Higgs Bundles and Geometric Structures on Manifolds

Higgs Bundles and Geometric Structures on Manifolds††This paper is a
contribution to the Special Issue on Geometry and Physics of Hitchin Systems.
The full collection is available at
https://www.emis.de/journals/SIGMA/hitchin-systems.html

Daniele ALESSANDRINI

D. Alessandrini

Ruprecht-Karls-Universitaet Heidelberg, INF 205, 69120, Heidelberg, Germany
<EMAIL_ADDRESS>https://www.mathi.uni-
heidelberg.de/~alessandrini/

Received September 28, 2018, in final form April 17, 2019; Published online
May 10, 2019

Geometric structures on manifolds became popular when Thurston used them in
his work on the geometrization conjecture. They were studied by many people
and they play an important role in higher Teichmüller theory. Geometric
structures on a manifold are closely related with representations of the
fundamental group and with flat bundles. Higgs bundles can be very useful in
describing flat bundles explicitly, via solutions of Hitchin’s equations.
Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to
construct geometric structures in some interesting cases. In this paper, we
will explain the main ideas behind this theory and we will survey some recent
results in this direction, which are joint work with Qiongling Li.

geometric structures; Higgs bundles; higher Teichmüller theory; Anosov
representations

57M50; 53C07; 22E40

##  1 Introduction

The theory of geometric structures on manifolds was introduced by Cartan and
Ehresmann in the 1920s, following the ideas given by Klein in his Erlangen
program. This theory became popular in the 1980s, when Thurston used it in the
statement of his Geometrization Conjecture. Since then, many people
contributed important results, see for example [7, 8, 10, 11, 12, 17, 20, 21].

Nowadays, geometric structures on manifols are also important in higher
Teichmüller theory, a research area that arose from the work of Goldman [21],
Choi–Goldman [11], Hitchin [27], Labourie [30], Fock–Goncharov [16]. They
studied some connected components of the character varieties of surface groups
in higher rank Lie groups which share many properties with Teichmüller spaces.
They are now called Hitchin components or higher Teichmüller spaces.

The first works which related geometric structures and higher Teichmüller
theory are Choi–Goldman [11] and Guichard–Wienhard [23], showing how the low-
rank Hitchin components can be used as parameter spaces of special geometric
structures on closed manifolds. These first results were then generalized by
Guichard–Wienhard [24] and Kapovich–Leeb–Porti [29], who show that Anosov
representations can often be used to construct geometric structures on closed
manifolds.

Higgs bundles are an important tool in higher Teichmüller theory because they
can be used to describe the topology of the character varieties (see for
example Hitchin [26, 27], Alessandrini–Collier [1]). Anyway, they were
initially believed to give very little information on the geometry of a single
representation. This point of view is changing, since we have now many
examples where the Higgs bundle can be used to give interesting information on
the geometric structures associated with a certain representation of a surface
group.

The main purpose of this survey paper is to explain these constructions. The
first ones were presented by Baraglia in his Ph.D. Thesis [9], more recent
ones are in Alessandrini–Li [2, 3, 5], and Collier–Tholozan–Toulisse [14]. The
main idea behind these constructions is that a geometric structure corresponds
to a section of a flat bundle which is transverse to the parallel foliation.
The holomorphic structure of the Higgs bundle helps to construct sections, and
the parallel foliation can be described by solving Hitchin’s equations.

I will initially describe the fundamental notions of the theory of geometric
structures on manifolds: the notion of geometry in the sense of Klein,
geometric manifolds, the relationship with the theory of domains of
discontinuity for Anosov representations, the relationship with
representations of fundamental groups of manifolds, and the deformation spaces
of geometric structures on a fixed topological manifold, see Section 2.

Then I will give an introduction to character varieties and their relationship
with the moduli space of flat bundles. I will also introduce the subspaces of
the character varieties that are most important in higher Teichmüller theory,
see Section 3.

After this, everything is ready to explain the relationship between geometric
structures and flat bundles, via a tool called the graph of a geometric
structure, see Section 4.

We will then enter in the main topic of the mini-course, using Higgs bundles
and solutions of Hitchin’s equations to describe flat bundles explicitly and
construct geometric structures. In Section 5 three simple examples are given
where this method works and allows us to construct hyperbolic structures,
complex projective structures and convex real projective structures on
surfaces.

Higgs bundles can only describe flat bundles on surfaces, but we also want to
describe flat bundles on higher-dimensional manifolds. See Section 6 for an
explanation on how flat bundles on manifolds of different dimension can be
related, and an exposition of interesting open problems in the theory of
geometric manifolds that are related with this issue.

We will finally see how geometric structures on higher-dimensional manifolds
can be constructed. As a warm-up, in Section 7 we consider the case of
$3$-dimensional manifolds, and we see how to construct the convex foliated
real projective structures and the anti-de Sitter structures on circle bundles
over surfaces.

In the last part, in Section 8, we will see how the technique works in the
case of manifolds of higher dimension. In this final case, the technical
details are more involved and will be mainly left out. We will see how to
construct real and complex projective structures on higher-dimensional
manifolds, and how this result has applications to the theory of domains of
discontinuity for Anosov representations.

This survey paper is based on the lecture notes for the mini-course “Higgs
bundles and geometric structures on manifolds” that I gave at the University
of Illinois at Chicago during the program “Workshop on the Geometry and
Physics of Higgs bundles II”, November 11–12, 2017. The mini-course was
targeted at graduate students and young post-docs with an interest in Gauge
Theory and Higgs bundles and this survey paper addresses the same public.

##  2 Geometric structures on manifolds

This section will be an introduction to the theory of geometric structures on
manifolds, their developing maps and holonomy representations. For more
details about this theory, see Thurston’s book [39] or Goldman’s notes [19].

###  2.1 Geometries

The theory of geometric structures on manifolds traces its origins back to
Felix Klein who, in his Erlangen program (1872) discussed what is geometry.
Klein’s idea is that geometry is the study of the properties of a space that
are invariant under the action of a certain group of symmetries. The main
examples he had in mind were the Euclidean geometry, where the space is
${\mathbb{R}}^{n}$ and the group is
$\operatorname{Isom}\big{(}{\mathbb{R}}^{n}\big{)}$, and the affine geometry,
where the space is ${\mathbb{R}}^{n}$ and the group is
$\operatorname{Aff}\big{(}{\mathbb{R}}^{n}\big{)}$. These geometries study
exactly the same space, but they focus on very different properties. Euclidean
geometry deals with lengths, angles, and circles, the notions that are
invariant under the group of isometries. These notions make no sense in affine
geometry, because they are not preserved by the affine group. Affine geometry,
instead, deals with ratios of lengths, parallelism and ellipses. Klein
emphasizes that when studying geometry, the symmetry group is as important as
the space. Let’s now give a definition in modern terms.

######  Definition 2.1.

A _geometry_ is a pair $(X,{\mathsf{G}})$, where ${\mathsf{G}}$, the _symmetry
group_ is a Lie group and $X$, the _model space_ , is a manifold endowed with
a transitive and effective action of ${\mathsf{G}}$. Recall that an action is
_effective_ if every $g\in{\mathsf{G}}{\setminus}\\{e\\}$ acts non-trivially
on $X$.

If $U\subset X$ is an open subset, we will say that a map $f\colon
U\rightarrow X$ is _locally in ${\mathsf{G}}$_ if for every connected
component $C$ of $U$, there exists $g\in{\mathsf{G}}$ such that
$f|_{C}=g|_{C}$.

For $x\in X$, the _isotropy group_ of $x$ in ${\mathsf{G}}$ is the subgroup

$\displaystyle{\mathsf{H}}=\operatorname{Stab}_{\mathsf{G}}(x)=\\{h\in{\mathsf{G}}\,|\,h(x)=x\\}.$

The isotropy group ${\mathsf{H}}$ is a closed subgroup of ${\mathsf{G}}$.
Since the action is transitive, the conjugacy class of the isotropy group does
not depend on the choice of the point $x$.

As an equivalent definition, a geometry can be defined as a pair
$({\mathsf{G}},{\mathsf{H}})$, where ${\mathsf{G}}$ is a Lie group and
${\mathsf{H}}$ is a closed subgroup of ${\mathsf{G}}$, up to conjugation. The
model space can then be reconstructed as the quotient
$X={\mathsf{G}}/{\mathsf{H}}$. From this description, we see that $X$ inherits
from ${\mathsf{G}}$ a structure of real analytic manifold such that the action
of ${\mathsf{G}}$ on $X$ is real analytic.

######  Example 2.2.

Classical examples of geometries are the _Euclidean geometry_
$\big{(}{\mathbb{R}}^{n},\operatorname{Isom}\big{(}{\mathbb{R}}^{n}\big{)}\big{)}$,
the _affine geometry_
$\big{(}{\mathbb{R}}^{n},\operatorname{Aff}\big{(}{\mathbb{R}}^{n}\big{)}\big{)}$
and the _real projective geometry_
$\big{(}{\mathbb{RP}}^{n},{\mathsf{PGL}}(n+1,{\mathbb{R}})\big{)}$. There are
many other examples which we will organize in families.

  1. 1.

A geometry is said to be _of Riemannian type_ if ${\mathsf{G}}$ acts on $X$
preserving a Riemannian metric. This happens if and only if the isotropy group
is compact. Examples are the _isotropic geometries_ (the Euclidean geometry
$\big{(}{\mathbb{R}}^{n},\operatorname{Isom}\big{(}{\mathbb{R}}^{n}\big{)}\big{)}$,
the _hyperbolic geometry_ $\big{(}{\mathbb{H}}^{n},{\mathsf{PO}}(1,n)\big{)}$
and the _spherical geometry_
$\big{(}{\mathbb{S}}^{n},{\mathsf{PO}}(n+1)\big{)}$), the geometries of
symmetric spaces and the geometries of Lie groups
($({\mathsf{G}},{\mathsf{G}})$, where $G$ acts on itself on the left).
Thurston’s eight $3$-dimensional geometries [39] are in this family.

  2. 2.

A geometry is said to be of _of pseudo-Riemannian type_ if ${\mathsf{G}}$ acts
on $X$ preserving a pseudo-Riemannian metric. Examples are many geometries
coming from the theory of relativity, such as the geometry of Minkowski space
$\big{(}{\mathbb{R}}^{n},{\mathsf{O}}(1,n-1)\ltimes{\mathbb{R}}^{n}\big{)}$,
of the anti-de Sitter space $\big{(}{\rm
AdS}^{n},{\mathsf{PO}}(2,n-1)\big{)}$, and of the de Sitter space $\big{(}{\rm
dS}^{n},{\mathsf{PO}}(1,n)\big{)}$.

  3. 3.

A geometry is said to be _of parabolic type_ if the isotropy group
${\mathsf{H}}$ is a parabolic subgroup of ${\mathsf{G}}$. Examples are the
real projective geometry
$\big{(}{\mathbb{RP}}^{n},{\mathsf{PGL}}(n+1,{\mathbb{R}})\big{)}$, the
_complex projective geometry_
$\big{(}{\mathbb{CP}}^{n},{\mathsf{PGL}}(n+1,{\mathbb{C}})\big{)}$, the
_conformal geometry_ $\big{(}{\mathbb{S}}^{n},{\mathsf{PO}}(1,n+1)\big{)}$,
the geometry of Grassmannians and of Flag manifolds.

######  Remark 2.3.

The notation $(X,{\mathsf{G}})$ for a geometry is, in most practical cases,
too cumbersome, hence we will usually denote the geometry just by $X$, when
this does not result in ambiguities. For example, we will often denote the
real projective geometry by ${\mathbb{RP}}^{n}$, instead of
$\big{(}{\mathbb{RP}}^{n},{\mathsf{PGL}}(n+1,{\mathbb{R}})\big{)}$. Similarly
for ${\mathbb{CP}}^{n}$, ${\mathbb{H}}^{n}$, ${\rm AdS}^{n}$, these symbols
will denote both the model space and the geometry.

###  2.2 Geometric manifolds

Every geometry can be used as a local model for geometric structures on
manifolds. This idea was introduced by Cartan and Ehresmann in the 1920s, and
it was made popular by Thurston around 1980, when he used it in the statement
of his geometrization conjecture (now Perelman’s theorem).

######  Definition 2.4.

Given a geometry $(X,G)$ and a manifold $M$ with $\dim(M)=\dim(X)$, an
_$(X,G)$ -structure_ on $M$ is a maximal atlas
$\mathcal{U}=\\{(U_{i},\varphi_{i})\\}$ where

  1. 1)

$\\{U_{i}\\}$ is an open cover of $M$,

  2. 2)

the functions

$\displaystyle\varphi_{i}\colon\ U_{i}\rightarrow X$

are homeomorphisms with the image, which is an open subset of $X$,

  3. 3)

the transition functions

$\displaystyle\varphi_{i}\circ\varphi_{j}^{-1}\colon\ \varphi_{j}(U_{i}\cap
U_{j})\rightarrow\varphi_{i}(U_{i}\cap U_{j})$

are locally in ${\mathsf{G}}$.

An _$(X,{\mathsf{G}})$ -manifold_ is a manifold endowed with an
$(X,{\mathsf{G}})$-structure.

An $(X,{\mathsf{G}})$-manifold is a real analytic manifold, because the
transition functions of the atlas are real analytic. Moreover, on an
$(X,{\mathsf{G}})$-manifold $M$, all the local properties of $X$ that are
preserved by ${\mathsf{G}}$ are given to $M$ by the atlas. For example, if
$(X,{\mathsf{G}})$ is of (pseudo-)Riemannian type, every
$(X,{\mathsf{G}})$-manifold inherits a (pseudo-)Riemannian metric from $X$.
Similarly, every manifold with a real or complex projective structure has a
well defined notion of projective line: some real or complex $1$-dimensional
submanifold that is mapped to a projective line by any chart. Moreover, given
$4$ points on such a projective line, it is possible to compute their cross-
ratio.

######  Example 2.5.

  1. 1.

For every geometry $(X,G)$, take $M=X$. The identity map is a global chart for
the tautological $(X,G)$-structure on $M$. Slightly more generally, if
$M\subset X$ is an open subset, again the identity map is a global chart for
an $(X,G)$-structure on $M$.

  2. 2.

Consider the Euclidean geometry:
$\big{(}{\mathbb{R}}^{n},\operatorname{Isom}\big{(}{\mathbb{R}}^{n}\big{)}\big{)}$.
Let $M$ be the torus

$\displaystyle M=T^{n}={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}.$

We can construct an atlas using the covering ${\mathbb{R}}^{n}\rightarrow M$:
every well covered open set is one of the $U_{i}$s, and every section of the
covering over such a $U_{i}$ is one of the $\varphi_{i}$s.

  3. 3.

Consider the hyperbolic geometry:
$\big{(}{\mathbb{H}}^{2},{\mathsf{PO}}(1,2)\big{)}$. Let $M$ be a closed
surface of genus $g\geq 2$. Recall that such a surface can be obtained by
gluing the sides of a $(4g)$-gon along the standard pattern
$a,b,a^{-1},b^{-1},c,d,c^{-1},d^{-1},\dots$, obtaining a cell complex with $1$
vertex, $2g$ edges and $1$ face. To put an ${\mathbb{H}}^{2}$-structure on
$M$, we first need to construct a regular $(4g)$-gon in ${\mathbb{H}}^{2}$
such that the sum of the internal angles of the polygon is $\frac{2\pi}{4g}$.
When the edges of the polygon are glued with the standard pattern, they give a
surface of genus $g$ with an ${\mathbb{H}}^{2}$-structure.

  4. 4.

