Moduli of Parabolic Connetions with Quadratic Differential

The Moduli Spaces of Parabolic Connections
with a Quadratic Differential
and Isomonodromic Deformations

Arata KOMYO

A. Komyo

Department of Mathematics, Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan<EMAIL_ADDRESS>
Received January 23, 2018, in final form October 03, 2018; Published online
October 13, 2018

In this paper, we study the moduli spaces of parabolic connections with a
quadratic differential. We endow these moduli spaces with symplectic
structures by using the fundamental 2-forms on the moduli spaces of parabolic
connections (which are phase spaces of isomonodromic deformation systems).
Moreover, we see that the moduli spaces of parabolic connections with a
quadratic differential are equipped with structures of twisted cotangent
bundles.

parabolic connection; quadratic differential; isomonodromic deformation;
twisted cotangent bundle

14D20; 34M56

##  1 Introduction

Let $C$ be a smooth projective curve of genus $g$ (where $g\geq 2$).
Narasimhan–Seshadri [20] showed that vector bundles on $C$ are stable if and
only if they arise from irreducible unitary representations of the fundamental
group of $C$. The moduli space of stable vector bundles on $C$ is equipped
with a natural symplectic structure. Although the complex structure on this
moduli space depends on the complex structure of $C$, the symplectic structure
of this moduli space depends only on the underlying topological surface of
$C$. This picture has been investigated by Atiyah–Bott [1] and Goldman [11].
There exist generalizations of this picture. One can consider the moduli space
of pairs $(E,\nabla)$ where $(E,\nabla)$ is a rank $r$ vector bundle on $C$
with a holomorphic connection $\nabla$. This moduli space is equipped with a
(holomorphic) symplectic structure. There exists an analytic isomorphism
between the moduli space of pairs $(E,\nabla)$ and the moduli space of
representations of the fundamental group of $C$ into
$\mathrm{GL}(r,\mathbb{C})$ by taking a holomorphic connection to its
monodromy representation. Considering the variation of this isomorphism when
deforming the curve, we can define the isomonodromic foliation on the moduli
space of triples $(C,E,\nabla)$. This foliation is transversal to the
fibration $(C,E,\nabla)\mapsto C$ of complementary dimension. There exists a
closed $2$-form on the moduli space of triples $(C,E,\nabla)$ such that the
kernel of the closed $2$-form coincides with the tangent spaces of leaves of
the foliation and this $2$-form induces a (holomorphic) symplectic structure
on the moduli space of pairs $(E,\nabla)$ over a fixed curve $C$. This
generalization has been investigated by Goldman [11], Hitchin [12], and
Simpson [21, 22]. Moreover, this generalized picture was generalized to the
singular setting by Iwasaki [17], Hitchin [13], Boalch [9], and Krichever
[19]. Remark that, in the logarithmic case, Inaba–Iwasaki–Saito [16] and Inaba
[15] have constructed the moduli scheme of triples $(C,E,\nabla)$ (satisfying
some stability condition) and showed that the closed 2-form on this moduli
scheme is algebraic.

We recall the definitions of Lagrangian triples and Hamiltonian data, which
are discussed in [3]. Let $p\colon X\rightarrow S$ be a smooth morphism of
smooth varieties. A $p$-connection is an $\mathcal{O}_{X}$-linear morphism
$\nabla_{S}\colon p^{*}\Theta_{S}\rightarrow\Theta_{X}$ such that
$dp\circ\nabla_{S}=\mathrm{id}_{p^{*}\Theta_{S}}$. Here $\Theta_{S}$ and
$\Theta_{X}$ are the tangent sheaves of $S$ and $X$, respectively. A
$p$-connection $\nabla_{S}$ is integrable if the corresponding map
$\Theta_{S}\rightarrow p_{*}\Theta_{X}$ commutes with brackets. Note that an
integrable $p$-connection $\nabla_{S}$ defines an action of $\Theta_{S}$ on
relative differential forms $\Omega_{X/S}$ by the Lie derivatives along
horizontal vector field $\nabla_{S}(\Theta_{S})$. A form
$\omega\in\Omega^{2}_{X/S}$ is $\nabla_{S}$-horizontal if $\omega$ is fixed by
the $\Theta_{S}$-action.

######  Definition 1.1.

Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and
$T^{*}=T^{*}(X)\rightarrow X$ be the cotangent bundle on $X$. A twisted
cotangent bundle on $X$ is a $T^{*}$-torsor $\pi_{\phi}\colon\phi\rightarrow
X$ (i.e., $\pi_{\phi}$ is a fibration equipped with a simple transitive action
of $T^{*}$ along the fibers) together with a symplectic form $\omega_{\phi}$
on $\phi$ such that $\pi_{\phi}$ is a polarization for $\omega_{\phi}$ (i.e.,
$\dim\phi=2\dim X$ and the Poisson bracket $\\{\cdot,\cdot\\}$ vanishes on
$\pi_{\phi}^{*}\mathcal{O}_{X}$) and for any $1$-form $\nu$ on an open set
$U\subset X$ one has
$t^{*}_{\nu}(\omega_{\phi})=\pi_{\phi}^{*}d\nu+\omega_{\phi}$ on
$\pi_{\phi}^{-1}(U)$. Here
$t_{\nu}\colon\pi_{\phi}^{-1}(U)\rightarrow\pi_{\phi}^{-1}(U)$;
$t_{\nu}(a)=a+\nu_{\pi(a)}$ is the translation by $\nu$.

For example, the map from the moduli space of pairs $(E,\nabla)$ to the moduli
space of vector bundles defined by $(E,\nabla)\mapsto E$ is a twisted
cotangent bundle on the moduli space of vector bundles (see [10, Lemma IV.4]
and [7, Section 4]). Note that this moduli space of vector bundles is a smooth
algebraic stack. We can define a twisted cotangent bundle on a smooth
algebraic stack in the same way.

######  Definition 1.2.

Let $S$ be a smooth variety. An $S$-Lagrangian triple consists of a morphism
$\pi\colon X\rightarrow Y$ of $S$-varieties $p_{X}\colon X\rightarrow S$ and
$p_{Y}\colon Y\rightarrow S$, a relative 2-form $\omega\in\Omega_{X/S}^{2}(X)$
and a $p_{X}$-connection $\nabla_{S}$ such that

  * (i)

$p_{X}$, $p_{Y}$ and $\pi$ are smooth surjective morphisms,

  * (ii)

the form $\omega$ is closed and non-degenerate,

  * (iii)

for any $s\in S$ the morphism $\pi_{s}\colon X_{s}\rightarrow Y_{s}$ is a
twisted cotangent bundle over $Y_{s}$, and

  * (iv)

$\nabla_{S}$ is integrable and $\omega$ is $\nabla_{S}$-horizontal.

######  Definition 1.3.

An $S$-Hamiltonian datum on an $S$-variety $p_{Y}\colon Y\rightarrow S$
consists of

  * (i)

a twisted cotangent bundle
$\big{(}\widetilde{X},\omega_{\widetilde{X}}\big{)}$,
$\tilde{\pi}\colon\widetilde{X}\rightarrow Y$ over $Y$. Put $X:=\widetilde{X}\
\mathrm{mod}\ p^{*}_{Y}\Omega_{S}^{1}$: this is a $\Theta_{Y/S}^{*}$-torsor
over $Y$; let $\widetilde{X}\xrightarrow{r}X\xrightarrow{\pi}Y$ be the
projections and

  * (ii)

a section $h\colon X\rightarrow\widetilde{X}$ of $r$ (called Hamiltonian)

such that for each $x\in X$ the form
$(\omega_{X})_{x}\in\bigwedge^{2}\Theta^{*}_{X,x}$ has rank $\dim X-\dim S$.
Here we put $\omega_{X}:=h^{*}\omega_{\widetilde{X}}$, which is a closed
2-form on $X$.

Remark that the twisted cotangent bundle $\widetilde{X}$ over $Y$ is
isomorphic to the fiber product $X\times_{S}T^{*}S$ as symplectic manifolds.
This isomorphism is given by the morphism
$\tilde{r}\colon\widetilde{X}\rightarrow$ $X\times_{S}T^{*}S$,
$\tilde{r}(\tilde{x})=(r(\tilde{x}),\tilde{x}-h(r(\tilde{x})))$. Here the
symplectic form on $X\times_{S}T^{*}S$ is equal to the sum of $\omega_{X}$ and
a standard symplectic form on $T^{*}S$. Now, we describe a construction of
$S$-Lagrangian triples from $S$-Hamiltonian data
$\big{(}\widetilde{X},\omega_{\widetilde{X}},\tilde{\pi},h\big{)}$. Let
$\pi\colon X\rightarrow Y$ be the map as in Definition 1.3. For each $s\in S$,
the map $\pi_{s}\colon X_{s}\rightarrow Y_{s}$ is a $T^{*}(Y_{s})$-torsor
induced by the $\Theta_{Y/S}^{*}$-torsor $X\rightarrow Y$. Let $\omega$ be the
image of $\omega_{X}$ under the natural morphism
$\Omega^{2}_{X}(X)\rightarrow\Omega^{2}_{X/S}(X)$. For the natural map
$\iota_{X_{s}}\colon X_{s}\rightarrow X$, the pull-back
$\iota^{*}_{X_{s}}\omega$ is a symplectic form on $X_{s}$. Let
$a\in\Omega^{1}_{Y_{s}}(U)$ be a local section over an open set $U\subset
Y_{s}$. We take a collection $\\{(U_{i},\tilde{a}_{i})\\}_{i}$ where
$\\{U_{i}\\}_{i}$ is an open covering of $U$ and
$\tilde{a}_{i}\in\Omega^{1}_{Y}|_{Y_{s}}(U_{i})$ such that
$\iota_{Y_{s}}^{*}(\tilde{a}_{i})=a|_{U_{i}}$, where $\iota_{Y_{s}}\colon
Y_{s}\rightarrow Y$ is the natural map. We can show that
$t_{a}^{*}(\iota^{*}_{X_{s}}h^{*}\omega_{\widetilde{X}})=(h|_{X_{s}})^{*}t_{\tilde{a}_{i}}^{*}(\iota^{*}_{\widetilde{X}_{s}}\omega_{\widetilde{X}})$
on $\pi^{-1}_{s}(U_{i})$, where
$\iota_{\widetilde{X}_{s}}\colon\widetilde{X}_{s}\rightarrow\widetilde{X}$ is
the natural map. (Here note that $\widetilde{X}$ is isomorphic to
$X\times_{S}T^{*}S$.) In particular, the right-hand side is independent of the
choice of a lift $\tilde{a}_{i}$ of $a|_{U_{i}}$. Then
$t_{a}^{*}(\iota^{*}_{X_{s}}\omega)-\iota^{*}_{X_{s}}\omega=(h|_{X_{s}})^{*}t_{\tilde{a}_{i}}^{*}(\iota^{*}_{\widetilde{X}_{s}}\omega_{\widetilde{X}})-(h|_{X_{s}})^{*}(\iota^{*}_{\widetilde{X}_{s}}\omega_{\widetilde{X}})=(h|_{X_{s}})^{*}(\tilde{\pi}|_{\widetilde{X}_{s}})^{*}d(\tilde{a}_{i})=\pi_{s}^{*}da$.
We have that $\pi_{s}\colon X_{s}\rightarrow Y_{s}$ is a twisted cotangent
bundle. Put $p_{X}:=p_{Y}\circ\pi$. The kernels of $(\omega_{X})_{x}$ for each
$x\in X$ form a subbundle of the tangent bundle $TX$, which is transversal to
fibers of $p_{X}$. Since the form
$(\omega_{X})_{x}\in\bigwedge^{2}\Theta^{*}_{X,x}$ has rank $\dim X-\dim S$
and $\omega_{X}$ is closed, this subbundle defines an integrable
$p_{X}$-connection $\nabla_{S}$. By the construction, $\omega$ is
$\nabla_{S}$-horizontal. Then $(\pi\colon X\rightarrow Y,\omega,\nabla_{S})$
is an $S$-Lagrangian triple. The purpose of this paper is to construct
$S$-Hamiltonian data
$\big{(}\widetilde{X},\omega_{\widetilde{X}},\tilde{\pi},h\big{)}$ from
$S$-Lagrangian triples $(\pi\colon X\rightarrow Y,\omega_{X},\nabla_{S})$ by
using concrete argument in the case of isomonodromic deformations. (There
exists a more abstract construction in [3] for a general case.) Now, following
[3], we describe that the Hamiltonian $h\colon X\rightarrow\widetilde{X}$ of
an $S$-Hamiltonian datum is locally given by local functions and the
integrable $p_{X}$-connection $\nabla_{S}$ associated to the $S$-Hamiltonian
datum has a description by these functions. Let $x$ be a point of $X$. Let
$y=\pi(x)$ and $s=p_{X}(x)$ be the projections of $x$. Let
$(t_{a})_{a=1,\ldots,\dim S}$ be local coordinates on a neighborhood of $s\in
S$ and $q_{i}$, $i=1,\ldots,\dim Y_{s}$, be functions on a neighborhood of
$y\in Y$ such that $(q_{i},t_{a})_{i,a}$ are local coordinates at $y$ on $Y$.
Here we denote pull-backs of local functions by the same notations as the
local functions for simplicity. Choose functions $h_{a}$ and $p_{i}$ on a
neighborhood of $h(x)\in\widetilde{X}$ such that

$\displaystyle\omega_{\widetilde{X}}=\sum\limits_{i=1}^{\dim
Y_{s}}dp_{i}\wedge dq_{i}+\sum\limits_{a=1}^{\dim S}dh_{a}\wedge dt_{a}.$

Then $(\boldsymbol{q}=(q_{i})_{i}$, $\boldsymbol{p}=(p_{i})_{i}$,
$\boldsymbol{t}=(q_{a})_{a})$ are local coordinates at $x$ on $X$. The
Hamiltonian $h\colon X\rightarrow\widetilde{X}$ is given by the functions
$h_{a}(\boldsymbol{q},\boldsymbol{p},\boldsymbol{t})$. Note that

$\displaystyle\omega_{X}=\sum\limits_{i=1}^{\dim Y_{s}}dp_{i}\wedge
dq_{i}+\sum\limits_{a=1}^{\dim
S}dh_{a}(\boldsymbol{q},\boldsymbol{p},\boldsymbol{t})\wedge dt_{a}.$

Put

$\displaystyle v_{h_{a}}=\partial_{t_{a}}+\sum\limits_{i=1}^{\dim
Y_{s}}\partial_{q_{i}}(h_{a}(\boldsymbol{q},\boldsymbol{p},\boldsymbol{t}))\partial_{p_{i}}-\partial_{p_{i}}(h_{a}(\boldsymbol{q},\boldsymbol{p},\boldsymbol{t}))\partial_{q_{i}}.$

We can check
$\omega_{X}(\partial_{p_{i}},v_{h_{a}})=\omega_{X}(\partial_{q_{i}},v_{h_{a}})=0$
easily. Moreover we have $\omega_{X}(\partial_{t_{b}},v_{h_{a}})=0$,
$a,b=1,\ldots,\dim S$, since for each $x\in X$ the form
$(\omega_{X})_{x}\in\bigwedge^{2}\Theta^{*}_{X,x}$ has rank $\dim X-\dim S$.
Then we have a description of $\nabla_{S}$ by
$h_{a}(\boldsymbol{q},\boldsymbol{p},\boldsymbol{t})$:
$\nabla_{S}(\partial_{t_{a}})=v_{h_{a}}$.

In this paper, we consider an $S$-Lagrangian triples $(\pi\colon X\rightarrow
Y,\omega_{X},\nabla_{S})$ associated to isomonodromic deformations of
parabolic connections (which are logarithmic connection with quasi-parabolic
structures). The isomonodromic deformations of parabolic connections have been
investigated by Inaba–Iwasaki–Saito [16] and Inaba [15]. In our case, $X$ is a
moduli space of pointed smooth projective curves and parabolic connections
(see [15, Theorem 2.1] and [16]), $Y$ is a moduli space of pointed smooth
projective curves and quasi-parabolic bundles admitting a parabolic
connection, and $S$ is a moduli space of pointed smooth projective curves. We
have projections $p_{X}\colon X\rightarrow S$, $p_{Y}\colon Y\rightarrow S$
and $\pi\colon X\rightarrow Y$. The moduli space $X$ has the relative
symplectic form $\omega$ over $S$ (see [15, Section 7]). The
$p_{X}$-connection $\nabla_{S}$ is given by the isomonodromic deformations of
parabolic connections (see [15, Proposition 8.1]). The main result of this
paper is to construct the corresponding twisted cotangent bundle
$\widetilde{X}$ over $Y$ by using computation of Čech cohomologies. We
construct the twisted cotangent bundle with the remark in mind: The twisted
cotangent bundle $\widetilde{X}$ over $Y$ is isomorphic to the fiber product
$X\times_{S}T^{*}S$.

Our argument is as follows. First, we consider the fiber product
$X\times_{S}T^{*}S$ (which called extended phase space, see [14, Section 7]).
The fiber product $X\times_{S}T^{*}S$ is the moduli space of (pointed smooth
projective curves and) parabolic connections with a quadratic differential. We
describe the tangent sheaf of $X\times_{S}T^{*}S$ and the symplectic form on
$X\times_{S}T^{*}S$ by the Čech cohomology (Propositions 3.1 and 3.6). Second,
we describe the cotangent sheaf $\Omega_{Y}^{1}$ by the Čech cohomology, and
we define an $\Omega_{Y}^{1}$-action on $X\times_{S}T^{*}S$ explicitly
(Definition 4.3). We show that by this $\Omega_{Y}^{1}$-action and the
symplectic form, $X\times_{S}T^{*}S$ is a twisted cotangent bundle over $Y$
(Theorem 4.4). The section $X\rightarrow X\times_{S}T^{*}S$ given by the zero
section of $T^{*}S\rightarrow S$ is the Hamiltonian of the Hamiltonian datum.

A twisted cotangent bundle over $Y$ is important for studying quantizations of
isomonodromic deformations. In fact, quantizations of isomonodromic
deformations may be described by using certain algebras of twisted
differential operators, which are quantizations of twisted cotangent bundles
(see [3, 7]). It is expected that the results of this paper are useful to
understand quantizations of isomonodromic deformations in the context of a
certain algebro-geometric way such as [15, 16].

The organization of this paper is as follows. In Section 2, we recall basic
definitions and basic facts on parabolic connections (in Section 2.1), Atiyah
algebras (in Section 2.2) and twisted cotangent bundles (in Section 2.3). In
Section 3, we treat moduli spaces of parabolic connections with a quadratic
differential. First, we describe the tangent sheaves of these moduli spaces in
terms of the hypercohomology of a certain complex. Second, we endow the moduli
spaces with symplectic structures. In Section 4, we see that the moduli spaces
of parabolic connections with a quadratic differential are equipped with
structures of twisted cotangent bundles.

