# Meromorphic Projective Structures,
Opers and Monodromy

_Titouan Sérandour_ 111Université Côte d’Azur, CNRS, Laboratoire J.-A.
Dieudonné, Parc Valrose, F-06108 Nice Cedex 2, France titouan.serandour@univ-
cotedazur.fr

(September 2023)

###### Abstract

The complex projective structures considered is this article are compact
curves locally modeled on $\mathbb{CP}^{1}$. To such a geometric object,
modulo marked isomorphism, the monodromy map associates an algebraic one: a
representation of its fundamental group into
$\operatorname{PGL}(2,\mathbb{C})$, modulo conjugacy. This correspondence is
neither surjective nor injective. Nonetheless, it is a local diffeomorphism
[Hejhal, 1975]. We generalize this theorem to projective structures admitting
poles (without apparent singularity and with fixed residues): the
corresponding monodromy map (including Stokes data) is a local biholomorphism.

Keywords and phrases: _meromorphic projective structure, monodromy, Stokes
phenomenon, quadratic differential, oper, meromorphic linear connection,
moduli space, irregular Riemann-Hilbert correspondence, isomonodromic
deformation_.

Mathematics Subject Classification 2020: 30Fxx, 34M40, 34M45.

###### Contents

  1. 0 Introduction
  2. 1 Meromorphic projective structures
    1. 1.1 Complex projective structures (without pole) and their monodromy map
    2. 1.2 The space of meromorphic projective structures
  3. 2 A smooth family of opers
    1. 2.1 Opers for the group $\operatorname{PGL}(2,\mathbb{C})$ over a smooth curve $C$
    2. 2.2 A unique meromorphic oper associated with a meromorphic projective structure without apparent singularity: the minimal birational model
    3. 2.3 Construction of a family of meromorphic $\operatorname{PGL}(2,\mathbb{C})$-opers
  4. 3 A smooth family of singular Riccati foliations
    1. 3.1 Riccati foliations and meromorphic rank $2$ connections
    2. 3.2 Projective structures and opers for the group $\operatorname{GL}(2,\mathbb{C})$
    3. 3.3 Formal data of rank $2$ connections and Riccati foliations
    4. 3.4 A smooth moduli space of singular Riccati foliations with fixed formal data
    5. 3.5 The irregular Riemann-Hilbert correspondence
    6. 3.6 Universal isomonodromic deformations
    7. 3.7 Construction of a family of singular Riccati foliations
  5. 4 The monodromy map is a local biholomorphism
    1. 4.1 Factorization of the monodromy map
    2. 4.2 Ehresmann transversality
    3. 4.3 Proof of the main theorem

##  0 Introduction

The projective structures considered is this article are surfaces locally
modeled on open subsets of the Riemann sphere $\mathbb{P}^{1}$. Since the
automorphism group of the latter is equal to the group of Möbius
transformations, this amounts to saying they are Riemann surfaces, but with a
constraint stronger than the mere holomorphy of the coordinate changes:
precisely, they must be linear fractional transformations.

Let us consider a projective structure $P$. The analytic continuations of an
initial projective chart $\varphi_{0}$ along paths traveling across the
surface give rise to a multivalued function, that can be uniformized by
passing to the universal covering. We thus obtain a holomorphic immersive
function $f$ with values in the Riemann sphere, called developing map, that
"globalizes" the initial chart. This developing map is equivariant with
respect to the monodromy representation

$\rho_{f}:\pi_{1}(P)\longrightarrow\operatorname{Aut}(\mathbb{P}^{1})\simeq\operatorname{PGL}(2,\mathbb{C})$

of $\varphi_{0}$. The latter "globalizes" the projective changes of
coordinates.

The developing map $f$, and in turn the associated representation $\rho_{f}$,
depend on the initial projective chart $\varphi_{0}$. However, the conjugacy
class of the monodromy representation does not. In this way, a projective
structure determines a conjugacy class of representations of its fundamental
group into $\operatorname{PGL}(2,\mathbb{C})$.

Given an oriented connected compact smooth real surface $S$, we generally
consider the set of projective structures on $S$ (i.e. inducing a
differentiable structure and an orientation compatible with the one of $S$),
each element of which defines a representation of the group $\pi_{1}(S)$, up
to conjugacy. This amounts to fix a genus $g$ and an orientation. Two
projective structures on $S$ can be isomorphic even if their monodromy
representations are not conjugated: in that sense, the monodromy is not an
invariant. To remedy this and define a true invariant, we introduce the notion
of an isomorphism (or equivalence) of marked projective structure: it is an
isomorphism that is moreover isotopic to the identity of $S$. We denote by
$\mathcal{P}(S)$ the set of equivalence classes of marked projective
structures on $S$. This allows us to define the monodromy map

$\operatorname{Mon}_{S}:\mathcal{P}(S)\longrightarrow\mathcal{R}(S):=\operatorname{Hom}(\pi_{1}(S),\operatorname{PGL}(2,\mathbb{C}))/\operatorname{PGL}(2,\mathbb{C}).$

To a geometric object, this map associate an algebraic one. It builds a bridge
between two types of mathematical objects of different natures, and we may
hope that adopting the most adequate perspective will make the resolution of
some problems easier, that is, by crossing the bridge if only it is possible
to do so in both directions. Having this picture in mind, two questions arise
naturally:

  1. 1.

_Under which condition an equivalence class of representations in
$\mathcal{R}(S)$ arises from a monodromy representation of a complex
projective structure on $S$?_

  2. 2.

_Are two marked projective structures defining equivalent monodromy
representations necessarily equivalent?_

The first question concerns the characterization of the image of the map
$\operatorname{Mon}_{S}$. It has been solved by Gallo, Kapovich and Marden
[GKM00] when $S$ has genus $g\geq 2$. The second question, of whether
$\operatorname{Mon}_{S}$ is one-to-one, has in general a negative answer.
Thus, the monodromy representation does not characterize completely the
complex projective structure to which it is associated. However, several
weaker injectivity results are known. For example, it appeared as early as in
the work of Poincaré that for all complex structure $C$ on $S$, the monodromy
map $\operatorname{Mon}_{S}$ is injective in restriction to the set of
projective structures compatible with $C$. Alternatively, one can endow the
sets $\mathcal{P}(S)$ and $\mathcal{R}(S)$ with a topology, or even with a
real or complex differentiable structure, and ask a local version of the
second question (and thus of the first one, in a sense) The monodromy map is
regular, and Hejhal showed in 1975 that if the genus $g$ of $S$ is at least
$2$, then it is a local $\mathcal{C}^{\infty}$-diffeomorphism [Hej75, Thm 1 p.
20]. A few years later, Earle and Hubbard showed via a different approach that
it is a local biholomorphism [Ear81, Hub81]. The genus $1$ case has to be
treated differently, but leads to a similar statement; we refer to [LMP09,
Sec. 1.4]. Let us mention that a new proof of Hejhal’s theorem for genus $2$
surfaces was given in [HL19, Sec. 8.4.2].

The main result of our paper (Theorem 4.3) is a generalization of Hejhal’s
theorem to meromorphic projective structures.

####  $(G,X)$-structures.

A complex projective structure can be equivalently defined as a
$(G,X)$-structure, for the group $G=\operatorname{Aut}(\mathbb{P}^{1})$ and
the variety $X=\mathbb{P}^{1}$. The monodromy map can be defined in the more
general setting of $(G,X)$-structures (for other $G$ and $X$) and the
arguments of Ehresmann and Thurston [Thu80, Prop. 5.1] show that it is also a
local homeomorphism. See also [Gol21, Thm. 7.2.1] and [Gol10, Sec. 7] and the
references therein.

#### Quadratic differentials.

Projective structures where studied as soon as in the xixth century in
relation with Schwarzian equations and their monodromy, and also with the
uniformization problem (see [dSG16]). Schwarzian equations are the homogeneous
linear equations of order two of the form

$y^{\prime\prime}+\frac{q(x)}{2}y=0,$

where $q$ is a meromorphic function over a domain of the complex plane. A
brief computation shows that the quotient $\varphi=y_{1}/y_{2}$ of two
linearly independent solutions of such an equation satisfies

$\mathcal{S}(\varphi):=\left(\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\right)^{2}=q$
(1)

where $\mathcal{S}(\varphi)$ is the Schwarzian derivative of $\varphi$. From
the basic properties of the latter, we deduce that two solutions differ by a
linear fractional transformation acting on the target: thus, such functions
$\varphi$ define a projective atlas.

Those basic properties are the following: first, the Schwarzian derivative of
an immersive function vanishes identically if and only if this function is the
restriction of a linear fractional transformation. Intuitively, this
derivative measures the failure of an immersion to be the restriction of a
linear fractional transformation. In addition, the composition rule
$\mathcal{S}(f\circ g)=(g^{\prime})^{2}\mathcal{S}(f)\circ g+\mathcal{S}(g)$
it satisfies, applied to projective changes of coordinates $g$ (for which
$\mathcal{S}(g)\equiv 0$), becomes exactly the change of variable formula for
global quadratic differential forms.

In fact, the set of projective structures on a given compact Riemann surface
$C$ is endowed with an affine space structure for the vector space of
quadratic differential forms on $C$. The projective charts of the translation
of $P$ by $\phi$ are defined as follows: if $x$ is a chart of $P$ with respect
to which $\phi=\frac{q(x)}{2}dx\otimes dx$, then the charts of $P+\phi$ are
the solutions of the equation (1). On the other hand, the difference between
two projective structures on $C$ is measured by a quadratic differential form
defined locally as the Schwarzian derivative of the difference between two
charts, one for each of those projective structures. It is this relation
between projective structures and quadratic differential forms that allows us
to endow the moduli space $\mathcal{P}(S)$ with a smooth complex structure
(more precisely: with an holomorphic affine bundle structure over the
Teichmüller space $\mathcal{T}(S)$, for the vector bundle of quadratic
differential forms).

Thus, a projective structure might be seen as a "global scalar differential
equation on a manifold", that is, as a connection.

#### Meromorphic projective structures and opers.

The viewpoint on projective structures that turns out to be the most fruitful
in the setting of this paper is the one of opers (as defined by Beilinson and
Drinfeld). To a projective structure on $S$, we can associate in a canonical
way a $G$-oper on the underlying Riemann surface $C$, for the group
$G=\operatorname{PGL}(2,\mathbb{C})$, and conversely. It is a triple
$(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ composed of a
$\mathbb{P}^{1}$-bundle $\pi$, a holomorphic foliation $\mathcal{F}$ on $Q$
transverse to the fibers of $\pi$ and a section $\sigma$ of the bundle $\pi$
transverse to the foliation $\mathcal{F}$. Such a foliation is said to be
Riccati with respect to $\pi$, because it is induced by a Riccati differential
equation in any local trivialization of this $\mathbb{P}^{1}$-bundle. Riccati
foliations play a preeminent role in the theory of holomorphic foliations.

A demonstration of Hejhal’s theorem based on this viewpoint was sketched by
Loray and Marín [LMP09]; the ideas implemented here are in fact due to
Ehresmann and Thurston. We have completed this demonstration and adapted it to
the case of meromorphic projective structures.

Let us mention that a generalization (a weak version) of Hejhal’s theorem for
$G$-opers (without singularity), valid for more general choices of group $G$,
was given by Sanders [San18, Thm. 6.3].

The relation between projective structures and quadratic differential forms is
also essential to the precise definition of projective structures with poles.
Let us consider a projective structure $P$ on $C$. A meromorphic projective
structure on $C$ can be defined as the translation of $P$ by a meromorphic
quadratic differential form $\phi$ on $C$ (away from the poles of the latter).
The poles $p_{i}$ of $P$, as well as their orders $n_{i}$, are by definition
the ones of $\phi$.

To meromorphic projective structures correspond meromorphic
$\operatorname{PGL}(2,\mathbb{C})$-opers, that is to say triples
$(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ as above, except $(\pi:Q\rightarrow
C,\mathcal{F})$ is now a singular Riccati foliation. This correspondence is
not bijective, though, in contrast with the non-singular case: two opers not
holomorphically equivalent can very well define the same meromorphic
projective structure. Nonetheless, there exists a unique minimal
$\operatorname{PGL}(2,\mathbb{C})$-oper associated with a meromorphic
projective structure without apparent singularity, minimal in the sense that
its polar divisor is minimal up to bimeromorphic gauge transformation.

Defining the monodromy map is a much more delicate task than in the non-
singular case. Let us denote by $\Sigma$ the set of poles of a meromorphic
projective structure $P$ on $C$. The monodromy representation of $P$ is
defined as the one of the non-singular projective structure induced by $P$ on
$C^{*}=C\smallsetminus\Sigma$. It includes the monodromy originating from the
topology of $S$, but also local monodromies, around each singular points. Then
comes, for poles of order at least $3$, the Stokes data, forming the so-called
generalized monodromy data. This additions are inspired by the theory of
meromorphic linear rank $2$ connections, of which Riccati foliations are the
projectivized versions. Note that the monodromy representation of a
meromorphic projective structure is also the one of the associated Riccati
foliation.

In fact, every singular Riccati foliation is the projectivization of
meromorphic rank $2$ connection $(E,\nabla)$, that is to say a global linear
differential system. Knowing this, we can use the existing works on linear
connections and deduce some analogous results regarding Riccati foliation.
Notably, it is generally in the context of linear connections that can be
found, in the literature, the constructions of moduli spaces and isomonodromic
deformations. We keep in mind, though, that in our work no mathematical
necessity imposes to speak about linear connections. In a similar way, a
$\operatorname{PGL}(2,\mathbb{C})$-oper lifts to a
$\operatorname{GL}(2,\mathbb{C})$-oper, that is to say a triple $(E,\nabla,L)$
composed of a rank $2$ vector bundle on $C$, a meromorphic connection $\nabla$
on $E$ and a line subbundle $L$ of $E$ satisfying a transversality condition.

The sort of "bridge" established by the monodromy map is well-known for linear
connections: it is the Riemann-Hilbert correspondence. The study of the
monodromy allows a qualitative study of differential equations in the complex
domain without having to solve them explicitly, which we can rarely achieve.
In the case of meromorphic connections, it is necessary, in order to establish
such a bijective correspondence, to include additional data to the monodromy
representation. Those are notably Stokes matrices, at poles of orders at least
$2$. The analysis of singularities reveals a profound dichotomy in the
behavior of solutions – and in turn in the local analytic classification of
equations – according to the orders of those singularities: equal to $1$ or at
least $2$ (this is known as the Stokes phenomenon, discovered by Stokes in the
xixth century). We say they are regular (or moderate, or tame) singularities
and irregular (or wild) singularities, respectively. The moduli spaces of
generalized monodromy data are called wild character varieties.

The same goes for meromorphic projective structures (but for pole orders at
most $2$ or at least $3$). Allegretti and Bridgeland constructed a smooth
(possibly non-Hausdorff) character variety
$\mathcal{X}^{*}(\mathbb{S},\mathbb{M})$ (using Fock-Goncharov coordinates
[FG06]) containing generalized monodromy data together with a framing, and
showed that the monodromy map

$\displaystyle\operatorname{Mon}_{\mathbb{S},\mathbb{M}}:\mathcal{P}^{*}(\mathbb{S},\mathbb{M})\longrightarrow\mathcal{X}^{*}(\mathbb{S},\mathbb{M})$
(2)

is holomorphic [AB20]. Here, $\mathcal{P}^{*}(\mathbb{S},\mathbb{M})$ denotes
the moduli space of (equivalence classes of) meromorphic projective structures
marked by $(\mathbb{S},\mathbb{M})$, signed, and without apparent singularity.
The marking of a meromorphic projective structure on $S$ by a real bordered
surface $\mathbb{S}$ with marked points $\mathbb{M}$ is a marking of the
surface obtained by the real blow-up of $S$ at each irregular singularity,
with marked points corresponding to regular singular points and Stokes
directions on this blown-up surface. The signing encodes a choice of fixed
point of the local monodromy at regular singularities, needed in the
definition of the monodromy map of Allegretti and Bridgeland (in order to
define the framings). Except in a few special cases (always in genus $0$ and
with a small number of poles counted with multiplicity),
$\mathcal{P}^{*}(\mathbb{S},\mathbb{M})$ is a smooth variety.

In this paper, we fix the value $\lambda_{-1}^{(i)}$ of the residue at each
pole $p_{i}$ (as well as a signing), even at irregular singularities. The
associated moduli space

$\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(\lambda_{-1}^{(i)}))$

is again smooth. In particular, the number of pole $d$ and their orders
$(n_{i})$ are fixed.

Several authors have developed independently some alternative constructions of
generalized monodromy data and of wild character varieties for linear
connections [BMM06, vdPS09, Boa14, BY15, PR15, CMR17, CMR18]. In order to
obtain a _smooth_ moduli space, it is necessary to impose some good conditions
on the monodromy data (stability, non-degeneracy…).

On the other hand, Inaba [Ina22] constructed a _smooth_ moduli space for
linear connections. Here too, in reality, it is necessary to restrict to
connections satisfying a stability condition. We can see
$\operatorname{GL}(2,\mathbb{C})$-opers as a special kind of connections,
satisfying a special condition: they admit a transverse line subbundle. We are
going to show that this condition implies in particular that they are stable
in the sense of Inaba. Now, the moduli space of Inaba (when the curve $C$, the
polar divisor and the formal data are fixed) is in bijection with a subset of
the wild character variety via the irregular Riemann-Hilbert correspondence.
Fixing moreover the trace connexion on the determinant bundle, we deduce a
smooth moduli space for $\operatorname{PGL}(2,\mathbb{C})$-opers, and we apply
the irregular Riemann-Hilbert correspondence, in its version for singular
Riccati foliations. This is how we obtain a smooth wild character variety

$\bar{\mathcal{R}}^{*}(\mathbb{S},\mathbb{M},(\lambda_{-1}^{(i)})),$

without having to formulate a condition directly on the generalized monodromy
data.