Consider the real projective geometry:
$\big{(}{\mathbb{RP}}^{2},{\mathsf{PGL}}(3,{\mathbb{R}})\big{)}$. The subgroup
${\mathsf{PO}}(1,2)<{\mathsf{PGL}}(3,{\mathbb{R}})$ acts on
${\mathbb{RP}}^{2}$ preserving a disc, this is the _Klein model_ of the
hyperbolic plane: there is a ${\mathsf{PO}}(1,2)$-equivariant map
$K\colon{\mathbb{H}}^{2}\rightarrow{\mathbb{RP}}^{2}$ with image this disc.
The ${\mathbb{H}}^{2}$-structure on $M$ constructed in point (3) induces an
${\mathbb{RP}}^{2}$-structure by composing the charts with the map $K$.

  5. 5.

Consider the complex projective geometry:
$\big{(}{\mathbb{CP}}^{1},{\mathsf{PGL}}(2,{\mathbb{C}})\big{)}$. The subgroup
${\mathsf{PSL}}(2,{\mathbb{R}})<{\mathsf{PGL}}(2,{\mathbb{C}})$ acts on
${\mathbb{CP}}^{1}$ preserving the upper half plane, this is the _Poincaré
model_ of the hyperbolic plane: the connected component
${\mathsf{PO}}_{0}(2,1)$ of ${\mathsf{PO}}(2,1)$ is isomorphic to
${\mathsf{PSL}}(2,{\mathbb{R}})$ in such a way that there is a
${\mathsf{PO}}_{0}(2,1)$-equivariant map
$P\colon{\mathbb{H}}^{2}\rightarrow{\mathbb{CP}}^{1}$ with image the upper
half plane. The ${\mathbb{H}}^{2}$-structure on $M$ constructed in point (3)
induces a ${\mathbb{CP}}^{1}$-structure by composing the charts with the map
$P$.b

###  2.3 Morphisms

######  Definition 2.6.

Given two $(X,{\mathsf{G}})$-manifolds $M$, $N$, a map $f\colon M\rightarrow
N$ is an _$(X,{\mathsf{G}})$ -map_ if for every $m\in M$, there exist charts
$(U,\varphi)$ for $M$ around $m$ and $(V,\psi)$ for $N$ around $f(m)$ such
that $f(U)\subset V$ and the composition

$\displaystyle\psi\circ f\circ\varphi^{-1}\colon\
\varphi(U)\rightarrow\psi(V)$

is locally in ${\mathsf{G}}$.

The $(X,{\mathsf{G}})$-maps are always real analytic local diffeomorphisms.
Composition of $(X,{\mathsf{G}})$-maps is an $(X,{\mathsf{G}})$-map, hence we
can form a category having the $(X,{\mathsf{G}})$-manifolds as objects and the
$(X,{\mathsf{G}})$-maps as arrows.

######  Definition 2.7.

An _$(X,{\mathsf{G}})$ -isomorphism_ is a diffeomorhism which is also an
$(X,{\mathsf{G}})$-map. An _$(X,{\mathsf{G}})$ -automorphism_ is an
isomorphism between an $(X,{\mathsf{G}})$-manifold and itself.

Notice that the inverse of an $(X,G)$-isomorphism is automatically an
$(X,{\mathsf{G}})$-map. If $M$ is an $(X,{\mathsf{G}})$-manifold, we will
denote its group of automorphisms by

$\displaystyle\operatorname{Aut}_{(X,{\mathsf{G}})}(M)=\\{f\colon M\rightarrow
M\,|\,f\text{ is an }(X,{\mathsf{G}})\text{-isomorphism}\\}.$

These groups can sometimes be understood:

######  Proposition 2.8.

$\displaystyle\operatorname{Aut}_{(X,{\mathsf{G}})}(X)={\mathsf{G}}.$

More generally, if $U\subset X$ is open, then

$\displaystyle\operatorname{Aut}_{(X,{\mathsf{G}})}(U)=\\{g\in{\mathsf{G}}\,|\,g(U)=U\\}.$

###  2.4 Kleinian geometric structures

The following proposition gives a tool that can be used to construct many
interesting manifolds carrying geometric structures.

######  Proposition 2.9.

Let $M$ be an $(X,{\mathsf{G}})$-manifold, and
$\Gamma<\operatorname{Aut}_{(X,{\mathsf{G}})}(M)$ be a subgroup acting
properly discontinuously and freely on $M$. Then $M/\Gamma$ is a manifold, and
there exists a unique $(X,{\mathsf{G}})$-structure on $M/\Gamma$ such that the
quotient $M\rightarrow M/\Gamma$ is an $(X,{\mathsf{G}})$-map.

Conversely, let $M$ be an $(X,{\mathsf{G}})$-manifold, and let
$\pi\colon\bar{M}\rightarrow M$ be a covering map. Then there exists a unique
$(X,{\mathsf{G}})$-structure on $\bar{M}$ such that $\pi$ is an
$(X,{\mathsf{G}})$-map.

######  Definition 2.10.

Let $\Gamma<{\mathsf{G}}$ be a discrete subgroup. A  _domain of discontinuity_
for $\Gamma$ is an open subset $\Omega\subset X$ that is $\Gamma$-invariant
and such that $\Gamma$ acts properly discontinuously on $\Omega$.

By applying Proposition 2.9, if $\Omega$ is a domain of discontinuity for
$\Gamma$ and $\Gamma$ acts freely on $\Omega$ (which is always true if
$\Gamma$ is torsion-free), then the quotient $\Omega/\Gamma$ is a manifold
with an $(X,{\mathsf{G}})$-structure.

######  Definition 2.11.

The geometric structures of the form $\Omega/\Gamma$ described above are
called _Kleinian $(X,{\mathsf{G}})$-structures_.

The theory of Anosov representations, introduced by Labourie [30] and
Guichard–Wienhard [24], gives methods for constructing interesting Kleinian
geometric structures. We will not give here the complete definition of Anosov
representations, we will only recall some of their properties. Let
${\mathsf{G}}$ be a semi-simple Lie group and $\Gamma$ be a Gromov-hyperbolic
group. Anosov representations $\rho\colon\Gamma\rightarrow{\mathsf{G}}$ are
defined with reference to a parabolic subgroup
${\mathsf{P}}\subset{\mathsf{G}}$, they will be called ${\mathsf{P}}$-Anosov
representations. One property of a ${\mathsf{P}}$-Anosov representation $\rho$
is the existence of a $\rho$-equivariant map

$\displaystyle\xi\colon\
\partial_{\infty}\Gamma\rightarrow{\mathsf{G}}/{\mathsf{P}}$

which must, by definition, satisfy some special properties. Here with
$\partial_{\infty}\Gamma$ we denote the boundary at infinity of $\Gamma$,
defined by Gromov [22] for hyperbolic groups. The ${\mathsf{P}}$-Anosov
representations form an open subset of the character variety (see Section 3):

$\displaystyle{\mathsf{P}}\text{-}\operatorname{Anosov}(\pi_{1}(S),{\mathsf{G}})\subset{\mathcal{X}}(\pi_{1}(S),{\mathsf{G}}).$

When $(X,{\mathsf{G}})$ is a geometry of parabolic type, whose isotropy group
might be different from ${\mathsf{P}}$, there is a very rich theory giving
sufficient conditions for a ${\mathsf{P}}$-Anosov representation to admit a
domain of discontinuity $\Omega\subset X$, which is, in the best cases, co-
compact. The domain $\Omega$ is defined using the map $\xi$. This theory was
founded by Guichard–Wienhard [24], and was improved and extended by
Kapovich–Leeb–Porti [29]. For an example of how this works, see Section 8.2.
In this way, it is possible to construct many examples of Kleinian geometric
structures on closed manifolds for geometries of parabolic type.

One limitation of this method is that even if we construct an
$(X,{\mathsf{G}})$-manifold $M=\Omega/\rho(\Gamma)$, we have no idea what the
topology of $M$ is. Other techniques are needed to get a good understanding of
these geometric manifolds, see for example Theorem 8.5.

###  2.5 Developing maps and holonomies

The Kleinian geometric structures are the easiest to understand, but not all
geometric structures are Kleinian. To work with general geometric structures,
we introduce here the tools of developing maps and holonomy representations.

######  Lemma 2.12.

Let $N$ be a simply-connected manifold $(X,{\mathsf{G}})$-manifold. Then $N$
admits a global $(X,{\mathsf{G}})$-map

$\displaystyle D\colon\ N\rightarrow X$

unique up to post-composition by an element of ${\mathsf{G}}$.

###### Proof.

Choose a point $n_{0}\in N$, and a chart $(U_{0},\varphi_{0})$ around $N$. We
will extend $\varphi_{0}\colon U_{0}\rightarrow X$ to a map $D$ defined on $N$
such that $D|_{U_{0}}=\varphi$. For every point $n\in N$, we will define the
value $D(n)\in X$ in the following way. Choose a path
$\gamma\colon[0,1]\rightarrow N$ such that $\gamma(0)=n_{0}$, $\gamma(1)=n$.
We can find charts $(U_{1},\varphi_{1}),\dots,(U_{k},\varphi_{k})$ and points
$t_{0},\dots,t_{k},s_{0},\dots,s_{k}\in[0,1]$ such that

  1. 1)

$0=t_{0}<t_{1}<s_{0}<t_{2}<s_{1}<t_{3}<\dots<s_{k-2}<t_{k}<s_{k-1}<s_{k}=1$,

  2. 2)

$\gamma([t_{0},s_{0}))=U_{0}\cap\gamma([0,1])$,

  3. 3)

$\gamma((t_{i},s_{i}))=U_{i}\cap\gamma([0,1])$,

  4. 4)

$\gamma((t_{k},s_{k}])=U_{k}\cap\gamma([0,1])$.

The path $\gamma((t_{i},s_{i-1}))$ is contained in a connected component
$C_{i}$ of $U_{i-1}\cap U_{i}$. There exists a $g_{i}\in{\mathsf{G}}$ such
that
$\varphi_{i-1}\circ\varphi_{i}^{-1}|_{\varphi_{i}(C_{i})}=g_{i}|_{\varphi_{i}(C_{i})}$.
We define

$\displaystyle D(n)=g_{1}\circ g_{2}\circ\dots\circ g_{k}\circ\varphi_{k}(n).$

Now it is necessary to show that the value $D(n)$ is independent on the choice
of the charts $(U_{i},\varphi_{i})$, with $i\geq 1$. This is an application of
the principle of unique analytic continuation. Then, we need to show that
$D(n)$ does not depend on the choice of the curve $\gamma$. This comes from
the fact that $M$ is simply-connected, hence every other curve
$\gamma^{\prime}$ is homotopic to $\gamma$ relatively to the end-points. This
defines a global $(X,{\mathsf{G}})$-map $D$, which depends only on the choice
of $(U_{0},\varphi_{0})$. The principle of unique analytic continuation gives
the uniqueness of $D$ up to an element of ${\mathsf{G}}$. ∎

Given an $(X,{\mathsf{G}})$-manifold $M$, we denote its universal covering by
$\widetilde{M}$. By Proposition 2.9, $\widetilde{M}$ inherits an
$(X,{\mathsf{G}})$-structure from $M$. Since $\widetilde{M}$ is simply-
connected, there is a global $(X,{\mathsf{G}})$-map

$\displaystyle D\colon\ \widetilde{M}\rightarrow X$

unique up to post-composition by an element of ${\mathsf{G}}$.

######  Definition 2.13.

The map $D$ is called the _developing map_ of the $(X,{\mathsf{G}})$-manifold
$M$.

The fundamental group $\pi_{1}(M)$ acts on $\widetilde{M}$ by deck
transformations. This action preserves the $(X,{\mathsf{G}})$-structure on
$\widetilde{M}$, hence we have the inclusion
$\pi_{1}(M)<\operatorname{Aut}_{(X,{\mathsf{G}})}(\widetilde{M})$. The
composition of an element $\gamma\in\pi_{1}(M)$ with the developing map $D$ is
again a developing map, hence there exists an element
$h(\gamma)\in{\mathsf{G}}$ such that

$\displaystyle D\circ\gamma=h(\gamma)\circ D.$

The map $h\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ is a group homomorphism,
and the formula above tells us that the developing map is $h$-equivariant.

######  Definition 2.14.

The group homomorphism $h$ is called the _holonomy representation_ of the
$(X,{\mathsf{G}})$-manifold $M$. The pair $(D,h)$ is called the _developing
pair_ of the $(X,{\mathsf{G}})$-manifold $M$.

######  Example 2.15.

In the case of a Kleinian geometric structure $M=\Omega/\Gamma$, for some
$\Omega\subset X$ and $\Gamma<{\mathsf{G}}$, the developing map is a covering

$\displaystyle D\colon\ \widetilde{M}\rightarrow\Omega$

and the holonomy representation is a homomorphism

$\displaystyle h\colon\ \pi_{1}(M)\rightarrow\Gamma$

such that $\ker(h)=\pi_{1}(\Omega)$.

If we change the developing map by post-composing it with an element
$g\in{\mathsf{G}}$, the holonomy representation changes by conjugation by $g$.
In other words, the group ${\mathsf{G}}$ acts on the developing pairs in the
following way:

$\displaystyle g\cdot(D,h)=\big{(}g\circ D,ghg^{-1}\big{)}.$

The developing pair $(D,h)$ of the $(X,{\mathsf{G}})$-manifold $M$ is well
defined up to this action of ${\mathsf{G}}$. The developing pair completely
determines the $(X,{\mathsf{G}})$-structure on $M$, as we will now see.

######  Definition 2.16.

Let $M$ be a manifold without a specified $(X,{\mathsf{G}})$-structure. We
will say that a pair $(D,h)$ is an $(X,{\mathsf{G}})$-_developing pair_ for
$M$ if

  1. 1)

$h$ is a representation $h\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$,

  2. 2)

$D$ is an $h$-equivariant local diffeomorphism.

Given an $(X,{\mathsf{G}})$-developing pair $(D,h)$ for $M$, we can construct
an $(X,{\mathsf{G}})$-structure in the following way: let $U$ be a simply-
connected open subset of $M$, and let $s\colon U\rightarrow\widetilde{M}$ be a
section of the universal covering. Assume that $U$ is small enough, so that
$s(U)$ is an open subset where $D$ is a diffeomorphism. Then $(U,D\circ s)$ is
a chart, and the collection of all the charts of this type forms an atlas for
a $(X,{\mathsf{G}})$-structure on $M$. This is the unique
$(X,{\mathsf{G}})$-structure on $M$ with developing pair $(D,h)$.

###  2.6 Parameter spaces

Given a fixed manifold $M$, we want to define a parameter space of all
$(X,{\mathsf{G}})$-structures on $M$.

######  Definition 2.17.

We will say that two $(X,{\mathsf{G}})$-structures on $M$ are _isotopic_ if
there exists a diffeomorphism $f\colon M\rightarrow M$ isotopic to the
identity, which is an $(X,{\mathsf{G}})$-isomorphism between the first
structure and the second.

We will denote by ${\mathcal{D}}_{(X,{\mathsf{G}})}(M)$ the set of all the
$(X,{\mathsf{G}})$-structures on $M$ up to isotopy. The topology on
${\mathcal{D}}_{(X,{\mathsf{G}})}(M)$ is given by the $C^{\infty}$-topology on
the corresponding developing maps. Let’s see this in more detail.

Consider the space $\operatorname{Dev}_{(X,{\mathsf{G}})}(M)$ of all
$(X,{\mathsf{G}})$-developing pairs $(D,h)$ for $M$. This space is endowed
with the $C^{\infty}$-topology on the developing maps. Given a sequence of
developing pairs $(D_{k},h_{k})$, it is easy to check that if the sequence
$(D_{k})$ converges to $D_{0}$ in the $C^{\infty}$-topology, then the sequence
$(h_{k})$ converges point-wise to $h_{0}$.

