# A canonical connection on bundles on Riemann surfaces and Quillen connection
on the theta bundle

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005<EMAIL_ADDRESS>and
Jacques Hurtubise  Department of Mathematics, McGill University, Burnside
Hall, 805 Sherbrooke St. W., Montreal, Que. H3A 2K6, Canada
<EMAIL_ADDRESS>
###### Abstract.

We investigate the symplectic geometric and also the differential geometric
aspects of the moduli space of connections on a compact connected Riemann
surface $X$. Fix a theta characteristic $K^{1/2}_{X}$ on $X$; it defines a
theta divisor on the moduli space ${\mathcal{M}}$ of stable vector bundles on
$X$ of rank $r$ degree zero. Given a vector bundle $E\,\in\,{\mathcal{M}}$
lying outside the theta divisor, we construct a natural holomorphic connection
on $E$ that depends holomorphically on $E$. Using this holomorphic connection,
we construct a canonical holomorphic isomorphism between the following two:

  1. (1)

the moduli space $\mathcal{C}$ of pairs $(E,\,D)$, where
$E\,\in\,{\mathcal{M}}$ and $D$ is a holomorphic connection on $E$, and

  2. (2)

the space ${\rm Conn}(\Theta)$ given by the sheaf of holomorphic connections
on the line bundle on $\mathcal{M}$ associated to the theta divisor.

The above isomorphism between $\mathcal{C}$ and ${\rm Conn}(\Theta)$ is
symplectic structure preserving, and it moves holomorphically as $X$ runs over
a holomorphic family of Riemann surfaces.

###### Key words and phrases:

Moduli space, theta bundle, holomorphic connections, Quillen connection

###### 2010 Mathematics Subject Classification:

14H60, 14D21

###### Contents

  1. 1 Introduction
  2. 2 Moduli space of stable vector bundles
    1. 2.1 Two torsors on a moduli space
    2. 2.2 An isomorphism of torsors
  3. 3 Another isomorphism of torsors
    1. 3.1 A canonical connection
    2. 3.2 A holomorphic isomorphism of torsors
  4. 4 The two isomorphisms of torsors coincide
  5. 5 Family of Riemann surfaces

##  1\. Introduction

Let $X$ be a compact connected Riemann surface of genus at least two. Let
$K^{1/2}_{X}$ be a square-root of the canonical line bundle $K_{X}$ of $X$; it
is called a theta characteristic of $X$. Let $\mathcal{M}$ denote the moduli
space of stable vector bundles on $X$ of rank $r$ and degree zero. It has the
theta divisor $D_{\Theta}$ defined by all $E$ such that $H^{0}(X,\,E\otimes
K^{1/2}_{X})\,\not=\,0$; the holomorphic line bundle on $\mathcal{M}$
corresponding to the divisor $D_{\Theta}$ is denoted by $\Theta$. The moduli
space $\mathcal{M}$ has a natural Kähler structure. The Kähler $2$-form on
$\mathcal{M}$ coincides with the symplectic form on the ${\rm U}(r)$ character
variety for $X$, [Go], [AB], once we identify this character variety with
$\mathcal{M}$ using [NS] (see the map $\psi_{U}$ below).

Let $\mathcal{C}$ denote the moduli space of holomorphic connections on $X$ of
rank $r$ such that the underlying holomorphic vector bundle is stable; it
projects to $\mathcal{M}$ by mapping elements to the underlying holomorphic
vector bundle. This $\mathcal{C}$ is a holomorphic torsor on $\mathcal{M}$ for
the holomorphic cotangent bundle $T^{*}\mathcal{M}$ (this means that the
fibers of $T^{*}\mathcal{M}$ act freely transitively on the fibers of
$\mathcal{C}$ over $\mathcal{M}$). This moduli space $\mathcal{C}$ is equipped
with a natural holomorphic symplectic structure [Go], [AB]. There is a natural
$C^{\infty}$ section

$\psi_{U}\,:\,{\mathcal{M}}\,\longrightarrow\,\mathcal{C}$

that sends any $E\,\in\,{\mathcal{M}}$ to the unique unitary flat connection
on $E$ [NS].

Let $\text{Conn}(\Theta)$ denote the holomorphic fiber bundle on
${\mathcal{M}}$ given by the sheaf of holomorphic connections on the line
bundle $\Theta$. There is a tautological holomorphic connection on the
pullback of $\Theta$ to $\text{Conn}(\Theta)$. The curvature of this
tautological holomorphic connection is a holomorphic symplectic form on
$\text{Conn}(\Theta)$. This $\text{Conn}(\Theta)$ is also a holomorphic torsor
on $\mathcal{M}$ for $T^{*}\mathcal{M}$. Although the line bundle $\Theta$
depends on the choice of the theta characteristic $K^{1/2}_{X}$, the
$T^{*}\mathcal{M}$–torsor $\text{Conn}(\Theta)$ does not depend on the choice
of the theta characteristic (see Remark 2.2).

There is a unique Hermitian connection on $\Theta$ whose curvature is the
Kähler form on $\mathcal{M}$ [Qu]. Let

$\psi_{Q}\,:\,{\mathcal{M}}\,\longrightarrow\,\text{Conn}(\Theta)$

be the corresponding $C^{\infty}$ section of the projection
$\text{Conn}(\Theta)\,\longrightarrow\,\mathcal{M}$.

Since both $\mathcal{C}$ and $\text{Conn}(\Theta)$ are torsors over
$\mathcal{M}$ for $T^{*}\mathcal{M}$, and they are equipped with the
$C^{\infty}$ sections $\psi_{U}$ and $\psi_{Q}$ respectively, there is a
unique $C^{\infty}$ isomorphism

$F\,:\,{\mathcal{C}}\,\longrightarrow\,\text{Conn}(\Theta)$

satisfying the following two conditions:

  * •

$F$ takes the section $\psi_{U}$ to $\psi_{Q}$, and

  * •

$F$ preserves the $T^{*}\mathcal{M}$–torsor structure up to the multiplicative
factor $2r$, meaning $F(E,\,D+v)\,=\,F(E,\,D)+2r\cdot v$, where
$E\,\in\,\mathcal{M}$ with $D$ a holomorphic connection on $E$ and
$v\,\in\,T^{*}_{E}{\mathcal{M}}\,=\,H^{0}(X,\,\text{End}(E)\otimes K_{X})$.

The following was proved in [BH] (recalled here in Theorem 2.1):

The above isomorphism $F$ is holomorphic, and it preserves the holomorphic
symplectic forms up to the factor $2r$, meaning the pullback, by $F$, of the
holomorphic symplectic form on $\text{Conn}(\Theta)$ coincides with $2r$ times
the holomorphic symplectic form on $\mathcal{C}$.

Take any holomorphic vector bundle $E\,\in\,{\mathcal{M}}$ such that
$H^{0}(X,\,E\otimes K^{1/2}_{X})\,=\,0$ (so $E$ lies outside the theta divisor
$D_{\Theta}$). We construct a natural holomorphic connection on $E$; see
Section 3.1. Unlike the unitary connection, it moves holomorphically as $E$
moves in a holomorphic family of vector bundles. In fact, this connection
moves holomorphically as the pair $(X,\,E)$ moves in a holomorphic family. Let

$\phi\,:\,{\mathcal{M}}\setminus
D_{\Theta}\,\longrightarrow\,{\mathcal{C}}\big{|}_{{\mathcal{M}}\setminus
D_{\Theta}}$

be the holomorphic section given by this natural holomorphic connection.

The holomorphic line $\Theta$ has a canonical trivialization outside the theta
divisor $D_{\Theta}$. This trivialization produces a holomorphic section of
the fiber bundle $\text{Conn}(\Theta)\,\longrightarrow\,\mathcal{M}$ outside
$D_{\Theta}$. Let

$\tau\,:\,{\mathcal{M}}\setminus
D_{\Theta}\,\longrightarrow\,\text{Conn}(\Theta)\big{|}_{{\mathcal{M}}\setminus
D_{\Theta}}$

be the section given by this canonical trivialization. Unlike the section
$\psi_{Q}$, this section $\tau$ is holomorphic.

Since both ${\mathcal{C}}\big{|}_{{\mathcal{M}}\setminus D_{\Theta}}$ and
$\text{Conn}(\Theta)\big{|}_{{\mathcal{M}}\setminus D_{\Theta}}$ are torsors
over $\mathcal{M}\setminus D_{\Theta}$ for the holomorphic cotangent bundle
$T^{*}({\mathcal{M}}\setminus D_{\Theta})$, and $\phi$ and $\tau$ are
holomorphic sections, there is a unique holomorphic isomorphism

$G\,:\,{\mathcal{C}}\big{|}_{{\mathcal{M}}\setminus
D_{\Theta}}\,\longrightarrow\,\text{Conn}(\Theta)\big{|}_{{\mathcal{M}}\setminus
D_{\Theta}}$

satisfying the following two conditions:

  * •

$G$ takes the section $\phi$ to $\tau$, and

  * •

$G$ preserves the $T^{*}({\mathcal{M}}\setminus D_{\Theta})$–torsor structures
up to the multiplicative factor $2r$; this means that
$G(E,\,D+v)\,=\,G(E,\,D)+2r\cdot v$, where $E\,\in\,{\mathcal{M}}\setminus
D_{\Theta}$, $D$ is a holomorphic connection on $E$ and
$v\,\in\,T^{*}_{E}{\mathcal{M}}\,=\,H^{0}(X,\,\text{End}(E)\otimes K_{X})$.

Our main result says the following (see Theorem 4.1):

######  Theorem 1.1.

The above isomorphism $G$ coincides with the restriction of the isomorphism
$F$ to the open subset ${\mathcal{C}}\big{|}_{{\mathcal{M}}\setminus
D_{\Theta}}$.

Theorem 1.1 has the following consequence (see Corollary 4.5):

######  Corollary 1.2.

The above holomorphic isomorphism $G$ extends to a holomorphic isomorphism

$G^{\prime}\,:\,{\mathcal{C}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\rm
Conn}(\Theta)$

over entire ${\mathcal{M}}$.

######  Remark 1.3.

We note that the isomorphism $G$ in Theorem 1.1 is constructed purely algebro-
geometrically. Hence the construction of its closure $G^{\prime}$ in Corollary
1.2 is purely algebro-geometric. On the other hand, the two $C^{\infty}$
sections $\psi_{U}$ and $\psi_{Q}$ mentioned earlier are not algebro-
geometric. Theorem 1.1 implies that given the input of the algebro-geometric
isomorphism $G^{\prime}$, any one of the two sections $\psi_{U}$ and
$\psi_{Q}$ determines the other uniquely.

As mentioned before, both ${\mathcal{C}}$ and $\text{Conn}(\Theta)$ are
equipped with holomorphic symplectic structures. Let $\Phi_{1}$ and $\Phi_{2}$
denote the holomorphic symplectic forms on ${\mathcal{C}}$ and
$\text{Conn}(\Theta)$ respectively. We prove the following relationship
between these two symplectic forms (see Corollary 4.6):

######  Corollary 1.4.

For the isomorphism $G^{\prime}$ in Corollary 1.2,

$(G^{\prime})^{*}\Phi_{2}\,=\,2r\cdot\Phi_{1}\,.$

Both $\phi$ and $\tau$ move holomorphically as $X$ moves in a holomorphic
family of Riemann surfaces. Therefore, Theorem 1.1 has the following
consequence (see Proposition 5.1):

######  Proposition 1.5.

The isomorphism $F$ moves holomorphically as $X$ moves in a holomorphic family
of Riemann surfaces.

It may be mentioned that Proposition 1.5 not immediate from the isomorphism of
[BH].

This is particularly useful as one of the questions inspiring this
investigation is the omnipresence of the determinantal line in questions
involving deformations of connections, that is isomonodromy; this manifests
itself in the role of tau-functions. The role of this line is somewhat
surprising; it is as if in a linear algebra problem, the main issue was the
determinant. Several papers have been devoted to this issue, notably by
Malgrange [Ma]. This paper and its predecessor [BH] can be viewed as a further
exploration of this issue; one is comparing one torsor ($\mathcal{C}$),
defined over the moduli space in terms of connections on a full Riemann
surface, and another (Conn) which is simply the natural locus for connections
on the determinant line; the first should contain much more information, but
for certain things, it does not.

##  2\. Moduli space of stable vector bundles

###  2.1. Two torsors on a moduli space

Let $X$ be a compact connected Riemann surface of genus $g$, with
$g\,\geq\,2$. We recall that a holomorphic vector bundle $V$ on $X$ is called
stable if

$\frac{\text{degree}(F)}{\text{rank}(F)}\,<\,\frac{\text{degree}(V)}{\text{rank}(V)}$

for all holomorphic subbundles $F\,\subsetneq\,V$ of positive rank. This
condition implies that any stable vector bundle is simple. Fix a positive
integer $r$. Let ${\mathcal{M}}$ denote the moduli space of stable vector
bundles on $X$ of rank $r$ and degree zero (see [Ne], [Si1] for the
construction of this moduli space).

