# Infinitesimal deformations of parabolic connections and parabolic opers

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>,
Sorin Dumitrescu  Université Côte d’Azur, CNRS, LJAD, France
<EMAIL_ADDRESS>,  Sebastian Heller  Institute of Differential Geometry,
Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover
<EMAIL_ADDRESS>and  Christian Pauly  Université Côte d’Azur, CNRS, LJAD,
France<EMAIL_ADDRESS>
###### Abstract.

We compute the infinitesimal deformations of quadruples of the form

$(X,\,S,\,E_{*},\,D),$

where $(X,\,S)$ is a compact Riemann surface with $n$ marked points, $E_{*}$
is a parabolic vector bundle on $X$ with parabolic structure over $S$, and $D$
is a parabolic connection on $E_{*}$. Using it we compute the infinitesimal
deformations of $(X,\,S,\,D)$, where $D$ is a parabolic ${\rm
SL}(r,{\mathbb{C}})$–oper on $(X,\,S)$. It is shown that the monodromy map,
from the moduli space of triples $(X,\,S,\,D)$, where $D$ is a parabolic ${\rm
SL}(r,{\mathbb{C}})$–oper on $(X,\,S)$, to the ${\rm
SL}(r,{\mathbb{C}})$–character variety of $X\setminus S$, is an immersion.

###### Key words and phrases:

Parabolic oper, logarithmic Atiyah bundle, isomonodromy, monodromy map

###### 2010 Mathematics Subject Classification:

14H60, 33C80, 53C07.

###### Contents

  1. 1 Introduction
  2. 2 Holomorphic connections and the Atiyah bundle
  3. 3 Infinitesimal deformations and isomonodromy
  4. 4 Infinitesimal deformations of parabolic opers
  5. 5 Monodromy of parabolic opers
  6. A Parabolic opers

##  1\. Introduction

Opers were introduced by Beilinson and Drinfeld [BD1], [BD2]. They have turned
out to be very important in diverse topics, for example in geometric Langlands
correspondence, nonabelian Hodge theory, some branches of mathematics physics,
differential equations et cetera; see [BF], [DFK+], [FT], [FG1], [FG2], [CS],
[Fr1], [Fr2], [KSZ], [MR], [BSY] and references therein.

Parabolic vector bundles were introduced by Mehta and Seshadri in [MS]. In
[BDP], ${\rm SL}(r,{\mathbb{C}})$–opers in the set-up of parabolic vector
bundles were introduced. The aim here is to further investigate the ${\rm
SL}(r,{\mathbb{C}})$–opers in the context of parabolic vector bundles.

Being inspired by the interesting work [Sa], we study the infinitesimal
deformations, and the monodromy map, of the parabolic ${\rm
SL}(r,{\mathbb{C}})$–opers. Computation of the infinitesimal deformations of
parabolic opers entails computation of the infinitesimal deformations of
quadruples of the form $(X,\,S,\,E_{*},\,D)$, where $(X,\,S)$ is a compact
Riemann surface with $n$ marked points, $E_{*}$ is a parabolic vector bundle
on $X$ with parabolic structure over $S$, and $D$ is a parabolic connection on
$E_{*}$.

We introduce the parabolic analog of the Atiyah bundle and the Atiyah exact
sequence. Given a parabolic vector bundle $E_{*}$ on $(X,\,S)$, its Atiyah
bundle $\text{At}(E_{*})$ fits in the short exact sequence of holomorphic
vector bundles

$0\,\longrightarrow\,\text{End}^{P}(E_{*})\,\longrightarrow\,{\rm
At}(E_{*})\,\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,0\,,$
(1.1)

where $\text{End}^{P}(E_{*})$ denotes the sheaf of quasiparabolic flag
preserving endomorphisms of $E_{*}$ (see (2.2) for the quasiparabolic flags);
the sequence in (1.1) is the Atiyah exact sequence in the set-up of parabolic
bundles (see Definition 2.3). We show the following:

  1. (1)

A holomorphic splitting of (1.1) produces a logarithmic connection on the
holomorphic vector bundle $E$ underlying $E_{*}$ such that the residues
preserve the quasiparabolic flags of $E_{*}$, and

  2. (2)

a parabolic connection on $E_{*}$ is a holomorphic splitting of (1.1) such
that the eigenvalues of the residues are given by the parabolic weights (see
(2.3) for the parabolic weights).

(See Lemma 2.4 and Corollary 2.5.)

We prove the following (see Lemma 3.1):

The infinitesimal deformations of the triple $(X,\,S,\,E_{*})$ are
parametrized by $H^{1}(X,\,{\rm At}(E_{*}))$.

Now let $D$ be a connection on the parabolic vector bundle $E_{*}$ over
$(X,\,S)$. We assume that the local monodromy of $D$ around each point of $S$
is semisimple (meaning diagonalizable). Let

$\mathcal{D}_{0}\,:\,\text{End}^{P}(E_{*})\,\longrightarrow\,\text{End}^{n}(E_{*})\otimes
K_{X}$

be the corresponding logarithmic connection on $\text{End}^{P}(E_{*})$, where
$\text{End}^{n}(E_{*})\,\subset\,\text{End}^{P}(E_{*})$ is the sheaf of
endomorphisms nilpotent with respect to the quasiparabolic flags of $E_{*}$.
We show that this operator $\mathcal{D}_{0}$ extends to a holomorphic
differential operator

$\mathcal{D}\,:\,{\rm
At}(E_{*})\,\longrightarrow\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

(see (2.24)). Let ${\mathcal{B}}_{\bullet}$ denote the following two-term
complex of sheaves on $X$

${\mathcal{B}}_{\bullet}\,\,:\,\,{\mathcal{B}}_{0}\,=\,{\rm
At}(E_{*})\,\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\,{\mathcal{B}}_{1}\,=\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$

where ${\mathcal{B}}_{i}$ is at the $i$-th position.

We prove the following (see Lemma 3.4):

The infinitesimal deformations of the quadruple $(X,\,S,\,E_{*},\,D)$ are
parametrized by the first hypercohomology
${\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})$.

Now assume that $D$ is a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper (the
definition of a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper is recalled in
Section 4). Since $D$ is a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper, the
local monodromy of $D$ around each point of $S$ is semisimple (see Lemma 4.2).
In (4.7) and (4.9) we construct the holomorphic vector bundles
$\text{At}_{X}(r)$ and
$\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ respectively
on $X$. Using $D$ we construct a differential operator

${\mathcal{D}}_{B}\,:\,\text{At}_{X}(r)\,\longrightarrow\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

(see (4.13)). Let ${\mathcal{C}}_{\bullet}$ denote the following two-term
complex of sheaves on $X$

${\mathcal{C}}_{\bullet}\,\,:\,\,{\mathcal{C}}_{0}\,=\,\text{At}_{X}(r)\,\stackrel{{\scriptstyle\mathcal{D}_{B}}}{{\longrightarrow}}\,{\mathcal{C}}_{1}\,=\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$

where ${\mathcal{C}}_{i}$ is at the $i$-th position.

We prove the following (see Theorem 4.5):

The space of all infinitesimal deformation of the triple $(X,\,S,\,D)$, where
$D$ is a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper on $(X,\,S)$, is given by
the hypercohomology ${\mathbb{H}}^{1}({\mathcal{C}}_{\bullet})$.

A reformulation of the above result is proved in Corollary 4.6.

In Theorem 5.1 we prove the following:

The monodromy map from the moduli space of triples $(X,\,S,\,D)$, where $D$ is
a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper on $(X,\,S)$, to the ${\rm
SL}(r,{\mathbb{C}})$–character variety for $X\setminus S$ is an immersion.

The isomonodromy condition defines a holomorphic foliation on the moduli space
of quadruples $(X,\,S,\,E_{*},\,D)$, where $D$ is a connection on the
parabolic vector bundle $E_{*}$ over $(X,\,S)$. The proof of Theorem 5.1
involves computing this foliation. This computation is carried out in Lemma
3.5.

In the appendix we give an alternative definition of a parabolic ${\rm
SL}(r,{\mathbb{C}})$–oper in terms of $\mathbb{R}$-filtered sheaves as
introduced and studied by Maruyama and Yokogawa. This definition is
conceptually closer to the definition of an ordinary ${\rm
SL}(r,{\mathbb{C}})$–oper and clarifies the one given in [BDP].

##  2\. Holomorphic connections and the Atiyah bundle

###  2.1. Atiyah bundle for parabolic bundles

Let $X$ be a compact connected Riemann surface. Fix a finite subset of $n$
distinct points

$S\,:=\,\\{x_{1},\,\cdots,\,x_{n}\\}\,\subset\,X.$ (2.1)

The reduced effective divisor $x_{1}+\ldots+x_{n}$ on $X$ will also be denoted
by $S$.

A quasiparabolic structure on a holomorphic vector bundle $E$ on $X$ is a
filtration of subspaces of the fiber $E_{x_{i}}$ of $E$ over $x_{i}$

$E_{x_{i}}\,=\,E_{i,1}\,\supset\,E_{i,2}\,\supset\,\cdots\,\supset\,E_{i,l_{i}}\,\supset\,E_{i,l_{i}+1}\,=\,0$
(2.2)

for every $1\,\leq\,i\,\leq\,n$. A parabolic structure on $E$ is a
quasiparabolic structure as above together with a string of real numbers

$0\,\leq\,\alpha_{i,1}\,<\,\alpha_{i,2}\,<\,\cdots\,<\,\alpha_{i,l_{i}}\,<\,1$
(2.3)

for every $1\,\leq\,i\,\leq\,n$. The above number $\alpha_{i,j}$ is called the
parabolic weight of the subspace $E_{i,j}$ in (2.2). The divisor $S$ is known
as the parabolic divisor. (See [MS], [MY], [Bh].)

A parabolic vector bundle is a holomorphic vector bundle $E$ with a parabolic
structure $(\\{E_{i,j}\\},\,\\{\alpha_{i,j}\\})$. For notational convenience,
$(E,\,(\\{E_{i,j}\\},\,\\{\alpha_{i,j}\\}))$ will be denoted by $E_{*}$.

######  Assumption 2.1.

Throughout we will work with the assumption that all the parabolic weights
$\alpha_{i,j}$ are rational numbers. We assume that each $x_{i}$ has at least
one nonzero parabolic weight. We also assume that $3({\rm
genus}(X)-1)+n\,>\,0$.

For any $1\,\leq\,i\,\leq\,n$ and $1\,\leq\,j\,\leq\,l_{i}+1$, let
${\mathcal{E}}_{i,j}\,\longrightarrow\,X$ be the holomorphic vector bundle
defined by the following short exact sequence of coherent analytic sheaves on
$X$:

$0\,\longrightarrow\,{\mathcal{E}}_{i,j}\,\longrightarrow\,E\,\longrightarrow\,E_{x_{i}}/E_{i,j}\,\longrightarrow\,0$
(2.4)

(see (2.2)). Since $E_{x_{i}}/E_{i,j}$ is supported on $x_{i}$, the subsheaf
${\mathcal{E}}_{i,j}$ of $E$ actually coincides with $E$ over the open subset
$X\setminus\\{x_{i}\\}\,\subset\,X$.

The space of all holomorphic sections, over an open subset $U\,\subset\,X$, of
a holomorphic vector bundle $V\,\longrightarrow\,X$ is denoted by
$\Gamma(U,\,V)$.

Let

$\text{End}^{P}(E_{*})\,\subset\,\text{End}(E)$ (2.5)

be the coherent analytic subsheaf such that for any open subset
$U\,\subset\,X$, the subspace

$\Gamma\left(U,\,\text{End}^{P}(E_{*})\right)\,\subset\,\Gamma(U,\,\text{End}(E))$

consists of all ${\mathcal{O}}_{U}$–linear homomorphisms
$s\,:\,E\big{|}_{U}\,\longrightarrow\,E\big{|}_{U}$ satisfying the condition
that

$s({\mathcal{E}}_{i,j}\big{|}_{U})\,\subset\,{\mathcal{E}}_{i,j}\big{|}_{U}$

for all $x_{i}\,\in\,U$ and all $1\,\leq\,j\,\leq\,l_{i}$, where
${\mathcal{E}}_{i,j}$ is defined in (2.4).

Let

$\text{End}^{n}(E_{*})\,\subset\,\text{End}^{P}(E_{*})$ (2.6)

be the coherent analytic subsheaf consists of all ${\mathcal{O}}_{U}$–linear
homomorphisms $s\,:\,E\big{|}_{U}\,\longrightarrow\,E\big{|}_{U}$ satisfying
the condition that for any open subset $U\,\subset\,X$,

$s({\mathcal{E}}_{i,j}\big{|}_{U})\,\subset\,{\mathcal{E}}_{i,j+1}\big{|}_{U}$

for all $x_{i}\,\in\,U$ and all $1\,\leq\,j\,\leq\,l_{i}$.

######  Remark 2.2.

It is customary to define $\text{End}^{P}(E_{*})$ as the subsheaf of
$\text{End}(E)$ that preserves the subspace $E_{i,j}\,\subset\,E_{x_{i}}$ for
all $x_{i}\,\in\,U$ and all $1\,\leq\,j\,\leq\,l_{i}$. Similarly,
$\text{End}^{n}(E_{*})$ is defined to be the subsheaf of
$\text{End}^{P}(E_{*})$ that takes any $E_{i,j}$ to $E_{i,j+1}$. While the
definitions in (2.5) and (2.6) are equivalent to these, we will see that the
definitions in (2.5) and (2.6) are more useful for our purpose.

Using the pairing $(\text{End}(E)\otimes{\mathcal{O}}(S))^{\otimes
2}\,\longrightarrow\,{\mathcal{O}}(2S)$ defined by trace

$A\otimes B\,\longmapsto\,\text{trace}(AB)\,,$

we have

$\text{End}^{P}(E_{*})^{*}\,=\,\text{End}^{n}(E_{*})\otimes{\mathcal{O}}(S)\,.$

For any integer $k\,\geq\,0$, let ${\rm Diff}^{k}(E,\,E)$ be the holomorphic
vector bundle on $X$ given by the sheaf of all holomorphic differential
operators, of order at most $k$, from $E$ to itself. We have the following
short exact sequence of holomorphic vector bundles on $X$

$0\,\longrightarrow\,{\rm
Diff}^{0}(E,\,E)\,=\,\text{End}_{{\mathcal{O}}_{X}}(E)\,\longrightarrow\,{\rm
Diff}^{1}(E,\,E)$ (2.7)
$\stackrel{{\scriptstyle\sigma_{0}}}{{\longrightarrow}}\,\text{End}_{{\mathcal{O}}_{X}}(E)\otimes
TX\,=\,E\otimes E^{*}\otimes TX\,\longrightarrow\,0\,,$

where $TX$ is the holomorphic tangent bundle of $X$ and $\sigma_{0}$ is the
symbol map on the first order differential operators. Let

${\rm Diff}^{1}_{P}(E,\,E)\,\subset\,{\rm Diff}^{1}(E,\,E)$ (2.8)

be the coherent analytic subsheaf consists of all differential operators
$D_{U}\,:\,E\big{|}_{U}\,\longrightarrow\,E\big{|}_{U}$, where $U\,\subset\,X$
is any open subset, satisfying the condition that for any
$s\,\in\,\Gamma(U,\,{\mathcal{E}}_{i,j})$,

$D_{U}(s)\,\in\,\Gamma(U,\,{\mathcal{E}}_{i,j})$

for all $x_{i}\,\in\,U$ and all $1\,\leq\,j\,\leq\,l_{i}$, where
${\mathcal{E}}_{i,j}$ is constructed in (2.4).

Comparing the definitions given in (2.8) and (2.5) we conclude that

${\rm
Diff}^{1}_{P}(E,\,E)\bigcap\text{End}_{{\mathcal{O}}_{X}}(E)\,=\,\text{End}^{P}(E_{*})\,;$
(2.9)

the above intersection takes place inside ${\rm Diff}^{1}(E,\,E)$ (see (2.7)
and (2.8)). Consider the subsheaf

$TX\otimes{\mathcal{O}}_{X}(-S)\,\subset\,TX\,=\,{\rm Id}_{E}\otimes
TX\,\subset\,\text{End}(E)\otimes TX\,,$ (2.10)

where $S$ is the divisor in (2.1). Define

${\rm At}(E_{*})\,:=\,{\rm
Diff}^{1}_{P}(E,\,E)\bigcap\sigma^{-1}_{0}(TX\otimes{\mathcal{O}}_{X}(-S))\,\subset\,{\rm
Diff}^{1}(E,\,E)\,,$ (2.11)

where $\sigma_{0}$ is the projection in (2.7) and ${\rm
Diff}^{1}_{P}(E,\,E)\,\subset\,{\rm Diff}^{1}(E,\,E)$ is the subsheaf in
(2.8). Now from (2.7) and (2.9) we get the following short exact sequence of
holomorphic vector bundles on $X$

$0\,\longrightarrow\,\text{End}^{P}(E_{*})\,\longrightarrow\,{\rm
At}(E_{*})\,\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,0\,,$
(2.12)

where $\sigma$ is the restriction of $\sigma_{0}$ to ${\rm
At}(E_{*})\,\subset\,{\rm Diff}^{1}(E,\,E)$. It is straightforward to check
that the homomorphism $\sigma$ in (2.12) is surjective; indeed, this follows
immediately from the fact that
$TX\otimes{\mathcal{O}}_{X}(-S)\,\subset\,\sigma_{0}({\rm
Diff}^{1}_{P}(E,\,E))$.

######  Definition 2.3.

The vector bundle ${\rm At}(E_{*})$ in (2.11) will be called the Atiyah bundle
for the parabolic bundle $E_{*}$, and the sequence in (2.12) will be called
the Atiyah exact sequence for the parabolic bundle $E_{*}$.

When $S$ is the zero divisor (meaning $n\,=\,0$ in (2.1)), then ${\rm
At}(E_{*})$ is the usual Atiyah bundle $\text{At}(E)$ for $E$, and (2.12) is
the usual Atiyah exact sequence for $E$. (See [At].)

###  2.2. Holomorphic connections on a parabolic bundle

The holomorphic cotangent bundle of $X$ will be denoted by $K_{X}$.

Let $V$ be a holomorphic vector bundle on $X$. A logarithmic connection on $V$
singular over $S$ is a holomorphic differential operator of order one

$D\,:\,V\,\longrightarrow\,V\otimes K_{X}\otimes{\mathcal{O}}_{X}(S)$

satisfying the Leibniz identity, which says that

$D(fs)\,=\,fD(s)+s\otimes df$ (2.13)

for any locally defined holomorphic function $f$ on $X$ and any locally
defined holomorphic section $s$ of $V$.

We note that any logarithmic connection on $V$ is flat because
$\Omega^{2,0}_{X}\,=\,0$.

Take a point $x_{i}\,\in\,S$. The fiber of $K_{X}\otimes{\mathcal{O}}_{X}(S)$
over $x_{i}$ is identified with $\mathbb{C}$ by the Poincaré adjunction
formula [GH, p. 146]. To explain this isomorphism

$(K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\mathbb{C}}\,,$
(2.14)

let $z$ be a holomorphic coordinate function on $X$ defined on an analytic
open neighborhood of $x_{i}$ such that $z(x_{i})\,=\,0$. Then we have the
isomorphism
${\mathbb{C}}\,\longrightarrow\,(K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}$
that sends any $c\,\in\,\mathbb{C}$ to
$c\cdot\frac{dz}{z}(x_{i})\,\in\,(K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}$.
It is straightforward to check that this map
${\mathbb{C}}\,\longrightarrow\,(K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}$ is
actually independent of the choice of the above holomorphic coordinate
function $z$.

