# Branched ${\rm SL}(r,{\mathbb{C}})$-opers

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

Branched projective structures were introduced by Mandelbaum [22], [23], and
opers were introduced by Beilinson and Drinfeld [2], [3]. We define the
branched analog of ${\rm SL}(r,{\mathbb{C}})$–opers and investigate their
properties. For the usual ${\rm SL}(r,{\mathbb{C}})$–opers, the underlying
holomorphic vector bundle is actually determined uniquely up to tensoring with
a holomorphic line bundle of order $r$. For the branched ${\rm
SL}(r,{\mathbb{C}})$–opers, the underlying holomorphic vector bundle depends
more intricately on the oper. While the holomorphic connection for a branched
${\rm SL}(r,{\mathbb{C}})$–oper is nonsingular, given a branched ${\rm
SL}(r,{\mathbb{C}})$–oper, we associate to it a certain holomorphic vector
bundle equipped with a logarithmic connection. This holomorphic vector bundle
in question supporting a logarithmic connection does not depend on the
branched oper. We characterize the branched ${\rm SL}(r,{\mathbb{C}})$–opers
in terms of the logarithmic connections on this fixed holomorphic vector
bundle.

###### Key words and phrases:

Opers, logarithmic connection, residue, branch structure

###### 2010 Mathematics Subject Classification:

53B15, 14H60, 14D21

###### Contents

  1. 1 Introduction
  2. 2 Branched SL($r$)–opers
    1. 2.1 Examples
  3. 3 Logarithmic connection from branched SL($r$)–opers
    1. 3.1 Homomorphism to a jet bundle
    2. 3.2 A logarithmic connection
  4. 4 Residues
  5. 5 Local monodromy
  6. 6 Characterizing the logarithmic connections
    1. 6.1 ${\rm SL}(r,{\mathbb{C}})$-opers with regular singularity and branched ${\rm SL}(r,{\mathbb{C}})$-opers

##  1\. Introduction

Opers were introduced by Beilinson and Drinfeld [2], [3]. For this they were
motivated by the earlier works of Drinfeld and Sokolov [10], [11]. Given a
semisimple complex Lie group $G$, a $G$–oper on a compact Riemann surface $X$
consists of

  * •

a holomorphic principal $G$–bundle $P$ on $X$ equipped with a holomorphic
connection $\nabla$, and

  * •

a holomorphic reduction of the structure group of $P$ to a Borel subgroup $B$
of $G$,

such that the reduction to $B$ satisfies an analogue of the Griffiths
transversality condition with respect to this connection $\nabla$, and
moreover the second fundamental form of the above reduction of structure group
to $B$, for the connection $\nabla$, satisfies certain nondegeneracy
conditions. Opers are useful in diverse topics, for example in geometric
Langlands correspondence, nonabelian Hodge theory, some branches of
mathematical physics, differential equations et cetera; see [4], [12], [18],
[16], [17], [13], [8], [14], [15], [21], [24], [7] and references therein.

An ${\rm SL}(2,{\mathbb{C}})$–oper on a compact Riemann surface $X$
corresponds to a complex projective structure on $X$ together with a theta
characteristic on $X$. We recall that a complex projective structure on $X$ is
given by a covering of $X$ by holomorphic coordinate charts such that all the
transition functions are restrictions of Möbius transformations [20]. During
the early seventies, Mandelbaum introduced, and studied, the notion of a
branched complex projective structure [22], [23]. Examples of branched complex
projective structures are provided by pulling back the usual complex
projective structures through a holomorphic ramified covering map; see [6] for
a recent study of branched complex projective structures. For higher
dimensional complex manifolds a branched analog of the more general notion of
holomorphic Cartan geometries was introduced in [5].

Our aim here is to introduce, and investigate, the branched analog of ${\rm
SL}(r,{\mathbb{C}})$–opers.

Let $X$ be a compact connected Riemann surface, and let $S\,\subset\,X$ be a
finite subset. The holomorphic cotangent bundle and the structure sheaf of $X$
are denoted by $K_{X}$ respectively ${\mathcal{O}}_{X}$.

A branched ${\rm SL}(r,{\mathbb{C}})$–oper on $X$, with branching over $S$, is
given by a triple of the form $(V,\,{\mathcal{F}},\,D)$, where

  * •

$V$ is a rank $r$ holomorphic vector bundle on $X$ such that
$\bigwedge^{r}V\,=\,{\mathcal{O}}_{X},$

  * •

${\mathcal{F}}$ is a filtration of holomorphic subbundles

$0\,=\,F_{0}\,\subset\,F_{1}\,\subset\,F_{2}\,\subset\,\cdots\,\subset\,F_{r-1}\,\subset\,F_{r}\,=\,V$
(1.1)

with ${\rm rank}(F_{i})\,=\,i$, for all $1\,\leq\,i\,\leq\,r$, and

  * •

$D$ is a holomorphic connection on $V$ such that
$D(F_{i})\,\subset\,F_{i+1}\otimes K_{X}$, for all $1\,\leq\,i\,\leq\,r-1$,
and $D$ induces the trivial connection on $\bigwedge^{r}V$,

such that for all $1\,\leq\,i\,\leq\,r-1$, the second fundamental form of
$F_{i}$ vanishes exactly on the reduced divisor $S$ (see Definition 2.1).

We recall that for a usual ${\rm SL}(r,{\mathbb{C}})$–oper, the second
fundamental forms of the filtration ${\mathcal{F}}$ are required to be nonzero
everywhere.

For usual ${\rm SL}(r,{\mathbb{C}})$–opers, i.e., when $S\,=\,0$, the
underlying holomorphic vector bundle $V$ supporting the holomorphic connection
is very special. To explain this, note that any holomorphic line bundle $\xi$
on $X$ of order $r$ has a tautological holomorphic connection $D_{\xi}$ which
is uniquely determined by the condition that the connection on $\xi^{\otimes
r}\,=\,{\mathcal{O}}_{X}$ induced by $D_{\xi}$ has trivial monodromy. For any
usual ${\rm SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$, note that

$(V\otimes\xi,\,{\mathcal{F}}\otimes\xi,\,D\otimes{\rm Id}_{\xi}+{\rm
Id}_{V}\otimes D_{\xi})$

is also a usual ${\rm SL}(r,{\mathbb{C}})$–oper. If we fix a usual ${\rm
SL}(r,{\mathbb{C}})$–oper $(V_{0},\,{\mathcal{F}}_{0},\,D_{0})$ on $X$, then
for any usual ${\rm SL}(r,{\mathbb{C}})$–oper
$(V^{\prime},\,{\mathcal{F}}^{\prime},\,D^{\prime})$ on $X$, there is a
holomorphic line bundle $\xi$ on $X$ of order $r$ such that
$V^{\prime}\,=\,V_{0}\otimes\xi$. Moreover, we have

${\mathcal{F}}^{\prime}\,=\,{\mathcal{F}}_{0}\otimes\xi\,;$

so the filtration for a usual ${\rm SL}(r,{\mathbb{C}})$–oper also does not
depend on the ${\rm SL}(r,{\mathbb{C}})$–oper. In the case of usual ${\rm
SL}(r,{\mathbb{C}})$–opers, the underlying holomorphic vector bundle $V$ in
(1.1) is a jet bundle, and the filtration $\mathcal{F}$ is the natural
filtration of a jet bundle.

However, for branched ${\rm SL}(r,{\mathbb{C}})$–opers the underlying vector
bundle $V$ supporting the holomorphic connection depends more intimately on
the oper, although each graded piece $F_{i+1}/F_{i}$ for the filtration
${\mathcal{F}}$ in (1.1) is actually independent of the branched ${\rm
SL}(r,{\mathbb{C}})$–oper. More precisely, $F_{i+1}/F_{i}$ is independent of
the branched ${\rm SL}(r,{\mathbb{C}})$–oper if $r$ is odd, and when $r$ is
even, the quotients $F_{i+1}/F_{i}$ are determined by the theta characteristic
on $X$ associated to the branched oper.

Denote the above quotient line bundle $F_{r}/F_{r-1}$ by $Q$. We prove that a
branched ${\rm SL}(r,{\mathbb{C}})$–oper produces a logarithmic connection on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$, where $J^{r-1}(Q)$ is the
$(r-1)$-th jet bundle of $Q$; see Proposition 3.5. Given the logarithmic
connection on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ associated to a
branched ${\rm SL}(r,{\mathbb{C}})$–oper, the branched ${\rm
SL}(r,{\mathbb{C}})$–oper can actually be recovered using the Hecke
transformations on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ and the
induced logarithmic connection on the Hecke transformations. The residues of
these logarithmic connection on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$,
and also the local properties of the logarithmic connection, are studied here.
Based on these studies, we characterize all logarithmic connections on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ that arise from branched ${\rm
SL}(r,{\mathbb{C}})$–opers; see Theorem 6.1.

The branched ${\rm SL}(r,{\mathbb{C}})$-opers are closely related with the
${\rm SL}(r,{\mathbb{C}})$-opers with regular singularity introduced by
Beilinson and Drinfeld; this is explained in Section 6.1, which was kindly
communicated to us by Edward Frenkel.

Once the branched ${\rm SL}(r,{\mathbb{C}})$–opers have been defined, it is
straight-forward to define the branched orthogonal and branched symplectic
opers. We have omitted this exercise.

##  2\. Branched SL($r$)–opers

As before, $X$ is a compact connected Riemann surface of genus $g$. The
holomorphic tangent bundle of $X$ is denoted by $TX$.

Let $V$ be a holomorphic vector bundle on $X$. A logarithmic connection on $V$
singular over a reduced effective divisor $S$ is a holomorphic differential
operator

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

satisfying the Leibniz identity which states that

$D(fs)\,=\,fD(s)+s\otimes df$ (2.1)

for any locally defined holomorphic function $f$ on $X$ and any locally
defined holomorphic section $s$ of $V$. If $S$ is the zero divisor, then $D$
is called a holomorphic connection [1]. Any logarithmic connection on $V$ is
integrable (same as flat) because $\Omega^{2}_{X}\,=\,0$. A criterion of Weil
and Atiyah says that $V$ admits a holomorphic connection if and only if the
degree of every indecomposable component of $V$ is zero [1, p. 203, Theorem
10], [25]. In particular, if $V$ is indecomposable, and
$\text{degree}(V)\,=\,0$, then $V$ admits a holomorphic connection.

Let $D$ be a logarithmic connection on $V$. For a holomorphic subbundle
$F\,\subset\,V$, consider the following composition of homomorphisms

$F\,\hookrightarrow\,V\,\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\,V\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\xrightarrow{\ q_{0}\otimes{\rm
Id}_{K_{X}\otimes{\mathcal{O}}_{X}(S)}\ }(V/F)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$ (2.2)

where $q_{0}\,:\,V\longrightarrow\,V/F$ is the natural quotient map. From
(2.1) it follows that this composition of homomorphisms is actually
${\mathcal{O}}_{X}$–linear, and hence it produces a holomorphic section

${\rm SF}(D,\,F)\,\in\,H^{0}(X,\,\text{Hom}(F,\,V/F)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,=\,H^{0}(X,\,F^{*}\otimes(V/F)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,,$ (2.3)

which is known as the second fundamental form of $F$ for the logarithmic
connection $D$.

If $D$ is a holomorphic connection on $V$, then the composition of
homomorphisms in (2.2) simplifies as

$F\,\hookrightarrow\,V\,\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\,V\otimes K_{X}\,\xrightarrow{\ q_{0}\otimes{\rm
Id}_{K_{X}}\ }(V/F)\otimes K_{X}\,,$ (2.4)

and it gives the following analogue of (2.3)

${\rm SF}(D,\,F)\,\in\,H^{0}(X,\,\text{Hom}(F,\,V/F)\otimes
K_{X})\,=\,H^{0}(X,\,F^{*}\otimes K_{X}\otimes(V/F))\,,$ (2.5)

which is the second fundamental form of $F$ for the holomorphic connection
$D$.

Assume that the rank $r$ of $V$ is at least two, and $D$ is, as before, a
holomorphic connection on $V$. Let

$0\,=\,F_{0}\,\subset\,F_{1}\,\subset\,F_{2}\,\subset\,\cdots\,\subset\,F_{r-1}\,\subset\,F_{r}\,=\,V$

be a filtration of holomorphic subbundles such that for all
$1\,\leq\,i\,\leq\,r-1$,

  * •

${\rm rank}(F_{i})\,=\,i$, and

  * •

$D(F_{i})\,\subset\,F_{i+1}\otimes K_{X}$.

Consequently, the second fundamental form ${\rm SF}(D,\,F_{i})$ in (2.5)
satisfies the following condition:

${\rm SF}(D,\,F_{i})(F_{i})\,\subset\,(F_{i+1}/F_{i})\otimes
K_{X}\,\subset\,(V/F_{i})\otimes K_{X}$

for all $1\,\leq\,i\,\leq\,r-1$. Therefore, ${\rm SF}(D,\,F_{i})$ produces a
holomorphic homomorphism

${\rm SF}(D,\,i)\,:\,F_{i}/F_{i-1}\,\longrightarrow\,(F_{i+1}/F_{i})\otimes
K_{X}$ (2.6)

for every $1\,\leq\,i\,\leq\,r-1$.

Assume that there is holomorphic line bundle ${\mathbf{L}}_{0}$ on $X$ such
that the line bundle $\text{Hom}(F_{i}/F_{i-1},\,(F_{i+1}/F_{i})\otimes
K_{X})\,=\,(F_{i+1}/F_{i})\otimes(F_{i}/F_{i-1})^{*}\otimes K_{X}$ is
holomorphically isomorphic to ${\mathbf{L}}_{0}$ for all
$1\,\leq\,i\,\leq\,r-1$. This implies that

$F_{i+1}/F_{i}\,=\,(F_{i}/F_{i-1})\otimes({\mathbf{L}}_{0}\otimes
TX)\,=\,(F_{i}/F_{i-1})\otimes{\mathbf{L}}\,,$ (2.7)

where ${\mathbf{L}}\,:=\,{\mathbf{L}}_{0}\otimes TX$. Then we have

$\det
V\,:=\,\bigwedge\nolimits^{r}V\,=\,\bigotimes_{j=1}^{r}F_{j}/F_{j-1}\,=\,F^{\otimes
r}_{1}\otimes{\mathbf{L}}^{{}^{\otimes\frac{r(r-1)}{2}}}\,=\,(F_{r}/F_{r-1})^{\otimes
r}\otimes\left({\mathbf{L}}^{{}^{\otimes\frac{r(r-1)}{2}}}\right)^{*}\,.$
(2.8)

Fix a reduced effective divisor

$S\,:=\,\sum_{k=1}^{d}x_{k}$ (2.9)

on $X$ of degree $d\,\geq\,0$; so $\\{x_{1},\,\cdots,\,x_{d}\\}$ are distinct
points of $X$. We will introduce branched $\text{SL}(r,{\mathbb{C}})$–opers on
$X$ with branching over $S$.

When $r$ is an even integer, we assume that $d$ is also even. If $r$ is an
even integer, fix a holomorphic line bundle $\mathcal{L}$ on $X$ of degree
$1+\frac{d}{2}-g\,\in\,\mathbb{Z}$ such that

${\mathcal{L}}^{{}^{\otimes 2}}\,=\,TX\otimes{\mathcal{O}}_{X}(S)\,;$ (2.10)

also, fix a holomorphic isomorphism of ${\mathcal{L}}^{{}^{\otimes 2}}$ with
$TX\otimes{\mathcal{O}}_{X}(S)$. Note that if $S$ is the zero divisor, then
the dual ${\mathcal{L}}^{*}$ is a theta characteristic on $X$.

