# Parabolic opers and differential operators

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>,
Niels Borne  Université Lille 1, Cité scientifique U.M.R. CNRS 8524, U.F.R. de
Mathématiques 59 655 Villeneuve d’Ascq Cédex, France<EMAIL_ADDRESS>lille1.fr ,  Sorin Dumitrescu  Université Côte d’Azur, CNRS, LJAD, France
<EMAIL_ADDRESS>,  Sebastian Heller  Beijing Institute of Mathematical
Sciences and Applications, Yanqi Island, Huairou District, Beijing 101408
<EMAIL_ADDRESS>and  Christian Pauly  Université Côte d’Azur, CNRS, LJAD,
France<EMAIL_ADDRESS>
###### Abstract.

Parabolic ${\rm SL}(r,\mathbb{C})$–opers were defined and investigated in
[BDP] in the set-up of vector bundles on curves with a parabolic structure
over a divisor. Here we introduce and study holomorphic differential operators
between parabolic vector bundles over curves. We consider the parabolic ${\rm
SL}(r,\mathbb{C})$–opers on a Riemann surface $X$ with given singular divisor
$S\,\subset\,X$ and with fixed parabolic weights satisfying the condition that
all parabolic weights at any $x_{i}\,\in\,S$ are integral multiples of
$\frac{1}{2N_{i}+1}$, where $N_{i}\,>\,1$ are fixed integers. We prove that
this space of opers is canonically identified with the affine space of
holomorphic differential operators of order $r$ between two natural parabolic
line bundles on $X$ (depending only on the divisor $S$ and the weights
$N_{i}$) satisfying the conditions that the principal symbol of the
differential operators is the constant function $1$ and the sub-principal
symbol vanishes identically. The vanishing of the sub-principal symbol ensures
that the logarithmic connection on the rank $r$ bundle is actually a
logarithmic ${\rm SL}(r,\mathbb{C})$–connection.

###### Key words and phrases:

Oper, parabolic bundle, differential operator, logarithmic connection

###### 2010 Mathematics Subject Classification:

14H60, 33C80, 53A55

###### Contents

  1. 1 Introduction
  2. 2 A rank two parabolic bundle
    1. 2.1 Parabolic bundles and parabolic connections
    2. 2.2 The parabolic Gunning bundle
    3. 2.3 Orbifold structure
  3. 3 Symmetric powers of parabolic bundle
    1. 3.1 Explicit description of some symmetric powers
    2. 3.2 Higher rank parabolic opers
  4. 4 Some properties of parabolic opers
  5. 5 Differential operators on parabolic bundles
    1. 5.1 Another description of differential operators on parabolic bundles
    2. 5.2 The symbol map
  6. 6 Parabolic opers and differential operators

##  1\. Introduction

After the seminal work of Drinfeld and Sokolov [DS1], [DS1], the notion of
opers was introduced by Beilinson and Drinfeld [BD1, BD2] as geometric
structures on Riemann surfaces that formalize the notion of ordinary
differential equations in a coordinate-free way. This broad formalism
encapsulates the classical notion of a Riccati equation, or equivalently that
of a complex projective structure on a Riemann surface, as being an ${\rm
SL}(2,\mathbb{C})$–oper. Since then the notion of oper turned out to be very
important, not only in the study of differential equations, but also in very
diverse topics, as for example, geometric Langlands correspondence, nonabelian
Hodge theory and also some branches of mathematical physics; see, for example,
[BF], [DFK+], [FT], [FG1], [FG2], [CS], [Fr1], [Fr2], [BSY] and references
therein. In contemporary research in mathematics and mathematical physics, the
study of opers and their applications have been firmly established as an
important topic, testified by the works of many. In particular, important
progress in the understanding of opers was carried out in [BD1, BD2, FG1, FG2,
AB, Wa, ABF, In, IIS1, IIS2].

In [BDP], three of the authors introduced and studied parabolic ${\rm
SL}(r,{\mathbb{C}})$–opers on curves in the set-up of parabolic vector bundles
as defined by Mehta and Seshadri, [MS], and also by Maruyama and Yokogawa
[MY].

Later on, being inspired by the works [AB, Sa], the infinitesimal deformations
of parabolic ${\rm SL}(r,{\mathbb{C}})$–opers and also the monodromy map for
parabolic ${\rm SL}(r,{\mathbb{C}})$–opers were studied in [BDHP]. It may be
mentioned that the appendix of [BDHP] provides 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 in [MY]. This
definition is conceptually closer to the definition of an ordinary ${\rm
SL}(r,{\mathbb{C}})$–oper and clarifies the one given in [BDP].

The objective of this article is to further investigate parabolic ${\rm
SL}(r,{\mathbb{C}})$–opers and to characterize them as a special class of
holomorphic differential operators on parabolic bundles. It should be recalled
that the relation between opers and differential operators is established and
well-known in the context of ordinary opers [BD1]. Here we introduce and study
holomorphic differential operators on parabolic vector bundles over Riemann
surfaces under the condition that at each point $x_{i}$ on the singular
divisor $S$ all the parabolic weights are integral multiples of
$\frac{1}{2N_{i}+1}$, with $N_{i}\,>\,1$ being an integer. Under this
assumption, the main result of the article, Theorem 6.2, proves that the space
of all parabolic ${\rm SL}(r,{\mathbb{C}})$–opers on $X$ with given singular
set $S\,:=\,\\{x_{1},\,\cdots,\,x_{n}\\}\,\subset\,X$ and fixed parabolic
weights integral multiples of $\frac{1}{2N_{i}+1}$ at each $x_{i}\,\in\,S$, is
canonically identified with the affine space of $r$-order holomorphic
differential operators between two natural parabolic line bundles on $X$
(depending only on $S$ and the weights $N_{i}$) having as principal symbol the
constant function $1$ and with vanishing sub-principal symbol. The vanishing
of the sub-principal symbol ensures that the logarithmic connection on the
rank $r$ bundle is indeed a logarithmic ${\rm SL}(r,{\mathbb{C}})$–connection.

The article is organized in the following way. Section 2 deals with parabolic
${\rm SL}(2,{\mathbb{C}})$–opers. In particular we introduce a rank two
parabolic bundle which is a parabolic version of the indigenous bundle (also
called Gunning bundle or uniformization bundle) introduced in [Gu] (see also
[De]); recall that this indigenous bundle introduced by Gunning is the rank
two holomorphic vector bundle associated to any ordinary ${\rm
SL}(2,{\mathbb{C}})$–oper (e.g. a complex projective structure) on a given
Riemann surface. It should be clarified that this parabolic analog of Gunning
bundle depends only on the divisor $S$ and the integers $N_{i}$. All parabolic
${\rm SL}(2,{\mathbb{C}})$–opers with given singular set $S$ and fixed weights
are parabolic connections on the same parabolic Gunning bundle.

Section 3 starts with an explicit description of several (parabolic) symmetric
powers of the rank two parabolic Gunning bundle constructed in Section 2; then
${\rm SL}(r,{\mathbb{C}})$–opers on a Riemann surface $X$, singular over
$S\,\subset\,X$, are defined (see Definition (3.3)). In this context
Proposition 3.6 proves that parabolic ${\rm SL}(r,{\mathbb{C}})$–opers on $X$
with weights equal to integral multiples of $\frac{1}{2N_{i}+1}$ at each
$x_{i}\,\in\,S$ are in natural bijection with invariant ${\rm
SL}(r,{\mathbb{C}})$–opers on a ramified Galois covering $Y$ over $X$ equipped
with an action of the Galois group. This Proposition 3.6 is a generalization
of Theorem 6.3 in [BDP] where a similar result was proved under the extra
assumption that $r$ is odd. The proof of Proposition 3.6 uses in an essential
way the correspondence studied in [Bi1], [Bo1], [Bo2], and also a result
(Corollary 2.6(3)) of Section 2 proving that, at each point of $S$, the
monodromy of any parabolic connection on the parabolic Gunning bundle is
semisimple.

Section 4 constructs the canonical parabolic filtration associated to any
parabolic ${\rm SL}(r,{\mathbb{C}})$–oper. This parabolic filtration depends
only on $S$ and the integers $N_{i}$. It is proved then that any parabolic
connection on the associated parabolic bundle satisfies the Griffith
transversality condition with respect to the above filtration (all
corresponding second fundamental forms are actually isomorphisms).

Section 5 defines and study several equivalent definitions for holomorphic
differential operators between parabolic vector bundles. Under the above
rationality assumption on the parabolic weights, Proposition 5.2 proves that
holomorphic differential operators between parabolic vector bundles are
canonically identified with the invariant holomorphic differential operators
between corresponding orbifold vector bundles on a ramified Galois covering
$Y$ over $X$ equipped with an action of the Galois group. We deduce the
construction of the principal symbol map defined on the space of differential
operators in the parabolic set-up (see Lemma 5.3).

The last Section focuses on the class of holomorphic differential operators
associated to ${\rm SL}(r,{\mathbb{C}})$–opers. These are holomorphic
differential operators between two parabolic line bundles over $X$ naturally
associated to the Gunning parabolic bundle (those line bundles only depend on
the divisor $S$ and the parabolic weights $N_{i}$). In this case the principal
symbol is the constant function $1$ and the sub-principal symbol map
(constructed in Lemma 6.1) defined on the space of parabolic differential
operators between the appropriate parabolic line bundles vanishes. Then the
main Theorem 6.2 stated above is proved.

##  2\. A rank two parabolic bundle

Let $X$ be a compact connected Riemann surface. Its canonical line bundle will
be denoted by $K_{X}$. 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 $S$.

If ${\rm genus}(X)\,=\,0$, we assume that $n\,\geq\,3$.

For any holomorphic vector bundle $E$ on $X$, and any $k\,\in\,\mathbb{Z}$,
the holomorphic vector bundle $E\otimes{\mathcal{O}}_{X}(kS)$ on $X$ will be
denoted by $E(kS)$.

Let us first start with the definition of a parabolic structure on a
holomorphic vector bundle over $X$ having $S$ as the parabolic divisor.

###  2.1. Parabolic bundles and parabolic connections

A quasiparabolic structure on a holomorphic vector bundle $E$ on $X$,
associated to the divisor $S$, 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 finite sequence of positive
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 number $\alpha_{i,j}$ is called the
parabolic weight of the corresponding subspace $E_{i,j}$ in (2.2) (see [MS],
[MY]).

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

A logarithmic connection on the holomorphic vector bundle $E$, singular over
$S$, is a holomorphic differential operator of order one

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

satisfying the Leibniz rule, meaning

$D(fs)\,=\,fD(s)+s\otimes df$ (2.4)

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

Recall that any logarithmic connection on $E$ over the Riemann surface is
necessarily flat. Indeed, the curvature ($2$-form) vanishes identically
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] which gives an isomorphism

${\mathcal{O}}_{X}(-x_{i})_{x_{i}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,(K_{X})_{x_{i}}.$
(2.5)

To describe this isomorphism, let $z$ be a holomorphic coordinate function on
$X$ defined on an analytic open neighborhood of $x_{i}$ such that
$z(x_{i})\,=\,0$. We have an isomorphism
${\mathcal{O}}_{X}(-x_{i})_{x_{i}}\,\longrightarrow\,(K_{X})_{x_{i}}$ that
sends $z$ to $dz(x_{i})$. It is straightforward to check that this map is
actually independent of the choice of the holomorphic local coordinate $z$ at
$x_{i}$.

Let $D\,:\,E\,\longrightarrow\,E\otimes K_{X}\otimes{\mathcal{O}}_{X}(S)$ be a
logarithmic connection on $E$. From (2.4) it follows that the composition of
homomorphisms

$E\,\xrightarrow{\,\ D\,\ }\,E\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\longrightarrow\,(E\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,E_{x_{i}}$
(2.6)

is ${\mathcal{O}}_{X}$–linear; the above isomorphism $(E\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))_{x_{i}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,E_{x_{i}}$
is given by the isomorphism in (2.5). Therefore, the composition of
homomorphisms in (2.6) produces a $\mathbb{C}$–linear homomorphism

${\rm Res}(D\,,x_{i})\,:\,E_{x_{i}}\,\longrightarrow\,E_{x_{i}}\,,$ (2.7)

which is called the residue of the logarithmic connection $D$ at $x_{i}$ (see
[De] for more details).

######  Remark 2.1.

The local monodromy of $D$ around $x_{i}$ is conjugated to

$\exp\left(-2\pi\sqrt{-1}\cdot{\rm Res}(D,\,x_{i})\right)\,\in\,{\rm
GL}(E_{x_{i}})$

[De].

Consider now $E$ with its parabolic structure
$E_{*}\,=\,(E,\,(\\{E_{i,j}\\},\,\\{\alpha_{i,j}\\}))$; see (2.2), (2.3).

A parabolic 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 (2.3)).

######  Remark 2.2.

The following necessary and sufficient condition for $E_{*}$ to admit a
parabolic connection was given in [BL]:

A parabolic vector bundle $E_{*}$ admits a parabolic connection if and only if
the parabolic degree of every direct summand of $E_{*}$ is zero [BL, p. 594,
Theorem 1.1].

###  2.2. The parabolic Gunning bundle

Choose a holomorphic line bundle ${\mathcal{L}}$ on $X$ such that
${\mathcal{L}}^{\otimes 2}$ is holomorphically isomorphic to $K_{X}$; also fix
a holomorphic isomorphism between ${\mathcal{L}}^{\otimes 2}$ and $K_{X}$.

We have $H^{1}(X,\,{\rm
Hom}({\mathcal{L}}^{*},\,{\mathcal{L}}))\,=\,H^{1}(X,\,K_{X})\,=\,H^{0}(X,\,{\mathcal{O}}_{X})^{*}\,=\,{\mathbb{C}}$
(Serre duality); note that here the chosen isomorphism between
${\mathcal{L}}^{\otimes 2}$ and $K_{X}$ is being used. Consequently, there is
a natural nontrivial extension $\widetilde{E}$ of ${\mathcal{L}}^{*}$ by
$\mathcal{L}$ that corresponds to

$1\,\in\,H^{1}(X,\,{\rm Hom}({\mathcal{L}}^{*},\,{\mathcal{L}})).$

So $\widetilde{E}$ fits in a short exact sequence of holomorphic vector
bundles

$0\,\longrightarrow\,{\mathcal{L}}\,\longrightarrow\,\widetilde{E}\,\xrightarrow{\,\,p_{0}\,\,}\,{\mathcal{L}}^{*}\,\longrightarrow\,0\,;$
(2.8)

this short exact sequence does not split holomorphically. Consider the
subsheaf ${\mathcal{L}}^{*}(-S)\,\subset\,{\mathcal{L}}^{*}$. Define

$E\,:=\,p^{-1}_{0}({\mathcal{L}}^{*}(-S))\,\subset\,\widetilde{E}\,,$

where $p_{0}$ is the projection in (2.8). From (2.8) we know that this $E$
fits in a short exact sequence of holomorphic vector bundles

$0\,\longrightarrow\,{\mathcal{L}}\,\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\,E\,\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\,{\mathcal{L}}^{*}(-S)\,\longrightarrow\,0\,;$ (2.9)

the projection $p$ in (2.9) is the restriction, to the subsheaf $E$, of
$p_{0}$ in (2.8).

######  Lemma 2.3.

Take any point $x\,\in\,S$. The fiber $E_{x}$ of $E$ (see (2.9)) over $x$
canonically decomposes as

$E_{x}\,=\,{\mathcal{L}}_{x}\oplus{\mathcal{L}}^{*}(-S)_{x}\,=\,{\mathcal{L}}_{x}\oplus{\mathcal{L}}_{x}\,.$

###### Proof.

Take $x\,\in\,S$. First we have the homomorphism

$\iota(x)\,:\,{\mathcal{L}}_{x}\,\longrightarrow\,E_{x}\,,$ (2.10)

where $\iota$ is the homomorphism in (2.9), which is evidently injective. On
the other hand, tensoring (2.8) with ${\mathcal{O}}_{X}(-S)$ and using the
natural map of it to (2.9) we have the commutative diagram

$\begin{matrix}0&\longrightarrow&{\mathcal{L}}(-S)&\stackrel{{\scriptstyle\iota^{\prime}}}{{\longrightarrow}}&\widetilde{E}(-S)&\stackrel{{\scriptstyle
p^{\prime}}}{{\longrightarrow}}&{\mathcal{L}}^{*}(-S)&\longrightarrow&0\\\
&&\,\,\,\Big{\downarrow}\psi^{\prime}&&\,\,\,\Big{\downarrow}\psi&&\,\,\,\,\Big{\downarrow}{\rm
Id}\\\
0&\longrightarrow&{\mathcal{L}}&\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}&E&\stackrel{{\scriptstyle
p}}{{\longrightarrow}}&{\mathcal{L}}^{*}(-S)&\longrightarrow&0,\end{matrix}$
(2.11)

where $\iota^{\prime}$ and $p^{\prime}$ are the restrictions of $\iota$ and
$p$ respectively. Note that the composition of maps

$\psi(x)\circ\iota^{\prime}(x)\,:\,{\mathcal{L}}(-S)_{x}\,\longrightarrow\,E_{x}$

in (2.11) is the zero homomorphism, because
$\psi^{\prime}(x)\,:\,{\mathcal{L}}(-S)_{x}\,\longrightarrow\,{\mathcal{L}}_{x}$
is the zero homomorphism and
$\psi\circ\iota^{\prime}\,=\,\iota\circ\psi^{\prime}$ by the commutativity of
(2.11). Since $\psi(x)\circ\iota^{\prime}(x)\,=\,0$, the homomorphism
$\psi(x)$ is given by a homomorphism

$q_{x}\,:\,\widetilde{E}(-S)_{x}/(\iota^{\prime}(x)({\mathcal{L}}(-S)_{x}))\,=\,{\mathcal{L}}^{*}(-S)_{x}\,\longrightarrow\,E_{x}\,.$
(2.12)

The homomorphism $q_{x}$ in (2.12) is injective, because $\psi(x)\,\not=\,0$.
From (2.10) and (2.12) we have

$\iota(x)\oplus
q_{x}\,:\,{\mathcal{L}}_{x}\oplus{\mathcal{L}}^{*}(-S)_{x}\,\longrightarrow\,E_{x}$
(2.13)

which is clearly an isomorphism.