Let $D_{V}\,:\,V\,\longrightarrow\,V\otimes K_{X}\otimes{\mathcal{O}}_{X}(S)$
be a logarithmic connection on $V$. Using the Leibniz identity in (2.13) it is
straightforward to deduce that the composition of homomorphisms

$V\,\xrightarrow{\,\ D_{V}\,\ }\,V\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\longrightarrow\,(V\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,V_{x_{i}}$
(2.15)

is ${\mathcal{O}}_{X}$–linear; the above isomorphism $(V\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,V_{x_{i}}$
is given by the isomorphism in (2.14). Therefore, the composition of
homomorphisms in (2.15) produces a $\mathbb{C}$–linear homomorphism

${\rm Res}(D_{V},\,x_{i})\,:\,V_{x_{i}}\,\longrightarrow\,V_{x_{i}}\,,$ (2.16)

which is called the residue of $D_{V}$ at $x_{i}$; see [De]. If
$\lambda_{1},\,\cdots,\,\lambda_{r}$ are the generalized eigenvalues of ${\rm
Res}(D_{V},\,x_{i})$ with multiplicity, where $r\,=\,\text{rank}(V)$, then the
generalized eigenvalues of the local monodromy of $D$ around $x_{i}$ are

$\exp(-2\pi\sqrt{-1}\lambda_{1}),\,\exp(-2\pi\sqrt{-1}\lambda_{1}),\,\cdots,\,\exp(-2\pi\sqrt{-1}\lambda_{r}$
(2.17)

[De].

We now recall another description of the logarithmic connections on $V$.
Consider the short exact sequence in (2.7)

$0\,\longrightarrow\,\text{End}(V)\,\longrightarrow\,{\rm
Diff}^{1}(V,\,V)\,\stackrel{{\scriptstyle\widehat{\sigma}_{V}}}{{\longrightarrow}}\,TX\otimes\text{End}(V)\,\longrightarrow\,0$
(2.18)

for $V$. Define

$\text{At}(V,S)\,:=\,\widehat{\sigma}^{-1}_{V}(TX\otimes{\mathcal{O}}_{X}(-S))\,\subset\,{\rm
Diff}^{1}(V,\,V)\,,$

where $TX\otimes{\mathcal{O}}_{X}(-S)\,\subset\,TX\otimes\text{End}(V)$ is the
subbundle defined as in (2.10) for $V$. So from (2.18) we have the short exact
sequence of holomorphic vector bundles

$0\,\longrightarrow\,\text{End}(V)\,\longrightarrow\,\text{At}(V,S)\,\stackrel{{\scriptstyle\sigma_{V}}}{{\longrightarrow}}\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,0\,,$
(2.19)

where $\sigma_{V}$ is the restriction of the projection $\widehat{\sigma}_{V}$
in (2.18) to the subsheaf $\text{At}(V,S)\,\subset\,{\rm Diff}^{1}(V,\,V)$. A
logarithmic connection on $V$ is a holomorphic splitting of the short exact
sequence in (2.19); in other words, a logarithmic connection on $V$ is a
holomorphic homomorphism of vector bundles

$h\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,\text{At}(V,S)$

such that $\sigma_{V}\circ h\,=\,{\rm Id}_{TX\otimes{\mathcal{O}}_{X}(-S)}$,
where $\sigma_{V}$ is the homomorphism in (2.19).

Take a parabolic vector bundle
$E_{*}\,=\,(E,\,(\\{E_{i,j}\\},\,\\{\alpha_{i,j}\\}))$; see (2.2), (2.3).

A connection on $E_{*}$ is a logarithmic connection $D$ on $E$, singular over
$S$, such that

  1. (1)

$\text{Res}(D,x_{i})(E_{i,j})\,\subset\,E_{i,j}$ for all
$1\,\leq\,j\,\leq\,l_{i}$, $1\,\leq\,i\,\leq\,n$ (see (2.2)), and

  2. (2)

the endomorphism of $E_{i,j}/E_{i,j+1}$ induced by $\text{Res}(D,x_{i})$
coincides with multiplication by the parabolic weight $\alpha_{i,j}$ for all
$1\,\leq\,j\,\leq\,l_{i}$, $1\,\leq\,i\,\leq\,n$.

(See [BL, Section 2.2].) A necessary and sufficient condition for $E_{*}$ to
admit a connection is given in [BL]. The condition in question says that
$E_{*}$ admits a connection if and only if the parabolic degree of every
direct summand of $E_{*}$ is zero [BL, p. 594, Theorem 1.1].

A holomorphic splitting of the Atiyah exact sequence in (2.12) (see Definition
2.3) is a holomorphic homomorphism of vector bundles

$\mathbf{h}\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,{\rm
At}(E_{*})$

such that

$\sigma\circ\mathbf{h}\,=\,{\rm Id}_{TX\otimes{\mathcal{O}}_{X}(-S)}\,,$
(2.20)

where $\sigma$ is the projection in (2.12).

######  Lemma 2.4.

Giving a holomorphic splitting $\mathbf{h}$ of the Atiyah exact sequence for
$E_{*}$ (see (2.20)) is equivalent to giving a logarithmic connection $D$ on
$E$ satisfying the following condition: for every $x_{i}\,\in\,S$,

${\rm Res}(D,\,x_{i})(E_{i,j})\,\subset\,E_{i,j}$

for all $1\,\leq\,j\,\leq\,l_{i}$ (see (2.2)), where ${\rm
Res}(D,\,x_{i})(E_{i,j})$ is constructed in (2.16).

###### Proof.

Let $\mathbf{h}$ be a holomorphic splitting of the Atiyah exact sequence in
(2.12). In other words,
$\mathbf{h}\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,{\rm
At}(E_{*})$ is a holomorphic homomorphism such that (2.20) holds. Recall from
(2.11) that ${\rm At}(E_{*})\,\subset\,{\rm Diff}^{1}_{P}(E,\,E)$. The
composition of homomorphisms

$TX\otimes{\mathcal{O}}_{X}(-S)\,\stackrel{{\scriptstyle\mathbf{h}}}{{\longrightarrow}}\,{\rm
At}(E_{*})\,\hookrightarrow\,{\rm Diff}^{1}_{P}(E,\,E)$

will be denoted by $\widetilde{\mathbf{h}}$. Let
$\widetilde{\mathbf{h}}^{\prime}$ denote the composition of homomorphisms

$TX\otimes{\mathcal{O}}_{X}(-S)\,\stackrel{{\scriptstyle\widetilde{\mathbf{h}}}}{{\longrightarrow}}\,{\rm
Diff}^{1}_{P}(E,\,E)\,\stackrel{{\scriptstyle\iota_{0}}}{{\hookrightarrow}}\,{\rm
Diff}^{1}(E,\,E)$ (2.21)

(see (2.8) for the inclusion map $\iota_{0}$). Since (2.20) holds, we conclude
that

$\sigma_{0}\circ\widetilde{\mathbf{h}}^{\prime}\,=\,{\rm
Id}_{TX\otimes{\mathcal{O}}_{X}(-S)}\,,$

where $\sigma_{0}$ is the projection in (2.7). Therefore,
$\widetilde{\mathbf{h}}^{\prime}$ is a logarithmic connection on $E$; this
logarithmic connection on $E$ will be denoted by $D$. Since

$\widetilde{\mathbf{h}}^{\prime}\,=\,\iota_{0}\circ\widetilde{\mathbf{h}}$

(see (2.21)), the residues of the above defined logarithmic connection $D$
satisfy the following condition: for every $x_{i}\,\in\,S$,

${\rm Res}(D,\,x_{i})(E_{i,j})\,\subset\,E_{i,j}$

for all $1\,\leq\,j\,\leq\,l_{i}$.

To prove the converse, let $D$ be a logarithmic connection on $E$ such that

${\rm Res}(D,\,x_{i})(E_{i,j})\,\subset\,E_{i,j}$ (2.22)

for all $x_{i}\,\in\,S$ and all $1\,\leq\,j\,\leq\,l_{i}$. So $D$ gives a
holomorphic homomorphism

$\mathbf{h}\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,\text{At}(E,S)$

(see (2.19) for $\text{At}(E,S)$) such that the composition of homomorphisms

$TX\otimes{\mathcal{O}}_{X}(-S)\,\stackrel{{\scriptstyle\mathbf{h}}}{{\longrightarrow}}\,\text{At}(E,S)\,\longrightarrow\,TX\otimes{\mathcal{O}}_{X}(-S)$

coincides with the identity map of $TX\otimes{\mathcal{O}}_{X}(-S)$; see
(2.19) for the above projection
$\text{At}(E,S)\,\longrightarrow\,TX\otimes{\mathcal{O}}_{X}(-S)$. The given
condition in (2.22) implies that

$\mathbf{h}(TX\otimes{\mathcal{O}}_{X}(-S))\,\subset\,{\rm
Diff}^{1}_{P}(E,\,E)\bigcap\text{At}(E,S)\,=\,{\rm At}(E_{*})\,;$

note that from (2.11) we have

${\rm At}(E_{*})\,:=\,{\rm
Diff}^{1}_{P}(E,\,E)\bigcap\sigma^{-1}_{0}(TX\otimes{\mathcal{O}}_{X}(-S))\,\subset\,\sigma^{-1}_{0}(TX\otimes{\mathcal{O}}_{X}(-S))\,=\,\text{At}(E,S)\,.$

Consequently, the homomorphism $\mathbf{h}$ gives a holomorphic splitting, as
in (2.20), of the Atiyah exact sequence for $E_{*}$. ∎

######  Corollary 2.5.

Giving a connection on the parabolic bundle $E_{*}$ is equivalent to giving a
holomorphic splitting $\mathbf{h}$ of the Atiyah exact sequence for $E_{*}$
(see (2.12)) such that the logarithmic connection $D$ on $E$ associated to
$\mathbf{h}$ (see Lemma 2.4) satisfies the following condition: for every
$x_{i}\,\in\,S$, the residue ${\rm Res}(D,\,x_{i})$ induces the endomorphism
$\alpha_{i,j}\cdot{\rm Id}_{E_{i,j}/E_{i,j+1}}$ of the quotient space
$E_{i,j}/E_{i,j+1}$ for all $1\,\leq\,j\,\leq\,l_{i}$.

###### Proof.

Note that from Lemma 2.4 we know that

${\rm Res}(D,\,x_{i})(E_{i,j})\,\subset\,E_{i,j}$

for all $x_{i}\,\in\,S$ and all $1\,\leq\,j\,\leq\,l_{i}$. Therefore, ${\rm
Res}(D,\,x_{i})$ induces an endomorphism of the quotient space
$E_{i,j}/E_{i,j+1}$.

The result follows from Lemma 2.4 and the definition of a connection on
$E_{*}$. ∎

###  2.3. A homomorphism associated to a connection

Let $E_{*}\,=\,(E,\,(\\{E_{i,j}\\},\,\\{\alpha_{i,j}\\}))$ be a parabolic
vector bundle on $X$ with parabolic divisor $S$. Let $D$ be a connection on
$E_{*}$. Using $D$ we will construct a first order holomorphic differential
operator

$\mathcal{D}_{0}\,:\,\text{End}^{P}(E_{*})\,\longrightarrow\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ (2.23)

(see (2.5) and (2.6) for $\text{End}^{P}(E_{*})$ and $\text{End}^{n}(E_{*})$
respectively).

To construct $\mathcal{D}_{0}$, take any holomorphic section
$\Phi\,\in\,\Gamma(U,\,\text{End}(E))$, where $U\,\subset\,X$ is any open
subset. Let

$A_{U}\,:\,\Gamma(U,\,E)\,\longrightarrow\,\Gamma(U,\,E\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))$

be the homomorphism defined by

$s\,\longmapsto\,D(\Phi(s))-(\Phi\otimes{\rm
Id}_{K_{X}\otimes{\mathcal{O}}_{X}(S)})(D(s))\,.$

This $A_{U}$ is evidently ${\mathcal{O}}_{U}$–linear. Hence we have

$A_{U}\,\in\,\Gamma(U,\,\text{End}(E)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,.$

From the properties of $D$ it follows that

$A_{U}\,\in\,\Gamma(U,\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,\subset\,\Gamma(U,\,\text{End}(E)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))$

if $\Phi\,\in\,\Gamma(U,\,\text{End}^{P}(E_{*}))$. The homomorphism
$\mathcal{D}_{0}$ in (2.23) is defined by $\Phi\,\longmapsto\,A_{U}$.

Recall from (2.12) that $\text{End}^{P}(E_{*})$ is a holomorphic subbundle of
${\rm At}(E_{*})$. We will now extend $\mathcal{D}_{0}$ in (2.23) to a first
order holomorphic differential operator

$\mathcal{D}\,:\,{\rm
At}(E_{*})\,\longrightarrow\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S).$ (2.24)

To construct $\mathcal{D}$, take holomorphic sections

$\Phi\,\in\,\Gamma(U,\,{\rm At}(E_{*}))\ \ \text{ and }\ \
v\,\in\,\Gamma(U,\,TX\otimes{\mathcal{O}}_{X}(-S)),$

where $U\,\subset\,X$ is any open subset. Denote

$w\,:=\,\sigma(\Phi)\,\in\,\Gamma(U,\,TX\otimes{\mathcal{O}}_{X}(-S)),$ (2.25)

where $\sigma$ is the projection in (2.12). Take any $s\,\in\,\Gamma(U,\,E)$.
So

$\Phi(s)\,\in\,\Gamma(U,\,E)$

(recall from (2.11) that ${\rm At}(E_{*})\,\subset\,{\rm Diff}^{1}(E,\,E)$),
and hence

$D(\Phi(s))\,\in\,\Gamma(U,\,E\otimes K_{X}\otimes{\mathcal{O}}_{X}(S))\,.$

Therefore, we have

$\langle D(\Phi(s)),\,v\rangle\,\in\,\Gamma(U,\,E)\,,$ (2.26)

where $\langle-,\,-\rangle$ is the natural duality pairing

$(K_{X}\otimes{\mathcal{O}}_{X}(S))\otimes(TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow\,{\mathcal{O}}_{X}\,.$
(2.27)

We have $\langle D(s),\,v\rangle\,\in\,\Gamma(U,\,E)$, so

$\Phi(\langle D(s),\,v\rangle)\,\in\,\Gamma(U,\,E)\,,$ (2.28)

where $\langle-,\,-\rangle$ is the pairing in (2.27). Consider the Lie bracket
of vector fields

$[v,\,w]\,\in\,\Gamma(U,\,TX\otimes{\mathcal{O}}_{X}(-S))\,,$

where $w$ is defined in (2.25). We have

$\langle D(s),\,[v,\,w]\rangle\,\in\,\Gamma(U,\,E)\,.$ (2.29)

Let $B_{U}\,:\,\Gamma(U,\,E)\,\longrightarrow\,\Gamma(U,\,E)$ be the
homomorphism defined by

$s\,\longmapsto\,\langle D(\Phi(s)),\,v\rangle-\Phi(\langle
D(s),\,v\rangle)-\langle D(s),\,[v,\,w]\rangle$

(see (2.26), (2.28) and (2.29)). This homomorphism $B_{U}$ is evidently
${\mathcal{O}}_{U}$–linear, and hence we have

$B_{U}\,\in\,\Gamma(U,\,\text{End}(E))\,.$ (2.30)

It is now straightforward to check that

$B_{U}\,\in\,\Gamma(U,\,\text{End}^{n}(E_{*}))\,\subset\,\Gamma(U,\,\text{End}(E))\,.$

The homomorphism $\mathcal{D}$ in (2.24) is uniquely defined by the following
property: For any open subset $U\,\subset\,X$ and sections

$\Phi\,\in\,\Gamma(U,\,{\rm At}(E_{*}))\ \ \text{ and }\ \
v\,\in\,\Gamma(U,\,TX\otimes{\mathcal{O}}_{X}(-S)),$

the equality

$\langle{\mathcal{D}}(\Phi),\,v\rangle\,=\,B_{U}$

holds, where $B_{U}$ is constructed in (2.30) from $\Phi$ and $v$ using $D$,
and $\langle-,\,-\rangle$ is the pairing in (2.27).

From the constructions of $\mathcal{D}$ and $\mathcal{D}_{0}$ (see (2.23)) it
follows immediately that the restriction of $\mathcal{D}$ to
$\text{End}^{P}(E_{*})\,\subset\,{\rm At}(E_{*})$ actually coincides with
$\mathcal{D}_{0}$.

##  3\. Infinitesimal deformations and isomonodromy

###  3.1. Infinitesimal deformations

Let $E_{*}$ be a parabolic vector bundle over $X$ with parabolic divisor $S$.
Then the infinitesimal deformations of $E_{*}$, keeping the pair $(X,\,S)$
fixed, are parametrized by $H^{1}(X,\,\text{End}^{P}(E_{*}))$, where
$\text{End}^{P}(E_{*})$ is defined in (2.5) [MS]. We recall that the
infinitesimal deformations of the pair $(X,\,S)$ are parametrized by
$H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))$.

######  Lemma 3.1.

The infinitesimal deformations of the triple

$(X,\,S,\,E_{*})$

are parametrized by $H^{1}(X,\,{\rm At}(E_{*}))$, where ${\rm At}(E_{*})$ is
defined in (2.11).

###### Proof.

The proof is very similar to that of [Ch2, p. 1413, Proposition 4.3] (see also
[Ch1]).

The lemma actually follows from [Ch2, p. 1413, Proposition 4.3] once we invoke
the correspondence between the parabolic bundles and the orbifold bundles.
This is explained below.

For any $x_{i}\,\in\,S$, let $N_{i}$ be the smallest positive integer such
that for all $1\,\leq\,j\,\leq\,l_{i}$,

$\alpha_{i,j}\,=\,\frac{m_{i,j}}{N_{i}}\,,$

where $m_{i,j}$ are nonnegative integers; see (2.3) and Assumption 2.1. There
is a ramified Galois covering

$\varphi\,:\,Y\,\longrightarrow\,X$ (3.1)

satisfying the following two conditions:

  * •

$\varphi$ is unramified over the complement $X\setminus S$, and

  * •

for every $x_{i}\,\in\,S$ and one (hence every) point
$y\,\in\,\varphi^{-1}(x_{i})$, the order of ramification of $\varphi$ at $y$
is $N_{i}$.

Such a covering $\varphi$ exists; see [Na, p. 26, Proposition 1.2.12] and
Assumption 2.1.

Let $\Gamma_{\varphi}\,=\,\text{Gal}(\varphi)\,\subset\,\text{Aut}(Y)$ be the
Galois group for the ramified covering $\varphi$, so
$X\,=\,Y/\Gamma_{\varphi}$. An equivariant vector bundle over $Y$ is a
holomorphic vector bundle $V\,\longrightarrow\,Y$ equipped with a lift of the
action of $\Gamma_{\varphi}$. In other words,

  * •

$\Gamma_{\varphi}$ acts holomorphically on the total space of $V$, and

  * •

the action of any $g\,\in\,\Gamma_{\varphi}$ on $V$ is a holomorphic
automorphism of the vector bundle $V$ over the automorphism $g$ of $Y$. In
particular, the projection map $V\,\longrightarrow\,Y$ is
$\Gamma_{\varphi}$–equivariant.

There is a natural equivalence of categories between the parabolic vector
bundles on $X$ whose parabolic weights at each $x_{i}$ are integral multiples
of $\frac{1}{N_{i}}$ and the equivariant vector bundles on $Y$ [Bi1], [Bo1],
[Bo2].

Let $F_{*}$ be a parabolic vector bundle on $X$ whose parabolic weights at
each $x_{i}$ are integral multiples of $\frac{1}{N_{i}}$. The holomorphic
vector bundle underlying $F_{*}$ will be denoted by $F$. Let $V$ be the
equivariant vector bundle on $Y$ corresponding to $F_{*}$. The action
$\Gamma_{\varphi}$ on $V$ produces a homomorphism from $\Gamma_{\varphi}$ to
the group $\text{Aut}(\varphi_{*}V)$ of holomorphic automorphisms of the
direct image $\varphi_{*}V$, over the identity map of $X$. Then we have

$F\,=\,(\varphi_{*}V)^{\Gamma_{\varphi}}\,\subset\,\varphi_{*}V\,,$

where $(\varphi_{*}V)^{\Gamma_{\varphi}}$ denotes the invariant part for the
action of $\Gamma_{\varphi}$ on $\varphi_{*}V$.

