Non-Abelian Hodge Theory and Related Topics

Non-Abelian Hodge Theory and Related Topics††This paper is a contribution to
the Special Issue on Integrability, Geometry, Moduli in honor of Motohico
Mulase for his 65th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Mulase.html

Pengfei HUANG †‡

P. Huang

† School of Mathematics, University of Science and Technology of China, Hefei
230026, China<EMAIL_ADDRESS>
‡ Laboratoire J.A. Dieudonné, Université Côte d’Azur, CNRS, 06108 Nice, France
<EMAIL_ADDRESS>
Received August 23, 2019, in final form April 08, 2020; Published online April
19, 2020

This paper is a survey aimed on the introduction of non-Abelian Hodge theory
that gives the correspondence between flat bundles and Higgs bundles. We will
also introduce some topics arising from this theory, especially some recent
developments on the study of the relevant moduli spaces together with some
interesting open problems.

non-Abelian Hodge theory; $\lambda$-connection; moduli space; conformal limit;
Hitchin section; oper; stratification; twistor space

14D20; 14D21; 32G20; 53C07; 57N80

##  1 Introduction

The non-Abelian Hodge theory, which is mainly based on the work of Corlette
[16] and Donaldson [21] on harmonic maps (flat bundles), and the work of
Hitchin [34] and Simpson [56] on Higgs bundles, gives a correspondence between
semisimple flat bundles over a compact Kähler manifold $X$ and polystable
Higgs bundles over the same manifold with vanishing Chern classes. This
correspondence can be generalized to a statement concerning $\lambda$-flat
bundles, a topic introduced by Deligne [17] and further developed by Simpson
[61, 62, 63]. More precisely, for arbitrary
$\lambda_{1},\lambda_{2}\in\mathbb{C}$, Mochizuki [45] established a
correspondence between the categories of polystable $\lambda_{1}$-flat bundles
with vanishing Chern classes and polystable $\lambda_{2}$-flat bundles with
vanishing Chern classes. In particular, when $\lambda_{1}=1$ and
$\lambda_{2}=0$, this correspondence recovers the original non-Abelian Hodge
correspondence. Therefore, in our setting, especially in this paper, non-
Abelian Hodge theory means a correspondence between flat bundles, Higgs
bundles and $\lambda$-flat bundles. All these objects are connected by the
existence of pluri-harmonic metrics (see Section 2.1), which produce a new
object: harmonic bundle.

All the work mentioned above arise from the study of a classical problem in
the non-Higgs setting: a correspondence between the existence of certain
special metrics on a vector bundle and the stability of that bundle.111We
usually call this correspondence the Kobayashi–Hitchin correspondence. This
correspondence builds a bridge between algebraic geometric side of stability
and differential geometric side of existence of pluri-harmonic metrics. The
study of this kind of problem can be dated back to Narasimhan and Seshadri’s
work on the stability of vector bundles [51]. Their theorem states that a
vector bundle over a compact Riemann surface is stable if and only if it
arises from an irreducible projectively unitary representation of the
fundamental group of that Riemann surface. The Narasimhan–Seshadri theorem was
latter reproved by Donaldson with a differential geometric method [19], which
relates the stability of vector bundles and the existence of certain special
metrics. This celebrating idea was later generalized to higher-dimensional
compact Kähler manifolds by Donaldson [20], Uhlenbeck and Yau [64].

There is a natural filed arising from non-Abelian Hodge theory, the study of
the corresponding moduli spaces of these objects. From Simpson’s work on the
construction of the moduli spaces [59, 60, 61], we have the following four
moduli spaces:

  * •

Betti moduli space $M_{\rm B}(X,r)$: the moduli space of rank $r$
representations $\pi_{1}(X)\\!\to\\!{\rm GL}(r,\mathbb{C})$;

  * •

de Rham moduli space $M_{\rm dR}(X,r)$: the moduli space of rank $r$ flat
bundles over $X$;

  * •

Dolbeault moduli space $M_{\rm Dol}(X,r)$: the moduli space of semistable rank
$r$ Higgs bundles over $X$ with vanishing Chern classes;

  * •

Hodge moduli space $M_{\rm Hod}(X,r)$: the moduli space of semistable rank $r$
$\lambda$-flat bundles over $X$ with vanishing Chern classes.

The study of these moduli spaces arising from non-Abelian Hodge theory shows
that they are also related. More precisely, the Riemann–Hilbert correspondence
implies that $M_{\rm B}(X,r)$ and $M_{\rm dR}(X,r)$ are analytic isomorphic.
Moreover, the Hodge moduli space $M_{\rm Hod}(X,r)$ has a fibration over
$\mathbb{C}$ such that the fibers over 0 and 1 are exactly $M_{\rm Dol}(X,r)$
and $M_{\rm dR}(X,r)$, respectively. The underlying topological spaces of
$M_{\rm Dol}(X,r)$ and $M_{\rm dR}(X,r)$ are homeomorphic, and are
$C^{\infty}$ isomorphic over the stable points.

This paper is a survey aimed on the introduction of non-Abelian Hodge theory
and some related topics, especially some recent developments on the study of
the moduli spaces $M_{\rm dR}(X,r)$, $M_{\rm Dol}(X,r)$ and $M_{\rm
Hod}(X,r)$. Meanwhile, some open problems will be introduced, these
conjectures are very interesting and important on the deep study of moduli
spaces. The non-Abelian Hodge theory first came to the world in 1980s, and
obtained a vast development over past decades. Many branches of mathematics
are shown to be related to this theory, and this theory also sheds light on
the study of other topics. During the development of this theory, there are
many good references written on the introduction of this theory from different
points of view, for example, [2, 3, 11, 14, 18, 29, 41, 54]. There are also
many important applications of this theory, for example, this theory is shown
to be a powerful tool on the study of higher Teichmüller theory [27, 65].
Recently, in [24] the authors applied the non-Abelian Hodge theory to study
the theory of vector valued modular forms and showed a conjecture on three-
termed inequality. For the application of this theory on the study of $M_{\rm
B}(X,r)$ (here $X$ is a compact Riemann surface), a recent survey paper by
Migliorini [43] introduces a kind of compactification of $M_{\rm B}(X,r)$ by
dual boundary complex. The geometric $P=W$ conjecture arises from this
compactification, which is a geometric analogue of the cohomological $P=W$
conjecture on the identification between the perverse filtration on the
cohomology of $M_{\rm Dol}(X,r)$, and the weight filtration on the cohomology
of $M_{\rm B}(X,r)$.

This survey is organized as follows. In the following section, we first give a
brief introduction to non-Abelian Hodge theory by introducing four objects:
Higgs bundles, flat bundles, $\lambda$-flat bundles, and harmonic bundles.
This is based on the work of Corlette [16], Donaldson [21], Hitchin [34],
Simpson [56] and Mochizuki [45]. Then we introduce a recent development of
this theory, that is the systematic work by Mochizuki on the Kobayashi–Hitchin
type correspondence between periodic monopoles and difference modules [47, 48,
49]. The next three sections can be treated as various applications of non-
Abelian Hodge theory, we will fix on compact Riemann surfaces. In the third
section, we introduce the conformal limit conjecture proposed by Gaiotto [26],
which identifies two objects in the corresponding moduli spaces. More
precisely, it relates the Hitchin section in $M_{\rm Dol}$, and the space of
opers in $M_{\rm dR}$. This conjecture was recently confirmed by the authors
of [22], and with some generalization by the authors of [15]. The fourth
section is aimed on the study of moduli spaces by generalizing the natural
$\mathbb{C}^{*}$-action on $M_{\rm Dol}$ to the $\mathbb{C}^{*}$-action on
$M_{\rm Hod}$. This is based on Simpson’s work [63], which gives the
Białynicki-Birula stratification of $M_{\rm Hod}(X,r)$ into locally closed
subsets. In particular, it gives the oper stratification of $M_{\rm dR}(X,r)$,
and recovers the Białynicki-Birula stratification of $M_{\rm Dol}(X,r)$. The
Hitchin sections and opers introduced in the third section, appears as a
special stratum of the stratifications of $M_{\rm Dol}$ and $M_{\rm dR}$,
respectively. Some interesting conjectures on moduli spaces proposed by
Simpson [63] are introduced in this section. Moreover, we give an explicit
description between the Harder–Narasimhan filtration and the Simpson
filtration for rank 3 flat bundles. The last section is aimed on the study of
twistor spaces. We first introduce the Hitchin’s twistor construction for
general hyper-Kähler manifolds, which applies in particular to the moduli
spaces with hyper-Kähelr structure. Then we introduce Deligne’s interpretation
of Hitchin twistor space in terms of flat $\lambda$-connections, which
produces a new twistor space which is analytic isomorphic to the old one. But
Deligne’s construction is from a different viewpoint, that is, by gluing the
moduli space $M_{\rm Hod}(X,r)$ over $X$ and the moduli space $M_{\rm
Hod}(\bar{X},r)$ over the conjugate $\bar{X}$. Finally, we introduce our
recent work [37] on a new interpretation of the Deligne–Hitchin twistor space.
It can be treated as a generalization of Deligne’s construction for the case
of compact Riemann surfaces. We define de Rham sections for this twistor
space, and show they have weight 1 property, which implies the twistor space
contains ample rational curves. As an application, we obtain a Torelli-type
theorem for this twistor space.

##  2 Non-Abelian Hodge theory

###  2.1 Higgs bundles, flat bundles and $\boldsymbol{\lambda}$-flat bundles

Let $(X,\omega)$ be a compact Kähler manifold, where $\omega$ is the Kähler
form. We have three different ways to define a harmonic bundle: from Higgs
bundles, from flat bundles, and from $\lambda$-flat bundles. The non-Abelian
Hodge theory tells us in fact they are equivalent, that is, the non-Abelian
Hodge theory gives a correspondence between these objects.

To begin this theory, we first give the following notations for reader’s
convenience:

  * •

$C^{\infty}(X,\mathbb{C})$: the space of smooth complex valued functions on
$X$;

  * •

$E$: a complex vector bundle over $X$;

  * •

$T^{*}_{\mathbb{C}}X$: complexified cotangent bundle, it has a decomposition

$\displaystyle T^{*}_{\mathbb{C}}X=(T^{*}X)^{1,0}\bigoplus(T^{*}X)^{0,1};$

  * •

$\Omega^{p,q}_{X}$: the exterior algebra bundle of $(p,q)$-type, that is,

$\displaystyle\Omega^{p,q}_{X}=\bigwedge\nolimits^{\\!p}\big{(}(T^{*}X)^{1,0}\big{)}\bigotimes\bigwedge\nolimits^{\\!q}\big{(}(T^{*}X)^{0,1}\big{)};$

  * •

$\mathcal{A}^{k}(X,E)$: the space of smooth complex $k$-forms on $X$ with
values in $E$, that is,

$\displaystyle\mathcal{A}^{k}(X,E)=C^{\infty}(X,E\bigotimes\bigwedge\nolimits^{\\!k}(T^{*}_{\mathbb{C}}X));$

  * •

$\mathcal{A}^{p,q}(X,E)$: the space of smooth $(p,q)$-forms on $X$ with values
on $E$, that is,

$\displaystyle\mathcal{A}^{p,q}(X,E)=C^{\infty}\big{(}X,E\bigotimes\Omega^{p,q}_{X}\big{)}.$

Then we have the decomposition

$\displaystyle\mathcal{A}^{k}(X,E)=\bigoplus_{p+q=k}\mathcal{A}^{p,q}(X,E).$

####  2.1.1 Higgs bundles and harmonic bundles

A holomorphic structure on $E$ is a $\mathbb{C}$-linear operator
$\bar{\partial}_{E}\colon\mathcal{A}^{0}(X,E)\to\mathcal{A}^{0,1}(X,E)$ such
that

  1. (1)

$\bar{\partial}_{E}(fs)=\bar{\partial}(f)s+f\bar{\partial}_{E}(s)$ for any
$f\in C^{\infty}(X,\mathbb{C})$ and $s\in\mathcal{A}^{0}(X,E)$,

  2. (2)

$(\bar{\partial}_{E})^{2}=0$;

where the second condition is obtained under the natural extension
$\bar{\partial}_{E}\colon\mathcal{A}^{p,q}(X,E)\to$ $\mathcal{A}^{p,q+1}(X,E)$
for any integers $p,q\geq 0$. Such a pair $\big{(}E,\bar{\partial}_{E}\big{)}$
is called a holomorphic vector bundle, sometimes we use a single $\mathcal{E}$
to denote a holomorphic vector bundle. By Koszul–Malgrange theorem, this is
equivalent to the usual definition of a holomorphic vector bundle by
holomorphic transition functions.

######  Definition 2.1.

A Higgs bundle over $X$ is a holomorphic vector bundle
$\big{(}E,\bar{\partial}_{E}\big{)}$ together with a map $\theta\colon E\to
E\otimes\Omega_{X}^{1}$ which is holomorphic and integrable, i.e.,
$\bar{\partial}_{E}(\theta)=0$ and $\theta\wedge\theta=0$. Such a map $\theta$
is called a Higgs field and the triple $(E,\bar{\partial}_{E},\theta)$ denotes
a Higgs bundle.

######  Definition 2.2.

A Higgs bundle $(E,\bar{\partial}_{E},\theta)$ is called stable (resp.
semistable) if for any proper coherent subsheaf $F$ of
$0<\operatorname{rk}(F)<\operatorname{rk}(E)$ and
$\theta(F)\subseteq\theta\otimes\Omega_{X}^{1}$ such that $E/F$ is torsion-
free, we have

$\displaystyle\mu(F)<(\text{resp.}\ \leq)\ \mu(E),$

where $\mu(E):=\frac{\deg(E)}{\operatorname{rk}(E)}$ denotes the slope of $E$,
and $\deg(E):=\int_{X}c_{1}(E)\cdot[\omega]^{n-1}$ is the degree of $E$. It is
called polystable if it is the direct sum of stable Higgs bundles of the same
slope as $\mu(E)$.

Given a Hermitian metric $h$ on the Higgs bundle
$\big{(}E,\bar{\partial}_{E},\theta\big{)}$, then $h$ and $\bar{\partial}_{E}$
uniquely determine an $(1,0)$-type operator $\partial_{E,h}$ such that
$\nabla_{h}:=\partial_{E,h}+\bar{\partial}_{E}$ is a unitary connection. Here
a connection $\nabla$ on a vector bundle $E$ is an operator
$\nabla\colon\mathcal{A}^{0}(X,E)\to\mathcal{A}^{1}(X,E)$ that satisfies the
Leibniz rule $\nabla(fs)=df\otimes s+f\nabla(s)$ for any $f\in
C^{\infty}(X,\mathbb{C})$ and $s\in\mathcal{A}^{0}(X,E)$. A connection
$\nabla$ is said to be unitary with respect to the metric $h$ if it preserves
this metric, i.e., if $dh(u,v)=h(\nabla(u),v)+h(u,\nabla(v))$ for any
$u,v\in\mathcal{A}^{0}(X,E)$. This unique unitary connection is called the
Chern connection, and its curvature $F_{h}:=\nabla_{h}\comp\nabla_{h}$ is
called the Chern curvature. A connection $\nabla$ on $E$ is said to be flat if
its curvature $F_{\nabla}:=\nabla\comp\nabla$ vanishes under the natural
extension $\nabla\colon\mathcal{A}^{k}(X,E)\to\mathcal{A}^{k+1}(X,E)$ for any
integer $k\geq 0$.

With $h$, the Higgs field $\theta$ determines an adjoint operator
$\theta_{h}^{\dagger}\in\mathcal{A}^{0,1}(X,\operatorname{End}(E))$ by

$\displaystyle h(\theta(u),v)=h\big{(}u,\theta_{h}^{\dagger}(v)\big{)}$

for any $u,v\in\mathcal{A}^{0}(X,E)$. The new obtained connection
$\mathbb{D}^{1}:=\partial_{E,h}+\bar{\partial}_{E}+\theta+\theta_{h}^{\dagger}$
is usually called a Hitchin–Simpson connection, and its curvature
$F_{(\bar{\partial}_{E},\theta,h)}:=\mathbb{D}^{1}\comp\mathbb{D}^{1}$ is
called a Hitchin–Simpson curvature.

######  Definition 2.3.

$h$ is called a pluri-harmonic metric on the Higgs bundle
$\big{(}E,\bar{\partial}_{E},\theta\big{)}$ if the Hitchin–Simpson connection
$\mathbb{D}^{1}:=\partial_{E,h}+\bar{\partial}_{E}+\theta+\theta_{h}^{\dagger}$
is flat, that is, if $\big{(}E,\mathbb{D}^{1}\big{)}$ is a flat bundle. A
Higgs bundle with a pluri-harmonic metric is called a harmonic bundle, and
denoted as $\big{(}E,\bar{\partial}_{E},\theta,h\big{)}$.

####  2.1.2 Flat bundles and harmonic bundles

A flat bundle over $X$ is a pair $(E,\nabla)$ that consists of a complex
vector bundle $E$ and a flat connection $\nabla$.

######  Definition 2.4.

A flat bundle $(E,\nabla)$ is called irreducible if it has no non-zero proper
flat subbundle, and it is called semisimple if it is a direct sum of
irreducible flat bundles.

Let $(E,\nabla)$ be a flat bundle over $X$ with a Hermitian metric $h$, then
$h$ induces a unique decomposition of $\nabla$:

$\displaystyle\nabla=\nabla_{h}+\Phi_{h}$

such that:

  * •

$\nabla_{h}$ is a unitary connection;

  * •

$\Phi_{h}\in\mathcal{A}^{1}(X,\operatorname{End}(E))$ is a self-adjoint
operator.

To see the decomposition, we first decompose $\nabla$ into its $(1,0)$ and
$(0,1)$ parts: $\nabla=\nabla^{1,0}+\nabla^{0,1}$. Then define a unique
$(0,1)$-type operator $\delta_{E,h}^{\prime\prime}$ such that
$\nabla^{1,0}+\delta_{E,h}^{\prime\prime}$ preserves $h$ and a unique
$(1,0)$-type operator $\delta_{E,h}^{\prime}$ such that
$\nabla^{0,1}+\delta_{E,h}^{\prime}$ preserves $h$. Finally, let

$\displaystyle\partial_{E,h}=\frac{1}{2}\big{(}\nabla^{1,0}+\delta_{E,h}^{\prime}\big{)},\qquad$
$\displaystyle\bar{\partial}_{E,h}=\frac{1}{2}\big{(}\nabla^{0,1}+\delta_{E,h}^{\prime\prime}\big{)},$
$\displaystyle\theta_{E,h}=\frac{1}{2}\big{(}\nabla^{1,0}-\delta_{E,h}^{\prime}\big{)},\qquad$
$\displaystyle\theta_{V,h}^{\dagger}=\frac{1}{2}\big{(}\nabla^{0,1}-\delta_{E,h}^{\prime\prime}\big{)},$

then $\nabla_{h}:=\partial_{E,h}+\bar{\partial}_{E,h}$ and
$\Phi_{h}:=\theta_{E,h}+\theta_{E,h}^{\dagger}$ is the desired decomposition.

######  Definition 2.5.

$h$ is called a pluri-harmonic metric on the flat bundle $(E,\nabla)$ if

$\big{(}\bar{\partial}_{E,h}+\theta_{E,h}\big{)}^{2}=0,$

that is, if $\big{(}E,\bar{\partial}_{E,h},\theta_{E,h}\big{)}$ is a Higgs
bundle. A flat bundle with a pluri-harmonic metric is called a harmonic
bundle, and denoted as $(E,\nabla,h)$.

####  2.1.3 $\boldsymbol{\lambda}$-flat bundles and harmonic bundles

For $\lambda\in\mathbb{C}$, the notion of flat $\lambda$-connection as the
interpolation of usual flat connection and Higgs field was suggested by
Deligne [17], illustrated by Simpson in [61] and further studied in [62, 63].

Let $\mathcal{E}=\big{(}E,\bar{\partial}_{E}\big{)}$ be a holomorphic vector
bundle over $X$. The following definitions of $\lambda$-flat bundles mainly
come from [61] and [45] (see also [37]).

