#  $L^{2}$-type invariants for complex smooth quasi-projective varieties – a
survey

Yongqiang Liu  Yongqiang Liu: The Institute of Geometry and Physics,
University of Science and Technology of China, 96 Jinzhai Road, Hefei Anhui
230026 China<EMAIL_ADDRESS>
(Date: May 20, 2024)

###### Abstract.

Let $X$ be a complex smooth quasi-projective variety with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. We survey recent
developments about the asymptotic behaviour of Betti numbers with any field
coefficients and the order of the torsion part of singular integral homology
of finite abelian covers of $X$ associated to $\nu$, known as the $L^{2}$-type
invariants. We give relations between $L^{2}$-type invariants, Alexander
invariants and cohomology jump loci. When $\nu$ is orbifold effective, we give
explicit formulas for $L^{2}$-invariants at homological degree one in terms of
geometric information of $X$. We also propose several related open questions
for hyperplane arrangement complement.

###### Key words and phrases:

Alexander polynomial, Mahler measure, jump loci, orbifold map, $L^{2}$-Betti
number, BNS-invariants, hyperplane arrangement

###### 2020 Mathematics Subject Classification:

Primary 14F45, 14H30, 32S20; Secondary 32S20

The author is supported by National Key Research and Development Project
SQ2020YFA070080, NSFC grant No. 12001511, the Project of Stable Support for
Youth Team in Basic Research Field CAS (YSBR-001), the starting grant from
University of Science and Technology of China, the project “Analysis and
Geometry on Bundles” of Ministry of Science and Technology of the People’s
Republic of China and Fundamental Research Funds for the Central Universities.
The author would like to thank the referee for a careful reading of the
manuscript and for several constructive suggestions.

##  1\. Introduction

There is a general principle to consider a classical invariant of a finite CW
complex and to define its analog for some covering space. This leads to the
$L^{2}$-type invariants. Atiyah [Ati76] introduced the notion of $L^{2}$-Betti
numbers in the context of a regular covering of a closed Riemannian manifold.
After that, there has been vast literature for the $L^{2}$-invariant theory,
see [Luc02]. The $L^{2}$-invariant theory has attracted considerable interest
and has relations to many other fields, such as operator theory and Algebraic
$K$-theory. A particular important result is Lück’s approximation theorem
[Luc94], which states that the $L^{2}$-Betti numbers of the universal cover of
a finite polyhedron can be found as limits of normalised Betti numbers of
finite sheeted normal coverings.

In this paper, we focus on a rather simple case of Lück’s approximation
problem, see e.g. Lück’s survey papers [Luc13, Luc16] on this topic. Let $X$
be a connected finite CW complex with a fixed epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Consider a descending
chain of subgroups

$\mathbb{Z}^{n}=\Gamma_{0}\supseteq\Gamma_{1}\supseteq\Gamma_{2}\supseteq\cdots$

such that $\Gamma_{j}$ is a sub-group in $\mathbb{Z}^{n}$ with finite index
and $\bigcap_{j=0}^{\infty}\Gamma_{j}=\\{0\\}$. Let $X_{j}$ denote the finite
covering of $X$ associated to the composed epimorphism $\nu$ with the quotient
map $\mathbb{Z}^{n}\twoheadrightarrow\mathbb{Z}^{n}/\Gamma_{j}$. We study the
asymptotic behaviour of the $i$-th Betti number with coefficients in a field
coefficient $\mathbb{K}$ and the logarithm of the cardinality of the torsion
subgroup of the $i$-th homology group with integral coefficients, i.e., the
two sequences

$\big{(}\frac{\dim
H_{i}(X_{j},\mathbb{K})}{|\mathbb{Z}^{n}/\Gamma_{j}|}\big{)}_{j\geq 0}$ and
$\big{(}\frac{\log|H_{i}(X_{j},\mathbb{Z})_{\mathrm{tor}}|}{|\mathbb{Z}^{n}/\Gamma_{j}|}\big{)}_{j\geq
0}.$

Here $|H_{i}(X_{j},\mathbb{Z})_{\mathrm{tor}}|$ denotes the order of the
torsion part of $H_{i}(X_{j},\mathbb{Z}).$

One can find the following approximation problem proposed by Lück in [Luc13,
Luc16].