Take a holomorphic family of compact Riemann surfaces equipped with $n$
ordered marked points. Assume that this family is parametrized by $T$, and
that there is a point $t_{0}\,\in\,T$ such that the fiber over $t_{0}$ is the
given pair $(X,\,S)$. Then the construction of the ramified Galois covering of
$X$, done in [Na, p. 26, Proposition 1.2.12], extends to produce a family of
ramified Galois coverings of all Riemann surfaces over an open neighborhood of
$t_{0}$ in $T$.

Let $V$ be the equivariant bundle on $Y$ corresponding to the parabolic vector
bundle $E_{*}$ in the lemma. Then ${\rm End}^{P}(E_{*})$ is the holomorphic
vector bundle underlying the parabolic bundle that corresponds to the
equivariant vector bundle $\text{End}(V)$ on $Y$. The action of
$\Gamma_{\varphi}$ on $Y$ produces an action of $\Gamma_{\varphi}$ on $TY$,
and $TX\otimes{\mathcal{O}}_{X}(-S)$ is the holomorphic line bundle underlying
the corresponding parabolic line bundle on $X$. The actions of
$\Gamma_{\varphi}$ on $V$ and $Y$ together produce an action of
$\Gamma_{\varphi}$ on $\text{At}(V)$. The Atiyah bundle ${\rm At}(E_{*})$ is
the holomorphic vector bundle underlying the parabolic bundle corresponding to
the equivariant vector bundle $\text{At}(V)$ on $Y$. This implies that

$H^{1}(X,\,{\rm At}(E_{*}))\,=\,H^{1}(Y,\,{\rm At}(V))^{\Gamma_{\varphi}}\,.$
(3.2)

In view of(3.2), the lemma follows immediately from [Ch2, p. 1413, Proposition
4.3]. ∎

Let

$0\,=\,H^{0}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow\,H^{1}\left(X,\,\text{End}^{P}(E_{*})\right)\,\stackrel{{\scriptstyle
p_{1}}}{{\longrightarrow}}\,H^{1}(X,\,{\rm At}(E_{*}))$ (3.3)
$\stackrel{{\scriptstyle
p_{2}}}{{\longrightarrow}}\,H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow\,H^{2}\left(X,\,\text{End}^{P}(E_{*})\right)\,=\,0$

be the long exact sequence of cohomologies associated to the short exact
sequence in (2.12); we have $H^{0}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,=\,0$
by Assumption 2.1. The projection $p_{2}$ in (3.3) is the forgetful map that
sends an infinitesimal deformation of the triple $(X,\,S,\,E_{*})$ to the
infinitesimal deformation of the pair $(X,\,S)$ obtained from it by simply
forgetting $E_{*}$. The injective homomorphism $p_{1}$ in (3.3) sends an
infinitesimal deformation of $E_{*}$ to the infinitesimal deformation of
$(X,\,S,\,E_{*})$ obtained from it by keeping the pair $(X,\,S)$ fixed.

######  Lemma 3.2.

Assume that the parabolic bundle $E_{*}$ has a connection $D$ such that the
local monodromy of $D$ around each point of $S$ is semisimple (meaning
diagonalizable). Then the local monodromy of every connection on $E_{*}$
around each point of $S$ is also semisimple.

###### Proof.

Take a ramified Galois covering

$\varphi\,:\,Y\,\longrightarrow\,X,$

as in (3.1), satisfying the following two conditions:

  * •

$\varphi$ is unramified over the complement $X\setminus S$, and

  * •

for every $x_{i}\,\in\,S$ and one (hence every) point
$y\,\in\,\varphi^{-1}(x_{i})$, the order of ramification of $\varphi$ at $y$
is $N_{i}$.

Let $V$ be the equivariant vector bundle on $Y$ corresponding to $E_{*}$. Then
$D$ corresponds to an equivariant holomorphic connection on $V$. The space of
all connections on $E_{*}$ is an affine space for the vector space
$H^{0}(X,\,\text{End}^{n}(E_{*})\otimes K_{X}\otimes{\mathcal{O}}_{X}(S))$,
where $\text{End}^{n}(E_{*})$ is defined in (2.6). On the other hand the space
of all equivariant holomorphic connections on $V$ is an affine space for the
vector space $H^{0}(Y,\,\text{End}(V)\otimes K_{Y})^{\Gamma}$, where $\Gamma$
is the Galois group for the covering map $\varphi$.

Since we have

$H^{0}(X,\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,=\,H^{0}(Y,\,\text{End}(V)\otimes
K_{Y})^{\Gamma},$

we conclude that every connection $D^{\prime}$ on $E_{*}$ is given by an
equivariant connection on $V$. This implies that the order of the local
monodromy of $D^{\prime}$ around any point $x_{i}\,\in\,S$ is a sub-multiple
of $N_{i}$. This implies that the local monodromy of $D^{\prime}$ around every
point of $S$ is semisimple. ∎

Let $D$ be a connection on the parabolic bundle $E_{*}$. We assume that the
local monodromy of $D$ around each point of $S$ is semisimple. As mentioned
above, the space of all connections on $E_{*}$ is an affine space for the
vector space $H^{0}(X,\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))$, where $\text{End}^{n}(E_{*})$ is defined
in (2.6). This implies that the infinitesimal deformations of the connection
$D$, keeping $(X,\,S,\,E_{*})$ fixed, are parametrized by
$H^{0}(X,\,\text{End}^{n}(E_{*})\otimes K_{X}\otimes{\mathcal{O}}_{X}(S))$.

Let ${\mathcal{A}}_{\bullet}$ be the following two-term complex of sheaves on
$X$

${\mathcal{A}}_{\bullet}\,\,:\,\,{\mathcal{A}}_{0}\,=\,\text{End}^{P}(E_{*})\,\stackrel{{\scriptstyle\mathcal{D}_{0}}}{{\longrightarrow}}\,{\mathcal{A}}_{1}\,=\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$ (3.4)

where $\mathcal{D}_{0}$ is the differential operator in (2.23), and
${\mathcal{A}}_{i}$ is at the $i$-th position.

The following lemma is a standard fact.

######  Lemma 3.3.

The infinitesimal deformations of the pair $(E_{*},\,D)$, keeping $(X,\,S)$
fixed, are parametrized by the first hypercohomology
${\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})$, where ${\mathcal{A}}_{\bullet}$
is the complex in (3.4).

The complex ${\mathcal{A}}_{\bullet}$ in (3.4) fits in the following short
exact sequence of complexes of sheaves on $X$

$\begin{matrix}&&0&&0\\\ &&\Big{\downarrow}&&\Big{\downarrow}\\\
&&0&\longrightarrow&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\
&&\Big{\downarrow}&&\,\,\,\,\,\,\Big{\downarrow}{\rm id}\\\
{\mathcal{A}}_{\bullet}&:&\text{End}^{P}(E_{*})&\stackrel{{\scriptstyle\mathcal{D}_{0}}}{{\longrightarrow}}&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\ &&\,\,\,\,\,\,\Big{\downarrow}{\rm
id}&&\Big{\downarrow}\\\ &&\text{End}^{P}(E_{*})&\longrightarrow&0\\\
&&\Big{\downarrow}&&\Big{\downarrow}\\\ &&0&&0\end{matrix}$

Let

$\longrightarrow\,H^{0}(X,\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,\stackrel{{\scriptstyle\alpha_{1}}}{{\longrightarrow}}\,{\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})\,\stackrel{{\scriptstyle\alpha_{2}}}{{\longrightarrow}}\,H^{1}\left(X,\,\text{End}^{P}(E_{*})\right)\,\longrightarrow$
(3.5)

be the long exact sequence of hypercohomologies associated to the above short
exact sequence of complexes. The homomorphism $\alpha_{2}$ in (3.5) sends an
infinitesimal deformation of the pair $(E_{*},\,D)$ to the infinitesimal
deformation of $E_{*}$ obtained from it by simply forgetting the connection.
The homomorphism $\alpha_{1}$ in (3.5) sends an infinitesimal deformation of
the connection $D$ to the infinitesimal deformation of the pair $(E_{*},\,D)$
obtained from it by keeping the parabolic bundle $E_{*}$ fixed.

Let ${\mathcal{B}}_{\bullet}$ denote the following two-term complex of sheaves
on $X$

${\mathcal{B}}_{\bullet}\,\,:\,\,{\mathcal{B}}_{0}\,=\,{\rm
At}(E_{*})\,\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\,{\mathcal{B}}_{1}\,=\,\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$ (3.6)

where $\mathcal{D}$ is the homomorphism in (2.24), and ${\mathcal{B}}_{i}$ is
at the $i$-th position.

######  Lemma 3.4.

The infinitesimal deformations of the quadruple

$(X,\,S,\,E_{*},\,D)$

are parametrized by the first hypercohomology
${\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})$, where ${\mathcal{B}}_{\bullet}$
is the complex in (3.6).

###### Proof.

The proof is very similar to the proof of [Ch2, p. 1415, Proposition 4.4] (see
also [Ch1]). In fact, it can also be deduced from [Ch2, p. 1415, Proposition
4.4], as done in the proof of Lemma 3.1. This is elaborated below.

Take the ramified Galois covering $(Y,\,\varphi)$ in (3.1). As in the proof of
Lemma 3.1, $V$ denotes the equivariant vector bundle on $Y$ corresponding to
the parabolic vector bundle $E_{*}$. The connection $D$ on $E_{*}$ corresponds
to a $\Gamma_{\varphi}$–invariant holomorphic connection on $V$, where
$\Gamma_{\varphi}$ is, as before, the Galois group for $\varphi$; this
$\Gamma_{\varphi}$–invariant holomorphic connection on $V$ will be denoted by
$D^{\prime}$. Let

${\mathcal{B}}^{\prime}_{\bullet}\,:\,{\mathcal{B}}^{\prime}_{0}\,:=\,\text{At}(V)\,\stackrel{{\scriptstyle\mathcal{D}^{\prime}}}{{\longrightarrow}}\,{\mathcal{B}}_{1}\,:=\,\text{End}(V)\otimes
K_{Y}$ (3.7)

be the complex in [Ch2, p. 1415, Proposition 4.4] for $D^{\prime}$; it is the
same complex as in (3.6) when there is no parabolic structure (meaning the
parabolic structure is trivial). We note that the differential operator
$\mathcal{D}^{\prime}$ in (3.7) is $\Gamma_{\varphi}$–equivariant, because the
connection $D^{\prime}$ on $V$ is $\Gamma_{\varphi}$–invariant. The
holomorphic vector bundle $\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ coincides with the holomorphic vector bundle
underlying the parabolic bundle that corresponds to the
$\Gamma_{\varphi}$–equivariant bundle $\text{End}(V)\otimes K_{Y}$ on $Y$. It
was noted in the proof of Lemma 3.1 that ${\rm At}(E_{*})$ is the holomorphic
vector bundle underlying the parabolic bundle corresponding to the equivariant
bundle $\text{At}(V)$ on $Y$. Moreover, the operator $\mathcal{D}$ in (3.6)
coincides with the one given by $\mathcal{D}^{\prime}$ on the
$\Gamma_{\varphi}$–invariant part of the direct image. These together imply
that

${\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,=\,{\mathbb{H}}^{1}({\mathcal{B}}^{\prime}_{\bullet})^{\Gamma_{\varphi}}\,,$
(3.8)

where
${\mathbb{H}}^{1}({\mathcal{B}}^{\prime}_{\bullet})^{\Gamma_{\varphi}}\,\subset\,{\mathbb{H}}^{1}({\mathcal{B}}^{\prime}_{\bullet})$
is the invariant part for the action of $\Gamma_{\varphi}$.

In view of (3.8), the lemma follows from [Ch2, p. 1415, Proposition 4.4]. ∎

The complex ${\mathcal{B}}_{\bullet}$ in (3.6) fits in the following short
exact sequence of complexes of sheaves on $X$

$\begin{matrix}&&0&&0\\\ &&\Big{\downarrow}&&\Big{\downarrow}\\\
{\mathcal{A}}_{\bullet}&:&\text{End}^{P}(E_{*})&\stackrel{{\scriptstyle\mathcal{D}_{0}}}{{\longrightarrow}}&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\
&&\Big{\downarrow}&&\,\,\,\,\,\,\Big{\downarrow}{\rm id}\\\
{\mathcal{B}}_{\bullet}&:&{\rm
At}(E_{*})&\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\ &&\Big{\downarrow}&&\Big{\downarrow}\\\
&&TX\otimes{\mathcal{O}}_{X}(-S)&\longrightarrow&0\\\
&&\Big{\downarrow}&&\Big{\downarrow}\\\ &&0&&0\end{matrix}$

where the vertical exact sequence in the left is the one in (2.12); see (3.4)
and (3.6) for ${\mathcal{A}}_{\bullet}$ and ${\mathcal{B}}_{\bullet}$
respectively. Let

$\longrightarrow\,{\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})\,\stackrel{{\scriptstyle\beta_{1}}}{{\longrightarrow}}\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,\stackrel{{\scriptstyle\beta_{2}}}{{\longrightarrow}}\,H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow$
(3.9)

be the long exact sequence of hypercohomologies associated to the above short
exact sequence of complexes. The homomorphism $\beta_{2}$ in (3.9) sends an
infinitesimal deformation of the quadruple $(X,\,S,\,E_{*},\,D)$ to the
infinitesimal deformation of $(X,\,S)$ obtained from it by simply forgetting
the pair $(E_{*},\,D)$. The homomorphism $\beta_{1}$ in (3.9) sends an
infinitesimal deformation of the pair $(E_{*},\,D)$ to the infinitesimal
deformation of the quadruple $(X,\,S,\,E_{*},\,D)$ obtained from it by keeping
the pair $(X,\,S)$ fixed.

###  3.2. Character variety and the monodromy map

Fix an integer $r\,\geq\,1$. Fix a base point $b_{0}\,\in\,X\setminus S$, and
denote by $\widetilde{\rm Hom}(\pi_{1}(X\setminus S,\,b_{0}),\,{\rm
GL}(r,{\mathbb{C}}))$ the space of all homomorphisms from $\pi_{1}(X\setminus
S,\,b_{0})$ to ${\rm GL}(r,{\mathbb{C}})$. Given any
$\rho\,\in\,\widetilde{\rm Hom}(\pi_{1}(X\setminus S,\,b_{0}),\,{\rm
GL}(r,{\mathbb{C}}))$, we consider ${\mathbb{C}}^{r}$ as a $\pi_{1}(X\setminus
S,\,b_{0})$–module by combining $\rho$ with the standard $r$–dimensional
representation of ${\rm GL}(r,{\mathbb{C}})$. We recall that $\rho$ is called
completely reducible if the $\pi_{1}(X\setminus S,\,b_{0})$–module
${\mathbb{C}}^{r}$ corresponding to $\rho$ is a direct sum of irreducible
$\pi_{1}(X\setminus S,\,b_{0})$–modules. Let

${\rm Hom}(\pi_{1}(X\setminus S,\,b_{0}),\,{\rm
GL}(r,{\mathbb{C}}))\,\subset\,\widetilde{\rm Hom}(\pi_{1}(X\setminus
S,\,b_{0}),\,{\rm GL}(r,{\mathbb{C}}))$

be the space of all completely reducible representations. The adjoint action
of ${\rm GL}(r,{\mathbb{C}})$ on itself produces an action of ${\rm
GL}(r,{\mathbb{C}})$ on ${\rm Hom}(\pi_{1}(X\setminus S,\,b_{0}),\,{\rm
GL}(r,{\mathbb{C}}))$. The quotient space

${\mathcal{R}}_{X}(r)\,=\,{\rm Hom}(\pi_{1}(X\setminus S,\,b_{0}),\,{\rm
GL}(r,{\mathbb{C}}))/{\rm GL}(r,{\mathbb{C}})$ (3.10)

is a normal quasiprojective variety defined over $\mathbb{C}$. (See [Si],
[LS], and references therein, for ${\mathcal{R}}_{X}(r)$.)

For another base point $b^{\prime}_{0}\,\in\,X\setminus S$, the two groups
$\pi_{1}(X\setminus S,\,b_{0})$ and $\pi_{1}(X\setminus S,\,b^{\prime}_{0})$
are identified up to inner automorphisms, and hence ${\mathcal{R}}_{X}(r)$
does not depend on the choice of the base point $b_{0}$. The complex structure
of $X$ does not play any role in the construction of ${\mathcal{R}}_{X}(r)$;
the space ${\mathcal{R}}_{X}(r)$ depends only on the topological surface
underlying $X\setminus S$.

A connection $D$ on a parabolic vector bundle $E_{*}$ is called completely
reducible if it is a direct sum of irreducible logarithmic connections.

Fix the dimensions of the subspaces $E_{i,j}$ and fix the parabolic weights
$\alpha_{i,j}$; see (2.2), (2.3). Let ${\mathcal{M}}_{X}(r)$ denote the moduli
space of pairs $(E_{*},\,D)$, where $E_{*}$ is a parabolic vector bundle of
rank $r$ on $X$ having the given parabolic type and $D$ is a completely
reducible connection on $E_{*}$; see [In], [IIS1], [IIS2], [Iw], [BBP] and
references therein for the moduli space.

Since any logarithmic connection on a holomorphic vector bundle on $X$ is
flat, we can associate a monodromy representation to any logarithmic
connection. Consequently, we have a holomorphic map

$\mathbb{M}_{X}\,:\,{\mathcal{M}}_{X}(r)\,\longrightarrow\,{\mathcal{R}}_{X}(r)\,,$
(3.11)

where ${\mathcal{R}}_{X}(r)$ is constructed in (3.10), that sends any
connection to its monodromy.

Let

$\varpi\,:\,X_{T}\,\longrightarrow\,T$ (3.12)

be a holomorphic family of compact connected Riemann surfaces parametrized by
a simply connected complex manifold $T$. For any $t\,\in\,T$, the fiber
$\varpi^{-1}(t)$ will be denoted by $X_{t}$. For $1\,\leq\,i\,\leq\,n$, let

$\phi_{i}\,\,:\,\,T\,\longrightarrow\,X_{T}$ (3.13)

be a holomorphic section such that
$\phi_{i}(T)\bigcap\phi_{j}(T)\,=\,\emptyset$ for all $i\,\not=\,j$. Fix a
base point $t_{0}\,\in\,T$. Denote $X_{t_{0}}$ by $X$, and also denote
$s_{i}(t_{0})$ by $x_{i}$ for every $1\,\leq\,i\,\leq\,n$. As before, denote
the subset $\\{x_{1},\,\cdots,\,x_{n}\\}\,\subset X$ by $S$. For any
$t\,\in\,T$, the subset $\\{\phi_{1}(t),\,\cdots,\,\phi_{n}(t)\\}\,\subset
X_{t}\,:=\,\varpi^{-1}(t)$ will be denoted by $S_{t}$.

Since the parameter space $T$ is simply connected, $\pi_{1}(X\setminus
S,\,b_{0})$ and $\pi_{1}(X_{t}\setminus S_{t},\,b_{t})$ are identified up to
inner automorphisms. This implies that the character variety
${\mathcal{R}}_{X_{t}}(r)$ constructed as in (3.10) is canonically identified
with ${\mathcal{R}}_{X}(r)$.

Let

${\mathcal{M}}^{T}(r)\,\longrightarrow\,T$ (3.14)

be the relative moduli space of parabolic bundles with connection for the
family $X_{T}$ in (3.12). In view of the above observation that
${\mathcal{R}}_{X_{t}}(r)$ is identified with ${\mathcal{R}}_{X}(r)$, the
monodromy maps $\mathbb{M}_{X}$ in (3.11) for points of $T$ actually fit
together to produce a holomorphic map

$\mathbb{M}\,:\,{\mathcal{M}}^{T}(r)\,\longrightarrow\,{\mathcal{R}}_{X}(r)\,.$
(3.15)

Let $E^{T}_{*}$ be a holomorphic family of parabolic bundles on $X_{T}$, and
let $D^{T}$ be a relative connection $E^{T}_{*}$. In other words, the pair
$(E^{T}_{*},\,D^{T})$ corresponds to a holomorphic section

$\Psi\,:\,T\,\longrightarrow\,{\mathcal{M}}^{T}(r)$ (3.16)

of the holomorphic family of moduli spaces in (3.14).

