# Canonical coordinates for moduli spaces of rank two irregular connections on
curves.

Arata Komyo and Frank Loray and Masa-Hiko Saito and Szilárd Szabó  Department
of Material Science, Graduate School of Science, University of Hyogo, 2167
Shosha, Himeji, Hyogo 671-2280, Japan<EMAIL_ADDRESS>Univ Rennes,
CNRS, IRMAR - UMR 6625, F-35000 Rennes, France<EMAIL_ADDRESS>Department of Data Science, Faculty of Business Administration, KobeGakuin
University, Minatojima, Chuou-ku, Kobe, 650-8586, Japan
<EMAIL_ADDRESS>Department of Algebra and Geometry, Insitute of
Mathematics, Faculty of Natural Sciences, Budapest University of Technology
and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study a geometric counterpart of the cyclic vector which
allow us to put a rank 2 meromorphic connection on a curve into a “companion”
normal form. This allow us to naturally identify an open set of the moduli
space of $\mathrm{GL}_{2}$-connections (with fixed generic spectral data, i.e.
unramified, non resonant) with some Hilbert scheme of points on the twisted
cotangent bundle of the curve. We prove that this map is symplectic, therefore
providing Darboux (or canonical) coordinates on the moduli space, i.e.
separation of variables. On the other hand, for $\mathrm{SL}_{2}$-connections,
we give an explicit formula for the symplectic structure for a birational
model given by Matsumoto. We finally detail the case of an elliptic curve with
a divisor of degree $2$.

###### Key words and phrases:

Moduli space of connections, cyclic vector, canonical coordinates

The first author is supported by JSPS KAKENHI: Grant Numbers JP17H06127 and
JP19K14506. The second author is supported by CNRS and Centre Henri Lebesgue,
program ANR-11-LABX-0020-0. The third author is supported by JSPS KAKENH:Grant
Number 17H06127, 22H00094 and 22K18669. The fourth author was supported by
_Lendület_ Low Dimensional Topology grant of the Hungarian Academy of Sciences
and by the grants K120697 and KKP126683 of NKFIH

##  1\. Introduction

In this paper, we introduce coordinates on the moduli spaces of rank 2
meromorphic connections on a Riemann surface, and we describe the symplectic
structures on the moduli spaces by the introduced coordinates. Finally, we
will have canonical coordinates on the moduli spaces. Our motivation is to
give explicit descriptions of the isomonodromic deformations of meromorphic
connections over a general Riemann surface. It is well-know that the
isomonodromic deformations have non-autonomous Hamiltonian descriptions (in
detail, see [39], [20], [14], for example). If we find explicit formulae for
the isomonodromic Hamiltonians, then we have explicit descriptions of
isomonodromic deformations. To find explicit formulae of Hamiltonians, it is
necessary to introduce canonical coordinates (which are also called Darboux
coordinates) on the moduli space of meromorphic connections. The present paper
is a first step to give explicit descriptions of isomonodromic deformations.

For the isomonodromic deformations of rank 2 projective connections with
regular singular points, there are some results of explicit descriptions. For
example, Okamoto considered non-autonomous Hamiltonian descriptions of
isomonodromic deformations on elliptic curves in [48] and [49]. Iwasaki
generalized for general Riemann surfaces in [28] and [29]. Here the
independent variables of the isomonodromic deformations are the position of
regular singular points on the Riemann surfaces. That is, they are
isomonodromic deformations of fixed Riemann surfaces. On the other hand, Kawai
[31] gave explicit descriptions of the isomonodromic Hamiltonians varying the
elliptic curve. Okamoto, Iwasaki, and Kawai in these papers introduced
canonical coordinates on (a generic part of) the moduli space of rank 2
meromorphic projective connections by using apparent singularities. For our
purpose, we take this strategy. That is, we will also introduce canonical
coordinates on the moduli space of meromorphic connections by using apparent
singularities. On the other hand, in this paper, we are interested in the
isomonodromic deformations of $\mathrm{GL}_{2}$-connections and of
$\mathrm{SL}_{2}$-connections. The coordinates using apparent singularities
are an analog of the separation of variables in the Hitchin system, which is a
birational map from the moduli space of stable Higgs bundles to the Hilbert
scheme of points on the cotangent bundle over the underlying curve of the
Higgs bundles (see [19] and [16]). This approach has been generalized to Higgs
bundles with unramified irregular singularities [38, Section 8.3], [52], [37,
Section 4.2]. The moduli spaces of Higgs bundles corresponding to the Painlevé
cases were analyzed from this perspective in [25], [26], [27]. Here this map
is a symplectomorphism of the open dense subsets of the moduli space. The
definition of the apparent singularities for general rank meromorphic
connections will be given in [50].

###  1.1. Our setting

Let $\nu$ be a positive integer. We set $I:=\\{1,2,\ldots,\nu\\}$. Let $C$ be
a compact Riemann surface of genus $g$ ($g\geq 0$), and $D=\sum_{i\in
I}m_{i}[t_{i}]$ be an effective divisor on $C$. Let $E$ be a vector bundle
over $C$ and $\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(D)$ be a
meromorphic connection acting on $E$. We assume that the leading term of the
expansion of a connection matrix of $\nabla$ at $t_{i}$ has distinct
eigenvalues. If $m_{i}=1$, then we assume that the difference of eigenvalues
of the residue matrix at $t_{i}$ is not integer. That is, $t_{i}$ is an
generic unramified irregular singular point of $\nabla$ or a non-resonant
regular singular point of $\nabla$.

When $C$ is the projective line and $E$ is the trivial bundle, the moduli
space of meromorphic connections has been studied by Boalch [7] and
Hiroe–Yamakawa [18]. This moduli space has the natural symplectic structure
coming from the symplectic structure on the (extended) coadjoint orbits. For
general $C$ and $E$, the moduli space of meromorphic connections (with quasi-
parabolic structures) has been studied by Inaba–Iwasaki–Saito [22, 23], Inaba
[21], and Inaba–Saito [24]. For general $C$ and $E$, the moduli space has also
the natural symplectic structure. In these papers, the symplectic form
described by a pairing of the hypercohomologies of some complex. This
description of the symplectic structure is an analog of the symplectic
structure of the moduli spaces of stable Higgs bundles due to Bottacin [8].
For the case where $\nabla$ has only regular singular points, Inaba showed
that this symplectic structure coincides with the pull-back of the Goldman
symplectic structure on the character variety via the Riemann–Hilbert map in
[21, the proof of Proposition 7.3].

Our purpose in this paper is to introduce canonical coordinates on the moduli
spaces of meromorphic connections. For this purpose, there are some
strategies. First one is to consider canonical coordinates on the products of
coadjoint orbits. This direction was studied by Jimbo–Miwa–Mori–Sato [30],
Harnad [17], and Woodhouse [53]. Sakai–Kawakami–Nakamura [32] and
Gaiur–Mazzocco–Rubtsov [15] gave some explicit formulae for the isomonodromic
Hamiltonians by the coordinates of this direction. Second one is to consider
the apparent singularities. As mentioned above, we take this strategies.

In this paper, we consider only the case where the rank of $E$ is two. Let $X$
be an irregular curve, which is described in Section 2.3. That is, $X$ is a
tuple of (i) a compact Riemann surface $C$, (ii) an effective divisor $D$ on
$C$, (iii) local coordinates around the support with $D$, and (iv) spectral
data of meromorphic connections at the support with $D$ (with data of residue
parts). Here, the spectral data is described in Section 2.3. We fix an
irregular curve $X$. That is, we fix spectral data of rank 2 meromorphic
connections at each point of the support with $D$. By applying elementary
transformations (which is also called Hecke modifications), we may change the
degree of the underlying vector bundle of a meromorphic connection freely. So
we assume that $\deg(E)=2g-1$. By this condition, the Euler characteristic of
the vector bundle $E$ is $1$ by the Riemann–Roch theorem. In this situation,
for generic meromorphic connections $(E,\nabla)$, we have
$\dim_{\mathbb{C}}H^{0}(C,E)=1$. So the global section of $E$ is uniquely
determined up to constant. This is convenient for the definition of the
apparent singularities. In this paper, we consider only meromorphic
connections with $\dim_{\mathbb{C}}H^{0}(C,E)=1$. Moreover we assume that
meromorphic connections $(E,\nabla)$ are irreducible. By this condition, the
definition of apparent singularities becomes simple.

###  1.2. $\mathrm{GL}_{2}$-connections

In the first part of this paper, we discuss on $\mathrm{GL}_{2}$-connections.
That is, we consider rank 2 meromorphic connections. We do not fix the
determinant bundles of the underlying vector bundles and the traces of
connections. Our purpose is to introduce canonical coordinates on the moduli
space of rank 2 meromorphic connections by using apparent singularities. When
$C$ is the projective line, many people introduced canonical coordinates on
the moduli space by using the apparent singularities ([48], [46], [10], [51],
[36], [9], and [34]). In this paper, we consider apparent singularities for
general Riemann surfaces.

Let $X$ be the fixed irregular curve. If $(E,\nabla)$ is a rank 2 meromorphic
connection such that $\deg(E)=2g-1$, $\dim_{\mathbb{C}}H^{0}(C,E)=1$, and
$(E,\nabla)$ is irreducible, then we can define apparent singularities for
$(E,\nabla)$. (In detail, see Definition 1 below). The apparent singularities
are the set of points $\\{q_{1},\ldots,q_{N}\\}$ on the underlying curve $C$.
Here we set $N:=4g-3+\deg(D)$. Let $M_{X}$ be the following moduli space

$M_{X}:=\left.\left\\{(E,\nabla)\ \middle|\ \begin{array}[]{l}\text{(i) $E$ is
a rank 2 vector bundle on $C$ with $\deg(E)=2g-1$}\\\ \text{(ii) $\nabla\colon
E\rightarrow E\otimes\Omega^{1}_{C}(D)$ is a connection}\\\ \text{(iii)
$(E,\nabla)$ is irreducible, and}\\\ \text{(iv) $\nabla$ has the fixed
spectral data in $X$}\end{array}\right\\}\right/\cong.$

This moduli space $M_{X}$ has a natural symplectic structure due to
Inaba–Iwasaki–Saito [22], Inaba [21], and Inaba–Saito [24]. We consider a
Zariski open subset $M_{X}^{0}$ of $M_{X}$ as follows:

$M^{0}_{X}:=\left.\left\\{(E,\nabla)\in M_{X}\ \middle|\
\begin{array}[]{l}\text{(i) $\dim_{\mathbb{C}}H^{0}(C,E)=1$},\\\ \text{(ii)
$q_{1}+\cdots+q_{N}$ is reduced, and}\\\ \text{(iii) $q_{1}+\cdots+q_{N}$ has
disjoint support with $D$}\end{array}\right\\}\right/\cong$

(in detail, see Section 3.1). The dimension of the moduli space $M^{0}_{X}$ is
$2N$ (Proposition 10). By taking apparent singularities, we have a map

$\displaystyle\operatorname{App}\colon M^{0}_{X}$
$\displaystyle\longrightarrow\mathrm{Sym}^{N}(C)$ $\displaystyle(E,\nabla)$
$\displaystyle\longmapsto\\{q_{1},q_{2},\ldots,q_{N}\\}.$

Remark that the dimension of $\mathrm{Sym}^{N}(C)$ is half of the dimension of
$M^{0}_{X}$. To introduce coordinates on $M^{0}_{X}$, it is necessary to find
further invariants of connections, that are customarily called _accessory
parameters_. To find these parameters, we introduce a twist of
$\Omega^{1}_{C}(D)$ by $c_{d}$, which is the first Chern class
$c_{1}(\det(E))\in H^{1}(C,\Omega^{1}_{C})$ of $E$. (In detail, Section 3.5
below). We denote by $\Omega_{C}^{1}(D,c_{d})$ the twist of
$\Omega^{1}_{C}(D)$. Let

$\pi_{c_{d}}\colon\boldsymbol{\Omega}(D,c_{d})\longrightarrow C$

the total space of $\Omega^{1}_{C}(D,c_{d})$. Let $\omega_{D,c_{d}}$ be the
rational 2-form on $\boldsymbol{\Omega}(D,c_{d})$ induced by the Liouville
symplectic form. This rational 2-form $\omega_{D,c_{d}}$ induces a symplectic
structure on $\boldsymbol{\Omega}(D,c_{d})\setminus\pi_{c_{d}}^{-1}(D)$. We
consider the symmetric product
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$. Let
$\sum_{j=1}^{N}\mathrm{pr}_{j}^{*}(\omega_{D,c_{d}})$ be the rational 2-form
on the product $\boldsymbol{\Omega}(D,c_{d})^{N}$. Here
$\mathrm{pr}_{j}\colon\boldsymbol{\Omega}(D,c_{d})^{N}\rightarrow\boldsymbol{\Omega}(D,c_{d})$
is the $j$-th projection. This rational 2-form
$\sum_{j=1}^{N}\mathrm{pr}_{j}^{*}(\omega_{D,c_{d}})$ induces a symplectic
structure on a generic part of
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$. We will define a map from
$M^{0}_{X}$ to $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ by the
following idea.

By the theory of apparent singularities discussed in Section 2.1, we have a
canonical inclusion morphism

$\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}\longrightarrow E.$

By this morphism, we have the connection $\nabla_{0}$ on
$\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}$ induced by a connection
$\nabla$ on $E$. Notice that $\nabla_{0}$ has simple poles at the apparent
singularities. By applying automorphisms on
$\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}$, we may normalize $\nabla_{0}$
as

$\nabla_{0}=\begin{pmatrix}\operatorname{d}&\beta\\\ 1&\delta\end{pmatrix},$

which is called a companion normal form (in detail, see Section 2.2 below).
Here $\operatorname{d}$ is the exterior derivative on $C$, $\beta\in
H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(2D+q_{1}+\cdots+q_{N}))$, and $\delta$ is
a connection on $(\Omega^{1}_{C}(D))^{-1}$, which has poles at the support of
$D$ and the apparent singularities $q_{1},\ldots,q_{N}$. Then we may define a
map

(1.1) $\displaystyle f_{\mathrm{App}}\colon M^{0}_{X}$
$\displaystyle\longrightarrow\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$
$\displaystyle(E,\nabla)$
$\displaystyle\longmapsto\\{(q_{j},\mathrm{res}_{q_{j}}(\beta)+\mathrm{tr}(\nabla)|_{q_{j}})\\}_{1\leq
j\leq N}.$

Here, notice that $\mathrm{res}_{q_{j}}(\beta)\in\Omega^{1}_{C}(D)|_{q_{j}}$
and $\mathrm{tr}(\nabla)|_{q_{j}}$ is justified by considering the twisted
cotangent bundle (in detail, see Definition 16 below). Remark that the
dimension of $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ is equal to the
dimension of $M^{0}_{X}$. A generic part of
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ has the natural symplectic
structure induced by the symplectic structure on the product
$(\boldsymbol{\Omega}(D,c_{d})\setminus\pi_{c_{d}}^{-1}(D))\times\cdots\times(\boldsymbol{\Omega}(D,c_{d})\setminus\pi_{c_{d}}^{-1}(D))$.
The first main theorem is the following:

######  Theorem A (Theorem 20 below).

The pull-back of the symplectic form on a generic part of
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ under the map (1.1) coincides
with the symplectic form on $M^{0}_{X}$.

If we take canonical coordinates on $\boldsymbol{\Omega}(D,c_{d})$, then we
have canonical coordinate on $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$,
since the symplectic structure on
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ is induced by the 2-form
$\sum_{j=1}^{N}\mathrm{pr}_{j}^{*}(\omega_{D,c_{d}})$. Then we have canonical
coordinates on $M^{0}_{X}$ by Theorem A. Detail of construction of concrete
canonical coordinates on $M^{0}_{X}$ is discussed in the paragraph after the
proof of Theorem 20 below.

In Section 5, we consider an example of this argument. We will calculate the
canonical coordinates for an elliptic curve and a divisor $D$ of length $2$.
The moduli space of rank 2 meromorphic connection with fixed trace connection
on an elliptic curve with two simple poles was studied in [41] and [12]. In
this paper, we will discuss the $\mathrm{GL}_{2}$-connection case.

###  1.3. $\mathrm{SL}_{2}$-connections

In the second part of this paper, we discuss on $\mathrm{SL}_{2}$-connections.
That is, we consider rank 2 meromorphic connections with fixed trace
connection $(L_{0},\nabla_{0})$. Here $L_{0}$ is a fixed line bundle on $C$ of
degree $2g-1$ and $\nabla_{0}\colon L_{0}\rightarrow
L_{0}\otimes\Omega^{1}_{C}(D)$ is a fixed connection. More precisely, we
consider rank 2 quasi-parabolic connections $(E,\nabla,\\{l^{(i)}\\})$,
defined in [24, Definition 2.1], with fixed trace connection
$(L_{0},\nabla_{0})$. Here the spectral data of $\nabla_{0}$ is determined by
the fixed irregular curve $X$. The quasi-parabolic structure $l^{(i)}$ at
$t_{i}$ induces a one dimensional subspace $l^{(i)}_{\mathrm{red}}$ of
$E|_{t_{i}}$, that is the restriction of $l^{(i)}$ to $t_{i}$ (without
multiplicity). Our moduli space is as follows:

$M_{X}(L_{0},\nabla_{0})_{0}:=\left.\left\\{(E,\nabla,\\{l^{(i)}\\})\
\middle|\ \begin{array}[]{l}\text{(i) $\nabla$ has the fixed spectral data in
$X$,}\\\ \text{(ii) $E$ is an extension of $L_{0}$ by $\mathcal{O}_{C}$,}\\\
\text{(iii) $\dim_{\mathbb{C}}H^{0}(C,E)=1$, and}\\\ \text{(iv)
$l_{\text{red}}^{(i)}\not\in\mathcal{O}_{C}|_{t_{i}}\subset\mathbb{P}(E)$ for
any $i$}\end{array}\right\\}\right/\cong,$

which is described in Section 4.2. Here $(E,\nabla,\\{l^{(i)}\\})$ are rank 2
quasi-parabolic connections on $(C,D)$ with fixed trace connection
$(L_{0},\nabla_{0})$. When $g=0$, we impose one more condition (in detail, see
the paragraph after the proof of Lemma 26 below). This moduli space also has a
natural symplectic structure. The dimension of the moduli space
$M_{X}(L_{0},\nabla_{0})_{0}$ is $2N_{0}$, where $N_{0}:=3g-3+\deg(D)$. For
$(E,\nabla,\\{l^{(i)}\\})\in M_{X}(L_{0},\nabla_{0})_{0}$, we can also define
apparent singularities (Section 4.2 below). The apparent singularities give an
element of $\mathbb{P}H^{0}(C,L_{0}\otimes\Omega_{C}^{1}(D))$. So we have a
map

$\pi_{\mathrm{App}}\colon
M_{X}(L_{0},\nabla_{0})_{0}\longrightarrow\mathbb{P}H^{0}(C,L_{0}\otimes\Omega_{C}^{1}(D)).$

For $(E,\nabla,\\{l^{(i)}\\})\in M(L_{0},\nabla_{0})_{0}$, we forget the
connection $\nabla$. So we have a quasi-parabolic bundle $(E,\\{l^{(i)}\\})$.
By taking the extension class for the quasi-parabolic bundle
$(E,\\{l^{(i)}\\})$, we have a map

$\pi_{\mathrm{Bun}}\colon
M_{X}(L_{0},\nabla_{0})_{0}\longrightarrow\mathbb{P}H^{1}(C,L_{0}^{-1}(-D)).$

Here the extension class is described in Section 4.1 below. We consider the
product

$\pi_{\mathrm{App}}\times\pi_{\mathrm{Bun}}\colon
M_{X}(L_{0},\nabla_{0})_{0}\longrightarrow\mathbb{P}H^{0}(C,L_{0}\otimes\Omega_{C}^{1}(D))\times\mathbb{P}H^{1}(C,L_{0}^{-1}(-D)).$

This map has been studied by Loray–Saito–Simpson [43], Loray–Saito [42],
Fassarella–Loray [12], Fassarella–Loray–Muniz [13], and Matsumoto [44].

Notice that $H^{1}(C,L_{0}^{-1}(-D))$ is isomorphic to the dual of
$H^{0}(C,L_{0}\otimes\Omega_{C}^{1}(D))$. Remark that

$\dim_{\mathbb{C}}\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))=\dim_{\mathbb{C}}\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))=N_{0}.$

Let us introduce the homogeneous coordinates
$\boldsymbol{a}=(a_{0}:\cdots:a_{N_{0}})$ on
$\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))\cong\mathbb{P}_{\boldsymbol{a}}^{N_{0}}$
and the dual coordinates $\boldsymbol{b}=(b_{0}:\cdots:b_{N_{0}})$ on

$\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))\cong\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))^{\vee}\cong\mathbb{P}_{\boldsymbol{b}}^{N_{0}}.$

We may define a $1$-form $\eta$ on
$\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}$
by

$\eta=\left(\text{constant}\right)\cdot\frac{a_{0}\,db_{0}+a_{1}\,db_{1}+\cdots+a_{N_{0}}\,db_{N_{0}}}{a_{0}b_{0}+a_{1}b_{1}+\cdots+a_{N_{0}}b_{N_{0}}}.$

(In detail, see Section 4.4). The $2$-form $d\eta$ gives an symplectic
structure on
$\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}\setminus\Sigma$.
Here we set

$\Sigma\colon(a_{0}b_{0}+a_{1}b_{1}+\cdots+a_{N_{0}}b_{N_{0}}=0)\subset\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}.$

The image of $M(L_{0},\nabla_{0})_{0}$ is contained in
$\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}\setminus\Sigma$.
(In detail, see Section 4.3). The second main theorem is the following:

######  Theorem B (Theorem 31 below).

We assume that the fixed spectral data satisfies the generic condition (4.8)
below. The pull-back of the symplectic form $d\eta$ on
$\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}\setminus\Sigma$
under the map $\pi_{\mathrm{App}}\times\pi_{\mathrm{Bun}}$ coincides with the
symplectic form on the moduli space $M_{X}(L_{0},\nabla_{0})_{0}$.

