# the associated map of the nonabelian Gauss-Manin connection

Ting Chen

###### Abstract.

The Gauss-Manin connection for nonabelian cohomology spaces is the
isomonodromy flow. We write down explicitly the vector fields of the
isomonodromy flow and calculate its induced vector fields on the associated
graded space of the nonabelian Hogde filtration. The result turns out to be
intimately related to the quadratic part of the Hitchin map.

##  1\. introduction

The variation of Hodge structures for families of complex Kähler manifolds has
been a much studied subject. Let $\pi:\mathscr{X}\to S$ be a proper
holomorphic submersion of connected complex manifolds. Ehresmann’s Lemma says
that it is a locally trivial fiber bundle with respect to its underlying
differentiable structure. In particular all the fibers of $\pi$ are
diffeomorphic. So if $s\in S$ and $X_{s}$ the fiber of $\pi$ over $s$,
$\mathscr{X}\to S$ can be viewed as a variation over $S$ of complex structures
on the underlying differentiable manifold of $X_{s}$.

Let $\pi^{k}:\mathscr{V}\to S$ be the corresponding vector bundle of
cohomologies whose fiber at $s\in S$ is $H^{k}(X_{s},\mathbb{C})$,
$k\in\mathbb{N}$. Since $\mathscr{X}\to S$ is locally trivial differentiably
(and therefore topologically), there is an induced local identification of
fibers of $\mathscr{V}\to S$. In another word, there is a flat connection on
the vector bundle $\mathscr{V}\to S$. This connection is called the Gauss-
Manin connection for the cohomologies of the family of complex Kähler
manifolds $\mathscr{X}\to S$.

From Hodge theory we know there is a natural Hodge filtration on the vector
bundle $\mathscr{V}\to S$: $\mathscr{V}=F^{0}\supset F^{1}\supset
F^{2}\ldots\supset F^{k}$. Let $\nabla$ be the Gauss-Manin connection,
Griffiths transversality theorem says that $\nabla(F^{p})\subset
F^{p-1}\otimes\Omega^{1}_{S}$, $1\leq p\leq k$. So if $gr\mathscr{V}$ is the
associated graded vector bundle of the filtered bundle $\mathscr{V}$, then the
induced map $gr\nabla$ of $\nabla$ on $gr\mathscr{V}$ will be
$\mathscr{O}_{S}$-linear. In fact, $gr\nabla$ is equal to a certain Kodaira-
Spencer map[3]. We call $gr\nabla$ the _associated map of the Gauss-Manin
connection_.

The above has a nonabelian analogue. Let $G$ be a complex algebraic group, $X$
a smooth algebraic curve over $\mathbb{C}$ of genus $g$. Let $Conn_{X}$ be the
moduli space of principal $G$-bundles over $X$ equipped with a flat
connection. If we denote as $H^{1}(X,G)$ the first Čech cohomology of $X$ with
coefficients the constant sheaf in $G$, then $Conn_{X}$ can be naturally
identified with $H^{1}(X,G)$, by considering the gluing data of flat
$G$-bundles. Since the group $G$ can be nonabelian, we call $Conn_{X}$ the
nonabelian cohomology space of $X$.

Let $\mathscr{M}_{g}$ be the moduli space of genus $g$ complex algebraic
curves. The universal curve $\mathscr{X}\to\mathscr{M}_{g}$ is (roughly) a
variation of complex structures of the underlying real surface, and the
universal moduli space of connections $\mathscr{C}onn\to\mathscr{M}_{g}$ is
the corresponding bundle of nonabelian cohomologies. For the same reason as
before there is a Gauss-Manin connection on the bundle
$\mathscr{C}onn\to\mathscr{M}_{g}$. The local trivialization that defines it
is often called the isomonodromy deformation, or the isomonodromy flow of
$\mathscr{C}onn$ over $\mathscr{M}_{g}$.

There is also a nonabelian analogue of Hodge filtration which was determined
by Carlos Simpson[8], using a generalized definition of filtration of spaces.
A vector space with filtration is equivalent, by the Rees construction, to a
locally free sheaf over $\mathbb{C}$ with a $\mathbb{C}^{\ast}$-action, and
with the fiber over $1$ being the vector space itself. To define the
nonabelian Hodge “filtration” on the space $\mathscr{C}onn$ therefore, it
would be reasonable to find a family of spaces over $\mathbb{C}$ whose fiber
over $1$ is $\mathscr{C}onn$, together with a $\mathbb{C}^{\ast}$-action on
the family. The way to do it in this case is to introduce the notion of
$\lambda$-connections on a principal $G$-bundle on $X$, for any
$\lambda\in\mathbb{C}$. It is a generalization of the notion of connections on
a $G$-bundle. In particular a $1$-connection is an ordinary connection, and a
$0$-connection is a so called _Higgs field_ , which is an object of much
interest to people in complex geometry and high energy physics. The moduli
space of principal $G$-bundles over $X$ together with a Higgs field is called
the Higgs moduli space over $X$, and denoted as $Higgs_{X}$. Simpson’s
definition of nonabelian Hodge filtration immediately implies that the
associated graded space of $Conn_{X}$ is $Higgs_{X}$. Then a question arises:
what is the associated map of the nonabelian Gauss-Manin connection on the
associated graded space? The answer is: it is a lifting111Here the word
lifting has a slightly more general meaning: it means a map of tangent vectors
in the opposite direction of the pushforward, without requiring its
composition with pushforward being identity. In fact, this lifting here
composed with pushforward is zero. of tangent vectors on the relative Higgs
moduli space $\mathscr{H}iggs\to\mathscr{M}_{g}$. On the other hand there is a
well-known Hitchin map from $Higgs_{X}$ to some vector spaces. The quadratic
part of the Hitchin maps also induce a lifting of tangent vectors on
$\mathscr{H}iggs\to\mathscr{M}_{g}$. The fact that the two liftings agree is
the content of our theorem.

######  Theorem 1.1.

The lifting of tangent vectors on $\mathscr{H}iggs\to\mathscr{M}_{g}$
representing the associated map of the nonabelian Gauss-Manin connection is
equal up to a constant multiple to the lifting of tangent vectors induced from
the quadratic Hitchin map.

Closely related results have been obtained in [1], where the authors apply
localization for vertex algebras to the Segal-Sugawara construction of an
internal action of the Virasoro algebra on affine Kac-Moody algebras to lift
twisted differential operators from the moduli of curves to the moduli of
curves with bundles. Their construction gives a uniform approach to several
phenomena describing the geometry of the moduli spaces of bundles over varying
curves, including a Hamiltonian description of the isomonodromy equations in
terms of the quadratic part of Hitchin’s system. Our result and proof are much
more elementary, avoiding the need for the vertex algebra machinery.