Using (2.5) and the given isomorphism between ${\mathcal{L}}^{\otimes 2}$ and
$K_{X}$ we have

${\mathcal{L}}^{*}(-S)_{x}\,=\,((K_{X})_{x}\otimes{\mathcal{L}}^{*}_{x})^{*}\otimes{\mathcal{O}}_{X}(-S)_{x}\,=\,({\mathcal{L}}^{*}_{x})^{*}\,=\,{\mathcal{L}}_{x}\,.$

Hence the isomorphism in (2.13) gives that
$E_{x}\,=\,{\mathcal{L}}_{x}\oplus{\mathcal{L}}^{*}(-S)_{x}\,=\,{\mathcal{L}}_{x}\oplus{\mathcal{L}}_{x}$.
∎

For each $x_{i}\,\in\,S$ (see (2.1)), fix

$c_{i}\,\in\,{\mathbb{R}}$ (2.14)

such that $c_{i}\,>\,1$. Using $\\{c_{i}\\}_{i=1}^{n}$ we will construct a
parabolic structure on the holomorphic vector bundle $E$ in (2.9).

For any $x_{i}\,\in\,S$, the quasiparabolic filtration of $E_{x_{i}}$ is the
following:

$0\,\subset\,{\mathcal{L}}^{*}(-S)_{x_{i}}\,\subset\,E_{x_{i}}$ (2.15)

(see Lemma 2.3). The parabolic weight of ${\mathcal{L}}^{*}(-S)_{x_{i}}$ is
$\frac{c_{i}+1}{2c_{i}+1}$; the parabolic weight of $E_{x_{i}}$ is
$\frac{c_{i}}{2c_{i}+1}$. The parabolic vector bundle defined by this
parabolic structure on $E$ will be denoted by $E_{*}$. Note that

$\text{par-
deg}(E_{*})\,=\,\text{degree}(E)+\sum_{i=1}^{n}\left(\frac{c_{i}+1}{2c_{i}+1}+\frac{c_{i}}{2c_{i}+1}\right)\,=\,-n+n\,=\,0\,;$
(2.16)

in fact the parabolic second exterior product is

$\det
E_{*}\,=\,\bigwedge\nolimits^{2}E_{*}\,=\,(\bigwedge\nolimits^{2}E)\otimes{\mathcal{O}}_{X}(S)\,=\,{\mathcal{O}}_{X}\,,$
(2.17)

where ${\mathcal{O}}_{X}$ is equipped with the trivial parabolic structure (no
nonzero parabolic weights).

######  Proposition 2.4.

  1. (1)

The holomorphic vector bundle $E$ in (2.9) is isomorphic to a direct sum of
holomorphic line bundles ${\mathcal{L}}\oplus{\mathcal{L}}^{*}(-S)$.

  2. (2)

The parabolic vector bundle $E_{*}$ in (2.15) is not isomorphic to a direct
sum of parabolic line bundles.

###### Proof.

Consider the short exact sequence in (2.9). Note that

$H^{1}(X,\,\text{Hom}({\mathcal{L}}^{*}(-S),\,{\mathcal{L}}))\,=\,H^{1}(X,\,K_{X}(S))\,=\,H^{0}(X,\,{\mathcal{O}}_{X}(-S))^{*}\,=\,0\,.$

Hence the short exact sequence in (2.9) splits holomorphically, and
$E\,=\,{\mathcal{L}}\oplus{\mathcal{L}}^{*}(-S)$. This proves the first
statement.

To prove the second statement by contradiction, assume that

$E_{*}\,=\,A_{*}\oplus B_{*}\,,$ (2.18)

where $A_{*}$ and $B_{*}$ are parabolic line bundles on $X$. Since

$\text{par-deg}(A_{*})+\text{par-deg}(B_{*})\,=\,\text{par-deg}(E_{*})\,=\,0$

(see (2.16)), at least one of $A_{*}$ and $B_{*}$ has nonnegative parabolic
degree. Assume that $\text{par-deg}(A_{*})\,\geq\,0$. Since the parabolic
degree of the quotient ${\mathcal{L}}^{*}(-S)$ in (2.9), equipped with the
parabolic structure induced by $E_{*}$, is negative (recall that $n\,\geq\,3$
if ${\rm genus}(X)\,=\,0$), there is no nonzero homomorphism from $A_{*}$ to
it (recall that $\text{par-deg}(A_{*})\,\geq\,0$). Consequently, the parabolic
subbundle $A_{*}\,\subset\,E_{*}$ in (2.18) coincides with the subbundle
$\mathcal{L}$ in (2.9) equipped with the parabolic structure induced by
$E_{*}$. This implies that the following composition of homomorphisms

$B\,\hookrightarrow\,E\,\longrightarrow\,E/{\mathcal{L}}\,=\,{\mathcal{L}}^{*}(-S)$

is an isomorphism, where $B$ denotes the holomorphic line bundle underlying
$B_{*}$ in (2.18). Therefore, the inclusion map $B\,\hookrightarrow\,E$ in
(2.18) produces a holomorphic splitting

$\rho\,:\,{\mathcal{L}}^{*}(-S)\,\longrightarrow\,E$ (2.19)

of (2.9). Since $\rho$ in (2.19) is given by (2.18), and the parabolic
subbundle $A_{*}\,\subset\,E_{*}$ in (2.18) coincides with the subbundle
$\mathcal{L}$ in (2.9) equipped with the parabolic structure induced by
$E_{*}$, it follows that for all $x\,\in\,S$,

$\rho({\mathcal{L}}^{*}(-S)_{x})\,=\,{\mathcal{L}}^{*}(-S)_{x}\,\subset\,E_{x}\,.$
(2.20)

Recall that the quasiparabolic structure of $E_{*}$ at $x$ is given by the
subspace ${\mathcal{L}}^{*}(-S)_{x}\,\subset\,E_{x}$ in Lemma 2.3, and
therefore ${\mathcal{L}}^{*}(-S)_{x}$ must lie in the image, in $E_{*}$, of
either $A_{*}$ or $B_{*}$.

From (2.20) it follows that $\rho$ in (2.19) satisfies the condition

$\rho({\mathcal{L}}^{*}(-S))\,\subset\,\psi(\widetilde{E}(-S))\,\subset\,E\,,$

where $\psi$ is the homomorphism in (2.11). Consequently, $\rho$ produces a
unique holomorphic homomorphism

$\alpha\,:\,{\mathcal{L}}^{*}(-S)\,\longrightarrow\,\widetilde{E}(-S)$

such that $\rho\,=\,\psi\circ\alpha$ on ${\mathcal{L}}^{*}(-S)$. This
homomorphism $\alpha$ evidently gives a holomorphic splitting of the top exact
sequence in (2.11), meaning $p^{\prime}\circ\iota\,=\,{\rm
Id}_{{\mathcal{L}}^{*}(-S)}$, where $p^{\prime}$ is the projection in (2.11).
After tensoring the above homomorphism $\alpha$ with ${\rm
Id}_{{\mathcal{O}}_{X}(S)}$ we get a homomorphism

${\mathcal{L}}^{*}\,=\,{\mathcal{L}}^{*}(-S)\otimes{\mathcal{O}}_{X}(S)\,\xrightarrow{\,\,\,\alpha\otimes{\rm
Id}_{{\mathcal{O}}_{X}(S)}\,\,\,}\,\widetilde{E}(-S)\otimes{\mathcal{O}}_{X}(S)\,=\,\widetilde{E}$

that splits holomorphically the short exact sequence in (2.8). But, as noted
earlier, the short exact sequence in (2.8) does not split holomorphically. In
view of this contradiction we conclude that there is no decomposition as in
(2.18). ∎

######  Remark 2.5.

Regarding Proposition 2.4(1) it should be clarified that although $E$ in (2.9)
is isomorphic to ${\mathcal{L}}\oplus{\mathcal{L}}^{*}(-S)$, there is no
natural isomorphism between them. Indeed, any two holomorphic splittings of
the short exact sequence (2.9) differ by an element of

$H^{0}(X,\,{\rm
Hom}({\mathcal{L}}^{*}(-S),\,{\mathcal{L}}))\,=\,H^{0}(X,\,K_{X}(S)).$

A holomorphic splittings of the short exact sequence (2.9) produces an
isomorphism of $E_{x}$ with ${\mathcal{L}}_{x}\oplus{\mathcal{L}}^{*}(-S)_{x}$
for any $x\,\in\,X$, but this isomorphism depends on the choice of the
splitting. This shows that Proposition 2.4(1) does not imply Lemma 2.3.

We recall that a parabolic connection on the parabolic vector bundle $E_{*}$
in (2.15) is a logarithmic connection $D_{0}\,:\,E\,\longrightarrow\,E\otimes
K_{X}(S)$ on $E$, singular over $S$, such that the following conditions hold:

  1. (1)

for any $x_{i}\,\in\,S$ the eigenvalues of the residue
$\text{Res}(D_{0},\,x_{i})$ of $D_{0}$ at $x_{i}$ are
$\frac{c_{i}+1}{2c_{i}+1}$ and $\frac{c_{i}}{2c_{i}+1}$ (see (2.14)).

  2. (2)

The eigenspace in $E_{x_{i}}$ for the eigenvalue $\frac{c_{i}+1}{2c_{i}+1}$ of
$\text{Res}(D_{0},\,x_{i})$ is the line

${\mathcal{L}}^{*}(-S)_{x}\,\subset\,E_{x_{i}}$

in Lemma 2.3.

Let $D_{0}\,:\,E\,\longrightarrow\,E\otimes K_{X}(S)$ be a logarithmic
connection on $E$. Take the holomorphic line subbundle
${\mathcal{L}}\,\subset\,E$ in (2.9), and consider the composition of
homomorphisms

${\mathcal{L}}\,\hookrightarrow\,E\,\xrightarrow{\,\,\,D_{0}\,\,}\,E\otimes
K_{X}(S)\,\xrightarrow{\,\,\,p\otimes{\rm
Id}_{K_{X}(S)}\,\,}{\mathcal{L}}^{*}(-S)\otimes K_{X}(S)\,=\,{\mathcal{L}}\,,$

where $p$ is the projection in (2.9); this composition of homomorphisms will
be denoted by ${\mathcal{S}}(D_{0},\,{\mathcal{L}})$. This homomorphism

${\mathcal{S}}(D_{0},\,{\mathcal{L}})\,:\,{\mathcal{L}}\,\longrightarrow\,{\mathcal{L}}$
(2.21)

is called the second fundamental form of the subbundle
$\mathcal{L}\,\subset\,E$ for the logarithmic connection $D_{0}$. We note that
${\mathcal{S}}(D_{0},\,{\mathcal{L}})$ is a constant scalar multiplication.

A parabolic connection on $E_{*}$ induces a holomorphic connection on $\det
E_{*}\,=\,{\mathcal{O}}_{X}$ (see (2.17)). Note that any holomorphic
connection on ${\mathcal{O}}_{X}$ is of the form $d+\omega$, where $d$ denotes
the de Rham differential and $\omega\,\in\,H^{0}(X,\,K_{X})$. A parabolic
connection $D_{0}$ on $E_{*}$ is called a parabolic
$\text{SL}(2,{\mathbb{C}})$–connection if the connection on $\det
E_{*}\,=\,{\mathcal{O}}_{X}$ induced by $D_{0}$ coincides with the trivial
connection $d$.

######  Corollary 2.6.

  1. (1)

The parabolic vector bundle $E_{*}$ in (2.15) admits a parabolic ${\rm
SL}(2,{\mathbb{C}})$–connection.

  2. (2)

For any parabolic connection $D_{0}$ on $E_{*}$, the second fundamental form
${\mathcal{S}}(D_{0},\,{\mathcal{L}})$ in (2.21) is an isomorphism of
$\mathcal{L}$.

  3. (3)

For any parabolic connection $D_{0}$ on $E_{*}$ the local monodromy of $D_{0}$
around any point of $S$ is semisimple.

###### Proof.

In view of Remark 2.2, from (2.16) and the second statement in Proposition 2.4
it follows immediately that $E_{*}$ admits a parabolic connection. Take a
parabolic connection $D_{0}$ on $E_{*}$. Let $d+\omega$ be the connection on
$\det E_{*}\,=\,{\mathcal{O}}_{X}$ induced by $D_{0}$, where
$\omega\,\in\,H^{0}(X,\,K_{X})$ and $d$ is the de Rham differential. Then
$D_{0}-\frac{1}{2}\omega\otimes{\rm Id}_{E}$ is a parabolic
$\text{SL}(2,{\mathbb{C}})$–connection on $E_{*}$.

For any parabolic connection $D_{0}$ on $E_{*}$, consider the second
fundamental form ${\mathcal{S}}(D_{0},\,{\mathcal{L}})$ in the second
statement. If ${\mathcal{S}}(D_{0},\,{\mathcal{L}})\,=\,0$, then $D_{0}$
produces a parabolic connection on the line subbundle
${\mathcal{L}}\,\subset\,E$ in (2.9) equipped with the parabolic structure
induced by $E_{*}$. But the parabolic degree of this parabolic line bundle is

$g-1+\sum_{i=1}^{n}\frac{c_{i}}{2c_{i}+1}\,>\,0.$

This implies that this parabolic line bundle does not admit any parabolic
connection. Hence we conclude that
${\mathcal{S}}(D_{0},\,{\mathcal{L}})\,\not=\,0$. This implies that
${\mathcal{S}}(D_{0},\,{\mathcal{L}})$ is an isomorphism of $\mathcal{L}$.

The local monodromy of $D_{0}$ around any $x\,\in\,S$ is conjugate to
$\exp\left(-2\pi\sqrt{-1}\cdot{\rm Res}(D_{0},\,x)\right)$ (see Remark 2.1).
Hence the eigenvalues of the local monodromy for $D_{0}$ around each
$x_{i}\,\in\,S$ are $\exp\left(-2\pi\sqrt{-1}\frac{c_{i}+1}{2c_{i}+1}\right)$
and $\exp\left(-2\pi\sqrt{-1}\frac{c_{i}}{2c_{i}+1}\right)$. This proves the
third statement. ∎

We will see in Corollary 4.2 that the endomorphism
${\mathcal{S}}(D_{0},\,{\mathcal{L}})$ in Corollary 2.6(2) is actually
independent of the parabolic connection $D_{0}$ on $E_{*}$.

######  Corollary 2.7.

Take any parabolic connection $D_{0}$ on $E_{*}$. There is no holomorphic line
subbundle of $E$ preserved by $D_{0}$.

###### Proof.

Let $L\,\subset\,E$ be a holomorphic line subbundle preserved by $D_{0}$.
Denoted by $L_{*}$ the parabolic line bundle defined by the parabolic
structure on $L$ induced by $E_{*}$. Since $D_{0}$ is a parabolic connection
on $E_{*}$, its restriction to $L$ is a parabolic connection on $L_{*}$.
Therefore, we have

$\text{par-deg}(L_{*})\,=\,0.$ (2.22)

Consider the parabolic structure on the quotient ${\mathcal{L}}^{*}(-S)$ in
(2.9) induced by $E_{*}$. Its parabolic degree is negative, and hence from
(2.22) we conclude that there is no nonzero parabolic homomorphism from
$L_{*}$ to it. Consequently, the subbundle $L\,\subset\,E$ coincides with the
subbundle $\mathcal{L}$ in (2.9). Since $L\,=\,\mathcal{L}$ is preserved by
$D_{0}$, the second fundamental form ${\mathcal{S}}(D_{0},\,{\mathcal{L}})$ in
(2.21) vanishes identically. But this contradicts Corollary 2.6(2). Hence
$D_{0}$ does not preserve any holomorphic line subbundle of $E$. ∎

Given a parabolic connection $D$ on $E_{*}$, consider its monodromy
representation

${\rm Mon}_{D}\,:\,\pi_{1}(X\setminus
S,\,y)\,\longrightarrow\,\text{GL}(2,{\mathbb{C}})\,,$

where $y\,\in\,X\setminus D$ is a base point. Corollary 2.7 implies that ${\rm
Mon}_{D}$ is irreducible, meaning the action of ${\rm
Mon}_{D}(\pi_{1}(X\setminus S,\,y))\,\subset\,\text{GL}(2,{\mathbb{C}})$ on
${\mathbb{C}}^{2}$ does not preserve any line.

###  2.3. Orbifold structure

In this subsection we assume that $\\{c_{i}\\}_{i=1}^{n}$ in (2.14) are all
integers; recall that $c_{i}\,>\,1$ for all $1\,\leq\,i\,\leq\,n$,

There is a ramified Galois covering

$\varphi\,:\,Y\,\longrightarrow\,X$ (2.23)

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 the ramification of $\varphi$ at
$y$ is $2c_{i}+1$.

