# Hyperbolicity in presence of a large local system

Yohan Brunebarbe  Institut de Mathématiques de Bordeaux, Université de
Bordeaux, 351 cours de la Libération, F-33405 Talence
<EMAIL_ADDRESS>
###### Abstract.

We prove that the projective complex algebraic varieties admitting a large
complex local system satisfy a strong version of the Green-Griffiths-Lang
conjecture.

##  1\. Introduction

Let $X$ be a (non-necessarily smooth nor irreducible, but reduced) proper
complex algebraic variety. Following Lang, we define three ‘special’ subsets
of $X$ that measure three different kind of hyperbolic behaviour:

  * •

The subset $\operatorname{Sp}(X)\subset X$ is defined as the union of the
(positive-dimensional) integral closed subvarieties of $X$ that are not of
general type.

  * •

The subset $\operatorname{Sp}_{ab}(X)\subset X$ is defined as the union of all
images of non-constant rational maps $A\dasharrow X$ from an abelian variety
$A$.

  * •

The subset $\operatorname{Sp}_{h}(X)\subset X$ is defined as the union of all
entire curves in $X$, i.e. the images of non-constant holomorphic maps
$\mathbb{C}\to X$.

It is not clear from their definition whether these special subsets are
Zariski-closed in $X$. The inclusions
$\operatorname{Sp}_{ab}(X)\subset\operatorname{Sp}(X)$ and
$\operatorname{Sp}_{ab}(X)\subset\operatorname{Sp}_{h}(X)$ always hold, see
Proposition 2.1.

The following conjecture is a strong version of conjectures of Green-Griffiths
[GG80] and Lang [Lan86].

######  Conjecture A.

Let $X$ be a projective complex algebraic variety. Then

  1. (1)

$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)$,

  2. (2)

$\operatorname{Sp}(X)$ is a closed algebraic subvariety of $X$,

  3. (3)

$\operatorname{Sp}(X)\neq X$ if and only if $X$ is of general type111To be
consistent with the conjecture, one says that a non-necessarily irreducible
projective variety is of generaly type if at least one of its irreducible
component is of general type..

The main result in this paper is the following (see [Zuo96, Yam10, CCE15,
JR20] for other hyperbolicity results in presence of a local system).

######  Theorem A.

Let $X$ be a projective complex algebraic variety. Assume that for every
integral closed subvariety $Z\hookrightarrow X$ there exists a complex local
system on $X$ whose pull-back to the normalization of $Z$ is not
isotrivial222A local system on an algebraic variety $X$ is called isotrivial
if it becomes trivial on a finite étale cover of $X$.. Then $X$ satisfies
Conjecture A.

In particular, Theorem A shows that Conjecture A holds if $X$ admits a large
complex local system, i.e. a complex local system $\mathcal{L}$ such that for
every integral closed subvariety $Z\hookrightarrow X$ the pull-back of
$\mathcal{L}$ to the normalization of $Z$ is not isotrivial333At least when
$X$ is normal, the assumptions of Theorem A are in fact equivalent to the
existence of a large complex local system on $X$, see Proposition 6.3..
Equivalently, the Galois étale cover $X^{\mathcal{L}}\to X$ associated to the
kernel of the monodromy representation of $\mathcal{L}$ does not have any
positive-dimensional compact complex subspaces, cf. Proposition 3.4. This
holds for example when the complex space $X^{\mathcal{L}}$ is Stein, and turns
out to be equivalent at least when $X$ is normal [Eys04, EKPR12]. Examples
include:

  1. (1)

Projective complex algebraic varieties admitting a finite morphism to an
abelian variety. In that case, Theorem A follows from works of Bloch [Blo26],
Ueno [Uen75], Ochiaia [Och77], Kawamata [Kaw80] and Yamanoi [Yam15a].

  2. (2)

Projective complex algebraic varieties admitting a (graded-polarizable)
variation of $\mathbb{Z}$-mixed Hodge structure with a finite period map. When
in addition the Hodge structures are pure, then it follows from works of
Griffiths and Schmid [GS69] that
$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)=\varnothing$.

In a nutshell, the proof of Theorem A consists in reducing the general case to
these two special cases by using general structure results from non-abelian
Hodge theory. A crucial input is also given by a result from [Bru] about the
behaviour of the different special subsets in a family of varieties of maximal
Albanese dimension.

#### Acknowledgements

We thank Marco Maculan for some useful discussions.

#### Conventions

A _complex algebraic variety_ is a separated reduced finite type
$\mathbb{C}$-scheme. One often makes no distinction between a complex
algebraic variety and the associated complex analytic space.
A _fibration_ between two normal complex algebraic varieties is a proper
surjective morphism $X\to Y$ with connected fibers.

##  2\. Generalities on special sets

We gather some easy properties of special sets for future reference.

######  Proposition 2.1.

Let $X$ be a proper complex algebraic variety. Then

$\operatorname{Sp}_{ab}(X)\subset\operatorname{Sp}(X)\text{ and
}\operatorname{Sp}_{ab}(X)\subset\operatorname{Sp}_{h}(X).$

###### Proof.

The inclusion $\operatorname{Sp}_{ab}(X)\subset\operatorname{Sp}(X)$ follows
from the fact that the image of a non-constant rational map $A\dasharrow X$
from an abelian variety $A$ is not of general type (this is an easy
consequence of a special case of Iitaka conjecture proved by Viehweg, cf.
[Vie83, Corollary IV]). On the other hand, for every rational map $Y\dasharrow
X$ with $Y$ smooth projective, there exists a sequence of blow-ups
$Y^{\prime}\to Y$ along smooth subvarieties such that the composite rational
map $Y^{\prime}\to Y\dasharrow X$ is defined everywhere. Note that the
exceptional locus of $Y^{\prime}\to Y$ is covered by rational curves. The
inclusion $\operatorname{Sp}_{ab}(X)\subset\operatorname{Sp}_{h}(X)$ follows,
since every abelian variety is covered by entire curves. ∎

######  Proposition 2.2.

Let $f:X\to Y$ be a finite étale cover between projective complex algebraic
varieties. Then, for any $\ast\in\\{\varnothing,ab,h\\}$,

$\operatorname{Sp}_{\ast}(X)=f^{-1}\left(\operatorname{Sp}_{\ast}(Y)\right).$

######  Proposition 2.3.

Let $f:X\to Y$ be a birational morphism between irreducible projective complex
algebraic varieties. Then, for any $\ast\in\\{\varnothing,ab,h\\}$,
$\operatorname{Sp}_{\ast}(X)$ is Zariski-dense in $X$ if, and only if,
$\operatorname{Sp}_{\ast}(Y)$ is Zariski-dense in $Y$.

###### Proof.

Let $Z\subset Y$ be a strict closed subvariety such that $f$ is an isomorphism
over $Y\backslash Z$. Then
$f\left(\operatorname{Sp}_{\ast}(X)\right)\backslash
Z=\operatorname{Sp}_{\ast}(Y)\backslash Z$, and the result follows. ∎

######  Definition 2.4.

Let $X\to Y$ be a proper morphism between projective complex algebraic
varieties. For any $\ast\in\\{\varnothing,ab,h\\}$, we let

$\operatorname{Sp}_{\ast}(X/Y):=\bigcup_{y\in
Y}\operatorname{Sp}_{\ast}(X_{y}).$

######  Proposition 2.5.

Let $f:X\to Y$ be a proper morphism between projective complex algebraic
varieties. Then, for any $\ast\in\\{\varnothing,ab,h\\}$,

$\operatorname{Sp}_{\ast}(X)\subset
f^{-1}\left(\operatorname{Sp}_{\ast}(Y)\right)\cup\operatorname{Sp}_{\ast}(X/Y).$

######  Lemma 2.6.

Consider a proper complex algebraic variety $X$ and a finite collection of
surjective morphisms $X\rightarrow S_{i}$, $i\in I$, where the $S_{i}$’s are
proper complex algebraic varieties. If the induced morphism
$p_{i}:X\rightarrow\prod_{i\in I}S_{i}$ is finite, then:

  1. (1)

If $\overline{\operatorname{Sp}_{h}(S_{i})}\neq S_{i}$ for every $i\in I$,
then $\overline{\operatorname{Sp}_{h}(X)}\neq X$.

  2. (2)

If $\overline{\operatorname{Sp}_{ab}(S_{i})}\neq S_{i}$ for every $i\in I$,
then $\overline{\operatorname{Sp}_{ab}(X)}\neq X$.