##  2 Preliminaries

###  2.1 Moduli space of stable parabolic connections

Following [15], we recall basic definitions and basic facts on parabolic
connections. Let $C$ be a smooth projective curve of genus $g$. We put

$\displaystyle T_{n}:=\\{(t_{1},\ldots,t_{n})\in C\times\cdots\times
C\,|\,t_{i}\neq t_{j}\ \text{for $i\neq j$}\\}$

for a positive integer $n$. For integers $e$, $r$ with $r>0$, we put

$\displaystyle N^{(n)}_{r}(e):=\bigg{\\{}\big{(}\nu^{(i)}_{j}\big{)}^{1\leq
i\leq n}_{0\leq j\leq
r-1}\in\mathbb{C}^{nr}\,\bigg{|}\,e+\sum_{i,j}\nu^{(i)}_{j}=0\bigg{\\}}.$

Take members $\boldsymbol{t}=(t_{1},\ldots,t_{n})\in T_{n}$ and
$\boldsymbol{\nu}=\big{(}\nu^{(i)}_{j}\big{)}_{1\leq i\leq n,0\leq j\leq
r-1}\in N^{(n)}_{r}(e)$.

######  Definition 2.1.

We say $\big{(}E,\nabla,\big{\\{}l_{*}^{(i)}\big{\\}}_{1\leq i\leq n}\big{)}$
is a $(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connection of rank $r$ and
degree $e$ over $C$ if

  * (1)

$E$ is a rank $r$ algebraic vector bundle on $C$,

  * (2)

$\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(t_{1}+\cdots+t_{n})$ is a
connection, that is, $\nabla$ is a $\mathbb{C}$-linear homomorphism of sheaves
satisfying $\nabla(fa)=a\otimes df+f\nabla(a)$ for $f\in\mathcal{O}_{C}$ and
$a\in E$, and

  * (3)

for each $t_{i}$, $l_{*}^{(i)}$ is a filtration $E|_{t_{i}}=l_{0}^{(i)}\supset
l_{1}^{(i)}\supset\cdots\supset l_{r}^{(i)}=0$ such that
$\dim\big{(}l_{j}^{(i)}/l_{j+1}^{(i)}\big{)}=1$ and $\big{(}{\sf
res}_{t_{i}}(\nabla)-\nu^{(i)}_{j}\mathrm{id}_{E|_{t_{i}}}\big{)}\big{(}l^{(i)}_{j}\big{)}\subset
l_{j+1}^{(i)}$ for $j=0,\ldots,r-1$.

######  Remark 2.2.

We have

$\displaystyle\deg E=\deg(\det(E))=-\sum^{n}_{i=1}\mathrm{tr}({\sf
res}_{t_{i}}(\nabla))=-\sum^{n}_{i=1}\sum^{r-1}_{j=0}\nu_{j}^{(i)}=e.$

######  Definition 2.3 ([15, Definition 2.3]).

Take an element $\boldsymbol{\nu}\in N^{(n)}_{r}(e)$. We call
$\boldsymbol{\nu}$ special if

  * (1)

$\nu_{j}^{(i)}-\nu_{k}^{(i)}\in\mathbb{Z}$ for some $i$ and $j\neq k$, or

  * (2)

there exists an integer $s$ with $1<s<r$ and a subset
$\\{j_{1}^{i},\ldots,j_{s}^{i}\\}\subset\\{0,\ldots,r-1\\}$ for each $1\leq
i\leq n$ such that
$\sum\limits_{i=1}^{n}\sum\limits^{s}_{k=1}\nu_{j^{i}_{k}}^{(i)}\in\mathbb{Z}$.

We call $\boldsymbol{\nu}$ generic if it is not special.

Take rational numbers
$0<\alpha^{(i)}_{1}<\alpha^{(i)}_{2}<\cdots<\alpha^{(i)}_{r}<1$ for
$i=1,\ldots,n$ satisfying
$\alpha^{(i)}_{j}\neq\alpha^{(i^{\prime})}_{j^{\prime}}$ for
$(i,j)\neq(i^{\prime},j^{\prime})$. We choose a sufficiently generic
$\boldsymbol{\alpha}=(\alpha_{j}^{(i)})$.

######  Definition 2.4.

A parabolic connection $\big{(}E,\nabla,\big{\\{}l^{(i)}_{*}\big{\\}}_{1\leq
i\leq n}\big{)}$ is $\boldsymbol{\alpha}$-stable (resp.
$\boldsymbol{\alpha}$-semistable) if for any proper nonzero subbundle
$F\subset E$ satisfying $\nabla(F)\subset
F\otimes\Omega^{1}_{C}(t_{1}+\cdots+t_{n})$, the inequality

$\displaystyle\frac{\deg
F+\sum\limits^{n}_{i=1}\sum\limits^{r}_{j=1}\alpha^{(i)}_{j}\dim\big{(}\big{(}F|_{t_{i}}\cap
l_{j-1}^{(i)}\big{)}/\big{(}F|_{t_{i}}\cap
l_{j}^{(i)}\big{)}\big{)}}{\operatorname{rank}F}$
$\displaystyle\qquad{}\underset{(\text{resp. $\leq$})}{<}\frac{\deg
E+\sum\limits^{n}_{i=1}\sum\limits^{r}_{j=1}\alpha^{(i)}_{j}\dim\big{(}l_{j-1}^{(i)}/l_{j}^{(i)}\big{)}}{\operatorname{rank}E}$

holds.

Let $\tilde{M}_{g,n}$ be a smooth algebraic scheme which is a certain covering
of the moduli stack of $n$-pointed smooth projective curves of genus $g$ over
$\mathbb{C}$ and take a universal family
$\big{(}\mathcal{C},\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{)}$ over
$\tilde{M}_{g,n}$.

######  Definition 2.5.

We denote the pull-back of $\mathcal{C}$ and $\tilde{\boldsymbol{t}}$ by the
morphism $\tilde{M}_{g,n}\times N^{(n)}_{r}(e)\rightarrow\tilde{M}_{g,n}$ by
the same character $\mathcal{C}$ and
$\tilde{\boldsymbol{t}}=\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}$.
Then
$D\big{(}\tilde{\boldsymbol{t}}\big{)}:=\tilde{t}_{1}+\cdots+\tilde{t}_{n}$
becomes an effective Cartier divisor on $\mathcal{C}$ flat over
$\tilde{M}_{g,n}\times N^{(n)}_{r}(e)$. We also denote by
$\tilde{\boldsymbol{\nu}}$ the pull-back of the universal family on
$N^{(n)}_{r}(e)$ by the morphism $\tilde{M}_{g,n}\times
N^{(n)}_{r}(e)\rightarrow N^{(n)}_{r}(e)$. We define a functor
$\mathcal{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$
from the category of locally noetherian schemes over $\tilde{M}_{g,n}\times
N^{(n)}_{r}(e)$ to the category of sets by

$\displaystyle\mathcal{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}(S):=\big{\\{}\big{(}E,\nabla,\big{\\{}l^{(i)}_{j}\big{\\}}\big{)}\big{\\}}/{\sim}$

for a locally noetherian scheme $S$ over $\tilde{M}_{g,n}\times
N^{(n)}_{r}(e)$, where

  * (1)

$E$ is a rank $r$ algebraic vector bundle on $\mathcal{C}_{S}$,

  * (2)

$\nabla\colon E\rightarrow
E\otimes\Omega^{1}_{\mathcal{C}_{S}/S}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}_{S}\big{)}$
is a relative connection,

  * (3)

for each $(\tilde{t}_{i})_{S}$, $l_{*}^{(i)}$ is a filtration by subbundles
$E|_{(\tilde{t}_{i})_{S}}=l_{0}^{(i)}\supset l_{1}^{(i)}\supset\cdots\supset
l_{r}^{(i)}=0$ such that $\big{(}{\sf
res}_{(\tilde{t}_{i})_{S}}(\nabla)-\big{(}\tilde{\nu}^{(i)}_{j}\big{)}_{S}\mathrm{id}_{E|_{t_{i}}}\big{)}\big{(}l^{(i)}_{j}\big{)}\subset
l_{j+1}^{(i)}$ for $j=0,\ldots,r-1$, and

  * (4)

for any geometric point $s\in S$,
$\dim\big{(}l_{j}^{(i)}/l_{j+1}^{(i)}\big{)}\otimes k(s)=1$ for any $i$, $j$
and $\big{(}E,\nabla,\big{\\{}l^{(i)}_{j}\big{\\}}\big{)}\otimes k(s)$ is
$\boldsymbol{\alpha}$-stable.

Here
$\big{(}E,\nabla,\big{\\{}l^{(i)}_{j}\big{\\}}\big{)}\sim\big{(}E^{\prime},\nabla^{\prime},\big{\\{}l^{\prime(i)}_{j}\big{\\}}\big{)}$
if there exist a line bundle $\mathcal{L}$ on $S$ and an isomorphism
$\sigma\colon E\xrightarrow{\sim}E^{\prime}\otimes\mathcal{L}$ such that
$\sigma|_{t_{i}}\big{(}l^{(i)}_{J}\big{)}=l^{\prime(i)}_{j}\otimes\mathcal{L}$
for any $i$, $j$ and the diagram

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\scriptstyle{\sigma}$$\textstyle{E\otimes\Omega^{1}_{\mathcal{C}/T}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\otimes\mathrm{id}}$$\textstyle{E^{\prime}\otimes\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla^{\prime}\otimes\mathcal{L}}$$\textstyle{E^{\prime}\otimes\Omega^{1}_{\mathcal{C}/T}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\otimes\mathcal{L}}$

commutes.

######  Theorem 2.6 ([15, Theorem 2.1]).

For the moduli functor
$\mathcal{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$,
there exists a fine moduli scheme

$\displaystyle
M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}\longrightarrow\tilde{M}_{g,n}\times
N^{(n)}_{r}(e)$

of $\boldsymbol{\alpha}$-stable parabolic connections of rank $r$ and degree
$e$, which is smooth and quasi-projective. The fiber
$M_{C_{x}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}}_{x},\boldsymbol{\nu}\big{)}$
over $(x,\boldsymbol{\nu})\in\tilde{M}_{g,n}\times N^{(n)}_{r}(e)$ is the
moduli space of $\boldsymbol{\alpha}$-stable
$\big{(}\tilde{\boldsymbol{t}}_{x},\boldsymbol{\nu}\big{)}$-parabolic
connections whose dimension is $2r^{2}(g-1)+nr(r-1)+2$ if it is non-empty.

###  2.2 Atiyah algebras

Following [4, Section 1], we recall the Atiyah algebra. Let $C$ be a smooth
projective curve, and $\Theta_{C}$ be the tangent sheaf. Let $E$ be a vector
bundle of rank $r$ on $C$. Put $\mathcal{D}_{E}=\mathcal{D}{\rm
iff}(E,E)=\bigcup_{i}\mathcal{D}_{i}$, $\mathcal{D}_{i}$ is the sheaf of
differential operators of degree $\leq i$ on $E$. We have
$\mathcal{D}_{i}/\mathcal{D}_{i-1}=\mathcal{E}{\rm nd}(E)\otimes
S^{i}(\Theta_{C})$ where $S^{i}(\Theta_{C})$ is the $i$-th symmetric product
of $\Theta_{C}$. Let
$\mathrm{symb}_{1}\colon\mathcal{D}_{1}\rightarrow\mathcal{E}{\rm
nd}(E)\otimes\Theta_{C}$ be the natural morphism
$\mathcal{D}_{1}\rightarrow\mathcal{D}_{1}/\mathcal{O}_{C}=\mathcal{E}{\rm
nd}(E)\otimes\Theta_{C}$.

######  Definition 2.7.

We define the Atiyah algebra of $E$ as

$\displaystyle\mathcal{A}_{E}=\\{\partial\in\mathcal{D}_{1}\,|\,\mathrm{symb}_{1}(\partial)\in\mathrm{id}_{E}\otimes\Theta_{C}\subset\mathcal{E}{\rm
nd}(E)\otimes\Theta_{C}\\}.$

Here, for $v\in\mathcal{D}_{1}$, $\mathrm{symb}_{1}(v)$ is the symbol of the
differential operator $v$.

We have inclusions $\mathcal{D}_{0}=\mathcal{E}{\rm
nd}(E)\subset\mathcal{A}_{E}\subset\mathcal{D}_{1}$ and the short exact
sequence

$\displaystyle 0\longrightarrow\mathcal{E}{\rm
nd}(E)\longrightarrow\mathcal{A}_{E}\xrightarrow{\mathrm{symb}_{1}}\Theta_{C}\longrightarrow
0.$

Fix a positive integer $n$. Let $D=t_{1}+\cdots+t_{n}$ be an effective divisor
of $C$ where $t_{1},\ldots,t_{n}$ are distinct points of $C$. We put
$\mathcal{A}_{E}(D):=\mathrm{symb}_{1}^{-1}(\Theta_{C}(-D))$. Then we have the
following exact sequence

$\displaystyle 0\longrightarrow\mathcal{E}{\rm
nd}(E)\longrightarrow\mathcal{A}_{E}(D)\xrightarrow{\mathrm{symb}_{1}}\Theta_{C}(-D)\longrightarrow
0.$

For a connection $\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(D)$, we
define a splitting

$\displaystyle\iota(\nabla)\colon\
\Theta_{C}(-D)\longrightarrow\mathcal{A}_{E}(D)$ (2.1)

as follows. Let $U$ be an affine open subset of $C$ where we have a
trivialization $E|_{U}\cong\mathcal{O}_{U}^{\oplus r}$. We denote by
$Af^{-1}df$ a connection matrix of $\nabla$ on $U$ where $f$ is a local
defining equation of $t_{i}$ and $A\in M_{r}(\mathcal{O}_{U})$. For an element
$g\frac{\partial}{\partial f}\in\Theta_{C}(-D)(U)$, we define the element
$\iota(\nabla)\big{(}g\frac{\partial}{\partial
f}\big{)}:=g\big{(}\frac{\partial}{\partial
f}+Af^{-1}\big{)}\in\mathcal{A}_{E}(D)(U)$, which gives a map
$\iota(\nabla)(U)\colon\Theta_{C}(-D)(U)\rightarrow\mathcal{A}_{E}(D)(U)$. By
this map, we obtain the splitting (2.1).

###  2.3 Twisted cotangent bundles

Following [2, Section 2], we recall the definition of $\Omega^{\geq
1}_{X}$-torsors and recall the correspondence between twisted cotangent
bundles and $\Omega^{\geq 1}_{X}$-torsors. Let $X$ be a smooth algebraic
variety over $\mathbb{C}$.

######  Definition 2.8.

Let $d\colon A^{n}\rightarrow A^{n+1}$ be a morphism of sheaves of abelian
groups on $X$, considered as length $2$ complex $A^{\bullet}$ supported in
degree $n$ and $n+1$. An $A^{\bullet}$-torsor is a pair $(\mathcal{F},c)$,
where $\mathcal{F}$ is an $A^{n}$-torsor and $c\colon\mathcal{F}\rightarrow
A^{n+1}$ is a map such that $c(a+\phi)=d(a)+c(\phi)$ for $a\in
A^{n},\phi\in\mathcal{F}$.

Let $\Omega^{\geq 1}_{X}:=\big{(}\Omega_{X}^{1}\rightarrow\Omega_{X}^{2{\rm
cl}}\big{)}$ be the truncated de Rham complex, where $\Omega_{X}^{2{\rm cl}}$
are closed $2$-forms on $X$. For example, let $T^{*}X\rightarrow X$ be the
cotangent bundle of $X$ and $\theta_{X}$ be the canonical 1-form on $T^{*}X$.
The cotangent sheaf $\Omega_{X}^{1}$ of $X$ is an $\Omega_{X}^{1}$-torsor,
which is trivial. For the $\Omega_{X}^{1}$-torsor $\Omega_{X}^{1}$, we define
a map $c\colon\Omega_{X}^{1}\rightarrow\Omega_{X}^{2{\rm cl}}$ as follows. Let
$U$ be a Zariski open set over $X$. We assigne $\gamma\in\Omega_{X}^{1}(U)$
(which is a section $\gamma\colon U\rightarrow T^{*}X$ of $T^{*}X\rightarrow
X$ on $U$) to $\gamma^{*}d\theta_{X}\in\Omega_{X}^{2{\rm cl}}(U)$. For
$\gamma,\gamma^{\prime}\in\Omega_{X}^{1}(U)$, we have
$c(\gamma+\gamma^{\prime})-c(\gamma)=(\gamma+\gamma^{\prime})^{*}d\theta_{X}-(\gamma)^{*}d\theta_{X}=d(\gamma+\gamma^{\prime})-d\gamma=d\gamma^{\prime}$.
Then the pair $(\Omega_{X}^{1},c)$ is an $\Omega^{\geq 1}_{X}$-torsor.

We recall the correspondence between twisted cotangent bundles and
$\Omega^{\geq 1}_{X}$-torsors. For any morphism $f\colon X\rightarrow Y$
between algebraic varieties, let $\Gamma(f)$ be the sheaf of set on $Y$ where
$\Gamma(f)(U)$ is the set of sections of $f$ over $U$ for each open set
$U\subset Y$. If $f\colon X\rightarrow Y$ is a $T^{*}Y$-torsor, then
$\Gamma(f)$ is an $\Omega_{Y}^{1}$-torsor. We consider a twisted cotangent
bundle $\pi_{\phi}\colon\phi\rightarrow X$. Then $\Gamma(\pi_{\phi})$ is an
$\Omega^{1}_{X}$-torsor. We define a map
$c\colon\Gamma(\pi_{\phi})\rightarrow\Omega_{X}^{2{\rm cl}}$ by
$c(\gamma):=\gamma^{*}(\omega_{\phi})$. We have
$c(a+\gamma)-c(\gamma)=\gamma^{*}t_{a}^{*}(\omega_{\phi})-\gamma^{*}(\omega_{\phi})=da$.
Then $(\Gamma(\pi_{\phi}),c)$ is an $\Omega^{\geq 1}_{X}$-torsor. Conversely,
for an $\Omega^{\geq 1}_{X}$-torsor $(\mathcal{F},c)$, let
$\pi_{\phi}\colon\phi\rightarrow X$ be the space of the torsor $\mathcal{F}$.
The symplectic form is defined as the unique form such that for a section
$\gamma\in\mathcal{F}$ of $\pi_{\phi}$ the corresponding isomorphism
$T^{*}X\xrightarrow{\sim}\phi$; $0\rightarrow\gamma$, identifies
$\omega_{\phi}$ with $\omega+\pi^{*}c(\gamma)$. Here $\omega$ is the canonical
symplectic form on the cotangent bundle $T^{*}X$.