######  Question 1.1.

(Approximation Problem)

  * (i)

Does the sequence converge?

  * (ii)

If yes, is the limit independent of the chain?

  * (iii)

If yes, what is the limit?

Luckily for us Question 1.1 has positive answers in our case as we explain
now. For a subgroup $\Gamma\subset\mathbb{Z}^{n}$ of finite index, we set

$\langle\Gamma\rangle=\min\left\\{\sum_{j=1}^{n}x_{j}^{2}\mid
x=(x_{1},\ldots,x_{n})\in\Gamma,x\neq 0\right\\}$

Let $X^{\nu,\Gamma}$ denote the covering space of $X$ associated to the
compositions of $\nu$ and the quotient map
$\mathbb{Z}^{n}\to\mathbb{Z}^{n}/\Gamma$. Consider the following limits:

$\alpha_{i}(X^{\nu},\mathbb{K})\coloneqq\lim_{\langle\Gamma\rangle\rightarrow\infty}\frac{\dim
H_{i}(X^{\nu,\Gamma},\mathbb{K})}{|\mathbb{Z}^{n}/\Gamma|}$

for any field coefficients $\mathbb{K}$ and

$\mathbf{M}_{i}(X^{\nu})\coloneqq\limsup_{\langle\Gamma\rangle\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,\Gamma},\mathbb{Z})_{\mathrm{tor}}|}{|\mathbb{Z}^{n}/\Gamma|}.$

It is known that the first limit always exists for any field coefficients and
does not depend on the choice of chains, see [Luc13, Theorem 3.2]. When
$\mathrm{char}(\mathbb{K})=0$, the well known Lück’s approximation theorem
[Luc94] shows that the first limit exists and does not depend on the choice of
chains. On the other hand, it was proved by Le [Le14] that the upper limit
$\mathbf{M}_{*}(X^{\nu})$ exists and can be computed by the Mahler measure of
the $i$-th integral Alexander polynomial. When $n=1$, the upper limit in the
definition of $\mathbf{M}_{*}(X^{\nu})$ can be replaced by the ordinary limit.

The $L^{2}$-type invariants are particular useful for the following Singer-
Hopf Conjecture.

######  Conjecture 1.2 (Singer-Hopf Conjecture).

Let $X$ be an aspherical closed manifold with real dimension $2N$. Then

$(-1)^{N}\chi(X)\geq 0.$

A connected CW complex is said to be aspherical if its universal cover is
contractible. Conjecture 1.2 is still wide open even in the realm of
aspherical complex smooth projective varieties. This is particularly
surprising as, in the projective case, one can use a large variety of tools
coming from algebraic geometry approaches to such conjecture. For the recent
developments in this direction, we refer to [AW23, AMW23, DL19, DL23, LMW21].

Inspired by these developments, we focus on the aforementioned $L^{2}$-type
invariants in the context of complex smooth quasi-projective varieties. These
$L^{2}$-type invariants have a deep connection with cohomology jump loci. The
cohomology jump loci of a complex smooth quasi-projective variety have been
intensively studied in recent years. In particular, the beautiful structure
theorem for cohomology jump loci put strong constraints for the homotopy type
of complex smooth quasi-projective variety, see Section 3.1 for a brief
introduction. We also refer the readers to Budur and Wang’s survey paper
[BW17] for a comprehensive background on this topic.

These results for cohomology jump loci can be used to prove many good
properties about the $L^{2}$-type invariants. For example, as an application
of structure theorem, we show that the upper limit in the definition of
$\mathbf{M}_{*}(X^{\nu})$ can be replaced by the ordinary limit when $X$ is a
smooth complex quasi-projective variety (see Corollary 3.2), which is not true
for general topological space. This result is new to the author’s knowledge.

