# A map between moduli spaces of connections

Frank Loray  and  Valente Ramírez  Univ Rennes, CNRS, IRMAR, UMR 6625,
F-35000 Rennes, France  frank.loray@univ-
<EMAIL_ADDRESS>
###### Abstract.

We are interested in studying moduli spaces of rank 2 logarithmic connections
on elliptic curves having two poles. To do so, we investigate certain
logarithmic rank 2 connections defined on the Riemann sphere and a
transformation rule to lift such connections to an elliptic curve. The
transformation is as follows: given an elliptic curve $C$ with elliptic
quotient $\pi\colon C\to\mathbb{P}^{1}$, and the logarithmic connection
$(E,\nabla)$ on $\mathbb{P}^{1}$, we may pullback the connection to the
elliptic curve to obtain a new connection $(\pi^{*}E,\pi^{*}\nabla)$ on $C$.
After suitable birational modifications we bring the connection to a
particular normal form. The whole transformation is equivariant with respect
to bundle automorphisms and therefore defines a map between the corresponding
moduli spaces of connections.

The aim of this paper is to describe the moduli spaces involved and compute
explicit expressions for the above map in the case where the target space is
the moduli space of rank 2 logarithmic connections on an elliptic curve $C$
with two simple poles and trivial determinant.

###### Key words and phrases:

Moduli spaces, parabolic connection, logarithmic connection, parabolic vector
bundle, parabolic Higgs bundle, elliptic curve

###### 2010 Mathematics Subject Classification:

Primary: 14D20, 32G34, 34M55 $\cdot$ Secondary: 14H52, 53D30

F.L. acknowledges the support of CNRS and the project Foliage
ANR-16-CE40-0008.

V.R. is supported by the grants PAPIIT IN-106217, CONACYT 219722, and the
PRESTIGE post-doc program (coordinated by Campus France and co-financed under
the Marie Curie Actions - COFUND of the FP7). He also acknowledges the support
of the Centre Henri Lebesgue ANR-11-LABX-0020-01.
ORCID ID: 0000-0002-4594-430X

##  1\. Introduction

Let $X$ be a compact complex curve, $E$ a rank 2 holomorphic vector bundle,
and $\nabla\colon E\to E\otimes\Omega^{1}_{X}(D)$ a connection having simple
poles at the (reduced) divisor $D=t_{1}+\ldots+t_{n}$. At each pole $t_{i}$,
consider the residue matrix $\operatorname{Res}_{t_{i}}(\nabla)$ and denote by
$\nu_{i}^{+},\nu_{i}^{-}$ its eigenvalues. Fixing the base curve $(X,D)$, the
spectral data $\bar{\nu}=(\nu_{1}^{\pm},\ldots,\nu_{n}^{\pm})$, the trace
connection $(\det E,\operatorname{tr}\nabla)$, and introducing weights
$\bar{\mu}$ for stability, we may construct the moduli space
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(X,D)$ of $\bar{\mu}$-semistable
$\bar{\nu}$-parabolic connections $(E,\nabla,\bar{\ell})$ using Geometric
Invariant Theory (GIT) [Nit93, IIS06]. This moduli space is a separated
irreducible quasi-projective variety of dimension $2N$, where $N=3g-3+n$ is
the dimension of deformation of the base curve, and $g$ is the genus of $X$.
This variety is moreover endowed with a holomorphic symplectic structure
(which is in fact algebraic) [Iwa92, Boa01, Ina13, IIS06].

Moduli spaces of connections over the Riemann sphere have been extensively
studied, in particular as these correspond to spaces of initial conditions for
Garnier systems. The elliptic case with one and two poles have been studied in
[Lor16] and [FL20], respectively.

Closely related, we have moduli spaces $\operatorname{Bun}^{\bar{\mu}}(X,D)$
of $\bar{\mu}$-semistable parabolic bundles, and a natural map (which we
denote $\operatorname{Bun}$) that assigns to a parabolic connection
$(E,\nabla,\bar{\ell})$ its underlying parabolic bundle $(E,\bar{\ell})$. This
correspondence is a Lagrangian fibration [LS15], and over the set of simple
bundles it defines an affine $\mathbb{C}^{N}$-bundle which is an affine
extension of the cotangent bundle of $\operatorname{Bun}^{\bar{\mu}}(X,D)$
[AL97a, AL97b].

Let $(C,T)$ be an elliptic curve with two marked points, and let $\iota$ be
the unique elliptic involution that permutes the marked points. Taking the
quotient by this involution defines an elliptic covering $\pi\colon
C\to\mathbb{P}^{1}$. Via this ramified covering we can pull bundles and
connections from $\mathbb{P}^{1}$ back to the elliptic curve $C$. This
correspondence, subject to some normalizations, defines a map between the
corresponding moduli spaces. In this paper we aim to study a particular map

$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\longrightarrow\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T),$

obtained in this way. The divisor $D$ above contains 5 points: the four branch
points of $\pi\colon C\to\mathbb{P}^{1}$ and the unique point
$t\in\mathbb{P}^{1}$ which satisfies $\pi^{-1}(t)=T$ (see Section 3.2 for the
explicit construction). This transformation was originally introduced in
[DL15], using the associated monodromy representations. There, it was shown to
be dominant and generically $2:1$. The same transformation rule induces also a
map between moduli spaces of parabolic bundles (which we denote by a lowercase
$\phi$), making the following diagram commute:

${\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)}$${\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)}$${\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)}$${\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T).}$$\scriptstyle{\Phi}$$\scriptstyle{\operatorname{Bun}}$$\scriptstyle{\operatorname{Bun}}$$\scriptstyle{\phi}$
(1.1)

The moduli spaces
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ and
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ have been explicitly
described in [LS15], as well as the fibration $\operatorname{Bun}$ between
them. The moduli space of parabolic bundles
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$ was later studied in
[Fer16]. Moreover, the latter paper also describes geometrically and in
coordinates the map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$.

The objective of this paper is to complete the explicit description of the
commutative diagram (1.1) by describing the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$,
endowing it with a coordinate system, and computing the map
$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
in such coordinates. In order to do so, we first study the associated map
$\Phi^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$
between monodromy representations.

###  1.1. Structure of the manuscript

Later in the present section we will make some clarifications about notation
and discuss related works.

In Section 2 we recall general facts and definitions about parabolic bundles
and connections, and about their moduli spaces.

In Section 3 we define explicitly the transformation that takes a connection
$\nabla$ on $(\mathbb{P}^{1},D)$ and returns a connection $\Phi(\nabla)$ on
$(C,T)$, thus defining the main object of study of the present paper: the map
$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
between moduli spaces. We also describe analogous transformations for
parabolic bundles, and monodromy representations. In this section we define
the weights $\bar{\mu}$ and spectral data $\bar{\nu}$ that we will use
throughout the present work.

The map $\Phi$ was originally defined for monodromy representations in [DL15].
In Section 4 we further discuss several properties of this map. Through the
Riemann-Hilbert correspondence, we conclude that the map $\Phi$ between moduli
spaces of connections enjoys analogous properties.

In Section 5 we define and explain the genericity assumptions assumed in the
statement of the main results.

The present work relies heavily on the constructions, results and ideas that
appear in [LS15, Fer16]. For the sake of the reader’s convenience, we provide
in Section 6 a brief survey of the results needed from these papers.

The main results of the present work are stated in Section 7. In Section 8 we
discuss some geometric properties of the map $\Phi$, before proving the main
results in Section 9. Finally, Section 10 contains additional results, with
the corresponding proofs, about the apparent map in the elliptic case.

Below we present a short summary of our results.

Let $C\subset\mathbb{P}^{2}$ be an elliptic curve given by the affine equation
$y^{2}=x(x-1)(x-\lambda)$, and $\pi\colon C\to\mathbb{P}^{1}$ the elliptic
quotient $(x,y)\mapsto x$. Let $t\in\mathbb{P}^{1}$ be a point different from
$0,1,\lambda,\infty$. We fix the divisors $D=0+1+\lambda+\infty+t$ on
$\mathbb{P}^{1}$, and $T=\pi^{*}(t)$ on $C$.

We begin with the map
${\Phi}^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$,
which is the topological counterpart of our map
$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
via the Riemann-Hilbert correspondence. It is proved in [DL15, Theorem 1] that
this map is dominant and generically $2:1$. In Section 4 this result is
extended by proving, in Theorem 4.1, that ${\Phi}^{\text{top}}$ is surjective
and a ramified cover between GIT spaces of representations; the ramification
and branch loci of ${\Phi}^{\text{top}}$ are described. Moreover, we show in
Theorem 4.1 that ${\Phi}^{\text{top}}$ is symplectic, up to a scalar factor of
$2$ (the symplectic form on the codomain is pulled back to twice the symlectic
form on the domain). As a consequence we obtain analogous results for the map
$\Phi$ (cf. Corollary 4.1 and Corollary 4.2).

######  Remark 1.1.

The fact that $\Phi$ is only symplectic up to a scalar factor is analogous to
the case of the quadratic transformation of the Painlevé VI equation. This
transformation is also induced by a ramified cover
$\mathbb{P}^{1}\to\mathbb{P}^{1}$ of degree two, and the induced map between
moduli spaces of connections is symplectic up to a factor of 2 [TOS05, Remark
3.1].

In order to describe the map $\Phi$ in coordinates, we construct in Section
7.1 a family of connections over $C$, denoted $\mathcal{U}_{C}$, birationally
parametrized by
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\stackrel{{\scriptstyle\sim}}{{\dasharrow}}\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\times\mathbb{C}^{2}$.
This family is the image under $\Phi$ of the universal family for
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ constructed in
[LS15, Section 5] (which will be discussed in Section 6.3). We can choose a
suitable set of generators
$\boldsymbol{\nabla}_{0},\boldsymbol{\Theta}_{z},\boldsymbol{\Theta}_{w}$ in
such a way that any element $\boldsymbol{\nabla}\in\mathcal{U}_{C}$ is given
by a unique combination

$\boldsymbol{\nabla}=\boldsymbol{\nabla}_{0}(u)+\kappa_{1}\boldsymbol{\Theta}_{z}(u)+\kappa_{2}\boldsymbol{\Theta}_{w}(u),\quad
u\in\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D),\quad(\kappa_{1},\kappa_{2})\in\mathbb{C}^{2}.$

The natural map into the moduli space
$\mathcal{U}_{C}\dashrightarrow\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
is a rational dominant map, generically 2:1. Using this family we are able to
give an explicit a birational equivalence

$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\,\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\,\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)\times\mathbb{C}^{2}.$

This gives a trivialization of the affine $\mathbb{C}^{2}$-bundle
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
over some open and dense subset of
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$. Furthermore, over this
dense set, it identifies the moduli space of _parabolic Higgs bundles_ with
the cotangent bundle $T^{*}\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
in a natural way. This identification is moreover symplectic.

Using the isomorphism
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)\cong\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$
constructed in [Fer16, Section 4.3], we obtain a coordinate system for the
moduli space of connections

$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\,\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\,\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}\times\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}.$

We have explicitly computed the map $\Phi$ in these coordinates. Computations
in coordinates appear throughout Section 9, and those corresponding to the map
$\Phi$ are given in Section 9.3. Moreover, we show that the 2-form
$\omega_{C}$ defining the symplectic structure of
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
given, in these coordinates, by

$\omega_{C}=d\kappa_{1}\wedge dz+d\kappa_{2}\wedge dw,$

which coincides, under our identification, with the canonical 2-form defining
the symplectic structure on
$T^{*}\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$. Moreover, we verify
through these computations that the map $\Phi$ is symplectic up to a constant
factor of $2$. By this we mean that the pullback $\Phi^{*}\omega_{C}$ is two
times $\omega_{\mathbb{P}^{1}}$, the symplectic form on
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$.

Unlike $\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$, the
moduli space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
singular. We describe in Section 9.6 the singular locus and describe the local
analytic type of such singularities, together with its symplectic structure.

In Section 10 we define an _apparent map_ for connections from the family
$\mathcal{U}_{C}$. This map is defined as the set of tangencies of the
connection with respect to two fixed subbundles. The image of this map belongs
to $\mathbb{P}^{2}\times\mathbb{P}^{2}$. This map is not well defined on the
moduli space, but after symmetrization, ie. after passing to the quotient
$\mathbb{P}^{2}\times\mathbb{P}^{2}\to\operatorname{Sym}^{2}(\mathbb{P}^{2})$,
we obtain a well defined map which we denote $\operatorname{App}_{C}$. Note
that this is a map between spaces of the same dimension, thus not a Lagrangian
fibration. The map is rational, dominant, and the generic fiber consists of
exactly 12 points (cf. Theorem 10.1).

Finally, inspired by the results of [LS15], we combine the maps
$\operatorname{App}$ and $\operatorname{Bun}$ to obtain a generically
injective map
$\operatorname{App}_{C}\times\operatorname{Bun}\colon\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\dashrightarrow\operatorname{Sym}^{2}(\mathbb{P}^{2})\times\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$,
showing that a generic connection is completely determined by its underlying
parabolic bundle together with its image under the apparent map (cf. Theorem
10.2).

###  1.2. Code repository

All the computations mentioned in the present work have been carried out using
the computer algebra system SageMath [Sage]. The code is available at the
following repository [GitHub].

###  1.3. Related work

It is well-known that compact Riemann surfaces of genus $g$ with $n$ punctures
are hyperelliptic for

$(g,n)=(2,0),\ (1,2),\ (1,1),\ \ \text{and}\ \ (1,0).$

It has been observed by W. Goldman in [Gol97, Theorem 10.2] that,
$\mathrm{SL}_{2}(\mathbb{C})$-representations of the fundamental group of
these surfaces, with parabolic representation around each puncture, are
invariant under the hyperelliptic involution; moreover, they come from the
orbifold quotient representations. From the Riemann-Hilbert correspondance,
this means that a similar result should hold true for logarithmic connections,
providing a dominant map between the corresponding moduli spaces of
connections. This has been studied in details in the genus 2 case in [HL19].
The genus 1 case has been considered much earlier in [Hit95] (see also
[LvdPU08]). For the genus 1 case with one puncture, the same results also
revealed to be true with arbitrary local monodromy at the puncture, which has
been studied in [Lor16].

The case studied here, 2 punctures on genus 1 curves, was first considered in
[DL15] for representations. There it was proved that the result of Goldman
recalled above, [Gol97, Theorem 10.2], extends as follows. Consider the unique
elliptic involution permuting the two punctures; then any
$\mathrm{SL}_{2}(\mathbb{C})$-representation whose image is Zariski dense, and
whose boundary components have image into the same conjugacy class, is
invariant under the involution and comes from a representation of the orbifold
quotient. The goal of the present paper was to provide the similar property
for logarithmic connections, and therefore complete the whole picture for
hyperelliptic curves. We note that similar constructions also hold within the
class of connections on the 4-punctured sphere (see [MV13]).

The present work relies strongly on several results from [LS15, Fer16], which
we discuss in Section 6.

Finally, we remark the following for the 2-punctured elliptic curve case. Let
$E$ be a rank 2 vector bundle over the elliptic curve $C$ of degree $d$. By
tensoring $E$ with a line bundle $L$, we can change the degree to any desired
value as long as it has the same parity as $d$. Therefore, the study of moduli
spaces of rank 2 connections falls into two cases: odd degree and even degree.
Usually the determinant of the bundle is fixed to be either $\mathcal{O}_{C}$
in the even case (as in the present paper), or
$\mathcal{O}_{C}(\mathbf{w}_{\infty})$, where
$\mathbf{w}_{\infty}\in\mathbb{C}$ is the identity element for the group
structure of $C$. The moduli space of connections on $C$ with two poles and
fixed determinant $\mathcal{O}_{C}(\mathbf{w}_{\infty})$ has already been
described in detail in [FL20], together with its symplectic structure and
apparent map. As pointed out in [Fer16], it is possible to pass from the
moduli space in the even degree case to that in the odd degree case. This is
done by one elementary transformation followed by a twist by a rank 1
connection of degree zero. However, the transformation is not canonical, and
this passage makes explicit computations hard to obtain.

###  1.4. A note about notation

In this text a curve will mean a nonsingular complex projective algebraic
curve, which is identified to its associated analytic object, namely, a
compact Riemann surface. We will also identify a vector bundle with its
associated locally free sheaf.