######  Definition 2.6.

  1. (1)

A _holomorphic $\lambda$-connection_ on $\mathcal{E}$ is a $\mathbb{C}$-linear
map $D^{\lambda}\colon\mathcal{E}\to\mathcal{E}\otimes\Omega_{X}^{1}$ that
satisfies the following $\lambda$-twisted Leibniz rule:

$\displaystyle D^{\lambda}(fs)=fD^{\lambda}s+\lambda s\otimes df,$

where $f$ and $s$ are holomorphic sections of $\mathcal{O}_{X}$ and $E$,
respectively. If
$\big{(}D^{\lambda}+\bar{\partial}_{E}\big{)}\comp\big{(}D^{\lambda}+\bar{\partial}_{E}\big{)}=0$
under the natural extension
$D^{\lambda}\colon\mathcal{E}\otimes\Omega_{X}^{p}\to\mathcal{E}\otimes\Omega_{X}^{p+1}$
for any integer $p\geq 0$, then we call $D^{\lambda}$ a (holomorphic) _flat
$\lambda$-connection_, and $\big{(}\mathcal{E},D^{\lambda}\big{)}$ a
(holomorphic) _$\lambda$ -flat bundle_.

  2. (2)

A _$C^{\infty}$ $\lambda$-connection_ on $E$ is a $\mathbb{C}$-linear map
$\mathbb{D}^{\lambda}\colon\mathcal{A}^{0}(X,E)\to\mathcal{A}^{1}(X,E)$ that
satisfies the following $\lambda$-twisted Leibniz rule:

$\displaystyle\mathbb{D}^{\lambda}(fs)=f\mathbb{D}^{\lambda}s+\lambda
s\otimes\partial f+s\otimes\bar{\partial}f,$

where $f\in C^{\infty}(X,\mathbb{C})$ and $s\in\mathcal{A}^{0}(X,E)$. If
$\mathbb{D}^{\lambda}\comp\mathbb{D}^{\lambda}=0$ under the natural extension
$\mathbb{D}^{\lambda}\colon\mathcal{A}^{p}(X,E)\to\mathcal{A}^{p+1}(X,E)$ for
any integer $p\geq 0$, then we call $\mathbb{D}^{\lambda}$ a ($C^{\infty}$)
_flat $\lambda$-connection_, and $\big{(}E,\mathbb{D}^{\lambda}\big{)}$ a
($C^{\infty}$) _$\lambda$ -flat bundle_.

Clearly, $\lambda=1$ and $0$ correspond to the usual flat connection and Higgs
field, respectively. If we work on a compact Riemann surface, then every
$\lambda$-connection is automatically flat. Giving a holomorphic flat
$\lambda$-connection $D^{\lambda}$ on $\mathcal{E}$ is equivalent to giving a
$C^{\infty}$ flat $\lambda$-connection $\mathbb{D}^{\lambda}$ on $E$. From now
on, we will denote a $\lambda$-flat bundle as
$\big{(}E,\bar{\partial}_{E},D^{\lambda}\big{)}$ in holomorphic category and
$\big{(}E,\mathbb{D}^{\lambda}\big{)}$ in $C^{\infty}$ category.

######  Definition 2.7.

A $\lambda$-flat bundle $\big{(}E,\mathbb{D}^{\lambda}\big{)}$ over $X$ is
called _stable_ (resp.  _semistable_) if for any $\lambda$-flat subbundle
$\big{(}V,\mathbb{D}^{\lambda}|_{V}\big{)}$ of
$0<\operatorname{rk}(V)<\operatorname{rk}(E)$, we have the following
inequality

$\displaystyle\mu(V)<(\text{resp.}\leq)\ \mu(E),$

where $\mu(E)$ is the slope of the bundle $E$ as in Definition 2.2. It is
_polystable_ if it decomposes as a direct sum of stable $\lambda$-flat bundles
with the same slope.

Let $\big{(}E,\mathbb{D}^{\lambda}\big{)}$ be a $\lambda$-flat bundle, and $h$
be a Hermitian metric on $E$. Then $h$ induces a unique decomposition of
$\mathbb{D}^{\lambda}$:

$\displaystyle\mathbb{D}^{\lambda}=\lambda\partial_{h}+\theta_{h}+\bar{\partial}_{h}+\lambda\theta_{h}^{\dagger}$

such that $\nabla_{h}:=\partial_{h}+\bar{\partial}_{h}$ is a unitary
connection and
$\Phi_{h}:=\theta_{h}+\theta_{h}^{\dagger}\in\mathcal{A}^{1}(X,\operatorname{End}(E))$
is a self-adjoint operator.

In fact, when $\lambda=0$, this is the trivial decomposition into different
types which defines a Higgs bundle structure, that is,
$\mathbb{D}^{0}=\bar{\partial}_{E}+\theta$ for $\bar{\partial}_{E}$ defines a
holomorphic structure on $E$ and $\theta$ defines a Higgs filed. When
$\lambda\neq 0$, we decompose $\mathbb{D}^{\lambda}$ into its $(1,0)$-part
$d_{E}^{\prime}$ and $(0,1)$-part $d_{E}^{\prime\prime}$ that defines a
holomorphic structure on $E$. From $h$ and $d_{E}^{\prime}$, we have a
(0,1)-operator $\delta_{h}^{\prime\prime}$ determined by the condition

$\displaystyle\lambda\partial
h(u,v)=h(d_{E}^{\prime}u,v)+h(u,\delta_{h}^{\prime\prime}v).$

Similarly, $h$ and $d_{E}^{\prime\prime}$ provides a $(1,0)$-operator
$\delta_{h}^{\prime}$ via the condition

$\displaystyle\bar{\partial}h(u,v)=h(d_{E}^{\prime\prime}u,v)+h(u,\delta_{h}^{\prime}v).$

One easily checks that

$\displaystyle\delta_{h}^{\prime}(fs)=f\delta^{\prime}_{h}(s)+\partial(f)\otimes
s,$
$\displaystyle\delta_{h}^{\prime\prime}(fs)=f\delta_{h}^{\prime\prime}(s)+\bar{\lambda}\bar{\partial}(f)\otimes
v$

for any $f\in C^{\infty}(X,\mathbb{C})$ and $s\in\mathcal{A}^{0}(X,E)$. We
introduce the following four operators

$\displaystyle\partial_{h}:=\frac{1}{1+|\lambda|^{2}}\big{(}\bar{\lambda}d_{E}^{\prime}+\delta_{h}^{\prime}\big{)},\qquad$
$\displaystyle\bar{\partial}_{h}:=\frac{1}{1+|\lambda|^{2}}\big{(}d_{E}^{\prime\prime}+\lambda\delta_{h}^{\prime\prime}\big{)},$
$\displaystyle\theta_{h}:=\frac{1}{1+|\lambda|^{2}}\big{(}d_{E}^{\prime}-\lambda\delta_{h}^{\prime}\big{)},\qquad$
$\displaystyle\theta_{h}^{\dagger}:=\frac{1}{1+|\lambda|^{2}}\big{(}\bar{\lambda}d_{E}^{\prime\prime}-\delta_{h}^{\prime\prime}\big{)}.$

They satisfy that

$\displaystyle d_{E}^{\prime}=\lambda\partial_{h}+\theta_{h},\qquad$
$\displaystyle
d_{E}^{\prime\prime}=\bar{\partial}_{h}+\lambda\theta_{h}^{\dagger},$
$\displaystyle\delta_{h}^{\prime}=\partial_{h}-\bar{\lambda}\theta_{h},\qquad$
$\displaystyle\delta_{h}^{\prime\prime}=\bar{\lambda}\bar{\partial}_{h}-\theta_{h}^{\dagger}.$

By direct calculation, $\partial_{h}$ and $\bar{\partial}_{h}$ obey the usual
Leibniz rule, so $D_{h}=\partial_{h}+\bar{\partial}_{h}$ is a usual
connection. Moreover, $D_{h}$ is a unitary connection with respect to $h$. By
definition, $\theta_{h}\in
C^{\infty}\big{(}X,\operatorname{End}(E)\otimes\Omega_{X}^{1,0}\big{)}$,
$\theta_{h}^{\dagger}\in
C^{\infty}\big{(}X,\operatorname{End}(E)\otimes\Omega_{X}^{0,1}\big{)}$, and
$\theta_{h}^{\dagger}$ is the adjoint of $\theta_{h}$ in the sense that
$h(\theta_{h}(u),v)=h\big{(}u,\theta_{h}^{\dagger}(v)\big{)}$.

######  Definition 2.8.

$h$ is called a _pluri-harmonic metric_ on the $\lambda$-flat bundle
$\big{(}E,\mathbb{D}^{\lambda}\big{)}$ if
$(\bar{\partial}_{h}+\theta_{h})^{2}=0$, that is, if
$\big{(}E,\bar{\partial}_{h},\theta_{h}\big{)}$ is a Higgs bundle. A
$\lambda$-flat bundle with a pluri-harmonic metric is called a _harmonic
bundle_ , and denoted as $\big{(}E,\mathbb{D}^{\lambda},h\big{)}$.

####  2.1.4 Non-Abelian Hodge correspondence

The non-Abelian Hodge theory, arises from the existence of pluri-harmonic
metric for the objects we introduced above. This is based on the work of
Donaldson [21] and Corlette [16] on harmonic maps (flat bundles), Hitchin [34]
and Simpson [56] on Higgs bundles, and Mochizuki [45] on $\lambda$-flat
bundles.

######  Theorem 2.9.

Let $(X,\omega)$ be a compact Kähler manifold. Then

  1. $(1)$

$($Donaldson [21], Corlette [16]) A flat bundle $(E,\nabla)$ over $X$ admits a
pluri-harmonic metric if and only if it is semisimple;

  2. $(2)$

$($Hitchin [34], Simpson [56]) A Higgs bundle
$\big{(}E,\bar{\partial}_{E},\theta\big{)}$ over $X$ admits a pluri-harmonic
metric if and only if it is polystable with vanishing Chern classes;

  3. $(3)$

$($Mochizuki [45]) A $\lambda$-flat bundle
$\big{(}E,\mathbb{D}^{\lambda}\big{)}$ $(\lambda\neq 0)$ over $X$ admits a
pluri-harmonic metric if and only if it is polystable with vanishing Chern
classes.

Moreover, in all above cases, the pluri-harmonic metric is unique up to scalar
multiplicities.

We give a brief description of above theorem here. By the work of Corlette and
Donaldson, for any given flat bundle $(E,\nabla)$ of rank $r$, let
$\rho\colon\pi_{1}(X)\to{\rm GL}(r,\mathbb{C})$ be the associated monodromy
representation. If $(E,\nabla)$ is semisimple, then there is a unique map
$h_{\rho}\colon\tilde{X}\to{\rm GL}(r,\mathbb{C})/O(r)$ which is
$\rho$-equivariant and harmonic (i.e., $h_{\rho}$ achieves the minimum of the
energy functional $E(h_{\rho}):=\int_{X}|dh_{\rho}|^{2}d\textrm{vol}$). We
call such harmonic map a harmonic metric. Then by a theorem of Siu and Sampson
on harmonic maps, over any compact Kähler manifold, each harmonic map is
pluri-harmonic, i.e., $D^{0,1}\big{(}(df_{\rho})^{0,1}\big{)}=0$, where $D$ is
the descend of the pull back of the canonical Levi-Civita connection on the
tangent bundle of the symmetric space ${\rm GL}(r,\mathbb{C})/O(r)$ by
$h_{\rho}$. Therefore, it induces a Higgs bundle structure. On the other hand,
by the work of Hitchin and Simpson, for any given Higgs bundle
$\big{(}E,\bar{\partial}_{E},\theta\big{)}$ of rank $r$. If it is polystable,
then it admits a Hermitian–Einstein metric $h$, i.e., the Hitchin–Simpson
curvature $F_{(\bar{\partial}_{E},\theta,h)}$ satisfies the
Hermitian–Yang–Mills equation
$\Lambda_{\omega}F_{(\bar{\partial}_{E},\theta,h)}=\alpha{\rm Id}$ for some
constant $\alpha$. Moreover, if such Higgs bundle has vanishing first and
second Chern characters, then a Bogomolov–Gieseker inequality will imply
$F_{(\bar{\partial}_{E},\theta,h)}=0$. Therefore, it induces a flat bundle
structure. This ideas were generalized by Mochizuki to study the case of
$\lambda$-flat bundles [45], where he considered this object over higher-
dimensional smooth complex projective varieties with a simple normal crossing
divisor. In [45], he built the Kobayashi–Hitchin correspondence between
polystable parabolic $\lambda$-flat bundles with vanishing first and second
parabolic Chern characters and tame harmonic bundles. Mochizuki’s method
relies on a $\varepsilon$-perturbation technique developed by himself on the
study of higher-dimensional Kobayashi–Hitchin correspondence between
polystable parabolic Higgs bundles with vanishing first and second parabolic
Chern characters and tame harmonic bundles [44].

Therefore, we obtain an equivalence between the categories of these objects:

######  Corollary 2.10 (non-Abelian Hodge correspondence).

Let $(X,\omega)$ be a compact Kähler manifold. Then for each
$\lambda\in\mathbb{C}$, we have one to one correspondence between the
equivalence classes of polystable $\lambda$-flat bundles with vanishing Chern
classes, the equivalence classes of polystable Higgs bundles with vanishing
Chern classes, and the equivalence classes of semisimple flat bundles, through
harmonic bundles.

######  Remark 2.11.

  1. (1)

The correspondence between harmonic $\lambda$-flat bundles and harmonic Higgs
bundles is given as follows:

$\displaystyle\begin{split}&\big{(}E,\mathbb{D}^{\lambda},h\big{)}\longmapsto\big{(}E,\bar{\partial}_{h},\theta_{h},h\big{)},\\\
&\big{(}E,\bar{\partial}_{E},\theta,h\big{)}\longmapsto\big{(}E,\bar{\partial}_{E}+\lambda\theta_{h}^{\dagger},\lambda\partial_{E,h}+\theta,h\big{)},\end{split}$

and the correspondence between harmonic $\lambda$-bundles and harmonic flat
bundles is given as follows:

$\displaystyle\big{(}E,\mathbb{D}^{\lambda},h\big{)}\longmapsto\big{(}E,\partial_{h}+\bar{\partial}_{h}+\theta_{h}+\theta_{h}^{\dagger},h\big{)},$
$\displaystyle(E,\nabla,h)\longmapsto\big{(}E,\bar{\partial}_{E,h}+\lambda\theta_{E,h}^{\dagger},\lambda\partial_{E,h}+\theta_{E,h},h\big{)}.$

  2. (2)

A natural direction of the study of non-Abelian Hodge correspondence is the
generalization to non-compact case. In regular case (tame case), Simpson
established the correspondence between tame harmonic bundles and parabolic
Higgs bundles for open curves [57]. The higher-dimensional generalization is
obtained by Biquard [6] for smooth divisor case and by Mochizuki [44, 45] for
general case. In irregular case (wild case), Biquard and Boalch built the
correspondence for curves [7]. Later Mochizuki generalized it to higher-
dimensional case [46]. Recently, the authors in [32] built the correspondence
to the context of projective varieties with Kawamata log terminal (brief as
klt) singularities.

  3. (3)

Another natural generalization is considering the corresponding case for real
Lie groups. In [10, 28], the authors considered the principal $G$-Higgs
bundles for real Lie group $G$. They studied the Kobayashi–Hitchin
correspondence for principal $G$-Higgs bundles, and therefore, they built the
non-Abelian Hodge correspondence for such objects.

  4. (4)

The non-Abelian Hodge theory for varieties over a field of characteristic $p$
was built by Ogus and Vologodsky in [52]. The $p$-adic version was suggested
by Faltings in [23] and finished by Abbes, Gros and Tsuji in [1].

Denote by $M_{\rm dR}(X,r)$ the coarse moduli space of completely reducible
flat bundles of rank $r$, and by $M_{\rm Dol}(X,r)$ the coarse moduli space of
semistable Higgs bundles of rank $r$ with vanishing Chern classes. Then by the
non-Abelian Hodge correspondence, we can obtain a map between moduli spaces:

$\displaystyle{\rm NAH}\colon\ M_{\rm Dol}(X,r)$ $\displaystyle\longrightarrow
M_{\rm dR}(X,r),$ $\displaystyle\big{[}E,\bar{\partial}_{E},\theta\big{]}$
$\displaystyle\longmapsto\big{[}E,\bar{\partial}_{E}+\theta^{\dagger_{h}},\partial_{E,h}+\theta\big{]},$

where $h$ is the unique pluri-harmonic metric of
$\big{[}E,\bar{\partial}_{E},\theta\big{]}$. We call such map the non-Abelian
Hodge map, it will be used in the next two sections.

###  2.2 Periodic monopoles and difference modules

The Kobayashi–Hitchin correspondence, which relates the stability of bundles
(i.e., Higgs bundles, flat bundles, $\lambda$-flat bundles) and the existence
of certain special metrics (i.e., Hermitian–Einstein metrics, pluri-harmonic
metrics), plays an important role on non-Abelian Hodge theory. Theorem 2.9 is
obtained under various work of Kobayashi–Hitchin correspondence.

Recently, Mochizuki built the Kobayashi–Hitchin type correspondence between
periodic monopoles and difference modules [47, 48, 49], as the non-Abelian
Hodge theory for monopoles with periodicity. Here we give a brief introduction
following these papers.

Let $\Gamma\subset\mathbb{R}^{3}$ be a lattice, with the quotient space
$\mathcal{M}:=\mathbb{R}^{3}/\Gamma$, which is equipped with a metric
$g_{\mathcal{M}}$ induced from the Euclidean metric of $\mathbb{R}^{3}$. Let
$Z\subset\mathcal{M}$ be a finite subset, let $(E,h)$ be a Hermitian vector
bundle over $\mathcal{M}\backslash Z$ with a unitary connection $\nabla$ and
an anti-self-adjoint endomorphism $\phi$, which is called a Higgs filed.

######  Definition 2.12.

The quadruple $(E,h,\nabla,\phi)$ is called a monopole on
$\mathcal{M}\backslash Z$ if it satisfies the Bogomolny equation:

$\displaystyle F(\nabla)-*\nabla\phi=0,$

where $F(\nabla)=\nabla\comp\nabla$ is the curvature of $\nabla$ and $*$ is
the Hodge star operator with respect to $g_{\mathcal{M}}$. Moreover, it is
called a periodic monopole (resp. doubly periodic monopole, resp. triply
periodic monopole) if $\Gamma\cong\mathbb{Z}$ (resp.
$\Gamma\cong\mathbb{Z}^{2}$, resp. $\Gamma\cong\mathbb{Z}^{3}$).

####  2.2.1 Periodic monopoles of GCK type and parabolic difference modules

In [47], Mochizuki studied and built the Kobayashi–Hitchin type correspondence
between irreducible periodic monopoles with GCK type singularity and stable
parabolic difference modules of degree 0. In this case, $\mathcal{M}\backslash
Z=\big{(}T^{1}\times\mathbb{R}^{2}\big{)}\backslash
Z=\big{(}T^{1}\times\mathbb{C}\big{)}\backslash Z$.

######  Definition 2.13.

A periodic monopole $(E,h,\nabla,\phi)$ on $\mathcal{M}\backslash Z$ is called
of GCK $($generalized Cherkis–Kapustin$)$ type if it satisfies the following
two conditions:

  1. (1)

each point of $P\in Z$ is of Dirac type singularity of the monopole, this is
equivalent to say, $|{\phi_{|}}_{Q}|_{h}=O\big{(}d(P,Q)^{-1}\big{)}$ for any
$Q$ close to $P$ [50];

  2. (2)

$|\phi|_{h}=O(\log|z|)$ and $|F(\nabla)|\to 0$ as $|z|\to\infty$, where $z$ is
the coordinate of $\mathbb{C}$.

Let $\lambda\in\mathbb{C}$, called a twistor parameter, let
$\Phi^{*}\in\operatorname{Aut}(\mathbb{C}[y])$ determined by
$\Phi^{*}(f(y))=f(y+\lambda)$.

