# Reductive Shafarevich Conjecture

Ya Deng, Katsutoshi Yamanoi

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Ya Deng, Katsutoshi Yamanoi. Reductive Shafarevich Conjecture. 2023. [https://hal.science/hal-04115373](https://hal.science/hal-04115373)

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Distributed under a Creative Commons Attribution 4.0 International License
# REDUCTIVE SHAFAREVICH CONJECTURE

by

Ya Deng & Katsutoshi Yamanoi

With an appendix joint with Ludmil Katzarkov

**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 essence, we prove the holomorphic convexity of the covering of a projective normal variety $X$, which corresponds to the intersection of kernels of reductive representations $\rho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$. Importantly, 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 prove that such Shafarevich morphism is algebraic in the function field level.

## Contents

<table><tr><td>0. Introduction</td><td>2</td></tr><tr><td>0.1. Shafarevich conjecture</td><td>2</td></tr><tr><td>0.2. Main theorems</td><td>3</td></tr><tr><td>0.3. Comparison with [Eys04] and Novelty</td><td>4</td></tr><tr><td>Convention and notation</td><td>5</td></tr><tr><td>Acknowledgments</td><td>5</td></tr><tr><td>1. Technical preliminary</td><td>5</td></tr><tr><td>1.1. Admissible coordinates</td><td>5</td></tr><tr><td>1.2. Tame and pure imaginary harmonic bundles</td><td>6</td></tr><tr><td>1.3. Positive currents on normal complex spaces</td><td>7</td></tr><tr><td>1.4. Holomorphic forms on complex normal spaces</td><td>7</td></tr><tr><td>1.5. The criterion for Kähler classes</td><td>8</td></tr><tr><td>1.6. Some criterion for Stein space</td><td>8</td></tr><tr><td>1.7. Some facts on moduli spaces of rank 1 local systems</td><td>8</td></tr><tr><td>1.8. Absolutely constructible subsets</td><td>9</td></tr><tr><td>1.9. Katzarkov-Eyssidieux reduction and canonical currents</td><td>10</td></tr><tr><td>1.10. The generalization of Katzarkov-Eyssidieux reduction to quasi-projective varieties</td><td>12</td></tr><tr><td>1.11. Simultaneous Sten factorization</td><td>12</td></tr><tr><td>2. Some non-abelian Hodge theories</td><td>13</td></tr><tr><td>2.1. Regular filtered Higgs bundles</td><td>13</td></tr><tr><td>2.2. Pullback of parabolic bundles</td><td>14</td></tr><tr><td>2.3. Functionality of pullback of regular filtered Higgs bundle</td><td>15</td></tr><tr><td>2.4. C*-action and R*-action on character varieties</td><td>17</td></tr><tr><td>3. Construction of the Shafarevich morphism</td><td>18</td></tr></table>

<footnote>**Key words and phrases.** — Reductive Shafarevich conjecture, Hamonic mapping to Bruhat-Tits buildings, period mappings and period domain, variation of Hodge structures.</footnote>
<table><tbody><tr><td>3.1. Factorizing through non-rigidity</td><td>18</td></tr><tr><td>3.2. Infinite monodromy at infinity</td><td>24</td></tr><tr><td>3.3. Construction of Shafarevich morphism (I)</td><td>27</td></tr><tr><td>3.4. Construction of Shafarevich morphism (II)</td><td>31</td></tr><tr><td>3.5. On the algebraicity of the Shafarevich morphism via L<sup>2</sup>-methods</td><td>36</td></tr><tr><td>4. Proof of the reductive Shafarevich conjecture</td><td>38</td></tr><tr><td>4.1. Reduction map of representation into algebraic tori</td><td>38</td></tr><tr><td>4.2. Some criterion for representation into tori</td><td>41</td></tr><tr><td>4.3. Eyssidieux-Simpson Lefschetz theorem and its application</td><td>42</td></tr><tr><td>4.4. A factorization theorem</td><td>46</td></tr><tr><td>4.5. Constructing Kähler classes via representations into non-archimedean fields</td><td>47</td></tr><tr><td>4.6. Holomorphic convexity associated with absolutely constructible subsets</td><td>52</td></tr><tr><td>4.7. Universal covering is Stein</td><td>53</td></tr><tr><td>Appendix A. Shafarevich conjecture for projective normal varieties</td><td>54</td></tr><tr><td>A.1. Absolutely constructible subset (II)</td><td>54</td></tr><tr><td>A.2. Reductive Shafarevich conjecture for normal projective varieties</td><td>55</td></tr><tr><td>References</td><td>56</td></tr></tbody></table>

# 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 $K \subset X$ the set

$$ \{x \in X : |f(x)| \le \sup_K |f|, \forall f \in \mathcal{O}(X)\}, $$

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 $f(x) \ne f(y)$. 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 (=Theorem 3.36 and Proposition 3.43).** — Let $X$ be a quasi-projective normal varieties, and let $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be a reductive representation. Then

(i) there exists a dominant holomorphic map $\text{sh}_\varrho : X \to \text{Sh}_\varrho(X)$ to a complex normal space $\text{Sh}_\varrho(X)$ whose general fibers are connected such that for any closed subvariety $Z \subset X$, $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)])$ is finite if and only if $\text{sh}_\varrho(Z)$ is a point.

Furthermore, when $X$ is smooth and $\varrho(\pi_1(X))$ is torsion free, there exists another smooth quasi-projective variety $X'$ containing $X$ as a Zariski dense open subset such that:

(ii) $\varrho$ extends to a reductive representation $\varrho_0 : \pi_1(X') \to \text{GL}_N(\mathbb{C})$;

(iii) the Shafarevich morphism $\text{sh}_{\varrho_0} : X' \to \text{Sh}_{\varrho_0}(X')$ exists, which is a holomorphic proper fibration;

(iv) $\text{sh}_\varrho = \text{sh}_{\varrho_0}|_X$; namely, we have the following commutative diagram:

<figure><img src="image_2.png" /></figure>

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

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

**Remark 0.2.** — It is noticeable that Theorems A.(i) to 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 Theorem A.(ii) is constructed in [Gri70, Theorem 9.5]. Additionally, Griffiths proved in [Gri70, Theorem 9.6] that the period mapping $p : X' \to \mathcal{D}/\Gamma$ associated with $\varrho_0$ is proper, where $\mathcal{D}$ 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 $\text{sh}_{\varrho_0} : X' \to \text{Sh}_{\varrho_0}(X')$ corresponds to the Stein factorization of the period mapping $p : X' \to \mathcal{D}/\Gamma$, and that Theorem A.(iv) holds.

We conjecture that $\text{Sh}_{\varrho_0}(X')$ is quasi-projective and $\text{sh}_{\varrho_0}$ is an algebraic morphism (cf. Conjecture 3.41). 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 Theorems A.(v) and A.(vi) as supporting evidence for our conjecture at the function field level. It is worth noting that Sommese proved Theorems A.(v) and A.(vi) when $\varrho$ underlies a $\mathbb{Z}$-VHS in [Som78], utilizing $\mathbb{L}^2$-methods. We adopt the same approach in [Som78] to prove Theorems 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 $M_B(X, \text{GL}_N(\mathbb{C}))$, where $X$ represents a projective normal variety.

**Theorem B (=Theorems 4.25 and A.3).** — Let $X$ be a projective normal variety, and let $\mathcal{C}$ be an absolutely constructible subset of $M_B(X, \text{GL}_N(\mathbb{C}))$ as defined in Definitions 1.16
and A.1. We assume that $\mathcal{C}$ is defined on $\mathbb{Q}$. Set $H := \cap_{\varrho} \ker \varrho$, where $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ ranges over all reductive representations such that $[\varrho] \in \mathcal{C}(\mathbb{C})$. Let $\tilde{X}$ be the universal covering of $X$, and denote $\tilde{X}_{\mathcal{C}} := \tilde{X}/H$. Then the complex space $\tilde{X}_{\mathcal{C}}$ is holomorphically convex. In particular, we have

(i) the covering of $X$ corresponding to the intersections of the kernels of all reductive representations of $\pi_1(X)$ in $\text{GL}_N(\mathbb{C})$ is holomorphically convex;

(ii) if $\pi_1(X)$ is a subgroup of $\text{GL}_N(\mathbb{C})$ whose Zariski closure is reductive, then the universal covering $\tilde{X}$ of $X$ is holomorphically convex.

When the representation is large, we have the following result.

**Theorem C (=Theorems 4.26 and A.5).** — Let $X$ and $\mathcal{C}$ be as described in Theorem B. If $\mathcal{C}$ is considered to be large, meaning that for any closed subvariety $Z$ of $X$, there exists a reductive representation $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in \mathcal{C}(\mathbb{C})$ and $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)])$ is infinite, then all intermediate coverings of $X$ between $\tilde{X}$ and $\tilde{X}_{\mathcal{C}}$ 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:

(a) The definition of absolutely constructible subsets (cf. Definitions 1.16 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.

(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. 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 representations of fundamental groups of singular normal varieties. Therefore, the extension of Theorem C to singular varieties allows for more practical applicability in the study of large representations.

**0.3. Comparison with [Eys04] and Novelty.** — It is worth noting that Eyssidieux [Eys04] does not explicitly require absolutely constructible subsets $\mathcal{C}$ 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.

— 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 Theorem 4.21). This application was briefly outlined in [Eys04, Proof of Proposition 5.4.6], but we present a more comprehensive and distinct approach, offering complete and thorough 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 Theorem A.(i). In [Bru23, Theorem B] Brunebarbe claims the existence of the Shafarevich morphism under a stronger assumption of monodromy at infinity and torsion-freeness of the representation, and he does not address the algebraicity of Shafarevich morphisms. It appears that certain arguments related to non-abelian Hodge theories in [Bru23] may have been overlooked (cf. Remark 3.33).

**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 $X^{\text{norm}}$.

— The bold letter Greek letter $\varrho$ (or $\tau$, $\sigma$...) denotes a family of finite reductive representations $\{\varrho_i : \pi_1(X) \to \text{GL}_N(K)\}_{i=1,...,k}$ 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 $\text{Im}[\Lambda^r V \to H^0(X, \Omega_X^r)] \neq 0$.

**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* will be used throughout the paper.

**Definition 1.1.** — (Admissible coordinates) Let $\overline{X}$ be a complex manifold and let $D$ be a simple normal crossing divisor. Let $x$ be a point of $\overline{X}$, and assume that $\{D_j\}_{j=1,...,l}$ be components of $D$ containing $x$. An *admissible coordinate* centered at $x$ is the tuple $(U; z_1, ..., z_n; \varphi)$ (or simply $(U; z_1, ..., z_n)$ if no confusion arises) where

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

— there is a holomorphic isomorphism $\varphi : U \to \mathbb{D}^n$ so that $\varphi(D_j) = (z_j = 0)$ for any $j = 1, ..., l$.
**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, $X = \overline{X} \setminus D$ 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 with $\bar{\partial}_E$ its complex structure, and $\theta: E \to E \otimes \Omega_X^1$ is a holomorphic one form with value in $\text{End}(E)$, 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 with respect to $h$, and by $\theta_h^\dagger$ the adjoint of $\theta$ with respect to $h$. We write $\theta^\dagger$ for $\theta_h^\dagger$ for short. 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 $(U; z_1, \dots, z_n)$ be an admissible coordinate centered at $p$. On $U$, we have the description:

$$ (1.1) \qquad \theta = \sum_{j=1}^{\ell} f_j d \log z_j + \sum_{k=\ell+1}^{n} f_k dz_k. $$

**Definition 1.4 (Tameness).** — Let $t$ be a formal variable. For any $j = 1, \dots, \ell$, the characteristic polynomial $\det(f_j - t) \in \mathcal{O}(U \setminus D)[t]$, 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 $\overline{X} \setminus D$, we prolong $E$ over $\overline{X}$ by a sheaf of $\mathcal{O}_{\overline{X}}$-module $\diamond E_h$ as follows:

$$ \diamond E_h(U) = \{ \sigma \in \Gamma(U \setminus D, E|_{U \setminus D}) \mid |\sigma|_h \lesssim \prod_{i=1}^{\ell} |z_i|^{-\epsilon} \text{ for all } \epsilon > 0 \}. $$

In [Moc07a] Mochizuki proved that $\diamond E_h$ is locally free and that $\theta$ extends to a morphism

$$ \diamond E_h \to \diamond E_h \otimes \Omega_{\overline{X}}(\log D), $$

which we still denote by $\theta$.

**Definition 1.5 (Pure imaginary).** — Let $(E, h, \theta)$ be a tame harmonic bundle on $\overline{X} \setminus D$. The residue $\text{Res}_{D_i}\theta$ 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 $\text{Res}_{D_i}\theta$ are all pure imaginary.

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

**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 $X := \overline{X} \setminus D$. 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.
**1.3. Positive currents on normal complex spaces.** — For this subsection, we refer to [Dem85] for more details.

**Definition 1.7.** — Let $Z$ be an irreducible normal complex space. A upper semi continuous function $\phi: Z \to \mathbb{R} \cup \{-\infty\}$ 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 $\phi|_U$ extends to a psh function on $B$.

A closed positive current with continuous potentials $\omega$ on $Z$ is specified by a data $\{U_i, \phi_i\}_i$ of an open covering $\{U_i\}_i$ 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 $dd^c\phi \ge \omega$ 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 $C^0 \cap PSH_Z/\text{Re}(O_Z)$.

**Definition 1.8.** — Assume $Z$ to be compact. The class of a closed positive current with continuous potentials is its image in $H^1(Z, \text{Re}(O_Z))$.

A class in $H^1(Z, \text{Re}(O_Z))$ 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 $H^1(Z, \text{Re}(O_Z)) = H^{1,1}(Z, \mathbb{R})$. Hence we use abuse of notation to write $H^{1,1}(Z, \mathbb{R})$ instead of $H^1(Z, \text{Re}(O_Z))$ in this paper.

**Lemma 1.9.** — 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 $T = dd^c\varphi$ 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

$$dd^c f_*\varphi = f_*T.$$

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 $f^*S = \frac{1}{\deg f}dd^c(f_*\varphi) \circ f$, the potential of $f^*S$ is continuous. It follows that $f^*S = T$ over the whole $X$. $\square$

**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 1.10.** — Let $X$ be a normal complex space. Let $(A_i)_{i \in I}$ 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 $\Omega_i^p$, is defined by local restrictions of holomorphic $p$-forms on the sets $\Omega_i$ above to $A_i^{\text{reg}}$, where $A_i^{\text{reg}}$ is the smooth locus of $A_i$.

The following fact will be used throughout the paper.

**Lemma 1.11.** — 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 1.10, 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 Ω ⊂ Cm (resp. Ω' ⊂ Cn).

— a holomorphic map $\tilde{f} : \Omega \to \Omega'$ such that $\tilde{f}|_A = f|_A$.

— A holomorphic p-form $\tilde{\omega}$ on $\Omega'$ 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 1.12** ([DHP22, Corollary 2.39]). — Let X be a compact normal Kähler space, $\omega$ a Kähler form on X, and $\alpha \in H_{BC}^{1,1}(X)$, then

(i) $\alpha$ is nef if and only if $\int_V \alpha^k \wedge \omega^{p-k} \ge 0$ for every analytic p-dimensional subvariety $V \subset X$ and for all $0 < k \le p$, and

(ii) $\alpha$ 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 $0 < k \le p$.

**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 1.13** ([Eys04, Proposition 4.1.1]). — Let X be a compact complex normal space and let $\pi: \tilde{X}' \to X$ 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 $\phi: \tilde{X}' \to \mathbb{R}_{\ge 0}$ such that $dd^c\phi \ge \pi^*T$. Then $\tilde{X}'$ 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 $M_B$, $M_{DR}$ and $M_{Dol}$ whose underlying real analytic groups are canonically isomorphic to M. The first is Betti moduli space $M_B := \text{Hom}(\pi_1(X), \mathbb{C}^*)$. The second is De Rham moduli space $M_{DR}$ 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 $M_{Dol}$ is moduli spaces of rank one Higgs bundles on X. Recall that $\text{Pic}^\tau(X)$ denotes the group of line bundles on X whose first Chern classes are torsion. We have

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

For any subset $S \subset M$, let $S_B$, $S_{Dol}$ and $S_{DR}$ denote the corresponding subsets of $M_B$, $M_{DR}$ and $M_{Dol}$.

**Definition 1.14** (Triple torus). — A triple torus is a closed, connected real analytic subgroup $N \subset M$ such that $N_B$, $N_{DR}$, and $N_{Dol}$ 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 $M_B^{\text{tor}}$ are defined over $\overline{\mathbb{Q}}$, and the points of $M_{DR}^{\text{tor}}$ and $M_{Dol}^{\text{tor}}$ are defined over $\overline{\mathbb{K}}$.

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 $X \to A$ 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 $\bar{K}$). 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 $\bar{K}$. We recall the following result in [Sim93, Lemma 2.1].

**Lemma 1.15 (Simpson).** — Let $N \subset M$ be a closed connected subgroup such that $N_B \subset M_B$ is complex analytic and $N_{\text{Dol}} \subset M_{\text{Dol}}$ is an algebraic subgroup. Then there is a connected abelian subvariety $P \subset A$, defined over $\bar{K}$, 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 $\bar{Q}$. The representation scheme of $\pi_1(X)$ is an affine $\bar{Q}$-algebraic scheme described by its functor of points:

$$R(X, G)(\text{Spec } A) := \text{Hom}(\pi_1(X), G(A))$$

for any $\bar{Q}$-algebra A. The character scheme of $\pi_1(X)$ with values in G is the affine scheme

$$M_B(X, G) := R(X, G) // G$$

If $G = \text{GL}_N$, we simply write $M_B(X, N) := M_B(X, \text{GL}_N)$. Simpson constructed a quasi-projective scheme $M_{\text{DR}}(X, G)$, and $M_{\text{Dol}}(X, G)$ over $\ell$. The $\mathbb{C}$-points of $M_{\text{DR}}(X, G)$ are in bijection with the equivalence classes of flat $G$-connections with reductive monodromy. There are natural isomorphisms

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

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

$$p_\sigma : M_{\text{DR}}(X, G) \to M_{\text{DR}}(X^\sigma, G^\sigma).$$

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

**Definition 1.16 (Absolutely constructible subset).** — A subset $\mathcal{C} \subset M_B(X, G)(\mathbb{C})$ is an absolutely constructible subset (resp. absolutely closed subset) if the following conditions are satisfied.

(i) $\mathcal{C}$ is the a $\bar{Q}$-constructible (resp. $\bar{Q}$-closed) subset of $M_B(X, G)$.

(ii) For each $\sigma \in \text{Aut}(\mathbb{C}/\bar{Q})$, there exists a $\bar{Q}$-constructible (resp. $\bar{Q}$-closed) set $\mathcal{C}_\sigma \subset M_B(X^\sigma, G^\sigma)(\mathbb{C})$ such that $\psi^{-1} \circ p_\sigma \circ \psi(\mathcal{C}) = \mathcal{C}^\sigma$.

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

**Remark 1.17.** — 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 $\psi(\mathcal{C})$ is $\bar{Q}$-constructible in $M_{\text{DR}}(X, G)(\mathbb{C})$. This revised definition allows for a broader range of applications, including quasi-projective varieties. In [Sim93, Eys04], the preservation of $\mathcal{C}(\mathbb{C})$ 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 $\mathcal{C}(\mathbb{C})$ 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 1.18 (Budur-Wang, Simpson).** — Let $X$ be a smooth projective variety over $\mathbb{C}$. If $\mathcal{C} \subset M_B(X, 1)(\mathbb{C})$ is an absolute constructible subset, then $\mathcal{C} = \bigcup_{i=1}^m N_i^\circ$ where each $N_i^\circ$ is a Zariski dense open subset of a torsion-translated subtori $N_i$ of $M_B(X, 1)$. 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 $M_B^0(X, 1) \simeq M_B(A, 1)$ of $M_B(A/P_i, 1)$. Here $M_B^0(X, 1)$ denotes the connected component of $M_B^0(X, 1)$ containing the identity.

Absolute constructible subsets are preserved by the following operations:

**Theorem 1.19 (Simpson).** — Let $f : Z \to X$ be a morphism between smooth projective varieties over $\mathbb{C}$ and let $g : G \to G'$ be a morphism between reductive groups over $\mathbb{Q}$. Consider the natural map $i : M_B(X, G) \to M_B(X, G')$ and $j : M_B(X, G) \to M_B(Z, G)$. Then for any absolutely constructible subsets $\mathcal{C} \subset M_B(X, G)(\mathbb{C})$ and $\mathcal{C}' \subset M_B(X, G')(\mathbb{C})$, we have $i(\mathcal{C})$, $i^{-1}(\mathcal{C}')$ and $j(\mathcal{C})$ are all absolutely constructible.

**Example 1.20.** — $M_B(X, G)(\mathbb{C})$, the isolated point in $M_B(X, G)(\mathbb{C})$, and the class of trivial representation in $M_B(X, G)(\mathbb{C})$ 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 1.21 (Katzarkov, Eyssidieux).** — Let $X$ be a projective normal variety, and let $K$ be a non-archimedean local field. Let $\varrho : \pi_1(X) \to \text{GL}_N(K)$ 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 $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)])$ is bounded if and only if $s_\varrho(Z)$ is a point. Moreover, if $X$ is smooth, then the above property holds for $\varrho(\text{Im}[\pi_1(Z) \to \pi_1(X)])$ 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 1.21.* — 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 $\mu^*\varrho : \pi_1(Y) \to \text{GL}_N(K)$ 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 $\mu^*\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(Y)])$.

If $Z$ is an irreducible component of a fiber of $\mu$. Note that $\mu^*\varrho(\text{Im}[\pi_1(Z) \to \pi_1(Y)]) = \{1\}$, 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 $\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(W^{\text{norm}})]$ is a finite index subgroup of $\pi_1(W^{\text{norm}})$. Therefore, $s_\varrho(Z)$ is a point if and only if $s_{\mu^*\varrho}(W)$ is a point, and if and only if $\mu^*\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(Y)]) = \varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)])$ is bounded, and if and only if $\varrho(\text{Im}[\pi_1(W^{\text{norm}}) \to \pi_1(X)])$ is bounded since $\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(W^{\text{norm}})]$ is a finite index subgroup of $\pi_1(W^{\text{norm}})$. $\square$

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 $\pi : X^{\text{sp}} \to X$ with the Galois group $H$, commonly
known as the spectral covering of X (cf. [CDY22, Definition 5.14]). This cover possesses holomorphic 1-forms $\eta_1, \dots, \eta_m \subset H^0(X^{\text{sp}}, \pi^*\Omega_X^1)$, which can be considered as the (1,0)-part of the complexified differential of the $\pi^*\varrho$-equivariant harmonic mapping from $X^{\text{sp}}$ 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 $a : X^{\text{sp}} \to A$ (cf. [CDY22, Definition 5.19]) induced by $\eta_1, \dots, \eta_m$ leads to the Katzarkov-Eyssidieux reduction map $s_{\pi^*\varrho} : X^{\text{sp}} \to S_{\pi^*\varrho}$ for $\pi^*\varrho$. Moreover, we have the following commutative diagram:

<figure><img src="image_3.png" /></figure>

Here $\sigma_\pi$ is also a finite ramified Galois cover with Galois group $H$. Note that there are one forms $\{\eta'_1, \dots, \eta'_m\} \subset H^0(A, \Omega_A^1)$ such that $a^*\eta'_i = \eta_i$. Consider the finite morphism $b : S_{\pi^*\varrho} \to A$. Then we define a positive (1,1)-current $T_{\pi^*\varrho} := b^* \sum_{i=1}^m i \eta'_i \wedge \bar{\eta}'_i$ on $S_{\pi^*\varrho}$. Note that $T_{\pi^*\varrho}$ is invariant under the Galois action $H$. Therefore, by Lemma 1.9 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 1.22 (Canonical current).** — The closed positive (1,1)-current $T_\varrho$ on $S_\varrho$ is called the canonical current of $\varrho$.