Such a ramified Galois covering $\varphi$ exists; see [Na, p. 26, Proposition
1.2.12]. Let

$\Gamma\,:=\,\text{Gal}(\varphi)\,=\,\text{Aut}(Y/X)\,\subset\,\text{Aut}(Y)$
(2.24)

be the Galois group for the Galois covering $\varphi$. A holomorphic vector
bundle $V\,\stackrel{{\scriptstyle q_{0}}}{{\longrightarrow}}\,Y$ is called an
orbifold bundle if $\Gamma$ acts on the total space of $V$ such that following
three conditions hold:

  1. (1)

The map $V\,\longrightarrow\,V$ given by the action of any element of $\Gamma$
on $V$ is holomorphic,

  2. (2)

the projection $q_{0}$ is $\Gamma$–equivariant, and

  3. (3)

the action of any $\gamma\,\in\,\Gamma$ on $V$ is a holomorphic automorphism
of the vector bundle $V$ over the automorphism $\gamma$ of $Y$.

Recall that the parabolic weights of $E_{*}$ at any $x_{i}\,\in\,S$ are
integral multiples of $\frac{1}{2c_{i}+1}$. Therefore, there is a unique, up
to an isomorphism, orbifold vector bundle $\mathcal{V}$ of rank two on $Y$
which corresponds to the parabolic vector bundle $E_{*}$ [Bi], [Bo1], [Bo2].
The action of $\Gamma$ on this $\mathcal{V}$ produces an action of $\Gamma$ on
the direct image $\varphi_{*}{\mathcal{V}}$. We have

$(\varphi_{*}{\mathcal{V}})^{\Gamma}\,=\,E\,.$ (2.25)

From (2.17) it follows that

$\det{\mathcal{V}}\,=\,\bigwedge\nolimits^{2}{\mathcal{V}}\,=\,{\mathcal{O}}_{Y}\,,$
(2.26)

and the action of $\Gamma$ on the orbifold bundle $\det{\mathcal{V}}$
coincides with the action of $\Gamma$ on ${\mathcal{O}}_{Y}$ given by the
action of $\Gamma$ on $Y$. Consider the subbundle $\mathcal{L}\,\subset\,E$ in
(2.9). Let

$\mathbf{L}\,\subset\,\mathcal{V}$ (2.27)

be the orbifold line subbundle corresponding to it. So the action of $\Gamma$
on $\mathcal{V}$ preserves the subbundle $\mathbf{L}$, and the subbundle

$(\varphi_{*}{\mathbf{L}})^{\Gamma}\,\subset\,(\varphi_{*}{\mathcal{V}})^{\Gamma}\,=\,E\,.$

coincides with $\mathcal{L}$.

The action of $\Gamma$ on $Y$ produces an action of $\Gamma$ on the canonical
bundle $K_{Y}$. For any automorphism $\gamma\,\in\,\Gamma$ consider its
differential $d\gamma\,:\,TY\,\longrightarrow\,\gamma^{*}TY$. The action of
$\gamma$ on $K_{Y}$ is given by $((d\gamma)^{*})^{-1}\,=\,(d\gamma^{-1})^{*}$.
Therefore, $K_{Y}$ is an orbifold line bundle.

######  Lemma 2.8.

The orbifold line bundle $\mathbf{L}^{\otimes 2}$ (see (2.27)) is isomorphic
to the orbifold line bundle $K_{Y}$.

###### Proof.

Let ${\mathcal{L}}_{*}$ denote the holomorphic line subbundle $\mathcal{L}$ in
(2.9) equipped with the parabolic structure on it induced by $E_{*}$. So the
underlying holomorphic line bundle for the parabolic bundle
${\mathcal{L}}_{*}\otimes{\mathcal{L}}_{*}$ is $K_{X}$, and the parabolic
weight at any $x_{i}\,\in\,S$ is $\frac{2c_{i}}{2c_{i}+1}$. Hence the orbifold
line bundle on $Y$ corresponding to
${\mathcal{L}}_{*}\otimes{\mathcal{L}}_{*}$ is

$(\varphi^{*}K_{X})\otimes{\mathcal{O}}_{Y}\left(\sum_{i=1}^{n}2c_{i}\varphi^{-1}(x_{i})_{\rm
red}\right)\,=\,K_{Y}$

equipped with the action of $\Gamma$ given by the action $\Gamma$ on $Y$,
where $\varphi^{-1}(x_{i})_{\rm red}$ is the reduced inverse image of $x_{i}$.
Since the orbifold line bundle $\mathbf{L}^{\otimes 2}$ corresponds to the
parabolic line bundle ${\mathcal{L}}_{*}\otimes{\mathcal{L}}_{*}$, the lemma
follows. ∎

From Lemma 2.8 it follows that $\mathbf{L}$ is an orbifold theta
characteristic on $Y$, and from (2.26) we have a short exact sequence of
orbifold bundles

$0\,\longrightarrow\,{\mathbf{L}}\,\longrightarrow\,\mathcal{V}\,\longrightarrow\,{\mathbf{L}}^{*}\,\longrightarrow\,0\,.$
(2.28)

######  Corollary 2.9.

The short exact sequence in (2.28) does not admit any $\Gamma$–equivariant
holomorphic splitting.

###### Proof.

If (2.28) has a $\Gamma$–equivariant holomorphic splitting, then $\mathcal{V}$
is a direct sum of orbifold line bundles. This would imply that the parabolic
vector bundle $E_{*}$ — that corresponds to $\mathcal{V}$ — is a direct sum of
parabolic line bundles. Therefore, from Proposition 2.4(2) it follows that
(2.28) does not admit any $\Gamma$–equivariant holomorphic splitting. ∎

Actually a stronger form of Corollary 2.9 can be proved using it.

######  Proposition 2.10.

The short exact sequence of holomorphic vector bundles in (2.28) does not
admit any holomorphic splitting.

###### Proof.

Assume that there is a holomorphic splitting

$\rho\,\,:\,\,{\mathbf{L}}^{*}\,\longrightarrow\,\mathcal{V}$

of the short exact sequence of holomorphic vector bundles in (2.28). Although
$\rho$ itself may not be $\Gamma$–equivariant, using it we will construct a
$\Gamma$–equivariant splitting. For any $\gamma\,\in\,\Gamma$, the composition
of homomorphisms

${\mathbf{L}}^{*}\,\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\,{\mathbf{L}}^{*}\,\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}\,{\mathcal{V}}\,\xrightarrow{\,\,\gamma^{-1}\,\,}\,\mathcal{V}\,,$

which will be denoted by $\rho[\gamma]$, is also a holomorphic splitting of
the short exact sequence of holomorphic vector bundles in (2.28). Now the
average

$\widetilde{\rho}\,:=\,\frac{1}{\\#\Gamma}\sum_{\gamma\in\Gamma}\rho[\gamma]\,\,:\,\,{\mathbf{L}}^{*}\,\longrightarrow\,\mathcal{V},$

where $\\#\Gamma$ is the order of $\Gamma$, is a $\Gamma$–equivariant
holomorphic splitting of the short exact sequence of holomorphic vector
bundles in (2.28). But this contradicts Corollary 2.9. Therefore, the short
exact sequence of holomorphic vector bundles in (2.28) does not admit any
holomorphic splitting. ∎

The $\Gamma$–invariant holomorphic connections on $\mathcal{V}$ correspond to
the parabolic connections on $E_{*}$. Moreover, the parabolic ${\rm
SL}(2,{\mathbb{C}})$–connections on $E_{*}$ correspond to the
$\Gamma$–invariant holomorphic connections $D_{V}$ on $\mathcal{V}$ that
satisfy the condition that the holomorphic connection on
$\det{\mathcal{V}}\,=\,{\mathcal{O}}_{Y}$ (see (2.26)) induced by $D_{V}$ is
the trivial connection on ${\mathcal{O}}_{Y}$ given by the de Rham
differential.

######  Lemma 2.11.

The orbifold vector bundle $\mathcal{V}$ admits ${\rm
SL}(2,{\mathbb{C}})$–oper connections. The parabolic ${\rm
SL}(2,{\mathbb{C}})$–connections on the parabolic bundle $E_{*}$ are precisely
the $\Gamma$–invariant ${\rm SL}(2,{\mathbb{C}})$–oper structures on the
orbifold bundle $\mathcal{V}$.

###### Proof.

From Proposition 2.10 it follows immediately that $\mathcal{V}$ admits ${\rm
SL}(2,{\mathbb{C}})$–oper connections. Now the second statement of the lemma
is deduced from the above observation that the parabolic ${\rm
SL}(2,{\mathbb{C}})$–connections on $E_{*}$ correspond to the
$\Gamma$–invariant holomorphic connections $D_{V}$ on $\mathcal{V}$ that
satisfy the condition that the holomorphic connection on
$\det{\mathcal{V}}\,=\,{\mathcal{O}}_{Y}$ induced by $D_{V}$ is the trivial
connection on ${\mathcal{O}}_{Y}$. ∎

##  3\. Symmetric powers of parabolic bundle

###  3.1. Explicit description of some symmetric powers

In Section 3.2 we will define parabolic ${\rm SL}(r,{\mathbb{C}})$–opers for
all $r\,\geq\,2$. The definition involves symmetric powers of the parabolic
vector bundle $E_{*}$ in (2.16). Keeping this in mind, we will explicitly
describe a few low degree symmetric powers of the parabolic vector bundle
$E_{*}$. This will done using the alternative description of parabolic bundles
— given by Maruyama and Yokogawa in [MY] (see also [Yo] and [BDHP, Appendix
A3]) — as filtered sheaves. This approach of [MY] is better suited for
handling the tensor product, symmetric product exterior product of parabolic
vector bundles.

First we will describe the second symmetric power $\text{Sym}^{2}(E_{*})$ of
the parabolic vector bundle $E_{*}$. Consider the rank three holomorphic
vector bundle $\text{Sym}^{2}(E)$, where $E$ is the vector bundle in (2.9).
Since $\text{Sym}^{2}(E)$ is a quotient of $E^{\otimes 2}$, any subspace of
$E^{\otimes 2}_{x}$ produces a subspace of $\text{Sym}^{2}(E)_{x}$. For each
$x_{i}\,\in\,S$, let

$B_{i}\,\subset\,\text{Sym}^{2}(E)_{x_{i}}\,=\,\text{Sym}^{2}(E_{x_{i}})$

be the subspace given by the image of

$E_{x_{i}}\otimes{\mathcal{L}}^{*}(-S)_{x_{i}}\,\subset\,E^{\otimes
2}_{x_{i}}$

in $\text{Sym}^{2}(E_{x_{i}})$, where
${\mathcal{L}}^{*}(-S)_{x_{i}}\,\subset\,E_{x_{i}}$ is the subspace in Lemma
2.3. Consider the unique holomorphic vector bundle $E^{2}$ of rank three on
$X$ that fits in the following short exact sequence of sheaves

$0\,\longrightarrow\,E^{2}\,\longrightarrow\,{\rm Sym}^{2}(E)(S)\,:=\,{\rm
Sym}^{2}(E)\otimes{\mathcal{O}}_{X}(S)$ (3.1)
$\longrightarrow\,\bigoplus_{i=1}^{n}\left(\text{Sym}^{2}(E)_{x_{i}}/B_{i}\right)\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,\longrightarrow\,0\,.$

The holomorphic vector bundle underlying the parabolic vector bundle
$\text{Sym}^{2}(E_{*})$ is $E^{2}$.

######  Lemma 3.1.

For every $x_{i}\,\in\,S$, the fiber $E^{2}_{x}$ fits in a natural exact
sequence

$0\,\longrightarrow\,{\mathcal{L}}^{\otimes
2}_{x_{i}}\,\longrightarrow\,E^{2}_{x_{i}}\,\longrightarrow\,B_{i}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$
$=\,(E_{x_{i}}\otimes{\mathcal{L}}^{*}(-S)_{x_{i}})\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,=\,(E\otimes{\mathcal{L}}^{*})_{x_{i}}\,\longrightarrow\,0\,.$

###### Proof.

Consider the commutative digram

$\begin{matrix}0&\longrightarrow&\text{Sym}^{2}(E)&\longrightarrow&{\rm
Sym}^{2}(E)(S)&\longrightarrow&\bigoplus_{i=1}^{n}\text{Sym}^{2}(E)(S)_{x_{i}}&\longrightarrow&0\\\
&&\,\,\Big{\downarrow}\mathbf{f}&&\,\,\,\,\,\Big{\downarrow}{\rm
Id}&&\Big{\downarrow}\\\ 0&\longrightarrow&E^{2}&\longrightarrow&{\rm
Sym}^{2}(E)(S)&\longrightarrow&\bigoplus_{i=1}^{n}\frac{\text{Sym}^{2}(E)(S)_{x_{i}}}{B_{i}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}}&\longrightarrow&0.\end{matrix}$

For any $x\,\in\,S$, the map
${\mathbf{f}}(x)\,:\,\text{Sym}^{2}(E)_{x}\,\longrightarrow\,E^{2}_{x}$ is
injective on the subspace ${\mathcal{L}}^{\otimes
2}_{x_{i}}\,\hookrightarrow\,\text{Sym}^{2}(E)_{x_{i}}$, and moreover
${\mathbf{f}}(x_{i})({\mathcal{L}}^{\otimes
2}_{x_{i}})\,\subset\,E^{2}_{x_{i}}$ coincides with
${\mathbf{f}}(x_{i})(\text{Sym}^{2}(E)_{x_{i}})$. Therefore, the subspace
${\mathcal{L}}^{\otimes 2}_{x_{i}}\,\hookrightarrow\,E^{2}_{x_{i}}$ in the
lemma is the image of the homomorphism ${\mathbf{f}}(x_{i})$.

For the map $E^{2}\,\longrightarrow\,{\rm Sym}^{2}(E)(S)\,:=\,{\rm
Sym}^{2}(E)\otimes{\mathcal{O}}_{X}(S)$ in (3.1), the image of $E^{2}_{x_{i}}$
is

$B_{i}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,=\,(E_{x_{i}}\otimes{\mathcal{L}}^{*}(-S)_{x_{i}})\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,=\,(E\otimes{\mathcal{L}}^{*})_{x_{i}}\,\subset\,{\rm
Sym}^{2}(E)(S)_{x_{i}}\,.$

This proves the lemma. ∎

For any $x_{i}\,\in\,S$, consider the subspace

${\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\,\subset\,B_{i}\,=\,({\mathcal{L}}_{x_{i}}\otimes{\mathcal{L}}^{*}(-S)_{x_{i}})\oplus{\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\,.$

Let

$\mathcal{F}_{i}\,\subset\,E^{2}_{x_{i}}$ (3.2)

be the inverse image of ${\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,\subset\,B_{i}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$
for the quotient map
$E^{2}_{x_{i}}\,\longrightarrow\,B_{i}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$ in
Lemma 3.1.

As mentioned before, the holomorphic vector bundle underlying the parabolic
vector bundle $\text{Sym}^{2}(E_{*})$ is $E^{2}$. The quasiparabolic
filtration of $E^{2}_{x_{i}}$, where $x_{i}\,\in\,S$, is the following:

${\mathcal{L}}^{\otimes
2}_{x_{i}}\,\,\subset\,\,\mathcal{F}_{i}\,\,\subset\,\,E^{2}_{x_{i}}\,,$ (3.3)

where ${\mathcal{L}}^{\otimes 2}_{x_{i}}$ and ${\mathcal{F}}_{i}$ are the
subspaces in Lemma 3.1 and (3.2) respectively. The parabolic weight of
${\mathcal{L}}^{\otimes 2}_{x_{i}}$ is $\frac{2c_{i}}{2c_{i}+1}$ and the
parabolic weight of $\mathcal{F}_{i}$ is $\frac{1}{2c_{i}+1}$; the parabolic
weight of $E^{2}_{x_{i}}$ is $0$.

The parabolic symmetric product $\text{Sym}^{3}(E_{*})$ is actually a little
easier to describe. The holomorphic vector bundle underlying the parabolic
vector bundle $\text{Sym}^{3}(E_{*})$ is the rank four vector bundle

$E^{3}\,:=\,(\text{Sym}^{3}(E))\otimes{\mathcal{O}}_{X}(S).$ (3.4)

For each $x_{i}\,\in\,S$, the decomposition of $E_{x_{i}}$ in Lemma 2.3 gives
the following decomposition of the fiber $E^{3}_{x_{i}}$:

$\left(({\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\otimes{\mathcal{L}}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)_{x_{i}}\otimes{\mathcal{L}}^{\otimes
2}_{x_{i}})\oplus({\mathcal{L}}^{\otimes
3}_{x_{i}})\right)\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,=\,E^{3}_{x_{i}}\,.$
(3.5)

The quasiparabolic filtration of $E^{3}_{x_{i}}$ is

$({\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}})\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,\subset\,\left(({\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\otimes{\mathcal{L}}_{x_{i}})\right)\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$
(3.6) $\subset\,\left(({\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\otimes{\mathcal{L}}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)_{x_{i}}\otimes{\mathcal{L}}^{\otimes
2}_{x_{i}})\right)\otimes{\mathcal{O}}_{X}(S)_{x_{i}}\,\subset\,E^{3}_{x_{i}}.$

The parabolic weight of ${\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}}\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$ is $\frac{c_{i}+2}{2c_{i}+1}$,
The parabolic weight of

$\left(({\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\otimes{\mathcal{L}}_{x_{i}})\right)\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$

is $\frac{c_{i}+1}{2c_{i}+1}$, the parabolic weight of
$\left(({\mathcal{L}}^{*}(-S)^{\otimes
3}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)^{\otimes
2}_{x_{i}}\otimes{\mathcal{L}}_{x_{i}})\oplus({\mathcal{L}}^{*}(-S)_{x_{i}}\otimes{\mathcal{L}}^{\otimes
2}_{x_{i}})\right)\otimes{\mathcal{O}}_{X}(S)_{x_{i}}$ is
$\frac{c_{i}}{2c_{i}+1}$, and the parabolic weight of $E^{3}_{x_{i}}$ is
$\frac{c_{i}-1}{2c_{i}+1}$.