A holomorphic family of the above type $(E^{T}_{*},\,D^{T})$, of parabolic
bundles with connection, is called isomonodromic if the composition
$\mathbb{M}\circ\Psi$ is a constant map, where $\mathbb{M}$ is the monodromy
map in (3.15) and $\Psi$ is the map in (3.16). This condition of being
isomonodromy defines a holomorphic foliation on ${\mathcal{M}}^{T}(r)$ which
is transversal to the holomorphic foliation given by the projection
${\mathcal{M}}^{T}(r)\,\longrightarrow\,T$ in (3.14). In other words, the
direct sum of these two distributions coincides with the full tangent bundle
$T{\mathcal{M}}^{T}(r)\,\longrightarrow\,{\mathcal{M}}^{T}(r)$, on the smooth
locus of ${\mathcal{M}}^{T}(r)$.

Let $D$ be a connection on a parabolic vector bundle $E_{*}$ over $X$ with
parabolic structure over $S$. We assume that the local monodromy of $D$ around
each point of $S$ is semisimple. The foliation given by isomonodromy produces
a homomorphism from the space of infinitesimal deformations of the pair
$(X,\,S)$ to the space of infinitesimal deformations of the quadruple
$(X,\,S,\,E_{*},\,D)$. In view of Lemma 3.4, this homomorphism, from the space
of infinitesimal deformations of $(X,\,S)$ to the space of infinitesimal
deformations of $(X,\,S,\,E_{*},\,D)$, is given by a homomorphism

$\gamma\
:\,H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,,$
(3.17)

where ${\mathcal{B}}_{\bullet}$ is the complex in (3.6).

From Corollary 2.5 we know that the connection $D$ produces a homomorphism

${\mathbf{h}}\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,{\rm
At}(E_{*})$

such that $\sigma\circ\mathbf{h}\,=\,{\rm
Id}_{TX\otimes{\mathcal{O}}_{X}(-S)}$, where $\sigma$ is the projection in
(2.12). This homomorphism ${\mathbf{h}}$ produces a homomorphism of complexes
of sheaves on $X$

$\begin{matrix}&&TX\otimes{\mathcal{O}}_{X}(-S)&\longrightarrow&0\\\
&&\,\,\,\Big{\downarrow}{\mathbf{h}}&&\Big{\downarrow}\\\
{\mathcal{B}}_{\bullet}&:&{\rm
At}(E_{*})&\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\end{matrix}$ (3.18)

We note that $\mathcal{D}\circ{\mathbf{h}}\,=\,0$ because the connection $D$
is flat. The homomorphism of complexes in (3.18) produces a homomorphism of
hypercohomologies

$\delta\
:\,H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,.$
(3.19)

######  Lemma 3.5.

The homomorphism $\gamma$ constructed in (3.17) coincides with the
homomorphism $\delta$ constructed in (3.19).

###### Proof.

The proof of Lemma 3.5 is similar to the proof of [Ch2, p. 1417, Proposition
5.1] (see also [Ch1]). In fact, as done in the proofs of Lemma 3.1 and Lemma
3.4, the lemma can be deduced from [Ch2, p. 1417, Proposition 5.1] using the
correspondence between the parabolic bundles and the orbifold bundles. We omit
the details. ∎

##  4\. Infinitesimal deformations of parabolic opers

We recall from [BDP] the definition of parabolic ${\rm
SL}(r,{\mathbb{C}})$-opers.

Consider the subset $S\,\subset\,X$ in (2.1); assume that the integer
$n\,=\,\\#S$ is even. Fix a holomorphic line bundle ${\mathbb{L}}$ on $X$ of
degree $-\frac{n}{2}$ such that ${\mathbb{L}}^{{}^{\otimes 2}}$ is isomorphic
to ${\mathcal{O}}_{X}(-S)$; also fix a holomorphic isomorphism

$\varphi_{0}\,:\,{\mathbb{L}}^{{}^{\otimes
2}}\,\longrightarrow\,{\mathcal{O}}_{X}(-S)$

of line bundles. Fix a holomorphic line bundle $K^{1/2}_{X}$ on $X$ of degree
$\text{genus}(X)-1$ such that $\left(K^{1/2}_{X}\right)^{\otimes 2}$ is
isomorphic to $K_{X}$, in other words, $K^{1/2}_{X}$ is a theta characteristic
on $X$; fix a holomorphic isomorphism

$I_{X}\,:\,\left(K^{1/2}_{X}\right)^{\otimes 2}\,\longrightarrow\,K_{X}$

of line bundles. Since

$H^{1}(X,\,\text{Hom}(K^{-1/2}_{X}\otimes{\mathbb{L}},\,K^{1/2}_{X}\otimes{\mathbb{L}}))\,=\,H^{1}(X,\,K_{X})\,=\,H^{0}(X,\,{\mathcal{O}}_{X})^{*}\,=\,\mathbb{C}$

(the isomorphism $H^{1}(X,\,K_{X})\,=\,H^{0}(X,\,{\mathcal{O}}_{X})^{*}$ is
given by Serre duality), we have a nontrivial extension

$0\,\longrightarrow\,K^{1/2}_{X}\otimes{\mathbb{L}}\,\longrightarrow\,\mathcal{E}\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,K^{-1/2}_{X}\otimes{\mathbb{L}}\,\longrightarrow\,0$
(4.1)

corresponding to
$1\,\in\,H^{1}(X,\,\text{Hom}(K^{-1/2}_{X}\otimes{\mathbb{L}},\,K^{1/2}_{X}\otimes{\mathbb{L}}))\,=\,\mathbb{C}$.

We will put a parabolic structure on the rank two vector bundle $\mathcal{E}$
in (4.1). Fix a function

$\mathbf{c}\,:\,S\,\longrightarrow\,\\{t\,\in\,{\mathbb{Z}}\,\mid\,t\,\geq\,2\\}\,;$
(4.2)

the integer ${\mathbf{c}}(x_{i})$ will also be denoted by $c_{i}$.

Equip the vector bundle $\mathcal{E}$ in (4.1) with the following parabolic
structure over $S$: For any $x_{i}\,\in\,S$, the quasiparabolic filtration of
$\mathcal{E}_{x_{i}}$ is

$(K^{1/2}_{X}\otimes{\mathbb{L}})_{x_{i}}\,\subset\,\mathcal{E}_{x_{i}}\,,$

where $K^{1/2}_{X}\otimes{\mathbb{L}}$ is the line subbundle in (4.1). The
parabolic weight of $(K^{1/2}_{X}\otimes{\mathbb{L}})_{x_{i}}$ is
$\frac{2c_{i}-1}{2c_{i}}$ and the parabolic weight of $\mathcal{E}_{x_{i}}$ is
$\frac{1}{2c_{i}}$ (see (4.2)). The resulting parabolic vector bundle of rank
two on $X$ will be denoted by $\mathcal{E}_{*}$. The parabolic degree of
$\mathcal{E}_{*}$ is zero. We note that the parabolic exterior product
$\bigwedge^{2}\mathcal{E}_{*}$ of $\mathcal{E}_{*}$ is the trivial holomorphic
line bundle on $X$ equipped with the trivial holomorphic structure; see [BDP,
Section 5], [Bi2] for the parabolic exterior product and the parabolic
symmetric product.

######  Remark 4.1.

It can be shown that the above rank two parabolic bundle $\mathcal{E}_{*}$ is
indecomposable. To prove this assume that $\mathcal{E}_{*}\,=\,L_{*}\oplus
L^{\prime}_{*}$, where $L_{*}$ and $L^{\prime}_{*}$ are parabolic line
bundles. Suppose that ${\rm par-deg}(L_{*})\,\geq\,{\rm par-
deg}(L^{\prime}_{*})$. Consider the composition of homomorphism

$L_{*}\,\hookrightarrow\,\mathcal{E}_{*}\,\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\,(K^{-1/2}_{X}\otimes{\mathbb{L}})_{*}\,,$

where $q$ is the projection in (4.1) and
$(K^{-1/2}_{X}\otimes{\mathbb{L}})_{*}$ is the quotient line bundle
$K^{-1/2}_{X}\otimes{\mathbb{L}}$ in (4.1) equipped with the parabolic
structure induced by $\mathcal{E}_{*}$. This composition of homomorphisms
vanishes identically, because ${\rm par-deg}(L_{*})\,>\,{\rm par-
deg}((K^{-1/2}_{X}\otimes{\mathbb{L}})_{*})$. This implies that the short
exact sequence in (4.1) splits. But the short exact sequence in (4.1) does not
split. In view of this contradiction we conclude that the parabolic bundle
$\mathcal{E}_{*}$ is indecomposable.

We note that a connection on a parabolic vector bundle $V_{*}$ induces
connections on $\bigwedge^{i}V_{*}$ and $\text{Sym}^{i}(V_{*})$ for every $i$.

Take any integer $r\,\geq\,2$. Let $\text{Sym}^{r-1}(\mathcal{E}_{*})$ be the
parabolic symmetric product of $\mathcal{E}_{*}$. We have
$\text{rank}(\text{Sym}^{r-1}(\mathcal{E}_{*}))\,=\,r$, and the parabolic
exterior product $\bigwedge^{r}\text{Sym}^{r-1}(\mathcal{E}_{*})$ is the
trivial holomorphic line bundle on $X$ equipped with the trivial parabolic
structure.

A parabolic ${\rm SL}(r,{\mathbb{C}})$–oper on $X$ is a connection $D$ on the
parabolic vector bundle $\text{Sym}^{r-1}(\mathcal{E}_{*})$ such that the
connection on $\bigwedge^{r}\text{Sym}^{r-1}(\mathcal{E}_{*})$ induced by $D$
coincides with the connection on ${\mathcal{O}}_{X}$ given by the de Rham
differential $d$; see [BDP, p. 511, Definition 5.2].

######  Lemma 4.2.

For any parabolic ${\rm SL}(r,{\mathbb{C}})$–oper connection $D$, the local
monodromy of $D$ around any point of $S$ is semisimple.

###### Proof.

First note that the parabolic $\mathcal{E}_{*}$ admits a connection, because
$\mathcal{E}_{*}$ is indecomposable (this was shown in Remark 4.1) and its
parabolic degree is zero [BL, p. 594, Theorem 1.1]. Since the two parabolic
weights at every $x_{i}\,\in\,S$ do not differ by an integer, we conclude that
for any connection on the parabolic $\mathcal{E}_{*}$ the local monodromy
around any point of $S$ is semisimple (see (2.17)). Therefore, the connection
on $\text{Sym}^{r-1}(\mathcal{E}_{*})$ induced by a connection on
$\mathcal{E}_{*}$ also has the property that the local monodromy around every
point of $S$ is semisimple. From this it follows that every connection on
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ has the property that the local monodromy
around every point of $S$ is semisimple; see Lemma 3.2. ∎

The subbundle $K^{1/2}_{X}\otimes{\mathbb{L}}\,\subset\,\mathcal{E}$ in (4.1)
equipped with the induced parabolic structure will be denoted by $F_{*}$. So
the holomorphic line bundle underlying $F_{*}$ is
$K^{1/2}_{X}\otimes{\mathbb{L}}$, and the parabolic weight of $F_{*}$ and any
$x_{i}\,\in\,S$ is $\frac{2c_{i}-1}{2c_{i}}$. The parabolic subbundle

$F_{*}\,\stackrel{{\scriptstyle\iota_{F}}}{{\hookrightarrow}}\,\mathcal{E}_{*}$
(4.3)

produces a filtration of parabolic subbundles of
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ in the following way. For any
$0\,\leq\,j\,\leq\,r-1$, consider the parabolic vector bundle
$(F_{*})^{\otimes(r-1-j)}\otimes\text{Sym}^{j}(\mathcal{E}_{*})$; by
definition, $(F_{*})^{\otimes 0}$ and $\text{Sym}^{0}(\mathcal{E}_{*})$
coincide with the trivial holomorphic line bundle with the trivial parabolic
structure (see [Bi2], [BDP] for the tensor product of parabolic vector
bundles). So the rank of
$(F_{*})^{\otimes(r-1-j)}\otimes\text{Sym}^{j}(\mathcal{E}_{*})$ is $j+1$.
This parabolic vector bundle
$(F_{*})^{\otimes(r-1-j)}\otimes\text{Sym}^{j}(\mathcal{E}_{*})$ will be
denoted by ${\mathcal{F}}^{(j+1)}_{*}$ (since $j+1$ is its rank). We note that

${\mathcal{F}}^{(1)}_{*}\,\subset\,{\mathcal{F}}^{(2)}_{*}\,\subset\,\cdots\,\subset\,{\mathcal{F}}^{(r-1)}_{*}\,\subset\,{\mathcal{F}}^{(r)}_{*}\,=\,\text{Sym}^{r-1}(\mathcal{E}_{*})$
(4.4)

is a filtration of parabolic subbundles of
$\text{Sym}^{r-1}(\mathcal{E}_{*})$. For any $1\,\leq\,i\,\leq\,r-1$, the
inclusion map

${\mathcal{F}}^{(i)}_{*}\,\hookrightarrow\,{\mathcal{F}}^{(i+1)}_{*}$

in (4.4) is constructed, in a straightforward way, using the inclusion map
$F_{*}\,\hookrightarrow\,\mathcal{E}_{*}$ in (4.3) together with the natural
projection
$\mathcal{E}_{*}\otimes\text{Sym}^{j}(\mathcal{E}_{*})\,\longrightarrow\,\text{Sym}^{j+1}(\mathcal{E}_{*})$.
More, precisely, we have

$(\iota_{F})^{\otimes(r-i)}\,\,:\,\,(F_{*})^{\otimes(r-i)}\,\hookrightarrow\,(F_{*})^{\otimes(r-i-1)}\otimes\mathcal{E}_{*},\,$

where $\iota_{F}$ is the inclusion map in (4.3). This implies that

${\mathcal{F}}^{(i)}_{*}\,=\,(F_{*})^{\otimes(r-i)}\otimes\text{Sym}^{i-1}(\mathcal{E}_{*})\,\hookrightarrow\,(F_{*})^{\otimes(r-i-1)}\otimes\mathcal{E}_{*}\otimes\text{Sym}^{i-1}(\mathcal{E}_{*})$
$\longrightarrow\,(F_{*})^{\otimes(r-i-1)}\otimes\text{Sym}^{i}(\mathcal{E}_{*})\,=\,{\mathcal{F}}^{(i+1)}_{*}\,.$

The above composition of homomorphisms produces the inclusion map
${\mathcal{F}}^{(i)}_{*}\,\subset\,{\mathcal{F}}^{(i+1)}_{*}$ in (4.4).

For any $1\,\leq\,j\,\leq\,r$, the holomorphic vector bundle of rank $j$
underlying the parabolic vector bundle ${\mathcal{F}}^{(j)}_{*}$ will be
denoted by ${\mathcal{F}}^{(j)}$. So the filtration in (4.4) produce a
filtration of holomorphic subbundles

${\mathcal{F}}^{(1)}\,\subset\,{\mathcal{F}}^{(2)}\,\subset\,\cdots\,\subset\,{\mathcal{F}}^{(r-1)}\,\subset\,{\mathcal{F}}^{(r)}$
(4.5)

of ${\mathcal{F}}^{(r)}$. Note that ${\mathcal{F}}^{(r)}$ is the holomorphic
vector bundle underlying the parabolic vector bundle
$\text{Sym}^{r-1}(\mathcal{E}_{*})$.

######  Remark 4.3.

It should be clarified that although the holomorphic vector bundle underlying
the parabolic bundle $\mathcal{E}_{*}$, namely the holomorphic vector bundle
$\mathcal{E}$, does not depend on the function $\mathbf{c}$ in (4.2), the
vector bundle ${\mathcal{F}}^{(r)}$ depends on $\mathbf{c}$ in general. It
should also be mentioned that the fact that the parabolic exterior product
$\bigwedge^{r}\text{Sym}^{r-1}(\mathcal{E}_{*})$ is the trivial holomorphic
line bundle equipped with the trivial parabolic structure does not imply that
$\bigwedge^{r}{\mathcal{F}}^{(r)}$ is the trivial holomorphic line bundle. In
fact we have
$\text{degree}\left(\bigwedge^{r}{\mathcal{F}}^{(r)}\right)\,<\,0$. Note that
$\text{degree}(\mathcal{E})\,=\,-n$.

######  Remark 4.4.

It is a straightforward computation to check that

$\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\,=\,{\mathcal{O}}_{X}(-\sum_{i=1}^{n}d_{i}x_{i})\,,$

where

$d_{i}\,=\,\sum_{k=0}^{r-1}\left(\frac{2k(c_{i}-1)+r-1}{2c_{i}}-\Big{\lfloor}{\frac{2k(c_{i}-1)+r-1}{2c_{i}}}\Big{\rfloor}\right)\,;$

the integral part of $b\,\in\,\mathbb{Q}$ is denoted $\lfloor{b}\rfloor$, so
$\lfloor{b}\rfloor\,\in\,\mathbb{Z}$ and $0\,\leq\,b-\lfloor{b}\rfloor\,<\,1$.
It is straightforward to check that each $d_{i}$ is an integer.

Consider the Atiyah bundle

$\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\subset\,\text{Diff}^{1}\left({\mathcal{F}}^{(r)},\,{\mathcal{F}}^{(r)}\right)$

constructed as in (2.11) for the parabolic bundle
$E_{*}\,=\,\text{Sym}^{r-1}(\mathcal{E}_{*})$, where ${\mathcal{F}}^{(r)}$ is
the vector bundle in (4.5) and $\text{Sym}^{r-1}(\mathcal{E}_{*})$ is the
parabolic vector bundle in (4.4). Let

$\text{At}^{\prime}_{X}(r)\,\subset\,\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\subset\,\text{Diff}^{1}\left({\mathcal{F}}^{(r)},\,{\mathcal{F}}^{(r)}\right)$
(4.6)

be the holomorphic subbundle of
$\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ constructed as
follows: The space of all holomorphic sections of $\text{At}^{\prime}_{X}(r)$
over any open subset $U\,\subset\,X$ is the space of all first order
differential operations

$D_{U}\,:\,\Gamma\left(U,\,\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\longrightarrow\,\Gamma\left(U,\,\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$

such that

  * •

$D_{U}\,\in\,\Gamma\left(U,\,\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)$,
and

  * •

$D_{U}\left({\mathcal{F}}^{(j)}\right)\,\subset\,{\mathcal{F}}^{(j)}$ for all
$1\,\leq\,j\,\leq\,r$, where ${\mathcal{F}}^{(j)}$ is the subbundle in (4.5).

Consider the subbundle

${\mathcal{O}}_{X}\,\subset\,\text{Hom}\left({\mathcal{F}}^{(r)},\,{\mathcal{F}}^{(r)}\right)\,=\,\text{Diff}^{0}\left({\mathcal{F}}^{(r)},\,{\mathcal{F}}^{(r)}\right)\,\subset\,\text{Diff}^{1}\left({\mathcal{F}}^{(r)},\,{\mathcal{F}}^{(r)}\right)$

given by pointwise multiplication. We note that

${\mathcal{O}}_{X}\,\subset\,\text{At}^{\prime}_{X}(r)\,,$

where $\text{At}^{\prime}_{X}(r)$ is the subsheaf in (4.6). Let

$\text{At}_{X}(r)\,:=\,\text{At}^{\prime}_{X}(r)/{\mathcal{O}}_{X}$ (4.7)

be the quotient bundle.