Note that the monodromy map for meromorphic projective structures differs from
the Riemann-Hilbert map, since the complex structure on $S$ is not fixed and
since we consider only connection of a special kind, namely opers.

Our main result is stated as follows:

####  Theorem 4.3

_Assume that if $g=0$, then $|\mathbb{M}|\geq 3$, and that if $g=1$, then
$|\mathbb{M}|\geq 1$. Then, the monodromy map_

$\operatorname{Mon}_{S,(n_{i}),(\lambda_{-1}^{(i)})}:\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(\lambda_{-1}^{(i)}))\longrightarrow\bar{\mathcal{R}}^{*}(\mathbb{S},\mathbb{M},(\lambda_{-1}^{(i)}))$

_is a local biholomorphism_.

Our theorem generalizes and unifies a number of analogous results that where
already known before in some specific cases, and which where obtained through
various techniques. Let us cite the works of Bakken [Bak77] (cf. [Sib75, p.
198]) in genus $g=0$, with a unique pole of order $n\geq 5$; Iwasaki [Iwa91,
Thm. 5.9] for projective structures admitting a fixed number of apparent
singularities; Luo [Luo93] for the case where all singularities are regular
(of order exactly $2$) with an index
$\theta\in\mathbb{C}\smallsetminus\mathbb{Z}$ (indices can vary freely in this
set), that is to say with non-trivial and non-parabolic local monodromy;
Hussenot Desenonges [HD19] for a case where all singularities are regular (of
order $2$) with a fixed index $\theta=0\in\mathbb{Z}$ (hence a parabolic, in
particular non-trivial, local monodromy); Gupta and Mj [GM21] for the case
where all singularities are irregular (and the index vary freely in
$\mathbb{C}$).

Recently, LeBarron Alley [All21] studied meromorphic cyclic
$\operatorname{SL}(n,\mathbb{C})$-opers admitting a unique pole on the Riemann
sphere and concluded that the associated monodromy map is a holomorphic
immersion if the pole order is a multiple of $n$.

#### Ingredients of the demonstration.

Beside the works of Allegretti, Bridgeland and Inaba, our demonstration rely
on the construction of universal isomonodromic deformations by Heu [Heu10].
Locally on $\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(\lambda_{-1}^{(i)}))$,
we construct an analytic family of meromorphic
$\operatorname{PGL}(2,\mathbb{C})$-opers that locally embeds in the universal
family of singular Riccati equations derived from Inaba’s work. Then, the
isomonodromic flow allows us to retract two nearby opers having the same
generalized monodromy data one on the other, leading to the local injectivity
of the monodromy map. 4.3 then follows from the holomorphy of the monodromy
map, thanks to the open mapping theorem.

#### About the image of the monodromy map.

A natural and related problem (which is the extension of question $1$), is to
determine the image of the monodromy map of meromorphic projective structures.
Let us mention that analogs of the Gallo-Kapovich-Marden theorem (with
variable residues) where recently proved in a series of work by Faraco, Gupta
and Mj [GM20, Gup21, FG21] and Nascimento [Nas21]. The case of branched
complex projective structures with fixed branch divisor was handled by Le Fils
[LF23].

#### An open question in symplectic geometry.

The question of whether the monodromy map of non-singular projective
structures is a symplectomorphism was addressed by Kawai [Kaw96] and Loustau
[Lou15] (see also [BKN17] and [San18]). Indeed, it is well-known since the
work of Goldman [Gol84, Gol04] that the character variety (i.e. the target of
the monodromy map) has a natural symplectic structure. On the other hand, the
space of projective structures possesses several cotangent symplectic
structures depending on the different Schwarzian parametrizations. Loustau
gave a necessary and sufficient condition on the parametrization which ensures
that the monodromy map is a symplectomorphism. Is the monodromy map of
_meromorphic_ projective structures a symplectomorphism? The works of [Iwa92]
and [Kor18] gave a positive answer to this question is some specific cases
involving only regular singularities.

#### Structure of the paper.

This article is organized as follows. The first section is dedicated to the
introduction of meromorphic projective structures, while section 2 present
their relation to meromorphic $\operatorname{PGL}(2,\mathbb{C})$-opers. In
particular, we describe their minimal birational models. In section 3, we show
that opers belong to Inaba’s smooth moduli space. Section 4 contains the proof
of the main theorem.

#### Acknowledgments.

This work was carried out during my PhD thesis [Sér22] at the University of
Rennes, in the laboratory IRMAR - UMR CNRS 6625; I was supported by the Centre
Henri Lebesgue program ANR-11-LABX-0020-0; I finished this article after I
integrated the laboratory UMPA - UMR 5669 CNRS, at ENS de Lyon. I would like
to express all my gratitude to my PhD advisor, Frank Loray, for its guidance
and for all the enriching discussions we had. I would also like to thank Tom
Bridgeland and Sorin Dumitrescu, who reported my thesis. Their careful reading
of my manuscript has led to several improvements. Finally, let me thank
Professor Michi-aki Inaba for answering in great details a question about one
of its papers, thus clarifying a result I use in the present article.

##  1 Meromorphic projective structures

We start this first section with an introduction to complex projective
structures without poles and their monodromy map. The Reader who is not yet
familiar with the subject might profitably consult the surveys [Dum09, LMP09]
and the book [dSG16]. In the first section, we explain in particular that the
set of projective structures on a fixed complex curve is an affine space for
the vector space of quadratic differentials. This is fundamental because it
gives rise to a structure of a complex manifold on the moduli space of
projective structures of genus $g$. Moreover, it allows us to define
meromorphic projective structures. We state Hejhal’s theorem, and in the last
section we recall the definition of the moduli space of marked meromorphic
projective structures of genus $g$ with poles of prescribed (in a somewhat
loose sense) orders, constructed by Allegretti and Bridgeland in [AB20].

###  1.1 Complex projective structures (without pole) and their monodromy map

Let $S$ be a connected $\mathcal{C}^{\infty}$-smooth oriented real surface.

######  Definition 1.1.

A _complex projective structure on S_ is a maximal atlas of charts mapping
open sets in $S$ into open sets in the complex projective line
$\mathbb{P}^{1}$ and such that the transition maps are _restrictions_ of
elements in the group of automorphisms of $\mathbb{P}^{1}$ as a complex
manifold (which we canonically identify with
$\operatorname{PGL}(2,\mathbb{C})$).

(It is always implicitly assumed that both the $\mathcal{C}^{\infty}$-smooth
structure and the orientation induced by this complex structure are the ones
initially fixed on $S$).

This is also the definition of a _$(G,X)$ -structure on $S$_, with
$G=\operatorname{PGL}(2,\mathbb{C})$ and $X=\mathbb{P}^{1}$.

Two projective structures on $S$ are said to be _isomorphic_ (resp. _marked
isomorphic_) if there exists an orientation-preserving diffeomorphism of $S$
(resp. a diffeomorphism of $S$ homotopic to the identity222Or isotopic, this
is equivalent. Moreover, the orientation is automatically preserved.) pulling
back any projective chart of one of them to a projective chart of the other.
We will denote by $\mathcal{P}(S)$ the set of marked isomorphism classes of
projective structures on $S$. The marking will be needed in the definition of
the monodromy map.

Every projective structure on $S$ induces a complex structure on $S$, hence
there is a forgetful map $\mathcal{P}(S)\rightarrow\mathcal{T}(S)$ (where
$\mathcal{T}(S)$ denotes the Teichmüller space of $S$) and we may as well
speak of a projective structure on a complex curve. Every complex structure is
induced by a projective one –meaning that the forgetful map is surjective– as
shown by the following example.

######  Example 1.2.

Let $U$ be an open subset of the complex projective line and assume that it is
invariant under a free and properly discontinuous action of a group of Möbius
transformations $\Gamma$. Then, the local inverses of the associated quotient
map $U\rightarrow U/\Gamma$, which is a covering map, form a projective atlas
on the quotient space.

The Poincaré-Koebe uniformization theorem ensures that any connected smooth
complex curve $C$ is biholomorphic to such a quotient space. The projective
structure induced on the latter can be pulled back to the curve $C$, giving a
projective structure compatible with the complex one. The resulting structure
is called the _canonical projective structure_ on $C$.

The standard atlas on $\mathbb{P}^{1}$ is also projective, and all projective
atlas on $\mathbb{P}^{1}$ are compatible with it. However, there is an
infinite number of distinct projective structures on $C$ as soon as it is
different from $\mathbb{P}^{1}$.

Let us denote by $u:\tilde{S}\longrightarrow S$ the universal cover of $S$,
and let $C$ be a complex structure on $S$.

######  Proposition 1.3 (See Lem. 1 in [Hub81]).

Given a projective structure $P$ on $C$:

  1. 1.

There exists an analytic map

$f:\tilde{C}\longrightarrow\mathbb{P}^{1}$

such that on each contractible open set $U\subset C$ on which the composition
$f\circ u^{-1}$ is defined (here, $u^{-1}$ denote any local inverse for $u$),
it is a projective chart. Any other such map is of the form $\sigma\circ f$
for some $\sigma\in\operatorname{PGL}(2,\mathbb{C})$. We say $f$ is _a
developing map_ for the projective structure $P$.

  2. 2.

To every developing map $f$ there corresponds a unique morphism

$\rho_{f}:\pi_{1}(C)\longrightarrow\operatorname{PGL}(2,\mathbb{C})$

(called _the monodromy representation_) such that $\rho_{f}(\gamma)\circ
f=f\circ\gamma$ (_equivariance_) and $\rho_{\sigma\circ
f}=\sigma\circ\rho_{f}\circ\sigma^{-1}$.

$U_{j}$
---
$U_{i}$
---
$P$
---
$U_{0}$
---
$\gamma$
---
$U_{n}$
---
$\varphi_{i}$
---
$\varphi_{j}$
---
$\varphi_{ij}$
---
$\mathbb{P}^{1}$
---
$\mathbb{P}^{1}$
---
$U_{1}$
---
Figure 1: If the $\varphi_{ij}:=\varphi_{i}\circ\varphi_{j}^{-1}$ denote the
change of coordinate charts of $P$ as well as the associated Möbius
transformations, then the monodromy along the loop $\gamma$ covered by the
charts $U_{0},\dots,U_{n}$ is obtained by composition of the $\varphi_{ij}$:
$\rho_{f}(\gamma)=\varphi_{n,(n-1)}\circ\cdots\circ\varphi_{2,1}\circ\varphi_{1,0}$.

A developing map can be constructed by analytic continuation of a projective
chart by other projective charts. In this sense, it globalizes a projective
chart, whereas the monodromy representation globalizes the transition maps. A
projective structure on $S$ can be described by the data of a
$\mathcal{C}^{\infty}$ immersion $f:\tilde{S}\rightarrow\mathbb{P}^{1}$
together with a representation
$\rho\in\operatorname{Hom}(\pi_{1}(S),\operatorname{PGL}(2,\mathbb{C}))$ with
respect to which it is equivariant: a so-called _development-holonomy pair_
333Here, ”holonomy” and ”monodromy” are synonyms. $(f,\rho)$.

######  Definition 1.4.

Let us denote $\mathcal{R}(S)$ the set of
$\operatorname{PGL}(2,\mathbb{C})$-conjugacy classes of representations of the
fundamental group of $S$ in $\operatorname{PGL}(2,\mathbb{C})$. Thanks to 1.3,
to a marked isomorphism class of projective structures we can associate the
conjugacy class of the monodromy representation of one of its developing maps.
This defines the so-called _monodromy map_

$\operatorname{Mon}_{S}:\mathcal{P}(S)\longrightarrow\mathcal{R}(S).$

Note that the marking is essential to define this map; it is also key to the
smoothness of the Teichmüller space, hence of the moduli space
$\mathcal{P}(S)$, that we will discuss latter.

######  Theorem 1.5 (Poincaré).

The monodromy map $Mon_{S}$ is injective in restriction to each fiber of the
forgetful map $\mathcal{P}(S)\rightarrow\mathcal{T}(S)$.

A short proof of this fact is given in [LMP09].

#### Quadratic differentials and affine structure.

Let $C$ denote a (connected) smooth curve (i.e. a Riemann surface). We have
seen that the set of complex projective structures over $C$ is non-empty (cf.
1.2). We can expect it to be large, since being a Möbius transformation is a
much stronger condition than being holomorphic. In fact, it can be endowed
with an affine space structure for the vector space $H^{0}(C,(T^{*}C)^{\otimes
2})$ of global holomorphic quadratic differentials (see [Hub81, Lem. 4] or
[Gun67, Cor. 2]). We recall bellow this affine structure and introduce some
notations. If $C$ is compact, the Riemann-Roch theorem implies that the above
vector space is of dimension $3g-3$ (resp. $1$, $0$) if $C$ is of genus $g\geq
2$ (resp. $g=1,0$) (see, for example, [Jos06, Cor. 5.4.2]).

Let $f$ be a locally injective holomorphic function of one complex variable.
The _Schwarzian derivative of $f$_ is

$\mathcal{S}(f)=\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{2}.$

Now let $f,g$ two locally injective holomorphic maps between open sets of the
complex plane. The key properties of the Schwarzian derivative are

  * •

$\mathcal{S}(f\circ g)=(g^{\prime})^{2}\mathcal{S}(f)\circ g+\mathcal{S}(g)$,

  * •

$\mathcal{S}(g)\equiv 0$ if and only if $g$ is the restriction of a Möbius
transformation, i.e. $g(x)=\frac{ax+b}{cx+d}$, $(a,b,c,d)\in\mathbb{C}^{4}$,
$ad-bc\neq 0$.

The Schwarzian derivative thus depends on the coordinate it is computed, so we
usually specify it by a subscript: $\mathcal{S}_{x}(f)$.

Let $P_{1}$ and $P_{2}$ be two projective structures on $C$, with local
coordinates $x_{1}$ and $x_{2}$, respectively, and denote444There is a slight
abuse here: it is understood that projective charts take there values in
$\mathbb{C}$ and that transition maps are invertible linear fractional
functions. This is an alternative definition of a projective atlas.
$\psi=x_{2}\circ x_{1}^{-1}$. Then, we define $\phi:=P_{2}-P_{1}$ locally as

$\phi:=\frac{\mathcal{S}_{x_{1}}(\psi)}{2}dx_{1}^{\otimes 2},$

where $\mathcal{S}_{x_{1}}(\psi)$ denotes the Schwarzian derivative of $\psi$
with respect to the variable $x_{1}$. Those local expressions glue together by
virtue of the basic properties of the Schwarzian derivative, defining a global
holomorphic quadratic differential. Conversely, given a projective structure
$P_{1}$ and a holomorphic quadratic differential $\phi$ written locally as
$\phi=\frac{q(x_{1})}{2}dx_{1}^{\otimes 2}$, the charts of $P_{2}:=P_{1}+\phi$
are defined to be the solutions of the equation

$\mathcal{S}_{x_{1}}(\varphi)=q$ (3)

in $\varphi$. Furthermore, according to a theorem of Schwarz, it is equivalent
to define projective charts as quotients $\varphi=y_{1}/y_{2}$ where $y_{1}$
and $y_{2}$ range over all independent solutions of the equation

$y^{\prime\prime}+\frac{q(x_{1})}{2}y=0.$ (4)

Thus, a projective structure may be interpreted as a global generalization of
such a second order homogeneous linear scalar equation. For each chart of $C$,
we get a coefficient $q$ and their collection forms a _projective Cartan
connection_ (see [Na19]). On the intersections, the corresponding second order
equations are _projectively equivalent_ , in the sense that their sets of
quotients of solutions coincide (see [dSG16, Prop. VIII.3.8]).

######  Remark 1.

An equation of the form (4) but with opposite sign before $q/2$ is sometimes
used in the literature, e.g. in [AB20]. Let us apply the projective change of
coordinate $\tilde{x_{1}}=ix_{1}$ to equation (4). We obtain an equation
projectively equivalent to $y^{\prime\prime}-\frac{q(\tilde{x_{1}})}{2}y=0$
(see [dSG16, Prop. VIII.3.8]).

The above affine structure endows the set of projective structures on a fixed
curve $C$ with a structure of a complex manifold. The aim of the following
paragraph is to describe a natural smooth complex structure on the whole
$\mathcal{P}(S)$, i.e. letting the complex structure on $S$ vary.

#### The moduli space $\mathcal{P}(X/B)$.

For any holomorphic family $f:X\rightarrow B$ of compact smooth curves, let us
denote

$\mathcal{P}(X/B):=\\{(t,P):t\in B\textmd{ and $P$ is a projective structure
on $C_{t}:=f^{-1}(t)$}\\}.$

Most of the time, there will be no ambiguity regarding the family $f$ to which
this notation refers. There is a holomorphic vector bundle

$\mathcal{Q}(X/B)\rightarrow B,$

defined as $\mathcal{Q}(X/B)=f_{*}((T^{*}_{X/B})^{\otimes 2})$ (where
$T_{X/B}$ stands for the vertical tangent bundle of $X$, relative to $f$),
whose fiber over a point $t\in B$ is the vector space of quadratic
differential forms $\mathcal{Q}(X/B)_{t}=H^{0}(C_{t},(T^{*}C_{t})^{\otimes
2})$.

