[thm] [assumpsion] [cor] [proposition] [dfn] [lem] [main] [rem]

https://ydeng.perso.math.cnrs.fr

https://sites.google.com/site/yamanoimath/

With an appendix joint with Ludmil Katzarkov

# Reductive Shafarevich conjecture

Ya Deng<EMAIL_ADDRESS>CNRS, Institut Élie Cartan de Lorraine,
Université de Lorraine, F-54000 Nancy, France.  Katsutoshi Yamanoi
<EMAIL_ADDRESS>Department of Mathematics, Graduate School of
Science,Osaka University, Toyonaka, Osaka 560-0043, Japan

(September 3, 2024)

###### Abstract

In this paper, we present a more accessible proof of Eyssidieux’s proof of the
reductive Shafarevich conjecture in 2004, along with several generalizations.
In a nutshell, we prove the holomorphic convexity of the covering of a
projective normal variety $X$, which corresponds to the intersection of
kernels of reductive representations $\varrho:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$. Our approach avoids the necessity of using the reduction
mod $p$ method employed in Eyssidieux’s original proof. Moreover, we extend
the theorems to singular normal varieties under a weaker condition of
absolutely constructible subsets, thereby answering a question by Eyssidieux,
Katzarkov, Pantev, and Ramachandran. Additionally, we construct the
Shafarevich morphism for reductive representations over quasi-projective
varieties unconditionally, and proving its algebraic nature at the function
field level.

###### keywords:

Reductive Shafarevich conjecture, Hamonic mapping to Bruhat-Tits buildings,
period mappings and period domain, variation of Hodge structures, Holomorphic
convexity

###### Contents

  1. 0 Introduction
    1. 0.1 Shafarevich conjecture
    2. 0.2 Main theorems
    3. 0.3 Comparison with [Eys04] and Novelty
  2. 1 Technical preliminary
    1. 1.1 Admissible coordinates
    2. 1.2 Tame and pure imaginary harmonic bundles
    3. 1.3 Positive currents on normal complex spaces
    4. 1.4 Holomorphic forms on complex normal spaces
    5. 1.5 The criterion for Kähler classes
    6. 1.6 Some criterion for Stein space
    7. 1.7 Some facts on moduli spaces of rank 1 local systems
    8. 1.8 Absolutely constructible subsets
    9. 1.9 Katzarkov-Eyssidieux reduction and canonical currents
    10. 1.10 The generalization of Katzarkov-Eyssidieux reduction to quasi-projective varieties
    11. 1.11 Simultaneous Sten factorization
  3. 2 Some non-abelian Hodge theories
    1. 2.1 Regular filtered Higgs bundles
    2. 2.2 Pullback of parabolic bundles
    3. 2.3 Functoriality of pullback of regular filtered Higgs bundle
    4. 2.4 $\mathbb{C}^{*}$-action and $\mathbb{R}^{*}$-action on character varieties
    5. 2.5 Pullback of reductive representations commutes with $\mathbb{C}^{*}$-action
  4. 3 Construction of the Shafarevich morphism
    1. 3.1 Factorizing through non-rigidity
    2. 3.2 Infinite monodromy at infinity
    3. 3.3 Construction of Shafarevich morphism (I)
    4. 3.4 Construction of Shafarevich morphism (II)
    5. 3.5 On the algebraicity of the Shafarevich morphism via $L^{2}$-methods
  5. 4 Proof of the reductive Shafarevich conjecture
    1. 4.1 Reduction map of representation into algebraic tori
    2. 4.2 Some criterion for representation into tori
    3. 4.3 Eyssidieux-Simpson Lefschetz theorem and its application
    4. 4.4 A factorization theorem
    5. 4.5 Constructing Kähler classes via representations into non-archimedean fields
    6. 4.6 Holomorphic convexity associated with absolutely constructible subsets
    7. 4.7 Universal covering is Stein
  6. A Shafarevich conjecture for projective normal varieties
    1. A.1 Absolutely constructible subset (II)
    2. A.2 Reductive Shafarevich conjecture for normal projective varieties

##  0 Introduction

###  0.1 Shafarevich conjecture

In his famous textbook “Basic Algebraic Geometry[Sha77, p 407], Shafarevich
raised the following tantalizing conjecture.

######  Conjecture 0.1 (Shafarevich).

Let $X$ be a complex projective variety. Then its universal covering is
holomorphically convex.

Recall that a complex normal space $X$ is _holomorphically convex_ if it
satisfies the following condition: for each compact $K\subset X$, its
_holomorphic hull_

$\left\\{x\in X\mid|f(x)|\leq\sup_{K}|f|,\forall f\in\mathcal{O}(X)\right\\},$

is compact. $X$ is _Stein_ if it is holomorphically convex and holomorphically
separable, i.e. for distinct $x$ and $y$ in $X$, there exists
$f\in\mathcal{O}(X)$ such that $f(x)\neq f(y)$. By the Cartan-Remmert theorem,
a complex space $X$ is holomorphically convex if and only if it admits a
proper surjective holomorphic mapping $\phi$ onto some Stein space.

The study of Conjecture 0.1 for smooth projective surfaces has been a subject
of extensive research since the mid-1980s. Gurjar-Shastri [GS85] and Napier
[Nap90] initiated this investigation, while Kóllar [Kol93] and Campana [Cam94]
independently explored the conjecture in the 1990s, employing the tools of
Hilbert schemes and Barlet cycle spaces. In 1994, Katzarkov discovered that
non-abelian Hodge theories developed by Simpson [Sim92] and Gromov-Schoen
[GS92] can be utilized to prove Conjecture 0.1. His initial work [Kat97]
demonstrated Conjecture 0.1 for projective varieties with nilpotent
fundamental groups. Shortly thereafter, he and Ramachandran [KR98]
successfully established Conjecture 0.1 for smooth projective surfaces whose
fundamental groups admit a faithful Zariski-dense representation in a
reductive complex algebraic group. Building upon the ideas presented in [KR98]
and [Mok92], Eyssidieux further developed non-abelian Hodge theoretic
arguments in higher dimensions. In [Eys04] he proved that Conjecture 0.1 holds
for any _smooth_ projective variety whose fundamental group possesses a
faithful representation that is Zariski dense in a reductive complex algebraic
group. This result is commonly referred to as the “ _Reductive Shafarevich
conjecture_. It is worth emphasizing that the work of Eyssidieux [Eys04] is
not only ingenious but also highly significant in subsequent research. It
serves as a foundational basis for advancements in the linear Shafarevich
conjecture [EKPR12] and the exploration of compact Kähler cases [CCE15]. More
recently, there have been significant advancements in the quasi-projective
setting by Green-Griffiths-Katzarkov [GGK22] and Aguilar-Campana [AC23],
particularly when considering the case of nilpotent fundamental groups.

###  0.2 Main theorems

The aim of this paper is to present a more comprehensive and complete proof of
Eyssidieux’s results on the reductive Shafarevich conjecture and its
associated problems, as originally discussed in [Eys04]. Additionally, we aim
to extend these results to the cases of quasi-projective and singular
varieties. Our first main result is the _unconditional_ construction of the
_Shafarevich morphism_ for reductive representations. Additionally, we
establish the algebraicity of the Shafarevich morphism at the function field
level.

######  Theorem A (=Theorems 3.39 and 3.46).

Let $X$ be a quasi-projective normal variety, and let
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ be a reductive representation.
Then

  1. (i)

there exists a dominant holomorphic map ${\rm sh}_{\varrho}:X\to{\rm
Sh}_{\varrho}(X)$ to a complex normal space ${\rm Sh}_{\varrho}(X)$ whose
general fibers are connected such that for any closed subvariety $Z\subset X$,
$\varrho({\rm Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(X)])$ is finite if and only
if ${\rm sh}_{\varrho}(Z)$ is a point.

Furthermore, when $X$ is a smooth, after we replace $X$ by some finite étale
cover and $\varrho$ by its pullback over the cover, there exists another
smooth quasi-projective variety $X^{\prime}$ containing $X$ as a Zariski dense
open subset such that:

  1. (ii)

$\varrho$ extends to a reductive representation
$\varrho_{0}:\pi_{1}(X^{\prime})\to{\rm GL}_{N}(\mathbb{C})$;

  2. (iii)

the Shafarevich morphism ${\rm sh}_{\varrho_{0}}:X^{\prime}\to{\rm
Sh}_{\varrho_{0}}(X^{\prime})$ exists, which is a holomorphic proper
fibration;

  3. (iv)

${\rm sh}_{\varrho}={\rm sh}_{\varrho_{0}}|_{X}$; namely, we have the
following commutative diagram:

${X}$${X^{\prime}}$${{\rm Sh}_{\varrho}(X)}$${{\rm
Sh}_{\varrho_{0}}(X^{\prime})}$$\scriptstyle{{\rm
sh}_{\varrho}}$$\scriptstyle{{\rm sh}_{\varrho_{0}}}$

  4. (v)

There exists a bimeromorphic map $h:{\rm Sh}_{\varrho}(X)\dashrightarrow Y$ to
a quasi-projective normal variety $Y$.

  5. (vi)

The composition $h\circ{\rm sh}_{\varrho}:X\dashrightarrow Y$ is a rational
map.

The holomorphic map ${\rm sh}_{\varrho}:X\to{\rm Sh}_{\varrho}(X)$ that
satisfies the properties in A.(i) will be called the _Shafarevich morphism_ of
$\varrho$.

The proof of A.(i) relies on a more technical result but with richer
information, cf. Theorem 3.20.

######  Remark 0.2.

It is noticeable that A.(i), A.(ii), A.(iii) and A.(iv) extend the previous
theorems of Griffiths [Gri70] when $\varrho$ underlies a
$\mathbb{Z}$-variation of Hodge structures. In this case, the representation
$\varrho_{0}$ in A.(ii) is constructed in [Gri70, Theorem 9.5]. Additionally,
Griffiths proved in [Gri70, Theorem 9.6] that the period mapping
$p:X^{\prime}\to\mathscr{D}/\Gamma$ associated with $\varrho_{0}$ is proper,
where $\mathscr{D}$ represents the period domain of the $\mathbb{C}$-VHS, and
$\Gamma$ denotes the monodromy group of $\varrho_{0}$. It can be easily
verified that the Shafarevich morphism ${\rm
sh}_{\varrho_{0}}:X^{\prime}\to{\rm Sh}_{\varrho_{0}}(X^{\prime})$ corresponds
to the Stein factorization of the period mapping
$p:X^{\prime}\to\mathscr{D}/\Gamma$, and that A.(iv) holds.

We conjecture that ${\rm Sh}_{\varrho_{0}}(X^{\prime})$ is quasi-projective
and ${\rm sh}_{\varrho_{0}}$ is an algebraic morphism (cf. Conjecture 3.44).
Our conjecture is motivated by Griffiths’ conjecture, which predicted the same
result when $\varrho$ underlies a $\mathbb{Z}$-VHS. Consequently, we can
interpret the results presented in A.(v) and A.(vi) as supporting evidence for
our conjecture at the function field level. It is worth noting that Sommese
proved A.(v) and A.(vi) when $\varrho$ underlies a $\mathbb{Z}$-VHS in
[Som78], utilizing $L^{2}$-methods. We adopt the same approach in [Som78] to
prove A.(v) and A.(vi). Griffiths’ conjecture was recently proved by Baker-
Brunebarbe-Tsimerman [BBT23] using o-minimal geometry.

Our second main result focuses on the holomorphic convexity of topological
Galois coverings associated with reductive representations of fundamental
groups within _absolutely constructible subsets_ of character varieties
$M_{\rm B}(\pi_{1}(X),{\rm GL}_{N})$, where $X$ represents a projective normal
variety.

######  Theorem B (=Theorems 4.25 and A.3).

Let $X$ be a projective normal variety, and let $\mathfrak{C}$ be an
absolutely constructible subset of $M_{\rm B}(\pi_{1}(X),{\rm
GL}_{N})(\mathbb{C})$ as defined in Definitions 1.17 and A.1. We assume that
$\mathfrak{C}$ is defined on $\mathbb{Q}$. Set $H:=\cap_{\varrho}\ker\varrho$,
where $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ ranges over all
reductive representations such that $[\varrho]\in\mathfrak{C}(\mathbb{C})$.
Let $\widetilde{X}$ be the universal covering of $X$, and denote
$\widetilde{X}_{\mathfrak{C}}:=\widetilde{X}/H$. Then the complex space
$\widetilde{X}_{\mathfrak{C}}$ is holomorphically convex. In particular, we
have

  1. (i)

the covering of $X$ corresponding to the intersections of the kernels of all
reductive representations of $\pi_{1}(X)$ in ${\rm GL}_{N}(\mathbb{C})$ is
holomorphically convex;

  2. (ii)

if $\pi_{1}(X)$ is a subgroup of ${\rm GL}_{N}(\mathbb{C})$ whose Zariski
closure is reductive, then the universal covering $\widetilde{X}$ of $X$ is
holomorphically convex.

For large representations, we have the following result.

######  Theorem C (=Theorems 4.26 and A.5).

Let $X$ and $\mathfrak{C}$ be as described in Theorem B. If $\mathfrak{C}$ is
_large_ , meaning that for any closed subvariety $Z$ of $X$, there exists a
reductive representation $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ such
that $[\varrho]\in\mathfrak{C}(\mathbb{C})$ and $\varrho({\rm
Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(X)])$ is infinite, then all intermediate
coverings of $X$ between $\widetilde{X}$ and $\widetilde{X}_{\mathfrak{C}}$
are Stein spaces.

In addition to employing new methods for the proof of Theorems B and C, it
yields a stronger result compared to [Eys04] in two aspects:

  1. (a)

The definition of absolutely constructible subsets (cf. Definitions 1.17 and
A.1) in our proof is more general than the one provided in [Eys04]. Our
definition allows for a broader range of applications, including the potential
extension of Conjecture 0.1 to quasi-projective varieties, which is currently
our ongoing project.

  2. (b)

Our result extends to the case where $X$ is a singular variety, whereas in
[Eys04], the result is limited to smooth varieties. This expansion of our
result answers a question raised by Eyssidieux, Katzarkov, Pantev and
Ramachadran in their celebrated work on linear Shafarevich conjecture for
smooth projective varieties (cf. [EKPR12, p. 1549]).

We remark that Theorem C is not a direct consequence of Theorem B. It is
important to note that Theorem C holds significant practical value in the
context of singular varieties. Indeed, finding a large representation over a
_smooth_ projective variety can be quite difficult. In practice, the usual
approach involves constructing large representations using the Shafarevich
morphism in Theorem A, resulting in large representations of fundamental
groups of _singular normal_ varieties. Therefore, the extension of Theorem C
to singular varieties allows for more practical applicability.

###  0.3 Comparison with [Eys04] and Novelty

It is worth noting that Eyssidieux [Eys04] does not explicitly require
absolutely constructible subsets $\mathfrak{C}$ to be defined over
$\mathbb{Q}$, although it may seem to be an essential condition (cf. Remark
3.13). Regarding Theorem A, it represents a new result that significantly
builds upon our previous work [CDY22]. While Theorem C is not explicitly
stated in [Eys04], it should be possible to derive it for smooth projective
varieties $X$ based on the proof provided therein. However, it is worth noting
that the original proof in [Eys04] is known for its notoriously difficult and
involved nature, with certain aspects outlined without sufficient detail. One
of the main goals of this paper is to provide a relatively accessible proof
for Theorem B by incorporating more detailed explanations. We draw inspiration
from some of the methods introduced in our recent work [CDY22], which aids in
presenting a more comprehensible proof. Our proofs of Theorems B and C require
us to apply Eyssidieux’s Lefschetz theorem from [Eys04]. We also owe many
ideas to Eyssidieux’s work in [Eys04] and frequently draw upon them without
explicit citation.

Despite this debt, there are some novelties in our approach, including:

  * •

An avoidance of the reduction mod $p$ method used in [Eys04].

  * •

A new and more canonical construction of the Shafarevich morphism that
incorporates both rigid and non-rigid cases, previously treated separately in
[Eys04].

  * •

The construction of the Shafarevich morphism for reductive representations
over quasi-projective varieties, along with a proof of its algebraic property
at the function field level.

  * •

A detailed exposition of the application of Simpson’s absolutely constructible
subsets to the proof of holomorphic convexity in Theorems B and C (cf. §§ 4.1
and 4.21). This application was briefly outlined in [Eys04, Proof of
Proposition 5.4.6], but we present a more comprehensive approach, providing
complete details.

The main part of this paper was completed in February 2023 and was
subsequently shared with several experts in the field in April for feedback.
During the revision process, it came to our attention that Brunebarbe [Bru23]
recently announced a result similar to A.(i). In [Bru23, Theorem B] Brunebarbe
claims the existence of the Shafarevich morphism under a stronger assumption
of infinite monodromy at infinity and torsion-freeness of the representation,
and he does not address the algebraicity of Shafarevich morphisms. It seems
that some crucial aspects of the arguments in [Bru23] need to be carefully
addressed, particularly those related to non-abelian Hodge theories may have
been overlooked (cf. Remark 3.36).

### Convention and notation

In this paper, we use the following conventions and notations:

  * •

Quasi-projective varieties and their closed subvarieties are assumed to be
positive-dimensional and irreducible unless specifically mentioned otherwise.
Zariski closed subsets, however, may be reducible.

  * •

Fundamental groups are always referred to as topological fundamental groups.

  * •

If $X$ is a complex space, its normalization is denoted by
$X^{\mathrm{norm}}$.

  * •

The bold letter Greek letter ${\bm{\varrho}}$ (or ${\bm{\tau}}$,
${\bm{\sigma}}$…) denotes a family of finite reductive representations
$\\{\varrho_{i}:\pi_{1}(X)\to{\rm GL}_{N}(K)\\}_{i=1,\ldots,k}$ where $K$ is
some non-archimedean local field or complex number field.

  * •

A _proper holomorphic fibration_ between complex spaces $f:X\to Y$ is
_surjective_ and each fiber of $f$ is _connected_.

  * •

Let $X$ be a compact normal Kähler space and let $V\subset
H^{0}(X,\Omega_{X}^{1})$ be a $\mathbb{C}$-linear subspace. The _generic rank_
of $V$ is the largest integer $r$ such that ${\rm Im}[\Lambda^{r}V\to
H^{0}(X,\Omega^{r}_{X})]\neq 0$.

  * •

For a quasi-projective normal variety $X$, we denote by $M_{\rm B}(X,N)$ the
character variety of the representations of $\pi_{1}(X)$ into ${\rm GL}_{N}$.
For any linear representation $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(K)$ where $K$
is some extension of $\mathbb{Q}$, we denote by $[\varrho]\in M_{\rm
B}(X,N)(K)$ the equivalent class of $\varrho$.

  * •

$\mathbb{D}$ denotes the unit disk in $\mathbb{C}$.

### Acknowledgments

We would like to thank Daniel Barlet, Michel Brion, Frédéric Campana, Philippe
Eyssidieux, Ludmil Katzarkov, Janos Kollár, Mihai Păun, Carlos Simpson, Botong
Wang and Mingchen Xia for answering our questions. The impact of Eyssidieux’s
work [Eys04] on this paper cannot be overstated. This work was completed
during YD’s visit at the University of Miami in February. He would like to
extend special thanks to Ludmil Katzarkov for the warm invitation and fruitful
discussions that ultimately led to the collaborative development of the joint
appendix.

##  1 Technical preliminary

###  1.1 Admissible coordinates

The following definition of _admissible coordinates_ introduced in [Moc07a]
will be used throughout the paper.

######  Definition 1.1.

(Admissible coordinates) Let ${X}$ be a complex manifold and let $D$ be a
simple normal crossing divisor. Let $x$ be a point of $D$, and assume that
$\\{D_{j}\\}_{j=1,\ldots,\ell}$ be components of $D$ containing $x$. An
_admissible coordinate_ centered at $x$ is the tuple
$(U;z_{1},\ldots,z_{n};\varphi)$ (or simply $(U;z_{1},\ldots,z_{n})$ if no
confusion arises) where

  * •

$U$ is an open subset of ${X}$ containing $x$.

  * •

there is a holomorphic isomorphism $\varphi:U\to\mathbb{D}^{n}$ so that
$\varphi(D_{j})=(z_{j}=0)$ for any $j=1,\ldots,\ell$.