The organization of the paper is as follows. In section 2 we give a detailed
definition of all the objects concerned and a precise statement of the
theorem. The rest of the sections are devoted to the proof. In section 3 we
recall the definition of Atiyah bundles and some of its properties that will
be useful in the proof. In section 4 we use deformation theory to write the
tangent spaces to $\mathscr{C}onn$ as certain hypercohomology spaces. Section
5 gives an explicit description of the lifting of tangent vector on
$\mathscr{C}onn\to\mathscr{M}_{g}$ given by the isomonodromy flow. Section 6
extend the isomonodromy lifting to the moduli space of $\lambda$-connections
for any $\lambda\neq 0$. Finally section 7 takes the limit of the lifting at
$\lambda=0$, which is precisely the associated map of the nonabelian Gauss-
Manin connection, and shows that it is equal up to a constant to the quadratic
Hitchin lifting of tangent vectors.

I would like to thank my advisor Ron Donagi for introducing me to the subject
and for many invaluable discussions.

##  2\. definitions and statement of the theorem

All objects and morphisms in this paper will be algebraic over $\mathbb{C}$,
unless otherwise mentioned.

###  2.1. Moduli space of connections and isomonodromy flow

Let $g$ be a natural number greater or equal to 2, so that a generic curve of
genus $g$ has no automorphisms. The moduli space of all genus $g$ curves is a
smooth Deligne-Mumford stack, but if we restrict to the curves that has no
automorphisms, the moduli space is actually a smooth scheme. Let
$\mathscr{M}_{g}$ be this scheme. In this paper we will ignore all the special
loci of the moduli spaces (as explained below) and focus on local behaviors
around generic points.

Let $G$ be a semisimple Lie group, $X$ a smooth curve of genus $g$. Let
$Bun_{X}$ be the coarse moduli space of regular stable $G$-bundles on $X$.
$Bun_{X}$ is also a smooth scheme[7]. The total space of the cotangent bundle
$T^{\ast}Bun_{X}$ is an open subscheme of the Higgs moduli space over $X$ [4].
However since we are only concerned with generic situations, we will use
$Higgs_{X}$ to denote the open subscheme $T^{\ast}Bun_{X}$.

Let $Conn_{X}$ be the moduli space of pairs $(P,\nabla)$, where $P$ is a
stable $G$-bundle on $X$, and $\nabla$ is a connection on $P$. $\nabla$ is
necessarily flat as the dimension of $X$ is equal to 1. $Conn_{X}$ is an
affine bundle on $Bun_{X}$ whose fiber over $P\in Bun_{X}$ is a torsor for
$T_{P}^{\ast}Bun_{X}$. So it is also a smooth scheme.

Let $\mathscr{C}onn\to\mathscr{M}_{g}$ be the relative moduli space of pairs
whose fiber at $X\in\mathscr{M}_{g}$ is $Conn_{X}$. Let $Irrep_{X}$ be the
space of all irreducible group homomorphisms $\pi_{1}(X)\to G$, $Irrep_{X}$ is
a smooth scheme[5]. There is also the relative space
$\mathscr{I}rrep\to\mathscr{M}_{g}$. The Riemann-Hilbert correspondence
$RH:Conn_{X}\to Irrep_{X}$ taking a flat connection to its monodromy is an
analytic(and therefore differentiable) inclusion. Let
$S\subset\mathscr{M}_{g}$ be a small neighborhood of $X$ in analytic topology.
By Ehresmann’s Lemma the family of curves over $S$ is a trivial family with
respect to the differentiable structure. This implies that the restriction of
$\mathscr{I}rrep$ over $S$ is a differentiable trivial family. The trivial
sections or trivial flows induce a flow on the restriction of $\mathscr{C}onn$
over $S$, by the Riemann-Hilbert correspondence. This flow is called the
isomonodromy flow of $\mathscr{C}onn$ over $\mathscr{M}_{g}$.

###  2.2. $\lambda$-connections and nonabelian Hodge filtration

As explained in the last section, $Conn_{X}$ is the nonabelian cohomology
space of $X$ with in coefficient $G$, and the isomonodromy flow on
$\mathscr{C}onn\to\mathscr{M}_{g}$ is the nonabelian Gauss-Manin connection.
To define Hodge filtration on $Conn_{X}$ one need to generalize the definition
of a filtration. A filtration on a vector space $V$ is equivalent, by the Rees
construction[4], to a locally free sheaf $W$ on $\mathbb{C}$ whose fiber at
$1\in\mathbb{C}$ is isomorphic to $V$, together with a
$\mathbb{C}^{\ast}$-action on $W$ compatible with the usual
$\mathbb{C}^{\ast}$-action on $\mathbb{C}$. The fiber of $W$ at
$0\in\mathbb{C}$ will be isomorphic to the associated graded vector space of
$V$.

This sheaf definition of filtrations can be generalized in an obvious way to
define filtrations on a space that is not a vector space. In our case the
space is $Conn_{X}$, and its Hodge filtration is constructed as follows.
$Conn_{X}$ parametrizes pairs ($P$,$\nabla$). Let $P$ also denote the sheaf of
sections of $P$ on $X$, $adP$ be the adjoint bundle of $P$ as well as the
sheaf of its sections, and $\mathfrak{g}$ the Lie algebra of $G$. A connection
$\nabla$ is a map of sheaves

$\nabla:P\to adP\otimes\Omega_{X}^{1}$

that after choosing local coordinates for $X$ and local trivialization for $P$
can be written as

$(\frac{\partial}{\partial x}+[A(x),\ ])\otimes dx$

where $A(x)$ is a $\mathfrak{g}$-valued function and the bracket means the
right multiplication action of $G$ on $\mathfrak{g}$. A $\lambda$-connection
on $P$ is defined to be a map of sheaves $\nabla_{\lambda}:P\to
adP\otimes\Omega_{X}^{1}$ that in local coordinates can be written as
$(\lambda\frac{\partial}{\partial x}+[A(x),\ ])\otimes dx$. Let the moduli
space of $\lambda$-connections be denoted as $\lambda Conn_{X}$. For
$\lambda\neq 0$, $\nabla\leftrightarrow\lambda\cdot\nabla$ is a bijection
between $Conn_{X}$ and $\lambda Conn_{X}$. For $\lambda=0$, the definition of
a $0$-connection agrees with that of a Higgs field. So $0Conn_{X}$ is just
$Higgs_{X}$.

Let $\mathcal{T}_{X}$ be the moduli space of all $\lambda$-connections for all
$\lambda\in\mathbb{C}$. There is a natural map $\mathcal{T}_{X}\to\mathbb{C}$
taking a $\lambda$-connection to $\lambda$, whose preimage at $1\in\mathbb{C}$
is $Conn_{X}$. In fact, Simpson showed that the nonabelian Hodge filtration of
$Conn_{X}$ is precisely the sheaf of sections of this map, with the
$\mathbb{C}^{\ast}$-action given by multiplication by $\lambda$ for
$\lambda\in\mathbb{C}^{\ast}$[4]. The $\mathbb{C}^{\ast}$-action is algebraic
and induces an isomorphism of $Conn_{X}$ and $\lambda Conn_{X}$.