More generally, let $\{\varrho_i : \pi_1(X) \to \text{GL}_N(K_i)\}_{i=1,\dots,k}$ be reductive representations where $K_i$ is a non-archimedean local field. We shall denote by the bolded letter $\varrho := \{\varrho_i\}_{i=1,\dots,k}$ be such family of representations. Let $s_\varrho : X \to S_\varrho$ be the Stein factorization of $(s_{\varrho_1}, \dots, s_{\varrho_k}) : X \to S_{\varrho_1} \times \dots \times S_{\varrho_k}$ where $s_{\varrho_i} : X \to S_{\varrho_i}$ denotes the reduction map associated with $\varrho_i$ with $p_i : S_\varrho \to S_{\varrho_i}$. $s_\varrho : X \to S_\varrho$ is called the reduction map for the family $\varrho$ of representations.

**Definition 1.23 (Canonical current II).** — The closed positive (1,1)-current $T_\varrho := \sum_{i=1}^k p_i^* T_{\varrho_i}$ on $S_\varrho$ is called the canonical current of $\varrho$.

**Lemma 1.24** ([Eys04, Lemme 1.4.9 & 3.3.10]). — Let $f : Z \to X$ be a morphism between projective normal varieties and let $\varrho := \{\varrho_i : \pi_1(X) \to \text{GL}_N(K_i)\}_{i=1,\dots,k}$ be a family of reductive representations where $K_i$ is a non-archimedean local field. Then we have

$$ (1.2) \qquad \begin{array}{ccc} Z & \xrightarrow{f} & X \\ \bigdownarrow_{s_{f^*\varrho}} & & \bigdownarrow_{s_\varrho} \\ S_{f^*\varrho} & \xrightarrow{\sigma_f} & S_\varrho \end{array} $$

where $\sigma_f$ is a finite morphism. Here $f^*\varrho = \{f^*\varrho_i\}_{i=1,\dots,k}$ denotes the pull back of the family of $\varrho := \{\varrho_i : \pi_1(X) \to \text{GL}_N(K_i)\}_{i=1,\dots,k}$. Moreover, the following properties hold:

(i) The local potential of $T_\varrho$ is continuous. In particular, for any closed subvariety $W \subset X$, we have

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

(ii) $T_{f^*\varrho} = \sigma_f^* T_\varrho$;

(iii) For every closed subvariety $\Xi \subset S_{f^*\varrho}$, $\{T_\varrho\}^{\dim \Xi} \cdot (\sigma_f(\Xi)) > 0$ if and only if $\{T_{f^*\varrho}\}^{\dim \Xi} > 0$. $\square$
Note that Lemma 1.24.(iii) is a consequence of the first two assertions.

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

**Proposition 1.25** ( [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 $u : \tilde{X} \to \Delta(G)$ be the associated the harmonic mapping, where $\tilde{X}^{\text{un}}$ is the universal covering of $X$. The function $\phi : \tilde{X}^{\text{un}} \to \mathbb{R}_{\ge 0}$ defined by

$$ \phi(x) = 2d^2(u(x), u(x_0)) $$

satisfies the following properties:

(1) $\phi$ descends to a function $\phi_\rho$ on $\tilde{X}_\rho = \tilde{X}^{\text{un}}/\ker(\rho)$.

(2) $dd^c\phi_\varrho \ge (s_\varrho \circ \pi)^*T_\varrho$, where we denote by $\pi : \tilde{X}_\varrho \to X$ the covering map.

(3) $\phi_\rho$ is locally Lipschitz;

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

(5) $dd^c\phi_\varrho^T \ge \nu^*T_\rho$, in particular, it is psh.

□

**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 1.21 to quasi-projective varieties. The theorem we established is stated below.

**Theorem 1.26** ( [CDY22, Theorem 0.10]). — Let $X$ be a complex smooth quasi-projective variety, and let $\varrho : \pi_1(X) \to \text{GL}_N(K)$ 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:

(a) the image $\rho(\text{Im}[\pi_1(T) \to \pi_1(X)])$ is a bounded subgroup of $G(K)$.

(b) For every irreducible component $T_o$ of $T$, the image $\rho(\text{Im}[\pi_1(T_o^{\text{norm}}) \to \pi_1(X)])$ is a bounded subgroup of $G(K)$.

(c) The image $s_\varrho(T)$ is a point.

This result plays a crucial role in this paper as it enables the construction of Shafarevich morphisms for reductive representations over quasi-projective varieties.

**1.11. Simultaneous Sten factorization.** —

**Lemma 1.27.** — Let $V$ be a smooth quasi-projective variety. For $i = 1, 2, \dots$, 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 $i_0 \in \mathbb{Z}_{\ge 2}$ such that for every $i \ge i_0$ 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

$$ E_i = \{(x, x') \in V \times V; p_i(x) = p_i(x')\}. $$

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 $(p_i, p_i)(x, x') = (p_i(x), p_i(x'))$ 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 $i \ge i_0$. 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 1.28.** — 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 $\lambda_1, \dots, \lambda_n \in \Lambda$ such that $f_\infty : V \to S_\infty$ is the quasi-Stein factorization of $(f_1, \dots, f_n) : V \to S_{\lambda_1} \times \dots S_{\lambda_n}$.

*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 1.27, this process should terminate to get $\lambda_1, \dots, \lambda_n \in \Lambda$. We let $S_\infty = W_n$, namely $f_\infty : V \to S_\infty$ is the Stein factorization of $(f_{\lambda_1}, \dots, f_{\lambda_n}) : V \to S_{\lambda_1} \times \dots \times S_{\lambda_n}$.

Now let $\lambda \in \Lambda$. Then by the construction, if $f_\infty(Z)$ is a point, then $(f_{\lambda_1}, \dots, f_{\lambda_n}, f_\lambda)(Z)$ is a point. In particular, $f_\lambda(Z)$ is a point. $\square$

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

**Theorem 1.29.** — Let $X, S$ be complex spaces and $f : X \to S$ be a morphism. Suppose a connected component $F$ of a fibre of $f$ is compact. Then, $F$ has an open neighborhood $V$ such that $f(V)$ is a locally closed analytic subvariety of $S$ and $V \to f(V)$ 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 $e : X \to Y$ which is a proper holomorphic fibration. $\square$

**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 theo-
ries over quasi-projective varieties. Our main objectives in this section are two-fold: firstly,
to establish the functoriality of pullback for regular filtered Higgs bundles (cf. Proposi-
tion 2.5); and secondly, to clarify the $\mathbb{C}^*$ and $\mathbb{R}^*$-action on the character varieties, follow-
ing [Moc06]. This section's significance lies in its essential role in establishing Proposi-
tions 3.12 and 3.32, which serves as a critical cornerstone in the proofs of Theorem A.

**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 $X = \overline{X} \setminus D$ 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 $(E_*, \theta)$ on $(\overline{X}, D)$ is holomorphic vector bundle $E$ on $X$, together with an $\mathbb{R}^\ell$-indexed filtration $aE$ (so-called *parabolic structure*) by locally free subsheaves of $j_*E$ such that

1. $a \in \mathbb{R}^\ell$ and $aE|_X = E$.

2. For $1 \le i \le \ell$, $a_{1+i}E = aE \otimes \mathcal{O}_X(D_i)$, where $\mathbf{1}_i = (0, \dots, 0, 1, 0, \dots, 0)$ with 1 in the $i$-th component.

3. $a+\epsilon E = aE$ for any vector $\epsilon = (\epsilon, \dots, \epsilon)$ with $0 < \epsilon \ll 1$.
4. The set of weights {$\mathbf{a}$} | $\mathbf{a}E/\mathbf{a}-\epsilon E \neq 0$ for any vector $\epsilon = (\epsilon, \dots, \epsilon)$ with $0 < \epsilon \ll 1$} is discrete in $\mathbb{R}^\ell$.

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

$$ \theta : \diamond E \to \Omega_{\bar{X}}^{\frac{1}{\ell}}(\log D) \otimes \diamond E $$

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

$$ (2.1) \qquad \theta(\mathbf{a}E) \subseteq \Omega_{\bar{X}}^{\frac{1}{\ell}}(\log D) \otimes \mathbf{a}E. $$

Denote $\mathbf{0}E$ by $\diamond E$, where $\mathbf{0} = (0, \dots, 0)$. When disregarding the Higgs field, $\mathbf{E}_*$ 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 $\mathcal{C}^\infty$ hermitian metric $h$ over $X$.

Let $U$ be an open subset of $\bar{X}$ with an admissible coordinate $(U; z_1, \dots, z_n)$ 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 $|\sigma|_h \in \mathcal{O}(\prod_{i=1}^\ell |z_i|^{-b_i})$ if there exists a positive number $C$ such that $|\sigma|_h \le C \cdot \prod_{i=1}^\ell |z_i|^{-b_i}$. For any $\mathbf{b} \in \mathbb{R}^\ell$, say $-\mathrm{ord}(\sigma) \le \mathbf{b}$ 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 $\mathbf{b}$, the sheaf $\mathbf{b}E$ is defined as follows:

$$ (2.2) \qquad \Gamma(U, \mathbf{b}E) := \{\sigma \in \Gamma(U\backslash D, E|_{U\backslash D}) \mid -\mathrm{ord}(\sigma) \le \mathbf{b}\}. $$

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

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

**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 conceptual and we shall make precise computations in next subsection.

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

$$ (2.3) \qquad \mathbf{b}L^\mathbf{a} := L \otimes \mathcal{O}_{\bar{X}} \left( \sum_{i=1}^{\ell} |a_i + b_i| D_i \right) $$

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

**Definition 2.3 (Locally abelian parabolic bundle).** — A parabolic sheaf $\mathbf{E}_*$ is a locally abelian parabolic bundle if, in a neighborhood of any point $x \in \bar{X}$ there is an isomorphism between $\mathbf{E}_*$ and a direct sum of parabolic line bundles.

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

$$ (2.4) \qquad f^*a := \left( \sum_{i=1}^{\ell} n_{i1}a_i, \dots, \sum_{i=1}^{\ell} n_{ik}a_i \right) \in \mathbb{R}^k. $$

Then $f^*(L_*^a)$ is defined by setting

$$ (2.5) \qquad \mathfrak{b}(f^*L)^{f^*a} := f^*L \otimes \mathcal{O}_{\overline{Y}} \left( \sum_{j=1}^{k} \lfloor \sum_{i=1}^{\ell} n_{ij}a_i + b_j \rfloor D'_j \right) $$

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

Let $\overline{X}$ be a compact complex manifold. Consider a locally abelian parabolic bundle $\mathcal{E}_*$ defined on $\overline{X}$. We can cover $\overline{X}$ with open subsets $U_1, \dots, U_m$, such that $\mathcal{E}_*|_{U_i}$ can be expressed as a direct sum of parabolic line bundles on each $U_i$.

Using this decomposition, we define the pullback $f^*(\mathcal{E}_*|_{U_i})$ as mentioned earlier. It can be verified that $f^*(\mathcal{E}_*|_{U_i})$ is compatible with $f^*(\mathcal{E}_*|_{U_j})$ whenever $U_i \cap U_j \neq \emptyset$. 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 $f^*\mathcal{E}_*$. In next section, we will see an explicit description of the pullback of regular filtered Higgs bundles induced by tame harmonic bundles.

**2.3. Factoriality 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 $v = (v_1, \dots, v_r)$ be a smooth frame of $E|_{X\backslash D}$. We obtain the $H(r)$-valued function $H(h, v)$ 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 $(z_1, \dots, z_n)$. Let $h$, $E$ and $v$ be as above.

**Definition 2.4.** — A smooth frame $v$ on $X\backslash D$ is called adapted up to log order, if the following inequalities hold over $X\backslash D$:

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

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 3.32.

**Proposition 2.5.** — Consider a morphism $f: \overline{Y} \to \overline{X}$ of smooth projective varieties $\overline{X}$ and $\overline{Y}$. Let $D$ and $D'$ be simple normal crossing divisors on $\overline{X}$ and $\overline{Y}$ respectively. Assume that $f^{-1}(D) \subset D'$. Let $(E, \theta, h)$ be a tame harmonic bundle on $X := \overline{X}\backslash D$. Let $(\mathcal{E}_*, \theta)$ be the regular filtered Higgs bundle defined in § 2.1. Consider the pullback of $f^*\mathcal{E}_*$ defined in § 2.2, which is also a parabolic bundle over $(\overline{Y}, D')$. Then

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

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

*Proof.* — Since this is a local result, we assume that $\overline{X} := \mathbb{D}^n$ and $D := \bigcup_{i=1}^{\ell} \{z_i = 0\}$. Let $\overline{Y} := \mathbb{D}^m$ and $D' := \bigcup_{j=1}^{k} \{w_j = 0\}$. Then, $f^*(z_i) = \prod_{j=1}^{k} w_j^{n_{ij}} g_i$ for some invertible functions $\{g_i\}_{i=1, \dots, \ell} \subset \mathcal{O}(\overline{Y})$.

By [Moc07a, Proposition 8.70], there exists a holomorphic frame $v = (v_1, \dots, v_r)$ of $\mathbb{C}E|_{\overline{X}}$ and $\{a_{ij}\}_{i=1, \dots, r; j=1, \dots, \ell} \subset [0, 1[$ such that if we put $\tilde{v}_i := v_i \cdot \prod_{j=1}^{\ell} |z_j|^{-a_{ij}}$, then for
the smooth frame $\tilde{v} = (\tilde{v}_1, \dots, \tilde{v}_r)$ over $X = \overline{X} \setminus D$, $H(h, \tilde{v})$ 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 $\mathbf{a}_i := (a_{i1}, \dots, a_{il}) \in \mathbb{R}^l$. Consider the parabolic line bundle $(L_i)^{\mathbf{a}_i}$ over $(\overline{X}, D)$ defined in (2.3), namely,

$$ (2.6) \qquad \mathbf{b}(L_i)^{\mathbf{a}_i} := L_i \otimes \mathcal{O}_{\overline{X}} \left( \sum_{j=1}^{\ell} |a_{ij} + b_j| D_j \right) $$

for any $\mathbf{b} \in \mathbb{R}^l$.

**Claim 2.6.** — The parabolic bundles $\mathbf{E}_*$ and $\oplus_{i=1}^r (L_i)^{\mathbf{a}_i}$ are the same. In particular, $\mathbf{E}_*$ is locally abelian.

*Proof.* — By (2.2), for any $\mathbf{b} \in \mathbb{R}^l$, any holomorphic section $\sigma \in \Gamma(\overline{X}, \mathbf{b}E)$ satisfies

$$ |\sigma|_h = \mathcal{O}(\prod_{j=1}^{\ell} |z_j|^{-b_j - \varepsilon}). $$

As $\mathbf{v}$ 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 $\mathbf{g} := (g_1, \dots, g_r)$. Since $H(h, \tilde{v})$ is adapted to log order, it follows that

$$ C^{-1}(-\sum_{j=1}^{\ell} \log |z_j|)^{-M} \cdot \sum_{i=1}^{r} |g_i|^2 \prod_{j=1}^{\ell} |z_j|^{2a_{ij}} \le \bar{g} H(h, \mathbf{v}) g^T = |\sigma|_h^2 = \mathcal{O}(\prod_{j=1}^{\ell} |z_j|^{-2b_i - \varepsilon}) $$

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, $\mathrm{ord}_{D_j} g_i \le -|b_j + a_{ij}|$. This proves that

$$ \mathbf{b}E \subset \oplus_{i=1}^r \mathbf{b}(L_i)^{\mathbf{a}_i}. $$

On the other hand, we consider any section $\sigma \in \Gamma(\overline{X}, \mathbf{b}(L_i)^{\mathbf{a}_i})$. Then $\sigma = gv_i$ for some meromorphic function $g$ defined over $\overline{X}$ such that $\mathrm{ord}_{D_j} g_i \le -|b_j + a_{ij}|$ by (2.6). Therefore, there exists some positive constant $C > 0$ such that

$$ |\sigma|_h^2 = |g|^2 |v_i|_h^2 \le C \prod_{j=1}^{\ell} |z_j|^{-2(b_j + a_{ij})} \cdot |\tilde{v}_i|_h^2 \cdot \prod_{j=1}^{\ell} |z_j|^{2a_{ij}} = C \prod_{j=1}^{\ell} |z_i|^{-2b_j} \cdot |\tilde{v}_i|_h^2 = \mathcal{O}(\prod_{i=1}^{\ell} |z_i|^{-b_i - \varepsilon}). $$

as $|\tilde{v}_i|_h^2 \le C(-\sum_{j=1}^{\ell} \log |z_j|)^M$ for some $C, M > 0$. This implies that

$$ \oplus_{i=1}^r \mathbf{b}(L_i)^{\mathbf{a}_i} \subset \mathbf{b}E. $$

The claim is proved.

<figure><img src="image_4.png" /></figure>

Consider the pullback $f^*\mathbf{v} := (f^*v_1, \dots, f^*v_m)$. Then it is a holomorphic frame of $f^*E|_Y$ where $Y := \overline{Y} \setminus D'$. Note that we have

$$ f^*\tilde{v}_i := f^*v_i \cdot \prod_{j=1}^{\ell} |f^*z_j|^{-a_{ij}} = f^*v_i \cdot \prod_{j=1}^{\ell} \prod_{q=1}^{k} |w_q|^{-n_{jq}a_{ij}} \cdot g'_i $$

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

$$ f^*\mathbf{a}_i := (\sum_{j=1}^{\ell} n_{j1} a_{ij}, \dots, \sum_{j=1}^{\ell} n_{jk} a_{ij}) \in \mathbb{R}^k. $$

Then we have

$$ f^*\tilde{v}_i := f^*v_i \cdot |\mathbf{w}^{-f^*\mathbf{a}_i}| \cdot g'_i. $$
Since $H(h, \tilde{v})$ is adapted to log order, it is easy to check that $H(f^*h, f^*\tilde{v})$ also is adapted to log order. Set $e_i := f^*v_i \cdot |w^{-f^*a_i}|$ for $i = 1, \dots, r$ and $e := (e_1, \dots, e_r)$. Then $e$ is a smooth frame for $f^*E|_Y$. Since $g'_i$ is invertible, it follows that $H(f^*h, e)$ 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

$$ (2.7) \qquad \tilde{E}_* = \bigoplus_{i=1}^r (f^*L_i)^{f^*a_i}, $$

where $(f^*L_i)^{f^*a_i}$ are parabolic line bundles defined by

$$ (2.8) \qquad \mathfrak{b}(f^*L_i)^{f^*a_i} := f^*L_i \otimes \mathcal{O}_{\bar{Y}} \left( \sum_{j=1}^k \lfloor \sum_{q=1}^\ell n_{qj} a_{iq} + b_j \rfloor D'_j \right). $$

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

$$ f^* \mathbf{E}_* := \bigoplus_{i=1}^r f^*(L_i)^{a_i} $$

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

$$ f^*(L_i)^{a_i} = f^*L_i \otimes \mathcal{O}_{\bar{Y}} \left( \sum_{j=1}^{\ell} \lfloor \sum_{q=1}^{\ell} n_{qj} a_{iq} + b_j \rfloor D'_j \right). $$

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

**2.4. $\mathbb{C}^*$-action and $\mathbb{R}^*$-action on character varieties.** — Let $\bar{X}$ be a smooth projective variety and let $D$ be a simple normal crossing divisor on $\bar{X}$. Write $X := \bar{X}\backslash D$. Fix an ample line bundle $L$ on $\bar{X}$. Let $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be a reductive representation. By Theorem 1.6, there is a tame pure imaginary harmonic bundle $(E, \theta, h)$ on $X$ such that $(E, \nabla_h + \theta + \theta_h^\dagger)$ is flat. Here $\nabla_h$ is the Chern connection of $(E, h)$ and $\theta_h^\dagger$ is the adjoint of $\theta$ with respect to $h$. Let $(\mathbf{E}_*, \theta)$ be the prolongation of $(E, \theta)$ on $\bar{X}$ defined in § 2.1. Then $(\mathbf{E}_*, \theta)$ is a $\mu_L$-polystable regular filtered Higgs bundle on $(\bar{X}, D)$ with trivial characteristic numbers. Therefore, for any $t \in \mathbb{C}^*$, $(\mathbf{E}_*, t\theta)$ be also $\mu_L$-polystable regular filtered Higgs bundle on $(\bar{X}, D)$ 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 $(\mathbf{E}_*, t\theta)$. 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 of $\theta$ with respect to $h_t$. Let us denote by $\varrho_t : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ 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 $M_B(X, N)$ and we shall denote it by

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

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 $\{\varrho_t : \pi_1(X) \to \text{GL}_N(\mathbb{C})\}_{t \in \mathbb{C}^*}$. By [Moc06, Proofs of Theorem 10.1 and Lemma 10.2] we have

**Lemma 2.7.** — The map

$$ \Phi : \mathbb{R}^* \to M_B(\pi_1(X), N) \\ t \mapsto [\varrho_t] $$

is continuous. $\Phi(\{t \in \mathbb{R}^* | |t| < 1\})$ is relatively compact in $M_B(\pi_1(X), N)$. $\square$
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 $\varrho_0 : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ and $g_n \in \text{GL}_N(\mathbb{C})$ such that $\lim_{n \to \infty} g_n^* \varrho_{t_n} = \varrho_0$ in the representation variety $\text{R}(\pi_1(X), \text{GL}_N(\mathbb{C}))$. 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 $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ 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 $(E_*, \theta)$ on $(\overline{X}, D)$ is $\mu_L$-polystable with trivial characteristic numbers. Then

$$ (E_*, \theta) = \bigoplus_{j \in \Lambda} (E_{j*}, \theta_j) \otimes \mathbb{C}^{m_j} $$

where $(E_{j*}, \theta_j)$ is $\mu_L$-stable regular filtered Higgs bundle with trivial characteristic numbers. Put $r(\varrho) := \sum_{j \in \Lambda} m_j$. Then $r(\varrho) \le \text{rank } E$. 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 $\varrho(\{t \in \mathbb{C}^* | |t| < 1\})$ is relatively compact, then there exists some $t_n \in \mathbb{R}^*$ which converges to zero such that $\lim_{t_n \to 0} \varrho_t$ exists, denoting by $\varrho_0$. Moreover, $\varrho_0$ corresponds to some tame harmonic bundle. There are two possibilities:

- For each $j \in \Lambda$, $(E_{j*}, t_n\theta_j)$ converges to some $\mu_L$-stable regular filtered Higgs sheaf. Then by [Moc06, Proposition 10.3], $\varrho_0$ corresponds to a $\mathbb{C}$-VHS.

- For some $j \in \Lambda$, $(E_{j*}, t_n\theta_j)$ 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 $(E_{j*}, \theta_j)$ 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 $\mathbb{C}$, a Zariski closed subset of $M_B(X, G)(\mathbb{C})$, where $G$ denotes a complex reductive group. If $\mathbb{C}$ is invariant under the action of $\mathbb{R}^*$ defined above, then each geometrically connected component of $\mathbb{C}(\mathbb{C})$ contains a class $[\varrho]$ such that $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ is a reductive representation that underlies a $\mathbb{C}$-variation of Hodge structures. $\square$

### 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 $\mathbb{C} \subset M_B(X, N)$ be a $\mathbb{Q}$-constructible subset. Since $M_B(X, N)(\mathbb{C})$ is an affine scheme defined over $\mathbb{Z}$ of finite type, $\mathbb{C}$ is defined over some number field $k$.

Let us utilize Lemma 1.28 and Theorem 1.26 to construct a reduction map $s_{\mathbb{C}} : X \to S_{\mathbb{C}}$ associated with $\mathbb{C}$, 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 1.28. Here $\tau: \pi_1(X) \to \text{GL}_N(K)$ 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 1.26.

