Section $10$ we found the right constant
to be $-l_1!...l_r!$
\bigskip
Consider now the correlator $\Phi $,
\[ \Phi = \sum _i \frac{\Pi _a (l_a\l_i -\l'_a)}{\Pi _{j\neq i} (\l_i-\l_j)}
\ e^{\l_i (t-\t )/\hbar } \ Z_i(e^t,\hbar ) \ Z_i(e^{\t }, -\hbar ), \]
in the Calabi-Yau case $l_1+...+l_r=n+1$ (see Section $9$ for a definition
of $Z_i$).
{\bf Proposition 11.4.} {\em $(1)$ The coefficients of the power series
$Z_i(q,\hbar) = \sum _d q^d C_i(d) $ are rational functions
\[ C_i(d)=\frac{ P_d^{(i)} }{d! \hbar ^d \Pi _{j\neq i}
\Pi _{m=1}^d (\l_i -\l_j +m\hbar )} \]
where $P_d^{(i)}$ is a polynomial in $(\hbar , \l, \l')$ of degree $(n+1)d$.
$(2)$ The polynomial coefficients $E_D(p)$ in $\Phi '$
are determined by their values
\[ E_D (\l_i +d\hbar )=\Pi _a (l_a\l_i -\l'_a) \ P_d^{(i)}(\hbar )
P_{D-d}^{(i)}(-\hbar ) \]
at $p=\l_i +d\hbar $, $i=0,...,n$, $d=0,...,D$.
$(3)$ The correlators $ Z_i(q, \hbar)$
satisfy the recursion relation
\[ Z_i(q,\hbar ) = 1+ \sum_d \frac{q^d}{\hbar ^d } \frac{R_{i,d}}{d!} +
\sum _d \sum _{j\neq i} \frac{q^d}{\hbar ^d}
\frac{ Coeff \ _i^j(d) }{\l_i-\l_j+d \hbar } \
Z_j (\frac{q}{\hbar }\frac{(\l_j-\l_i)}{d}, \frac{(\l_j-\l_i)}{d}) \]
where $R_{i,d}=R_{i,d}^{(0)}\hbar ^d + R_{i,d}^{(1)}\hbar ^{d-1} + ...$
is a polynomial of $(\hbar ,\l ,\l')$ of degree $d$, and
\[ Coeff\ _i^j(d)= \
\frac{\Pi _a \Pi _{m=1}^{l_a d} (l_a\l_i-\l'_a +m(\l_j-\l_i)/d)}
{d!\Pi _{\a \neq i}\ _{m=1}^d\ _{(\a,m)\neq (j,d)}
(\l_i-\l_{\a } +m(\l_j-\l_i)/d)} \ .\]
For any given $\{ R_{i,d} \}$ these recursion relations have a unique solution
$\{ Z_i \} $.
$(4)$ Consider the class $\cal{P}$ of solutions $\{ Z_i \}$ to these recursion
relations which give rise to polynomial coefficients
$E_d$ in the corresponding $\Phi $. A solution from $\cal{P}$ is
uniquely determined by the first two coefficients
$R_{i,d}^{(0)}, R_{i,d}^{(1)}, i=0,...,n, d=1,...,\infty ,$
of its initial condition (that is by the first two terms in the
expansion of $Z_i= Z_i^{(0)} + Z_i^{(1)}/\hbar + ... $
as power series in $1/\hbar $).
$(5)$ The class $\cal{P}$ is invariant with respect to the following
operations:
(a) simultaneous multiplication $Z_i \mapsto f(q) Z_i $ by a power series
of $q$ with $f(0)=1$;
(b) changes $ Z_i (q,\hbar) \mapsto e^{ \l_i f(q)/\hbar }
Z_i(q e^{f(q)},\hbar ) $ with $f(0)=0$;
(c) multiplication $Z_i \mapsto \exp (C f(q)/\hbar ) Z_i$ where $C$
is a linear function of $(\l, \l')$ and $f(0)=0$.}
\bigskip
{\em Proof.} $(3)$ We have
\[ Z_i =1+ \sum _{d>0} q^d [ \sum _{k=0}^{d-1} \hbar ^{-k-1} \int _{Y_{2,d}}
E'_d e_0^*(\phi _i) (-c)^k ] + \sum _{d>0} q^d \hbar ^{-d} \int _{Y_{2,d}}
E'_d e_0^*(\phi _i) \frac{(-c)^d}{\hbar + c} \]
where the integrals of the last sum have zero contributions from the type (ii)
fixed point components (Lemmas $9.7, 10.1$). Thus these integrals have a
recursive expression identical to those of Sections $9$ and $10$. The terms
of the double sum constitute the initial condition $\{ R_{i,d} \} $. The
recursion relations have the form of the decomposition of rational functions
of $\hbar $ (coefficients at powers of $Q=q/\hbar $) into the sum of simple
fractions in the case when degrees of numerators exceed degrees of
denominators. This proves existence and uniqueness of solutions to the
recursion relations.
$(1)$ follows directly from the form and topological meaning of the recursion
relations.
$(2)$ follows from the definition of $\Phi $ in terms of $Z_i$.
$(4)$ Perturbation theory: Suppose that two solutions from the class $\cal{P}$
have the same initial condition up to the order $(d-1)$ inclusively. Then
$(2)$ shows that corresponding $E_k$ for these solutions coincide for
$k<d$ and
the variation $\d E_d (p) $ vanishes at $p=\l_i + k\hbar $ for $0<k<d$.
This means that the polynomial $\d E_d$ is divisible by
$\Pi _j \Pi _{m=1}^{d-1} (p-\l_j -m\hbar)$. On the other hand $(1)$ and $(2)$
imply that the variation $\d R_{i,d}$ of the initial condition satisfies
\[ \d R_{i,d} (\hbar ) \ \Pi _a (l_a \l_i -\l'_a) \Pi _{j\neq i} \Pi _{m=1}^d
(\l_i -\l_j + m\hbar ) = \d E_d |_{p= \l_i +\hbar d } \]
(since $R_{i,0}=1$) and thus $\d R_{i,d} $ is divisible by $\hbar ^{d-1}$.
Since $\d R_{i,d}$ is a degree $d$ polynomial, it leaves only the possibility
\[ \d R_{i,d} = \d R_{i,d}^{(0)} \hbar ^d + \d R_{i,d}^{(1)} \hbar ^{d-1} .\]
Thus if two class $\cal{P}$ solutions coincide in orders $\hbar ^{0},
\hbar ^{-1}$ then $\d R_{i,d} =0$, and thus the very solutions coincide.
$(5)$ The operations (a),(b),(c) give rise to the operations of type (i)-(iii)
for corresponding polynomials $E_d$. Thus it suffices to show that the
operations (a),(b),(c) transform a solution
$\{ Z_i \} $ of the recursion relations to another solution.
Consider in our recursion relation the coefficient
$\hbar ^{-d}q^d\ Coeff \ _i^j(d)$
responsible for the simple
fraction with the denominator $(\l_i -\l_j +d\hbar)$. The operations (a), (b),
(c)
cause respectively the following modifications in this coefficient:
\[ q^d\mapsto f(q)q^d/f(Q), \]
\[ q^d\mapsto q^d \exp \{ \frac{\l_i f(q)}{\hbar } + d f(q) -
\frac{d\l_j f(Q)}{(\l_j-\l_i)} \} , \]
\[ q^d\mapsto q^d \exp \{ C\frac{f(q)}{\hbar} +
C\frac{df(Q)}{\l_i-\l_j} \} , \]
where $Q=(\l_j-\l_i)q/d\hbar = q - (\l_i -\l_j + d \hbar ) q/ \hbar d $.
In the case of the change (b), additionally, the argument $q$ in $Z_j$
on the RHS of the recursion relation gets an extra-factor
$\exp [ f(q)-f(Q) ]$.
The difference $Q-q$ and the exponents vanish at $\hbar = (\l_j-\l_i)/d$.
This means that
\[ \frac{q^d}{l_i -\l_j + d\hbar } \mapsto \frac{q^d}{\l_i-\l_j +d\hbar }
+ \ \text{terms without the pole} .\]
The latter terms give contributions to a new initial condition, while the
coefficient \newline $\hbar ^{-d} q^d\ Coeff \ _i^j(d)$ does not change.
It is easy to see that the required properties of the initial condition
(that the degree of $R_{i,d}(\hbar ) $ does not exceed $d$ and $R_{i,0} =1$)
are also satisfied under our assumptions about $f$ (for those contributions
involve $\hbar $ only in the combination $q/\hbar $).
\bigskip
Let us consider now the hypergeometric series
\[ Z_i^* = \sum _{d=0}^{\infty } q^d \
\frac{\Pi _{a=1}^r \Pi _{m=1}^{l_a d} (l_a\l_i -\l'_a +m\hbar)}
{\Pi _{\a =0}^n \Pi _{m=1}^d (\l_i -\l_{\a }+m\hbar)} \]
where $l_1+...+l_r=n+1$.
It is straightforward to see that $\{ Z_i^* \} $ satisfy the recursion
relations of Proposition $11.4 (3)$ (see the proof of Proposition $9.10$)
and that the formulas of Proposition $11.4 (2)$ generate corresponding
\[ \Phi ^* = \frac{1}{2\pi i} \oint \ e^{p(t-\t )/\hbar }
\sum _{d=0}^{\infty } e^{d\t }
\frac{\Pi _{a=1}^r \Pi _{m=0}^{l_a d} (l_a p - \l'_a -m\hbar)}
{\Pi _{i=0}^n \Pi _{m=0}^d (p-\l_i -m\hbar)} dp \]
with polynomial numerators. Thus $\{ Z_i \} $ is a solution from the class
$\cal{P}$.
Computation of the first two terms in the initial condition gives
\[ Z_i^*\ ^{(0)} = f(q)=\sum _{d=0}^{\infty }
\frac{(l_1d)!...(l_rd)!}{(d!)^{n+1}} \ q^d \ ,\]
\[ Z_i^*\ ^{(1)} = \l_i \sum _a l_a [g_{l_a}(q)- g_1(q)] +
(\sum _{\a } \l_{\a }) g_1(q) - \sum _a \l'_a g_{l_a}(q) \]
where
\[ g_l=\sum _{d=1}^{\infty } q^d \frac{\Pi _a (l_a d)!}{(d!)^{n+1}}
\ (\sum _{m=1}^{ld} \frac{1}{m} ) \ .\]
\bigskip
Let us compare these initial conditions with those for $\{ Z_i \}$.
{\bf Proposition 11.5.} $Z_i^{(0)}=1, \ Z_i ^{(1)} =0$.
{\em Proof.} The first statement follows from the definition of $Z_i$ while
the second means that $\int _{Y_{2,d}} E'_d e_0^*(\phi _i) =0$ for all $d>0$.
It is due to the fact that the class $E'_d e_0^*(\phi _i)$
is a pull-back from $Y_{1,d}$.
(In fact we have just repeated an argument proving $(5)$ from Section $5$
and thus the proposition can be deduced from general properties of quantum
cohomology.)
\bigskip
Combining the last two propositions we arrive to the following
\bigskip
{\bf Theorem 11.6.} {\em The hypergeometric solution $\{ Z_i^*(q,\hbar) \} $
coincides with the solution $\{ Z_i(Q,\hbar) \} $ up to transformations
(a),(b),(c). More precisely, perform the following operations
with $\{ Z_i \}$
1) put
\[ Q=q\exp \{ \sum _a l_a [g_{l_a}(q) - g_1(q)]/f(q) \} \ ,\]
2) multiply $Z_i (Q(q),\hbar )$ by
\[ \exp \{ \frac{1}{f(q) \hbar } [\sum_{a} (l_a\l_i- \l'_a) g_{l_a} (q)
-(\sum_{\a} (\l_i-\l_{\a })) g_1(q)] \} ,\]
3) multiply all $Z_i$ simultaneously by $f(q)$.
Then the resulting functions coincide with hypergeometric
series $Z_i^*(q,\hbar)$.}
{\em Proof.} The three steps correspond to consecutive applications of
operations of type (b),(c) and (a) to $\{ Z_i \} $ and transform the
initial condition of Proposition $11.5$ to that for $\{ Z_i^*\} $.
According to Proposition $11.4$ this transforms the whole solution
$\{ Z_i \}$ to $\{ Z_i^* \} $.
\bigskip
Consider the solutions
\[ s_i = \ e^{\l_i T/\hbar } \ Z_i (e^T, \hbar ) \]
to the equivariant quantum cohomology differential equations.
{\bf Corollary 11.7.} {\em The operations
1) change $T=t+ \sum _{\a } l_a [g_{l_a} (e^t) -g_1(e^t)]/f(e^t) $,
2) multiplication by
\[ f(e^t)\exp \{ [g_1(e^t) (\sum _{\a } l_{\a }) - \sum _a \l'_a g_{l_a}(e^t)]
/(\hbar f(e^t)) \} \]
transform $\{ s_i \} $ to the hypergeometric solutions
\[ s_i^* = e^{pt/\hbar } \sum _d e^{dt}
\frac{\Pi _a \Pi _{m=1}^{l_a d} (l_a p -\l'_a +m\hbar)}
{\Pi _{\a } \Pi _{m=1}^d (p -\l_{\a } +m\hbar)} \ |_{p=\l_i } \]
of the differential equation
\[ \Pi _{\a} (\hbar \frac{d}{dt} -\l_{\a } ) s^* = e^t \
\Pi_a \Pi_{m=1}^{l_a} (\hbar l_a \frac{d}{dt} -\l'_a +m\hbar ) \ s^* \ .\]
For $\l'=0 ,\ \l_0+...+\l_n=0$ the solutions $s_i^*$ have the following
integral representation:
\[ \int _{\Gamma ^n\subset \{ F_0(u)=e^t \} }
\frac{u_0^{\l_0}...u_n^{\l_n} \ du_0\wedge ... \wedge du_n }
{ \ F_1(u) \ ... \ F_r(u) \ dF_0} \]
where
\[ F_1=(1-u_1-...-u_{l_1}), \ F_2=(1-u_{l_1+1}-...-u_{l_1+l_2}), \ ...,
F_r=(1-u_{l_1+...+l_{r-1}+1}-...-u_{l_1+...+l_r}) \ \]
and $F_0=u_0...u_n$. }
\bigskip
{\bf Corollary 11.8.} {\em The hypergeometric class $S^*(t,\hbar )\in
H^*({\Bbb C} P^n)={\Bbb C} [P]/(P^{n+1})$,
\[ S^*= e^{Pt/\hbar } \sum_d e^{dt} \frac{\Pi_a \Pi_{m=0}^{l_ad}
(l_aP+m\hbar )}{\Pi_{m=1}^d (P+m\hbar)^{n+1} } \]
whose $n+1-r$ non-zero components are solutions to the Picard-Fuchs equation
\[ (\frac{d}{dt})^{n+1-r} s^* = l_1...l_r e^t \Pi_a \Pi_{m=1}^{l_a-1}
(l_a \frac{d}{dt} + m) s^* \]
for the integrals
\[ \int _{\gamma ^{n-r} \subset X_t'} \frac{du_0\wedge ...\wedge du_n}
{dF_0\wedge dF_1\wedge ... \wedge dF_r} \ ,\]
(here $X_t'=\{ (u_0,...,u_n) | F_0(u)=e^t, F_1(u)=0, ..., F_r(u)=0 \} $)
are obtained from the class $S$ (describing the quantum cohomology
$\cal D$-module for the Calabi-Yau complete intersection
$X^{n-r}\subset {\Bbb C} P^n$),
\[ S=e^{PT/\hbar } \sum_d e^{dT} (e_0)_* (\frac{E_d}{\hbar+c_1^{(0)}}) ,\]
by the change
\[ T=t + \sum _a l_a [g_{l_a}(e^t)-g_1(e^t)]/f(e^t) \]
followed by the multiplication by $f(e^t)$.}
{\em Proof.} Corollary $11.7$ shows that for $\l'=0, \sum \l_{\a }=0$
these change and multiplication transform the corresponding equivariant
classes $S'$ and $S'\ ^*$ to one another. The class $-p$ in the formula for
$s_i^*$ in Corollary $11.7$ is the equivariant Chern class of the Hopf line
bundle over ${\Bbb C} P^n$. In the limit $\l =0$ it becomes $-P$ while $S'$ and
$S'\ ^*$ transform to their non-equivariant counterparts $S$ and $S^*$.
{\em Remarks.} 1) Notice that the components $S_0^*$ and $S_1^*$ in
\[ S^*=l_1...l_r[P^r S_0^* (t)+ P^{r+1} S_1^*(t) + ... + P^n S_n^*(t)] \]
are exactly $f(e^t)$ and $tf(e^t)+\sum_a l_a [g_{l_a }(e^t) - g_1(e^t)]$
respectively. Thus the inverse transformation from $S^*$ to $S$ consists in
division by $S_0^*$ followed by the change $T= S_1^*(t)/S_0^*(t)$ in complete
accordance with the recipe \cite{COGP, BVS, HG1} based on the mirror
conjecture.
2) According to \cite{B} the $(n-r)$-dimensional manifolds $X_t'$ admit a
Calabi-Yau compactification to the family $\bar{X}_t'$ of
{\em mirror manifolds } of the Calabi-Yau complete intersection
$X^{n-r}\subset {\Bbb C} P^n$. The Picard-Fuchs differential equation from
Corollary $11.8$ describes variations of complex structures for $\bar{X}'$.
This proves the mirror conjecture (described in detail in \cite{BVS})
for projective Calabi-Yau complete intersections and confirms the enumerative
predictions about rational curves and quantum cohomology algebras made there
(and in some other papers) on the basis of the mirror conjecture.
3) The description \cite{AM} of the
quantum cohomology algebra of a Calabi-Yau $3$-fold in terms of the numbers
$n_d$ of rational curves of all degrees $d$ (see for instance \cite{HG1} for
the description of the corresponding class $S$ in these terms) has been
rigorously justified in \cite{M}. Combining these results with Corollary
$11.8$ we arrive to the theorem formulated in the introduction.
\bigskip
{\em Proof of The Main Lemma.}
In our construction of the map $\mu: L_d\to L'_d$ we will denote $L_d$
the moduli space of stable maps $C\to {\Bbb C} P^n\times {\Bbb C} P^1$ of bidegree
$(d,1)$ with no marked points (it also has dimension $d(n+1)+n$).
The construction works for any given number
of marked points but produces a map which is the composition of $\mu $ with
the forgetful map. In this form it applies to the submanifold of stable maps
with two marked points confined over $0$ and $\infty $ in ${\Bbb C} P^1$ (this
submanifold is what we denoted $L_d$ in the formulation of The Main Lemma).
Let $\psi: C\to {\Bbb C} P^n \times {\Bbb C} P^1 $ be a stable genus $0$ map of bidegree
$(d,1)$. Then $C=C_0 \cup C_1 ... \cup C_r$ where $C_0$ is isomorphic to
${\Bbb C} P^1$
and $\psi | C_0$ maps $C_0$ onto the graph of a degree $d'\leq d$ map ${\Bbb C} P^1
\to {\Bbb C} P^n$, and for $i=1,...,r$ the bidegree $(d_i,0)$ map $\psi | C_i$
sends $C_i$ into the slice ${\Bbb C} P^n \times \{ p_i \} $ where $p_i \neq p_j$
and $d_1+...+d_r=d-d'$.
The map $\mu :L_d \to L'_d$ assigns to $[\psi ]$ the $(n+1)$-tuples
$(f_0 g : f_1 g : ... : f_n g)$ of polynomials ($=$ binary forms) on
${\Bbb C} P^1$ where
$g$ is the polynomial of degree $d-d'$ with roots $(p_1,...,p_r)$ of
multiplicities $(d_1,...,d_r)$ and the tuples $(f_0:...:f_n)$ of degree $d'$
polynomials (with no common roots, including $\infty $)
is the one that describes the map $\psi | C_0$.
In order to prove that the map $\mu $ is regular let us give it another, more
invariant description.
Denote $\hat{L}_d$ the moduli space of bidegree $(d,1)$ stable maps with
an extra-marked point and pull back to $\hat{L}_d$ the line bundle
\[ H:= Hom (\pi _1^* {\cal O} _{{\Bbb C} P^n} (1), \pi _2^* {\cal O} _{{\Bbb C} P^1} (d)) \]
by the evaluation map $e: \hat{L}_d \to {\Bbb C} P^n\times {\Bbb C} P^1$ (where $\pi _i$
are projections to the factors). Consider the push-forward sheaf
$H^0:=R^0\pi _* e^* (H)$
of the locally free sheaf $e^* H$ along the forgetful map $\pi :\hat{L}_d\to
L_d$. To a small neighborhood $U\subset L_d$, it assigns the ${\cal O} _U$
-module $H^0(\pi ^{-1}(U), e^* H)$ of sections of $e^* H$.
{\em Claim.} {\em $1$) $H^0$ is a rank $1$ locally free sheaf on $L_d$.
$2$) The fiber at $[\psi ]$ of the corresponding line bundle can be
identified with
\[ H^0(C_0, (\psi | C_0)^*(H) \otimes {\cal O} (-[p_1])^{\otimes d_1} ...
\otimes {\cal O} (-[p_r])^{\otimes d_r}) .\]
$3$) The kernel of the natural map
\[ h: H^0(C, \psi ^*\pi _1^*({\cal O}_{{\Bbb C} P^n}(1))) \to
H^0(C, \psi ^*\pi _2^*({\cal O}_{{\Bbb C} P^1}(d))) =
H^0({\Bbb C} P^1, {\cal O} (d)) \]
defined by a nonzero vector in this fiber consists of the sections
vanishing identically on $C_0$.}
Using this, we pick $n+1 $ independent sections of ${\cal O}_{{\Bbb C} P^n}(1)$
(that is homogeneous coordinates on ${\Bbb C} P^n$), define corresponding
sections of $e^* \pi _1^* {\cal O}_{{\Bbb C} P^n} (1)$ and apply the map $h$.
By this we obtain a degree $1$ map from the total space of the line bundle
$H^0$ to the linear space ${\Bbb C} ^{n+1} \otimes H^0({\Bbb C} P^1, {\cal O} (d))$. Since
the homogeneous coordinates on ${\Bbb C} P^n$ nowhere vanish simultaneously,
we obtain a natural map
\[ L_d \to L'_d = Proj ({\Bbb C} ^{n+1}\otimes H^0({\Bbb C} P^1, {\cal O} (d))) \]
which sends $[\psi ]$ to $(f_0 g:...:f_n g)$ and conclude that $\mu $ is
regular.
The remaining statements of The Main Lemma are proved by looking at
localizations of the equivariant class $p$ at the $S^1\times T^{n+1}$-fixed
points in $L'_d$ and $L_d$ (in this paragraph we use the notation $L_d$ for
the same space as in the formulation of The Main Lemma).
The fixed points in $L'_d$ are represented by
the vector-monomials $(0:...:0:x^{d'}:0:...:0)$ where $p$ localizes to
$\l_i + d'\hbar $. A fixed point in $L_d$ is represented by $\psi $ with
$\psi (C_0) = (0: ... :0:1: 0:...:0)$, $r=2$, $p_0=0$, $p_1=\infty $ and
the maps $\psi | C_k : C_k \to {\Bbb C} P^n$, $k=1,2$ representing $T^{n+1}$-fixed
points respectively in $Y_{2,d'}$ and $Y_{2,d-d'}$ such that their (say)
second marked points are mapped to the point $\psi (C_0)$.
This implies that the class $\mu^*(p)$ localizes to $\l_i+d'\hbar $ at such a
fixed point and thus the pull back of $p$ to the fixed point set
\[ \{ [\psi ]\in Y_{2,d'}\times Y_{2,d-d'} | (e_2\times e_2) ([\psi ])
\in \Delta \subset Y\times Y \]
of the $S^1$-action on $L_d$ coincides with the pull back through the common
marked point of the $T^{n+1}$-equivariant class $p+d'\hbar$ on the diagonal
$\Delta = {\Bbb C} P^n$. Now localizations of $\int _{L_d} e^{p(t-\t )} {\cal E} _d $
to the fixed points of $S^1$-action identify the form of the correlator
$\Phi $ given in The Main Lemma with the definition of $\Phi $ as
the convolution of $S'(t,\hbar )$ and $S'(\t ,-\hbar )$.
\bigskip
In order to justify the {\em claim} we need to compute the space of global
sections of the sheaf $e^* (H)$ over the formal neighborhood of the fiber
$\pi ^{-1} ([\psi ])$ of the forgetful map $\pi : \hat{L}_d \to L_d$.
The fiber itself is isomorphic to the tree-like genus $0$ curve $C$.
Let $(x_j,y_j), j=1,...,N\geq r$ be some local parameters on irreducible
components of $C$ near the singular points such that $\varepsilon_j = x_jy_j$ are local
coordinates on the {\em orbifold} $L_d$ near $[\psi ]$ (one should add
some local coordinates $\varepsilon'$ on the stratum $\varepsilon_1=...\varepsilon_N=0$ of stable maps
$C\to {\Bbb C} P^n$ in order to construct a complete
local coordinate system on $L_d$). Such a description of local coordinates on
$L_d$ follows from the very construction of the moduli spaces of stable maps
to ample manifolds; we refer the reader to \cite{Kn, BM} for details.
A line bundle over the neighborhood of $C\subset \hat{L}_d$ can be specified
by the set
\[ u_j (x_j^{\pm 1}, \varepsilon ),
v_j (y_j^{\pm 1}, \varepsilon ), \ j=1,...,N, \]
of non-vanishing functions describing transition maps between trivializations
of the bundle inside and outside the neighborhoods
(with local coordinates $(x_j,y_j,\varepsilon_1,...,\hat{\varepsilon _j},...,\varepsilon_N,\varepsilon')$)
of the double points.
Let us consider first the following model case. Suppose that $C$
consists of $r+1$ irreducible components $(C_0, C_1,...,C_r)$ such that each
$C_j$ with $j>0$ intersects $C_0$ at some point $p_j$. Let $x_j$ be
the local parameter on $C_0$ near $p_j$, and the line bundle
(of the degree $-d_j\leq 0$ on $C_j$) be specified by $v_j=y_j^{-d_j}$.
In the neighborhood of $p_j$ a section of such a bundle is given
by a function $s(x_j,y_j,\hat{\varepsilon _j})$ satisfying
\[ s=y_j^{-d_j}s_j(y_j^{-1}, \varepsilon) \]
where the function $s_j$ represents the section in the trivialization
over the neighborhood of $C_j-p_j$. Here $\hat{\varepsilon_j}$ means that $\varepsilon_j$
is excluded from the set of coordinates $\varepsilon $ (remember that $\varepsilon_j=x_jy_j$).
This implies that $s_j = \varepsilon _j^{d_j} f_j(y_j^{-1}\varepsilon_j, \varepsilon)$ where $f_j$
is some regular function. Thus this section in the neighborhood of $p\in C_0$
is given by a function $ s(x_j, \varepsilon ) = x_j^{d_j}f_j(x_j,\varepsilon)$ with zero of
order $d_j$ at $x_j=0$, and the restriction of this section to the
neighborhood of $C_j$ is determined by $s$.
In other words, the ${\Bbb C} [[\varepsilon ]]$-module of global sections in the
formal neighborhood of $C$ identifies with the module of global sections
on $C_0$ for the line bundle given by the loops $x_j^{-d_j} u_j$
instead of $u_j$ (this corresponds to the subtraction of the divisor
$\sum d_j [p_j]$.
The more general situation where $v_j$ is the product of $y_j^{-d_j}$ with
an invertible function $w_j(y_j, x_j ,\hat{\varepsilon_j })$
preserves the above conclusion with
$w^{-1}s=x_j^{d_j} f_j(x_j,\varepsilon )$ instead of $s$.
Obviously, the above computation bears dependence on additional parameters.
Now we apply our model computation to the neighborhood of a general tree-like
curve $C$ {\em inductively} by decomposing the tree into simpler ones
starting from
the root component $C_0$. We conclude that the ${\Bbb C} [[\varepsilon]] $-module of
sections of the bundle $e^*(H)$ is identified with the module of sections
of some line bundle
over the product of $C_0$ with the polydisk with coordinates
$(\varepsilon_1,...,\varepsilon_r, ..., \varepsilon_N, \varepsilon')$,
and that this line bundle is $e^*(H)$ for $C_0$
(given by the loops $u_j$ in our current
notations) twisted by the loops $x_j^{-d_j}$
in the punctured neighborhoods of the points $(p_1,...,p_r)$, where
$(d_1,...,d_r)$ are the degrees of the maps $\psi |C_j: C_j \to {\Bbb C} P^n$
(in the notations of the {\em claim} so that $d_1+...+d_r=d-d'$).
This implies that the ${\Bbb C} [[\varepsilon ]]$-module ${\cal H}^0$
of global sections can be identified with the
module of those global sections of the degree $d-d'$ locally free sheaf
$(\psi |C_0)^* (H) \otimes {\Bbb C} [[\varepsilon ]]$ which have zeroes of order
$d_j$ at $p_j$ for $j=1,...,r$. In particular
1) ${\cal H}^0$ is a free ${\Bbb C} [[\varepsilon ]]$-module of rank $1$,
2) ${\cal H}^0 \otimes _{{\Bbb C} [[\varepsilon ]] } ({\Bbb C} [[\varepsilon ]]/(\varepsilon))$ is the
$1$-dimensional space $H^0|_{[\psi]} $ described in the {\em claim}, and
3) non-zero vectors in $H^0|_[\psi]$ represent sections of $\psi ^*(H)$ over
$C$ non-zero on $C_0$ (and thus their product with a non-zero on $C_0$
section of
$\psi ^*\pi _1^*({\cal O} _{{\Bbb C} P^n} (1))$ can not vanish identically on $C_0$.)
Factorization by the discrete group $Aut (\psi )$ preserves $(1-3)$ with
${\Bbb C} [[\varepsilon ]]$ replaced by ${\Bbb C} [[\varepsilon ]]^{Aut (\psi )}$.
\newpage
|
1996-03-22T06:20:40 | 9603 | alg-geom/9603016 | fr | https://arxiv.org/abs/alg-geom/9603016 | [
"alg-geom",
"math.AG"
] | alg-geom/9603016 | Jean-Marc Drezet | Jean-Marc Dr\'ezet | Espaces abstraits de morphismes et mutations | 41 pages, LaTeX | null | null | null | null | A transformation of morphisms of sheaves, called mutation, is used to build
new moduli spaces of morphisms.
| [
{
"version": "v1",
"created": "Thu, 21 Mar 1996 10:23:18 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Drézet",
"Jean-Marc",
""
]
] | alg-geom | \section{Introduction}
\subsection{Vari\'et\'es de modules de morphismes}
Soient \m{X} une vari\'et\'e alg\'egrique projective sur le corps
des nombres complexes, et \m{{\cal E}}, \m{{\cal F}} des faisceaux
alg\'ebriques coh\'erents sur \m{X}. Soit
$$W = \mathop{\rm Hom}\nolimits({\cal E},{\cal F}) .$$
Alors le groupe alg\'ebrique
$$G = Aut({\cal E})\times Aut({\cal F})$$
agit d'une fa\c con \'evidente sur \m{W}. Si deux morphismes sont dans
la m\^eme \m{G}-orbite, leurs noyaux sont isomorphes, ainsi que
leurs conoyaux. C'est pourquoi il peut \^etre int\'eressant, pour
d\'ecrire certaines vari\'et\'es de modules de faisceaux, de
construire de bons quotients d'ouverts \m{G}-invariants de \m{W} par
\m{G}. On s'int\'eresse au cas particulier suivant : soient \m{r}, \m{s} des
entiers positifs, \m{{\cal E}_1,\ldots,{\cal E}_r,},
\m{{\cal F}_1,\ldots,{\cal F}_s} des faisceaux
coh\'erents sur \m{X}, qui sont {\em simples}, c'est-\`a-dire que leurs
seuls endomorphismes sont les homoth\'eties. On suppose aussi que
$$\mathop{\rm Hom}\nolimits({\cal E}_i,{\cal E}_{i'}) = \lbrace 0 \rbrace \ \ {\rm si \ } i > i' \ , \
\mathop{\rm Hom}\nolimits({\cal F}_j,{\cal F}_{j'}) = \lbrace 0 \rbrace \ \ {\rm si \ } j > j', $$
$$\mathop{\rm Hom}\nolimits({\cal F}_j,{\cal E}_i) = \lbrace 0 \rbrace \ \
{\rm pour \ tous \ } i,j .$$
Soient \m{M_1,\ldots,M_r}, \m{N_1,\ldots,N_s} des espaces vectoriels
complexes de dimension finie. On suppose que
$${\cal E} = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}({\cal E}_i\otimes M_i) \ , \
{\cal F} = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}({\cal F}_l\otimes N_l) .$$
Les \'el\'ements de \m{W} sont appel\'es {\em morphismes de type} \m{(r,s)}.
Le groupe \m{G} n'est pas r\'eductif en g\'en\'eral.
On a consid\'er\'e dans
\cite{dr_tr} le probl\`eme de l'existence de bon quotients d'ouverts
\m{G}-invariants de \m{\mathop{\rm Hom}\nolimits({\cal E},{\cal F})}.
On introduit une notion de {\em semi-stabilit\'e} pour les
mor-\break phismes de type \m{(r,s)} qui d\'epend du choix d'une suite
\m{(\lambda_1,\ldots,\lambda_r,}\m{\mu_1,\ldots,\mu_s)} de nombres
rationnels positifs tels que
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq r}\lambda_i\dim(M_i) =
\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq s}\mu_l\dim(N_l) = 1.$$
On appelle cette suite une {\em polarisation} de l'action de \m{G}.
Il existe un bon quotient de l'ouvert des points
semi-stables pour certaines valeurs de
\m{(\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s)}
(ces r\'esultats sont rappel\'es au \paragra~\hskip -2pt 3).
Pour traiter ce genre de probl\`eme de la mani\`ere la plus g\'en\'erale,
on le traduit d'abord en termes
d'alg\`ebre lin\'eaire (c'est-\`a-dire que ce qu'on \'etudie est une action
particuli\`ere d'un certain groupe non r\'eductif sur un espace vectoriel de
dimension finie). C'est ce qui est fait dans \cite{dr_tr} et ici.
Les morphismes de type \m{(2,1)} sont utilis\'es dans \cite{dr2} pour
d\'ecrire certaines vari\'et\'es de modules de faisceaux semi-stables
sur \proj{2}. Dans un certain nombre de travaux (cf. par exemple
\cite{miro}, \cite{oko}) des faisceaux semi-stables ou des faisceaux
d'id\'eaux de sous-vari\'et\'es de l'espace projectif sont d\'ecrits
comme conoyaux de morphismes de type \m{(r,s)}.
\subsection{Mutations de morphismes}
Le but du pr\'esent article est de d\'ecrire et d'\'etudier certaines
transformations, appel\'ees {\em mutations}, associant \`a un morphisme de
type \m{(r,s)} un autre morphisme, pouvant \^etre d'un autre type
(mais la somme \m{r+s} reste constante). On obtient en quelque sorte
une correspondance entre deux espaces de morphismes \m{W} et \m{W'}, sur
lesquels agissent respectivement les groupes en g\'en\'eral non
r\'eductifs \m{G} et \m{G'}. Ceci permet
de d\'efinir une bijection de l'ensemble des \m{G}-orbites
d'un ouvert de \m{W} sur l'ensemble des \m{G'}-orbites d'un ouvert de
\m{W'}. La forme que prend une mutation dans le language des morphismes
de faisceaux est explicit\'ee au \paragra~\hskip -2pt 1.3 (dans la description du
chapitre 5). On donnera toutefois une d\'efinition plus abstraite et
plus g\'en\'erale de ce qu'est une mutation dans le chapitre 6.
On associe de mani\`ere naturelle
\`a chaque polarisation \m{\sigma} de l'action de \m{G} sur \m{W} une
polarisation \m{\sigma'} de l'action de \m{G'} sur \m{W'}.
Dans certains cas on montre qu'un point de \m{W} est semi-stable
relativement \`a \m{\sigma} si et seulement si la mutation de ce
point est semi-stable relativement \`a \m{\sigma'}. Ceci
permet de prouver que les quotients correspondants sont isomorphes. On
peut ainsi \'etendre les r\'esultats de \cite{dr_tr} \`a d'autres
polarisations. Par exemple, dans \cite{dr2} on prouve dans certains cas
l'existence de bon quotients (lesquels sont isomorphes \`a des vari\'et\'es
de modules de faisceaux semi-stables sur \m{\proj{2}}). On
n'a pas besoin dans ce cas d'un th\'eor\`eme d'existence d'un quotient par
un groupe non r\'eductif, car la vari\'et\'e de modules existe d\'ej\`a.
Ces exemples de bons quotients ne peuvent pas \^etre directement retrouv\'es
\`a partir de \cite{dr_tr}, mais en utilisant des mutations, on peut se
ramener aux cas trait\'es dans \cite{dr_tr}. Autre exemple, les morphismes
$${\cal O}(-2)\oplus{\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{n+2}$$
sur \m{\proj{n}}. L'application directe de \cite{dr_tr} ne fournit
aucun quotient non vide. En utilisant des mutations,
on peut trouver plusieurs types de quotients.
La d\'efinition des mutations s'introduit naturellement lorsqu'on \'etudie
les faisceaux semi-stables sur \m{\proj{n}} au moyen des
suites spectrales de Beilinson g\'en\'eralis\'ees. On associe une telle suite
spectrale \`a un faisceau coh\'erent \m{{\cal E}}
sur \m{\proj{n}} et \`a une {\em base d'h\'elice \m{\sigma}
de fibr\'es exceptionnels } sur \m{\proj{n}} (cf \cite{dr1} et
\cite{go_ru}). Si le diagramme de Beilinson correspondant est suffisamment
simple, on obtient une suite exacte
$$0\longrightarrow \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}(E_i\otimes\cx{m_i})
\longrightarrow\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}(F_l\otimes\cx{n_l})\longrightarrow{\cal E}\longrightarrow 0 ,$$
la base d'h\'elice \m{\sigma} \'etant
\m{(E_1,\ldots,E_r,F_1,\ldots,F_s)} (et donc \m{r + s = n+1}). On peut,
en changeant judicieusement la base d'h\'elice, obtenir d'autres
repr\'esentations semblables de \m{{\cal E}}. On peut changer de
base d'h\'elice en faisant subir \`a celle dont on part une s\'erie de
transformations \'el\'ementaires appel\'ees {\em mutations}, d'o\`u la
terminologie employ\'ee pour les transformations de morphismes \'etudi\'ees
ici.
Pour la d\'efinition, les propri\'et\'es et
l'usage des suites spectrales de Beilinson
g\'en\'eralis\'ees sur les espaces
projectifs, voir \cite{dr1}, \cite{dr2}, \cite{dr3}, \cite{dr_lp},
\cite{go_ru}.
\subsection{Plan des chapitres suivants}
Dans le chapitre 2 on donne un exemple simple et bien connu de mutations
dans le cas des morphismes de type (1,1). C'est ce type de r\'esultats
qu'il s'agit de g\'en\'eraliser.
\medskip
Dans le chapitre 3 on rappelle certains r\'esultats de \cite{dr_tr},
concernant les quotients d'espa-\break ces de morphismes de type \m{(r,s)}.
Le th\'eor\`eme 3.1 d\'ecrit ce qu'on sait des quotients d'espaces
de morphismes de type (2,1).
\medskip
Dans le chapitre 4 on rappelle la d\'efinition des suites spectrales de
Beilinson g\'en\'erali-\break s\'ees sur les espaces projectifs et on
d\'ecrit des mutations de morphismes de type
\m{(r,s)} obtenues en utilisant les suites spectrales de
Beilinson g\'en\'eralis\'ees sur les espaces projectifs.
\medskip
Dans le chapitre 5 on g\'en\'eralise un peu ce qui pr\'ec\`ede.
On donne la d\'efinition des mutations de morphismes
en termes de faisceaux. Plus pr\'ecis\'ement on montre que si un
faisceau coh\'erent peut \^etre repr\'esent\'e comme conoyau d'un
morphisme injectif de faisceaux, on peut dans certaines conditions le
repr\'esenter aussi comme conoyau d'un morphisme injectif d'un autre
type. Les r\'esultats du \paragra~\hskip -2pt 5.1 sont plus g\'en\'eraux que ce qui est
n\'ecessaire ici. Dans le \paragra~\hskip -2pt 5.2 on donne des applications aux morphismes
de type \m{(r,s)}. Voici un exemple du type de r\'esultat obtenu : soit
$$\Phi : \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}({\cal E}_i\otimes M_i) \longrightarrow
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}({\cal F}_l\otimes N_l)$$
un morphisme injectif, \m{\cal U} son conoyau
et \m{p} un entier tel que \m{0\leq p\leq r-1}. On suppose que pour
\m{p+1\leq j\leq r} le morphisme canonique
$${\cal E}_j\longrightarrow \mathop{\rm Hom}\nolimits({\cal E}_j,{\cal F}_1)^*\otimes {\cal F}_1$$
est injectif. Soit \m{{\cal G}_j} son conoyau. Soit
$$f_p : \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(\mathop{\rm Hom}\nolimits({\cal E}_j,{\cal F}_1)^*\otimes M_j)\longrightarrow N_1$$
l'application lin\'eaire d\'eduite de $\Phi$. On suppose que $f_p$ est
surjective. Alors on montre que sous certaines hypoth\`eses
il existe une suite exacte
$$0\longrightarrow \biggl(\bigoplus_{1\leq i\leq p}({\cal E}_i\otimes M_i)\biggr)\oplus
({\cal F}_1\otimes\ker(f_p))\longrightarrow
\biggr(\bigoplus_{p<j\leq r}({\cal G}_j\otimes M_j)\biggl)\oplus
\biggr(\bigoplus_{2\leq l\leq s}({\cal F}_l\otimes N_l)\biggl)\longrightarrow{\cal U}\longrightarrow 0.$$
On a donc associ\'e \`a un morphisme de type \m{(r,s)} un morphisme de
type
\noindent\m{(p+1,r+s-p-1)}.
\medskip
Dans le chapitre 6 on d\'efinit les mutations dans un cadre plus abstrait.
On d\'efinit des actions de groupes sur des espaces vectoriels
model\'ees sur les cas \'etudi\'es dans le chapitre pr\'ec\'edent.
Une mutation est dans ce cas une correspondance entre une telle action d'un
groupe $G$ sur un espace vectoriel $V$ et une autre action d'un groupe $G'$
sur un espace vectoriel $V'$, de telle sorte qu'on ait une bijection
\ \m{V^0/G\simeq {V'}^0/G'}, pour des ouverts ad\'equats $V^0$ et
\m{{V'}^0} non vides de $V$ et $V'$ respectivement.
\medskip
Dans le chapitre 7, on applique les r\'esultats qui pr\'ec\`edent dans le
but de trouver d'autres cas o\`u on sait d\'efinir des bons quotients
d'espaces de morphismes de type \m{(r,s)}. Dans le cas des morphismes
de type (2,1), le th\'eor\`eme 7.6 obtenu \'etend les r\'esultats du
th\'eor\`eme 3.1.
\medskip
Dans le chapitre 8 on donne des exemples d'applications des r\'esultats
pr\'ec\'edents.
\bigskip
{\bf Remerciements.} L'auteur tient \`a remercier G. Trautmann pour de
nombreuses discussions
qui l'ont beaucoup aid\'e, ainsi que l'Universit\'e de Kaiserslautern pour
son hospitalit\'e durant la r\'ealisation d'une partie de ce travail.
\vfil
\eject
\section{Un exemple simple}
Les r\'esultats de ce chapitre sont d\'emontr\'es dans \cite{dr2}. Soient
\m{L}, \m{M} et \m{N} des espaces vectoriels complexes de dimension finie,
avec \ \m{\dim(L)\geq 3}. On pose \
\m{q = \dim(L)}, \m{m = \dim(M)}, \m{n = \dim(N)}.
Les applications lin\'eaires
$$L\otimes M\longrightarrow N$$
sont appel\'ees des \m{L}-{\em modules de Kronecker}. Soit
$$W=\mathop{\rm Hom}\nolimits(L\otimes M,N).$$
Sur \m{W} op\`ere de mani\`ere \'evidente le groupe alg\'ebrique r\'eductif
$$G=(GL(M)\times GL(N))/\cx{*}.$$
L'action de \ \m{SL(M)\times SL(N)} \ sur \m{\projx{W}} se lin\'earisant de
fa\c con \'evidente, on a une notion de point {\em (semi-)stable} de
\m{\projx{W}} (au sens de la g\'eom\'etrie invariante). On montre que si
\m{f\in W}, \m{f} est semi-stable (resp.
stable) si et seulement si pour tous sous-espaces vectoriels \m{M'} de
\m{M} et \m{N'} de \m{N}, tels que \m{M'\not = \lbrace 0\rbrace},
\m{N'\not =N}, et \ \m{f(L\otimes M')\subset N'}, on a
$$\q{\dim(N')}{\dim(M')}\geq\q{\dim(N)}{\dim(M)}\ \ {\rm (resp.}
\ > \ {\rm)}.$$
Soit \m{W^{ss}} (resp. \m{W^s}) l'ouvert des points semi-stables (resp.
stables) de \m{W}. Alors il existe un bon quotient (resp. un quotient
g\'eom\'etrique)
$$N(L,M,N) = W^{ss}//G \ \ \ {\rm (resp. } \ \ N_s(L,M,N)=W^s/G \ {\rm)},$$
\m{N(L,M,N)} est projective, et \m{N_s(L,M,N)} est un ouvert lisse de
\m{N(L,M,N)}.
On pose \ \m{m'=qm-n}. On suppose que \m{m'>0}. Soit \m{M'} un
espace vectoriel complexe de dimension \m{m'}. Soit \ \m{f:L\otimes M\longrightarrow N} \
un \m{L}-module de Kronecker surjectif. Alors \ \m{\dim(\ker(f))=m'}. Soient
$$f' : L^*\otimes\ker(f)\longrightarrow H_0$$
la restriction de l'application
$$tr\otimes I_{H_0} : L^*\otimes L\otimes H_0\longrightarrow H_0$$
(\m{tr} d\'esignant l'application trace), et
$$A(f) : L^*\otimes H_0^*\longrightarrow\ker(f)^*$$
l'application lin\'eaire d\'eduite de \m{f'}, qu'on peut voir comme un
\'el\'ement de
\noindent\m{W' = \mathop{\rm Hom}\nolimits(L^*\otimes H_0^*,M')}, en utilisant un
isomorphisme \ \m{\ker(f)^*\simeq M'}. Soit \m{W_0} l'ouvert de \m{W}
constitu\'e des applications surjectives, \m{W'_0} l'ouvert analogue de
\m{W'}, et
\noindent\m{G' = (GL(H_0^*)\times GL(M'))/\cx{*}}. On d\'emontre
ais\'ement la
\begin{xprop}
1 - En associant \m{A(f)} \`a \m{f} on d\'efinit une bijection
$$W_0/G \ \simeq \ W'_0/G'.$$
\noindent 2 - La bijection pr\'ec\'edente induit un isomorphisme
$$N(L,M,N)\ \simeq \ N(L^*,H_0^*,M')$$
(induisant un isomorphisme \ \m{N_s(L,M,N)\ \simeq \ N_s(L^*,H_0^*,M')}).
\end{xprop}
\section{Vari\'et\'es de modules de morphismes de type $(r,s)$}
On rappelle ici le probl\`eme des vari\'et\'es de modules de morphismes
de type $(r,s)$ abord\'e dans \cite{dr_tr}.
En termes de faisceaux, on consid\`ere des morphismes
$${\cal E} = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}({\cal E}_i\otimes M_i) \longrightarrow
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}({\cal F}_l\otimes N_l) = {\cal F} ,$$
et l'action du groupe alg\'ebrique
$$G = Aut({\cal E})\times Aut({\cal F})$$
\ sur l'espace vectoriel \m{W} de tous ces morphismes.
Il est pr\'ef\'erable de g\'en\'eraliser ce probl\`eme en termes
d'alg\`ebre lin\'eaire.
\subsection{D\'efinition abstraite de \m{W}}
Soient \m{r,s} des entiers positifs,
\m{H_{li}}, \m{A_{ji}}, \m{B_{ml}}, \m{1\leq i\leq j\leq r},
\m{1\leq l\leq m\leq s} des espaces vectoriels de dimension finie (qui
jouent le r\^ole de \m{\mathop{\rm Hom}\nolimits({\cal E}_i,{\cal F}_l)},
\m{\mathop{\rm Hom}\nolimits({\cal E}_i,{\cal E}_j)} et \m{\mathop{\rm Hom}\nolimits({\cal F}_l,{\cal F}_m)}
respectivement). On suppose que \ \m{A_{ii} = \cx{}}
\ pour \m{1\leq i\leq r} et \ \m{B_{ll} = \cx{}} \
pour \m{1\leq l\leq s}. Pour \m{1\leq i\leq j\leq k\leq r} et
\m{1\leq l\leq m\leq n\leq s} on se donne des applications lin\'eaires
(appel\'ees {\it compositions})
$$H_{lj}\otimes A_{ji}\longrightarrow H_{li},$$
$$A_{kj}\otimes A_{ji}\longrightarrow A_{ki},$$
$$B_{ml}\otimes H_{li}\longrightarrow H_{mi},$$
$$B_{nm}\otimes B_{ml}\longrightarrow B_{nl}.$$
On suppose que si \m{i=j} les deux premi\`eres applications sont les
identit\'es, ainsi que la se-\break conde si \m{j=k}, la quatri\`eme si
\m{m=n} et les troisi\`eme et cinqui\`eme si \m{l=m}. Ces applications
jouent le r\^ole de la composition des morphismes dans le cas des
faisceaux. On suppose qu'elles sont toutes surjectives et qu'elles
v\'erifient les propri\'et\'es usuelles qu'on attend des applications
habituelles de composition. Cela signifie que les diagrammes suivants
sont commutatifs (les fl\`eches \'etant les fl\`eches \'evidentes) :
$$\diagram{
A_{kj}\otimes A_{ji}\otimes A_{ih} & \hfl{}{} & A_{ki}\otimes A_{ih} \cr
\vfl{}{} & & \vfl{}{}\cr
A_{kj}\otimes A_{jh} & \hfl{}{} & A_{kh} \cr
}\ \ \ \
\diagram{
H_{lk}\otimes A_{kj}\otimes A_{ji} & \hfl{}{} & H_{lj}\otimes A_{ji} \cr
\vfl{}{} & & \vfl{}{}\cr
H_{lk}\otimes A_{ki} & \hfl{}{} & H_{li}\cr
}$$
$$\diagram{
B_{ml}\otimes H_{lj}\otimes A_{ji} & \hfl{}{} & H_{mj}\otimes A_{ji} \cr
\vfl{}{} & & \vfl{}{}\cr
B_{ml}\otimes H_{li} & \hfl{}{} & H_{mi}\cr
}\ \ \ \
\diagram{
B_{nm}\otimes B_{ml}\otimes H_{li} & \hfl{}{} & B_{nl}\otimes H_{li} \cr
\vfl{}{} & & \vfl{}{}\cr
B_{nm}\otimes H_{mi} & \hfl{}{} & H_{ni}\cr
}$$
$$\diagram{
B_{on}\otimes B_{nm}\otimes B_{ml} & \hfl{}{} & B_{om}\otimes B_{ml} \cr
\vfl{}{} & & \vfl{}{}\cr
B_{on}\otimes B_{nl} & \hfl{}{} & B_{ol}\cr
}$$
On supposera aussi que les applications
$$H_{li}^* \otimes A_{ji}\longrightarrow H_{lj}^* \ \ , \ \
H_{mi}^*\otimes B_{ml}\longrightarrow H_{li}^*$$
induites par les applications de composition sont surjectives. Soient
\m{M_i, 1\leq i\leq r,}
\break\m{N_l, 1\leq l\leq s} des espaces vectoriels de
dimension finie. On notera
$$m_i = \dim(M_i), \ n_l = \dim(N_l), \ 1\leq i\leq r, 1\leq l\leq s.$$
On veut \'etudier l'espace vectoriel
$$W = \bigoplus_{1\leq i\leq r,1\leq l\leq s}
\mathop{\rm Hom}\nolimits(H_{li}^*\otimes M_i,N_l) .$$
\subsection{D\'efinition du groupe \m{G}}
Soit \m{G_L} l'ensemble des matrices
$$g = \pmatrix{g_1 & 0 & . & . & . & 0 \cr
u_{21} & g_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & u_{ij} & & . & . \cr
u_{r1} & . & . & . & . & g_r \cr
} ,$$
avec \ \m{g_i\in GL(M_i)}, et pour \ \m{1\leq j < i \leq r},
$$u_{ij}\in \mathop{\rm Hom}\nolimits(A_{ij}^*\otimes M_j,M_i) = \mathop{\rm Hom}\nolimits(M_j,A_{ij}\otimes M_i) .$$
Soit \m{G_R} l'ensemble des matrices
$$h = \pmatrix{h_1 & 0 & . & . & . & 0 \cr
v_{21} & h_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & v_{lm} & & . & . \cr
v_{s1} & . & . & . & . & h_s \cr
} ,$$
avec \ \m{h_l\in GL(N_l)}, et pour \ \m{1\leq m < l \leq s},
$$v_{lm}\in \mathop{\rm Hom}\nolimits(B_{lm}^*\otimes N_m,N_l) = \mathop{\rm Hom}\nolimits(N_m,B_{lm}\otimes N_l) .$$
On d\'efinit une loi de composition, not\'ee \m{*}, de la fa\c con
suivante : si
$$ u_{kj}\in \mathop{\rm Hom}\nolimits(A_{kj}^*\otimes M_j,M_k) \
\ \ {\rm et} \ \ u_{ji}\in \mathop{\rm Hom}\nolimits(A_{ji}^*\otimes M_i,M_j),$$
alors
$$u_{kj}*u_{ji}\in L(A_{ki}^*\otimes M_i,M_k) $$
est la composition
$$A_{ki}^*\otimes M_i\hfl{u_{ji}}{} A_{ki}^*
\otimes A_{ji}\otimes M_j\hfl{}{}A_{kj}^*\otimes M_j\hfl{u_{kj}}{}M_k ,$$
ou l'application du milieu est induite par la composition
$$A_{kj}\otimes A_{ji}\longrightarrow A_{ki}.$$
On d\'efinit une structure de groupe sur \m{G_L} de la fa\c con suivante :
si \m{g,g' \in G_L}, avec
$$g = \pmatrix{g_1 & 0 & . & . & . & 0 \cr
u_{21} & g_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & u_{ij} & & . & . \cr
u_{r1} & . & . & . & . & g_r \cr
} \ , \
g' = \pmatrix{g'_1 & 0 & . & . & . & 0 \cr
u'_{21} & g'_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & u'_{ij} & & . & . \cr
u'_{r1} & . & . & . & . & g'_r \cr
} ,$$
alors
$$g'g = \pmatrix{g''_1 & 0 & . & . & . & 0 \cr
u''_{21} & g''_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & u''_{ij} & & . & . \cr
u''_{r1} & . & . & . & . & g''_r \cr
} ,$$
avec
$$g''_i = g'_i\circ g_i \ \ (1\leq i\leq r),$$
$$u''_{ij} = u'_{ij}\circ g_j + \sum_{1\leq k<i-j} u'_{i,j+k}*u_{j+k,j}
+ g'_i\circ u_{ji} \ \ (1\leq j < i \leq r) .$$
La v\'erification qu'on obtient ainsi une structure de groupe sur
\m{G_L} est imm\'ediate. On d\'efinit une structure de groupe analogue
sur \m{G_R}. Soit
$$G = G_L\times G_R .$$
\subsection{D\'efinition de l'action de \m{G} sur \m{W}}
On va d\'efinir une action \`a gauche de \m{G_L} sur \m{W} et une action
\`a droite de \m{G_R} sur \m{W}. L'action de \m{G} sur \m{W} en d\'ecoule :
si \m{(g,h)\in G} et \m{w\in W}, on a
$$(g,h).w \ = \ h.w.g^{-1} .$$
Soit \ \m{w = (\phi_{li})_{1\leq i\leq r,1\leq l\leq s}\in W} (donc
\m{\phi_{il}} est un application lin\'eaire
\ \m{H_{li}^*\otimes M_i\longrightarrow N_l}). Soit
$$g = \pmatrix{g_1 & 0 & . & . & . & 0 \cr
u_{21} & g_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & u_{ij} & & . & . \cr
u_{r1} & . & . & . & . & g_r \cr
} $$
un \'el\'ement de \m{G_L}. Alors \m{w.g =
(\phi'_{li})_{1\leq i\leq r,1\leq l\leq s}}, o\`u
$$\phi'_{li} = \mathop{\hbox{$\displaystyle\sum$}}\limits_{i\leq j\leq r}\psi_{ijl},$$
\m{\psi_{iil}} \'etant la composition
$$M_i\otimes H_{li}^*\hfl{g_i}{}M_i\otimes H_{li}^*\hfl{\phi_{li}}{}N_l$$
et, si \m{i<j\leq r}, \m{\psi_{ijl}} la composition
$$M_i\otimes H_{li}^*\hfl{u_{ji}}{}M_j\otimes A_{ji}\otimes H_{li}^*
\hfl{}{}M_j\otimes H_{lj}^*\hfl{\phi_{lj}}{}N_l,$$
l'application du milieu \'etant induite par la composition \
\m{H_{lj}\otimes A_{ji}\longrightarrow H_{li}}.
L'action de \m{G_R} est analogue.
\subsection{Notions de (semi-)stabilit\'e}
On veut d\'efinir une notion de {\em (semi-)stabilit\'e} pour les points
de \m{W}. On ne peut pas appliquer la g\'eom\'etrie invariante si
\m{r>1} ou \m{s>1} car le groupe \m{G} n'est pas r\'eductif. On va d\'efinir
deux sous-groupes canoniques de \m{G}.
Soit \m{H_L} (resp. \m{G_{L,red}} ) le sous-groupe de \m{G_L} form\'e des
\'el\'ements
$$\pmatrix{g_1 & 0 & . & . & . & 0 \cr
u_{21} & g_2 & . & . & . & 0 \cr
. & . & . & & & . \cr
. & & & . & & . \cr
. & & u_{ij} & & . & . \cr
u_{r1} & . & . & . & . & g_r \cr
} $$
tels que \ \m{g_i = I_{M_i}} \ pour \m{1\leq i\leq r} (resp. \
\m{u_{ij} = 0} \ pour \m{1\leq j < i\leq r}).
Alors \m{H_L} est un sous-groupe unipotent normal maximal
de \m{G_L}, \m{G_{L,red}} est un sous-groupe r\'eductif de \m{G_L}
et l'inclusion \m{G_{L,red}\subset G_L} induit un isomorphisme
\m{G_{L,red}\simeq G_L/H_L}. On d\'efinit de m\^eme les sous-groupes \m{H_R}
et \m{G_{R,red}} de \m{G_R}.
Maintenant soient
$$H = H_L\times H_R \ , \ G_{red} = G_{L,red}\times G_{R,red} .$$
Alors \m{H} est un sous-groupe unipotent normal maximal de \m{G} et
\m{G_{red}} est un sous-groupe r\'eductif de \m{G}.
L'action de \m{G_{red}} sur \m{W} est un cas particulier des actions
trait\'ees dans \cite{king}. Soient
\m{\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s} des nombres
rationnels positifs tels que
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq r}\lambda_im_i = \mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq s}\mu_ln_l.$$
\begin{defin}
On dit qu'un \'el\'ement \m{(\phi_{li})} de \m{W} est
\m{G_{red}}-semi-stable ( resp. \m{G_{red}}-stable) relativement \`a
\m{(\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s)} si la propri\'et\'e
suivante est v\'erifi\'ee : soient \m{M'_i\subset M_i},
\m{N'_l\subset N_l} des sous-espaces vectoriels tels
que l'un au moins des \m{N'_l} soit distinct de \m{N_l} et que
pour \m{1\leq i\leq r}, \m{1\leq l\leq s}, on ait
$$\phi_{li}(H_{li}^*\otimes M'_i)\subset N'_l.$$
Alors on a
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq r}\lambda_i\dim(M'_i)\ \leq \
\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq s}\mu_l\dim(N'_l) \ \ \ {\rm (resp. \ } <{\rm )} \ .$$
\end{defin}
\begin{defin}
On dit qu'un \'el\'ement \m{x} de \m{W} est
\m{G}-semi-stable ( resp. \m{G}-stable) relativement \`a
\m{(\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s)} si tous les
points de l'orbite \m{H.x} sont \m{G_{red}}-semi-stables ( resp.
\m{G_{red}}-stables) relativement \`a
\m{(\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s)}.
\end{defin}
On note \m{W^{ss}(\lambda_1,\ldots,\lambda_r,\mu_1,\ldots,\mu_s)}
(resp. \m{W^{s}(\lambda_1,\ldots,\lambda_r,\mu_1,\ldots,\mu_s)}), ou
plus simplement \m{W^{ss}} (resp. \m{W^{s}})
si aucune confusion n'est \`a craindre, l'ouvert
de \m{W} constitu\'e des points
\m{G}-semi-stables ( resp. \m{G}-stables) relativement \`a
\m{(\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s)}.
\subsection{Cas d'existence d'un bon quotient projectif}
On donne dans \cite{dr_tr} des conditions suffisantes portant sur
\m{\lambda_1,\ldots,\lambda_r,} \m{\mu_1,\ldots,\mu_s}, pour qu'il existe
un bon quotient
$$\pi : W^{ss}\longrightarrow M = M(\lambda_1,\ldots,\lambda_r,\mu_1,\ldots,\mu_s)$$
par \m{G} avec \m{M} projective. Dans ce cas \m{M} est normale et la
restriction de \m{\pi}
$$W^{s}\longrightarrow M^s = \pi(W^s)$$
est un quotient g\'eome\'trique. Le r\'esultat le plus g\'en\'eral est assez
compliqu\'e. Rappelons simplement ici le cas des morphismes de type
\m{(2,1)}, le seul qu'on utilisera ici (dans le \paragra~\hskip -2pt 8).
Il faut d'abord d\'efinir certaines constantes. Soit \m{k>0}
un entier. Soient
$$\tau : H_{11}^*\otimes A_{21}\longrightarrow H_{12}^*$$
l'application lin\'eaire d\'eduite de la composition \
\m{H_{12}\otimes A_{21}\longrightarrow H_{11}}, et
$$\tau_k = \tau_1\otimes I_{\scx{k}} :
H_{11}^*\otimes(A_{21}\otimes\cx{k})\longrightarrow H_{12}^*\otimes\cx{k}.$$
Soit \m{\cal K} l'ensemble des sous-espaces vectoriels propres
\ \m{K\subset A_{21}\otimes\cx{k}} \
tels que pour tout sous-espace propre \m{F\subset\cx{k}}, \m{K} ne soit pas
contenu dans \m{A_{21}\otimes F}. Alors posons
$$c(\tau, k) = \mathop{\hbox{$\sup$}}\limits_{K\in{\cal K}}(\q{\mathop{\rm codim}\nolimits(\tau_k(H_{11}^*\otimes K)}
{\mathop{\rm codim}\nolimits(K)}).$$
Dans le cas des morphismes de type \m{(2,1)}, les notions de
semi-stabilit\'e sont d\'efinies \`a partir de triplets
$$(\lambda_1,\lambda_2,\q{1}{n_1})$$
tels que
\ \m{\lambda_1 m_1 + \lambda_2 m_2 = 1}.
Elles d\'ependent donc essentiellement d'un param\`etre. Le r\'esultat
suivant est d\'emontr\'e dans \cite{dr_tr} :
\begin{xtheo}
Il existe un bon quotient projectif
\m{W^{ss}//G} d\`es que
$$\q{\lambda_2}{\lambda_1}>\dim(A_{21}) \ \ \ {\rm et} \ \
\lambda_2\geq \q{\dim(A_{21})}{n_1} c(\tau,m_2).$$
\end{xtheo}
\subsection{Dualit\'e}
La notion de dualit\'e est claire dans le contexte des morphismes de
faisceaux (en supposant qu'ils sont localement libres). Au lieu
d'\'etudier des morphismes
\ \m{{\cal E}\longrightarrow{\cal F}} \
on consid\`ere les morphismes transpos\'es
\ \m{{\cal F}^*\longrightarrow{\cal E}^*}.
Dans le cas g\'en\'eral, on pose \m{r'=s, s'=r},
\noindent\m{A'_{ij}=B_{s+1-j,s+1-i}},
\m{B'_{lm}=A_{r+1-m,r+1-l}},
\noindent\m{H'_{li}=H_{s+1-i,r+1-l}}, les compositions
sont les m\^emes. On prend \m{M'_i = N_{s+1-i}^*},
\noindent\m{N'_l=M_{r+1-l}^*}.
L'espace associ\'e \m{W'} est isomorphe \`a \m{W}, le facteur
\m{\mathop{\rm Hom}\nolimits(M_i\otimes H_{li},N_l)} s'identifiant \`a \m{\mathop{\rm Hom}\nolimits(M'_{s+1-l}\otimes
H'_{r+1-i,s+1-l},N'_{r+1-i})}. Le groupe \m{G'} est le m\^eme (sauf pour
l'ordre des facteurs, c'est-\`a-dire \m{G'_L = G_R} et \m{G'_R = G_L}). Les
actions des groupes sont bien s\^ur les m\^emes.
\bigskip
\bigskip
\section{Mutations d\'efinies \`a l'aide de la suite spectrale de
Beilinson g\'en\'eralis\'ee}
\subsection{Rappels sur les suites spectrales de Beilinson
g\'en\'eralis\'ees sur les espaces projectifs}
Les d\'efinitions et propri\'et\'es de base des h\'elices de fibr\'es exceptionnels sur
\m{\proj{n}} se trouvent dans \cite{go_ru}.
\subsubsection{H\'elices de fibr\'es exceptionnels sur \m{\proj{n}}}
Une {\em h\'elice} $\gamma = (E_i)_{i \in Z \hskip -4pt Z}$ de fibr\'es exceptionnels
sur \m{\proj{n}} poss\`ede les propri\'et\'es suivantes :
\medskip
\noindent 1) C'est une suite {\em p\'eriodique}, c'est-\`a-dire qu'on a
\ \m{E_{i+n+1} \simeq E_i(n+1)} \ pour tout entier \m{i}.
\medskip
\noindent 2) On a \ \m{\chi(E_i,E_j) = 0} \ si \ \m{j < i \leq j+n} .
\medskip
\noindent 3) Pour tout entier \m{i}, le morphisme canonique
$$ev : E_{i-1}\otimes \mathop{\rm Hom}\nolimits(E_{i-1},E_i)\longrightarrow E_i, \ \ \
\rm{( resp. \ } ev^* : E_i\longrightarrow E_{i+1}\otimes\mathop{\rm Hom}\nolimits(E_i,E_{i+1})^* \rm{ )}$$
\noindent est surjectif (resp. injectif) et son noyau (resp. conoyau) est un
fibr\'e exceptionnel $E$ (resp. $F$). De plus, la suite p\'eriodique de fibr\'es vectoriels
bas\'ee sur
$$(E_{i-2},E,E_{i-1},E_{i+1},\ldots,E_{i+n-2})
\ \ \ {\rm (resp. } \ \
(E_{i-1},E_{i+1},F,E_{i+2},\ldots,E_{i+n-1})\ {\rm)}$$
est une h\'elice
(le terme $E$ (resp. $F$) \'etant d'indice \m{i-1} (resp. \m{i+1})).
Cette h\'elice s'appelle {\em mutation \`a gauche} (resp. {\em mutation \`a
droite} de $\gamma$ en \m{E_i}, et est not\'ee \m{L_{E_i}(\gamma)} (resp.
\m{R_{E_i}(\gamma)}). Le fibr\'e exceptionnel \m{E} (resp. \m{F}) est not\'e
\m{L_\gamma(E_i)} (resp. \m{R_\gamma(E_i)}).
\medskip
\noindent 4) On pose \ \m{L^2_{E_i}(\gamma) = L_E\circ L_{E_i}(\gamma)}.
C'est une h\'elice ayant pour base une suite de la forme
$$(E_{i-3},E',E_{i-2},E_{i-1},E_{i+1},\ldots,E_{i+n-3}).$$
On d\'efinit de
m\^eme \m{L^p_{E_i}(\gamma)} pour tout entier p tel que \
\m{1\leq p < n}. C'est la suite infinie p\'eriodique bas\'ee sur une
suite du type
$$(E_{i-p-1},E^{(p)},E_{i-p},\ldots,E_{i-1},E_{i+1},\ldots,
E_{i+n-p-1}),$$
\m{E^{(p)}} \'etant un fibr\'e exceptionnel et d'indice \m{i-p}. En
particulier, \m{L^{n-1}_{E_i}(\gamma)} est bas\'ee sur la suite
$$(E_{i-n},E^{(n-1)},E_{i-n+1},\ldots,E_{i-1}). $$
L'h\'elice
$$L^n_{E_i}(\gamma) = L_{E^{(n-1)}}\circ L^{(n-1)}(\gamma)$$
est bas\'ee sur une suite du type
$$(E^{(n)},E_{i-n},\ldots,E_{i-1}),$$
\m{E^{(n)}} \'etant un fibr\'e exceptionnel , d'indice \m{i-n}. Alors on a
$$E^{(n)} \simeq E_{i-n-1},$$
c'est-\`a-dire que \m{L^n_{E_i}(\gamma)} est \'egale \`a \m{\gamma} \`a un
d\'ecalage pr\`es. On notera
$$L^p_\gamma(E_i) = E^{(p)}, $$
et en consid\'erant les mutations \`a droite on d\'efinit de m\^eme les
fibr\'es exceptionnels \m{R^p_\gamma(E_i)}.
\medskip
\noindent 5) On a \ $L_{E_i}\circ L^2_{E_{i+1}}(\gamma) = L^2_{E_{i+1}}\circ
L_{E_i}(\gamma)$ .
\bigskip
On a bien s\^ur des propri\'et\'es analogues \`a 4) et 5) concernant les
mutations \`a droite. L'h\'elice la plus simple est
$$({\cal O}(i))_{i \in Z \hskip -4pt Z}$$
et toutes les h\'elices de fibr\'es exceptionnels connues
peuvent s'obtenir en partant de cette h\'elice et en lui faisant subir une
suite finie de mutations et une translation des indices.
Une {\em base} de l'h\'elice
\m{\gamma = (E_i)_{i \in Z \hskip -4pt Z}} de fibr\'es exceptionnels sur \m{\proj{n}} est
une suite
$$\sigma = (E_i,\ldots,E_{i+n})$$
extraite de \m{\gamma}. A cause de la propri\'et\'e 1-, \m{\gamma} peut
\^etre reconstitu\'ee \`a partir de \m{\sigma}. Les
notions de mutations \`a droite et \`a gauche s'\'etendent de mani\`ere
\'evidente aux bases d'h\'elice. Si \m{i< j< n}, on note
$$L_{j+1}^1(\sigma) =
(E_i,\ldots, E_{j-1}, L^1_\gamma(E_{j+1}), E_j,E_{j+2},\ldots, E_n),$$
$$R_j^1(\sigma) =
(E_i,\ldots, E_{j-1}, E_{j+1}, R^1_\gamma(E_j),E_{j+2},\ldots, E_n).$$
Plus g\'en\'eralement, si \m{1\leq p\leq j-i}, on pose
$$L_{j+1}^p(\sigma) =
(E_i,\ldots, E_{j-p}, L^p_\gamma(E_{j+1}),E_{j-p+1},\ldots,
E_j,E_{j+2},\ldots, E_n),$$
et si \m{1\leq q\leq n-j-1},
$$R_j^q(\sigma) = (E_i,\ldots, E_{j-1}, E_{j+1},\ldots, E_{j+q},
R^q_\gamma(E_j),E_{j+q+1},\ldots, E_n).$$
\subsubsection{Suite spectrale de Beilinson g\'en\'eralis\'ee}
\subsubsubsection{4.1.2.1 -}{D\'efinition}
Soit \ \m{\sigma = (E_0,\ldots,E_n)} \ une base d'h\'elice sur \m{\proj{n}}.
On associe \`a \m{\sigma} une autre base d'h\'elice, dite {\em duale} de
\m{\sigma}, et not\'ee
$$\sigma^* = (E_{\sigma 0},\ldots,E_{\sigma n}),$$
d\'efinie par
$$E_{\sigma p} = L^p_\sigma(E_{p})^*(-n-1) = R^{n-p}_\sigma(E_{p})^* .$$
C'est une base d'une autre h\'elice que \m{\gamma}.
Si \m{\gamma} est l'h\'elice engendr\'ee par \m{\sigma}, on note
\m{\gamma^*} l'h\'elice engendr\'ee par \m{\sigma^*}.
On montre qu'il existe
une r\'esolution canonique de la diagonale
\m{\Delta} de \ \m{\proj{n}\times\proj{n}} :
$$0\longrightarrow E_0\timex E_{\sigma 0}\longrightarrow\cdots\longrightarrow E_{n}\timex E_{\sigma,n}
\longrightarrow{\cal O}_{\Delta}\longrightarrow 0 .$$
On en d\'eduit, pour tout faisceau coh\'erent \m{{\cal E}} sur \m{\proj{n}}, une
suite spectrale \m{E_r^{pq}} de faisceaux coh\'erents sur \m{\proj{n}},
convergeant vers \m{{\cal E}} en degr\'e \m{0} et vers \m{0} en tout autre
degr\'e, et dont les termes \m{E_1^{p,q}} pouvant \'eventuellement \^etre
non nuls sont les
$$E_1^{p,q} = E_{p+n}\otimes H^q(E_{\sigma,p+n}\otimes{\cal E}) \ ,
\ -n \leq p\leq 0 \ , \ 0\leq q\leq n .$$
On en d\'eduit le {\em complexe de Beilinson} :
$$0\longrightarrow{\cal F}_{-n}\longrightarrow{\cal F}_{-n+1}\longrightarrow\ldots\longrightarrow{\cal F}_{n-1}\longrightarrow
{\cal F}_n\longrightarrow 0,$$
o\`u \ \m{{\cal F}_i = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+q=i} E_1^{pq}}. Il est exact en degr\'e
diff\'erent
de \m{0}, et sa cohomologie en degr\'e \m{0} est isomorphe \`a \m{{\cal E}}.
\bigskip
\subsubsubsection{4.1.2.2 -}{Bases duales et mutations}
Soit \ \m{\sigma = (E_0,\ldots,E_n)} \ une base d'h\'elice, et \m{j} un
entier tel que \ \m{1\leq j< n}. Alors on a
$$L_{j+1}(\sigma)^* = R_j(\sigma^*),
\ \ \ R_j(\sigma)^* = L_{j+1}(\sigma^*).$$
\bigskip
\subsection{Mutations de morphismes}
Soient \m{r,s} des entiers positifs, et \m{n=r+s-1}.
Soit
$$\sigma = (E_1,\ldots,E_r,F_1,\ldots,F_s)$$
une base d'h\'elice de fibr\'es exceptionnels sur
$\proj{n}$, \m{\gamma} l'h\'elice engendr\'ee par \m{\sigma}.
Pour \m{1\leq i\leq r}, le morphisme canonique de fibr\'es vectoriels
$$E_i\longrightarrow F_1\otimes \mathop{\rm Hom}\nolimits(E_i,F_1)^*$$
est surjectif, et son conoyau \m{G_i} est un fibr\'e exceptionnel.
On va d\'efinir une suite \m{\sigma_r}, \m{\sigma_{r-1}}, \m{\ldots},
\m{\sigma_0} de bases d'h\'elice par
$$\sigma_r=\sigma,$$
et si \m{p} est un entier tel que \m{0\leq p\leq r-1},
$$\sigma_p = R_{p}(\sigma_{p+1}).$$
On a
$$\sigma_p =
(E_1,\ldots,E_p,F_1,G_{p+1},\ldots,G_r,F_2,\ldots, F_s).$$
Il d\'ecoule du \paragra~\hskip -2pt 3.2.2 qu'on a
$$\sigma_p^* = (E_{\sigma 0},\ldots,E_{\sigma,p-1},
L^p_{\gamma^*}(E_{\sigma r}),E_{\sigma p},\ldots, E_{\sigma n}).$$
En utilisant la suite spectrale de Beilinsion g\'en\'eralis\'ee associ\'ee
\`a \m{\sigma}, on d\'emontre ais\'ement ce qui suit : soit ${\cal U}$ un
faisceau coh\'erent sur $\proj{n}$ tel que
$$H^j({\cal U}\otimes E_{\sigma i}) = \lbrace 0\rbrace$$
si \m{0\leq i<r} et \m{j\not = n-i-1}, ou \m{r\leq i\leq n} et
\m{j\not = n-i}. On pose
$$M_i = H^{n-i}({\cal U}\otimes E_{\sigma,i-1}) \ \ \ {\rm pour\ \ }
1\leq i\leq r,$$
$$N_l = H^{n-r-l+1}({\cal U}\otimes E_{\sigma,l+r-1}) \ \ \ {\rm pour\ \ }
1\leq l\leq s,$$
de telle sorte que le diagramme de Beilinson de \m{\cal U} a l'allure
suivante
$$\matrix{
0 & . & . & . & 0 & 0 & . & . & . & 0 \cr
M_1 & & & & 0 & 0 & . & . & . & 0 \cr
0 & . & & & . & . & & & & . \cr
. & & . & & . & . & & & & . \cr
. & & & . & 0 & 0 & & & & . \cr
0 & . & . & 0 & M_r & N_1 & 0 & . & . & 0 \cr
. & & & & 0 & 0 & . & & & . \cr
. & & & & . & . & & . & & . \cr
. & & & & . & . & & & . & . \cr
0 & & & & . & . & & & & N_s \cr
}$$
Alors il existe une suite exacte
$$0\longrightarrow\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}(E_i\otimes M_i) \hfl{\Phi}{}
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}(F_l\otimes N_l)\longrightarrow{\cal U}\longrightarrow 0.$$
Soit \m{p} un entier tel que \m{0\leq p\leq r-1}. On note
$$f_p : \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(\mathop{\rm Hom}\nolimits(E_j,F_1)^*\otimes M_j)\longrightarrow N_1$$
l' application lin\'eaire d\'eduite de \m{\Phi}.