In the ordinary Hodge theory, if one uses the sheaf definition of filtrations,
then the associated map of the Gauss-Manin connection is obtained as follows.
Start with the Gauss-Manin connection on $\mathscr{V}\to S$, the local
trivialization by the flat sections gives a lifting of tangent vectors

$L:T_{s}S\to T_{v}\mathscr{V}$

for $s\in S$ and $v\in\mathscr{V}$ s.t. $\pi^{k}(v)=s$. The lifting $L$ is a
spliting of $\pi^{k}_{\ast}$, i.e. it satisfies

$\pi^{k}_{\ast}\circ L=id_{T_{s}S}$

Let $\mathscr{W}\to\mathbb{C}$ be the sheaf associated to the Hodge filtration
on $\mathscr{V}\to S$. The fiber of $\mathscr{W}$ at $1$ is $\mathscr{V}\to
S$, and denote the fiber over $\lambda$ as
$\pi^{k}_{\lambda}:\mathscr{V}_{\lambda}\to S$. The action of
$\lambda\in\mathbb{C}^{\ast}$ induces an isomorphism of $\mathscr{V}$ and
$\mathscr{V}_{\lambda}$. So the local trivialization of $\mathscr{V}\to S$
induces a local trialization of $\mathscr{V}_{\lambda}\to S$ via this
isomorphism. Let $L_{\lambda}:T_{s}S\to T_{v_{\lambda}}\mathscr{V}_{\lambda}$
be the induced lifting on $\mathscr{V}_{\lambda}\to S$ _multiplied by
$\lambda$_. $L_{\lambda}$ satisfies

$\pi^{k}_{\lambda\ast}\circ L_{\lambda}=\lambda\cdot id_{T_{s}S}$

$L_{\lambda}$ is defined for all $\lambda\neq 0$. For a fixed vector
$\vec{t}\in T_{s}S$, the images of $\vec{t}$ under all the $L_{\lambda}$,
$\lambda\neq 0$ gives a vector field on the total space of $\mathscr{W}$ away
from $\mathscr{V}_{0}$, which is the fiber over $0\in\mathbb{C}$. The
continuous limit of that vector field on $\mathscr{V}_{0}$ exist, and
therefore defines a lifting $L_{0}:T_{s}S\to T_{v_{0}}\mathscr{V}_{0}$ on
$\mathscr{V}_{0}\to S$. $L_{0}$ satisfies

$\pi^{k}_{0\ast}\circ L_{0}=0$

i.e. the images of $\vec{t}\in T_{s}S$ under $L_{0}$ is a vectors field on the
_fiber_ $V_{0,s}$ of $\mathscr{V}_{0}$ over $s$. This vector field is in fact
linear and defines a linear map on $V_{0,s}$. Also $\mathscr{V}_{0}$ is
identified with $gr\mathscr{V}$. From these we see $L_{0}$ really gives a
vector bundle map $gr\mathscr{V}\to gr\mathscr{V}\otimes\Omega^{1}_{S}$, and
that map is the associated map of the Gauss-Manin that we started with.

So in nonabelian Hodge theory, in order to calculate the associated map of the
nonabelian Gauss-Manin connection, we will start with the lifting $L$ induced
from the isomonodromy flow on $\mathscr{C}onn\to\mathscr{M}_{g}$(by a slight
abuse of notation we will use the same notations for the liftings, the meaning
should be clear from the context), and try to find the associated limit
lifting $L_{0}$. Specifically, let $\mathscr{T}\to\mathscr{M}_{g}$ be the
relative moduli space whose fiber at $X\in\mathscr{M}_{g}$ is
$\mathcal{T}_{X}$. $\mathscr{T}$ maps to $\mathbb{C}$ and the fiber at
$\lambda$ is the relative moduli space of $\lambda$-connections, which is
denoted $\lambda\mathscr{C}onn$. There is clearly also a
$\mathbb{C}^{\ast}$-action on $\mathscr{T}$ compatible with the
$\mathbb{C}^{\ast}$-action on $\mathbb{C}$. Let $L_{\lambda}$ be analogously
the lifting on $\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ induced by the
lifting $L$ via the $\mathbb{C}^{\ast}$-action and multiplied by $\lambda$.
Then the limit lifting $L_{0}$ will be the associated map that we want to
calculate. It will again be a vertical lifting, i.e. the images of $L_{0}$
will be vectors tangent to the fibers $Higgs_{X}$ of
$\mathscr{H}iggs\to\mathscr{M}_{g}$, $X\in\mathscr{M}_{g}$.

###  2.3. Quadratic Hitchin map and statement of the theorem

$Higgs_{X}$ has a symplectic structure as it is equal to $T^{\ast}Bun_{X}$.
Let $<\ ,\ >$ be the Killing form on the Lie algebra $\mathfrak{g}$ of $G$,
the quadratic Hitchin map is

$qh:Higgs_{X}\to H^{0}(X,\Omega^{\otimes 2})$
$(P,\theta)\mapsto<\theta,\theta>$

where $\theta\in H^{0}(X,adP\otimes\Omega^{1}_{X})$ is a 0-connection or a
Higgs field. One can define a lifting of tangent vectors associated to $qh$

$L_{qh}:T_{X}\mathscr{M}_{g}\to T_{(P,\theta)}Higgs_{X}$ $f\mapsto
H_{qh^{\ast}f}|_{(P,\theta)}$

where $f\in T_{X}\mathscr{M}_{g}\cong H^{1}(X,TX)$ is viewed as a linear
function on $H^{0}(X,\Omega^{\otimes 2})$ by Serre duality, and
$H_{qh^{\ast}f}$ is the Hamiltonian vector field of $qh^{\ast}f$ on
$Higgs_{X}$.

The theorem can now be more precisely stated as:

######  Theorem 2.1 (precise version of Theorem 1.1).

The limit lifting of tangent vectors $L_{0}$ associated to the isomonodromy
lifting $L$ is equal to $\frac{1}{2}L_{qh}$.

##  3\. Atiyah bundles

Before starting to prove the theorem, we recall here some facts about Atiyah
bundles which will be used later. As before let $X$ be a smooth curve of genus
$g$, $G$ a semisimple Lie group, $p:P\to X$ a principal $G$-bundle over $X$.

###  3.1. Atiyah bundle and its sections

Let $TP$ be the tangent bundle over $P$. $G$ acts on $P$ and has an induced
action on $TP$. The action is free and compatible with the vector bundle
structure of $TP\to P$, so the quotient will be a vector bundle $TP/G\to
P/G=X$. This vector bundle over $X$ is called the Atiyah bundle associated to
$P$, and denoted as $A_{P}$.

