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# Monodromies of Projective Structures on Surface of Finite-type

Genyle Nascimento  Universidade Federal Fluminense
Campus Gragoatá, Rua Alexandre Moura 8 - São Domingos
24210-200, Niterói, Rio de Janeiro
Brazil<EMAIL_ADDRESS>
###### Abstract.

We characterize the monodromies of projective structures with fuchsian-type
singularities. Namely, any representation from the fundamental group of a
Riemann surface of finite-type in $PSL_{2}(\mathbb{C})$ can be represented as
the holonomy of branched projective structure with fuchsian-type singularities
over the cusps. We made a geometrical/topological study of all local conical
projective structures whose Schwarzian derivative admits a simple pole at the
cusp. Finally, we explore isomonodromic deformations of such projective
structures and the problem of minimizing angles.

###### 2021 Mathematics Subject Classification:

Primary 30F99, 32S65; Secondary 14H60, 53A55

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Proof of the Existence Theorem
  4. 4 Isomonodromic Deformations
  5. 5 Branching order

##  1\. Introduction

This theory has its roots in the study of automorphic functions and
differential equations by Klein [17, Part 1], Poincaré [26], Riemann [27], and
others in the late nineteenth century (see Hejhal’s works [16], [15] for
further historical discussion and references).

A complex projective structure on an oriented surface is a distinguished
system of local coordinates modeled in $\mathbb{CP}^{1}$ in such a way that
transition maps extend to homographies, i.e., lie in $PSL_{2}(\mathbb{C})$.
Branched projective structures on closed orientable surfaces are given by
atlases where local charts are finite branched coverings and transition maps
lie in $PSL_{2}(\mathbb{C})$.

We know that if $S$ is a surface with a projective structure modeled in the
projective space $\mathbb{CP}^{1}$, then there is a pair $(dev,\rho)$ (unique
up to the action of automorphisms of $\mathbb{CP}^{1}$), where
$dev:\tilde{S}\rightarrow\mathbb{CP}^{1}$, defined on the universal covering
of $S$ is a projective immersion equivariant with respect to the homomorphism
$\rho:\pi_{1}(S)\rightarrow Aut(\mathbb{CP}^{1})$. Two projective structures
are equivalent if the developing maps $dev$ differ by a homography.

We use a point of view for projective structure that is a description as in
which case the monodromy is conjugated in Goldman’s thesis ([12]), that
associated a projective structure to a triple $(\pi,\mathcal{F},\sigma)$ where
let $\rho:\pi_{1}(S)\rightarrow Aut(\mathbb{CP}^{1})$ be a representation,
there exists a natural bijection between equivalence classes of complex
projective structures with monodromy $\rho$ in $S$ and sections $\sigma$ of
the $\mathbb{CP}^{1}$-bundle $S\times_{\rho}\mathbb{CP}^{1}$, suspension of
$\rho$, that are transversal to the foliation $\mathcal{F}$ obtained by
quotienting the horizontal foliation of $\tilde{S}\times\mathbb{CP}^{1}$.

A natural question about complex projective structures and their monodromy
representations is to describe which representations can be realized as
monodromy of a projective structure. In the case of closed surfaces of genus
$g\geq 2$, Gallo-Kapovich-Marden [11, 2000] showed that non-elementary
representations are monodromies of projective structures with at most one
branch point.

They do not prescribe the complex structure in advance, rather it is
determined as part of the solution. The need to introduce a branch point in
the Theorem is however reminiscent of the need for "apparent singularities" in
the theory about linear ordinary differential equations on Riemann surfaces
introduced by Poincaré.

Also in [11, 2000], they listed some open problems that we treat here about
complex projective structures:

######  Problem 1.

Prove and/or explore the existence and non-uniqueness of complex projective
structures with given monodromy in punctured surfaces.

######  Problem 2.

Make precise and optimize the connection between branching divisors and
monodromy. Namely, compute the function
$d:Hom(\pi_{1}(S),PSL_{2}(\mathbb{C}))\rightarrow\mathbb{Z}$, where $d(\rho)$
is the smallest integer for which there exists a branched complex projective
structure with branching divisor of degree d and monodromy $\rho$.

The problem of building complex projective structures on surfaces of finite-
type had already been explored by Poincaré through his studies in solving
linear differential equations to come at the Uniformization Theorem even
though this was not his initial goal.

We define a singularity of Fuchsian-type as a point such that around it there
is a map that, up to local holomorphic coordinate change, is given by
$z^{\alpha}$, $\alpha\in\mathbb{C}^{*}$, or $\log z+\frac{1}{z^{n}}$,
$n\in\mathbb{N}$. We define a singular projective structure of Fuchsian-type
in $S$ as projective structures where a finite number of the singularities of
this type are allowed. These singularities are the same considered by Fuchs in
his studies about differential equations ([28]).

We know that every projective structure on a surface has a subjacent complex
structure. If we consider a surface of finite-type $S^{*}:=S\setminus P$,
where $S$ is a compact Riemann surface and $P$ is a finite subset
$\\{p_{1},\ldots,p_{k}\\}$ of $S$, the complex structure extends in a unique
way to $S$.

To obtain a result, analogous to the Gallo-Kapovich-Marden’s Theorem to
Riemann surfaces of finite-type, we will prove:

######  Theorem 1.1.

Let $S$ be a compact Riemann surface of any genus and
$\\{p_{1},p_{2},\ldots,p_{k}\\}\subset S$ a finite subset with $S\setminus
S^{*}=\\{p_{1},\ldots,p_{k}\\}$. Given a representation
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ there exists a singular
projective structure of the Fuchsian-type in $S$ with monodromy $\rho$.

We prescribe the monodromy, but differently of Gallo-Kapovich-Marden, we
prescribe a complex structure before building the projective structure. We do
not control the local models, i.e., they cannot be optimal, it can have a
finite number of singularities with trivial local monodromy to exist outside
the cusps $S\setminus S^{*}$, i.e., branch points, and angle excesses at the
cusps.

The proof uses techniques from algebraic geometry precisely that ruled
surfaces have sections and ideas developed by Loray and Marín [18]. Let
$\pi:P\rightarrow S$ be a $\mathbb{P}^{1}$-bundle over $S$ associated to the
monodromy representation $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$
of a singular projective structure of Fuchsian-type in $S$. This
$\mathbb{P}^{1}$-bundle is equipped with a Riccati foliation
$\mathcal{F}_{\rho}$ (see Brunella [1]) with the same monodromy obtained by
compactification the suspension of the representation $\rho$. The developing
map of the projective structure defined in $S$ defines a non-trivial
holomorphic section $\sigma$ of $\mathbb{P}^{1}$-bundle $\pi$ and non-
invariant by $\mathcal{F}_{\rho}$. After, we will study the relation between
isomonodromic projective structures and flipping of a fiber.

This result goes back to a work by Loray and Pereira [19] that restricted to
projective surfaces, it is possible to build transversely projective
foliations with prescribed monodromy. The approach is similar to ours,
although they use other tools such as Deligne’s work on the Riemann-Hilbert
problem is used to build a meromorphic plane connection in a rank 2 vector
bundle whose projectivization gives the $\mathbb{CP}^{1}$-bundle and build a
meromorphic section generally transversal to the foliation using also fiber
bundle’s theory.

In the presence of branching, or excess we show that the solution, using the
surgery of moving branch points, given in Theorem 1.1 is not rigid: we can
isomonodromically deform the structure if we have the models $\log
z+\frac{1}{z^{n}}$, $n\geq 2$ e $z^{\alpha}$, $\Re\alpha>1$ .

We generalize this surgery when one of the singularities involved is of
Fuchsian-type using the topological/geometrical description of projective
charts around the singularities and we will show the inverse surgery of moving
branch points with one of the singularities has non-parabolic monodromy and
excess angles.

We extended the notion of branching order at singular points of Fuchsian-type,
in fact, it will follow from Theorem 1.1 that around each singular point $p$
of the projective structure of Fuchsian-type $\sigma$ with given monodromy
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ and the projective charts
are defined by $z^{\alpha+n_{p}}$, $0<\Re\alpha\leq 1$ or $\log
z+\frac{1}{z^{n_{p}}}$. We define as $n_{p}\in\mathbb{Z}$ the branching order
at each singular point $p$ and the sum $e(\sigma)=\sum_{p\in S}n_{p}$ as
branching order of the projective structure $\sigma$.

We obtain the following result for representations that monodromies of
projective structures of Fuchsian-type not of minimum branching order:

######  Theorem 1.2.

Let $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ be a representation.
A projective structure of Fuchsian-type with monodromy $\rho$ has an odd
branching order if and only if:

  1. (1)

$w_{2}(P)$ is even and the number of cusps with non-trivial local monodromy is
odd; or

  2. (2)

$w_{2}(P)$ is odd and the number of cusps with non-trivial local monodromy is
even.

where $w_{2}(P)$ is the 2nd Stiefel-Whitney class of the bundle
$\pi:P\rightarrow S$.

Minimizing the branching order of a singular projective structure of Fuchsian-
type in a surface $S$ with monodromy $\rho$ is equivalent minimizing the index
$tang(\mathcal{F}_{\rho},\sigma(S))$ (see Brunella [1][Section 2.2]) of a
Riccati foliation and a section $\sigma$ of the $\mathbb{CP}^{1}$-bundle with
a specific compactification of $\mathbb{CP}^{1}$-bundle that defines a
projective structure.

We define $d(\rho)=\min\\{e(\sigma):\sigma$ is a projective structure of
Fuchsian-type with monodromy $\rho\\}$. For the Theorem cases, we will
necessarily have $d(\rho)\geq 1$. Given a representation
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$, we can rewrite the
Problem 2 as what is the minimum branching order of a projective structure of
Fuchsian-type with this monodromy representation? For the Theorem 1.2 cases,
do we have $d(\rho)=1$? And in other cases do we have $d(\rho)=0$?

Independently, Gupta [14] has also shown a version to the Gallo-Kapovich-
Marden’s Theorem with an analogous statement by using techniques from
hyperbolic geometry in dimension 3. One of the biggest differences with our
work is a restrictive hypothesis in monodromy representations that will not
provide branch points outside the cusps.

Calsamiglia, Deroin, and Francaviglia in [7] proved that two-branched
projective structures on compact surfaces with the same quasi-Fuchsian
holonomy and same branching order are related by moving branch points. It
would be interesting to do the same for the case of projective structures with
Fuchsian-type singularities. As in [7], we can use the surgery debubbling for
reduce the branching order.