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

<figure><img src="image_5.png" /></figure>

**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 all non-archimedean local field $L$ and all reductive representation $\tau: \pi_1(X) \to \text{GL}_N(L)$, the image $\tau(\text{Im}[\pi_1(F) \to \pi_1(X)])$ is a bounded subgroup of $\text{GL}_N(L)$.

*Proof.* — By our construction $s_{\tau} = e_{\tau} \circ s_{\mathfrak{C}}$, so $s_{\tau}(F)$ is a single point. Hence by Theorem 1.26, $\tau(\text{Im}[\pi_1(F) \to \pi_1(X)])$ is bounded. $\square$

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 $\{v(f(b)) | b \in B\}$ 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 $Y(K)$. $\square$

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 $\varrho: \pi_1(X) \to \text{GL}_N(K)$ be a (un)bounded representation. Then its semisimplification $\varrho^{ss}: \pi_1(X) \to \text{GL}_N(\bar{K})$ is also (un)bounded.

*Proof.* — Note that there exists some $g \in \text{GL}_N(\bar{K})$ such that

$$ (3.1) \qquad g\varrho g^{-1} = \begin{bmatrix} \varrho_1 & a_{12} & \cdots & a_{1n} \\ 0 & \varrho_2 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \varrho_n \end{bmatrix} $$

where $\varrho_i: \pi_1(X) \to \text{GL}_{N_i}(\bar{K})$ 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

$$ \varrho^{ss} = \begin{bmatrix} \varrho_1 & 0 & \cdots & 0 \\ 0 & \varrho_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \varrho_n \end{bmatrix} $$

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 $\text{GL}_{N_i}(L)$. Since all maximal compact subgroups of $\text{GL}_{N_i}(L)$ are conjugate to $\text{GL}_{N_i}(\mathcal{O}_L)$, then there exists $g_i \in \text{GL}_{N_i}(L)$ such that $g_i \varrho_i g_i^{-1} : \pi_1(X) \to \text{GL}_{N_i}(\mathcal{O}_L)$. Define

$$ (3.2) \qquad \tau := \begin{bmatrix} g_1 & 0 & \cdots & 0 \\ 0 & g_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_n \end{bmatrix} \begin{bmatrix} \varrho_1 & a_{12} & \cdots & a_{1n} \\ 0 & \varrho_2 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \varrho_n \end{bmatrix} \begin{bmatrix} g_1^{-1} & 0 & \cdots & 0 \\ 0 & g_2^{-1} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_n^{-1} \end{bmatrix} $$

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

$$ \tau = \begin{bmatrix} g_1 \varrho_1 g_1^{-1} & h_{12} & \cdots & h_{1n} \\ 0 & g_2 \varrho_2 g_2^{-1} & \cdots & h_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_n \varrho_n g_n^{-1} \end{bmatrix} $$

such that $g_i \varrho_i g_i^{-1} : \pi_1(X) \to \text{GL}_N(\mathcal{O}_L)$ is irreducible. Write

$$ \tau_1 := \begin{bmatrix} g_1 \varrho_1 g_1^{-1} & 0 & \cdots & 0 \\ 0 & g_2 \varrho_2 g_2^{-1} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_n \varrho_n g_n^{-1} \end{bmatrix} $$

and

$$ \tau_2 := \begin{bmatrix} 0 & h_{12} & \cdots & h_{1n} \\ 0 & 0 & \cdots & h_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix} $$

Note that $\tau_2$ is not a group homomorphism but only a map from $\pi_1(X)$ to $\text{GL}_N(L)$.

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 $M(A+B) \ge \min\{M(A), M(B)\}$.

Let $x_1, \dots, x_m$ 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 $\min_{i=1,\dots,m} M(g_i \varrho_i g_i^{-1}(x_i)) \ge 0$. It follows that $M(\tau_1(x_i)) \ge 0$ for each $x_i$. Then for any $x = x_{i_1} \cdots x_{i_l}$,

$$ M(\tau(x)) = M\left(\sum_{j_1, \dots, j_\ell = 1, 2} \tau_{j_1}(x_{i_1}) \cdots \tau_{j_\ell}(x_{i_\ell})\right) \\ \ge \min_{j_1, \dots, j_\ell = 1, 2; \#\{k | j_k = 2\} < n} \{M(\tau_{j_1}(x_{i_1}) \cdots \tau_{j_\ell}(x_{i_\ell}))\}. $$

Note that $\tau_{j_1}(x_{i_1}) \cdots \tau_{j_\ell}(x_{i_\ell}) = 0$ if $\#\{k | j_k = 2\} \ge n$ since $\tau_2(x_i)$ is nipotent. Hence

$$ M(\tau(x)) \ge \min_{j_1, \dots, j_\ell = 1, 2; \#\{k | j_k = 2\} < n} \{M(\tau_{j_1}(x_{i_1}) \cdots \tau_{j_\ell}(x_{i_\ell}))\}. $$

Since $M(\tau_1(x_i)) \ge 0$ for each $x_i$, it follows that $M(\tau_{j_1}(x_{i_1}) \cdots \tau_{j_\ell}(x_{i_\ell}) \ge (n-1)C$ if $\#\{k | j_k = 2\} < n$. Therefore, $M(\tau(x)) \ge (n-1)C$ 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. $\square$

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 $M_B(X, N)$ are in one-to-one correspondence with the conjugate classes of reductive representations $\pi_1(X) \to \text{GL}_N(K)$. More precisely, if $\{\varrho_i : \pi_1(X) \to \text{GL}_N(K)\}_{i=1,2}$ are two linear representations such that $[\varrho_1] = [\varrho_2] \in M_B(X, N)(K)$, then the semisimplification of $\varrho_1$ and $\varrho_2$ are conjugate. $\square$

The following result is thus a consequence of Lemma 3.5.
**Lemma 3.7.** — Let $K$ be a non-archimedean local field. Let $x \in M_B(X, N)(K)$. If $\{\varrho_i : \pi_1(X) \to GL_N(\bar{K})\}_{i=1,2}$ are two linear representations such that $[\varrho_1] = [\varrho_2] = x \in M_B(X, N)(\bar{K})$, 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 $GL_N$, 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 $\varrho_1^{ss} : \pi_1(X) \to GL_N(\bar{K})$ of $\varrho_2^{ss} : \pi_1(X) \to GL_N(\bar{K})$ are conjugate by an element $g \in GL_N(\bar{K})$. Therefore, there exists a finite extension $L$ of $K$ such that $\varrho_i^{ss}$ and $\varrho_i$ are all defined in $L$ and $g \in GL_N(L)$. 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. $\square$

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 $\varrho : \pi_1(X) \to GL_N(\bar{K})$ (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 $M_B(X, N)$. Let $F \subset X$ be a connected Zariski closed subset such that $s_{\mathfrak{C}}(F)$ is a point in $S_{\mathfrak{C}}$. Let $\{\tau_i : \pi_1(X) \to GL_N(\mathbb{C})\}_{i=1,2}$ be reductive representations such that $[\tau_1]$ and $[\tau_2]$ are in the same geometric connected component of $\mathfrak{C}(\mathbb{C})$. Then $\tau_1 \circ \iota : \pi_1(F) \to GL_N(\mathbb{C})$ and $\tau_2 \circ \iota : \pi_1(F) \to GL_N(\mathbb{C})$ are conjugate, where $\iota : \pi_1(F) \to \pi_1(X)$ is the natural map. In other words, if $\mathfrak{C}'$ is a geometric connected component of $\mathfrak{C}$, then $j(\mathfrak{C}')$ is zero-dimensional, where $j : M_B(X, N) \to M_B(F, N)$ is the natural morphism of character varieties.

*Proof.* — Though $F$ might not be normal, it is a finite CW-complex by [Loj64] and thus $\pi_1(F)$ is also finitely presented. Let $M_X$ (resp. $M$) be the moduli space of representations of $\pi_1(X)$ (resp. $\pi_1(F^{\text{norm}})$) in $GL_N$. 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 $R_X(L) = \text{Hom}(\pi_1(X), GL_N(L))$ (resp. $R(L) = \text{Hom}(\pi_1(F), GL_N(L))$) for any field $L/\mathbb{Q}$. Then we have

$$ (3.3) \qquad \begin{tikzcd} R_X \arrow[r, "\pi"] \arrow[d, "\iota"] & M_X \arrow[d, "j"] \\ R \arrow[r, "p"] & M \end{tikzcd} $$

where $\pi : R_X \to M_X$ and $p : R \to M$ are the GIT quotients which are both surjective. For any field extension $K/\mathbb{Q}$ and any $\varrho \in R_X(K)$, we write $[\varrho] := \pi(\varrho) \in M_X(K)$. Write $\mathfrak{R} := \pi^{-1}(\mathfrak{C})$ which is a constructible subset defined over some number field $k$. Then $\tau_i \in \mathfrak{R}(\mathbb{C})$.

**Claim 3.10.** — Let $\mathfrak{R}'$ be any geometric irreducible component of $\mathfrak{R}$. Then $j \circ \pi(\mathfrak{R}')$ is zero dimensional.

*Proof.* — Assume, for the sake of contradiction, that $j \circ \pi(\mathfrak{R}')$ is positive-dimensional. If we replace $k$ by a finite extension, we may assume that $\mathfrak{R}'$ is defined over $k$. Since $M$ is affine $\mathbb{Q}$-scheme of finite type, it follows that there exist a $k$-morphism $\psi : M \to \mathbb{A}^1$ such that the image $\psi \circ j \circ \pi(\mathfrak{R}')$ is Zariski dense in $\mathbb{A}^1$. After replacing $k$ by a finite extension, we can find a locally closed irreducible curve $C \subset \mathfrak{R}'$ such that the restriction $\psi \circ j \circ \pi|_C : C \to \mathbb{A}^1$ 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 $p$ be a prime ideal of the ring of integer $\mathfrak{O}_k$ 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 $\psi \circ j \circ \pi|_C$ :
$C \to \mathbb{A}^1$. 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 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 $\psi \circ j([\varrho]) \notin \psi(M_0)$. Note that $[\varrho \circ \iota] = j([\varrho])$ by (3.3). Hence $[\varrho \circ \iota] \notin M_0$ which implies that $\varrho \circ \iota \notin R_0$. By definition of $R_0$, $\varrho \circ \iota$ is unbounded.

Let $\varrho^{ss}: \pi_1(X) \to \text{GL}_N(\bar{L})$ be the semisimplification of $\varrho$. Then $[\varrho] = [\varrho^{ss}] \in \mathfrak{C}(\bar{L})$ by Lemma 3.6. Therefore, $[\varrho \circ \iota] = [\varrho^{ss} \circ \iota] \in \text{M}(\bar{L})$ by (3.3). By Lemma 3.7, $\varrho^{ss} \circ \iota: \pi_1(F) \to \text{GL}_N(\bar{L})$ is also unbounded. Since $\pi_1(F)$ is finitely generated, there exist a finite extension $L'$ of $L$ such that $\varrho^{ss}$ is defined over $L'$. However, by our definition of $s_{\mathfrak{C}}: X \to S_{\mathfrak{C}}$, $\varrho^{ss} \circ \iota$ is always bounded. We obtain a contradiction and thus $j \circ \pi(\mathfrak{R}')$ 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 $\varrho'$: $\pi_1(X) \to \text{GL}_N(L)$ as well by [Kem78, Corollary 4.3]. By Lemma 3.6 $\varrho'$ is reductive and $[\varrho'] = [\varrho]$. Hence $[\varrho' \circ \iota] = [\varrho \circ \iota] \notin M_0$. Therefore, $\varrho' \circ \iota: \pi_1(F) \to \text{GL}_N(L)$ 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, $\varrho' \circ \iota$ is always bounded. We obtain a contradiction and thus $j \circ \pi(\mathfrak{R}')$ is zero dimensional. $\square$

Let $\tau_i: \pi_1(X) \to \text{GL}_N(\mathbb{C})_{i=1,2}$ be reductive representations such that $[\tau_1]$ and $[\tau_2]$ are contained in the same connected component $\mathfrak{C}'$ of $\mathfrak{C}(\mathbb{C})$. We aim to prove that $j(\mathfrak{C}')$ is a point in $M(\mathbb{C})$.

Consider an irreducible component $\mathfrak{C}''$ of $\mathfrak{C}'$. We can choose an irreducible component $Z$ of $\pi^{-1}(\mathfrak{C}'')$ such that $\pi(Z)$ is dense in $\mathfrak{C}''$. 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, $j(\mathfrak{C}'')$ is also a point in $M(\mathbb{C})$.

Consequently, $j(\mathfrak{C}')$ is a point in $M(\mathbb{C})$. As a result, we have $[\tau_1 \circ \iota] = j([\tau_1]) = j([\tau_2]) = [\tau_2 \circ \iota]$. Since both $\tau_1 \circ \iota$ and $\tau_2 \circ \iota$ are reductive, according to Lemma 3.6, they are conjugate to each other. We have established the proposition. $\square$

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 $M_B(X, N)(\mathbb{C})$. Then we have

$$ (3.4) \qquad \cap_{[\varrho] \in \mathfrak{C}} \ker \varrho = \cap_{[\varrho] \in \bar{\mathfrak{C}}} \ker \varrho $$

where $\varrho$'s are reductive representations of $\pi_1(X)$ into $\text{GL}_N(\mathbb{C})$.

*Proof.* — Let $M_X$ be the moduli space of representation of $\pi_1(X)$ in $\text{GL}_N$. Let $R_X$ be the affine scheme of finite type such that $R(L) = \text{Hom}(\pi_1(X), N)(L)$ 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 $\text{GL}_N(\mathbb{C})$-invariant subset where $\text{GL}_N(\mathbb{C})$ 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 $\text{GL}_N(\mathbb{C})$. 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 $\text{GL}_N(\mathbb{C})$-invariant. Define $Z := \cap_{\gamma \in H} Z_\gamma$. Then $Z$ is also $\text{GL}_N(\mathbb{C})$-invariant. Therefore, $\pi(Z)$ is also a Zariski closed subset of $M$. Note that $\mathfrak{C} \subset \pi(Z)$. Therefore, $\bar{\mathfrak{C}} \subset \pi(Z)$. Note that for any reductive $\varrho: \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in \pi(Z)$, we have $\varrho(\gamma) = 1$ for any $\gamma \in H$. It follows that (3.4) holds. $\square$

Lastly, let us prove the main result of this subsection. This result will serve as a crucial cornerstone in the proofs of Theorems A to C.
**Proposition 3.12.** — Let $X$ be a smooth quasi-projective variety. Let $\mathfrak{C}$ be a constructible subset of $M_B(X, N)(\mathbb{C})$, defined over $\mathbb{Q}$, such that $\mathfrak{C}$ is invariant under $\mathbb{R}^*$-action (for example, $\mathfrak{C}$ is an absolutely constructible subset as defined in Definition 1.16 when $X$ is compact). When $X$ is non-compact, we further assume that $\mathfrak{C}$ is closed. Then there exist reductive representations $\{\sigma_i^{\text{VHS}} : \pi_1(X) \to \text{GL}_N(\mathbb{C})\}_{i=1,\dots,m}$ such that each $\sigma_i^{\text{VHS}}$ underlies a $\mathbb{C}$-VHS, and for any connected Zariski closed subset $Z$ of $X$ with $s_{\mathfrak{C}}(Z)$ a point, the following properties hold:

(i) Let $\sigma = \bigoplus_{i=1}^m \sigma_i^{\text{VHS}}$. Then $\sigma(\text{Im}[\pi_1(Z) \to \pi_1(X)])$ is discrete in $\prod_{i=1}^m \text{GL}_N(\mathbb{C})$.

(ii) For each reductive $\tau : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ with $[\tau] \in \mathfrak{C}(\mathbb{C})$, $\iota^*\tau$ is conjugate to some $\iota^*\sigma_i^{\text{VHS}}$, where $\iota : Z \to X$ is the inclusion map.

(iii) For every $i = 1, \dots, m$, we have

$$ (3.5) \qquad \cap_{[\varrho] \in \mathfrak{C}} \ker \varrho \subset \ker \sigma_i^{\text{VHS}} $$

where $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ varies among all reductive representations such that $[\varrho] \in \mathfrak{C}(\mathbb{C})$.

*Proof.* — Let $\mathfrak{C}_1, \dots, \mathfrak{C}_\ell$ be all geometric connected components of $\mathfrak{C}$ which are defined over $\mathbb{Q}$. We can pick reductive representations $\{\varrho_i : \pi_1(X) \to \text{GL}_N(\mathbb{Q})\}_{i=1,\dots,\ell}$ such that $[\varrho_i] \in \mathfrak{C}_i(\mathbb{Q})$. Since $\pi_1(X)$ is finitely generated, we can enlarge the number field $k$ such that $k$ is a finite Galois extension of $\mathbb{Q}$ and $\rho_i : \pi_1(X) \to \text{GL}_N(k)$ for every $\rho_i$.

Let $\text{Ar}(k)$ be all archimedean places of $k$ with $w_1$ the identity map. Since $k$ is a Galois extension of $\mathbb{Q}$, for any $w \in \text{Ar}(k)$ there exists $a \in \text{Gal}(k/\mathbb{Q})$ such that $w = w_1 \circ a$. Note that $\mathfrak{C}$ is defined over $\mathbb{Q}$. Then $\mathfrak{C}$ is invariant under the conjugation $\sigma$. Therefore, for any $w : k \to \mathbb{C} \in \text{Ar}(k)$, letting $\varrho_{i,w} : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be the composition $w \circ \varrho_i$, $[\varrho_{i,w}] \in \mathfrak{C}(\mathbb{C})$.

For any $t \in \mathbb{R}^*$, we consider the $\mathbb{R}^*$-action $\varrho_{i,w,t} : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ 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 $[t^*\varrho_{i,w,t}] = [t^*\varrho_{i,w}]$ for any $t \in \mathbb{R}^*$.

We first assume that $X$ is compact. According to [Sim92], $\lim_{t \to 0} [\varrho_{i,w,t}]$ exists, and there exists a reductive $\varrho_{i,w}^{\text{VHS}} : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho_{i,w}^{\text{VHS}}] = \lim_{t \to 0} [\varrho_{i,w,t}]$. Moreover, $\varrho_{i,w}^{\text{VHS}}$ underlies a $\mathbb{C}$-VHS. Therefore, $[t^*\varrho_{i,w}] = \lim_{t \to 0} [t^*\varrho_{i,w,t}] = [t^*\varrho_{i,w}^{\text{VHS}}]$. Since $[\varrho_{i,w,t}] \in \mathfrak{C}(\mathbb{C})$ for any $t \in \mathbb{R}^*$, it follows that $[\varrho_{i,w}^{\text{VHS}}] \in \bar{\mathfrak{C}}(\mathbb{C})$. By eq. (3.4), we conclude (3.5).

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 $\varrho_{i,w}^{\text{VHS}} : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that

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

- $[\varrho_{i,w}^{\text{VHS}}]$ and $[\varrho_{i,w}]$ are in the same geometric connected component of $\mathfrak{C}(\mathbb{C})$.

Note that (3.5) is satisfied automatically. By Proposition 3.9, $t^*\varrho_{i,w}$ is conjugate to $t^*\varrho_{i,w}^{\text{VHS}}$.

In summary, we construct reductive representations $\varrho_{i,w}^{\text{VHS}}$ $i = 1, \dots, k$; $w \in \text{Ar}(k)$ in both compact and non-compact cases. Each of these representations underlies a $\mathbb{C}$-VHS and satisfies $[t^*\varrho_{i,w}] = [t^*\varrho_{i,w}^{\text{VHS}}]$ and (3.5).

Let $v$ be any non-archimedean place of $k$ and write $k_v$ to be the non-archimedean completion of $k$ with respect to $v$. Write $\varrho_{i,v} : \pi_1(X) \to \text{GL}_N(k_v)$ 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

$$ \iota^* \varrho_i : \pi_1(Z) \to \text{GL}_N(\mathcal{O}_k). $$
Note that $GL_N(\mathcal{O}_k) \to \prod_{w \in \text{Ar}(k)} \text{GL}_N(\mathbb{C})$ is a discrete subgroup by [Zim84, Proposition 6.1.3]. It follows that for

$$ \prod_{w \in \text{Ar}(k)} \iota^* \varrho_i : \pi_1(Z) \to \prod_{w \in \text{Ar}(k)} \text{GL}_N(\mathbb{C}), $$

its image is discrete.

Since $\iota^* \varrho_{i,w}^{\text{VHS}}$, if follows that $\prod_{w \in \text{Ar}(k)} \iota^* \varrho_{i,w}^{\text{VHS}} : \pi_1(Z) \to \text{GL}_N(\mathbb{C})$ has discrete image. Consider the product representation of $\varrho_{i,w}^{\text{VHS}}$

$$ \sigma := \prod_{i=1}^{\ell} \prod_{w \in \text{Ar}(k)} \varrho_{i,w}^{\text{VHS}} : \pi_1(X) \to \prod_{i=1}^{\ell} \prod_{w \in \text{Ar}(k)} \text{GL}_N(\mathbb{C}). $$

Then $\sigma$ underlies a $\mathbb{C}$-VHS and $\iota^*\sigma : \pi_1(Z) \to \prod_{i=1}^{\ell} \prod_{w \in \text{Ar}(k)} \text{GL}_N(\mathbb{C})$ has discrete image.

Let $\tau : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ 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 $[\iota^*\tau] = [\iota^*\varrho_{i,w_1}] = [\iota^*\varrho_{i,w_1}^{\text{VHS}}]$. Hence $\iota^*\tau$ also underlies a $\mathbb{C}$-VHS, which is conjugate to a direct factor of $\iota^*\sigma$. The proposition is proved. $\square$

**Remark 3.13.** — In the proof of Proposition 3.12, we take the Galois conjugate of $\mathfrak{C} \subset M_B(X, N)$ under $a \in \text{Gal}(k/\mathbb{Q})$. If $\mathfrak{C}$ is not defined over $\mathbb{Q}$, it is not known that $a(\mathfrak{C}) \subset M_B(X, N)$ 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 $X := A\{0\}$, where $A$ is an abelian surface. Its universal covering $\tilde{X}$ is $\mathbb{C}^2 - \Gamma$, where $\Gamma$ is a lattice in $\mathbb{C}^2$. Then $\tilde{X}$ 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 $\bar{X}$ be a projective compactification of $X$. We say a subset $M \subset M_B(X, N)(\mathbb{C})$ has infinite monodromy at infinity if for any holomorphic map $\gamma : \mathbb{D} \to \bar{X}$ with $\gamma^{-1}(\bar{X} \setminus X) = \{0\}$, there exists a reductive $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in M$ and $\gamma^*\varrho : \pi_1(\mathbb{D}^*) \to \text{GL}_N(\mathbb{C})$ 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 $M \subset M_B(X, N)(\mathbb{C})$ has infinite monodromy at infinity, then $f^*M \subset M_B(Y, N)(\mathbb{C})$ also has infinite monodromy at infinity.