Alors, si $f_p$ est surjective, on peut montrer que la suite spectrale
de Beilinson g\'en\'eralis\'ee associ\'ee a \m{\sigma_p},
appliqu\'ee \`a ${\cal U}$, donne une suite exacte
$$0\longrightarrow \biggl(\bigoplus_{1\leq i\leq p}(E_i\otimes M_i)\biggr)\oplus
(F_1\otimes\ker(f_p))\longrightarrow
\biggr(\bigoplus_{p<j\leq r}(G_j\otimes M_j)\biggl)\oplus
\biggr(\bigoplus_{2\leq l\leq s}(F_l\otimes N_l)\biggl)\longrightarrow{\cal U}\longrightarrow 0.$$
On a aussi bien s\^ur un \'enonc\'e r\'eciproque.
Ce r\'esultat va \^etre g\'en\'eralis\'e dans le chapitre suivant.
\bigskip
\bigskip
\section{Mutations en termes de morphismes de faisceaux}
On va d'abord d\'emontrer deux r\'esultats, qu'on appliquera ensuite \`a
la d\'efinition des mutations de morphismes de type \m{(r,s)}. On \'etudie
des faisceaux coh\'erents pouvant \^etre repr\'esent\'es comme conoyaux
de morphismes injectifs de faisceaux d'un certain type. Une \'etude
similaire pourrait sans doute \^etre faite sur les noyaux.
\subsection{R\'esultats g\'en\'eraux}
Soient \m{{\cal E}}, \m{{\cal E}'}, \m{{\cal F}}, \m{{\cal F}'} et \m{\Gamma} des faisceaux
coh\'erents sur une vari\'et\'e projective irr\'eductible \m{X}, avec
\m{\Gamma} simple. On
suppose que le morphisme canonique
$$ev : \Gamma\otimes \mathop{\rm Hom}\nolimits(\Gamma,{\cal F})\longrightarrow {\cal F}$$
est surjectif. Soit \m{{\cal E}_0} son noyau. On suppose que le morphisme
canonique
$$ev^* : {\cal E}'\longrightarrow \Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*$$
est injectif. Soit \m{{\cal F}_0} son conoyau. On suppose enfin que
$$\mathop{\rm Hom}\nolimits({\cal E}',{\cal E}_0) = \mathop{\rm Ext}\nolimits^1({\cal E}',{\cal E}_0) = \mathop{\rm Ext}\nolimits^1({\cal F}_0,{\cal F}') =
\mathop{\rm Ext}\nolimits^1({\cal E},{\cal E}_0) = \lbrace 0\rbrace.$$
De la suite exacte
$$0\longrightarrow{\cal E}_0\longrightarrow\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal F})\longrightarrow{\cal F}\longrightarrow 0$$
on d\'eduit un isomorphisme
$$\mathop{\rm Hom}\nolimits({\cal E}',{\cal F}) \ \simeq \ \mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*,\mathop{\rm Hom}\nolimits(\Gamma,{\cal F})).$$
Si \ \m{\lambda\in\mathop{\rm Hom}\nolimits(\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*,\mathop{\rm Hom}\nolimits(\Gamma,{\cal F}))}, le morphisme
\ \m{{\cal E}'\longrightarrow{\cal F}} \ correspondant est la compos\'ee
$${\cal E}'\hfl{ev^*}{}\Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\hfl{I_\Gamma\otimes\lambda}{}
\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal F})\hfl{ev}{}{\cal F}.$$
\bigskip
\begin{xprop}
Soit
$$\Phi : {\cal E}\oplus{\cal E}'\longrightarrow{\cal F}\oplus{\cal F}'$$
un morphisme injectif de faisceaux, et \m{\cal U} son conoyau. Soit
$$\lambda : \mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\longrightarrow\mathop{\rm Hom}\nolimits(\Gamma,{\cal F})$$
l'application lin\'eaire d\'eduite du morphisme \ \m{{\cal E}'\longrightarrow{\cal F}} \ d\'efini
par \m{\Phi}.
\medskip
\noindent 1 - On suppose que \m{\lambda} est surjective et que
$$\mathop{\rm Ext}\nolimits^1({\cal E},\Gamma) = \lbrace 0\rbrace.$$
Alors il existe une suite exacte
$$0\longrightarrow {\cal E}\oplus{\cal E}_0\oplus(\Gamma\otimes\ker(\lambda))\longrightarrow{\cal F}_0\oplus{\cal F}'
\longrightarrow{\cal U}\longrightarrow 0.$$
\medskip
\noindent 2 - On suppose que \m{\lambda} est injective et que
$$\mathop{\rm Ext}\nolimits^1(\Gamma,{\cal F}_0) = \mathop{\rm Ext}\nolimits^1(\Gamma,{\cal F}') = \lbrace 0\rbrace.$$
Alors il existe une suite exacte
$$0\longrightarrow {\cal E}\oplus{\cal E}_0\longrightarrow(\Gamma\otimes\mathop{\rm coker}\nolimits(\lambda))\oplus{\cal F}_0\oplus{\cal F}'
\longrightarrow{\cal U}\longrightarrow 0.$$
\end{xprop}
\noindent{\em D\'emonstration}. On consid\`ere le morphisme
$$A : {\cal E}'\longrightarrow (\Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*)\oplus{\cal F}' = {\cal A}$$
dont la premi\`ere composante est \m{ev^*} et la seconde provient de
\m{\Phi}. On a un diagramme commutatif avec lignes et colonnes exactes :
\bigskip
\begin{picture}(360,230)
\put(135,220){$0$}
\put(280,220){$0$}
\put(10,170){$0$}
\put(55,170){${\cal E}'$}
\put(115,170){$\Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*$}
\put(280,170){${\cal F}_0$}
\put(365,170){$0$}
\put(10,120){$0$}
\put(55,120){${\cal E}'$}
\put(135,120){$\cal A$}
\put(270,120){$\mathop{\rm coker}\nolimits(A)$}
\put(365,120){$0$}
\put(135,70){${\cal F}'$}
\put(280,70){${\cal F}'$}
\put(135,20){$0$}
\put(280,20){$0$}
\put(137,184){\vector(0,1){30}}
\put(137,134){\vector(0,1){30}}
\put(137,84){\vector(0,1){30}}
\put(137,34){\vector(0,1){25}}
\put(282,184){\vector(0,1){30}}
\put(282,134){\vector(0,1){30}}
\put(282,84){\vector(0,1){30}}
\put(282,34){\vector(0,1){30}}
\put(56,163){\line(0,-1){29}}
\put(58,163){\line(0,-1){29}}
\put(70,173){\vector(1,0){35}}
\put(295,173){\vector(1,0){63}}
\put(23,173){\vector(1,0){28}}
\put(204,173){\vector(1,0){67}}
\put(23,123){\vector(1,0){28}}
\put(70,123){\vector(1,0){56}}
\put(90,126){A}
\put(152,123){\vector(1,0){111}}
\put(323,123){\vector(1,0){35}}
\put(150,73){\line(1,0){122}}
\put(150,71){\line(1,0){122}}
\end{picture}
\bigskip
Puisque \ \m{\mathop{\rm Ext}\nolimits^1({\cal F}_0,{\cal F}') = \lbrace 0\rbrace}, on a un isomorphisme
$$\mathop{\rm coker}\nolimits(A) \ \simeq \ {\cal F}'\oplus{\cal F}_0 .$$
On suppose maintenant que les hypoth\`eses de 1- sont v\'erifi\'ees. Soit
$$\pi : \Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\longrightarrow{\cal F}$$
le morphisme compos\'e
$$\Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\hfl{I_\Gamma\otimes\lambda}{}
\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal F})\hfl{ev}{}{\cal F}.$$
Alors on a
$$\ker(\pi)\simeq{\cal E}_0\oplus(\ker(\lambda)\otimes\Gamma).$$
Soit \m{\cal V} le conoyau du morphisme injectif
$${\cal E}'\longrightarrow{\cal F}\oplus{\cal F}'$$
d\'eduit de \m{\Phi}. On a un diagramme commutatif avec lignes et colonnes exactes :
\bigskip
\begin{picture}(360,230)
\put(135,220){$0$}
\put(280,220){$0$}
\put(10,170){$0$}
\put(55,170){${\cal E}'$}
\put(123,170){${\cal F}\oplus{\cal F}'$}
\put(280,170){$\cal V$}
\put(365,170){$0$}
\put(10,120){$0$}
\put(55,120){${\cal E}'$}
\put(135,120){$\cal A$}
\put(262,120){${\cal F}'\oplus{\cal F}_0$}
\put(365,120){$0$}
\put(125,70){$\ker(\pi)$}
\put(270,70){$\ker(\pi)$}
\put(135,20){$0$}
\put(280,20){$0$}
\put(137,184){\vector(0,1){30}}
\put(137,134){\vector(0,1){30}}
\put(137,84){\vector(0,1){30}}
\put(137,34){\vector(0,1){25}}
\put(140,149){$\pi\oplus I_{{\cal F}'}$}
\put(282,184){\vector(0,1){30}}
\put(282,134){\vector(0,1){30}}
\put(282,84){\vector(0,1){30}}
\put(282,34){\vector(0,1){30}}
\put(56,163){\line(0,-1){29}}
\put(58,163){\line(0,-1){29}}
\put(70,173){\vector(1,0){45}}
\put(295,173){\vector(1,0){63}}
\put(23,173){\vector(1,0){28}}
\put(170,173){\vector(1,0){101}}
\put(23,123){\vector(1,0){28}}
\put(70,123){\vector(1,0){56}}
\put(90,126){A}
\put(152,123){\vector(1,0){103}}
\put(312,123){\vector(1,0){46}}
\put(160,73){\line(1,0){102}}
\put(160,71){\line(1,0){102}}
\end{picture}
\bigskip
On a une suite exacte
$$0\longrightarrow{\cal E}\longrightarrow{\cal V}\longrightarrow{\cal U}\longrightarrow 0,$$
et l'inclusion \ \m{{\cal E}\subset{\cal V}} \ se rel\`eve en un morphisme
injectif
$${\cal E}\longrightarrow{\cal F}'\oplus{\cal F}_0$$
(car \ \m{\mathop{\rm Ext}\nolimits^1({\cal E},{\cal E}_0) = \mathop{\rm Ext}\nolimits^1({\cal E},\Gamma) = \lbrace 0\rbrace}). On
note \m{\cal W} le conoyau de ce morphisme. On
a alors un diagramme commutatif avec lignes et colonnes exactes, dont
la ligne verticale du milieu provient du diagramme pr\'ec\'edent :
\bigskip
\begin{picture}(360,230)
\put(135,220){$0$}
\put(280,220){$0$}
\put(10,170){$0$}
\put(55,170){${\cal E}$}
\put(133,170){$\cal V$}
\put(280,170){$\cal U$}
\put(365,170){$0$}
\put(10,120){$0$}
\put(55,120){${\cal E}$}
\put(120,120){${\cal F}'\oplus{\cal F}_0$}
\put(280,120){$\cal W$}
\put(365,120){$0$}
\put(125,70){$\ker(\pi)$}
\put(270,70){$\ker(\pi)$}
\put(135,20){$0$}
\put(280,20){$0$}
\put(137,184){\vector(0,1){30}}
\put(137,134){\vector(0,1){30}}
\put(137,84){\vector(0,1){30}}
\put(137,34){\vector(0,1){25}}
\put(282,184){\vector(0,1){30}}
\put(282,134){\vector(0,1){30}}
\put(282,84){\vector(0,1){30}}
\put(282,34){\vector(0,1){30}}
\put(56,163){\line(0,-1){29}}
\put(58,163){\line(0,-1){29}}
\put(70,173){\vector(1,0){55}}
\put(295,173){\vector(1,0){63}}
\put(23,173){\vector(1,0){28}}
\put(147,173){\vector(1,0){123}}
\put(23,123){\vector(1,0){28}}
\put(70,123){\vector(1,0){44}}
\put(164,123){\vector(1,0){106}}
\put(298,123){\vector(1,0){60}}
\put(160,73){\line(1,0){102}}
\put(160,71){\line(1,0){102}}
\end{picture}
\bigskip
On en d\'eduit une suite exacte
$$0\longrightarrow{\cal E}\oplus\ker(\pi)\longrightarrow{\cal F}'\oplus{\cal F}_0\longrightarrow{\cal U}\longrightarrow 0.$$
Ceci d\'emontre 1-.
Supposons maintenant que les hypoth\`eses de 2- soient v\'erifi\'ees.
Soient
$${\cal B} = (\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal F}))\oplus{\cal F}'.$$
On consid\`ere le morphisme injectif
$$B : {\cal E}'\longrightarrow{\cal B}$$
dont la premi\`ere composante est la compos\'ee
$${\cal E}'\hfl{ev^*}{}\Gamma\otimes\mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\hfl{\lambda}{}
\Gamma\otimes\mathop{\rm Hom}\nolimits(\Gamma,{\cal F})$$
et dont la seconde provient de \m{\Phi}. On a un diagramme commutatif avec lignes et colonnes exactes :
\bigskip
\begin{picture}(360,230)
\put(135,220){$0$}
\put(280,220){$0$}
\put(10,170){$0$}
\put(55,170){${\cal E}'$}
\put(133,170){$\cal A$}
\put(266,170){${\cal F}_0\oplus{\cal F}'$}
\put(365,170){$0$}
\put(10,120){$0$}
\put(55,120){${\cal E}'$}
\put(133,120){$\cal B$}
\put(84,125){$B$}
\put(265,120){$\mathop{\rm coker}\nolimits(B)$}
\put(365,120){$0$}
\put(110,70){$\Gamma\otimes\mathop{\rm coker}\nolimits(\lambda)$}
\put(255,70){$\Gamma\otimes\mathop{\rm coker}\nolimits(\lambda)$}
\put(135,20){$0$}
\put(280,20){$0$}
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\put(137,164){\vector(0,-1){30}}
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\end{picture}
\bigskip
Puisque \ \m{\mathop{\rm Ext}\nolimits^1(\Gamma,{\cal F}_0) = \mathop{\rm Ext}\nolimits^1(\Gamma,{\cal F}') =
\lbrace 0\rbrace}, on a un isomorphisme
$$\mathop{\rm coker}\nolimits(B) \ \simeq \ (\Gamma\otimes\mathop{\rm coker}\nolimits(\lambda))\oplus{\cal F}_0\oplus{\cal F}'.$$
Le carr\'e commutatif
$$\diagram{
{\cal E}' & \hfl{B}{} & {\cal B} \cr
\vfl{}{} & & \vfl{}{ev\oplus I_{{\cal F}'}} \cr
{\cal E}\oplus{\cal E}' & \hfl{\Phi}{} & {\cal F}\oplus{\cal F}' \cr
}$$
induit un morphisme surjectif
$$\rho : \mathop{\rm coker}\nolimits(B)\longrightarrow\mathop{\rm coker}\nolimits(\Phi) = {\cal U},$$
et une suite exacte
$$0\longrightarrow{\cal E}_0\longrightarrow\ker(\rho)\longrightarrow{\cal E}\longrightarrow 0.$$
Comme \ \m{\mathop{\rm Ext}\nolimits^1({\cal E},{\cal E}_0)=\lbrace 0\rbrace}, on a un isomorphisme
$$\ker(\rho) \ \simeq \ {\cal E}\oplus{\cal E}_0.$$
On a donc une suite exacte
$$0\longrightarrow{\cal E}\oplus{\cal E}_0\longrightarrow
(\Gamma\otimes\mathop{\rm coker}\nolimits(\lambda))\oplus{\cal F}_0\oplus{\cal F}'\longrightarrow 0.$$
Ceci d\'emontre 2-. $\Box$
\bigskip
Les cas particuliers \ \m{{\cal E}_0 = 0} \ ou \m{{\cal F}_0 = 0} suivants seront
utilis\'es par la suite. Soit \m{M} un espace vectoriel de dimension finie.
\begin{xcoro}
1 - On suppose que \ \m{\mathop{\rm Ext}\nolimits^1({\cal E},\Gamma) = \lbrace 0\rbrace}. Soient
$$\Phi : {\cal E}\oplus{\cal E}'\longrightarrow(\Gamma\otimes M)\oplus{\cal F}'$$
un morphisme injectif induisant une surjection
$$\lambda : \mathop{\rm Hom}\nolimits({\cal E}',\Gamma)^*\longrightarrow M,$$
et \ \m{{\cal U}=\mathop{\rm coker}\nolimits(\Phi)}. Alors il existe une
suite exacte
$$0\longrightarrow{\cal E}\oplus(\Gamma\otimes\ker(\lambda))\longrightarrow{\cal F}_0\oplus{\cal F}'\longrightarrow
{\cal U}\longrightarrow 0.$$
\medskip
\noindent 2- On suppose que \ \m{\mathop{\rm Ext}\nolimits^1(\Gamma,{\cal F}') = \lbrace 0\rbrace}.
Soient$$\Phi : {\cal E}\oplus(\Gamma\otimes M)\longrightarrow{\cal F}\oplus{\cal F}'$$
un morphisme injectif induisant une injection
$$\lambda : M\longrightarrow\mathop{\rm Hom}\nolimits(\Gamma,{\cal F}),$$
et \ \m{{\cal U}=\mathop{\rm coker}\nolimits(\Phi)}.
Alors il existe une
suite exacte
$$0\longrightarrow{\cal E}\oplus{\cal E}_0\longrightarrow(\Gamma\otimes\mathop{\rm coker}\nolimits(\lambda))\oplus{\cal F}'\longrightarrow
{\cal U}\longrightarrow 0.$$
\end{xcoro}
\subsection{Applications}
Soient \m{X} une vari\'et\'e projective, \m{r,s} des entiers positifs, et
\m{{\cal E}_1,\ldots,{\cal E}_r,},\m{{\cal F}_1,\ldots,{\cal F}_s} des
faisceaux coh\'erents simples sur \m{X} tels que
$$\mathop{\rm Hom}\nolimits({\cal E}_i,{\cal E}_{i'}) = 0 \ \ {\rm si \ } i > i' \ , \
\mathop{\rm Hom}\nolimits({\cal F}_j,{\cal F}_{j'}) = 0 \ \ {\rm si \ } j > j', $$
$$\mathop{\rm Hom}\nolimits({\cal F}_j,{\cal E}_i) = \lbrace 0 \rbrace \ \
{\rm pour \ tous \ } i,j .$$
On suppose que pour \m{1\leq i\leq r} le morphisme canonique
$${\cal E}_i\longrightarrow \mathop{\rm Hom}\nolimits({\cal E}_i,{\cal F}_1)^*\otimes {\cal F}_1$$
est injectif. Soit \m{{\cal G}_i} son conoyau. Du corollaire 5.2, 1-, on
d\'eduit la
\begin{xprop}
1 - Soient
$$\Phi : \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}({\cal E}_i\otimes M_i) \longrightarrow
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}({\cal F}_l\otimes N_l)$$
un morphisme injectif, \m{\cal U} son conoyau
et \m{p} un entier tel que \m{0\leq p\leq r-1}. On suppose que
$$\mathop{\rm Ext}\nolimits^1({\cal G}_j,{\cal F}_l) = \mathop{\rm Ext}\nolimits^1({\cal E}_i,{\cal F}_1) =
\lbrace 0\rbrace$$
pour \m{p+1\leq j\leq r}, \m{1\leq i\leq p} et \m{2\leq l\leq s}.
Soit
$$f_p : \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(\mathop{\rm Hom}\nolimits({\cal E}_j,{\cal F}_1)^*\otimes M_j)\longrightarrow N_1$$
l'application lin\'eaire d\'eduite de $\Phi$. On suppose que $f_p$ est
surjective. Alors il existe une suite exacte
$$0\longrightarrow \biggl(\bigoplus_{1\leq i\leq p}({\cal E}_i\otimes M_i)\biggr)\oplus
({\cal F}_1\otimes\ker(f_p))\longrightarrow
\biggr(\bigoplus_{p<j\leq r}({\cal G}_j\otimes M_j)\biggl)\oplus
\biggr(\bigoplus_{2\leq l\leq s}({\cal F}_l\otimes N_l)\biggl)\longrightarrow{\cal U}\longrightarrow 0.$$
\end{xprop}
\bigskip
Du corollaire 5.2, 2-, on d\'eduit la
\begin{xprop}
Soient \m{P_1} un espace vectoriel de dimension finie,
$$\Psi : \biggl(\bigoplus_{1\leq i\leq p}({\cal E}_i\otimes M_i)\biggr)\oplus
({\cal F}_1\otimes P_1)\longrightarrow
\biggr(\bigoplus_{p<j\leq r}({\cal G}_j\otimes M_j)\biggl)\oplus
\biggr(\bigoplus_{2\leq l\leq s}({\cal F}_l\otimes N_l)\biggl)$$
un morphisme injectif et \m{\cal U} son conoyau. On suppose que
$$\mathop{\rm Ext}\nolimits^1({\cal F}_1,{\cal E}_i) = \mathop{\rm Ext}\nolimits^1({\cal E}_i,{\cal E}_j) = \mathop{\rm Ext}\nolimits^1({\cal F}_1,{\cal F}_l) =
\lbrace 0\rbrace$$
pour \m{1\leq i\leq p}, \m{p+1\leq j\leq r}, \m{2\leq l\leq s}. Soit
$$g : P_1\longrightarrow \bigoplus_{p+1\leq j\leq r}\biggl(
\mathop{\rm Hom}\nolimits({\cal F}_1,{\cal G}_j)\otimes M_j\biggr)$$
l'application lin\'eaire d\'eduite de \m{\Psi}. On suppose \m{g} injective.
Alors il existe une suite exacte
$$0\longrightarrow\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}({\cal E}_i\otimes M_i) \longrightarrow ({\cal F}_1\otimes\mathop{\rm coker}\nolimits(g))\oplus
\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}({\cal F}_l\otimes N_l)\biggr)\longrightarrow{\cal U}\longrightarrow 0.$$
\end{xprop}
\vfil
\eject
\section{Mutations abstraites}
\subsection{Espaces abstraits de morphismes}
\subsubsection{D\'efinition g\'en\'erale}
Soient \m{\X{1}}, \m{\X{2}}, \m{\X{3}}, \m{\X{4}}, \m{M} , \m{H_L}, \m{H_R}
des espaces vectoriels sur un corps commutatif \m{k}, de dimension finie,
avec
$$\dim(M) < \dim(\X{2}).$$
On pose
$$W = (\X{1}\otimes M)\oplus(\X{2}\otimes M)\oplus \X{3}\oplus\X{4}.$$
Soient \m{\GG{0}}, \m{\GG{1}}, \m{\GG{2}} des groupes. On suppose que :
\begin{itemize}
\item[] $\GG{0}$ op\`ere lin\'eairement \`a gauche sur \m{\X{3}}, \m{\X{4}},
\m{H_R}.
\item[] $\GG{1}$ op\`ere lin\'eairement \`a droite sur \m{\X{1}}, \m{\X{3}},
\m{H_L}.
\item[] $\GG{2}$ op\`ere lin\'eairement \`a droite sur \m{\X{2}}, \m{\X{4}},
et \`a gauche sur \m{H_L}.
\end{itemize}
On suppose que ces actions sont {\em compatibles}, c'est-\`a-dire que
si deux de ces groupes \m{G_\alpha}, \m{G_\beta}, op\`erent sur un
m\^eme espace vectoriel \m{Z}, \`a gauche et \`a droite respectivement,
on a, pour tous \m{g_\alpha\in G_\alpha},\m{g_\beta\in G_\beta} et
\m{z\in Z},
$$g_\alpha(zg_\beta) = (g_\alpha z)g_\beta.$$
On suppose aussi que le groupe \m{\lbrace1,-1\rbrace} est contenu dans
\m{\GG{1}}, \m{\GG{2}} et \m{\GG{3}}, et agit comme on le pense sur les
espaces vectoriels sur lesquels ces groupes agissent (c'est-\`a-dire
que \m{-1} agit par multiplication par \m{-1}).
Soient
$$\g{3} : H_R\otimes \X{1}\longrightarrow \X{3},$$
$$\g{4} : H_R\otimes \X{2}\longrightarrow \X{4},$$
$$\g{1} : \X{2}\otimes H_L\longrightarrow \X{1},$$
$$\g{2} : \X{4}\otimes H_L\longrightarrow \X{3}$$
des applications lin\'eaires. On suppose que le diagramme suivant \m{(D)}
est commutatif :
$$\diagram{
H_R\otimes \X{2}\otimes H_L & \hfl{I_{H_R}\otimes \g{1}}{} & H_R\otimes \X{1}\cr
\vfl{\g{4}\otimes I_{H_L}}{} & & \vfl{}{\g{3}} \cr
\X{4}\otimes H_L & \hfl{\g{2}}{} & \X{3}
}$$
On suppose aussi que ces applications lin\'eaires sont compatibles avec
l'action des groupes. Par exemple \m{\GG{1}} op\`ere \`a droite sur \m{\X{1}}
et \m{H_L}, donc pour tous \m{\gg{1}\in \GG{1}}, \m{h_L\in H_L} et \m{y_1\in
\X{2}}
on a
$$\g{1}(y_1\otimes (h_L\gg{1})) = \g{1}(y_1\otimes h_L).\gg{1}.$$
De m\^eme, \m{\GG{2}} op\`ere \`a droite sur \m{\X{2}} et \`a gauche sur
\m{H_L}, donc pour tous \m{\gg{2}\in \GG{2}}, \m{h_L\in H_L} et \m{y_1\in
\X{2}}
on a
$$\g{1}(y_1\gg{2}\otimes h_L) = \g{1}(y_1\otimes \gg{2}h_L).$$
On suppose aussi que \m{\g{4}} est surjective, et que l'application
lin\'eaire
$$\ov{\g{1}} : H_L\longrightarrow \X{2}^*\otimes \X{1}$$
d\'eduite de \m{\g{1}} est injective.
\begin{defin}
On appelle {\em espace abstrait de morphismes} et on note \m{\Theta}
la donn\'ee de \m{\X{1}}, \m{\X{3}}, \m{\X{2}}, \m{\X{4}}, \m{H_L}, \m{H_R},
\m{\GG{0}}, \m{\GG{1}}, \m{\GG{2}}, \m{\g{1}}, \m{\g{2}}, \m{\g{3}} et
\m{\g{4}}.
L'espace vectoriel
$$W = (\X{1}\otimes M)\oplus(\X{2}\otimes M)\oplus \X{3}\oplus \X{4}$$
est {\em l'espace total} de \m{\Theta}.
\end{defin}
\subsubsection{Dictionnaire}
Si on \'etudie les morphismes
$${\cal E}\oplus{\cal E}'\longrightarrow(\Gamma\otimes M)\oplus{\cal F}',$$
l'espace abstrait de morphismes associ\'e est d\'efini par
$$\X{1} = \mathop{\rm Hom}\nolimits({\cal E},\Gamma), \ \X{2} = \mathop{\rm Hom}\nolimits({\cal E}',\Gamma),$$
$$\X{3} = \mathop{\rm Hom}\nolimits({\cal E},{\cal F}'), \ \X{4} = \mathop{\rm Hom}\nolimits({\cal E}',{\cal F}'),$$
$$H_L = \mathop{\rm Hom}\nolimits({\cal E},{\cal E}'), \ H_R = \mathop{\rm Hom}\nolimits(\Gamma,{\cal F}'),$$
$$\GG{0} = Aut({\cal F}'), \ \GG{1} = Aut({\cal E}), \ \GG{2} = Aut({\cal E}'),$$
les applications \m{\g{1}}, \m{\g{2}}, \m{\g{3}}, \m{\g{4}}
\'etant les compositions des morphismes. On a dans ce cas
$$W = \mathop{\rm Hom}\nolimits({\cal E}\oplus{\cal E}',(\Gamma\otimes M)\oplus{\cal F}').$$
\subsection{Groupes associ\'es}
On va construire deux nouveaux groupes associ\'es \`a \m{\Theta} :
\m{G_L} et \m{G_R}. Le groupe \m{G_L} est constitu\'e des matrices
$$\pmatrix{
\gg{1} & 0\cr h_L & \gg{2}}$$
avec \m{\gg{1}\in \GG{1}}, \m{\gg{2}\in \GG{2}}, \m{h_L\in H_L}.
La loi de groupe de \m{G_L} est
$$\pmatrix{\gg{1} & 0\cr h_L & \gg{2}}.\pmatrix{\gg{1}' & 0\cr h_L' &
\gg{2}'} =
\pmatrix{\gg{1}\gg{1}' & 0\cr h_L\gg{1}'+ \gg{2}h_L' & \gg{2}\gg{2}'}.$$
Le groupe \m{G_R} est constitu\'e des matrices
$$\pmatrix{
g_M & 0\cr \lambda & g_0}$$
avec \m{g_M\in GL(M)}, \m{g_0\in \GG{0}}, \m{\lambda\in M^*\otimes H_R}.
La loi de groupe de \m{G_R} est
$$\pmatrix{g_M & 0\cr \lambda & g_0}.\pmatrix{g_M' & 0\cr \lambda' & g_0'} =
\pmatrix{g_Mg_M' & 0\cr \lambda g_M'+ g_0\lambda' & g_0g_0'}$$
(\m{GL(M)} agit de mani\`ere \'evidente \`a droite sur le premier facteur
de \m{M^*\otimes H_R}, et \m{\GG{0}} \`a gauche sur le deuxi\`eme facteur).
Dans le cas du \paragra~\hskip -2pt 6.1, on a
$$G_L=Aut({\cal E}\oplus{\cal E}'), \ G_R=Aut((\Gamma\otimes M)\oplus{\cal F}').$$
\subsection{Actions des groupes associ\'es sur l'espace de morphismes}
Le groupe \m{G_L} op\`ere \`a droite sur \m{W} : si \m{\phi_1\in \X{1}\otimes M},
\m{\phi_2\in \X{2}\otimes M}, \m{x_2\in \X{3}}, \m{y_2\in \X{4}}, \m{\gg{1}\in
\GG{1}},
\m{\gg{2}\in \GG{2}} et \m{h_L\in H_L} on a
$$\pmatrix{\phi_1 &\phi_2\cr x_2 & y_2}\pmatrix{\gg{1} & 0\cr h_L & \gg{2}} =
\pmatrix{\phi_1\gg{1}+(\g{1}\otimes I_M)(\phi_2\otimes h_L) & \phi_2 \gg{2} \cr
x_2\gg{1} + \g{2}(y_2\otimes h_L) & y_2\gg{2}}.$$
Le groupe \m{G_R} op\`ere \`a gauche sur \m{W} : si \m{\phi_1\in \X{1}\otimes M},
\m{\phi_2\in \X{2}\otimes M}, \m{x_2\in \X{3}}, \m{y_2\in \X{4}}, \m{g_0\in \GG{0}},
\m{g_M\in GL(M)} et \m{\lambda\in M^*\otimes H_R} on a
$$\pmatrix{g_M & 0\cr \lambda & g_0}\pmatrix{\phi_1 &\phi_2\cr x_2 & y_2} =
\pmatrix{(I_{\X{1}}\otimes g_M)(\phi_1) & (I_{\X{2}}\otimes g_M)(\phi_2)\cr
g_0x_2+\g{3}(\pline{\lambda,\phi_1}) &
g_0y_2+\g{4}(\pline{\lambda,\phi_2})}.$$
Les actions de ces groupes sont compatibles, c'est-\`a-dire que si
\m{g_L\in G_L}, \m{g_R\in G_R} et \m{w\in W}, on a
$$g_R(wg_L) = (g_Rw)g_L.$$
C'est pourquoi on parlera abusivement du groupe \ \m{G_L\times G_R}
\ ou d'un de ses sous-groupes et de son action sur \m{W} (au lieu
d'utiliser par exemple le groupe \m{G_L^{op}\times G_R}).
On note \m{H} le \leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$} sous-groupe\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$} \ de \ \m{G_L\times G_R} constitu\'e
des paires
$$\pmatrix{\pmatrix{1 & 0 \cr h_L & 1},
\pmatrix{1 & 0 \cr \lambda & 1}},$$
(o\`u \ \m{h_l\in H_L}, \m{\lambda\in M^*\otimes H_R}).
\subsection{Mutation d'un espace abstrait de morphismes}
Soit \m{M'} un \m{k}-espace vectoriel tel que
$$\dim(M') = \dim(\X{2})-\dim(M).$$
On va d\'efinir un nouvel espace abstrait de morphismes \m{D(\Theta)}
associ\'e \`a \m{\Theta}. Posons
$$\X{1}' = H_R, \ \ \X{2}' = \X{2}^*, \ \ \X{3}' = \X{3}, \ \
\X{4}' = \mathop{\rm coker}\nolimits(\ov{\g{1}}) = (\X{2}^*\otimes\X{1})/H_L,$$
$$H'_R = \X{1}, \ \ H'_L = \ker(\g{4})\subset H_R\otimes\X{2}.$$
Soient
$$\g{1}' : \X{2}'\otimes H'_L\longrightarrow \X{1}'$$
la restriction de la contraction
$$\X{2}^*\otimes\X{2}\otimes H_R\longrightarrow H_R,$$
$$\g{3}' = \g{3} : H'_R\otimes\X{1}'\longrightarrow\X{3}',$$
et
$$\g{4}' : H'_R\otimes\X{2}'\longrightarrow\X{4}'$$
la projection
$$\X{2}^*\otimes\X{1}\longrightarrow(\X{2}^*\otimes\X{1})/H_L.$$
La d\'efinition de \m{\g{2}'} est un peu plus compliqu\'ee. On a un
diagramme commutatif, o\`u la ligne du haut et la colonne de gauche sont
commutatives :
\bigskip
\bigskip
\begin{picture}(360,230)
\put(90,220){$0$}
\put(10,170){$0$}
\put(55,170){$\ker(\g{4})\otimes H_L$}
\put(168,170){$\ker(\g{4})\otimes\X{2}^*\otimes \X{1}$}
\put(301,170){$\ker(\g{4})\otimes\mathop{\rm coker}\nolimits(\ov{\g{1}})$}
\put(430,170){$0$}
\put(55,120){$H_R\otimes\X{2}\otimes H_L$}
\put(176,120){$H_R\otimes\X{2}\otimes\X{2}^*\otimes \X{1}$}
\put(75,70){$\X{4}\otimes H_L$}
\put(188,70){$\X{3}$}
\put(90,20){$0$}
\put(92,214){\vector(0,-1){30}}
\put(92,164){\vector(0,-1){30}}
\put(92,114){\vector(0,-1){30}}
\put(92,64){\vector(0,-1){25}}
\put(192,164){\vector(0,-1){30}}
\put(194,94){$\phi$}
\put(192,114){\vector(0,-1){30}}
\put(133,173){\vector(1,0){30}}
\put(410,173){\vector(1,0){15}}
\put(23,173){\vector(1,0){28}}
\put(273,173){\vector(1,0){22}}
\put(138,129){$I\otimes\ov{\g{1}}$}
\put(133,123){\vector(1,0){40}}
\put(136,78){$\g{2}$}
\put(128,72){\vector(1,0){55}}
\end{picture}
\bigskip
Le morphisme \m{\phi} est la contraction de \m{\X{2}\otimes\X{2}^*}, suivie
de \m{\g{3}}. La commutativit\'e du carr\'e du bas d\'ecoule de celle du
carr\'e \m{(D)} du \paragra~\hskip -2pt 6.1.1. Il en d\'ecoule que \m{\phi} s'annule
sur \ \m{\ker(\g{4})\otimes H_L}, et induit donc une application lin\'eaire
$$\g{2}' : \X{4}'\otimes H_L' = \mathop{\rm coker}\nolimits(\ov{\g{1}})\otimes\ker(\g{4})
\longrightarrow\X{3}=\X{3}'.$$
Il est ais\'e de voir que l'analogue du carr\'e \m{(D)} du \paragra~\hskip -2pt 6.1.1
est commutatif. Il est clair que \m{\g{1}'} induit une injection
$$\ov{\g{1}'} : H'_L\longrightarrow\X{2}'^*\otimes\X{1}',$$
(c'est l'inclusion \ \m{\ker(\g{4})\subset H_R\otimes\X{2}}), et que \m{\g{4}'}
est surjective. On pose
$$\GG{0}' = \GG{1}^{op}, \ \GG{1}' = \GG{0}^{op}, \ \GG{2}' = \GG{2}^{op}.$$
Les actions de ces groupes se d\'eduisent imm\'ediatement de celles des
groupes \m{\GG{0}},\m{\GG{1}} et \m{\GG{2}}. Par exemple, \m{\GG{1}} agit \`a
droite
sur \m{H_L} et \m{\X{1}}, et cette action est compatible avec
$$\g{1} : \X{2}\otimes H_L\longrightarrow\X{1}.$$
On obtient donc une action \`a droite de \m{\GG{1}} sur
\m{(\X{2}^*\otimes\X{1})/H_L}, c'est-\`a-dire une action \`a gauche de
\m{\GG{0}'} sur \m{\X{4}'}.
\begin{defin}
On note \m{D(\Theta)} l'espace abstrait de morphismes d\'efini par
\m{\X{1}'}, \m{\X{2}'}, \m{\X{3}'}, \m{\X{4}'}, \m{H'_L}, \m{H'_R},
\m{\GG{0}'}, \m{\GG{1}'}, \m{\GG{2}'}, \m{\g{1}'}, \m{\g{2}'}, \m{\g{3}'} et
\m{\g{4}'}. On l'appelle la {\em mutation de \m{\Theta}}.
\end{defin}
\begin{xprop}
On a \ \ \m{D(D(\Theta)) = \Theta}.
\end{xprop}
Imm\'ediat. $\Box$
On d\'efinit comme pour \m{\Theta} les \leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$} groupes\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$} \ \m{G'_L\times G'_R}
et \m{H'} correspondant \`a \m{D(\Theta)}.
\subsection{Mutation des morphismes}
On note \m{W'} l'espace total de \m{D(\Theta)}, c'est-\`a-dire
$$W' = (\X{1}'\otimes M')\oplus (\X{2}'\otimes M') \oplus\X{3}'\oplus\X{4}'.$$
On note \m{W^0} l'ouvert de \m{W} constitu\'e des
$$\pmatrix{\phi_1 & \phi_2\cr \x{3} & \x{4}}$$
tels que l'application lin\'eaire
$$\ov{\phi_2} : \X{2}^*\longrightarrow M$$
d\'eduite de \m{\phi_2} soit surjective. On d\'efinit de m\^eme l'ouvert
\m{W'^0} de \m{W'}.
Rappelons que la projection
$$\X{2}^*\otimes\X{1}\longrightarrow(\X{2}^*\otimes\X{1})/H_L$$
n'est autre que \m{\g{4}'}. De m\^eme, la projection
$$\X{2}'^*\otimes\X{1}'\longrightarrow({\X{2}'}^*\otimes\X{1}')/H'_L$$
n'est autre que \m{\g{4}}. Si \ \m{\phi_2\in\X{2}\otimes M}, on notera
\m{q(\phi_2)} l'application lin\'eaire
$$\ov{\phi_2}\otimes I_{\X{1}} : \X{2}^*\otimes\X{1}\longrightarrow M\otimes\X{1}.$$
On d\'efinit de m\^eme, pour tout \ \m{\phi_2'\in\X{2}'\otimes M'}
l'application lin\'eaire
$$q'(\phi_2') : \X{2}\otimes H_R = {\X{2}'}^*\otimes\X{1}'\longrightarrow M'\otimes\X{1}'.$$
Soit
$$w = \pmatrix{\phi_1 & \phi_2\cr \x{3} & \x{4}}\in W^0.$$
On va en d\'eduire un \'el\'ement de \m{W'^0} (pas de mani\`ere unique).
On choisit d'abord un isomorphisme
$$\ker(\ov{\phi_2})^*\simeq M'.$$
On note \m{\phi_2'} l'\'el\'ement de \ \m{\X{2}'\otimes M'} \ provenant de
l'application lin\'eaire
$$\ov{\phi_2'} : {\X{2}'}^* = \X{2}\longrightarrow\ker(\ov{\phi_2})^*=M',$$
qui est la transpos\'ee de l'inclusion de \m{\ker(\ov{\phi_2})} dans
\m{\X{2}^*}.
Soit
$$u\in\g{4}^{-1}(-\x{4})\subset H_R\otimes\X{2}.$$
Notons que \m{u} est d\'efini \`a un \'el\'ement pr\`es de
\ \m{\ker(\g{4})=H'_L}.
Soit
$$\phi'_1 = q'(\phi_2')(u) \in \X{1}'\otimes M'.$$
On peut aussi \'ecrire
$$\phi'_1 = \pline{\phi_2',u}. $$
Soient
$$\alpha\in q(\phi_2)^{-1}(\phi_1)\subset \X{2}^*\otimes\X{1},$$
et
$$\x{4}' = \g{4}'(\alpha)\in\X{4}'.$$
Notons que \m{\alpha} est d\'efini \`a un \'el\'ement pr\`es de
$$\ker(\ov{\phi_2})\otimes\X{1} = H'_R\otimes {M'}^*.$$
Soient enfin
$$\x{3}' = \x{3}+\g{3}(\pline{\alpha,u}).$$
et
$$z(w,u,\alpha) = \pmatrix{\phi'_1 & \phi_2'\cr \x{3}' & \x{4}'}
\in W'^0.$$
On emploiera aussi la notation
$$z(w) = z(w,u,\alpha)$$
bien que cet \'el\'ement de \m{W'^0} ne d\'epende pas uniquement de \m{w}.
\begin{xprop}
Soit \m{w\in W^0}. Les \'el\'ements \m{z(w,u,\alpha)}, pour tous les choix
possibles de \m{u} et \m{\alpha}, constituent une \m{H'}-orbite de \m{W'^0}.
\end{xprop}
\noindent{\em D\'emonstration}. On v\'erifie ais\'ement que si on remplace
\m{u} par \m{u+h'_L} et \m{\alpha} par \m{\alpha+\psi} (avec
\m{h'_L\in H'_L} et \m{\psi\in M^*\otimes H_R}), l'\'el\'ement obtenu de
\m{W'} est
$$\pmatrix{1 & 0 \cr \psi & 1}\pmatrix{\phi_1' & \phi_2' \cr \x{3}' &
\x{4}'}\pmatrix{1 & 0 \cr h'_L & 1}.$$
$\Box$
\begin{xprop}
Pour tout \m{w\in W^0}, on a
$$z(z(w)) \ \in \ (G_L\times G_R)w.$$
\end{xprop}
\noindent{\em D\'emonstration}. On part de
$$w = \pmatrix{\phi_1 & \phi_2 \cr \x{3} & \x{4}} \in W^0,$$
et on prend comme pr\'ec\'edemment \ \m{u\in\g{4}^{-1}(-\x{4})},
\m{\alpha\in q(\phi_2)^{-1}(\phi_1)} \ pour d\'efinir
$$z(w) = \pmatrix{\phi_1 & \phi_2' \cr \x{3}' & \x{4}'} \in {W'}^0.$$
On cherche maintenant \ \m{u'\in\g{4}'^{-1}(-\x{4}')} \ et \
\m{\alpha\in q(\phi_2')^{-1}(\phi'_1)} \ pour d\'efinir \m{z(z(w))}.
On a
$$\g{4}'(-\alpha) = - \x{4}',$$
donc on peut prendre
$$u' = - \alpha.$$
D'autre part, on a
$$q'(\phi_2') = \phi'_1,$$
et on peut prendre
$$\alpha' = u.$$
Soit
$$z(z(w)) = \pmatrix{\phi_1 & \phi_2'' \cr \x{3}'' & \x{4}''} \in {W}^0$$
l'\'el\'ement de \m{W^0} d\'efini par \m{u'} et \m{\alpha}. On a
\'evidemment \ \m{\phi_2'' = \phi_2}, et
$$\phi_1'' = q(\phi_2)(u') = -q(\phi_2)(\alpha) = -\phi_1,$$
$$\x{4}'' = \g{4}(\alpha') = \g{4}(u) = -\x{4},$$
$$\x{3}'' = \x{3}' + \g{3}(\pline{\alpha',u'}) =
\x{3} + \g{3}(\pline{\alpha,u}) - \g{3}(\pline{\alpha,u}) = \x{3}.$$
Donc
\begin{eqnarray*}
z(z(w)) & = & \pmatrix{-\phi_1 & \phi_2 \cr \x{3} & -\x{4}} \cr
& = & \pmatrix{-1 & 0 \cr 0 & 1}
\pmatrix{\phi_1 & \phi_2 \cr \x{3} & \x{4}}\pmatrix{1 & 0 \cr 0 & -1}\cr
\end{eqnarray*}
$\Box$
\begin{xprop}
Soient \m{w_1,w_2\in W^0} des points qui sont dans la m\^eme \
\m{(G_L\times G_R)}-orbite. Alors \m{z(w_1)} et \m{z(w_2)} sont dans la
m\^eme \ \m{(G'_L\times G'_R)}-orbite.
\end{xprop}
\noindent{\em D\'emonstration}. On v\'erifie ais\'ement que c'est vrai si
$$w_2 = \pmatrix{g_M & 0 \cr 0 & g_0}w_1\pmatrix{g_1 & 0 \cr 0 & g_2}.$$
Il reste \`a traiter les cas
$$w_2 = \pmatrix{1 & 0 \cr \psi & 1}w_1,$$
ou
$$w_2 = w_1\pmatrix{1 & 0 \cr h_l & 1},$$
avec \m{h_l\in H_L}, \m{\psi\in M^*\otimes H_R}. On ne traitera que le premier
cas, le second \'etant analogue. Posons
$$w_1 = \pmatrix{\phi_1 & \phi_2 \cr \x{3} & \x{4}}.$$
Alors on a
$$w_2 = \pmatrix{\phi_1 + (\g{1}\otimes I_M)(\phi_2\otimes h_L) & \phi_2 \cr
\x{3} + \g{2}(\x{4}\otimes h_L) & \x{4}}.$$
On suppose que
$$z(w_1) = \pmatrix{\phi'_1 & \phi_2' \cr \x{3}' & \x{4}'}$$
est d\'efini par \ \m{u_1\in H_R\otimes X_2} \ et \
\m{\alpha_1\in\X{2}^*\otimes\X{1}}. On va chercher des \'el\'ements \m{u_2},
\m{\alpha_2} convenables pour d\'efinir \m{z(w_2)}. On doit avoir
\ \m{u_2\in\g{4}^{-1}(-\x{4})}, donc on peut prendre
$$u_2 = u_1.$$
On doit avoir
$$q(\phi_2)(\alpha_2) = \phi_1 + (\g{1}\otimes I_M)(\phi_2\otimes h_L).$$
Posons
$$\alpha_2 = \alpha_1+\alpha_0.$$
On doit donc avoir
$$q(\phi_2)(\alpha_0) = (\g{1}\otimes I_M)(\phi_2\otimes h_L).$$
Pour cela il suffit de prendre
$$\alpha_0 = h_L$$
(vu comme \'el\'ement de \ \m{\X{2}^*\otimes\X{1}}, \`a l'aide de
\m{\ov{\g{1}}}). On a alors
$$z(w_2) = \pmatrix{\phi_1'' & \phi_2' \cr \x{3}'' & \x{4}''},$$
avec
$$\phi_1'' = q'(\phi_2')(u_2) = q'(\phi_2')(u_1) = \phi_1',$$
$$\x{4}'' = \g{4}(\alpha_2) = \g{4}(\alpha_1+\alpha_0)
= \g{4}(\alpha_1) = \x{4}',$$
\begin{eqnarray*}
\x{3}'' & = & \x{3}+\g{3}(\pline{\alpha_2,u_2}) \cr
& = & \x{3}'-\g{2}(h_L\otimes\g{4}(u_2))+
\g{3}(\pline{\alpha_0,u_2})\cr
\end{eqnarray*}
Mais on a
$$\g{2}(h_L\otimes\g{4}(u_2))=\g{3}(\pline{\alpha_2,u_2}),$$
si on se souvient que \ \m{\alpha_0=\ov{\g{1}}(h_L)}, \`a cause du
diagramme commutatif \m{(D)} du \paragra~\hskip -2pt 6.1.1. On a donc
\ \m{\x{3}''=\x{3}'} \ et finalement
$$z(w_1)=z(w_2).$$
$\Box$
\subsection{Th\'eor\`emes d'isomorphisme}
Le th\'eor\`eme suivant d\'ecoule imm\'ediatement des r\'esultats du
\paragra~\hskip -2pt 6.5 :
\begin{xtheo}
L'application associant \`a l'orbite d'un point \m{w} de \m{W^0} l'orbite
de \m{z(w)} d\'efinit une bijection
$$D_\Theta : W^0/(G_L\times G_R) \ \simeq \ {W'}^0/(G_L'\times G_R').$$
\end{xtheo}
\bigskip
On suppose maintenant que les groupes \m{\GG{1}}, \m{\GG{2}}, \m{\GG{3}},
sont alg\'ebriques sur \m{k} et que leurs actions sont alg\'ebriques. Il
est alors clair par construction que pour tout \m{w\in W^0}, il
existe un voisinage de Zariski \m{U} de \m{w} dans \m{W^0} et un
voisinage de Zariski \m{U'} de \m{z(w)} dans \m{{W'}^0} tels que
\m{D_\Theta} se rel\`eve en un morphisme \ \m{U\longrightarrow U'} et que
\m{D_{D(\Theta)}} se rel\`eve en un morphisme \ \m{U'\longrightarrow U}. On en
d\'eduit ais\'ement le r\'esultat suivant :
\begin{xtheo}
Soit \m{U} un ouvert \m{(G_L\times G_R)}-invariant de \m{W^0} tel qu'il
existe un bon quotient \m{U//(G_L\times G_R^{op})}. Soit \m{U'}
l'ensemble des points de \m{{W'}^0} au dessus de
\noindent
\m{D_\Theta(U/((G_L\times G_R))}. Alors \m{U'} est un ouvert
\m{(G'_L\times G'_R)}-invariant de \m{{W'}^0}, et il existe un bon
quotient \m{U'//(G'_L\times {G'_R}^{op})}, qui est isomorphe \`a
\m{U//(G_L\times G_R^{op})}.
\end{xtheo}
\bigskip
\bigskip
\section{Mutations de morphismes de type \m{(r,s)}}
\subsection{Espaces de morphismes abstraits associ\'es}
On applique les r\'esultats du \paragra~\hskip -2pt 6 aux cas d\'ecrits au \paragra~\hskip -2pt 3.
Commen\c cons par d\'ecrire la situation en termes de morphismes de
faisceaux. On s'int\'eresse aux morphismes
$$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}({\cal E}_i\otimes M_i)\longrightarrow\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq l\leq s}({\cal F}_l\otimes N_l).$$
Soit $p$ un entier tel que \m{0\leq p\leq r-1}. On pose
$${\cal E}=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p}({\cal E}_i\otimes M_i), \
{\cal E}'=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}({\cal E}_j\otimes M_j),$$
$$\Gamma={\cal F}_1, \ M = N_1, \ {\cal F}'=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}({\cal F}_l\otimes N_l).,$$
de sorte que les morphismes pr\'ec\'edents peuvent s'\'ecrire sous la
forme
$${\cal E}\oplus{\cal E}'\longrightarrow(\Gamma\otimes M)\oplus{\cal F}',$$
comme dans le \paragra~\hskip -2pt 5 et le \paragra~\hskip -2pt 6.
On reprend maintenant le language du \paragra~\hskip -2pt 3. On supposera que
$$n_1 < \mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq r}\dim(H_{1i})m_i.$$
On pose
$$X_1 = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p}(H_{1i}\otimes M_i^*), \
X_2 = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}\otimes M_j^*), $$
$$X_3 = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p, 2\leq l\leq s}(H_{li}\otimes M_i^*\otimes N_l), \
X_4 = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r, 2\leq l\leq s}(H_{lj}\otimes M_j^*\otimes N_l), $$
$$H_L=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p,p+1\leq j\leq r}(A_{ji}\otimes M_i^*\otimes M_j), \
M=N_1, \ H_R=\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}(B_{l1}\otimes N_l).$$
On d\'efinit de m\^eme les groupes \m{\GG{0}}, \m{\GG{1}}, \m{\GG{2}}, et
les applications \m{\g{1}},\m{\g{2}},\m{\g{3}}, et \m{\g{4}}. On obtient
ainsi un espace abstrait de morphismes not\'e \m{\Theta_p}, d'espace
total de morphismes not\'e \m{W_p}. Il est clair que
$$W_p = W = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r, 1\leq s\leq l}(H_{li}\otimes M_i^*\otimes N_l),$$
mais en g\'en\'eral
$$W_p^0\not = W_q^0$$
si \m{p\not = q}.
Soit
$$w=(\phi_{li})_{1\leq i\leq r,1\leq s\leq s}\in W.$$
Alors, par d\'efinition, on a \ \m{w\in W_p^0} \ si et seulement si
l'application lin\'eaire
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\phi_j :
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}^*\otimes M_j)\longrightarrow N_1$$
est surjective. On a donc
$$W_{r-1}^0\subset W_{r-2}^0\subset\cdots\subset W_0^0\subset W.$$
\subsection{Description des mutations d'espaces de morphismes abstraits}
L'espace de morphismes abstraits \m{D(\Theta_p)} correspond \`a un
espace de morphismes de type \m{(r+s-p-1,p+1)}, dont nous allons d\'ecrire
le dual (cf. \paragra~\hskip -2pt 3.6), qui est donc un espace de morphismes de type
\m{(p+1,r+s-p-1)}. Il est d\'efini par les espaces vectoriels
\m{M^{(p)}_1,\ldots,M^{(p)}_r},\m{N^{(p)}_1,\ldots,N^{(p)}_s},
\m{A^{(p)}_{ji},B^{(p)}_{ml},H^{(p)}_{li}}, et par les compositions
ad\'equates. On a
$$M^{(p)}_i=M_i \ \ {\rm si} \ 1\leq i\leq p,$$
$$\dim(M^{(p)}_{p+1}) = \biggl(\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}m_j\dim(H_{1j})
\biggr)-n_1,$$
$$N^{(p)}_i=M_{p+i} \ \ {\rm si} \ 1\leq i\leq r-p,\ \ \
N^{(p)}_l = N_{l-r+p+1} \ \ {\rm si} \ r-p+1\leq l\leq r+s-p-1,$$
$$A^{(p)}_{ji}=A_{ji} \ \ {\rm si} \ 1\leq i\leq j\leq p,\ \ \
A^{(p)}_{p+1,i}=H_{1i} \ \ {\rm si} \ 1\leq i\leq p,$$
$$B^{(p)}_{lm} = A_{l+p,m+p} \ \ {\rm si} \ 1\leq m\leq l\leq r-p,$$
$$B^{(p)}_{lm} = B_{l-r+p+1,m-r+p+1} \ \ {\rm si} \ r-p+1\leq m\leq l
\leq r+s-p-1,$$
$$B^{(p)}_{lm} = \ker(B_{l-r+p+1,1}\otimes H_{1,m+p}\longrightarrow H_{l-r+p+1,m+p})\ \ \ \
\ \ $$
$$\ \ \ \ \ \ \ \ {\rm si} \ r-p+1\leq l\leq r+s-p-1, \ 1\leq m\leq r - p,$$
$$H^{(p)}_{li} = (H_{1i}\otimes H^*_{1,l+p})/A_{l+p,i} \ \ {\rm si} \
1\leq i\leq p, 1\leq l\leq r-p,$$
$$H^{(p)}_{li} = H_{l-r+p+1,i} \ \ {\rm si} \ 1\leq i\leq p,
r-p+1\leq l\leq r+s-p-1,$$
$$H^{(p)}_{l,p+1} = H^*_{1,l+p} \ \ {\rm si} \ 1\leq l\leq r-p,\ \ \
H^{(p)}_{l,p+1} = B_{l-r+p+1,1} \ \ {\rm si} \ r-p+1\leq l\leq r+s-p-1.$$
La description des compositions est laiss\'ee au lecteur. On note
$$W'(p) = \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p+1,1\leq l\leq r+s-p-1}
L(H^{(p)*}_{li}\otimes M^{(p)}_i,N^{(p)}_l)$$
l'espace total de morphismes de \m{D(\Theta_p)}, \m{W'_0(p)} l'ouvert
correspondant, \m{G(p)} le groupe agissant sur \m{W'(p)}. Le th\'eor\`eme
6.5 dit qu'on a une bijection
$$W^0_p/G\ \simeq\ W'_0(p)/G(p).$$
\subsection{Description des mutations de morphismes}
Soit
$$w = (\phi_{li})_{1\leq i\leq r,1\leq l\leq s}\in W_p^0\subset
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r,1\leq l\leq s}L(H_{li}^*\otimes M_i,N_l),$$
et
$$\phi_1 = (\phi_{1i})_{1\leq i\leq p}, \ \ \phi_2 =
(\phi_{1j})_{p+1\leq j\leq r}, \ \ x_3 = (\phi_{li})_{1\leq i\leq p,
2\leq l\leq s}, \ \ x_4 = (\phi_{li})_{p+1\leq i\leq r,
2\leq l\leq s}.$$
On va d\'ecrire
$$z(w) = \pmatrix{\phi'_1 & \phi'_2\cr x'_3 & x'_4}.$$
On construit d'abord les \'el\'ements \m{u} et \m{\alpha} du \paragra~\hskip -2pt 6.5.
On doit prendre pour \m{u} un \'el\'ement de
$$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r,2\leq l\leq s}(H_{1j}\otimes B_{l1}\otimes M_j^*\otimes N_l)$$
tel que le diagramme suivant soit commutatif :
\bigskip
\begin{picture}(360,100)
\put(100,90){$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r,2\leq l\leq s}(H_{lj}^*\otimes M_j)$}
\put(90,10){$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r,2\leq l\leq s}
(H_{1j}^*\otimes B_{l1}^*\otimes M_j)$}
\put(320,90){$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}N_l$}
\put(225,92){\vector(1,0){85}}
\put(120,65){\vector(0,-1){40}}
\put(225,25){\vector(3,2){80}}
\put(245,95){$-x_4$}
\put(250,50){$u$}
\end{picture}
\bigskip
On prend pour \m{\alpha} un \'el\'ement de
$$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p,p+1\leq j\leq r}(H_{1j}^*\otimes H_{1i}\otimes M_i^*\otimes M_j)$$
tel que le diagramme suivant soit commutatif :
\bigskip
\begin{picture}(360,100)
\put(100,90){$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p}(H_{li}^*\otimes M_i)$}
\put(320,90){$N_1$}
\put(280,10){$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{lj}^*\otimes M_j)$}
\put(200,92){\vector(1,0){110}}
\put(325,25){\vector(0,1){55}}
\put(170,75){\vector(3,-2){95}}
\put(220,45){$\alpha$}
\put(330,50){$\ov{\phi_2}$}
\end{picture}
\bigskip
Dans le premier diagramme, la fl\`eche verticale est induite par les
compositions \break \m{B_{l1}\otimes H_{1j}\longrightarrow H_{lj}}, et dans le second, la
fl\`eche horizontale provient de \m{\phi_1}.
D\'eterminons maintenant \m{\phi'_1}, \m{\phi'_2}, \m{x'_3} et \m{x'_4}.
On prend \ \m{M'=\ker(\ov{\phi_2})}, qui est donc un quotient de
\ \m{\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{lj}^*\otimes M_j)}. Alors \m{x'_4} est l'image
de \m{\alpha} dans
$$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p,p+1\leq j\leq r}\Biggl(\biggl((H_{1j}^*\otimes H_{1i})
/A_{ji}\biggr)\otimes M_i^*\otimes M_j\Biggr)\ = \
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p,1\leq l\leq r-p}(H^{(p)}_{li}\otimes M^{(p)*}_i\otimes
N^{(p)}_l),$$
\m{\phi'_1} est la restriction \`a \m{\ker(\ov{\phi_2})} de
l'aaplication lin\'eaire
$$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}^*\otimes M_j)\longrightarrow \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}
(B_{l1}\otimes N_l)$$
provenant de \m{u} et \m{\phi'_2} est l'inclusion
$$\ker(\ov{\phi_2})\subset \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{lj}^*\otimes M_j).$$
Pour obtenir \m{x'_3}, on fait la somme de \m{x_3} et de la compos\'ee
$$\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq r}(H_{1i}^*\otimes M_i)\hfl{\alpha}{}
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}^*\otimes M_j)\hfl{u}{}\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}
(B_{l1}\otimes N_l).$$
\subsection{Polarisations associ\'ees}
Soit \m{(\lambda_1,\ldots,\lambda_r,\mu_1,\ldots,\mu_s)} une polarisation
de l'action de \m{G} sur \m{W}. On va en d\'eduire une polarisation de
l'action de \m{G(p)} sur \m{W'(p)}.
Soient \ \m{M'_i\subset M_i}, \m{N'_l\subset N_l} ,\m{1\leq i\leq r},
\m{1\leq l\leq s} des sous-espaces vectoriels. On pose
$$m'_i=\dim(M'_i), \ \ n'_l=\dim(N'_l),$$
$${M^{(p)}}'_i=M'_i\ \ {\rm si} \ 1\leq i\leq p,$$
$${N^{(p)}}'_l=M'_{l-p} \ \ {\rm si} \ 1\leq l\leq r-p,\ \ \
{N^{(p)}}'_l=N'_{l-r+p+1} \ \ {\rm si} r-p+1\leq l\leq r+s-p-1,$$
$${m^{(p)}}'_i=\dim({M^{(p)}}'_i)\ \ {\rm si} \ 1\leq i\leq p, \ \ \
{m^{(p)}}'_{p+1} = \biggl(\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\dim(H_{1j})m'_j\biggr)-
n'_1,$$
$${n^{(p)}}'_l=\dim({N^{(p)}}'_l)\ \ {\rm si} \ 1\leq l\leq r+s-p-1.$$
On d\'efinit une suite
\m{(\alpha'_1,\ldots,\alpha'_{p+1},}\m{\beta'_1,\ldots,\beta'_{r+s-p-1})} de
nombres rationnels par les identit\'es
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq r}\lambda_im'_i-\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq s}\mu_ln'_l =
\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq p+1}\alpha'_i{m^{(p)}}'_i-\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq r+s-p-1}
\beta'_l{n^{(p)}}'_l.$$
On a donc
$$\alpha'_i=\lambda_i\ \ {\rm si}\ 1\leq i\leq p,\ \ \
\alpha'_{p+1} = \mu_1,$$
$$\beta'_l=\mu_1\dim(H_{1,l+p})-\lambda_{l+p} \ \ {\rm si} \
1\leq l\leq r-p,$$
$$\beta'_l=\mu_{l+r-p+1} \ \ {\rm si} \ r-p+1\leq l\leq r+s-p-1.$$
On normalise ensuite, pour obtenir la suite
\m{(\alpha_1,\ldots,\alpha_{p+1},}\m{\beta_1,\ldots,\beta_{r+s-p-1})}
v\'erifiant
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq p+1}\alpha_i\dim(M^{(p)}_i) =
\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq r+s-p-1}\beta_l\dim(N^{(p)}_l) = 1.$$
On a donc
$$\alpha_i = \q{\alpha'_i}{c}, \ \ \ \beta_l = \q{\beta'_l}{c},$$
avec
$$c=\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq p}\lambda_im_i+\mu_1\biggl(
(\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}m_j\dim(H_{1j}))-n_1\biggr).$$
On appelle
\m{(\alpha_1,\ldots,\alpha_{p+1},}\m{\beta_1,\ldots,\beta_{r+s-p-1})}
la {\em polarisation associ\'ee} \`a
\noindent
\m{(\lambda_1,\ldots,\lambda_r,}\m{\mu_1,\ldots,\mu_s)}. C'est une
polarisation de l'action de \m{G(p)} sur \m{W'(p)}. On supposera que les
\m{\alpha_i} et les \m{\beta_l} sont positifs.
\subsection{Comparaison des (semi-)stabilit\'es}
On veut comparer la (semi-)stabilit\'e d'un \'el\'ement de \m{W^0} avec
celle des \'el\'ements de \m{W'_0(p)} associ\'es.
\begin{xprop}
On suppose que
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq p}\lambda_im_i\ \leq \ \mu_1.$$
Si \ \m{w\in W^0} \ n'est pas \m{G}-(semi-)stable relativement \`a
\m{(\lambda_1,\ldots,\lambda_r,}\m{\mu_1,\ldots,\mu_s)}, alors \break
\m{z(w)\in W'_0(p)} \ n'est pas \m{G(p)}-(semi-)stable relativement \`a
\m{(\alpha_1,\ldots,\alpha_{p+1},}\m{\beta_1,\ldots,\beta_{r+s-p-1})}.
\end{xprop}
{\em\noindent D\'emonstration.} On ne traitera que le cas de la
semi-stabilit\'e, la stabilit\'e \'etant analogue. Posons \
\m{w=(\phi{li})_{1\leq i\leq r,1\leq l\leq s}}. Soient \
\m{M'_i\subset M_i}, \m{N'_l\subset N_l} \ des sous-espaces vectoriels
tels que
$$\epsilon=\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq r}\lambda_i\dim(M'_i)-
\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq l\leq s}\mu_l\dim(N'_l)>0$$
et \ \m{\phi_{li}(H^*_{li}\otimes M'_i)\subset N'_l} \ pour \m{1\leq i\leq r},
\m{1\leq l\leq s}. On peut supposer que
$$N'_1=\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\phi_{1j}(H^*_{1j}\otimes M'_j).$$
En effet, supposons que
$$k=\mathop{\rm codim}\nolimits_{N'_1}(\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\phi_{1j}(H^*_{1j}\otimes M'_j))>0.$$
On a, en posant \ \m{m'_i=\dim(M'_i)}, \m{n'_l=\dim(N'_l)},
\begin{eqnarray*}
\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq i\leq r}\lambda_im'_i-\mu_1(n'_1-k) & = &
\epsilon-\mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq p}\lambda_im'_i+k\mu_1+
\mathop{\hbox{$\displaystyle\sum$}}\limits_{2\leq l\leq s}\mu_ln'_l\cr
& > & k\mu_1 - \mathop{\hbox{$\displaystyle\sum$}}\limits_{1\leq i\leq p}\lambda_im'_i > 0\cr
\end{eqnarray*}
par hypoth\`ese. On peut donc au besoin remplacer
$$(M'_1,\ldots,M'_r,N'_1,\ldots,N'_s)$$
par
$$(0,\ldots,0,M'_{p+1},\ldots,M'_r,
\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\phi_{1j}(H^*_{1j}\otimes M'_j),0,\ldots,0).$$
Soient \ \m{{M^{(p)}_i}'=M'_i\subset M^{(p)}_i} \ pour \m{1\leq i\leq p},
$${M^{(p)}_{p+1}}'=\ker(\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\phi_{1j})\cap
\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}^*\otimes M'_j)\biggr)\ \subset \
M^{(p)}_{p+1}=\ker(\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\phi_{1j}),$$
$${N^{(p)}_l}'=M'_{l+p}\subset N^{(p)}_l \ \ {\rm si} \ 1\leq l\leq r-p,
\ \ \ {N^{(p)}_l}'= N'_{l-r+p+1} \ \ {\rm si} \ r-p+1\leq l\leq r+s-p-1.$$
Il faut s'arranger pour trouver
$$z(w)=\pmatrix{\phi'_1 & \phi'_2 \cr x'_3 & x'_4}=
(\psi_{li})_{1\leq i\leq p+1,1\leq l\leq r+s-p-1}$$
de telle sorte que
$$\psi_{li}(H^{(p)*}_{li}\otimes{M^{(p)}_i}')\subset {N^{(p)}_l}' \ \
{\rm pour} \ 1\leq i\leq p+1, 1\leq l\leq r+s-p-1.$$
On peut prendre \m{u} tel que
$$u\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r,2\leq l\leq s}
(H_{1j}^*\otimes B_{l1}^*\otimes M'_j)\biggr)
\ \subset \ \mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{2\leq l\leq s}N'_l,$$
et \m{\alpha} tel que
$$\alpha\biggl(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{1\leq i\leq p}(H_{1i}^*\otimes M'_i)\biggr) \ \subset
\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}^*\otimes M'_j)$$
(car \ \m{N'_1 = \ov{\phi_2}(\mathop{\hbox{$\displaystyle\bigoplus$}}\limits_{p+1\leq j\leq r}(H_{1j}^*\otimes M'_j))} ).
Dans ce cas \m{z(w)} poss\`ede les propri\'et\'es voulues. $\Box$
\medskip
\begin{xcoro}
Si
$$\mu_1 \ \geq \ \q{1}{n_1+1}$$
et si \m{w\in W^0_p} est tel que \m{z(w)} n'est pas \m{G(p)}-(semi-)stable
relativement \`a
\noindent
\m{(\alpha_1,\ldots,\alpha_{p+1},}\m{\beta_1,\ldots,\beta_{r+s-p-1})},
alors \m{w} n'est pas \m{G}-(semi-)stable relativement \`a
\noindent\m{(\lambda_1,\ldots,\lambda_r,}\m{\mu_1,\ldots,\mu_s)}
\end{xcoro}
{\em\noindent D\'emonstration}. On applique la proposition pr\'ec\'edente
\`a la mutation inverse. $\Box$
\bigskip
Le r\'esultat suivant est imm\'ediat :
\begin{xprop}
Si
$$\mu_1 \ < \ \q{\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\lambda_jm_j}{n_1-1},$$
alors on a \ \m{W^{ss}\subset W^0_p}.
\end{xprop}
\medskip
\begin{xcoro}
Si
$$\mu_1 \ > \ \q{\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\lambda_jm_j}{n_1+1},$$
alors on a \ \m{W'(p)^{ss}\subset W'_0(p)}.
\end{xcoro}
\medskip
On en d\'eduit le
\begin{xtheo}
Si
$${\rm Max}(\q{1}{n_1+1},1-\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\lambda_jm_j) \
\leq \ \mu_1 < \q{\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\lambda_jm_j}{n_1-1},$$
alors il existe un bon quotient \ \m{W'(p)^{ss}//G(p)} \ si et seulement
si il existe un bon quotient \ \m{W^{ss}//G}, et dans ce cas les deux
quotients sont isomorphes.
\end{xtheo}
\bigskip
{\em\noindent Cas particuliers :}
\noindent 1 - Si \m{p=0}, la condition du th\'eor\`eme pr\'ec\'edent se
r\'eduit \`a
$$\mu_1 \ \geq \ \q{1}{n_1+1}.$$
\medskip
\noindent 2 - Si \m{s=1}, la condition du th\'eor\`eme pr\'ec\'edent se
r\'eduit \`a
$$\mathop{\hbox{$\displaystyle\sum$}}\limits_{p+1\leq j\leq r}\lambda_jm_j \ \geq \ \q{n_1-1}{n_1}.$$
\medskip
\noindent 3 - Si \m{p=0} et \m{s=1}, la condition du th\'eor\`eme
pr\'ec\'edent est toujours v\'erifi\'ee.
\bigskip
On suppose maintenant que \m{r=2} et \m{s=1}. Soient
$$\tau^* : H_{12}\otimes A_{21}\longrightarrow H_{11}$$
la composition, et
$$\tau : H_{11}^*\otimes A_{21}\longrightarrow H_{12}^*$$
l'application d\'eduite de \m{\tau}. On d\'eduit de ce qui pr\'ec\`ede
l'am\'elioration suivante du th\'eor\`eme 3.1 :
\begin{xtheo}
Si \ \m{r=2} \ et \ \m{s=1}, il existe un bon quotient projectif \
\m{W^{ss}//G} \ dans chacun des deux cas suivants :
\noindent 1 - On a
$$\q{\lambda_2}{\lambda_1}>\dim(A_{21}) \ \ \ {\rm et} \ \
\lambda_2\geq \q{\dim(A_{21})}{n_1} c(\tau,m_2).$$
\noindent 2 - On a
$$\lambda_1<\q{\dim(H_{11})}{n_1}, \ \ \lambda_2<\q{\dim(H_{12})}{n_1},
\ \ \lambda_2\dim(A_{21})-\lambda_1>
\q{\dim(A_{21})\dim(H_{12})-\dim(H_{11})}{n_1}, $$
et
$$\dim(H_{11})-\lambda_1n_1\geq c(\tau^*,m_1)\dim(A_{21}).$$
\end{xtheo}
\bigskip
\bigskip
\section{Applications}
\subsection{Vari\'et\'es de modules extr\'emales sur \proj{2}}
On note \m{Q} (resp. \m{Q_2}) le fibr\'e vectoriel sur \m{\proj{2}}
conoyau du morphisme canonique
$${\cal O}(-1)\longrightarrow{\cal O}\otimes H^0({\cal O}(1))^* \ {\rm \ (resp. \ }
{\cal O}(-2)\longrightarrow{\cal O}\otimes H^0({\cal O}(2))^* \ {\rm \ )}.$$
Soient \m{m_1,m_2,n} des entiers positifs. On consid\`ere des morphismes
$$(*) \ \ \ \ (Q^*\otimes\cx{m_1})\oplus(Q_2^*\otimes\cx{m_2})\longrightarrow{\cal O}\otimes\cx{n}.$$
Une polarisation est dans ce cas une suite \m{(\lambda_1,\lambda_2,\mu_1)}
de nombres rationnels positifs tels que
$$\lambda_1m_1+\lambda_2m_2 = \mu_1n = 1.$$
Cette polarisation est enti\`erement d\'etermin\'ee par le rapport
$$\rho = \q{\lambda_2}{\lambda_1}.$$
Supposons que \ \m{n < 3m_1+6m_2}. Alors les mutations des morphismes
pr\'ec\'edents sont du type
$$(**) \ \ \ \
{\cal O}\otimes\cx{3m_1+6m_2-n}\longrightarrow({\cal O}(1)\otimes\cx{m_1})\oplus({\cal O}(2)\otimes\cx{m_2}).$$
La polarisation associ\'ee est \m{(\alpha_1,\beta_1,\beta_2)}, avec
$$\alpha_1=\q{1}{3m_1+6m_2-n}, \ \beta_1=\q{3-n\lambda_1}{3m_1+6m_2-n}, \
\beta_2=\q{6-n\lambda_2}{3m_1+6m_2-n}.$$
Elle est enti\`erement d\'etermin\'ee par le rapport
$$\rho' = \q{\beta_1}{\beta_2} = \q{3m_2\rho+3m_1-n}
{(6m_2-n)\rho+6m_1}.$$
Supposons que \ \m{m_1=1}. Alors on sait d'apr\`es \cite{dr_tr} construire
un bon quotient de l'ouvert des morphismes \m{G}-semi-stables de type
\m{(**)} d\`es que
$$\rho' > 3.$$
\subsubsection{Exemple 1}
Soit \m{m} un entier, \m{m\geq 0}. On consid\`ere des morphismes du
type
$$Q^*\oplus(Q_2^*\otimes\cx{5m+2})\longrightarrow{\cal O}\otimes\cx{29m+14}.$$
Si
$$\rho = \q{29}{12}+\epsilon,$$
avec \m{0<\epsilon\ll 1}, on sait d'apr\`es \cite{dr3} qu'un bon
quotient de l'ouvert des points \m{G}-semi-stables existe, et est
isomorphe \`a la vari\'et\'e de modules \m{M(4m+2, -2m-1, 2(m+1)^2)} des
faisceaux semi-stables au sens de Gieseker-Maruyama, de rang \m{4m+2} et de
classes de Chern \m{-2m-1} et \m{2(m+1)^2} sur \proj{2}. Dans \cite{dr3},
on ne donne pas de construction intrins\`eque du quotient
(on sait d\'ej\`a que \m{M(4m+2, -2m-1, 2(m+1)^2)} existe). Les
r\'esultats de \cite{dr_tr} ne permettent pas non plus de donner
directement l'existence du quotient. On peut cependant y parvenir en
utilisant le th\'eor\`eme 7.4, car dans ce cas
$$\rho' = 3 + \q{144\epsilon(m+1)}{29m+14+\epsilon(12m-24)} > 3.$$
On obtient des morphismes du type
$${\cal O}\otimes\cx{m+1}\longrightarrow{\cal O}(1)\oplus({\cal O}(2)\otimes\cx{5m+2}).$$
\subsubsection{Exemple 2}
Soit \m{m} un entier, \m{m\geq 0}. On consid\`ere des morphismes du
type
$$Q^*\oplus(Q_2^*\otimes\cx{17m+8})\longrightarrow{\cal O}\otimes\cx{99m+49}.$$
Si
$$\rho = \q{99}{41}+\epsilon,$$
avec \m{0<\epsilon\ll 1}, on sait d'apr\`es \cite{dr3} qu'un bon
quotient de l'ouvert des points \m{G}-semi-stables existe, et est
isomorphe \`a la vari\'et\'e de modules
\noindent
\m{M(7(2m+1), -4(2m+1), 32m^2+37m+11)}. Le th\'eor\`eme 7.4 permet de
construire le quotient, en utilisant les morphismes
$${\cal O}\otimes\cx{3m+2}\longrightarrow{\cal O}(1)\oplus({\cal O}(2)\otimes\cx{17m+8}).$$
\subsection{Un exemple sur \proj{n}}
On consid\`ere les morphismes
$$(\phi_1,\phi_2) : {\cal O}(-2)\oplus{\cal O}(-1)\longrightarrow{\cal O}\otimes\cx{n+2}$$
sur \proj{n}. Une polarisation est dans ce cas un triplet
\m{(\lambda_1,\lambda_2,\mu_1)} de nombres rationnels positifs tel que
$$\mu_1=\q{1}{n+2}, \ \lambda_1+\lambda_2 = 1.$$
Il revient au m\^eme de se donner
$$\rho = \q{\lambda_2}{\lambda_1}.$$
On sait construire des bons quotients (en utilisant les r\'esultats de
\cite{dr_tr})) d\`es que
$$\rho > n+1.$$
Mais dans ce cas le quotient est vide ! En effet, il existe toujours un
sous-espace vectoriel \ \m{H\subset\cx{n+2}} \ de dimension \m{n+1} tel
que \ \m{\mathop{\rm Im}\nolimits(\phi_2)\subset {\cal O}\otimes H}. On doit donc avoir, si
\m{(\phi_1,\phi_2)}
est \m{G}-semi-stable relativement \`a \m{(\lambda_1,\lambda_2,\mu_1)},
$$\lambda_2 - \q{n+1}{n+2} \leq 0,$$
c'est-\`a-dire \ \m{\rho\leq n+1}.