###### Proof.

Let $\mathbb{C}\rightarrow X$ be a non-constant holomorphic map. Since the
morphism $f:X\rightarrow\prod_{i\in I}S_{i}$ is finite, at least one of the
map $\mathbb{C}\rightarrow S_{i}$ obtained by composing $\mathbb{C}\rightarrow
X$ with one of the map $p_{i}:X\rightarrow S_{i}$ is non-constant. Therefore
$\overline{\operatorname{Sp}_{h}(X)}\subset\cup_{i\in
I}\,p_{i}^{-1}(\overline{\operatorname{Sp}_{h}(S_{i})})$. Since the $p_{i}$’s
are surjective, it follows that $\overline{\operatorname{Sp}_{h}(X)}\neq X$.
The proof of the second item is similar. ∎

##  3\. Generalities on large local systems

###  3.1. Monodromy groups

Let $k$ be a field. Let $\mathcal{L}$ be a $k$-local system on a connected
complex space $X$. For any $x\in X$, one has the corresponding monodromy
representation $\pi_{1}(X,x)\to\operatorname{GL}(\mathcal{L}_{x})$. By
definition, the monodromy group of $\mathcal{L}$ is the image of the monodromy
representation, and the algebraic monodromy group of $\mathcal{L}$ is the
Zariski-closure of the image of the monodromy representation. Differents
points in $X$ yield isomorphic groups, respectively $k$-algebraic groups.

###  3.2. Equivalent definitions

######  Definition 3.1.

A local system $\mathcal{L}$ on a projective complex algebraic variety $X$ is
called large if for every integral closed subvariety $Z\hookrightarrow X$, the
pull-back of $\mathcal{L}$ to the normalization of $Z$ is not isotrivial.

If $f:X\to Y$ is a dominant morphism between two irreducible normal complex
algebraic varieties, the image of the induced morphism of groups
$f_{\ast}:\pi_{1}(X)\to\pi_{1}(Y)$ has finite index in $\pi_{1}(Y)$ [Cam91].
Therefore, a local system $\mathcal{L}$ on a projective complex algebraic
variety $X$ is large if, and only if, for any non-constant morphism $f:Y\to X$
from an irreducible normal projective complex algebraic variety $Y$ the local
system $f^{-1}\mathcal{L}$ is not isotrivial. With this observation, the
following results are immediate.

######  Proposition 3.2.

Let $X$ be a projective complex algebraic variety and $\mathcal{L}$ be a large
local system on $X$. Let $f:Y\to X$ be a finite morphism from another
projective complex algebraic variety $Y$. Then the pull-back local system
$f^{-1}\mathcal{L}$ on $Y$ is large.

######  Proposition 3.3.

Let $f:Y\to X$ be a finite étale morphism between projective complex algebraic
varieties. Let $\mathcal{L}$ be a complex local system on $X$. Then
$\mathcal{L}$ is large if, and only if, the pull-back local system
$f^{-1}\mathcal{L}$ on $Y$ is large.

The following result is useful to prove that a local system is large.

######  Proposition 3.4 (compare with [Kol93, Proposition 2.12]).

Consider a local system $\mathcal{L}$ on a connected projective complex
algebraic variety $X$, and denote by $X^{\mathcal{L}}\to X$ the associated
connected covering space. Then the local system $\mathcal{L}$ is large if, and
only if, the complex analytic space $X^{\mathcal{L}}$ does not contain any
positive dimensional compact complex subspaces.

###### Proof.

This is essentially the same proof as in [Kol93, Proposition 2.12]. Let
$Y\subset X^{\mathcal{L}}$ be an irreducible compact complex subspace with
normalization $\bar{Y}$. The induced holomorphic map $\bar{Y}\to X$ has
discrete (hence finite) fibres. Let $Z\subset X$ be the image of $Y$ and
$\bar{Z}$ denote its normalization. Then the monodromy group of
$\mathcal{L}_{|\bar{Z}}$ is isomorphic to the Galois group of
$\bar{Y}/\bar{Z}$, in particular it is finite. This show that (up to deck
transformations of $X^{\mathcal{L}}\to X$) there is a one-to-one
correspondence between irreducible compact complex subspaces of
$X^{\mathcal{L}}$ and irreducible compact complex subspaces $Z\subset X$ such
that the pull-back of $\mathcal{L}$ to the normalization $\bar{Z}$ is
isotrivial. ∎

###  3.3. Shafarevitch morphisms

We will use in several places the existence of the Shafarevitch morphism
associated to a complex local system.

######  Theorem 3.5.

Let $\mathcal{L}$ be a complex local system on a projective normal complex
algebraic variety. Then there exists a surjective morphism with connected
fibres
$\operatorname{sh}_{X}^{\mathcal{L}}:X\rightarrow\operatorname{Sh}_{X}^{\mathcal{L}}$
onto a projective normal algebraic variety, unique up to unique isomorphism,
such that the following property holds: for any connected normal projective
complex algebraic variety $Z$ and any morphism $f:Z\rightarrow X$, the
composite map $\operatorname{sh}_{X}^{\mathcal{L}}\circ
f:Z\rightarrow\operatorname{Sh}_{X}^{\mathcal{L}}$ is constant if, and only
if, the local system $f^{\ast}\mathcal{L}$ is isotrivial. Moreover, when the
monodromy group of $\mathcal{L}$ is torsion-free, there exists a complex local
system $\mathcal{M}$ on $\operatorname{Sh}_{X}^{\mathcal{L}}$ such that
$\mathcal{L}=\left(\operatorname{sh}_{X}^{\mathcal{L}}\right)^{\ast}\mathcal{M}$.
In particular, the local system $\mathcal{M}$ is large.

The map
$\operatorname{sh}_{X}^{\mathcal{L}}:X\rightarrow\operatorname{Sh}_{X}^{\mathcal{L}}$
is called the Shafarevitch morphism associated to $\mathcal{L}$. The theorem
above is proved in [Eys04] when $X$ is smooth projective and the complex local
system $\mathcal{L}$ is semisimple, and more generally in [EKPR12] only
assuming that $X$ is smooth projective. See also [CCE15] for the case where
$X$ is a compact Kähler manifold. The existence of the Shafarevitch morphism
when $X$ is only normal is proved by applying the following result to a
desingularization of $X$.

######  Proposition 3.6.

Let $X^{\prime}\to X$ be a surjective morphism with connected fibres between
normal projective complex algebraic varieties. Let $\mathcal{L}$ be a complex
local system on $X$ and
$\operatorname{sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}:X^{\prime}\rightarrow\operatorname{Sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}$
be the Shafarevitch morphism associated to
$\mathcal{L}^{\prime}:=\nu^{-1}\mathcal{L}$. Then there is a (unique)
factorization

and the induced morphism
$X\rightarrow\operatorname{Sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}$ is the
Shafarevitch morphism associated to $\mathcal{L}$.

###### Proof.

Let $F$ be an irreducible component of the normalization of a fiber of $\nu$.
Since the induced morphism $F\to X$ is constant, the restriction of
$\mathcal{L}$ to $F$ is trivial, therefore $F$ is mapped to a point by the
composition $F\to
X^{\prime}\rightarrow\operatorname{Sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}$.
Since the fibers of $\nu$ are connected and $X$ is normal, this shows that
$\operatorname{sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}:X^{\prime}\rightarrow\operatorname{Sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}$
factorizes through a map
$X\rightarrow\operatorname{Sh}_{X^{\prime}}^{\mathcal{L}^{\prime}}$. The easy
verification that this is the Shafarevitch morphism associated to
$\mathcal{L}$ is left to the reader. ∎

####  3.3.1. The abelian case

We recall a construction of the Shafarevitch morphism when the monodromy group
is abelian. Let $X$ be a smooth projective complex algebraic variety. Let
$\mathcal{L}$ be a complex local system on $X$ with abelian monodromy, and let
$\pi_{1}(X)\to\Gamma$ denote its monodromy representation. Since by assumption
$\Gamma$ is abelian, the monodromy representation factorizes through the
abelianization $\pi_{1}(X)\to\operatorname{H}_{1}(X,\mathbb{Z})$ of
$\pi_{1}(X)$. Let $\operatorname{Alb}(X)\to A_{\mathcal{L}}$ be the quotient
of $\operatorname{Alb}(X)$ by the biggest abelian subvariety $T$ of
$\operatorname{Alb}(X)$ such that
$\pi_{1}(T)\subset\ker(H_{1}(X,\mathbb{Z})\to\Gamma)$. Then the Shafarevich
morphism associated to $\mathcal{L}$ coincides with the Stein factorization of
the composition of the Albanese morphism $X\to\operatorname{Alb}(X)$ with
$\operatorname{Alb}(X)\to A_{\mathcal{L}}$.

####  3.3.2. The solvable case

We recall a construction of the Shafarevitch morphism when the monodromy group
is solvable, cf. [CCE15, Théorème 4.4].