We are going to deal with a lot of objects that are defined over the elliptic
curve $C$, and analogous objects defined over $\mathbb{P}^{1}$. In order to
avoid confusion, we will try to use bold typography for objects in $C$ that
have a counterpart in $\mathbb{P}^{1}$ (eg. $\boldsymbol{\nabla}$ and
$\nabla{}$).

Throughout this work we will use $\Phi$ to denote the transformation described
in Section 3.2, which takes a connection defined over $(\mathbb{P}^{1},D)$,
and returns a connection over $(C,T)$. We use the same symbol for the
analogous transformation acting on parabolic Higgs bundles. The corresponding
map between parabolic bunldes is denoted by $\phi$, and that between monodromy
representations $\Phi^{\text{top}}$. All these maps are generically $2:1$, and
we denote the corresponding Galois involutions, permuting the two sheets of
the cover, by $\Psi$, $\psi$ and $\Psi^{\text{top}}$, respectively.

Finally, we remark that we write $\mathbb{P}^{1}_{z}$ whenever we want to make
explicit the fact that the space $\mathbb{P}^{1}$ is endowed with an affine
coordinate $z\in\mathbb{C}$. This will allow us to distinguish different
occurrences of $\mathbb{P}^{1}$. Similar for affine spaces such as
$\mathbb{C}^{2}_{(c_{1},c_{2})}$.

###  1.5. Acknowledgements

We would like to thank Thiago Fassarella and Néstor Fernandez Vargas for many
valuable discussions on this topic. We also thank Nicolas Tholozan who helped
us to understand the action of $\Phi^{\text{top}}$ on the symplectic $2$-form
on the monodromy side.

##  2\. General aspects about parabolic bundles and connections

Let $X$ be a smooth projective complex curve and $D=t_{1}+\ldots+t_{n}$ a
reduced divisor. A _quasi-parabolic bundle_ of rank 2 on $(X,D)$ is a pair
$(E,\bar{\ell})$, where $E$ is a holomorphic vector bundle of rank 2 over $X$,
and $\bar{\ell}=\\{\ell_{1},\ldots,\ell_{n}\\}$ a collection of rank 1
subspaces $\ell_{i}\subset E|_{t_{i}}$. A _parabolic bundle_ is a quasi-
parabolic bundle endowed with a vector of weights
$\bar{\mu}=(\mu_{1},\ldots,\mu_{n})$, where $\mu_{i}\in[0,1]$. We will usually
omit the vector $\bar{\mu}$ in the notation and denote a parabolic bundle
simply by $(E,\bar{\ell})$.

A _logarithmic connection_ on $X$ with poles at $D$ is a pair $(E,\nabla)$,
where $E$ is a holomorphic vector bundle over $X$, and $\nabla\colon E\to
E\otimes\Omega^{1}_{X}(D)$ is a $\mathbb{C}$-linear map satisfying Leibniz’
rule. The eigenvalues of the residue $\operatorname{Res}_{t_{i}}(\nabla)$,
$\nu_{i}^{+},\nu_{i}^{-}$ are called the _local exponents_ , and the
collection $\bar{\nu}=(\nu_{1}^{\pm},\ldots,\nu_{n}^{\pm})$ is the spectral
data. We have the following equality known as _Fuchs’ relation_ :

$\sum_{i=1}^{n}(\nu_{i}^{+}+\nu_{i}^{-})+\deg E=0.$ (2.1)

######  Definition 2.1.

Let $\bar{\nu}$ be fixed spectral data and $\bar{\mu}$ a fixed vector of
weights. A $\bar{\nu}$-_parabolic connection_ of rank 2 on $(X,D)$ is a triple
$(E,\nabla,\bar{\ell})$ where $(E,\bar{\ell})$ is a rank 2 parabolic bundle
and $(E,\nabla)$ is a logarithmic connection with poles on $D$, such that at
each subspace $\ell_{i}$ the residue $\operatorname{Res}_{t_{i}}(\nabla)$ acts
by multiplication by $\nu_{i}^{+}$.

We remark that the difference between two connections is an
$\mathcal{O}_{X}$-linear operator (known as a _Higgs field_), and the space of
connections with a fixed parabolic structure is a finite dimensional affine
space.

######  Definition 2.2.

A _parabolic Higgs bundle_ of rank 2 on $(X,D)$ is a triple
$(E,\Theta,\bar{\ell})$, where $(E,\bar{\ell})$ is a parabolic vector bundle,
$\Theta\colon E\to E\otimes\Omega^{1}_{X}(D)$ is a $\mathcal{O}_{X}$-linear
map, and such that for each $t_{i}\in D$ the residue
$\operatorname{Res}_{t_{i}}(\Theta)$ is nilpotent with null space given by
$\ell_{i}$.

We now introduce the notion of $\bar{\mu}$-semistability.

######  Definition 2.3.

Let $(E,\bar{\ell})$ be a rank 2 parabolic bundle and
$\bar{\mu}=(\mu_{1},\ldots,\mu_{n})\in[0,1]^{n}$ its weight vector. We define
the $\bar{\mu}$-_parabolic degree_ of a line subbundle $L\subset E$ as

$\deg E-2\deg L+\sum_{\ell_{i}\not\subset L}\mu_{i}-\sum_{\ell_{i}\subset
L}\mu_{i}.$

The parabolic bundle $(E,\bar{\ell})$ is said to be $\bar{\mu}$-_semistable_
($\bar{\mu}$-_stable_) if the parabolic degree is non-negative (resp.
positive) for every subline bundle $L$.
A parabolic connection $(\nabla,E,\bar{\ell})$ is said to be
$\bar{\mu}$-_semistable_ ($\bar{\mu}$-_stable_) if the parabolic degree is
non-negative (resp. positive) for every subline bundle invariant by the
connection.

In order to define moduli spaces it is convenient to fix the determinant
bundle $\det(E)$, and the trace $\operatorname{tr}(\nabla)$ in the case of
connections. These choices will not appear explicitly in the notation, but we
always assume these objects have been defined and fixed. The moduli space does
not depend on the choice of the prescribed determinant, as we can freely
change it by twisting by a line bundle and performing elementary
transformations. This is further explained in Section 2.1.

######  Definition 2.4.

We denote by $\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(X,D)$ the moduli
space of $\bar{\mu}$-semistable $\bar{\nu}$-parabolic connections on $(X,D)$,
where the determinant bundle and trace connection equal some fixed pair
$(L,\eta)$. Similarly, we denote by $\operatorname{Higgs}^{\bar{\mu}}(X,D)$
the moduli space of $\bar{\mu}$-semistable parabolic Higgs bundles with given
trace and determinant bundle, and by $\operatorname{Bun}^{\bar{\mu}}(X,D)$ the
moduli space of $\bar{\mu}$-semistable parabolic bundles with a given
determinant bundle.

From now on, unless otherwise specified, connections, Higgs fields and bundles
are assumed to have trivial determinant and zero trace.

######  Remark 2.1.

The precise spectral data and weight vectors to be used throughout this text
are given in Definition 3.1. In Lemma 3.1 it is proved that, for such spectral
data, any $\bar{\nu}$-parabolic connection is $\bar{\mu}$-semistable.
Therefore, the moduli space of $\bar{\mu}$-semistable $\bar{\nu}$-parabolic
connections defined above coincides with the moduli space
$\operatorname{Con}_{\bar{\nu}}(X,D)$ of $\bar{\nu}$-parabolic connections.

######  Remark 2.2.

As can be seen in Definition 2.3, a parabolic connection is semistable if and
only if every subbundle invariant by the connection has non-negative parabolic
degree. Thus it is possible for a connection to be semistable while the
underlying parabolic bundle is not. These are exceptional cases and will be
excluded from our definition of a generic connection.

###  2.1. Twists and elementary transformations

In the previous section we have defined the moduli spaces of parabolic bundles
and connections by restricting ourselves to bundles with a prescribed
determinant. By imposing this condition we do not loose any generality.
Indeed, the determinant bundle can be arbitrarily modified by a suitable
sequence of elementary transformations and twists by rank 1 connections. In
this section we briefly describe these transformations for the special case of
rank 2.

A twist by a line bundle $L$ is simply the transformation
$(E,\bar{\ell})\mapsto(E\otimes L,\bar{\ell})$. Twisting by a rank 1
connection is defined as follows.

######  Definition 2.5.

Let $(E,\nabla,\bar{\ell})$ be a parabolic connection on $(X,D)$ with spectral
data $\bar{\nu}$. The twist by a rank 1 connection $(L,\xi)$ on $(X,D)$ with
spectral data $\bar{\vartheta}$ is defined to be the parabolic connection

$(E,\nabla,\bar{\ell})\otimes(L,\xi)=(E\otimes
L,\nabla\otimes\xi,\bar{\ell}).$

The spectral data over a pole $p\in D$ transforms as
$(\nu^{+},\nu^{-})\mapsto(\nu^{+}+\vartheta,\nu^{-}+\vartheta)$.

Let us now define the elementary transformations. We refer the reader to
[Mac07, Section 6] for a detailed discussion and equivalent ways of defining
these transformations.

######  Definition 2.6.

Let $(E,\bar{\ell})$ be a parabolic bundle over $(X,D)$, and consider a point
$p\in D$. The parabolic structure $\bar{\ell}$ provides a rank 1 subspace of
$\ell_{p}\subset E|_{p}$. Let us regard $\ell_{p}$ as a sky-scrapper sheaf
supported at $p$. We define a new parabolic bundle
$(E^{-},\bar{\ell}^{\prime})=\operatorname{elem}^{-}_{p}(E,\bar{\ell})$ as
follows. The underlying vector bundle is characterized by the exact sequence
of locally free sheaves

$0\longrightarrow E^{-}\longrightarrow E\longrightarrow
E/\ell_{p}\longrightarrow 0.$

The parabolic structure is unchanged outside of $p$. The new parabolic
direction $\ell^{-}_{p}\subset E^{-}|_{p}$ is defined by the kernel of the
inclusion $E^{-}\to E$. We remark that
$\det(E^{-})=\det(E)\otimes\mathcal{O}_{X}(-p)$. Thus this is called a
negative elementary transformation.

We define
$(E^{+},\bar{\ell}^{\prime})=\operatorname{elem}^{+}_{p}(E,\bar{\ell})$ as a
twist by the line bundle $\mathcal{O}_{X}(p)$ followed by a negative
elementary transformation at $p$. Thus $E^{+}$ fits into the exact sequence

$0\rightarrow E^{+}\rightarrow E(p)\rightarrow E/\ell_{p}\rightarrow 0.$

The parabolic direction $\ell^{+}_{p}\subset E^{+}|_{p}$ is once again defined
by the kernel of $E^{+}\to E(p)$. This time
$\det(E^{+})=\det(E)\otimes\mathcal{O}_{X}(p)$, and we call this a positive
elementary transformation.

From the perspective of the ruled surface $\mathbb{P}(E)$, the parabolic
structure $\ell_{p}$ is nothing but a point on the projectivized fiber
$F=\mathbb{P}(E|_{p})$. The elementary transformation
$\operatorname{elem}^{+}_{p}(E,\bar{\ell})$ corresponds to the birational
transformation of the total space $\mathbb{P}(E)$ given by blowing-up the
point $\ell_{p}\in F$, and then contracting the total transform of the fiber
$F$. The point resulting from this contraction gives the new parabolic
direction. Since the two transformations
$\operatorname{elem}^{-}_{p}(E,\bar{\ell})$ and
$\operatorname{elem}^{+}_{p}(E,\bar{\ell})$ differ only by twisting by a line
bundle, they coincide projectively. Namely, they define the same birational
transformation on the ruled surface $\mathbb{P}(E)$.

Finally, we remark that a connection $\nabla$ on $E$ induces a connection
$\nabla^{\prime}$ on the subsheaf $E^{-}\subset E$. Over $p$ the residual
eigenvalues are changed by the rule
$(\nu^{+},\nu^{-})\mapsto(\nu^{-}+1,\nu^{+})$. This means that the eigenvalue
associated to the parabolic direction $\ell^{\prime}_{p}$ is now $\nu^{-}+1$.
In a positive elementary transformation we must twist by a rank 1 connection
on $\mathcal{O}_{X}(p)$ having a unique pole at $p$ with eigenvalue $-1$.
Therefore, under a positive elementary transformation, the eigenvalues are
transformed as $(\nu^{+},\nu^{-})\mapsto(\nu^{-},\nu^{+}-1)$.

##  3\. The pullback map

Let $C\subset\mathbb{P}^{2}$ be an elliptic curve such that in some fixed
affine chart it is given by the equation

$y^{2}=x(x-1)(x-\lambda),\quad\lambda\in\mathbb{C}\setminus\\{0,1\\}.$

This curve is endowed with the elliptic involution
$\iota\colon(x,y)\mapsto(x,-y)$. With respect to the group structure of $C$,
this involution is precisely $\mathbf{p}\mapsto-\mathbf{p}$. The quotient of
$C$ under this involution gives rise to the elliptic quotient $\pi\colon
C\to\mathbb{P}^{1}$. This is a $2:1$ cover ramified over the 2-torsion points
$\mathbf{w}_{0},\mathbf{w}_{1},\mathbf{w}_{\lambda},\mathbf{w}_{\infty}$,
which are the points on $C$ that satisfy $x=0,1,\lambda,\infty$, respectively.

Let us choose a point $t\in\mathbb{P}^{1}\setminus\\{0,1,\lambda,\infty\\}$,
and let $\pi^{-1}(t)=\\{\mathbf{t}_{1},\mathbf{t}_{2}\\}$. We define the
following divisors of $\mathbb{P}^{1}$:

$W=0+1+\lambda+\infty,\quad D=W+t.$

We define analogous divisors for $C$:

$\mathbf{W}=\mathbf{w}_{0}+\mathbf{w}_{1}+\mathbf{w}_{\lambda}+\mathbf{w}_{\infty},\quad
T=\mathbf{t}_{1}+\mathbf{t}_{2},\quad\mathbf{D}=\mathbf{W}+T.$

Now, let us fix the spectral data and weights to use throughout the text. We
remark that for the most part we will work with
$\mathfrak{sl}_{2}(\mathbb{C})$-connections. Therefore the spectral data will
always satisfy $\nu^{-}_{i}=-\nu^{+}_{i}$.

######  Definition 3.1.

Let $\nu$ any complex number such that $2\nu\not\in\mathbb{Z}$, and choose
$\mu$ a real number $0<\mu<1$. When working with parabolic bundles over
$(C,T)$, we define the spectral data $\bar{\boldsymbol{\nu}}=(\pm\nu,\pm\nu)$
and the weight vector $\bar{\boldsymbol{\mu}}=(\mu,\mu)$. For working with
bundles over $(\mathbb{P}^{1},D)$, we define the vectors
$\bar{\nu}=\left(\pm\frac{1}{4},\pm\frac{1}{4},\pm\frac{1}{4},\pm\frac{1}{4},\pm\nu\right)$
and
$\bar{\mu}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\mu\right)$.

###  3.1. The map $\phi$ on parabolic bundles

The map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
was originally introduced in [Fer16, Section 6.1] (there it is denoted by an
uppercase $\Phi$). We repeat here the construction for the reader’s
convenience. We refer the reader to the aforementioned paper for extra details
and proofs.

Consider a semistable parabolic bundle $(E,\bar{\ell})$ on
$(\mathbb{P}^{1},D)$, with weight vector as defined in Definition 3.1, namely,
$\bar{\mu}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\mu\right)$,
for some $0<\mu<1$. We recall that through this text we deal with degree zero
bundles, thus $\det(E)=\mathcal{O}_{\mathbb{P}^{1}}$. In order to define the
image $\phi(E,\bar{\ell})$ we proceed as follows.

  1. (1)

Pullback $(E,\bar{\ell})$ to $C$ using the elliptic cover
$\pi\colon\mathbb{P}^{1}\to C$. This defines a bundle $E^{\prime}=\pi^{*}E$
with parabolic structure supported over $\mathbf{D}=\mathbf{W}+T$. Such bundle
is semistable with respect to the weights
$\bar{\mu}^{\prime}=(1,1,1,1,\mu,\mu)$.

  2. (2)

Perform a positive elementary transformation (cf. Definition 2.6) at each of
the parabolic points in the divisor
$\mathbf{W}=\mathbf{w}_{0}+\mathbf{w}_{1}+\mathbf{w}_{\lambda}+\mathbf{w}_{\infty}$.
This defines a new bundle $E^{\prime\prime}$ of degree 4 (in fact
$\det(E^{\prime\prime})=\mathcal{O}_{C}(\mathbf{W})=\mathcal{O}_{C}(4\mathbf{w}_{\infty})$).
The parabolic bundle is semistable with respect to
$\bar{\mu}^{\prime\prime}=(0,0,0,0,\mu,\mu)$.