######  Definition 2.14.

A difference module on $\mathbb{C}$ (or called $\lambda$-difference module) is
a $\mathbb{C}[y]$-module $\mathbf{V}$ with a complex linear automorphism
$\Phi^{*}$ such that $\Phi^{*}(fs)=\Phi^{*}(f)\cdot\Phi^{*}(s)$ for any
$f\in\mathbb{C}[y]$ and $s\in\mathbf{V}$. It is torsion-free if it is torsion-
free as a $\mathbb{C}[y]$-module. It is of finite type if it is finitely
generated over the algebra
$\mathscr{A}_{\lambda}=\bigoplus_{n\in\mathbb{Z}}\mathbb{C}[y](\Phi^{*})^{n}$
and $\dim_{\mathbb{C}[y]}\mathbf{V}\otimes\mathbb{C}(y)<\infty$, where
$\mathscr{A}_{\lambda}$ has the product determined by $\Phi^{*}\cdot
f(y)=f(y+\lambda)\cdot\Phi^{*}$.

Now let $\mathbf{V}$ be a torsion-free $\lambda$-difference module of finite
type.

######  Definition 2.15.

A parabolic structure at finite place of $\mathbf{V}$ consists of the
followings:

  1. (1)

A free sub $\mathbb{C}[y]$-module $V\subset\mathbf{V}$ such that
$\mathscr{A}_{\lambda}\cdot V=\textbf{V}$ and
$V\otimes_{\mathbb{C}[y]}\mathbb{C}(y)=\mathbf{V}\otimes_{\mathbb{C}[y]}\mathbb{C}(y)$;

  2. (2)

A function $m\colon\mathbb{C}\to\mathbb{Z}_{\geq 0}$ with
$\sum\limits_{x\in\mathbb{C}}m(x)<\infty$, we assume
$V\otimes_{\mathbb{C}[y]}\mathbb{C}[y]_{D}=(\Phi^{*})^{-1}(V)\otimes_{\mathbb{C}[y]}\mathbb{C}[y]_{D}$
for $D:=\\{x\in\mathbb{C}\,|\,m(x)>0\\}$, where $\mathbb{C}[y]_{D}$ denotes
the localization of $\mathbb{C}[y]$ with respect to $y-x$ ($x\in D$);

  3. (3)

For each $x\in\mathbb{C}$, there is a sequence of real numbers
$\mathbb{\alpha}_{x}:=\\{0\leq\alpha_{x,1}<\cdots<\alpha_{x,m(x)}<1\\}$, when
$m(x)=0$, it is assumed to be empty;

  4. (4)

For each $x\in\mathbb{C}$, there is a sequence of lattices
$\mathbf{L}_{x}:=\big{\\{}L_{x,0}=V\otimes_{\mathbb{C}[y]}\mathbb{C}[\\![y-x]\\!],L_{x,i}\subset
V\otimes_{\mathbb{C}[y]}\mathbb{C}(\\!(y-x)\\!)(1\leq i\leq
m(x)-1),L_{x,m(x)}=(\Phi^{*})^{-1}(V)\otimes_{\mathbb{C}[y]}\mathbb{C}[\\![y-x]\\!]\big{\\}}$.

We use the quadruple $(V,m,\\{\alpha_{x},\mathbf{L}_{x}\\}_{x\in\mathbb{C}})$
to denote it.

Let $V$ be a torsion-free $\lambda$-difference module of finite type and let
$\widehat{\mathbf{V}}:=\mathbf{V}\otimes_{\mathbb{C}[y]}\mathbb{C}\big{(}\\!\big{(}y^{-1}\big{)}\\!\big{)}$
be its formal completion at $\infty$.

######  Definition 2.16.

A parabolic structure at $\infty$ of $\mathbf{V}$ is a filtered bundle
$\mathscr{F}_{*}\widehat{\mathbf{V}}=\big{(}\mathscr{F}_{a}\widehat{\mathbf{V}}\,|\,a\in\mathbb{R}\big{)}$
over $\widehat{\mathbf{V}}$ such that:

  1. (1)

For each $a\in\mathbb{R}$, $\mathscr{F}_{a}\widehat{\mathbf{V}}$ is a sub
$\mathbb{C}\big{[}\\!\big{[}y^{-1}\big{]}\\!\big{]}$-module of
$\widehat{\mathbf{V}}$ with
$\mathscr{F}_{a}\widehat{\mathbf{V}}\otimes_{\mathbb{C}[y]}\mathbb{C}\big{(}\\!\big{(}y^{-1}\big{)}\\!\big{)}=\widehat{\mathbf{V}}$;

  2. (2)

$\mathscr{F}_{a}\widehat{\mathbf{V}}\subset\mathscr{F}_{b}\widehat{\mathbf{V}}$
if $a\leq b$;

  3. (3)

$\mathscr{F}_{a+n}\widehat{\mathbf{V}}=y^{n}\mathscr{F}_{a}\widehat{\mathbf{V}}$
for any $a\in\mathbb{R}$ and $n\in\mathbb{Z}$;

  4. (4)

For any $a\in\mathbb{R}$, there exists $\varepsilon>0$ such that
$\mathscr{F}_{a}\widehat{\mathbf{V}}=\mathscr{F}_{a+\varepsilon}\widehat{\mathbf{V}}$.

$\mathscr{F}_{*}\widehat{\mathbf{V}}$ is called good if we have a
decomposition

$\displaystyle\mathscr{F}_{*}\big{(}\widehat{\mathbf{V}}\otimes\mathbb{C}\big{(}\\!\big{(}y^{-1/p}\big{)}\\!\big{)}\big{)}=\bigoplus_{c\in
p^{-1}\mathbb{Z}}\bigoplus_{d\in\mathbb{C}^{*}}\bigoplus_{e\in
S(p)}\mathscr{F}_{*}\widehat{\mathbf{V}}_{c,d,e}$

such that
$\big{(}c^{-1}y^{-c}\Phi^{*}-(1+e)\text{Id}\big{)}\mathscr{F}_{a}\widehat{\mathbf{V}}_{c,d,e}\subset
y^{-1}\mathscr{F}_{a}\widehat{\mathbf{V}}_{c,d,e}$ for any $a\in\mathbb{R}$,
here $p\in\mathbb{Z}_{>0}$ and
$S(p):=\big{\\{}\sum\limits_{i=1}^{p-1}e_{i}y^{-i/p}\,|\,e_{i}\in\mathbb{C}\big{\\}}$.
A parabolic $\lambda$-difference module is a torsion-free $\lambda$-difference
module of finite type with a parabolic structure at finite place and a good
parabolic structure at $\infty$. We denote it as the following:

$\displaystyle\big{(}\mathbf{V},(V,m,\\{\alpha_{x},\mathbf{L}_{x}\\}_{x\in\mathbb{C}}),\mathscr{F}_{*}\widehat{\mathbf{V}}\big{)}.$

For simplicity, we will just use $(\mathbf{V},\bullet)$ to denote such an
object.

######  Definition 2.17.

The degree of a parabolic $\lambda$-difference module $(\mathbf{V},\bullet)$
is given by

$\displaystyle\deg(\mathbf{V},\bullet):=\deg(\mathscr{F}_{0}\mathcal{V})-\sum_{-1<a\leq
0}a\dim_{\mathbb{C}}\mathrm{Gr}_{a}^{\mathscr{F}}\big{(}\widehat{\mathbf{V}}\big{)}-\sum_{c\in\mathbb{Q}}\frac{c}{2}r(c)$
$\displaystyle\hphantom{\deg(\mathbf{V},\bullet):=}{}+\sum_{x\in\mathbb{C}}\sum_{i}(1-\alpha_{x,i})\deg(L_{x,i},L_{x,i-1}),$

where $\mathscr{F}_{0}\mathcal{V}$ is the
$\mathcal{O}_{\mathbb{P}^{1}}$-modules associated to the free
$\mathbb{C}[y]$-module $V$,
$\mathrm{Gr}_{a}^{\mathscr{F}}\big{(}\widehat{\mathbf{V}}\big{)}:=\mathscr{F}_{a}\widehat{\mathbf{V}}/\mathscr{F}_{<a}\widehat{\mathbf{V}}$,
$r(c):=\sum\limits_{d\in\mathbb{C}^{*}}\sum\limits_{e\in
S(p)}\dim_{\mathbb{C}(\\!(y^{-1/p})\\!)}\widehat{\mathbf{V}}_{c,d,e}$ and if
$m(x)>0$, then for $1\leq i\leq m(x)$,

$\displaystyle\deg(L_{x,i},L_{x,i}):=\dim_{\mathbb{C}}(L_{x,i}/(L_{x,i}\cap
L_{x,i-1}))-\dim_{\mathbb{C}}(L_{x,i-1}/(L_{x,i}\cap L_{x,i-1})).$

The slope is given by

$\displaystyle\mu(\mathbf{V},\bullet):=\deg(\mathbf{V},\bullet)/\operatorname{rk}_{\mathbb{C}[y]}(V).$

For any $\mathbb{C}(y)$-subspace
$\widetilde{\mathbf{V}}^{\prime}\subset\widetilde{\mathbf{V}}:=\mathbf{V}\otimes\mathbb{C}(y)$
that is $\Phi^{*}$-invariant, i.e.,
$\phi^{*}\big{(}\widetilde{\mathbf{V}}^{\prime}\big{)}\subset\widetilde{\mathbf{V}}^{\prime}$,
set $V^{\prime}=\widetilde{\mathbf{V}}^{\prime}\cap V$ and
$\mathbf{V}^{\prime}=\mathscr{A}\cdot V^{\prime}$, then there is an induced
structure on $\mathbf{V}^{\prime}$:

$\displaystyle L_{x,i}^{\prime}:=L_{x,i}\cap
V^{\prime},\qquad\widehat{\mathbf{V}}^{\prime}:=\mathbf{V}^{\prime}\otimes_{\mathbb{C}[y]}\mathbb{C}\big{(}\\!\big{(}y^{-1}\big{)}\\!\big{)},\qquad\mathscr{F}_{a}\widehat{\mathbf{V}}^{\prime}:=\mathscr{F}_{a}\widehat{\mathbb{V}}\cap\widehat{\mathbf{V}}^{\prime}.$

This structure is a parabolic structure at finite place and a good parabolic
structure at $\infty$, we denote the sub parabolic $\lambda$-difference module
as
$\big{(}\mathbf{V}^{\prime},m,\\{\alpha_{x},\mathbf{L}_{x}^{\prime}\\}_{x\in\mathbb{C}},\mathscr{F}_{*}\widehat{\mathbf{V}}^{\prime}\big{)}$,
for simplicity, we denote it as $(\mathbf{V}^{\prime},\bullet)$.

######  Definition 2.18.

A parabolic $\lambda$-difference module
$\big{(}\mathbf{V},(V,m,\\{\alpha_{x},\mathbf{L}_{x}\\}_{x\in\mathbb{C}}),\mathscr{F}_{*}\widehat{\mathbf{V}}\big{)}$
is called stable (resp. semistable) if for any non-zero proper
$\mathbb{C}(y)$-subspace $\widetilde{\mathbf{V}}^{\prime}$ that is
$\Phi^{*}$-invariant, we have $\mu(\mathbf{V}^{\prime},\bullet)<(\text{resp.\
$\leq$})\ \mu(\mathbf{V},\bullet)$.

In [47], Mochizuki built the following Kobayashi–Hitchin type correspondence
between periodic monopoles and $\lambda$-difference modules, as the analogue
of the correspondence between harmonic bundles and (poly-)stable
$\lambda$-flat bundles:

######  Theorem 2.19 ([47]).

For each $\lambda\in\mathbb{C}$, there is an one to one correspondence between
irreducible periodic monopoles of $GCK$ type and stable parabolic
$\lambda$-difference modules of degree $0$.

####  2.2.2 Meromorphic doubly periodic monopoles and parabolic
$\boldsymbol{q}$-difference modules

As we have defined, when $\Gamma\cong\mathbb{Z}^{2}$,
$\mathcal{M}=\mathbb{R}^{3}/\Gamma=T^{2}\times\mathbb{R}$ and
$Z\subset\mathcal{M}$ a finite subset, a monopole $(E,h,\nabla,\phi)$ on
$\mathcal{M}\backslash Z$ is called a doubly periodic monopole. In [48],
Mochizuki studied and built the Kobayashi–Hitchin type correspondence between
irreducible meromorphic doubly periodic monopoles and stable $q$-difference
modules of degree 0.

######  Definition 2.20.

A doubly periodic monopole $(E,h,\nabla,\phi)$ on $\mathcal{M}\backslash Z$ is
called meromorphic if each point of $Z$ is of Dirac type singularity and if
the curvature $F(\nabla)$ is bounded with respect to $h$ and $g_{\mathcal{M}}$
when $|t|\to\infty$, where $t$ is the coordinate for $\mathbb{R}$.

Let $q\in\mathbb{C}^{*}$, called a twistor parameter, let
$\Phi\in\operatorname{Aut}\big{(}\mathbb{C}\big{[}y,y^{-1}\big{]}\big{)}$
determined by $\Phi(f(y))=f(qy)$.

######  Definition 2.21.

A difference module on $\mathbb{C}^{*}$ (or called $q$-difference module) is a
$\mathbb{C}\big{[}y,y^{-1}\big{]}$-module $\mathbf{V}$ with a complex linear
automorphism $\Phi^{*}$ such that $\Phi^{*}(fs)=\Phi^{*}(f)\cdot\Phi^{*}(s)$
for any $f\in\mathbb{C}\big{[}y,y^{-1}\big{]}$ and $s\in\mathbf{V}$. It is
torsion-free if it is torsion-free as a
$\mathbb{C}\big{[}y,y^{-1}\big{]}$-module.

Let
$\mathscr{A}_{q}:=\bigoplus_{n\in\mathbb{Z}}\mathbb{C}\big{[}y,y^{-1}\big{]}(\Phi^{*})^{n}$
be an algebra determined by $\Phi^{*}\cdot f(y)=f(qy)\cdot\Phi^{*}$, and let
$\mathbf{V}$ be a torsion-free $q$-difference module.

######  Definition 2.22.

A parabolic structure at finite place of $\mathbf{V}$ consists of the
followings:

  1. (1)

A free sub $\mathbb{C}\big{[}y,y^{-1}\big{]}$-module $V\subset\mathbf{V}$ such
that $\mathscr{A}_{q}\cdot V=\textbf{V}$ and
$V\otimes_{\mathbb{C}[y,y^{-1}]}\mathbb{C}(y)=\mathbf{V}\otimes_{\mathbb{C}[y,y^{-1}]}\mathbb{C}(y)$;

  2. (2)

A finite subset $D\subset\mathbb{C}^{*}$ such that
$V(*D)=(\Phi^{*})^{-1}(V)(*D)$, where
$V(*D):=V\otimes_{\mathbb{C}[y,y^{-1}]}\mathbb{C}[y,y^{-1}](*D)$;

  3. (3)

For each $x\in D$, there is a sequence of real numbers
$\mathbb{\alpha}_{x}:=\\{0\leq\alpha_{x,1}<\cdots<\alpha_{x,m(x)}<1\\}$, when
$m(x)=0$, it is assumed to be empty;

  4. (4)

For each $x\in D$, there is a sequence of lattices
$\mathbf{L}_{x}:=\big{\\{}L_{x,0}=V\otimes_{\mathbb{C}[y,y^{-1}]}\mathbb{C}[\\![y-x]\\!],L_{x,i}\subset
V\otimes_{\mathbb{C}[y,y^{-1}]}\mathbb{C}(\\!(y-x)\\!)(1\leq i\leq
m(x)-1),L_{x,m(x)}=(\Phi^{*})^{-1}(V)\otimes_{\mathbb{C}[y,y^{-1}]}\mathbb{C}[\\![y-x]\\!]\big{\\}}$.

We denote it as the quadruple $(V,D,\\{\alpha_{x},\mathbf{L}_{x}\\}_{x\in
D})$.

######  Definition 2.23.

A good parabolic structure at $\infty$ of $\mathbf{V}$ is the pair
$\big{(}\mathscr{F}_{*}{\mathbf{V}_{|}}_{\widehat{0}},\mathscr{F}_{*}{\mathbf{V}_{|}}_{\widehat{\infty}}\big{)}$,
where $\mathscr{F}_{*}{\mathbf{V}_{|}}_{\widehat{0}}$ is a good filtered
bundle over
${\mathbf{V}_{|}}_{\widehat{0}}:=\mathbf{V}\otimes\mathbb{C}(\\!(y)\\!)$ and
$\mathscr{F}_{*}{\mathbf{V}_{|}}_{\widehat{\infty}}$ is a good filtered bundle
over
${\mathbf{V}_{|}}_{\widehat{\infty}}:=\mathbf{V}\otimes\mathbb{C}\big{(}\\!\big{(}y^{-1}\big{)}\\!\big{)}$.
A parabolic $q$-difference module is a $q$-difference module $\mathbf{V}$ with
a parabolic structure at finite place and a good parabolic structure at
$\infty$, we denote it as
$\big{(}\mathbf{V},(V,D\\{\alpha_{x},\mathbf{L}_{x}\\}_{x\in
D}),\mathscr{F}_{*}{\mathbf{V}_{|}}_{\widehat{0}},\mathscr{F}_{*}{\mathbf{V}_{|}}_{\widehat{\infty}}\big{)}$,
for simplicity, we denote it as $(\mathbf{V},\bullet)$.

The degree and slope of a parabolic $q$-difference module
$(\mathbf{V},\bullet)$ are given by

$\displaystyle\deg(\mathbf{V},\bullet):=\deg(\mathscr{F}_{0}\mathcal{V})+\sum_{x\in
D}\sum_{i=1}^{m(x)}(1-\alpha_{x,i})\deg(L_{x,i},L_{x,i-1})$
$\displaystyle\hphantom{\deg(\mathbf{V},\bullet):=}{}-\sum_{c\in\mathbb{Q}}\frac{c}{2}\big{(}\big{(}\dim_{\mathbb{C}(\\!(y)\\!)}({\mathbb{V}_{|}}_{\widehat{0}})_{c}\big{)}+\dim_{\mathbb{C}(\\!(y^{-1})\\!)}(({\mathbf{V}_{|}}_{\widehat{\infty}})_{c})\big{)},$

and
$\mu(\mathbf{V},\bullet):=\frac{\deg(\mathbf{V},\bullet)}{\operatorname{rk}_{\mathbb{C}[y,y^{-1}]}(V)}$,
respectively, where each $\big{(}{\mathbb{V}_{|}}_{\widehat{0}}\big{)}_{c}$ is
a term associated to the slope decomposition
${\mathbf{V}_{|}}_{\widehat{0}}=\bigoplus\limits_{c\in\mathbb{Q}}\big{(}{\mathbb{V}_{|}}_{\widehat{0}}\big{)}_{c}$
that is $\Phi^{*}$-invariant and for each $c=l/k\in\mathbb{Q}$, there exists
$\mathbb{C}[\\![y]\\!]$-lattice
$L_{c}\subset({\mathbb{V}_{|}}_{\widehat{0}})_{c}$ with
$y^{l}(\Phi^{*})^{k}L_{c}=L_{c}$. It is called stable (resp. semistable) if
for any non-zero proper $q$-difference $\mathbb{C}(y)$-subspace
$\widetilde{\mathbf{V}}^{\prime}$ of
$\widetilde{\mathbf{V}}:=\mathbf{V}\otimes\mathbb{C}(y)$ with the naturally
induced parabolic structure, we have
$\mu(\mathbf{V}^{\prime},\bullet)<(\text{resp. $\leq$})\
\mu(\mathbf{V},\bullet)$.

In [48], Mochizuki built the following Kobayashi–Hitchin type correspondence
between doubly periodic monopoles and $q$-difference modules:

######  Theorem 2.24 ([48]).

For each $q\in\mathbb{C}^{*}$, there is an one to one correspondence between
irreducible meromorphic doubly periodic monopoles and stable parabolic
$q$-difference modules of degree $0$.