######  Proposition 3.7.

Let $\mathcal{L}$ be a complex local system on a normal projective complex
algebraic variety $X$. Let
$\pi_{1}(X)\to\Gamma\subset\mathrm{Gl}(n,\mathbb{C})$ be its monodromy
representation. Assume that its monodromy group $\Gamma$ is solvable. If
moreover the derived group $[\Gamma,\Gamma]$ is nilpotent, then the
Shafarevitch morphism associated to $\mathcal{L}$ is equal to the Shafarevitch
morphism associated to the representation $\pi_{1}(X)\to\Gamma^{ab}$. In
particular, $X$ admits a finite morphism to an abelian variety.

###### Proof.

Thanks to Proposition 3.6, one can assume that $X$ is smooth. Then the
Shafarevitch morphism of $\pi_{1}(X)\to\Gamma^{ab}$ is the Stein factorization
of a morphism from $X$ to an abelian variety $T$. Let $Z$ be a smooth
projective complex algebraic variety and $Z\to X$ a morphism. Assume that the
composition $f:Z\to X\to T$ is constant. Then, up to replace $Z$ by a finite
étale cover, one can assume that the induced homomorphism
$\pi_{1}(Z)\to\Gamma^{ab}$ is trivial, so that
$f_{\ast}\pi_{1}(Z)\subset[\Gamma,\Gamma]$. Thanks to Lemma 3.8 below,
$f_{\ast}\left(\pi_{1}(Z)/C^{k}\pi_{1}(Z)\right)=f_{\ast}\pi_{1}(Z)/C^{k}f_{\ast}\pi_{1}(Z)$
has finite image for every positive integer $k$. Since
$f_{\ast}\pi_{1}(Z)\subset[\Gamma,\Gamma]$ and $[\Gamma,\Gamma]$ is nilpotent
by assumption, it follows that $C^{k}f_{\ast}\pi_{1}(Z)=\\{0\\}$ for $k\gg 1$.
Therefore $f_{\ast}\pi_{1}(Z)$ is finite. ∎

######  Lemma 3.8.

Let $f:X\to Y$ be a morphism between two smooth projective complex algebraic
varieties. If the induced $\mathbb{Q}$-linear map
$f_{\ast}:\operatorname{H}_{1}(X,\mathbb{Q})\to\operatorname{H}_{1}(Y,\mathbb{Q})$
is zero, then $f_{\ast}\left(\pi_{1}(X)/C^{k}\pi_{1}(X)\right)$ has finite
image in $\pi_{1}(Y)/C^{k}\pi_{1}(Y)$ for every positive integer $k$. Here,
for every group $G$, $\\{C^{k}G\\}_{k\geq 0}$ denote the descending central
series of $G$, defined by $C^{0}G=G$ and $C^{k+1}=[G,C^{k}G]$ for every
integer $k\geq 0$.

In Proposition 3.7, the condition that $[\Gamma,\Gamma]$ is nilpotent is true
up to replacing $X$ by a finite étale cover defined from a finite index
subgroup of $\Gamma$ thanks to the following observation.

######  Proposition 3.9.

If $\Gamma$ is a subgroup of $\mathrm{Gl}(n,\mathbb{C})$ whose Zariski-closure
is connected and solvable, then $[\Gamma,\Gamma]$ is nilpotent.

###### Proof.

Let $\operatorname{G}$ denote the Zariski-closure of $\Gamma$ in
$\mathrm{Gl}(n,\mathbb{C})$. Since $\operatorname{G}$ is a connected solvable
algebraic group, its derived group $[\operatorname{G},\operatorname{G}]$ is a
connected nilpotent algebraic group. It follows that
$[\Gamma,\Gamma]\subset[\operatorname{G},\operatorname{G}]$ is nilpotent. ∎

####  3.3.3. The case of variations of Hodge structure with discrete monodromy

We recall the construction of the Shafarevtich morphism when the complex local
system underlies a variation of Hodge structure with discrete monodromy.

######  Proposition 3.10.

Let $X$ be a connected normal projective complex algebraic variety and
$\mathcal{L}$ a large complex local system on $X$. Assume that $\mathcal{L}$
underlies a polarized complex variation of pure Hodge structure
$(\mathcal{L},\mathcal{F}^{\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{3.0}\end{picture}\
},h)$. Assume moreover that the monodromy group $\Gamma$ of $\mathcal{L}$ is
discrete, so that the associated period map induces a holomorphic map
$X\to\Gamma\backslash\mathcal{D}$. Then the Shafarevitch morphism associated
to $\mathcal{L}$ coincide with the Stein factorization of the proper
holomorphic map $X\to\Gamma\backslash\mathcal{D}$.

###### Proof.

See for example [CCE15, Proposition 3.5] and its proof. Besides the
algebraicity of the image of the period map, the key point is that every
horizontal holomorphic map $W\to\mathcal{D}$ from a compact complex manifold
$W$ is constant, cf. [GS69, Corollary 8.3]. Therefore, a polarized complex
variation of pure Hodge structure
$(\mathcal{L},\mathcal{F}^{\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{3.0}\end{picture}\
},h)$ on a compact analytic space is trivial if, and only if, the underlying
local system $\mathcal{L}$ is trivial. ∎

######  Corollary 3.11.

Same assumptions as in the proposition. Then the complex local system
$\mathcal{L}$ is large if, and only if, the period map
$X\to\Gamma\backslash\mathcal{D}$ associated to $(\mathcal{L},\mathcal{F},h)$
is finite.

######  Proposition 3.12.

Let $X$ be a normal projective complex variety and $\mathcal{L}$ a large
complex local system. Assume that $\mathcal{L}$ underlies a polarized complex
variation of pure Hodge structure with discrete monodromy. Then

$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)=\varnothing.$

###### Proof.

Thanks to [GS69, Corollary 9.4], every horizontal holomorphic map
$\mathbb{C}\to\mathcal{D}$ is constant. This implies that
$\operatorname{Sp}_{h}(X)=\varnothing$. A fortiori,
$\operatorname{Sp}_{ab}(X)=\varnothing$ thanks to Lemma 2.1. Finally,
$\operatorname{Sp}(X)=\varnothing$ is a reformulation of the fact that any
smooth projective complex variety that admits a polarized complex variation of
pure Hodge structure with discrete monodromy and a generically finite period
map is of general type, see e.g. [CCE15, Proposition 3.5]. ∎

###  3.4. Three canonical decompositions

######  Proposition 3.13.

Let $X$ be a normal irreducible projective complex algebraic variety
supporting a large complex local system $\mathcal{L}$. Up to replacing $X$
with a finite étale cover, there exists a surjective morphism with connected
fibers $f:X\to Y$ onto a normal irreducible projective complex algebraic
variety $Y$ such that:

  * •

$Y$ admits a large complex local system with torsion-free monodromy and a
semisimple algebraic monodromy group;

  * •

the monodromy of the restriction of $\mathcal{L}$ to the normalization of any
fiber of $f$ is solvable.

###### Proof.

Consider the monodromy representation
$\pi_{1}(X)\to\operatorname{G}(\mathbb{C})$ of $\mathcal{L}$, where
$\operatorname{G}$ is the algebraic monodromy group of $\mathcal{L}$. Up to
replacing $X$ with a finite étale cover, one can assume that
$\operatorname{G}$ is connected. Let $\operatorname{N}$ be the (solvable)
radical of $\operatorname{G}$, so that $\operatorname{N}$ is a (connected)
solvable complex algebraic group. Let $\operatorname{H}$ be the quotient of
$\operatorname{G}$ by $\operatorname{N}$, so that $\operatorname{H}$ is a
connected semisimple complex algebraic group. The induced representation
$\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ has a Zariski-dense image, and
replacing $X$ with a finite étale cover, one can assume that it has torsion-
free image thanks to Selberg Lemma.
Let $f:X\to Y$ denote the Shafarevitch morphism associated to
$\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$. In particular, $Y$ is a normal
irreducible projective complex algebraic variety and $f$ is surjective with
connected fibres. Since the representation
$\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ has torsion-free image, it
factorizes through the homomorphism $\pi_{1}(X)\to\pi_{1}(Y)$, cf. Theorem
3.5. The induced homomorphism $\pi_{1}(Y)\to\operatorname{H}(\mathbb{C})$
corresponds to a large complex local system with a semisimple algebraic
monodromy group.
Let $F$ be the normalization of an irreducible component of a fiber of $f$.
Since the induced morphism $F\to Y$ is constant, the induced homomorphism
$\pi_{1}(F)\to\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ has finite image. But
the image of $\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ is torsion-free,
hence the image of $\pi_{1}(F)\to\operatorname{H}(\mathbb{C})$ is in fact
trivial. Therefore it is contained in $\operatorname{N}(\mathbb{C})$, from
what it follows that it is solvable. ∎

######  Proposition 3.14.

Let $X$ be a normal irreducible projective complex algebraic variety
supporting a large complex local system $\mathcal{L}$. Up to replacing $X$
with a finite étale cover, there exists a surjective morphism with connected
fibers $f:X\to Y$ onto a normal irreducible projective complex algebraic
variety $Y$ such that:

  * •

$Y$ admits a large complex local system with torsion-free monodromy and a
reductive algebraic monodromy group;

  * •

the monodromy of the restriction of $\mathcal{L}$ to the normalization of any
fiber of $f$ is unipotent.