The holomorphic cotangent bundle of $X$ will be denoted by $K_{X}$. A
holomorphic connection on a holomorphic vector bundle $E$ on $X$ is a
holomorphic differential operator of order one

${\mathcal{D}}_{E}\,:\,E\,\longrightarrow\,E\otimes K_{X}$

satisfying the Leibniz identity, which says that $D(f\cdot s)\,=\,f\cdot
D(s)+s\otimes df$, where $s$ is any locally defined holomorphic section of $E$
and $f$ is any locally defined holomorphic function on $X$ [At]. Holomorphic
connections on $X$ are flat because there are no nonzero $(2,\,0)$–forms on a
Riemann surface.

Let $\mathcal{C}$ denote the moduli space of all holomorphic connections on
$X$ of rank $r$ such that the underlying holomorphic vector bundle is stable
[Si1], [Si2]. In other words, $\mathcal{C}$ parametrizes all isomorphism
classes of pairs of the form $(E,\,{\mathcal{D}}_{E})$, where $E$ is a stable
holomorphic vector bundle on $X$ of rank $r$ and degree zero and
${\mathcal{D}}_{E}$ is a holomorphic connection on $E$.

Any indecomposable holomorphic vector bundle on $X$ of degree zero admits a
holomorphic connection [At, p. 203, Proposition 19], [We], in particular, any
stable vector bundle on $X$ of degree zero admits a holomorphic connection.
Let

$\varphi\,:\,{\mathcal{C}}\,\longrightarrow\,\mathcal{M}\,,\ \
(E,\,{\mathcal{D}}_{E})\,\longmapsto\,E$ (2.1)

be the forgetful map that forgets the holomorphic connection; as noted above,
the map $\varphi$ is surjective. Any two holomorphic connections on
$E\,\in\,\mathcal{M}$ differ by an element of

$H^{0}(X,\,\text{End}(E)\otimes K_{X})\,=\,T^{*}_{E}\mathcal{M}\,.$

In fact, $\mathcal{C}$ is a holomorphic torsor over $\mathcal{M}$ for the
holomorphic cotangent bundle $T^{*}\mathcal{M}$. This means that there is a
holomorphic action

$\delta\,:\,{\mathcal{C}}\times_{\mathcal{M}}T^{*}\mathcal{M}\,\longrightarrow\,{\mathcal{C}}$

of $T^{*}\mathcal{M}$ on $\mathcal{C}$ such that the map of fiber products

${\mathcal{C}}\times_{\mathcal{M}}T^{*}\mathcal{M}\,\longrightarrow\,{\mathcal{C}}\times_{\mathcal{M}}{\mathcal{C}}\,,\
\ (a,\,b)\,\longmapsto\,(\delta(a,\,b),\,a)$

is an isomorphism.

Any stable holomorphic vector bundle on $X$ of degree zero admits a unique
holomorphic connection whose monodromy representation is unitary [NS, p.
560–561, Theorem 2]. Therefore, the projection $\varphi$ in (2.1) has a
$C^{\infty}$ section

$\psi_{U}\,:\,{\mathcal{M}}\,\longrightarrow\,\mathcal{C}$ (2.2)

that sends any stable vector bundle $E\,\in\,\mathcal{M}$ to the unique
holomorphic connection on $E$ whose monodromy representation is unitary. This
section $\psi_{U}$ is not holomorphic.

There is a natural holomorphic symplectic form on $\mathcal{C}$

$\Phi_{1}\,\in\,H^{0}({\mathcal{C}},\,\Omega^{2}_{\mathcal{C}})$ (2.3)

[Go], [AB]. It is known that this holomorphic $2$–form $\Phi_{1}$ is algebraic
[Bi2].

We shall now construct another holomorphic torsor over $\mathcal{M}$ for the
holomorphic cotangent bundle $T^{*}\mathcal{M}$.

Fix a theta characteristic $K^{1/2}_{X}$ on $X$. So $K^{1/2}_{X}$ is a
holomorphic line bundle on $X$ of degree $g-1$ such that $K^{1/2}_{X}\otimes
K^{1/2}_{X}$ is holomorphically isomorphic to the holomorphic cotangent bundle
$K_{X}$. Fix a holomorphic isomorphism of $K^{1/2}_{X}\otimes K^{1/2}_{X}$
with $K_{X}$. Let

$D_{\Theta}\,:=\,\\{E\,\in\,\mathcal{M}\,\mid\,H^{0}(X,\,E\otimes
K^{1/2}_{X})\,\not=\,0\\}\,\subset\,\mathcal{M}$ (2.4)

be the theta divisor on $\mathcal{M}$ (see [La]). Note that by Riemann–Roch we
have

$\dim H^{0}(X,\,E\otimes K^{1/2}_{X})-\dim H^{1}(X,\,E\otimes
K^{1/2}_{X})\,=\,\text{degree}(E\otimes K^{1/2}_{X})-r(g-1)$
$=\,r(g-1)-r(g-1)\,=\,0\,.$

So $H^{1}(X,\,E\otimes K^{1/2}_{X})\,\not=\,0$ if and only if
$E\,\in\,D_{\Theta}$. The holomorphic line bundle
${\mathcal{O}}_{\mathcal{M}}(D_{\Theta})$ on $\mathcal{M}$ will be denoted by
$\Theta$.

Let $\text{At}(\Theta)\,\longrightarrow\,\mathcal{M}$ be the Atiyah bundle for
$\Theta$. It fits in the short exact sequence of holomorphic vector bundles

$0\,\longrightarrow\,{\mathcal{O}}_{\mathcal{M}}\,\longrightarrow\,\text{At}(\Theta)\,\longrightarrow\,T{\mathcal{M}}\,\longrightarrow\,0$
(2.5)

over $\mathcal{M}$ (see [At, p. 187, Theorem 1]). For $i\,\geq\,0$, let ${\rm
Diff}^{1}_{\mathcal{M}}(\Theta,\,\Theta)$ be the holomorphic vector bundle on
${\mathcal{M}}$ given by the sheaf of holomorphic differential operators from
$\Theta$ to itself. We note that $\text{At}(\Theta)\,=\,{\rm
Diff}^{1}_{\mathcal{M}}(\Theta,\,\Theta)$, and the exact sequence in (2.5)
coincides with the sequence

$0\,\longrightarrow\,{\rm
Diff}^{0}_{\mathcal{M}}(\Theta,\,\Theta)\,=\,{\mathcal{O}}_{\mathcal{M}}\,\longrightarrow\,{\rm
Diff}^{1}_{\mathcal{M}}(\Theta,\,\Theta)\,\longrightarrow\,T{\mathcal{M}}\,\longrightarrow\,0\,,$

where the projection to $T{\mathcal{M}}$ is the symbol map. Consider the dual
exact sequence of (2.5)

$0\,\longrightarrow\,T^{*}{\mathcal{M}}\,\longrightarrow\,\text{At}(\Theta)^{*}\,\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\,{\mathcal{O}}^{*}_{\mathcal{M}}\,=\,{\mathcal{O}}_{\mathcal{M}}\,\longrightarrow\,0\,.$
(2.6)

Let
$1_{\mathcal{M}}\,:\,{\mathcal{M}}\,\longrightarrow\,{\mathcal{O}}_{\mathcal{M}}$
be the section given by the constant function $1$ on $\mathcal{M}$. Now define

$\text{At}(\Theta)^{*}\,\supset\,\alpha^{-1}(1_{\mathcal{M}}({\mathcal{M}}))\,=:\,\text{Conn}(\Theta)\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,{\mathcal{M}}\,,$ (2.7)

where $\alpha$ is the projection in (2.6). Holomorphic sections of
$\text{Conn}(\Theta)$ over an open subset $U\,\subset\,{\mathcal{M}}$ are
identified with the holomorphic connections on $\Theta\big{|}_{U}$. From (2.6)
it follows immediately that $\text{Conn}(\Theta)$ is a holomorphic torsor over
$\mathcal{M}$ for the holomorphic cotangent bundle $T^{*}\mathcal{M}$.

For any open subset $U\,\subset\,{\mathcal{M}}$, all $C^{\infty}$ maps
$s\,:\,U\,\,\longrightarrow\,\text{Conn}(\Theta)$ such that $q\circ
s\,=\,\text{Id}_{U}$, where $q$ is the projection in (2.7), are in bijection
with the $C^{\infty}$ connections $D_{U}$ on the holomorphic line bundle
$\Theta\big{|}_{U}$ such that the $(0,\,1)$-part of $D_{U}$ coincides with the
Dolbeault operator given by the holomorphic structure on $\Theta\big{|}_{U}$.
This condition on $D_{U}$ is equivalent to the condition that $D_{U}(\gamma)$
is a $C^{\infty}$ section of $(\Theta\otimes K_{X})\big{|}_{U}$ for every
holomorphic section $\gamma$ of $\Theta_{U}$. Such a connection $D_{U}$ on
$\Theta\big{|}_{U}$ is holomorphic if and only if the corresponding section
$s$ of the projection $q$ is holomorphic.

There is a natural Kähler form $\omega_{\mathcal{M}}$ on $\mathcal{M}$ [AB];
this form $\omega_{\mathcal{M}}$ coincides with the symplectic form on the
irreducible unitary character variety ${\rm Hom}^{\rm ir}(\pi_{1}(X),\,{\rm
U}(r))/{\rm U}(r)$ that was constructed by Goldman [Go]; here ${\rm Hom}^{\rm
ir}(\pi_{1}(X),\,{\rm U}(r))$ denotes the space of all homomorphisms
$\rho\,:\,\pi_{1}(X)\,\longrightarrow\,{\rm U}(r)$ such that the standard
action of $\rho(\pi_{1}(X))$ on ${\mathbb{C}}^{r}$ does not preserve any
nonzero proper subspace of ${\mathbb{C}}^{r}$. More precisely, we have

$\omega_{\mathcal{M}}\,=\,\psi^{*}_{U}\Phi_{1}\,,$ (2.8)

where $\psi_{U}$ and $\Phi_{1}$ are as in (2.2) and (2.3) respectively.

Quillen constructed an explicit Hermitian structure on the line bundle
$\Theta$ with the property that the curvature of the corresponding Chern
connection on $\Theta$ coincides with the Kähler form $\omega_{\mathcal{M}}$
in (2.8) [Qu] (see also [BGS1], [BGS2], [BGS3]). As a corollary, the de Rham
cohomology class for $\omega_{\mathcal{M}}$ is integral. We note that there is
at most one Hermitian connection on $\Theta$ whose curvature is
$\omega_{\mathcal{M}}$. In other words, the Chern connection of the Hermitian
structure on $\Theta$ constructed in [Qu] is the unique Hermitian connection
whose curvature is $\omega_{\mathcal{M}}$. It should be clarified that this
condition — that the curvature is $\omega_{\mathcal{M}}$ — does not determine
the Hermitian structure on $\Theta$ uniquely; any two Hermitian structures on
$\Theta$ satisfying this condition differ by a constant scalar multiplication.
However, the Hermitian connection is unique. Let $\nabla^{Q}$ denote the
unique Hermitian connection on $\Theta$ whose curvature is
$\omega_{\mathcal{M}}$. So $\nabla^{Q}$ produces a $C^{\infty}$ section

$\psi_{Q}\,:\,{\mathcal{M}}\,\longrightarrow\,\text{Conn}(\Theta)$ (2.9)

of the holomorphic fibration $q$ in (2.7).

There is a holomorphic symplectic form

$\Phi_{2}\,\in\,H^{0}(\text{Conn}(\Theta),\,\Omega^{2}_{\text{Conn}(\Theta)})$
(2.10)

on $\text{Conn}(\Theta)$ which can be described as follows. The holomorphic
line bundle $q^{*}\Theta$, where $q$ is the projection in (2.7), has a
tautological holomorphic connection (see [BHS, p. 372, Proposition 3.3],
[BB]). The curvature of this tautological holomorphic connection on
$q^{*}\Theta$ is the $2$–form $\Phi_{2}$ in (2.10).

###  2.2. An isomorphism of torsors

In this subsection a result from [BH] will be recalled.