Construct the holomorphic vector bundle
$\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ by
substituting, in (2.6), the parabolic vector bundle
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ in place of $E_{*}$. Let

$\text{ad}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\subset\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$

be the holomorphic subbundle of co-rank one given by the intersection of
$\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ with the sheaf
of endomorphisms of ${\mathcal{F}}^{(r)}$ of trace zero. So using the natural
inclusion map
${\mathcal{O}}_{X}(-S)\,\hookrightarrow\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$
defined by pointwise multiplication, we have

$\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,=\,\text{ad}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\oplus{\mathcal{O}}_{X}(-S)\,.$
(4.8)

Let

$\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\subset\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$

be the subbundle defined by imposing the condition that the subbundle
${\mathcal{F}}^{(i)}$ in (4.5) is mapped to ${\mathcal{F}}^{(i+1)}$ for all
$1\,\leq\,i\,\leq\,r-1$. In other words, a locally defined holomorphic section
$s$ of the vector bundle $\text{End}^{n}(\text{Sym}^{r-1}(\mathcal{E}_{*}))$
is a locally defined section of
$\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ if and only
if $s\left({\mathcal{F}}^{(i)}\right)\,\subset\,{\mathcal{F}}^{(i+1)}$ for
every $1\,\leq\,i\,\leq\,r-1$. Define the intersection

$\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,:=\,\text{ad}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\bigcap\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,;$
(4.9)

this intersection is taking place inside
$\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$. From (4.8) we
have

$\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,=\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\oplus{\mathcal{O}}_{X}(-S)\,.$
(4.10)

Now let $D$ be a connection on the parabolic vector bundle
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ defining a parabolic ${\rm
SL}(r,{\mathbb{C}})$–oper on $X$. In other words, the connection on
$\bigwedge^{r}\text{Sym}^{r-1}(\mathcal{E}_{*})$ induced by $D$ coincides with
the connection on ${\mathcal{O}}_{X}$ given by the de Rham differential $d$.
Consider the first order holomorphic differential operator

${\mathcal{D}}\,:\,\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\longrightarrow\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ (4.11)

constructed as in (2.24) from $D$; substitute
$(\text{Sym}^{r-1}(\mathcal{E}_{*}),\,D)$ in place of $(E_{*},\,D)$ in (2.24).
The restriction of the differential operator ${\mathcal{D}}$ to the subbundle
$\text{At}^{\prime}_{X}(r)$ constructed in (4.6) has its image contained in

$\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\subset\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

(see (4.10)), in other words,

${\mathcal{D}}(\text{At}^{\prime}_{X}(r))\,\subset\,\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,.$ (4.12)

Furthermore, the differential operator ${\mathcal{D}}$ takes the subbundle
${\mathcal{O}}_{X}\,\subset\,\text{At}^{\prime}_{X}(r)$ (see (4.7)) to

$K_{X}\,=\,{\mathcal{O}}_{X}(-S)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\subset\,\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,;$

see (4.10) for the subbundle
${\mathcal{O}}_{X}(-S)\,\subset\,\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$.
In fact, the restriction of ${\mathcal{D}}$ to
${\mathcal{O}}_{X}\,\subset\,\text{At}^{\prime}_{X}(r)$ coincides with the de
Rham differential $d$. Consequently, using (4.12) and the decomposition in
(4.10), the differential operator
${\mathcal{D}}\big{|}_{\text{At}^{\prime}_{X}(r)}$ in (4.11) produces a
differential operator

${\mathcal{D}}_{B}\,:\,\text{At}_{X}(r)\,\longrightarrow\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,.$ (4.13)

Let ${\mathcal{C}}_{\bullet}$ denote the following two-term complex of sheaves
on $X$

${\mathcal{C}}_{\bullet}\,\,:\,\,{\mathcal{C}}_{0}\,=\,\text{At}_{X}(r)\,\stackrel{{\scriptstyle\mathcal{D}_{B}}}{{\longrightarrow}}\,{\mathcal{C}}_{1}\,=\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$ (4.14)

where $\mathcal{D}_{B}$ is the homomorphism in (4.13), and ${\mathcal{C}}_{i}$
is at the $i$-th position.

######  Theorem 4.5.

The space of all infinitesimal deformation of the triple $(X,\,S,\,D)$, where
$D$ is a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper on $X$, is given by the
hypercohomology

${\mathbb{H}}^{1}({\mathcal{C}}_{\bullet}),\,$

where ${\mathcal{C}}_{\bullet}$ is the complex in (4.14).

###### Proof.

As in (3.1), take a ramified Galois covering

$\varphi\,:\,Y\,\longrightarrow\,X$ (4.15)

satisfying the following two conditions:

  * •

$\varphi$ is unramified over the complement $X\setminus S$, and

  * •

for every $x_{i}\,\in\,S$ and each point $y\,\in\,\varphi^{-1}(x_{i})$, the
order of ramification of $\rho$ at $y$ is $c_{i}\,=\,\mathbf{c}(x_{i})$ (see
(4.2)).

Such a covering $\varphi$ exists by [Na, p. 26, Proposition 1.2.12]. As
before, denote by $\Gamma_{\varphi}$ the Galois group $\text{Gal}(\varphi)$ of
the map $\varphi$. From [BDP, p. 514, Theorem 6.3] we know that the parabolic
${\rm SL}(r,{\mathbb{C}})$–oper $D$ on $X$ corresponds to a $\Gamma$–invariant
${\rm PSL}(r,{\mathbb{C}})$–oper $\mathbb{D}$ on $Y$. We note that there is a
natural bijection between the ${\rm PSL}(r,{\mathbb{C}})$–opers on $Y$ and
each connected component of the ${\rm SL}(r,{\mathbb{C}})$–opers on $Y$. We
will recall a description of this ${\rm PSL}(r,{\mathbb{C}})$–oper
$\mathbb{D}$ on $Y$ corresponding to $D$.

We first note that the parabolic vector bundle
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ defines a holomorphic parabolic principal
$\text{SL}(r,{\mathbb{C}})$–bundle on $X$, because
$\bigwedge^{r}\text{Sym}^{r-1}(\mathcal{E}_{*})$ is the trivial parabolic line
bundle; see [BBN1], [BBN2], [BBP] for the parabolic analog of principal
bundles. Using the quotient map
$\text{SL}(r,{\mathbb{C}})\,\longrightarrow\,\text{PSL}(r,{\mathbb{C}})$, a
parabolic principal $\text{SL}(r,{\mathbb{C}})$–bundle produces parabolic
principal $\text{PSL}(r,{\mathbb{C}})$–bundle. Let

$\widetilde{\mathbb{P}}_{*}\,\longrightarrow\,X$ (4.16)

denote the parabolic principal $\text{PSL}(r,{\mathbb{C}})$–bundle given by
the parabolic principal $\text{SL}(r,{\mathbb{C}})$–bundle on $X$ defined by
$\text{Sym}^{r-1}(\mathcal{E}_{*})$.

Fix a Borel subgroup

$B\,\subset\,\text{PSL}(r,{\mathbb{C}})\,.$ (4.17)

The filtration of parabolic vector bundles
$\\{{\mathcal{F}}^{(j)}_{*}\\}_{j=1}^{r-1}$ in (4.4) produces a reduction of
structure group of the parabolic principal $\text{PSL}(r,{\mathbb{C}})$–bundle
$\widetilde{\mathbb{P}}_{*}$ in (4.16) to the subgroup $B$ of
$\text{PSL}(r,{\mathbb{C}})$ in (4.17). Let

$\widetilde{\mathbb{P}}(B)_{*}\,\subset\,\widetilde{\mathbb{P}}_{*}$ (4.18)

be this reduction of structure group to $B$.

The parabolic principal $\text{PSL}(r,{\mathbb{C}})$–bundle
$\widetilde{\mathbb{P}}_{*}$ in (4.16) corresponds to an equivariant
holomorphic principal $\text{PSL}(r,{\mathbb{C}})$–bundle on $Y$ [BBN2],
[BBN1]; let

${\mathbb{P}}(r)\,\longrightarrow\,Y$ (4.19)

be the equivariant holomorphic principal $\text{PSL}(r,{\mathbb{C}})$–bundle
corresponds to $\widetilde{\mathbb{P}}_{*}$. We note that ${\mathbb{P}}(r)$ is
the unique holomorphic principal $\text{PSL}(r,{\mathbb{C}})$–bundle on $Y$
underlying the $\text{PSL}(r,{\mathbb{C}})$–opers on $Y$. We will briefly
recall a description of ${\mathbb{P}}(r)$.

Take a theta characteristic $K^{1/2}_{Y}$ on $Y$. Let

$0\,\longrightarrow\,K^{1/2}_{Y}\,\longrightarrow\,W\,\longrightarrow\,\left(K^{1/2}_{Y}\right)^{*}\,\longrightarrow\,0$
(4.20)

be the nontrivial extension corresponding to

$1\,\in\,{\mathbb{C}}\,=\,H^{0}(Y,\,{\mathcal{O}}_{Y})^{*}\,=\,H^{1}(Y,\,K_{Y})\,=\,H^{1}(Y,\,\text{Hom}((K^{1/2}_{Y})^{*},\,K^{1/2}_{Y}))\,.$

The holomorphic principal $\text{PSL}(r,{\mathbb{C}})$–bundle
${\mathbb{P}}(r)$ in (4.19) coincides with the holomorphic principal
$\text{PSL}(r,{\mathbb{C}})$–bundle defined by the projective bundle
${\mathbb{P}}\left(\text{Sym}^{r-1}(W)\right)$. (By $\mathbb{P}(V)$ we denote
the projective bundle defined by the spaces of lines in the fibers of $V$.)
While the vector bundle $\text{Sym}^{r-1}(W)$ depends on the choice of the
theta characteristic $K^{1/2}_{Y}$ when $r$ is even (the vector bundle
$\text{Sym}^{r-1}(W)$ does not depend on the choice of $K^{1/2}_{Y}$ when $r$
is odd), the projective bundle ${\mathbb{P}}\left(\text{Sym}^{r-1}(W)\right)$
is actually independent of the choice of the theta characteristic
$K^{1/2}_{Y}$. Indeed, if we replace $K^{1/2}_{Y}$ by $K^{1/2}_{Y}\otimes\xi$,
where $\xi$ is a holomorphic line bundle on $Y$ of order two, then $W$ gets
replaced by $W\otimes\xi$, and hence $\text{Sym}^{r-1}(W)$ gets replaced by
$\text{Sym}^{r-1}(W)\otimes\xi^{\otimes(r-1)}$.

Since any holomorphic automorphism of $Y$ takes a theta characteristic on $Y$
to a (possibly different) theta characteristic on $Y$, from the above property
of ${\mathbb{P}}\left(\text{Sym}^{r-1}(W)\right)$ it follows immediately that
${\mathbb{P}}(r)$ is an equivariant holomorphic principal
$\text{PSL}(r,{\mathbb{C}})$–bundle on $Y$. In other words, the action of
$\Gamma_{\varphi}$ on $Y$ lifts to an holomorphic action of $\Gamma_{\varphi}$
on ${\mathbb{P}}(r)$ that commutes with the action of
$\text{PSL}(r,{\mathbb{C}})$ on the principal bundle ${\mathbb{P}}(r)$.

In particular, ${\mathbb{P}}(W)$ is an equivariant vector bundle. The action
of $\Gamma_{\varphi}$ on ${\mathbb{P}}(W)$ preserves the holomorphic section
of the projective bundle ${\mathbb{P}}(W)$ given by the line subbundle
$K^{1/2}_{Y}$ in (4.20).

The connection $D$ on the parabolic vector bundle
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ produces a connection on the parabolic
principal $\text{PSL}(r,{\mathbb{C}})$–bundle $\widetilde{\mathbb{P}}_{*}$ on
$X$ given by ${\mathbb{P}}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$.
This connection on $\widetilde{\mathbb{P}}_{*}$ in turn produces a
$\Gamma_{\varphi}$–invariant holomorphic connection on the principal
$\text{PSL}(r,{\mathbb{C}})$–bundle ${\mathbb{P}}(r)$ on $Y$. This
$\Gamma_{\varphi}$–invariant holomorphic connection on ${\mathbb{P}}(r)$ will
be denoted by

$D^{Y}.$ (4.21)

Any holomorphic connection on ${\mathbb{P}}(r)$ is a
$\text{PSL}(r,{\mathbb{C}})$–oper on $Y$ [BD1], [BD2], [Fr1]. In particular,
$D^{Y}$ in (4.21) is a $\text{PSL}(r,{\mathbb{C}})$–oper on $Y$. Given an
oper, a complex of sheaves was constructed by Sanders in [Sa]. We will briefly
recall his construction of complex of sheaves for the
$\text{PSL}(r,{\mathbb{C}})$–oper $D^{Y}$ in (4.21).

Let

${\mathcal{W}}_{1}\,\subset\,{\mathcal{W}}_{2}\,\subset\,\cdots\,\subset\,{\mathcal{W}}_{r-1}\,\subset\,{\mathcal{W}}_{r}\,=\,\text{Sym}^{r-1}(W)$

be the filtration of holomorphic subbundles, where
${\mathcal{W}}_{j}\,=\,(K^{1/2}_{Y})^{\otimes(r-j)}\otimes\text{Sym}^{j-1}(W)$
(see (4.20)); in particular ${\rm rank}({\mathcal{W}}_{j})\,=\,j$. Let

${\mathbb{P}}({\mathcal{W}}_{1})\,\subset\,{\mathbb{P}}({\mathcal{W}}_{2})\,\subset\,\cdots\,\subset\,{\mathbb{P}}({\mathcal{W}}_{r-1})\,\subset\,{\mathbb{P}}({\mathcal{W}}_{r})\,=\,{\mathbb{P}}(\text{Sym}^{r-1}(W))$
(4.22)

be the corresponding filtration of projective bundles; recall that
$\mathbb{P}(V)$ denotes the projective bundle defined by the spaces of lines
in the fibers of $V$. The filtration in (4.22) produces a holomorphic
reduction of structure group

${\mathbb{P}}(r)_{B}\,\subset\,{\mathbb{P}}(r)$ (4.23)

to the Borel subgroup $B$ in (4.17), where ${\mathbb{P}}(r)$ is the
holomorphic principal $\text{PSL}(r,{\mathbb{C}})$–bundle in (4.19).

Recall that ${\mathbb{P}}(W)$ is an equivariant vector bundle, and the action
of $\Gamma_{\varphi}$ on ${\mathbb{P}}(W)$ preserves the holomorphic section
of ${\mathbb{P}}(W)$ given by the line subbundle $K^{1/2}_{Y}$ in (4.20).
Therefore, from the construction of the filtration in (4.22) it follows
immediately that the action of $\Gamma_{\varphi}$ on
${\mathbb{P}}(\text{Sym}^{r-1}(W))$ preserves each projective subbundle
${\mathbb{P}}({\mathcal{W}}_{j})$. Consequently, the action of
$\Gamma_{\varphi}$ on ${\mathbb{P}}(r)$ preserves the reduction of the
structure group ${\mathbb{P}}(r)_{B}$ in (4.23). The principal $B$–bundle
${\mathbb{P}}(r)_{B}$ in (4.23) in fact corresponds to the reduction
$\widetilde{\mathbb{P}}(B)_{*}$ in (4.18).

The holomorphic reduction ${\mathbb{P}}(r)_{B}$ in (4.23) coincides with the
holomorphic reduction of structure group of ${\mathbb{P}}(r)$ to the subgroup
$B\,\subset\,\text{PSL}(r,{\mathbb{C}})$ that appears in the definition of a
$\text{PSL}(r,{\mathbb{C}})$–oper on $Y$. Let

$\text{At}({\mathbb{P}}(r)_{B})\,\longrightarrow\,Y$

be the Atiyah bundle for the holomorphic principal $B$–bundle
${\mathbb{P}}(r)_{B}$ (see [At]). Let

$\text{At}({\mathbb{P}}(r))\,\longrightarrow\,Y$

be the Atiyah bundle for the principal $\text{PSL}(r,{\mathbb{C}})$–bundle
${\mathbb{P}}(r)$ in (4.19). We have

$\text{At}({\mathbb{P}}(r)_{B})\,\subset\,\text{At}({\mathbb{P}}(r))$ (4.24)

because of the reduction of structure group in (4.23).

Let $\text{ad}({\mathbb{P}}(r))\,\longrightarrow\,Y$ be the adjoint bundle of
the holomorphic principal $\text{PSL}(r,{\mathbb{C}})$–bundle
${\mathbb{P}}(r)$. We recall that $\text{ad}({\mathbb{P}}(r))$ is the
holomorphic vector bundle associated to the principal
$\text{PSL}(r,{\mathbb{C}})$–bundle ${\mathbb{P}}(r)$ for the adjoint action
of $\text{PSL}(r,{\mathbb{C}})$ on its Lie algebra. We will describe
$\text{ad}({\mathbb{P}}(r))$ explicitly. Let

$\gamma\,:\,{\mathbb{P}}(\text{Sym}^{r-1}(W))\,\longrightarrow\,Y$

be the natural projection. Let

$T_{\gamma}\,\subset\,T{\mathbb{P}}(\text{Sym}^{r-1}(W))$

be the relative holomorphic tangent bundle for the projection $\gamma$,
meaning $T_{\gamma}$ is the kernel of the differential
$d\gamma\,:\,T{\mathbb{P}}(\text{Sym}^{r-1}(W))\,\longrightarrow\,\gamma^{*}TY$
of $\gamma$. Then we have

$\text{ad}({\mathbb{P}}(r))\,=\,\gamma_{*}T_{\gamma}\,.$ (4.25)

Given any holomorphic connection on the principal
$\text{PSL}(r,{\mathbb{C}})$–bundle ${\mathbb{P}}(r)$, there is a holomorphic
differential operator of order one from $\text{At}({\mathbb{P}}(r))$ to
$\text{ad}({\mathbb{P}}(r))\otimes K_{Y}$ [Ch2, p. 1415, (1)], [Ch2, p. 1415,
Proposition 4.4]. Consider the differential operator

$\text{At}({\mathbb{P}}(r))\,\longrightarrow\,\text{ad}({\mathbb{P}}(r))\otimes
K_{Y}$

corresponding to the connection $D^{Y}$ in (4.21). Let

$\widetilde{D}^{Y}\,:\,\text{At}({\mathbb{P}}(r)_{B})\,\longrightarrow\,\text{ad}({\mathbb{P}}(r))\otimes
K_{Y}$ (4.26)

be its restriction to the subbundle $\text{At}({\mathbb{P}}(r)_{B})$ (see
(4.24)).