Choose a point $m\in M$, and consider the group
$\operatorname{Diffeo}_{0}(M,m)$ of all diffeomorphisms of $M$ that fix the
point $m$ and are isotopic to the identity. Every element of this group can be
lifted in a unique way to a diffeomorphism of $\widetilde{M}$ that fixes the
fiber over $m$. In this way, $\operatorname{Diffeo}_{0}(M,m)$ acts on
$\widetilde{M}$. The group $\operatorname{Diffeo}_{0}(M,m)\times G$ acts on
$\operatorname{Dev}_{(X,{\mathsf{G}})}(M)$ in the following way:

$\displaystyle(f,g)\cdot(D,h)=\big{(}g\circ D\circ f,ghg^{-1}\big{)}.$

We have that

$\displaystyle{\mathcal{D}}_{(X,{\mathsf{G}})}(M)=\operatorname{Dev}_{(X,{\mathsf{G}})}(M)/\operatorname{Diffeo}_{0}(M,m)\times
G.$

In this way, the parameter space of $(X,{\mathsf{G}})$-structures on $M$
inherits the quotient topology.

##  3 Representations and flat bundles

In this section, we review the correspondence between conjugacy classes of
representations of the fundamental group of a manifold and isomorphism classes
of flat bundles. We tried to keep the required Lie theory to a minimum,
anyway, for all the Lie-theoretical notions, the reader can refer to [25, 28,
36].

###  3.1 Character varieties

Let $\Gamma$ be a finitely generated group. Here, the most interesting case is
when $\Gamma$ is the fundamental group of a closed manifold, but for the
moment it can be arbitrary. Let ${\mathsf{G}}$ be a reductive Lie group with
Lie algebra ${\mathfrak{g}}$. We will denote by
$\operatorname{Hom}(\Gamma,{\mathsf{G}})$ the set of all representations
(i.e., group homomorphisms) of $\Gamma$ in ${\mathsf{G}}$, endowed with the
topology of point-wise convergence of representations.

######  Definition 3.1.

A _reductive representation_ of $\Gamma$ in ${\mathsf{G}}$ is a representation
$\rho\colon\Gamma\rightarrow{\mathsf{G}}$ such that the induced action on
${\mathfrak{g}}$ given by the adjoint representation is completely reducible.

######  Example 3.2.

If ${\mathsf{G}}$ is a linear group, then $\rho$ is reductive if and only if
it is completely reducible.

We will denote by $\operatorname{Hom}^{*}(\Gamma,{\mathsf{G}})$ the subspace
of all reductive representations of $\Gamma$ in ${\mathsf{G}}$. The group
${\mathsf{G}}$ acts on $\operatorname{Hom}^{*}(\Gamma,{\mathsf{G}})$ by
conjugation, and the action is proper. We will denote the quotient by this
action by

$\displaystyle{\mathcal{X}}(\Gamma,{\mathsf{G}})=\operatorname{Hom}^{*}(\Gamma,{\mathsf{G}})/{\mathsf{G}}.$

######  Definition 3.3.

The space ${\mathcal{X}}(\Gamma,{\mathsf{G}})$ is called the _character
variety_ of $\Gamma$ in ${\mathsf{G}}$.

Character varieties are Hausdorff topological spaces. They are in general not
manifolds since they can have singularities, but they are always locally
contractible.

When $\Gamma=\pi_{1}(S)$, for a closed orientable surface $S$ of genus $g\geq
2$ and ${\mathsf{G}}$ is a real Lie group, there are results describing the
topology of some connected components of the character varieties.

######  Example 3.4.

  1. 1.

When ${\mathsf{G}}={\mathsf{PSL}}(2,{\mathbb{R}})$, Goldman [18] used a
topological invariant, the Euler number, to classify the connected components
of the character variety: it has $4g-3$ connected components corresponding to
the values of the Euler number from $2-2g$ to $2g-2$. Moreover Goldman proved
that a representation in ${\mathsf{PSL}}(2,{\mathbb{R}})$ is discrete and
faithful if and only if it has Euler number $\pm(2g-2)$. Such representations
are called _Fuchsian representations_ , and they form two connected components
of the character variety, each of whom is a copy of the Teichmüller space
${\mathcal{T}}(S)$ of the surface. Hitchin [26] described the topology of all
the connected components with non-zero Euler number.

  2. 2.

Similarly, when ${\mathsf{G}}={\mathsf{PGL}}(2,{\mathbb{R}})$, the set of
discrete and faithful representations, again called Fuchsian representations,
forms a connected component of the character variety which is a copy of the
Teichmüller space ${\mathcal{T}}(S)$ of the surface. This component is then
homeomorphic to ${\mathbb{R}}^{6g-6}$, and it is also denoted by
$\operatorname{Hit}(S,2)$, see below.

  3. 3.

Consider now the case when ${\mathsf{G}}={\mathsf{PGL}}(n,{\mathbb{R}})$. A
_Fuchsian representation_ in ${\mathsf{PGL}}(n,{\mathbb{R}})$ is defined as
the composition of a Fuchsian representation in
${\mathsf{PGL}}(2,{\mathbb{R}})$ with the irreducible representation
${\mathsf{PGL}}(2,{\mathbb{R}})\rightarrow{\mathsf{PGL}}(n,{\mathbb{R}})$.
This construction gives an embedding of the Teichmüller space
${\mathcal{T}}(S)$ in
${\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(n,{\mathbb{R}}))$, whose image is
called the _Fuchsian locus_. Hitchin [27] proved that the connected component
of the character variety containing the Fuchsian locus is homeomorphic to
${\mathbb{R}}^{(n^{2}-1)(2g-2)}$. This component is called the _Hitchin
component_ , and denoted by $\operatorname{Hit}(S,n)$. Labourie [30] described
the geometry of the representations in this component. The Hitchin components
share many properties with the Teichmüller spaces, hence they are sometimes
called _higher Teichmüller spaces_ , and they give the name to higher
Teichmüller theory.

  4. 4.

Hitchin [27] defined special components in the character varieties of all
split real simple Lie groups ${\mathsf{G}}$. They are homeomorphic to
${\mathbb{R}}^{\dim({\mathsf{G}})(2g-2)}$.

  5. 5.

The Euler number can be generalized to representations into all Lie groups of
Hermitian type, in this case it is called the Toledo number, see Toledo [40].
Representations with maximal value of the Toledo number are called _maximal
representations_ , and they form a union of connected components in the
corresponding character varieties. This is another way to generalize Fuchsian
representations to higher rank Lie groups. For
${\mathsf{G}}={\mathsf{Sp}}(4,{\mathbb{R}})$ and
${\mathsf{PSp}}(4,{\mathbb{R}})$, an explicit description of the topology of
the maximal components was determined in a joint work with Brian Collier [1].

When ${\mathsf{G}}$ is a complex Lie group, we don’t have explicit
descriptions of connected components of character varieties. But there are at
least some special open subsets that are particularly interesting. For
example, in the character variety of
${\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(2,{\mathbb{C}}))$ we have the open
subset of _quasi-Fuchsian representations_ , denoted by
$\operatorname{QFuch}(S)$. Quasi-Fuchsian representations can be defined as
those representations whose action on ${\mathbb{CP}}^{1}$ is topologically
conjugate to the action of a Fuchsian representation on ${\mathbb{CP}}^{1}$.
The open subset $\operatorname{QFuch}(S)$ is homeomorphic to
${\mathbb{R}}^{12g-12}$.

Hitchin components generalize Teichmüller spaces to higher rank Lie groups. In
a similar way, there are some special open subsets of the character varieties
of a simple complex Lie group ${\mathsf{G}}$ that generalize the space of
quasi-Fuchsian representations. We will call it the space of _quasi-Hitchin
representations_. To define them, consider the open subset

$\displaystyle{\mathsf{B}}\text{-}\operatorname{Anosov}(\pi_{1}(S),{\mathsf{G}})\subset{\mathcal{X}}(\pi_{1}(S),{\mathsf{G}})$

consisting of all ${\mathsf{B}}$-Anosov representations, where ${\mathsf{B}}$
is the Borel subgroup of ${\mathsf{G}}$. The space of quasi-Hitchin
representations is then defined as the connected component of
${\mathsf{B}}\text{-}\operatorname{Anosov}(\pi_{1}(S),{\mathsf{G}})$
containing the Hitchin component of the split real form of ${\mathsf{G}}$. For
the group ${\mathsf{PGL}}(n,{\mathbb{C}})$, we will denote the space of quasi-
Hitchin representations by $\operatorname{QHit}(S,n)$.

###  3.2 Flat bundles

Let $M$ be a manifold, and let $X$ be a manifold endowed with an effective
action of ${\mathsf{G}}$.

######  Definition 3.5.

A _fiber bundle_ on $M$ with _structure group_ ${\mathsf{G}}$ and _fiber_ $X$
(also called a _${\mathsf{G}}$ -bundle_ with fiber $X$) is a manifold $B$ with
a smooth map $\pi\colon B\rightarrow M$ and a maximal ${\mathsf{G}}$-atlas for
$\pi$. Recall that a _${\mathsf{G}}$ -atlas_ is a set of charts
$\\{(U_{i},\varphi_{i})\\}$ where the $U_{i}$s are open subsets of $M$ which
cover $M$, and

$\displaystyle\varphi_{i}\colon\ \pi^{-1}(U_{i})\rightarrow U_{i}\times X$

is a diffeomorphism that intertwines $\pi|_{U_{i}}$ and the projection on the
first factor. The $\varphi_{i}$s must be _${\mathsf{G}}$ -compatible_ in the
following sense: the maps

$\displaystyle\varphi_{i}\circ\varphi_{j}^{-1}\colon\ (U_{i}\cap U_{j})\times
X\rightarrow(U_{i}\cap U_{j})\times X$

are of the form $\varphi_{i}\circ\varphi_{j}^{-1}(m,x)=(m,t_{ij}(m)x)$, where
$t_{ij}(m)\in{\mathsf{G}}$. The functions

$\displaystyle t_{ij}\colon\ U_{i}\cap U_{j}\rightarrow{\mathsf{G}}$

are called the _transition functions_ of the atlas.

It is interesting to remark that the bundle is determined up to isomorphism by
the transition functions, and that the transition functions don’t depend at
all on the space $X$. This has the following consequence: if $X,Y$ are
manifolds with effective actions of ${\mathsf{G}}$, then a bundle $B$ with
structure group ${\mathsf{G}}$ and fiber $X$, determines a bundle $B(Y)$ with
the same structure group and fiber $Y$. The bundle $B(Y)$ is defined as the
bundle with fiber $Y$ having the same transition functions as the bundle $B$.

More generally, given a group homomorphism
$q\colon{\mathsf{G}}\rightarrow{\mathsf{H}}$, assume that $X$ is a manifold
with an effective ${\mathsf{G}}$-action, $Y$ is a manifold with an effective
${\mathsf{H}}$-action, and $B$ is a ${\mathsf{G}}$-bundle with fiber $X$. We
can apply the construction given above by composing the transition functions
of $B$ with the homomorphism $q$. This produces an ${\mathsf{H}}$-bundle
$B(Y)$ with fiber $Y$.

######  Definition 3.6.

The bundle $B(Y)$ is called the _associated bundle_ to $B$ with fiber $Y$.

######  Example 3.7.

  1. 1.

The most important example is the special case when $X={\mathsf{G}}$, acting
on itself on the left. A bundle with structure group and fiber ${\mathsf{G}}$
is called a _principal_ ${\mathsf{G}}$-bundle.

  2. 2.

Another fundamental example is the case when the Lie group ${\mathsf{G}}$ is a
_linear group_ , i.e., when it can be embedded as a Lie subgroup of
${\mathsf{GL}}(n,{\mathbb{R}})$ or ${\mathsf{GL}}(n,{\mathbb{C}})$. In this
case it admits an effective linear action on $V={\mathbb{R}}^{n}$ or
${\mathbb{C}}^{n}$, and a ${\mathsf{G}}$-bundle with fiber $V$ is called a
_vector bundle_. Starting from every bundle $B$ with structure group
${\mathsf{G}}$ and some fiber, we can construct the associated vector bundle
$B(V)$.

  3. 3.

Similarly, a _projective group_ is a Lie group ${\mathsf{G}}$ that can be
embedded as a Lie subgroup of ${\mathsf{PGL}}(n+1,{\mathbb{R}})$ or
${\mathsf{PGL}}(n+1,{\mathbb{C}})$. In this case it admits an effective
projective action on $P={\mathbb{RP}}^{n}$ or ${\mathbb{CP}}^{n}$, and a
${\mathsf{G}}$-bundle with fiber $P$ is called a _projective bundle_. Starting
from every bundle $B$ with structure group ${\mathsf{G}}$ and some fiber, we
can construct the associated projective bundle $B(P)$.

There is a close relationship between vector bundles and projective bundles.
Assume that ${\mathsf{G}}$ is a linear group, $X={\mathbb{R}}^{n+1}$ or
${\mathbb{C}}^{n+1}$, ${\mathsf{H}}$ is the corresponding projectivized group
and $Y={\mathbb{RP}}^{n}$ or ${\mathbb{CP}}^{n}$. From every vector bundle $E$
with structure group ${\mathsf{G}}$, we can construct the associated
projective bundle $E(Y)$. We will denote $E(Y)$ by ${\mathbb{P}}(E)$, the
_projectivized bundle_ of $E$.

######  Definition 3.8.

A _flat structure_ on a ${\mathsf{G}}$-bundle $B$ is an atlas of $B$
satisfying the additional condition that all transition functions are locally
constant, and maximal among all atlases satisfying this additional condition.
A bundle with a flat structure is called a _flat bundle_ , or a _local
system_.

The flat structure is just a special atlas, hence, as explained above, it does
not depend on the fiber. It is thus possible to construct _associated bundles_
and to replace a flat bundle $B$ with fiber $X$ by a flat bundle $B(Y)$ with
fiber $Y$.

If ${\mathsf{G}}$ is a linear group acting on a vector space
$V={\mathbb{R}}^{n}$ or ${\mathbb{C}}^{n}$, starting from every flat bundle
$B$ with fiber $X$, we can change fiber and construct the vector bundle
$B(V)$, with a flat structure. A flat structure on a vector bundle can be
described by a _flat connection_ , i.e., a connection with vanishing curvature
form. This description is useful for doing computations.

A flat bundle $B$ with fiber $X$ has a well defined foliation, called the
_parallel foliation_ , that can be described in local charts: in
$\pi^{-1}(U_{i})$, for every $x\in X$ there is a local leaf given by
$\varphi_{i}^{-1}(U_{i}\times\\{x\\})$. The local foliations defined by the
different charts all match up, giving rise to a global foliation.

A _local parallel section_ of a flat bundle is a section $s\colon U\rightarrow
B$ defined on an open subset $U$, which is locally constant when restricted to
every chart of the flat structure. In other words, it is a section whose image
is contained in a leaf of the parallel foliation. Similarly, given a curve
$\gamma\colon[0,1]\rightarrow M$, a  _parallel section along_ $\gamma$ is a
section along $\gamma$ which is locally constant in the charts.

Using parallel sections, we can define the _parallel transport operator_ along
a curve $\gamma\colon[0,1]\rightarrow M$: it is an operator

$\displaystyle P_{\gamma}\colon\
\pi^{-1}(\gamma(0))\rightarrow\pi^{-1}(\gamma(1))$

defined in the following way: given $x_{0}\in\pi^{-1}(\gamma(0))$, there
exists a unique parallel section $s$ along $\gamma$ such that $s(0)=x_{0}$. We
define $P_{\gamma}(x_{0})=s(1)$. The parallel transport $P_{\gamma}$ only
depends on the homotopy class of $\gamma$ relative to the end points.