Let

$\delta\,:\,{\mathcal{C}}\times_{\mathcal{M}}T^{*}{\mathcal{M}}\,\longrightarrow\,{\mathcal{C}}\
\ \text{ and }\ \ \eta\,:\,{\rm
Conn}(\Theta)\times_{\mathcal{M}}T^{*}{\mathcal{M}}\,\longrightarrow\,{\rm
Conn}(\Theta)$ (2.11)

be the holomorphic $T^{*}{\mathcal{M}}$–torsor structures on $\mathcal{C}$ and
${\rm Conn}(\Theta)$ respectively. Let

${\textbf{m}}\,:\,T^{*}{\mathcal{M}}\,\longrightarrow\,T^{*}{\mathcal{M}}\,,\
\ v\,\longmapsto\,2r\cdot v$ (2.12)

be the multiplication by $2r$.

######  Theorem 2.1 ([BH, Proposition 2.3, Theorem 3.1]).

There is a unique holomorphic isomorphism

$F\,:\,{\mathcal{C}}\,\longrightarrow\,{\rm Conn}(\Theta)$

such that

  1. (1)

$\varphi\,=\,q\circ F$, where $\varphi$ and $q$ are the projections in (2.1)
and (2.7) respectively,

  2. (2)

$F\circ\psi_{U}\,=\,\psi_{Q}$, where $\psi_{U}$ and $\psi_{Q}$ are the
sections in (2.2) and (2.9) respectively, and

  3. (3)

$F\circ\delta\,=\,\eta\circ(F\times{\textbf{m}})$ as maps from
${\mathcal{C}}\times_{\mathcal{M}}T^{*}{\mathcal{M}}$ to ${\rm Conn}(\Theta)$,
where $\delta$, $\eta$ and m are the maps in (2.11) and (2.12).

Moreover, $F^{*}\Phi_{2}\,=\,2r\cdot\Phi_{1}$, where $\Phi_{1}$ and $\Phi_{2}$
are the symplectic forms in (2.3) and (2.10) respectively.

There is a unique $C^{\infty}$ isomorphism
${\mathcal{C}}\,\longrightarrow\,{\rm Conn}(\Theta)$ that satisfies the three
conditions in the first part of Theorem 2.1. The content of the first part of
Theorem 2.1 is that this $C^{\infty}$ isomorphism is actually holomorphic. The
second part of Theorem 2.1 says that this isomorphism is compatible, up to the
factor $2r$, with the symplectic structures on $\mathcal{C}$ and ${\rm
Conn}(\Theta)$.

######  Remark 2.2.

For a different choice of the theta characteristic on $X$, the corresponding
theta line bundle on ${\mathcal{M}}$ differs from $\Theta$ by a line bundle of
order two on ${\mathcal{M}}$. Any line bundle of finite order has a
tautological flat holomorphic connection. This implies that the
$T^{*}{\mathcal{M}}$–torsor ${\rm Conn}(\Theta)$ is actually independent of
the choice that theta characteristic on $X$.

##  3\. Another isomorphism of torsors

###  3.1. A canonical connection

For $i\,=\,1,\,2$, let

$p_{i}\,:\,X\times X\,\longrightarrow\,X$ (3.1)

be the projection to the $i$–th factor. Let

$\Delta\,:=\,\\{(x,\,x)\,\mid\,x\,\in\,X\\}\,\subset\,X\times X$

be the diagonal divisor. We shall identify $\Delta$ with $X$ via the map
$x\,\longmapsto\,(x,\,x)$. Using the Poincaré adjunction formula, the
restriction of the holomorphic line bundle ${\mathcal{O}}_{X\times X}(\Delta)$
to $\Delta$ is identified with the normal bundle of the divisor
$\Delta\,\subset\,X\times X$, which in turn is identified with $TX$ using the
identification of $\Delta$ with $X$. However this isomorphism between
${\mathcal{O}}_{X\times X}(\Delta)\big{|}_{\Delta}$ and $TX$ changes by
multiplication by $-1$ under the involution $(x,\,y)\,\longmapsto\,(y,\,x)$ of
$X\times X$. In other words, this involution acts by multiplication by $-1$ on
${\mathcal{O}}_{X\times X}(\Delta)\big{|}_{\Delta}$.

Using the isomorphism between ${\mathcal{O}}_{X\times
X}(\Delta)\big{|}_{\Delta}$ and $TX$, the restriction of
$(p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta)$ to $\Delta$ is identified with $K_{X}\otimes
TX\,=\,{\mathcal{O}}_{X}$. It should be clarified that this isomorphism
changes by multiplication by $-1$ under the involution
$(x,\,y)\,\longmapsto\,(y,\,x)$ of $X\times X$.

Take any $E\,\in\,{\mathcal{M}}\setminus D_{\Theta}$, where $D_{\Theta}$ is
constructed in (2.4). Since

$H^{0}(X,\,E\otimes K^{1/2}_{X})\,=\,0\,=\,H^{1}(X,\,E\otimes K^{1/2}_{X})\,,$

using Serre duality, we have

$H^{0}(X,\,E^{*}\otimes K^{1/2}_{X})\,=\,H^{1}(X,\,E\otimes
K^{1/2}_{X})^{*}\,=\,0\,=\,H^{0}(X,\,E\otimes
K^{1/2}_{X})^{*}\,=\,H^{1}(X,\,E^{*}\otimes K^{1/2}_{X})\,.$ (3.2)

Consider the short exact sequence of coherent sheaves on $X\times X$

$0\,\longrightarrow\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes
K^{1/2}_{X}))\,\longrightarrow\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes
K^{1/2}_{X}))\otimes{\mathcal{O}}_{X\times X}(\Delta)$
$\longrightarrow\,\text{End}(E)\big{|}_{\Delta}\,\longrightarrow\,0\,;$

recall that
$(p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta)\big{|}_{\Delta}\,=\,{\mathcal{O}}_{X}$, and note that
$(p^{*}_{1}E)\otimes(p^{*}_{2}E^{*})\big{|}_{\Delta}\,=\,\text{End}(E)$ (the
identification between $X$ and $\Delta$ is being used). Let

$H^{0}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes K^{1/2}_{X})))$
$\longrightarrow\,H^{0}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes
K^{1/2}_{X}))\otimes{\mathcal{O}}_{X\times X}(\Delta))$
$\stackrel{{\scriptstyle
h_{E}}}{{\longrightarrow}}\,H^{0}(X,\,\text{End}(E))\,\longrightarrow\,H^{1}(X\times
X,\,(p^{*}_{1}(E\otimes K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes
K^{1/2}_{X})))$ (3.3)

be the corresponding long exact sequence of cohomologies. Using Künneth
formula,

$H^{0}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes
K^{1/2}_{X})))\,=\,H^{0}(X,\,E\otimes K^{1/2}_{X})\otimes
H^{0}(X,\,E^{*}\otimes K^{1/2}_{X})$

and

$H^{1}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes K^{1/2}_{X})))$ (3.4)
$=\,\bigoplus_{j=0}^{1}H^{j}(X,\,E\otimes K^{1/2}_{X})\otimes
H^{1-j}(X,\,E^{*}\otimes K^{1/2}_{X})\,.$

Hence invoking (3.2) we conclude that

$H^{k}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes K^{1/2}_{X})))\,=\,0$ (3.5)

for $k\,=\,0,\,1$. Consequently, the homomorphism $h_{E}$ in the exact
sequence in (3.3) is an isomorphism. Now define

$\beta_{E}\,:=\,h^{-1}_{E}(\text{Id}_{E})\,\in\,H^{0}(X\times
X,\,(p^{*}_{1}(E\otimes K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes
K^{1/2}_{X}))\otimes{\mathcal{O}}_{X\times X}(\Delta))\,,$ (3.6)

where $\text{Id}_{E}\,\in\,H^{0}(X,\,\text{End}(E))$ is the identity
endomorphism.

It was noted earlier that
$(p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta)\big{|}_{\Delta}\,=\,{\mathcal{O}}_{X}$. We shall now show that

$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{2\Delta}\,=\,{\mathcal{O}}_{2\Delta}\,.$ (3.7)

To prove (3.7), take any holomorphic coordinate function
$z\,:\,U\,\longrightarrow\,\mathbb{C}$ on some analytic open subset $U$ of
$X$. Take a holomorphic section

$s_{z}\,\in\,H^{0}(U,\,K^{1/2}_{X}\big{|}_{U})$

such that $s_{z}\otimes s_{z}\,=\,dz\,\in\,H^{0}(U,\,K_{U})$; note that there
are exactly two such sections, and they differ by multiplication by $-1$. Now
we have

$\frac{1}{z\circ p_{1}-z\circ
p_{2}}(p^{*}_{1}s_{z})\otimes(p^{*}_{2}s_{z})\,\in\,H^{0}(U\times
U,\,((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{U\times U})\,.$

Of course, this section $\frac{1}{z\circ p_{1}-z\circ
p_{2}}(p^{*}_{1}s_{z})\otimes(p^{*}_{2}s_{z})$ depends on the coordinate
function $z$. However, it is straight-forward to check that the restriction of
the section to $(2\Delta)\bigcap(U\times U)$

$\frac{1}{z\circ p_{1}-z\circ
p_{2}}(p^{*}_{1}s_{z})\otimes(p^{*}_{2}s_{z})\big{|}_{(2\Delta)\cap(U\times
U)}$

is actually independent of the choice of the holomorphic coordinate function
$z$. Consequently, the locally defined sections of the form $\frac{1}{z\circ
p_{1}-z\circ p_{2}}(p^{*}_{1}s_{z})\otimes(p^{*}_{2}s_{z})$ patch together
compatibly to define a canonical section of
$(p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta)\big{|}_{2\Delta}$. Let

$\sigma_{0}\,\in\,H^{0}(2\Delta,\,(p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))$ (3.8)

be this canonical section. This section $\sigma_{0}$ produces the isomorphism
in (3.7) between
$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{2\Delta}$ and ${\mathcal{O}}_{2\Delta}$ by sending any
locally defined section $f$ of ${\mathcal{O}}_{2\Delta}$ to the locally
defined section $f\cdot\sigma_{0}$ of
$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{2\Delta}$.

We note that the restriction of the section $\sigma_{0}$ in (3.8) to
$\Delta\,\subset\,2\Delta$ coincides with the section given by the
trivialization of
$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{\Delta}$ (the trivialization of
$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{\Delta}$ was obtained earlier using the Poincaré
adjunction formula). Consider the section $\beta_{E}$ in (3.6). It is easy to
see that there is a unique section

$\widehat{\beta}_{E}\,\in\,H^{0}(2\Delta,\,(p^{*}_{1}E)\otimes(p^{*}_{2}E^{*}))\,,$

over $2\Delta$, such that

$\beta_{E}\big{|}_{2\Delta}\,=\,\widehat{\beta}_{E}\otimes\sigma_{0}\,.$ (3.9)

Indeed, $(\sigma_{0})^{-1}$ is a section of
$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))^{*}$ over $2\Delta$. Now define

$\widehat{\beta}_{E}\,:=\,(\beta_{E}\big{|}_{2\Delta})\otimes(\sigma_{0})^{-1}\,,$

and consider it as a section of
$((p^{*}_{1}E)\otimes(p^{*}_{2}E^{*}))\big{|}_{2\Delta}$ using the duality
pairing

$(((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{2\Delta})\otimes(((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))^{*}\big{|}_{2\Delta})\,\longrightarrow\,{\mathcal{O}}_{2\Delta}\,.$

Since $h_{E}(\beta_{E})\,=\,\text{Id}_{E}$ (see (3.6)), and the restriction of
$\sigma_{0}$ to $\Delta\,\subset\,2\Delta$ coincides with the section of
$((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta))\big{|}_{\Delta}$ given by its trivialization, we conclude that

$\widehat{\beta}_{E}\big{|}_{\Delta}\,=\,\text{Id}_{E}$

using the natural identification of
$((p^{*}_{1}E)\otimes(p^{*}_{2}E^{*}))\big{|}_{\Delta}\,\longrightarrow\,\Delta$
with $\text{End}(E)\,\longrightarrow\,X$. Consequently, the section
$\widehat{\beta}_{E}$ in (3.9) defines a holomorphic connection on the
holomorphic vector bundle $E$, following the idea of Grothendieck of defining
a connection as an extension, to the first order neighborhood of the diagonal,
of the isomorphism of the two pullbacks on the diagonal (see [De, p. 6,
2.2.4]). This holomorphic connection on $E$ defined by $\widehat{\beta}_{E}$
will be denoted by

$\widehat{\beta}^{\prime}_{E}\,.$ (3.10)

For notational convenience, let

${\mathcal{M}}^{0}\,:=\,{\mathcal{M}}\setminus D_{\Theta}$ (3.11)

denote the complement of $D_{\Theta}$ in ${\mathcal{M}}$. Define

${\mathcal{C}}^{0}\,:=\,\varphi^{-1}({\mathcal{M}}^{0})\,\subset\,{\mathcal{C}}\,,$

where $\varphi$ is the projection in (2.1). Let

$\widehat{\varphi}\,:\,{\mathcal{C}}^{0}\,\longrightarrow\,{\mathcal{M}}^{0}$
(3.12)

be the restriction of the map $\varphi$ to ${\mathcal{C}}^{0}$. We note that
${\mathcal{C}}^{0}$ is a holomorphic torsor over ${\mathcal{M}}^{0}$ for
$T^{*}{\mathcal{M}}^{0}$.