######  Definition 2.1.

A branched $\text{SL}(r,{\mathbb{C}})$–oper over $X$ with branching over $S$
is a triple

$(V,\,{\mathcal{F}},\,D)\,,$

where

  * •

$V$ is a rank $r$ holomorphic vector bundle on $X$ such that
$\bigwedge^{r}V\,=\,{\mathcal{O}}_{X}$,

  * •

${\mathcal{F}}$ is a filtration of holomorphic subbundles

$0\,=\,F_{0}\,\subset\,F_{1}\,\subset\,F_{2}\,\subset\,\cdots\,\subset\,F_{r-1}\,\subset\,F_{r}\,=\,V$
(2.11)

with ${\rm rank}(F_{i})\,=\,i$, for all $1\,\leq\,i\,\leq\,r$, and

  * •

$D$ is a holomorphic connection on $V$,

such that the following five conditions hold:

  1. (1)

if $r$ is even, then
$F_{1}\,=\,\left({\mathcal{L}}^{{}^{\otimes(r-1)}}\right)^{*}$, where
$\mathcal{L}$ is a line bundle as in (2.10), and if $r$ is odd, then
$F_{1}\,=\,\left(\left(TX\otimes{\mathcal{O}}_{X}(S)\right)^{{}^{\otimes\frac{r-1}{2}}}\right)^{*}$,

  2. (2)

$F_{j+1}/F_{j}\,\simeq\,F_{1}\otimes\left(TX\otimes{\mathcal{O}}_{X}(S)\right)^{\otimes
j}$, for all $0\,\leq\,j\,\leq\,r-1$,

  3. (3)

the connection on $\bigwedge^{r}V\,=\,{\mathcal{O}}_{X}$ induced by the
connection $D$ on $V$ coincides with the trivial connection given by the de
Rham differential $d$ on ${\mathcal{O}}_{X}$,

  4. (4)

$D(F_{i})\,\subset\,F_{i+1}\otimes K_{X}$, for all $1\,\leq\,i\,\leq\,r-1$,
and

  5. (5)

the homomorphism in (2.6)

${\rm SF}(D,\,i)\,:\,F_{i}/F_{i-1}\,\longrightarrow\,(F_{i+1}/F_{i})\otimes
K_{X}$ $=\,(F_{i}/F_{i-1})\otimes TX\otimes{\mathcal{O}}_{X}(S)\otimes
K_{X}\,=\,(F_{i}/F_{i-1})\otimes{\mathcal{O}}_{X}(S)$ (2.12)

coincides with the natural inclusion map of $F_{i}/F_{i-1}$ into
$(F_{i}/F_{i-1})\otimes{\mathcal{O}}_{X}(S)$.

######  Remark 2.2.

Note that the above condition that
$F_{j+1}/F_{j}\,=\,F_{1}\otimes\left(TX\otimes{\mathcal{O}}_{X}(S)\right)^{\otimes
j}$ implies that $(F_{i+1}/F_{i})\,=\,(F_{i}/F_{i-1})\otimes
TX\otimes{\mathcal{O}}_{X}(S)$. The fifth condition that ${\rm SF}(D,\,i)$ is
the natural inclusion map coincides with the condition that ${\rm SF}(D,\,i)$
is given by tensoring with the section of ${\mathcal{O}}_{X}(S)$ defined by
the constant function $1$ on $X$.

######  Remark 2.3.

Comparing (2.12) and (2.7) we conclude that in Definition 2.1, the holomorphic
line bundle ${\mathcal{O}}_{X}(S)$ plays the role of ${\mathbf{L}}_{0}$ in
(2.7). So in Definition 2.1, the holomorphic line bundle
${\mathcal{O}}_{X}(S)\otimes TX$ plays the role of ${\mathbf{L}}$ in (2.8).
The second condition in Definition 2.1, that
$F_{j+1}/F_{j}\,\simeq\,F_{1}\otimes\left(TX\otimes{\mathcal{O}}_{X}(S)\right)^{\otimes
j}$, is obtained from (2.7), and the first condition in Definition 2.1 is
motivated by (2.8), because $\bigwedge^{r}V\,=\,{\mathcal{O}}_{X}$.

######  Remark 2.4.

Any two choices of the holomorphic line bundle $\mathcal{L}$ in (2.10) differ
by tensoring with a line bundle on $X$ of order two. Any holomorphic line
bundles $\xi$ of order two on $X$ has a unique holomorphic connection
$D_{\xi}$ satisfying the following condition: Any holomorphic isomorphism
between $\xi^{\otimes 2}$ and ${\mathcal{O}}_{X}$ takes the connection
$D_{\xi}\otimes{\rm Id}_{\xi}+{\rm Id}_{\xi}\otimes D_{\xi}$ on $\xi^{\otimes
2}$ to the connection on ${\mathcal{O}}_{X}$ given by the de Rham differential
$d$. If $W$ is a holomorphic vector bundle on $X$ equipped with a holomorphic
connection $D_{W}$, then $D_{\xi}$ and $D_{W}$ together produce a holomorphic
connection $D_{W}\otimes{\rm Id}_{\xi}+{\rm Id}_{W}\otimes D_{\xi}$ on
$W\otimes\xi$. When $r$ is even, for any two choices of the holomorphic line
bundle $\mathcal{L}$ in (2.10), a priori, we have two different spaces of
branched $\text{SL}(r,{\mathbb{C}})$–opers over $X$, with branching over $S$,
given by Definition 2.1. Using the above observation we conclude that these
two spaces are actually canonically identified.

When $S$ in (2.9) is the zero divisor, a branched
$\text{SL}(r,{\mathbb{C}})$–oper is a usual $\text{SL}(r,{\mathbb{C}})$–oper
[2], [3]. In that special (unbranched) case, the underlying holomorphic vector
bundle $V$ of an $\text{SL}(r,{\mathbb{C}})$–oper does not depend on the oper
in the following sense: The vector bundle $V$ is the jet bundle
$J^{r-1}({\mathcal{L}}^{{}^{\otimes(r-1)}}\otimes\xi)$, where ${\mathcal{L}}$
is a fixed holomorphic line bundle on $X$ such that ${\mathcal{L}}^{2}\,=\,TX$
while $\xi$ is a holomorphic line bundle on $X$ of order $r$.

Contrary to the unbranched case, when $\text{degree}(S)\,=\,d\,>\,0$, the
underlying holomorphic vector bundle $V$ of a $\text{SL}(r,{\mathbb{C}})$–oper
with branching over $S$ generally depends on the branched oper. Nevertheless
the successive quotients for the filtration $\mathcal{F}$ of $V$ are clearly
independent of the branched $\text{SL}(r,{\mathbb{C}})$–oper. In Section 3 we
will show that a branched $\text{SL}(r,{\mathbb{C}})$–oper
$(V,\,{\mathcal{F}},\,D)$ with branching over $S$ gives rise to a logarithmic
connection on a certain holomorphic vector bundle over $X$ obtained by
performing Hecke transformation on $V$ over the points of $S$. The holomorphic
vector bundle in question turns out to be actually independent of the branched
$\text{SL}(r,{\mathbb{C}})$–oper.

Take a branched $\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$
over $X$ with branching over $S$. Let $D^{*}$ be the holomorphic connection on
the dual vector bundle $V^{*}$ induced by the connection $D$. The filtration
$\mathcal{F}$ of $V$ in (2.11) produces a filtration of $V^{*}$ by holomorphic
subbundles as follows: Consider the dual homomorphisms of the inclusion maps
in (2.11)

$V^{*}\,=\,F^{*}_{r}\,\twoheadrightarrow\,F^{*}_{r-1}\,\twoheadrightarrow\,\cdots\,\twoheadrightarrow\,F^{*}_{2}\,\twoheadrightarrow\,F^{*}_{1}\,\twoheadrightarrow\,F^{*}_{0}\,=\,0\,;$

the subbundles of $V^{*}$ giving the filtration are the kernels of the above
projections $V^{*}\,\twoheadrightarrow\,F^{*}_{i}$. This filtration by
holomorphic subbundles of $V^{*}$ will be denoted by ${\mathcal{F}}^{*}$.

######  Lemma 2.5.

The above triple $(V^{*},\,{\mathcal{F}}^{*},\,D^{*})$ is a branched ${\rm
SL}(r,{\mathbb{C}})$–oper over $X$ with branching over $S$.

###### Proof.

As in (2.4), take a holomorphic subbundle $F\,\subset\,V$. So we have the
holomorphic subbundle $(V/F)^{*}\,\subset\,V^{*}$. Note that the quotient
bundle $V^{*}/((V/F)^{*})$ is identified with $F^{*}$. Let

${\rm
SF}(D^{*},\,(V/F)^{*})\,\in\,H^{0}(X,\,\text{Hom}((V/F)^{*},\,V^{*}/((V/F)^{*}))\otimes
K_{X})$ $=\,H^{0}(X,\,\text{Hom}((V/F)^{*},\,F^{*})\otimes
K_{X})\,=\,H^{0}(X,\,(V/F)\otimes F^{*}\otimes K_{X})$

be the second fundamental form of $(V/F)^{*}\,\subset\,V^{*}$ for the
connection $D^{*}$ on $V^{*}$ (see (2.5)). Then it is straightforward to check
that

${\rm SF}(D^{*},\,(V/F)^{*})\,=\,{\rm SF}(D,\,F)\,,$ (2.13)

where ${\rm SF}(D,\,F)$ is constructed in (2.5). The lemma is a
straightforward consequence of (2.13). ∎

###  2.1. Examples

We now give some examples of branched ${\rm SL}(r,{\mathbb{C}})$–oper.

Let

$\varpi\,:\,X\,\longrightarrow\,Y$

be a nonconstant holomorphic map between compact connected Riemann surfaces
such that all the branch points of $\varpi$ have branch number $1$, and let
$(V,\,{\mathcal{F}},\,D)$ be a usual ${\rm SL}(r,{\mathbb{C}})$–oper over $Y$.
Then the pullback $(\varpi^{*}V,\,\varpi^{*}{\mathcal{F}},\,\varpi^{*}D)$ is a
branched ${\rm SL}(r,{\mathbb{C}})$–oper over $X$. The branching divisor for
$(\varpi^{*}V,\,\varpi^{*}{\mathcal{F}},\,\varpi^{*}D)$ coincides with the
branching divisor for the map $\varpi$.

Let $\varpi\,:\,X\,\longrightarrow\,{\mathbb{C}}{\mathbb{P}}^{1}$ be any
nonconstant holomorphic map such that all the branch points of $\varpi$ have
branch number $1$. Then pulling back the standard ${\rm
SL}(2,{\mathbb{C}})$–oper on ${\mathbb{C}}{\mathbb{P}}^{1}$ by $\varpi$ we get
a branched ${\rm SL}(2,{\mathbb{C}})$–oper over $X$.

More generally, let $V$ be a rank two holomorphic vector bundle on $X$ with
$\bigwedge^{2}V\,=\,{\mathcal{O}}_{X}$, and let $D$ be a holomorphic
connection on $V$ such that the induced holomorphic connection on
$\bigwedge^{2}V$ coincides with the holomorphic connection on
${\mathcal{O}}_{X}$ given by the de Rham differential $d$. Let $L\,\subset\,V$
be a holomorphic line subbundle satisfying the following two conditions:

  1. (1)

the second fundamental form ${\rm SF}(D,\,L)$ is not identically zero, and

  2. (2)

all the zeros of ${\rm SF}(D,\,L)$ are of order one.

Then $(V,\,L,\,D)$ is a branched ${\rm SL}(2,{\mathbb{C}})$–oper on $X$. The
branching divisor for $(V,\,L,\,D)$ coincides with the vanishing divisor of
${\rm SF}(D,\,L)$. The previous example, given by the pull-back of the
standard ${\rm SL}(2,{\mathbb{C}})$–oper on ${\mathbb{C}}{\mathbb{P}}^{1}$,
actually corresponds to the special case where
$V\,=\,\varpi^{*}{\mathcal{O}}^{\oplus
2}_{{\mathbb{C}}{\mathbb{P}}^{1}}\,=\,{\mathcal{O}}^{\oplus 2}_{X}$, $D$ is
the trivial connection on it, and $L$ is the pull-back, by $\varpi$, of the
tautological line subbundle of ${\mathcal{O}}^{\oplus
2}_{{\mathbb{C}}{\mathbb{P}}^{1}}$.

Let $(V,\,L,\,D)$ be a branched ${\rm SL}(2,{\mathbb{C}})$–oper over $X$. Then
the holomorphic line subbundle $L\,\subset\,V$ produces a filtration of
holomorphic subbundles $\\{F_{i}\\}_{i=1}^{r}$ of the symmetric product
$\text{Sym}^{r-1}(V)$ whose $i$-th term $F_{i}$ is the image of
$L^{\otimes(r-i)}\otimes V^{\otimes(i-1)}$ in $\text{Sym}^{r-1}(V)$. The
triple $(\text{Sym}^{r-1}(V),\,\\{F_{i}\\}_{i=1}^{r},\,\widetilde{D})$, where
$\widetilde{D}$ is the holomorphic connection on $\text{Sym}^{r-1}(V)$ induced
by $D$, is a branched ${\rm SL}(r,{\mathbb{C}})$–oper over $X$. Its branching
divisor coincides with that of $(V,\,L,\,D)$.

##  3\. Logarithmic connection from branched SL($r$)–opers

###  3.1. Homomorphism to a jet bundle

As in Definition 2.1, let

$(V,\,{\mathcal{F}},\,D)$ (3.1)

be a branched $\text{SL}(r,{\mathbb{C}})$–oper over $X$, with branching over
$S$, where ${\mathcal{F}}$ stands for the filtration in (2.11) of holomorphic
subbundles of $V$.

In this subsection we will construct another filtered holomorphic vector
bundle from the branched $\text{SL}(r,{\mathbb{C}})$–oper
$(V,\,{\mathcal{F}},\,D)$ in (3.1).

For notational convenience, let

$Q\,:=\,F_{r}/F_{r-1}\,=\,F_{1}\otimes\left(TX\otimes{\mathcal{O}}_{X}(S)\right)^{\otimes(r-1)}$
(3.2)

denote the quotient line bundle in (2.11); the second statement in Definition
2.1 gives the above isomorphism. We will construct a holomorphic homomorphism

$\Phi\,:\,V\,\longrightarrow\,J^{r-1}(Q)$ (3.3)

to the $(r-1)$-th order jet bundle of the line bundle $Q$ in (3.2); see [20],
[6] for jet bundles.

Let

$q\,:\,V\,\longrightarrow\,Q\,=\,V/F_{r-1}$ (3.4)

be the natural quotient map. Take any $x\,\in\,X$ and also take any
$v\,\in\,V_{x}$. Consider a simply connected open neighborhood
$x\,\in\,U\,\subset\,X$ of $x$, and denote by

$\widetilde{v}\,\in\,H^{0}(U,\,V)$

the unique flat section of $V\big{|}_{U}$, for the connection $D$, such that
$\widetilde{v}(x)\,=\,v$. Restricting the section
$q(\widetilde{v})\,\in\,H^{0}(U,\,Q)$ to the $(r-1)$-th order infinitesimal
neighborhood of $x$, where $q$ is the quotient map in (3.4), we get an element

$\widetilde{v}^{\prime}\,:=\,q(\widetilde{v})\big{|}_{rx}\,\in\,J^{r-1}(Q)_{x}\,,$

where $rx$ is the nonreduced divisor with multiplicity $r$. The map $\Phi$ in
(3.3) sends $v$ to this element $\widetilde{v}^{\prime}\,\in\,J^{r-1}(Q)_{x}$
constructed from $v$ using the connection $D$. The homomorphism $\Phi$ is
evidently holomorphic.

We have the natural short exact sequence of jet bundles

$0\,\longrightarrow\,Q\otimes K^{\otimes
k}_{X}\,\longrightarrow\,J^{k}(Q)\,\longrightarrow\,J^{k-1}(Q)\,\longrightarrow\,0$
(3.5)

for all $k\,\geq\,1$. For $0\,\leq\,j\,\leq\,r$, let $H_{j}$ be the kernel of
the projection $J^{r-1}(Q)\,\longrightarrow\,J^{r-1-j}(Q)$ obtained by
iterating the projection in (3.5); we use the convention that $J^{k}(W)\,=\,0$
if $k\,<\,0$. So ${\rm rank}(H_{j})\,=\,j$, and we have the short exact
sequence of holomorphic vector bundles

$0\,\longrightarrow\,H_{j}\,\longrightarrow\,J^{r-1}(Q)\,\longrightarrow\,J^{r-1-j}(Q)\,\longrightarrow\,0$
(3.6)

on $X$. From (3.6) it follows that the quotient $H_{j}/H_{j-1}$ coincides with
the kernel of the projection $J^{r-j}(Q)\,\longrightarrow\,J^{r-j-1}(Q)$.
Therefore, from (3.5) it follows that

$H_{j}/H_{j-1}\,=\,Q\otimes K^{\otimes(r-j)}_{X}$ (3.7)

for all $1\,\leq\,j\,\leq\,r$.