A choice of a section $s$ of the map
$\operatorname{pr}_{1}:\mathcal{P}(X/B)\rightarrow B$ furnishes a "choice of
origin" in each fibers, which in turn provides an identification

$\displaystyle\mathcal{P}(X/B)$ $\displaystyle\longrightarrow\mathcal{Q}(X/B)$
$\displaystyle(t,P)$
$\displaystyle\longmapsto(t,P-\operatorname{pr}_{2}(s(t)))$

(here, $\operatorname{pr}_{i}$ denotes the restriction to $\mathcal{P}(X/B)$
of the projection on the $i$-th factor). Two such sections $s_{1}$ and $s_{2}$
will induce the same complex structure on $\mathcal{P}(X/B)$ if and only if
their difference $s_{2}-s_{1}$ is a holomorphic section of $\mathcal{Q}(X/B)$.
Hubbard showed there is a natural complex structure on $\mathcal{P}(X/B)$,
though, which is the only one such that firstly the map
$\mathcal{P}(X/B)\rightarrow B$ is holomorphic and secondly the _analytic
families of projective structures_ given by its holomorphic sections are
induced by relative projective structures on $f$ [Hub81, Prop. 1]. The map
$\mathcal{P}(X/B)\rightarrow B$ is a holomorphic affine bundle for the vector
bundle $\mathcal{Q}(X/B)\rightarrow B$.

######  Definition 1.6.

A _relative projective atlas_ on a holomorphic family $f:X\rightarrow B$ of
(compact or not) smooth curves is an atlas $(U_{i},\varphi_{i})$ for $X$ such
that for each $t\in B$, the restriction $(U_{i}\cap C_{t},\varphi_{i|C_{t}})$
(where the range of $\varphi_{i|C_{t}}$ is understood to be appropriately
restricted as well) forms a projective atlas on $C_{t}$. A _relative
projective structure_ is a maximal relative projective atlas.

######  Example 1.7.

The Teichmüller space of genus $1$ curves is isomorphic to the upper half-
plane $\mathbb{H}$. Consider the first projection
$p_{1}:\mathbb{H}\times\mathbb{C}\rightarrow\mathbb{H}$. A relative projective
atlas on the holomorphic family

$f:(\mathbb{H}\times\mathbb{C})/\braket{(\tau,z+1),(\tau,z+\tau)}\longrightarrow\mathbb{H}$

of elliptic curves induced by $p_{1}$ is given by local inverses of the
quotient map

$\mathbb{H}\times\mathbb{C}\rightarrow(\mathbb{H}\times\mathbb{C})/\braket{(\tau,z+1),(\tau,z+\tau)},$

composed with the projection on the second factor $\mathbb{C}$, regarded as
embedded in $\mathbb{P}^{1}$.

The aforementioned natural complex structure on $\mathcal{P}(X/B)$ makes the
projection $\mathcal{P}(X/B)\rightarrow B$ into a holomorphic affine bundle
for the vector bundle $\mathcal{Q}(X/B)$, with the action
$(t,\phi)\cdot(t,P)=(t,P+\phi)$. Note that the key ingredient in Hubbard’s
proof is the existence, at least locally over the base $B$, of a relative
projective structure. It is worth mentioning that not every holomorphic family
of curves carry a relative projective structure555In contrast, the constant
rank theorem provides any holomorphic families of curves with a relative
complex atlas.. For example, as showed by Zhao in [Zha19], a non-isotrivial
holomorphic family of compact curves over a compact curve $B$ cannot support
any relative projective structures. Hubbard verified the existence of a
relative projective structure on a family of genus $g\geq 2$ curves under the
assumption that the base $B$ is Stein.

Assume that $B=\mathcal{T}(S)$ is the Teichmüller space of marked curves (i.e.
marked Riemann surfaces) on the compact smooth surface $S$, and
$f:X\rightarrow B$ is the universal holomorphic family of curves over $B$.
Then, the bundle $\mathcal{Q}(X/B)\rightarrow B$ is trivial, because the
Teichmüller space is contractible and Stein. Moreover, in this case we can
reinterpret elements $(t,P)\in\mathcal{P}(X/B)$ as follow: $P$ is a marked
projective structure, with its marking being the only one compatible with the
marking of $C_{t}$. Then, there is a natural bijection
$\mathcal{P}(S)\simeq\mathcal{P}(X/B)$ (remember that marked complex curves
have no non-trivial automorphisms; this rigidity is key to the smoothness of
Teichmüller spaces and the existence of universal family of curves).

In the genus $1$ case, the section $s$ of
$\mathcal{P}(S)\rightarrow\mathcal{T}(S)$ provided by the uniformization
theorem (as in 1.7) induces the natural complex structure on $\mathcal{P}(S)$.
However, this is false as soon as $g\geq 2$ (note that if $g=1$ the monodromy
representation of the canonical projective structure is complex, whereas if
$g\geq 2$ it is real). A way to construct a relative projective structure in
the latter case is to use the simultaneous uniformization theorem of Bers.
This leads to a so-called _quasi-Fuchsian section_ ; for more details about
this, see the discussion in [Dum09, Sec. 3.3].

The existence of a relative projective structure locally over the base of an
arbitrary holomorphic family of compact curves may then be deduced from the
universal property of Teichmüller spaces.

######  Proposition 1.8.

Let $f:X\rightarrow B$ be a holomorphic family of compact smooth curves. For
any point $t_{0}$ in $B$, there exists a neighborhood $B_{t_{0}}$ of $t_{0}$
in $B$ such that the holomorphic family
$f_{t_{0}}:f^{-1}(B_{t_{0}})\rightarrow B_{t_{0}}$ induced by $f$ supports a
relative projective structure.

#### Hejhal’s theorem.

Gunning showed that a compact smooth curve $C$ admits an _affine structure_
666A projective structure such that, in coordinates, transition maps belong to
the group $\operatorname{Aff}(\mathbb{C})$. if and only if its genus is $1$
(this is false for meromorphic projective structures, as it will become
apparent latter). Moreover, every projective structure on a genus $1$ curve
can be reduced to an affine structure (see [Gun66, Cor. 3 p. 173, p. 192] or
[LMP09, Ex. 1.7]). This implies that the monodromy representation of any
developing map of a projective structure on a surface of genus $g\geq 2$ have
non-commutative image [Hub81, Cor. p. 260] (and are irreducible). After
choosing a set of generators of the group $\pi_{1}(S)$, the set of
representations
$\operatorname{Hom}(\pi_{1}(S),\operatorname{PGL}(2,\mathbb{C}))$ can be
identified with an analytic space in $\operatorname{PGL}(2,\mathbb{C})^{2g}$.
The open subset of representations with non-commutative image in
$\operatorname{Hom}(\pi_{1}(S),\operatorname{PGL}(2,\mathbb{C}))$ is a sub-
manifold of $\operatorname{PGL}(2,\mathbb{C})^{2g}$ and its quotient
$\mathcal{R}^{\operatorname{nc}}(S)$ modulo conjugacy has a unique structure
of complex manifold such that the quotient map is analytic [Hub81, Prop. 4].
It is of complex dimension $6g-6$, just as $\mathcal{P}(S)$.

######  Theorem 1.9.

_([Hej75, Thm. 1 p. 20], [Ear81, Cor. 2], [Hub81, Thm. p. 272])_ If $S$ has
genus $g\geq 2$, then the monodromy map

$\operatorname{Mon}_{S}^{g\geq
2}:\mathcal{P}(S)\longrightarrow\mathcal{R}^{\operatorname{nc}}(S)$

is a local biholomorphism.

In the genus $1$ case, the monodromy representation always belongs to the set
$\mathcal{A}(S):=\operatorname{Hom}(\pi_{1}(S),\operatorname{Aff}(\mathbb{C}))/\sim$
of affine representation modulo conjugacy. It is also possible to show that
the monodromy map
$\operatorname{Mon}_{S}^{g=1}:\mathcal{P}(S)\longrightarrow\mathcal{A}(S)$ is
a local biholomorphism (see [LMP09, Sec. 1.4]). Here, the _affine distortion_
$f^{\prime\prime}/f^{\prime}$ can play the role of the Schwarzian derivative.

In the genus $0$ case, there is only one projective structure and it is simply
connected, so it would not make sense to seek for a similar statement.

###  1.2 The space of meromorphic projective structures

The definition of a meromorphic projective structure on $C$ relies on the
affine space structure on the set of complex projective structures over a
complex curve.

######  Definition 1.10.

A _meromorphic projective structure $P$ on a complex curve $C$_ is a
projective structure $P^{*}$ on the complement $C^{*}=C\smallsetminus\Sigma$
of a finite subset $\Sigma\subset C$, such that given a holomorphic projective
structure $P_{0}$ on $C$, the quadratic differential $\phi=P^{*}-P_{0|C^{*}}$
on $C^{*}$ extends to a meromorphic quadratic differential on $C$, that is to
say a meromorphic section of $(T^{*}C)^{\otimes 2}$.

Such a structure $P^{*}$ is therefore determined by a pair $(P_{0},\phi)$.
Though it is not unique, every other pair $(\tilde{P_{0}},\tilde{\phi})$
describing $P$ satisfies
$(\tilde{P_{0}},\tilde{\phi})=(P_{0}+\phi_{\operatorname{hol}},\phi-\phi_{\operatorname{hol}})$
for some holomorphic quadratic differential $\phi_{\operatorname{hol}}$. Hence
the condition on $\phi$ is independent of $P_{0}$: points in $\Sigma$ are
called _poles_ of the structure $P$ and the _pole order_ of $P$ at a point can
be defined as the pole order at that point of one of its so-called _polar
differentials_ $\phi$. Hence if $n_{p}$ is the order of $p\in\Sigma$, we can
define the (effective) _polar divisor_ of $P$ by
$D_{P}=\sum_{p\in\Sigma}n_{p}p$.

#### The monodromy representation.

The monodromy representation of a meromorphic projective structure $P$ is
well-defined, up to $\operatorname{PGL}(2,\mathbb{C})$-conjugacy, as the one
of $P^{*}$ in (the notations of 1.10). It is a class of representation of the
fundamental group of the punctured curve $C\smallsetminus\Sigma$. The _local
monodromy_ around a pole is the image of a loop encircling it once in the
counterclockwise direction.

######  Remark 2.

If $(y_{1},y_{2})$ is a fundamental system of solutions of a second order
homogeneous scalar linear differential equation, then

$\begin{pmatrix}y_{1}&y_{2}\\\ y_{1}^{\prime}&y_{2}^{\prime}\end{pmatrix}$

is a fundamental solution of its companion systems, and reciprocally. As a
consequence, the monodromy matrices of the scalar equation and its companion
system (both acting by right multiplication) around a pole are the same, say
$M$.

The monodromy of the projective structure associated to an equation of the
form (4), however, is given in the homogeneous coordinates $[y_{1}:y_{2}]$ by
the projectivization of the _transpose_ ${}^{\top}M$. But it is a classical
fact that a matrix with complex coefficients is always conjugated to its
transpose (in particular, a matrix has the same eigenvalues as its transpose),
thus the local monodromy of the projective structure is, up to conjugacy, the
projectivization of the local monodromy of the associated scalar equation and
of that of its companion linear system.

#### Regular singularities.

This terminology comes from second order scalar differential equations (and
their companion first order linear differential systems). The term _regular
singular_ refers to points where solutions of the scalar equation (or the
linear system) exhibit at most polynomial growth on some sectors (see [Ily]).
For the scalar equation (4) (and by definition for projective structures),
this is equivalent to $q$ having a pole order at most $2$ (we also say it is a
_Fuchsian singularity_), but not for linear systems. On the one hand, a
singularity with _Poincaré rank_ equal to $0$ is regular singular. On the
other, only the _minimal_ Poincaré rank (minimal in the equivalence class of
the system up to meromorphic gauge transformation777cf. 3.2.) at a regular
singularity must be equal to $0$. Singularities that are not regular are of
course called _irregular singularities_. Though this terminology is well-
established, some authors prefer the adjectives _tame_ (or _moderate_) and
_wild_ , which are less confusing.

A regular singularity is called _apparent_ 888cf. [Gup21, Sec. 2.3] for a
review of other definitions appearing in the literature. if $P^{*}$ has
trivial local monodromy, else it is called _logarithmic_.

Apparent singularities correspond exactly to _branch points_ of the projective
structure, i.e. points for which there exists an integer $k\in\mathbb{N}$ such
that projective charts read $\varphi(x)=x^{k}$ in some local coordinates.
About branch points, see also 2.12. In the present paper, we will mainly focus
on projective structures without branch points. Branched projective structures
where studied by Mandelbaum [Man72].

Let $p$ be a regular singularity of $P$, and write
$\phi=\frac{q(x)}{2}dx^{\otimes 2}$ for a polar differential in some
coordinate chart $x$ centered at $p$. We define the _residue_ 999Here, we
follow the terminology and notations of [dSG16, Sec. IX.1], which are a
slightly different from the ones in [AB20]: our $q$ is equal to their
$-2\varphi$, hence their definitions of ”leading coefficient”, ”residue” and
”exponent” does not match ours. (of order $2$) of $P$ at $p$ to be

$\operatorname{res}_{2}(p)=\lim_{x\to 0}x^{2}q(x).$ (5)

This definition is independent of the polar differential and of the coordinate
chart. Latter, we will also call "residue at $p$" a quantity different from
$\operatorname{res}_{2}(p)$, but related (see 13).

The _index_ $\theta(p)$ of $P$ at $p$, is then defined up to a multiplication
by $-1$ as

$\theta(p)=\pm\sqrt{1-2\operatorname{res}_{2}(p)},$

in other words, writing the Schwarzian equation (3) for a projective chart
$\varphi$ around $p$ (in a complex coordinate $x$ centered at $p$) we have

$\mathcal{S}_{x}(\varphi)=q(x)=\frac{1-\theta(p)^{2}}{2x^{2}}+o(x^{-2}).$

Projective charts and monodromy around regular singularities may be studied
according to the value of the index $\theta(p)$. For a detailed analysis, the
reader may refer to [dSG16, Sec. IX.1].

Consider the companion system of the second order equation (4) (cf. 2), or
more naturally equation (16),

$dY+\begin{pmatrix}0&-1\\\ q(x)/2&0\end{pmatrix}Ydx=0.$ (6)

Up to conjugacy, the projectivization of the monodromy of this system
corresponds to the local monodromy of the projective structure.

######  Proposition 1.11 (cf. [AB20], Lem. 5.1).

Assume that the system (6) has a regular singularity at $z=0$. Then, the
eigenvalues of its monodromy around $z=0$ are

$\xi^{\pm}=\exp\left(2i\pi\left(\frac{1\pm\theta}{2}\right)\right),$

where $\theta=\theta(0)$ is defined up to a multiplication by $-1$ as above.
In particular, the eigenvalues $\xi^{\pm}$ are distinct if and only if
$\pm\theta\in\mathbb{C}\smallsetminus\mathbb{Z}$.

Several possibilities occur for the local monodromy of the projective
structure, depending on the value of the index:

  * •

if $\pm\theta\in\mathbb{C}\smallsetminus\mathbb{Z}$, it is non-trivial and
non-parabolic,

  * •

if $\pm\theta\in\mathbb{Z}$, it is trivial or parabolic.

######  Remark 3.

In order to construct the _monodromy framed local system_ of a projective
structure, Allegretti and Bridgeland needed to chose a point of
$\mathbb{P}^{1}$ fixed by the local monodromy of each regular singularity.
Such a choice is encoded by the notion of _signed_ meromorphic projective
structure: a signing is just a choice of sign "$+$" or "$-$" at some of the
regular singularities. In [AB20, Prop. 8.4], the signing considered is the one
given by a "choice of sign" of the index
$\theta(p)=\pm\sqrt{1-2\operatorname{res}_{2}(p)}$ at each regular singularity
$p$, i.e. a determination of the square root. The square root branches over
$\theta=0$, so that no sign is chosen in this case, which is fine because the
local monodromy is then parabolic [dSG16, Sec. IX.1], hence it has a unique
fixed point. However, the square root does not branch over
$\theta\in\mathbb{Z}\smallsetminus\\{0\\}$, hence a choice is made here
although it is not really needed.

Those points of the moduli space of projective structures where parabolic
local monodromy occur (we always exclude apparent singularities) causes a
failure in the local injectivity of the monodromy map.

In the introduction, we have mentioned several works generalizing Hejhal’s
theorem for meromorphic projective structures. Luo [Luo93] considered the case
where all singularities are regular (of order exactly $2$) with an index
$\theta\in\mathbb{C}\smallsetminus\mathbb{Z}$, thus excluding non-trivial and
non-parabolic local monodromy. On the other hand, Hussenot Desenonges [HD19]
considered a case where all singularities are regular (of order $2$) with an
index $\theta=0\in\mathbb{Z}$ (hence a parabolic local monodromy). Thus, Luo
does not fix the indices but excludes parabolic local monodromies, whereas
Hussenot Desenonges considers only parabolic local monodromies but fixes the
indices. In the present paper, we will fix the values of the indices and we
will not exclude parabolic local monodromies.

As for Gupta and Mj [GM21], they dealt with projective structures with only
irregular singularities, hence this issue does not arise in their work.