##  3 Moduli scheme of parabolic connections
with a quadratic differential

In this section, we study the moduli space of parabolic connections with a
quadratic differential, which is generalization of the moduli space of
parabolic connections studied by Inaba–Iwasaki–Saito [16] and Inaba [15]. In
Section 3.2, we describe the (algebraic) tangent sheaf of this moduli space in
terms of the hypercohomology of a certain complex by generalization of the
description of the tangent sheaf of the moduli space of parabolic connections
in [15, 16, 18]. Moreover, we describe the analytic tangent sheaf in terms of
the hypercohomology of a certain analytic complex as in [15, Section 7]. This
description is more simple than the algebraic one. In Section 3.3, we recall
the description of the vector fields associated to the isomonodromic
deformations in terms of the description of the (algebraic) tangent sheaf as
in [14, Section 6] and [18, Section 3.3]. In Section 3.4, we show that the
moduli space of parabolic connections with a quadratic differential is endowed
with a symplectic structures. This is the main purpose of this section. In
Section 3.5, we consider moduli spaces of parabolic connections with a
quadratic differential as extended phase spaces of isomonodromic deformations.
The classical trick of turning a time dependent Hamiltonian flow into an
autonomous one by adding variables is well-known. In this trick, the space
given by adding the variables to a phase space is called an extended phase
space. (Hamiltonians of isomonodromic deformations are time dependent.)

Hurtubise [14] also studied the moduli space of connections with a quadratic
differential. In [14, Section 7], the moduli space of connections with a
quadratic differential is decomposed locally into a product of the symplectic
manifolds: the moduli space of connections on a fixed curve and the cotangent
bundle of the moduli space of curves. This local decomposition is given by
using the isomonodromic deformation of the “background connection”, which is
discussed in [14, Section 6]. Then we can show that the moduli space of
connections with a quadratic differential is endowed with a symplectic
structure locally. Moreover, in [14] the section of the map from the moduli
space of connections with a quadratic differential to the moduli space of
connections is defined by using the Hamiltonians defined in [14, Section 6].
On the other hand, in our argument we use the ordinary isomonodromic
deformation instead of the isomonodromic deformation of the background
connection. Then we have an algebraic symplectic structure on the moduli space
of (parabolic) connections with a quadratic differential globally. Our
corresponding section of the map from the moduli space of connections with a
quadratic differential to the moduli space of connections is defined by using
the zero section of the map from the moduli space of curves with a quadratic
differential to the moduli space of curves.

###  3.1 Moduli space of stable parabolic connections with a quadratic
differential

Let $T^{*}\tilde{M}_{g,n}$ be the total space of the cotangent bundle of
$\tilde{M}_{g,n}$. We denote by
$\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$
the fiber product of $T^{*}\tilde{M}_{g,n}\times N^{(n)}_{r}(e)$ and
$M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$
over $\tilde{M}_{g,n}\times N^{(n)}_{r}(e)$:

$\textstyle{\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{T^{*}\tilde{M}_{g,n}\times
N^{(n)}_{r}(e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{M}_{g,n}\times
N^{(n)}_{r}(e).}$

We call the fiber product
$\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$
the moduli space of $\boldsymbol{\alpha}$-stable parabolic connections with a
quadratic differential. If we take a zero section of
$T^{*}\tilde{M}_{g,n}\rightarrow\tilde{M}_{g,n}$, then we have an inclusion

$\displaystyle
M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}\longrightarrow\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}.$

Let $(C,\boldsymbol{t})\in\tilde{M}_{g,n}$. The tangent space of
$\tilde{M}_{g,n}$ at $(C,\boldsymbol{t})$ is isomorphic to
$H^{1}(C,\Theta_{C}(-D(\boldsymbol{t})))$. By the Serre duality, the cotangent
space at $(C,\boldsymbol{t})$ is isomorphic to $H^{0}\big{(}C,\Omega^{\otimes
2}_{C}(D(\boldsymbol{t}))\big{)}$, which is the space of (global) quadratic
differentials on $(C,\boldsymbol{t})$.

###  3.2 Infinitesimal deformations

For simplicity, we put
$\widehat{M}:=\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$
and
$\mathcal{C}_{\widehat{M}}:=\mathcal{C}\times_{\tilde{M}_{g,n}}\widehat{M}$.
Let
$\big{(}\tilde{E},\tilde{\nabla},\big{\\{}\tilde{l}^{(i)}_{j}\big{\\}},\tilde{\psi}\big{)}$
be a universal family on $\mathcal{C}_{\widehat{M}}$. Let
$\mathcal{A}_{\tilde{E}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}$
be the relative Atiyah algebra which is the extension

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}{\rm
nd}\big{(}\tilde{E}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}_{\tilde{E}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{symb}_{1}}$$\textstyle{\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota\big{(}\tilde{\nabla}\big{)}}$$\textstyle{0,}$

where
$\iota\big{(}\tilde{\nabla}\big{)}\colon\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\rightarrow\mathcal{A}_{\tilde{E}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}$
is the $\mathcal{O}_{X}$-linear section of $\mathrm{symb}_{1}$ associated to
the relative connection $\tilde{\nabla}$. We put

$\displaystyle\widetilde{\mathcal{F}}^{0}:=\big{\\{}s\in\mathcal{E}{\rm
nd}\big{(}\tilde{E}\big{)}\,\big{|}\,s|_{\tilde{t}_{i}\times\widehat{M}}\big{(}\tilde{l}^{(i)}_{j}\big{)}\subset\tilde{l}^{(i)}_{j}\text{
for any $i$, $j$}\big{\\}}\qquad\text{and}$
$\displaystyle\mathcal{F}^{0}:=\big{\\{}s\in\mathcal{A}_{\tilde{E}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\,\big{|}\,(s-\iota\big{(}\tilde{\nabla}\big{)}\circ\mathrm{symb}_{1}(s))|_{\tilde{t}_{i}\times\widehat{M}}\big{(}\tilde{l}^{(i)}_{j}\big{)}\subset\tilde{l}^{(i)}_{j}\text{
for any $i$, $j$}\big{\\}}.$

Then we have an extension

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathcal{F}}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{symb}_{1}}$$\textstyle{\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota\big{(}\tilde{\nabla}\big{)}}$$\textstyle{0.}$

We put

$\displaystyle\widetilde{\mathcal{F}}^{1}:=\big{\\{}s\in\mathcal{E}{\rm
nd}\big{(}\tilde{E}\big{)}\otimes\Omega^{1}_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\,\big{|}\,{\sf
res}_{\tilde{t}_{i}\times
M_{\mathcal{C}/T}^{\boldsymbol{\alpha}}(\tilde{\boldsymbol{t}},r,e)}(s)\big{(}\tilde{l}^{(i)}_{j}\big{)}\subset\tilde{l}^{(i)}_{j+1}\text{
for any $i$, $j$}\big{\\}}\qquad\text{and}$
$\displaystyle\mathcal{F}^{1}:=\widetilde{\mathcal{F}}^{1}\oplus\Omega_{\mathcal{C}_{\widehat{M}}/\widehat{M}}^{\otimes
2}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}.$

We define a homomorphism
$d_{\tilde{\nabla}}\colon\widetilde{\mathcal{F}}^{0}\rightarrow\widetilde{\mathcal{F}}^{1}$
as $s\mapsto\tilde{\nabla}\circ s-s\circ\tilde{\nabla}$ and we define a
homomorphism
$d_{\tilde{\psi}}\colon\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\rightarrow\Omega_{\mathcal{C}_{\widehat{M}}/\widehat{M}}^{\otimes
2}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}$ as follows. Take an
affine open covering $\\{U_{\alpha}\\}$ of $\mathcal{C}_{\widehat{M}}$ such
that we can take trivializations of
$\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}$
and $\Omega_{\mathcal{C}_{\widehat{M}}/\widehat{M}}^{\otimes
2}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}$ on each $U_{\alpha}$.
For an element $a\partial/\partial
f_{\alpha}\in\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}(U_{\alpha})$,
we define a homomorphism on $U_{\alpha}$ by

$\displaystyle a\frac{\partial}{\partial
f_{\alpha}}\longmapsto\left(\frac{\partial\psi_{U_{\alpha}}}{\partial
f_{\alpha}}a+2\psi_{U_{\alpha}}\frac{\partial a}{\partial
f_{\alpha}}\right)df_{\alpha}\otimes
df_{\alpha}\in\Omega_{\mathcal{C}_{\widehat{M}}/\widehat{M}}^{\otimes
2}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}(U_{\alpha}),$ (3.1)

where $\tilde{\psi}|_{U_{\alpha}}=\psi_{U_{\alpha}}df_{\alpha}\otimes
df_{\alpha}$. By the homomorphism on each $U_{\alpha}$, we can define a
homomorphism $d_{\tilde{\psi}}$. We define a complex $\mathcal{F}^{\bullet}$
by the differential
$d_{\mathcal{F}^{\bullet}}=(d_{\tilde{\nabla}},d_{\tilde{\psi}})\circ\big{(}\mathrm{Id}-\iota\big{(}\tilde{\nabla}\big{)}\circ\mathrm{symb}_{1},\mathrm{symb}_{1}\big{)}$:

$\textstyle{\mathcal{F}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{Id}-\iota(\tilde{\nabla})\circ\mathrm{symb}_{1},\mathrm{symb}_{1})}$$\scriptstyle{d_{\mathcal{F}^{\bullet}}}$$\textstyle{\widetilde{\mathcal{F}}^{0}\oplus\Theta_{\mathcal{C}_{\widehat{M}}/\widehat{M}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(d_{\tilde{\nabla}},d_{\tilde{\psi}})}$$\textstyle{\widetilde{\mathcal{F}}^{1}\oplus\Omega_{\mathcal{C}_{\widehat{M}}/\widehat{M}}^{\otimes
2}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\hbox to0.0pt{.\hss}}$

######  Proposition 3.1.

We put
$\widehat{M}=\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$
and
$M=M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}$.
Let $\mathcal{F}_{M}^{0}$, $\widetilde{\mathcal{F}}_{M}^{0}$, and
$\widetilde{\mathcal{F}}_{M}^{1}$ be the pull-backs of $\mathcal{F}^{0}$,
$\widetilde{\mathcal{F}}^{0}$, and $\widetilde{\mathcal{F}}^{1}$ by the
natural immersion $\mathcal{C}_{M}\rightarrow\mathcal{C}_{\widehat{M}}$,
respectively. There exist canonical isomorphisms

$\displaystyle\hat{\varsigma}\colon\
\Theta_{\widehat{M}/N^{(n)}_{r}(e)}\xrightarrow{\,\sim\,}\mathbb{R}^{1}(\pi_{\widehat{M}})_{*}(\mathcal{F}^{\bullet}),$
$\displaystyle\tilde{\varsigma}\colon\
\Theta_{M/N^{(n)}_{r}(e)}\xrightarrow{\,\sim\,}\mathbb{R}^{1}(\pi_{M})_{*}\big{(}\mathcal{F}_{M}^{0}\rightarrow\widetilde{\mathcal{F}}_{M}^{1}\big{)},\qquad\text{and}$
$\displaystyle\varsigma\colon\ \Theta_{M/(\tilde{M}_{g,n}\times
N^{(n)}_{r}(e))}\xrightarrow{\,\sim\,}\mathbb{R}^{1}(\pi_{M})_{*}\big{(}\widetilde{\mathcal{F}}_{M}^{0}\rightarrow\widetilde{\mathcal{F}}_{M}^{1}\big{)},$

where $\pi_{\widehat{M}}\colon\mathcal{C}_{\widehat{M}}\rightarrow\widehat{M}$
and $\pi_{M}\colon\mathcal{C}_{M}\rightarrow M$ are the natural morphisms.

###### Proof.

We show the existence of the isomorphism $\hat{\varsigma}$. For the existence
of the isomorphisms $\tilde{\varsigma}$ and $\varsigma$, see the proof of [18,
Proposition 3.2] and the proof of [15, Theorem 2.1]. We take an affine open
set $\widehat{U}\subset\widehat{M}$. Let
$\big{(}\tilde{E},\tilde{\nabla},\big{\\{}\tilde{l}^{(i)}_{j}\big{\\}},\tilde{\psi}\big{)}$
be the family on $\mathcal{C}\times_{\tilde{M}_{g,n}}\widehat{U}$. We take an
affine open covering $\mathcal{C}_{\widehat{U}}=\bigcup_{\alpha}U_{\alpha}$
such that
$\phi_{\alpha}\colon\tilde{E}|_{U_{\alpha}}\xrightarrow{\sim}\mathcal{O}^{\oplus
r}_{U_{\alpha}}$ for any $\alpha$,
$\sharp\\{i\,|\,\tilde{t}_{i}|_{\mathcal{C}_{U}}\cap
U_{\alpha}\neq\varnothing\\}\leq 1$ for any $\alpha$ and
$\sharp\\{\alpha\,|\,\tilde{t}_{i}|_{\mathcal{C}_{U}}\cap
U_{\alpha}\neq\varnothing\\}\leq 1$ for any $i$. Take a relative tangent
vector field
$v\in\Theta_{\widehat{M}/N^{(n)}_{r}(e)}\big{(}\widehat{U}\big{)}$. The field
$v$ corresponds to a member
$\big{(}(C_{\epsilon},\boldsymbol{t}_{\epsilon},\psi_{\epsilon}),\big{(}E_{\epsilon},\nabla_{\epsilon},\big{\\{}(l_{\epsilon})^{(i)}_{j}\big{\\}}\big{)}\big{)}\in\widehat{M}(\operatorname{Spec}\mathcal{O}_{\widehat{U}}[\epsilon])$
such that
$\big{(}(C_{\epsilon},\boldsymbol{t}_{\epsilon},\psi_{\epsilon}),\big{(}E_{\epsilon},\nabla_{\epsilon},\big{\\{}(l_{\epsilon})^{(i)}_{j}\big{\\}}\big{)}\big{)}\otimes\mathcal{O}_{\widehat{U}}[\epsilon]/(\epsilon)\cong\big{(}\big{(}\mathcal{C}_{\widehat{U}},\boldsymbol{t}_{\widehat{U}},\tilde{\psi}\big{)},\big{(}\tilde{E},\tilde{\nabla},\big{\\{}\tilde{l}^{(i)}_{j}\big{\\}}\big{)}\big{)}$,
where
$\mathcal{O}_{\widehat{U}}[\epsilon]=\mathcal{O}_{\widehat{U}}[t]/\big{(}t^{2}\big{)}$.
Here,

  * •

$\psi_{\epsilon}\in H^{0}\big{(}C_{\epsilon},\Omega^{\otimes
2}_{C_{\epsilon}/\widehat{U}}\big{(}\log\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}_{\mathcal{O}_{\widehat{U}}[\epsilon]}\big{)}\big{)}\big{)}$,
and

  * •

$\nabla_{\epsilon}\colon E_{\epsilon}\rightarrow
E_{\epsilon}\otimes\Omega^{1}_{C_{\epsilon}/\widehat{U}}\big{(}\log\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}_{\mathcal{O}_{\widehat{U}}[\epsilon]}\big{)}\big{)}$
is a connection,

where we define the sheaf
$\Omega^{1}_{C_{\epsilon}/\widehat{U}}\big{(}\log\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}_{\mathcal{O}_{\widehat{U}}[\epsilon]}\big{)}\big{)}$
as the coherent subsheaf of
$\Omega^{1}_{C_{\epsilon}/\widehat{U}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}_{\mathcal{O}_{\widehat{U}}[\epsilon]}\big{)}$
locally generated by $f^{-1}df$ and $d\epsilon$ for a local defining equation
$f$ of
$D\big{(}\tilde{\boldsymbol{t}}\big{)}_{\mathcal{O}_{\widehat{U}}[\epsilon]}$
which is the pull-back of $D(\tilde{\boldsymbol{t}})$ by the morphism
$C_{\epsilon}\rightarrow\mathcal{C}_{\widehat{U}}\rightarrow\mathcal{C}$. Set
$U_{\alpha}^{\epsilon}:=U_{\alpha}\times\operatorname{Spec}\mathcal{O}_{\widehat{U}}[\epsilon]$.
Let

$\displaystyle\mu_{\alpha\beta}(\epsilon)\colon\
U_{\alpha\beta}\times\operatorname{Spec}\mathcal{O}_{\widehat{U}}[\epsilon]\xrightarrow{\,\sim\,}U_{\alpha\beta}\times\operatorname{Spec}\mathcal{O}_{\widehat{U}}[\epsilon]$
(3.2)

be an isomorphism associated to the first-order deformation $C_{\epsilon}$ of
$\mathcal{C}_{\widehat{U}}$. The isomorphism $\mu_{\alpha\beta}(\epsilon)$
satisfies

$\displaystyle\mu_{\alpha\beta}(\epsilon)^{*}(\epsilon)=\epsilon,\qquad\mu_{\alpha\beta}(\epsilon)^{*}(f)=f+\epsilon
d_{\alpha\beta}f,\qquad\text{for}\quad f\in\mathcal{O}_{U_{\alpha\beta}},$

for some
$d_{\alpha\beta}\in\Theta_{\mathcal{C}_{\widehat{U}}}(-D)(U_{\alpha\beta})$.
We describe $d_{\alpha\beta}$ as
$d_{\alpha\beta}=\frac{\partial\mu_{\alpha\beta}(\epsilon)}{\partial\epsilon}\frac{\partial}{\partial
f_{\alpha}}\in\Theta_{\mathcal{C}_{\widehat{U}}}(-D)(U_{\alpha\beta})$. Here,
$f_{\alpha}$ is a local defining equation of
$\tilde{t}_{i}|_{\mathcal{C}_{\widehat{U}}}\cap U_{\alpha}$. Set
$\phi_{\alpha}^{\epsilon}\colon
E_{\epsilon}|_{U^{\epsilon}_{\alpha}}\cong\mathcal{O}^{\oplus
r}_{U_{\alpha}^{\epsilon}}$. There is an isomorphism

