# On the moduli of logarithmic connections on elliptic curves

Thiago Fassarella1, Frank Loray2 and Alan Muniz1,3 1Universidade Federal
Fluminense – UFF, Rua Alexandre Moura 8, Niterói, RJ, Brazil  2Univ Rennes,
CNRS, IRMAR - UMR 6625, F-35000 Rennes, France  3Universidade Federal do
Espírito Santo – UFES, Av. Fernando Ferrari 514, Vitória, ES, Brasil
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We describe moduli spaces of logarithmic rank $2$ connections on elliptic
curves with $n\geq 1$ poles and generic residues. In particular, we generalize
a previous work by the first and second named authors. Our main approach is to
analyze the underlying parabolic bundles; their stability and instability play
a major role.

###### Key words and phrases:

logarithmic connection, parabolic structure, elliptic curve, apparent
singularities, symplectic structure.

###### 2020 Mathematics Subject Classification:

Primary 34M55; Secondary 14D20, 32G20, 32G34.

The second author is supported by CNRS, and ANR-16-CE40-0008 project
“Foliage”. This work was conducted during the postdoctoral periods of the
third author at UFES and UFF, when he was supported in part by the Coordenação
de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance
Code 001. The authors also thank Brazilian-French Network in Mathematics and
CAPES-COFECUB project MA 932/19.

###### Contents

  1. 1 Introduction
  2. 2 Parabolic vector bundles
  3. 3 Logarithmic connections
    1. 3.1 Connections on $E_{1}$
    2. 3.2 Description of $\mathfrak{Con}^{\nu}_{st}$
    3. 3.3 Connections with unstable parabolic bundles.
    4. 3.4 The classical Painlevé case
  4. 4 Fuchsian systems with $n+1$ poles
    1. 4.1 The affine bundle of Fuchsian systems
  5. 5 Symplectic structure
    1. 5.1 Computation of the symplectic structure
  6. 6 Apparent map

##  1\. Introduction

In this paper, we investigate the geometry of certain moduli spaces of
connections on complex elliptic curves $C$. We will consider pairs
$(E,\nabla)$ where $E\rightarrow C$ is a rank $2$ vector bundle and
$\nabla\colon E\rightarrow E\otimes\Omega^{1}_{C}(D)$ is a logarithmic
connection with (reduced) polar divisor $D=t_{1}+\cdots+t_{n}$ with $n\geq 1$.
We also prescribe the following data:

  * •

The eigenvalues $(\nu_{i}^{+},\nu_{i}^{-})$ of ${\rm Res}_{t_{i}}(\nabla)$,
for each $i=1,\dots,n$, such that:

    1. (1)

$\nu_{1}^{\epsilon_{1}}+\cdots+\nu_{n}^{\epsilon_{n}}\notin\mathbb{Z}$, for
any $\epsilon_{i}\in\\{+,-\\}$;

    2. (2)

and $\nu_{i}^{+}\neq\nu_{i}^{-}$, for $i=1,\dots,n$;

  * •

A trace connection $(L,\zeta)$, i.e. $\det(E)=L$ and ${\rm tr}(\nabla)=\zeta$;

In particular $\sum_{i}\nu_{i}^{+}+\nu_{i}^{-}=-\deg(L)$ which is called the
Fuchs relation. Once we have fixed this data, we can define the moduli space
$\mathfrak{Con}^{\nu}$ of those pairs $(E,\nabla)$ up to isomorphism. With the
generic condition on the eigenvalues, every connection $(E,\nabla)$ is
irreducible and the moduli $\mathfrak{Con}^{\nu}$ can be constructed as a GIT
quotient. It follows from [12] that $\mathfrak{Con}^{\nu}$ is a smooth
irreducible quasi-projective variety of dimension $2n$, equipped with a
holomorphic symplectic structure.

It is natural to consider the forgetful map $\pi\colon(E,\nabla)\mapsto(E,{\bf
p})$ which associates to a connection an underlying quasi-parabolic bundle.
Given a choice of signs $\epsilon_{i}\in\\{+,-\\}$ for each $i=1,\dots,n$, the
parabolic data ${\bf
p}^{\epsilon}(\nabla)=(p_{1}^{\epsilon_{1}}(\nabla),\ldots,p_{n}^{\epsilon_{n}}(\nabla))$
consists of the $\nu_{i}^{\epsilon_{i}}$-eigenspace
$p_{i}^{\epsilon_{i}}\subset E|_{t_{i}}$ for ${\rm Res}_{t_{i}}(\nabla)$ at
each pole; these are well-defined since $\nu_{i}^{+}\neq\nu_{i}^{-}$. There
exist $2^{n}$ underlying quasi-parabolic structures for each connection,
according to the choice of $\epsilon=(\epsilon_{1},\ldots,\epsilon_{n})$
giving rise to $2^{n}$ forgetful maps
$\pi_{\epsilon}\colon(E,\nabla)\mapsto(E,{\bf p}^{\epsilon}(\nabla))$.

In Section 2 we study the quasi-parabolic bundles $(E,{\bf p})$ over $(C,D)$
that admit a connection $\nabla$ with prescribed trace and eigenvalues,
compatible with parabolic directions. The major difference from the case $n=2$
investigated in [8] is that, when $n$ is odd, there exist pairs $(E,\nabla)$
such that all the underlying quasi-parabolic structures $(E,{\bf
p}^{\epsilon}(\nabla))$ are not $\mu$-semistable for any choice of weights; it
occurs for the item 3 of Lemma 2.4. Following the study of stability, we
describe a wall-crossing phenomenon in Lemma 2.7 and Lemma 2.8.

In Section 3 we study the logarithmic connections. We investigate
$\mathfrak{Con}^{\nu}$ via the forgetful map to an underlying quasi-parabolic
structure. We are especially concerned with the $\mu$-stability of these
quasi-parabolic bundles. It turns out that there exists an open subset of
$\mathfrak{Con}^{\nu}$ where the underlying vector bundle is $E_{1}$, the
unique indecomposable vector bundle of degree one, with given determinant. We
call this open subset ${\rm Con}^{\nu}$ and consider the map

$\displaystyle\mathrm{Par}\colon{\rm Con}^{\nu}$
$\displaystyle\longrightarrow{{(\mathbb{P}^{1}\times\mathbb{P}^{1})\times\cdots\times(\mathbb{P}^{1}\times\mathbb{P}^{1})}}$
$\displaystyle(E,\nabla)$ $\displaystyle\longmapsto\left(\
p_{1}^{+}(\nabla),p_{1}^{-}(\nabla),\ldots,p_{n}^{+}(\nabla),p_{n}^{-}(\nabla)\
\right)$

that associates to each connection all its residual eigenspaces (i.e. with
respect to all eigenvalues $\nu_{i}^{+}$ and $\nu_{i}^{-}$). The image of
$\mathrm{Par}$ is contained in $S^{n}$, where $S$ is the complement of the
diagonal in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We show that this map is an
isomorphism (cf. Theorem 3.3):

######  Theorem A.

The map $\mathrm{Par}\colon{\rm Con}^{\nu}\rightarrow S^{n}$ is an
isomorphism.

We also study the open subset
$\mathfrak{Con}^{\nu}_{st}\subset\mathfrak{Con}^{\nu}$ formed by pairs
$(E,\nabla)$ which admit a $\mu$-stable parabolic structure $(E,{\bf
p}^{\epsilon}(\nabla))$ for some $\epsilon$ and some weight vector $\mu$. We
call

$\mathcal{Z}_{n}=\mathfrak{Con}^{\nu}\setminus\mathfrak{Con}^{\nu}_{st}$

its complement. We describe $\mathcal{Z}_{n}$ in Theorem 3.10, it is empty for
$n$ even and has four irreducible components which are isomorphic to
$\mathbb{C}^{n}$ for $n$ odd. Assuming
$\nu_{i}^{+}-\nu_{i}^{-}\notin\\{0,1,-1\\}$ for $i\in\\{1,\cdots,n\\}$, we see
that the large subset $\mathfrak{Con}^{\nu}_{st}$ admits an open covering
given by open subsets isomorphic to $S^{n}$. It leads to the following
characterization of $\mathfrak{Con}^{\nu}_{st}$ (cf. Theorem 3.8):

######  Theorem B.

Assume that $\nu_{i}^{+}-\nu_{i}^{-}\notin\\{0,1,-1\\}$ for
$i\in\\{1,\cdots,n\\}$. Then $\mathfrak{Con}^{\nu}_{st}$ is obtained by gluing
a finite number of copies of $S^{n}$ via birational maps

$\Psi_{J,I}^{\delta,\epsilon}\colon S^{n}\dashrightarrow S^{n}.$

Moreover, if $\epsilon=\delta$, then $\Psi_{J,I}^{\delta,\epsilon}$ preserves
the fibers of $\pi_{\epsilon}$.

In the remainder of our work, we do a finer analysis of the open subset ${\rm
Con}^{\nu}$. Let ${\rm Bun}$ denote the moduli space of parabolic vector
bundles whose underling vector bundle is $E_{1}$; it is isomorphic to
$(\mathbb{P}^{1})^{n}$. It follows from Theorem A that the affine bundle ${\rm
Con}^{\nu}\to{\rm Bun}$ does not depend on the eigenvalues. In Section 4, we
describe the affine bundle ${\rm Con}^{\nu}\rightarrow{\rm Bun}$ via Fuchsian
systems; this construction yields a vector bundle $\mathcal{E}$ whose
projectivization compactifies ${\rm Con}^{\nu}$. Although
$\mathbb{P}(\mathcal{E})$ does not depend on the eigenvalues, the boundary
divisor is determined by by $\nu_{i}=\nu_{i}^{+}-\nu_{i}^{-}$, $i=1,\dots,n$
(cf. Theorem 4.3):

######  Theorem C.

The moduli space ${\rm Con}^{\nu}$ has compactification $\overline{{\rm
Con}^{\nu}}=\mathbb{P}(\mathcal{E})$, where the boundary divisor is isomorphic
to $\mathbb{P}\mathrm{Higgs}$, the projectivization of the space of Higgs
fields on $E_{1}$. Moreover, the inclusion
$\mathbb{P}\mathrm{Higgs}\hookrightarrow\mathbb{P}(\mathcal{E})$ is
determined, up to automorphisms of $\mathbb{P}(\mathcal{E})$, by
$\left(\nu_{1},\dots,\nu_{n}\right)$.

In Section 5, we deal with the symplectic structure of the moduli space and
compute the explicit expression in the main chart ${\rm Con}^{\nu}\simeq
S^{n}$, see (5.8). We show that $\left(\nu_{1},\dots,\nu_{n}\right)$ is
detected by the symplectic structure (cf. Corollary 5.9):

######  Theorem D.

If there exists fiber preserving symplectic isomorphism

${\left({\rm Con}^{\nu},\omega\right)}$${\left({\rm
Con}^{\tilde{\nu}},\tilde{\omega}\right)}$${{\rm Bun}}$${{\rm
Bun}}$$\displaystyle{\pi_{+}}$$\displaystyle{\Phi}$$\displaystyle{\sim}$$\displaystyle{\pi_{+}}$$\displaystyle{\phi}$$\displaystyle{\sim}$

then there exists a permutation $\sigma$ of $n$ elements such that
$\tilde{\nu}_{k}=\nu_{\sigma(k)}$ for every $k\in\\{1,\cdots,n\\}$.

In Theorem D, we have a fixed underlying space of parabolic bundles, ${\rm
Bun}$, and we can recover $\left(\nu_{1},\dots,\nu_{n}\right)$ from the
symplectic structure. The main result of [7] is a Torelli type result which
asserts that the moduli space of parabolic bundles determines the punctured
curve $(C,D)$. We wonder if the same is true for ${\rm Con}^{\nu}$, or
$\mathfrak{Con}^{\nu}$. This belief is based on some results in the
literature, see [25, 5, 4].

In Section 6, we conclude the paper by studying the Apparent map. A global
section of $E_{1}$ plays the role of a cyclic vector for connection $\nabla$,
which yields a second order ODE on $C$; the map ${\rm App}$ assigns to
$\nabla$ the apparent singular points of this equation, see [21]. It leads to
an interesting result about the birational geometry of $\overline{{\rm
Con}^{\nu}}$ (Theorem 6.2):

######  Theorem E.

The map $\pi_{+}\times\operatorname{App}$ induces a birational map

$\pi_{+}\times\operatorname{App}\colon\overline{\rm{Con}^{\nu}}\dashrightarrow{\rm
Bun}\times|\mathcal{O}_{C}(w_{\infty}+D)|$

whose indeterminacy locus is contained in
$\overline{\rm{Con}^{\nu}}\backslash\rm{Con}^{\nu}$. Moreover, given
$(E,{\bf{p}})\in{\rm Bun}$, the rank of

$\left.(\pi_{+}\times\operatorname{App})\right|_{\pi_{+}^{-1}(E,{\bf{p}})}\colon\pi_{+}^{-1}(E,{\bf{p}})\longrightarrow{\\{(E,{\bf{p}})\\}\times}|\mathcal{O}_{C}(w_{\infty}+D)|$

coincides with the cardinality of the set $\\{i\mid
p_{i}\not\subset\mathcal{O}_{C}\\}$.

Logarithmic connections on elliptic curves have being investigated by several
authors. First, they were studied in the works of Okamoto and later by Kawai
in [17] in relation with isomonodromic deformations. In particular, the
symplectic form of [17, Theorem 1], when restricted to a fixed punctured curve
$(C,D)$, must coincide to that one we give in (5.8). However, as [17] is
dealing with analytic differential equations written in terms of Weierstrass
zeta functions, it is not so easy to relate the computations with ours. More
recently, our moduli space has been considered and proved to be related to
some moduli spaces of logarithmic connections on $\mathbb{P}^{1}$ for the
special cases $n=1$ (see [19]), and $n=2$ with special eigenvalues (see [20]).

######  Remark 1.1 (Notation and convention).

Throughout the text $C$ will denote a genus one curve and
$D=t_{1}+\dots+t_{n}$ will be a reduced divisor on $C$. Let $w_{\infty}\in C$
be a point and let $w_{0},w_{1}$ and $w_{\lambda}$ be the torsion points of
the elliptic curve $(C,w_{\infty})$. In the construction of
$\mathfrak{Con}^{\nu}$, we will assume (without loss of generality) that the
determinant is $\mathcal{O}_{C}(w_{\infty})$ and that
$D+w_{0}+w_{1}+w_{\lambda}$ is reduced.

##  2\. Parabolic vector bundles

Let $C$ be an elliptic curve with $w_{\infty}\in C$ being its distinguished
point. A rank two quasi-parabolic vector bundle $(E,{\bf p})$ on
$\left(C,D\right)$, $D=t_{1}+\dots+t_{n}$, consists of a holomorphic vector
bundle $E$ of rank two on $C$ and a collection ${\bf
p}=\\{p_{1},\dots,p_{n}\\}$ of $1$-dimensional linear subspaces $p_{i}\subset
E_{t_{i}}$. We refer to the points $t_{i}$ as parabolic points, and to the
subspace $p_{i}\subset E_{t_{i}}$ as the parabolic direction of $E$ at
$t_{i}$.

A triple $(E,{\bf p};\mu)$ of a quasi-parabolic vector bundle and an $n$-tuple
$\mu=(\mu_{1},\dots,\mu_{n})$ of real numbers in the interval $(0,1)$ is
called parabolic vector bundle of rank two. We often write $(E,{\bf p})$ for a
parabolic vector bundle when the choice of the weight $\mu$ is clear.

Let $(E,{\bf p};\mu)$ be a parabolic vector bundle and let $L\subset E$ be a
line subbundle then we define

${\rm Stab}_{\mu}(L):=\deg E-2\deg L+\sum_{p_{k}\neq
L_{t_{k}}}\mu_{k}-\sum_{p_{k}=L_{t_{k}}}\mu_{k}.$

We say that $(E,{\bf p};\mu)$ is semistable if ${\rm Stab}_{\mu}(L)\geq 0$
holds for every $L\subset E$. It is stable if the strict inequality holds for
every line subbundle $L\subset E$. We call ${\rm Stab}_{\mu}(L)$ the parabolic
stability of $L\subset E$ with respect to $\mu$.

We denote by $\operatorname{Bun}_{w_{\infty}}^{\mu}$ the moduli space of
semistable parabolic vector bundles $(E,{\bf p};\mu)$ on $(C,D)$ with $\det
E=\mathcal{O}_{C}(w_{\infty})$. In this case, either $E\simeq L\oplus
L^{-1}(w_{\infty})$ or $E\simeq E_{1}$, where $E_{1}$ is the unique non
trivial extension

$0\longrightarrow\mathcal{O}_{C}\longrightarrow
E_{1}\longrightarrow\mathcal{O}_{C}(w_{\infty})\longrightarrow 0.$

If there exists $L\subset E$ such that ${\rm Stab}_{\mu}(L)$ is zero then the
weights lie on the hyperplane

$H(d,I):=\left\\{\mu\mathrel{\Big{|}}1-2d+\sum_{k\notin I}\mu_{k}-\sum_{k\in
I}\mu_{k}=0\right\\}$

where $d=\deg L$ and $I\subset\\{1,\dots,n\\}$ denotes the set of indices of
those parabolic directions $p_{k}\subset L_{t_{k}}$. A connected component of
the complement in $(0,1)^{n}$ of all these hyperplanes $H(d,I)$ is called a
chamber. If $\mu$ and $\tilde{\mu}$ belong to the same chamber then
$\operatorname{Bun}_{w_{\infty}}^{\mu}=\operatorname{Bun}_{w_{\infty}}^{\tilde{\mu}}$,
see for example [22] or [6, Lemma 2.7].

In the next result, we define an interesting chamber; the underlying vector
bundle is fixed and the corresponding moduli space is a product of projective
lines.

######  Proposition 2.1.

The following assertions hold:

  1. (1)

The set $\mathfrak{C}:=\\{\mu\in(0,1)^{n}\mid\sum_{k=1}^{n}\mu_{k}<1\\}$ is a
chamber.

  2. (2)

If $\mu\in\mathfrak{C}$ then $\operatorname{Bun}_{w_{\infty}}^{\mu}=\\{(E,{\bf
p})\mid E=E_{1}\\}$. Moreover, it is isomorphic to $(\mathbb{P}^{1})^{n}$.

###### Proof.

Note that $\mathfrak{C}$ is convex, hence connected. Then (1) follows from
proving that $\mathfrak{C}$ does not intersect any wall $H(d,I)$. This is
straightforward and we leave it to the reader.

Now we prove (2). Recall that $\det E=\mathcal{O}_{C}(w_{\infty})$ implies
that either $E=E_{1}$ or $E=L\oplus L^{-1}(w_{\infty})$ with $\deg L\geq 1$.
The later cannot occur. Indeed,

${\rm Stab}_{\mu}(L)=1-2\deg L+\sum_{p_{k}\neq
L_{t_{k}}}\mu_{k}-\sum_{p_{k}=L_{t_{k}}}\mu_{k}\leq-2\deg L<0$

and $E$ cannot be $\mu$-semistable for $\mu\in\mathfrak{C}$. Hence $E=E_{1}$
and each parabolic bundle is completely determined by

$(p_{1},\dots,p_{n})\in\mathbb{P}\left(E_{1}|_{t_{1}}\right)\times\dots\times\mathbb{P}\left(E_{1}|_{t_{n}}\right)\simeq(\mathbb{P}^{1})^{n}.$

Thus we get the desired isomorphism. ∎

For a weight vector $\mu=(\mu_{1},\dots,\mu_{n})\in(0,1)^{n}$ and a subset
$I\subset\\{1,\dots,n\\}$ of even cardinality, we consider the map
$\varphi_{I}\colon(0,1)^{n}\longrightarrow(0,1)^{n}$ defined by

$\varphi_{I}(\mu):=(\mu^{\prime}_{1},\dots,\mu^{\prime}_{n})\in(0,1)^{n}$

where $\mu^{\prime}_{i}=\mu_{i}$ if $i\not\in I$, and
$\mu^{\prime}_{i}=1-\mu_{i}$ if $i\in I$. Note that $\varphi_{I}$ is
continuous and preserves the walls $H(d,J)$. Then the image of $\mathfrak{C}$
by $\varphi_{I}$ yields a new chamber

(2.1)
$\displaystyle{\mathfrak{C}}_{I}:=\left\\{\mu\in(0,1)^{n}\mathrel{\Big{|}}\sum_{k\notin
I}\mu_{k}-\sum_{k\in I}\mu_{k}+|I|<1\right\\}$

where $|I|$ is the cardinality of $I$. When $I=\emptyset$ then
${\mathfrak{C}}_{I}=\mathfrak{C}$.