######  Example 1.12.

The family of Gauss hypergeometric equations

$x(x-1)\frac{d^{2}y}{dx^{2}}+[(\alpha+\beta+1)x-\gamma]\frac{dy}{dx}+\alpha\beta
y=0,\textmd{ with $\alpha,\beta,\gamma\in\mathbb{C}$ and $x\in\mathbb{C}$}$

induces a family of meromorphic projective structures on $\mathbb{P}^{1}$ with
$3$ regular singularities at $0$, $1$ and $\infty$ [dSG16, Sec. IX.2].

In fact, every projective structure of this type is induced by a
hypergeometric equation. Projective charts take the form

$\varphi(x)=f(x)^{\theta}\textmd{ or
}\varphi(x)=f(x)^{\theta}+\operatorname{log}(f(x))$

where $\theta\in\mathbb{C}$ depends on $(\alpha,\beta,\gamma)$, and $f$ is a
local complex coordinate around a singularity $x_{0}$, $f(x_{0})=0$.

######  Example 1.13.

The Heun equations induces projective structures on $\mathbb{P}^{1}$ with $4$
regular singularities (see [dSG16, Sec. IX.3.1]).

######  Example 1.14.

An example on the torus is provided by the Lamé equation (see [dSG16,
IX.3.3]).

The subsequent paragraph is devoted to the description of a moduli space of
meromorphic projective structures.

#### The moduli space $\mathcal{P}(X/B;\mathcal{D})$.

Let $d\in\mathbb{N}$ be an integer and $(n_{i})\in\mathbb{N}^{d}$ be a
collection of integers and

$\textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}}$

a holomorphic family of compact smooth curves, together with $d$ disjoint
holomorphic sections $p_{1},\dots,p_{d}$. We can define an effective divisor
$\mathcal{D}$ as $\mathcal{D}:=\sum_{i=1}^{d}n_{i}\mathcal{D}_{i}$ with
$\mathcal{D}_{i}=\operatorname{im}(p_{i})$. Its restriction to
$C_{t}=f^{-1}(t)$, for some $t\in B$, will be denoted
$D_{t}=\sum_{i=1}^{d}n_{i}p_{i}(t)$. There is a holomorphic vector bundle

$\mathcal{Q}(X/B;\mathcal{D})\rightarrow B$

whose fiber over a point $t\in B$ is the vector space
$\mathcal{Q}(X/B;\mathcal{D})_{t}=H^{0}(C_{t},(T^{*}C_{t})^{\otimes
2}(D_{t}))$ of meromorphic quadratic differentials on $C_{t}$ having poles of
order at most $n_{i}$ at the points $p_{i}(t)$ and no other pole. As a
consequence of the Riemann-Roch theorem,

$\dim(H^{0}(C_{t},(T^{*}C_{t})^{\otimes 2}(D_{t})))=3g-3+\sum_{i=1}^{d}n_{i}.$
(7)

Let us denote $\mathcal{P}(X/B)_{t}$ the fiber of the projection of
$\mathcal{P}(X/B)$ to $B$. Then, define $\mathcal{P}(X/B;\mathcal{D})$ as the
affine bundle (for the vector bundle $\mathcal{Q}(X/B;\mathcal{D})$) whose
fiber over $t$ is the quotient

$\mathcal{P}(X/B;\mathcal{D})_{t}:=(\mathcal{P}(X/B)_{t}\times
Q(X/B;\mathcal{D})_{t})/\sim$

for the equivalence relation

$((t,P_{1}),\phi_{1})\sim((t,P_{2}),\phi_{2})\Leftrightarrow(P_{2},\phi_{2})=(P_{1}+\phi_{\operatorname{hol}},\phi_{1}-\phi_{\operatorname{hol}})$

for some $\phi_{\operatorname{hol}}\in Q(X/B)$. The resulting quotient space
is in bijection with the set

$\displaystyle\\{(t,P):~{}$ $\displaystyle t\in B\textmd{ and $P$ is a
meromorphic projective structure on $C_{t}$}$ $\displaystyle\textmd{having
poles of order at most $n_{i}$ at the points $p_{i}(t)$ and no other
pole}\\}.$

The space $\mathcal{Q}(X/B;\mathcal{D})_{t}$ acts transitively and freely on
equivalence classes by
$(\tilde{\phi},t)\cdot[(t,P),\phi]=[(t,P),\phi+\tilde{\phi}]$.

A _family of meromorphic projective structures_ on $f$ relative to the divisor
$\mathcal{D}$ is defined to be a holomorphic section of this affine bundle
$\mathcal{P}(X/B;\mathcal{D})\rightarrow B$. It induces a relative projective
structure on the restriction of $f$ to $X\smallsetminus\mathcal{D}$ [AB20,
Sec. 7.3].

The subset corresponding to projective structures with poles of orders exactly
$(n_{i})$ forms a dense open set

$\mathcal{P}(X/B;\mathcal{D},(n_{i}))\subset\mathcal{P}(X/B;\mathcal{D})$

(see [AB20, Lem. 8.1]).

As usual, denote by $S$ a smooth oriented compact real surface of genus $g$.
If $B=\mathcal{T}(S,d)$ is the Teichmüller space of marked curves (i.e. marked
Riemann surfaces) on $S$ equipped with $d$ marked points, and if $f$ is the
universal holomorphic family of marked curves, we chose the sections $p_{i}$
such that for $t\in\mathcal{T}(S,d)$, the $p_{i}(t)$ correspond to marked
points on $C_{t}$.

#### Marked meromorphic projective structures.

Take a meromorphic projective structure $P$ on $C$ with pole orders $(n_{i})$,
and perform a real oriented blow-up101010Locally, the blow down is:
$\mathbb{R}_{\geq 0}\times S^{1}\longrightarrow\mathbb{R}^{2}\text{,
}(r,\theta)\longmapsto(r\operatorname{cos}(\theta),r\operatorname{sin}(\theta))$.
at each irregular singularity. Consider one of them, say $p_{i}$. Then, the
$n_{i}-2$ corresponding _asymptotic horizontal directions_ (see [AB20]) of any
polar differential of $P$ at $p_{i}$ define $n_{i}-2$ points on the boundary
component of the resulting blown-up surface $\mathbb{S}_{P}$ lying over
$p_{i}$. Together with the set $\mathbb{P}_{P}$ of points in the interior of
$\mathbb{S}_{P}$ corresponding to regular singularities (the _punctures_),
they form the set $\mathbb{M}_{P}\subset\mathbb{S}_{P}$ of _marked points_
(cf. Figure 2).

A _marked bordered surface_ is defined to be a pair $(\mathbb{S},\mathbb{M})$
consisting of a compact, connected, oriented, smooth surface with boundary
$\mathbb{S}$ and a finite non-empty set $\mathbb{M}\subset\mathbb{S}$ of
marked points such that each boundary component of $\mathbb{S}$ contains at
least one marked point (we refer to the definition given in [AB20, Sec. 3.6]).
Let $\mathbb{S}^{\prime}$ denote the surface obtained by taking the real
oriented blow-up of $\mathbb{S}$ at each puncture: this replaces punctures
with boundary components containing no marked points. A marked bordered
surface $(\mathbb{S},\mathbb{M})$ is determined up to isomorphism by its genus
and a collection of non-negative integers $\\{k_{1},\dots,k_{d}\\}$ giving the
number of marked points on the $d$ boundary components of
$\mathbb{S}^{\prime}$. For example, $(\mathbb{S}_{P},\mathbb{M}_{P})$ is a
marked bordered surface determined by the genus of $P$ and the numbers

$k_{i}=\begin{cases*}0\text{ if }n_{i}\leq 2\\\ n_{i}-2\text{
otherwise.}\end{cases*}$ (8)

Figure 2: The surface $\mathbb{S}_{P}$ with marked points $\mathbb{M}_{P}$.

A _marking of the meromorphic projective structure $P$ by a marked bordered
surface $(\mathbb{S},\mathbb{M})$_ is defined to be an isotopy class (relative
to the boundaries and the marked points) of orientation preserving
$\mathcal{C}^{\infty}$-diffeomorphisms
$\theta:\mathbb{S}\rightarrow\mathbb{S}_{P}$ such that
$\theta(\mathbb{M})=\mathbb{M}_{P}$.111111The terminology is a little
confusing here: we just defined a ”marking” of $P$ by a ”marked” bordered
surface. The ”marking” of $P$ refers to $\theta$, whereas ”marked” bordered
surface is a surface $\mathbb{S}$ with marked points $\mathbb{M}$.

######  Definition 1.15.

A _marked meromorphic projective structure_ is a triple $(C,P,\theta)$, where
$C$ is a complex curve, $P$ is a meromorphic projective structure on $C$, and
$\theta$ is a marking of $P$ by a marked bordered surface
$(\mathbb{S},\mathbb{M})$.

Fix a marked bordered surface $(\mathbb{S},\mathbb{M})$. Two meromorphic
projective structures marked by $(\mathbb{S},\mathbb{M})$,
$(C_{i},P_{i},\theta_{i})$, $i=1,2$, are said to be _equivalent_ if there
exists a biholomorphism $C_{1}\rightarrow C_{2}$ that preserves the projective
structures and whose lift to the blown-up surfaces commutes with the markings.

#### The moduli space $\mathcal{P}(\mathbb{S},\mathbb{M})$.

Let $(\mathbb{S},\mathbb{M})$ be a fixed marked bordered surface determined by
a genus $g$ and a non-empty collection of non-negative integers
$\\{k_{1},\dots,k_{d}\\}$. From now on, suppose $B=\mathcal{T}(S,d)$. Let
$\mathcal{D}$ be the divisor on $X$ defined by

$\mathcal{D}:=\sum_{i=1}^{d}(k_{i}+2)\mathcal{D}_{i}.$

Thanks to [AB20, Lem. 8.1, Prop. 8.2], we know that there is a dense subset
$\mathcal{V}$ of $\mathcal{P}(X/B;\mathcal{D})$ and a covering map

$\displaystyle
M:\mathcal{P}(\mathbb{S},\mathbb{M})\rightarrow\mathcal{V}\subset\mathcal{P}(X/B;\mathcal{D})$
(9)

whose fibers over a point $[(t,P),\phi]$ is the set of markings of $P$ by
$(\mathbb{S},\mathbb{M})$ inducing (by blowing down the boundary component of
$\mathbb{S}$) the marking of $S$ corresponding to $t$. This covering space
$\mathcal{P}(\mathbb{S},\mathbb{M})$ is in bijection with the set of
equivalence classes of marked meromorphic projective structures marked by
$(\mathbb{S},\mathbb{M})$.

######  Remark 4.

The moduli space $\mathcal{P}(\mathbb{S},\mathbb{M})$ was introduced by
Allegretti and Bridgeland in [AB20]. We would like to emphasize that the
marked bordered surface $(\mathbb{S},\mathbb{M})$ does not always determine
the pole orders of the projective structures (the exact orders of regular
singularities are not specified). Thus, if $p_{i}\in C$ corresponds to a point
in the interior of $\mathbb{S}$, the open subset $\mathcal{V}$ contains
projective structures with a simple pole at $p_{i}$ and others with a double
pole at $p_{i}$. Projective structures with simple poles appear in the
boundary of spaces of projective structures with double poles.

This is natural for Allegretti and Bridgeland because of the result of
Bridgeland and Smith [BS15] which identifies the corresponding unions of
components of moduli spaces of meromorphic quadratic differentials with spaces
of stability conditions.

However, in this paper we will soon fix the residues of the meromorphic
projective structures (see 3.10), hence the orders of regular poles as well.

Allegretti and Bridgeland showed the following proposition.

######  Proposition 1.16 (Proposition 8.2 of [AB20]).

Assume that if $g=0$, then $|\mathbb{M}|\geq 3$. Then
$\mathcal{P}(\mathbb{S},\mathbb{M})$ is a complex manifold of dimension

$6g-6+\sum_{i=1}^{d}(k_{i}+3).$

According to the relation (8), special cases satisfying $g=0$ and
$|\mathbb{M}|\leq 2$ correspond to non-singular projective structures and
cases when $(n_{i})$ is equal, up to a permutation of entries, to $(1)$,
$(2)$, $(3)$, $(4)$, $(1,1)$, $(1,2)$, $(2,2)$, $(1,3)$, $(2,3)$ and $(3,3)$
(about those special cases, see [AB20, Sec. 6.4]).

#### Submanifolds of $\mathcal{P}(\mathbb{S},\mathbb{M})$.

Let us denote by $\mathbb{P}$ the set of marked points points in the interior
of $\mathbb{S}$ (i.e. those corresponding to regular singularities). There is
a map

$R:\mathcal{P}(\mathbb{S},\mathbb{M})\longrightarrow\mathbb{C}^{\mathbb{P}}$

sending a class of marked meromorphic projective structures to the
corresponding residue at each regular singularity. This map is a holomorphic
submersion [AB20, Prop. 8.4]. We introduce the following submanifolds of
$\mathcal{P}(\mathbb{S},\mathbb{M})$. First, assigning the values
$(\mu_{i})\in\mathbb{C}^{\mathbb{P}}$ for the residues leads to

$\mathcal{P}(\mathbb{S},\mathbb{M},(\mu_{i})):=R^{-1}((\mu_{i})_{i}).$

Second, if
$Z=\cup_{\theta\in\mathbb{N}}\\{z\in\mathbb{C}:z-(1-\theta^{2})/2=0\\}$ (which
is a discrete set of points of the real line in $\mathbb{C}$), then

$\mathcal{P}^{\bullet}(\mathbb{S},\mathbb{M}):=R^{-1}(\mathbb{C}^{\mathbb{P}}\smallsetminus
Z^{\mathbb{P}}),$

is a dense open set in $\mathcal{P}(\mathbb{S},\mathbb{M})$ corresponding to
projective structures having non-trivial and non-parabolic local monodromies
around regular singular points. Third, there is an open subset

$\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M})$

in $\mathcal{P}(\mathbb{S},\mathbb{M})$ corresponding to marked projective
structures without apparent singularity [AB20, Lem. 8.3] (it contains
$\mathcal{P}^{\bullet}(\mathbb{S},\mathbb{M})$).

Latter, we will introduce a new notion of "residue" which apply not only to
regular singularities but also to irregular ones, and we will consider the
submanifolds of $\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M})$ defined by fixing
those residues (cf. 3.10).

Let us mention the work of Billon [Bil22], who constructed smooth moduli
spaces of branched projective structures, i.e. meromorphic projective
structures having only apparent singularities.

##  2 A smooth family of opers

We have seen already that a complex projective structure may be described in
several ways: as a projective atlas, as a development-holonomy pair or as a
projective connection. Yet another viewpoint turns out to be the most
appropriate to develop our arguments: _opers_. A
_$\operatorname{PGL}(2,\mathbb{C})$ -opers_ is a Riccati foliation of a
$\mathbb{P}^{1}$-bundle over a curve, equipped with a transverse section. In
the meromorphic case, this leads a priori to consider singular Riccati
foliations of a $\mathbb{P}^{1}$-bundle, equipped with a section that might be
tangent or might go through singularities of the foliation. However, in
contrast with the holomorphic case, it is not uniquely determined by the
projective structure: there can be different birational models [Bru15]. As
observed by Loray and Pereira [LP07], there is however a unique birational
model minimizing the polar divisor and such that the section does not
intersect the singular set of the foliation. In the second section, we deduce
that if the projective structure has no apparent singularity, then the above
ambiguity can be resolved, so that there is a bijection between the set of
meromorphic projective structures without apparent singularity on a curve $C$
and the set of meromorphic $\operatorname{PGL}(2,\mathbb{C})$-oper on $C$ with
minimal polar divisor. Finally, in the third section, we construct a smooth
family of such opers.

###  2.1 Opers for the group $\operatorname{PGL}(2,\mathbb{C})$ over a smooth
curve $C$

Let us define opers, first in the case without poles.

######  Definition 2.1.

An _oper for the group $\operatorname{PGL}(2,\mathbb{C})$ on $C$_ (or
$\operatorname{PGL}(2,\mathbb{C})$-oper on $C$) is defined to be a triple
$(\pi:Q\rightarrow C,\mathcal{F},\sigma)$, where

  * •

$\pi:Q\rightarrow C$ is a holomorphic $\mathbb{P}^{1}$-bundle over C,

  * •

$\mathcal{F}$ is a regular holomorphic foliation on $Q$ transverse to the
fibers of $\pi$ (i.e. a Riccati foliation, cf. below),

  * •

$\sigma:C\rightarrow Q$ is a holomorphic section of the bundle, transverse to
the foliation $\mathcal{F}$.

$C$
---
$Q$
---
$\sigma$
---

$\longrightarrow$
---

$\mathcal{F}$
---
$\pi$
---
Figure 3: A $\operatorname{PGL}(2,\mathbb{C})$-oper on $C$.

This definition makes sense for non-compact curves $C$ as well. This
terminology was introduced by Beilinson and Drinfeld [BD05] (see also [Fre07])
and is a particular case of a much more general class of objects, including
opers for groups such as $\operatorname{GL}(2,\mathbb{C})$ or
$\operatorname{SL}(2,\mathbb{C})$. Another formulation of the above definition
will be given later, in terms of equivalence classes of opers for the group
$\operatorname{GL}(2,\mathbb{C})$ (cf. 3.7).

#### Opers and projective structures.

Let $C$ be a smooth curve, and $P$ be a projective structure on $C$, with its
projective atlas denoted by $(U_{i},\varphi_{i})_{i}$.