###  3.3 Monodromy

Let $\pi\colon B\rightarrow M$ be a flat ${\mathsf{G}}$-bundle with fiber $X$.
Given a base point $m_{0}\in M$, we can use a chart to identify the fiber
$\pi^{-1}(m_{0})$ with $X$. If we change chart, this identification changes by
the action of an element of ${\mathsf{G}}$. Now the parallel transport
$P_{\gamma}$ along a loop $\gamma$ based at $m_{0}$ is a map $P_{\gamma}\colon
X\rightarrow X$, which agrees with the action of an element of ${\mathsf{G}}$,
hence we can write $P_{\gamma}\in{\mathsf{G}}$. Since $P_{\gamma}$ only
depends on the homotopy class of $\gamma$, we get a map

$\displaystyle P\colon\ \pi_{1}(M,m_{0})\rightarrow{\mathsf{G}}.$

This map behaves well under composition of loops, hence it is a
representation. If we change the chart around $m_{0}$, the representation
changes by conjugation by an element of ${\mathsf{G}}$.

######  Definition 3.9.

The representation $P$ is called the _monodromy representation_ of the bundle.
We will say that a flat bundle is _reductive_ if the monodromy representation
is reductive.

We will denote by $\operatorname{Flat}(M,G,X)$ the space of all reductive flat
$G$-bundles with fiber $X$ up to isomorphism. If $X$, $Y$ are manifolds with
effective actions of ${\mathsf{G}}$, the associated bundle construction gives
a natural bijection
$\operatorname{Flat}(M,G,X)\rightarrow\operatorname{Flat}(M,G,Y)$. Hence, we
can suppress the $X$ in the notation, and consider the space
$\operatorname{Flat}(M,G)$, parametrizing reductive flat
${\mathsf{G}}$-bundles with any fixed fiber $X$.

The monodromy representation gives a map

$\displaystyle{\mathcal{P}}\colon\
\operatorname{Flat}(M,G)\rightarrow{\mathcal{X}}(\pi(M),G).$

######  Proposition 3.10.

The map ${\mathcal{P}}$ is a bijection.

###### Proof.

The inverse map is given by the following construction. Let
$\rho\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ be a representation. This gives
an action of $\pi_{1}(M)$ on $\widetilde{M}\times X$, acting on the first
factor by deck transformations and on the second factor via $\rho$. This
action is properly discontinuous and free because the action on the first
factor has these properties. Hence, we can construct the manifold

$\displaystyle X_{\rho}=\big{(}\widetilde{M}\times X\big{)}/\pi_{1}(M).$

The projection on the first factor induces a map $p\colon X_{\rho}\rightarrow
M$ which turns $X_{\rho}$ into a ${\mathsf{G}}$-fiber bundle with fiber $X$.
Moreover, the product $\widetilde{M}\times X$ induces a flat structure on the
bundle. It is easy to show that the flat bundle $X_{\rho}$ has monodromy
$\rho$, and that any other flat ${\mathsf{G}}$-bundle with fiber $X$ and
monodromy $\rho$ is isomorphic to $X_{\rho}$. ∎

##  4 The graph of a geometric structure

In this section, we will see how geometric structures correspond to flat
bundles with a transverse section. The flat bundle encodes the holonomy
representation of the geometric structure, and the transverse section encodes
the developing map. This construction is described in detail in Goldman’s
notes [19].

###  4.1 Sections and equivariant maps

Let $\rho\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ be a representation, and
consider the space $\operatorname{Equiv}(\rho,X)$ of smooth $\rho$-equivariant
maps from the universal covering $\widetilde{M}$ to $X$, endowed with the
$C^{\infty}$-topology. Let $B$ be the a flat bundle over $M$ with fiber $X$
and holonomy $\rho$, and consider the space $\Gamma(M,B)$ of smooth sections
of $B$, endowed with the $C^{\infty}$-topology.

######  Proposition 4.1.

There is a natural homeomorphism between $\Gamma(M,B)$ and
$\operatorname{Equiv}(\rho,X)$.

###### Proof.

Recall first that $B$ is isomorphic to $X_{\rho}$, the bundle defined in the
proof of Proposition 3.10. From the construction of $X_{\rho}$, we can see
that the pull-back of $X_{\rho}$ to $\widetilde{M}$ is isomorphic to a product
$\widetilde{M}\times X$, with the product flat structure.

A section $s\in\Gamma(M,X_{\rho})$ can be pulled back to a section
$\widetilde{s}$ of $\widetilde{M}\times X$. A section of a product bundle is
just a map $\widetilde{s}\colon\widetilde{M}\rightarrow X$. The fact that
$\widetilde{s}$ is a pull-back tells us that this map is $\rho$-equivariant.
This gives a map between $\Gamma(M,X_{\rho})$ and
$\operatorname{Equiv}(\rho,X)$.

To find the inverse of this map, just notice that a $\rho$-equivariant map
$f\colon\widetilde{M}\rightarrow X$ is a section of the product bundle
$\widetilde{M}\times X$. The fact that $f$ is $\rho$-equivariant implies that
it passes to the quotient, giving a section $[f]$ of $X_{\rho}$.

To check that the maps are continuous, we can work locally on small open sets
of $M$ which are well covered by the universal covering. ∎

###  4.2 Transverse sections

Let $\rho\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ be a representation, and $B$
be the flat bundle over $M$ with fiber $X$ and holonomy $\rho$.

######  Definition 4.2.

A section $s\in\Gamma(M,B)$ is _transverse_ if it is transverse to the
parallel foliation of the bundle.

######  Proposition 4.3.

A section $s\in\Gamma(M,B)$ is transverse if and only if the corresponding
$\rho$-equivariant map is

  1. $1)$

an immersion if $\dim(M)\leq\dim(X)$,

  2. $2)$

a submersion if $\dim(M)\geq\dim(X)$.

In particular, if $\dim(M)=\dim(X)$, then $s$ is transverse if and only if the
corresponding $\rho$-equivariant map is a local diffeomorphism.

###### Proof.

Let $f\colon\widetilde{M}\rightarrow X$ be the corresponding
$\rho$-equivariant map, and let $\pi\colon\widetilde{M}\rightarrow M$ denote
the universal covering. Let $v\in T_{x}\widetilde{M}$, and
$v^{\prime}=d\pi[v]\in T_{\pi(x)}M$. Then the differential of $f$ vanishes at
$v$ if and only if the differential of $s$ at $v^{\prime}$ is tangent to the
parallel foliation. ∎

######  Definition 4.4.

If $\dim(M)=\dim(X)$, a _graph of an $(X,{\mathsf{G}})$-structure_ is a pair
$(B,s)$ where $B$ is a flat bundle over $M$ with fiber $X$ and
$s\in\Gamma(M,B)$ is a transverse section.

Graphs of $(X,{\mathsf{G}})$-structures correspond to
$(X,{\mathsf{G}})$-developing pairs, which determine
$(X,{\mathsf{G}})$-structures on $M$.

Let’s see this more explicitly in the case of real or complex projective
structures. Let ${\mathbb{K}}$ be ${\mathbb{R}}$ or ${\mathbb{C}}$, and
consider the geometry ${\mathbb{KP}}^{n}$. Given a representation
$\rho\colon\pi_{1}(M)\rightarrow{\mathsf{PGL}}(n+1,{\mathbb{K}})$, we want to
construct a ${\mathbb{KP}}^{n}$-structure on $M$ with holonomy $\rho$. To do
this, we need to consider the flat bundle $B$ over $M$ with fiber
${\mathbb{KP}}^{n}$ and holonomy $\rho$, and construct a transverse section of
$B$.

This becomes more concrete when $\rho$ lifts to a representation
$\bar{\rho}\colon\pi_{1}(M)\rightarrow{\mathsf{GL}}(n+1,{\mathbb{K}})$. In
this case, there is a flat vector bundle $E$ with holonomy $\bar{\rho}$ such
that the projectivized bundle ${\mathbb{P}}(E)$ is isomorphic to $B$. The flat
structure on $E$ is described by a flat connection $\nabla$. A section of $B$
is the same thing as a line subbundle of $E$. The next proposition shows how
it is possible to verify whether a section of $B$ is transverse with a
computation in local coordinates involving the derivatives with reference to
the flat connection on $E$.

######  Proposition 4.5.

Let $E$ be a flat vector bundle of rank $n+1$ over $M$, and $L\subset E$ be a
line subbundle. Then $L$ is a transverse section of ${\mathbb{P}}(E)$ if and
only if for every $m\in M$ there exists a coordinate neighborhood $U$ of $m$
$($with coordinates $x_{1},\dots,x_{k}$, where $k=\dim(M))$ and a local non-
vanishing section $s\colon U\rightarrow L$ such that the local vector fields

$\displaystyle s,\nabla_{\\!\\!\\!\frac{\partial}{\partial
x_{1}}}s,\dots,\nabla_{\\!\\!\\!\frac{\partial}{\partial x_{k}}}s$

satisfy one of the following conditions:

  1. $1)$

are linearly independent on $U$ if $\dim(M)\leq n$,

  2. $2)$

span every fiber over $U$ if $\dim(M)\geq n$.

###  4.3 The holonomy map

If ${\mathsf{G}}$ is reductive, we can consider the subspace

$\displaystyle{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(M)\subset{\mathcal{D}}_{(X,{\mathsf{G}})}(M)$

of all $(X,{\mathsf{G}})$-structures on $M$ with reductive holonomy. This
subspace has a natural map to the character variety, given by the holonomy
representation:

$\displaystyle\mathrm{Hol}\colon\
{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(M)\rightarrow{\mathcal{X}}(\pi_{1}(M),{\mathsf{G}}).$

######  Theorem 4.6 (Thurston’s holonomy principle).

If $M$ is a closed manifold, the map $\mathrm{Hol}$ is open and it has
discrete fiber.

###### Proof.

See Goldman [19]. The openness of the map $\mathrm{Hol}$ can be proved easily
using graphs of $(X,{\mathsf{G}})$-structures. ∎

When $M$ is closed, the map $\mathrm{Hol}$ is very often a local
homeomorphism, but not always (for a counterexample, see Baues [10]). This
issue needs to be better understood:

######  Question 4.7 (refined Thurston’s holonomy principle).

  1. 1.

Is it true that the map $\mathrm{Hol}$ is always a branched local
homeomorphism?

  2. 2.

What are some sufficient conditions for it to be a local homeomorphism?

Other important questions are raised by the fact that the map $\mathrm{Hol}$
is in general neither injective nor surjective.

######  Question 4.8.

Let $\rho\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ be a representation. Is
there an $(X,{\mathsf{G}})$-structure on $M$ with holonomy $\rho$? And in the
affirmative case, how many are there?

A complete answer to Question 4.8 is known only in very special cases, for
example for ${\mathbb{CP}}^{1}$-structures on closed surfaces (see
Gallo–Kapovich–Marden [17], Goldman [20], Baba [7, 8]). In the case of
${\mathbb{RP}}^{2}$-structures on closed surfaces a partial answer is given in
Choi–Goldman [12]. The character varieties are much easier to understand than
the parameter spaces ${\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(M)$, hence, if we
can obtain a better understanding of Question 4.8, we can use our knowledge
about representations to understand parameter spaces of geometric structures.

A plan to answer these questions can be the following: given a representation
$\rho$, we construct the corresponding flat bundle, and we then try to
understand all the possible transverse sections. An obstacle is that even for
representations that we know very well, we don’t always understand the
corresponding flat bundle well enough to see the transverse sections. This is
the point when Higgs bundles can be very useful: they can give an explicit
description of the flat bundle.

##  5 How to use Higgs bundles?

We will show in some simple examples how Higgs bundles can be used to
construct geometric structures on manifolds. The flat connection can be
expressed in terms of solutions of Hitchin’s equations and the transverse
section can be constructed from the study of the holomorphic structure of the
vector bundle. This idea first appeared in Baraglia’s Ph.D. Thesis [9].

###  5.1 $\boldsymbol{{\mathsf{SL}}(2,{\mathbb{R}})}$-Higgs bundles

In this subsection we will describe all the
${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundles. We will use this description in
Sections 5.2, 5.3, 7.2, and 8.1. Let $\Sigma$ be a closed Riemann surface.

######  Definition 5.1.

An _${\mathsf{SL}}(2,{\mathbb{R}})$ -Higgs bundle_ on $\Sigma$ is a tuple
$(E,Q,\omega,\varphi)$, where

  * 1)

$E$ is a holomorphic vector bundle on $\Sigma$ of rank $2$,

  * 2)

$Q\colon E\rightarrow E^{*}$ is a holomorphic symmetric
${\mathbb{C}}$-bilinear form,

  * 3)

$\omega\in H^{0}\big{(}\Sigma,\Lambda^{2}E\big{)}$ is a holomorphic
${\mathbb{C}}$-volume form such that $Q$ has volume $1$,

  * 4)

$\varphi\in H^{0}(\Sigma,\operatorname{End}(E)\otimes K)$ is $Q$-symmetric and
satisfies $\operatorname{tr}(\varphi)=0$ (the _Higgs field_).

The first three conditions say that $(E,Q,\omega)$ is a rank $2$ vector bundle
with an ${\mathsf{SO}}(2,{\mathbb{C}})$-structure. In particular,
$\Lambda^{2}E={\mathcal{O}}$.

The structure of such an ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle can be
made more explicit. This description was done by Hitchin [26], who started
from a different definition of ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundles.
Consider the set of $Q$-isotropic vectors:

$\displaystyle\operatorname{Iso}(Q)=\\{v\in E\,|\,Q(v,v)=0\\}.$

In every fiber, this set is the union of two lines. $E$ has two line
subbundles whose total spaces are given by:

$\displaystyle
L_{+}=\left\\{v\in\operatorname{Iso}(Q)\,|\,\forall\,w\in\operatorname{Iso}(Q){\setminus}\operatorname{Span}(v),\
i\frac{\omega(v,w)}{Q(v,w)}>0\right\\},$ $\displaystyle
L_{-}=\left\\{v\in\operatorname{Iso}(Q)\,|\,\forall\,w\in\operatorname{Iso}(Q){\setminus}\operatorname{Span}(v),\
i\frac{\omega(v,w)}{Q(v,w)}<0\right\\}.$

Hence, we have $E=L_{+}\oplus L_{-}$. The condition
$\Lambda^{2}E={\mathcal{O}}$ now says that $L_{+}=L_{-}^{-1}$. To simplify the
notation, we write $L=L_{+}$, $L^{-1}=L_{-}$. The Higgs bundle can be written
as

$\displaystyle E=L\oplus L^{-1},\qquad Q=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\omega=\frac{i}{\sqrt{2}}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\qquad\varphi=\begin{pmatrix}0&a\\\ b&0\end{pmatrix},$

where $a\in H^{0}\big{(}\Sigma,L^{2}K\big{)}$, $b\in
H^{0}\big{(}\Sigma,L^{-2}K\big{)}$. The condition for the Higgs bundle to be
poly-stable is that:

  1. 1.

If $\deg(L)>0$, then $b\neq 0$.

  2. 2.

If $\deg(L)<0$, then $a\neq 0$.

  3. 3.

If $\deg(L)=0$, then $a,b\neq 0$ or $a=b=0$.

In the case when $a=b=0$, the Higgs bundle is strictly poly-stable, in all
other cases it is stable. These conditions impose a restriction to the degree
of $L$ for a poly-stable ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle:
$|\deg(L)|\leq g-1$ (Milnor–Wood inequality).