$\displaystyle\varphi_{\alpha}\colon\
E_{\epsilon}|_{U^{\epsilon}_{\alpha}}\xrightarrow[\,\sim\,]{\,\phi_{\alpha}^{\epsilon}\,}\mathcal{O}^{\oplus
r}_{U_{\alpha}^{\epsilon}}\xrightarrow[\,\sim\,]{\,\phi_{\alpha}^{-1}\,}\tilde{E}|_{U_{\alpha}}\otimes\mathcal{O}_{\widehat{U}}[\epsilon]$

such that
$\varphi_{\alpha}\otimes\mathcal{O}_{\widehat{U}}[\epsilon]/(\epsilon)\colon
E_{\epsilon}\otimes\mathcal{O}_{\widehat{U}}[\epsilon]/(\epsilon)|_{U_{\alpha}}\xrightarrow{\sim}\tilde{E}|_{U_{\alpha}}\otimes\mathcal{O}_{\widehat{U}}[\epsilon]/(\epsilon)=\tilde{E}|_{U_{\alpha}}$
is the given isomorphism and that
$\varphi_{\alpha}|_{t_{i}\otimes\mathcal{O}_{\widehat{U}}[\epsilon]}((l_{\epsilon})_{j}^{(i)})=\tilde{l}_{j}^{(i)}|_{U_{\alpha}\times\operatorname{Spec}\mathcal{O}_{\widehat{U}}[\epsilon]}$
if $\tilde{t}_{i}|_{\mathcal{C}_{\widehat{U}}}\cap U_{\alpha}\neq\varnothing$.
Put

$\displaystyle\theta_{\alpha\beta}(\epsilon)\colon\ \mathcal{O}^{\oplus
r}_{U_{\alpha\beta}^{\epsilon}}\xrightarrow[\,\sim\,]{(\phi^{\epsilon}_{\beta})^{-1}|_{U_{\alpha\beta}^{\epsilon}}}E_{\epsilon}|_{U^{\epsilon}_{\alpha\beta}}\xrightarrow[\,\sim\,]{\phi_{\alpha}^{\epsilon}|_{U^{\epsilon}_{\alpha\beta}}}\mathcal{O}^{\oplus
r}_{U_{\alpha\beta}^{\epsilon}},$

which is an element of $\mathcal{E}{\rm nd}\big{(}\mathcal{O}^{\oplus
r}_{U_{\alpha\beta}^{\epsilon}}\big{)}(U_{\alpha\beta}^{\epsilon})$. We denote
$\theta_{\alpha\beta}(\epsilon)$ by

$\displaystyle\theta_{\alpha\beta}(\epsilon)=\tilde{\theta}_{\alpha\beta}+\epsilon\frac{\partial\theta_{\alpha\beta}(\epsilon)}{\partial\epsilon},\qquad\text{where}\quad\tilde{\theta}_{\alpha\beta},\frac{\partial\theta_{\alpha\beta}(\epsilon)}{\partial\epsilon}\in\mathcal{E}{\rm
nd}\big{(}\mathcal{O}^{\oplus r}_{U_{\alpha\beta}}\big{)}(U_{\alpha\beta}).$

Set

$\displaystyle\eta_{\alpha\beta}:=\frac{\partial\theta_{\alpha\beta}(\epsilon)}{\partial\epsilon}\big{(}\tilde{\theta}_{\alpha\beta}\big{)}^{-1}\in\mathcal{E}{\rm
nd}\big{(}\mathcal{O}^{\oplus r}_{U_{\alpha\beta}}\big{)}(U_{\alpha\beta}).$

We define elements $u_{\alpha\beta}\in\mathcal{F}^{0}(U_{\alpha\beta})$ and
$(v_{\alpha},w_{\alpha})\in\mathcal{F}^{1}(U_{\alpha})$ by

$\displaystyle u_{\alpha\beta}:=\
({\phi}_{\alpha}|_{U_{\alpha\beta}})^{-1}\circ\epsilon(d_{\alpha\beta}+\eta_{\alpha\beta})\circ{\phi}_{\alpha}|_{U_{\alpha\beta}},$
$\displaystyle
v_{\alpha}:=(\varphi_{\alpha}\otimes\mathrm{id})\circ\nabla_{\epsilon}|_{U_{\alpha}^{\epsilon}}\circ\varphi_{\alpha}^{-1}-\tilde{\nabla}|_{U_{\alpha}^{\epsilon}}\qquad\text{mod
$d\epsilon$},$ $\displaystyle
w_{\alpha}:=\psi_{\epsilon}|_{U_{\alpha}^{\epsilon}}-\tilde{\psi}|_{U_{\alpha}^{\epsilon}}\qquad\text{mod
$d\epsilon$},$

respectively. We can see that

$\displaystyle
u_{\beta\gamma}-u_{\alpha\gamma}+u_{\alpha\beta}=0,\qquad\text{and}\qquad
d_{\mathcal{F}^{\bullet}}(u_{\alpha\beta})=(v_{\beta},w_{\beta})-(v_{\alpha},w_{\alpha}).$

Then $[\\{u_{\alpha\beta}\\},\\{(v_{\alpha},w_{\alpha})\\}]$ determines an
element $\sigma_{\widehat{U}}(v)$ of
${\mathbf{H}}^{1}\big{(}\mathcal{F}_{\widehat{U}}^{\bullet}\big{)}$. We can
check that $v\mapsto\sigma_{\widehat{U}}(v)$ determines an isomorphism

$\displaystyle\Theta_{\widehat{M}/N^{(n)}_{r}(e)}\big{(}\widehat{U}\big{)}\xrightarrow{\,\sim\,}{\mathbf{H}}^{1}\big{(}\mathcal{F}_{\widehat{U}}^{\bullet}\big{)},\qquad
v\longmapsto\sigma_{\widehat{U}}(v).$

We denote by $\hat{\varsigma}_{\widehat{U}}$ this isomorphism. The isomorphism
$\hat{\varsigma}_{\widehat{U}}$ induces the desired isomorphism
$\hat{\varsigma}$. ∎

We describe the analytic tangent sheaf in terms of the hypercohomology of a
certain analytic complex. Let $\boldsymbol{\nu}$ be an element of
$N^{(n)}_{r}(e)$. Put
$\widehat{M}_{\boldsymbol{\nu}}=\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}$,
which is the fiber of $\boldsymbol{\nu}$ under
$\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}\rightarrow
N^{(n)}_{r}(e)$. Let
$j\colon\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}\setminus\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}_{\widehat{M}_{\boldsymbol{\nu}}}\rightarrow\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}$
be the canonical inclusion. Let
$\widehat{\boldsymbol{V}}:=\operatorname{Ker}\tilde{\nabla}^{{\rm
an}}|_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}\setminus\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}_{\widehat{M}_{\boldsymbol{\nu}}}}$
be the locally constant sheaf of the locally free
$(\pi_{\widehat{M}_{\boldsymbol{\nu}}}\circ
j)^{-1}\mathcal{O}_{\widehat{M}_{\boldsymbol{\nu}}}$-module associated to the
relative analytic connection $\tilde{\nabla}^{{\rm an}}$ on
$\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}\setminus\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}_{\widehat{M}_{\boldsymbol{\nu}}}$,
where
$\pi_{\widehat{M}_{\boldsymbol{\nu}}}\colon\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}\rightarrow\widehat{M}_{\boldsymbol{\nu}}$
is the natural map.

Assume that $\boldsymbol{\nu}$ is generic. We define a complex
$\big{(}\widehat{\mathcal{F}}^{\bullet}\big{)}^{{\rm an}}$ by

$\displaystyle\big{(}\widehat{\mathcal{F}}^{\bullet}\big{)}^{{\rm an}}\colon\
j_{*}\big{(}\mathcal{E}{\rm
nd}\big{(}\widehat{\boldsymbol{V}}\big{)}\big{)}\oplus\Theta_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}/\widehat{M}_{\boldsymbol{\nu}}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\xrightarrow{\
d_{\tilde{\psi}}\circ\,\mathrm{pr}_{2}\ }\Omega^{\otimes
2}_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}/\widehat{M}_{\boldsymbol{\nu}}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)},$

where $\mathrm{pr}_{2}$ is the second projection. We have the following
commutative diagram

$\textstyle{j_{*}\big{(}\mathcal{E}{\rm
nd}\big{(}\widehat{\boldsymbol{V}}\big{)}\big{)}\oplus\Theta_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}/\widehat{M}_{\boldsymbol{\nu}}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{\tilde{\psi}}\circ\,\mathrm{pr}_{2}}$$\textstyle{\Omega^{\otimes
2}_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}/\widehat{M}_{\boldsymbol{\nu}}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}\mathcal{F}^{0}\big{)}^{{\rm
an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(d_{\mathcal{F}^{\bullet}})^{{\rm
an}}}$$\textstyle{\big{(}\mathcal{F}^{1}\big{)}^{{\rm an}}.}$

We can show that the homomorphism $\operatorname{Ker}d_{\tilde{\nabla}^{{\rm
an}}}|_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}}\rightarrow
j_{*}\big{(}\mathcal{E}{\rm nd}\big{(}\widehat{\boldsymbol{V}}\big{)}\big{)}$
is an isomorphism and the homomorphism $d_{\tilde{\nabla}^{{\rm
an}}}\colon\big{(}\widetilde{\mathcal{F}}^{0}\big{)}^{{\rm
an}}\rightarrow\big{(}\widetilde{\mathcal{F}}^{1}\big{)}^{{\rm an}}$ is
surjective as in the proof of [15, Proposition 7.3]. Then we have the
following proposition.

######  Proposition 3.2.

If $\boldsymbol{\nu}$ is generic, then we have

$\displaystyle\mathbb{R}^{1}(\pi_{\widehat{M}_{\boldsymbol{\nu}}})_{*}\big{(}(\mathcal{F}^{\bullet})^{{\rm
an}}\big{)}\xrightarrow{\,\sim\,}\mathbb{R}^{1}(\pi_{\widehat{M}_{\boldsymbol{\nu}}})_{*}\big{(}\big{(}\widehat{\mathcal{F}}^{\bullet}\big{)}^{{\rm
an}}\big{)},$

where
$\pi_{\widehat{M}_{\boldsymbol{\nu}}}\colon\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}\rightarrow\widehat{M}_{\boldsymbol{\nu}}$
is the natural map.

###  3.3 Isomonodromic deformations

Let $\boldsymbol{\nu}$ be an element of $N^{(n)}_{r}(e)$. Put
$M_{\boldsymbol{\nu}}=M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}$
which is the fiber of $\boldsymbol{\nu}$ under
$M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}\rightarrow
N^{(n)}_{r}(e)$. Let
$j\colon\mathcal{C}_{M_{\boldsymbol{\nu}}}\setminus\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}_{M_{\boldsymbol{\nu}}}\rightarrow\mathcal{C}_{M_{\boldsymbol{\nu}}}$
be the canonical inclusion. Let $\operatorname{Ker}\tilde{\nabla}^{{\rm
an}}|_{\mathcal{C}_{M_{\boldsymbol{\nu}}}\setminus\\{\tilde{t}_{1},\ldots,\tilde{t}_{n}\\}_{M_{\boldsymbol{\nu}}}}$
be the locally constant sheaf of the locally free
$(\pi_{M_{\boldsymbol{\nu}}}\circ
j)^{-1}\mathcal{O}_{M_{\boldsymbol{\nu}}}$-module associated to the relative
analytic connection $\tilde{\nabla}^{{\rm an}}$ on
$\mathcal{C}_{M_{\boldsymbol{\nu}}}\setminus\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}_{M_{\boldsymbol{\nu}}}$,
where
$\pi_{M_{\boldsymbol{\nu}}}\colon\mathcal{C}_{M_{\boldsymbol{\nu}}}\rightarrow
M_{\boldsymbol{\nu}}$ is the natural map.

######  Definition 3.3.

For $\pi_{\boldsymbol{\nu}}\colon
M_{\boldsymbol{\nu}}\rightarrow\tilde{M}_{g,n}$, we say a complex foliation
$\mathcal{F}$ is a foliation determined by the isomonodromic deformations if

  * (1)

$\mathcal{F}$ is transverse to each fiber
$(M_{\boldsymbol{\nu}})_{t}=\pi_{\boldsymbol{\nu}}^{-1}(t)$,
$t\in\tilde{M}_{g,n}$, and

  * (2)

for each leaf $l$ on $M_{\boldsymbol{\nu}}$, the restriction of the local
system $j_{*}\big{(}\operatorname{Ker}\tilde{\nabla}^{{\rm
an}}|_{\mathcal{C}_{M_{\boldsymbol{\nu}}}\setminus\big{\\{}\tilde{t}_{1},\ldots,\tilde{t}_{n}\big{\\}}}\big{)}|_{\mathcal{C}\times_{\tilde{M}_{g,n}}l}$
is constant.

Let $\mu\colon\pi_{\boldsymbol{\nu}}^{*}\Theta_{\tilde{M}_{g,n}}\rightarrow
R^{1}(\pi_{M_{\boldsymbol{\nu}}})_{*}\big{(}\Theta_{\mathcal{C}_{M_{\boldsymbol{\nu}}}/M_{\boldsymbol{\nu}}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\big{)}$
be the Kodaira–Spencer map, where $\pi_{M_{\boldsymbol{\nu}}}\colon$
$\mathcal{C}_{M_{\boldsymbol{\nu}}}\rightarrow M_{\boldsymbol{\nu}}$ is the
natural morphism. We define a splitting $\mathfrak{D}$ of the tangent map
$\Theta_{M_{\boldsymbol{\nu}}}\rightarrow\pi_{\boldsymbol{\nu}}^{*}\big{(}\Theta_{\tilde{M}_{g,n}}\big{)}$
as follows

$\displaystyle\mathfrak{D}\colon\
\pi_{\boldsymbol{\nu}}^{*}\big{(}\Theta_{\tilde{M}_{g,n}}\big{)}\longrightarrow\Theta_{M_{\boldsymbol{\nu}}}\cong\mathbb{R}^{1}(\pi_{M_{\boldsymbol{\nu}}})_{*}\big{(}\mathcal{F}_{M_{\boldsymbol{\nu}}}^{0}\rightarrow\widetilde{\mathcal{F}}_{M_{\boldsymbol{\nu}}}^{1}\big{)},\qquad
v\longmapsto\big{[}\big{\\{}\iota\big{(}\tilde{\nabla}\big{)}(d_{\alpha\beta})\big{\\}},\\{0\\}\big{]},$

where $[\\{d_{\alpha\beta}\\}]$ is a description of $\mu(v)$ by the Čech
cohomology. Here, we take an affine open covering $\\{U_{\alpha}\\}$.

######  Proposition 3.4 ([14, Section 6], [15, Section 8], [18, Section 3.3]).

The subsheaf
$\mathfrak{D}\big{(}\pi^{*}\big{(}\Theta_{\tilde{M}_{g,n}}\big{)}\big{)}$
determines the foliation determined by the isomonodromic deformations.

We can take a natural lift
$\widehat{\mathfrak{D}}\colon\hat{\pi}_{\boldsymbol{\nu}}^{*}\big{(}\Theta_{T^{*}\tilde{M}_{g,n}}\big{)}\rightarrow\Theta_{\widehat{M}_{\boldsymbol{\nu}}}$
of
$\mathfrak{D}\colon\pi_{\boldsymbol{\nu}}^{*}\big{(}\Theta_{\tilde{M}_{g,n}}\big{)}\rightarrow\Theta_{M_{\boldsymbol{\nu}}}$
as follows. We define a complex $\mathcal{G}^{\bullet}$ by

$\displaystyle\Theta_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}/\widehat{M}_{\boldsymbol{\nu}}}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}=:\mathcal{G}^{0}\xrightarrow{\
d_{\tilde{\psi}}\ }\mathcal{G}^{1}:=\Omega^{\otimes
2}_{\mathcal{C}_{\widehat{M}_{\boldsymbol{\nu}}}/\widehat{M}_{\boldsymbol{\nu}}}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)},$
(3.3)

where $d_{\tilde{\psi}}$ is defined by (3.1). Then we can show that
$\hat{\pi}_{\boldsymbol{\nu}}^{*}\Theta_{T^{*}\tilde{M}_{g,n}}\cong\mathbb{R}^{1}(\pi_{\widehat{M}_{\boldsymbol{\nu}}})_{*}(\mathcal{G}^{\bullet})$.
We define a lift
$\widehat{\mathfrak{D}}\colon\hat{\pi}_{\boldsymbol{\nu}}^{*}\Theta_{T^{*}\tilde{M}_{g,n}}\rightarrow\Theta_{\widehat{M}_{\boldsymbol{\nu}}}$
of $\mathfrak{D}$ by the following homomorphism

$\displaystyle\widehat{\mathfrak{D}}\colon\
{\mathbf{H}}^{1}\big{(}\mathcal{G}^{\bullet}_{\widehat{U}}\big{)}\longrightarrow{\mathbf{H}}^{1}\big{(}\mathcal{F}_{\widehat{U}}^{\bullet}\big{)},\qquad[\\{d_{\alpha\beta}\\},\\{w_{\alpha}\\}]\longmapsto\big{[}\big{\\{}\iota\big{(}\tilde{\nabla}\big{)}(d_{\alpha\beta})\big{\\}},\\{(0,w_{\alpha})\\}\big{]}.$
(3.4)

###  3.4 Symplectic structure

First, we recall the canonical symplectic structure $\omega_{\tilde{M}_{g,n}}$
on $T^{*}\tilde{M}_{g,n}$. Let $U$ be an affine open set of
$T^{*}\tilde{M}_{g,n}$ and let $(\mathcal{C}_{U},\tilde{\psi})$ be a family of
curves and quadratic differentials on $U$. Let
$\psi_{\alpha}df_{\alpha}^{\otimes 2}$ be the restriction of $\tilde{\psi}$ on
an affine open set $U_{\alpha}\subset\mathcal{C}_{U}$. Let $\mu_{\alpha\beta}$
be the isomorphism (3.2): $f_{\alpha}=\mu_{\alpha\beta}(f_{\beta})$. We define
a 1-form $\theta_{\tilde{M}_{g,n}}$ on $T^{*}\tilde{M}_{g,n}$ by

$\displaystyle\theta_{\tilde{M}_{g,n}}\colon\
{\mathbf{H}}^{1}\big{(}\mathcal{G}^{\bullet}_{U}\big{)}\longrightarrow
H^{1}\big{(}\Omega^{1}_{\mathcal{C}_{U}/U}\big{)},\qquad[\\{d_{\alpha\beta}\\},\\{w_{\alpha}\\}]\longmapsto\left[\left\\{d_{\beta\alpha}\psi_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}df_{\alpha}\right\\}\right],$

where $\mathcal{G}_{U}$ is the complex
$d_{\tilde{\psi}}\colon\Theta_{\mathcal{C}_{U}/U}\big{(}{-}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}\rightarrow\Omega^{\otimes
2}_{\mathcal{C}_{U}/U}\big{(}D\big{(}\tilde{\boldsymbol{t}}\big{)}\big{)}$.
The 1-form $\theta_{\tilde{M}_{g,n}}$ is the canonical 1-form on the cotangent
bundle $T^{*}\tilde{M}_{g,n}$. Let $d\theta_{\tilde{M}_{g,n}}$ be the exterior
differential of $\theta_{\tilde{M}_{g,n}}$. The 2-form
$d\theta_{\tilde{M}_{g,n}}$ gives the symplectic form on the cotangent bundle
$T^{*}\tilde{M}_{g,n}$.