*Proof.* — We take projective compactification $\bar{X}$ and $\bar{Y}$ of $X$ and $Y$ respectively such that $f$ extends to a morphism $\bar{f} : \bar{Y} \to \bar{X}$. Let $\gamma : \mathbb{D} \to \bar{Y}$ be any holomorphic map with $\gamma^{-1}(\bar{Y} \setminus Y) = \{0\}$. Then $\bar{f} \circ \gamma : \mathbb{D} \to \bar{X}$ satisfies $(\bar{f} \circ \gamma)^{-1}(\bar{X} \setminus X) = \{0\}$ as $f$ is proper. Then by Definition 3.14 there exists a reductive $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in M$ and $\gamma^*(f^*\varrho) = (f \circ \gamma)^*\varrho : \pi_1(\mathbb{D}^*) \to \text{GL}_N(\mathbb{C})$ has infinite image. The lemma follows. $\square$

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 $\bar{X}$, where $D := \bar{X} \setminus X$ is a simple normal crossing divisor. A set $M \subset M_B(X, N)(\mathbb{C})$ has infinite monodromy at infinity is equivalent to the following: for any $x \in D$, there exists an admissible coordinate $(U; z_1, \dots, z_n)$ centered at $x$ with $U \cap D = (z_1 \cdots z_k = 0)$ such that for any $k$-tuple $(i_1, \dots, i_k) \in \mathbb{Z}_{k>0}^k$, there exists a reductive $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ 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 $f: \mathbb{D} \to \bar{X}$ with $\gamma^{-1}(D) = \{0\}$, let $x := f(0)$ which lies on $D$. We take an admissible coordinate $(U; z_1, \dots, z_n)$ centered at $x$ in the lemma. Then $f(\mathbb{D}_2\varepsilon) \subset U$ for some small $\varepsilon > 0$. Then $f(t) = (f_1(t), \dots, f_n(t))$ with $f_1(0) = \dots = f_k(0) = 0$ and $f_i(0) \neq 0$ for $i = k+1, \dots, n$. Denote by $m_i := \text{ord}_0 f_i$ the vanishing order of $f_i(t)$ at 0. Consider the anti-clockwise loop $\theta \mapsto \varepsilon e^{i\theta}$ which generates $\pi_1(\mathbb{D}_\varepsilon)$. Then $\theta \mapsto f(\varepsilon e^{i\theta})$ is homotopy equivalent to $\gamma_1^{m_1} \cdots \gamma_k^{m_k}$ in $\pi_1(U\setminus D)$. If $M$ has infinite monodromy at infinity, by Definition 3.14 there exists a reductive $\varrho: \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in M(\mathbb{C})$ and $f^*\varrho(e^{i\theta}) \neq 0$. This is equivalent to that $\varrho(\gamma_1^{m_1} \cdots \gamma_k^{m_k}) \neq 0$. The lemma is proved. $\square$

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 $\varrho: \pi_1(X) \to \text{GL}_N(\mathbb{C})$ is a reductive representation with $\varrho(\pi_1(X))$ is torsion free. Then we can find a birational morphism $\mu: \bar{X}_0 \to \bar{X}$ by taking a sequence of blowing-ups with smooth centers such that

(i) $\mu$ is an isomorphism over $X$;

(ii) there exists a Zariski open set $X' \subset \bar{X}_0$ containing $X$ such that $\varrho$ extends to a representation $\varrho_0$ over $\pi_1(X')$;

(iii) $\varrho_0: \pi_1(X') \to \text{GL}_N(\mathbb{C})$ 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 $\pi_1(\bar{X} \setminus \bigcup_{i=2}^m D_i) \to \text{GL}_N(\mathbb{C})$. We replace $X$ by $\bar{X} \setminus \bigcup_{i=2}^m D_i$. To prove the proposition, we will use induction as follows.

We first define an index $i(x)$ of $x \in D$ by setting $i(x) := \#\{j | x \in D_j\}$. 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 $\bar{X}_0 \to \bar{X}$ required in the proposition such that we can extend $\varrho$ on some Zariski dense open set $X'$ of $\bar{X}_0$ containing $X$, and achieve the following: for any point $x$ in the new boundary $\bar{X}_0 \setminus X'$, $\varrho$ has infinite monodromy at $x$ if $i(x) \le k-1$. Here the index of $x$ is computed with respect to the new boundary divisor $\bar{X}_0 \setminus X'$.

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

By induction, we can assume for any $x \in D$ with $i(x) \le k-1$, $\varrho$ 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 $D_1 \cup \dots \cup D_m$ by a natural stratification. For any $J \subset \{1, \dots, m\}$, define

$$D_J := \{x \in D_1 \cup \dots \cup D_m \mid x \in D_j \Leftrightarrow j \in J\}.$$

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 $D_{\{1, \dots, k\}}$. Pick any point $x \in Z$. We choose an admissible coordinate $(U; z_1, \dots, z_n)$ centered at $x$ with $D_i \cap U = (z_i = 0)$ for $i = 1, \dots, k$ and $D_j \cap U = \emptyset$ for $j = k+1, \dots, n$. Let $\gamma_i$ be the anti-clockwise loop around the origin in the i-th factor of $U \setminus D \simeq (\mathbb{D}^*)^k \times \mathbb{D}^{n-k}$. By our assumption, there exists $(i_1, \dots, i_k) \in \mathbb{Z}_{>0}^k$ such that $\varrho(\gamma_1^{i_1} \cdots \gamma_k^{i_k}) = 0$.
**Claim 3.18.** — If there are some $k$-tuple $(j_1, \dots, j_k) \in \mathbb{Z}_{>0}^k$ such that $\varrho(\gamma_1^{j_1} \cdots \gamma_k^{j_k}) = 0$, then $(j_1, \dots, j_k) = \ell(i_1, \dots, i_k)$ for some $\ell > 0$.

*Proof.* — Assume that the claim does not hold. After reordering $1, \dots, k$, we can assume that

$$ \frac{j_1}{i_1} = \cdots = \frac{j_{\ell-1}}{i_{\ell-1}} < \frac{j_\ell}{i_\ell} \le \cdots \le \frac{j_k}{i_k} $$

for some $\ell \in \{2, \dots, k\}$. Then $i_1(j_1, \dots, j_k) - j_1(i_1, \dots, i_k) = (0, \dots, 0, i'_\ell, \dots, i'_k)$ with $i'_\ell, \dots, i'_k \in \mathbb{Z}_{>0}$. As $\varrho(\gamma_1^{i_1} \cdots \gamma_k^{i_k}) = 0$, it follows that $\varrho(\gamma_\ell^{i'_\ell} \cdots \gamma_k^{i'_k}) = 0$. Let us define a holomorphic map

$$ g: \mathbb{D} \to U \\ t \mapsto (\frac{1}{2}, \dots, \frac{1}{2}, t^{i'_{\ell}}, \dots, t^{i'_{k}}). $$

Then we have $1 \le i(g(0)) \le k - 1$. The loop $\theta \mapsto g(\frac{1}{2}e^{i\theta})$ is homotopy to $\gamma_\ell^{i'_\ell} \cdots \gamma_k^{i'_k}$. Hence $g^*\varrho: \pi_1(\mathbb{D}^*) \to \text{GL}_N(\mathbb{C})$ is trivial. As we assume that $\varrho$ has infinite monodromy at $g(0)$, a contradiction is obtained. The claim follows. $\square$

After reordering $1, \dots, k$, we can assume that $i_1 = \cdots = i_{\ell-1} < i_\ell \le \dots \le i_k$. Then we have $2 \le \ell \le k + 1$. 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 $(i_1, \dots, i_k)$ with $\frac{1}{\text{gcd}(i_1, \dots, i_k)}(i_1, \dots, i_k)$. This allows us to assume that $\text{gcd}(i_1, \dots, i_k) = 1$. Let $\bar{Z}$ be the closure of $Z$. Then it is a smooth, connected, closed subvariety of codimension $k$ contained in $D_1 \cap \dots \cap D_k$. We proceed by performing the blow-up of $\bar{Z}$. Let $D'_0$ represent the exceptional divisor resulting from the blow-up, and we denote the strict transform of $D_i$ as $D'_i$. It is important to note that $D'_1 \cap \dots \cap D'_k \cap D'_0 = \emptyset$.

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

$$ D'_J := \{x \in D'_0 | x \in D'_J \Leftrightarrow j \in J\}, $$

and

$$ D'_\emptyset := D'_0 \setminus (D'_1 \cup \dots \cup D'_k). $$

Note that for any $x \in D'_J$, its index $i(x) = 1 + \#J$. We can verify that

$$ S_k := \{x \in D'_0 | i(x) \le k\} = \cup_{J \subset \{1, \dots, k\}} D'_J. $$

**Claim 3.19.** — For any point $y$ in $S_k$, $\varrho$ has infinite monodromy at $y$ if and only if $y \notin D_{\{\ell, \dots, k\}}$. Here we make the convention that $\{\ell, \dots, k\} = \emptyset$ if $\ell = k + 1$.

*Proof.* — Write $J = \{j_2, \dots, j_p\}$. We make the convention that $J = \emptyset$ if $p = 1$. Let $(V; w_1, \dots, w_n)$ be an admissible coordinate centered at $y$ with $V \cap D'_0 = (w_1 = 0)$, $V \cap D'_{j_i} = (w_i = 0)$ for $i = 2, \dots, p$ and $V \cap D'_q = \emptyset$ for other irreducible components $D'_q$ of the boundary divisor. Let $\gamma'_i$ 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 $\gamma'_1 \sim \gamma_1 \cdots \gamma_k$, and $\gamma'_i \sim \gamma_{j_i}$ for $i = 2, \dots, p$. Here “~” stands for homotopy equivalent. Then for any $p$-tuple $(q_1, \dots, q_p) \in \mathbb{Z}_{>0}^p$, writing $(\gamma'_1)^{q_1} \cdots (\gamma'_p)^{q_p} \sim \gamma_1^{q_1} \cdots \gamma_k^{q_k}$. An easy computation shows that $(n_1, \dots, n_k)$ is never linear to $(i_1, \dots, i_k)$. By Claim 3.18 we conclude that $\varrho((\gamma'_1)^{j_1} \cdots (\gamma'_k)^{j_k}) \ne 0$ if $J \ne \{\ell, \dots, k\}$. The claim is proved. $\square$

By the above claim, there are two possibilities:

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

(a) there exists a Zariski open set $X' \subset \bar{X}_0$ containing $X$ such that $\varrho$ extends to a representation $\varrho_0$ over $\pi_1(X')$;

(b) for any point $x \in \mu^{-1}(\bar{Z})$ with $i(x) \le k$, $\varrho_0$ at infinite monodromy at $x$.
Case 2: $\#J = k - 1$. In this case, $J = \{2, \dots, k\}$. Pick any point $y \in D'_J$. Let $(V; w_1, \dots, w_n)$ be an admissible coordinate centered at $y$ with $V \cap D'_0 = (w_1 = 0)$ and $V \cap D'_J = (w_j = 0)$ for $j = 2, \dots, k$. Let $\gamma'_i$ 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 $\gamma'_1 \sim \gamma_1 \cdots \gamma_k$, and $\gamma'_i \sim \gamma_i$ for $i = 2, \dots, k$. Then for the $k$-tuple $(j_1, \dots, j_k) := (i_1, i_2 - i_1, \dots, i_k - i_1) \in \mathbb{Z}_{>0}^k$, we have $(\gamma'_1)^{j_1} \cdots (\gamma'_k)^{j_k} \sim \gamma_1^{j_1} \cdots \gamma_k^{j_k}$. Therefore, $\varrho((\gamma'_1)^{j_1} \cdots (\gamma'_k)^{j_k}) = 0$. In this case $j_1 + \cdots + j_k < i_1 + \cdots + i_k$. Next, we proceed to blow up the closure $\overline{D'_J}$ of $D'_J$ and iterate the algorithm described above. This iterative process will terminate after a finite number of steps, resulting in a birational morphism $\mu: \overline{X}' \to \overline{X}$ 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. $\square$

**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 $M_B(X, N)(\mathbb{C})$ defined over $\mathbb{Q}$ that is invariant under $\mathbb{R}^*$-action.

**Theorem 3.20.** — Let $X$ be a smooth quasi-projective variety. Let $\mathcal{C}$ be a constructible subset of $M_B(X, N)(\mathbb{C})$, defined over $\mathbb{Q}$, such that $\mathcal{C}$ is invariant under $\mathbb{R}^*$-action. When $X$ is non-compact, we make two additional assumptions:

— $\mathcal{C}$ is closed;

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

Then there is a proper surjective fibration $\mathfrak{sh}_\mathcal{C}: X \to \mathfrak{Sh}_\mathcal{C}(X)$ over a normal complex variety $\mathfrak{Sh}_\mathcal{C}(X)$ such that for any closed subvariety $Z$ of $X$, $\mathfrak{Sh}_\mathcal{C}(Z)$ is a point if and only if $\varrho(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$ is finite for any reductive representation $\varrho: \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ such that $[\varrho] \in \mathcal{C}$. When $X$ is compact, $\mathfrak{Sh}_\mathcal{C}(X)$ is projective.

**Proof.** — By Proposition 3.12, there exist reduction representations $\{\sigma_i^{\mathrm{VHS}} : \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})\}_{i=1,\dots,m}$ such that each $\sigma_i^{\mathrm{VHS}}$ underlies a $\mathbb{C}$-VHS, and for any connected Zariski closed subset $Z$ with $s_\mathcal{C}(Z)$ a point, the following properties hold:

(a) Let $\sigma = \bigoplus_{i=1}^m \sigma_i^{\mathrm{VHS}}$. Then $\sigma(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$ is discrete in $\prod_{i=1}^m \mathrm{GL}_N(\mathbb{C})$.

(b) For each reductive $\tau: \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ with $[\tau] \in \mathcal{C}(\mathbb{C})$, $\iota^*\tau$ is conjugate to some $\iota^*\sigma_i^{\mathrm{VHS}}$, where $\iota: Z \to X$ is the inclusion map.

(c) We have the following inclusion:

$$ (3.6) \qquad \cap_{[\varrho] \in \mathcal{C}(\mathbb{C})} \ker \varrho \subset \ker \sigma_i^{\mathrm{VHS}} $$

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

Define $H := \cap_{\varrho} \ker \varrho \cap \ker \sigma$, where $\varrho: \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ ranges over all reductive representation such that $[\varrho] \in \mathcal{C}$. By (3.6) we have $H = \cap_{\varrho} \ker \varrho$, where $\varrho: \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ ranges over all reductive representation such that $[\varrho] \in \mathcal{C}(\mathbb{C})$. Denote by $\tilde{X}_H := \tilde{X}/H$. Let $\mathcal{D}$ be the period domain associated with the $\mathbb{C}$-VHS $\sigma$ and let $p: \tilde{X}_H \to \mathcal{D}$ be the period mapping. We define a holomorphic map

$$ (3.7) \qquad \Psi: \tilde{X}_H \to S_\mathcal{C} \times \mathcal{D}, \\ z \mapsto (s_\mathcal{C} \circ \pi_H(z), p(z)) $$

where $\pi_H: \tilde{X}_H \to X$ denotes the covering map and $s_\mathcal{C}: X \to S_\mathcal{C}$ is the reduction map defined in Definition 3.1.

**Lemma 3.21.** — Each connected component of any fiber of $\Psi$ is compact.

**Proof.** — Fix a reference point $o \in \mathcal{D}$. Note that there is a complex Lie group $G_0$ which is a subgroup of a complex general linear group which acts on $\mathcal{D}$ transitively such that the fixator of $o$ is a compact subgroup $V$. Hence $\mathcal{D} = G_0/V$. For any $t \in S_\mathcal{C}$, let $F$
be a connected component of the fiber $(s_{\mathcal{C}})^{-1}(t)$. By Item (a) the monodromy $\Gamma := \sigma(\mathrm{Im}[\pi_1(F) \to \pi_1(X)])$ is a discrete subgroup of $G_0$.

Given that $F$ is singular and may not be normal or irreducible, there seems to be a gap in the existing literature regarding the definition of the period mapping for the restricted $\mathbb{C}$-VHS over $F$. In this paper, we aim to address this issue and provide a precise definition.

To begin, we utilize the fact that $F$ is a Zariski closed subset of $X$ and refer to [Hof09, Theorem 4.5] for the existence of a tubular neighborhood $U$ of $F$ in $X$ such that $F$ is a deformation retraction of $U$. Consequently, we have an isomorphism $\pi_1(F) \to \pi_1(U)$. Therefore, $\Gamma = \sigma(\mathrm{Im}[\pi_1(U) \to \pi_1(X)])$.

Next, we restrict the $\mathbb{C}$-VHS $\sigma$ over $U$ in order to define a period mapping denoted as $p_U : U \to \mathcal{D}/\Gamma$. We define the period mapping for $F$ as the restriction of $p_U$ to $F$, resulting in $p_U|_F : F \to \mathcal{D}/\Gamma$.

**Claim 3.22.** — The period mapping $F \to \mathcal{D}/\Gamma$ is proper.

*Proof.* — Let $F_o$ be an irreducible component of $F$. We take an embedded resolution of singularities $\mu : Y \to X$ of $F_o$ such that the strict transform $E$ of $F_o$ is smooth and $\mu|_E : E \to F_o$ is birational. Denote $\Gamma_1 := \sigma(\mathrm{Im}[\pi_1(E) \to \pi_1(X)]) \subset \Gamma$. Then

<figure><img src="image_6.png" /></figure>

To prove that $F_o \to \mathcal{D}/\Gamma$ is proper, it suffices to prove that $E \to \mathcal{D}/\Gamma$ is proper. Let $\bar{X}$ be a smooth projective compactification such that $D := \bar{X} \setminus X$ is a simple normal crossing divisor. Given that $E \to X$ is a proper morphism, we can take a smooth projective compactification $\bar{E}$ of $E$ such that

— $D_E := \bar{E} \setminus E$ is a simple normal crossing divisor;

— there is a morphism $j : \bar{E} \to \bar{X}$ such that $j^{-1}(D) = D_E$.

We shall prove that $j^*\sigma : \pi_1(E) \to \prod_{i=1}^m \mathrm{GL}_N(\mathbb{C})$ has infinite monodromy at infinity.

Let $\gamma : \mathbb{D} \to \bar{E}$ be any holomorphic map such that $\gamma^{-1}(D_E) = \{0\}$. Then $(j \circ \gamma)^{-1}(D) = \{0\}$. As we assumed that $\mathcal{C}(\mathbb{C})$ has infinite monodromy at infinity, there exists a reductive representation $\tau : \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ such that $[\tau] \in \mathcal{C}(\mathbb{C})$ and $(j \circ \gamma)^*\tau(\pi_1(\mathbb{D}^*))$ is infinite. By Item (b), $j^*\tau$ is conjugate to some $j^*\sigma_i^{\mathrm{VHS}}$. As $\sigma_i^{\mathrm{VHS}}$ is a direct factor of $\sigma$, it follows that $(j \circ \gamma)^*\sigma(\pi_1(\mathbb{D}^*))$ is infinite. Therefore, $j^*\sigma$ has infinite monodromy at infinity.

By a theorem of Griffiths (cf. [CMP17, Corollary 13.7.6]), we conclude that both $E \to \mathcal{D}/\Gamma_1$ and $E \to \mathcal{D}/\Gamma$ are proper. Therefore, $F_o \to \mathcal{D}/\Gamma$, and thus $F \to \mathcal{D}/\Gamma$ is proper. $\square$

Consider any connected component $Z$ of a fiber in $F \to \mathcal{D}/\Gamma$, which is guaranteed to be compact according to Claim 3.22. We have that $\sigma(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)]) \subset V \cap \Gamma$, where $V$ denotes the fixator of $o \in \mathcal{D}$ under the action of $G_0$ on $\mathcal{D}$. Notably, $V$ is compact, and $\Gamma$ is discrete. As a result, it follows that $\sigma(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$ is finite.

**Claim 3.23.** — $\mathrm{Im}[\pi_1(Z) \to \pi_1(X)] \cap H$ is a finite index subgroup of $\mathrm{Im}[\pi_1(Z) \to \pi_1(X)]$.

*Proof.* — By Item (b) and (3.6), we have

$$ (3.8) \qquad \ker \sigma \cap \mathrm{Im}[\pi_1(F) \to \pi_1(X)] = H \cap \mathrm{Im}[\pi_1(F) \to \pi_1(X)]. $$

Since $\sigma(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$ is finite, $\ker \sigma \cap \mathrm{Im}[\pi_1(Z) \to \pi_1(X)]$ is a finite index subgroup of $\mathrm{Im}[\pi_1(Z) \to \pi_1(X)]$. The claim follows from the above inequality. $\square$

Pick any connected component $Z_0$ of $\pi_H^{-1}(Z)$. Note that $\mathrm{Aut}(Z_0/Z) = \frac{\mathrm{Im}[\pi_1(Z) \to \pi_1(X)]}{\mathrm{Im}[\pi_1(Z) \to \pi_1(X)] \cap H}$. By Claim 3.23, $\mathrm{Aut}(Z_0/Z)$ is finite and thus $Z_0$ is compact. Then $\pi_H^{-1}(Z)$ is a disjoint union of compact subvarieties which are finite étale Galois covers of $Z$ under $\pi_H$ with the Galois group $\frac{\mathrm{Im}[\pi_1(Z) \to \pi_1(X)]}{\mathrm{Im}[\pi_1(Z) \to \pi_1(X)] \cap H}$.
Note that $\Psi(Z_0)$ is the point $(t, o) \in S_{\mathcal{C}} \times \mathcal{D}$ where $\Psi$ is defined in (3.7). Pick a reference point $z \in Z_0$. Then for any $\gamma \in \pi_1(X, z)/H \simeq \text{Aut}(\tilde{X}_H/X)$, $\gamma \cdot Z_0 \neq Z_0$ if and only if $\gamma \notin \text{Im}[\pi_1(Z) \to \pi_1(X)] \cdot H$. Note that $p(\gamma \cdot Z_0) = \sigma(\gamma) \cdot p(Z_0) = \sigma(\gamma) \cdot o$ for every $\gamma \in \pi_1(X, z)/H$, where $p: \tilde{X}_H \to \mathcal{D}$ is the period mapping. It follows that

$$ (3.9) \qquad \Psi^{-1}(t, o) = \cup_{\{\gamma \in \pi_1(X,z)/H | \sigma(\gamma) \in V\}} \gamma \cdot Z_0. $$

Hence each connected component of any fiber of $\Psi$ is compact. Lemma 3.21 is proved. $\square$

As we see in (3.9), each fiber of $\Psi$ might have infinite connected components unless $\sigma(\pi_1(X))$ is discrete. By Theorem 1.29 the set $\tilde{S}_H$ of connected components of fibres of $\Psi$ can be endowed with the structure of a normal complex space such that $\Psi = g \circ \text{sh}_H$ where $\text{sh}_H: \tilde{X}_H \to \tilde{S}_H$ is a proper holomorphic fibration and $g: \tilde{S}_H \to \tilde{S}_{\mathcal{C}} \times \mathcal{D}$ is a holomorphic map. In Claim 3.28 we will prove that each fiber of $g$ is discrete.

**Claim 3.24.** — $\text{sh}_H$ contracts every compact irreducible subvariety of $\tilde{X}_H$.

*Proof.* — If $Z \subset \tilde{X}_H$ is a compact irreducible subvariety, then $W := \pi_H(Z)$ is also a compact irreducible subvariety in $X$ with $\dim Z = \dim W$. Hence $\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(W^{\text{norm}})]$ is a finite. Note that $\rho(\text{Im}[\pi_1(Z) \to \pi_1(X)]) \subset \varrho(\text{Im}[\pi_1(\tilde{X}_H) \to \pi_1(X)]) = \{1\}$ for any reductive representation $\varrho: \pi_1(X) \to \text{GL}_N(\mathbb{C})$ such that $[\varrho] \in \mathcal{C}(\mathbb{C})$. Since for any non archimedean local field $K$ there is an abstract isomorphism $\bar{K} \simeq \mathbb{C}$, it follows that for any reductive $\tau: \pi_1(X) \to \text{GL}_N(K)$ such that $[\tau] \in \mathcal{C}(K)$, one has $\tau(\text{Im}[\pi_1(Z) \to \pi_1(X)]) = \{1\}$. Therefore, $\tau(\text{Im}[\pi_1(W^{\text{norm}}) \to \pi_1(X)])$ is finite which is thus bounded. Hence $W$ is contained in a fiber $F$ of $s_{\mathcal{C}}$ by Definition 3.1 and Theorem 1.26. Note that $\sigma(\text{Im}[\pi_1(Z) \to \pi_1(X)]) = \{1\}$. Hence $\sigma(\text{Im}[\pi_1(W^{\text{norm}}) \to \pi_1(X)])$ is also finite. Recall that $\sigma$ underlies a $\mathbb{C}$-VHS such that $\Gamma := \sigma(\text{Im}[\pi_1(F) \to \pi_1(X)])$ is discrete. It follows that $W$ under the period map $F \to \mathcal{D}/\Gamma$ of $\sigma$ is a point. Hence $Z$ is contracted by $\Psi$. The claim follows. $\square$

By (3.9), we can see that $\text{Aut}(\tilde{X}_H/X) = \pi_1(X)/H$ acts on $\tilde{S}_H$ and such an action is equivariant for the proper holomorphic fibration $\text{sh}_H: \tilde{X}_H \to \tilde{S}_H$.