###  1.4. The organization of this paper

In Section 2, the apparent singularities for a generic rank 2 meromorphic
connection are defined. After the definition of the apparent singularities, we
will discuss on the companion normal form of a generic rank 2 meromorphic
connection. We will use this companion normal form when we will introduce
canonical coordinates. In Section 3, first, we will describe our moduli space
of rank 2 meromorphic connections. Second, we will discuss on tangent spaces
of the moduli space of rank 2 meromorphic connections. We will recall that the
tangent spaces at a meromorphic connection are isomorphic to a hypercohomology
of the complex defined by the meromorphic connection. After that, we will
describe a natural symplectic structure on the moduli space of rank 2
meromorphic connections. Section 3.3 and Section 3.4 are preliminaries of the
proof of the first main theorem. In Section 3.5, we will give the map from a
generic part of the moduli space to
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ and will show the first main
theorem.

In Section 4, we will consider rank 2 meromorphic connections with fixed trace
connection. First, to describe the bundle map $\pi_{\mathrm{Bun}}$, we recall
the moduli space of stable quasi-parabolic bundles with fixed determinant.
Second, we will describe our moduli space of rank 2 meromorphic connections
with fixed trace connection. Third, we will describe the map
$\pi_{\mathrm{App}}$ defined by considering the apparent singularities. In
Section 4.4, we will recall a natural symplectic structure on the moduli space
of rank 2 meromorphic connections with fixed trace connection, and will show
the second main theorem.

In Section 5, we will apply the argument in Section 2 and Section 3 to the
case of an elliptic curve with a divisor $D$ of length $2$. When $D$ is
reduced, this amounts to two logarithmic singularities, otherwise to an
irregular singularity. It is remarkable that using our approach these two
cases can be studied completely similarly.

In Section 6, we will provide a method for obtaining canonical coordinates
$\tilde{p}_{j}\in\boldsymbol{\Omega}(D,c_{d})_{|q_{i}}$ for generic
$(E,\nabla)\in M^{0}_{X}$ by introducing a section $s\in H^{0}(C,\det(E))$ and
$\gamma\in H^{0}(C,\Omega^{1}_{C}(D))$. We will utilize an open set
$U_{0}=C\setminus\\{s=0,\gamma=0\\}$ and the trivialization of $E_{|U_{0}}$ to
define $\tilde{p}_{j}\in\Omega^{1}_{C}(D)_{|q_{j}}$. This method can be also
used for constructing a meromorphic connection $\nabla_{1}\colon
E\longrightarrow E\otimes\Omega^{1}_{C}(D(s))$ for a given $s\in
H^{0}(C,\det(E))$, where $D(s)$ denotes the zero divisor of $s$. In Theorem
41, we will provide an alternative proof of the birationality of
$f_{\mathrm{App}}$ (cf. Proposition 17) by utilizing the Higgs fields
$\nabla-\nabla_{1}$ and the BNR correspondence [3]. This approach may shed new
light on the relationship between the canonical coordinates of the moduli
spaces of connections and the moduli spaces of Higgs bundles. (cf. [50]).

### Acknowledgments

The authors would like to warmly thank Michi-aki Inaba and Takafumi Matsumoto
for useful discussions. The first, third, and fourth authors would like to
thank Frank Loray for his hospitality at IRMAR, Univ. Rennes.

##  2\. Companion normal form

Let $C$ be a compact Riemann surface of genus $g$ ($g\geq 0$), and $D$ be an
effective divisor on $C$. We assume $4g-3+n>0$ where $n=\deg(D)$. We consider
a rank 2 meromorphic connection

(2.1) $\nabla:E\longrightarrow E\otimes\Omega^{1}_{C}(D)$

on $C$ where $\deg(E)=2g-1$.

When $g=0$, Diarra–Loray have given companion normal forms of the rank 2
meromorphic connections in [9]. By the companion normal forms, we may
construct a universal family of the rank 2 meromorphic connections on some
generic part of the moduli space of rank 2 meromorphic connections. This
universal family is useful to describe the isomonodromic deformations [34].
The purpose of this section is to give companion normal forms of rank 2
meromorphic connections when $g\geq 0$. For this purpose, first, we will
introduce the apparent singularities for (generic) rank 2 meromorphic
connections.

###  2.1. Apparent singularities

First we assume that $\dim_{\mathbb{C}}H^{0}(C,E)=1$ for the rank 2
meromorphic connection (2.1). This assumption holds for a generic vector
bundle of the rank 2 meromorphic connection with $\deg(E)=2g-1$. For an
element of $H^{0}(C,E)$, we define the sequence of $\mathbb{C}$-linear maps

(2.2) $\varphi_{\nabla}\colon\mathcal{O}_{C}\longrightarrow E\xrightarrow{\
\nabla\ }E\otimes\Omega^{1}_{C}(D)\longrightarrow
E/\mathcal{O}_{C}\otimes\Omega^{1}_{C}(D).$

This composition $\varphi_{\nabla}$ is an $\mathcal{O}_{C}$-linear map. From
now on we assume that $\varphi_{\nabla}\neq 0$. This assumption holds for
every $(E,\nabla)$, provided that the eigenvalues of the residues are chosen
generically (see Remark 4 below). We call the global section in $H^{0}(C,E)$
in (2.2) the cyclic vector.

Let us now define $E_{0}\subset E$ as the rank $2$ locally free subsheaf
spanned by $\mathcal{O}_{C}$ and

$\operatorname{Im}\left\\{\nabla|_{\mathcal{O}_{C}}\otimes\operatorname{Id}_{(\Omega^{1}_{C}(D))^{-1}}\colon(\Omega^{1}_{C}(D))^{-1}\to
E\right\\}.$

This construction gives rise to a short exact sequence of coherent sheaves

$0\longrightarrow\mathcal{O}_{C}\longrightarrow
E_{0}\longrightarrow(\Omega^{1}_{C}(D))^{-1}\longrightarrow 0.$

We claim that this sequence splits, i.e.

(2.3) $E_{0}\cong\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}.$

Indeed, equivalence classes of extensions of $(\Omega^{1}_{C}(D))^{-1}$ by
$\mathcal{O}_{C}$ are classified by the group

$\operatorname{Ext}^{1}((\Omega^{1}_{C}(D))^{-1},\mathcal{O}_{C})=\operatorname{Ext}^{1}(\mathcal{O}_{C}(-D),\Omega^{1}_{C})\cong
H^{0}(C,\mathcal{O}_{C}(-D))^{\vee}=0,$

where we have used Grothendieck–Serre duality. We denote by

(2.4) $\phi_{\nabla}\colon E_{0}\longrightarrow E.$

the canonical inclusion morphism, and define the meromorphic connection

(2.5) $\nabla_{0}=\phi_{\nabla}^{*}(\nabla)$

on $E_{0}$. We note that the polar divisor of $\nabla_{0}$ is $D+B$ where

(2.6) $B=\mathrm{div}(\varphi_{\nabla}).$

We note that

(2.7) $\deg(B)=4g-3+n.$

From now on, moreover, we assume that $B$ is reduced, with support disjoint
from $D$. In different terms, in view of (2.7), we have

$B=q_{1}+\cdots+q_{4g-3+n}$

where $q_{i}\neq q_{j}$ once $i\neq j$ and $q_{i}\notin D$ for all $i$.

######  Definition 1.

Assume that $\varphi_{\nabla}\neq 0$ and $\mathrm{div}(\varphi_{\nabla})$ is
reduced, with support disjoint from $D$. We call the points of the support
$\\{q_{1},\ldots,q_{4g-3+n}\\}$ of $\mathrm{div}(\varphi_{\nabla})$ the
apparent singularities of $(E,\nabla)$.

###  2.2. Companion normal form

The desired companion normal form is a normal form of $\nabla_{0}$ in (2.5).
So the companion normal form is given by normalization of $\nabla_{0}$ by
applying automorphisms on $\mathcal{O}_{C}\oplus(\Omega^{1}_{C}(D))^{-1}$. To
give the companion normal form, first, we describe a decomposition of
$\nabla_{0}$ relative to (2.3):

$\nabla_{0}=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}$

where

$\left\\{\begin{matrix}\alpha&:&\mathcal{O}_{C}\longrightarrow\Omega^{1}_{C}(D+B)\hfill(\text{connection})\\\
\beta&:&(\Omega^{1}_{C}(D))^{-1}\longrightarrow\Omega^{1}_{C}(D+B)\hfill(\mathcal{O}_{C}\text{-linear})\\\
\gamma&:&\mathcal{O}_{C}\longrightarrow\mathcal{O}_{C}(B)\hfill(\mathcal{O}_{C}\text{-linear})\\\
\delta&:&(\Omega^{1}_{C}(D))^{-1}\longrightarrow(\Omega^{1}_{C}(D))^{-1}\otimes\Omega^{1}_{C}(D+B)\hskip
28.45274pt(\text{connection})\end{matrix}\right.$

This form is unique only up to pre-composition by an element of the
automorphism group $\mathrm{Aut}(E_{0})$ of $E_{0}$. Elements of
$\mathrm{Aut}(E_{0})$ are described as follows:

$\begin{pmatrix}\lambda_{1}&F\\\ 0&\lambda_{2}\end{pmatrix},$

where $\lambda_{1},\lambda_{2}\in\mathbb{C}^{*}$ and $F\in
H^{0}(C,\Omega^{1}_{C}(D))$. It follows by construction that $\nabla_{0}$
admits no pole in restriction to $\mathcal{O}_{C}$ over the divisor $B$, so
that actually we have

$\left\\{\begin{matrix}\alpha&:&\mathcal{O}_{C}\longrightarrow\Omega^{1}_{C}(D)\hskip
28.45274pt(\text{connection})\\\
\gamma&:&\mathcal{O}_{C}\longrightarrow\mathcal{O}_{C}\hfill(=\text{identity})\end{matrix}\right.$

The action of an automorphism of the form

(2.8) $\begin{pmatrix}1&F\\\ 0&1\end{pmatrix},\ \ \ F\in
H^{0}(C,\Omega^{1}_{C}(D))$

transforms $\alpha$ into $\alpha-F\gamma$ (without affecting $\gamma$).
Therefore, there exists a unique choice $F$ such that
$\alpha=\operatorname{d}$ is the trivial connection on $\mathcal{O}_{C}$. We
thus get the unique companion normal form

(2.9) $\nabla_{0}=\begin{pmatrix}\operatorname{d}&\beta\\\
1&\delta\end{pmatrix}.$

Notice that the same companion normal form is obtained simply by taking the
generator $\varphi_{\nabla}(1)$ for the second factor of (2.3), and the action
of the automorphism (2.8) in the above argument simply amounts to switching to
this particular generator.

###  2.3. Spectral data

Now we consider the polar part of the meromorphic connection (2.1) at each
point of the support of $D$. We impose some conditions on the polar parts. To
describe the conditions, we introduce the notion of irregular curves with
residues. Let $\nu$ be a positive integer. We set $I:=\\{1,2,\ldots,\nu\\}$.
Let $\mathfrak{h}$ be the Cartan subalgebra

$\mathfrak{h}=\left\\{\begin{pmatrix}h_{1}&0\\\ 0&h_{2}\end{pmatrix}\
\middle|\ h_{1},h_{2}\in\mathbb{C}\right\\}$

of the Lie algebra $\mathfrak{gl}_{2}(\mathbb{C})$. Let $\mathfrak{h}_{0}$ be
the regular locus of $\mathfrak{h}$.

######  Definition 2.

We say $X=(C,D,\\{z_{i}\\}_{i\in I},\\{\boldsymbol{\theta}_{i}\\}_{i\in
I},\boldsymbol{\theta}_{\mathrm{res}})$ is an irregular curve with residues if

  * (i)

$C$ is a compact Riemann surface of genus $g$,

  * (ii)

$D=\sum_{i\in I}m_{i}[t_{i}]$ is an effective divisor on $C$.

  * (iii)

$z_{i}$ is a generator of the maximal ideal of $\mathcal{O}_{C,t_{i}}$,

  * (iv)

$\boldsymbol{\theta}_{i}=(\theta_{i,-m_{i}},(\theta_{i,-m_{i}+1},\ldots,\theta_{i,-2}))\in\mathfrak{h}_{0}\times\mathfrak{h}^{m_{i}-2}$,
and

  * (v)

$\boldsymbol{\theta}_{\mathrm{res}}=(\theta_{1,-1},\theta_{2,-1},\ldots,\theta_{\nu,-1})$,
where $\theta_{i,-1}\in\mathfrak{h}$, such that
$\sum_{i=1}^{\nu}\mathrm{tr}(\theta_{i,-1})=-(2g-1)$.

We set

$\theta_{i,-1}=\begin{pmatrix}\theta_{i,-1}^{-}&0\\\
0&\theta_{i,-1}^{+}\end{pmatrix}\quad\text{for each $i\in I$}.$

We assume that $\sum_{i=1}^{\nu}\theta^{\pm}_{i,-1}\not\in\mathbb{Z}$ whatever
are the signs $\pm$, and, if $m_{i}=1$, then
$\theta^{+}_{i,-1}-\theta^{-}_{i,-1}\not\in\mathbb{Z}$.

For an irregular curve with residues $X$, we set

(2.10)
$\omega_{i}(X):=\theta_{i,-m_{i}}\frac{\operatorname{d}\\!z_{i}}{z_{i}^{m_{i}}}+\theta_{i,-m_{i}+1}\frac{\operatorname{d}\\!z_{i}}{z_{i}^{m_{i}-1}}+\cdots+\theta_{i,-2}\frac{\operatorname{d}\\!z_{i}}{z_{i}^{2}}+\theta_{i,-1}\frac{\operatorname{d}\\!z_{i}}{z_{i}}$

and $\mathcal{O}_{m_{i}[t_{i}]}:=\mathcal{O}_{C,t_{i}}/(z_{i}^{m_{i}})$. For
an irregular curve with residues $X$ and a meromorphic connection $(E,\nabla)$
in (2.1), we set $E|_{m_{i}[t_{i}]}:=E\otimes\mathcal{O}_{m_{i}[t_{i}]}$. Let

$\nabla|_{m_{i}[t_{i}]}\colon E|_{m_{i}[t_{i}]}\longrightarrow
E|_{m_{i}[t_{i}]}\otimes\Omega^{1}_{C}(D)$

be the morphism induced by $\nabla$.

######  Definition 3.

We call $(E,\nabla)$ a rank $2$ meromorphic connection over an irregular curve
with residues $X$ if

  * (i)

$E$ is a rank $2$ vector bundle of degree $2g-1$ on $C$,

  * (ii)

$\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(D)$ is a connection, and

  * (iii)

there exists an isomorphism $\varphi_{m_{i}[t_{i}]}\colon
E|_{m_{i}[t_{i}]}\rightarrow\mathcal{O}^{\oplus 2}_{m_{i}[t_{i}]}$ for each
$i\in I$ such that

$(\varphi_{m_{i}[t_{i}]}\otimes
1)\circ\nabla|_{m_{i}[t_{i}]}\circ\varphi^{-1}_{m_{i}[t_{i}]}=\operatorname{d}+\,\omega_{i}(X).$

Here $\omega_{i}(X)$ is defined in (2.10).

We call $\omega_{i}(X)$ the spectral data of $(E,\nabla)$ and call the
submodule $\varphi_{m_{i}[t_{i}]}^{-1}(\mathcal{O}_{m_{i}[t_{i}]}\oplus 0)$ of
$E|_{m_{i}[t_{i}]}$ the quasi-parabolic structure of $(E,\nabla)$ at $t_{i}$.

From now on, by a connection we will mean a rank $2$ meromorphic connection
over a fixed irregular curve with residues $X$. So we impose the condition
(iii) of Definition 3 on the polar parts of the meromorphic connection
$\nabla$ in (2.1) at the points of the support of $D$. This condition means
that the polar parts of $\nabla$ at $t_{i}$ are diagonalizable with
eigenvalues equal to the diagonal entries of $\omega_{i}(X)$ for
$i=1,2,\ldots,n$.

######  Remark 4.

In Definition 3, we impose the condition that
$\sum_{i=1}^{\nu}\theta^{\pm}_{i,-1}\not\in\mathbb{Z}$ whatever are the signs
$\pm$. By this assumption and the argument as in [35, Proposition 6], we have
that $(E,\nabla)$ is irreducible. Then some arguments become simple. For
example, $\varphi_{\nabla}=0$ if and only if the free subsheaf
$\mathcal{O}_{C}$ of $E$ is a proper $\nabla$-invariant subbundle. So we have
that $\varphi_{\nabla}\neq 0$. Moreover, $(E,\nabla)$ is automatically stable
(described in Section 3.1 below).

###  2.4. The polar parts of $\delta$

We fix an irregular curve with residues $X$. Let $(E,\nabla)$ be a rank $2$
meromorphic connection over $X$ and $\nabla_{0}$ be the companion normal form
for $(E,\nabla)$. We consider the $(2,2)$-entry $\delta$ of this companion
normal form $\nabla_{0}$.

It immediately follows from (2.9) that the connection $\delta$ coincides with
the trace connection $\operatorname{tr}(\nabla_{0})$ on
$\det(E_{0})=(\Omega^{1}_{C}(D))^{-1}$. It is further related to the trace
connection $\operatorname{tr}(\nabla)$ by

$\delta=\operatorname{tr}(\nabla_{0})=\operatorname{tr}(\nabla)+\frac{\operatorname{d}\\!\varphi_{\nabla}}{\varphi_{\nabla}}.$

######  Lemma 5.

  1. (1)

The polar part of $\delta$ over $D$ is determined by the spectral data;

  2. (2)

The polar part of $\delta$ over $B$ is logarithmic with residue $+1$;

  3. (3)

$\delta$ is determined by the irregular curve with residues $X$ up to adding a
holomorphic $1$-form of $C$.

###### Proof.

The polar part of $\delta$ at $t_{i}$ is equal to
$\operatorname{tr}(\omega_{i})$, showing the first assertion. In view of our
assumption $q_{j_{1}}\neq q_{j_{2}}$ for ${j_{1}}\neq{j_{2}}$, the second
assertion is classical. Let now $\delta,\delta^{\prime}$ be the
$(2,2)$-entries of companion normal forms $\nabla_{0},\nabla_{0}^{\prime}$ of
connections $\nabla,\nabla^{\prime}$ satisfying the conditions of Definition
3. By the first part, $\delta-\delta^{\prime}$ is then a global holomorphic
$1$-form of $C$. ∎

As a consequence of the lemma and by
$\operatorname{dim}_{\mathbb{C}}H^{0}(C,\Omega_{C}^{1})=g$, the possible
values for $\delta$ represent $g$ free parameters for a meromorphic connection
over $X$.

###  2.5. The polar parts of $\beta$

Next we consider the $(1,2)$-entry $\beta$ of the companion normal form
$\nabla_{0}$ of a meromorphic connection $\nabla$ over the irregular curve
with residues $X$. By the condition $\gamma=1$ in (2.9), $\beta$ accounts for
the determinant of the characteristic polynomial of the residues. By
Definition 3, the eigenvalues of the connection matrix of $\nabla$ are
differentials (of the first kind) with a pole of order at most $m_{i}$ at
$t_{i}$. The same condition then holds for $\nabla_{0}$ too, because it only
differs from $\nabla$ by elementary modifications at points $q_{j}\neq t_{i}$.
As the determinant of a $2\times 2$ matrix is a quadratic expression of the
eigenvalues, we see that $\beta$ must be a quadratic differential with poles
of order at most $2m_{i}$ at $t_{i}$. Over $B$, a similar argument shows that
$\beta$ has poles of order at most $2$.

Let us fix local coordinate charts $z_{i}$ centered at the pole $t_{i}$. One
may then expand $\beta$ into Laurent series:

$\beta=\left(\beta_{i,-2m_{i}}z_{i}^{-2m_{i}}+\cdots+\beta_{i,-2}z_{i}^{-2}+O(z_{i}^{-1})\right)(\operatorname{d}\\!z_{i})^{\otimes
2}.$

Notice that for given $\beta$ the coefficient $\beta_{i,-2}$ is independent of
the chosen coordinate chart $z_{i}$, however the other coefficients depend on
$z_{i}$. We also fix local coordinate charts $z_{j}$ centered at the apparent
singularity $q_{j}$, and have a similar expansion

$\beta=\left(\beta_{j,-2}z_{j}^{-2}+\beta_{j,-1}z_{j}^{-1}+O(z_{j}^{0})\right)(\operatorname{d}\\!z_{j})^{\otimes
2}.$

Analogously to Lemma 5, we therefore find

######  Lemma 6.

  1. (1)

The coefficients $\beta_{i,-2m_{i}},\ldots,\beta_{i,-2}$ are uniquely
determined by the irregular curve with residues $X$ (and the holomorphic
coordinate $z_{i}$);

  2. (2)

We have $\beta_{j,-2}=0$.