######  Proposition 3.5.

Let $v=[(\\{d_{\alpha\beta}\\},\\{w_{\alpha}\\})]$ and
$v^{\prime}=[(\\{d^{\prime}_{\alpha\beta}\\},\\{w^{\prime}_{\alpha}\\})]$ be
elements of ${\mathbf{H}}^{1}\big{(}\mathcal{G}_{U}^{\bullet}\big{)}$. The
pairing

$\displaystyle{\mathbf{H}}^{1}\big{(}\mathcal{G}_{U}^{\bullet}\big{)}\otimes{\mathbf{H}}^{1}\big{(}\mathcal{G}_{U}^{\bullet}\big{)}\longrightarrow{\mathbf{H}}^{2}\big{(}\Omega^{\bullet}_{\mathcal{C}_{U}/U}\big{)},$
$\displaystyle v\otimes w\longmapsto[\\{2\cdot d_{\beta\alpha}\circ
d^{\prime}_{\beta\gamma}\circ\psi_{\beta}\\},\\{-d_{\beta\alpha}\circ
w_{\beta}^{\prime}-w_{\alpha}\circ d^{\prime}_{\alpha\beta}\\}]$

coincides with the symplectic form $d\theta_{\tilde{M}_{g,n}}$.

###### Proof.

Let
$D_{v}\colon\mathcal{O}_{U_{\alpha\beta}}\rightarrow\mathcal{O}_{U_{\alpha\beta}}$
be a derivation corresponding to $v$. We compute the 2-form
$d\theta_{\tilde{M}_{g,n}}(v,v^{\prime})$ as follows

$\displaystyle
D_{v}\theta_{\tilde{M}_{g,n}}(v^{\prime})-D_{v^{\prime}}\theta_{\tilde{M}_{g,n}}(v)+\theta_{\tilde{M}_{g,n}}([v,v^{\prime}])$
$\displaystyle\qquad{}=D_{v^{\prime}}(\mu_{\beta\alpha})D_{v}\left(\psi_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}\right)df_{\alpha}-D_{v}(\mu_{\beta\alpha})D_{v^{\prime}}\left(\psi_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}\right)df_{\alpha}$
$\displaystyle\qquad{}=D_{v^{\prime}}(\mu_{\beta\alpha})\psi_{\alpha}\frac{\partial
D_{v}(\mu_{\alpha\beta})}{\partial
f_{\beta}}df_{\alpha}-D_{v}(\mu_{\beta\alpha})\psi_{\alpha}\frac{\partial
D_{v^{\prime}}(\mu_{\alpha\beta})}{\partial f_{\beta}}df_{\alpha}$
$\displaystyle\qquad\quad{}+D_{v^{\prime}}(\mu_{\beta\alpha})\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}D_{v}(\psi_{\alpha}df_{\alpha})-D_{v}(\mu_{\beta\alpha})\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}D_{v^{\prime}}(\psi_{\alpha}df_{\alpha})$
$\displaystyle\qquad{}=-d^{\prime}_{\alpha\beta}\psi_{\alpha}\frac{\partial
d_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}+d_{\alpha\beta}\psi_{\alpha}\frac{\partial
d^{\prime}_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}-d^{\prime}_{\alpha\beta}w_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}+d_{\alpha\beta}w^{\prime}_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}.$

We add the exterior differential of
$d^{\prime}_{\alpha\beta}d_{\alpha\beta}\psi_{\alpha}$ to the formula above

$\displaystyle-d^{\prime}_{\alpha\beta}\psi_{\alpha}\frac{\partial
d_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}+d_{\alpha\beta}\psi_{\alpha}\frac{\partial
d^{\prime}_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}-d^{\prime}_{\alpha\beta}w_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}+d_{\alpha\beta}w_{\alpha}\frac{\partial\mu_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}+d\left(d^{\prime}_{\alpha\beta}d_{\alpha\beta}\psi_{\alpha}\right)$
$\displaystyle\qquad{}=d_{\alpha\beta}\left(d^{\prime}_{\alpha\beta}d\psi_{\alpha}+2\psi_{\alpha}\frac{\partial
d^{\prime}_{\alpha\beta}}{\partial
f_{\beta}}df_{\beta}\right)-d^{\prime}_{\alpha\beta}w_{\alpha}df_{\alpha}+d_{\alpha\beta}w^{\prime}_{\alpha}df_{\alpha}$
$\displaystyle\qquad{}=-d_{\beta\alpha}w^{\prime}_{\beta}df_{\beta}-d^{\prime}_{\alpha\beta}w_{\alpha}df_{\alpha}.$

By the isomorphism
$H^{1}\big{(}\Omega^{1}_{\mathcal{C}_{U}/U}\big{)}\cong{\mathbf{H}}^{2}\big{(}\Omega^{\bullet}_{\mathcal{C}_{U}/U}\big{)}$,
we have this proposition. ∎

######  Proposition 3.6.

Take a point $\boldsymbol{\nu}\in N^{(n)}_{r}(e)$. Let
$\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}(\tilde{\boldsymbol{t}},r,e)_{\boldsymbol{\nu}}$
be the fiber of $\boldsymbol{\nu}$ under the composition
$\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}\rightarrow
N^{(n)}_{r}(e)$. Then the fiber
$\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}$
has an algebraic symplectic structure.

We can obtain the above proposition by the following two propositions.

######  Proposition 3.7.

There is a non-degenerate relative $2$-form $\omega\in
H^{0}\big{(}\widehat{M},\Omega^{2}_{\widehat{M}/N^{(n)}_{r}(e)}\big{)}$.

###### Proof.

We set
$\eta(s):=s-\iota\big{(}\tilde{\nabla}\big{)}\circ\mathrm{symb}_{1}(s)\in\mathcal{E}{\rm
nd}\big{(}\tilde{E}\big{)}$, where $s\in\mathcal{F}^{0}$. For

$\displaystyle
v=[(\\{u_{\alpha,\beta}\\},\\{(v_{\alpha},w_{\alpha})\\})]\in{\mathbf{H}}^{1}\big{(}\mathcal{C}\times_{T}\widehat{U},\mathcal{F}_{\widehat{U}}^{\bullet}\big{)}\qquad\text{and}$
$\displaystyle
w=[(\\{u_{\alpha,\beta}^{\prime}\\},\\{(v_{\alpha}^{\prime},w_{\alpha}^{\prime})\\})]\in{\mathbf{H}}^{1}\big{(}\mathcal{C}\times_{T}\widehat{U},\mathcal{F}_{\widehat{U}}^{\bullet}\big{)},$

we put

$\displaystyle\omega_{1}(v,w)=[(\\{\operatorname{Tr}(\eta(u_{\alpha\beta})\circ\eta(u_{\beta\gamma}^{\prime}))\\},-\\{\operatorname{Tr}(\eta(u_{\alpha\beta})\circ
v_{\beta}^{\prime})-\operatorname{Tr}(v_{\alpha}\circ\eta(u^{\prime}_{\alpha\beta}))\\})]\qquad\text{and}$
(3.5)
$\displaystyle\omega_{2}(v,w)=[\\{2\cdot\mathrm{symb}_{1}(u_{\beta\alpha})\circ\mathrm{symb}_{1}(u^{\prime}_{\beta\gamma})\circ\psi_{\beta}\\},{-}\\{\mathrm{symb}_{1}(u_{\beta\alpha})\circ
w_{\beta}^{\prime}+w_{\alpha}\circ\mathrm{symb}_{1}(u^{\prime}_{\alpha\beta})\\}].\\!$

For each affine open subset $\widehat{U}\subset\widehat{M}$, we define a
pairing

$\displaystyle{\mathbf{H}}^{1}\big{(}\mathcal{C}\times_{T}\widehat{U},\mathcal{F}_{\widehat{U}}^{\bullet}\big{)}\otimes{\mathbf{H}}^{1}\big{(}\mathcal{C}\times_{T}\widehat{U},\mathcal{F}_{\widehat{U}}^{\bullet}\big{)}\longrightarrow{\mathbf{H}}^{2}\big{(}\mathcal{C}\times_{T}\widehat{U},\Omega_{\mathcal{C}\times_{T}\widehat{U}/\widehat{U}}^{\bullet}\big{)}\cong
H^{0}\big{(}\mathcal{O}_{\widehat{U}}\big{)},$ $\displaystyle v\otimes
w\longmapsto\omega_{1}(v,w)+\omega_{2}(v,w),$

where we consider in Čech cohomology with respect to an affine open covering
$\\{U_{\alpha}\\}$ of $\mathcal{C}\times_{T}U$, $\\{u_{\alpha\beta}\\}\in
C^{1}\big{(}\mathcal{F}^{0}\big{)}$, $\\{(v_{\alpha},w_{\alpha})\\}\in
C^{0}\big{(}\mathcal{F}^{1}\big{)}$ and so on. This pairing determines a
pairing

$\displaystyle\omega\colon\
\mathbb{R}^{1}(\pi_{\widehat{M}})_{*}(\mathcal{F}^{\bullet})\otimes\mathbb{R}^{1}(\pi_{\widehat{M}})_{*}(\mathcal{F}^{\bullet})\longrightarrow\mathcal{O}_{\widehat{M}}.$

By the same argument as in the proof of [15, Proposition 7.2], $\omega$ is
skew symmetric and non-degenerate. ∎

######  Proposition 3.8.

For the $2$-form constructed in Proposition 3.7, we have $d\omega=0$.

###### Proof.

Let $\Theta_{\widehat{M}_{\boldsymbol{\nu}}}^{\text{initial}}$ be the
subbundle of $\Theta_{\widehat{M}_{\boldsymbol{\nu}}}$ consisted by the images
of the tangent morphism
$\Theta_{\widehat{M}_{\boldsymbol{\nu}}/T^{*}\tilde{M}_{g,n}}\rightarrow\Theta_{\widehat{M}_{\boldsymbol{\nu}}}$
and let $\Theta_{\widehat{M}_{\boldsymbol{\nu}}}^{\text{IMD}}$ be the
subbundle of $\Theta_{\widehat{M}_{\boldsymbol{\nu}}}$ consisted by the images
of
$\widehat{\mathfrak{D}}\big{(}\hat{\pi}_{\boldsymbol{\nu}}^{*}\big{(}\Theta_{T^{*}\tilde{M}_{g,n}}\big{)}\big{)}\rightarrow\Theta_{\widehat{M}_{\boldsymbol{\nu}}}$.
We take an affine open set $\widehat{U}\subset\widehat{M}_{\boldsymbol{\nu}}$.
We have a canonical decomposition

$\displaystyle{\mathbf{H}}^{1}\big{(}\mathcal{F}_{\widehat{U}}^{\bullet}\big{)}\longrightarrow\Theta^{\text{initial}}_{\widehat{U}}\oplus\Theta^{\text{IMD}}_{\widehat{U}},\qquad
v=[\\{u_{\alpha\beta}\\},\\{(v_{\alpha},w_{\alpha})\\}]\longmapsto
v_{\text{initial}}+v_{\text{IMD}},$

where

$\displaystyle
v_{\text{initial}}=[\\{\eta(u_{\alpha\beta})\\},\\{(v_{\alpha},0)\\}]\qquad\text{and}\qquad
v_{\text{IMD}}=\big{[}\big{\\{}\iota\big{(}\tilde{\nabla}\big{)}\circ\mathrm{symb}_{1}(u_{\alpha\beta})\big{\\}},\\{(0,w_{\alpha})\\}\big{]}.$

We may assume that $\boldsymbol{\nu}$ is generic. Let $\widehat{U}$ be an
affine open set of $\widehat{M}_{\boldsymbol{\nu}}$ and let
$\big{(}\tilde{E},\tilde{\nabla},\big{\\{}\tilde{l}^{(i)}_{j}\big{\\}},\tilde{\psi}\big{)}$
be the family on $\mathcal{C}\times_{\tilde{M}_{g,n}}\widehat{U}$. We take an
affine open covering $\mathcal{C}_{\widehat{U}}=\bigcup_{\alpha}U_{\alpha}$
such that
$\phi_{\alpha}\colon\tilde{E}|_{U_{\alpha}}\xrightarrow{\sim}\mathcal{O}^{\oplus
r}_{U_{\alpha}}$ for any $\alpha$,
$\sharp\\{i\,|\,\tilde{t}_{i}|_{\mathcal{C}_{U}}\cap
U_{\alpha}\neq\varnothing\\}\leq 1$ for any $\alpha$ and
$\sharp\\{\alpha\,|\,\tilde{t}_{i}|_{\mathcal{C}_{U}}\cap
U_{\alpha}\neq\varnothing\\}\leq 1$ for any $i$. If we replace $U_{\alpha}$
sufficiently smaller, there exists a sheaf $E_{\alpha}$ on $U_{\alpha}$ such
that $E_{\alpha}|_{U_{\alpha}\cap
U_{\beta}}\cong\big{(}\pi^{-1}_{\widehat{M}_{\boldsymbol{\nu}}}\mathcal{O}_{\widehat{M}_{\boldsymbol{\nu}}}|_{U_{\alpha}\cap
U_{\beta}}\big{)}^{\oplus r^{2}}$ for any $\beta\neq\alpha$ and an isomorphism
$\phi_{\alpha}\colon
j_{*}\big{(}\widehat{\boldsymbol{V}}\big{)}|_{U_{\alpha}}\xrightarrow{\sim}E_{\alpha}$.
Here the local system $\widehat{\boldsymbol{V}}$ is defined in Section 3.2.
For each $\alpha$, $\beta$, we put

$\displaystyle\varphi_{\alpha\beta}\colon\ E_{\beta}|_{U_{\alpha}\cap
U_{\beta}}\xrightarrow{\phi_{\beta}^{-1}}j_{*}\big{(}\widehat{\boldsymbol{V}}\big{)}|_{U_{\alpha}\cap
U_{\beta}}\xrightarrow{\phi_{\alpha}}E_{\alpha}|_{U_{\alpha}\cap U_{\beta}}.$

For each $\alpha$, $\beta$, let $\mu_{\alpha\beta}\colon
U_{\alpha\beta}\rightarrow U_{\alpha\beta}$ be an isomorphism such that the
glueing scheme of the collection
$(U_{\alpha},U_{\alpha\beta},\mu_{\alpha\beta})$ is isomorphic to
$\mathcal{C}_{\widehat{U}}$.

We consider a vector field $v\in
H^{0}\big{(}\widehat{U},\Theta_{\widehat{U}}\big{)}$. Then $v$ corresponds to
a derivation
$D_{v}\colon\mathcal{O}_{\widehat{U}}\rightarrow\mathcal{O}_{\widehat{U}}$
which naturally induces a morphism

$\displaystyle D_{v}\colon\ \mathcal{H}{\rm om}(E_{\beta}|_{U_{\alpha}\cap
U_{\beta}},E_{\alpha}|_{U_{\alpha}\cap
U_{\beta}})\longrightarrow\mathcal{H}{\rm om}(E_{\beta}|_{U_{\alpha}\cap
U_{\beta}},E_{\alpha}|_{U_{\alpha}\cap U_{\beta}}).$

The isomorphism
$\Theta_{\widehat{M}_{\boldsymbol{\nu}}}\cong\mathbb{R}^{1}(\pi_{\widehat{M}_{\boldsymbol{\nu}}})_{*}\big{(}\big{(}\widehat{\mathcal{F}}^{\bullet}\big{)}^{{\rm
an}}\big{)}$ is given by

$\displaystyle\Theta_{\widehat{M}_{\boldsymbol{\nu}}}\ni
v\longmapsto\big{[}\big{\\{}\big{(}\phi_{\alpha}^{-1}\circ
D_{v}(\varphi_{\alpha\beta})\circ\phi_{\beta},D_{v}(\mu_{\alpha\beta})\big{)}\big{\\}},\\{D_{v}(\psi|_{U_{\alpha}})\\}\big{]}\in\mathbb{R}^{1}(\pi_{\widehat{M}_{\boldsymbol{\nu}}})_{*}\big{(}\big{(}\widehat{\mathcal{F}}^{\bullet}\big{)}^{{\rm
an}}\big{)},$

and the 2-form $\omega(u,v)=\omega_{1}(u,v)+\omega_{2}(u,v)$,
$u,v\in\Theta_{\widehat{M}}$, is given by

$\displaystyle\omega_{1}(u,v)=[\\{\operatorname{Tr}\left(D_{u_{\text{initial}}}(\varphi_{\alpha\beta})D_{v_{\text{initial}}}(\varphi_{\beta\gamma})\varphi_{\gamma\alpha}\right)\\}]\qquad\text{and}$
$\displaystyle\omega_{2}(u,v)=[\\{2D_{u_{\text{IMD}}}(\mu_{\beta\alpha})D_{v_{\text{IMD}}}(\mu_{\beta\alpha})\tilde{\psi}_{\beta}|_{U_{\alpha\beta}}\\},$
$\displaystyle\hphantom{\omega_{2}(u,v)=[}{}\\{-D_{u_{\text{IMD}}}(\mu_{\beta\alpha})D_{v_{\text{IMD}}}(\tilde{\psi}_{\beta}|_{U_{\alpha\beta}})-D_{u_{\text{IMD}}}(\tilde{\psi}_{\alpha}|_{U_{\alpha\beta}})D_{v_{\text{IMD}}}(\mu_{\alpha\beta})\\}].$

Since the image of $\Theta^{\text{IMD}}_{\widehat{U}}$ under the tangent
morphism of $\widehat{M}_{\boldsymbol{\nu}}\rightarrow M_{\boldsymbol{\nu}}$
determines the foliation determined by the isomonodromic deformations, we can
show that

$\displaystyle
d\omega_{1}(u,v,w)=d\omega_{1}(u_{\text{initial}},v_{\text{initial}},w_{\text{initial}}).$

We have
$d\omega_{1}(u_{\text{initial}},v_{\text{initial}},w_{\text{initial}})=0$ by
[15, Proposition 7.3]. We can also show that $d\omega_{2}(u,v,w)=0$. Then we
have the closeness of $\omega=\omega_{1}+\omega_{2}$. ∎

###  3.5 Extended phase spaces of isomonodromic deformations

######  Proposition 3.9.

The morphism
$\hat{\pi}_{\boldsymbol{\nu}}\colon\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}\rightarrow
T^{*}\tilde{M}_{g,n}$ is a Poisson map.