A poly-stable Higgs bundle where $L$ has maximal possible degree
($\deg(L)=g-1$) is called a  _Fuchsian Higgs bundle_ , and they correspond to
Fuchsian representations. The stability condition $b\neq 0$ forces $L$ to be a
square root of $K$ (we will write $L=K^{\frac{1}{2}}$). The section $b$ is a
constant, and, up to gauge transformations we can assume $b=1$. The section
$a$ is a quadratic differential, we will write $a=q_{2}\in
H^{0}\big{(}\Sigma,K^{2}\big{)}$.

Let $H$ be the Hermitian metric on $E$ that solves Hitchin’s equations.

######  Proposition 5.2 ([2, Theorem 3.1]).

If an ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle is stable, then

$\displaystyle H=\begin{pmatrix}h&0\\\ 0&h^{-1}\end{pmatrix},$

for some real positive $h\in\Gamma(\Sigma,\bar{L}\otimes L)$.

###### Proof.

Consider the Higgs field $Q^{-1}\varphi^{T}Q\in
H^{0}(\Sigma,\operatorname{End}(E)\otimes K)$. Then the metric
$\bar{Q}^{T}\big{(}H^{T}\big{)}^{-1}Q$ is a solution of Hitchin’s equations
for the Higgs bundle $\big{(}E,Q^{-1}\varphi^{T}Q\big{)}$. The fact that
$\varphi$ is $Q$-symmetric means that $\varphi=Q^{-1}\varphi^{T}Q$, hence
$H=\bar{Q}^{T}\big{(}H^{T}\big{)}^{-1}Q$. This, plus the condition $\det(H)=1$
implies the statement. ∎

Let $\ell$ be a local holomorphic frame for $L$. Denote by $\ell^{\prime}$ the
dual holomorphic frame on $L^{-1}$. The pair $(\ell,\ell^{\prime})$ is a local
frame for $E$. In this local frame, we can write the flat connection given by
the solutions of Hitchin’s equations in the following way:

$\displaystyle\nabla=d+H^{-1}\partial
H+\varphi+H^{-1}\bar{\varphi}^{T}H=d+\begin{pmatrix}-\partial\log
h&a+h^{2}\bar{b}\\\ b+h^{-2}\bar{a}&\partial\log h\end{pmatrix}.$

The real structure is given by

$\displaystyle\tau\colon\ E\ni\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix}{\longrightarrow}\begin{pmatrix}0&h\\\
h^{-1}&0\end{pmatrix}\begin{pmatrix}\bar{v_{1}}\\\
\bar{v_{2}}\end{pmatrix}=\begin{pmatrix}h\bar{v_{2}}\\\
h^{-1}\bar{v_{1}}\end{pmatrix}\in E.$

And the real locus is given by

$\displaystyle E_{\mathbb{R}}=\\{v\in E\,|\,\tau(v)=v\\}.$

###  5.2 Hyperbolic structures on surfaces

Now we show the simplest example of how to use Higgs bundles to construct
geometric structures with given holonomy. We start with a Fuchsian
representation
$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PSL}}(2,{\mathbb{R}})$, and we want
to construct an ${\mathbb{H}}^{2}$-structure with holonomy $\rho$. We will
first construct a ${\mathbb{CP}}^{1}$-structure with holonomy $\rho$, and we
will then verify that this ${\mathbb{CP}}^{1}$-structure is actually an
${\mathbb{H}}^{2}$-structure.

We choose a complex structure $\Sigma$ on $S$ and we consider the
${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle $(E,\varphi)$ corresponding to a
lift of $\rho$ to ${\mathsf{SL}}(2,{\mathbb{R}})$. Since $\rho$ is Fuchsian,
we know that

$\displaystyle E=K^{\frac{1}{2}}\oplus
K^{-\frac{1}{2}},\qquad\varphi=\begin{pmatrix}0&q_{2}\\\
1&0\end{pmatrix},\qquad q_{2}\in H^{0}\big{(}\Sigma,K^{2}\big{)}.$

To construct a ${\mathbb{CP}}^{1}$-structure on $\Sigma$, we need to choose a
line subbundle, and prove that it gives a transverse section of the
projectivized bundle ${\mathbb{P}}(E)$. We can choose $K^{\frac{1}{2}}$ as a
subbundle. We will use the transversality condition from Proposition 4.5.
Given a local section $s$ of $K^{\frac{1}{2}}$, we can compute the
derivatives:

$\displaystyle s=\begin{pmatrix}1\\\ 0\end{pmatrix},\ \ \ \ \
\nabla_{\\!\\!\\!\frac{\partial}{\partial z}}s=\begin{pmatrix}-\partial\log
h\\\ 1\end{pmatrix},\ \ \ \ \
\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s=\begin{pmatrix}0\\\
h^{-2}\bar{q_{2}}\end{pmatrix}.$

Here we computed the derivatives in the complex directions
$\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial\bar{z}}$, but to
apply Proposition 4.5 we need to transform into derivatives in the real
directions. This gives the following modified condition: the section
$K^{\frac{1}{2}}$ is transverse if and only if

$\displaystyle\forall\,A,B\in{\mathbb{C}},\qquad
A\nabla_{\\!\\!\\!\frac{\partial}{\partial
z}}s+\bar{A}\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s+Bs=0\qquad\Rightarrow\qquad
A=B=0.$

Substituting, we see that the section $K^{\frac{1}{2}}$ is transverse if and
only if

$\displaystyle\forall\,A,B\in{\mathbb{C}},\qquad\begin{cases}-A\partial\log
h+B&=0,\\\ A+\bar{A}h^{-2}\bar{q_{2}}&=0\end{cases}\qquad\Rightarrow\qquad
A=B=0.$

If $A\neq 0$, the second equation is equivalent to

$\displaystyle\frac{A}{\bar{A}}=-h^{-2}\bar{q_{2}}.$

This cannot be satisfied because of the following lemma:

######  Lemma 5.3 (Hitchin [26]).

In the above setup, we have

$\displaystyle\big{|}h^{-2}\bar{q_{2}}\big{|}<1.$

###### Proof.

If $q_{2}=0$, this is obvious. Otherwise, it was proven by Hitchin [26]
applying the maximum principle. ∎

We have found a graph of a ${\mathbb{CP}}^{1}$-structure
$\big{(}{\mathbb{P}}(E),K^{\frac{1}{2}}\big{)}$. We denote by
$D\colon\widetilde{\Sigma}\rightarrow{\mathbb{CP}}^{1}$ the corresponding
developing map. We can now check that the image of this map never meets
${\mathbb{RP}}^{1}$, this is because we wrote the real structure $\tau$
explicitly, and it is easy to check that $K^{\frac{1}{2}}$ is never in the
real locus:

$\displaystyle\tau\begin{pmatrix}1\\\ 0\end{pmatrix}=\begin{pmatrix}0\\\
h^{-1}\end{pmatrix}.$

Hence we have a developing map

$\displaystyle D\colon\ \widetilde{\Sigma}\rightarrow{\mathbb{H}}^{2}.$

The holonomy is in ${\mathsf{PSL}}(2,{\mathbb{R}})$ and hence this
${\mathbb{CP}}^{1}$-structure is actually an ${\mathbb{H}}^{2}$-structure.

The map $D$ actually coincides with the harmonic map to the symmetric space
coming from solving Hitchin’s equations. The fact that $D$ is a local diffeo
was proved by Sampson [34], Wolf [42] and Hitchin [26]. The proof given here
is Hitchin’s proof.

The case when $q_{2}=0$ is the easiest, but it is particularly interesting.
Fuchsian Higgs bundles with $q_{2}=0$ are called _uniformizing Higgs bundles_
, because they give an alternative proof of a version of the uniformization
theorem. This was done by Hitchin [26] with essentially the same proof we give
here, but without mentioning geometric structures.

######  Theorem 5.4 (uniformization theorem).

Every complex structure $\Sigma$ on a closed surface $S$ admits a conformal
Riemannian metric of constant curvature $-1$.

###### Proof.

Choose a square root $K^{\frac{1}{2}}$ of the canonical bundle, and take the
uniformizing Higgs bundle with that square root. The equivariant map $D$
constructed above is now conformal: to see this, notice that

$\displaystyle\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s=\begin{pmatrix}0\\\
h^{-2}\bar{q_{2}}\end{pmatrix}=0.$

Hence, the pull-back of the hyperbolic metric on ${\mathbb{H}}^{2}$ is
conformal, and it has curvature $-1$. ∎

###  5.3 Almost-Fuchsian representations

Given a Fuchsian representation in the character variety
${\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(2,{\mathbb{R}}))$, we want to deform
it in
$\operatorname{QFuch}(S)\subset{\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(2,{\mathbb{C}}))$,
the space of quasi-Fuchsian representations. These representations have a very
interesting geometry, and they are holonomies of some very special
${\mathbb{CP}}^{1}$-structures called the quasi-Fuchsian
${\mathbb{CP}}^{1}$-structures.

######  Definition 5.5.

Consider a homeomorphism
$f\colon{\mathbb{CP}}^{1}\rightarrow{\mathbb{CP}}^{1}$ that topologically
conjugates the action of a Fuchsian representation with the action of a quasi-
Fuchsian representation $\rho$. Then the open subset
$f\big{(}{\mathbb{H}}^{2}\big{)}$ is a domain of discontinuity for $\rho$, and
$S=f\big{(}{\mathbb{H}}^{2}\big{)}/\rho(\pi_{1}(S))$ is a surface with a
${\mathbb{CP}}^{1}$-structure which is called a _quasi-Fuchsian
${\mathbb{CP}}^{1}$-structure_.

We would like to see the quasi-Fuchsian ${\mathbb{CP}}^{1}$-structures in
terms of Higgs bundles, but we are not able to do this in full generality. We
can see this for a special open subset of the quasi-Fuchsian representations,
which is called the space of almost-Fuchsian representations. The material in
this section is part of a joint work with Qiongling Li [3].

Let’s start with a uniformizing Higgs bundle

$\displaystyle\left(K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}},\
\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\right).$

We can deform this Higgs bundle for ${\mathsf{SL}}(2,{\mathbb{R}})$ to a Higgs
bundle for ${\mathsf{SL}}(2,{\mathbb{C}})$ by changing the holomorphic
structure of the vector bundle. Consider the vector bundle

$\displaystyle E=K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}}$

endowed with the following holomorphic structure:

$\displaystyle\bar{\partial}_{E}=\bar{\partial}+\begin{pmatrix}0&0\\\
\beta&0\end{pmatrix},$

with $\beta\in\Omega^{0,1}\big{(}\Sigma,K^{-1}\big{)}$. In the formula,
$\bar{\partial}$ is the standard holomorphic structure of the direct sum,
which is modified by adding a correction term. Such a bundle is an extension

$\displaystyle 0\rightarrow K^{-\frac{1}{2}}\rightarrow E\rightarrow
K^{\frac{1}{2}}\rightarrow 0.$

These extensions are classified by the Dolbeault cohomology class $[\beta]\in
H^{1}\big{(}\Sigma,K^{-1}\big{)}$ a space isomorphic, by Serre’s duality, to
the dual of the space of quadratic differentials on $\Sigma$. Different
$\beta$s in the same cohomology class give rise to isomorphic vector bundles.
The choice of the representative $\beta$ in the class corresponds to a choice
of a non-holomorphic section $K^{\frac{1}{2}}\rightarrow E$, whose image is
the non-holomorphic subbundle appearing in the direct sum.

We consider now the Higgs bundle $(E,\varphi)$, where

$\displaystyle E=\big{(}K^{\frac{1}{2}}\oplus
K^{-\frac{1}{2}},\bar{\partial}_{E}\big{)},\qquad\varphi=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}.$

The Higgs bundles of this form are parametrized by the pair
$(\Sigma,[\beta])$. For every quasi-Fuchsian representation $\rho$, there
exists a lift
$\bar{\rho}\colon\pi_{1}(S)\rightarrow{\mathsf{SL}}(2,{\mathbb{C}})$ and a
pair $(\Sigma,[\beta])$ such that the flat connection of the corresponding
Higgs bundle has monodromy $\bar{\rho}$ (see [35]). This pair is not unique in
general. Moreover, not all the pairs $(\Sigma,[\beta])$ give rise to a quasi-
Fuchsian monodromy. It is an open problem to distinguish them:

######  Question 5.6.

Given a complex structure $\Sigma$, how can we characterize the classes
$[\beta]{\in}H^{1}\\!\big{(}\Sigma{,}K^{-1}\\!\big{)}$ such that the flat
connection of the corresponding Higgs bundle has quasi-Fuchsian monodromy?

Answering this question was our initial motivation for trying to construct the
quasi-Fuchsian ${\mathbb{CP}}^{1}$-structures using Higgs bundles, but, as
explained above, we still cannot construct all of them.

Let’s fix now a pair $(\Sigma,[\beta])$. To construct a
${\mathbb{CP}}^{1}$-structure on $\Sigma$, we need to choose a line subbundle.
We choose the holomorphic subbundle $K^{-\frac{1}{2}}$, and we then have to
verify the transversality conditions.

Denote by $H$ the solutions of Hitchin’s equations for the corresponding Higgs
bundle. We can choose the representative $\beta$ in the Dolbeault cohomology
class in a way such that the non-holomorphic subbundle $K^{\frac{1}{2}}$ is
$H$-orthogonal to the holomorphic subbundle $K^{-\frac{1}{2}}$. With this
choice, we can write $H$ as

$\displaystyle H=\begin{pmatrix}h^{-1}&0\\\ 0&h\end{pmatrix}.$

We can now write the flat connection:

$\displaystyle\nabla=d+\begin{pmatrix}-\partial\log
h&h^{2}\big{(}\bar{1}+\bar{\beta}\big{)}\\\ 1+\beta&\partial\log
h\end{pmatrix}.$

Given a local section $s$ of $K^{-\frac{1}{2}}$, we can compute the
derivatives:

$\displaystyle s=\begin{pmatrix}0\\\
1\end{pmatrix},\qquad\nabla_{\\!\\!\\!\frac{\partial}{\partial
z}}s=\begin{pmatrix}h^{2}\bar{\beta}\\\ \partial\log
h\end{pmatrix},\qquad\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s=\begin{pmatrix}h^{2}\\\
0\end{pmatrix}.$

As in the previous subsection, the transversality condition from Proposition
4.5 is equivalent to the following condition: the section $K^{-\frac{1}{2}}$
is transverse if and only if

$\displaystyle\forall\,A,B\in{\mathbb{C}},\qquad
A\nabla_{\\!\\!\\!\frac{\partial}{\partial
z}}s+\bar{A}\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s+Bs=0\qquad\Rightarrow\qquad
A=B=0.$

Substituting, we see that the section $K^{-\frac{1}{2}}$ is transverse if and
only if

$\displaystyle\forall\,A,B\in{\mathbb{C}},\qquad\begin{cases}Ah^{2}\bar{\beta}+\bar{A}h^{2}=0,\\\
A\ \partial\log h+B=0\end{cases}\qquad\Rightarrow\qquad A=B=0.$

If $A\neq 0$, the first equation is equivalent to

$\displaystyle\frac{\bar{A}}{A}=\bar{\beta}.$

If $|\beta|<1$, this cannot be satisfied, hence the section is transverse. The
condition $|\beta|<1$ is well known, see Uhlenbeck [41]:

######  Definition 5.7.

A representation
$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PGL}}(2,{\mathbb{C}})$ is called
_almost-Fuchsian_ if it is the projectivization of the monodromy of the flat
connection of a Higgs bundle associated with a pair $(\Sigma,[\beta])$, with
$|\beta|<1$.

Almost-Fuchsian representations are a special type of quasi-Fuchsian
representations having very good analytic properties. Summarizing, we find the
following:

######  Theorem 5.8 (Alessandrini–Li [3]).

Let $\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PGL}}(2,{\mathbb{C}})$ be an
almost-Fuchsian representation corresponding to the Higgs bundle $(E,\varphi)$
defined by the pair $(\Sigma,[\beta])$. Then the holomorphic line subbundle
$K^{-\frac{1}{2}}\subset E$ induces a quasi-Fuchsian
${\mathbb{CP}}^{1}$-structure with holonomy $\rho$.