We have the holomorphic map

$\phi\,:\,{\mathcal{M}}^{0}\,\longrightarrow\,{\mathcal{C}}^{0}\,,\ \
E\,\longmapsto\,(E,\,\widehat{\beta}^{\prime}_{E})\,,$ (3.13)

where $\widehat{\beta}^{\prime}_{E}$ is the holomorphic connection in (3.10).
So $\phi$ is a holomorphic section of the projection $\widehat{\varphi}$ in
(3.12), meaning
$\widehat{\varphi}\circ\phi\,=\,\text{Id}_{{\mathcal{M}}^{0}}$.

###  3.2. A holomorphic isomorphism of torsors

Define

$\text{Conn}(\Theta)^{0}\,:=\,q^{-1}({\mathcal{M}}^{0})\,\subset\,\text{Conn}(\Theta)\,,$

where $q$ is the projection in (2.7), and ${\mathcal{M}}^{0}$ is the Zariski
open subset of $\mathcal{M}$ in (3.11). Let

$\widehat{q}\,:\,\text{Conn}(\Theta)^{0}\,\longrightarrow\,{\mathcal{M}}^{0}$
(3.14)

be the restriction of the map $q$ to $\text{Conn}(\Theta)^{0}$. We note that
$\text{Conn}(\Theta)^{0}$ is a holomorphic torsor over ${\mathcal{M}}^{0}$ for
$T^{*}{\mathcal{M}}^{0}$.

The restriction of the line bundle
$\Theta\,=\,{\mathcal{O}}_{\mathcal{M}}(D_{\Theta})$ to ${\mathcal{M}}^{0}$
has a tautological isomorphism with the trivial line bundle
${\mathcal{O}}_{{\mathcal{M}}^{0}}$. Therefore, the trivial holomorphic
connection on ${\mathcal{O}}_{{\mathcal{M}}^{0}}$, defined by the de Rham
differential, produces a holomorphic connection on the restriction
$\Theta\big{|}_{{\mathcal{M}}^{0}}$. Let

$\tau\,:\,{\mathcal{M}}^{0}\,\longrightarrow\,\text{Conn}(\Theta)^{0}$ (3.15)

be the holomorphic section of the projection $\widehat{q}$ in (3.14) given by
this tautological connection on $\Theta\big{|}_{{\mathcal{M}}^{0}}$.

Let

$\delta^{0}\,:\,{\mathcal{C}}^{0}\times_{{\mathcal{M}}^{0}}T^{*}{\mathcal{M}}^{0}\,\longrightarrow\,{\mathcal{C}}^{0}\
\ \text{ and }\ \ \eta^{0}\,:\,{\rm
Conn}(\Theta)^{0}\times_{{\mathcal{M}}^{0}}T^{*}{\mathcal{M}}^{0}\,\longrightarrow\,{\rm
Conn}(\Theta)^{0}$ (3.16)

be the restrictions of the maps $\delta$ and $\eta$ in (2.11). So $\delta^{0}$
and $\eta^{0}$ give the $T^{*}{\mathcal{M}}^{0}$–torsor structures on
${\mathcal{C}}^{0}$ and ${\rm Conn}(\Theta)^{0}$ respectively. Similarly,

${\textbf{m}}^{0}\,:\,T^{*}{\mathcal{M}}^{0}\,\longrightarrow\,T^{*}{\mathcal{M}}^{0}\,,\
\ v\,\longmapsto\,2r\cdot v$ (3.17)

is the restriction of the map in (2.12).

######  Lemma 3.1.

There is a unique holomorphic isomorphism

$G\,\,:\,\,{\mathcal{C}}^{0}\,\longrightarrow\,{\rm Conn}(\Theta)^{0}$

such that

  1. (1)

$\widehat{\varphi}\,=\,\widehat{q}\circ G$, where $\widehat{\varphi}$ and
$\widehat{q}$ are the projections in (3.12) and (3.14) respectively,

  2. (2)

$G\circ\phi\,=\,\tau$, where $\phi$ and $\tau$ are the sections in (3.13) and
(3.15) respectively, and

  3. (3)

$G\circ\delta^{0}\,=\,\eta^{0}\circ(G\times{\textbf{m}}^{0})$ as maps from
${\mathcal{C}}^{0}\times_{{\mathcal{M}}^{0}}T^{*}{\mathcal{M}}^{0}$ to ${\rm
Conn}(\Theta)^{0}$, where $\delta^{0}$, $\eta^{0}$ and ${\textbf{m}}^{0}$ are
the maps in (3.16) and (3.17).

###### Proof.

This is straightforward. For any stable vector bundle
$E\,\in\,{\mathcal{M}}^{0}$ and any

$\nu\,\in\,H^{0}(X,\,\text{End}(E)\otimes
K_{X})\,=\,T^{*}_{E}{\mathcal{M}}^{0},$

define

$G(\delta^{0}(\phi(E),\,\nu))\,=\,\eta^{0}(\tau(E),\,2r\cdot\nu)\,.$

Then $G$ is evidently a well defined map from ${\mathcal{C}}^{0}$ to ${\rm
Conn}(\Theta)^{0}$. It is holomorphic because $\phi$, $\tau$, $\delta^{0}$ and
$\eta^{0}$ are all holomorphic maps. This map $G$ satisfies all the three
conditions in the lemma. The uniqueness of $G$ is evident. ∎

##  4\. The two isomorphisms of torsors coincide

The following theorem is the main result proved here.

######  Theorem 4.1.

The restriction of the isomorphism $F$ in Theorem 2.1 to the open subset
${\mathcal{C}}^{0}\,\subset\,{\mathcal{C}}$ coincides with the isomorphism $G$
in Lemma 3.1.

###### Proof.

In view of the first condition in both Theorem 2.1 and Lemma 3.1, we get a map

$\Gamma_{0}\,:\,{\mathcal{C}}^{0}\,\longrightarrow\,T^{*}{\mathcal{M}}^{0}\,,\
\ z\,\longmapsto\,F(z)-G(z)\,.$ (4.1)

In other words, $F(z)\,=\,\eta^{0}(G(z),\,\Gamma_{0}(z))$. This map
$\Gamma_{0}$ is holomorphic because both $F$ and $G$ are so. Take any
$E\,\in\,{\mathcal{M}}^{0}$ and any

$\alpha\,\in\,\widehat{\varphi}^{-1}(E)\,\subset\,{\mathcal{C}}^{0}\,,$

where $\widehat{\varphi}$ is the projection in (3.12), and also take any

$\nu\,\in\,H^{0}(X,\,\text{End}(E)\otimes
K_{X})\,=\,T^{*}_{E}{\mathcal{M}}^{0}\,.$

Now from the third condition in both Theorem 2.1 and Lemma 3.1, we have

$\Gamma_{0}(\alpha+\nu)\,=\,F(\alpha+\nu)-G(\alpha+\nu)\,=\,F(\alpha)-G(\alpha)+2r\cdot\nu-2r\cdot\nu\,=\,\Gamma_{0}(\alpha)\,.$

Consequently, the map $\Gamma_{0}$ in (4.1) produces a holomorphic $1$–form

$\Gamma\,\in\,H^{0}({\mathcal{M}}^{0},\,T^{*}{\mathcal{M}}^{0})$ (4.2)

that sends any $E\,\in\,{\mathcal{M}}^{0}$ to
$\Gamma_{0}(\alpha)\,\in\,T^{*}_{E}{\mathcal{M}}^{0}$ with
$\alpha\,\in\,\widehat{\varphi}^{-1}(E)$; as shown above, $\Gamma_{0}(\alpha)$
is independent of the choice of $\alpha$. In other words,
$\Gamma_{0}\,=\,\Gamma\circ\widehat{\varphi}$, where $\widehat{\varphi}$ is
the projection in (3.12).

The following proposition would be used in the proof of Theorem 4.1.

######  Proposition 4.2.

The holomorphic $1$–form $\Gamma$ on ${\mathcal{M}}^{0}$ in (4.2) is a
meromorphic $1$–form on ${\mathcal{M}}$, and its order of pole at the divisor
$D_{\Theta}\,=\,{\mathcal{M}}\setminus{\mathcal{M}}^{0}$ is at most one, or
equivalently,

$\Gamma\,\in\,H^{0}({\mathcal{M}},\,(T^{*}{\mathcal{M}})\otimes\Theta)\,=\,H^{0}({\mathcal{M}},\,(T^{*}{\mathcal{M}})\otimes{\mathcal{O}}_{\mathcal{M}}(D_{\Theta}))\,.$

###### Proof of Proposition 4.2.

Let ${\mathbb{W}}\,\longrightarrow\,{\mathcal{M}}$ be a holomorphic torsor for
$T^{*}{\mathcal{M}}$ and $s$ a holomorphic section of ${\mathbb{W}}$ over the
open subset ${\mathcal{M}}^{0}\,=\,{\mathcal{M}}\setminus D_{\Theta}$. Then
the meromorphicity of $s$ is defined by choosing holomorphic trivializations
of ${\mathbb{W}}$ on open neighborhoods, in ${\mathcal{M}}$, of points of
$D_{\Theta}$ (a trivialization of a torsor is just a holomorphic section of
it). Such a trivialization of $\mathbb{W}$ over $U\,\subset\,{\mathcal{M}}$
turns $s$ into a holomorphic $1$–form on $U\cap{\mathcal{M}}^{0}$; define
$s\big{|}_{U}$ to be meromorphic if this holomorphic $1$–form on
$U\cap{\mathcal{M}}^{0}$ is meromorphic near $D_{\Theta}\bigcap
U\,\subset\,U$. Since any two holomorphic trivializations, over $U$, of the
torsor ${\mathbb{W}}$ differ by a holomorphic $1$–form on $U$, this definition
of the meromorphicity of $s\big{|}_{U}$ does not depend on the choice of the
trivialization of ${\mathbb{W}}\big{|}_{U}$. For the same reason, the order of
pole at $D_{\Theta}$ of a meromorphic section $s$ of ${\mathbb{W}}$ of the
above type is also well-defined.

Let $\varpi_{1}$ be the smooth $(1,0)$–form on ${\mathcal{M}}^{0}$ given by
$\phi-\psi_{U}\big{|}_{{\mathcal{M}}^{0}}$, where $\phi$ (respectively,
$\psi_{U}$) is the section of the $T^{*}{\mathcal{M}}^{0}$–torsor
${\mathcal{C}}^{0}$ (respectively, $T^{*}{\mathcal{M}}$–torsor
${\mathcal{C}}$) constructed in (3.13) (respectively, (2.2)). Let $\varpi_{2}$
be the smooth $(1,0)$–form on ${\mathcal{M}}^{0}$ given by
$\tau-\psi_{Q}\big{|}_{{\mathcal{M}}^{0}}$, where $\tau$ (respectively,
$\psi_{Q}$) is the section of the $T^{*}{\mathcal{M}}^{0}$–torsor
$\text{Conn}(\Theta)^{0}$ (respectively, $T^{*}{\mathcal{M}}$–torsor
$\text{Conn}(\Theta)$) constructed in (3.15) (respectively, (2.9)). It can be
shown that

$\Gamma\,=\,2r\cdot\varpi_{1}-\varpi_{2}\,.$ (4.3)

Indeed, using the third property in Theorem 2.1 and the third property in
Lemma 3.1 we have

$2r\cdot\varpi_{1}-\varpi_{2}\,=\,2r(\phi-\psi_{U}\big{|}_{{\mathcal{M}}^{0}})-(\tau-\psi_{Q}\big{|}_{{\mathcal{M}}^{0}})\,=\,G(\phi)-G(\psi_{U}\big{|}_{{\mathcal{M}}^{0}})-\tau+\psi_{Q}\big{|}_{{\mathcal{M}}^{0}}$
$=\,\psi_{Q}\big{|}_{{\mathcal{M}}^{0}}-G(\psi_{U}\big{|}_{{\mathcal{M}}^{0}})\,=\,F(\psi_{U}\big{|}_{{\mathcal{M}}^{0}})-G(\psi_{U}\big{|}_{{\mathcal{M}}^{0}})\,=\,\Gamma\,.$

Both $\psi_{U}$ and $\psi_{Q}$ are smooth sections over entire
${\mathcal{M}}$. The holomorphic section $\tau$ of $\text{Conn}(\Theta)^{0}$
is a meromorphic section of $\text{Conn}(\Theta)$ with a pole of order one at
$D_{\Theta}$. Indeed, this follows immediately from the fact that the
holomorphic connection on the line bundle
$\Theta\big{|}_{{\mathcal{M}}^{0}}\,=\,{\mathcal{O}}_{{\mathcal{M}}^{0}}$ over
${\mathcal{M}}^{0}$, given by the canonical holomorphic trivialization of
$\Theta\big{|}_{{\mathcal{M}}^{0}}$ (the holomorphic connection is defined by
the de Rham differential), is actually a logarithmic connection on $\Theta$
over ${\mathcal{M}}$. In view of these, using (4.3) we conclude the following:

  * •

$\Gamma$ is a meromorphic $1$–form on ${\mathcal{M}}$ if and only if the
section $\phi$ of ${\mathcal{C}}^{0}$ in (3.13) is meromorphic, and

  * •

if $\Gamma$ is meromorphic, and the order of its pole at $D_{\Theta}$ is more
than one, then the order of the pole of $\Gamma$ at $D_{\Theta}$ coincides
with the order of pole of $\phi$ at $D_{\Theta}$, in particular, the order of
the pole of $\phi$ at $D_{\Theta}$ is more than one.