  3. (3)

Tensor the previous bundle with the line bundle
$\mathcal{O}_{C}(-2\mathbf{w}_{\infty})$. This new parabolic bundle has
trivial determinant, and continues to be semistable for
$\bar{\mu}^{\prime\prime}=(0,0,0,0,\mu,\mu)$.

  4. (4)

Because of the nullity of the weights over $\mathbf{W}$, we may simply forget
the parabolic structure over $\mathbf{W}$ to recover a parabolic bundle with
trivial determinant over $(C,T)$, which is semistable for the weight vector
$\bar{\boldsymbol{\mu}}=(\mu,\mu)$.

We denote the final parabolic bundle by $\phi(E,\bar{\ell})$. This defines a
transformation from parabolic bundles of degree zero over
$(\mathbb{P}^{1},D)$, to parabolic bundles with trivial determinant over
$(C,T)$. The weight vectors are precisely those in Definition 3.1. This
transformation induces a map between the corresponding moduli spaces, which is
also denoted $\phi$.

###  3.2. The map $\Phi$ on connections

The same construction, with minor additions, can be adapted to define a
transformation between parabolic connections. Below we make such steps
explicit.

Let $\nabla$ be a rank 2 connection on $\mathbb{P}^{1}$, defined over a
degree-zero bundle $E$, having simple poles over the divisor $D$, and with
spectral data given by
$\bar{\nu}=\left(\pm\frac{1}{4},\pm\frac{1}{4},\pm\frac{1}{4},\pm\frac{1}{4},\pm\nu\right)$.
The following series of transformations defines the map $\Phi$. We remark that
the underlying bundles and weight vectors are the same as in Section 3.1.

  1. (1)

Pullback $\nabla$ to $C$ using $\pi$. This gives a connection $\pi^{*}\nabla$
on $E^{\prime}=\pi^{*}E$ with poles on $\mathbf{D}$. Locally, the connection
near $\mathbf{t}_{1},\mathbf{t}_{2}$ looks like $\nabla$ around $t$. This is
not the case around the ramification points $\mathbf{w}_{k}$, but we know this
construction multiplies the residual eigenvalues by a factor of two. Therefore
the spectral data is given by
$\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\nu,\pm\nu\right)$.

  2. (2)

Perform a positive elementary transformation (cf. Definition 2.6) for each
pole in the divisor $\mathbf{W}$. This gives a new connection on some bundle
$E^{\prime\prime}$ of degree 4. The spectral data over the points
$\mathbf{t}_{i}$ is unchanged, and the new spectral data at the
$\mathbf{w}_{k}$ is $\nu^{+}_{k}=-\frac{1}{2}$, $\nu_{k}^{-}=-\frac{1}{2}$
(not an $\mathfrak{sl}_{2}(\mathbb{C})$-connection).

  3. (3)

Tensor with the rank 1 connection
$(\mathcal{O}_{C}(-2\mathbf{w}_{\infty}),\xi)$, where $\xi$ is a suitable rank
1 connection with simple poles on $\mathbf{W}$ having residue $\frac{1}{2}$ at
each of them (no poles on $T$)111 The connection $\xi$ on
$\mathcal{O}_{C}(-2\mathbf{w}_{\infty})$ is the pullback under the elliptic
cover $\pi\colon C\to\mathbb{P}^{1}$ of the rank 1 connection on
$\mathcal{O}_{\mathbb{P}^{1}}(-\infty)$ given by
$d+\frac{1}{4}\left(\frac{dP}{P}\right)$, where $P(x)=x(x-1)(x-\lambda)$. The
former connection has residue $\frac{1}{4}$ at $x=0,1,\lambda,\infty$.
Therefore $\xi$ has residue $\frac{1}{2}$ at the branch points
$\mathbf{w}_{0},\mathbf{w}_{1},\mathbf{w}_{\lambda},\mathbf{w}_{\infty}$, as
desired. . By construction, the bundle
$E^{\prime\prime}\otimes\mathcal{O}_{C}(-2\mathbf{w}_{\infty})$ has trivial
determinant and the residual eigenvalues at $\mathbf{w}_{k}$ are all zero. The
fact that the poles of the original connection over $\mathbb{P}^{1}$ have
semisimple monodromy around each pole in $W$ implies that this new connection
over $C$ has trivial monodromy around each point $\mathbf{w}_{k}$. Thus this
connection is in fact holomorphic at each point in $\mathbf{W}$.

  4. (4)

Since the final connection is holomorphic at $\mathbf{W}$, we may forget these
points from the divisor of poles and consider it as a connection defined on
$(C,T)$ with spectral data $\bar{\nu}=(\pm\nu,\pm\nu)$.

We denote the last connection by $\Phi(\nabla)$.

######  Lemma 3.1.

Let $\nabla$ be any connection on $(\mathbb{P}^{1},D)$ with spectral data
$\bar{\nu}$. Then $\Phi(\nabla)$ is $\bar{\mu}$-semistable (and has spectral
data $\bar{\nu}$). Therefore, the correspondence $\nabla\mapsto\Phi(\nabla)$
induces a map

$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\longrightarrow\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T).$

We will use the same notation for the geometric transformation defined by
steps (1)–(4) and the induced map between moduli spaces.

###### Proof.

First, we remark that, by definition, a connection can only be unstable if it
is reducible. By our choice of spectral data $\bar{\nu}$, any connection
$\nabla$ on $(\mathbb{P}^{1},D)$ is irreducible and therefore
$\bar{\mu}$-stable. On the other hand, $\Phi(\nabla)$ (or any connection
having spectral data $\bar{\boldsymbol{\nu}}$ on $C$) must be
$\bar{\boldsymbol{\mu}}$-semistable. To see this, assume that $\Phi(\nabla)$
has an invariant line bundle $L$. Because $2\nu\not\in\mathbb{Z}$, Fuchs’
relation (2.1) implies that the restriction to the invariant line bundle $L$
must have spectral data $\\{+\nu,-\nu\\}$, and $\deg(L)=0$, and so the
$\bar{\boldsymbol{\mu}}$-parabolic degree of $L$ (cf. Definition 2.3) is
$\mu-\mu=0$. This implies that the connection (and also the underlying
parabolic bundle) is strictly $\bar{\boldsymbol{\mu}}$-semistable in the
reducible case. ∎

We remark that a typical element of
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ has a trivial underlying
bundle (as will be discussed in Section 5). On the other hand, a typical
element of $\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$ (for example a
$\bar{\boldsymbol{\mu}}$-stable parabolic bundle) is such that its underlying
bundle can be written as $E^{\prime\prime}=\mathbf{L}\oplus\mathbf{L}^{-1}$,
where $\mathbf{L}$ is a rank 1 bundle on $C$ of degree zero [Fer16,
Proposition 4.5]. For such case, Figure 1 shows the effect of steps (1) and
(2) on the projectivization of the bundles involved.

Figure 1. Steps of the transformation $\Phi$. The canonical sections
corresponding to $\mathbf{L},\mathbf{L}^{-1}$ in
$\mathbb{P}(\mathbf{L}\oplus\mathbf{L}^{-1})$ come from a _multisection_
$S_{\Sigma}$ in
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}})$.
This is further explained in Remark 8.1.

###  3.3. The map $\Phi^{\text{top}}$ on monodromy representations

The map $\Phi$ is also defined in a more topological setting. This was
originally introduced in [DL15]. We repeat here the construction presented in
Section 4 of the cited paper. Below we continue to use the spectral data
$\bar{\nu}$ and $\bar{\boldsymbol{\nu}}$ introduced in Definition 3.1.

Let us define $\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)$ as the
subspace of

$\operatorname{Rep}(\mathbb{P}^{1},D):=\operatorname{Hom}(\pi_{1}(\mathbb{P}^{1}\setminus
D),\,\mathrm{SL}_{2}(\mathbb{C}))\,/\,\mathrm{SL}_{2}(\mathbb{C}),$

consisting of those representations which are compatible with the spectral
data $\bar{\nu}$ around the punctures $D$. Explicitly, we define

$\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D):=\left\\{(M_{0},M_{1},M_{t},M_{\lambda},M_{\infty})\in\mathrm{SL}_{2}(\mathbb{C})^{5}\
;\ \begin{array}[]{c}M_{0}M_{1}M_{t}M_{\lambda}M_{\infty}=I\\\
\mathrm{trace}(M_{i})=0\ \text{for}\ i\not=t\\\
\mathrm{trace}(M_{t})=2\cos(2\pi\nu)\end{array}\right\\}/\sim\,,$

where two representations are equivalent, $(M_{i})\sim(M_{i}^{\prime})$, if
and only if there exists $M\in\mathrm{SL}_{2}(\mathbb{C})$ such that
$M_{i}^{\prime}=MM_{i}M^{-1}$ for all $i\in D$. These representations are
always irreducible, therefore the space defined above coincides with the GIT
quotient of the action of $\mathrm{SL}_{2}(\mathbb{C})$ on
$\operatorname{Hom}(\pi_{1}(\mathbb{P}^{1}\setminus
D),\,\mathrm{SL}_{2}(\mathbb{C}))$. In this way, it admits a structure of
smooth irreducible affine variety of complex dimension $4$. In a similar
fashion, we define for the twice-punctured torus

$\widetilde{\operatorname{Rep}}_{\bar{\boldsymbol{\nu}}}(C,T):=\left\\{(A,B,C_{1},C_{2})\in\mathrm{SL}_{2}(\mathbb{C})^{4}\
;\ \ \ \begin{array}[]{c}AB=C_{1}BAC_{2}\\\
\mathrm{trace}(C_{i})=2\cos(2\pi\nu)\end{array}\right\\}/\sim\,.$

This quotient however is not Hausdorff, as there are reducible
representations; it is an algebraic stack. A Hausdorff quotient

$\Pi_{\text{GIT}}\colon\widetilde{\operatorname{Rep}}_{\bar{\boldsymbol{\nu}}}(C,T)\twoheadrightarrow\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T),$

is obtained by GIT, and $\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
an irreducible affine variety of complex dimension $4$. This quotient
$\Pi_{\text{GIT}}$ is obtained by identifying triangular representations with
their diagonal part; this gives rise to a singular locus for the affine
variety $\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$. We now define a
map between the above two spaces. Given a representation
$(M_{i})\in\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)$, we can perform
the following.

  1. (1)

Pullback $(M_{i})$ via the elliptic cover $\pi\colon C\to\mathbb{P}^{1}$. This
defines a representation $\pi^{*}(M_{i})$ of the fundamental group
$\pi_{1}(C\setminus\mathbf{W}\cup T)$ with local monodromy $-I$ at the
$\mathbf{W}$-punctures.

  2. (2)

Twist $\pi^{*}(M_{i})$ by the central representation
$\pi_{1}(C\setminus\mathbf{W})\to\\{\pm I\\}$ with local monodromy $-I$ at
each of the punctures.

  3. (3)

This last representation has trivial monodromy around the
$\mathbf{W}$-punctures, and so we can regard it as an element of
$\widetilde{\operatorname{Rep}}_{\bar{\boldsymbol{\nu}}}(C,T)$, or better of
the quotient $\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$.

The above procedure provides a well-defined map

$\Phi^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T),$

and is the analogue of the map $\Phi$ on the monodromy side. This claim is
justified by the next remark.

######  Remark 3.1.

Let $(E,\nabla,\bar{\ell})$ be a parabolic rank 2 connection on a marked curve
$(X,D)$. An elementary transformation over a pole $p\in D$ does not change the
monodromy representation associated with $\nabla$ (this defines an isomorphism
outside the poles, where the monodromy is computed). On the other hand,
twisting $\nabla$ by a rank 1 connection with residue $\frac{1}{2}$ at $p\in
D$, scales the local monodromy around $p$ by a factor of $-1$. Indeed, with
respect to a suitable local coordinate $z$ for which $p$ is given by $z=0$,
the connection $\nabla$ is given by $\nabla=d+A\frac{dz}{z}$, for a constant
matrix $A$. Such a twist transforms the connection into
$\nabla^{\prime}=d+\left(A+\frac{1}{2}I\right)\frac{dz}{z}$. Since the matrix
$\frac{1}{2}I$ is scalar, it evidently commutes with $A$, and so the local
monodromy is multiplied by the matrix: $\operatorname{exp}\left(2\pi
i\,\frac{1}{2}I\right)=-I$.

The steps defining the map $\Phi$ in Section 3.2 can now be clearly matched to
those used to define $\Phi^{\text{top}}$ in the present section. Moreover, the
Riemann-Hilbert correspondance provides us with a commutative diagram

$\textstyle{\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{RH}$$\textstyle{\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\Phi}^{\text{top}}}$$\textstyle{\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{RH}}$$\textstyle{\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T).}$
(3.1)

The horizontal maps are the Rieman-Hilbert correspondences associating to a
connection its monodromy representation. We remark that these are complex
analytic isomorphisms, but they are transcendental. Note that indeed, in the
space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$,
connections on decomposable bundles which are reducible but not diagonal are
also identified to their diagonal reduction (these connections are said to be
$s$-equivalent). Thus we’ve made equivalent identifications on each side and
$\boldsymbol{RH}$ gives an isomorphism of complex analytic spaces.

##  4\. A topological approach: the monodromy side

In this section we will discuss a few important facts about the map
$\Phi^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$
defined in Section 3.3. Through the Riemann-Hilbert correspondence (3.1), we
obtain analogous facts for the map
$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.
The subsequent sections are devoted to further describing the map $\Phi$, and
will not discuss monodromy representations any further.

The map
$\Phi^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$
is studied in [DL15, Section4], and it is proved that it is dominant and
generically a $2:1$ map. Here, we want to provide a more precise result,
namely:

######  Theorem 4.1.

Assume $2\nu\not\in\mathbb{Z}$. Then the map
${\Phi}^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$
is a branched covering of degree $2$ (in particular surjective). The branch
locus consists of diagonal representations, forming the $2$-dimensional
singular set of $\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$. The Galois
involution of the covering

${\Psi}^{\textnormal{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)$

fixes the ramification locus, which consists of dihedral representations. In
particular we have an isomorphism
$\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)/{\Psi}^{\textnormal{top}}$.

######  Remark 4.1.

If we consider the analogous map

$\tilde{\Phi}^{\text{top}}\colon\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\widetilde{\operatorname{Rep}}_{\bar{\boldsymbol{\nu}}}(C,T)$

at the level of isomorphism classes of representations, not onto the GIT
quotient, then it is no longer surjective. As we shall see in the proof of
Theorem 4.1 below, strictly triangular representation (i.e. non diagonal ones)
are precisely missing in the image of $\tilde{\Phi}^{\text{top}}$; however,
this image defines a section of the GIT quotient $\Pi_{\text{GIT}}$, so that
${\Phi}^{\text{top}}$ is indeed surjective on the GIT quotient.