Finally, we will describe the parabolic symmetric product
$\text{Sym}^{4}(E_{*})$. Consider the rank five vector bundle

$\text{Sym}^{4}(E)(2S)\,=\,(\text{Sym}^{4}(E))\otimes{\mathcal{O}}_{X}(2S)\,.$

Using Lemma 2.3, the fiber $\text{Sym}^{4}(E)(2S)_{x_{i}}$, where
$x_{i}\,\in\,S$, decomposes into a direct sum of lines. More precisely, as in
(3.5),

$\text{Sym}^{4}(E)(2S)_{x_{i}}\,=\,(({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-S))_{x_{i}}$ (3.7)
$\oplus(({\mathcal{L}}^{*})^{\otimes 2}\otimes{\mathcal{L}}^{\otimes
2})_{x_{i}}\oplus({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3}(S))_{x_{i}}\oplus({\mathcal{L}}^{\otimes 4}(2S))_{x_{i}}.$

Let $E^{4}$ denote the vector bundle of rank five defined by the following
short exact sequence of sheaves:

$0\,\longrightarrow\,E^{4}\,\stackrel{{\scriptstyle\mathbf{h}}}{{\longrightarrow}}\,\text{Sym}^{4}(E)(2S)\,\longrightarrow\,$
(3.8)
$\bigoplus_{i=1}^{n}{\mathcal{Q}}_{i}\,=\,\bigoplus_{i=1}^{n}\frac{\text{Sym}^{4}(E)(2S)_{x_{i}}}{(({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
2}\otimes{\mathcal{L}}^{\otimes 2})_{x_{i}}}\,\longrightarrow\,0,$

where

${\mathcal{Q}}_{i}\,:=\,\frac{\text{Sym}^{4}(E)(2S)_{x_{i}}}{(({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
2}\otimes{\mathcal{L}}^{\otimes 2})_{x_{i}}}.$ (3.9)

The holomorphic vector bundle underlying the parabolic vector bundle
$\text{Sym}^{4}(E_{*})$ is $E^{4}$ defined in (3.8).

######  Lemma 3.2.

For every $x_{i}\,\in\,S$, the fiber $E^{4}_{x_{i}}$ fits in the following
short exact sequence of vector spaces:

$0\,\longrightarrow\,({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes
4}(S))_{x_{i}}\,\longrightarrow\,E^{4}_{x_{i}}$
$\xrightarrow{\,\,\rho_{i}\,\,}\,(({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
2}\otimes{\mathcal{L}}^{\otimes 2})_{x_{i}}\,\longrightarrow\,0\,.$

###### Proof.

The projection

$\rho_{i}\,:\,E^{4}_{x_{i}}\,\longrightarrow\,(({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
2}\otimes{\mathcal{L}}^{\otimes 2})_{x_{i}}$

in the lemma is given by the homomorphism ${\mathbf{h}}(x_{i})$ in (3.8). To
describe the homomorphism

$({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes
4}(S))_{x_{i}}\,\longrightarrow\,E^{4}_{x_{i}}$

in the lemma, we consider the commutative diagram of homomorphisms

$\begin{matrix}0&\longrightarrow&\text{Sym}^{4}(E)(S)&\longrightarrow&{\rm
Sym}^{4}(E)(2S)&\longrightarrow&\bigoplus_{i=1}^{n}\text{Sym}^{4}(E)(2S)_{x_{i}}&\longrightarrow&0\\\
&&\,\,\Big{\downarrow}\mathbf{f}&&\Big{\downarrow}&&\Big{\downarrow}\\\
0&\longrightarrow&E^{4}&\longrightarrow&{\rm
Sym}^{4}(E)(2S)&\longrightarrow&\bigoplus_{i=1}^{n}{\mathcal{Q}}_{i}&\longrightarrow&0\end{matrix}$

where ${\mathcal{Q}}_{i}$ is defined in (3.9). Let

${\mathbf{f}}(x_{i})\,:\,\text{Sym}^{4}(E)(S)_{x_{i}}\,\longrightarrow\,E^{4}_{x_{i}}$
(3.10)

be the restriction of it to $x_{i}\,\in\,S$. As in (3.7), we have the
decomposition

$\text{Sym}^{4}(E)(S)_{x_{i}}\,=\,(({\mathcal{L}}^{*})^{\otimes
4}(-3S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-2S))_{x_{i}}$ $\oplus(({\mathcal{L}}^{*})^{\otimes
2}\otimes{\mathcal{L}}^{\otimes
2}(-S))_{x_{i}}\oplus({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes 4}(S))_{x_{i}}.$

The subspace

$(({\mathcal{L}}^{*})^{\otimes
4}(-3S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
2}\otimes{\mathcal{L}}^{\otimes
2}(-S))_{x_{i}}\,\subset\,\text{Sym}^{4}(E)(S)_{x_{i}}$

is the kernel of the homomorphism ${\mathbf{f}}(x_{i})$ in (3.10). The
restriction of ${\mathbf{f}}(x_{i})$ to the subspace

$({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes
4}(S))_{x_{i}}\,\subset\,\text{Sym}^{4}(E)(S)_{x_{i}}$

is injective. Therefore, ${\mathbf{f}}(x_{i})$ gives the homomorphism

$({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes
4}(S))_{x_{i}}\,\longrightarrow\,E^{4}_{x_{i}}$

in the lemma. It is evident that the quotient map
$E^{4}_{x_{i}}\,\longrightarrow\,E^{4}_{x_{i}}/(({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes 4}(S))_{x_{i}})$ coincides with
$\rho_{i}$. ∎

Define the subspaces

${\mathcal{F}}^{i}_{3}\,:=\,\rho^{-1}_{i}((({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}})\,\subset\,{\mathcal{F}}^{i}_{4}\,:=\,\rho^{-1}_{i}((({\mathcal{L}}^{*})^{\otimes
4}(-2S))_{x_{i}}\oplus(({\mathcal{L}}^{*})^{\otimes
3}\otimes{\mathcal{L}}(-S))_{x_{i}})\,\subset\,E^{4}_{x_{i}}$ (3.11)

where $\rho_{i}$ is the homomorphism in Lemma 3.2.

As mentioned before, the holomorphic vector bundle underlying the parabolic
vector bundle $\text{Sym}^{4}(E_{*})$ is $E^{4}$. The quasiparabolic
filtration of $E^{4}_{x_{i}}$ is

$({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\,\subset\,({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes
4}(S))_{x_{i}}\,\subset\,{\mathcal{F}}^{i}_{3}\,\subset\,{\mathcal{F}}^{i}_{4}\,\subset\,E^{4}_{x_{i}}$

(see Lemma 3.2 and (3.11)). The parabolic weight of
$({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes 3})_{x_{i}}$ is
$\frac{2c_{i}}{2c_{i}+1}$, the parabolic weight of
$({\mathcal{L}}^{*}\otimes{\mathcal{L}}^{\otimes
3})_{x_{i}}\oplus({\mathcal{L}}^{\otimes 4}(S))_{x_{i}}$ is
$\frac{2c_{i}-1}{2c_{i}+1}$, the parabolic weight of ${\mathcal{F}}^{i}_{3}$
is $\frac{2}{2c_{i}+1}$, the parabolic weight of ${\mathcal{F}}^{i}_{4}$ is
$\frac{1}{2c_{i}+1}$ and the parabolic weight of $E^{4}_{x_{i}}$ is $0$.

###  3.2. Higher rank parabolic opers

For any $r\,\geq\,2$, consider the parabolic vector bundle of rank $r$ defined
by the symmetric product $\text{Sym}^{r-1}(E_{*})$ of the parabolic vector
bundle $E_{*}$ in (2.15). Since $\det E_{*}\,=\,{\mathcal{O}}_{X}$ (see
(2.17)), it follows that

$\det\text{Sym}^{r-1}(E_{*})\,=\,\bigwedge\nolimits^{r}\text{Sym}^{r-1}(E_{*})\,=\,{\mathcal{O}}_{X},$
(3.12)

where ${\mathcal{O}}_{X}$ is equipped with the trivial parabolic structure (no
nonzero parabolic weights).

A parabolic ${\rm SL}(r,{\mathbb{C}})$–connection on $\text{Sym}^{r-1}(E_{*})$
is a parabolic connection on $\text{Sym}^{r-1}(E_{*})$ satisfying the
condition that the induced parabolic connection on
$\det\text{Sym}^{r-1}(E_{*})\,=\,{\mathcal{O}}_{X}$ is the trivial connection.

Two parabolic ${\rm SL}(r,{\mathbb{C}})$–connections on
$\text{Sym}^{r-1}(E_{*})$ are called equivalent if they differ by a
holomorphic automorphism of the parabolic bundle $\text{Sym}^{r-1}(E_{*})$. If
$D_{1}$ is a parabolic ${\rm SL}(r,{\mathbb{C}})$–connection on
$\text{Sym}^{r-1}(E_{*})$, and $D_{2}$ is another parabolic connection on
$\text{Sym}^{r-1}(E_{*})$ equivalent to $D_{1}$, then $D_{2}$ is clearly a
parabolic ${\rm SL}(r,{\mathbb{C}})$–connection. Indeed, this follows
immediately from the fact that the holomorphic automorphisms of a holomorphic
line bundle $\mathbb{L}$ on $X$ act trivially on the space of all logarithmic
connections on $\mathbb{L}$.

######  Definition 3.3.

A parabolic $\text{SL}(r,{\mathbb{C}})$–oper on $X$ is an equivalence class of
parabolic ${\rm SL}(r,{\mathbb{C}})$–connections on $\text{Sym}^{r-1}(E_{*})$.

######  Remark 3.4.

It should be clarified that the class of parabolic
$\text{SL}(r,{\mathbb{C}})$–opers in Definition 3.3 is different from the
class in [BDP] (see [BDP, p. 504, Definition 4.1] and [BDP, p. 511, Definition
5.2]). Indeed, the parabolic vector bundle $E_{*}$ in (2.16) is different from
the one in [BDP] (see [BDP, p. 497, (3.4)], [BDP, p. 497, (3.5)]). In fact the
underlying rank two bundles are different and the parabolic weights are also
different. In the nonparabolic case there is only one class of
$\text{SL}(r,{\mathbb{C}})$–opers. Roughly speaking, parabolic
$\text{SL}(r,{\mathbb{C}})$–opers can be considered as equivariant opers and
the two classes of parabolic $\text{SL}(r,{\mathbb{C}})$–opers arise because
of two different types of equivariant structures.

######  Proposition 3.5.

  1. (1)

The parabolic vector bundle ${\rm Sym}^{r-1}(E_{*})$ admits a parabolic ${\rm
SL}(r,{\mathbb{C}})$–connection.

  2. (2)

For any parabolic connection $D_{r}$ on ${\rm Sym}^{r-1}(E_{*})$, the local
monodromy of $D_{r}$ around any $x_{i}\,\in\,S$ is semisimple.

###### Proof.

Any parabolic connection on $E_{*}$ induces a parabolic connection on
$\text{Sym}^{r-1}(E_{*})$. Moreover, a parabolic ${\rm
SL}(2,{\mathbb{C}})$–connection on $E_{*}$ induces a parabolic ${\rm
SL}(r,{\mathbb{C}})$–connection on $\text{Sym}^{r-1}(E_{*})$. Therefore, from
Corollary 2.6(1) it follows that $\text{Sym}^{r-1}(E_{*})$ admits a parabolic
connection on $E_{*}$.

Let $D_{2}$ be a parabolic ${\rm SL}(2,{\mathbb{C}})$–connection on $E_{*}$.
Denote by $D_{r}$ the parabolic connection on $\text{Sym}^{r-1}(E_{*})$
induced by $D_{2}$. From Corollary 2.6(3) we know that the local monodromy of
$D_{2}$ around any $x_{i}\,\in\,S$ is semisimple. Since the local monodromy of
$D_{r}$ around any $x_{i}\,\in\,S$ is simply the $(r-1)$-th symmetric product
of the local monodromy of $D_{2}$ around $x_{i}\,\in\,S$, and the local
monodromy of $D_{2}$ around $x_{i}\,\in\,S$ is semisimple, it follows that the
local monodromy of $D_{r}$ around $x_{i}\,\in\,S$ is semisimple.

We have shown that $\text{Sym}^{r-1}(E_{*})$ admits a parabolic connection for
which the local monodromy around any $x_{i}\,\in\,S$ is semisimple. On the
other hand, the space of parabolic connections on $\text{Sym}^{r-1}(E_{*})$ is
an affine space for the vector space

$H^{0}(X,\,\text{End}^{n}(\text{Sym}^{r-1}(E_{*}))\otimes K_{X}(S)),$

where

$\text{End}^{n}(\text{Sym}^{r-1}(E_{*}))\,\subset\,\text{End}(\text{Sym}^{r-1}(E_{*}))$
(3.13)

is the subsheaf defined by the sheaf of endomorphisms nilpotent with respect
to the quasiparabolic filtrations of $\text{Sym}^{r-1}(E_{*})$ over $S$.
Consequently, using Remark 2.1 it follows that for every parabolic connection
$D^{\prime}_{r}$ on $\text{Sym}^{r-1}(E_{*})$ the local monodromy of
$D^{\prime}_{r}$ around any $x_{i}\,\in\,S$ is semisimple. ∎

In the rest of this section we assume that $c_{i}$, $1\,\leq\,i\,\leq\,n$, in
(2.14) are integers. Take a ramified Galois covering
$\varphi\,:\,Y\,\longrightarrow\,X$ as in (2.23). As in Section 2, let
$\mathcal{V}$ denote the orbifold bundle on $Y$ corresponding to the parabolic
bundle $E_{*}$ on $X$. The action of the Galois group
$\Gamma\,=\,\text{Gal}(\varphi)$ on $\mathcal{V}$ produces an action of
$\Gamma$ on $\text{Sym}^{r-1}({\mathcal{V}})$. A holomorphic connection on
$\text{Sym}^{r-1}({\mathcal{V}})$ is called equivariant if it is preserved by
the action of $\Gamma$ on $\text{Sym}^{r-1}({\mathcal{V}})$.

From (3.12) it follows immediately that

$\det\text{Sym}^{r-1}({\mathcal{V}})\,=\,\bigwedge\nolimits^{r}\text{Sym}^{r-1}({\mathcal{V}})\,=\,{\mathcal{O}}_{Y}.$

An ${\rm SL}(r,{\mathbb{C}})$–connection on $\text{Sym}^{r-1}({\mathcal{V}})$
is a holomorphic connection $D^{\prime}_{r}$ on
$\text{Sym}^{r-1}({\mathcal{V}})$ such that the connection on
$\det\text{Sym}^{r-1}({\mathcal{V}})\,=\,{\mathcal{O}}_{Y}$ induced by
$D^{\prime}_{r}$ coincides with the trivial connection on ${\mathcal{O}}_{Y}$.
Two equivariant ${\rm SL}(r,{\mathbb{C}})$–connections on
$\text{Sym}^{r-1}({\mathcal{V}})$ are called equivalent if they differ by a
holomorphic $\Gamma$–equivariant automorphism of
$\text{Sym}^{r-1}({\mathcal{V}})$.

######  Proposition 3.6.

There is a natural bijection between the parabolic ${\rm
SL}(r,{\mathbb{C}})$–opers on $X$ and the equivalence classes of equivariant
${\rm SL}(r,{\mathbb{C}})$–connections on ${\rm Sym}^{r-1}({\mathcal{V}})$.

###### Proof.

Let $D_{2}$ be a parabolic connection on $E_{*}$. Since the local monodromy of
$D_{2}$ around any $x_{i}\,\in\,S$ is semisimple, it corresponds to an
equivariant holomorphic connection $\widehat{D}_{2}$ on $\mathcal{V}$. Let
$\widehat{D}_{r}$ be the equivariant connection on
$\text{Sym}^{r-1}({\mathcal{V}})$ induced by $\widehat{D}_{2}$. As before,
$D_{r}$ denotes the parabolic connection on $\text{Sym}^{r-1}(E_{*})$ induced
by $D_{2}$. Therefore, $\widehat{D}_{r}$ corresponds to $D_{r}$.

The holomorphic vector bundle underlying the parabolic bundle
$\text{Sym}^{r-1}(E_{*})$ is denoted by $\text{Sym}^{r-1}(E_{*})_{0}$ [MY]. As
in (3.13), let

$\text{End}^{n}(\text{Sym}^{r-1}(E_{*}))\,\subset\,\text{End}(\text{Sym}^{r-1}(E_{*})_{0})$

be the coherent analytic subsheaf consisting of all locally defined sections
$s$ of the endomorphism bundle $\text{End}(\text{Sym}^{r-1}(E_{*})_{0})$
satisfying the condition that $s(x)$ is nilpotent with respect to the quasi-
parabolic filtration of $\text{Sym}^{r-1}(E_{*})_{x}$, for all $x\,\in\,S$
lying in the domain of $s$. Recall that any parabolic connection on
$\text{Sym}^{r-1}(E_{*})$ is of the form $D_{r}+\theta$ for some

$\theta\,\in\,H^{0}(X,\,\text{End}^{n}(\text{Sym}^{r-1}(E_{*}))\otimes
K_{X}(S)).$

We have

$H^{0}(X,\,\text{End}^{n}(\text{Sym}^{r-1}(E_{*}))\otimes
K_{X}(S))\,=\,H^{0}(Y,\,\text{End}(\text{Sym}^{r-1}({\mathcal{V}})))^{\Gamma}.$
(3.14)

Also the space of all equivariant holomorphic connections on ${\rm
Sym}^{r-1}({\mathcal{V}})$ is an affine space for
$H^{0}(Y,\,\text{End}(\text{Sym}^{r-1}({\mathcal{V}})))^{\Gamma}$.