Therefore, to prove the proposition it suffices to show the following two:

  1. (1)

the section $\phi$ of ${\mathcal{C}}^{0}$ is meromorphic, and

  2. (2)

the order of pole of $\phi$ at $D_{\Theta}$ is one.

These will be proved by giving a global construction of $\phi$.

It is known that there is no Poincaré vector bundle over
$X\times{\mathcal{M}}$ [Ra, p. 69, Theorem 2]. However, there is a canonical
algebraic vector bundle over $X\times X\times{\mathcal{M}}$ whose fiber over
$X\times X\times\\{E\\}$ is $E\boxtimes
E^{*}\,=\,(p^{*}_{1}E)\otimes(p^{*}_{2}E^{*})$ for every
$E\,\in\,{\mathcal{M}}$, where $p_{1}$ and $p_{2}$ are the projections in
(3.1). This canonical vector bundle on $X\times X\times{\mathcal{M}}$, which
we shall denote by ${\mathcal{E}}$, can be constructed as a descended bundle
from the product of $X\times X$ with the quot scheme. The reason that the
corresponding vector bundle descends is that the action of the multiplicative
group ${\mathbb{C}}^{*}$ on $E\boxtimes E^{*}$, induced by the scalar
multiplications on $E$, is the trivial action. The restriction of the vector
bundle ${\mathcal{E}}\,\longrightarrow\,X\times X\times{\mathcal{M}}$ to

$\Delta\times{\mathcal{M}}\,\subset\,X\times X\times{\mathcal{M}}$

coincides with the universal endomorphism bundle over $X\times{\mathcal{M}}$.
Let

${\mathcal{V}}\,:=\,{\mathcal{E}}\big{|}_{\Delta\times{\mathcal{M}}}\,\longrightarrow\,\Delta\times{\mathcal{M}}\,=\,X\times{\mathcal{M}}$
(4.4)

be the universal endomorphism bundle. So we have
${\mathcal{V}}\big{|}_{X\times\\{E\\}}\,=\,\text{End}(E)$ for all
$E\,\in\,\mathcal{M}$.

Let $q_{12}\,:\,X\times X\times{\mathcal{M}}\,\longrightarrow\,X\times X$ be
the projection to the first two factors in the Cartesian product. Let

$q_{2}\,:\,X\times X\times{\mathcal{M}}\,\longrightarrow\,X\,,\ \
(x,\,y,\,E)\,\longmapsto\,y$

be the projection to the second factor. Let

$J\,:\,X\times X\times{\mathcal{M}}\,\longrightarrow\,{\mathcal{M}}\,,\ \
(x,\,y,\,E)\,\longmapsto\,E$ (4.5)

be the projection to the third factor. For notational convenience, the
holomorphic line bundle
$q_{12}^{*}((p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X}))$ on $X\times
X\times{\mathcal{M}}$ will be denoted by $\mathcal{K}$; recall that
$K^{1/2}_{X}$ is a theta characteristic on $X$.

Consider the vector bundle

${\mathcal{E}}\otimes{\mathcal{K}}\otimes q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta)\,\longrightarrow\,X\times X\times{\mathcal{M}}\,.$ (4.6)

It fits in the following short exact sequence of coherent sheaves on $X\times
X\times{\mathcal{M}}$:

$0\,\longrightarrow\,{\mathcal{E}}\otimes{\mathcal{K}}\,\longrightarrow\,{\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta)\,\longrightarrow\,{\mathcal{V}}\,\longrightarrow\,0\,,$ (4.7)

where $\mathcal{V}$ is defined in (4.4), and it is supported on
$\Delta\times{\mathcal{M}}\,=\,X\times{\mathcal{M}}\,\subset\,X\times
X\times{\mathcal{M}}$. Recall from Section 3.1 that the restriction of
$(p^{*}_{1}K^{1/2}_{X})\otimes(p^{*}_{2}K^{1/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(\Delta)$ to $\Delta\,\subset\,X\times X$ is identified with
${\mathcal{O}}_{X}$; so, the restriction of
${\mathcal{E}}\otimes{\mathcal{K}}\otimes q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta)$ to $\Delta\times\mathcal{M}$ is identified with
${\mathcal{E}}\big{|}_{\Delta\times{\mathcal{M}}}\,=\,\mathcal{V}$. Now
consider the long exact sequence of direct images, for the projection $J$ in
(4.5), corresponding to the short exact sequence of sheaves in (4.7):

$0\,\longrightarrow\,J_{*}({\mathcal{E}}\otimes{\mathcal{K}})\,\longrightarrow\,J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\,\longrightarrow\,J_{*}{\mathcal{V}}$ (4.8)
$\longrightarrow\,R^{1}J_{*}({\mathcal{E}}\otimes{\mathcal{K}})\,\longrightarrow\,\ldots\,.$

First note that $J_{*}({\mathcal{E}}\otimes{\mathcal{K}})\,=\,0$, because for
every $E\,\in\,{\mathcal{M}}^{0}$, we have

$H^{0}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes K^{1/2}_{X})))\,=\,0$

(see (3.5)). Also, $J_{*}{\mathcal{V}}\,=\,{\mathcal{O}}_{\mathcal{M}}$,
because every stable vector bundle is simple. Consequently, from (4.8) we have
the exact sequence

$0\,\longrightarrow\,J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\,\longrightarrow\,{\mathcal{O}}_{\mathcal{M}}\longrightarrow\,R^{1}J_{*}({\mathcal{E}}\otimes{\mathcal{K}})\,.$
(4.9)

Next we note that

$H^{1}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes K^{1/2}_{X})))\,=\,0$

for all $E\,\in\,{\mathcal{M}}^{0}$ (see (3.5)). Also, for a general point
$E\,\in\,D_{\Theta}$, using (3.4) it follows that

$\dim H^{1}(X\times X,\,(p^{*}_{1}(E\otimes
K^{1/2}_{X}))\otimes(p^{*}_{2}(E^{*}\otimes K^{1/2}_{X})))\,=\,1\,.$

Consequently, the support of $R^{1}J_{*}({\mathcal{E}}\otimes{\mathcal{K}})$
is the divisor $D_{\Theta}$, and the rank of the sheaf

$R^{1}J_{*}({\mathcal{E}}\otimes{\mathcal{K}})\,\longrightarrow\,D_{\Theta}$

is one.

Let $1_{\mathcal{M}}$ be the section of ${\mathcal{O}}_{\mathcal{M}}$ given by
the constant function $1$ on $\mathcal{M}$. Since
$R^{1}J_{*}({\mathcal{E}}\otimes{\mathcal{K}})$ is supported on $D_{\Theta}$,
from (4.9) we conclude the following:

  * •

The restriction $1_{\mathcal{M}}$ to
${\mathcal{M}}^{0}\,=\,{\mathcal{M}}\setminus
D_{\Theta}\,\subset\,{\mathcal{M}}$ is a holomorphic section of

$(J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta)))\big{|}_{{\mathcal{M}}^{0}}\,\longrightarrow\,{\mathcal{M}}^{0}$

(more precisely, $1_{\mathcal{M}}\big{|}_{{\mathcal{M}}^{0}}$ is the image of
a holomorphic section of $(J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times X}(\Delta)))\big{|}_{{\mathcal{M}}^{0}}$);
this section of $(J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times X}(\Delta)))\big{|}_{{\mathcal{M}}^{0}}$
given by $1_{\mathcal{M}}$ will be denoted by $1^{\prime}_{\mathcal{M}}$.

  * •

The above defined $1^{\prime}_{\mathcal{M}}$ is a meromorphic section of
$J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times X}(\Delta))$ with a pole of order one on
$D_{\Theta}$.

In other words, we have

$1^{\prime}_{\mathcal{M}}\,\in\,H^{0}({\mathcal{M}},\,J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\otimes{\mathcal{O}}_{\mathcal{M}}(D_{\Theta}))\,.$ (4.10)

Now using the projection formula we have

$J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\otimes{\mathcal{O}}_{\mathcal{M}}(D_{\Theta})\,=\,J_{*}({\mathcal{E}}\otimes{\mathcal{K}}\otimes(q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\otimes J^{*}{\mathcal{O}}_{\mathcal{M}}(D_{\Theta}))\,,$

and hence $1^{\prime}_{\mathcal{M}}$ in (4.10) defines a section

$1^{\prime\prime}_{\mathcal{M}}\,\in\,H^{0}(X\times
X\times{\mathcal{M}},\,{\mathcal{E}}\otimes{\mathcal{K}}\otimes(q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\otimes J^{*}{\mathcal{O}}_{\mathcal{M}}(D_{\Theta}))\,.$ (4.11)

For every $E\,\in\,{\mathcal{M}}^{0}$, the section $\beta_{E}$ in (3.6)
coincides with the restriction $1^{\prime\prime}_{\mathcal{M}}\big{|}_{X\times
X\times\\{E\\}}$, where $1^{\prime\prime}_{\mathcal{M}}$ is the section in
(4.11). Now from the construction in (3.13) of the section $\phi$ of the
$T^{*}{\mathcal{M}}^{0}$–torsor ${\mathcal{C}}^{0}$ it follows that

  1. (1)

$\phi$ is meromorphic, and

  2. (2)

the order of pole, at $D_{\Theta}$, of $\phi$ is one.

As noted before, Proposition 4.2 follows from these two. ∎

Continuing with the proof Theorem 4.1, let

$\Psi\,\,:\,\,{\mathcal{M}}\,\longrightarrow\,J(X)\,=\,\text{Pic}^{0}(X)$
(4.12)

be the determinant map $E\,\longmapsto\,\bigwedge^{r}E$. The image of the
pullback homomorphism

$(d\Psi)^{*}\,\,:\,\,\Psi^{*}T^{*}J(X)\,\longrightarrow\,T^{*}{\mathcal{M}}\,,$

where $d\Psi$ is the differential of $\Psi$, has a canonical direct summand;
we shall now recall a description of this direct summand.

As in (4.4), let ${\mathcal{V}}\,\longrightarrow\,X\times{\mathcal{M}}$ be the
universal endomorphism bundle, and let

${\mathcal{V}}^{0}\,\subset\,\mathcal{V}$

be the universal endomorphism bundle of trace zero. There is a natural
decomposition into traceless and trace components:

${\mathcal{V}}\,=\,{\mathcal{V}}^{0}\oplus{\mathcal{O}}_{X\times\mathcal{M}}\,;$
(4.13)

the above inclusion map
${\mathcal{O}}_{X\times\mathcal{M}}\,\hookrightarrow\,\mathcal{V}$ is defined
by $f\,\longmapsto\,f\cdot\text{Id}$. Let

$P\,:\,X\times{\mathcal{M}}\,\longrightarrow\,{\mathcal{M}}\ \ \text{ and
}p\,:\,X\times{\mathcal{M}}\,\longrightarrow\,X$ (4.14)

be the natural projections. Then we have

$T^{*}{\mathcal{M}}\,=\,P_{*}({\mathcal{V}}\otimes p^{*}K_{X})\,,$

where $P$ and $p$ are the projections in (4.14). Consequently, the
decomposition in (4.13) produces a holomorphic decomposition

$T^{*}{\mathcal{M}}\,=\,P_{*}({\mathcal{V}}^{0}\otimes p^{*}K_{X})\oplus
P_{*}p^{*}K_{X}\,;$ (4.15)

we note that $P_{*}p^{*}K_{X}$ is the trivial holomorphic vector bundle

${\mathcal{M}}\times H^{0}(X,\,K_{X})\,\longrightarrow\,\mathcal{M}$

with fiber $H^{0}(X,\,K_{X})$. Tensoring (4.15) with $\Theta$ we obtain

$(T^{*}{\mathcal{M}})\otimes\Theta\,=\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta\oplus({\mathcal{M}}\times
H^{0}(X,\,K_{X}))\otimes\Theta\,.$

This produces a decomposition

$H^{0}({\mathcal{M}},\,(T^{*}{\mathcal{M}})\otimes\Theta)\,=\,H^{0}({\mathcal{M}},\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta)\oplus(H^{0}({\mathcal{M}},\,\Theta)\otimes
H^{0}(X,\,K_{X}))$ $=\,H^{0}({\mathcal{M}},\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta)\oplus H^{0}(X,\,K_{X})\,;$ (4.16)

the last equality follows from the fact that
$H^{0}({\mathcal{M}},\,\Theta)\,=\,\mathbb{C}$ [BNR, p. 169, Theorem 2]. We
note that the inclusion map

$H^{0}(X,\,K_{X})\,=\,H^{0}(J(X),\,T^{*}J(X))\,\hookrightarrow\,H^{0}({\mathcal{M}},\,(T^{*}{\mathcal{M}})\otimes\Theta)$

in (4.16) coincides with the pullback of $1$–forms on $J(X)$ to
${\mathcal{M}}$ by the projection $\Psi$ in (4.12).