####  2.2.3 Meromorphic triply periodic monopoles and parbolic difference
modules
on elliptic curves

When $\Gamma\cong\mathbb{Z}^{3}$, $\mathcal{M}=\mathbb{R}^{3}/\Gamma=T^{3}$
and $Z\subset\mathcal{M}$ a finite subset, a monopole $(E,h,\nabla,\phi)$ on
$\mathcal{M}\backslash Z$ is called a triply periodic monopole. In [49],
Mochizuki studied and built the Kobayashi–Hitchin type correspondence between
meromorphic triply periodic monopoles and stable difference modules on
elliptic curves of degree 0.

######  Definition 2.25.

A triply periodic monopole $(E,h,\nabla,\phi)$ on $\mathcal{M}\backslash Z$ is
called meromorphic if each point of $Z$ is of Dirac singularity.

Let $\Gamma_{0}\subset\mathbb{C}$ be a lattice and let
$T:=\mathbb{C}/\Gamma_{0}$ be the elliptic curve, let $a\in T$, called a
twistor parameter, let $\Phi\in\operatorname{Aut}(T)$ given by $\Phi(z)=z+a$,
let $D\subset T$ be a finite subset.

######  Definition 2.26.

A parabolic difference module on the elliptic curve $T$ (or parabolic
$a$-difference module) consists of the followings:

  1. (1)

A locally free $\mathcal{O}_{T}$-module $V$ and an isomorphism of
$\mathcal{O}_{T}$-modules $V(*D)\cong(\Phi^{*})^{-1}(V)(*D)$;

  2. (2)

For each $x\in D$, there is a sequence of real numbers
$\mathbb{\alpha}_{x}:=\\{0\leq\alpha_{x,1}<\cdots<\alpha_{x,m(x)}<1\\}$, when
$m(x)=0$, it is assumed to be empty;

  3. (3)

For each $x\in D$, there is a sequence of lattices
$\mathbf{L}_{x}:=\big{\\{}L_{x,0}=V_{x},L_{x,i}\subset V(*D)_{x}(1\leq i\leq
m(x)-1),L_{x,m(x)}=(\Phi^{*})^{-1}(V)_{x}\big{\\}}$.

We denote it as the quadruple
$\big{(}V,D,\\{\alpha_{x},\mathbf{L}_{x}\\}_{x\in D}\big{)}$, and for
simplicity, we denote it as $(V,\bullet)$. The degree and slope of
$(V,\bullet)$ are given by

$\displaystyle\deg(V,\bullet):=\deg(V)+\sum_{x\in
D}\sum_{i=1}^{m(x)}(1-\alpha_{x,i})\deg(L_{x,i},L_{x,i-1})$

and $\mu(V,\bullet):=\frac{\deg(V,\bullet)}{\operatorname{rk}(V)}$,
respectively. It is called stable (resp. semistable) if for any non-zero
proper submodule $V^{\prime}\subset V$ such that
$V^{\prime}(*D)\cong(\Phi^{*})^{-1}(V^{\prime})(*D)$ with the naturally
induced parabolic structure, we have $\mu(V^{\prime},\bullet)<(\text{resp.
$\leq$})\ \mu(V,\bullet)$.

In [49], Mochizuki built the following Kobayashi–Hitchin type correspondence
between triply periodic monopoles and $a$-difference modules:

######  Theorem 2.27 ([49]).

For each $a\in\mathbb{C}/\Gamma_{0}$, there is an one to one correspondence
between irreducible meromorphic triply periodic monopoles and stable parabolic
$a$-difference modules of degree $0$.

##  3 Conformal limits and Gaiotto’s conjecture

There are many interesting applications of the non-Abelian Hodge theory. Here
we give a brief introduction as the beginning of this section. Throughout this
section, $X$ will be denoted as a compact Riemann surface of genus $g\geq 2$.
Let $\mathcal{X}$ be the underlying smooth oriented surface, and denoted by
$K_{X}$ the canonical line bundle.

One important application of the non-Abelian Hodge theory is to study the
theory for real Lie groups. Then non-Abelian Hodge theory is an important tool
to find special connected components of the Betti moduli space222This moduli
space is usually called character variety or representation variety by people
work on surface group representation theory and higher Teichmüller theory,
e.g., [14, 39, 41, 65].
$M_{\mathrm{B}}(\mathcal{X},G):=\mathrm{Hom}(\pi_{1}(\mathcal{X}),G)/\\!\\!/G$
that parametrizes representations of $\pi_{1}(\mathcal{X})$ into the real Lie
group $G$. We will give more details in Remark 3.1.

Another important application of the non-Abelian Hodge theory is to study the
relation between certain irreducible components333In fact, we will see in the
next section that they are closed stata of the stratifications given by
$\mathbb{C}^{*}$-action on the corresponding moduli spaces. of $M_{\rm
Dol}(X,r)$ and of $M_{\rm dR}(X,r)$. As holomorphic symplectic manifolds [34],
both $M_{\rm Dol}(X,r)$ and $M_{\rm dR}(X,r)$ have special holomorphic
Lagrangian submanifolds that can be parametrized by holomorphic differentials
on $X$. These are the Hitchin section in $M_{\rm Dol}$ (it is the image of the
Hitchin base under a section of the Hitchin map) and the space of opers in
$M_{\rm dR}$.

In this section, we will introduce a famous conjecture by Gaiotto that
predicts the Hitchin section in the Dolbeault side and the space of opers in
the de Rham side can be identified by a map called conformal limit. Recently,
this conjecture is confirmed by the authors of [22]. In [15], by studying
conformal limit, the authors generalize this result to any strata of the
stratification of moduli spaces given by $\mathbb{C}^{*}$-action.

###  3.1 $\boldsymbol{\mathbb{C}^{*}}$-action on flat
$\boldsymbol{\lambda}$-connections

The Dolbeault moduli space $M_{\rm Dol}(X,r)$ admits a natural
$\mathbb{C}^{*}$-action [34, 58]:

$\displaystyle
t\cdot\big{[}\big{(}E,\bar{\partial}_{E},\theta\big{)}\big{]}:=\big{[}\big{(}E,\bar{\partial}_{E},t\theta\big{)}\big{]}.$

This action is well-defined because it does not change the stability and
vanishing of Chern classes. In [61], Simpson showed the existence of the
coarse moduli space $M_{\rm Hod}(X,r)$ of semistable $\lambda$-flat bundles
over $X$ of rank $r$ and with vanishing Chern classes. This moduli space
admits a fibration $\pi\colon M_{\rm Hod}(X,r)\to\mathbb{C}$ such that
$\pi^{-1}(1)=M_{\rm dR}(X,r)$ and $\pi^{-1}(0)=M_{\rm Dol}(X,r)$. Denoted by
$M_{{\rm Hod}}^{\lambda}(X,r):=\pi^{-1}(\lambda)$ the fiber over
$\lambda\in\mathbb{C}$.

Moreover, the $\mathbb{C}^{*}$-action on $M_{\rm Dol}(X,r)$ extends to an
action of $\mathbb{C}^{*}$ on $M_{\rm Hod}(X,r)$:

$\displaystyle
t\cdot\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}:=\big{[}E,\bar{\partial}_{E},tD^{\lambda},t\lambda\big{]}.$

This action is well-defined and sends a flat $\lambda$-connection to a flat
$t\lambda$-connection, so the fixed points must lie inside the fiber at
$\lambda=0$, i.e., in $M_{\rm Dol}(X,r)$. By [58, Lemma 4.1], the fixed points
in $M_{\rm Dol}(X,r)$ are those Higgs bundles having the structure of systems
of Hodge bundles,444In many contexts, these fixed points are called complex
variations of Hodge structure. that is, of the form

$\displaystyle\left[E=\bigoplus_{p=1}^{k}E^{p},\,\bar{\partial}_{E}=\begin{pmatrix}\bar{\partial}_{E^{1}}&&\\\
&\ddots&\\\ &&\bar{\partial}_{E^{k}}\end{pmatrix},\,\theta\colon E^{p}\to
E^{p-1}\otimes\Omega_{X}^{1}\right].$

Following the notations in [63], let $P$ be the subset consisting all the
fixed points, and let $P=\coprod\limits_{\alpha}P_{\alpha}$ be the union of
its connected components. This fixed point set will play an important role on
the stratifications of moduli spaces, which will be described in the next
section.

###  3.2 Hitchin sections and opers

Denote by $M_{\rm Dol}({\rm SL}(r,\mathbb{C}))$ the moduli space of semistable
${\rm SL}(r,\mathbb{C})$-Higgs bundles over $X$. There is a proper map, called
the Hitchin map

$\displaystyle h\colon\ M_{\rm Dol}({\rm SL}(r,\mathbb{C}))$
$\displaystyle\longrightarrow
B:=\bigoplus_{i=2}^{r}H^{0}\big{(}X,K_{X}^{i}\big{)},$
$\displaystyle\big{[}E,\bar{\partial}_{E},\theta\big{]}$
$\displaystyle\mapsto{\rm char}(\theta):=(q_{2}(\theta),\dots,q_{r}(\theta)),$

with each $q_{i}(\theta)=\operatorname{Tr}\big{(}\wedge^{i}\theta\big{)}$.
This map defines a completely integrable system [34]. Fix a line bundle $L$
such that $L^{r}\cong K_{X}^{\frac{r(r-1)}{2}}$, there are $r^{2g}$ choices of
such $L$, and each choice makes the Hitchin base $B$ parametrizes a section
called the Hitchin section

$\displaystyle s_{h}\colon\ B\longrightarrow M_{\rm Dol}({\rm
SL}(r,\mathbb{C})).$

More explicitly, to a point $(q_{2},\dots,q_{r})\in B$ one associates a ${\rm
SL}(r,\mathbb{C})$-Higgs bundle

$\displaystyle\left[\big{(}E,\bar{\partial}_{E}\big{)}=L\oplus\big{(}L\otimes
K_{X}^{-1}\big{)}\oplus\big{(}L\otimes
K_{X}^{-2}\big{)}\oplus\cdots\oplus\big{(}L\otimes
K_{X}^{-(r-1)}\big{)},\vphantom{\begin{pmatrix}0&q_{2}&q_{3}&\cdots&q_{r}\\\
a_{1}&0&q_{2}&\cdots&q_{r-1}\\\ &a_{2}&0&\ddots&\vdots\\\
&&\ddots&\ddots&q_{2}\\\ &&&a_{r-1}&0\\\ \end{pmatrix}}\right.$
$\displaystyle\left.\qquad\theta=\begin{pmatrix}0&q_{2}&q_{3}&\cdots&q_{r}\\\
a_{1}&0&q_{2}&\cdots&q_{r-1}\\\ &a_{2}&0&\ddots&\vdots\\\
&&\ddots&\ddots&q_{2}\\\ &&&a_{r-1}&0\\\ \end{pmatrix}\right],$

where each $a_{i}=\frac{i(r-i)}{2}$. For each choice of $L$, the image
$s_{h}(B)$ of the Hitchin base $B$ consists a connected component of $M_{\rm
Dol}({\rm SL}(r,\mathbb{C}))$, called a Hitchin component, and denoted as
${\rm Hit}_{r}$. This is related to special connected components of Betti
moduli space (see the following remark).

######  Remark 3.1.

The Betti moduli space $M_{\rm B}(\mathcal{X},{\rm
PSL}(2,\mathbb{R}))\\!:=\\!\mathrm{Hom}(\pi_{1}(\mathcal{X}),{\rm
PSL}(2,\mathbb{R}))/\\!\\!/{\rm PSL}(2,\mathbb{R})$ is the coarse moduli space
of representations of $\pi_{1}(\mathcal{X})$ into ${\rm PSL}(2,\mathbb{R})$.
In this Betti side, a Hitchin component usually means a connected component of
$M_{\rm B}(\mathcal{X},{\rm PSL}(2,\mathbb{R}))$ that consists entirely the
discrete and faithful representations $\rho\colon\pi_{1}(\mathcal{X})\to{\rm
PSL}(2,\mathbb{R})$. In [30], by using Euler number as invariant, Goldman
showed that $M_{\rm B}(\mathcal{X},{\rm PSL}(2,\mathbb{R}))$ possesses $4g-3$
connected components with the corresponding Euler numbers from $2-2g$ to
$2g-2$. Among these components it has two Hitchin components, as copies of the
Teichmüller spaces $\mathrm{Teich}(\mathcal{X})$ and
$\mathrm{Teich}\big{(}\bar{\mathcal{X}}\big{)}$, respectively. Moreover, each
Hitchin component is homeomorphic to $\mathbb{R}^{6g-6}$.

In [35], Hitchin introduced the $r$-Fuchsian representations ($r>2$):
representations $\rho\colon\pi_{1}(\mathcal{X})\allowbreak\to{\rm
PSL}(r,\mathbb{R})$ that are the composition of a discrete and faithful
representation $\rho_{0}\colon\pi_{1}(\mathcal{X})\allowbreak\to{\rm
PSL}(2,\mathbb{R})$ and the unique irreducible representation $i\colon{\rm
PSL}(2,\mathbb{R})\to{\rm PSL}(r,\mathbb{R})$. A connected component of
$M_{\rm B}(\mathcal{X},{\rm PSL}(r,\mathbb{R}))$ that contains $r$-Fuchsian
representations would be still called a Hitchin component. By applying the
Higgs bundle theory, Hitchin showed that $M_{\rm B}(\mathcal{X},{\rm
PSL}(r,\mathbb{R}))$ possesses 3 connected components when $r$ is odd, and 6
connected components when $r$ is even. Among these components it has 1 Hitchin
component when $r$ is odd, and 2 Hitchin components when $r$ is even.
Moreover, each Hitchin component is homeomorphic to
$\mathbb{R}^{(2g-2)(r^{2}-1)}$.

A further step in [35] is the study of special connected components (i.e.,
Hitchin components) of $M_{\rm B}\big{(}\mathcal{X},G^{r}\big{)}$ for any
split real form $G^{r}$ of a complex simple Lie group $G$. Hitchin showed this
moduli space has Hitchin components homeomorphic to
$\mathbb{R}^{(2g-2)\dim(G^{r})}$.

A natural generalization of Hitchin’s theory is to consider more general real
non-compact semisimple Lie groups, then the study of some spacial components
of the Betti moduli space leads to a new filed: higher Teichmüller theory.
These special connected components are usually called higher Teichmüller
spaces. For example, when the Lie groups are non-compact of Hermitian type,
then higher Teichmüller spaces correspond to those connected components with
maximal Toledo invariant. The corresponding representations are called maximal
representations, they are showed to be Anosov by Labourie [39]. In particular,
they are discrete and faithful. Recently, higher Teichmüller spaces are found
for some special real Lie groups, e.g., [12, 4]. Moreover, there are many good
references on the introduction and the development of this theory from
different points of view (e.g., [2, 13, 14, 27, 65]).

######  Example 3.2.

When $r=2$, we fix a line bundle $L$ such that $L^{2}\cong K_{X}$, we will
write it as $K_{X}^{\frac{1}{2}}$. Then the corresponding Higgs bundles in
${\rm Hit}_{2}$ have the form

$\left(\big{(}E,\bar{\partial}_{E}\big{)}=K_{X}^{\frac{1}{2}}\oplus
K_{X}^{-\frac{1}{2}},\,\theta=\begin{pmatrix}0&q_{2}\\\
1&0\end{pmatrix}\right),$

so the Hitchin section is parametrized by the quadratic differentials
$q_{2}\in H^{0}\big{(}X,K_{X}^{2}\big{)}$. Moreover, for any $q_{2}$, this
Higgs bundle is stable, since the only $\theta$-invariant proper subbundle is
$K_{X}^{-\frac{1}{2}}$. ${\rm Hit}_{2}$ describes the Teichmüller space ${\rm
Teich}(\mathcal{X})$ in terms of Higgs bundles, and the quadratic differential
$q_{2}$ measures the non-conformality of the harmonic diffeomorphism
$(\mathcal{X},g_{0})\to(\mathcal{X},g)$ [41].

For any $r$, each Higgs bundle $\big{[}E,\bar{\partial}_{E},\theta\big{]}$ in
${\rm Hit}_{r}$ is stable. Now let $g_{t}$ be a gauge transformation given by

$\displaystyle g_{t}=\begin{pmatrix}t^{\frac{r-1}{2}}&&\\\ &\ddots&\\\
&&t^{-\frac{r-1}{2}}\end{pmatrix},$

then

$\displaystyle\lim_{t\to 0}g_{t}\cdot(t\cdot\theta)\cdot
g_{t}^{-1}=\begin{pmatrix}0&&&\\\ a_{2}&0&&\\\ &\ddots&\ddots&\\\
&&a_{r-1}&0\end{pmatrix}.$

As the limit of the $\mathbb{C}^{*}$-action on
$\big{[}E,\bar{\partial}_{E},\theta\big{]}$, the Higgs bundle

$\displaystyle\left(\big{(}E,\bar{\partial}_{E}\big{)}=L\oplus\big{(}L\otimes
K_{X}^{-1}\big{)}\oplus\big{(}L\otimes
K_{X}^{-2}\big{)}\oplus\cdots\oplus\big{(}L\otimes
K_{X}^{-(r-1)}\big{)},\vphantom{\begin{pmatrix}0&&&\\\ a_{2}&0&&\\\
&\ddots&\ddots&\\\ &&a_{r-1}&0\end{pmatrix}}\right.$
$\displaystyle\left.\qquad{}\theta^{\prime}=\begin{pmatrix}0&&&\\\
a_{2}&0&&\\\ &\ddots&\ddots&\\\ &&a_{r-1}&0\end{pmatrix}\right)$

is stable. Therefore, $\big{[}E,\bar{\partial}_{E},\theta\big{]}$ is also
stable, since stability is an open condition.

For each $L$, the Higgs bundle $s_{h}(0,\dots,0)$ corresponds to the image of
the zero point $(0,\dots,0)\allowbreak\in B$ under the Hitchin section $s_{h}$
is called a uniformizing Higgs bundle. We also call it a Fuchsian point, since
the associated flat bundle corresponds to the Fuchsian representation that
uniformizes the Riemann surface itself [35]. In fact, in [35], Hitchin showed
that for each choice of $L$ such that $L^{r}\cong K_{X}^{\frac{r(r-1)}{2}}$,
Higgs bundles in the Hitchin section ${\rm Hit}_{r}$ have real monodromy
representations

$\displaystyle\rho\colon\ \pi_{1}(X)\to{\rm SL}(r,\mathbb{R}),$

that is, elements in ${\rm Hit}_{r}$ are ${\rm SL}(r,\mathbb{R})$-Higgs
bundles.

Another object we will introduce is oper, as a special object in the de Rham
moduli space $M_{\rm dR}$.

For any complex connected reductive Lie group $G$, the notion of $G$-oper was
introduced and studied by Beilinson and Drinfeld in [5]. Here for our study of
the correspondence between Hitchin sections and opers, we just consider the
case when $G={\rm SL}(r,\mathbb{C})$.

######  Definition 3.3.

A ${\rm SL}(r,\mathbb{C})$-oper over $X$ is a triple $(E,\nabla,F^{\bullet})$,
where $E$ is a holomorphic vector bundle of rank $r$ equipped with an
isomorphism $\Lambda^{r}E\cong\mathcal{O}_{X}$, $\nabla\colon E\to
E\otimes_{\mathcal{O}_{X}}K_{X}$ is a holomorphic flat connection, and
$F^{\bullet}$ is a filtration given by holomorphic subbundles of $E$:

$\displaystyle 0=F^{r}\subseteq F^{r-1}\subseteq\cdots\subseteq F^{1}\subseteq
F^{0}=E$

satisfies the following three conditions:

  * (1)

the filtration is of full flag, i.e., each graded term $E^{i}:=F^{i}/F^{i+1}$
is a line bundle;

  * (2)

Griffiths transversality: $\nabla\colon F^{i}\to
F^{i-1}\otimes_{\mathcal{O}_{X}}K_{X}$ for $i=1,\dots,r$;

  * (3)

the induced map $\theta\colon E^{i}\to E^{i-1}\otimes_{\mathcal{O}_{X}}K_{X}$
is an isomorphism for $i=1,\dots,r$.