Each $\varphi_{I}$ admits a modular realization as an elementary
transformation, which we now describe. Consider the following exact sequence
of sheaves

$0\longrightarrow
E^{\prime}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}E\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\bigoplus_{i\in
I}(E_{t_{i}}/p_{i})\longrightarrow 0$

where for each (local) section $s$ of $E$ we define
$\beta(s)=(\beta_{1}(s),\dots,\beta_{n}(s))$ by
$\beta_{j}(s)=s(t_{j})\pmod{p_{j}}$ if $s$ is defined at $t_{j}$, and
$\beta_{j}(s)=0$ otherwise. Then $E^{\prime}$ is a vector bundle of rank two
such that

$\det E^{\prime}=\det E\otimes\mathcal{O}_{C}\left(-\sum_{i\in
I}t_{i}\right).$

In particular, $E^{\prime}$ has degree $1-|I|$. We define a natural quasi-
parabolic structure for $E^{\prime}$ as follows. If $i\not\in I$ then
$\alpha_{t_{i}}\colon E^{\prime}_{t_{i}}\longrightarrow E_{t_{i}}$ is an
isomorphism and

$p_{i}^{\prime}=(\alpha_{t_{i}})^{-1}(p_{i})\subset E^{\prime}_{t_{i}}$

is the parabolic direction at $t_{i}$. If $i\in I$ we define
$p_{i}^{\prime}=\ker(\alpha_{t_{i}})$ as the parabolic direction at $t_{i}$.
This operation corresponds to the birational transformation of ruled surfaces
$\mathbb{P}(E)\dashrightarrow\mathbb{P}(E^{\prime})$ obtained by blowing-up
the points $p_{i}\in\mathbb{P}(E_{t_{i}})$ and then blowing-down the strict
transforms of the fibers $\mathbb{P}(E_{t_{i}})$ to the points
$p^{\prime}_{i}\in\mathbb{P}(E^{\prime}_{t_{i}})$, $i\in I$. This is well-
defined since the $p_{i}$ lie on different fibers.

$\displaystyle\mathbb{P}(E)$$\displaystyle\mathbb{P}(E^{\prime})$ Figure 1.
Elementary transformation

Since $|I|$ is even, we can fix a square root $L_{0}$ of the line bundle
$\mathcal{O}_{C}\left(\sum_{i\in I}t_{i}\right)$, i.e.

$L_{0}^{2}=\mathcal{O}_{C}\\!\left(\sum_{i\in I}t_{i}\right).$

This gives a correspondence

$\mathrm{elm}_{I}\colon\left(E,{\bf
p}\right)\longmapsto\left(E^{\prime}\otimes L_{0},{\bf p^{\prime}}\right)$

between quasi-parabolic vector bundles on $\left(C,D\right)$ which have
$\mathcal{O}_{C}(w_{\infty})$ as determinant line bundle.

The reader can check that if $\left(E,{\bf p}\right)$ is semistable with
respect to $\mu$, then $\mathrm{elm}_{I}\left(E,{\bf p}\right)$ is semistable
with respect to $\varphi_{I}(\mu)$. We conclude that the correspondence
$\mathrm{elm}_{I}$ defines an isomorphism between moduli spaces

$\mathrm{elm}_{I}\colon\operatorname{Bun}_{w_{\infty}}^{\mu}\longrightarrow\operatorname{Bun}_{w_{\infty}}^{\varphi_{I}(\mu)}.$

######  Definition 2.2.

Given $I\subset\\{1,\dots,n\\}$ of even cardinality, let
$\mu\in\mathfrak{C}_{I}$. We will denote

${\rm Bun}^{I}=\operatorname{Bun}_{w_{\infty}}^{\mu}.$

When $I$ is the empty set, we write simply ${\rm Bun}$ instead of ${\rm
Bun}^{\emptyset}$; it corresponds to the moduli space of parabolic vector
bundles whose underlying vector bundle is $E_{1}$.

######  Remark 2.3.

From Proposition 2.1, we conclude that ${\rm
Bun}^{I}\simeq(\mathbb{P}^{1})^{n}$ for any $I\subset\\{1,\dots,n\\}$ of even
cardinality.

A quasi-parabolic vector bundle $(E,{\bf p})$ is called decomposable if there
exist $(L,{\bf p^{\prime}})$ and $(M,{\bf p^{\prime\prime}})$ such that
$(E,{\bf p})\simeq(L,{\bf p^{\prime}})\oplus(M,{\bf p^{\prime\prime}})$ as
quasi-parabolic vector bundles. Otherwise it is called indecomposable. Note
that $(E,{\bf p})$ can be indecomposable with $E$ decomposable as a vector
bundle.

######  Lemma 2.4.

Let $(E,{\bf p})$ be a rank two indecomposable quasi-parabolic bundle, over
$(C,D)$, with $\det E=\mathcal{O}_{C}(w_{\infty})$. Then one of the following
holds:

  1. (1)

$E$ is indecomposable, i.e. $E=E_{1}$;

  2. (2)

$E=L\oplus L^{-1}(w_{\infty})$ and $2\leq 2\deg L\leq n$;

  3. (3)

$E=L\oplus L^{-1}(w_{\infty})$ with $L^{2}=\mathcal{O}_{C}(D+w_{\infty})$,
hence $2\deg L=n+1$. Moreover, every parabolic direction lies on
$L^{-1}(w_{\infty})$ except for one that lies outside both subbundles.

###### Proof.

When $E=E_{1}$ we have nothing to prove. So suppose that $E=L\oplus
L^{-1}(w_{\infty})$. Since $L\oplus L^{-1}(w_{\infty})\simeq M\oplus
M^{-1}(w_{\infty})$ with $M=L^{-1}(w_{\infty})$ we can assume $\deg L=s\geq
1$.

To decompose $(E,{\bf p})$ we need to find an embedding of
$L^{-1}(w_{\infty})$ in $E$ passing through every direction that does not lie
on $L$. Note that this is the same as finding an automorphism of $E$ that
sends every direction outside $L$ to $(0:1)$. Let $p_{j}=(u_{j}:1)$ denote the
parabolic direction over $t_{j}$ which is outside $L$. Recall that

${\rm End}(E)=\left\\{\begin{pmatrix}\alpha&\beta\\\
0&\delta\end{pmatrix}\mathrel{\Big{|}}\alpha,\delta\in\mathbb{C},\,\beta\in{\rm
H^{0}}(C,L^{2}(-w_{\infty}))\right\\}.$

If $2s\geq n+2$ then $h^{0}(L^{2}(-w_{\infty}-D+t_{j}))=2s-n\geq 2$ and we are
free to choose $\beta_{j}$ that vanishes on $t_{i}$ for $i\neq j$ and such
that $\beta_{j}(t_{j})=-u_{j}$. Thus, choosing
$\beta=\sum_{j=1}^{n}\beta_{j}$, $\alpha=1$ and $\delta=1$, the corresponding
automorphism sends any direction $p_{j}$ outside $L$ to $(0:1)$.

Now set $2s=n+1$. By the same argument as above, to show that $(E,{\bf p})$ is
decomposable, we need to find a section $\beta_{j}$ of ${\rm
H^{0}}(C,L^{2}(-w_{\infty}))$ that vanishes on $t_{i}$ for $i\neq j$ and such
that $\beta_{j}(t_{j})=-u_{j}$, for each $j\in\\{1,\dots,n\\}$. We can find
$\beta_{j}$ as required if $L^{2}(-w_{\infty}-D)\neq\mathcal{O}_{C}$. Indeed,
assume $L^{2}(-w_{\infty}-D+t_{j})\simeq\mathcal{O}_{C}(x_{j})$ with
$x_{j}\neq t_{j}$, and take any section $\alpha_{j}$ of
$L^{2}(-w_{\infty}-D+t_{j})$ with $\alpha_{j}(t_{j})\neq 0$. The desired
section is defined as $\beta_{j}=-\frac{u_{j}}{\alpha_{j}(t_{j})}\alpha_{j}$.
Hence $(E,{\bf p})$ is decomposable when
$L^{2}(-w_{\infty}-D)\neq\mathcal{O}_{C}$.

If $L^{2}=\mathcal{O}_{C}(D+w_{\infty})$ we can apply the same argument for
$D-t_{1}$ instead of $D$ to find an embedding of $L^{-1}(w_{\infty})$ passing
through $n-1$ parabolic directions outside $L$. In particular, if $(E,{\bf
p})$ is indecomposable then there exists no parabolic direction on $L$ and
this finishes the proof. ∎

######  Remark 2.5.

The parabolic bundles in the third case of Lemma 2.4 have a peculiar property:
they are never $\mu$-semistable, whatever is $\mu$. Indeed, for $E=L\oplus
L^{-1}(w_{\infty})$ with $2\deg L=n+1$, no parabolic direction lying on $L$,
and any weight $\mu$, we have

${\rm Stab}_{\mu}(L)=-n+\sum_{j=1}^{n}\mu_{j}<0.$

We can give a partial converse to this fact. Note that if $E=E_{1}$ then any
quasi-parabolic bundle is stable for $\mu\in\mathfrak{C}$.

######  Lemma 2.6.

Let $(E,{\bf p})$ be an indecomposable quasi-parabolic bundle such that
$E=L\oplus L^{-1}(w_{\infty})$ and $p_{j}$ lies outside $L$, for every $j$. If
$2\leq 2\deg L\leq n$ then there exists $I\subset\\{1,\dots,n\\}$, with
$|I|=2\deg L$, such that $(E,{\bf p})\in{\rm Bun}^{I}$.

###### Proof.

Given that every direction $p_{j}$ lies outside $L$, we may find an embedding
of $L^{-1}(w_{\infty})$ that passes through some of these directions. Any
subset of directions with cardinality $2\deg L-1$ admits at most one embedding
of $L^{-1}(w_{\infty})$ passing through them. Since $(E,{\bf p})$ is
indecomposable, such embedding cannot pass through all $p_{j}$. In particular,
we can find $k\in\\{2\deg L,\dots,n\\}$ such that no embedding of
$L^{-1}(w_{\infty})$ passes trough the directions indexed by
$I=\\{1,\dots,2\deg L-1,k\\}$. It is straightforward to verify that $(E,{\bf
p})\in{\rm Bun}^{I}$. ∎

Until now we have only considered a rank two $E$ and its line subbundles
$L\subset E$. But a more general setting will be suitable for the next
results; we may allow subsheaves that are not saturated. We will consider
general morphisms $L\rightarrow E$ that do not, necessarily, lead to an
embedding of $L$ in $E$. Recall that, over a curve, being a subbundle means
that there exist an injective morphism $L\hookrightarrow E$ whose cokernel is
also a line bundle, i.e. $L$ is a saturated subsheaf of $E$. For a general
morphism $\phi\colon L\rightarrow E$ this does not need to be true. However,
we can factor out a divisor $Z$ where $\phi$ vanishes, leading to an injective
morphism $L(Z)\hookrightarrow E$. For details, see [9, Chapter 2, Proposition
5]. On the other hand, given a subbundle $L\subset E$ we can produce a
morphism $L(-Z)\rightarrow E$ that vanishes on the fibers over the support of
$Z$.

Given a morphism $\phi\colon L\rightarrow E$ we say that its image passes
through $p_{j}\subset E_{t_{j}}$ if $\phi_{t_{j}}(L_{t_{j}})\subset p_{j}$.

######  Lemma 2.7.

Let $I\subset\\{1,\dots,n\\}$ have cardinality $2k+2$ with $k\geq 0$ and fix
$\mu\in\mathfrak{C}_{I}$. Then $(E_{1},{\bf p})$ is not $\mu$-semistable if
and only if there exists a line bundle $L$ of degree $\deg L=-k$ and a
morphism $L\rightarrow E_{1}$ whose image passes through $p_{j}$ for all $j\in
I$.

###### Proof.

Fix $\mu=(\mu_{1},\cdots,\mu_{n})\in\mathfrak{C}_{I}$. First recall that
$\mu=\varphi_{I}(\mu^{\prime})$ for some $\mu^{\prime}\in\mathfrak{C}$. Then
$(E_{1},{\bf p})$ is not $\mu$-semistable if there exists a subbundle
$L\subset E_{1}$ such that

$Stab_{\mu}(L)=1-2\deg L+\sum\pm\mu_{j}=1-2\deg
L+|I|-2|A|+\sum\pm\mu^{\prime}_{j}<0$

where $A\subset I$ is the set of $j\in I$ such that $p_{j}$ lie on $L$. Since
$\mu^{\prime}\in\mathfrak{C}$ we have $|\sum\pm\mu^{\prime}_{j}|<1$. Then we
need

$1-2\deg L+|I|-2|A|\leq-1.$

Let $k\geq 0$ be defined by $|I|=2k+2$ and define $u=\deg L+k$. Then we
rewrite the inequality as $2k+2\leq u+|A|$. From $A\subset I$ and $\deg L\leq
0$ we have that $0\leq u\leq k$.

Next we will produce a degree $-k$ line bundle and a morphism to $E_{1}$ such
that its image passes through every direction $p_{j}^{\epsilon}$ with $j\in
I$. If $u=0$, i.e. $\deg L=-k$, then $A=I$ and we are done. If $u>0$ or
equivalently $\deg L>-k$, the argument is more involved.

Fix $J\subset A$ such that $|J|=2k+2-u$ and let $Z=\sum_{j\in I\setminus
J}t_{j}$. Considering the inclusions $L\hookrightarrow E_{1}$ and
$\mathcal{O}_{C}(-Z)\hookrightarrow\mathcal{O}_{C}$ we define a map by the
composition

$\phi\colon L\otimes\mathcal{O}_{C}(-Z)\longrightarrow
E_{1}(-Z)\longrightarrow E_{1},$

with the property that it gives the same directions over $D-Z$ and vanishes
over $Z$. Hence the image of $\phi$ passes through every direction from $I$.

Conversely, suppose that there exists a degree $-k$ line bundle $L$ and a
nontrivial morphism $\phi\colon L\rightarrow E_{1}$ passing through every
$p_{j}$, $j\in I$. Let $Z$ be the zero divisor of $\phi$ and consider the
reduction $\phi^{\prime}\colon L(Z)\rightarrow E_{1}$. As $\phi^{\prime}$ is
injective, $L(Z)$ is a subbundle of $E_{1}$. Hence it has non-positive degree,
i.e. $\deg Z\leq k$. On the other hand, we have that $p_{j}$ lie on $L(Z)$ for
every $j$ such that $t_{j}\not\in{\rm Supp}Z$. If $A$ is given as above, we
have $|A|\geq 2k+2-\deg Z_{red}$, hence

$1-2\deg L(Z)+|I|-2|A|\leq-1+2(\deg Z_{red}-\deg Z)\leq-1$

This completes the proof. ∎

We now see the real advantage of switching to this slightly more general
setting. The previous lemma describes a wall-crossing phenomenon. And the next
lemma can be used to describe geometrically the space of quasi-parabolic
bundles that become unstable when we cross a wall.

######  Lemma 2.8.

Let $n=2k+2$ for some $k\geq 0$. Let $V\subset(\mathbb{P}^{1})^{n}$ be the
locus of points that correspond to quasi-parabolic bundles $(E_{1},{\bf p})$
satisfying the following property: there exist a line bundle $L$ of degree
$-k$ and a morphism $\phi\colon L\longrightarrow E_{1}$ whose image passes
through ${\bf p}$. Then $V$ is a hypersurface of degree $(2,\dots,2)$.

###### Proof.

Let $\pi_{j}\colon(\mathbb{P}^{1})^{n}\longrightarrow(\mathbb{P}^{1})^{n-1}$
be the projection given by forgetting the $j$th component and let $h_{j}$ be
the class of a fiber of $\pi_{j}$. Then we only need to show that $V\cap
h_{j}=2$ for every $j$. Up to permuting indices we only need to consider
$j=2k+2$.

If $k=0$ the result follows from [27, Proposition 3.3]. Indeed, for each
degree $0$ line bundle $L\in{\rm Pic}^{0}(C)$ there exists a unique map
$\phi\colon L\rightarrow E_{1}$ and the map
$L\mapsto\phi_{t_{1}}(L)\in\mathbb{P}^{1}$ is generically $2:1$. Then, for a
generic direction $p_{1}$, there exist two choices for $L\in{\rm Pic}^{0}(C)$
such that $\phi_{t_{1}}(L)\subset p_{1}$. Therefore, $(p_{1},p_{2})\in V$ if
and only if $p_{2}$ one of the directions defined by these line bundles, i.e.
$V\cap h_{2}=2$.

Now we consider $k\geq 1$. We will show that we can reduce to the previous
case. Fix $p_{1},\dots,p_{2k}$ generic directions. By generic we mean that
there exists no subbundle of degree at least $1-k$ passing through these
directions. Let $L\in{\rm Pic}^{-k}(C)$ be any line bundle. To give a map
$\phi\colon L\longrightarrow E_{1}$ passing through $p_{1},\dots,p_{2k}$ is
equivalent to giving a map $L\longrightarrow E^{\prime}$, where $E^{\prime}$
is obtained by elementary transformation with respect to $p_{1},\dots,p_{2k}$.
Indeed, we have

$0\longrightarrow
E^{\prime}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}E_{1}\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\bigoplus_{j=1}^{2k}(E_{1})_{t_{j}}/p_{j}\longrightarrow
0$

and $\beta\circ\phi=0$ if and only if there exists $\phi^{\prime}\colon
L\longrightarrow E^{\prime}$ such that $\phi=\alpha\circ\phi^{\prime}$.
Nonetheless, this is equivalent to giving a map

$\phi^{\prime}\otimes 1\colon L\otimes M\longrightarrow E^{\prime}\otimes M$

where $M$ is a line bundle such that
$M^{2}=\mathcal{O}_{C}(t_{1}+\dots+t_{2k})$.

Since $p_{1},\dots,p_{2k}$ are generic, $E^{\prime}$ is indecomposable. In
particular, $E^{\prime}\otimes M=E_{1}$. We then apply the same argument of
the case $k=0$ to the directions $p_{2k+1}$ and $p_{2k}$ to show that $V\cap
h_{2k+2}=2$. ∎

######  Definition 2.9.

Let $n\geq 2$ be an integer and let $I\subset\\{1,\dots,n\\}$ be a subset of
even cardinality. We will denote by $\Gamma_{I}\subset(\mathbb{P}^{1})^{n}$
the subvariety that parameterizes quasi-parabolic bundles $(E_{1},{\bf p})$
that are not $\mu$-semistable for $\mu\in\mathfrak{C}_{I}$.

######  Corollary 2.10.

The subvariety $\Gamma_{I}\subset(\mathbb{P}^{1})^{n}$ is an hypersurface of
degree $(d_{1},\dots,d_{n})$, where $d_{i}=2$ if $i\in I$ and $d_{i}=0$
otherwise.

###### Proof.

Note that the formation of $\Gamma_{I}$ depends only on the directions indexed
by $I$. Then it will be a product $\Gamma_{I}\simeq
V\times(\mathbb{P}^{1})^{n-|I|}$. Therefore, we may reduce to the case
$\Gamma_{I}=V$, i.e. $|I|=n$ and the conclusion follows from Lemma 2.7 and
Lemma 2.8. ∎

######  Remark 2.11.

A quasi-parabolic bundle $(E_{1},{\bf p})$ is not $\mu$-semistable for
$\mu\in\mathfrak{C}_{I}$ if and only if $\mathrm{elm}_{I}(E_{1},{\bf
p})=(E,{\bf p^{\prime}})$ is not $\mu^{\prime}$-semistable for
$\mu^{\prime}=\varphi_{I}(\mu)\in\mathfrak{C}$. The later occurs if and only
if $E$ splits. Therefore $\Gamma_{I}$ corresponds, via $\mathrm{elm}_{I}$, to
the locus in $\operatorname{Bun}^{I}$ of those quasi-parabolic bundles whose
underlying vector bundles split.

##  3\. Logarithmic connections

A logarithmic connection on a rank two vector bundle $E$ over $C$ with polar
divisor $D=t_{1}+\cdots+t_{n}$ is a $\mathbb{C}$-linear map

$\nabla\colon E\longrightarrow E\otimes\Omega_{C}^{1}(D)$

satisfying the Leibniz rule

$\nabla(fs)=s\otimes df+f\nabla(s)$

for (local) sections $s$ of $E$ and $f$ of $\mathcal{O}_{C}$. If $t\in C$ is a
pole for $\nabla$ and $U\subset C$ is a small trivializing neighborhood of
$t$, we write $\nabla|_{U}=d+A$ where
$d\colon\mathcal{O}_{C}\longrightarrow\Omega_{C}^{1}$ is the exterior
derivative and $A$ is a $2\times 2$ matrix whose coefficients are $1$-forms
with at most simple poles on $t$. Note that $A$ depends on the trivialization,
but its similarity class does not. Then the residue endomorphism

${\rm Res}_{t}(\nabla):={\rm Res}_{t}(A)\in{\rm End}(E_{t})$

is well defined. Let $\nu_{k}^{+}$ and $\nu_{k}^{-}$ be the eigenvalues of
${\rm Res}_{t_{k}}(\nabla)$. The data

$\nu=(\nu_{1}^{+},\nu_{1}^{-},...,\nu_{n}^{+},\nu_{n}^{-})\in\mathbb{C}^{2n}$

are called the eigenvalues of $\nabla$. The induced trace connection

${\rm tr}(\nabla)\colon\det(E)\rightarrow\det(E)\otimes\Omega^{1}_{C}(D)$

satisfies ${\rm Res}_{t_{k}}({\rm tr}(\nabla))=\nu_{k}^{+}+\nu_{k}^{-}$ and
Residue Theorem yields the Fuchs relation:

$\deg E+\sum_{k=1}^{n}(\nu_{k}^{+}+\nu_{k}^{-})=0.$

######  Remark 3.1.

Hereafter we will fix the following data:

  1. (1)

A $2n$-tuple of complex numbers
$\nu=(\nu_{1}^{+},\nu_{1}^{-},...,\nu_{n}^{+},\nu_{n}^{-})$ satisfying the
Fuchs relation

$1+\sum_{k=1}^{n}(\nu_{k}^{+}+\nu_{k}^{-})=0$

and the generic condition:
$\nu_{1}^{\epsilon_{1}}+\cdots+\nu_{n}^{\epsilon_{n}}\notin\mathbb{Z}$ for any
$\epsilon_{k}\in\\{+,-\\}$, to avoid reducible connections; and
$\nu_{k}^{+}\neq\nu_{k}^{-}$ for all $k\in\\{1,\dots,n\\}$, so that the
residues have distinguished eigenspaces;

  2. (2)

A fixed trace connection
$\zeta\colon\mathcal{O}_{C}(w_{\infty})\rightarrow\mathcal{O}_{C}(w_{\infty})\otimes\Omega^{1}_{C}(D)$
satisfying

${\rm Res}_{t_{k}}(\zeta)=\nu_{k}^{+}+\nu_{k}^{-}$

for all $k=1,...,n$.