On emploie maintenant le th\'eor\`eme 7.4, et on consid\`ere donc les
morphismes
$${\cal O}\otimes\cx{\q{n(n+3)}{2}}\longrightarrow Q_2\oplus Q,$$
o\`u, comme dans le \paragra~\hskip -2pt 8.1, \m{Q} (resp. \m{Q_2}) d\'esigne le
le fibr\'e vectoriel sur \m{\proj{n}} conoyau du morphisme canonique
$${\cal O}(-1)\longrightarrow{\cal O}\otimes H^0({\cal O}(1))^* \ {\rm \ (resp. \ }
{\cal O}(-2)\longrightarrow{\cal O}\otimes H^0({\cal O}(2))^* \ {\rm \ )}.$$
En utilisant les r\'esultats de \cite{dr_tr} on parvient \`a construire
un bon quotient projectif d\`es que
$$\rho \geq 1 - \q{2n}{(n+1)(n+4)}.$$
Les valeurs {\em singuli\`eres} de \m{\rho} sont par d\'efinition celles
pour lesquelles la \m{G}-semi-stabilit\'e n'implique pas la
\m{G}-stabilit\'e. Ces valeurs sont exactement les nombres
$$\rho_k = \q{k}{n+2-k}$$
pour \m{1\leq k\leq n+1} . Dans ce cas un morphisme \m{(\phi_1,\phi_2)}
$G$-semi-stable non $G$-stable est construit de la fa\c con suivante :
on consid\`ere un sous-espace vectoriel \ \m{H\subset\cx{n+2}} \ de
dimension \m{k}, et on prend pour \m{\phi_2} un morphisme tel que
\ \m{\mathop{\rm Im}\nolimits(\phi_2)\subset{\cal O}\otimes H} \ et que $H$ soit le plus petit
sous-espace vectoriel ayant cette propri\'et\'e. On prend pour \m{\phi_1}
un morphisme tel que l'application lin\'eaire induite
$$H^0({\cal O}(2))^*\longrightarrow\cx{n+2}$$
soit surjective.
On obtient donc au total \ \m{2\lbrack\q{n}{2}\rbrack +2} \ quotients
distincts et non vides, dont \ \m{\lbrack\q{n}{2}\rbrack} \ sont
singuliers. Ils sont de dimension \
\m{\q{(n+2)(n^2+3n-2)}{2}}, sauf celui correspondant \`a \m{\rho_{n+1}},
qui est de dimension \m{\q{n(n+3)}{2}}.
\bigskip
On peut g\'en\'eraliser ce qui pr\'ec\`ede et obtenir des bons quotients
projectifs d'espaces de morphismes du type
$${\cal O}(-p-q)\oplus{\cal O}(-p)\longrightarrow{\cal O}\otimes\cx{n+2}$$
sur \proj{n}, \m{p,q} \'etant des entiers positifs.
\bigskip
\bigskip
|
1996-03-04T06:20:24 | 9603 | alg-geom/9603002 | en | https://arxiv.org/abs/alg-geom/9603002 | [
"alg-geom",
"math.AG"
] | alg-geom/9603002 | Alice Silverberg | A. Silverberg and Yu. G. Zarhin | Connectedness extensions for abelian varieties | LaTeX2e | null | null | null | null | Suppose $A$ is an abelian variety over a field $F$, and $\ell$ is a prime not
equal to the characteristic of $F$. Let $F_{\Phi,\ell}(A)$ denote the smallest
extension of $F$ such that the Zariski closure of the image of the $\ell$-adic
representation associated to $A$ is connected. Serre introduced this field, and
proved that when $F$ is a finitely generated extension of ${\mathbf Q}$,
$F_{\Phi,\ell}(A)$ does not depend on the choice of $\ell$. In this paper we
study extensions $F_{\Phi,\ell}(B)/F$ for twists $B$ of a given abelian
variety, especially when the abelian varieties are of Weil type.
| [
{
"version": "v1",
"created": "Fri, 1 Mar 1996 22:03:02 GMT"
},
{
"version": "v2",
"created": "Sun, 3 Mar 1996 21:54:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Silverberg",
"A.",
""
],
[
"Zarhin",
"Yu. G.",
""
]
] | alg-geom | \section{Introduction}
Suppose ${A}$ is an abelian variety defined over a field $F$,
$\ell$ is a prime number, and $\ell \neq \mathrm{char}(F)$.
Let $F^s$ denote a separable closure of $F$, let
$T_\ell({A}) = {\displaystyle \lim_\leftarrow {A}_{\ell^r}}$
(the Tate module), let
$V_\ell({A}) = T_\ell({A}) \otimes_{{\mathbf Z}_\ell}{\mathbf Q}_\ell$, and let
$\rho_{{A},\ell}$
denote the $\ell$-adic representation
$$\rho_{{A},\ell} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(T_\ell({A})) \subseteq
\mathrm{Aut}(V_\ell({A})).$$
If $L$ is an extension of $F$ in $F^s$, let
$G_{L,{A}}$ denote the image of $\mathrm{Gal}(F^s/L)$ under $\rho_{{A},\ell}$.
Let ${\mathfrak G}_\ell(F,{A})$
denote the algebraic envelope of the image of $\rho_{{A},\ell}$,
i.e., the Zariski closure of $G_{F,{A}}$ in
$\mathrm{Aut}(V_\ell({A})) \cong \mathrm{GL}_{2d}({\mathbf Q}_\ell)$,
where $d = \mathrm{dim}({A})$.
Let $F_{\Phi,\ell}({A})$ be the smallest extension $F'$ of $F$ such that
${\mathfrak G}_\ell(F',{A})$ is connected. We call this extension the
$\ell$-connectedness extension, or connectedness extension.
The algebraic group ${\mathfrak G}_\ell(F,{A})$
and the field $F_{\Phi,\ell}({A})$ were
introduced by Serre (\cite{serre},
\cite{resume}, \cite{serremotives}), who
proved that if $F$ is a global field or a finitely
generated extension of ${\mathbf Q}$,
then $F_{\Phi,\ell}({A})$ is independent of $\ell$
(see also \cite{LPinv}, \cite{LPmathann}, \cite{LP}). In such cases,
we will denote the field $F_{\Phi,\ell}({A})$ by $F_\Phi({A})$. For
every integer $n \ge 3$ we have
$$F_{\Phi}({A}) \subseteq F({A}_n)$$
(see \cite{Borovoi}, \cite{Borovoisb}, Proposition 3.6 of
\cite{Chi}, and \cite{conn}). Larsen and Pink \cite{LP}
recently proved that for every integer $n \ge 3$,
$$F_\Phi({A}) = \bigcap_{\mathrm{prime}~~p \geq n} F({A}_p).$$
In \cite{conn} we found conditions for the connectedness
of ${\mathfrak G}_\ell(F,{A})$, while in \cite{connexts} we used connectedness
extensions and Serre's $\ell$-in\-de\-pen\-dence results
to obtain $\ell$-in\-de\-pen\-dence results for the intersection of
${\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$ with the torsion subgroup of the center of
$\mathrm{End}({A}) \otimes {\mathbf Q}$.
Let $F(\mathrm{End}({A}))$
denote the smallest extension of $F$ over which all the endomorphisms
of ${A}$ are defined. Then
(see Proposition 2.10 of \cite{conn}),
$$F(\mathrm{End}({A})) \subseteq F_{\Phi,\ell}({A}).$$
Therefore, ${\mathfrak G}_\ell(F,{A})$ fails to be connected when the ground field is
not a field of definition for the endomorphisms of ${A}$.
For example, if $F$ is a subfield of ${\mathbf C}$, and
${A}$ is an elliptic curve over $F$ with complex multiplication
by an imaginary quadratic field $K$ which is not contained in $F$, then
$F \neq KF = F(\mathrm{End}({A})) \subseteq F_{\Phi,\ell}({A})$.
More generally, if ${A}$ is an abelian variety
of CM-type, and ${\tilde K}$ is the
reflex CM-field, then $F(\mathrm{End}({A})) \supseteq {\tilde K}$; if
${\tilde K}$ is not contained in $F$ then
$F \neq F(\mathrm{End}({A})) \subseteq F_{\Phi,\ell}({A})$.
It is therefore natural to enlarge
the ground field $F$ so that it is a field of definition for the
endomorphisms of ${A}$.
By enlarging the ground field, we may assume that
$F = F(\mathrm{End}({A})) = F_{\Phi,\ell}({A})$. We then consider the $F$-forms
${B}$ of ${A}$ such that $F = F(\mathrm{End}({B}))$. For such ${B}$, we describe
the connectedness extensions
$F_{\Phi,\ell}({B})/F$ (see \S\ref{exclasses}, especially
Theorem \ref{discond} and Corollary \ref{discondcor}).
Properties of Mumford-Tate groups given in \S\ref{notation} allow
us to obtain explicit information about the connectedness extensions
$F_{\Phi,\ell}({B})/F$ under additional conditions (see
Theorems \ref{twistX} and \ref{twistE}).
Our conditions in Theorems \ref{twistX} and \ref{twistE}
are based on Weil's philosophy in \cite{WeilHodge} whereby
exceptional Hodge classes arise from certain abelian varieties
that have a CM-field embedded in their endomorphism algebras.
In \S \ref{examples} we use the results of \S\ref{exclasses}
to explicitly compute non-trivial connectedness extensions in
special cases.
\noindent {\bf Acknowledgments:}
The authors would like to thank the Mathematische Institut der
Universit\"at Erlangen-N\"urnberg for its hospitality.
\section{Definitions, notation, and lemmas}
\label{notation}
Let ${\mathbf Z}$, ${\mathbf Q}$, and ${\mathbf C}$ denote respectively the integers, rational
numbers, and complex numbers. If $r$ is an integer, then
${\mathbf Q}(r)$ denotes the rational Hodge structure of weight $-2r$ on ${\mathbf Q}$ (see
\S 1 of \cite{sln900}). If $a$ and $b$ are integers, let $(a,b)$
denote the greatest common divisor of $a$ and $b$.
If $F$ is a field, let $F^s$ denote a separable closure of $F$ and let
${\bar F}$ denote an algebraic closure of $F$. If ${A}$ is an abelian variety
over a field $F$, write $\mathrm{End}_F({A})$
for the set of endomorphisms of ${A}$ which are defined over $F$, let
$\mathrm{End}({A}) = \mathrm{End}_{F^s}({A})$, and let $\mathrm{End}^0({A}) = \mathrm{End}({A}) \otimes_{\mathbf Z} {\mathbf Q}$.
Let $Z_{A}$ denote the center of $\mathrm{End}({A})$.
If $G$ is an algebraic
group, let $G^0$ denote the identity connected component.
\begin{lem}[Lemma 2.7 of \cite{conn}]
\label{conncomp}
If ${A}$ is an abelian variety over a field $F$, $L$ is a finite
extension of $F$ in $F^s$, and $\ell$ is a prime number, then
$${\mathfrak G}_\ell(L,{A}) \subseteq {\mathfrak G}_\ell(F,{A})
\text{ and } {\mathfrak G}_\ell(L,{A})^0 = {\mathfrak G}_\ell(F,{A})^0.$$
In particular, if ${\mathfrak G}_\ell(F,{A})$
is connected, then ${\mathfrak G}_\ell(F,{A}) = {\mathfrak G}_\ell(L,{A})$.
\end{lem}
\begin{lem}
\label{conncomplem}
Suppose ${A}$ and ${B}$ are abelian varieties over a field $F$, $L$ is a finite
extension of $F$ in $F^s$, $\ell$ is a prime number,
$\ell \neq \mathrm{char}(F)$, ${\mathfrak G}_\ell(F,{A})$ is
connected, and ${A}$ and ${B}$ are isomorphic over $L$. Then:
\begin{enumerate}
\item[{(i)}] ${\mathfrak G}_\ell(F,{B})^0 = {\mathfrak G}_\ell(F,{A})$, and
\item[{(ii)}]${\mathfrak G}_\ell(L,{B})$ is connected, i.e.,
$F_{\Phi,\ell}({B}) \subseteq L$.
\end{enumerate}
\end{lem}
\begin{proof}
Since ${A}$ and ${B}$ are isomorphic over $L$, and ${\mathfrak G}_\ell(F,{A})$ is
connected, we have
$${\mathfrak G}_\ell(L,{B}) = {\mathfrak G}_\ell(L,{A}) = {\mathfrak G}_\ell(F,{A}) = {\mathfrak G}_\ell(F,{A})^0$$
$$= {\mathfrak G}_\ell(L,{A})^0 = {\mathfrak G}_\ell(L,{B})^0 = {\mathfrak G}_\ell(F,{B})^0,$$
using Lemma \ref{conncomp}. The result follows.
\end{proof}
\begin{prop}
\label{endaut}
Suppose ${A}$ and ${B}$ are abelian varieties over a field $F$, $L$ is a field
extension of $F$ in $F^s$, and $f: {A} \to {B}$ is an isomorphism defined over
$L$. Suppose that for every $\sigma \in \mathrm{Gal}(F^s/F)$, the element
$f^{-1}\sigma(f)$ of $\mathrm{Aut}({A})$ commutes with
every element of $\mathrm{End}_L({A})$. Then $\mathrm{End}_F({A}) \cong \mathrm{End}_F({B})$.
\end{prop}
\begin{proof}
Define an isomorphism $\varphi : \mathrm{End}_L({A}) \to \mathrm{End}_L({B})$ by
$\varphi(\beta) = f\beta f^{-1}$. For every $\beta \in \mathrm{End}_L({A})$ and $\sigma
\in \mathrm{Gal}(F^s/F)$, we have
$f^{-1}\sigma (f)\beta = \beta f^{-1}\sigma(f)$.
Therefore, $\sigma(f\sigma^{-1}(\beta) f^{-1}) = f\beta f^{-1}$.
Thus, $\beta \in \mathrm{End}_F({A})$ if and only if $f\beta f^{-1} \in \mathrm{End}_F({B})$.
In other words, the restriction of $\varphi$ to $\mathrm{End}_F({A})$ induces an
isomorphism onto $\mathrm{End}_F({B})$.
\end{proof}
As a corollary we have the following result. See also Lemma 5.1 of
\cite{szisog}.
\begin{cor}
\label{cocy}
Suppose ${A}$ is an abelian variety over a field $F$. If an element
of $H^1(\mathrm{Gal}(F^s/F),\mathrm{Aut}({A}))$ is represented by a cocycle $c$
with values
in the center of $\mathrm{End}^0({A})$, and ${B}$ is the twist of ${A}$ by $c$,
then $\mathrm{End}_F({A}) \cong \mathrm{End}_F({B})$.
\end{cor}
\begin{proof}
The cocycle $c$ defines an isomorphism $f : {A} \to {B}$ such that for every
$\sigma \in \mathrm{Gal}(F^s/F)$, $f^{-1}\sigma(f) = c(\sigma)$. We apply
Proposition \ref{endaut}.
\end{proof}
\begin{lem}
\label{cocychar}
Suppose ${A}$ is an abelian variety over a field $F$,
$c$ is a cocycle on $\mathrm{Gal}(F^s/F)$ with values in $\mathrm{Aut}({A})$,
${B}$ is the twist of ${A}$ by $c$,
and $F = F(\mathrm{End}({A})) = F(\mathrm{End}({B}))$.
Then $c$ is a character with values in $Z_{A}^\times$,
where $Z_{A}$ denotes the center of $\mathrm{End}({A})$.
\end{lem}
\begin{proof}
Since $\mathrm{Gal}(F^s/F)$ acts trivially on $\mathrm{End}({A})$,
the cocycle $c$ is a homomorphism. Let $f : {A} \to {B}$ be the
isomorphism induced by $c$. Then $c(\sigma) = f^{-1}\sigma(f)$
for every $\sigma \in \mathrm{Gal}(F^s/F)$. Since
$F = F(\mathrm{End}({A})) = F(\mathrm{End}({B}))$, it easily follows that
$c(\sigma) \in Z_{A}$ and $c(\sigma)^{-1} \in Z_{A}$.
\end{proof}
\begin{rem}
\label{twistyrem}
If an abelian variety ${B}$ over $F$ is the twist of an abelian variety
${A}$ by $c \in H^1(\mathrm{Gal}(F^s/F),\mathrm{Aut}({A}))$ then one may easily check that
the Galois module ${B}(F^s)$ is the twist by $c$ of the Galois module
${A}(F^s)$, and therefore the Galois module $V_\ell({B})$ is the twist by
$c$ of the Galois module $V_\ell({A})$.
\end{rem}
We define the Mumford-Tate group of a complex abelian variety ${A}$
(see \S2 of \cite{Ribet} or \S6 of \cite{Izv}). If ${A}$ is a complex abelian
variety, let $V = H_1({A}({\mathbf C}),{\mathbf Q})$ and consider the Hodge decomposition
$V \otimes {\mathbf C} = H_1({A}({\mathbf C}),{\mathbf C}) = H^{-1,0} \oplus H^{0,-1}$.
Define a homomorphism $\mu : {\mathbf G}_m \to GL(V)$ as follows. For
$z \in {\mathbf C}$, let $\mu(z)$ be the automorphism of $V \otimes {\mathbf C}$ which is
multiplication by $z$ on $H^{-1,0}$ and is the identity on $H^{0,-1}$.
\begin{defn}
The {\em Mumford-Tate group} $MT_{A}$ of ${A}$ is the smallest
algebraic subgroup of $GL(V)$, defined over ${\mathbf Q}$, which after extension of
scalars to ${\mathbf C}$ contains the image of $\mu$.
\end{defn}
It follows from the definition that $MT_{A}$ is connected.
Define a homomorphism $\varphi : {\mathbf G}_m \times {\mathbf G}_m \to GL(V)$
as follows. For $z, w \in {\mathbf C}$, let $\varphi(z,w)$ be the automorphism of
$V \otimes {\mathbf C}$ which is multiplication by $z$ on $H^{-1,0}$ and is
multiplication by $w$ on $H^{0,-1}$. Then $MT_{A}$ can also be defined as
the smallest algebraic subgroup of $GL(V)$, defined over ${\mathbf Q}$, which after
extension of scalars to ${\mathbf C}$ contains the image of $\varphi$. The equivalence
of the definitions follows easily from the fact that $H^{-1,0}$ is the complex
conjugate of $H^{0,-1}$. (See \S 3 of \cite{serrereps}, where $MT_{A}$ is called
the Hodge group. See also \S 6 of \cite{Izv}.)
If ${A}$ is an abelian variety over a subfield $F$ of ${\mathbf C}$, we fix an
embedding of ${\bar F}$ in ${\mathbf C}$. This gives an identification of
$V_\ell({A})$ with $H_1({A},{\mathbf Q})\otimes{\mathbf Q}_\ell$, and allows us to view
$MT_{A} \times {\mathbf Q}_\ell$ as a linear ${\mathbf Q}_\ell$-algebraic subgroup of
$GL(V_\ell({A}))$. Let
$$MT_{{A},\ell} = MT_{A} \times_{\mathbf Q} {\mathbf Q}_\ell.$$
Then $MT_{{A}}({\mathbf Q}_\ell) = MT_{{A},\ell}({\mathbf Q}_\ell)$.
The Mumford-Tate conjecture for abelian varieties (see \cite{serrereps})
may be reformulated as the equality of ${\mathbf Q}_\ell$-algebraic groups,
${\mathfrak G}_\ell(F,{A})^0 = MT_{{A},\ell}$.
\begin{thm}[Piatetski-Shapiro \cite{ps}, Deligne \cite{sln900}, Borovoi
\cite{Borovoisb}]
\label{psdcor}
If ${A}$ is an abelian variety over a finitely generated extension $F$ of
${\mathbf Q}$, then ${\mathfrak G}_\ell(F,{A})^0 \subseteq MT_{{A},\ell}$.
\end{thm}
In \S\ref{exclasses} it will be helpful to use a slightly
different version of the Mumford-Tate group, as defined by Deligne
(see p.~43 and pp.~62--63 of \cite{sln900}). We will denote this group
${\widetilde{MT}}_{A}$. (See also pp.~466--467 of \cite{Milne} for a
comparison between $MT_{A}$ and ${\widetilde{MT}}_{A}$.)
Letting $V^*$ be the dual of $V$, then
$T = V^{\otimes p} \otimes (V^*)^{\otimes q} \otimes {\mathbf Q}(r)$ has a Hodge
structure of weight $q - p - 2r$. If $\nu \in {\mathbf G}_m$, let $\nu$ act on
${\mathbf Q}(1)$ as $\nu^{-1}$, and we obtain a canonical action of
$GL(V) \times {\mathbf G}_m$ on $T$. (Note that $V^* \cong V \otimes {\mathbf Q}(1)$,
since $V$ is a polarized Hodge structure of weight $-1$.)
\begin{defn}
The group ${\widetilde{MT}}_{A}$ is the subgroup of
$GL(V) \times {\mathbf G}_m$ consisting of the elements which fix all
rational tensors of bidegree $(0,0)$ belonging to any $T$.
\end{defn}
\begin{lem}[Proposition 3.4 of \cite{sln900}]
\label{murem}
The algebraic group ${\widetilde{MT}}_{A}$ is the smallest algebraic subgroup of
$GL(V) \times {\mathbf G}_m$ defined over ${\mathbf Q}$ which, after extension of
scalars to ${\mathbf C}$, contains the image of
$(\mu,{\mathrm{id}}) : {\mathbf G}_m \to GL(V) \times {\mathbf G}_m$.
\end{lem}
If $F$ is a field and $\ell$ is a prime
number different from $\mathrm{char}(F)$, let
$$\chi_\ell : \mathrm{Gal}(F^s/F) \to {\mathbf Z}_\ell^\times \subset {\mathbf Q}_\ell^\times$$
denote the cyclotomic character. If $r$ is an integer, then the
$\mathrm{Gal}(F^s/F)$-module ${\mathbf Q}_\ell(r)$ is the ${\mathbf Q}_\ell$-vector space ${\mathbf Q}_\ell$
with Galois action defined by the character $\chi_\ell^r$. We have
${\mathbf Q}_\ell(r) = {\mathbf Q}(r) \otimes_{\mathbf Q} {\mathbf Q}_\ell$ (see \S 1 of \cite{sln900}).
Suppose ${A}$ is an abelian variety over $F$. Let $V_\ell = V_\ell({A})$ and
let $V_\ell^*$ be the dual of $V_\ell$.
If $\nu \in {\mathbf G}_m$, let $\nu$ act on
${\mathbf Q}_\ell(1)$ as $\nu^{-1}$. We obtain a canonical action of
$GL(V_\ell) \times {\mathbf G}_m$ on
$V_\ell^{\otimes p} \otimes (V_\ell^*)^{\otimes q} \otimes {\mathbf Q}_\ell(r)$.
Define
$${\tilde \rho}_{{A},\ell} : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}(V_\ell) \times {\mathbf Q}_\ell^\times
= \mathrm{Aut}(V_\ell) \times {\mathbf G}_m({\mathbf Q}_\ell)$$
by ${\tilde \rho}_{{A},\ell}(\sigma) =
(\rho_{{A},\ell}(\sigma),\chi_\ell^{-1}(\sigma))$.
\begin{defn}
Let ${\tilde {\mathfrak G}}_\ell(F,{A})$ denote the smallest
${\mathbf Q}_\ell$-algebraic subgroup of
$$GL(V_\ell) \times {\mathbf G}_m$$ whose group of ${\mathbf Q}_\ell$-points contains
the image of ${\tilde \rho}_{{A},\ell}$.
\end{defn}
If ${A}$ is a complex abelian variety, then a polarization on ${A}$ (i.e.,
the imaginary part of a Riemann form) produces an element $E$ of
$\mathrm{Hom}(\wedge^2V,{\mathbf Q}(1))$ which is a rational tensor of bidegree $(0,0)$.
If ${A}$ is an abelian variety over an arbitrary field $F$, then a
polarization on ${A}$ defined over $F$ defines a $\mathrm{Gal}(F^s/F)$-invariant
element $E_\ell$ of $\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell(1))$
(since the Weil pairing is $\mathrm{Gal}(F^s/F)$-equivariant).
If ${\bar F}$ is a
subfield of ${\mathbf C}$, and we fix a polarization on ${A}$ defined over $F$,
then the line generated by $E_\ell$ in $\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell(1))$
is the extension of scalars to ${\mathbf Q}_\ell$ of the line generated by $E$
in $\mathrm{Hom}(\wedge^2V,{\mathbf Q}(1))$.
(See p.~237 of \cite{MumfordAV}, especially the last sentence.)
The following result implies that the projection map
$GL(V) \times {\mathbf G}_m \to GL(V)$ induces an isomorphism from
${\widetilde{MT}}_{A}$ onto $MT_{A}$. Since we were not able to find a proof
in the literature, we have included one for the benefit of the
reader.
\begin{prop}
\label{MTtilde}
If ${A}$ is a complex abelian variety, then there exists a (unique)
character $\gamma : MT_{A} \to {\mathbf G}_m$ such that
${\widetilde{MT}}_{A}$ is the graph of $\gamma$.
\end{prop}
\begin{proof}
Let $p_1$ and $p_2$ denote the
projection maps from $GL(V) \times {\mathbf G}_m$ onto $GL(V)$ and ${\mathbf
G}_m$,
respectively. By Lemma \ref{murem}, $MT_{A}$ is the image of
${\widetilde{MT}}_{A}$
under $p_1$.
Fix a polarization on ${A}$. The polarization generates a line $D$ in the
${\mathbf Q}$-vector space $\mathrm{Hom}(\wedge^2V,{\mathbf Q}(1))$, on which ${\widetilde{MT}}_{A}$ acts
trivially. Let $D(-1) = D \otimes {\mathbf Q}(-1)$, a line in
$\mathrm{Hom}(\wedge^2V,{\mathbf Q})$. Since ${\widetilde{MT}}_{A}$ acts trivially on $D$,
${\widetilde{MT}}_{A}$ acts on $D(-1)$ via $p_2$. Let
$$B = \{\alpha \in GL(V) : \alpha D(-1) \subseteq D(-1) \}$$
and let the character $\gamma : B \to \mathrm{Aut}(D(-1)) = {\mathbf G}_m$
be induced by the action of $GL(V)$ on $\mathrm{Hom}(\wedge^2V,{\mathbf Q})$.
The action of $GL(V) \times {\mathbf G}_m$ on $\mathrm{Hom}(\wedge^2V,{\mathbf Q})$ factors
through $GL(V)$. Therefore $MT_{A} \subseteq B$, and we have a
commutative diagram
\downdiag{{\widetilde{MT}}_{A}}{MT_{A}}{{\mathbf G}_m}{p_1}{p_2}{\gamma}
which gives the desired result.
\end{proof}
\begin{prop}
\label{rtilde}
If ${A}$ is an abelian variety over a field $F$, $\ell$ is a prime number,
and $\ell \ne \mathrm{char}(F)$, then there exists a
(unique) character $\gamma_\ell : {\mathfrak G}_\ell(F,{A}) \to {\mathbf G}_m$ such that
\begin{enumerate}
\item[{(i)}]
${\tilde {\mathfrak G}}_\ell(F,{A})$ is the graph of $\gamma_\ell$,
\item[{(ii)}]
the restriction of $\gamma_\ell$ to $G_{F,{A}}$ is $\chi_\ell^{-1}$,
\item[{(iii)}]
if ${\bar F}$ is a subfield of ${\mathbf C}$, then $\gamma_\ell = \gamma$ on
$MT_{{A},\ell} \cap {\mathfrak G}_\ell(F,{A})$.
\end{enumerate}
\end{prop}
\begin{proof}
Let $\pi_1$ and $\pi_2$ denote
the projection maps from $GL(V_\ell) \times {\mathbf G}_m$ onto $GL(V_\ell)$
and
${\mathbf G}_m$, respectively. By the definitions, ${\mathfrak G}_\ell(F,{A})$ is the image
of ${\tilde {\mathfrak G}}_\ell(F,{A})$ under $\pi_1$.
Fix a polarization on ${A}$ defined over $F$. The
polarization generates a line $D_\ell$ in the ${\mathbf Q}_\ell$-vector space
$\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell(1))$.
Let $D_\ell(-1) = D_\ell \otimes {\mathbf Q}_\ell(-1)$, a line in
$\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell)$. Since the Weil pairing is
$\mathrm{Gal}(F^s/F)$-equivariant,
$\mathrm{Gal}(F^s/F)$ acts trivially on $D_\ell$. Therefore
${\tilde {\mathfrak G}}_\ell(F,{A})$ acts trivially on $D_\ell$, and acts via $\pi_2$ on
$D_\ell(-1)$. Let
$$B_\ell =
\{\alpha \in GL(V_\ell) : \alpha D_\ell(-1) \subseteq D_\ell(-1) \}$$
and let the character
$\gamma_\ell : B_\ell \to \mathrm{Aut}(D_\ell(-1)) = {\mathbf G}_m$
be induced by the
action of $GL(V_\ell)$ on $\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell)$.
The action of $GL(V_\ell) \times {\mathbf G}_m$ on
$\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell)$ factors
through the action of $GL(V_\ell)$. Therefore
${\tilde {\mathfrak G}}_\ell(F,{A}) \subseteq B_\ell$, and we have a
commutative diagram
\downdiag{{\tilde {\mathfrak G}}_\ell(F,{A})}{{\mathfrak G}_\ell(F,{A})}{{\mathbf
G}_m}{\pi_1}{\pi_2}{\gamma_\ell}
which gives (i).
Since the restriction of $\pi_2$ to $G_{F,{A}}$ is $\chi_\ell^{-1}$, we have
(ii). Now suppose ${\bar F}$ is a subfield of ${\mathbf C}$.
Using the fixed polarization, define $D$, $D(-1)$,
$B$, and $\gamma$ as in the proof of Theorem \ref{MTtilde}. Then
$B_\ell = B \times_{\mathbf Q} {\mathbf Q}_\ell$, and therefore
$MT_{{A},\ell} \subseteq B_\ell$. Since
$\gamma$ (respectively, $\gamma_\ell$) is induced by the action of
$GL(V)$ on $\mathrm{Hom}(\wedge^2V,{\mathbf Q})$ (respectively, $GL(V_\ell)$ on
$\mathrm{Hom}(\wedge^2V_\ell,{\mathbf Q}_\ell)$), and $V_\ell = V \otimes_{\mathbf Q} {\mathbf Q}_\ell$,
we have (iii).
\end{proof}
Write ${\widetilde{MT}}_{{A},\ell}$ for the ${\mathbf Q}_\ell$-algebraic subgroup
${\widetilde{MT}}_{A} \times_{\mathbf Q} {\mathbf Q}_\ell$ of $GL(V_\ell) \times {\mathbf G}_m$.
Then ${\widetilde{MT}}_{{A}}({\mathbf Q}_\ell) = {\widetilde{MT}}_{{A},\ell}({\mathbf Q}_\ell)$.
We state a reformulation of Theorem \ref{psdcor}, which we will use in
\S\ref{exclasses}.
\begin{thm}
\label{psdtilde}
If ${A}$ is an abelian variety over a finitely generated extension $F$ of
${\mathbf Q}$, then
${\tilde {\mathfrak G}}_\ell(F,{A})^0 \subseteq {\widetilde{MT}}_{{A},\ell}$.
\end{thm}
\begin{proof}
The result follows directly from Theorem \ref{psdcor} and Propositions
\ref{MTtilde} and \ref{rtilde}.
\end{proof}
\section{Connectedness extensions}
\label{exclasses}
\begin{thm}
\label{discond}
Suppose ${A}$ is an abelian variety over a field $F$, $\ell$ is a prime
number not equal to $\mathrm{char}(F)$,
$$c : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}_F({A}) \subseteq \mathrm{Aut}(V_\ell({A}))$$
is a homomorphism,
${B}$ is the twist of ${A}$ by the cocycle determined by $c$,
and $$F = F(\mathrm{End}({A})) = F_{\Phi,\ell}({A}).$$
Then:
\begin{enumerate}
\item[{(i)}] $c$ induces an isomorphism
$$\mathrm{Gal}(F_{\Phi,\ell}({B})/F) \cong
\mathrm{Im}(c)/(\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)),$$
\item[{(ii)}] ${\mathfrak G}_\ell(F,{B})$ is connected if and only
if $\mathrm{Im}(c) \subseteq {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$,
\item[{(iii)}] if $M$ is the abelian extension of $F$ in
$F^s$ cut out by $c$, then $c$ induces an isomorphism
$$\mathrm{Gal}(M/F_{\Phi,\ell}({B})) \cong
\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell).$$
\end{enumerate}
\end{thm}
\begin{proof}
By Lemma \ref{conncomplem}ii, $F_{\Phi,\ell}({B}) \subseteq M$.
The character $c$ induces isomorphisms
$$\mathrm{Gal}(M/F) \cong \mathrm{Im}(c)$$
and
$$\mathrm{Gal}(M/F_{\Phi,\ell}({B})) \cong
\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,{B})^0({\mathbf Q}_\ell).$$
By Lemma \ref{conncomplem}i, we have
${\mathfrak G}_\ell(F,{B})^0 \cong {\mathfrak G}_\ell(F,{A})$, and the result follows.
\end{proof}
\begin{cor}
\label{discondcor}
Suppose ${A}$ is an abelian variety over a field $F$, $\ell$ is a prime
number not equal to $\mathrm{char}(F)$, ${B}$ is the twist of ${A}$ by a cocycle
$$c : \mathrm{Gal}(F^s/F) \to \mathrm{Aut}({A}) \subseteq \mathrm{Aut}(V_\ell({A})),$$
and $$F = F(\mathrm{End}({A})) = F_{\Phi,\ell}({A}) = F(\mathrm{End}({B})).$$
Then:
\begin{enumerate}
\item[{(i)}] $c$ is a character with values in $Z_{A}^\times$
(where $Z_{A}$ denotes the center of $\mathrm{End}({A})$),
\item[{(ii)}] $c$ induces an isomorphism
$$\mathrm{Gal}(F_{\Phi,\ell}({B})/F) \cong
\mathrm{Im}(c)/(\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)),$$
\item[{(iii)}] ${\mathfrak G}_\ell(F,{B})$ is connected if and only
if $\mathrm{Im}(c) \subseteq {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell)$,
\item[{(iv)}] if $M$ is the abelian extension of $F$ in
$F^s$ cut out by $c$, then $c$ induces an isomorphism
$$\mathrm{Gal}(M/F_{\Phi,\ell}({B})) \cong
\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_\ell).$$
\end{enumerate}
\end{cor}
\begin{proof}
By Lemma \ref{cocychar} and the assumption that
$F = F(\mathrm{End}({B}))$, we have (i). The result now follows from
Theorem \ref{discond}.
\end{proof}
Suppose $A$ is an abelian variety defined over a field $F$ of characteristic
zero, $k$ is a CM-field, $\iota : k \hookrightarrow \mathrm{End}_F^0(A)$
is an embedding, and $C$ is an algebraically
closed field containing $F$. Let $\mathrm{Lie}(A)$ be the tangent space of $A$
at the origin,
an $F$-vector space. If $\sigma$ is an embedding of $k$ into $C$, let
$$n_\sigma = \mathrm{dim}_C\{t \in \mathrm{Lie}(A)\otimes_F C :
\iota(\alpha)t = \sigma(\alpha)t {\text{ for all }} \alpha \in k\}.$$
Write ${\bar \sigma}$ for the composition of $\sigma$ with the involution
complex conjugation of $k$.
\begin{defn}
If $A$ is an abelian variety over an algebraically closed field $C$ of
characteristic zero, $k$ is a CM-field, and
$\iota : k \hookrightarrow \mathrm{End}^0(A)$ is an embedding,
we say $(A,k,\iota)$ is {\em of Weil type}
if $n_\sigma = n_{\bar \sigma}$ for all embeddings $\sigma$ of $k$ into
$C$.
\end{defn}
Although we do not use this fact, we remark that $(A,k,\iota)$
is of Weil type if and only if $\iota$ makes $\mathrm{Lie}(A) \otimes_F C$ into a free
$k \otimes_{\mathbf Q} C$-module (see p.~525 of \cite{Ribet} for the case where $k$
is an imaginary quadratic field). Using the
semisimplicity of the $F$-algebra $k \otimes_{\mathbf Q} F$ and the
$C$-algebra $k \otimes_{\mathbf Q} C$, one may easily deduce that $\iota$ makes
$\mathrm{Lie}(A) \otimes_F C$ into a free $k \otimes_{\mathbf Q} C$-module
if and only if $\iota$ makes $\mathrm{Lie}(A)$ into a free $k \otimes_{\mathbf Q} F$-module.
Suppose $(A,k,\iota)$ is of Weil type, and we have an element of
$$H^1(\mathrm{Gal}({\bar F}/F),\mathrm{Aut}(A))$$ which is represented by a cocycle $c$ with
values in the center of $\mathrm{End}^0(A)$. Let $B$ be the twist of $A$ by $c$,
and let $\varphi$ be the isomorphism from $\mathrm{End}_F(A)$ to $\mathrm{End}_F(B)$
obtained in Corollary \ref{cocy} and Proposition \ref{endaut}. Since
$(A,\iota)$ and
$(B,\varphi\circ \iota)$ are isomorphic over $C$, it follows that
$(B,k,\varphi\circ \iota)$ is of Weil type.
Note that if $(A,k,\iota)$ is of Weil type, then $\mathrm{dim}(A)$ is
divisible by $[k:{\mathbf Q}]$.
\begin{thm}
\label{twistX}
Suppose $A$ is an abelian variety over a finitely generated
extension $F$ of ${\mathbf Q}$,
$\ell$ is a prime number, $k$ is a CM-field,
$\iota$ is an embedding of $k$ into the center of
$\mathrm{End}^0(A)$ such that $(A, k, \iota)$ is of Weil type,
$c: \mathrm{Gal}({\bar F}/F) \to k^\times$ is a character
of finite order $n$,
$r = 2\mathrm{dim}(A)/[k:{\mathbf Q}] \in {\mathbf Z}$,
$M$ is the ${\mathbf Z}/n{\mathbf Z}$-extension of $F$ cut out by $c$, and
$B$ is the twist of $A$ by $c$.
Suppose
$F = F(\mathrm{End}(A))$, $\iota \circ c$ takes values in $\mathrm{Aut}(A)$,
$r$ is even, and $n$ does not divide $r$. Then
\begin{enumerate}
\item[{(i)}] $F = F(\mathrm{End}(B))$,
\item[{(ii)}]
either $F \ne F_{\Phi}(A)$ or $F \ne F_{\Phi}(B)$,
\item[{(iii)}] if $F_{\Phi}(A) = F$, then $F_{\Phi}(B) \subseteq M$
and $[M:F_{\Phi}(B)]$ divides $(n,2r)$,
\item[{(iv)}] if $F_{\Phi}(A) = F$ and $(n,2r) = 2$, then
$[M:F_{\Phi}(B)] = 2$.
\end{enumerate}
\end{thm}
\begin{proof}
The Galois module $V_{\ell}(B)$ is the twist of
$V_{\ell}(A)$ by $c$ (see Remark \ref{twistyrem}).
By applying Corollary \ref{cocy} to the cocycle induced
by $c$, we deduce (i) and we obtain an isomorphism $\varphi$
from $\mathrm{End}_F(A)$ onto $\mathrm{End}_F(B)$ such that $(B,k,\varphi\circ\iota)$
is of Weil type.
Let $k_\ell = k \otimes {\mathbf Q}_\ell$. For ${U} = A$ or $B$, let
$$W_{U} =
\mathrm{Hom}_{\mathbf Q}(\wedge^{r}_{k} H_1({U},{\mathbf Q}),{\mathbf Q}({\frac{r}{2}})), \quad
W_{{U},\ell} =
\mathrm{Hom}_{{\mathbf Q}_\ell}(\wedge^{r}_{k_\ell} V_\ell({U}),{\mathbf Q}_\ell({\frac{r}{2}})),$$
where $\mathrm{Hom}_E$ means homomorphisms of $E$-vector spaces, if $E$ is a field.
Then $W_{U}$ is a one-dimensional $k$-vector space and $W_{{U},\ell}$ is a
free
rank-one
$k_\ell$-module. The elements of $W_{U}$ are called Weil classes for ${U}$.
Since $V_{\ell}({U})=H_1({U},{\mathbf Q})\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}$, we have
$W_{{U},\ell}=W_{U}\otimes_{{\mathbf Q}}{\mathbf Q}_{\ell}.$
Consider the action of the Galois group $\mathrm{Gal}({\bar F}/F)$.
The Galois module
$W_{B,\ell}$ is the twist of the Galois module $W_{A,\ell}$
by the character $c^{-r}$.
Since $n$ does not divide $r$, this is a non-trivial twist,
so the Galois modules
$W_{B,\ell}$ and $W_{A,\ell}$ cannot be simultaneously trivial.
By pp.~52--54 of \cite{sln900} (see also Lemma 2.8 of \cite{moonzar}
and p.~423 of \cite{WeilHodge}), the elements of $W_{U}$ are Hodge classes
(since we are dealing with abelian varieties of Weil type).
Since ${\widetilde MT}_{U}$ acts trivially on the Hodge classes,
${\widetilde MT}_{{U},\ell}({\mathbf Q}_\ell)$ acts trivially on
$W_{{U},\ell} = W_{U} \otimes_{\mathbf Q} {\mathbf Q}_\ell$.
Suppose now that ${\mathfrak G}_\ell(F,A)$ and ${\mathfrak G}_\ell(F,B)$ are both
connected. Then ${\tilde {\mathfrak G}}_\ell(F,A)$ and ${\tilde {\mathfrak G}}_\ell(F,B)$ are both
connected (by Proposition \ref{rtilde}i).
It follows from Theorem \ref{psdtilde} that
${\tilde {\mathfrak G}}_\ell(F,{U}) \subseteq {\widetilde MT}_{{U},\ell}$.
Therefore, $W_{B,\ell}$ and $W_{A,\ell}$ are both trivial
as $\mathrm{Gal}({\bar F}/F)$-modules. This is a contradiction. We therefore have (ii).
Suppose that $F_{\Phi}(A) = F$. Then ${\mathfrak G}_\ell(F,A)$ is connected,
so ${\mathfrak G}_\ell(F,B)$ is disconnected. By Lemma \ref{conncomplem},
${\mathfrak G}_\ell(M,B)$ is connected. Therefore, $F_{\Phi}(B) \subseteq M$.
By Corollary \ref{discondcor}iv, $$\mathrm{Gal}(M/F_{\Phi}(B)) \cong
\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,A)({\mathbf Q}_\ell).$$
Let $\mu_s(k)$ denote the group of $s$-th roots of unity in $k^\times$.
We have $$\mathrm{Im}(c) = \mu_n(k) \cong {\mathbf Z}/n{\mathbf Z}.$$
Suppose $\alpha \in \mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,A)({\mathbf Q}_\ell)$.
Then $\alpha^n = 1$. By Theorem \ref{psdcor} and the facts that
${\mathfrak G}_\ell(F,A) = {\mathfrak G}_\ell(F,A)^0$ and
$\alpha \in \mathrm{End}^0(A)$, we have $\alpha \in MT_A({\mathbf Q})$.
Applying the character
$\gamma$ of Theorem \ref{MTtilde}, we have that $\gamma(\alpha)$
is an $n$-th root of unity in ${\mathbf Q}^\times$, and therefore
$\gamma(\alpha)$ is $1$ or $-1$.
By the definition of $W_{A,\ell}$, $\alpha$ acts on $W_{A,\ell}$
as multiplication by $\alpha^{-r}\gamma(\alpha)^{-r/2}$.
Since $\alpha \in MT_{A,\ell}({\mathbf Q}_\ell)$,
$\alpha$ acts trivially on $W_{A,\ell}$. Therefore
$\alpha^{-r}\gamma(\alpha)^{-r/2} = 1$, so
$\alpha^{2r} = 1$.
Let $t= (n,2r)$.
Then
$$\mathrm{Gal}(M/F_{\Phi}(B)) \cong
\mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,A)({\mathbf Q}_\ell) \subseteq
\mu_n(k) \cap \mu_{2r}(k) = \mu_t(k).$$
Therefore,
$[M:F_{\Phi}(B)]$ divides $t$.
Since ${\mathfrak G}_\ell(F,{A})$ contains the homotheties
${\mathbf G}_m$ (see 2.3 of \cite{serrereps}), we have
$-1 \in {\mathfrak G}_\ell(F,{A})({\mathbf Q}_{\ell})$.
So $-1 \in \mathrm{Im}(c) \cap {\mathfrak G}_\ell(F,{A})({\mathbf Q}_{\ell})$
if and only if $-1 \in \mathrm{Im}(c)$, i.e., if and only if $n$ is even.
Thus if $t = 2$, then
$$\mathrm{Gal}(M/F_{\Phi}(B)) \cong \{\pm 1\}.$$
\end{proof}
\begin{thm}
\label{twistE}
Suppose $X$ and $Y$ are abelian varieties over a finitely generated
extension $F$ of ${\mathbf Q}$, $\ell$ is a prime number, $\mathrm{Hom}(X,Y) = 0$,
$F = F(\mathrm{End}(X)) = F(\mathrm{End}(Y))$, $k$ is a CM-field,
$[k:{\mathbf Q}] = 2\mathrm{dim}(Y)$, and
$\mathrm{dim}(X) = t\mathrm{dim}(Y)$ for some odd positive integer $t$.
Suppose $\iota_X$
and $\iota_Y$ are embeddings of $k$ into $\mathrm{End}^0(X)$ and $\mathrm{End}^0(Y)$,
respectively, and
$(X \times Y, k, \iota_X \times \iota_Y)$
is of Weil type. Suppose $c$ is the non-trivial
character associated to a quadratic extension $M$ of $F$,
let $Y^c$ denote the twist of $Y$ by $c$, let $A = X \times Y$, and
let $B = X \times Y^c$. Then
\begin{enumerate}
\item[{(i)}] $F = F(\mathrm{End}(B))$,
\item[{(ii)}]
either $F(\mathrm{End}(A)) \ne F_{\Phi}(A)$ or $F(\mathrm{End}(B)) \ne F_{\Phi}(B)$,
\item[{(iii)}] if $F_{\Phi}(A) = F$, then $F_{\Phi}(B) = M$.
\end{enumerate}
\end{thm}
\begin{proof}
We have $F = F(\mathrm{End}(A))$.
Since $\mathrm{Hom}(X,Y) = 0$, we have $\mathrm{End}^0(A) = \mathrm{End}^0(X) \oplus \mathrm{End}^0(Y)$
and $\mathrm{Aut}(A) = \mathrm{Aut}(X) \times \mathrm{Aut}(Y)$. Consider the cocycle $c$ that
sends $\sigma \in \mathrm{Gal}({\bar F}/F)$ to $(1, c(\sigma)) \in
\mathrm{Aut}(X) \times \mathrm{Aut}(Y) = \mathrm{Aut}(A)$. All
the values of $c$ are of the form $(1, \pm 1)$, and therefore
belong to the center of
$\mathrm{End}^0(A)$. The abelian variety $B$ ( $ = X \times Y^c$) is the twist
of $A$ ( $ = X \times Y$) by $c$. By Corollary \ref{cocy} we have
(i), and we obtain an isomorphism $\varphi$ from $\mathrm{End}_F(A)$ onto
$\mathrm{End}_F(B)$ such that $(B, k, \varphi\circ(\iota_X \times \iota_Y))$ is of
Weil type.
Let $k_\ell = k \otimes {\mathbf Q}_\ell$.
We have
$$V_\ell(A) = V_\ell(X) \oplus V_\ell(Y), \quad
V_\ell(B) = V_\ell(X) \oplus V_\ell(Y^c).$$
Viewing the Tate modules as free $k_\ell$-modules, we have
$$\wedge^{t+1}_{k_{\ell}} V_\ell(A) = \wedge^{t}_{k_\ell}V_\ell(X) \otimes
_{k_{\ell}} V_\ell(Y), \quad
\wedge^{t+1}_{k_{\ell}} V_{\ell}(B)= \wedge^{t}_{k_\ell}V_\ell(X)
\otimes_{k_{\ell}} V_\ell(Y^c).$$ For
${U} = A$ or $B$, let
$$W_{{U},\ell} =
\mathrm{Hom}_{{\mathbf Q}_\ell}(\wedge^{t+1}_{k_\ell} V_\ell({U}),{\mathbf Q}_\ell({\frac{t+1}{2}})).$$
The Galois module $V_{\ell}(Y^{c})$ is the twist of the Galois module
$V_{\ell}(Y)$ by the character $c$ (see Remark \ref{twistyrem}),
and the Galois module
$W_{B,\ell}$ is the twist of the Galois module $W_{A,\ell}$ by
$c^{-1}=c$. Since $c$ is non-trivial, the Galois modules
$W_{B,\ell}$ and $W_{A,\ell}$ cannot be simultaneously trivial.
As in the proof of Theorem \ref{twistX}, it follows that
${\mathfrak G}_\ell(F,A)$ and ${\mathfrak G}_\ell(F,B)$ cannot both be connected.
If $F_{\Phi}(A) = F$, then ${\mathfrak G}_\ell(F,A)$ is connected, so
${\mathfrak G}_\ell(F,B)$ is disconnected. By Lemma \ref{conncomplem},
${\mathfrak G}_\ell(M,B)$ is connected, and so $F_{\Phi}(B)$ must be the
quadratic extension $M$ of $F$.
\end{proof}
\begin{rem}
Suppose $F$ is a subfield of ${\mathbf C}$, $Y$ is an elliptic curve over
$F$ with complex multiplication by an
imaginary quadratic field $K$, and $X$ is an absolutely simple
$3$-dimensional abelian
variety over $F$ with $K$ embedded in its endomorphism algebra.
Then we can always ensure (by taking complex
conjugates if necessary) that the two embeddings of
$K$ into ${\mathbf C}$ occur with the same multiplicity in the action of $K$ on the
tangent space of the $4$-dimensional abelian variety $A = X \times Y$.
Note that the hypotheses of Theorem \ref{twistE} (or of
Theorem \ref{twistX}) cannot be simultaneously
satisfied with $\mathrm{dim}(A) < 4$. In Example \ref{bigex} we exhibit
$4$-dimensional abelian
varieties satisfying the hypotheses of Theorem \ref{twistE}.
\end{rem}
\section{Examples}
\label{examples}
Using Theorems \ref{twistX} and \ref{twistE}, we can construct
examples of abelian varieties $B$ such that ${\mathfrak G}_\ell(F,B)$
is disconnected, and compute the connectedness extensions.
\subsection{Example}
Let $k = {\mathbf Q}(\sqrt{-3})$ and let $K$ be the CM-field which
is the compositum of
${\mathbf Q}(\sqrt{-3})$ with the maximal totally real subfield $L$ of
${\mathbf Q}(\zeta_{17})$. Then
$$\mathrm{Gal}({K}/{\mathbf Q}) \cong \mathrm{Gal}({k}/{\mathbf Q}) \times \mathrm{Gal}({L}/{\mathbf Q}) \cong
{\mathbf Z}/2{\mathbf Z} \times {\mathbf Z}/8{\mathbf Z}.$$
Let $\Psi$ be the subset of ${\mathbf Z}/2{\mathbf Z} \times {\mathbf Z}/8{\mathbf Z} \cong
\mathrm{Gal}({K}/{\mathbf Q})$ defined by
$$\Psi =
\{(0,0), (0,1), (0,4), (0,7), (1,2), (1,3), (1,5), (1,6)\}.$$
Let ${\mathcal O}_{K}$ denote the ring of integers of $K$. Let
$(A,\iota_{K})$ be an $8$-dimensional CM abelian variety of
CM-type $(K,\Psi)$
constructed from the lattice ${\mathcal O}_{K}$ as in Theorem 3
on p.~46 of \cite{st}, and defined over a number field $F$ (this can
be done by Proposition 26 on p.~109 of \cite{st}).
Then $A$ is absolutely simple, by the choice of $\Psi$ and
Proposition 26 on p.~69 of \cite{st}, and
$\mathrm{End}(A) = {\mathcal O}_{K}$ (see Proposition 6 on p.~42 of \cite{st}). Further,
the reflex field of $(K,\Psi)$ is $K$.
Take the number field $F$ to be
sufficiently large so that $F_\Phi(A) = F$. Let
$\iota$ be the restriction of $\iota_{K}$ to $k$. By
the definition of $\Psi$, if
$\sigma \in \mathrm{Gal}(k/{\mathbf Q}) = \mathrm{Hom}(k,{\mathbf C})$
then $n_\sigma = 4$. Therefore
$(A,k,\iota)$ is of Weil type.
Let $c: \mathrm{Gal}({\bar F}/F) \to k^\times$
be a non-trivial cubic character associated to a cubic extension
$M$ of $F$, and let $B$ denote
the twist of $A$ by $c$. Applying Theorem \ref{twistX}iii
with $n = 3$ and $r = 8$, then $F(\mathrm{End}(B)) = F$
and $F_\Phi(B) = M$.
\subsection{Example}
\label{bigex}
Let $J$ be the Jacobian of the genus $3$ curve $$y^7 = x(1-x),$$
and let $E$ be the elliptic curve $X_0(49)$. A model for $E$ is given
by the equation
$$y^2 + xy = x^3 - x^2 - 2x - 1.$$
Let $d$ be a non-zero square-free integer. If $d \ne 1$
let $E^{(d)}$ be the twist of $E$ by the non-trivial character of
${\mathbf Q}(\sqrt{d})$, and if $d = 1$ let
$E^{(d)} = E$.
Let $$A = J \times E, \qquad A^{(d)} = J \times E^{(d)}.$$
The abelian varieties $A^{(d)}$ are defined over ${\mathbf Q}$. Let $\zeta_7$ be
a primitive seventh root of unity and let
$${K} = {\mathbf Q}(\zeta_7), \quad L_d = {K}(\sqrt{d}), \quad {\text{ and }} \quad
k = {\mathbf Q}(\sqrt{-7}).$$
If $d = 1$ or $-7$ then ${K} = L_d$; otherwise, $[L_d:{K}] = 2$.
The abelian variety $J$ is a simple abelian variety with complex
multiplication by ${K}$, and the elliptic curves $E^{(d)}$ have complex
multiplication by the subfield $k$ of ${K}$.
We have $\mathrm{Gal}({K}/{\mathbf Q}) = \{\sigma_1,...,\sigma_6\}$
where $\sigma_i(\zeta_7) = \zeta_7^i$.
The CM-type of $J$ is $({K}, \{\sigma_1, \sigma_2, \sigma_3\})$
(see p.~34 of \cite{Lang} or \S15.4.2 of \cite{st}), and the
reflex CM-type is $({K}, \{\sigma_4, \sigma_5, \sigma_6\})$
(see \S8.4.1 of \cite{st}).
We can identify $\mathrm{End}(A^{(d)})$ with the direct sum of
$\mathrm{End}(J)$ and $\mathrm{End}(E^{(d)})$. By Proposition 30 on p.~74 of \cite{st},
the smallest extension of ${\mathbf Q}$ over which all the elements of $\mathrm{End}(J)$ are
defined is the reflex CM-field of the CM-type of $J$, which is ${K}$.
Similarly, $ k$ is
the smallest extension of ${\mathbf Q}$ over which all the elements of $\mathrm{End}(E^{(d)})$
are defined. We therefore have
$${K} = {\mathbf Q}(\mathrm{End}(A^{(d)})).$$
Next, we will prove that $L_d = {\mathbf Q}_{\Phi}(A^{(d)})$.
Write ${\mathcal O}_\Omega$ for the ring of integers of a number field $\Omega$.
If $q$ is a prime number, let ${\mathcal O}_q = {\mathcal O}_\Omega \otimes {\mathbf Z}_q$.
\begin{claim}
\label{powerofp}
If ${K}'$ is a finite abelian extension of ${K}$ which is unramified away
from
the primes above $7$, then $[{K}':{K}]$ is a power of $7$.
\end{claim}
\begin{proof}
We have $-1 - \zeta_7 = (1 - \zeta_7^2)/(\zeta_7 - 1) \in {\mathcal O}_{K}^\times$.
Let ${\mathcal P}$ be the prime ideal of $K$ above $7$. The reduction map
$${\mathcal O}_{K}^\times \to ({\mathcal O}_{K}/{\mathcal P})^\times \cong ({\mathbf Z}/7{\mathbf Z})^\times$$
is surjective, since $-1 - \zeta_7$ maps to $-2$, a generator of
$({\mathbf Z}/7{\mathbf Z})^\times$. Moreover, the class number of ${K}$ is one.
Therefore by class field theory, there is no
non-trivial abelian extension of ${K}$ of degree prime to $7$ and
unramified away from the primes above ${\mathcal P}$.
\end{proof}
\begin{claim}
\label{Kitself}
If $p$ is a prime and $p \equiv 3 \pmod{7}$,
then the only field ${K}'$ such that
\begin{enumerate}
\item[{(i)}] ${K} \subseteq {K}' \subseteq {K}(A_p)$, and
\item[{(ii)}] ${K}'/{K}$ is unramified away from the primes above $7$,
\end{enumerate}
\noindent is ${K}$ itself.
\end{claim}
\begin{proof}
Since ${K}$ is a field of definition for the endomorphisms of the CM
abelian varieties $J$ and $E$, the extension ${K}(A_n)/{K}$ is abelian for
every integer $n$ (see Corollary 2 on p.~502 of \cite{serretate}).
Suppose $p$ and ${K}'$ satisfy the hypotheses of Lemma \ref{Kitself}.
Let $I_p \subseteq \mathrm{Gal}({K}(J_p)/{K})$ be the inertia subgroup at $p$.
We will first show
\begin{equation}
\label{claimb}
\#(I_p) = {\frac{p^6 - 1}{p^2 + p + 1}}.
\end{equation}
The image of ${\mathcal O}_p^\times$ in $\mathrm{Gal}({K}(J_p)/{K})$ under the Artin map of
class field theory is $I_p$, and we have natural homomorphisms
$$\mathrm{Gal}({K}(J_p)/{K}) \hookrightarrow
\mathrm{Aut}_{{\mathcal O}_{K}}(J_p) \cong
({\mathcal O}_{K}/p{\mathcal O}_{K})^\times \cong {\mathcal O}_p^\times/(1 + p{\mathcal O}_p).$$
We therefore obtain maps
\begin{equation}
\label{maps}
{\mathcal O}_p^\times \to I_p \hookrightarrow {\mathcal O}_p^\times/(1 + p{\mathcal O}_p).
\end{equation}
Since the first map of (\ref{maps}) is surjective, the order of $I_p$ is
the order of the image of the composition.
Since $p \equiv 3 \pmod{7}$, we know that $p$ is inert in ${K}/{\mathbf Q}$, so
$({\mathcal O}_{K}/p{\mathcal O}_{K})^\times$ ($\cong {\mathcal O}_p^\times/(1 + p{\mathcal O}_p)$) is a cyclic
group of
order $p^6 - 1$. Since the greatest common divisor of $p^6 - 1$ and
$p^3(p^2 + p + 1)$ is $p^2 + p + 1$, equation
(\ref{claimb}) will be proved when we show that the composition
of maps in (\ref{maps})
sends $u \in {\mathcal O}_p^\times$ to $u^{-p^3(p^2+p+1)} \pmod{1 + p{\mathcal O}_p}$.
We can view elements of $\mathrm{Gal}({K}/{\mathbf Q})$ as automorphisms of ${\mathcal O}_p^\times$.
Proposition 7.40 on p.~211 of \cite{shimura} implies that the image of $u$
is of the form $\alpha(u)/\eta(u) \pmod{1 + p{\mathcal O}_p}$
where $\eta(u) = \sigma_4(u)\sigma_5(u)\sigma_6(u)$ and
$\alpha(u) \in {K}^\times$. Write ${K}_{\mathbf A}^\times$ for
the idele group of ${K}$, and for
each archimedean prime $\lambda$ of ${K}$, define
a Gr\"ossencharacter
$\psi_\lambda : {K}_{\mathbf A}^\times \to {\mathbf C}^\times$ by
$\psi_\lambda(x) = (\alpha(x)/\eta(x))_\lambda$.
View ${\mathcal O}_p^\times$ as a subgroup of ${K}_{\mathbf A}^\times$.
Since $J$ has good reduction outside $7$, we have
$\psi_\lambda({\mathcal O}_p^\times) = 1$,
by Theorem 7.42 of \cite{shimura}.
For $u \in {\mathcal O}_p^\times$, we have
$1 = \psi_\lambda(u) = \alpha(u)_\lambda = \alpha(u)$.
Therefore the image of $u$ in ${\mathcal O}_p^\times/(1 + p{\mathcal O}_p)$ is
$1/\eta(u) \pmod{1 + p{\mathcal O}_p}$. Since $p$ is inert in ${K}/{\mathbf Q}$, we have
$\mathrm{Gal}({K}/{\mathbf Q}) \cong \mathrm{Gal}(({\mathcal O}_{K}/p)/({\mathbf Z}/p)) = D_p$, where $D_p$ is
the decomposition group at $p$. The latter group is a cyclic group of
order $6$ generated by the Frobenius element,
and we compute that
$$\sigma_4(u) \equiv u^{p^4}, \quad \sigma_5(u) \equiv u^{p^5}, \quad
{\text{and}} \quad \sigma_6(u) \equiv u^{p^3} \quad \pmod{1 + p{\mathcal O}_p}$$
(since $p^4 \equiv 4 \pmod{7}$, $p^5 \equiv 5 \pmod{7}$,
and $p^3 \equiv 6 \pmod{7}$). Therefore
$$1/\eta(u) \equiv u^{-p^3(p^2+p+1)} \pmod{1 + p{\mathcal O}_p},$$
as desired.
We have
$$\mathrm{Gal}({K}(E_p)/{K}) \hookrightarrow
\mathrm{Aut}_{{\mathcal O}_k}(E_p) \cong ({\mathcal O}_k/p{\mathcal O}_k)^{\times}.$$
The order of $({\mathcal O}_k/p{\mathcal O}_k)^{\times}$ is $p^2 - 1$, which is not
divisible by $7$. Therefore $[{K}(A_p):{K}(J_p)]$ is not divisible by $7$.
By Lemma \ref{powerofp}, $[{K}':{K}]$ is a power of $7$. Therefore
${K}' \subseteq {K}(J_p)$. Since ${K}'/{K}$ is unramified
at $p$, we have
$I_p \subseteq \mathrm{Gal}({K}(J_p)/{K}')$.
Suppose ${K}' \neq {K}$. Then
$\#(I_p)$ divides $(p^6 - 1)/7$. By (\ref{claimb}), $(p^6 - 1)/(p^2 + p + 1)$
divides $(p^6 - 1)/7$. Therefore $7$ divides $p^2 + p + 1$, which contradicts
the assumption that $p \equiv 3 \pmod{7}$. Therefore, ${K}' = {K}$.
\end{proof}
Suppose $p$ and $q$ are distinct odd primes, and $p \equiv 3 \pmod{7}$.
Let ${K}' = K(A_p) \cap K(A_q)$.
Since $A$ has good reduction outside $7$, the extension
${K}'/K$ is unramified away from the primes above $7$.
By Lemma \ref{Kitself}, we have ${K}' = K$.
As mentioned in the introduction, for
every integer $n \ge 3$ we have
$$K_{\Phi}(A) \subseteq K(A_n).$$
We therefore obtain
$$K = K_\Phi(A) = {\mathbf Q}_\Phi(A).$$
It follows from Theorem \ref{twistE} that
$$L_d = {\mathbf Q}_{\Phi}(A^{(d)}).$$
Note that Shioda (see Theorem 4.4 of \cite{shioda}) proved the Hodge
Conjecture for $A$, and therefore
also for $A^{(d)}$. Thus, the Weil classes
on $A^{(d)}$ are algebraic. It follows easily that $L_d$ is the smallest
extension of ${\mathbf Q}$ over which all the algebraic cycle classes on all powers
of $A^{(d)}$ are defined.
\begin{remss}
If ${A}$ is an abelian variety over a finitely generated extension $F$ of ${\mathbf Q}$,
and if the (as yet unproved) Tate Conjecture is true for all powers of ${A}$
over $F_\Phi({A})$,
then the field $F_\Phi({A})$ is the smallest extension of $F$
over which all the algebraic cycle classes on all powers of ${A}$ are
defined.
\end{remss}
|
1996-03-14T06:20:30 | 9603 | alg-geom/9603011 | en | https://arxiv.org/abs/alg-geom/9603011 | [
"alg-geom",
"math.AG"
] | alg-geom/9603011 | Scott R. Nollet | Scott Nollet | The Hilbert Schemes of Degree Three Curves are Connected | 20 pages, Latex | null | null | null | null | In this paper we show that the Hilbert scheme $H(3,g)$ of locally
Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected.
In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many
irreducible components (roughly $-g/3$ of them). To show connectedness, we
classify the curves (giving particular attention to the triple lines),
determine the irreducible components, and give flat families over $\Aone$ to
show that the components meet. As a byproduct, we find that there are curves
which lie in the closure of each irreducible component.
| [
{
"version": "v1",
"created": "Thu, 14 Mar 1996 00:42:06 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Nollet",
"Scott",
""
]
] | alg-geom | \section{Introduction}
In his thesis \cite{HC}, Hartshorne showed that the (full) Hilbert scheme
for projective subschemes with a fixed Hilbert polynomial is connected.
Often one studies certain subsets of the Hilbert scheme which parametrize
subschemes satisfying a certain property. For example, one can consider the
Hilbert scheme of smooth curves in ${\Bbb P}^3$. The smooth curves of degree $9$
and genus $10$ afford an example in which this Hilbert scheme is {\it not}
connected (see \cite{H}, IV, ex. 6.4.3). It is not known for which properties
the corresponding Hilbert scheme is connected.
\par
In the present paper, we are interested in the Hilbert scheme $H(d,g)$
of locally Cohen-Macaulay curves in ${\Bbb P}^3$ of degree $d$ and arithmetic
genus $g$. By work of several authors \cite{GSP,OS,S}, it is known that
$H(d,g)$ is nonempty precisely when $d>0, g={1 \over 2}(d-1)(d-2)$ or
$d > 1, g \leq {1 \over 2}(d-2)(d-3)$. In a recent paper \cite{MDP3},
Martin-Deschamps and Perrin prove that $H(d,g)$ is
reduced only when $d=2$ or $g={1 \over 2}(d-1)(d-2)$ or $g={1 \over 2}(d-2)(d-3)$ or
$(d,g)=(3,-1)$. For all other $(d,g)$ pairs for which $H(d,g)$ is nonempty,
there is a nonreduced irreducible component corresponding to curves which
are extremal in the sense that their Rao modules have the largest possible
dimension.
\par
It can be gleaned from several sources \cite{MDP3,E1,E2} that $H(d,g)$ is
irreducible precisely in the cases $d=2$ or $g > {1 \over 2}(d-3)(d-4)+1$ or
$(d,g) \in \{(4,1),(3,1),(3,0),(3,-1)\}$. In particular, $H(d,g)$ is
connected in these cases. In the present paper we show that
$H(3,g)$ is connected for all $g$. This is the first interesting
case in the sense that these Hilbert schemes have several irreducible
components. Curiously, there certain extremal curves lie in the closure
of each irreducible component. \par
The paper is organized as follows. In the first section, we review
several results of Banica and Forster \cite{BF} on multiplicity structures
on smooth curves in a smooth threefold and classify space curves
of degree two as an example. We also briefly review the extremal curves
studied by Martin-Deschamps and Perrin. In the second section, we
classify the multiplicity three structures on a line. This is used in
the third section, where we classify all locally Cohen-Macaulay curves of
degree $3$ in ${\Bbb P}^3$. In particular, the irreducible components
of the Hilbert scheme are determined. We also produce some flat families of
triple lines, which show that the Hilbert scheme is connected.
\par
In this paper, we work over an algebraically closed field $k$ of arbitrary
characteristic. $S=k[x,y,z,w]$ denotes the homogeneous coordinate ring of
${\Bbb P}^3_{k}$. If $V \subset S$ is a closed subvariety, then $S_V$ denotes
the the homogeneous coordinate ring $S/I_V$ of $V$. We often use the
abbreviation CM to mean locally Cohen-Macaulay. $H(d,g)$ denotes the
Hilbert scheme of locally Cohen-Macaulay curves in ${\Bbb P}^3$ of degree
$d$ and arithmetic genus $g$.
\section{Preliminaries}
In this section we review the results of Banica and Forster \cite{BF} on
multiplicity structures on smooth curves in smooth threefolds. As an
example, we give the classification of double lines in ${\Bbb P}^3$, which
will be used in section two when we classify the triple lines in ${\Bbb P}^3$.
We also recall a few notions from linkage theory and summarize the results
of Martin-Deschamps and Perrin on extremal curves.
\begin{defn}{\em
If $Y$ is a scheme, then a {\it locally Cohen-Macaulay multiplicity
structure} $Z$ {\it on} $Y$ is a locally Cohen-Macaulay scheme $Z$
which contains $Y$ and has the same support as $Y$. For short, we
simply say that $Z$ is a {\it multiplicity structure} on $Y$.
\em}
\end{defn}
In \cite{BF}, Banica and Forster consider a smooth curve $Y$ inside
a smooth threefold $X$. Starting with a multiplicity structure
$Y \subset Z \subset X$, they define a filtration on $Z$ as follows.
Let $Y^{(i)}$ denote the subscheme of $X$ defined by ${I_Y}^i$.
Let $Z_i$ denote the subscheme of $X$ obtained from $Z \cap Y^{(i)}$
by removing the embedded points. This gives the (unique) largest
Cohen-Macaulay subscheme contained in $Z \cap Y^{(i)}$. If $k$
is the smallest integer such that $Z \subset Y^{(k)}$, we obtain the
{\it Cohen-Macaulay filtration} for $Y \subset Z$
$$Y=Z_1 \subset Z_2 \subset \dots \subset Z_k=Z.$$
\par
Letting ${\cal I}_i={\cal I}_{Z_i}$, there are sheaves $L_j = {\cal I}_j/{{\cal I}_{j+1}}$
associated to this filtration. For any $i,j \geq 1$, it turns out that
${{\cal I}_i}{{\cal I}_j} \subset {\cal I}_{i+j}$, and hence the $L_j$ are ${\cal O}_Y$-modules.
In fact, the $L_j$ are shown to be locally free ${\cal O}_Y$-modules. Further,
there are induced multiplication maps $ L_i \otimes L_j \rightarrow L_{i+j} $, which
are generically surjective (because ${\cal I}_j = {\cal I}_Z + {{\cal I}_Y}^i$ on an open set).
In particular, we get generically surjective maps $L_1^{\otimes j} \rightarrow L_j$.
\par
{}From the above, we see that if $Z$ is a multiplicity structure on $Y$,
then there is a filtration $\{Z_j\}$ and exact sequences
$$0 \rightarrow {\cal I}_{Z_{j+1}} \rightarrow {\cal I}_{Z_j} \rightarrow L_j \rightarrow 0$$
where the $L_j$ are vector bundles on $Y$. If $Y$ is connected, the
multiplicity of $Z$ is defined by $\mu(Z)=\dim_K({\cal O}_{Z,\eta})$, where
$\eta$ is the generic point of $Y$ and $K={\cal O}_{Y,\eta}$ is the function
field of $Y$. The sequences above show that
$\mu(Z)=1+\sum_{j=1}^{k}{\mathop{\rm rank} L_j}.$
\begin{rmk}{\em
The above constructions can be carried out in $Z$ (instead of $X$),
and would yield the same filtration as above. Thus the Cohen-Macaulay
filtration is well-defined for abstract (non-embedded) multiplicity structures.
\em}
\end{rmk}
Now we use the fact that $X$ is a smooth threefold. In this case, the
conormal sheaf ${\cal I}_Y/{{\cal I}_Y^2}$ is a rank two bundle on $Y$. Since the
surjection ${\cal I}_Y \rightarrow L_1$ factors through the conormal bundle, we see
that $L_1$ has rank zero, one, or two. If the rank is zero,
then $L_1=0$ and the generically surjective maps show that all the $L_j=0$,
hence $Z=Y$. If the rank is two, then the surjection ${\cal I}_Y/{{\cal I}_Y^2} \rightarrow L_1$
becomes an isomorphism, hence $Y^{(2)} \subset Z_2$ and $Z$ has generic
embedding dimension three. \par
We are mainly interested in the case rank$(L_1)=1$, in which case we
say the extension $Y \subset Z$ is {\it quasi-primitive}. In this case, the
generically surjective maps $L^{\otimes j} \rightarrow L_j$ show that there are
divisors $D_j$ on $Y$ such that $L_j=L^j(D_j)$. The multiplication maps
$L_i \otimes L_j \rightarrow L_{i+j}$ show that $D_i + D_j \leq D_{i+j}$ for all
$i,j \geq 1$ (define $D_1=0$). We say that $(L,D_2,\dots D_k)$
is the {\it type} of the extension $Z$. \par
\begin{rmk}{\em
The condition that $Y \subset Z$ be quasi-primitive is equivalent to
the condition that the generic embedding dimension of $Z$ is $2$.
If $z \in Z$ is a point where the embedding dimension is $2$, there is
an open neighborhood $U$ about $z$ and a smooth surface $S \subset U$ such that
$Z \subset S$. $Y$ is Cartier on $S$. If $t$ is a local equation for $Y$,
then $t^i$ gives a local equation for $Z_i$ on $S$.
\em}
\end{rmk}
Having reviewed the theory of multiplicity structures on curves, we
present as an example the simplest case, the double structures on a
line in ${\Bbb P}^3$.
\begin{prop}\label{2line}
Let $Y$ be the line $\{x=y=0\}$ in ${\Bbb P}^3$ and let $a \geq -1$
be an integer. Let $f$ and $g$ be two homogeneous polynomials of degree
$a+1$ which have no common zeros along $Y$. Then $f$ and $g$ define a
surjection $u: {\cal I}_Y \rightarrow {\cal O}_Y(a)$ by $x \mapsto f, y \mapsto g$. The
kernel of $u$ gives the ideal sheaf of a multiplicity two structure $Z$
on $Y$. Further, we have \\
(a) $p_a(Z)=-1-a$ \\
(b) $H^1_*({\cal I}_Z) \cong (S/(x,y,f,g))(a)$. \\
(c) $I_Z=(x^2,xy,y^2,xg-yf)$. \\
(d) If $f^\prime,g^\prime$ define another two structure $Z^\prime$, then
$Z=Z^\prime$ if and only if there exists $c \in k^*$ such that
$f^\prime=cf \mathop{\rm mod} I_Y$ and $g^\prime=cg \mathop{\rm mod} I_Y$. \\
(e) Each multiplicity two structure $Z$ on $Y$ arises by the construction
above. \\
\end{prop}
\begin{pf}
This can be found in work of Migliore \cite{M} and also by work of
Martin-Deschamps and Perrin (\cite{MDP1}, IV, example 6.9) in the context
of linkage theory.
{}From the above theory of multiplicity structures, we see that giving a double
structure $Z$ on $Y$ is equivalent to finding a surjection $u:{\cal I}_Y \rightarrow {\cal L}$,
where ${\cal L}$ is a line bundle on $Y$. Since such a surjection must factor
through ${\cal I}_Y/{{\cal I}_Y^2} \cong {\cal O}_Y(-1)^2$, we see that ${\cal L} \cong {\cal O}_Y(a)$
with $a \geq -1$ and that the map is given by two polynomials $f,g$ of
degree $a+1$.
\end{pf}
\begin{rmk}\label{smooth}{\em
If $Z$ is a double line from proposition \ref{2line} above, the exact sequence
$$0 \rightarrow {\cal O}_Y(a) \rightarrow {\cal O}_Z \rightarrow {\cal O}_Y \rightarrow 0$$
shows that $Z$ has local embedding dimension two at each point. In fact, $Z$ is
contained in a smooth (global) surface of degree $a+2$. To see this, one
can choose the polynomials $f,g$ in the variable $z,w$. Since $f$ and
$g$ have no common zeros along $Y$, the surface with equation $xg-yf$ is
smooth along $Y$. When $a=-1$,
this surface is a (smooth) plane which contains $Y$. When $a \geq 0$,
we note that the general surface of degree $a+2$ which contains $Y^{(2)}$ is
smooth away from $Y$. If we intersect these open conditions in
${{\Bbb P}}H^0({\cal I}_Z(a+2))$, we find that there are surfaces of degree $a+2$
containing $Z$ which are smooth everywhere. The general surface containing
$Z$ of higher degree will have a finite number of singularities along $Y$.
\em}
\end{rmk}
\begin{cor}\label{2linehilb}
Description of $H(2,g)$: \\
(a) If $g>0$, then $H(2,g)$ is empty. \\
(b) If $g=0$, then $H(2,g)$ is irreducible of dimension 8. All curves
in $H(2,g)$ are planar, and the general member is a smooth conic.
The reduced reducible curves (two lines meeting at a point) form
an irreducible family of dimension 7, and the multiplicity two structures
on a line form an irreducible family of dimension 5. \\
(c) If $g=-1$, then $H(2,g)$ is irreducible of dimension 8. The general curve
is a union of two skew lines. The multiplicity two
structures on a line form an irreducible family of dimension 7. \\
(d) If $g<-1$, then $H(2,g)$ is irreducible of dimension 5-2g. all
curves are multiplicity two structures on a line with $a=-1-g$ \\
\end{cor}
\begin{pf}
(a) is known, since ${{1 \over 2}}(d-1)(d-2)$ is an upper bound on the genus of
locally Cohen-Macaulay curves (see \cite{GSP} or \cite{OS}). The descriptions
of the families of reduced curves is standard. To describe the moduli for the
double lines of genus $g \leq 0$, we use proposition \ref{2line}. The choice
of the line $Y$ is given by a $4$-dimensional (irreducible) Grassman variety.
Given the line $Y$, the multiplicity structure $Z$ is uniquely determined
by the open set of $(f,g) \in H^0({\cal O}_Y(a+1))^2/{k^*}$ where $f$ and $g$ have
no common multiple. This is an irreducible choice of dimension $2a+3 = 1-2g$.
Adding the choice of the line $Y$ gives an irreducible family of dimension
$5-2g$.
\end{pf}
In their excellent book \cite{MDP1}, Martin-Deschamps and Perrin build
a strong foundation for linkage theory of locally Cohen-Macaulay
curves in ${\Bbb P}^3$.