Since the aforementioned limits for these $L^{2}$-type invariants exist, we
rephrase Question 1.1(iii) as follows.

######  Question 1.3.

How to compute these $L^{2}$-type invariants in terms of the algebraic
geometric information of $X$, when $X$ is a smooth complex quasi-projective
variety?

In the case of hyperplane arrangement complement, one can ask if these
$L^{2}$-type invariants are combinatorially determined. Based on Lück’s
approximation theorem [Luc94], a related question is whether the fundamental
group of hyperplane arrangement complement is residually finite?

The purpose of this survey is to provide a gentle introduction to these
concepts and results, with an emphasis on applications and open questions. In
section 2, we recall the definition of Alexander type invariants and
cohomology jump loci and study their relations with $L^{2}$-type invariants.
In section 3, we focus on the case when $X$ is a complex smooth quasi-
projective variety. In section 3.1, we recall the structure theorem about
cohomology jump loci and its applications to $L^{2}$-type invariants. Section
3.2 is devoted to cohomology jump loci and $L^{2}$-type invariants at
homological degree one. Section 4 is devoted to hyperplane arrangements. We
propose many open questions regarding if these invariants are combinatorially
determined. Section 5 is devoted for proofs.

Notations.

  1. (1)

$\mathbb{K}$ is assumed to be an algebraically closed filed in the rest of the
paper and $\mathrm{char}(\mathbb{K})$ denotes the characteristic of the field
$\mathbb{K}$.

  2. (2)

For any prime number $p\geq 2$, let $\overline{\mathbb{F}}_{p}$ denote the
algebraic closure of the finite field $\mathbb{F}_{p}$ with $p$ elements.

##  2\. Alexander invariants and cohomology jump loci

Let $X$ be a connected finite CW complex with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. We study $L^{2}$-type
invariants associated to the pair $(X,\nu)$.

###  2.1. Alexander type invariants

Let $X^{\nu}$ denote the covering space of $X$ associated to the epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. The group of covering
transformations of the covering space $X^{\nu}$ is isomorphic to
$\mathbb{Z}^{n}$ and acts on it. By choosing fixed lifts of the cells of $X$
to $X^{\nu}$, we obtain a free basis for the cellular chain complex of
$X^{\nu}$ as $\mathbb{Z}[\mathbf{t}^{\pm}]$-modules, where
$\mathbb{Z}[\mathbf{t}^{\pm}]$ is the multi-variable Laurentiu polynomial ring
$\mathbb{Z}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]=\mathbb{Z}[\mathbb{Z}^{n}]$. So
the cellular chain complex of $X^{\nu}$, $C_{*}(X^{\nu},\mathbb{Z})$, is a
bounded complex of finitely generated free
$\mathbb{Z}[\mathbf{t}^{\pm}]$-modules:

(2.1) $\cdots\to
C_{i+1}(X^{\nu},\mathbb{Z})\overset{\partial_{i}}{\to}C_{i}(X^{\nu},\mathbb{Z})\overset{\partial_{i-1}}{\to}C_{i-1}(X^{\nu},\mathbb{Z})\overset{\partial_{i-2}}{\to}\cdots\overset{\partial_{0}}{\to}C_{0}(X^{\nu},\mathbb{Z})\to
0.$

######  Definition 2.1.

The $i$-th integral Alexander homology $H_{i}(X^{\nu},\mathbb{Z})$ of the pair
$(X,\nu)$ is by definition the $i$-th homology of $C_{*}(X^{\nu},\mathbb{Z})$
as complex of $\mathbb{Z}[\mathbf{t}^{\pm}]$-modules. Similarly, for any field
$\mathbb{K}$, the $i$-th Alexander homology with $\mathbb{K}$-coefficient
$H_{i}(X^{\nu},\mathbb{K})$ of the pair $(X,\nu)$ is by definition the $i$-th
homology of $C_{*}(X^{\nu},\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{K}$ as
complex of $\mathbb{K}[\mathbf{t}^{\pm}]$-modules, where
$\mathbb{K}[\mathbf{t}^{\pm}]$ denotes the multi-variable Laurent polynomial
ring $\mathbb{K}[t^{\pm}_{1},\ldots,t^{\pm}_{n}].$