Then we define the moduli space:

$\mathfrak{Con}^{\nu}=\left\\{(E,\nabla)\mathrel{\Big{|}}\begin{matrix}\nabla\text{
has eigenvalue }\nu,\\\ \det E=\mathcal{O}_{C}(w_{\infty}),\,{\rm
tr}\nabla=\zeta\end{matrix}\right\\}{/\sim}$

where $\sim$ stands for $S$-equivalence. In fact, the condition (1) on $\nu$
implies that $\sim$ may be thought as equivalence up to isomorphism.

Algebraic constructions of moduli spaces of connections goes back to the works
of Simpson and, in the logarithmic case, Nitsure in [23]. In our setting, it
is more convenient to refer to the works of Inaba, Iwasaki and Saito [13], and
more precisely Inaba [12]. Indeed, under our generic assumption on $\nu$, each
connection $\nabla$ on $E$ defines a unique parabolic structure, by selecting
the eigenspace $p_{k}\subset E|_{t_{k}}$ associated to $\nu_{k}^{+}$ at each
pole $t_{k}$; therefore, $\mathfrak{Con}^{\nu}$ can equivalently be viewed as
the moduli space of parabolic connections as considered in the work [12] of
Inaba. Then it follows from [12, Theorem 2.1, Proposition 5.2] that it is
quasi-projective and irreducible of dimension $2n$. Moreover, [12, Theorem
2.2] shows that it is moreover smooth. In fact, in order to fit with the
stability condition [12, Definition 2.2], we set
$\alpha_{1}^{(k)}=\frac{1-\mu_{k}}{2}$ and
$\alpha_{2}^{(k)}=\frac{1+\mu_{k}}{2}$; our moduli space therefore corresponds
to the fiber $\det^{-1}(L,\zeta)$ of the determinant map considered at the
beginning of [12, Section 5]. When $\nu_{k}^{+}=\nu_{k}^{-}$ for some $k$,
there exist connections with scalar residue (apparent singular point) which
give rise to a singular locus in the moduli space; the role of the parabolic
structure in [12] is to get a smooth moduli space even in that case.

Then the moduli space $\mathfrak{Con}^{\nu}$ is a smooth irreducible quasi-
projective variety of dimension $2n$, provided that the condition (1) on $\nu$
is satisfied. The case $n=1$ is not covered by [12], but it follows from [19];
we will discuss this case at the end of this section.

In the study of $\mathfrak{Con}^{\nu}$, it is useful to consider the quasi-
parabolic bundles underlying a connection. Let us assume
$\nu_{k}^{+}\neq\nu_{k}^{-}$ for all $k\in\\{1,\dots,n\\}$. Given a connection
$(E,\nabla)$, we associate, for each $k=1,\dots,n$, a pair of “positive” and
“negative” eigenspaces of ${\rm Res}_{t_{k}}(\nabla)$

$p_{k}^{+}(\nabla),p_{k}^{-}(\nabla)\in\mathbb{P}(E_{t_{k}})$

corresponding to the eigenvalues $\nu_{k}^{+}$ and $\nu_{k}^{-}$ respectively.

Given an $n$-tuple $\epsilon=(\epsilon_{1},\cdots,\epsilon_{n})$, where each
$\epsilon_{i}\in\\{+,-\\}$, we denote

${\bf
p}^{\epsilon}(\nabla)=\\{p_{1}^{\epsilon_{1}}(\nabla),\cdots,p_{n}^{\epsilon_{n}}(\nabla)\\}$

and consider $(E,{\bf p}^{\epsilon}(\nabla))$ the quasi-parabolic vector
bundle defined by these directions.

######  Remark 3.2.

The hypothesis that $\nu_{1}^{a_{1}}+\cdots+\nu_{n}^{a_{n}}\notin\mathbb{Z}$,
for every $a\in\\{+.-\\}^{n}$, ensures that $(E,{\bf p}^{\epsilon}(\nabla))$
is an indecomposable quasi-parabolic bundle. Indeed, if $(L,{\bf p})$ is rank
one direct summand of $(E,{\bf p}^{\epsilon}(\nabla))$ then the residues of
the induced connection on $L$ are either $\nu_{j}^{+}$ or $\nu_{j}^{-}$; their
sum is $-\deg L$, see [8, Corollary 2.3].

One can then ask for the stability of these quasi-parabolic bundles with
respect to some weight. We define

$\mathfrak{Con}^{\nu}_{st}=\left\\{(E,\nabla)\in\mathfrak{Con}^{\nu}\mathrel{\Big{|}}\begin{matrix}\exists\,\epsilon\in\\{+,-\\}^{n}\text{
and }\exists\,I\subset\\{1,\dots,n\\}\\\ \text{such that }(E,{\bf
p}^{\epsilon}(\nabla))\in{\rm Bun}^{I}\end{matrix}\right\\}.$

Recall that $|I|$ is always assumed to be even. It follows that

$\mathfrak{Con}^{\nu}=\mathfrak{Con}^{\nu}_{st}\sqcup\mathcal{Z}_{n}$

where $\mathcal{Z}_{n}$ denotes the complement of $\mathfrak{Con}^{\nu}_{st}$.
Our aim in the next subsections is to describe these varieties. We will show
that $\mathfrak{Con}^{\nu}_{st}$ can be covered by simple open subsets and
that $\mathcal{Z}_{n}$ falls in two cases: either $n$ is even and
$\mathcal{Z}_{n}=\emptyset$ or $n$ is odd and $\mathcal{Z}_{n}$ has four
connected components, each one is a quotient of $\mathbb{C}^{n+1}$ by a free
affine action of the additive group $(\mathbb{C},+)$, hence isomorphic to
$\mathbb{C}^{n}$.

###  3.1. Connections on $E_{1}$

Our main building block in the description of $\mathfrak{Con}^{\nu}_{st}$ is
the space defined by

${\rm Con}^{\nu}=\left\\{(E,\nabla)\in\mathfrak{Con}^{\nu}\mid
E=E_{1}\right\\}.$

Note that every underlying quasi-parabolic bundle lies in ${\rm Bun}={\rm
Bun}^{\emptyset}$, see Proposition 2.1 and Definition 2.2. The same
proposition shows that ${\rm Bun}\simeq(\mathbb{P}^{1})^{n}$. We will see that
${\rm Con}^{\nu}$ has a similar description.

Let $\Delta\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be the diagonal and let
$S:=(\mathbb{P}^{1}\times\mathbb{P}^{1})\backslash\Delta$ be its complement.
Then we define a map

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\rm Par}\colon{\rm
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}\hbox{${\quad\quad(E_{1},\nabla)}$} }}\pgfsys@invoke{\lxSVG@closescope
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This map is in fact an isomorphism.

######  Theorem 3.3.

The map $\mathrm{Par}\colon{\rm Con}^{\nu}\rightarrow S^{n}$ is an
isomorphism.

We will postpone the proof of this theorem to Section 4, right after
Proposition 4.2. Now we can prove a direct consequence.

######  Corollary 3.4.

${\rm Con}^{\nu}$ is an affine variety.

###### Proof.

Since the diagonal $\Delta\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ supports
an ample divisor, its complement, $S$, is affine. Therefore ${\rm
Con}^{\nu}\simeq S^{n}$ is also affine. ∎

In the next subsection we will see that $S^{n}$ is a local model for
$\mathfrak{Con}^{\nu}_{st}$.

###  3.2. Description of $\mathfrak{Con}^{\nu}_{st}$

From the definition, $\mathfrak{Con}^{\nu}_{st}$ is the space of (isomorphism
classes of) connections $(E,\nabla)$ for which there exist
$I\subset\\{1,\dots,n\\}$, with $|I|$ even, and $\epsilon\in\\{-,+\\}^{n}$
such that $(E,{\bf p}^{\epsilon}(\nabla))\in{\rm Bun}^{I}$, i.e. $(E,{\bf
p}^{\epsilon}(\nabla))$ is $\mu$-stable for any $\mu\in\mathfrak{C}_{I}$. For
each $I$ and $\epsilon$ we define

${\rm
Con}_{I,\epsilon}^{\nu}:=\left\\{(E,\nabla)\in\mathfrak{Con}^{\nu}\mid(E,{\bf
p}^{\epsilon}(\nabla))\in{\rm Bun}^{I}\right\\},$

hence we get a decomposition

(3.1) $\displaystyle\mathfrak{Con}^{\nu}_{st}=\bigcup\limits_{I,\epsilon}{\rm
Con}_{I,\epsilon}^{\nu}.$

Note that ${\rm Con}_{\emptyset,\epsilon}^{\nu}={\rm Con}^{\nu}$ for any
$\epsilon$. Next we will see that, for generic $\nu$, each ${\rm
Con}_{I,\epsilon}^{\nu}$ is isomorphic to $S^{n}$. More precisely, we will
prove that ${\rm Con}_{I,\epsilon}^{\nu}$ is isomorphic to ${\rm
Con}^{\lambda}$, for some eigenvalue $\lambda$ to be determined. Consider

$\pi_{\epsilon}\colon{\rm Con}_{I,\epsilon}^{\nu}\longrightarrow{\rm Bun}^{I}$

the forgetful morphism.

######  Proposition 3.5.

The map $\mathrm{elm}_{I}$ induces a fiber-preserving isomorphism
$\Phi_{I}^{\epsilon}$:

${{\rm Con}_{I,\epsilon}^{\nu}}$${{\rm Con}^{\lambda}}$${{\rm Bun}^{I}}$${{\rm
Bun}}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\Phi_{I}^{\epsilon}}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\mathrm{elm}_{I}}$

for
$\lambda=(\lambda_{1}^{+},\lambda_{1}^{-},\cdots,\lambda_{n}^{+},\lambda_{n}^{-})$
defined as follows. If $k\not\in I$ then $\lambda_{k}^{+}=\nu_{k}^{+}$ and
$\lambda_{k}^{-}=\nu_{k}^{-}$, and if $k\in I$ then

$\lambda_{k}^{\delta_{k}}=\nu_{k}^{\epsilon_{k}}-\frac{1}{2},\text{ and
}\lambda_{k}^{\epsilon_{k}}=\nu_{k}^{\delta_{k}}+\frac{1}{2},$

where $\\{\delta_{k},\epsilon_{k}\\}=\\{+,-\\}$.

###### Proof.

Given $(E,\nabla)\in{\rm Con}_{I,\epsilon}^{\nu}$ we will perform an
elementary transformation centered in ${\bf p}^{\epsilon}(\nabla)$. Recall
that $\mathrm{elm}_{I}$ sends $(E,{\bf p}^{\epsilon}(\nabla))$ to
$\Big{(}E^{\prime}\otimes L,{\bf p^{\prime}}\Big{)}$ where $E^{\prime}$ is
obtained from the exact sequence

$0\longrightarrow
E^{\prime}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}E\longrightarrow\bigoplus_{i\in
I}\left(E_{t_{i}}/p_{i}^{\epsilon_{i}}(\nabla)\right)\longrightarrow 0,$

and $L$ is a square root $\mathcal{O}_{C}\left(\sum_{i\in I}t_{i}\right)$. The
pullback connection $\alpha^{*}(\nabla)$ has the following property, which can
be verified in local coordinates. For $k\not\in I$, the eigenvalue at $t_{k}$
are the same $\\{\nu_{k}^{+},\nu_{k}^{-}\\}$. For $k\in I$, the eigenvalues
are

$(\nu_{k}^{\delta_{k}}+1,\nu_{k}^{\epsilon_{k}})$

and $\ker(\alpha_{t_{k}})$ corresponds to $\nu_{k}^{\delta_{k}}+1$.

Now let $\xi\colon L\rightarrow L\otimes\Omega_{C}^{1}(D)$ be the rank one
connection defined as follows. Let $\\{U_{i}\\}$ be a trivializing cover for
$E,E^{\prime}$ and $L$ and let $\\{G_{ij}\\},\\{G_{ij}^{\prime}\\}$ and
$\\{h_{ij}\\}$ be the respective cocycles. Assume further that $\det
G_{ij}=h_{ij}^{2}\det G_{ij}^{\prime}$ for every pair $(i,j)$. Then

$\xi|_{U_{i}}=d-\frac{1}{2}{\rm tr}\left(\alpha_{i}^{-1}d\alpha_{i}\right),$

where $\alpha_{i}$ is the local expression for $\alpha$. Note that ${\rm
tr}(\alpha^{*}(\nabla)\otimes\xi)={\rm tr}(\nabla)$ and

${\rm Res}_{t_{k}}\xi=\begin{cases}-\frac{1}{2},&k\in I;\\\ 0,&k\not\in
I.\end{cases}$

The map $\Phi_{I}^{\epsilon}$ is then defined as

$\Phi_{I}^{\epsilon}(E,\nabla)=(E^{\prime}\otimes
L,\alpha^{*}(\nabla)\otimes\xi).$

Since it can be reversed by the same process, we have the isomorphism.
Moreover, the diagram in the statement commutes from the construction of
$\Phi_{I}^{\epsilon}$. ∎

To give an isomorphism ${\rm Con}_{I,\epsilon}^{\nu}\simeq S^{n}$ we just need
to require that $\lambda$ satisfies the hypothesis of Theorem 3.3, i.e.
$\lambda_{k}^{+}\neq\lambda_{k}^{-}$ and in Remark 3.1.

######  Corollary 3.6.

If, for every $k=1,\dots,n$, $\nu_{k}^{+}-\nu_{k}^{-}\notin\\{0,1,-1\\}$ then,
for every $I\subset\\{1,\dots,n\\}$, with $|I|$ even, and every
$\epsilon\in\\{+,-\\}^{n}$,

${\rm Con}_{I,\epsilon}^{\nu}\simeq S^{n}.$

###### Proof.

For $k\not\in I$ we have $\lambda_{k}^{+}-\lambda^{-}=\nu_{k}^{+}-\nu_{k}^{-}$
and for $k\in I$ we have
$\lambda_{k}^{+}-\lambda^{-}=\nu_{k}^{-}-\nu_{k}^{+}\pm 1$. The result follows
from Proposition 3.5 and Theorem 3.3 once we have
$\nu_{k}^{+}-\nu_{k}^{-}\notin\\{0,1,-1\\}$. ∎

Hereafter we will also assume that $\nu_{k}^{+}-\nu_{k}^{-}\notin\\{0,1,-1\\}$
for every $k=1,\dots,n$. We will also denote by $\pi_{\epsilon}$ the
projection $\pi_{\epsilon}\colon S^{n}\rightarrow(\mathbb{P}^{1})^{n}$ that
makes the following diagram commute

${{\rm Con}_{I,\epsilon}^{\nu}}$${S^{n}}$${{\rm
Bun}^{I}}$${(\mathbb{P}^{1})^{n}}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\pi_{\epsilon}}$

and we define
$\widetilde{\Gamma}_{I,\epsilon}=\pi_{\epsilon}^{-1}(\Gamma_{I})$, where
$\Gamma_{I}$ is the hypersurface from Definition 2.9.

######  Proposition 3.7.

Let $I,J\subset\\{1,\dots,n\\}$ with even cardinalities and fix
$\epsilon\in\\{+,-\\}^{n}$. Then

${\rm Con}^{\nu}_{I,\epsilon}\setminus{\rm Con}_{J,\epsilon}^{\nu}\simeq{\rm
Con}^{\nu}_{J,\epsilon}\setminus{\rm
Con}_{I,\epsilon}^{\nu}\simeq\widetilde{\Gamma}_{I\Delta J,\epsilon}$

where $I\Delta J=(I\cup J)\setminus(I\cap J)$ is their symmetric difference.

###### Proof.

Note that $|I\Delta J|$ is even and
$\mathrm{elm}_{I}\circ\mathrm{elm}_{J}=\mathrm{elm}_{I\Delta J}$; hence
$\Phi_{I}^{\epsilon}$ gives an isomorphism

${\rm Con}^{\nu}_{I,\epsilon}\setminus{\rm Con}_{J,\epsilon}^{\nu}\simeq{\rm
Con}^{\lambda}\setminus{\rm Con}_{I\Delta J,\epsilon}^{\lambda}.$

An element of ${\rm Con}^{\lambda}\setminus{\rm Con}_{I\Delta
J,\epsilon}^{\lambda}$ is a connection whose underlying quasi-parabolic bundle
$(E_{1},\textbf{p}^{\epsilon}({\nabla}))$ is stable for the weights
$\mu\in\mathfrak{C}$ but its image under $\mathrm{elm}_{I\Delta J}$ is
unstable or, equivalently, it is $\mu$-stable but $\varphi_{I\Delta
J}(\mu)$-unstable. Therefore, using Theorem 3.3 and Remark 2.11 we get

${\rm Par}({\rm Con}^{\lambda}\setminus{\rm Con}_{I\Delta
J,\epsilon}^{\lambda})=\widetilde{\Gamma}_{I\Delta J,\epsilon}.$

∎

######  Theorem 3.8.

Assume and $\nu_{1}^{a_{1}}+\cdots+\nu_{n}^{a_{n}}\notin\mathbb{Z}$, for any
$a_{k}\in\\{+,-\\}$, and that $\nu_{k}^{+}-\nu_{k}^{-}\notin\\{0,1,-1\\}$ for
$k\in\\{1,\cdots,n\\}$. Then $\mathfrak{Con}^{\nu}_{st}$ is obtained by gluing
a finite number of copies of $S^{n}$ via birational maps

$\Psi_{J,I}^{\delta,\epsilon}\colon S^{n}\dashrightarrow S^{n}.$

Moreover, if $\epsilon=\delta$, then $\Psi_{J,I}^{\delta,\epsilon}$ preserves
the fibers of $\pi_{\epsilon}$.

###### Proof.

Since $\mathfrak{Con}^{\nu}_{st}=\cup_{I,\epsilon}{\rm
Con}_{I,\epsilon}^{\nu}$, (3.1), we may give local charts ${\rm
Par}\circ\Phi_{I}^{\epsilon}\colon{\rm Con}^{\nu}_{I,\epsilon}\rightarrow
S^{n}$. Note that ${\rm Con}^{\nu}_{I,\epsilon}\cap{\rm Con}^{\nu}_{J,\delta}$
is Zariski open. Then the maps $\Psi_{J,I}^{\delta,\epsilon}$ are defined by
extending the transition maps. They fit in the following diagram.

${S^{n}}$${{\rm Con}^{\lambda}}$${{\rm Con}^{\nu}_{I,\epsilon}}$${{\rm
Con}^{\nu}_{I,\epsilon}\cap{\rm Con}^{\nu}_{J,\delta}}$${{\rm
Con}^{\nu}_{J,\delta}}$${{\rm
Con}^{\rho}}$${S^{n}}$${(\mathbb{P}^{1})^{n}}$${{\rm Bun}}$${{\rm
Bun}^{I}}$${{\rm Bun}^{J}}$${{\rm
Bun}}$${(\mathbb{P}^{1})^{n}}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\Psi_{J,I}^{\delta,\epsilon}}$$\displaystyle{\rm
Par}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\Phi_{I}^{\epsilon}}$$\subset$$\subset$$\displaystyle{\Phi_{J}^{\delta}}$$\displaystyle{\pi_{\delta}}$$\displaystyle{\pi_{\delta}}$$\displaystyle{\rm
Par}$$\displaystyle{\pi_{\delta}}$

If $\delta=\epsilon$ then we can complete the diagram. By abuse of notation,
we may write

${\rm Bun}^{IJ}={\rm Bun}^{I}\cap{\rm Bun}^{J}$

the space parabolic bundles in ${\rm Bun}^{I}$ that are also $\mu$-stable for
$\mu\in\mathfrak{C}_{J}$ and vice-versa. Thus we get a birational map ${\rm
Bun}^{I}\dashrightarrow{\rm Bun}^{J}$ extending the identity on ${\rm
Bun}^{IJ}$. Hence we get

${S^{n}}$${S^{n}}$${(\mathbb{P}^{1})^{n}}$${(\mathbb{P}^{1})^{n}}$$\displaystyle{\pi_{\epsilon}}$$\displaystyle{\Psi_{J,I}^{\epsilon,\epsilon}}$$\displaystyle{\pi_{\epsilon}}$

∎

###  3.3. Connections with unstable parabolic bundles.

Now we describe the space $\mathcal{Z}_{n}$ of (isomorphism classes of)
connections such that every underlying parabolic bundle is not semistable. We
will show that it falls in two cases:

  1. (1)

If $n$ is even then $\mathcal{Z}_{n}=\emptyset$;

  2. (2)

If $n$ is odd then $\mathcal{Z}_{n}$ has four connected components, each
isomorphic to $\mathbb{C}^{n}$.