###  5.4 Convex real projective structures

######  Definition 5.9.

An ${\mathbb{RP}}^{2}$-structure on a closed surface $S$ is said to be a
_convex ${\mathbb{RP}}^{2}$-structure_ if the developing map

$\displaystyle D\colon\ \widetilde{S}\rightarrow{\mathbb{RP}}^{2}$

is a diffeomorphism with an open convex subset of ${\mathbb{RP}}^{2}$.

Examples of convex ${\mathbb{RP}}^{2}$-structures were given in Example 2.5,
where we have seen that every ${\mathbb{H}}^{2}$-structure on $S$ produces
such an ${\mathbb{RP}}^{2}$-structure via the Klein model.

The subset of ${\mathcal{D}}_{{\mathbb{RP}}^{2}}(S)$ consisting of convex real
projective structures will be denoted by
${\mathcal{D}}^{\mathrm{conv}}_{{\mathbb{RP}}^{2}}(S)$. The holonomy of these
structures is always reductive, hence we have

$\displaystyle\mathrm{Hol}\colon\
{\mathcal{D}}^{\mathrm{conv}}_{{\mathbb{RP}}^{2}}(S)\rightarrow{\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(3,{\mathbb{R}}))).$

Goldman [21] proved that
${\mathcal{D}}^{\mathrm{conv}}_{{\mathbb{RP}}^{2}}(S)$ is connected, hence the
image of $\mathrm{Hol}$ lies in the Hitchin component
$\operatorname{Hit}(S,3)$. Choi–Goldman [11] proved that $\mathrm{Hol}$ gives
a homeomorphism between ${\mathcal{D}}^{\mathrm{conv}}_{{\mathbb{RP}}^{2}}(S)$
and $\operatorname{Hit}(S,3)$. This gives a nice geometric interpretation of
the Hitchin component as the parameter space of convex
${\mathbb{RP}}^{2}$-structures on the surface.

In Baraglia’s thesis [9], he shows how to see these convex
${\mathbb{RP}}^{2}$-structures using Higgs bundles. Every
$\rho\in\operatorname{Hit}(S,3)$ admits a lift to a representation
$\bar{\rho}\colon\pi_{1}(S)\rightarrow{\mathsf{SL}}(3,{\mathbb{R}})$. By a
theorem of Loftin [33] and Labourie [31], there exists a complex structure
$\Sigma$ and a cubic differential $q_{3}\in H^{0}\big{(}\Sigma,K^{3}\big{)}$
such that the representation $\bar{\rho}$ is the monodromy of the flat
connection of the Higgs bundle $(E,\varphi)$, where

$\displaystyle E=K\oplus{\mathcal{O}}\oplus
K^{-1},\qquad\varphi=\begin{pmatrix}0&0&q_{3}\\\ 1&0&0\\\ 0&1&0\end{pmatrix}.$

Baraglia [9] proved that the solution $H$ of Hitchin’s equations for this
Higgs bundle is diagonal:

$\displaystyle H=\begin{pmatrix}h^{-1}&0&0\\\ 0&1&0\\\ 0&0&h\end{pmatrix}.$

To construct an ${\mathbb{RP}}^{2}$-structure on $\Sigma$, we can choose the
section given by the line subbundle ${\mathcal{O}}$. We can verify in the
usual way that this section is transverse, hence it gives an
${\mathbb{RP}}^{2}$-structure, which can be checked to be convex.

When $q_{3}=0$, the representation takes values in ${\mathsf{SO}}(1,2)$, and
the convex set is precisely an ellipsoid, the Klein model of the hyperbolic
plane.

##  6 Higher-dimensional manifolds

One limitation of the method described in the previous section is that Higgs
bundles can only describe flat bundles on surfaces. We would like to apply
similar methods to construct geometric structures on higher-dimensional
manifolds, but we need to find a good way to describe the flat bundle. This is
possible in some special cases, when the representation factors through a
surface group.

###  6.1 Sections of the holonomy map

Let $N$ be a closed manifold and ${\mathsf{G}}$ be a reductive Lie group.
Consider the character variety

$\displaystyle{\mathcal{X}}(\pi_{1}(N),{\mathsf{G}}).$

Sometimes, it is possible to find special open subsets
${\mathcal{U}}\subset{\mathcal{X}}(\pi_{1}(N),{\mathsf{G}})$ which parametrize
geometric structures on $N$. To give a meaning to this, we first need to find
a manifold $X$ with a transitive and effective action of ${\mathsf{G}}$ and
$\dim(X)=\dim(N)$. Consider then the holonomy map

$\displaystyle\mathrm{Hol}\colon\
{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(N)\rightarrow{\mathcal{X}}(\pi_{1}(N),{\mathsf{G}}).$

We want to find an open subset
${\mathcal{U}}\subset{\mathcal{X}}(\pi_{1}(N),{\mathsf{G}})$ and a map

$\displaystyle T\colon\
{\mathcal{U}}\rightarrow{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(N)$

such that $\mathrm{Hol}\circ T=\mathrm{Id}_{\mathcal{U}}$. Such a map $T$ is a
section of the holonomy map on ${\mathcal{U}}$. Finding such a $T$ gives a
geometric interpretation to the open subset ${\mathcal{U}}$: it becomes a
parameter space for a special subset of $(X,{\mathsf{G}})$-structures on $N$.

######  Example 6.1.

In the previous section, we have seen some very interesting examples of this
construction:

$\displaystyle{\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(2,{\mathbb{R}}))\supset\operatorname{Hit}(S,2)\rightarrow{\mathcal{D}}_{{\mathbb{H}}^{2}}(S)={\mathcal{T}}(S),$
$\displaystyle{\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(2,{\mathbb{C}}))\supset\operatorname{QFuch}(S)\rightarrow{\mathcal{D}}_{{\mathbb{CP}}^{1}}(S),$
$\displaystyle{\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(3,{\mathbb{R}}))\supset\operatorname{Hit}(S,3)\rightarrow{\mathcal{D}}^{\mathrm{conv}}_{{\mathbb{RP}}^{2}}(S)\subset{\mathcal{D}}_{{\mathbb{RP}}^{2}}(S).$

If we want to find more examples like these, the hypothesis that
$\dim(X)=\dim(N)$ becomes a serious problem: for some groups ${\mathsf{G}}$ we
don’t have homogeneous spaces of the correct dimension. To relax this
condition, we will look for geometric structures on another closed manifold
$M$. At this point, we don’t even need that $N$ is a manifold: the role of
$\pi_{1}(N)$ will be played by a finitely generated group $\Gamma$. Consider
the character variety

$\displaystyle{\mathcal{X}}(\Gamma,{\mathsf{G}}).$

We want to use an open subset of it to parametrize
$(X,{\mathsf{G}})$-structures on a closed manifold $M$ (with
$\dim(M)=\dim(X)$) which is related with $\Gamma$ by a group homomorphism
$\alpha\colon\pi_{1}(M)\rightarrow\Gamma$. This group homomorphism induces a
map

$\displaystyle\alpha^{*}\colon\
{\mathcal{X}}(\Gamma,{\mathsf{G}})\ni\rho\rightarrow\rho\circ\alpha\in{\mathcal{X}}(\pi_{1}(M),{\mathsf{G}}).$

We want to find an open subset
${\mathcal{U}}\subset{\mathcal{X}}(\Gamma,{\mathsf{G}})$ and a map

$\displaystyle T\colon\
{\mathcal{U}}\rightarrow{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(M)$

such that $\mathrm{Hol}\circ T=\alpha^{*}|_{\mathcal{U}}$. Finding such a map
$T$ gives a geometric interpretation to the open subset ${\mathcal{U}}$ as a
parameter space for a special subset of $(X,{\mathsf{G}})$-structures on $M$.

Many examples of this scenario come from the theory of domains of
discontinuity for Anosov representation in geometries of parabolic type (see
the discussion at the end of Section 2.3, Guichard and Wienhard [24] and
Kapovich, Leeb and Porti [29]). Assume that ${\mathsf{G}}$ is semi-simple,
${\mathsf{P}}\subset{\mathsf{G}}$ is a parabolic subgroup, $\Gamma$ is Gromov-
hyperbolic and torsion-free, ${\mathcal{U}}$ is a connected component of
${\mathsf{P}}\text{-}\operatorname{Anosov}(\pi_{1}(S),{\mathsf{G}})$. Then, we
need to choose a geometry $(X,{\mathsf{G}})$ of parabolic type which is in a
special relation with ${\mathsf{P}}$, in a way that the theory of domains of
discontinuity guarantees the existence of a co-compact domain of discontinuity
$\Omega_{\rho}\subset X$ for all the ${\mathsf{P}}$-Anosov representations.
Under these hypotheses, the topology of the manifold
$M=\Omega_{\rho}/\rho(\Gamma)$ does not depend on $\rho$, and the map

$\displaystyle T\colon\
{\mathcal{U}}\ni\rho\rightarrow\Omega_{\rho}/\rho(\Gamma)\in{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(M)$

has all the properties listed above. In this way, we give a geometric
interpretation to many open connected subsets of Anosov representations as
parametrizing $(X,{\mathsf{G}})$-structures on a closed manifold $M$. For an
example where these hypotheses are satisfied, see Section 8.2.

Even if we know the group $\Gamma$, we usually have no idea what the topology
of $M$ is:

######  Question 6.2.

For some connected component ${\mathcal{U}}$ of
${\mathsf{P}}\text{-}\operatorname{Anosov}(\pi_{1}(S),{\mathsf{G}})$,
understand the map

$\displaystyle T\colon\
{\mathcal{U}}\rightarrow{\mathcal{D}}^{*}_{(X,{\mathsf{G}})}(M).$

The first step is to determine the topology of $M$.

###  6.2 Transverse maps and transverse submanifolds

Let $\Gamma$ be a finitely generated group, and
$\rho\colon\Gamma\rightarrow{\mathsf{G}}$ a representation.

######  Definition 6.3.

Let $M$ be a manifold. A representation
$\bar{\rho}\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ _factors through $\rho$_
if there exists a group homomorphism $\alpha\colon\pi_{1}(M)\rightarrow\Gamma$
such that $\bar{\rho}=\rho\circ\alpha$.

Assume now that $\Gamma=\pi_{1}(N)$ for some manifold $N$. If $N$ is
aspherical (for example, if $N$ is a surface), then every group homomorphism
$\alpha\colon\pi_{1}(M)\rightarrow\pi_{1}(N)$ is induced by a smooth map
$f\colon M\rightarrow N$, i.e., $\alpha=f_{*}$. If $N$ is not aspherical, this
is not automatic.

######  Definition 6.4.

Let $\rho\colon\pi_{1}(N)\rightarrow{\mathsf{G}}$ be a representation. A
representation $\bar{\rho}\colon\pi_{1}(M)\rightarrow{\mathsf{G}}$ _strongly
factors through $\rho$_ if there exists a smooth map $f\colon M\rightarrow N$
such that $\bar{\rho}=\rho\circ f_{*}$.

In this case, the representation $\bar{\rho}$ is the monodromy of a flat
bundle $\bar{p}\colon\bar{B}\rightarrow M$, with fiber $X$, and $\rho$ is the
monodromy of a flat bundle $p\colon B\rightarrow N$, with fiber $X$, where
$(X,{\mathsf{G}})$ is a geometry. The former bundle is isomorphic to the pull-
back of the latter by the map $f$:

$\displaystyle\bar{B}=f^{*}B.$

Consider the following commutative diagram:

$\displaystyle\begin{matrix}\bar{B}&\stackrel{{\scriptstyle
f_{+}}}{{{\longrightarrow}}}&B\\\ \scriptstyle{\bar{p}}\Big{\downarrow}\
&&\scriptstyle{p}\Big{\downarrow}\ \\\ M&\stackrel{{\scriptstyle
f}}{{{\longrightarrow}}}&N.\end{matrix}$

######  Proposition 6.5.

There is a homeomorphism between $\Gamma(M,\bar{B})$, the space of smooth
sections of $\bar{B}$, and the space of smooth functions $s\colon M\rightarrow
B$ satisfying $p\circ s=f$, endowed with the $C^{\infty}$ topology. The
homeomorphism is given by

$\displaystyle\Gamma(M,\bar{B})\ni\bar{s}\rightarrow s=\bar{s}\circ f_{+}\in
C^{\infty}(M,B),$

where $f_{+}\colon\bar{B}\rightarrow B$ is the map given by the pull-back.

Now let’s change perspective and think that we don’t know the map $f\colon
M\rightarrow N$ in advance. Let’s just start from a map $s\colon M\rightarrow
B$. From $s$, we can construct a map $f=p\circ s\colon M\rightarrow N$, a
representation $\bar{\rho}=\rho\circ f_{*}$, a flat bundle $\bar{B}=f^{*}B$
and a section $\bar{s}\in\Gamma(M,f^{*}B)$. We will call the section $\bar{s}$
the _tautological section_ , because it is just a reinterpretation of the map
$s$ as section of a bundle.

######  Definition 6.6.

We will say that a smooth map $M\rightarrow B$ is a _transverse map_ if it is
transverse to the parallel foliation of the flat bundle $B$.

######  Proposition 6.7.

The map $s$ is a transverse map if and only if the tautological section
$\bar{s}\in\Gamma(M,f^{*}B)$ is a transverse section.

We can summarize the constructions in this subsection with the following
proposition.

######  Proposition 6.8.

Let $\rho\colon\pi_{1}(N)\rightarrow{\mathsf{G}}$, and $B$ be a flat bundle
with holonomy $\rho$ and fiber $X$, where $(X,{\mathsf{G}})$ is a geometry.
Every $(X,{\mathsf{G}})$-structure on some manifold $M$ with holonomy that
strongly factors through $\rho$ comes from a transverse map $M\rightarrow B$.

From this proposition we see that we can construct geometric structures on a
manifold of higher dimension than $N$, by only understanding the parallel
foliation of a flat bundle over $N$.

Recall that, when $\dim(M)=\dim(X)$, transverse maps are always immersions. An
interesting special case is when the transverse map is an embedding. In that
case we can confuse the map with its image, a submanifold of $B$. Even more
interesting is the case when the submanifold is a subbundle of $B$ (not
necessarily with structure group ${\mathsf{G}}$).

######  Definition 6.9.

A _transverse submanifold of $B$_ is a submanifold that is transverse to the
parallel foliation of $B$. A _transverse subbundle_ is a subbundle that is a
transverse submanifold.

Transverse submanifolds and transverse subbundles of $B$ can be constructed
without any a priori knowledge of $M$, for example as zero loci of systems of
equations defined on $B$. In the case of a transverse subbundle, we can hope
to get an explicit description of its topology.

This discussion suggests a reformulation of Question 6.2:

######  Question 6.10.

With the notation of Question 6.2, add the hypothesis that $\Gamma=\pi_{1}(N)$
is the fundamental group of a closed aspherical manifold $N$. Is it true that
$M$ is homeomorphic to a fiber bundle over $N$?

This question generalizes a conjecture by Dumas and Sanders:

######  Conjecture 6.11 (Dumas–Sanders [15]).

Let ${\mathsf{G}}$ be a simple complex Lie group and
$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{G}}$ be a quasi-Hitchin
representation. Consider a geometry $(X,{\mathsf{G}})$ of parabolic type, and
assume that $\rho$ has a co-compact domain of discontinuity $\Omega\in X$
coming from the construction of Kapovich, Leeb and Porti [29]. Then the
manifold $\Omega/\rho(\pi_{1}(S))$ admits a continuous fiber bundle map to the
surface $S$.