  3. (3)

$\beta$ is determined by the irregular curve with residues $X$ up to adding a
section of $(\Omega^{1}_{C})^{\otimes 2}(D)$.

###### Proof.

The coefficients $\beta_{i,-2m_{i}},\ldots,\beta_{i,-2}$ all admit homogeneous
quadratic expressions in terms of the eigenvalues of
$\boldsymbol{\theta}_{i},\boldsymbol{\theta}_{\mathrm{res}}$, therefore they
are determined by them. Conversely, the coefficients
$\beta_{i,-2m_{i}},\ldots,\beta_{i,-2}$ determine the polar part of the
eigenvalues. It is classical that for an apparent singularity of $\nabla_{0}$,
one of the two eigenvalues of the residue must vanish. This implies that for
every $q\in B$ the product of the eigenvalues of
$\operatorname{res}_{q}(\nabla_{0})$ vanishes. As this latter product gives
the leading (second) order term $\beta_{j,-2}$, we get the second assertion.
The last part follows from the first two as in Lemma 5. ∎

As a consequence of the lemma and by
$\operatorname{dim}_{\mathbb{C}}H^{0}(C,(\Omega^{1}_{C})^{\otimes
2}(D))=3g-3+n$, the possible values for $\beta$ represent $3g-3+n$ free
parameters for a connection $\nabla$ on $X$ having apparent singularities at a
fixed reduced divisor $B$ of length $N$.

From now on, we set $\beta_{j,-1}=\zeta_{j}$, so that we have the expansion

(2.11) $\beta=\zeta_{j}\frac{(\operatorname{d}\\!z_{j})^{\otimes
2}}{z_{j}}+\beta^{(j)}$

for some local holomorphic quadratic differential $\beta^{(j)}$. Notice that
$\zeta_{j}$ depends on the coordinate $z_{j}$, however the element
$\zeta_{j}\operatorname{d}\\!z_{j}\in\Omega_{C}^{1}|_{q_{j}}$ of the fiber of
the holomorphic cotangent (or canonical) bundle over $q_{j}$ does not depend
on it. As a matter of fact, since $\beta$ belongs to an affine space modelled
over $H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(D))$ (and in order to be consistent
with the decomposition (2.3)), it is even more rigourous to consider
$\zeta_{j}\operatorname{d}\\!z_{j}$ as elements of the fiber
$\Omega^{1}_{C}(D)|_{q_{j}}$, using the inclusion
$\Omega^{1}_{C}\subset\Omega^{1}_{C}(D)$. In the sequel we will consider them
to be such elements. It will turn out that these quantities
$\zeta_{j}\operatorname{d}\\!z_{j}$ are closely related to accessory
parameters.

###  2.6. Determination of $\beta$ and $\delta$ in terms of $\zeta$

Fix a reduced divisor $B$ of length $N$ on $C$ with support disjoint from $D$.
In Subsections 2.4, 2.5 we have found that (normal forms of) meromorphic
connections with residue on $X$ that have apparent singularities at $B$ can be
described by an affine space of complex dimension $g+3g-3+n=N$ ($g$ coming
from the choice of $\delta$ and $3g-3+n$ from the choice of $\beta$). In this
section, we provide a description of such connections in terms of analogs of
separated variables. Namely, it will turn out that generically the data of
$\delta,\beta$ is equivalent to the $N$-tuple
$(\zeta_{1}\operatorname{d}\\!z_{1},\ldots,\zeta_{N}\operatorname{d}\\!z_{N})$.

The fact that singular points are apparent over $B$ imposes further
constraints on $\beta$ and $\delta$. This constraint gives $1$ linear
condition for each point $q_{j}$ and we can expect that these constraints fix
$\beta$ and $\delta$ uniquely in terms of the data
$(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})_{j=1}^{N}$. In fact, this is true
for the genus $g=0$ case (see [9]) and we will show in Lemma 7 that this is
also true for _generic_ choices of
$(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})_{j=1}^{N}$ if $g>0$.

In fact, the data of $\zeta_{j}\operatorname{d}\\!z_{j}$ can be interpreted as
a certain quasi-parabolic structure over $B$. Indeed, at a point $q_{j}$ and
with respect to the decomposition (2.3), the residue of $\nabla_{0}$ reads as

$\mathrm{res}_{q_{j}}\nabla_{0}=\begin{pmatrix}0&\zeta_{j}\operatorname{d}\\!z_{j}\\\
0&1\end{pmatrix}.$

So, the vector $\begin{pmatrix}\zeta_{j}\operatorname{d}\\!z_{j}\\\
1\end{pmatrix}$ is an eigenvector with respect to eigenvalue $1$ and the map
$\phi_{\nabla}$ (see (2.4)) is just the positive elementary transformation
with respect to these parabolic directions at all points $q_{j}$. In summary,
the data of all values $\zeta_{j}\operatorname{d}\\!z_{j}$ is equivalent to
the data of a quasi-parabolic structure of $E_{0}$ over $B$ (i.e., a line in
the fiber of $E_{0}$ over each $q_{j}$) distinct from the destabilizing
subbundle $\mathcal{O}_{C}\subset E_{0}$ for every $j$.

Let us denote by $\boldsymbol{\Omega}(D)$ the total space of the line bundle
$\Omega^{1}_{C}(D)$.

######  Lemma 7.

For generic data
$(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})_{j}\in\mathrm{Sym}^{4g-3+n}(\boldsymbol{\Omega}(D))$
there exist unique $\beta$ and $\delta$ as above such that the corresponding
$\nabla_{0}$ has apparent singular points at all the points $q_{j}$ ($1\leq
j\leq N:=4g-3+n$), and such that the Laurent expansion (2.11) is fulfilled.

###### Proof.

Let us consider $(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})_{j}$ such that
$q_{j}$’s are pair-wise distinct, and do not intersect the support of $D$.
Given one point $(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})$, we can
diagonalize the residue $\mathrm{res}_{q_{i}}\nabla_{0}$ by conjugating by a
triangular matrix

(2.12) $\begin{pmatrix}1&\zeta_{j}\operatorname{d}\\!z_{j}\\\
0&1\end{pmatrix}^{-1}\begin{pmatrix}0&\beta\\\
1&\delta\end{pmatrix}\begin{pmatrix}1&\zeta_{j}\operatorname{d}\\!z_{j}\\\
0&1\end{pmatrix}+\begin{pmatrix}1&\zeta_{j}\operatorname{d}\\!z_{j}\\\
0&1\end{pmatrix}^{-1}d\begin{pmatrix}1&\zeta_{j}\operatorname{d}\\!z_{j}\\\
0&1\end{pmatrix}$
$=\begin{pmatrix}-\zeta_{j}\operatorname{d}\\!z_{j}&\beta-\zeta_{j}\delta\otimes\operatorname{d}\\!z_{j}-\zeta_{j}^{2}\operatorname{d}\\!z_{j}^{\otimes
2}\\\
1&\delta+\zeta_{j}\operatorname{d}\\!z_{j}\end{pmatrix}=\begin{pmatrix}0&0\\\
0&\frac{dz_{j}}{z_{j}}\end{pmatrix}+\text{holomorphic}$

where $z_{j}$ stands for a local coordinate at $q_{j}$. Then the elementary
transformation $\phi_{\nabla}$ is locally equivalent to the conjugacy by
$\begin{pmatrix}1&0\\\ 0&z_{j}^{-1}\end{pmatrix}$ yielding

(2.13)
$\begin{pmatrix}-\zeta_{j}\operatorname{d}\\!z_{j}&\frac{\beta-\zeta_{j}\delta\otimes\operatorname{d}\\!z_{j}-\zeta_{j}^{2}\operatorname{d}\\!z_{j}^{\otimes
2}}{z_{j}}\\\
z_{j}&\delta+\zeta_{j}\operatorname{d}\\!z_{j}-\frac{dz_{j}}{z_{j}}\end{pmatrix}.$

The apparent point condition is therefore equivalent to saying that
$\beta-\zeta_{j}\delta\otimes\operatorname{d}\\!z_{j}-\zeta_{j}^{2}\operatorname{d}\\!z_{j}^{\otimes
2}$ is (holomorphic and) vanishing at $q_{j}$. This condition is linear on
$\beta$ and $\delta$ and rewrites

(2.14)
$\underbrace{\beta-\zeta_{j}\delta\otimes\operatorname{d}\\!z_{j}}_{\text{holomorphic}}|_{q_{j}}=\zeta_{j}^{2}\operatorname{d}\\!z_{j}^{\otimes
2}|_{q_{j}},$

where the right hand side does not involve $\beta$ and $\delta$. If we assume
that $(q_{1},\ldots,q_{N})$ lies in the image of the map $\operatorname{App}$
(see (1.2)), then the normal form of any $(E,\nabla)$ in the preimage produces
a solution $(\delta_{0},\beta_{0})$. Fixing such solutions, by Lemmas 5, 6 we
may rewrite

$\left\\{\begin{matrix}\beta&=&\beta_{0}+b_{1}\nu_{1}+\cdots+b_{N-g}\nu_{N-g}\hfill\\\
\delta&=&\delta_{0}+d_{1}\omega_{1}+\cdots+d_{g}\omega_{g}\end{matrix}\right.$

where $(\omega_{l})_{l=1}^{g}$, $(\nu_{k})_{k=1}^{N-g}$ are respective bases
of $H^{0}(C,\Omega^{1}_{C})$ and $H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(D))$.
Using these expressions, the constraint that $q_{j}$ is an apparent
singularity can be rewritten as a linear system consisting of $N$ equations in
the $N$ variables $b_{k}$, $d_{l}$. The condition to uniquely determine
$\beta$ and $\delta$ in terms of the data
$(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})$ is that the following determinant
does not vanish

(2.15)
$\det\begin{pmatrix}\nu_{1}(q_{1})&\cdots&\nu_{N-g}(q_{1})&\zeta_{1}\operatorname{d}\\!z_{1}\omega_{1}(q_{1})&\cdots&\zeta_{1}\operatorname{d}\\!z_{1}\omega_{g}(q_{1})\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
\nu_{1}(q_{N})&\cdots&\nu_{N-g}(q_{N})&\zeta_{N}\operatorname{d}\\!z_{N}\omega_{1}(q_{N})&\cdots&\zeta_{N}\operatorname{d}\\!z_{N}\omega_{g}(q_{N})\end{pmatrix}$

Of course, it is sufficient for our purpose to check that we can find some
$(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})$’s such that this determinant does
not vanish, so that it will be generically non vanishing. If we set
$\zeta_{1}=\cdots=\zeta_{N-g}=0$, then the matrix has a zero block of
dimension $(N-g)\times g$ in the top right corner, and the determinant factors
as

$\zeta_{N-g+1}\operatorname{d}\\!z_{N-g+1}\cdots\zeta_{N}\operatorname{d}\\!z_{N}\cdot\det\begin{pmatrix}\nu_{1}(q_{1})&\cdots&\nu_{N-g}(q_{1})\\\
\vdots&\ddots&\vdots\\\
\nu_{1}(q_{N-g})&\cdots&\nu_{N-g}(q_{N-g})\end{pmatrix}\cdot\det\begin{pmatrix}\omega_{1}(\tilde{q}_{1})&\cdots&\omega_{g}(\tilde{q}_{1})\\\
\vdots&\ddots&\vdots\\\
\omega_{1}(\tilde{q}_{g})&\cdots&\omega_{g}(\tilde{q}_{g})\end{pmatrix}$

where $\tilde{q}_{j}=q_{j+N-g}$. After setting
$\zeta_{N-g+1}=\cdots=\zeta_{N}=1$, it is enough to find $q_{j}$’s such that
the two smaller determinants are non zero. To conclude the proof, let us
denote by $L$ any of the two lines bundles $\Omega^{1}_{C}$ or
$(\Omega^{1}_{C})^{\otimes 2}(D)$, and by $\mu_{1},\ldots,\mu_{N^{\prime}}$ a
corresponding basis of $H^{0}(C,L)$. Then we want to prove that the image of
the curve by the evaluation map

$C\stackrel{{\scriptstyle\text{ev}}}{{\longrightarrow}}\mathbb{P}^{N^{\prime}-1}\
;\ q\mapsto(\mu_{1}(q):\ldots:\mu_{N^{\prime}}(q))$

is not contained in some hyperplane, i.e. that we can find
$q_{1},\ldots,q_{N^{\prime}}\in C$ such that the image is not contained in
some hyperplane. But this is true, otherwise, we would have a linear relation
between $\mu_{1},\ldots,\mu_{N^{\prime}}$ contradicting that they form a
basis. ∎

######  Remark 8.

In the previous proof, the locus of $q_{j}$’s for which
$\det(\omega_{i}(\tilde{q}_{j}))_{i,j}$ vanishes correspond to the Brill-
Noether locus for divisor $\tilde{q}_{1}+\cdots+\tilde{q}_{g}$.

######  Lemma 9.

When $g=0$, any data
$(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})_{j}\in\mathrm{Sym}^{n-3}(\boldsymbol{\Omega}(D))$
gives rise to unique $\beta$ and $\delta$ such that the corresponding
$\nabla_{0}$ has apparent singular points at all $q_{j}$’s. However, for
$g>0$, there always exist data $(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})_{j}$
such that the determinant (2.15) vanishes.

###### Proof.

When $g=0$, this directly follows from [9] (a consequence of Lagrange
interpolation). When $g>0$, fix generic $q_{j}$’s and let $\omega\in
H^{0}(C,\Omega^{1}_{C}(D))$. If we set $\zeta_{j}:=\omega(q_{j})$, then the
last colum of (2.15) is just the evaluation of the section
$\omega\otimes\omega_{g}\subset H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(D))$ at
$q_{1},\cdots,q_{4g-3+n}$ and is therefore a linear combination of the
$3g-3+n$ first colums. ∎

##  3\. Symplectic structure and canonical coordinates

We fix an irregular curve with residues $X=(C,D,\\{z_{i}\\}_{i\in
I},\\{\boldsymbol{\theta}_{i}\\}_{i\in
I},\boldsymbol{\theta}_{\mathrm{res}})$. As usual, we use the notation
$N:=4g+n-3$, where $g$ is the genus of $C$ and $n=\deg(D)$. We will consider
the moduli space $M_{X}$ of rank 2 meromorphic connections over $X$. This
moduli space is constructed in [24, Theorem 2.1] and carries a natural
symplectic structure described in [24, Proposition 4.1]. The purpose of this
section is to give canonical coordinates on an open subset of $M_{X}$ with
respect to this symplectic structure. First we describe the moduli space
$M_{X}$.

###  3.1. Moduli spaces

Let $(E,\nabla)$ be a rank 2 meromorphic connection over $X$. Then, the
subsheaf

$l^{(i)}:=\varphi_{m_{i}[t_{i}]}^{-1}(\mathcal{O}_{m_{i}[t_{i}]}\oplus
0)\subset E_{m_{i}[t_{i}]}.$

equips $(E,\nabla)$ with a canonical quasi-parabolic structure at each
$t_{i}$. So we may consider $(E,\nabla)$ as a quasi-parabolic connection
$(E,\nabla,\\{l^{(i)}\\})$ defined in [21, Definition 1.1] and [24, Definition
2.1]. A stability condition for quasi-parabolic connections is introduced in
[21, Definition 2.1] and [24, Definition 2.2]. The moduli space of stable
quasi-parabolic connections is constructed in [21, Theorem 2.1] and [24,
Theorem 2.1]. In our situation, any rank $2$ meromorphic connections over $X$
are irreducible (see Remark 4). So our objects are automatically stable
objects. We omit the stability condition of the quasi-parabolic connections.

Let $M_{X}$ be the moduli space of rank 2 meromorphic connections over the
irregular curve with residues $X$. If $(E,\nabla)\in M_{X}$ satisfies
$\dim_{\mathbb{C}}H^{0}(C,E)=1$, then we have a unique
$\mathcal{O}_{C}$-morphism $\varphi_{\nabla}$ in (2.2). The
$\mathcal{O}_{C}$-morphism $\varphi_{\nabla}$ is nonzero, since $(E,\nabla)$
is irreducible. So we may define the divisor $\mathrm{div}(\varphi_{\nabla})$
in (2.6) for $(E,\nabla)$. We set

$M_{X}^{0}:=\left\\{(E,\nabla)\in M_{X}\ \middle|\
\begin{array}[]{l}\text{$\dim_{\mathbb{C}}H^{0}(C,E)=1$,}\\\
\text{$\mathrm{div}(\varphi_{\nabla})$ is reduced, and}\\\
\text{$\mathrm{div}(\varphi_{\nabla})$ has disjoint support with
$D$}\end{array}\right\\}.$

Next we recall the natural symplectic structure on $M_{X}$.

###  3.2. Symplectic structure

We will describe the natural symplectic structure on $M_{X}$ via Čech
cohomology. This is defined in [21, Proposition 7.2] and [24, Proposition
4.1]. This is analog of the symplectic form on the moduli space of stable
Higgs bundles in [8]. This description of the symplectic structure is useful
to comparing this symplectic structure with the Goldman symplectic structure
on the character variety via the Riemann–Hilbert map (for example, see [21,
the proof of Proposition 7.3] and [4, Theorem 3.2]). Moreover, this
description of the symplectic structure is useful to describe the
isomonodromic deformations (for example, see [5, Proposition 4.3], [6,
Proposition 4.4], and [33, Proposition 3.8]).

First we recall the description of the tangent space of $M_{X}$ at
$(E,\nabla)\in M_{X}$ in terms of the hypercohomology of a certain complex
([21, the proof of Theorem 2.1] and [24, the proof of Proposition 4.1]). We
consider $(E,\nabla)$ as a quasi-parabolic connection
$(E,\nabla,\\{l^{(i)}\\})$. We define a complex ${\mathcal{F}}^{\bullet}$ for
$(E,\nabla,\\{l^{(i)}\\})$ by

(3.1) $\displaystyle{\mathcal{F}}^{0}:=\left\\{s\in{\mathcal{E}}nd(E)\
\middle|\ s|_{m_{i}t_{i}}(l^{(i)})\subset l^{(i)}\text{ for any $i$}\right\\}$
$\displaystyle{\mathcal{F}}^{1}:=\left\\{s\in{\mathcal{E}}nd(E)\otimes\Omega^{1}_{C}(D)\
\middle|\ s|_{m_{i}t_{i}}(l^{(i)})\subset l^{(i)}\otimes\Omega^{1}_{C}\text{
for any $i$}\right\\}$
$\displaystyle\nabla_{{\mathcal{F}}^{\bullet}}\colon{\mathcal{F}}^{0}\longrightarrow{\mathcal{F}}^{1};\quad\nabla_{{\mathcal{F}}^{\bullet}}(s)=\nabla\circ
s-s\circ\nabla.$

Then we have an isomorphism between the tangent space
$T_{(E,\nabla,\\{l^{(i)}\\})}M_{X}$ and
${\mathbf{H}}^{1}(\mathcal{F}^{\bullet})$.