Note that in the last case $\dim\mathcal{Z}_{n}=n$. Let us make our first
reduction.

######  Lemma 3.9.

If $n$ is even then $\mathcal{Z}_{n}$ is empty and if $n$ is odd then

(3.2) $\mathcal{Z}_{n}=\\{(E,\nabla)\in\mathfrak{Con}^{\nu}\mid E=L\oplus
L^{-1}(w_{\infty})\text{ with }L^{2}=\mathcal{O}_{C}(D+w_{\infty})\\}.$

###### Proof.

Suppose that $\mathcal{Z}_{n}\neq\emptyset$ and let
$(E,\nabla)\in\mathcal{Z}_{n}$. We will show that $E=L\oplus
L^{-1}(w_{\infty})$ with $L^{2}=\mathcal{O}_{C}(D+w_{\infty})$. In particular,
$n$ must be odd.

We know that $(E,{\bf p}^{\epsilon}(\nabla))$ is indecomposable for any
$\epsilon$, see Remark 3.2. Then we may apply Lemma 2.4. We cannot have
$E=E_{1}$ since it would imply $(E_{1},\nabla)\in{\rm Con}^{\nu}$; hence
$E=L\oplus L^{-1}(w_{\infty})$. In this case, we may take $\epsilon$ such that
$p_{k}^{\epsilon_{k}}(\nabla)\not\subset L_{t_{k}}$ for every $k=1,\dots,n$.
Note that, by Lemma 2.6, the case $2\deg L\leq n$ is not possible. Therefore
Lemma 2.4 implies that $L^{2}=\mathcal{O}_{C}(D+w_{\infty})$.

Conversely, any connection on $L\oplus L^{-1}(w_{\infty})$, with
$L^{2}=\mathcal{O}_{C}(D+w_{\infty})$, represents a point in
$\mathcal{Z}_{n}$, see Remark 2.5. ∎

Next we will describe the connections in $\mathcal{Z}_{n}$. In order to do so,
we compute the logarithmic Atiyah class $\phi^{A}_{E}\in{\rm End}(E)^{\vee}$
whose vanishing establishes the existence of a connection with prescribed
residues, see [3]. Let $T\in{\rm End}(E)$ then $\phi^{A}_{E}$ is defined by

$\phi^{A}_{E}(T)=\phi^{0}_{E}(T)+\sum_{j=1}^{n}{\rm
tr}\left(A_{j}T(t_{j})\right)$

where $A_{j}$ is the residue endomorphism over $t_{j}$ and $\phi^{0}_{E}$ is
the classical Atiyah class, see [2]. In our case,

$A_{j}=\begin{pmatrix}u_{j}&a_{j}\\\
v_{j}&b_{j}\end{pmatrix}\begin{pmatrix}\nu_{j}^{+}&0\\\
0&\nu_{j}^{-}\end{pmatrix}\begin{pmatrix}b_{j}&-a_{j}\\\
-v_{j}&u_{j}\end{pmatrix}$

where $u_{j}b_{j}-a_{j}v_{j}=1$. Here we may take local coordinates around
each $t_{j}$ such that $L$ and $L^{-1}(w_{\infty})$ correspond to $(1:0)$ and
$(0:1)$, respectively.

Note that any direction $p_{k}^{\epsilon_{k}}(\nabla)$ lies outside $L$,
otherwise there would exist a choice of parabolic directions ${\bf
p}^{\epsilon}(\nabla)$ such that $(E,{\bf p}^{\epsilon}(\nabla))$ is
decomposable and this would force a relation on eigenvalues $\nu$. Indeed, we
can find an embedding of $L^{-1}(w_{\infty})$ passing through $n-1$ directions
away from $L$. Then we suppose, without loss of generality, that our
directions are as in the Figure 2. In particular $(u_{j},v_{j})=(0,1)$ and
$a_{j}=-1$ for $j\geq 2$, and $u_{1}v_{1}\neq 0$. Up to applying a diagonal
automorphism of $E$ we suppose that $u_{1}=v_{1}=1$, i.e.
$p_{1}^{+}(\nabla)=(1:1)$.

$t_{1}$$t_{2}$$t_{3}$$t_{n}$$L^{-1}(w_{\infty})$$L$$p_{2}^{+}$$p_{3}^{+}$$p_{n}^{+}$$p_{2}^{-}$$p_{3}^{-}$$p_{n}^{-}$$p_{1}^{-}$$p_{1}^{+}$$\cdots$$\cdots$
Figure 2. Possible configuration of directions for
$(E,\nabla)\in\mathcal{Z}_{n}$.

Note that ${\rm End}(E)$ is generated (as a vector space) by the identity,
nilpotent endomorphisms and the projection to $L$. For the identity,
$\phi^{A}_{E}(1_{E})$ gives the Fuchs relation that we already know is valid.
Let $\beta\in{\rm H^{0}}(C,L^{2}(-w_{\infty}))$ and define

$P(\beta):=\phi^{A}_{E}\left(\begin{pmatrix}0&\beta\\\
0&0\end{pmatrix}\right)=b_{1}\beta(t_{1})(\nu_{1}^{+}-\nu_{1}^{-})+\sum_{j\geq
2}b_{j}\beta(t_{j})(\nu_{j}^{+}-\nu_{j}^{-}).$

For $j\geq 2$ let $\beta_{j}\in{\rm H^{0}}(C,L^{2}(-w_{\infty}))$ with the
following property: $\beta_{j}(t_{k})=0$ if $k\neq 1,j$ and
$\beta_{j}(t_{j})=1$. These sections are unique. In particular,
$\beta_{j}(t_{1})\neq 0$ and we have

$P(\beta_{j})=b_{1}\beta_{j}(t_{1})(\nu_{1}^{+}-\nu_{1}^{-})+b_{j}(\nu_{j}^{+}-\nu_{j}^{-})$

Note that the image of evaluation map ${\rm
H^{0}}(C,L^{2}(-w_{\infty}))\rightarrow\mathbb{C}^{n}$,
$\beta\mapsto(\beta(t_{1}),\dots,\beta(t_{n}))$, has dimension $(n-1)$; hence
the images of the $\beta_{j}$ define basis. Therefore $P(\beta)=0$ for every
$\beta\in{\rm H^{0}}(C,L^{2}(-w_{\infty}))$ if and only if $P(\beta_{j})=0$
for $j\geq 2$, i.e. the $\nu_{j}^{-}$ direction is

$(-1:b_{j})=\left(\nu_{j}^{+}-\nu_{j}^{-}:b_{1}\beta_{j}(t_{1})(\nu_{1}^{+}-\nu_{1}^{-})\right)$

For the projection to $L$ we have

$\phi^{A}_{E}\left(\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\right)=\deg
L+\sum_{j=1}^{n}u_{j}b_{j}\nu_{j}^{+}-a_{j}v_{j}\nu_{j}^{-}=b_{1}(\nu_{1}^{+}-\nu_{1}^{-})+\frac{n+1}{2}+\sum_{j=1}^{n}\nu_{j}^{-}=0.$

This implies that the directions over $t_{1}$ are $p_{1}^{+}(\nabla)=(1:1)$
and

$p_{1}^{-}(\nabla)=\left(b_{1}-1:b_{1}\right)=\left(\frac{n+1}{2}+\sum_{j\geq
2}^{n}\nu_{j}^{-}+\nu_{j}^{+}:\frac{n+1}{2}+\sum_{j=1}^{n}\nu_{j}^{-}\right)$

Therefore the residues are completely independent of the isomorphism class of
$(E,\nabla)$, i.e. the residues of every connection in $\mathcal{Z}_{n}$ are,
up to ${\rm Aut}(E)$, in the above configuration. Also note that any two
connections with these residues differ by an element of ${\rm
Hom}(E,E\otimes\Omega_{C})$ with vanishing trace. From this discussion we can
prove the following result.

######  Theorem 3.10.

Let $n$ be an odd integer. Then $\mathcal{Z}_{n}$ has four connected
components, each of them being isomorphic to $\mathbb{C}^{n}$.

###### Proof.

First note that there exist precisely four possibilities for the underlying
vector bundle of a connection in $\mathcal{Z}_{n}$. Indeed, Lemma 3.9 shows
that any such vector bundle is $E=L\oplus L^{-1}(w_{\infty})$ where $L$ is
such that $L^{2}=\mathcal{O}_{C}(D+w_{\infty})$. Twisting by $2$-torsion line
bundles yields four non-isomorphic possibilities for $L$. Hence four non-
isomorphic possibilities for $E$. Therefore $\mathcal{Z}_{n}$ has four
connected components.

Fix one such $E$ and denote $\mathcal{Z}_{n}^{E}$ the corresponding component
of $\mathcal{Z}_{n}$. Up to the action of ${\rm Aut}(E)$, we can fix a
configuration of directions as in Figure 2 so that we may only consider
connections on $E$ that have this configuration. Note that since
$L^{2}=\mathcal{O}_{C}(D+w_{\infty})$ the stabilizer of such configuration in
${\rm Aut}(E)$ is a copy of the additive group $(\mathbb{C},+)$ generated by

$\begin{pmatrix}1&\beta\\\ 0&1\end{pmatrix}$

where $\beta\in H^{0}(L^{2}(-w_{\infty}))\setminus\\{0\\}$ vanishes on $D$. On
the other hand, if we fix a connection $\nabla_{0}$ on $E$, for any other
connection $\nabla$, the difference $\nabla-\nabla_{0}\in{\rm
Hom}(E,E\otimes\Omega_{C})$ is a holomorphic Higgs field. Since $\nabla$ and
$\nabla_{0}$ must have the same trace, this Higgs field is traceless. Thus we
have an isomorphism

${\mathcal{Z}_{n}^{E}}$${{\rm
Higgs}_{0}(E)/(\mathbb{C},+)}$${\left[\nabla\right]}$${\left[\nabla-\nabla_{0}\right]}$

where ${\rm Higgs}_{0}(E)$ is the space of traceless Higgs fields. Note that
${\rm Higgs}_{0}(E)\simeq\mathbb{C}^{n+1}$.

We now explicitly describe the action of $(\mathbb{C},+)$ on ${\rm
Higgs}_{0}(E)$. Locally, we can write

$\nabla_{0}=d+\begin{pmatrix}a_{0}&b_{0}\\\ c_{0}&d_{0}\end{pmatrix}\text{ and
}\varphi=\begin{pmatrix}a_{1}&b_{1}\\\ 0&-a_{1}\end{pmatrix}.$

Then the action of $t\in\mathbb{C}$ is given by

$\displaystyle t\cdot\varphi$ $\displaystyle=\begin{pmatrix}1&-t\beta\\\
0&1\end{pmatrix}\left[\begin{pmatrix}0&td\beta\\\
0&0\end{pmatrix}+\begin{pmatrix}a_{0}+a_{1}&b_{0}+b_{1}\\\
c_{0}&d_{0}-a_{1}\end{pmatrix}\begin{pmatrix}1&t\beta\\\
0&1\end{pmatrix}\right]-\begin{pmatrix}a_{0}&b_{0}\\\
c_{0}&d_{0}\end{pmatrix}=$ $\displaystyle=\begin{pmatrix}1&-2t\beta\\\
0&1\end{pmatrix}\begin{pmatrix}a_{1}&b_{1}\\\
0&-a_{1}\end{pmatrix}+\begin{pmatrix}-tc_{0}\beta&t[(a_{0}-d_{0})\beta+d\beta]-t^{2}c_{0}\beta^{2}\\\
0&tc_{0}\beta\end{pmatrix}.$

In particular, the action is given by affine transformations. Also note that
$c_{0}=0$ would force an integer relation
$\nu_{1}^{\epsilon_{1}}+\cdots+\nu_{n}^{\epsilon_{n}}\in\mathbb{Z}$ which is
not possible; hence the action is free. This concludes the proof since the
quotient of an affine free action of $(\mathbb{C},+)$ on $\mathbb{C}^{n+1}$
must be isomorphic to $\mathbb{C}^{n}$, see [24, Corollary 7]. ∎

In addition, notice that given $(E,\nabla)$ representing a point in
$\mathcal{Z}_{n}$ we may perform an elementary transformation centered at $n$
parabolic directions outside $L$ (see (3.2)), then $E$ can be transformed into
$E_{0}\otimes L_{i}$, where $L_{i}^{2}=\mathcal{O}_{C}$ and $E_{0}$
corresponds to the unique, up to isomorphism, nontrivial extension

$0\longrightarrow\mathcal{O}_{C}\longrightarrow
E_{0}\longrightarrow\mathcal{O}_{C}\longrightarrow 0.$

We conclude that any connection representing a point in $\mathcal{Z}_{n}$ can
be obtained from a connection over an indecomposable vector bundle, with
trivial determinant line bundle, by performing an elementary transformation
centered in $n$ parabolic directions which lie in the unique maximal
subbundle. Note that there exist $n$ directions to choose and $\dim{\rm
Higgs}_{0}(E_{0}\otimes L_{i})=1$ giving dimension $n+1$ for the space of
connections on $E_{0}\otimes L_{i}$. Taking the quotient by the action of the
automorphism group of the corresponding vector bundle, we get dimension $n$.

###  3.4. The classical Painlevé case

Here it is worth to compare the case $n=1$ with the Painlevé case, i.e. those
logarithmic connections over the $4$-punctured Riemann sphere. We assume that
$D=w_{\infty}$ and let $(\nu,-\nu-1)$, $\nu\in\mathbb{C}\setminus\mathbb{Z}$,
denote the eigenvalues. The moduli space $\mathfrak{Con}^{\nu}$ parametrizes
$(E,\nabla)$, where $\nabla$ is a logarithmic connection over $C$ with a
single logarithmic pole over the point $w_{\infty}$, i.e. it is a Lamé
connection. The whole moduli space writes

$\mathfrak{Con}^{\nu}={\rm Con}^{\nu}\sqcup\mathcal{Z}_{1}$

where ${\rm Con}^{\nu}\simeq S$ is formed by $(E,\nabla)$ with $E=E_{1}$. It
follows from Theorem 3.10 that $\mathcal{Z}_{1}$ is a union of four
irreducible curves isomorphic to $\mathbb{C}$:

(3.3) $\mathcal{Z}_{1}^{i}=\\{(E,\nabla)\in\mathfrak{Con}^{\nu}\mid E=L\oplus
L^{-1}(w_{\infty})\text{ with }L=\mathcal{O}_{C}(w_{i})\\}$

for $i\in\\{0,1,\lambda,\infty\\}$, where $w_{i}$ are Weierstrass points.

We let ${\rm Con}^{\theta}(\mathbb{P}^{1},T)$, $T=0+1+\lambda+\infty$, denote
the moduli space of logarithmic connections over the $4$-punctured Riemann
sphere $(\mathbb{P}^{1},T)$, with eigenvalue

$\theta=\left(\pm\frac{1}{4},\pm\frac{1}{4},\pm\frac{1}{4},\theta^{\pm}\right)\;;\;\;\theta^{+}=\frac{\nu}{2}-\frac{1}{4}\;,\;\;\theta^{-}=-\theta^{+}-1.$

The determinant line bundle and the trace connection on it are fixed, i.e. we
fix a rank one connection $\zeta$ on $\mathcal{O}_{\mathbb{P}^{1}}(1)$ with
residue $-1$ at $\infty$ and for any $(E,\nabla)\in{\rm
Con}^{\theta}(\mathbb{P}^{1},T)$ we have

$(\det E,{\rm tr}\nabla)=(\mathcal{O}_{\mathbb{P}^{1}}(1),\zeta).$

The eigenvalue $\theta$ is chosen so that we can construct a map

$F\colon{\rm
Con}^{\theta}(\mathbb{P}^{1},T)\longrightarrow\mathfrak{Con}^{\nu}$

which turns out to be an isomorphism (see [19]), and which we now describe.
Given

$(E,\nabla)\in{\rm Con}^{\theta}(\mathbb{P}^{1},T)$

we first take its pullback $(\pi^{*}E,\pi^{*}\nabla)$ via the $2$-cover
$\pi:C\rightarrow\mathbb{P}^{1}$, the underlying vector bundle has determinant
$\det(\pi^{*}E)=\mathcal{O}_{C}(2w_{\infty})$. After $\pi^{*}$ the eigenvalues
are multiplied by $2$ and by performing an elementary transformation ${\rm
elem}_{0,1,\lambda}$ over the parabolic points $0,1,\lambda\in\mathbb{P}^{1}$,
centered on directions corresponding to eigenvalues $1/2$, we get

$(E^{\prime},\nabla^{\prime})={\rm elm}_{0,1,\lambda}(\pi^{*}E,\pi^{*}\nabla)$

with $\det E^{\prime}=\mathcal{O}_{C}(-w_{\infty})$, recall that
$w_{0}+w_{1}+w_{\lambda}\sim 3w_{\infty}$. The transformed connection
$\nabla^{\prime}$ has apparent singularities over $0,1,\lambda$, thus they
disappear after twist by a suitable rank one connection $\zeta_{0}$ on
$\mathcal{O}_{C}(w_{\infty})$. More precisely, we may fix $\zeta_{0}$ with
residues

$-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\text{ and }\;\;\frac{1}{2}$

over $0,1,\lambda$ and $\infty$ to get

$F(E,\nabla):=(E^{\prime},\nabla^{\prime})\otimes(\mathcal{O}_{C}(w_{\infty}),\zeta_{0})\in\mathfrak{Con}^{\nu}.$

We now describe the preimage of $\mathcal{Z}_{1}$ via $F$. For any
$(E,\nabla)\in{\rm Con}^{\theta}(\mathbb{P}^{1},T)$ the underlying vector
bundle is always
$E=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)$, see
[19]. Given two parabolic directions outside
$\mathcal{O}_{\mathbb{P}^{1}}(1)$, we can choose an embedding of
$\mathcal{O}_{\mathbb{P}^{1}}\rightarrow E$ passing through them, and there is
at most one direction lying in $\mathcal{O}_{\mathbb{P}^{1}}(1)$. For each
$i\in\\{0,1,\lambda\\}$ we let $p_{i}^{+}(\nabla)$ denote the parabolic
direction with respect to the eigenvalue $1/2$, we define

$A^{i}=\\{(E,\nabla)\in{\rm Con}^{\theta}(\mathbb{P}^{1},T)\mid
p_{i}^{+}(\nabla)\in\mathcal{O}_{\mathbb{P}^{1}}(1)\\}$

and let $A^{\infty}$ be the locus of $(E,\nabla)\in{\rm
Con}^{\theta}(\mathbb{P}^{1},T)$ such that there is an embedding of
$\mathcal{O}_{\mathbb{P}^{1}}$ passing through the three directions
$p_{0}^{+}(\nabla),p_{1}^{+}(\nabla),p_{\lambda}^{+}(\nabla)$.

######  Claim 1.

We claim that $F$ sends $A^{i}$ to $\mathcal{Z}_{1}^{i}$.

Let us assume $i=0$, other cases are similar. Pulling back $(E,\nabla)\in
A^{0}$ to the elliptic curve we get

$(\pi^{*}E,\pi^{*}\nabla)=(\mathcal{O}_{C}\oplus\mathcal{O}_{C}(2w_{\infty}),\pi^{*}\nabla)$

with one parabolic direction $p_{0}^{+}(\pi^{*}\nabla)$ lying in
$\mathcal{O}_{C}(2w_{\infty})$. We may assume that $\mathcal{O}_{C}$ pass
through $p_{1}^{+}(\nabla)$ and $p_{\lambda}^{+}(\nabla)$, thus after
elementary transformation ${\rm elem}_{0,1,\lambda}$ we get

$(E^{\prime},\nabla^{\prime})=(\mathcal{O}_{C}(-w_{0})\oplus\mathcal{O}_{C}(2w_{\infty}-w_{1}-w_{\lambda}),\nabla^{\prime})$

and twisting by $(\mathcal{O}_{C}(w_{\infty}),\zeta_{0})$ we obtain

$F(E,\nabla)=(\widetilde{E},\nabla^{\prime}\otimes\zeta_{0})$

where $\widetilde{E}\simeq L\oplus L^{-1}(w_{\infty})$, with
$L=\mathcal{O}_{C}(w_{0})$. Consequently, $F(E,\nabla)\in\mathcal{Z}_{1}^{0}$.
This proves the claim.