The parabolic connection $D_{r}+\theta$, where
$\theta\,\in\,H^{0}(X,\,\text{End}^{n}(\text{Sym}^{r-1}(E_{*}))\otimes
K_{X}(S))$, corresponds to the equivariant connection
$\widehat{D}_{r}+\widehat{\theta}$ on $\text{Sym}^{r-1}({\mathcal{V}})$, where
$\widehat{\theta}\,\in\,H^{0}(Y,\,\text{End}(\text{Sym}^{r-1}({\mathcal{V}})))^{\Gamma}$
corresponds to $\theta$ by the isomorphism in (3.14). Also, parabolic
automorphisms of $\text{Sym}^{r-1}(E_{*})$ are identified with the
$\Gamma$–equivariant automorphisms of $\mathcal{V}$. Now the proposition
follows from (3.14), Proposition 3.5 and Definition 3.3. ∎

The above Proposition 3.6 is a generalization of Theorem 6.3 in [BDP] where a
similar statement was proved under the extra assumption that $r$ is odd.

##  4\. Some properties of parabolic opers

Consider the vector bundle $E$ in (2.9). Let

$\text{End}^{n}(E_{*})\,\subset\,\text{End}(E)$ (4.1)

be the coherent analytic subsheaf defined by the conditions that
$s(E_{x})\,\subset\,{\mathcal{L}}^{*}(-S)_{x}$ and
$s({\mathcal{L}}^{*}(-S)_{x})\,=\,0$ for all $x\,\in\,S$ lying in the domain
of the local section $s$ of $\text{End}(E)$ (see Lemma 2.3). Take any

$\phi\,\,\in\,\,H^{0}(X,\,\text{End}^{n}(E_{*})\otimes K_{X}(S))\,.$

Let

$\widehat{\phi}\,\,:\,\,{\mathcal{L}}\,\,\longrightarrow\,{\mathcal{L}}^{*}(-S)\otimes
K_{X}(S)\,=\,{\mathcal{L}}$ (4.2)

be the homomorphism given by the following composition of homomorphisms:

${\mathcal{L}}\,\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\,E\,\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\,E\otimes
K_{X}(S)\,\xrightarrow{\,\,p\otimes{\rm
Id}_{K_{X}(S)}\,}\,{\mathcal{L}}^{*}(-S)\otimes K_{X}(S)\,=\,{\mathcal{L}}\,,$

where $\iota$ and $p$ are the homomorphisms in (2.9); recall that
${\mathcal{L}}^{\otimes 2}\,=\,K_{X}$.

######  Proposition 4.1.

For every $\phi\,\in\,H^{0}(X,\,{\rm End}^{n}(E_{*})\otimes K_{X}(S))$ the
homomorphism $\widehat{\phi}$ constructed from it in (4.2) vanishes
identically.

###### Proof.

Tensoring the diagram in (2.11) with $K_{X}(S)$ we have the following
commutative diagram

$\begin{matrix}0&\longrightarrow&{\mathcal{L}}\otimes
K_{X}&\longrightarrow&\widetilde{E}\otimes
K_{X}&\longrightarrow&{\mathcal{L}}&\longrightarrow&0\\\
&&\Big{\downarrow}&&\,\,\,\Big{\downarrow}q&&\Big{\downarrow}\\\
0&\longrightarrow&{\mathcal{L}}\otimes K_{X}(S)&\longrightarrow&E\otimes
K_{X}(S)&\longrightarrow&{\mathcal{L}}&\longrightarrow&0.\end{matrix}$ (4.3)

Take any $\phi\,\in\,H^{0}(X,\,\text{End}^{n}(E_{*})\otimes K_{X}(S))$.
Consider the composition of homomorphisms

$\widetilde{E}(-S)\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,E\,\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\,E\otimes
K_{X}(S)\,,$

where $\psi$ is the homomorphism in (2.11), and denote this composition by
$\widetilde{\phi}$. From (4.3), (4.1) and the construction of the
decomposition in Lemma 2.3 it follows that the image of this homomorphism
$\widetilde{\phi}\,:\,\widetilde{E}(-S)\,\longrightarrow\,E\otimes K_{X}(S)$
is contained in the image of the homomorphism $q$ in (4.3); in other words,
the subsheaf $\phi\circ\psi(\widetilde{E}(-S))\,\subset\,E\otimes K_{X}(S)$
lies in the image of the homomorphism

$\psi\otimes{\rm Id}_{K_{X}(S)}\,\,:\,\,\widetilde{E}(-S)\otimes
K_{X}(S)\,=\,\widetilde{E}\otimes K_{X}\,\longrightarrow\,E\otimes
K_{X}(S)\,.$

Consequently, $\phi$ produces a homomorphism

$\phi^{\prime}\,\,:\,\,\widetilde{E}(-S)\,\,\longrightarrow\,\,\widetilde{E}\otimes
K_{X}\,.$ (4.4)

More precisely, $\phi^{\prime}$ is determined uniquely by the condition

$\widetilde{\phi}\,=\,(\psi\otimes{\rm Id}_{K_{X}(S)})\circ\phi^{\prime}.$

Let

$\phi^{\prime\prime}\,\,:\,\,\mathcal{L}(-S)\,\,\longrightarrow\,\,\mathcal{L}$
(4.5)

denote the following composition of homomorphisms

$\mathcal{L}(-S)\,\stackrel{{\scriptstyle\iota^{\prime}}}{{\longrightarrow}}\,\widetilde{E}(-S)\,\stackrel{{\scriptstyle\phi^{\prime}}}{{\longrightarrow}}\,\widetilde{E}\otimes
K_{X}\,\xrightarrow{\,p_{0}\otimes{\rm
Id}_{K_{X}}\,}\,{\mathcal{L}}^{*}\otimes K_{X}\,=\,{\mathcal{L}}\,,$

where $\iota^{\prime}$ and $p_{0}$ are the homomorphisms in (2.11) and (2.8)
respectively. To prove the proposition it suffices to show that
$\phi^{\prime\prime}$ in (4.5) vanishes identically.

Take any $x_{i}\,\in\,S$. Since

$q(\phi^{\prime}(x_{i})(\widetilde{E}(-S)_{x_{i}}))\,=\,\phi(\psi(x_{i})(\widetilde{E}(-S)_{x_{i}}))\,=\,\phi({\mathcal{L}}^{*}(-S)_{x_{i}})\,=\,0\,,$

where $\psi$, $\phi^{\prime}$ and $q$ are the homomorphisms in (2.11), (4.4)
and (4.3) respectively, we conclude that

$\phi^{\prime}(x_{i})(\widetilde{E}(-S)_{x_{i}})\,\subset\,({\mathcal{L}}\otimes
K_{X})_{x_{i}}\,\subset\,(\widetilde{E}\otimes K_{X})_{x_{i}}\,,$ (4.6)

where $\phi^{\prime}$ is the homomorphism in (4.4) and
${\mathcal{L}}\,\subset\,\widetilde{E}$ is the subbundle in (2.8).

Furthermore, it can be shown that

$\phi^{\prime}(x_{i})(\mathcal{L}(-S)_{x_{i}})\,=\,0\,;$ (4.7)

see (2.11) for the subspace
$\mathcal{L}(-S)_{x_{i}}\,\subset\,\widetilde{E}(-S)_{x_{i}}$. Indeed, this
again follows from (2.11), (4.3), (4.1) and the construction of the
decomposition in Lemma 2.3.

In view of (4.6) and (4.7), the homomorphism $\phi^{\prime\prime}$ in (4.5)
vanishes at each $x_{i}$. Therefore, $\phi^{\prime\prime}$ produces a
homomorphism

$\phi^{\prime\prime\prime}\,\,:\,\,\mathcal{L}(-S)\,\,\longrightarrow\,\,\mathcal{L}(-S)\,.$
(4.8)

Consider the image
$\phi^{\prime}(\mathcal{L}(-S))\,\subset\,\widetilde{E}\otimes K_{X}$, where
$\phi^{\prime}$ is the homomorphism in (4.4). If the homomorphism
$\phi^{\prime\prime\prime}$ in (4.8) in nonzero, then this subsheaf
$\phi^{\prime}(\mathcal{L}(-S))$ produces a holomorphic splitting of the top
short exact sequence in (2.11) tensored with $K_{X}$. Indeed, in that case the
homomorphism $p^{\prime}\otimes{\rm Id}_{K_{X}}$ (see (2.11) for $p^{\prime}$)
maps $\phi^{\prime}(\mathcal{L}(-S))$ surjectively to
$\mathcal{L}^{*}(-S)\otimes K_{X}\,=\,\mathcal{L}(-S)$ and hence
$\phi^{\prime}(\mathcal{L}(-S))$ gives a holomorphic splitting of the short
exact sequence

$0\,\longrightarrow\,{\mathcal{L}}(-S)\otimes
K_{X}\,\longrightarrow\,\widetilde{E}(-S)\otimes
K_{X}\,\longrightarrow\,{\mathcal{L}}^{*}(-S)\otimes
K_{X}\,\longrightarrow\,0$

obtained from the top exact sequence in (2.11) by tensoring it with $K_{X}$. A
holomorphic splitting of the above exact sequence produces a holomorphic
splitting of the top short exact sequence in (2.11). But the exact sequence in
(2.8) does not split holomorphically, which implies that the top short exact
sequence in (2.11) does not split holomorphically. This implies that
$\phi^{\prime\prime\prime}\,=\,0$ (see (4.8)), and hence
$\phi^{\prime\prime}\,=\,0$ (see (4.5)). As noted before, to prove the
proposition it is enough to show that $\phi^{\prime\prime}$ vanishes
identically. This completes the proof. ∎

######  Corollary 4.2.

The endomorphism
${\mathcal{S}}(D_{0},\,{\mathcal{L}})\,:\,{\mathcal{L}}\,\longrightarrow\,{\mathcal{L}}$
in Corollary 2.6(2) does not depend on the parabolic connection $D_{0}$.

###### Proof.

The space of parabolic connections on $E_{*}$ is an affine space for the
vector space $H^{0}(X,\,{\rm End}^{n}(E_{*})\otimes K_{X}(S))$. Note that for
any parabolic connection $D$ on $E_{*}$ and any $\phi\,\in\,H^{0}(X,\,{\rm
End}^{n}(E_{*})\otimes K_{X}(S))$, we have

${\mathcal{S}}(D+\phi,\,{\mathcal{L}})\,=\,{\mathcal{S}}(D,\,{\mathcal{L}})+\widehat{\phi},$

where $\widehat{\phi}$ is constructed in (4.2) from $\phi$. Therefore, from
Proposition 4.1 it follows immediately that
${\mathcal{S}}(D+\phi,\,{\mathcal{L}})\,=\,{\mathcal{S}}(D,\,{\mathcal{L}})$.
∎

As before, let ${\mathcal{L}}_{*}$ denote the holomorphic line bundle
$\mathcal{L}$ in (2.9) equipped with the parabolic structure on it induced by
$E_{*}$ for the inclusion map $\iota$ in 2.9. We denote by
$E_{*}/{\mathcal{L}}_{*}$ the quotient line bundle $E/{\mathcal{L}}$ in (2.9)
equipped with the parabolic structure on it induced by $E_{*}$. So from (2.9)
we have a short exact sequence of parabolic bundles

$0\,\longrightarrow\,{\mathcal{L}}_{*}\,\longrightarrow\,E_{*}\,\longrightarrow\,E_{*}/{\mathcal{L}}_{*}\,\longrightarrow\,0\,.$
(4.9)

For notational convenience, both $\text{Sym}^{0}(E_{*})$ and
$({\mathcal{L}}_{*})^{0}$ will denote the trivial holomorphic line bundle
${\mathcal{O}}_{X}$ equipped with the trivial parabolic structure (no nonzero
parabolic weights). Since $\text{Sym}^{r-1}(E_{*})$ is a quotient of
$(E_{*})^{\otimes(r-1)}$, we have a natural homomorphism of parabolic bundles

$\tau_{j}\,\,:\,\,\text{Sym}^{j-1}(E_{*})\otimes({\mathcal{L}}_{*})^{r-j}\,\longrightarrow\,\text{Sym}^{r-1}(E_{*})$

for every $1\,\leq\,j\,\leq\,r$ (see (4.9)). This $\tau_{j}$ is an injective
homomorphism, and its image is a parabolic subbundle of
$\text{Sym}^{r-1}(E_{*})$. Let

${\mathcal{F}}^{j}_{*}\,\,:=\,{\rm
Image}(\tau_{j})\,\subset\,\text{Sym}^{r-1}(E_{*})$

be the parabolic subbundle; its rank is $j$. So we have a filtration of
parabolic subbundles

$0\,=\,{\mathcal{F}}^{0}_{*}\,\subset\,{\mathcal{F}}^{1}_{*}\,\subset\,{\mathcal{F}}^{2}_{*}\,\subset\,\cdots\,\subset\,{\mathcal{F}}^{r-1}_{*}\,\subset\,{\mathcal{F}}^{r}_{*}\,=\,\text{Sym}^{r-1}(E_{*}).$
(4.10)

The holomorphic vector bundle underlying any ${\mathcal{F}}^{i}_{*}$ will be
denoted by ${\mathcal{F}}^{i}_{0}$.

For any $1\,\leq\,j\,\leq\,r$, the quotient parabolic line bundle
${\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*}$ in (4.10) actually has the
following description:

${\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*}\,=\,({\mathcal{L}}_{*})^{r-j}\otimes(E_{*}/{\mathcal{L}}_{*})^{j-1}\,.$
(4.11)

Indeed, this follows immediately from (4.9); by convention,
$(E_{*}/{\mathcal{L}}_{*})^{0}$ is the trivial line bundle ${\mathcal{O}}_{X}$
with the trivial parabolic structure. It can be shown that

$({\mathcal{L}}_{*})^{*}\,=\,E_{*}/{\mathcal{L}}_{*}.$ (4.12)

Indeed, from (2.17) it follows that
${\mathcal{L}}_{*}\otimes(E_{*}/{\mathcal{L}}_{*})\,=\,\det E_{*}$ is the
trivial line bundle ${\mathcal{O}}_{X}$ with the trivial parabolic structure,
and hence (4.12) holds. Therefore, from (4.11) it follows that

$\text{par-
deg}({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*})\,=\,(2j-r-1)\cdot\text{par-
deg}(E_{*}/{\mathcal{L}}_{*})\,=\,(2j-r-1)\cdot\left(1-g-n+\sum_{i=1}^{n}\frac{c_{i}+1}{2c_{i}+1}\right)\,,$
(4.13)

where $g\,=\,\text{genus}(X)$. Now from (4.10) and (4.13) it is deduced that

$\text{par-deg}({\mathcal{F}}^{j}_{*})\,=\,\sum_{i=1}^{j}\text{par-
deg}({\mathcal{F}}^{i}_{*}/{\mathcal{F}}^{i-1}_{*})\,=\,j(r-j)\cdot\left(g-1+\sum_{i=1}^{n}\frac{c_{i}}{2c_{i}+1}\right)\,.$
(4.14)

######  Lemma 4.3.

Let $D$ be any parabolic connection on the parabolic bundle ${\rm
Sym}^{r-1}(E_{*})$. Then the following two hold:

  1. (1)

For any $1\,\leq\,j\,\leq\,r-1$, the parabolic subbundle
${\mathcal{F}}^{j}_{*}$ in (4.10) is not preserved by $D$.

  2. (2)

$D({\mathcal{F}}^{j}_{0})\,\subset\,{\mathcal{F}}^{j+1}_{0}\otimes K_{X}(S)$,
where ${\mathcal{F}}^{i}_{0}$ is the holomorphic vector bundle underlying
${\mathcal{F}}^{i}$, for all $1\,\leq\,j\,\leq\,r-1$.

###### Proof.

From (4.14) it follows that $\text{par-deg}({\mathcal{F}}^{j}_{*})\,\not=\,0$
(in fact, $\text{par-deg}({\mathcal{F}}^{j}_{*})\,>\,0$) for all
$1\,\leq\,j\,\leq\,r-1$. Consequently, $D$ does not preserve
${\mathcal{F}}^{j}_{*}$.