Then, the coordinate changes
$\varphi_{ij}:=\varphi_{i}\circ\varphi_{j}^{-1}:\varphi_{j}(U_{i}\cap
U_{j})\rightarrow\varphi_{i}(U_{i}\cap U_{j})$ are restrictions of Möbius
transformations and satisfy, on $\varphi_{j}(U_{i}\cap U_{k}\cap U_{j})$, the
cocycle relation $\varphi_{ik}\circ\varphi_{kj}=\varphi_{ij}$. Whence, we can
define a map $g_{ij}$ with value in $\operatorname{PGL}(2,\mathbb{C})$: to
$x\in U_{i}\cap U_{j}$ we associate the automorphism $g_{ij}(x)$ of
$\mathbb{P}^{1}$ whose restriction equals $\varphi_{ij}$. The $g_{ij}$ also
satisfy a cocycle relation, hence they define a $\mathbb{P}^{1}$-bundle

$\pi:Q=\sqcup_{i}(U_{i}\times\mathbb{P}^{1})/((j,x,y)\sim(i,x,g_{ij}(x)\cdot
y))\longrightarrow C.$

By construction, the $g_{ij}$ are locally constant thus the transition
functions of the bundle $\pi$ do not depend on the variable $x$ and it makes
sense to speak of the horizontal foliation on a trivial chart of $\pi$. This
foliation descends to a foliation $\mathcal{F}$ on $P$ transverse to the
fibers of $\pi$.

Furthermore, the graph of each projective chart $\varphi_{i}$ defines a local
section on $U_{i}\times\mathbb{P}^{1}$. The open cover and the gluing data of
the bundle $\pi$ correspond exactly to those of the projective atlas of $P$,
so that projective charts define a global section $\sigma$ of $\pi$. This
section is transverse to the foliation $\mathcal{F}$ because the derivatives
of the $\varphi_{i}$ never vanish.

$U_{i}$
---
$\varphi_{i}$
---

$\longrightarrow$
---

$\longrightarrow$
---

$U_{i}\times\mathbb{P}^{1}$
---
$U_{j}$
---
$\varphi_{j}$
---
$U_{j}\times\mathbb{P}^{1}$
---
$\longmapsto$
---
$(x,g_{ji}(x)\cdot y)$
---
Figure 4: The gluing construction.

Finally, we have constructed a $\operatorname{PGL}(2,\mathbb{C})$-oper
$(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ on $C$.

Let us describe the inverse construction. Given the triple $(\pi:Q\rightarrow
C,\mathcal{F},\sigma)$, we can recover the projective structure $P$ in the
following way. First, the flow box theorem ensures that on the trivialization
charts there exists a bundle isomorphism sending the Riccati foliation to the
horizontal one. (Concretely, this bundle isomorphism may be described using
the sharp $3$-transitivity of the action of the group of Möbius
transformations on $\mathbb{P}^{1}$: choose three distinct leaves and send
them to the constant sections $\\{y=0\\}$, $\\{y=1\\}$ and $\\{y=\infty\\}$).
Then, restricted to the trivializing open sets, the graph of the section
$\sigma$ defines a projective charts. Finally, the compatibility between
charts follows from the gluing data of the bundle. (This projective structure
induces the complex structure $C$ because the section is analytic).

Let us formulate this correspondence as a proposition.

######  Proposition 2.2.

The above correspondence defines a bijection between the set of projective
structures on $C$ and the set $\operatorname{PGL}(2,\mathbb{C})$-opers on $C$.

Note that the conjugacy class of the monodromy representations of
$\mathcal{F}$ and the associated projective structure are the same, because
transition maps of the projective atlas are the same as cocycles of the
bundle.

The isomorphism class of the couple $(\pi:Q\rightarrow C,\mathcal{F})$ can be
recovered from the data of the complex curve $C$ together with its monodromy
representation up to conjugacy. This is the classical Riemann-Hilbert
correspondence, which in this context amounts to the suspension of the
monodromy representation.

In a way analogous to 2.2, meromorphic projective structures correspond to
_meromorphic opers_. The rest of the present section is devoted to the
definition of those objects and to the clarification of their relation to
meromorphic projective structures.

#### Singular Riccati foliations.

Let $\pi:Q\rightarrow C$ be a holomorphic $\mathbb{P}^{1}$-bundle. A singular
holomorphic foliation $\mathcal{F}$ on $Q$ is said to be _Riccati_ with
respect to $\pi$ if it is transverse to the generic fiber of $\pi$. Let
$(U,x)$ be a coordinate chart of $C$ such that the bundle is trivial over $U$,
i.e. $\pi^{-1}(U)\simeq U\times\mathbb{P}^{1}$, and choose a standard
coordinate $y$ on $\mathbb{P}^{1}$. Then, is these coordinates the foliation
$\mathcal{F}$ is described by a Pfaffian equation of the form

$d(x)dy+[a(x)y^{2}+b(x)y+c(x)]dx=0,$ (10)

where $a,b,c$ and $d$ are holomorphic functions and $d$ is non-identically
vanishing (a detailed study of Riccati foliations is exposed in [Bru15, Chap.
4.1]; see also [dSG16, Prop. VIII.1.1]). A fiber of $\pi$ over a point
corresponding to a zero of $d$ is invariant by $\mathcal{F}$, whereas fibers
over points where $d$ is non-zero are transverse to $\mathcal{F}$. The set of
singularities $\operatorname{sing}(\mathcal{F})$ of the foliation is contained
in the union of invariant fibers. If $F$ is an invariant fiber, then the set
$\operatorname{sing}(\mathcal{F})\cap F$ contains one or two elements since
they are obtained by solving a quadratic equation.

Dividing by $d$, we obtain a Riccati equation (hence the terminology):

$\mathcal{F}_{|U}:dy+y^{2}\alpha+y\beta+\gamma=0,$ (11)

where $\alpha,\beta$ and $\gamma$ are meromorphic $1$-forms on $U$. Note that
equation (11) also determines $\mathcal{F}$: the leaves are obtained as the
graphs of its solutions – this is possible only outside fibers corresponding
to the poles of $\alpha,\beta$ and $\gamma$, but then the resulting foliation
extends uniquely outside a codimension $1$ analytic subset (this is basic
foliation theory, see for instance [Lor12, Prop. 27] or [IY08, Thm. 2.20]).
Invariant fibers correspond exactly to the poles of $\alpha,\beta$ and
$\gamma$.

######  Definition 2.3.

To a singular Riccati foliation $(\pi:Q\rightarrow C,\mathcal{F})$, we can
associate a _polar divisor_ on $C$ defined as the direct image by $\pi$ of the
tangency divisor between $\mathcal{F}$ and the foliation induced by the fibers
of $\pi$.

A Möbius transformation fixes the point at infinity in the projective line if
and only if it is affine. Recall that the affine group is generated by
translations and dilatations by a non-zero complex number. Affine holomorphic
gauge transformations act on the Riccati equation as follow. A pullback by a
gauge transformation of the form $y=a(x)\tilde{y}$ transforms the Riccati
equation (11) into

$d\tilde{y}+a\tilde{y}^{2}\alpha+\tilde{y}\left(\beta+\frac{da}{a}\right)+\frac{\gamma}{a}=0$
(12)

whereas a pullback by a gauge transformation of the form $y=\tilde{y}+b(x)$
transforms it into

$d\tilde{y}+\tilde{y}^{2}\alpha+\tilde{y}(\beta+2b\alpha)+(b^{2}\alpha+db+b\beta+\gamma)=0.$
(13)

The following proposition gives the relation between Riccati foliations and
the quadratic differentials associated with projective structures.

######  Proposition 2.4 (see Prop. 2.1 in [LMP09]).

Let $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ be a
$\operatorname{PGL}(2,\mathbb{C})$-oper defining a projective structure $P$ on
a smooth curve $C$. Let $x$ be a coordinate chart of $C$ defined on an open
set $U$ over which the bundle $\pi$ is trivial121212In fact, it is trivial on
any proper open subset $U\subsetneq C$, because $\pi$ is the projectivization
of a rank $2$ vector bundle, which is trivial on any non-compact curve by
Grauert-Röhrl theorem.. Then

  * (i)

there exists a unique coordinate chart $y$ on
$\mathbb{C}_{y}\subset\mathbb{P}^{1}$ in which the triple takes the normal
form

$\begin{cases}\mathcal{F}_{|U}:dy+\left(y^{2}+\frac{q(x)}{2}\right)dx=0\textmd{
with $q\in\mathcal{O}(U)$}\\\ \sigma_{|U}:\\{y=\infty\\}.\end{cases}$ (14)

  * (ii)

Moreover, on $U$, the projective charts of the structure $P$ are exactly the
solutions of the Schwarzian equation $\mathcal{S}_{x}(\varphi)=q$.

Note that, after the substitution $y=-1/\tilde{y}$, we see that the same is
true for the form

$\begin{cases}\mathcal{F}_{|U}:d\tilde{y}+\left(\frac{q(x)}{2}\tilde{y}^{2}+1\right)dx=0\textmd{
with $q\in\mathcal{O}(U)$}\\\ \sigma_{|U}:\\{\tilde{y}=0\\}.\end{cases}$ (15)

which is the projectivization $\tilde{y}=\tilde{y}_{1}/\tilde{y}_{2}$ of the
companion system of equation (4),

$d\tilde{Y}+\begin{pmatrix}0&-1\\\ q(x)/2&0\end{pmatrix}\tilde{Y}dx=0,$ (16)

with the section $\sigma$ corresponding to the line subbundle
$L=\braket{{}^{t}(0,1)}$. See [AB20, Rk. 3.3(ii)] for the corresponding
computation directly on the rank $2$ connection side.

###### Proof of 2.4.

Let $U$ be an open set of $C$ such that the bundle $\pi$ is trivial over $U$.

_(i)_ By transitivity of Möbius transformations, there exist trivializing
coordinates $(x,y)$ on $\pi^{-1}(U)\simeq U\times\mathbb{P}^{1}$ in which
$\sigma_{|U}=\\{y=\infty\\}$. As above, the Pfaffian equation reads

$\mathcal{F}_{|U}:dy+y^{2}\alpha+y\beta+\gamma=0,$

where $\alpha,\beta$ and $\gamma$ are holomorphic $1$-forms on $U$, with
$\alpha$ non-vanishing (due to the transversality between the section and the
foliation). The action of affine holomorphic gauge transformations (i.e. those
fixing the section at infinity) on the Riccati equation is such that $\alpha$
and $\beta$ can be arbitrarily chosen, except $\alpha$ must remain non-
vanishing (see equations (12) and (13)). After such a choice of $\alpha$ and
$\beta$ is made, no non-trivial gauge transformation leaves them unchanged.
Hence $\gamma$ is uniquely determined by the projective structure $P$. Choose
$\alpha=dx$, $\beta=0$ and denote $\gamma=q/2dx$. (The factor $1/2$ is a
matter of convention: if it wasn’t written here, a factor $2$ would pop up in
the Schwarzian equation of point _(ii)_ anyway). In practice, one can first
normalize the quadratic coefficient to $1$ via $y=a(x)\tilde{y}$, then cancel
the linear coefficient via $y=\tilde{y}+b(x)$ (see equations (12) and (13)).

_(ii)_ Just as in the proof of 2.2, we can find coordinates $(x,y)$ on
$U\times\mathbb{P}^{1}$ in which the foliation $\mathcal{F}$ is given by
$y^{\prime}=0$ (i.e. horizontal) and the section $\sigma$ defines a projective
chart $\varphi=\sigma_{|U}$. After a gauge transformation sending the section
to infinity

$(x,y)\longmapsto(x,y_{1}):=\left(x,\frac{1}{y-\varphi}\right),$

the equation takes on the form $y_{1}^{\prime}-\varphi^{\prime}y_{1}^{2}=0$.
Next, we apply change of the $y_{1}$ coordinate preserving
$\\{y_{1}=\infty\\}$: those are exactly the elements of the affine group.
First, the transformation

$(x,y_{1})\longmapsto(x,y_{2}):=\left(x,-\varphi^{\prime}y_{1}\right)$

followed by a multiplication of the equation by $-\varphi^{\prime}$ normalizes
the coefficient of $y_{2}^{2}$ to $1$ without perturbing the coefficient of
$y_{2}^{\prime}$. Second, the transformation

$(x,y_{2})\longmapsto(x,y_{3}):=\left(x,y_{2}-\frac{\varphi^{\prime\prime}}{2\varphi^{\prime}}\right)$

annihilates the coefficient of $y_{3}$ and we get

$y_{3}^{\prime}+y_{3}^{2}+\frac{\mathcal{S}_{x}(\varphi)}{2}=0.$

The uniqueness in _(i)_ yields $\mathcal{S}_{x}(\varphi)=q$.

Finally, two solutions of the Schwarzian equation differ from a Möbius
transformation (compare (1.1)), hence the conclusion. ∎

#### Meromorphic $\operatorname{PGL}(2,\mathbb{C})$-opers.

######  Definition 2.5.

A _meromorphic $\operatorname{PGL}(2,\mathbb{C})$-oper on $C$_ is a triple of
the form $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$, where

  * •

$\pi:Q\rightarrow C$ is a holomorphic $\mathbb{P}^{1}$-bundle over C,

  * •

$\mathcal{F}$ is a singular holomorphic foliation on $Q$ transverse to the
generic fibers of $\pi$ (a singular Riccati foliation on $Q$),

  * •

$\sigma:C\rightarrow Q$ is a holomorphic section of the bundle, transverse to
the foliation $\mathcal{F}$.

$C$
---
$\sigma$
---

$\longrightarrow$
---

$Q$
---
$\pi$
---
$\mathcal{F}$
---
Figure 5: A meromorphic $\operatorname{PGL}(2,\mathbb{C})$-oper on $C$.

The _polar divisor_ of a meromorphic $\operatorname{PGL}(2,\mathbb{C})$-oper
on $C$ is defined to be the one of the associated singular Riccati foliation
(cf. 2.3).

#### "Singular" triples $(\pi,\mathcal{F},\sigma)$ associated with a
meromorphic projective structure.

Let $P$ be a meromorphic projective structure on $C$ described by a pair
$(P_{0},\phi)$, following the notations of 1.10. Consider the trivial
$\mathbb{P}^{1}$-bundle over $C$, a standard coordinate $y$ on
$\mathbb{P}^{1}$, and the holomorphic section $\sigma=\\{y=\infty\\}$. Suppose
that, in the charts of a projective atlas $(U_{i},x_{i})$ of $P_{0}$, the
quadratic differential reads $\phi=\frac{q_{i}(x_{i})}{2}dx_{i}^{\otimes 2}$.
Define

$\mathcal{F}_{i}:dy+\left(y^{2}+\frac{q_{i}(x_{i})}{2}\right)dx_{i}=0.$ (17)

Since $x_{i}$ is a projective chart for $P_{0}$, those expressions glue
together to give a singular Riccati foliation $\mathcal{F}$ on the trivial
$\mathbb{P}^{1}$-bundle over $C$. We get a triple $(\pi,\mathcal{F},\sigma)$,
where $\pi$ is the trivial bundle, which by 2.4 _defines_ the projective
structure $P=P_{0}+\phi$ in the sense that we can recover a projective atlas
outside of the polar locus just as in 2.2.

Consider a triple $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$, where $\pi$ is a
holomorphic $\mathbb{P}^{1}$-bundle on $C$, $\mathcal{F}$ is a singular
Riccati foliation on $\pi$ and $\sigma$ is a holomorphic section of $\pi$
generically transverse to $\mathcal{F}$. We say that the triple _defines_ a
meromorphic projective structure $P$ on $C$ if we can recover a projective
atlas outside of the polar locus just as in 2.2.

However, in contrast with the latter situation, triples not holomorphically
equivalents may define the same meromorphic projective structure on $C$. The
object we are dealing with here is really _the triple $(\pi:Q\longrightarrow
C,\mathcal{F},\sigma)$ modulo gauge transformations of the bundle that are
bimeromorphic and such that their indeterminacy locus are contained in the
fibers above poles of the projective structure_.

######  Proposition 2.6.

2.4 holds for meromophic projective structures, if we allow bimeromorphic
gauge transformations and replace $q\in\mathcal{O}(U)$ by
$q\in\mathcal{M}(U)$.

Note that, when the triple is locally expressed in a form (14), but with $q$
possibly having a pole at $x=0$, we see by the change of variable $Y=1/y$ that
$q$ admits a pole of order at least $1$ if and only if
$\sigma:=\\{y=\infty\\}$ intersects the singular set of $\mathcal{F}$. (This
is the unique singular point on the invariant fiber). Hence the section
$\sigma$ is merely _generically_ transverse to the foliation $\mathcal{F}$: it
is transverse above points in $C^{*}$, and it goes through singularities of
$\mathcal{F}$ above poles of $P$. Thus, we are not dealing with an oper. We
can reduce to the situation where the section is _everywhere_ transverse to
the foliation by applying bimeromorphic gauge transformations of the bundle,
though. It is the object of 2.8 in the next section.

###  2.2 A unique meromorphic oper associated with a meromorphic projective
structure without apparent singularity: the minimal birational model

######  Definition 2.7.

Let $\pi:Q\rightarrow C$ be a $\mathbb{P}^{1}$-bundle and $q\in Q$. The blow
up of $q$, composed with the contraction (which is possible thanks to
Castelnuovo’s criterion) of the proper (or strict) transform of the fiber
$\pi^{-1}(\pi(q))$ is called the _elementary transformation centered at $q$_
and denoted by $\operatorname{elem}_{q}$.