The following proposition would be used in the proof of Theorem 4.1.

######  Proposition 4.3.

For the projections $P$ and $p$ in (4.14),

$H^{0}({\mathcal{M}},\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta)\,=\,0\,,$

where ${\mathcal{V}}^{0}$ is the subbundle in (4.13).

###### Proof of Proposition 4.3.

If $r\,=\,1$, then ${\mathcal{V}}^{0}\,=\,0$, and hence in this case the
proposition is obvious. Hence in the proof we assume that $r\,\geq\,2$. The
proof proceeds by showing that the sections must vanish on the projective
spaces lying inside ${\mathcal{M}}$ given by Hecke transforms.

Let

$H\,:\,{\mathbb{P}}\,\longrightarrow\,X\times{\mathcal{M}}$ (4.17)

be the universal projective bundle; so for any
$(x,\,E)\,\in\,X\times{\mathcal{M}}$, the inverse image $H^{-1}(x,\,E)$ is the
space of all hyperplanes in the fiber $E_{x}$; in particular, $\mathbb{P}$ is
a holomorphic fiber bundle over $X\times{\mathcal{M}}$ with the projective
space ${\mathbb{C}}{\mathbb{P}}^{r-1}$ as the typical fiber. Let

$T_{H}\,=\,\text{kernel}(dH)\,\longrightarrow\,{\mathbb{P}}$

be the (holomorphic) relative tangent bundle for the projection $H$ in (4.17),
where $dH$ is the differential of the map $H$. We note that
${\mathcal{V}}^{0}$ in (4.13) is the following direct image:

$H_{*}T_{H}\,=\,{\mathcal{V}}^{0}\,\longrightarrow\,X\times{\mathcal{M}}\,.$

Given any element $(x,\,E)\,\in\,X\times\mathcal{M}$, along with a hyperplane
$S\,\subset\,E_{x}$, let $F(x,E,S)$ be the holomorphic vector bundle over $X$
whose sheaf of sections fits in the short exact sequence of coherent sheaves
on $X$

$0\,\longrightarrow\,F(x,E,S)\,\longrightarrow\,E\,\longrightarrow\,E_{x}/S\,\longrightarrow\,0\,;$

the above sheaf $E_{x}/S$ is the torsion sheaf supported at the point $x$ and
its stalk at $x$ is the quotient line $E_{x}/S$.

Let $\mathcal{N}$ denote the moduli space of stable vector bundles over $X$ of
rank $r$ and degree $-1$. Using the above construction of $F(x,E,S)$, we get a
rational map

$\xi\,:\,{\mathbb{P}}\,\dasharrow\,X\times\mathcal{N}\,,\ \
(x,\,E,\,S)\,\longmapsto\,(x,\,F(x,E,S))\,,$

which is called the Hecke morphism [NR1], [NR2]. It is known that there is a
nonempty Zariski open subset

${\mathcal{U}}\,\subset\,{\mathbb{P}}$

such that the pair $(\xi,\,{\mathcal{U}})$ satisfies the following conditions:

  1. (1)

The rational map $\xi$ is actually defined as a map on $\mathcal{U}$; the
restriction of $\xi$ to $\mathcal{U}$ will be denoted by $\widehat{\xi}$.

  2. (2)

The codimension of the complement ${\mathbb{P}}\setminus\mathcal{U}$ is at
least two (see the proof of [NR2, Proposition 5.4]).

  3. (3)

The map $\widehat{\xi}\,:\,{\mathcal{U}}\,\longrightarrow\,\xi({\mathcal{U}})$
defines a holomorphic fiber bundle over $\xi({\mathcal{U}})$ with the
projective space ${\mathbb{C}}{\mathbb{P}}^{r-1}$ as the typical fiber [NR2,
p. 411, Proposition 6.8].

  4. (4)

The relative tangent bundle $T_{H}$ on $\mathcal{U}$ coincides with
$\Omega_{\widehat{\xi}}\otimes(p\circ H)^{*}K_{X}$, where
$\Omega_{\widehat{\xi}}$ is the relative cotangent bundle for the map
$\widehat{\xi}$ (see [Bi1, p. 265, (2.7)]).

It may be clarified that the above vector bundle $\Omega_{\widehat{\xi}}$ is
the cokernel of the pullback homomorphism

$d(\widehat{\xi})^{*}\,:\,\widehat{\xi}^{*}T^{*}(\xi({\mathcal{U}}))\,\longrightarrow\,T^{*}{\mathcal{U}}\,.$

Take a point

$z\,=\,(x,\,W)\,\in\,\widehat{\xi}({\mathcal{U}})\,\subset\,X\times\mathcal{N}\,.$
(4.18)

Let

${\mathbb{F}}_{z}\,=\,\widehat{\xi}^{-1}(z)\,\subset\,{\mathcal{U}}$

be the fiber of $\widehat{\xi}$ over $z$; as mentioned in (3) above, this
fiber is isomorphic to ${\mathbb{C}}{\mathbb{P}}^{r-1}$. We will compute the
restriction of the line bundle $(P\circ H)^{*}\Theta$ to
${\mathbb{F}}_{z}\,\cong\,{\mathbb{C}}{\mathbb{P}}^{r-1}$, where $P$ and $H$
are the projections in (4.14) and (4.17) respectively.

Let $P(W_{x})$ be the projective space that parametrizes all the lines in the
fiber $W_{x}$ of the vector bundle $W$ in (4.18). Let
$L_{0}\,\longrightarrow\,P(W_{x})$ be the tautological line bundle of degree
$-1$; the fiber of $L_{0}$ over any line $\zeta\,\subset\,W_{x}$ is $\zeta$
itself. The inverse image ${\mathbb{F}}_{z}\,=\,\widehat{\xi}^{-1}(z)$ is
identified with this projective space $P(W_{x})$. For any
$\zeta\,\in\,P(W_{x})$, the corresponding element
$(x,\,E,\,S)\,\in\,{\mathbb{F}}_{z}\,\subset\,{\mathbb{P}}$ is uniquely
determined by the following condition: The holomorphic vector bundle $E$ fits
in the short exact sequence of sheaves on $X$

$0\,\longrightarrow\,W\,\longrightarrow\,E\,\longrightarrow\,{\mathcal{Q}}\,\longrightarrow\,0\,,$

where $\mathcal{Q}$ is a torsion sheaf of degree one supported at $x$, and the
kernel of the homomorphism of fibers $W_{x}\,\longrightarrow\,E_{x}$, given by
the above homomorphism $W\,\longrightarrow\,E$ of sheaves, is the line
$\zeta$, while the subspace $S\,\subset\,E_{x}$ is the image of this
homomorphism $W_{x}\,\longrightarrow\,E_{x}$.

To describe the fiber ${\mathbb{F}}_{z}$ globally, let $\Pi_{1}$
(respectively, $\Pi_{2}$) be the projection of $X\times P(W_{x})$ to $X$
(respectively, $P(W_{x})$). On $X\times P(W_{x})$ we have the holomorphic
vector bundle ${\mathcal{W}}$ which is defined by the short exact sequence of
sheaves on $X\times P(W_{x})$

$0\,\longrightarrow\,{\mathcal{W}}^{*}\,\longrightarrow\,\Pi^{*}_{1}W^{*}\,\longrightarrow\,\iota^{x}_{*}L^{*}_{0}\,\longrightarrow\,0\,,$

where $\iota^{x}\,:\,P(W_{x})\,\longrightarrow\,X\times P(W_{x})$ is the
embedding defined by $y\,\longmapsto\,(x,\,y)$. From this exact sequence it
follows that $\mathcal{W}$ fits in the short exact sequence of sheaves

$0\,\longrightarrow\,\Pi^{*}_{1}W\,\longrightarrow\,{\mathcal{W}}\,\longrightarrow\,\iota^{x}_{*}L_{0}\,\longrightarrow\,0$
(4.19)

on $X\times P(W_{x})$. The map $H\big{|}_{{\mathbb{F}}_{z}}$ coincides with
the classifying map

${\mathbb{F}}_{z}\,\longrightarrow\,{\mathcal{U}}\,\subset\,{\mathcal{M}}$

for the above holomorphic family of vector bundles ${\mathcal{W}}$ on $X$
parametrized by $P(W_{x})\,=\,{\mathbb{F}}_{z}$.

Now, tensoring the exact sequence in (4.19) with $\Pi^{*}_{1}K^{1/2}_{X}$, and
then taking the long exact sequence of direct images with respect to the
projection $\Pi_{2}$, we have the exact sequence of sheaves on $P(W_{x})$

$0\,\longrightarrow\,\Pi_{2*}(\Pi^{*}_{1}(W\otimes
K^{1/2}_{X}))\,\longrightarrow\,\Pi_{2*}({\mathcal{W}}\otimes\Pi^{*}_{1}K^{1/2}_{X})\,\longrightarrow\,L_{0}$
$\,\longrightarrow\,R^{1}\Pi_{2*}(\Pi^{*}_{1}(W\otimes
K^{1/2}_{X}))\,\longrightarrow\,R^{1}\Pi_{2*}({\mathcal{W}}\otimes\Pi^{*}_{1}K^{1/2}_{X})\,\longrightarrow\,0\,;$
(4.20)

note that the restriction of $\Pi^{*}_{1}K^{1/2}_{X}$ to the image of the
embedding $\iota^{x}$ is a trivial line bundle, and also note that

$R^{1}\Pi_{2*}((\iota^{x}_{*}L_{0})\otimes\Pi^{*}_{1}K^{1/2}_{X})\,=\,0$

because the support of $(\iota^{x}_{*}L_{0})\otimes\Pi^{*}_{1}K^{1/2}_{X}$ is
finite over $P(W_{x})$.

Since $H\big{|}_{{\mathbb{F}}_{z}}$ coincides with the classifying map for the
above holomorphic family of vector bundles ${\mathcal{W}}$ on $X$ parametrized
by $P(W_{x})$, it follows that the pulled back line bundle $((P\circ
H)^{*}\Theta)\big{|}_{{\mathbb{F}}_{z}}$, where $P$ is the projection in
(4.14), is identified with the line bundle

$((P\circ
H)^{*}\Theta)\big{|}_{{\mathbb{F}}_{z}}\,=\,\det(\Pi_{2*}({\mathcal{W}}\otimes\Pi^{*}_{1}K^{1/2}_{X}))^{*}\otimes\det(R^{1}\Pi_{2*}({\mathcal{W}}\otimes\Pi^{*}_{1}K^{1/2}_{X}))$
(4.21)

(see [Ko, Ch. V, § 6] for the construction of determinant bundle). For any
exact sequence of coherent sheaves

$0\,\longrightarrow\,A_{1}\,\longrightarrow\,A_{2}\,\longrightarrow\,\ldots\,\longrightarrow\,A_{m}\,\longrightarrow\,0$

on a complex manifold $Y$, we have
$\bigotimes_{i=1}^{m}(\det(A_{i}))^{(-1)^{i}}\,=\,{\mathcal{O}}_{Y}$ [Ko, p.
165, Proposition (6.9)]. Consequently, from (4.20) and (4.21) we conclude that

$((P\circ H)^{*}\Theta)\big{|}_{{\mathbb{F}}_{z}}\,=\,L^{*}_{0}\,,$

because both $\Pi_{2*}(\Pi^{*}_{1}(W\otimes K^{1/2}_{X}))$ and
$R^{1}\Pi_{2*}(\Pi^{*}_{1}(W\otimes K^{1/2}_{X}))$ are trivial vector bundles.
In other words, the degree of the line bundle $(P\circ H)^{*}\Theta$
restricted to ${\mathbb{F}}_{z}\,=\,P(W_{x})$ is $1$.