### Acknowledgments

This paper is the result of the author’s PhD thesis that to thank the advisor
G. Calsamiglia for introduce this subject and for help and orientation. The
author also acknowledge the financial support from Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code
001 and CAPES-Mathamsud. I am grateful to the following institutions for the
very nice working conditions provided: École Normale Supérieure de Paris and
Universidade Federal Fluminense (UFF). I am thankful to B. Deroin, T.
Fassarella and A. Muniz for their comments.

##  2\. Preliminaries

###  2.1. Projective Structures

On a surface S, a complex projective structure is defined by an atlas
$\\{U_{i},f_{i}\\}$ of homeomorphisms $f_{i}:U_{i}\rightarrow V_{i}$,
$V_{i}\subset\mathbb{CP}^{1}$, where the transition maps $f_{i}=\phi_{ij}\circ
f_{j}$ are restrictions of Möbius transformations $\phi_{ij}\in
PGL_{2}(\mathbb{C})$.

We can define of alternative way: let $\tilde{S}\rightarrow(S,z_{0})$ be a
universal covering of $S$ based on $z_{0}$. A pair $(dev,\rho)$ where
$dev:\tilde{S}\rightarrow\mathbb{CP}^{1}$ is a local homeomorphism equivariant
to respect the monodromy representation $\rho:\pi_{1}(S,z_{0})\rightarrow
PGL_{2}(\mathbb{C})$ defines a complex projective structure on $S$. Two
projective structures on $S$ where the developing maps differ by homography
are equivalent.

A Riemann surface $S^{*}$ is of finite-type if it is biholomorphic to
$S^{*}:=S\setminus P$, where $S$ is a compact Riemann surface and $P$ is a
finite subset $\\{p_{1},\ldots,p_{k}\\}$ of $S$, we call $p_{i}$ of cusps.

We define a singular projective structure in $S$ as a complex projective
structure in $S^{*}=S\setminus\\{p_{1},\ldots,p_{k}\\}$, where
$\\{p_{1},\ldots,p_{k}\\}\subset S$ is a finite subset of $S$ and each $p_{i}$
is called the singularity of the structure.

The restriction of a projective structure to an open subset $U\subset S^{*}$
produces a projective structure in $U$ and we can consider the structures as a
germ in the local ring and consider equivalence of germs of projective
structures around their singularities.

The monodromy of a singularity is the monodromy of the restriction of the
projective structure to a disk around it.

######  Example 2.1.

In $S^{*}=\mathbb{CP}^{1}\setminus\\{0,\infty\\}$ with non-trivial monodromy,
we can build projective charts as the branches of the multivalued map
$z^{\alpha}$ with $\alpha\in\mathbb{C}\setminus\mathbb{Z}$ fixed and monodromy
around the cusps conjugate to $w\mapsto e^{2\pi i\alpha}w$, as well as, the
branches of $\log z+\frac{1}{z^{n}}$ will also define a singular projective
structure with monodromy $w\mapsto w+2\pi i$.

We define a singularity of Fuchsian-type as a point such that around it there
is a map that, up to local holomorphic coordinate change, is given by
$z^{\alpha}$, $\alpha\in\mathbb{C}^{*}$, or $\log z+\frac{1}{z^{n}}$,
$n\in\mathbb{N}$. We define a singular projective structure of Fuchsian-type
in $S$ as projective structures where only singularities of this type are
allowed.

We remark that singularities of Fuchsian-type with trivial monodromy have a
simple topological description that comes from branched coverings. In
particular, a branch point is a singularity of Fuchsian-type.

In an analytic approach, a projective structure is represented by a quadratic
differential on a Riemann surface, which is extracted from the Schwarzian
derivative.

Considering differential equations with poles corresponding to singular
projective structures, works by Fuchs and later by Schwarz give meaning to the
nomenclatures used by Poincaré such as “Fuchsian functions” and “Fuchsian
groups”. For equivalence between complex projective structure (compatible with
the complex structure) in S and second-order linear differential equation and
more details can be seen in [28, Chapitres 8,9].

We say that a reduced linear differential equation with a $h$ meromorphic
coefficient

(1) $\frac{d^{2}u}{dz^{2}}+hu=0$

is Fuchsian on $z=z_{0}$ if $h$ has a maximum of a double pole in $z_{0}$.

The chart $w$ of the projective structure around $z_{0}$ is the quotient
$w=\frac{u_{1}}{u_{2}}$ of two independent solutions of the equation (1)
around $z_{0}$, or better, as a solution of Schwarzian equation

(2)
$S_{z}(w):=\Bigg{\\{}\left(\frac{w^{\prime\prime}z)}{w^{\prime}(z)}\right)^{\prime}-\frac{1}{2}\left(\frac{w^{\prime\prime}(z)}{w^{\prime}(z)}\right)^{2}\Bigg{\\}}=2h,$

where $h$ is the coefficient of the equation (1). Then, we will say that the
projective structure around $z_{0}$ has a Fuchsian-type singularity in
$z_{0}$. A meromorphic quadratic differential defined by the Schwarzian
derivative of projective charts has the form

(3)
$\Bigg{\\{}\frac{1-\alpha^{2}}{2z^{2}}+\sum_{n\geq-1}b_{n}z^{n}\Bigg{\\}}dz^{2},$

in local coordinates around each singularity of Fuchsian-type,
$\alpha,b_{n}\in\mathbb{C}$. Conversely, the quotient of two linearly
independent solutions of Schwarzian equation (2) defines a projective chart of
Fuchsian-type.

Fuchs-Schwarz [28, Théorème IX.1.1.] resolves the Schwarzian equation in the
neighborhood of a double pole which the quotient of solutions are, in local
coordinate around the pole, $y^{\alpha}$, $\alpha\in\mathbb{C}^{*}$ and
$\frac{1}{y^{\alpha}}+\log y$ if $\alpha\in\mathbb{N}$.

About the projective structure of Fuchsian-type over the three-punctured
sphere, we can see as the quotient of solutions of a Schwarzian equation
explicitly.

######  Theorem 2.2.

([28])Given
$\alpha_{0},\alpha_{1},\alpha_{\infty}\in\mathbb{C}\setminus\mathbb{Z}$, the
Schwarzian given by

(4)
$\Bigg{\\{}\frac{1-\alpha_{0}^{2}}{2z^{2}}+\frac{1-\alpha_{1}^{2}}{2(z-1)^{2}}-\frac{\alpha_{0}^{2}+\alpha_{1}^{2}-\alpha_{\infty}^{2}-1}{2z(z-1)}\Bigg{\\}}dz^{2}$

defines the only projective structure in
$\mathbb{CP}^{1}\setminus\\{0,1,\infty\\}$ with charts projectively equivalent
to $z\mapsto z^{\alpha_{i}}$, $i=0,1,\infty$, at the cusps. If
$\alpha_{0},\alpha_{1},\alpha_{\infty}\in\mathbb{Z}$, the Laurent series
expansion around $z_{i}\in\\{0,1,\infty\\}$ of the Schwarzian given by

$\Bigg{\\{}\frac{1-\alpha_{i}^{2}}{2(z-z_{i})^{2}}+\sum_{n\geq-1}a_{n}^{(i)}(z-z_{i})^{n}\Bigg{\\}}dz^{2}$

is the meromorphic quadratic differential of a branched projective structure
if and only if $a_{n}^{(i)}$ satisfies the indicial equation
$A_{\alpha_{i}}(a_{-1}^{(i)},\ldots,a_{\alpha_{i}-1}^{(i)})=0$ where
$A_{\alpha_{i}}$ is a polynomial with coefficients in $\mathbb{C}$. Otherwise,
the charts around the cusps $z_{i}$ is projectively equivalent to
$z\mapsto\log z+\frac{1}{z^{\alpha_{i}}}$.

Since any three points in $\mathbb{CP}^{1}$ can be taken to $0,1$ and $\infty$
by a Möbius transformation, then the projective structures with three
singularities on $\mathbb{CP}^{1}$ are completely determined by their indexes.

Let
$[\gamma_{0}],[\gamma_{1}],[\gamma_{\infty}]\in\pi_{1}(\mathbb{CP}^{1}\setminus\\{0,1,\infty\\})$
be loops around each $i$, for $i=0,1,\infty$, which have the same base point,
satisfy $[\gamma_{0}]\cdot[\gamma_{1}]\cdot[\gamma_{\infty}]=Id$. Since this
relation, the monodromy representation
$\rho:\pi_{1}(\mathbb{CP}^{1}\setminus\\{0,1,\infty\\})\rightarrow
PSL_{2}(\mathbb{C})$ must satisfy

(5)
$\rho([\gamma_{0}])\cdot\rho([\gamma_{1}])\cdot\rho([\gamma_{\infty}])=Id.$

The transformation $\rho([\gamma_{i}])$ is a local monodromy around
$i=0,1,\infty$ and conjugate in $PSL_{2}(\mathbb{C})$, in the non-parabolic
case, to $w\mapsto e^{2\pi i\alpha_{i}}w$, $\alpha_{i}\in\mathbb{C}$. At the
cusps with parabolic monodromy is conjugate to $w\mapsto w+2\pi i$. The
relation (5) is equivalent to
$\alpha_{0}+\alpha_{1}+\alpha_{\infty}\in\mathbb{Z}$ and the representation
$\rho$ induces a complex projective structure in
$\mathbb{CP}^{1}\setminus\\{0,1,\infty\\}$.

###  2.2. Compactification of $\mathbb{CP}^{1}$-bundles

Different from Gallo-Kapovich-Marden, we prescribe a complex structure before
building the projective structure. It is allowed a finite number of
singularities with trivial local monodromy to exist outside the cusps
$S\setminus S^{*}$, i.e., branch points. As it was done on Goldman’s thesis in
[12], we will build these projective structures through sections of the
$\mathbb{CP}^{1}$-bundle obtained from the suspension of the given
representation.

We denote by $S\times_{\rho}\mathbb{CP}^{1}$ the suspension of a
representation $\rho:\pi_{1}(S)\rightarrow PSL_{2}(\mathbb{C})$, the
construction can be found in [8, Chapter 5].

Let $S^{*}=S\setminus\\{p_{1},\ldots,p_{k}\\}$ be a Riemann surface of finite-
type and $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ a
representation, we can compactify the suspension
$S^{*}\times_{\rho}\mathbb{CP}^{1}$ as a fiber bundle over $S$ provided with a
Riccati (possibly singular) foliation111A foliation $\mathcal{F}$ on a compact
connected surface $X$ is called Riccati foliation if there exists a
$\mathbb{CP}^{1}$-bundle $\pi:X\rightarrow B$ (possibly with singular fibers)
whose generic fiber is tranverse to $\mathcal{F}$. , where the fibers over the
cusps are invariant curves and each contains one or two singularities of
foliation.