An elementary transformation is a birational transformation, and its inverse
is an elementary transformation. The image of a $\mathbb{P}^{1}$-bundle by an
elementary transformation is again a holomorphic $\mathbb{P}^{1}$-bundle.
Every $\mathbb{P}^{1}$-bundle over $C$ can be obtained from the trivial bundle
$C\times\mathbb{P}^{1}\rightarrow C$ by some successive elementary
transformations (see [IS96, p. 229]). All birational transformations between
$\mathbb{P}^{1}$-bundles over $C$ are obtained by composing elementary
transformations.

For instance, consider the trivial bundle
$\mathbb{D}\times\mathbb{P}^{1}\rightarrow\mathbb{D}$, with coordinate $x$ on
the disc, homogeneous coordinates $[y_{1}:y_{2}]$ on the projective line.
Then,
$\operatorname{elem}_{(0,[y_{1}:0])}(x,[y_{1},y_{2}])=(x,[xy_{1},y_{2}])$.

Let $\sigma$ be a section of the $\mathbb{P}^{1}$-bundle $\pi$. After the
elementary transformation $\operatorname{elem}_{q}$, the self-intersection
number of the strict transform $\tilde{\sigma}$ of $\sigma$ satisfies

$\tilde{\sigma}\cdot\tilde{\sigma}=\begin{cases}\sigma\cdot\sigma+1\text{ if
}q\notin\sigma\\\ \sigma\cdot\sigma-1\text{ if }q\in\sigma.\end{cases}$

######  Proposition 2.8.

Let $P$ be a meromorphic projective structure on $C$, and let
$(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ be a triple defining it. There
exists a $\operatorname{PGL}(2,\mathbb{C})$-oper
$(\tilde{\pi}:\tilde{Q}\rightarrow C,\tilde{\mathcal{F}},\tilde{\sigma})$
defining $P$ where $\tilde{\pi}$ is bimeromorphically equivalent to $\pi$ via
elementary gauge transformations sending $\mathcal{F}$ to
$\tilde{\mathcal{F}}$ and $\sigma$ to $\tilde{\sigma}$ (the point here is that
$\tilde{\sigma}$ is _everywhere_ transverse to $\tilde{\mathcal{F}}$).

###### Proof.

Suppose $P$ admits at least one pole, otherwise there is nothing to prove.
Start by applying 2.6 for a local coordinate $x$ centered at a pole $p$ of $P$
of order $n\in\mathbb{N}^{*}$. Sticking to the same notations, we get

$\mathcal{F}_{|U}:dy+\left(y^{2}+\frac{\tilde{q}(x)}{2x^{n}}\right)dx=0\textmd{
with $\tilde{q}:=x^{n}q\in\mathcal{O}(U)$ and $\tilde{q}(0)\neq 0$.}$

As we have already noticed, the only point where $\sigma$ goes through a
singularity of the foliation $\mathcal{F}_{|U}$ is $(x,y)=(0,\infty)$. The
idea is to apply an appropriate number of elementary transformations centered
at this point.

Apply first the gauge transformation $Y=1/y$, then multiply by $-Y^{2}$ before
performing $l\in\mathbb{N}$ elementary gauge transformations
$Y=x^{l}\tilde{Y}$. After a division by $x^{n}$, we get the Pfaffian equation

$\tilde{\mathcal{F}}_{|U}:x^{l}d\tilde{Y}+\left[-x^{2l-n}\frac{\tilde{q}(x)}{2}\tilde{Y}^{2}+lx^{l-1}\tilde{Y}-1\right]dx=0,$

so that fixing $l=(n+1)/2$ or $n/2$ depending if $n$ is odd or even,
respectively, we get $-dx=0$ at $(x,\tilde{Y})=(0,0)$. This $1$-form is both
non-zero and transverse to the pull-back $\tilde{\sigma}:=\\{\tilde{Y}=0\\}$
of the section $\sigma$.

Applying this kind of transformations for every poles of $P$ leads to the
desired oper $(\tilde{\pi}:\tilde{Q}\rightarrow
C,\tilde{\mathcal{F}},\tilde{\sigma})$. The projective structure associated to
this new triple is the same as that of the initial one, because elementary
transformations are biholomorphic outside of the fibers containing the points
at which they are centered. ∎

or
---
$U$
---
$\sigma=\\{y=\infty\\}$
---

$\longrightarrow$
---

$\dashrightarrow$
---

$\longrightarrow$
---

$\\{y=0\\}$
---
$x=0$
---
$x=0$
---
$U\times\mathbb{P}^{1}$
---

$\longrightarrow$
---

$x=0$
---
$n$ is odd
---
$n$ is even
---
Figure 6: 2.8.

An oper defining a meromorphic projective structure (with at least one pole)
is not unique modulo bimeromorphic gauge transformation. If we add a condition
on the order of tangency between the foliation $\mathcal{F}$ and the fibers of
$\pi$, though, we get a uniqueness statement.

Just as the transversality between the section and the foliation, the order of
tangency between the foliation and the fibers can be changed by a
bimeromorphic gauge transformation.

######  Remark 5.

Consider a singular Riccati foliation $(\pi:Q\rightarrow C,\mathcal{F})$, and
assume it has an invariant fiber. Chose a coordinate system $(x,y)$ on a local
trivialization of the bundle such that the set $\\{x=0\\}$ corresponds to an
invariant fiber. By $3$-transitivity of the action of Möbius transformations
on the projective line, we can assume without loss of generality that $(0,0)$
is a singular point of the foliation and that $(0,\infty)$ is not. Then, there
exists an integer $m\in\mathbb{N}^{*}$ and holomorphic functions $a,b$ and $c$
such that the Pfaffian equation defining locally the foliation reads

$x^{m}dy+(a(x)y^{2}+b(x)y+c(x))dx=0,$

with $a(0)\neq 0$ and $c(0)=0$. By a holomorphic gauge transformation, we can
normalize $a$ to $1$ while preserving $\\{y=0\\}$ and $\\{y=\infty\\}$ (cf.
equation (12)). This changes $b$ and $c$ to, say, $\tilde{b}$ and $\tilde{c}$,
respectively. After the meromorphic transformation
$(x,y):=(x,x^{k}\tilde{y})$, $k\in\mathbb{Z}$, we get

$x^{m+k}d\tilde{y}+[x^{2k}\tilde{y}^{2}+(kx^{m+k-1}+\tilde{b}(x)x^{k})y+\tilde{c}(x)]dx=0,$

with $\tilde{b},\tilde{c}$ holomorphic ($\tilde{c}(0)=0$). If $k>0$, this
transformation (which also preserves $\\{y=0\\}$ and $\\{y=\infty\\}$ away
from $\\{x=0\\}$) corresponds to an elementary transformations centered at
$(0,\infty)$, and we see from the latter equation that the order of vanishing
of the coefficient before $d\tilde{y}$ at $0$ has either decreased or remained
equal to $m$ after $k=1$ elementary transformation (depending on the order of
vanishing of $\tilde{b}$ and $\tilde{c}$, and after division by the
appropriate power of $x$). If $k<0$, we are applying elementary
transformations centered at $(0,0)$. If $k=-1$ we see (after a multiplication
by $x^{2}$) that the pole order of the Riccati equation has increased from $m$
to $m+1$.

This proves that _an elementary transformation centered _at a singular point_
of the foliation decreases or leaves unchanged the order of the pole of the
Riccati equation, whereas an elementary transformation centered _outside_ of
the singular locus of the foliation on an invariant fiber always increase the
order of the pole of the Riccati equation._

A birational model of $(\pi:Q\rightarrow C,\mathcal{F})$ minimizing the polar
divisor of the foliation is not unique (see [Bru15]).

######  Proposition 2.9.

Let $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ be a triple defining a
meromorphic projective structure on $C$.

  * (i)

Suppose $\mathcal{F}$ reads locally around a singularity centered at $x=0$ as
in 2.6, with $q$ having a pole of order $n\in\mathbb{N}^{*}$. Then, applying
elementary gauge transformations centered at a point in $\\{x=0\\}$ we can get
a $1$-form defining the foliation having poles of order
$\lceil\frac{n}{2}\rceil$ (where $\lceil\cdot\rceil$ denotes the ceiling
function). This is the minimum order that can be achieved by applying
meromorphic gauge transformations.

  * (ii)

Suppose $\mathcal{F}$ is defined locally in a coordinate $x$ by a $1$-form
admitting a pole of order $m$ at $x=0$, then $q$ (2.6) has a pole of order at
most $2m$.

###### Proof.

_(i)_ Recall that any birational gauge transformation is the composition of
elementary transformations. Consider the normal form (14). The only
singularity of the foliation $\mathcal{F}$ being $(x,y)=(0,\infty)$, the idea
(cf. 5) is to apply an elementary transformation centered at this point, and
repeat the process as long as it is possible, say $k\in\mathbb{N}$ times.
Pulling-back the $1$-form via $(x,\tilde{y})\mapsto(x,y):=(x,x^{-k}\tilde{y})$
leads to

$2x^{n-k}d\tilde{y}+\left[2x^{n-2k}\tilde{y}^{2}-kx^{n-1-k}\tilde{y}+\tilde{q}(x)\right]dx=0,$

with $\tilde{q}(x):=x^{n}q(x)\in\mathcal{O}(U)$ and $\tilde{q}(0)\neq 0$.
Whence the result.

_(ii)_ This can be seen by carrying out the computations in the proof of
2.4(i) but with $a$ and $b$ meromorphic and possibly admitting a pole at
$x=0$. ∎

In fact, as observed by Loray and Pereira in [LP07, Sec. 3.2]:

######  Proposition 2.10.

Let $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ be a singular Riccati foliation
$\mathcal{F}$, together with a section $\sigma$ generically transverse to
$\mathcal{F}$. There is a unique birational model minimizing the polar divisor
such that the section does not intersect the singular set of the foliation.

If we further assume that the corresponding projective structure has no
apparent singularity, then by 2.12 (bellow) the section of the above minimal
model cannot be tangent to the foliation, hence it defines an oper. In
conclusion, we get a correspondence similar to 2.2 for meromorphic projective
structures without apparent singularity on the smooth curve $C$.

######  Proposition 2.11.

The above correspondence defines a bijection between the set of meromorphic
projective structures without apparent singularity on $C$ and the set of
meromorphic $\operatorname{PGL}(2,\mathbb{C})$-oper $(\pi:Q\rightarrow
C,\mathcal{F},\sigma)$ such that the polar divisor of $\mathcal{F}$ is
minimal.

#### Apparent singularities, branch points and tangency between the section
and the foliation.

2.9 allows us to characterize apparent singularities and give there relation
to branch points (see also [dSG16, Prop. IX.1.2]).

######  Proposition 2.12.

Let $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ be a
$\operatorname{PGL}(2,\mathbb{C})$-oper. Suppose the associated projective
structure admits an apparent singularity at a point $p\in C$. Then, there
exist elementary gauge transformations desingularizing the foliation over $p$,
while possibly introducing a tangency point between $\sigma$ and
$\mathcal{F}$. The converse is true.

###### Proof.

Suppose first that the section $\sigma$ is tangent to $\mathcal{F}$ at some
point in the fiber over a point $p\in C$ along which the foliation is regular.
Then, there are holomorphic functions $a,b$ and an integer
$k\in\mathbb{N}^{*}$ such that, in a bundle chart $U_{x}\times\mathbb{C}_{y}$
such that $p=0$ and $\sigma=\\{y=0\\}$,

$\mathcal{F}_{|U}:dy+[a(x)y^{2}+b(x)y+x^{k-1}]dx=0.$

Here, $\operatorname{tang}(\mathcal{F},\sigma)=k-1$. The composition

$(x,y)\longmapsto(x,\tilde{y}):=(x,y/x^{k})$

of $k$ elementary gauge transformations (the first one being centered at
$(x,y)=(0,0)$) leads to

$d\tilde{y}+\left[a(x)x^{k}\tilde{y}^{2}+\left(\frac{k}{x}+b(x)\right)\tilde{y}+\frac{1}{x}\right]dx=0.$

We see, after multiplication by $x$, that the section is now transverse to the
foliation around $x=0$. Moreover, a calculation similar to the proof of 2.4
shows that the projective structure has a pole of order at most $2$ at $x=0$.
But the local monodromy is trivial since we started with a non-singular
foliation.

Conversely: Thanks to the 2.9, the foliation may be described by a $1$-form
with a pole of order at most $1$. This falls into the non-degenerate case of
[Bru15, Prop. 4.2]. Furthermore, the monodromy being trivial, we can perform
(see [Bru15, page 44 and 45]) a change of variable (not necessarily a gauge
transformation preserving $\sigma$) to get, for some $n\in\mathbb{Z}$

$dy-\frac{ny}{x}dx=0.$

Then, a composition of elementary transformations (the first one being
centered at $(x,y)=(0,0)$) leads to

$dy=0,$

which is non-singular. Finally, the corresponding section may be tangent to
the foliation. We can’t ensure the transversality.

Other types of singularities (in [Bru15]) cannot be desingularized. Indeed,
there monodromy are non-trivial. Hence, apparent singularities are exactly
those that can be desingularized. ∎

Around those points, the projective charts take the form $\varphi(x)=x^{k-1}$
up to a change of coordinate: they are branch points.

###  2.3 Construction of a family of meromorphic
$\operatorname{PGL}(2,\mathbb{C})$-opers

We use de notations of section 1.2.

######  Lemma 2.13.

For all point $\tau_{0}\in\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(n_{i}))$,
there exists an open neighborhood $U$ of $\tau_{0}$ and a holomorphic family
of curves $F:\mathcal{C}\rightarrow U$, together with

  * •

a holomorphic $\mathbb{P}^{1}$-bundle $\Pi:\mathcal{Q}\rightarrow\mathcal{C}$,

  * •

a dimension $1$ singular foliation $\mathcal{F}$ on $\mathcal{Q}$,

  * •

and a holomorphic section $\mathfrak{S}$ of $\Pi$

such that $F\circ\Pi$ is a first integral of $\mathcal{F}$, and in restriction
to any fiber $C_{\tau}:=F^{-1}(\tau)$ of the family $F$, the triple
$(\Pi_{|C_{\tau}},\mathcal{F}_{|Q_{\tau}},\mathfrak{S}_{|C_{\tau}})$ is the
unique $\operatorname{PGL}(2,\mathbb{C})$-oper with minimal polar divisor on
$C_{\tau}$ associated to meromorphic projective structures corresponding to
$\tau$.

###### Proof.

Recall from section 1.2 that $B$ denotes the Teichmüller space
$\mathcal{T}(S,d)$ and $f:X\rightarrow B$ denotes the universal family of
marked curves. Pick a point
$\tau_{0}\in\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(n_{i}))$ and denote by
$b_{0}$ its image in $B$. Replacing $B$ by an open ball, we know from 1.8 that
there exists a relative projective structure (without pole) $\alpha$ on
$f:X\rightarrow B$. It induces a holomorphic section $s_{0}$ of
$\mathcal{P}(X/B;\mathcal{D})\rightarrow B$. This section in turn induces a
bijection

$\displaystyle\Phi_{s_{0}}:\mathcal{P}(X/B;\mathcal{D})$
$\displaystyle\longrightarrow\mathcal{Q}(X/B;\mathcal{D})$
$\displaystyle[(t,P),\phi]$
$\displaystyle\longmapsto(t,(P+\phi)-(P_{s_{0}(t)}+\phi_{s_{0}(t)}))\text{ if
}s_{0}(t)=[(t,P_{s_{0}(t)}),\phi_{s_{0}(t)}].$

Then, define $U$ as the preimage of $B$ in
$\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(n_{i}))$ and form the fibred
product

$\mathcal{C}:=U\times_{B}X$

and define $F:\mathcal{C}\rightarrow U$ as the pullback of $f$.

$\textstyle{\mathcal{C}=U\times_{B}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M_{|U}}$$\textstyle{\mathcal{P}(X/B;\mathcal{D},(n_{i}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{P}(X/B;\mathcal{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{s_{0}}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}}$$\textstyle{\mathcal{Q}(X/B;\mathcal{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

For all $\tau\in U$, let us denote
$t=\operatorname{pr}_{1}\circ\Phi_{s_{0}}\circ M(\tau)$ and
$\phi_{\tau}:=\operatorname{pr}_{2}\circ\Phi_{s}\circ M(\tau)$
($\operatorname{pr}_{i}$ means the projection to the $i$-th factor). The
projective structure $P_{t}:=(P_{s_{0}(t)}+\phi_{s_{0}(t)})+\phi_{\tau}$ on
$C_{t}$ induced by the above relative projective structure is also a
projective structure $P_{\tau}$ on $C_{\tau}:=F^{-1}(\tau)$ (by construction
of the fibered product). This defines a holomorphic family of meromorphic
projective structure on $U$ (this is the tautological family: roughly
speaking, it is $\tau$ on the fiber over $\tau$).

Denote by $\tilde{D}$ and $\tilde{\alpha}$ the pullbacks of $\mathcal{D}$ and
$\alpha$ on $\mathcal{C}$, respectively. According to [AB20, Lem. 7.4], the
holomorphic family defined in the preceding paragraph induces a relative
projective structure without poles on the family of curves
$\mathcal{C}\smallsetminus\tilde{\mathcal{D}}\rightarrow U$ and subtracting
(the restriction of) $\tilde{\alpha}$ to it gives the holomorphic family of
relative quadratic differentials (see [AB20, Prop. 7.3]) corresponding to
$\phi_{\tau}$.