**Claim 3.25.** — $\pi_1(X)/H$ acts on $\tilde{S}_H$ analytically and properly discontinuous. Namely, for any point $x$ of $\tilde{S}_H$, there exists an open neighborhood $U_x$ of $x$ such that the set

$$ \{\gamma \in \pi_1(X)/H \mid \gamma \cdot U_x \cap U_x \neq \emptyset\} $$

is finite.

*Proof.* — By the proof of Lemma 3.21 and the construction of $\tilde{S}_H$, we note that there is a connected component of some fiber $F$ of $s_{\mathcal{C}}: X \to S_{\mathcal{C}}$ such that for some connected component $Z$ of some fiber of $F \to \mathcal{D}/\Gamma$, one has $\text{sh}_H^{-1}(x) = Z_0$ where $Z_0$ is some connected component of $\pi_H^{-1}(Z)$. Here $\Gamma$ is the image $\sigma(\text{Im}[\pi_1(F) \to \pi_1(X)])$ which is discrete. Note that $Z_0 \to Z$ is a finite Galois étale cover of degree.

As a consequence of [Hof09, Theorem 4.5], there is a connected open neighborhood $U$ of $Z$ such that $\pi_1(Z) \to \pi_1(U)$ is isomorphic. In particular, by Claim 3.23 $\frac{\text{Im}[\pi_1(U) \to \pi_1(X)]}{\text{Im}[\pi_1(U) \to \pi_1(X)] \cap H}$ is finite. Therefore, $\pi_H^{-1}(U)$ is a disjoint union of open sets $\{U_\alpha\}_{\alpha \in I}$ such that

— For each $U_\alpha$, $U_\alpha \to U$ is a finite unramified covering under $\pi_H$.

— Each $U_\alpha$ contains exactly one connected component of $\pi_H^{-1}(Z)$.

We may assume that $Z_0 \subset U_{\alpha_1}$ for some $\alpha_1 \in I$. Recall that for any $\gamma \in \pi_1(X, z)/H \simeq \text{Aut}(\tilde{X}_H/X)$, $\gamma \cdot Z_0 \neq Z_0$ if and only if $\gamma \notin \text{Im}[\pi_1(Z) \to \pi_1(X)] \cdot H$. Therefore, $\gamma \cdot U_{\alpha_1} \cap U_{\alpha_1} \neq \emptyset$ if and only if $\gamma \in \text{Im}[\pi_1(Z) \to \pi_1(X)] \cdot H$. Recall that $\frac{\text{Im}[\pi_1(Z) \to \pi_1(X)]}{\text{Im}[\pi_1(Z) \to \pi_1(X)] \cap H}$ is finite. Therefore, there are only finitely many $\gamma \in \pi_1(X, z)/H$ such that $\gamma \cdot U_{\alpha_1} \cap U_{\alpha_1} \neq \emptyset$. We proved that $\pi_1(X)/H$ acts on $\tilde{S}_H$ properly discontinuous.
Take any $g \in \pi_1(X)/H$. We can consider $g$ as an analytic automorphism of $\tilde{X}_H$. According to Claim 3.24, $\mathrm{sh}_H \circ g : \tilde{X}_H \to \tilde{S}_H$ contracts each fiber of the proper holomorphic fibration $\mathrm{sh}_H : \tilde{X}_H \to \tilde{S}_H$. As a result, it induces a holomorphic map $\tilde{g} : \tilde{S}_H \to \tilde{S}_H$ such that we have the following commutative diagram:

$$ \begin{tikzcd} \tilde{X}_H \arrow[r, "g"] & \tilde{X}_H \arrow[d, "\mathrm{sh}_H"] \arrow[d, "\mathrm{sh}_H"'] \\ \tilde{S}_H \arrow[r, "\tilde{g}"] & \tilde{S}_H \end{tikzcd} $$

It is evident that $\tilde{g} : \tilde{S}_H \to \tilde{S}_H$ corresponds precisely to the action of $g$ on $\tilde{S}_H$. Thus, we conclude that $g$ is an analytic automorphism. Hence, the claim is proved. $\square$

Let $\nu : \pi_1(X)/H \to \mathrm{Aut}(\tilde{S}_H)$ be the induced homomorphism and denote by $\Gamma_0 := \nu(\pi_1(X)/H)$. By Claim 3.25, the quotient $\mathrm{Sh}_\mathcal{C}(X) := \tilde{S}_H/\Gamma_0$ is a complex normal space, and it is compact if $X$ is compact. Since $\mathrm{sh}_H : \tilde{X} \to \tilde{S}_H$ is $\nu$-equivariant, it induces a surjective proper fibration $\mathrm{sh}_\mathcal{C} : X \to \mathrm{Sh}_\mathcal{C}(X)$ from $X$ to a compact normal complex space $\mathrm{Sh}_\mathcal{C}(X)$.

**Claim 3.26.** — For any closed subvariety $Z \subset X$, $\mathrm{sh}_\mathcal{C}(Z)$ is a point if and only if $\mathcal{P}(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$ is finite for any reductive representation $\mathcal{P} : \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ such that $[\mathcal{P}] \in \mathbb{C}$.

*Proof.* — *Proof of “$\Leftarrow$”:* Let $f : Y \to Z$ be a desingularization. Then for any non archimedean local field $K$ and any reductive $\tau : \pi_1(X) \to \mathrm{GL}_N(K)$ with $[\tau] \in \mathbb{C}(K)$, $f^*\tau(\mathrm{Im}[\pi_1(Y) \to \pi_1(X)])$ is also finite which is thus bounded. Hence $f(Y)$ is contained in some fiber $F$ of $s_\mathcal{C}$ by Definition 3.1. By Items (a) and (b), we have $f^*\sigma(\pi_1(Y))$ is finite. Since $f^*\sigma$ has discrete monodromy, $Y$ is mapped to one point by the period mapping $Y \to \mathcal{D}/\Gamma$ of $f^*\sigma$ where $\Gamma := f^*\sigma(\pi_1(Y))$ which is discrete. Therefore, $Z$ is contracted by $\mathrm{sh}_\mathcal{C} : X \to \mathrm{Sh}_\mathcal{C}(X)$.

*Proof of “$\Rightarrow$”:* Assume that $Z \subset X$ is a closed subvariety such that $Z$ is contracted by $\mathrm{sh}_\mathcal{C}$. Then for any connected component $Z'$ of $\pi_H^{-1}(Z)$, it is contracted by $\Psi$. By Lemma 3.21 $Z'$ is contained in some compact subvariety of $\tilde{X}_H$. Since $Z'$ is closed in $\tilde{X}_H$, it is also compact. Hence $Z' \to Z$ is a finite étale cover. It follows that $\rho(\mathrm{Im}[\pi_1(Z') \to \pi_1(X)])$ is a finite index subgroup of $\rho(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$. Note that for $\rho(\mathrm{Im}[\pi_1(Z') \to \pi_1(X)]) = \{1\}$ any reductive $\mathcal{P} : \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ with $[\mathcal{P}] \in \mathbb{C}$. It implies that $\rho(\mathrm{Im}[\pi_1(Z) \to \pi_1(X)])$ is finite for all such $\mathcal{P}$. The claim is proved. $\square$

Therefore, we construct the desired proper surjective holomorphic fibration $\mathrm{sh}_\mathcal{C} : X \to \mathrm{Sh}_\mathcal{C}(X)$. For the remaining part of the proof, we assume that $X$ is compact and focus on establishing the projectivity of $\mathrm{Sh}_\mathcal{C}(X)$.

**Claim 3.27.** — There exists a finite index normal subgroup $N$ of $\pi_1(X)/H$ such that its action on $\tilde{S}_H$ does not have fixed point.

*Proof.* — Since $\sigma(\pi_1(X)) = \sigma(\pi_1(X)/H)$ is finitely generated and linear, by Malcev's theorem, $\pi_1(X)/H$ has a finite index normal subgroup $N$ such that $\sigma(N)$ is torsion free.

Let $x \in \tilde{S}_H$ be an arbitrary point. By Lemma 3.21, we can find a connected component $F$ of some fiber of $s_\mathcal{C} : X \to S_\mathcal{C}$ and a connected component $Z$ of a fiber of the period mapping $F \to \mathcal{D}/\Gamma$ such that $x$ corresponds to some connected component $Z_0$ of $\pi_H^{-1}(Z)$. Here, $\Gamma := \sigma(\mathrm{Im}[\pi_1(F) \to \pi_1(X)])$ which is discrete. Furthermore, for any $\gamma \in \pi_1(X)/H$, $\gamma \cdot Z_0 \neq Z_0$ if and only if $\gamma \notin \mathrm{Im}[\pi_1(Z) \to \pi_1(X)] \cdot H$.

Therefore, if $\nu(\gamma)(x) = x$ for some $\gamma \in N$, it is equivalent to that $\gamma \cdot Z_0 = Z_0$. We thus have $\gamma \in \mathrm{Im}[\pi_1(Z) \to \pi_1(X)] \cdot H$. Note that $\sigma(\mathrm{Im}([\pi_1(Z) \to \pi_1(X)]))$ is finite. Therefore, $\sigma(\gamma) = 1$. It follows from (3.8) that $\gamma \in H \cap \mathrm{Im}[\pi_1(Z) \to \pi_1(X)]$. Hence $\gamma$ is the identity element in $\pi_1(X)/H$. This implies that the action of $N$ on $\tilde{S}_H$ is fixed-point-free. $\square$
Let $Y := \tilde{X}_H/N$. Then $Y \to X$ is a finite Galois étale cover with $Aut(\tilde{X}_H/Y) = N$. Since $\mathrm{sh}_H : \tilde{X}_H \to \tilde{S}_H$ is $\nu$-equivariant, the group $N$ gives rise to a surjective proper fibration $Y \to \tilde{S}_H/\nu(N)$ over a complex normal space $\tilde{S}_H/\nu(N)$. By Claims 3.22 and 3.27, $\nu(N)$ acts on $\tilde{S}_H$ properly continuous and freely and thus the covering $\tilde{S}_H \to \tilde{S}_H/\Gamma$ is étale.

**Claim 3.28.** — Each fiber of $g : \tilde{S}_H \to S_\mathcal{C} \times \mathcal{D}$ is discrete.

*Proof.* — Let $(t, o) \in S_\mathcal{C} \times \mathcal{D}$ be any point. Let $Z$ be any component of the fiber $\Psi^{-1}((t, o))$. By Theorem 1.29, $Z$ has an open neighborhood $U$ such that $\Psi(U)$ is a locally closed analytic subvariety of $S_\mathcal{C} \times \mathcal{D}$ and $\Psi|_U : U \to \Psi(U)$ is proper. Therefore, for the Stein factorization $U \to V \xrightarrow{\pi_V} \Psi(U)$ of $\Psi|_U$, $U \to V$ coincides with $\mathrm{sh}_H|_U : U \to \mathrm{sh}_H(U)$ and $\pi_V : V \to \Psi(U)$ is finite. Therefore, $V$ is an open subset of $\tilde{S}_H$ and $\pi_V : V \to \Psi(U)$ coincides with $g|_V : V \to S_\mathcal{C} \times \mathcal{D}$. Therefore, $V \cap g^{-1}((t, o)) = V \cap (\pi_V)^{-1}(t, o)$ is finite. The claim follows. $\square$

In [Gri70], Griffiths discovered a so-called canonical bundle $K_\mathcal{D}$ on the period domain $\mathcal{D}$, which is invariant under $G_0$. It is endowed with a $G_0$-invariant smooth metric $h_\mathcal{D}$ whose curvature is positive-definite in the horizontal direction. As $\phi : \tilde{S}_H \to \mathcal{D}$ is $\nu(N)$-equivariant, it follows that $\phi^*K_\mathcal{D}$ descends to a line bundle on the quotient $W := \tilde{S}_H/\nu(N)$, denoted by $L_G$. The smooth metric $h_\mathcal{D}$ induces a smooth metric on $L_G$ whose curvature form is denoted by $T$. Let $x \in W$ be a smooth point of $W$ and let $v \in T_{\tilde{S}_H,x}$. Then $|v|_\omega^2 > 0$ if $d\phi(v) \neq 0$.

**Claim 3.29.** — $\mathrm{Sh}_\mathcal{C}(X)$ is a projective normal variety.

*Proof.* — Note that $S_\tau$ is a projective normal variety. We take an ample line bundle $L$ over $S_\tau$. Recall that there is a line bundle $L_G$ on $W$ equipped with a smooth metric $h_G$ such that its curvature form is $T$. Denote by $f : W \to S_\mathcal{C}$ the natural morphism induced by $g : \tilde{S}_H \to S_\mathcal{C} \times \mathcal{D}$. Let $\mu : W' \to W$ be a resolution of singularities of $W$. We take a smooth metric $h$ on $L$ such that its curvature form $i\Theta_h(L)$ is Kähler. Therefore, for the line bundle $\mu^*(L_G \otimes f^*L)$ equipped with the smooth metric $\mu^*(h \otimes f^*h_G)$, its curvature form is strictly positive at some points of $W'$ as $g : \tilde{S}_H \to S_\mathcal{C} \times \mathcal{D}$ is discrete by Claim 3.28. By Demailly’s holomorphic Morse inequality or Siu’s solution for the Grauert-Riemenschneider conjecture, $\mu^*(h \otimes f^*h_G)$ is a big line bundle and thus $W'$ is a Moishezon manifold. Hence $W$ is a Moishezon variety. Moreover, one can check that for irreducible positive dimensional closed subvariety $Z$ of $W$, there exists a smooth point $x$ in $Z$ such that it has a neighborhood $\Omega$ which can be lifted to the covering $\tilde{S}_H$ and $g|_\Omega : \Omega \to S_\mathcal{C} \times \mathcal{D}$ is immersive. It follows that $(if^*\Theta_h(L) + T)|_\Omega$ is strictly positive. Note that

$$(L_G \otimes f^*L)^{\dim Z} \cdot [Z] = \int_{Z^{\mathrm{reg}}} (if^*\Theta_h(L) + T)^{\dim Z} > 0.$$

By the Nakai-Moishezon criterion for Moishezon varieties (cf. [Kol90, Theorem 3.11]), $L_G \otimes f^*L$ is ample and thus $W$ is projective. Recall that $\mathrm{Sh}_\mathcal{C}(X) := \tilde{S}_H/\Gamma_0$ is a quotient of $W := \tilde{S}_H/\nu(N)$ by the finite group $\Gamma_0/\nu(N)$. Therefore, $\mathrm{Sh}_\mathcal{C}(X)$ is also projective. $\square$

We accomplish the proof of the theorem. $\square$

**Definition 3.30 (Shafarevich morphism).** — The proper morphism $\mathrm{sh}_\mathcal{C} : X \to \mathrm{Sh}_\mathcal{C}(X)$ in Theorem 3.20 is called the Shafarevich morphism of $\mathcal{C}$.

**3.4. Construction of Shafarevich morphism (II).** — In the previous subsection, we established the existence of the Shafarevich morphism associated with a constructible subset of $M_B(X, N)(\mathbb{C})$ defined over $\mathbb{Q}$ that are invariant under $\mathbb{R}^*$-action. In this section, we focus on proving an existence theorem for the Shafarevich morphism associated with a single reductive representation, based on Theorem 3.20. Initially, we assume that the representation has infinite monodromy at infinity. However, we will subsequently employ Proposition 3.17 to remove this assumption and establish the more general result.
**Proposition 3.31.** — Let $X$ be a quasi-projective normal variety. Let $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be a reductive representation. Assume that $\varrho$ has infinite monodromy at infinity if $X$ is non-compact. Then there exists a proper surjective holomorphic fibration $\text{sh}_\varrho : X \to \text{Sh}_\varrho(X)$ onto a complex normal space $\text{Sh}_\varrho(X)$ such that for any closed subvariety $Z \subset X$, $\varrho(\text{Im}[\pi_1(Z^{\text{norm}}) \to \pi_1(X)])$ is finite if and only if $\text{sh}_\varrho(Z)$ is a point. If $X$ is compact, then $\text{Sh}_\varrho(X)$ is projective.

We first prove the following crucial result.

**Proposition 3.32.** — Let $X$ be a smooth quasi-projective variety. Let $f : Z \to X$ be a proper morphism from a smooth quasi-projective variety $Z$. Let $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be a reductive representation. Define $M := j_Z^{-1}\{1\}$, where 1 stands for the trivial representation, and $j_Z : M_B(X, N) \to M_B(Z, N)$ is the natural morphism of $\mathbb{Q}$-scheme. Then $M$ is a closed subscheme of $M_B(X, N)$ defined over $\mathbb{Q}$ such that $M(\mathbb{C})$ is invariant under $\mathbb{C}^*$-action.

*Proof.* — We take a smooth projective compactification $\overline{X}$ (resp. $\overline{Z}$) of $X$ (resp. $Z$) such that $D := \overline{X} \setminus X$ (resp. $D_Z := \overline{Z} \setminus Z$) is a simple normal crossing divisor and $f$ extends to a morphism $\tilde{f} : \overline{X} \to \overline{Z}$. Note that the morphism $j_Z$ is a $\mathbb{Q}$-morphism between affine schemes of finite type $M_B(X, N)$ and $M_B(Z, N)$ defined over $\mathbb{Q}$. $M$ is thus a closed subscheme of $M_B(X, N)$ defined over $\mathbb{Q}$. Let $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be a reductive representation such that $[\varrho] \in M(\mathbb{C})$. 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 $\nabla_h + \theta + \theta_h^\dagger$. By definition, $f^*\varrho$ is a trivial representation. Therefore, $f^*\varrho$ corresponds to a trivial harmonic bundle $(\oplus^N\mathcal{O}_Z, 0, h_0)$ where $h_0$ is the canonical metric for the trivial vector bundle $\oplus^N\mathcal{O}_Z$ with zero curvature. By the unicity theorem in [Moc06, Theorem 1.4], $(\oplus^N\mathcal{O}_Z, 0, h_0)$ coincides with $(f^*E, f^*\theta, f^*h)$ with some obvious ambiguity of $h_0$. Therefore, $f^*E = \oplus^N\mathcal{O}_Z$ and $f^*\theta = 0$. In particular, the regular filtered Higgs bundle $(\tilde{E}_*, \tilde{\theta})$ on $(\overline{Z}, D_Z)$ induced by the prolongation of $(f^*E, f^*\theta, f^*h)$ using norm growth defined in § 2.1 is trivial; namely we have $a\tilde{E} = \mathcal{O}_{\overline{Z}}^N \otimes \mathcal{O}_{\overline{Z}}(\sum_{i=1}^\ell a_i D_i')$ for any $a = (a_1, \dots, a_m) \in \mathbb{R}^\ell$ and $\tilde{\theta} = 0$. Here we write $D_Z = \sum_{i=1}^\ell D_i'$.

Let $(\mathbf{E}_*, \theta)$ 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 $(f^*\mathbf{E}_*, f^*\theta)$, which also forms a regular filtered Higgs bundle on $(\overline{Z}, D_Z)$ with trivial characteristic numbers. By virtue of Proposition 2.5, we deduce that $(f^*\mathbf{E}_*, f^*\theta) = (\tilde{E}_*, \tilde{\theta})$. Consequently, it follows that $(f^*\mathbf{E}_*, f^*\theta)$ is trivial. Hence $(f^*\mathbf{E}_*, tf^*\theta)$ is trivial for any $t \in \mathbb{C}^*$.

Fix some ample line bundle $L$ on $\overline{Z}$. It is worth noting that for any $t \in \mathbb{C}^*$, $(\mathbf{E}_*, t\theta)$ 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 $(\mathbf{E}_*, t\theta)$. By Proposition 2.5 once again, the regular filtered Higgs bundle $(f^*\mathbf{E}_*, tf^*\theta)$ is the prolongation of the tame harmonic bundle $(f^*E, tf^*\theta, f^*h_t)$ using norm growth defined in § 2.1. Since $(f^*\mathbf{E}_*, tf^*\theta)$ is trivial for any $t \in \mathbb{C}^*$, by the unicity theorem in [Moc06, Theorem 1.4] once again, it follows that $(\oplus^N\mathcal{O}_Z, 0, h_0)$ coincides with $(f^*E, tf^*\theta, f^*h_t)$ with some obvious ambiguity of $h_0$. 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 $f^*(\nabla_{h_t} + t\theta + \bar{t}\theta_{h_t}^\dagger)$. Therefore, $f^*\varrho_t$ is a trivial representation.

However, it is worth noting that $\varrho_t$ might not be reductive as $(E, t\theta, h_t)$ might not be pure imaginary. Let $\varrho_t^{ss}$ be the semisimplification of $\varrho_t$. Then $[\varrho_t] = [\varrho_t^{ss}]$. Since $f^*\varrho_t$ is a trivial representation, then $f^*\varrho_t^{ss}$ is trivial. The proposition is proved. $\square$

**Remark 3.33.** — It is important to note that, unlike the projective case, the proof of Proposition 3.32 becomes considerably more involved when $X$ is quasi-projective. This complexity arises from the utilization of the functoriality of pullback of regular filtered Higgs bundles, which is established in Proposition 2.5. Lemma 3.32 plays a crucial role in the proof of Proposition 3.31 as it allows us to remove the condition of $\mathbb{R}^*$-invariance in
**Theorem 3.20.** However, we remark that Proposition 3.32 is claimed without a proof in the proof of [Bru23, Lemma 9.3].

*Proof of Proposition 3.31.* — Step 1: We assume that $X$ is smooth. Let $f: Z \to X$ be a proper morphism from a smooth quasi-projective variety $Z$. Then $j_Z: M_B(X, N) \to M_B(Z, N)$ is a morphism of $\mathbb{Q}$-scheme. Define

$$ (3.10) \qquad \mathcal{C} := \bigcap_{\{f:Z \to X | f^*\varrho=1\}} j_Z^{-1}\{1\}, $$

where 1 stands for the trivial representation, and $f: Z \to X$ ranges over all proper morphisms from smooth quasi-projective varieties $Z$ to $X$. Then $\mathcal{C}$ is a zariski closed subset defined over $\mathbb{Q}$ and by Proposition 3.32 $\mathcal{C}(\mathbb{C})$ is invariant under $\mathbb{C}^*$-action. Note that $[\varrho] \in \mathcal{C}(\mathbb{C})$. As we assume that $\varrho$ has infinite monodromy at infinity, conditions in Theorem 3.20 are fulfilled. Therefore, we apply Theorem 3.20 to conclude that the Shafarevich morphism $\mathrm{sh}_{\mathcal{C}}: X \to \mathrm{Sh}_{\mathcal{C}}(X)$ exists. It is a proper holomorphic fibration over a complex normal space.

**Claim 3.34.** — For any proper morphism $f: Z \to X$ from a smooth quasi-projective variety $Z$, $f^*\varrho(\pi_1(Z))$ is finite if and only if $\mathrm{sh}_{\mathcal{C}}(Z)$ is a point.