There are local models, can be found at Brunella [1], that allow us to
compactify the suspension and its foliation over the cusps.

Before that, we need a previous result that allows us to paste the local
models around the cusps to the suspension.

Let $\mathbb{D}=\\{z\in\mathbb{C}\ |\ |z|<1\\}$ the unit disk centered on the
source and $\mathbb{D}^{*}=\mathbb{D}\setminus\\{0\\}$. We will denote by
$(\mathbb{D}^{*},\mathcal{F},\pi)$ a Riccati foliation $\mathcal{F}$ defined
in $\mathbb{D}^{*}\times\mathbb{CP}^{1}$ with $\pi$ a $\mathbb{CP}^{1}$-bundle
transversal to the foliation $\mathcal{F}$.

######  Proposition 2.3.

Let $(\mathbb{D}^{*},\mathcal{F}_{1},\pi_{1})$ and
$(\mathbb{D}^{*},\mathcal{F}_{2},\pi_{2})$ be Riccati foliations. There is a
biholomorphism
$\phi:\mathbb{D}^{*}\times\mathbb{CP}^{1}\rightarrow\mathbb{D}^{*}\times\mathbb{CP}^{1}$
that takes leaves of $\mathcal{F}_{1}$ to leaves $\mathcal{F}_{2}$ and such
that $\pi_{1}$ and $\pi_{2}$ are equivalent bundles if and only if the
representations of holonomy are analytically conjugate.

This is just a modified statement of Theorem 2 [8] p. 98. For a complete proof
with the details and modifications can be found in [25, Proposição 2.1] (see
also note after Theorem 2 [8] p. 99).

######  Lemma 2.4.

Every suspension $S^{*}\times_{\rho}\mathbb{CP}^{1}$ admits a compactification
$\pi:\overline{S^{*}\times_{\rho}\mathbb{CP}^{1}}\rightarrow S$,
$\mathbb{CP}^{1}$-bundle over $S$, provided with a Riccati foliation
$\mathcal{F}_{\rho}$ with invariant fibers over the cusps with non-trivial
monodromy.

###### Proof.

There is a regular "horizontal" foliation on
$S^{*}\times_{\rho}\mathbb{CP}^{1}$. Let $D_{i}$ be a disk (image of a disk of
complex plan by a chart of complex structure of $S$) on $S$ around of $p_{i}$,
we have that the foliation over $D_{i}\setminus\\{p_{i}\\}$ is determined by
$\rho(\partial D_{i})\in PSL_{2}(\mathbb{C})$. We choose a biholomorphism that
maps $p_{i}$ to $0$ and $D_{i}$ to $\mathbb{D}$.

We can choose on $\mathbb{D}\times\mathbb{CP}^{1}$ a singular Riccati
foliation with any prescribed monodromy, where in coordinates
$(z,w)\in\mathbb{D}\times\mathbb{CP}^{1}$ of fiber bundle trivialization
around of invariant fiber, the foliation will be generated by a meromorphic
1-form defined in $\mathbb{D}\times\mathbb{CP}^{1}$ rational in the variable
$w$ (or, dually generated by vector fields). For each monodromy, we will
choose the following models:

  1. (1)

In the case of non-parabolic monodromy, conjugated to $w\mapsto e^{2\pi
i\alpha_{i}}w$, the vector field $z\dfrac{\partial}{\partial
z}+\alpha_{i}w\dfrac{\partial}{\partial w}$ and the 1-form $\alpha_{i}wdz-
zdw=0$, $\alpha_{i}\in\mathbb{C}$, or

  2. (2)

In the case of parabolic monodromy, conjugated to $w\mapsto w+1$, the vector
field $z\dfrac{\partial}{\partial z}+\dfrac{\partial}{\partial w}$ and the
1-form $dz-zdw=0$, or

  3. (3)

In the case of trivial monodromy, conjugated to the identity, the vector field
$z\dfrac{\partial}{\partial z}+mw\dfrac{\partial}{\partial w}$ and the 1-form
$mwdz-zdw=0$, for some $m\in\mathbb{N}$.

Then, locally the monodromies are the same, so by the Proposition 2.3 the
foliations over $D_{i}\setminus\\{p_{i}\\}$ are biholomorphic, so we can glue
and obtain a singular Riccati foliation in all
$\overline{S^{*}\times_{\rho}\mathbb{CP}^{1}}$ where over the cusps with non-
trivial monodromy has invariant fibers in $\\{z=0\\}$ and one or two
singularities.

In the non-parabolic case, the singularities of foliation are $(0,0)$ and
$(0,\infty)$ with separatrix $\\{w=0\\}$ and $\\{w=\infty\\}$, and in the
parabolic case has a saddle-node singularity in $(0,\infty)$ and a separatrix
$\\{w=\infty\\}$. ∎

######  Remark 2.5.

In that compactification, the fibers over the cusps with non-trivial monodromy
are always invariant by $\mathcal{F}_{\rho}$. In the cusps of trivial
monodromy, only in the case $m=0$ in (3) we’d have a compactification given in
the neighborhood these points by product foliation without invariant fibers
and singularities.

###  2.3. Flippings and existence of holomorphic sections

We will show that there is a holomorphic section generically transversal to
the foliation for the bundle obtained from the Lemma 2.4. This is necessary
for describing the singularities of the projective structures obtained through
holomorphic sections of $\overline{S^{*}\times_{\rho}\mathbb{CP}^{1}}$,

Recall that a $\mathbb{CP}^{1}$-bundle, suspension of a representation
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$, has an invariant
holomorphic section if and only if $\rho$ has fixed points. Each fixed point
determines an invariant holomorphic section transporting the fixed point
through the holonomy of the foliation. Therefore, we have at most two fixed
points for non-trivial representations and therefore we will have at most two
invariant sections.

We say that a representation $\rho:\pi_{1}(S)\rightarrow PSL_{2}(\mathbb{C})$
is elementary if the action of $Im(\rho)$ on $\mathbb{H}^{3}$ fixes one point
or two in $\mathbb{H}^{3}\cup\partial\mathbb{H}^{3}$, otherwise, we call it
non-elementary. If the representation is non-elementary, the
$\mathbb{CP}^{1}$-bundle will not have an invariant section.

We will show that if the monodromy of a Riccati foliation in a
$\mathbb{CP}^{1}$-bundle over a Riemann surface $S$ is non-trivial, then the
fiber bundle has at least three holomorphic sections to guarantee the
existence of at least one non-invariant. We use a result by Tsen ([2] p. 140)
that affirms that $\mathbb{CP}^{1}$-bundle
$\overline{S^{*}\times_{\rho}\mathbb{CP}^{1}}$ has a holomorphic section.

The monodromy representation of a Riccati foliation gives a complete
description of the foliation module birational isomorphisms, according to [1,
Chapter 4].

When the monodromy representation is non-parabolic (including the trivial
case) the foliation around an invariant fiber is $(\alpha+n)wdz-zdw=0$ and
$\alpha wdz-zdw=0$, $\alpha\in\mathbb{C}$, $n\in\mathbb{Z}$, and these
foliations are related to each other by flipping (or elementary
transformation) of that fiber, i.e., related through a sequence of blowings up
at the singular points and contractions of invariant fibers. Flipping the
fiber does not change the monodromy $w\mapsto e^{2\pi i\alpha}w$ around of
fiber. Similarly, when the monodromy is parabolic the foliations are $dz-
zdw=0$ and $(nw+z^{n})dz-zdw=0$, $n\in\mathbb{N}$, and they are related by
flipping of the invariant fiber.

For showing that there exists infinitely many sections in
$\mathbb{CP}^{1}$-bundles, it follows immediately of next result:

######  Theorem 2.6.

([18]) The composition of a finite number of flippings in a trivial bundle
$S\times\mathbb{CP}^{1}$ gives a $\mathbb{CP}^{1}$-bundle over a compact
Riemann surface $S$ and every $\mathbb{CP}^{1}$-bundle over $S$ can be
obtained of this way.

So we can take the images of infinitely many sections of the trivial bundle
over $S$ by the composition of flippings assured by the above theorem and
therefore we have at least a non-invariant section between them.

###  2.4. Projective structures with prescribed monodromy

To prove the Theorem 1.1, we calculate the projective charts by projecting the
section along of the leaves in a fiber transversal to the foliation. This
construction was already known for the case without branch points and it can
be extended in a similar way when there are branch points. It can be
calculated through the image of $\sigma(S)$ by local first integral $h$
composing with the inverse of local first integral restricted to a transversal
fiber $F_{1}$. Therefore, the local submersions that define the regular
foliation restricted to the curve $\sigma(S)$ define charts of branched
projective structure on $S$. The tangency points between $\sigma(S)$ and the
foliation produce the critical points of the charts.

A priori, we do not have this control at the cusps with singularities of
Fuchsian-type. We can perform this, if the surface is finite-type, after the
compactification of the suspension $S^{*}\times_{\rho}\mathbb{CP}^{1}$, the
foliation becomes a singular foliation. We can only use the construction above
when the section does not pass through the singular points of the foliation,
because in these points there exist no local submersion. However, we can
extend the construction to the points where there is a closed meromorphic
1-form that defines the foliation locally, such as, for example, the form
$\omega=\frac{dz}{z}+\lambda\frac{dw}{w}$ is a closed meromorphic and has a
Liouvillian first integral $h(z,w)=zw^{\lambda}$. Therefore, around a singular
point the foliation $\mathcal{F}_{\rho}$ comes in a closed meromorphic form
that has a Liouvillian first integral, and the projective chart can be
calculated in the same way above.

Finally, the developing map is the local inverse of the holonomy germ $f$
between the transversal fiber $F_{1}$ and the section.

In the next proposition, we will prove that the choice of the local model in
the compactification, up to birational isomorphism, is related to the section
of the $\mathbb{CP}^{1}$-bundle passes through the singular points of the
foliation or not.

######  Proposition 2.7.

Let $\mathcal{F}$ be a Riccati foliation in
${\mathbb{D}}\times\mathbb{CP}^{1}$ and a non-trivial section $\sigma$ and
non-invariant by foliation that intersects a singularity. Then,

(i)The tangency order of the image of $\sigma$ by a flipping decreases 1 with
the initial section with foliation, the Liouvillian first integral also
changes, but the coordinate chart around the singularity of projective
structure does not change.

(ii)There is one only model, up to flippings, where the section does not
intersect the singularities of foliation.