For any $1\,\leq\,j\,\leq\,r-2$, and any $2\,\leq\,k\,\leq\,r-j$, consider the
parabolic line bundle

$({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*})^{*}\otimes({\mathcal{F}}^{j+k}_{*}/{\mathcal{F}}^{j+k-1}_{*})\,=\,(({\mathcal{L}}_{*})^{r-j}\otimes(E_{*}/{\mathcal{L}}_{*})^{j-1})^{*}\otimes(({\mathcal{L}}_{*})^{r-j-k}\otimes(E_{*}/{\mathcal{L}}_{*})^{j+k-1})$
$=\,({\mathcal{L}}_{*})^{r-j-k-(r-j)}\otimes(E_{*}/{\mathcal{L}}_{*})^{j+k-1-(j-1)}\,=\,({\mathcal{L}}_{*})^{-k}\otimes(E_{*}/{\mathcal{L}}_{*})^{k}\,=\,(E_{*}/{\mathcal{L}}_{*})^{2k};$

see (4.11) and (4.12) for the above isomorphisms. The holomorphic line bundle
underlying the parabolic line bundle
$({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*})^{*}\otimes({\mathcal{F}}^{j+k}_{*}/{\mathcal{F}}^{j+k-1}_{*})\,=\,(E_{*}/{\mathcal{L}}_{*})^{2k}$
will be denoted by $\xi_{r,k}$. We have

$\text{degree}(\xi_{r,k})\,=\,2k\cdot\text{degree}(E/{\mathcal{L}})+\sum_{i=1}^{n}\left[\frac{2k(c_{i}+1)}{2c_{i}+1}\right]$
$=\,2k(1-g-n)+kn+\sum_{i=1}^{n}\left[\frac{k}{2c_{i}+1}\right]\,=\,k(2-2g-n)++\sum_{i=1}^{n}\left[\frac{k}{2c_{i}+1}\right]\,,$

where $[t]\,\in\,\mathbb{Z}$ denotes the integral part of $t$, meaning
$0\,\leq\,t-[t]\,<\,1$. This implies that

$\text{degree}(\xi_{r,k})\,<\,2-2g-n\,=\,-\text{degree}(K_{X}(S))$

(recall that $n\,\geq\,3$ if $g\,=\,0$), and hence
$\text{degree}(\xi_{r,k}\otimes K_{X}(S))\,<\,0$. Consequently, we have

$H^{0}(X,\,\xi_{r,k}\otimes K_{X}(S))\,=\,0\,.$

This implies that

$H^{0}(X,\,({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*})^{*}\otimes({\mathcal{F}}^{j+k}_{*}/{\mathcal{F}}^{j+k-1}_{*})\otimes
K_{X}(S))\,=\,0\,.$ (4.15)

From (4.15) it is deduced that the following composition of homomorphisms

${\mathcal{F}}^{j}_{0}\,\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\,{\mathcal{F}}^{r}_{0}\otimes
K_{X}(S)\,\longrightarrow\,({\mathcal{F}}^{r}_{0}/{\mathcal{F}}^{j+1}_{0})\otimes
K_{X}(S)$ (4.16)

vanishes identically, where ${\mathcal{F}}^{\ell}_{0}$ is the holomorphic
vector bundle underlying the parabolic bundle ${\mathcal{F}}^{\ell}_{*}$. To
see this, observe that the parabolic vector bundle

$\text{Hom}({\mathcal{F}}^{j}_{*},\,({\mathcal{F}}^{r}_{*}/{\mathcal{F}}^{j+1}_{*})\otimes
K_{X}(S))\,=\,({\mathcal{F}}^{r}_{*}/{\mathcal{F}}^{j+1}_{*})\otimes
K_{X}(S)\otimes({\mathcal{F}}^{j}_{*})^{*}\,=\,({\mathcal{F}}^{r}_{*}/{\mathcal{F}}^{j+1}_{*})\otimes({\mathcal{F}}^{j}_{*})^{*}\otimes
K_{X}(S)$

has a filtration of parabolic subbundles such that the successive quotients
are

$({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*})^{*}\otimes({\mathcal{F}}^{j+k}_{*}/{\mathcal{F}}^{j+k-1}_{*})\otimes
K_{X}(S)\,,\ \ \,2\,\leq\,k\,\leq\,r-j.$

So (4.15) implies that the composition of homomorphisms in (4.16) vanishes
identically. Since the composition of homomorphisms in (4.16) vanishes
identically we have

$D({\mathcal{F}}^{j}_{*})\,\subset\,{\mathcal{F}}^{j+1}_{*}$

for all $1\,\leq\,j\,\leq\,r-1$. ∎

From (4.11) it follows that for any $1\,\leq\,j\,\leq\,r-1$, the parabolic
line bundle

$({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*})^{*}\otimes({\mathcal{F}}^{j+1}_{*}/{\mathcal{F}}^{j}_{*})\,=\,(E_{*}/{\mathcal{L}}_{*})\otimes{\mathcal{L}}^{*}_{*}\,=\,(E_{*}/{\mathcal{L}}_{*})^{\otimes
2}$

is $TX(-S)\,=\,K_{X}(S)^{*}$ equipped with the parabolic weight
$\frac{1}{2c_{i}+1}$ at each $x_{i}\,\in\,S$ (see (4.12) for the above
isomorphism). Therefore, from Lemma 4.3(2) we conclude that for any parabolic
connection $D$ on the parabolic bundle ${\rm Sym}^{r-1}(E_{*})$, the second
fundamental forms for the parabolic subbundles in (4.10) are given by a
collection of holomorphic homomorphisms

$\psi(D,j)\,\,\in\,\,H^{0}(X,\,\text{Hom}({\mathcal{F}}^{j}_{*}/{\mathcal{F}}^{j-1}_{*},\,{\mathcal{F}}^{j+1}_{*}/{\mathcal{F}}^{j}_{*})\otimes
K_{X}(S))\,=\,H^{0}(X,\,{\mathcal{O}}_{X})$ (4.17)

with $1\,\leq\,j\,\leq\,r-1$.

######  Corollary 4.4.

For each $1\,\leq\,j\,\leq\,r-1$, the section $\psi(D,j)$ in (4.17) is a
nonzero constant.

###### Proof.

From Lemma 4.3(1) it follows immediately that $\psi(D,j)\,\not=\,0$. ∎

##  5\. Differential operators on parabolic bundles

In this section we will describe differential operators between parabolic
vector bundles. As before, fix a compact Riemann surface $X$ and a reduced
effective divisor $S\,=\,\sum_{i=1}^{n}x_{i}$ on it; if
$\text{genus}(X)\,=\,0$, then assume that $n\,\geq\,3$. For each point
$x_{i}\,\in\,S$ fix an integer $N_{i}\,\geq\,2$. We will consider parabolic
bundles on $X$ with parabolic structure on $S$ such that all the parabolic
weights at each $x_{i}\,\in\,S$ are integral multiplies of $1/N_{i}$.

There is a ramified Galois covering

$\varphi\,:\,Y\,\longrightarrow\,X$ (5.1)

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 the ramification of $\varphi$ at
$y$ is $N_{i}$.

Such a ramified Galois covering $\varphi$ exists; see [Na, p. 26, Proposition
1.2.12]. Let

$\Gamma\,:=\,\text{Gal}(\varphi)\,:=\,\text{Aut}(Y/X)\,\subset\,\text{Aut}(Y)$
(5.2)

be the Galois group for $\varphi$. So the restriction

$\varphi^{\prime}\,:=\,\varphi\big{|}_{Y^{\prime}}\,:\,Y^{\prime}\,:=\,Y\setminus\varphi^{-1}(S)\,\,\longrightarrow\,\,X^{\prime}\,:=\,X\setminus
S$ (5.3)

is an étale Galois covering with Galois group $\Gamma$.

As before, a holomorphic vector bundle $V$ on $Y$ is called an orbifold bundle
if $\Gamma$ acts on $V$ as holomorphic bundle automorphisms over the action of
$\Gamma$ on $Y$.

Consider the trivial vector bundle

${\mathbb{C}}[\Gamma]_{Y}\,:=\,Y\times{\mathbb{C}}[\Gamma]\,\longrightarrow\,Y\,,$
(5.4)

where ${\mathbb{C}}[\Gamma]$ is the group algebra for $\Gamma$ with
coefficients in $\mathbb{C}$. The usual action of $\Gamma$ on
${\mathbb{C}}[\Gamma]$ and the Galois action of $\Gamma$ on $Y$ together
produce an action of $\Gamma$ on $Y\times{\mathbb{C}}[\Gamma]$. This action
makes $Y\times{\mathbb{C}}[\Gamma]\,=\,{\mathbb{C}}[\Gamma]_{Y}$ an orbifold
bundle on $Y$. Let

${\mathcal{E}}_{*}\,\,\longrightarrow\,\,X$ (5.5)

be the corresponding parabolic vector bundle on $X$ with parabolic structure
on $S$ [Bi], [Bo1], [Bo2]. The action of $\Gamma$ on the vector bundle
${\mathbb{C}}[\Gamma]_{Y}$ in (5.4) produces an action of $\Gamma$ on its
direct image $\varphi_{*}{\mathbb{C}}[\Gamma]_{Y}$ over the trivial action of
$\Gamma$ on $X$. We have

${\mathcal{E}}_{0}\,=\,(\varphi_{*}{\mathbb{C}}[\Gamma]_{Y})^{\Gamma}\,\subset\,\varphi_{*}{\mathbb{C}}[\Gamma]_{Y}\,,$
(5.6)

where $(\varphi_{*}{\mathbb{C}}[\Gamma]_{Y})^{\Gamma}$ is the
$\Gamma$–invariant part, and ${\mathcal{E}}_{0}$ is the holomorphic vector
bundle underlying the parabolic bundle ${\mathcal{E}}_{*}$ in (5.5).

It can be shown that the holomorphic vector bundle
${\mathcal{E}}_{0}\,=\,(\varphi_{*}{\mathbb{C}}[\Gamma]_{Y})^{\Gamma}$ is
identified with $\varphi_{*}{\mathcal{O}}_{Y}$. Indeed, there is a natural
$\Gamma$–equivariant isomorphism

$\varphi_{*}{\mathbb{C}}[\Gamma]_{Y}\,\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\,(\varphi_{*}{\mathcal{O}}_{Y})\otimes_{\mathbb{C}}{\mathbb{C}}[\Gamma]\,;$

it is in fact given by the projection formula. Therefore, the natural
isomorphism

$\varphi_{*}{\mathcal{O}}_{Y}\,\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\,((\varphi_{*}{\mathcal{O}}_{Y})\otimes_{\mathbb{C}}{\mathbb{C}}[\Gamma])^{\Gamma}$

(any complex $\Gamma$–module $M$ is naturally identified with
$(M\otimes_{\mathbb{C}}{\mathbb{C}}[\Gamma])^{\Gamma}$) produces an
isomorphism

$\varphi_{*}{\mathcal{O}}_{Y}\,\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\,(\varphi_{*}{\mathbb{C}}[\Gamma]_{Y})^{\Gamma}.$
(5.7)

The direct image $\varphi_{*}{\mathcal{O}}_{Y}$ has a natural parabolic
structure which we will now describe.

Take any $x_{i}\,\in\,S$. Fix an analytic open neighborhood $U\,\subset\,X$ of
$x_{i}$ such that $U\bigcap S\,=\,x_{i}$. Let
${\mathcal{U}}\,:=\,\varphi^{-1}(U)\,\subset\,Y$ be the inverse image. The
restriction of $\varphi$ to $\mathcal{U}$ will be denoted by
$\widetilde{\varphi}$. Let $\widetilde{D}_{i}\,:=\,\varphi^{-1}(x_{i})_{\rm
red}\,\subset\,Y$ be the reduced inverse image. For all $k\,\in\,[1,\,N_{i}]$,
define the vector bundle

$V_{k}\,:=\,\widetilde{\varphi}_{*}{\mathcal{O}}_{\mathcal{U}}(-(N_{i}-k))\widetilde{D}_{i})\,\longrightarrow\,U\,.$

So we have a filtration of subsheaves of
$V_{N_{i}}\,=\,(\varphi_{*}{\mathcal{O}}_{Y})\big{|}_{U}$:

$0\,\subset\,V_{1}\,\subset\,V_{2}\,\subset\,\cdots\,\subset\,V_{N_{i}-1}\,\subset\,V_{N_{i}}\,=\,(\varphi_{*}{\mathcal{O}}_{Y})\big{|}_{U}\,.$

The restriction of this filtration of subsheaves to $x_{i}$ gives a filtration
of subspaces

$0\,\subset\,(V_{1})^{\prime}_{x_{i}}\,\subset\,(V_{2})^{\prime}_{x_{i}}\,\subset\,\cdots\,\subset\,(V_{N_{i}-1})^{\prime}_{x_{i}}\,\subset\,(V_{N_{i}})_{x_{i}}\,=\,(\varphi_{*}{\mathcal{O}}_{Y})_{x_{i}}$
(5.8)

of the fiber $(\varphi_{*}{\mathcal{O}}_{Y})_{x_{i}}$. We note that
$(V_{k})^{\prime}_{x_{i}}$ in (5.8) is the image, in the fiber
$(\varphi_{*}{\mathcal{O}}_{Y})_{x_{i}}$, of the fiber $(V_{k})_{x_{i}}$ over
$x_{i}$ of the vector bundle $V_{k}$.

The parabolic structure on $\varphi_{*}{\mathcal{O}}_{Y}$ is defined as
follows. The parabolic divisor is $S$. The quasiparabolic filtration over any
$x_{i}\,\in\,S$ is the filtration of $(\varphi_{*}{\mathcal{O}}_{Y})_{x_{i}}$
constructed in (5.8). The parabolic weight of the subspace $(V_{k})_{x_{i}}$
in (5.8) is $\frac{N_{i}-k}{N_{i}}$. The resulting parabolic vector bundle is
identified with ${\mathcal{E}}_{*}$ in (5.5); recall from (5.6) and (5.7) that
${\mathcal{E}}_{0}$ is identified with $\varphi_{*}{\mathcal{O}}_{Y}$.

The trivial connection on the trivial vector bundle
${\mathbb{C}}[\Gamma]_{Y}\,:=\,Y\times{\mathbb{C}}[\Gamma]$ in (5.4) is
preserved by the action of the Galois group $\Gamma$ on
${\mathbb{C}}[\Gamma]_{Y}$. Therefore, this trivial connection produces a
parabolic connection on the corresponding parabolic vector bundle
${\mathcal{E}}_{*}$ in (5.5). This parabolic connection on ${\mathcal{E}}_{*}$
will be denoted by $\nabla^{\mathcal{E}}$.

Using the isomorphism between ${\mathcal{E}}_{0}$ and
$\varphi_{*}{\mathcal{O}}_{Y}$ (see (5.6) and (5.7)), the logarithmic
connection on ${\mathcal{E}}_{0}$ defining the above parabolic connection
$\nabla^{\mathcal{E}}$ on ${\mathcal{E}}_{*}$ produces a logarithmic
connection on $\varphi_{*}{\mathcal{O}}_{Y}$. This logarithmic connection on
$\varphi_{*}{\mathcal{O}}_{Y}$ given by $\nabla^{\mathcal{E}}$ is easy to
describe. To describe it, take the de Rham differential
$d\,:\,{\mathcal{O}}_{Y}\,\longrightarrow\,K_{Y}$ on $Y$. Let

$\varphi_{*}d\,:\,\varphi_{*}{\mathcal{O}}_{Y}\,\longrightarrow\,\varphi_{*}K_{Y}$
(5.9)

be its direct image. On the other hand, using the projection formula, the
natural homomorphism

$K_{Y}\,\hookrightarrow\,K_{Y}\otimes{\mathcal{O}}_{Y}(\varphi^{-1}(S)_{\rm
red})\,=\,\varphi^{*}(K_{X}\otimes{\mathcal{O}}_{X}(S))\,.$

produces a homomorphism

$\varphi_{*}K_{Y}\,\,\longrightarrow\,\,\varphi_{*}(\varphi^{*}(K_{X}\otimes{\mathcal{O}}_{X}(S)))\,=\,(\varphi_{*}{\mathcal{O}}_{Y})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,.$

Combining this with $\varphi_{*}d$ in (5.9) we obtain homomorphisms

$\varphi_{*}{\mathcal{O}}_{Y}\,\longrightarrow\,\varphi_{*}K_{Y}\,\,\longrightarrow\,\,(\varphi_{*}{\mathcal{O}}_{Y})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,.$

This composition of homomorphisms
$\varphi_{*}{\mathcal{O}}_{Y}\,\longrightarrow\,(\varphi_{*}{\mathcal{O}}_{Y})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ defines a logarithmic connection on
$\varphi_{*}{\mathcal{O}}_{Y}$. This logarithmic connection coincides with the
one that defines the above constructed parabolic connection
$\nabla^{\mathcal{E}}$ on ${\mathcal{E}}_{*}$.

The parabolic connection $\nabla^{\mathcal{E}}$ on ${\mathcal{E}}_{*}$ defines
a nonsingular holomorphic connection ${\nabla}^{\prime}$ on

${\mathcal{E}}^{\prime}_{0}\,:=\,{\mathcal{E}}_{0}\big{|}_{X^{\prime}}\,=\,\varphi_{1*}{\mathcal{O}}_{Y^{\prime}}$

over $X^{\prime}$ (see (5.3)). For any holomorphic vector bundle $V^{\prime}$
on $X^{\prime}$, note that

$J^{k}(V^{\prime}\otimes{\mathcal{E}}^{\prime}_{0})\,=\,J^{k}(V^{\prime})\otimes{\mathcal{E}}^{\prime}_{0}$
(5.10)

for all $k\,\geq\,0$. To see this isomorphism, for any $x\,\in\,X^{\prime}$
and $u\,\in\,({\mathcal{E}}^{\prime}_{0})_{x}$, let $\widetilde{u}$ denote the
unique flat section of ${\mathcal{E}}^{\prime}_{0}$ for the connection
$\nabla^{\prime}$, defined on any simply connected open neighborhood of $x$,
such that $\widetilde{u}(x)\,=\,u$. Now the homomorphism

$J^{k}(V^{\prime})\otimes{\mathcal{E}}^{\prime}_{0}\,\longrightarrow\,J^{k}(V^{\prime}\otimes{\mathcal{E}}^{\prime}_{0})$

that sends any $v\otimes u$ to the image of $v\otimes\widetilde{u}$, where
$v\,\in\,J^{k}(V^{\prime})_{x}$ and $u\,\in\,({\mathcal{E}}^{\prime}_{0})_{x}$
with $x\,\in\,X^{\prime}$, is evidently an isomorphism.