Using the above properties of $(\xi,\,{\mathcal{U}})$ we are in a position to
complete the proof of the proposition.

We have

$H^{0}({\mathcal{M}},\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta)\,=\,H^{0}(X\times{\mathcal{M}},\,{\mathcal{V}}^{0}\otimes(p^{*}K_{X})\otimes(P^{*}\Theta))\,,$
(4.22)

because
$P_{*}({\mathcal{V}}^{0}\otimes(p^{*}K_{X})\otimes(P^{*}\Theta))\,=\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta$ by the projection formula. Next we have

$H^{0}(X\times{\mathcal{M}},\,{\mathcal{V}}^{0}\otimes(p^{*}K_{X})\otimes(P^{*}\Theta))\,=\,H^{0}({\mathbb{P}},\,T_{H}\otimes((p\circ
H)^{*}K_{X})\otimes(P\circ H)^{*}\Theta)$
$=\,H^{0}({\mathcal{U}},\,T_{H}\otimes((p\circ H)^{*}K_{X})\otimes(P\circ
H)^{*}\Theta)\,;$ (4.23)

the first equality follows from the fact that $H_{*}(T_{H}\otimes((p\circ
H)^{*}K_{X})\otimes(P\circ
H)^{*}\Theta)\,=\,{\mathcal{V}}^{0}\otimes(p^{*}K_{X})\otimes(P^{*}\Theta)$
(by the projection formula), and the second equality follows from the fact
that the codimension of the complement ${\mathbb{P}}\setminus\mathcal{U}$ is
at least two.

As before, take a fiber ${\mathbb{F}}_{z}\,=\,\widehat{\xi}^{-1}(z)$ of the
map $\widehat{\xi}$. As shown above, ${\mathbb{F}}_{z}$ is identified with the
projective space $P(W_{x})$, and the restriction of $T_{H}$ (respectively,
$(P\circ H)^{*}\Theta$) to ${\mathbb{F}}_{z}$ is isomorphic to
$T^{*}{\mathbb{F}}_{z}$ (respectively, ${\mathcal{O}}_{{\mathbb{F}}_{z}}(1)$);
note that the restriction of $(p\circ H)^{*}K_{X}$ to $\widehat{\xi}^{-1}(z)$
is a trivial line bundle. Consequently, the restriction of
$T_{H}\otimes((p\circ H)^{*}K_{X})\otimes(P\circ H)^{*}\Theta$ to
${\mathbb{F}}_{z}$ is isomorphic to
$T^{*}{\mathbb{F}}_{z}\otimes{\mathcal{O}}_{{\mathbb{F}}_{z}}(1)$.

Next we note that the holomorphic vector bundle
$T^{*}{\mathbb{F}}_{z}\otimes{\mathcal{O}}_{{\mathbb{F}}_{z}}(1)$ on the
projective space ${\mathbb{F}}_{z}\,=\,P(W_{x})$ is semistable of negative
degree (its degree is $-1$), and hence the vector bundle
$T^{*}{\mathbb{F}}_{z}\otimes{\mathcal{O}}_{{\mathbb{F}}_{z}}(1)$ does not
have any nonzero holomorphic section. This implies that

$H^{0}({\mathcal{U}},\,T_{H}\otimes((p\circ H)^{*}K_{X})\otimes(P\circ
H)^{*}\Theta)\,=\,0\,.$

Consequently, from (4.23) and (4.22) we now conclude that

$H^{0}({\mathcal{M}},\,P_{*}({\mathcal{V}}^{0}\otimes
p^{*}K_{X})\otimes\Theta)\,=\,0\,.$

This completes the proof of Proposition 4.3. ∎

We continue with the proof of Theorem 4.1. Combining (4.16) with Proposition
4.3, it follows that we are reduced to the trace component:

$H^{0}({\mathcal{M}},\,(T^{*}{\mathcal{M}})\otimes\Theta)\,=\,\\{\Psi^{*}\omega\,\mid\,\omega\,\in\,H^{0}(J(X),\,T^{*}J(X))\\}\,=\,H^{0}(X,\,K_{X})\,,$
(4.24)

where $\Psi$ is the projection in (4.12).

Let

$\Gamma^{\prime}\,\in\,H^{0}(J(X),\,T^{*}J(X))\,=\,H^{0}(X,\,K_{X})$ (4.25)

be the $1$–form corresponding to the section $\Gamma$ in Proposition 4.2 for
the isomorphism in (4.24).

The proof of Theorem 4.1 will be completed using the following lemma.

######  Lemma 4.4.

The $1$–form $\Gamma^{\prime}$ on $J(X)$ in (4.25) is invariant under the
holomorphic involution

$\iota_{J}\,:\,J(X)\,\longrightarrow\,J(X)$

defined by $L\,\longmapsto\,L^{*}$.

###### Proof of Lemma 4.4.

Let $\iota_{\mathcal{M}}\,:\,{\mathcal{M}}\,\longrightarrow\,{\mathcal{M}}$ be
the holomorphic involution defined by $E\,\longmapsto\,E^{*}$. Note that

$\iota_{J}\circ\Psi\,=\,\Psi\circ\iota_{\mathcal{M}}\,,$

where $\Psi$ is constructed in (4.12) and $\iota_{J}$ is defined in the
statement of the lemma. By Serre duality,

$H^{k}(X,\,E^{*}\otimes K^{1/2}_{X})\,=\,H^{1-k}(X,\,E\otimes
K^{1/2}_{X})^{*}$ (4.26)

for $k\,=\,0,\,1$. This implies that the above involution
$\iota_{\mathcal{M}}$ preserves the divisor $D_{\Theta}$ defined in (2.4).
Since $D_{\Theta}$ is preserved by $\iota_{\mathcal{M}}$, the involution
$\iota_{\mathcal{M}}$ has a tautological lift to the line bundle
$\Theta\,=\,{\mathcal{O}}_{\mathcal{M}}(D_{\Theta})$. Let

$\iota_{\Theta}\,\,:\,\,\Theta\,\longrightarrow\,\Theta$ (4.27)

be the resulting involution of $\Theta$ over the involution
$\iota_{\mathcal{M}}$ of $\mathcal{M}$. This involution $\iota_{\Theta}$ of
$\Theta$ produces a holomorphic involution of the complex manifold ${\rm
Conn}(\Theta)$ constructed in (2.7). The involution of ${\rm Conn}(\Theta)$
constructed this way will be denoted by $\iota_{T}$. We note that

${\iota}_{\mathcal{M}}\circ q\,=\,q\circ\iota_{T}\,,$

where $q$ is the projection in (2.7). This implies that the involution
$\iota_{T}$ preserves the open subset ${\rm
Conn}(\Theta)^{0}\,=\,q^{-1}({\mathcal{M}}\setminus D_{\Theta})$ in (3.14).

The involution $\iota_{T}$ (respectively, ${\iota}_{\mathcal{M}}$) defines an
action of ${\mathbb{Z}}/2\mathbb{Z}$ on ${\rm Conn}(\Theta)$ (respectively,
${\mathcal{M}}$). The section $\tau$ in (3.15) is evidently
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, for the actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\rm Conn}(\Theta)^{0}$ and
${\mathcal{M}}^{0}$.

For any $E\,\in\,\mathcal{M}$, the fiber $\Theta_{E}$ of $\Theta$ over $E$ is
the line $\bigwedge^{\rm top}H^{0}(X,\,E\otimes
K^{1/2}_{X})^{*}\otimes\bigwedge^{\rm top}H^{1}(X,\,E\otimes K^{1/2}_{X})$.
Using (4.26) we get an isomorphism of $\Theta_{E}$ with the fiber
$\Theta_{E^{*}}$. Also, the involution $\iota_{\Theta}$ of $\Theta$ in (4.27)
produces an isomorphism of $\Theta_{E}$ with $\Theta_{E^{*}}$. These two
isomorphisms between $\Theta_{E}$ and $\Theta_{E^{*}}$ actually coincide.

The Kähler form $\omega_{\mathcal{M}}$ on $\mathcal{M}$ (see (2.8)) is clearly
preserved by the involution $\iota_{\mathcal{M}}$ of $\mathcal{M}$. From this
it can be deduced that the section $\psi_{Q}$ in (2.9) is
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, for the above actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\rm Conn}(\Theta)$ and ${\mathcal{M}}$.
Indeed, the section $\psi_{Q}$ corresponds to the unique Hermitian connection
on $\Theta$ whose curvature is the Kähler form $\omega_{\mathcal{M}}$. In
other words, the section $\psi_{Q}$ is uniquely determined by
$\omega_{\mathcal{M}}$. Therefore, the section $\psi_{Q}$ in (2.9) is
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, because $\omega_{\mathcal{M}}$ is
preserved by $\iota_{\mathcal{M}}$.

Given a holomorphic connection $\nabla$ on a holomorphic vector bundle $E$,
the dual vector bundle $E^{*}$ is equipped with the dual connection
$\nabla^{*}$. Therefore, we have a holomorphic involution

$\iota_{\mathcal{C}}\,:\,{\mathcal{C}}\,\longrightarrow\,{\mathcal{C}}\,,\ \
(E,\,\nabla)\,\longmapsto\,(E^{*},\,\nabla^{*})\,.$

The involution $\iota_{\mathcal{C}}$ gives an action of
${\mathbb{Z}}/2\mathbb{Z}$ on $\mathcal{C}$. The projection $\varphi$ in (2.1)
is clearly ${\mathbb{Z}}/2\mathbb{Z}$–equivariant, for the actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\mathcal{C}}$ and ${\mathcal{M}}$. In
particular, $\iota_{\mathcal{C}}$ preserves the Zariski open subset
${\mathcal{C}}^{0}$ in (3.12). Since the dual of a unitary connection on $E$
is a unitary connection on $E^{*}$, the section $\psi_{U}$ in (2.2) is
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, for the actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\mathcal{C}}$ and ${\mathcal{M}}$.

Let $\widehat{\iota}\,:\,X\times
X\times{\mathcal{M}}\,\longrightarrow\,X\times X\times{\mathcal{M}}$ be the
holomorphic involution defined by
$(x,\,y,\,E)\,\longmapsto\,(y,\,x,\,\iota_{\mathcal{M}}(E))\,=\,(y,\,x,\,E^{*})$.
This involution naturally lifts to an involution of the vector bundle
${\mathcal{E}}\otimes{\mathcal{K}}\otimes q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta)$ in (4.6). The earlier mentioned involution $\iota_{\Theta}$ of the
line bundle $\Theta$ produces an action of ${\mathbb{Z}}/2\mathbb{Z}$ on the
pullback $J^{*}\Theta$, where $J$ is the projection in (4.5). These actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\mathcal{E}}\otimes{\mathcal{K}}\otimes
q^{*}_{12}{\mathcal{O}}_{X\times X}(\Delta)$ and $J^{*}\Theta$ together
produce an action of ${\mathbb{Z}}/2\mathbb{Z}$ on the tensor product

${\mathcal{E}}\otimes{\mathcal{K}}\otimes(q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\otimes J^{*}\Theta\,\longrightarrow\,X\times
X\times{\mathcal{M}}\,.$

The section $1^{\prime\prime}_{\mathcal{M}}$ in (4.11) of this tensor product
is anti-invariant for the above action of ${\mathbb{Z}}/2\mathbb{Z}$ on
${\mathcal{E}}\otimes{\mathcal{K}}\otimes(q^{*}_{12}{\mathcal{O}}_{X\times
X}(\Delta))\otimes J^{*}\Theta$ (meaning the nontrivial element of
${\mathbb{Z}}/2\mathbb{Z}$ acts as multiplication by $-1$). From this it
follows that the section $\phi$ in (3.13) is
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, for the actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\mathcal{C}}^{0}$ and ${\mathcal{M}}^{0}$.