Let

$X_{0}\,:=\,X\setminus\\{x_{1},\,\cdots,\,x_{d}\\}\,=\,X\setminus
S\,\subset\,X$ (3.8)

be the complement of $S$ in (2.9).

The triple $(V,\,{\mathcal{F}},\,D)$ in (3.1) defines a usual
$\text{SL}(r,{\mathbb{C}})$–oper over the open subset $X_{0}\,\subset\,X$ in
(3.8). The following proposition is well-known for the usual
$\text{SL}(r,{\mathbb{C}})$–opers.

######  Proposition 3.1.

The restriction $\Phi\big{|}_{X_{0}}$ of the homomorphism in (3.3) to $X_{0}$
is an isomorphism
$V\big{|}_{X_{0}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,J^{r-1}(Q)\big{|}_{X_{0}}$.

For all $0\,\leq\,j\,\leq\,r$, the homomorphism $\Phi\big{|}_{X_{0}}$ takes
the subbundle $F_{j}\big{|}_{X_{0}}$ (defined in (2.11)) isomorphically to the
subbundle $H_{j}\big{|}_{X_{0}}$ (constructed in (3.6)).

From Proposition 3.1 it follows that when $S$ is the zero divisor, then

$F_{r}\,=\,V\,=\,J^{r-1}(Q)\,.$

The following is a consequence of Proposition 3.1.

######  Corollary 3.2.

The homomorphism of ${\mathcal{O}}_{X}$–modules $\Phi$ in (3.3) is injective.
It satisfies the condition

$\Phi(F_{j})\,\subset\,H_{j}$

for all $0\,\leq\,j\,\leq\,r$, where $F_{j}$ and $H_{j}$ are defined in (2.11)
and (3.6) respectively.

###### Proof.

Since $\Phi\big{|}_{X_{0}}$ is injective it follows immediately that $\Phi$ is
injective. Since

$\Phi\big{|}_{X_{0}}(F_{j}\big{|}_{X_{0}})\,\subset\,H_{j}\big{|}_{X_{0}}$

(see Proposition 3.1), and $X_{0}$ is a dense subset of $X$, we conclude that
$\Phi(F_{j})\,\subset\,H_{j}$. ∎

From Corollary 3.2 it follows that $\Phi$ produces a grading preserving
holomorphic homomorphism from the graded vector bundle
$\bigoplus_{j=1}^{r}F_{j}/F_{j-1}$ in (2.11) to the graded vector bundle
$\bigoplus_{j=1}^{r}H_{j}/H_{j-1}$ in (3.6). For any $1\,\leq\,j\,\leq\,r$,
let

$\Phi_{j}\,:\,F_{j}/F_{j-1}\,\longrightarrow\,H_{j}/H_{j-1}\,=\,Q\otimes
K^{\otimes(r-j)}_{X}$ (3.9)

be the homomorphism induced by $\Phi$; see (3.7) for the isomorphism in (3.9).

For any $1\,\leq\,j\,\leq\,r$, note that $Q\otimes
K^{\otimes(r-j)}_{X}\otimes{\mathcal{O}}_{X}(-(r-j)S)$ is actually a subsheaf
of $Q\otimes K^{\otimes(r-j)}_{X}$ because $S$ is an effective divisor. The
following proposition identifies this subsheaf $Q\otimes
K^{\otimes(r-j)}_{X}\otimes{\mathcal{O}}_{X}(-(r-j)S)$ with the image of the
homomorphism $\Phi_{j}$.

######  Proposition 3.3.

For each $1\,\leq\,j\,\leq\,r$, the image of the homomorphism $\Phi_{j}$ in
(3.9) is the subsheaf

$Q\otimes
K^{\otimes(r-j)}_{X}\otimes{\mathcal{O}}_{X}(-(r-j)S)\,\subset\,Q\otimes
K^{\otimes(r-j)}_{X}\,=\,H_{j}/H_{j-1}\,.$

###### Proof.

Recall the homomorphisms ${\rm SF}(D,\,i)$ in statement (5) of Definition 2.1.
Take any $1\,\leq\,k\,\leq\,r-1$. Consider the composition of homomorphisms

$({\rm SF}(D,\,r-1)\otimes{\rm
Id}_{K^{\otimes(r-1-k)}_{X}})\circ\cdots\circ({\rm SF}(D,\,k+1)\otimes{\rm
Id}_{K_{X}})\circ{\rm SF}(D,\,k)\,:\,F_{k}/F_{k-1}$
$\longrightarrow\,(F_{r}/F_{r-1})\otimes K^{\otimes(r-k)}_{X}\,=\,Q\otimes
K^{\otimes(r-k)}_{X}\,=\,(F_{k}/F_{k-1})\otimes{\mathcal{O}}_{X}((r-k)S)\,;$
(3.10)

note that since $(F_{i+1}/F_{i})\otimes
K_{X}\,=\,(F_{i}/F_{i-1})\otimes{\mathcal{O}}_{X}(S)$ for all
$1\,\leq\,i\,\leq\,r-1$ (see Remark 2.2) if follows that
$(F_{r}/F_{r-1})\otimes
K^{\otimes(r-k)}_{X}\,=\,(F_{k}/F_{k-1})\otimes{\mathcal{O}}_{X}((r-k)S)$.
From statement (5) in Definition 2.1 we know that the composition of
homomorphisms in (3.10) coincides with the natural inclusion map

$F_{k}/F_{k-1}\,\hookrightarrow\,(F_{k}/F_{k-1})\otimes{\mathcal{O}}_{X}((r-k)S)\,.$

In other words, if $\textbf{1}\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}((r-k)S))$ is
the section given by the constant function $1$ on $X$, then the composition of
homomorphisms in (3.10) coincides with the homomorphism

${\rm
Id}_{F_{k}/F_{k-1}}\otimes\textbf{1}\,:\,F_{k}/F_{k-1}\,\longrightarrow\,(F_{k}/F_{k-1})\otimes{\mathcal{O}}_{X}((r-k)S)\,.$

On the other hand, we have $Q\otimes K^{\otimes(r-k)}_{X}\,=\,H_{k}/H_{k-1}$
(see (3.7)). Therefore, the image of the composition of homomorphisms in
(3.10) is the subsheaf

$(H_{k}/H_{k-1})\otimes{\mathcal{O}}_{X}(-(r-k)S)\,=\,Q\otimes
K^{\otimes(r-k)}_{X}\otimes{\mathcal{O}}_{X}(-(r-k)S)\,\subset\,Q\otimes
K^{\otimes(r-k)}_{X}\,.$

This completes the proof. ∎

For $0\,\leq\,j\,\leq\,r$, define

$\widehat{H}_{j}\,:=\,H_{j}\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\subset\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,,$

where $H_{j}$ is constructed in (3.6). So the filtration
$\\{H_{j}\\}_{j=0}^{r}$ of $J^{r-1}(Q)$ produces a filtration of holomorphic
subbundles

$0\,=\,\widehat{H}_{0}\,\subset\,\widehat{H}_{1}\,\subset\,\cdots\,\subset\,\widehat{H}_{r-1}\,\subset\,\widehat{H}_{r}\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$
(3.11)

of $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$.

The next result is deduced using Proposition 3.3 and Corollary 3.2.

######  Corollary 3.4.

The natural inclusion map
$I\,:\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\hookrightarrow\,J^{r-1}(Q)$
factors through $\Phi$ in (3.3), meaning there is a unique homomorphism

${\mathbf{i}}\,:\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\longrightarrow\,V$

such that $I\,=\,\Phi\circ{\mathbf{i}}$. For all $0\,\leq\,j\,\leq\,r$,

${\mathbf{i}}(\widehat{H}_{j})\,=\,F_{j}\,,$

where $\widehat{H}_{j}$ and $F_{j}$ are as in (3.11) and (2.11), and moreover,
$\Phi(F_{j})\,=\,H_{j}$.

###### Proof.

By Proposition 3.3, the image of the injective homomorphism $\Phi_{j}$ in
(3.9) is the subsheaf

$Q\otimes
K^{\otimes(r-j)}_{X}\otimes{\mathcal{O}}_{X}(-(r-j)S)\,\subset\,Q\otimes
K^{\otimes(r-j)}_{X}\,=\,H_{j}/H_{j-1}\,.$

Consequently, we have the following inclusions:

$\widehat{H}_{j}/\widehat{H}_{j-1}\,=\,(H_{j}/H_{j-1})\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\hookrightarrow\,F_{j}/F_{j-1}$
(3.12)
$=\,(H_{j}/H_{j-1})\otimes{\mathcal{O}}_{X}(-(r-j)S)\,\hookrightarrow\,H_{j}/H_{j-1}\,,$

where the isomorphism
$F_{j}/F_{j-1}\,=\,(H_{j}/H_{j-1})\otimes{\mathcal{O}}_{X}(-(r-j)S)$ is given
by $\Phi_{j}$. In view of Corollary 3.2, from these inclusion maps we conclude
that the homomorphism $\Phi$ in (3.3) satisfies the following:

$\widehat{H}_{r}\,:=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\stackrel{{\scriptstyle{\mathbf{i}}^{\prime}}}{{\hookrightarrow}}\,\Phi(V)\,\subset\,J^{r-1}(Q)\,.$
(3.13)

Since the coherent analytic sheaf $V$ is identified with the coherent analytic
sheaf $\Phi(V)$ using the isomorphism $\Phi$ between them. the homomorphism
${\mathbf{i}}^{\prime}$ produces the homomorphism $\mathbf{i}$ in the
statement of the corollary, in other words, ${\mathbf{i}}$ is defined by the
equality ${\mathbf{i}}^{\prime}\,=\,\Phi\circ{\mathbf{i}}$.

The map ${\mathbf{i}}^{\prime}$ in (3.13) evidently takes $\widehat{H}_{j}$ in
(3.11) into $\Phi(F_{j})$. Also, the inclusion map
$\Phi(V)\,\hookrightarrow\,J^{r-1}(Q)$ in (3.13) takes $\Phi(F_{j})$ into the
subbundle $H_{j}\,\subset\,J^{r-1}(Q)$ in (3.6). ∎

In view of Corollary 3.4, the filtration of holomorphic subbundles in (3.11)
fits in the following filtration of coherent analytic subsheaves:

$0\,=\,\widehat{H}_{0}\,\subset\,\widehat{H}_{1}\,\subset\,\cdots\,\subset\,\widehat{H}_{r-1}\,\subset\,\widehat{H}_{r}\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\subset\,V\,.$
(3.14)

###  3.2. A logarithmic connection

Take a branched ${\rm SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$ as
in (3.1). The following proposition shows that the holomorphic connection $D$
produces a logarithmic connection on the holomorphic vector bundle
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$.

######  Proposition 3.5.

The holomorphic connection $D\,:\,V\,\longrightarrow\,V\otimes K_{X}$ in (3.1)
sends the subsheaf
$\widehat{H}_{r}\,:=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\subset\,V$
in (3.14) to the subsheaf

$\big{(}J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-2)S)\otimes
K_{X}\big{)}\bigcap(V\otimes K_{X})\,\subset\,V\otimes K_{X}\,.$

In other words, $D$ produces a logarithmic connection, singular over $S$, on
the holomorphic vector bundle $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$.

###### Proof.

Recall that
$V\big{|}_{X_{0}}\,=\,J^{r-1}(Q)\big{|}_{X_{0}}\,=\,(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-2)S))\big{|}_{X_{0}}$,
where $X_{0}$ is the open subset in (3.8). So we need to investigate the
connection $D$ only around the points of $S$. Take any point

$x^{\prime}\,\in\,S\,.$

Fix a holomorphic splitting of the filtration $\\{F_{i}\\}_{i=0}^{r}$ of $V$
in (2.11) over a sufficiently small analytic neighborhood $U$ of $x^{\prime}$;
this subset $U$ is chosen such that $U\bigcap S\,=\,x^{\prime}$. So we have a
holomorphic isomorphism

$V\big{|}_{U}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\bigoplus_{i=1}^{r}(F_{i}/F_{i-1})\big{|}_{U}\,=:\,\bigoplus_{i=1}^{r}{\mathcal{F}}_{i}$
(3.15)

using the notation ${\mathcal{F}}_{i}\,:=\,(F_{i}/F_{i-1})\big{|}_{U}$; it
should be clarified that there is no natural isomorphism. We would express the
holomorphic connection $D\big{|}_{U}$ in (3.1) in terms of the decomposition
in (3.15). From the fifth condition in Definition 2.1 it follows that
$D\big{|}_{U}$ has the following expression in terms of the decomposition in
(3.15):

$D\big{|}_{U}\,=\,\begin{pmatrix}D_{1}&\alpha_{1,2}&\alpha_{1,3}&\alpha_{1,4}&\cdots&\alpha_{1,r-2}&\alpha_{1,r-1}&\alpha_{1,r}\\\
\gamma_{1}&D_{2}&\alpha_{2,3}&\alpha_{2,4}&\cdots&\alpha_{2,r-2}&\alpha_{2,r-1}&\alpha_{2,r}\\\
0&\gamma_{2}&D_{3}&\alpha_{3,4}&\cdots&\alpha_{3,r-2}&\alpha_{3,r-1}&\alpha_{3,r}\\\
0&0&\gamma_{3}&D_{4}&\cdots&\alpha_{4,r-2}&\alpha_{4,r-1}&\alpha_{4,r}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&\cdots&D_{r-2}&\alpha_{r-2,r-1}&\alpha_{r-2,r}\\\
0&0&0&0&\cdots&\gamma_{r-2}&D_{r-1}&\alpha_{r-1,r}\\\
0&0&0&0&\cdots&0&\gamma_{r-1}&D_{r}\\\ \end{pmatrix}$ (3.16)

where $D_{i}$ (the entry at the $i\times i$-th position of the matrix) is a
holomorphic connection on ${\mathcal{F}}_{i}$, and $\gamma_{i}$ (the entry at
the $(i+1)\times i$-th position) is a section

$\gamma_{i}\,\in\,H^{0}(U,\,\text{Hom}({\mathcal{F}}_{i},\,{\mathcal{F}}_{i+1})\otimes(K_{X}\big{|}_{U})\otimes{\mathcal{O}}_{U}(-x^{\prime}))\,;$
(3.17)

in other words, $\gamma_{i}$ is a holomorphic homomorphism
${\mathcal{F}}_{i}\,\longrightarrow\,{\mathcal{F}}_{i+1}\otimes(K_{X}\big{|}_{U})$
that vanishes at the point $x^{\prime}\,\in\,U$. For any $j\,>\,i$, we have

$\alpha_{i,j}\,\in\,H^{0}(U,\,\text{Hom}({\mathcal{F}}_{j},\,{\mathcal{F}}_{i})\otimes(K_{X}\big{|}_{U}))\,.$

The entry at the $i\times j$-th position of the matrix in (3.16) is zero if
$i\,>\,j+1$. It may be mentioned that by choosing the splitting in (3.15)
carefully it is possible to make $\alpha_{i,j}\,=\,0$ for all $j\,>\,i$; but
this will not be needed here.