###  1.2 Tame and pure imaginary harmonic bundles

Let $\overline{X}$ be a compact complex manifold, $D=\sum_{i=1}^{\ell}D_{i}$
be a simple normal crossing divisor, $X=\overline{X}\backslash D$ be the
complement of $D$ and $j:X\to\overline{X}$ be the inclusion.

######  Definition 1.2 (Higgs bundle).

A _Higgs bundle_ on $X$ is a pair $(E,\theta)$ where $E$ is a holomorphic
vector bundle, and $\theta:E\to E\otimes\Omega^{1}_{X}$ is a holomorphic one
form with value in ${\rm\small End}(E)$, called the _Higgs field_ , satisfying
$\theta\wedge\theta=0$.

Let $(E,\theta)$ be a Higgs bundle over a complex manifold $X$. Suppose that
$h$ is a smooth hermitian metric of $E$. Denote by $\nabla_{h}$ the Chern
connection of $(E,h)$, and by $\theta^{\dagger}_{h}$ the adjoint of $\theta$
with respect to $h$. We write $\theta^{\dagger}$ for $\theta^{\dagger}_{h}$
for short if no confusion arises. The metric $h$ is _harmonic_ if the
connection $\nabla_{h}+\theta+\theta^{\dagger}$ is flat.

######  Definition 1.3 (Harmonic bundle).

A harmonic bundle on $X$ is a Higgs bundle $(E,\theta)$ endowed with a
harmonic metric $h$.

Let $(E,\theta,h)$ be a harmonic bundle on $X$. Let $p$ be any point of $D$,
and $(U;z_{1},\ldots,z_{n})$ be an admissible coordinate centered at $p$. On
$U$, we have the description:

$\displaystyle\theta=\sum_{j=1}^{\ell}f_{j}d\log
z_{j}+\sum_{k=\ell+1}^{n}f_{k}dz_{k}.$ (1.1)

######  Definition 1.4 (Tameness).

Let $t$ be a formal variable. For any $j=1,\dots,\ell$, the characteristic
polynomial $\det(f_{j}-t)\in\mathcal{O}(U\backslash D)[t],$ is a polynomial in
$t$ whose coefficients are holomorphic functions. If those functions can be
extended to the holomorphic functions over $U$ for all $j$, then the harmonic
bundle is called _tame_ at $p$. A harmonic bundle is _tame_ if it is tame at
each point.

For a tame harmonic bundle $(E,\theta,h)$ over $\overline{X}\backslash D$, we
prolong $E$ over $\overline{X}$ by a sheaf of
$\mathcal{O}_{\overline{X}}$-module ${}^{\diamond}\\!E_{h}$ as follows:

$\displaystyle{}^{\diamond}\\!E_{h}(U)=\\{\sigma\in\Gamma(U\backslash
D,E|_{U\backslash
D})\mid|\sigma|_{h}\lesssim{\prod_{i=1}^{\ell}|z_{i}|^{-\varepsilon}}\ \
\mbox{for all}\ \varepsilon>0\\}.$

In [Moc07a] Mochizuki proved that ${}^{\diamond}\\!E_{h}$ is locally free and
that $\theta$ extends to a morphism

${}^{\diamond}\\!E_{h}\to{}^{\diamond}\\!E_{h}\otimes\Omega^{1}_{\overline{X}}(\log
D),$

which we still denote by $\theta$.

######  Definition 1.5 (Pure imaginary).

Let $(E,h,\theta)$ be a tame harmonic bundle on $\overline{X}\backslash D$.
The residue ${\rm Res}_{D_{i}}\theta$ induces an endomorphism of
${}^{\diamond}\\!E_{h}|_{D_{i}}$. Its characteristic polynomial has constant
coefficients, and thus the eigenvalues are all constant. We say that
$(E,\theta,h)$ is _pure imaginary_ if for each component $D_{j}$ of $D$, the
eigenvalues of ${\rm Res}_{D_{i}}\theta$ are all pure imaginary.

One can verify that Definition 1.5 does not depend on the compactification
$\overline{X}$ of $\overline{X}\backslash D$.

######  Theorem 1.6 (Mochizuki [Moc07b, Theorem 25.21]).

Let $\overline{X}$ be a projective manifold and let $D$ be a simple normal
crossing divisor on $\overline{X}$. Let $(E,\theta,h)$ be a tame pure
imaginary harmonic bundle on $X:=\overline{X}\backslash D$. Then the flat
bundle $(E,D_{h}+\theta+\theta^{\dagger})$ is semi-simple. Conversely, if
$(V,\nabla)$ is a semisimple flat bundle on $X$, then there is a tame pure
imaginary harmonic bundle $(E,\theta,h)$ on $X$ so that
$(E,\nabla_{h}+\theta+\theta^{\dagger})\simeq(V,\nabla)$. Moreover, when
$\nabla$ is _simple_ , then any such harmonic metric $h$ is unique up to
positive multiplication.

The following important theorem by Mochizuki will be used throughout the
paper.

######  Theorem 1.7.

where $Y$ is smooth and $X$ is normal. For any reductive representation
$\varrho:\pi_{1}(Y)\to{\rm GL}_{N}(K)$, where $K$ is a non-archimedean local
field of characteristic zero or a complex number field, the pullback
$f^{*}\varrho:\pi_{1}(X)\to{\rm GL}_{N}(K)$ is also reductive.

###### Proof.

If $K$ is a non-archimedean local field of characteristic zero, then there is
an abstract embedding $K\hookrightarrow\mathbb{C}$. Therefore, it suffices to
prove the theorem for $K=\mathbb{C}$.

Let $\mu:X^{\prime}\to X$ be a desingularization of $X$. By [Moc07b, Theorem
25.30], $(f\circ\mu)^{*}\varrho:\pi_{1}(X^{\prime})\to{\rm
GL}_{N}(\mathbb{C})$ is reductive. Since
$\mu_{*}:\pi_{1}(X^{\prime})\to\pi_{1}(X)$ is surjective, it follows that
$(f\circ\mu)^{*}\varrho(\pi_{1}(X^{\prime}))=f^{*}\varrho(\pi_{1}(X))$. Hence,
$f^{*}\varrho$ is also reductive. ∎

###  1.3 Positive currents on normal complex spaces

For this subsection, we refer to [Dem85] for more details.

######  Definition 1.8.

Let $Z$ be an irreducible normal complex space. A upper semi continuous
function $\phi:Z\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is plurisubharmonic if
it is not identically $-\infty$ and every point $z\in Z$ has a neighborhood
$U$ embeddable as a closed subvariety of the unit ball $B$ of some
$\mathbb{C}^{M}$ in such a way that $\left.\phi\right|_{U}$ extends to a psh
function on $B$.

A closed positive current with continuous potentials $\omega$ on $Z$ is
specified by a data $\left\\{U_{i},\phi_{i}\right\\}_{i}$ of an open covering
$\left\\{U_{i}\right\\}_{i}$ of $Z$, a continuous psh function $\phi_{i}$
defined on $U_{i}$ such that $\phi_{i}-\phi_{j}$ is pluriharmonic on
$U_{i}\cap U_{j}$.

A closed positive current with continuous potentials $Z$ is a Kähler form iff
its local potentials can be chosen smooth and strongly plurisubharmonic.

A psh function $\phi$ on $Z$ is said to satisfy ${\rm d}{\rm d}^{\rm
c}\phi\geq\omega$ iff $\phi-\phi_{i}$ is psh on $U_{i}$ for every $i$.

In other words, a closed positive current with continuous potentials is a
section of the sheaf $C^{0}\cap PSH_{Z}/\operatorname{Re}\left(O_{Z}\right)$.

######  Definition 1.9.

Assume $Z$ to be compact. The class of a closed positive current with
continuous potentials is its image in
$H^{1}\left(Z,\operatorname{Re}\left(O_{Z}\right)\right)$.

A class in $H^{1}\left(Z,\operatorname{Re}\left(O_{Z}\right)\right)$ is said
to be Kähler if it is the image of a Kähler form.

To make contact with the usual terminology observe that if $Z$ is a compact
Kähler manifold
$H^{1}\left(Z,\operatorname{Re}\left(O_{Z}\right)\right)=H^{1,1}(Z,\mathbb{R})$.
Hence we use abuse of notation to write $H^{1,1}(Z,\mathbb{R})$ instead of
$H^{1}\left(Z,\operatorname{Re}\left(O_{Z}\right)\right)$ in this paper.

######  Lemma 1.10.

Let $f:X\to Y$ be a Galois cover with Galois group $G$, where $X$ and $Y$ are
both irreducible normal complex space. Let $T$ be a positive $(1,1)$-current
on $X$ with continuous potential. Assume that $T$ is invariant under $G$. Then
there is a closed positive $(1,1)$-current $S$ on $Y$ with continuous
potential such that $T=f^{*}S$.

###### Proof.

Since the statement is local, we may assume that $T={\rm d}{\rm d}^{\rm
c}\varphi$ such that $\varphi\in C^{0}(X)$. Define a function on $Y$ by

$f_{*}\varphi(y):=\sum_{x\in f^{-1}(y)}\varphi(x)$

here the sums are counted with multiplicity. By [Dem85, Proposition 1.13.(b)],
we know that $f_{*}\varphi$ is a psh function on $Y$ and

${\rm d}{\rm d}^{\rm c}f_{*}\varphi=f_{*}T.$

One can see that $f_{*}\varphi$ is also continuous. Define a current
$S:=\frac{1}{\deg f}f_{*}T$. Since $T$ is $G$-invariant, it follows that
$f^{*}S=T$ outside the branch locus of $f$. Since $f^{*}S=\frac{1}{\deg f}{\rm
d}{\rm d}^{\rm c}(f_{*}\varphi)\circ f$, the potential of $f^{*}S$ is
continuous. It follows that $f^{*}S=T$ over the whole $X$. ∎

###  1.4 Holomorphic forms on complex normal spaces

There are many ways to define holomorphic forms on complex normal spaces. For
our purpose of the current paper, we use the following definition in [Fer70].

######  Definition 1.11.

Let $X$ be a normal complex space. Let $\left(A_{i}\right)_{i\in I}$ be an
open finite covering of $X$ such that each subset $A_{i}$ is an analytic
subset of some open subset $\Omega_{i}\subset\mathbb{C}^{N_{i}}$. The space of
holomorphic $p$-forms, denoted by $\Omega_{X}^{p}$, is defined by local
restrictions of holomorphic $p$-forms on the sets $\Omega_{i}$ above to
$A_{i}^{\rm reg}$, where $A_{i}^{\rm reg}$ is the smooth locus of $A_{i}$.

The following fact will be used throughout the paper.

######  Lemma 1.12.

Let $f:X\to Y$ be a holomorphic map between normal complex spaces. Then for
any holomorphic $p$-form $\omega$ on $Y$, $f^{*}\omega$ a holomorphic $p$-form
on $Y$.

###### Proof.

By Definition 1.11, for any $x\in X$, there exist

  * •

a neighborhood $A$ (resp. $B$) of $x$ (resp. $f(x)$) such that $A$ (resp. $B$)
is an analytic subset of some open $\Omega\subset\mathbb{C}^{m}$ (resp.
$\Omega^{\prime}\subset\mathbb{C}^{n}$).

  * •

a holomorphic map $\tilde{f}:\Omega\to\Omega^{\prime}$ such that
$\tilde{f}|_{A}=f|_{A}$.

  * •

A holomorphic $p$-form $\tilde{\omega}$ on $\Omega^{\prime}$ such that
$\omega=\tilde{\omega}|_{B}$.

Therefore, we can define
$f^{*}\omega|_{A}:=\tilde{f}^{*}\tilde{\omega}|_{\Omega}$. One can check that
this does not depend on the choice of local embeddings of $X$ and $Y$. ∎

###  1.5 The criterion for Kähler classes

We will need the following extension of the celebrated Demailly-Păun’s theorem
[DP04] on characterization of Kähler classes on complex normal Kähler spaces
by Das-Hacon-Păun in [DHP22].

######  Theorem 1.13 ([DHP22, Corollary 2.39]).

Let $X$ be a compact normal Kähler space, $\omega$ a Kähler form on $X$, and
$\alpha\in H_{\rm BC}^{1,1}(X)$, then $\alpha$ is Kähler if and only if
$\int_{V}\alpha^{k}\wedge\omega^{p-k}>0$ for every analytic $p$-dimensional
subvariety $V\subset X$ and for all $0<k\leq p$.

###  1.6 Some criterion for Stein space

We require the following criterion for the Stein property of a topological
Galois covering of a compact complex normal space.

######  Proposition 1.14 ([Eys04, Proposition 4.1.1]).

Let $X$ be a compact complex normal space and let
$\pi:\widetilde{X}^{\prime}\to X$ be some topological Galois covering. Let $T$
be a positive current on $X$ with continuous potential such that $\\{T\\}$ is
a Kähler class. Assume that there exists a continuous plurisubharmonic
function $\phi:\widetilde{X}^{\prime}\to\mathbb{R}_{\geq 0}$ such that ${\rm
d}{\rm d}^{\rm c}\phi\geq\pi^{*}T$. Then $\widetilde{X}^{\prime}$ is a Stein
space.

###  1.7 Some facts on moduli spaces of rank 1 local systems

For this subsection we refer the readers to [Sim93] for a systematic
treatment. Let $X$ be a smooth projective variety defined over a field
$K\subset\mathbb{C}$. Let $M=M(X)$ denote the moduli space of complex local
systems of rank one over $X$. We consider $M$ as a real analytic group under
the operation of tensor product. There are three natural algebraic groups
$M_{\mathrm{B}}$, ${\rm M}_{\rm DR}$ and ${\rm M}_{\rm Dol}$ whose underlying
real analytic groups are canonically isomorphic to $M$. The first is Betti
moduli space $M_{\mathrm{B}}:={\rm Hom}(\pi_{1}(X),\mathbb{C}^{*})$. The
second is De Rham moduli space $M_{\mathrm{DR}}$ which consists of pairs
$(L,\nabla)$ where $L$ is a holomorphic line bundle on $X$ and $\nabla$ is an
integrable algebraic connection on $L$. The last one $M_{\text{Dol }}$is
moduli spaces of rank one Higgs bundles on $X$. Recall that ${\rm
Pic}^{\tau}(X)$ denotes the group of line bundles on $X$ whose first Chern
classes are torsion. We have

$M_{\text{Dol }}={\rm Pic}^{\tau}(X)\times H^{0}(X,\Omega_{X}^{1})$

For any subset $S\subset M$, let $S_{\rm B}$, $S_{\rm Dol}$ and $S_{\rm DR}$
denote the corresponding subsets of $M_{\mathrm{B}}$, ${\rm M}_{\rm DR}$ and
${\rm M}_{\rm Dol}$.

######  Definition 1.15 (Triple torus).

A triple torus is a closed, connected real analytic subgroup $N\subset M$ such
that $N_{\mathrm{B}},N_{\mathrm{DR}}$, and $N_{\text{Dol }}$ are algebraic
subgroups defined over $\mathbb{C}$. We say that a closed real analytic
subspace $S\subset M$ is a translate of a triple torus if there exists a
triple torus $N\subset M$ and a point $v\in M$ such that $S=\\{v\otimes w,w\in
N\\}$. Note that, in this case, any choice of $v\in M$ will do.

We say that a point $v\in M$ is torsion if there exists an integer $a>0$ such
that $v^{\otimes a}=1$. Let $M^{\text{tor }}$ denote the set of torsion
points. Note that for a given integer $a$, there are only finitely many
solutions of $v^{\otimes a}=1$. Hence, the points of
$M_{\mathrm{B}}^{\text{tor }}$ are defined over $\overline{\mathbb{Q}}$, and
the points of $M_{\mathrm{DR}}^{\text{tor }}$ and $M_{\mathrm{Dol}}^{\text{tor
}}$ are defined over $\overline{K}$.

We say that a closed subspace $S$ is a torsion translate of a triple torus if
$S$ is a translate of a triple torus $N$ by an element $v\in M^{\text{tor }}$.
This is equivalent to asking that $S$ be a translate of a triple torus, and
contain a torsion point.

Let $A$ be the Albanese variety of $X$ (which can be defined as
${H}^{0}({X},\Omega_{{X}}^{1})^{*}/{H}_{1}({X},\mathbb{Z}))$. Let
${X}\rightarrow{A}$ be the map from ${X}$ into ${A}$ given by integration
(from a basepoint, which will be suppressed in the notation but assumed to be
defined over $\overline{K})$. Pullback of local systems gives a natural map
from $M(A)$ to $M(X)$, which is an isomorphism

$M(A)\cong M^{0}(X),$

where $M^{0}(X)$ is the connected component of $M(X)$ containing the trivial
rank one local system. The Albanese variety $A$ is defined over
$\overline{{K}}$. We recall the following result in [Sim93, Lemma 2.1].

######  Lemma 1.16 (Simpson).

Let $N\subset M$ be a closed connected subgroup such that
$N_{\mathrm{B}}\subset M_{\mathrm{B}}$ is complex analytic and
$N_{\mathrm{Dol}}\subset M_{\mathrm{Dol}}$ is an algebraic subgroup. Then
there is a connected abelian subvariety ${P}\subset{A}$, defined over
$\overline{K}$, such that $N$ is the image in $M$ of $M(A/{P})$. In
particular, $N$ is a triple torus in $M$.

###  1.8 Absolutely constructible subsets

In this section we will recall some facts on _absolutely constructible
subsets_ (resp. _absolutely closed subsets_) introduced by Simpson in [Sim93,
§6] and later developed by Budur-Wang [WB20].

Let $X$ be a smooth projective variety defined over a subfield $\ell$ of
$\mathbb{C}$. Let $G$ be a reductive group defined over $\bar{\mathbb{Q}}$.
The representation scheme of $\pi_{1}(X)$ is an affine
$\bar{\mathbb{Q}}$-algebraic scheme described by its functor of points:

$R(X,G)(\textrm{Spec}\,A):={\rm Hom}(\pi_{1}(X),G(A))$

for any $\bar{\mathbb{Q}}$-algebra $A$. The character scheme of $\pi_{1}(X)$
with values in $G$ is the finite type affine scheme $M_{\rm
B}(X,G):=R(X,G)\mathbin{/\mkern-6.0mu/}G,$ where
“$\mathbin{/\mkern-6.0mu/}$denotes the GIT quotient. If $G={\rm GL}_{N}$, we
simply write $M_{\rm B}(X,N):=M_{\rm B}(X,{\rm GL}_{N})$. Simpson constructed
a quasi-projective scheme $M_{\rm DR}(X,G)$, and $M_{\rm Dol}(X,G)$ over
$\ell$. The $\mathbb{C}$-points of $M_{\rm DR}(X,G)$ are in bijection with the
equivalence classes of flat $G$-connections with reductive monodromy. There
are natural isomorphisms

$\psi:M_{\rm B}(X,G)(\mathbb{C})\to M_{\rm DR}(X,G)(\mathbb{C})$

such that $\psi$ is an isomorphism of complex analytic spaces. For each
automorphism $\sigma\in\operatorname{Aut}(\mathbb{C}/\mathbb{Q})$, let
$X^{\sigma}:=X\times_{\sigma}\mathbb{C}$ be the conjugate variety of $X$,
which is also smooth projective. There is a natural map

$\begin{gathered}p_{\sigma}:M_{\mathrm{DR}}(X,G)\rightarrow
M_{\mathrm{DR}}\left(X^{\sigma},G^{\sigma}\right).\end{gathered}$

Let us now introduce the following definition of absolutely constructible
subsets.

######  Definition 1.17 (Absolutely constructible subset).

A subset $\mathfrak{C}\subset{M}_{\rm B}(X,G)(\mathbb{C})$ is an _absolutely
constructible subset_ (resp. _absolutely closed subset_) if the following
conditions are satisfied.

  1. (i)

$\mathfrak{C}$ is the a $\bar{\mathbb{Q}}$-constructible (resp.
$\bar{\mathbb{Q}}$-closed) subset of $M_{\rm B}(X,G)$.

  2. (ii)

For each $\sigma\in\operatorname{Aut}(\mathbb{C}/\mathbb{Q})$, there exists a
$\bar{\mathbb{Q}}$-constructible (resp. $\bar{\mathbb{Q}}$-closed) set
$\mathfrak{C}^{\sigma}\subset M_{\rm
B}\left(X^{\sigma},G^{\sigma}\right)(\mathbb{C})$ such that $\psi^{-1}\circ
p_{\sigma}\circ\psi(\mathfrak{C})=\mathfrak{C}^{\sigma}$.