###### Proof.

Let
$\hat{\pi}_{\boldsymbol{\nu}}^{t}\colon\Theta_{\widehat{M}_{\boldsymbol{\nu}}}\rightarrow\pi^{*}\Theta_{T^{*}\tilde{M}_{g,n}}$
be the tangent morphism. We denote by
$\tilde{\xi}\colon\Theta_{T^{*}\tilde{M}_{g,n}}\rightarrow\Omega^{1}_{T^{*}\tilde{M}_{g,n}}$
and
$\xi\colon\Theta_{\widehat{M}_{\boldsymbol{\nu}}}\rightarrow\Omega^{1}_{\widehat{M}_{\boldsymbol{\nu}}}$
the homomorphisms induced by the symplectic structures on
$T^{*}\tilde{M}_{g,n}$ and $\widehat{M}_{\boldsymbol{\nu}}$, respectively. The
assertion follows from that the following diagram

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.41177pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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(3.6)

is commutative and
$\hat{\pi}_{\boldsymbol{\nu}}^{t}\circ\widehat{\mathfrak{D}}=\mathrm{id}$.
Here, $\widehat{\mathfrak{D}}$ is the homomorphism (3.4). ∎

Let $\mu_{1},\ldots\mu_{3g-3+n}$ be local vector fields on an affine open
subset $U\subset\tilde{M}_{g,n}$. Let $h_{i}$ be a linear function on
$T^{*}\tilde{M}_{g,n}$ corresponding to the local vector field $\mu_{i}$ on
$U$. Assume that $\\{h_{i},h_{j}\\}_{\tilde{M}_{g,n}}=0$ for
$i,j=1,\ldots,3g-3+n$ and $dh_{1}\wedge\cdots\wedge dh_{3g-3+n}$ is not
identically $0$, where $\\{\cdot,\cdot\\}_{\tilde{M}_{g,n}}$ is the Poisson
bracket associated to the symplectic structure $\omega_{\tilde{M}_{g,n}}$. Put
$\widehat{U}=\big{(}\hat{\pi}_{\boldsymbol{\nu}}\circ
p_{\tilde{M}_{g,n}}\big{)}^{-1}(U)$, where
$\hat{\pi}_{\boldsymbol{\nu}}\colon\widehat{M}_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}\rightarrow
T^{*}\tilde{M}_{g,n}$ and $p_{\tilde{M}_{g,n}}\colon
T^{*}\tilde{M}_{g,n}\rightarrow\tilde{M}_{g,n}$. Let
$\omega_{T^{*}\tilde{M}_{g,n}}$ be the symplectic structure on
$T^{*}\tilde{M}_{g,n}$. We define a Hamiltonian $E_{i}$ on $\widehat{U}$ as
$\hat{\pi}^{*}h_{i}$ for $i=1,\ldots,3g-3+n$. Let $a_{i}$
$(i=1,\ldots,3g-3+n)$ be constants. We call the Hamiltonian vector field on
$\widehat{U}$ associated to $\sum\limits_{i=1}^{3g-3+n}a_{i}\mu_{i}$ the
vector field $\sum\limits_{i=1}^{3g-3+n}a_{i}\\{\cdot,E_{i}\\}$ on
$\widehat{U}$.

######  Proposition 3.10.

First, the Hamiltonians $E_{i}$ satisfy $\\{E_{i},E_{j}\\}=0$ for
$i,j=1,\ldots,3g-3+n$. In particular, the functions $E_{i}$ are conserved
quantities associated to the Hamiltonian vector fields. Second, the
restriction of the Hamiltonian vector field associated to
$\sum\limits_{i=1}^{3g-3+n}a_{i}\mu_{i}$ to the common level surface
$E_{1}=0,\dots,E_{3g-3+n}=0$ in $\widehat{U}$, is coincide with
$\mathfrak{D}\Big{(}\sum\limits_{i=1}^{3g-3+n}a_{i}\mu_{i}\Big{)}$, which is a
vector field associated to isomonodromic deformations. Here we consider the
vector field $\sum\limits_{i=1}^{3g-3+n}a_{i}\mu_{i}$ as an element of
$\pi_{\boldsymbol{\nu}}^{*}(\Theta_{\tilde{M}_{g,n}})\big{(}\widehat{U}\big{)}$.

###### Proof.

Let $\\{\cdot,\cdot\\}_{\tilde{M}_{g,n}}$ be the Poisson bracket associated to
the symplectic structure $\omega_{T^{*}\tilde{M}_{g,n}}$ on
$T^{*}\tilde{M}_{g,n}$. Let $v_{h_{i}}$ be the element
$\hat{\pi}_{\boldsymbol{\nu}}^{*}(\Theta_{T^{*}\tilde{M}_{g,n}})\big{(}\widehat{U}\big{)}$
defined by the vector field $\\{\cdot,h_{i}\\}_{\tilde{M}_{g,n}}$ on
$\hat{\pi}\big{(}\widehat{U}\big{)}\subset\tilde{M}_{g,n}$. In other words,
$\omega_{T^{*}\tilde{M}_{g,n}}(v_{h_{i}},v)=dh_{i}(v)$ for any
$v\in\Theta_{U}$. Put
$v_{h_{i}}=\big{[}\big{\\{}d^{h_{i}}_{\alpha\beta}\big{\\}},\big{\\{}w^{h_{i}}_{\alpha}\big{\\}}\big{]}\in{\mathbf{H}}^{1}\big{(}\mathcal{G}_{\widehat{U}}^{\bullet}\big{)}$,
where $\mathcal{G}^{\bullet}$ is the complex (3.3). Put
$v^{\text{IMD}}_{h_{i}}=\big{[}\big{\\{}\iota\big{(}\tilde{\nabla}\big{)}(d^{h_{i}}_{\alpha\beta})\big{\\}},\big{\\{}(0,w^{h_{i}}_{\alpha})\big{\\}}\big{]}\\!$
$\in{\mathbf{H}}^{1}\big{(}\mathcal{F}_{\widehat{U}}^{\bullet}\big{)}$. By the
diagram (3.6), we have
$\omega\big{(}v^{\text{IMD}}_{h_{i}},v\big{)}=dE_{i}(v)$ for any
$v\in\Theta_{\widehat{U}}$, that is,
$v^{\text{IMD}}_{h_{i}}=\\{\cdot,E_{i}\\}$, which is the Hamiltonian vector
field associated to $\mu_{i}$.

Note that
$\\{E_{i},E_{j}\\}=\omega\big{(}v^{\text{IMD}}_{h_{i}},v^{\text{IMD}}_{h_{j}}\big{)}=\omega_{T^{*}\tilde{M}_{g,n}}(v_{h_{i}},v_{h_{j}})=\\{h_{i},h_{j}\\}_{\tilde{M}_{g,n}}$.
By the assumption that the linear functions $h_{i}$ satisfy
$\\{h_{i},h_{j}\\}_{\tilde{M}_{g,n}}=0$, we have $\\{E_{i},E_{j}\\}=0$. The
common level surface $E_{1}=\cdots=E_{3g-3+n}=0$ is
$M_{\mathcal{C}/\tilde{M}_{g,n}}^{\boldsymbol{\alpha}}\big{(}\tilde{\boldsymbol{t}},r,e\big{)}_{\boldsymbol{\nu}}$.
On this common level surface, the vector field associated to the Hamiltonian
vector field of $\mu_{i}$ is
$\big{[}\big{\\{}\iota\big{(}\tilde{\nabla}\big{)}\big{(}d^{h_{i}}_{\alpha\beta}\big{)}\big{\\}},\\{0\\}\big{]}\in{\mathbf{H}}^{1}\big{(}\mathcal{F}^{0}_{\widehat{U}}\rightarrow\widetilde{\mathcal{F}}^{1}_{\widehat{U}}\big{)}$,
which is a vector field associated to the isomonodromic deformations. ∎

##  4 Moduli stack of stable parabolic connections
with a quadratic differential and twisted cotangent bundle

Let $C$ be a smooth projective curve of genus $g$, $g\geq 2$. The map from the
moduli space of pairs $(E,\nabla)$ to the moduli space of vector bundles
defined by $(E,\nabla)\mapsto E$ is a twisted cotangent bundle on the moduli
space of vector bundles. Here, $E$ is a rank $r$ vector bundle on the fixed
curve $C$ and $\nabla$ is a holomorphic connection on $E$. This twisted
cotangent bundle has been investigated by Faltings, Ben-Zvi–Biswas, and Ben-
Zvi–Frenkel (see [10, Section 4], [8, Section 5], [5, Section 5], and [7,
Section 4.1]). Moreover, Ben-Zvi–Biswas and Ben-Zvi–Frenkel studied on a
twisted cotangent bundle on the moduli space of pairs $(C,E)$ (see [5, Section
6] and [7, Section 4.3]). In [5, 6], Ben-Zvi and Biswas have introduced
extended connections, which are generalization of holomorphic connections. We
can define a natural map from the moduli space of extended connections to the
moduli space of pairs $(C,E)$. This map is generalization of the map
$(E,\nabla)\mapsto E$ and has been investigated in [5, Section 6] and [7,
Section 4.3]. In this section, we consider parabolic connections instead of
holomorphic connections and study the moduli space of parabolic connections
with a quadratic differential instead of the moduli space of extended
connections. The purpose of this section is to show that the moduli space of
parabolic connections with a quadratic differential is equipped with structure
of a twisted cotangent bundles. In Section 4.1, we consider the moduli stack
corresponding to the moduli scheme considered in the previous section. We
introduce the moduli stack of pointed smooth projective curves and quasi-
parabolic bundles. We consider the cotangent bundle of this moduli stack. We
describe the tangent sheaf of the total space of this cotangent bundle and the
canonical symplectic form on this cotangent bundle. In Section 4.2, we
consider a map from the moduli stack of parabolic connections with a quadratic
differential to the moduli stack of pointed smooth projective curves and
quasi-parabolic bundles. We endow this map with structure of a twisted
cotangent bundle. In Section 4.3, we introduce extended parabolic connections,
which are generalization of parabolic connections and also extended
connections. We consider a relation between parabolic connections with a
quadratic differential (which are also generalization of parabolic
connections) and extended parabolic connections.

In this section, we assume that $\boldsymbol{\nu}$ is generic. If
$\boldsymbol{\nu}$ is generic, then any
$(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connection is irreducible. So
all $(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connections are stable.

###  4.1 Moduli stack of stable parabolic connections with a quadratic
differential

Let $\mathfrak{M}_{g,n}$ be the moduli stack of $n$-pointed smooth projective
curves of genus $g$, where $n$-points consist of distinct points. Let
$\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$ be the moduli stack of
collections $((C,\boldsymbol{t},\psi),(E,\nabla,\boldsymbol{l}))$, where
$(C,\boldsymbol{t})$, $\boldsymbol{t}=(t_{1},\ldots,t_{n})$, is an $n$-pointed
smooth projective curve of genus $g$ over $\mathbb{C}$ where
$t_{1},\ldots,t_{n}$ are distinct points, $\psi$ is an element of
$H^{0}\big{(}C,\Omega_{C}^{\otimes 2}(D(\boldsymbol{t}))\big{)}$, and
$(E,\nabla,\boldsymbol{l})$ is a $(\boldsymbol{t},\boldsymbol{\nu})$-parabolic
connection of rank $r$ and of degree $e$ on $C$. Let
$\Theta_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}$ be the tangent
complex of $\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$, that is, for
each smooth map $f_{U}\colon
U\rightarrow\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$ from a scheme
$U$, the pull-back
$f_{U}^{*}\Theta_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}$ is
$\Theta_{U/\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}\rightarrow\Theta_{U}$
considered as a length $2$ complex supported in degree $-1$ and $0$. Here
$\Theta_{U/\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}:=\Delta^{*}\big{(}\Theta_{(U\times_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}U)/U}\big{)}$,
where $U\rightarrow
U\times_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}U$ is the
diagonal. Let $\Theta_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu}),x}$
be the fiber of $\Theta_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})}$
over a point $x=((C,\boldsymbol{t},\psi),(E,\nabla,\boldsymbol{l}))$ of
$\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$. Then
$H^{0}(\Theta_{\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu}),x})$ is
isomorphic to ${\mathbf{H}}^{1}\big{(}\mathcal{F}_{x}^{\bullet}\big{)}$. Here,
we recall the complex $\mathcal{F}_{x}^{\bullet}$:

$\displaystyle\mathcal{F}_{x}^{0}:=\big{\\{}s\in\mathcal{A}_{E}(D(\boldsymbol{t}))\,|\,(s-\iota(\nabla)\circ\mathrm{symb}_{1}(s))|_{t_{i}}\big{(}l^{(i)}_{j}\big{)}\subset
l^{(i)}_{j}\text{ for any $i$, $j$}\big{\\}},$
$\displaystyle\widetilde{\mathcal{F}}_{x}^{1}:=\big{\\{}s\in\mathcal{E}{\rm
nd}(E)\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))\,|\,{\sf
res}_{t_{i}}(s)\big{(}l^{(i)}_{j}\big{)}\subset l^{(i)}_{j+1}\text{ for any
$i$, $j$}\big{\\}},$
$\displaystyle\mathcal{F}^{1}_{x}:=\widetilde{\mathcal{F}}_{x}^{1}\oplus\Omega_{C}^{\otimes
2}(D(\boldsymbol{t}));\text{ and
}d_{\mathcal{F}^{\bullet}}:=(d_{\nabla},d_{\psi})\circ\big{(}\mathrm{Id}-\iota\big{(}\tilde{\nabla}\big{)}\circ\mathrm{symb}_{1},\mathrm{symb}_{1}\big{)}\colon\mathcal{F}_{x}^{0}\longrightarrow\mathcal{F}^{1}_{x},$

where
$d_{\nabla}\colon\widetilde{\mathcal{F}}^{0}\rightarrow\widetilde{\mathcal{F}}^{1}$,
$s\mapsto\nabla\circ s-s\circ\nabla$ and
$d_{\psi}\colon\Theta_{C}(-D(\boldsymbol{t}))\rightarrow\Omega_{C}^{\otimes
2}(D(\boldsymbol{t}))$ defined by (3.1). The pairing
${\mathbf{H}}^{1}\big{(}\mathcal{F}_{x}^{\bullet}\big{)}\otimes{\mathbf{H}}^{1}\big{(}\mathcal{F}_{x}^{\bullet}\big{)}\rightarrow{\mathbf{H}}^{2}\big{(}\Omega_{C}^{\bullet}\big{)}$
defined by (3.5) gives a symplectic structure on
$\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$.

######  Definition 4.1.

Let $(C,\boldsymbol{t})$ be an $n$-pointed smooth projective curve of genus
$g$ over $\mathbb{C}$ where $t_{1},\ldots,t_{n}$ are distinct points. We say
$(E,\boldsymbol{l})$, $\boldsymbol{l}=\big{\\{}l^{(i)}_{*}\big{\\}}_{1\leq
i\leq n}$, is a quasi-parabolic bundle of rank $r$ and of degree $e$ on
$(C,\boldsymbol{t})$ if $E$ is a rank $r$ algebraic vector bundle of degree
$e$ on $C$, and for each $t_{i}$, $l^{(i)}_{*}$ is a filtration
$E|_{t_{i}}=l_{0}^{(i)}\supset l_{1}^{(i)}\supset\cdots\supset l_{r}^{(i)}=0$
such that $\dim\big{(}l_{j}^{(i)}/l_{j+1}^{(i)}\big{)}=1$, $j=0,1,\ldots,r-1$.

Let $\mathfrak{P}_{g,n}(r,e)$ be the moduli stack of pairs
$((C,\boldsymbol{t}),(E,\boldsymbol{l}))$, where $(C,\boldsymbol{t})$
($\boldsymbol{t}=(t_{1},\ldots,t_{n})$) is an $n$-pointed smooth projective
curve of genus $g$ over $\mathbb{C}$ where $t_{1},\ldots,t_{n}$ are distinct
points, and $(E,\boldsymbol{l})$ is a quasi-parabolic bundle of rank $r$ and
of degree $e$ on $(C,\boldsymbol{t})$. We have a projection
$\mathfrak{P}_{g,n}(r,e)\rightarrow\mathfrak{M}_{g,n}$. Let
$\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$ be the substack defined by the
condition where a quasi-parabolic bundle admits a
$(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connection. Let
$\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ and
$\pi_{\mathfrak{M}_{g,n}}$ be the following morphisms:

$\displaystyle\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\colon\
\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})\longrightarrow\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu}),\qquad((C,\boldsymbol{t},\psi),(E,\nabla,\boldsymbol{l}))\longmapsto((C,\boldsymbol{t}),(E,\boldsymbol{l})),$
$\displaystyle\pi_{\mathfrak{M}_{g,n}}\colon\
\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})\longrightarrow\mathfrak{M}_{g,n},\qquad((C,\boldsymbol{t}),(E,\boldsymbol{l}))\longmapsto(C,\boldsymbol{t}).$

Let $\Theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ be the tangent complex
of $\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$. Let
$\Theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu}),p}$ be the fiber of
$\Theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\\!\\!$ over a point
$p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$ of
$\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$.