Now consider the trivial bundle
$\Pi_{0}:\mathcal{C}\times\mathbb{P}^{1}\rightarrow\mathcal{C}$. Chose
moreover a standard coordinate $y$ on $\mathbb{P}^{1}$ and consider the
section $\mathfrak{S}_{\infty}:=\\{y=\infty\\}$ of this trivial bundle. From
the pair $(P_{\tau},\phi_{\tau})$, we know (see Equation 17) how to construct
a triple $(\Pi_{0|C_{\tau}}:C_{\tau}\times\mathbb{P}^{1}\rightarrow
C_{\tau},\mathcal{F}_{\tau},\mathfrak{S}_{\infty|C_{\tau}})$ describing the
projective structure associated to $\tau$.

The polar divisor on $C_{\tau}$ of the Riccati foliation $\mathcal{F}_{\tau}$
is not necessarily minimal, and $\mathfrak{S}_{\infty|C_{\tau}}$ is not
necessarily everywhere transverse to $\mathcal{F}_{\tau}$. To get an oper with
minimal polar divisor, we apply elementary transformations along the locus of
the divisor $\mathcal{D}$ (this is a family version of Propositions 2.8, 2.9
and 2.10). Each component $\mathcal{D}_{i}$ is an hypersurface in
$\mathcal{C}$. An elementary transformation centered at
$\mathfrak{S}_{\infty}(\mathcal{D}_{i})$ can be described as follows: first
blow-up $\mathfrak{S}_{\infty}(\mathcal{D}_{i})$ and then contract the strict
transform of $\Pi_{0}^{-1}(\mathcal{D}_{i})$. We apply elementary
transformations until we get an oper above $C_{\tau}$ for all $\tau\in U$.

By construction, the resulting objects satisfy the requirements of the
statement. ∎

This lemma remains true if we replace
$\mathcal{P}^{\circ}(\mathbb{S},\mathbb{M},(n_{i}))$ by one of its
submanifolds.

##  3 A smooth family of singular Riccati foliations

Let $C$ be a smooth curve and $E\rightarrow C$ be a rank $k$ holomorphic
vector bundle, that we will frequently denote simply by $E$. Its sheaf of
holomorphic sections is a locally free sheaf of $\mathcal{O}_{C}$-modules of
rank $k$, that we denote by $E$ as well. The sheaf of
$\mathcal{O}_{C}$-modules associated with a divisor $D$ on $C$ is denoted by
$\mathcal{O}_{C}(D)$, and it is defined by setting

$U\longmapsto\mathcal{O}_{C}(D)(U):=\\{\text{$f:U\rightarrow\mathbb{C}$ $|$
$f$ is meromorphic and $\operatorname{div}(f)\geq-D$}\\}.$

We denote
$\Omega^{1}_{C}(D):=\Omega^{1}_{C}\otimes_{\mathcal{O}_{C}}\mathcal{O}_{C}(D)$
the sheaf of meromorphic $1$-forms with polar divisor "at most" $D$.

######  Definition 3.1.

A _meromorphic connection $\nabla$ on the holomorphic vector bundle $E$_ is a
$\mathbb{C}$-linear morphism of sheaf

$\nabla:E\longrightarrow E\otimes_{\mathcal{O}_{C}}\Omega^{1}_{C}(D)$

for some effective divisor $D$, satisfying the _Leibniz rule_ , i.e. for any
open set $U\subset C$, any function $f\in\mathcal{O}_{C}(U)$ and any local
section $s$ of $E$ over $U$, $\nabla(fs)=f\nabla s+s\otimes df$.

Note that because $C$ has dimension $1$, the connection is automatically flat.

Often, in the context of moduli spaces, it is the couple $(E,\nabla)$ that we
call a connection. We say that the connection $(E,\nabla)$ has rank $k$,
referring to the rank of the vector bundle $E$.

Consider for instance the rank $2$ case. Concretely, over an open set
$U\subset C$ where the bundle is trivial and for a given local frame
$(e_{1},e_{2})$, such a connection is represented by a matrix $1$-form
$\Omega=(\omega_{ij})_{ij}\in\operatorname{Mat}(2,\Omega^{1}_{C}(D)(U))$
called the _connection matrix_ , whose coefficients are defined by the
relations $\nabla e_{j}=\sum_{i=1}^{2}e_{i}\otimes\omega_{ij}$. Conversely,
given a matrix $1$-form $\Omega$ on $U$, we can define a connection $\nabla$
on the trivial bundle $U\times\mathbb{C}^{2}$ as follows. The image of a
section $s=\sum_{j=1}^{2}y_{j}e_{j}$ is of course defined by

$\nabla s:=\sum_{i=1}^{2}e_{i}\otimes\left(dY+\Omega Y\right)_{i},$

where $Y:=\begin{pmatrix}y_{1}\\\ y_{2}\end{pmatrix}$. Thus, we may abusively
write

$\nabla=d+\Omega$

if we remember that this expression depends on the local frame that we have
chosen. Whenever we pullback by a holomorphic _gauge transformation_
$\tilde{Y}\mapsto G\tilde{Y}$, the connection matrix becomes

$\tilde{\Omega}=G^{-1}\Omega G+G^{-1}dG.$

A collection of matrix $1$-forms – one for each local trivialization of $E$ –
satisfying this gluing property represents a unique connection on $E$. Note
that since $G^{-1}dG$ is holomorphic, $G$ conjugates the principal part of
$\Omega$ to the principal part of $\tilde{\Omega}$. The poles and pole orders
of the connection are defined to be those of its matrices. We can chose $D$ to
be the effective _polar divisor_ of $\nabla$, though strictly speaking our
definition does not require this.

The _Poincaré rank_ of $\nabla$ at a pole of order $m$ is defined to be $m-1$.
It might change after a meromorphic gauge transformation. The _minimal_
Poincaré rank in the equivalence class of the system up to meromorphic gauge
transformation is sometimes called the _true_ Poincaré rank.

######  Definition 3.2.

The polar divisor of a meromorphic connection $(E,\nabla)$ on $C$ is said to
be _minimal_ if the pole order at each pole cannot be decreased by a
meromorphic gauge transformation.

######  Definition 3.3.

We say that a connection $(E,\nabla)$ is _reducible_ if there exists an
_invariant subbundle_ , i.e. a subbundle $0\subsetneq L\subsetneq E$
satisfying $\nabla(L)\subset L\otimes\Omega^{1}_{C}(D)$. Otherwise, we say it
is _irreducible_.

#### Fuchs relation.

The _trace connection_ $(\operatorname{det}(E),\operatorname{tr}(\nabla))$ is
the connection induced by $(E,\nabla)$ on the determinant bundle of $E$. In
local trivialization coordinates, its connection matrices are the traces of
the matrices of $\nabla$. We say a connection is _trace-free_ if its trace is
isomorphic to the horizontal connection $d$ on the trivial line bundle
$\mathcal{O}_{C}$.

The following relation holds for arbitrary rank and is a consequence of the
residue formula (for a proof, see [IY08, Cor. 17.35]).

######  Proposition 3.4 (_Fuchs relation_).

If $\nabla$ is a meromorphic connection on a smooth vector bundle $E$ over a
smooth compact curve $C$, then

$\sum_{p\in
C}\operatorname{res}_{p}(\operatorname{tr}(\nabla))+\operatorname{deg}(E)=0,$
(18)

where the degree of $E$ is by definition the degree of its determinant bundle.

###  3.1 Riccati foliations and meromorphic rank $2$ connections

Let $(E,\nabla)$ be a rank $2$ meromorphic connection on $C$. Consider a non-
zero local section $s=\sum_{j=1}^{2}y_{j}e_{j}$ of $E$ that is flat for
$\nabla$, i.e. $\nabla s=0$. As usual, $(e_{1},e_{2})$ is a local frame of $E$
and $\Omega=(\omega_{ij})_{ij}$ denotes the connection matrix of $\nabla$ with
respect to this frame. Assume that $y_{2}\neq 0$ and denote $y=y_{1}/y_{2}$.
Computing the logarithmic derivative of $y$ leads to the Riccati equation

$dy+\omega_{2,1}y^{2}+(\omega_{2,2}-\omega_{1,1})y-\omega_{1,2}=0.$

This equation defines the projectivization
$\mathbb{P}(E,\nabla):=(\mathbb{P}(E),\mathbb{P}(\nabla))$ of the connection
$(E,\nabla)$, i.e. a Riccati foliation on the $\mathbb{P}^{1}$-bundle
$\mathbb{P}(E)$.

Thus, to a rank $2$ connection we can associate its trace and its
projectivization,

$(E,\nabla)\longmapsto\begin{cases*}(\det(E),\operatorname{tr}(\nabla))\\\
(\mathbb{P}(E),\mathbb{P}(\nabla)).\end{cases*}$

Now, can we always find a rank $2$ connection with prescribed trace and
projectivization? In other words, is it possible to _lift_ a given Riccati
foliation $(Q,\mathcal{F})$ to a rank $2$ meromorphic connection $(E,\nabla)$
with a given trace $(L,\nabla_{L})$? If it exists, such a lift is not unique.
Indeed, $\mathbb{P}(E,\nabla)=\mathbb{P}(\tilde{E},\tilde{\nabla})$ if and
only if there exists a rank $1$ meromorphic connection
$(L^{\prime},\nabla_{L^{\prime}})$ such that
$(\tilde{E},\tilde{\nabla})=(E\otimes
L^{\prime},\nabla\otimes\nabla_{L^{\prime}})$. This can be seen by comparing
their cocycles and connection matrices: if the cocycles of $E$ and
$L^{\prime}$ with respect to a common open cover $(U_{i})$ of $C$ are
respectively $(G_{ij})$ and $(g_{ij})$, then $E\otimes L^{\prime}$ have
cocycles $G_{ij}\cdot g_{ij}$. If $\nabla=d+\Omega_{i}$ and
$\nabla_{L^{\prime}}=d+\omega_{i}$ with respect to some trivialization charts,
then $\nabla\otimes\nabla_{L^{\prime}}=d+(\Omega_{i}+\omega_{i}I)$. We call
this operation a _twist_ of $(E,\nabla)$ by the rank $1$ meromorphic
connection $(L^{\prime},\nabla_{L^{\prime}})$. For any rank $1$ meromorphic
connection $(L^{\prime},\nabla_{L^{\prime}})$,

$\begin{cases*}\operatorname{det}(E\otimes
L^{\prime})=\operatorname{det}(E)\otimes L^{\prime\otimes 2}\\\
\operatorname{tr}(\nabla\otimes\nabla_{L^{\prime}})=\operatorname{tr}(\nabla)\otimes\operatorname{tr}(\nabla_{L^{\prime}})^{\otimes
2}.\end{cases*}$

Consequently, such a lift $(E,\nabla)$ is unique up to a twist by a tensor
square root of the rank $1$ trivial connection $(\mathcal{O}_{C},d)$. Note
that this square root is necessarily a _holomorphic_ connection. If $g$ is the
genus of $C$, there are $2^{2g}$ such elements (indeed, they correspond via
the Riemann-Hilbert correspondence –which preserves the group structures– to
the $2^{2g}$ elements of $\operatorname{Hom}(\pi_{1}(C),\\{\pm 1\\})$), whence
at most $2^{2g}$ lifts.

Let us return to the question of the existence of such a lift. Every
$\mathbb{P}^{1}$-bundle on a curve is the projectivization of a rank $2$
vector bundle (see [Mar70, Intro.]), and this vector bundle is unique up to a
tensor product with a line bundle $L^{\prime}$. Let $E$ be a vector bundle
such that $\mathbb{P}(E)=Q$. We are looking for a line bundle $L^{\prime}$
such that $\det(E\otimes L^{\prime})=\det(E)\otimes\det(L^{\prime})^{\otimes
2}=L$, that is, a square root of $L\otimes\det(E)^{\otimes(-1)}$ in the Picard
group $\operatorname{Pic}(C)$. Such a square root exists if and only if the
degree of $L\otimes\det(E)^{\otimes(-1)}$ is even. As for the connection
$\nabla_{L}$ and the Riccati foliation $\mathcal{F}$, suppose they are given
locally over each open set of an open cover $(U_{i})$ as $d+\delta_{i}$ and
$dy+y^{2}\alpha_{i}+y\beta_{i}+\gamma_{i}=0$, respectively. Then, the
connection on $E\otimes L^{\prime}$ defined locally by

$dY+\begin{pmatrix}\frac{\delta_{i}-\beta_{i}}{2}&-\gamma_{i}\\\
\alpha_{i}&\frac{\delta_{i}+\beta_{i}}{2}\end{pmatrix}Y=0$

has trace $\nabla_{L}$ and projectivization $\mathcal{F}$. In conclusion, we
get the following proposition.

######  Proposition 3.5.

Let $(L,\nabla_{L})$ and $(Q,\mathcal{F})$ be a rank $1$ meromorphic
connection and a singular Riccati foliation on $C$, respectively. Let $E$ be a
rank $2$ vector bundle on $C$ such that $\mathbb{P}(E)=Q$. There exists a rank
$2$ meromorphic connection on $C$ with trace $(L,\nabla_{L})$ and
projectivization $(Q,\mathcal{F})$ if and only if

$\deg(L)-\deg(E)\equiv 0~{}\operatorname{mod}~{}2.$

This lift is unique up to a twist by a tensor square root of the rank $1$
trivial connection $(\mathcal{O}_{C},d)$.

######  Remark 6.

Since we are interested in projective structures, there is a priori no reason
for us to speak about linear connections. However, some results we will use
are formulated in the literature in the setting of linear connection.

###  3.2 Projective structures and opers for the group
$\operatorname{GL}(2,\mathbb{C})$

#### Some properties of $\mathbb{P}^{1}$-bundles.

Let us first recall some useful facts regarding $\mathbb{P}^{1}$-bundles and
their sections; proofs can be found in Maruyama’s book [Mar70, Chap. I].

######  Facts 3.6.

Let $E$ be a holomorphic rank $2$ vector bundle on a genus $g$ curve $C$, and
denote $Q=\mathbb{P}(E)$ its projectivization. Every $\mathbb{P}^{1}$-bundle
on $C$ arises in this way. To give a (holomorphic) section of $Q$ is
equivalent to give a line subbundle of $E$, and there always exists one.
Degrees of line subbundles of $E$ are bounded above; a line subbundle of $E$
is said to be _maximal_ if its degree is maximal. We denote the maximal degree
by $M(E)$.

Let $\sigma$ be a section of $Q$ and $L_{\sigma}$ be the corresponding line
subbundle of $E$. The self-intersection number of $\sigma$ is equal to the
degree of its normal bundle $N_{\sigma}$, which satisfies
$\sigma^{*}(N_{\sigma})=\operatorname{det}(E)\otimes
L_{\sigma}^{\otimes(-2)}$. Thus,

$\sigma\cdot\sigma=\deg(E)-2\deg(L_{\sigma}).$

Maximal line subbundles of $E$ are in bijective correspondence with _minimal_
sections of $Q$, i.e. with section with minimal self-intersection number
$N(E):=\deg(E)-2M(E)$. In fact, since this number depends only on $Q$, we also
denote it by $N(Q)$. It is at most $g$. The parity of $\sigma\cdot\sigma$, for
any section $\sigma$, is the same as the parity of $N(Q)$.

In fact $N(Q)\equiv\operatorname{deg}(E)$ mod $2$ and this parity also
determines the topological class of the $\mathbb{S}^{2}$-bundle subjacent to
$\mathbb{P}(E)$ (see [GM89], [Lor14], [LMP09, Sec. 2.1]).

If $\sigma$ is minimal and $\sigma\cdot\sigma=N(Q)<0$, then any other section
$\sigma^{\prime}$ has a self-intersection number
$\sigma^{\prime}\cdot\sigma^{\prime}\geq-N(Q)$ (there is a gap in the possible
values).

Finally, we have that

  1. 1.

if $N(Q)<0$, then there is a unique maximal line subbundle of $E$,

  2. 2.

if $N(Q)=0$ and $E$ is indecomposable, then again there is a unique maximal
line subbundle of $E$,

  3. 3.

if $N(Q)=0$, $E$ is decomposable but $E\not\simeq L\oplus L$ for any line
subbundle $L$ of $E$, then there are only two maximal line subbundles
$L_{1},L_{2}$ of $E$, and in fact $E=L_{1}\oplus L_{2}$,

  4. 4.

if $E=L\oplus L$ for some line subbundle $L$ of $E$, then $E$ has infinitely
many maximal line subbundles, all of which are isomorphic to $L$.

In fact, if $N(Q)=0$, $\deg(E)=0$ and $E$ is decomposable, we have $E=L\oplus
L^{-1}$ for some line subbundle $L$ and the two last cases correspond
respectively to $L^{\otimes 2}\not\simeq\mathcal{O}_{C}$ and $L^{\otimes
2}\simeq\mathcal{O}_{C}$.

######  Remark 7.

A line subbundle $L$ of $E$ is invariant by $\nabla$ if and only if the
corresponding section of $\mathbb{P}(E)$ is invariant by the foliation induced
by $\nabla$.