##  4\. Fuchsian systems with $n+1$ poles

Given $(E_{1},\nabla)\in{\rm Con}^{\nu}$, we can associate a
$\mathfrak{sl}_{2}$-connection on the trivial bundle
$\mathcal{O}_{C}\oplus\mathcal{O}_{C}$ by performing an elementary
transformation for a particular choice of directions over the $2$-torsion
points $w_{0}$, $w_{1}$ and $w_{\lambda}$. We begin the section establishing
this correspondence. The process creates new singularities which are apparent,
i.e. they become regular points after one elementary transformation.

We have that $t\in C$ is an apparent singular point for $\nabla$ if the
residual part ${\rm Res}_{t}\nabla$ has $\\{\frac{1}{2},-\frac{1}{2}\\}$ as
eigenvalues and the $\frac{1}{2}$-eigenspace of ${\rm Res}_{t}\nabla$ is also
invariant by the constant part of the connection matrix.

Recall that $D^{\prime}=w_{0}+w_{1}+w_{\lambda}+D$ is also reduced, see Remark
1.1. Let $(\nu_{1},\dots,\nu_{n})\in(\mathbb{C}^{\ast})^{n}$ and fix an
eigenvalue $\sigma$ by setting $\left(\frac{1}{2},-\frac{1}{2}\right)$ over
$w_{0},w_{1}$ and $w_{\lambda}$, and
$\left(\frac{\nu_{j}}{2},-\frac{\nu_{j}}{2}\right)$ over $t_{j}$,
$j=1,\dots,n$, i.e.

(4.1)
$\sigma=\left(\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{\nu_{1}}{2},-\frac{\nu_{1}}{2},\dots,\frac{\nu_{n}}{2},-\frac{\nu_{n}}{2}\right).$

We denote by ${\rm Syst}^{\sigma}(C,D^{\prime})$ the moduli space of Fuchsian
systems (i.e. logarithmic $\mathfrak{sl}_{2}$-connections on the trivial
bundle $\mathcal{O}_{C}\oplus\mathcal{O}_{C}$) having pole divisor
$D^{\prime}$, eigenvalue $\theta$ and such that:

  * •

the three singular points $w_{0},w_{1}$ and $w_{\lambda}$ are apparent;

  * •

over $w_{0}$, $w_{1}$ and $w_{\lambda}$ the corresponding
$\frac{1}{2}$-eigenspaces are $(1:0)$, $(1:1)$ and $(0:1)$ respectively.

$\mathcal{O}_{C}(w_{0}-w_{\infty})$$\mathcal{O}_{C}(w_{1}-w_{\infty})$$\mathcal{O}_{C}(w_{\lambda}-w_{\infty})$$w_{0}$$w_{1}$$w_{\lambda}$$\mathcal{O}_{C}$
Figure 3. Sections of $\mathbb{P}(E_{1})$

######  Proposition 4.1.

There is an isomorphism of moduli spaces

${\rm Con}^{\nu}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
Syst}^{\sigma}(C,D^{\prime})$

where $\displaystyle\nu_{j}=\nu^{+}_{j}-\nu^{-}_{j}$ for $j=1,\dots,n$.

###### Proof.

Let $(E_{1},\nabla)\in{\rm Con}^{\nu}$. First note that the intersection
configuration in Figure 3 tells us that, after the elementary transformation,
$\mathcal{O}_{C}(w_{0}-w_{\infty}),\mathcal{O}_{C}(w_{1}-w_{\infty}),\mathcal{O}_{C}(w_{\lambda}-w_{\infty})\subset
E_{1}$ become three disjoint copies of $\mathcal{O}_{C}(-w_{\infty})$. Hence
the elementary transformed of $(E_{1},\nabla)$ is $(E,\nabla^{\prime})$ where
$E=\mathcal{O}_{C}(-w_{\infty})\oplus\mathcal{O}_{C}(-w_{\infty})$ and the
eigenvalues of $\nabla^{\prime}$ are the same as $\nabla$ at the $t_{k}$,
$k=1,\dots,n$, and the eigenvalues at $w_{0},w_{1}$ and $w_{\lambda}$ are
equal to $(1,0)$. Consider the connection $(\mathcal{O}_{C}(w_{\infty}),\xi)$
such that

${\rm
Res}_{w_{0}}\xi=Res_{w_{1}}\xi=Res_{w_{\lambda}}\xi=-\frac{1}{2}\quad\text{and}\quad{\rm
Res}_{t_{k}}\xi=-\frac{\nu_{k}^{+}+\nu_{k}^{-}}{2}.$

It exists since $\sum_{k=1}^{n}\nu_{k}^{+}+\nu_{k}^{-}=-1$ by Fuchs relation.
Therefore

$(E_{1},\nabla)\longmapsto(\mathcal{O}_{C}\oplus\mathcal{O}_{C},\nabla^{\prime}\otimes\xi)\in{\rm
Syst}^{\sigma}(C,D^{\prime})$

is our desired isomorphism. ∎

Consider the space $\operatorname{Bun}_{0}$ of parabolic bundles $(E,{\bf p})$
over $(C,D^{\prime})$ such that $E$ is the trivial bundle, the parabolic
directions over $w_{0},w_{1}$ and $w_{\lambda}$ are $(1:0)$, $(1:1)$ and
$(0:1)$, respectively, and we let the parabolic directions over
$D=t_{1}+\cdots+t_{n}$ vary. We have a natural identification
$\operatorname{Bun}_{0}=(\mathbb{P}^{1})^{n}$ where each copy of
$\mathbb{P}^{1}$ parametrizes parabolic directions over $t_{j}$.

The elementary transformation used in the proof yields an alternative proof
for the isomorphism ${\rm Bun}\simeq(\mathbb{P}^{1})^{n}$ in Proposition 2.1.
Indeed, $(E_{1},{\bf p})$ corresponds to
$(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf p^{\prime}})$ and, up to
automorphism, we can fix the directions over $w_{0},w_{1}$ and $w_{\lambda}$
so that it defines an element of $\operatorname{Bun}_{0}$; in particular, this
gives us the identification

(4.2) $\displaystyle{\rm Bun}\simeq\operatorname{Bun}_{0}.$

###  4.1. The affine bundle of Fuchsian systems

The moduli space ${\rm Syst}^{\sigma}(C,D^{\prime})$ is an affine bundle of
rank $n$ over $\operatorname{Bun}_{0}$. We will describe its trivializations
over the Zariski open sets

$U_{0}=\prod_{j=1}^{n}\left\\{(z_{j}:w_{j})\in\mathbb{P}^{1}\mid
w_{j}=1\right\\}\quad{\text{a}nd}\quad
U_{\infty}=\prod_{j=1}^{n}\left\\{(z_{j}:w_{j})\in\mathbb{P}^{1}\mid
z_{j}=1\right\\},$

and then give the affine transition map.

In order to make explicit computations, we will consider that $C$ is given by
the affine equation $y^{2}=x(x-1)(x-\lambda)$, with
$\lambda\in\mathbb{C}\setminus\\{0,1\\}$, and $w_{\infty}\in C$ will be the
point at infinity. In particular, the torsion points are $w_{0}=(0,0)$,
$w_{1}=(1,0)$ and $w_{\lambda}=(\lambda,0)$. To begin with, let us fix a basis
of meromorphic one-forms with at most simple poles on $D^{\prime}$. Consider
the holomorphic one-form
$\omega=\frac{dx}{2y}=\frac{dy}{3x^{2}-2(1+\lambda)x+\lambda}$ and define

$\displaystyle\phi_{0}=\frac{(1-\lambda)x}{y}\omega,\;\displaystyle\phi_{1}=\frac{-\lambda(x-1)}{y}\omega\text{
and }\theta_{j}=\frac{x_{j}(x_{j}x-\lambda)}{x_{j}y-y_{j}x}\omega,$

where $t_{j}=(x_{j},y_{j})$. These $n+3$ one-forms give us the desired basis,
and their residues (and constant term at $w_{i}$) are given as follows in
terms of local coordinate $y$:

  * •

at $w_{0}$, we have $x=O(y^{2})$, and:

$\displaystyle\omega=\frac{dy}{\lambda}+O(y),$ $\displaystyle\phi_{0}=O(y),$
$\displaystyle\phi_{1}=\frac{dy}{y}+O(y),$
$\displaystyle\theta_{j}=-\frac{dy}{y}-\frac{y_{j}}{\lambda x_{j}}dy+O(y);$

  * •

at $w_{1}$, we have $x=1+O(y^{2})$, and:

$\displaystyle\omega=\frac{dy}{1-\lambda}+O(y),$
$\displaystyle\phi_{0}=\frac{dy}{y}+O(y),$ $\displaystyle\phi_{1}=O(y),$
$\displaystyle\theta_{j}=\frac{x_{j}(x_{j}-\lambda)}{y_{j}}dy+O(y);$

  * •

at $w_{\lambda}$, we have $x=\lambda+O(y^{2})$, and:

$\displaystyle\omega=\frac{dy}{\lambda(\lambda-1)}+O(y),$
$\displaystyle\phi_{0}=-\frac{dy}{y}+O(y),$
$\displaystyle\phi_{1}=-\frac{dy}{y}+O(y),$
$\displaystyle\theta_{j}=-\frac{x_{j}(x_{j}-1)}{y_{j}}dy+O(y);$

  * •

at $t_{j}$, the $1$-forms are holomorphic except for $\theta_{j}$ which has a
simple pole with ${\rm Res}_{t_{j}}\theta_{j}=1$.

Therefore, any Fuchsian system $\nabla\in{\rm Syst}^{\sigma}(C,D^{\prime})$
can be written as

$\nabla=d+\begin{pmatrix}a_{0}&b_{0}\\\
c_{0}&-a_{0}\end{pmatrix}\phi_{0}+\begin{pmatrix}a_{1}&b_{1}\\\
c_{1}&-a_{1}\end{pmatrix}\phi_{1}+\begin{pmatrix}a_{2}&b_{2}\\\
c_{2}&-a_{2}\end{pmatrix}\omega+\sum_{j=1}^{n}\begin{pmatrix}\alpha_{j}&\beta_{j}\\\
\gamma_{j}&-\alpha_{j}\end{pmatrix}\theta_{j},$

and, as we impose apparent singularities at $w_{0}$, $w_{1}$ and
$w_{\lambda}$, we get

$\displaystyle a_{0}$ $\displaystyle=-\sum_{j=1}^{n}\alpha_{j},$
$\displaystyle b_{0}$ $\displaystyle=\frac{1}{2}+\sum_{j=1}^{n}\alpha_{j},$
$\displaystyle c_{0}$ $\displaystyle=\frac{1}{2}-\sum_{j=1}^{n}\alpha_{j},$
$\displaystyle a_{1}$ $\displaystyle=\frac{1}{2}+\sum_{j=1}^{n}\alpha_{j},$
$\displaystyle b_{1}$ $\displaystyle=-\frac{1}{2}-\sum_{j=1}^{n}\alpha_{j},$
$\displaystyle c_{1}$ $\displaystyle=\sum_{j=1}^{n}\gamma_{j},$ $\displaystyle
a_{2}$
$\displaystyle=\sum_{j=1}^{n}\frac{y_{j}}{2}\left[\frac{2\alpha_{j}+\beta_{j}-\gamma_{j}}{x_{j}-1}+\frac{\gamma_{j}}{x_{j}}-\frac{\beta_{j}}{x_{j}-\lambda}\right],$
$\displaystyle b_{2}$
$\displaystyle=\sum_{j=1}^{n}\frac{y_{j}}{(x_{j}-\lambda)}\beta_{j},$
$\displaystyle c_{2}$
$\displaystyle=\sum_{j=1}^{n}\frac{y_{j}}{x_{j}}\gamma_{j}.$

In particular, $\nabla$ is completely determined by the residues

${\rm Res}_{t_{j}}(\nabla)=\begin{pmatrix}\alpha_{j}&\beta_{j}\\\
\gamma_{j}&-\alpha_{j}\end{pmatrix}.$

######  Proposition 4.2.

If $\nu_{j}\neq 0$ for $j=1,\dots,n$ then the map ${\rm Par}\colon{\rm
Syst}^{\sigma}(C,D^{\prime})\rightarrow S^{n}$ that associates to $\nabla$ the
eigenspaces of ${\rm Res}_{t_{j}}(\nabla)$, $j=1,\dots,n$, is an isomorphism.

###### Proof.

We know that $\nabla$ is determined by ${\rm Res}_{t_{j}}(\nabla)$,
$j=1,\dots,n$. On the other hand,

$\displaystyle\begin{pmatrix}\alpha_{j}&\beta_{j}\\\
\gamma_{j}&-\alpha_{j}\end{pmatrix}=\frac{1}{zv-wu}\begin{pmatrix}z&u\\\
w&v\end{pmatrix}\cdot\begin{pmatrix}\displaystyle\frac{\nu_{j}}{2}&0\\\
0&\displaystyle-\frac{\nu_{j}}{2}\end{pmatrix}\cdot\begin{pmatrix}v&-u\\\
-w&z\end{pmatrix}$

where $(z:w)$ and $(u:v)$ are the eigenspaces associated to
$\frac{\nu_{j}}{2}$ and $-\frac{\nu_{j}}{2}$, respectively. Note that
$S=\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus\\{zv-wu=0\\}$, hence the
isomorphism is clear. ∎

As a consequence we can prove Theorem 3.3.

###### Proof of Theorem 3.3 .

Now it is immediate to see that Proposition 4.1 and Proposition 4.2 combined
give an isomorphism by the composition

${\rm Con}^{\nu}\longrightarrow{\rm
Syst}^{\sigma}(C,D^{\prime})\longrightarrow S^{n}.$

∎

####  4.1.1. Trivialization over $U_{0}$

Given ${\bf z}=(z_{1},\dots,z_{n})\in U_{0}$ we define $\nabla_{0}$ as the
connection which has $(z_{j}:1)$ as eigenspaces corresponding to
$\frac{\nu_{j}}{2}$ and has $(1:0)$ as eigenspaces corresponding to
$-\frac{\nu_{j}}{2}$. It can be written as

$\nabla_{0}=d+\begin{pmatrix}a_{0}&b_{0}\\\
c_{0}&-a_{0}\end{pmatrix}\phi_{0}+\begin{pmatrix}a_{1}&b_{1}\\\
0&-a_{1}\end{pmatrix}\phi_{1}+\begin{pmatrix}a_{2}&b_{2}\\\
0&-a_{2}\end{pmatrix}\omega+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\begin{pmatrix}-1&2z_{j}\\\
0&1\end{pmatrix}\theta_{j}.$

with

$\begin{array}[]{lll}\displaystyle
a_{0}=\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
b_{0}=\frac{1}{2}-\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
c_{0}=\frac{1}{2}+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\\\ \displaystyle
a_{1}=\frac{1}{2}-\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
b_{1}=-\frac{1}{2}+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\\\ \displaystyle
a_{2}=\sum_{j=1}^{n}\frac{\nu_{j}y_{j}}{2}\left[\frac{z_{j}-1}{x_{j}-1}-\frac{z_{j}}{x_{j}-\lambda}\right]&\displaystyle
b_{2}=\sum_{j=1}^{n}\nu_{j}y_{j}\frac{z_{j}}{(x_{j}-\lambda)}\end{array}$

Now if $\nabla$ is any connection which has ${\bf z}$ as positive eigenspaces
then the difference $\Theta=\nabla-\nabla_{0}$ is a Higgs field which is
nilpotent with respect to ${\bf z}$. This means that $\Theta$ is a strongly
parabolic Higgs field over $(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf z})$.
We shall fix a basis $\\{\Theta_{1}^{0},\cdots,\Theta_{n}^{0}\\}$ for the
space of strongly parabolic Higgs fields such that

${\rm Res}_{t_{j}}\Theta_{j}^{0}=\begin{pmatrix}z_{j}&-z_{j}^{2}\\\
1&-z_{j}\end{pmatrix}$

and $\Theta_{j}^{0}$ is regular elsewhere. So, we define

$\displaystyle\Theta_{j}^{0}$ $\displaystyle=\begin{pmatrix}-z_{j}&z_{j}\\\
-z_{j}&z_{j}\end{pmatrix}\phi_{0}+\begin{pmatrix}z_{j}&-z_{j}\\\
1&-z_{j}\end{pmatrix}\phi_{1}+A_{j}\cdot\omega+\begin{pmatrix}z_{j}&-z_{j}^{2}\\\
1&-z_{j}\end{pmatrix}\theta_{j}$

where

$A_{j}=\begin{pmatrix}\frac{y_{j}}{2}\left[\frac{-(z_{j}-1)^{2}}{x_{j}-1}+\frac{1}{x_{j}}+\frac{z_{j}^{2}}{x_{j}-\lambda}\right]&-\frac{y_{j}z_{j}^{2}}{x_{j}-\lambda}\\\
\frac{y_{j}}{x_{j}}&-\frac{y_{j}}{2}\left[\frac{-(z_{j}-1)^{2}}{x_{j}-1}+\frac{1}{x_{j}}+\frac{z_{j}^{2}}{x_{j}-\lambda}\right]\end{pmatrix}.$

This matrix $A_{j}$ has been chosen to assure apparent singularities over
$w_{0},w_{1}$ and $w_{\lambda}$.

Any $\mathfrak{sl}_{2}$-connection $(E,\nabla)\in{\rm
Syst}^{\sigma}(C,D^{\prime})$ having $(z_{j}:1)$ as parabolic direction (over
$t_{j}$) corresponding to $\frac{\nu_{j}}{2}$ can be written as

$\nabla=\nabla_{0}+r_{j}\Theta_{j}^{0}$

for suitable $r_{j}\in\mathbb{C}$. It has $(z_{j}r_{j}-\nu_{j}:r_{j})$ as the
complementary direction corresponding to $-\frac{\nu_{j}}{2}$.

####  4.1.2. Trivialization over $U_{\infty}$

Similarly, we define $\nabla_{\infty}$ having $(1:w_{j})$ as eigenspaces
corresponding to $\frac{\nu_{j}}{2}$ and $(0:1)$ as eigenspaces corresponding
to $-\frac{\nu_{j}}{2}$:

$\nabla_{\infty}=d+\begin{pmatrix}a_{0}&b_{0}\\\
c_{0}&-a_{0}\end{pmatrix}\phi_{0}+\begin{pmatrix}a_{1}&b_{1}\\\
c_{1}&-a_{1}\end{pmatrix}\phi_{1}+\begin{pmatrix}a_{2}&0\\\
c_{2}&-a_{2}\end{pmatrix}\omega+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\begin{pmatrix}\displaystyle
1&\displaystyle 0\\\ \displaystyle
2w_{j}&\displaystyle-1\end{pmatrix}\theta_{j}$

with

$\begin{array}[]{lll}\displaystyle
a_{0}=-\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
b_{0}=\frac{1}{2}+\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
c_{0}=\frac{1}{2}-\sum_{j=1}^{n}\frac{\nu_{j}}{2}\\\ \displaystyle
a_{1}=\frac{1}{2}+\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
b_{1}=-\frac{1}{2}-\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
c_{1}=\sum_{j=1}^{n}\nu_{j}w_{j}\\\ \displaystyle
a_{2}=\sum_{j=1}^{n}\frac{\nu_{j}y_{j}}{2}\left[\frac{(1-w_{j})}{x_{j}-1}+\frac{w_{j}}{x_{j}}\right]&&\displaystyle
c_{2}=\sum_{j=1}^{n}\frac{y_{j}}{x_{j}}\nu_{j}w_{j}\end{array}$

And the strongly parabolic Higgs fields are defined by

$\displaystyle\Theta_{j}^{\infty}$
$\displaystyle=\begin{pmatrix}-w_{j}&w_{j}\\\
-w_{j}&w_{j}\end{pmatrix}\phi_{0}+\begin{pmatrix}w_{j}&-w_{j}\\\
w_{j}^{2}&-w_{j}\end{pmatrix}\phi_{1}+B_{j}\cdot\omega+\begin{pmatrix}w_{j}&-1\\\
w_{j}^{2}&-w_{j}\end{pmatrix}\theta_{j}$

where

$B_{j}=\begin{pmatrix}\frac{y_{j}}{2}\left[\frac{-(1-w_{j})^{2}}{x_{j}-1}+\frac{w_{j}^{2}}{x_{j}}+\frac{1}{x_{j}-\lambda}\right]&\frac{-y_{j}}{(x_{j}-\lambda)}\\\
\frac{y_{j}w_{j}^{2}}{x_{j}}&-\frac{y_{j}}{2}\left[\frac{-(1-w_{j})^{2}}{x_{j}-1}+\frac{w_{j}^{2}}{x_{j}}+\frac{1}{x_{j}-\lambda}\right]\end{pmatrix}.$

Given a Fuchsian system $\nabla\in{\rm Syst}^{\sigma}(C,D^{\prime})$ with
$(1:w_{j})$ as parabolic direction (over $t_{j}$) corresponding to
$\frac{\nu_{j}}{2}$, we can write

$\nabla=\nabla_{\infty}+s_{j}\Theta_{j}^{\infty}$

for suitable $s_{j}\in\mathbb{C}$. The complementary direction corresponding
to $-\frac{\nu_{j}}{2}$ is $(s_{j}:w_{j}s_{j}+\nu_{j})$.