Take holomorphic vector bundles $V^{\prime}$ and $W^{\prime}$ on a nonempty
Zariski open subset $U\,\subset\,X^{\prime}$. Recall that a holomorphic
differential operator of order $k$ from $V^{\prime}$ to $W^{\prime}$ is a
holomorphic homomorphism $J^{k}(V^{\prime})\,\longrightarrow\,W^{\prime}$. Let

$D^{\prime}\,:\,J^{k}(V^{\prime})\,\longrightarrow\,W^{\prime}$

be a holomorphic differential operator of order $k$ from $V^{\prime}$ to
$W^{\prime}$ on $U$.

We will show that $D^{\prime}$ extends to a holomorphic differential operator

$\widetilde{D^{\prime}}\,:\,J^{k}(V^{\prime}\otimes{\mathcal{E}}^{\prime}_{0})\,\longrightarrow\,W^{\prime}\otimes{\mathcal{E}}^{\prime}_{0}$
(5.11)

from $V^{\prime}\otimes{\mathcal{E}}^{\prime}_{0}$ to
$W^{\prime}\otimes{\mathcal{E}}^{\prime}_{0}$ over $U$. To construct
$\widetilde{D^{\prime}}$, using the isomorphism in (5.10) we have

$J^{k}(V^{\prime}\otimes{\mathcal{E}}^{\prime}_{0})\,=\,J^{k}(V^{\prime})\otimes{\mathcal{E}}^{\prime}_{0}\,\,\,\xrightarrow{\,\,D^{\prime}\otimes{\rm
Id}_{{\mathcal{E}}^{\prime}_{0}}\,\,}\,\,\,W^{\prime}\otimes{\mathcal{E}}^{\prime}_{0}\,.$

This homomorphism is the one in (5.11).

Let $V_{*}$ and $W_{*}$ be parabolic vector bundles on $X$. Denote the
restrictions $V_{0}\big{|}_{X^{\prime}}$ and $W_{0}\big{|}_{X^{\prime}}$ by
$V^{\prime}$ and $W^{\prime}$ respectively. The holomorphic vector bundle
underlying the parabolic tensor product $V_{*}\otimes{\mathcal{E}}_{*}$
(respectively, $W_{*}\otimes{\mathcal{E}}_{*}$) will be denoted by
$(V_{*}\otimes{\mathcal{E}}_{*})_{0}$ (respectively,
$(W_{*}\otimes{\mathcal{E}}_{*})_{0}$), where ${\mathcal{E}}_{*}$ is the
parabolic bundle in (5.5).

######  Definition 5.1.

A holomorphic differential operator of order $k$ from $V_{*}$ to $W_{*}$ over
an open subset $\widetilde{U}\,\subset\,X$ is a holomorphic homomorphism

$D^{\prime}\,:\,J^{k}(V^{\prime})\,\longrightarrow\,W^{\prime}$

over $U\,:=\,\widetilde{U}\bigcap X^{\prime}$ such that the homomorphism

$\widetilde{D^{\prime}}\,:\,J^{k}(V^{\prime}\otimes{\mathcal{E}}^{\prime}_{0})\,\longrightarrow\,W^{\prime}\otimes{\mathcal{E}}^{\prime}_{0}$

in (5.11) extends to a holomorphic homomorphism
$J^{k}((V_{*}\otimes{\mathcal{E}}_{*})_{0})\,\longrightarrow\,(W_{*}\otimes{\mathcal{E}}_{*})_{0}$
over entire $\widetilde{U}$.

It is straightforward to check that the above definition does not depend on
the choice of the map $\varphi$.

We denote by ${\rm Diff}^{k}_{X}(V_{*},\,W_{*})$ the sheaf of holomorphic
differential operators of order $k$ from $V_{*}$ to $W_{*}$. Define

${\rm DO}^{k}_{P}(V_{*},\,W_{*})\,\,:=\,\,H^{0}(X,\,{\rm
Diff}^{k}_{X}(V_{*},\,W_{*}))$

to be the space of all holomorphic differential operators of order $k$ from
$V_{*}$ to $W_{*}$ over $X$.

Let $\mathbb{V}$ and $\mathbb{W}$ denote the orbifold vector bundles on $Y$
corresponding to the parabolic vector bundles $V_{*}$ and $W_{*}$
respectively. Consider the space

${\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})\,:=\,H^{0}(Y,\,\text{Hom}(J^{k}(\mathbb{V}),\,\mathbb{W}))$

of holomorphic differential operators of order $k$ from $\mathbb{V}$ to
$\mathbb{W}$ over $Y$. Then the actions of $\Gamma$ on $\mathbb{V}$ and
$\mathbb{W}$ together produce an action of $\Gamma$ on ${\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})$. Let

$H^{0}(Y,\,\text{Hom}(J^{k}(\mathbb{V}),\,\mathbb{W}))^{\Gamma}\,=\,{\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})^{\Gamma}\,\subset\,{\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})$

be the space of all $\Gamma$–invariant differential operators of order $k$
from $\mathbb{V}$ to $\mathbb{W}$.

######  Proposition 5.2.

There is a natural isomorphism

${\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})^{\Gamma}\,\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\,{\rm
DO}^{k}_{P}(V_{*},\,W_{*})\,.$

###### Proof.

We will first prove that

$\varphi_{*}\mathbb{V}\,=\,(V_{*}\otimes{\mathcal{E}}_{*})_{0}\,,$ (5.12)

where ${\mathcal{E}}_{*}$ is the parabolic bundle in (5.5) and
$(V_{*}\otimes{\mathcal{E}}_{*})_{0}$ is the vector bundle underlying the
parabolic vector bundle $V_{*}\otimes{\mathcal{E}}_{*}$. To prove (5.12),
first note that

$\varphi_{*}\mathbb{V}\,=\,\left(\varphi_{*}({\mathbb{V}}\otimes{\mathbb{C}}[\Gamma]_{Y})\right)^{\Gamma}\,,$
(5.13)

where ${\mathbb{C}}[\Gamma]_{Y}$ is the orbifold bundle in (5.4). Since
$\mathcal{E}_{*}$ and $V_{*}$ correspond to the orbifold bundles
${\mathbb{C}}[\Gamma]_{Y}$ and $\mathbb{V}$ respectively, the parabolic bundle
corresponding to the orbifold bundle
${\mathbb{V}}\otimes{\mathbb{C}}[\Gamma]_{Y}$ is
$V_{*}\otimes{\mathcal{E}}_{*}$. In particular, we have

$\left(\varphi_{*}({\mathbb{V}}\otimes{\mathbb{C}}[\Gamma]_{Y})\right)^{\Gamma}\,=\,(V_{*}\otimes{\mathcal{E}}_{*})_{0}\,.$

This and (5.13) together give the isomorphism in (5.12).

Let $D\,:\,\mathbb{V}\,\longrightarrow\,\mathbb{W}$ be a holomorphic
differential operator of order $k$ on $Y$. Taking its direct image for the map
$\varphi$, we have

$\varphi_{*}D\,\,:\,\,\varphi_{*}\mathbb{V}\,\,\longrightarrow\,\,\varphi_{*}\mathbb{W}\,.$

Now if $D\,\in\,{\rm DO}^{k}(\mathbb{V},\,\mathbb{W})^{\Gamma}$, then clearly

$\varphi_{*}D((\varphi_{*}\mathbb{V})^{\Gamma})\,\,\subset\,\,(\varphi_{*}\mathbb{W})^{\Gamma}\,.$

Let

$D_{\varphi}\,\,:=\,\,(\varphi_{*}D)\big{|}_{(\varphi_{*}\mathbb{V})^{\Gamma}}\,\,:\,\,(\varphi_{*}\mathbb{V})^{\Gamma}\,\,\longrightarrow\,\,(\varphi_{*}\mathbb{W})^{\Gamma}$

be the restriction of $\varphi_{*}D$ to
$(\varphi_{*}\mathbb{V})^{\Gamma}\,\subset\,\varphi_{*}\mathbb{V}$.

Using (5.12) it is now straightforward to check that $D_{\varphi}$ defines a
holomorphic differential operator of order $k$ from the parabolic bundle
$V_{*}$ to $W_{*}$. The corresponding homomorphism
$J^{k}((V_{*}\otimes{\mathcal{E}}_{*})_{0})\,\longrightarrow\,(W_{*}\otimes{\mathcal{E}}_{*})_{0}$
in Definition 3.3 is given by $\varphi_{*}D$ using the isomorphism in (5.12).

The isomorphism in the proposition sends any $D\,\in\,{\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})^{\Gamma}$ to $D_{\varphi}\,\in\,{\rm
DO}^{k}_{P}(V_{*},\,W_{*})$ constructed above from $D$.

For the inverse map, given any $\mathbf{D}\,\in\,{\rm
DO}^{k}_{P}(V_{*},\,W_{*})$, consider the homomorphism

$J^{k}((V_{*}\otimes{\mathcal{E}}_{*})_{0})\,\longrightarrow\,(W_{*}\otimes{\mathcal{E}}_{*})_{0}$

in Definition 3.3 given by the differential operator $\mathbf{D}$. Using the
isomorphism in (5.12) it produces a holomorphic differential operator from
$\mathbb{V}$ to $\mathbb{W}$. This differential operator is evidently fixed by
the action of $\Gamma$ on ${\rm DO}^{k}(\mathbb{V},\,\mathbb{W})$. ∎

###  5.1. Another description of differential operators on parabolic bundles

We will give an alternative description of the holomorphic differential
operators between two parabolic vector bundles. Let ${\rm
Diff}^{k}_{Z}(A,\,B)$ denote the sheaf of holomorphic differential operators
of order $k$ from a holomorphic vector bundle $A$ on a complex manifold $Z$ to
another holomorphic vector bundle $B$ on $Z$. The sheaf ${\rm
Diff}^{k}_{Z}({\mathcal{O}}_{Z},\,{\mathcal{O}}_{Z})\,=\,J^{k}({\mathcal{O}}_{Z})^{*}$
has both left and right ${\mathcal{O}}_{Z}$–module structures, and

${\rm Diff}^{k}_{Z}(A,\,B)\,\,=\,\,B\otimes_{{\mathcal{O}}_{Z}}{\rm
Diff}^{k}_{Z}({\mathcal{O}}_{Z},\,{\mathcal{O}}_{Z})\otimes_{{\mathcal{O}}_{Z}}A^{*}\,.$
(5.14)

We have a short exact sequence of holomorphic vector bundles

$0\,\longrightarrow\,{\rm
Diff}^{k}_{Z}({\mathcal{O}}_{Z},\,{\mathcal{O}}_{Z})\,\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\,{\rm
Diff}^{k+1}_{Z}({\mathcal{O}}_{Z},\,{\mathcal{O}}_{Z})\,\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}\,\text{Sym}^{k+1}(TZ)\,\longrightarrow\,0\,,$
(5.15)

where $\eta$ is the symbol map. The homomorphism

${\rm Id}_{B}\otimes\alpha\otimes{\rm
Id}_{A^{*}}\,:\,B\otimes_{{\mathcal{O}}_{Z}}{\rm
Diff}^{k}_{Z}({\mathcal{O}}_{Z},\,{\mathcal{O}}_{Z})\otimes_{{\mathcal{O}}_{Z}}A^{*}\,\longrightarrow\,B\otimes_{{\mathcal{O}}_{Z}}{\rm
Diff}^{k+1}_{Z}({\mathcal{O}}_{Z},\,{\mathcal{O}}_{Z})\otimes_{{\mathcal{O}}_{Z}}A^{*}\,,$

where $\alpha$ is the homomorphism in (5.15), coincides with the natural
inclusion map

${\rm Diff}^{k}_{Z}(A,\,B)\,\hookrightarrow\,{\rm Diff}^{k+1}_{Z}(A,\,B).$

The holomorphic differential operators between two parabolic vector bundles
will be described along the above line.

Consider the pair $(Y,\,\varphi)$ in (5.1). The action of
$\Gamma\,=\,\text{Gal}(\varphi)$ on $Y$ produces an action of $\Gamma$ on
${\mathcal{O}}_{Y}$. This action of $\Gamma$ on ${\mathcal{O}}_{Y}$ induces an
action of $\Gamma$ on $J^{k}({\mathcal{O}}_{Y})$, which in turn induces an
action of $\Gamma$ on the dual vector bundle
$J^{k}({\mathcal{O}}_{Y})^{*}\,=\,{\rm
Diff}^{k}_{Y}({\mathcal{O}}_{Y},\,{\mathcal{O}}_{Y})$. As mentioned before,
${\rm Diff}^{k}_{Y}({\mathcal{O}}_{Y},\,{\mathcal{O}}_{Y})$ is equipped with
left and right ${\mathcal{O}}_{Y}$–module structures. These module structures
are $\Gamma$–equivariant. Let ${\mathcal{J}}^{k}_{*}$ denote the parabolic
vector bundle on $X$ associated to the orbifold vector bundle
$J^{k}({\mathcal{O}}_{Y})^{*}\,=\,{\rm
Diff}^{k}_{Y}({\mathcal{O}}_{Y},\,{\mathcal{O}}_{Y})$ on $Y$.

Note that the rank of ${\mathcal{J}}^{k}_{*}$ is $k+1$. The parabolic line
bundle ${\mathcal{J}}^{0}_{*}$ is the trivial line bundle ${\mathcal{O}}_{X}$
equipped with the trivial parabolic structure. The underlying holomorphic
vector bundle for the parabolic bundle ${\mathcal{J}}^{1}_{*}$ is
${\mathcal{O}}_{X}\oplus TX(-S)$. The quasiparabolic filtration of
${\mathcal{J}}^{1}_{*}$ over any point $x_{i}\,\in\,S$ is

$TX(-S)_{x_{i}}\,\subset\,({\mathcal{O}}_{X})_{x_{i}}\oplus
TX(-S)_{x_{i}}\,=\,({\mathcal{J}}^{1}_{0})_{x_{i}}\,.$

The parabolic weight of $TX(-S)_{x_{i}}$ is $\frac{1}{N_{i}}$ and the
parabolic weight of $({\mathcal{J}}^{1}_{0})_{x_{i}}$ is $0$. Let

$TX(-S)_{*}\,\,\longrightarrow\,X$ (5.16)

denote the parabolic line bundle defined by $TX(-S)$ equipped with the
parabolic weight $\frac{1}{N_{i}}$ at each $x_{i}\,\in\,S$. So

${\mathcal{J}}^{1}_{*}\,=\,TX(-S)_{*}\oplus{\mathcal{O}}_{X},$

where ${\mathcal{O}}_{X}$ has the trivial parabolic structure.

Using the homomorphism $\alpha$ in (5.15) for $Y$ and $j\,=\,k$ we see that
${\mathcal{J}}^{j}_{*}$ is a parabolic subbundle of ${\mathcal{J}}^{j+1}_{*}$
for all $j\,\geq\,0$. Consequently, we have filtration of parabolic subbundles

${\mathcal{J}}^{0}_{*}\,\subset\,{\mathcal{J}}^{1}_{*}\,\subset\,\cdots\,\subset\,{\mathcal{J}}^{k-1}_{*}\,\subset\,{\mathcal{J}}^{k}_{*}$
(5.17)

for all $k\,\geq\,0$ such that each successive quotient is a parabolic line
bundle.

We will describe the quotient parabolic line bundle
${\mathcal{J}}^{j}_{*}/{\mathcal{J}}^{j-1}_{*}$ in (5.17) for all
$1\,\leq\,j\,\leq\,k$.

The holomorphic line bundle underlying the parabolic bundle
${\mathcal{J}}^{j}_{*}/{\mathcal{J}}^{j-1}_{*}$ is

$(TX)^{\otimes
j}(-jS)\otimes{\mathcal{O}}_{X}\left(\sum_{i=1}^{n}\left[\frac{j}{N_{i}}\right]x_{i}\right)\,,$

where $\left[\frac{j}{N_{i}}\right]\,\in\,\mathbb{Z}$ is the integral part of
$\frac{j}{N_{i}}$, and its parabolic weight at any $x_{i}\,\in\,S$ is
$\frac{j}{N_{i}}-\left[\frac{j}{N_{i}}\right]$. Indeed, from (5.15) we know
that the parabolic line bundle ${\mathcal{J}}^{j}_{*}/{\mathcal{J}}^{j-1}_{*}$
corresponds to the orbifold line bundle $(TY)^{\otimes j}$ on $Y$. On the
other hand, the parabolic line bundle $TX(-S)_{*}$ defined in (5.16)
corresponds to the orbifold line bundle $TY$. Therefore, we have

${\mathcal{J}}^{j}_{*}/{\mathcal{J}}^{j-1}_{*}\,=\,TX(-S)^{\otimes j}_{*}\,.$
(5.18)

The above description of ${\mathcal{J}}^{j}_{*}/{\mathcal{J}}^{j-1}_{*}$
follows immediately from (5.18).

The $\Gamma$–equivariant left and right ${\mathcal{O}}_{Y}$–module structures
on ${\rm Diff}^{k}_{Y}({\mathcal{O}}_{Y},\,{\mathcal{O}}_{Y})$ produces left
and right ${\mathcal{O}}_{X}$–module structures on ${\mathcal{J}}^{k}_{*}$.