Now we recall this isomorphism. We take an analytic (or affine) open covering
$C=\bigcup_{\alpha}U_{\alpha}$ such that
$E|_{U_{\alpha}}\cong\mathcal{O}^{\oplus 2}_{U_{\alpha}}$ for any $\alpha$,
$\sharp\\{i\mid t_{i}\cap U_{\alpha}\neq\emptyset\\}\leq 1$ for any $\alpha$
and $\sharp\\{\alpha\mid t_{i}\cap U_{\alpha}\neq\emptyset\\}\leq 1$ for any
$i$. Take a tangent vector $v\in T_{(E,\nabla,\\{l^{(i)}\\})}M_{X}$. The field
$v$ corresponds to an infinitesimal deformation
$(E_{\epsilon},\nabla_{\epsilon},\\{l_{\epsilon}^{(i)}\\})$ of
$(E,\nabla,\\{l^{(i)}\\})$ over $C\times\mathrm{Spec}\,\mathbb{C}[\epsilon]$
such that
$(E_{\epsilon},\nabla_{\epsilon},\\{l_{\epsilon}^{(i)}\\})\otimes\mathbb{C}[\epsilon]/(\epsilon)\cong(E,\nabla,\\{l^{(i)}\\})$,
where $\mathbb{C}[\epsilon]=\mathbb{C}[t]/(t^{2})$. There is an isomorphism

$\varphi_{\alpha}\colon
E_{\epsilon}|_{U_{\alpha}\times\mathrm{Spec}\,\mathbb{C}[\epsilon]}\xrightarrow{\sim}\mathcal{O}^{\oplus
2}_{U_{\alpha}\times\mathrm{Spec}\,\mathbb{C}[\epsilon]}\xrightarrow{\sim}E|_{U_{\alpha}}\otimes\mathbb{C}[\epsilon]$

such that $\varphi_{\alpha}\otimes\mathbb{C}[\epsilon]/(\epsilon)\colon
E_{\epsilon}\otimes\mathbb{C}[\epsilon]/(\epsilon)|_{U_{\alpha}}\xrightarrow{\sim}E|_{U_{\alpha}}\otimes\mathbb{C}[\epsilon]/(\epsilon)=E|_{U_{\alpha}}$
is the given isomorphism and that $\varphi_{\alpha}|_{t_{i}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]}(l_{\epsilon}^{(i)})=l^{(i)}|_{U_{\alpha}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]}$ if $t_{i}\cap U_{\alpha}\neq\emptyset$.
We put

$\displaystyle u_{\alpha\beta}:=$ $\displaystyle\
\varphi_{\alpha}\circ\varphi_{\beta}^{-1}-\mathrm{id}_{E|_{U_{\alpha\beta}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]}},$ $\displaystyle v_{\alpha}:=$
$\displaystyle\
(\varphi_{\alpha}\otimes\mathrm{id})\circ\nabla_{\epsilon}|_{U_{\alpha}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]}\circ\varphi^{-1}_{\alpha}-\nabla|_{U_{\alpha}\times\mathop{\rm
Spec}\nolimits\mathbb{C}[\epsilon]}.$

Then $\\{u_{\alpha\beta}\\}\in C^{1}((\epsilon)\otimes{\mathcal{F}}^{0})$,
$\\{v_{\alpha}\\}\in C^{0}((\epsilon)\otimes{\mathcal{F}}^{1})$ and we have
the cocycle conditions

$u_{\beta\gamma}-u_{\alpha\gamma}+u_{\alpha\beta}=0\quad\text{and}\quad\nabla\circ
u_{\alpha\beta}-u_{\alpha\beta}\circ\nabla=v_{\beta}-v_{\alpha}.$

So $[(\\{u_{\alpha\beta}\\},\\{v_{\alpha}\\})]$ determines an element of
${\mathbf{H}}^{1}(\mathcal{F}^{\bullet})$. This correspondence gives an
isomorphism between the tangent space $T_{(E,\nabla,\\{l^{(i)}\\})}M_{X}$ and
${\mathbf{H}}^{1}(\mathcal{F}^{\bullet})$.

We define a pairing

(3.2)
$\displaystyle{\mathbf{H}}^{1}({\mathcal{F}}^{\bullet})\otimes{\mathbf{H}}^{1}({\mathcal{F}}^{\bullet})$
$\displaystyle\longrightarrow{\mathbf{H}}^{2}(\mathcal{O}_{C}\xrightarrow{d}\Omega_{C}^{1})\cong\mathbb{C}$
$\displaystyle[(\\{u_{\alpha\beta}\\},\\{v_{\alpha}\\})]\otimes[(\\{u_{\alpha\beta}^{\prime}\\},\\{v_{\alpha}^{\prime}\\})]$
$\displaystyle\longmapsto[(\\{\mathrm{tr}(u_{\alpha\beta}\circ
u_{\beta\gamma}^{\prime})\\},-\\{\mathrm{tr}(u_{\alpha\beta}\circ
v_{\beta}^{\prime})-\mathrm{tr}(v_{\alpha}\circ
u^{\prime}_{\alpha\beta})\\})],$

considered in Čech cohomology with respect to an open covering
$\\{U_{\alpha}\\}$ of $C$, $\\{u_{\alpha\beta}\\}\in
C^{1}({\mathcal{F}}^{0})$, $\\{v_{\alpha}\\}\in C^{0}({\mathcal{F}}^{1})$ and
so on. This pairing gives a nondegenerate 2-form on the moduli space $M_{X}$.
This fact follows from the Serre duality and the five lemma:

(3.3)
$\textstyle{H^{0}(\mathcal{F}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{0}(\mathcal{F}^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{\mathbf{H}}^{1}(\mathcal{F}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}(\mathcal{F}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}(\mathcal{F}^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}(\mathcal{F}^{1})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathcal{F}^{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{H}}^{1}(\mathcal{F}^{\bullet})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathcal{F}^{1})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathcal{F}^{0})^{\vee}\hbox
to0.0pt{.\hss}}$

We denote by $\omega$ the nondegenerate 2-form on $M_{X}$. This 2-form
$\omega$ is a symplectic structure. That is, we have $d\omega=0$ (see [21,
Proposition 7.3] and [24, Proposition 4.2]).

We get as a consequence:

######  Proposition 10.

The dimension of $M_{X}^{0}$ is equal to $2N$, where $N=4g-3+n$.

###### Proof.

By irreducibility of $(E,\nabla)$ and Schur’s lemma we have

${\mathbf{H}}^{0}(\mathcal{F}^{\bullet})\cong\mathbb{C}.$

On a Zariski open subset of $M_{X}$, the underlying quasi-parabolic vector
bundle $(E,\\{l^{(i)}\\}_{i})$ is irreducible, so we also have

$H^{0}(\mathcal{F}^{0})\cong\mathbb{C}.$

Clearly, we have
$\operatorname{deg}(\mathcal{F}^{0})=-\operatorname{length}(D)$. From
Riemann–Roch we find

$\displaystyle\operatorname{dim}_{\mathbb{C}}H^{1}(\mathcal{F}^{0})$
$\displaystyle=\operatorname{dim}_{\mathbb{C}}H^{0}(\mathcal{F}^{0})+4(g-1)-\operatorname{deg}(\mathcal{F}^{0})$
$\displaystyle=4g-3+n=N.$

By Serre duality and Euler characteristic count applied to the hypercohomology
long exact sequence (3.3), we get the statement. ∎

###  3.3. Trivializations of $E$

Our purpose is to give canonical coordinates of $M_{X}^{0}$ with respect to
the symplectic form (3.2). To do it, we will calculate the Čech cohomology by
taking trivializations of $E$. To simplify the calculation, we take
trivializations of $E$ by using

$\phi_{\nabla}\colon E_{0}\xrightarrow{\ \subset\ }E,$

whose cokernel defines the apparent singularities. In this section, we will
discuss construction of the trivializations of $E$ by using $\phi_{\nabla}$.

We take $(E,\nabla,\\{l^{(i)}\\})\in M_{X}^{0}$. Let
$\\{(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})\\}_{j=1,2,\ldots,N}$ be the
point on $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))$ corresponding to
$(E,\nabla,\\{l^{(i)}\\})$. We assume that the point
$\\{(q_{j},\zeta_{j}\operatorname{d}\\!z_{j})\\}_{j=1,2,\ldots,N}$ is generic
in the sense of Lemma 7. Let $U^{\mathrm{an}}_{q_{j}}$ be an analytic open
subset of $C$ such that $q_{j}\in U^{\mathrm{an}}_{q_{j}}$ and
$U^{\mathrm{an}}_{t_{i}}$ be an analytic open subset of $C$ such that
$t_{i}\in U^{\mathrm{an}}_{t_{i}}$. We assume that $U^{\mathrm{an}}_{q_{j}}$
and $U^{\mathrm{an}}_{t_{i}}$ are small enough. We take an analytic coordinate
$z_{j}$ on $U^{\mathrm{an}}_{q_{j}}$ such that it is independent of the moduli
space $M_{X}^{0}$. We denote also by $q_{j}$ the complex number so that the
point $q_{j}$ on $C$ is defined by $z_{j}-q_{j}=0$.

######  Definition 11.

Let $\\{U_{\alpha}\\}_{\alpha}$ be an analytic open covering of $C$:
$C=\bigcup_{\alpha}U_{\alpha}$ such that

  * (i)

$\sharp\\{i\mid t_{i}\cap U_{\alpha}\neq\emptyset\\}\leq 1$ for any $\alpha$,
and $\sharp\\{\alpha\mid t_{i}\cap U_{\alpha}\neq\emptyset\\}\leq 1$ for any
$i$,

  * (ii)

$\sharp\\{j\mid q_{j}\cap U_{\alpha}\neq\emptyset\\}\leq 1$ for any $\alpha$,
and $\sharp\\{\alpha\mid q_{j}\cap U_{\alpha}\neq\emptyset\\}\leq 1$ for any
$j$,

  * (iii)

$\Omega^{1}_{C}(D)$ is free on $U_{\alpha}$ for any $\alpha$, that is,
$\Omega^{1}_{C}(D)|_{U_{\alpha}}\cong\mathcal{O}_{U_{\alpha}}$,

  * (iv)

$U_{\alpha_{t_{i}}}=U^{\mathrm{an}}_{t_{i}}$ and
$U_{\alpha_{q_{j}}}=U^{\mathrm{an}}_{q_{j}}$.

Here we denote by $\alpha_{t_{i}}$ the index $\alpha$ such that $t_{i}\in
U_{\alpha}$, and by $\alpha_{q_{j}}$ the index $\alpha$ such that $q_{j}\in
U_{\alpha}$.

We fix trivializations
$\omega_{\alpha}\colon\mathcal{O}_{U_{\alpha}}\xrightarrow{\ \sim\
}\Omega^{1}_{C}(D)|_{U_{\alpha}}$ of $\Omega^{1}_{C}(D)$. We assume that
$\omega_{\alpha}$ is independent of the moduli space $M_{X}^{0}$. By using
$\omega_{\alpha}$, we have
$\omega^{-1}_{\alpha}\colon\mathcal{O}_{U_{\alpha}}\xrightarrow{\ \sim\
}(\Omega^{1}_{C}(D))^{-1}|_{U_{\alpha}}$. By the trivializations, we have
trivializations $\varphi_{\alpha}^{\mathrm{norm}}\colon\mathcal{O}^{\oplus
2}_{U_{\alpha}}\xrightarrow{\ \sim\ }E_{0}|_{U_{\alpha}}$ of $E_{0}$. Assume
that the connection matrices $A_{\alpha}^{\mathrm{norm}}$ of $\nabla_{0}$
associated to $\varphi_{\alpha}^{\mathrm{norm}}$ are

(3.4) $A_{\alpha}^{\mathrm{norm}}=\begin{pmatrix}0&\beta_{\alpha}\\\
\gamma_{\alpha}&\delta_{\alpha}\end{pmatrix},$

where $\beta_{\alpha},\delta_{\alpha}\in\Omega_{C}^{1}(D+B)|_{U_{\alpha}}$ are
determined by $\\{(q_{j},\mathrm{res}_{q_{j}}(\beta))\\}_{j=1,2,\ldots,N}$
(see Lemma 7). The 1-form $\gamma_{\alpha}\in\Omega_{C}^{1}(D)|_{U_{\alpha}}$
is the image of $1$ under the composition

$\mathcal{O}_{U_{\alpha}}\xrightarrow{\ \sim\
}(\Omega^{1}_{C}(D))^{-1}\otimes\Omega^{1}_{C}(D)|_{U_{\alpha}}\xrightarrow{\omega_{\alpha}\otimes
1}\mathcal{O}_{U_{\alpha}}\otimes\Omega^{1}_{C}(D).$

In particular, $\gamma_{\alpha}$ is independent of the moduli space
$M_{X}^{0}$ for any $\alpha$. The polar part of
$A_{\alpha_{t_{i}}}^{\mathrm{norm}}$ at $t_{i}$ is independent of the moduli
space $M_{X}^{0}$ for any $i$. We set

(3.5)
$\zeta_{j}:=\frac{\mathrm{res}_{q_{j}}(\beta)}{\gamma_{\alpha_{q_{j}}}|_{q_{j}}}\in\mathbb{C}\quad\text{
for $j=1,2,\ldots,N$.}$

Here $\beta\in H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(2D+B))$ is the
$(1,2)$-entry of (2.9). Notice that
$\beta|_{U_{\alpha}}=\beta_{\alpha}\gamma_{\alpha}$, where $\beta_{\alpha}$
and $\gamma_{\alpha}$ are in (3.4). So we have

$\mathrm{res}_{q_{j}}(A_{\alpha_{q_{j}}}^{\mathrm{norm}})=\begin{pmatrix}0&\zeta_{j}\\\
0&1\end{pmatrix}\quad\text{ for $j=1,2,\ldots,N$.}$

######  Definition 12.

We define other trivializations
$\varphi_{\alpha}^{\mathrm{App},0}\colon\mathcal{O}^{\oplus
2}_{U_{\alpha}}\xrightarrow{\ \sim\ }E_{0}|_{U_{\alpha}}$ of $E_{0}$ for each
$\alpha$ as follows:

  * (i)

When $\alpha=\alpha_{q_{j}}$, we take a trivialization
$\varphi_{\alpha}^{\mathrm{App},0}$ as

$\varphi_{\alpha}^{\mathrm{App},0}=\varphi_{\alpha}^{\mathrm{norm}}\circ\begin{pmatrix}1&\zeta_{j}\\\
0&1\end{pmatrix}.$

Note that this triangular matrix appeared in (2.12).

  * (ii)

Otherwise, we take a trivialization $\varphi_{\alpha}^{\mathrm{App},0}$ as
$\varphi_{\alpha}^{\mathrm{App},0}=\varphi_{\alpha}^{\mathrm{norm}}$.

Let $A_{\alpha}^{\mathrm{App},0}$ be the connection matrix of $\nabla_{0}$
associated to $\varphi_{\alpha}^{\mathrm{App},0}$, that is,

$(\varphi_{\alpha}^{\mathrm{App},0})^{-1}\circ(\phi_{\nabla}^{*}\nabla)\circ\varphi_{\alpha}^{\mathrm{App},0}=\operatorname{d}+A_{\alpha}^{\mathrm{App},0}.$

We have that

$A_{\alpha}^{\mathrm{App},0}=\begin{cases}\begin{pmatrix}-\zeta_{j}\gamma_{\alpha}&\beta_{\alpha}-\zeta_{j}\delta_{\alpha}-\zeta_{j}^{2}\gamma_{\alpha}\\\
\gamma_{\alpha}&\delta_{\alpha}+\zeta_{j}\gamma_{\alpha}\end{pmatrix}&\text{when
$\alpha=\alpha_{q_{j}}$}\\\ \begin{pmatrix}0&\beta_{\alpha}\\\
\gamma_{\alpha}&\delta_{\alpha}\end{pmatrix}&\text{otherwise}.\end{cases}$

We have

$\mathrm{res}_{q_{j}}(A_{\alpha_{q_{j}}}^{\mathrm{App},0})=\begin{pmatrix}0&0\\\
0&1\end{pmatrix}\quad\text{ for $j=1,2,\ldots,N$.}$

Now we define trivializations of $E$ by using $\phi_{\nabla}\colon
E_{0}\rightarrow E$ in (2.4) and the trivialization of $E_{0}$ in Definition
12.

######  Definition 13.

Now we define trivialization
$\varphi^{\mathrm{App}}_{\alpha}\colon\mathcal{O}^{\oplus
2}_{U_{\alpha}}\xrightarrow{\ \sim\ }E|_{U_{\alpha}}$ of $E$ for the open
covering $\\{U_{\alpha}\\}_{\alpha}$ in Definition 11 as follows.

  * (i)

When $\alpha=\alpha_{q_{j}}$, we take a trivialization
$\varphi^{\mathrm{App}}_{\alpha}$ so that

$(\varphi^{\mathrm{App}}_{\alpha})^{-1}\circ\phi_{\nabla}|_{U_{\alpha}}\circ\varphi^{\mathrm{App},0}_{\alpha}=\begin{pmatrix}1&0\\\
0&z_{j}-q_{j}\end{pmatrix}.$

  * (ii)

When $\alpha=\alpha_{t_{i}}$, we take
$g^{t_{i}}_{\alpha}\in\mathrm{Aut}(\mathcal{O}_{U_{\alpha}}^{\oplus 2})$ so
that the polar part of
$(g^{t_{i}}_{\alpha})^{-1}A_{\alpha}^{\mathrm{norm}}g^{t_{i}}_{\alpha}$ is
diagonal at $m_{i}[t_{i}]$. We take a trivialization
$\varphi^{\mathrm{App}}_{\alpha}$ as

$\varphi^{\mathrm{App}}_{\alpha}=\phi_{\nabla}|_{U_{\alpha}}\circ\varphi_{\alpha}^{\mathrm{norm}}\circ
g^{t_{i}}_{\alpha}.$

Here remark that $\phi_{\nabla}|_{U_{\alpha}}$ is invertible. Since the polar
part of $A_{\alpha_{t_{i}}}^{\mathrm{norm}}$ at $t_{i}$ is independent of the
moduli space $M_{X}^{0}$, we may assume that $(g^{t_{i}}_{\alpha})_{<m_{i}}$
is independent of the moduli space $M_{X}^{0}$. Here we define
$(g^{t_{i}}_{\alpha})_{<m_{i}}$ so that
$g^{t_{i}}_{\alpha}=(g^{t_{i}}_{\alpha})_{<m_{i}}+O(z_{i}^{m_{i}})$.

  * (iii)

Otherwise, we take a trivialization $\varphi^{\mathrm{App}}_{\alpha}$ so that

$(\varphi^{\mathrm{App}}_{\alpha})^{-1}\circ\phi_{\nabla}|_{U_{\alpha}}\circ\varphi^{\mathrm{norm}}_{\alpha}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}.$

Since $\phi_{\nabla}|_{U_{\alpha}}$ is invertible in this case,
$\varphi^{\mathrm{App}}_{\alpha}=\phi_{\nabla}|_{U_{\alpha}}\circ\varphi^{\mathrm{norm}}_{\alpha}$.

Let $A_{\alpha}$ be the connection matrix of $\nabla$ associated to
$\varphi^{\text{App}}_{\alpha}$, that is

$(\varphi^{\text{App}}_{\alpha})^{-1}\circ\nabla\circ\varphi^{\text{App}}_{\alpha}=\operatorname{d}+A_{\alpha}.$

We have that

(3.6)
$A_{\alpha}=\begin{cases}\begin{pmatrix}-\zeta_{j}\gamma_{\alpha}&\frac{\beta_{\alpha}-\zeta_{j}\delta_{\alpha}-\zeta_{j}^{2}\gamma_{\alpha}}{z_{j}-q_{j}}\\\
(z_{j}-q_{j})\gamma_{\alpha}&\delta_{\alpha}+\zeta_{j}\gamma_{\alpha}-\frac{dz_{j}}{z_{j}-q_{j}}\end{pmatrix}&\text{when
$\alpha=\alpha_{q_{j}}$}\\\ \omega_{i}(X)+[\text{holo.\ part}]&\text{when
$\alpha=\alpha_{t_{i}}$}\\\ \begin{pmatrix}0&\beta_{\alpha}\\\
\gamma_{\alpha}&\delta_{\alpha}\end{pmatrix}&\text{otherwise}.\end{cases}$

Here $\omega_{i}(X)$ is the 1-form defined in (2.10). The connection matrix
$A_{\alpha_{q_{j}}}$ on $U_{\alpha_{q_{j}}}$ appeared in (2.13). The
connection matrix $A_{\alpha_{q_{j}}}$ has no pole at $q_{j}$ for any
$j=1,2,\ldots,N$, since $\beta_{\alpha},\delta_{\alpha}$ are determined by
Lemma 7. We have considered diagonalization of the polar part of the
connection $(E,\nabla)$ at each $t_{i}$. The reason why we consider
diagonalization of the polar parts is that we use the connection matrix (3.6)
to calculate an infinitesimal deformation of $(E,\nabla)$. So we will
calculate variations of the transition functions with respect to the
trivializations in Definition 13 and variations of the connection matrices
(3.6). These are elements of $\mathcal{F}^{0}$ and $\mathcal{F}^{1}$ of (3.1),
respectively. To be elements of $\mathcal{F}^{0}$ and $\mathcal{F}^{1}$, we
need the compatibility with the quasi-parabolic structure. However, this
compatibility follows directly from diagonalization of the polar parts.