Perhaps the most important result there is the structure theorem
for even linkage classes (see \cite{MDP1},IV,theorem 5.1). It states that
if ${\cal L}$ is an even linkage class of curves which are not arithmetically
Cohen-Macaulay, then there exists a curve $C_0 \in {\cal L}$ such that any other
curve $C \in {\cal L}$ is obtained from $C_0$ be a sequence of ascending elementary
double links (see \cite{MDP1}, III, definition 2.1) followed by a deformation
with constant cohomology through curves in ${\cal L}$. In particular, $C_0$ achieves
the smallest degree and genus among curves in ${\cal L}$. $C_0$ is called a
{\it minimal curve} for ${\cal L}$.
\par
A practical aspect of \cite{MDP1} (see chapter IV) is an algorithm for
finding a minimal curve associated to a finite length graded $S$-module.
If $M$ is such a module, there exists a minimal graded free resolution
$$0 \rightarrow L_4 \stackrel{\sigma_4}{\rightarrow} L_3 \stackrel{\sigma_3}{\rightarrow}
L_2 \stackrel{\sigma_2}{\rightarrow} L_1 \stackrel{\sigma_1}{\rightarrow} L_0 \rightarrow M \rightarrow 0$$
which sheafifies to an exact sequence of direct sums of line bundles.
Let ${\cal N}_0=\mathop{\rm ker} \tilde{\sigma_2}$ and ${\cal E}_0=\mathop{\rm ker} \tilde{\sigma_3}$. The
algorithm of Martin-Deschamps and Perrin gives a way to split ${\cal L}_2$ into
${\cal P} \oplus {\cal Q}$, where ${\cal P}$ and ${\cal Q}$ are also direct sums of lines bundles.
If $C_0$ is a minimal curve, then there exists an integer $h_0$ and exact
sequences
$$0 \rightarrow {\cal P} \rightarrow {\cal N}_0 \rightarrow {\cal I}_{C_0}(h_0) \rightarrow 0$$
$$0 \rightarrow {\cal E}_0 \rightarrow {\cal Q} \rightarrow {\cal I}_{C_0}(h_0) \rightarrow 0.$$
One consequence of this is that if
$$0 \rightarrow L_4 \stackrel{\theta}{\rightarrow} L_3 \rightarrow Q \rightarrow I_{C_0} \rightarrow 0$$
is a minimal graded $S$-resolution for the total ideal of the minimal curve
$C_0$, then $L_3^\vee \stackrel{\theta^\vee}{\rightarrow} L_4^\vee$ begins a minimal
resolution for $M^*$. \par
In a later paper \cite{MDP2}, Martin-Deschamps and Perrin
tackled the problem of bounding the dimensions of the Rao module of a curve
in terms of the degree and genus. For a curve $C$, define the
{\it Rao function} $\rho_C$ by $\rho_C(n)=h^1({\cal I}_C(n))$. $r_a$ (resp. $r_o$)
is the smallest (resp. largest) value $n$ for which $\rho_C(n) \neq 0$.
Their bounds can be stated as follows (see \cite{MDP2}, theorem 2.5 and
corollary 2.6).
\begin{prop}\label{raobound}
Let $C \subset {\Bbb P}^3$ be a curve of degree $d$ and genus $g$. Set
$$l=d-2,a={1 \over 2}(d-2)(d-3)-g.$$
Then $a \geq 1,l \geq 0$, and the Rao function is bounded by \\
(1) $r_a \geq -a+1$. \\
(2) $\rho_C(n) \leq a$ for $0 \leq n \leq l$.\\
(3) $r_o \leq a+l-1$. \\
\end{prop}
The question of sharpness for these bounds has a nice answer. Not only can
equalities in (1), (2) and (3) be realized individually, but they can be
realized by one curve. Martin-Deschamps and Perrin call such a curve
{\it extremal}. These curves are characterized in theorem \ref{extremal}
below (see \cite{MDP3}, proposition 0.6 and theorem 1.1). In theorem
\ref{nilcomp} which follows (see \cite{MDP3}, theorem 4.2 and theorem 4.3),
it is shown that for $d \geq 3$, these curves form a nonreduced irreducible
component of $H(d,g)$. A finite length graded $S$-module $M$ is said to be a
{\it Koszul module parametrized by} $a \geq 1$ {\it and} $l \geq 0$ if
$M$ is isomorphic to a complete intersection module $S/(f_1,f_2,f_3,f_4)$
with $\deg f_1 = \deg f_2 = 1$, $\deg f_3=a$ and $\deg f_4=a+l$.
\begin{ther}\label{extremal}
Characterization of extremal curves:\\
(a) Fix $a \geq 1,l \geq 0$, and let $M$ be a Koszul module parametrized by
$a$ and $l$. Then any minimal curve for the even linkage class ${\cal L}(M)$ is
an extremal curve of degree $d=l+2$ and genus $g=-a+{1 \over 2}(d-2)(d-3)$.\\
(b) Conversely, let $C \subset {\Bbb P}^3$ be an extremal curve of degree
$d \geq 2$ and genus $g < {1 \over 2}(d-2)(d-3)$. If $l=d-2$ and
$a={1 \over 2}(d-2)(d-3)-g$, then $C$ is a minimal curve for a Koszul module
parametrized by $a$ and $l$.
\end{ther}
\begin{ther}\label{nilcomp}
Let $d \geq 3$ and $g < {1 \over 2}(d-2)(d-3)$. Then the family of extremal
curves gives an irreducible component of the Hilbert scheme
$H(d,g)$ of dimension ${3 \over 2}d(d-3)+9-2g$. This component is
nonreduced except when $(d,g)=(3,-1)$.
\end{ther}
\section{Triple Lines in ${\Bbb P}^3$}
In this section we classify the multiplicity three structures on a fixed
line $Y \subset {\Bbb P}^3$. If $W$ a quasiprimitive multiplicity three
structure of type $(L,D_2)$, then we have two exact sequences
\begin{equation}\label{filt1}
0 \rightarrow {\cal I}_Z \rightarrow {\cal I}_Y \rightarrow {\cal O}_Y(a) \rightarrow 0
\end{equation}
\begin{equation}\label{filt2}
0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \rightarrow {\cal O}_Y(2a+b) \rightarrow 0
\end{equation}
where $Z$ is one of the multiplicity two structures on $Y$ described in
proposition \ref{2line}, $L = {\cal O}_Y(a), a \geq -1, \deg D_2=b \geq 0$.
We loosely say that $W$ is of {\it type} $(a,b)$.
\par
In classifying the triple lines of type $(a,b)$, we will handle the case
$a = -1$ separately. This is because the corresponding double line $Z$ is a
complete intersection when $a=-1$, while this is not the case for $a > 0$.
\begin{prop}\label{constr1}
Let $Y \subset {\Bbb P}^3$ be the line
$\{x=y=0\}$ and let $Z$ be the multiplicity two structure $\{x=y^2=0\}$ on
$Y$. Let $p,q$ be two homogeneous polynomials of degrees $b-1,b$ which have
no common zeros along $Y$. Then $p$ and $q$ define a surjection
$u:{\cal I}_Z \rightarrow {\cal O}_Y(b-2)$ by $x \mapsto p, y^2 \mapsto q$. The kernel of $u$
is the ideal sheaf of multiplicity three structure $W$ on $Y$.
Further, we have \\
(a) $p_a(W)=1-b$ \\
(b) $H^1_*({\cal I}_W) \cong (S/(x,y,p,q))(b-2)$ \\
(c) $I_W=(x^2,xy,y^3,xq-y^2p)$ \\
(d) If $p^\prime,q^\prime$ define another three structure $W^\prime$, then
$W=W^\prime$ if and only if there exists $c \in k^*$ such that
$p^\prime=cp \mathop{\rm mod} I_Y$ and $q^\prime=cq \mathop{\rm mod} I_Y$. \\
(e) $W$ is quasiprimitive with second CM filtrant $Z$, unless
$b=1$ and $q=0$, in which case $W=Y^{(2)}$. \\
\end{prop}
\begin{pf}
Since $Z$ is a global complete intersection with total ideal $(x,y^2)$,
$I_Z \otimes S_Y = I_Z/{I_ZI_Y} \cong S_Y(-1) \oplus S_Y(-2)$
is freely generated by the images of $x$ and $y^2$. The map
$x \mapsto {\overline p}, y^2 \mapsto {\overline q}$ defines
a graded homomorphism $\phi:I_Z \rightarrow I_Z/{I_ZI_Y} \rightarrow S_Y(b-2)$. Since
$({\overline p},{\overline q})$ form a regular sequence in $S_Y$, the kernel
of the map $I_Z/{I_ZI_Y} \rightarrow S_Y(b-2)$ is given by the Koszul relation
$x{\overline q}-y^2{\overline p}$, hence
$\mathop{\rm ker} \phi=(I_ZI_Y,xq-y^2p)=(x^2,xy,y^3,xq-y^2p)$. The cokernel
$\mathop{\rm coker} \phi=S_Y(b-2)/({\overline p},{\overline q}) \cong (S/(x,y,p,q))(b-2)$
has finite length, hence $\phi$ sheafifies to a surjection
$u: {\cal I}_Z \rightarrow {\cal O}_Y(b-2)$. \par
Letting $W$ be the subscheme defined by ${\cal I}_W=\mathop{\rm ker} u$, we have an exact
sequence
$$0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \rightarrow {\cal O}_Y(b-2) \rightarrow 0.$$
Since $H^0_*(u)=\phi$ and $H^1_*({\cal I}_Z)=0$, we immediately deduce properties
(b) and (c). The snake lemma provides a second exact sequence
$$0 \rightarrow {\cal O}_Y(b-2) \rightarrow {\cal O}_W \rightarrow {\cal O}_Z \rightarrow 0,$$
which shows that $W$ is supported on $Y$ and that $\mathop{\rm depth} {\cal O}_W \geq 1$,
hence $W$ is a CM multiplicity three structure on $Y$ with $p_a(W)=1-b$.
If the polynomials $p^\prime,q^\prime$ also define $W$ by the construction
above, then $(q,p)$ and $(q^\prime,p^\prime)$ generate the same principal
$S_Y$-submodule of $S_Y(-1) \oplus S_Y(-2)$, hence property (d) holds. \par
If $q=0$, then $p$ must be a unit, as otherwise $p$ and $q$ will have common
zeros along $Y$. In this case, we see that $b-1 = \deg p =0$ and that
$I_W=I_Y^2$, whence $W=Y^{(2)}$. If $q \neq 0$, we use part (c) to see that
$I_W+I_Y^2=(x^2,xy,y^2,xq)=(I_Y^2,xq)$. At the points on
$Y$ where $q \neq 0$, this ideal is simply $(y^2,x)$, so the cokernel
of the inclusion $(I_Y^2,xq) \subset (y^2,x)$ has finite support. Since
the latter ideal defines the multiplicity two structure $Z$ on $Y$, we see
that $Z$ is the second CM filtrant for $W$, proving part (e).
\end{pf}
\begin{cor}\label{3line-1}
Triple lines of type $a=-1,b \geq 0$: Let $W$ be a quasiprimitive
multiplicity three structure on a line $Y$ of type $(-1,b)$ or the second
infinitesimal neighborhood $Y^{(2)}$. Then, after a suitable change of
coordinates, $W$ is constructed by proposition \ref{constr1}. The family of
such multiplicity three structures is irreducible of dimension $5+2b$.
\end{cor}
\begin{pf}
If $W=Y^{(2)}$ is given by the construction in taking $b=1,q=0,p=1$.
If $W$ is quasiprimitive, then we have the exact sequence \ref{filt2},
and the construction above gives all surjections $u:{\cal I}_Z \rightarrow {\cal O}_Y(b-2)$.
To parametrize this family, we first choose the double line $Z$ (an
irreducible choice of dimension $5$ by corollary \ref{2linehilb}), and then
we must choose $(p,q) \in H^0({\cal O}_Y(b-1)) \times H^0({\cal O}_Y(b))/k^*$, which
is an irreducible choice of dimension $2b$. Thus the family is irreducible
of dimension $5+2b$.
\end{pf}
\begin{prop}\label{constr2}
Let $Z \subset {\Bbb P}^3$ be the double line with total ideal
$I_Z=(I_Y^2,xg-yf)$, where $Y$ is the line $\{x=y=0\}$ and $f,g$ are
homogeneous polynomials of degree $a+1$ having no common zeros along $Y$,
as in proposition \ref{2line}. Let $p$ and $q$ be homogeneous polynomials of
degrees $b,3a+b+2$ having no common
zeros along $Y$. Then $p$ and $q$ define a surjection $u:{\cal I}_Z \rightarrow {\cal O}_Y(2a+b)$
by $x^2 \mapsto pf^2, xy \mapsto pfg, y^2 \mapsto pg^2$ and
$xg-yf \mapsto q$. The kernel of $u$ is the ideal sheaf of a quasiprimitive
multiplicity three structure on $Y$ with second Cohen-Macaulay filtrant $Z$.
Further, we have \\
(a) $p_a(W)=-2-3a-b$ \\
(b) $I_W=(I_Y^3,x(xg-yf),y(xg-yf),p(xg-yf)-ax^2-bxy-cy^2)$, where $a,b,c$
are chosen so that $q=af^2+bfg+cg^2 \mathop{\rm mod} I_Y$.\\
(c) If $p^\prime,q^\prime$ define the multiplicity three structure
$W^\prime$, then $W=W^\prime$ if and only if there exists $d \in k^*$ such
that $p^\prime=dp \mathop{\rm mod} I_Y$ and $q^\prime=dq \mathop{\rm mod} I_Y$. \\
\end{prop}
\begin{pf}
The ideal $I_Z=(x^2,xy,y^2,xg-yf)$ has $S$-presentation
$$S(-a-3)^2 \oplus S(-3)^2 \stackrel{\varphi}{\rightarrow} S(-a-2) \oplus S(-2)^3 \rightarrow
I_Z \rightarrow 0$$
given by the matrix
$$\varphi=\left(\begin{array}{cccc}
y & 0 & -g & 0 \\
-x & y & f & -g \\
0 & -x & 0 & f \\
0 & 0 & x & y \\
\end{array}\right).$$
Tensoring with $S_Y$, we see that $I_Z/{I_Z}{I_Y} = \mathop{\rm coker} {\varphi \otimes S_Y}$
is isomorphic to $S_Y(-a-2) \oplus (f^2,fg,g^2)(2a)$,
where ${\overline x}^2,{\overline x}{\overline y},{\overline y}^2$ are
identified with $f^2,fg,g^2$. Making this identification, we have an inclusion
$I_Z/{I_Z}{I_Y} \subset S_Y(-a-2) \oplus S_Y(2a)$ whose cokernel has finite
length. It follows that the sheafification of $I_Z/{I_Z}{I_Y}$ is isomorphic
to ${\cal O}_Y(-a-2) \oplus {\cal O}_Y(2a)$, freely generated by $xg-yf$ and an element
$e$ such that $ef^2={\overline x}^2,efg={\overline x}{\overline y}$ and
$eg^2={\overline y}^2$. \par
The polynomials $p$ and $q$ give a graded homomorphism
$$\phi:I_Z \rightarrow I_Z/{I_Z}{I_Y} \subset S_Y(-a-2) \oplus S_Y(2a)
\stackrel{({\overline q},{\overline p})}{\longrightarrow} S_Y(2a+b).$$
The kernel of the map $({\overline q},{\overline p})$ is given by the
Koszul relation $qe-p(xg-yf)$. Since $f$ and $g$ are relatively prime of
degree $a+1$, the map $S_Y(-2a-2)^3 \rightarrow S_Y$ given by $(f^2,fg,g^2)$ is
surjective in degrees $\geq 3(a+1)-1$, and hence there exist
$(a,b,c)$ such that $q=af^2+bfg+cg^2 \mathop{\rm mod} I_Y$ (because $\deg q \geq 3a+2$).
We can now write $(ax^2+bxy+cy^2-p(xg-yf))=I_Z/{I_Z}{I_Y} \cap \mathop{\rm ker} ({\overline
q},{\overline p})$, and hence
$$\mathop{\rm ker} \phi=(x^3,x^2y,xy^2,y^3,x(xg-yf),y(xg-yf),ax^2+bxy+cy^2-p(xg-yf)).$$
The cokernel of $\phi$ is of finite length, so $\phi$ sheafifies to a
surjection $u:{\cal I}_Z \rightarrow {\cal O}_Y(2a+b)$.
\par
Letting $W$ be the subscheme whose ideal sheaf is the kernel of $u$, we get
an exact sequence
$$0 \rightarrow {\cal O}_Y(2a+b) \rightarrow {\cal O}_W \rightarrow {\cal O}_Z \rightarrow 0$$
which shows that $\mathop{\rm Supp} W=Y$ and $\mathop{\rm depth} {\cal O}_W \geq 1$, hence $W$ is a
multiplicity three structure on $Y$. Since $p_a(Z)=-1-a$, the exact sequence
shows that $p_a(W)=-2-3a-b$. The exact sequence
$$0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \rightarrow {\cal O}_Y(2a+b) \rightarrow 0$$ shows that $I_W=\mathop{\rm ker} \phi$.
If $p^\prime$ and $q^\prime$ define $W^\prime$ by the above construction and
$W^\prime=W$, then $eq^\prime-(xg-yf)p^\prime$ generates the same
$S_Y$-submodule of
$I_Z/{I_Z}{I_Y} \subset S_Y(-a-2) \oplus S_Y(2a)$ as $eq-(xg-yf)p$. Since
$e$ and $xg-yf$ are free generators, it follows that there exists
$d \in k^*$ such that $p^\prime=dp \mathop{\rm mod} I_Y$ and $q^\prime=dq \mathop{\rm mod} I_Y$.
This proves (a),(b) and (c). \par
{}From part (b), we find that $I_W+I_Y^2=(I_Y^2,h(xg-yf))$. The cokernel of
the inclusion $(I_Y^2,h(xg-yf)) \subset I_Z$ is supported on the zeros of
$h$ along $Y$. Since $Z$ has no embedded points, the second CM filtrant of $W$
is $Z$ and the extension $Y \subset W$ is quasi-primitive. \par
\end{pf}
\begin{rmk}\label{coh}{\em
Note that if $u:{\cal I}_Z \rightarrow {\cal O}_Y(2a+b)$ is the surjection above, then
$H^1_*(u)$ is the zero map. Indeed, $H^1_*({\cal I}_Z)$ is generated in degree
$-a$ by proposition \ref{2line} while $H^1({\cal O}_Y(a+b))=0$. In particular,
we have an exact sequence
$$0 \rightarrow \mathop{\rm coker} \phi \rightarrow M_W \rightarrow M_Z \rightarrow 0$$
which shows that the Rao module $M_W$ is 2-generated. Since all curves of
degree $\leq 2$ have Rao modules which are zero or principal, $W$ is a
minimal curve. Further, the exact sequences
\ref{filt1} and \ref{filt2} now give that
$$h^2({\cal I}_W(l))=h^1({\cal O}_Y(l))+h^1({\cal O}_Y(a+l))+h^1({\cal O}_Y(2a+b+l))$$
for all $l \in {\bf Z}$. \par
{}From the total ideal $I_W$ given in part (b), one can compute a minimal
graded $S$-resolution for $I_W$, which has the form
\begin{equation}\label{wres}
S(-a-b-4) \oplus S(-a-5)^2 \stackrel{\theta}{\rightarrow} S(-a-b-3)^2 \oplus S(-a-4)^4
\rightarrow S(-a-b-2) \oplus S(-3)^4
\end{equation}
This resolution determines $h^0({\cal I}_W(l))$ for all $l \in {\bf Z}$. Combining
with the dimensions $h^2({\cal I}_W(l))$ found above, all the $h^i({\cal I}_W(l))$ can
be computed. \par
The machinery for minimal curves of Martin-Deschamps and Perrin shows that
$\theta^\vee$ begins a minimal resolution for $M_W^*$. Completing this
resolution and dualizing the last map gives a presentation for $M_W$.
Carrying this out (we suppress the calculation here), one finds that
$H^1_*({\cal I}_W) \cong \mathop{\rm coker} \psi$, where
$$\psi: S(2a+b-1)^2 \oplus S(a+b-1)^2 \oplus S(a-1)^2 \rightarrow S(2a+b) \oplus S(a)$$
is the map given by the matrix
$$\left(\begin{array}{cccccc}
x & y & fp & gp & -gc & -fa-gb \\
0 & 0 & x & y & f & g \\
\end{array}\right) $$
Here $a,b$ and $c$ are chosen as in part (b) of the proposition.
\em}\end{rmk}
\begin{rmk}{\em
In the case when $b=0$, ${\overline p}$ must be a unit. It follows that
the generators $x(xg-yf)$ and $y(xg-yf)$ are not needed for the total
ideal $I_W$ (see also \cite{B}, p. 24). In this case it is clear that $W$ is
the unique triple line supported on $Y$ and contained in the surface
defined by $ax^2+bxy+cy^2-p(xg-yf)$.
\em}\end{rmk}
\begin{cor}\label{3line0}
Triple lines of type $a,b \geq 0$: Let $W$ be a quasi-primitive multiplicity
three structure on a line $Y \subset {\Bbb P}^3$ of type $(a,b)$ with
$a,b \geq 0$. Then $W$ arises from the construction of proposition
\ref{constr2} after a suitable change of coordinates. The family of these
triple lines is irreducible of dimension $10+5a+2b$.
\end{cor}
\begin{pf}
Since $W$ is of type $(a,b)$ with $a \geq 0$, there is an exact sequence
$$0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \stackrel{u}{\rightarrow} {\cal O}_Y(2a+b) \rightarrow 0$$
where $Z$ is a double line of type $a \geq 0$. By Proposition \ref{2line},
we may change coordinates so that $I_Z=(x^2,xy,y^2,xf-yg)$ where $f,g$ are
homogeneous polynomials of degree $a+1$ with no common zeros along $Y$.
As in the proof of \ref{constr2} above, ${\cal I}_Z \otimes {\cal O}_Y \cong
{\cal O}_Y(2a) \oplus {\cal O}_Y(-a-2)$ freely generated by $e$ and the image of $xg-yf$,
where $ef^2=x^2,efg=xy$ and $eg^2=y^2$. From this it is clear that such a
map $u$ is given by homogeneous polynomials $p,q$ of degrees $b,3a+b+2$
which have no common zero along $Y$, and hence $W$ arises by the construction
of proposition \ref{constr2}.
\end{pf}
\section{The Hilbert Scheme}
In this section we describe the Hilbert scheme $H(3,g)$ of locally
Cohen-Macaulay curves of degree $3$ and arithmetic genus $g \leq 1$.
In particular, we classify all CM curves of degree $3$ and describe
the irreducible components of $H(3,g)$. We also show that certain
extremal curves lie in the closure of each irreducible component, hence
that $H(3,g)$ is connected. We begin with the curves of genus
$-1 \leq g \leq 1$, which have been described elsewhere. \par
\begin{prop}\label{irred}
For $-1 \leq g \leq 1$, the Hilbert scheme $H(3,g)$ is smooth and irreducible
of dimension $12$.
\end{prop}
\begin{pf}
$H(3,1)$ consists of arithmetically Cohen-Macaulay curves, hence
we can apply a theorem of Ellingsrud \cite{E2}. That $H(3,0)$ and $H(3,-1)$
are smooth and irreducible of dimension $12$ is part of \cite{MDP3}, theorem
4.1. $H(3,0)$ consists of arithmetically Cohen-Macaulay curves and
$H(3,-1)$ consists of extremal curves.
\end{pf}
For $g \leq -2$, the Hilbert scheme is not irreducible, and more work is
required to show connectedness. Our first task is to describe how the
unions of double lines and reduced lines fit in with the irreducible families
of triple lines.
\begin{prop}\label{famI}
Fix $g \leq -2$. Then \\
(a) The family of curves $W = Z \cup_{2P} L$ formed by taking the union
of a double line $Z$ with $p_a(Z)=g-1$ and a line $L$ which meets $Z$ in
a double point form an irreducible family of dimension $9-2g$. \\
(b) The family of curves $W = Z \cup_P L$ formed by taking the union
of a double line $Z$ with $p_a(Z)=g$ and a line $L$ which meets $Z$ in
a reduced point form an irreducible family of dimension $8-2g$. \\
(c) The family of curves $W$ which are triple lines of type $(-1,1-g)$
form an irreducible family of dimension $7-2g$. \\
Each curve above is an extremal curve, hence is a minimal curve for a
complete intersection module with parameters $l=1$ and $a=-g$. The families
(b) and (c) lie in the closure of the family (a).
\end{prop}
\begin{pf}
Let $W=Z \cup_{2P} L$ be a curve from family (a) above. After a change of
coordinates we may write $I_L=(x,z)$ and $I_Z=(x^2,xy,y^2,xg-yf)$.
We have an exact sequence
$$0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \oplus {\cal I}_L \stackrel{\pi}{\rightarrow} {\cal I}_{2P} \rightarrow 0$$
where $2P=Z \cap L$ denotes the double point. Noting that
$I_L+I_Z=(x,z,y^2,yf)$ and that $I_{2P}=(x,z,y^2)$ ($2P$ is a complete
intersection), we see that $H_0^*(\pi)$ is surjective. Since $H^1_*({\cal I}_{2P})$
vanishes in positive degree, we conclude that $r_o(W)=r_o(Z)=-g$ and that
$\rho_W(1)=-g$. $Z(x^2,y^2z)$ links $W$ to $W^\prime=Z^\prime \cup_{2Q} L$,
which is also from family (a). Applying the argument above and using the
isomorphism $M_{W^\prime}^* \cong M_W(1)$ shows that $r_a(W)=1+g$ and
$\rho_W(0)=-g$. Thus $W$ is extremal. \par
To parametrize this family of curves, one first chooses the double line
$Z$ (an irreducible choice of dimension $7-2g$ by corollary
\ref{2linehilb}, since $p_a(Z)=g-1$), then a point $P \in Z$ ($1$ parameter),
and finally a line $L$ through $P$ lying in the tangent plane to $Z$ at $P$
($1$ parameter). This shows that this family is irreducible of dimension
$9-2g$. \par
The argument for $W=Z \cup_P L$ is similar. We choose suitable
coordinates and write $I_L=(x,z), I_Z=(x^2,xy,y^2,xg-yf)$. We have an exact
sequence
$$0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \oplus {\cal I}_L \stackrel{\pi}{\rightarrow} {\cal I}_P \rightarrow 0$$
where $P=Z \cap L$. Writing $I_{P}=(x,y,z)$ and $I_L+I_Z=(x,z,y^2,yf)$ we
see that $\dim \mathop{\rm coker} H^0_*(\pi(l))=1$ for $1 \leq l \leq -g=\deg yf-1$.
It follows again that $r_o(W)=r_o(Z)-1=-g$ and $\rho_W(1)=\rho_Z(1)-1=-g$.
$Z(x^2,y^2z)$ links $W$ to a curve $W^\prime$ from family (b), so we find
that $W$ is extremal. \par
To parametrize these curves, we first choose a double line $Z$ (an
irreducible choice of dimension $5-2g$, since $p_a(Z)=g$), and then
choose a general line $L$ which meets $Z$ ($3$ parameters). This shows
that (b) is an irreducible family of dimension $8-2g$. \par
If $W$ is a triple line of type $(-1,1-g)$, then from corollary \ref{3line-1}
we have $M_W=(S/(x,y,p,q))(-1-g)$ where $\deg p = 1-g$ and $\deg q = 2-g$.
It follows that $W$ is extremal. The family of such triple lines $W$ is
irreducible of dimension $7-2g$ by corollary \ref{3line-1}.
\par
By theorem \ref{nilcomp}, the scheme $H_{\gamma,\rho}$ of extremal curves
is irreducible of dimension $9-2g$ when $g \leq -2$. It follows that
family (a) gives the general member of the family, and that families (b)
and (c) lie in the closure.
\end{pf}
\begin{prop}\label{famII}
Fix $g \leq -2$. Then\\
(a) The family of curves $W = Z \cup L$ formed by taking the union of a
double line $Z$ with $p_a(Z)=g+1$ and a disjoint line $L$ form an irreducible
family of dimension $7-2g$. \\
(b) The family of curves $W$ which are triple lines of type $(0,-2-g)$ form
an irreducible family of dimension $6-2g$. \\
The curves above are all minimal, and each curve in family (b) is obtained
from curves in family (a) by a deformation which preserves cohomology.
\end{prop}
\begin{pf}
Let $W = Z \cup L$ be a curve from family (a) above. We begin by computing the
total ideal and cohomology for $W$. In suitable coordinates, we may write
$I_L=(z,w),I_Z=(x^2,xy,y^2,xg-yf)$ with $g,f \in k[z,w]$. In particular,
$xg-yf \in I_L$ and hence $J=((x,y)^2(z,w),xg-yf) \subset I_L \cap I_Z$.
One can compute that the minimal graded $S$-resolution of $J$ is of the form
$$S(g-3)\oplus S(-5)^2 \rightarrow S(g-2)^2\oplus S(-4)^7 \rightarrow S(g-1)\oplus S(-3)^6 $$
and hence $J$ is the total ideal for $W$. Comparing with resolution
\ref{wres} (with $a=0,b=-2-g$) shows that $W$ has the same Hilbert function
as a curve in family (b). Moreover, the exact sequence
$$0 \rightarrow {\cal I}_W \rightarrow {\cal I}_Z \oplus {\cal I}_L \rightarrow {\cal O} \rightarrow 0$$
shows that $h^2({\cal I}_W(l))=h^2({\cal I}_L(l))+h^1({\cal I}_Z(l))=h^1({\cal O}_L(l))+
h^1({\cal O}_Y(l))+h^1({\cal O}_Y(-g-2+l))$. This agrees with the second cohomology
dimensions found in remark \ref{coh} for triple lines of type $(0,-2-g)$,
hence the dimensions of the cohomology groups for families (a) and (b)
are the same. This same exact sequence also shows that $M_W$ is
2-generated. \par
The family (a) is parametrized by first choosing $Z$ (an irreducible
choice of dimension $3-2g$ by corollary \ref{2linehilb}) and then choosing
a general line $L$ ($4$ parameters), hence the family (a) is irreducible
of dimension $7-2g$. Family (b) is irreducible of dimension $6-2g$ by
corollary \ref{3line0}. These curves are minimal because they are of
degree three and their Rao modules are 2-generated. \par
Let $W$ be a triple line from family (b). By corollary \ref{3line0}, we
can change coordinates and write $I_W=((x,y)^3,xq,yq,hq-ax^2-bxy-cy^2)$, where
$q=xg-yf$ is a quadric surface containing the underlying second CM filtrant
$Z$. By remark \ref{smooth}, $q$ may be chosen to be the equation of a
smooth quadric $Q$. We may choose $z$ and $w$ so that $q=xz-yw$. \par
On the smooth quadric $Q$ the family of lines $L_t=Z(x+wt,y+zt)$ give a
flat family over ${\Bbb A}^1$ with $L_0=Y$. $D_t=L_t \cup Y$ forms a flat family
such that $D_0=Z$ is the double line $Z$ on $Q$ supported on $Y$, the second
Cohen-Macaulay filtrant of $W$. Writing this family as
$D \subset {\Bbb P}^3 \times {\Bbb A}^1 \stackrel{\pi}{\rightarrow} {\Bbb A}^1$, we see by
By Grauert's theorem, $\pi_*({\cal I}_D(-g))$ is locally free on ${\Bbb A}^1$, hence
globally free. In particular, if $s_1 \in {\cal I}_{D_1}(-g)$ is
the equation of a smooth surface containing $D_1=L_1 \cup Y$, we can find a
section $s_t$ extending $s_1$ such that $s_0=hq-ax^2-bxy-cy^2$. \par
Now consider the family $C_t=S_t \cap (Y^{(2)} \cup L_t)$. Let $U \subset
{\Bbb A}^1$
be the open set where $C_t$ is locally Cohen-Macaulay. For $t \neq 0$, $C_t$
is the disjoint union of a double line on $Y$ and the line $L_t$. The
ideal of $C_t$ is given by $I_t=((x,y)^2(x+wt,y+zt),s_t)$. Note that
$x^2(y+zt)-xy(x+wt)=xqt \in I_t$ and similarly $yqt \in I_t$.
Flattening over $U$, we must add $xq$ and $yq$ to $I_t$. In particular, the
limit ideal $I_0$ contains $((x,y)^3,xq,yq,s_0)$, and hence gives $W$.
\end{pf}
\begin{prop}\label{g-2}
The Hilbert scheme $H(3,-2)$ consists of the following pair of irreducible
components:\\
(a) The irreducible family $H_{-1}$ of dimension $13$ from proposition
\ref{famI}.\\
(b) The closure $H_0$ of the irreducible family of sets of three disjoint
lines.
This closure is $12$-dimensional and contains the curves from proposition
\ref{famII}.
\end{prop}
\begin{pf}
Let $W \in H(3,-2)$ be a curve. Since $g<0$, $W$ is not integral, hence
is reducible or nonreduced. If $W$ is reduced, then $W$ is the union of
$3$ disjoint lines (because any union of a conic and line has $g \geq -1$),
hence lies in family (b). From corollary \ref{2linehilb}, any double line with
arithmetic genus $-1$ is a limit of disjoint unions of two lines. Adding
another disjoint line to this deformation, we see that the disjoint union
of a double line $Z$ of genus $-1$ and a reduced line $L$ lies in the
closure of the family of three disjoint lines. Thus family (b) is
irreducible. \par
If $W$ is not reduced, then $\deg \mathop{\rm Supp} W <3$. If $\deg \mathop{\rm Supp} W=1$, then
$W$ is a triple structure on a line $Y$ of arithmetic genus $-2$, which can
only have type $(-1,3)$ or $(0,0)$ ($W$ must be quasiprimitive, since
otherwise $W=Y^{(2)}$, which satisfies $p_a(W)=0$). By propositions \ref{famI}
and \ref{famII}, these lie in families (a) and (b) respectively. If
$\deg \mathop{\rm Supp} W=2$, then $W$ is a union of a double line $Z$ and a reduced
line $L$ (the support of $W$ cannot be an irreducible conic, since a multiple
conic has degree $\geq 4$). $Z$ can meet $L$ in a scheme of length $0$, $1$,
or $2$, hence $W$ lies in family (a) or family (b). \par
Family (a) cannot lie in the closure of family (b) by reason of dimension.
Family (b) cannot lie in the closure of family (a) by reason of semicontinuity;
the curves in family (a) are extremal, while the curves in family (b) are not
(If $C$ is in family (b), one checks that $h^1({\cal I}_C(-1))=0$).
\end{pf}
\begin{prop}\label{g-3}
Let $g \leq -3$. Then the Hilbert scheme $H(3,g)$ consists of the following
irreducible components: \\
(a) The irreducible family of dimension $9-2g$ from proposition \ref{famI},
which we now denote $H_{-1}$.\\
(b) The closure of the irreducible family of dimension $7-2g$ from
proposition \ref{famII}, which we now denote $H_0$.\\
(c) for each $0 < a < (-2-g)/3$, the closure of the irreducible family
$H_a$ of dimension $14-2g-a$ consisting of triple lines of type
$(a,-2-3a-g)$.\\
\end{prop}
\begin{pf}
Let $C \in H(3,g)$. Then $C$ is not integral because $g \leq -3$. If $C$ were
reduced, it would be a union of $3$ lines (these have genus $\geq -2$, hence
are ruled out) or the union of a conic and a line (which has $\geq -1$, hence
is ruled out). Thus $C$ is not reduced and $\dim \mathop{\rm Supp} C < 3$.
If $C$ has support of degree $2$, the support cannot be irreducible, since
a multiplicity structure on a conic has degree at least $4$. Hence the
support of $C$ consists of two lines, and all possible configurations are
covered in families (a) and (b) above. If $C$ has support of degree $1$,
then $C$ is a triple line and corollaries \ref{3line-1} and \ref{3line0}
show that $C$ is among the families listed above. \par
Now we show that the $H_i$ are irreducible components. Let $-1 \leq i < j \leq
(-2-g)/3$. $H_i$ is not contained in the closure of $H_j$ because
$\dim H_i > \dim H_j$. On the other hand, semicontinuity shows that
$H_j$ is not contained in the closure of $H_i$. Indeed, from corollaries
\ref{3line-1} and \ref{3line0}, we see that the Rao
module for a triple line of type $(a,b)$ as a generator of minimal degree
$-2a-b$, and hence a minimal degree generator for the Rao module
of a curve in $H_a$ occurs in degree $g+2+a$. This shows that for
$C \in H_i$ we have $h^0({\cal O}_C(g+2+i)) \neq 0$ while for $C \in H_j$ we have
$h^0({\cal O}_C(g+2+i))=0$. Hence there can be no specialization from a family
of curves in $H_i$ to a curve in $H_j$.
\end{pf}
\begin{prop}\label{spec}
For each $a \geq 0$ and $b \geq 0$, there exists a flat family
$W \subset {\Bbb P}^3 \times {\Bbb A}^1$ whose general member $W_t$ is a triple
line of type $(a,b)$ for $t \neq 0$ and whose special member $W_0$ is
a triple line of type $(-1,3a+b+3)$.
\end{prop}
\begin{pf}
Consider the family defined by the ideal $I_t$ with generators
$$x^3,x^2y,xy^2,y^3,x(xz^{a+1}-tyw^{a+1}),y(xz^{a+1}-tyw^{a+1})$$
$$z^bt^2(xz^{a+1}-tyw^{a+1})-x^2w^{a+b}.$$
We flatten this family over $t$ by adding to the ideal those elements
which are multiples of $t$. Let $A,B,C$ denote the last three generators
given for the ideal. Then we must add
$$D=(w^{a+b}A+z^{a+1}C)/t=-xyw^{2a+b+1}+z^{a+b+1}t(xz^{a+1}-tyw^{a+1})$$
to the ideal. We must also add
$$E=(w^{2a+b+1}B+z^{a+1}D)/t=-y^2w^{3a+b+2}+z^{2a+b+2}(xz^{a+1}-tyw^{a+1})$$
to the ideal. Setting $t=0$, we find that the limit ideal $I_0$ contains
the generators
$$x^3,x^2y,xy^2,y^3,xyz^{a+1},x^2w{a+b},xyw^{2a+b+1},xz^{3a+b+3}-y^2w^{3a+b+2}.$$
It follows that the saturation of $I_0$ contains the ideal
$$(x^2,xy,y^3,xz^{3a+b+3}-y^2w^{3a+b+2}),$$
but this is the total ideal of a triple structure of type $(-1,3a+b+3)$ by
corollary \ref{3line-1}. On the other hand, corollary \ref{3line0}
shows that the ideal $I_t$ for $t \neq 0$ is the total ideal of a triple
line of type $(a,b)$. This gives the flat family $W$.
\end{pf}
\begin{rmk}{\em
The commutative algebra in the proof above was inspired by a geometric
example of Robin Hartshorne. He gave an example of a deformation of
three disjoint lines to a triple line of type $(-1,3)$ by deforming
the unique quadric containing the three lines to a double plane while
at the same time bringing the lines together.
\em}\end{rmk}
\begin{ther}
The Hilbert scheme $H(3,g)$ is connected is connected whenever it is nonempty.
\end{ther}
\begin{pf}
By proposition \ref{irred}, it suffices to consider the case $g \leq -2$.
In this case $H(3,g)$ has irreducible components $\{H_a\}_{a \geq -1}$ by
propositions \ref{g-2} and \ref{g-3}. Let $H_a$ be one of these components
with $a \geq 0$. Choosing $b=-2-3a-g$, proposition \ref{spec} gives a
family of triple lines whose general member lies in $H_a$ and whose special
member lies in $H_{-1}$.
\end{pf}
\begin{rmk}{\em
The proof of proposition \ref{spec} shows that a triple line $W$ with total
ideal $(x^2,xy,y^3,xz^{1-g}-y^2w^{-g})$ lies in the closure of each irreducible
component of $H(3,g)$.
\em}\end{rmk}
\begin{ex}{\em
Hartshorne has shown that the Hilbert scheme $H(4,0)$ is also connected.
Here we give an independent proof using the methods of this paper. $H(4,0)$
has two irreducible components (\cite{MDP3},$\S 4$): $H_1=$ the extremal
curves (these have Rao module of Koszul type parametrized by
$a=1$ and $l=2$) and $H_2=$ the curves with Rao module $k$ in degree $1$.
We will give a specialization from quadruple lines in $H_2$ to quadruple
lines in $H_1$.
\par
Let $Y$ be the line $\{x=y=0\}$ and $W$ be the planar triple line with total
ideal $I_W=(x,y^3)$. As in proposition \ref{3line-1}, a pair $(h,k)$ of
homogeneous polynomials of degrees $1$ and $3$ with no common zeros along
$Y$ determines a map $I_W \stackrel{\phi}{\rightarrow} S_Y$ by $x \mapsto h,
y \mapsto k$ which sheafifies to a surjection $u$. $\mathop{\rm ker} u={\cal I}_T$ defines a
multiplicity four line $Z_1$ such that $p_a(Z_1)=0$,
$I_{Z_1}=(x^2,xy,y^4,xk-y^2h)$, and $H^1_*({\cal I}_{Z_1}) \cong S/(x,y,h,k)$.
It follows that $Z_1 \in H_1$.
\par
Letting $V$ be a quasiprimitive multiplicity three structure of type $(-1,1)$
on $Y$, proposition \ref{3line-1} shows that we may write
$I_V=(x^2,xy,xq-y^2)$, with $q \not\in I_Y$ ($p$ is unit in this case).
As in the proof of proposition \ref{3line0}, a pair $(f,g)$ of forms with
no common zeros along $Y$ determines a map $I_V \rightarrow S_Y(-1)$ by
$x^2 \mapsto 0,xy \mapsto f,xq-y^2 \mapsto g$ which sheafifies to a
surjection $w$. $\mathop{\rm ker} w={\cal I}_{Z_2}$ defines a multiplicity four line $Z_2$ such
that $Z_2$ such that $p_a(Z_2)=0$,$I_{Z_2}=(x^2,xy^2,xyq-y^3,gxy-f(xq-y^2))$
and $H^1_*({\cal I}_{Z_2}) \cong S/(x,y,f,g))(-1)$, hence $Z_2 \in H_2$.
\par
Now consider the ideal
$$I_t=(x^2,xy^2,ty^3-xyz,xyw-tz(y^2t-xz))$$
in the ring $k[t][x,y,z,w]$. For $t \neq 0$, this gives the total ideal
of a curve in $H_2$ (see $Z_2$ above). Flattening over $t$, we add to this
ideal the multiples of $t$. Letting $A,B,C$ denote the last three generators
listed, we add
$$D=(wB+zC)/t=y^3w+xz^3-ty^2z$$
$$E=(zA+yB)/t=y^4$$
to $I_t$. Setting $t=0$, it follows that
$$(x^2,xy^2,xyz,xyw,y^3w+xz^3,y^4) \subset I_0$$
and hence $(x^2,xy,y^4,y^3w+xz^3) \subset {\overline I_0}.$ This ideal gives
a multiplicity four line in $H_1$ (see $Z_1$ above). This shows that
$H(4,0)$ is connected.
\em}\end{ex}
\begin{rmk}{\em
The results in this paper raise several questions: \\
(1) Can each locally Cohen-Macaulay curve $C \subset {\Bbb P}^3$ be deformed
with constant cohomology to a quasiprimitive multiple line?\\
(2) Can each multiplicity structure on a line be deformed to an extremal
multiplicity structure on the same line? \\
(3) Is $H(d,g)$ connected for all $(d,g)$?\\
The answers are yes when $d=2$ and $d=3$. Positive answers to (1) and (2)
would give a positive answer to (3).
\em}\end{rmk}
|
1996-08-13T11:10:09 | 9603 | alg-geom/9603009 | en | https://arxiv.org/abs/alg-geom/9603009 | [
"alg-geom",
"math.AG"
] | alg-geom/9603009 | Richard Earl | Richard Earl | A Note on the Cohomology of Moduli of Rank Two Stable Bundles | 6 pages, no figures. (A few typos have been corrected.) LaTeX 2.09 | null | null | null | null | The rational cohomology of the moduli space of rank two, odd degree stable
bundles over a curve (of genus g > 1) has been studied intensely in recent
years and in particular the invariant subring generated by Newstead's
generators alpha, beta, gamma. Several authors have independently found a
minimal complete set of relations for this subring. Their methods are very
different from the methods originally employed by Kirwan to prove Mumford's
conjecture -- that relations derived from the vanishing Chern classes of a
particular rank 2g-1 bundle are a complete set of relations for the entire
cohomology ring. This note contains two theorems which readily follow from
Kirwan's original calculations. We rederive the above result showing that the
first three invariant Mumford relations generate the relation ideal of the
invariant subring. Secondly we prove a stronger version of Mumford's conjecture
and show that the relations coming from the first vanishing Chern class
generate the relation ideal of the entire cohomology ring as a
Q[alpha,beta]-module. (Only a few typos have been amended in this revised
version).
| [
{
"version": "v1",
"created": "Tue, 12 Mar 1996 16:49:47 GMT"
},
{
"version": "v2",
"created": "Wed, 13 Mar 1996 13:34:40 GMT"
},
{
"version": "v3",
"created": "Tue, 13 Aug 1996 09:07:11 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Earl",
"Richard",
""
]
] | alg-geom | \section{Introduction}
In recent years the cohomology ring of the moduli space ${\cal
N}_{g}(2,d)$ of rank two and odd degree $d$ stable bundles over a Riemann surface $M$ of genus $g \geq 2$ has been extensively studied \cite{B}, \cite{KN}, \cite{ST}, \cite{Z}. The subring $H^{*}({\cal N}_{g}(2,d);{\bf Q})^{\Gamma}$ of $H^{*}({\cal N}_{g}(2,d);{\bf Q})
$ which is invariant under the induced action of the mapping class group $\Gamma$ of $M$ has also been much studied and has been shown to play a central role in the ring structure of $H^{*}({\cal N}_{g}(2,d);{\bf Q})$.\\
\indent In 1991 Zagier \cite{Z} began a study of certain relations in
the invariant cohomology ring. These are defined recursively in terms
of Newstead's generators $\alpha,\beta,\gamma$ \cite{N} by
\begin{equation}
(r+1)\zeta_{r+1} = \alpha \zeta_{r} + r \beta \zeta_{r-1} + 2 \gamma
\zeta_{r-2} \label{A}
\end{equation}
with $\zeta_{0}=1$ and $\zeta_{r}=0$ for $r<0$. Each of the authors
\cite{B}, \cite{KN}, \cite{ST}, \cite{Z} showed that $\zeta_{r}$ is a
relation in $H^{*}({\cal N}_{g}(2,d);{\bf Q})^{\Gamma}$ for $r \geq g$ and
that $\zeta_{g}, \zeta_{g+1}, \zeta_{g+2}$ generate
the relation ideal of the invariant cohomology ring. King and Newstead
further proved a decomposition theorem \cite[Prop.2.5]{KN}, originally
conjectured by Mumford, describing $H^{*}({\cal N}_{g}(2,d);{\bf Q})$ in
terms of $H^{*}({\cal N}_{k}(2,d);{\bf Q})^{\Gamma}$ $(k \leq g)$ and
exterior powers of $H^{3}({\cal N}_{g}(2,d);{\bf Q}).$\\
\indent The methods employed by the authors \cite{B}, \cite{KN},
\cite{ST}, \cite{Z} differ greatly from Kirwan's original proof
\cite[$\S$2]{K} of Mumford's conjecture. Mumford introduced relations
in $H^{*}({\cal N}_{g}(2,d);{\bf Q})$ which are constructed from the
vanishing Chern classes of a rank $2g-1$ bundle $\pi_{!}V$ over ${\cal
N}_{g}(2,d)$ and conjectured that these relations are complete. Zagier
\cite[$\S$6]{Z} showed that the relations $\zeta_{r}, r \geq g$ form a subset of the Mumford relations and for this reason we will refer to the vanishing $\zeta_{r}$ as the Zagier-Mumford relations.\\
\indent The purpose of this note is two-fold. We will firstly
rederive the result that the first three Zagier-Mumford relations form
a minimal complete set for the invariant cohomology. The second result is to
prove a subsequent and stronger version of Mumford's conjecture;
namely we will show that the relations constructed solely from the
first vanishing Chern class $c_{2g}(\pi_{!}V)$ freely generate the
relation ideal of $H^{*}({\cal N}_{g}(2,d);{\bf Q})$ as a ${\bf Q}[\alpha,\beta]$-module. Both results follow easily
from Kirwan's calculations in \cite[$\S$2]{K}. Partly the aim of this
note is to demonstrate the power of the methods of \cite{K} which
currently is the only approach to have generalised to the rank three
case \cite{E}.\\
\indent For ease of notation we will from now on write ${\cal N}$ for
${\cal N}_{g}(2,d)$ and write ${\cal N}_{0}$ for the moduli space of rank
two odd degree stable bundles of fixed determinant. Also we write $\bar{g}$
for $g-1$ and $[2g]$ for the set $\{1,...,2g\}$.
\section{Kirwan's Approach}
\indent Let ${\cal C}$ denote the space of all holomorphic structures on a
fixed $C^{\infty}$ complex vector bundle ${\cal E}$ over $M$ of rank two and
odd degree $d$ and let ${\cal G}_{c}$ denote the group of all $C^{\infty}$
complex automorphisms of ${\cal E}$. We may then identify ${\cal N}$ with
the quotient ${\cal C}^{s}/{\cal G}_{c}$ where ${\cal C}^{s} \subset {\cal C}$ is the open subset
consisting of stable holomorphic structures. Let ${\cal G}$ denote the gauge group of
all $C^{\infty}$ automorphisms of ${\cal E}$ which are unitary with
respect to a fixed Hermitian structure and let $\overline{{\cal G}}$ denote
the quotient of ${\cal G}$ by its $U(1)$ centre.\\
\indent Then $H^{*}({\cal N};{\bf Q})$ is naturally isomorphic to
$H^{*}_{\overline{{\cal G}}}({\cal C}^{s};{\bf Q})$ \cite[9.1]{AB}, and Atiyah and Bott
show further \cite[thm.7.14]{AB} that the restriction map
\begin{equation}
H^{*}(B\overline{\cal G};{\bf Q}) \cong H^{*}_{\overline{{\cal G}}}({\cal C};{\bf Q}) \rightarrow
H^{*}_{\overline{{\cal G}}}({\cal C}^{s};{\bf Q}) \cong H^{*}({\cal N};{\bf Q}) \label{B}
\end{equation}
is surjective. They construct a rank two ${\cal G}$-equivariant
holomorphic bundle ${\cal V}$ over ${\cal C} \times M$ and define generators
\begin{equation}
a_{1}, a_{2}, f_{2} \mbox{ and } b_{1}^{s}, b_{2}^{s} \quad (s \in
[2g]) \label{gen}
\end{equation}
for $H^{*}_{\cal G}({\cal C};{\bf Q}) = H^{*}(B{\cal G};{\bf Q})$ by
taking the slant products
\[
c_{r}({\cal V}) = a_{r} \otimes 1 + \sum_{s=1}^{2g} b_{r}^{s} \otimes
e_{s} + f_{r} \otimes \omega
\]
where $e_{1},...,e_{2g}$ is a fixed basis for $H^{1}(M;{\bf Q})$ and
$\omega$ is the standard generator for $H^{2}(M;{\bf Q})$. The only
relations amongst the generators (\ref{gen}) are that $a_{1}, a_{2},
f_{2}$ commute with everything and that the $b_{i}^{s}$ anticommute
amongst themselves.\\
\indent Rather than (\ref{B}) we shall consider the surjection
\begin{equation}
H^{*}_{\overline{{\cal G}}}({\cal C};{\bf Q}) \otimes {\bf Q}[a_{1}] \cong H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow
H^{*}_{\cal G}({\cal C}^{s};{\bf Q}) \cong H^{*}({\cal N};{\bf Q}) \otimes {\bf Q}[a_{1}]. \label{C}
\end{equation}
The images of (\ref{gen}) under this map form generators for
$H^{*}_{\cal G}({\cal C}^{s};{\bf Q})$ which we will also refer to as
$a_{1}, a_{2}, f_{2}, b_{1}^{s},$ $b_{2}^{s}$ and the relations among
these restrictions form the kernel of (\ref{C}). In order to study
this kernel we introduce a ${\cal G}$-perfect stratification of ${\cal C}$ due to
Shatz \cite{Sh}.\\
\indent
\indent Any unstable holomorphic bundle $E$ over $M$ of rank $n$ and degree $d$
has a canonical filtration (or flag) \cite[p.221]{HN} \label{filt}
which in the rank $n=2$ case is a line subbundle $L$ of $E$ of
degree $d_{1}$ such that $d_{1} > d/2$. We define the type of $E$ to be $(d_{1},d-d_{1})$ and define the
type of a stable bundle to be $\mu_{0} = (d/2,d/2)$. The stratum ${\cal C}_{\mu}
\subseteq {\cal C}$ is the set of all
holomorphic vector bundles of type $\mu$ and we construct a total
order $\preceq$ on the set of types by writing $(\mu_{1},\mu_{2}) \preceq
(\nu_{1},\nu_{2})$ if $\mu_{1} \leq \nu_{1}$.\\
\indent Kirwan's proof of Mumford's conjecture is based upon a set of
completeness criteria for a set $\cal R$ of relations in $H^{*}_{\cal G}({\cal C};{\bf Q})$
\cite[Prop.1]{K}. These criteria involve finding for each $\mu \neq
\mu_{0}$ relations ${\cal R}_{\mu} \subseteq {\cal R}$ which in a
technical sense correspond to the stratum ${\cal C}_{\mu}$. We introduce here
similar completeness criteria for the invariant cohomology:
\begin{prop}
\label{ICC}
(Invariant Completeness Criteria) Let ${\cal R}$
be a subset of the kernel of the restriction map
\begin{equation}
H^{*}_{\cal G}({\cal C};{\bf Q})^{\Gamma} \rightarrow H^{*}_{\cal G}({\cal C}^{s};{\bf Q}). \label{Q23}
\end{equation}
Suppose that for each unstable type $\mu$ there is a subset ${\cal
R}_{\mu}$ of the ideal generated by ${\cal R}$ in $H^{*}_{\cal G}({\cal C};{\bf Q})^{\Gamma}$ such
that the restriction of ${\cal R}_{\mu}$ to $H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})$
is zero when $\nu \prec \mu$ and when $\nu = \mu$ equals the ideal
generated by $e_{\mu}$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})^{\Gamma}$, where $e_{\mu}$
denotes the equivariant Euler class of the normal bundle to ${\cal C}_{\mu}$ in
${\cal C}$. Then ${\cal R}$ generates the kernel of the restriction map
(\ref{Q23}) as an ideal of $H^{*}_{\cal G}({\cal C};{\bf Q})^{\Gamma}.$
\end{prop}
{\bf PROOF:} We include now the main points in the proof of the above
proposition. However the only difference between this proof and the
argument of \cite[prop.4]{E} is to observe that ${\cal C}_{\mu}$ is
$\Gamma$-invariant and hence $e_{\mu} \in H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})^{\Gamma}$.\\
\indent For $\mu$ an unstable type let $\mu-1$ denote
the type previous to $\mu$ with respect to $\preceq$ and define
$V_{\mu} = \bigcup_{\nu \preceq \mu} {\cal C}_{\nu}.$ Then $V_{\mu}$ is an open
subset of ${\cal C}$ which contains ${\cal C}_{\mu}$ as a closed submanifold.\\
\indent Let $d_{\nu}$ denote the complex codimension of ${\cal C}_{\nu}$ in
${\cal C}$. For any given $i \geq 0$ there are only finitely many $\nu \in
{\cal M}$ such that $2d_{\nu} \leq i$ \cite[7.16]{AB} and so for each $i \geq 0$ there
exists some $\mu$ such that
\[
H_{{\cal G}}^{i}({\cal C};{\bf Q}) = H_{{\cal G}}^{i}(V_{\mu};{\bf Q}).
\]
Hence it is enough to show that for each $\mu$ the image in
$H^{*}_{\cal G}(V_{\mu};{\bf Q})^{\Gamma}$ of the ideal generated by ${\cal R}$ contains the
image in $H^{*}_{\cal G}(V_{\mu};{\bf Q})^{\Gamma}$ of the kernel of (\ref{Q23}). Note that the
above is clearly true for $\mu=\mu_{0}$ as $V_{\mu_{0}}={\cal C}^{s}.$ We will
proceed by induction with respect to $\preceq$.\\
\indent Assume now that $\mu \neq \mu_{0}$ and that $\zeta \in H^{*}_{\cal G}({\cal C};{\bf Q})^{\Gamma}$ lies in the kernel of (\ref{Q23}). Suppose that the image of $\zeta$ in $H^{*}_{\cal G}(V_{\mu-1};{\bf Q})$ is in the
image of the ideal generated by ${\cal R}.$ We may, without any loss of
generality, assume that the image of $\zeta$ in $H^{*}_{\cal G}(V_{\mu-1};{\bf Q})$ is
zero. Then by the exactness of the Thom-Gysin sequence
\[
\cdot \cdot \cdot \rightarrow H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q})
\rightarrow H^{*}_{\cal G}(V_{\mu};{\bf Q})
\rightarrow H^{*}_{\cal G}(V_{\mu-1};{\bf Q}) \rightarrow \cdot \cdot \cdot
\]
there exists an
element $\eta \in H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q})$ which is
mapped to the image of $\zeta$ in $H^{*}_{\cal G}(V_{\mu};{\bf Q})$ by the Thom-Gysin
map. The composition
\[
H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}(V_{\mu};{\bf Q})
\rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})
\]
is given by multiplication by $e_{\mu}$ which is
not a zero-divisor in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ \cite[p.569]{AB}. Hence the
restriction of $\zeta$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ is $\eta e_{\mu}$ and by
our initial observation $\eta
\in H^{*-2d_{\mu}}_{{\cal G}}({\cal C}_{\mu};{\bf Q})^{\Gamma}.$ By hypothesis there exists
$\theta$ in ${\cal R}_{\mu}$ whose image in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$
equals $\eta e_{\mu}.$\\
\indent Kirwan shows \cite[p.867]{K} that the direct sum of
restriction maps
\begin{equation}
H^{*}_{\cal G}(V_{\mu};{\bf Q}) \rightarrow \bigoplus_{\nu \preceq \mu}
H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q}) \label{sumres}
\end{equation}
is injective. The images of $\theta$ and $\zeta$ under (\ref{sumres})
are equal and hence the restrictions of $\theta$ and $\zeta$ to
$H^{*}_{\cal G}(V_{\mu};{\bf Q})$ are the same, completing the proof. $\indent \Box$
\section{The Mumford and Zagier-Mumford Relations}
\indent The group $\overline{\cal G}$ acts freely on ${\cal C}^{s}$ and the $U(1)$-centre of
${\cal G}$ acts as scalar multiplication on the fibres of ${\cal V}$. The projective
bundle of ${\cal V}$ descends to a holomorphic projective bundle over
${\cal N} \times M$ which is the projective bundle of a universal
holomorphic bundle $V$ of rank two and odd degree \cite[p.580]{AB}.\\
\indent The bundle $V$ is universal in the sense that the restriction
of $V$ to $[E] \times M$ for any class $[E] \in {\cal N}$ is
isomorphic to $E$. Note V is not unique; we may tensor $V$ by the pullback
of a line bundle over ${\cal N}$ to produce a second bundle with the same
universal property. However we may normalise $V$ \cite[p.857, p.877]{K} by
requiring the relation
\begin{equation}
f_{2} = (d - 2 \bar{g})a_{1} + \sum_{s=1}^{g}
b_{1}^{s} b_{1}^{s+g}. \label{norm}
\end{equation}
\indent Let $\pi: {\cal N} \times M \to {\cal N}$ be the first projection. When $d
= 4g-3$ then any $E \in {\cal C}^{s}$ has slope $\mu(E) = d/n > 2\bar{g}$ and thus
\cite[lemma 5.2]{N2} $H^{1}(M,E) = 0$. Hence $\pi_{!}V$ is a genuine vector
bundle over ${\cal N}$ of rank $2g-1$ with fibre $H^{0}(M,E)$ over $[E] \in
{\cal N}$. We know from \cite[prop.9.7]{AB} that
\[
H^{*}_{\cal G}({\cal C}^{s};{\bf Q}) \cong H^{*}({\cal N}_{0};{\bf Q}) \otimes {\bf Q} [a_{1}] \otimes
\Lambda^{*}\{b_{1}^{1},...,b_{1}^{2g}\}.
\]
The Mumford relations $c_{r,S}$ $(r \geq 2g, S \subseteq [2g])$ are
then defined by writing
\begin{equation}
c_{r}(\pi_{!}V) = \sum_{S \subseteq [2g]} c_{r,S} \prod_{s \in S}
b_{1}^{s}, \label{first}
\end{equation}
where each $c_{r,S}$ is written in terms of generators for $H^{*}({\cal
N}_{0};{\bf Q}) \otimes {\bf Q} [a_{1}]$, namely $a_{1}$ and Newstead's
generators $\alpha, \beta, \psi_{s}$. In terms of
the generators (\ref{gen}) these are given by
\[
\alpha = 2f_{2} - da_{1},\indent \beta = (a_{1})^{2} - 4a_{2}, \indent
\psi_{s} = 2b_{2}^{s}.
\]
Kirwan's proof of Mumford's conjecture \cite[$\S$2]{K} shows that the
Mumford relations together with the normalising relation (\ref{norm})
form a complete set of relations for $H^{*}({\cal N}_{0};{\bf Q})$. Following
Kirwan \cite[p.871]{K} we reformulate the definition (\ref{first}) and
write
\[
\Psi(t) = \sum_{r=0}^{\infty} c_{r}(\pi_{V}) t^{2g-1-r} = \sum_{r=-\infty}^{\bar{g}}
(\sigma_{r}^{0} + \sigma_{r}^{1} t) (t^{2} + a_{1} t + a_{2})^{r},
\indent \sigma_{r}^{k} = \sum_{S \subseteq [2g]} \sigma_{r,S}^{k}
\prod_{s \in S} b_{1}^{s}.
\]
We will also refer to $\sigma_{r,S}^{k} (k=0,1, r<0, S \subseteq
[2g])$ as the Mumford relations. (Note $\sigma_{r}^{0}$ and
$\sigma_{r}^{1}$ differ slightly from Kirwan's terms $\sigma_{r}$ and
$\tau_{r}$.) This new formulation will prove more
convenient when we need to determine the restrictions of the Mumford
relations to various strata. Atiyah and Bott define generators
\[
a_{1}^{1}, a_{1}^{2}, \mbox{ and } b_{1}^{1,s}, b_{1}^{2,s}
\]
for $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ via the isomorphism \cite[prop.7.12]{AB}
\[
H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}) \cong H^{*}_{{\cal G}(1,d_{1})}({\cal C}(1,d_{1})^{ss};{\bf Q}) \otimes
H^{*}_{{\cal G}(1,d_{2})}({\cal C}(1,d_{2})^{ss};{\bf Q}).
\]
In terms of these generators the crucial calculation of Kirwan in her
proof of Mumford's conjecture is:
\begin{lemma}[Kirwan]
\rm{\cite[pp.871-873]{K}} {\em Let $\mu = (d_{1},d_{2})$ and write $D = d_{2} - 2g +1$. Then the
restrictions $\sigma_{D,S}^{k,\mu}$ of $\sigma_{D,S}^{k}$ $(k=0,1)$ in
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ are given by
\begin{equation}
\sigma_{D,S}^{0,\mu} = \frac{(-1)^{g\bar{g}/2}}{2^{2g} g!} \left( \prod_{s
\not \in S} (b_{1}^{2,s} - b_{1}^{1,s}) \right) a_{1}^{1} e_{\mu}, \indent
\sigma_{D,S}^{1,\mu} = \frac{(-1)^{g\bar{g}/2}}{2^{2g} g!} \left( \prod_{s
\not \in S} (b_{1}^{2,s} - b_{1}^{1,s}) \right) e_{\mu}. \label{Q61}
\end{equation}}
\end{lemma}
This calculation plays a major role in the following two theorems.
\begin{thm}
(\cite{B}, \cite{KN}, \cite{ST}, \cite{Z}.)
Each of the sets
\begin{eqnarray}
\{ \zeta_{g}, \zeta_{g+1}, \zeta_{g+2} \}, \label{Q43}\\
\{ c_{2g,[2g]}, c_{2g+1,[2g]}, c_{2g+2,[2g]} \}, \label{Q44} \\
\{ \sigma_{-1,[2g]}^{1}, \sigma_{-1,[2g]}^{0}, \sigma_{-2,[2g]}^{1}
\}, \label{Q45}
\end{eqnarray}
forms a minimal complete set of relations for the invariant cohomology
ring $H^{*}({\cal N}_{0};{\bf Q})^{\Gamma}$.
\end{thm}
{\bf PROOF:} From \cite[$\S$6]{Z} we know that the relations (\ref{Q43}) and
(\ref{Q44}) generate the same ideal of $H^{*}_{\cal G}({\cal C};{\bf Q})^{\Gamma}$. Since
\begin{eqnarray*}
c_{2g,[2g]} & = & \sigma_{-1,[2g]}^{1},\\
c_{2g+1,[2g]} & = & \sigma_{-1,[2g]}^{0} -a_{1} \sigma_{-1,[2g]}^{1},\\
c_{2g+2,[2g]} & = & \sigma_{-2,[2g]}^{1} -a_{1} \sigma_{-1,[2g]}^{0}
+((a_{1})^{2} -a_{2}) \sigma_{-1,[2g]}^{1},
\end{eqnarray*}
we can see that the relations (\ref{Q45}) also generate the same
ideal.\\
\indent Let $\mu = (d_{1},d_{2})$. Now $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})^{\Gamma}$ is generated by
\[
a_{1}^{1}, a_{1}^{2}, \xi_{1,1}^{1,1}, \xi_{1,1}^{1,2}+ \xi_{1,1}^{2,1},
\xi_{1,1}^{2,2},
\]
where $\xi_{1,1}^{i,j} = \sum_{s=1}^{g} b_{1}^{i,s}
b_{1}^{j,s+g}$. On restriction to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$
\[
a_{1} \mapsto a_{1}^{1} + a_{1}^{2}, \qquad b_{1}^{s} \mapsto b_{1}^{1,s}
+ b_{1}^{2,s}, \qquad f_{2} \mapsto d_{1}a_{1}^{2} + d_{2}a_{1}^{1} +
\xi_{1,1}^{1,2} + \xi_{1,1}^{2,1}.
\]
Hence $e_{\mu}H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})^{\Gamma}$ is generated by
\begin{equation}
\sum_{ \begin{array}{c} \mbox{\scriptsize $S \subseteq [g]$} \\[-2pt]
\mbox{ \scriptsize $|S| = k$} \end{array}} \hspace{-4mm}
\left( \prod_{s \in S}
b_{1}^{1,s}b_{1}^{1,s+g}-b_{1}^{2,s}b_{1}^{2,s+g} \right) (a_{1}^{1})^{i}
e_{\mu} \label{mess}
\end{equation}
for $i = 0,1,$ $0 \leq k \leq g$ and the restrictions of $a_{1}$,
$f_{2}$ and $\xi_{1,1} = \sum_{s=1}^{g} b_{1}^{s} b_{1}^{s+g}$. Let $P(S)$
denote the set of partitions $S$ into two sets $S_{1},S_{2}$. We then
see from lemma 2 that (\ref{mess}) above is the restriction of
\begin{equation}
\frac{1}{2^{k}} \hspace{-4mm} \sum_{ \begin{array}{c}
\mbox{\scriptsize $S \subseteq [g]$} \\[-2pt]
\mbox{ \scriptsize $|S| = k$} \end{array}} \hspace{-3mm}
\sum_{P(S)} (\pm)
\left( \prod_{s \in S_{1}} b_{1}^{s} \right)
\left( \prod_{s \in S_{2}} b_{1}^{s+g} \right)
\sigma^{i}_{D,[2g] - (S_{2} \cup (S_{1}+g))} \label{mess2}
\end{equation}
where the sign $(\pm)$ depends on the particular partition of
$S$. Thus by proposition \ref{ICC} the relations (\ref{mess2}) above
for $D < 0,$ $i=0,1,$ $0 \leq k \leq g$ generate the invariant relation ideal of
\[
H^{*}_{\cal G}({\cal C}^{s};{\bf Q}) \cong H^{*}({\cal N}_{0};{\bf Q}) \otimes
\Lambda^{*}\{b_{1}^{1}, \ldots ,b_{1}^{2g}\} \otimes {\bf Q}[a_{1}].
\]
In particular the relations
\[
\left\{ \sigma_{D,[2g]}^{i} : D<0,i=0,1 \right\}
\]
generate $H^{*}({\cal N}_{0};{\bf Q})^{\Gamma}$. It follows from (\ref{A})
that the sets (\ref{Q43}),
(\ref{Q44}) and (\ref{Q45}) each form a complete set of relations for
the invariant cohomology ring of ${\cal N}_{0}$.\\
\indent Minimality then follows easily. Suppose that for some $\eta,
\theta \in H^{*}({\cal N}_{0};{\bf Q})^{\Gamma}$ we have
\begin{equation}
\sigma_{-2,[2g]}^{1} + \eta \sigma_{-1,[2g]}^{0} + \theta
\sigma_{-1,[2g]}^{1} = 0. \label{Q58}
\end{equation}
So $\eta$ has degree 2 and $\theta$ has degree 4.\\
\indent Let $\mu = (2\bar{g}+1,2\bar{g})$. Restricting equation (\ref{Q58}) to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ we find from (\ref{Q61}) that
\begin{equation}
\eta_{\mu} a_{1}^{1} + \theta_{\mu} = 0 \label{Q59}
\end{equation}
since $e_{\mu}$ is not a zero-divisor in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$. The
restriction map
\begin{equation}
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}) \label{Q60}
\end{equation}
is given by
\[
a_{1} \mapsto a_{1}^{1} + a_{1}^{2}, \indent a_{2} \mapsto a_{1}^{1}
a_{1}^{2}, \indent f_{2} \mapsto 2\bar{g} a_{1}^{1} + (2\bar{g}+1) a_{1}^{2}
+ \xi_{1,1}^{1,2} + \xi_{1,1}^{2,1},
\]
\[
b_{1}^{s} \mapsto b_{1}^{1,s} + b_{1}^{2,s}, \indent b_{2}^{s} \mapsto
a_{1}^{2} b_{1}^{1,s} + a_{1}^{1} b_{1}^{2,s}.
\]
From (\ref{Q59}) we see that $\eta_{\mu}$ and $\theta_{\mu}$ are both
zero. Since the restriction map (\ref{Q60}) is injective in degrees 4
and less, we have that $\eta$ and $\theta$ are both zero -- which
contradicts (\ref{Q58}).\\
\indent Similarly the equation
\[
\sigma_{-1,[2g]}^{0} + \eta \sigma_{-1,[2g]}^{1} = 0
\]
has no solutions for $\eta \in H^{2}_{{\cal G}}({\cal C};{\bf Q})$. \indent $\Box$
\section{Mumford's Conjecture}
The final result of this note is a stronger version of Mumford's
conjecture as proven by Kirwan \cite[$\S$2]{K} and which is
confusingly also referred to as Mumford's conjecture.\\
\indent The Poincar\'{e} polynomial of the relation ideal of
$H^{*}({\cal N}_{0};{\bf Q})$ equals \cite[p.593]{AB}
\[
\frac{t^{2g}(1+t)^{2g}}{(1-t^{2})(1-t^{4})}.
\]
Now there are ${2g \choose r}$ relations of the form $c_{2g,S}$ of degree
$2g+r$; $\alpha$ has degree two, $\beta$ has degree four and neither
are nilpotent in $H^{*}_{\cal G}({\cal C};{\bf Q})$. This strongly suggests:
\begin{thm}
The relation ideal of $H^{*}({\cal N}_{0};{\bf Q})$ is freely generated as a
${\bf Q}[\alpha,\beta]$-module by the Mumford relations $c_{2g,S}$ for $S
\subseteq [2g]$.
\end{thm}
{\bf PROOF:} Define
\[
\bar{\alpha} = \alpha - \sum_{s=1}^{g} b_{1}^{s} b_{1}^{s+g} = 2f_{2} - da_{1}
- \sum_{s=1}^{g} b_{1}^{s} b_{1}^{s+g}.
\]
We will show that the relations $c_{2g,S}$ generate the relation ideal
of $H^{*}_{\cal G}({\cal C}^{s};{\bf Q})$ as a ${\bf Q}[\bar{\alpha},\beta]$-module. As $\bar{\alpha}$ restricts to
$\alpha$ in $H^{*}({\cal N}_{0};{\bf Q})$ then this is equivalent to
the above result. It will be sufficient to prove that
\begin{equation}
\sum_{S \subseteq [2g]} \lambda_{S}(\bar{\alpha},\beta) c_{2g,S} = 0
\indent \lambda_{S}(\bar{\alpha},\beta) \in {\bf Q}[\bar{\alpha},\beta] \label{Q53}
\end{equation}
in $H^{*}_{\cal G}({\cal C};{\bf Q})$ if and only if $\lambda_{S}(\bar{\alpha},\beta) = 0$
for each $S \subseteq [2g].$\\
\indent Let $\mu = (2\bar{g}+1,2\bar{g})$. Then from lemma 2 we know that the
restriction of $c_{2g,S} = \sigma_{-1,S}^{1}$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
\frac{(-1)^{g\bar{g}/2}}{2^{2g} g!} \left( \prod_{s
\not \in S} (b_{1}^{2,s} - b_{1}^{1,s}) \right) e_{\mu}.
\]
If we restrict equation (\ref{Q53}) to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ and recall that
$e_{\mu}$ is not a zero-divisor in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ \cite[p.569]{AB} we obtain
\[
\sum_{S \subseteq [2g]} \lambda_{S}(\bar{\alpha}_{\mu},\beta_{\mu}) \left( \prod_{s \not
\in S} (b_{1}^{2,s}-b_{1}^{1,s}) \right) = 0.
\]
Now the restrictions of $\bar{\alpha}$ and $\beta$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equal
\begin{equation}
\bar{\alpha}_{\mu} = (a_{1}^{2}-a_{1}^{1}) - \sum_{s=1}^{g}
(b_{1}^{1,s} - b_{1}^{2,s})(b_{1}^{1,s+g}-b_{1}^{2,s+g}), \indent
\beta_{\mu} = (a_{1}^{2} - a_{1}^{1})^{2}. \label{Q54}
\end{equation}
By comparing the coefficients of $\prod_{s \in S} (b_{1}^{2,s} -
b_{1}^{1,s})$ for each $S \subseteq [2g]$ we see that
\[
\lambda_{S}(\bar{\alpha}_{\mu},\beta_{\mu}) = 0 \indent S \subseteq [2g].
\]
\indent Consider the restriction map
\begin{equation}
{\bf Q}[\bar{\alpha},\beta] \rightarrow {\bf Q}[\bar{\alpha}_{\mu},\beta_{\mu}]. \label{Q55}
\end{equation}
From the expressions (\ref{Q54}) of $\bar{\alpha}_{\mu}$ and $\beta_{\mu}$
we can see that the kernel of the restriction map (\ref{Q55}) is
the ideal of ${\bf Q}[\bar{\alpha},\beta]$ generated by
\[
(\bar{\alpha}^{2} - \beta)^{g+1}.
\]
However $\bar{\alpha}^{2} - \beta$ is not a zero-divisor in
$H^{*}_{\cal G}({\cal C};{\bf Q})$. So we can assume without any loss of generality that for
some $S,$ $\lambda_{S}$ is either zero or not in the ideal generated by
$\bar{\alpha}^{2}-\beta$. For this $S$ we have $\lambda_{S} = 0$ since
$\lambda_{S}(\bar{\alpha}_{\mu},\beta_{\mu}) = 0$. Inductively we can see that
\[
\lambda_{S}(\bar{\alpha},\beta) = 0 \mbox{ for } S \subseteq [2g]. \indent \Box
\]
|
1996-03-25T06:20:23 | 9603 | alg-geom/9603018 | en | https://arxiv.org/abs/alg-geom/9603018 | [
"alg-geom",
"math.AG"
] | alg-geom/9603018 | Dan Abramovich | Dan Abramovich and Johan de Jong | Smoothness, Semistability, and Toroidal Geometry | 12 pages (in large font)., LATEX 2e (in latex 2.09 compatibility
mode) | null | null | null | null | We provide a new proof of the following result:
Let $X$ be a variety of finite type over an algebraically closed field $k$ of
characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a
modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular
variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings.
Needless to say, this theorem is a weak version of Hironaka's well known
theorem on resolution of singularities. Our proof has the feature that it
builds on two standard techniques of algebraic geometry: semistable reduction
for curves, and toric geometry.
Another proof of the same result was discovered independently by F. Bogomolov
and T. Pantev. The two proofs are similar in spirit but quite different in
detail.
| [
{
"version": "v1",
"created": "Sun, 24 Mar 1996 01:02:33 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Abramovich",
"Dan",
""
],
[
"de Jong",
"Johan",
""
]
] | alg-geom | \section{INTRODUCTION}
\subsection{Statement} We provide a new proof of the following result:
\begin{th}[Hironaka]\label{resolution}
Let $X$ be a variety of finite type over an algebraically closed field $k$
of characteristic 0, let $Z\subset X$ be a proper closed subset.
There exists a modification $f:X_1 \rightarrow} \newcommand{\dar}{\downarrow X$,
such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 =
f^{-1}(Z)_{\mbox{\small red}}$
is a strict divisor of normal crossings.
\end{th}
\begin{rem}
Needless to say, this theorem is a weak version of Hironaka's well known
theorem on resolution of singularities. Our proof has the feature
that it builds on two standard techniques of algebraic geometry: semistable
reduction for curves, and toric geometry.
\end{rem}
\begin{rem}
Another proof
of the same result was
discovered independently by F. Bogomolov and T. Pantev \cite{bp}.
The two proofs are similar in spirit but quite different in detail.