Then, for any two parabolic bundles $V_{*}$ and $W_{*}$ over $X$, it follows
from Proposition 5.2 and (5.14) that ${\rm Diff}^{k}_{X}(V_{*},\,W_{*})$
coincides with the holomorphic vector bundle underlying the parabolic tensor
product

$W_{*}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}}^{k}_{*}\otimes_{{\mathcal{O}}_{X}}V^{*}_{*}\,;$

in other words, we have

${\rm
Diff}^{k}_{X}(V_{*},\,W_{*})\,=\,(W_{*}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}}^{k}_{*}\otimes_{{\mathcal{O}}_{X}}V^{*}_{*})_{0}\,.$

###  5.2. The symbol map

Consider the quotient map

$\gamma\,\,:\,\,{\mathcal{J}}^{k}_{*}\,\longrightarrow\,{\mathcal{J}}^{k}_{*}/{\mathcal{J}}^{k-1}_{*}\,=\,TX(-S)^{\otimes
k}_{*}$

(see (5.17), (5.18)). It produces a map

$\sigma\,:=\,({\rm Id}_{W_{*}}\otimes\gamma\otimes{\rm
Id}_{V^{*}_{*}})_{0}\,:\,{\rm
Diff}^{k}_{X}(V_{*},\,W_{*})\,=\,(W_{*}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}}^{k}_{*}\otimes_{{\mathcal{O}}_{X}}\otimes
V^{*}_{*})_{0}$ (5.19) $\longrightarrow\,(W_{*}\otimes TX(-S)^{\otimes
k}_{*}\otimes V^{*}_{*})_{0}\,=\,(TX(-S)^{\otimes k}_{*}\otimes{\rm
Hom}(V_{*},\,W_{*})_{*})_{0}\,.$

The above homomorphism $\sigma$ is the symbol map of differential operators
between parabolic bundles.

Take any $\widehat{D}\,\in\,{\rm DO}^{k}_{P}(V_{*},\,W_{*})$. Denote by
$\mathbb{V}$ (respectively, $\mathbb{W}$) the orbifold bundle on $Y$
corresponding to $V_{*}$ (respectively, $W_{*}$), and let

$D\,\in\,{\rm DO}^{k}(\mathbb{V},\,\mathbb{W})^{\Gamma}$

be the invariant differential operator given by $\widehat{D}$ using
Proposition 5.2. Let

$\sigma(\widehat{D})\,\in\,H^{0}(X,\,(TX(-S)^{\otimes k}_{*}\otimes{\rm
Hom}(V_{*},\,W_{*})_{*})_{0})$

be the symbol of $\widehat{D}$ (see (5.19)). Let

$\sigma(D)\,\in\,H^{0}(Y,\,\text{Hom}(\mathbb{V},\,\mathbb{W})\otimes(TY)^{\otimes
k})$

be the symbol of $D$. We have

$\sigma(D)\,\in\,H^{0}(Y,\,\text{Hom}(\mathbb{V},\,\mathbb{W})\otimes(TY)^{\otimes
k})^{\Gamma}$

because $D$ is fixed by the action of $\Gamma$ on ${\rm
DO}^{k}(\mathbb{V},\,\mathbb{W})$. The proof of the following lemma is
straightforward.

######  Lemma 5.3.

The parabolic vector bundle $TX(-S)^{\otimes k}_{*}\otimes{\rm
Hom}(V_{*},\,W_{*})_{*}$ on $X$ corresponds to the orbifold vector bundle
$\text{Hom}(\mathbb{V},\,\mathbb{W})\otimes(TY)^{\otimes k}$ on $Y$. The
natural isomorphism

$H^{0}(X,\,(TX(-S)^{\otimes k}_{*}\otimes{\rm
Hom}(V_{*},\,W_{*})_{*})_{0})\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,H^{0}(Y,\,{\rm
Hom}(\mathbb{V},\,\mathbb{W})\otimes(TY)^{\otimes k})^{\Gamma}$

takes the symbol $\sigma(\widehat{D})$ to the symbol $\sigma(D)$.

##  6\. Parabolic opers and differential operators

Recall the short exact sequence in (4.9) and the isomorphism in (4.12). For
notational convenience, $({\mathcal{L}}_{*})^{*}\,=\,E_{*}/{\mathcal{L}}_{*}$
will be denoted by ${\mathcal{L}}^{-1}_{*}$. For any $j\,\leq\,1$, the
parabolic line bundle $({\mathcal{L}}_{*})^{\otimes j}$ (respectively,
$({\mathcal{L}}^{*}_{*})^{\otimes j}$) will be denoted by
${\mathcal{L}}^{j}_{*}$ (respectively, ${\mathcal{L}}^{-j}_{*}$). Also,
${\mathcal{L}}^{0}_{*}$ will denote the trivial line bundle
${\mathcal{O}}_{X}$ with the trivial parabolic structure.

We note that

${\mathcal{L}}^{-2}_{*}\,=\,TX(-S)_{*}\,,$ (6.1)

where $TX(-S)_{*}$ is the parabolic line bundle in (5.16). From (5.18) and
(6.1) it follows that

${\mathcal{J}}^{j}_{*}/{\mathcal{J}}^{j-1}_{*}\,=\,{\mathcal{L}}^{-2j}_{*}$
(6.2)

for all $j\,\geq\,1$.

For any integer $r\,\geq\,2$, consider the space of parabolic differential
operators of order $r$

${\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*})\,:=\,H^{0}(X,\,{\rm
Diff}^{r}_{X}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*}))$

from ${\mathcal{L}}^{1-r}_{*}$ to ${\mathcal{L}}^{r+1}_{*}$. Let

$\sigma\,:\,{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*})\,\longrightarrow\,({\mathcal{L}}^{r+1}_{*}\otimes(TX(-S)_{*})^{\otimes
r}\otimes{\mathcal{L}}^{r-1}_{*})_{0}$ (6.3)
$=\,({\mathcal{L}}^{r+1}_{*}\otimes{\mathcal{L}}^{-2r}_{*}\otimes{\mathcal{L}}^{r-1}_{*})_{0}\,=\,({\mathcal{L}}^{0}_{*})_{0}\,=\,{\mathcal{O}}_{X}$

be the symbol map constructed in (5.19) (see (6.2) for the isomorphism used in
(6.3)).

Let

$\widetilde{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*})\,\subset\,{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*})$ (6.4)

be the affine subspace consisting of parabolic differential operators whose
symbol is the constant function $1$.

The following Lemma constructs the sub-principal symbol of the operator:

######  Lemma 6.1.

There is a natural map

$\Psi\,:\,\widetilde{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*})\,\longrightarrow\,H^{0}(X,\,K_{X})\,.$

###### Proof.

As in (2.27), let $\mathbf{L}$ denote the orbifold line bundle on $Y$
corresponding to ${\mathcal{L}}$. So the parabolic bundle
${\mathcal{L}}^{1-r}_{*}$ (respectively, ${\mathcal{L}}^{r+1}_{*}$)
corresponds to the orbifold line bundle $\mathbf{L}^{1-r}$ (respectively,
$\mathbf{L}^{r+1}$). Take any

$D\,\in\,\widetilde{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*}).$

Now Proposition 5.2 says that $D$ corresponds to a $\Gamma$–invariant
holomorphic differential operator of order $r$ from $\mathbf{L}^{1-r}$ to
$\mathbf{L}^{r+1}$. Let

${\mathcal{D}}\,\in\,{\rm
DO}^{r}(\mathbf{L}^{1-r},\,\mathbf{L}^{r+1})^{\Gamma}$ (6.5)

be the $\Gamma$–invariant differential operator corresponding to $D$. As the
orbifold bundle $\mathbf{L}^{2}$ is isomorphic to $TY$ (see Lemma 2.8), the
symbol of ${\mathcal{D}}$ is a section of ${\mathcal{O}}_{Y}$. Since the
symbol of $D$ is the constant function $1$, from Lemma 5.3 it follows that the
symbol of $\mathcal{D}$ is the constant function $1$ on $Y$.

We will now show that a differential operator ${\mathbf{D}}\,\in\,{\rm
DO}^{r}(\mathbf{L}^{1-r},\,\mathbf{L}^{r+1})$ of symbol $1$ produces a section

$\theta_{\mathbf{D}}\,\in\,H^{0}(Y,\,K_{Y})\,.$ (6.6)

Consider the short exact sequence of jet bundles

$0\,\longrightarrow\,\mathbf{L}^{1-r}\otimes K^{\otimes
r}_{Y}\,=\,\mathbf{L}^{r+1}\,\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}\,J^{r}(\mathbf{L}^{1-r})\,\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}\,J^{r-1}(\mathbf{L}^{1-r})\,\longrightarrow\,0$
(6.7)

(see Lemma 2.8 for the above isomorphism) together with the homomorphism

${\mathbf{D}}^{\prime}\,:\,J^{r}(\mathbf{L}^{1-r})\,\longrightarrow\,\mathbf{L}^{r+1}$

defining the given differential operator $\mathbf{D}$. Since the symbol of
$\mathbf{D}$ is $1$, we have

${\mathbf{D}}^{\prime}\circ\mu\,=\,{\rm Id}_{\mathbf{L}^{r+1}}\,,$

where $\mu$ is the homomorphism in (6.7). Therefore, ${\mathbf{D}}^{\prime}$
produces a holomorphic splitting of the short exact sequence in (6.7). Let

$\tau\,:\,J^{r-1}(\mathbf{L}^{1-r})\,\longrightarrow\,J^{r}(\mathbf{L}^{1-r})$
(6.8)

be the holomorphic homomorphism given by this splitting of the short exact
sequence in (6.7), so $\tau$ is uniquely determined by the following two
conditions:

  * •

$\nu\circ\tau\,=\,{\rm Id}_{J^{r-1}(\mathbf{L}^{1-r})}$, where $\nu$ is the
projection in (6.7), and

  * •

${\rm
image}(\tau)\,=\,\text{kernel}({\mathbf{D}}^{\prime})\,\subset\,J^{r}(\mathbf{L}^{1-r})$.

Next consider the following natural commutative diagram of homomorphisms of
jet bundles:

$\begin{matrix}&&0&&0&&0\\\
&&\Big{\downarrow}&&\Big{\downarrow}&&\Big{\downarrow}\\\
0&\longrightarrow&\mathbf{L}^{1-r}\otimes K^{\otimes
r}_{Y}\,=\,\mathbf{L}^{r+1}&\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}&J^{r}(\mathbf{L}^{1-r})&\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}&J^{r-1}(\mathbf{L}^{1-r})&\longrightarrow&0\\\
&&\Big{\downarrow}&&\,\,\,\Big{\downarrow}\varpi&&\Big{\|}\\\
0&\longrightarrow&J^{r-1}(\mathbf{L}^{1-r})\otimes
K_{Y}&\longrightarrow&J^{1}(J^{r-1}(\mathbf{L}^{1-r}))&\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}&J^{r-1}(\mathbf{L}^{1-r})&\longrightarrow&0\\\
&&\Big{\downarrow}&&\,\,\,\Big{\downarrow}\zeta&&\\\
0&\longrightarrow&J^{r-2}(\mathbf{L}^{1-r})\otimes
K_{Y}&\stackrel{{\scriptstyle=}}{{\longrightarrow}}&J^{r-2}(\mathbf{L}^{1-r})\otimes
K_{Y}&&\\\ &&\Big{\downarrow}&&\Big{\downarrow}&&\\\ &&0&&0&&\end{matrix}$
(6.9)

where the horizontal sequences are the natural jet sequences, and the vertical
sequence in the left is the jet sequence tensored with $K_{Y}$; the
homomorphism $\varpi$ is the natural homomorphism of jet bundles. The
homomorphism $\zeta$ in (6.9) is constructed as follows: We have the natural
homomorphism

$h_{1}\,:\,J^{1}(J^{r-1}(\mathbf{L}^{1-r}))\,\longrightarrow\,J^{r-1}(\mathbf{L}^{1-r}).$

On the other hand, we have the composition of homomorphisms

$J^{1}(J^{r-1}(\mathbf{L}^{1-r}))\,\longrightarrow\,J^{1}(J^{r-2}(\mathbf{L}^{1-r}))\,\longrightarrow\,J^{r-1}(\mathbf{L}^{1-r}),$

which will be denoted by $h_{2}$. Now, we have $\zeta\,=\,h_{1}-h_{2}$; note
that $J^{r-2}(\mathbf{L}^{1-r})\otimes K_{Y}$ is a subbundle of
$J^{r-1}(\mathbf{L}^{1-r})$.

Next consider the homomorphism

$\varpi\circ\tau\,:\,J^{r-1}(\mathbf{L}^{1-r})\,\longrightarrow\,J^{1}(J^{r-1}(\mathbf{L}^{1-r}))\,,$

where $\tau$ and $\varpi$ are the homomorphisms in (6.8) and (6.9)
respectively. We have

$\alpha\circ(\varpi\circ\tau)\,=\,{\rm Id}_{J^{r-1}(\mathbf{L}^{1-r})}\,,$
(6.10)

where $\alpha$ is the projection in (6.9), because (6.9) is a commutative
diagram.

From (6.10) it follows immediately that $\varpi\circ\tau$ gives a holomorphic
splitting of the bottom exact sequence in (6.9). But a holomorphic splitting
of the bottom exact sequence in (6.9) is a holomorphic connection on
$J^{r-1}(\mathbf{L}^{1-r})$.

Let $\nabla$ denote the holomorphic connection on $J^{r-1}(\mathbf{L}^{1-r})$
given by $\varpi\circ\tau$. The holomorphic connection on
$\bigwedge^{r}J^{r-1}(\mathbf{L}^{1-r})\,=\,{\mathcal{O}}_{Y}$ (see Lemma 2.8)
induced by $\nabla$ will be denoted by $\nabla^{0}$. So the connection
$\nabla^{0}$ is of the form

$\nabla^{0}\,=\,d+\theta_{\mathbf{D}}\,,$

where $\theta_{\mathbf{D}}\,\in\,H^{0}(Y,\,K_{Y})$ and $d$ is the de Rham
differential on ${\mathcal{O}}_{Y}$. This $\theta_{\mathbf{D}}$ is the
holomorphic $1$-form in (6.6).

By the construction of it, the form $\theta_{\mathbf{D}}$ vanishes identically
if and only if the above connection $\nabla$ on $J^{r-1}(\mathbf{L}^{1-r})$
induces the trivial connection on
$\bigwedge^{r}J^{r-1}(\mathbf{L}^{1-r})\,=\,{\mathcal{O}}_{Y}$. Therefor
$\theta_{\mathbf{D}}$ should be seen as a sub-principal symbol.

Consider $\theta_{\mathcal{D}}\,\in\,H^{0}(Y,\,K_{Y})$ (as in (6.6)) for the
differential operator $\mathcal{D}$ in (6.5). Since $\mathcal{D}$ is
$\Gamma$–invariant, we know that $\theta_{\mathcal{D}}$ is also
$\Gamma$–invariant. On the other hand,

$H^{0}(Y,\,K_{Y})^{\Gamma}\,=\,H^{0}(X,\,K_{X})\,.$

The element of $H^{0}(X,\,K_{X})$ corresponding to $\theta_{\mathcal{D}}$ will
be denoted by $\theta^{\prime}_{\mathcal{D}}$.

Now we have a map

$\Psi\,:\,\widetilde{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*})\,\longrightarrow\,H^{0}(X,\,K_{X})$

that sends any $D$ to $\theta^{\prime}_{\mathcal{D}}$ constructed above from
$D$. ∎

The following main Theorem deals with the space of all parabolic ${\rm
SL}(r,{\mathbb{C}})$–opers on $X$ (see Definition 3.3) with given singular set
$S\,:=\,\\{x_{1},\,\cdots,\,x_{n}\\}\,\subset\,X$ and fixed integers
$c_{i}\,=\,N_{i}$ (see (2.14)).

######  Theorem 6.2.

The space of all parabolic ${\rm SL}(r,{\mathbb{C}})$–opers on $X$ is
identified with the inverse image

$\Psi^{-1}(0)\,\subset\,\widetilde{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*}),$

where $\Psi$ is the map in Lemma 6.1.

###### Proof.

This theorem will be proved using Proposition 3.6, Proposition 5.2, Lemma 5.3
and Lemma 6.1.

As before, fix a ramified Galois covering

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

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 $2N_{i}+1$.

As before, $\Gamma$ denotes $\text{Aut}(Y/X)$. Parabolic ${\rm
SL}(r,{\mathbb{C}})$–opers on $X$ are in a natural bijective correspondence
with the equivariant ${\rm SL}(r,{\mathbb{C}})$–opers on $Y$ (see Proposition
3.6). Equivariant ${\rm SL}(r,{\mathbb{C}})$–opers on $Y$ are in a natural
bijective correspondence with the subspace of ${\mathcal{D}}\,\in\,{\rm
DO}^{r}(\mathbf{L}^{1-r},\,\mathbf{L}^{r+1})^{\Gamma}$ (see (6.5)) defined by
all invariant differential operators $D$ satisfying the following two
conditions:

  * •

the symbol of $D$ is the constant function $1$, and

  * •

the element in $H^{0}(Y,\,K_{Y})$ corresponding to $D$ (see (6.6)) vanishes
(this is equivalent to the vanishing of the sub-principal symbol of $D$; see
[BD1, p. 13]).

(See Proposition 5.2 and Lemma 5.3.)

This subspace of ${\rm DO}^{r}(\mathbf{L}^{1-r},\,\mathbf{L}^{r+1})^{\Gamma}$
is in a natural bijective correspondence with

$\Psi^{-1}(0)\,\subset\,\widetilde{\rm
DO}^{r}_{P}({\mathcal{L}}^{1-r}_{*},\,{\mathcal{L}}^{r+1}_{*}),$

where $\Psi$ is the map in Lemma 6.1. ∎

## Acknowledgements

We are very grateful to the referee for helpful comments. This work has been
supported by the French government through the UCAJEDI Investments in the
Future project managed by the National Research Agency (ANR) with the
reference number ANR2152IDEX201. The first author is partially supported by a
J. C. Bose Fellowship, and school of mathematics, TIFR, is supported by
12-R$\&$D-TFR-5.01-0500.

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