###### Proof.

According to Remark 4.1, we will consider the map $\tilde{\Phi}^{\text{top}}$
between sets of isomorphism classes of representations. It is proved in [DL15,
Section4] that the map $\tilde{\Phi}^{\text{top}}$ is explicitely given by

$\tilde{\Phi}^{\text{top}}\colon(M_{0},M_{1},M_{t},M_{\lambda},M_{\infty})\ \
\ \longmapsto\ \ \ \left\\{\begin{array}[]{ccc}A&=&M_{1}M_{t}M_{\lambda},\\\
B&=&M_{\lambda}M_{\infty},\\\ C_{1}&=&M_{t},\\\
C_{2}&=&M_{\infty}M_{t}M^{-1}_{\infty}.\end{array}\right.$ (4.1)

and is generically two-to-one with Galois involution

${\Psi}^{\text{top}}\colon(M_{0},M_{1},M_{t},M_{\lambda},M_{\infty})\ \ \
\longmapsto\ \ \ (-M_{0},-M_{1},M_{t},-M_{\lambda},-M_{\infty}).$ (4.2)

More precisely, it is proved in [DL15, Theorem6] that a representation
$(A,B,C_{1},C_{2})$ is in the image of $\tilde{\Phi}^{\text{top}}$ provided
that $A$ and $B$ generate an irreducible group: in that case, there exists a
matrix $M\in\mathrm{SL}_{2}(\mathbb{C})$, unique up to a sign, such that

$M^{-1}AM=A^{-1},\ \ \ M^{-1}BM=B^{-1}\ \ \ \text{and}\ \ \
M^{-1}C_{1}M=C_{2}.$ (4.3)

Moreover, $M^{2}=-I$ and
$(A,B,C_{1},C_{2})=\tilde{\Phi}^{\text{top}}(M_{0},M_{1},M_{t},M_{\lambda},M_{\infty})$
for

$\begin{array}[]{ccc}M_{0}&=&-AM,\\\ M_{1}&=&ABC_{2}^{-1}M,\\\
M_{t}&=&C_{1},\\\ M_{\lambda}&=&-BM,\\\ M_{\infty}&=&M.\end{array}$ (4.4)

Clearly, ${\Psi}^{\text{top}}$ defines an involutive isomorphism of
$\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)$ and we have
$\tilde{\Phi}^{\text{top}}\circ{\Psi}^{\text{top}}=\tilde{\Phi}^{\text{top}}$.
Then to prove the statement of the Theorem, it is enough to prove:

  * •

a representation $(A,B,C_{1},C_{2})$ is in the image of
$\tilde{\Phi}^{\text{top}}$ if, and only if, it is either irreducible or
diagonal; therefore, the restriction
$\Pi_{\text{GIT}}\colon\text{image}(\tilde{\Phi}^{\text{top}})\to\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$
will be one-to-one;

  * •

a diagonal representation has exactly one preimage by
$\tilde{\Phi}^{\text{top}}$ which is dihedral; these are the fixed points of
${\Psi}^{\text{top}}$;

  * •

an irreducible representation has two distinct preimages by
$\tilde{\Phi}^{\text{top}}$; equivalently, a preimage cannot be fixed by
${\Psi}^{\text{top}}$.

To complete the proof, we need to consider now the case where $A$ and $B$
generate a reducible, therefore triangular group. A necessary condition to be
in the image of $\tilde{\Phi}^{\text{top}}$ is that the representation
commutes (up to conjugacy) with the action of the elliptic involution
$\iota:C\to C$ which is as follows (see [DL15, Section4]):

$(A,B,C_{1},C_{2})\ \ \ \stackrel{{\scriptstyle\iota^{*}}}{{\mapsto}}\ \ \
(A^{-1},B^{-1},C_{2},C_{1}).$

We are going to prove that for all irreducible and diagonal representations,
there exists an $M$ like above so that the representation is in the image of
${\Phi}^{\text{top}}$; on the other hand, non-abelian triangular
representations never commute with the involution $\iota^{*}$, and cannot be
in the image.

Consider first the case where $A$ and $B$ (are triangular and) do not commute;
then we cannot conjugate simultaneously $A$ and $B$ to their inverse and the
representation does not commute with $\iota^{*}$. Indeed, one of the two
matrices $A$ and $B$ must have non zero trace, otherwise they commute; if
$\operatorname{tr}(A)\not=0$, then the conjugacy with $A^{-1}$ must permute
the two eigenvectors. One of them is common with $B$, and if we can
simultaneously conjugate $B$ to $B^{-1}$, then they share the two
eigendirections, and therefore commute, contradiction. So the representation
$(A,B,C_{1},C_{2})$ is not in the image of ${\Phi}^{\text{top}}$ in this case;
let us show that the representation is triangular in that case, i.e. $C_{1}$
and $C_{2}$ are also triangular. Indeed, rewriting the relation

$\underbrace{ABA^{-1}B^{-1}}_{[A,B]}=C_{1}\underbrace{(BA)C_{2}(BA)^{-1}}_{\tilde{C}_{2}}$

and, since traces satisfy
$\operatorname{tr}(\tilde{C}_{2})=\operatorname{tr}(C_{1})$ and
$\operatorname{tr}(C_{1}\tilde{C}_{2})=2$, we deduce (see for instance [Lor16,
Section 6.2]) that $C_{1}$ and $\tilde{C}_{2}$ generate also a reducible
group, i.e. there is a common eigenvector for $C_{1}$, $\tilde{C}_{2}$ and
$[A,B]$, which must coincide with the common eigenvector of $A$ and $B$. We
therefore conclude, when $A$ and $B$ do not commute, that the representation
is triangular, it does not commute with the involution $\iota^{*}$, and is not
in the image of ${\Phi}^{\text{top}}$.

Assume now that $A$ and $B$ commute, and at least one of them has trace
$\not=2$. Therefore, we can assume that they are diagonal, and denote by
$z_{0},z_{\infty}\in\mathbb{P}^{1}$ the two common eigendirections. The
relation writes

$(BA)^{-1}C_{1}(BA)=(C_{2})^{-1}.$

Denote by $z_{i}^{+},z_{i}^{-}\in\mathbb{P}^{1}$ the two eigendirections of
$C_{i}$, with $z_{1}^{+}$ and $z_{2}^{+}$ corresponding to the same
eigenvalue. If $z_{0},z_{\infty},z_{1}^{+},z_{1}^{-}$ are pairwise distinct,
then we have the following cross-ratio equivalences:

$(z_{0},z_{\infty},z_{1}^{+},z_{1}^{-})\sim(z_{0},z_{\infty},z_{2}^{-},z_{2}^{+})\sim(z_{\infty},z_{0},z_{2}^{+},z_{2}^{-})$

where the left equivalence comes from the relation, and the second one is the
cross-ratio’s invariance under double transpositions. It follows that one can
find a matrix $M\in\mathrm{SL}_{2}(\mathbb{C})$, unique up to a sign,
permuting $z_{0}$ and $z_{\infty}$, and sending $z_{1}^{\pm}$ to
$z_{2}^{\pm}$, i.e. satisfying (4.3), and therefore (4.4); the two choices
provide the two preimages of the irreducible representation
$(A,B,C_{1},C_{2})$ by ${\Phi}^{\text{top}}$. If now $A,B$ and $C_{1}$ share a
common eigenvector, say $z_{\infty}=z_{1}^{+}$, then the representation is
triangular; the relation yields $z_{\infty}=z_{1}^{+}=z_{2}^{-}$ and a $M$
satisfying (4.3) must permute

$z_{0}\leftrightarrow z_{\infty},\ \ \ z_{1}^{+}\leftrightarrow z_{2}^{+},\ \
\ \text{and}\ \ \ z_{1}^{-}\leftrightarrow z_{2}^{-}.$

there exists such a $M$ if, and only if we also have
$z_{0}=z_{1}^{-}=z_{2}^{+}$, i.e. the representation is diagonal. We therefore
conclude in this case that either the representation is irreducible and in the
image of ${\Phi}^{\text{top}}$, or it is triangular and can be in the image of
${\Phi}^{\text{top}}$ only if it is diagonal; we will check later that
diagonal representations are indeed in the image of ${\Phi}^{\text{top}}$.

Assume now that $A$ and $B$ commute, and one of them is parabolic (i.e.
$\operatorname{tr}=\pm 2$ but not in the center). Then we have only one common
eigenvector $z_{0}=z_{\infty}$ for $A$ and $B$. Again, if
$z_{\infty},z_{1}^{+},z_{1}^{-}$ are pairwise distinct, then one can find a
matrix $M\in\mathrm{SL}_{2}(\mathbb{C})$, unique up to a sign, fixing
$z_{\infty}$, and sending $z_{1}^{\pm}$ to $z_{2}^{\pm}$, i.e. satisfying
(4.3). Again, the representation $(A,B,C_{1},C_{2})$ is irreducible and has
two preimages by ${\Phi}^{\text{top}}$. On the other hand, when $z_{\infty}$
is also fixed by $C_{1}$, say $z_{\infty}=z_{1}^{+}$, then the representation
is reducible non-abelian, the relation gives $z_{0}=z_{1}^{-}=z_{2}^{+}$, and
we cannot find a $M$ fixing $z_{\infty}$ and sending $z_{1}^{+}$ to
$z_{2}^{+}$ for instance, so we are not in the image of ${\Phi}^{\text{top}}$.
Finally, if $A,B=\pm I$, then the relation shows that the representation is
abelian (cyclic) and therefore diagonal; it is in the image of
${\Phi}^{\text{top}}$.

To finish the proof, note that the ramification locus of ${\Phi}^{\text{top}}$
is given by the fixed points of the involution ${\Psi}^{\text{top}}$, i.e. by
those $(M_{0},M_{1},M_{t},M_{\lambda},M_{\infty})$ for which there exists
$M\in\mathrm{SL}_{2}(\mathbb{C})$ such that

$M^{-1}M_{t}M=M_{t},\ \ \ \text{and}\ \ \ M^{-1}M_{i}M=-M_{i}\ \ \text{for}\ \
i=0,1,\lambda,\infty.$

This forces the representation to be dihedral, i.e.
$(M_{0},M_{1},M_{t},M_{\lambda},M_{\infty})$ of the form

$\left(\ \begin{pmatrix}0&a_{0}\\\
-a_{0}^{-1}&0\end{pmatrix},\begin{pmatrix}0&a_{1}\\\
-a_{1}^{-1}&0\end{pmatrix},\begin{pmatrix}a_{t}&0\\\
0&a_{t}^{-1}\end{pmatrix},\begin{pmatrix}0&a_{\lambda}\\\
-a_{\lambda}^{-1}&0\end{pmatrix},\begin{pmatrix}0&a_{\infty}\\\
-a_{\infty}^{-1}&0\end{pmatrix}\ \right)$

with $a_{1}a_{\infty}=a_{0}a_{t}a_{\lambda}$ and $M$ diagonal with eigenvalues
$\pm\sqrt{-1}$. By diagonal conjugacy, one can normalize $a_{\infty}=1$ and
check that these representations are parametrized by $(a_{0},a_{\lambda})$ (we
can fix $a_{t}=e^{2i\pi\nu}$). After lifting on $C$ we get the diagonal
representation

$(A,B,C_{1},C_{2})=\left(\ \begin{pmatrix}-a_{0}&0\\\
0&-a_{0}^{-1}\end{pmatrix},\begin{pmatrix}a_{\lambda}&0\\\
0&a_{\lambda}^{-1}\end{pmatrix},\begin{pmatrix}a_{t}&0\\\
0&a_{t}^{-1}\end{pmatrix},\begin{pmatrix}a_{t}^{-1}&0\\\ 0&a_{t}\end{pmatrix}\
\right)$

and we conclude that ${\Psi}^{\text{top}}$ is one-to-one between above
dihedral representations on $\mathbb{P}^{1}$ and diagonal ones on $C$.

To conclude, $\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$ is precisely
the quotient of the smooth variety
$\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)$ divided by the action of
the involution ${\Psi}^{\text{top}}$. The locus of fixed points has
codimension $2$ and gives rise to a singular set in the quotient. ∎

######  Corollary 4.1.

The map constructed in Section 3.2,

${\Phi}\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\longrightarrow\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T),$

is a $2:1$ branched covering, branching over the locus of decomposable
connections. Moreover, this locus is the singular set for
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.

###### Proof.

The Riemann-Hilbert correspondence is one-to-one between isomorphism classes
of connections and conjugacy classes of representations. On
$(\mathbb{P}^{1},D)$, all connections (representations) are irreducible and
the GIT quotient therefore coincide with isomorphism (conjugacy) classes. The
Riemann-Hilbert map is an analytic isomorphism in this case. On $(C,T)$, all
connections (representations) are semistable, as was shown in Lemma 3.1. Again
the Riemann-Hilbert map induces an homeomorphism between the GIT quotients
which is analytic in restriction to the stable part. Clearly, the strictly
semistable locus of
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$,
which has codimension $2$, must be singular, since, locally around a point in
the strictly semistable locus, the complement of this locus is not simply
connected. ∎

We now proceed to discuss the symplectic structure of the spaces involved.
There is a natural holomorphic symplectic structure on the spaces of
representations arising from works of Atyiah-Bott and Goldman. In the compact
case, it is defined as follows. Let $S$ be a compact surface (without
boundary), and denote by $\operatorname{Rep}(S)$ the space of representations
$\rho:\pi_{1}(S)\to\mathrm{SL}_{2}(\mathbb{C})$, up to conjugacy. The tangent
space at an irreducible representation $\rho\in\operatorname{Rep}(S)$
identifies with the cohomology group
$H^{1}(\pi_{1}(S),\mathfrak{sl}_{2}(\mathbb{C})_{\mathrm{Ad}_{\rho}})$ where
$\mathfrak{sl}_{2}(\mathbb{C})_{\mathrm{Ad}_{\rho}}$ is the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{C})$, viewed as a $\pi_{1}(S)$-module via the
adjoint action of $\rho$ (see [Gol84, Section1.2]). Equivalently, one can
define the tangent space via the de Rham cohomology by
${H}^{1}_{dR}(S,E_{\rho})$, where $E_{\rho}$ is the flat
$\mathfrak{sl}_{2}(\mathbb{C})$-bundle whose monodromy is given by the adjoint
representation $\mathrm{Ad}_{\rho}$ on the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{C})$.

Now, there is a natural bilinear map on the tangent space which can be
defined, in the de Rham setting, by

$\begin{matrix}&\text{\tiny{cup product}}&&\text{\tiny Killing form}&\\\
H^{1}_{dR}(S,E_{\rho})\otimes
H^{1}_{dR}(S,E_{\rho})&\longrightarrow&H^{2}_{dR}(S,E_{\rho}\otimes
E_{\rho})&\longrightarrow&H^{2}_{dR}(S,\mathbb{C}).\end{matrix}$ (4.5)

Combining with the canonical isomorphism

$H^{2}_{dR}(S,\mathbb{C})\longrightarrow\mathbb{C},\quad\text{given
by}\quad\eta\longmapsto\int_{S}\eta,$ (4.6)

we obtain the pairing

$\begin{matrix}H^{1}_{dR}(S,E_{\rho})\otimes
H^{1}_{dR}(S,E_{\rho})&\longrightarrow&\mathbb{C}\\\
(\alpha,\beta)&\longmapsto&\displaystyle\int_{S}\mathrm{trace}(\alpha\wedge\beta),\end{matrix}$

see [Gol84, Section1.8] for more details. In this way, we get a holomorphic
$2$-form $\omega$ on $\operatorname{Rep}(S)$, which turns out to be closed and
non-degenerate, i.e.

$d\omega=0\ \ \ \text{and}\ \ \
\underbrace{\omega\wedge\cdots\wedge\omega}_{\text{half the dimension of
}\operatorname{Rep}(S)}\not=0,$

(cf. main theorem in [Gol84, Section1.7]). The non-compact case, where $S$ has
a non-empty boundary $\partial S$, was considered by Iwasaki in [Iwa92,
Section3] (see also [Iwa02, Section5] for more details, including explicit
computations for the punctured sphere case). There we have to deal with
relative cohomology, i.e. cohomology of pairs $(S,\partial S)$ in order to
take into account the fixed conjugacy classes at punctures. We have a map
$\partial:\operatorname{Rep}(S)\to\operatorname{Rep}(\partial S)$ associating
to a representation $\rho$, the conjugacy classes at punctures; our space of
representations correspond to the fibers of this map and we have to restrict
$H^{1}_{dR}(S,E_{\rho})$ (closed $1$-forms with compact support) to some
subspace. But at the end, the definition in the de Rham setting is the same
and Iwasaki proves that this is symplectic in restriction to
$\partial$-fibers. Going back to our map $\Phi^{\text{top}}$, let us simply
denote by $\omega_{\mathbb{P}^{1}}^{IG}$ and $\omega_{C}^{IG}$ the Goldman-
Iwasaki symplectic $2$-form on the corresponding side.

######  Proposition 4.1.

The map ${\Phi}^{\text{top}}$ of Theorem 4.1 is symplectic, up to a constant
factor of $2$. Namely, it satisfies

$\left({\Phi}^{\text{top}}\right)^{*}\omega_{C}^{IG}=2\,\omega_{\mathbb{P}^{1}}^{IG}.$

###### Proof.

This fact is true in general whenever we have a covering map $\pi:X\to Y$
between Riemann surfaces. In the construction of the Goldman-Iwasaki bracket
(4.5), everything commutes with the base change. In the last step (4.6), where
we integrate on $Y$ instead of $X$, the result is multiplied by the degree
$\deg(\pi\colon X\to Y)$. The ramified case is analogous using relative
cohomology. In our case we have $\pi\colon C\to\mathbb{P}^{1}$ with
$\deg(\pi)=2$. ∎

From the main theorem of [Iwa92], the Riemann-Hilbert map
$\operatorname{RH}\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(X,D)\to\operatorname{Rep}_{\bar{\nu}}(X,D)$
is symplectic, i.e. pulls back the Goldman-Iwasaki bracket $\omega^{IG}$ to
the Atiyah-Bott symplectic structure $\omega^{AB}$ on the moduli space of
connections. In particular, our pullback map $\Phi$ is also symplectic (up to
a multiplicative factor of $2$) and we will be able to compute the symplectic
structure on
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
from the known symplectic structure on
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ (for the
latter, see [LS15] and references therein).