  * •

the normalization of every fiber of $f$ admits a finite morphism to an abelian
variety.

###### Proof.

The two first assertions are proved exactly as in the proof of Proposition
3.13, replacing the solvable radical $\operatorname{N}$ of $\operatorname{G}$
with the unipotent radical $\operatorname{U}$ of $\operatorname{G}$. For the
last assertion, let $F$ be the normalization of an irreducible component of a
fiber of $f$. Since the monodromy of the restriction of $\mathcal{L}$ to $F$
is unipotent, one can apply Proposition 3.6 and Proposition 3.7 to infer that
$F$ admits a finite morphism to an abelian variety. The statement follows by
taking a product over the irreducible components. ∎

######  Proposition 3.15.

Let $X$ be a normal irreducible projective complex algebraic variety
supporting a large complex local system $\mathcal{L}$. Assume that the
algebraic monodromy group $\operatorname{G}$ is reductive (equivalently, the
complex local system $\mathcal{L}$ is semisimple). Up to replacing $X$ with a
finite étale cover, there exists a surjective morphism with connected fibers
$f:X\to Y$ onto a normal irreducible projective complex algebraic variety $Y$
and a morphism to an abelian variety $g:X\to A$ such that:

  * •

$Y$ admits a large complex local system with torsion-free monodromy and a
semisimple algebraic monodromy group;

  * •

the monodromy of the restriction of $\mathcal{L}$ to the normalization of any
fiber of $f$ is commutative.

  * •

the induced morphism $(f,g):X\to Y\times A$ is finite

###### Proof.

Consider the monodromy representation
$\rho:\pi_{1}(X)\to\operatorname{G}(\mathbb{C})$ of $\mathcal{L}$, where
$\operatorname{G}$ is the algebraic monodromy group of $\mathcal{L}$. Up to
replacing $X$ with a finite étale cover, one can assume that
$\operatorname{G}$ is connected. Then
$\operatorname{T}:=\operatorname{G}/[\operatorname{G},\operatorname{G}]$ is a
torus and $\operatorname{H}:=\operatorname{G}/Z(\operatorname{G})$ is a
connected semisimple algebraic group. Let
$\rho_{\operatorname{T}}:\pi_{1}(X)\to\operatorname{T}(\mathbb{C})$ and
$\rho_{\operatorname{H}}:\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ denote the
induced representations. Since $\rho$ has Zariski-dense image, both
$\rho_{\operatorname{T}}$ and $\rho_{\operatorname{H}}$ have Zariski-dense
image. Since $\rho$ is large and the canonical morphism
$\operatorname{G}\to\operatorname{T}\times\operatorname{H}$ is an isogeny, the
morphism
$\operatorname{sh}_{X}^{\rho_{\operatorname{T}}}\times\operatorname{sh}_{X}^{\rho_{\operatorname{H}}}:X\rightarrow\operatorname{Sh}_{X}^{\rho_{\operatorname{T}}}\times\operatorname{Sh}_{X}^{\rho_{\operatorname{H}}}$
is finite. Up to replacing $X$ with a finite étale cover, one can assume that
the image of $\rho_{\operatorname{H}}$ is torsion-free, so that
$\rho_{\operatorname{H}}$ factorizes through
$\operatorname{Sh}_{X}^{\rho_{\operatorname{H}}}$. Therefore,
$\operatorname{Sh}_{X}^{\rho_{\operatorname{H}}}$ admits a large complex local
system with a semisimple algebraic monodromy group. On the other hand,
recalling the construction of the Shafarevitch morphism in the abelian case
(cf. Proposition 3.6 and section 3.3.1), it follows that
$\operatorname{Sh}_{X}^{\rho_{\operatorname{T}}}$ admits a finite morphism to
an abelian variety $A_{\rho_{\operatorname{T}}}$.
This proves the first and the third assertions if one lets $f:X\to Y$ be the
morphism
$\operatorname{sh}_{X}^{\rho_{\operatorname{H}}}:X\rightarrow\operatorname{Sh}_{X}^{\rho_{\operatorname{H}}}$
and $g:X\to A$ be the composition of
$\operatorname{sh}_{X}^{\rho_{\operatorname{T}}}:X\rightarrow\operatorname{Sh}_{X}^{\rho_{\operatorname{T}}}$
with $\operatorname{Sh}_{X}^{\rho_{\operatorname{T}}}\to
A_{\rho_{\operatorname{T}}}$.
Let $F$ be the normalization of an irreducible component of a fiber of $f$.
Since the induced morphism $F\to Y$ is constant, the induced homomorphism
$\pi_{1}(F)\to\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ has finite image. But
the image of $\pi_{1}(X)\to\operatorname{H}(\mathbb{C})$ is torsion-free,
hence the image of $\pi_{1}(F)\to\operatorname{H}(\mathbb{C})$ is in fact
trivial. Therefore it is contained in $\operatorname{T}(\mathbb{C})$, from
what it follows that it is commutative. ∎

###  3.5. Algebraic varieties with a large complex local system.

The following statement collect some known results on algebraic varieties
supporting a large complex local system.

######  Theorem 3.16.

Let $X$ be a connected normal projective complex algebraic variety with a
large complex local system $\mathcal{L}$.

  1. (1)

Assume that the monodromy group of $\mathcal{L}$ is solvable (equivalently the
algebraic monodromy group of $\mathcal{L}$ is solvable). Then, up to a finite
étale cover, $X$ is isomorphic to the product of an abelian variety by a
variety of general type.

  2. (2)

Assume that $X$ is Brody-special444A proper complex algebraic variety $X$ is
called Brody-special if there exists a Zariski-dense entire curve
$\mathbb{C}\to X$.. Then, up to a finite étale cover, $X$ is isomorphic to an
abelian variety.

  3. (3)

Assume that the algebraic monodromy group of $\mathcal{L}$ is semisimple. Then
$X$ is of general type.

  4. (4)

Assume that $X$ is weakly-special555Following Campana [Cam04], a proper
complex algebraic variety $X$ is called weakly special if it does not admit a
finite étale cover $X^{\prime}$ with a rational dominant map
$X^{\prime}\dasharrow Y$ to a positive dimensional variety of general type
$Y$.. Then, up to a finite étale cover, $X$ is isomorphic to an abelian
variety.

###### Proof.

For the first item, up to replacing $X$ by a finite étale cover, one can
assume that the derived group of the monodromy group is nilpotent, cf.
Proposition 3.9. It follows from Proposition 3.7 that $A$ admits a finite
morphism to an abelian variety. Therefore, thanks to a result of Kawamata
[Kaw81, Theorem 13], after passing to another finite étale cover, X is
biholomorphic to a product $B\times X^{\prime}$ of an abelian variety $B$ and
a projective variety of general type $X^{\prime}$ whose dimension is equal to
the Kodaira dimension $\kappa(X)$ of $X$. The second item is due to Yamanoi,
see [Yam10] and [Yam15b, Theorem 2.17]. The third item is due to Zuo [Zuo96],
see also [CCE15, Théorème 6.3] for an alternative proof. For the last item, it
follows from Proposition 3.13 that, up to replacing $X$ with a finite étale
cover, there exists a surjective morphism with connected fibers $f:X\to Y$
onto a normal irreducible projective complex algebraic variety $Y$ such that:

  * •

$Y$ admits a large complex local system with a semisimple algebraic monodromy
group, and

  * •

the monodromy of the restriction of $\mathcal{L}$ to the normalization of any
fiber of $f$ is solvable.

Thanks to the third item, $Y$ is of general type. Since $X$ is weakly-special,
it follows that $Y$ is a point and that $\mathcal{L}$ has solvable monodromy.
But then the result follows from the first item. ∎

##  4\. The special subsets coincide

######  Theorem 4.1.

Let $X$ be a projective complex algebraic variety supporting a large complex
local system $\mathcal{L}$. If $X$ is not of general type, then
$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)=X$.