Let $\text{ad}({\mathbb{P}}(r)_{B})\,\longrightarrow\,Y$ be the adjoint bundle
for the holomorphic principal $B$–bundle ${\mathbb{P}}(r)_{B}$. Note that we
have

$\text{ad}({\mathbb{P}}(r)_{B})\,\subset\,\text{ad}({\mathbb{P}}(r))$

because of the reduction of structure group in (4.23). Recall the description
of $\text{ad}({\mathbb{P}}(r))$ in (4.25). For any $y\,\in\,Y$, the subspace

$\text{ad}({\mathbb{P}}(r)_{B})_{y}\,\subset\,\text{ad}({\mathbb{P}}(r))_{y}$

consists of all holomorphic vector fields $v$ on
${\mathbb{P}}(\text{Sym}^{r-1}(W))_{y}$ satisfying the following condition:
for any $1\,\leq\,j\,\leq\,r-1$, and any

$z\,\in\,{\mathbb{P}}({\mathcal{W}}_{j})_{y}\,\subset\,{\mathbb{P}}(\text{Sym}^{r-1}(W))_{y}$

(see (4.22)),

$v(z)\,\in\,T_{z}{\mathbb{P}}({\mathcal{W}}_{j})_{y}\,,$

note that
$T_{z}{\mathbb{P}}({\mathcal{W}}_{j})_{y}\,\subset\,T_{z}{\mathbb{P}}(\text{Sym}^{r-1}(W))_{y}$
because
${\mathbb{P}}({\mathcal{W}}_{j})_{y}\,\subset\,{\mathbb{P}}(\text{Sym}^{r-1}(W))_{y}$.
Let

$R_{n}(\text{ad}({\mathbb{P}}(r)_{B}))\,\subset\,\text{ad}({\mathbb{P}}(r)_{B})$

be the subbundle given by the nilpotent radical bundle; so for any
$y\,\in\,Y$, the subspace
$R_{n}(\text{ad}({\mathbb{P}}(r)_{B}))_{y}\,\subset\,\text{ad}({\mathbb{P}}(r)_{B})_{y}$
is the nilpotent radical. So, for any $y\,\in\,Y$, we have

$R_{n}(\text{ad}({\mathbb{P}}(r)_{B}))_{y}\,=\,[\text{ad}({\mathbb{P}}(r)_{B})_{y},\,\text{ad}({\mathbb{P}}(r)_{B})_{y}]\,.$

Let

$[R_{n}(\text{ad}({\mathbb{P}}(r)_{B}),\,R_{n}(\text{ad}({\mathbb{P}}(r)_{B})]\,\subset\,R_{n}(\text{ad}({\mathbb{P}}(r)_{B})$

be the commutator. For any $y\,\in\,Y$, we have

$[R_{n}(\text{ad}({\mathbb{P}}(r)_{B}),\,R_{n}(\text{ad}({\mathbb{P}}(r)_{B})]_{y}\,=\,[R_{n}(\text{ad}({\mathbb{P}}(r)_{B})_{y},\,R_{n}(\text{ad}({\mathbb{P}}(r)_{B})_{y}]\,.$

Let

$\text{ad}_{1}({\mathbb{P}}(r)_{B})\,:=\,[R_{n}(\text{ad}({\mathbb{P}}(r)_{B}),\,R_{n}(\text{ad}({\mathbb{P}}(r)_{B})]^{\perp}\,\subset\,\text{ad}({\mathbb{P}}(r))$
(4.27)

be the annihilator of
$[R_{n}(\text{ad}({\mathbb{P}}(r)_{B})),\,R_{n}(\text{ad}({\mathbb{P}}(r)_{B}))]$
for the fiberwise Killing form on the adjoint bundle
$\text{ad}({\mathbb{P}}(r))$. The vector bundle
$\text{ad}_{1}({\mathbb{P}}(r)_{B})$ has the following description in terms of
the isomorphism in (4.25). For any $y\,\in\,Y$, a holomorphic vector field $v$
on ${\mathbb{P}}(\text{Sym}^{r-1}(W))_{y}$ lies in the fiber
$\text{ad}_{1}({\mathbb{P}}(r)_{B})_{y}$ if and only if the following
condition holds: for any $1\,\leq\,j\,\leq\,r-1$ and any
$z\,\in\,{\mathbb{P}}({\mathcal{W}}_{j})_{y}$,

$v(z)\,\in\,T_{z}{\mathbb{P}}({\mathcal{W}}_{j+1})_{y}\,.$

Since the connection $D^{Y}$ in (4.21) is a $\text{PSL}(r,{\mathbb{C}})$–oper
on $Y$, the image of the differential operator $\widetilde{D}^{Y}$ in (4.26)
is contained in the subbundle

$\text{ad}_{1}({\mathbb{P}}(r)_{B})\otimes
K_{Y}\,\subset\,\text{ad}({\mathbb{P}}(r))\otimes K_{Y}$

defined in (4.27). Therefore, $\widetilde{D}^{Y}$ defines a differential
operator

${D}^{Y}_{1}\,:\,\text{At}({\mathbb{P}}(r)_{B})\,\longrightarrow\,\text{ad}_{1}({\mathbb{P}}(r)_{B})\otimes
K_{Y}\,;$ (4.28)

see [Sa, (5.8)]. Let ${\mathcal{H}}_{\bullet}$ be the following two-term
complex of sheaves on $Y$

${\mathcal{H}}_{\bullet}\,\,:\,\,{\mathcal{H}}_{0}\,=\,\text{At}({\mathbb{P}}(r)_{B})\,\stackrel{{\scriptstyle
D^{Y}_{1}}}{{\longrightarrow}}\,{\mathcal{H}}_{1}\,=\,\text{ad}_{1}({\mathbb{P}}(r)_{B})\otimes
K_{Y}\,,$ (4.29)

where $D^{Y}_{1}$ is the homomorphism in (4.28), and ${\mathcal{H}}_{i}$ is at
the $i$-th position.

The space of all infinitesimal deformations of the
$\text{PSL}(r,{\mathbb{C}})$–oper $(Y,\,D^{Y})$ is given by the
hypercohomology ${\mathbb{H}}^{1}({\mathcal{H}}_{\bullet})$, where
${\mathcal{H}}_{\bullet}$ is the complex constructed in (4.29) [Sa, Theorem
5.9].

We noted earlier that the action of $\Gamma_{\varphi}$ on ${\mathbb{P}}(r)$
preserves the reduction of structure group ${\mathbb{P}}(r)_{B}$ in (4.23).
Since ${\mathbb{P}}(r)_{B}$ is an equivariant bundle, we conclude that that
both $\text{At}({\mathbb{P}}(r)_{B})$ and $\text{ad}({\mathbb{P}}(r)_{B})$ are
equivariant vector bundles. Hence $R_{n}(\text{ad}({\mathbb{P}}(r)_{B}))$ in
(4.27) is an equivariant subbundle of $\text{ad}({\mathbb{P}}(r)_{B})$, which
in turn implies that $\text{ad}_{1}({\mathbb{P}}(r)_{B})$ in (4.27) is an
equivariant subbundle of $\text{ad}({\mathbb{P}}(r))$; the fiberwise Killing
form on $\text{ad}({\mathbb{P}}(r))$ is evidently
$\Gamma_{\varphi}$–invariant. The operator ${D}^{Y}_{1}$ in (4.28) is
$\Gamma_{\varphi}$–equivariant, because the connection $D^{Y}$ in (4.21) is
invariant under the action of $\Gamma_{\varphi}$ on ${\mathbb{P}}(r)$.
Consequently, the complex ${\mathcal{H}}_{\bullet}$ in (4.29) is
$\Gamma_{\varphi}$–equivariant. Therefore, the group $\Gamma_{\varphi}$ acts
on the hypercohomology ${\mathbb{H}}^{1}({\mathcal{H}}_{\bullet})$. Let

${\mathbb{H}}^{1}({\mathcal{H}}_{\bullet})^{\Gamma_{\varphi}}\,\subset\,{\mathbb{H}}^{1}({\mathcal{H}}_{\bullet})$
(4.30)

be the invariant part for the action of $\Gamma_{\varphi}$.

We have

$(\varphi_{*}\text{At}({\mathbb{P}}(r)_{B}))^{\Gamma_{\varphi}}\,=\,\text{At}_{X}(r)$

and

$(\varphi_{*}(\text{ad}_{1}({\mathbb{P}}(r)_{B})\otimes
K_{Y}))^{\Gamma_{\varphi}}\,=\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

(see (4.14) for $\text{At}_{X}(r)$ and
$\text{ad}^{n}_{1}(\text{Sym}^{r-1}(\mathcal{E}_{*}))\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$). Moreover, the differential operator
${D}^{Y}_{1}$ in (4.28) gives the differential operator $\mathcal{D}_{B}$ in
(4.14). Therefore, we conclude that

${\mathbb{H}}^{1}({\mathcal{C}}_{\bullet})\,=\,{\mathbb{H}}^{1}({\mathcal{H}}_{\bullet})^{\Gamma_{\varphi}}\,,$

where ${\mathcal{C}}_{\bullet}$ is the complex in (4.14) (see (4.30)). Now the
theorem follows from the earlier mentioned result of [Sa] that the
infinitesimal deformations of the $\text{PSL}(r,{\mathbb{C}})$–oper
$(Y,\,D^{Y})$ are parametrized by ${\mathbb{H}}^{1}({\mathcal{H}}_{\bullet})$.
∎

Consider the short exact sequence of holomorphic vector bundles on $X$

$0\,\longrightarrow\,{\mathcal{O}}_{X}\,\longrightarrow\,\text{At}^{\prime}_{X}(r)\,\longrightarrow\,\text{At}_{X}(r)\,\longrightarrow\,0$
(4.31)

in (4.7). We will show that it splits holomorphically.

Take any

$\delta\,\in\,\Gamma\left(U,\,\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)\,\subset\,\Gamma\left(U,\,\text{Diff}^{1}\left({\mathcal{F}}^{(r)},\,{\mathcal{F}}^{(r)}\right)\right)$

(see (4.6)), where $U\,\subset\,X$ is an open subset. Then $\delta$ produces a
holomorphic differential operator

$\widetilde{\delta}\,\in\,\Gamma\left(U,\,\text{Diff}^{1}\left(\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)},\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\right)\right)$
(4.32)

which is constructed as follows: Take any

$s\,=\,s_{1}\wedge\cdots\wedge
s_{r}\,\in\,\Gamma\left(U,\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\right)$
(4.33)

where $s_{i}\,\in\,\Gamma\left(U,\,{\mathcal{F}}^{(r)}\right)$ for all
$1\,\leq\,i\,\leq\,r$. Now define

$\widetilde{\delta}(s)\,:=\,\sum_{j=1}^{r}s_{1}\wedge\cdots\wedge
s_{j-1}\wedge\delta(s_{j})\wedge s_{j+1}\wedge\cdots\wedge
s_{r}\,\in\,\Gamma\left(U,\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\right)\,.$

It is straightforward to check that $\widetilde{\delta}(s)$ is indeed
independent of the choice of the decomposition of of the section $s$ in
(4.33).

Let

$\eta\,\,:\,\,\text{At}^{\prime}_{X}(r)\,\longrightarrow\,TX\otimes{\mathcal{O}}_{X}(-S)$
(4.34)

be the restriction of the natural projection
$\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\longrightarrow\,TX\otimes{\mathcal{O}}_{X}(-S)$
(see (2.12)) to the subbundle
$\text{At}^{\prime}_{X}(r)\,\subset\,\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$
in (4.6).

We recall from Remark 4.4 that
$\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\,=\,{\mathcal{O}}_{X}(-\sum_{i=1}^{n}d_{i}x_{i})$.
Therefore, the de Rham differential $d$ on ${\mathcal{O}}_{X}$ produces a
logarithmic connection on
${\mathcal{O}}_{X}(-\sum_{i=1}^{n}d_{i}x_{i})\,=\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}$.
Let

$\widetilde{d}\,\,:\,\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\,\longrightarrow\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(\sum_{i=1}^{n}x_{i})$ (4.35)

be the logarithmic connection on $\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}$
given by the de Rham differential.

Let

$\text{At}^{0}_{X}(r)\,\subset\,\text{At}^{\prime}_{X}(r)$ (4.36)

be the holomorphic subbundle whose holomorphic sections over any open subset
$U\,\subset\,X$ consist of all

$\delta\,\in\,\Gamma(U,\,\text{At}^{\prime}_{X}(r))$

satisfying the following condition:

$\widetilde{\delta}(s)\,=\,\langle\widetilde{d}(s),\,\eta(\delta)\rangle$

for all
$s\,\in\,\Gamma\left(U,\,\bigwedge\nolimits^{r}{\mathcal{F}}^{(r)}\right)$,
where $\widetilde{\delta}$ is constructed in (4.32) from $\delta$ and $\eta$
is the homomorphism in (4.34), while and $\langle-,\,-\rangle$ is the duality
pairing in (2.27) and $\widetilde{d}$ is constructed in (4.35).

The composition of homomorphisms

$\text{At}^{0}_{X}(r)\,\hookrightarrow\,\text{At}^{\prime}_{X}(r)\,\longrightarrow\,\text{At}_{X}(r)$

(see (4.36) and (4.31) for these homomorphisms) is evidently an isomorphism.
Therefore, the holomorphic subbundle $\text{At}^{0}_{X}(r)$ in (4.36) produces
a holomorphic splitting of the short exact sequence in (4.31).

Let $D$ be a connection on $\text{Sym}^{r-1}(\mathcal{E}_{*})$ defining a
parabolic ${\rm SL}(r,{\mathbb{C}})$–oper on $X$. Consider the differential
operator ${\mathcal{D}}$ in (4.11) constructed from the ${\rm
SL}(r,{\mathbb{C}})$–oper $D$. Clearly, we have

${\mathcal{D}}\left(\text{At}^{0}_{X}(r)\right)\,\subset\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$

where $\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ and
$\text{At}^{0}_{X}(r)$ are constructed in (4.9) and (4.36) respectively;
recall from (4.12) that

${\mathcal{D}}\left(\text{At}^{\prime}_{X}(r)\right)\,\subset\,\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

and $\text{End}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$
decomposes as in (4.10).

Consequently, the complex of sheaves ${\mathcal{C}}_{\bullet}$ in (4.14) is
equivalent to the following complex ${\mathcal{C}}^{\prime}_{\bullet}$ of
sheaves on $X$

${\mathcal{C}}^{\prime}_{\bullet}\,\,:\,\,{\mathcal{C}}^{\prime}_{0}\,=\,\text{At}^{0}_{X}(r)\,\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\,{\mathcal{C}}^{\prime}_{1}\,=\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,.$ (4.37)

Hence Theorem 4.5 gives the following:

######  Corollary 4.6.

The space of all infinitesimal deformation of the triple $(X,\,S,\,D)$, where
$D$ is a parabolic ${\rm SL}(r,{\mathbb{C}})$–oper on $X$, is given by the
hypercohomology

${\mathbb{H}}^{1}({\mathcal{C}}^{\prime}_{\bullet}),\,$

where ${\mathcal{C}}^{\prime}_{\bullet}$ is the complex in (4.37).

##  5\. Monodromy of parabolic opers

Consider a family of $n$–pointed Riemann surfaces

$\\{(X_{T}\,\stackrel{{\scriptstyle\varpi}}{{\longrightarrow}}\,T),\,(\phi_{1},\,\cdots,\,\phi_{n})\\}$
(5.1)

as in (3.12) and (3.13). Assume that this family is locally universal. The
relative canonical line bundle on $X_{T}$ for the projection $\varpi$ will be
denoted by $K_{\varpi}$.

Fix a holomorphic line bundle $\mathbf{L}$ on $X_{T}$ such that
${\mathbf{L}}\otimes{\mathbf{L}}$ is holomorphically isomorphic to
$K_{\varpi}\otimes{\mathcal{O}}_{X_{T}}(-\sum_{i=1}^{n}\phi_{i}(T))$; such a
line bundle $\mathbf{L}$ exists locally with respect to $T$. Fix a holomorphic
isomorphism between ${\mathbf{L}}\otimes{\mathbf{L}}$ and
$K_{\varpi}\otimes{\mathcal{O}}_{X_{T}}(-\sum_{i=1}^{n}\phi_{i}(T))$. Fix a
function

$\mathbf{c}\,:\,\\{1,\,\cdots,\,n\\}\,\longrightarrow\,\\{t\,\in\,{\mathbb{Z}}\,\mid\,t\,\geq\,2\\}$

as in (4.2).

Fix the parabolic structure to be that of an ${\rm SL}(r,{\mathbb{C}})$–oper.
As in (3.14),

$\beta\,\,:\,\,{\mathcal{M}}^{T}(r)\,\longrightarrow\,T$ (5.2)

is the corresponding relative moduli space of parabolic bundles with
connection. Let

${\mathcal{O}}^{T}(r)\,\hookrightarrow\,{\mathcal{M}}^{T}(r)$ (5.3)

be the locus of parabolic ${\rm SL}(r,{\mathbb{C}})$–opers.

Consider the monodromy map

$\mathbb{M}\,:\,{\mathcal{M}}^{T}(r)\,\longrightarrow\,{\mathcal{R}}_{X}(r)$

in (3.15). Let

$\mathbb{M}^{0}\,:\,{\mathcal{O}}^{T}(r)\,\longrightarrow\,{\mathcal{R}}_{X}(r)$
(5.4)

be the restriction of $\mathbb{M}$ to the subspace ${\mathcal{O}}^{T}(r)$ in
(5.3).

We will prove that the holomorphic map $\mathbb{M}^{0}$ in (5.4) is an
immersion. Our proof is modeled on the proof of Sanders that $\mathbb{M}^{0}$
is an immersion under the assumption that $n\,=\,0$ (parabolic points are
absent); see [Sa, Theorem 6.3].

Take a point $t_{0}\,\in\,T$. Denote the Riemann surface
$\varpi^{-1}(t_{0})\,=\,X_{t_{0}}$ by $X$. Denote the divisor
$\sum_{i=1}^{n}\phi_{i}(t_{0})$ on $X$ by $S$. We have

$T_{t_{0}}T\,=\,H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S));$

recall that $(\varpi,\,\\{\phi_{i}\\}_{i=1}^{n})$ is a locally complete
family. Take any parabolic bundle with connection

$(E_{*},\,D)\,\in\,\beta^{-1}(t_{0})$

on $X$, where $\beta$ is the projection in (5.2). From Lemma 3.4 it follows
that

$T_{(E_{*},D)}{\mathcal{M}}^{T}(r)\,=\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet}),$
(5.5)

where ${\mathcal{B}}_{\bullet}$ is the complex in (3.6). Let

$T_{\beta}\,\subset\,T{\mathcal{M}}^{T}(r)$

be the relative tangent bundle for the projection $\beta$ in (5.2). From Lemma
3.3 it follows that
$(T_{\beta})_{(E,D)}\,=\,{\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})$, where
${\mathcal{A}}_{\bullet}$ is the complex in (3.4). Consider the homomorphism

$\delta\,:\,T_{t_{0}}T\,=\,H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))\,\longrightarrow\,T{\mathcal{M}}^{T}(r)\,=\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})$

in (3.19). Recall that the two holomorphic foliations on
${\mathcal{M}}^{T}(r)$, one given by the isomonodromy condition and the other
given by the projection $\beta$ in (5.2), are transversal. Therefore, from
Lemma 3.5 and (5.5) we know that

${\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,=\,(T_{\beta})_{(E_{*},D)}\oplus\delta(T_{t_{0}}T)\,=\,{\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})\oplus\delta(H^{1}(X,\,TX\otimes{\mathcal{O}}_{X}(-S))).$
(5.6)

Let

$\Phi\,:\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,\longrightarrow\,{\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})$
(5.7)

be the projection corresponding to the decomposition in (5.6). We will now
describe this homomorphism $\Phi$ explicitly.

The connection $D$ on $E_{*}$ gives a holomorphic decomposition

$\text{At}(E_{*})\,=\,\text{End}^{P}(E_{*})\oplus(TX\otimes{\mathcal{O}}_{X}(-S))$
(5.8)

(see Lemma 2.4). As in (3.18),
${\mathbf{h}}\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,{\rm
At}(E_{*})$ is the homomorphism given by the decomposition in (5.8). Consider
$\mathcal{D}$ constructed in (2.24). Recall that

$\mathcal{D}\circ{\mathbf{h}}\,=\,0$

in (3.18). Consequently, the decomposition in (5.8) produces a homomorphism
$\mathcal{P}$ of complexes

$\begin{matrix}{\mathcal{B}}_{\bullet}&:&{\rm
At}(E_{*})&\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\
\,\,\,\,\,\Big{\downarrow}{\mathcal{P}}&&\,\,\,\Big{\downarrow}p&&\Big{\|}\\\
{\mathcal{A}}_{\bullet}&:&\text{End}^{P}(E_{*})&\stackrel{{\scriptstyle\mathcal{D}_{0}}}{{\longrightarrow}}&\text{End}^{n}(E_{*})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\end{matrix}$ (5.9)

where $p$ is the projection given by the decomposition in (5.8). Let

${\mathcal{P}}_{*}\,\,:\,\,{\mathbb{H}}^{1}({\mathcal{B}}_{\bullet})\,\longrightarrow\,{\mathbb{H}}^{1}({\mathcal{A}}_{\bullet})$
(5.10)

be the homomorphism of hypercohomologies corresponding to the homomorphism of
complexes $\mathcal{P}$ in (5.9). The homomorphism ${\mathcal{P}}_{*}$ in
(5.10) evidently coincides with the projection $\Phi$ in (5.7).