###  3.4. Descriptions of the cocycles of an infinitesimal deformation

Let $\boldsymbol{\Omega}(D)\rightarrow C$ be the total space of
$\Omega^{1}_{C}(D)$. By the argument as in Lemma 6, we may define a map

(3.7) $\displaystyle f_{\mathrm{App},0}\colon M_{X}^{0}$
$\displaystyle\longrightarrow\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))$
$\displaystyle(E,\nabla)$
$\displaystyle\longmapsto\left\\{(q_{j},\mathrm{res}_{q_{j}}(\beta))\right\\}_{j=1,2,\ldots,N}.$

Here $\beta\in H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(2D+B))$ is the
$(1,2)$-entry of (2.9) and
$\mathrm{res}_{q_{j}}(\beta)\in\Omega_{C}^{1}(D)|_{q_{j}}$. We take an
analytic open subset $V$ of $M_{X}^{0}$. For the analytic open subset $V$, we
assume that we may define a composition

$\displaystyle V\longrightarrow f_{\mathrm{App},0}(V)$
$\displaystyle\longrightarrow\mathrm{Sym}^{N}(\mathbb{C}_{(q,\zeta)}^{2})$
$\displaystyle(E,\nabla)\longmapsto\left\\{(q_{j},\mathrm{res}_{q_{j}}(\beta))\right\\}_{j=1,2,\ldots,N}$
$\displaystyle\longmapsto\left\\{(q_{j},\zeta_{j})\right\\}_{j=1,2,\ldots,N},$

where $\zeta_{j}$ is defined in (3.5), and the image of $V$ under the
composition is isomorphic to some analytic open subset of
$\mathbb{C}_{(q,\zeta)}^{2N}$. Let $U_{(q,\zeta)}$ be such an analytic open
subset of $\mathbb{C}_{(q,\zeta)}^{2N}$. So we have a map

(3.8) $\displaystyle M_{X}^{0}\supset V$ $\displaystyle\longrightarrow
U_{(q,\zeta)}\subset\mathbb{C}_{(q,\zeta)}^{2N}$ $\displaystyle(E,\nabla)$
$\displaystyle\longmapsto(q_{1},\ldots,q_{N},\zeta_{1},\ldots,\zeta_{N}),$

which are coordinates that we will use in this subsection. We consider the
family of $(E,\nabla,\\{l^{(i)}\\})$ parametrized by $U_{(q,\zeta)}$ such that
this family induces the inverse map of the map $V\rightarrow U_{(q,\zeta)}$.
Here this family is constructed by Lemma 7. By using the trivializations
$\\{\varphi^{\text{App}}_{\alpha}\\}_{\alpha}$ of $E$ in Definition 13, we
have transition functions and connection matrices of the family of
$(E,\nabla,\\{l^{(i)}\\})$ parametrized by $U_{(q,\zeta)}$. Indeed, the
transition function is

(3.9)
$B_{\alpha\beta}:=(\varphi^{\text{App}}_{\alpha}|_{U_{\alpha\beta}})^{-1}\circ\varphi^{\text{App}}_{\beta}|_{U_{\alpha\beta}}\colon\mathcal{O}^{\oplus
2}_{U_{\alpha\beta}}\longrightarrow\mathcal{O}^{\oplus 2}_{U_{\alpha\beta}},$

and the connection matrix is as in (3.6).

Let $(q_{j},\zeta_{j})_{j}$ be a point on $U_{(q,\zeta)}$. The purpose of this
subsection is to describe the tangent map

(3.10) $\displaystyle T_{(q_{j},\zeta_{j})_{j}}\mathbb{C}_{(q,\zeta)}^{2N}$
$\displaystyle\longrightarrow T_{(E,\nabla,\\{l^{(i)}\\})}M_{X}^{0}\cong{\bf
H}^{1}(\mathcal{F}^{\bullet})$ $\displaystyle v$
$\displaystyle\longmapsto[(\\{u_{\alpha\beta}(v)\\},\\{v_{\alpha}(v)\\})]$

induced by the inverse map of (3.8). For this purpose, we will calculate the
variations of the transition functions and the connection matrices
parametrized by $U_{(q,\zeta)}$ with respect to the tangent vector $v$ in
$U_{(q,\zeta)}\subset\mathbb{C}_{(q,\zeta)}^{2N}$. By using these variations,
we will calculate the cocycles
$(\\{u_{\alpha\beta}(v)\\},\\{v_{\alpha}(v)\\})$ of the infinitesimal
deformation of $(E,\nabla,\\{l^{(i)}\\})$ with respect to $v$.

First, we calculate $u_{\alpha\beta}(v)\in\mathcal{F}^{1}(U_{\alpha\beta})$.
We consider the variation of $B_{\alpha\beta}$ in (3.9) by $v$:

$B_{\alpha\beta}(\mathrm{id}+\epsilon
B_{\alpha\beta}^{-1}v(B_{\alpha\beta}))\colon\mathcal{O}^{\oplus
2}_{U_{\alpha\beta}}\longrightarrow\mathcal{O}^{\oplus
2}_{U_{\alpha\beta}}\otimes\mathbb{C}[\epsilon].$

Then $u_{\alpha\beta}(v)$ has the following description:

(3.11)
$u_{\alpha\beta}(v)=\varphi^{\text{App}}_{\beta}|_{U_{\alpha\beta}}\circ\left(B_{\alpha\beta}^{-1}v(B_{\alpha\beta})\right)\circ(\varphi^{\text{App}}_{\beta}|_{U_{\alpha\beta}})^{-1}.$

######  Lemma 14.

Let $I_{\mathrm{cov}}$ be the set of the indices of the open covering
$\\{U_{\alpha}\\}$ in Definition 11. We set
$I^{t}_{\mathrm{cov}}=\\{\alpha_{t_{1}},\ldots,\alpha_{t_{\nu}}\\}$ and
$I^{q}_{\mathrm{cov}}=\\{\alpha_{q_{1}},\ldots,\alpha_{q_{N}}\\}$, which are
subsets of $I_{\mathrm{cov}}$. For $v\in
T_{(E,\nabla,\\{l^{(i)}\\})}M_{X}^{0}$, we have the equality

(3.12) $u_{\alpha\beta}(v)=\begin{cases}0&\alpha,\beta\in
I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup I^{q}_{\mathrm{cov}})\\\
\varphi^{\mathrm{App}}_{\alpha_{q_{j}}}|_{U_{\alpha\alpha_{q_{j}}}}\circ\begin{pmatrix}0&\frac{v(\zeta_{j})}{z_{j}-q_{j}}\\\
0&\frac{v(q_{j})}{z_{j}-q_{j}}\end{pmatrix}\circ(\varphi^{\mathrm{App}}_{\alpha_{q_{j}}}|_{U_{\alpha\alpha_{q_{j}}}})^{-1}&\alpha\in
I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup
I^{q}_{\mathrm{cov}}),\beta=\alpha_{q_{j}}\in I^{q}_{\mathrm{cov}}\\\
\varphi^{\mathrm{App}}_{\alpha_{t_{i}}}|_{U_{\alpha\alpha_{t_{i}}}}\circ\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)\circ(\varphi^{\mathrm{App}}_{\alpha_{t_{i}}}|_{U_{\alpha\alpha_{t_{i}}}})^{-1}&\alpha\in
I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup
I^{q}_{\mathrm{cov}}),\beta=\alpha_{t_{i}}\in
I^{t}_{\mathrm{cov}},\end{cases}$

and we have that

(3.13)
$(g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})=O(z_{i}^{m_{i}}).$

###### Proof.

Let $\alpha\in I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup
I^{q}_{\mathrm{cov}})$. If $\beta\in
I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup I^{q}_{\mathrm{cov}})$,
then we have the following equalities:

$\displaystyle B_{\alpha\beta}$
$\displaystyle=(\varphi^{\text{App}}_{\alpha}|_{U_{\alpha\beta}})^{-1}\circ\varphi^{\text{App}}_{\beta}|_{U_{\alpha\beta}}$
$\displaystyle=(\varphi^{\mathrm{norm}}_{\alpha}|_{U_{\alpha\beta}})^{-1}\circ(\phi_{\nabla}|_{U_{\alpha\beta}})^{-1}\circ\phi_{\nabla}|_{U_{\alpha\beta}}\circ\varphi^{\mathrm{norm}}_{\beta}|_{U_{\alpha\beta}}$
$\displaystyle=(\varphi^{\mathrm{norm}}_{\alpha}|_{U_{\alpha\beta}})^{-1}\circ\varphi^{\mathrm{norm}}_{\beta}|_{U_{\alpha\beta}}=\begin{pmatrix}1&0\\\
0&((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})\end{pmatrix}.$

Here $\omega^{-1}_{\alpha}$ is a trivialization
$\mathcal{O}_{U_{\alpha}}\xrightarrow{\cong}(\Omega^{1}_{C}(D))^{-1}|_{U_{\alpha}}$
for any $\alpha$. Since
$((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})$ is
independent of the moduli space $M_{X}^{0}$, we have $v(B_{\alpha\beta})=0$.
So $u_{\alpha\beta}(v)=0$.

If $\beta=\alpha_{q_{j}}$, then we have the following equalities:

(3.14) $\displaystyle B_{\alpha\alpha_{q_{j}}}$
$\displaystyle=(\varphi^{\text{App}}_{\alpha}|_{U_{\alpha\alpha_{q_{j}}}})^{-1}\circ\varphi^{\text{App}}_{\alpha_{q_{j}}}|_{U_{\alpha\alpha_{q_{j}}}}$
$\displaystyle=(\varphi^{\mathrm{App},0}_{\alpha}|_{U_{\alpha\alpha_{q_{j}}}})^{-1}\circ(\phi_{\nabla}|_{U_{\alpha\alpha_{q_{j}}}})^{-1}\circ\phi_{\nabla}|_{U_{\alpha\alpha_{q_{j}}}}\circ\varphi^{\mathrm{App},0}_{\alpha_{q_{j}}}|_{U_{\alpha\alpha_{q_{j}}}}\circ\begin{pmatrix}1&0\\\
0&\frac{1}{z_{j}-q_{j}}\end{pmatrix}$
$\displaystyle=(\varphi^{\mathrm{App},0}_{\alpha}|_{U_{\alpha\alpha_{q_{j}}}})^{-1}\circ\varphi^{\mathrm{App},0}_{\alpha_{q_{j}}}|_{U_{\alpha\alpha_{q_{j}}}}\circ\begin{pmatrix}1&0\\\
0&\frac{1}{z_{j}-q_{j}}\end{pmatrix}$
$\displaystyle=(\varphi^{\mathrm{norm}}_{\alpha}|_{U_{\alpha\alpha_{q_{j}}}})^{-1}\circ\varphi^{\mathrm{norm}}_{\alpha_{q_{j}}}|_{U_{\alpha\alpha_{q_{j}}}}\circ\begin{pmatrix}1&\zeta_{j}\\\
0&1\end{pmatrix}\begin{pmatrix}1&0\\\ 0&\frac{1}{z_{j}-q_{j}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&0\\\
0&((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})\end{pmatrix}\begin{pmatrix}1&\frac{\zeta_{j}}{z_{j}-q_{j}}\\\
0&\frac{1}{z_{j}-q_{j}}\end{pmatrix}.$

So we have

$B_{\alpha\alpha_{q_{j}}}^{-1}v(B_{\alpha\alpha_{q_{j}}})=\begin{pmatrix}1&-\zeta_{j}\\\
0&z_{j}-q_{j}\end{pmatrix}\begin{pmatrix}0&\frac{v(\zeta_{j})(z_{j}-q_{j})+\zeta_{j}v(q_{j})}{(z_{j}-q_{j})^{2}}\\\
0&-\frac{-v(q_{j})}{(z_{j}-q_{j})^{2}}\end{pmatrix}=\begin{pmatrix}0&\frac{v(\zeta_{j})}{z_{j}-q_{j}}\\\
0&\frac{v(q_{j})}{z_{j}-q_{j}}\end{pmatrix}.$

If $\beta=\alpha_{t_{i}}$, then we have the following equalities:

$\displaystyle B_{\alpha\alpha_{t_{i}}}$
$\displaystyle=(\varphi^{\text{App}}_{\alpha}|_{U_{\alpha\alpha_{t_{i}}}})^{-1}\circ\varphi^{\text{App}}_{\alpha_{t_{i}}}|_{U_{\alpha\alpha_{t_{i}}}}$
$\displaystyle=(\varphi^{\mathrm{norm}}_{\alpha}|_{U_{\alpha\alpha_{t_{i}}}})^{-1}\circ(\phi_{\nabla}|_{U_{\alpha\alpha_{t_{i}}}})^{-1}\circ\phi_{\nabla}|_{U_{\alpha\alpha_{t_{i}}}}\circ\varphi^{\mathrm{norm}}_{\alpha_{t_{i}}}|_{U_{\alpha\alpha_{t_{i}}}}\circ
g^{t_{i}}_{\alpha_{t_{i}}}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})\end{pmatrix}\circ
g^{t_{i}}_{\alpha_{t_{i}}}.$

So we have
$B_{\alpha\alpha_{t_{i}}}^{-1}v(B_{\alpha\alpha_{t_{i}}})=(g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})$.
Since $(g^{t_{i}}_{\alpha})_{<m_{i}}$ is independent of the moduli space
$M^{0}_{X}$, we have that $v(g^{t_{i}}_{\alpha_{t_{i}}})=O(z_{i}^{m_{i}})$.
Finally, we have the statement of the lemma. ∎

Next we calculate $v_{\alpha}(v)\in\mathcal{F}^{1}(U_{\alpha})$ for $v\in
T_{(E,\nabla,\\{l^{(i)}\\})}M^{0}_{X}$. This is given by calculating the
variation of the connection matrix $A_{\alpha}$ in (3.6) with respect to $v$.
So we have

(3.15)
$v_{\alpha}(v)=\begin{cases}\varphi^{\text{App}}_{\alpha}\circ\begin{pmatrix}-v(\zeta_{j})\gamma_{\alpha}&v\left(\frac{\beta_{\alpha}-\zeta_{j}\delta_{\alpha}-\zeta_{j}^{2}\gamma_{\alpha}}{z_{j}-q_{j}}\right)\\\
-v(q_{j})\gamma_{\alpha}&v(\mathrm{tr}(A_{\alpha_{q_{j}}}))+v(\zeta_{j})\gamma_{\alpha}\end{pmatrix}\circ(\varphi^{\text{App}}_{\alpha})^{-1}&\text{when
$\alpha=\alpha_{q_{j}}$}\\\
\varphi^{\text{App}}_{\alpha}\circ\begin{pmatrix}0&v(\beta_{\alpha})\\\
0&v(\mathrm{tr}(A_{\alpha}))\end{pmatrix}\circ(\varphi^{\text{App}}_{\alpha})^{-1}&\text{when
$\alpha\in I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup
I^{q}_{\mathrm{cov}})$}\end{cases}.$

Here remark that $\gamma_{\alpha}$ is independent of the moduli space
$M_{X}^{0}$ for any $\alpha$. When $\alpha=\alpha_{t_{i}}$, we have that
$v_{\alpha}(v)$ is holomorphic at $t_{i}$.

###  3.5. Canonical coordinates

Now we introduce canonical coordinates on $M_{X}^{0}$ with respect to the
symplectic form (3.2). We recall that we have set $N:=4g+n-3$.

Let $\pi\colon\boldsymbol{\Omega}(D)\rightarrow C$ and
$\pi_{0}\colon\boldsymbol{\Omega}\rightarrow C$ be the total spaces of
$\Omega^{1}_{C}(D)$ and $\Omega^{1}_{C}$, respectively. The total space
$\boldsymbol{\Omega}$ has the Liouville symplectic form
$\omega_{\text{Liouv}}$. Since we have an isomorphism

$\pi_{0}^{-1}(C\setminus\mathrm{Supp}(D))\xrightarrow{\ \sim\
}\pi^{-1}(C\setminus\mathrm{Supp}(D)),$

the Liouville symplectic form induces a symplectic form
$\pi^{-1}(C\setminus\mathrm{Supp}(D))$. Let
$\pi_{N}\colon\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))\rightarrow\mathrm{Sym}^{N}(C)$
be the map induced by the map $\pi\colon\boldsymbol{\Omega}(D)\rightarrow C$.
We set

$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))_{0}:=\left\\{\\{q_{1},\ldots,q_{N}\\}\in\pi_{N}^{-1}(\mathrm{Sym}^{N}(C\setminus\mathrm{Supp}(D)))\
\middle|\ q_{j_{1}}\neq q_{j_{2}}\ (j_{1}\neq j_{2})\right\\}.$

Then $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))_{0}$ has the induced symplectic
form from the Liouville symplectic form.

######  Remark 15.

We have a map $f_{\mathrm{App},0}\colon
M_{X}^{0}\rightarrow\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))_{0}$, which is
described in (3.7). Notice that $M_{X}^{0}$ and
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))_{0}$ have symplectic forms. But by
the explicit calculation as below, we realize that this map
$f_{\mathrm{App},0}$ does not preserve these symplectic structures. So
$f_{\mathrm{App},0}$ does not give canonical coordinates directly. To give
canonical coordinates, we have to modify the map $f_{\mathrm{App},0}$ as
follows.

We twist $\boldsymbol{\Omega}(D)$ by a class in $H^{1}(C,\Omega^{1}_{C})$ as
follows. Let $c_{d}$ be the image of the line bundle $\mathrm{det}(E)$ under
the morphism

$H^{1}(C,\mathcal{O}^{*}_{C})\xrightarrow{\ \operatorname{d}\log\
}H^{1}(C,\Omega^{1}_{C})\cong\operatorname{Ext}_{C}^{1}(T_{C},\mathcal{O}_{C}).$

Let $\mathcal{A}_{C}(c_{d})$ be the sheaf produced by the Atiyah sequence on
$C$ with respect to $c_{d}$, that is, $\mathcal{A}_{C}(c_{d})$ is given by the
extension

(3.16)
$0\longrightarrow\mathcal{O}_{C}\longrightarrow\mathcal{A}_{C}(c_{d})\longrightarrow
T_{C}\longrightarrow 0$

with respect to $c_{d}\in H^{1}(C,\Omega_{C}^{1})$. Then,
$\mathcal{A}_{C}(c_{d})$ is naturally a Lie-algebroid, called the Atiyah
algebroid of the $\mathbb{G}_{m}$-principal bundle
$\operatorname{Tot}(T_{C})\setminus 0$, where $0$ stands for the $0$-section;
for details, see [40, Section 3.1.2]. We denote by
$\mathrm{symb}_{1}\colon\mathcal{A}_{C}(c_{d})\rightarrow T_{C}$ the morphism
in (3.16). We consider the subsheaf $T_{C}(-D)\subset T_{C}$. We set
$\mathcal{A}_{C}(c_{d},D):=\mathrm{symb}_{1}^{-1}T_{C}(-D)$, which is an
extension

$0\longrightarrow\mathcal{O}_{C}\longrightarrow\mathcal{A}_{C}(c_{d},D)\longrightarrow
T_{C}(-D)\longrightarrow 0.$

Let $\Omega^{1}_{C}(D,c_{d})$ be the twisted cotangent bundle over $C$ with
respect to $\mathcal{A}_{C}(c_{d},D)$, that is,

$\Omega^{1}_{C}(D,c_{d})=\left\\{\phi\in\mathcal{A}_{C}(c_{d},D)^{\vee}\
\middle|\ \langle\phi,1_{\mathcal{A}_{C}(c_{d},D)}\rangle=1\right\\}.$

We denote by

$\pi_{c_{d}}\colon\boldsymbol{\Omega}(D,c_{d})\longrightarrow C$

the total space of the twisted cotangent bundle $\Omega^{1}_{C}(D,c_{d})$, and
a generic element of this affine bundle by $(q,\tilde{p})$ in analogy with
classical notation $(q,p)$ for points of $\boldsymbol{\Omega}(D)$. For each
$(E,\nabla,\\{l^{(i)}\\})\in M_{X}^{0}$, we have
$(\mathrm{det}(E),\mathrm{tr}(\nabla))$. The connection $\mathrm{tr}(\nabla)$
on the line bundle $\mathrm{det}(E)$ is considered as a global section of
$\boldsymbol{\Omega}(D,c_{d})\rightarrow C$, which is the total space of the
twisted cotangent bundle with respect to $\mathrm{det}(E)$. The global section
$\mathrm{tr}(\nabla)$ gives a diffeomorphism

$\displaystyle\boldsymbol{\Omega}(D)\longrightarrow\boldsymbol{\Omega}(D,c_{d});\quad(q,p)\longmapsto(q,p+\mathrm{tr}(\nabla)).$

Notice that $\mathrm{tr}(\nabla)$ does depend on $M_{X}^{0}$. So this morphism
depends on $M_{X}^{0}$. Moreover, it is not a morphism of vector bundles.

######  Definition 16.

We define the _accessory parameter_ associated to $(E,\nabla)$ at $q_{j}$ by

$\tilde{p}_{j}=\mathrm{res}_{q_{j}}(\beta)+\mathrm{tr}(\nabla)|_{q_{j}},$

where $\beta\in H^{0}(C,(\Omega^{1}_{C})^{\otimes 2}(2D+B))$ is the
$(1,2)$-entry of (2.9) and
$\mathrm{res}_{q_{j}}(\beta)\in\Omega^{1}_{C}(D)|_{q_{j}}$. The $N$-tuple
$\left\\{\left(q_{j},\tilde{p}_{j}\right)\right\\}_{j=1,2,\ldots,N}$ will be
called _canonical coordinates_ of $(E,\nabla)$. We let $f_{\mathrm{App}}$ be
the map

$\displaystyle f_{\mathrm{App}}\colon M_{X}^{0}$
$\displaystyle\longrightarrow\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$
$\displaystyle(E,\nabla,\\{l^{(i)}\\})$
$\displaystyle\longmapsto\left\\{\left(q_{j},\tilde{p}_{j}\right)\right\\}_{j=1,2,\ldots,N}.$

Notice that the map $f_{\mathrm{App},0}$ in (3.7) is defined by using only
$\mathrm{res}_{q_{j}}(\beta)$. The reason why we consider the twisted
cotangent bundle $\boldsymbol{\Omega}(D,c_{d})$ is to justify
$\mathrm{tr}(\nabla)|_{q_{j}}$. The next proposition shows that the quantities
introduced in the definition may indeed be called coordinates.

######  Proposition 17.

The map $f_{\mathrm{App}}$ introduced in Definition 16 is birational.

###### Proof.

It follows from Proposition 10 that the dimensions of the source and target of
$f_{\mathrm{App}}$ agree. We therefore need to show two things: first, that
$f_{\mathrm{App}}$ is rational, and second, that it admits an inverse over a
Zariski open subset of $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$.