###### Proof.

Suppose non-parabolic case with model $\alpha wdz-zdw=0$,
$\alpha\in\mathbb{C}$. We consider the case where the section passes through
one of the singular points of the foliation
$\sigma:\mathbb{D}\longrightarrow\mathbb{D}\times\mathbb{CP}^{1}$ given by
$\sigma(z)=(z,\sigma_{1}(z))=(z,z^{n}\phi(z))$, $n\geq 1$ and $\phi(0)\neq 0$.
We have that $F=\pi^{-1}(0)$ is a invariant fiber by the foliation whose
monodromy is given by $e^{2\pi i\alpha}w$. The fixed points of the monodromy,
$w=0$ and $w=\infty$, represent the separatrices of the foliation which pass
through singular points $(0,0)$ and $(0,\infty)$.

Let $h(z,w)=z^{\alpha}w^{-1}$ be a multi-valued first integral of the
foliation in a neighborhood $U$ of $(0,\sigma_{1}(0))$. We remark that the
(multi-valued) graphs of $w=cz^{\alpha}$, $c\in\mathbb{C}$, are the leaves.
Let us study the projection of $\sigma\cap U$ along the leaves in transversal
fiber to a foliation $F_{1}=\pi^{-1}(z_{1})$, $z_{1}\in\mathbb{D}^{*}$.

If $\sigma_{1}(0)=0$, it follows that
$f(z)=z_{1}^{\alpha}\cdot\frac{\sigma_{1}(z)}{z^{\alpha}}$. As $\sigma_{1}(z)$
is a holomorphic germ with $\sigma_{1}(0)=0$, we can rewrite as
$\sigma_{1}(z)=z^{n}\phi(z)$, $n\geq 1$ and $\phi(0)\neq 0$, up to an
automorphism of $\mathbb{CP}^{1}$,
$f(z)=(z\tilde{\phi}(z))^{n-\alpha}=(k(z))^{n-\alpha}$ where $\tilde{\phi}$ is
the only holomorphic solution in the neighborhood of $0$ of the equation
$\tilde{\phi}(z)^{n-\alpha}=\phi(z)$, $\phi(0)\neq 0$, and $k$ is a invertible
germ. Thus, $f(k^{-1}(z))=z^{n-\alpha}$.

If $\sigma_{1}(0)=\infty$, in the analogous way, we obtain the coordinate
chart $z^{n+\alpha}$.

For the parabolic case, the foliation is induced by $\omega=dz-zdw$. We have
to $F=\pi^{-1}(0)$ is the invariant fiber through the foliation whose
monodromy is given by $w+1$. It has a saddle-node in the point $(0,\infty)$
whose weak separatrix is $F$ and the strong separatrix is transversal to $F$.

Let $h(z,w)=\log z-w$ be the holomorphic first integral of the foliation in a
neighborhood $U$ of $(0,\sigma_{1}(0))$. At the same way as above, we obtain
$f(z)=\log z_{1}-\log z+\sigma_{1}(z)$.

We affirm that there exists diffeomorphism germ $w$ such that
$\log\left(w(z)e^{\frac{1}{w(z)^{n}}}\right)=\log\left(ze^{-\sigma_{1}\left({z}\right)}\right)$
where $n$ satisfies
$\sigma_{1}\left({z}\right)=\frac{1}{z^{n}}\cdot\sigma_{2}(z)$,
$\sigma_{2}(0)\neq 0$.

In fact, we put $w(z)=zh(z)$ $\log
h(z)+\frac{1}{z^{n}h^{n}}=-\frac{\sigma_{2}(z)}{z^{n}}.$ We take
$F(z,\zeta)=z^{n}\log\zeta+\frac{1}{\zeta^{n}}+\sigma_{2}(z)$, where
$F(z,h(z))\equiv 0$. Since $\frac{\partial
F}{\partial\zeta}(0,h(0))=-\frac{n}{h(0)^{n+1}}\neq 0$, since
$\frac{1}{h(0)^{n}}=-\sigma_{2}(0)\neq 0$, therefore $h(0)\neq 0$.

By Implicit Function Theorem, there will exist $h(z)$ holomorphic in the
neighborhood of $0$, thus $w(z)$ is a invertible germ ($w^{\prime}(0)=h(0)\neq
0$). Thus, $dev(w^{-1}(z))=\log z+\frac{1}{z^{n}}$.

(i)After one blowing up ($z=z$ e $w=zy$) and one contraction of the fiber
$\\{z=0\\}$, we obtain a section $\beta(z)=(z,z^{n-1}\phi(z))$, thus decreases
1 in the tangency order of the initial section with the foliation. Already
foliation after the flipping becomes $(\alpha-1)ydz-zdy=0$ with first integral
$z^{\alpha-1}y^{-1}$ different of the initial. But we obtain the same
projective chart equal to $z^{n-\alpha}$.

(ii)The flipping given by the composition of $n$ blowings up and contractions
sends the section $\sigma$ in the section $\beta(z)=(z,\phi(z))$, $\phi$
biholomorphism germ, $\phi(z)\neq 0$, that is, a transversal section to the
foliation and follows of i) that it’s the only flipping that happens this.

We can use the same idea for the parabolic case. ∎

Therefore, we can choose a local model to compactify the suspension over
$S^{*}$ such that the section is transversal to the foliation around invariant
fibers, i.e, it doesn’t intersect the foliation’s singularities.

##  3\. Proof of the Existence Theorem

###### Proof of the Theorem 1.1.

At the $\mathbb{CP}^{1}$-bundle
$\pi:\overline{S^{*}\times_{\rho}\mathbb{CP}^{1}}\rightarrow S$, by the
discussion in the previous section $\pi$ has at least one non-trivial, and
non-invariant holomorphic section $\sigma$. We will study two cases: regular
points and cusps.

1st case: Regular points.

At regular points on the surface, we will obtain, up to appropriate
coordinates changing, complex projective charts or branched coverings.

In fact, at a regular point $p=(z_{0},w_{0})\in S^{*}\times\mathbb{CP}^{1}$ of
a non-trivial and non-invariant section $\sigma$ we have that
$\mathcal{F}_{\rho}\ \vbox{ \halign{#\cr\smash{$-$}\crcr$\pitchfork$\crcr} }\
\pi$. We analyze two cases: whether $\sigma$ is transversal to
$\mathcal{F}_{\rho}$ at $p$ or not.

By introducing coordinates $(z,w)\in S^{*}\times\mathbb{CP}^{1}$ centered on
$p$, in a neighborhood of $p$ to foliation $\mathcal{F}_{\rho}$ is regular, we
can think as "horizontal" foliation $\frac{\partial}{\partial z}$.

Putting $\sigma(z):=(z,\sigma_{1}(z))$, let $U$ be a neighborhood of
$(0,\sigma_{1}(0))$ in $\mathbb{D}\times\mathbb{CP}^{1}$, $h(z,w)=z$ the
holomorphic first integral of $\mathcal{F}_{\rho}$ in $U$ and
$F_{1}=\pi^{-1}(z_{1})$ a fiber near to $0$. The restriction of $h$ to
$F_{1}\cap U$ is a diffeomorphism and
$\left({\left.\kern-1.2pth\vphantom{\big{|}}\right|_{F_{1}\cap
U}}\right)^{-1}\circ h(\sigma(z))=(z_{1},f(z))$, then $f(z)=\sigma_{1}(z)$.

In this case, $f$ is holomorphic and if the foliation is transversal to the
section in $(0,\sigma_{1}(0))$, we obtain that projective chart around to $0$
is a homeomorphism. Otherwise, $\sigma_{1}^{\prime}(z)=0.$ We can rewrite as
$\sigma_{1}(z)=z^{n}\phi(z)$, $\phi(0)\neq 0$, or better,
$\sigma_{1}(z)=(z\tilde{\phi}(z))^{n}=(k(z))^{n}$, $n-1$ is the tangency order
of the section with the foliation and since $k^{\prime}(0)\neq 0$, so $k$ is a
invertible holomorphic germ.

Thus, $f(k^{-1}(z))=z^{n}$, i.e., ramified covering with $n$ sheets.

2nd case: Cusps

At the points $\\{p_{1},\ldots,p_{k}\\}$ we will obtain, up to appropriate
coordinates changing, singular projective charts $z\mapsto z^{\alpha}$,
$\alpha\in\mathbb{C}^{*}$ , if the monodromy around the point is non-parabolic
and when it’s parabolic it will be$z\mapsto\log z+\frac{1}{z^{n}}$,
$n\in\mathbb{N}$.

The foliations used in Lemma 2.4 have Liouvillian first integrals. We shall
separate in parabolic, non-parabolic, and trivial cases.

In coordinates $(z,w)\in\mathbb{D}\times\mathbb{CP}^{1}$, we consider the case
where the section
$\sigma:\mathbb{D}\longrightarrow\mathbb{D}\times\mathbb{CP}^{1}$ given by
$\sigma(z)=(z,\sigma_{1}(z))$ doesn’t pass through singularities of the
foliation, i.e., $\sigma_{1}(0)\neq 0,\infty$.

(i)Non-parabolic Monodromy

The foliation is induced by $\omega=\alpha wdz-zdw$. We have that
$F=\pi^{-1}(0)$ is a invariant fiber by the foliation whose monodromy is given
by $e^{2\pi i\alpha}w$. The fixed points of the monodromy, $w=0$ and
$w=\infty$, represent the separatrices of the foliation which pass through
singular points $(0,0)$ and $(0,\infty)$.

Let $h(z,w)=z^{\alpha}w^{-1}$ be a multi-valued first integral defined at
$\mathbb{D}\times\mathbb{CP}^{1}$ of the foliation in a neighborhood $U$ of
$(0,\sigma_{1}(0))$. We remark that the (multi-valued) graphs of
$w=cz^{\alpha}$, $c\in\mathbb{C}$, are the leaves. Let us study the projection
of $\sigma\cap U$ along the leaves in transversal fiber to a foliation
$F_{1}=\pi^{-1}(z_{1})$, $z_{1}\in\mathbb{D}^{*}$.