######  Corollary 4.2.

The map constructed in Section 3.2

${\Phi}\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\longrightarrow\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$

is symplectic up to a factor of $2$:

${\Phi}^{*}\omega_{C}^{AB}=2\,\omega_{\mathbb{P}^{1}}^{AB}.$

##  5\. Genericity assumptions

In this section we will briefly describe the geometry of the moduli spaces of
parabolic bundles we work with. We will explain which families of bundles are
particularly special, and define a _generic bundle_ to be one not belonging to
these families. For our own convenience, we will first define these special
families in terms of coordinate descriptions of the moduli spaces of parabolic
bundles. Only later we describe such families intrinsically.

We begin with bundles over $\mathbb{P}^{1}$. Let
$\lambda,t\in\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ be different points, and
let $D$ be the divisor $D=0+1+\lambda+\infty+t$. We are interested in
$\bar{\mu}$-semistable parabolic bundles $(E,\bar{\ell})$ of degree zero over
the marked curve $(\mathbb{P}^{1},D)$.

According to [Fer16, Proposition 6.1] (which is a direct consequence of the
results in [LS15, Section 6.1]), the moduli space
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$, for the weights fixed in
Definition 3.1, is isomorphic to a Del Pezzo surface of degree 4 which we
denote $\mathcal{S}$. We refer the reader to [LS15, Section 3] for the
explicit construction of moduli spaces of rank 2 parabolic bundles over
$\mathbb{P}^{1}$ as projective varieties. An explicit bridge between
$\mathcal{S}$ and parabolic bundles is given, in the generic case, by Remark
6.1.

The surface $\mathcal{S}$ is a smooth projective surface that is obtained by
blowing-up 5 particular points $D_{i}\in\mathbb{P}^{2}$. It is well-known that
this surface $\mathcal{S}$ has exactly 16 rational curves of self-intersection
$-1$. Namely, the five exceptional divisors from the blow-up $E_{i}$, the
strict transform of the conic $\Pi$ passing through the five points, and the
strict transform of the 10 lines $L_{ij}$ passing through every possible pair
$(D_{i},D_{j})$. The five points $D_{i}$ in the smooth conic $\Pi$ are in the
same position as the five points $0,1,\lambda,t,\infty\in\mathbb{P}^{1}$.

Let us take the above geometric description of
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ as the base for our
definition of genericity. An intrinsic interpretation will follow.

######  Definition 5.1.

We will say that a parabolic bundle is generic in
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\cong\mathcal{S}$ if it lies
outside the union of the 16 $(-1)$-curves $\\{\Pi,E_{i},L_{ij}\\}$. A
parabolic connection in
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ will be called
generic if the underlying parabolic bundle is $\bar{\mu}$-semistable and
generic. We denote by $\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$
and $\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)^{0}$ the
open subsets of generic bundles and connections, respectively.

######  Proposition 5.1.

A parabolic bundle $(E,\bar{\ell})$ of degree zero belongs to
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$ if and only if all the
following conditions are true:

  * •

$E=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}$.

  * •

There exists no subbundle $L=\mathcal{O}_{\mathbb{P}^{1}}\hookrightarrow E$,
such that $\ell_{i},\ell_{j}\subset L$ for any pair $i\neq j$.

  * •

There exists no subbundle $L=\mathcal{O}_{\mathbb{P}^{1}}(-1)\hookrightarrow
E$, such that $\ell_{i},\ell_{j},\ell_{k},\ell_{m}\subset L$ for pairwise
different $i,j,k,m$.

The proof of this proposition follows immediately from [LS15, Table 1, Section
6.1], where the non-generic families are $\Pi,E_{i},L_{ij}$ are explicitly
described. In the work just cited these families are described for parabolic
bundles of degree $-1$, but translation to bundles of degree zero is done in a
straightforward manner by performing one positive elementary transformation.

We know from [LS15] that the _coarse moduli space_ of indecomposable parabolic
bundles is a non-separated variety obtained by gluing together a finite number
of spaces $\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ for suitable
choices of weight vectors $\bar{\mu}$. As the weights vary, the bundles in the
special families $\\{\Pi,E_{i},L_{ij}\\}$ may become unstable, and new bundles
that were previously unstable are now semistable. However, the bundles
represented in $\mathcal{S}\setminus\\{\Pi,E_{i},L_{ij}\\}$ are always stable
and thus common to every chart. In particular, this means that the set
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$ coincides with the set
of closed points in the non-separated coarse moduli space of indecomposable
bundles.

Let us now move on to parabolic bundles over $(C,T)$. As shown in [Fer16,
Theorem A], the moduli space
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$ is isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. Indeed, an explicit isomorphism
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)\rightarrow\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$
is constructed in [Fer16, Section 4.3]. We briefly recall the construction
here. The moduli space of semistable rank 2 vector bundles with trivial
determinant, $\operatorname{Bun}(C)$ is canonically isomorphic to the quotient
of $\operatorname{Jac}(C)$ by the elliptic involution [Tu93]. The map
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)\rightarrow\mathbb{P}^{1}_{z}$,
defining the first coordinate, is given by forgetting the parabolic structure
on $(E,\bar{\ell})$, and mapping $E$ to
$\operatorname{Jac}(C)/\pi\cong\mathbb{P}^{1}$ via the above isomorphism. Let
us call this map $\mathrm{Tu}$. The second coordinate is obtained by
precomposing $\mathrm{Tu}$ with the following automorphism $\varphi$ of
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$: Given $(E,\bar{\ell})$,
perform a positive elementary transformation above each of the two poles
$\mathbf{t}_{i}\in T$, and a twist by $\mathcal{O}_{C}(-\mathbf{w}_{\infty})$.
The resulting parabolic bundle $\varphi(E,\bar{\ell})$ has trivial determinant
and thus belongs to $\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$. This
map is in fact an involution, and in coordinates $(z,w)$ it is, by definition,
precisely the map $\varphi\colon(z,w)\mapsto(w,z)$.

The map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
has been described using the above coordinate system. It is proved in [Fer16,
Section 6.3] that it transforms the special $(-1)$-curves of $\mathcal{S}$ to
either horizontal or vertical lines defined by $z=0,1,\lambda,\infty$, and
$w=0,1,\lambda,\infty$. Our definition of a generic bundle in
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$ will exclude these special
lines, together with the curve described below.

######  Definition 5.2.

We denote by
$\boldsymbol{\Sigma}\subset\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
the set of strictly $\bar{\mu}$-semistable parabolic bundles. We also denote
$\Sigma=\phi^{-1}(\boldsymbol{\Sigma})\subset\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$.

######  Definition 5.3.

We will say that a parabolic bundle is generic in
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)\cong\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$
if it lies outside the following loci:

  * •

The union of the 8 lines $z=0,1,\lambda,\infty$, and $w=0,1,\lambda,\infty$,

  * •

The strictly $\bar{\mu}$-semistable locus $\boldsymbol{\Sigma}$,

A parabolic connection in
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
will be called generic if the underlying parabolic bundle is
$\bar{\mu}$-semistable and generic. We denote by
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}$ and
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0}$
the open subsets of generic bundles and connections, respectively.

It is natural to exclude the curve $\boldsymbol{\Sigma}$ from our definition
of genericity, since such bundles are not $\bar{\mu}$-stable. In fact, because
of the following theorem, this curve plays a crucial role in the description
of the moduli space of parabolic connections.

######  Theorem 5.1 (Theorem B and Theorem 6.4 in [Fer16]).

The map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
is a $2:1$ ramified cover. The branch locus is precisely the strictly
semistable locus $\boldsymbol{\Sigma}$, and the ramification locus coincides
with $\Sigma$. Moreover, both curves are isomorphic to the elliptic curve $C$.

The vertical and horizontal lines $z=0,1,\lambda,\infty$, and
$w=0,1,\lambda,\infty$, in $\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$ are
excluded mostly for technical reasons, since we are interested in the map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$,
and these lines are the images of the families of non-generic bundles over
$\mathbb{P}^{1}$. However, these families do represent parabolic bundles with
very specific properties. For example, if a parabolic bundle
$(\mathbf{E},\bar{\ell})$ satisfies $z\in\\{0,1,\lambda,\infty\\}$, then
either $\mathbf{E}=\mathbf{L}\oplus\mathbf{L}$, with $\mathbf{L}$ a torsion
bundle, or $\mathbf{E}=\mathbf{E}_{0}\otimes\mathbf{L}$, where
$\mathbf{E}_{0}$ is the unique non-trivial extension of $\mathcal{O}_{C}$ by
$\mathcal{O}_{C}$, and $\mathbf{L}$ is a torsion bundle. See [Tu93] and
[Fer16, Section 4.3].

Figure 2 shows some of the curves in
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ which are excluded from the
definiton of a generic bundle, and how these curves are transformed under
$\phi$.

Later on, once we perform computations in coordinates, we will sometimes need
to exclude the vertical line $\boldsymbol{\Lambda}=\\{z=t\\}$, which appears
as a polar divisor in some of our formulas. The pre-image
$\phi^{-1}(\boldsymbol{\Lambda})$ corresponds to another vertical line
$\Lambda\subset\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$.

Figure 2. Non-generic curves on $\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$
and their counterparts in
$\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$.

##  6\. Recap of previously known results

In this section we will further recall several facts from [LS15, Fer16] in
order to make our results precise and to put them into context. We restrict
ourselves to the cases that are relevant to us. We refer the reader to the
original papers cited for a detailed treatment and for more general cases.
Definitions and basic results about parabolic bundles and connections have
already been presented in Section 2.

###  6.1. Moduli spaces of parabolic bundles

In the previous section we have mentioned that the moduli space
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ is isomorphic to a Del
Pezzo surface, which we continue to denote $\mathcal{S}$. Below we present two
coordinate systems that can be used to describe the set
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$ of generic parabolic
bundles. For later convenience we present them as two remarks.

######  Remark 6.1.

The first coordinate system relies on Proposition 5.1. According to this
proposition, a generic parabolic bundle has a trivial underlying bundle, and
no two parabolics lie on a same trivial subline bundle. Consider such a
generic bundle and let us introduce an affine coordinate $\zeta$ on the
projectivized fibers of the (trivial) bundle. After a fractional linear
transformation, we may assume that the parabolic structures over the points
$0,1,\infty$ are given by $\zeta=0,1,\infty$, respectively. Under this
situation, any parabolic bundle is completely determined by two parameters
$u_{\lambda},u_{t}\in\mathbb{P}^{1}$, which represent the value of the
$\zeta$-coordinate for the parabolic structures over $\lambda$ and $t$,
respectively. This assignment defines a birational map
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\dashrightarrow\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$,
which provides a coordinate system. The surface $\mathcal{S}$ is recovered by
blowing-up the points $(0,0),(1,1),(\lambda,t)$ and $(\infty,\infty)$ in
$\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$.

######  Remark 6.2.

The second system is based on the fact that the Del Pezzo surface
$\mathcal{S}$ is defined as the blow-up of $\mathbb{P}^{2}$ at five points, so
there is a canonical birational map
$\mathbb{P}^{2}\dashrightarrow\mathcal{S}$. Since
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$ excludes (together with
other curves) the exceptional divisors of the blow-up, this map defines a one-
to-one map between an open subset of $\mathbb{P}^{2}$ and
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$. Fixing homogeneous
coordinates $[b_{0}:b_{1}:b_{2}]$, the space $\mathbb{P}^{2}_{b}$ defines a
coordinate system for $\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$.
An explicit interpretation of these coordinates can be found (for bundles of
degree $-1$) in [LS15, Section 3.6].

Now, let $C\subset\mathbb{P}^{2}$ be an elliptic curve such that in some fixed
affine chart it is given by the equation

$y^{2}=x(x-1)(x-\lambda),\quad\lambda\in\mathbb{C}\setminus\\{0,1\\}.$

As described in Section 3.1, a parabolic bundle $(E,\bar{\ell})$ on
$\mathbb{P}^{1}$ can be lifted to the elliptic curve using the elliptic
covering $\pi\colon C\to\mathbb{P}^{1}$. After a series of birational
transformations, we obtain a parabolic bundle on $C$ with parabolic structure
supported over the divisor $T=\pi^{*}(t)$. This defines a map

$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\longrightarrow\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$

between moduli spaces. The map $\phi$ is a $2:1$ ramified covering (as was
already stated in Theorem 5.1). The domain space is the Del Pezzo surface
$\mathcal{S}$ discussed above, and the target space is proved to be isomorphic
to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ in [Fer16, Theorem A] (see Section 5
for a brief description of this coordinate system).

######  Definition 6.1.

We define $\psi\in\operatorname{Aut}(\mathcal{S})$ as the involution of
$\mathcal{S}\cong\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ which
permutes the two sheets of the map $\phi$ and fixes every point in the
ramification divisor $\Sigma$.

The above involution is a lift of a de Jonquières automorphism of
$\mathbb{P}^{2}_{b}$ (a birational automorphism of degree 3 that preserves a
pencil of lines through a point, and a pencil of conics through four other
points). The curve $\Sigma\subset\mathcal{S}$ is, according to Theorem 5.1,
the ramification locus of $\phi$. Therefore, it corresponds to the curve of
fixed points of $\psi$. This involution is further discussed in [Fer16,
Section 6.4].

######  Remark 6.3.

The moduli space $\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$ is endowed
with an involution $\psi$, in such a way that the quotient of
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)/\psi$ is isomorphic to
$\operatorname{Bun}^{\bar{\mu}}(C,T)$. We have a similar situation in Theorem
4.1, where
$\operatorname{Rep}_{\bar{\nu}}(\mathbb{P}^{1},D)/\Psi^{\text{top}}$ is
identified to $\operatorname{Rep}_{\bar{\boldsymbol{\nu}}}(C,T)$. Our
description of
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
will be analogous, and so it is clear the the Galois involution $\psi$ (and
its counterpart $\Psi$) will play a crucial role in the present work.

The involution $\psi$, which is defined above as an automorphism of the moduli
space $\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$, can also be defined
by an explicit geometric transformation acting directly on bundles (i.e. not
just on equivalence classes of bundles). Because of the importance of $\psi$
in the sequel, we describe such geometric construction below.

######  Remark 6.4.

Consider a parabolic bundle $(E,\bar{\ell})$ on $(\mathbb{P}^{1},D)$ of degree
zero. We recall that the divisor $D$ is given by $D=0+1+\lambda+\infty+t$.
Consider the following transformations.

  1. (1)

Perform a positive elementary transformation (cf. Definition 2.6) at each of
the parabolic points in the divisor $W=0+1+\lambda+\infty$. This defines a new
bundle of degree 4.

  2. (2)

Tensor the previous bundle with the line bundle
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$. This bundle now has degree zero.

The bundle obtained by the above steps is again a parabolic bundle on
$(\mathbb{P}^{1},D)$ of degree zero. It represents a class in the moduli space
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$, and so this transformation
defines a self-map of the moduli space. It was proved in [Fer16, Proposition
6.5] that such map is precisely the involution $\psi$. In other words, the
transformation defined by steps (1) and (2) above coincides with the Galois
involution of the covering
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$.

######  Remark 6.5.

The above construction naturally extends to parabolic connections if we twist
by the rank 1 connection $(\mathcal{O}_{\mathbb{P}^{1}}(-2),\eta)$ in the last
step, where $\eta$ is the unique rank 1 connection having poles at $W$ with
residue $\frac{1}{2}$ at each pole. We denote this transformation by $\Psi$.
As the lemma below shows, this corresponds to the Galois involution of the
ramified cover $\Phi$ between moduli spaces of connections.

######  Lemma 6.1.

The transformation $\Psi$ defined by steps (1) and (2) above, twisting by the
rank 1 connection $(\mathcal{O}_{\mathbb{P}^{1}}(-2),\eta)$ in the second
step, coincides with the Galois involution of the covering
$\Phi\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.