  3. (iii)

$\mathfrak{C}(\mathbb{C})$ is preserved by the action of $\mathbb{R}^{*}$
defined in § 2.4.

######  Remark 1.18.

Note that this definition is significantly weaker than the notion of
absolutely constructible sets defined in [Sim93, Eys04], as it does not
consider moduli spaces of semistable Higgs bundles with trivial characteristic
numbers, and it does not require that $\psi(\mathfrak{C})$ is
$\bar{\mathbb{Q}}$-constructible in ${M}_{\rm DR}(X,G)(\mathbb{C})$. This
revised definition allows for a broader range of applications, including
quasi-projective varieties. In [Sim93, Eys04], the preservation of
$\mathfrak{C}(\mathbb{C})$ under the action of $\mathbb{C}^{*}$ is a necessary
condition. It is important to emphasize that our definition only requires
$\mathbb{R}^{*}$-invariance, which is weaker than $\mathbb{C}^{*}$-invariance.
Our definition corresponds to the _absolutely constructible subset_ as defined
in [WB20, Definition 6.3.1], with the additional condition that
$\mathfrak{C}(\mathbb{C})$ is preserved by the action of $\mathbb{R}^{*}$.

By [WB20, Theorem 9.1.2.(2) & Proposition 7.4.4.(2)] we have the following
result, which generalizes [Sim93].

######  Theorem 1.19 (Budur-Wang, Simpson).

Let $X$ be a smooth projective variety over $\mathbb{C}$. If
$\mathfrak{C}\subset M_{\rm B}({X},1)(\mathbb{C})$ is an absolute
constructible subset, then $\mathfrak{C}=\cup_{i=1}^{m}N^{\circ}_{i}$ where
each $N^{\circ}_{i}$ is a Zariski dense open subset of a torsion-translated
subtori $N_{i}$ of $M_{\rm B}({X},1)$. Moreover, let $A$ be the Albanese
variety of $X$. Then there are abelian subvarieties $P_{i}\subset A$ such that
$N_{i}$ is the torsion translate of the image in $M^{0}_{\rm B}(X,1)\simeq
M_{\rm B}(A,1)$ of $M_{\rm B}(A/P_{i},1)$. Here $M^{0}_{\rm B}(X,1)$ denotes
the connected component of $M^{0}_{\rm B}(X,1)$ containing the identity. ∎

Absolute constructible subsets are preserved by the following operations:

######  Theorem 1.20 (Simpson).

Let $f:Z\to X$ be a morphism between smooth projective varieties over
$\mathbb{C}$ and let $g:G\to G^{\prime}$ be a morphism of reductive groups
over $\bar{\mathbb{Q}}$. Consider the natural map $i:M_{\rm B}(X,G)\to M_{\rm
B}(X,G^{\prime})$ and $j:M_{\rm B}(X,G)\to M_{\rm B}(Z,G)$. Then for any
absolutely constructible subsets $\mathfrak{C}\subset M_{\rm
B}(X,G)(\mathbb{C})$ and $\mathfrak{C}^{\prime}\subset M_{\rm
B}(X,G^{\prime})(\mathbb{C})$, we have $i(\mathfrak{C})$,
$i^{-1}(\mathfrak{C}^{\prime})$ and $j(\mathfrak{C})$ are all absolutely
constructible. ∎

######  Example 1.21.

$M_{\rm B}(X,G)(\mathbb{C})$, the isolated point in $M_{\rm
B}(X,G)(\mathbb{C})$, and the class of trivial representation in $M_{\rm
B}(X,G)(\mathbb{C})$ are all absolutely constructible.

In this paper, absolutely constructible subsets are used to prove the
holomorphic convexity of some topological Galois covering of $X$ in Theorems B
and C. It will not be used in the proof of Theorem A.

###  1.9 Katzarkov-Eyssidieux reduction and canonical currents

For this subsection, we refer to the papers [Eys04, §3.3.2] or [CDY22] for a
comprehensive and systematic treatment.

######  Theorem 1.22 (Katzarkov, Eyssidieux).

Let $X$ be a projective normal variety, and let $K$ be a non-archimedean local
field. Let $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(K)$ be a reductive
representation. Then there exists a fibration $s_{\varrho}:X\to S_{\varrho}$
to a normal projective space, such that for any subvariety $Z$ of $X$, the
image $\varrho({\rm Im}\,[\pi_{1}(Z^{\rm norm})\to\pi_{1}(X)])$ is bounded if
and only if $s_{\varrho}(Z)$ is a point. Moreover, if $X$ is smooth, then the
above property holds for $\varrho({\rm Im}\,[\pi_{1}(Z)\to\pi_{1}(X)])$
without requiring the normalization of $Z$.

We will call the above $s_{\varrho}$ the (Katzarkov-Eyssidieux) reduction map
for $\varrho$. When $X$ is smooth this theorem is proved by Katzarkov [Kat97]
and Eyssidieux [Eys04]. It is easier to derive the singular case from their
theorem.

###### Proof of Theorem 1.22.

Let $\mu:Y\to X$ be a resolution of singularities. Since $X$ is normal,
$\mu_{*}:\pi_{1}(Y)\to\pi_{1}(X)$ is surjective and thus
$\mu^{*}\varrho:\pi_{1}(Y)\to{\rm GL}_{N}(K)$ is reductive. By the original
theorem of Katzarkov-Eyssidieux, there exists a surjective proper fibration
$s_{\mu^{*}\varrho}:Y\to S_{\mu^{*}\varrho}$ such that, for any closed
subvariety $Z\subset Y$, $s_{\mu^{*}\varrho}(Z)$ is a point if and only if
$\mu^{*}\varrho({\rm Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(Y)])$.

If $Z$ is an irreducible component of a fiber of $\mu$. Note that
$\mu^{*}\varrho({\rm Im}[\pi_{1}(Z)\to\pi_{1}(Y)])=\\{1\\}$, it follows that
$s_{\mu^{*}\varrho}(Z)$ is a point by the proper of Katzarkov-Eyssidieux.
Since each fiber of $\mu$ is connected, $s_{\mu^{*}\varrho}$ contracts each
fiber of $\mu$ to a point, and it thus descends to a morphism
$s_{\varrho}:X\to S_{\mu^{*}\varrho}$ such that
$s_{\mu^{*}\varrho}=s_{\varrho}\circ\mu$.

Let $W\subset X$ be any closed subvariety. Then there exist a closed
subvariety $Z\subset Y$ such that $\mu(Z)=W$. Note that ${\rm
Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(W^{\rm norm})]$ is a finite index subgroup
of $\pi_{1}(W^{\rm norm})$. Therefore, $s_{\varrho}(Z)$ is a point if and only
if $s_{\mu^{*}\varrho}(W)$ is a point. This condition is equivalent to
$\mu^{*}\varrho({\rm Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(Y)])=\varrho({\rm
Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(X)])$ being bounded. In turn, this is
equivalent to $\varrho({\rm Im}[\pi_{1}(W^{\rm norm})\to\pi_{1}(X)])$ being
bounded since ${\rm Im}[\pi_{1}(Z^{\rm norm})\to\pi_{1}(W^{\rm norm})]$ is a
finite index subgroup of $\pi_{1}(W^{\rm norm})$. ∎

We will outline the construction of certain _canonical_ positive closed
$(1,1)$-currents over $S_{\varrho}$. As demonstrated in the proof of [Eys04],
we can establish the existence of a finite ramified Galois cover denoted by
$\pi:X^{\\!\rm sp}\to X$ with the Galois group $H$, commonly known as the
_spectral covering of $X$_ (cf. [CDY22, Definition 5.14]). This cover
possesses holomorphic 1-forms ${\eta_{1},\ldots,\eta_{m}}\subset
H^{0}(X^{\\!\rm sp},\pi^{*}\Omega_{X}^{1})$, which can be considered as the
$(1,0)$-part of the complexified differential of the
$\pi^{*}\varrho$-equivariant harmonic mapping from $X^{\\!\rm sp}$ to the
Bruhat-Tits building of $G$. These particular 1-forms, referred to as the
_spectral one-forms_ (cf. [CDY22, Definition 5.16]), play a significant role
in the proof of Theorems B and C. Consequently, the Stein factorization of the
_partial Albanese morphism_ $a:X^{\\!\rm sp}\to A$ (cf. [CDY22, Definition
5.19]) induced by ${\eta_{1},\ldots,\eta_{m}}$ leads to the Katzarkov-
Eyssidieux reduction map $s_{\pi^{*}\varrho}:X^{\\!\rm sp}\to
S_{\pi^{*}\varrho}$ for $\pi^{*}\varrho$. Moreover, we have the following
commutative diagram:

${X^{\\!\rm
sp}}$${X}$${S_{\pi^{*}\varrho}}$${S_{\varrho}}$${A}$$\scriptstyle{\pi}$$\scriptstyle{s_{\pi^{*}\varrho}}$$\scriptstyle{a}$$\scriptstyle{s_{\varrho}}$$\scriptstyle{b}$$\scriptstyle{\sigma_{\pi}}$

Here $\sigma_{\pi}$ is also a finite ramified Galois cover with Galois group
$H$. Note that there are one forms
$\\{\eta_{1}^{\prime},\ldots,\eta_{m}^{\prime}\\}\subset
H^{0}(A,\Omega_{A}^{1})$ such that $a^{*}\eta_{i}^{\prime}=\eta_{i}$. Consider
the finite morphism $b:S_{\pi^{*}\varrho}\to A$. Then we define a positive
$(1,1)$-current
$T_{\pi^{*}\varrho}:=b^{*}\sum_{i=1}^{m}i\eta_{i}^{\prime}\wedge\overline{\eta_{i}}^{\prime}$
on $S_{\pi^{*}\varrho}$. Note that $T_{\pi^{*}\varrho}$ is invariant under the
Galois action $H$. Therefore, by Lemma 1.10 there is a positive closed
$(1,1)$-current $T_{\varrho}$ defined on $S_{\varrho}$ with continuous
potential such that $\sigma_{\pi}^{*}T_{\varrho}=T_{\pi^{*}\varrho}$.

######  Definition 1.23 (Canonical current).

The closed positive $(1,1)$-current $T_{\varrho}$ on $S_{\varrho}$ is called
the _canonical current_ of $\varrho$.

More generally, let $\\{\varrho_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(K_{i})\\}_{i=1,\ldots,k}$ be reductive representations where $K_{i}$
is a non-archimedean local field. We shall denote by the bolded letter
$\bm{\varrho}:=\\{\varrho_{i}\\}_{i=1,\ldots,k}$ be such family of
representations. Let $s_{\bm{\varrho}}:X\to S_{{\bm{\varrho}}}$ be the Stein
factorization of $(s_{\varrho_{1}},\ldots,s_{\varrho_{k}}):X\to
S_{\varrho_{1}}\times\cdots\times S_{\varrho_{k}}$ where $s_{\varrho_{i}}:X\to
S_{\varrho_{i}}$ denotes the reduction map associated with $\varrho_{i}$ and
$p_{i}:S_{{\bm{\varrho}}}\to S_{\varrho_{i}}$ is the induced finite morphism.
$s_{\bm{\varrho}}:X\to S_{{\bm{\varrho}}}$ is called the _reduction map_ for
the family ${{\bm{\varrho}}}$ of representations.

######  Definition 1.24 (Canonical current II).

The closed positive $(1,1)$-current
$T_{{\bm{\varrho}}}:=\sum_{i=1}^{k}p_{i}^{*}T_{\varrho_{i}}$ on
$S_{\bm{\varrho}}$ is called the canonical current of ${\bm{\varrho}}$.

######  Lemma 1.25 ([Eys04, Lemme 1.4.9 & 3.3.10]).

Let $f:Z\to X$ be a morphism between projective normal varieties and let
${\bm{\varrho}}:=\\{\varrho_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(K_{i})\\}_{i=1,\ldots,k}$ be a family of reductive representations
where $K_{i}$ is a non-archimedean local field. Then we have

${Z}$${X}$${S_{f^{*}{\bm{\varrho}}}}$${S_{{\bm{\varrho}}}}$$\scriptstyle{f}$$\scriptstyle{s_{f^{*}{\bm{\varrho}}}}$$\scriptstyle{s_{{\bm{\varrho}}}}$$\scriptstyle{\sigma_{f}}$
(1.2)

where $\sigma_{f}$ is a _finite_ morphism. Here
$f^{*}{\bm{\varrho}}=\\{f^{*}\varrho_{i}\\}_{i=1,\ldots,k}$ denotes the pull
back of the family of ${\bm{\varrho}}:=\\{\varrho_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(K_{i})\\}_{i=1,\ldots,k}$. Moreover, the following properties hold:

  1. (i)

The local potential of $T_{{\bm{\varrho}}}$ is continuous. In particular, for
any closed subvariety $W\subset X$, we have

$\\{T_{\bm{\varrho}}\\}^{\dim W}\cdot W=\int_{W}T_{\bm{\varrho}}^{\dim W}\geq
0.$

  2. (ii)

$T_{f^{*}{\bm{\varrho}}}=\sigma_{f}^{*}T_{\bm{\varrho}}$;

  3. (iii)

For every closed subvariety $\Xi\subset S_{f^{*}{\bm{\varrho}}}$,
$\\{T_{\bm{\varrho}}\\}^{\dim\Xi}\cdot(\sigma_{f}(\Xi))>0$ if and only if
$\\{T_{f^{*}{\bm{\varrho}}}\\}^{\dim\Xi}\cdot\Xi>0$. ∎

Note that 1.25.(iii) is a consequence of the first two assertions.

The current $T_{\varrho}$ will serve as a lower bound for the complex hessian
of plurisubharmonic functions constructed by the method of harmonic mappings.

######  Proposition 1.26 ([Eys04, Proposition 3.3.6, Lemme 3.3.12]).

Let $X$ be a projective normal variety and let $\varrho:\pi_{1}(X)\to G(K)$ be
a Zariski dense representation where $K$ is a non archimedean local field and
$G$ is a reductive group. Let $x_{0}\in\Delta(G)$ be an arbitrary point. Let
$u:\widetilde{X}\to\Delta(G)$ be the associated the harmonic mapping, where
$\widetilde{X}$ is the universal covering of $X$. The function
$\phi:\widetilde{X}\rightarrow\mathbb{R}_{\geq 0}$ defined by

$\phi(x)=2d^{2}\left(u(x),u(x_{0})\right)$

satisfies the following properties:

  1. (a)

$\phi$ descends to a function $\phi_{\varrho}$ on
$\widetilde{X_{\varrho}}=\widetilde{X}/\operatorname{ker}\left(\varrho\right)$.

  2. (b)

${\rm d}{\rm d}^{\rm
c}\phi_{\varrho}\geq(s_{\varrho}\circ\pi)^{*}T_{\varrho}$, where we denote by
$\pi:\widetilde{X}_{\varrho}\to X$ the covering map.

  3. (c)

$\phi_{\varrho}$ is locally Lipschitz;

  4. (d)

Let $T$ be a normal complex space and $r:\widetilde{X}_{\varrho}\to T$ a
proper holomorphic fibration such that
$s_{\varrho}\circ\pi:\widetilde{X}_{\varrho}\rightarrow S_{\varrho}$
factorizes via a morphism $\nu:T{\rightarrow}S_{\varrho}$. The function
$\phi_{\varrho}$ is of the form $\phi_{\varrho}=\phi_{\varrho}^{T}\circ r$
with $\phi_{\varrho}^{T}$ being a continuous plurisubharmonic function on $T$;

  5. (e)

${\rm d}{\rm d}^{\rm c}\phi^{T}_{\varrho}\geq\nu^{*}T_{\varrho}$.∎

###  1.10 The generalization of Katzarkov-Eyssidieux reduction to quasi-
projective varieties

In our work [CDY22] on hyperbolicity of quasi-projective varieties, we
extended Theorem 1.22 to quasi-projective varieties. The theorem we
established is stated below.

######  Theorem 1.27 ([CDY22, Theorem 0.10]).

Let $X$ be a complex smooth quasi-projective variety, and let
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(K)$ be a reductive representation where $K$
is non-archimedean local field. Then there exists a quasi-projective normal
variety $S_{\varrho}$ and a dominant morphism $s_{\varrho}:X\to S_{\varrho}$
with connected general fibers, such that for any connected Zariski closed
subset $T$ of $X$, the following properties are equivalent:

  1. (a)

the image $\varrho({\rm Im}[\pi_{1}(T)\to\pi_{1}(X)])$ is a bounded subgroup
of $G(K)$.

  2. (b)

For every irreducible component $T_{o}$ of $T$, the image $\varrho({\rm
Im}[\pi_{1}(T_{o}^{\rm norm})\to\pi_{1}(X)])$ is a bounded subgroup of $G(K)$.

  3. (c)

The image $s_{\varrho}(T)$ is a point.∎

This result plays a crucial role in the proof of Theorem A.

###  1.11 Simultaneous Sten factorization

######  Lemma 1.28.

Let $V$ be a smooth quasi-projective variety. For $i=1,2,\ldots$, let $W_{i}$
be normal quasi-projective varieties such that

  * •

there exist dominant morphisms $p_{i}:V\to W_{i}$, and

  * •

there exist dominant morphisms $q_{i}:W_{i}\to W_{i-1}$ such that $q_{i}\circ
p_{i}=p_{i-1}$.

Then there exists $i_{0}\in\mathbb{Z}_{\geq 2}$ such that for every $i\geq
i_{0}$ and every subvariety $Z\subset V$, if $p_{i-1}(Z)$ is a point, then
$p_{i}(Z)$ is a point.

###### Proof.

Let $E_{i}\subset V\times V$ be defined by

$E_{i}=\\{(x,x^{\prime})\in V\times V;p_{i}(x)=p_{i}(x^{\prime})\\}.$

Then $E_{i}\subset V\times V$ is a Zariski closed set. Indeed,
$E_{i}=(p_{i},p_{i})^{-1}(\Delta_{i})$, where $(p_{i},p_{i}):V\times V\to
W_{i}\times W_{i}$ is the morphism defined by
$(p_{i},p_{i})(x,x^{\prime})=(p_{i}(x),p_{i}(x^{\prime}))$ and
$\Delta_{i}\subset W_{i}\times W_{i}$ is the diagonal. Now by $q_{i}\circ
p_{i}=p_{i-1}$, we have $E_{i}\subset E_{i-1}$. By the Noetherian property,
there exists $i_{0}$ such that $E_{i+1}=E_{i}$ for all $i\geq i_{0}$. Then the
induced map $p_{i+1}(V)\to p_{i}(V)$ is injective. Hence if $p_{i-1}(Z)$ is a
point, then $p_{i}(Z)$ is a point. ∎

######  Lemma 1.29.

Let $V$ be a quasi-projective normal variety and let $(f_{\lambda}:V\to
S_{\lambda})_{\lambda\in\Lambda}$ be a family of morphisms into quasi-
projective varieties $S_{\lambda}$. Then there exist a normal projective
variety $S_{\infty}$ and a morphism $f_{\infty}:V\to S_{\infty}$ such that

  * •

for every subvariety $Z\subset V$, if $f_{\infty}(Z)$ is a point, then
$f_{\lambda}(Z)$ is a point for every $\lambda\in\Lambda$, and

  * •

there exist $\lambda_{1},\ldots,\lambda_{n}\in\Lambda$ such that
$f_{\infty}:V\to S_{\infty}$ is the quasi-Stein factorization of
$(f_{1},\ldots,f_{n}):V\to S_{\lambda_{1}}\times\cdots S_{\lambda_{n}}$.

###### Proof.

We take $\lambda_{1}\in\Lambda$. Let $p_{1}:V\to W_{1}$ be the quasi-Stein
factorization of $f_{\lambda_{1}}:V\to S_{\lambda_{1}}$.

Next we take (if it exists) $\lambda_{2}\in\Lambda$ such that for the quasi-
Stein factorization $p_{2}:V\to W_{2}$ of
$(s_{\lambda_{1}},s_{\lambda_{2}}):X\to S_{\lambda_{1}}\times
S_{\lambda_{2}}$, there exists a subvariety $Z\subset V$ such that $p_{1}(Z)$
is a point, but $p_{2}(Z)$ is not a point.