The decomposition of $V\big{|}_{U}$ in (3.15) produces a holomorphic
decomposition of the vector bundle
$(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))\big{|}_{U}$. To see this,
consider the intersection of coherent analytic subsheaves of $V\big{|}_{U}$

${\mathcal{G}}_{i}\,:=\,{\mathcal{F}}_{i}\bigcap\big{(}J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\big{)}\big{|}_{U}\,=\,{\mathcal{F}}_{i}\bigcap\big{(}J^{r-1}(Q\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(r-1)x^{\prime})\big{)}\,\subset\,V\big{|}_{U}\,;$
(3.18)

note that both ${\mathcal{F}}_{i}$ and
$J^{r-1}(Q\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(r-1)x^{\prime})$ are
subsheaves of $V\big{|}_{U}$ (see Corollary 3.4 and (3.15)). Then we have a
holomorphic decomposition of
$J^{r-1}(Q\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(r-1)x^{\prime})$

$J^{r-1}(Q\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(r-1)x^{\prime})\,=\,\bigoplus_{i=1}^{r}{\mathcal{G}}_{i}$
(3.19)

into a direct sum of holomorphic line bundles on $U$. Indeed, the natural
homomorphism of coherent analytic sheaves

$\bigoplus_{i=1}^{r}{\mathcal{G}}_{i}\,\longrightarrow\,J^{r-1}(Q\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(r-1)x^{\prime})$

is clearly surjective; it is also injective because it is injective over the
open subset $U\setminus\\{x^{\prime}\\}\,\subset\,U$ and the coherent analytic
sheaf $\bigoplus_{i=1}^{r}{\mathcal{G}}_{i}$ is torsionfree. From (3.12) it
follows immediately that

  * •

${\mathcal{G}}_{i}\,=\,{\mathcal{F}}_{i}\otimes{\mathcal{O}}_{U}(-(i-1)x^{\prime})$,
and

  * •

the inclusion map ${\mathcal{G}}_{i}\,\hookrightarrow\,{\mathcal{F}}_{i}$ (see
(3.18)) coincides with the natural inclusion map

${\mathcal{F}}_{i}\otimes{\mathcal{O}}_{U}(-(i-1)x^{\prime}))\,\hookrightarrow\,{\mathcal{F}}_{i}\,.$

Now it is straight-forward to check that the connection operator
$D\big{|}_{U}$ in (3.16) on $\bigoplus_{i=1}^{r}{\mathcal{F}}_{i}$ produces a
holomorphic differential operator

$\bigoplus_{i=1}^{r}{\mathcal{G}}_{i}\,\longrightarrow\,\left(\bigoplus_{i=1}^{r}{\mathcal{G}}_{i}\right)\otimes(K_{X}\big{|}_{U})\otimes{\mathcal{O}}_{U}(x^{\prime})\,.$

To see this, first note that the section $\gamma_{i}$ in (3.16) produces a
holomorphic homomorphism

${\mathcal{G}}_{i}\,\longrightarrow\,{\mathcal{G}}_{i+1}\otimes(K_{X}\big{|}_{U})$

because the homomorphism
$\gamma_{i}\in\,H^{0}(U,\,\text{Hom}({\mathcal{F}}_{i},\,{\mathcal{F}}_{i+1})\otimes(K_{X}\big{|}_{U}))$
in (3.17) vanishes at the point $x^{\prime}$. Secondly, for any $j\,>\,i$, the
section $\alpha_{i,j}$ in (3.16) produces a section

$\widetilde{\alpha}_{i,j}\,\in\,H^{0}(U,\,\text{Hom}({\mathcal{G}}_{j},\,{\mathcal{G}}_{i})\otimes(K_{X}\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(j-i)x^{\prime}))\,,$
(3.20)

in particular,
$\widetilde{\alpha}_{i,j}\,\in\,H^{0}(U,\,\text{Hom}({\mathcal{G}}_{j},\,{\mathcal{G}}_{i})\otimes(K_{X}\big{|}_{U}))$.
Thirdly, the connection operator

$D_{i}\,:\,{\mathcal{F}}_{i}\,\longrightarrow\,{\mathcal{F}}_{i}\otimes(K_{X}\big{|}_{U})$

in (3.16) produces a first order holomorphic differential operator

${\mathcal{G}}_{i}\,\longrightarrow\,{\mathcal{G}}_{i}\otimes(K_{X}\big{|}_{U})\otimes{\mathcal{O}}_{U}(x^{\prime})$

that satisfies the Leibniz identity, because $D_{i}$ itself satisfies the
Leibniz identity.

Consequently, $D$ sends the subsheaf
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\subset\,V$ in (3.14) to the
subsheaf

$\big{(}J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-2)S)\otimes
K_{X}\big{)}\bigcap(V\otimes K_{X})\,\subset\,V\otimes K_{X}\,.$

Since

$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-2)S)\otimes
K_{X}\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,,$

this implies that $D$ produces a logarithmic connection on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$. ∎

######  Remark 3.6.

Consider the holomorphic connection $D_{i}$ in (3.16) on the holomorphic
vector bundle ${\mathcal{F}}_{i}$ on $U$. The connection on
${\mathcal{G}}_{i}\,:=\,{\mathcal{F}}_{i}\bigcap(J^{r-1}(Q\big{|}_{U})\otimes{\mathcal{O}}_{U}(-(r-1)x^{\prime}))$
induced by $D_{i}$ is singular at $x^{\prime}$ if $i\,\geq\,2$. Therefore, the
singular locus of the logarithmic connections on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ constructed in Proposition 3.5
is exactly $S$.

Take any branched $\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$
as in (3.1). Let $\mathcal{D}$ denote the logarithmic connection on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ constructed in Proposition 3.5
from $(V,\,{\mathcal{F}},\,D)$. The following is a straight-forward
consequence of the construction of $\mathcal{D}$.

######  Corollary 3.7.

  1. (1)

The logarithmic connection $\mathcal{D}$ on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ constructed in Proposition 3.5
satisfies the condition

${\mathcal{D}}(\widehat{H}_{i})\,\subset\,\widehat{H}_{i+1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

for all $1\,\leq\,i\,\leq\,r-1$, where $\\{\widehat{H}_{j}\\}_{j=0}^{r}$ is
the filtration of $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ in (3.11).

  2. (2)

The second fundamental form of the subbundle

$\widehat{H}_{i}\big{|}_{X_{0}}\,\subset\,(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))\big{|}_{X_{0}}$

for the holomorphic connection ${\mathcal{D}}\big{|}_{X_{0}}$ is nowhere zero
on $X_{0}$ for all $1\,\leq\,i\,\leq\,r-1$, where $X_{0}$ is the open subset
in (3.8).

###### Proof.

Over the open subset $X_{0}$ in (3.8), we have

$J^{r-1}(Q)\big{|}_{X_{0}}\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\big{|}_{X_{0}}\,=\,V\big{|}_{X_{0}}$

(see Proposition 3.1). This isomorphism takes
$F_{j}\big{|}_{X_{0}}\,\subset\,V\big{|}_{X_{0}}$ isomorphically to
$H_{j}\big{|}_{X_{0}}\,=\,\widehat{H}_{j}\big{|}_{X_{0}}$ (see Proposition
3.1). It also takes the holomorphic connection $D\big{|}_{X_{0}}$ to
${\mathcal{D}}\big{|}_{X_{0}}$; this is an immediate consequence of the
construction of ${\mathcal{D}}$. Since $X_{0}$ is dense in $X$, the first
statement now follows from the facts that $D(F_{i})\,\subset\,F_{i+1}\otimes
K_{X}$ for all $1\,\leq\,i\,\leq\,r-1$ (see the fourth statement in Definition
2.1), while the second statement follows from the fact that the second
fundamental form of the subbundle $F_{i}\,\subset\,V$, for the holomorphic
connection $D$, is nowhere zero on $X_{0}$ (see the fifth statement in
Definition 2.1). ∎

For any $1\,\leq\,i\,\leq\,r-1$, let

${\rm
SF}(\mathcal{D},\,\widehat{H}_{i})\,\in\,H^{0}(X,\,\text{Hom}(\widehat{H}_{i},\,\widehat{H}_{r}/\widehat{H}_{i})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))$

be the second fundamental form of the subbundle $\widehat{H}_{i}$ in (3.11)
for the logarithmic connection $\mathcal{D}$ on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,=\,\widehat{H}_{r}$ constructed
in Proposition 3.5; see (2.3) for the second fundamental form. From Corollary
3.7(1) we know that ${\rm SF}(\mathcal{D},\,\widehat{H}_{i})$ lies in the
image of the natural inclusion map

$H^{0}(X,\,\text{Hom}(\widehat{H}_{i},\,\widehat{H}_{i+1}/\widehat{H}_{i})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))\,\hookrightarrow\,H^{0}(X,\,\text{Hom}(\widehat{H}_{i},\,\widehat{H}_{r}/\widehat{H}_{i})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))$

given by the inclusion map
$\widehat{H}_{i+1}\,\hookrightarrow\,\widehat{H}_{r}$. So we get a
homomorphism

${\rm
SF}(\mathcal{D},\,i)\,\in\,H^{0}(X,\,\text{Hom}(\widehat{H}_{i}/\widehat{H}_{i-1},\,\widehat{H}_{r}/\widehat{H}_{i})\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))$ (3.21)

for every $1\,\leq\,i\,\leq\,r-1$ given by these ${\rm
SF}(\mathcal{D},\,\widehat{H}_{i})$.

For the filtration of $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$, from
(3.12) and (3.7) it follows immediately that

$\widehat{H}_{j}/\widehat{H}_{j-1}\,=\,Q\otimes
K^{\otimes(r-j)}_{X}\otimes{\mathcal{O}}_{X}(-(r-1)S)$ (3.22)

for all $1\,\leq\,j\,\leq\,r$. Therefore, ${\rm SF}(\mathcal{D},\,i)$ in
(3.21) is a holomorphic section

${\rm SF}(\mathcal{D},\,i)\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}(S))$ (3.23)

for every $1\,\leq\,i\,\leq\,r-1$.

The following lemma is a refinement of Corollary 3.7(2).

######  Lemma 3.8.

The holomorphic section ${\rm SF}(\mathcal{D},\,i)$ in (3.23) coincides with
the section of ${\mathcal{O}}_{X}(S)$ given by the constant function $1$ on
$X$.

###### Proof.

In the proof of Corollary 3.7 it was observed that

$(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S),\,\\{\widehat{H}_{i}\\}_{i=1}^{r},\,{\mathcal{D}})\big{|}_{X_{0}}\,=\,(V,\,\\{F_{i}\\}_{i=1}^{r},\,D)\big{|}_{X_{0}}\,,$

where $X_{0}$ the open subset $X_{0}$ in (3.8). Therefore, the lemma follows
from the fifth statement in Definition 2.1 and Remark 2.2. ∎

Using (3.5) it is deduced that

$\det J^{r-1}(Q)\,:=\,\bigwedge\nolimits^{r}J^{r-1}(Q)\,=\,Q^{\otimes
r}\otimes K^{\otimes r(r-1)/2}_{X}\,.$

Therefore, from (3.2) it follows that

$\det J^{r-1}(Q)\,=\,F^{\otimes r}_{1}\otimes(TX)^{\otimes
r(r-1)/2}\otimes{\mathcal{O}}_{X}(r(r-1)S)\,.$

Now the expression of $F_{1}$ in the first statement in Definition 2.1 gives
that

$\det J^{r-1}(Q)\,=\,{\mathcal{O}}_{X}\left(\frac{r(r-1)}{2}S\right)\,.$

Hence we have the following:

$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))\,=\,{\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)\,.$
(3.24)

Take any branched $\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$
as in (3.1). Let $\mathcal{D}$ denote the logarithmic connection on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ constructed in Proposition 3.5
from $(V,\,{\mathcal{F}},\,D)$.

######  Lemma 3.9.

The logarithmic connection on
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ induced by the logarithmic
connection $\mathcal{D}$ on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$
coincides with the logarithmic connection on
${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ given by the de Rham
differential $d$ (it sends $f$ to $df$, in particular, constant functions are
covariant constant), once $\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$
is identified with ${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ using
(3.24).

###### Proof.

Consider the homomorphism $\mathbf{i}$ in Corollary 3.4. Let

$\wedge^{r}{\mathbf{i}}\,:\,\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))\,=\,{\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)\,\longrightarrow\,\det
V\,=\,{\mathcal{O}}_{X}$

be the corresponding homomorphism of the top exterior products. This
homomorphism $\bigwedge^{r}{\mathbf{i}}$ evidently coincides with the natural
inclusion map of ${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ into
${\mathcal{O}}_{X}$. Now, since $D$ induces the connection on $\det V$ given
by the de Rham differential (see the third statement in Definition 2.1), and
$D$ restricts to $\mathcal{D}$ on
${\mathbf{i}}(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$, the lemma
follows. ∎

In Section 6 we will determine all the logarithmic connections on the rank $r$
vector bundle $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ that arise from
branched $\text{SL}(r,{\mathbb{C}})$–opers on $X$ with branching over $S$.

##  4\. Residues

Take a point $y\,\in\,S$. The fiber of $K_{X}\otimes{\mathcal{O}}_{X}(S)$ over
$y$ is identified with $\mathbb{C}$ by the Poincaré adjunction formula [19, p.
146]. To explain this isomorphism

$(K_{X}\otimes{\mathcal{O}}_{X}(S))_{y}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,{\mathbb{C}}\,,$
(4.1)

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

Let $D_{W}\,:\,W\,\longrightarrow\,W\otimes K_{X}\otimes{\mathcal{O}}_{X}(S)$
be a logarithmic connection on a holomorphic vector bundle $W$ on $X$.
Consider the composition of homomorphisms

$W\,\xrightarrow{\,\ D_{W}\,\ }\,W\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)\,\longrightarrow\,(W\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S))_{y}\,=\,W_{y}\,.$

This composition of homomorphisms is evidently ${\mathcal{O}}_{X}$–linear and
therefore is given by an endomorphism

${\rm Res}(D_{W},\,y)\,:\,W_{y}\,\longrightarrow\,W_{y}\,;$

called the residue of $D_{W}$ at $y$.

If $r\,=\,\text{rank}(W)$ and $\lambda_{1},\,\cdots,\,\lambda_{r}$ are the
generalized eigenvalues, with multiplicity, of ${\rm Res}(D_{W},\,y)$, then
the generalized eigenvalues of the local monodromy of $D_{W}$ around $y$ are

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

see [9].

For convenience we use the notation

$E\,:=\,\widehat{H}_{r}\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ (4.2)

(see (3.11)).

Take a branched $\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$ as
in (3.1). As before, let $\mathcal{D}$ denote the logarithmic connection on
$E$ (see (4.2)) constructed in Proposition 3.5 from $(V,\,{\mathcal{F}},\,D)$.
Recall from Remark 3.6 that the singular locus of $\mathcal{D}$ is exactly
$S$. The following lemma describes the residues of $\mathcal{D}$.

######  Lemma 4.1.

For any point $x^{\prime}\,\in\,S$, let

${\rm Res}({\mathcal{D}},\,x^{\prime})\,\in\,{\rm End}(E_{x^{\prime}})$

be the residue of ${\mathcal{D}}$ at $x^{\prime}$. Then the eigenvalues of
${\rm Res}({\mathcal{D}},\,x^{\prime})$ are the integers
$\\{0,\,1,\,\cdots,\,r-2,\,r-1\\}$, and the multiplicity of each of them is
one. For any $0\,\leq\,i\,\leq\,r-1$, the eigenspace of ${\rm
Res}({\mathcal{D}},\,x^{\prime})$ for the eigenvalue $i$ is contained in the
subspace

$(\widehat{H}_{i+1})_{x^{\prime}}\,\subset\,E_{x^{\prime}}$

(see (3.11) and (4.2)).