###### Proof.

Thanks to Lemma 2.1, it is sufficient to prove that
$\operatorname{Sp}_{ab}(X)=X$. It is harmless to assume that $X$ is
irreducible. Note also that one can freely replace $X$ with any projective
complex algebraic variety $X^{\prime}$ not of general type and such that there
exists a finite surjective morphism $X^{\prime}\to X$. Indeed, the pull-back
to $X^{\prime}$ of the local system $\mathcal{L}$ is still large thanks to
Proposition 3.2, whereas $\operatorname{Sp}_{ab}(X^{\prime})=X^{\prime}$
implies $\operatorname{Sp}_{ab}(X)=X$. In particular, one can assume that $X$
is normal. We will also freely replace $X$ by any finite étale cover.
Up to replacing $X$ with a finite étale cover, one can assume that there
exists a fibration $f:X\to Y$ as in Proposition 3.13. The normal projective
variety $Y$ admits a large complex local system with a semisimple algebraic
group, hence it is of general type thanks to Theorem 3.16. Since by assumption
$X$ is not of general type, the (geometric) generic fibre of $f$ is a
positive-dimensional variety which is not of general type.
Let $C$ be an irreducible component of a fibre of $f$. Thanks to Theorem 4.2
below, $C$ is not of general type. On the other hand, by definition of $f$,
the restriction of $\mathcal{L}$ to the normalization of $C$ is a large
complex local system with solvable monodromy. Therefore, thanks to Theorem
3.16, the normalization of $C$ is, up to a finite étale cover, a product of a
positive dimensional abelian variety by a variety of general type. This proves
that the fibers of $f$ are covered by images of abelian varieties by finite
maps, hence a fortiori $\operatorname{Sp}_{ab}(X)=X$. ∎

######  Theorem 4.2 (Nakayama, [Nak04, Theorem VI.4.3]).

Let $X\to S$ be a projective surjective morphism with connected fibres from a
normal complex analytic variety onto a smooth curve and $0\in S$. Let
$X_{0}=\cup_{i\in I}\Gamma_{i}$ the decomposition into irreducible components.
If there is at least one irreducible component $\Gamma_{j}$ which is of
general type (i.e. its desingularisation is such), then for $s\in S$ general
the fiber $X_{s}$ is of general type.

######  Corollary 4.3.

Let $X$ be a projective complex algebraic variety supporting a large complex
local system $\mathcal{L}$. Then:

$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X).$

###### Proof.

It follows from Theorem 4.1 that
$\operatorname{Sp}(X)\subset\operatorname{Sp}_{ab}(X)$, hence the equality
thanks to Lemma 2.1. Moreover, given an entire curve $\mathbb{C}\to X$, the
normalization $Z$ of the Zariski-closure of its image in $X$ is connected and
Brody-special. Since the induced morphism $Z\to X$ is finite, the pull-back of
$\mathcal{L}$ to $Z$ is still a large complex local system (cf. Proposition
3.2). Therefore a finite étale cover of $Z$ is isomorphic to an abelian
variety thanks to Theorem 3.16. This proves the inclusion
$\operatorname{Sp}_{h}(X)\subset\operatorname{Sp}_{ab}(X)$, hence the equality
thanks to Lemma 2.1. ∎

##  5\. A non-Archimedean detour

###  5.1. Katzarkov-Zuo reductions.

The following result is due to Eyssidieux [Eys04, Proposition 1.4.7], based on
former works of Katzarkov and Zuo [Kat97, Zuo96, Zuo99].

######  Theorem 5.1.

Let $k$ be a non-Archimedean local field666A local field is assumed to be
locally compact by definition. Therefore, a non-Archimedean local field is
either a finite extension of $\mathbb{Q}_{p}$ for some prime $p$ or a field of
formal Laurent series $\mathbb{F}_{q}((T))$ over a finite field. and
$\operatorname{G}$ be a reductive algebraic group over $k$. Let $X$ be a
connected normal compact Kähler analytic space and
$\rho:\pi_{1}(X)\rightarrow\operatorname{G}(k)$ a representation with Zariski-
dense image. Then there exists a surjective holomorphic map with connected
fibres $\sigma_{X}^{\rho}:X\rightarrow S_{X}^{\rho}$ onto a connected normal
compact Kähler analytic space such that the following property holds: for any
connected normal compact complex analytic space $Z$ and any holomorphic map
$f:Z\rightarrow X$, the composition $\sigma_{X}^{\rho}\circ f:Z\rightarrow
S_{X}^{\rho}$ is constant if, and only if, the representation $f^{\ast}\rho$
has bounded image.

Observe that a fibration $\sigma_{X}^{\rho}:X\rightarrow S_{X}^{\rho}$ with
this property is unique, up to unique isomorphism. It is called the Katzarkov-
Zuo reduction of $(X,\rho)$. Its existence is proved in [Eys04] for a smooth
$X$. However, one can argue as in Proposition 3.6 to prove its existence more
generally when $X$ is normal.

The following result is a key ingredient in the proof of Theorem A.

######  Theorem 5.2.

Let $k$ be a non-Archimedean local field and $\operatorname{G}$ be a reductive
algebraic group over $k$. Let $X$ be a connected normal compact Kähler
analytic space and $\rho:\pi_{1}(X)\rightarrow\operatorname{G}(k)$ a
representation with Zariski-dense image. Assume moreover that
$\operatorname{G}/k$ is absolutely simple and that $\rho$ is a large
representation with unbounded image. Then:

  1. (1)

the Katzarkov-Zuo reduction $\sigma_{X}^{\rho}:X\rightarrow S_{X}^{\rho}$ is
birational.

  2. (2)

$\overline{\operatorname{Sp}_{ab}(X)}\neq X$.

The first assertion is a result of Zuo [Zuo96]. It is a key ingredient in both
Zuo’s paper [Zuo96] and Yamanoi’s paper [Yam10]. We give an alternative and
arguably simpler argument below. The rest of this section is devoted to the
proof of this theorem.

###  5.2. A construction of the Katzarkov-Zuo reduction

We give an alternative construction of the Katzarkov-Zuo reduction based on a
construction of Klingler [Kli03].

Let $k$ be a non-Archimedean local field and $\operatorname{G}$ be a reductive
algebraic group over $k$. Let $\mathscr{B}(\operatorname{G},k)$ be the Bruhat-
Tits building of $\operatorname{G}/k$, $W$ the Weyl group of
$\operatorname{G}/k$ and $W^{aff}$ the affine Weyl group of
$\operatorname{G}/k$. Let $\mathcal{A}$ be the affine real vector space on
which the apartments of $\mathscr{B}(\operatorname{G},k)$ are modelled, and
let $\mathcal{A}_{\mathbb{C}}$ be its complexification. The group $W^{aff}$
acts on $\mathcal{A}$ by affine reflections.

Let $X$ be a Riemannian manifold and $f:X\to\mathscr{B}(\operatorname{G},k)$
be a continuous map. A point $x\in X$ is called regular for $f$ if there is an
apartment in $\mathscr{B}(\operatorname{G},k)$ that contains a sufficiently
small neighborhood of $x$ [GS92, p.225]. Otherwise $x$ is called singular. The
map $f$ is called harmonic if for every point $x\in X$, there exists a small
ball $B$ centered at $x$ on which $f$ minimizes the energy relatively to
$f_{|\partial B}$ [GS92, p.232].

Assume now that $X$ is a connected compact Kähler manifold and let
$\rho:\pi_{1}(X)\rightarrow\operatorname{G}(k)$ be a representation with
Zariski-dense image. Thanks to [GS92, Theorem 7.8, Lemma 8.1] there exists a
Lipschitz harmonic $\rho$-equivariant pluriharmonic map
$f:\tilde{X}\to\mathscr{B}(\operatorname{G},k)$ from the universal covering of
$X$ (with finite energy since $X$ is compact). The subset
$R(\tilde{X},\rho)\subset\tilde{X}$ of regular points for $f$ is a
$\pi_{1}(X)$-invariant open subset of $\tilde{X}$, and one denotes by
$R(X,\rho)$ its image in $X$. The Hausdorff codimension of its complementary
$S(X,\rho)\subset X$ is at least $2$ [GS92, Theorem 6.4]. Moreover $f$ is
pluriharmonic, i.e. $\partial\bar{\partial}f=0$ on $R(\tilde{X},\rho)$[GS92,
Theorem 7.3], and for any holomorphic map $g:Y\to X$ from a connected compact
Kähler manifold $Y$ with universal covering $\tilde{Y}$, the representation
$g^{\ast}\rho:\pi_{1}(Y)\rightarrow\operatorname{G}(k)$ is reductive (i.e. the
Zariski-closure of its image is a reductive group) and the composition
$f\circ\tilde{g}:\tilde{Y}\to\mathscr{B}(\operatorname{G},k)$ is
$g^{\ast}\rho$-equivariant and pluriharmonic [Eys04, Corollaire 1.3.8].

Klingler explains in [Kli03, section 2.2.2] the construction of a complex
local system $F(X,\rho)$ with finite monodromy on $R(X,\rho)$ that corresponds
intuitively to the pull-back by $f$ of the complexified tangent bundle of the
building $\mathscr{B}(\operatorname{G},k)$. We briefly recall the construction
and refer to loc. cit. for the details.