####  4.1.3. Transition matrix

From the trivializations on $U_{0}$ and $U_{\infty}$ we may compute a
transition affine transformation for the affine bundle ${\rm
Syst}^{\sigma}(C,D^{\prime})$. Since $U_{0}\cup U_{\infty}$ covers the basis
$\operatorname{Bun}_{0}\simeq(\mathbb{P}^{1})^{n}$ minus a subvariety of
codimension two, then the bundle structure of ${\rm
Syst}^{\sigma}(C,D^{\prime})$ is determined by this affine transformation.

In order to make this affine transformation explicit we note that

$\Theta_{j}^{\infty}=w_{j}^{2}\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right),$

and

$\nabla_{0}\left(\frac{1}{w}\right)=d+\begin{pmatrix}a_{0}&b_{0}\\\
c_{0}&-a_{0}\end{pmatrix}\phi_{0}+\begin{pmatrix}a_{1}&b_{1}\\\
0&-a_{1}\end{pmatrix}\phi_{1}+\begin{pmatrix}a_{2}&b_{2}\\\
0&-a_{2}\end{pmatrix}\omega+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\begin{pmatrix}-1&\frac{2}{w_{j}}\\\
0&1\end{pmatrix}\theta_{j}.$

with

$\begin{array}[]{lll}\displaystyle
a_{0}=\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
b_{0}=\frac{1}{2}-\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
c_{0}=\frac{1}{2}+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\\\ \displaystyle
a_{1}=\frac{1}{2}-\sum_{j=1}^{n}\frac{\nu_{j}}{2}&\displaystyle
b_{1}=-\frac{1}{2}+\sum_{j=1}^{n}\frac{\nu_{j}}{2}\\\ \displaystyle
a_{2}=\sum_{j=1}^{n}\frac{y_{j}\nu_{j}}{2w_{j}}\left[\frac{1-w_{j}}{x_{j}-1}-\frac{1}{x_{j}-\lambda}\right]&\displaystyle
b_{2}=\sum_{j=1}^{n}\frac{y_{j}}{(x_{j}-\lambda)}\frac{\nu_{j}}{w_{j}}\end{array}$

Hence

(4.3)
$\nabla_{\infty}=\nabla_{0}\left(\frac{1}{w_{1}},\dots,\frac{1}{w_{n}}\right)+\sum_{j=1}^{n}\nu_{j}w_{j}\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right).$

Over the intersection $U_{0}\cap U_{\infty}$ we have

$\displaystyle(s_{1},\dots,s_{n})$
$\displaystyle\mapsto\nabla_{\infty}(w_{1},\dots,w_{n})+\sum_{j=1}^{n}s_{j}\Theta_{j}^{\infty}(w_{j})=$
$\displaystyle=\nabla_{0}\left(\frac{1}{w_{1}},\dots,\frac{1}{w_{n}}\right)+\sum_{j=1}^{n}\left(\nu_{j}w_{j}+s_{j}w_{j}^{2}\right)\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right)=$
$\displaystyle=\nabla_{0}(z_{1},\dots,z_{n})+\sum_{j=1}^{n}\left(\nu_{j}w_{j}+s_{j}w_{j}^{2}\right)\Theta_{j}^{0}(z_{j})=$
$\displaystyle=\nabla_{0}(z_{1},\dots,z_{n})+\sum_{j=1}^{n}r_{j}\Theta_{j}^{0}(z_{j})$

which gives us the transition affine transformation

(4.4) $\begin{pmatrix}r_{1}\\\ \vdots\\\
r_{n}\end{pmatrix}=\begin{pmatrix}w_{1}^{2}&\dots&0\\\ \vdots&\ddots&\vdots\\\
0&\dots&w_{n}^{2}\end{pmatrix}\cdot\begin{pmatrix}s_{1}\\\ \vdots\\\
s_{n}\end{pmatrix}+\begin{pmatrix}\nu_{1}w_{1}\\\ \vdots\\\
\nu_{n}w_{n}\end{pmatrix}.$

Note that if we choose new trivializations such that
$r^{\prime}_{j}=\frac{\rho_{j}}{\nu_{j}}r_{j}$ and
$s^{\prime}_{j}=\frac{\rho_{j}}{\nu_{j}}s_{j}$, this transition map becomes

$\begin{pmatrix}r_{1}^{\prime}\\\ \vdots\\\
r_{n}^{\prime}\end{pmatrix}=\begin{pmatrix}w_{1}^{2}&\dots&0\\\
\vdots&\ddots&\vdots\\\
0&\dots&w_{n}^{2}\end{pmatrix}\cdot\begin{pmatrix}s_{1}^{\prime}\\\ \vdots\\\
s_{n}^{\prime}\end{pmatrix}+\begin{pmatrix}\rho_{1}w_{1}\\\ \vdots\\\
\rho_{n}w_{n}\end{pmatrix}.$

Hence the affine bundles ${\rm Con}^{\nu}\xrightarrow{\pi_{+}}{\rm Bun}$ are
all isomorphic, not depending on $\nu$; this agrees with Theorem 3.3. The
transition map (4.4) can also be written as

(4.5) $\begin{pmatrix}\lambda\\\ r_{1}\\\ \vdots\\\
r_{n}\end{pmatrix}=\begin{pmatrix}1&0&\dots&0\\\
\nu_{1}w_{1}&w_{1}^{2}&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\nu_{n}w_{n}&0&\dots&w_{n}^{2}\end{pmatrix}\cdot\begin{pmatrix}\lambda\\\
s_{1}\\\ \vdots\\\ s_{n}\end{pmatrix}$

and can be thought, owing to (4.2), as the transition matrix of a vector
bundle $\mathcal{E}$ over ${\rm Bun}$ that parameterizes logarithmic
$\lambda$-connections: triples $(E_{1},\lambda,\nabla)$ where $\nabla\colon
E_{1}\rightarrow E_{1}\otimes\Omega_{C}(D)$ and $\lambda\in\mathbb{C}$ such
that

$\nabla(fs)=\lambda s\otimes df+f\nabla(s),$

see [26, Section 4]. If $\lambda\neq 0$ then $\frac{1}{\lambda}\nabla$ is an
usual logarithmic connection and if $\lambda=0$ then $\nabla$ is a Higgs
field. We note that the complement of $U_{0}\cup U_{\infty}$ is a subvariety
of codimension two, hence the transition matrix given in (4.5) determines
$\mathcal{E}$.

The projectivization of $\mathcal{E}$ provides a compactification
$\overline{{\rm Con}^{\nu}}$ for the moduli space ${\rm Con}^{\nu}$; in the
boundary we get the projectivized moduli space of Higgs fields

$\mathbb{P}\mathrm{Higgs}\simeq\overline{{\rm Con}^{\nu}}\setminus{\rm
Con}^{\nu}.$

The inclusion $\mathbb{P}\mathrm{Higgs}\hookrightarrow\mathbb{P}(\mathcal{E})$
comes from the natural extension

(4.6) $0\longrightarrow T^{*}{\rm
Bun}{\longrightarrow}\mathcal{E}{\longrightarrow}\mathcal{O}_{{\rm
Bun}}\longrightarrow 0$

where the first map is the inclusion of Higgs fields and the last is the
projection $(E_{1},\lambda,\nabla)\mapsto\lambda$. We saw that $\mathcal{E}$
does not depend on $\nu$. However, this extension, hence the inclusion
$\mathbb{P}\mathrm{Higgs}\hookrightarrow\mathbb{P}(\mathcal{E})$, is
determined by the exponents $\nu$.

######  Theorem 4.3.

The moduli space ${\rm Con}^{\nu}$ has compactification $\overline{{\rm
Con}^{\nu}}=\mathbb{P}(\mathcal{E})$, where the boundary divisor is isomorphic
to $\mathbb{P}\mathrm{Higgs}$, the projectivization of the space of Higgs
fields on $E_{1}$. Moreover, the inclusion
$\mathbb{P}\mathrm{Higgs}\hookrightarrow\mathbb{P}(\mathcal{E})$ is
determined, up to automorphisms of $\mathbb{P}(\mathcal{E})$, by
$\left(\nu_{1},\dots,\nu_{n}\right)$.

###### Proof.

The inclusion
$\phi\colon\mathbb{P}\mathrm{Higgs}\hookrightarrow\mathbb{P}(\mathcal{E})$
comes from the projectivization of (4.6). We will show that the isomorphism
class of this extension is determined by $\left(\nu_{1},\dots,\nu_{n}\right)$.
This is equivalent to determining $\phi$, up to composing with an automorphism
of $\mathbb{P}(\mathcal{E})$ as a $\mathbb{P}^{n}$-bundle over ${\rm Bun}$.
The proof will follow by computing the extension class $e^{\nu}\in H^{1}({\rm
Bun},T^{*}{\rm Bun})$, see [2, p. 185].

Over $U_{0}$, the map $\mathcal{O}_{{\rm Bun}}\rightarrow\mathcal{E}^{\nu}$,
given by $h\mapsto h\nabla_{0}$, defines a splitting; the same is true for
$\nabla_{\infty}$ over $U_{\infty}$. Therefore $e^{\nu}$ is represented, on
$U_{0}\cap U_{\infty}$, by

$\nabla_{\infty}-\nabla_{0}=\sum_{j=1}^{n}\nu_{j}w_{j}\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right),$

see the equation in display (4.3). Any cohomologous cocycle must be

$\displaystyle\sum_{j=1}^{n}\nu_{j}w_{j}\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right)+\sum_{j=1}^{n}b_{j}(w_{1},\dots
w_{n})\Theta_{j}^{\infty}(w_{j})-\sum_{j=1}^{n}a_{j}\left(\frac{1}{w_{1}},\dots,\frac{1}{w_{n}}\right)\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right)=$
$\displaystyle=\sum_{i=1}^{n}\left[\nu_{j}w_{j}+b_{j}(w_{1},\dots
w_{n})w_{j}^{2}-a_{j}\left(\frac{1}{w_{1}},\dots,\frac{1}{w_{n}}\right)\right]\Theta_{j}^{0}\left(\frac{1}{w_{j}}\right)$

for some holomorphic functions $a_{i}$ and $b_{i}$. In particular, the linear
terms $\nu_{j}w_{j}$ are left unchanged, whence the extension is completely
determined by $\left(\nu_{1},\dots,\nu_{n}\right)$. ∎

Returning to the affine bundle ${\rm Con}^{\nu}\xrightarrow{\pi_{+}}{\rm
Bun}$, we have seen that it does not distinguish $\nu$. However, if we
consider ${\rm Con}^{\nu}$ with the inherited symplectic structure from
$\mathfrak{Con}^{\nu}$, the picture is different; this will be the topic of
the next section. The trivializations in (4.4) will be important to bring the
symplectic form to the Darboux normal form.

##  5\. Symplectic structure

It follows from the work of Iwasaki [15] that moduli spaces of logarithmic
connections on curves carry a natural symplectic structure. In fact, there
exists a $2$-form defined on a larger moduli space where the poles are allowed
to move on $C$ inducing a Poisson structure whose kernel defines the
isomonodromic deformations, therefore a Hamiltonian system. Moreover, through
the Riemann-Hilbert correspondence, Iwasaki proved in [16] that this $2$-form
coincides with the symplectic structure on moduli spaces of representations
that originated through the works of Goldman. In fact, Iwasaki considers
moduli spaces of Sturm-Liouville operators rather than connections. In order
to define the symplectic structure globally on our moduli space, the approach
of Arinkin in [1] and Inaba-Iwasaki-Saito in [13] is more convenient. Let us
briefly recall how the tangent space and the symplectic structure are
constructed there. Later we will give an explicit formula for the $2$-form on
an open set of our moduli space.

First we recall the following result from [11, Section 15].

######  Lemma 5.1.

Assume that the connection $(E,\nabla)$ has a simple pole at $t\in C$ with
eigenvalues $\\{\nu^{+},\nu^{-}\\}$, and let
$z\colon(C,t)\rightarrow(\mathbb{C},0)$ be a local coordinate. Then there
exists a local trivialization of the vector bundle $\Phi\colon
E|_{(C,t)}\rightarrow(\mathbb{C},0)\times\mathbb{C}^{2}$ such that
$\Phi_{*}\nabla$ is one of the following models:

(5.1) $d+\begin{pmatrix}\nu^{+}&0\\\ 0&\nu^{-}\end{pmatrix}\frac{dx}{x},$

or, in the resonant case $\\{\nu^{+},\nu^{-}\\}=\\{\nu,\nu+n\\}$,
$n\in\mathbb{Z}_{\geq 0}$

(5.2) $d+\begin{pmatrix}\nu&x^{n}\\\ 0&\nu+n\end{pmatrix}\frac{dx}{x}.$

Moreover, in the resonant case $\nu^{+}-\nu^{-}\in\boldsymbol{Z}$, the model
(5.2) is generic among the two, i.e. occurs for an open set of the
deformations of $(E,\nabla)$.

Consider a first order deformation $(E_{\epsilon},\nabla_{\epsilon})$ of the
(generic) connection $(E,\nabla)$ with the same eigenvalues and same
determinant/trace. Consider a covering $\\{U_{i}\\}$ of $C$ such that the
restriction of $\nabla$ is either trivial, or can be normalized like in Lemma
5.1. Since the local models (5.1) with $\nu^{+}-\nu^{-}\not\in\mathbb{Z}$, and
(5.2) are stable under small deformations with fixed eigenvalues, we deduce
that there exist bundle isomorphisms

$\Phi_{i}\colon E_{\epsilon}|_{U_{i}}\rightarrow E|_{U_{i}}\text{ such that
}\nabla_{\epsilon}|_{U_{i}}=(\Phi_{i})^{*}\nabla|_{U_{i}}.$

Therefore, on $U_{i}\cap U_{j}$, the comparison
$\Phi_{ij}=\Phi_{i}^{-1}\circ\Phi_{j}$ defines automorphisms of $E|_{U_{i}\cap
U_{j}}$ preserving the connection $\nabla|_{U_{i}\cap U_{j}}$. Moreover, one
can choose $\Phi_{i}$, $\Phi_{j}$, and therefore $\Phi_{ij}$, preserving the
determinant/trace. One can check that $\\{\Phi_{ij}\\}$ defines an element of
$H^{1}(\mathrm{SL}(E,\nabla))$, the sheaf of bundle automorphisms preserving
$\nabla$ and acting trivially on $\det(E)$.

Note that $\Phi_{i}=\Phi_{i}^{0}+\epsilon\Phi_{i}^{1}$ where
$\\{\Phi_{i}^{0}\\}$ is a global automorphism of $E$ that preserves $\nabla$,
i.e. $\nabla=(\Phi_{i}^{0})^{*}\nabla$. It follows that $\Phi_{ij}$ is
cohomologous to $\text{id}+\epsilon\phi_{ij}$ and we get an element
$\\{\phi_{ij}\\}$ of $H^{1}(\mathfrak{sl}(E,\nabla))$, the sheaf of trace-free
endomorphisms preserving $\nabla$. Therefore

$T_{(E,\nabla)}\mathfrak{Con}^{\nu}\simeq H^{1}(\mathfrak{sl}(E,\nabla)).$

Observe that $\mathfrak{sl}(E,\nabla)$ is a subsheaf of
$\mathfrak{sl}(E,\textbf{p})$ the sheaf of endomorphisms of the quasi-
parabolic bundle $(E,\textbf{p})$ with trace zero. Also note that $\nabla$
induces a natural (logarithmic) connection on the vector bundle
$\mathfrak{sl}(E)$, namely
$\nabla^{1}\colon\phi\mapsto\nabla\circ\phi-\phi\circ\nabla$. We then
construct a complex of sheaves of abelian groups

(5.3) $\displaystyle
0\longrightarrow\mathfrak{sl}(E,\nabla)\longrightarrow\mathfrak{sl}(E,\textbf{p})\stackrel{{\scriptstyle\nabla^{1}}}{{\longrightarrow}}\mathfrak{sl}(E,\textbf{p})\otimes\Omega^{1}_{C}(D)^{\text{nil}}\longrightarrow
0$

where the rightmost member corresponds to parabolic Higgs fields with
nilpotent residues. The two sheaves on the right correspond to
$\mathcal{F}^{0}$ and $\mathcal{F}^{1}$ in [13, 12, 14, 18]. Note that
$\nabla^{1}$ is not $\mathcal{O}_{C}$-linear and $\mathfrak{sl}(E,\nabla)$ is
not an $\mathcal{O}_{C}$-module. Next we show that for a generic connection
the deformation complex (5.3) is indeed exact.

######  Lemma 5.2.

For a generic connection $(E,\nabla)$ in the moduli space, the sequence in
display (5.3) is exact.

###### Proof.

First note that the kernel of $\nabla^{1}$ is precisely
$\mathfrak{sl}(E,\nabla)$ and the leftmost map is the inclusion. Thus we only
need to show that $\nabla^{1}$ is surjective, for a generic $\nabla$. It will
follow from a direct computation with local models.

Outside the polar locus, $(E,\nabla)$ is locally trivial and the map
$\nabla^{1}$ may be written as

$\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}\longmapsto\begin{pmatrix}da&db\\\
dc&-da\end{pmatrix}$

so that local surjectivity of $\nabla^{1}$ at a regular point follows from
local integration of holomorphic $1$-forms. On the other hand, at a non
resonant pole, i.e. $\nu^{+}-\nu^{-}\not\in\mathbb{Z}$, the connection
$(E,\nabla)$ is locally like (5.1) and the map $\nabla^{1}$ may be written as

$\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}\longmapsto\begin{pmatrix}da&db+(\nu^{+}-\nu^{-})b\frac{dx}{x}\\\
dc+(\nu^{-}-\nu^{+})c\frac{dx}{x}&-da\end{pmatrix}$

where $c(0)=0$ (parabolic condition); for the local surjectivity, we just have
to check that a matrix of $1$-forms

$\begin{pmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{pmatrix}$

is in the image if and only if $\alpha$ and $\gamma$ are holomorphic
(parabolic and nilpotent condition), and $\beta$ has a simple pole, which is
straightforward. Finally, consider a resonant pole of the form (5.2). Then the
map $\nabla^{1}$ may be written as

$\begin{pmatrix}a&b\\\
c&-a\end{pmatrix}\longmapsto\begin{pmatrix}da+cx^{n}\frac{dx}{x}&db-(nb+2ax^{n})\frac{dx}{x}\\\
dc+nc\frac{dx}{x}&-da-cx^{n}\frac{dx}{x}\end{pmatrix}.$

To prove the surjectivity in that case, we have to successively integrate

$\left\\{\begin{matrix}dc+nc\frac{dx}{x}&=\gamma\\\
da+cx^{n}\frac{dx}{x}&=\alpha\\\
db-(nb+2ax^{n})\frac{dx}{x}&=\beta\end{matrix}\right.$

In the first equation, $c(x)$ can be derived such that $c(0)=0$
(parabolicity); therefore we can integrate the second one and find $a(x)$ up
to a constant; finally, for a good choice of $a(0)$, the last equation admits
a solution $b(x)$. ∎

Hereafter we will assume that $(E,\textbf{p})$ is $\mu$-stable for some weight
$\mu$ and we will denote by $\mathcal{B}$ the corresponding moduli space of
parabolic bundles.

Note that $(E,\textbf{p})$ is simple, i.e.
$H^{0}(\mathfrak{sl}(E,\textbf{p}))=0$, then we get
$H^{1}(\mathfrak{sl}(E,\textbf{p})\otimes\Omega^{1}_{C}(D)^{\text{nil}})=0$ by
Serre duality. The long exact sequence of cohomology of (5.3) gives

(5.4) $0\longrightarrow
H^{0}(\mathfrak{sl}(E,\textbf{p})\otimes\Omega^{1}_{C}(D)^{\text{nil}})\longrightarrow
H^{1}(\mathfrak{sl}(E,\nabla))\rightarrow
H^{1}(\mathfrak{sl}(E,\textbf{p}))\longrightarrow 0$

In the middle we get the tangent space to the moduli space of connections at
$(E,\nabla)$. On the left-hand side, we get the space of Higgs fields, which
does not depend on $\nabla$. And on the right-hand side we have
$H^{1}(\mathfrak{sl}(E,\textbf{p}))$ that is the tangent space to
$\mathcal{B}$ at $(E,\bf{p})$. In particular, the image of $\\{\phi_{ij}\\}$
in $H^{1}(\mathfrak{sl}(E,\textbf{p}))$, where we omit the connection and just
keep track of the parabolic data, corresponds to the underlying first order
deformation of the parabolic bundle.