We consider infinitesimal deformations of
$p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$. We put

$\displaystyle\widetilde{\mathcal{H}}_{p}^{0}:=\big{\\{}s\in\mathcal{E}{\rm
nd}(E)\,|\,s|_{t_{i}}\big{(}l^{(i)}_{j}\big{)}\subset l^{(i)}_{j}\text{ for
any $i$, $j$}\big{\\}}\qquad\text{and}$
$\displaystyle\widetilde{\mathcal{H}}_{p}^{1}:=\big{\\{}s\in\mathcal{E}{\rm
nd}(E)\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))\,|\,{\sf
res}_{t_{i}}(s)\big{(}l^{(i)}_{j}\big{)}\subset l^{(i)}_{j+1}\text{ for any
$i$, $j$}\big{\\}}.$

Note that
$\big{(}\widetilde{\mathcal{H}}_{p}^{0}\big{)}^{*}\otimes\Omega^{1}_{C}\cong\widetilde{\mathcal{H}}_{p}^{1}$.
Put

$\displaystyle\mathcal{H}_{p}^{0}:=\big{\\{}s\in\mathcal{A}_{E}(D(\boldsymbol{t}))\subset\mathcal{E}{\rm
nd}_{\mathbb{C}}(E)\,|\,s|_{t_{i}}\big{(}l^{(i)}_{j}\big{)}\subset
l^{(i)}_{j}\text{ for any $i$, $j$}\big{\\}}$

and
$\mathcal{H}_{p}^{1}:=\big{(}\mathcal{H}_{p}^{0}\big{)}^{*}\otimes\Omega^{1}_{C}$.
Then we have exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathcal{H}}_{p}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}_{p}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{symb}_{1}}$$\textstyle{\Theta_{C}(-D(\boldsymbol{t}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$and
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\otimes
2}_{C}(D(\boldsymbol{t}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathcal{H}_{p}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{\widetilde{\mathcal{H}}_{p}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We take an affine open covering $\\{U_{i}\\}$ of $C$ so that we can take a
trivialization $\phi_{i}\colon E|_{U_{i}}\cong\mathcal{O}_{U_{i}}^{\oplus r}$
of $E$ on each $U_{i}$ and the restriction of $\mathcal{H}_{p}^{1}$ to $U_{i}$
is $\mathcal{O}_{U_{i}}$-isomorphic to the direct sum
$\big{(}\widetilde{\mathcal{H}}_{p}^{1}\big{)}_{U_{i}}\oplus\Omega^{\otimes
2}_{C}(D(\boldsymbol{t}))_{U_{i}}$. We fix trivializations of $E$. On
$U_{i}\cap U_{i}$, the transformation
$\big{(}\mathcal{H}_{p}^{1}\big{)}_{U_{i}}\rightarrow\big{(}\mathcal{H}_{p}^{1}\big{)}_{U_{j}}$
is given by

$\displaystyle\left(\Phi_{i}(f_{i})df_{i},\phi_{i}(f_{i})df_{i}\otimes
df_{i}\right)\longmapsto(\Phi_{j}(f_{i})df_{i},\phi_{j}(f_{i})df_{i}\otimes
df_{i})$
$\displaystyle\qquad{}:=\left((\theta_{ij}^{-1}\Phi_{i}(f_{i})\theta_{ij})df_{i},\phi_{i}(f_{i})df_{i}\otimes
df_{i}+\operatorname{Tr}\left(\theta_{ij}^{-1}\Phi_{i}(f_{i})\frac{\partial\theta_{ij}}{\partial
f_{i}}\right)df_{i}\otimes df_{i}\right),$ (4.1)

where $\theta_{ij}:=\phi_{i}\circ\phi^{-1}_{j}\colon\mathcal{O}_{U_{i}\cap
U_{j}}^{\oplus r}\rightarrow\mathcal{O}_{U_{i}\cap U_{j}}^{\oplus r}$ is a
transition function of $E$. Then
$H^{0}\big{(}\Theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu}),p}\big{)}$ is
isomorphic to $H^{1}\big{(}\mathcal{H}_{p}^{0}\big{)}$, and
$H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$ is the dual of
$H^{1}\big{(}\mathcal{H}_{p}^{0}\big{)}$. The vector space
$H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$ is the space of 1-forms at $p$.

Put $p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$. Let $\widehat{\Phi}_{p}$ be
an element of $H^{0}\big{(}\mathcal{H}^{1}_{p}\big{)}$, which is described by
$(\Phi_{p},\phi_{p})$ locally, where
$\Phi_{p}df\in\widetilde{\mathcal{H}}^{1}_{p}$ and $\phi_{p}df\otimes
df\in\Omega_{C}^{\otimes 2}(D(\boldsymbol{t}))$. We consider infinitesimal
deformations of $\big{(}p,\widehat{\Phi}_{p}\big{)}$. For
$\widehat{\Phi}_{p}$, we define a complex
$d^{0}\big{(}\widehat{\Phi}_{p}\big{)}\colon\mathcal{H}^{0}_{p}\rightarrow\mathcal{H}^{1}_{p}$
as follows. For each affine open set $U\subset C$, we define the image of
$a_{U}\partial/\partial f_{U}+\eta_{U}\in\mathcal{H}^{0}_{p}(U)$ as

$\displaystyle\left(\Phi_{p}df_{U}\circ\eta_{U}-\eta_{U}\circ\Phi_{p}df_{U}-\frac{\partial(a_{U}\Phi_{p})}{\partial
f_{U}}df_{U},\right.$
$\displaystyle\left.\qquad\operatorname{Tr}\left(\frac{\partial\eta_{U}}{\partial
f_{U}}\Phi_{p}df_{U}\otimes
df_{U}\right)-a_{U}\frac{\partial\phi_{p}}{\partial f_{U}}df_{U}\otimes
df_{U}-2\frac{\partial a_{U}}{\partial f_{U}}\phi_{p}df_{U}\otimes
df_{U}\right).$

We can show that this homomorphism on each $U$ gives a homomorphism
$d^{0}\big{(}\widehat{\Phi}_{p}\big{)}\colon\mathcal{H}^{0}_{p}\rightarrow\mathcal{H}^{1}_{p}$.
We consider the first hypercohomology
${\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}$ of
$d^{0}\big{(}\widehat{\Phi}_{p}\big{)}\colon\mathcal{H}^{0}_{p}\rightarrow\mathcal{H}^{1}_{p}$.
By the Čech cohomology, an element of
${\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}$ is described by
$\big{[}\big{\\{}a_{ij}\partial/\partial
f_{i}+\eta_{ij}\big{\\}},\big{\\{}(\hat{v}_{i},\hat{w}_{i})\big{\\}}\big{]}$,
where

$\displaystyle a_{jk}\frac{\partial f_{i}}{\partial f_{j}}-a_{ik}+a_{ij}=0,$
$\displaystyle\left(\theta_{ji}^{-1}\eta_{jk}\theta_{ji}+a_{jk}\theta_{ji}^{-1}\frac{\partial\theta_{ji}}{\partial
f_{j}}\right)-\eta_{ik}+\eta_{ij}=0,\qquad\text{and}$
$\displaystyle\left(\theta_{ji}^{-1}\hat{v}_{j}\theta_{ji},\hat{w}_{j}+\operatorname{Tr}\left(\theta_{ji}^{-1}\hat{v}_{j}\frac{\partial\theta_{ji}}{\partial
f_{j}}df_{j}\right)\right)-(\hat{v}_{i},\hat{w}_{i})=d^{0}\big{(}\widehat{\Phi}_{p}\big{)}(\eta_{ij})$

for some affine open covering $\\{U_{i}\\}_{i}$ of $C$. Infinitesimal
deformations of $\big{(}p,\widehat{\Phi}_{p}\big{)}$ are parametrized by
${\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}$. Then the fiber of
the tangent sheaf of the moduli stack of pairs
$\big{(}((C,\boldsymbol{t}),(E,\boldsymbol{l})),\widehat{\Phi}\big{)}$ (where
$\widehat{\Phi}\in H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$) at a point
$\big{(}p,\widehat{\Phi}_{p}\big{)}$ is isomorphic to
${\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}$. Moreover, we define
a paring
${\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}\otimes{\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}\rightarrow{\mathbf{H}}^{2}\big{(}\Omega^{\bullet}_{C}\big{)}$
by

$\displaystyle\big{[}\big{(}\big{\\{}a_{ij}\partial/\partial
f_{i}+\eta_{ij}\big{\\}},\big{\\{}\big{(}\hat{v}_{i},\hat{w}_{i}\big{)}\big{\\}}\big{)}\big{]}\otimes\big{[}\big{(}\big{\\{}a^{\prime}_{ij}\partial/\partial
f_{i}+\eta^{\prime}_{ij}\big{\\}},\big{\\{}\big{(}\hat{v}^{\prime}_{i},\hat{w}^{\prime}_{i}\big{)}\big{\\}}\big{)}\big{]}$
$\displaystyle\qquad{}\longmapsto\big{[}\big{(}\big{\\{}\operatorname{Tr}\big{(}\eta_{ji}\big{(}a^{\prime}_{jk}\Phi_{j}\big{)}\big{)}+\operatorname{Tr}\big{(}\big{(}a_{ji}\Phi_{j}\big{)}\eta^{\prime}_{jk}\big{)}-2a_{ji}a^{\prime}_{jk}\phi_{j}\big{\\}},$
$\displaystyle\qquad\qquad{}-\big{\\{}{-}\operatorname{Tr}\big{(}\eta_{ji}\hat{v}^{\prime}_{j}\big{)}+\big{(}a_{ji}\hat{w}_{j}^{\prime}\big{)}-\operatorname{Tr}\big{(}\hat{v}_{i}\eta^{\prime}_{ij}\big{)}+\big{(}\hat{w}_{i}a^{\prime}_{ij}\big{)}\big{\\}}\big{)}\big{]}.$

######  Proposition 4.2.

This pairing gives a symplectic structure on the moduli stack of pairs
$\big{(}((C,\boldsymbol{t}),$ $(E,\boldsymbol{l})),\widehat{\Phi}\big{)}$.

###### Proof.

We define a 1-form $\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ by

$\displaystyle\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\colon\
{\mathbf{H}}^{1}\big{(}\mathcal{H}^{\bullet}_{p}\big{)}\longrightarrow
H^{1}\big{(}\Omega^{1}_{C}\big{)},$
$\displaystyle\big{[}\big{\\{}a_{ij}\partial/\partial
f_{i}+\eta_{ij}\big{\\}},\big{\\{}\big{(}\hat{v}_{i},\hat{w}_{i}\big{)}\big{\\}}\big{]}\longmapsto\left[\operatorname{Tr}(\eta_{ij}\Phi_{i}df_{i})+a_{ji}\phi_{i}\left(\frac{\partial\mu_{ij}}{\partial
f_{i}}df_{j}\right)\right]$

for each $(\Phi_{i}df_{i},\phi_{i}df_{i}\otimes df_{i})$. This 1-form
$\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ is the canonical 1-form on
the cotangent bundle of $\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$. Let
$d\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ be the exterior
differential of $\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$. The
2-form $d\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ gives the
symplectic form on the cotangent bundle of
$\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$. We compute the 2-form
$d\theta_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ as follows

$\displaystyle\operatorname{Tr}\big{(}D_{v^{\prime}}(\theta_{ij})D_{v}\big{(}\theta_{ij}^{-1}\Phi_{i}df_{i}\big{)}-D_{v}(\theta_{ij})D_{v^{\prime}}\big{(}\theta_{ij}^{-1}\Phi_{i}df_{i}\big{)}\big{)}+d(a_{ij}\operatorname{Tr}(\eta_{ij}^{\prime}\Phi_{i}))$
$\displaystyle\qquad{}=\operatorname{Tr}\big{(}{-}D_{v^{\prime}}(\theta_{ij})\theta_{ij}^{-1}D_{v}(\theta_{ij})\theta_{ij}^{-1}\Phi_{i}df_{i}+D_{v}(\theta_{ij})\theta_{ij}^{-1}D_{v^{\prime}}(\theta_{ij})\theta_{ij}^{-1}\Phi_{i}df_{i}$
$\displaystyle\qquad\quad{}+D_{v^{\prime}}(\theta_{ij})\theta_{ij}^{-1}D_{v}(\Phi_{i}df_{i})-D_{v}(\theta_{ij})\theta_{ij}^{-1}D_{v^{\prime}}(\Phi_{i}df_{i})\big{)}+d\big{(}a_{ij}\operatorname{Tr}(\eta_{ij}^{\prime}\Phi_{i})\big{)}$
$\displaystyle\qquad{}=\operatorname{Tr}\big{(}{-}\eta_{ij}\big{(}[\Phi_{i}df_{i},\eta^{\prime}_{ij}]-d(a_{ij}\Phi_{i})\big{)}-\eta_{ij}\hat{v}^{\prime}_{i}+\eta^{\prime}_{ij}\hat{v}_{i}\big{)}+a_{ij}\operatorname{Tr}(d(\eta_{ij}^{\prime})\Phi_{i})$
$\displaystyle\qquad{}=-\operatorname{Tr}(\eta_{ij}\hat{v}^{\prime}_{j})+\operatorname{Tr}(\eta_{ij}^{\prime}\hat{v}_{i})+a_{ij}\operatorname{Tr}(d(\eta_{ij}^{\prime})\Phi_{i})$
$\displaystyle\qquad{}=\operatorname{Tr}(\eta_{ji}\hat{v}^{\prime}_{j})+\operatorname{Tr}(\eta_{ij}^{\prime}\hat{v}_{i})-a_{ij}\operatorname{Tr}\left(\theta_{ji}^{-1}\hat{v}^{\prime}_{j}\frac{\partial\theta_{ji}}{\partial
f_{i}}\right)+a_{ij}\operatorname{Tr}(d(\eta_{ij}^{\prime})\Phi_{i})$

and

$\displaystyle
D_{v^{\prime}}(\mu_{ji})D_{v}\left(\phi_{i}\frac{\partial\mu_{ij}}{\partial
f_{j}}\right)df_{i}-D_{v}(\mu_{ji})D_{v^{\prime}}\left(\phi_{i}\frac{\partial\mu_{ij}}{\partial
f_{j}}\right)df_{i}+d\left(a^{\prime}_{ji}a_{ji}\phi_{j}\right)$
$\displaystyle\qquad{}=a_{ij}\left(-a^{\prime}_{ij}d\psi_{i}-2\phi_{i}\frac{\partial
a^{\prime}_{ij}}{\partial
f_{i}}df_{i}\right)-a^{\prime}_{ij}\hat{w}_{i}df_{i}+a_{ij}\hat{w}^{\prime}_{i}df_{i}$
$\displaystyle\qquad{}=a_{ij}\hat{w}^{\prime}_{j}df_{j}-a^{\prime}_{ij}\hat{w}_{i}df_{i}+a_{ij}\operatorname{Tr}\left(\theta^{-1}_{ji}\hat{v}^{\prime}_{j}\frac{\partial\theta_{ji}}{\partial
f_{i}}\right)-a_{ij}\operatorname{Tr}(d(\eta_{ij}^{\prime})\Phi_{i}).$

Then we have this proposition. ∎

Put $p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$. Let $\nabla$ be a connection:
$\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))$. For a
connection $\nabla$, we define a decomposition of
$H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$ as follows

$\displaystyle\begin{split}&H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}\longrightarrow
H^{0}\big{(}\widetilde{\mathcal{H}}_{p}^{1}\big{)}\oplus
H^{0}\big{(}\Omega^{\otimes 2}_{C}(D(\boldsymbol{t}))\big{)},\\\
&\widehat{\Phi}\longmapsto\big{(}\kappa\big{(}\widehat{\Phi}\big{)},\widehat{\Phi}-\psi\big{(}\nabla,\kappa\big{(}\widehat{\Phi}\big{)}\big{)}\big{)}.\end{split}$

Here $\psi\big{(}\nabla,\kappa\big{(}\widehat{\Phi}\big{)}\big{)}\in
H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$ is defined as follows. We take an
affine open covering $\\{U_{i}\\}$ of $C$ such that on $U_{i}$ the connection
$\nabla|_{U_{i}}$ is described by $d+A_{i}df_{i}$ and the Higgs field
$\kappa\big{(}\widehat{\Phi}\big{)}|_{U_{i}}$ is described by
$\Phi_{i}df_{i}$. On each $U_{i}$, we define an element
$\psi\big{(}\nabla,\kappa\big{(}\widehat{\Phi}\big{)}\big{)}|_{U_{i}}$ as

$\displaystyle\psi\big{(}\nabla,\kappa\big{(}\widehat{\Phi}\big{)}\big{)}|_{U_{i}}=\left(\Phi_{i}df_{i},\operatorname{Tr}\left(\Phi_{i}A_{i}+\frac{1}{2}\Phi_{i}\Phi_{i}\right)df_{i}\otimes
df_{i}\right)\in\mathcal{H}_{p}^{1}(U_{i}),$

which gives an element
$\psi\big{(}\nabla,\kappa\big{(}\widehat{\Phi}\big{)}\big{)}\in
H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$.

###  4.2 Moduli stack as twisted cotangent bundle

Let $\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}$ be
the sheaf of set on $\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$ where
$\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}(U)$ is the
set of sections of $\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ over $U$
for each smooth map $U\rightarrow\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$.
Here $U$ is a scheme. We take a section $\sigma$ and put
$\sigma(p)=(\nabla_{p},\psi_{p})$, where $\nabla_{p}\colon E\rightarrow
E\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))$ is a connection such that
$(E,\boldsymbol{l},\nabla_{p})$ is a
$(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connection of rank $r$ and of
degree $e$ on $C$, and $\psi_{p}\in H^{0}\big{(}C,\Omega_{C}^{\otimes
2}(D(\boldsymbol{t}))\big{)}$ for $p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$.

######  Definition 4.3.

For a 1-form $\widehat{\Phi}$ on $\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$,
we define a translation by

$\displaystyle\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}\longrightarrow\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)},$
$\displaystyle\sigma(p)=(\nabla_{p},\psi_{p})\longmapsto
t_{\widehat{\Phi}}(\sigma)(p):=\big{(}\nabla_{p}+\kappa\big{(}\widehat{\Phi}_{p}\big{)},\psi_{p}+\big{(}\widehat{\Phi}_{p}-\psi\big{(}\nabla_{p},\kappa\big{(}\widehat{\Phi}_{p}\big{)}\big{)}\big{)}\big{)}.$
(4.2)

By this translation (4.2), we have an
$\Omega^{1}_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$-torsor structure on
$\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}$.

######  Theorem 4.4.

Assume that $\boldsymbol{\nu}$ is generic. Let $\omega$ be the symplectic form
on $\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$. We define a map
$c\colon\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}\rightarrow\Omega_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}^{2{\rm
cl}}$ by $c(\gamma)=\gamma^{*}(\omega)$ for
$\gamma\in\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}$.
Then for any
$\widehat{\Phi}\in\Omega_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}^{1}$ we
have $c(t_{\widehat{\Phi}}(\gamma))=d\big{(}\widehat{\Phi}\big{)}+c(\gamma)$.
That is,
$\big{(}\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)},c\big{)}$
is an $\Omega_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}^{\geq 1}$-torsor.

By this theorem and the argument as in Section 2.3, the morphism
$\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\colon\widehat{\mathfrak{M}}_{g,n}(r,e,\boldsymbol{\nu})$
$\rightarrow\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$ is equipped with
structure of a twisted cotangent bundle.

###### Proof.

Put $p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$. Let $v$ be an element of
$H^{1}\big{(}\mathcal{H}^{0}_{p}\big{)}$. We take a section $\sigma$ and put
$\sigma(p)=(\nabla_{p},\psi_{p})$, where $\nabla_{p}\colon E\rightarrow
E\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))$ is a connection and $\psi_{p}\in
H^{0}\big{(}C,\Omega_{C}^{\otimes 2}(D(\boldsymbol{t}))\big{)}$.