We can prove this conjecture in some special cases, see Section 8.2.

In the following we will assume that $N$ is a closed surface. In this case we
can use Higgs bundles to describe the bundle $B$ and construct geometric
structures on higher-dimensional manifolds by finding transverse subbundles of
$B$.

##  7 Geometric structures on circle bundles over surfaces

We will now show examples where we can construct transverse subbundles which
are $3$-dimensional manifolds. In this case they are circle bundles over
surfaces. The case of manifolds of dimension higher than $3$ is more
complicated, and it will be treated in the next section.

###  7.1 Convex-foliated real projective structures

Let’s consider the Hitchin component

$\displaystyle\operatorname{Hit}(S,4)\subset{\mathcal{X}}(\pi_{1}(S),{\mathsf{PGL}}(4,{\mathbb{R}})).$

We can here see an example of the ideas explained in Section 6.1 about how to
interpret this component as parameter space for a special subset of
${\mathbb{RP}}^{3}$-structures on $T^{1}S$, the unit tangent bundle of the
surface.

Guichard–Wienhard [23] proved that every $\rho\in\operatorname{Hit}(S,4)$ has
a co-compact domain of discontinuity $\Omega_{\rho}\subset{\mathbb{RP}}^{3}$
which has two connected components
$\Omega_{\rho}=\Omega^{+}_{\rho}\cup\Omega^{-}_{\rho}$. One of the two (say
$\Omega^{+}_{\rho}$) has the property that the quotient
$\Omega_{\rho}^{+}/\rho(\pi_{1}(S))$ is homeomorphic to $T^{1}S$, and that its
${\mathbb{RP}}^{3}$-structure is of a special type, called a convex foliated
${\mathbb{RP}}^{3}$-structure.

This gives a map

$\displaystyle\operatorname{Hit}(S,4)\rightarrow{\mathcal{D}}_{(X,{\mathsf{G}})}\big{(}T^{1}S\big{)}$

that they prove to be a homeomorphism onto a connected component of
${\mathcal{D}}_{(X,{\mathsf{G}})}\big{(}T^{1}S\big{)}$ containing exactly all
convex foliated ${\mathbb{RP}}^{3}$-structures on $T^{1}S$ ([23]).

We can construct some of these structures using Higgs bundles. In this way we
can see some properties of these structures that where not known before.

Every $\rho\in\operatorname{Hit}(S,4)$ admits a lift to a representation
$\bar{\rho}\colon\pi_{1}(S)\rightarrow{\mathsf{SL}}(4,{\mathbb{R}})$. By a
theorem of Labourie [32], there exists a complex structure $\Sigma$, a square
root $K^{\frac{1}{2}}$ of the canonical bundle and differentials $q_{3}$,
$q_{4}$ with $q_{3}\in H^{0}\big{(}\Sigma,K^{3}\big{)}$ and $q_{4}\in
H^{0}\big{(}\Sigma,K^{4}\big{)}$ such that the representation $\bar{\rho}$ is
the monodromy of the flat connection of the following Higgs bundle:

$\displaystyle E=K^{\frac{3}{2}}\oplus K^{\frac{1}{2}}\oplus
K^{-\frac{1}{2}}\oplus
K^{-\frac{3}{2}},\qquad\varphi=\begin{pmatrix}0&0&q_{3}&q_{4}\\\
1&0&0&q_{3}\\\ 0&1&0&0\\\ 0&0&1&0\end{pmatrix}.$

In Baraglia’s thesis [9], he considered the special case when the image of
$\rho$ is contained in ${\mathsf{PSp}}(4,{\mathbb{R}})$. In terms of Higgs
bundles, this corresponds to the case when $q_{3}=0$. This belongs to a
special type of Higgs bundles called cyclic Higgs bundles, and Baraglia proved
that, in this case, the solution $H$ of Hitchin’s equations is diagonal

$\displaystyle H=\begin{pmatrix}h_{1}&0&0&0\\\ 0&h_{2}&0&0\\\ 0&0&h_{3}&0\\\
0&0&0&h_{4}\end{pmatrix}.$

We can then write the real structure

$\displaystyle\tau\colon\ E\ni\begin{pmatrix}v_{1}\\\ v_{2}\\\ v_{3}\\\
v_{4}\end{pmatrix}\rightarrow\begin{pmatrix}h_{4}\bar{v_{4}}\\\
h_{3}\bar{v_{3}}\\\ h_{2}\bar{v_{2}}\\\ h_{1}\bar{v_{1}}\end{pmatrix}\in E.$

The real locus of $E$ is the real vector bundle

$\displaystyle\operatorname{Re}(E)=\\{v\in E\,|\,\tau(v)=v\\}.$

We want to construct ${\mathbb{RP}}^{3}$-structures, hence we set
$X={\mathbb{RP}}^{3}$. The flat bundle $B$ with monodromy $\rho$ and fiber $X$
is $B={\mathbb{P}}(\operatorname{Re}(E))$.

We now want to find a transverse subbundle. Consider

$\displaystyle M={\mathbb{P}}\left(\left\\{\begin{pmatrix}0\\\ v_{2}\\\
v_{3}\\\ 0\end{pmatrix}\in
B\,\Big{|}\,v_{2}=h_{3}\bar{v_{3}}\right\\}\right).$

This is a circle bundle over $\Sigma$, isomorphic to the unit tangent bundle.
To check that it is transverse, we will put local coordinates on $M$, using a
local holomorphic coordinate $z$ on $\Sigma$, and a real coordinate $\theta$
on the circle fiber. For every $s$ local section, we compute the derivatives

$\displaystyle\nabla_{\\!\\!\\!\frac{\partial}{\partial
z}}s,\qquad\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s,\qquad\nabla_{\\!\\!\\!\frac{\partial}{\partial\theta}}s.$

We can then check the transversality condition:

$\displaystyle\forall\,A\in{\mathbb{C}},\ \forall\,B,C\in{\mathbb{R}},\qquad
A\nabla_{\\!\\!\\!\frac{\partial}{\partial
z}}s+\bar{A}\nabla_{\\!\\!\\!\frac{\partial}{\partial\bar{z}}}s+B\nabla_{\\!\\!\\!\frac{\partial}{\partial\theta}}s+Cs=0\qquad\Rightarrow\qquad
A=B=C=0.$

After the transversality condition has been verified, Baraglia also proves
that these ${\mathbb{RP}}^{3}$-structures are convex foliated. In this way, he
obtains the following new result:

######  Theorem 7.1 (Baraglia [9]).

For every convex foliated ${\mathbb{RP}}^{3}$-structure on $T^{1}S$ with
holonomy in ${\mathsf{PSp}}(4,{\mathbb{R}})$, there exists a map
$T^{1}S\rightarrow S$ which is a circle bundle and has the property that every
circle fiber is a projective line for the ${\mathbb{RP}}^{3}$-structure.

This statement is completely geometric, but there is no known geometric proof
of it, the only proof is the one with Higgs bundles. We conjecture that the
same statement is true in general, for all convex foliated
${\mathbb{RP}}^{3}$-structures also when the holonomy is not restricted to be
in ${\mathsf{PSp}}(4,{\mathbb{R}})$.

Working with Qiongling Li, we explored and modified this construction. With
the same Higgs field as before, we noticed that the other choice of subbundle

$\displaystyle M^{\prime}={\mathbb{P}}\left(\left\\{\begin{pmatrix}v_{1}\\\
0\\\ 0\\\ v_{4}\end{pmatrix}\in
B\,\Big{|}\,v_{1}=h_{4}\bar{v_{4}}\right\\}\right)$

is also a transverse subbundle, and gives the other
${\mathbb{RP}}^{3}$-structure we discussed before, the one given by
$\Omega_{\rho}^{-}/\rho(\pi_{1}(S))$. This is a circle bundle over $S$ with
Euler class $6g-6$.

We then changed the Higgs field, considering the case when $q_{4}=0$, but
$q_{3}\neq 0$. For this kind of Higgs bundles, the solution of Hitchin’s
equations are again diagonal (see Collier–Li [13]), and the construction can
be applied in a similar way.

Moreover, since the transversality condition is an open condition, we can also
understand the cases when at least one of $q_{3}$, $q_{4}$ is small enough:

######  Theorem 7.2 (Alessandrini–Li, work in progress).

Let $\rho\in\operatorname{Hit}(S,4)$ be a representation such that the
corresponding Higgs bundle has the form given above, with the additional
hypothesis that at least one between $q_{3}$ and $q_{4}$ is small enough. Then
the two ${\mathbb{RP}}^{3}$-structures $M=\Omega^{+}/\rho(\pi_{1}(S))$ and
$M^{\prime}=\Omega^{-}/\rho(\pi_{1}(S))$ have the property that there exist
maps $M\rightarrow S$ and $M^{\prime}\rightarrow S$ which are circle bundles
with the property that every circle fiber is a projective line for the
${\mathbb{RP}}^{3}$-structure.

###  7.2 Closed anti-de Sitter 3-manifolds

Consider a symmetric bilinear form $Q$ on ${\mathbb{R}}^{4}$ of signature
$(2,2)$. This form is preserved by the group ${\mathsf{O}}(2,2)$ which has
four connected components. The connected component of the identity is called
${\mathsf{SO}}_{0}(2,2)$.

On ${\mathbb{RP}}^{3}$, this form defines the open subset

$\displaystyle{\rm
AdS}^{3}=\big{\\{}[v]\in{\mathbb{RP}}^{3}\,|\,Q(v,v)>0\big{\\}}.$

Let ${\mathsf{PO}}(2,2)$ be the projectivization of ${\mathsf{O}}(2,2)$, and
${\mathsf{PO}}_{0}(2,2)$ the connected component of the identity. The geometry
$\big{(}{\rm AdS}^{3},{\mathsf{PO}}(2,2)\big{)}$ is called the $3$-dimensional
_anti-de Sitter geometry_ , and it is a geometry of pseudo-Riemannian type,
carrying an invariant pseudo-Riemannian metric of signature $(2,1)$.

We can construct the bilinear form $Q$ in the following special way. Consider
the vector space ${\mathbb{R}}^{2}$ with a volume form $\omega$. This volume
form is preserved by the group ${\mathsf{SL}}(2,{\mathbb{R}})$. On the tensor
product ${\mathbb{R}}^{4}={\mathbb{R}}^{2}\otimes{\mathbb{R}}^{2}$ we have a
bilinear form $Q=\omega\otimes\omega$. This bilinear form is symmetric and it
has signature $(2,2)$. This construction shows us how
${\mathsf{SL}}(2,{\mathbb{R}})\times{\mathsf{SL}}(2,{\mathbb{R}})$ acts on
${\mathbb{R}}^{4}$ preserving $Q$. This gives a homomorphism

$\displaystyle{\mathsf{SL}}(2,{\mathbb{R}})\times{\mathsf{SL}}(2,{\mathbb{R}})\rightarrow{\mathsf{SO}}_{0}(2,2),$

which induces an isomorphism

$\displaystyle{\mathsf{PSL}}(2,{\mathbb{R}})\times{\mathsf{PSL}}(2,{\mathbb{R}})\rightarrow{\mathsf{PO}}_{0}(2,2).$

We will consider a representation
$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PO}}_{0}(2,2)$ which can be lifted to
a representation
$\bar{\rho}\colon\pi_{1}(S)\rightarrow{\mathsf{SL}}(2,{\mathbb{R}})\times{\mathsf{SL}}(2,{\mathbb{R}})$.
The corresponding Higgs bundle can be written as the tensor product of two
Higgs bundles for ${\mathsf{SL}}(2,{\mathbb{R}})$: given two Higgs bundles for
${\mathsf{SL}}(2,{\mathbb{R}})$

$\displaystyle E_{1}=L_{1}\oplus L_{1}^{-1},\qquad Q_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\omega_{1}=\frac{i}{\sqrt{2}}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},$ $\displaystyle\varphi_{1}=\begin{pmatrix}0&a_{1}\\\
b_{1}&0\end{pmatrix},\qquad H_{1}=\begin{pmatrix}h_{1}^{-1}&0\\\
0&h_{1}\end{pmatrix},$ $\displaystyle E_{2}=L_{2}\oplus L_{2}^{-1},\qquad
Q_{2}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\omega_{2}=\frac{i}{\sqrt{2}}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},$ $\displaystyle\varphi_{2}=\begin{pmatrix}0&a_{2}\\\
b_{2}&0\end{pmatrix},\qquad H_{2}=\begin{pmatrix}h_{2}^{-1}&0\\\
0&h_{2}\end{pmatrix},$

we can form their tensor product

$\displaystyle E=E_{1}\otimes E_{2}=L_{1}L_{2}\oplus L_{1}L_{2}^{-1}\oplus
L_{1}^{-1}L_{2}\oplus
L_{1}^{-1}L_{2}^{-1},\qquad\varphi=\begin{pmatrix}0&a_{2}&a_{1}&0\\\
b_{2}&0&0&a_{1}\\\ b_{1}&0&0&a_{2}\\\ 0&b_{1}&b_{2}&0\end{pmatrix}.$

The solutions of Hitchin’s equations for this Higgs bundle are given by

$\displaystyle H=\begin{pmatrix}h_{1}^{-1}h_{2}^{-2}&&&\\\
&h_{1}^{-1}h_{2}&&\\\ &&h_{1}h_{2}^{-1}&\\\ &&&h_{1}h_{2}\end{pmatrix}.$

We now want to construct an ${\rm AdS}^{3}$-structure on a $3$-manifold with
this holonomy. To do so, we will first construct an
${\mathbb{RP}}^{3}$-structure, then we verify that the image of the developing
map lies inside ${\rm AdS}^{3}$ by writing the bilinear form
$Q=\omega\otimes\omega$ explicitly. Since the developing map goes to ${\rm
AdS}^{3}$ and the holonomy is in ${\mathsf{PO}}(2,2)$, the structure we are
constructing is actually an ${\rm AdS}^{3}$-structure.

We consider the following subbundle:

$\displaystyle M={\mathbb{P}}\big{(}\operatorname{Re}\big{(}L_{1}L_{2}\oplus
L_{1}^{-1}L_{2}^{-1}\big{)}\big{)}.$

We then start to verify the transversality condition in the usual way. But we
notice that the condition is not always verified.

To state the result we need to recall that the solutions of Hitchin’s
equations for an ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle describe an
equivariant harmonic map to the hyperbolic plane. So we have the two harmonic
maps

$\displaystyle f_{1},f_{2}\colon\
\widetilde{\Sigma}\rightarrow{\mathbb{H}}^{2}.$

We will denote by $\widetilde{g_{1}}$, $\widetilde{g_{2}}$ the two pull-backs
of the hyperbolic metric to $\widetilde{\Sigma}$. These tensors are
$\pi_{1}(\Sigma)$-invariant, hence they define two tensors $g_{1}$, $g_{2}$ on
$\Sigma$. These symmetric tensors are called the _pull-back metrics_ , even
though they are not always Riemannian metrics, they can be degenerate at some
points.

######  Definition 7.3.

The ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle
$(E_{1},Q_{1},\omega_{1},\varphi_{1})$ _dominates_
$(E_{2},Q_{2},\omega_{2},\varphi_{2})$ if

$\displaystyle g_{1}-g_{2}>0,$

i.e., if the symmetric tensor $g_{1}-g_{2}$ is positive definite.

######  Theorem 7.4 (Alessandrini–Li [2]).

The subbundle $M$ is a transverse subbundle if and only if the
${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle
$(E_{1},Q_{1},\omega_{1},\varphi_{1})$ dominates
$(E_{2},Q_{2},\omega_{2},\varphi_{2})$.