We can check that the map
$H^{0}(\mathfrak{sl}(E,\textbf{p})\otimes\Omega^{1}_{C}(D)^{\text{nil}})\hookrightarrow
H^{1}(\mathfrak{sl}(E,\nabla))$ of (5.4) is the derivative of the natural
action $\nabla\mapsto\nabla+\theta$ for a global parabolic Higgs field on the
affine space of parabolic connections on $(E,\textbf{p})$. Indeed, if we write
$\nabla_{\epsilon}=\nabla+\epsilon\theta$ and
$\Phi_{i}=\text{id}+\epsilon\phi_{i}$, then we get

$\displaystyle(\nabla+\epsilon\theta)|_{U_{i}}=(\Phi_{i})^{*}\nabla$
$\displaystyle=(\text{id}-\epsilon\phi_{i})\circ\nabla\circ(\text{id}+\epsilon\phi_{i})=$
$\displaystyle=\nabla+\epsilon(\nabla\circ\phi_{i}-\phi_{i}\circ\nabla)=\nabla+\epsilon\nabla^{1}(\phi_{i})$

The first order deformation is therefore encoded in $\\{\phi_{j}-\phi_{i}\\}$,
which, by construction, has zero image in
$H^{1}(\mathfrak{sl}(E,\textbf{p}))$. But the image of $\theta$ in
$H^{1}(\mathfrak{sl}(E,\nabla))$ given by (5.4) gives exactly the same
formula: we first integrate $\nabla^{1}\phi_{i}=\theta|_{U_{i}}$ and then
associate $\\{\phi_{j}-\phi_{i}\\}$.

One can define a bilinear map on $T_{(E,\nabla)}\mathfrak{Con}^{\nu}$ as
follows:

(5.5) ${H^{1}(\mathfrak{sl}(E,\nabla))\times
H^{1}(\mathfrak{sl}(E,\nabla))}$${H^{2}(\mathbb{C})\simeq\mathbb{C};}$${(\\{\phi_{ij}\\},\\{\psi_{ij}\\})}$${\\{u_{ijk}={\rm
tr}(\phi_{ij}\psi_{jk})\\}.}$$\displaystyle{\displaystyle\omega_{\rm Con}}$

One can choose the covering $\\{U_{i}\\}$ such that $(E,\nabla)$ is trivial
over $U_{i}\cap U_{j}$. Observe that $\phi_{ij}\psi_{jk}$ defines a section of
$\mathfrak{gl}(E,\nabla)$ for which trace and determinant are constant;
indeed, in a trivialization of $(E,\nabla)$ on $U_{i}\cap U_{j}\cap U_{k}$,
the section $\phi_{ij}\psi_{jk}$ is constant. This is why $u_{ijk}$ is a
section of the constant sheaf $\mathbb{C}$. One can show that this bilinear
form is non degenerate, and defines a closed $2$-form $\omega_{\rm Con}$ on
$\mathfrak{Con}^{\nu}$ which is symplectic:

$d\omega_{\rm Con}\equiv 0\text{ and }\omega^{n}_{\rm
Con}=\underbrace{\omega_{\rm Con}\wedge\cdots\wedge\omega_{\rm Con}}_{n\
\text{times}}\neq 0,$

i.e. $\omega^{n}_{\rm Con}$ defines a holomorphic volume form, a
trivialization of the canonical bundle. We refer to [13, Section 6] for
details and proofs.

######  Remark 5.3.

For the non generic resonant case

$d+\begin{pmatrix}\nu&0\\\ 0&\nu+n\end{pmatrix}\frac{dx}{x},\ \ \
n\in\mathbb{Z}_{\geq 0}$

the sequence (5.3) is not exact anymore: $\nabla^{1}$ is not surjective. This
is why the description of the symplectic structure is more complicated in [1,
13, 12, 14, 18]. They use hypercohomology to overcome this difficulty. In our
case, we want to understand the symplectic structure at a generic point of the
moduli space, so it is not necessary to consider this kind of models.

In order to describe the symplectic structure $\omega_{\rm Con}$, we will make
a reduction step. We will see that, locally, we can split
$T_{(E,\nabla)}\mathfrak{Con}^{\nu}$ as a direct sum of two distinguished
subspaces: one concerning Higgs fields and the other being the tangent space
to a Lagrangian submanifold. This will allow us to compare $\omega_{\rm Con}$
with the natural symplectic form for Higgs fields.

Consider

${\rm Higgs}(E,{\bf
p})=H^{0}(\mathfrak{sl}(E,\textbf{p})\otimes\Omega^{1}_{C}(D)^{\text{nil}})$

the space of Higgs fields for $E$ with nilpotent residues preserving ${\bf
p}$. We define a pairing

(5.6) ${{\rm Higgs}(E,{\bf p})\times
H^{1}(\mathfrak{sl}(E,\textbf{p}))}$${H^{1}(\Omega^{1}_{C})\simeq\mathbb{C}}$${(\Theta,\\{\phi_{ij}\\})}$${\\{{\rm
tr}(\Theta\cdot\phi_{ij})\\}}$$\displaystyle{\displaystyle\eta}$

One proves, see [15], that $\eta$ is a perfect pairing via identifying
$H^{1}(\mathfrak{sl}(E,\textbf{p}))^{\vee}\simeq T^{*}_{(E,{\bf
p})}\mathcal{B}$ and $\text{Higgs}(E,\textbf{p})$. In other words, pairing
$\eta$ corresponds to the Liouville form on $T^{*}\mathcal{B}$.

Consider the following diagram

(5.7) ${H^{1}(\mathfrak{sl}(E,\nabla))\times
H^{1}(\mathfrak{sl}(E,\nabla))}$${H^{2}(\mathbb{C})}$${\text{Higgs}(E,\textbf{p})\times
H^{1}(\mathfrak{sl}(E,\nabla))}$${\text{Higgs}(E,\textbf{p})\times
H^{1}(\mathfrak{sl}(E,\textbf{p}))}$${H^{1}(\Omega^{1})}$$\displaystyle{\displaystyle\omega_{\rm
Con}}$$\displaystyle{\displaystyle\eta}$$\displaystyle{\displaystyle\sim}$$\displaystyle{\displaystyle\delta}$

where we have used the maps from (5.4) and $\delta$ is the connecting morphism
coming from the long exact sequence associated to

$0\longrightarrow\mathbb{C}\longrightarrow\mathcal{O}_{C}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\Omega^{1}_{C}\longrightarrow 0.$

Then we reconcile $\omega_{\rm Con}$ and the pairing $\eta$.

######  Lemma 5.4.

The diagram (5.7) is commutative.

###### Proof.

Let $(\theta,\\{\psi_{ij}\\})$ be an element of
$\text{Higgs}(E,\textbf{p})\times H^{1}(\mathfrak{sl}(E,\nabla))$. As $\theta$
is a parabolic Higgs field, one can locally write
$\theta|_{U_{i}}=\nabla^{1}(\varphi_{i})$ and the upper map yields
$(\\{\varphi_{j}-\varphi_{i}\\},\\{\psi_{ij}\\})\in
H^{1}(\mathfrak{sl}(E,\nabla))^{2}$, which is sent to
$\\{\text{tr}((\varphi_{j}-\varphi_{i})\circ\psi_{jk})\\}\in
H^{2}(\mathbb{C})$. On the other hand, by first going down, the pairing $\eta$
from (5.6) gives $\\{\text{tr}(\theta\circ\psi_{ij})\\}\in H^{1}(\Omega^{1})$.
To compute the image by $\delta$, we first remark that we have

$\displaystyle\text{tr}(\theta\circ\psi_{ij})$
$\displaystyle=\text{tr}(\nabla^{1}(\varphi_{i})\circ\psi_{ij})=$
$\displaystyle=\text{tr}(\nabla^{1}(\varphi_{i}\circ\psi_{ij}))-\text{tr}(\varphi_{i}\circ\nabla^{1}(\psi_{ij}))=$
$\displaystyle=\text{tr}(\nabla^{1}(\varphi_{i}\circ\psi_{ij}))=$
$\displaystyle=d(\text{tr}(\varphi_{i}\circ\psi_{ij}))$

Here, we have used that $\nabla^{1}(\psi_{ij})=0$ and the properties
$\nabla^{1}(F\circ G)=\nabla^{1}(F)\circ G+F\circ\nabla^{1}(G)$ and
$\text{tr}(\nabla^{1}(F))=d(\text{tr}(F))$, which can be deduced from a local
computation in matrix form. Therefore, we have

$\displaystyle\delta(\\{\text{tr}(\theta\circ\psi_{ij})\\})$
$\displaystyle=\\{\text{tr}(\varphi_{i}\circ\psi_{ij})+\text{tr}(\varphi_{j}\circ\psi_{jk})-\text{tr}(\varphi_{i}\circ\psi_{ik})\\}=$
$\displaystyle=\\{\text{tr}(\varphi_{i}\circ(\psi_{ij}-\psi_{ik}))+\text{tr}(\varphi_{j}\circ\psi_{jk})\\}=$
$\displaystyle=\\{\text{tr}((\varphi_{j}-\varphi_{i})\circ\psi_{jk})\\}$

and get the same result. ∎

Given a choice of directions $\epsilon\in\\{+.-\\}^{n}$, let $\mathcal{C}$
denote the open subset of $\mathfrak{Con}^{\nu}$ composed by the connections
$(E,\nabla)$ such that $(E,{\bf p}^{\epsilon}(\nabla))$ belongs to
$\mathcal{B}$. Recall that a subvariety is called Lagrangian the restriction
of symplectic form vanishes identically.

Assume now that we are given a local section $\nabla_{0}\colon
B\rightarrow\mathcal{C}$ on an open subset $B\subset\mathcal{B}$. Then we can
define an isomorphism to the moduli space of parabolic Higgs bundles

$\Psi_{\nabla_{0}}\colon\left.\mathcal{C}\right|_{B}\longrightarrow\left.{\rm
Higgs}\right|_{B}$

by setting $(E,\nabla)\mapsto(E,\nabla-\nabla_{0}(E,\textbf{p}))$; the inverse
being given by $(E,\varphi)\mapsto(E,\nabla_{0}(E,\textbf{p})+\varphi)$.
Recall that (5.6) identifies Higgs fields and the cotangent space to
$\mathcal{B}$. This yields an identification between the moduli space of
parabolic Higgs bundles and the cotangent bundle $T^{*}\mathcal{B}$ which,
itself, carries the Liouville symplectic structure that we will call
$\omega_{\text{Higgs}}$.

Recall that $\Psi_{\nabla_{0}}$ is a symplectomorphism if
$\Psi^{*}_{\nabla_{0}}\omega_{\text{Higgs}}=\omega_{\text{Con}}$. On the other
hand, we say that the section $\nabla_{0}$ is Lagrangian if so is its image.

######  Proposition 5.5.

The isomorphism $\Psi_{\nabla_{0}}$ is a symplectomorphism if and only if the
section $\nabla_{0}\colon B\longrightarrow\mathcal{C}$ is Lagrangian.

Before proving this proposition, let us recall some basic property of the
Liouville form $\omega_{\text{Liouville}}$ on $T^{*}M$ for an arbitrary
manifold $M$. If $(p_{1},\ldots,p_{n})$ are coordinates on $M$, and if we
denote by $(q_{1},\ldots,q_{n})$ the coordinates on the fibers such that
$dp_{i}$ corresponds to the section $q_{i}\equiv 1$ and $q_{j}\equiv 0$ for
$j\not=i$, then $\omega_{\text{Liouville}}=\sum_{i=1}^{n}dp_{i}\wedge dq_{i}$.
Given a section $\alpha\colon M\longrightarrow T^{*}M$, we can define an
affine bundle transformation by

$\psi_{\alpha}\colon T^{*}M\longrightarrow T^{*}M;\
(p,q)\longmapsto(p,q+\alpha(p)),$

where $T^{*}M$ is viewed as an affine bundle.

######  Lemma 5.6.

Let $M$, $\alpha$ and $\psi_{\alpha}$ as above. Then the following are
equivalent:

  * •

$\psi_{\alpha}$ is a symplectomorphism,

  * •

$\alpha(M)$ is Lagrangian (as a submanifold),

  * •

$d\alpha=0$ (i.e. closed as a $1$-form).

###### Proof of Proposition 5.5.

We want to prove that $\Psi_{\nabla_{0}}$ is a symplectomorphism; then we only
need to show that
$\Psi^{*}_{\nabla_{0}}\omega_{\text{Higgs}}=\omega_{\text{Con}}$ at an
arbitrary point $(E,\varphi)\in\text{Higgs}$. As the vector bundle Higgs
identifies by (5.6) to $T^{*}\mathcal{B}$, we can use Lemma 5.6 to compose
with an affine transformation sending $\varphi$ to the zero Higgs field. Now,
the tangent space decomposes as

$T_{(E,0)}\text{Higgs}=\text{Higgs}(E,\textbf{p})\oplus
T_{(E,\textbf{p})}\mathcal{B}$

and the symplectic form $\omega_{\text{Higgs}}$ may be written as

$\omega_{\text{Higgs}}(u_{1}+v_{1},u_{2}+v_{2})=\eta(u_{1},v_{2})-\eta(u_{2},v_{1})$

where $\eta$ is the pairing given by (5.6). Now, the tangent space at
$(E,\nabla)$ decomposes as

$T_{(E,\nabla)}\mathcal{C}=\text{Higgs}(E,\textbf{p})\oplus
T_{(E,\nabla)}\nabla_{0}(B)=\text{Higgs}(E,\textbf{p})\oplus
D\nabla_{0}(T_{(E,\textbf{p})}{\mathcal{B}})$

and note that the differential of $\Psi_{\nabla_{0}}^{-1}$ is the block
diagonal $D\Psi_{\nabla_{0}}^{-1}=id\oplus D\nabla_{0}$. Now
$\omega_{\text{Con}}$ may be written as

$\displaystyle\omega_{\rm
Con}\\!\left(u_{1}+D\nabla_{0}v_{1},u_{2}+D\nabla_{0}v_{2}\right)$
$\displaystyle=\omega_{\rm
Con}\\!\left(u_{1},D\nabla_{0}v_{2}\right)-\omega_{\rm
Con}\\!\left(u_{2},D\nabla_{0}v_{1}\right)+$
$\displaystyle\quad\quad+\omega_{\rm Con}(u_{1},u_{2})+\omega_{\rm
Con}\\!\left(D\nabla_{0}v_{1},D\nabla_{0}v_{2}\right)$
$\displaystyle=\eta\\!\left(u_{1},D\nabla_{0}v_{2}\right)-\eta\\!\left(u_{2},D\nabla_{0}v_{1}\right)+\omega_{\rm
Con}\\!\left(D\nabla_{0}v_{1},D\nabla_{0}v_{2}\right)$
$\displaystyle=\Psi^{*}_{\nabla_{0}}\omega_{\text{Higgs}}\\!\left(u_{1}+v_{1},u_{2}+v_{2}\right)+\omega_{\rm
Con}\\!\left(D\nabla_{0}v_{1},D\nabla_{0}v_{2}\right)$

Indeed, it follows from the diagram (5.7). To prove the vanishing of
$\omega_{\rm Con}(u_{1},u_{2})$, note that by (5.4) the image of
$\text{Higgs}(E,\textbf{p})$ in $H^{1}(\mathfrak{sl}(E,\textbf{p}))$ is zero.
Hence by taking the lower path in (5.7) the pairing $\eta$ yields $0\in
H^{1}(\Omega^{1})$.

Therefore $\Psi^{*}_{\nabla_{0}}\omega_{\text{Higgs}}=\omega_{\text{Con}}$ if
and only if $\omega_{\rm
Con}\\!\left(D\nabla_{0}v_{1},D\nabla_{0}v_{2}\right)=0$, i.e. $\nabla_{0}(B)$
is Lagrangian. ∎

In our situation, we can compute this symplectic structure explicitly on a
Zariski open set. Indeed, let $\mathcal{C}\rightarrow\mathcal{B}$ be given by
the choice of directions
$\epsilon=(\epsilon_{1},\dots,\epsilon_{n})\in\\{-,+\\}^{n}$ and let
$\delta=-\epsilon$ be the opposite choice. One can verify that a fiber of
$(E,\nabla)\mapsto(E,{\bf p}^{\delta}(\nabla))$ is a rational section of
$(E,\nabla)\mapsto(E,{\bf p}^{\epsilon}(\nabla))$. The reduction is symplectic
if the chosen section is Lagrangian, wherever it is defined. This is ensured
if the projection is Lagrangian, which is covered by the following result.

######  Lemma 5.7.

The forgetful map $\mathcal{C}\longrightarrow\mathcal{B};\
(E,\nabla)\mapsto(E,\textbf{p}^{\epsilon}(\nabla))$ is Lagrangian, i.e. its
fibers are Lagrangian subvarieties.

###### Proof.

The differential of the forgetful map at $(E,\nabla)$ determines the dual to
the map $\text{Higgs}(E,\textbf{p})\longrightarrow
H^{1}(\mathfrak{sl}(E,\nabla))$ from (5.4). Then we want to prove that the
pairing $\omega_{\rm Con}$ of (5.5) is zero on a pair
$(\\{\phi_{ij}\\},\\{\psi_{ij}\\})$ where each entry is in the image of
$\text{Higgs}(E,\textbf{p})$. We showed that this is true in the proof of the
last proposition. ∎

We now apply this construction to exhibit the symplectic form on ${\rm
Con}^{\nu}$ via explicit computations with Fuchsian systems.

###  5.1. Computation of the symplectic structure

We have seen that choosing a Lagrangian section $\nabla_{0}$ of
$\pi_{\epsilon}\colon{\rm Con}^{\nu}\rightarrow{\rm Bun}$ allows us to reduce
the computation of the symplectic structure of $\mathcal{\rm Con}^{\nu}$ to
that of Higgs moduli space (see Proposition 5.5). We can apply this to the
setting of Fuchsian systems. The section $\nabla_{0}$ constructed in section
4.1.1 corresponds, via the isomorphism of Proposition 4.1, to the fiber of
$\pi_{-\epsilon}$ over the point
$(\infty,\ldots,\infty)\in\mathrm{Bun}\simeq(\mathbb{P}^{1})^{n}$; it is
therefore Lagrangian. Let us compute the symplectic structure on the moduli
space of Higgs bundles. For this, we consider the chart:

$\mathbb{C}^{2n}\longrightarrow\mathrm{Higgs}(C,D);\
(z_{1},\ldots,z_{n},r_{1},\ldots,r_{n})\longmapsto(E_{1},\Theta)$

where $\Theta$ is obtained by applying elementary transformations at $w_{0}$,
$w_{1}$, $w_{\lambda}$ to

$r_{1}\Theta^{0}_{1}+\cdots+r_{n}\Theta^{0}_{n}$

as defined in section 4.1.1. The image coincides with the restriction
$\mathrm{Higgs}(C,D)|_{U_{0}}$ of Higgs fields on parabolic bundles belonging
to the chart $U_{0}\subset\mathrm{Bun}$ of section 4.1.1. We want now to
compute the isomorphism $\mathrm{Higgs}(C,D)|_{U_{0}}\simeq T^{*}U_{0}$ so
that the symplectic form can be easily deduced from the Liouville form.

Here we follow the ideas of the proof of [10, Proposition 6.1]. Given a
parabolic vector bundle $(E_{1},{\bf p})$ in $U_{0}$, or equivalently
$(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf p^{\prime}})$ defined by
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$, Serre Duality (5.6) provides a
perfect pairing

$\left<\cdot,\cdot\right>\colon{\rm Higgs}(E,{\bf p})\times
H^{1}(\mathfrak{sl}(E,\textbf{p}))\longrightarrow\mathbb{C}$

which allows us to identify the fiber of ${\rm Higgs}$ at $(E_{1},{\bf p})$
with the dual of the tangent space

$T_{(E_{1},{\bf p})}U_{0}\simeq H^{1}(\mathfrak{sl}(E_{1},\textbf{p})).$

Let us compute first this latter identification.