The first assertion is trivial, because the construction of the apparent
singularities $q_{j}$ and their accessory parameters $\tilde{p}_{j}$ follow
from algebraic arguments on certain Zariski open subsets.

The key statement is existence of a generic inverse. This is now a variant of
Lemma 7. Namely, fixing generic
$\left\\{\left(q_{j},\tilde{p}_{j}\right)\right\\}_{j=1,2,\ldots,N}$, we must
find a unique $(\delta,\beta)$. Since we have
$\delta=\mathrm{tr}(\nabla_{0})$, we get the expression

$\tilde{p}_{j}=\zeta_{j}\operatorname{d}\\!z_{j}+\delta-\frac{\operatorname{d}\\!z_{j}}{z_{j}}.$

An algebraic manipulation shows that the constraint (2.14) expressing that the
singularity at $q_{j}$ be apparent is equivalent to the holomorphicity and
vanishing of the expression

(3.17)
$\beta+\delta\left(\tilde{p}_{j}+\frac{\operatorname{d}\\!z_{j}}{z_{j}}\right)-\left(\tilde{p}_{j}+\frac{\operatorname{d}\\!z_{j}}{z_{j}}\right)^{2}.$

We now study these conditions by taking the Laurent expansion of this
expression with respect to $z_{j}$. We first observe that it clearly admits a
pole of order at most $2$ at $q_{j}$, because $q_{j}\neq t_{i}$. Since
$\delta$ has a simple pole with residue $1$, the term of degree $-2$ is

$\left(\operatorname{d}\\!z_{j}\right)^{\otimes
2}-\left(\operatorname{d}\\!z_{j}\right)^{\otimes 2}=0.$

So the pole is automatically at most simple.

For the study of the residue, we need to introduce some notation: let us write

$\displaystyle\delta_{0}$
$\displaystyle=\frac{\operatorname{d}\\!z_{j}}{z_{j}}+\delta_{0}^{(j)}$
$\displaystyle\beta_{0}$
$\displaystyle=\zeta_{j}\frac{\left(\operatorname{d}\\!z_{j}\right)^{\otimes
2}}{z_{j}}+\beta_{0}^{(j)}$

for a holomorphic rank $1$ connection $\delta_{0}^{(j)}$ and a holomorphic
quadratic differential $\beta_{0}^{(j)}$ on $U_{q_{j}}$. Then, the degree $-1$
part of (3.17) is (up to a global factor $\operatorname{d}\\!z_{j}$)

$\zeta_{j}\operatorname{d}\\!z_{j}+\tilde{p}_{j}+\left(\delta-\frac{\operatorname{d}\\!z_{j}}{z_{j}}\right)-2\tilde{p}_{j}=0$

by the definition of $\tilde{p}_{j}$.

Finally, to deal with the vanishing constraint, we make use of the same basis
expansions for $\delta$ and $\beta$ as in Lemma 7. Then, the conditions read
as

$\sum_{k=1}^{N-g}b_{k}\nu_{k}(q_{j})+\tilde{p}_{j}\sum_{l=1}^{g}d_{l}\omega_{l}(q_{j})=\left(\tilde{p}_{j}\right)^{\otimes
2}-\delta_{0}^{(j)}(q_{j})\tilde{p}_{j}-\beta_{0}^{(j)}(q_{j}).$

Now, the determinant of this linear system of $N$ equations (for $1\leq j\leq
N$) in $N$ variables $b_{1},\ldots,b_{N-g},d_{1},\ldots,d_{g}$ agrees with the
determinant studied in Lemma 7, up to replacing each occurrence of
$\zeta_{j}\operatorname{d}\\!z_{j}$ by $\tilde{p}_{j}$. The end of the proof
then follows word by word the method of Lemma 7. ∎

######  Remark 18.

The expression (3.17) has variables $\tilde{p}_{j}$ in the twisted cotangent
sheaf rather than the ordinary cotangent sheaf. The quadratic polynomial of
$\tilde{p}_{j}$ can be viewed as the characteristic polynomial of the
connection matrix of $\nabla_{0}$. Thus, in a sense the vanishing condition on
(3.17) may be interpreted as the requirement that $\tilde{p}_{j}$ lie on the
_quantum spectral curve_ of $\nabla_{0}$, see e.g. [11].

By taking a local trivialization of $\mathrm{det}(E)$, we have a concrete
description of the map $f_{\mathrm{App}}$. Now we will discuss on such a
description of $f_{\mathrm{App}}$. The description discussed below is useful
for the proof of Theorem 20 below. Let $(E,\nabla,\\{l^{(i)}\\})\in
M_{X}^{0}$. As a local trivialization of $\mathrm{det}(E)$, we take the
isomorphism

(3.18)
$\mathrm{det}(\varphi_{\alpha_{q_{j}}}^{\mathrm{App}})\colon\mathcal{O}_{U_{\alpha_{q_{j}}}}\longrightarrow\mathrm{det}(E)|_{U_{\alpha_{q_{j}}}},$

which is the determinant of the trivialization in Definition 13. Notice that
the composition

$\mathcal{O}_{U_{\alpha_{q_{j}}}}\xrightarrow{\omega_{\alpha_{q_{j}}}^{-1}}(\Omega^{1}_{C}(D))^{-1}|_{U_{\alpha_{q_{j}}}}\xrightarrow{\mathrm{det}(\phi_{\nabla})|_{U_{\alpha_{q_{j}}}}}\mathrm{det}(E)|_{U_{\alpha_{q_{j}}}}\xrightarrow{\mathrm{det}(\varphi_{\alpha_{q_{j}}}^{\mathrm{App}})^{-1}}\mathcal{O}_{U_{\alpha_{q_{j}}}}$

coincides with
$(z_{j}-q_{j})\colon\mathcal{O}_{U_{\alpha_{q_{j}}}}\rightarrow\mathcal{O}_{U_{\alpha_{q_{j}}}}$.
Let
$\mathrm{tr}(A_{\alpha_{q_{j}}})\in\Omega^{1}_{C}(D)|_{U_{\alpha_{q_{j}}}}$ be
the connection matrix of $(\mathrm{det}(E),\mathrm{tr}(\nabla))$ on
$U_{\alpha_{q_{j}}}$ with respect to the local trivialization
$\mathrm{det}(\varphi_{\alpha_{q_{j}}}^{\mathrm{App}})$. Then, by using (3.5),
the map $f_{\mathrm{App}}$ has the following description:

$\displaystyle
f_{\mathrm{App}}\colon(E,\nabla,\\{l^{(i)}\\})\longmapsto\left\\{\left(q_{j},\zeta_{j}\gamma_{\alpha_{q_{j}}}|_{q_{j}}+\mathrm{tr}(A_{\alpha_{q_{j}}})|_{q_{j}}\right)\right\\}_{j=1,2,\ldots,N},$

Here
$\zeta_{j}\gamma_{\alpha_{q_{j}}}|_{q_{j}}+\mathrm{tr}(A_{\alpha_{q_{j}}})|_{q_{j}}$
is an element of $\Omega^{1}_{C}(D)|_{q_{j}}$. We set

(3.19)
$p_{j}:=\mathrm{res}_{q_{j}}\left(\frac{\zeta_{j}\gamma_{\alpha_{q_{j}}}}{z_{j}-q_{j}}\right)+\mathrm{res}_{q_{j}}\left(\frac{\mathrm{tr}(A_{\alpha_{q_{j}}})}{z_{j}-q_{j}}\right),$

which is the image of
$\zeta_{j}\gamma_{\alpha_{q_{j}}}|_{q_{j}}+\mathrm{tr}(A_{\alpha_{q_{j}}})|_{q_{j}}$
under the isomorphism $\Omega^{1}_{C}(D)|_{q_{j}}\cong\mathbb{C}$.

######  Remark 19.

This $p_{j}$ is just the evaluation of the $(2,2)$-entry of the connection
matrix $A_{\alpha_{q_{j}}}$ in (3.6) at $q_{j}$. Note that the $(2,1)$-entry
of this connection matrix $A_{\alpha_{q_{j}}}$ at $q_{j}$ vanishes. So $p_{j}$
is an “eigenvalue” of $\nabla$ at $q_{j}$. (On the other hand, $\zeta_{j}$ is
an “eigenvector” of $\nabla_{0}$ at $q_{j}$). This fact means that the
coordinates $(q_{j},p_{j})_{j}$ are an analog of the coordinates on the moduli
space of (parabolic) Higgs bundles given as in [16] and [19]. The coordinates
on the moduli space of (parabolic) Higgs bundles are by using the BNR
correspondence [3]. (See Section 6).

Let
$\pi_{c_{d},N}\colon\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))\rightarrow\mathrm{Sym}^{N}(C)$
be the map induced by the map
$\pi_{c_{d}}\colon\boldsymbol{\Omega}(D,c_{d})\rightarrow C$. We set

$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))_{0}:=\left\\{\\{(q_{j},\tilde{p}_{j})\\}_{j=1}^{N}\in\pi_{c_{d},N}^{-1}(\mathrm{Sym}^{N}(C\setminus\mathrm{Supp}(D)))\
\middle|\ q_{j_{1}}\neq q_{j_{2}}\ (j_{1}\neq j_{2})\right\\}.$

Then $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))_{0}$ has the induced
symplectic form from the Liouville symplectic form. Notice that by
construction the image of $M_{X}^{0}$ under the map $f_{\mathrm{App}}$ is
contained in $\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))_{0}$.

######  Theorem 20.

Let $\omega$ be the symplectic form on $M_{X}^{0}$ defined by (3.2). The pull-
back of the symplectic form on
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))_{0}$ under the map

$f_{\mathrm{App}}\colon
M_{X}^{0}\longrightarrow\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))_{0}$

in Definition 16 coincides with $\omega$.

###### Proof.

Let $V$ be an analytic open subset of $M_{X}^{0}$ as in Section 3.4. Moreover,
we assume that we may define a composition

$\displaystyle V\longrightarrow f_{\mathrm{App}}(V)$
$\displaystyle\longrightarrow\mathrm{Sym}^{N}(\mathbb{C}_{(q,p)}^{2})$
$\displaystyle(E,\nabla)\longmapsto f_{\mathrm{App}}(E,\nabla)$
$\displaystyle\longmapsto\left\\{(q_{j},p_{j})\right\\}_{j=1,2,\ldots,N},$

where $p_{j}$ is defined in (3.19), and the image of $V$ under the composition
is isomorphic to some analytic open subset of $\mathbb{C}_{(q,p)}^{2N}$. Let
$U_{(q,p)}$ be such an analytic open subset of $\mathbb{C}_{(q,p)}^{2N}$. We
denote by $f_{2}$ the map

$\displaystyle M_{X}^{0}\supset V$ $\displaystyle\longrightarrow
U_{(q,p)}\subset\mathbb{C}_{(q,\zeta)}^{2N}$ $\displaystyle(E,\nabla)$
$\displaystyle\longmapsto(q_{1},\ldots,q_{N},p_{1},\ldots,p_{N}).$

We consider the following maps

$\textstyle{U_{(q,\zeta)}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{\sim}$$\scriptstyle{f_{2}}$$\textstyle{U_{(q,p)}\hbox
to0.0pt{.\hss}}$

Here $f_{1}\colon V\xrightarrow{\sim}U_{(q,\zeta)}$ is the isomorphism (3.8).
The symplectic structure on $U_{(q,p)}$ induced by the symplectic structure on
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D,c_{d}))$ is
$\sum^{N}_{j=1}dp_{j}\wedge dq_{j}$. We will show that

$(f_{1}^{-1})^{*}(\omega|_{V})=(f_{2}\circ
f_{1}^{-1})^{*}\left(\sum^{N}_{j=1}dp_{j}\wedge dq_{j}\right).$

Let $v,v^{\prime}$ be elements of $T_{(q_{j},\zeta_{j})_{j}}U_{(q,\zeta)}$ for
$(q_{j},\zeta_{j})_{j}\in U_{(q,\zeta)}$. We will use the description of the
tangent map (3.10) of $f^{-1}_{1}\colon U_{(q,\zeta)}\rightarrow V$. That is,
we calculate $(f_{1}^{-1})^{*}(\omega|_{V})$ by applying the descriptions
(3.12) and (3.15) of $u_{\alpha\beta}(v)$ and $v_{\alpha}(v)$, respectively.

First we consider
$\\{u_{\alpha\beta}(v)u_{\beta\gamma}(v^{\prime})\\}_{\alpha\beta\gamma}$.
Remark that $U_{\alpha_{q_{j_{1}}}}\cap U_{\alpha_{q_{j_{2}}}}=\emptyset$ for
any $j_{1}$ and $j_{2}$, $U_{\alpha_{t_{i_{1}}}}\cap
U_{\alpha_{t_{i_{2}}}}=\emptyset$ for any $i_{1}$ and $i_{2}$, and
$U_{\alpha_{q_{j}}}\cap U_{\alpha_{t_{i}}}=\emptyset$ for any $j$ and $i$.
Then we have $u_{\alpha\beta}u_{\beta\gamma}=0$ by Lemma 14. So we may take a
representative of the class in the pairing (3.2) so that

$[-\\{\mathrm{tr}(u_{\alpha\beta}(v)\circ
v_{\beta}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)\circ
u_{\alpha\beta}(v^{\prime}))\\}_{\alpha\beta}]\in
H^{1}(C,\Omega^{1}_{C})\cong\mathbb{C}.$

Now we calculate $\mathrm{tr}(u_{\alpha\beta}(v)\circ
v_{\beta}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)\circ
u_{\alpha\beta}(v^{\prime}))$. If $\alpha\in
I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup I^{q}_{\mathrm{cov}})$ and
$\beta=\alpha_{q_{j}}$, then, by applying (3.12) and (3.15), we have the
following equalities

(3.20)
$\displaystyle\mathrm{tr}(u_{\alpha\alpha_{q_{j}}}(v)v_{\alpha_{q_{j}}}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)u_{\alpha\alpha_{q_{j}}}(v^{\prime}))$
$\displaystyle=\mathrm{tr}\left(\begin{pmatrix}0&\frac{v(\zeta_{j})}{z_{j}-q_{j}}\\\
0&\frac{v(q_{j})}{z_{j}-q_{j}}\end{pmatrix}\begin{pmatrix}*&*\\\
-v^{\prime}(q_{j})\gamma_{\alpha_{q_{j}}}&v^{\prime}(\mathrm{tr}(A_{\alpha_{q_{j}}}))+v^{\prime}(\zeta_{j})\gamma_{\alpha_{q_{j}}}\end{pmatrix}\right)$
$\displaystyle\qquad-\mathrm{tr}\left(\begin{pmatrix}*&*\\\
0&v(\mathrm{tr}(A_{\alpha}))\end{pmatrix}\begin{pmatrix}0&\frac{v^{\prime}(\zeta_{j})}{z_{j}-q_{j}}\\\
0&\frac{v^{\prime}(q_{j})}{z_{j}-q_{j}}\end{pmatrix}\right)$
$\displaystyle=-\frac{v(\zeta_{j})v^{\prime}(q_{j})\gamma_{\alpha_{q_{j}}}}{z_{j}-q_{j}}+\frac{v(q_{j})\left(v^{\prime}(\mathrm{tr}(A_{\alpha_{q_{j}}}))+v^{\prime}(\zeta_{i})\gamma_{\alpha_{q_{j}}}\right)}{z_{j}-q_{j}}-\frac{v^{\prime}(q_{j})\left(v(\mathrm{tr}(A_{\alpha}))\right)}{z_{j}-q_{j}}$
$\displaystyle=-\frac{\left(v(\zeta_{j})\gamma_{\alpha_{q_{j}}}+v(\mathrm{tr}(A_{\alpha}))\right)v^{\prime}(q_{j})}{z_{j}-q_{j}}+\frac{v(q_{j})\left(v^{\prime}(\mathrm{tr}(A_{\alpha_{q_{j}}}))+v^{\prime}(\zeta_{i})\gamma_{\alpha_{q_{j}}}\right)}{z_{j}-q_{j}}.$

Now we consider the difference between $v(\mathrm{tr}(A_{\alpha_{q_{j}}}))$
and $v(\mathrm{tr}(A_{\alpha}))$. So we consider infinitesimal deformation of
$(\mathrm{det}(\nabla),\mathrm{tr}(\nabla))$. We have that

$\mathrm{det}(B_{\alpha\alpha_{q_{j}}})=\mathrm{det}\left(\begin{pmatrix}1&0\\\
0&((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})\end{pmatrix}\begin{pmatrix}1&\frac{\zeta_{j}}{z_{j}-q_{j}}\\\
0&\frac{1}{z_{j}-q_{j}}\end{pmatrix}\right)=\frac{((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})}{z_{j}-q_{j}}.$

Here $B_{\alpha\alpha_{q_{j}}}$ is calculated in (3.14). Set

(3.21)
$u^{\mathrm{det}}_{\alpha\alpha_{q_{j}}}(v):=\mathrm{det}(B_{\alpha\alpha_{q_{j}}})^{-1}v(\mathrm{det}(B_{\alpha\alpha_{q_{j}}}))=\frac{v(q_{j})}{z_{j}-q_{j}}.$

Here remark that
$((\omega_{\alpha}^{-1})^{-1}\circ\omega_{\alpha_{q_{j}}}^{-1})$ is
independent of the moduli space $M_{X}^{0}$. We have a cocycle condition

$v(\mathrm{tr}(A_{\alpha_{q_{j}}}))-v(\mathrm{tr}(A_{\alpha}))=\mathrm{tr}(\nabla)\circ
u^{\mathrm{det}}_{\alpha\alpha_{q_{j}}}-u^{\mathrm{det}}_{\alpha\alpha_{q_{j}}}\circ\mathrm{tr}(\nabla).$

So we have

$v(\mathrm{tr}(A_{\alpha_{q_{j}}}))-v(\mathrm{tr}(A_{\alpha}))=\operatorname{d}\left(\frac{v(q_{j})}{z_{j}-q_{j}}\right)=-\frac{v(q_{j})\operatorname{d}\\!z_{j}}{(z_{j}-q_{j})^{2}}.$

By applying this difference to (3.20), we have that

(3.22)
$\displaystyle\mathrm{tr}(u_{\alpha\alpha_{q_{j}}}(v)v_{\alpha_{q_{j}}}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)u_{\alpha\alpha_{q_{j}}}(v^{\prime}))$
$\displaystyle=-\frac{\left(v(\zeta_{j})\gamma_{\alpha_{q_{j}}}+v(\mathrm{tr}(A_{\alpha_{q_{j}}}))\right)v^{\prime}(q_{j})}{z_{j}-q_{j}}+\frac{v(q_{j})\left(v^{\prime}(\mathrm{tr}(A_{\alpha_{q_{j}}}))+v^{\prime}(\zeta_{i})\gamma_{\alpha_{q_{j}}}\right)}{z_{j}-q_{j}}-\frac{v(q_{j})v^{\prime}(q_{j})\operatorname{d}\\!z_{j}}{(z_{j}-q_{j})^{3}}.$

So we may extend the 1-form

$\mathrm{tr}(u_{\alpha\alpha_{q_{j}}}(v)v_{\alpha_{q_{j}}}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)u_{\alpha\alpha_{q_{j}}}(v^{\prime}))$

from $U_{\alpha\alpha_{q_{j}}}$ to $U_{\alpha_{q_{j}}}$ by (3.22). Then we
have a meromorphic 1-form defined on $U_{\alpha_{q_{j}}}$, which has a pole at
$q_{j}$. We denote by $\omega_{\alpha_{q_{j}}}(v,v^{\prime})$ the meromorphic
1-form defined on $U_{\alpha_{q_{j}}}$.

Next we consider the case where $\alpha\in
I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup I^{q}_{\mathrm{cov}})$ and
$\beta=\alpha_{t_{i}}$. We have the following equalities

$\displaystyle\mathrm{tr}(u_{\alpha\alpha_{t_{i}}}(v)v_{\alpha_{t_{i}}}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)u_{\alpha\alpha_{t_{i}}}(v^{\prime}))$
$\displaystyle=\mathrm{tr}\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})v^{\prime}(A_{\alpha_{t_{i}}})\right)-\mathrm{tr}\left(\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(A_{\alpha})g^{t_{i}}_{\alpha_{t_{i}}}\right)(g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v^{\prime}(g^{t_{i}}_{\alpha_{t_{i}}})\right)$

We have the cocycle condition

$\displaystyle
v(A_{\alpha_{t_{i}}})-(g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(A_{\alpha})(g^{t_{i}}_{\alpha_{t_{i}}})$
$\displaystyle=(\operatorname{d}+A_{\alpha_{t_{i}}})\circ\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)-\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)\circ(\operatorname{d}+A_{\alpha_{t_{i}}})$
$\displaystyle=\operatorname{d}\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)+\left[\,A_{\alpha_{t_{i}}},\,\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)\,\right].$

By this condition, we have

(3.23)
$\displaystyle\mathrm{tr}(u_{\alpha\alpha_{t_{i}}}(v)v_{\alpha_{t_{i}}}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)u_{\alpha\alpha_{t_{i}}}(v^{\prime}))$
$\displaystyle=\mathrm{tr}\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})v^{\prime}(A_{\alpha_{t_{i}}})\right)-\mathrm{tr}\left(v(A_{\alpha_{t_{i}}})(g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v^{\prime}(g^{t_{i}}_{\alpha_{t_{i}}})\right)$
$\displaystyle\qquad+\mathrm{tr}\left(\left(\operatorname{d}\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)+\left[\,A_{\alpha_{t_{i}}},\,\left((g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v(g^{t_{i}}_{\alpha_{t_{i}}})\right)\right]\right)(g^{t_{i}}_{\alpha_{t_{i}}})^{-1}v^{\prime}(g^{t_{i}}_{\alpha_{t_{i}}})\right)$

So we may extend the 1-form

$\mathrm{tr}(u_{\alpha\alpha_{t_{i}}}(v)v_{\alpha_{t_{i}}}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)u_{\alpha\alpha_{t_{i}}}(v^{\prime}))$

from $U_{\alpha\alpha_{t_{i}}}$ to $U_{\alpha_{t_{i}}}$ by (3.23). Since we
have the vanishing of the lower terms (3.13), the extended 1-form defined on
$U_{\alpha_{t_{i}}}$ is holomorphic. We denote by
$\omega_{\alpha_{t_{i}}}(v,v^{\prime})$ the holomorphic 1-form defined on
$U_{\alpha_{t_{i}}}$.