###### Proof.

Let $L$ be a holomorphic line bundle on the open neighborhood $U\,\subset\,X$
of $x^{\prime}$ (as in (3.15)) equipped with a holomorphic connection $D_{L}$.
Then for any integer $k$, the differential operator
$D_{L}\big{|}_{U\setminus\\{x^{\prime}\\}}$ on
$L\big{|}_{U\setminus\\{x^{\prime}\\}}\,=\,(L\otimes{\mathcal{O}}_{U}(kx^{\prime}))\big{|}_{U\setminus\\{x^{\prime}\\}}$
extends to a logarithmic connection on
$L\otimes{\mathcal{O}}_{U}(kx^{\prime})$ over $U$. In fact it coincides with
the logarithmic connection on $L\otimes{\mathcal{O}}_{U}(kx^{\prime})$ given
by the holomorphic connection $D_{L}$ on $L$ and the logarithmic connection on
${\mathcal{O}}_{U}(kx^{\prime})$ defined by the de Rham differential $d$. The
residue, at $x^{\prime}$, of this logarithmic connection on
$L\otimes{\mathcal{O}}_{U}(kx^{\prime})$ is $-k$; indeed, this follows
immediately from the fact that the residue of the logarithmic connection on
${\mathcal{O}}_{U}(kx^{\prime})$ defined by the de Rham differential $d$ is
$-k$. In view of this, the lemma follows from the expression of $D\big{|}_{U}$
in (3.16) in terms of the direct sum of line bundles ${\mathcal{G}}_{i}$ in
(3.19). Recall the observation in the proof of Proposition 3.5 that the
section $\gamma_{i}$ in (3.17) produces a holomorphic homomorphism

${\mathcal{G}}_{i}\,\longrightarrow\,{\mathcal{G}}_{i+1}\otimes(K_{X}\big{|}_{U}).$

Therefore, $\gamma_{i}$ does not contribute to the residue ${\rm
Res}({\mathcal{D}},\,x^{\prime})$. Since $\widetilde{\alpha}_{i,j}$ in (3.20)
does not have any pole as a homomorphism from ${\mathcal{G}}_{j}$ to
${\mathcal{G}}_{i}\otimes(K_{X}\big{|}_{U})$, it also does not contribute to
the residue ${\rm Res}({\mathcal{D}},\,x^{\prime})$.

Therefore, the residue of ${\mathcal{D}}$ at $x^{\prime}$ is given by the
residues on the logarithmic connections on ${\mathcal{G}}_{j}$,
$1\,\leq\,j\,\leq\,r$, induced by the holomorphic connections $D_{j}$ on
${\mathcal{F}}_{j}$ in (3.16). From the above observation on the residue of
the logarithmic connection on $L\otimes{\mathcal{O}}_{U}(kx^{\prime})$ we know
that the residues on this logarithmic connection ${\mathcal{G}}_{j}$ is $j-1$.
This proves the lemma. ∎

We will end this section on residues by noting some general properties of it.

Let $W\,\longrightarrow\,X$ be a holomorphic vector bundle, and let $D_{W}$ be
a logarithmic connection on $W$, singular at $y\,\in\,X$. Assume that the
residue ${\rm Res}(D_{W},\,y)$ of $D_{W}$ at $y$ is semisimple, meaning ${\rm
Res}(D_{W},\,y)$ is diagonalizable. Let $\lambda_{1},\,\cdots,\,\lambda_{b}$
be the eigenvalues of ${\rm Res}(D_{W},\,y)$ (they need not be of multiplicity
one). For any $1\,\leq\,i\,\leq\,b$, let

$W^{i}_{y}\,\subset\,W_{y}$

be the eigenspace of ${\rm Res}(D_{W},\,y)\,\in\,{\rm End}(W_{y})$ for the
eigenvalue $\lambda_{i}$. For any given $1\,\leq\,k\,<\,b$, consider the
following natural homomorphisms

$W\,\longrightarrow\,W_{y}\,\longrightarrow\,W_{y}\Big{/}\left(\bigoplus_{i=1}^{k}W^{i}_{y}\right)\,;$

both are natural quotient maps. The kernel of this composition of
homomorphisms will be denoted by $\widetilde{W}$. So $\widetilde{W}$ is a
torsion-free coherent analytic sheaf on $X$ that fits in the following short
exact sequence of coherent analytic sheaves on $X$:

$0\,\longrightarrow\,\widetilde{W}\,\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\,W\,\stackrel{{\scriptstyle
q_{1}}}{{\longrightarrow}}\,W_{y}\Big{/}\left(\bigoplus_{i=1}^{k}W^{i}_{y}\right)\,\longrightarrow\,0\,.$
(4.3)

The following is straight-forward to check.

######  Lemma 4.2.

The logarithmic connection $D_{W}\,:\,W\,\longrightarrow\,W\otimes
K_{X}\otimes{\mathcal{O}}_{X}(y)$ sends the subsheaf
$\widetilde{W}\,\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}\,W$ to
$\widetilde{W}\otimes K_{X}\otimes{\mathcal{O}}_{X}(y)$. Hence $D_{W}$ induces
a logarithmic connection on $\widetilde{W}$.

The logarithmic connection on $\widetilde{W}$ induced by $D_{W}$ will be
denoted by $\widetilde{D}$.

We will now describe the residue of $\widetilde{D}$ at the singular point $y$.

Let

$0\,\longrightarrow\,\text{kernel}(\phi(y))\,\longrightarrow\,\widetilde{W}_{y}\,\xrightarrow{\
\phi(y)\ }\,W_{y}\,\stackrel{{\scriptstyle
q_{1}}}{{\longrightarrow}}\,\text{cokernel}(\phi(y))\,=\,W_{y}\Big{/}\left(\bigoplus_{i=1}^{k}W^{i}_{y}\right)\,\longrightarrow\,0$
(4.4)

be the exact sequence of vector spaces obtained by restricting, to the point
$y$, the short exact sequence of coherent analytic sheaves in (4.3). Note that
the homomorphism of fibers of vector bundles corresponding to an injective
homomorphism of coherent analytic sheaves need not be injective, so
$\text{kernel}(\phi(y))$ may be nonzero.

We will now show that there is a canonical isomorphism

$\left(\bigoplus_{i=k+1}^{b}W^{i}_{y}\right)\otimes(K_{X})_{y}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\text{kernel}(\phi(y))\,.$
(4.5)

To prove (4.5), take any
$w\,\in\,\left(\bigoplus_{i=k+1}^{b}W^{i}_{y}\right)\otimes(K_{X})_{y}$. Using
the isomorphism $(K_{X})_{y}\,=\,{\mathcal{O}}_{Y}(-y)_{y}$ (see (4.1)), we
have

$w\,\in\,\left(\bigoplus_{i=k+1}^{b}W^{i}_{y}\right)\otimes{\mathcal{O}}_{Y}(-y)_{y}\,=\,\bigoplus_{i=k+1}^{b}(W^{i}\otimes{\mathcal{O}}_{Y}(-y))_{y}\,.$

Now take a holomorphic section defined on an analytic open neighborhood
$U\,\subset\,X$

$s\,\in\,H^{0}(U,\,(W\big{|}_{U})\otimes{\mathcal{O}}_{U}(-y))$

such that $s(y)\,=\,w$. We note that $q_{1}(s)\,=\,0$, where $q_{1}$ is the
projection in (4.3). Therefore, from (4.3) it follows that $s$ is the image of
a holomorphic section of $\widetilde{W}\big{|}_{U}$ under the homomorphism
$\phi$ in (4.3). Let

$\widetilde{s}\,\in\,H^{0}(U,\,\widetilde{W}\big{|}_{U})$

be the unique holomorphic section such that $\phi(\widetilde{s})\,=\,s$. It
can be shown that the evaluation

$\widetilde{s}(y)\,\in\,\widetilde{W}_{y}$

is independent of the choice of the above section
$s\,\in\,H^{0}(U,\,(W\big{|}_{U})\otimes{\mathcal{O}}_{U}(-y))$ satisfying
$s(y)\,=\,w$. Indeed, for another holomorphic section

$t\,\in\,H^{0}(U,\,(W\big{|}_{U})\otimes{\mathcal{O}}_{U}(-y))$

with $t(y)\,=\,w$, we have

$s-t\,\in\,H^{0}(U,\,(W\big{|}_{U})\otimes{\mathcal{O}}_{U}(-2y))\,,$

and hence

$\widetilde{s}-\widetilde{t}\,\in\,H^{0}(U,\,(\widetilde{W}\big{|}_{U})\otimes{\mathcal{O}}_{U}(-y))\,,$
(4.6)

where $\widetilde{t}\,\in\,H^{0}(U,\,\widetilde{W}\big{|}_{U})$ is the unique
section for which $\phi(\widetilde{t})\,=\,t$. From (4.6) it follows
immediately that $\widetilde{s}(y)\,=\,\widetilde{t}(y)$, and hence
$\widetilde{s}(y)\,\in\,\widetilde{W}_{y}$ is independent of the choice of the
section $s\,\in\,H^{0}(U,\,(W\big{|}_{U})\otimes{\mathcal{O}}_{U}(-y))$
satisfying $s(y)\,=\,w$.

For the homomorphism $\phi(y)$ in (4.4) we have

$\phi(y)(\widetilde{s}(y))\,=\,0\,,$

because
$\phi(\widetilde{s})\,=\,s\,\in\,H^{0}(U,\,(W\big{|}_{U})\otimes{\mathcal{O}}_{U}(-y))$.
Therefore, from (4.4) we conclude that

$\widetilde{s}(y)\,\in\,\text{kernel}(\phi(y))\,.$

The isomorphism in (4.5) sends any

$w\,\in\,\left(\bigoplus_{i=k+1}^{b}W^{i}_{y}\right)\otimes(K_{X})_{y}$

to $\widetilde{s}(y)\,\in\,\text{kernel}(\phi(y))$ constructed above from it.

The following lemma is a straight-forward consequence of the construction of
residue of a logarithmic connection.

######  Lemma 4.3.

The residue ${\rm Res}(\widetilde{D},\,y)$ of the logarithmic connection
$\widetilde{D}$ on $\widetilde{W}$ (see Lemma 4.2) has the following
properties:

  1. (1)

The eigenvalues of ${\rm Res}(\widetilde{D},\,y)$ are
$\\{\lambda_{i}\\}_{i=1}^{k}\bigcup\\{\lambda_{i}+1\\}_{i=k+1}^{b}$.

  2. (2)

For any $k+1\,\leq\,i\,\leq\,b$, the eigenspace of ${\rm
Res}(\widetilde{D},\,y)$ for the eigenvalue $\lambda_{i}+1$ is the subspace
$W^{i}_{y}\otimes(K_{X})_{y}\,\subset\,\widetilde{W}_{y}$ (see (4.5) and
(4.4)).

  3. (3)

For any $1\,\leq\,i\,\leq\,k$, the eigenspace of ${\rm
Res}(\widetilde{D},\,y)$ for the eigenvalue $\lambda_{i}$ is taken
isomorphically to the eigenspace $W^{i}_{y}$ by the homomorphism $\phi(y)$ in
(4.4).

##  5\. Local monodromy

A logarithmic connection has local monodromy around a singular point of the
connection. The two holomorphic vector bundles $V$ and

$E\,:=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ (5.1)

are identified over $X_{0}$ (3.8) (see Corollary 3.4), and this identification
takes the holomorphic connection ${\mathcal{D}}\big{|}_{X_{0}}$ on
$E\big{|}_{X_{0}}$ in Lemma 4.1 to the holomorphic connection
$D\big{|}_{X_{0}}$ on $V\big{|}_{X_{0}}$ in (3.1). Now $D\big{|}_{X_{0}}$ does
not have local monodromy around any point $x^{\prime}\,\in\,S$ because $D$ is
a holomorphic connection on $V$. Hence the logarithmic connection
${\mathcal{D}}$ has trivial local monodromy around every point of $S$, as
well. In this section we will reformulate this condition of vanishing of local
monodromies.

Let

${\mathbb{D}}\,:\,E\,\longrightarrow\,E\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ (5.2)

be a logarithmic connection on $E$ (see (5.1)) singular over $S$ that
satisfies the following four conditions:

  1. (1)

${\mathbb{D}}(\widehat{H}_{i})\,\subset\,\widehat{H}_{i+1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$, for all $1\,\leq\,i\,\leq\,r-1$, where
$\\{\widehat{H}_{i}\\}_{i=0}^{r}$ is the filtration of
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ in (3.11). This implies that the
second fundamental forms of the subbundles $\widehat{H}_{i}$ produce a section

${\rm SF}(\mathbb{D},\,i)\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}(S))$

as in (3.23) for every $1\,\leq\,i\,\leq\,r-1$.

  2. (2)

For all $1\,\leq\,i\,\leq\,r-1$, the above section ${\rm
SF}(\mathbb{D},\,i)\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}(S))$ coincides with the
section of ${\mathcal{O}}_{X}(S)$ given by the constant function $1$ on $X$.

  3. (3)

For every $x^{\prime}\,\in\,S$, the eigenvalues of ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ are the integers
$\\{0,\,1,\,\cdots,\,r-2,\,r-1\\}$. Note that the multiplicity of each of the
eigenvalues is one.

  4. (4)

For all $0\,\leq\,i\,\leq\,r-1$ and every $x^{\prime}\,\in\,S$, the eigenspace
of ${\rm Res}({\mathbb{D}},\,x^{\prime})$ for the eigenvalue $i$ is contained
in the subspace

$(\widehat{H}_{i+1})_{x^{\prime}}\,\subset\,E_{x^{\prime}}$

(see (3.11) and (4.2)).

In other words, ${\mathbb{D}}$ shares all the properties of ${\mathcal{D}}$
stated in Lemma 4.1, Corollary 3.7 and Lemma 3.8.

######  Remark 5.1.

Note that we did not impose the condition that all the local monodromies of
$\mathbb{D}$ are trivial, despite the fact that $\mathcal{D}$ enjoys this
property (this was shown above).

Let

$L_{x^{\prime}}(i)\,\subset\,E_{x^{\prime}}$ (5.3)

be the eigenline for the eigenvalue $0\,\leq\,i\,\leq\,r-1$ of the residue
${\rm Res}({\mathbb{D}},\,x^{\prime})$ at $x^{\prime}\,\in\,S$. So we have a
decomposition of the fiber $E_{x^{\prime}}$

$E_{x^{\prime}}\,=\,\bigoplus_{i=0}^{r-1}L_{x^{\prime}}(i)\,,$ (5.4)

where $L_{x^{\prime}}(i)$ is the subspace in (5.3). The above condition that
the eigenspace of ${\rm Res}({\mathbb{D}},\,x^{\prime})$ for the eigenvalue
$i$ is contained in the subspace
$(\widehat{H}_{i+1})_{x^{\prime}}\,\subset\,E_{x^{\prime}}$ implies that

$(\widehat{H}_{j+1})_{x^{\prime}}\,=\,\bigoplus_{i=0}^{j}L_{x^{\prime}}(i)$

for all $0\,\leq\,j\,\leq\,r-1$; so we have

$L_{x^{\prime}}(j)\,=\,(\widehat{H}_{j+1})_{x^{\prime}}/(\widehat{H}_{j})_{x^{\prime}}\,.$
(5.5)

We recall from (3.22) that for the filtration of $E$ given in (3.11),

$\widehat{H}_{j}/\widehat{H}_{j-1}\,=\,Q\otimes
K^{\otimes(r-j)}_{X}\otimes{\mathcal{O}}_{X}(-(r-1)S)$

for all $1\,\leq\,j\,\leq\,r$.

Take any point $x^{\prime}\,\in\,S$, and also take any integer
$2\,\leq\,j\,\leq\,r$. We will construct, from ${\mathbb{D}}$, an element

$M_{j}({\mathbb{D}},\,x^{\prime})\,\in\,{\rm
Hom}((\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-1)},\,(\widehat{H}_{j-1}/\widehat{H}_{j-2})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-2)})$
(5.6) $=\,{\rm
Hom}((\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}},\,(\widehat{H}_{j-1}/\widehat{H}_{j-2})_{x^{\prime}})\otimes(K_{X})_{x^{\prime}}\,=\,(K^{\otimes
2}_{X})_{x^{\prime}}\,,$

where $T_{x^{\prime}}X$ is the holomorphic tangent space to $X$ at
$x^{\prime}$; see (3.22) for the equality

${\rm
Hom}((\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}},\,(\widehat{H}_{j-1}/\widehat{H}_{j-2})_{x^{\prime}})\,=\,(K_{X})_{x^{\prime}}$

in (5.6).