In fact, $TP$ is isomorphic to the fiber product of $P$ and $A_{P}$ over $X$.
So any section $t$ of $A_{P}$ over $X$ has a unique lift $\tilde{t}$ that
makes the diagram commute Dashto dashdash¿

The lift $\tilde{t}$ can be viewed as a vector field on $P$ which is
$G$-invariant. Conversely, any $G$-invariant vector field on $P$ defines a
section $t$ in the quotient bundle. Therefore sections of $A_{P}$ over $X$ are
the same as $G$-invariant vector fields on $P$.

###  3.2. Atiyah sequence

The sequence of tangent bundles associated to $P\to X$ is:

$0\to T_{P/X}\to TP\to p^{\ast}TX\to 0$

$G$ acts on the sequence, and the quotient is

$0\to adP\to A_{P}\to TX\to 0$

This quotient sequence is called the Atiyah sequence of $A_{P}$. We will
denote the map $A_{P}\to TX$ also as $p_{\ast}$.

###  3.3. Relation to connections

If $\nabla$ is a connection on $P$, then $\nabla$ must be flat since the
dimension of $X$ is 1. So over a small open subset $U\subset X$, there is a
natural trivialization of $P$ associated to $\nabla$

$\tau:U\times F\longrightarrow P|_{U}$

given by the flat sections of $\nabla$. Here F denotes a torsor for $G$.

The local trivialization gives a local section $\tilde{s}_{U}:p^{\ast}TU\to
TP|_{U}$, which is the composition

(1)
$p^{\ast}TU\xrightarrow{\tau^{-1}_{\ast}}p_{U}^{\ast}TU\xrightarrow{(id,0)}p_{U}^{\ast}TU\oplus
p_{F}^{\ast}TF\xrightarrow{\tau_{\ast}}TP|_{U}$

where $p_{U}$ and $p_{F}$ are the projections of $U\times F$ to $U$ and $F$.

Since $\tilde{s}_{U}$ is canonically associated to $\nabla$, so for two such
open subsets $U,V$, $\tilde{s}_{U}$ and $\tilde{s}_{V}$ agree on their
intersection. So there is a well-defined map $\tilde{s}:p^{\ast}TU\to TP$.
Since $\tau$ is $G$-invariant and the map $(id,0)$ is obviously $G$-invariant,
$\tilde{s}_{U}$ is $G$-invariant. So $\tilde{s}$ is $G$-invariant, and gives a
map $s:TX\to A_{P}$. The map $(id,0)$ in the definition of $\tilde{s}_{U}$
implies that $s$ is a splitting of $p_{\ast}:A_{P}\to TX$, i.e. $p_{\ast}\circ
s=id_{TX}$. We can also say that $s$ is a splitting of the Atiyah sequence.

To summarize, for any connection $\nabla$ on $P$ there is uniquely associated
a splitting $s$ of the Atiyah sequence of $P$. $s$ is locally defined as the
splitting $(id,0)$ with $P$(and therefore $A_{P}$) locally trivialized by
$\nabla$.

##  4\. tangent spaces

Now we start to prove the theorem. In this section we will identify the
tangent spaces of $\mathscr{C}onn$ and more generally $\lambda\mathscr{C}onn$
as some hypercohomology spaces, so that we may write down the isomonodromy
lifting $L$ and the extened liftings $L_{\lambda}$ explicitly in the next two
sections.

The tangent space to a moduli space at a regular point is identified with the
infinitesimal deformations of the object corresponding to that point. So we
are really looking at infinitesimal deformations of the objects parametrized
by $\mathscr{C}onn$, which are triples ($X$,$P$,$\nabla$). We start with
deformations of pairs ($X$,$P$).

###  4.1. Deformation of pairs

From Deformation Theory, the following two propositions are well-known.

######  Proposition 4.1.1.

The tangent space to $\mathscr{M}_{g}$ at a point $X$ is naturally isomorphic
to $H^{1}(X,TX)$.

######  Proposition 4.1.2.

The tangent space to $Bun_{X}$ at a point $P$ is naturally isomorphic to
$H^{1}(X,adP)$.

Let $\mathscr{B}un$ be the moduli space of pairs ($X$,$P$). We expect that
generically the tangent space at a point ($X$,$P$) would satisfy

$0\to H^{1}(X,adP)\to T_{(X,P)}\mathscr{B}un\to H^{1}(X,TX)\to 0$

On the other hand since the Atiyah sequence of $P$ $0\to adP\to A_{P}\to TX\to
0$ induces

$0\to H^{1}(X,adP)\to H^{1}(X,A_{P})\to H^{1}(X,TX)\to 0$

It is natural to guess that

######  Proposition 4.1.3.

$T_{(X,P)}\mathscr{B}un$ is naturally isomorphic to $H^{1}(X,A_{P})$.

###### Proof.

the proof is a combination of the usual proofs for Proposition 4.1.1 and
Proposition 4.1.2. Let $\\{U_{i}\\}_{i\in I}$ be an Čech covering of $X$,
$P_{\epsilon}\to X_{\epsilon}$ a family of principal $G$-bundles over
$D_{\epsilon}=\mathbb{C}[\epsilon]/(\epsilon^{2})$, which restrict to $P\to X$
over the closed point. Over each $U_{i}$, let

$\phi_{i}:P|_{U_{i}}\times D_{\epsilon}\to P_{\epsilon}|_{U_{i}}\ \ \ \ \ \
(\phi_{i}^{\vee}:\mathscr{O}_{P|_{U_{i}}}\otimes\mathbb{C}[\epsilon]/(\epsilon^{2})\leftarrow\mathscr{O}_{P_{\epsilon}|_{U_{i}}})$

be an isomorphism of $G$-bundles. So it is compatible with the $G$-actions and
descends to an isomorphism

$\iota_{i}:U_{i}\times D_{\epsilon}\to X_{\epsilon}|_{U_{i}}\ \ \ \ \ \
(\iota_{i}^{\vee}:\mathscr{O}_{U_{i}}\otimes\mathbb{C}[\epsilon]/(\epsilon^{2})\leftarrow\mathscr{O}_{X_{\epsilon}|_{U_{i}}})$

Over $U_{ij}=U_{i}\cap U_{j}$, the transition functions are related as in the
commutative diagram

Let $\xi_{ij}\in\Gamma(U_{ij},TX)$ be the vector field on $U_{ij}$ such that
$(\iota_{j}^{-1}\circ\iota_{i})^{\vee}=Id+\epsilon\xi_{ij}$, and
$\eta_{ij}\in\Gamma(P|_{U_{ij}},TP)$ be the vector field on $P|_{U_{ij}}$ such
that $(\phi_{j}^{-1}\circ\phi_{i})^{\vee}=Id+\epsilon\eta_{ij}$. Because
$\phi_{i}$ is $G$-invariant, $\eta_{ij}$ is $G$-invariant. So one can view it
as $\eta_{ij}\in\Gamma(U_{ij},A_{P})$. $(\eta_{ij})_{i,j\in I}$ form a Čech
1-cochain on $X$ with coefficients in $A_{P}$.