#### Opers for the group $\operatorname{GL}(2,\mathbb{C})$.

######  Definition 3.7.

A _(meromorphic) oper for the group $\operatorname{GL}(2,\mathbb{C})$ on a
complex curve $C$_ is a triple $(E,\nabla,L)$, where $E$ is a rank $2$
holomorphic vector bundle, $\nabla$ is a holomorphic (resp. meromorphic)
connection on $E$ and $L\subset E$ is a line subbundle satisfying a _non-
degeneracy_ (or _strictness_) _condition_ : the $\mathcal{O}_{C}$-linear map
$\varphi$ defined as the composite

$\displaystyle\varphi:L\xrightarrow{\iota}E\xrightarrow{\nabla}E\otimes\Omega^{1}_{C}\xrightarrow{q\otimes\operatorname{id}}(E/L)\otimes\Omega^{1}_{C}$
(19) $\displaystyle(\text{resp. }$
$\displaystyle\varphi:L\xrightarrow{\iota}E\xrightarrow{\nabla}E\otimes\Omega^{1}_{C}(D)\xrightarrow{q\otimes\operatorname{id}}(E/L)\otimes\Omega^{1}_{C}(D))$
(20)

(where $\iota$ is the inclusion, $q$ is the quotient map and $D$ is an
effective divisor on $C$) is an isomorphism.

A $\operatorname{PGL}(2,\mathbb{C})$-oper on $C$ may then be alternatively
defined as an equivalence class of $\operatorname{GL}(2,\mathbb{C})$-opers
under the following twist relation. Two
$\operatorname{PGL}(2,\mathbb{C})$-opers $(E_{1},\nabla_{1},L_{1})$ and
$(E_{2},\nabla_{2},L_{2})$ are considered to be equivalent if there is a rank
$1$ holomorphic connection $(L,\nabla_{L})$ and an isomorphism

$(E_{1},\nabla_{1})\simeq(E_{2},\nabla_{2})\otimes(L,\nabla_{L})$

sending $L_{1}$ to $L_{2}\otimes L$.

The non-degeneracy condition tells us that $L$ is nowhere invariant by
$\nabla$, which is exactly the transversality condition of the section
$\sigma$ induced by $L$ on $\mathbb{P}(E,\nabla)$. Another condition called
the _Griffiths transversality_ is asked in the general definition of a
$\operatorname{GL}(k,\mathbb{C})$-oper in greater dimensions $k\geq 2$, but we
don’t need to consider it here since it is automatically satisfied in rank
$k=2$.

######  Remark 8.

This alternative definition of a $\operatorname{PGL}(2,\mathbb{C})$-oper on
$C$ is equivalent to 2.5 (and 2.1, in the holomorphic case). Indeed, as we
will show in the following paragraphs, the Riccati foliation associated to a
$\operatorname{PGL}(2,\mathbb{C})$-oper must be the projectivization of a rank
$2$ linear connection. However, our definitions of meromorphic opers are a
little tighter than those suggested by Allegretti and Bridgeland since we
impose that the map $\varphi$ in (20) is an isomorphism (compare [AB20, Rmk
3.3(i)]).

#### A self-intersection formula.

Let $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ be a meromorphic
$\operatorname{PGL}(2,\mathbb{C})$-oper with the Riccati foliation having
polar divisor $D=\sum_{i=1}^{d}m_{i}p_{i}$ on the genus $g$ curve $C$. Because
the section $\sigma$ is transverse to $\mathcal{F}$, we have, according to
Brunella [Bru15, Prop. 2.2] (see also [Lor14, Prop. 25]),

$\sigma\cdot\sigma=2-2g-\operatorname{deg}(D).$ (21)

From this formula, we deduce in particular that the parity of $\deg(E)$, which
is the same as the parity of $\sigma\cdot\sigma$, is also equal to the parity
of $\deg(D)$. Another consequence is that

$\sigma\cdot\sigma\geq 0\Leftrightarrow\begin{cases*}g=0,~{}\deg(D)=0,1\text{
or }2\\\ g=1,~{}\deg(D)=0.\end{cases*}$ (22)

Thus, apart from those four special values for $(g,\deg(D))$, the self-
intersection number of $\sigma$ is negative. This implies that $\sigma$ is the
unique minimal section of $Q$ and the only one with negative self-intersection
number.

######  Remark 9.

Let us denote by $P$ the meromorphic projective structure described by the
oper $(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ and by $\mathbb{S}_{P}$ the
associated bordered surface with marked points $\mathbb{M}_{P}$. Assume, as in
1.16, that if $g=0$ then $|\mathbb{M}_{P}|\leq 3$. Then, by 2.9, this excludes
the above three special values when $g=0$ and $\deg(D)=0,1$ or $2$. The fourth
one, $(g,\deg(D))=(1,0)$, is not excluded but corresponds to a projective
structure without pole; our main theorem (4.3) is already known in this case
(cf. subsection 1.1) so we might exclude it as well.

#### Lift of a $\operatorname{PGL}(2,\mathbb{C})$-oper to a
$\operatorname{GL}(2,\mathbb{C})$-oper.

Suppose we want to _lift_ a $\operatorname{PGL}(2,\mathbb{C})$-oper
$(\pi:Q\rightarrow C,\mathcal{F},\sigma)$ with polar divisor $D$ to a
$\operatorname{GL}(2,\mathbb{C})$-oper on $C$. If $(E,\nabla)$ is a lift of
$(Q,\mathcal{F})$, then $\sigma$ corresponds to a unique line subbundle $L$ of
$E$ and by 7, $(E,\nabla,L)$ is a $\operatorname{GL}(2,\mathbb{C})$-oper on
$C$ whose equivalence class under the twist relation is $(\pi:Q\rightarrow
C,\mathcal{F},\sigma)$.

From the self-intersection formula (21) combined with 3.5, the possible
choices of trace connection for $(E,\nabla,L)$ depend on the parity of the
degree of the polar divisor $D$. If it is even, then so is $\deg(E)$ and
$(\pi:Q\rightarrow C,\mathcal{F})$ can be lifted to a rank $2$ connection with
determinant having any even degree; a trace-free rank $2$ connection, for
example. If $\deg(D)$ is odd, however, the determinant bundle must have an odd
degree. Here is an example of a valid choice for the trace connection. Note
that in this case, the $\operatorname{PGL}(2,\mathbb{C})$-oper has at least
one pole, let us choose one, say $p_{i}$. We would like to choose a trace
connection with a single simple pole at $p_{i}$, so that the corresponding
lifts still have polar divisor $D$ (and it is minimal if it was for
$(\pi:Q\rightarrow C,\mathcal{F})$). By the Fuchs relation, such a connection
must have residue $-1$ at $p_{i}$, hence trivial local monodromy around
$p_{i}$. By the Riemann-Hilbert correspondence, there exists a unique rank $1$
connection $(\mathcal{O}_{C}(p_{i}),\zeta)$ having a single pole of order $1$
at $p_{i}$ and a trivial global monodromy.

Recall that once we have made a choice of a trace connection, a rank $2$ lift
$(E,\nabla)$ is unique up to a twist by one of the $2^{2^{g}}$ tensor square
root of the rank $1$ trivial connection.

###  3.3 Formal data of rank $2$ connections and Riccati foliations

We denote by $\mathbb{C}[[x]]$ the ring of formal power series in the variable
$x$, $\mathbb{C}((x))=\mathbb{C}[[x]][x^{-1}]$ its quotient field (the field
of formal Laurent series in the variable $x$), and $\mathbb{C}\\{x\\}$ the
ring of convergent power series in the variable $x$ (those with non-zero
radius of convergence at $0$).

#### Local formal invariants.

Here, we work around a singular point, hence the concept of connection (or
formal connection) is unnecessary. We thus consider formal rank $2$ linear
systems in the complex variable $x$ around a possible pole at $x=0$.

To give a formal classification of such systems is to describe the orbits
under the action of formal holomorphic gauge transformations. The following
theorem tells us there are "good" representatives in each orbits, called
formal normal forms. These in turn provides formal invariants of a system.

######  Theorem 3.8 (Section 2 of [DL22]).

Let $dY+\Omega Y=0$ be a rank $2$ linear differential system with

$\Omega=\left(\frac{A_{-m}}{x^{m}}+\frac{A_{-m+1}}{x^{m-1}}+\cdots+\frac{A_{-1}}{x}+A(x)\right)dx$

where $A_{-m},\dots,A_{-1}\in\operatorname{Mat}(2,\mathbb{C})$ are constant
matrices, $A_{-m}\neq 0$ and $A\in\operatorname{Mat}(2,\mathbb{C}\\{x\\})$ is
holomorphic. Assume that the pole order $m\in\mathbb{N}^{*}$ at $x=0$ is
_minimal_ , i.e. cannot be decreased by a meromorphic gauge transformation of
the system.

Then, there exists a formal holomorphic gauge transformation
$G\in\operatorname{GL}(2,\mathbb{C}[[x]])$ such that

$\tilde{\Omega}=G^{-1}\Omega G+G^{-1}dG$

is of one of the following formal normal forms.

  * •

Regular case ($m=1$).

$\displaystyle\tilde{\Omega}=\begin{pmatrix}\lambda^{+}&0\\\
0&\lambda^{-}\end{pmatrix}dx\text{~{}~{} or
~{}~{}}\tilde{\Omega}=\begin{pmatrix}\lambda^{+}&x^{k}\\\
0&\lambda^{-}\end{pmatrix}dx,$ (23)

for some non-negative integer $k\in\mathbb{N}$ and
$\lambda^{\pm}=\frac{\lambda_{-1}^{\pm}}{x}$, where
$\lambda^{\pm}_{-1}\in\mathbb{C}$ are the eigenvalues of $A_{-1}$, such that
in the second case $\lambda^{+}_{-1}=\lambda^{-}_{-1}+k$.

Moreover, the latter formal normal form is unique, while the first one is
unique up to the permutation of $\lambda^{+}$ and $\lambda^{-}$.

  * •

Irregular ($m\geq 2$) and unramified ($A_{-m}$ is semi-simple) case.

$\displaystyle\tilde{\Omega}=\begin{pmatrix}\lambda^{+}&0\\\
0&\lambda^{-}\end{pmatrix}dx,$ (24)

for some
$\lambda^{\pm}=\frac{\lambda_{-m}^{\pm}}{x^{m}}+\cdots+\frac{\lambda_{-1}^{\pm}}{x}$
and $\lambda_{i}^{\pm}\in\mathbb{C}$ ($\lambda_{-m}^{\pm}$ are the eigenvalues
of $A_{-m}$).

Moreover, this formal normal form is unique up to the permutation of
$\lambda^{+}$ and $\lambda^{-}$.

  * •

Irregular ($m\geq 2$) and ramified ($A_{-m}$ is not semi-simple) case.

$\displaystyle\tilde{\Omega}=\begin{pmatrix}\alpha&\beta\\\
x\beta&\alpha-\frac{1}{2x}\end{pmatrix}dx,$ (25)

for some
$\begin{cases}\alpha=\frac{\alpha_{-m}}{x^{m}}+\cdots+\frac{\alpha_{-2}}{x^{2}}+\frac{\alpha_{-1}}{x},~{}~{}\alpha_{i}\in\mathbb{C}\\\
\beta=\frac{\beta_{-m}}{x^{m}}+\cdots+\frac{\beta_{-2}}{x^{2}},~{}~{}\beta_{i}\in\mathbb{C}.\end{cases}$

Moreover, this formal normal form is unique up to the multiplication of
$\beta$ by $-1$.

###### Proof complement.

A version of this theorem is proven in [DL22, Sec. 2], allowing twists in
addition to gauge transformations. This assumption can be circumvented in the
following way. First, it is unnecessary if $\Omega$ has trace-free principal
part (or merely trace-free leading coefficient $A_{-m}$). But we can always
reduce to this case, by working separately on each terms of the decomposition

$\left[\Omega-\frac{\operatorname{trace}(\Omega^{<0})}{2}I_{2}\right]+\frac{\operatorname{trace}(\Omega^{<0})}{2}I_{2},$

where $\Omega^{<0}$ is the principal part of $\Omega$. Indeed, the first term
of this sum has trace-free principal part, while the last term is in the
center of $\operatorname{GL}(2,\mathbb{C}[[x]])$ and is thus invariant by
formal holomorphic gauge transformation (such a transformation acts by
conjugacy on the negative part of the system’s matrix). ∎

######  Remark 10.

One must be careful here, not to confuse eigenspaces of the matrix $\Omega$
with subspaces invariant by $d+\Omega$, or with subspaces generated by
solutions of the system $dY+\Omega Y=0$.

We call the coefficients $(\lambda^{+},\lambda^{-})$ and $(\alpha,\beta)$ in
the above normal forms, considered up to a permutation of $\lambda^{+}$ and
$\lambda^{-}$ and up to the multiplication of $\beta$ by $-1$, the _basic
formal data_ of the system. Under the assumption that the pole order $m$ is
minimal, the basic formal data of a system together with the value of $m$
fully characterize the formal structure of the system $dY+\Omega Y=0$ (i.e.
they determine its equivalence class under formal holomorphic gauge
equivalence), except in the regular case131313More precisely: in the regular
_resonant_ case, i.e. when the eigenvalues of $A_{-1}$ satisfy
$|\lambda^{-}-\lambda^{+}|\in\mathbb{N}$, the system can be formally
equivalent to either of the two normal forms (23). In the regular non-resonant
case, however, only the first one is possible (see [IY08, Thm. 16.15])..

######  Remark 11.

Since $G$ is formally holomorphic, so is $G^{-1}dG$. Therefore, the gauge
transformation $G$ acts by conjugacy on the principal part $\Omega^{<0}$ of
the system’s matrix. As a consequence, the trace
$\operatorname{tr}(\Omega^{<0})$ is a formal invariant of the system.
Moreover, it is fully determined by the system’s basic formal data.

In the regular resonant case, applying the meromorphic gauge transformation
(which is a composition of elementary gauge transformations in rank $2$)

$G=\begin{pmatrix}x^{-k}&0\\\ 0&1\end{pmatrix}$

leads to a connection matrix of the form

$\tilde{\tilde{\Omega}}=\begin{pmatrix}\lambda^{-}_{-1}&1\\\
0&\lambda^{-}_{-1}\end{pmatrix}\frac{dx}{x}.$

Furthermore, in the irregular ramified case, applying the ramification
$x=z^{2}$ and the gauge transformation

$G=\begin{pmatrix}1&1\\\ z&-z\end{pmatrix}$

leads to a connection matrix of the form

$\tilde{\tilde{\Omega}}=\begin{pmatrix}\lambda^{+}&0\\\
0&\lambda^{-}\end{pmatrix}dz$

with $\lambda^{\pm}=2z(\alpha\pm z\beta)=2(\pm x\beta+z\alpha)$. This ramified
formal normal form is unique up to the permutation of $\lambda^{+}$ and
$\lambda^{-}$.

With those additional transformations, the formal normal forms of 3.8 are all
_Hukuhara-Turrittin ramified formal normal forms_. As the above relations
shows, their diagonal elements (up to a permutation) can be deduced from the
basic formal data, and conversely. Thus, it is just another way to encode the
basic formal data, that we will use from now on and call _basic formal data_
as well.

In the literature, the system $dY+\Omega Y=0$ is often considered up to formal
_meromorphic_ gauge transformations. The normal forms obtained in this way are
also Hukuharra-Turrittin ramified formal normal forms (see [Sib90, Thm.
6.8.1]), but only the set of their diagonal elements modulo $\mathbb{Z}dz/z$
is a formal meromorphic invariant of the system. Those diagonal elements,
called "generalized eigenvalues" by Inaba in [Ina22], thus lead to a looser
notion of formal invariants that we do not use.

#### Local formal invariants for singular Riccati foliations.

Let $\mathcal{F}$ be a singular Riccati foliation on the trivial
$\mathbb{P}^{1}$-bundle over a disc $0\in\mathbb{D}\subset\mathbb{C}$ defined
by the equation

$\mathcal{F}:dy+[a(x)y^{2}+b(x)y+c(x)]\frac{dx}{x^{m}}=0,~{}~{}(x,y)\in\mathbb{D}_{x}\times\mathbb{P}^{1}_{y},$
(26)

were $a,b,c\in\mathbb{C}\\{x\\}$ are holomorphic functions. Assume that the
pole order $m\in\mathbb{N}^{*}$ at $x=0$ is _minimal_ , i.e. cannot be
decreased by a meromorphic gauge transformation.

Locally, this Riccati equation is the projectivization of a unique trace-free
rank $2$ system (this is just the local part of the proof of 3.5). After
projectivization of the Hukuhara-Turrittin ramified formal normal form of this
system, we immediately deduce a ramified formal normal forms for the Riccati
equation (see also [LPT16, Prop. 5.1]). In each case (regular, irregular
unramified and irregular ramified), the normal form only depends on a
coefficient $\lambda=\lambda^{-}-\lambda^{+}\in\mathbb{C}[z^{-1}]$, well-
defined up to a multiplication by $-1$, that we call the _basic formal data_
of the Riccati equation.

We may assume without loss of generality that $\\{y=0\\}$ is transverse to
$\mathcal{F}$ (if not, then we can send any transverse section to infinity by
a holomorphic gauge transformation of the trivial bundle). Then, there is a
unique meromorphic affine gauge transformation such that

$\mathcal{F}:dy+\left(y^{2}+\frac{g(x)}{x^{n}}\right)dx=0$

with $g\in\mathcal{O}^{*}(\mathbb{D})$ holomorphic and non-vanishing and
$m\leq n\leq 2m$ (the computation is just a meromorphic version of the proofs