###### Proof.

This is a direct consequence of the analogous result for the map
$\Phi^{\text{top}}$ between the spaces of monodromy representations discussed
in Section 4. The Galois involution of $\Phi^{\text{top}}$ is given in (4.2).
It corresponds to the transformation that preserves $M_{t}$ and transforms
$M_{i}\mapsto-M_{i}$, for $i=0,1,\lambda,\infty$. In the definition of $\psi$
above, step (1) does not change the monodromy representation, while step (2)
changes the local monodromy precisely by a factor of $-1$ at the points
$0,1,\lambda,\infty$ (cf. Remark 3.1). ∎

###  6.2. Moduli spaces of connections over $\mathbb{P}^{1}$

Recall that the space
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ carries a
natural symplectic structure in such a way that the map
$\operatorname{Bun}\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$
is a Lagrangian fibration. In [LS15] it is shown that the so-called apparent
map defines a dual Lagrangian fibration. Given a connection $\nabla$ on a
bundle $E$ and a rank 1 subbundle $L\subset E$, the apparent map is defined by
the zero divisor of the composite map

${L}$${E}$${E\otimes\Omega^{1}_{\mathbb{P}^{1}}(D)}$${(E/L)\otimes\Omega^{1}_{\mathbb{P}^{1}}(D).}$$\scriptstyle{\nabla}$

Note that the apparent map is defined geometrically as the set of points of
tangency between the Riccati foliation defined by $\nabla$ on $\mathbb{P}(E)$
and the section induced by the subbundle $L$.

For a generic connection of degree $-1$, the underlying bundle is
$E=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)$. This
bundle has a unique trivial subbundle $L=\mathcal{O}_{\mathbb{P}^{1}}$, which
provides a canonical choice for the apparent map. In this case we obtain a
rational map

$\operatorname{App}\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\dashrightarrow|\mathcal{O}_{\mathbb{P}^{1}}(n-3)|\cong\mathbb{P}^{n-3},$
(6.1)

where $n$ denotes the number of singularities (in our particular case $n=5$).
For generic connections of degree zero the underlying bundle is
$\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}$, and we may
perform an elementary transformation to replace it by
$\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)$. After
this, we may proceed as above. This extends the definition of the apparent map
to bundles of degree zero.

The Lagrangian fibrations provide a description of the geometric structure of
the space of connections. Indeed, the map

$\operatorname{App}\times\operatorname{Bun}\colon\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\dashrightarrow\mathbb{P}^{2}\times\mathbb{P}^{2},$
(6.2)

when restricted to the space of generic connections
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)^{0}$, and under
a simple assumption on the residual eigenvalues, is an open embedding [LS15,
Theorem 4.2]. Moreover, a suitable compactification of the space of generic
bundles makes the above map an isomorphism.

###  6.3. A universal family of connections

Another result that we try to imitate is the construction of an explicit
universal family for
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$. By universal
family we mean an algebraic family
$\mathcal{U}=\\{(E_{\theta},\nabla_{\theta},\ell_{\theta})\\}$ with the
property that the natural map
$\mathcal{U}\rightarrow\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$,
which assigns to each parabolic connection its isomorphism class, is dominant
and injective. Thus every generic element of the moduli space is represented
by a unique connection
$(E_{\theta},\nabla_{\theta},\ell_{\theta})\in\mathcal{U}$. We describe such a
family below.

As pointed out in Remark 6.1, a generic parabolic bundle $(E,\bar{\ell})$ of
degree zero and polar divisor $D=0+1+\lambda+\infty+t$ has
$E=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}$ as
underlying bundle, and we may assume the parabolic structure is given by
$\bar{\ell}=(0,1,u_{\lambda},\infty,u_{t})$, for some
$u_{\lambda},u_{t}\in\mathbb{P}^{1}$. This defines a coordinate system on
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$, which we continue to use
below.

For each pair $(u_{\lambda},u_{t})$ with $u_{\lambda},u_{t}\neq\infty$, we
define a parabolic connection $\nabla_{0}(u_{\lambda},u_{t})$ and two Higgs
bundles $\Theta_{i}(u_{\lambda},u_{t})$, $i=1,2$, compatible with the
parabolic structure $(u_{\lambda},u_{t})$ by the following explicit formulas.

$\displaystyle\nabla_{0}(u_{\lambda},u_{t})$
$\displaystyle=\frac{1}{4}\begin{pmatrix}-1&0\\\ -2-4\nu&1\\\
\end{pmatrix}\frac{dx}{x}+\frac{1}{4}\begin{pmatrix}1+4\nu&-4\nu\\\
2+4\nu&-1-4\nu\\\ \end{pmatrix}\frac{dx}{x-1}$
$\displaystyle+\frac{1}{4}\begin{pmatrix}-1&2u_{\lambda}\\\ 0&1\\\
\end{pmatrix}\frac{dx}{x-\lambda}+\nu\begin{pmatrix}-1&2u_{t}\\\ 0&1\\\
\end{pmatrix}\frac{dx}{x-t},$

$\Theta_{1}(u_{\lambda},u_{t})=\begin{pmatrix}0&0\\\ 1-u_{t}&0\\\
\end{pmatrix}\frac{dx}{x}+\begin{pmatrix}u_{t}&-u_{t}\\\ u_{t}&-u_{t}\\\
\end{pmatrix}\frac{dx}{x-1}+\begin{pmatrix}-u_{t}&u_{t}^{2}\\\ -1&u_{t}\\\
\end{pmatrix}\frac{dx}{x-t},$

$\Theta_{2}(u_{\lambda},u_{t})=\begin{pmatrix}0&0\\\ 1-u_{\lambda}&0\\\
\end{pmatrix}\frac{dx}{x}+\begin{pmatrix}u_{\lambda}&-u_{\lambda}\\\
u_{\lambda}&-u_{\lambda}\\\
\end{pmatrix}\frac{dx}{x-1}+\begin{pmatrix}-u_{\lambda}&u_{\lambda}^{2}\\\
-1&u_{\lambda}\\\ \end{pmatrix}\frac{dx}{x-\lambda}.$

######  Remark 6.6.

The connection $\nabla_{0}$ is characterized by the fact that it is the unique
connection (compatible with the given parabolic structure) such that the
divisor of the apparent map $\operatorname{App}$ in (6.1) is precisely
$\lambda+t$. Since the apparent map is a Lagrangian fibration [LS15], it
follows that the section
$\nabla_{0}\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\dashrightarrow\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$
is Lagrangian.

In [LS15, Section 5.1], the authors show that any connection on a generic
parabolic bundle defined by parameters $u_{\lambda},u_{t}$ can be written
uniquely as

$\nabla=\nabla_{0}(u_{\lambda},u_{t})+c_{1}\Theta_{1}(u_{\lambda},u_{t})+c_{2}\Theta_{2}(u_{\lambda},u_{t}),$
(6.3)

for some $(c_{1},c_{2})\in\mathbb{C}^{2}$. This follows from the fact that for
any $(u_{\lambda},u_{t})$ the Higgs bundles $\Theta_{i}(u_{\lambda},u_{t})$
are linearly independent over $\mathbb{C}$. Note that the above description
defines a birational map

$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)\dashrightarrow\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}\times\mathbb{C}^{2}_{(c_{1},c_{2})}.$
(6.4)

######  Definition 6.2.

We define the universal family over $\mathbb{P}^{1}$ as the explicit algebraic
family of connections
$\mathcal{U}=\\{(E_{\theta},\nabla_{\theta},\ell_{\theta})\\}$, where
$\theta=(u_{\lambda},u_{t},c_{1},c_{2})\in\mathbb{C}^{4}$ and

  * •

$E_{\theta}=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}$
for all $\theta$,

  * •

$\nabla_{\theta}=\nabla_{0}(u_{\lambda},u_{t})+c_{1}\Theta_{1}(u_{\lambda},u_{t})+c_{2}\Theta_{2}(u_{\lambda},u_{t})$,

  * •

$\ell_{\theta}=(0,1,u_{\lambda},\infty,u_{t})$.

Note that we now have three isomorphic spaces that can be naturally identified
to each other: the parameter space
$\mathbb{C}_{u_{\lambda}}\times\mathbb{C}_{u_{t}}\times\mathbb{C}^{2}_{(c_{1},c_{2})}$,
the above family $\mathcal{U}$, and the open set
$U\subset\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ which
is the image of the natural map
$\mathcal{U}\rightarrow\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$
assigning to each connection its isomorphism class in the moduli space.

The above identifications give a coordinate chart on a Zariski open subset of
the moduli space of connections. Moreover, we obtain, over the space of
generic bundles, a trivialization of the affine $\mathbb{C}^{2}$-bundle
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)^{0}\to\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$.
Indeed, if we consider the space of generic bundles
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}$ as a subset of
$\mathbb{C}_{u_{\lambda}}\times\mathbb{C}_{u_{t}}$, equation (6.3) implies
that
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)^{0}\cong\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)^{0}\times\mathbb{C}^{2}_{(c_{1},c_{2})}$.

We remark that in the coordinates defined by the map (6.4), the symplectic
structure is given by the 2-form

$\omega_{\mathbb{P}^{1}}=dc_{1}\wedge du_{t}+dc_{2}\wedge du_{\lambda}.$

In general, the moduli space of parabolic Higgs bundles is naturally
identified with the total space of the cotangent bundle to the moduli space of
parabolic bundles [AL97a, AL97b]. As explained in [LS15, Section 5.1], under
this correspondence we have $\displaystyle\Theta_{1}\mapsto
du_{t},\,\Theta_{2}\mapsto du_{\lambda}$. Note that the 1-form
$\omega_{\mathbb{P}^{1}}$ given above corresponds with the canonical
symplectic form on the cotangent bundle
$T^{*}\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$, using coordinates
$(u_{\lambda},u_{t})$ to describe
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$, and coordinates
$(c_{1},c_{2})$ to describe the cotangent space at a given point
$(u_{\lambda},u_{t})$.

##  7\. Statement of the main results

We begin with the map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
between moduli spaces of parabolic bundles. Our main objective is to describe
the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
and the map obtained by extending $\phi$ to the moduli spaces of connections
(which we denote $\Phi$). We know little about the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$,
except it is an affine $\mathbb{C}^{2}$-bundle over the space
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$. However, we know
beforehand from Proposition 4.1 that the extended map $\Phi$ is dominant and
generically $2:1$, and if follows from Theorem 4.1 that the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
can be obtained as a quotient of
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ by the Galois
involution $\Psi$ associated to the map $\Phi$.

###  7.1. A “double” universal family of connections

Recall that $\Psi$ acts on parabolic connections on $(\mathbb{P}^{1},D)$, as
is defined in Remark 6.5, by performing positive elementary transformations at
$0,1,\lambda,\infty$, and a twist by a rank 1 connection. Moreover, Lemma 6.1
shows that such transformation corresponds to the Galois involution of the map
$\Phi$ defined in Section 3.2. Namely, $\Psi$ is not the identity and it
satisfies $\Phi\circ\Psi=\Phi$.

We can use the transformation $\Phi$ to pull the universal family of
connections $\mathcal{U}$, described in Definition 6.2, from $\mathbb{P}^{1}$
to $C$. We do this by simply applying $\Phi$ to every element
$(E_{\theta},\nabla_{\theta},\ell_{\theta})\in\mathcal{U}$. Explicitly, we
pullback each connection to $C$ using the elliptic cover $\pi\colon
C\to\mathbb{P}^{1}$, we then perform elementary transformations at the torsion
points $\mathbf{w}_{k}$, and finally we twist by a rank 1 connection on
$\mathcal{O}_{C}(-2\mathbf{w}_{\infty})$. This defines a family of connections
on $(C,T)$, which is also parametrized by
$\theta\in\mathbb{C}_{u_{\lambda}}\times\mathbb{C}_{u_{t}}\times\mathbb{C}^{2}_{(c_{1},c_{2})}$.

######  Definition 7.1.

We define the double universal family over $C$ as the explicit algebraic
family of connections
$\mathcal{U}_{C}=\\{\Phi(E_{\theta},\nabla_{\theta},\ell_{\theta})\\}$, where
$(E_{\theta},\nabla_{\theta},\ell_{\theta})\in\mathcal{U}$.

This family of connections represents every generic class in the moduli space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$,
but each class has two representatives in $\mathcal{U}_{C}$. This follows from
the fact that we already know from Corollary 4.1 that $\Phi$, as a map between
moduli spaces of connections, is dominant and generically $2:1$. In this way,
the family $\mathcal{U}_{C}$ is not birational to the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$,
but a double cover of it. Thus we call it a “double” universal family.

It is important to note that (the image of) the original basis
$\nabla_{0}(u_{\lambda},u_{t})$, $\Theta_{1}(u_{\lambda},u_{t})$,
$\Theta_{2}(u_{\lambda},u_{t})$ in (6.3) is not the most suitable for
describing the family $\mathcal{U}_{C}$. Indeed, these are not equivariant
with respect to the Galois involutions $\Psi,\psi$, which are fundamental in
the description of
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
(cf. Remark 6.3). Let us clarify the above claim. Given a parabolic bundle
defined by $u=(u_{\lambda},u_{t})$, it is not true in general that
$\Psi\nabla_{0}(u)=\nabla_{0}(\psi u)$ nor $\Psi\Theta_{1}(u)=\Theta_{1}(\psi
u)$ (incidentally, $\Theta_{2}$ is always equivariant). We now seek for a
basis that is equivariant with respect to these involutions. This will descend
as a basis to the quotient
$\mathcal{U}/\Psi\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.

######  Remark 7.1.

The connection $\Psi\nabla_{0}(u)$ is a parabolic connection whose underlying
parabolic structure is given by $\psi u$. The latter connection can be
expressed uniquely as a combination $\nabla_{0}(\psi u)+c_{1}\Theta_{1}(\psi
u)+c_{2}\Theta_{2}(\psi u)$, for some coefficients $c_{1},c_{2}$. The claim
that the basis is not equivariant means that the coefficients $c_{1},c_{2}$
are not both zero. Equivalently, we could state that $\Psi\nabla_{0}(\psi
u)\neq\nabla_{0}(u)$, even though both left hand side and right hand side are
connections with parabolic structure given by $\psi(\psi u)=u$.

Note that because $\Psi\nabla_{0}(\psi u)$ is a connection with underlying
parabolic structure $u$, the mean
$\frac{1}{2}(\nabla_{0}(u)+\Psi\nabla_{0}(\psi u))$ also has $u$ as its
parabolic structure. Moreover, this average is equivariant with respect to
$(\Psi,\psi)$.

######  Definition 7.2.

We define the following elements of $\mathcal{U}$:

$\nabla_{0}^{\psi}(u)=\frac{1}{2}(\nabla_{0}(u)+\Psi\nabla_{0}(\psi
u)),\quad\Theta_{i}^{\psi}(u)=\frac{1}{2}(\Theta_{i}(u)+\Psi\Theta_{i}(\psi
u)),\quad i=1,2,$

and of $\mathcal{U}_{C}$:

$\boldsymbol{\nabla}_{0}^{\psi}(u)=\Phi(\nabla_{0}^{\psi}(u)),\quad\text{and}\quad\boldsymbol{\Theta}_{i}^{\psi}(u)=\Phi(\Theta_{i}^{\psi}(u)),\quad
i=1,2,$

and we call them the equivariant bases for $\mathcal{U}$ and
$\mathcal{U}_{C}$, respectively.

By construction, all the above are equivariant with respect to $(\Psi,\psi)$.
The underlying parabolic structure of $\nabla_{0}^{\psi}(u)$ is precisely $u$,
and the parabolic structure of $\boldsymbol{\nabla}_{0}(u)$ is $\Phi(u)$.

By combining $\boldsymbol{\Theta}_{1}^{\psi}$ and
$\boldsymbol{\Theta}_{2}^{\psi}$ in a suitable manner, we arrive to a new
equivariant basis
$\boldsymbol{\nabla}_{0}^{\psi},\boldsymbol{\Theta}_{z},\boldsymbol{\Theta}_{w}$,
which is more convenient. The exact expression for these elements can be found
in Definition 9.2. We shall refer to this as the _canonical basis_ (it comes
from the canonical identification of the moduli space of parabolic Higgs
bundles and the cotangent bundle of
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$, cf. Section 9.4). This is
the basis we will use to describe the family $\mathcal{U}_{C}$ (it is only
well defined for generic connections). Each generic element of
$\mathcal{U}_{C}$ can be written uniquely as a linear combination
$\boldsymbol{\nabla}_{0}^{\psi}(u)+\kappa_{1}\boldsymbol{\Theta}_{z}(u)+\kappa_{2}\boldsymbol{\Theta}_{w}(u)$.
The change of basis acts on the coefficients in such a way that the map
$(c_{1},c_{2})\mapsto(\kappa_{1},\kappa_{2})$ is given by an affine
transformation which depends rationally on $u$.