\end{rem}
\subsection{Structure of the proof}
\begin{enumerate} \item As in \cite{dj}} \newcommand{\do}{\cite{d-o}, we choose a projection $X\das P$ of relative
dimension 1, and apply semistable reduction to obtain a model $X'\rightarrow} \newcommand{\dar}{\downarrow B$ over a
suitable Galois base change $B\rightarrow} \newcommand{\dar}{\downarrow P$, with Galois group $G$.
\item We apply induction on the dimension to $B$ , therefore we may assume that
$B$ is smooth, and that the discriminant locus of $X'\rightarrow} \newcommand{\dar}{\downarrow B$ is a $G$-strict
divisor of normal crossings.
\item A few auxiliary blowups make the quotient $X'/G$ toroidal.
\item Theorem 11* of \cite{te} about toroidal resolutions finishes the
argument.
\end{enumerate}
\subsection{What we do not show}
\subsubsection{Canonicity}
Our proof has the drawback that the resolution
is noncanonical. Some of the steps are not easily carried out in practice, and
in fact we almost always blow up in the smooth locus. However, it follows
from the proof, that if $C\subset X_{ns}$ is a curve, then we can guarantee
that $X'\rightarrow} \newcommand{\dar}{\downarrow X$ is an isomorphism in a neighborhood of $C$. We do expect a
slight modification of our argument to give equivariant resolution of
singularities in the case where a finite group acts.
\subsubsection{Positive characteristic} \label{char-p-bad}
There is one crucial point where the proof fails if ${\operatorname{char}}\ k =
p>0$. This is at the point where we claim that the quotient $X'/G$ is
toroidal. Given $x\in X'$ and $g\in {\operatorname{Stab}}\ x$ of order $p$, the action of $g$
on
${\cal{O}}} \newcommand{\I}{{\cal{I}}_{X',x}$ is unipotent. Thus even if $X'$ is toroidal, we cannot guarantee
that the quotient is toroidal as well. It might happen that
the quotient is toroidal by accident, and it would be interesting to see to
what extent such accidents can be encouraged to happen.
In any case, the quotient step goes through if $p \not| \#G$. See remark
\ref{char-p} for a discussion of a bound on $\#G$. As a result, there exists a
function
$$M:\left\{\mbox{\parbox{6cm}{\begin{center}varieties with subvarieties
\end{center}}}\right\} \longrightarrow} \newcommand{\das}{\dashrightarrow {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$$
which is bounded on any bounded
family, and which is ``describable'' in a geometrically meaningful way ($M$ for
``multi-genus''), such
that whenever $p>M([X\supset Z])$, our proof goes through for the pair
$X\supset Z$. We were informed by T. Scanlon and E. Hrushovski that the
existence of a function $M$ which is bounded on bounded families is known, by
an application of the compactness theorem in model theory, for any resolution
process, in particular Hironaka's. A proof of this was given in \cite{ek}.
\subsection{Acknowledgements} We would like to thank B. Hassett, Y. Karshon,
S. Katz, D. Rohrlich, T. Scanlon, and M. Spivakovsky for discussions relevant
to the subject of this paper.
\subsection{Terminology}
We recall some definitions; we restrict ourselves to the case
of varieties over $k$.
A {\bf modification} is a proper birational morphism of irreducible varieties.
An {\bf alteration} $a:B_1\rightarrow} \newcommand{\dar}{\downarrow B$ is a proper, surjective, generically finite
morphism of irreducible varieties, see \cite[2.20]{dj}.
The alteration $a$ is a {\bf Galois
alteration} if there is a finite group $G\subset {\operatorname{Aut}}_B (B_1)$ such that
the associated morphism $B_1/G\rightarrow} \newcommand{\dar}{\downarrow B$ is birational, compare \cite[5.3]{dj2}.
Let a finite group $G$ act on a (possibly reducible) variety $Z$. Let $Z=\cup
Z_i$ be the decomposition of $Z$ into irreducible components. We say that
{\bf $Z$ is $G$-strict} if the union of translates $\cup_{g\in G} g(Z_i)$ of
each component $Z_i$ is
a normal variety. We simply say that $Z$ is
{\bf strict} if it is $G$-strict for the trivial group, namely every $Z_i$ is
normal.
A divisor $D\subset X$ is called a {\bf divisor of normal crossings} if \'etale
locally at every point it is the zero set of $u_1\cdots u_k$ where
$u_1,\ldots,u_k$ is part of a regular system of parameters. Thus a strict
divisor of normal crossings is what is usually called a divisor of strict
normal crossings,
i.e., all components of $D$ are nonsigular.
An open embedding $U\hookrightarrow X$ is called a {\bf toroidal embedding} if
locally in the \'etale topology (or classical topology in case $k=\bfc$,
or formally) it is
isomorphic to a torus embedding $T \hookrightarrow V$, (see \cite{te}, II\S
1). If $D=X\setminus U$, we will sometimes denote this toroidal embedding by
$(X,D)$. A finite group action
$G\subset {\operatorname{Aut}}(U\hookrightarrow X)$ is said to be {\bf toroidal} if the
stabilizer of every point is identified on the appropriate neighborhood with a
subgroup of the torus $T$. We say that a toroidal action is {\bf $G$-strict} if
$X\setminus U$ is $G$-strict. In particular the toroidal embedding itself is
said to be strict if $X\setminus U$ is strict. This is the same as the notion
of {\bf
toroidal embedding without self-intersections} in \cite{te}.
The fundamental theorem about toroidal embeddings we will exploit is the
following:
\begin{th}[\cite{te}, II \S 2, Theorem 11$^*$,
p. 94]\label{mumford} For any strict
to\-ro\-idal e\-m\-be\-dding
$ U\hookrightarrow X$
there exists a canonical
sheaf of ideals $\I \subset {\cal{O}}} \newcommand{\I}{{\cal{I}}_X$ such that the blowup $B_\I(X)$ is
nonsingular.
\end{th}
(See \cite{te}, II \S 2, definition 1 for the notion of canonical
modification.)
\section{The proof}
\subsection{Reduction steps}
We argue by induction on $d=\dim X$. The case $d=1$ is given
by normalizing $X$.
As in \cite{dj}, 3.6-3.10 we may assume that
\begin{enumerate} \item \em $X$ is projective and normal, and
\item $Z$ is the support of a divisor.
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
Recall that according to \cite{dj}, lemma 3.11 there is a modification $X'\rightarrow} \newcommand{\dar}{\downarrow
X$ and a generically smooth
morphism $f:X'\rightarrow} \newcommand{\dar}{\downarrow P={\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^{d-1}$ satisfying the following properties:
\begin{enumerate}
\setcounter{enumi}{\value{cumulative}} \it
\item The modification $X'\rightarrow} \newcommand{\dar}{\downarrow X$ is
isomorphic in a neighborhood of $Z$,
\item every fiber of $f$ is of pure dimension 1,
\item the smooth locus of $f$ is dense in every fiber, and
\item the morphism $f_{|_{Z}}:Z\rightarrow} \newcommand{\dar}{\downarrow P$ is finite.
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
Since the construction in \cite{dj}} \newcommand{\do}{\cite{d-o}\ is obtained via a general
projection, we can guarantee that
\begin{enumerate}
\setcounter{enumi}{\value{cumulative}}
\item {\it
the generic fiber of $f:X' \rightarrow} \newcommand{\dar}{\downarrow P$ is
a geometrically connected curve} (this follows from \cite{jou}, 6.3(4), see
\cite{a-pluri}, 4.2).
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
We now replace $X$ by $X'$.
Applying Lemma 3.13 of \cite{dj}, we may choose a divisor $D\subset X$
mapping
finitely to $P$, which meets every component of every fiber of $f$ in at least
3 smooth points. We may replace $Z$ by $Z\cup D$ (see \cite[3.9]{dj}), and
therefore we have
\begin{enumerate}\it
\setcounter{enumi}{\value{cumulative}}
\item $Z$ meets every component of every fiber of $f$ in at least
3 smooth points.
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
Let $\Delta_X\subset P$ be the discriminant locus of the map $X\rightarrow} \newcommand{\dar}{\downarrow P$, and
let
$\Delta_Z\subset P$ be the discriminant locus of the map $Z\rightarrow} \newcommand{\dar}{\downarrow P$. Let
$\Delta=\Delta_Z\cup \Delta_X$. We can think of $\Delta$ as the discriminant
locus of the pair $Z\subset X\rightarrow} \newcommand{\dar}{\downarrow P$.
The inductive assumption gives us a resolution of singularities of
$\Delta\subset P$. Thus we may replace $X,Z,P$ and
assume that $P$ is smooth and $\Delta$ is a strict divisor of normal crossings.
Let $\nu:X^{\rm nor}\rightarrow} \newcommand{\dar}{\downarrow X$ be the normalization. By \cite{d-o}, the discriminant
locus
of $\nu^{-1}Z\subset X^{\rm nor}\rightarrow} \newcommand{\dar}{\downarrow P$ is a divisor $\Delta'$ contained in
$\Delta$. We
replace $X$ by $X^{\rm nor}$ and $\Delta$ by $\Delta'$. Thus we may assume in
addition to {\bf 1} - {\bf \thecumulative}, that
\begin{enumerate}\it
\setcounter{enumi}{\value{cumulative}}
\item $P$ is smooth, and
\item $\Delta$ is a strict normal crossings divisor.
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
\subsection{Stable reduction}
\renewcommand{\theenumi}{{\bf\alph{enumi}}}
We are now ready to perform stable reduction.
We follow \cite{dj}, 3.18-3.21, but see Remark \ref{opmerking} below.
Let $j:U_0 = P\setminus\Delta\hookrightarrow P$.
Let $a_U: U_0' \rightarrow} \newcommand{\dar}{\downarrow U_0$ be an \'etale Galois cover
which splits the projection
$Z_{U_0} =j^{-1}(U_0)\cap Z \rightarrow} \newcommand{\dar}{\downarrow U_0$ into $n$ sections, and trivializes
the 3-torsion subgroup in the
relative Jacobian of $X_{U_0}\rightarrow} \newcommand{\dar}{\downarrow U_0$. Let $G$ be the
Galois group of this cover. Let $P^\sharp$ be the normalization of $P$ in the
function field of $U_0'$. Let $g$ be the
genus of the generic fiber of $X\rightarrow} \newcommand{\dar}{\downarrow P$. Let us write
$\overline{{}_3{\bf M}_{g,n}}$ for the Deligne-Mumford compactification
of the moduli scheme of $n$-pointed genus $g$ curves with an
abelian level 3 structure, see \cite{DM},
\cite[2.3.7]{PJ}, \cite[2.24]{dj} and references therein.
There is a `universal'
stable $n$-pointed curve over $\overline{{}_3{\bf M}_{g,n}}$.
We take the closure of the graph of the morphism
$U_0'\rightarrow} \newcommand{\dar}{\downarrow \overline{{}_3{\bf M}_{g,n}}$ in
$P^\sharp\times \overline{{}_3{\bf M}_{g,n}}$ and obtain a
modification $P'\rightarrow} \newcommand{\dar}{\downarrow P^\sharp$ and a family of
stable pointed curves $X' \rightarrow} \newcommand{\dar}{\downarrow P'$. Note that $P'\to P^\sharp$
blows up outside of $U_0'$. We perform a $G$-equivariant blow up
of $P'$ to ensure that we have a morphism $r : X'\to X\times_PP'$,
see \cite[3.18, 3.19 and 7.6]{dj}. Again this blow up has center
outside $U_0'$. In summary:
\begin{situ}\label{situatie}
There is a Galois
alteration $a:P'\rightarrow} \newcommand{\dar}{\downarrow P$, with Galois group $G$, and a modification
$r:X'\rightarrow} \newcommand{\dar}{\downarrow X\times_P P'$ satisfying:
\begin{enumerate} \it
\item the morphism $a$ is finite \'etale over $U_0$,
\item the morphism $r$ is an isomorphism over the open set $U_0\times_P P'$,
\item there are $n$ sections $\sigma_i:P'\rightarrow} \newcommand{\dar}{\downarrow X'$ such that the proper
transform $Z'$
of $Z$ is the union of their images, and
\item $(X'\rightarrow} \newcommand{\dar}{\downarrow P',\sigma_1,\ldots,\sigma_n)$ is a stable pointed curve of genus
$g$.
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
\end{situ}
\begin{rem}\label{opmerking}
The results of \cite{d-o} imply that the curve $X'$ exists
over the variety $P^\sharp$. We suspect
that the morphism $r$ exists over $P^\sharp$ as well. If this is
true, the following step is redundant.
See \cite{a-pluri} for a similar construction
using Kontsevich's space of stable maps.
\end{rem}
We want to change the situation such that we get $P'=P^\sharp$.
This we can do as follows. Take a blow up $\beta:P_1\to P$ such that
the strict transform $P_1'$ of $P'$ with respect to $\beta$ is finite
flat over $P_1$, see \cite{RG}. By our inductive assumption, we may assume that
$P_1$ is nonsingular and that the inverse image $\Delta_1$ of $\Delta$
in $P_1$ is a divisor with normal crossings. The Galois covering
$U_0'\to U_0$ pulls back to a Galois covering of $P_1\setminus \Delta_1$,
and we get a ramified normal covering $P_1^\sharp$ of $P_1$.
It follows that $P_1^\sharp$ maps to the strict transform $P_1'$.
Hence it is clear that if we replace $P$ by $P_1$, $\Delta$ by $\Delta_1$,
$X'$ by $X'\times_P' P_1^\sharp$, then the morphism $r$ of \ref{situatie}
exists over $P_1^\sharp$.
\renewcommand{\thecumulative}{{\bf\alph{cumulative}}}
We now have in addition to ({\bf a}) - (\thecumulative),
\begin{enumerate}\it
\setcounter{enumi}{\value{cumulative}}
\item $P$ is smooth, $\Delta$ is a strict divisor of normal crossings, and
$a:P'\rightarrow} \newcommand{\dar}{\downarrow P$ is finite.
\setcounter{cumulative}{\value{enumi}}
\end{enumerate}
Since stable reduction over a normal base is unique (see \cite{d-o}, 2.3)
we have that the action of $G$ lifts to $X'$. Note that the $G$-action
does not preserve the order of the sections.
Note also that since the local fundamental groups of $P\setminus\Delta$
are abelian, so is the stabilizer in $G$ of any point in $P'$. The same
is true for points on $X'$. This property will therefore remain true for
any equivariant modification of $P'$, as well as for points on $X'$.
\subsection{Local description}\label{locdescr}
Denote by $q:P'\rightarrow} \newcommand{\dar}{\downarrow P$ the quotient map. Let $p\in P $ and let $p'\in
q^{-1}(p)$. Let $s_1,\ldots,s_{d-1}\in {\cal O}_{P,p}$ be a regular system
of parameters on $P$ at $p$ such that $\Delta_x= V(s_1\cdots s_r)$.
Recall that the stabilizer of $p'$ is abelian; this actually
follows from {\bf (a)-(e)} above as the morphism
$P'\rightarrow} \newcommand{\dar}{\downarrow P$ is ramified only
along the divisor of normal crossings $\Delta$.
Writing $t_i^n = s_i$ for suitable $n$, we
identify a formal neighbourhood of $p'$ in $P'$ as a quotient of the smooth
formal scheme $P''={\operatorname{Spf }}\ k[[t_1,\ldots t_{r},s_{r+1},\ldots,s_{d-1}]]$ by the
finite
group $({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}/n{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})^r$ acting by $n$-th roots of unity on the $t_i$.
Thus a formal neighbourhood
of $p'$ in $P'$ is the quotient of $P''$ by a suitable subgroup $H\subset
({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}/n{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})^r$. This implies that $({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}/n{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})^r/H$ is identified
with the stabilizer of $p'$ in $G$.
Denote by $X''= P''\times_{P'} X'\rightarrow} \newcommand{\dar}{\downarrow P''$ the
pulled back stable pointed curve. As
$({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}/n{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})^r/H \subset G $ acts on $X'$ over
$P'$, we get an action of $({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}/n{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})^r$ on $X''$ over $P''$.
Let $x\in X''$ be a closed point lying over $p'$, and let $G_x$ be the
stabilizer of $x$ in $({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}/n{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})^r$. There are two cases:
\renewcommand{\theenumi}{{\bf\roman{enumi}}}
\begin{enumerate}
\item\label{smooth} (Smooth case) Here $x$ is a smooth point of the
morphism $X''\to P''$. In this case the completion of $X''$ at $x$ is
isomorphic to the formal spectrum of the $k[[t_1,\ldots
t_{r},s_{r+1},\ldots,s_{d-1}]]$-algebra
$$k[[t_1,\ldots t_{r},s_{r+1},\ldots,s_{d-1}]][[x]].$$
Here we chose some coordinate $x$ along
the fiber such that $G_x$ acts by a character $\psi_x$ on $x$.
There are two cases with respect to the position of the sections
$Z''\subset X''$:
\begin{enumerate} \item The point $x\in Z''$. In this case, since $Z''$ is
invariant under the action of $G_x$, we can choose the coordinate $x$ so that
$Z'' = V(x)$.
\item\label{totorify} The point $x\not\in Z''$. Note that the coordinate $x$
is not
uniquely chosen, and therefore the locus $x=0$ is not uniquely determined.
However, in case $\psi_x$ is nontrivial if we want to see $G_x$ as acting
through the torus for some toroidal structure on $X''$ at $x$
it is necessary to include a locus like $V(x)$ in the boundary.
\end{enumerate}
\item\label{nodal} (Node case) Here $x$ is a node of the fiber of
$X''\to P''$ over $p''$. In this case the completion of $X''$ at $x$ is
isomorphic to the formal spectrum of the $k[[t_1,\ldots
t_{r},s_{r+1},\ldots,s_{d-1}]]$-algebra
$$k[[t_1,\ldots t_{r},s_{r+1},\ldots,s_{d-1}]]
[[x,y]]/(xy - t_1^{k_1}\cdots t_r^{k_r}).$$
We may choose the coordinates $x,y$ such that there
is a subgroup $G_d$ of $G_x$ of index at most $2$ such that $G_d$ acts
by characters on $x,y$, but elements of $G_x$ not in $G_d$
switch the fiber components $x=0$ and $y=0$.
\end{enumerate}
We would like to have the stabilizers acting toroidally on $X''$ in such a way
that the
quotient $X$ becomes strict toroidal.
\subsection{Making the group act toroidally}
There are two issues we need to resolve: in case (\ref{nodal}) above, we want
to modify so that
$G_x/G_d$ disappears from the picture. In case (\ref{totorify}) we
need to modify so that the stratum $x=0$ is not necessary for the toroidal
description of the $G$-action.
In order to describe a global modification we go back to our stable pointed
curve $f:X'\rightarrow} \newcommand{\dar}{\downarrow P'$.
\subsubsection{Separating branches along nodes}\label{Sepban}
Let $S=\operatorname{Sing} f$ be the singular scheme of the projection $f$.
Let $Y' = B_S(X')\rightarrow} \newcommand{\dar}{\downarrow X'$ be the blowup of $X'$
along $S$. Let $Y''$ be the fiber product
$Y'\times_{P'} X''$, and let $y''\in Y''$. We remark that neither
$Y'$ nor $Y''$ is normal in general.
We want to give a local description of $Y'$ and $Y''$. We use the
notation $X_{/x}$ to denote the completion of $X$ at the closed point
$x\in X(k)$, and similar for the other varieties occuring below.
We have $P'_{/p'}={\operatorname{Spf }}\ R$ where $R$ is the ring
$(k[[t_1,\ldots t_{r},s_{r+1},\ldots,s_{d-1}]])^H$, with $H$ as in
\ref{locdescr} and we
have $X'_{/x'}= {\operatorname{Spf }}\ R[[x,y]]/(xy-h)$ for some monomial $h\in R\subset
k[[t_1,\ldots t_{r},s_{r+1},\ldots,s_{d-1}]]$.
Finally $S=V(x,y)$ scheme theoretically.
{From} this it is easy to read off the following local description,
using that blowing up commutes with completion
in a suitable manner.
\renewcommand{\theenumi}{{\bf\roman{enumi}'}}
\begin{enumerate}
\item\label{smooth'} (Smooth case) $Y''_{/y''}\simeq X''_{/x''}$ as above.
\item\label{nodal'} (Node case) $Y''_{/y''}\simeq {\operatorname{Spf }}\ k[[t_1,\ldots
t_{r},s_{r+1},\ldots,s_{d-1}]][[x,z]]/(xz^2 - t_1^{l_1}\cdots
t_r^{l_r})$. The stabilizer $G_{y''}$ acts diagonally on $t_i, x,z$ (i.e.,
it acts via characters on these elements).
\item\label{double'} (Double case) $Y''_{/y''}\simeq {\operatorname{Spf }}\ k[[t_1,\ldots
t_{r},s_{r+1},\ldots,s_{d-1}]][[x,z]]/(z^2 - t_1^{l_1}\cdots
t_r^{l_r})$. The stabilizer $G_{y''}$ acts
diagonally, but as
in (\ref{totorify}) the coordinate $x$ may not be determined.
\end{enumerate}
The descriptions above determine the local structure of $Y'$ also.
Indeed, $Y'_{/y'}$ is simply the quotient of $Y''_{/y''}$ by the
group $H$ which acts trivially on the coordinate(s) $x$ (and $z$).
We are actually interested in a local description of the normalization
$Y=(Y')^{\rm nor}$ of $Y'$.
Let $D\subset Y=(Y')^{\rm nor}$ be the union of the
inverse image of $\Delta$ and of $Z$ in $Y$.
{From} the local descriptions given above it is already clear
that $Y\setminus D \hookrightarrow Y$ is a strict toroidal embedding.
Moreover, $D$ is $G$-strict, since $\Delta$ is
strict and since the blowup $Y\rightarrow} \newcommand{\dar}{\downarrow X'$ separates fiber components.
However, the action of $G$ on the pair $(Y\setminus D, Y)$
is not yet toroidal: indeed, in case (\ref{smooth'}) and
in case (\ref{double'})
if the character $\psi_x$ is nontrivial, we have a problem.
More precisely, this is the situation explained in (\ref{totorify}).
\subsubsection{Torifying a pre-toroidal action}\label{Torlta}
Here we show how to do one canonical blow up $Y_1\rightarrow} \newcommand{\dar}{\downarrow Y$
(analogousely to \cite{te}, II \S 2) which makes the action of $G$ toroidal.
The situation $(Y, D, G)$ we reached at the end of
Section \ref{Sepban} is summarized by the conditions in Defintion
\ref{pretoroidal} below.
\begin{rem}
We expect that this discussion should be of interest in a more general
setting.
\end{rem}
\begin{pdfn}\label{pretoroidal} Let $U=Y\setminus D\subset Y$ be a toroidal
embedding, $G\subset
{\operatorname{Aut}}(U\subset
Y)$ a finite subgroup, such that $D$ is $G$-strict. For any point $y\in Y$
denote the stabilizer of $y$ by $G_y$. We say that the action of $G$ is {\bf
pre-toroidal} if at every point $y\in Y$, either $G_y$ acts toroidally at $y$,
or the following situation holds:
\begin{itemize}
\item There exists an isomorphism $\epsilon: Y_{/y}\cong {\operatorname{Spf }}\ R[[x]]$,
\item where ${\operatorname{Spf }} R$ is the completion of a toroidal embedding
$T_0\hookrightarrow Y_0$ at a point $y_0\in Y_0$,
\item where $D_{/y}$ corresponds to
$(Y_0\setminus T_0)_{/y_0}\times_{{\operatorname{Spf }}\ k} {\operatorname{Spf }}\ k[[x]]$,
\item where $G_y$ acts toroidally
on $T_0\hookrightarrow Y_0$ fixing $y_0$
and
\item $G_y$ acts on the coordinate $x$ via a character $\psi_x$ such that the
isomorphism $\epsilon$ of completions is $G_y$-equivariant. In other words,
$G_y$ acts
toroidally on $$(T_0)_{/y_0}\times_{{\operatorname{Spf }}\ k}{\operatorname{Spf }}\ k[[x, x^{-1}]]\quad \subset
\quad
Y_{/y}.$$
\end{itemize}
\end{pdfn}
Analogousely to definition 1 of \cite{te}, II \S 2, we define pre-canonical
ideals:
\begin{pdfn}
Let $G\subset {\operatorname{Aut}}(U\subset Y)$ be a pre-toroidal action. A $G$-equi\-va\-riant
ideal sheaf
$\I\subset {\cal{O}}} \newcommand{\I}{{\cal{I}}_Y$ is said to be {\bf pre-canonical} if the following holds:
For any $y, y'$ lying on
the same stratum, and any isomorphism $\alpha: O_{Y_{/y}} \rightarrow} \newcommand{\dar}{\downarrow O_{Y_{/y'}}$
preserving the strata and inducing an isomorphism carrying $G_y$ to $G_{y'}$,
we have $\alpha(\I_{/y}) =\I_{/y'}$.
If $\I$ is pre-canonical, we say that the normalized blowup
$b:(B_\I Y)^{\rm nor}\rightarrow} \newcommand{\dar}{\downarrow Y$ is a pre-canonical blowup.
\end{pdfn}
\begin{pdfn}
We say that a pre-canonical blowup $\tilde{Y} \rightarrow} \newcommand{\dar}{\downarrow Y$ {\bf torifies} $Y$ if
$G$ acts toroidally on $(b^{-1} U \subset \tilde{Y})$.
\end{pdfn}
\begin{th}\label{opblazen}Let $G\subset {\operatorname{Aut}}(U\subset Y)$ be a pre-toroidal
action. Then there
exists a canonical choice of a pre-canonical ideal sheaf $\I_G$ such that the
pre-canonical blowup $b:(B_{\I_G} Y)^{\rm nor}\rightarrow} \newcommand{\dar}{\downarrow Y$ torifies $Y$.
\end{th}
The theorem follows immediately from the affine case below:
\begin{prp}\label{affineopblazen}
Let $T_0\subset X_0$ be an affine torus embedding, $X_0 = {\operatorname{Spec }}\
R$.
Let $G\subset T_0$ be a finite subgroup of $T_0$, let $p_0\in X_0$ be
a fixed point of the action of $G$, and let $\psi_x$ be a character
of $G$. Consider the torus embedding of $T=T_0\times {\operatorname{Spec }}\ k[x,x^{-1}]$
into $X= X_0\times {\operatorname{Spec }}\ k[x]$, where we let $G$ act
on $x$ via the character $\psi_x$. Assume that the map
$G\to T$ induced from this is injective. Write $p=(p_0,0)\in X$ and
write $D=(X_0\setminus T_0) \times {\operatorname{Spec }}\ k[x]$.
There is a canonical $T$-equivariant ideal $I_G\subset R[x]$,
satisfying the following:
Let $b:X_1 = (B_{I_G}X)^{\rm nor}\rightarrow} \newcommand{\dar}{\downarrow X$, the normalization of the blowup of $X$
along $I_G$. Let $U_1=b^{-1}(T_0\times {\operatorname{Spec }}\ k[x])$. Then
$U_1\hookrightarrow X_1$ is toroidal and $G$ acts toroidally on
$U_1\hookrightarrow X_1$.
If $X_0', T_0', G', p_0'$ and $\psi'_x$ is a second set of such data,
and if we have an isomorphism of completions
$$ \varphi : X_{/p}\cong X'_{/p'}, $$
which induces isomorphisms $G\cong G'$ and
$D_{/p}\cong D'_{/p'}$, then
$\varphi$ pulls back $I_G$ to the ideal $I_{G'}$.
Furthermore, if $q_0$ is any point of $X_0$ and if $G_q\subset G$ is
the stabilizer of $q$ in $G$, then the stalk of $I_G$ at $q$
is the same as the stalk of $I_{G_q}$ at $q$.
\end{prp}
\begin{rem} The ideal $I_G$ is
called the {\bf torific ideal} of the situation $G\subset {\operatorname{Aut}}(T_0\times
{\operatorname{Spec }}\ k[x] \subset X)$.
\end{rem}
{\bf Proof.}
For any monomial $t\in R[x]$ let $\chi_t $ be the associated
character. We restrict these characters to $G$ and obtain a character
$\psi_t: G\rightarrow} \newcommand{\dar}{\downarrow k^\ast$.
Define $M_x=\{t | \psi_t = \psi_x\}$, the set of mononials on which
$G$ acts as it acts on $x$. Notice that the character $\psi_x$ of $G$
is uniquely determined by the data $G\to {\operatorname{Aut}}(X_{/p},D_{/p})$.
Define the torific ideal $I_G= \left< M_x\right >$, the ideal generated by
$M_x$. We see that it satisfies the variance property of the
proposition with respect to isomorphisms $\varphi$. We leave
the localization property at the end of the proposition as an
excercise to the reader.
Define $b:X_1 \rightarrow} \newcommand{\dar}{\downarrow X$ as in the proposition.
Since $G$ is a subgroup of $T_0$, we have that $M_x$ contains a monomial in
$R$. Therefore the ideal $I_G$
is generated by $x$ and a number of monomials $t_1,\ldots,t_m\in R\cap M_x$.
The blow up has a chart associated to each of the generators
$x, t_1,\ldots, t_m$. On the chart ``$x\neq 0$'' we have that the
inverse image of $D$ contains the inverse image
of $D\cup V(x)$, and hence the action of $G$ is toroidal, being
toroidal with respect to $T\subset X$ on $X$. The other charts ``$t_i\neq 0$''
can be described as the spectra of the rings
$$\tilde R= R[x][u, \{s_j\}_{j\not=i}]/(ut_i-x, s_jt_i-t_j, h_\alpha),$$
that is $u = x/t_i$ and $s_j =t_j/t_i$. Since there are no
relations between $x$ and $t_j$ we can take the $h_\alpha$ to be certain
polynomials in $R[s_j]$ (the ideal
generated by the $t_i$ is flat over $k[x]$). Note that $G$ fixes the element
$u$ in this algebra.
Thus it follows that the normalization of the ring $\tilde R$
is of the form $R'[u]$, where $G$ acts trivially on $u$, $R'$ is
the ring associated to an affine torus embedding and $G$ acts toroidally
on ${\operatorname{Spec }}\ R'$.
\qed
\subsubsection{Strictness}\label{striktheid}
We return to the triple $(Y, D, G)$ we obtained at the end of \ref{Sepban},
in particular we have the $G$-equivariant map $Y\to P'$ and the Galois
alteration $P'\to P$ with group $G$.
Denote by $b: Y_1 \rightarrow} \newcommand{\dar}{\downarrow Y$ the torifying blowup obtained by normalizing the
blowup at the torific ideal of $Y$, as in Theorem \ref{opblazen}.
It remains to check that the
divisor $b^{-1}(D)\subset Y_1$ is $G$-strict, so that the quotient is a
strict toroidal embedding.
First, although a coordinate $x$ for the pre-toroidal action is not globally
defined, we may always find such coordinates on Zariski neighborhoods of the
relevant points. Let $y\in Y$ be such a point. Let
$\epsilon: Y_{/y}\cong {\operatorname{Spf }}\ R[[x]]$ be as in Defintion \ref{pretoroidal}.
Let $x'\in O_{Y,y}$ be an element of the local ring of $Y$ at
$y$ that is congruent to $\epsilon^{-1}(x)$ up to a high power of the
maximal ideal. Then the element
$$x''=\sum_{g\in G_y} g(x') \psi_x^{-1}(g) / |G_y| \in O_{Y,y}$$
transforms according to $\psi_x$ under the action of $G_y$ and
is congruent to $\epsilon^{-1}(x)$ up to a high power of the
maximal ideal. We may then change the isomorphism $\epsilon$ so
that the element $x''$ will correspond to the coordinate $x$.
Therefore, we may assume that there is a $G_y$-invariant
Zariski open neighbourhood $W=W_y$ of $y$ and a function
$x\in \Gamma(W, {\cal O})$ that transforms according to the character
$\psi_x$ under the group $G_y$, and giving rise to the local
coordinate for a suitably chosen isomorphism $\epsilon$ as
in Definition \ref{pretoroidal}. After possibly shrinking
$W$, we get that $W$ is strict toroidal with respect
to the divisor $D_W= (W\cap D) \cup V(x)$. If we choose $W$ sufficiently
small, we may assume that the conical polyhedral complex of $(W, D_W)$ a
single cone $\sigma_W$ (see \cite{te}, II \S 1 definition 5, through p. 71).
Now, to show that $b^{-1}({D})$ is strict, it suffices to show this on
an affine neighborhood of any point. This follows immediately from
theorem $1^*$ of \cite{te}, applied to $(W,D_W)$.
At this point it should be remarked that we could get away without
$G$-strictness: using the constructions in \cite{te}, I \S2 lemmas 1-3 on pages
33-35, and \cite{dj}, 7.2, it is not difficult to construct a $G$-equivariant
blowup which is a $G$-strict toroidal embedding. Still it is of interest to
know that $b^{-1}(D)\subset Y_1$ is already $G$-strict.
Let $E\subset b^{-1}(D)$ be a component
and let $g\in G$ be an element such that $g(E)\cap E\not= \emptyset$.
We have to show that $g(E)=E$. In the case that $b(E)$ is a component
of $D$, this follows immediately from the $G$-strictness of $D$,
see end of \ref{Sepban}; the difficult case is that of a component
that is collapsed under $b$.
Let $y$ be a general point of the intersection $g(E)\cap E$,
and let $y=b(y)$. Let $D_1,\ldots, D_s$ be the components of
$D$ at $y$.
The components $g^{-1}(D_i)$ are the components of
$D$ at $g^{-1}(y)$.
After reordering, we may assume that $D_1,\ldots, D_r$ are the components
which contain $b(E)$, for some $r\leq s$. As $y\in g(E)\cap E$ we
get $y\in g(b(E))\cap b(E)$, hence $y\in
g(D_i)\cap D_i$
for all $i=1,\ldots, r$. By $G$-strictness of $D$, we get
$g(D_i)=D_i$ for $i=1,\ldots,r$.
Let $\sigma_y$ be the cone corresponding to the
toroidal structure $D_{W_y}$ at $y$ of the second paragraph above.
The divisors $D_i$ correspond to rays $\tau_{D_i,y}$ in the boundary of
$\sigma_y$. The divisor $E$ corresponds to a
ray $\tau_{E,y}$ in $\sigma_y$, belonging to the polyhedral
decomposition associated to the given blowup.
For the Zariski open neighbourhood $W_{g^{-1}(y)}$ we take
$g^{-1}(W_y)$ and for the divisor $D_{W_{g^{-1}(y)}}$ we take
$g^{-1}(D_{W_y})$. Hence we get an identification
$\sigma_y\cong \sigma_{g^{-1}(y)}$ given by $g^{-1}$.
Under this identification the ray $\tau_{E,y}$ is mapped
to the ray $\tau_{g^{-1}(E), g^{-1}(y)}$. Note that
$E$ passes through $g^{-1}(y)$, hence we have the
ray $\tau_{E, g^{-1}(y)}$ in $\sigma_{g^{-1}(y)}$.
We have to show that $\tau_{E,g^{-1}(y)}=\tau_{g^{-1}(E), g^{-1}(y)}$.
Let $M_y$ be the group of Cartier divisors supported along $D_{W_y}$ (see
\cite{te}, II \S 1, definition 3). After
shrinking $W_y$ we may assume that $M_y$ is the free abelian group generated by
the divisors of
rational functions
$f_j\in \Gamma(W_y,{\cal O}), j=1,\ldots,m$ and the function $x$ on $W_y$.
We may consider
$f_j$ and $x$ as rational functions on $Y_1$ as well.
The ray $\tau_{E,y}$ is determined by the order of vanishing
of the functions $f_j$ and $x$ along $E$ on $Y_1$,
let us call these orders $n_j$ and $n$.
Similarly, $\tau_{E,g^{-1}(y)}$ (resp.\ $\tau_{g^{-1}(E), g^{-1}(y)}$)
is determined by the order of vanishing of the functions
$g^*(f_j)$ and $g^*(x)$ along $E$ (resp.\ $g^{-1}(E)$), let
us call these orders $n'_j$ and $n'$ (resp.\
$n''_j$ and $n''$).
It is clear that $n_j=n''_j$ and $n=n''$.
Notice that $g^*(f_j)$ are supported along $g^{-1}(D_i)$ in $W_{g^{-1}(y)}$.
Here $g^{-1}(D_i) = D_i$ for $i\leq r$, so that $f_j/g^*f_j$
has order 0 along $D_i, i\leq r$.
Since the vanishing along $E$ may be computed on
$b^{-1}(W_y\cap W_{g^{-1}(y)})$, we see that $n_j=n_j'$.
We are through if we show that $n=n'$; this is equivalent to
showing that the order of vanishing of $x$ along
$E$ is the same as the order of vanishing of $g^{-1}(x)$ along $E$.
The component $E$ is exceptional for the morphism $b$.
The torifying blowup $b$ over $W_y$ blows up inside $V(x)$
only. Hence we see that $b(E)\cap W_y$ is contained in
$V(x)$, i.e., $n>0$. The same argument applied to $g^{-1}(x)$
on $W_{g^{-1}(y)}$ works to show that $n'>0$. For a general
point $z\in b(E)$, we see that both $x$ and $g^{-1}(x)$
define a local coordinate that can be used to define the
local toroidal structure around $z$, as in Definition
\ref{pretoroidal}. Therefore, we need only to show that
the valuation of $x$ along $E$ does not depend on the choice
of the parameter $x$ as in Definition \ref{pretoroidal}.
Let $\epsilon: Y_{/z}\cong {\operatorname{Spf }}\ R[[x]]$ be as in
Definition \ref{pretoroidal}. Any other $x'\in R[[x]]$
that gives a local coordinate for some
pretoroidal structure is of the form
$$x'= u x + r_0 + \sum_{j\geq 2} r_j x^j, $$
with $u, r_0, r_j\in R$ and where $u$ is a unit and
$r_0$ is in the maximal ideal of $R$. The element
$u$ being a unit,
we may divide by it without changing the
orders of vanishing on $Y_1$. Thus we may assume $u=1$.
Consider the family of automorphisms $\varphi_t$,
$t\in [0,1]$ of $R[[x]]$ given by $\varphi_t(r)=r$, $r\in R$ and
$$ \varphi_t(x)=x+t\Big(r_0+ \sum_{j\geq 2} r_j x^j\Big).$$
These are automorphisms that occur in Proposition
\ref{affineopblazen}. Hence these act on the
formal completion $Y_1^\wedge$ of $Y_1$ along $b^{-1}(z)$,
in view of Proposition \ref{affineopblazen}.
Since they form a continuous family (acting continuously
on the charts described in the proof of Proposition
\ref{affineopblazen}), they will fix (formal) components
of $b^{-1}(D_{/z})$ such as
$E^\wedge$ in $Y_1^\wedge$. Hence the orders
of vanishing of the functions $\varphi_t(x)$ along
the (formal) component $E^\wedge$ are all the same.
In particular, we get the equality for $x$ and $x'$, as
desired.
\subsection{Conclusion of proof}
If we combine the results of \ref{Sepban}, \ref{Torlta}
and \ref{striktheid} then we see that we may assume our
Galois alteration $(Y_1, D_1, G)$
of $(X,Z)$ is such that $Y\setminus D_1\hookrightarrow Y_1$ is a $G$-strict
toroidal embedding.
Hence the quotient $(Y_1\setminus D_1)/G\hookrightarrow Y_1/G$ is a
strict toroidal embedding, and $Y_1/G\to X$ is a modification.
Hence we may replace $(X, Z)$ by $(Y_1/G, D_1/G)$.
By Theorem \ref{mumford}, there exists a (toroidal)
resolution of singularities of this pair.
This ends the proof of Theorem \ref{resolution}.
\qed
\begin{rem}\label{char-p}
If ${\operatorname{char}}\ k = p>0$ the proof goes through if it works on $P$, and if the
exponent of the Galois group $G$ is small enough. Since $G$ can be taken as the
Galois group of the torsion on a generalized Jacobian, we can bound it in terms
of $g+n$, where $g$ is the relative genus and $n$ is the degree of the divisor
$Z$. Given a family of varieties of finite type $X\rightarrow} \newcommand{\dar}{\downarrow S$, we
can
make the constructions in the proof uniform over $S$. Roughly speaking,
after replacing
$S$ by a dense open, there is a sequence of projections $X\das
X_1\das\cdots\das S$ and relative divisors $D_i \subset X_i$ which will do the
job. Therefore the order of all groups involved is bounded in terms of the
relative genus of $X_i\das X_{i+1}$ and the degree of $D_i\das X_{i+1}$. Thus
there is a ``geometrically meaningful'' function $M$, as
described in \ref{char-p-bad}.
\end{rem}
|
1996-03-31T05:56:27 | 9603 | alg-geom/9603013 | en | https://arxiv.org/abs/alg-geom/9603013 | [
"alg-geom",
"math.AG"
] | alg-geom/9603013 | Fernando Torres | Rainer Fuhrmann, and Fernando Torres | On curves over finite fields with many rational points | LaTex2e, 10 pages | null | null | null | null | We study arithmetical and geometrical properties of {\it maximal curves},
that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of
$\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a
hypothesis on non-gaps at rational points we prove that maximal curves are
$\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.
| [
{
"version": "v1",
"created": "Thu, 14 Mar 1996 19:11:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Fuhrmann",
"Rainer",
""
],
[
"Torres",
"Fernando",
""
]
] | alg-geom | \section{Introduction}
Goppa in \cite{Go} showed how to construct linear codes from curves
defined over finite fields. One of the main features of these codes is
the fact that one can state a lower bound for the minimum distance of the
codes. In fact, let $C_X(D,G)$ be a Goppa code defined over a curve $X$
over the finite field $\mathbb F_q$ with $q$ elements, where
$D=P_1+\ldots+P_n$, $P_i\in X(\mathbb F_q)$ for each $i$ and $G$ is a
$\mathbb F_q$-rational divisor on $X$. Then it is known
that the minimum distance $d$ of $C_X(D,G)$ satisfies
$$
d\ge n - {\rm deg}(G).
$$
Certainly this bound is meaningful only if $n$ is large enough. This
provides
motivation for the study of curves over finite fields with many rational
points.
The purpose of this paper is to study {\it maximal curves}, that is,
curves $X$ over $\mathbb F_q$ whose number of rational points
$\#X(\mathbb F_q)$ reaches the Hasse-Weil upper bound. In this case one
knows that $q$ must be a square.
Let $k$ be the finite field with $q^2$ elements,
where $q$ is a power of a prime $p$. Let $X$ be a projective, connected,
non-singular
algebraic curve defined over $k$ which is maximal, that is, $\#X(k)$
satisfies
\begin{equation}\label{h}
\#X(k)=q^2 + 2gq + 1.
\end{equation}
Let $P\in X(k)$ and set ${\mathcal D}= g^{n+1}_{q+1}$ the $k$-linear system on
$X$ defined by the divisor $(q+1)P$. Then $n\ge 1$, and ${\mathcal D}$ is
independent of
$P$. In fact ${\mathcal D}$ is a simple base-point-free linear system on $X$
(Corollary \ref{R-Sti}, Remark \ref{rem-sem} (ii)). This allow us to apply
St\"ohr-Voloch's approach concerning Weierstrass point theory over finite
fields \cite{S-V}. Moreover, the dimension $n+1$ of ${\mathcal D}$ and the genus
$g$ are related by Castelnuovo's genus bound for curves in projective
spaces (\cite{C}, \cite[p.116]{ACGH}, \cite[Corollary 2.8]{Ra}).
It is known that $2g\le (q-1)q$
(\cite[V.3.3]{Sti}), and that the Hermitian curve is the unique
maximal curve whose genus is $(q-1)q/2$
\cite{R-Sti}. Furthermore in \cite{F-T} we proved the following stronger
bounds for the genus, namely
$$
4g\le (q-1)^2\qquad {\rm or}\qquad 2g=(q-1)q.
$$
Moreover by using the already mentioned Castelnuovo's bound one can prove
that $4g>(q-1)^2$ if and only if $n=1$. Therefore, we assume from
now on that $n\ge 2$.
The Hermitian curve is a particular case of the following
type of curves. Let $m$ be a positive divisor of $q+1$, and let consider
\begin{equation*}
y^q + y = x^m.\tag{${\mathcal H}_{m,q}$}
\end{equation*}
These curves are maximal (\cite[Thm. 1]{G-V}) and have very
remarkable properties (see e.g \cite{G-V}, \cite{Sch}).
Under a hypothesis on non-gaps at rational points we prove that
maximal curves are $k$-isomorphic to ${\mathcal H}_{m,q}$ for some $m\in \mathbb Z^+$.
\begin{theorem}\label{result1}
Let $X$ be a maximal curve of genus $g>0$. Assume
that there exists $P_0\in X(k)$ such that the first non-gap
$m_1$ at $P_0$ satisfies
$$
nm_1 \le q+1,
$$
where $n+1$ is the dimension of the complete linear system
defined by $(q+1)P_0$. Then one of the following possibilities is
satisfied
\begin{enumerate}
\item[(i)] $nm_1=q+1$, $2g=(m_1-1)(q-1)$, and $X$ is $k$-isomorphic to
${\mathcal H}_{m_1,q}$.
\item[(ii)] $nm_1=q$.
\end{enumerate}
\end{theorem}
{}From this theorem and a result due to Lewittes (see inequality \ref{le})
we obtain an analogous of
the main result in \cite{F-T}:
\begin{corollary}\label{result2}
Let $X$ be a curve satisfying the hypotheses of Theorem \ref{result1}.
Let $t\ge 1$ be an integer, and suppose that the genus $g$ of $X$ satisfies
$$
(q-1)(\frac{q+1}{t+1}-1)< 2g \le (q-1)(\frac{q+1}{t}-1).
$$
Then one of the following conditions is satisfied
\begin{enumerate}
\item[(i)] $t=n$, $2g=(q-1)(\frac{q+1}{t}-1)=(q-1)(m_1-1)$.
\item[(ii)] $t\ge n$, $2g\le
(q-1)(\frac{q+1}{n}-1)=(q-1)(\frac{m_1(q+1)}{q}-1)$.
\end{enumerate}
\end{corollary}
\begin{remarke}\label{conj}
In case $nm_1=q$ the authors actually conjecture that then $2g=(m_1-1)q$,
and $X$ is $k$-isomorphic to a curve whose plane model is given by
$F(y)=x^{q+1}$, where $F(y)$ is a $\mathbb F_p$-linear polynomial of
degree $m_1$. But we have not yet been able to prove
this. We notice that the veracity of this conjecture implies $t=n$ and
$2g=(\frac{q}{t}-1)q=(m_1-1)q$ in the statement (ii) of the above corollary.
\end{remarke}
\section{Preliminaries}
Throughout this paper we use the following notation:
\begin{itemize}
\item $k$ denotes the finite field with $q^2$ elements, where
$q$ is a power of a prime $p$. ${\bar k}$ denotes its algebraic closure.
\item By a curve we mean a projective, connected,
non-singular algebraic curve defined over $k$.
\item The symbol $X(k)$ (resp. ${k}(X)$) stands for the set
of $k$-rational points (resp. the field of k-rational functions) of a curve
$X$.
\item If $x\in {k}(X)$, ${\rm div}(x)$ (resp. ${\rm div}_\infty(x)$) denotes
the divisor (resp. the polar divisor) of $x$.
\item Let $P$ be a point of a curve. $v_P$ (resp. $H(P)$)) stands
for the valuation (resp. the Weierstrass semigroup) associated to $P$. We
denote by $m_i(P)$ the $i$th non-gap at $P$.
\item Let $D$ be a divisor on $X$ and $P\in X$. We denote by
${\rm deg}(D)$ the degree of $D$, by ${\rm Supp}(D)$ the support of $D$ and by
$v_P(D)$ the coefficient of $P$ in $D$. If $D$ is a $k$-divisor, we set
$$
L(D):= \{f\in {k}(X): {\rm div}(f)+D \succeq 0\},
$$
and $\ell(D):= {\rm dim}_k L(D)$. The symbol ``$\sim$" denotes
module linear equivalence.
\item The symbol $g^r_d$ stands for a linear system of dimension $r$
and degree $d$.
\end{itemize}
\subsection{Weierstrass points.}\label{wp} We summarize some results from
\cite{S-V}.
Let $X$ be a curve of genus $g$, ${\mathcal D}=g^r_d$ a base-point-free $k$-linear
system on $X$. Then associated to $P\in X$ we have the hermitian
$P$-invariants $j_0(P)=0<j_1(P)<\ldots<j_r(P)\le d$ of ${\mathcal D}$ (or simply
the $({\mathcal D},P)$-orders). This sequence is the same for all but finitely many
points. These finitely many points $P$, where exceptional $({\mathcal D},P)$-orders
occur, are called the ${\mathcal D}$-Weierstrass points of $X$. (If ${\mathcal D}$
is generated by a canonical divisor, we obtain the usual
Weierstrass points of $X$.) Associated to
${\mathcal D}$ there exists a divisor $R$ supporting the ${\mathcal D}$-Weierstrass points. Let
$\epsilon_0<\epsilon_1<\ldots<\epsilon_r$ denote the $({\mathcal D},P)$-orders for
a generic $P\in X$. Then
\begin{equation}\label{ineq1}
\epsilon_i \le j_i(P),
\end{equation}
for $P\in X$, for each $i$, and
\begin{equation}\label{degR}
{\rm deg}(R)= (\epsilon_1+\ldots+\epsilon_r)(2g-2)+(r+1)d.
\end{equation}
Associated to ${\mathcal D}$ we also have a divisor $S$ whose support contains
$X(k)$. Its degree is
given by
\begin{equation*}
{\rm deg}(S)=(\nu_1+\ldots+\nu_{r-1})(2g-2)+(q^2+r)d,
\end{equation*}
where the $\nu_i's$ is a subsequence of the $\epsilon_i's$. More
precisely there exists an integer $I$ with $0<I\le r$ such that
$\nu_i=\epsilon_i$ for $i<I$ and $\nu_i=\epsilon_{i+1}$ otherwise.
Moreover for $P\in X(k)$ we have
\begin{equation}\label{ineq2}
v_P(S)\ge \sum_{i=1}^{r}(j_i(P)-\nu_{i-1}),
\end{equation}
and
\begin{equation}\label{ineq3}
\nu_i\le j_{i+1}(P)-j_1(P),
\end{equation}
for each $i$.
\subsection{Maximal curves}
Let $X$ be a maximal curve of genus $g$. In this
section we study some arithmetical and geometrical properties of $X$. To
begin with we have the following basic result which is containing in the
proof of \cite[Lemma 1]{R-Sti}. For the sake of completeness we state a
proof of it.
\begin{lemma1}\label{mult}
The Frobenius map ${\rm Fr}_{\mathcal J}$ (relative to $k$)
of the Jacobian ${\mathcal J}$ of $X$ acts
just as multiplication by $(-q)$ on ${\mathcal J}$.
\end{lemma1}
\begin{proof}
All the facts concerning Jacobians can be found in \cite[VI,
\S3]{L}. Let $\ell\not= p$ be a prime and let
$T_{\ell}({\mathcal J})$ be the
Tate module of ${\mathcal J}$. Then the characteristic polynomial $P({\rm Fr}_{\mathcal J})(t)$ of
the
action ${\rm Fr}_{\mathcal J}$ on $T_{\ell}({\mathcal J})$ is equal to $t^ {2g}L(1/t)$ where $L(t)$ denotes
the numerator of the Zeta function of $X$. Since $X$ satisfies
(\ref{h}), $L(t)=\prod_{i=1}^{2g}(1+qt)$ and thus $P({\rm Fr}_{\mathcal J})(t)= (t+q)^{2g}$.
Now we know that ${\rm Fr}_{\mathcal J}$ is diagonalizable \cite[Thm. 2]{Ta} and all its
eingenvalues are $-q$. This means that ${\rm Fr}_{\mathcal J}$ acts just as multiplication
by $-q$ on $T_{\ell}({\mathcal J})$. Finally since the natural homomorphism of $\mathbb
Z$-algebras $$
{\rm End}({\mathcal J})\to {\rm End}(T_{\ell}({\mathcal J}))
$$
is injective, the proof follows.
\end{proof}
Now fix $P_0\in X(k)$, and consider the map
$f=f^{P_0}: X\to {\mathcal J}$ given by $P\to [P-P_0]$. We have
$$
f\circ {\rm Fr}_{X} = {\rm Fr}_{\mathcal J}\circ f,
$$
where ${\rm Fr}_{X}$ denotes the Frobenius morphism of $X$ relative to $k$. Hence
from the above equality and Lemma \ref{mult} we get
\begin{corollary1}\label{frob}
$$
{\rm Fr}_{X}(P)+qP \sim (q+1)P_0.
$$
\end{corollary1}
{}From this corollary it follows immediately the following:
\begin{corollary1}[[R-Sti, Lemma 1{]}]\label{R-Sti}
Let $P_0, P_1 \in X(k)$. Then $(q+1)P_1\sim (q+1)P_0$.
\end{corollary1}
Now, let consider the linear system ${\mathcal D} = g^{n+1}_{q+1}:=
|(q+1)P_0|$. Corollary \ref{R-Sti} says that ${\mathcal D}$ is a $k$-invariant of
the curve. In
particular its dimension $n+1$ is independent of $P\in X(k)$. Moreover
from Corollary \ref{R-Sti} we have that $q+1\in H(P_0)$ and hence
${\mathcal D}$ is base-point-free. Consequently we can apply \cite{S-V} to ${\mathcal D}$.
\begin{thm1}\label{thm1} With notation as in \S\ref{wp} (for ${\mathcal D}$) we have:
\begin{enumerate}
\item[(i)] $\epsilon_{n+1}=\nu_n=q$;
\item[(ii)] $j_{n+1}(P)=q+1$ if $P\in X(k)$ and $j_{n+1}(P)=q$, otherwise;
\item[(iii)] $j_1(P)=1$ for each $P\in X$.
\end{enumerate}
\end{thm1}
\begin{proof} Statement (iii) for $P\in X(k)$ follows from (i), (ii) and
inequality (\ref{ineq3}). From Corollary \ref{frob} it follows (ii) and
$\epsilon_{n+1}=q$. Furthermore it also follows that $j_1(P)=1$ for
$P\not\in X(k)$: for let $P'\in X$ such that ${\rm Fr}_{X}(P')=P$; then $P+qP'=
{\rm Fr}_{X}(P')+ qP'\sim (q+1)P_0$.
Now we are going to prove that $\nu_n=\epsilon_{n+1}$. Let $P\in
X\setminus \{P_0\}$.
Corollary \ref{frob} says that $\pi({\rm Fr}_{X}(P))$ belongs to the osculating
hyperplane at $P$, where $\pi$ stands for the morphism associated to
${\mathcal D}$. $\pi$ can be defined by a base $\{f_0,f_1,\ldots,f_{n+1}\}$ of
$L((q+1)P_0)$, where $v_P(f_i)\ge 0$ for each $i$. Let $x$ be a
separating variable of $k(X)\mid k$. Then by
\cite[Prop. 1.4(c), Corollary 1.3]{S-V} the rational function
$$
w:= {\rm det} \begin{pmatrix}
f_0\circ{\rm Fr}_{X} & \ldots & f_{n+1}\circ{\rm Fr}_{X}\\
D^{\epsilon_0}_x f_0 & \ldots & D^{\epsilon_0}_x f_{n+1}\\
\vdots & \vdots & \vdots\\
D^{\epsilon_n}_x f_0 & \ldots & D^{\epsilon_n}_x f_{n+1}
\end{pmatrix}
$$
satisfies $w(P)=0$ for each generic point $P$. Let $I$ be the smallest
integer such that the row $(f_0\circ{\rm Fr}_{X},\ldots,f_{n+1}\circ{\rm Fr}_{X})$ is a
linear combination of the vectors $(D^{\epsilon_i}_x
f_0,\ldots,D^{\epsilon_i}_x f_{n+1})$ with $i=0,\ldots,I$. Then according
to \cite[Prop. 2.1]{S-V} we find
$$
\{\nu_0<\ldots<\nu_n\}=\{\epsilon_0<\ldots<\epsilon_{I-1}
<\epsilon_{I+1}<\ldots<\epsilon_{n+1}\}.
$$
This concludes the proof.
\end{proof}
\begin{remark}\label{rem-sem} Let $X$ be a maximal curve.
\begin{enumerate}
\item[(i)] We claim that
$\nu_1=\epsilon_1=1$ (that is, the number $I$
in the above proof is bigger than one). In fact, suppose that $\nu_1>1$.
Then, due to the fact that $j_1(P)=1$ for each $P$, we can apply the proof
of
\cite[Thm. 1]{H-V} to conclude that $\#X(k)=(q+1)(q^2-1)-(2g-2)$. Then
from (\ref{h}) we must have $2g=(q-1)q$ and hence by \cite{F-T}, $n=1$, a
contradiction.
\item[(ii)] Let $P\in X(k)$. Due to Corollary \ref{R-Sti}
the fact that $j_1(P)=1$ and $j_{n+1}(P)=q+1$ is equivalent to have
$q,q+1\in H(P)$. This was noticed for some $P_0\in X(K)$ in \cite[Prop.
1]{Sti-X}.
\item[(iii)] Let $P\in X\setminus X(k)$. If $P \in X(\mathbb F_{q^4})$,
then $q-1\in H(P)$. If $P\not\in X(\mathbb F_{q^4})$, then $q\in H(P)$.
In fact, set $i:= \min \{j\in \mathbb Z^+: {\rm Fr}_{X}^j(P)=P\}$. Now applying
$\left({\rm Fr}_{X}^{i-1}\right)_*$ (see \cite[IV, Ex. 2.6]{Har}) to the
equivalence in Corollary \ref{frob} we get
$$
{\rm Fr}_{X}(P)+(q-1)P \sim q{\rm Fr}_{X}^{i-1}(P).
$$
Now the remarks follows from the fact that ${\rm Fr}_{X}^{i-1}(P)={\rm Fr}_{X}(P)$
if and only if $P\in X(\mathbb F_{q^4})$.
\end{enumerate}
\end{remark}
In particular the above remark (ii) implies that ${\mathcal D}$ is simple. Thus the
genus $g$ of $X$ and the dimension $n+1$ of ${\mathcal D}$ are related by
Castelnuovo's genus bound for curves in projective spaces (\cite{C},
\cite[p. 116]{ACGH}, \cite[Corollary 2.8]{Ra}). Thus
\begin{equation}\label{cast}
2g\le c(n,q):=M(q-n+e)\le
\begin{cases}
(2q-n)^2/4n & \text{if $n$ is even}\\
((2q-n)^2-1)/4n & \text{if $n$ is odd},
\end{cases}
\end{equation}
where $M$ is the biggest integer $\le q/n$ and $e:= q-Mn$.
We can also bound $g$ by using non-gaps at $P_0\in X(k)$. In fact,
Lewittes
\cite[Thm. 1(b)]{Le} proved that
$$
\#X(k)\le q^2m_1(P) +1,
$$
and hence from (\ref{h})
we conclude that
\begin{equation}\label{le}
2g\le q(m_1(P)-1).
\end{equation}
\begin{prop1}\label{ra1}
The following statements are equivalent:
\begin{enumerate}
\item[(i)] $\pi: X\rightarrow \mathbb P^{n+1}$ is a closed embedding,
i.e. $X$ is $k$-isomorphic to $\pi(X)$.
\item[(ii)] $\forall P \in X(\mathbb F_{q^4}): \pi(P)\in \mathbb
P^{n+1}(k) \Leftrightarrow P \in X(k)$.
\item[(iii)] $\forall P \in X(\mathbb F_{q^4}): q \in H(P)$.
\end{enumerate}
\end{prop1}
\begin{proof}
Let $P\in X$. Since $j_1(P)=1$ (cf. Theorem \ref{thm1} (iii)) we know
already that $\pi(X)$ is non-singular at all the branches centered at
$P$. Thus $\pi$ is an embedding if and only if $\pi$ is injective.
\begin{claim*}
If $\#\pi^{-1}(\pi(P))\ge 2$, then $P\in X(\mathbb F_{q^4})\setminus
X(k)$ and $\pi(P)\in \mathbb P^{n+1}(k)$.
\end{claim*}
\begin{proof} {\it (Claim).}
{}From Corollary \ref{frob} it follows that $\pi^{-1}(\pi(P)\subseteq
\{P,\pi(P)\}$. Analogically we have $\pi^{-1}(\pi({\rm Fr}_{X}(P)))\subseteq
\{{\rm Fr}_{X}(P), {\rm Fr}_{X}^2(P)\}$. Thus if $\#\pi^{-1}(\pi(P))\ge 2$, then $P$
cannot be rational and ${\rm Fr}_{X}^2(P)=P$, i.e. $P\in X(\mathbb
F_{q^4})\setminus X(k)$. Furthermore we have
$\pi(P)=\pi({\rm Fr}_{X}(P))={\rm Fr}_{X}(\pi(P))$, i.e. $\pi(P)\in \mathbb P^{n+1}(k)$.
\end{proof}
{}From this claim the equivalence (i) $\Leftrightarrow$ (ii) follows
immediately. As to the implication (i) $\Rightarrow$ (iii)
we know that $\dim |{\rm Fr}_{X}(P)+qP-P-{\rm Fr}_{X}(P)|=\dim |{\rm Fr}_{X}(P)+qP| - 2$
(Corollary \ref{frob} and \cite[Prop.3.1(b)]{Har}), i.e. $q\in H(P)$.
Finally we want to conclude that $\pi$ is an embedding from
(iii). According to the above claim it is sufficient to show that
$\pi^{-1}(\pi(P))=\{P\}$ for $P \in X(\mathbb F_{q^4})$. Let $P\in X(\mathbb
F_{q^4})$. Because of Corollary \ref{frob} we know that
$\pi^{-1}(\pi(P))\subseteq \{P, {\rm Fr}_{X}(P)\}$. Since $q\in H(P)$, there is a
divisor $D\in |qP|$ with $P\notin {\rm Supp}(D)$. In particular
$$
{\rm Fr}_{X}(P)+D\sim {\rm Fr}_{X}(P)+qP\sim (q+1)P_0.
$$
Thus $\pi^{-1}(\pi({\rm Fr}_{X}(P)))\subseteq {\rm Supp}({\rm Fr}_{X}(P)+D)$. So either
$\pi(P)\neq \pi({\rm Fr}_{X}(P))$ or $P={\rm Fr}_{X}(P)$. In both cases we have
$\pi^{-1}(\pi(P))=\{P\}$. This means altogether that $\pi$ is injective
and so indeed a closed embedding.
\end{proof}
\begin{prop1}\label{ra2}
Suppose that $\pi: X\rightarrow \mathbb P^{n+1}$ is a closed embedding.
Assume furthermore that there exist $r, s \in H(P_0)$ such that all
non-gaps at $P_0$ less than or equal to $q+1$ are generated by $r$ and
$s$. Then $H(P_0)$ is generated by $r$ and $s$. In particular the genus
of $X$ is equal to $(r-1)(s-1)/2$.
\end{prop1}
\begin{proof}
Let $x, y\in k(X)$ with ${\rm div}_\infty(x)=sP_0$ and ${\rm div}_\infty(y)=rP_0$. Since $q, q+1
\in H(P_0)$, the numbers $r$ and $s$ are coprime. Let $\pi_2:
X\rightarrow \mathbb P^2$, $P\mapsto (1:x(P):y(P))$. Then the curves $X$
and $\pi_2(X)$ are birational and $\pi_2(X)$
is a plane curve given by an equation of type
$$
x^r+\beta y^s+\sum_{is+jr<nm} \alpha_{ij}x^iy^j=0,
$$
where $\beta,\alpha_{ij}\in k$ and $\beta\neq 0$. We are going to prove that
$\pi_2(P)$ is a non-singular point
of $\pi_2(X)$ for $P\neq P_0$. From this follows by \cite[Ch. 7]{Ful} that
$g=1/2(r-1)(s-1)$. Then by Jenkins \cite{J} we have $H(P_0)=\langle
r,s\rangle$.
Let $1,f_1,\ldots,f_{n+1}$ be a basis of $L((q+1)P_0)$, where
$n+1:=\dim |(q+1)P_0|$. Then there exist polynomials $F_i(T_1,T_2)\in
k[T_1,T_2]$ for $i=1,\ldots,n+1$ such that
$$
f_i=F_i(x,y)\ \mbox{on}\ X\qquad \mbox{for}\qquad i=1,\ldots, n+1.
$$
Consider the maps $\pi |(X\setminus\{P_0\}): X\setminus\{P_0\}\rightarrow
\mathbb A^{n+1}$ given by $P\mapsto (f_1(P),\ldots,f_{n+1}(P))$;
$\pi_2 |(X\setminus\{P_0\}): X\setminus\{P_0\}\rightarrow \mathbb A^2$,
$P\mapsto (x(P),y(P))$; and $\phi : \mathbb A^2\rightarrow \mathbb
A^{n+1}$, given by $(p_1,p_2)\mapsto
(F_1(p_1,p_2),\ldots,F_{n+1}(p_1,p_2))$.
Then the following diagram is commuting
$$
\Atriangle[X\setminus\{P_0\}`{\Bbb A}^2`{\Bbb A}^r;\pi_2`\pi`\phi].
$$
Thus we have for a point $P$ of
$X\setminus\{P_0\}$ and the corresponding local rings assigned to
$\pi(P), \pi_2(P)$ the commutative diagram
$$
\Atriangle[O_{\pi(X),\pi(P)}`O_{\pi_2(X),\pi_2(P)}`O_{X,P};f`c`h],
$$
where $h$ is injective since $k(X)=k(x,y)$, and $c$ is an isomorphism by
assumption. Thus $\pi_2{X}$ is non-singular at $\pi_2{P}$.
\end{proof}
\section{Proofs of Theorem 0.1 and Corollary 0.2}
Set $m:=m_1$. Recall that $n+1$ is by definition the dimension of
${\mathcal D}:=|(q+1)P|$ for any $P\in X(k)$. Let $\pi$ be the morphism associated
to ${\mathcal D}$. By Remark \ref{rem-sem} (ii) we have $nm \ge q$, and hence by the
hypothesis on $m$ we get
$$
nm\in \{q,q+1\}.
$$
\subsection{Case: $nm=q+1$.}
\begin{prop1}\label{prop1} Let $X$ be a maximal curve of genus $g$.
Assume there exists $P_0\in X$ such that $nm_1(P_0)=q+1$. Then
$$
2g=(q-1)(m_1-1).
$$
\end{prop1}
\begin{proof} Since $m,q\in H(P_0)$ and $\gcd(m,q)=1$, then $2g\le
(m-1)(q-1)$ (see e.g. Jenkis \cite{J}). Now, $\pi$ can be
defined by $(1:y:\ldots:y^{n-1}:x:y^n)$ where $x, y \in k(X)$ such that
\begin{equation}\label{x,y}
{\rm div}_\infty(x)=qP_0\qquad {\rm and}\qquad {\rm div}_\infty(y)=mP_0.
\end{equation}
Let $P\in X\setminus \{P_0\}$. From the proof of \cite[Thm. 1.1]{S-V}, we
have that
\begin{equation}\label{vals}
v_P(y),\ldots,nv_P(y)
\end{equation}
are $({\mathcal D},P)$-orders. Thus by considering
a non-ramified point for $y:X\to \mathbb P^1$, and by
(\ref{ineq1}) we find
$$
\epsilon_i = i,\qquad {\rm for}\ \ i=1,\ldots, n.
$$
\begin{lemma1}\label{type-sem}
There are at most two types of $({\mathcal D},P)$-orders for $P\in X(k)$:
\begin{enumerate}
\item[(i):] $0,1,m,\ldots,(n-1)m,q+1$. Hence $w_1:=v_P(R)=
\frac{n((n-1)m-n-1)}{2}+2$.
\item[(ii):] $0,1,\ldots,n,q+1$. Hence $w_2:=v_P(R)=1$.
\end{enumerate}
Moreover, the set of the ${\mathcal D}$-Weierstrass points of $X$ coincides with
the set of $k$-rational points.
\end{lemma1}
\begin{proof} The statement on $v_P(R)$ follows from \cite[Thm. 1.5]{S-V}.
Let $P\in X(k)$. By Theorem \ref{thm1} we know that $1$ and $q+1$ are
$({\mathcal D},P)$-orders. We consider two cases:
\smallskip
(1) $v_P(y)=1$: With (\ref{vals}) this implies statement (ii).
\smallskip
(2) $v_P(y)>1$: From (\ref{vals}) it follows $nv_P(y)=q+1$ and then we
obtain statement (i).
\smallskip
Let $P\in X\setminus X(k)$. By Theorem \ref{thm1} we have that
$j_{n+1}(P)=q$. If $v_P(y)>1$, then from (\ref{vals}) we get
$nv_P(x)=q=mn-1$
and hence $n=1$. Since by hypothesis $n>1$ then $v_P(y)=1$. This finish
the proof of the lemma.
\end{proof}
Let $T_1$ (resp. $T_2$) denote the number of points $P\in X(k)$ whose
$({\mathcal D},P)$-orders are of type (i) (resp. type (ii)) in Lemma
\ref{type-sem}. Thus by (\ref{degR}) we have
$$
{\rm deg}(R)= (n(n+1)/2+q)(2g-2)+(n+2)(q+1)= w_1 T_1 + T_2,
$$
and by Riemann-Hurwitz applied to $y:X\to \mathbb P^1$
$$
2g-2=-2m + (m-1)T_1.
$$
Consequently, since $T_1+T_2 = \#X(k)= q^2+2gq+1$, from the above two
equations we obtain Proposition \ref{prop1}.
\end{proof}
Now we are going to prove the uniqueness part of the result. To begin
with we generalize \cite[Lemma 5]{R-Sti}.
\begin{lemma1}\label{galois}
Let $X$ be a curve satisfying the hypotheses of Proposition \ref{prop1}.
Take
$y$ as in (\ref{x,y}). Then $k(X)\mid k(y)$ is a Galois cyclic extension.
\end{lemma1}
\begin{proof} Consider $y:X\to \mathbb
P^1(\bar k)$ as a map of degree $m=m_1$. From the proof of Lemma
\ref{type-sem} we see that $y$ has $(q+1)$
ramified points. Moreover, all of them are rational and totally ramified.
\begin{claim*} Let $P\in k\cup \{\infty\}$ such that $\#
y^{-1}(P)=m$. Then $y^{-1}(P)\subseteq X(k)$.
\end{claim*}
\begin{proof} {\it (Claim).} Let $P_1, \ldots, P_r\in k\cup \{\infty\}$
which are not ramified for $y$. Then $r\le q^2-q$. Let $n_i=\#
y^{-1}(P_i)\le m$. Since $2g=(q-1)(m-1)$ by Proposition \ref{prop1}, then
we have
$$
(q^2-q)m=\#X(k)-q-1= \sum_{i=1}^{r} n_i,
$$
from where it follows that $r=q^2-q$ and $n_i=m$ for each $i$.
\end{proof}
Now it follows that $k(X)\mid k(y)$ is Galois as in the proof of
\cite[Lemma 5]{R-Sti}. It is cyclic because there exists rational points
that are totally ramified for $y$.
\end{proof}
\begin{prop1}\label{prop2} Let $X$ be a curve as in Proposition
\ref{prop1}. Then $X$ is $k$ isomorphic to ${\mathcal H}_{m_1,q}$.
\end{prop1}
\begin{proof} Let $y$ be as in (\ref{x,y}).
\begin{claim}\label{eq} $X$ has a model plane given by an equation of type
$$
f(y)=v^m,
$$
where $f\in k[T]$ with ${\rm deg}(f)=q$, $f(0)=0$, and $v \in L(qP_0)$.
\end{claim}
\begin{proof} {\it (Claim \ref{eq}.)} We know that $k(X)\mid k(y)$ is
cyclic (Lemma \ref{galois}). Let $\sigma$ be a generator of $k(X)\mid
k(y)$. Set $V:= L(qP_0)$, $U:= L((n-1)mP_0)$. Then $\sigma\mid V\in {\rm
Aut}(V)$ and $\sigma\mid U = {\rm id}\mid U$. Since $p\nmid m$ we then
have that $\sigma\mid V$ is diagonalizable with an eigenvalue $\lambda$ a
primitive $m$-root of unity in $k$. Let $v\in V\setminus U$ be the
corresponding eigenvector for $\lambda$. Now since ${\rm Norm}_{k(X)\mid
k(y)}(v)= -v^m$ and since $v\in L(qP_0)$ we conclude the existence of
$f\in k[T]$ such that $f(y)=v^m$ and ${\rm deg}(f)=q$. Finally from the
fact that $y$ has exactly $(q+1)$ rational points as totally ramified
points, it follows that $f$ splits into linear factors in $k[T]$. Hence
we can assume $f(0)=0$.
\end{proof}
Now from the claim in the proof of Lemma \ref{galois}, Claim \ref{eq} and
$nm=q+1$ it follows that
$f^n(\alpha)-f^{nq}(\alpha)=0$ for $\alpha\in k$, and hence we obtain
\begin{equation*}
f^n(T) \equiv f^{nq}(T) \pmod{T^{q^2}-T}\tag{$*$}.
\end{equation*}
Set $f(T)=\sum_{i=1}^{q}a_iT^i$, $f^n(T)=\sum_{i=1}^{nq}b_i T^i$.
\begin{claim}\label{eq1}
$a_1\neq 0$, $a_i=0$ for $2\le i\le q-1$.
\end{claim}
\begin{proof} {\it (Claim \ref{eq1}).} $a_1\neq 0$ follows from $(*)$ and
$f(0)=0$. Suppose that $\{2\le i\le q-1: a_i\neq 0 \}\neq \emptyset$. Set
$$
t:={\rm min}\{1\le i\le q-1: a_i\neq 0\}\qquad \mbox{and} \qquad j:={\rm
max}\{1\le i\le q-1: a_i\neq 0\}.
$$
Due to the facts: multiplication by $q$ gives an automorphism of
$\mathbb Z/(q^2-1)\mathbb Z$, and $n-1+qt<q^2$ we then get
$b_{n-1+qj}=b^q_{(n-1)q+j}=na^{n-1}_q a_j\neq 0$. Then $nq={\rm deg}(f)\ge
n-1+qj$ implies together with $2n\le q+1$ that
\begin{equation*}
j+n-1<q \tag{$\dagger$}.
\end{equation*}
Then from $(*)$ we have
$b_{t+n-1}=na_ta^{n-1}_1\neq 0$.
Then again by $(*)$ and by $(\dagger)$ it follows that
$b_{q(t+n-1)}=b^q_{t+n-1}\neq 0$ which
implies $nq={\rm deg}(f^n)\ge q(t+n-1)$. But this contradicts to $t\ge
2$. Thus $a_i=0$ for $2\le i\le q-1$ and we are done.
\end{proof}
Write $f(T)=aT^q+bT$, $a,b\in k^*$. By Claim \ref{eq} we have that
$$
f(k)\subseteq \{\beta^m : \beta\in k\}=\cup{i=q}^{n-1} \xi^{im}\mathbb F_q,
$$
where $\xi$ is a primitive element of $k$. Now since $f(k)$ is a one
dimensional $k$-space, it follows that there exists $i\in\{0,\ldots,n-1\}$
such that $f(k)=\xi^{im}\mathbb F_q$. Set $x_1:= \xi^{-i}x$, $y_1:=
\epsilon y$, with $\epsilon$ being the unique element of $k^*$ such that
$$
{\rm Trace}^k_{\mathbb F_q}(\epsilon \alpha)=\xi^{-im}f(\alpha) \forall
\alpha \in k.
$$
These functions fulfil $y^q_1+y_1=x^m_1$ and we finish the proof of
Proposition \ref{prop1}.
\end{proof}
Now the proof of Theorem \ref{result1} (i) follows from the last two
propositions.
\begin{proof} {\it (Corollary \ref{result2}).} By Theorem \ref{result1}
we have two possibilities:
\smallskip
(1) $nm_1=q+1$: Then $2g=(m_1-1)(q-1)$ and statement (i) follows from the
hypothesis on $g$.
\smallskip
(2) $nm_1=q$: Here from $(q-1)(\frac{q+1}{t+1}-1)<2g$, (\ref{le}) and
$n<q$ we found $t\ge n$. The remaining part of (ii) follows from $2g\le
(q-1)(\frac{q+1}{t}-1)$.
\end{proof}
\begin{corollary1}\label{result3} Let $X$ be a maximal curve with genus
$g=(q-1)^2/4$. Then one of the following possibilities is satisfied:
\begin{enumerate}
\item[(i)] $X$ is $k$-isomorphic to ${\mathcal H}_{\frac{q+1}{2},q}$, or
\item[(ii)] For every point $P\in X(k)$ the first three non-gaps at $P$
are $\{q-1,q,q+1\}$.
\end{enumerate}
\end{corollary1}
\begin{proof}
{}From the hypothesis on $g$ and from (\ref{cast}) (applied to
$g^{n+1}_{q+1}$) we conclude $n+1\le 3$. Then by \cite{F-T} we must have
$n+1=3$. Let $P\in X(k)$ and let $m,q,q+1$ be the first three non-gaps.
Then $2m\ge q+1$ due to (\ref{le}). Moreover, $g\le g'$, where $g'$ is
the genus of the semigroup $\langle m,q,q+1\rangle $. We bound from above
$g'$ according to Selmer \cite[\S3.II]{Sel}. From that reference it
follows that $g'$ will be larger if $\gcd(m,q)=1$. So let assume this.
Define $s, t$ by $q+1=sq-tm$, $1<s<m$, $t>0$. Write $m=us+r$, $0\le r<s$.
Then we have ([p.6 loc. cit.])
$$
2g'= (m-1)(q-1) - ut(m-s+r).
$$
Hence by the hypothesis on $g$ we then have
$$
2ut(m-s+r)\le (q-1)(2m-q-1),
$$
and now it is easy to see that $2m=(q+1)$ or $m=q-1$. The first case
for some point $P\in X(k)$, and Theorem \ref{result1} (i) imply the
result.
\end{proof}
\subsection{Case: $nm_1=q$.}
As in the proof of Lemma \ref{type-sem} here one also has $\epsilon_i=\nu_i$
for $i=0,1\ldots, n-1$; $\epsilon_n=n$. However we cannot apply
\cite[Thm. 1.5]{S-V} to compute $v_P(R)$ for $P\in X(k)$.