$(\eta_{ij})_{i,j\in I}$ is closed because it comes from transition functions
$\phi_{j}^{-1}\circ\phi_{i}$. Any closed cochain $(\eta_{ij})_{i,j\in I}$
comes from some $D_{\epsilon}$ family of pairs. Also for a fixed
$D_{\epsilon}$ family of pairs, a different choice of $\phi_{i}$’s will result
in a cocycle differing from $(\eta_{ij})_{i,j\in I}$ by an exact cocycle. And
any exact cocycle is the result of different choices of $\phi_{i}$’s.
Therefore the infinitesimal deformations of ($X$,$P$) are in natural
correspondence with $H^{1}(X,A_{P})$, which proves the proposition. ∎

###  4.2. Deformation of triples

Now we come to the infinitesimal deformations of a triple ($X$,$P$,$\nabla$).
First a notation related to the connection $\nabla$. As discussed in section
3.3, a connection $\nabla$ on $P$ is equivalent to a splitting of the Atiyah
sequence Let $\hat{s}\in H^{0}(X,A_{P}\otimes\Omega_{X}^{1})$ denote the
global section associated to the splitting map $s$. We see that
$\hat{s}\mapsto 1$ under the map $H^{0}(X,A_{P}\otimes\Omega_{X}^{1})\to
H^{0}(X,TX\otimes\Omega_{X}^{1})\cong H^{0}(X,\mathscr{O}_{X})$.

To find the deformation of the triple ($X$,$P$,$\nabla$), let
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) be a family of triples
over $D_{\epsilon}$ starting with it. Let $s_{\epsilon}$ be the family of
sections corresponding to $\nabla_{\epsilon}$. As in the proof of Proposition
4.1.3, let $\\{U_{i}\\}_{i\in I}$ again be an Čech covering of $X$, and
$\phi_{i}$, $\iota_{i}$, $i\in I$ defined in the same way. Let
$s_{i}:TU_{i}\to A_{P}|_{U_{i}}$ and $\sigma_{i}:TU_{i}\to adP|_{U_{i}}$ be
sections such that the following diagram commutate: The target space of
$\sigma_{i}$ is $adP$ instead of $A_{P}$, because $p_{\ast}\circ s=id$ for all
$s$, so $\sigma_{i}$, being the derivative of $s$ (locally on $U_{i}$, under
the trivialization of the family $\phi_{i}$), projects to $0$ under
$p_{\ast}$.

A deformation of the triple should contain the information about the
deformation of the pair ($X$,$P$) as well as the deformation of $\nabla$. So
the data associated to the infinitesimal family
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) should be the pair:

$(\eta_{ij})_{i,j\in I},(\sigma_{i})_{i\in I}$

where $(\eta_{ij})_{i,j\in I}$ is defined in section 4.1.3 and shown to
characterize the deformation of the pair ($X$,$P$), and $(\sigma_{i})_{i\in
I}$ describe the deformation of $\nabla$.

The data $((\eta_{ij})_{i,j\in I},(\sigma_{i})_{i\in I})$ looks like a
1-cocycle in defining the hypercohomology of some complex of sheaves. Recall
that the tangent space to $Higgs_{X}$ at a point $(P,\theta)$ is
$\mathbb{H}^{1}(X,adP\xrightarrow{[\ ,\theta]}adP\otimes\Omega^{1}_{X})$. We
will prove an analogous result about the tangent spaces to $\mathscr{C}onn$.

On $U_{ij}$, the transition relations are expressed in the following diagram:

Since $(\iota_{j}^{-1}\circ\iota_{i})^{\vee}=Id+\epsilon\xi_{ij}$ and
$(\phi_{j}^{-1}\circ\phi_{i})^{\vee}=Id+\epsilon\eta_{ij}$, we can write down
the two horizontal maps more explicitly. $\forall\ Y+\epsilon Y_{1}\in
TU_{ij}\times D_{\epsilon}$, its image $Y^{\prime}+\epsilon Y_{1}^{\prime}$
under $d(\iota_{j}^{-1}\circ\iota_{i})$ is determined by: for any function $f$
on $U_{ij}$,

$(Y^{\prime}+\epsilon Y_{1}^{\prime})(f)=(I+\epsilon\xi_{ij})(Y+\epsilon
Y_{1})(I-\epsilon\xi_{ij})(f)$

After simplification we get $Y^{\prime}=Y,Y_{1}^{\prime}=Y_{1}+[\xi_{ij},Y]$,
where the bracket is the Lie bracket of vector fields on $U_{ij}$. Similarly
$\forall\ Z+\epsilon Z_{1}\in A_{P}|_{U_{ij}}\times D_{\epsilon}$ (by section
3.1 it can be viewed as a $G$-invariant vector field on $P|_{U_{ij}}$), we get
$d(\phi_{j}^{-1}\circ\phi_{i})(Z+\epsilon
Z_{1})=Z+\epsilon(Z_{1}+[\eta_{ij},Z])$, where the bracket is the Lie bracket
of ($G$-invariant) vector fields on $P|_{U_{ij}}$.

The diagram is commutative, i.e. $\forall\ Y+\epsilon Y_{1}\in TU_{ij}\times
D_{\epsilon}$

$d(\phi_{j}^{-1}\circ\phi_{i})\circ(s_{i}+\epsilon\sigma_{i})(Y+\epsilon
Y_{1})=(s_{j}+\epsilon\sigma_{j})\circ
d(\iota_{j}^{-1}\circ\iota_{i})(Y+\epsilon Y_{1})$

After simplification we get

$s_{i}(Y)=s_{j}(Y)$ (2)
$(\sigma_{j}-\sigma_{i})(Y)=[\eta_{ij},s_{i}(Y)]-s_{j}([\xi_{ij},Y])$

So if we use $\hat{\sigma}_{i}\in H^{0}(X,adP\otimes\Omega_{X}^{1})$ to denote
the global section associated to $\sigma_{i}$, the pair

$((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$

is a hyper Čech 1-cochain on $X$ with coefficients in

$A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X}$

where the map $[\ ,\hat{s}]$ is defined as: if
$\hat{s}=s^{\prime}\otimes\omega$, where $s^{\prime}\in
H^{0}(X,A_{P}),\omega\in H^{0}(X,\Omega_{X}^{1})$, then $[\ ,\hat{s}]:=[\
,s^{\prime}]\otimes\omega-s^{\prime}\otimes[p_{\ast}(\ ),\omega]$.