As pointed out earlier, the map
$\mathcal{U}_{C}\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
is dominant and generically $2:1$. Let us denote by $\boldsymbol{\Psi}$ the
transformation that permutes the fibers. Note that $\Psi$ and
$\boldsymbol{\Psi}$ represent essentially the same transformation, except that
$\Psi$ is acting on connections on $(\mathbb{P}^{1},D)$, and
$\boldsymbol{\Psi}$ on connections on $(C,T)$. We obtain the following
commutative diagram

${\mathcal{U}}$${\mathcal{U}_{C}}$${\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)}$${\mathcal{U}}$${\mathcal{U}_{C}}$${\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T).}$$\scriptstyle{\cong}$$\scriptstyle{\psi}$$\scriptstyle{\boldsymbol{\Psi}}$$\scriptstyle{2:1}$$\scriptstyle{id}$$\scriptstyle{\cong}$$\scriptstyle{2:1}$

We conclude that

$\mathcal{U}_{C}/\boldsymbol{\Psi}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T).$
(7.1)

Since $\mathcal{U}_{C}$ is isomorphic to $\mathcal{U}$, and the latter is
birational to $\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$,
we can further write

$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)/\Psi\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T),$

as anounced at the beginning of this section. This fact is analogous to the
description of $\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$ as the
quotient $\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)/\psi$ in Remark
6.3. However, we remark that, unlike the case of parabolic bundles, the fixed-
point set of $\Psi$ has codimension bigger than one, making the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
singular. This is discussed in Theorem 7.2 below, and was previously announced
in Corollary 4.1.

Because of equivariance of the basis $\boldsymbol{\nabla}_{0}^{\psi}$,
$\boldsymbol{\Theta}_{z}$, $\boldsymbol{\Theta}_{w}$, we have that
$\boldsymbol{\Psi}$ acts only on the first factor of
$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\times\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}$.
Namely,

$\boldsymbol{\Psi}(\boldsymbol{\nabla}_{0}^{\psi}(u)+\kappa_{1}\boldsymbol{\Theta}_{z}(u)+\kappa_{2}\boldsymbol{\Theta}_{w}(u))=\boldsymbol{\nabla}_{0}^{\psi}(\psi
u)+\kappa_{1}\boldsymbol{\Theta}_{z}(\psi
u)+\kappa_{2}\boldsymbol{\Theta}_{w}(\psi u),$

and so, in coordinates,
$\boldsymbol{\Psi}\colon(u,\kappa_{1},\kappa_{2})\mapsto(\psi
u,\kappa_{1},\kappa_{2})$. We conclude that

$\mathcal{U}_{C}/\boldsymbol{\Psi}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)/\psi\times\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}.$
(7.2)

From the description of the moduli space of parabolic bundles in [Fer16], we
have that

$\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)/\psi\cong\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)\,\cong\,\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}.$
(7.3)

Finally, combining (7.1)–(7.3) we obtain

$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}\times\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}.$
(7.4)

This gives a coordinate system, as well as a local trivialization of the
affine $\mathbb{C}^{2}$-bundle
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$,
over some open and dense subset of
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$.

We recall the reader that
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0}$
denotes the open subset of _generic_ connections, as specified in Definition
5.3 (see also Definition 5.1 for connections on $\mathbb{P}^{1}$).

######  Theorem 7.1.

The family of connections $\boldsymbol{\nabla}_{0}^{\psi}$ in Definition 7.2
defines a global section
$\boldsymbol{\nabla}_{0}^{\psi}\colon\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}\to\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0}$,
thus identifying the affine bundle
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0}\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}$
to the vector bundle
$\operatorname{Higgs}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}$.
Moreover, the section is Lagrangian making the above identification symplectic
with respect to the natural symplectic structures on the moduli spaces of
connections and Higgs bundles. Finally, the latter vector bundle is
algebraically trivial, namely

$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0}\cong\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}\times\mathbb{C}^{2}.$

That the section $\boldsymbol{\nabla}_{0}^{\psi}$ is Lagrangian is proved in
Section 9.4. The proof that such a section is well defined over the space
$\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}$, and that the affine
bundle $\operatorname{Bun}$ is algebraically trivial is given in Section 9.5.

######  Corollary 7.1.

Under the canonical identification between the spaces
$T^{*}\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$ and
$\operatorname{Higgs}^{\bar{\boldsymbol{\mu}}}(C,T)$, we have the
correspondence $dz\mapsto\boldsymbol{\Theta}_{z}$,
$dw\mapsto\boldsymbol{\Theta}_{w}$, and so the symplectic structure of
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
given by the 2-form

$\omega_{C}=d\kappa_{1}\wedge dz+d\kappa_{2}\wedge dw.$

The proof of this corollary can be found in Section 9.4.

######  Remark 7.2.

The open space of generic parabolic connections
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0}$
is smooth. Therefore the map $\Phi$ restricted to
$\Phi^{-1}(\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)^{0})\subset\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)^{0}$
is smooth. An important contribution of the present paper is providing
explicit expressions for this map in terms of the coordinates defined by
(7.4). The formulas are given in Section 9.

Using the coordinate expression of $\Phi$, we have further verified that
$\Phi^{*}\omega_{C}=2\omega_{\mathbb{P}^{1}}$, where
$\omega_{\mathbb{P}^{1}}=dc_{1}\wedge du_{t}+dc_{2}\wedge du_{\lambda}$ is the
symplectic 2-form on
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$. This is a
restatement of the fact that the map $\Phi$ is symplectic up to a constant
factor, which was already shown in Corollary 4.2.

The full space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
singular. The coordinate system used above fails to describe the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
precisely because the latter is singular over $\boldsymbol{\Sigma}$. Recall
that our description of the moduli space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
based on the fact that such space can be realized as the quotient of
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ by the
involution $\Psi$.

######  Theorem 7.2.

The set of fixed points
$\operatorname{Fix}(\Psi)\subset\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)^{0}$
is a codimension 2 subvariety. In fact, it defines a subbundle of rank 1 of
the affine $\mathbb{C}^{2}$ bundle $\operatorname{Bun}|_{\Sigma}$. A
connection
$(\boldsymbol{\nabla},\mathbf{E},\\{\ell_{1},\ell_{2}\\})\in\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$
belongs to $\Phi(\operatorname{Fix}(\Psi))$ if and only if it decomposes as a
direct sum

$\big{(}\boldsymbol{\nabla},\mathbf{E},\bar{\ell}\big{)}\,=\,\big{(}\eta,\mathbf{L},\ell\big{)}\oplus\iota^{*}\big{(}\eta,\mathbf{L},\ell\big{)},$

where $\eta$ is a rank 1 connection with a single pole at either
$\mathbf{t}_{1}$ or $\mathbf{t}_{2}$, $\mathbf{L}$ is a line bundle of degree
zero, and $\iota\colon C\to C$ is the elliptic involution.

Evidently, the connection
$\big{(}\boldsymbol{\nabla},\mathbf{E},\bar{\ell}\big{)}$ has each direct
summand of $\mathbf{E}$ as an invariant subbundle. However, the parabolic
degree of $(\mathbf{L},\ell)$ is zero (cf. Definition 2.3). There are no more
invariant subbundles and so the connection is (strictly) semistable, which
means that it does belong to the moduli space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.

We remark that the fact that the fixed-point set of $\Psi$ has codimension
bigger than one causes the space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ to
have singularities at $\Phi(\operatorname{Fix}(\Psi))$. The next theorem gives
the local description of the singular set.

######  Theorem 7.3.

Around a generic point of the singular locus $\Phi(\operatorname{Fix}(\Psi))$,
the moduli space
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$ is
locally isomorphic to the hypersurface in $\mathbb{C}^{5}_{\bar{x}}$ given by
the equation $x_{0}^{2}=x_{1}x_{2}$. Thus, locally, the singularities look
like the product of a quadratic conic singularity and a bidisk. Moreover, the
symplectic structure is given by

$\tilde{\omega}=\frac{dx_{1}\wedge dx_{2}}{4x_{0}}+dx_{3}\wedge dx_{4}.$

Since, in the above coordinates, the singular locus is given by
$x_{0}=x_{1}=x_{2}=0$, we conclude that the symplectic form restricted to the
singular locus is $dx_{3}\wedge dx_{4}$. Thus the singular locus is a
symplectic subvariety of
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.

The proof of the last two theorems can be found in Section 9.6.

##  8\. Geometric description of a generic parabolic bundle

In this section we want to describe some geometric aspects of parabolic
bundles in $\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)^{0}$.

We begin with a generic parabolic connection $(E,\nabla,\bar{\ell})$ of degree
zero and polar divisor $D=0+1+\lambda+\infty+t$ over $\mathbb{P}^{1}$. As
pointed out in Proposition 5.1, in the generic case we can assume
$E=\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}$.
Introducing an affine coordinate $\zeta$ on the fibers of $\mathbb{P}(E)$, the
parabolic structure is given by $\bar{\ell}=(0,1,u_{\lambda},\infty,u_{t})$,
for some
$u_{\lambda},u_{t}\in\mathbb{P}^{1}\setminus\\{0,1,\lambda,\infty\\}$. We
denote by $\sigma_{\infty}$ the constant horizontal section $\zeta=\infty$.
This section plays a central role, since it is used to define the apparent map
and the universal family $\mathcal{U}$ (cf. Section 6.2 and Section 6.3). The
birational involution $\psi$ defined in Remark 6.4 preserves the trivial
bundle $E$ and transforms $\sigma_{\infty}$ into a section $\sigma_{\psi}$ of
self-intersection $+2$. This section passes through the parabolic points over
$0,1,\lambda$ but not $\infty$. The sections $\sigma_{\infty}$ and
$\sigma_{\psi}$ intersect (transversally) over a unique point $x=p$.

Now we apply the transformation $\Phi$ to obtain a parabolic connection on
$(C,T)$ with trivial determinant. This time the underlying vector bundle
$\mathbf{E}$ will not be trivial, and it depends on the parabolic structure of
$(E,\bar{\ell})$. Generically, it is of the form
$\mathbf{E}=\mathbf{L}\oplus\mathbf{L}^{-1}$, where $\mathbf{L}$ is a line
bundle of degree zero. As such, there exists a unique point $\mathbf{p}_{1}\in
C$ such that $\mathbf{L}=\mathcal{O}_{C}(\mathbf{p}_{1}-\mathbf{w}_{\infty})$.
The other summand is given by
$\mathbf{L}^{-1}=\mathcal{O}_{C}(\mathbf{p}_{2}-\mathbf{w}_{\infty})$, in such
a way that $\mathbf{p}_{1},\mathbf{p}_{2}$ is a pair of points in involution.
Moreover, these points project to the point $p$ defined at the end of the last
paragraph (ie. $\pi^{-1}(p)=\\{\mathbf{p}_{1},\mathbf{p}_{2}\\}$). The bundle
$\mathbf{E}$ contains two sections $S_{\infty},S_{\psi}$, which are the images
of $\sigma_{\infty},\sigma_{\psi}$, respectively. These sections are exchanged
by an automorphism of the bundle $\mathbf{E}$, which we denote
$\boldsymbol{\Psi}$, making the following diagram commute

${E}$${\mathbf{E}}$${E}$${\mathbf{E}.}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$$\scriptstyle{\boldsymbol{\Psi}}$$\scriptstyle{\Phi}$

######  Remark 8.1.

The subbundles $\mathbf{L},\mathbf{L}^{-1}$ do not come from subbundles of $E$
in $\mathbb{P}^{1}$. Rather, there exists a curve $S_{\Sigma}$ in
$\mathbb{P}(E)\cong\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{\zeta}$ of bidegree
$(2,2)$ projecting down to $\mathbb{P}^{1}_{x}$ as a double cover ramified at
the points $x=0,1,\lambda,\infty$ (hence isomorphic to the elliptic curve $C$
itself). In fact, this curve is defined by the fixed points of the
transformation $\psi$. Once pulled back to $C$ via $\pi$, this curve splits
into two different sections $S_{\Sigma}^{+},S_{\Sigma}^{-}$ of
$\mathbb{P}(\pi^{*}(E))$ that intersect over the points
$\mathbf{w}_{0},\mathbf{w}_{1},\mathbf{w}_{\lambda},\mathbf{w}_{\infty}$.
After the elementary transformations dictated by $\phi$ (step (2) in Section
3.1), the sections no longer intersect in $\mathbb{P}(\mathbf{E})$. The action
of $\phi$ on $S_{\Sigma}$ can be seen in Figure 1 at the end of Section 3.

Figure 3. The sections $\sigma_{\infty}$ and $\sigma_{\psi}$ under $\Phi$.

The curve $S_{\Sigma}$ passes through the parabolic points above
$0,1,\lambda,\infty$. In general, it does not pass through the parabolic above
$t$. When it does, the parabolic structure is unchanged by $\psi$. Therefore
the fixed points of
$\psi\in\operatorname{Birat}(\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}})$
are precisely those for which $(t,u_{t})\in S_{\Sigma}$. Moreover, if the
curve $S_{\Sigma}$ passes through the parabolic point above $t$ in
$\mathbb{P}^{1}$, then, after performing the transformation $\phi$, the
parabolics over $T$ are in either $\mathbf{L}$ or $\mathbf{L}^{-1}$. The
elliptic involution permutes the summands $\mathbf{L}\oplus\mathbf{L}^{-1}$.
Since the parabolic bundles that come from $\mathbb{P}^{1}$ via $\phi$ are
invariant under the elliptic involution, we conclude that each direct summand
contains exactly one parabolic point. In such case we have a direct sum
decomposition
$(\mathbf{E},\\{\ell_{1},\ell_{2}\\})=(\mathbf{L},\\{\ell_{1}\\})\oplus(\mathbf{L}^{-1},\\{\ell_{2}\\})$.
Thus we conclude that the generic elements of $\boldsymbol{\Sigma}$ are
precisely the decomposable parabolic bundles.

##  9\. Computations in coordinates

Recall that in Section 6.3 we discussed how to define a coordinate sytem
$\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}\times\mathbb{C}^{2}_{(c_{1},c_{2})}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$,
using the basis $\nabla_{0},\Theta_{1},\Theta_{2}$ of the universal family
$\mathcal{U}$.

In this section we will first define the canonical basis
$\boldsymbol{\nabla}_{0}^{\psi},\boldsymbol{\Theta}_{z},\boldsymbol{\Theta}_{w}$,
and use it to establish a coordinate system
$\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}\times\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$.
We obtain the following diagram:

${\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)}$${\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}\times\mathbb{C}^{2}_{(c_{1},c_{2})}}$${\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)}$${\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}\times\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}.}$$\scriptstyle{\Phi}$$\scriptstyle{\Phi}$

The horizontal arrows are birational isomorphisms. In order to describe the
left vertical map $\Phi$ between moduli spaces, we will compute explicitly the
right arrow $\Phi$ in the given coordinates.

First, let us split the map $\Phi$ as follows:
$\Phi=(\phi,\mathrm{T}_{\phi})$, where
$\phi\colon\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}\to\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$
is the map between moduli spaces of parabolic bundles, and $\mathrm{T}_{\phi}$
defines fiberwise an affine map
$\mathbb{C}_{(c_{1},c_{2})}\to\mathbb{C}_{(\kappa_{1},\kappa_{2})}$. The map
$\phi\colon\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)\to\operatorname{Bun}^{\bar{\boldsymbol{\mu}}}(C,T)$
has already been explicitly described both in coordinates and in geometric
terms in [Fer16].

######  Proposition 9.1 ([Fer16]).

The map
$\phi\colon\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}\dashrightarrow\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$
is given by $(u_{\lambda},u_{t})\mapsto(z,w)$, where

$z=\frac{\lambda(u_{\lambda}-1)}{u_{\lambda}-\lambda},\quad w=\frac{\lambda
u_{t}(\lambda u_{t}-tu_{\lambda}+t-\lambda-u_{t}+u_{\lambda})}{t\lambda
u_{t}-t\lambda u_{\lambda}-tu_{t}u_{\lambda}+\lambda u_{t}u_{\lambda}-\lambda
u_{t}+tu_{\lambda}}.$ (9.1)

###  9.1. The involution $\psi$ and families of non-generic bundles in
coordinates

The next proposition describes the involution $\psi$ in terms of the
$(u_{\lambda},u_{t})$ coordinates. In particular, we have that $\psi$ fixes
the variable $u_{\lambda}$ thus acting only on $u_{t}$. We will use the
notation $(u_{\lambda},\overline{u}_{t})$ to denote the image of
$(u_{\lambda},u_{t})$ under $\psi$. We remark that the value of
$\overline{u}_{t}$ does depend on $u_{\lambda}$, but as long as $u_{\lambda}$
is fixed, the map $u_{t}\mapsto\overline{u}_{t}$ is an involution (i.e.
$\overline{\overline{u}}_{t}=u_{t}$), hence the choice of this notation.