If one can show that $\pi: X\rightarrow \mathbb P^{n+1}$ is a closed
embedding, then from Proposition \ref{ra2} we would have $2g=q(m_1(P)-1)$
for $P\in X(k)$.
\begin{remark} The hypothesis on the first non-gap of Theorem
\ref{result1} is necessary. In fact, consider the curve from Serre's list
(see \cite[\S4]{Se}) over $\mathbb F_{25}$, $g=3$. Then it is maximal. Let
$m,5,6$
be the first non-gaps at $P\in X(\mathbb F_{25})$. If $m=3$ from Theorem
\ref{result1} (i) we would have $g=4$. Thus $m=4$.
\end{remark}
|
1995-12-15T06:20:17 | 9511 | alg-geom/9511010 | en | https://arxiv.org/abs/alg-geom/9511010 | [
"alg-geom",
"math.AG"
] | alg-geom/9511010 | Vd | Valerii V. Dolotin | On discriminants of polylinear forms | 12 pages, latex | null | null | null | null | In this paper we propose a conseptual framework for the observed properties
of discriminants of polylinear forms. The connection with classical problems of
linear algebra is shown. A new class of algebraic varieties (hypergrassmanians)
is introduced, the particular case of which are grassman manifolds. An
algorithm is given for computing the discriminants of polylinear forms of
"boundary format" (hypergrassmanian analogues of plukker coordinates). The
algorithm for computing the discriminants of polylinear forms of general
formats is outlined.
| [
{
"version": "v1",
"created": "Mon, 20 Nov 1995 11:23:57 GMT"
},
{
"version": "v2",
"created": "Fri, 15 Dec 1995 02:29:11 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Dolotin",
"Valerii V.",
""
]
] | alg-geom | \section{Introduction}
It is convenient (and instructive) to keep for certain objects of
multilinear algebra the terminology from their counterparts in linear
algebra (see [1]). So for the set of elements of $d-$linear form will
be used the term "$d-$dimensional matrix", and for its discriminant
(see definition below) we can use the term "determinant" with the notion
of "minors" also making sense.
The investigation of determinants of multidimensional
matrices may be useful already in the linear algebra (see Section 1), which
essentially deals with sets of rectangular matrices. A good example
(Section 1.1) is
a 3-dimensional interpretation of the theory of kronekker pairs, which
in this context obtains a straightforward generalization. It is
also interesting to look for a generalization of eigenvalue theory.
The theory of eigenvalues in its different versions is
equivalent to the investigation of matrices of type $A+\lambda B$,
where $A$ and $B$ is a paire of $n\times n$ matrices, and
invariants of this pair, which can be expressed in terms of
$GL(n)\times GL(n)$-action on $n\times n\times 2$ form (3-dimensional
matrix). In a similar way the
``multidimansional eigenvalue theory'' is reduced to the invariants of
3-dimensional matrices of larger formats.
We know, that in the case of bilinear forms (2-dimansional matrices)
the determinant (as one polynomial) is defined only for square matrices.
So in order to specify the size of 2-dimensional matrix which has a
unique expression (determinant) as a characteristic of its degeneracy,
it is enough to give one number $n$ - the number of rows or columns.
For polylinear forms the difference from bilinear ones we can
illustrate on the case of 3 dimensions.
Take a 3-dimensional matrix with elements
$a_{i_1i_2i_3}$, where $1\le i_1\le n_1$, $1\le i_2\le n_2$, $1\le i_3\le n_3$.
Let us fix $n_1$ and $n_2$. Then the determinant is defined for matrices
with $n_3$ satisfiing the following inequality (see [1]):
\begin{equation}
n_1-n_2+1\le n_3\le n_1+n_2-1
\end{equation}
So the size ("format") of $d$-dimensional matrices which have the determinant
is defined in general by $d$ parameters while in 2-dimentional case by only
1. But in d-dimensianl case for $d>2$ there also exists a class of
matrices the size of which is described by $d-1$ parameters. These are so
called ``matrices of boundary format'', the size of which corresponds
to the equality in $(0.1)$. In 2-dimensional case the matrices, the
degeneracy of which can not be characterized by one expression, are
rectangular matrices. In higher dimensional case these are ``matrices
of grassman format'', those which in case $d=3$ do not satisfy $(0.1)$.
It this paper we give a set of properties of matrices of boundary and
grassman formats, which show that they are the proper
generalizations if square and rectangular matrices correspondingly.
In particular, the condition for rectangular matrices of beeing of
corank 1 is that the determinants of all maximal square submatrices
(maximal minors) are 0. The corresponding first degeneracy condition for
d-dimensioanl matrices, defined here as the condition of beeing of
corank 1 (see Section 2.1), is that the determinants of all maximal
submatrices of boundary format are 0. To a rectangular matrix one can
put into correspondense a set of vectors - its rows or columns.
Then the condition for
beeing of corank 1 for rectangular matrix has a geometric
interpretation as the linear dependence of these vectors
(1-dimensional matrices). To a d-dimensional matrix with elements
$a_{i_1...i_d}$ one can put into correspondence a set of "slices" in
k-th direction, which are $(d-1)$-dimensional matrices with elements
$a_{i_1...\widehat i_k...i_d}$. Then the condition of
beeing of corank 1 for d-dimensional matrices of grassman format can be
expressed geometrically in terms of singularities of the intersection
of the span of these slices with the submanifold of
$(d-1)-$dimensional matrices of corank 1. The remarkable fact, making
the notion of corank 1 matrices well defined, is that this singularity
condition does not depend on the direction of slicing of our
matrix (the number k above).
Consider the problem of finding the kernel of a lenear combination
$S(\lambda):=\lambda_1 A_1+...+\lambda_k A_k$ of
k rectangular matrices of size $m\times n$. This kernel will be an
$(n-m)$-dimensional subspace, i.e. an element of $G_{m,n}$. Changing
$(\lambda_1,...,\lambda_k)$ we get an k-parametric subset in
$G_{m,n}$. In the case when $k=n-m+1$ the image of this subset via
plukker embedding of $G_{m,n}$ will be a Veronese manifold. The
Veronese manifolds which can be obtained in this way are called here
``proper Veronese manifolds''. The condition for a proper Veronese manifold to
be degenerate can be expressed in to ways:\\
$1)$ as the condition that there exists such $(\lambda_1,...,\lambda_k)$
that all plukker coordinates of the kernel of $S(\lambda)$ (which
are $m\times m$ minors of $S(\lambda)$) are equal to 0, or as the
singularity of the intersection of $span(A_1,...,A_k)$ with the
submanifold $M^{'}_{mn}$ of degenerate $m\times n$ matrices\\
$2)$ as the condition that the determinant of 3-dimensional matrix of
size $m\times n\times (n-m+1)$ made of the elements of
$A_1,...,A_{n-m+1}$ is 0.
In the case of grassman format, when $k>n-m+1$, the condition of the
existence of the intersection of $span(A_1,...,A_k)$ with $M^{'}_{mn}$
is that the determinants of all $m\times n\times (n-m+1)$ submatrices
of the corresponding 3-dimensional matrix are equal to 0. It happens
that the similar fact holds in general d-dimensional case. This allows
to interprete the determinants of maximal submatrices of boundary
format of matrix of grassman format as multidimensional anologues
of plukker coordinates and to consider the anologue of plukker map on
the space of d-dimensional matrices of grassman format:
$M_{n_1...n_d}\to \mbox{${\bf P}$} ^{\left( \matrix{n\cr m}\right)}$, putting into
correspondence for a matrix the set of its maximal minors of boundary
format. As in 2-dimensional case here arises the fundamental problem
to find the relations between the minors of multidimensional matrix,
the analogues of plukker relations, i.e. to discribe the image of
$M_{n_1...n_d}$ as an algebraic manifold.
For studiing the discriminants of polylinear forms there is a fundamental
question about the algorithm of explicit calculation of these discriminants.
In Section 3 we develop a technique which gives an algorithm of calculating
the discriminants of $d-$linear forms of boundary format (``hyperplukker
koordinates''). This technique happens to be basic for calculating
the discriminants of $d-$linear forms of general format.
In Section 4 we outline and give an example of this general algorithm.
\begin{Definition}
{\rm If $p(x_1,...,x_m)=\displaystyle
\sum^{}_{i_1\le{...}\le{i_n}}c_{i_1{...}{i_{n}}}x_{i_{1}}
{...}x_{i_{n}}$ is a homogeneous polynomial of degree $n$,
then the set of values of coefficients $c$ is called {\em discriminantal}
if the system of equations
\begin{equation}
\frac{\partial{p(x_1,...,x_m)}}{\partial{x_i}}=0,\quad i=1,...,m
\end{equation}
has a solution $(x_1,...,x_m)\in{(\mbox{${\bf C}$}^{*})^m}$.\\
In case when the discriminantal set is an algebraic submanifold of codimension
$1$ in the space of coefficients, it is called {\em the discriminant} of $p$,
denoted by $D(p)$.}
\end{Definition}
Now we will consider a particular case of this definition.
Let $V_{n_1},...,V_{n_d}$ be a set of linear vector spaces, such that
$dim(V_{n_i})=n_i$.
Let $a\in (V_{n_1}\otimes ...\otimes V_{n_d})^*$ be a $d-$linear
form. For a set of vectors $x^{(k)}\in V_{n_k}$, $k=1,...,d$ with coordinates
$x^{(k)}=(x^{(k)}_1,...,x^{(k)}_{n_k})$ in a chosen basis the value of
the form on $\otimes^d_{k=1}x^{(k)}$ is a polinomial of $d$ sets of
variables $x^{(1)},...,x^{(d)}$ of degree $d$
$$a(x^{(1)},...,x^{(d)})=p(x^{(1)},...,x^{(d)})$$
and the system $(0.2)$ becomes
\begin{equation}
\frac{\partial{a (x^{(1)},...,x^{(d)})}}{\partial{x^{(k)}_i}}=0,
\quad i=1,...,n_k, \quad k=1,...,d
\end{equation}
The coefficients $a_{i_1...i_d}$ of this form are the elements of
$d-$dimensional rectangular $n_1\times ...\times n_d$ matrix.
\begin{Definition}
{\rm The discriminant of the polinomial $a(x^{(1)},...,x^{(d)})$ is
called {\em the determinant} of $n_1\times ...\times n_d$ matrix
$(a_{i_1...i_d})$ and denoted by $det(a)$.}
\end{Definition}
{\bf Example.} Let $a\in (U_{n}\otimes V_{n})^*$. Then its
coefficients form a usual $n\times n$ square matrix. Then
$h=\displaystyle\sum^{}_{1\le i,j\le n} x_i a_{ij} y_j$.
The system $(0.2)$ in this case becomes:
\begin{equation}
\sum_{j=1}^n a_{ij} y_j=0,\quad i=1,...,n\quad
\sum_{i=1}^n x_ia_{ij}=0,\quad j=1,...,n
\end{equation}
Note, that this system contains the system of linear homogeneous
equations as well as its conjugate. For the discriminants of
polylinear forms this property will become essential.
There is a natural action of the group $GL_{n_1}\times ...\times GL_{n_d}$
on $\otimes^d_{j=1}V_{n_j}$ with the induced action on
$(\otimes^d_{j=1}V_{n_j})^*$. Since the system $(0.2)$ for
$p=a(x^{(1)},...,x^{(d)})$ is invariant under this action we have
\begin{Proposition}\label{P1}
The determinant of $n_1\times ...\times n_d$ matrix is invariant under
the action of $GL_{n_1}\times ...\times GL_{n_d}$.
\end{Proposition}
{\bf Notation} Denote $M_{n_1...n_d}:=\displaystyle
(\otimes^d_{j=1}V_{n_j})^{*}$.
\setcounter{section}{0}
\section{Problems of linear algebra and determinants of
$3-$dimensional matricies}
\subsection{General setting}
Let $M_{nm}$,where $n\le m$, be a linear space of $n\times
m$ matricies. Let $M^{\prime}_{nm}\subset M_{nm}$ be a
submanifold of matricies of rank $n-1$. Let $A_1,...,A_k\in M_{nm}$
be a set of $n\times m$ matrices. From their elements we can
make an $n\times m\times k$ matrix of coefficients of 3-linear
form $(a_{i_{1}i_{2}i_3})$. There is $1-1$ correspondence between
linear subspaces $span(A_1,...,A_k)\subset M_{nm}$ for different
choices of $A_1,...,A_k$ and the orbits
of the corresponding forms $a (A_1,...,A_k)$ under
the action of $GL_k$ on $M_{nmk}=(V_n\otimes V_m\otimes V_k)^*$.
\begin{example}
Let $A$ and $B$ be $n\times n$ matrices.
\begin{Proposition}\label{P1}
Let $det(B)\ne 0$. Then the following statements are aquivalent:\\
1) $det(a_{i_{1}i_{2}i_{3}}(A,B))=0$\\
2) $D(det(A+zB))=0$ or
$D(det(AB^{-1}-zI))=0$, i.e. the characteristic polinomial of
$AB^{-1}$ has multiple roots\\
where $D(p(z))$ denotes the discriminant of polinomial $p(z)$.
\end{Proposition}
\end{example}
\begin{example}
Let $A$ and $B$ be $n\times (n+1)$ matrices.
\begin{Proposition}
The determinant of $n\times (n+1)\times 2$ matrix
$det(a_{i_{1}i_{2}i_{3}}(A,B))\ne 0$ iff the pair $(A,B)$ is
``kronekker'', i.e. by the action of $GL_n\times GL_{n+1}$ it can be
reduced to the following canonical form\\
$$A=\left( \matrix{
1 & \ &{0} &0 \cr
\ & \ddots & \ &\vdots\cr
0 & \ &1 &0
}\right),\quad
B=\left( \matrix{
0 & 1 & \ & 0 \cr
\vdots & \ & \ddots & \ \cr
0 & 0 & \ & 1
}\right)$$
\end{Proposition}
\end{example}
\subsection{Veronese manifolds}
Let $k=m-n+1$, $A_1,...,A_k\in M_{nm}$.
Let $S(z_1,...,z_k):=z_1 A_1+...+z_k A_k$ be a point of subspace
$span(A_1,...,A_k)\subset M_{nm}$.
Let $\Delta(z_1,...,z_k):=(\Delta_{i_{1}...i_{m}}(S))_{(i_1,...,i_k)\in
\left( \matrix{[m]\cr n}\right)}\in \mbox{${\bf P}$} ^{\left( \matrix{n\cr
m}\right)}$ be a vector with components - $n\times n$ minors of $S$,
the image of $S(z)$ via Plukker embedding.
The map $\varphi : (z_1,...,z_k)\mapsto \Delta(z_1,...,z_k)$ gives a
$k-$parametric submanifold ${\cal{V}}(A_1,...,A_k)$ in
$\mbox{${\bf P}$} ^{\left( \matrix{m\cr n}\right)}$.
\begin{Proposition}
For $A_1,...,A_k$ in general position the intersection
$M^{\prime}_{nm}\cap span(A_1,...,A_k)$ is empty.
\end{Proposition}
\begin{Proposition}
For $A_1,...,A_k$ in general position the manifold
${\cal{V}}(A_1,...,A_k):=\varphi (\mbox{${\bf C}$}^k)\in \mbox{${\bf P}$} ^{\left(
\matrix{m\cr n}\right)}$ is a veronese manifold.
\end{Proposition}
The veronese manifolds obtained in this way we call {\em proper}.
\begin{Theorem}
The following statements are equivalent:\\
1) the intersection $M^{\prime}_{nm}\cap span(A_1,...,A_k)$ is not empty\\
2) $det(a_{i_{1}i_{2}i_{3}}(A_1,...,A_k))=0$\\
3) the manifold ${\cal{V}}(A_1,...,A_k)$ is singular\\
4) ${\cal{V}}(A_1,...,A_k)$ belongs to a hyperplane in $\mbox{${\bf P}$} ^{\left(
\matrix{m\cr n}\right)}$
\end{Theorem}
Let $(x_1,...,x_n),(y_1,...,y_m),(z_1,...,z_k)$ be the coordinates in
$V_n,V_m$ and $V_k$ correspondingly.
For $a=a(A_1,...,A_k)\in M_{nmk}=V_n\otimes V_m\otimes V_k$ the
system $(0.2)$, where $p=a (x,y,z)$, contains a subsystem
\begin{equation}
\frac{\partial{a(x,y,z)}}{\partial{x_{i}}}=
\sum^m_{j=1}(z_1A_1+...+z_kA_k)_{ij}y_j=S(z)y=0,\quad i=1,...,n
\end{equation}
To a given $z\in (\mbox{${\bf C}$}^*)^k$ we can put into correspondence a subspace
$Ker(S(z))\subset V_m$ of solutions of (1.5). If $rank(S(z))=n$ then
$Ker(S(z))$ has dimension $(m-n)$. If $rank(S(z))<n$
then $Ker(S(z))$ has dimension greate then $(m-n)$.
\begin{Proposition}
$det(a(A_1,...,A_k))=0$ iff there are values of $z\in (\mbox{${\bf C}$}^*)^k$
such that the dimension of the space of solutions of corresponding
system (1.5) is greate then $(m-n)$ and the $\varphi$-
image of the set of such $z$ is exactly the set of singular points of
${\cal{V}}(A_1,...,A_k)$.
\end{Proposition}
\section{Hyperveronese and hypergrassmanian manifolds}
\subsection{On the rank of polylinear forms}
For a given form $a\in M_{n_1...n_d}$ and an integer number $1\le k\le
d$ we can put into correspondence a set of forms ${a}^{(k)}_{i_k}\in
M_{n_1...\widehat{n}_k...n_d},\quad i_k=1,...,n_k$, such that
$({a}^{(k)}_{i_k})_{i_{1}...\widehat{i}_{k}...i_{d}}=
a_{i_{1}...i_{d}}$.
For a given $k$ there is $1-1$ correspondence between the orbits
of the forms $a$ under
the action of $GL_{n_k}$ on $M_{n_1...n_d}$
and the corresponding linear subspaces
$span({a}^{(k)}_1,...,{a}^{(k)}_{n_k})\subset M_{n_1...\widehat{n}_k...n_d}$.
We say, that the {\em $GL_{n_k}$-orbit of $a$ has an intersection} with
a submanifold in $M_{n_1...\widehat{n}_k...n_d}$ if
$span({a}^{(k)}_1,...,{a}^{(k)}_{n_k})$ has an intersection
with this submanifold.
{\bf Notation.} Denote by $M'_{n_1n_2}$ the set of $n_1\times n_2$ matrices of
corank 1.
For a multiindex $n_1...n_d$ let $\widehat{n}_k:=n_1...n_{k-1}n_{k+1}...n_d$.
Now by induction on $d$ we can introduce the following definition.
\begin{Definition}
A subset of the space $M_{n_1...n_d}$ is called the set
of forms of corank $1$ (denoted by $M'_{n_1...n_d}$) if for $a
\in M'_{n_1...n_d}$ and any "direction" $k=1,...,d$
the intersection $span({a}^k_1,...,{a}^k_{n_k})
\cap M^{'}_{\widehat{n}_k}$
is not in general position (i.e when $a$ comes onto $M'_{n_1...n_d}$ the
topology of this intersection changes).
\end{Definition}
\begin{Theorem}
The set $M'_{n_1...n_d}$ is the discriminantal set for the system (0.3).
\end{Theorem}
As it is shown in Section 2.2 $M'_{n_1...n_d}$ is an algebraic manifold.
\begin{Definition}
{\rm The space $M_{n_{1}...n_{d}}$ is called the space of {\it inner format}
if for any $a\in M_{n_{1}...n_{d}}$ and any "direction" $k=1,...,d$
the intersection of $GL_{n_k}$ orbit of $a$
with the submanifold $M'_{\widehat{n}_k}$ of $(d-1)-$ linear
forms of corank $1$ is not empty.\\
Otherwise the format $n_1...n_d$ is called {\it grassmanian}. The
grassmanian format $n_1...n_d$ for which $M'_{n_1...n_d}$ has
codimension 1 is called {\it boundary}.}
\end{Definition}
The term "grassmanian format" is motivated by the fact that a certain
(see Section 2.2) factorization of the space $M_{n_{1}...n_{d}}$ gives
an algebraic manifold which in case $d=2$ is the grassman
manifold $G_{n_1,n_2}$.
{\bf Example}
Consider the case of 3-linear forms of format $2\times 2\times n$ for
$n\ge 4$. These are fograssmanoundary format. For a given $a
\in M_{22n}$ take its $GL_{n}$-orbit, i.e. the set of linear
combinations $S(z):=A_1z_1+...+A_nz_n$, where
$2\times 2$ matrices $A_1,...,A_n$ are $2\times 2$ slices of $a$.
The intersection of $span(A_1,...,A_n)$ with
the submanifold $M^{'}_{22}$ of degenerate $2\times 2$ matrices
corresponds to the set of such $(z_1,...,z_n)$ that $det(S(z))=0$.
The expression $det(S(z))$ is a quadratic form on $z$. The
intersection of $span(A_1,...,A_n)$ with $M^{'}_{22}$ is described in
terms of the rank of this quadratic form. Then the notion of the corank of our
$3-$linear form $a (A_1,...,A_n)$ can be formulated in terms of
the rank of quadratic form $det(S(z))$ as follows:
\begin{Proposition}
The corank of $2\times 2\times n$ form $a (A_1,...,A_n)$ is equal
to 1 iff the rank of quadratic form $det(S(z))$ is equal to 2.
\end{Proposition}
\subsection{Proper hyperveronese manifolds}
For a set of integers $n_1,...,n_d$ let $m_d:=n_1+...+n_d+1-d$.
Take an $n_1\times ...\times n_r \times n_{r+1}\times m_{r+1}$ matrix
$(a_{i_1..i_{r+1}j})$. Denote $T_i:=a^{(r+1)}_i,\quad i=1,...,n_{r+1}$
the $n_1\times ...\times n_r\times m_{r+1}$ "slices" of $a$.
For $(z_1,...,z_{n_{r+1}})\in {(\mbox{${\bf C}$}^{*})^{n_{r+1}}}$ let
$S(z_1,...,z_{n_{r+1}}):=z_1T_1+...+z_{n_{r+1}}T_{n_{r+1}}$ be points of
$span(T_1,...,T_{n_{r+1}})$. Denote by $\Delta_{j_1...j_{m_r}}(S)$,
where $(j_1,...,j_{m_r})\in \left(\matrix{[m_{r+1}]\cr m_r}\right)$ the
$n_1\times ...\times n_r \times m_r$ minors of $S$.
Then the problem of finding the intersection of $span(T_1,...,T_{n_{r+1}})
\cap M'_{n_1...n_rm_{r+1}}$ is the problem of solving the system:
\begin{equation}
\Delta_{j_1...j_{m_r}}(S)=0,
\quad (j_1,...,j_{m_r})\in \left(\matrix{[m_{r+1}]\cr m_r}\right)
\end{equation}
\begin{Proposition}
The system (6) has a nonzero solution $z\in (\mbox{${\bf C}$}^*)^{n_{r+1}}$ iff $a$
belongs to a submanifold of codimension 1.
\end{Proposition}
According to Definitions 2.1.1 and 2.1.2 this means that the format
$n_1\times ...\times n_{r+1}\times m_{r+1}$ is boundary grassmanian.
The set $\Delta=(\Delta_{j_1...j_{m_r}})$ of minors gives us the components of
a vector in $\mbox{${\bf P}$}^{\left(\matrix{[m_{r+1}] \cr m_r}\right)}$. So we have
a map:
$$\phi :(C^*)^{n_{r+1}}\to \mbox{${\bf P}$}^{\left( \matrix{m_{r+1}\cr m_r}\right)}$$
$$(z_1,...,z_{n_{r+1}})\mapsto \Delta (z)$$
the image of which is a manifold $\cal{V}$ parametrized by
$(z_1,...,z_{n_{r+1}})$.
\begin{Theorem}
The following statements are equivalent:\\
1) $det(a_{i_1...i_{r+1}j})=0$\\
2) the manifold $\cal{V}$ is singular\\
3) the system
$$\Delta_{j_1...j_{m_r}}(S(z))=0,\quad (j_1,...,j_{m_r})\in
\left( \matrix{[m_{r+1}]\cr m_r}\right)$$
has solutions in $(\mbox{${\bf C}$}^*)^{n_{r+1}}$ and $\phi$ gives a $1-1$ correspondence
between the solutions of the system and the singular points of $\cal{V}$.
\end{Theorem}
Compairing this statement with Teorem 1.2.1 we are lead to the following:
\begin{Definition}
{\rm For $(a)\in M_{n_1...n_{r+1}m_{r+1}}$ the manifold $\cal{V}$ is called
{\it proper hyperveronese manifold}}.
\end{Definition}
\subsection{Hypergrassmanians}
Let $(a_{i_1...i_rj})\in M_{n_1...n_rm}$ where $m>m_r=1+n_1+...+n_r-r$.
The $n_1\times ...\times n_r\times m_r$
minors $\Delta_{j_1...j_{m_r}}$ of $(a)$ are invariants of $GL_{n_1}\times
...\times GL_{n_r}$ action on $M_{n_1...n_rm}=(V_{n_1}\otimes ...\otimes
V_{n_r}\otimes V_{m})^*$.
For $1\le k\le r$ take the set $a^{(k)}_1,...,a^k_{n_k}$ of
$n_1\times ...\times {\widehat{n}_k}\times ...\times m$
"slices" of $(a)$ in $k-$th direction (see Section 2.1).
\begin{Proposition}
The intersection $span(a^{(k)}_1,...,a^{(k)}_{n_k})\cap M'_{\widehat{n}_k}$
is not empty iff
$$\Delta_{j_1...j_{m_r}}(a)=0,\quad for all (j_1,...,j_{m_r})\in
\left( \matrix {[m]\cr m_r}\right).$$
\end{Proposition}
This implies that the map
$${\cal{P}}: M_{n_1...n_rm}\to \mbox{${\bf P}$}^{\left( \matrix{m\cr m_r}\right)}$$
$$(a)\mapsto \Delta (a)$$
has the kernel $M'_{n_1...n_rm}$, induces an injection of the open stratum
of the space of orbits $M_{n_1...n_rm}/GL_{n_1}\times ...\times GL_{n_r}$
into $\mbox{${\bf P}$}^{\left( \matrix {[m]\cr m_r}\right)}$ and its image is a projective
algebraic manifold ${\cal{G}}_{n_1...n_r,m}$.
\begin{Definition}
{\rm The manifold ${\cal{G}}_{n1...n_r,m}$ is called {\it hypergrassmanian}.}
\end{Definition}
So the coordinates on a hypergrassmanian (the open stratum of the factor space
of the space $M_{n_1...n_rm}$ of forms of grassmanian format), as in the
particular case (for $r=1$) of grassman manifolds, are given by the minors
of boundary format ("hyperplukker coordinates").
\section{Algorithm}
Here we give an algorithm for computing the determinants of
matrices of boundary format (``hyperplukker coordinates'').
\subsection{Basic example}
Let $P_0,...,P_{d}$ be a sequence of ordered sets
$P_k$, such that $|P_k|=k+1$.\\
{\bf Notation.} Denote by $\cal{C}$ the space of
sequences $q:=(q_1,q_2,...,q_d)$, where $q_k\in{P_k}$.
For a pair of such sequences $q^{'},q^{''}\in{\cal{C}}$ we say that
$q'\le{q^{''}}$, if $\exists K<{d}$, such that
$q^{'}_k\le{q^{''}_k}$ in $P_k$, for $k>{K}$.
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The sequence $P_0,...,P_4$ can be represented
in the form of a diagram on Fig.1, where $(k+1)-$th row represents
the elements of the set $P_k$ ordered from the left to the right.
A sequence
$q$ may be represented in the form of a diagram on Fig.2, where
$q_k$ is represented by a cross in the $(k+1)-$th row.
For a given $q_{k+1}\in{P_{k+1}}$ there is a unique
ordering-preserving injection
$f^q_{k}:P_k\hookrightarrow{P_{k+1}}$, such that ${q_{k+1}}${\put(5,3)
{\makebox(0,0){$\in$}}}{\put(0,3){\makebox(0,0){$\not$}}}
\quad${f^q_{k}(P_k)}$ and vise versa.
For $q$ represented by Fig.2 the corresponding sequence
$f^q:={(f^q_1,...,f^q_4)}$ of injections may be represented in
form of a diagram on Fig.3.
Then each sequence $q$ defines a sequence $p(q):=(p_1,...,p_d)$, where
$p_k\in{P_k}$, such that $p_{k+1}=f^q_{k}(p_k)$. The sequence
$p$, corresponding to the sequence $q$ on Fig.2, may be represented
in form of a diagram on Fig.4, where the path goes through the
elements $p_k$. The rule for drawing a path, corresponding to a
given sequence $q$ may be formulated as follows:
\begin{em}
if $q_{k+1}\in{P_{k+1}}$ lies (on the diagram) to the left from $p_k$,
then we have to go from $p_k$ to the right closest to it point
of $P_{k+1}$, if $q_{k+1}\in{P_{k+1}}$ lies to the right from
$p_k$, then we have to go from $p_k$ to the left closest
point of $P_{k+1}$.
\end{em}
On the other hand, for a given $q_k$ the corresponding
$f^q_k:P_k\to{P_{k+1}}$ defines a partition of $P_k$ into two
parts $P^{-}_{k}(q)$ and $P^{+}_{k}(q)$, where
$$P^{-}_{k}(q):=\{{p_k\in{P_k}|f^q_k(p_k)<q_{k+1}}\}$$
$$P^{+}_{k}(q):=\{{p_k\in{P_k}|f^q_k(p_k)>q_{k+1}}\}$$
The sequence of partitions, corresponding to the sequence of
ordering pre- serving injections
$f^q=(f^q_0,f^q_1,f^q_2,f^q_3)$ on Fig.3, may be represented in form of
a diagram on Fig.5, where in the $k-$th row the elements
of $P^{-}_{k}(q)$ are represented by characters ``1'' and the
the elements of $P^{+}_{k}(q)$ are represented by characters ``2''.
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{\bf Notation.}Denote by $j(q)$ the ordinal number of the element
$p_{d}(q)\in{P_{d}}$ with respect to the ordering on $P_{d}$.
For a given pair $q^{'},q^{''}\in{\cal{C}}$ denote by
$[i_1,...,i_d,j](q^{'},q^{''})$ the sequence of indecies
$(i_1,...,i_d,j)$ such that $$j=j(Q)
{\quad}and{\quad}i_{k+1}={\left\{\begin{array}{ll}
1,\quad{{\rm if \ }p_{k}(q^{'})\in{P^{-}_{k}(q^{''})}}\\
2,\quad{{\rm if \ }p_{k}(q^{'})\in{P^{+}_{k}(q^{''})}}
\end{array}\right.}$$
For example, for $q^{'}=q^{''}=q$ from Fig.2 we can write $(i_1,...,i_4,j)$
using the diagram on Fig.6, which may be viewed as a ``superposition''
of Fig.4 and Fig.5, $(i_1,...,i_4,j)=[i_1,i_2,i_3,i_4,j](q,q)=(2,1,2,2,4)$.
\begin{Proposition}\label{P1}
$$\prod^{}_{q\in{\cal{C}}}{a_{[i_1,...,i_d,j](q,q)}}$$
is a monomial of discriminant of polylinear form of
dimension ${\underbrace{2\times{2}\times{...}\times{2}}_{\mbox
{\scriptsize{d\ {times}}}}}$
$\times{(d+1)}$ with coefficients $(a_{i_1,...,i_d,j})$.
\end{Proposition}
This monomial will be called {\it diagonal{\ }monomial\/}.
\subsection{General algorithm}
Let $n_1,...,n_d$ be a sequense of positive integers and set $n_0=1$.
Denote $m_k:=n_0+n_1+...+n_k-k$.
Let $P_0,...,P_d$ be a sequence of ordered sets, such that
$|P_k|=m_k$. Let $T_1,...,T_d$ be a sequence of ordered sets,
such that $|T_k|=n_k$.\\
{\bf Notation.} Denote by ${\cal{C}}_k:=\{Q_k\subset P_k\
:\ |Q_k|=m_k-m_{k-1}\}$ and by $\cal{C}$ the space of sequences
$Q:=(Q_1,...,Q_d)$, where $Q_k\in {\cal{C}}_k$.
For a pair
$S^{'},S^{''}\subset{P_k}$, where $S^{'}=(s^{'}_1,...,s^{'}_{n_k}),\
S^{''}=(s^{''}_1,...,s^{''}_{n_k})$ and the elements of $S^{'}$ and
$S^{''}$ are written in the order induced by the ordering on $P_k$,
we say that $S^{'}\le{S^{''}}$ if $\exists N<n_k$, such that $s^{'}
_n\le{s^{''}_n}$ for $n>N$. This gives the ordering of ${\cal{C}}_k$.
For a pair of sequences $Q^{'},Q^{''}\in{\cal{C}}$ we say that
$Q^{'}\le{Q^{''}}$ if $\exists K<d$, such that $Q^{'}_k\le{Q^{''}_k}$,
for $k>K$.
For a given $Q_{k+1}\in {\cal{C}}_{k+1}$ there exists a unique
order preserving injection $f^Q_k:P_k\hookrightarrow P_{k+1}\setminus
Q_{k+1}$
Then each sequence
$Q\in{\cal{C}}$ defines a sequence $p(Q):=(p_0,p_1,...,p_d)$, where
$p_k\in P_k$, such that $p_{k+1}=f^Q_k(p_k)$ which is caled {\em $Q-$path}.
An example of a $Q-$path is shown on Fig.4.
On the other hand, since $P_k$ are ordered, then a given $Q_k$
defines a partition $R_k=(R^1_k,...,R^{n_k}_k)$ of $P_k\setminus Q_k$
into $n_k$ subsets $R^i_k$.
Then any injection $\phi : P_{k-1}\hookrightarrow P_{k}\setminus
Q_k$ induce a map $g_k: P_{k-1}\to T_k$ as follows:
\begin{equation}
if p\in \phi^{-1}(R^i_k)\quad then g_k(p)=i\in T_k
\end{equation}
where we write number $i$ instead of the $i-$th element of $T_k$.
\begin{Definition}
{\rm For a given $Q_k\in {\cal{C}}_k$ such a map will be called
{\it $Q-$admissible} and the sequence $g^Q=(g_1,...,g_d)$ of $Q-$admissible
maps is called a {\it $Q-$diagram}. If in a $Q-$diagram all $g_k$ are induced
by
the order preserving injections $f^Q_k$, then this $Q-$diagram is called
{\it initial}}.
\end{Definition}
An example of the initial $Q-$diagram is shown on Fig.5.
Now let us take a pair $Q^{'},Q^{''}\in\cal{C}$. Then for $Q^{'}$
we take the $Q^{'}-$path $p(Q^{'})=(p_0,p_1,...,p_d)$ and for $Q^{''}$
we choose a $Q^{''}-$diagram $g^{Q^{''}}=(g_1,...,g_d)$ from the set of
$Q^{''}-$admissible diagrams. This gives us a sequence of indecies:
\begin{equation}
I(Q^{'},g^{Q^{''}})=(i_1,...,i_d), \quad where
i_k:=g^{Q^{''}}(p^{Q^{'}}_{k-1}).
\end{equation}
\begin{Definition}
{\rm The pair $(p^{Q^{'}},g^{Q^{''}})$ is called {\it $Q^{'}-$path over a
$Q^{''}-$diagram}.}
\end{Definition}
An example of this construction is shown on Fig.6.
{\bf Notation.} Denote by ${\cal{C}}^{(k)}$ the space of subsequences
$Q^{(k)}:=(Q_{k+1},...,Q_{d})$.
\begin{Definition}
{\rm For a given $Q^{(k)}\in{{\cal{C}}^{(k)}}$ two sequences
$Q^{'},Q^{''}\in{\cal{C}}$ will be called {\em $Q^{(k)}-$conjugate}
if $Q^{'}_m=Q^{''}_m=Q^{(k)}_m$ for $m>k$.
For a given $Q^{(k)}\in{{\cal{C}}^{(k)}}$ and $p:=(p_1,...,p_d)$
two sequences $Q^{'},Q^{''}
\in{\cal{C}}$ will be called {\it $(Q^{(k)},p_k)-$conjugate}, if
$Q^{'}_m=Q^{''}_m=Q^{(k)}_m$ for $m>k$, and $p(Q^{'})_k=
p(Q^{''})_k=p_k\in{P_k}$.}
\end{Definition}
{\bf Notation.} Denote by ${\cal{D}}{(Q^{(k)})}$ the set of all
$Q^{(k)}-$conjugate sequences and by ${\cal{D}}{(Q^{(k)},p_k)}$
the set of all $(Q^{(k)},p_k)-$conjugate sequences in $\cal{C}$.
Set by definition ${\cal{D}}(Q^{(d)}):=\cal{C}$.
\begin{Proposition}
For a given $k\le d$ $$|{\cal{D}}{(Q^{(k)},p_k)}|=\frac{(m_k-1)!}
{(n_1-1)!...(n_k-1)!}$$ for any $p_k\in P_k$.
\end{Proposition}
On each ${\cal{D}}{(Q^{(k)},p_k)}$ there is an ordering induced by
the ordering on $\cal{C}$.\\
{\bf Notation.} For a given $k$ denote by ${\cal{L}}_{k}$
the set of ordinal numbers,
ennumerating the elements of each ${\cal{D}}{(Q^{(k)},p_k)}$.
{\em Remark:} Proposition 2.1 implies, that
$|{\cal{L}}_{0}|=1$ and $|{\cal{L}}_{1}|=1$.\\
Then for a given $Q\in {\cal{C}}$ we
have a sequence of integer numbers $L(Q):=(l_1,...,
l_d)$, where $l_k (Q)$ is the ordinal number of $Q$
in ${\cal{D}}(Q^{(k)},p_k)$.\\
{\bf Notation.} For a given $Q^{(k)}$ and $l\in
{\cal{L}}_{k-1}$ denote by ${\cal{E}}(Q^{(k)},l)$ the
set of sequences $Q$, such that $l_{k-1}(Q)=l$.
Denote by $S_{{\cal{C}}_k}$ the group of permutations of the elements of
${\cal{C}}_k$.
Then $S_{{\cal{C}}_k}$ acts on ${\cal{E}}(Q^{(k)},l)$ as follows:
for $\sigma\in S_{{\cal{C}}_k}$ and a given $Q=(Q_1,...,Q_k,Q^{(k)})
\in {\cal{E}}(Q^{(k)},l)$, $$\sigma Q=(Q_1,...,
\sigma Q_k,Q^{(k)}).$$
If to each ${\cal{E}}(Q^{(k)},l_{k-1})$ we put into correspondence a group
$S(Q^{(k)},l_{k-1})\cong S_{{\cal{C}}_k}$ with the action discribed above,
then on the whole ${\cal{C}}$ we have the action of the group
$$\Sigma:=\prod^{d-1}_{k=0}\ \prod^{}_{Q^{(k+1)}}
\ \prod^{}_{l_{k}}{S(Q^{(k+1)},l_{k})}$$
Each $Q\in{\cal{C}}$ defines the following subgroup $\Sigma_Q$ of
$\Sigma$:$$\Sigma_Q=S_{{\cal{C}}_1}(Q)\times
...\times{S_{{\cal{C}}_d}}(Q)$$ where $S_{{\cal{C}}_k}(Q)=S(Q^{(k)},
\l_{k-1}(Q))$. $\Sigma_Q$ acts on $Q$ componentwise: for $\tau=
(\tau_1,...,\tau_d)\in \Sigma_Q$
$$\tau Q=(\tau_{1}Q_1,...,\tau_d Q_d).$$
The role of the group $\Sigma$ is analogous to the role of
symmetric group in calculation of the determinant of $n\times n$ matrix.
\begin{Proposition}
If $Q\in\cal{C}$, $\sigma\in\Sigma$ and $\sigma_Q\in\Sigma_Q\subset
\Sigma$ is the $\Sigma_Q$-component of $\sigma$ then
$$\sigma Q=\sigma_Q Q$$.
\end{Proposition}
For $\sigma\in \Sigma$ denote $$sign(\sigma):=\prod^{d-1}_{k=0}
\ \prod^{}_{Q^{(k+1)}}
\ \prod^{}_{l_{k}}{sign(\sigma(Q^{(k+1)},l_{k}))}$$ where
$sign(\sigma(Q^{(k+1)},l_{k}))$ is the signature of permutation
$\sigma(Q^{(k+1)},l_{k})\in S(Q^{(k+1)},l_{k})$.
Let us put into correspondence to each $Q^{(k)}\in {\cal{C}}^{(k)}$
and $l_k\in{\cal{L}}_k$ a set $G(Q^{(k)},l_k)$
of $Q_{k+1}$-admissible functions $g_{k+1}$ and take their direct product:
$$\Gamma:=\prod^{d-1}_{k=0}\prod^{}_{Q^{(k)}}\prod^{}_{l_{k}}{G(Q^{(k)},l_k)}$$
Denote by $j(Q)$ the ordinal number of the element $p_{d}(Q)\in P_d$
with respect to the ordering on $P_d$.
For given $Q\in\cal{C}$, $\sigma\in\Sigma$ and $\gamma\in\Gamma$ denote
by $[i_1,...,i_d](Q,\sigma Q,\gamma)$ the sequence of indecies
$(i_1,...,i_d,j)$, where $$j=j(Q),{\quad} i_k=g_k(p_{k-1}(Q)){\quad}
and{\quad}g_k=\gamma(Q^{(k)},l_{k}(Q))\in{G((\sigma Q)^{(k)},l_k(Q))}.$$
\begin{Theorem}
If $\Omega=(a_{i_1...,i_dj})$ is a polylinear form of
dimension $n_1\times{...}\times{n_d}\times{m_d}$, then its discriminant
$$D_{\Omega}=\sum^{}_{\sigma\in\Sigma}{sign(\sigma)}{\sum^{}_{\gamma
\in\Gamma}}\ {\prod^{}_{Q\in\cal{C}}}{a_{[i_1,...,i_d,j]
(Q,\sigma Q,\gamma)}}$$
\end{Theorem}
{\bf Example.} {\em Let $d=1$, $n_1=n$}. \\Then $m_1=n$,
${\cal{C}}={\cal{C}}_1=\{{Q_1\subset P_1:\ |Q|=n-1}\}$ and each $Q$ is
defined by the value of $j(Q)$. Since
$|{\cal{L}}_0|=1$, then for each $Q=Q_1$ the set $G(Q^{(0)})$
consists of only one element $g^Q$, such that the ordinal number
of $g^{Q}(p_0)$ in $T_1$ is equal to $j(Q)$, and $\Gamma=\prod^{}
_{Q^{(0)}}G(Q^{(0)})$ consists of only one element $\gamma=\prod^{}_
{Q\in\cal{C}}g^{Q}$. Also $\Sigma=S_{{\cal{C}}_1}\cong S_n$.
Then for $\Omega=(a_{ij})_{1\le{i,j}\le{n}}$
$$D_{\Omega}=\sum^{}_{\sigma\in S_n}{sign(\sigma)}\prod^{}_{Q\in\cal{C}}
{a_{[i,j](Q,\sigma Q,\gamma)}}=
\sum^{}_{\sigma\in S_n}{sign(\sigma)}\prod^{n}_{j=1}{a_{\sigma
j,j}}$$ is the determinant of the square matrix $(a_{ij})_
{1\le{i,j}\le{n}}$.
\section{Closed determinant}
\begin{Definition}
{\rm For $n_1\times {...}\times n_d$ $d-$dimensional
matrix $(a_{i_1...i_d})$ the product of all its minors
(including the determinant) is called
the {\it closed determinant} (denoted by $Det(a)$)}
\end{Definition}
The term "closed" comes from the fact that as an algebraic manifold $Det(a)$
corresponds to the projectively dual to the closure of the
$(\mbox{${\bf C}$}^*)^{n_1+...+n_d}$ orbit of $(1,...,1)\otimes {...}\otimes (1,...,1)\in
V_{n_1}\otimes{...}\otimes V_{n_d}$.
{\bf Example.} {\em $2\times 2\times 2$ matrix}.
Let $(a_{i_1i_2i_3})_{i_1,i_2,i_3=1,2}$ be a $2\times 2\times 2$ matrix.
Then its closed determinant
$$Det(a)=a_{111}a_{112}a_{121}a_{122}a_{211}a_{212}a_{221}a_{222}\times$$
$$\times (a_{111}a_{122}-a_{121}a_{112})(a_{211}a_{222}-a_{221}a_{212})
(a_{111}a_{212}-a_{211}a_{112})(a_{121}a_{222}-a_{221}a_{122})
(a_{111}a_{221}-a_{211}a_{121})(a_{112}a_{222}-a_{212}a_{122})\times$$
$$\times
(a^2_{111}a^2_{222}+a^2_{112}a^2_{221}+a^2_{121}a^2_{212}+a_{211}^2a_{122}^2
-2a_{111}a_{121}a_{212}a_{222}-2a_{111}a_{211}a_{122}a_{222}
-2a_{111}a_{112}a_{221}a_{222}-2a_{121}a_{221}a_{112}a_{212}
-2a_{211}a_{221}a_{112}a_{122}-2a_{212}a_{211}a_{121}a_{122}
+4a_{111}a_{221}a_{212}a_{122}+4a_{121}a_{211}a_{112}a_{222})$$
\begin{Proposition}
The degree of the closed determinant of $d-$dimensional matrix of format
$n_1\times {...}\times n_d$ is equal to the degree of the (ordinary)
determinant of $(d+1)-$dimensional matrix of boundary format $n_1\times
{...}\times n_d\times (1+n_1+...+n_d-d)$.
\end{Proposition}
Let us take for each initial $Q-$diagram the $Q-$path on it (see Definition
3.2.1).
The corresponding set of indecies will be denoted by $I(Q)$.
\begin{Theorem}
Let $(a_{i_1...i_d})$ be a $d-$dimensional matrix.
The monomial
\begin{equation}
\prod^{}_{Q\in \cal C}a_{I(Q)}
\end{equation}
is a monomial of the closed determinant $Det(a)$.
\end{Theorem}
\begin{Definition}
{\rm The monomial} $\prod^{}_{Q\in {\cal C}}a_{I(Q)}$ {\rm is called {\it
diagonal}}.
\end{Definition}
There is an algorithm given in terms of paths over $Q-$diagrams (of which
the evidencies of existence are Proposition 4.1 and Theorem 4.1) of
computing closed determinants of polylnear forms, which will be published
separately. Here we give an example of this procedure.\\
{\bf Example.} {\em $2\times 2\times 2$ matrix}.
As a tool for our calculation let us draw the set of initial $Q-$diagrams
with corresponding $Q-$paths on them
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\end{center}
The diagrams are grouped into four rows with three pairs in each row.
Let us ennumerate the diagrams by triples of numbers $(q_1,q_2,q_3)$, where
$q_3=1,2,3,4$ is the row number, $q_2=1,2,3$ is the
number of the group and $q_1=1,2$ is the number inside the group. For a
diagram with a number $(q_1,q_2,q_3)$ its rows represent the functions
$g_1^{q_1,q_2,q_3}:P_0\to {1,2}, g_2^{q_1,q_2,q_3}:P_1\to {1,2},
g_3^{q_1,q_2,q_3}:P_2\to {1,2}$ (where $P_0, P_1, P_2$ are ordered sets
from Section 3.2 such that $|P_0|=1, |P_1|=2, |P_2|=3$), so to say
``function $g_k^{q_1,q_2,q_3}$'' is the same as to say ``the $k-$th row
of the diagram $(q_1,q_2,q_3)$'' and vice versa. Each triple $(q_1,q_2,q_3)$
is just a sequence $Q$ of $1-$element subsets of $P_1,...,P_3$ (see Section
3.2), so here we can say ``$(q_1,q_2,q_3)$-diagram'' instead of
``$Q-$diagram for $Q=(q_1,q_2,q_3)$''.
The set of $Q-$paths on Fig.7 gives us the diagonal monomial
$$a_{111}a_{112}a_{121}a_{122}a_{211}a_{212}a_{221}a_{222}
a_{111}a_{122}a_{211}a_{222}a_{111}a_{212}a_{121}a_{222}
a_{111}a_{221}a_{112}a_{222}\times$$
$$\times a^2_{111}a^2_{222}$$
of $Det(2\times 2\times 2)$. The rest of monomials is obtained as
$Q-$paths over diagrams obtained from initial ones by
permutations of the following type:
i) Permutations of odd type. These are permutations of rows between
different diagrams. The generators are:
1) for a given $(q_2,q_3)$ and $\tau\in S_2$ the action of $\tau$
on $g_1^{\bullet,q_2,q_3}$ (first rows of diagrams $(1,q_2,q_3)$
and $(2,q_2,q_3)$) is the following
\begin{equation}
\tau (g_1^{q_1,q_2,q_3})=g_1^{\tau (q_1),q_2,q_3}
\end{equation}
2) for a given $(q_3)$ and $\tau \in S_3$ the action of $\tau$
on $g_1^{\bullet,\bullet,q_3}$ is the following
\begin{equation}
\tau (g_2^{\bullet,q_2,q_3})=g_2^{\bullet,\tau(q_2),q_3}
\end{equation}
and permutation $\tau$ for $q_3=2$ has to be chosen the same as for
$q_3=3$. In other words we have to permute the corresponding second
rows of diagrams with $q_3=2$ and $q_3=3$ synchronously, so the permutations
of second rows are the elements of the group $S_3\times S_3\times S_3$.\\
We assign to an odd type permutation the sign wich correspond to its parity.
ii) Permutations of even type. These are permutations of elements in the
rows (elements of $P_k$). The generators are:
for a given $(q_3)$ permute the elements in second rows (elements of sets
$P_1$)
of $(\bullet,2,q_3)-$diagrams (the $\bullet$ means that this permutation does
not depend on the value of $q_1$) so that these permutations for diagrams with
$q_3=2$ and $q_3=3$ coincide (synchronization condition).\\
We assign to even type permutations positive sign.
The permutations of different (odd and even) types commute.
For a given permutation the sign of the monomial computed from $Q-$paths
over $permutation(Q)-$diagrams equals to the sign of odd type component
of the permutation. As a prelude to the general algorithm we can remark
that "synchronization" of permutations on the space of $Q-$diagrams takes
place for the sets of permutations which have the same domaine of values
of $g_3:P_2\to T_3$.
As soon as we can compute the closed determinant for a form of a given
format, its determinant is computed as the quotient of the closed
determinant and the product of all its minors, which are the
determinants of submatricies of smaller formats.
|
1995-11-16T06:20:14 | 9511 | alg-geom/9511009 | en | https://arxiv.org/abs/alg-geom/9511009 | [
"alg-geom",
"math.AG"
] | alg-geom/9511009 | Misha Verbitsky | Misha Verbitsky | Cohomology of compact hyperkaehler manifolds and its applications | 12 pages, LaTeX 2.09 | GAFA vol. 6 no. 4 pp. 601--612 (1996) | null | null | null | This article contains a compression of results from alg-geom/9501001, with
most proofs omitted. We prove that every two points of the connected moduli
space of holomorphically symplectic manifolds can be connected with so-called
``twistor lines'' -- projective lines holomorphically embedded to the moduli
space and corresponding to the hyperk\"ahler structures. This has interesting
implications for the geometry of compact hyperk\"ahler manifolds and of
holomorphic vector bundles over such manifolds.
| [
{
"version": "v1",
"created": "Thu, 16 Nov 1995 01:15:11 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Verbitsky",
"Misha",
""
]
] | alg-geom | \section{Lie algebra action.}
We refer to \cite{_main_} for details of definitions
and missing proofs.
A hyperk\"ahler manifold is a Riemannian manifold
$M$ equipped with three complex structures $I$,
$J$ and $K$, such that $I\circ J=-J\circ I=K$ and $M$
is K\"ahler with respect to $I$,
$J$ and $K$. Relations between $I$, $J$ and $K$
imply that there is is an action of quaternions in its tangent
space. Consequently, there is a multiplicative action of
$SU(2)$ on the algebra of differential forms. This action
commutes with Laplacian. Hence there is a canonical
action of $SU(2)$ on cohomology of $M$.
Let $M$ be a complex manifold which admits a hyperk\"ahler
structure. A simple linear-algebraic argument implies that
$M$ is equipped with a holomorphic symplectic form. Calabi-Yau theorem
shows that, conversely, every compact holomorphically symplectic
K\"ahler manifold admits a hyperk\"ahler structure, which is uniquely
defined by these data. Further on, we consider only holomorphically symplectic
manifolds which are compact and of K\"ahler type.
For simplisity of statements, we assume also that
\[ \dim H^{2,0}(M)=1, \mbox{\ \ and \ \ }H^1(M) =0, \]though these
assumptions are not necessarily for most results.
\hfill
The algebraic structure on $H^*(M)$ is studied using the
general theory of Lefschetz-Frobenius algebras, introduced
in \cite{_Lunts-Loo_}.
Let $A= \bigoplus\limits^{2d}_{i=0} A_i$
be a graded commutative associative algebra over a field of
characteristic zero. Let $H\in End(A)$ be a linear endomorphism
of $A$ such that for all $\eta \in A_i$, $H(\eta)= (i-d) \eta$.
For all $a\in A_2$, denote by $L_a:\;\; A\longrightarrow A$ the linear map which
associates with $x\in A$ the element $ax\in A$.
The triple $(L_a, H, \Lambda_a)\in End(A)$ is called {\bf a Lefschetz triple}
if
\[ [ L_a, \Lambda_a] = H,\;\; [ H, L_a ] = 2 L_a, \;\;
[ H, \Lambda_a] = -2 \Lambda_a.
\]
A Lefschetz triple establishes a representation of the
Lie algebra $\goth{sl}(2)$ in the space $A$. For cohomology
algebras, this representation
arises as a part of Lefschetz theory.
In a Lefschetz triple, the endomorphism $\Lambda_a$ is
uniquely defined by the element $a\in A_2$
(\cite{_Bou:Lie_} VIII \S 3). For arbitrary
$a\in A_2$, $a$ is called {\bf of Lefschetz type} if the
Lefschetz triple $(L_a, H, \Lambda_a)$ exists. If $A= H^*(X)$
where $X$ is a compact complex manifold of K\"ahler type,
then all K\"ahler classes $\omega\in H^2(M)$ are elements
of Lefschetz type. As one can easily check,
the set $S\subset A_2$ of all elements of Lefschetz type
is Zariski open in $A_2$.
\hfill
\definition
A Lefschetz-Frobenius algebra is a
Frobenius graded supercommutative algebra which admits a
Lefschetz triple.
\hfill
\definition
Let $A$ be a Lefschetz-Frobenius algebra.
The structure Lie algebra ${\goth g}(A)\subset End(A)$ is a Lie subalgebra
of $End(A)$ generated by $L_a$, $\Lambda_a$, for all
elements of Lefschetz type $a\in S$.
\hfill
Let $M$ be a compact hyperk\"ahler manifold with the complex
structures $I$, $J$, $K$. Consider the K\"ahler forms
$\omega_I$, $\omega_J$, $\omega_K$ associated with these
complex structures. Let $\rho_I:\; \goth{sl}(2)\longrightarrow End(H^*(M))$,
$\rho_J:\; \goth{sl}(2)\longrightarrow End(H^*(M))$,
$\rho_K:\; \goth{sl}(2)\longrightarrow End(H^*(M))$
be the corresponding Lefschetz homomorphisms.
Let $\goth a\subset End(H^*(M))$ be the minimal
Lie subalgebra which contains images of $\rho_I$,
$\rho_J$, $\rho_K$. The algebra $\goth a$ was computed
explicitely in \cite{_so5_on_cohomo_}.
\hfill
\theorem \label{_so_5_Theorem_}
(\cite{_so5_on_cohomo_})
The Lie algebra $\goth a$ is naturally isomorphic to $\goth{so}(4,1)$.
\hfill
This statement can be regarded as a ``hyperk\"ahler Lefschetz theorem''.
Indeed, its proof parallels the proof of Lefschetz theorem.
\hfill
Using \ref{_so_5_Theorem_}, we compute the structure Lie algebra
of $H^*(M)$.
\hfill
\theorem \label{_structure_alge_for_coho_hyperkahe_Theorem_}
(\cite{_main_}, Theorem 11.1)
Let $M$ be a compact holomorphically symplectic manifold. Assume
that $\dim H^{2,0}(M)$
Let $n= \dim(H^2(M))$. Let ${\goth g}(A)$ be a structure Lie algebra
for $A=H^*(M)$. Then ${\goth g}(A)$ is isomorphic to
$\goth{so}(4, n-2)$.\footnote{This isomorphism can be made canonical.
The Lie algebra ${\goth g}(A)$ is isomorphic to $\goth{so}(V\oplus \c H)$
where $V$ is the linear space $H^2(M, {\Bbb R})$ equipped with the
natural pairing of a signature $(3,n-3)$ (\cite{_Beauville_}
Remarques, p. 775; see also \ref{_H-R_form_defi_Theorem_}), and $\c H$
is 2-dimensional vector space with hyperbolic quadratic form.}
\hfill
Let $H^*_r(M)$ be a sub-algebra of $H^*(M)$ generated by $H^2(M)$.
It is easy to see that ${\goth g}(A)$ acts on $H^*_r(M)$, and $H^*_r(M)$
is an irreducible representation of ${\goth g}(A)$. Moreover, multiplicative
structure in $H^*_r(M)$ is easily recovered from an action of ${\goth g}(A)$.
Using the general knowledge of representations of $\goth{so}(n)$,
we obtain exact knowledge of the multiplicative structure
of $H^*_r(M)$. In particular, we obtain the
following theorem.
\hfill
\theorem \label{_S^*H^2_is_H^*M_intro-Theorem_}
(\cite{_main_}, Theorem 15.2)
Let $\dim_{\Bbb C} M=2n$.
Then
\[\bigg\{\begin{array}{lr}
H^{2i}_r(M)\cong S^i H^2(M)&
\mbox{\ \ for $i\leq n$, and}\\
H^{2i}_r(M)\cong S^{2n-i} H^2(M) &
\mbox{\ \ for $i\geq n$ }
\end{array}
\]
\section{The Riemann-Hodge pairing.}
Let $M$ be a compact holomorphically symplectic manifold
f K\"ahler type, satisfying
\[ \dim H^{2,0}(M)=1. \]
In \cite{_Beauville_} Remarques, p. 775, Beauville introduces
canonical 2-form on $H^2(M)$, of signature
$(n-3,3)$, where $n=\dim H^2(M)$. Throughout the paper \cite{_main_},
this form was called {\bf the Riemann-Hodge
pairing}.\footnote{More accurately, this form should be
called Bogomolov-Beauville form.
The author was unaware of Beauville's
remark, and did not understand the part of Bogomolov's
paper (\cite{_Bogomolov_}) where this form is also introduced.}
In \cite{_main_}, this form was described via the action of
$SU(2)$ on $H^2(M)$.
\hfill
Let $\omega$ be a K\"ahler class on $M$
such that \[\int_M \omega^{\dim_{\Bbb C} M}=1,\] and $(I,J, K, (\cdot,\cdot))$
be the corresponding hyperk\"ahler structure. Let
\[ (\cdot,\cdot)_{Her}:\; H^2(M,{\Bbb C})\times H^2(M,{\Bbb C})\longrightarrow {\Bbb C}\]
be a positively Hermitian form on second cohomology of $M$ which
corresponds to the Riemannian structure $(\cdot,\cdot)$.
Let $H^2(M) = H^{inv}(M)\oplus H^{+}(M)$ be a decomposition
such that
$H^{inv}(M)$ consists of all $SU(2)$-invariant 2-forms,
and $H^{+}(M)$ is the complementary $SU(2)$-invariant subspace.
Let $(\cdot,\cdot)_{\c H}$ be the form which is equal to
$(\cdot,\cdot)_{Her}$ on $H^{+}(M)$ and $-(\cdot,\cdot)_{Her}$
on $H^{inv}(M)$
\hfill
\theorem \label{_H-R_form_defi_Theorem_}
(\cite{_main_}, Theorem 6.1, cf. \cite{_Beauville_} Remarques, p. 775)
The form $(\cdot,\cdot)_{\c H}$ is independent
from the choice of the complex and K\"ahler structure on $M$.
\hfill
The form $(\cdot,\cdot)_{\c H}$ is used in the proof
of \ref{_structure_alge_for_coho_hyperkahe_Theorem_}.
\hfill
Let $\rho_I:\; \goth{u}(1)\longrightarrow End(H^*(M))$ be a map for which
$z\in \goth{u}(1)$ acts on $H^{p,q}(M)$ by $(p-q)z$.
Clearly, the action of $\goth{u}(1)$ on $H^2(M)$ respects the form
$(\cdot,\cdot)_{\c H}$.
Let ${\goth g}_M\subset End(H^*(M))$ be a Lie algebra generated by the images
of $\rho_I$ for all complex structures $I$ on $M$.
Let $V$ denote the linear space $H^2(M)$ equipped with
bilinear form $(\cdot,\cdot)_{\c H}$.
By \ref{_H-R_form_defi_Theorem_}, the action of ${\goth g}_M$ on
$V$ preserves $(\cdot,\cdot)_{\c H}$.
This defines a Lie algebra homomorphism
$\Gamma:\;{\goth g}_M \longrightarrow \goth{so}(V)$. The following theorem
is a chief tool in proving the Mirror Conjecture
for a compact holomorphically symplectic manifold.
\hfill
\theorem \label{_embe_Mum_Tate_to_so_Theorem_}
The map $\Gamma:\; {\goth g}_M \longrightarrow \goth{so}(V)$ is an isomorphism.
{\bf Proof:} \cite{_main_}, Theorem 13.1, 13.2. $\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
The Lie algebra ${\goth g}(A)\subset End(H^*(M))$ is equipped with a natural
grading, induced by the grading on $H^*(M)= \oplus H^i(M)$.
Let $k$ be the one-dimensional Lie subalgebra of $End(H^*(M))$
spanned by $Id$.
\hfill
\theorem \label{_Mum_Tate_is_g_0_Theorem_}
(\cite{_main_}, Theorem 13.2)
The Lie subalgebra \[ {\goth g}_M\oplus k\subset End(H^*(M))\]
coinsides with the grading-zero part of ${\goth g}(A)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We have a {\bf period map}
$P_c:\; Comp \longrightarrow {\Bbb P}H^2(M, {\Bbb C})$ associating a line
$H^{2,0}_I(M) \subset H^2(M, {\Bbb C})$ to a complex structure $I$.
Complexifying $H^2(M, {\Bbb R})$, we can consider
$(\cdot,\cdot)_{\c H}$ as a complex-linear,
complex-valued form on $H^2(M, {\Bbb R})$. For all $I\in Comp$,
$P_c(I)$ belongs to a conic hypersurface $C \subset {\Bbb P}H^2(M, {\Bbb C})$,
\[ C = \{ l \;\; |\;\; (l,l)_{\c H} =0\}. \]
Torelli principle (proved by Bogomolov in the case of holomorphically
symplectic manifolds, \cite{_Bogomolov_})
implies that $P_c:\; Comp \longrightarrow C$ is etale.
\hfill
Let $\underline{\c H}= \oplus H^{p,q}(M)$ be a variation of Hodge structures
(VHS) on Comp associated with the total cohomology space of $M$.
\ref{_embe_Mum_Tate_to_so_Theorem_} implies that there exist a VHS
$\c H$ on $C$, such that $\underline {\c H}$
is a pullback of a variation of Hodge structures $\c H$:
$\underline {\c H} = P^*_c(\c H)$. Let $G_M$ be the Lie group
associated with ${\goth g}_M$,
$G_M = Spin\bigg(H^2(M, {\Bbb R}), (\cdot,\cdot)_{\c H}\bigg)$.
The set $C$ is equipped with a natural action of a group $G_M$.
This group also acts in the total cohomology space $H^*(M)$
of $M$. This defines an equivariant structure in the
bundle $\c H$. The chief idea used in the proof
of Mirror Symmetry is the following theorem:
\hfill
\theorem \label{_VHS_equi_Intro_Theorem_}
The VHS $\c H$ is $G_M$-equivariant, under the natural
action of $G_M$ on $C$ and $\c H$.
{\bf Proof:} See \cite{_V:Mirror_}, Theorem 2.2.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
To make this statement more explicit, we recall that the variation
of Hodge structures is a flat bundle, equipped with a real structure
and a holomorphic filtration (Hodge filtration),
which is complementary to its complex adjoint filtration.
Then, \ref{_VHS_equi_Intro_Theorem_} says
that the action of $G_M$ on $\c H$
maps flat sections to flat sections, and preserves the real structure
and the Hodge filtration.
\section{The twistor lines.}
The main technical tool used in the text of \cite{_main_} is
results about (coarse, marked) moduli space $Comp$ of complex structures
on a holomorphically symplectic manifold $M$.
Let $\omega$ be a K\"ahler class on $M$ and $\c H=(I,J,K, (\cdot,\cdot))$
be the corresponding hyperk\"ahler structure. Then,
for every triple of real numbers $(a,b,c), a^2+b^2+c^2=1$,
the operator $aI+bJ+cK$ defines an integrable complex
structure%
\footnote{This complex structure is
called {\bf a complex structure induced by a hyperk\"ahler
structure}.}
on $M$.
Identifying the set of such triples
with ${\Bbb C} P^1$, we obtain a map
${\Bbb C} P^1\stackrel{i_{\c H}}{\hookrightarrow} Comp$
where $Comp$ is a connected component of the coarse
moduli space of $M$.
The following claim is easy.
\hfill
\claim
The map $i_{\c H}$ is a
holomorphic embedding of complex analytic varieties.
\hfill
Let $P:\; Comp \longrightarrow C$ be the period map, assigning
to a complex structure $I$ a line $H^{2,0}(M,I)$.
Let $C\subset {\Bbb P}^1(H^2(M,{\Bbb C})= P(Comp)$.
According to \cite{_Beauville_}, $P$ is etale.
The projective line $i_{\c H}({\Bbb C} P^1)\subset Comp$ is called
{\bf a twistor line}, and is denoted by $R_{\c H}$.
The following theorem was, regrettably, omitted in \cite{_main_},
though all necessary tools were developed for its proof.
For conceptual understanding of our argument, this
theorem is indispensable.
\hfill
\theorem \label{_twistor_connect_Theorem_}
Let $I_1, I_2\in Comp$. Then there exist a sequence of
intersecting twistor lines which connect $I_1$ with $I_2$.
{\bf Proof:}
To prove \ref{_twistor_connect_Theorem_}$'$, we have to show that
a set $\tilde{\c L_0}$ of all twistor lines $i_{\c H_0}({\Bbb C} P^1)$
which are connected to $i_{\c H}({\Bbb C} P^1)$ with
intersecting twistor lines is open.
Since $P:\; Comp \longrightarrow C$ is etale, it suffices to show that
$I_1, I_2$ can be connected with twistor lines $l_i$
such that $P(l_i)$ intersect $P(l_{i+1})$.
With every twistor line $R_{\c H}$, we associate
a 3-dimensional plane $\ell_{\c H}\subset H^2(M,{\Bbb R})$ which is
spanned by the K\"ahler classes $\omega_I$, $\omega_J$, $\omega_K$.
A linear algebraic argument shows that the twistor
lines $R_{\c H_1}$ and $R_{\c H_2}$ intersect if
and only if $\dim (\ell_{\c H_1}\cap \ell_{\c H_1})\geq 2$.
Hence we need to show that
\hfill
{\bf \ref{_twistor_connect_Theorem_}$'$}
Each pair of twistor lines $R_{\c H}$, $R_{\c H'}$
can be connected with
a sequence of twistor lines $R_{\c H}=R_{\c H_1}$, ...,
$R_{\c H_n}=R_{\c H'}$ such that
$\dim (\ell_{\c H_i}\cap \ell_{\c H_{i+1}})\geq 2$.
\hfill
\lemma \label{_Kah_open_Lemma_}
Let $H$ be a hyperk\"ahler structure on $M$, $i_{\c H}({\Bbb C} P^1)\subset Comp$
be the set of all induced complex structures, and $Kah(\c H)$ be the set
of all K\"ahler classes corresponding to $L\in i_{\c H}({\Bbb C} P^1)$. Then
$Kah(\c H)$ is open in $H^2(M, {\Bbb R})$.
{\bf Proof:} \cite{_main_}, Claim 6.6 $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $\c L$ be the space of all triples
$\omega_I$, $\omega_J$, $\omega_K$ in $H^2(M)$ which are orthonormal
with respect to the pairing $(\cdot,\cdot)_{\c H}$ of
\ref{_H-R_form_defi_Theorem_}, and $Hyp$ be the connected
component of the set of all hyperk\"ahler structures.
Let $P_h:\; Hyp \longrightarrow \c L$ be the natural period map.
Comparing dimensions and using Calabi-Yau, we observe
that $P_h$ is etale. Let $\c L_0$ be the
space of twistor lines corresponding to $\tilde{L_0}$.
Using \ref{_Kah_open_Lemma_}, we find that the differential
of $P_h\restrict{\c L_0}$ is surjective. Therefore,
$\c L_0$ is open in $\c L$, and $\tilde{\c L_0}$ is open
in the set of all twistor lines. This proves
\ref{_twistor_connect_Theorem_}. $\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\section{An outline of proofs.}
Let $(I,J, K, (\cdot,\cdot))$ be a hyperk\"ahler structure
on $M$. One can check that the cohomology classes
$\omega_I$, $\omega_J$, $\omega_K\in H^2(M,{\Bbb R})$ are orthogonal
with respect to the pairing $(\cdot,\cdot)$. Let $Hyp$ be the classifying
space of the hyperk\"ahler structures on $M$. Let
$P_{hyp}:Hyp\longrightarrow H^2(M)\times H^2(M)\times H^2(M)$
be the map which associates with the hyperk\"ahler structure
$\c H= (I, J, K, (\cdot,\cdot))$ the triple
$(\omega_I,\omega_J,\omega_K)$. Then the image of $P_{hyp}$ in
$H^2(M)\times H^2(M)\times H^2(M)$ satisfies
\begin{equation}\label{_image_of_P_hyp_Equation_}
\forall (x,y,z)\in im P_{hyp}\;\; \bigg |\;\;
\begin{array}{l}
(x,y)_{\c H}=(x,z)_{\c H}=(y,z)_{\c H}=0,\\[3mm]
(x,x)_{\c H}=(y,y)_{\c H}=(z,z)_{\c H},
\end{array}
\end{equation}
where $(\cdot,\cdot)_{\c H}$ is the canonical pairing
defined above.
Let $D\subset H^2(M)\times H^2(M)\times H^2(M)$ be the set
defined by the equations \eqref{_image_of_P_hyp_Equation_}.
Using Torelli theorem and Calabi-Yau, we prove the following
statement:
\hfill
\theorem\label{_image_of_P_hyp_Theorem_}
The image of $P_{hyp}$ is Zariski dense in $D$.
\hfill
\ref{_image_of_P_hyp_Theorem_} shows that all algebraic relations
which are true for \[ (x,y,z)\in P_{hyp}(Hyp) \] are true
for all $(x,y,z)\in D$. Computing the Lie algebra $\goth a$
as in \ref{_so_5_Theorem_}, we obtain a number of relations
between $x,y,z\in H^2(M)$ which hold for all $(x,y,z)\in Im(P_{hyp})$.
Using the density argument, we obtain that these relations
are universally true. This idea leads to the proof of
\ref{_structure_alge_for_coho_hyperkahe_Theorem_}.
\hfill
The proof of \ref{_H-R_form_defi_Theorem_} is deduced from the
standard period argument and \ref{_twistor_connect_Theorem_}.
Let $\c H$ be a hyperk\"ahler structure corresponding to $I$ and $\omega$.
Clearly from the definition, the form $(\cdot,\cdot)_{\c H}$
depends only from the twistor line $\c H$, and not from the
choice of particular $I$ and $\omega$. A computation shows that
$(\cdot,\cdot)_{\c H}$ depends from $P(I)$ and not from $\omega$.
Using the fact that $C$ is all connected with twistor lines
(\ref{_twistor_connect_Theorem_}\footnote{In \cite{_main_}, we
proved a slightly weaker statement, which still suffices to prove
\ref{_H-R_form_defi_Theorem_}.}), we prove that
$(\cdot,\cdot)_{\c H}$is independent from $\c H$.
\section{Implications.}
This section contains implications of our results.
\hfill
{\bf 5.1 Mirror symmetry.} (\cite{_V:Mirror_})
Using \ref{_embe_Mum_Tate_to_so_Theorem_}
and \ref{_embe_Mum_Tate_to_so_Theorem_}, we compute
the variation of Hodge structures corresponding to the
universal VHS over the moduli space $Comp$. In
\cite{Verbitsky:Symplectic_II_}, it is proven that
for ``sufficiently generic'' deformation $W$ of a given
compact holomorphically symplectic manifold $M$, the manifold
$W$ admits no closed holomorphic curves. Therefore, using the definition
of quantum cohomology from \cite{_Kontsevich-Manin_}, we
can easily compute the quantum variation of Frobenius algebras.
Comparing these computations, we find
that Mirror Conjecture is true for holomorphically symplectic manifolds,
which are Mirror self-dual.
In proof of Mirror Symmetry, we use the fact that
tangent bundle $TM$ of a holomorphically symplectic manifold
is isomorphic to the cotangent bundle $\Omega^1(M)$ thereof. For every
Calabi-Yau manifold $M$, $\dim M =n$, the Serre's duality induces an
isomorphism\footnote{Canonical up to a choice of a
non-degenerate section of $\Omega^n(M)$.}
\begin{equation}\label{_Yukawa_isomo_Equation_}
H^{p}(\Omega^q(M)) \cong H^{p}(\Lambda^{n-q}(TM))
\end{equation}
beweeen cohomology of the holomorphic differential forms
and cohomology of exterrior powers of holomorphic tangent bundle.
Using the isomorphism $TM\cong \Omega^1(M)$, we interpret
the isomorphism \eqref{_Yukawa_isomo_Equation_} as a map $\eta$
from the total cohomology space $H^*(M)$ to itself. A linear-algebraic
check ensures that this map is involutive. A slightly less elementary
consideration shows that $\eta:\; H^*(M)\longrightarrow H^*(M)$ belongs to the Lie
group $G\subset End(H^*(M))$ corresponding to the Lie algebra ${\goth g}(A)$
from \ref{_structure_alge_for_coho_hyperkahe_Theorem_}.
Clearly, Yukawa multiplication is equal to the cup-product
in cohomology twisted by $\eta$. This gives a way to
describe Yukawa product explicitely in terms of Lie algebra action.
\hfill
{\bf 5.2 Twistor paths.}
\definition
Let $M$ be a holomorphically symplectic manifold,
$Comp$ be its moduli space, $P_0$, ... $P_n\subset Comp$ be
a sequence of twistor lines, supplied with an intersection point
$x_{i+1}\in P_i\cap P_{i+1}$ for each $i$. We say that
$\gamma= P_0, ..., P_n, x_1, ..., x_n$ is
a {\bf twistor path}. Let $I$, $I'\in Comp$.
We say that $\gamma$ is {\bf a twistor path
connecting $I$ to $I'$} if $I\in P_0$ and $I'\in P_n$.
The lines $P_i$ are called {\bf the edges},
and the points $x_i$ {\bf the vertices}
of a twistor path.
\hfill
\ref{_twistor_connect_Theorem_} proves that
every two points $I$, $I'$ in $Comp$ are connected
with a twistor path. Clearly, each twistor path
induces a diffeomorphism $\mu_\gamma:\; (M,I)\longrightarrow (M,I')$.
We are interested in algebro-geometrical properties of this
diffeomorphism.
\hfill
For every hyperk\"ahler structure $\c H$ on $M$, let
${\goth g}_{\c H}\subset End(H^*(M))$ be the corresponding $\goth{su}(2)$
embedded to $End(H^*(M))$. Let $H^*(M)^{{\goth g}_{\c H}}$ be the
${\goth g}_{\c H}$-invariant part of $H^*(M)$.
Let $I\in Comp$ and $\c H$ be a hyperk\"ahler structure
which induces $I$. We say that $I$ is {\bf of general type
with respect to $\c H$} if
\[ H^*(M)^{{\goth g}_{\c H}}\cap H^*(M, {\Bbb Z}) =
\oplus H^{p,p} \cap H^*(M, {\Bbb Z}).
\]
In \cite{Verbitsky:Symplectic_II_}, we prove that for every hyperk\"ahler
structure, all induced complex structures are of general type,
except may be a countable number thereof. Results
of \cite{_Verbitsky:Hyperholo_bundles_} and
\cite{Verbitsky:Symplectic_II_} can be compressed down to
the following statement.
\hfill
\theorem\label{_generi_implication_Theorem_}
Let $\c H$ be a hyperk\"ahler structure on $M$ and $I$ be an
induced complex structure of general type.
(i) (\cite{Verbitsky:Symplectic_II_})
Let $N$ be a closed complex analytic subset of $(M, I)$. Then
$N$ is complex analytic with respect to $J$, for all induced
complex structures%
\footnote{In \cite{Verbitsky:Symplectic_II_},
such subsets are called {\bf trianalytic}.}%
$J$.
(ii) (\cite{_Verbitsky:Hyperholo_bundles_})
Let $Bun_I$ be the tensor category of
polystable\footnote{Polystable means direct sum of stable.
Stability is understood in the sense of Takemoto -- Mumford.}
holomorphic vector bundles of slope $0$ over $(M,I)$. For arbitrary
induced complex structure $J$, there exist a natural
injective tensor functor
$\Phi_{I\rightarrow J}:\; Bun_I\longrightarrow Bun_J$, which is an equivalence
of $J$ is of general type with respect to $\c H$.
For $I, J, J'$ being induced complex structures and
$I$, $J$ of general type, we have
\[ \Phi_{I\rightarrow J}\circ \Phi_{J\rightarrow J'}
= \Phi_{I\rightarrow J'}.
\]
{\bf Remark on proof of
\ref{_generi_implication_Theorem_} (ii):}
\ref{_generi_implication_Theorem_} (ii) is an implication
of the following result from \cite{_Verbitsky:Hyperholo_bundles_}.
Let $B$ be a polystable bundle on a holomorphically symplectic
Kaehler manifold $M$. We associate with the Kaehler structure
on $M$ a canonical hyperkaehler structure $\c H$
as in Calabi-Yau theorem. Assume that the first and second
Chern classes of stable summands of $B$ are
invariant under the natural action of $SU(2)$
in cohomology.