###  3.3 Conformal limits and Gaiotto’s conjecture

As holomorphic Lagrangian submanifolds of moduli spaces, both Hitchin section
and the space of opers are parametrized by the Hitchin base
$\bigoplus\limits_{i=2}^{r}H^{0}\big{(}X,K_{X}^{i}\big{)}$ (details can be
found in [15, Section 2.7]). Moreover, they appear as closed subsets of the
moduli spaces [5, 15, 34]. The non-Abelian Hodge correspondence relates the
moduli spaces $M_{\rm Dol}$ and $M_{\rm dR}$, however, it doesn’t relate the
Hitchin section and the space of opers. One would hope to find a map to relate
these two Lagrangian submanifolds, and if possible, gives an identification
between them. Gaiotto’s conjecture arises from this consideration.

Let $\big{[}E,\bar{\partial}_{E},\theta\big{]}\in M_{\rm Dol}(X,r)$ be a
(poly)stable Higgs bundle with the pluri-harmonic metric denoted as $h$. Then
for any $R>0$, the $\mathbb{C}^{*}$-action gives us a family of (poly)stable
Higgs bundles $\big{[}E,\bar{\partial}_{E},R\theta\big{]}$ with the
corresponding pluri-harmonic metrics denoted as $h_{R}$. Each
$\big{[}E,\bar{\partial}_{E},R\theta\big{]}$ determines a family of flat
connections

$\displaystyle
D_{R,\lambda}:=\bar{\partial}_{E}+\partial_{E,h_{R}}+\lambda^{-1}R\theta+\lambda
R\theta^{\dagger}_{h_{R}}$

parametrized by $\lambda\in\mathbb{C}^{*}$. If we fix the product
$\hslash=\lambda R^{-1}$, then we can obtain a family of flat connections

$\displaystyle
D_{R,\hslash}:=\bar{\partial}_{E}+\partial_{E,h_{R}}+\hslash^{-1}\theta+\hslash
R^{2}\theta^{\dagger}_{h_{R}}$

parametrized by $R\in\mathbb{R}^{+}$. The study of the limit
$\lim\limits_{R\to 0}D_{R,\hslash}$ (if exits) of the flat connection
$D_{R,\hslash}$ is an interesting question, this is the following definition:

######  Definition 3.4.

For a (poly)stable Higgs bundle $\big{[}E,\bar{\partial}_{E},\theta\big{]}\in
M_{\rm Dol}(X,r)$, the limit

$\displaystyle\lim_{R\to 0}\big{(}E,\bar{\partial}_{E}+\hslash
R^{2}\theta^{\dagger}_{h_{R}},\partial_{E,h_{R}}+\hslash^{-1}\theta\big{)}$

if exists, is called its conformal limit.

We can interpret this flat connection in terms of $\lambda$-connections and
its $\mathbb{C}^{*}$-action introduced in Section 3.1. For each $R>0$, every
stable Higgs bundle $\big{[}E,\bar{\partial}_{E},\theta\big{]}$ determines a
stable Higgs bundle $\big{[}E,\bar{\partial}_{E},R\theta\big{]}$, thus a
$\lambda$-flat bundle $\big{[}E,\bar{\partial}_{E}+\lambda
R\theta^{\dagger}_{h_{R}},\lambda\partial_{E,h_{R}}+R\theta\big{]}$ by non-
Abelian Hodge correspondence. In fact, this $\lambda$-flat bundle is obtained
from $\lambda$ acts on the Higgs bundle $\big{[}E,\bar{\partial}_{E}+\lambda
R\theta^{\dagger}_{h_{R}},\partial_{E,h_{R}}+\lambda^{-1}R\theta\big{]}$:

$\displaystyle\big{[}E,\bar{\partial}_{E}+\lambda
R\theta^{\dagger}_{h_{R}},\lambda\partial_{E,h_{R}}+R\theta\big{]}=\lambda\cdot\big{[}E,\bar{\partial}_{E}+\lambda
R\theta^{\dagger}_{h_{R}},\partial_{E,h_{R}}+\lambda^{-1}R\theta\big{]}.$

Again by non-Abelian Hodge correspondence, this Higgs bundle corresponds to a
flat bundle, its flat connection is exactly $D_{R,\hslash}$.

For special choice of the conformal constant $\hslash$ and the initial Higgs
bundle $\big{[}E,\bar{\partial}_{E},\theta\big{]}$, its confromal limit
recovers the non-Abelian Hodge map ${\rm NAH}$ defined in Section 2.1. In
fact, when $\hslash=1$ and $\big{[}E,\bar{\partial}_{E},\theta\big{]}$ is a
fixed point of the $\mathbb{C}^{*}$-action, then

$\displaystyle\lim_{R\to 0}D_{R,1}={\rm
NAH}\big{(}\bar{\partial}_{E},\theta\big{)}.$

In [26], Gaiotto proposed the following conjecture that relates the Hitchin
component and the locus of opers by the conformal limits:

######  Conjecture 3.5.

For any Higgs bundle in the Hitchin component, its conformal limit exists and
is an oper. Moreover, it gives a biholomorphism between the Hitchin component
and the space of opers.

In the paper of [22], the authors confirmed this conjecture as the following
theorem:

######  Theorem 3.6 ([22]).

For any simple and simply connected Lie group $G$. If
$\big{(}E,\bar{\partial}_{E},\theta\big{)}$ is a $G$-Higgs bundle in the
Hitchin component, then its conformal limit exists and lies in the space of
opers. Moreover, it gives a biholomorphism between the Hitchin component and
the space of opers.

In fact, the locus ${\rm NAH}(s_{h}(B))$ of the Hitchin section under the non-
Abelian Hodge correspondence in $M_{\rm dR}(G)$ intersects with the space
${\rm Op}(r)$ of opers transversely at the uniformizing point
$s_{h}(0,\dots,0)$ [40]. Theorem 3.6 tells us the conformal limits give an
identification between them.

In [15], the authors studied the conformal limit for general Higgs bundle
$\big{[}E,\bar{\partial}_{E},\theta\big{]}\in M_{\rm Dol}$ such that the
limiting point
$\big{[}E,\bar{\partial}_{0},\theta_{0}\big{]}:=\lim\limits_{t\to
0}t\cdot[E,\bar{\partial}_{E},\theta]$ is stable. They obtained the following
general conformal limit correspondence:

######  Theorem 3.7 ([15]).

For any Higgs bundle $\big{[}E,\bar{\partial}_{E},\theta\big{]}\in M_{\rm
Dol}(\mathrm{SL}(r,\mathbb{C}))$ such that the limiting point
$u:=\lim\limits_{t\to 0}t\cdot\big{[}E,\bar{\partial}_{E},\theta\big{]}\in
P_{\alpha}$ is stable, its conformal limit always exists. Moreover, it gives a
biholomorphism between $G_{\alpha}^{0}(u)$ and $G_{\alpha}^{1}(u)$.

Here $G_{\alpha}^{0}$ and $G_{\alpha}^{1}$ stand for the strata in $M_{\rm
Dol}$ and $M_{\rm dR}$ that correspond to the connected component $P_{\alpha}$
of the fixed point set. Details on this related to the stratifications of
moduli spaces given by $\mathbb{C}^{*}$-action will be described in the next
section.

This result generalizes Theorem 3.6 since the limiting point of any element in
the Hitchin section is stable.

##  4 Stratification of moduli spaces

Throughout this section, $X$ denotes a compact Riemann surface.

Following the description in Section 3.1, the natural $\mathbb{C}^{*}$-action
on $M_{\rm Dol}(X,r)$ extends to an action of $\mathbb{C}^{*}$ on $M_{\rm
Hod}(X,r)$. By a work of Simpson [63], for each
$\big{[}E,\bar{\partial}_{E},D^{\lambda}\big{]}\in M_{\rm Hod}(X,r)$, the
limit $\lim\limits_{t\to
0}t\cdot\big{[}E,\bar{\partial}_{E},D^{\lambda}\big{]}$ exists and as a fixed
point lies in $P$. Therefore, this action gives a Białynicki-Birula
stratification of $M_{\rm Hod}$ into locally closed subsets (see Section 4.2).
Restricting the stratification to the fiber over $0$, it recovers the
classical Białynicki-Birula stratification of $M_{\rm Dol}$. Restricting the
stratification to the fiber over $1$, we will have a stratification of $M_{\rm
dR}$ into locally closed subsets, and moreover, the space of opers, appears as
a special stratum. This stratification is new to us, provides a new direction
on the study of $M_{\rm dR}$.

On the other hand, in the same paper, Simpson showed each flat bundle over $X$
admits a filtration satisfies the Griffiths transversality and such that the
induced graded Higgs bundle is semistable (see Section 4.1). And moreover, the
limit of the $\mathbb{C}^{*}$-action on that flat bundle, as a fixed point in
$M_{\rm Dol}$, is S-equivalent to the graded Higgs bundle. This property
provides a possibility on the description of certain flat bundles (e.g.,
Theorem 4.11), and moreover, plays an important role on the study of $M_{\rm
dR}$ (e.g., on the proof of Conjecture 4.8 for rank 2).

This section can be treated as an application of non-Abelian Hodge theory to
the study of the de Rham moduli space $M_{\rm dR}$. In this section, we will
describe the stratifications of moduli spaces given by
$\mathbb{C}^{*}$-action. Meanwhile, some conjectures related to the study of
$M_{\rm dR}$ will be introduced.

###  4.1 Simpson filtrations on flat bundles

For the $\mathbb{C}^{*}$-action on the Dolbeault moduli space $M_{\rm
Dol}(X,r)$. Since the Hitchin map $h\colon\\!M_{\rm
Dol}(X,r)\\!\\!\allowbreak\to\bigoplus\limits_{i=1}^{r}H^{0}\big{(}X,K_{X}^{i}\big{)}$
is proper and $\mathbb{C}^{*}$-equivariant, for any
$\big{[}E,\bar{\partial}_{E},\theta\big{]}\in M_{\rm Dol}(X,r)$, the limit
$\lim\limits_{t\to 0}t\cdot\big{[}E,\bar{\partial}_{E},\theta\big{]}$ exists
and as a fixed point of this action.

There is no analogue of the Hitchin map for $M_{\rm Hod}(X,r)$. However, for
each $\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}\in M_{\rm
Hod}(X,r)$, the limit $\lim\limits_{t\to
0}t\cdot\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$ of the
$\mathbb{C}^{*}$-action still exists as a fixed point and lies in some
$P_{\alpha}$ [63]. In particular, the limit $\lim\limits_{t\to
0}t\cdot(E,\nabla)$ of a flat bundle $(E,\nabla)$ exists as a fixed point.
Moreover, this limit can be described by the existence of a special filtration
of this flat bundle. This filtration, is found by Simpson, we will call it a
Simpson filtration throughout the whole paper.

######  Definition 4.1 ([37, 63]).

Let $E$ be a vector bundle over $X$ with flat connection $\nabla\colon
E\rightarrow E\otimes_{\mathcal{O}_{X}}\Omega^{1}_{X}$. A decreasing
filtration $F^{\bullet}$ of $E$ by strict subbundles

$\displaystyle E=F^{0}\supset F^{1}\supset\cdots\supset F^{k}=0$

is called a _Simpson filtration_ if it satisfies the following two conditions:

  1. (1)

Griffiths transversality: $\nabla\colon F^{p}\rightarrow
F^{p-1}\otimes_{\mathcal{O}_{X}}\Omega^{1}_{X}$ for $p=1,\dots,k$;

  2. (2)

graded-semistability: the associated graded Higgs bundle
$(\mathrm{Gr}_{F}(E),\mathrm{Gr}_{F}(\nabla))$, where
$\mathrm{Gr}_{F}(E)=\bigoplus\limits_{p=0}^{k-1}E^{p}$ with
$E^{p}=F^{p}/F^{p+1}$ and $\mathrm{Gr}_{F}(\nabla)\colon E^{p}\rightarrow
E^{p-1}\otimes_{\mathcal{O}_{X}}\Omega^{1}_{X}$ induced from $\nabla$, is a
semistable Higgs bundle.

Such a triple $\big{(}E,\nabla,F^{\bullet}\big{)}$ is called a partial oper.

Simpson proved the following nice theorem [63].

######  Theorem 4.2.

Let $(E,\nabla)$ be a flat bundle over a smooth projective curve $X$.

  1. $(1)$

There exist Simpson filtrations $F^{\bullet}$ on $(E,\nabla)$, this means any
flat bundle has partial oper structure.

  2. $(2)$

Let $F_{1}^{\bullet}$ and $F_{2}^{\bullet}$ be two Simpson filtrations on
$(E,\nabla)$, then the associated graded Higgs bundles
$(\mathrm{Gr}_{F_{1}}(E),\mathrm{Gr}_{F_{1}}(\nabla))$ and
$(\mathrm{Gr}_{F_{2}}(E),\mathrm{Gr}_{F_{2}}(\nabla))$ are $S$-equivalent.

  3. $(3)$

$(E,\nabla,F^{\bullet})$ is graded-stable if and only if the Simpson
filtration is unique $($up to indices translation$)$.

  4. $(4)$

$\lim\limits_{t\rightarrow
0}t\cdot(E,\nabla)=[\mathrm{Gr}_{F}(E),\mathrm{Gr}_{F}(\nabla)]\in M_{\rm
Dol}(X,r)$.

In [63], Simpson gave a wonderful iterated process to show the existence of
Simpson filtration for $(E,\nabla)$. We now sketch how it works.

Suppose $(E,\nabla)$ admits a filtration

$\displaystyle F^{\bullet}\colon\ 0\subset F^{k-1}\subset\cdots\subset
F^{0}=E$

that satisfies the Griffiths transversality $\nabla(F^{p})\subset
F^{p-1}\otimes\Omega_{X}^{1}$, and such that the associated Higgs bundle
$(V,\theta):=({\rm Gr}_{F}(E),{\rm Gr}_{F}(\nabla))$ is not semistable. To see
the existence of such filtration, we can begin with the trivial filtration
$0\subset F^{0}=E$, the graded Higgs bundle will be $({\rm Gr}_{F}(E),{\rm
Gr}_{F}(\nabla))=(E,0)$. Then applying the following iteration process, we can
always have a such filtration. Take $H\subset(V,\theta)$ to be the maximal
destabilizing subsheaf, which is known being unique and a subbundle of $V$,
and the quotient $V/H$ is also a subbundle of $E$. Since $H$ is unique, it
must be a fixed point of the $\mathbb{C}^{*}$-action. Therefore, $H$ has a
structure of system of Hodge bundles, and as a sub-system of Hodge bundles of
$(V,\theta)$, that is, $H=\bigoplus H^{p}$ with each $H^{p}=H\cap{\rm
Gr}_{F}^{p}(E)\subset F^{p}(E)/F^{p+1}(E)$ being a strict subbundle.

The new filtration $G^{\bullet}$ is defined as

$\displaystyle
G^{p}:=\operatorname{Ker}\left(E\to\frac{E/F^{p}(E)}{H^{p-1}}\right).$

It satisfies the Griffiths traversality since $\theta(H^{p})\subset
H^{p-1}\otimes\Omega_{X}^{1}$, and it fits into the exact sequence

$\displaystyle 0\to{\rm Gr}_{F}^{p}(E)/H^{p}\to{\rm Gr}_{G}^{p}(E)\to
H^{p-1}\to 0.$

If the new resulting graded Higgs bundle $({\rm Gr}_{G}(E),{\rm
Gr}_{G}(\nabla))$ is still not semistable, then we continue this process to
obtain a new graded Higgs bundle. By introducing three bounded invariants,
Simpson showed that the iteration process will strictly decrease these
invariants in lexicographic order. Therefore, after a finite step, we will
find a filtration such that the associated graded Higgs bundle is semistable.

######  Corollary 4.3 ([63]).

Let $X$ be a smooth projective curve and let
$\big{[}E,\bar{\partial}_{E},D^{\lambda}\big{]}\in M_{\rm Hod}(X,r)$ be any
$\lambda$-flat bundle $(\lambda\neq 0)$ in the moduli space, then

$\displaystyle\lim_{t\to
0}t\cdot\big{[}E,\bar{\partial}_{E},D^{\lambda}\big{]}=\lim_{t\to
0}t\cdot\big{[}E,\bar{\partial}_{E},\lambda^{-1}D^{\lambda}\big{]}\in M_{\rm
Dol}(X,r),$

where $\big{(}E,\bar{\partial}_{E},\lambda^{-\lambda}D^{\lambda}\big{)}$ is
the flat bundle $(1$-bundle$)$ associated to
$\big{(}E,\bar{\partial}_{E},D^{\lambda}\big{)}$.

By definition, the filtration for a $\mathrm{GL}(r,\mathbb{C})$-oper is a
special Simpson filtration.

######  Corollary 4.4.

Every oper $(E,\nabla,F^{\bullet})$ over a smooth projective curve $X$ of
$g\geq 2$ is graded stable, in particular, $(E,\nabla)$ has $F^{\bullet}$ as
the only Simpson filtration $($up to indices translation$)$.

###### Proof.

Let $\big{(}\bigoplus\limits_{i=0}^{r-1}E^{i},\theta\big{)}$ be the associated
graded Higgs bundle, where each $E^{i}$ is a line bundle and each
$\theta|_{E^{i}}\colon E^{i}\to E^{i-1}\otimes\Omega_{X}^{1}$ is an
isomorphism. This means

$\displaystyle\deg\big{(}E^{i}\big{)}=\deg\big{(}E^{i-1}\big{)}+2g-2=\cdots=\deg\big{(}E^{0}\big{)}+i(2g-2),$

note that each $\theta$-invariant non-zero proper subbundle of
$\big{(}\bigoplus\limits_{i=0}^{r-1}E^{i},\theta\big{)}$ has the form
$\bigoplus\limits_{i=0}^{k}E^{i}$ ($0\leq k<r-1$), with

$\displaystyle\deg\left(\bigoplus_{i=0}^{k}E^{k}\right)=(k+1)\deg\big{(}E^{0}\big{)}+k(k+1)(g-1),$

so

$\displaystyle\mu\left(\bigoplus_{i=1}^{k}E^{i}\right)=\deg\big{(}E^{0}\big{)}+k(g-1)<\deg\big{(}E^{0}\big{)}+(r-1)(g-1)=\mu(E),$

this means $(E,\nabla,F^{\bullet})$ is graded stable. In particular, by (3) of
Theorem 4.2, $F^{\bullet}$ is the only Simpson filtration for $(E,\nabla)$. ∎

######  Remark 4.5.

The non-uniqueness of the Simpson filtration is easy to see. In fact, any
irreducible rank 2 flat bundle of degree 0 with the underlying vector bundle
strictly semistable which is an extension of a degree 0 line bundle admits
more than one Simpson filtration. One is the trivial filtration, and the other
one has two terms with the first term the extension of line bundle. The two
resulting graded Higgs bundles are automatically semistable and $S$-equivalent
to each other, as a unique representative point in the Dolbeault moduli space,
which parametrizes the limit point of the $\mathbb{C}^{*}$-action.