*Proof.* — *Proof of “$\Leftarrow$”: this follows from the fact that $[\varrho] \in \mathcal{C}(\mathbb{C})$ and Theorem 3.20.*

*Proof of “$\Rightarrow$”:* we take a finite étale cover $Y \to Z$ such that $f^*\varrho(\mathrm{Im}[\pi_1(Y) \to \pi_1(Z)])$ is trivial. Denote by $g: Y \to X$ the composition of $f$ with $Y \to Z$. Then $g$ is proper and $g^*\varrho = 1$. Let $\tau: \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ be any reductive representation such that $[\tau] \in \mathcal{C}(\mathbb{C})$. Then $g^*\tau = 1$ by (3.10). It follows that $f^*\tau(\pi_1(Z))$ is finite. The lemma is proved. $\square$

Let $\mathrm{sh}_{\varrho}: X \to \mathrm{Sh}_{\varrho}(X)$ be $\mathrm{sh}_{\mathcal{C}}: X \to \mathrm{Sh}_{\mathcal{C}}(X)$. The proposition is proved if $X$ is smooth.

**Step 2:** *We does not assume that $X$ is smooth.* We take a desingularization $\mu: Y \to X$. Then $\mu^*\varrho: \pi_1(Y) \to \mathrm{GL}_N(\mathbb{C})$ is also a reductive representation. By Lemma 3.16 it also has infinite monodromy at infinity when $X$ is non-compact. Based on the first step, the Shafarevich morphism $\mathrm{sh}_{\mu^*\varrho}: Y \to \mathrm{Sh}_{\mu^*\varrho}(Y)$ exists, which is a surjective proper holomorphic fibration. Let $Z$ be an irreducible component of a fiber of $\mu$. Then $\mu^*(\pi_1(Z)) = \{1\}$. It follows that $\mathrm{sh}_{\mu^*\varrho}(Z)$ is a point. Note that each fiber of $\mu$ is connected as $X$ is normal. It follows that each fiber of $\mu$ is contracted to a point by $\mathrm{sh}_{\mu^*\varrho}$. Therefore, there exists a dominant holomorphic map $\mathrm{sh}_{\varrho}: X \to \mathrm{Sh}_{\mu^*\varrho}(Y)$ with connected general fibers such that we have the following commutative diagram:

$$ (3.11) \qquad \begin{tikzcd} Y \arrow[r] \arrow[d, "\mu"] & \mathrm{sh}_{\mu^*\varrho} \arrow[d, "\mathrm{sh}_{\varrho}"] \\ X \arrow[r, "\mathrm{sh}_{\varrho}"] & \mathrm{Sh}_{\mu^*\varrho}(Y) \end{tikzcd} $$

**Claim 3.35.** — For any closed subvariety $Z \subset X$, $\mathrm{sh}_{\varrho}(Z)$ is a point if and only if $\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)]$ is finite.

*Proof.* — Let us choose an irreducible component $W$ of $\mu^{-1}(Z)$ which is surjective onto $Z$. Since $\mathrm{Im}[\pi_1(W^{\mathrm{norm}}) \to \pi_1(Z^{\mathrm{norm}})]$ is a finite index subgroup of $\pi_1(Z^{\mathrm{norm}})$, and $\mu_*: \pi_1(Y) \to \pi_1(X)$ is surjective, it follows that $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is finite if and only if $\mu^*\varrho(\mathrm{Im}[\pi_1(W^{\mathrm{norm}}) \to \pi_1(Y)])$ is finite.

*Proof of $\Rightarrow$:* Note that $\mathrm{sh}_{\mu^*\varrho}(W)$ is a point and thus $\mu^*\varrho(\mathrm{Im}[\pi_1(W^{\mathrm{norm}}) \to \pi_1(Y)])$ is finite. Hence $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is finite.

*Proof of $\Leftarrow$:* Note that $\mu^*\varrho(\mathrm{Im}[\pi_1(W^{\mathrm{norm}}) \to \pi_1(Y)])$ is finite. Therefore, $\mathrm{sh}_{\mu^*\varrho}(W)$ is a point and thus $\mathrm{sh}_{\varrho}(Z)$ is a point by (3.11). $\square$
Let us write $\mathrm{Sh}_{\varrho}(X) := \mathrm{Sh}_{\mu^*\varrho}(Y)$. Then $\mathrm{sh}_{\varrho}: X \to \mathrm{Sh}_{\varrho}(X)$ is the Shafarevich morphism associated with $\varrho: \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$. $\square$

The condition in Proposition 3.31 that $\varrho$ has infinite monodromy at infinity poses significant practical limitations for further applications. However, we can overcome this drawback by utilizing Proposition 3.17, which allows us to eliminate this requirement.

**Theorem 3.36.** — Let $X$ be a non-compact, quasi-projective normal varieties, and let $\varrho : \pi_1(X) \to \mathrm{GL}_N(\mathbb{C})$ be a reductive representation. Then there exists a dominant holomorphic map $\mathrm{sh}_{\varrho}: X \to \mathrm{Sh}_{\varrho}(X)$ to a complex normal space $\mathrm{Sh}_{\varrho}(X)$ whose general fibers are connected such that for any closed subvariety $Z \subset X$, $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is finite if and only if $\mathrm{sh}_{\varrho}(Z)$ is a point.

*Proof.* — By Step 2 of the proof of Proposition 3.31, it suffices to prove the theorem for $X$ being a smooth variety. Therefore, we can replace $X$ by its desingularization and replace $\varrho$ by its pullback over this smooth model. Since $\varrho(\pi_1(X))$ is residually finite by Malcev's theorem, we can find a finite étale cover $\nu_0 : \tilde{X} \to X$ such that $\nu_0^*\varrho$ is torsion free.

**Claim 3.37.** — There are partial compactifications $X'$ (resp. $\tilde{X}'$) of $X$ (resp. $\tilde{X}$) such that

— $\tilde{X}'$ and $X'$ are quasi-projective normal varieties;

— $\nu_0 : \tilde{X} \to X$ extends to a finite morphism $\nu' : \tilde{X}' \to X'$;

— $\pi^*\varrho$ extends to a reductive representation $\varrho' : \pi_1(\tilde{X}') \to \mathrm{GL}_N(\mathbb{C})$ that has infinite monodromy at infinity.

*Proof.* — Let $\overline{X}$ be a smooth projective compactification of $X$. Then there exists a smooth projective variety $\overline{X}_1$ that compactifies $\overline{X}$, and a surjective generically finite morphism $\nu_1 : \overline{X}_1 \to \overline{X}$ that extends $\nu_0$.

By utilizing Proposition 3.17, after replacing $\overline{X}_1$ by some birational modification, there exists a simple normal crossing divisor $D \subset \overline{X}_1$ such that we have $\overline{X}_1 := \overline{X}_1 \setminus D \supset \overline{X}$ and $\nu^*\varrho$ extends to a representation $\varrho_1 : \pi_1(\overline{X}_1) \to \mathrm{GL}_N(\mathbb{C})$ that has infinite monodromy at infinity.

The morphism $\nu_1 : \overline{X}_1 \to \overline{X}$ is not necessarily finite, but the restriction $\nu_1|_{\tilde{X}} : \tilde{X} \to X$ is a finite étale cover. By applying Hironaka-Raynaud-Gruson's flattening theorem, we can find a birational morphism $\overline{X}_1 \to \overline{X}$ that is isomorphic over $X$ such that for the base change $\overline{X}_1 \times_{\overline{X}} \overline{X}_1 \to \overline{X}_1$, the main component denoted as $(\overline{X}_1 \times_{\overline{X}} \overline{X}_1)_{\mathrm{main}}$ which dominates $\overline{X}_1$, is flat over $\overline{X}_1$. Let $\overline{X}$ be the normalization of $(\overline{X}_1 \times_{\overline{X}} \overline{X}_1)_{\mathrm{main}}$.

<figure><img src="image_7.png" /></figure>

Then $\nu$ is a finite morphism. Let's define $D' := \mu^{-1}(D)$. Now, consider the pullback $\mu^*\varrho_1 : \pi_1(\overline{X} \setminus D') \to \mathrm{GL}_N(\mathbb{C})$. By Lemma 3.15, we observe that it has infinite monodromy at infinity. Consequently, $(\mu_0 \circ \nu)^*\varrho$ has infinite monodromy at each point of $D'$ in the sense of Lemma 3.16. Next, we assert that $\nu^{-1}(\nu(D')) = D'$.

To establish the claim, we need to show that $\mu_0^*\varrho$ has infinite monodromy at each point of $\nu(D')$. Assume, for the sake of contradiction, that there exists $q \in \nu(D')$ such that $\mu_0^*\varrho$ does not have infinite monodromy at $q$. Let $p' \in D'$ be such that $\nu(p') = q$. Then there exists a holomorphic map $f : \mathbb{D} \to \overline{X}_1$ such that

— $f(\mathbb{D}^*) \subset X$ and $f(0) = q$;

— $f^*(\mu_0^*\varrho)(\pi_1(\mathbb{D}^*)) = \{1\}$;
**Claim 3.39.** — For any closed subvariety $Z$ of $X$, $c(Z)$ is a point if and only if $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(Y')])$ is finite.

*Proof.* — Let $Y := f^{-1}(X)$ and $f_0 := f|_Y$. Note that $f_0 : Y \to X$ is a Galois cover with Galois group $G$. We have $h^*\varrho'|_{\pi_1(Y)} = f_0^*\varrho$. Now, consider any closed subvariety $Z$ of $X$. There exists an irreducible closed subvariety $W$ of $Y$ such that $f_0(W) = Z$. Let $\bar{W}$ be the closure of $W$ in $Y'$, which is an irreducible closed subvariety of $Y'$.

Observe that $c(Z)$ is a point if and only if $\mathrm{sh}_{h^*\varrho'}(\bar{W})$ is a point, which is equivalent to $h^*\varrho'(\mathrm{Im}[\pi_1(\bar{W}^{\mathrm{norm}}) \to \pi_1(Y')])$ being finite. Furthermore, this is equivalent to $f_0^*\varrho(\mathrm{Im}[\pi_1(W^{\mathrm{norm}}) \to \pi_1(Y)])$ being finite since $h^*\varrho'|_{\pi_1(Y)} = f_0^*\varrho$. Since $\mathrm{Im}[\pi_1(W^{\mathrm{norm}}) \to \pi_1(Z^{\mathrm{norm}})]$ is a finite index subgroup of $\pi_1(Z^{\mathrm{norm}})$, the above condition is equivalent to $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ being finite. $\square$

Let $f := \mathrm{sh}_{\varrho}$ and $Q := \mathrm{Sh}_{\varrho}(X)$. This concludes our construction of the Shafarevich morphism of $\varrho$. Therefore, our theorem is proven. $\square$

**Corollary 3.40.** — Let $X$ be a quasi-projective normal variety. Let $\Sigma$ be a (non-empty) set of reductive representations $\varrho : \pi_1(X) \to \mathrm{GL}_{N_p}(\mathbb{C})$. If $X$ is non-compact, we assume additionally that each $\varrho$ has infinite monodromy at infinity. Then there is a proper surjective holomorphic fibration $\mathrm{sh}_{\Sigma} : X \to \mathrm{Sh}_{\Sigma}(X)$ onto a complex normal space such that for closed subvariety $Z \subset X$, $\mathrm{sh}_{\Sigma}(Z)$ is a point if and only if $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is finite for every $\varrho \in \Sigma$. Moreover, $\mathrm{Sh}_{\Sigma}(X)$ is a projective normal variety if $X$ is compact.

*Proof.* — By Proposition 3.31, for each $\varrho \in \Sigma$, there exists a surjective proper holomorphic fibration $\mathrm{sh}_{\varrho} : X \to \mathrm{Sh}_{\varrho}(X)$ onto a complex normal space $\mathrm{Sh}_{\varrho}(X)$. By [Car60], there exists a surjective proper holomorphic fibration $\mathrm{sh}_{\Sigma} : X \to \mathrm{Sh}_{\Sigma}(X)$ onto a complex normal space $\mathrm{Sh}_{\Sigma}(X)$ and holomorphic maps $e_{\varrho} : \mathrm{Sh}_{\Sigma}(X) \to \mathrm{Sh}_{\varrho}(X)$ such that

(a) $\mathrm{sh}_{\varrho} = e_{\varrho} \circ \mathrm{sh}_{\Sigma}$;

(b) for any $y \in \mathrm{Sh}_{\Sigma}(X)$, we have $\mathrm{sh}_{\Sigma}^{-1}(y) = \cap_{\varrho \in \Sigma} \mathrm{sh}_{\varrho}^{-1}(e_{\varrho}(y))$.

Let $Z$ be a closed subvariety of $X$. If $\mathrm{sh}_{\Sigma}(Z)$ is a point, then $\mathrm{sh}_{\varrho}(Z)$ is a point for any $\varrho \in \Sigma$ by Item (a). It follows that $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is finite for every $\varrho \in \Sigma$.

Conversely, if $\varrho(\mathrm{Im}[\pi_1(Z^{\mathrm{norm}}) \to \pi_1(X)])$ is finite for every $\varrho \in \Sigma$, then $\mathrm{sh}_{\varrho}(Z)$ is a point for any $\varrho \in \Sigma$. By Item (a), $\mathrm{sh}_{\Sigma}(Z)$ is a point. The corollary is proved. $\square$

**3.5. On the algebraicity of the Shafarevich morphism via $L^2$-methods.** — In Theorem 3.36, when $X$ is compact, we proved that the image $\mathrm{Sh}_{\varrho}(X)$ is projective. In general, as mentioned in Remark 0.2, we propose the following conjecture.

**Conjecture 3.41 (Algebraicity of Shafarevich morphism)**

Let $X$, $\varrho$ and $\mathrm{sh}_{\varrho} : X \to \mathrm{Sh}_{\varrho}(X)$ be as in Theorem 3.36. Then $\mathrm{Sh}_{\varrho}(X)$ is a quasi-projective normal variety and $\mathrm{sh}_{\varrho} : X \to \mathrm{Sh}_{\varrho}(X)$ is an algebraic morphism.

This conjecture seems to be a difficult problem, with the special case when $\varrho$ arises from a $\mathbb{Z}$-VHS known as a long-standing Griffiths conjecture. In this paper, we provide confirmation of such expectations at the function field level, inspired by the work of Sommese [Som75, Som78].

We first recall the definition of (bi)meromorphic maps of complex spaces $X$ and $Y$ (in the sense of Remmert) with a few exceptional convenience. Let $X^\circ$ be an open subset of $X$ such that $X\backslash X^\circ$ is a nowhere-dense analytic subset and suppose that a holomorphic mapping $f : X^\circ \to Y$ has been given. Then $f : X \dashrightarrow Y$ is called a meromorphic mapping if the closure $\Gamma_f$ of the graph of $f$ in $X \times Y$ is an analytic subset of $X \times Y$ and if the projection $\Gamma_f \to X$ is a proper mapping. If additionally, there is a Zariski dense open subset $X' \subset X$ such that $f|_{X'} : X' \to f(X')$ is a biholomorphism, then $f$ is called *bimeromorphic*. Note that we do not require that $\Gamma_f$ is proper over $Y$.

We present a result that is derived from [Som78, Proposition I], where the proof utilizes an elegant application of the Hörmander-Andreotti-Vesentini $L^2$-estimate.
**Proposition 3.42.** — Let $X$ be a smooth quasi-projective variety and let $f : X \to Y$ be a proper surjective holomorphic map onto a normal complex space $Y$. Let $\bar{X}$ be a smooth projective compactification of $X$ such that $\bar{X}\setminus X$ is a simple normal crossing divisor. Assume that there exists a holomorphic line bundle $L$ on $Y$ equipped with a smooth hermitian metric $h$ satisfying the following property:

(a) $f^*L$ extends to an algebraic line bundle $\mathcal{L}$ on $\bar{X}$.

(b) $\mathcal{L}$ has $L^2$-poles with respect to $f^*h$, i.e., for any point $x$ in the smooth locus of $D$, it has an admissible coordinate $(U; z_1, \dots, z_n)$ centered at $x$ with $D \cap U = (z_1 = 0)$ such that $\mathcal{L}|_U$ is trivialized by a section $s \in \Gamma(U, \mathcal{L})$ and $\int_{U\setminus D} |z_1|^N |s|^{2f^*h} idz_1 \wedge d\bar{z}_1 \wedge \dots \wedge idz_n \wedge d\bar{z}_n < \infty$ for some integer $N \ge 1$.

(c) The curvature $i\Theta_h(L)$ is semipositive everywhere and strictly positive at a general smooth point of $Y$.

Then there exists a bimeromorphic map $h : Y \dashrightarrow W$ to a quasi-projective variety $W$ such that $h \circ f : X \dashrightarrow W$ is rational.

As the paper by Sommese [Som78] is rather involved, for the readers’ convenience, we recall briefly the ideas of the proof in [Som78].

*Sketch of the proof.* — After taking successive generic hyperplane sections on $\bar{X}$, we assume that there exists a proper surjective generically finite holomorphic map $g : Z \to Y$ from a complete Kähler manifold $Z$. By Item (c) we can choose an open set $U \subset Y^{\text{reg}}$ such that

(a) $g^{-1}(U) = \cup_{i=1}^m U_i$ with $g|_{U_i} : U_i \to U$ is a biholomorphism;

(b) $i\Theta_h(L)$ is strictly positive at $U$.

We fix a point $y \in U$ and let $z_i$ be the unique point in $U_i$ such that $g(z_i) = y$. By applying the Hörmander $L^2$-estimate, we can prove that there exists integer $N_0 \ge 1$ such that for any $N \ge N_0$, the global $L^2$-sections $L^2(Z, K_Z \otimes g^*L^{\otimes N})$ generates 1-jets at points $z_1, \dots, z_m$, where $g^*L^{\otimes N}$ is equipped with the metric $g^*h^{\otimes N}$. For any $e \in L^2(Z, K_Z \otimes g^*L^{\otimes N})$, the trace map induces a section on $\tilde{e} \in L^2(Z, K_Y \otimes L^{\otimes N})$, where $K_Y$ is the Grauert-Riemenschneider sheaf of $Y$ (cf. [Som78, Lemma II-A]). Therefore, when $N \ge N_0$, the sections $L^2(Y, K_Y \otimes L^{\otimes N})$ generating 1-jet at $y$. We then choose a finite set of sections in $L^2(Y, K_Y \otimes L^{\otimes N})$ generating 1-jets at $y$. It thus induces a meromorphic map $h : Y \dashrightarrow \mathbb{P}^N$ such that $h$ is immersive at a neighborhood of $y$.

On the other hand, by [Som78, Lemma I-C], $f^*L^2(Y, K_Y \otimes L^{\otimes N})$ extends to a meromorphic section of $\Omega_X^k \otimes \mathcal{L}^{\otimes N}$, where $k := \dim Y$. Therefore, by [Som75, Lemma I-E] there is a meromorphic map $p : \bar{X} \dashrightarrow \mathbb{P}^N$ such that $h \circ f = p|_X$.

<figure><img src="image_8.png" /></figure>

By the Chow theorem, $p$ is rational. Let $W$ be the image of $p$ which is a projective variety. Then $\dim W = \dim Y$ and $h(Y) \subset W$. Since $h$ is immersive at one point. It follows that there is a Zariski dense open set $Y^\circ$ such that $h|_{Y^\circ} : Y^\circ \to h(Y^\circ)$ is a biholomorphism. Therefore, $h : Y \dashrightarrow W$ is a bimeromorphic map. $\square$

Let us apply Proposition 3.42 to study the property of the Shafarevich morphism constructed in Theorem 3.36.

**Proposition 3.43.** — Let $X$ be a non-compact smooth quasi-projective variety and $\varrho : \pi_1(X) \to \text{GL}_N(\mathbb{C})$ be a reductive representation such that $\varrho(\pi_1(X))$ is torsion free. Then there exists a bimeromorphic map $h : \text{Sh}_\varrho(X) \dashrightarrow Y$ to a quasi-projective normal variety $Y$ such that $h \circ \text{sh}_\varrho : X \dashrightarrow Y$ is rational.
*Proof.* — Based on Theorem 3.36, we may assume that $\varrho$ has infinite monodromy at infinity. To maintain consistency with the proof of Theorem 3.20, we will adopt the notations used therein. For the readers' convenience, we draw a commutative diagram below.

<figure><img src="image_9.png" /></figure>

As $\varrho(\pi_1(X))$ is torsion free, by the proof of Claim 3.27, we know that $\pi_1(X)/H$ acts on $\tilde{S}_H$ is discretely continuous and free. Therefore, $\tilde{S}_H \to \text{Sh}_\mathcal{E}(X) := \tilde{S}_H/\Gamma_0$ is an étale covering.

Recall that the the canonical bundle $K_\mathcal{D}$ of the period domain $\mathcal{D}$ is equipped with a $G_0$-invariant smooth metric $h_\mathcal{D}$. This metric has positive-definite curvature in the horizontal direction. Since the morphism $\phi: \tilde{S}_H \to \mathcal{D}$ is equivariant, the pullback $\phi^*K_\mathcal{D}$ descends to a line bundle on the quotient $\text{Sh}_\mathcal{E}(X)$, denoted by $L_G$. The line bundle $L_G$ is equipped with a metric $h_G$, and the curvature form of $(L_G, h_G)$ is semipositive everywhere.

Similarly, for the period mapping $p: \tilde{X}_H \to \mathcal{D}$ associated with the $C$-VHS underlied by $\sigma$, the pullback $p^*K_\mathcal{D}$ descends to a line bundle on $X$ that is equal to $\text{sh}_\mathcal{E}^*L_G$. It is well-known that $\text{sh}_\mathcal{E}^*L_G$ extends to an algebraic line bundle $\mathcal{L}_1$ over $\bar{X}$, known as the Deligne extension, which has $L^2$-poles with respect to $\text{sh}_\mathcal{E}^*h_G$ (cf. [Den23, §2.1]).

Considering that $s_\mathcal{E}: X \to S_\mathcal{E}$, as defined in Definition 3.1, is an algebraic morphism, we can assume it extends to a morphism $\bar{s}_\mathcal{E}: \bar{X} \to \bar{S}_\mathcal{E}$, where $\bar{S}_\mathcal{E}$ is a projective compactification of $S_\mathcal{E}$. Let us choose an ample line bundle $L_0$ on $\bar{S}_\mathcal{E}$, equipped with a smooth metric $h_0$ that has positive-definite curvature. Denoting $L := f^*L_0 \otimes L_G$, with the associated smooth metric $h := f^*h_0 \otimes h_G$, we observe that the algebraic line bundle $\mathcal{L} := \mathcal{L}_1 \otimes \bar{s}_\mathcal{E}^*L_0$ on $\bar{X}$ is an extension of $\text{sh}_\mathcal{E}^*L$, which has $L^2$-poles with respect to $\text{sh}_\mathcal{E}^*h$.

According to Claim 3.28, we know that $g: \tilde{S}_H \to \mathcal{D} \times S_\mathcal{E}$ has discrete fibers. Hence, at a general point on the regular locus of $\text{Sh}_\mathcal{E}(X)$, the curvature of $h$ is strictly positive. Consequently, the conditions in Proposition 3.42 are satisfied, leading us to conclude the proposition. $\square$

**4. Proof of the reductive Shafarevich conjecture**

The goal of this section is to provide proofs for Theorems B and C when $X$ is a smooth projective variety. It is important to note that our methods differs significantly from the approach presented in [Eys04], although we do follow the general strategy outlined in that work.

In this section, we will use the notation $\mathcal{DG}$ to denote the derived group of any given group $G$. Throughout the section, our focus is on non-archimedean local fields with characteristic zero. More precisely, we consider finite extensions of $\mathbb{Q}_p$ for some prime $p$.

**4.1. Reduction map of representation into algebraic tori.** — Let $X$ be a smooth projective variety. Let $a: X \to A$ be the Albanese morphism of $X$.

**Lemma 4.1.** — Let $P \subset A$ be an abelian subvariety of the Albanese variety $A$ of $X$ and $K$ be a non-archimedean local field. If $\tau: \pi_1(X) \to \text{GL}_1(K)$ factors through $\sigma: \pi_1(A/P) \to \text{GL}_1(K)$, then the Katzarkov-Eyssidieux reduction map $s_\tau: X \to S_\tau$ factors through the Stein factorization of the map $q: X \to A/P$.