The deformation $(z_{1},\ldots,z_{j}+s,\ldots,z_{n})$ of the $j^{\text{th}}$
component of the parabolic structure can be realized as follows. Cut out the
curve $C$ into two open sets $U_{j}=C-\\{t_{j}\\}$ and $V_{j}$ a small disk
around $t_{j}$. Then we can paste the restrictions of
$(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf p^{\prime}})$ to these open sets
by means of the following gauge transformation over the restriction

$\left.\left(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf
p^{\prime}}\right)\right|_{U_{j}\cap
V_{j}}\longrightarrow\left.\left(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf
p^{\prime}}\right)\right|_{U_{j}\cap V_{j}};\ \begin{pmatrix}z\\\
w\end{pmatrix}\longmapsto\begin{pmatrix}1&s\\\
0&1\end{pmatrix}\begin{pmatrix}z\\\ w\end{pmatrix}.$

This clearly shifts the parabolic direction $(z_{j}:1)\mapsto(z_{j}+s:1)$
without moving other parabolic directions. After derivating, we obtain that
the infinitesimal deformation $\frac{\partial}{\partial z_{j}}$ corresponds to
the gauge transformation $\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\in
H^{0}(U_{j}\cup V_{j},\mathfrak{sl}(\mathcal{O}_{C}\oplus\mathcal{O}_{C},{\bf
p^{\prime}}))$, which can better be viewed as a cocyle with respect to the
covering $(U_{j},V_{j})$. We can now compute the Serre Duality: the pairing of
the Higgs field $\Theta^{0}_{i}$ with the above cocycle is defined by

$\mathrm{tr}\left(\Theta^{0}_{k}\cdot\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right)=-z_{k}\phi_{0}+\phi_{1}+\frac{y_{k}}{x_{k}}\omega+\theta_{k}$

a holomorphic $1$-form on $U_{j}\cap V_{j}$. Viewed as a cocycle in
$H^{1}(\Omega)$, one can give a meromorphic resolution by considering it as
the difference between the zero $1$-form on $U_{j}$ and the meromorphic
$1$-form $-z_{k}\phi_{0}+\phi_{1}+\frac{y_{k}}{x_{k}}\omega+\theta_{k}$ on
$V_{j}$. The residue at $t_{j}$ gives the pairing:

$\left<\Theta^{0}_{k},\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right>=\mathrm{Res}_{t_{j}}\\!\left(-z_{k}\phi_{0}+\phi_{1}+\frac{y_{k}}{x_{k}}\omega+\theta_{k}\right)=\delta_{j}^{k}$

where $\delta_{j}^{k}$ is the Kronecker symbol. We deduce that
$\Theta^{0}_{j}$ is dual to $\frac{\partial}{\partial z_{j}}$, and the
symplectic structure on $\mathrm{Higgs}\ni\sum_{j}r_{j}\Theta_{j}^{0}$ is
given by

$\omega_{\rm Higgs}=dz_{1}\wedge dr_{1}+\cdots+dz_{n}\wedge dr_{n}.$

Now, because our section $\nabla_{0}\colon U_{0}\rightarrow\mathrm{Con}$ is
Lagrangian, we can apply Proposition 5.5 and deduce the symplectic structure
on $\mathrm{Con}\ni\nabla_{0}+\sum_{j}r_{j}\Theta_{j}^{0}$, namely:

(5.8) $\omega_{\rm Con}=dz_{1}\wedge dr_{1}+\cdots+dz_{n}\wedge dr_{n}.$

The same construction shows, mutatis mutandis, that the symplectic form on
$\nabla_{\infty}+\sum_{j}s_{j}\Theta_{j}^{\infty}$ is

(5.9) $\omega_{\rm Con}=-(dw_{1}\wedge ds_{1}+\cdots+dw_{n}\wedge ds_{n}).$

Next we will see how $\omega_{\rm Con}$ encodes the eigenvalue $\nu$. First we
consider only maps that preserves the affine bundle structure.

######  Theorem 5.8.

If there exists a bundle symplectic isomorphism

${\left(\mathcal{\rm Con}^{\nu},\omega\right)}$${\left(\mathcal{\rm
Con}^{\tilde{\nu}},\tilde{\omega}\right)}$${{\rm
Bun}}$$\displaystyle{\pi_{+}}$$\displaystyle{\Phi}$$\displaystyle{\sim}$$\displaystyle{\pi_{+}}$

then $\tilde{\nu}=\nu$ and $\Phi$ is the identity.

###### Proof.

Since $\pi_{+}\circ\Phi=\pi_{+}$, it preserves $\pi_{+}^{-1}(U_{0})$ and
$\pi_{+}^{-1}(U_{\infty})$. Then we may write the map in coordinates. For
${\rm Con}^{\nu}$ we will consider the two coordinate patches:

$(\pi_{+}^{-1}(U_{0}),(z,r))\text{ and }(\pi_{+}^{-1}(U_{\infty}),(w,s))$

where $z_{j}=\frac{1}{w_{j}}$ and $r_{j}=s_{j}w_{j}^{2}+2\nu_{j}w_{j}$. For
${\rm Con}^{\tilde{\nu}}$ we consider the corresponding open sets with
coordinates $(z,\tilde{r})$ and $(w,\tilde{s})$, respectively, where
$z_{j}=\frac{1}{w_{j}}$ and
$\tilde{r}_{j}=\tilde{s}_{j}w_{j}^{2}+2\tilde{\nu}_{j}w_{j}$. In both cases,
$\pi_{+}$ is the projection onto the first $n$ coordinates.

Therefore we can express $\Phi$ as

$\displaystyle\left.\Phi\right|_{\pi_{+}^{-1}(U_{0})}$
$\displaystyle=\left(z_{1},\cdots,z_{n},\phi^{0}_{1}(z,r),\cdots,\phi^{0}_{n}(z,r)\right)=(z,\tilde{r}),$
$\displaystyle\left.\Phi\right|_{\pi_{+}^{-1}(U_{\infty})}$
$\displaystyle=\left(w_{1},\cdots,w_{n},\phi^{\infty}_{1}(w,s),\cdots,\phi^{\infty}_{n}(w,s)\right)=(w,\tilde{s}),$

where $\phi^{0}_{j}(z,r)$ and $\phi^{\infty}_{j}(w,s)$ are holomorphic
functions on $\mathbb{C}^{2n}$ satisfying the compatibility conditions:

(5.10)
$\phi^{\infty}_{j}(\cdots,w_{k},\cdots,s_{k},\cdots)w_{j}^{2}+\tilde{\nu_{j}}w_{j}=\phi^{0}_{j}\left(\cdots,\frac{1}{w_{k}},\cdots,s_{k}w_{k}^{2}+\nu_{k}w_{k},\cdots\right),\quad\forall
w\in(\mathbb{C}^{\ast})^{n}.$

The condition that $\Phi$ is symplectic, $\Phi^{\ast}\tilde{\omega}=\omega$,
translates to the coordinate charts via the $2$-forms in display (5.8) and
(5.9). In particular, $\phi^{0}_{j}$ and $\phi^{\infty}_{j}$ must satisfy

$\frac{\partial\phi^{0}_{j}}{\partial
r_{k}}=\frac{\partial\phi^{\infty}_{j}}{\partial s_{k}}=\delta^{j}_{k},$

where $\delta^{j}_{k}$ is the Kronecker symbol. Therefore, there exist
holomorphic functions $c_{j}^{0}(z)$ and $c_{j}^{\infty}(w)$ such that

$\displaystyle\phi^{0}_{j}(z,r)=r_{j}+c_{j}^{0}(z)\text{ and
}\phi^{\infty}_{j}(w,s)=s_{j}+c_{j}^{\infty}(w).$

Substituting these expressions in (5.10) yields

(5.11)
$c_{j}^{0}\left(\frac{1}{w_{1}},\cdots,\frac{1}{w_{n}}\right)=c_{j}^{\infty}(w)w_{j}^{2}+(\tilde{\nu_{j}}-\nu_{j})w_{j},\quad\forall
w\in(\mathbb{C}^{\ast})^{n}.$

It follows that $c_{j}^{0}$ extends as a holomorphic function on $U_{0}\cup
U_{\infty}$. On the other hand, the complement of $U_{0}\cup U_{\infty}$ in
$(\mathbb{P}^{1})^{n}$ has codimension $2$, which implies that $c_{j}^{0}$
extends to the whole space by Hartogs’s Theorem. Thus $c_{j}^{0}$ is constant
and so is $c_{j}^{\infty}$, by the same argument. Thus, the equation (5.11)
implies that

$c_{j}^{0}=c_{j}^{\infty}=0\text{ and }\tilde{\nu_{j}}=\nu_{j}.$

∎

We can use the map $\mathrm{Par}\colon{\rm Con}^{\nu}\rightarrow S^{n}$ given
by Theorem A to push $\omega_{\rm Con}$ forward to $S^{n}$. The opposite
parabolic structure of our universal family
$\nabla=\nabla_{0}+\sum_{j}r_{j}\Theta_{j}^{0}$ is given by

$(\zeta_{j}:1):=(z_{j}r_{j}-\nu_{j}:r_{j}).$

Therefore, if we denote by $(z_{j},\zeta_{j})$ the coordinates on the
$j^{\text{th}}$ factor
$S\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus\\{z_{j}=\zeta_{j}\\}$,
then we have

$\zeta_{j}=z_{j}-\frac{\nu_{j}}{r_{j}}\ \ \ \Leftrightarrow\ \ \
r_{j}=\frac{\nu_{j}}{z_{j}-\zeta_{j}}$

which yields after substitution:

$\omega_{\nu}=\sum_{i=1}^{n}\nu_{i}\frac{dz_{i}\wedge
d\zeta_{i}}{(z_{i}-\zeta_{i})^{2}}.$

Using this identification we can prove the following corollary.

######  Corollary 5.9.

If there exists fiber preserving symplectic isomorphism

${\left({\rm Con}^{\nu},\omega\right)}$${\left({\rm
Con}^{\tilde{\nu}},\tilde{\omega}\right)}$${{\rm Bun}}$${{\rm
Bun}}$$\displaystyle{\pi_{+}}$$\displaystyle{\Phi}$$\displaystyle{\sim}$$\displaystyle{\pi_{+}}$$\displaystyle{\phi}$$\displaystyle{\sim}$

then there exists a permutation $\sigma$ of $n$ elements such that
$\tilde{\nu}_{k}=\nu_{\sigma(k)}$ for every $k\in\\{1,\cdots,n\\}$.

###### Proof.

Note that the case where $\phi$ is the identity is ensured by Theorem 5.8. In
particular, the identity on ${\rm Bun}$ has a unique lifting which is the
identity on ${\rm Con}^{\nu}$. It follows that, a general $\Phi$ is determined
by the map $\phi$ on the base. Indeed, if $\Phi_{1}$ and $\Phi_{2}$ lift
$\phi$, then $\Phi_{2}\circ\Phi_{1}^{-1}$ lifts the identity, hence
$\Phi_{2}=\Phi_{1}$. The proof of the statement will follow from a reduction
to this case.

Up to composing with ${\rm Par}$, we may prove the result for

$\Phi\colon(S^{n},\omega_{\nu})\longrightarrow(S^{n},\omega_{\tilde{\nu}}).$

Any automorphism $\phi$ of $(\mathbb{P}^{1})^{n}$ has the form

$\phi(z)=\left(\varphi_{1}(z_{\sigma(1)}),\cdots,\varphi_{n}(z_{\sigma(n)})\right)$

where $\sigma$ is a permutation of $n$ elements and $\varphi_{i}$ are Möbius
transformations. Now define

$\Xi(z,\zeta)=\left(\varphi_{1}(z_{\sigma(1)}),\cdots,\varphi_{n}(z_{\sigma(n)}),\varphi_{1}(\zeta_{\sigma(1)}),\cdots,\varphi_{n}(\zeta_{\sigma(n)})\right).$

We will show that $\Xi$ is an automorphism of $S^{n}$ such that
$\Xi^{\ast}\omega_{\nu^{\sigma}}=\omega_{\nu}$, where
$\nu_{j}^{\sigma}=\nu_{\sigma(j)}$. Recall that any Möbius transformation is a
composition of basic transformations $t\mapsto\alpha t$, $t\mapsto t+1$ and
$t\mapsto 1/t$. Therefore we can factorize each $\varphi_{j}$ to show that

$\left(\frac{d\varphi_{j}(z_{\sigma(j)})\wedge
d\varphi_{j}(\zeta_{\sigma(j)})}{(\varphi_{j}(z_{\sigma(j)})-\varphi_{j}(\zeta_{\sigma(j)}))^{2}}\right)=\frac{dz_{\sigma(j)}\wedge
d\zeta_{\sigma(j)}}{\left(z_{\sigma(j)}-\zeta_{\sigma(j)}\right)^{2}}$

for all $j\in\\{1,\cdots,n\\}$, and this shows that
$\Xi^{*}(\omega_{\nu})=\omega_{\nu^{\sigma}}$.

Now consider the $\Phi$ the unique extension of $\phi$ such that
$\Phi^{\ast}\omega_{\tilde{\nu}}=\omega_{\nu}$. We have that
$\Phi\circ\Xi^{-1}$ lifts the identity and
$\left(\Phi\circ\Xi^{-1}\right)^{\ast}\\!\omega_{\tilde{\nu}}=\omega_{\nu^{\sigma}}$.
Thus the result follows from Theorem 5.8. ∎

##  6\. Apparent map

Given a connection $\nabla\colon E_{1}\rightarrow
E_{1}\otimes\Omega^{1}_{C}(D)$ we can define an $\mathcal{O}_{C}$-linear map:

$\varphi_{\nabla}\colon\mathcal{O}_{C}\stackrel{{\scriptstyle
s}}{{\hookrightarrow}}E_{1}\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}E_{1}\otimes\Omega_{C}^{1}(D)\longrightarrow(E_{1}/\mathcal{O}_{C})\otimes\Omega_{C}^{1}(D)$

where the last arrow is defined by the quotient map from $E_{1}$ to
$E_{1}/s(\mathcal{O}_{C})\simeq\mathcal{O}_{C}(w_{\infty})$ (denoted
$E_{1}/\mathcal{O}_{C}$ for short). The zero divisor of $\varphi_{\nabla}$
defines an element of the linear system

$Z(\varphi_{\nabla})\in|\mathcal{O}_{C}(w_{\infty}+D)|.$

Since $\deg D=n$ and $\Omega_{C}^{1}\simeq\mathcal{O}_{C}$, as $C$ has genus
one, then

$|\mathcal{O}_{C}(w_{\infty}+D)|=\mathbb{P}\\!\left(H^{0}(C,(E_{1}/\mathcal{O}_{C})\otimes\Omega_{C}^{1}(D))\right)\simeq\mathbb{P}^{n}.$

Hence we may define a rational map

$\operatorname{App}\colon{\rm
Con}^{\nu}\dashrightarrow|\mathcal{O}_{C}(w_{\infty}+D)|$

which associates $(E_{1},\nabla)$ to $Z(\varphi_{\nabla})$. Under a generic
hypothesis on the eigenvalues we show that $\operatorname{App}$ is in fact a
morphism. It turns out that this hypothesis is also necessary as we prove in
the following lemma.

######  Lemma 6.1.

The rational map $\operatorname{App}\colon{\rm
Con}^{\nu}\dashrightarrow|\mathcal{O}_{C}(w_{\infty}+D)|$ is a morphism if and
only if $\nu_{1}^{\epsilon_{1}}+\cdots+\nu_{n}^{\epsilon_{n}}\neq 0$ for any
$\epsilon_{k}\in\\{+,-\\}$.

###### Proof.

The indeterminacy locus of $\operatorname{App}$ is composed by the connections
$\nabla$ mapped to zero in $(E_{1}/\mathcal{O}_{C})\otimes\Omega_{C}^{1}(D)$
which means that there exists a meromorphic $1$-form $\xi$ with poles at most
on $D$ such that

$\nabla(s)=s\xi.$

In such case we have for each $j=1,\dots,n$ that $s(t_{j})\in
p_{j}^{\epsilon_{j}}(\nabla)$ with $\epsilon_{j}\in\\{+,-\\}$. And we also
have that $\nabla$ restricts to a logarithmic connection on $\mathcal{O}_{C}$
whence

$\nu_{1}^{\epsilon_{1}}+\cdots+\nu_{n}^{\epsilon_{n}}=0.$

Conversely, suppose that there exist $\delta_{j}\in\\{+,-\\}$, $j=1,\dots n$,
such that $\nu_{1}^{\delta_{1}}+\cdots+\nu_{n}^{\delta_{n}}=0$. On the one
hand there exists a meromorphic $1$-form $\xi$ with simple poles only on $D$
and prescribed residues

$Res_{t_{j}}\xi=\nu_{j}^{\delta_{j}}.$

On the other hand let $\nabla$ be a logarithmic connection such that
$s(t_{j})\in p_{j}^{\delta_{j}}(\nabla)$. It follows that $\nabla(s)-s\xi$ is
a holomorphic section of $E_{1}\otimes\Omega_{C}^{1}$. Since
$H^{0}(C,E_{1}\otimes\Omega_{C}^{1})$ is generated by $s\omega$ (recall that
$\omega$ is a holomorphic one-form on $C$) it follows that

$\nabla(s)=s\,(\xi+c\,\omega)$

for some $c\in\mathbb{C}$. Hence $\varphi_{\nabla}=0$. ∎

We may also consider the forgetful map

$\pi_{+}\colon{\rm Con}^{\nu}\longrightarrow{\rm
Bun}\simeq(\mathbb{P}^{1})^{n}$

which sends $(E_{1},\nabla)$ to the parabolic vector bundle $(E_{1},{\bf
p}^{+}(\nabla))$. We also denote $\pi_{+}$ the natural extension of this map
to $\overline{{Con}^{\nu}}$. Combining these two maps we can give a birational
model for $\overline{{Con}^{\nu}}$ as we prove in the following result.

######  Theorem 6.2.

If $\nu_{1}^{\epsilon_{1}}+\cdots+\nu_{n}^{\epsilon_{n}}\neq 0$ for any
$\epsilon_{k}\in\\{+,-\\}$ then the map $\pi_{+}\times\operatorname{App}$
induces a birational map

$\pi_{+}\times\operatorname{App}\colon\overline{{Con}^{\nu}}\dashrightarrow{\rm
Bun}\times|\mathcal{O}_{C}(w_{\infty}+D)|$

whose indeterminacy locus is contained in
$\overline{{Con}^{\nu}}\backslash{Con}^{\nu}$. Moreover, given
$(E,{\bf{p}})\in\operatorname{Bun}_{0}$ the rank of

$\left.(\pi_{+}\times\operatorname{App})\right|_{\pi_{+}^{-1}(E,{\bf{p}})}\colon\pi_{+}^{-1}(E,{\bf{p}})\longrightarrow|\mathcal{O}_{C}(w_{\infty}+D)|$

coincides with the cardinality of the set $\\{j\mid
p_{j}\not\subset\mathcal{O}_{C}\\}$.

###### Proof.

Let $s$ denote the section $\mathcal{O}_{C}\hookrightarrow E_{1}$. For any
logarithmic connection $\nabla$ on $E_{1}$ with poles on $D=t_{1}+\dots+t_{n}$
we have that $\operatorname{App}(\nabla)$ is given by the zero divisor of

$\varphi_{\nabla}=\nabla(s)\wedge s\in
H^{0}(C,\det(E_{1})\otimes\Omega_{C}(D))$

which is the same as defining

$\operatorname{App}(\nabla)=[\varphi_{\nabla}]\in\mathbb{P}\\!\left(H^{0}(C,\det(E_{1})\otimes\Omega_{C}(D))\right).$

By Lemma 6.1 the condition on the eigenvalues implies that
$\operatorname{App}$, hence $\operatorname{Bun}\times\operatorname{App}$, is
regular on ${Con}^{\nu}$. Therefore the indeterminacy locus must lie only in
the boundary divisor.

Now fix $(E_{1},{\bf{p}}=\\{p_{1},\dots,p_{n}\\})\in{\rm Bun}$. We will
compute the rank of
$\left.\operatorname{App}\right|_{\pi_{+}^{-1}(E_{1},{\bf{p}})}$. Consider
$H^{0}(\mathfrak{sl}(E_{1})\otimes\Omega_{C}(t_{j}))$ the space of traceless
Higgs fields with simple pole on $t_{j}$. Since its dimension is three, there
exists a unique (up to scalar multiplication) strongly parabolic Higgs field
$\Theta^{p}_{j}$ with respect to $\bf{p}$, i.e. $Res_{t_{j}}(\Theta^{p}_{j})$
is nilpotent with image $p_{j}$ and has no other poles. Let $\nabla_{0}$ be a
connection in $\pi_{+}^{-1}(E_{1},{\bf{p}})$. Then any other
$\lambda$-connection $\nabla\in\pi_{+}^{-1}(E_{1},{\bf{p}})$ can be written in
a unique way as

$\nabla=c_{0}\nabla_{0}+\sum_{j=1}^{n}c_{j}\Theta^{p}_{j},$

for some $(c_{0}:\dots:c_{n})\in\mathbb{P}^{n}$. Hence

$\varphi_{\nabla}=c_{0}\nabla_{0}(s)\wedge
s+\sum_{j=1}^{n}c_{j}\Theta^{p}_{j}(s)\wedge s.$

Suppose that $\Theta^{p}_{j}(s)\wedge s\neq 0$ for each $j$. Since they have
one pole each and at different points these sections are linearly independent.
Since $H^{0}(C,\mathcal{O}_{C}(w_{\infty}+D))$ has dimension $n+1$ they form a
basis together with $\nabla_{0}(s)\wedge s$ and the Apparent map restricted to
this fiber is an isomorphism. Indeed, if

$c_{0}\nabla_{0}(s)\wedge s+\sum_{j=1}^{n}c_{j}\Theta^{p}_{j}(s)\wedge
s=\left(c_{0}\nabla_{0}(s)+\sum_{j=1}^{n}c_{j}\Theta^{p}_{j}(s)\right)\wedge
s=0$

then, by Lemma 6.1, $c_{0}=0$ and it follows that $c_{1}=\dots=c_{n}=0$ since
the $\Theta^{p}_{j}(s)\wedge s$ are linearly independent.

Now suppose that there exists $j\in\\{1,\dots,n\\}$ such that
$\Theta^{p}_{j}(s)\wedge s=0$. This occurs if and only if $\Theta^{p}_{j}(s)$
is holomorphic at $t_{j}$ which in turn means that $s(t_{j})\in p_{j}$.