We take an affine open covering $\\{U_{i}\\}$ of $C$ such that elements of
$H^{1}\big{(}\mathcal{H}^{0}_{p}\big{)}$ are described by the Čech cohomology:
$v=[\\{a_{ij}\partial/\partial f_{i}+\eta_{ij}\\}]$, where
$a_{ij}\partial/\partial f_{i}+\eta_{ij}\in\mathcal{H}^{0}_{p}(U_{i}\cap
U_{j})$. Let $\nabla_{p}(v,\epsilon)$, $\widehat{\Phi}_{p}(v,\epsilon)$, and
$\psi_{p}(v,\epsilon)$ be the infinitesimal deformations of $\nabla_{p}$,
$\widehat{\Phi}_{p}$, and $\psi_{p}$ associated to $v$ over
$\operatorname{Spec}\mathbb{C}[\epsilon]$, respectively, where
$\epsilon^{2}=0$. We take local descriptions of the connection, the Higgs
field, and the quadratic differentials on $U_{i}$ as follows. The connection
$\nabla_{p}(v,\epsilon)$ and the Higgs field
$\kappa\big{(}\widehat{\Phi}_{p}\big{)}(v,\epsilon)$ are described as
$d+A_{i}df_{i}+\epsilon v_{i}$ $\mathrm{mod}\ d\epsilon$ and
$\Phi_{i}df_{i}+\epsilon\hat{v}_{i}$ $\mathrm{mod}\ d\epsilon$ on $U_{i}$,
respectively. Moreover, on $U_{i}$ the quadratic differentials
$\psi_{p}(v,\epsilon)$ and
$\big{(}\widehat{\Phi}_{p}-\psi\big{(}\nabla_{p},\kappa\big{(}\widehat{\Phi}_{p}\big{)}\big{)}\big{)}(v,\epsilon)$
are described by $\psi_{i}df_{i}\otimes df_{i}+\epsilon w_{i}df_{i}\otimes
df_{i}$ and $\phi_{i}df_{i}\otimes df_{i}+\epsilon\hat{w}_{i}df_{i}\otimes
df_{i}$ $\mathrm{mod}\ d\epsilon$ on $U_{i}$, respectively.

We decompose $\widehat{\Phi}_{p}=\widehat{\Phi}_{1}+\widehat{\Phi}_{2}$, where
$\widehat{\Phi}_{1}=\psi\big{(}\nabla_{p},\widehat{\Phi}_{p}\big{)}$ and
$\widehat{\Phi}_{2}=\widehat{\Phi}_{p}-\psi\big{(}\nabla_{p},\widehat{\Phi}_{p}\big{)}$.
Then we can compute the exterior differentials of $\widehat{\Phi}_{1}$ and
$\widehat{\Phi}_{2}$ as follows

$\displaystyle
d\widehat{\Phi}_{1}(v,v^{\prime})=\big{[}\big{(}\big{\\{}\operatorname{Tr}(\eta_{ji}(a^{\prime}_{jk}\Phi_{j}))+\operatorname{Tr}((a_{ji}\Phi_{j})\eta^{\prime}_{jk})-\operatorname{Tr}(2a_{ji}a^{\prime}_{jk}A_{j}\Phi_{j}+a_{ji}a^{\prime}_{jk}\Phi_{j}\Phi_{j})\big{\\}}$
$\displaystyle\hphantom{d\widehat{\Phi}_{1}(v,v^{\prime})=}{}-\big{\\{}{-}\operatorname{Tr}(\eta_{ji}\hat{v}^{\prime}_{j})+\operatorname{Tr}\big{(}(a_{ji}A_{j})\hat{v}_{j}^{\prime}+(a_{ji}\Phi_{j})v_{j}^{\prime}+(a_{ji}\Phi_{j})\hat{v}_{j}^{\prime}\big{)}$
$\displaystyle\hphantom{d\widehat{\Phi}_{1}(v,v^{\prime})=}{}-\operatorname{Tr}(\hat{v}_{i}\eta^{\prime}_{ij})+\operatorname{Tr}\big{(}(a^{\prime}_{ij}A_{i})\hat{v}_{i}+(a^{\prime}_{ij}\Phi_{i})v_{i}+(a^{\prime}_{ij}\Phi_{i})\hat{v}_{i}\big{)}\big{\\}}\big{)}\big{]},$
$\displaystyle
d\widehat{\Phi}_{2}(v,v^{\prime})=\big{[}\big{(}\\{-2a_{ji}a^{\prime}_{jk}\phi_{j}\\},-\\{a_{ji}\hat{w}^{\prime}_{j}+\hat{w}_{i}a^{\prime}_{ij}\\}\big{)}\big{]}.$

On the other hand, the symplectic form is computed as follows. Put
$c(\gamma)_{1}:=\gamma^{*}(\omega_{1})$ and
$c(\gamma)_{2}:=\gamma^{*}(\omega_{2})$, where $\omega_{1}$ and $\omega_{2}$
are defined by (3.5) in the proof of Proposition 3.7. We have

$\displaystyle
c(t_{\widehat{\Phi}}(\gamma))_{1}(v,v^{\prime})=\big{[}\big{(}\big{\\{}\operatorname{Tr}\big{(}(\eta_{ij}-a_{ij}A_{i}-a_{ij}\Phi_{i})\circ(\eta^{\prime}_{jk}-a^{\prime}_{jk}A_{j}-a^{\prime}_{jk}\Phi_{j})\big{)}\big{\\}},$
$\displaystyle\hphantom{c(t_{\widehat{\Phi}}(\gamma))_{1}(v,v^{\prime})=}{}-\big{\\{}\operatorname{Tr}\big{(}(\eta_{ij}-a_{ij}A_{i}-a_{ij}\Phi_{i})\circ(v^{\prime}_{j}+\hat{v}^{\prime}_{j})\big{)}$
$\displaystyle\hphantom{c(t_{\widehat{\Phi}}(\gamma))_{1}(v,v^{\prime})=}{}-\operatorname{Tr}\big{(}(v_{i}+\hat{v}_{i})\circ(\eta^{\prime}_{ij}-a^{\prime}_{ij}A_{i}-a^{\prime}_{ij}\Phi_{i})\big{)}\big{\\}}\big{)}\big{]}$
$\displaystyle\hphantom{c(t_{\widehat{\Phi}}(\gamma))_{1}(v,v^{\prime})}{}=\big{[}\big{(}\big{\\{}{-}\operatorname{Tr}\big{(}(\eta_{ji}-a_{ji}A_{j}-a_{ji}\Phi_{j})\circ(\eta^{\prime}_{jk}-a^{\prime}_{jk}A_{j}-a^{\prime}_{jk}\Phi_{j})\big{)}\big{\\}},$
$\displaystyle\hphantom{c(t_{\widehat{\Phi}}(\gamma))_{1}(v,v^{\prime})=}{}-\big{\\{}{-}\operatorname{Tr}\big{(}(\eta_{ji}-a_{ji}A_{j}-a_{ji}\Phi_{j})\circ(v^{\prime}_{j}+\hat{v}^{\prime}_{j})\big{)}$
$\displaystyle\hphantom{c(t_{\widehat{\Phi}}(\gamma))_{1}(v,v^{\prime})=}{}-\operatorname{Tr}\big{(}(v_{i}+\hat{v}_{i})\circ(\eta^{\prime}_{ij}-a^{\prime}_{ij}A_{i}-a^{\prime}_{ij}\Phi_{i})\big{)}\big{\\}}\big{)}\big{]}$

and

$\displaystyle
c(\gamma)_{1}(v,v^{\prime})=\big{[}\big{(}\big{\\{}\operatorname{Tr}\big{(}(\eta_{ij}-a_{ij}A_{i})\circ(\eta^{\prime}_{jk}-a^{\prime}_{jk}A_{j})\big{)}\big{\\}},$
$\displaystyle\hphantom{c(\gamma)_{1}(v,v^{\prime})=}{}-\big{\\{}\operatorname{Tr}\big{(}(\eta_{ij}-a_{ij}A_{i})\circ
v^{\prime}_{j}\big{)}-\operatorname{Tr}\big{(}v_{i}\circ(\eta^{\prime}_{ij}-a^{\prime}_{ij}A_{i})\big{)}\big{\\}}\big{)}\big{]}$
$\displaystyle\hphantom{c(\gamma)_{1}(v,v^{\prime})}{}=\big{[}\big{(}\big{\\{}{-}\operatorname{Tr}\big{(}(\eta_{ji}-a_{ji}A_{j})\circ(\eta^{\prime}_{jk}-a^{\prime}_{jk}A_{j})\big{)}\big{\\}},$
$\displaystyle\hphantom{c(\gamma)_{1}(v,v^{\prime})=}{}-\big{\\{}{-}\operatorname{Tr}\big{(}(\eta_{ji}-a_{ji}A_{j})\circ
v^{\prime}_{j}\big{)}-\operatorname{Tr}(v_{i}\circ\big{(}\eta^{\prime}_{ij}-a^{\prime}_{ij}A_{i})\big{)}\big{\\}}\big{)}\big{]}.$

Then we obtain

$\displaystyle
c(t_{\tilde{\Phi}}(\gamma))_{1}(v,v^{\prime})-c(\gamma)_{1}(v,v^{\prime})=d\widehat{\Phi}_{1}(v,v^{\prime}).$

Moreover, we also can show that

$\displaystyle
c(t_{\tilde{\Phi}}(\gamma))_{2}(v,v^{\prime})-c(\gamma)_{2}(v,v^{\prime})=d\widehat{\Phi}_{2}(v,v^{\prime}).$

Then we obtain that
$\big{(}\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)},c\big{)}$
is an $\Omega_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}^{\geq 1}$-torsor. ∎

###  4.3 Extended parabolic connections

Let $(C,\boldsymbol{t})$, $\boldsymbol{t}=t_{1}+\cdots+t_{n}$, be an
$n$-pointed smooth projective curve of genus $g$ over $\mathbb{C}$ where
$t_{1},\ldots,t_{n}$ are distinct points. Put
$D(\boldsymbol{t})=t_{1}+\cdots+t_{n}$. We describe a description of
$(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connection with a quadratic
differential in terms of a “integral kernel” on $C\times C$ as in [5] and [6].
Let $p_{1}\colon C\times C\rightarrow C$ and $p_{2}\colon C\times C\rightarrow
C$ be the first and second projections, respectively. Put
$\mathcal{O}_{C}(*D(\boldsymbol{t})):=\varinjlim_{m}\mathcal{O}_{C}(mD(\boldsymbol{t}))$,
and
$\Omega^{1}_{C}(*D(\boldsymbol{t})):=\Omega^{1}_{C}\otimes\mathcal{O}_{C}(*D(\boldsymbol{t}))$.
Let $\mathcal{E}{\rm nd}^{0}(E)\subset\mathcal{E}{\rm nd}(E)$ be the subbundle
of traceless endmorphisms of $E$. We define sheaves
$\mathcal{K}_{D(\boldsymbol{t})}(E)$ on $C\times C$ as
$\mathcal{K}_{D(\boldsymbol{t})}(E):=p_{1}^{*}\big{(}E\otimes\Omega^{1}_{C}(*D(\boldsymbol{t}))\big{)}\otimes
p_{2}^{*}\big{(}E^{*}\otimes\Omega^{1}_{C}\big{)}(2\Delta)$, where
$\Delta\subset C\times C$ is the diagonal. We have a natural injective
morphism $\Omega_{C}^{\otimes 2}(*D(\boldsymbol{t}))\otimes\mathcal{E}{\rm
nd}^{0}(E)\rightarrow\mathcal{K}_{D(\boldsymbol{t})}(E)|_{3\Delta}$. We define
a sheaf $\mathcal{E}{\rm xConn}_{D(\boldsymbol{t})}(E)$ on $3\Delta$ by

$\displaystyle\mathcal{E}{\rm
xConn}_{D(\boldsymbol{t})}(E)=\big{\\{}s\in\mathcal{K}_{D(\boldsymbol{t})}(E)|_{3\Delta}/(\Omega_{C}^{\otimes
2}(*D(\boldsymbol{t}))\otimes\mathcal{E}{\rm
nd}^{0}(E))\,|\,s|_{\Delta}=\mathrm{Id}_{E}\big{\\}}.$

Note that we can consider $s|_{2\Delta}$ as a connection $s|_{2\Delta}\colon
E\rightarrow E\otimes\Omega^{1}_{C}(*D(\boldsymbol{t}))$. We consider elements
of $\mathcal{E}{\rm xConn}_{D(\boldsymbol{t})}(E)$ as pairs of connections and
quadratic differentials on $C$ locally. Let $U_{i}$ and $U_{j}$ be open sets
of $C$. Let $(A_{i},a_{i})$ be an elements of $\mathcal{E}{\rm
xConn}_{D(\boldsymbol{t})}(E)$ on $U_{i}$. Here $A_{i}df_{i}$ is a connection
matrix on $U_{i}$ and $a_{i}df_{i}\otimes df_{i}\in
H^{0}\big{(}U_{i},\Omega_{C}^{\otimes 2}(D(\boldsymbol{t}))\big{)}$. The
transformation of the pair is the following

$\displaystyle(A_{i}df_{i},a_{i}df_{i}\otimes df_{i})$
$\displaystyle\quad{}\longmapsto\left(\theta_{ij}^{-1}A_{i}\theta_{ij}df_{i}+\theta_{ij}^{-1}\frac{d\theta_{ij}}{df_{i}}df_{i},\left(a_{i}+\operatorname{Tr}\left(\theta_{ij}^{-1}A_{i}\frac{d\theta_{ij}}{df_{i}}\right)+\frac{1}{2}\operatorname{Tr}\left(\theta_{ij}^{-1}\frac{d^{2}\theta_{ij}}{df_{i}^{2}}\right)\right)df_{i}\otimes
df_{i}\right)$

on $U_{i}\cap U_{j}$. We may define an
$\mathcal{H}_{(C,\boldsymbol{t},E,\boldsymbol{l})}^{1}$-action on
$\mathcal{E}{\rm xConn}_{D(\boldsymbol{t})}(E)$ for any parabolic structures
$\boldsymbol{l}$ by (4.1).

Let $\nabla\colon E\rightarrow E\otimes\Omega_{C}^{1}(D(\boldsymbol{t}))$ be a
connection. We can define a global section $\nabla_{\text{Ex}}$ of
$\mathcal{E}{\rm xConn}_{D(\boldsymbol{t})}(E)$ associated to $\nabla$ as
follows. Take a trivialization of the locally free sheaf $E$ on an open set
$U_{i}$ of $C$. Let $A_{i}df_{i}$ be the connection matrix of $\nabla$ on
$U_{i}$. We define $\nabla_{\text{Ex}}|_{U_{i}}\in\mathcal{E}{\rm
xConn}_{D}(E)(U_{i})$ as

$\displaystyle\left(A_{i}df_{i},\frac{1}{2}\operatorname{Tr}\left(d(A_{i})\otimes
df_{i}\right)+\frac{1}{2}\operatorname{Tr}\left(A_{i}df_{i}\otimes
A_{i}df_{i}\right)\right),$

where $d$ is the exterior derivative. Let
$\big{(}\big{(}E,\nabla,\big{\\{}l_{j}^{(i)}\big{\\}}\big{)},\psi\big{)}$ be a
$(\boldsymbol{t},\boldsymbol{\nu})$-parabolic connection with a quadratic
differential. For $(\nabla,\psi)$, we can define a global section
$\nabla_{\text{Ex}}+\psi$ of $\mathcal{E}{\rm xConn}_{D(\boldsymbol{t})}(E)$
by the above construction. We call this description
$\big{(}E,\nabla_{\text{Ex}}+\psi,\big{\\{}l_{j}^{(i)}\big{\\}}\big{)}$ a
$(\boldsymbol{t},\boldsymbol{\nu})$-extended parabolic connection.

By the identification a parabolic connection with a quadratic differential as
an extended parabolic connection, we obtain another
$\Omega^{1}_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$-torsor structure on
$\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}$. We take
a section $\sigma$ of $\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$ and put
$\sigma(p)=(\nabla_{p},\psi_{p})$, where $\nabla_{p}\colon E\rightarrow
E\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))$ is a connection and $\psi_{p}\in
H^{0}\big{(}C,\Omega_{C}^{\otimes 2}(D(\boldsymbol{t}))\big{)}$ for
$p=((C,\boldsymbol{t}),(E,\boldsymbol{l}))$. For a connection
$\nabla_{p}\colon E\rightarrow E\otimes\Omega^{1}_{C}(D(\boldsymbol{t}))$ and
$\Phi_{p}\in H^{0}\big{(}\widetilde{\mathcal{H}}_{p}^{1}\big{)}$, we define
$\psi^{\prime}(\nabla_{p},\Phi_{p})\in H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$
as follows. We take an affine open covering $\\{U_{i}\\}$ of $C$ such that on
$U_{i}$ the connection $\nabla_{p}|_{U_{i}}$ is described by $d+A_{i}df_{i}$
and the Higgs field $\Phi_{p}|_{U_{i}}$ is described by $\Phi_{i}df_{i}$. On
each $U_{i}$, we define an element
$\psi^{\prime}(\nabla_{p},\Phi_{p})|_{U_{i}}$ as

$\displaystyle\psi^{\prime}(\nabla_{p},\Phi_{p})|_{U_{i}}=\left(\Phi_{i}df_{i},\operatorname{Tr}\left(\Phi_{i}A_{i}df_{i}\otimes
df_{i}+\frac{1}{2}\Phi_{i}\Phi_{i}df_{i}\otimes
df_{i}+\frac{1}{2}d(\Phi_{i})\otimes
df_{i}\right)\right)\in\mathcal{F}_{P_{\boldsymbol{\nu}}}^{1}(U_{i}),$

which gives an element $\psi^{\prime}(\nabla_{p},\Phi_{p})\in
H^{0}\big{(}\mathcal{H}_{p}^{1}\big{)}$. For a 1-form $\widehat{\Phi}$ on
$\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})$, we define a translation by

$\displaystyle t^{\prime}_{\widehat{\Phi}}\colon\
\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}\longrightarrow\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)},$
$\displaystyle\hphantom{t^{\prime}_{\widehat{\Phi}}\colon}{}\
\sigma(p)=(\nabla_{p},\psi_{p})\longmapsto
t^{\prime}_{\widehat{\Phi}}(\sigma)(p):=\big{(}\nabla_{p}+\kappa\big{(}\widehat{\Phi}_{p}\big{)},\psi_{p}-\big{(}\widehat{\Phi}_{p}-\psi^{\prime}\big{(}\nabla_{p},\kappa\big{(}\widehat{\Phi}_{p}\big{)}\big{)}\big{)}\big{)}.$

By the translation, we have another
$\Omega^{1}_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}$-torsor structure on
$\Gamma\big{(}\pi_{\mathfrak{P}_{g,n}(r,e,\boldsymbol{\nu})}\big{)}$.

### Acknowledgements

The author is supported by Grant-in-Aid for JSPS Research Fellows Number
18J00245. He is grateful to the anonymous referees’ suggestions which helped
to improve the paper.

## References

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