From all these it follows that $\Gamma_{0}$ (constructed in (4.1)) is
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, for the actions of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\mathcal{C}}^{0}$ and
$T^{*}{\mathcal{M}}^{0}$; the action of ${\mathbb{Z}}/2\mathbb{Z}$ on
$T^{*}{\mathcal{M}}^{0}$ is induced by the action of
${\mathbb{Z}}/2\mathbb{Z}$ on ${\mathcal{M}}^{0}$ constructed using the above
involution $\iota_{\mathcal{M}}$. Since $\Gamma_{0}$ is
${\mathbb{Z}}/2\mathbb{Z}$–equivariant, it follows that $\Gamma$ in
Proposition 4.2 is ${\mathbb{Z}}/2{\mathbb{Z}}$–invariant for the action on
$(T^{*}{\mathcal{M}})\otimes\Theta$ constructed using the actions of
${\mathbb{Z}}/2{\mathbb{Z}}$ on $T^{*}\mathcal{M}$ and $\Theta$ (given by
$\iota_{\Theta}$ in (4.27)). This immediately implies that $\Gamma^{\prime}$
in (4.25) is left invariant under the involution $\iota_{J}$ of $J(X)$. This
completes the proof of Lemma 4.4. ∎

Continuing with the proof of Theorem 4.1, we note that
$\iota^{*}_{J}\alpha\,=\,-\alpha$ for all
$\alpha\,\in\,H^{0}(J(X),\,T^{*}J(X))$, where $\iota_{J}$ is the involution in
Lemma 4.4. Hence from Lemma 4.4 it follows immediately that
$\Gamma^{\prime}\,=\,0$. In view of (4.24), this implies that $\Gamma$ in
Proposition 4.2 vanishes identically. Hence $\Gamma_{0}$ in (4.1) vanishes
identically. Therefore, we conclude that the restriction, to the open subset
${\mathcal{C}}^{0}\,\subset\,\mathcal{C}$, of the isomorphism $F$ in Theorem
2.1 coincides with the isomorphism $G$ in Lemma 3.1. This completes the proof
of Theorem 4.1. ∎

Theorem 2.1 and Theorem 4.1 together give the following:

######  Corollary 4.5.

The holomorphic isomorphism $G\,:\,{\mathcal{C}}^{0}\,\longrightarrow\,{\rm
Conn}(\Theta)^{0}$ in Lemma 3.1 extends to a holomorphic isomorphism

$G^{\prime}\,:\,{\mathcal{C}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\rm
Conn}(\Theta)\,.$

###### Proof.

Since $F$ in Theorem 2.1 is a holomorphic isomorphism from ${\mathcal{C}}$ to
${\rm Conn}(\Theta)$, this follows from Theorem 4.1. ∎

The isomorphism $G^{\prime}$ in Corollary 4.5 has the following property:

######  Corollary 4.6.

For the isomorphism $G^{\prime}$ in Corollary 4.5,

$(G^{\prime})^{*}\Phi_{2}\,=\,2r\cdot\Phi_{1}\,,$

where $\Phi_{1}$ and $\Phi_{2}$ are the symplectic forms in (2.3) and (2.10)
respectively.

###### Proof.

The final part of Theorem 2.1 says that $F^{*}\Phi_{2}\,=\,2r\cdot\Phi_{1}$.
Since $F\,=\,G^{\prime}$, the result follows from this. ∎

######  Corollary 4.7.

The image of the section $\phi$ in (3.13) is a Lagrangian submanifold of
${\mathcal{C}}^{0}$ equipped with the symplectic form
$\Phi_{1}\big{|}_{{\mathcal{C}}^{0}}$ in (2.3).

###### Proof.

In Corollary 4.6 we saw that $G^{\prime}$ is symplectic structure preserving
(up to the factor $2r$). The image of the section $\tau$ in (3.15) is clearly
a Lagrangian submanifold of $\text{Conn}(\Theta)^{0}$ with respect to the
symplectic form $\Phi_{2}\big{|}_{{\rm Conn}(\Theta)^{0}}$ in (2.10) (the
trivial connection is flat). Since
$G(\phi({\mathcal{M}}^{0}))\,=\,\tau({\mathcal{M}}^{0})$, and $G$ is
symplectic structure preserving up to the factor $2r$, from the above
observation — that the image of $\tau$ is a Lagrangian submanifold of
$\text{Conn}(\Theta)^{0}$ with respect to the symplectic form
$\Phi_{2}\big{|}_{{\rm Conn}(\Theta)^{0}}$ — it follows immediately that
$\phi({\mathcal{M}}^{0})$ is a Lagrangian submanifold of ${\mathcal{C}}^{0}$
with respect to the symplectic form $\Phi_{1}\big{|}_{{\mathcal{C}}^{0}}$ in
(2.3). ∎

##  5\. Family of Riemann surfaces

Let $\mathcal{T}$ be a connected complex manifold, and let

$F\,:\,{\mathcal{X}}_{\mathcal{T}}\,\longrightarrow\,\mathcal{T}$

be a holomorphic family of compact connected Riemann surfaces of genus $g$,
with $g\,\geq\,2$, parametrized by $\mathcal{T}$, and equipped with a theta
characteristic ${\mathbb{L}}$. This means that ${\mathbb{L}}$ is a holomorphic
line bundle over ${\mathcal{X}}_{\mathcal{T}}$, and there is a given
holomorphic isomorphism

$I\,:\,{\mathbb{L}}\otimes{\mathbb{L}}\,\longrightarrow\,K_{F}\,,$

where $K_{F}\,\longrightarrow\,{\mathcal{X}}_{\mathcal{T}}$ is the relative
holomorphic cotangent bundle for the project $F$; in other words, $K_{F}$ is
the cokernel of the dual of the differential $dF$

$(dF)^{*}\,:\,F^{*}T^{*}{\mathcal{T}}\,\longrightarrow\,T^{*}{\mathcal{X}}_{\mathcal{T}}\,.$

For each point $t\,\in\,{\mathcal{T}}$, the compact Riemann surface
$F^{-1}(t)$ will be denoted by ${\mathcal{X}}_{t}$. The holomorphic line
bundle ${\mathbb{L}}\big{|}_{{\mathcal{X}}_{t}}$ on ${\mathcal{X}}_{t}$ will
be denoted by ${\mathbb{L}}_{t}$.

Let

$\gamma\,:\,{\mathcal{M}}_{\mathcal{T}}\,\longrightarrow\,{\mathcal{T}}$

be the relative moduli space of stable vector bundles of rank $r$ and degree
zero. So for any $t\,\in\,\mathcal{T}$, the fiber $\gamma^{-1}(t)$ is the
moduli space of stable vector bundles on ${\mathcal{X}}_{t}$ of rank $r$ and
degree zero. Let

$\Theta_{\mathcal{T}}\,\longrightarrow\,{\mathcal{M}}_{\mathcal{T}}$

be the relative theta bundle constructed using the relative theta
characteristic $\mathbb{L}$. So $\Theta_{\mathcal{T}}$ corresponds to the
reduced effective divisor on ${\mathcal{M}}_{\mathcal{T}}$ defined by all
$(t,\,E)$, where $t\,\in\,{\mathcal{T}}$ and $E\,\in\,\gamma^{-1}(t)$, such
that $H^{0}({\mathcal{X}}_{t},\,E\otimes{\mathbb{L}}_{t})\,\not=\,0$.

Let

$q_{\mathcal{T}}\,:\,\text{Conn}^{r}(\Theta_{\mathcal{T}})\,\longrightarrow\,{\mathcal{M}}_{\mathcal{T}}$
(5.1)

be the holomorphic fiber bundle over ${\mathcal{M}}_{\mathcal{T}}$ defined by
the sheaf of relative holomorphic connections on $\Theta_{\mathcal{T}}$. So
for any $t\,\in\,\mathcal{T}$, the fiber $(q_{\mathcal{T}})^{-1}(t)$ is
$\text{Conn}(\Theta)$ in (2.7) for $X\,=\,{\mathcal{X}}_{t}$. The holomorphic
fiber bundle in (5.1) has a $C^{\infty}$ section

$\widehat{\psi}_{Q}\,:\,{\mathcal{M}}_{\mathcal{T}}\,\longrightarrow\,\text{Conn}^{r}(\Theta_{\mathcal{T}})$
(5.2)

given by the Chern connection associated to the Quillen metric on
$\Theta_{\mathcal{T}}$ [Qu]; so for each $t\,\in\,\mathcal{T}$, the
restriction of $\widehat{\psi}_{Q}$ to $\gamma^{-1}(t)$ is the section
$\psi_{Q}$ in (2.9) for the Riemann surface $X\,=\,{\mathcal{X}}_{t}$.

Let

${\varphi}_{\mathcal{T}}\,:\,{\mathcal{C}}_{\mathcal{T}}\,\longrightarrow\,{\mathcal{M}}_{\mathcal{T}}$
(5.3)

be the moduli space of relative holomorphic connections; the fiber of
${\varphi}_{\mathcal{T}}$ over any $(t,\,E)$, where $t\,\in\,{\mathcal{T}}$
and $E\,\in\,\gamma^{-1}(t)$, is the space of all holomorphic connections on
the stable vector bundle $E\,\longrightarrow\,{\mathcal{X}}_{t}$, in
particular, this fiber is an affine space for
$H^{0}({\mathcal{X}}_{t},\,\text{End}(E)\otimes T^{*}{\mathcal{X}}_{t})$.

The holomorphic fiber bundle in (5.3) has a $C^{\infty}$ section

$\widehat{\psi}_{U}\,\,:\,\,{\mathcal{M}}_{\mathcal{T}}\,\longrightarrow\,{\mathcal{C}}_{\mathcal{T}}$
(5.4)

that sends any stable vector bundle of degree zero to the unique holomorphic
connection on it whose monodromy is unitary; so for each
$t\,\in\,\mathcal{T}$, the restriction of $\widehat{\psi}_{U}$ to
$\gamma^{-1}(t)$ is the section $\psi_{U}$ in (2.2) for the Riemann surface
$X\,=\,{\mathcal{X}}_{t}$.

For each $t\,\in\,{\mathcal{T}}$, there is a natural holomorphic isomorphism

$F_{t}\,\,:\,\,(\gamma\circ{\varphi}_{\mathcal{T}})^{-1}(t)\,\longrightarrow\,(\gamma\circ
q_{\mathcal{T}})^{-1}(t)$

(${\varphi}_{\mathcal{T}}$ and $q_{\mathcal{T}}$ are the projections in (5.3)
and (5.1) respectively) that takes the image of the section
$\widehat{\psi}_{U}$ (constructed in (5.4)) to the image of the section
$\widehat{\psi}_{Q}$ constructed in (5.2) (see Theorem 2.1). These
isomorphisms $\\{F_{t}\\}_{t\in\mathcal{T}}$ together define a $C^{\infty}$
isomorphism

$\widehat{F}\,\,:\,\,{\mathcal{C}}_{\mathcal{T}}\,\longrightarrow\,\text{Conn}^{r}(\Theta_{\mathcal{T}})\,;$
(5.5)

the restriction of $\widehat{F}$ to
$(\gamma\circ{\varphi}_{\mathcal{T}})^{-1}(t)$ is the above holomorphic
isomorphism $F_{t}$ for every $t\,\in\,\mathcal{T}$.

######  Proposition 5.1.

The $C^{\infty}$ isomorphism $\widehat{F}$ in (5.5) is holomorphic.

###### Proof.

For every $t\,\in\,\mathcal{T}$, consider the holomorphic isomorphism

$G^{\prime}_{t}\,\,:\,\,(\gamma\circ{\varphi}_{\mathcal{T}})^{-1}(t)\,\longrightarrow\,(\gamma\circ
q_{\mathcal{T}})^{-1}(t)$

in Corollary 4.5; so $G^{\prime}_{t}$ is $G^{\prime}$ in Corollary 4.5 for
${\mathcal{X}}_{t}\,=\,X$. These isomorphisms combine together to define an
isomorphism

$\widehat{G}^{\prime}\,\,:\,\,{\mathcal{C}}_{\mathcal{T}}\,\longrightarrow\,\text{Conn}^{r}(\Theta_{\mathcal{T}})\,;$

the restriction of $\widehat{G}^{\prime}$ to
$(\gamma\circ{\varphi}_{\mathcal{T}})^{-1}(t)$ is the above holomorphic
isomorphism $G^{\prime}_{t}$ for every $t\,\in\,\mathcal{T}$. From the
construction of the isomorphism $G$ in Lemma 3.1 it follows immediately that
$G$ depends holomorphically on the Riemann surface $X$. Note that both the
sections $\phi$ and $\tau$, constructed in (3.13) and (3.15) respectively,
depend holomorphically on the Riemann surface. Therefore, its extension
$G^{\prime}$ also depends holomorphically on the Riemann surface $X$.
Consequently, the above isomorphism $\widehat{G}^{\prime}$ is holomorphic.

Now, Theorem 4.1 implies that $\widehat{G}^{\prime}$ coincides with
$\widehat{F}$ in (5.5). Hence the map $\widehat{F}$ is holomorphic. ∎

## Acknowledgements

The first-named author thanks Centre de Recherches Mathématiques, Montreal,
for hospitality. He is partially supported by a J. C. Bose Fellowship. The
second-named author is supported by an NSERC Discovery grant.

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