To construct $M_{j}({\mathbb{D}},\,x^{\prime})$, first note that the fiber
${\mathcal{O}}_{X}((j-1)S)_{x^{\prime}}$ is identified with
$(T_{x^{\prime}}X)^{\otimes(j-1)}$ using the Poincaré adjunction formula (see
(4.1)). This produces an isomorphism

$\eta\,:\,(\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}}\otimes{\mathcal{O}}_{X}((j-1)S)_{x^{\prime}}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,(\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-1)}$
(5.7)

Take any

$w\,\in\,(\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-1)}\,.$
(5.8)

Using (5.5) and $\eta$ (constructed in (5.7)) we have

$(\widehat{H}_{j}/\widehat{H}_{j-1})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-1)}\,=\,L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{X}((j-1)x^{\prime})_{x^{\prime}}\,.$

Let

$w^{\prime}\,\in\,L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{X}((j-1)x^{\prime})_{x^{\prime}}$

be the element that corresponds to $w$ (see (5.8)) by this isomorphism.

Now we choose a holomorphic section

$\widetilde{w}\,\in\,H^{0}\left(U,\,(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})\right)\,,$
(5.9)

defined on some sufficiently small analytic neighborhood $U\,\subset\,X$ of
$x^{\prime}$, such that

$\widetilde{w}(x^{\prime})\,=\,w^{\prime}\,.$

Let $\widehat{\mathbb{D}}$ be the logarithmic connection on
$(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})$ induced by the
logarithmic connection ${\mathbb{D}}\big{|}_{U}$ on $E\big{|}_{U}$ and the
logarithmic connection on ${\mathcal{O}}_{U}((j-1)x^{\prime})$ given by the de
Rham differential $d$. Consider the residue ${\rm
Res}(\widehat{\mathbb{D}},\,x^{\prime})\,\in\,{\rm
End}(E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}})$ of
the logarithmic connection $\widehat{\mathbb{D}}$ at the point $x^{\prime}$.
It can be shown that the line

$L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}}\,\subset\,E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}}$

(see (5.4)) is contained in the eigenspace of ${\rm
Res}(\widehat{\mathbb{D}},\,x^{\prime})$ for the eigenvalue $0$. Indeed, ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ acts on $L_{x^{\prime}}(j-1)$ as
multiplication by $j-1$ and the logarithmic connection on
${\mathcal{O}}_{U}((j-1)x^{\prime})$ given by the de Rham differential $d$ has
the property that its residue at $x^{\prime}$ is $1-j$. Hence
$L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}}$ is
contained in the eigenspace of ${\rm Res}(\widehat{\mathbb{D}},\,x^{\prime})$
for the eigenvalue $0$. Actually,
$L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}}$ is
the eigenspace of ${\rm Res}(\widehat{\mathbb{D}},\,x^{\prime})$ for the
eigenvalue $0$.

Since the section $\widetilde{w}$ in (5.9) satisfies the condition
$\widetilde{w}(x^{\prime})\,\in\,L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}}$,
from the above property of ${\rm Res}(\widehat{\mathbb{D}},\,x^{\prime})$ that
$L_{x^{\prime}}(j-1)\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})_{x^{\prime}}$ is
contained in the eigenspace of ${\rm Res}(\widehat{\mathbb{D}},\,x^{\prime})$
for the eigenvalue $0$ it follows that

${\mathbb{D}}(\widetilde{w})\,\in\,H^{0}\big{(}U,\,(E\big{|}_{U})\otimes
K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})\big{)}\,,$ (5.10)

where $K_{U}\,:=\,K_{X}\big{|}_{U}$.

The decomposition of $E_{x^{\prime}}$ in (5.4) gives a decomposition

$E_{x^{\prime}}\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}\,=\,\bigoplus_{i=0}^{r-1}L_{x^{\prime}}(i)\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}\,.$
(5.11)

Let

$\beta_{j-2}(w)\,\in\,L_{x^{\prime}}(j-2)\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}$
(5.12)

be the component of ${\mathbb{D}}(\widetilde{w})(x^{\prime})\,\in\,(E\otimes
K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}$ (see (5.10)) in

$L_{x^{\prime}}(j-2)\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}\,\subset\,E_{x^{\prime}}\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}$

with respect to the decomposition in (5.11). Since
$L_{x^{\prime}}(j-2)\,=\,({\widehat{H}}_{j-1})_{x^{\prime}}/({\widehat{H}}_{j-2})_{x^{\prime}}$
(see (5.5)), and
${\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}\,=\,(T_{x^{\prime}}X)^{\otimes(j-1)}$
(see (4.1)), the element $\beta_{j-2}(w)$ in (5.12) is also an element

$\beta_{j-2}(w)\,\in\,({\widehat{H}}_{j-1}/{\widehat{H}}_{j-2})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-2)}.$
(5.13)

The map $M_{j}({\mathbb{D}},\,x^{\prime})$ in (5.6) sends the element $w$ in
(5.8) to $\beta_{j-2}(w)$ constructed in (5.13).

But we need to show that this map is well-defined in the sense that
$\beta_{j-2}(w)$ depends only on $w$, in other words, $\beta_{j-2}(w)$ is
independent of the choice of the section $\widetilde{w}$ in (5.9). The
following lemma shows that $\beta_{j-2}(w)$ depends only on $w$.

######  Lemma 5.2.

The element $\beta_{j-2}(w)$ constructed in (5.13) does not depend on the
choice of the section $\widetilde{w}$ in (5.9).

###### Proof.

We may replace $\widetilde{w}$ in (5.9) by $\widetilde{w}+t$, where

$t\,\in\,H^{0}\big{(}U,\,(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})\big{)}$

with $t(x^{\prime})\,=\,0$, where $E$ is defined in (4.2). Let

$\beta^{t}_{j-2}(w)\,\in\,({\widehat{H}}_{j-1}/{\widehat{H}}_{j-2})_{x^{\prime}}\otimes(T_{x^{\prime}}X)^{\otimes(j-2)}$

be the element constructed as in (5.13) after substituting $\widetilde{w}+t$
in place of $\widetilde{w}$ in the construction of $\beta_{j-2}(w)$. To prove
the lemma we need to show that

$\beta^{t}_{j-2}(w)\,=\,\beta_{j-2}(w)\,.$ (5.14)

To prove (5.14), first note that

$t\,\in\,H^{0}\big{(}U,\,(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})\big{)}\,\subset
H^{0}\big{(}U,\,(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})\big{)}$
(5.15)

because of the given condition that $t(x^{\prime})\,=\,0$.

Let $\widehat{\mathbb{D}}_{1}$ be the logarithmic connection on
$(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})$ given by the
logarithmic connection ${\mathbb{D}}\big{|}_{U}$ on $E\big{|}_{U}$ and the
logarithmic connection on ${\mathcal{O}}_{U}((j-2)x^{\prime})$ given by the de
Rham differential $d$. We note that $\widehat{\mathbb{D}}_{1}$ is simply the
restriction of the logarithmic connection $\widehat{\mathbb{D}}$ to the
subsheaf

$(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})\,\subset\,(E\big{|}_{U})\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})\,.$

From (5.15) we have

$\widehat{\mathbb{D}}_{1}(t)\,\in\,H^{0}\big{(}U,\,(E\big{|}_{U})\otimes
K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime})\big{)}\,.$

Using the isomorphism in (4.1), the evaluation, at $x^{\prime}$, of this
section $\widehat{\mathbb{D}}_{1}(t)$ is considered as an element of

$\widehat{\mathbb{D}}_{1}(t)(x^{\prime})\,\in\,E_{x^{\prime}}\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}\,=\,E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}\,.$
(5.16)

The decomposition in (5.4) produces a decomposition

$E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}\,=\,\bigoplus_{i=0}^{r-1}L_{x^{\prime}}(i)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}\,.$
(5.17)

The residue of the logarithmic connection $\widehat{\mathbb{D}}_{1}$ at
$x^{\prime}$

${\rm
Res}(\widehat{\mathbb{D}}_{1},\,x^{\prime})\,\in\,\text{End}(E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}})$

preserves the decomposition in (5.17). Moreover, ${\rm
Res}(\widehat{\mathbb{D}}_{1},\,x^{\prime})$ acts on the subspace

$L_{x^{\prime}}(i)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}\,\subset\,E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}$

in (5.17) as multiplication by $i-j+2$. Indeed, the residue ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ acts on $L_{x^{\prime}}(i)$ as multiplication
by $i$ (see (5.3)), and the residue, at $x^{\prime}$, of the logarithmic
connection on ${\mathcal{O}}_{U}((j-2)x^{\prime})$ given by the de Rham
differential $d$ is $2-j$. Consequently, ${\rm
Res}(\widehat{\mathbb{D}}_{1},\,x^{\prime})$ acts on
$L_{x^{\prime}}(i)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}$ as
multiplication by $i-j+2$. This implies that

${\rm
Res}(\widehat{\mathbb{D}}_{1},\,x^{\prime})(E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}})\,\subset\,\bigoplus_{i\in\\{0,\cdots,r-1\\}\setminus\\{j-2\\}}L_{x^{\prime}}(i)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}\,,$
(5.18)

and $\text{kernel}({\rm
Res}(\widehat{\mathbb{D}}_{1},\,x^{\prime}))\,=\,L_{x^{\prime}}(j-2)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}$.

On the other hand, the evaluation
$\widehat{\mathbb{D}}_{1}(t)(x^{\prime})\,\in\,E_{x^{\prime}}\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}$
in (5.16) satisfies the identity

$\widehat{\mathbb{D}}_{1}(t)(x^{\prime})\,=\,{\rm
Res}(\widehat{\mathbb{D}}_{1},\,x^{\prime})\big{(}(t)(x^{\prime})\big{)}\,.$

Therefore, from (5.18) it follows that

$\widehat{\mathbb{D}}_{1}(t)(x^{\prime})\,\in\,\bigoplus_{i\in\\{0,\cdots,r-1\\}\setminus\\{j-2\\}}L_{x^{\prime}}(i)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}\,.$
(5.19)

Recall that $\beta_{j-2}(w)$ in (5.12) is the component of
${\mathbb{D}}(\widetilde{w})(x^{\prime})$ in

$L_{x^{\prime}}(j-2)\otimes(K_{U}\otimes{\mathcal{O}}_{U}((j-1)x^{\prime}))_{x^{\prime}}\,=\,L_{x^{\prime}}(j-2)\otimes{\mathcal{O}}_{U}((j-2)x^{\prime})_{x^{\prime}}$

with respect to the decomposition in (5.11). Therefore, from (5.19) it follows
immediately that the equality in (5.14) holds. As noted before, (5.14)
completes the proof. ∎

The map $M_{j}({\mathbb{D}},\,x^{\prime})$ in (5.6) is defined by sending any
element $w$ as in (5.8) to $\beta_{j-2}(w)$ constructed in (5.13) from $w$.
Lemma 5.2 ensures that it is well-defined.

######  Remark 5.3.

Let $D^{\prime}$ be a logarithmic connection singular at a point $x$, such
that residue at $x$ is semisimple. If $\lambda$ is an eigenvalue of the local
monodromy of $D^{\prime}$ around $x$, then $\lambda\,=\,\exp(2\pi\sqrt{-1}b)$,
where $b$ is an eigenvalue of the residue $\text{Res}(D^{\prime},\,x)$ [9].
For any point $x^{\prime}\,\in\,S$, the eigenvalues of the residue
$\text{Res}({\mathbb{D}},\,x^{\prime})$ of the connection ${\mathbb{D}}$ in
(5.2) are integers. Therefore, we conclude that $1$ is the only eigenvalue of
the local monodromy of ${\mathbb{D}}$ around the point $x^{\prime}$. In other
words, the local monodromy of ${\mathbb{D}}$ around $x^{\prime}$ is a
unipotent automorphism. This local monodromy is given by

$(M_{2}({\mathbb{D}},\,x^{\prime}),\,M_{3}({\mathbb{D}},\,x^{\prime}),\,\cdots,\,M_{r}({\mathbb{D}},\,x^{\prime}))\,\in\,((K^{\otimes
2}_{X})_{x^{\prime}})^{\oplus(r-1)}\,,$

where the elements $M_{j}({\mathbb{D}},\,x^{\prime})$ are constructed in
(5.6). The elements $M_{j}({\mathbb{D}},\,x^{\prime})$ will be studied in the
next section.

##  6\. Characterizing the logarithmic connections

As in (5.2), let

${\mathbb{D}}\,:\,E\,:=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\longrightarrow\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-2)S)\otimes
K_{X}$ (6.1)

satisfying the following four conditions:

  1. (1)

${\mathbb{D}}(\widehat{H}_{i})\,\subset\,\widehat{H}_{i+1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$, for all $1\,\leq\,i\,\leq\,r-1$, where
$\\{\widehat{H}_{i}\\}_{i=0}^{r}$ is the filtration of
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ in (3.11). This implies that the
second fundamental forms of the subbundles $\widehat{H}_{i}$ produce a section

${\rm SF}(\mathbb{D},\,i)\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}(S))$ (6.2)

as in (3.23) for every $1\,\leq\,i\,\leq\,r-1$.

  2. (2)

For all $1\,\leq\,i\,\leq\,r-1$, the section ${\rm SF}(\mathbb{D},\,i)$ in
(6.2) coincides with the section of ${\mathcal{O}}_{X}(S)$ given by the
constant function $1$ on $X$.

  3. (3)

For every $x^{\prime}\,\in\,S$, the eigenvalues of ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ are the integers

$\\{0,\,1,\,\cdots,\,r-2,\,r-1\\},$ (6.3)

with the multiplicity of each of them being one.

  4. (4)

For all $0\,\leq\,i\,\leq\,r-1$ and every $x^{\prime}\,\in\,S$, the eigenspace
of ${\rm Res}({\mathbb{D}},\,x^{\prime})$ for the eigenvalue $i$ is contained
in the subspace $(\widehat{H}_{i+1})_{x^{\prime}}\,\subset\,E_{x^{\prime}}$ in
(4.2).

######  Theorem 6.1.

There is a branched ${\rm SL}(r,{\mathbb{C}})$–oper

$(V,\,{\mathcal{F}},\,D)$

such that the logarithmic connection ${\mathbb{D}}$ in (6.1) coincides with
the logarithmic connection on $E$ associated to $(V,\,{\mathcal{F}},\,D)$ by
Proposition 3.5 if and only if the following two conditions hold:

  1. (1)

The logarithmic connection on
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ induced by the logarithmic
connection $\mathbb{D}$ on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$
coincides with the logarithmic connection on
${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ given by the de Rham
differential $d$, once $\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ is
identified with ${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ using
(3.24).

  2. (2)

$M_{j}({\mathbb{D}},\,x^{\prime})\,=\,0$, for all $2\,\leq\,j\,\leq\,r$ and
every $x^{\prime}\,\in\,S$, where $M_{j}({\mathbb{D}},\,x^{\prime})$ are
constructed in (5.6).

###### Proof.

If there is a branched $\text{SL}(r,{\mathbb{C}})$–oper
$(V,\,{\mathcal{F}},\,D)$ such that ${\mathbb{D}}$ coincides with the
logarithmic connection on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$
associated to $(V,\,{\mathcal{F}},\,D)$ by Proposition 3.5, then from Lemma
3.9 we know that the logarithmic connection on
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ induced by $\mathbb{D}$ on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ coincides with the logarithmic
connection on ${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ given by the
de Rham differential $d$, once
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ is identified with
${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ using (3.24).

Therefore, we assume that the logarithmic connection on
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ induced by $\mathbb{D}$ on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ coincides with the logarithmic
connection on ${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ given by the
de Rham differential $d$, after
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ is identified with
${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ using (3.24).

To prove the theorem we need to show the following: There is a branched
$\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$ such that
${\mathbb{D}}$ coincides with the logarithmic connection on
$E\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ associated to
$(V,\,{\mathcal{F}},\,D)$ by Proposition 3.5 if and only if
$M_{j}({\mathbb{D}},\,x^{\prime})\,=\,0$ for all $2\,\leq\,j\,\leq\,r$.

In Proposition 3.5 we constructed a logarithmic connection on $E$ from a
branched $\text{SL}(r,{\mathbb{C}})$–oper. The above statement will be proved
by establishing an inverse of this construction in Proposition 3.5.