###  4.2 Stratifications of moduli spaces

Following [63], we introduce the following set:

$\displaystyle
G_{\alpha}:=\big{\\{}\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}\in
M_{\rm Hod}(X,r)\,\big{|}\,\lim_{t\to
0}t\cdot\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}\in
P_{\alpha}\big{\\}},$

then these $G_{\alpha}$ gives a Białynicki-Birula type stratification of the
Hodge moduli space $M_{\rm Hod}(X,r)$

$\displaystyle M_{\rm Hod}(X,r)=\bigcup_{\alpha}G_{\alpha}$

into locally closed subsets. There is a natural projection $p_{\alpha}\colon
G_{\alpha}\to P_{\alpha}$ by taking the limit of the $\mathbb{C}^{*}$-action.
Restricting the stratification to the fiber over each $\lambda\in\mathbb{C}$
gives the stratification of $M_{{\rm Hod}}^{\lambda}(X,r)(X,r)$

$\displaystyle M_{{\rm
Hod}}^{\lambda}(X,r)(X,r)=\bigcup_{\alpha}G_{\alpha}^{\lambda}:=\bigcup_{\alpha}\Big{(}G_{\alpha}\bigcap\pi^{-1}(\lambda)\Big{)}$

into locally closed subsets. In particular, taking $\lambda=0$ and $1$, we
have the stratifications of $M_{\rm Dol}(X,r)$ and $M_{\rm dR}(X,r)$

$\displaystyle M_{\rm Dol}(X,r)=\bigcup_{\alpha}G_{\alpha}^{0},\qquad M_{\rm
dR}(X,r)=\bigcup_{\alpha}G_{\alpha}^{1}$

into locally closed subsets. The first one is in fact the Białynicki-Birula
stratification of $M_{\rm Dol}(X,r)$ given by the $\mathbb{C}^{*}$-action. The
second one is called the oper stratification of $M_{\rm dR}(X,r)$, since the
space of opers appears as a special stratum (Theorem 4.2 and Corollary 4.3).
The projection $p_{\alpha}$ restricts on fibers gives projections
$p_{\alpha}^{0}\colon G_{\alpha}^{0}\to P_{\alpha}$ and $p_{\alpha}^{1}\colon
G_{\alpha}^{1}\to P_{\alpha}$. By Białynicki-Birula theory, over each point
$u\in P_{\alpha}$, the fiber
$G_{\alpha}^{0}(u):=\big{(}p_{\alpha}^{0}\big{)}^{-1}(u)$ is an affine space.
Moreover, in [15], by applying the conformal limit techniques, the authors
showed each $G_{\alpha}^{1}(u):=\big{(}p_{\alpha}^{1}\big{)}^{-1}(u)$ is also
affine [15, Corollary 1.5].

In [63], Simpson showed that over any point $u$ which is a stable system of
Hodge bundles in the fixed point set, the fiber $G_{\alpha}^{1}(u)$ is a
Lagrangian submanifold of $M_{\rm dR}$. In [15], the authors showed the
Lagrangian property also holds for $M_{\rm Dol}(X,r)$, that is, for each
stable $u$, the fiber $G_{\alpha}^{0}(u)$ is a Lagrangian submanifold of
$M_{\rm Dol}$. Moreover, by studying the conformal limits, they showed each
fiber $G_{\alpha}^{0}(u)$ is in fact biholomorphic to $G_{\alpha}^{1}(u)$ (see
Theorem 3.7), which generalizes the conformal limit correspondence between
Hitchin section and the space of opers. As we have seen in the last section
that the locus ${\rm NAH}(s_{h}(B))$ intersects with ${\rm Op}(r)$
transversely at $s_{h}(0,\dots,0)$, the authors of [15] showed that under the
non-Abelian Hodge correspondence, each fiber of the Białynicki-Birula
stratification and intersects with the fiber of the oper stratification
transversely at the base point. More explicitly, for each stable point $u\in
P_{\alpha}$, the image ${\rm NAH}\big{(}G_{\alpha}^{0}(u)\big{)}$ of the fiber
$G_{\alpha}^{0}(u)$ under non-Abelian Hodge correspondence intersects with the
fiber $G_{\alpha}^{1}(u)$ transversely at ${\rm NAH}(u)$.

The non-Abelian Hodge correspondence shares no light on the study of $M_{\rm
dR}$ from the study of $M_{\rm Dol}$, since $M_{\rm dR}$ and $M_{\rm Dol}$
share very few similarities as algebraic spaces. However, the strata
$G_{\alpha}^{1}$ play an important role on the understanding of $M_{\rm dR}$.
Especially the Lagrangian property for each fiber of $G_{\alpha}^{1}$ induces
a natural question on the relationship between these Lagrangian fibers:

######  Conjecture 4.6 (foliation conjecture, [63]).

When varying $\alpha$, these Lagrangian fibers of $p_{\alpha}^{1}\colon
G_{\alpha}^{1}\allowbreak\to P_{\alpha}$ fit together to provide a smooth
foliation of $M_{\rm dR}(X,r)$ with each leaf closed.

This conjecture is still open, one progress was recently made by the authors
of [42] for the case of moduli space of rank 2 parabolic connections on
$\mathbb{P}^{1}$ minus 4 points.

This closedness property for $M_{\rm Dol}(X,r)$ is clearly not right, since
any fiber contained in the compact nilpotent cone would be not closed. If
fact, if it is closed, as a subset of a compact space, it is compact also, as
it is affine, this couldn’t happen.

We give a more explicit explanation here, define the following indexed sets by
the limit of $\mathbb{C}^{*}$-action:

$\displaystyle
D_{\alpha}^{0}:=\big{\\{}\big{[}E,\bar{\partial}_{E},\theta\big{]}\in M_{\rm
Dol}(X,r)\,|\,\lim_{t\to\infty}t\cdot\big{[}E,\bar{\partial}_{E},\theta\big{]}\in
P_{\alpha}\big{\\}}.$

Then by Hausel’s thesis [33], these sets fit together into the nilpotent cone:

$\displaystyle h^{-1}(0)=\bigcup_{\alpha}D_{\alpha}^{0},$ (4.1)

which is a deformation retraction of the whole moduli space. Let $u\in
P_{\alpha}$ be a fixed point such that the whole fiber
$G_{\alpha}^{0}(u)=\big{\\{}[E,\bar{\partial}_{E},\theta]\,|\,\lim\limits_{t\to
0}t\cdot\big{[}E,\bar{\partial}_{E},\theta\big{]}=u\big{\\}}$ is contained in
the nilpotent cone, that is, $G_{\alpha}^{0}(u)\subseteq h^{-1}(0)$. Take any
$v\in G_{\alpha}^{0}(u)$ that is not a fixed point, as $t\cdot v\in
G_{\alpha}^{0}(u)$ for all $t\in\mathbb{C}^{*}$ and $G_{\alpha}^{0}(u)$ is
closed, both $\lim\limits_{t\to 0}t\cdot v$ and
$\lim\limits_{t\to\infty}t\cdot v$ lie in $G_{\alpha}^{0}(u)$. Note the first
limit is the fixed point $u$, and the second limit is also a fixed point by
(4.1). By definition, $G_{\alpha}^{0}(u)$ can contain only one fixed point,
$u$, this means the two limits of a non-fixed point should coincide, this
couldn’t happen since the fixed point sets are ordered by the energy
functional.

Following this idea, with a discussion with Simpson, he told me the following
pure algebraic-geometric result:

######  Lemma 4.7.

Let $\mathbb{G}_{m}$ the multiplicative group, and $Y$ be any algebraic
variety. Suppose $\mathbb{G}_{m}$ acts on $Y$ with open dense orbit isomorphic
to $\mathbb{G}_{m}$, and such that the two endpoints are identified. Then
there does not exist an ample linearized line bundle on $Y$.

###### Proof.

Let $y$ be any point lies in the open dense orbit and let $z$ be an extra
point not in the orbit, denote by $\mathcal{O}_{y}$ the orbit. Then by
assumption, $\mathcal{O}_{y}$ and $\\{z\\}$ are the only two orbits of this
action, and moreover, $\lim\limits_{t\to 0}t\cdot
y=\lim\limits_{t\to\infty}t\cdot y=z$. Suppose $Y$ has an ample
$\mathbb{G}_{m}$-linearized line bundle $L$, then there exists an invariant
section of $L^{\otimes n}$ for some $n\in\mathbb{Z}_{+}$. Hence the linear
action of $\mathbb{G}_{m}$ on the fiber $L_{\lim\limits_{t\to 0}t\cdot y}$ has
positive weight, while it acts on the fiber $L_{\lim\limits_{t\to\infty}t\cdot
y}$ has negative weight, which is a contradiction. ∎

With this lemma, we can see in another way why $G_{\alpha}^{0}(u)$ could not
have two endpoints of the $\mathbb{C}^{*}$-action on a non-fixed point
identified. Since the Dolbeault moduli space $M_{\rm Dol}$ has an ample
$\mathbb{G}_{m}$-linearized line bundle (see Simpson’s construction of the
moduli space [59, 60]), it could not have such $G_{\alpha}^{0}(u)$ inside with
the property. Therefore, $G_{\alpha}^{0}(u)$ could not have a point with two
endpoints of the $\mathbb{C}^{*}$-action identified, this means
$G_{\alpha}^{0}(u)$ could not be closed.

Let $U(X,r)$ be the moduli space of semistable vector bundles over $X$ of
degree 0 and rank $r$, this is known to be an irreducible variety. It
naturally embeds into $M_{\rm Dol}(X,r)$ as Higgs bundles with zero Higgs
field. Moreover, it appears as a connected component of the fixed point set
$P$, let $P_{0}$ denotes this component. On the de Rham side, the
corresponding stratum $G_{0}^{1}$ in $M_{\rm dR}$ is the unique open stratum
that consists of flat bundles of the form
$\big{(}E,\bar{\partial}_{E},\partial+\varphi\big{)}$, where
$\big{(}E,\bar{\partial}_{E}\big{)}$ is a polystable vector bundle,
$\varphi\in H^{0}\big{(}X,\operatorname{End}(E)\otimes\Omega_{X}^{1}\big{)}$,
and $\partial$ is the unique unitary flat connection. For each such flat
bundle, the Simpson filtration is trivial. On the Dolbeault side, the
corresponding stratum $G_{0}^{0}$ is a dense open subset of $M_{\rm
Dol}(X,r)$, and can be identified with the cotangent bundle $T^{*}U(X,r)$. The
strata $G_{0}^{1}$ and $G_{0}^{0}$ are usually called the lowest strata. And
if we take $u=\big{[}E,\bar{\partial}_{E},0\big{]}\in P_{0}$, then
$G_{0}^{0}(u)=H^{0}\big{(}X,\operatorname{End}(E)\otimes\Omega_{X}^{1}\big{)}\subseteq
M_{\rm Dol}(X,r)$, the space of Higgs fields on $E$. This fiber is closed if
and only if $E$ is very stable, i.e., there is no non-zero nilpotent Higgs
field on $E$ (see also [53]).

The space $P_{u}$ of uniformizing Higgs bundles (see last section) corresponds
to the stratum of opers $G_{u}^{1}$ in $M_{\rm dR}$ and the stratum
$G_{u}^{0}$ of Hitchin component in $M_{\rm Dol}$, moreover, $G_{u}^{1}$ is
closed in $M_{\rm dR}$, since $G_{u}^{0}$ is closed in $M_{\rm Dol}$ [63]. We
will call them the oper stratum and the Hitchin stratum, respectively.

In [63], Simpson proposed another method to study the behaviour of the
stratifications. Let $M$ be a (quasi-)projective variety with a stratification
of locally closed subsets $M=\coprod\limits_{\alpha\in\Lambda}G_{\alpha}$, we
call this stratification nested if there is a partial order $(\Lambda,\leq)$
such that

$\displaystyle\overline{G_{\alpha}}=\coprod_{\beta\leq\alpha}G_{\beta},$

this implies the partial order is defined as

$\displaystyle\beta\leq\alpha\Longleftrightarrow
G_{\beta}\subseteq\overline{G_{\alpha}}.$

######  Conjecture 4.8 (nestedness conjecture).

The stratifications for $M_{\rm Dol}(X,r)$ and $M_{\rm dR}(X,r)$ are both
nested, and the arrangements for both stratifications are the same.

Simpson himself studied and showed it for rank 2 case by using the beautiful
techniques of deformation theory, but for the higher rank case, it is still an
open problem.

At least from the proof of nestedness conjecture of rank 2 case, we can see
that the oper stratum (the highest stratum) $G_{u}^{1}$ is the stratum of
minimal dimension among all the strata, based on this, Simpson proposed
another conjecture [63]:

######  Conjecture 4.9 (oper stratum conjecture).

The oper stratum $G_{u}^{1}$ is the unique closed stratum and the unique
stratum with minimal dimension.

In [37], by classifying irreducible components of the fixed point set $P$ of
the $\mathbb{C}^{*}$-action on $M_{\rm Dol}(X,r)$, we partially confirmed this
conjecture:

######  Theorem 4.10 ([37]).

The oper stratum is the unique closed stratum with minimal dimension.

We still do not know if there exists a closed stratum of strictly higher
dimension. Moreover, we showed the Hitchin stratum is also the unique closed
stratum of minimal dimension, but it is not the unique closed stratum.

In [31], the authors considered the relation between the Białynicki-Birula
stratification and the Shatz stratification of $M_{\rm Dol}(X,3)$, where the
Shatz stratification is given by Harder–Narasimhan type of the underlying
vector bundles of the Higgs bundles. With this inspiration, we can consider
the relation between the oper stratification given by Simpson filtrations and
the Shatz stratification of $M_{\rm dR}(X,3)$ given by Harder–Narasimhan type
of the underlying vector bundles.

For a flat bundle $(E,\nabla)$ of rank 3, suppose $E$ is not a stable vector
bundle. Then the Harder–Narasimhan type of $E$ can be given as follows:

  1. 1.

Type $(1,2)$, that is, the Harder–Narasimhan filtration is given by
$0\subsetneq H^{1}\subsetneq E$ with $\operatorname{rk}\big{(}H^{1}\big{)}=1$
and $\deg\big{(}H^{1}\big{)}=d_{1}$. In this case, $H_{1}\subseteq E$ is the
maximal destabilizing subsheaf, so $d_{1}>0$.

  2. 2.

Type $(2,1)$, that is, the Harder–Narasimhan filtration is given by
$0\subsetneq H^{1}\subsetneq E$ with $\operatorname{rk}\big{(}H^{1}\big{)}=2$
and $\deg\big{(}H^{1}\big{)}=d_{1}$. As in (1), $H_{1}\subseteq E$ is the
maximal destabilizing subsheaf, so $d_{1}>0$.

  3. 3.

Type $(1,1,1)$, that is, the Harder–Narasimhan filtration is given by
$0\subsetneq H^{1}\subsetneq H^{2}\subsetneq E$ with
$\operatorname{rk}\big{(}H^{1}\big{)}=1$, $\deg\big{(}H^{1}\big{)}=d_{1}$ and
$\operatorname{rk}\big{(}H^{2}\big{)}=2$,
$\deg\big{(}H^{2}/H^{1}\big{)}=d_{2}$. In this case, $H_{1}\subseteq E$ is the
maximal destabilizing subsheaf, and $H_{2}/H_{1}\subseteq E/H_{1}$ is the
maximal destabilizing subsheaf, hence $d_{1}>0$ and $d_{1}+d_{2}>0$.

######  Theorem 4.11.

Let $(E,\nabla)\in M_{\rm dR}(X,3)$ be an irreducible flat bundle of rank $3$
that is graded-stable, and such that $E$ is not a stable vector bundle. Then
its Simpson filtration is determined by its Harder–Narasimhan filtration as
follows:

  * $(1)$

If the Harder–Narasimhan type of $E$ is $(1,2)$ as above. Let $I\subset
E/H^{1}$ be the sub line bundle by saturating the subsheaf
$\theta\big{(}H^{1}\big{)}\otimes K_{X}^{-1}\subset E/H^{1}$, where
$\theta\colon H^{1}\to E/H^{1}\otimes K_{X}$ is the non-zero map induced by
$\nabla$, then

    * $(1.1)$

$0<d_{1}<g-1$ $\&$ $d_{1}-2g+2\leq\deg(I)<-d_{1}$, the Simpson filtration
coincides with the Harder–Narasimhan filtration. Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\big{[}H^{1}\oplus
E/H^{1},\theta\big{]}.$

    * $(1.2)$

$0<d_{1}<g-1$ $\&$ $-d_{1}<\deg(I)\leq-\frac{d_{1}}{2}$ or
$g-1<d_{1}\leq\frac{4g-4}{3}$ $\&$
$d_{1}-2g+2\leq\deg(I)\leq-\frac{d_{1}}{2}$, in either case, the Simpson
filtration is given by

$\displaystyle 0\subsetneq H^{1}\subsetneq F^{1}\subsetneq E,$

where $F^{1}=\operatorname{Ker}\big{(}E\to\frac{E/H^{1}}{I}\big{)}\subset E$
is a rank $2$ subbundle. Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\left[H^{1}\oplus
I\oplus\frac{E/H^{1}}{I},\begin{pmatrix}0&0&0\\\ \varphi_{1}&0&0\\\
0&\varphi_{2}&0\end{pmatrix}\right],$

where $\varphi_{1}\colon H^{1}\to I\otimes K_{X}$ is induced by $\theta$ and
$\varphi_{2}\colon I\to\frac{E/H^{1}}{I}\otimes K_{X}$ is induced by
$\nabla\colon F^{1}\to E\otimes K_{X}$.

  * $(2)$

If the Harder–Narasimhan type of $E$ is $(2,1)$ as above. Let $\theta\colon
H^{1}\to E/H^{1}\otimes K_{X}$ be the non-zero induced map and let
$N:=\operatorname{Ker}\big{(}H^{1}\to E/H^{1}\otimes K_{X}\big{)}\subset
H^{1}$ be the sub line bundle, then

    * $(2.1)$

$0<d_{1}<g-1$ $\&$ $2d_{1}-2g+2\leq\deg(N)<0$, the Simpson filtration
coincides with the Harder–Narasimhan filtration. Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\big{[}H^{1}\oplus
E/H^{1},\theta\big{]}.$

    * $(2.2)$

$0<d_{1}<g-1$ $\&$ $0<\deg(N)\leq\frac{d_{1}}{2}$ or
$g-1<d_{1}\leq\frac{4g-4}{3}$ $\&$
$2d_{1}-2g+2\leq\deg(N)\leq\frac{d_{1}}{2}$, in either case, the Simpson
filtration is given by

$\displaystyle 0\subsetneq N\subsetneq G^{1}\subsetneq E,$

where $G_{1}:=\operatorname{Ker}\big{(}E\to E/H^{1}\big{)}\subset E$ is a rank
$2$ subbundle. Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\left[N\oplus H^{1}/N\oplus
E/H^{1},\begin{pmatrix}0&0&0\\\ \psi_{1}&0&0\\\
0&\psi_{2}&0\end{pmatrix}\right],$

where $\psi_{1}\colon N\to H^{1}/N\otimes K_{1}$ and $\psi_{2}\colon
H^{1}/N\to E/H^{1}\otimes K_{X}$ are induced by $\nabla\colon N\to
H^{1}\otimes K_{X}$.

  * $(3)$

If the Harder–Narasimhan type of $E$ is $(1,1,1)$ as above. Let $I\subset
E/H^{1}$ be the sub line bundle as defined in $(1)$ and $N\subset H^{2}$ be
the sub line bundle as defined in $(2)$.

    * $(3.1)$

$\max\\{-d_{1},2d_{2}-d_{1}\\}<\deg(I)\leq d_{2}$, the Simpson filtration is
given by

$\displaystyle 0\subsetneq H^{1}\subsetneq F^{1}\subsetneq E,$

where $F^{1}=\operatorname{Ker}\big{(}E\to\frac{E/H^{1}}{I}\big{)}\subset E$
is a rank $2$ subbundle. Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\left[H^{1}\oplus
I\oplus\frac{E/H^{1}}{I},\begin{pmatrix}0&0&0\\\ \varphi_{1}&0&0\\\
0&\varphi_{2}&0\end{pmatrix}\right],$

where $\varphi_{1}\colon H^{1}\to I\otimes K_{X}$ is induced by $\theta$ in
$(1)$ and $\varphi_{2}\colon I\to\frac{E/H^{1}}{I}\otimes K_{X}$ is induced by
$\nabla\colon F^{1}\to E\otimes K_{X}$. In particular, if $\deg(I)=d_{2}$,
then $F^{1}=H^{2}$, that is, the Simpson filtration coincides with the
Hardar–Narasimhan filtration.

    * $(3.2)$

$d_{1}-2g+2\leq\deg(I)<\min\\{-d_{1},2d_{2}-d_{1}\\}$,

      * $(3.2.1)$

$d_{2}<0$, then the Simpson filtration is given by

$\displaystyle 0\subsetneq H^{1}\subsetneq E.$

Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\big{[}H^{1}\oplus
E/H^{1},\theta\big{]},$

where $\theta\colon H^{1}\to E/H^{1}\otimes K_{X}$ is induced from
$\nabla\colon H^{1}\to E\otimes K_{X}$.