In fact,
$\left({\left.\kern-1.2pth\vphantom{\big{|}}\right|_{F_{1}}}\right)^{-1}\circ(h(\sigma(z))=\left({\left.\kern-1.2pth\vphantom{\big{|}}\right|_{F_{1}}}\right)^{-1}\circ\left(\frac{z^{\alpha}}{\sigma_{1}(z)}\right)=\left(z_{1},z_{1}^{\alpha}\cdot\frac{\sigma_{1}(z)}{z^{\alpha}}\right)$.
Thus, $f(z)=z_{1}^{\alpha}\cdot\frac{\sigma_{1}(z)}{z^{\alpha}}$. Since
$\sigma_{1}(z)$ is a holomorphic germ with $\sigma_{1}(0)\neq 0$, up to an
automorphism of $\mathbb{CP}^{1}$, f is
$\frac{\sigma_{1}(z)}{z^{\alpha}}=(z\tilde{\phi}(z))^{-\alpha}=(k(z))^{-\alpha}$.
We remark that the equation $\tilde{\phi}(z)^{-\alpha}=\sigma_{1}(z),\
\sigma_{1}(0)\neq 0$, admits only one solution
$\tilde{\phi}(z)=e^{-\frac{1}{\alpha}\log(\sigma_{1}(z))}$ holomorphic in the
neighborhood of $0$ with $k$ a invertible germ, since
$k^{\prime}(0)=\tilde{\phi}(0)\neq 0$. Thus,
$f(k^{-1}(\frac{1}{z}))=z^{\alpha}$.

(ii)Parabolic Monodromy

We have that $f(z)=\log z_{1}-\log z+\sigma_{1}(z)$, as above, with
coordinates appropriate changing, we have $dev(w^{-1}(z))=\log z$.

(iii)Trivial Monodromy

The foliation is induced by $\omega=mwdz-zdw$ where $m\in\mathbb{N}$. The
fiber $F=\pi^{-1}(0)$ is invariantand the monodromy around it is the identity.
The singular points of the foliation are $(0,0)$ and $(0,\infty)$.

In an analogous way to the previous cases, we obtain $f(z)=z^{n-m}$, when
$\sigma_{1}(0)\neq\infty$.

∎

######  Remark 3.1.

The case of trivial monodromy around a cusp, the charts are as in the case of
the regular points where the section is not transversal to
$\mathcal{F}_{\rho}$. In fact, if in Lemma 2.4 we choose the model $dw=0$
instead of $mwdz-zdw=0$, for some $m\in\mathbb{Z}$, the foliation (also the
first integral) would extend holomorphically at the cusps with trivial
monodromy.

Let $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ a representation. We
obtain a dictionary between a triple $(\pi,\mathcal{F}_{\rho},\sigma)$, where
$\pi:P\rightarrow S$ is a $\mathbb{P}^{1}$-bundle equipped with a Riccati
foliation $\mathcal{F}_{\rho}$ and $\sigma$ is a holomorphic section of $\pi$
generically transversal to $\mathcal{F}_{\rho}$, and a singular projective
structure of Fuchsian-type in the Riemann surface $S$ with monodromy $\rho$.

##  4\. Isomonodromic Deformations

In this section, we study geometry and topology of the local structures around
the cusps for deform continuously projective structures on surfaces of finite-
type that preserve the holonomy.

###  4.1. Geometry and topology of Fuchsian-type singularities

To answer the problem of Gallo-Kapovich-Marden about non-uniqueness of
projective structure on surfaces of finite-type, singularities of Fuchsian-
type have complicated behaviors and it was studied to assist in surgeries like
moving branch points.

We denote by $\mathbb{D}^{*}$ a deleted open neighborhood of a cusp, up to
biholomorphism, and $p:T\rightarrow\mathbb{D}^{*}$ universal covering of
$\mathbb{D}^{*}$ where $T=\\{x\in\mathbb{C}\ |\ \Re x<0\\}$ with $p(x)=e^{x}$,
where $\Re x$ represents the real part of $x$. Let $f$ be a multi-valued
function, we take $\tilde{f}$ a lifting of $f$ to universal covering.

######  Definition 4.1.

The degree of a multi-valued map
$f\colon\mathbb{D}^{*}\dashrightarrow\mathbb{D}^{*}$ is the maximum number of
preimage of each $z\in\mathbb{D}^{*}$ by $\tilde{f}$ restricted to a
fundamental domain.

We remark that for a multi-valued map $f$ of degree $1$, $\tilde{f}$ is
injective on each fundamental domain.

######  Example 4.2.

The multi-valued map $z^{\frac{3}{2}}$ defines a projective chart around a
cusp with monodromy $w\mapsto-w$ and developing map $e^{\frac{3}{2}x}$ defined
in $T$. The points of $\\{z\in\mathbb{D}^{*}\ |\ \Im\ z<0\\}$, where $\Im x$
represents the imaginary part of $x$, has a preimage in the fundamental domain
$T_{0}=\\{x\in T\ |\ 0<\Im x<2\pi\\}$, while in $\\{z\in\mathbb{D}^{*}\ |\ \Im
z>0\\}$ has two preimages in $T_{0}$, therefore this map has degree 2.

Given a local non-parabolic monodromy conjugate to $w\mapsto e^{2\pi
i\alpha}w$, when $\Re\alpha\neq 0$. The projective structure defined around
one of the fixed points of this monodromy, which we will assume to be the
origin, can be thought as a sector of $\mathbb{CP}^{1}\setminus\\{0,\infty\\}$
centered on $0$ with angle opening $2\pi\Re\alpha$ and length sides $1$ and
$e^{-2\pi\Im\alpha}$ identified by the transformation $w\mapsto e^{2\pi
i\alpha}w$.

Geometrically, two points $u+iv,u^{\prime}+iv^{\prime}\in T$ have the same
image by $dev$ if and only if $(u^{\prime},v^{\prime})=(u,v)-\beta\mathbb{Z}$,
where $\beta=\frac{2\pi i}{\alpha}$.

We put $\alpha=a+ib$. The semi-plane $au-bv<0$ is decomposed into
biholomorphic strips to the disk minus the radius $[0,1)$ by $dev$ and this
decomposition is given by parallel lines to $av+bu=0$ and equidistant with
distance $\frac{2\pi}{|\alpha|}$.

Figure 1. Decomposition of $dev(x)=e^{\alpha x}$, $\Re\alpha>0$

We change the universal covering of $\mathbb{D}^{*}$ such that the new
fundamental domain is given by a strip whose boundary consists of lines
$bu+av=0$ and $bu+av=2\pi|\alpha|$. Therefore, the maximum number of preimages
of $z\in\mathbb{D}^{*}$ for $dev$ restricted to fundamental domain is
$\lceil\Re\alpha\rceil$ \- degree of $z^{\alpha}$.

In the case $\Re\alpha=0$, the fundamental domain $T_{0}$ covers a ring
$A=\\{z\in\mathbb{C}\ |\ e^{-2\pi b}<|z|<1\\}$ through $dev(x)=e^{ibx}$. The
action of $dev$ is defined by the translation $w\mapsto w-\frac{2\pi}{b}$
where the semi-plane $v>0$ will be decomposed by the lines parallel to $u=0$
and equidistant with distance $\frac{2\pi}{|b|}$, see figure 2. In that case,
the projective structure can be seen as the ring $A$ with the boundary lines
identified by the transformation $w\mapsto e^{-2\pi b}w$, which is
topologically a torus.

Figure 2. Decomposition of $dev(x)=e^{\alpha x}$, $\Re\alpha=0$

We prove that two actions in the universal covering of $\mathbb{D}^{*}$
classify projective structures of type $z^{\alpha}$, $\Re\alpha>0$. This cover
all charts of type $z^{\alpha}$, $\Re\alpha\neq 0$.

######  Proposition 4.3.

The projective structure defined by the branches of $z^{\alpha}$,
$\Re\alpha>0$, in $\mathbb{D}^{*}$ is represented by a pair of vectors $(2\pi
i,\frac{2\pi i}{\alpha})$, where $x\mapsto x+2\pi i$ and $x\mapsto
x+\frac{2\pi i}{\alpha}$ are in $\pi_{1}(\mathbb{D}^{*})$ acting in T.
Conversely, this pair defines the projective structure defined by branches of
$z^{\alpha}$ in $\mathbb{D}^{*}$.

###### Proof.

The first assumption follows by the discussion above. Conversely, given the
pair $(2\pi i,\frac{2\pi i}{\alpha})$, we establish that $2\pi i$ is the
vector of $\pi_{1}(\mathbb{D}^{*})$ action in T and $\frac{2\pi i}{\alpha}$ is
the vector of equivalence action of $dev$ by the monodromy representation.

This pair is associated with the structure coming from the branches of
$z^{\alpha}$, if we show that there is a biholomorphism
$\tilde{\phi}:T\rightarrow T$ such that $\tilde{\phi}(t+2\pi
i)=\tilde{\phi}(t)+2\pi i$ and $dev=e^{\alpha x}\circ\tilde{\phi}$ but, if we
put $\tilde{\phi}=id$ and the result follows. ∎

We recall that translation structure on a surface is defined as an atlas such
that the coordinate changes are translations. The branches of $\log z$ and
$\log z+\frac{1}{z}$ define different translation structures in
$\mathbb{D}^{*}$, for example. We will show that these structures and their
pull-backs by covering maps of degree $\geq 2$ provide us with a list of
translation structures in $\mathbb{D}^{*}$ modulo projective equivalence.

######  Proposition 4.4.

Translation structures in $\mathbb{D}^{*}$ induced by the pull-back of $\log
z$ by the map $z\mapsto z^{n}$, $n\in\mathbb{N}$ , $n\geq 2$, are projectively
equivalent to those induced by $\log z$.

The translation structure defined by $\log z$ can be seen as an infinite
cylinder with one end.

We know that a branch of $\log z$ is injective in its domain, the same occurs
with $\log z+\frac{1}{z}$. We use some ideas from Section 2.2 of [6] where
local models of poles of meromorphic forms that induce translation structures
on compact Riemann surfaces were studied.

Let $U_{R}=\\{z\in\mathbb{C}\mid|z|>R\\}$ and $V_{R}$ be the Riemann surface
obtained after removing from $U_{R}$ the $\pi$-neighborhood of the real half-
line $\mathbb{R}^{-}$, and identifying the lines $-i\pi+\mathbb{R}^{-}$ and
$i\pi+\mathbb{R}^{-}$ by the translation $z\mapsto z+2\pi i$.

We choose the usual determination of $\log z$ in
$\mathbb{C}\setminus\mathbb{R}^{-}$ restricted to $U_{R^{\prime}}$, we obtain
the map $z\mapsto z+\log z$ well-defined from
$U_{R^{\prime}}\setminus\mathbb{R}^{-}$ to $\mathbb{C}$.

######  Proposition 4.5.

The map $z+\log z$ extends to a injective holomorphic map
$f:U_{R^{\prime}}\rightarrow V_{R}$, if $R^{\prime}$ is large enough.