Consider the holomorphic vector bundle

${\mathcal{W}}\,:=\,J^{r-1}(Q)\,=\,E\otimes{\mathcal{O}}_{X}((r-1)S)$ (6.4)

on $X$ (see (4.2)). The logarithmic connection $\mathbb{D}$ on $E$ and the
logarithmic connection on ${\mathcal{O}}_{X}((r-1)S)$ given by the de Rham
differential $d$ together produce a logarithmic connection

$\widetilde{\mathbb{D}}\,:\,{\mathcal{W}}\,\longrightarrow\,{\mathcal{W}}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ (6.5)

on the holomorphic vector bundle ${\mathcal{W}}$ in (6.4). At any point
$x^{\prime}\,\in\,S$, the residue of the logarithmic connection on
${\mathcal{O}}_{X}((r-1)S)$ given by the de Rham differential $d$ is $1-r$. On
the other hand, the eigenvalues of ${\rm Res}({\mathbb{D}},\,x^{\prime})$ are
given to be $\\{0,\,1,\,\cdots,\,r-2,\,r-1\\}$ (see (6.3)). Therefore, the
eigenvalues of the residue $\text{Res}(\widetilde{\mathbb{D}},\,x^{\prime})$
of $\widetilde{\mathbb{D}}$ at $x^{\prime}$ are
$\\{1-r,\,2-r,\,\cdots,\,-1,\,0\\}$. We note that the multiplicity of every
eigenvalue of $\text{Res}(\widetilde{\mathbb{D}},\,x^{\prime})$ is one.

For each $x^{\prime}\,\in\,S$, let

$\ell_{x^{\prime}}(0)\,\subset\,{\mathcal{W}}_{x^{\prime}}$

be the eigenspace, for the eigenvalue $0$, of
$\text{Res}(\widetilde{\mathbb{D}},\,x^{\prime})$; so $\ell_{x^{\prime}}(0)$
is a line in ${\mathcal{W}}_{x^{\prime}}$. Let ${\mathcal{W}}_{1}$ be the
holomorphic vector bundle of rank $r$ on $X$ defined by the following short
exact sequence of coherent analytic sheaves on $X$:

$0\,\longrightarrow\,{\mathcal{W}}_{1}\,\longrightarrow\,{\mathcal{W}}\,\longrightarrow\,\bigoplus_{x^{\prime}\in
S}{\mathcal{W}}_{x^{\prime}}/\ell_{x^{\prime}}(0)\,\longrightarrow\,0\,.$
(6.6)

From Lemma 4.2 we know that the logarithmic connection
${\mathbb{D}}\,:\,{\mathcal{W}}\,\longrightarrow\,{\mathcal{W}}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ preserves the subsheaf ${\mathcal{W}}_{1}$
in (6.6). Let

${\mathbb{D}}_{1}\,:\,{\mathcal{W}}_{1}\,\longrightarrow\,{\mathcal{W}}_{1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

be the logarithmic connection on ${\mathcal{W}}_{1}$ induced by
${\mathbb{D}}$. Since the eigenvalues of ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ are $\\{0,\,-1,\,\cdots,\,2-r,\,1-r\\}$, and
$\ell_{x^{\prime}}(0)$ is the eigenspace of ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ for the eigenvalue $0$, from Lemma 4.3 it
follows that the eigenvalues of ${\rm Res}({\mathbb{D}}_{1},\,x^{\prime})$ are
$\\{0,\,-1,\,\cdots,\,2-r\\}$. The eigenvalue $0$ of ${\rm
Res}({\mathbb{D}}_{1},\,x^{\prime})$ has multiplicity two, while the rest of
the eigenvalues of ${\rm Res}({\mathbb{D}}_{1},\,x^{\prime})$ are of
multiplicity one.

For each $x^{\prime}\,\in\,S$, let

$H_{1}(x^{\prime})\,\subset\,({\mathcal{W}}_{1})_{x^{\prime}}$

be the eigenspace of ${\rm Res}({\mathbb{D}}_{1},\,x^{\prime})\,\in\,{\rm
End}(({\mathcal{W}}_{1})_{x^{\prime}})$ for the eigenvalue $0$. As noted
above, we have $\dim H_{1}(x^{\prime})\,=\,2$. Imitating (6.6) we define
${\mathcal{W}}_{2}$. More precisely, let ${\mathcal{W}}_{2}$ be the
holomorphic vector bundle of rank $r$ on $X$ defined by the following short
exact sequence of coherent analytic sheaves on $X$:

$0\,\longrightarrow\,{\mathcal{W}}_{2}\,\longrightarrow\,{\mathcal{W}}_{1}\,\longrightarrow\,\bigoplus_{x^{\prime}\in
S}({\mathcal{W}}_{1})_{x^{\prime}}/H_{1}(x^{\prime})\,\longrightarrow\,0\,.$
(6.7)

From Lemma 4.2 we know that the logarithmic connection
${\mathbb{D}}_{1}\,:\,{\mathcal{W}}_{1}\,\longrightarrow\,{\mathcal{W}}_{1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$ preserves the subsheaf ${\mathcal{W}}_{2}$
(6.7). Let

${\mathbb{D}}_{2}\,:\,{\mathcal{W}}_{2}\,\longrightarrow\,{\mathcal{W}}_{2}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

be the logarithmic connection on ${\mathcal{W}}_{2}$ induced by
${\mathbb{D}}_{1}$. Since the eigenvalues of ${\rm
Res}({\mathbb{D}}_{1},\,x^{\prime})$ are $\\{0,\,-1,\,\cdots,\,2-r\\}$, from
Lemma 4.3 it follows that the eigenvalues of ${\rm
Res}({\mathbb{D}}_{2},\,x^{\prime})$ are $\\{0,\,-1,\,\cdots,\,3-r\\}$. The
eigenvalue $0$ of ${\rm Res}({\mathbb{D}}_{2},\,x^{\prime})$ has multiplicity
three.

For each $x^{\prime}\,\in\,S$, let
$H_{2}(x^{\prime})\,\subset\,({\mathcal{W}}_{2})_{x^{\prime}}$ be the
eigenspace of ${\rm Res}({\mathbb{D}}_{2},\,x^{\prime})$ for the eigenvalue
$0$. Define the holomorphic vector bundle ${\mathcal{W}}_{3}$ by the short
exact sequence of coherent analytic sheaves

$0\,\longrightarrow\,{\mathcal{W}}_{3}\,\longrightarrow\,{\mathcal{W}}_{2}\,\longrightarrow\,\bigoplus_{x^{\prime}\in
S}({\mathcal{W}}_{2})_{x^{\prime}}/H_{2}(x^{\prime})\,\longrightarrow\,0\,.$

We now proceed inductively. To explain this, for $2\,\leq\,j\,\leq\,r-2$,
suppose that we have constructed a holomorphic vector bundle
${\mathcal{W}}_{j}$, and a logarithmic connection ${\mathbb{D}}_{j}$ on it,
such that the following conditions hold:

  * •

For each $x^{\prime}\,\in\,S$, the eigenvalues of ${\rm
Res}({\mathbb{D}}_{j},\,x^{\prime})$ are $\\{0,\,-1,\,\cdots,\,j+1-r\\}$.

  * •

The multiplicity of the eigenvalue zero of ${\rm
Res}({\mathbb{D}}_{j},\,x^{\prime})$ is $j+1$.

Let

$H_{j}(x^{\prime})\,\subset\,({\mathcal{W}}_{j})_{x^{\prime}}$

be the eigenspace of ${\rm Res}({\mathbb{D}}_{j},\,x^{\prime})$ for the
eigenvalue $0$. Then define the holomorphic vector bundle
${\mathcal{W}}_{j+1}$ by the short exact sequence of coherent analytic sheaves

$0\,\longrightarrow\,{\mathcal{W}}_{j+1}\,\longrightarrow\,{\mathcal{W}}_{j}\,\longrightarrow\,\bigoplus_{x^{\prime}\in
S}({\mathcal{W}}_{j})_{x^{\prime}}/H_{j}(x^{\prime})\,\longrightarrow\,0\,.$
(6.8)

From Lemma 4.2 we know that the logarithmic connection ${\mathbb{D}}_{j}$ on
${\mathcal{W}}_{j}$ preserves the subsheaf
${\mathcal{W}}_{j+1}\,\subset\,{\mathcal{W}}_{j}$ in (6.8); the logarithmic
connection on ${\mathcal{W}}_{j+1}$ induced by ${\mathbb{D}}_{j}$ is denoted
by ${\mathbb{D}}_{j+1}$. Since the eigenvalues of ${\rm
Res}({\mathbb{D}}_{j},\,x^{\prime})$ are $\\{0,\,-1,\,\cdots,\,j+1-r\\}$, from
Lemma 4.3 it follows that the eigenvalues of ${\rm
Res}({\mathbb{D}}_{j+1},\,x^{\prime})$ are $\\{0,\,-1,\,\cdots,\,j+2-r\\}$.
The eigenvalue $0$ of ${\rm Res}({\mathbb{D}}_{j+1},\,x^{\prime})$ has
multiplicity $j+2$.

Proceeding inductively, we finally obtain the following:

  1. (1)

a holomorphic vector bundle ${\mathcal{W}}_{r-1}$ on $X$ of rank $r$, and

  2. (2)

a logarithmic connection ${\mathbb{D}}_{r-1}$ on ${\mathcal{W}}_{r-1}$ whose
singular locus is contained in $S$, and for each point $x^{\prime}\,\in\,S$,
the residue

${\rm Res}({\mathbb{D}}_{r-1},\,x^{\prime})\,\in\,{\rm
End}(({\mathcal{W}}_{r-1})_{x^{\prime}})$ (6.9)

is nilpotent (meaning, zero is the only eigenvalue of it).

The next step in the proof of the theorem is to prove the following
proposition.

######  Proposition 6.2.

Take any point $x^{\prime}\,\in\,S$. Then

${\rm Res}({\mathbb{D}}_{r-1},\,x^{\prime})\,=\,0$

(see (6.9)) if and only if $M_{j}({\mathbb{D}},\,x^{\prime})\,=\,0$, for all
$2\,\leq\,j\,\leq\,r$, where $M_{j}({\mathbb{D}},\,x^{\prime})$ are
constructed in (5.6).

###### Proof.

Consider the holomorphic vector bundle $\mathcal{W}$ in (6.4). Let

$\varphi\,:\,{\mathcal{W}}_{r-1}\,\longrightarrow\,{\mathcal{W}}$ (6.10)

be the following composition of homomorphisms

${\mathcal{W}}_{r-1}\,\longrightarrow\,{\mathcal{W}}_{r-2}\,\longrightarrow\,\cdots\,\longrightarrow\,{\mathcal{W}}_{2}\,\longrightarrow\,{\mathcal{W}}_{1}\,\longrightarrow\,{\mathcal{W}}\,;$

see (6.8), (6.7) and (6.6) for the above homomorphisms. We note that the
homomorphism $\varphi$ in (6.10) is an isomorphism over the open subset
$X_{0}$ in (3.8). We will now show that any holomorphic subbundle
${\mathcal{V}}\,\subset\,{\mathcal{W}}$ produces a holomorphic subbundle of
${\mathcal{W}}_{r-1}$. To prove this, let

${\mathcal{V}}^{\prime}\,\subset\,{\mathcal{W}}_{r-1}$

be the coherent analytic subsheaf uniquely defined by the following condition:
A holomorphic section $\sigma\,\in\,H^{0}(U,\,{\mathcal{W}}_{r-1})$ over some
analytic open subset $U\,\subset\,X$ is a section of ${\mathcal{V}}^{\prime}$
if and only if the restriction of $\varphi(\sigma)$ to the complement
$U\setminus(U\bigcap S)$ is a section of the subbundle
${\mathcal{V}}\,\subset\,{\mathcal{W}}$. It is straight-forward to check that
${\mathcal{V}}^{\prime}$ is a holomorphic subbundle of ${\mathcal{W}}_{r-1}$.

For $0\,\leq\,j\,\leq\,r$, consider the holomorphic subbundle

$H_{j}\,\subset\,J^{r-1}(Q)\,=\,{\mathcal{W}}$

(see (3.6) and (6.4)). Let

${\mathcal{E}}_{j}\,\subset\,{\mathcal{W}}_{r-1}$

be the holomorphic subbundle corresponding to $H_{j}$. So we have the
filtration of holomorphic subbundles

$0\,=\,{\mathcal{E}}_{0}\,\subset\,{\mathcal{E}}_{1}\,\subset\,{\mathcal{E}}_{2}\,\subset\,\cdots\,\subset\,{\mathcal{E}}_{r-1}\,\subset\,{\mathcal{E}}_{r}\,=\,{\mathcal{W}}_{r-1}$
(6.11)

of ${\mathcal{W}}_{r-1}$. Note that we have

$\varphi({\mathcal{E}}_{j})\,\subset\,H_{j}$ (6.12)

for all $0\,\leq\,j\,\leq\,r$, where $\varphi$ is the homomorphism in (6.10).
Therefore, $\varphi$ produces a homomorphism

$\varphi_{j}\,:\,{\mathcal{E}}_{j}/{\mathcal{E}}_{j-1}\,\longrightarrow\,H_{j}/H_{j-1}\,=\,Q\otimes
K^{\otimes(r-j)}_{X}$ (6.13)

for all $1\,\leq\,j\,\leq\,r$; see (3.7) for the isomorphism in (6.13).

Recall that the logarithmic connection $\mathbb{D}$ on
$E\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ satisfies the following
condition: For any $x^{\prime}\,\in\,S$ and any $0\,\leq\,i\,\leq\,r-1$, the
eigenspace of ${\rm Res}({\mathbb{D}},\,x^{\prime})$ for the eigenvalue $i$ is
contained in the subspace
$(\widehat{H}_{i+1})_{x^{\prime}}\,\subset\,E_{x^{\prime}}$ (see (3.11) and
(4.2)). This condition implies that the logarithmic connection
$\widetilde{\mathbb{D}}$ on ${\mathcal{W}}$ in (6.5) has the following
property: For any $x^{\prime}\,\in\,S$ and any $0\,\leq\,i\,\leq\,r-1$, the
eigenspace of ${\rm Res}(\widetilde{\mathbb{D}},\,x^{\prime})$ for the
eigenvalue $i-r+1$ is contained in the subspace
$(H_{i+1})_{x^{\prime}}\,\subset\,{\mathcal{W}}_{x^{\prime}}$ in (3.6). Using
this and (6.12) it follows that ${\rm Res}({\mathbb{D}}_{r-1},\,x^{\prime})$
in (6.9) preserves the filtration of subspaces

$0\,=\,({\mathcal{E}}_{0})_{x^{\prime}}\,\subset\,({\mathcal{E}}_{1})_{x^{\prime}}\,\subset\,({\mathcal{E}}_{2})_{x^{\prime}}\,\subset\,\cdots\,\subset\,({\mathcal{E}}_{r-1})_{x^{\prime}}\,\subset\,({\mathcal{E}}_{r})_{x^{\prime}}\,=\,({\mathcal{W}}_{r-1})_{x^{\prime}}$
(6.14)

obtained from (6.11). Since ${\rm Res}({\mathbb{D}}_{r-1},\,x^{\prime})$ is a
nilpotent endomorphism, and it preserves the filtration in (6.14), we conclude
that

${\rm
Res}({\mathbb{D}}_{r-1},\,x^{\prime})(({\mathcal{E}}_{i})_{x^{\prime}})\,\subset\,({\mathcal{E}}_{i-1})_{x^{\prime}}$

for all $1\,\leq\,i\,\leq\,r$. It also follows from the construction of
${\mathcal{W}}_{r-1}$ that

${\rm
Res}({\mathbb{D}}_{r-1},\,x^{\prime})(({\mathcal{E}}_{i})_{x^{\prime}})\bigcap({\mathcal{E}}_{i-2})_{x^{\prime}}\,=\,0$

for all $2\,\leq\,i\,\leq\,r$.