######  Proposition 9.2.

The involution
$\psi\in\operatorname{Birat}(\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}})$
is given by $(u_{\lambda},u_{t})\mapsto(u_{\lambda},\overline{u}_{t})$, where
$\overline{u}_{t}$ is given by

$\overline{u}_{t}=\frac{tu_{\lambda}(\lambda u_{t}-tu_{\lambda}+t-\lambda-
u_{t}+u_{\lambda})}{t\lambda u_{t}-t\lambda
u_{\lambda}-tu_{t}u_{\lambda}+\lambda u_{t}u_{\lambda}-\lambda
u_{t}+tu_{\lambda}}.$ (9.2)

###### Proof.

Let $(z,w)=\phi(u_{\lambda},u_{t})$. We know that $(z,w)$ has another preimage
under $\phi$, which by definition is $\psi(u_{\lambda},u_{t})$. Therefore, we
need to solve $\phi(u^{\prime}_{\lambda},u^{\prime}_{t})=(z,w)$ for
$u^{\prime}_{\lambda},u^{\prime}_{t}$. From (9.1) we can see that $z$ is
uniquely determined by $u_{\lambda}$, and that fixing the value for $w$
imposes a quadratic condition on $u_{t}$. Solving the equation we recover
(9.2). ∎

######  Remark 9.1.

The polynomial

$P_{\Pi}=t\lambda u_{t}-t\lambda u_{\lambda}-tu_{t}u_{\lambda}+\lambda
u_{t}u_{\lambda}-\lambda u_{t}+tu_{\lambda},$ (9.3)

which appears as the denominator of (9.2) defines a rational curve in
$\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$, which corresponds
to the rational curve $\Pi\subset\mathcal{S}$ introduced in Section 5 (and
corresponds to the conic $b_{1}^{2}-b_{0}b_{2}$ in the birational model
$\mathbb{P}^{2}_{b}$ discussed in Remark 6.2). The involution $\psi$ permutes
this curve with the line $u_{t}=\infty$, thus the former appears as a pole of
$\overline{u}_{t}$. These rational curves correspond to the two $(-1)$-curves
in the Del Pezzo surface $\mathcal{S}$ that are mapped by $\phi$ to the
horizontal line $w=\infty$. Therefore, the polynomial $\Pi$ also appears in
the denominator of the formula for $w$ in (9.1). Indeed, comparing (9.1) to
(9.2), we can see that
$w=\frac{\lambda}{tu_{\lambda}}\,u_{t}\,\overline{u}_{t}$.

The last equation above shows that we can write the product
$u_{t}\overline{u}_{t}$ in terms of $w$. Moreover, we can also write
$u_{t}+\overline{u}_{t}$ in terms of $z,w$. Explicitly,

$u_{t}+\overline{u}_{t}=\frac{zw+(z-w)t-\lambda}{z-\lambda},\quad
u_{t}\overline{u}_{t}=\frac{wt(z-1)}{z-\lambda}.$

Finally, we can also express $u_{\lambda}$ as a fractional linear
transformation of $z$. This shows that any rational function on
$u_{\lambda},u_{t}$ that is invariant under $\psi$ can be expressed as a
rational function on $z,w$.

######  Remark 9.2.

In the chart $\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$ the
curve $\Sigma$ which was defined as the ramification locus of $\phi$ is
defined by the zeros of the following polynomial:

$P_{\Sigma}=t\lambda u_{t}^{2}-2t\lambda
u_{t}u_{\lambda}-tu_{t}^{2}u_{\lambda}+\lambda
u_{t}^{2}u_{\lambda}+t^{2}u_{\lambda}^{2}-\lambda
u_{t}^{2}-t^{2}u_{\lambda}+t\lambda
u_{\lambda}+2tu_{t}u_{\lambda}-tu_{\lambda}^{2}.$ (9.4)

######  Remark 9.3.

In Section 5 we have defined
$\boldsymbol{\Lambda}\subset\mathbb{P}^{1}_{z}\times\mathbb{P}^{1}_{w}$ to be
the vertical line given by $z=t$, and $\Lambda$ its preimage
$\Lambda=\phi^{-1}(\boldsymbol{\Lambda})\subset\operatorname{Bun}^{\bar{\mu}}(\mathbb{P}^{1},D)$.
In our chart $\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$, the
curve $\Lambda$ is defined by the vertical line
$u_{\lambda}=\lambda(1-t)/(\lambda-t)$, or equivalently, by the zeros of the
polynomial

$P_{\Lambda}=\lambda u_{\lambda}-tu_{\lambda}+\lambda t-\lambda.$ (9.5)

The special curves discussed in the above remarks can be seen in Figure 2,
Section 5.

###  9.2. Geometry of the apparent map on $\mathbb{P}^{1}$

Before we move on, let us recall the geometric picture of the universal family
$\mathcal{U}$ on $\mathbb{P}^{1}$. As usual, we consider parabolic bundles on
$(\mathbb{P}^{1},D)$ with trivial underlying bundle. We assume that, with
respect to an affine coordinate $\zeta$ on
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}})$,
the parabolic structure is given by
$\bar{\ell}=(0,1,u_{\lambda},\infty,u_{t})$ (cf. Remark 6.1). For each
parabolic bundle represented in
$\mathbb{P}^{1}_{u_{\lambda}}\times\mathbb{P}^{1}_{u_{t}}$, we define a
connection $\nabla_{0}(u_{\lambda},u_{t})$. This connection is characterized
as the unique connection compatible with the parabolic structure such that the
divisor of the apparent map is given by
$\operatorname{App}(\nabla_{0})=\lambda+t$. We recall that the apparent map is
defined by the tangencies of the Riccati foliation with the section
$\sigma_{\infty}=\\{\zeta=\infty\\}$. We will exploit the fact that every
connection $\nabla\in\mathcal{U}$ is completely determined by its parabolic
structure and its image under the apparent map.

Consider the transformation $\psi$ as defined in Remark 6.4. For generic
bundles, $\psi$ acts on the trivial bundle in such a way that it exchanges
$\sigma_{\infty}$ with the section $\sigma_{\psi}$ of self-intersection $+2$.

######  Proposition 9.3.

Assume $(u_{\lambda},u_{t})$ defines a generic parabolic bundle. Then $\psi$
transforms $\sigma_{\infty}$ into the section
$\sigma_{\psi}\colon\mathbb{P}^{1}\to\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}})\cong\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{\zeta}$
defined by

$\sigma_{\psi}(x)=\left(x,\,\frac{u_{\lambda}(1-\lambda)x}{(u_{\lambda}-\lambda)x-\lambda(u_{\lambda}-1)}\right).$

###### Proof.

Recall that $\psi$ is the transformation obtained by performing elementary
transformations at the parabolic points above the divisor
$W=0+1+\lambda+\infty$, and subsequently performing a twist by the bundle
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$ (cf. Remark 6.4). Since $\sigma_{\infty}$
is a section of degree zero (constant) passing only through the parabolic
above $x=\infty$, we conclude that $\sigma_{\psi}$ must be a section of self-
intersection $+2$ passing through the parabolics above $0,1,\lambda$ (but not
$\infty$). A simple computation shows that there is a unique such section and
it is given by the expression above. ∎

######  Remark 9.4.

Let $\nabla\in\mathcal{U}$ be defined by the parabolic structure
$(u_{\lambda},u_{t})$ and divisor $Z$ for the apparent map. Then $\Psi\nabla$
is the unique connection with parabolic structure
$\psi(u_{\lambda},u_{t})=(u_{\lambda},\overline{u}_{t})$ and whose tangencies
with the section $\sigma_{\psi}$ are exactly given by the divisor $Z$.

In order to be more explicit, let us denote $\operatorname{App}_{\infty}$ the
usual apparent map with respect to the constant section $\sigma_{\infty}$. Now
that we have a formula for $\sigma_{\psi}$, we can compute the tangencies of
this section with a given connection. Let us detail this construction.

######  Definition 9.1.

Given a connection $\nabla\in\mathcal{U}$, define the vector
$v(x)=(1,\sigma_{\psi}(x))^{\top}$. Let $v_{1}=v(x)$ and $v_{2}=\nabla v_{1}$.
We define $\operatorname{App}_{\psi}(\nabla)$ as the numerator of the rational
expression $\det\big{(}v_{1},v_{2}\big{)}$. We call the map
$\nabla\mapsto\operatorname{App}_{\psi}(\nabla)$ the _apparent map_ with
respect to $\sigma_{\psi}$.

Explicit expressions for $\operatorname{App}_{\infty}$ in terms of the
variables $u_{\lambda},u_{t},c_{1},c_{2}$ are given in [LS15, Section 6]. We
omit those for $\operatorname{App}_{\psi}$ here since they are considerably
more intricate. Explicit formulas can be found in the repository [GitHub].

###  9.3. The base change map

Recall that we have split the map $\Phi$ between moduli spaces of connections
as $\Phi=(\phi,\mathrm{T}_{\phi})$, where $\phi$ is the map between moduli
spaces of parabolic bundles and $\mathrm{T}_{\phi}$ defines maps
$\mathbb{C}^{2}_{(c_{1},c_{2})}\to\mathbb{C}^{2}_{(\kappa_{1},\kappa_{2})}$,
between the fibers.

In this and the next section we describe the map $\mathrm{T}_{\phi}$. Since it
an affine map between the fibers, it can be expressed as a $3\times 3$ matrix.
Furthermore, it can be understood as an affine change of coordinates. We do
this in two steps. First, in this section, we go from the original basis (used
to describe $\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$ in
[LS15] and given explicitly in Section 6.3) to the an intermediate basis
called the equivariant basis. In the next section we define the canonical
basis (which is the one we will use to describe
$\operatorname{Con}^{\bar{\boldsymbol{\mu}}}_{\bar{\boldsymbol{\nu}}}(C,T)$),
and compute the base change from the equivariant basis to the canonical one.

For the reader’s convenience we repeat Definition 7.2 here. The equivariant
basis $\nabla_{0}^{\psi}(u),\Theta_{1}^{\psi}(u),\Theta_{2}^{\psi}(u)$ are the
elements of $\mathcal{U}$ given by:

$\nabla_{0}^{\psi}(u)=\frac{1}{2}(\nabla_{0}(u)+\Psi\nabla_{0}(\psi
u)),\quad\Theta_{i}^{\psi}(u)=\frac{1}{2}(\Theta_{i}(u)+\Psi\Theta_{i}(\psi
u)),\quad i=1,2.$

These satisfy the equivariant property
$\Psi\nabla_{0}^{\psi}(u)=\nabla_{0}^{\psi}(\psi u)$,
$\Psi\Theta_{i}^{\psi}(u)=\Theta_{i}^{\psi}(\psi u)$.

In order to compute the base change, we must describe in
$(c_{1},c_{2})$-coordinates the action of the involution $\Psi$ on
$\operatorname{Con}^{\bar{\mu}}_{\bar{\nu}}(\mathbb{P}^{1},D)$. To do so we
will exploit the idea presented in Remark 9.4.

Let $\nabla\in\mathcal{U}$ have parabolic structure
$\operatorname{Bun}(\nabla)=(u_{\lambda},u_{t})$, and assume that with respect
to the original basis for $\mathcal{U}$ it is written as
$\nabla=\nabla_{0}(u_{\lambda},u_{t})+c_{1}\Theta_{1}(u_{\lambda},u_{t})+c_{2}\Theta_{2}(u_{\lambda},u_{t}).$
Following Remark 9.4, we seek for the unique connection $\nabla^{\prime}$ such
that

$\operatorname{Bun}(\nabla^{\prime})=(u_{\lambda},\overline{u}_{t}),\quad\text{and}\quad\operatorname{App}_{\psi}(\nabla^{\prime})=\operatorname{App}_{\infty}(\nabla).$

The connection $\nabla^{\prime}$ is precisely the image of $\nabla$ under
$\Psi$. A straightforward computation allows us to find coefficients
$c^{\prime}_{1},c^{\prime}_{2}$ as functions of
$u_{\lambda},u_{t},c_{1},c_{2}$, such that

$\nabla^{\prime}=\nabla_{0}(u_{\lambda},\overline{u}_{t})+c^{\prime}_{1}\Theta_{1}(u_{\lambda},\overline{u}_{t})+c^{\prime}_{2}\Theta_{2}(u_{\lambda},\overline{u}_{t}).$

If we hold the parabolic structure fixed, the coefficients
$c^{\prime}_{1},c^{\prime}_{2}$ are affine functions of $c_{1},c_{2}$. This
means that there exists a $3\times 3$ matrix
$\mathrm{T}_{\psi}(u_{\lambda},\overline{u}_{t})$ such that

$\begin{pmatrix}1\\\ c^{\prime}_{1}\\\
c^{\prime}_{2}\end{pmatrix}=\mathrm{T}_{\psi}(u_{\lambda},\overline{u}_{t})\begin{pmatrix}1\\\
c_{1}\\\ c_{2}\end{pmatrix}.$

The involution $\Psi$ decomposes as $\Psi=(\psi,\mathrm{T}_{\psi})$. The
symbol $\mathrm{T}_{\psi}$ has been chosen, in analogy to $\mathrm{T}_{\phi}$.
The former gives the fiberwise action of the involution $\Psi$, while the
latter is the fiberwise action of the map $\Phi$.

######  Proposition 9.4.

The matrix $\mathrm{T}_{\psi}=\mathrm{T}_{\psi}(u_{\lambda},u_{t})$ is given
as follows:

$\mathrm{T}_{\psi}=\begin{pmatrix}1&0&0\\\
\mathrm{T}_{10}/\delta&\mathrm{T}_{11}/\delta&0\\\
\mathrm{T}_{20}/\delta&\mathrm{T}_{21}/\delta&1\\\ \end{pmatrix},$

where $\delta,\mathrm{T}_{ij}$ are functions of $u_{\lambda},u_{t}$ given by

$\displaystyle\delta$
$\displaystyle=t(t-1)(t-\lambda)u_{\lambda}(u_{\lambda}-1)(u_{\lambda}-\lambda),$
$\displaystyle\mathrm{T}_{10}$ $\displaystyle=-2\nu(t\lambda u_{t}-t\lambda
u_{\lambda}-tu_{t}u_{\lambda}+\lambda u_{t}u_{\lambda}-\lambda
u_{t}+tu_{\lambda})(t\lambda-tu_{\lambda}+\lambda u_{\lambda}-\lambda),$
$\displaystyle\mathrm{T}_{20}$ $\displaystyle=\nu
t(t-1)(2\lambda^{2}u_{t}-2t\lambda u_{\lambda}+tu_{\lambda}^{2}-\lambda
u_{\lambda}^{2}+t\lambda-\lambda^{2}-2\lambda u_{t}+2\lambda u_{\lambda}),$
$\displaystyle\mathrm{T}_{11}$ $\displaystyle=-(t\lambda u_{t}-t\lambda
u_{\lambda}-tu_{t}u_{\lambda}+\lambda u_{t}u_{\lambda}-\lambda
u_{t}+tu_{\lambda})^{2},$ $\displaystyle\mathrm{T}_{21}$
$\displaystyle=-t(t-1)(-\lambda^{2}u_{t}^{2}+2t\lambda
u_{t}u_{\lambda}-t\lambda u_{\lambda}^{2}-tu_{t}u_{\lambda}^{2}+\lambda
u_{t}u_{\lambda}^{2}-t\lambda u_{t}$
$\displaystyle\phantom{=t(t-1)(}+\lambda^{2}u_{t}+\lambda u_{t}^{2}-2\lambda
u_{t}u_{\lambda}+tu_{\lambda}^{2}).$

Note that the factors that appear in $\mathrm{T}_{10}$ are $P_{\Pi}$ and
$P_{\Lambda}$ (defined in Section 9.1), and that
$\mathrm{T}_{11}=-P_{\Pi}^{2}$.