###### Proof.

See [6, Section 2.2] or [25, Proposição 4.3] for more details. ∎

We conclude that $\log z+\frac{1}{z}$ is also injective when restricted to a
deleted neighborhood of the origin. It remains to show that $f$ is surjective
in a neighborhood of infinity and to conclude through Proposition 4.5 that
this chart, defined in the neighborhood of origin, is topologically $V_{R}$.
In fact, we have

######  Proposition 4.6.

The map $f:U_{R^{\prime}}\rightarrow V_{R}$ is surjective in a neighborhood of
infinity, i.e., for $Z\in V_{R}$ with large enough modulus, there exists $z\in
U_{R^{\prime}}$ such that $f(z)=Z$.

###### Proof.

See [6, Section 2.2] or [25, Proposição 4.4] for more details.

∎

######  Proposition 4.7.

The projective structure on $\mathbb{D}^{*}$ given by the pull-back of $\log
z+\frac{1}{z}$ by $z\mapsto z^{n}$ is projectively equivalent to $\log
z+\frac{1}{z^{n}}$.

Then $\log z+\frac{1}{z^{n}}$, in terms of projective structure, is a suitable
rotating and rescaling covering of $V_{R}$ of order $n$.

###  4.2. Generalization of surgery

Moving branch points is a surgery that consists of deformation of branched
local projective charts; it can be thought as a configuration analogous to
Schiffer’s variations in Riemann surface’s theory, as [22]. These movements
were introduced by Tan in [29] for projective structures with simple branch
points and then generalized in [7] for higher- order branch points. Schiffer
variations, in particular the moving branch points, produce deformations of
the projective structure without changing the monodromy representation but, in
general, do not preserve the underlying complex structure.

Let $S$ be a closed Riemann surface with a singular projective structure of
Fuchsian-type with developing map $dev$. Let $p$ be a singularity of Fuchsian-
type.

######  Definition 4.8.

We define a pair of twins embedding in $S$ as a pair of embedded curves
$\gamma=\\{\gamma_{1},\gamma_{2}\\}$ starting from $p$ such that there is a
determination of developing map around $\gamma_{1}\cup\gamma_{2}$ which maps
$\gamma_{1}$, $\gamma_{2}$ into a simple curve
$\hat{\gamma}\subset\mathbb{CP}^{1}$.

According to the study of degree of $dev(x)=e^{\alpha x}$ done in the previous
section, we have that there are twin curves for $\Re\alpha>1$. In the case of
parabolic monodromy, we saw in the previous section that $\log z$ and $\log
z+\frac{1}{z}$ are injective in $\mathbb{D}^{*}$, so they don’t have twins.
For $n\geq 2$, $\log z+\frac{1}{z^{n}}$ can be seen as a branched covering of
order $n$ of $\log z+\frac{1}{z}$, as it was done in Proposition 4.7 and then
the preimage of a segment will have $n$ copies in $\mathbb{D}^{*}$ for $\log
z+\frac{1}{z^{n}}$ are the candidate twins in this model.

We’ll describe the moving branch points as it was done in [7]. Let $p$ be a
branch point of $S$, we take twin curves $\gamma_{1}$, $\gamma_{2}$ starting
from $p$ with endpoints $q_{1}$ and $q_{2}$. We denote by $\alpha$ and $\beta$
angles in $p$, and $\theta_{i}$ the angles in $q_{i}$, $i=1,2$, where
$\theta_{i}=2\pi$ , if $q_{i}$ is a regular point. A new branched projective
structure in $S$ will be obtained by cutting $S$ along
$\gamma_{1}\cup\gamma_{2}$ and pasting the copies according to the
identifications made in Figure 3.

$\ast$$q_{1}$$\theta_{1}$$\gamma_{1}$$\circ$$p$$\alpha$$\beta$$\gamma_{2}$$\ast$$q_{2}$$\theta_{2}$cut$\circ$$p_{1}$$||$$\ast$$q_{2}$$|$$\circ$$p_{2}$$|$$\ast$$q_{1}$$||$paste$\circ$$p_{1}$$\beta$$\gamma_{2}^{\prime}$$\ast$q$\theta_{1}+\theta_{2}$$\gamma_{1}^{\prime}$$\circ$$p_{2}$$\alpha$
Figure 3. Moving branch points

After this process, we obtain two new twin curves $\gamma^{\prime}_{1}$ and
$\gamma^{\prime}_{2}$ starting from a point $q$ with total angle
$\theta_{1}+\theta_{2}$ and the endpoints $p_{1}$ and $p_{2}$ of the new twins
have angles $\beta$ and $\alpha$, respectively. Note that the image of
$\gamma^{\prime}_{1}$ and $\gamma^{\prime}_{2}$ by the developing map is the
same as $\gamma_{1}$ and $\gamma_{2}$, that is, the surgery does not change
the image by dev, then we have a pair of twins embedded into the new
structure, and we will return to the initial structure if we move the points
along of that pair.

Note that the angles $\alpha$, $\theta_{i}$ and $\beta$ are multiples of
$2\pi$ and if $p$ is a single branch point, then $\alpha=\beta=2\pi$. This
process describes locally a continuous deformation in the space of classes of
branched projective structures over $S$ with monodromy representation fixed.
In addition to being used to collapse branch points, the process can be used
to change the position of branch points and to separate a higher-order branch
point into several branch points of lower order.

Calsamiglia, Deroin, and Francaviglia in [7] prove that two-branched
projective structures in compact surfaces with the same quasi-Fuchsian
holonomy and the same degree of branching are related by a movement of branch
points, so it is possible to use this surgery to show the non-uniqueness of
projective structures with the same monodromy representation.

A cone-angle $\theta$ is produced from a sector of angle $\theta$ by
identification of their boundaries by an isometry. Then the singularities with
cone-angle, called conical singularities, still have the same notion of angle
as they have in the case of branch points. So the surgery to move branch
points will work in the same way.

In [30], Troyanov characterized orientable compact surfaces with conical
singularities. The invariants that represent the opening of the cone are real
numbers and, he obtained a classification of these surfaces. More precisely,
given $p_{1},\ldots,p_{k}\in S$ and $\theta_{1},\ldots,\theta_{k}>0$, if
$\chi(S)+\sum_{i=1}^{k}(2\pi-\theta_{i})<0$ (respectively, $=0$ or $=1$), then
there exists a hyperbolic metric (respectively, Euclidean or spherical) in
$S\setminus\\{p_{1},\ldots,p_{k}\\}$ with a conical angle $\theta_{i}$ in
$p_{i}$.

Now, we will prove the inverse surgery for the case $\Re\alpha>1$, that is,
when the degree of multi-valued function $z^{\alpha}$ is at least 2. Given two
twin curves starting of $z^{\alpha}$, we remove an angle $2\pi$, that we see
in a fundamental domain, it would be to remove one of the biholomorphic strips
of the disk minus a radius through of $dev(x)=e^{\alpha x}$ and glue in the
perpendicular way to the boundary of strip.

######  Proposition 4.9.

Let $\gamma_{1}$, $\gamma_{2}$ be a pair of twins that start from a
singularity $p$ of type $z^{\alpha}$, with $\Re\alpha>1$, forming a sector
with angle $2\pi$ and end-points $q_{1}$ and $q_{2}$ are regular points. The
inverse surgery of the movement that removes the angle $2\pi$ in $p$ results
in a simple branch point where start two twin curves whose end-points are a
singularity of type $z^{\alpha-1}$ and a regular point.

$\ast$$q_{1}$$2\pi$$\gamma_{1}$$\circ$p$z^{\alpha}$$2\pi$$\gamma_{2}$$\ast$$q_{2}$$2\pi$cut$\circ$$p_{1}$$2\pi$$||$$\ast$$q_{2}$$|$$\circ$$p_{2}$$|$$\ast$$q_{1}$$||$paste$\circ$$p_{1}$$2\pi$$\gamma_{2}^{\prime}$$\ast$q$4\pi$$\gamma_{1}^{\prime}$$\circ$$p_{2}$$z^{\alpha-1}$
Figure 4. Inverse moving branch points in $z^{\alpha}$

###### Proof.

After the cut and paste process, we identify two regular points $q_{1}$ and
$q_{2}$, making a simple branch point $q$. We still need to show that removing
an angle $2\pi$ of $z^{\alpha}$ will result in a singularity of type
$z^{\alpha-1}$.

We use the decomposition of the universal covering of $\mathbb{D}^{*}$ as in
Figure 1 and we note the degree of $dev$ is $\lceil\Re\alpha\rceil$.

We will remove a biholomorphic strip from the disk minus the radius by $dev$
and all its copies via the action of fundamental group
$\pi_{1}(\mathbb{D}^{*})$ and define a relation in the lines $av+bu=2\pi l$,
$l\in\mathbb{Z}$, given by $u+iv\sim u+iv+j\beta$, $j\in\mathbb{Z}$. This
identifies the boundaries of strips in the direction of vector
$\beta=\frac{2\pi i}{\alpha}$.

The initial $dev$ is given by $D(x)=e^{\alpha x}$. Since the family of lines
are twins of the projective structure in $\mathbb{D}^{*}$ we see that
$D(0)=D(\beta)$ and it follows from the equivalence of $D$ by the monodromy
representation $\rho:\pi_{1}(\mathbb{D}^{*})\rightarrow PSL_{2}(\mathbb{C})$
given by $\rho([\gamma])=e^{2\pi i\alpha}w$ that $D(2\pi i)=D(0+2\pi
i)=D(\beta)\cdot e^{2\pi i\alpha}$, here we use the action $x\mapsto x+2\pi i$
of fundamental group in $T$.

Note that $D(\beta+w)=D(\beta)\cdot e^{2\pi i\alpha}$, where $w=2\pi i-\beta$.
We affirm that $D$ is equivalent for the monodromy representation $\rho$ and
the new action of the fundamental group is given by $x\mapsto x+w$. We just
need to show that, $\forall x\in T$, $D(x+w)=D(x)\cdot e^{2\pi i\alpha}$. In
fact, $D(x+w)=D(x+2\pi i-\beta)=D(x-\beta)\cdot e^{2\pi i\alpha}=D(x)\cdot
e^{2\pi i\alpha}$, since $D(x-\beta)=D(x)$.

We will obtain a new domain $T^{\prime}$ of the covering application and a new
developing map $D_{1}$ equivariant with respect to $\rho$ with the new action
of the fundamental group and the images will coincide with the initial
developing map at the respective paste points.

The domain $T^{\prime}$ is simply connected, its quotient by the action of
$x\mapsto x+w$ is homeomorphic to $\mathbb{D}^{*}$ and therefore can be taken
as a universal cover of $\mathbb{D}^{*}$.