      * $(3.2.2)$

$d_{2}>0$,

        * $\bullet$

$2(d_{1}+d_{2})-2g+2\leq\deg(N)<0$, the Simpson filtration is given by

$\displaystyle 0\subsetneq H^{2}\subsetneq E.$

Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\big{[}H^{2}\oplus
E/H^{2},\theta\big{]},$

where $\theta\colon H^{2}\to E/H^{2}\otimes K_{X}$ is induced from
$\nabla\colon H^{2}\to E\otimes K_{X}$.

        * $\bullet$

$0<\deg(N)\leq d_{1}$, the Simpson filtration is given by

$\displaystyle 0\subsetneq N\subsetneq H^{2}\subsetneq E.$

Hence

$\displaystyle\lim_{t\to 0}t\cdot(E,\nabla)=\left[N\oplus H^{2}/N\oplus
E/H^{2},\begin{pmatrix}0&0&0\\\ \varphi_{1}&0&0\\\
0&\varphi_{2}&0\end{pmatrix}\right],$

where $\psi_{1}\colon N\to H^{2}/N\otimes K_{X}$ and $\theta$ in $(1)$ and
$\psi_{2}\colon H^{2}/N\to E/H^{2}\otimes K_{X}$ are induced by $\nabla\colon
N\to H^{2}\otimes K_{X}$. In particular, if $\deg(N)=d_{1}$, then $N=H^{1}$,
that is, the Simpson filtration coincides with the Hardar–Narasimhan
filtration.

###### Proof.

(1) and (2) are dual to each other, here we just prove (1). Since
$H^{1}\subset E$ is the maximal destabilizing subbundle, we have
$\mu\big{(}H^{1}\big{)}=d_{1}>0$, and since $I$ is the sub line bundle of the
semistable bundle $E/H^{1}$, we have
$\mu(I)\leq\mu\big{(}E/H^{1}\big{)}=-\frac{d_{1}}{2}$. On the other hand, the
induced map $\theta\colon H^{1}\to I\otimes K_{X}$ is non-zero, which gives
$\deg(I)\geq d_{1}-2g+2$. These give the maximal bound of $d_{1}$ and
$\deg(I)$ as follows:

$\displaystyle 0<d_{1}\leq\frac{4g-4}{3},\qquad
d_{1}-2g+2\leq\deg(I)\leq-\frac{d_{1}}{2}.$

Consider the induced graded Higgs bundle $\big{(}H^{1}\oplus
E/H^{1},\theta\big{)}$, it is graded stable if and only if $\mu(H^{1}\oplus
I)<0$, that is, $\deg(I)<-d_{1}$. Therefore, if the conditions in (1.1) are
satisfied, the Simpson filtration is $0\subsetneq H^{1}\subsetneq E$, with the
associated gr-stable Higgs bundle $\big{(}H^{1}\oplus E/H^{1},\theta\big{)}$.
When $\big{(}H^{1}\oplus E/H^{1},\theta\big{)}$ is not semistable, then its
maximal destabilizing subbundle is $H^{1}\oplus I$, which should satisfy
$\mu(H^{1}\oplus I)>0$. By Simpson’s iteration process, the next filtration is

$\displaystyle 0\subsetneq H^{1}\subsetneq F^{1}\subsetneq E,$

where $F^{1}=\operatorname{Ker}\big{(}E\to\frac{E/H^{1}}{I}\big{)}\subset E$
is a rank 2 subbundle. The associated graded Higgs bundle is

$\left(H^{1}\oplus I\oplus\frac{E/H^{1}}{I},\begin{pmatrix}0&0&0\\\
\varphi_{1}&0&0\\\ 0&\varphi_{2}&0\end{pmatrix}\right),$

with $\varphi_{1}\colon H^{1}\to I\otimes K_{X}$ is induced by $\theta$ and
$\varphi_{2}\colon I\to\frac{E/H^{1}}{I}\otimes K_{X}$ is induced by
$\nabla\colon F^{1}\to E\otimes K_{X}$. It is gr-stable if and only if
$\mu\big{(}\frac{E/H^{1}}{I}\big{)}<0$, which can be divided into the two
kinds of bounds for $d_{1}$ and $\deg(I)$ as in (1.2), in either case, the
Simpson filtration is $0\subsetneq H^{1}\subsetneq F^{1}\subsetneq E$. But if
the associated graded Higgs bundle is not gr-semistable, we should have
$\mu\big{(}\frac{E/H^{1}}{I}\big{)}>0$, that is, $\deg(I)>-d_{1}$, in this
case, $d_{1}$ and $\deg(I)$ have bounds
$0<d_{1}<g-1,d_{1}-2g+2\leq\deg(I)<-d_{1}$. By Simpson’s iteration process,
the next filtration is $0\subsetneq H^{1}\subsetneq E$, which comes back to
the case (1.1), the associated graded Higgs bundle will be stable, and the
iteration process stop here. Therefore, we finish the proof of (1).

(3) Since $H^{1}\subset E$ is the maximal destabilizing subsheaf and
$H^{2}/H^{1}\subset E/H^{1}$ is the maximal destabilizing subsheaf. And the
induced morphisms $H^{1}\to I\otimes K_{X}$ and $H^{2}/N\to E/H^{2}\otimes
K_{X}$ are both non-zero, we have the maximal bounds of $\deg(I)$ and
$\deg(N)$ as follows:

$\displaystyle d_{1}-2g+2\leq\deg(I)\leq d_{2},\qquad
2(d_{1}+d_{2})-2g+2\leq\deg(N)\leq d_{1}.$

By the uniqueness of maximal destabilizing subsheaf, $\deg(I)=d_{2}$ if and
only if $I=H^{2}/H^{1}$, and $\deg(N)=d_{1}$ if and only if $N=H^{1}$.

Look at the graded Higgs bundle $\big{(}H^{1}\oplus E/H^{1},\theta\big{)}$,
where $\theta\colon H^{1}\to E/H^{1}\otimes K_{X}$ is the induced map from
$\nabla\colon H^{1}\to E\otimes K_{X}$. It is stable if and only if
$\mu\big{(}H^{2}/H^{1}\big{)}<0$ and $\mu\big{(}H^{1}\oplus I\big{)}<0$, that
is, $d_{2}<0$ and $\deg(I)<-d_{1}$, this is the case (3.2.1). When it is not
stable, then the maximal destabilizing subsheaf should have positive slope,
the possible maximal destabilizing subsheaf is $H^{1}\oplus I$ or
$H^{2}/H^{1}$:

  1. (a)

If $\mu\big{(}H^{1}\oplus I\big{)}>\mu\big{(}H^{2}/H^{1}\big{)}$ and
$\mu\big{(}H^{1}\oplus I\big{)}>0$, that is,
$\max\\{2d_{2}-d_{1},-d_{1}\\}<\deg(I)\leq d_{2}$, then $\big{(}H^{1}\oplus
E/H^{1},\theta\big{)}$ has maximal destabilizing subsheaf $H^{1}\oplus I$. By
Simpson’s iteration, the next filtration is $0\subsetneq H^{1}\subsetneq
F^{1}\subsetneq E$ for
$F^{1}=\operatorname{Ker}\big{(}E\to\frac{E/H^{1}}{I}\big{)}\subset E$ a rank
2 subbundle. Easy to see that its associated graded Higgs bundle is stable.

  2. (b)

If $\mu\big{(}H^{2}/H^{1}\big{)}>\mu\big{(}H^{1}\oplus I\big{)}$ and
$\mu\big{(}H^{2}/H^{1}\big{)}>0$, that is, $d_{2}>0$ and
$d_{1}-2g+2\leq\deg(I)<2d_{2}-d_{1}$, then $\big{(}H^{1}\oplus
E/H^{1},\theta\big{)}$ has maximal destabilizing subsheaf $H^{2}/H^{1}$. By
Simpson’s iteration, the next filtration is $0\subsetneq H^{2}\subsetneq E$,
but we should discuss the stability of the graded Higgs bundle
$\big{(}H^{2}\oplus E/H^{2},\theta^{\prime}\big{)}$:

     * •

it is stable if and only if $\mu(N)<0$, that is, when $d_{2}>0$,
$d_{1}-2g+2\leq\deg(I)<2d_{2}-d_{1}$ and $2(d_{1}+d_{2})-2g+2\leq\deg(N)<0$,
the Simpson filtration is $0\subsetneq H^{2}\subsetneq E$;

     * •

if it is not semistable, then its maximal destabilizing subsheaf is $N$ and
should satisfy $\mu(N)>0$, and by Simpson’s iteration, the next filtration is
$0\subsetneq N\subsetneq H^{2}\subsetneq E$. Its associated Higgs bundle is
stable, so the iteration stops.

Combining all the above, we obtain the statement (3). ∎

For each $\alpha$, let $\big{(}G_{\alpha}^{1}\big{)}^{\rm VHS}\subset
G_{\alpha}^{1}$ be the subset that consists of the polarized $\mathbb{C}$-VHS
(complex variations of Hodge structure). It identifies to those Higgs bundles
having the structure of systems of Hodge bundles (i.e., the fixed point set
$P_{\alpha}\subset G_{\alpha}^{0}$) by the non-Abelian Hodge correspondence.
Therefore, $\big{(}G_{\alpha}^{1}\big{)}^{\rm VHS}={\rm NAH}(P_{\alpha})$.
Simpson guessed in [63] that points in
$G_{\alpha}^{1}\backslash\big{(}G_{\alpha}^{1}\big{)}^{\rm VHS}$ do not relate
to the points in $G_{\alpha}^{0}$ via the non-Abelian Hodge correspondence,
this is the following conjecture:

######  Conjecture 4.12.

$\big{(}G_{\alpha}^{1}\big{)}^{\rm VHS}=G_{\alpha}^{1}\bigcap{\rm
NAH}\big{(}G_{\alpha}^{0}\big{)}$.

For each polarized $\mathbb{C}$-VHS $(E,\nabla)$ such that the corresponding
monodromy representation is irrreducible, then the Simpson filtration
coincides with its Hodge filtration, this provides a method to construct its
Hodge filtration.

##  5 Twistor structures arising from non-Abelian Hodge
correspondence

Another important application of non-Abelian Hodge theory is the study of
Hitchin’s construction of twistor spaces for hyper-Kähler manifolds applied to
the moduli spaces with hyper-Kähler structure (see Section 5.1). This
construction of twistor structures is interpreted by Deligne as a gluing of
two moduli spaces, the resulting twistor space is shown being analytic
isomorphic to the original one (see Section 5.2). When we focus on compact
Riemann surfaces, then Deligne’s gluing can be generalized to a gluing of two
moduli spaces induced from any element of the outer automorphism group of the
fundamental group (see Section 5.3).

In this section, we will introduce the twistor construction from Hitchin and
Deligne’s viewpoints. Then we will provide a new interpretation based on a
recent work [37].

###  5.1 Hitchin’s twistor construction

In [36], the authors provide a construction of twistor space for any hyper-
Kähler manifold. Let $M$ be a heper-Kähler manifold with three complex
structures $(I,J,K=IJ)$. The stereographic projection $\mathbb{P}^{1}\to
S^{2}$,

$\displaystyle\lambda=u+iv\mapsto\left(x=\frac{1-|\lambda|^{2}}{1+|\lambda|^{2}},\,y=\frac{2u}{1+|\lambda|^{2}},\,z=\frac{2v}{1+|\lambda|^{2}}\right)$

defines a family of complex structures $I_{\lambda}:=xI+yJ+zK$ on $M$.

The twistor space ${\rm TW}(M)$ of $M$ is a $C^{\infty}$ trivialization ${\rm
TW}(M)\cong M\times\mathbb{P}^{1}$. $I_{\lambda}$ determines an almost complex
structure $\mathcal{I}$ on ${\rm TW}(M)$, let $a=m\times\lambda\in{\rm
TW}(M)$, then

$\displaystyle\mathcal{I}\colon\ T_{a}{\rm TW}(M)=T_{m}M\oplus
T_{\lambda}\mathbb{P}^{1}\to T_{m}M\oplus T_{\lambda}\mathbb{P}^{1}$

is given by $(I_{\lambda},I_{0})$, where $I_{0}\colon
T_{\lambda}\mathbb{P}^{1}\to T_{\lambda}\mathbb{P}^{1}$ is the usual complex
structure on $\mathbb{P}^{1}$ given by $I_{0}(v)=iv$. In fact, $\mathcal{I}$
is integrable [36, 55], thus the twistor space ${\rm TW}(M)$ is a complex
manifold of dimension $\dim_{\mathbb{C}}({\rm TW}(M))=1+\dim_{\mathbb{C}}(M)$.

The twistor space ${\rm TW}(M)$ has the following properties:

  * (1)

the projection $\pi\colon{\rm TW}(M)\to\mathbb{P}^{1}$ is holomorphic;

  * (2)

there is an antilinear involution $\sigma\colon{\rm TW}(M)\to{\rm TW}(M)$,
$(m,\lambda)\mapsto\big{(}m,-\bar{\lambda}^{-1}\big{)}$, which covers the
antipodal involution
$\sigma_{\mathbb{P}^{1}}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$,
$\lambda\mapsto-\bar{\lambda}^{-1}$, so it gives a real structure on ${\rm
TW}(M)$;

  * (3)

for any $m\in M$, the section $\\{m\\}\times\mathbb{P}^{1}\subset{\rm TW}(M)$
is holomorphic and $\sigma$-invariant, we call it a preferred section, in some
papers like [36, 55], it is called a twistor line;

  * (4)

weight 1 property: the normal bundle along any preferred section is isomorphic
to $\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus\dim_{\mathbb{C}}(M)}$.

######  Proposition 5.1.

Preferred sections are $\sigma$-invariant. Moreover, locally, preferred
sections are the only $\sigma$-invariant holomorphic sections.

If we denoted by ${\rm Pre}$ the set of preferred sections in the Douaby
moduli space ${\rm Sec}$ of holomorphic sections. Then “locally” means there
exists an open neighborhood $U$ of ${\rm Pre}$ in ${\rm Sec}$ such that
preferred sections are the only $\sigma$-invariant sections in $U$ [38]. To my
knowledge, whether Proposition 5.1 holds globally is unknown:

######  Question 5.2.

Are the preferred sections the only $\sigma$-invariant sections?

Moreover, let $p\colon{\rm TW}(M)\to M$ be the natural projection, then the
twistor space ${\rm TW}(M)$ admits a natural hermitian metric
$g:=p^{*}g_{M}+\pi^{*}g_{\rm FS}$, where $g_{M}$ is the hyper-Kähler metric on
$M$ and $g_{\rm FS}$ is the Fubini–Study metric on $\mathbb{P}^{1}$. This
metric makes the twistor space ${\rm TW}(M)$ into a balanced manifold [38]
(i.e., the associated fundamental form
$\omega(\bullet,\bullet):=g(\mathcal{I}\bullet,\bullet)$ satisfies the weak
closedness condition $d(\omega^{\dim_{\mathbb{C}}M})=0$).

Let $X$ be a complex projective variety, denote by $M_{\rm dR}^{\rm sm}(X,r)$
and $M_{\rm Dol}^{\rm sm}(X,r)$ the open subsets of smooth points of the
moduli spaces $M_{\rm dR}(X,r)$ and $M_{\rm Dol}(X,r)$, respectively. From
non-Abelian Hodge theory, we have $C^{\infty}$ isomorphism $M_{\rm dR}^{\rm
sm}(X,r)\cong M_{\rm Dol}^{\rm sm}(X,r)$ coming from the homeomorphism between
the underlying topological spaces [59, 60, 61]. Let $M^{\rm sm}(X,r)$ be the
underlying differentiable manifold. Then by the work of Hitchin ([34] for
1-dimensional base manifold) and Fujiki ([25] for higher-dimensional Kähler
manifolds as the base), $M^{\rm sm}(X,r)$ carries two complex structures $I$
and $J$ from the complex manifolds $\big{(}M_{\rm Dol}^{\rm sm}(X,r),I\big{)}$
and $\big{(}M_{\rm dR}^{\rm sm}(X,r),J\big{)}$. Moreover, $IJ=:K$ gives the
third complex structure, which makes$\big{(}M^{\rm sm}(X,r),I,J,K\big{)}$ has
the structure of a hyper-Kähler manifold. Hence Hitchin’s idea for
constructing the twistor space of hyper-Kähler manifolds can be applied to
$M^{\rm sm}(X,r)$. Therefore, we obtain a twistor space for $M^{\rm sm}(X,r)$,
denoted as ${\rm TW}\big{(}M^{\rm sm}(X,r)\big{)}$, and called the Hitchin
twistor space.

###  5.2 Deligne’s interpretation

Deligne’s original aim was to understand the Hitchin twistor space ${\rm
TW}\big{(}M^{\rm sm}(X,r)\big{)}$ via $\lambda$-connections. The idea is
gluing the Hodge moduli space $M_{\rm Hod}(X,r)$ over $X$ and the Hodge moduli
space $M_{\rm Hod}\big{(}\bar{X},r\big{)}$ over its conjugate chart $\bar{X}$
to obtain a twistor space ${\rm TW}_{\rm DH}(X,r)$ that is analytic isomorphic
to ${\rm TW}\big{(}M^{\rm sm}(X,r)\big{)}$. This was described and further
studied by Simpson, he interpreted $M_{\rm Hod}(X,r)$ as the Hodge filtration
on the non-Abelian de Rham cohomology $M_{\rm dR}(X,r)$. He showed the
Griffiths transversality and the regularity of the Gauss–Manin connection for
this filtration [61]. Here we introduce their ideas based on Simpson’s papers
[61, 62].

Let $X$ be a complex projective variety with a fixed base point $x$. The
complex conjugation gives a map $\varphi\colon X^{\rm top}\to\bar{X}^{\rm
top}$ between the underlying topological spaces, which induces an isomorphism

$\displaystyle\varphi_{*}\colon\
\pi_{1}(X,x)\to\pi_{1}\big{(}\bar{X},\bar{x}\big{)}.$ (5.1)

As the notations used in the last section, let $M_{\rm Hod}(X,r)$ be the
moduli space of semistable $\lambda$-flat bundles of rank $r$ with vanishing
Chern classes, let $M_{\rm B}(X,\mathrm{GL}(r,\mathbb{C}))$ be the moduli
space of representations. The Riemann–Hilbert correspondence gives the
analytic isomorphism [59]:

$\displaystyle M_{\rm dR}(X,r)\cong M_{\rm B}(X,r).$ (5.2)

The $\mathbb{C}^{*}$-action on $M_{\rm Hod}(X,r)$ gives the algebraic
isomorphism:

$\displaystyle M_{\rm Hod}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}$
$\displaystyle\cong M_{\rm dR}(X,r)\times\mathbb{C}^{*},$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\leftrightarrow\big{(}\big{[}E,\bar{\partial}_{E},\lambda^{-1}D^{\lambda}\big{]},\lambda\big{)}.$
(5.3)

The gluing isomorphism $\mathbf{d}_{\varphi}\colon M_{\rm
Hod}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}\to M_{\rm
Hod}(\bar{X},r)\times_{\mathbb{C}}\mathbb{C}^{*}$ is given as follows:

First by isomorphisms (5.1)–(5.3), any point
$\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$ in $M_{\rm
Hod}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}$ determines a representation
$\rho\big{(}\lambda^{-1}D^{\lambda}\big{)}\comp\varphi_{*}^{-1}\colon\pi_{1}\big{(}\bar{X},\bar{x}\big{)}\to{\rm
GL}(r,\mathbb{C})$, where
$\rho\big{(}\lambda^{-1}D^{\lambda}\big{)}\colon\pi_{1}(X,x)\to{\rm
GL}(r,\mathbb{C})$ is the monodromy corresponds to the flat connection
$\lambda^{-1}D^{\lambda}$. Then by the conjugate version of isomorphisms
(5.2), $\rho\big{(}\lambda^{-1}D^{\lambda}\big{)}\comp\varphi_{*}^{-1}$
corresponds to a flat bundle over $\bar{X}$. Finally, the conjugate version of
(5.3) at the fiber $\lambda^{-1}$ gives a $\lambda^{-1}$-flat bundle over
$\bar{X}$. The obtained $\lambda^{-1}$-flat bundle over $\bar{X}$ is
$\big{[}E,\lambda^{-1}D^{\lambda},\lambda^{-1}\bar{\partial}_{E},\lambda^{-1}\big{]}$.
Therefore, the gluing isomorphism is

$\displaystyle\mathbf{d}_{\varphi}\colon\ M_{\rm
Hod}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}$ $\displaystyle\cong M_{\rm
Hod}\big{(}\bar{X},r\big{)}\times_{\mathbb{C}}\mathbb{C}^{*},$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\leftrightarrow\big{[}E,\lambda^{-1}D^{\lambda},\lambda^{-1}\bar{\partial}_{E},\lambda^{-1}\big{]},$

which covers the map
$\mathbb{C}^{*}\to\mathbb{C}^{*},\lambda\mapsto\lambda^{-1}$. Therefore, we
obtained a space ${\rm TW}_{\rm DH}(X,r)$, called the Deligne–Hitchin twistor
space, together with a fibration ${\rm TW}_{\rm DH}(X,r)\to\mathbb{P}^{1}$
which extends the projection $\pi\colon M_{\rm Hod}(X,r)\to\mathbb{C}$. The
fibers of this fibration are $M_{\rm Dol}(X,r)$ at $\lambda=0$, $M_{\rm
Dol}\big{(}\bar{X},r\big{)}$ at $\lambda=\infty$, and analytic isomorphic to
$M_{\rm dR}(X,r)$ at $\lambda\neq 0,\infty$.