######  Proposition 4.2.1.

$T_{(X,P,\nabla)}\mathscr{C}onn$ is naturally isomorphic to
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X})$.

###### Proof.

To any $D_{\epsilon}$ family of triples
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) is associated a hyper
1-cochain $((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$ by the above
discussion. It is closed because of three facts: first, $(\eta_{ij})_{i,j\in
I}$ is a closed Čech 1-cochain with coefficients in $A_{P}$ \- it’s closed
again because it comes from the transition function
$\phi_{j}^{-1}\circ\phi_{i}$; second, because of (2); third, the complex
$A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X}$ has only two nonzero
terms. These three facts imply that $((\eta_{ij})_{i,j\in
I},(\hat{\sigma}_{i})_{i\in I})$ is closed. Any closed hyper 1-cochain comes
from some $D_{\epsilon}$ family of triples. Also for a fixed $D_{\epsilon}$
family of triples, a different choice of the $\phi_{i}$’s will result in a
hyper cocycle differing from $((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in
I})$ by an exact hyper cocycle. And any exact hyper cocycle is the result of
different choices of the $\phi_{i}$’s. Therefore the infinitesimal
deformations of ($X$,$P$,$\nabla$) are in natural correspondence with
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X})$,
which is what we need to prove. ∎

###  4.3. Tangent spaces to $\lambda\mathscr{C}onn$

Let $\lambda\in\mathbb{C}$ be a fixed complex number. For the moduli space
$\lambda\mathscr{C}onn$ of triples ($X$,$P$,$\nabla_{\lambda}$) where
$\nabla_{\lambda}$ is a $\lambda$-connection, the statement about its tangent
spaces is completely analogous to that when $\lambda=1$.

For a $\lambda$-connection $\nabla_{\lambda}$ on $P$, $\lambda\neq 0$,
$\frac{1}{\lambda}\nabla_{\lambda}$ is an ordinary connection, therefore
corresponds to a splitting $s_{\frac{1}{\lambda}\nabla_{\lambda}}$ of the
Atiyah sequence of $P$. Let $s_{\lambda}=\lambda\cdot
s_{\frac{1}{\lambda}\nabla_{\lambda}}$, so $s_{\lambda}$ is a
“$\lambda$-splitting” of the Atiyah sequence of $P$, i.e. $p_{\ast}\circ
s_{\lambda}=\lambda\cdot id_{TX}$. Therefore to any $\lambda$-connection
$\nabla_{\lambda}$($\lambda\neq 0$) is associated a $\lambda$-splitting of the
Atiyah bundle. Notice that this is true for $\lambda=0$ as well, as a
0-splitting of the Atiyah bundle of $P$ is exactly a Higgs field on $P$.

Let $\hat{s}_{\lambda}\in H^{0}(X,A_{P}\otimes\Omega_{X}^{1})$ be the global
section associated to $s_{\lambda}$, we see $\hat{s}_{\lambda}\mapsto\lambda$
under the map $H^{0}(X,A_{P}\otimes\Omega_{X}^{1})\to
H^{0}(X,TX\otimes\Omega_{X}^{1})\cong H^{0}(X,\mathscr{O}_{X})$. The arguments
in the last subsection can be repeated with slight changes (replace 1 by
$\lambda$ at appropriate places) to give the following statement.

######  Proposition 4.3.1.

$T_{(X,P,\nabla_{\lambda})}\lambda\mathscr{C}onn$ is naturally isomorphic to
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}_{\lambda}]}adP\otimes\Omega^{1}_{X})$, $\forall\lambda\in\mathbb{C}$

### Remark

When $\lambda=0$, the result agrees with the previous results about tangent
spaces to the Higgs moduli space.

##  5\. isomonodromy vector field

The nonabelian Gauss-Manin connection on $\mathscr{C}onn\to\mathscr{M}_{g}$ is
the isomonodromy flow. The local trivialization of
$\mathscr{C}onn\to\mathscr{M}_{g}$ given by the flow induces a lifting of
tangent vectors $L:T_{X}\mathscr{M}_{g}\to T_{(X,P,\nabla)}\mathscr{C}onn$. We
have identified these tangent spaces as (hyper)cohomology spaces in the last
section, now we will write down the map $L$ as a map of cohomology spaces. We
start with a useful fact about an isomonodromy family of connections.

###  5.1. Universal connection of an isomonodromy family

In [5] Inaba et al. constructed the moduli space of triples
($X$,$P$,$\nabla$), and a universal $G$-bundle on the universal curve with a
universal connection. Though they did it for a special case(rank 2 parabolic
vector bundle on $\mathbb{P}^{1}$ with 4 points), the more general case can be
done similarly. The universal connection, when restricted to an isomonodromy
family of triples, has the following important property.

######  Proposition.

If ($X_{t}$,$P_{t}$,$\nabla_{t}$) is an isomonodromy family of triples over a
complex line $D=Spec(\mathbb{C}[t])$, then the restriction of the universal
connection on $P_{t}$(viewed as a $G$-bundle over the total space of $X_{t}$)
is flat.

###### Proof.

If we only look at the underlying differentiable structure, the isomonodromy
family over $D=\mathbb{C}[t]$ is a trivial family of triples. The trivial
family structure gives a flat connection on $P_{t}$, which must be equal to
the restriction of the universal connection on $P_{t}$ since they are equal on
each fiber of the family. ∎

###  5.2. Isomonodromy lifting of tangent vectors

For $\forall\lambda\in\mathbb{C}$, let $\pi_{\lambda}$ be the projection:

$\pi_{\lambda}:\lambda\mathscr{C}onn\to\mathscr{M}_{g}$
$(X,P,\nabla_{\lambda})\mapsto X$

From the proof of Proposition 4.1.3 and the discussions in front of
Proposition 4.2.1 it is not hard to see that the differential of
$\pi_{\lambda}$ is induced from the map $(p_{\ast},0)$ of complexes of sheaves

The lifting of tangent vectors induced from the isomonodromy flow is a
splitting of the map $\pi_{1\ast}$ Notice that the splitting map $s:TX\to
A_{P}$ associated to $\nabla$ gives a map of the complexes The diagram is
commutative because $[\ ,\hat{s}]\circ s$ is basically bracketing $\hat{s}$
with itself and therefore equal to 0. The map of complexes $(s,0)$ is
obviously a splitting of the map $(p_{\ast},0)$.

The map $(s,0)$ of the complexes of sheaves induce a map on the first
hypercohomology, which we denote as $H^{1}(s)$.

######  Proposition 5.2.1.

The isomonodromy lifting $L$ is equal to

$H^{1}(s):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}]}adP\otimes\Omega^{1}_{X})$

###### Proof.

At a point ($X$,$P$,$\nabla$) of $\mathscr{C}onn$, let
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) be an isomonodromy family
of triples over $D_{\epsilon}$ starting with it. Again let $\\{U_{i}\\}_{i\in
I}$ be an Čech covering of $X$.