Let $x\in R(\tilde{X},\rho)$, so that there exists an isometric embedding
$i:\mathcal{A}\subset\mathscr{B}(\operatorname{G},k)$ and a neighborhood $B$
of $x$ in $\tilde{X}$ such that the map
$f_{|B}:B\to\mathscr{B}(\operatorname{G},k)$ factorizes through a
pluriharmonic map $h:B\to\mathcal{A}$. The map $h$ is well-defined up to the
action of $W^{aff}$ on $\mathcal{A}$. Since $h$ is pluriharmonic, by taking
the $(1,0)$-part of the complexification of its differential, one obtains a
$\mathbb{C}$-linear map $\mathcal{A}_{\mathbb{C}}^{\vee}\to\Omega^{1}_{B}$,
well-defined up to the action of $W$ on $\mathcal{A}_{\mathbb{C}}^{\vee}$.
Globalizing, the map $f$ defines a real local system with monodromy
representation $\pi_{1}(R(X,\rho))\to
W^{aff}\subset\operatorname{Aut}(\mathcal{A})$. By composing with the
homomorphism $W^{aff}\to W$, one obtains a real local system
$F_{\mathbb{R}}(X,\rho)$ corresponding to the monodromy representation
$\pi_{1}(R(X,\rho))\to W\subset\operatorname{Aut}(\mathcal{A})$, and the
derivative of $f$ yields a real one-form $\mu^{\mathbb{R}}_{X}$ with values in
$F_{\mathbb{R}}(X,\rho)$. We denote by $F(X,\rho)$ the complex local system
associated to $F_{\mathbb{R}}(X,\rho)$. Since $h$ is pluriharmonic, the
complexification of $\mu^{\mathbb{R}}_{X}$ is a holomorphic one-form $\mu_{X}$
with values in $F(X,\rho)$.

Since $S(X,\rho)$ is a closed analytic subset of $X$ [Eys04, Proposition
1.3.3] distinct from $X$ and since the monodromy of $F(X,\rho)$ is finite,
there exists a normal ramified Galois covering $p:Z\to X$ with Galois group
$\Lambda\subset W$ such that the complex local system $p^{\ast}F(X,\rho)$ on
$p^{-1}(R(X,\rho))$ is trivial, hence extends to $Z$. Since $f$ is Lipschitz,
the holomorphic one-form $\mu_{Z}=p^{\ast}\mu_{X}$ on $p^{-1}(R(X,\rho))$ is
bounded, hence it extends as a holomorphic one-form
$\mu_{Z}\in\operatorname{H}^{0}(Z,\Omega^{1}_{Z}\otimes_{\mathbb{C}}\mathcal{A})$.
Moreover, $\mu_{Z}$ is the zero one-form if, and only if, the map $f$ is
constant.

Let $Z\to\operatorname{Alb}(Z)$ denote the Albanese morphism of $Z$ (we refer
to [Eys11, Section 3.4.2] for the definitions of holomorphic forms and
Albanese morphism for normal Kähler spaces). The map
$Z\to\operatorname{Alb}(Z)$ is an initial object in the category of
holomorphic maps $Z\to T$ where $T$ is a compact complex torus. In particular,
$\operatorname{Alb}(Z)$ is equipped with a $\Lambda$-action such that
$Z\to\operatorname{Alb}(Z)$ is $\Lambda$-equivariant. Moreover, the induced
$\mathbb{C}$-linear $\Lambda$-equivariant map
$\operatorname{H}^{0}(\operatorname{Alb}(Z),\Omega^{1})\to\operatorname{H}^{0}(Z,\Omega^{1})$
is an isomorphism.

There is a the largest subtorus $B$ of $\operatorname{Alb}(Z)$ such that the
composition of
$\mu_{Z}:\mathcal{A}^{\vee}\to\operatorname{H}^{0}(Z,\Omega^{1}_{Z})$ with
$\operatorname{H}^{0}(\operatorname{Alb}(Z),\Omega^{1})\to\operatorname{H}^{0}(B,\Omega^{1})$
is zero. Since $B$ is preserved by the action of $\Lambda$, we get a
commutative diagram:

It is easy to check that the Stein factorization of the holomorphic map
$X\to\left(\operatorname{Alb}\left(Z\right)/B\right)/\Lambda$ is the
Katzarkov-Zuo reduction of $(X,\rho)$.

###  5.3. A lemma

######  Lemma 5.3.

Let $k$ be a non-Archimedean local field. Let $\operatorname{G}$ be an
absolutely simple $k$-algebraic group. Let $\Gamma\subset\operatorname{G}(k)$
be a Zariski-dense and unbounded subgroup. Let $\Delta\subset\Gamma$ be a
normal subgroup. If $\Delta$ is bounded in $\operatorname{G}(k)$, then
$\Delta$ is finite.

###### Proof.

We briefly recall a class of compactifications of Bruhat-Tits buildings
introduced in [Ber90] and generalized in [RTW10].

Up to replacing $k$ with a finite extension, one can assume that
$\operatorname{G}$ is $k$-isotropic. (One can even assume for simplicity that
$\operatorname{G}/k$ is split.) Let $\mathscr{B}(\operatorname{G},k)$ be the
Euclidean Bruhat-Tits building associated to $\operatorname{G}/k$ equipped
with its natural $\operatorname{G}(k)$-action by isometries. For any
$\operatorname{G}(k)$-conjugacy class $t$ of parabolic subgroups of
$\operatorname{G}$, there exists a continuous,
$\operatorname{G}(k)$-equivariant map
$\theta_{t}:\mathscr{B}(\operatorname{G},k)\to\mathrm{Par}_{t}(\operatorname{G})^{an}$
which is a homeomorphism onto its image. Here
$\mathrm{Par}_{t}(\operatorname{G})$ denotes the connected component of type
$t$ in the proper $k$-algebraic variety $\mathrm{Par}(\operatorname{G})$ of
all parabolic subgroups in $\operatorname{G}$ (on which $\operatorname{G}$
acts by conjugation). The superscript an means that we pass from the
$k$-variety $\mathrm{Par}_{t}(\operatorname{G})$ to the Berkovich $k$-analytic
space associated with it. Note that the space
$\mathrm{Par}(\operatorname{G})^{an}$ is compact since
$\mathrm{Par}(\operatorname{G})$ is projective. Let
$\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ denote the closure of the
image of $\theta_{t}$.

Two parabolic subgroups $\operatorname{P}$ and $\operatorname{Q}$ of
$\operatorname{G}$ are called osculatory if their intersection
$\operatorname{P}\cap\operatorname{Q}$ is also a parabolic subgroup of
$\operatorname{G}$. Moreover, each parabolic subgroup
$\operatorname{P}\in\mathrm{Par}(\operatorname{G})$ defines a closed
osculatory subvariety $\mathrm{Osc}_{t}(P)$ of
$\mathrm{Par}_{t}(\operatorname{G})$, namely the one consisting of all
parabolics of type $t$ whose intersection with $\operatorname{P}$ is a
parabolic subgroup. Then $\operatorname{P}$ is $t$-relevant if it is maximal
among all parabolic $k$-subgroups defining the same osculatory subvariety. It
is readily seen that each parabolic subgroup is contained in a unique
$t$-relevant one.

The space $\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ admits a
stratification in disjoint locally closed subspaces that are indexed by the
$t$-relevant parabolic $k$-subgroups in $\operatorname{G}$. If
$x\in\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ belongs to the stratum
indexed by the $t$-relevant parabolic $k$-subgroup $\operatorname{Q}$, then
the subgroup of $\operatorname{G}(k)$ that fixes $x$ is contained in
$\operatorname{Q}(k)$, cf. [RTW10, Theorem 4.11]. In particular, a subgroup of
$\operatorname{G}(k)$ that fixes a point in the boundary
$\partial\overline{\mathscr{B}}_{t}(\operatorname{G},k)=\overline{\mathscr{B}}_{t}(\operatorname{G},k)\backslash\mathscr{B}_{t}(\operatorname{G},k)$
cannot be Zariski-dense in $\operatorname{G}$.

Let us finish the proof of the lemma. Since $\Delta$ is normal in $\Gamma$,
its Zariski-closure $\bar{\Delta}^{Zar}$ is normal in
$\bar{\Gamma}^{Zar}=\operatorname{G}$. Since $\operatorname{G}$ is simple, it
follows that $\bar{\Delta}^{Zar}$ is either equal to $\operatorname{G}$ or is
a finite group.