Similarly, we take (if it exists) $\lambda_{3}\in\Lambda$ such that for the
Stein factorization $p_{3}:V\to W_{3}$ of
$(f_{\lambda_{1}},f_{\lambda_{2}},f_{\lambda_{3}}):V\to S_{\lambda_{1}}\times
S_{\lambda_{2}}\times S_{\lambda_{3}}$, there exists a subvariety $Z\subset V$
such that $p_{2}(Z)$ is a point, but $p_{3}(Z)$ is not a point.

We repeat this process forever we may continue. However by Lemma 1.28, this
process should terminate to get $\lambda_{1},\ldots,\lambda_{n}\in\Lambda$. We
let $S_{\infty}=W_{n}$, namely $f_{\infty}:V\to S_{\infty}$ is the Stein
factorization of $(f_{\lambda_{1}},\ldots,f_{\lambda_{n}}):V\to
S_{\lambda_{1}}\times\cdots\times S_{\lambda_{n}}$.

Now let $\lambda\in\Lambda$. Then by the construction, if $f_{\infty}(Z)$ is a
point, then $(f_{\lambda_{1}},\ldots,f_{\lambda_{n}},f_{\lambda})(Z)$ is a
point. In particular, $f_{\lambda}(Z)$ is a point. ∎

We also need the following generalized Stein factorization proved by Henri
Cartan in [Car60, Theorem 3].

######  Theorem 1.30.

Let $X,S$ be complex spaces and $f:X\rightarrow S$ be a morphism. Suppose a
connected component $F$ of a fibre of $f$ is compact. Then, $F$ has an open
neighborood $V$ such that $f(V)$ is a locally closed analytic subvariety $S$
and $V\rightarrow f(V)$ is proper.

Suppose furthermore that $X$ is normal and that every connected component $F$
of a fibre of $f$ is compact. The set $Y$ of connected components of fibres of
$f$ can be endowed with the structure of a normal complex space such that $f$
factors through the natural map $e:X\rightarrow Y$ which is a proper
holomorphic fibration. ∎

##  2 Some non-abelian Hodge theories

In this section, we will build upon the previous work of Simpson [Sim92],
Iyer-Simpson [IS07, IS08], and Mochizuki [Moc07a, Moc06] to further develop
non-abelian Hodge theories over quasi-projective varieties. We begin by
establishing the functoriality of pullback for regular filtered Higgs bundles
(cf. Proposition 2.5). Then we clarify the $\mathbb{C}^{*}$ and
$\mathbb{R}^{*}$-action on the character varieties of smooth quasi-projective
varieties, following [Moc06]. Lastly, we prove Proposition 2.9, which
essentially states that the natural morphisms of character varieties induced
by algebraic morphisms commute with the $\mathbb{C}^{*}$-action. This
section’s significance lies in its essential role in establishing Propositions
3.35 and 3.12, which serves as a critical cornerstone of the whole paper.

###  2.1 Regular filtered Higgs bundles

In this subsection, we recall the notions of regular filtered Higgs bundles
(or parabolic Higgs bundles). For more details refer to [Moc06]. Let
$\overline{X}$ be a complex manifold with a reduced simple normal crossing
divisor $D=\sum_{i=1}^{\ell}D_{i}$, and let $X=\overline{X}\backslash D$ be
the complement of $D$. We denote the inclusion map of $X$ into $\overline{X}$
by $j$.

######  Definition 2.1.

A _regular filtered Higgs bundle_ $(\bm{E}_{*},\theta)$ on $(\overline{X},D)$
is holomorphic vector bundle $E$ on $X$, together with an
$\mathbb{R}^{\ell}$-indexed filtration ${}_{\bm{a}}E$ (so-called parabolic
structure) by locally free subsheaves of $j_{*}E$ such that

  1. 1.

$\bm{a}\in\mathbb{R}^{\ell}$ and ${}_{\bm{a}}E|_{X}=E$.

  2. 2.

For $1\leq i\leq\ell$,
${}_{\bm{a}+\bm{1}_{i}}E={}_{\bm{a}}E\otimes\mathcal{O}_{X}(D_{i})$, where
$\bm{1}_{i}=(0,\ldots,0,1,0,\ldots,0)$ with $1$ in the $i$-th component.

  3. 3.

${}_{\bm{a}+\bm{\epsilon}}E={}_{\bm{a}}E$ for any vector
$\bm{\epsilon}=(\epsilon,\ldots,\epsilon)$ with $0<\epsilon\ll 1$.

  4. 4.

The set of weights {$\bm{a}$ | ${}_{\bm{a}}E/_{\bm{a}-\bm{\epsilon}}E\not=0$ for any vector $\bm{\epsilon}=(\epsilon,\ldots,\epsilon)$ with $0<\epsilon\ll 1$} is discrete in $\mathbb{R}^{\ell}$.

  5. 5.

There is a $\mathcal{O}_{\overline{X}}$-linear map, so-called Higgs field,

$\theta:{}^{\diamond}\\!E\to\Omega_{\overline{X}}^{1}(\log
D)\otimes{}^{\diamond}\\!E$

such that $\theta\wedge\theta=0$, and

$\displaystyle\theta(_{\bm{a}}E)\subseteq\Omega_{\overline{X}}^{1}(\log
D)\otimes{}_{\bm{a}}E.$ (2.1)

Denote ${}_{\bm{0}}E$ by ${}^{\diamond}\\!E$, where $\bm{0}=(0,\ldots,0)$.
When disregarding the Higgs field, $\bm{E}_{*}$ is referred to as a _parabolic
bundle_. By the work of Borne-Vistoli the parabolic structure of a parabolic
bundle is _locally abelian_ , _i.e._ it admits a local frame compatible with
the filtration (see e.g. [IS07]).

A natural class of regular filtered Higgs bundles comes from prolongations of
tame harmonic bundles. We first recall some notions in [Moc07a, §2.2.1]. Let
$E$ be a holomorphic vector bundle with a smooth hermitian metric $h$ over
$X$.

Let $U$ be an open subset of $\overline{X}$ with an admissible coordinate
$(U;z_{1},\ldots,z_{n})$ with respect to $D$. For any section
$\sigma\in\Gamma(U\backslash D,E|_{U\backslash D})$, let $|\sigma|_{h}$ denote
the norm function of $\sigma$ with respect to the metric $h$. We denote
$|\sigma|_{h}\mathcal{O}(\prod_{i=1}^{\ell}|z_{i}|^{-b_{i}})$ if there exists
a positive number $C$ such that $|\sigma|_{h}\leq
C\cdot\prod_{i=1}^{\ell}|z_{i}|^{-b_{i}}$. For any
$\bm{b}\in\mathbb{R}^{\ell}$, say $-\mbox{ord}(\sigma)\leq\bm{b}$ means the
following:

$|\sigma|_{h}=\mathcal{O}(\prod_{i=1}^{\ell}|z_{i}|^{-b_{i}-\varepsilon})$

for any real number $\varepsilon>0$ and $0<|z_{i}|\ll 1$. For any $\bm{b}$,
the sheaf ${}_{\bm{b}}E$ is defined as follows:

$\displaystyle\Gamma(U,{}_{\bm{b}}E):=\\{\sigma\in\Gamma(U\backslash
D,E|_{U\backslash D})\mid-\mbox{ord}(\sigma)\leq\bm{b}\\}.$ (2.2)

The sheaf ${}_{\bm{b}}E$ is called the prolongment of $E$ by an increasing
order $\bm{b}$. In particular, we use the notation ${}^{\diamond}E$ in the
case $\bm{b}=(0,\ldots,0)$.

According to Simpson [Sim90, Theorem 2] and Mochizuki [Moc07a, Theorem 8.58],
the above prolongation gives a regular filtered Higgs bundle.

######  Theorem 2.2 (Simpson, Mochizuki).

Let $\overline{X}$ be a complex manifold and $D$ be a simple normal crossing
divisor on $\overline{X}$. If $(E,\theta,h)$ is a tame harmonic bundle on
$\overline{X}\backslash D$, then the corresponding filtration ${}_{\bm{b}}E$
defined above defines a regular filtered Higgs bundle $(\bm{E}_{*},\theta)$ on
$(\overline{X},D)$. ∎

###  2.2 Pullback of parabolic bundles

In this subsection, we introduce the concept of pullback of parabolic bundles.
We refer the readers to [IS07, IS08] for a more systematic treatment. We avoid
the language of Deligne-Mumford stacks in [IS07, IS08]. This subsection is
conceptional and we shall make precise computations in next subsection.

A parabolic line bundle is a parabolic sheaf $F$ such that all the
${}_{\bm{a}}F$ are line bundles. An important class of examples is obtained as
follows: let $L$ be a line bundle on $\overline{X}$, if
${\bm{a}}=(a_{1},\ldots,a_{\ell})$ is a $\mathbb{R}^{\ell}$-indexed, then we
can define a parabolic line bundle denoted $L^{\bm{a}}_{*}$ by setting

$\displaystyle_{\bm{b}}L^{\bm{a}}:=L\otimes\mathcal{O}_{\overline{X}}\left(\sum_{i=1}^{\ell}\lfloor
a_{i}+b_{i}\rfloor D_{i}\right)$ (2.3)

for any ${\bm{b}}\in\mathbb{R}^{\ell}$.

######  Definition 2.3 (Locally abelian parabolic bundle).

A parabolic sheaf $\bm{E}_{*}$ is a _locally abelian parabolic bundle_ if, in
a neighborhood of any point $x\in\overline{X}$ there is an isomorphism between
$\bm{E}_{*}$ and a direct sum of parabolic line bundles.

Let $f:\overline{Y}\to\overline{X}$ be a holomorphic map of complex manifolds.
Let $D^{\prime}=\sum_{j=1}^{k}D_{j}^{\prime}$ and $D=\sum_{i=1}^{\ell}D_{i}$
be simple normal crossing divisors on $\overline{Y}$ and $\overline{X}$
respectively. Assume tht $f^{-1}(D)\subset D^{\prime}$. Denote by $n_{ij}={\rm
ord}_{D^{\prime}_{j}}f^{*}D_{i}\in\mathbb{Z}_{\geq 0}$. Let $L$ be a line
bundle on $\overline{X}$ and let $L^{\bm{a}}_{*}$ be the parabolic line bundle
defined in (2.3). Set

$\displaystyle
f^{*}\bm{a}:=(\sum_{i=1}^{\ell}n_{i1}a_{i},\ldots,\sum_{i=1}^{\ell}n_{ik}a_{i})\in\mathbb{R}^{k}.$
(2.4)

Then $f^{*}(L^{\bm{a}}_{*})$ is defined by setting

$\displaystyle_{\bm{b}}(f^{*}L)^{f^{*}\bm{a}}:=f^{*}L\otimes\mathcal{O}_{\overline{Y}}\left(\sum_{j=1}^{k}\lfloor\sum_{i=1}^{\ell}n_{ij}a_{i}+b_{j}\rfloor
D^{\prime}_{j}\right)$ (2.5)

for any ${\bm{b}}\in\mathbb{R}^{k}$.

Let $\overline{X}$ be a compact complex manifold. Consider a locally abelian
parabolic bundle $\bm{E}_{*}$ defined on $\overline{X}$. We can cover
$\overline{X}$ with open subsets $U_{1},\ldots,U_{m}$, such that
$\bm{E}_{*}|_{U_{i}}$ can be expressed as a direct sum of parabolic line
bundles on each $U_{i}$.

Using this decomposition, we define the pullback $f^{*}(\bm{E}_{*}|_{U_{i}})$
as in (2.5). It can be verified that $f^{*}(\bm{E}_{*}|_{U_{i}})$ is
compatible with $f^{*}(\bm{E}_{*}|_{U_{j}})$ whenever $U_{i}\cap
U_{j}\neq\varnothing$. This allows us to extend the local pullback to a global
level, resulting in the definition of the pullback of a locally abelian
parabolic bundle denoted by $f^{*}\bm{E}_{*}$. In next section, we will see an
explicit description of the pullback of regular filtered Higgs bundles induced
by tame harmonic bundles.

###  2.3 Functoriality of pullback of regular filtered Higgs bundle

We recall some notions in [Moc07a, §2.2.2]. Let $X$ be a complex manifold, $D$
be a simple normal crossing divisor on $X$, and $E$ be a holomorphic vector
bundle on $X\backslash D$ such that $E|_{X\backslash D}$ is equipped with a
hermitian metric $h$. Let $\bm{v}=(v_{1},\ldots,v_{r})$ be a smooth frame of
$E|_{X\backslash D}$. We obtain the $H(r)$-valued function $H(h,\bm{v})$
defined over $X\backslash D$,whose $(i,j)$-component is given by
$h(v_{i},v_{j})$.

Let us consider the case $X=\mathbb{D}^{n}$, and $D=\sum_{i=1}^{\ell}D_{i}$
with $D_{i}=(z_{i}=0)$. We have the coordinate $(z_{1},\ldots,z_{n})$. Let
$h$, $E$ and $\bm{v}$ be as above.

######  Definition 2.4.

A smooth frame $\bm{v}$ on $X\backslash D$ is called _adapted up to log order_
, if the following inequalities hold over $X\backslash D$:

$C^{-1}(-\sum_{i=1}^{\ell}\log|z_{i}|)^{-M}\leq H(h,\bm{v})\leq
C(-\sum_{i=1}^{\ell}\log|z_{i}|)^{M}$

for some positive numbers $M$ and $C$.

The goal of this subsection is to establish the following result concerning
the functoriality of the pullback of a regular filtered Higgs bundle. This
result will play a crucial role in proving Proposition 3.35.

######  Proposition 2.5.

Consider a morphism $f:\overline{Y}\to\overline{X}$ of smooth projective
varieties $\overline{X}$ and $\overline{Y}$. Let $D$ and $D^{\prime}$ be
simple normal crossing divisors on $\overline{X}$ and $\overline{Y}$
respectively. Assume that $f^{-1}(D)\subset D^{\prime}$. Let $(E,\theta,h)$ be
a tame harmonic bundle on $X:=\overline{X}\backslash D$. Let
$(\bm{E}_{*},\theta)$ be the regular filtered Higgs bundle defined in § 2.1.
Consider the pullback of $f^{*}\bm{E}_{*}$ defined in § 2.2, which is also a
parabolic bundle over $(\overline{Y},D^{\prime})$. Then

  1. (i)

$f^{*}\bm{E}_{*}$ is the prolongation $\tilde{E}_{*}$ of $f^{*}E$ using the
norm growth with respect to the metric $f^{*}h$ as defined in (2.2).

  2. (ii)

$(f^{*}\bm{E}_{*},f^{*}\theta)$ is a filtered regular Higgs bundle.

###### Proof.

Since this is a local result, we assume that $\overline{X}:=\mathbb{D}^{n}$
and $D:=\bigcup_{i=1}^{\ell}\left\\{z_{i}=0\right\\}$. Let
$\overline{Y}:=\mathbb{D}^{m}$ and
$D^{\prime}:=\bigcup_{j=1}^{k}\left\\{w_{j}=0\right\\}$. Then,
$f^{*}\left(z_{i}\right)=\prod_{j=1}^{k}w_{j}^{n_{ij}}g_{i}$ for some
invertible functions
$\\{g_{i}\\}_{i=1,\ldots,\ell}\subset\mathcal{O}(\overline{Y})$.

By [Moc07a, Proposition 8.70], there exists a holomorphic frame
$\bm{v}=(v_{1},\ldots,v_{r})$ of ${}^{\diamond}\\!E|_{\overline{X}}$ and
$\\{a_{ij}\\}_{i=1,\ldots,r;j=1,\ldots,\ell}\subset\mathbb{R}$ such that if we
put $\tilde{v}_{i}:=v_{i}\cdot\prod_{j=1}^{\ell}|z_{j}|^{-a_{ij}},$ then for
the smooth frame $\widetilde{\bm{v}}=(\tilde{v}_{1},\ldots,\tilde{v}_{r})$
over $X=\overline{X}\backslash D$, $H(h,\widetilde{\bm{v}})$ is adapted to log
order in the sense of Definition 2.4.

Define $L_{i}$ to be the sub-line bundle of ${}^{\diamond}\\!E$ generated by
$v_{i}$. Write $\bm{a}_{i}:=(a_{i1},\ldots,a_{i\ell})\in\mathbb{R}^{\ell}$.
Consider the parabolic line bundle $(L_{i})^{\bm{a}_{i}}_{*}$ over
$(\overline{X},D)$ defined in (2.3), namely,

$\displaystyle_{\bm{b}}(L_{i})^{\bm{a}_{i}}:=L_{i}\otimes\mathcal{O}_{\overline{X}}\left(\sum_{j=1}^{\ell}\lfloor
a_{ij}+b_{j}\rfloor D_{j}\right)$ (2.6)

for any $\bm{b}\in\mathbb{R}^{\ell}$.

######  Claim 2.6.

The parabolic bundles $\bm{E}_{*}$ and
$\oplus_{i=1}^{r}(L_{i})^{\bm{a}_{i}}_{*}$ are the same. In particular,
$\bm{E}_{*}$ is locally abelian.

###### Proof.

By (2.2), for any $\bm{b}\in\mathbb{R}^{\ell}$, any holomorphic section
$\sigma\in\Gamma(\overline{X},{}_{\bm{b}}E)$ satisfies

$|\sigma|_{h}=\mathcal{O}(\prod_{j=1}^{\ell}|z_{j}|^{-b_{j}-\varepsilon}).$

As $\bm{v}$ is a frame for ${}^{\diamond}\\!E$, one can write
$\sigma=\sum_{i=1}^{r}g_{i}v_{i}$ where $g_{i}$ is a holomorphic function
defined on $X$. Write $\bm{g}:=(g_{1},\ldots,g_{r})$. Since
$H(h,\widetilde{\bm{v}})$ is adapted to log order, it follows that

$C^{-1}(-\sum_{j=1}^{\ell}\log|z_{j}|)^{-M}\cdot\sum_{i=1}^{r}|g_{i}|^{2}\prod_{j=1}^{\ell}|z_{j}|^{2a_{ij}}\leq\overline{\bm{g}}H(h,\bm{v})\bm{g}^{T}=|\sigma|_{h}^{2}=\mathcal{O}(\prod_{j=1}^{\ell}|z_{j}|^{-2b_{i}-\varepsilon})$

for any $\varepsilon>0$. Hence for each $i$ and any $\varepsilon>0$ we have

$|g_{i}|^{2}=\mathcal{O}(\prod_{j=1}^{\ell}|z_{j}|^{-2(b_{j}+a_{ij})-\varepsilon}).$

Therefore, ${\rm ord}\,_{D_{j}}g_{i}\leq-\lfloor b_{j}+a_{ij}\rfloor$. This
proves that

${}_{\bm{b}}E\subset\oplus_{i=1}^{r}{}_{\bm{b}}(L_{i})^{\bm{a}_{i}}.$

On the other hand, we consider any section
$\sigma\in\Gamma(\overline{X},{}_{\bm{b}}(L_{i})^{\bm{a}_{i}})$. Then
$\sigma=gv_{i}$ for some meromorphic function $g$ defined over $\overline{X}$
such that ${\rm ord}\,_{D_{j}}g_{i}\leq-\lfloor b_{j}+a_{ij}\rfloor$ by (2.6).
Therefore, there exists some positive constant $C>0$ such that

$|\sigma|_{h}^{2}=|g|^{2}|v_{i}|_{h}^{2}\leq
C\prod_{j=1}^{\ell}|z_{j}|^{-2(b_{j}+a_{ij})}\cdot|\tilde{v}_{i}|_{h}^{2}\cdot\prod_{j=1}^{\ell}|z_{j}|^{2a_{ij}}=C\prod_{j=1}^{\ell}|z_{i}|^{-2b_{j}}\cdot|\tilde{v}_{i}|_{h}^{2}=\mathcal{O}(\prod_{i=1}^{\ell}|z_{i}|^{-b_{i}-\varepsilon}).$

as $|\tilde{v}_{i}|_{h}^{2}\leq C(-\sum_{j=1}^{\ell}\log|z_{j}|)^{M}$ for some
$C,M>0$. This implies that