Over $U_{i}$, Let

$\tau_{i,\epsilon}:X_{\epsilon}|_{U_{i}}\times F\to P_{\epsilon}|_{U_{i}}$

be the trivialization of $P_{\epsilon}|_{U_{i}}$ over $X_{\epsilon}|_{U_{i}}$
determined by the flat universal connection (see section 5.1) on
$P_{\epsilon}|_{U_{i}}$, and $\tau_{i}$ be its restriction at $\epsilon=0$.

Let

$\iota_{i}:U_{i}\times D_{\epsilon}\to X_{\epsilon}|_{U_{i}}$

be an isomorphism and define

$\phi_{i}:P|_{U_{i}}\times D_{\epsilon}\to P_{\epsilon}|_{U_{i}}$

as the composition

$P|_{U_{i}}\times
D_{\epsilon}\xrightarrow{(\tau_{i}^{-1},id_{D_{\epsilon}})}U_{i}\times
D_{\epsilon}\times
F\xrightarrow{(\iota_{i},id_{F})}X_{\epsilon}|_{U_{i}}\times
F\xrightarrow{\tau_{i,\epsilon}}P_{\epsilon}|_{U_{i}}$

Let $\xi_{ij}$, $\eta_{ij}$, $s_{\epsilon}$, $s_{i}$ and $\sigma_{i}$ be all
defined as before in the proofs of proposition 4.1.3 and section 4.2. Notice
that since the local trivializations of the $G$-bundles are canonically given
by the flat universal connection, $\tau_{i,\epsilon}$ and $\tau_{j,\epsilon}$
agree on $U_{ij}$, i.e. on $U_{ij}$

$\tau_{i,\epsilon}=\tau_{j,\epsilon}$ $\tau_{i}=\tau_{j}$

Therefore over $U_{ij}$, the transition map $\phi_{j}^{-1}\circ\phi_{i}$ fits
in the diagram In another word with the local trivializations
$(\tau_{i}^{-1},id_{D_{\epsilon}})$ and $(\tau_{j}^{-1},id_{D_{\epsilon}})$,
the transition map $\phi_{j}^{-1}\circ\phi_{i}$ corresponds to
$(\iota_{j}^{-1}\circ\iota_{i},id_{F})$. Let
$(\phi_{j}^{-1}\circ\phi_{i})^{\prime}$ and $\eta_{ij}^{\prime}$ be
$(\phi_{j}^{-1}\circ\phi_{i})$ and $\eta_{ij}$ under the local
trivializations, then

$(\phi_{j}^{-1}\circ\phi_{i})^{\prime}=(\iota_{j}^{-1}\circ\iota_{i},id_{F})$

and therefore

$Id+\epsilon\eta_{ij}^{\prime}=(Id+\epsilon\xi_{ij},Id_{F})$

Comparing the coefficients of $\epsilon$ we get

$\eta_{ij}^{\prime}=(\xi_{ij},0)$

According to the last paragraph in section 3.3, we see this means precisely
that $\eta_{ij}=s(\xi_{ij})$.

With $\phi_{i}:P|_{U_{i}}\times D_{\epsilon}\to P_{\epsilon}|_{U_{i}}$ defined
as above, $s_{\epsilon}|_{U_{i}}:TX_{\epsilon}|_{U_{i}}\to
A_{P_{\epsilon}}|_{U_{i}}$ correspond to the section $s_{i}:TU_{i}\times
D_{\epsilon}\to A_{P}|_{U_{i}}\times D_{\epsilon}$ constant along
$D_{\epsilon}$, i.e. $\sigma_{i}=0$.

Therefore $\hat{\sigma}_{i}=0$, and the pair

$(\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I}$

is exactly the hyper 1-cocycle which is the image of $(\xi_{ij},0)$ under the
map $H^{1}(s)$, which finishes the proof. ∎

##  6\. extended isomonodromy lifting

The associated lifting $L_{\lambda}$ is obtained by extending the isomonodromy
lifting $L$ to $\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ by the
$\mathbb{C}^{\ast}$-action, and multiplying by $\lambda$. For a fixed
$\lambda$, $\lambda\neq 0$, the $\mathbb{C}^{\ast}$-action gives an
isomorphism

$\mathscr{C}onn\leftrightarrow\lambda\mathscr{C}onn$
$\nabla\leftrightarrow\lambda\cdot\nabla$

The induced lifting on $\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ by $L$ via
the isomorphism, called the extended isomonodromy lifting, can be written very
similarly as $L$. In the same way that the splitting map $s$ associated to a
connection $\nabla$ induces a map $H^{1}(s)$ of hypercohomologies, the
$\lambda$-splitting map $s_{\lambda}$ associated to a $\lambda$-connection
$\nabla_{\lambda}$ induces a map of the corresponding hypercohomology spaces,
which will be denoted as $H^{1}(s_{\lambda})$.

######  Proposition 6.1.

The extended isomonodromy lifting of tangent vector on
$\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ is given by:

$\frac{1}{\lambda}H^{1}(s_{\lambda}):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}_{\lambda}]}adP\otimes\Omega^{1}_{X})$

###### Proof.

Since the map of moduli spaces is $\nabla\mapsto\lambda\cdot\nabla$ (or
$s\mapsto\lambda s$, $\hat{s}\mapsto\lambda\hat{s}$), the induced map on the
tangent spaces $T_{(X,P,\nabla)}\mathscr{C}onn\to
T_{(X,P,\lambda\nabla)}\lambda\mathscr{C}onn$ is

$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}]}adP\otimes\Omega^{1}_{X})\xrightarrow{H^{1}(id,\lambda)}\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\lambda\hat{s}]}adP\otimes\Omega^{1}_{X})$

where $(id,\lambda)$ is the map of complexes of sheaves and
$H^{1}(id,\lambda)$ is the induced map on hypercohomology.

So to get the corresponding lifting on $\lambda\mathscr{C}onn$, i.e. to make
the following diagram commutate, the vertical map on the right must be
$\frac{1}{\lambda}H^{1}(\lambda s)$. ∎

Since $L_{\lambda}$ is the extended isomonodromy lifting multiplied by
$\lambda$, $L_{\lambda}=H^{1}(s_{\lambda})$. $L_{\lambda}$ is a
$\lambda$-lifting of tangent vectors.