Let $\big{[}E,\bar{\partial}_{E},\theta,h\big{]}\in M_{\rm Dol}(X,r)$ be a
harmonic Higgs bundle with the pluri-harmonic metric $h$, then it determines a
holomorphic section $p\colon\mathbb{P}^{1}\to{\rm TW}_{\rm DH}(X,r)$:

$\displaystyle\lambda\longmapsto\big{[}E,\bar{\partial}_{E}+\lambda\theta_{h}^{\dagger},\lambda\partial_{E,h}+\theta,\lambda\big{]},$

where $\theta_{h}^{\dagger}$ and $\partial_{E,h}$ are the unique operators
determined by $(\theta,h)$ and $\big{(}\bar{\partial}_{E},h\big{)}$,
respectively. This section is called a preferred section.

We can also define an antilinear involution $\sigma\colon{\rm TW}_{\rm
DH}(X,r)\to{\rm TW}_{\rm DH}(X,r)$ that covers the antipodal involution
$\sigma_{\mathbb{P}^{1}}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$. This map is
defined by gluing the antiholomorphic ismorphisms $\sigma_{\rm Hod,X}\colon
M_{\rm Hod}(X,r)\to M_{\rm Hod}\big{(}\bar{X},r\big{)}$ and $\sigma_{\rm
Hod,\bar{X}}\colon M_{\rm Hod}\big{(}\bar{X},r\big{)}\to M_{\rm Hod}(X,r)$,
where

$\displaystyle\sigma_{\rm Hod,X}\colon\ M_{\rm Hod}(X,r)$ $\displaystyle\to
M_{\rm Hod}\big{(}\bar{X},r\big{)},$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\mapsto\big{[}\bar{E}^{*},\overline{\bar{\partial}_{E}}^{*},\overline{D^{\lambda}}^{*},-\bar{\lambda}\big{]}.$

By the existence of pluri-harmonic metric for (poly-)stable $\lambda$-flat
bundles with vanishing Chern classes (see Theorem 2.10), take $h$ being the
pluri-harmonic metric associated to
$\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$. Then we have the
isomorphism

$\displaystyle\big{[}\bar{E}^{*},\overline{\bar{\partial}_{E}}^{*},\overline{D^{\lambda}}^{*},-\bar{\lambda}\big{]}\cong\big{[}E,\delta_{h}^{\prime},-\delta_{h}^{\prime\prime},-\bar{\lambda}\big{]},$

where the operators $\delta_{h}^{\prime}$ and $\delta_{h}^{\prime\prime}$ can
be found in the second section.

Therefore, by means of pluri-harmonic metric, we can write the involution
$\sigma$ as follows:

$\displaystyle\sigma\colon\ {\rm TW}_{\rm DH}(X,r)$ $\displaystyle\to{\rm
TW}_{\rm DH}(X,r),$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\mapsto\big{[}E,\bar{\lambda}^{-1}\delta_{h}^{\prime\prime},-\bar{\lambda}^{-1}\delta_{h}^{\prime},-\bar{\lambda}^{-1}\big{]}.$

In fact, $\sigma$ is the product of the following 3 involutions [62]:

  * (1)

an antiholomorphic involution

$\displaystyle C\colon\ {\rm TW}_{\rm DH}(X,r)$ $\displaystyle\to{\rm TW}_{\rm
DH}(X,r),$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\mapsto\big{[}E,\bar{\lambda}^{-1}\overline{D^{\lambda}},\bar{\lambda}^{-1}\overline{\bar{\partial}_{E}},\bar{\lambda}^{-1}\big{]}$

obtained by gluing complex conjugations of $\lambda$-flat bundles;

  * (2)

a holomorphic involution

$\displaystyle D\colon\ {\rm TW}_{\rm DH}(X,r)$ $\displaystyle\to{\rm TW}_{\rm
DH}(X,r),$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\mapsto\big{[}E^{*},\big{(}\bar{\partial}_{E}\big{)}^{*},\big{(}D^{\lambda}\big{)}^{*},\lambda\big{]}$

obtained by gluing duals of $\lambda$-flat bundles;

  * (3)

a holomorphic involution

$\displaystyle N\colon\ {\rm TW}_{\rm DH}(X,r)$ $\displaystyle\to{\rm TW}_{\rm
DH}(X,r),$
$\displaystyle\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}$
$\displaystyle\mapsto\big{[}E,\bar{\partial}_{E},-D^{\lambda},-\lambda\big{]}$

obtained by $-1\in\mathbb{C}^{*}$ acts on ${\rm TW}_{\rm DH}(X,r)$.

By definition, any point in the twistor space determines a unique preferred
section. Therefore, the set of preferred sections gives a homeomorphism

$\displaystyle{\rm TW}_{\rm DH}(X,r)\cong M_{\rm
Dol}(X,r)\times\mathbb{P}^{1},$

which is a $C^{\infty}$ isomorphism over smooth points [61]:

$\displaystyle{\rm TW}_{\rm DH}^{\rm sm}(X,r)\cong M_{\rm Dol}^{\rm
sm}(X,r)\times\mathbb{P}^{1}.$

######  Proposition 5.3.

Preferred sections are $\sigma$-invariant. Moreover, locally, preferred
sections are the only $\sigma$-invariant holomorphic sections.

Preferred sections are invariant under the antilinear involution $\sigma$ is
easy to see. Here “locally” means that for any given preferred section
$p\colon\mathbb{P}^{1}\to{\rm TW}_{\rm DH}(X,r)$, there exists an open
neighbourhood $U\subseteq{\rm TW}_{\rm DH}(X,r)$ of $p$ such that preferred
sections are the only $\sigma$-invariant sections in $U$. This is true because
the normal bundle along any preferred section is isomorphic to
$\big{(}\mathcal{O}_{\mathbb{P}^{1}}(1)\big{)}^{\oplus\dim_{\mathbb{C}}M_{\rm
dR}(X,r)}$.

In [61], Simpson asked the following question:

######  Question 5.4.

Does Proposition 5.3 hold globally? That is, are the preferred sections the
only $\sigma$-invariant sections?

In [61, 62], Simpson showed this question is true for twistor space of rank 1
bundles. Recently, in [9], the authors constructed holomorphic
$\sigma$-invariant but not preferred sections for twistor space of rank 2
bundles over compact Riemann surface of $g\geq 2$.

Now we give a brief conclusion on above two different kinds of constructing
the twistor spaces. In Hitchin’s constructing, we first have a hyper-Kähler
manifold (the moduli spaces are hyper-Kähler) $M^{\rm sm}(X,r)$, then the
twistor space has a naturally structure of $M^{\rm
sm}(X,r)\times\mathbb{P}^{1}$. The complex structure is induced from a family
of complex structures (given by the quaternionic structure) on $M^{\rm
sm}(X,r)$ and the natural complex structure on $\mathbb{P}^{1}$. While from
Deligne’s interpretation, the twistor space is obtained by gluing the moduli
spaces $M_{\rm Hod}(X,r)$ over $X$ and $M_{\rm Hod}\big{(}\bar{X},r\big{)}$
over the conjugate $\bar{X}$. This gluing is obtained via the algebraic
isomorphism between $M_{\rm Hod}^{\lambda}$ ($\lambda\neq 0$) and $M_{\rm
dR}$, the isomorphism of fundamental groups induced from the complex conjugate
map $X\to\bar{X}$, and the analytic isomorphism $M_{\rm dR}\cong M_{\rm B}$
given by the Riemann–Hilbert correspondence. Preferred sections give the
twistor space the product structure, and the weight 1 property implies the
quaternionic structure. This viewpoint shows that a quaternionic structure on
$M^{\rm sm}(X,r)$ is equivalent to the weight 1 property of a preferred
section [62].

Moreover, the twistor spaces arising from these two different methods are
analytic isomorphic:

######  Theorem 5.5 ([61]).

The twistor space ${\rm TW}^{\rm sm}_{\rm DH}(X,r)$ is analytic isomorphic to
${\rm TW}(M^{\rm sm}(X,r))$.

###  5.3 A new interpretation

In [37], we concentrated on the twistor space of Riemann surface case, we
generalized Deligne’s construction for any element $\gamma$ of the outer
automorphism group of the fundamental group of the Riemann surface to obtain
the $\gamma$-twistor space, we give a brief introduction here (for more
details and proofs, see [37]).

Let $X$ be a compact Riemann surface of genus $g\geq 2$ and let $\mathcal{X}$
be the underlying smooth surface, so $X$ can be wrote as $X=(\mathcal{X},I)$
for a complex structure $I$ that defines the Riemann surface structure of $X$.
The fundamental group of $\mathcal{X}$ is given by

$\displaystyle\pi_{1}(\mathcal{X})=\bigg{\langle}\alpha_{1},\beta_{1},\dots,\alpha_{g},\beta_{g}\colon\prod_{i=1}^{g}\alpha_{i}\beta_{i}\alpha_{i}^{-1}\beta_{i}^{-1}=1\bigg{\rangle}.$

Take $G=\mathrm{GL}(r,\mathbb{C})$ and let $M_{\mathrm{B}}(\mathcal{X},G)$ be
the coarse moduli space of rank $r$ representations of $\pi_{1}(\mathcal{X})$
into $G$.

When $\lambda\neq 0$, we have the analytic isomorphism

$\displaystyle\mu_{\lambda}\colon\ M^{\lambda}_{{\rm
Hod}}(X)\xlongrightarrow{\cong}M_{\mathrm{B}}(\mathcal{X},G$

given by the composition of the algebraic isomorphism $M^{\lambda}_{{\rm
Hod}}(X)\cong M_{\mathrm{dR}}(X,r)$ and the Riemann–Hilbert correspondence
$M_{\mathrm{dR}}(X,r)\cong M_{\mathrm{B}}(\mathcal{X},G)$, where the first
algebraic isomorphism is given by rescaling the twistor parameter $\lambda$
(does not change the holomorphic structures of the underlying bundles).

The outer automorphism group

$\displaystyle\Gamma_{\mathcal{X}}:=\mathrm{Out}(\pi_{1}(\mathcal{X}))=\operatorname{Aut}(\pi_{1}(\mathcal{X}))/\mathrm{Inn}(\pi_{1}(\mathcal{X}))$

acts on $M_{\mathrm{B}}(\mathcal{X},G)$. It is known that
$\Gamma_{\mathcal{X}}$ is isomorphic to the extended mapping class group

$\displaystyle{\rm
Mod}^{\diamondsuit}(\mathcal{X}):=\pi_{0}(\mathrm{Diff}(\mathcal{X}))=\mathrm{Diff}(\mathcal{X})/\mathrm{Diff}_{0}(\mathcal{X})$

that acts on the Teichmüller space ${\rm Teich}(\mathcal{X})$ of
$\mathcal{X}$. To define the Deligne gluing, we first define an action of
$\Gamma_{\mathcal{X}}$ on ${\rm Teich}(\mathcal{X})\times
M_{\mathrm{B}}(\mathcal{X},G)$ as follows: for
$f\in\mathrm{Diff}(\mathcal{X})$ such that the equivalence class $[f]\in{\rm
Mod}^{\diamondsuit}(\mathcal{X})$ is nontrivial, there is a induced
isomorphism

$\displaystyle f_{*}\colon\
\pi_{1}(\mathcal{X},x)\rightarrow\pi_{1}(\mathcal{X},f(x))$

for $x\in\mathcal{X}$ such that $[f_{*}]\in\Gamma_{\mathcal{X}}$ is
nontrivial, the action of $f$ on ${\rm Teich}(\mathcal{X})$ maps
$X=(\mathcal{X},I)$ to $X^{\prime}=(\mathcal{X},I^{\prime})$ where $I$ is a
complex structure on $\mathcal{X}$ and $I^{\prime}$ is another complex
structure induced by $f$, and the action of $f_{*}$ on
$M_{\mathrm{B}}(\mathcal{X},G)$ maps a representation
$\rho\colon\pi_{1}(\mathcal{X},x)\to G$ to another representation
$\rho^{\prime}=\rho\circ f_{*}^{-1}\colon\pi_{1}(\mathcal{X},f(x))\to G$, thus
the action of $\Gamma_{\mathcal{X}}$ on ${\rm Teich}(\mathcal{X})\times
M_{\mathrm{B}}(\mathcal{X},G)$ sends $(X,\rho)$ to
$(X^{\prime},\rho^{\prime})$.

Choose $\gamma\in\Gamma_{\mathcal{X}}$. For
$\big{[}E,\bar{\partial}_{E},D^{\lambda},\lambda\big{]}\in M_{{\rm
Hod}}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}$, we have an $(X,\rho)\in{\rm
Teich}(\mathcal{X})\times M_{\mathrm{B}}(\mathcal{X},G)$ via the isomorphism
$\mu_{\lambda}$, then the $\gamma$-action maps $(X,\rho)$ to
$(X^{\prime},\rho^{\prime})$ which corresponds to some
$\big{[}E^{\prime},\bar{\partial}_{E^{\prime}},D^{{}^{\prime}\lambda^{-1}},\lambda^{-1}\big{]}\in
M_{{\rm Hod}}(X^{\prime},r)\times_{\mathbb{C}}\mathbb{C}^{*}$ via
$\mu_{\lambda}^{-1}$ “at the point $\lambda^{-1}\in\mathbb{C}^{*}$.

Therefore, we define an analytic isomorphism called the Deligne isomorphsim

$\displaystyle\mathbf{d}_{\gamma}\colon\ M_{{\rm
Hod}}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}\rightarrow M_{{\rm
Hod}}\big{(}X^{\prime},r\big{)}\times_{\mathbb{C}}\mathbb{C}^{*}$

that covers the map $\mathbb{C}^{*}\to\mathbb{C}^{*}$,
$\lambda\mapsto\lambda^{-1}$. Now we can use this isomorphism $d_{\gamma}$ to
glue together two analytic spaces $M_{{\rm Hod}}(X,r)$ and $M_{{\rm
Hod}}(X^{\prime},r)$ along their open sets. The resulting space is denoted by
$\mathrm{TW}^{\gamma}(X,r)$, called the $\gamma$-twistor space, it has a
fibration ${\rm TW}^{\gamma}(X,r)\to\mathbb{P}^{1}$. Obviously, this
construction is independent of the choice of representative of $\gamma$ up to
isomorphism. In conclusion, our construction is along the following diagram:

$\textstyle{\mathcal{M}_{\rm Hod}(X,r)|_{\lambda}\
(\lambda\in\mathbb{C}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong_{\rm
alg}}$$\scriptstyle{\cong_{\rm an}\
\mathbf{d}_{\gamma}}$$\textstyle{\mathcal{M}_{\rm
Hod}(X^{\prime},r)|_{\lambda^{-1}}\
(\lambda\in\mathbb{C}^{*})}$$\textstyle{\mathcal{M}_{\rm
dR}(X,r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong_{\rm
an}}$$\textstyle{\mathcal{M}_{\rm
dR}(X^{\prime},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong_{\rm
alg}}$$\textstyle{\mathcal{M}_{\rm
B}(\mathcal{X},r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet\comp
f_{*}^{-1}}$$\textstyle{\mathcal{M}_{\rm
B}(\mathcal{X},r).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong_{\rm
an}}$

Taking $X=(\mathcal{X},I)$ and $X^{\prime}=(\mathcal{X},-I)$, let
$\varphi\colon X\to X^{\prime},x\mapsto\bar{x}$ be the complex conjugate map,
and $\varphi^{\prime}\colon\mathcal{X}\to\mathcal{X}$ be the associated map on
underlying smooth surface. Then $\varphi^{\prime}$ is not an identity map and
lies in a non-trivial class of ${\rm Mod}^{\diamondsuit}(\mathcal{X})$. This
induces an isomorphism
$\varphi_{*}\colon\pi_{1}(\mathcal{X},x)\to\pi_{1}(\mathcal{X},\varphi^{\prime}(x))$,
which lies in a non-trivial class of the outer automorphism group
$\mathrm{Out}(\pi_{1}(\mathcal{X}))$. Therefore, the Deligne–Hitchin twistor
space $\mathrm{TW}_{\mathrm{DH}}(X,r)$ can be thought as a special
$\gamma$-twistor space.

Fix a point $\big{[}E,\bar{\partial}_{E},D^{\lambda_{0}},\lambda_{0}\big{]}\in
M_{\rm Hod}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}$ for some fixed
$\lambda_{0}\in\mathbb{C}^{*}$, then $u$ determines a holomorphic section
$s_{\lambda_{0}}\colon\mathbb{C}^{*}\to M_{\rm
Hod}(X,r)\times_{\mathbb{C}}\mathbb{C}^{*}$ as follows:

$\displaystyle\lambda\longmapsto\big{[}E,\bar{\partial}_{E},\lambda\lambda_{0}^{-1}D^{\lambda_{0}},\lambda\big{]}.$

This section can be extended to a holomorphic section $\mathbb{P}^{1}\to{\rm
TW}^{\gamma}(X,r)$ of the $\gamma$-twistor space by Simpson’s Theorem 4.2 (see
also [37]), the extended section is still denoted as $s_{\lambda_{0}}$, called
the de Rham section of ${\rm TW}^{\gamma}(X,r)$.

The following property is obtained in [37], where the Torelli-type theorem for
the $\gamma$-twistor space is obtained by applying the techniques in [8],
where the authors obtained the property for the Deligne–Hitchin twistor space.

######  Theorem 5.6.

  1. $(1)$

The de Rham section $s_{\lambda_{0}}$ also has the weight $1$ property, that
is, we have the following isomorphism:

$\displaystyle
N_{s_{\lambda_{0}}}\simeq\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus\dim_{\mathbb{C}}(M_{\rm
dR}(X,r))},$

where $N_{s_{\lambda_{0}}}$ denotes the normal bundle. In particular, the
$\gamma$-twistor space $\mathrm{TW}^{\gamma}(X,r)$ contains ample rational
curves.

  2. $(2)$

$($Torelli-type theorem$)$ Let $X$, $Y$ be two compact Riemann surfaces with
genus $g\geq 3$. If we have the analytical isomorphism of $\gamma$-twistor
spaces $\mathrm{TW}^{\gamma}(X,r)\cong\mathrm{TW}^{\gamma}(Y,r)$, then as
Riemann surfaces, either $X\cong Y$, or $X\cong Y^{\prime}$, where
$Y^{\prime}$ is the Riemann surface determined by $Y$ and $\gamma$.

### Acknowledgements

The author would like to thank his thesis supervisor Professor Carlos Simpson
for much help on the understanding of this subject and for the kind guidance
and useful discussions, and thank Professor Jiayu Li for continuous
encouragement. Moreover, the author would like to express his deep
appreciation to the anonymous referees for many valuable suggestions. The
author is partial supported by China Scholarship Council (No. 201706340032).

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