Now set $E_{*}$ to be the rank $r$ parabolic vector bundle
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ in (4.4), and let $D$ be a connection on
$\text{Sym}^{r-1}(\mathcal{E}_{*})$ such that

$\left(\text{Sym}^{r-1}(\mathcal{E}_{*}),\,D\right)\,\in\,{\mathcal{O}}^{T}(r)\bigcap\beta^{-1}(t_{0})$

(see (5.3) and (5.2)). From Corollary 4.6 we know that

$T_{(\text{Sym}^{r-1}(\mathcal{E}_{*}),D)}{\mathcal{O}}^{T}(r)\,=\,{\mathbb{H}}^{1}({\mathcal{C}}^{\prime}_{\bullet})\,,$

where ${\mathcal{C}}^{\prime}_{\bullet}$ is the complex in (4.37). We have the
following two homomorphisms $P$ and $Q$ of complexes

$\begin{matrix}{\mathcal{C}}^{\prime}_{\bullet}&:&{\mathcal{C}}^{\prime}_{0}\,=\,\text{At}^{0}_{X}(r)&\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}&{\mathcal{C}}^{\prime}_{1}\,=\,\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\
\,\,\,\,\Big{\downarrow}Q&&\,\,\,\,\,\,\Big{\downarrow}Q_{0}&&\,\,\,\,\,\,\Big{\downarrow}Q_{1}\\\
\widetilde{\mathcal{B}}_{\bullet}&:&\widetilde{\mathcal{B}}_{0}\,=\,\text{At}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)&\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}&\widetilde{\mathcal{B}}_{1}\,=\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\\\
\,\,\,\,\Big{\downarrow}P&&\,\,\,\,\,\,\Big{\downarrow}P_{0}&&\,\,\,\,\,\,\Big{\downarrow}P_{1}\\\
\widetilde{\mathcal{A}}_{\bullet}&:&\widetilde{\mathcal{A}}_{0}\,=\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)&\stackrel{{\scriptstyle\mathcal{D}_{0}}}{{\longrightarrow}}&\widetilde{\mathcal{A}}_{1}\,=\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\end{matrix}$ (5.11)

where

  * •

$\widetilde{\mathcal{B}}_{\bullet}$ is the complex in (3.6), with
$\left(\text{Sym}^{r-1}(\mathcal{E}_{*}),\,D\right)$ substituted in place in
$(E_{*},\,D)$,

  * •

$\widetilde{\mathcal{A}}_{\bullet}$ is the complex in (3.4), with
$\left(\text{Sym}^{r-1}(\mathcal{E}_{*}),\,D\right)$ substituted in place in
$(E_{*},\,D)$,

  * •

${\mathcal{C}}^{\prime}_{\bullet}$ is the complex in (4.37),

  * •

for $i\,=\,0,\,1$, the homomorphism
${\mathcal{C}}^{\prime}_{i}\,\longrightarrow\,\widetilde{\mathcal{B}}_{i}$ in
(5.11) is the natural inclusion map, and

  * •

the homomorphism $P$ is the homomorphism $\mathcal{P}$ in (5.9) with
$\left(\text{Sym}^{r-1}(\mathcal{E}_{*}),\,D\right)$ substituted in place in
$(E_{*},\,D)$. So $P_{1}$ is the identity map.

Let

$(P\circ
Q)_{*}\,\,:\,\,{\mathbb{H}}^{1}({\mathcal{C}}^{\prime}_{\bullet})\,\longrightarrow\,{\mathbb{H}}^{1}(\widetilde{\mathcal{A}}_{\bullet})$
(5.12)

be the homomorphism of hypercohomologies induced by the homomorphism $P\circ
Q$ in (5.11).

Since the homomorphism ${\mathcal{P}}_{*}$ in (5.10) coincides with the
projection $\Phi$ in (5.7), to prove that the map $\mathbb{M}^{0}$ in (5.4) is
an immersion, it suffices to show that the homomorphism $(P\circ Q)_{*}$ in
(5.12) is injective.

Let

$q\,:\,\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\longrightarrow\,$
$\left(\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\right)\big{/}P_{1}Q_{1}\left(\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\right)$
$=\,\left(\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

be the quotient map. From the commutativity of (5.11) it follows immediately
that the composition of maps

$q\circ{\mathcal{D}}_{0}\circ(P_{0}\circ
Q_{0})\,:\,\text{At}^{0}_{X}(r)\,\longrightarrow\,\left(\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

vanishes on the subbundle
$\text{At}^{0}_{X}(r)\bigcap\text{End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\,\subset\,\text{At}^{0}_{X}(r)$
(see (2.12) for the subbundle
$\text{End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$ of ${\rm
At}(\text{Sym}^{r-1}(\mathcal{E}_{*}))$). Therefore,
$q\circ{\mathcal{D}}_{0}\circ(P_{0}\circ Q_{0})$ produces a homomorphism

$\mathcal{S}\,:\,TX\otimes{\mathcal{O}}_{X}(-S)\,\longrightarrow\,\left(\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,.$

This homomorphism $\mathcal{S}$ coincides with the second fundamental form of
the reduction of structure group $\widetilde{\mathbb{P}}(B)_{*}$ in (4.18) for
the connection on $\widetilde{\mathbb{P}}_{*}$ given by $D$. We know that this
second fundamental form is everywhere nonzero, because $D$ is a parabolic
$\text{SL}(r,{\mathbb{C}})$–oper. Consequently, $\mathcal{S}$ is everywhere
nonzero. This implies that the homomorphism $P_{0}\circ Q_{0}$ in (5.11) is
injective.

Since $P_{0}\circ Q_{0}$ in (5.11) is injective, it follows that the kernel of
the homomorphism $(P\circ Q)_{*}$ in (5.12) is the quotient of a subspace of
$H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}P_{0}Q_{0}\left(\text{At}^{0}_{X}(r)\right)\right)$.
More precisely, let

$\mathbf{V}\,\subset\,H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}P_{0}Q_{0}(\text{At}^{0}_{X}(r))\right)$

be the subspace consisting of all sections $s$ such that
${\mathcal{D}}_{0}(s)\,=\,0$; note that from the commutativity of the diagram
in (5.11) it follows that ${\mathcal{D}}_{0}$ produces a homomorphism

$H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}P_{0}Q_{0}(\text{At}^{0}_{X}(r))\right)\,\longrightarrow$
$H^{0}\left(X,\,\left(\text{End}^{n}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}\text{ad}^{n}_{1}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\right)\,.$

Let

$\mathbf{W}\,\subset\,\mathbf{V}$

be the subspace consisting of all sections $s$ such that there is a section

$\widetilde{s}\,\in\,H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)$

satisfying the following two conditions:

  * •

${\mathcal{D}}_{0}(\widetilde{s})\,=\,0$, where ${\mathcal{D}}_{0}$ is the
homomorphism in (5.11), and

  * •

$\widetilde{s}$ projects to $s$ under the natural map

$H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\right)\,\longrightarrow\,H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}P_{0}Q_{0}(\text{At}^{0}_{X}(r))\right)\,.$

Then we have

${\rm kernel}((P\circ Q)_{*})\,\,=\,\,\mathbf{V}/\mathbf{W}\,,$ (5.13)

where $(P\circ Q)_{*}$ is the homomorphism in (5.12).

Now

${\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}P_{0}Q_{0}(\text{At}^{0}_{X}(r))\,=\,{\mathcal{O}}_{X}\oplus{\mathcal{W}}\,,$

where ${\mathcal{W}}$ admits a filtration of holomorphic subbundles such that
every successive quotient is of the form
$(TX\otimes{\mathcal{O}}_{X}(-S))^{\otimes m}$, $m\,\geq\,1$. From Assumption
2.1 it follows that

$H^{0}(X,(TX\otimes{\mathcal{O}}_{X}(-S))^{\otimes m})\,=\,0$

for all $m\,\geq\,1$. Hence

$H^{0}\left(X,\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)\big{/}P_{0}Q_{0}(\text{At}^{0}_{X}(r))\right)\,=\,H^{0}\left(X,\,{\mathcal{O}}_{X}\right)\,.$

But ${\mathcal{O}}_{X}\,\subset\,{\rm
End}^{P}\left(\text{Sym}^{r-1}(\mathcal{E}_{*})\right)$, and
${\mathcal{D}}_{0}(H^{0}(X,\,{\mathcal{O}}_{X}))\,=\,0$. Consequently, we have

$H^{0}\left(X,\,{\mathcal{O}}_{X}\right)\,\subset\,\mathbf{W},$

where $\mathbf{W}$ is the subspace in (5.13). Therefore, from (5.13) it
follows that

${\rm kernel}((P\circ Q)_{*})\,=\,0\,.$ (5.14)

In other words, the homomorphism $(P\circ Q)_{*}$ is injective.

As noted above, the map $\mathbb{M}^{0}$ in (5.4) is an immersion if the
homomorphism $(P\circ Q)_{*}$ in (5.12) is injective. Therefore, we have
proved the following:

######  Theorem 5.1.

The map $\mathbb{M}^{0}$ in (5.4) is an immersion.

##  Appendix A Parabolic opers

The purpose of this appendix is to recall some results by K. Yokogawa [Yo] on
Hom-sheaves, tensor products and extension classes of parabolic bundles and to
give an alternative definition of a parabolic ${\rm SL}(r)$-oper [BDP] which
is conceptually closer to the definition of an ordinary ${\rm SL}(r)$-oper.

###  A.1. Correspondence: flags and $\mathbb{R}$-filtered sheaves

We first recall the correspondence between a parabolic vector bundle as
defined in section 2.1 and an $\mathbb{R}$-filtered sheaf
$\\{E_{t}\\}_{t\in\mathbb{R}}$ as introduced and studied in [MY], [Yo], [BY].
Using the notation of section 2.1 we define for $t\in[0,1]$ the vector bundle
$E_{t}$ by the following equalities

$\begin{matrix}E^{i}_{t}&=&E&\,\ {\rm for}\,\ 0\leq t\leq\alpha_{i,1}\\\
E^{i}_{t}&=&\mathrm{ker}(E\rightarrow E_{x_{i}}/E_{i,j})&\,\ {\rm for}\,\
\alpha_{i,j-1}<t\leq\alpha_{i,j}\\\ E^{i}_{t}&=&E(-x_{i})&\,\ {\rm for}\,\
\alpha_{i,l_{i}}<t\leq 1\\\ E_{t}&=&\bigcap_{i=1}^{n}E^{i}_{t}&\end{matrix}$

We extend to $\mathbb{R}$ by the formula $E_{t+1}=E_{t}(-S)$. We also denote
this $\mathbb{R}$-filtered sheaf by $E_{*}$. Note that $E_{t}\subset
E_{t^{\prime}}$ for any $t\geq t^{\prime}$.

We recall that the family $E_{*}$ is left-continuous, meaning that for any
$t\in\mathbb{R}$

$\lim_{s\to t\atop{s<t}}E_{s}=E_{t}.$

The family $E_{*}$ is not right-continuous and we will denote for any
$t\in\mathbb{R}$

$E_{t+}\,:=\,\lim_{s\to t\atop{s>t}}E_{s}.$

Then $E_{t+}$ is a subsheaf of $E_{t}$ and the quotient $E_{t}/E_{t+}$ is a
torsion-sheaf supported at the parabolic divisor $S$.

###  A.2. Special structure and shifts

Every vector bundle $E$ can be considered as a parabolic vector bundle $E_{*}$
with the special structure, defined either by the properties
$l_{i}=1,\alpha_{i,1}=0$ for $1\leq i\leq n$, or equivalently by the
equalities

$E_{t}\,:=\,E\qquad\text{for}\ t\,\in\,]-1,\,0].$

Given a parabolic vector bundle $E_{*}$ and an n-tuple
$\underline{\beta}\,=\,(\beta_{1},\,\ldots,\,\beta_{n})\,\in\,\mathbb{R}^{n}$
we define the shift $E[\underline{\beta}]_{*}$ by the equalities

$E[\underline{\beta}]_{t}\,=\,\bigcap_{i=1}^{n}E^{i}_{t+\beta_{i}}\qquad\text{for}\
t\,\in\,\mathbb{R}.$

Also, in order to simplify notation, we define for $\beta\in\mathbb{R}$ the
shift of $E[\beta]_{*}$ as $E[\beta]_{*}=E[\underline{\beta}]_{*}$ with
$\beta_{i}=\beta$ for every $i$.

###  A.3. Tensor products, Hom-sheaves and duals

Given two parabolic vector bundles $E_{*}$ and $E^{\prime}_{*}$ we define
their parabolic tensor product (see e.g. [Yo] section 3) by the formula

$(E_{*}\otimes E^{\prime}_{*})_{t}:=\sum_{s\in\mathbb{R}}E_{s}\otimes
E^{\prime}_{t-s}\subset E_{0}\otimes E^{\prime}_{0}(*S).$

Here $E_{0}\otimes E^{\prime}_{0}(*S)$ denotes the (non-coherent) sheaf of
rational sections of $E_{0}\otimes E^{\prime}_{0}$ admitting arbitrary poles
at the parabolic divisor $S$. We note that it is enough to consider the sum
for $s$ running over a subinterval of $\mathbb{R}$ of length $1$ (because of
the invariance of the tensor product $E_{s}\otimes E^{\prime}_{t-s}$ under the
shift $s\mapsto s+1$) and that only a finite number of subsheaves
$E_{s}\otimes E^{\prime}_{t-s}$ occur.

Before defining the parabolic Hom-sheaf of two parabolic vector bundles
$E_{*}$ and $E^{\prime}_{*}$, we define the sheaf $Hom(E_{*},E^{\prime}_{*})$
as the subsheaf of $Hom(E_{0},\,E^{\prime}_{0})$ consisting of parabolic
homomorphisms, i.e., homomorphisms
$f\,:\,E_{0}\,\longrightarrow\,E^{\prime}_{0}$ satisfying

$f(E_{t})\,\subset\,E^{\prime}_{t}$

for any $t\,\in\,[0,\,1[$ and thus for any $t\,\in\,\mathbb{R}$. Note that if
$E_{*}$ and $E^{\prime}_{*}$ are vector bundles, then
$Hom(E_{*},\,E^{\prime}_{*})$ is also a vector bundle. Now, we define the
parabolic Hom-sheaf $Hom(E_{*},\,E^{\prime}_{*})_{*}$ by the formula

$Hom(E_{*},\,E^{\prime}_{*})_{t}\,:=\,Hom(E_{*},\,E^{\prime}[t]_{*})$

for any $t\,\in\,\mathbb{R}$.

The parabolic dual $E^{\vee}_{*}$ of a parabolic vector bundle $E_{*}$ is by
definition

$E^{\vee}_{*}\,:=\,Hom(E_{*},\,\mathcal{O}_{*})_{*},$

where $\mathcal{O}_{*}$ denotes the trivial bundle with the special structure.

The above definitions of parabolic tensor products, Hom-sheaves and duals
extend the standard operations on vector bundles, when considering a vector
bundle as a parabolic vector bundle with its special structure. Also, the
following relations are easy to check :

$E[\underline{\beta}]_{*}\otimes
E^{\prime}[\underline{\beta^{\prime}}]_{*}=E\otimes
E^{\prime}[\underline{\beta}+\underline{\beta^{\prime}}]_{*}$
$E[\underline{\beta}]_{*}^{\vee}=E^{\vee}[-\underline{\beta}]_{*}$
$E^{\vee}_{*}\otimes E^{\prime}_{*}=Hom(E_{*},E^{\prime}_{*})_{*}$

###  A.4. Cohomology of a parabolic bundle

Given a parabolic vector bundle $E_{*}$ over the curve $X$ we define the
cohomology of $E_{*}$ as the cohomology of the vector bundle $E_{0}$

$H^{i}(X,E_{*})=H^{i}(X,E_{0}).$

###  A.5. Parabolic subbundles and parabolic degree

We say that $E^{\prime}_{*}$ is a parabolic subbundle of $E_{*}$ if there is
an injective parabolic homomorphism

$E^{\prime}_{*}\hookrightarrow E_{*}$

with torsion-free cokernel, or equivalently, for any $t\in\mathbb{R}$, the
subsheaf $E^{\prime}_{t}$ is a subbundle of $E_{t}$.

The parabolic degree of a parabolic bundle $E_{*}$ is defined as

$\mathrm{pardeg}(E_{*})=\int_{0}^{1}\deg(E_{t})dt+n\mathrm{rk}(E_{*}),$

where $n$ is the number of parabolic points. We have the following formulae :

$\mathrm{pardeg}(E_{*}\otimes
E^{\prime}_{*})=\mathrm{rk}(E^{\prime}_{*})\mathrm{pardeg}(E_{*})+\mathrm{rk}(E_{*})\mathrm{pardeg}(E^{\prime}_{*}),$
$\mathrm{pardeg}(E[\underline{\beta}]_{*})=\mathrm{pardeg}(E_{*})-\mathrm{rk}(E_{*})\sum_{i=1}^{n}\beta_{i}.$

###  A.6. Canonical injections and quasi-isomorphisms

Given a parabolic line bundle $L_{*}$ and a vector
$\underline{\gamma}\in\mathbb{R}^{n}$ with $\gamma_{i}\geq 0$ for all $i$, we
have a canonical parabolic injection

$\iota:L_{*}\longrightarrow L[-\underline{\gamma}]_{*}$

induced by the natural inclusions $L_{t}\subset L_{t-\gamma_{i}}$.

######  Definition A.1.

We say that a parabolic homomorphism between two parabolic line bundles

$\varphi:L_{*}\longrightarrow M_{*}$

is a quasi-isomorphism, if there exists a vector
$\underline{\gamma}\in\mathbb{R}^{n}$ with $0\leq\gamma_{i}<1$ and a parabolic
isomorphism $M_{*}\cong L[-\underline{\gamma}]_{*}$ such that via this
isomorphism $\varphi$ identifies with the canonical injection $\iota$. In that
case we say that $\varphi$ is a quasi-isomorphism of weight
$\underline{\gamma}\in\mathbb{R}^{n}$.

###  A.7. Extensions of parabolic bundles

Given two parabolic vector bundles $E_{*}$ and $E^{\prime}_{*}$ we say that
the parabolic vector bundle $F_{*}$ is an extension of $E_{*}$ by
$E^{\prime}_{*}$ if there exists a short exact sequence of parabolic
homomorphisms

$0\longrightarrow E^{\prime}_{*}\longrightarrow F_{*}\longrightarrow
E_{*}\longrightarrow 0.$

By [Yo] Lemma 1.4 and Lemma 3.6 the isomorphism classes of extensions of
$E_{*}$ by $E^{\prime}_{*}$ are in one-to-one correspondence with the
cohomology space
$\mathrm{Ext}^{1}(E_{*},E^{\prime}_{*})=H^{1}(X,Hom(E_{*},E^{\prime}_{*}))$.

###  A.8. Connections on parabolic bundles

Given a parabolic vector bundle $E_{*}$ with parabolic divisor $S\subset X$ we
define a connection $\nabla_{*}$ on $E_{*}$ as a $\mathbb{C}$-linear
homomorphism between the parabolic bundles $E_{*}$ and $E_{*}\otimes
K[-1]_{*}$

$\displaystyle\nabla_{*}:E_{*}\longrightarrow E_{*}\otimes K[-1]_{*}$ (A.1)

such that for every $t\in\mathbb{R}$ the map
$\nabla_{t}:E_{t}\rightarrow(EK[-1])_{t}=E_{t}K(S)$ is a logarithmic
connection with poles at the parabolic divisor $S$ and for any $t\leq
t^{\prime}$ we have a commutative diagram

$\textstyle{E_{t^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{t^{\prime}}K(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{t}K(S)}$

where the vertical maps are the natural inclusions.

On the trivial parabolic bundle $\mathcal{O}_{*}$ there is a natural
connection given by the de Rham differentiation and which is denoted by
$d_{*}$

$d_{*}:\mathcal{O}_{*}\rightarrow K[-1]_{*}$

and, fixing an integer $n$, is given for $t\in]n-1,n]$ by differentiation of
regular functions having zeros or poles of order $n$ at $S$

$d_{t}:\mathcal{O}_{t}=\mathcal{O}(-nS)\rightarrow K[-1]_{t}=K(-(n-1)S).$

We now describe the properties of $\nabla_{*}$ in terms of the parabolic
structure given by the flags at the parabolic divisor.