##  7\. limit lifting at $\lambda=0$

The continuous limit of $L_{\lambda}$ at $\lambda=0$ is a 0-lifting
$L_{0}:T_{X}\mathscr{M}_{g}\to T_{(X,P,\nabla_{0})}\mathscr{H}iggs$. Since
$L_{\lambda}=H^{1}(s_{\lambda})$, by continuity $L_{0}$ is equal to

$H^{1}(s_{0}):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$

where $s_{0}$ is the 0-splitting of the Atiyah bundle of $P$ associated to the
0-connection(or Higgs field) $\nabla_{0}$ on $P$. Because $\pi_{0\ast}\circ
H^{1}(s_{0})=0$, so in fact $H^{1}(s_{0})$ can be written as

$H^{1}(s_{0}):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$

The images of a vector in $\vec{t}\in T_{X}\mathscr{M}_{g}$ under
$H^{1}(s_{0})$ form a vector field on the fiber $Higgs_{X}$ of $\pi_{0}$.

Recall that the quadratic Hitchin map on $Higgs_{X}$ is

$qh:Higgs_{X}\to H^{0}(X,\Omega^{\otimes 2})$
$(P,s_{0})\mapsto<\hat{s}_{0},\hat{s}_{0}>$

and its associated lifting of tangent vectors is

$L_{qh}:H^{1}(X,TX)\to\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$ $f\mapsto
H_{qh^{\ast}f}|_{(P,s_{0})}$

The main theorem (Theorem 2.1) is that $H^{1}(s_{0})$ is equal to
$\frac{1}{2}L_{qh}$. To prove it we need two lemmas. For the first lemma, Let
$((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$ be a representative of
an arbitrary element $v\in\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$. Because on $U_{ij}$,
$<\hat{s}_{0},\hat{\sigma}_{j}-\hat{\sigma}_{i}>=<\hat{s}_{0},[\eta_{ij},\hat{s}_{0}]>=-<[\hat{s}_{0},\hat{s}_{0}],\eta_{ij}>=0$,
therefore

$<\hat{s}_{0},\hat{\sigma}_{i}>=<\hat{s}_{0},\hat{\sigma}_{j}>$

Let $<\hat{s}_{0},\hat{\sigma}>\in H^{0}(X,\Omega^{\otimes 2})$ denote the
resulting global quadratic differential form.

######  Lemma 7.1.

Using the above notations, the differential of the map $qh$ is equal to:

$qh_{\ast}:\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})\to H^{0}(X,\Omega^{\otimes 2})$
$v\mapsto 2<\hat{s}_{0},\hat{\sigma}>$

###### Proof.

Let $\\{U_{i}\\}_{i\in I}$ be the Čech covering of the curve $X$,
$(P_{\epsilon},s_{\epsilon})$ the family of Higgs bundles over $D_{\epsilon}$
that correspond to $v$, i.e. for some $\phi_{i}:P|_{U_{i}}\times
D_{\epsilon}\to P_{\epsilon}|_{U_{i}}$, some $s_{i}:TU_{i}\to adP|_{U_{i}}$
and the given $\sigma_{i}:TU_{i}\to adP|_{U_{i}}$, the diagram is commutative.
Because
$qh:(P_{\epsilon},s_{\epsilon})\mapsto<\hat{s}_{\epsilon},\hat{s}_{\epsilon}>$,
and that over $U_{i}$,
$<\hat{s}_{\epsilon},\hat{s}_{\epsilon}>=<\hat{s}_{i}+\epsilon\hat{\sigma}_{i},\hat{s}_{i}+\epsilon\hat{\sigma}_{i}>=<\hat{s}_{i},\hat{s}_{i}>+2<\hat{s}_{i},\hat{\sigma}_{i}>\epsilon$,
so
$qh:(P_{\epsilon},s_{\epsilon})\mapsto<\hat{s}_{0},\hat{s}_{0}>+2<\hat{s}_{0},\hat{\sigma}>\epsilon$.
Taking the coefficient of $\epsilon$, we see that $qh_{\ast}$ maps $v$ to
$2<\hat{s}_{0},\hat{\sigma}>$. ∎

For the second lemma, let $\omega_{H}$ be the symplectic 2-form on
$Higgs_{X}$, $((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$ and
$((\eta^{\prime}_{ij})_{i,j\in I},(\hat{\sigma}^{\prime}_{i})_{i\in I})$
representatives of two vectors
$v,v^{\prime}\in\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$.

######  Lemma 7.2.

Let $\int:H^{1}(X,\Omega^{1}_{X})\to\mathbb{C}$ be the canonical map, then

$\omega_{H}(v,v^{\prime})=\int(\eta_{ij}\sqcup\hat{\sigma}^{\prime}_{i}+\eta^{\prime}_{ij}\sqcup\hat{\sigma}_{i})$

where $\sqcup$ means the cup product $\cup$ of Čech cochains composed with the
Killing form $<\ ,\ >$.

###### Proof.

see [6] Proposition 7.12. ∎

######  Theorem.

$H^{1}(s_{0})$ is equal to $\frac{1}{2}L_{qh}$.

###### Proof.

$\forall f\in H^{1}(X,TX)$, we want to show that $L_{qh}(f)=2H^{1}(s_{0})(f)$.
Let $((\eta^{\prime}_{ij})_{i,j\in I},(\hat{\sigma}^{\prime}_{i})_{i\in I})$
be a representative of an element $v\in\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$. Using Lemma 7.1,

$\omega_{H}(L_{qh}(f),v)=d(qh^{\ast}f)(v)=df(qh_{\ast}v)=df(2<\hat{s}_{0},\sigma>)=f(2<\hat{s}_{0},\sigma>)$

Using Lemma 7.2,

$\omega_{H}(H^{1}(s_{0})(f),v)=\omega_{H}((s_{0}(f),0),(\eta_{ij},\sigma_{i}))=f(<\hat{s}_{0},\sigma>)$

So $L_{qh}(f)=2H^{1}(s_{0})(f)$, $\forall f\in H^{1}(X,TX)$. Therefore
$H^{1}(s_{0})=\frac{1}{2}L_{qh}$. ∎

## References

  * [1] Ben-Zvi, David; Frenkel, Edward Geometric realization of the Segal-Sugawara construction. Topology, geometry and quantum field theory, 46 C97, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004
  * [2] José Bertin, Introduction to Hodge Theory, AMS Bookstore, 2002.
  * [3] P. Griffiths, Periods of integrals on algebraic manifolds II (Local Study of the Period Mapping) Amer. J. Math. , 90 (1968) pp. 808 C865
  * [4] Hitchin, N. J. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59 C126.
  * [5] M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlev e equation of type V I, Part I , Publ. Res. Inst. Math. Sci. 42, no. 4 (2006), 987 C1089. (math.AG/0309342).
  * [6] Markman, Eyal Spectral curves and integrable systems. Compositio Math. 93 (1994), no. 3, 255 C290.
  * [7] Mumford, David Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Stockholm, 1962)
  * [8] C. Simpson, The Hodge filtration on nonabelian cohomology, Algebraic geometry Santa Cruz 1995, Proc. Sympos. Pure Math., 62, Part 2, A.M.S. (1997), 217-281.