$\oplus_{i=1}^{r}{}_{\bm{b}}(L_{i})^{\bm{a}_{i}}\subset{}_{\bm{b}}E.$

The claim is proved. ∎

Consider the pullback $f^{*}\bm{v}:=(f^{*}v_{1},\ldots,f^{*}v_{m})$. Then it
is a holomorphic frame of $f^{*}E|_{Y}$ where $Y:=\overline{Y}\backslash
D^{\prime}$. Note that we have

$f^{*}\tilde{v}_{i}:=f^{*}v_{i}\cdot\prod_{j=1}^{\ell}|f^{*}z_{j}|^{-a_{ij}}=f^{*}v_{i}\cdot\prod_{j=1}^{\ell}\prod_{q=1}^{k}|w_{q}|^{-n_{jq}a_{ij}}\cdot
g^{\prime}_{i}$

for some invertible holomorphic function
$g^{\prime}_{i}\in\mathcal{O}(\overline{Y})$. Similar to (2.4), we set

$f^{*}\bm{a}_{i}:=(\sum_{j=1}^{\ell}n_{j1}a_{ij},\ldots,\sum_{j=1}^{\ell}n_{jk}a_{ij})\in\mathbb{R}^{k}.$

Then we have

$f^{*}\tilde{v}_{i}:=f^{*}v_{i}\cdot|\bm{w}^{-f^{*}\bm{a}_{i}}|\cdot
g^{\prime}_{i}.$

Since $H(h,\widetilde{\bm{v}})$ is adapted to log order, it is easy to check
that $H(f^{*}h,f^{*}\widetilde{\bm{v}})$ also is adapted to log order. Set
$e_{i}:=f^{*}v_{i}\cdot|\bm{w}^{-f^{*}\bm{a}_{i}}|$ for $i=1,\ldots,r$ and
$\bm{e}:=(e_{1},\ldots,e_{r})$. Then $\bm{e}$ is a smooth frame for
$f^{*}E|_{Y}$. Since $g_{i}^{\prime}$ is invertible, it follows that
$H(f^{*}h,\bm{e})$ is also adapted to log order. Consider the prolongation
$(\tilde{E}_{*},\tilde{\theta})$ of the tame harmonic bundle
$(f^{*}E,f^{*}\theta,f^{*}h)$ using the norm growth as defined in (2.2).
Applying the result from Claim 2.6 to $(f^{*}E,f^{*}\theta,f^{*}h)$, we can
conclude that the parabolic bundle $\tilde{E}_{*}$ is given by

$\displaystyle\tilde{E}_{*}=\oplus_{i=1}^{r}(f^{*}L_{i})^{f^{*}\bm{a}_{i}}_{*},$
(2.7)

where $(f^{*}L_{i})^{f^{*}\bm{a}_{i}}_{*}$ are parabolic line bundles defined
by

$\displaystyle_{\bm{b}}(f^{*}L_{i})^{f^{*}\bm{a}_{i}}:=f^{*}L_{i}\otimes\mathcal{O}_{\overline{Y}}\left(\sum_{j=1}^{k}\lfloor\sum_{q=1}^{\ell}n_{qj}a_{iq}+b_{j}\rfloor
D^{\prime}_{j}\right).$ (2.8)

On the other hand, by our definition of pullback of parabolic bundles and
Claim 2.6, we have

$f^{*}\bm{E}_{*}:=\oplus_{i=1}^{r}f^{*}(L_{i})^{\bm{a}_{i}}_{*}$

where $f^{*}(L_{i})^{\bm{a}_{i}}_{*}$ is the pullback of parabolic line bundle
$(L_{i})^{\bm{a}_{i}}_{*}$ defined in (2.5). By performing a straightforward
computation, we find that

$f^{*}(L_{i})^{\bm{a}_{i}}_{*}=f^{*}L_{i}\otimes\mathcal{O}_{\overline{Y}}\left(\sum_{j=1}^{\ell}\lfloor\sum_{q=1}^{\ell}n_{qj}a_{iq}+b_{j}\rfloor
D^{\prime}_{j}\right).$

This equality together with (2.7) and (2.8) yields
$\tilde{E}_{*}=f^{*}\bm{E}_{*}.$ We prove our first assertion. The second
assertion can be deduced from the first one, combined with Theorem 2.2. ∎

###  2.4 $\mathbb{C}^{*}$-action and $\mathbb{R}^{*}$-action on character
varieties

Consider a smooth projective variety $\overline{X}$ equipped with a simple
normal crossing divisor $D$. We define $X$ as the complement of $D$ in
$\overline{X}$. Additionally, we fix an ample line bundle $L$ on
$\overline{X}$. Let $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ be a
reductive representation.

According to Theorem 1.6, there exists a tame pure imaginary harmonic bundle
$(E,\theta,h)$ on $X$ such that $(E,\nabla_{h}+\theta+\theta_{h}^{\dagger})$
is flat, with the monodromy representation being precisely $\varrho$. Here
$\nabla_{h}$ is the Chern connection of $(E,h)$ and $\theta_{h}^{\dagger}$ is
the adjoint of $\theta$ with respect to $h$. Let $(\bm{E}_{*},\theta)$ be the
prolongation of $(E,\theta)$ on $\overline{X}$ defined in § 2.1. By [Moc06,
Theorem 1.4], $(\bm{E}_{*},\theta)$ is a $\mu_{L}$-polystable regular filtered
Higgs bundle on $(\overline{X},D)$ with trivial characteristic numbers.
Therefore, for any $t\in\mathbb{C}^{*}$, $(\bm{E}_{*},t\theta)$ be also
$\mu_{L}$-polystable regular filtered Higgs bundle on $(\overline{X},D)$ with
trivial characteristic numbers. By [Moc06, Theorem 9.4], there is a
pluriharmonic metric $h_{t}$ for $(E,t\theta)$ adapted to the parabolic
structures of $(\bm{E}_{*},t\theta)$. Then $(E,t\theta,h_{t})$ is a harmonic
bundle and thus the connection
$\nabla_{h_{t}}+t\theta+\bar{t}\theta_{h_{t}}^{\dagger}$ is flat. Here
$\nabla_{h_{t}}$ is the Chern connection for $(E,h_{t})$ and
$\theta_{h_{t}}^{\dagger}$ is the adjoint ot $\theta$ with respect to $h_{t}$.
Let us denote by $\varrho_{t}:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ the
monodromy representation of
$\nabla_{h_{t}}+t\theta+\bar{t}\theta_{h_{t}}^{\dagger}$. It should be noted
that the representation $\varrho_{t}$ is well-defined up to conjugation. As a
result, the $\mathbb{C}^{*}$-action is only well-defined over $M_{\rm B}(X,N)$
and we shall denote it by

$t.[\varrho]:=[\varrho_{t}]\quad\mbox{for any }t\in\mathbb{C}^{*}.$

It is important to observe that unlike the compact case, $\varrho_{t}$ is not
necessarily reductive in general, even if the original representation
$\varrho$ is reductive. However, if $t\in\mathbb{R}^{*}$, $(E,t\theta)$ is
also pure imaginary and by Theorem 1.6, $\varrho_{t}$ is reductive.
Nonetheless, we can obtain a family of (might not be semisimple)
representations $\\{\varrho_{t}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{t\in\mathbb{C}^{*}}$. By [Moc06, Proofs of Theorem
10.1 and Lemma 10.2] we have

######  Lemma 2.7.

The map

$\displaystyle\Phi:\mathbb{R}^{*}$ $\displaystyle\to M_{\rm B}(\pi_{1}(X),N)$
$\displaystyle t$ $\displaystyle\mapsto[\varrho_{t}]$

is continuous. $\Phi(\\{t\in\mathbb{R}^{*}\mid|t|<1\\})$ is relatively compact
in $M_{\rm B}(\pi_{1}(X),N)$.∎

Note that Lemma 2.7 can not be seen directly from [Moc06, Lemma 10.2] as he
did not treat the character variety in his paper. Indeed, based on Uhlenbeck’s
compactness in Gauge theory, Mochizuki’s proof can be read as follows: for any
$t_{n}\in\mathbb{R}^{*}$ converging to $0$, after subtracting to a
subsequence, there exists some $\varrho_{0}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$ and $g_{n}\in{\rm GL}_{N}(\mathbb{C})$ such that
$\lim\limits_{n\to\infty}g_{n}^{*}\varrho_{t_{n}}=\varrho_{0}$ in the
representation variety $R(\pi_{1}(X),{\rm GL}_{N})(\mathbb{C})$. Moreover, one
can check that $\varrho_{0}$ corresponds to some tame pure imaginary harmonic
bundle, and thus by Theorem 1.6 it is reductive (cf. [BDDM22] for a more
detailed study). For this reason, we can see that it will be more practical to
work with $\mathbb{R}^{*}$-action instead of $\mathbb{C}^{*}$-action as the
representations we encounter are all reductive.

When $X$ is compact, Simpson proved that $\lim_{t\to 0}\Phi(t)$ exists and
underlies a $\mathbb{C}$-VHS. However, it is current unknown in the quasi-
projective setting. Instead, Mochizuki proved that, we achieve a
$\mathbb{C}$-VHS after finite steps of deformations. Let us recall it briefly
and the readers can refer to [Moc06, §10.1] for more details.

Let $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ be a semisimple
representation. Then there exists a tame and pure imaginary harmonic bundle
$(E,\theta,h)$ corresponding to $\varrho$. Then the induced regular filtered
Higgs bundle $(\bm{E}_{*},\theta)$ on $(\overline{X},D)$ is
$\mu_{L}$-polystable with trivial characteristic numbers. Hence we have a
decomposition

$(\bm{E}_{*},\theta)=\oplus_{j\in\Lambda}(\bm{E}_{j*},\theta_{j})\otimes\mathbb{C}^{m_{j}}$

where $(\bm{E}_{j*},\theta_{j})$ is $\mu_{L}$-stable regular filtered Higgs
bundle with trivial characteristic numbers. Put
$r(\varrho):=\sum_{j\in\Lambda}m_{j}$. Then $r(\varrho)\leq{\rm rank}\,E$. For
any $t\in\mathbb{R}^{*}$, we know that $(E,t\theta)$ is still tame and pure
imaginary and thus $\varrho_{t}$ is also reductive. Since
$\varrho(\\{t\in\mathbb{C}^{*}\mid|t|<1\\})$ is relatively compact, then there
exists some $t_{n}\in\mathbb{R}^{*}$ which converges to zero such that
$\lim_{t_{n}\to 0}[\varrho_{t_{n}}]$ exists, denoting by $[\varrho_{0}]$.
Moreover, $\varrho_{0}$ corresponds to some tame harmonic bundle. There are
two possibilities:

  * •

For each $j\in\Lambda$, $(\bm{E}_{j*},t_{n}\theta_{j})$ converges to some
$\mu_{L}$-stable regular filtered Higgs sheaf (cf. [Moc06, p. 96] for the
definition of convergence). Then by [Moc06, Proposition 10.3], $\varrho_{0}$
underlies a $\mathbb{C}$-VHS.

  * •

For some $j\in\Lambda$, $(\bm{E}_{j*},t_{n}\theta_{j})$ converges to some
$\mu_{L}$-semistable regular filtered Higgs sheaf, but not $\mu_{L}$-stable.
Then by [Moc06, Lemma 10.4] $r(\varrho)<r(\varrho_{0})$. In other words,
letting $\varrho_{i}$ be the representation corresponding to
$(\bm{E}_{j*},\theta_{j})$ and $\varrho_{i,t}$ be the deformation under
$\mathbb{C}^{*}$-action. Then $\lim_{n\to\infty}\varrho_{i,t_{n}}$ exists,
denoted by $\varrho_{i,0}$. Then $\varrho_{i,0}$ corresponds to some tame
harmonic bundle, and thus also a $\mu_{L}$-polystable regular filtered Higgs
bundle which is not stable. In this case, we further deform $\varrho_{0}$
until we achieve Case 1.

In summary, Mochizuki’s result implies the following, which we shall refer to
as _Mochizuki’s ubiquity_ , analogous to the term _Simpson’s ubiquity_ for the
compact case (cf. [Sim91]).

######  Theorem 2.8.

Let $X$ be a smooth quasi-projective variety. Consider $\mathfrak{C}$, a
Zariski closed subset of $M_{\rm B}(X,G)(\mathbb{C})$, where $G$ denotes a
complex reductive group. If $\mathfrak{C}$ is invariant under the action of
$\mathbb{R}^{*}$ defined above, then each geometrically connected component of
$\mathfrak{C}(\mathbb{C})$ contains a $\mathbb{C}$-point $[\varrho]$ such that
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ is a reductive representation
that underlies a $\mathbb{C}$-variation of Hodge structure. ∎

###  2.5 Pullback of reductive representations commutes with
$\mathbb{C}^{*}$-action

In this section, we prove that the $\mathbb{C}^{*}$-action on character
varieties commutes with the pullback.

######  Proposition 2.9.

Let $f:Y\to X$ be a morphism of smooth quasi-projective varieties. If
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ is a reductive representation,
then for any $t\in\mathbb{C}^{*}$, we have

$\displaystyle f^{*}(t.[\varrho])=t.[f^{*}\varrho].$ (2.9)

###### Proof.

Let $\overline{X}$ and $\overline{Y}$ be smooth projective compactifications
of $X$ and $Y$ such that $D:=\overline{X}\backslash X$ and
$D^{\prime}:=\overline{Y}\backslash Y$ are simple normal crossing divisors. We
may assume that $f$ extends to a morphism $f:\overline{Y}\to\overline{X}$.

By Theorem 1.6, there is a tame pure imaginary harmonic bundle $(E,\theta,h)$
on $X$ such that $\varrho$ is the monodromy representation of the flat
connection $\nabla_{h}+\theta+\theta_{h}^{\dagger}$. Then $f^{*}\varrho$ is
the monodromy representation of
$f^{*}(\nabla_{h}+\theta+\theta_{h}^{\dagger})$, which is the flat connection
corresponding to the harmonic bundle $(f^{*}E,f^{*}\theta,f^{*}h)$.

Let $(\bm{E}_{*},\theta)$ be the induced regular filtered Higgs bundle on
$(\overline{X},D)$ by $(E,\theta,h)$ defined in § 2.1. According to §§ 2.2 and
2.3 we can define the pullback $(f^{*}\bm{E}_{*},f^{*}\theta)$, which also
forms a regular filtered Higgs bundle on $(\overline{Y},D^{\prime})$ with
trivial characteristic numbers.

Fix some ample line bundle $L$ on $\overline{X}$. It is worth noting that for
any $t\in\mathbb{C}^{*}$, $(\bm{E}_{*},t\theta)$ is $\mu_{L}$-polystable with
trivial characteristic numbers. By [Moc06, Theorem 9.4], there is a
pluriharmonic metric $h_{t}$ for $(E,t\theta)$ adapted to the parabolic
structures of $(\bm{E}_{*},t\theta)$. Recall that in § 2.4, $\varrho_{t}$ is
defined to be the monodromy representation of the flat connection
$\nabla_{h_{t}}+t\theta+\bar{t}\theta_{h_{t}}^{\dagger}$. It follows that
$f^{*}\varrho_{t}$ is the monodromy representation of the flat connection
$f^{*}(\nabla_{h_{t}}+t\theta+\bar{t}\theta_{h_{t}}^{\dagger})$.

By virtue of Proposition 2.5, the regular filtered Higgs bundle
$(f^{*}\bm{E}_{*},tf^{*}\theta)$ is the prolongation of the tame harmonic
bundle $(f^{*}E,tf^{*}\theta,f^{*}h_{t})$ using norm growth defined in § 2.1.
By the definition of $\mathbb{C}^{*}$-action, $(f^{*}\varrho)_{t}$ is the
monodromy representation of the flat connection
$\nabla_{f^{*}h_{t}}+tf^{*}\theta+\bar{t}(f^{*}\theta)_{f^{*}h_{t}}^{\dagger}$,
which is equal to
$f^{*}(\nabla_{h_{t}}+t\theta+\bar{t}\theta_{h_{t}}^{\dagger})$. It follows
that $(f^{*}\varrho)_{t}=f^{*}\varrho_{t}$. This concludes (2.9). ∎

As a direct consequence of Proposition 2.9, we have the following result.

######  Corollary 2.10.

Let $f:Y\to X$ be a morphism of smooth quasi-projective varieties. Let
$M\subset M_{\rm B}(X,N)(\mathbb{C})$ be a subset which is invariant by
$\mathbb{C}^{*}$-action (or $\mathbb{R}^{*}$-action). Then for the morphism
$f^{*}:M_{\rm B}(X,N)\to M_{\rm B}(Y,N)$ between character varieties, $f^{*}M$
is also invariant by $\mathbb{C}^{*}$-action (or $\mathbb{R}^{*}$-action). ∎

##  3 Construction of the Shafarevich morphism

The aim of this section is to establish the proofs of Theorem A. Additionally,
the techniques developed in this section will play a crucial role in § 4
dedicated to the proof of the reductive Shafarevich conjecture.

###  3.1 Factorizing through non-rigidity

In this subsection, $X$ is assumed to be a smooth quasi-projective variety.
Let $\mathfrak{C}\subset M_{\rm B}(X,N)(\mathbb{C})$ be a
$\bar{\mathbb{Q}}$-constructible subset. Since $M_{\rm B}(X,N)$ is an finite
type affine scheme defined over $\mathbb{Q}$, $\mathfrak{C}$ is defined over
some number field $k$.

Let us utilize Lemmas 1.29 and 1.27 to construct a reduction map
$s_{\mathfrak{C}}:X\to S_{\mathfrak{C}}$ associated with $\mathfrak{C}$, which
allows us to factorize non-rigid representations into those underlying
$\mathbb{C}$-VHS with discrete monodromy.

######  Definition 3.1.

The _reduction map_ $s_{\mathfrak{C}}:X\to S_{\mathfrak{C}}$ is obtained
through the simultaneous Stein factorization of the reductions
$\\{s_{\tau}:X\to S_{\tau}\\}_{[\tau]\in\mathfrak{C}(K)}$, employing Lemma
1.29. Here $\tau:\pi_{1}(X)\to{\rm GL}_{N}(K)$ ranges over all reductive
representations with $K$ a non-archimedean local field containing $k$ such
that $[\tau]\in\mathfrak{C}(K)$ and $s_{\tau}:X\to S_{\tau}$ is the reduction
map constructed in Theorem 1.27.

Note that $s_{\mathfrak{C}}$ is a dominant morphism with connected general
fibers and we have the following diagram.

${X}$${S_{\mathfrak{C}}}$${S_{\tau}}$$\scriptstyle{s_{\mathfrak{C}}}$$\scriptstyle{s_{\tau}}$$\scriptstyle{e_{\tau}}$

The reduction map $s_{\mathfrak{C}}:X\to S_{\mathfrak{C}}$ employs the
following crucial property, thanks to Theorem 1.27.

######  Lemma 3.2.

Let $F\subset X$ be a connected Zariski closed subset such that
$s_{\mathfrak{C}}(F)$ is a single point in $S_{\mathfrak{C}}$. Then for any
non-archimedean local field $L$ and any reductive representation
$\tau:\pi_{1}(X)\to{\rm GL}_{N}(L)$, the image
$\tau(\mathrm{Im}[\pi_{1}(F)\to\pi_{1}(X)])$ is a bounded subgroup of ${\rm
GL}_{N}(L)$.

###### Proof.

By our construction $s_{\tau}=e_{\tau}\circ s_{\mathfrak{C}}$, so
$s_{\tau}(F)$ is a single point. Hence by Theorem 1.27,
$\tau(\mathrm{Im}[\pi_{1}(F)\to\pi_{1}(X)])$ is bounded. ∎

Recall the following definition in [KP23, Definition 2.2.1].

######  Definition 3.3 (Bounded set).

Let $K$ be a non-archimedean local field. Let $X$ be an affine $K$-scheme. A
subset $B\subset X(K)$ is _bounded_ if for every $f\in K[X]$, the set
$\\{v(f(b))\mid b\in B\\}$ is bounded below, where $v:K\to\mathbb{R}$ is the
valuation of $K$.

We have the following lemma in [KP23, Fact 2.2.3].

######  Lemma 3.4.

If $B\subset X(K)$ is closed, then $B$ is bounded if and only if $B$ is
compact with respect to the analytic topology of $X(K)$. If $f:X\to Y$ is a
morphism of affine $K$-schemes of finite type, then $f$ carries bounded
subsets of $X(K)$ to bounded subsets in $Y(K)$. ∎

We will establish a lemma that plays a crucial role in the proof of
Proposition 3.9 and is also noteworthy in its own regard.