The above observations on ${\rm Res}({\mathbb{D}}_{r-1},\,x^{\prime})$ combine
together to give the following:

For every $x\,\in\,S$, the residue ${\rm
Res}({\mathbb{D}}_{r-1},\,x^{\prime})$ gives an element

$\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})\,\in\,\bigoplus_{i=1}^{r-1}{\rm
Hom}({\mathcal{E}}_{i+1}/{\mathcal{E}}_{i},\,{\mathcal{E}}_{i}/{\mathcal{E}}_{i-1})_{x^{\prime}}\,.$
(6.15)

This $\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})$ has the property that ${\rm
Res}({\mathbb{D}}_{r-1},\,x^{\prime})\,=\,0$ if and only if
$\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})\,=\,0$.

From (6.13) and the construction of ${\mathcal{W}}_{r-1}$ it follows that

${\mathcal{E}}_{i}/{\mathcal{E}}_{i-1}\,=\,(H_{i}/H_{i-1})\otimes{\mathcal{O}}_{X}(-(r-i)S)\,=\,Q\otimes
K^{\otimes(r-i)}_{X}\otimes{\mathcal{O}}_{X}(-(r-i)S)$

for all $1\,\leq\,i\,\leq\,r$. From this we conclude that

${\mathcal{E}}_{i+1}/{\mathcal{E}}_{i}\,=\,({\mathcal{E}}_{i}/{\mathcal{E}}_{i-1})\otimes
TX\otimes{\mathcal{O}}_{X}(S)\,.$ (6.16)

The isomorphism in (6.16) implies that for any $x^{\prime}\,\in\,S$ and all
$1\,\leq\,i\leq\,k-1$, we have

${\rm
Hom}({\mathcal{E}}_{i+1}/{\mathcal{E}}_{i},\,{\mathcal{E}}_{i}/{\mathcal{E}}_{i-1})_{x^{\prime}}\,=\,(K_{X}\otimes{\mathcal{O}}_{X}(-S))_{x^{\prime}}\,=\,(K^{\otimes
2}_{X})_{x^{\prime}}$

(see (4.1) for the last isomorphism). Therefore,
$\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})$ in (6.15) can be considered as
an element

$\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})\,\in\,\bigoplus_{i=1}^{r-1}(K^{\otimes
2}_{X})_{x^{\prime}}\,=\,((K^{\otimes 2}_{X})_{x^{\prime}})^{\oplus(r-1)}\,.$

For any $1\,\leq\,i\,\leq\,r-1$, the element of $(K^{\otimes
2}_{X})_{x^{\prime}}$ in the $i$–th component of
$\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})$, with respect to the above
decomposition, coincides with $M_{i+1}({\mathbb{D}},\,x^{\prime})$ constructed
in (5.6). It was noted earlier that ${\rm
Res}({\mathbb{D}}_{r-1},\,x^{\prime})\,=\,0$ if and only if
$\mathbb{R}({\mathbb{D}}_{r-1},\,x^{\prime})\,=\,0$. Therefore, the proof of
the proposition is complete. ∎

Continuing with the proof of Theorem 6.1, first assume that there is a
branched $\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$ such that
${\mathbb{D}}$ coincides with the logarithmic connection on
$E\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ associated to
$(V,\,{\mathcal{F}},\,D)$ by Proposition 3.5.

It is straight-forward to check that the construction of the triple
$({\mathcal{W}}_{r-1},\,\\{{\mathcal{E}}_{i}\\}_{i=0}^{r},\,{\mathbb{D}}_{r-1})$
from $(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S),\,{\mathbb{D}})$ is the
inverse of the construction of
$(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S),\,{\mathbb{D}})$ from
$(V,\,{\mathcal{F}},\,D)$. More precisely,
$({\mathcal{W}}_{r-1},\,\\{{\mathcal{E}}_{i}\\}_{i=0}^{r},\,{\mathbb{D}}_{r-1})$
coincides with $(V,\,{\mathcal{F}},\,D)$. In particular, ${\mathbb{D}}_{r-1}$
is a holomorphic connection on ${\mathcal{W}}_{r-1}$, as $D$ is a holomorphic
connection on $V$. Now from Proposition 6.2 we conclude that
$M_{j}({\mathbb{D}},\,x^{\prime})\,=\,0$ for all $2\,\leq\,j\,\leq\,r$ and
every $x^{\prime}\,\in\,S$.

To prove the converse, assume that

$M_{j}({\mathbb{D}},\,x^{\prime})\,=\,0$ (6.17)

for all $2\,\leq\,j\,\leq\,r$ and every $x^{\prime}\,\in\,S$, where
$M_{j}({\mathbb{D}},\,x^{\prime})$ are constructed in (5.6). We will show that
there is a branched $\text{SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$
such that ${\mathbb{D}}$ coincides with the logarithmic connection on
$E\,=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ associated to
$(V,\,{\mathcal{F}},\,D)$ by Proposition 3.5.

Since (6.17) holds, from Proposition 6.2 we know that the logarithmic
connection ${\mathbb{D}}_{r-1}$ on ${\mathcal{W}}_{r-1}$ is actually a
holomorphic connection. Consider the filtration
$\\{{\mathcal{E}}_{i}\\}_{i=0}^{r}$ of ${\mathcal{W}}_{r-1}$ in (6.11). It can
be shown that

${\mathbb{D}}_{r-1}({\mathcal{E}}_{i})\,\subset\,{\mathcal{E}}_{i+1}\otimes
K_{X}$ (6.18)

for all $0\,\leq\,i\,\leq\,r-1$. Indeed, we have

$({\mathcal{W}}_{r-1},\,\\{{\mathcal{E}}_{i}\\}_{i=0}^{r},\,{\mathbb{D}}_{r-1})\big{|}_{X_{0}}\,=\,(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S),\,\\{\widehat{H}_{i}\\}_{i=0}^{r},\,{\mathbb{D}})$
(6.19)

over the nonempty open subset $X_{0}$ in (3.8); the filtration
$\\{\widehat{H}_{i}\\}_{i=0}^{r}$ is constructed in (3.11). So (6.18) follows
from the given condition that

${\mathbb{D}}(\widehat{H}_{i})\,\subset\,\widehat{H}_{i+1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$

for all $0\,\leq\,i\,\leq\,r-1$.

In view of (6.18), the second fundamental form of
${\mathcal{E}}_{i}\,\subset\,{\mathcal{W}}_{r-1}$ for the holomorphic
connection ${\mathbb{D}}_{r-1}$ produces a homomorphism

$\Psi_{i}\,\in\,H^{0}(X,\,\text{Hom}({\mathcal{E}}_{i}/{\mathcal{E}}_{i-1},\,{\mathcal{E}}_{i+1}/{\mathcal{E}}_{i})\otimes
K_{X})$

for every $1\,\leq\,i\,\leq\,r-1$. Now using the isomorphism in (6.16) we
conclude that

$\Psi_{i}\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}(S))\,.$ (6.20)

Recall the given condition that for all $1\,\leq\,i\,\leq\,r-1$, the section
${\rm SF}(\mathbb{D},\,i)$ in (6.2) is given by the constant function $1$ on
$X$. Therefore, from the isomorphism in (6.19) we conclude that the section
$\Psi_{i}$ in (6.20) coincides with the section of ${\mathcal{O}}_{X}(S)$
given by the constant function $1$ on $X$. From this it follows that

$({\mathcal{W}}_{r-1},\,\\{{\mathcal{E}}_{i}\\}_{i=0}^{r},\,{\mathbb{D}}_{r-1})$

is a branched $\text{SL}(r,{\mathbb{C}})$–oper. The logarithmic connection on
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ that corresponds to the branched
$\text{SL}(r,{\mathbb{C}})$–oper
$({\mathcal{W}}_{r-1},\,\\{{\mathcal{E}}_{i}\\}_{i=0}^{r},\,{\mathbb{D}}_{r-1})$
by Proposition 3.5 coincides with $\mathbb{D}$, because the construction
$({\mathcal{W}}_{r-1},\,\\{{\mathcal{E}}_{i}\\}_{i=0}^{r},\,{\mathbb{D}}_{r-1})$
from $\mathbb{D}$ is the inverse of the construction in Proposition 3.5. This
completes the proof of Theorem 6.1. ∎

Let

${\mathbb{D}}\,:\,E\,:=\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)\,\longrightarrow\,J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-2)S)\otimes
K_{X}$ (6.21)

be a logarithmic connection satisfying the following five conditions:

  1. (1)

The logarithmic connection on
$\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ induced by the logarithmic
connection $\mathbb{D}$ on $J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$
coincides with the logarithmic connection on
${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ given by the de Rham
differential $d$, once $\det(J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S))$ is
identified with ${\mathcal{O}}_{X}\left(-\frac{r(r-1)}{2}S\right)$ using
(3.24).

  2. (2)

${\mathbb{D}}(\widehat{H}_{i})\,\subset\,\widehat{H}_{i+1}\otimes
K_{X}\otimes{\mathcal{O}}_{X}(S)$, for all $1\,\leq\,i\,\leq\,r-1$, where
$\\{\widehat{H}_{i}\\}_{i=0}^{r}$ is the filtration of
$J^{r-1}(Q)\otimes{\mathcal{O}}_{X}(-(r-1)S)$ in (3.11). This implies that the
second fundamental forms of the subbundles $\widehat{H}_{i}$ produce a section

${\rm SF}(\mathbb{D},\,i)\,\in\,H^{0}(X,\,{\mathcal{O}}_{X}(S))$ (6.22)

as in (3.23) for every $1\,\leq\,i\,\leq\,r-1$.

  3. (3)

For all $1\,\leq\,i\,\leq\,r-1$, the section ${\rm SF}(\mathbb{D},\,i)$ in
(6.22) coincides with the section of ${\mathcal{O}}_{X}(S)$ given by the
constant function $1$ on $X$.

  4. (4)

For every $x^{\prime}\,\in\,S$, the eigenvalues of ${\rm
Res}({\mathbb{D}},\,x^{\prime})$ are the integers
$\\{0,\,1,\,\cdots,\,r-2,\,r-1\\}$, with the multiplicity of each of them
being one.

  5. (5)

For all $0\,\leq\,i\,\leq\,r-1$ and every $x^{\prime}\,\in\,S$, the eigenspace
of ${\rm Res}({\mathbb{D}},\,x^{\prime})$ for the eigenvalue $i$ is contained
in the subspace $(\widehat{H}_{i+1})_{x^{\prime}}\,\subset\,E_{x^{\prime}}$ in
(4.2).

In the proof of Theorem 6.1 we constructed the following:

  1. (1)

a holomorphic vector bundle ${\mathcal{W}}_{r-1}$ on $X$ of rank $r$, and

  2. (2)

a logarithmic connection ${\mathbb{D}}_{r-1}$ on ${\mathcal{W}}_{r-1}$ whose
singular locus is contained in $S$, and for each point $x^{\prime}\,\in\,S$,
the residue

${\rm Res}({\mathbb{D}}_{r-1},\,x^{\prime})\,\in\,{\rm
End}(({\mathcal{W}}_{r-1})_{x^{\prime}})$

is nilpotent (see (6.9)).

If the residue ${\rm Res}(\nabla,\,y)$ of a logarithmic connection $\nabla$ at
a point $y$ is nilpotent, then the local monodromy of $\nabla$ around $y$ is
trivial if and only if ${\rm Res}(\nabla,\,y)\,=\,0$. Therefore, Proposition
6.2 has the following immediate corollary:

######  Corollary 6.3.

The connection ${\mathbb{D}}_{r-1}$ on ${\mathcal{W}}_{r-1}$ is nonsingular
(meaning it is a holomorphic connection) if and only if the local monodromy of
${\mathbb{D}}_{r-1}$ around each point of $S$ it trivial.

In view of Corollary 6.3, from the last part of the proof of Theorem 6.1 (the
part after Proposition 6.2) we have the following:

######  Proposition 6.4.

There is a branched ${\rm SL}(r,{\mathbb{C}})$–oper $(V,\,{\mathcal{F}},\,D)$
such that the logarithmic connection ${\mathbb{D}}$ in (6.21) coincides with
the logarithmic connection on $E$ associated to $(V,\,{\mathcal{F}},\,D)$ by
Proposition 3.5 if and only if the local monodromy of $\mathbb{D}$ around each
point $S$ is trivial.

###  6.1. ${\rm SL}(r,{\mathbb{C}})$-opers with regular singularity and
branched ${\rm SL}(r,{\mathbb{C}})$-opers

This subsection was communicated to us by Edward Frenkel.

In this subsection we relate the branched ${\rm SL}(r,{\mathbb{C}})$-opers
with the ${\rm SL}(r,{\mathbb{C}})$-opers with regular singularity and trivial
monodromy introduced by Frenkel and Gaitsgory in [16, Section 2.9].

Let $G$ be a connected simple affine algebraic group over $\mathbb{C}$, and
let $X$ be a smooth projective algebraic curve over ${\mathbb{C}}$. As before,
$S$ is a finite subset of $X$. Beilinson and Drinfeld have introduced in [2,
Section 4] and [3, Section 3.8] the notion of a $G$–oper on $X$ with regular
singularity at $S$. We recall that it is an ordinary $G$–oper on the
complement of $S$ in $X$, whose restriction to the formal punctured disc
$D^{\times}_{x_{k}}$ at each point $x_{k}\,\in\,S$ is a $G$–oper on the
punctured disc with regular singularity; the details are in [3, Section 3.8.8]
(see also [16, Section 2.4]). Further, Beilinson and Drinfeld have defined the
residue of such an oper. This residue is a point of the geometric invariant
theoretic quotient ${\mathfrak{g}}/G\simeq{\mathfrak{h}}/W$, where
${\mathfrak{g}}$ is the Lie algebra of $G$,
${\mathfrak{h}}\,\subset\,{\mathfrak{g}}$ is its Cartan subalgebra, and $W$ is
the corresponding Weyl group $N_{G}(T)/T$.

Denote by $\varpi$ the natural projection
${\mathfrak{h}}\,\longrightarrow\,{\mathfrak{h}}/W$. For a dominant integral
coweight $\check{\lambda}\,\in\,{\mathfrak{h}}$, denote by

${\rm Op}^{{\rm RS},\varpi(-\check{\lambda}-\check{\rho})}_{\mathfrak{g}}$

the space of $G$–opers with regular singularity on a punctured disc with fixed
residue $\varpi(-\check{\lambda}-\check{\rho})$.

Frenkel and Gaitsgory have defined in [16, Section 2.9], following Beilinson
and Drinfeld (unpublished), a subspace

${\rm Op}^{\check{\lambda},{\rm reg}}_{\mathfrak{g}}\,\subset\,{\rm Op}^{{\rm
RS},\varpi(-\check{\lambda}-\check{\rho})}_{\mathfrak{g}}$ (6.23)

consisting of all those $G$–opers whose monodromy around the origin is
trivial. So, in particular, ${\rm Op}^{0,{\rm reg}}_{\mathfrak{g}}$ is
naturally identified with the space of regular $G$–opers on the disc. It may
be mentioned that opers of this kind play an important rôle in the geometric
Langlands correspondence (see [17] and [15, Section 9.6]).

The following proposition is due to E. Frenkel.

######  Proposition 6.5.

There is a natural isomorphism between the space of all branched ${\rm
SL}(r,{\mathbb{C}})$–opers on $X$ with branching over $S$ and the space of all
${\rm SL}(r,{\mathbb{C}})$–opers on $X$ with regular singularity at $S$
satisfying the following two conditions:

  1. (1)

the restriction to the punctured formal disc around each point $x_{k}\,\in\,S$
has residue $\varpi(-2\check{\rho})$, and

  2. (2)

this oper belongs to the subspace ${\rm Op}^{\check{\rho},{\rm
reg}}_{\mathfrak{g}}$ in (6.23).

The proof of Proposition 6.5 follows directly by comparing Definition 2.1 and
Theorem 6.1 with the definition of the subspace ${\rm Op}^{\check{\rho},{\rm
reg}}_{\mathfrak{g}}$ in [16].

## Acknowledgements

We are very grateful to Edward Frenkel for communicating Section 6.1. We thank
the referee for helpful comments.

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