*Proof.* — As $\tau = q^*\sigma$, if follows that for each connected component $F$ of the fiber of $q: X \to A/P$, $\tau(\pi_1(F)) = \{1\}$. Therefore, $F$ is contracted by $s_\tau$. The lemma follows. $\square$
**Lemma 4.2.** — Let $P \subset A$ be an abelian subvariety of $A$. Let $N$ be a Zariski dense open set of the image $j: M_B(A/P, 1) \to M_B(A, 1)$ where we consider $M_B(A/P, 1)$ and $M_B(A, 1)$ as algebraic tori defined over $\mathbb{Q}$. Then there are non-archimedean local fields $K_i$ and a family of representations $\tau := \{\tau_i : \pi_1(X) \to \text{GL}_1(K_i)\}_{i=1,\dots,m}$ such that

- $\tau_i \in N(K_i)$, where we use the natural identification $M_B^0(X, 1) \simeq M_B(A, 1)$. Here $M_B^0(X, 1)$ denotes the connected component of $M_B^0(X, 1)$ containing the trivial representation.

- The reduction map $s_\tau : X \to S_\tau$ is the Stein factorization of $X \to A/P$.

- For the canonical current $T_\tau$ defined over $S_\tau$, $\{T_\tau\}$ is a Kähler class.

**Proof.** — Let $e_1, \dots, e_m$ be a basis of $\pi_1(A/P) \simeq H_1(A/P, \mathbb{Z})$. Note that $\mathbb{Q}$-scheme $M_B(A/P, 1) \simeq (\mathbb{Q}^\times)^m$. Denote by $S \subset U(1) \cap \mathbb{Q}$ the set of roots of unity. Then $S$ is Zariski dense in $\mathbb{Q}^\times$. Since $j^{-1}(N)$ is a Zariski dense open set of $M_B(A/P, 1)$, it follows that there are $\{a_{ij}\}_{i,j=1,\dots,m} \in \mathbb{Q}^\times$ and representations $\{\varrho_i : \pi_1(A/P) \to \mathbb{Q}^\times\}_{i=1,\dots,m}$ defined by $\rho_i(e_j) = a_{ij}$ such that

- $[\varrho_i] \in j^{-1}(N)(\mathbb{Q})$;

- If $i = j$, $a_{ij} \in \mathbb{Q}^\times \setminus U(1)$;

- If $i \neq j$, $a_{ij} \in S$.

Consider a number field $k_i$ containing $a_{i1}, \dots, a_{im}$ endowed with a discrete non-archimedean valuation $v_i : k_i \to \mathbb{R}$ such that $v_i(a_{ii}) \neq 0$. Then $v_i(a_{ij}) = 0$ for every $j \neq i$. Indeed, for every $j \neq i$, since $a_{ij}$ is a root of unity, there exists $\ell \in \mathbb{Z}_{>0}$ such that $a_{ij}^\ell = 1$. It follows that $0 = v(a_{ij}^\ell) = \ell v(a_{ij})$. Let $K_i$ be the non-archimedean local field which is the completion of $k_i$ with respect to $v_i$. It follows that each $\rho_i : \pi_1(A/P) \to K_i^\times$ is unbounded. Consider $\nu_i : \pi_1(A/P) \to \mathbb{R}$ by composing $\rho_i$ with $v_i : K_i^\times \to \mathbb{R}$. Then $\{\nu_1, \dots, \nu_m\} \subset H^1(A/P, \mathbb{R})$ is a basis for the $\mathbb{R}$-linear space $H^1(A/P, \mathbb{R})$. It follows that $\nu_i(e_j) = \delta_{ij}$ for any $i, j$. Let $\eta_i \in H^0(A/P, \Omega_{A/P}^1)$ be the (1,0)-part of the Hodge decomposition of $\nu_i$. Therefore, $\{\eta_1, \dots, \eta_m\}$ spans the $\mathbb{C}$-linear space $H^0(A/P, \Omega_{A/P}^1)$. Hence $\sum_{i=1}^m i\eta_i \wedge \bar{\eta}_i$ is a Kähler form on $A/P$. Let $\tau_i : \pi_1(X) \to K_i^\times$ be the composition of $\varrho_i$ with $\pi_1(X) \to \pi_1(A/P)$.

Let $q : A \to A/P$ be the quotient map. Let $P'$ the largest abelian subvariety of $A$ such that $q^*\eta_i|_{P'} \equiv 0$ for each $i$. Since $\{\eta_1, \dots, \eta_m\}$ spans $H^0(B, \Omega_B^1)$, it follows that $P' = P$. Therefore, the reduction map $s_\tau : X \to S_\tau$ is the Stein factorization of $X \to A/P$ with $g : S_\tau \to A/P$ be the finite morphism. According to Definition 1.23, $T_\tau = g^* \sum_{i=1}^m i\eta_i \wedge \bar{\eta}_i$. Since $\sum_{i=1}^m i\eta_i \wedge \bar{\eta}_i$ is a Kähler form on $A/P$, it follows that $\{T_\tau\}$ is a Kähler class by Theorem 1.12. The lemma is proved. $\square$

**Corollary 4.3.** — Let $X$ be a smooth projective variety. If $\mathcal{C} \subset M_B(X, 1)$ is an absolutely constructible subset. Consider the reduction map $s_{\mathcal{C}} : X \to S_{\mathcal{C}}$ defined in Definition 3.1. Then there is a family of representations $\varrho := \{\rho_i : \pi_1(X) \to \text{GL}_1(K_i)\}_{i=1,\dots,\ell}$ where $K_i$ are non-archimedean local fields such that

- For each $i = 1, \dots, \ell$, $\rho_i \in \mathcal{C}(K_i)$;

- The reduction map $s_{\varrho} : X \to S_{\varrho}$ of $\varrho$ coincides with $s_{\mathcal{C}}$.

- For the canonical current $T_{\varrho}$ defined over $S_{\mathcal{C}}$, $\{T_{\varrho}\}$ is a Kähler class.

**Proof.** — Let $A$ be the Albanese variety of $X$. Since $\mathcal{C} \subset M_B(X, 1)$ is an absolute constructible subset, by Theorem 1.18, there are abelian subvarieties $P_i \subset A$ and torsion points $v_i \in M_B(X, 1)(\mathbb{Q})$ such that $\mathcal{C} = \bigcup_{i=1}^m v_i N_i^\circ$; where $N_i$ is the image in $M_B^0(X, 1) \simeq M_B(A, 1)$ of the natural morphism $M_B(A/P_i, 1) \to M_B(A, 1)$ and $N_i^\circ$ is a Zariski dense open subset of $N_i$. Let $k$ be a number field such that $v_i \in M_B(X, 1)(k)$ for each $i$.

**Claim 4.4.** — Denote by $P := \bigcap_{i=1}^m P_i$. Then $s_{\mathcal{C}} : X \to S_{\mathcal{C}}$ is the Stein factorization of $X \to A/P$.

**Proof.** — Let $\tau : \pi_1(X) \to \text{GL}_1(K)$ be a reductive representation with $K$ a non-archimedean local field such that $\tau \in \mathcal{C}(K)$. Note that the reduction map $s_\tau$ is the same
if we replace $K$ by a finite extension. We thus can assume that $k \subset K$. Note that there exists some $i \in \{1, \dots, \ell\}$ such that $[v_i^{-1} \cdot \tau] \in N_i(K)$. Write $\varrho := v_i^{-1} \cdot \tau$. Since $v_i$ is a torsion element, it follows that $v_i(\pi_1(X))$ is finite, and thus the reduction map $s_\varrho$ coincides with $s_\tau$. Since $\varrho$ factors through $\pi_1(A/P_i) \to \text{GL}_1(K)$, by Lemma 4.1 $s_\varrho$ factors through the Stein factorization of $X \to A/P_i$. Hence $s_\varrho$ factors through the Stein factorization of $X \to A/P$. By Definition 3.1, it follows that $s_\mathfrak{C} : X \to \text{S}\mathfrak{C}$ factors through the Stein factorization of $X \to A/P$.

Fix any $i$. By Lemma 4.2 there are non-archimedean local fields $K_j$ and a family of reductive representations $\tau := \{\tau_j : \pi_1(X) \to \text{GL}_1(K_j)\}_{j=1,\dots,n}$ such that

- $\tau_j \in N_i^\circ(K_j)$.

- The reduction map $s_\tau : X \to S_\tau$ is the Stein factorization of $X \to A/P_i$.

- For the canonical current $T_\tau$ over $S_\tau$, $\{T_\tau\}$ is a Kähler class.

We can replace $K_i$ by a finite extension such that $k \subset K_i$ for each $K_i$. Then $v_i \cdot \tau_i \in \mathfrak{C}(K_i)$ for every $i$. Note that the Katzarkov-Eyssidieux reduction map $s_{v_i \cdot \tau_j} : X \to S_{v_i \cdot \tau_j}$ coincides with $s_{\tau_j} : X \to S_{\tau_j}$. Therefore, the Stein factorization of $X \to A/P_i$ factors through $s_\mathfrak{C}$. Since this holds for each $i$, it follows that the Stein factorization $X \to A/P_1 \times \dots \times A/P_m$ factors through $s_\mathfrak{C}$. Note that the Stein factorization $X \to A/P_1 \times \dots \times A/P_m$ coincides with the Stein factorization of $X \to A/P$. Therefore, the Stein factorization of $X \to A/P$ factors through $s_\mathfrak{C}$. The claim is proved. $\square$

By the above arguments, for each $i$, there exists a family of reductive representations into non-archimedean local fields $\varrho_i := \{\varrho_{ij} : \pi_1(X) \to \text{GL}_1(K_{ij})\}_{j=1,\dots,k_i}$ such that

- $\varrho_{ij} \in \mathfrak{C}(K_{ij})$

- $s_{\varrho_i} : X \to S_{\varrho_i}$ is the Stein factorization of $X \to A/P_i$

- For the canonical current $T_{\varrho_i}$ defined over $S_{\varrho_i}$, $\{T_{\varrho_i}\}$ is a Kähler class.

By the above claim, we know that $s_\mathfrak{C} : X \to S_\mathfrak{C}$ is the Stein factorization of $X \to S_{\varrho_1} \times \dots \times S_{\varrho_m}$. Then for the representation $\varrho := \{\varrho_{ij} : \pi_1(X) \to \text{GL}_1(K_{ij})\}_{i=1,\dots,m; j=1,\dots,k_i}$, $s_\varrho : X \to S_\varrho$ is the Stein factorization of $X \to A/P$ hence $s_\varrho$ coincides with $s_\mathfrak{C}$. Moreover, the canonical current $T_\varrho = \sum_{i=1}^m g_i^* T_{\varrho_i}$ where $g_i : S_\mathfrak{C} \to S_{\varrho_i}$ is the natural map. As $S_\mathfrak{C} \to S_{\varrho_1} \times \dots \times S_{\varrho_m}$ is finite, by Theorem 1.12 $\{T_\varrho\}$ is Kähler. $\square$

Let us prove the main result in this subsection.

**Theorem 4.5.** — Let $X$ be a smooth projective variety and let $T$ be an algebraic tori defined over some number field $k$. Let $\mathfrak{C} \subset M_B(X, T)(\mathbb{C})$ be an absolutely constructible subset. Consider the reduction map $s_\mathfrak{C} : X \to S_\mathfrak{C}$. Then there is a family of reductive representations $\tau := \{\tau_i : \pi_1(X) \to T(K_i)\}_{i=1,\dots,N}$ where $K_i$ are non-archimedean local fields containing $k$ such that

- For each $i = 1, \dots, N$, $[\tau_i] \in \mathfrak{C}(K_i)$;

- The reduction map $s_\tau : X \to S_\tau$ of $\tau$ coincides with $s_\mathfrak{C}$.

- For the canonical current $T_\tau$ over $S_\tau$ defined in Definition 1.23, $\{T_\tau\}$ is a Kähler class.

*Proof.* — We replace $k$ by a finite extension such that $T$ is split over $k$. Then we have $T \simeq \mathbb{G}_{m,k}^\ell$. Note that this does not change the reduction map $s_\mathfrak{C} : X \to S_\mathfrak{C}$. We take $p_i : T \to \mathbb{G}_{m,k}$ to be the $i$-th projection which is a $k$-morphism. It induces a morphism of $k$-schemes $\psi_i : M_B(X, T) \to M_B(X, \text{GL}_1)$. By Theorem 1.19, $\mathfrak{C}_i := \psi_i(\mathfrak{C})$ is also an absolutely constructible subsets. Consider the reduction maps $s_{\mathfrak{C}_i} : X \to S_{\mathfrak{C}_i}$.

**Claim 4.6.** — $s_\mathfrak{C} : X \to S_\mathfrak{C}$ is the Stein factorization of $X \to S_{\mathfrak{C}_1} \times \dots \times S_{\mathfrak{C}_\ell}$.

*Proof.* — Let $\varrho : \pi_1(X) \to T(K)$ be any reductive representation where $K$ is a non-archimedean local field containing $k$ such that $[\varrho] \in \mathfrak{C}(K)$. Write $\varrho_i = p_i \circ \varrho : \pi_1(X) \to \text{GL}_1(K)$. Then $[\varrho_i] = \psi_i([\varrho]) \in \mathfrak{C}_i(K)$. Note that for any subgroup $\Gamma \subset \pi_1(X)$, $\varrho(\Gamma)$ is bounded if and only if $\varrho_i(\Gamma)$ is bounded for any $i$. Therefore, $s_\varrho : X \to S_\varrho$ is the Stein
factorization of $X \to S_{\varrho_1} \times \cdots \times S_{\varrho_l}$. Hence $s_{\mathfrak{C}} : X \to S_{\mathfrak{C}}$ factors through the Stein factorization of $X \to S_{\mathfrak{C}_1} \times \cdots \times S_{\mathfrak{C}_l}$.

On the other hand, consider any $\varrho_i \in \mathfrak{C}_i(K)$ where $K$ is a non-archimedean local field containing $k$. Then there is a finite extension $L$ of $K$ such that

- there is a reductive representation $\varrho : \pi_1(X) \to T(L)$ with $[\varrho] \in \mathfrak{C}(L)$;

- $p_i \circ \varrho = \varrho_i$.

By the above argument, $s_{\varrho_i} : X \to S_{\varrho_i}$ factors through $s_{\varrho} : X \to S_{\varrho}$. Note that $s_{\varrho}$ factors through $s_{\mathfrak{C}}$. It follows that the Stein factorization of $X \to S_{\mathfrak{C}_1} \times \cdots \times S_{\mathfrak{C}_l}$ factors through $s_{\mathfrak{C}}$. The claim is proved. □

We now apply Corollary 4.3 to conclude that for each $i$, there exists a family of reductive representations into non-archimedean local fields $\varrho_i := \{\varrho_{ij} : \pi_1(X) \to \text{GL}_1(K_{ij})\}_{j=1,\dots,k_i}$ such that

- $\varrho_{ij} \in \mathfrak{C}_i(K_{ij})$;

- The reduction map $s_{\varrho_i} : X \to S_{\varrho_i}$ of $\varrho_i$ coincides with $s_{\mathfrak{C}_i} : X \to S_{\mathfrak{C}_i}$;

- for the canonical current $T_{\varrho_i}$ defined over $S_{\varrho_i}$, $\{T_{\varrho_i}\}$ is a Kähler class.

Denote by $\varrho := \{\varrho_{ij}\}_{i=1,\dots,l; j=1,\dots,k_i}$. Then $s_{\varrho} : X \to S_{\varrho}$ coincides with $s_{\mathfrak{C}} : X \to S_{\mathfrak{C}}$ by the above claim. Then $T_{\varrho}$ is a Kähler class.

By the definition of $\mathfrak{C}_i$, we can find a finite extension $L_{ij}$ of $K_{ij}$ such that

- there is a reductive representation $\tau_{ij} : \pi_1(X) \to T(L_{ij})$ with $[\tau_{ij}] \in \mathfrak{C}(L_{ij})$;

- $p_i \circ \tau_{ij} = \varrho_{ij}$.

Therefore, for the family $\tau := \{\tau_{ij}\}_{i=1,\dots,l; j=1,\dots,k_i}$, $s_{\tau} : X \to S_{\tau}$ coincides with $s_{\mathfrak{C}}$ by the above claim. Note that for any $i, j$, there exists an morphism $e_{ij} : S_{\tau_{ij}} \to S_{\varrho_{ij}}$ such that $s_{\varrho_{ij}} : X \to S_{\varrho_{ij}}$ factors through $e_{ij}$. We also note that $e_{ij}^* T_{\varrho_{ij}} \le T_{\tau_{ij}}$ for the canonical currents. It follows that $T_{\varrho} \le T_{\tau}$ (note that $S_{\tau} = S_{\varrho} = S_{\mathfrak{C}}$). Therefore, $\{T_{\tau}\}$ is a Kähler class. We prove the theorem. □

**4.2. Some criterion for representation into tori.** — We recall a lemma in [CDY22, Lemma 5.3].

**Lemma 4.7.** — Let $G$ be an almost simple algebraic group over the non-archimedean local field $K$. Let $\Gamma \subset G(K)$ be a finitely generated subgroup so that

- it is a Zariski dense subgroup in $G$,

- it is not contained in any bounded subgroup of $G(K)$.

Let $\Upsilon$ be a normal subgroup of $\Gamma$ which is bounded. Then $\Upsilon$ must be finite.

This lemma enables us to prove the following result.

**Lemma 4.8.** — Let $G$ be a reductive algebraic group over the non-archimedean local field $K$ of characteristic zero. Let $X$ be a projective manifold and let $\varrho : \pi_1(X) \to G(K)$ be a Zariski dense representation. If $\varrho(D\pi_1(X))$ is bounded, then after replacing $K$ by some finite extension, for the reductive representation $\tau : \pi_1(X) \to G/\mathcal{D}G(K)$ which is the composition of $\varrho$ with $G \to G/\mathcal{D}G$, the reduction map $s_{\tau} : X \to S_{\tau}$ coincides with $s_{\varrho} : X \to S_{\varrho}$.

*Proof.* — Since $G$ is reductive, then after replacing $K$ by a finite extension, there is an isogeny $G \to H_1 \times \cdots \times H_k \times T$, where $H_i$ are almost simple algebraic groups over $K$ and $T = G/\mathcal{D}G$ is an algebraic tori over $K$. Write $G' := H_1 \times \cdots \times H_k \times T$. We denote by $\varrho' : \pi_1(X) \to G'(K)$ the induced representation by the above isogeny.

**Claim 4.9.** — The Katzarkov-Eyssidieux reduction map $s_{\varrho} : X \to S_{\varrho}$ coincides with $s_{\varrho'} : X \to S_{\varrho'}$.
*Proof.* — It suffices to prove that, for any subgroup $\Gamma$ of $\pi_1(X)$, $\varrho(\Gamma)$ is bounded if and only if $\varrho'(\Gamma)$ is bounded. Note that we have the following short exact sequence of algebraic groups

$$0 \to \mu \to G \to G' \to 0$$

where $\mu$ is finite. Then we have

$$0 \to \mu(K) \to G(K) \xrightarrow{f} G'(K) \to H^1(K, \mu),$$

where $H^1(K, \mu)$ is the Galois cohomology. Note that $\mu(K)$ is finite. Since $K$ is a finite extension of some $\mathbb{Q}_p$, it follows that $H^1(K, \mu)$ is also finite. Therefore, $f:G(K) \to G'(K)$ has finite kernel and cokernel. Therefore, $\varrho(\Gamma)$ is bounded if and only if $\varrho'(\Gamma)$ is bounded. $\square$

Set $\Gamma := \varrho'(\pi_1(X))$ and $\Upsilon := \varrho'(\mathcal{D}\pi_1(X))$. Let $\Upsilon_i \subset H_i(K)$ and $\Gamma_i$ be the image of $\Upsilon$ and $\Gamma$ under the projection $G(K) \to H_i(K)$. Then $\Gamma_i$ is Zariski dense in $H_i$ and $\Upsilon_i \subsetneq \Gamma_i$ is also bounded. Furthermore, $\mathcal{D}\Gamma_i = \Upsilon_i$.

**Claim 4.10.** — $\Gamma_i$ is bounded for every $i$.

*Proof.* — Assuming a contradiction, let's suppose that some $\Gamma_i$ is unbounded. Since $\Upsilon_i \subsetneq \Gamma_i$ and $\Upsilon_i$ is bounded, we can refer to Lemma 4.7 which states that $\Upsilon_i$ must be finite. We may replace $X$ with a finite étale cover, allowing us to assume that $\Upsilon_i$ is trivial. Consequently, $\Gamma_i$ becomes abelian, which contradicts the fact that $\Gamma_i$ is Zariski dense in the almost simple algebraic group $H_i$. $\square$

Based on the previous claim, it follows that the induced representations $\tau_i : \pi_1(X) \to H_i(K)$ are all bounded for every $i$. Consequently, they do not contribute to the reduction map of $s_{\varrho'} : X \to S_{\varrho'}$. Therefore, the only contribution to $s_{\varrho'}$ comes from $\tau : \pi_1(X) \to T(K)$, where $\tau$ is the composition of $\varrho : \pi_1(X) \to G(K)$ and $G(K) \to T(K)$.

According to Claim 4.9, we can conclude that $s_{\varrho}$ coincides with the reduction map $s_{\tau} : X \to S_{\tau}$ of $\tau : \pi_1(X) \to T(K)$. This establishes the lemma. $\square$

**4.3. Eyssidieux-Simpson Lefschetz theorem and its application.** — Let $X$ be a compact Kähler manifold and let $V \subset H^0(X, \Omega_X^1)$ be a $\mathbb{C}$-subspace. Let $a : X \to \mathcal{A}_X$ be the Albanese morphism of $X$. Note that $a^* : H^0(\mathcal{A}_X, \Omega_{\mathcal{A}_X}^1) \to H^0(X, \Omega_X^1)$ is an isomorphism. Write $V' := (a^*)^{-1}(V)$. Define $B(V) \subset \mathcal{A}_X$ to be the largest abelian subvariety of $\mathcal{A}_X$ such that $\eta|_{B(V)} = 0$ for every $\eta \in V'$. Set $\mathcal{A}_{X,V} := \mathcal{A}_X/B(V)$. The partial Albanese morphism associated with $V$ is the composition of $a$ with the quotient map $\mathcal{A}_X \to \mathcal{A}_{X,V}$, denoted by $g_V : X \to \mathcal{A}_{X,V}$. Note that there exists $V_0 \subset H^0(\mathcal{A}_{X,V}, \Omega_{\mathcal{A}_{X,V}}^1)$ with $\dim_{\mathbb{C}} V_0 = \dim_{\mathbb{C}} V$ such that $g_V^*V_0 = V$. Let $\widetilde{\mathcal{A}_{X,V}} \to \mathcal{A}_{X,V}$ be the universal covering and let $X_V$ be $X \times_{\mathcal{A}_{X,V}} \widetilde{\mathcal{A}_{X,V}}$. Note that $V_0$ induces a natural linear map $\widetilde{\mathcal{A}_{X,V}} \to V_0^*$. Its composition with $X_V \to \widetilde{\mathcal{A}_{X,V}}$ and $g_V^* : V_0 \to V$ gives rise to a holomorphic map

Let $f : X \to S$ be the Stein factorization of $g_V : X \to \mathcal{A}_{X,V}$ with $q : S \to \mathcal{A}_{X,V}$ the finite morphism. Set $\mathbb{V} := q^*V_0$.

**Definition 4.11.** — $V$ is called *perfect* if for any closed subvariety $Z \subset S$ of dimension $d \ge 1$, one has $\mathrm{Im}[\Lambda^d\mathbb{V} \to H^0(Z, \Omega_Z^d)] \ne 0$.