For $\alpha\in I_{\mathrm{cov}}\setminus(I^{t}_{\mathrm{cov}}\cup
I^{q}_{\mathrm{cov}})$, we set $\omega_{\alpha}(v,v^{\prime})=0$. By (3.22)
and (3.23), we have a meromorphic coboundary
$\\{\omega_{\alpha}(v,v^{\prime})\\}_{\alpha}$ of

$\\{\mathrm{tr}(u_{\alpha\beta}(v)\circ
v_{\beta}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)\circ
u_{\alpha\beta}(v^{\prime}))\\}_{\alpha\beta}.$

So we have

$\displaystyle H^{1}(C,\Omega_{C}^{1})$ $\displaystyle\xrightarrow{\ \cong\
}\mathbb{C}$ $\displaystyle[-\\{\mathrm{tr}(u_{\alpha\beta}(v)\circ
v_{\beta}(v^{\prime}))-\mathrm{tr}(v_{\alpha}(v)\circ
u_{\alpha\beta}(v^{\prime}))\\}_{\alpha\beta}]$
$\displaystyle\longmapsto\sum_{x\in
C}-\mathrm{res}_{x}\left(\omega_{\alpha}(v,v^{\prime})\right).$

By taking the residues of the right hand sides of (3.22) and (3.23), we have
that

$\displaystyle-\sum_{x\in
C}\mathrm{res}_{x}\left(\omega_{\alpha}(v,v^{\prime})\right)$
$\displaystyle=\sum_{j=1}^{N}\mathrm{res}_{q_{j}}\left(\frac{\left(v(\zeta_{j})\gamma_{\alpha_{q_{j}}}+v(\mathrm{tr}(A_{\alpha_{q_{j}}}))\right)v^{\prime}(q_{j})}{z_{j}-q_{j}}\right)$
$\displaystyle\qquad-\sum_{j=1}^{N}\mathrm{res}_{q_{j}}\left(\frac{v(q_{j})\left(v^{\prime}(\mathrm{tr}(A_{\alpha_{q_{j}}}))+v^{\prime}(\zeta_{i})\gamma_{\alpha_{q_{j}}}\right)}{z_{j}-q_{j}}\right)$
$\displaystyle=\sum_{j=1}^{N}\left(v(p_{j})v^{\prime}(q_{j})-v(q_{j})v^{\prime}(p_{j})\right)=\left(\sum^{N}_{j=1}dp_{j}\wedge
dq_{j}\right)(v,v^{\prime}).$

Here remark that $\gamma_{\alpha}$ is independent of the moduli space
$M_{X}^{0}$ for any $\alpha$. ∎

By the map $f_{\mathrm{App}}$, we have concrete canonical coordinates as
follows. We take an analytic open subset $V$ of $M^{0}_{X}$ at a point
$(E,\nabla)$, which is small enough. We define functions $q_{j}$ and $p_{j}$
($j=1,2,\ldots,N$) on $V$ as follows. (So, here, the notation $q_{j}$ has a
double meaning). Let $U_{\alpha_{q_{j}}}$ be an analytic open subset of $C$
such that $U_{\alpha_{q_{j}}}$ contains the apparent singularity $q_{j}$ of
the point $(E,\nabla)$ and is small enough. Let $q_{j}^{\prime}$ be the
apparent singularity of each $(E^{\prime},\nabla^{\prime})\in V$, where
$q_{j}^{\prime}\in U_{\alpha_{q_{j}}}$. First we take a local coordinate
$z_{j}$ on $U_{\alpha_{q_{j}}}$. By evaluating the apparent singularity
$q_{j}^{\prime}$ by the local coordinate $z_{j}$ for each
$(E^{\prime},\nabla^{\prime})\in V$, we have a function $q_{j}\colon
V\rightarrow\mathbb{C}$. Second, let $(E_{V},\nabla_{V})$ be a vector bundles
on $C\times V$, which is a family of vector bundles on $C$ parametrized by
$V$. We take a trivialization of $\det(E_{V})$ on $U_{\alpha_{q_{j}}}\times V$
which depends on only $q_{j}\colon V\rightarrow\mathbb{C}$ (which is described
in (3.18)). We take the connection matrix of $\mathrm{tr}(\nabla_{V})$ with
respect to the local trivialization. Let
$\boldsymbol{\Omega}(D,c_{d})_{V}\rightarrow C\times V$ be the relative
twisted cotangent bundle over $V$ with respect to the family of line bundles
$\det(E_{V})$ on $C\times V$. We have an identification between
$\boldsymbol{\Omega}(D,c_{d})_{V}$ and $\boldsymbol{\Omega}(D)\times V$ on
$U_{\alpha_{q_{j}}}\times V$ that depends only on $q_{j}\colon
V\rightarrow\mathbb{C}$. By evaluating
$\mathrm{res}_{q_{j}^{\prime}}(\beta^{\prime})+\mathrm{tr}(\nabla^{\prime})|_{q_{j}^{\prime}}$
by the identification
$\boldsymbol{\Omega}(D,c_{d})|_{q_{j}^{\prime}}\cong\boldsymbol{\Omega}(D)|_{q_{j}^{\prime}}\cong\mathbb{C}$
for each $(E^{\prime},\nabla^{\prime})\in V$, we have a function $p_{j}\colon
V\rightarrow\mathbb{C}$. This is just (3.19). That is, this is the following
composition:

$\displaystyle V$ $\displaystyle\longrightarrow U_{\alpha_{q_{j}}}\times
V\longrightarrow\boldsymbol{\Omega}(D,c_{d})_{V}|_{U_{\alpha_{q_{j}}}\times
V}\longrightarrow\boldsymbol{\Omega}(D)|_{U_{\alpha_{q_{j}}}}\longrightarrow\mathbb{C}$
$\displaystyle(E^{\prime},\nabla^{\prime})$
$\displaystyle\longmapsto(q_{j}^{\prime},(E^{\prime},\nabla^{\prime}))\longmapsto\left((\zeta_{j}\gamma_{\alpha_{q_{j}}})_{V}+\mathrm{tr}(\nabla_{V})\right)|_{(q_{j}^{\prime},(E^{\prime},\nabla^{\prime}))}$
$\displaystyle\qquad\longmapsto\left(\zeta_{j}^{\prime}\gamma_{\alpha_{q_{j}}}+\mathrm{tr}(A^{\prime}_{\alpha_{q_{j}}})\right)|_{q_{j}^{\prime}}\longmapsto\mathrm{res}_{q^{\prime}_{j}}\left(\frac{\zeta^{\prime}_{j}\gamma_{\alpha_{q_{j}}}+\mathrm{tr}(A^{\prime}_{\alpha_{q_{j}}})}{z_{j}-q^{\prime}_{j}}\right).$

By Theorem 20, the symplectic structure on $V$ has the following description:
$\sum_{j=1}^{N}\operatorname{d}\\!p_{j}\wedge\operatorname{d}\\!q_{j}$.

######  Remark 21.

We set

$p_{j}^{0}:=\mathrm{res}_{q_{j}}\left(\frac{\zeta_{j}\gamma_{\alpha_{q_{j}}}}{z_{j}-q_{j}}\right)\in\mathbb{C}.$

If $g=0$, then
$\mathrm{res}_{q_{j}}\left(\frac{\mathrm{tr}(A_{\alpha_{q_{j}}})}{z_{j}-q_{j}}\right)$
depends on only $q_{j}$. So we have
$\sum^{N}_{j=1}\operatorname{d}\\!p_{j}\wedge\operatorname{d}\\!q_{j}=\sum^{N}_{j=1}\operatorname{d}\\!p^{0}_{j}\wedge\operatorname{d}\\!q_{j}$.
Here the symplectic form
$\sum^{N}_{j=1}\operatorname{d}\\!p^{0}_{j}\wedge\operatorname{d}\\!q_{j}$ is
induced by the symplectic form on
$\mathrm{Sym}^{N}(\boldsymbol{\Omega}(D))_{0}$.

######  Remark 22.

In general,
$\sum^{N}_{j=1}\operatorname{d}\\!p_{j}\wedge\operatorname{d}\\!q_{j}\not=\sum^{N}_{j=1}\operatorname{d}\\!p^{0}_{j}\wedge\operatorname{d}\\!q_{j}$,
that is,

(3.24)
$\sum_{j}\operatorname{d}\\!\left(\mathrm{res}_{q_{j}}\left(\frac{\mathrm{tr}(A_{\alpha_{q_{j}}})}{z_{j}-q_{j}}\right)\right)\wedge\operatorname{d}\\!q_{j}$

does not vanish. This is related to the determinant map

$\displaystyle M_{X}^{0}$ $\displaystyle\longrightarrow
M^{\mathrm{rk}=1}_{X}(\boldsymbol{\nu}_{\textrm{res}})$
$\displaystyle(E,\nabla,\\{l^{(i)}\\})$
$\displaystyle\longmapsto(\mathrm{det}(E),\mathrm{tr}(\nabla)).$

The 2-form (3.24) comes from

$\left[\\{u^{\mathrm{det}}_{\alpha\beta}(v)u^{\mathrm{det}}_{\beta\gamma}(v^{\prime})\\},-\\{u^{\mathrm{det}}_{\alpha\beta}(v)v^{\prime}(\mathrm{tr}(A_{\beta}))-v(\mathrm{tr}(A_{\alpha}))u^{\mathrm{det}}_{\alpha\beta}(v^{\prime})\\}\right]\in{\mathbf{H}}^{2}(\mathcal{O}_{C}\rightarrow\Omega^{1}_{C}).$

Here $u^{\mathrm{det}}_{\alpha\beta}(v)$ is defined as in (3.21). This class
gives rise to the $2$-form on $M_{X}^{0}$ which is just the pull-back of the
natural symplectic form on
$M^{\mathrm{rk}=1}_{X}(\boldsymbol{\theta}_{\textrm{res}})$ under the
determinant map. The determinant map is not degenerate in general. So the
class (3.24) does not vanish in general.

##  4\. Symplectic structure on the moduli space with fixed trace connection

In this section, we consider the moduli spaces of rank 2 quasi-parabolic
connections with fixed trace connection. When the effective divisor $D$ is
reduced, this moduli space is detailed in [1], [42] (when $g=0$), [12], [13]
(when $g=1$), and [44] (when $g\geq 1$). The moduli spaces of rank 2 quasi-
parabolic connections with fixed trace connection has a natural symplectic
structure described as in Section 3.2. The purpose of this section is to give
coordinates on some generic part of the moduli space and to describe the
natural symplectic structure by using the coordinates. As in the case where
the effective divisor $D$ is reduced ([42], [12], [13], [44]), we may define
the map forgetting connections and the apparent map. These maps are from a
generic part of the moduli space to projective spaces. These maps will give
our coordinates on the generic part of the moduli space. First we describe
these maps.

###  4.1. Moduli space of quasi-parabolic bundles with fixed determinant

To describe the map forgetting connections, we recall the moduli space of
quasi-parabolic bundles. The moduli space of (quasi-)parabolic bundles was
introduced in Mehta–Seshadri [45]. Yokogawa generalized this notion to
(quasi-)parabolic sheaves and studied their moduli [54].

Let $\nu$ be a positive integer. Set $I:=\\{1,2,\ldots,\nu\\}$. Let $C$ be a
compact Riemann surface of genus $g$, and $D=\sum_{i\in I}m_{i}[t_{i}]$ be an
effective divisor on $C$. We assume $3g-3+n>0$ where
$n=\operatorname{length}(D)$. Let $z_{i}$ be a generator of the maximal ideal
of $\mathcal{O}_{C,t_{i}}$. We fix a line bundle $L_{0}$ with
$\deg(L_{0})=2g-1$.

######  Definition 23.

We say $(E,\\{l^{(i)}\\})$ a rank $2$ quasi-parabolic bundle with determinant
$L_{0}$ over $(C,D)$ if

  * (i)

$E$ is a rank $2$ vector bundle of degree $2g-1$ on $C$ with $\det(E)\cong
L_{0}$, and

  * (ii)

$E|_{m_{i}[t_{i}]}\supset l^{(i)}\supset 0$ is a filtration by free
$\mathcal{O}_{m_{i}[t_{i}]}$-modules such that
$E|_{m_{i}[t_{i}]}/l^{(i)}\cong\mathcal{O}_{m_{i}[t_{i}]}$ and
$l^{(i)}\cong\mathcal{O}_{m_{i}[t_{i}]}$ for any $i\in I$.

We fix weights $\boldsymbol{w}=(w_{1},\ldots,w_{\nu})$ such that
$w_{i}\in[0,1]$ for any $i\in I$. When $g=0$, we assume that $(w_{i})_{i\in
I}$ satisfies

(4.1)
$w_{1}=\cdots=w_{\nu}\quad\text{and}\quad\frac{1}{\deg(D)}<w_{i}<\frac{1}{\deg(D)-2}.$

When $g\geq 1$, we assume that $(w_{i})_{i\in I}$ satisfy

(4.2) $0<w_{i}\ll 1.$

######  Definition 24.

Let $(E,\\{l^{(i)}\\})$ be a rank $2$ quasi-parabolic bundle with determinant
$L_{0}$. Let $L$ be a line subbundle of $E$. We define the
$\boldsymbol{w}$-stability index of $L$ to be the real number

$\mathrm{Stab}_{\boldsymbol{w}}(L):=\deg(E)-2\deg(L)+\sum_{i\in
I}w_{i}\left(m_{i}-2\,\mathrm{length}(l_{i}\cap L|_{m_{i}[t_{i}]})\right).$

######  Definition 25.

A rank $2$ quasi-parabolic bundle $(E,\\{l^{(i)}\\})$ is
$\boldsymbol{w}$-stable if for any subbundle $L\subset E$, the inequality
$\mathrm{Stab}_{\boldsymbol{w}}(L)>0$ holds.

We say that a quasi-parabolic bundle $(E,\\{l^{(i)}\\})$ is decomposable if
there exists a decomposition $E=L_{1}\oplus L_{2}$ such that
$l^{(i)}=l^{(i)}_{1}$ or $l^{(i)}=l^{(i)}_{2}$ for any $i\in I$, where we set
$l^{(i)}_{1}:=l^{(i)}\cap(L_{1}|_{m_{i}[t_{i}]})$ and
$l^{(i)}_{2}:=l^{(i)}\cap(L_{2}|_{m_{i}[t_{i}]})$. We say that
$(E,\\{l^{(i)}\\})$ is undecomposable if $(E,\\{l^{(i)}\\})$ is not
decomposable. A free $\mathcal{O}_{m_{i}[t_{i}]}$-submodule $l^{(i)}$ of
$E|_{m_{i}[t_{i}]}$ induces a one dimensional subspace
$l^{(i)}_{\mathrm{red}}$ of $E|_{t_{i}}$, that is the restriction of $l^{(i)}$
to $t_{i}$ (without multiplicity).

######  Lemma 26.

Let $(E,\\{l^{(i)}\\})$ be a rank $2$ quasi-parabolic bundle with determinant
$L_{0}$. If

  * (i)

$E$ is an extension of ${L_{0}}$ by $\mathcal{O}_{C}$ (when $g=0$, moreover we
assume that $(E,\\{l^{(i)}\\})$ is undecomposable)

  * (ii)

$\dim_{\mathbb{C}}H^{1}(C,E)=0$

  * (iii)

$l_{\text{red}}^{(i)}\not\in\mathcal{O}_{C}|_{t_{i}}\subset\mathbb{P}(E)$ for
any $i$,

then $(E,\\{l^{(i)}\\})$ is $\boldsymbol{w}$-stable.

###### Proof.

When $g=0$, we have this statement from [35, Proposition 46] by the condition
(4.1). When $g\geq 1$, we have that $E$ is stable, that is, $\deg(E)-2\deg(L)$
is a positive integer for any line subbundle $L\subset E$. This claim follows
from the same argument as in [44, Lemma 4.2]. Since $0<w_{i}\ll 1$ in (4.2),
we have that $\mathrm{Stab}_{\boldsymbol{w}}(L)>0$. ∎

Let $P_{(C,D)}^{\boldsymbol{w}}$ be a moduli space of $\boldsymbol{w}$-stable
quasi-parabolic bundles constructed in [54]. Let
$P_{(C,D)}^{\boldsymbol{w}}(L_{0})$ be the fiber of $L_{0}$ under the
determinant map

$P_{(C,D)}^{\boldsymbol{w}}\longrightarrow\mathrm{Pic}_{C}^{2g-1};\quad(E,\\{l^{(i)}\\})\longmapsto\det(E).$

We set

$P_{(C,D)}(L_{0})_{0}:=\left\\{(E,\\{l^{(i)}\\})\ \middle|\
\begin{array}[]{l}\text{$(E,\\{l^{(i)}\\})$ is rank $2$ quasi-parabolic bundle
over $(C,D)$ such that}\\\ \text{(i) $\det(E)\cong L_{0}$,\quad(ii) $E$ is an
extension of ${L_{0}}$ by $\mathcal{O}_{C}$,}\\\ \text{(iii)
$\dim_{\mathbb{C}}H^{1}(C,E)=0$,\quad(iv)
$l_{\text{red}}^{(i)}\not\in\mathcal{O}_{C}|_{t_{i}}\subset\mathbb{P}(E)$ for
any $i$,}\\\ \text{(v) $(E,\\{l^{(i)}\\})$ is undecomposable (when
$g=0$)}\end{array}\right\\}.$

By Lemma 26, we have an inclusion

$P_{(C,D)}(L_{0})_{0}\subset P_{(C,D)}^{\boldsymbol{w}}(L_{0}).$

For $(E,\\{l^{(i)}\\})\in P_{(C,D)}(L_{0})_{0}$, we have an extension

(4.3) $0\longrightarrow\mathcal{O}_{C}\longrightarrow E\longrightarrow
L_{0}\longrightarrow 0.$

Since $\dim_{\mathbb{C}}H^{1}(C,E)=0$, we have that
$\dim_{\mathbb{C}}H^{0}(C,E)=1$. So the injection
$\mathcal{O}_{C}\xrightarrow{\subset}E$ in (4.3) is unique up to a constant.

######  Definition 27.

Let $(E,\\{l^{(i)}\\})\in P_{(C,D)}(L_{0})_{0}$. We take an affine open
covering $\\{U_{\alpha}\\}_{\alpha}$ of $C$, i.e.
$C=\bigcup_{\alpha}U_{\alpha}$. Let
$\\{\varphi_{\alpha}^{\mathrm{Ext}}\\}_{\alpha}$ be trivializations
$\varphi_{\alpha}^{\mathrm{Ext}}\colon\mathcal{O}_{U_{\alpha}}^{\oplus
2}\rightarrow E|_{U_{\alpha}}$ of the underlying vector bundle $E$ such that

  * (i)

the composition

$\displaystyle\mathcal{O}_{U_{\alpha}}$
$\displaystyle\longrightarrow\mathcal{O}_{U_{\alpha}}^{\oplus
2}\xrightarrow{\varphi_{\alpha}^{\mathrm{Ext}}}E|_{U_{\alpha}}$ $\displaystyle
f$ $\displaystyle\longmapsto(f,0)$

is just the inclusion $\mathcal{O}_{C}\subset E$ of the extension (4.3) for
any $\alpha$, and

  * (ii)

the image of the composition

$\displaystyle\mathcal{O}_{U_{\alpha}}$
$\displaystyle\longrightarrow\mathcal{O}_{U_{\alpha}}^{\oplus
2}\xrightarrow{\varphi_{\alpha}^{\mathrm{Ext}}}E|_{U_{\alpha}}\longrightarrow
E|_{m_{i}[t_{i}]}$ $\displaystyle f$ $\displaystyle\longmapsto(0,f)$

generates the submodule $l^{(i)}\subset E|_{m_{i}[t_{i}]}$ for each $i$ and
$\alpha$ where $t_{i}\in U_{\alpha}$.

Notice that the claim that we may take $\varphi_{\alpha}^{\mathrm{Ext}}$ which
satisfies the condition (ii) of Definition 27 follows from the condition that
$l_{\text{red}}^{(i)}\not\in\mathcal{O}_{C}|_{t_{i}}\subset\mathbb{P}(E)$ for
any $i$.