Then there exist a unique
holomorphic connection on $B$ which is holomorphic
under each of complex structures induced by $\c H$.
This lets one identify the categories
of polystable bundles for different complex structures $L$
induced by $\c H$, provided that $L$ is of general type with respect
to $\c H$.
\hfill
\definition
Let $I$, $J\in Comp$ and $\gamma= P_0, ... P_n$ be a twistor path from
$I$ to $J$, which corresponds to the hyperk\"ahler structures
$\c H_0$, ..., $\c H_n$. We say that $\gamma$ is admissible
if $I$ is of general type with respect to $P_0$, $J$ to $P_n$,
and all vertices of $\gamma$ are of general type with respect
to the corresponding edges.
\hfill
\corollary \label{_admi_twi_impli_Corollary_}
Let $I$, $J\in Comp$, and $\gamma$ be admissible twistor path from
$I$ to $J$.
(i) Let $\mu_\gamma:\; (M, I) \longrightarrow (M, J)$ be the
corresponding diffeomorphism. Then, for every
complex analytic subset $N \subset (M, I)$,
$\mu_\gamma(N)$ is complex analytic with respect
to $J$, for all induced complex structures.
(ii) There exist a natural
isomorphism of tensor cetegories
\[ \Phi_{\gamma}:\; Bun_I\longrightarrow Bun_J.\]
{\bf Proof:} Follows from \ref{_generi_implication_Theorem_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
To sum it up, whenever we can connect two complex structures
by an admissible twistor path, these complex structures are
quite similar from algebro-geometrical point of view.
There is a cohomological criterion of existence of
admissible twistor path,
which is proven in the similar fashion to
\ref{_twistor_connect_Theorem_}.
\hfill
\newcommand{{\Bbb Q}}{{\Bbb Q}}
For $I\in Comp$, denore by $\mbox{NS}(I, {\Bbb Q})$ the space
$H^{1,1}(M, I)\cap H^2(M, {\Bbb Q})\subset H^2(M)$.
Let $Q\subset H^2(M, {\Bbb Q})$ be a subspace of $H^2(M, {\Bbb Q})$.
Let
\[ Comp_Q:= \{ I\in Comp \;\; | \;\; \mbox{NS}(I, {\Bbb Q}) =Q\}. \]
\hfill
\theorem\label{_admi_exi_Theorem_}
Let $\c H$, $\c H'$
be hyperk\"ahler structures, and
$I$, $I'$ be complex structures of general type to and induced by
$\c H$, $\c H'$. Assume that $\mbox{NS}(I, {\Bbb Q}) = \mbox{NS}(I', {\Bbb Q}) =Q$,
and $I$, $I'$ lie in the same connected component of $Comp_Q$.
Then $I$, $I'$ can be connected by an admissible path.
{\bf Proof:} Follows the proof of \ref{_twistor_connect_Theorem_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
For general $Q$, we have no control over the number of connected
components of $Comp_Q$ (unless global Torelli theorem is proven),
and therefore we cannot directly apply
\ref{_admi_exi_Theorem_} to obtain results from algebraic
geometry.\footnote{Exception is K3 surface, where Torelli holds.
For K3, $Comp_Q$ is connected for all $Q\subset H^2(M, {\Bbb Q})$.}
However, when $Q=\emptyset$, $Comp_Q$ is clearly
connected and open in $Comp$, assuming that $Comp$ is connected,
which we assumed. On the other hand, for $I\in Comp_\emptyset$,
and every $\c H$ inducing $I$, $I$ is of general type with respect
to $\c H$ (this is essentially an implication of
\ref{_embe_Mum_Tate_to_so_Theorem_}). This proves the following corollary.
\hfill
\corollary
Let $I$, $I'\in Comp_\emptyset$. Then $I$ can be connected to $I'$
by an admissible twistor path.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
{\bf Remark:}
We obtain that for all $I\in Comp_\emptyset$, the closed complex
analytic subsets of $(M, I)$ have the same real analytic
structure, and categories of polystable holomorphic vector
bundles are isomorphic. There are non-trivial polystable
holomorphic vector bundles over such manifolds (tangent bundle
and its tensor powers come to mind). It is not completely clear
if manifolds $(M, I)$ with $I\in Comp_\emptyset$ have any closed
complex analytic subvarieties, except points.
\hfill
{\bf 5.3 Generalization of $(\cdot,\cdot)_{\c H}$.}
Unlike the (otherwise clearly superior)
approach used by Beauville and Bogomolov, our way of constructing
the form $(\cdot,\cdot)_{\c H}$ lends itself to an immediate
generalization.
Let ${\goth g}_0(A)$ be the grading-zero part of ${\goth g}(A)$ computed in
\ref{_Mum_Tate_is_g_0_Theorem_}, and $H^*(M)^{{\goth g}_0(A)}$
be the space of all vectors invariant under ${\goth g}_0(A)$.
Let $H^*_{\bf r}(M)$ be a subalgebra of cohomology
generated by $H^2(M)$ and $H^*(M)^{{\goth g}_0(A)}$.%
\footnote{There are only two known series of
compact hyperk\"ahler manifolds: Hilbert schemes of
Artinian sheaves on K3 surfaces, and Hilbert schemes
of Artinian sheaves on compact 2-dimensional tori, factorized by free
action of a compact torus. In both cases, the cohomology algebra
is computed by Nakajima (\cite{_Nakajima_}). It seems reasonable
to conjecture that, in either of these cases, $H^*(M)=H^*_{\bf r}(M)$.}
Let $\c H$ be a hyperk\"ahler structure on $M$. Consider the
corresponding action of $SU(2)$ on $H^*(M)$.
Let $H^i(M)= \oplus_w H^i_w(M)$ be an isotypic decomposition
of $H^i(M)$ corresponding to this action. By definition, $H^i_w(M)$ is a
direct sum of isomorphic $SU(2)$-representation of weight $w$,
where $w$, $0\leq w \leq i$ runs through the natural numbers
of the same parity as $i$. Let $(\cdot,\cdot)_{Her}$
be the Hermitian metrics on cohomology induced by the Riemannian
structure on $M$, and $(\cdot,\cdot)_{\c H}$ be the pairing which
is equal to $(-1)^{\frac{i-w}{2}} (\cdot,\cdot)_{Her}$ on $H^i_w(M)$.
\hfill
\theorem
Consider restriction of $(\cdot,\cdot)_{\c H}$ to $H^*_{\bf r}(M)$.
This restriction $(\cdot,\cdot)_{\c H}$ is non-degenerate and
independent on $\c H$ (up to a constant multiplier).
\hfill
{\bf Proof:} For $i=2$, this statement coinsides with the statement
of \ref{_H-R_form_defi_Theorem_}. For
general $i$, the proof is essentially linear-algebraic and
identical to the proof of \cite{_main_}, Theorem 6.1.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
{\bf Acknowledgements:}
I am grateful to my advisor David Kazhdan for warm support
and encouragement, P. Deligne and F. Bogomolov for
their suggestions and correstions, D. Kaledin and T. Pantev
for stimulating discussions. Also I owe to
Julie Lynch and IP Press for providing me with
employment.
\hfill
|
1995-11-30T06:20:10 | 9511 | alg-geom/9511019 | en | https://arxiv.org/abs/alg-geom/9511019 | [
"alg-geom",
"math.AG"
] | alg-geom/9511019 | N. Mohan Kumar | N. Mohan Kumar | Construction of rank two bundles on ${\bf P}^4$ in positive
characteristic | Latex. Bibtex is necessary to get the references right. See the note
at the beginining of text | null | null | null | null | Many examples of rank two bundles on ${\bf P}^4$ are constructed in positive
characteristic. Construction depends on constructing certain special bundles on
${\bf P}^3$ which is shown to be equivalent to constructing bundles on ${\bf
P}^4$ (in any characteristic). In fact this is true for projective spaces of
any dimension.
| [
{
"version": "v1",
"created": "Wed, 29 Nov 1995 17:29:48 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kumar",
"N. Mohan",
""
]
] | alg-geom | \section{Introduction}
Vector bundles on Projective spaces have been the subject of many
papers and many results in this direction are known. For a
somewhat dated account, the reader may see \cite{OSS}. One of the most
interesting problems in this area is the study of small rank bundles
on Projective spaces. For example, a conjecture by R.~Hartshorne
\cite{rHartshorne} states that there are no small rank vector bundles
on Projective spaces, other than direct sum of line bundles. The
solution to this tantalising problem still seems remote, though very
many results are known. Let me restrict my attention to rank 2 bundles
for the moment. Many interesting bundles of rank 2 are known over
Projective spaces of dimension 2 and 3. But over $\P^4$, essentially
the only interesting one known is the well known Horrocks-Mumford
bundle \cite{HorMum}. There are also some intersting ones in
characteristic 2, discovered by Tango \cite{Tango} and G.~Horrocks
\cite{gHorrocks}.
In this paper, we shall deal with this problem and prove a criterion
relating bundles on $\P^{n+1}$ to bundles on $\P^n$. This condition
on certain bundles over $\P^n$
is necessary and sufficient for the existence of bundles on
$\P^{n+1}$. Though this criterion is not very pleasant, it allows you
to restrict your attention to bundles just on $\P^n$ to construct
bundles on $\P^{n+1}$. Using this criterion (which has nothing to do
with the characteristic of the field), we construct many rank two
bundles on $\P^4$ over a field of positive characteristic. (For the
Chern classes of these bundles, see section \ref{chern}).
Unfortunately we have not been able to extend our construction to
complex numbers, though I feel it should be possible. The
construction follows closely what we did in \cite{MK}.
\section{Genral remarks}\label{remarks}
Let me start with a word about notation. We will have to deal with
maps, $\phi: M\to M\otimes L$ often in this article, where $M$ is a
sheaf and $L$, a line bundle. Then it makes sense to talk about
$\phi\otimes {\rm Id\/}: M\otimes L'\to M\otimes L\otimes L'$ for any
line bundle $L'$. We will denote this map also for brevity by $\phi$.
Also it makes sense to talk about $\phi^i: M\to M\otimes L^i$, by
composing $\phi$. So we will talk about $\phi$ as an {\em
endomorphism}, though strictly speaking, it is not. It also make sense
to say when such a map is {\em nilpotent}, by saying that $\phi^n=0$
for some $n$.
We make the following transparent observation:
\begin{rmk}
If $\pi: X\to Y$ is a finite map then the category of sheaves on $Y$
which are $\pi_*{\cal O}_X$ modules is the same as the category of shevaes
of the form $\pi_*{\cal F}$ where ${\cal F}$ is a sheaf on $X$ (with the
appropriate homomorphisms).
\end{rmk}
A typical case where we plan to apply this is when $X$ is the
$m^{th}$ order thickening of $\P^n\subset\P^{n+1}$ and $Y=\P^n$ with
$\pi$ the projection from a point in $\P^{n+1}$ away from this
hyperplane. We had applied this in \cite{MK} in a similar but
slightly different context.
So let $X$ be the $m^{th}$ order thickening of $\P^n\subset\P^{n+1}$
and $Y=\P^n$ and $\pi: X\to Y$ the projection from a point away from
this hyperplane.
\begin{enumerate}\label{obv1}
\item Then $\pi_*{\cal O}_X=\oplus_{i=0}^{m-1} {\cal O}_Y(-i)$. Thus we see that
giving a sheaf on $X$ is equivalent to giving a sheaf ${\cal F}$ on $Y$ and
an endomorphism $\phi: {\cal F}\to{\cal F}(1)$ with $\phi^m=0$.
\item Let $(E_1,\phi_1)$ and $(E_2,\phi_2)$ be two such sheaves on
$Y$. Giving a map $\psi:E_1\to E_2$ which commutes with the
$\phi_i$'s is
equivalent to giving a map between the corresponding sheaves on $X$.
\item We want to apply this to the special case when $E_1$ arises as
the restriction of a direct sum of line bundles say $F$ on
$\P^{n+1}$. We will write for clarity $G$, for $F$ restricted to
$\P^n$. Then we see that
$$E_1=\pi_*(F\mid_X)=\oplus_{i=0}^{m-1} G(-i)$$ as before. $\phi_1$ can
be identified with the map which just shifts the blocks. That is to
say the $\phi_1$ takes $G(-i)$ to the corresponding $G(-i)$ in
$E_1(1)$ as identitiy. (Of course $G(-m+1)$ goes to zero).
\item Thus giving a map from $F$ as above to the sheaf corresponding
to $(E,\phi)$ is just giving a map $\theta: G\to E$. Because then we
get for any $i$ a map $G(-i)\to E$ by taking the induced map
$G(-i)\to E(-i)$ and then composing it with $\phi^i$.
\end{enumerate}
Now let $E$ be any vector bundle on $\P^{n+1}$ and let $Y=\P^n$ be any
hyperplane. Then by Quillen--Suslin Theorem [{\it e.g.\/},\ see\cite{Q}], E
restricted to the complement of this hyperplane is free. Thus, if we
denote by $F={\cal O}_{\P^{n+1}}^r$ where $r=\rank E$, then we have an exact
sequence,
$$0\to E(-i)\to F\to {\cal F}\to 0$$
for some integer $i$ and ${\cal F}$ is a coherent sheaf on some $X$ as
above ($m^{th}$ order thickening of the hyperplane for some $m$).
Since we will be primarily interested in deciding when such an $E$ is
a direct sum of line bundles, we may as well rename $E(-i)$ by $E$.
Using $\pi$ as above we get a coherent sheaf $\pi_*{\cal F}=M$ on $Y$. But
${\cal F}$ has homological dimension one and therefore by
Auslander-Buchsbaum Theorem [{\it e.g.\/},\ see \cite{Mat}], $M$ is a vector
bundle. As above, we also have a nilpotent map $\phi:M\to M(1)$.
Letting $G={\cal O}_Y^r$ we also have a map $\psi: G\to M$ since the
distinguished $r$ sections of ${\cal F}$ give $r$ sections of $M$. Further
by the surjectivity of the above map from $F\to{\cal F}$, we see that
$$\phi(M(-1))+\psi(G)=M.$$ What we want to state is the converse of
this remark.
\begin{lemma}\label{extension}
Let $Y=\P^n$, $G$ be a rank $r$ bundle on $Y$ which is a direct sum
of line bundles. Assume $M$ is a vector bundle on $Y$ with
a nilpotent map $\phi: M\to M(1)$ and a map $\psi: G\to M$ such that
$\phi(M(-1))+\psi(G)=M$. Then there exists a vector bundle $E$ of
rank $r$ on $\P^{n+1}$ and an exact sequence,
\begin{equation}\label{rem1}
0\to E\to F\to {\cal F}\to 0
\end{equation}
where $F$ is direct sum of $r$ line bundles on $\P^{n+1}$ with
$F\mid_{\P^n}=G$ and $\pi_*{\cal F}=M$.
\end{lemma}
\begin{proof}
Proof is obvious using the remarks above.
\end{proof}
\begin{rmk}
The above lemma can be also thought of as a criterion for extending
vector bundles from $\P^n$ to $\P^{n+1}$. In other words, given a vector
bundle $E$ of rank $r$ on $\P^n$, it can be extended to $\P^{n+1}$ as a
vector bundle if and only if there exists a vector bundle $M$ over
$\P^n$ with a nilpotent endomorphism $\phi: M\to M(1)$, a map $\psi:
G\to M$ where $G$ is a direct sum of $r$ line bundles and an exact
sequence,
$$0\to E\to M(-1)\oplus G\stackrel{(\phi,\psi)}{\longrightarrow} M\to
0.$$
\end{rmk}
\begin{lemma}\label{nonsplit}
Let the notation be as in the above lemma and assume $n\geq 2$. If
$M$ is not a direct sum of line bundles then $E$ is not a direct sum
of line bundles. Conversely, if $r\leq n$, and $M$ is a direct sum of
line bundles, then so is $E$.
\end{lemma}
\begin{proof}
Assume $M$ is not a direct sum of line bundles. Then by Horrock's
criterion. [{\it e.g.\/},\ see\cite{OSS}], $H^i(M(l))\neq 0$ for some $i$, with
$0<i<n$ and some $l\in\Z$. $H^j(F(l))=0\quad \forall j, 0<j\leq n$,
since $F$ is a direct sum of line bundles. Therefore from our exact
sequence \ref{rem1}, $H^i({\cal F}(l))=H^{i+1}(E(l))$. Also
$H^i({\cal F}(l))=H^i(M(l))$ since $\pi$ is a finite map from ${\rm supp}\, {\cal F}\to
Y$. Thus $H^{i+1}(E(l))\neq 0$ and since $0<i+1<n+1$, we see that $E$
is not a direct sum of line bundles.
Conversely, assume that $M$ is a direct sum of line bundles. Exactly
as before, we get $H^i(E(l))=0 \quad\forall l, 1<i\leq n$. By
duality, $H^i(E^*(l))=0\quad\forall l, 0<i<n$. Thus by the Syzygy
theorem \cite{GE} rank of $E^*=r>n$ or $E^*$ is a direct sum of
line bundles.
\end{proof}
\section{The Construction}
In this section, we will outline the construction of bundles $M$ on
$\P^3$ as described above. More generally, let $d\in\Z$ be any
integer. We will construct a bundle $M$ on $\P^3$ with a nilpotent
endomorphism $\phi: M\to M(d)$ and a map $\psi: G\to M$ where $G$ is
the direct sum of two line bundles such that $\phi(M(-d))+\psi(G)=M$
and $M$ is not a direct sum of line bundles, over a field of positive
characteristic.
\begin{rmk}
The only intersting cases are $d=-1,0,1$. If we have $M$'s for these
values, then by taking the pull back of these by finite maps
$\P^3\to\P^3$, we can construct bundles for all $d$. The case $d=0$
was treated in \cite{MK}.
\end{rmk}
Let $p>0$ be the characteristic of our algebraically closed field.
Choose positive numbers $N,k,l$ so that $N-k,N-l$ both positive,
$4pkl> d^2$ and
$p(k+l)=(p-1)N+d$. Let $x,y,z,t$ be the homogeneous co-ordinates of
$\P^3$. Let $A=x^kz^{N-k}+y^lt^{N-l}$. Let $C_i$ be the curve defined
by the vanishing of $x^{pk}, y^{pl}$ and $A^i$, for $1\leq i\leq p$.
Let $C$ be the curve (line) defined by $x=y=0$.
\begin{claim}
\begin{enumerate}
\item $C_i$'s are local complete intersection curves for $1\leq i\leq
p$ and $C_p$ is a complete intersection of $x^{pk}, y^{pl}$.
\item $\omega_{C_i}\cong {\cal O}_{C_i}((i-1)N+d-4)$ for $1\leq i\leq p$
where $\omega$ as usual denote the dualising sheaf.
\item Thus by Serre Construction [{\it e.g.\/},\ see\cite{OSS}], if we denote by
$L_i={\cal O}_Y(-(i-1)N-d)$ for all $i$, then we have exact sequences,
$$0\to L_i\stackrel{\alpha_i}{\longrightarrow}
M_i\stackrel{\beta_i}{\longrightarrow} {\cal I}_{C_i}\to 0,$$
where $M_i$ are rank two vector bundles on $Y$ for $1\leq i\leq p$.
In fact we can arrange these extensions to fit into the following
commutative diagrams,
\[
\begin{array}{ccccccccc}
0&\to& L_i&{\stackrel{\alpha_i}{\longrightarrow}}&
M_i&{\stackrel{\beta_i}{\longrightarrow}}& {\cal I}_{C_i}&\to& 0\\
&&\downarrow\cdot A&&\downarrow\eta_i&&\downarrow&&\\
0&\to& L_{i-1}&{\stackrel{\alpha_{i-1}}{\longrightarrow}}&
M_{i-1}&{\stackrel{\beta_{i-1}}{\longrightarrow}}& {\cal I}_{C_{i-1}}&\to& 0
\end{array}
\]
where $L_i\to L_{i-1}$ is multiplication by $A$ and ${\cal I}_{C_i}\to
{\cal I}_{C_{i-1}}$ is the natural inclusion of ideals.
\item There exists a nilpotent endomorphism $\phi:M_1\to M_1(d)$
given as follows: Notice that $L_1={\cal O}_Y(-d)$. So we can identify
${\cal I}_{C_1}\subset L_1(d)$ and then define $\phi=\alpha_1\beta_1$.
\item $M_i/\eta_{i+1}(M_{i+1})$ is annihilated by $A$,
$1\leq i<p$. Thus the natural map $M_i\otimes{\cal O}_Y(-N)\to M_i$ got by
multiplication by $A$ factors through $\eta_{i+1}$.
\item We have maps $g_i: L_{i+1}\to M_i(-d)$ for $1\leq i\leq p$ by
lifting $A^i$. {\it i.e.\/},\ the composite $\beta_i\circ g_i$ is just given by
the element $A^i\in{\cal I}_{C_i}$. We can arrange these maps so that
$\eta_i\circ g_i=g_{i-1}\circ A$ and $\phi\circ g_1=\alpha_1\circ A$.
\setcounter{try}{\theenumi}
\end{enumerate}
\end{claim}
The claim is proved exactly as in \cite{MK}. In fact the whole
construction is similar to that in \cite{MK}. So let me defer these
proofs and continue the construction assuming the above claim. Let
$${\cal L}=L_p\oplus L_{p-1}(-d)\oplus \cdots \oplus L_2(-d(p-2))$$
and
$${\cal M}=M_p\oplus M_{p-1}(-d)\oplus\cdots\oplus M_1(-d(p-1))$$
We have a map $f:{\cal L}\to{\cal M}$ given by sending
$(x_p,x_{p-1},\ldots,x_2)\in{\cal L}$ to
$$(-\alpha_p(x_p),-\alpha_{p-1}(x_{p-1})+g_{p-1}(x_p),\ldots,
-\alpha_2(x_2)+g_2(x_3), g_1(x_2))\in{\cal M}$$
Let the cokernel be called $M$.
\begin{claim}
\begin{enumerate}
\addtocounter{enumi}{\thetry}
\item $M$ is a rank $p+1$ vector bundle on $Y$.
\setcounter{try}{\theenumi}
\end{enumerate}
\end{claim}
We have an endomorphism $\theta: {\cal M}\to {\cal M}(d)$ given by,
$$(x_p,x_{p-1},\ldots, x_1)\mapsto (0, \eta_p(x_p),
\ldots,\eta_3(x_3),\eta_2(x_2)+ \phi(x_1))$$
Since $\phi^2=0$, one can easily see that $\theta^{p+1}=0$.
\begin{claim}
\begin{enumerate}
\addtocounter{enumi}{\thetry}
\item $\theta$ descends to a nilpotent endomorphism, $\varphi:M\to M(d)$
\setcounter{try}{\theenumi}
\end{enumerate}
\end{claim}
We have the natural map $\psi:M_p\to M$. Notice that since $C_p$ is a
complete intersection, $M_p$ is the direct sum of two line bundles.
Finally we have,
\begin{claim}
\begin{enumerate}
\addtocounter{enumi}{\thetry}
\item $$\psi(M_p)+\varphi(M(-d))=M$$
\item $M_1$ is not a direct sum of line bundles and hence neither
is $M$.
\setcounter{try}{\theenumi}
\end{enumerate}
\end{claim}
By taking $d=1$ in the above construction, we get a rank 2 bundle $E$
on $\P^4$ by lemma \ref{extension}. Since $M$ is not a direct sum of
line bundles, by lemma \ref{nonsplit}, $E$ is not a direct sum of line
bundles.
\subsection{Computation of Chern classes}\label{chern}
Our vector bundle $E$ on $\P^4$ is given by the exact sequence,
$$0\to E\to {\cal O}(-pk)\oplus{\cal O}(-pl)\to {\cal F}\to 0$$
where $M=\pi_*{\cal F}$ is the vector bundle on $\P^3$ as we have
constructed.
Notice that we are looking at the case $d=1$. So
to compute the Chern classes of $E$, we may as well restrict to a
general linear space of dimension 2, since rank of $E$ is 2. On this
$\P^2$ we will compute the class of $E$ in $K_0$. We have our
distinguished $\P^3\subset\P^4$ and the curve $C\subset\P^3$. So by
choosing our linear space generally, we may assume that it does not
meet this curve. Then we have $\P^1\subset\P^2$, after intersecting
with this linear space. We will denote by the same letters
restrictions of all our vector bundles. Since ${\cal I}_{C_i}={\cal O}_{\P^1}$
now, we see that $[{\cal M}_i]=[{\cal O}]+[L_i]$ in $K_0(\P^1)$. Thus,
$$[M]=[{\cal O}]+[{\cal O}(-1)]+\ldots+[{\cal O}(-(p-1))]+[L_1(-(p-1))]$$
$$=p[{\cal O}]+[{\cal O}(-{p(p+1)\over 2})].$$
Thus on $\P^2$, we see that,
$$[{\cal F}]=p[{\cal O}]-p[{\cal O}(-1)]+[{\cal O}(-{p(p+1)\over 2})]-[{\cal O}(-1-{p(p+1)\over 2})].$$
So we get $[E]$ to be,
$$[{\cal O}(-pk)]+[{\cal O}(-pl)]-p[{\cal O}]+p[{\cal O}(-1)]-[{\cal O}(-{p(p+1)\over 2})]+
[{\cal O}(-1-{p(p+1) \over 2})].$$
Now an easy computation will show that,
$$c_1(E)=-1-p(k+l+1)$$
$$c_2(E)=p(p+1)(k+l)+p^2kl$$
For instance, taking $p=2$ and $k=l=1$, we get,
$$c_1(E)=-7, \quad c_2(E)=16.$$
By choosing appropriate $k,l$, one can construct vectorbundles with
$c_1^2>4c_2$ for example, in any characteristic, $p>0$. For instance,
let $s\geq 1$ be any integer and $k=1, l=ps-s$. Then $N=ps+1$ and the
corresponding rank two vector bundle has
$$c_1^2-4c_2=\alpha s^2+\beta s+\gamma$$
where $\alpha,\beta,\gamma$ depend only on $p$ and
$\alpha=p^2(p-1)^2>0$. Thus by choosing $s>>0$, we can make the
vector bundle to be of the required type.
\section{Proofs of the claims}
\begin{enumerate}
\item $C_p$ is a complete itersection is clear, since $A^p$ is in the
ideal generated by $x^{pk}$ and $y^{pl}$. To check the rest, we need
only look at points where either $z\neq 0$ or $t\neq 0$. If $z\neq 0$
one sees immediately that $x^{pk}\in (y^{pl}, A^i)$.
\item This is done by descending induction on $i$. For $i=p$, since
$C_p$ is a complete intersection of $x^{pk},y^{pl}$, this is obvious.
So assume result proved for all $p\geq i>1$. Then we have an exact
seqence,
$$0\to {\cal O}_{C_{i-1}}(-N)\to {\cal O}_{C_i}\to {\cal O}_{C_1}\to 0$$
which we dualise to get,
$$0\to \omega_{C_1}\to \omega_{C_i}\to \omega_{C_{i-1}}(N)\to 0$$
Since we already know from 1) that the last term is a line bundle on
$C_{i-1}$ and then the proof is clear.
\item This is just Serre construction.
Assume we have constructed the exact sequences upto $i-1$ with the
commutative diagrams, the first one is just by the usual Serre
construction. By taking the natural map $L_i\to L_{i-1}$ given by
multiplication by $A$, we get a map,
$$H^0({\cal O}_{C_i})=\Ext^1({\cal I}_{C_i}, L_i)\to
\Ext^1({\cal I}_{C_i}, L_{i-1})=H^0({\cal O}_{C_i}(N))$$
which is just multiplication by $A$. We also have a natural map,
induced from the inclusion, ${\cal I}_{C_i}\subset {\cal I}_{C_{i-1}}$,
$$H^0({\cal O}_{C_{i-1}})=\Ext^1({\cal I}_{C_{i-1}}, L_{i-1})\to
\Ext^1({\cal I}_{C_i}, L_{i-1})=H^0({\cal O}_{C_i}(N))$$
In this case also, it is clear that the element `1'$\in
H^0({\cal O}_{C_{i-1}})$ goes to `A'$\in H^0({\cal O}_{C_i}(N))$, which is also
the image of `1'$\in H^0({\cal O}_{C_i})$ by multiplication by $A$. But
these 1's give extensions as desired and the commutative diagram as
desired.
\item This is obvious.
\item Notice that outside $\{A=0\}$, since multiplication by $A$ and
natural inclusions of ideal sheaves are isomorphisms, $\eta_{i+1}$ is
also an isomorphism. So we need to verify the claim at points on
$A=0$. For such a point, which is not on $C$,
${\cal I}_{C_{i+1}}\hookrightarrow {\cal I}_{C_i}$ is an isomorphism. So the
cokernel of $\eta_{i+1}$ is the same as the cokernel of $\cdot A$, so
claim is proved for such points. Now let $p\in C$. Then near $p$,
${\cal I}_{C_i}=(z,A^i)$, where $z=x^{pk}$ or $y^{pl}$ at $p$. Also
${\cal I}_{C_{i+1}}=(z,A^{i+1})$. Pick a basis for $M_i$ and $M_{i+1}$,
which go to $z, A^i, A^{i+1}$. Then $\eta_{i+1}$ is represented by a
matrix of the form $(v_1,v_2)$, where $v_i\in M_i$ and
$$v_1=(1,0)+\lambda\alpha_i(1),\quad v_2=(0,A)+\mu\alpha_i(1),$$
where $\lambda,\mu\in{\cal O}_p$. But the fact that
$\eta_{i+1}\circ\alpha_{i+1}=\alpha_i\circ A$ implies immediately that
the cokernel of $\eta_{i+1}$ is in fact isomorphic to ${\cal O}_p/A{\cal O}_p$.
(In fact, this argument shows that $M_i/\eta_{i+1}M_{i+1}$ is a line
bundle on the hypersurface $A=0$. Moreover, one can even write down
exactly this line bundle, though we will not use that fact.) Thus the
map $M_i\otimes {\cal O}_Y(-N)\to M_i$, got by multiplication by $A$,
factors through $\eta_{i+1}$.
\item This follows essentially from the fact that
$H^1(L_i^{-1}\otimes L_{i-1}(-d))=0$. We will construct the $g$'s by
induction. By the stated vanishing, we have $g_1:L_2\to M_1(-d)$ by
lifting $A\in{\cal I}_{C_1}$. Clearly $\phi\circ g_1=\alpha_1\circ A$. So
assume we have constructed $g_{i-1}$ with the required property. So we
have $g_{i-1}\circ A: L_{i+1}\to M_{i-1}(-d)$. This is just the
composite,
$$L_{i+1}=L_i\otimes{\cal O}_Y(-N)\stackrel{g_{i-1}\otimes
1}{\longrightarrow}
M_{i-1}(-d)\otimes{\cal O}_Y(-N)\stackrel{A}{\longrightarrow} M_{i-1}(-d).$$
Now by the previous claim, we see that the last map factors through
$\eta_i$. So we get a map $g_i:L_{i+1}\to M_i(-d)$ such that $\eta_i
g_i=g_{i-1}\circ A$. To compute $\beta_i g_i$ we may compose it with
the natural inclusion of ideals and then it is just
$$\beta_{i-1}\eta_i g_i=\beta_{i-1} g_{i-1}\circ A=A^{i-1}\circ
A=A^i.$$
\item We must show that $f$ is injective at every point. So let $P\in
C$. Then $\alpha_i$'s are all zero. So if $f(x_p,\ldots,x_2)=0$ at
$P$, then $g_i(x_i)=0$. But since near $P$, $A^{i-1}$ is part of a
minimal set of generator of ${\cal I}_{C_{i-1}}$ and thus $g_i(x_i)=0$
implies $x_i=0$ at this point for all $i$. Now let $P\not\in C$. Then
$\alpha_i$'s are injective at this point. If $f(x_p,\ldots, x_2)=0$
at $P$, we will use descending induction to prove that all the
$x_i$'s are zero. Clearly $\alpha_p(x_p)=0$ implies $x_p=0$. Assume
we have proved $x_p=\dots= x_k=0$. Then by looking at the definition
of $f$, we see that $\alpha_{k-1}(x_{k-1})=0$ and thus $x_{k-1}=0$.
\item We should show that ${\rm Im\ }\theta\circ f\subset {\rm Im\ }
f$.
\[
\begin{array}{ll}
\theta\circ f(x_p,\dots,x_2)&\\
=\theta(-\alpha_p(x_p),
-\alpha_{p-1}(x_{p-1})+g_p(x_p), \ldots, -\alpha_2(x_2)+g_3(x_3),
g_2(x_2))&\\
= f(0,Ax_p,\ldots,Ax_3)&
\end{array}
\]
\item For this it clearly suffices to prove that
$${\cal M}'={\rm Im\ }f({\cal L})+{\rm Im\ }\theta({\cal M}(-d)) +M_p={\cal M}.$$
So let $b=(b_p,\dots, b_1)\in{\cal M}$.
First let us look at a point
$P\not\in C$. We will show inductively that there exists a
$c_i\in{\cal M}'$ such that for all $j\geq i$, the $j^{th}$ coordinate of
$b-c_i$ is zero. We may clearly take $c_p=(b_p,0,\ldots,0)$. So by
induction we may assume that $b_j=0$ for $j>i$. Since $P\not\in C$,
we see that $\alpha_i(L_i)+\eta_{i+1}(M_{i+1})=M_i$ at $P$. So we may
write $b_i=\alpha_i(s)+\eta_{i+1}(t)$. Let us first look at the case
when $i\geq 2$. Consider
$$c_i=f(0,\ldots, 0,-s,0,\ldots,0)+\theta(0,\dots,0, t,\dots,0)\in{\cal M}'$$
By our definition of $f,\theta$, we can easily see that $b-c_i$ has
all coordinates upto $i$ zero. Next look at the case when $i=1$.
Again since $P\not\in C$, we see that
$\phi(M_1(-d))+\eta_2(M_2)=M_1$. Thus we can write
$b_1=\phi(s)+\eta_2(t)$. Let $c_1=\theta(0,\dots,0,t,s)\in{\cal M}'$ and
we are done.
Now let us look at points $P\in C$. Now we will show inductively that
there exists $c_i\in{\cal M}'$ such that $b-c_i$ has $j^{th}$ coordinate
zero for all $j\leq i$. For $i=1$, we have
$g_2(L_2(d))+\eta_2(M_2)=M_1$ at $P\in C$. So $b_1=g_2(s)+\eta_2(t)$.
Take $c_1=f(0,\ldots,0,s)+\theta(0,\ldots,0,t,0)\in{\cal M}'$. So assume
that $b_j=0$ for $j<i$. Again let us first look at the case when
$i<p$. Again we have $g_{i+1}(L_{i+1}(1))+\eta_{i+1}(M_{i+1})=M_i$ at
$P\in C$. Therefore we may write $b_i=g_{i+1}(s)+\eta_{i+1}(t)$.
Consider
$c_i=f(0,\ldots,s,\ldots,0)+\theta(0,\ldots,t,\ldots,0)\in{\cal M}'$. One
easily checks that $b-c_i$ has all coordinates zero upto the
$i^{th}$ by using our definition of $f,\theta$. Finally assume $i=p$.
But $(b_p,0,\dots,0)$ clearly belongs to ${\cal M}'$ and thus we are done.
\item If $M_1$ is a direct sum of line bundles, we get ${\cal I}_{C_1}$
is a complete intersection, say of $f,g$ of degrees $a,b$. Then we see
by our Koszul exact sequence, $a+b=d$. Also degree of $C_1=ab$ by
Bezout's theorem. Since $C_1$ is supported along the line $C$, we may
compute its degree by computing the length of ${\cal O}_{C_1}$ at the
generic point of $C$. Easy to see that this is $pkl$. Thus
$pkl=ab\leq d^2/4$. This contradicts our choice of $N,k,l$.
Thus ${\cal M}$ is not a direct sum of line bundles. Since ${\cal L}$ is a
direct sum of line bundles, this implies that $M$ is also not a direct
sum of line bundles.
\end{enumerate}
\section{Characteristic zero case}
In this section, we will analyse our construction in characteristic
zero. So assume that our base field is $\C$, the complex numbers from
now on. Let $E$ be a vector bundle on $\P^3$ with a nilpotent
endomorphism $\phi: E\to E(d)$ for a fixed integer $d$ and let $F$ be
the direct sum of two line bundles. Let $\psi:F\to E$ be a
homomorphism such that $\phi(E(-d))+\psi(F)=E$. Assume further that
$E$ is not a direct sum of line bundles and $\rank E=k$. Assume that
we have chosen $E$ with the smallest possible rank.
\begin{enumerate}
\item {\em We may assume that rank of $\phi$ is $k-1$}
If not rank of $\phi\leq k-2$ and since $F$ has rank two, we see that
$E=\phi(E(-d))\oplus\psi(F).$ Then easy to see that the vector bundle
$\phi(E)$ also has all the properties, with $\phi\psi:F(-d)\to\phi(E)$
replacing $\psi$. So by minimality of ranks we have proved the claim.
(Note that $E$ is not a direct sum of line bundles implies $\phi(E)$ is
also not a direct sum of line bundles since $F$ is.)
Thus we see that $\ker\phi$ has rank one, $\phi^{k-1}\neq 0$ and
$\phi^k=0$.
\item {\em We may assume $\phi^{k-1}(E)$ is an ideal sheaf defining a
complete intersection curve}
By the previous step, $\phi^{k-1}(E)$ is a rank one torsion free sheaf
and hence isomorphic to $I(l)$ for some ideal sheaf $I$ of height
bigger than or equal to 2 and $l$ an integer. Since we may twist $E$
and $F$ by ${\cal O}(-l)$, we may assume that $l=0$. So we have an exact
sequence,
\begin{equation}\label{basic1}
0\to M\to E\to I\to 0
\end{equation}
where $M=\ker\phi^{k-1}$. Since $\psi(F)$ maps onto $E/\phi(E(-d))$
and since $\phi(E(-d))$ is contained in $M$ we see that $F$ maps onto
$I$. So if $I$ is a proper ideal we would be done. If not $I={\cal O}$ and
then since $F$ is rank two, this surjection must split. So $F$ is
isomorphic to ${\cal O}\oplus{\cal O}(n)$ for some $n$ and the map $E\to{\cal O}$
also splits. Thus $E=M\oplus{\cal O}$. Notice that $\psi{\cal O}(n)\subset M$.
We have
$$M=(\phi(E(-d))+\psi({\cal O}(n))+\psi({\cal O}))\cap M=\phi(E(-d))+\psi({\cal O}(n))$$
which in turn is equal to
$$\phi(M(-d))+\phi({\cal O}(-d))+\psi({\cal O}(n)).$$
Thus replacing $(E,\phi,\psi)$ by $M$, $\phi: M(-d)\to M$, the
restriction of $\phi$ and
$$\psi':{\cal O}(-d)\oplus{\cal O}(n)\to M\quad \psi'(x,y)=\phi(x)+\psi(y),$$
we get an example with smaller rank. This contradicts the minimality
of rank assumption.
The fact that $I$ has homological dimension one implies
that $M$ is a vector bundle.
\item {\em rank $E\geq 3$.}
Clearly rank is bigger than 1, since $E$ is not a direct sum of line
bundles. So if the
assertion is false, then rank must be two. But since $E$ maps onto a
complete intersection ideal, by Serre's construction we see that $E$
must be a direct sum of line bundles.
{\em From now on we assume that the rank of $E$ is three.}
We have the basic diagram,
\begin{equation}
\begin{array}{ccccccccc}\label{basic2}
0&\rightarrow&M&\rightarrow&E&\rightarrow&I&\rightarrow&0\\
&&\theta\uparrow&&\psi\uparrow&&id\uparrow&&\\
0&\rightarrow&L&\rightarrow&F&\rightarrow&I&\rightarrow&0
\end{array}
\end{equation}
where $L$ is the determinant of $F$. Notice that, since $I$ is a
proper ideal, $\psi$ and hence $\theta$ are both inclusions.
\item {\em $M/\phi(E(-d))$ is not supported along divisors}
Since $\phi$ has rank two, $M/\phi(E(-d))$ is a torsion sheaf. If it
is supported along divisors, choose $Z\subset {\rm supp}\,(M/\phi(E(-d)))$, a
reduced irreducible divisor. Then one sees that
$E/\phi(E(-d))\otimes{\cal O}_Z$ has rank at least two. We of course have a
surjection using $\psi$ from $F\otimes{\cal O}_Z$ to this sheaf. This
implies that this surjection must be an isomorphism since
$E/\phi(E(-d))\otimes{\cal O}_Z$ has rank at least two over $Z$ and
$F_{\mid Z}$ is a vector bundle of rank two. Thus $E\otimes{\cal O}_Z$ is a
direct summ of $F\otimes{\cal O}_Z$ and a line bundle which must be of the
form ${\cal O}_Z(l)$ for some $l$ by determinant considerations. Thus
$E\otimes{\cal O}_Z$ is a direct sum of line bundles. Now one can see
easily that $H^1(E(*))=H^2(E(*))=0$ using the fact $H^1({\cal O}_Z(*))=0$.
So by Horrocks criterion, we see that $E$ is also a direct sum of line
bundles leading to a contradiction.
Thus we see that $\det E=\det M=\det\phi(E(-d))$. But the kernel of
$\phi:E(-d)\to E$ is a rank one sheaf which is reflexive and hence a
line bundle, say $A$. So $\det E(-d)=A\otimes\det\phi(E(-d))$ and
hence $A={\cal O}(-3d)$.
Notice that $A\subset M(-d)$ and $M(-d)/A\cong J(l)$ for some ideal
sheaf $J$
of height bigger than one and $l$ an integer. Also $\phi$ restricts
to a nilpotent endomorphism $M\to M(d)$ and one easily sees that it is
obtained by going to $J(l)$ and composing it with some embedding of
$J(l)$ in $A(d)$. In particular, $l\leq -2d$. On the other hand we have a
natural map $I=E/M\to M(d)/A(2d)=J(l+2d)$ induced by $\phi$ and this is
injective. So $l+2d\geq 0$ and thus $l=-2d$. Thus $\det M={\cal O}(-3d)$ and
hence $\det E=\det M={\cal O}(-3d)$.
So we have the next basic exact sequence,
\begin{equation}\label{basic3}
0\rightarrow{\cal O}(-d)\rightarrow M(d)\rightarrow J\rightarrow 0
\end{equation}
\item {\em $J$ is a proper local complete intersection ideal of a curve}
We only need to show that $J$ is a proper ideal since $M$ is a rank
two vector bundle. If $J={\cal O}$, then $M(d)={\cal O}(-d)\oplus{\cal O}$.
The image of $\phi(E)$ in $J$ is $I$. We have
$$M(d)= E(d)\cap M(d)=(\phi(E)+\psi(F(d)))\cap M(d)$$
$$=\phi(E)+\psi(F(d))\cap M(d)=\phi(E)+\theta(L(d))$$
In particular, $J=I+{\rm image}(L(d))$. If $J={\cal O}$ then, we see that
$F\oplus L(d)$ surjects onto $J$ and thus one of them must be ${\cal O}$
(Any three non-trivial polynomials in $\P^3$ have a common zero). This
is impossible since $I$ is non-trivial and $L$ is the determinant of
$F$ unless $d=a+b$ where $F={\cal O}(-a)\oplus{\cal O}(-b)$. This implies in
particular that $d>0$ and going to the commutative diagram
(\ref{basic2}) above, using the fact that $M={\cal O}(-d)\oplus{\cal O}(-2d)$
and $\theta$ is injective, we get an exact sequence,
$$0\to F\stackrel{\psi}{\longrightarrow} E\to {\cal O}(-2d)\to 0,$$
which implies that $E$ is a direct sum of line bundles. We see $J$
is a proper ideal of height at least two and since $M$ surjects onto
it, it must be a local complete intersection ideal of height two. The
image of $L(d)$ corresponds to an element $G\in J$ of degree
$c=a+b-d.$ We have $J=I+(G)$. We have an exact sequence,
\begin{equation}\label{basic5}
0\rightarrow I\rightarrow J\rightarrow T\rightarrow 0\label{basic4}
\end{equation}
where $T={\cal O}_X(-c)$, with $X={\rm supp}\, T$. Let us also denote
by $C$, the curve defined by $I$ and $D$ that defined by $J$. Then $D$
is a closed subscheme of $C$.
\item {\em $X$ is a local complete intersection curve}
First we show that it has no components of dimension two. If it
did, say $Z$, an irreducible hypersurface, restricting to $Z$, we see
that $M\otimes{\cal O}_Z$ surjects onto ${\cal O}_Z(-c)$ and then
$M\otimes{\cal O}_Z$ must be isomorphic to ${\cal O}_Z(-c)\oplus{\cal O}_Z(c)$.
Again as before we get by Horrock's criterion, that
$M\cong{\cal O}(c)\oplus{\cal O}(-c)$ and since it is supposed to surject onto
$J$ and $c\geq 0$ or $-c\geq 0$, we reach a contradiction. To check
that it is a local complete intersection curve, let us do it locally.
If $x\notin C$ then since $D\subset C$, we see that $J$ and $I$ are
locally ${\cal O}$ and if the inclusion were not an isomorphism, the
cokernel $T$ would be supported on a hypersurface which we have seen
is not the case. So $X_{\rm red}\subset C$. If $x\in C$ and not in $D$
clearly we are done since $I$ is a local complete intersection. So let
us look at a point $x\in D$. Since $I$ and $J$ are local complete
intersections and $J/I$ is principal, we may choose generators so that
$J=(f,g)$ and $I=(f,gh)$ and then $X$ is defined by $(f,h)$ and we are
done.
Dualising the above exact sequence (\ref{basic5}), we get,
$$ \sext^1(T, {\cal O})\rightarrow\sext^1(J,{\cal O})\rightarrow\sext^1(I,{\cal O})\rightarrow
\sext^2(T, {\cal O})\rightarrow 0$$
Since $X$ is a local complete intersection, we get that the first
term is zero. Also $\sext^2(T, {\cal O})\cong\omega_X(c+4)$.
{}From our basic exact sequences \ref{basic1}, \ref{basic3} it is easy
to see that
$\sext^1(I,{\cal O})\cong {\cal O}_C(a+b)$ and $\sext^1(J,{\cal O})\cong {\cal O}_D(d)$.
Consider the diagram,
\[
\begin{array}{ccccccccc}
0&\rightarrow&{\cal O}(-(a+b))&\rightarrow&F&\rightarrow&I&\rightarrow&0\\
&&G\downarrow&&\phi\psi\downarrow&&\downarrow&&\\
0&\rightarrow&{\cal O}(-d)&\rightarrow&M(d)&\rightarrow&J&\rightarrow&0
\end{array}
\]
This diagram is commutative. The only fact we need to show is that the
vertical arrow on the left is multiplication by $G$. But the map $\psi$
restricted to $L={\cal O}(-(a+b))$ maps to $M$ and its image in $J$ is $G$.
Since $\phi:M\to M(d)$ is got by going to $J(-d)$ and including it in
the kernel (of $\phi: M(d)\to M(2d)$), ${\cal O}(-d)$, we see that
$\phi\psi$ restricted to $L$ is precisely multiplication by $G$.
The fact that the vertical map on the left is multiplication by $G$
implies by dualising, that the natural map ${\cal O}_D(d)\to {\cal O}_C(a+b)$ is
multiplication by $G$. So the cokernel is isomorphic to ${\cal O}_D(a+b)$,
since the image of $G$ generates $J$ in ${\cal O}_C$. So we get that
$\omega_X(c+4)\cong{\cal O}_D(a+b)$. In particular $X=D$. So $C$ is
set-theoretically the same as $D$. Thus $G^2\in I$. Let $I=(f,g)$.
Write $G^2=Af+Bg$.
\item {\em $A, B, f, g$ have no common zeroes in $\P^3$}
If $p$ is such a zero, since $f,g$ vanish there, $p\in D$. Locally at
that point, we see that $J$ is generated by one of $f,g$ and $G$. Let
us assume without loss of generality, that $(f,G)=J$. Then $I=(f,
G^2)$. So $(f,g)=(f, Af+Bg)=(f,Bg)$ and thus $B$ must be a unit
there. That means $B$ does not vanish at $p$.
\item {\em The ideal sheaf defined by $f,g$ in $G^2=0$ is a line bundle}
This is obvious by the earlier remark and a local checking.
\begin{lemma}\cite{Pesk}
Let $Y\subset\P^3$ be the hypersurface defined by $G=0$ and $Y'$ that
defined by $G^2=0$. Then any line bundle on $Y'$ is trivial. In other
words, the natural map $\Pic \P^3\to\Pic Y'$ is an isomorphism.
\end{lemma}
\begin{proof} Proof is essentially due to Ellingsrud {\em et. al.} One
can first reduce to the case when $Y$ is a reduced irreducible
hypersurface as follows. First we may assume that $Y$ is irreducible
(not necessarily reduced). For this let $Y_1, Y_2,\ldots ,Y_n$ be the
scheme theoretic irreducible components of $Y'$. Let $L\in\Pic Y'$
such that its restriction to all the $Y_i$'s are of the form
${\cal O}_{Y_i}(n_i)$ for some $n_i$. Then restricting to the intersection
$Y_i\cap Y_j$ we see that $n_i$ is a constant independent of $i$.
Twisting by the negative of this line bundle, we are reduced to
proving that if $L$ restricted to all $Y_i$ is ${\cal O}_{Y_i}$, then
$L={\cal O}_{Y'}$. The proof is standard boot strapping. Assume we have
proved that $L$ restricted to $Z=Y_1\cup Y_2\cup\ldots Y_k$ is trivial
($\cong{\cal O}_Z$). We want to show that $L$ restricted to $Z'=Z\cup
Y_{k+1}$ is also trivial. We have the natural exact sequence,
$$
0\to {\cal O}_{Y_{k+1}}(-Z)\to{\cal O}_{Z'}\to{\cal O}_Z\to 0
$$
Tensoring this by $L$ and noting that $L_{\mid Z}\cong{\cal O}_Z$ and $L$
restricted to $Y_{k+1}$ is trivial and hence the kernel has no first
cohomology, we see that the section $`1'\in H^0({\cal O}_Z)$ can be lifted
to a section of $L$ restricted to $Z'$. Easy to see that this section
generates this line bundle by using the fact that $Z\cap
Y_{k+1}\neq\emptyset$.
So let us assume that $Y$ is irreducible defined by $G=0$. By the
${\cal O}^{*}$ exact sequences one can see that $\Pic Y'\to\Pic Y$ is
injective. Thus we reduce to the case of $Y$ a reduced irreducible
surface. Notice that $\Pic Y/(\Z=\Pic \P^3)$ is torsion free. Since
$H^1({\cal O}_{Y})=0$, it follows that the natural map
$$\Pic Y=H^1({\cal O}_Y^*)\stackrel{dlog}{\longrightarrow}
H^1(\Omega^1_{Y})$$
is injective [see \cite{SGA6}, Th\'{e}or\`{e}me 4.7, ({\romannumeral
3})]. So we only need to understand the map
$$
H^1(\Omega^1_{Y'}\otimes{\cal O}_Y)\to H^1(\Omega^1_Y).
$$
We have the fundamental exact sequence,
$$ {\cal O}_{\P^3}(-Y')\otimes{\cal O}_{Y'}\to \Omega^1_{\P^3}\otimes{\cal O}_{Y'}\to
\Omega^1_{Y'}\to 0$$
When we restrict this to $Y$, we see that the first map is zero, since
it is given essentially by the derivative of $G^2$, which is zero
modulo $G$.
Thus,
$\Omega^1_{Y'}\otimes{\cal O}_Y\cong\Omega^1_{\P^3}\otimes{\cal O}_Y.$ So we
have,
\begin{equation}\label{one}
H^1(\Omega^1_{Y'}\otimes{\cal O}_Y)\cong
H^1(\Omega^1_{\P^3}\otimes{\cal O}_Y)\cong \C
\end{equation}
We have the standard commutative diagram,
\[
\begin{array}{ccccc}
\Pic\P^3&\hookrightarrow&\Pic Y'&\hookrightarrow&\Pic Y\\
\downarrow&&\downarrow&&\downarrow\\
H^1(\Omega^1_{\P^3})&\to&H^1(\Omega^1_{Y'})&\to&H^1(\Omega^1_Y)
\end{array}
\]
{}From the various inclusions, we see that if $\alpha\in\Pic Y'$, we
must show that its image in $H^1(\Omega_Y)$ comes from $\Pic\P^3$. By
chasing the diagram, and using the equation (\ref{one}) we see that
there exists a {\em complex number},
$z$, such that $\alpha=zH$ where $H$ is the tautological class in
$\P^3$. By restricting to a general map from a smooth curve to $Y'$
and noting that then the classes correspond to degrees of the
corresponding line bundles on this curve, we get
$z\in\Q$. But now using the torsion freeness of $\Pic Y/\Z$, we see
that $\alpha\in\Pic \P^3$. Thus we get the result.
\end{proof}
Putting all these together, we see that $I=(f,g)=(G^2,h)$ for some
$h$. But then $J=(f, g,G)=(G,h)$ is a complete intersection. Let
$\deg h=e$ (where $e=a$ or $b$). $M(d)={\cal O}(-e)\oplus {\cal O}(-c)$. Then
let us look at the map $\theta$. It is a map from ${\cal O}(-a-b)\to
{\cal O}(-e-d)+{\cal O}(-c-d)$. But $-c-d=-a-b$ and thus the quotient must be
either a line bundle or must be a line bundle direct sum a sheaf ${\cal F}$
supported along a divisor. In the former case, since we also know
that it is the quotient of $E$ by $\psi(F)$, we get $E$ to be a direct
sum of line bundles, which we have assumed is not the case. In the
latter case one can see that $E$ can not be a vector bundle at any
point of intersection of ${\rm supp}\, {\cal F}$ and the curve defined by $f,g$.
This is also a contradiction. This finishes the proof that such
examples can not exist.
\end{enumerate}
Thus in characteristic zero, any such $E$ must have rank at least 4.
There are several technical difficulties which must be overcome to
analyse these cases (say of rank 4). The analysis which we have
carried out will be discussed elsewhere.
|
1995-11-03T06:20:18 | 9511 | alg-geom/9511002 | en | https://arxiv.org/abs/alg-geom/9511002 | [
"alg-geom",
"math.AG"
] | alg-geom/9511002 | Dr. Mueller-Stach | Stefan M\"uller-Stach | Constructing Indecomposable Motivic Cohomology Classes on Algebraic
Surfaces | Latex 2e, 34 pages | null | null | null | null | We describe a method to construct indecomposable classes in Bloch's higher
Chow group $CH^2(X,1)$ on algebraic surfaces over the complex numbers via
transcendental methods and apply it to obtain examples on K3-surfaces and some
surfaces of general type.
| [
{
"version": "v1",
"created": "Thu, 2 Nov 1995 13:15:32 GMT"
}
] | 2014-10-24T00:00:00 | [
[
"Müller-Stach",
"Stefan",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a smooth algebraic surface over ${\Bbb C}$. It is well known that
$K_0(X) \otimes {\Bbb Q} \cong \oplus CH^p(X) \otimes {\Bbb Q}$
as a consequence of Grothendieck's
Riemann-Roch theorem. S. Bloch has generalized this to higher algebraic
K-theory in \cite{Bl1}, see also \cite{Lev}. For example one obtains
$$K_1(X) \otimes {\Bbb Q} \cong \bigoplus_p CH^p(X,1)\otimes {\Bbb Q} $$
The groups $CH^p(X,n)$ are called {\it higher Chow groups}. \\
The purpose of this paper is to give an explicit way to construct
classes in $CH^2(X,1)$ that are non-trivial modulo the image of the natural
map
$$\gamma: {\rm Pic}(X) \otimes {\Bbb C}^* \to CH^2(X,1)$$
and modulo torsion. We call such classes {\it indecomposable}.
Cycles in $CH^2(X,1)$ can be constructed
by finding curves $Z_i $ in $X$ with rational functions $f_i$ on them that
satisfy $\sum {\rm div}(f_i)=0$ on $X$. By general conjectures
(see section 7), the cokernel of $\gamma$ is expected to be a countable
group on any smooth surface.\\
We will construct indecomposable elements in $CH^2(X,1)$
on general quartic K3-surfaces that contain a line
and on some special quintic surfaces of general type.
Other examples over the complex numbers have been constructed by M. Nori
(unpublished) on abelian surfaces, by A. Collino in \cite{Col} on Jacobian
varieties, and by C. Voisin and C. Oliva
on K3 surfaces in the unpublished work \cite{Voi2}.
The first examples ever given were over
number fields, by A. Beilinson in \cite{Be1}, other examples
on products of modular curves are contained in \cite{Fla}, \cite{Mil}
and probably at other places. \\
Our method consists of deforming the
complex structure of the pair $(X,Z)$ and studying the variation of mixed
Hodge structures associated to the open complements. The ideas of this
technique in the case of ordinary Chow groups are similar to those
developed in \cite{BMS} and in \cite{Voi3}, but eventually go back
to the fundamental idea of Griffiths to show the non-triviality of
cohomology classes on the general member of a family of varieties by
showing that their derivatives are non-zero.
The advantage of our method is that it is not restricted to surfaces with
trivial canonical bundle and that it is
very simple to apply in situations where some geometry is known, in
particular if an explicit family or a degeneration to a singular
configuration can be written down.\\
\ \\
Let us now describe the contents of this paper. In chapter 2 we first recall
the definition of Bloch's higher Chow groups
$CH^p(X,n)$ from \cite{Bl1} together with some of their basic properties.
Then we sketch the construction
of Deligne-Beilinson cohomology (\cite{Be1},\cite{EV})
and compare various definitions for the Chern class maps
$$c_{p,n}: CH^p(X,n) \to H^{2p-n}_{\cal D}(X_{\rm an},{\Bbb Z}(p)) $$
($X_{\rm an}$ denotes the underlying analytic space, where we assume that $X$
is
defined over ${\Bbb C}$) due to Beilinson, Bloch, Gillet and others. In
particular we recall the explicit integral that computes $c_{2,1}$ and
study the relation between the extension of mixed Hodge structures given
by the Gysin sequence attached to the support of a given cycle
$Z \in CH^p(X,1)$ and $c_{p,1}(Z)$, using the description of Deligne
cohomology of smooth, projective varieties as ${\rm Ext^1}$ in the category
of mixed Hodge structures. \\
In chapter 3 we present the circle of ideas around the deformation
theory of Deligne-Beilinson cohomology classes and the rigidity of
Chern classes from higher Chow groups. This is essentially due to Bloch and
Beilinson, see \cite{Be1}. It implies in particular that
$$c_{p,n}: CH^p(X,n) \otimes {\Bbb Q} \to H^{2p-n}_{\cal D}(X_{\rm an},{\Bbb Q}(p)) $$
has a countable image for $n \ge 2$ if $X$ is smooth and proper over ${\Bbb C}$.
In the case $n=1$ this is not true anymore, instead we can show the
following:
\begin{proposition} Let $X$ be a smooth and projective variety over ${\Bbb C}$.
Then for all $p$
the image of $c_{p,1}$ in $H^{2p-1}_{\cal D}(X,{\Bbb Q}(p))/Hg^{p-1,p-1}(X) \otimes
{\Bbb C}/{\Bbb Q}(1)$ is countable.
\end{proposition}
Here $ Hg^{i,i}(X) \subset H^{2i}(X,{\Bbb Z}(i))$
denotes the set of Hodge classes.
This seems to be a well known fact, but I
could not find a proof in the literature, so I decided to include one here
for the sake of completeness. A similar result holds in the case $n=0$,
stating that the Griffiths group has countable image in the intermediate
Jacobian $J^p(X)$ modulo the maximal abelian subvariety $J^p_a(X)$, see
\cite{Sh}. \\
\ \\
In chapter 4 these results are applied to the study of $CH^2(X,1)$ for an
algebraic surface $X$ over ${\Bbb C}$.
We prove the following result, which was communicated by H. Esnault:
\begin{proposition} Let $X$ be smooth, projective over ${\Bbb C}$. Then
$CH^2(X,1)$ decomposes iff ${\cal F}^2_{\Bbb Z} = {\cal F}^1_{\Bbb Z} \wedge
{\cal F}^1_{\Bbb Z}$ and $H^1(X,{\cal F}^2_{\Bbb Z}) \otimes {\Bbb Q}=0$.
\end{proposition}
The sheaves ${\cal F}^p_{\Bbb Z}$ have holomorphic p-forms with log-poles and
${\Bbb Z}(p)-$periods as sections. A precise formulation can be found in the text.
The proof uses Gersten-Quillen resolutions. Then we discuss the relations
with Bloch's conjecture on the Chow groups of zero cycles on surfaces
mentioned already above and the result of Esnault-Levine (\cite{EL}) about
the relation between the decomposability of $CH^p(X,1)$ and the injectivity
of the cycle maps $c_{r,0}$ for $d-p+1 \le r \le d$, where $d={\rm dim}(X)$.
In the remaining part of the chapter we prepare the setup of variations
of mixed Hodge structures associated to a family of cycles $Z_t \in
CH^2(X_t,1)$.
Let us explain the necessary deformation theory. Assume we look at a
smooth, proper deformation $f:{\cal X} \to S$ of $X$ with $S$ a smooth
and quasiprojective variety, a base point $0 \in S$ such that $f^{-1}(0)=X$
and a normal crossing divisor ${\cal Z}$ in ${\cal X}$ containing out of two smooth
components ${\cal Z}_1,{\cal Z}_2$ such that ${\cal Z}_1$ and
${\cal Z}_2$ (resp.
${\cal Z}_1 \cap {\cal Z}_2$) are smooth of relative dimension one (resp.
zero) over $S$ and restrict to $Z_1$ and $Z_2$ over the central fiber.
We get an exact diagram:
$$\matrix{ 0 & \to & T_X(log Z) & \to & T_{{\cal X}}(log({\cal Z}))|_X
& \to & f^*T_{S,0} & \to & 0 \cr
& & \downarrow & & \downarrow & & || & & \cr
0 & \to & T_X & \to & T_{ {\cal X}}|X & \to & f^*T_{S,0} & \to & 0 }
$$
The {\it logarithmic
Kodaira-Spencer map} is defined as the coboundary map
$$ T_{S,0} \longrightarrow H^1(X,T_X(log Z)) $$
Let us denote the image of $T_{S,0} \to H^1(X,T_X(log Z))$ by $W(log)$ and
the further image in $H^1(X,T_X)$ by $W$.\\
Our main result then is:
\begin{theorem} {\bf (CRITERION FOR INDECOMPOSABILITY)}: \\
Let $X$ be a smooth projective surface over ${\Bbb C}$.
Assume we are given two smooth and connected curves $Z_1$ and $Z_2$
on $X$ intersecting
transversally and nontrivial rational functions $f_i$ on $Z_i$ ($i=1,2$),
such that ${\rm div}(f_1)+{\rm div}(f_2)=0$ as a zero-cycle on $X$. Denote by
$Z=Z_1 \otimes f_1 +Z_2 \otimes f_2$ the resulting cycle in
$CH^2(X,1)=H^1(X,{\cal K}_2)$ and suppose the following
conditions hold:\\
(1) $Z$ also defines a cycle in Bloch's higher Chow group
$CH^1(|Z|,1)$ - again denoted by $Z$- and as such is not equivalent to
$Z_1 \otimes a_1 + Z_2 \otimes a_2$ with $a_1,a_2 \in {\Bbb C}^*$. \\
(2) There exist a smooth, proper deformation $f:{\cal X} \to S$ with $S$ a smooth
and quasiprojective variety, a base point $0 \in S$ such that $f^{-1}(0)=X$
and the following properties hold:\\
(a) The situation in (1) deforms together with $X$: There exists a normal
crossing divisor ${\cal Z}={\cal Z}_1+{\cal Z}_2 \subset {\cal X}$ with ${\cal Z}|_X=Z_1+Z_2$,
consisting out of two smooth components ${\cal Z}_1,{\cal Z}_2$
such that ${\cal Z}_1$ and ${\cal Z}_2$ (resp.
${\cal Z}_1 \cap {\cal Z}_2$) are smooth of relative dimension one (resp.
zero) over $S$. Furthermore there exist rational functions $F_i$ on ${\cal Z}_i$
such that their restriction to each fiber $X_t:=f^{-1}(t)$ satisfy
${\rm div}(F_{1,t})+ {\rm div}(F_{2,t})=0$ as a zero-cycle in $X_t$ and
therefore define classes
$ Z_t=Z_{1,t} \otimes F_{1,t}+Z_{2,t} \otimes F_{2,t}$
in $CH^2(X_t,1)$ and in $CH^1(|Z_t|,1)$ for all $t \in S$.\\
(b) The cup-product map
$$H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(-Z)) \to H^1(X,\Omega^1_X)
/\oplus_i H^0(Z_i,{\cal O}_{Z_i}) $$
has no left kernel.\\
(c) If $W(log) \subset H^1(X,T_X(logZ))$ denotes the image of
the logarithmic Kodaira-Spencer map in
$H^1(X,T_X(logZ))$,
then $W(log)$ contains the image of the natural map
$$H^1(X,T_X(-Z)) \to H^1(X,T_X(logZ))$$ \\
(d) For $t$ outside a countable number of proper analytic subsets of $S$,
$Z_{1,t}$ and $Z_{2,t}$ generate ${\rm NS}(X_t) \otimes {\Bbb Q}$.\\
{\bf Then:} $Z_t$ is non-torsion in $CH^2(X_t,1)/{\rm Pic}(X_t) \otimes {\Bbb C}^*$ for
$t$ outside a countable number of proper analytic subsets of $S$.
\end{theorem}
\ \\
Note that (2b) and (2d) together imply that the Picard number of $X_t$ is
not maximal for $t$ outside a countable number of proper analytic subsets of
$S$. Instead of the assumptions (2b) and (2c) we can also state a weaker
assumption (3) to get a sharper result:
\ \\
{\bf VARIANT:} \\
{\it Assume (1),(2a),(2d) of the above main theorem and additionally the
following instead of (2b),(2c):\\
(3)If $W(log) \subset H^1(X,T_X(logZ))$ denotes the image of
the logarithmic Kodaira-Spencer map in
$H^1(X,T_X(logZ))$, then the following map has no left kernel:
$$H^0(X,\Omega^2_X(logZ)) \otimes W(log) \to H^1(X,\Omega^1_X(logZ))
$$
{\bf Then:} $Z_t$ is non-torsion in $CH^2(X_t,1)/{\rm Pic}(X_t) \otimes {\Bbb C}^*$ for
$t$ outside a countable number of proper analytic subsets of $S$.}\\
\ \\
We prove both statements in chapter 5.
In chapter 6 we apply this result and Nori's connectivity theorem to study
some examples:\\
{\bf Example 1:}
\begin{theorem}
Let $X \subset \P^3$ be a general hypersurface of degree
$d \ge 5$. Then the Chern class map $CH^2(X,1) \otimes {\Bbb Q} \to
H^3_{\cal D}(X,{\Bbb Q}(2)) $ has image isomorphic to $H^3_{\cal D}(\P^3,{\Bbb Q}(2))
\cong {\Bbb C}/{\Bbb Q}(1)$.
\end{theorem}
This result should be seen as a generalization of the classical
Noether-Lefschetz theorem:
\begin{theorem}(Noether-Lefschetz) \\
Let $X \subset \P^3$ be a general hypersurface of degree
$d \ge 4$. Then $CH^1(X) \otimes {\Bbb Q} $ is isomorphic to
$H^2_{\cal D}(\P^3,{\Bbb Q}(1)) \cong {\Bbb Q} $.
\end{theorem}
Both results can be shown using Nori's connectivity theorem \cite{Nori}
by the methods
of \cite{GM}. This was also observed by S.Bloch, M.Nori and C.Voisin
for example in \cite{Voi1}. We mention a slightly more general statement:
\begin{theorem}
Let $(Y,{\cal O}(1))$ be a smooth and projective polarized variety
of dimension $n+h$, $X \subset Y$ a
general complete intersection of dimension $n$ and
multidegree $(d_1,..,d_h)$ with $min(d_i)$ sufficiently large. Furthermore
assume that $1 \le p \le n$. Then:
$${\rm Image}( CH^p(X,1) \otimes {\Bbb Q} \to H^{2p-1}_{\cal D}(X,{\Bbb Q}(p))) $$
$$ \subset {\rm Image}(H^{2p-1}_{\cal D}(Y,{\Bbb Q}(p)) \to H^{2p-1}_{\cal D}(X,{\Bbb Q}(p))) $$
\end{theorem}
This result - already in the case of projective space - is somehow of
negative nature, because it destroys some obvious conjectures about the
image of higher Chern classes, see \cite{Voi1}. It leaves open the possibility
to construct examples on quartic K3-surfaces, also done in the paper
\cite{Voi2}. \\
{\bf Example 2:} We can show that the general quartic K3 surface $X$
containing a line has indecomposable $CH^2(X,1)$: Let $N$ be the
irreducible component
of smooth quartic surfaces containg a line such that some point $0 \in N$
corresponds to the Fermat quartic surface with the equation
$X_0=\{x \mid x_0^4+x_1^4-x_2^4-x_3^4=0 \} $.
Our cycles are obtained as follows:\\
A quartic surface $X$ that contains a line $G$ also contains an elliptic pencil
cut out by the residual elliptic curves of all hyperplane sections through
$G$. For a finite number of elements $E$ in this pencil, 2 of the 3
intersections points $P_1,P_2,P_3$ of $E$ and $G$ have the property that
$2(P_1-P_2)$ is rationally equivalent to zero. We show this in one example by
giving the explicit hyperelliptic map from $E$ to $\P^1$
ramified at those two points. Hence there is a
rational function $f_{1}$ on $E$ with zero divisor $2(P_1-P_2)$ and a rational
function $f_2$ on $G$ with zero divisor $2(P_2-P_1)$.
This construction can be extended to an irreducible
component $N$ of the Noether-Lefschetz locus of surfaces containing a
line. We obtain a cycle $E_t \otimes f_{1,t}+G_t \otimes f_{2,t} \in
CH^2(X_t,1)$ on a suitable covering $S$ of that component that respects the
choices of ordering of the chosen points.
\begin{theorem} Let $X_t$ be a general member of this family.
Then $CH^2(X_t,1)$ is not decomposable.
\end{theorem}
It is even true that the
general quartic K3-surface has indecomposable $CH^2(X,1)$, and this can be
proved by the same method as in the next example. The cycles used there were
used also by C. Oliva and C. Voisin in \cite{Voi2} on quartics. \\
In order to verify the assumptions of the main theorem we make use of
the Green- Gotzmann theorem and Griffiths' description of cohomology
groups of hypersurfaces via residues of differential forms, as
described in \cite{1594}. Assumption (2d) will hold for $t$ general on
such a component by Noether-Lefschetz theory.\\
With some more work, using a monodromy argument of H. Clemens which
was also used by A. Collino in \cite{Col}, one can probably prove infinite
generation for $CH^2(X_t,1)$ of a general quartic hypersurface
containing a line. Since this was also proved in \cite{Col} and \cite{Voi2}
we refrain from presenting it here. Obviously the idea would be to study the
monodromy around a countable set of loci on the parameter space
of the surfaces $X_t$.
\newpage
{\bf Example 3:} One can even obtain some examples of general
type: Look at the Shioda hypersurface of degree 5:
$$ X=\{x \in \P^3 \mid x_0x_1^4+x_1x_2^4+x_2x_0^4+x_3^5=0 \} $$
It has an automorphism $\sigma$ of order 65, given by
$$ \sigma: (x_0:x_1:x_2:x_3) \mapsto
(\zeta^{16}x_0:\zeta^{-4}x_1:\zeta x_2:x_3)$$
where $\zeta$ is a 65-th root of unity.
Shioda proves that the Picard group of $X$ is of rank one. Let $Z_1:=
X \cap H_3$ and $Z_2= X \cap H_0$,
where $H_i$ are the linear hyperplane sections $H_i=\{x_i=0\}$.
Then $Z_1 \cap Z_2$ intersect in two points (one
with multiplicity 4), called $P$ and $Q$ and one can show that $52P$ and
$52Q$ are rationally equivalent on both curves. $Z_2$ is not smooth, but
we construct a deformation $(X_t)_{t\in {\Bbb C}}$ and curves
$Z_{i,t}$ that are smooth for $t \neq 0$.
Taking a maximal irreducible
component $N$ of quintic surfaces such that it contains all $X_t$
and the cycle
$Z$ deforms along a suitable covering $S$ of $N$, the assumptions
of the theorem can be checked in the same way as in the previous example.
We obtain therefore:
\begin{theorem} A general member $X_t$ of the irreducible components of
quintics
that deform to the Shioda hypersurface $X$ and that preserve the given
cycle $Z \in CH^2(X,1)$ has indecomposable $CH^2(X_t,1)$.
\end{theorem}
In the remaining chapter 7 we sketch some ideas around these problems and
formulate some open problems. In particular we think that it would be
very convenient to have a good theory for singular surfaces in order to
get shorter proofs for indecomposability by degeneration methods.
\section{Higher Chow Groups and Chern Classes }
\subsection{Bloch's Higher Chow Groups }
Let $X$ be a quasiprojective variety over a field $k$. Define
$$\Delta^n := {\rm Spec \,}(k[T_0,...,T_n]/\sum T_i=1 ) $$
Then $\Delta^n \cong {\Bbb A}^n_k$ is affine n-space and by setting the coordinates
$t_i=1$ one obtains $(n+1)$ linear hypersurfaces in $\Delta^n$ called
codimension one faces. By iterating this one gets codimension $(n-m)$-faces
isomorphic to $\Delta^m$ for every $m<n$ inside of $\Delta^n$. These are
parametrized by strictly increasing maps $\rho:\{1,...,m\} \to \{1,...,n\}$.
Higher Chow groups are defined as the homology groups of a chain complex. Let
$Z^p(X,n) \subset Z^p(X \times \Delta^n)$ the subset of cycles
of codimension $p$ that meet all faces $X \times \Delta^m$ again in
codimension $p$ for $m<n$. Let $\partial_i: Z^p(X,n) \to Z^p(X,n-1)$ be the
restriction map to the $i-$th codimension one face for $i=0,...,n$ and let
$\partial=\sum (-1)^i \partial_i $. Then the homology of the complex
$$...\to Z^p(X,n+1) \to Z^p(X,n) \to Z^p(X,n-1) \to ...$$
at position $n$ is denoted by $CH^p(X,n)$ \cite{Bl1}. We will need the
following facts about higher Chow groups: \\
(1) There is also a cubical version: Here let $\square^n:=(\P^1 \setminus
\{1\})^n $ with coordinates $t_i$ and codimension one faces
obtained by setting $t_i=0,\infty$. The rest of the definition is completely
analogous except that one has to divide out degenerate cycles
and it is known \cite{Bl1} that both complexes are quasiisomorphic.\\
(2) The groups $CH^*(X,*)$ are covariant for proper maps and contravariant
for flat maps.\\
(3) If $W \subset X$ is a codimension $r$ subvariety, then one has
localization
$$...\to CH^*(X,n) \to CH^*(X \setminus W,n) \to CH^{*-r}(W,n-r) \to
CH^*(X,n-1) \to ... $$ \\
(4) $CH^*(X,0)=CH^*(X)$ are the usual Chow groups.\\
(5) If $X$ is smooth, there is a product \cite{Bl1}
$$CH^p(X,q) \otimes CH^r(X,s) \to CH^{p+r}(X,q+s)$$
which can be easily defined using the cubical version. Thus it is
possible to define an action of correspondences on higher Chow groups.\\
(6) There exist cycle classes to Deligne-Beilinson cohomology \cite{Bl2}
in case $k$ is a field of characteristic zero: If we fix an embedding
$\sigma: k \hookrightarrow {\Bbb C}$ and denote by $X_{an}$
the associated complex analytic space then we have maps
$$c_{p,n}: CH^p(X,n) \to H^{2p-n}_{\cal D}(X_{an},{\Bbb Z}(p))$$
They will be discussed in section (2.3).
\\
(7) There is a Riemann-Roch formula $K_n(X) \otimes {\Bbb Q} =\bigoplus_p CH^p(X,n)
\otimes {\Bbb Q}$, see \cite{Bl1} and \cite{Lev2}.\\
(8) Suslin's theorem \cite{Sus}: For $k$ itself: $CH^n({\rm Spec \,}(k),n)=K_n^M(k)$
(Milnor K-theory).\\
(9) If $X$ is smooth and proper then $CH^1(X,1)=k^*$ and $CH^1(X,n)=0$
for $n \ge 2$. \\
(10) For $X$ smooth, we have $CH^p(X,1) =H^{p-1}(X,{{\cal K}}_p)$ where
${\cal K}_p$ is Quillen's K-theory Zariski sheaf associated to the presheaf
$U \mapsto K_p({\cal O}(U)) $. Remember Bloch's formula $CH^p(X)=H^p(X,{\cal K}_p)$.\\
{\it Proof:} (for (10)) Consider the diagram
$$\matrix{ Z^p(X,2) & \to & Z^p(X,1) & \to & Z^p(X) \cr
\downarrow N & & \downarrow N & & || \cr
{\oplus}_{x \in X^{(p-2)}} K_2(k(x)) & \to &
\oplus_{x \in X^{(p-1)}} k(x)^* & \to & Z^p(X) }
$$
Here $N$ denotes the norm map. The inverse map is given by taking the graph
of rational functions on codimension $(p-1)-$subvarieties. To show that
the maps are mutually inverse to each other one needs only to show that
the graph of the norm of a cycle in $X \times {\Bbb A}^1$ is equivalent to
the cycle modulo $Z^2(X,2)$. This can be shown explicitely: First
reduce to the case where $X$ is a point and then use the explicit
formulas in \cite{Sus}.\hspace*{\fill}\hbox{$\square$}
{\it Remark:} By similar methods $CH^p(X,2) \to H^{p-2}(X,{\cal K}_p)$ is surjective
and an isomorphism for $p=2$.
\subsection{Deligne-Beilinson Cohomology}
\label{Hg}
Let $X$ be any scheme of finite type over ${\Bbb C}$. Then there exists a twisted
duality theory in the sense of Bloch-Ogus consisting of Deligne homology
and cohomology groups (\cite{EV},\cite{Gil},\cite{Ja2})
$$H^i_{\cal D}(X,{\Bbb Z}(j)),\quad H^{\cal D}_i(X,{\Bbb Z}(j)) $$
satisfying the axioms in \cite{BO}. We just mention the duality isomorphism
$$ H^i_{{\cal D},Z}(X,{\Bbb Z}(j)) \cong H^{\cal D}_{2d-i}(Z,{\Bbb Z}(d-j))$$
($d=\dim(X)$) for $X$ smooth and $Z \subset X$ a closed subvariety.