Let $\mathcal{F}\subset\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ be the
subset of points that are fixed by the induced action of $\Delta$. It is
compact, since the action of $\operatorname{G}(k)$ on
$\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ is continuous and
$\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ is compact. Moreover,
$\mathcal{F}$ is non-empty since $\Delta$ is bounded by assumption. Since
$\Delta$ is a normal subgroup of $\Gamma$, the action of $\Gamma$ on
$\overline{\mathscr{B}}_{t}(\operatorname{G},k)$ stabilizes $\mathcal{F}$. If
$\mathcal{F}$ was contained in $\mathscr{B}_{t}(\operatorname{G},k)$, then
$\Gamma$ would fix the barycenter of $\mathcal{F}$. This would be in
contradiction with the assumption that $\Gamma$ is unbounded. Therefore,
$\mathcal{F}$ meets the boundary
$\partial\overline{\mathscr{B}}_{t}(\operatorname{G},k)$. By the preceding
discussion, it follows that $\bar{\Delta}^{Zar}$ cannot be equal to
$\operatorname{G}$, so that it is necessary a finite group. This proves that
$\Delta$ is finite. ∎

###  5.4. Proof of Theorem 5.2

Let us now turn to the proof of Theorem 5.2. We keep the notations introduced
in section 5.2. We can assume without loss of generality that $X$ is smooth.
Let $X_{s}$ be a general fiber of the Katzarkov-Zuo reduction
$\sigma_{X}^{\rho}:X\rightarrow S_{X}^{\rho}$, so that there is an exact
sequence of groups:

$\pi_{1}(X_{s})\to\pi_{1}(X)\to\pi_{1}(S_{X}^{\rho})\to 1.$

Let $\Gamma$ be the image of $\rho:\pi_{1}(X)\to\operatorname{G}(k)$ and
$\Delta$ be the image of the composition
$\pi_{1}(X_{s})\to\pi_{1}(X)\to\operatorname{G}(k)$. The assumption of Lemma
5.3 are fulfilled, hence $\Delta$ is finite. Since the representation $\rho$
is large, this is only possible if $X_{s}$ has dimension zero. This proves
that $\sigma_{X}^{\rho}$ is birational.

Let $Z\to X$ denote the Galois covering associated to the pair $(X,\rho)$,
whose construction is recalled in section 5.2. By construction, $Z$ is a
connected normal compact Kähler space.

Let us prove that $Z$ is of general type, using an argument of Zuo [Zuo96].
Since the induced map $Z\to X$ is finite surjective, the image of the
homomorphism $\pi_{1}(Z)\to\pi_{1}(X)$ has finite index in $\pi_{1}(X)$, hence
the image of the induced representation
$\rho_{|Z}:\pi_{1}(Z)\to\operatorname{G}(k)$ is also Zariski-dense and
unbounded. Moreover, $\rho_{|Z}$ is also a large representation, hence it
follows from the preceding paragraph that the Katzarkov-Zuo reduction of
$(Z,\rho_{|Z})$ is birational. But by construction, the image of the
Katzarkov-Zuo reduction of $(Z,\rho_{|Z})$ admits a finite morphism to a
compact complex torus. Therefore, assuming by contradiction that $Z$ is not of
general type, there would exist a finite étale cover $Z^{\prime}\to Z$ and a
non-trivial fibration $f:Z^{\prime}\to Y$ whose general fibre is a compact
complex torus [Kaw81, Theorem 23]. Consider the induced representation
$\rho_{|Z^{\prime}}:\pi_{1}(Z^{\prime})\to\operatorname{G}(k)$. It is again
large since $Z^{\prime}\to Z$ is finite. Let $F$ be a general fibre of $f$.
There is an exact sequence of groups
$\pi_{1}(F)\to\pi_{1}(Z^{\prime})\to\pi_{1}(Y)$. Since $\pi_{1}(F)$ is normal
in $\pi_{1}(Z^{\prime})$ and the image of $\pi_{1}(Z^{\prime})$ is Zariski-
dense in $\operatorname{G}$, the Zariski-closure $\operatorname{H}$ of the
image of $\pi_{1}(F)$ in $\operatorname{G}$ is a normal algebraic subgroup of
$\operatorname{G}$. But $\pi_{1}(F)$ is abelian, hence $\operatorname{H}$ is
abelian too. Since $\operatorname{G}$ is absolutely simple, $\operatorname{H}$
is necessarily a finite subgroup of $\operatorname{G}$. This is a
contradiction, since $\rho_{|Z^{\prime}}$ is large.

Let $a:A\dasharrow X$ be a non-constant rational map from an abelian variety.
Since $X$ posseds a large local system, it has no rational curves. It follows
that $a$ is defined everywhere. Since the fundamental group of $A$ is abelian,
the Zariski-closure $\operatorname{H}$ of the image of
$\rho_{|A}:\pi_{1}(A)\to\operatorname{G}(k)$ is a torus. Therefore, the
Bruhat-Tits building $\mathscr{B}(\operatorname{H},k)$ is an Euclidean space.
On the other hand, if $\tilde{A}$ is a universal covering of $A$, the
composition $\tilde{A}\to\tilde{X}\to\mathscr{B}(\operatorname{G},k)$ is
$\rho_{|A}$-equivariant and pluriharmonic [Eys04, Corollaire 1.3.8]. By
unicity (up to translation) of the harmonic map associated to a reductive
representation, it follows that the composition
$\tilde{A}\to\tilde{X}\to\mathscr{B}(\operatorname{G},k)$ takes values in a
totally geodesic embedded copy of $\mathscr{B}(\operatorname{H},k)$ in
$\mathscr{B}(\operatorname{G},k)$. Since the Weyl group of
$\operatorname{H}/k$ is trivial, it follows that the complex local system
$F(A,\rho_{|A})$ introduced in section 5.2 is trivial.

Assume that the image of $a$ is not contained in $S(X,\rho)$. By construction,
the pull-back to $R(A,\rho_{|A})$ along $a$ of the complex local system
$F(X,\rho)$ is equal to $F(A,\rho_{|A})$. Since the local system
$F(A,\rho_{|A})$ is trivial, there is a rational map $A\dashrightarrow Z$
lifting $a:A\to X$. It follows that $\operatorname{Sp}_{ab}(X)$ is contained
in the union of $S(X,\rho)$ and the image of $\operatorname{Sp}_{ab}(Z)$. But
$Z$ is a projective variety of general type with a generically finite map to
an abelian variety. It follows from a result of Yamanoi [Yam15a, Corollary 1]
that $\overline{\operatorname{Sp}_{ab}(Z)}\neq Z$, so that
$\overline{\operatorname{Sp}_{ab}(X)}\neq X$ by the preceding discussion.

##  6\. Proof of Theorem A

###  6.1. A first reduction

######  Definition 6.1.

Let $X$ be a projective complex algebraic variety. A collection of complex
local systems $\\{\mathcal{L}_{i}\\}_{i\in I}$ on $X$ is large if for every
integral closed subvariety $Z\hookrightarrow X$ the pull-back to the
normalization of $Z$ of one of the $\mathcal{L}_{i}$’s is not isotrivial.
Equivalently, for any non-constant morphism $f:Y\to X$ from an irreducible
normal projective complex algebraic variety $Y$, there exists $i\in I$ such
that the local system $f^{-1}\mathcal{L}_{i}$ is not isotrivial.

In particular, a complex local system $\mathcal{L}$ is large when the
collection $\\{\mathcal{L}\\}$ is large. Proposition 3.2 has the following
immediate generalization:

######  Proposition 6.2.

Let $X$ be a projective complex algebraic variety and
$\\{\mathcal{L}_{i}\\}_{i\in I}$ a large collection of complex local systems
on $X$. Let $f:Y\to X$ be a finite morphism from another projective complex
algebraic variety $Y$. Then the collection $\\{f^{-1}\mathcal{L}_{i}\\}_{i\in
I}$ of complex local systems on $Y$ is large.

######  Proposition 6.3.

Let $X$ be a projective normal complex algebraic variety. If $X$ admits a
large collection of complex local systems, then $X$ admits a large complex
local system.

###### Proof.

For any complex local system $\mathcal{L}$, we have the associated
Shafarevitch morphism $X\to\operatorname{Sh}_{X}^{\mathcal{L}}$. If
$\\{\mathcal{L}_{i}\\}_{i\in I}$ is a large collection of complex local
systems on $X$, then the morphism $X\to\prod_{i\in
I}\operatorname{Sh}_{X}^{\mathcal{L}_{i}}$ does not contract any subvariety,
therefore it is finite. By Noetherianity, there exists finitely many local
systems $\mathcal{L}_{i_{1}},\cdots,\mathcal{L}_{i_{N}}$ such that the
morphism $X\to\prod_{k=1}^{N}\operatorname{Sh}_{X}^{\mathcal{L}_{i_{k}}}$ is
finite. It follows that the complex local system
$\bigoplus_{k=1}^{N}\mathcal{L}_{i_{k}}$ is large. ∎

In view of Proposition 6.2 and Proposition 6.3, the proofs of Theorem 4.1 and
Corollary 4.3 immediately generalize as follows:

######  Theorem 6.4.

Let $X$ be a projective complex algebraic variety supporting a large
collection of complex local systems. Then
$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)$, and
$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)=X$ if
$X$ is not of general type.