The terminology of “perfect $V$” in Definition 4.11 is called “SSKB factorisable” in [Eys04, Lemme 5.1.6].

Let us recall the following Lefschetz theorem by Eyssidieux, which is a generalization of previous work by Simpson [Sim92]. This theorem plays a crucial role in the proofs of Theorems B and C.
**Theorem 4.12** ([Eys04, Lemme 5.1.22]). — Let $X$ be a compact Kähler normal space and let $V \subset H^0(X, \Omega_X^1)$ be a subspace. Assume that

$$ \mathrm{Im}[\Lambda^{\dim V}V \to H^0(X, \Omega_X^{\dim V})] = \eta \neq 0. $$

Set $(\eta = 0) = \cup_{i=1}^k Z_i$ where $Z_i$ are proper closed subvarieties of $X$. For each $Z_i$, denote by $V_i := \mathrm{Im}[V \to H^0(Z_i, \Omega_{Z_i})]$. Assume that $V_i$ is perfect for each $i$. Then there are two possibilities which exclude each other:

- either $V$ is perfect;

- or for the holomorphic map $\tilde{g}_V : X_V \to V^*$ defined as (4.1), $(X_V, \tilde{g}_V^{-1}(t))$ is 1-connected for any $t \in V^*$; i.e. $\tilde{g}_V^{-1}(t)$ is connected and $\pi_1(\tilde{g}_V^{-1}(t)) \to \pi_1(X_V)$ is surjective.

We need the following version of the Castelnuovo-De Franchis theorem.

**Theorem 4.13 (Castelnuovo-De Franchis).** — Let $X$ be a compact Kähler normal space and let $W \subset H^0(X, \Omega_X)$ be the subspace of dimension $d \ge 2$ such that

- $\mathrm{Im}(\Lambda^d W \to H^0(X, \Omega_X^d)) = 0$;

- for every hyperplane $W' \subset W$, $\mathrm{Im}(\Lambda^{d-1}W' \to H^0(X, \Omega_X^{d-1})) \neq 0$.

Then there is a projective normal variety $S$ of dimension $d-1$ and a fibration $f: X \to S$ such that $W \subset f^*H^0(S, \Omega_S)$.

To apply Theorem 4.13, we need to show the existence of a linear subspace $W \subset H^0(X, \Omega_X)$ as in the theorem.

**Lemma 4.14.** — Let $X$ be a projective normal variety and let $V \subset H^0(X, \Omega_X)$. Let $r$ be the largest integer such that $\mathrm{Im}[\Lambda^r V \to H^0(X, \Omega_X^r)] \neq 0$. Assume that $r < \dim_{\mathbb{C}} V$. There exists $W \subset H^0(X, \Omega_X)$ such that

(i) $2 \le \dim W \le r + 1$.

(ii) $\mathrm{Im}[\Lambda^{\dim W}W \to H^0(X, \Omega_X^{\dim W})] = 0$;

(iii) for every hyperplane $W' \subset W$, we always have $\mathrm{Im}[\Lambda^{\dim W^{-1}}W' \to H^0(X, \Omega_X^{\dim W^{-1}})] \neq 0$.

*Proof.* — By our assumption there exist $\{\omega_1, \dots, \omega_r\} \subset V$ such that $\omega_1 \wedge \dots \wedge \omega_r \neq 0$. Let $W_0 \subset V$ be the subspace generated by $\{\omega_1, \dots, \omega_r\}$. Since $r < \dim_{\mathbb{C}} V$, there exists $\omega \in V \setminus W_0$.

Pick a point $x \in X$ such that $\omega_1 \wedge \dots \wedge \omega_r(x) \neq 0$. Then there exists a coordinate system $(U; z_1, \dots, z_n)$ centered at $x$ such that $dz_i = \omega_i$ for $i = 1, \dots, r$. Write $\omega = \sum_{i=1}^n a_i(z)dz_i$. By our choice of $r$, we have $\omega_1 \wedge \dots \wedge \omega_r \wedge \omega = 0$. It follows that

- $a_j(z) = 0$ for $j = r + 1, \dots, n$;

- at least one of $a_1(z), \dots, a_r(z)$ is not constant.

Let $k+1$ be the transcendental degree of $\{1, a_1(z), \dots, a_r(z)\} \subset \mathbb{C}(U)$. Then $k \ge 1$. We assume that $1, a_1(z), \dots, a_k(z)$ is linearly independent for the transcendental extension $\mathbb{C}(U)/\mathbb{C}$. One can check by an easy linear algebra that the subspace $W$ generated $\{\omega_1, \dots, \omega_k, \omega\}$ is an element of $E$. The lemma is proved. $\square$

**Lemma 4.15.** — Let $X$ be a projective normal variety and let $V \subset H^0(X, \Omega_X)$. Let $r$ be the largest integer such that $\mathrm{Im}[\Lambda^r V \to H^0(X, \Omega_X^r)] \neq 0$, which will be called generic rank of $V$. Consider the partial Albanese morphism $g_V : X \to \mathcal{A}_{X,V}$ induced by $V$. Let $V_0 \subset H^0(\mathcal{A}_{X,V}, \Omega_{\mathcal{A}_{X,V}}^1)$ be the linear subspace such that $g_V^*V_0 = V$. Let $f: X \to S$ be the Stein factorization of $g_V$ with $q: S \to \mathcal{A}_{X,V}$ the finite morphism. Consider $\mathbb{V} := q^*V_0$. Assume that

$$ \mathrm{Im}[\Lambda^{\dim \mathbb{V}} \mathbb{V} \to H^0(\mathbb{V}, \Omega_{\mathbb{V}}^{\dim \mathbb{V}})] \neq 0 $$

for every proper closed subvariety $Z \subsetneq S$. Then there are two possibilities.
— either

$$ \mathrm{Im}[\Lambda^{\dim S} \mathbb{V} \to H^0(S, \Omega_S^{\dim S})] \neq 0; $$

— or $r = \dim_{\mathbb{C}} V$.

*Proof.* — Assume that both

$$ \mathrm{Im}[\Lambda^{\dim S} \mathbb{V} \to H^0(S, \Omega_S^{\dim S})] = 0, $$

and $r < \dim_{\mathbb{C}} V$. Therefore, $r < \dim S \le \dim X$. By Lemma 4.14 there is a subspace $W \subset V$ with $\dim_{\mathbb{C}} W = k + 1 \le r + 1$ such that $\mathrm{Im}[\Lambda^{\dim W} W \to H^0(X, \Omega_{\dim W}^1)] = 0$, and for any subspace $W' \subsetneq W$, we always have $\mathrm{Im}[\Lambda^{\dim W'} W' \to H^0(X, \Omega_{\dim W'}^1)] \neq 0$. By our assumption, we have $\dim_{\mathbb{C}} W \le \dim X$. By Theorem 4.13, there is a fibration $p : X \to B$ with $B$ a projective normal variety with $\dim B = \dim W - 1 \le \dim X - 1$ such that $W \subset p^*H^0(B, \Omega_B^1)$. In particular, the generic rank of the forms in $W$ is $\dim W - 1$. Consider the partial Albanese morphism $g_W : X \to \mathcal{A}_{X,W}$ associated with $W$. We shall prove that $p$ can be made as the Stein factorisation of $g_W$.

Note that each fiber of $p$ is contracted by $g_W$. Therefore, we have a factorisation $X \xrightarrow{p} B \xrightarrow{h} \mathcal{A}_{X,W}$. Note that there exists a linear space $W_0 \subset H^0(\mathcal{A}_{X,W}, \Omega_{\mathcal{A}_{X,W}}^1)$ such that $W = g_W^*W_0$. If $\dim h(B) < \dim B$, then the generic rank of $W$ is less or equal to $\dim h(B)$. This contradicts with Theorem 4.13. Therefore, $\dim h(B) = \dim B$. Let $X \xrightarrow{p'} B' \to \mathcal{A}_{X,W}$ be the Stein factorisation of $g_W$. Then there exists a birational morphism $\nu : B \to B'$ such that $p' = \nu \circ p$. We can thus replace $B$ by $B'$, and $p$ by $p'$.

Recall that $f : X \to S$ is the Stein factorisation of the partial Albanese morphism $g_V : X \to \mathcal{A}_{X,V}$ associated with $V$. As $g_W$ factors through the natural quotient map $\mathcal{A}_{X,V} \to \mathcal{A}_{X,W}$, it follows that $p : X \to B$ factors through $X \xrightarrow{f} S \xrightarrow{\nu} B$.

Assume that $\dim S = \dim B$. Then $\nu$ is birational. Since $\dim B = \dim W - 1$ and the generic rank of $W$ is $\dim W - 1$, it follows that

$$ \mathrm{Im}[\Lambda^{\dim S} \mathbb{V} \to H^0(S, \Omega_S^{\dim S})] \neq 0. $$

This contradicts with our assumption at the beginning. Hence $\dim S > \dim B$.

Let $Z$ be a general fiber of $\nu$ which is positive-dimensional. Since $W \subset p^*H^0(B, \Omega_B^1)$, and we have assumed that the generic rank of $\mathbb{V}$ is less than $\dim S$, it follows that the generic rank of $\mathrm{Im}[\mathbb{V} \to H^0(Z, \Omega_Z^1)]$ is less than $\dim Z$. This implies that

$$ \mathrm{Im}[\Lambda^{\dim Z} \mathbb{V} \to H^0(Z, \Omega_Z^{\dim Z})] = 0, $$

which contradicts with our assumption. Therefore we obtain a contradiction. The lemma is proved. $\square$

**Remark 4.16.** — Let $Y$ be a normal projective variety. Let $\mathcal{Q} = \{\varrho_i : \pi_1(Y) \to \mathrm{GL}_N(K_i)\}_{i=1,...,k}$ be a family of reductive representations where $K_i$ are non-archimedean local field. Let $\pi : X \to Y$ be a Galois cover dominating all spectral covers induced by $\varrho_i$. Let $V \subset H^0(X, \Omega_X)$ be the set of all spectral forms (cf. § 1.9 for definitions). We use the same notations as in Lemma 4.15. Considering Katzarkov-Eyssidieux reduction maps $s_{\mathcal{Q}} : Y \to S_{\mathcal{Q}}$ and $s_{\pi^*\mathcal{Q}} : X \to S_{\pi^*\mathcal{Q}}$. One can check that, for every closed subvariety $Z \subset S_{\mathcal{Q}}$, $\{T_{\mathcal{Q}}^{\dim Z}\} \cdot Z > 0$ if and only if for any closed subvariety $W \subset S_{\pi^*\mathcal{Q}}$ dominating $Z$ under $\sigma_{\pi} : S_{\pi^*\mathcal{Q}} \to S_{\mathcal{Q}}$ defined in (1.2), one has

$$ \mathrm{Im}[\Lambda^{\dim W} \mathbb{V} \to H^0(W, \Omega_W^{\dim W})] \neq 0. $$

In particular, $V$ is perfect if and only if $\{T_{\mathcal{Q}}\}$ is a Kähler class by Theorem 1.12.

**Theorem 4.17.** — Let $X$ be a smooth projective variety and let $\mathcal{Q} := \{\varrho_i : \pi_1(X) \to \mathrm{GL}_N(K_i)\}_{i=1,...,k}$ be a family of reductive representations where $K_i$ is a non-archimedean local field. Let $S_{\mathcal{Q}} : X \to S_{\mathcal{Q}}$ be the Katzarkov-Eyssidieux reduction map. Let $T_{\mathcal{Q}}$ be the canonical (1,1)-current on $S_{\mathcal{Q}}$ associated with $\mathcal{Q}$ defined in Definition 1.23. Denote by $H_i$ the Zariski closure of $\varrho_i(\pi_1(X))$. Assume that for any proper closed subvariety $\Sigma \subsetneq S$, one has $\{T_{\mathcal{Q}}\}^{\dim \Sigma} \cdot \Sigma > 0$. Then
- either $\{T_{\mathcal{Q}}\}^{\dim S_{\mathcal{Q}} \cdot S_{\mathcal{Q}}} > 0$;

- or the reduction map $s_{\sigma_i} : X \to S_{\sigma_i}$ coincides with $s_{\varrho_i} : X \to S_{\varrho_i}$ for each $i$, where $\sigma_i : \pi_1(X) \to (H_i/\mathcal{D}H_i)(K_i)$ is the composition of $\varrho_i$ with the group homomorphism $H_i \to H_i/\mathcal{D}H_i$.

*Proof.* — Assume that $\{T_{\mathcal{Q}}\}^{\dim S_{\mathcal{Q}} \cdot S_{\mathcal{Q}}} = 0$. Let $Y \to X$ be a Galois cover which dominates all spectral covers of $\varrho_i$. We pull back all the spectral one forms on $Y$ to obtain a subspace $V \subset H^0(Y, \Omega_Y^1)$. Consider the partial Albanese morphism $g_V : Y \to \mathcal{A}_{Y,V}$ associated to $V$, then $s_{\pi^*\varrho} : Y \to S_{\pi^*\varrho}$ is its Stein factorization with $q : S_{\pi^*\varrho} \to \mathcal{A}_{Y,V}$ the finite morphism. Note that there is a $\mathbb{C}$-linear subspace $W \subset H^0(S_{\pi^*\varrho}, \Omega_{S_{\pi^*\varrho}}^1)$ such that $s_{\pi^*\varrho}^*W = V$.

$$ \begin{tikzcd} Y \arrow[r, "\pi"] \arrow[d, "s_{\pi^*\varrho}"] & X \arrow[d, "s_{\varrho}"] \\ s_{\pi^*\varrho_i} \arrow[r, "\sigma_{\pi}", "\cong"] \arrow[d, "q_i"] & S_{\varrho} \arrow[d, "p_i"] \\ S_{\pi^*\varrho_i} & S_{\varrho_i} \arrow[l, "\cong"] \arrow[d, "s_{\varrho_i}"] \end{tikzcd} $$

Note that $\sigma_{\pi}$ is finite surjective morphism. By Lemma 1.24 we have $T_{\pi^*\varrho} = \sigma_{\pi}^*T_{\varrho}$. By our assumption, for an proper closed subvariety $\Xi \subsetneq S_{\varrho}$, one has $\{T_{\varrho}\}^{\dim \Xi} \cdot \Xi > 0$. Hence for an proper closed subvariety $\Xi \subsetneq S_{\pi^*\varrho}$, one has $\{T_{\pi^*\varrho}\}^{\dim \Xi} \cdot \Xi > 0$. According to Remark 4.16, this implies that

$$ \mathrm{Im}[\Lambda^{\dim \Xi} W \to H^0(\Xi, \Omega_{\Xi}^{\dim \Xi})] \neq 0. $$

Since $\{T_{\varrho}\}^{\dim S \cdot S} = 0$, it follows that $\{T_{\pi^*\varrho}\}^{\dim S_{\pi^*\varrho} \cdot S_{\pi^*\varrho}} = 0$. This implies that

$$ \mathrm{Im}[\Lambda^{\dim S_{\pi^*\varrho} W} \to H^0(S_{\pi^*\varrho}, \Omega_{S_{\pi^*\varrho}}^{\dim S_{\pi^*\varrho}})] = 0. $$

Let $r$ be the generic rank $V$. According to Remark 4.16, we have $r = \dim S_{\pi^*\varrho} - 1$. By Lemma 4.15, we have $r = \dim_{\mathbb{C}} V$. Therefore, $\mathrm{Im}[\Lambda^r V \to H^0(Y, \Omega_Y^r)] \simeq \mathbb{C}$.

**Claim 4.18.** — For any non-zero $\eta \in \mathrm{Im}[\Lambda^r V \to H^0(Y, \Omega_Y^r)]$, each irreducible component $Z'$ of $(\eta = 0)$ satisfies that $s_{\pi^*\varrho}(Z')$ is a proper subvariety of $S_{\pi^*\varrho}$.

*Proof.* — Assume that this is not the case. Let $Z \to Z'$ be a desingularization. Set $V' := \mathrm{Im}[V \to H^0(Z, \Omega_Z^1)]$. Denote by $r'$ the generic rank of $V'$. Then $r' < r$ as $Z'$ is an irreducible component of $(\eta = 0)$. Write $\iota : Z \to Y$ and $g : Z \to X$ for the natural map. Then the Katzarkov-Eyssidieux reduction $s_{g^*\varrho} : Z \to S_{g^*\varrho}$ associated with $g^*\varrho$ is the Stein factorization of the partial Albanese morphism $g_{V'} : Z \to \mathcal{A}_{Z,V'}$. We have the diagram

$$ \begin{tikzcd} Z \arrow[r, "\iota"] \arrow[d, "s_{g^*\varrho}"] & Y \arrow[d, "s_{\varrho}"] & X \arrow[d, "s_{\varrho}"] \\ S_{g^*\varrho} \arrow[r, "\sigma_{\iota}", "\cong"] & S_{\pi^*\varrho} \arrow[r, "\sigma_{g}", "\cong"] & S_{\varrho} \arrow[u, "\cong"] \arrow[d, "s_{\varrho}"] \end{tikzcd} $$

such that $\sigma_{\iota}$ is a finite surjective morphism as we assume that $s_{\pi^*\varrho}(Z') = S_{\pi^*\varrho}$. Let $\Sigma \subsetneq S_{g^*\varrho}$ be a proper closed subvariety. Let $\Sigma' := \sigma_g(\Sigma)$. Since $\{T_{\varrho}\}^{\dim \Sigma'} \cdot \Sigma' > 0$ by our assumption, by Lemma 1.24 $\{T_{g^*\varrho}\}^{\dim \Sigma} \cdot \Sigma > 0$. By Remark 4.16, it follows that the generic rank $r'$ of $V'$ is equal to $\dim S_{g^*\varrho} - 1 = \dim S_{\pi^*\varrho} - 1$. This contradicts with the fact that $r' < r = \dim S_{\pi^*\varrho} - 1$. The claim is proved. $\square$

By the above claim, $s_{\pi^*\varrho}(Z')$ is a proper subvariety of $S_{\pi^*\varrho}$. Therefore, we have $\{T_{\varrho}\}^{\dim S_{g^*\varrho} \cdot S_{g^*\varrho}} > 0$. Hence for each irreducible component $Z$ of $(\eta = 0)$, $\mathrm{Im}[V \to H^0(Z, \Omega_Z^1)]$ is perfect by Remark 4.16 once again. We can apply Theorem 4.12 to conclude that for the holomorphic map $\tilde{g}_V : Y_V \to V^*$ defined as (4.1), $(Y_V, \tilde{g}_V^{-1}(t))$ is 1-connected
for any $t \in V^*$. For the covering $Y_V \to Y$, we know that $\mathrm{Im}[\pi_1(Y_V) \to \pi_1(Y)]$ contains the derived subgroup $\mathcal{D}\pi_1(Y)$ of $\pi_1(Y)$. Then $\pi^*\varrho_i(\mathrm{Im}[\pi_1(Y_V) \to \pi_1(Y)])$ contains $\pi^*\varrho_i(\mathcal{D}\pi_1(Y))$. On the other hand, since $(Y_V, \tilde{g}_V^{-1}(t))$ is 1-connected for any $t \in V^*$, it follows that $\pi^*\varrho_i(\mathrm{Im}[\pi_1(\tilde{g}_V^{-1}(t)) \to \pi_1(Y)])$ contains $\pi^*\varrho_i(\mathcal{D}\pi_1(Y))$. Note that $V$ is consists of all the spectral forms of $\pi^*\varrho_i$ for all $i$, hence each $\pi^*\varrho_i$-equivariant harmonic mapping $u_i$ vanishes over each connected component $p^{-1}(\tilde{g}_V^{-1}(t))$ where $p : \tilde{Y} \to Y$ is the universal covering. Then $\pi^*\varrho_i(\mathrm{Im}[\pi_1(\tilde{g}_V^{-1}(t)) \to \pi_1(Y)])$ fixes a point $P$ in the Bruhat-Tits building, which implies that it is bounded. Therefore, $\pi^*\varrho_i(\mathcal{D}\pi_1(Y))$ is also bounded. Note that the image of $\pi_1(Y) \to \pi_1(X)$ is a finite index subgroup of $\pi_1(X)$. Hence $\varrho_i(\mathcal{D}\pi_1(X))$ is also bounded for each $\rho_i$. The theorem then follows from Lemma 4.8. $\square$

**4.4. A factorization theorem.** — As an application of Theorem 4.17, we will prove the following factorization theorem which partially generalizes previous theorem by Corlette-Simpson [CS08]. This result is also a warm-up for the proof of Theorem 4.21.

**Theorem 4.19.** — Let $X$ be a smooth projective variety and let $G$ be an almost simple algebraic group defined over $K$. Assume that $\rho : \pi_1(X) \to G(K)$ is a Zariski dense representation such that for any morphism $f : Z \to X$ from any positive dimensional smooth projective variety $Z$ to $X$ which is birational to the image, the Zariski closure of $f^*\rho(\pi_1(Z))$ is a semisimple algebraic group. Then after we replace $X$ by a finite étale cover and a birational modification, there is an algebraic fiber space $f : X \to Y$ and a big and Zariski dense representation $\tau : \pi_1(Y) \to G(K)$ such that $f^*\tau = \rho$. Moreover, $\dim Y \le \mathrm{rank}_K G$.

*Proof.* — We know that there after we replace $X$ by a finite étale cover and a birational modification, there are an algebraic fiber space $f : X \to Y$ over a smooth projective variety $Y$ and a big and Zariski dense representation $\tau : \pi_1(Y) \to G(K)$ such that $f^*\tau = \rho$. We will prove that $\dim Y \le \mathrm{rank}_K G$.

**Claim 4.20.** — The $(1,1)$-class $\{T_\tau\}$ on $S_\tau$ is Kähler, where $T_\tau$ is the canonical current on $S_\tau$ associated to $\tau$.

*Proof.* — By Theorem 1.12, it is equivalent to prove that for any closed subvariety $\Sigma \subset S_\tau$, $\int_\Sigma \{T_\tau\}^{\dim \Sigma} > 0$. We will prove it by induction on $\dim \Sigma$.

**Induction.** Assume that for every closed subvariety $\Sigma \subset S_\tau$ of dimension $\le r-1$, $\{T_\tau\}^{\dim \Sigma} \cdot \Sigma > 0$.

Let $\Sigma$ be any closed subvariety of $S_\tau$ with $\dim \Sigma = r$. Let $Z$ be a desingularization of any irreducible component in $s_\tau^{-1}(\Sigma)$ which is surjective over $\Sigma$. Denote by $f : Z \to Y$.

$$ \begin{array}{ccc} Z & \xrightarrow{f} & Y \\ \big|_{S_{f^*\tau}} & & \big|_{S_\tau} \\ S_{f^*\tau} & \xrightarrow{\sigma_f} & S_\tau \end{array} $$

By Lemma 1.24, $\sigma_f$ is a finite morphism whose image is $\Sigma$ and $T_{f^*\tau} = \sigma_f^*T_\tau$.

We first prove the induction for $\dim \Sigma = 1$. In this case $\dim S_{f^*\tau} = 1$. Since the spectral forms associated to $f^*\tau$ are not constant, it follows that $T_{f^*\tau}$ is big. By Lemma 1.24, $\{T_\tau|_{\Sigma}\}$ is big. Therefore, we prove the induction when $\dim \Sigma = 1$.

Assume now the induction holds for closed subvariety $\Sigma \subset S_\tau$ with $\dim \Sigma \le r-1$. Let us deal with the case $\dim \Sigma = r$. By Lemma 1.24 and the induction, we know that for any closed proper positive dimensional subvariety $\Xi \subset S_{f^*\tau}$, we have $\{T_{f^*\tau}\}^{\dim \Xi} \cdot \Xi > 0$. Note that the conditions in Theorem 4.17 for $f^*\tau$ is fulfilled. Therefore, there are two possibilities:

— either $\{T_{f^*\tau}\}^r \cdot S_{f^*\tau} > 0$;