######  Lemma A.2.

Consider a connection $\nabla_{*}$ on $E_{*}$ as defined in (A.1). Then the
logarithmic connection $\nabla_{0}$ on $E_{0}$ obtained by putting $t=0$
satisfies

$\displaystyle\mathrm{Res}(\nabla_{0},x_{i})(E_{i,j})\subset E_{i,j}$ (A.2)

for any $i=1,\ldots,n$ and any $j=1,\ldots,l_{i}$. Conversely, any logarithmic
connection $\nabla_{0}$ on $E_{0}$ satisfying (A.2) gives rise to a connection
$\nabla_{*}$ on $E_{*}$.

The proof of this lemma is standard and therefore left to the reader.

###  A.9. Parabolic connections on parabolic bundles

Consider a connection $\nabla_{*}$ on a parabolic bundle $E_{*}$ as defined in
(A.1). Then for any $t\in\mathbb{R}$ we can consider the residue at $x_{i}\in
S$ of the logarithmic connection $\nabla_{t}$

$\mathrm{Res}(\nabla_{t},x_{i})\in\mathrm{End}_{\mathbb{C}}((E_{t})_{x_{i}}).$

Since $\nabla_{t}$ preserves all subsheaves $E_{t^{\prime}}$ for
$t^{\prime}\geq t$ we obtain by passing to the quotient
$(E_{t}/E_{t+})_{x_{i}}$ a linear map

$\overline{\mathrm{Res}}(\nabla_{t},x_{i})\in\mathrm{End}_{\mathbb{C}}((E_{t}/E_{t+})_{x_{i}}).$

and therefore an endomorphism, which we simply denote by
$\overline{\mathrm{Res}}(\nabla_{t})$, of the torsion sheaf $E_{t}/E_{t+}$.

With this notation, we can now define a parabolic connection on a parabolic
bundle.

######  Definition A.3.

Let $E_{*}$ be a parabolic bundle. We say that a connection $\nabla_{*}$ on
$E_{*}$ is a parabolic connection if for any $t\in\mathbb{R}$

$\overline{\mathrm{Res}}(\nabla_{t})=t\mathrm{Id}.$

We leave it as an exercise to the reader to check that this definition is
equivalent via the correspondence of section A.1 to the definition given in
section 2.2, i.e. for any parabolic point $x_{i}\in S$ the residue of the
logarithmic connection $\nabla_{0}$ acts as $\alpha_{i,j}\mathrm{Id}$ on the
quotient space $E_{i,j}/E_{i,j+1}$.

We also mention the following useful facts, whose proofs are standard.

Let $(E_{*},\nabla_{*})$ and $(E^{\prime}_{*},\nabla^{\prime}_{*})$ be
parabolic vector bundles equipped with parabolic connections. Then

  * •

the parabolic tensor product $E_{*}\otimes E^{\prime}_{*}$ is naturally
equipped with the tensor product connection
$(\nabla\otimes\nabla^{\prime})_{*}$, which is also parabolic.

  * •

the parabolic symmetric power $\mathrm{Sym}^{m}E_{*}$ is naturally equipped
with the symmetric power connection $(\mathrm{Sym}^{m}\nabla)_{*}$, which is
also parabolic.

  * •

the de Rham differentiation $d_{*}$ on the trivial parabolic bundle
$\mathcal{O}_{*}$ is a parabolic connection.

######  Remark A.4.

Note that, when considering parabolic bundles from the “flag”-point of view,
the relation between the parabolic weights of $E_{*}$ and those of its
symmetric powers $\mathrm{Sym}^{m}E_{*}$ is quite complicated (as one needs to
take fractional parts and reorder them in order to obtain an increasing
sequence of parabolic weights in the interval $[0,1[$). Therefore the
“$\mathbb{R}$-filtered sheaf”-point of view is more adapted when considering
symmetric powers, as we will need to do in the sequel.

With this notation we can reformulate the following existence theorem.

######  Theorem A.5 ([BL]).

The parabolic bundle $E_{*}$ admits a parabolic connection $\nabla_{*}$ if and
only if any direct summand of $E_{*}$ has parabolic degree equal to $0$.

###  A.10. Parabolic ${\rm SL}(2)$-opers

We now define the parabolic analogue of the Gunning bundle. Given the
parabolic weights $\alpha_{i,1},\alpha_{i,2}$ for $1\leq i\leq n$ satisfying
the inequalities (2.3) and the additional assumption
$\alpha_{i,1}+\alpha_{i,2}=1$ for all $i$, we introduce the real numbers

$\beta_{i}=\alpha_{2,i}-\alpha_{1,i}\in]0,1[$

and we define

$\underline{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{n})\in\mathbb{R}^{n}.$

Let $K$ be the canonical bundle of the curve $X$. Then we define the canonical
parabolic bundle by

$K^{par}_{*}=K[-\underline{\beta}]_{*},$

where we equip $K$ with the special structure. We also define a parabolic
theta-characteristic $\theta^{par}_{*}$ as a parabolic line bundle satisfying

$\theta^{par}_{*}\otimes\theta^{par}_{*}=K^{par}_{*}.$

One can easily check that the parabolic line bundle $\theta^{par}_{*}$ (resp.
$\left(\theta^{par}_{*}\right)^{-1}$) corresponds via the above correspondence
to a line bundle $M$ (resp. $Q$) satisfying $M^{2}=K(-S)$ (resp.
$Q^{2}=K^{-1}(-S)$) with parabolic weight $\alpha_{i,2}$ (resp.
$\alpha_{i,1}$) at the parabolic point $x_{i}$ for $1\leq i\leq n$.

We define the parabolic Gunning bundle $\mathcal{G}^{par}_{*}$ as the unique
non-split parabolic extension

$0\longrightarrow\theta^{par}_{*}\longrightarrow\mathcal{G}^{par}_{*}\longrightarrow\left(\theta^{par}_{*}\right)^{-1}\longrightarrow
0.$ (A.3)

We note that the space of parabolic extensions of
$\left(\theta^{par}_{*}\right)^{-1}$ by $\theta^{par}_{*}$ is one-dimensional,
since

$\displaystyle\dim\mathrm{Ext}^{1}(\left(\theta^{par}_{*}\right)^{-1},\theta^{par}_{*})$
$\displaystyle=$ $\displaystyle\dim
H^{1}(\theta^{par}_{*}\otimes\theta^{par}_{*})$ $\displaystyle=$
$\displaystyle\dim H^{1}(K[-\underline{\beta}]_{*})=\dim H^{1}(K)=1.$

Clearly, the parabolic bundle $\mathcal{G}^{par}_{*}$ has trivial parabolic
determinant since

$\mathrm{det}\
\mathcal{G}^{par}_{*}=\theta^{par}_{*}\otimes\left(\theta^{par}_{*}\right)^{-1}=\mathcal{O}_{*}.$

It is easy to check that the exact sequence (A.3) equals the exact sequence
(3.4) in [BDP] defining the underlying parabolic bundle of a parabolic ${\rm
SL}(2)$-oper.

Furthermore, any parabolic connection $\nabla_{*}$ on $\mathcal{G}^{par}_{*}$
induces a second fundamental form, which is a $\mathcal{O}_{X}$-linear
parabolic homomorphism

$\psi:\theta^{par}_{*}\longrightarrow(\theta^{par}_{*})^{-1}\otimes
K[-1]_{*}\cong\theta^{par}_{*}[-1+\underline{\beta}].$

The same argument as in the non-parabolic case shows that $\psi\not=0$, since
$\psi=0$ would imply the existence of a parabolic connection on the parabolic
bundle $\theta^{par}_{*}$ having parabolic degree
$g-1+\frac{1}{2}(\sum_{i=1}^{n}\beta_{i})>0$, which contradicts Theorem A.5.
Thus $\psi$ is a quasi-isomorphism of weight $1-\underline{\beta}$.

###  A.11. Parabolic ${\rm SL}(r)$-opers

With the above introduced notation we give a new definition of a parabolic
${\rm SL}(r)$-oper.

######  Definition A.6.

A parabolic ${\rm SL}(r)$-oper is a triple
$(E_{*},{E_{\bullet}}_{*},\nabla_{*})$ consisting of a rank-$r$ parabolic
vector bundle $E_{*}$, a filtration ${E_{\bullet}}_{*}$ of $E_{*}$ by
parabolic subbundles

$0=E_{0*}\subset E_{1*}\subset E_{2*}\subset\ldots\subset E_{r-1*}\subset
E_{r*}=E_{*}$

with $\mathrm{rk}(E_{i*})=i$ and a parabolic connection $\nabla_{*}$ on
$E_{*}$ satisfying the following conditions

  * •

$\det(E_{*},\nabla_{*})=(\mathcal{O}_{*},d_{*})$

  * •

$\nabla_{*}(E_{i*})\subset E_{i+1*}\otimes K[-1]_{*}$ for any $i=1,\ldots,r-1$

  * •

There exists a vector $\underline{\beta}\in\mathbb{R}^{n}$ with
$0<\beta_{i}<1$ such that for any $i=1,\ldots,r-1$ the parabolic homomorphisms
induced by $\nabla_{*}$ between parabolic line bundles

$\left(E_{i*}/E_{i-1*}\right)\longrightarrow\left(E_{i+1*}/E_{i*}\right)\otimes
K[-1]_{*}$

are quasi-isomorphisms of weight $1-\underline{\beta}$ (see Definition (A.1)).

######  Remark A.7.

If $r=2$ one can easily show that any parabolic ${\rm SL}(2)$-oper is of the
form $(\mathcal{G}^{par}_{*},\mathcal{G}^{par}_{\bullet*},\nabla_{*})$, where
$\mathcal{G}^{par}_{*}$ is the parabolic Gunning bundle introduced in section
A.10, $\mathcal{G}^{par}_{\bullet*}$ is given by the exact sequence (A.3) and
$\nabla_{*}$ is any parabolic connection satisfying $\det\nabla_{*}=d_{*}$.

We now show that, similar to the non-parabolic case, the underlying parabolic
bundle of a parabolic ${\rm SL}(r)$-oper is a parabolic symmetric power of the
parabolic Gunning bundle.

######  Theorem A.8.

Let $(E_{*},{E_{\bullet}}_{*},\nabla_{*})$ be a parabolic ${\rm SL}(r)$-oper
associated to the vector $\underline{\beta}\in\mathbb{R}^{n}$. Then, up to
tensor product with an $r$-torsion parabolic line bundle, we have an
isomorphism between parabolic bundles

$E_{*}\cong\mathrm{Sym}^{r-1}\mathcal{G}^{par}_{*},$

where $\mathcal{G}^{par}_{*}$ is the parabolic Gunning bundle associated to
the vector $\underline{\beta}\in\mathbb{R}^{n}$. Moreover, under this
isomorphism the filtration ${E_{\bullet}}_{*}$ corresponds to the natural
filtration of $\mathrm{Sym}^{r-1}\mathcal{G}^{par}_{*}$.

###### Proof.

In order to simplify the notation we introduce the parabolic line bundles
$Q_{i*}=E_{i*}/E_{i-1*}$ for $i=1,\ldots,r$. Then the quasi-isomorphisms
$Q_{i*}\rightarrow Q_{i+1*}\otimes K[-1]_{*}$ correspond to isomorphisms

$Q_{i*}=Q_{i+1*}\otimes K[-\underline{\beta}]_{*}$

for $i=1,\ldots,r-1$. Iterating these formulae we can express all line bundles
in terms of $Q_{r*}$

$Q_{r-i*}=Q_{r*}\otimes K^{i}[-i\underline{\beta}].$

Since $\det E_{*}=\mathcal{O}_{*}$, we obtain that the parabolic tensor
product of all $Q_{i*}$ equals $\mathcal{O}_{*}$, which leads to the
isomorphism

$Q_{r*}^{r}=K^{-\frac{r(r-1)}{2}}[\frac{r(r-1)}{2}\underline{\beta}].$

We choose a parabolic theta-characteristic $\theta^{par}_{*}$, i.e., a
parabolic line bundle satisfying
$(\theta^{par}_{*})^{2}=K^{par}=K[-\underline{\beta}]$. Then the above
equality is equivalent to saying that $Q_{r*}$ and
$(\theta^{par}_{*})^{-(r-1)}$ differ by an $r$-torsion line bundle. So, after
tensorizing $E_{*}$ and consequently all quotient line bundles $Q_{i*}$ by
this $r$-torsion line bundle, we can assume that
$Q_{r*}=(\theta^{par}_{*})^{-(r-1)}$. From the above formulae, we immediately
obtain that

$Q_{r-i*}=(\theta^{par}_{*})^{-(r-1)+2i}$

for $i=0,\ldots,r-1$. Next we will show that the natural exact sequence

$0\longrightarrow Q_{i*}\longrightarrow E_{i+1*}/E_{i-1*}\longrightarrow
Q_{i+1*}\longrightarrow 0$

is the unique non-split parabolic extension of $Q_{i+1*}$ by $Q_{i*}$. By
section A.7 parabolic extensions are parameterized by
$\mathrm{Ext}^{1}(Q_{i+1}*,Q_{i*})$ and we have

$\displaystyle\dim\mathrm{Ext}^{1}(Q_{i+1*},Q_{i*})$ $\displaystyle=$
$\displaystyle\dim H^{1}(Q_{i+1*}^{\vee}\otimes Q_{i*})$ $\displaystyle=$
$\displaystyle\dim H^{1}(K[-\underline{\beta}]_{*})=\dim H^{1}(K)=1.$

We now show that $E_{i+1*}/E_{i-1*}$ is non-split. Suppose on the contrary
that we have a direct sum decomposition

$E_{i+1*}/E_{i-1*}=Q_{i*}\oplus Q_{i+1*}.$

We claim that this splitting implies that the exact sequence

$0\longrightarrow E_{i*}\longrightarrow E_{i+1*}\longrightarrow
Q_{i+1*}\longrightarrow 0$ (A.4)

also splits. To see that, we consider the long exact sequence

$\cdots\longrightarrow\mathrm{Ext}^{1}(Q_{i+1*},E_{i-1*})\longrightarrow\mathrm{Ext}^{1}(Q_{i+1*},E_{i*})\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}\mathrm{Ext}^{1}(Q_{i+1*},Q_{i*})\longrightarrow\cdots$

where $\mu$ is induced by the push-out under the map $E_{i*}\rightarrow
Q_{i*}$. Thus, to show that the exact sequence (A.4) splits, it will be enough
to show that $\mu$ is injective. But $\mathrm{Ext}^{1}(Q_{i+1*},E_{i-1*})=0$,
since $E_{i-1*}$ can be constructed by a series of a successive extensions of
$Q_{j*}$’s for $j\leq i-1$ and we have

$\mathrm{Ext}^{1}(Q_{i+1*},Q_{j*})=0\qquad\text{for all}\ j\leq i-1.$

Thus $E_{i+1*}=E_{i*}\oplus Q_{i+1*}$ and after projecting from $E_{i+1*}$
onto $E_{i*}$ the connection $\nabla_{*}$ restricts to a connection on
$E_{i*}$. But, for $i\leq r-1$ we have $\mathrm{pardeg}(E_{i*})>0$, which is
the desired contradiction by Theorem A.5.

Finally, we invoke Theorem 4.7 [JP] to conclude that, since the rank-$2$
parabolic bundles $E_{i+1*}/E_{i-1*}$ are the unique non-split parabolic
extensions of $Q_{i+1*}$ by $Q_{i*}$ for all $i=1,\ldots,r-1$, the underlying
parabolic vector bundle $E_{*}$ is unique up to isomorphism. On the other
hand, it is easily checked that the parabolic symmetric power
$\mathrm{Sym}^{r-1}\mathcal{G}^{par}_{*}$ also satisfies these properties,
hence by uniqueness both parabolic vector bundles $E_{*}$ and
$\mathrm{Sym}^{r-1}\mathcal{G}^{par}_{*}$ are isomorphic.

Note that Theorem 4.7 [JP] deals with non-parabolic opers, but its extension
to parabolic opers is straightforward. ∎

######  Remark A.9.

The last theorem shows that the above definition of parabolic ${\rm
SL}(r)$-oper coincides with [BDP] Definition 5.2.

## References

  * [At] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
  * [BBN1] V. Balaji, I. Biswas and D. S. Nagaraj, Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. J. 53 (2001), 337–367.
  * [BBN2] V. Balaji, I. Biswas and D. S. Nagaraj, Ramified G-bundles as parabolic bundles, J. Ramanujan Math. Soc. 18 (2003), 123–138.
  * [BBP] V. Balaji, I. Biswas and Y. Pandey, Connections on parahoric torsors over curves, Publ. Res. Inst. Math. Sci. 53 (2017), 551–585.
  * [BD1] A. Beilinson and V. G. Drinfeld, Opers, arXiv:0501398.
  * [BD2] A. Beilinson and V. G. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, (1991).
  * [BF] D. Ben-Zvi and E. Frenkel, Spectral curves, opers and integrable systems, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 87–159.
  * [Bh] U. N. Bhosle, Parabolic sheaves on higher-dimensional varieties, Math. Ann. 293 (1992), 177–192.
  * [Bi1] I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J. 88 (1997), 305–325.
  * [Bi2] I. Biswas, Parabolic ample bundles, Math. Ann. 307 (1997), 511–529.
  * [BDP] I. Biswas, S. Dumitrescu and C. Pauly, Parabolic ${\rm SL}_{r}$-opers, Illinois J. Math. 64 (2020), 493–517.
  * [BL] I. Biswas and M. Logares, Connection on parabolic vector bundles over curves, Inter. Jour. Math. 22 (2011), 593–602.
  * [BSY] I. Biswas, L. P. Schaposnik and M. Yang, Generalized B-opers. Symmetry Integrability Geom. Methods Appl. 16 (2020), Article 041.
  * [BY] H. Boden and K. Yokogawa, Moduli spaces of parabolic Higgs bundles and parabolic $K(D)$ pairs over smooth curves: I, Inter. Jour. Math. 7 (1996), 573–598.
  * [Bo1] N. Borne, Fibrés paraboliques et champ des racines, Int. Math. Res. Not. IMRN, 16, Art. ID rnm049, 38, (2007).
  * [Bo2] N. Borne, Sur les représentations du groupe fondamental d’une variété privée d’un diviseur à croisements normaux simples, Indiana Univ. Math. Jour. 58 (2009), 137–180.
  * [Ch1] T. Chen, The associated map of the nonabelian Gauss–Manin connection, Thesis (Ph.D.)–University of Pennsylvania (2012), https://www.math.upenn.edu/grad/dissertations/ChenThesis.pdf.
  * [Ch2] T. Chen, The associated map of the nonabelian Gauss–Manin connection, Cent. Eur. Jour. Math. 10 (2012), 1407–1421.
  * [CS] B. Collier and A. Sanders, (G,P)-opers and global Slodowy slices, Adv. Math. 377 (2021), Paper No. 107490, 43 pp.
  * [De] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970.
  * [DFK+] O. Dumitrescu, L. Fredrickson, G. Kydonakis, R. Mazzeo, M. Mulase and A. Neitzke, From the Hitchin section to opers through nonabelian Hodge, J. Differential Geom. 117 (2021), 223–253.
  * [Fr1] E. Frenkel, Gaudin model and opers, Infinite dimensional algebras and quantum integrable systems, 1–58, Progr. Math., 237, Birkhäuser, Basel, 2005.
  * [Fr2] E. Frenkel, Lectures on the Langlands program and conformal field theory, Frontiers in number theory, physics, and geometry. II, 387–533, Springer, Berlin, 2007.
  * [FG1] E. Frenkel and D. Gaitsgory, Local geometric Langlands correspondence and affine Kac-Moody algebras, Algebraic geometry and number theory, 69–260, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006.
  * [FG2] E. Frenkel and D. Gaitsgory, Weyl modules and opers without monodromy, Arithmetic and geometry around quantization, 101–121, Progr. Math., 279, Birkhäuser Boston, Boston, MA, 2010.
  * [FT] E. Frenkel and C. Teleman, Geometric Langlands correspondence near opers, J. Ramanujan Math. Soc. 28 (2013), 123–147.
  * [GH] P. Griffiths and J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.
  * [In] M. Inaba, Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, J. Algebraic Geom. 22 (2013), 407–480.
  * [IIS1] M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé, 103–167, Sémin. Congr., 14, Soc. Math. France, Paris, 2006.