Using the classification obtained in Proposition 4.3, we have that after the
surgery, they are given by $(w,\beta)=\left(2\pi
i\left(\frac{\alpha-1}{\alpha}\right),\frac{2\pi i}{\alpha}\right)$ and the
linear transformation $L:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ given by
$L(t)=\frac{\alpha}{\alpha-1}t$ takes them to a pair of the form $(2\pi
i,\beta^{\prime})$ where $\beta^{\prime}=\frac{2\pi i}{\alpha-1}$ and
therefore the structure obtained is equivalent biholomorphically to
$z^{\alpha-1}$ in $\mathbb{D}^{*}$ with new developing map is $D_{1}\circ
L^{-1}$.

∎

We can conclude that the rigid models, that is, those that do not have twins
path (do not have excess angle) and the models $\log z+\frac{1}{z^{n}}$,
$n\geq 2$ and $z^{\alpha}$, $\Re\alpha>1$ have twins and candidates to be
deformed isomonodromically.

##  5\. Branching order

In this section, we explore the Problem 2 posed by Gallo-Kapovich-Marden about
minimizing angles on projective structures. Let
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ be a representation, what
is the minimum branching order of a projective structure of Fuchsian-type with
this monodromy representation?

We obtain a result with a existence of obstruction for prescribing local
models without branch points and angle excess in the cusps of a compact
Riemann surface.

###  5.1. Algebro-geometric interpretations of projective structures of
Fuchsian-type

There are exactly two oriented topologically $\mathbb{S}^{2}$-bundle over the
closed Riemann surface $S$ and they are distinguished by the 2nd Stiefel-
Whitney class $w_{2}(P)$ of the bundle $\pi:P\rightarrow S$, $\sigma^{2}\equiv
w_{2}(P)(mod\ 2)$, where $\sigma$ is section of $\pi$. Then, the parity of the
self-intersection $\sigma^{2}$ depends only on the bundle: $\sigma^{2}$ is
even if the bundle is diffeomorphic to the trivial bundle and it is odd,
otherwise.

######  Proposition 5.1.

Let $\sigma$ and $\sigma^{\prime}$ be two holomorphic sections of holomorphic
$\mathbb{CP}^{1}$-bundles on a compact Riemann surface that have the same 2nd
class of Stiefel-Whitney. We have to

$\sigma^{2}\equiv\sigma^{\prime 2}(mod\ 2).$

In particular, holomorphic sections of the same $\mathbb{CP}^{1}$-bundle have
self-intersection with the same parity.

Let $\pi:P\rightarrow S$ be a $\mathbb{CP}^{1}$-bundle with a Riccati
foliation, after a flipping of an invariant fiber we get another
$\mathbb{CP}^{1}$-bundle $\pi^{\prime}:P^{\prime}\rightarrow S$ also equipped
with an equivalent birationally Riccati foliation equivalent to $\pi$. In
fact, flipping changes the topological class of the bundle:

######  Proposition 5.2.

The second Stiefel-Whitney classes $w_{2}(P)$ and $w_{2}(P^{\prime})$ have
different parities.

###### Proof.

Let $\sigma$ be holomorphic section of fiber bundle $\pi:P\rightarrow S$,
after a flipping on an invariant fiber, if we consider a blow-up at a point
outside the section, after flipping we have the new section $\tilde{\sigma}$
de $\pi^{\prime}:P^{\prime}\rightarrow S$ has self-intersection
$\tilde{\sigma}^{2}=\sigma^{2}+1$, and if we consider blow-up at a point in
the section, after flipping we have that the new section has self-intersection
$\sigma^{2}-1$. ∎

In general, the intersection numbers of holomorphic sections are either all
even, or all odd: $\sigma^{2}\mod 2$ is the topological invariant of the
bundle. Then, at the same compactification, two holomorphic sections have the
same parity of tangency order with the foliation.

Let $\pi:P\rightarrow S$ be a $\mathbb{CP}^{1}$-bundle over $S$ associated to
the monodromy representation $\rho:\pi_{1}(S^{*})\rightarrow
PSL_{2}(\mathbb{C})$ of a projective structure of Fuchsian-type in $S$. This
$\mathbb{CP}^{1}$-bundle is equipped with a Riccati foliation
$\mathcal{F}_{\rho}$ (see Brunella [1][Section 4.1]) with the same monodromy
obtained by compactification the suspension of the representation $\rho$. The
developing map of the projective structure defined in $S$ defines a non-
trivial holomorphic section $\sigma$ of $\pi$ non-invariant by
$\mathcal{F}_{\rho}$.

We obtain a formula that relates topological invariants of the surface with
the tangency order of Riccati foliation with the holomorphic section (see [1,
p. 22]) of the suspension and its self-intersection.

######  Proposition 5.3.

Under the conditions above, the self-intersection of $\sigma(S)$ in $P$ is

$\sigma(S)\cdot\sigma(S)=tang(\mathcal{F}_{\rho},\sigma(S))+\chi(S)-k_{0},$

where $k_{0}$ represents the number of fibers invariant by foliation
$\mathcal{F}_{\rho}$.

###### Proof.

It’s a consequence of Brunella’s formula [1]: the cotangent bundle of a
Riccati foliation is

$T_{\mathcal{F_{\rho}}}^{*}=\pi^{*}(K_{S})\otimes\mathcal{O}_{P}\left(\sum_{j=1}^{n}k_{j}F_{j}\right),$

where $K_{S}$ is the canonical bundle and $F_{1},\ldots,F_{n}$ are the
$\mathcal{F_{\rho}}$-invariant fibres of multiplicity $k_{1},\ldots,k_{n}.$
Since after the compactification, there are $k_{0}$
$\mathcal{F_{\rho}}$-invariant fibers with multiplicity 1, we have
$T_{\mathcal{F}_{\rho}}.\sigma=2-2g-k_{0}$ with the formula
$T_{\mathcal{F_{\rho}}}\cdot\sigma=\sigma\cdot\sigma-
tang(\mathcal{F}_{\rho},\sigma)$ (see [1, Proposition 2.2]), the result
follows.

∎

######  Remark 5.4.

We can prove this Proposition with the same ideas of Proposition 11.2.2 of
[11], for complete proof see [25, Teorema 5.3].

######  Corollary 5.5.

Let $\sigma$ and $\sigma^{\prime}$ be two non-trivial holomorphic sections of
compactified bundle over $S$ associated to monodromy representation
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$. Then,

$tang(\mathcal{F}_{\rho},\sigma(S))\equiv
tang(\mathcal{F}_{\rho},\sigma^{\prime}(S))\mod 2,$

where $\mathcal{F}_{\rho}$ is Riccati foliation of compactified bundle.

###### Proof.

It follows immediately from the propositions 5.1 and 5.3. ∎

###  5.2. Minimum branching order

Let $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ be a representation.
Minimizing the branching order of a projective structure of Fuchsian-type on
$S$ with monodromy $\rho$ is equivalent to minimizing the index
$tang(\mathcal{F}_{\rho},\sigma(S))$ of a Riccati foliation and a section
$\sigma$ of the $\mathbb{CP}^{1}$-bundle with a specific compactification that
defines a projective structure.

We recall that a branched projective structure induces a complex structure and
thus angles on S. Unbranched points are called regular and the total angle
around them is $2\pi$. The cone-angle around a point p whose branching order
is $n_{p}\geq 2$ is $2\pi n_{p}$. The branching divisor of $\sigma$ is the
divisor $\sum_{p\in S}(n_{p}-1)p$. Its degree $\sum_{p\in S}(n_{p}-1)$ is
called the total branching order of $\sigma$.

We extend the notion of branching order to singular points of Fuchsian-type.
In fact, it will follow from Theorem 1.1 that around each singular point $p$
of the projective structure $\sigma$ with given monodromy $\rho$ the
projective charts are defined by $z^{\alpha+n_{p}}$, $0<\Re\alpha\leq 1$ or
$\log z+\frac{1}{z^{n_{p}}}$. We define $n_{p}\in\mathbb{Z}$ as the branching
order at each singular point $p$ and the sum $e(\sigma)=\sum_{p\in S}n_{p}$ as
the branching order of projective structure $\sigma$. We also define

$d(\rho)=\min\\{e(\sigma):\sigma\ \text{is a projective structure of Fuchsian-
type with monodromy}\rho\\}.$

Gallo-Kapovich-Marden proved that $d(\rho)=0$ for all liftable non-elementary
representations $\rho$ and $d(\rho)=1$ for all non-liftable non-elementary
representations $\rho$.

We can see the sum $e(\sigma)$ as a tangency order of a Riccati foliation with
sections of fiber bundles from compactification of suspension of a
representation $\rho$. We fix a complex structure on $S$, it follows from the
proof of the Existence Theorem that $n_{p}$ are tangency orders of foliation
with the section:

$e(\sigma)=tang(\mathcal{F}_{\rho}^{min},\sigma(S)),$

where $\mathcal{F}_{\rho}^{min}$ is the foliation provided the
compactification which the local models are:

  * •

$\alpha wdz-zdw=0$, $\alpha\in\mathbb{C}^{*}$ e $0\leq\Re\alpha<1$, at the
cusps with non-parabolic monodromy;

  * •

$zdw-dz=0$ at the cusps with parabolic monodromy;

  * •

$dw=0$ at the cusps with trivial monodromy.

In that compactification, $e(\sigma)=tang(\mathcal{F}_{\rho}^{min},\sigma)=0$
if and only if the section $\sigma$ is transversal to
$\mathcal{F}_{\rho}^{min}$. For this reason, we’ll call it minimum
compactification.

The Theorem 1.2 is about representations that are not realized as monodromy of
projective structures of Fuchsian-type with minimal branching order. For the
these cases, we will necessarily have $d(\rho)\geq 1$, that is, these cases do
not realize projective structures without angle excesses.

First, it follows from the formula $\sigma^{2}\equiv w_{2}(P)(mod\ 2)$ and
Proposition 5.3.

######  Lemma 5.6.

Let $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ be a monodromy
representation of a projective structure of Fuchsian-type $\sigma$ on $S$, we
have:

$w_{2}(P)+k_{0}\equiv e(\sigma)\mod 2$

where $w_{2}(P)$ is 2nd Stiefel-Whitney class of the minimum compactification
$\mathcal{F}_{\rho}^{min}$ and $k_{0}$ represents the number of points with
non-trivial local monodromy.