Now we define a map

(4.4) $P_{(C,D)}(L_{0})_{0}\longrightarrow\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))$

as follows. Let $\\{\varphi_{\alpha}^{\mathrm{Ext}}\\}_{\alpha}$ be the
trivializations in Definition 27. We have the transition matrices

$B_{\alpha\beta}:=(\varphi_{\alpha}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}\circ\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\colon\mathcal{O}_{U_{\alpha\beta}}^{\oplus
2}\longrightarrow\mathcal{O}_{U_{\alpha\beta}}^{\oplus 2}.$

We represent $B_{\alpha\beta}$ as a matrix:

(4.5) $B_{\alpha\beta}=\begin{pmatrix}1&b^{12}_{\alpha\beta}\\\
0&b^{22}_{\alpha\beta}\end{pmatrix}.$

Remark that $\\{b^{22}_{\alpha\beta}\\}_{\alpha\beta}$ is a multiplicative
cocycle which defines the fixed line bundle $L_{0}$. We take a meromorphic
coboundary

(4.6) $\\{b^{22}_{\alpha}\\}_{\alpha}\quad\left(\text{where
}b^{22}_{\alpha\beta}=\frac{b^{22}_{\alpha}}{b^{22}_{\beta}}\right)$

of the multiplicative cocycle $\\{b^{22}_{\alpha\beta}\\}_{\alpha\beta}$. By
using the coboundary $\\{b^{22}_{\alpha}\\}_{\alpha}$, we define a cocycle

(4.7) $b^{\text{Bun}}_{\alpha\beta}:=b^{12}_{\alpha\beta}b^{22}_{\alpha},$

which gives a class $[\\{b^{\text{Bun}}_{\alpha\beta}\\}]\in
H^{1}(C,L_{0}^{-1}(-D))$. Then we have a map (4.4):

$(E,\\{l^{(i)}\\})\longmapsto\overline{[\\{b^{\text{Bun}}_{\alpha\beta}\\}]}.$

###  4.2. Moduli space of quasi-parabolic connections with fixed trace
connection

Now we recall the moduli space of quasi-parabolic connections. We fix an
irregular curve with residues
$X=(C,D,\\{z_{i}\\},\\{\boldsymbol{\theta}_{i}\\},\boldsymbol{\theta}_{\mathrm{res}})$
defined in Definition 2. Moreover we assume that

(4.8) $\sum_{i\in I}\theta_{i,-1}^{-}\neq 0.$

Let
$({L_{0}},\nabla_{L_{0}}\colon{L_{0}}\rightarrow{L_{0}}\otimes\Omega^{1}_{C}(D))$
be a rank $1$ connection on $C$ with degree $2g-1$ such that the polar part of
$\nabla_{L_{0}}$ at $t_{i}$ is $\mathrm{tr}(\omega_{i}(X))$.

######  Definition 28.

We say $(E,\nabla,\lambda,\\{l^{(i)}\\})$ a rank $2$ quasi-parabolic
$\lambda$-connection over $X$ with fixed trace connection
$({L_{0}},\nabla_{L_{0}})$ if

  * (i)

$E$ is a rank $2$ vector bundle on $C$ with $\det(E)\cong L_{0}$,

  * (ii)

$\lambda\in\mathbb{C}$ and $\nabla\colon E\rightarrow
E\otimes\Omega^{1}_{C}(D)$ is a $\lambda$-connection that is,
$\nabla(fs)=\lambda s\otimes df+f\nabla(s)$ for any $f\in\mathcal{O}_{C}$ and
$s\in E$, and

  * (iii)

$\nabla(s_{1})\wedge
s_{2}+s_{1}\wedge\nabla(s_{2})=\lambda\nabla_{L}(s_{1}\wedge s_{2})$ for
$s_{1},s_{2}\in E$,

  * (iv)

$E|_{m_{i}[t_{i}]}\supset l^{(i)}\supset 0$ is a filtration by free
$\mathcal{O}_{m_{i}[t_{i}]}$-modules such that, for any $i\in I$,

    * –

$E|_{m_{i}[t_{i}]}/l^{(i)}\cong\mathcal{O}_{m_{i}[t_{i}]}$ and
$l^{(i)}\cong\mathcal{O}_{m_{i}[t_{i}]}$,

    * –

$\nabla|_{m_{i}[t_{i}]}(l^{(i)})\subset l^{(i)}\otimes\Omega^{1}_{C}(D)$, and

    * –

the image of $(E|_{m_{i}[t_{i}]}/l^{(i)})\oplus l^{(i)}$ under
$\mathrm{Gr}_{i}(\nabla)-\lambda\cdot\omega_{i}(X)$ is contained in

$\left((E|_{m_{i}[t_{i}]}/l^{(i)})\oplus l^{(i)}\right)\otimes\Omega^{1}_{C}.$

Here $\mathrm{Gr}_{i}(\nabla)$ is the induced morphism

$\mathrm{Gr}_{i}(\nabla)\colon(E|_{m_{i}[t_{i}]}/l^{(i)})\oplus
l^{(i)}\longrightarrow\left((E|_{m_{i}[t_{i}]}/l^{(i)})\oplus
l^{(i)}\right)\otimes\Omega^{1}_{C}(D).$

Notice that, if $\lambda=0$, then $\nabla$ is an $\mathcal{O}_{C}$-morphism,
which is called a Higgs field. So $(E,\nabla,\lambda,\\{l^{(i)}\\})$ is called
a (trace free) quasi-parabolic Higgs bundle when $\lambda=0$. We consider only
rank $2$ quasi-parabolic $\lambda$-connections
$(E,\nabla,\lambda,\\{l^{(i)}\\})$ over $X$ with $({L_{0}},\nabla_{L_{0}})$
such that the underlying quasi-parabolic bundle $(E,\\{l^{(i)}\\})$ is in the
moduli space $P_{(C,D)}(L_{0})_{0}$.

We define the moduli spaces $\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}$
and $M_{X}(L_{0},\nabla_{L_{0}})_{0}$ as follows:

$\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}=\left.\left\\{\begin{array}[]{l}(E,\nabla,\lambda,\\{l^{(i)}\\})\\\
\text{quasi-parabolic $\lambda$-connection}\\\ \text{over $X$ with trace
$({L_{0}},\nabla_{L_{0}})$}\end{array}\ \middle|\
\begin{array}[]{l}(E,\\{l^{(i)}\\})\in
P_{(C,D)}(L_{0})_{0}\end{array}\right\\}\right/\cong$

and

$M_{X}(L_{0},\nabla_{L_{0}})_{0}=\left.\left\\{\begin{array}[]{l}(E,\nabla,\lambda,\\{l^{(i)}\\})\\\
\text{quasi-parabolic $\lambda$-connection}\\\ \text{over $X$ with trace
$({L_{0}},\nabla_{L_{0}})$}\end{array}\ \middle|\
\begin{array}[]{l}(E,\\{l^{(i)}\\})\in P_{(C,D)}(L_{0})_{0}\\\ \text{and
}\lambda\neq 0\end{array}\right\\}\right/\cong.$

###  4.3. Maps from the moduli space

Now we describe two maps: the forgetful map $\pi_{\mathrm{Bun}}$ forgetting
connections and the apparent map $\pi_{\mathrm{App}}$. First we consider the
composition

$\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}\longrightarrow
P_{(C,D)}(L_{0})_{0}\longrightarrow\mathbb{P}H^{1}(C,L_{0}^{-1}(-D)).$

Here the first map is the forgetful map, and the second map is (4.4). We
denote by

$\pi_{\mathrm{Bun}}\colon\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}\longrightarrow\mathbb{P}H^{1}(C,L_{0}^{-1}(-D)).$

the composition.

Second we define a map

(4.9)
$\pi_{\mathrm{App}}\colon\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}\longrightarrow\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))$

as follows. Let $(E,\nabla,\lambda,\\{l^{(i)}\\})$ be a point on
$\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}$. Let
$\\{\varphi_{\alpha}^{\mathrm{Ext}}\\}_{\alpha}$ be the trivializations in
Definition 27. Let $A_{\alpha}$ be the connection matrix of the
$\lambda$-connection $\nabla$ with respect to
$\varphi_{\alpha}^{\mathrm{Ext}}$, that is,

$\lambda\operatorname{d}+A_{\alpha}:=(\varphi_{\alpha}^{\mathrm{Ext}})^{-1}\circ\nabla\circ\varphi_{\alpha}^{\mathrm{Ext}}\colon\mathcal{O}_{U_{\alpha}}^{\oplus
2}\longrightarrow(\Omega^{1}_{U_{\alpha}}(D))^{\oplus 2}.$

We denote the matrix $A_{\alpha}$ as follows:

(4.10) $A_{\alpha}=\begin{pmatrix}a_{\alpha}^{11}&a^{12}_{\alpha}\\\
a^{21}_{\alpha}&a^{22}_{\alpha}\end{pmatrix}.$

By the condition (ii) in Definition 27 and the condition (iv) in Definition
28, the polar part of the connection matrix $A_{\alpha}$ at $t_{i}$ is a lower
triangular matrix, that is, the Laurent expansion of $A_{\alpha}$ at $t_{i}$
is as follows:

(4.11) $A_{\alpha}=\begin{pmatrix}\lambda\nu_{i}^{-}&0\\\
*&\lambda\nu_{i}^{+}\end{pmatrix}\frac{1}{z_{i}^{m_{i}}}+[\text{ holo. part
}].$

Here $\nu_{i}^{-},\nu_{i}^{+}\in\Omega^{1}_{C}(D)|_{m_{i}[t_{i}]}$ are defined
so that

$\lambda\cdot\omega_{i}(X)=\begin{pmatrix}\lambda\nu_{i}^{-}&0\\\
0&\lambda\nu_{i}^{+}\end{pmatrix}.$

By using the coboundary $\\{b^{22}_{\alpha}\\}_{\alpha}$ in (4.6), we define
cocycles

(4.12) $a^{\text{App}}_{\alpha}:=a^{21}_{\alpha}(b^{22}_{\alpha})^{-1},$

which give a class $[\\{a^{\text{App}}_{\alpha}\\}]\in
H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))$. Then we have a map (4.9):

$(E,\nabla,\lambda,\\{l^{(i)}\\})\longmapsto\overline{[\\{a^{\text{App}}_{\alpha}\\}]}.$

Finally, we have a map

(4.13)
$(\pi_{\mathrm{App}},\pi_{\mathrm{Bun}})\colon\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}\longrightarrow\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))\times\mathbb{P}H^{1}(C,L_{0}^{-1}(-D)).$

We consider the natural pairing

(4.14) $H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))\times
H^{1}(C,L_{0}^{-1}(-D))\longrightarrow
H^{1}(C,\Omega^{1}_{C})\cong\mathbb{C}.$

######  Lemma 29.

Let
$(E,\nabla,\lambda,\\{l^{(i)}\\})\in\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}$.
Let $a^{\text{App}}_{\alpha}$ and $b^{\text{Bun}}_{\alpha\beta}$ be the
cocycles in (4.7) and in (4.12), respectively. Then we have

$[\\{b^{\mathrm{Bun}}_{\alpha\beta}\cdot
a^{\mathrm{App}}_{\beta}\\}]=\lambda\cdot\sum_{i\in I}\theta_{i,-1}^{-}.$

Here the left hand side is the pairing (4.14).

###### Proof.

Let $B_{\alpha\beta}$ be the transition function in (4.5). Let $A_{\alpha}$ be
the connection matrix in (4.10). Then we have

$\lambda\cdot\operatorname{d}\\!B_{\alpha\beta}+A_{\alpha}B_{\alpha\beta}=B_{\alpha\beta}A_{\beta}.$

By comparing the $(1,1)$-entries of the both hand sides, we have

$a_{\alpha}^{11}-a_{\beta}^{11}=b^{\mathrm{Bun}}_{\alpha\beta}\cdot
a^{\mathrm{App}}_{\beta}.$

By (4.11) and the isomorphism $H^{1}(C,\Omega^{1}_{C})\cong\mathbb{C}$, we
have $[\\{b^{\mathrm{Bun}}_{\alpha\beta}\cdot
a^{\mathrm{App}}_{\beta}\\}\\}]=\lambda\cdot\sum_{i}\theta_{i,-1}^{-}$. ∎

Set

$N_{0}:=\dim_{\mathbb{C}}\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))=3g+n-3.$

Let us introduce the homogeneous coordinates
$\boldsymbol{a}=(a_{0}:\cdots:a_{N_{0}})$ on
$\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))\cong\mathbb{P}_{\boldsymbol{a}}^{N_{0}}$
and the dual coordinates $\boldsymbol{b}=(b_{0}:\cdots:b_{N_{0}})$ on

$\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))\cong\mathbb{P}H^{0}(C,{L_{0}}\otimes\Omega^{1}_{C}(D))^{\vee}\cong\mathbb{P}_{\boldsymbol{b}}^{N_{0}}.$

Let
$\Sigma\subset\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}$
be the incidence variety whose defining equation is given by
$\sum_{j}a_{j}b_{j}=0$. By Lemma 29, we have that

$\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}\setminus
M_{X}({L_{0}},\nabla_{L_{0}})_{0}\xrightarrow{\
(\pi_{\mathrm{App}},\pi_{\mathrm{Bun}})\ }\Sigma.$

######  Remark 30.

Loray–Saito (for $g=0$) and Matsumoto (for $g\geq 1$) discussed on the
birationality of the map (4.13). They showed the birationality of the map
(4.13) when $D$ is a reduced effective divisor ([42, Theorem 4.3] for $g=0$
and [44, Theorem 4.5] for $g\geq 1$). In these cases, quasi-parabolic
connections have only simple poles. But we may apply the arguments in [42,
Theorem 4.3] and in [44, Theorem 4.5] to our cases where quasi-parabolic
connections admit generic unramified irregular singular points. So we can
reconstruct
$(E,\nabla,\lambda,\\{l^{(i)}\\})\in\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}$
from an element of

$\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))_{0}\times\mathbb{P}H^{0}(C,L_{0}\otimes\Omega^{1}_{C}(D)).$

Here we set

$\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))_{0}:=\left\\{b\in\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))\
\middle|\ \begin{array}[]{l}\text{{\rm The extension $E$ corresponding to
$b$}}\\\ \text{{\rm satisfies
$\dim_{\mathbb{C}}H^{1}(C,E)=0$}}\end{array}\right\\}.$

Then we have isomorphisms

$\widetilde{M}_{X}({L_{0}},\nabla_{L_{0}})_{0}\cong\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))_{0}\times\mathbb{P}H^{0}(C,L_{0}\otimes\Omega^{1}_{C}(D))$

and

$M_{X}({L_{0}},\nabla_{L_{0}})_{0}\cong\mathbb{P}H^{1}(C,L_{0}^{-1}(-D))_{0}\times\mathbb{P}H^{0}(C,L_{0}\otimes\Omega^{1}_{C}(D))\setminus\Sigma.$

###  4.4. Symplectic structure and explicit description

Now we recall the natural symplectic structure on
$M_{X}({L_{0}},\nabla_{L_{0}})_{0}$. We define a complex
${\mathcal{F}}_{0}^{\bullet}$ for $(E,\frac{1}{\lambda}\nabla,\\{l^{(i)}\\})$
by

$\displaystyle{\mathcal{F}}_{0}^{0}:=\left\\{s\in{\mathcal{E}}nd(E)\ \middle|\
\mathrm{tr}(s)=0,\,s|_{m_{i}t_{i}}(l^{(i)})\subset l^{(i)}\text{ for any
$i$}\right\\}$
$\displaystyle{\mathcal{F}}^{1}_{0}:=\left\\{s\in{\mathcal{E}}nd(E)\otimes\Omega^{1}_{C}(D)\
\middle|\ \mathrm{tr}(s)=0,\,s|_{m_{i}t_{i}}(l^{(i)})\subset
l^{(i)}\otimes\Omega^{1}_{C}\text{ for any $i$}\right\\}$
$\displaystyle\nabla_{{\mathcal{F}}^{\bullet}}\colon{\mathcal{F}}_{0}^{0}\longrightarrow{\mathcal{F}}_{0}^{1};\quad\nabla_{{\mathcal{F}}_{0}^{\bullet}}(s)=(\frac{1}{\lambda}\nabla)\circ
s-s\circ(\frac{1}{\lambda}\nabla).$

We define the following morphism

(4.15)
$\displaystyle{\mathbf{H}}^{1}({\mathcal{F}}_{0}^{\bullet})\otimes{\mathbf{H}}^{1}({\mathcal{F}}_{0}^{\bullet})\longrightarrow{\mathbf{H}}^{2}(\mathcal{O}_{C}\xrightarrow{d}\Omega_{C}^{1})\cong\mathbb{C}$

as in (3.2). This pairing gives the symplectic form on
$M_{X}({L_{0}},\nabla_{L_{0}})_{0}$. We denote by $\omega_{0}$ the symplectic
form.

The maps $\pi_{\mathrm{App}}$ and $\pi_{\mathrm{Bun}}$ give coordinates on
$M_{X}({L_{0}},\nabla_{L_{0}})_{0}$ (see Remark 30). Now we describe the
symplectic structure (4.15) by using the coordinates on
$M_{X}({L_{0}},\nabla_{L_{0}})_{0}$. We define a 1-form $\eta$ on
$\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}$
as follows:

$\eta:=\left(-\sum_{i}\theta_{i,-1}^{-}\right)\cdot\frac{a_{0}\,db_{0}+a_{1}\,db_{1}+\cdots+a_{N_{0}}\,db_{N_{0}}}{a_{0}b_{0}+a_{1}b_{1}+\cdots+a_{N_{0}}b_{N_{0}}}.$

######  Theorem 31.

Assume that $\sum_{i\in I}\theta_{i,-1}^{-}\neq 0$. Let
$\omega_{\boldsymbol{a},\boldsymbol{b}}$ be the $2$-form on
$\mathbb{P}^{N_{0}}_{\boldsymbol{a}}\times\mathbb{P}^{N_{0}}_{\boldsymbol{b}}$
defined by $\omega_{\boldsymbol{a},\boldsymbol{b}}=d\eta$. The pull-back of
$\omega_{\boldsymbol{a},\boldsymbol{b}}$ under the map

$M_{X}({L_{0}},\nabla_{L_{0}})_{0}\xrightarrow{\
(\pi_{\mathrm{App}},\pi_{\mathrm{Bun}})\
}\mathbb{P}_{\boldsymbol{a}}^{N_{0}}\times\mathbb{P}_{\boldsymbol{b}}^{N_{0}}$

coincides with the symplectic form $\omega_{0}$ on
$M_{X}({L_{0}},\nabla_{L_{0}})_{0}$.

###### Proof.

Let $v,v^{\prime}\in
T_{(E,\frac{1}{\lambda}\nabla,\\{l^{(i)}\\})}M_{X}({L_{0}},\nabla_{L_{0}})_{0}$.
We have the isomorphism

$T_{(E,\frac{1}{\lambda}\nabla,\\{l^{(i)}\\})}M_{X}({L_{0}},\nabla_{L_{0}})_{0}\xrightarrow{\
\cong\ }{\mathbf{H}}^{1}({\mathcal{F}}_{0}^{\bullet}).$

Let $u_{\alpha\beta}(v)$ and $v_{\alpha}(v)$ be cocycles such that the class
$[\\{u_{\alpha\beta}(v)\\}_{\alpha\beta},\\{v_{\alpha}(v)\\}_{\alpha}]$ is the
image of $v$ under the isomorphism. We calculate $u_{\alpha\beta}(v)$ and
$v_{\alpha}(v)$ by using the trivialization
$\\{\varphi_{\alpha}^{\mathrm{Ext}}\\}_{\alpha}$ as follows:

(4.16) $\displaystyle u_{\alpha\beta}(v)$
$\displaystyle=\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\left(B_{\alpha\beta}^{-1}v(B_{\alpha\beta})\right)\circ(\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}$
$\displaystyle=\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\begin{pmatrix}0&v(b^{12}_{\alpha\beta})\\\
0&0\end{pmatrix}\circ(\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}$
$\displaystyle=\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}}\circ\begin{pmatrix}0&\frac{v(b^{\text{Bun}}_{\alpha\beta})}{b^{22}_{\alpha}}\\\
0&0\end{pmatrix}\circ(\varphi_{\beta}^{\mathrm{Ext}}|_{U_{\alpha\beta}})^{-1}$

and

(4.17) $\displaystyle v_{\alpha}(v)$
$\displaystyle=\varphi_{\alpha}^{\mathrm{Ext}}\circ
v\left(\frac{1}{\lambda}A_{\alpha}\right)\circ(\varphi_{\alpha}^{\mathrm{Ext}})^{-1}$
$\displaystyle=\varphi_{\alpha}^{\mathrm{Ext}}\circ\begin{pmatrix}v(a_{\alpha}^{11}/\lambda)&v(a^{12}_{\alpha}/\lambda)\\\
v(a^{21}_{\alpha}/\lambda)&v(a^{22}_{\alpha}/\lambda)\end{pmatrix}\circ(\varphi_{\alpha}^{\mathrm{Ext}})^{-1}$