Very important for our purposes is also the weak purity statement under the
same assumptions:
The groups $H^i_{{\cal D},Z}(X,{\Bbb Z}(j))$ vanish for $i < 2r$ where $r$ denotes the
codimension of $Z$. Recall that for $X$ smooth one has an exact sequence
$$0 \to {H^{i-1}(X,{\Bbb C}) \over {F^j + H^{i-1}(X,{\Bbb Z})}} \to
H^i_{\cal D}(X,{\Bbb Z}(j)) \to F^j \cap H^i(X,{\Bbb Z}) \to 0 $$
where the notation $F^j \cap H^i(X,{\Bbb Z})$ denotes the set of all classes
$\alpha \in H^i(X,{\Bbb Z})$ such that $\alpha \otimes {\Bbb C} \in F^jH^i(X,{\Bbb C})$. If
$X$ is smooth and proper we have additionally for $i=2j$:
$$0 \to J^j(X) \to H^{2j}_{\cal D}(X,{\Bbb Z}(j)) \to F^j \cap H^{2j}(X,{\Bbb Z}) \to 0 $$
where $J^j(X)$ is the intermediate Jacobian. For $i < 2j$ and $X$ smooth
and proper, the group $F^j \cap H^i(X,{\Bbb Z})$ is equal to the torsion
subgroup of $ H^i(X,{\Bbb Z})$. By the isomorphism
$${\rm Ext}^1_{\rm MHS}({\Bbb Z}(-j),H^{i-1}(X,{\Bbb Z}))=
{H^{i-1}(X,{\Bbb C}) \over {F^j + H^{i-1}(X,{\Bbb Z})}}$$
both of the statements above can be subsumed into the exactness of the
sequence
$$0 \to {\rm Ext}^1_{\rm MHS}({\Bbb Z}(-j),H^{i-1}(X,{\Bbb Z})) \to H^i_{\cal D}(X,{\Bbb Z}(j)) \to
{\rm Hom}_{\rm MHS}({\Bbb Z}(-j),H^i(X,{\Bbb Z})) \to 0 $$
for $X$ smooth and proper. See \cite{Be2}.
\subsection{Cycle Classes}
Let $k$ be a field of characteristic zero. Fix some embedding $\sigma: k
\hookrightarrow {\Bbb C}$ and let $X_{an}$ be the associated complex analytic
space. We give several definitions of cycle classes
$$c_{p,n}: CH^p(X,n) \to H^{2p-n}_{\cal D}(X_{an},{\Bbb Z}(p))$$
which are in fact equivalent. There is Bloch's definition
\begin{definition} (\cite{Bl2})
\end{definition}
This is somehow the most general definition since it only uses some
functoriality and weak purity of Deligne cohomology
and can also be applied to get cycle classes
to \'etale cohomology. $X$ is just assumed to be quasiprojective over $k$.
A definition for all $c_{p,n}$ was given first by Beilinson
\begin{definition} (\cite{Be1},\cite{Gil},see \cite{Sch} for a survey)
\end{definition}
Again $X$ is quasiprojective over $k$. The definition uses simplicial schemes
and the axioms of Bloch and Ogus. The following definition is due to
Deninger and Scholl
\begin{definition} (\cite{DS},\cite{Ja1})
\end{definition}
Let $X$ be smooth and projective over $k$.
Here for $n \ge 1$ a cycle class
$$c_{p,n}: CH^p(X,n) \otimes {\Bbb Q} \to
{\rm Ext}^1_{{\Bbb Q}-{\rm MHS}}({\Bbb Q}(-p),H^{2p-n-1}(X,{\Bbb Q}))$$
is defined by giving an explicit extension.
\ \\
These definition coincide by \cite{Bl2} for the first two definitions
and \cite{DS} for the first and third definition. If $n=1$ we can give more
detailed descriptions of the cycle class (\cite{Bl3},\cite{Lev}):\\
Assume that $X$ is smooth and projective over ${\Bbb C}$ and that
$H^{2p-1}(X,{\Bbb Z})$ is torsionfree (otherwise work over ${\Bbb Q}$). Then we
have by property (10) of higher Chow groups that $CH^p(X,1)=H^{p-1}(X,{\cal K}_p)$
and a class in this group is given by $Z=\sum Z_i \otimes f_i$ where the
$Z_i$ are integral subschemes of codimension $(p-1)$ and $f_i$ are
rational functions on each $Z_i$ with $\sum \div(f_i)=0$ as a cycle on $X$.
Then write $\div(f_i)=\partial \gamma_i$ for real analytic $(2d-2p+1)-$
dimensional chains on $Z_i$. In fact define $\gamma_i=f_i^{-1}(u_i)$ with
$u_i$ the standard path on the real axis from $0$ to $\infty$.
Let $\gamma=\cup \gamma_i$. We have $\partial
\gamma =0$ by $\sum \div(f_i)=0$ and even more $Z$ defines a class in
$F^pH^{2p-1}_{|Z|}(X,{\Bbb Z})$ and therefore $\gamma$ has zero cohomology
class in $H^{2p-1}(X,{\Bbb Z})$ (by torsion-freeness). Thus
$\gamma =\partial \Gamma$ for some
chain $\Gamma \subset X$ of dimension $2d-2p+2$. The rational maps
$f_i:Z_i \to \P^1$ are such that $\gamma_i $ is the preimage
of a path connecting $0$ and $\infty$ on $\P^1$ and hence allows to choose a
branch of logarithm on $Z_i \setminus \gamma_i$. The functional
$$ \alpha \mapsto \sum_i \int_{Z_i -\gamma_i} \log(f_i) \cdot \alpha +
(2\pi \sqrt{-1}) \int_\Gamma \alpha $$
defines a functional on $(2p-2)$ forms on $X$ and therefore via Poincar\'e
duality a class in $H^{2p-2}(X,{\Bbb C})$. The choices we made let it become well
defined only in the quotient group $H^{2p-1}_{\cal D}(X,{\Bbb Z}(p))$.
\ \\
The definition via extensions can also be described partially by the
following construction:\\
If we set $U=X \setminus \cup Z_i$, then one has a long exact sequence
$$ H^{2p-2}_{|Z|}(X,{\Bbb Z}) \to H^{2p-2}(X,{\Bbb Z}) \to H^{2p-2}(U,{\Bbb Z})
\to H^{2p-1}_{|Z|}(X,{\Bbb Z}) \to H^{2p-1}(X,{\Bbb Z}) $$
A cycle $Z \in CH^p(X,1)$ gives rise to a class in $F^p \cap
H^{2p-1}_{|Z|}(X,{\Bbb Z})$ and hence gives an extension
$$ 0 \to H^{2p-2}(X,{\Bbb Z})/Hg^{p-1,p-1}(X) \to {\Bbb E} \to {\Bbb Z}(-p) \to 0 $$
Here $Hg^{i,i}=F^i \cap H^{2i}(X,{\Bbb Z})$ with the meaning defined in \ref{Hg}, in
particular $Hg^{1,1}={\rm NS}(X)$.
This extension class is the image of $c_{p,1}(Z)$ under the map
$${\rm Ext}^1({\Bbb Z}(-p),H^{2p-2}(X,{\Bbb Z})) \to
{\rm Ext}^1({\Bbb Z}(-p),H^{2p-2}(X,{\Bbb Z})/Hg^{p-1,p-1}(X))$$
{\it Remark:} Once we believe in the existence of regulator maps also for the local
situation, there is another way to describe the cycle classes for $n=1$
(\cite{E2}):\\
Let ${\cal K}_p$ the Quillen K-theory sheaf (see above) and ${\cal H}_{\cal D}^p(p)$ be the
sheafified Deligne-Beilinson cohomology
(with presheaf $U \mapsto H^p_{\cal D}(U,{\Bbb Z}(p))$) both
viewed as sheaves in Zariski topology.
Then there is a Leray spectral sequence (as in \cite{BO}) arising
from changing from Zariski to the analytic site
$$E_2^{p,q}(r)=H^p_{\rm Zar}(X,{\cal H}_{\cal D}^q(r)) \Longrightarrow
H^{p+q}_{{\cal D},\rm an}(X,{\Bbb Z}(r)) $$
and we get an edge morphism
$H^{p-1}(X,{\cal H}_{\cal D}^p(p)) \to H^{2p-1}_{\cal D}(X,{\Bbb Z}(p))$
as a byproduct. The existence of regulator maps on affine schemes implies
a regulator map of sheaves
$${\rm reg}: {\cal K}_p \to {\cal H}_{\cal D}^p(p)$$
The composition
$CH^p(X,1) \to H^{p-1}(X,{\cal K}_p) \to H^{p-1}(X,{\cal H}_{\cal D}^p(p))
\to H^{2p-1}_{\cal D}(X,{\Bbb Z}(p))$
is also equivalent to the cycle classes defined above.\\
\section{Deformations and Rigidity of Cycle Classes}
Let $Y$ be a reduced quasiprojective scheme over ${\Bbb C}$ and $A$ a ring with
${\Bbb Q} \subset A \subset {\Bbb R}$. Let $\epsilon_A$ be the map
$\epsilon_A: H^k_{\cal D}(Y,A(p)) \to F^p \cap H^k(Y,A(p))$.
\begin{lemma} (\cite{Be1}, 1.6.6.1.)\\
(a) If $p > min(k,\dim(Y))$, then $\epsilon_A \equiv 0$.\\
(b) If $Y=X \times S$ with $X$ smooth, projective and $S$ smooth, affine and
$k < 2p-\dim(S)$ then also $\epsilon_A \equiv 0$.
\end{lemma}
{\it Proof:} In both cases $F^p \cap H^k(Y,A(p))=0$ by type reasoning.\hspace*{\fill}\hbox{$\square$}
\begin{lemma} (\cite{Be1}, 1.6.6.2.)\\
Let $X$ be projective, $S$ smooth affine, $Y=X \times S$ and given
$s_1,s_2 \in S$. Assume
$k \le 2p-2$ and a class $\alpha \in H^k_{\cal D}(Y,A(p))$ is given. Then:
$$\alpha|_{X \times \{s_1\}} = \alpha|_{X \times \{s_2\}} \in
H^k_{\cal D}(X,A(p))$$
\end{lemma}
{\it Proof:} $S$ is an affine curve wlog. By the lemma above
$\epsilon_A(\alpha)=0$, d.h. $\alpha \in {H^{k-1}(Y,{\Bbb C}) \over {F^p
\oplus H^{k-1}(Y,A(p))}}$. But Betti classes are rigid. \hspace*{\fill}\hbox{$\square$}
\begin{corollary} (\cite{Be1}, 2.3.4.)\\
Let $X$ be smooth, projective over ${\Bbb C}$ and $n \ge 2$.
Then the image of
$c_{p,n}:CH^p(X,n) \otimes {\Bbb Q} \to H^{2p-n}_{\cal D}(X,{\Bbb Q}(p))$ is countable.
\end{corollary}
{\it Proof:} There exists an algebraically closed, countable field $L \subset {\Bbb C}$
such that $X = X_0 \otimes_L {\Bbb C}$ for some $L-$variety $X_0$. Hence
$CH^p(X_0,n)$ is countable and therefore the image of
$c_{p,n}: CH^p(X_0,n)\otimes {\Bbb Q} \to H^{2p-n}_{\cal D}(X_{an},{\Bbb Q}(p)) $.
It remains to show that the cycle classes from $X$ and $X_0$
have the same image in $H^{2p-n}_{\cal D}(X_{an},{\Bbb Q}(p)) $.
Choose a smooth affine scheme $S=Spec(R)$ for some $L-$algebra $R$ and a
L-rational point $0\in S$, such that the geometric general fiber in
$X_0 \times S$
is given by $X_{\Bbb C}$ . Given a cycle $Z \in CH^p(X,n) \otimes {\Bbb Q}$,
there is a spreading
${\cal Z} \in CH^p(X_0 \times S,n) \otimes {\Bbb Q}$
with Deligne class
$\alpha \in H^{2p-n}_{\cal D}(X_{an} \times S_{an},{\Bbb Q}(p))$. In particular this
means that $Z={\cal Z}|_X$, the restriction to the geometric general fiber.\\
By the lemma above $\alpha|_X=\alpha|_{X_0}$ and hence $c_{p,n}(Z)$ is also
contained in the countable image of
$c_{p,n}(CH^p(X_0,n) \otimes {\Bbb Q}) \subset H^{2p-n}_{\cal D}(X_{an},{\Bbb Q}(p))$. \hspace*{\fill}\hbox{$\square$}
Now consider the cases $k=2p-1,2p$. Let $X$ be again smooth and projective
over ${\Bbb C}$ and $S$ an affine, smooth complex curve with good
compactification ${\overline S}=S \cup \Sigma $. Then:\\
\begin{lemma}
(a) If $n=1$, then $$F^p \cap H^{2p-1}(X \times S,A(p)) =
H^{p-1,p-1}(X,A(p-1)) \oplus \bigoplus_\Sigma A(-1)
$$ where the summation ranges over all divisors at infinity.\\
(b) (see \cite{ES}.) If $n=0$, then $$F^p \cap H^{2p}(X \times S,A(p)) =
F^p \cap H^{2p}(X \times {\overline S},A(p))
\subset $$
$$ \subset H^{p,p}(X,A(p)) \oplus
H^{p-1,p}(X,{\Bbb C})\otimes H^0({\overline S},
\Omega^1_{\overline S}(log \Sigma)) \oplus
H^{p,p-1}(X,{\Bbb C})\otimes H^1({\overline S},{\cal O}_{\overline S}) $$
\end{lemma}
{\it Proof:} In both cases
$F^{p+1} \cap H^{k}(X \times S,A(p)) = 0$ and the weight $k$ piece arises
exactly from the compactification of $X \times S$. This is the only weight
in (b). The claim follows therefore from K\"unneth decomposition. (a) follows
in the same way but there is only the weight $k+1=2p$. \hspace*{\fill}\hbox{$\square$}
{\it Remark:} For
$k=2p-1$, $\epsilon(\alpha)$ is only determined by the residues around
boundary divisors.
\\
If $k=2p$, then $\epsilon(\alpha)=\alpha_1+\alpha_2+\alpha_3$,
with $\alpha_1 \in H^{p,p}(X,A(p))$ the cohomology class of a restriction
to the general fiber.
$\alpha_2$ and $\alpha_3$ are complex conjugate.\\
\begin{corollary}
Let $X$ be smooth, projective over ${\Bbb C}$ and $n=1$.
Then the image of the truncated cycle class
$$c_{p,1}^{\rm tr}:CH^p(X,1) \otimes {\Bbb Q} \to
H^{2p-1}_{\cal D}(X_{an},{\Bbb Q}(p))/{\rm Hg}^{p-1,p-1}(X) \otimes {\Bbb C}/{\Bbb Q}(1)$$
is countable.
\end{corollary}
{\it Proof:} Let $S$ be a smooth, affine and connected curve.
Consider the product map
$$ CH^{p-1}(X) \otimes CH^1(S,1) \to CH^p(X \times S,1) $$
respectively its Deligne cohomology version
$$ \gamma: H^{2p-2}_{\cal D}(X_{\rm an},{\Bbb Z}(p-1)) \otimes H^1_{\cal D}(S_{\rm an},{\Bbb Z}(1))
\to H_{\cal D}^{2p-1}(X_{\rm an} \times S_{\rm an},{\Bbb Z}(p)) $$
Therefore
$H_{\cal D}^{2p-1}(X_{\rm an} \times S_{\rm an},{\Bbb Z}(p))/ {\rm Image}(\gamma)$
becomes equal to
$${ H^{2p-2}(X \times S,{\Bbb C}^*) \over
{F^p + {\rm Hg}^{p-1,p-1}(X) \otimes {\Bbb C}^*
+{\rm other \quad terms}}}$$
If we mod out by the image of $\gamma$, we get a restriction map (the other
terms restrict to zero) for every $t \in S$:
$$ r_t: {H_{\cal D}^{2p-1}(X_{\rm an} \times S_{\rm an},{\Bbb Z}(p)) \over
{\rm Image}(\gamma) }
\to H^{2p-1}_{\cal D}(X_{an},{\Bbb Z}(p))/{\rm Hg}^{p-1,p-1}(X) \otimes {\Bbb C}^*$$
with the important property that the truncated cycle class $c_{p,1}^{\rm tr}$
factors through it. Therefore
we again have rigidity: Given $\alpha \in
H_{\cal D}^{2p-1}(X_{\rm an} \times S_{\rm an},{\Bbb Z}(p))/ {\rm Image}(\gamma)$
one has after tensoring with ${\Bbb Q}$:
$\alpha|_{X \times \{s_1\}} = \alpha|_{X \times \{s_2\}}$ in
$H^{2p-1}_{\cal D}(X_{an},{\Bbb Q}(p))/{\rm Hg}^{p-1,p-1}(X) \otimes {\Bbb C}/{\Bbb Q}(1)$, since
$H_{\cal D}^{2p-1}(X_{\rm an} \times S_{\rm an},{\Bbb Z}(p))/ {\rm Image}(\gamma)$
can be represented by Betti classes.
Now the same argument as in the proof of Cor. 3.3. can be applied to
deduce the countability.
\hspace*{\fill}\hbox{$\square$}
\section{Theory of $CH^2(X,1)$}
\subsection{Decomposability}
Let $X$ be a smooth and projective variety over an algebraically closed field
of characteristic zero. There is a natural map
$$ \gamma: {\rm Pic}(X) \otimes k^* \longrightarrow CH^2(X,1) $$
which in the cubical version of higher Chow groups can be described by
sending $D \otimes a $ to $D \times \{a\} \subset X \times \square^1 $ for
$D$ an integral subscheme and $a \in k$ and extending linearly. \\
Alternatively the cup-product map
${\cal O}_X^* \otimes_{\Bbb Z} {\cal O}_X^* \to {\cal K}_2 $ gives rise to the map
$$H^1(X,{\cal O}_X^*) \otimes H^0(X,{\cal O}_X^*) \to H^1(X,{\cal K}_2) $$
Equally consider the product map
$$CH^1(X,0) \otimes CH^1(X,1) \to CH^2(X,1) $$
and it is an easy exercise to show that all three definitions coincide.
\begin{definition} We say that $CH^2(X,1)$ is decomposable, if the map
$ \gamma $ has torsion cokernel. More generally
we will say that $CH^p(X,1) $ is decomposable if the map
$CH^{p-1}(X) \otimes CH^1(X,1) \to CH^p(X,1) $ has torsion cokernel.
\end{definition}
The cokernel is a birational invariant in that case.
About the kernel we note that the composed map
${\rm Pic}^0(X) \otimes k^* \to CH^2(X,1) \to H^3_{\cal D}(X_{an},{\Bbb Z}(2)) $
is zero, however we do not know in which way the map
${\rm Pic}^0(X) \otimes k^* \to CH^2(X,1)$ itself behaves.
\subsection{Criteria for Decomposability}
Let $X$ be as in the previous section and $k={\Bbb C}$.
Let us fix some notations:\\
Denote by ${\cal F}^p_{\Bbb Z}$ the Zariski sheaf associated to the presheaf
that associates to each open set $U$ the vector space of holomorphic
$p-$forms with ${\Bbb Z}(p)-$ periods and logarithmic poles along a
desingularization of $X \setminus U$. One has an exact sequence
$$0 \to {\cal H}^{p-1}_{\rm DR}({\Bbb C}/{\Bbb Z}(p)) \to {\cal H}^p_{\cal D}(p) \to {\cal F}^p_{\Bbb Z} \to 0 $$
Here ${\cal H}^{p-1}_{\rm DR}({\Bbb C}/{\Bbb Z}(p))$ is the sheaf associated to the
presheaf
$U \mapsto H^{p-1}(U,{\Bbb C}/{\Bbb Z}(p))$ and
${\cal H}^p_{\cal D}(p)$ is the sheafified Deligne-Beilinson cohomology as explained
before. We then have ${\cal F}^1_{\Bbb Z}={\rm dlog}{\cal O}^*_X$ and define
${\cal C}:= {\cal F}^2_{\Bbb Z} /{\cal F}^1_{\Bbb Z} \wedge {\cal F}^1_{\Bbb Z} $. One can show that the sheaves
${\cal H}^{p-1}_{\rm DR}({\Bbb C}/{\Bbb Z}(p)), {\cal H}^p_{\cal D}(p)$ and ${\cal F}^p_{\Bbb Z}$ admit
Gersten-Quillen type resolutions which we will use below.
The following is
private communication by H. Esnault and partially explained in \cite{E1}.
\begin{theorem} $CH^2(X,1)$ decomposes if
${\cal C}=0$ and $H^1(X,{\cal F}^2_{\Bbb Z})\otimes {\Bbb Q}=0$.
\end{theorem}
{\it Proof:}
Consider the surjective map
$\alpha: {\cal K}_2 \to {\cal F}^1_{\Bbb Z} \wedge {\cal F}^1_{\Bbb Z}$ induced by the
dlog map. By assumption $H^1(X,{\cal F}^1_{\Bbb Z} \wedge {\cal F}^1_{\Bbb Z}) \otimes
{\Bbb Q}=0$ and therefore it is sufficient to show that $H^1(X,{\cal K}_2^0)$
decomposes where ${\cal K}_2^0:={\rm Ker}(\alpha)$.
Gersten-Quillen resolutions for all three sheaves form a
commutative diagram
$$\matrix{ {\cal K}^2_0 & \to & K_2^0({\Bbb C}(X)) & \to &
\oplus_{X^{(1)}} {\Bbb C}/{\Bbb Z}(1) & & \cr
\downarrow && \downarrow && \downarrow && \cr
{\cal K}_2 & \to & K_2({\Bbb C}(X)) & \to &
\oplus_{X^{(1)}} K_1({\Bbb C}(D)) & \to & \oplus_{X^{(2)}} {\Bbb Z} \cr
\downarrow && \downarrow && \downarrow && || \cr
{\cal F}^2_{\Bbb Z} & \to & F^2_{\Bbb Z}({\Bbb C}(X)) & \to &
\oplus_{X^{(1)}} F^1_{\Bbb Z}({\Bbb C}(D)) & \to & \oplus_{X^{(2)}} {\Bbb Z} }
$$
where the third column is the direct sum of exact sequences
$$0 \to {\Bbb C}/{\Bbb Z}(1) \to {\Bbb C}(D)^* {\buildrel {\rm dlog} \over \longrightarrow}
F^1_{\Bbb Z}({\Bbb C}(D)) \to 0 $$
Thus we have a surjective map ${\rm Pic(X)} \otimes {\Bbb C}^* \to
H^1(X,{\cal K}_2^0)$
and the assertion follows. \hspace*{\fill}\hbox{$\square$}
{\it Remark:} The statement in the theorem holds also in the reverse direction.
Bloch conjectures that for a complex algebraic surface
with $p_g=0$ the kernel of the Albanese map $CH^2_0(X) \to {\rm Alb}(X)$
is zero. The conjecture holds if $X$ is not of general type by \cite{BKL}.
If $X$ is of general type $p_g=0$ implies also $q(X)=0$ and the conjecture
has been verified for Godeaux- and Barlow surfaces for example.
Note that if $p_g >0$, then by \cite{Mum} quite
the contrary happens. If
Bloch's conjecture holds, it implies that $CH^2(X,1)$ decomposes
if $p_g(X)=0$ by \cite{BS}.
There is a more general statement in \cite{EL}:
\begin{theorem} (\cite{EL}) \\
Let $X$ be a smooth algebraic variety over ${\Bbb C}$.
Suppose that the cycle maps
$$c_{p,0}: CH^p(X) \to H^{2p}_{\cal D}(X,{\Bbb Z}(p))$$
are injective for $d-s \le p \le d=\dim(X)$ for some $s \ge 0$.
Then $CH^p(X,1)$ is decomposable for $0 \le p \le s+1$.
\end{theorem}
\subsection{Criteria for Non-Decomposability and Main Theorem }
In this section assume for simplicity that $X$ is an algebraic surface over
${\Bbb C}$. To show that $CH^2(X,1)$ is not decomposable we use cycle class maps to
Deligne cohomology. Note that the image of
$$c_{1,2}:CH^2(X,1) \to H^3_{\cal D}(X,{\Bbb Z}(2)) $$
when restricted to ${\rm Pic}(X) \otimes {\Bbb C}^*$ is contained in
${\rm NS}(X) \otimes {\Bbb C}^*$ considered as a subgroup of $H^3_{\cal D}(X,{\Bbb Z}(2))$.
Therefore it will be enough to show that there are classes not contained in
that subgroup modulo torsion. In fact the image of
$CH^2(X,1)/{\rm Pic}(X) \otimes {\Bbb C}^*$ in
$H^3_{\cal D}(X,{\Bbb Z}(2))/{\rm NS}(X) \otimes {\Bbb C}^*$
is at most countable by Corollary (3.5).\\
In the last section we will construct examples and the
following theorem provides the necessary technical tool. We use the following
notation: If $Z$ is a divisor with normal crossings on $X$ and
smooth components $Z_i$, then let
$\Omega^p_X(logZ)$ be the sheaf of holomorphic p-forms with log-poles along
$Z$ and $T_X(logZ)$ be the dual of $\Omega^1_X(logZ)$.
There is an exact sequence
$$0 \to \Omega^1_X \to \Omega^1_X(logZ) \to \oplus {\cal O}_{Z_i} \to 0 $$
and a commutative diagram where we define
${\cal G}:=\Omega^2_X(logZ)/\Omega^2_X$:
$$\matrix{\Omega^2_X &\to &W_1 \Omega^2_X(logZ) &\to &\oplus \Omega^1_{Z_i}
\cr
|| & & \downarrow & & \downarrow \cr
\Omega^2_X & \to & \Omega^2_X(logZ) & \to & {\cal G} \cr
& & \downarrow & & \downarrow \cr
& & {\Bbb C}^k & = & {\Bbb C}^k }
$$
where ${\Bbb C}^k$ is supported on the $k$ points in the singular locus of $Z$
consisting of the intersection points in $Z_i \cap Z_j$.
The cup product with the Kodaira-Spencer class
$H^0(X,\Omega^2_X) \otimes H^1(X,T_X) \to H^1(X,\Omega^1_X) $ and
the natural map
$H^1(X,T_X(logZ)) \to H^1(X,T_X)$ induce a commutative diagram
$$\matrix{
H^0(X,\Omega^2_X) \otimes H^1(X,T_X(logZ)) & \to & H^1(X,\Omega^1_X) \cr
\downarrow & & \downarrow \cr
H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(logZ)) & \to &
H^1(X,\Omega^1_X(logZ)) }
$$
Let us explain the necessary deformation theory. Assume we look at a
smooth, proper deformation $f:{\cal X} \to S$ of $X$ with $S$ a smooth
and quasiprojective variety, a base point $0 \in S$ such that $f^{-1}(0)=X$
and a normal crossing divisor ${\cal Z}$ in ${\cal X}$ containing out of two smooth
components ${\cal Z}_1,{\cal Z}_2$ such that ${\cal Z}_1$ and
${\cal Z}_2$ (resp. ${\cal Z}_1 \cap {\cal Z}_2$)
are smooth of relative dimension one (resp.
zero) over $S$ and restrict to $Z_1$ and $Z_2$ over the central fiber.
We get an exact diagram
$$\matrix{ 0 & \to & T_X(log Z) & \to & T_{{\cal X}}(log({\cal Z}))|_X
& \to & f^*T_{S,0} & \to & 0 \cr
& & \downarrow & & \downarrow & & || & & \cr
0 & \to & T_X & \to & T_{{\cal X}}|X & \to & f^*T_{S,0} & \to & 0 }
$$
The {\it logarithmic
Kodaira-Spencer map} is defined as the coboundary map
$$ T_{S,0} \longrightarrow H^1(X,T_X(log Z)) $$
Let us denote the image of $T_{S,0} \to H^1(X,T_X(log Z))$ by $W(log)$ and
the further image in $H^1(X,T_X)$ by $W$. \\
Before stating the theorem let us desribe one of the higher Chow groups of
the projective (however reducible) variety $|Z|$, the
support of $Z$. For simplicity we will
assume that $|Z|$ consists of two smooth components $|Z|=Z_1+Z_2$ with $k$
intersection points in $Z_1 \cap Z_2$.
\begin{proposition}. $CH^1(|Z|,1) \cong H^3_{{\cal D},|Z|}(X,{\Bbb Z}(2))$ and there
is an exact sequence
$$0 \to ({\Bbb C}^*)^{\oplus 2} \to CH^1(|Z|,1)
{\buildrel \tau \over \to}
{\rm Ker}({\Bbb Z}^k \to \oplus {\rm Pic}(Z_i)) \to 0 $$
where ${\Bbb Z}^k$ is supported on $Z_1 \cap Z_2$.
\end{proposition}
{\it Proof:} next section. \hspace*{\fill}\hbox{$\square$}
Let us denote the map $CH^1(|Z|,1) \to {\rm Ker}({\Bbb Z}^{k} \to
\oplus {\rm Pic}(Z_i))$ given in the proposition by $\tau$. Now we are
ready to state the main result and a stronger variant of it:
{\bf MAIN THEOREM (CRITERION FOR INDECOMPOSABILITY):} \\
{\it Let $X$ be a smooth projective surface over ${\Bbb C}$.
Assume we are given two smooth and connected curves $Z_1$ and $Z_2$
on $X$ intersecting
transversally and nontrivial rational functions $f_i$ on $Z_i$ ($i=1,2$),
such that ${\rm div}(f_1)+{\rm div}(f_2)=0$ as a zero-cycle on $X$. Denote by
$Z=Z_1 \otimes f_1 +Z_2 \otimes f_2$ the resulting cycle in
$CH^2(X,1)=H^1(X,{\cal K}_2)$ and suppose the following
conditions hold:\\
(1) $Z$ also defines a cycle in Bloch's higher Chow group
$CH^1(|Z|,1)$ - again denoted by $Z$- and as such is not equivalent to
$Z_1 \otimes a_1 + Z_2 \otimes a_2$ with $a_1,a_2 \in {\Bbb C}^*$. \\
(2) There exist a smooth, proper deformation $f:{\cal X} \to S$ with $S$ a smooth
and quasiprojective variety, a base point $0 \in S$ such that $f^{-1}(0)=X$
and the following properties hold:\\
(a) The situation in (1) deforms together with $X$: There exists a normal
crossing divisor ${\cal Z}={\cal Z}_1+{\cal Z}_2 \subset {\cal X}$ with ${\cal Z}|_X=Z_1+Z_2$,
consisting out of two smooth components ${\cal Z}_1,{\cal Z}_2$
such that ${\cal Z}_1$ and ${\cal Z}_2$ (resp.
${\cal Z}_1 \cap {\cal Z}_2$) are smooth of relative dimension one (resp.
zero) over $S$. Furthermore there exist rational functions $F_i$ on ${\cal Z}_i$
such that their restriction to each fiber $X_t:=f^{-1}(t)$ satisfy
${\rm div}(F_{1,t})+ {\rm div}(F_{2,t})=0$ as a zero-cycle in $X_t$ and
therefore define classes
$ Z_t=Z_{1,t} \otimes F_{1,t}+Z_{2,t} \otimes F_{2,t}$
in $CH^2(X_t,1)$ and in $CH^1(|Z_t|,1)$ for all $t \in S$.\\
(b) The cup-product map
$$H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(-Z)) \to H^1(X,\Omega^1_X)
/\oplus_i H^0(Z_i,{\cal O}_{Z_i}) $$
has no left kernel.\\
(c) If $W(log) \subset H^1(X,T_X(logZ))$ denotes the image of
the logarithmic Kodaira-Spencer map in
$H^1(X,T_X(logZ))$,
then $W(log)$ contains the image of the natural map
$$H^1(X,T_X(-Z)) \to H^1(X,T_X(logZ))$$\\
(d) For $t$ outside a countable number of proper analytic subsets of $S$,
$Z_{1,t}$ and $Z_{2,t}$ generate ${\rm NS}(X_t) \otimes {\Bbb Q}$.\\
{\bf Then:} $Z_t$ is non-torsion in $CH^2(X_t,1)/{\rm Pic}(X_t) \otimes {\Bbb C}^*$ for
$t$ outside a countable number of proper analytic subsets of $S$.}
\ \\ \\
{\bf VARIANT:} \\
{\it Assume (1),(2a),(2d) of the above main theorem and additionally the
following instead of (2b),(2c):\\
(3)If $W(log) \subset H^1(X,T_X(logZ))$ denotes the image of
the logarithmic Kodaira-Spencer map in
$H^1(X,T_X(logZ))$, then the following map has no left kernel:
$$H^0(X,\Omega^2_X(logZ)) \otimes W(log) \to H^1(X,\Omega^1_X(logZ))
$$
{\bf Then:} $Z_t$ is non-torsion in $CH^2(X_t,1)/{\rm Pic}(X_t) \otimes {\Bbb C}^*$ for
$t$ outside a countable number of proper analytic subsets of $S$.}\\
\ \\ \\
{\it Remark:} The assumption that we have only two components is not necessary but
simplifies the proof and suffices for the applications. (2b) and (2d)
imply necessarily that $p_g(X) \ge 1$, since it follows from
(2b) and (2d) that the Picard number of $X_t$ is not maximal for $t$ outside
a countable number of analytic subsets of $S$.
(2b) and (2c) imply (3) (see proof of the main theorem) and therefore the
variant is the more general formulation.
\section{Proof of the Main Theorem }
\subsection{Auxiliary Results in Hodge Theory}
Let $S$ be a smooth complex variety.
\begin{definition} (\cite{SZ})\\
A graded polarized ${\Bbb Z}-$variation of mixed Hodge structures (${\Bbb Z}$-VMHS)
on $S$ is a local system ${\cal V}$ on
$S$ of ${\Bbb Z}-$modules of finite rank with the following data:\\
(a) An increasing filtration ...$\subset {\cal W}_k \subset {\cal W}_{k+1} \subset
{\cal V} \otimes_{\Bbb Z} {\Bbb Q} $ by local systems over ${\Bbb Q}$. \\
(b) A decreasing filtration $...\subset {\cal F}^{p+1} \subset {\cal F}^p \subset
... \subset {\cal F}^0={\cal V} \otimes_{\Bbb Z} {\cal O}_S$ by holomorphic vector bundles.\\
(c) The flat connection $\nabla$ on ${\cal V}$ satisfies $\nabla {\cal F}^p \subset
\Omega^1_S \otimes {\cal F}^{p-1}$.\\
(d) The ${\cal F}-$filtration induces on the local systems ${\rm Gr}^W_k {\cal V}_{\Bbb Q}:=
{\cal W}_k/{\cal W}_{k-1}$ a pure variation of polarized Hodge structures on $S$.
\end{definition}
The graded pieces of the connection we denote by $\nabla^p$:
$$\nabla^p: {\cal F}^p/{\cal F}^{p+1} \to \Omega^1_S \otimes {\cal F}^{p-1}/{\cal F}^p $$
In particular there are examples where such VMHS come from a geometric
situation, for example in the situation of the theorem:
Assume we are given an explicit
smooth, proper deformation $f:{\cal X} \to S$ with $S$ a smooth
complex variety and there exist a cycle
${\cal Z} \in CH^2({\cal X},1)$ with ${\cal Z}|_X=Z$ consisting out of two
smooth components ${\cal Z}_1,{\cal Z}_2$ intersecting transversally,
such that ${\cal Z}_1$ and ${\cal Z}_2$ (resp.
${\cal Z}_1 \cap {\cal Z}_2$) are smooth of relative dimension one (resp.
zero) over $S$ and such that its restriction to each fiber
$X_t:=f^{-1}(t)$ defines classes $ Z_t \in H^1(X_t,{\cal K}_2)$.
Then it is a result in \cite{SZ} that
the cohomology groups $H^2(X_t \setminus Z_t,{\Bbb Z})$ form an
admissible ${\Bbb Z}$-VMHS over $S$. \\
If all fibers are algebraic surfaces, there are no holomorphic 3-forms, hence
${\cal F}^3=0$ and ${\cal F}^2$
is the holomorphic subbundle with fibers $H^0(X_t,\Omega^2_{X_t}(logZ_t))$.
The graded piece $\nabla^2: {\cal F}^2 \to \Omega^1_S
\otimes {\cal F}^1/{\cal F}^2$ can be described in the stalk at $0 \in S$ by the maps
$$
\nabla^2: H^0(X,\Omega^2_X(logZ)) \to H^1(X,\Omega^1_X(logZ)) \otimes
W(log)^*
$$
where $W(log) $ is the logarithmic Kodaira-Spencer image of $T_{S,0}$.
Bringing $W(log)$ to the other side we get the cup-product maps
$$
H^0(X,\Omega^2_X(logZ)) \otimes W(log) \to H^1(X,\Omega^1_X(logZ))
$$
\begin{lemma} In this situation, assume that \\
$H^0(X,\Omega^2_X(logZ)) \otimes W(log) \to H^1(X,\Omega^1_X(logZ)) $ has
no left kernel.
\\
{\bf Then:} $F^2 \cap H^2(U_t,{\Bbb Q})=0$ for $t$ outside a countable number
of analytic subsets of $S$.
\end{lemma}
{\it Proof:} Assume there exists a nonzero
class $\lambda_0 \in F^2 \cap H^2(U_0,{\Bbb Q})$ that
is supported over some germ of an analytic subvariety $S(\lambda_0) \subset
S$ containing $0 \in S$, i.e. there exists a holomorphic section
$\Lambda \in \Gamma(S(\lambda_0),{\cal F}^2)$ such that $\lambda_t=\Lambda|{X_t}
\in F^2 \cap H^2(U_t,{\Bbb Q})$ for all $t \in S(\lambda_0)$. Let $T_0 \subset
T_{S,0}$ be the holomorphic tangent space to $S(\lambda_0)$ at $0 \in S$.
Since $\lambda_0$ is an integral class,
$$\nabla^2(\lambda_0) \in {\rm Hom}(T_{S(\lambda_0),0},({\cal F}^1/{\cal F}^2)_0) $$
satisfies $\nabla^2(\lambda_0)(T_0)=0$. Therefore -
if $W_0 \subset W(log)$ denotes the subspace corresponding to $T_0$
under the logarithmic Kodaira-Spencer map - we have that
$${\Bbb C} \cdot \lambda_0 \otimes W_0 \mapsto 0 \in H^1(X,\Omega_X^1(logZ)) $$
i.e. ${\Bbb C} \cdot \lambda_0$ is contained in the left kernel of $\nabla^2$
with respect to $W_0$. By the assumption, however, this implies that $W_0$
is a proper subspace of $W(log)$ and hence also $T_0$ is a proper subspace
of $T_{S,0}$. It follows that $S(\lambda_0)$ is a proper subvariety of
$S$. The countability follows from the countability of the
groups $H^2(U_t,{\Bbb Q})$. \hspace*{\fill}\hbox{$\square$}
\subsection{Proof of Proposition 3.4.}
We use the notations and assumptions of the main theorem and let
$k$ be the number of intersection points of $Z_1$ and $Z_2$. \\
{\it Proof:} (of Prop. 3.4.)\\
Let $X_0:=X \setminus Z_{sing}$ and $U=X \setminus Z$,
where $Z=Z_1 \cup Z_2$.
By weak purity $H^i_{{\cal D},|Z_{sing}|}(X,{\Bbb Z}(2))=0$
for $i \le 3$. Therefore $H^i_{{\cal D}}(X,{\Bbb Z}(2)) =H^i_{{\cal D}}(X_0,{\Bbb Z}(2))$ for
$i \le 2$ und $H^3_{{\cal D}}(X,{\Bbb Z}(2)) \subset H^3_{{\cal D}}(X_0,{\Bbb Z}(2))$. Note that
$H^4_{{\cal D},|Z_{sing}|}(X,{\Bbb Z}(2)) \cong {\Bbb Z}^k$. Also let
$K:={\rm Ker}(H^4_{{\cal D},|Z_{sing}|}(X,{\Bbb Z}(2)) \to H^4_{{\cal D}}(X,{\Bbb Z}(2))) $ \\
There is a diagram
$$\matrix{
&& 0 && 0 &&\cr
&& \downarrow && \downarrow \cr
H^2_{{\cal D}}(U,{\Bbb Z}(2)) & \to &H^3_{{\cal D},|Z|}(X,{\Bbb Z}(2)) &
\to & H^3_{{\cal D}}(X,{\Bbb Z}(2)) &\to & H^3_{{\cal D}}(U,{\Bbb Z}(2)) \cr
|| & & \downarrow & & \downarrow & & || \cr
H^2_{{\cal D}}(U,{\Bbb Z}(2)) & \to &H^3_{{\cal D},|Z|}(X_0,{\Bbb Z}(2)) &
\to & H^3_{{\cal D}}(X_0,{\Bbb Z}(2)) &\to & H^3_{{\cal D}}(U,{\Bbb Z}(2)) \cr
&& \downarrow && \downarrow && \cr
&& K & = & K && }
$$
Let
$D_i=Z_i \setminus Z_{sing}$. The $D_i$ are smooth disjoint divisors in $X_0$
and one has an isomorphism $H^3_{{\cal D},|Z|}(X_0,{\Bbb Z}(2)) \cong
\oplus H^1_{{\cal D}}(D_i,{\Bbb Z}(1))$. These groups can be computed via the
exact sequences
$$ 0 \to {\Bbb C}^* \to H^1_{{\cal D}}(D_i,{\Bbb Z}(1)) \to F^1 \cap H^1(D_i,{\Bbb Z}) \to 0
$$
for $i=1,2$. The localization sequence
$$ 0 \to {\Bbb C}^*=H^1_{\cal D}(Z_i,{\Bbb Z}(1)) \to H^1_{{\cal D}}(D_i,{\Bbb Z}(1)) \to
H^0_{\cal D}(Z_{\rm sing},{\Bbb Z}(0)) \cong {\Bbb Z}^k \to H^2_{\cal D}(Z_i,{\Bbb Z}(1)) $$
identifies $F^1 \cap H^1(D_i,{\Bbb Z})$ with ${\rm Ker}({\Bbb Z}^k \to
{\rm Pic}(Z_i))$. Using the fact that \\
${\rm Ker}(\oplus_i {\rm Ker}({\Bbb Z}^k \to {\rm Pic}(Z_i)) \to K) =
{\rm Ker}({\Bbb Z}^k \to \oplus_i {\rm Pic}(Z_i) )$
we have proved the exactness of
$$0 \to ({\Bbb C}^*)^{\oplus 2} \to H^3_{{\cal D},|Z|}(X,{\Bbb Z}(2)) \to {\rm Ker}({\Bbb Z}^{k}
\to \oplus {\rm Pic}(Z_i)) \to 0 $$
and it remains to show that $CH^1(|Z|,1) \cong H^3_{{\cal D},|Z|}(X,{\Bbb Z}(2))$.
But the $D_i$ are smooth and hence $CH^1(D_i,1) \cong H^1_{{\cal D}}(D_i,{\Bbb Z}(1))$
and the localization sequence for
$D_i \subset Z_i$ with complement $Z_{sing}$ gives an exact sequence
$$0 \to ({\Bbb C}^*) \cong CH^1(Z_i,1) \to CH^1(D_i,1) \to CH^0(Z_{\rm sing},0)
\cong {\Bbb Z}^{k} \to CH^1(Z_i) \to \ldots $$
and the claim follows in the same way as for $H^3_{{\cal D},|Z|}(X,{\Bbb Z}(2))$. \hspace*{\fill}\hbox{$\square$}
\subsection{Proof of the Main Theorem}
We use the notations as in the theorem.\\
{\bf Proof of the main theorem:} By the proposition above we have a sequence
$$0 \to ({\Bbb C}^*)^{\oplus 2} \to CH^1(|Z|,1) {\buildrel \tau \over \to}
{\rm Ker}({\Bbb Z}^{k} \to \oplus {\rm Pic}(Z_i)) \to 0 $$
and assumption (1) is equivalent to $\tau(Z) \neq 0$. The complex
$$ H^2_{{\cal D}}(X,{\Bbb Z}(2)) \to H^2_{{\cal D}}(U,{\Bbb Z}(2)) \to H^3_{{\cal D},|Z|}(X,{\Bbb Z}(2))
\to H^3_{{\cal D}}(X,{\Bbb Z}(2)) \to H^3_{{\cal D}}(U,{\Bbb Z}(2)) $$
has the subcomplex
$$ H^1(X,{\Bbb C}^*) \to H^1(U,{\Bbb C}^*) \to ({\Bbb C}^*)^2 \to
{\rm NS(X)} \otimes {\Bbb C}^* \to 0 $$
by assumption (2d). Here we have assumed that $Z_1$ and $Z_2$ generate
${\rm NS}(X) \otimes {\Bbb Q}$, because by assumption (2d) we can always choose
a general deformation of $X$ that satisfies this property and
also the other assumptions of
the theorem, since they are of generic nature. We get a diagram:
$$\matrix{
CH^2(U,2) & \to & CH^1(Z,1) & \to & CH^2(X,1) & \to & CH^2(U,1) \cr
\downarrow && \downarrow \tau && \downarrow && \downarrow \cr
{H^2_{{\cal D}}(U,{\Bbb Z}(2))\over H^1(U,{\Bbb C}^*)} &
{\buildrel \alpha \over \to} & {\rm Ker}({\Bbb Z}^{k} \to
\oplus {\rm Pic}(Z_i)) & \to &
{H^3_{{\cal D}}(X,{\Bbb Z}(2)) \over {\rm NS}(X) \otimes {\Bbb C}^*} & \to
& H^3_{{\cal D}}(U,{\Bbb Z}(2)) }
$$
Since $H^2_{{\cal D}}(U,{\Bbb Z}(2))/ H^1(U,{\Bbb C}^*)=F^2 \cap H^2(U,{\Bbb Z})$,
it is enough to show that the image of
$\alpha: F^2 \cap H^2(U,{\Bbb Z}) \to {\rm Ker}({\Bbb Z}^{k} \to
\oplus {\rm Pic}(Z_i)) $ does
not contain $\tau(Z)$. Note that the kernel is a free group, so it
will be enough to show that $F^2 \cap H^2(U_t,{\Bbb Q})=0 $
for general $t$ by the following argument.\\
{\it Claim:} The cup-product map
$H^0(X,\Omega^2_X(logZ)) \otimes W(log) \to H^1(X,\Omega_X^1(logZ)) $
has no left kernel.\\
Taking this for granted we are finished with the proof
by Lemma (5.2).\\
Hence it remains to prove the claim:\\
Look at the commutative diagram that exists by assumption (2c):
$$\matrix{
H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(-Z)) & \to &
H^1(X,\Omega^1_X)/\oplus H^0(Z_i,{\cal O}_{Z_i}) \cr
\downarrow & & \bigcap \cr
H^0(X,\Omega^2_X(logZ)) \otimes W(log) & \to & H^1(X,\Omega^1_X(logZ)) }
$$
By assumption (2b) the upper map has no left kernel and therefore also the
lower map, since $H^1(X,\Omega^1_X)/\oplus H^0(Z_i,{\cal O}_{Z_i}) \to
H^1(X,\Omega^1_X(logZ))$ is an injection. \hspace*{\fill}\hbox{$\square$}
\section{Examples}
\subsection{Sufficiently Ample General
Complete Intersections in Arbitrary Varieties}
The result that follows has several roots. First of all there is the
Noether-Lefschetz theorem stated in the introduction, which says that a
general hypersurface of degree $d \ge 4$ in $\P^3_{{\Bbb C}}$ has ${\rm Pic}(X)
\cong {\Bbb Z}$. Then there is the theorem of Green-Voisin (\cite{Gre}), saying
that a general hypersurface of degree $d \ge 6$ in $\P^4_{{\Bbb C}}$ has torsion
Abel-Jacobi invariants. In 1989 we started to generalize this
together with M. Green by using
a sophisticated version of residue representations of differential forms
via global sections of adjoint linear systems. But at the same time M. Nori
came up with his fantastic connectivity theorem from \cite{Nori}, which gave
a much shorter proof of the following result:
\begin{theorem} \cite{GM} \\
Let $(Y,{\cal O}(1))$ be a smooth and projective polarized variety
of dimension $n+h$, $X \subset Y$ a
general complete intersection of dimension $n$ and
multidegree $(d_1,..,d_h)$ with $min(d_i)$ sufficiently large. Furthermore
assume that $0 \le p \le n-1$. Then:
$${\rm Image}( CH^p(X) \otimes {\Bbb Q} \to H^{2p}_{\cal D}(X,{\Bbb Q}(p))) $$
$$={\rm Image}(CH^p(Y) \otimes {\Bbb Q} \to H^{2p}_{\cal D}(X,{\Bbb Q}(p))) $$
modulo possibly the image of a certain subtorus of $J^p(Y)_{\Bbb Q}$, which
vanishes if the generalized Hodge conjecture holds.
\end{theorem}
As always {\sl general} means for all points in the moduli space outside
a countable set of proper analytic subvarieties, which is sometimes
also called {\sl very general}.
The generalization to higher Chow groups can be proved with the same method
(this was also observed by S.Bloch, M.Nori and C. Voisin and
is mentioned in \cite{Voi1}):
\begin{theorem}
Let $(Y,{\cal O}(1))$ be a smooth and projective polarized variety
of dimension $n+h$, $X \subset Y$ a
general complete intersection of dimension $n$ and
multidegree $(d_1,..,d_h)$ with $min(d_i)$ sufficiently large. Furthermore
assume that $1 \le p \le n$. Then:
$${\rm Image}( CH^p(X,1) \otimes {\Bbb Q} \to H^{2p-1}_{\cal D}(X,{\Bbb Q}(p))) $$
$$ \subset {\rm Image}(H^{2p-1}_{\cal D}(Y,{\Bbb Q}(p)) \to H^{2p-1}_{\cal D}(X,{\Bbb Q}(p))) $$
\end{theorem}
{\it Proof:}
Let $S:=\prod_i^h \P(H^0(Y,{\cal O}_Y(d_i)))$ and $f:B \to S$ the universal
complete intersection. If $g: T \to S$ is any smooth morphism, we write
also $g: B_T=B \times_S T \to B$ for the base change and $A_T=Y \times T$.
If $Z_s$ is a cycle on $X=X_s$
for $s \in S$, we can find a cycle ${\cal Z} \in CH^p(B_T,1)$
for some smooth morphism
$g: T \to S$ with the property that for some $t \in g^{-1}(s)$ we have
${\cal Z} \cap X_t=Z_s$ and such that $g$ is smooth and finite.
By Nori's theorem \cite{Nori} we have for $1 \le p \le n$
that the restriction homomorphism
$$ i^*: H^{2p-1}_{\cal D}(A_T,{\Bbb Q}(p)) \longrightarrow H^{2p-1}_{\cal D}(B_T,{\Bbb Q}(p))$$
is an isomorphism. Therefore there is a cohomology class
$\alpha \in H^{2p-1}_{\cal D}(A_T,{\Bbb Q}(p))$ such that $i^* \alpha =c_{p,1}({\cal Z})$.
The commutative diagram of restriction maps
$$\matrix{ H^{2p-1}_{\cal D}(A_T,{\Bbb Q}(p)) & \to & H^{2p-1}_{\cal D}(B_T,{\Bbb Q}(p)) \cr
\downarrow & & \downarrow \cr
H^{2p-1}_{\cal D}(Y,{\Bbb Q}(p)) & \to & H^{2p-1}_{\cal D}(X_s,{\Bbb Q}(p)) }
$$
shows that $\alpha|_{X_t}$ lies in the image of
$H^{2p-1}_{\cal D}(Y,{\Bbb Q}(p))$ for every $t$ with $g(t)=s$
and therefore the theorem is proved. \hspace*{\fill}\hbox{$\square$}
\begin{corollary}
Let $X \subset \P^3$ be a general hypersurface of degree
$d \ge 5$. Then: The Chern class $CH^2(X,1) \otimes {\Bbb Q} \to
H^3_{\cal D}(X,{\Bbb Q}(2)) $ has image isomorphic to $H^3_{\cal D}(\P^3,{\Bbb Q}(2)) \cong
{\Bbb C}/{\Bbb Q}(1)$.
\end{corollary}
{\it Proof:} Let $B_T$ be any smooth base change of the universal
hyperplane section $f: B \to S$ with $S=\P(H^0(\P^3,{\cal O}_{\P^3}(d)))$.
By \cite{Par} we have
$H^3_{\cal D}(\P^3 \times T,{\Bbb Q}(2)) \cong H^{3}_{\cal D}(B_T,{\Bbb Q}(2))$ for
$d \ge 5$. This is the required equality in \cite{GM}. \hspace*{\fill}\hbox{$\square$}
\subsection{K3 Surfaces}
\subsubsection{A Deformation of the Fermat Quartic}
Here we give an explicit example satifying the assumptions of the
main theorem:\\
Consider the Fermat quartic surface
$$X=\{x_0^4+x_1^4-x_2^4-x_3^4=0\} \subset \P^3$$
and the linear forms
$$L_0=x_1-x_2, \quad L_1=x_0-x_3, \quad L_2=x_1+x_3, \quad L_3=x_0-x_2 $$
in $\P^3$. We look at the family of quartics
$$F_t(x_0:x_1:x_2:x_3)=x_0^4+x_1^4-x_2^4-x_3^4 + 2t L_0L_1L_2L_3 $$
and define $X_t:=\{ x \in \P^3 \mid F_t(x)=0 \}$.\\
The following line on $X$ is important:
$$G=\{x_1-x_2=x_0-x_3=0\}
$$
Planes containing it define a pencil of planes in $\P^3$:
$$H_\lambda=\{x_1-x_2-\lambda(x_0-x_3)=0\}
$$
The residual curves $E_{\lambda,t}$ to the intersection of $H_\lambda$ with
$X_t$
are elliptic curves and their equations can be computed as follows: The
coordinate transformation $(x_0:x_1:x_2:x_3) \mapsto ((x:y:z),\lambda)$,
where
$$x_0=x+z,x_1=y+\lambda z,x_2=y-\lambda z ,x_3=x-z$$
defines a rational map (blowing up along $G$) and we obtain the new
equations
$$F_t(x,y,z,\lambda)=8z(x^3+\lambda y^3+z^2(x+\lambda^3 y) +\lambda t z
L_2 L_3) =:8z F'(x,y,z,\lambda)$$
where $L_2(x,y,z,\lambda)=x+y+(\lambda-1)z,
L_3(x,y,z,\lambda)=x-y+(\lambda+1)z$
and hence for the strict transforms we get the following equations in
$\P^2 \times \P^1$:
$$X_t=\{(x,y,z,\lambda) \mid x^3+\lambda y^3+z^2(x+\lambda^3 y)
+\lambda t z L_2(x,y,z,\lambda) L_3(x,y,z,\lambda) =0 \}$$
defining an elliptic pencil with fibers $E_{\lambda,t}$ for every $t$.\\
The strict transform of $G$ is $G=\{z=0\}$ and $E_{\lambda,t} \cap G =
\{P_1=(a:1:0),P_2=(a\zeta:1:0),P_3=(a\zeta^2:1:0)\}$ independent of $t$ with
$a^3:=-\lambda$ and $\zeta=e^{2\pi \sqrt{-1}/3} $. To compute the tangent
lines $T_i$ of $E_{\lambda,t}$ at $P_i$ we compute the partial derivatives
$$ {\partial F'_t \over \partial x} = 3x^2 +z^2 + \lambda t z (L_2+L_3)$$
$$ {\partial F'_t \over \partial y} = 3\lambda y^2 +\lambda^3 z^2
+ \lambda t z (L_3-L_2)$$
$$ {\partial F'_t \over \partial z} = 2z(x+\lambda^3 y) + \lambda t
L_2(x,y,z,\lambda) L_3(x,y,z,\lambda) + \lambda t z
((\lambda-1)L_3+(\lambda+1)L_2) $$
Therefore we get the equations of the tangent lines
$$T_1=\{ 3a^2 x + 3 \lambda y +\lambda t (a^2-1) z=0 \} $$
$$T_2=\{ 3a^2 \zeta^2 x + 3 \lambda y +\lambda t (a^2 \zeta^2 -1) z =0 \} $$
$$T_3=\{ 3a^2 \zeta^4 x + 3 \lambda y +\lambda t (a^2 \zeta^4- 1) z =0 \}$$
with intersection $T_1 \cap T_2 \cap T_3 =\{(-\lambda t:t:3) \} =:
\{P_{\lambda,t} \} $. The projection from $P_{\lambda,t}$ onto $G=\{z=0\}$
defines a hyperelliptic morphism onto $G$ if and only if
$P_{\lambda,t} \in E_{\lambda,t}$. This is one algebraic condition
we will compute:
$$0= F'(P_{\lambda,t})=t \lambda (\lambda-1) F''_t(\lambda) $$
with $F''_t=\lambda(t^2-18t+9)+4t^2-6t-18$. Solving $F''_t(\lambda)=0$
we get
$$\lambda(t) = -{4t^2-6t-18 \over t^2-18t+9} $$
\subsubsection{Non-Trivial Cycles}
If $\lambda=\lambda(t)$, the hyperelliptic map of the preceeding section
$E_{\lambda(t),t} \to G$ defines
a rational function $f_1$ with zero divisor $2(P_1)-2(P_2)$ as a cycle on
$E_{\lambda(t),t}$. There is always a rational function $f_2$ on $G$
with zero divisor $2(P_2)-2(P_1)$ as a quotient of 2 squares. Let
$Z:=E_{\lambda(t),t} \otimes f_{1,t} + G \otimes f_{2,t} \in CH^2(X_t,1)$.
Note that the cycle $Z_t$ is only defined over a covering of $\P^1$
since the choice of two of the 3 intersection points with an order is only
determined up to permutations. \\
However in order to verify the assumptions in the main theorem,
in particular (2c), we have to extend the base space of our deformation. Let
$N \subset \P H^0(\P^3,{\cal O}_{\P^3}(4))$ be an irreducible
component of the Noether-Lefschetz locus of quartics containing a line and
containing the deformation of the Fermat quartic in section (6.2.1).
After some smooth base change $g: S \to N$ we get a family of lines
$G_t$, elliptic curves $E_t$ in $X_t$, rational functions $f_{1,t}$ on $E_t$,
$f_{2,t}$on $G_t$ and cycles $Z_t=E_t \otimes f_{1,t} + G_t \otimes f_{2,t}
\in CH^2(X_t,1)$ well-defined on $S$ that extend the cycles
$E_{\lambda(t),t}$ and $G=G_0$ above. $S$ can be chosen in such a way that
assumption (2a) of the criterion is satisfied.
The cycles even define classes also denoted
by $Z_t$ in $CH^1(|Z_t|,1)$ that map to $CH^2(X_t,1)$ under the natural map
$CH^1(|Z_t|,1) \to CH^2(X_t,1)$. \\
$W(log) \subset H^1(X,T_X(logZ))$ is a codimension one subspace,
since the deformations of the pair $(X_t,Z_t)$
that preserve the cycle in $CH^2(X_t,1)$ map generically
finite onto
the deformations of $X$ itself: on every generic elliptic fibration
only a finite number of fibers will have rationally equivalent intersection
points with a line, as we checked in the example above. Note that
it is sufficient to check such an assertion at one point of the moduli space.
Thus $W(log) \to W$ is an isomorphism.
$W$ is isomorphic to the tangent space to $N$
at the point $X$ by Noether-Lefschetz theory. The following diagram is
therefore exact and commutative:
$$\matrix{ 0 \to {\Bbb C} & \to & H^1(X,T_X(log Z)) & \to & H^1(X,T_X) &
\to & {\Bbb C}^2 & \to & 0 \cr
&& \bigcup && \bigcup &&&& \cr
&& W(log) & {\buildrel \cong \over \to} & W &&&& }
$$
\begin{lemma} The cycle
$Z_t=E_{t} \otimes f_{1,t} + G_t \otimes f_{2,t}$ in
$CH^2(X_t,1)/{\rm Pic}(X_t)\otimes {\Bbb C}^*$ is nontrivial modulo torsion
for $t$ outside a countable union of proper analytic subsets of $S$.
\end{lemma}
{\it Proof:} We have to verify the assumptions of the main theorem.
To simplify the notation, we assume for the moment that $t=0$ and let
$E=E_0,G=G_0$ and $Z=Z_0$. $Z$ will also denote the effective cycle $E+G$.
To check (1), it is sufficient to remark that $\tau(Z)$ corresponds to the
element $(2,-2,0)$ in ${\rm Ker}({\Bbb Z}^3 \to {\rm Pic}(E) \oplus {\rm Pic}(G))$
and therefore does not decompose. Note that if we write
$\tau(Z)$ as $(2,-2,0)$, we have chosen an ordering of our curves $Z_1$ and
$Z_2$ (as we did in the proof of prop. 3.4.), and then the map
$\tau: CH^1(|Z|,1) \to {\Bbb Z}^3$ is given by $Z_1 \otimes h_1+
Z_2 \otimes h_2 \mapsto {\rm div}(h_1)=-{\rm div}(h_2)$.\\
Assumption 2(a) is clear by construction.\\
To prove (2b), we look at the following graded Artinian ring $R_*=
\oplus R_d$ with
$$R_d={{H^0(\P^3,{\cal O}_{\P^3}(d))} \over J_{F,d}}$$
In the example the Jacobian ideal $J_{F,d}$ is the d-th graded piece of the
homogenous ideal generated by the monomials $x_i^3$.
One has the following isomorphisms first proved by Griffiths (see
\cite{1594}, page 44):
$$H^1(X,\Omega_X^1)_{\rm pr} \cong R_4,\quad H^{0,2}(X) \cong R_8 \cong {\Bbb C}$$
$$H^1(X,T_X(-Z)) \cong R_3, R_0 \cong {\Bbb C}, R_1 \cong H^0(\Omega^2_X(logZ))
$$
Further there is Macaulay duality: $R_k^* \cong R_{8-k}$, see op.cit.\\
In this language we have to prove that
$$R_1 \otimes R_3 \longrightarrow R_4/{\Bbb C} \cdot g $$
has no left kernel, where $g$ is a quartic polynomial representing the
extra cohomology class of $G$ (note that the cohomology class of $Z=G+E$ is
the hyperplane class and therefore not primitive). In other words we have to
prove that
$$R_1 \longrightarrow R_3^* \otimes R_4/ {\Bbb C} \cdot g $$
is injective. After dualizing this means that
$$m: V/J_{F,4} \otimes R_3 \longrightarrow R_7 $$
is surjective, where $V \subset H^0(\P^3,{\cal O}_{\P^3}(4))$ is the codimension
one linear subspace dual to ${\Bbb C} \cdot g$. $V$ contains $J_{F,4}$ and is
therefore basepointfree. The surjectivity of $m$ then follows from
the theorem of Green-Gotzmann (\cite{1594}, pg.74),
which has the following corollary:\\
Let $V \subset H^0(\P^3,{\cal O}_{\P^3}(d))$ be a linear basepointfree subspace
of codimension $c$, then the map
$$m : V \otimes H^0(\P^3,{\cal O}_{\P^3}(k-d)) \to H^0(\P^3,{\cal O}_{\P^3}(k))$$
is surjective if $k \ge d+c$. \\
We apply this result for $k=7,d=4,c=1$ thus proving (2b).\\
To prove (2c), consider the sheaf ${\cal G}:=\Omega^2_X(logZ)/\Omega^2_X$, which
we can also denote by $\omega_Z$. One has an exact sequence
$$0 \to \Omega^1_G \oplus \Omega^1_E \to {\cal G} \to {\Bbb C}^{\oplus 3} \to 0$$
which, since $G$ is rational, has the same $H^0$ sequence of vector spaces as
the one coming from
$$0 \to \Omega^1_E \to \Omega^1_E(log G) \to {\Bbb C}^{\oplus 3}\to 0 $$
and the coboundary maps are equal.
Hence here $H^0({\cal G}) \cong H^0(E,\Omega^1_E(log G))$.
The cycle
$Z \in CH^1(|Z|,1)$ defines a non-zero element $\alpha \in H^0({\cal G})$ with
$\alpha={1 \over {2\pi i}} \cdot {df_1 \over f_1}$. There is an induced map
$$H^0({\cal G}) \otimes H^1(X,T_X(logZ)) \to H^1(E,{\cal O}_E) $$
(since $G$ is rational) and $W(log)$ equals the right annihilator
of $\alpha$ under that map (this follows, because $W(log) $ is a codimension
1 subspace of $H^1(X,T_X(logZ))$ that annihilates $\alpha$ and therefore is
equal to the annihilator, since the latter also has codimension 1). \\
To show (2c) it is sufficient to look at the
following diagram (note that the right column is exact but not the left one):
$$\matrix{
H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(-Z)) & \to &
H^1(X,\Omega^1_X)/{\Bbb C}^2 \cr
\downarrow & & \downarrow \cr
H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(logZ)) & \to &
H^1(X,\Omega^1_X(logZ)) \cr
\downarrow & & \downarrow \cr
H^0({\cal G}) \otimes H^1(X,T_X(logZ)) &\to &H^1(E,{\cal O}_E)
}
$$
It shows that the image of
$H^0(X,\Omega^2_X(logZ)) \otimes H^1(X,T_X(-Z)) \to H^1(E,{\cal O}_E)$ is zero,
hence the image of $H^1(X,T_X(-Z))$ in $H^1(X,T_X(logZ))$
annihilates $\alpha$ and therefore is a subspace of $W(log)$.
This finishes the proof of (2c). \\
To prove (2d), look at the multiplication map
$$W \otimes H^1(X,\Omega^1_X) \to H^2(X,{\cal O}_X) \cong {\Bbb C} $$
$W$ annihilates the cycle classes of the curves $G$ and $E$ and the
restriction to the quotient
$$W \otimes {H^1(X,\Omega^1_X) \over {{\Bbb C} [G] \oplus {\Bbb C} [E]}}
\to {\Bbb C} $$
is a perfect pairing by the Hodge Riemann bilinear relations. It follows
that no class in $ {H^1(X,\Omega^1_X) \over {{\Bbb C} [G] \oplus {\Bbb C} [E]}} $
survives in a general deformation direction of $W$ by the same argument
as in the proof of lemma (5.2.). Therefore for a general
surface $X_t$ one has that $NS(X_t) \otimes {\Bbb Q}$ is generated by the two
elements $[G]$ and $[E]$. \hspace*{\fill}\hbox{$\square$}
{\it Remark:} With some more work, using a monodromy argument of H. Clemens which
was also used by A. Collino in \cite{Col}, one can probably prove infinite
generation for the image of $CH^2(X_t,1)$ for a general quartic hypersurface
containing a line. Since for oother examples
this was proved in \cite{Col} and \cite{Voi2}
we refrain from presenting it here. Obviously the idea would be to study the
monodromy around a countable set of loci on the parameter space
of the surfaces $X_t$.
\subsection{Examples of General Type}
In the previous section we have studied some special quartic surfaces and
have seen that on can apply the main theorem. Now we would like to show
that one can do a similar construction on some quintic hypersurfaces
$X$ in $\P^3$ and again apply the main theorem to get cycles that are
indecomposable in $CH^2(X,1)$. Note that they have to be very special
surfaces in view of the result (6.2).\\
To construct examples of general type, we
look at the Shioda hypersurface of degree 5 from \cite{Sh}:
$ X=\{x \in \P^3 \mid x_0x_1^4+x_1x_2^4+x_2x_0^4+x_3^5=0 \} $.
It has an automorphism $\sigma$ of order 65, given by
$(x_0:x_1:x_2:x_3) \mapsto (\zeta^{16}x_0:\zeta^{-4}x_1:\zeta x_2:x_3)$
where $\zeta$ is a 65-th root of unity.
Shioda proves that the Picard group of $X$ is of rank one, by showing
that $\sigma$ acts irreducibly on $H^2_{\rm pr}(X,{\Bbb Q})$. \\
Let us look at the 1-parameter family
$$X_t=\{ x \in \P^3 \mid
F_t(x)=x_0 x_1^4+x_1 x_2^4 + x_2 x_0^4 + x_3^5 + tx_3 x_1^4 =0 \}
$$
for $t \in \P^1_{\Bbb C}$. Furthermore let $H_i=\{x_i=0\}$ be the coordinate
hyperplanes and define the curves $Z_{1,t}:=X_t \cap H_3$ and
$Z_{2,t}:=X_t \cap H_0$. Finally we set $P:=(0:1:0:0)$, $Q:=(0:0:1:0)$ and
$R:=(1:0:0:0)$.
\begin{proposition}
(a) $Z_{1,t}$ is smooth for all $t$, $Z_{2,t}$ is smooth for $t \neq 0,
\infty$ and $Z_{2,0}$ is a rational curve with a point of multiplicity 4 at
$P$.\\
(b) $P,Q \in Z_{1,t} \cap Z_{2,t}$.\\
(c) for all $t$: $4P=4Q$ in $CH_0(Z_{2,t})$.\\
(d) for all $t$: $13P=13Q$ in $CH_0(Z_{1,t})$.
\end{proposition}
{\it Proof:} (a) We omit the subscripts t, if there is no ambiguity.
$Z_1 =\{ (x_0:x_1:x_2) \in \P^2 \mid
x_0 x_1^4+x_1 x_2^4 + x_2 x_0^4 =0 \}$ independent of $t$,
which is smooth by the Jacobian criterion. $Z_{2,t}$ depends on $t$:
$ Z_{2,t} =\{ (x_1:x_2:x_3) \in \P^2 \mid
x_1 x_2^4 + x_3^5 + t x_3 x_1^4 =0 \}$. The gradient is therefore
$ (x_2^4 + 4t x_3 x_1^3 : 4 x_2^3 x_1 : 5 x_3^4 + t x_1^4 )$, which is
nowhere zero for $t \neq 0,\infty$.
For $t=0$ one gets a 4-fold point at $P$.
$Z_{2,0}$ is rational by the parametrization $(t_0:t_1) \mapsto
(0: -t_0^5: t_1^5 : t_1^4 t_0) $. This proves (a).\\
(b) One computes
$Z_1 \cap Z_2 = X_t \cap H_0 \cap H_3 =
\{x \in \P^3 \mid x_0=x_3=0,\quad x_1 x_2^4=0 \}$. Hence
$Z_2 \cdot H_3 = Q + 4 P $ as cycles on $\P^3$ with multiplicity counted.
(b) follows.\\
(c) We obtain $Z_2 \cap H_1 = X_t \cap H_0 \cap H_1 =
\{x \in \P^3 \mid x_0=x_1=0,\quad x_3^5=0 \}$. Hence
$Z_2 \cdot H_1 = 5Q $ as a cycle on $\P^3$. But $Z_2 \cdot H_3 $ is
rationally equivalent to $Z_2 \cdot H_1$ on $Z_2$. Together with (b), we
obtain that $Q+4P=5Q$ and hence $4P=4Q$ in $CH_0(Z_{2,t})$ for all $t$.\\
(d) $Z_{1,t} \cap H_2 = X_t \cap H_3 \cap H_2 =
\{x \in \P^3 \mid x_2=x_3=0,\quad x_0 x_1^4=0 \}$. It follows that
$Z_1 \cdot H_2 = P+4R$ as cycles. Finally we compute
$Z_1 \cdot H_1 = R+4 Q$ by symmetry. Together we get that
$Q+4P=P+4R=R+4Q$ in $CH_0(Z_1)$. Solving these equations gives that
$13P=13Q=13R$ in $CH_0(Z_1)$.\hspace*{\fill}\hbox{$\square$}
\begin{corollary}
For all $t$ , $52P$ and
$52Q$ are rationally equivalent on both curves. This
defines rational functions $f_i$ on $Z_i$ with
${\rm div}(f_1)=52P-52Q=-{\rm div}(f_2)$ as a zero cycle on
$X_t$ for all $t$ and hence a cycle $Z_t \in
CH^2(X_t,1)$.
\end{corollary}
As in the previous example we now choose a maximal irreducible
component $N \subset
\P(H^0(\P^3,{\cal O}(5)))$, containing the 1-parameter family above and
such that the properties of the cycles $Z_i$
extend along a suitable \'etale covering $S$ of $N$ and therefore
cycles $Z_t \in CH^2(X_t,1)$ are well defined for all $t \in S$.
The assumptions of the main theorem can be checked in the
same way as in the previous example:
\begin{theorem}
For $t \in S$ general, $Z_t$ is indecomposable in $CH^2(X_t,1)$.
\end{theorem}
{\bf Proof:} Again the assumptions (1),(2a) are easy to verify. To prove (2b)
we use the same method as above. Assume $t=0$ and look at the Shioda
hypersurface. Here $K_X={\cal O}_X(1)$ and $K_X+Z={\cal O}_X(3)$. Again let $R_*$
denote the Griffiths' Jacobian ring. One has $H^0(X,K_X+Z)=R_3$ and
$H^{1,1}_{\rm pr}=R_6$. One also computes that $H^1(X,T_X(-Z)) = R_3$.
Note that $Z_1$ and $Z_2$ have the same cohomology class. Therefore we have
to show that $R_3 \otimes R_3 \to R_6$ has no left kernel, or dually that
$R_6 \otimes R_3 \to R_9 $ is surjective, which is obvious. This proves
(2b). The proof of (2c) proceeds in the same way as above with the necessary
modifications. Finally (2d) holds since it holds on the Shioda surface $X_0$
and therefore the Picard number is also one on a general surface parametrized
by $S$. \hspace*{\fill}\hbox{$\square$}
\section{Further Remarks}
First let us imitate a method of
Bloch (\cite{Bl3}) to show what structure the groups that are involved
should have. Note that $CH^2(X,1) \cong H^1(X,{\cal K}_2)$ via the
Gersten-Quillen resolution.
Bloch's method is to study $H^1(X,{\cal K}_2)$ infinitesimally
by giving an infinitesimal version of the sheaf ${\cal K}_2$. Instead of ${\Bbb C}$ let
the ground field be $k$ and look at the local $k-$ algebra of dual numbers
$k[\epsilon]/\epsilon^2$. Imitating Bloch's construction in the case of
$CH^2(X)$ , it makes sense to speak of the formal tangent space
$TH^1(X,{\cal K}_2)$
to $H^1(X,{\cal K}_2)$ by defining it as the group $H^1(X,{\cal K}_{2,\epsilon})$ where
${\cal K}_{2,\epsilon}$ is this case given by the absolute differentials
$\Omega^1_{X/{\Bbb Q}}$. This is also nicely explained in Murre's lecture in
\cite{1594}, chapter V. From the exact sequence
$$ 0 \to {\cal O}_X \otimes_k \Omega^1_{k/{\Bbb Q}} \to
\Omega^1_{X/{\Bbb Q}} \to \Omega^1_{X/k} \to 0
$$
we get a long exact sequence
$$H^1(X,{\cal O}_X) \otimes_k \Omega^1_{k/{\Bbb Q}} \to TH^1(X,{\cal K}_2) \to
H^1(X,\Omega^1_{X/k}) \to \ldots $$
$$ \ldots \to H^2(X,{\cal O}_X) \otimes_k \Omega^1_{k/{\Bbb Q}} \to TCH^2(X) \to
T{\rm Alb}(X)$$
We see two principles, however without giving a correct proof:
If $k={\Bbb C}$ and $p_g(X)=0$ or $k={\Bbb Q}$, then the Albanese map should
be injective (Bloch's conjecture on $CH^2(X)$) and if $k={\Bbb C}$ and
$X$ has maximal Picard number, then the map ${\rm Pic}(X) \otimes {\Bbb C}^* \to
CH^2(X,1)$ should be surjective modulo torsion. But there is a caveat:
In \cite{Ram} there is an example of a submotive of
a Hilbert modular surface that has maximal Picard number but
indecomposable $CH^2(X,1)$. Over
number fields the situation seems to be different, we refer to
\cite{Ram}, \cite{Mil} and \cite{Fla}. I was told that also \cite{Mil}
is probably a counterexample. See also the discussion in \cite{Ras}.
\\
Another hope would be to give a direct computation
of the Chern class map $c_{2,1}$ in terms of integrals and to show that
for some special value of $t$, the Deligne classes of $Z_t$ are not torsion
modulo ${\rm NS}(X_t) \otimes {\Bbb C}^*$.\\
{\bf Question:} Does a direct computation of the integral imply that already
on the Shioda surface $X$ the cycle $Z$ is indecomposable? Note that
$X$ is defined over ${\Bbb Q}$ and a positive answer would contribute to the study
of the conjectures of Bloch and Beilinson. \\
It would also be nice to prove directly (without use of
Deligne cohomology) that $CH^2(X,1)/{\rm Pic}(X) \otimes {\Bbb C}^*$ is
only countable, for example by Hilbert scheme methods.
This was suggested to me by C. Voisin.\\
Finally one would like to study higher Chow groups of degenerations of
algebraic varieties, for example the degeneration of a quartic K3 surface
into a tetrahedron of 4 planes as we described in section (6.2.1). We would
like to know whether already a general member of that one-parameter
degeneration has an indecomposable $CH^2(X_t,1)$. This can probably be
done by using another slight variant of our
main theorem and checking the assumptions on the variety $X_{\infty}$, by
showing that there are no two-forms of a certain logarithmic type.
More general this should lead to a good understanding of the mixed motive of
a variety like the tetrahedron (consisting out of linear varieties with
certain combinatorial data) in the sense of \cite{Ja1} and relating it
to the smooth case via deformation theory as developed in chapter 3.\\
{\bf Acknowledgements}\\
It is a pleasure to thank H\'el\`ene Esnault for her encouragement and
support during this project and for the opportunity to
include theorem 4.2. into this paper. \\
Special thanks go to Spencer Bloch, Mark Green, Marc Levine, Wayne Raskind,
Claire Voisin and James Lewis for some discussions and proof reading
and to Alberto Collino for
showing me his paper \cite{Col} at an early stage.\\
Financial support during this project was provided by the DFG
Forschergruppe in Essen and the DFG Schwerpunkt Komplexe Mannigfaltigkeiten.
The University of Chicago and the University of Leiden have offered support
for visits in October 1993 and October 1994 .
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