######  Lemma 3.5.

Let $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(K)$ be a (un)bounded representation.
Then its semisimplification $\varrho^{ss}:\pi_{1}(X)\to{\rm GL}_{N}(\bar{K})$
is also (un)bounded.

###### Proof.

Note that there exists some $g\in{\rm GL}_{N}(\bar{K})$ such that

$g\varrho g^{-1}=\
\left[\begin{array}[]{cccc}\varrho_{1}&a_{12}&\cdots&a_{1n}\\\
0&\varrho_{2}&\cdots&a_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\varrho_{n}\end{array}\right]$ (3.1)

where $\varrho_{i}:\pi_{1}(X)\to{\rm GL}_{N_{i}}(\bar{K})$ is an irreducible
representation such that $\sum_{i=1}^{N}N_{i}=N$. Note that $g\varrho g^{-1}$
is unbounded if and only if $\varrho$ is unbounded. Hence we may assume at the
beginning that $\varrho$ has the form of (3.1). The semisimplification of
$\varrho$ is defined by

$\varrho^{ss}=\left[\begin{array}[]{cccc}\varrho_{1}&0&\cdots&0\\\
0&\varrho_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\varrho_{n}\end{array}\right]$

It is obvious that if $\varrho$ is bounded, then $\varrho^{ss}$ is bounded.

Assume now $\varrho^{ss}$ is bounded. Then each $\varrho_{i}$ is bounded. Let
$L$ be a finite extension of $K$ such that $\varrho$ is defined over $L$. Then
$\varrho_{i}(\pi_{1}(X))$ is contained in some maximal compact subgroup of
${\rm GL}_{N_{i}}(L)$. Since all maximal compact subgroups of ${\rm
GL}_{N_{i}}(L)$ are conjugate to ${\rm GL}_{N_{i}}(\mathcal{O}_{L})$, then
there exists $g_{i}\in{\rm GL}_{N_{i}}(L)$ such that
$g_{i}\varrho_{i}g_{i}^{-1}:\pi_{1}(X)\to{\rm GL}_{N_{i}}(\mathcal{O}_{L})$.
Define

$\tau:=\left[\begin{array}[]{cccc}g_{1}&0&\cdots&0\\\ 0&g_{2}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&g_{n}\end{array}\right]\left[\begin{array}[]{cccc}\varrho_{1}&a_{12}&\cdots&a_{1n}\\\
0&\varrho_{2}&\cdots&a_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\varrho_{n}\end{array}\right]\left[\begin{array}[]{cccc}g_{1}^{-1}&0&\cdots&0\\\
0&g_{2}^{-1}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&g_{n}^{-1}\end{array}\right]$ (3.2)

which is conjugate to $\varrho$, and is thus unbounded. Then $\tau$ can be
written as

$\tau=\
\left[\begin{array}[]{cccc}g_{1}\varrho_{1}g_{1}^{-1}&h_{12}&\cdots&h_{1n}\\\
0&g_{2}\varrho_{2}g_{2}^{-1}&\cdots&h_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&g_{n}\varrho_{n}g_{n}^{-1}\end{array}\right]$

such that $g_{i}\varrho_{i}g_{i}^{-1}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathcal{O}_{L})$ is irreducible. Write

$\tau_{1}:=\left[\begin{array}[]{cccc}g_{1}\varrho_{1}g_{1}^{-1}&0&\cdots&0\\\
0&g_{2}\varrho_{2}g_{2}^{-1}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&g_{n}\varrho_{n}g_{n}^{-1}\end{array}\right]$

and

$\tau_{2}:=\left[\begin{array}[]{cccc}0&h_{12}&\cdots&h_{1n}\\\
0&0&\cdots&h_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{array}\right]$

Note that $\tau_{2}$ is not a group homomorphism but only a map from
$\pi_{1}(X)$ to ${\rm GL}_{N}(L)$.

For any matrix $B$ with values in $L$, we shall write $v(B)$ the matrix whose
entries are the valuation of the corresponding entries in $B$ by
$v:L\to\mathbb{R}$. Let us define $M(B)$ the lower bound of the entries of
$v(B)$. Then for another matrix $A$ with values in $L$, one has
$M(A+B)\geq\min\\{M(A),M(B)\\}$.

Let $x_{1},\ldots,x_{m}$ be a generator of $\pi_{1}(X)$. Let $C$ be the lower
bound of the entries of $v(h_{ij}(x_{k}))$. We assume that $C<0$, or else it
is easy to see that $\tau$ is bounded. Note that
$\min_{i=1,\ldots,m}M(g_{i}\varrho_{i}g_{i}^{-1}(x_{i}))\geq 0$. It follows
that $M(\tau_{1}(x_{i}))\geq 0$ for each $x_{i}$. Then for any
$x=x_{i_{1}}\cdots x_{i_{\ell}}$,

$\displaystyle M(\tau(x))$
$\displaystyle=M(\sum_{j_{1},\ldots,j_{\ell}=1,2}\tau_{j_{1}}(x_{i_{1}})\cdots\tau_{j_{\ell}}(x_{i_{\ell}}))$
$\displaystyle\geq\min_{{j_{1},\ldots,j_{\ell}=1,2}}\\{M(\tau_{j_{1}}(x_{i_{1}})\cdots\tau_{j_{\ell}}(x_{i_{\ell}}))\\}.$

Note that $\tau_{j_{1}}(x_{i_{1}})\cdots\tau_{j_{\ell}}(x_{i_{\ell}})=0$ if
$\\#\\{k\mid j_{k}=2\\}\geq n$ since $\tau_{2}(x_{i})$ is nipotent. Hence

$M(\tau(x))\geq\min_{{j_{1},\ldots,j_{\ell}=1,2};\\#\\{k\mid
j_{k}=2\\}<n}\\{M(\tau_{j_{1}}(x_{i_{1}})\cdots\tau_{j_{\ell}}(x_{i_{\ell}}))\\}.$

Since $M(\tau_{1}(x_{i}))\geq 0$ for each $x_{i}$, it follows that
$M(\tau_{j_{1}}(x_{i_{1}})\cdots\tau_{j_{\ell}}(x_{i_{\ell}})\geq(n-1)C$ if
$\\#\\{k\mid j_{k}=2\\}<n$. Therefore, $M(\tau(x))\geq(n-1)C$ for any
$x\in\pi_{1}(X)$. $\tau$ is thus bounded. Since $\varrho$ is conjugate to
$\tau$, $\varrho$ is also bounded. We finish the proof of the lemma. ∎

We recall the following facts of character varieties.

######  Lemma 3.6.

Let $K$ be an _algebraically closed_ field of characteristic zero. Then the
$K$-points $M_{\rm B}(X,N)$ are in one-to-one correspondence with the
conjugate classes of reductive representations $\pi_{1}(X)\to{\rm GL}_{N}(K)$.
More precisely, if $\\{\varrho_{i}:\pi_{1}(X)\to{\rm GL}_{N}(K)\\}_{i=1,2}$
are two linear representations such that $[\varrho_{1}]=[\varrho_{2}]\in
M_{\rm B}(X,N)(K)$, then the semisimplification of $\varrho_{1}$ and
$\varrho_{2}$ are conjugate. ∎

The following result is thus a consequence of Lemma 3.5.

######  Lemma 3.7.

Let $K$ be a non-archimedean local field. Let $x\in M_{\rm B}(X,N)(K)$. If
$\\{\varrho_{i}:\pi_{1}(X)\to{\rm GL}_{N}(\bar{K})\\}_{i=1,2}$ are two linear
representations such that $[\varrho_{1}]=[\varrho_{2}]=x\in M_{\rm
B}(X,N)(\bar{K})$, then $\varrho_{1}$ is bounded if and only if $\varrho_{2}$
is bounded. In other words, for the GIT quotient $\pi:R(X,N)\to M_{B}(X,N)$
where $R(X,N)$ is the representation variety of $\pi_{1}(X)$ into ${\rm
GL}_{N}$, for any $x\in M_{B}(X,N)(\bar{K})$, the representations in
$\pi^{-1}(x)\subset R(X,N)(\bar{K})$ are either all bounded or all unbounded.

###### Proof.

By the assumption and Lemma 3.6, we know that the semisimplificaitons
$\varrho_{1}^{ss}:\pi_{1}(X)\to{\rm GL}_{N}(\bar{K})$ of
$\varrho_{2}^{ss}:\pi_{1}(X)\to{\rm GL}_{N}(\bar{K})$ are conjugate by an
element $g\in{\rm GL}_{N}(\bar{K})$. Therefore, there exists a finite
extension $L$ of $K$ such that $\varrho_{i}^{ss}$ and $\varrho_{i}$ are all
defined in $L$ and $g\in{\rm GL}_{N}(L)$. Hence $\varrho_{1}^{ss}$ is bounded
if and only if $\varrho_{2}^{ss}$ is bounded. By Lemma 3.5, we know that
$\varrho_{i}^{ss}$ is bounded if and only if $\varrho_{i}$ is bounded.
Therefore, the lemma follows. ∎

We thus can make the following definition.

######  Definition 3.8 (Class of bounded representations).

Let $K$ be a non-archimedean local field of characteristic zero. A point $x\in
M_{B}(X,N)(\bar{K})$ is called _a class of bounded representations_ if there
exist $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\bar{K})$ (thus any $\varrho$ by
Lemma 3.7) such that $[\varrho]=x$ and $\varrho$ is bounded.

######  Proposition 3.9.

Let $X$ be a smooth quasi-projective variety and let $\mathfrak{C}$ be a
$\bar{\mathbb{Q}}$-constructible subset of $M_{\rm B}(X,N)$. Let $\iota:F\to
X$ be a morphism from a quasi-projective _normal_ variety $F$ such that
$s_{\mathfrak{C}}\circ\iota(F)$ is a point. Let $\\{\tau_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{i=1,2}$ be reductive representations such that
$[\tau_{1}]$ and $[\tau_{2}]$ are in the same geometric connected component of
$\mathfrak{C}(\mathbb{C})$. Then $\tau_{1}\circ\iota$ is conjugate to
$\tau_{2}\circ\iota$. In other words, $j(\mathfrak{C})$ is zero-dimensional,
where $j:M_{\rm B}(X,N)\to M_{\rm B}(F,N)$ is the natural morphism of
character varieties induced by $\iota:F\to X$.

###### Proof.

Let $M_{X}$ (resp. $M$) be the moduli space of representations of $\pi_{1}(X)$
(resp. $\pi_{1}(F^{\mathrm{norm}})$) in ${\rm GL}_{N}$. Note that $M_{X}$ and
$M$ are both affine schemes of finite type defined over ${\mathbb{Q}}$. Let
$R_{X}$ (resp. $R$) be the affine scheme of finite type defined over
$\mathbb{Q}$ such that $R_{X}(L)={\rm Hom}(\pi_{1}(X),{\rm GL}_{N}(L))$ (resp.
$R(L)={\rm Hom}(\pi_{1}(F),{\rm GL}_{N}(L))$) for any field $L/\mathbb{Q}$.
Then we have

${R_{X}}$${M_{X}}$${R}$${M}$$\scriptstyle{\pi}$$\scriptstyle{\iota^{*}}$$\scriptstyle{j}$$\scriptstyle{p}$
(3.3)

where $\pi:R_{X}\to M_{X}$ and $p:R\to M$ are the GIT quotient that are both
surjective. For any field extension $K/\mathbb{Q}$ and any $\varrho\in
R_{X}(K)$, we write $[\varrho]:=\pi(\varrho)\in M_{X}(K)$. Let
$\mathfrak{R}:=\pi^{-1}(\mathfrak{C})$ that is a constructible subset defined
over some number field $k$. Then $\tau_{i}\in\mathfrak{R}(\mathbb{C})$.

######  Claim 3.10.

Let $\mathfrak{R}^{\prime}$ be any geometric irreducible component of
$\mathfrak{R}$. Then $j\circ\pi(\mathfrak{R}^{\prime})$ is zero dimensional.

###### Proof.

Assume, for the sake of contradiction, that $j\circ\pi(\mathfrak{R}^{\prime})$
is positive-dimensional. If we replace $k$ by a finite extension, we may
assume that $\mathfrak{R}^{\prime}$ is defined over $k$. Since $M$ is an
affine $\mathbb{Q}$-scheme of finite type, it follows that there exist a
$k$-morphism $\psi:M\rightarrow\mathbb{A}^{1}$ such that the image $\psi\circ
j\circ\pi(\mathfrak{R}^{\prime})$ is Zariski dense in $\mathbb{A}^{1}$. After
replacing $k$ by a finite extension, we can find a locally closed irreducible
curve $C\subset\mathfrak{R}^{\prime}$ such that the restriction $\psi\circ
j\circ\pi|_{C}:C\rightarrow\mathbb{A}^{1}$ is a generically finite
$k$-morphism. We take a Zariski open subset $U\subset\mathbb{A}^{1}$ such that
$\psi\circ j\circ\pi$ is finite over $U$. Let $\mathfrak{p}$ be a prime ideal
of the ring of integer $\mathcal{O}_{k}$ and let $K$ be its non-archimedean
completion. In the following, we shall work over $K$.

Let $x\in U(K)$ be a point, and let $y\in C(\bar{K})$ be a point over $x$.
Then $y$ is defined over some extension of $K$ whose extension degree is
bounded by the degree of $\psi\circ
j\circ\pi|_{C}:C\rightarrow\mathbb{A}^{1}$. Note that there are only finitely
many such field extensions. Hence there exists a finite extension $L/K$ such
that the points over $U(K)$ are all contained in $C(L)$. Since
$U(K)\subset\mathbb{A}^{1}(L)$ is unbounded, the image $\psi\circ
j\circ\pi(C(L))\subset\mathbb{A}^{1}(L)$ is unbounded.

Write $p:R\to M$ be the GIT quotient. Let $R_{0}$ be the set of bounded
representations in $R(L)$. Recall that by [Yam10], $M_{0}:=p(R_{0})$ is
compact in $M(L)$ with respect to analytic topology, hence $M_{0}$ is bounded
by Lemma 3.4. By Lemma 3.4 once again, $\psi(M_{0})$ is a bounded subset in
$\mathbb{A}^{1}(L)$. Recall that $\psi\circ
j\circ\pi(C(L))\subset\mathbb{A}^{1}(L)$ is unbounded. Therefore, there exists
$\varrho\in C(L)$ such that $\psi\circ j([\varrho])\not\in\psi(M_{0})$. Note
that $[\varrho\circ\iota]=j([\varrho])$ by (3.3). Hence
$[\varrho\circ\iota]\not\in M_{0}$ which implies that
$\varrho\circ\iota\not\in R_{0}$. By definition of $R_{0}$,
$\varrho\circ\iota$ is unbounded.

Let $\varrho^{ss}:\pi_{1}(X)\to{\rm GL}_{N}(\bar{L})$ be the
semisimplification of $\varrho$. Then
$[\varrho]=[\varrho^{ss}]\in\mathfrak{C}(\bar{L})$ by Lemma 3.6. Therefore,
$[\varrho\circ\iota]=[\varrho^{ss}\circ\iota]\in M(\bar{L})$ by (3.3). By
Lemma 3.7, $\varrho^{ss}\circ\iota:\pi_{1}(F)\to{\rm GL}_{N}(\bar{L})$ is also
unbounded. Note that $\varrho^{ss}\circ\iota$ is reductive by Theorem 1.7.
Since $\pi_{1}(F)$ is finitely generated, there exist a finite extension
$L^{\prime}$ of $L$ such that $\varrho^{ss}$ is defined over $L^{\prime}$.
However, by Lemma 3.2, $\varrho^{ss}\circ\iota$ is always bounded. We obtain a
contradiction and thus $j\circ\pi(\mathfrak{R}^{\prime})$ is zero dimensional.

We can also apply [Kem78, Corollary 4.3] instead of Lemma 3.7. As $\varrho\in
C(L)$, its image $[\varrho]\in M_{X}(L)$. Consider the fiber
$\pi^{-1}([\varrho])$ which is a $L$-variety. Its closed orbit is defined over
$L$ by Galois descent. As $\pi^{-1}([\varrho])$ contains the $L$-point
$\varrho$, the closed orbit in $\pi^{-1}([\varrho])$ has an $L$-point
$\varrho^{\prime}:\pi_{1}(X)\to{\rm GL}_{N}(L)$ as well by [Kem78, Corollary
4.3]. By Lemma 3.6 $\varrho^{\prime}$ is reductive and
$[\varrho^{\prime}]=[\varrho]$. Hence
$[\varrho^{\prime}\circ\iota]=[\varrho\circ\iota]\not\in M_{0}$. Therefore,
$\varrho^{\prime}\circ\iota:\pi_{1}(F)\to{\rm GL}_{N}(L)$ is unbounded by our
definition of $M_{0}$. However, by the definition of $s_{\mathfrak{C}}:X\to
S_{\mathfrak{C}}$ in Definition 3.1, $\varrho^{\prime}\circ\iota$ is always
bounded. We obtain a contradiction and thus $j\circ\pi(\mathfrak{R}^{\prime})$
is zero dimensional. ∎

Let $\\{\tau_{i}:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})\\}_{i=1,2}$ be
reductive representations such that $[\tau_{1}]$ and $[\tau_{2}]$ are
contained in the same connected component $\mathfrak{C}^{\prime}$ of
$\mathfrak{C}(\mathbb{C})$. We aim to prove that $j(\mathfrak{C}^{\prime})$ is
a point in $M(\mathbb{C})$.

Consider an irreducible component $\mathfrak{C}^{\prime\prime}$ of
$\mathfrak{C}^{\prime}$. We can choose an irreducible component $Z$ of
$\pi^{-1}(\mathfrak{C}^{\prime\prime})$ such that $\pi(Z)$ is dense in
$\mathfrak{C}^{\prime\prime}$. It follows that $Z$ is an irreducible component
of $\mathfrak{R}(\mathbb{C})$. By Claim 3.10, we know that $j\circ\pi(Z)$ is a
point in $M(\mathbb{C})$. Thus, $j(\mathfrak{C}^{\prime\prime})$ is also a
point in $M(\mathbb{C})$.

Consequently, $j(\mathfrak{C}^{\prime})$ is a point in $M(\mathbb{C})$. As a
result, we have
$[\tau_{1}\circ\iota]=j([\tau_{1}])=j([\tau_{2}])=[\tau_{2}\circ\iota]$. By
Theorem 1.7, $\tau_{1}\circ\iota$ and $\tau_{2}\circ\iota$ are reductive, and
according to Lemma 3.6, they are conjugate to each other. We have established
the proposition.

∎

We will need the following lemma on the intersection of kernels of
representations.

######  Lemma 3.11.

Let $X$ be a quasi-projective normal variety and let $\mathfrak{C}$ be a
constructible subset of $M_{\rm B}(X,N)(\mathbb{C})$. Then we have

$\displaystyle\cap_{[\varrho]\in\mathfrak{C}}\ker\varrho=\cap_{[\varrho]\in\overline{\mathfrak{C}}}\ker\varrho,$
(3.4)

where $\varrho$’s are reductive representations of $\pi_{1}(X)$ into ${\rm
GL}_{N}(\mathbb{C})$.

###### Proof.

Let $M_{X}$ be the moduli space of representation of $\pi_{1}(X)$ in ${\rm
GL}_{N}$. Let $R_{X}$ be the affine scheme of finite type such that $R(L)={\rm
Hom}(\pi_{1}(X),N)(L)$ for any field $\mathbb{Q}\subset L$. We write
$M:=M_{X}(\mathbb{C})$ and $R:=R_{X}(\mathbb{C})$. Then the GIT quotient
$\pi:R\to M$ is a surjective morphism. It follows that
$\pi^{-1}(\mathfrak{C})$ is a ${\rm GL}_{N}(\mathbb{C})$-invariant subset
where ${\rm GL}_{N}(\mathbb{C})$ acts on $R$ by the conjugation. Define
$H:=\cap_{[\varrho]\in\mathfrak{C}}\ker\varrho$, where $\varrho$’s are
reductive representations of $\pi_{1}(X)$ into ${\rm GL}_{N}(\mathbb{C})$.
Pick any $\gamma\in H$. Then the set $Z_{\gamma}:=\\{\varrho\in
R\mid\varrho(\gamma)=1\\}$ is a Zariski closed subset of $R$. Moreover,
$Z_{\gamma}$ is ${\rm GL}_{N}(\mathbb{C})$-invariant. Define
$Z:=\cap_{\gamma\in H}Z_{\gamma}$. Then $Z$ is also ${\rm
GL}_{N}(\mathbb{C})$-invariant. Therefore, $\pi(Z)$ is also a Zariski closed
subset of $M$. Note that $\mathfrak{C}\subset\pi(Z)$. Therefore,
$\overline{\mathfrak{C}}\subset\pi(Z)$. Note that for any reductive
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ such that
$[\varrho]\in\pi(Z)$, we have $\varrho(\gamma)=1$ for any $\gamma\in H$. It
follows that (3.4) holds. ∎

Lastly, let us prove the main result of this subsection. This result will
serve as a crucial cornerstone in the proofs of Theorems B, A and C.