Thus, the sum of angle excesses $e(\sigma)$ has the same parity as
$w_{2}(P)+k_{0}$. Since minimum compactification only depends on the
monodromy, so does its 2nd Stiefel-Whitney class, and $k_{0}$ represents the
number of cusps with non-trivial local monodromy. We conclude that the parity
of the sum of angle excesses only depends on the monodromy.

If a representation $\rho$ is the monodromy of a projective structure without
angle excess, we have that the 2nd Stiefel-Whitney class of minimum
compactification has the same parity as the number of invariant fibers by
foliation.

Let $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ be a representation
of fundamental group of surface of finite-type
$S^{*}=S\setminus\\{p_{1},\ldots,p_{k}\\}$, where $S$ is a closed surface of
genus $g\geq 1$, we consider a presentation of $\pi_{1}(S^{*})$:

$\langle a_{i},b_{i},c_{j},\ i=1,\ldots,g,\ j=1,\ldots,k\ |\
\prod_{i=1}^{g}[a_{i},b_{i}]\prod_{j=1}^{k}c_{j}=Id\rangle,$

where $[a_{i},b_{i}]=a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}$ is the commutator of
$a_{i}$ and $b_{i}$, in this presentation we can define the representation
$\rho$ as $\rho(a_{i})=A_{i}$, $\rho(b_{i})=B_{i}$ and $\rho(c_{j})=C_{j}$,
where $A_{i},B_{i}$ and $C_{j}$ are elements of $PSL_{2}(\mathbb{C})$ that
satisfy

$\prod_{i=1}^{g}[A_{i},B_{i}]\prod_{j=1}^{k}C_{j}=Id.$

For each generator $a_{i},b_{i}$ and $c_{j}$, $\rho$ can lift in two ways,

$\pm\tilde{A}_{i},\pm\tilde{B}_{i}\ \mbox{e}\ \pm\tilde{C}_{j}\in
SL_{2}(\mathbb{C}),$

whose projetivizations give the Möbius transformations of $A_{i},B_{i}$ and
$C_{j}$, respectively, we choose a sign for each element and the product

(6) $\prod_{i=1}^{g}[\tilde{A}_{i},\tilde{B}_{i}]\prod_{j=1}^{k}\tilde{C}_{j}$

can be $\pm Id$. For the choices where the product gives $Id$ the
representation lifts to $SL_{2}(\mathbb{C})$, if it gives $-Id$, the
representation does not lift to $SL_{2}(\mathbb{C})$.

In the case of genus $0$, the presentation of $\pi_{1}(S^{*})$ there are not
$a_{i},b_{i}$, only $c_{j}$, $j=1,\ldots,k$, satisfying
$\prod_{j=1}^{k}c_{j}=Id$.

######  Remark 5.7.

Every representation $\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$
lifts to $SL_{2}(\mathbb{C})$, because the group $\pi_{1}(S^{*})$ is free. The
question here is the representation lifts if we prescribe the local models (e.
g. minimal angles) at the cusps or not. Namely, if we have a minimal angle at
a point, then we choose a lift.

######  Proposition 5.8.

The parity of the 2nd Stiefel-Whitney class of the minimum compactification
changes depending on whether the representation lifts to $SL_{2}(\mathbb{C})$
or not.

###### Proof.

Let $\pi:P\rightarrow S$ be a $\mathbb{CP}^{1}$-bundle with minimum
compactification, after a flipping of an invariant fiber we get another
$\mathbb{CP}^{1}$-bundle $\pi^{\prime}:P^{\prime}\rightarrow S$ with a
birationally equivalent Riccati foliation. It follows from Proposition 5.2
that flipping changes the topological class of bundles and therefore the 2nd
Stiefel-Whitney classes $w_{2}(P)$ and $w_{2}(P^{\prime})$ have distinct
parities.

If $\rho$ lifts to $SL_{2}(\mathbb{C})$, we have that each generator of
$\pi_{1}(S^{*})$ lifts to a matrix in $SL_{2}(\mathbb{C})$, where the product
given by the equation (6) of these matrices is $Id$. The matrices related to
the local monodromy representations come from linear differential equations
with simple poles used to projectivize and thus obtain the local Riccati
model. When we do one flipping the sign of that matrix will change, changing
the compactification and therefore the product of all matrices is $-Id$ in
that compactification the representation $\rho$ does not lift.

We will obtain two families that alternate parity when making a flipping: in
one of the families, the parity is even and is odd in the others. ∎

We obtain a version analogous to Theorem 3.10 of Goldman’s thesis [12]:

######  Proposition 5.9.

At the minimum compactification, the representation
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ lifts to
$SL_{2}(\mathbb{C})$ if and only if $w_{2}(P)$ is even.

###### Proof.

Build a path in the character variety

$\displaystyle Hom(\pi_{1}(S^{*}),PSL_{2}(\mathbb{C}))/PSL_{2}(\mathbb{C})$

between one representation $\rho$ which lifts and the trivial representation,
that also lifts, preserving the lifting relation expressed in the equation (6)
equal to $Id$.

Using the continuity of 2nd class Stiefel-Whitney, we can deduce that it will
be constant along the path, and therefore equals to zero. ∎

######  Corollary 5.10.

At the minimum compactification, the representation
$\rho:\pi_{1}(S^{*})\rightarrow PSL_{2}(\mathbb{C})$ lifts to
$SL_{2}(\mathbb{C})$ if and only if $e(\sigma)\equiv k_{0}\mod 2$.

###### Proof.

If the representation $\rho$ lifts, it follows from the Proposition 5.9 that
$w_{2}(P)$ is even and using the Lemma 5.6 will follow that $e(\sigma)\equiv
k_{0}\mod 2$. Similarly, $e(\sigma)$ is shown to have parity other than
$k_{0}$ when the representation $\rho$ doesn’t lift. ∎

###### Proof of Theorem 1.2.

Suppose $tang(\mathcal{F}_{\rho}^{min},\sigma(S))$ is odd. If the $w_{2}(P)$
is even, then $\sigma^{2}\equiv w_{2}(P)\equiv 0\mod 2$ and therefore it
follows from the Proposition 5.3 that
$tang(\mathcal{F}_{\rho}^{min},\sigma(S))$ $\equiv k_{0}\mod 2$, so $k_{0}$ is
odd. Similarly, if $w_{2}(P)$ is odd, it follows that $k_{0}$ is even.

The other implication follows immediately from Lemma 5.6 above. ∎

## References

  * [1] M. Brunella, Birational Geometry of Foliations, IMPA Monographs, Springer, 2015.
  * [2] W. Barth, C. Peters & A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin, 1984.
  * [3] D. Gallo, M. Kapovich & A. Marden, The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces, Ann. of Math. 151 (2000), p. 625-704.
  * [4] W. Goldman, Discontinuous Groups and the Euler class, PhD Thesis, University of California, Berkeley, 1980.
  * [5] D. Allegretti & T. Bridgeland, The monodromy of meromorphic projective structures, preprint, arXiv:1802.02505.
  * [6] C. Boissy, Moduli space of meromorphic differentials with marked horizontal separatrices, preprint, arXiv:1507.00555.
  * [7] G. Calsamiglia, B. Deroin & S. Francaviglia, Branched Projective Structures with Fuchsian Holonomy, Geom. Topol. 18.1 (2014), p. 379-446.
  * [8] C. Camacho & A. Lins Neto, Geometric Theory of Foliations, Birkhäuser, 1985.
  * [9] D. Dumas, Complex Projective Structures, Handbook of Teichmuller Theory, vol 2, Eur. Math. Soc, Zurich, (2009), p. 455-508.
  * [10] V. Fock & A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Etudes Sci. (2006), 103, 1-211.
  * [11] D. Gallo, M. Kapovich & A. Marden, The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces, Ann. of Math. 151 (2000), p. 625-704.
  * [12] W. Goldman, Discontinuous Groups and the Euler class, PhD Thesis, University of California, Berkeley, 1980.
  * [13] W. Goldman, Projective Structure with Fuchsian Holonomy, J. Diff. Geom. 25 (1987), p. 297-326.
  * [14] S. Gupta, Monodromy Groups of $\mathbb{CP}^{1}$-structures on Punctured Surfaces, preprint, ArXiv:1909.10771v1
  * [15] D. Hejhal, Monodromy Groups and Linearly Polymorphic Functions, Acta Math. 135 (1975), p. 1-55.
  * [16] D. Hejhal, Monodromy groups and Poincaré series, Bull. Amer. Math. Soc., 84(3):339-376, 1978.
  * [17] F. Klein, Ausgewählte Kapital aus der Theorie der linearen Differentialgleichungen zweiter Ordnung, volume 1. Göttingen, 1891.
  * [18] F. Loray & D. Marín, Projective Structures and Projective Bundles over Compact Riemann Surfaces, Asterisque 323, (2009), p. 223-252.
  * [19] F. Loray & J. V. Pereira, Transversely projective foliations on surfaces: existence of normal forms and prescription of the monodromy, Internat. J. Math., 18:6 (2007), 723-747.
  * [20] R. Mandelbaum, Branched Structures on Riemann Surfaces, Trans. Amer. Math. Soc. 163, (1972), p. 261-275.
  * [21] M. Maruyama, On Classification of Ruled Surfaces, Lectures in Mathematics, Kinokuniya Book-Store Co. Ltd, Tokyo, 1970.
  * [22] S. Nag, Schiffer Variation and Complex Structure and coordinates for Teichmüller Spaces, Proc. Indian Acad. Sci. Math. Sci. 94.2-3 (1985), p. 111-122.
  * [23] M. Nagata, On self-intersection number of a section on a ruled surface, Nagoya Math J., vol. 37 (1970), p. 191-196.
  * [24] Z. Nehari,The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc., 55:545-551, (1949).
  * [25] J. G. Nascimento Santana, Estruturas projetivas em superfícies de tipo finito, PhD Thesis-Universidade Federal Fluminense, Niterói, 2020.
  * [26] H. Poincaré, Sur les groupes des équations linéaires, Acta Math., 4(1):201-312, 1884\.
  * [27] B. Riemann, Vorlesungen über die hypergeometrische Reihe, In Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge, p. 667-692. Springer-Verlag, Berlin, 1990.
  * [28] H. P. de Saint-Gervais, Uniformisation des Surfaces de Riemann. Retour sur un Théorème Centenaire, ENS Editions, Lyon, 2010.
  * [29] S. P. Tan, Branched-Structures on Surfaces with Prescribed Real Holonomy, Math. Ann. 300.4 (1994), p. 649-667.
  * [30] M. Troyanov, Les Surfaces Euclidiennes à Singularités Coniques, L’Enseignement Mathématique 32 (1986), p. 79-94.