In the theory of anti-de Sitter $3$-manifolds, there exists a necessary and
sufficient condition for the representation $\rho$ to be the holonomy of an
anti-de Sitter structure on a closed manifold. It was shown by Tholozan [37]
that this condition is equivalent to the existence of a complex structure
$\Sigma$ on $S$ such that $(E_{1},Q_{1},\omega_{1},\varphi_{1})$ dominates
$(E_{2},Q_{2},\omega_{2},\varphi_{2})$. It follows that with Higgs bundles we
can construct all closed anti-de Sitter $3$-manifolds with holonomy that lifts
to ${\mathsf{SL}}(2,{\mathbb{R}})\times{\mathsf{SL}}(2,{\mathbb{R}})$.

Our main motivation for this work was to use the special parametrization that
the Higgs bundle give to the manifold $M$ to explicitly compute invariants of
the anti-de Sitter structure, such as the volume. The computation of the
volume of the closed anti-de Sitter $3$-manifolds was an open problem that was
solved shortly before us by Tholozan [38].

######  Theorem 7.5 (Tholozan [38], Alessandrini–Li [2]).

The volume of the anti-de Sitter structure constructed on $M$ is

$\displaystyle\operatorname{Vol}(M)=\pi^{2}\left|\deg(L_{1})+\deg(L_{2})\right|.$

##  8 Projective structures with Hitchin
or quasi-Hitchin holonomies

In the previous section we presented examples of constructions of geometric
structures on $3$-dimensional manifolds. Now we will see how to apply the
method to higher-dimensional manifolds. The general strategy is similar, but
the technical details are more complicated.

Hitchin and quasi-Hitchin representations act on odd-dimensional real and
complex projective spaces admitting a co-compact domains of discontinuity
(Guichard–Wienhard [24]). The quotient of this domain is a closed manifold
with a projective structure. The holomorphic structure of the Higgs bundles
and the solutions of Hitchin’s equations help us to construct the same
projective structure using Higgs bundles, in this way we can determine the
topology of the manifold. This is joint work with Qiongling Li [5].

Another construction of geometric structures on higher-dimensional manifolds
using Higgs bundles was done by Collier, Tholozan and Toulisse [14]. They
constructed photon structures whose holonomy factors through maximal
representations in ${\mathsf{O}}(2,n)$. This work was not discussed during the
mini-course for lack of time.

###  8.1 Construction of real and complex projective structures

Consider a representation
$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PGL}}(2n,{\mathbb{R}})$ in the
Fuchsian locus of the Hitchin component $\operatorname{Hit}(S,2n)$. Recall
from Example 3.4 that such a representation is the composition of a Fuchsian
representation in ${\mathsf{PGL}}(2,{\mathbb{R}})$ with the irreducible
representation
${\mathsf{PGL}}(2,{\mathbb{R}})\rightarrow{\mathsf{PGL}}(2n,{\mathbb{R}})$. We
now want to construct ${\mathbb{RP}}^{2n-1}$ and
${\mathbb{CP}}^{2n-1}$-structures with holonomy that factors through $\rho$.
Let’s start with the corresponding uniformizing Higgs bundle for
${\mathsf{SL}}(2,{\mathbb{R}})$:

$\displaystyle E=K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}},\qquad
Q=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\omega=\frac{i}{\sqrt{2}}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},$ $\displaystyle\varphi=\begin{pmatrix}0&0\\\
1&0\end{pmatrix},\qquad H=\begin{pmatrix}h^{-1}&0\\\ 0&h\end{pmatrix}.$

The composition with the irreducible representation corresponds, in terms of
Higgs bundles, to the symmetric tensor product

$\displaystyle S(E)=\operatorname{Symm}^{2n-1}(E).$

To make our formulae more explicit and more readable, we will write them only
for $n=3$, but similar formulae work for every $n$

$\displaystyle S(E)=K^{\frac{5}{2}}\oplus K^{\frac{3}{2}}\oplus
K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}}\oplus K^{-\frac{3}{2}}\oplus
K^{-\frac{5}{2}},\qquad S(\varphi)=\begin{pmatrix}0&0&0&0&0&0\\\
1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&0\end{pmatrix}.$

The solution of Hitchin’s equations and the real structure are as usual given
by

$\displaystyle S(H)=\begin{pmatrix}h_{1}&\\\ &\ddots\\\
&&h_{6}\end{pmatrix},\qquad\tau\colon\ S(E)\ni\begin{pmatrix}v_{1}\\\
\vdots\\\ v_{6}\end{pmatrix}\rightarrow\begin{pmatrix}h_{6}\bar{v_{6}}\\\
\vdots\\\ h_{1}\bar{v_{1}}\end{pmatrix}\in S(E).$

We will consider the subbundles defined by the following equations:

$\displaystyle
U^{\mathbb{C}}={\mathbb{P}}\left(\left\\{\begin{pmatrix}h_{1}^{-\frac{1}{2}}t_{1}\\\
\vdots\\\
h_{6}^{-\frac{1}{2}}t_{6}\end{pmatrix}\,\middle|\,t_{1}\bar{t_{2}}+t_{3}\bar{t_{4}}+t_{5}\bar{t_{6}}=0\right\\}\right),\qquad
U^{\mathbb{R}}=U^{\mathbb{C}}\cap{\mathbb{P}}\left(\operatorname{Re}(E)\right).$

Similar formulae define these subbundles for every $n$.

######  Theorem 8.1 (Alessandrini–Li [5]).

The subbundle $U^{\mathbb{C}}$ is a transverse subbundle of ${\mathbb{P}}(E)$,
hence it supports a ${\mathbb{CP}}^{2n-1}$-structure whose holonomy factors
through $\rho$.

The subbundle $U^{\mathbb{R}}$ is a transverse subbundle of
${\mathbb{P}}(\operatorname{Re}(E))$, hence it supports an
${\mathbb{RP}}^{2n-1}$-structure whose holonomy factors through $\rho$.

This method of constructing projective structures has the merit that we can
see explicitly the topology of the manifold that supports the structure.
Consider the spaces

$\displaystyle F^{\mathbb{R}}=T^{1}{\mathbb{RP}}^{n-1},\qquad
F^{\mathbb{C}}=\big{(}T^{1}{\mathbb{S}}^{2n-1}\big{)}/{\mathsf{U}}(1),$

where ${\mathsf{U}}(1)$ acts on the unit sphere in a complex vector space
${\mathbb{S}}^{2n-1}\subset{\mathbb{C}}^{n}$ by component-wise multiplication
by a complex number; this action is then lifted to the unit tangent bundle
using the differential. Both spaces carry an action of ${\mathsf{SO}}(2)$: on
$T^{1}{\mathbb{RP}}^{n-1}$, the action of ${\mathsf{SO}}(2)$ is given by the
geodesic flow, which is periodic. Similarly, ${\mathsf{SO}}(2)$ acts via the
geodesic flow on $T^{1}{\mathbb{S}}^{2n-1}$, and this action commutes with the
action of ${\mathsf{U}}(1)$, hence it descends to an action on the quotient.

######  Theorem 8.2 (Alessandrini–Li [5]).

Let $P$ be a principal ${\mathsf{SO}}(2)$-bundle with Euler class $2g-2$. Then

  1. $1)$

$U^{\mathbb{R}}\simeq P\big{(}F^{\mathbb{R}}\big{)}$, for $n\geq 3$,

  2. $2)$

$U^{\mathbb{C}}\simeq P\big{(}F^{\mathbb{C}}\big{)}$, for $n\geq 2$.

Since the Euler number completely determines a principal
${\mathsf{SO}}(2)$-bundle, this result completely determines the topology of
the manifolds $U^{\mathbb{R}}$ and $U^{\mathbb{C}}$.

######  Remark 8.3.

We can apply the same technique to other representations, namely the ones
which are composition of a Fuchsian representation in
${\mathsf{SL}}(2,{\mathbb{R}})$ with the diagonal representation
${\mathsf{SL}}(2,{\mathbb{R}})\rightarrow{\mathsf{SL}}(2n,{\mathbb{R}})$.
Again we can find transverse subbundles and construct ${\mathbb{RP}}^{2n-1}$
and ${\mathbb{CP}}^{2n-1}$-structures on these manifolds [5]. This part is
less interesting though, since the same construction can be done in a
completely geometric way, without using Higgs bundles at all, thanks to the
special geometry of the diagonal representation. The interesting thing about
the result for the irreducible representation of
${\mathsf{SL}}(2,{\mathbb{R}})$ is that it is very hard to see these
transverse subbundles using only geometry.

###  8.2 Domains of discontinuity

Let ${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{C}}$, $\Gamma$ be a Gromov-
hyperbolic group and
$\rho\colon\Gamma\rightarrow{\mathsf{PGL}}(2n,{\mathbb{K}})$ be a
representation which is ${\mathsf{P}}$-Anosov, where ${\mathsf{P}}$ is the
stabilizer of an $(n-1)$-dimensional projective subspace. The space
${\mathsf{G}}/{\mathsf{P}}$ can be identified with the Grassmannian
$\operatorname{Gr}\big{(}n,{\mathbb{K}}^{2n}\big{)}$, which parametrizes the
$n$-dimensional linear subspaces of ${\mathbb{K}}^{2n}$. The Anosov property
gives us the $\rho$-equivariant map

$\displaystyle\xi\colon\
\partial_{\infty}\Gamma\rightarrow\operatorname{Gr}\big{(}n,{\mathbb{K}}^{2n}\big{)}.$

Guichard–Wienhard [24] used this map to construct a co-compact domain of
discontinuity for the action of $\rho$ in ${\mathbb{KP}}^{2n-1}$. We first
define the $\rho$-equivariant compact subset

$\displaystyle
K^{\mathbb{K}}_{\rho}=\bigcup_{t\in\partial_{\infty}\Gamma}[\xi(t)]\subset{\mathbb{KP}}^{2n-1},$

which is the complement of the $\rho$-equivariant open subset

$\displaystyle\Omega^{\mathbb{K}}_{\rho}={\mathbb{KP}}^{2n-1}{\setminus}K.$

######  Theorem 8.4 (Guichard–Wienhard [24]).

If $\rho$ is a $P$-Anosov representation in ${\mathsf{PGL}}(2n,{\mathbb{K}})$,
then $\rho$ acts on $\Omega^{\mathbb{K}}_{\rho}$ properly discontinuously and
co-compactly.

If $\Gamma$ is torsion-free, we can construct the quotient manifold
$M^{\mathbb{K}}=\Omega^{\mathbb{K}}_{\rho}/\rho(\Gamma)$, a closed manifold
carrying a ${\mathbb{KP}}^{2n-1}$-structure. The topology of $M^{\mathbb{K}}$
is constant when $\rho$ varies in a connected component ${\mathcal{U}}$ of
${\mathsf{P}}$-$\operatorname{Anosov}(\Gamma,{\mathsf{PGL}}(2n,{\mathbb{K}}))$.
In this way, we get a map

$\displaystyle T\colon\
{\mathcal{U}}\rightarrow{\mathcal{D}}^{*}_{{\mathbb{KP}}^{2n-1}}\big{(}M^{\mathbb{K}}\big{)},$

which gives an example of the construction described in Section 6.1.

Now let’s assume that $\Gamma=\pi_{1}(S)$ is a surface group, and that the
connected component ${\mathcal{U}}$ we have chosen is the Hitchin component
$\operatorname{Hit}(S,2n)$ when ${\mathbb{K}}={\mathbb{R}}$, and the space of
quasi-Hitchin representations when ${\mathbb{K}}={\mathbb{C}}$. In these cases
we can understand the topology of $M^{\mathbb{K}}$, using our construction
with Higgs bundles in the previous section.

######  Theorem 8.5 (Alessandrini–Li [5]).

For $n\leq 63$, $M^{\mathbb{K}}$ is diffeomorphic to $U^{\mathbb{K}}$.

###### Proof.

For a representation $\rho$ in the Fuchsian locus, we constructed a
${\mathbb{KP}}^{2n-1}$-structure on the manifold $U^{\mathbb{K}}$. We can
explicitly compute the developing map of this structure, and we can prove that
this structure is isomorphic to the structure
$\Omega^{\mathbb{K}}_{\rho}/\rho(\pi_{1}(S))$ if and only if a certain
explicit $n\times n$ matrix is positive definite. We then used the computer to
check whether the matrix is actually positive definite. Unfortunately, since
with the computer we could only check finitely many values of $n$, we stopped
after $n=63$. ∎

We believe the result to be true for every value of $n$, but we don’t have a
general proof yet. Together with Theorem 8.2, this result completely describes
the topology of $M^{\mathbb{K}}$. Moreover, this result describes some
interesting properties of the geometry of the projective structure on
$\Omega^{\mathbb{K}}_{\rho}/\rho(\pi_{1}(S))$ when $\rho$ is close enough to
the Fuchsian locus. In particular, Theorem 8.5 proves that, for $n\leq 63$,
the manifold $M^{\mathbb{C}}$, is diffeomorphic to a fiber bundle over the
surface: this result proves Conjecture 6.11 by Dumas and Sanders in this
special case.

The topology of the manifold $M^{\mathbb{R}}$ was also studied by Guichard and
Wienhard (announced in [24, Remark 11.4(ii)]). They also saw that it is
diffeomorphic to a fiber bundle over the surface with fiber $F^{\mathbb{R}}$.

More recent work about Conjecture 6.11 uses different methods, which don’t
involve Higgs bundles. In a joint work with Qiongling Li [4], we proved the
following theorem.

######  Theorem 8.6 (Alessandrini–Li [4]).

Let $\Omega$ be the domain of discontinuity described in [24] of a quasi-
Hitchin representation $\rho$, where:

  1. $1)$

$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PGL}}(2n,{\mathbb{C}})$, and
$\Omega\subset{\mathbb{CP}}^{2n-1}$, or

  2. $2)$

$\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PGL}}(n,{\mathbb{C}})$, and
$\Omega\subset\mathcal{F}_{1,n-1}$, the partial flag manifold parametrizing
flags made of lines and hyperplanes.

Then, for every $n$, the quotient $M=\Omega/\pi_{1}(S)$ is homeomorphic to the
total space of a continuous fiber bundle over $S$.

This theorem proves Conjecture 6.11, for infinitely many cases. The method we
use for the proof has broader scope than the method we used to prove Theorem
8.5, but it does not allow us to completely understand the topology of the
fiber, nor it gives information about the geometric structures.

In a joint work with Maloni and Wienhard [6], we proved Conjecture 6.11 in
another case:

######  Theorem 8.7 (Alessandrini–Maloni–Wienhard).

Let $\rho\colon\pi_{1}(S)\rightarrow{\mathsf{PSp}}(4,{\mathbb{C}})$ be a
quasi-Hitchin representation, and let $\Omega$ be the domain of discontinuity
described in [24] of $\rho$ in $\mathrm{Lag}\big{(}{\mathbb{C}}^{4}\big{)}$,
the Lagrangian Grassmannian of ${\mathbb{C}}^{4}$. Then the quotient
$M=\Omega/\pi_{1}(S)$ is homeomorphic to the total space of a continuous fiber
bundle over $S$.

The work continues giving an explicit description of the fiber. Conjecture
6.11 is still open in general, and it is the subject of active research.

### Acknowledgements

I am grateful to Qiongling Li for the collaboration that brought many of the
results surveyed here, to Steve Bradlow, Brian Collier, John Loftin and Anna
Wienhard for interesting discussions about this topic and to the anonymous
referees for their useful comments on the first draft of the paper. The mini-
course was funded by the UIC NSF RTG grant DMS-1246844, L.P. Schaposnik’s UIC
Start up fund, and NSF DMS 1107452, 1107263, 1107367 “RNMS: GEometric
structures And Representation varieties” (the GEAR Network).

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