######  Proposition 3.12.

Let $X$ be a smooth quasi-projective variety. Let $\mathfrak{C}$ be a
constructible subset of $M_{\rm B}(X,N)(\mathbb{C})$, defined over
$\mathbb{Q}$, such that $\mathfrak{C}$ is invariant under
$\mathbb{R}^{*}$-action. When $X$ is non-compact, we further assume that
$\mathfrak{C}$ is closed. Then there exist reductive representations
$\\{\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{i=1,\ldots,m}$ such that each
$\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$ underlies a $\mathbb{C}$-VHS, and for
a morphism $\iota:Z\to X$ from any quasi-projective _normal_ variety $Z$ with
$s_{\mathfrak{C}}\circ\iota(Z)$ being a point, the following properties hold:

  1. (i)

For $\sigma:=\oplus_{i=1}^{m}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$,
$\iota^{*}\sigma(\pi_{1}(Z))$ is discrete in $\prod_{i=1}^{m}{\rm
GL}_{N}(\mathbb{C})$.

  2. (ii)

For each reductive representation $\tau:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$
with $[\tau]\in\mathfrak{C}(\mathbb{C})$, $\iota^{*}\tau$ is conjugate to some
$\iota^{*}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$.

  3. (iii)

For each $\sigma_{i}^{\scaleto{V\\!H\\!S}{4pt}}$, there exists a reductive
representation $\tau:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ with
$[\tau]\in\mathfrak{C}(\mathbb{C})$ such that $\iota^{*}\tau$ is conjugate to
$\iota^{*}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$.

  4. (iv)

For every $i=1,\ldots,m$, we have

$\displaystyle\cap_{[\varrho]\in\mathfrak{C}}\ker\varrho\subset\ker\sigma_{i}^{{\scaleto{V\\!H\\!S}{4pt}}}$
(3.5)

where $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ varies among all
reductive representations such that $[\varrho]\in\mathfrak{C}(\mathbb{C})$.

###### Proof.

Let $\mathfrak{C}_{1},\ldots,\mathfrak{C}_{\ell}$ be all geometric connected
components of $\mathfrak{C}$ which are defined over $\bar{\mathbb{Q}}$. We can
pick reductive representations $\\{\varrho_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(\bar{\mathbb{Q}})\\}_{i=1,\ldots,\ell}$ such that
$[\varrho_{i}]\in\mathfrak{C}_{i}(\bar{\mathbb{Q}})$ for every $i$. Since
$\pi_{1}(X)$ is finitely generated, there exists a number field $k$ which is a
Galois extension of $\mathbb{Q}$ such that $\varrho_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(k)$ for every $\varrho_{i}$.

Let ${\rm Ar}(k)$ be all archimedean places of $k$ with $w_{1}$ the identity
map. Then for any $w\in{\rm Ar}(k)$ there exists $a\in{\rm Gal}(k/\mathbb{Q})$
such that $w=w_{1}\circ a$. Note that $\mathfrak{C}$ is defined over
$\mathbb{Q}$. Then $\mathfrak{C}$ is invariant under the conjugation $a$.
Therefore, for any $w:k\to\mathbb{C}$ in ${\rm Ar}(k)$, letting
$\varrho_{i,w}:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ be the composition
$w\circ\varrho_{i}$, we have $[\varrho_{i,w}]\in\mathfrak{C}(\mathbb{C})$.

For any $t\in\mathbb{R}^{*}$, we consider the $\mathbb{R}^{*}$-action
$\varrho_{i,w,t}:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ of $\varrho_{i,w}$
defined in § 2.4. Then $\varrho_{i,w,t}$ is also reductive by the arguments in
§ 2.4. Since we assume that $\mathfrak{C}(\mathbb{C})$ is invariant under
$\mathbb{R}^{*}$-action, it follows that
$[\varrho_{i,w,t}]\in\mathfrak{C}(\mathbb{C})$. By Lemma 2.7,
$[\varrho_{i,w,t}]$ is a continuous deformation of $[\varrho_{i,w}]$. Hence
they are in the same geometric connected component of
$\mathfrak{C}(\mathbb{C})$, and by Proposition 3.9 we conclude that
$[\iota^{*}\varrho_{i,w,t}]=[\iota^{*}\varrho_{i,w}]$ for any
$t\in\mathbb{R}^{*}$.

We first assume that $X$ is compact. According to [Sim92], $\lim\limits_{t\to
0}[\varrho_{i,w,t}]$ exists, and there exists a reductive
$\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$ such that
$[\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}]=\lim\limits_{t\to
0}[\varrho_{i,w,t}]$. Moreover, $\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}$
underlies a $\mathbb{C}$-VHS. Therefore,
$[\iota^{*}\varrho_{i,w}]=\lim\limits_{t\to
0}[\iota^{*}\varrho_{i,w,t}]=[\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}]$.
Since $[\varrho_{i,w,t}]\in\mathfrak{C}(\mathbb{C})$ for any
$t\in\mathbb{R}^{*}$, it follows that
$[\varrho_{i,w}^{{\scaleto{V\\!H\\!S}{4pt}}}]\in\overline{\mathfrak{C}}(\mathbb{C})$.
By eq. 3.4, we conclude

$\displaystyle\cap_{[\varrho]\in\mathfrak{C}}\ker\varrho\subset\ker\varrho_{i,w}^{{\scaleto{V\\!H\\!S}{4pt}}}.$
(3.6)

Assume now $X$ is non-compact. As we assume that $\mathfrak{C}$ is closed and
invariant under $\mathbb{R}^{*}$-action, by Theorem 2.8, we can choose a
reductive representation
$\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$ such that

  * •

it underlies a $\mathbb{C}$-VHS;

  * •

$[\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}]$ and $[\varrho_{i,w}]$ are in the
same geometric connected component of $\mathfrak{C}(\mathbb{C})$.

Note that (3.6) is satisfied automatically. By Proposition 3.9, we have
$[\iota^{*}\varrho_{i,w}]=[\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}]$.

In summary, we construct reductive representations
$\\{\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{i=1,\ldots,k;w\in{\rm Ar}(k)}$ in both compact and
non-compact cases. Each of these representations underlies a $\mathbb{C}$-VHS
and satisfies
$[\iota^{*}\varrho_{i,w}]=[\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}]$
and (3.6).

Let $v$ be any non-archimedean place of $k$ and $k_{v}$ be the non-archimedean
completion of $k$ with respect to $v$. Write $\varrho_{i,v}:\pi_{1}(X)\to{\rm
GL}_{N}(k_{v})$ the induced representation from $\varrho_{i}$. By the
construction of $s_{\mathfrak{C}}$, it follows that
$\iota^{*}\varrho_{i,v}(\pi_{1}(Z))$ is bounded. Therefore, we have a
factorization

$\iota^{*}\varrho_{i}:\pi_{1}(Z)\to{\rm GL}_{N}(\mathcal{O}_{k}).$

Note that ${\rm GL}_{N}(\mathcal{O}_{k})\to\prod_{w\in{\rm Ar}(k)}{\rm
GL}_{N}(\mathbb{C})$ is a discrete subgroup by [Zim84, Proposition 6.1.3]. It
follows that for the product representation

$\prod_{w\in{\rm Ar}(k)}\iota^{*}\varrho_{i,w}:\pi_{1}(Z)\to\prod_{w\in{\rm
Ar}(k)}{\rm GL}_{N}(\mathbb{C}),$

its image is discrete.

Since $Z$ is normal, by Theorem 1.7, both $\iota^{*}\varrho_{i,w}$ and
$\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}$ are reductive. Recall that
$[\iota^{*}\varrho_{i,w}]=[\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}]$,
it follows that $\iota^{*}\varrho_{i,w}$ is conjugate to
$\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}$ by Lemma 3.6.
Consequently, $\prod_{w\in{\rm
Ar}(k)}\iota^{*}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}:\pi_{1}(Z)\to{\rm
GL}_{N}(\mathbb{C})$ has discrete image. Consider the product representation
of $\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}$

$\sigma:=\prod_{i=1}^{\ell}\prod_{w\in{\rm
Ar}(k)}\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}:\pi_{1}(X)\to\prod_{i=1}^{\ell}\prod_{w\in{\rm
Ar}(k)}{\rm GL}_{N}(\mathbb{C}).$

Then $\sigma$ underlies a $\mathbb{C}$-VHS and
$\iota^{*}\sigma:\pi_{1}(Z)\to\prod_{i=1}^{\ell}\prod_{w\in{\rm Ar}(k)}{\rm
GL}_{N}(\mathbb{C})$ has discrete image.

Let $\tau:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ be any reductive
representation such that $[\tau]\in\mathfrak{C}(\mathbb{C})$. Then
$[\tau]\in\mathfrak{C}_{i}(\mathbb{C})$ for some $i$. By Proposition 3.9, it
follows that
$[\iota^{*}\tau]=[\iota^{*}\varrho_{i,w_{1}}]=[\iota^{*}\varrho_{i,w_{1}}^{\scaleto{V\\!H\\!S}{4pt}}].$
By Theorems 1.7 and 3.6 once again, $\iota^{*}\tau$ is conjugate to
$\iota^{*}\varrho_{i,w_{1}}^{\scaleto{V\\!H\\!S}{4pt}}$. The proposition is
proved if we let $\\{\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{i=1,\ldots,m}$ be
$\\{\varrho_{i,w}^{\scaleto{V\\!H\\!S}{4pt}}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{i=1,\ldots,\ell;w\in{\rm Ar}(k)}$. ∎

######  Remark 3.13.

In the proof of Proposition 3.12, we take the Galois conjugate of
$\mathfrak{C}\subset M_{\rm B}(X,N)$ under $a\in{\rm Gal}(k/\mathbb{Q})$. If
$\mathfrak{C}$ is not defined over $\mathbb{Q}$, it is not known that
$a(\mathfrak{C})\subset M_{\rm B}(X,N)$ is $\mathbb{R}^{*}$-invariant. This is
why we include the assumption that $\mathfrak{C}$ is defined over $\mathbb{Q}$
in our proof, whereas Eyssidieux disregarded such a condition in [Eys04]. It
seems that this condition should also be necessary in [Eys04].

###  3.2 Infinite monodromy at infinity

When considering a non-compact quasi-projective variety $X$, it is important
to note that the Shafarevich conjecture fails in simple examples. For
instance, take $X:=A\backslash\\{0\\}$, where $A$ is an abelian surface. Its
universal covering $\widetilde{X}$ is $\mathbb{C}^{2}-\Gamma$, where $\Gamma$
is a lattice in $\mathbb{C}^{2}$. Then $\widetilde{X}$ is not holomorphically
convex. Therefore, additional conditions on the fundamental groups at infinity
are necessary to address this issue.

######  Definition 3.14 (Infinity monodromy at infinity).

Let $X$ be a quasi-projective normal variety and let $\overline{X}$ be a
projective compactification of $X$. We say a subset $M\subset M_{\rm
B}(X,N)(\mathbb{C})$ _has infinite monodromy at infinity_ if for any
holomorphic map $\gamma:\mathbb{D}\to\overline{X}$ with
$\gamma^{-1}(\overline{X}\setminus X)=\\{0\\}$, there exists a reductive
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ such that $[\varrho]\in M$ and
$\gamma^{*}\varrho:\pi_{1}(\mathbb{D}^{*})\to{\rm GL}_{N}(\mathbb{C})$ has
infinite image.

Note that Definition 3.14 does not depend on the projective compactification
of $X$.

######  Lemma 3.15.

Let $f:Y\to X$ be a _proper_ morphism between quasi-projective normal
varieties. If $M\subset M_{\rm B}(X,N)(\mathbb{C})$ has infinite monodromy at
infinity, then $f^{*}M\subset M_{\rm B}(Y,N)(\mathbb{C})$ also has infinite
monodromy at infinity.

###### Proof.

We take projective compactification $\overline{X}$ and $\overline{Y}$ of $X$
and $Y$ respectively such that $f$ extends to a morphism
$\bar{f}:\overline{Y}\to\overline{X}$. Let $\gamma:\mathbb{D}\to\overline{Y}$
be any holomorphic map with $\gamma^{-1}(\overline{Y}\setminus Y)=\\{0\\}$.
Then $\bar{f}\circ\gamma:\mathbb{D}\to\overline{X}$ satisfies
$(\bar{f}\circ\gamma)^{-1}(\overline{X}\setminus X)=\\{0\\}$ as $f$ is proper.
Then by Definition 3.14 there exists a reductive $\varrho:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$ such that $[\varrho]\in M$ and
$\gamma^{*}(f^{*}\varrho)=(f\circ\gamma)^{*}\varrho:\pi_{1}(\mathbb{D}^{*})\to{\rm
GL}_{N}(\mathbb{C})$ has infinite image. The lemma follows. ∎

We have a precise local characterization of a representation with infinite
monodromy at infinity.

######  Lemma 3.16.

Consider a smooth quasi-projective variety $X$ along with a smooth projective
compactification $\overline{X}$, where $D:=\overline{X}\backslash X$ is a
simple normal crossing divisor. A set $M\subset M_{\rm B}(X,N)(\mathbb{C})$
has infinite monodromy at infinity is equivalent to the following: for any
$x\in D$, there exists an admissible coordinate $(U;z_{1},\ldots,z_{n})$
centered at $x$ with $U\cap D=(z_{1}\cdots z_{k}=0)$ such that for any
$k$-tuple $(i_{1},\ldots,i_{k})\in\mathbb{Z}_{>0}^{k}$, there exists a
reductive $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ such that
$[\varrho]\in M(\mathbb{C})$ and
$\varrho(\gamma_{1}^{i_{1}}\cdots\gamma_{k}^{i_{k}})\neq 0$, where
$\gamma_{i}$ is the anti-clockwise loop around the origin in the i-th factor
of $U\setminus D\simeq(\mathbb{D}^{*})^{k}\times\mathbb{D}^{n-k}$. For such
condition we will say that $\varrho$ _has infinite monodromy at $x$_.

###### Proof.

For any holomorphic map $f:\mathbb{D}\to\overline{X}$ with
$f^{-1}(D)=\\{0\\}$, let $x:=f(0)$ which lies on $D$. We take an admissible
coordinate $(U;z_{1},\ldots,z_{n})$ centered at $x$ in the lemma. Then
$f({\mathbb{D}_{2\varepsilon}})\subset U$ for some small $\varepsilon>0$. We
can write $f(t)=(f_{1}(t),\ldots,f_{n}(t))$ such that
$f_{1}(0)=\cdots=f_{k}(0)=0$ and $f_{i}(0)\neq 0$ for $i=k+1,\ldots,n$. Denote
by $m_{i}:={\rm ord}_{0}f_{i}$ the vanishing order of $f_{i}(t)$ at $0$.
Consider the anti-clockwise loop $\gamma$ defined by $\theta\mapsto\varepsilon
e^{i\theta}$ which generates $\pi_{1}(\mathbb{D}_{2\varepsilon}^{*})$. Then
$f\circ\gamma$ is homotopy equivalent to
$\gamma_{1}^{m_{1}}\cdots\gamma_{k}^{m_{k}}$ in $\pi_{1}(U\backslash D)$. If
$M$ has infinite monodromy at infinity, by Definition 3.14 there exists a
reductive $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ such that
$[\varrho]\in M(\mathbb{C})$ and $f^{*}\varrho(\gamma)\neq 0$. This is
equivalent to that $\varrho(\gamma_{1}^{m_{1}}\cdots\gamma_{k}^{m_{k}})\neq
0$. The lemma is proved. ∎

Definition 3.14 presents a stringent condition that is not be practically
applicable in many situations. To address this issue, we establish the
following result:

######  Proposition 3.17.

Assume that $\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ is a reductive
representation with $\varrho(\pi_{1}(X))$ is torsion free. Then we can find a
birational morphism $\mu:\overline{X}_{0}\to\overline{X}$ by taking a sequence
of blowing-ups with smooth centers such that

  1. (i)

$\mu$ is an isomorphism over $X$;

  2. (ii)

there exists a Zariski open set $X^{\prime}\subset\overline{X}_{0}$ containing
$X$ such that $\varrho$ extends to a representation $\varrho_{0}$ over
$\pi_{1}(X^{\prime})$;

  3. (iii)

$\varrho_{0}:\pi_{1}(X^{\prime})\to{\rm GL}_{N}(\mathbb{C})$ has infinite
monodromy at infinity.

###### Proof.

Write $D=\sum_{i=1}^{m}D_{i}$ into sum of irreducible components. We first
look at smooth points of $D$. If there exists some irreducible component
$D_{1}$ of $D$ such that the local monodromy of $\varrho$ around $D_{1}$ is
finite (which is thus trivial as $\varrho(\pi_{1}(X))$ is assumed to be
torsion-free), then $\varrho$ extends across the irreducible component of
$D_{1}$. It follows that $\varrho$ extends to a representation
$\pi_{1}(\overline{X}\backslash\cup_{i=2}^{m}D_{i})\to{\rm
GL}_{N}(\mathbb{C})$. We replace $X$ by
$\overline{X}\backslash\cup_{i=2}^{m}D_{i}$. To prove the proposition, we will
use induction as follows.

We first define an index $i(x)$ of $x\in D$ by setting $i(x):=\\#\\{j\mid x\in
D_{j}\\}$. This depends on the compactification of $X$, and the index is
computed with respect to the new boundary divisor if we blow-up the boundary
$D$.

Induction. Assume that there exists an algorithm of the blowing-ups
$\overline{X}_{0}\to\overline{X}$ required in the proposition such that we can
extend $\varrho$ on some Zariski dense open set $X^{\prime}$ of
$\overline{X}_{0}$ containing $X$, and achieve the following: for any point
$x$ in the new boundary $\overline{X}_{0}\backslash X^{\prime}$, $\varrho$ has
infinite monodromy at $x$ if $i(x)\leq k-1$. Here the index of $x$ is computed
with respect to the new boundary divisor $\overline{X}_{0}\backslash
X^{\prime}$.

We know that $k=2$ can be achieved by the above argument without blowing-up
$\overline{X}$.

By induction, we can assume for any $x\in D$ with $i(x)\leq k-1$, $\varrho$
always has infinite monodromy at $x$ in the sense of Lemma 3.16.

We will work on points in $D$ of index $k$ at which $\varrho$ has infinite
monodromy. We need to cover $D_{1}\cup\ldots\cup D_{m}$ by a natural
stratification. For any $J\subset\\{1,\ldots,m\\}$, define

$D_{J}:=\left\\{x\in D_{1}\cup\ldots\cup D_{m}\mid x\in D_{j}\Leftrightarrow
j\in J\right\\}.$

Note that for any point $x\in D_{J}$, its index $i(x)=\\#J$. It is worth
noting that for any connected component $Z$ of $D_{J}$, for each two points
$x,y\in Z$, $\varrho$ has infinite monodromy at $x$ if and only if it has
infinite monodromy at $y$. Therefore, we only have to deal with finitely many
strata whose points have index is $k$.