With the preceding results at hand, we first reduce Theorem A to the following
result.

######  Theorem 6.5.

Let $X$ be a projective complex algebraic variety supporting a large
collection of complex local systems. If $X$ is of general type, then
$\operatorname{Sp}_{ab}(X)$ is not Zariski-dense in $X$.

###### Proof of Theorem A, assuming Theorem 6.5.

Recalling Theorem 6.4, we need only to check that
$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X)$ is
Zariski-closed in $X$. There is nothing to prove when $\dim X=0$, so that one
can assume that $\dim X>0$. By Noetherian induction, let us assume that the
result holds for all strict subvarieties of $X$. If $X$ is not of general
type, then the result follows from Theorem 6.4. Otherwise, if $X$ is of
general type, then $S:=\operatorname{Sp}_{ab}(X)$ is not Zariski-dense in $X$
thanks to Theorem 6.5. Denoting by $\bar{S}$ its Zariski-closure in $X$,
observe that $\operatorname{Sp}_{ab}(\bar{S})=\operatorname{Sp}_{ab}(X)=S$.
Therefore, $\operatorname{Sp}_{ab}(\bar{S})$ is Zariski-dense in $\bar{S}$.
Since $\bar{S}$ is a strict subvariety of $X$, it follows by Noetherian
induction that $\bar{S}$ is not of general type. Thanks to Theorem 6.4,
$\bar{S}$ is covered by images of abelian varieties, so that $S=\bar{S}$. ∎

###  6.2. Proof of Theorem 6.5.

Let $X$ be a projective complex algebraic variety supporting a large
collection of complex local systems. Assume that $X$ is of general type. Our
goal in this section is to prove that $\operatorname{Sp}_{ab}(X)$ is not
Zariski-dense in $X$.

Let $X=\cup_{i}X_{i}$ be the decomposition of $X$ in its irreducible
components. Since $\operatorname{Sp}_{ab}(X_{i})\subset X_{i}$ for every $i$,
it is sufficient to prove that $\operatorname{Sp}_{ab}(X_{i})$ is not Zariski
dense in $X_{i}$ for at least one of the $X_{i}$’s which is of general type.
Therefore one can assume from now on that $X$ is irreducible.

Let $\nu:\bar{X}\to X$ denote the normalization of $X$ and $Z\subset X$ the
non-normal locus of $X$. Then
$\nu(\operatorname{Sp}_{ab}(\bar{X}))\subset\operatorname{Sp}_{ab}(X)$ since
$\nu$ is finite, and
$\operatorname{Sp}_{ab}(X)\subset\nu(\operatorname{Sp}_{ab}(\bar{X}))\cup Z$
since $\nu$ is an isomorphism onto its image outside $\nu^{-1}(Z)$. Therefore,
$\operatorname{Sp}_{ab}(X)$ is Zariski-dense in $X$ if, and only if,
$\operatorname{Sp}_{ab}(\bar{X})$ is Zariski-dense in $\bar{X}$. In view of
Proposition 6.2 and Proposition 6.3, $\bar{X}$ admits a large complex local
system. As a consequence, one can assume from now on that $X$ is normal
irreducible projective complex algebraic variety with a large complex local
system. We denote by $\rho:\pi_{1}(X)\to\operatorname{G}(\mathbb{C})$ the
corresponding monodromy representation, with $\operatorname{G}$ its algebraic
monodromy group.

Note also that it is harmless to replace $X$ with a finite étale cover, cf.
Proposition 2.2 and Proposition 3.3. In particular, one can assume that the
algebraic monodromy group $\operatorname{G}$ is a connected algebraic group.

####  6.2.1. The semisimple case.

Let us first finish the proof under the additional assumption that the
algebraic monodromy group $\operatorname{G}$ is semisimple. Thanks to [Eys04],
there exist finitely many representations $\rho_{i}$ such that:

  1. (1)

Every $\rho_{i}$ is of the form $\pi_{1}(X)\rightarrow\operatorname{G}(k)$,
with $k$ a non-Archimedean local field and $G$ an absolutely simple
$k$-algebraic group, and $\rho_{i}$ has Zariski-dense and unbounded image;

  2. (2)

If $X\to S_{X}^{\rho_{i}}$ denote the Katzarkov-Zuo reduction of $\rho_{i}$,
then the restriction of $\mathcal{L}$ to the normalization of any fiber of
$X\to\prod_{i}S_{X}^{\rho_{i}}$ underlies a polarized variation of pure Hodge
structures with discrete monodromy.

Let $f:X\to Y$ denote the Stein factorization of the morphism
$X\to\prod_{i}S_{X}^{\rho_{i}}$. Note that for every $i$ the induced morphism
$Y\to S_{X}^{\rho_{i}}$ is surjective, since its precomposition $X\to
S_{X}^{\rho_{i}}$ with the canonical morphism $X\to Y$ is surjective. Thanks
to Theorem 5.2 and Proposition 2.3,
$\overline{\operatorname{Sp}_{ab}(S_{X}^{\rho_{i}})}\neq S_{X}^{\rho_{i}}$ for
every $i\in I$. Using Proposition 2.6, it follows that
$\overline{\operatorname{Sp}_{ab}(Y)}\neq Y$.

Therefore, in view of Proposition 2.5, it is sufficient to prove that
$\operatorname{Sp}_{ab}(X/Y)$ is not Zariski-dense in $X$. Since $X$ is
normal, there exists a Zariski-dense open $Y^{o}$ of $Y$ over which the
(geometric) fibers of $f$ are normal. Moreover, by construction, any such
fiber $X_{y}$ admits a large complex local system that underlies a polarized
variation of pure Hodge structures with discrete monodromy. It follows from
Proposition 3.12 that $\operatorname{Sp}_{ab}(X_{y})=\varnothing$ for any
$y\in Y^{o}$. This proves that $\operatorname{Sp}_{ab}(X/Y)$ is contained in
$f^{-1}(Y\backslash Y^{o})$, hence it is not Zariski-dense in $X$. This
concludes the proof.

####  6.2.2. The reductive case.

Assume that the algebraic monodromy group $\operatorname{G}$ is reductive.
Thanks to Proposition 3.15, replacing $X$ with a finite étale cover, we can
assume that there exists a surjective morphism with connected fibers $f:X\to
Y$ onto a normal irreducible projective complex algebraic variety $Y$ such
that:

  * •

$Y$ admits a large complex local system with torsion-free monodromy and a
semisimple algebraic monodromy group;

  * •

for $y$ in a Zariski-dense open subset of $Y$, the fiber $X_{y}$ admits a
finite morphism to an abelian variety.

Let $X^{\prime}\to X$ be a projective desingularization of $X$. Thanks to
Proposition 2.3, it is equivalent to prove that
$\operatorname{Sp}_{ab}(X^{\prime})$ is not Zariski-dense in $X^{\prime}$. The
composition $X^{\prime}\to X\to Y$ is still a fibration, and there exists a
Zariski-dense open subset $Y^{o}$ of $Y$ such that $X^{\prime}_{y}$ is a
desingularization of $X_{y}$ for every $y\in Y^{o}$. In particular,
$X^{\prime}_{y}$ has maximal Albanese dimension (since $X_{y}$ admits a finite
morphism to an abelian variety) and is of general type (since $X^{\prime}$ is
of general type by assumption).

The set $\operatorname{Sp}_{ab}(Y)$ is not Zariski-dense in $Y$ thanks to the
preceding section. Therefore, thanks to Proposition 2.5, it is sufficient to
prove that $\operatorname{Sp}_{ab}(X^{\prime}/Y)$ is not Zariski-dense in
$X^{\prime}$. Since one can freely shrink $Y$ to make the morphism
$X^{\prime}\to Y$ smooth, this is a consequence of the following result.

######  Theorem 6.6 (cf. [Bru]).

Let $f:X\to Y$ be a smooth projective surjective morphism with connected
fibers between smooth complex algebraic varieties. Assume that the fibers of
$f$ over a Zariski-dense open subset of $Y$ are of general type and of maximal
Albanese dimension. Then $\operatorname{Sp}_{ab}(X/Y)$ is not Zariski-dense in
$X$.

####  6.2.3. The general case.

For the general case, we can argue exactly as in the reductive case, making
use Proposition 3.14 instead of Proposition 3.15.

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