Without loss of generality, we may assume that $\varrho$ does not have
infinite monodromy at the points of a connected component $Z$ of the strata
$D_{\\{1,\ldots,k\\}}$. Pick any point $x\in Z$. We choose an admissible
coordinate $(U;z_{1},\ldots,z_{n})$ centered at $x$ with $D_{i}\cap
U=(z_{i}=0)$ for $i=1,\ldots,k$ and $D_{j}\cap U=\varnothing$ for
$j=k+1,\ldots,n$. Let $\gamma_{i}$ be the anti-clockwise loop around the
origin in the i-th factor of $U\backslash
D\simeq(\mathbb{D}^{*})^{k}\times\mathbb{D}^{n-k}$. By our assumption, there
exists $(i_{1},\ldots,i_{k})\in\mathbb{Z}_{>0}^{k}$ such that
$\varrho(\gamma_{1}^{i_{1}}\cdots\gamma_{k}^{i_{k}})=0$.

######  Claim 3.18.

If there are some $k$-tuple $(j_{1},\ldots,j_{k})\in\mathbb{Z}_{>0}^{k}$ such
that $\varrho(\gamma_{1}^{j_{1}}\cdots\gamma_{k}^{j_{k}})=0$, then
$(j_{1},\ldots,j_{k})=\ell(i_{1},\ldots,i_{k})$ for some $\ell>0$.

###### Proof.

Assume that the claim does not hold. After reordering $1,\ldots,k$, we can
assume that

$\frac{j_{1}}{i_{1}}=\cdots=\frac{j_{\ell-1}}{i_{\ell-1}}<\frac{j_{\ell}}{i_{\ell}}\leq\cdots\leq\frac{j_{k}}{i_{k}}$

for some $\ell\in\\{2,\ldots,k\\}$. Then
$i_{1}(j_{1},\ldots,j_{k})-j_{1}(i_{1},\ldots,i_{k})=(0,\cdots,0,i_{\ell}^{\prime},\cdots,i_{k}^{\prime})$
with $i_{\ell}^{\prime},\ldots,i_{k}^{\prime}\in\mathbb{Z}_{>0}$. As
$\varrho(\gamma_{1}^{i_{1}}\cdots\gamma_{k}^{i_{k}})=0$, it follows that
$\varrho(\gamma_{\ell}^{i_{\ell}^{\prime}}\cdots\gamma_{k}^{i^{\prime}_{k}})=0$.
Let us define a holomorphic map

$\displaystyle g:\mathbb{D}$ $\displaystyle\to U$ $\displaystyle t$
$\displaystyle\mapsto(\frac{1}{2},\ldots,\frac{1}{2},t^{i_{\ell}^{\prime}},\ldots,t^{i_{k}^{\prime}}).$

Then we have $1\leq i(g(0))\leq k-1$. The loop $\theta\mapsto
g(\frac{1}{2}e^{i\theta})$ is homotopy to
$\gamma_{\ell}^{i^{\prime}_{\ell}}\cdots\gamma_{k}^{i^{\prime}_{k}}$. Hence
$g^{*}\varrho:\pi_{1}(\mathbb{D}^{*})\to{\rm GL}_{N}(\mathbb{C})$ is trivial.
As we assume that $\varrho$ has infinite monodromy at $g(0)$, a contradiction
is obtained. The claim follows. ∎

After reordering $1,\ldots,k$, we can assume that
$i_{1}=\cdots=i_{\ell-1}<i_{\ell}\leq\ldots\leq i_{k}$. Then we have
$2\leq\ell\leq k+1$. Here we make the convention that $i_{1}=\cdots=i_{k}$ if
$\ell=k+1$. Since $\varrho(\pi_{1}(X))$ is torsion-free, we can replace the
tuple $(i_{1},\ldots,i_{k})$ with $\frac{1}{{\rm
gcd}(i_{1},\ldots,i_{k})}(i_{1},\ldots,i_{k})$. This allows us to assume that
${\rm gcd}(i_{1},\ldots,i_{k})=1$. Let $\overline{Z}$ be the closure of $Z$.
Then it is a smooth, connected, closed subvariety of codimension $k$ contained
in $D_{1}\cap\ldots\cap D_{k}$. We proceed by performing the blow-up of
$\overline{Z}$. Let $D^{\prime}_{0}$ represent the exceptional divisor
resulting from the blow-up, and we denote the strict transform of $D_{i}$ as
$D^{\prime}_{i}$. It is important to note that $D^{\prime}_{1}\cap\ldots\cap
D^{\prime}_{k}\cap D_{0}^{\prime}=\varnothing$.

For any $J\subset\\{1,\ldots,k\\}$, we set

$D^{\prime}_{J}:=\left\\{x\in D_{0}^{\prime}\mid x\in
D^{\prime}_{j}\Leftrightarrow j\in J\right\\},$

and

$D_{\varnothing}^{\prime}:=D_{0}^{\prime}\setminus(D_{1}^{\prime}\cup\ldots\cup
D^{\prime}_{k}).$

Note that for any $x\in D^{\prime}_{J}$, its index $i(x)=1+\\#J$. We can
verify that

$\mathcal{S}_{k}:=\\{x\in D_{0}^{\prime}\mid i(x)\leq
k\\}=\cup_{J\subset\\{1,\ldots,k\\}}D_{J}^{\prime}.$

######  Claim 3.19.

For any point $y$ in $\mathcal{S}_{k}$, $\varrho$ has infinite monodromy at
$y$ if and only if $y\notin D_{\\{\ell,\ldots,k\\}}$. Here we make the
convention that $\\{\ell,\ldots,k\\}=\varnothing$ if $\ell=k+1$.

###### Proof.

Write $J=\\{j_{2},\ldots,j_{p}\\}$. We make the convention that
$J=\varnothing$ if $p=1$. Let $(V;w_{1},\ldots,w_{n})$ be an admissible
coordinate centered at $y$ with $V\cap D^{\prime}_{0}=(w_{1}=0)$, $V\cap
D^{\prime}_{j_{i}}=(w_{i}=0)$ for $i=2,\ldots,p$ and $V\cap
D_{q}^{\prime}=\varnothing$ for other irreducible components $D_{q}^{\prime}$
of the boundary divisor. Let $\gamma^{\prime}_{i}$ be the anti-clockwise loop
around the origin in the i-th factor of
$(\mathbb{D}^{*})^{p}\times\mathbb{D}^{n-p}$. We can see that
$\gamma_{1}^{\prime}\sim\gamma_{1}\cdots\gamma_{k}$, and
$\gamma_{i}^{\prime}\sim\gamma_{j_{i}}$ for $i=2,\ldots,p$. Here “$\sim$stands
for homotopy equivalent. Then for any $p$-tuple
$(q_{1},\ldots,q_{p})\in\mathbb{Z}_{>0}^{p}$, writing
$(\gamma^{\prime}_{1})^{q_{1}}\cdots(\gamma^{\prime}_{p})^{q_{p}}\sim\gamma_{1}^{n_{1}}\cdots\gamma_{k}^{n_{k}}$.
An easy computation shows that $(n_{1},\ldots,n_{k})$ is never linear to
$(i_{1},\ldots,i_{k})$. By Claim 3.18 we conclude that
$\varrho((\gamma^{\prime}_{1})^{j_{1}}\cdots(\gamma^{\prime}_{k})^{j_{k}})\neq
0$ if $J\neq\\{\ell,\ldots,k\\}$. The claim is proved. ∎

By the above claim, there are two possibilites:

Case 1: $\\#J<k-1$. In this case, for each $x\in\mathcal{S}_{k}$, one has
$i(x)\leq k-1$. By our induction, we can perform a further sequence of
blowing-ups with smooth centers in the boundary to obtain a birational
morphism $\mu:\overline{X}^{\prime}\to\overline{X}$ such that

  1. (a)

there exists a Zariski open set $X^{\prime}\subset\overline{X}_{0}$ containing
$X$ such that $\varrho$ extends to a representation $\varrho_{0}$ over
$\pi_{1}(X^{\prime})$;

  2. (b)

for any point $x\in\mu^{-1}(\overline{Z})$ with $i(x)\leq k$, $\varrho_{0}$ at
infinite monodromy at $x$.

Case 2: $\\#J=k-1$. In this case, $J=\\{2,\ldots,k\\}$. Pick any point $y\in
D^{\prime}_{J}$. Let $(V;w_{1},\ldots,w_{n})$ be an admissible coordinate
centered at $y$ with $V\cap D^{\prime}_{0}=(w_{1}=0)$ and $V\cap
D^{\prime}_{j}=(w_{j}=0)$ for $j=2,\ldots,k$. Let $\gamma^{\prime}_{i}$ be the
anti-clockwise loop around the origin in the i-th factor of
$(\mathbb{D}^{*})^{k}\times\mathbb{D}^{n-k}$. We can see that
$\gamma_{1}^{\prime}\sim\gamma_{1}\cdots\gamma_{k}$, and
$\gamma_{i}^{\prime}\sim\gamma_{i}$ for $i=2,\ldots,k$. Then for the $k$-tuple
$(j_{1},\ldots,j_{k}):=(i_{1},i_{2}-i_{1},\ldots,i_{k}-i_{1})\in\mathbb{Z}_{>0}^{k}$,
we have
$(\gamma^{\prime}_{1})^{j_{1}}\cdots(\gamma^{\prime}_{k})^{j_{k}}\sim\gamma_{1}^{i_{1}}\cdots\gamma_{k}^{i_{k}}$.
Therefore,
$\varrho((\gamma^{\prime}_{1})^{j_{1}}\cdots(\gamma^{\prime}_{k})^{j_{k}})=0$.
In this case $j_{1}+\cdots+j_{k}<i_{1}+\cdots+i_{k}$. Next, we proceed to blow
up the closure $\overline{D_{J}^{\prime}}$ of $D_{J}^{\prime}$ and iterate the
algorithm described above. This iterative process will terminate after a
finite number of steps, resulting in a birational morphism
$\mu:\overline{X}^{\prime}\to\overline{X}$ that satisfies the properties
described in Items (a) and (b). We repeat this algorithm of blowing-up for all
other connected components $Z$ of $D_{J}$ with $|J|=k$ where $\varrho$ does
not have infinite monodromy at points in $Z$. By establishing and proving the
induction, we complete the proof of the proposition. ∎

###  3.3 Construction of Shafarevich morphism (I)

We will construct the Shafarevich morphism for smooth quasi-projective
varieties $X$ associated to a constructible subsets of $M_{\rm
B}(X,N)(\mathbb{C})$ defined over $\mathbb{Q}$ that is invariant under
$\mathbb{R}^{*}$-action.

######  Theorem 3.20.

Let $X$ be a smooth quasi-projective variety. Let $\mathfrak{C}$ be a
constructible subset of $M_{\rm B}(X,N)(\mathbb{C})$, defined over
$\mathbb{Q}$, such that $\mathfrak{C}$ is invariant under
$\mathbb{R}^{*}$-action. When $X$ is non-compact, we make two additional
assumptions:

  * •

$\mathfrak{C}$ is closed;

  * •

$\mathfrak{C}$ has infinite monodromy at infinity in the sense of Definition
3.14.

Then there is a proper surjective holomorphic fibration ${\rm
sh}_{\mathfrak{C}}:X\to{\rm Sh}_{\mathfrak{C}}(X)$ over a _normal complex
space_ ${\rm Sh}_{\mathfrak{C}}(X)$ such that for any closed subvariety $Z$ of
$X$, ${\rm Sh}_{\mathfrak{C}}(Z)$ is a point if and only if $\varrho({\rm
Im}[\pi_{1}(Z)\to\pi_{1}(X)])$ is finite for any reductive representation
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ such that
$[\varrho]\in\mathfrak{C}$. When $X$ is compact, ${\rm Sh}_{\mathfrak{C}}(X)$
is projective.

###### Proof.

We will divide the proof into two steps. The first step is dedicated to
constructing ${\rm sh}_{\mathfrak{C}}:X\to{\rm Sh}_{\mathfrak{C}}(X)$. In the
second step, we will prove the projectivity of ${\rm Sh}_{\mathfrak{C}}(X)$
when $X$ is compact.

Step 1: constructing the Shafarevich morphism. By Proposition 3.12, there
exist reduction representations
$\\{\sigma^{{\scaleto{V\\!H\\!S}{4pt}}}_{i}:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})\\}_{i=1,\ldots,m}$ that underlie $\mathbb{C}$-VHS such
that, for a morphism $\iota:Z\to X$ from any quasi-projective _normal_ variety
$Z$ with $s_{\mathfrak{C}}\circ\iota(Z)$ being a point, the following
properties hold:

  1. (a)

For $\sigma:=\oplus_{i=1}^{m}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$, the image
$\iota^{*}\sigma(\pi_{1}(Z))$ is discrete in $\prod_{i=1}^{m}{\rm
GL}_{N}(\mathbb{C})$.

  2. (b)

For each reductive $\tau:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ with
$[\tau]\in\mathfrak{C}(\mathbb{C})$, $\iota^{*}\tau$ is conjugate to some
$\iota^{*}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$. Moreover, for each
$\sigma_{i}^{\scaleto{V\\!H\\!S}{4pt}}$, there exists some reductive
representation $\tau:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ with
$[\tau]\in\mathfrak{C}(\mathbb{C})$ such that $\iota^{*}\tau$ is conjugate to
$\iota^{*}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$.

  3. (c)

We have the following inclusion:

$\displaystyle\cap_{[\varrho]\in\mathfrak{C}(\mathbb{C})}\ker\varrho\subset\ker\sigma_{i}^{{\scaleto{V\\!H\\!S}{4pt}}}$
(3.7)

where $\varrho$ varies in all reductive representations such that
$[\varrho]\in\mathfrak{C}(\mathbb{C})$.

Define $H:=\cap_{\varrho}\ker\varrho\cap\ker\sigma$, where
$\varrho:\pi_{1}(X)\to{\rm GL}_{N}(\mathbb{C})$ ranges over all reductive
representation such that $[\varrho]\in\mathfrak{C}(\mathbb{C})$. By (3.7) we
have $H=\cap_{\varrho}\ker\varrho$, where $\varrho:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$ ranges over all reductive representation such that
$[\varrho]\in\mathfrak{C}(\mathbb{C})$. Denote by
$\widetilde{X}_{H}:=\widetilde{X}/H$. Let $\mathscr{D}$ be the period domain
associated with the $\mathbb{C}$-VHS $\sigma$ and let
$p:\widetilde{X}_{H}\to\mathscr{D}$ be the period mapping. We define a
holomorphic map

$\displaystyle\Psi:\widetilde{X}_{H}$ $\displaystyle\to
S_{\mathfrak{C}}\times\mathscr{D},$ (3.8) $\displaystyle z$
$\displaystyle\mapsto(s_{\mathfrak{C}}\circ\pi_{H}(z),p(z))$

where $\pi_{H}:\widetilde{X}_{H}\to X$ denotes the covering map and
$s_{\mathfrak{C}}:X\to S_{\mathfrak{C}}$ is the reduction map defined in
Definition 3.1.

######  Lemma 3.21.

Each connected component of any fiber of $\Psi$ is compact.

###### Proof of Lemma 3.21.

It is equivalent to prove that for any $(t,o)\in
S_{\mathfrak{C}}\times\mathscr{D}$, any connected component of
$\Psi^{-1}(t,o)$ is compact. We fix any $t\in S_{\mathfrak{C}}$.

Step 1: we first assume that each irreducible component of
$(s_{\mathfrak{C}})^{-1}(t)$ is normal. Let $F$ be an irreducible component of
$(s_{\mathfrak{C}})^{-1}(t)$. Then the natural morphism $\iota:F\to X$ is
proper. By Item (a), $\Gamma:=\sigma({\rm Im}\,[{\pi_{1}(F)}\to\pi_{1}(X)])$
is a discrete subgroup of $\prod_{i=1}^{m}{\rm GL}_{N}(\mathbb{C})$.

######  Claim 3.22.

The period mapping $F\to\mathscr{D}/\Gamma$ is proper.

###### Proof.

Although $F$ might be singular, we can still define its period mapping since
it is normal. The definition is as follows: we begin by taking a resolution of
singularities $\mu:E\to F$. Since $F$ is normal, each fiber of $\mu$ is
connected, and we have $\Gamma=\sigma({\rm Im}[\pi_{1}(E)\to\pi_{1}(X)])$. It
is worth noting that $\mathscr{D}/\Gamma$ exists as a complex normal space
since $\Gamma$ is discrete. Now, consider the period mapping
$E\to\mathscr{D}/\Gamma$ for the $\mathbb{C}$-VHS induced $\mu^{*}\sigma$.
This mapping then induces a holomorphic mapping $F\to\mathscr{D}/\Gamma$,
which satisfies the following commutative diagram:

${E}$${F}$${\mathscr{D}/\Gamma}$$\scriptstyle{\mu}$

The resulting holomorphic map $F\to\mathscr{D}/\Gamma$ is the period mapping
for the $\mathbb{C}$-VHS on $F$ induced by $\sigma|_{\pi_{1}(F)}$. To
establish the properness of $F\to\mathscr{D}/\Gamma$, it suffices to prove
that $E\to\mathscr{D}/\Gamma$ is proper. Let $\overline{X}$ be a smooth
projective compactification such that $D:=\overline{X}\backslash X$ is a
simple normal crossing divisor. Given that $E\to X$ is a proper morphism, we
can take a smooth projective compactification $\overline{E}$ of $E$ such that

  * •

the complement $D_{E}:=\overline{E}\setminus E$ is a simple normal crossing
divisor;

  * •

there exists a morphism $j:\overline{E}\to\overline{X}$ such that
$j^{-1}(D)=D_{E}$.

We aim to prove that $j^{*}\sigma:\pi_{1}(E)\to\prod_{i=1}^{m}{\rm
GL}_{N}(\mathbb{C})$ has infinite monodromy at infinity.

Consider any holomorphic map $\gamma:\mathbb{D}\to\overline{E}$ such that
$\gamma^{-1}(D_{E})=\\{0\\}$. Then $(j\circ\gamma)^{-1}(D)=\\{0\\}$. As we
assume that $\mathfrak{C}(\mathbb{C})$ has infinite monodromy at infinity,
there exists a reductive representation $\tau:\pi_{1}(X)\to{\rm
GL}_{N}(\mathbb{C})$ such that $[\tau]\in\mathfrak{C}(\mathbb{C})$ and
$(j\circ\gamma)^{*}\tau(\pi_{1}(\mathbb{D}^{*}))$ is infinite. Using Item (b),
it follows that $j^{*}\tau$ and $j^{*}\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$
are conjugate to each other as $E$ is smooth quasi-projective. As
$\sigma^{\scaleto{V\\!H\\!S}{4pt}}_{i}$ is a direct factor of $\sigma$, it
follows that $(j\circ\gamma)^{*}\sigma(\pi_{1}(\mathbb{D}^{*}))$ is also
infinite. Hence, we conclude that $j^{*}\sigma$ has infinite monodromy at
infinity.

By a theorem of Griffiths (cf. [CMP17, Corollary 13.7.6]), we conclude that
$E\to\mathscr{D}/\Gamma$ is proper. Therefore, $F\to\mathscr{D}/\Gamma$ is
proper. ∎

Take any point $o\in\mathscr{D}$. Note that there is a real Lie group $G_{0}$
which acts holomorphically and transitively on $\mathscr{D}$. Let $V$ be the
compact subgroup that fixes $o$. Thus, we have $\mathscr{D}=G_{0}/V$. Now, let
$Z$ be any connected component of the fiber of $F\to\mathscr{D}/\Gamma$ over
$[o]$. According to Claim 3.22, $Z$ is guaranteed to be compact. We have that
$\sigma({\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)])\subset V\cap\Gamma$. Notably,
$V$ is compact, and $\Gamma$ is discrete. As a result, it follows that
$\sigma({\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)])$ is finite.

######  Claim 3.23.

${\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)]\cap H$ is a finite index subgroup of
${\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)]$.

###### Proof.

By Item (b) and (3.7), we have

$\displaystyle\ker\sigma\cap{\rm Im}\,[\pi_{1}(F)\to\pi_{1}(X)]=H\cap{\rm
Im}\,[\pi_{1}(F)\to\pi_{1}(X)].$ (3.9)

Since $\sigma({\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)])$ is finite,
$\ker\sigma\cap{\rm Im}\,[\pi_{1}(Z)\to\pi_{1}(X)]$ is a finite index subgroup
of ${\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)]$. The claim follows from (3.9). ∎

Pick any connected component $Z_{0}$ of $\pi_{H}^{-1}(Z)$. Note that ${\rm
Aut}(Z_{0}/Z)=\frac{{\rm Im}\,[{\pi_{1}(Z)}\to\pi_{1}(X)]}{{\rm