With the above free basis for $C_{*}(X^{\nu},\mathbb{Z})$, $\partial_{i}$ can
be written down as a matrix with entries in $\mathbb{Z}[\mathbf{t}^{\pm}]$.
Note that $\mathbb{Z}[\mathbf{t}^{\pm}]$ is a Noetherian UFD. Let
$\Delta_{i}(X^{\nu})$ denote the greatest common divisor of all non-zero
$(\mathrm{rank}\partial_{i}\times\mathrm{rank}\partial_{i})$-minors of
$\partial_{i}$. When $\partial_{i}=0$, $\Delta_{i}(X^{\nu})=1$ by convention.

######  Definition 2.2.

$\Delta_{i}(X^{\nu})$ is called the $i$-th integral Alexander polynomial of
the pair $(X,\nu)$.

Note that $\Delta_{i}(X^{\nu})$ is defined uniquely up to multiplication with
a unit in $\mathbb{Z}[\mathbf{t}^{\pm}]$.

######  Remark 2.3.

The complex $C_{*}(X^{\nu},\mathbb{Z})$ determines
$\alpha_{*}(X^{\nu},\mathbb{K})$ and $\mathbf{M}_{*}(X^{\nu})$. In fact, for
any $\Gamma\subset\mathbb{Z}^{n}$ with finite index, consider
$\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]$ as a quotient ring of
$\mathbb{Z}[\mathbf{t}^{\pm}]$. Then $H_{*}(X^{\nu,\Gamma},\mathbb{Z})$ can be
computed by the following complex (viewed as a complex of abelian groups):

$C_{*}(X^{\nu},\mathbb{Z})\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma].$

So it is no surprise that the $L^{2}$-type invariants have a deep connection
with the Alexander invariants.

Next we recall the definition of Mahler measure. Let
$h\in\mathbb{Z}[\mathbf{t}^{\pm}]$ be a nonzero polynomial. The Mahler measure
of $h$ is defined by

$\mathbf{M}(h)\coloneqq\int_{(S^{1})^{n}}\log|h(s)|\mathrm{d}s,$

where $\mathrm{d}s$ indicates the integration with respect to the normalized
Haar measure, and $(S^{1})^{n}$ is the multiplicative subgroup of
$n$-dimensional complex space $\mathbb{C}^{n}$ consisting of all vectors
$(s_{1},\ldots,s_{n})$ with $|s_{1}|=\cdots=|s_{n}|=1$. When $n=1$ Mahler
measure has the following equivalent definition.

######  Definition 2.4.

[Mah60, Mah62] For a non-zero polynomial $h(t)\in\mathbb{Z}[\mathbf{t}^{\pm}]$
with $n=1$, assume that

$h(t)=a_{d}t^{d}+a_{d-1}t^{d-1}+\cdots+a_{0}=a_{d}\cdot\prod_{j=1}^{d}(t-\alpha_{j})$
with $a_{d}\neq 0$, $a_{0}\neq 0$ and $\alpha_{j}\in\mathbb{C}$.

We define the Mahler measure of $h$ to be

$\mathbf{M}(h)=\log(|a_{d}|\cdot\prod_{j=1}^{d}\max\\{1,|\alpha_{j}|\\}).$

Recall that an element $h\in\mathbb{Z}[\mathbf{t}^{\pm}]$ is called a
generalized cyclotomic polynomial [Sch15, page 47] if it is of the form
$h(t_{1},\ldots,t_{n})=\Phi(\prod_{i=1}^{n}t_{i}^{a_{i}})$ (up to units in
$\mathbb{Z}[\mathbf{t}^{\pm}]$), where
$0\neq(a_{1},\cdots,a_{n})\in\mathbb{Z}^{n}$ and $\Phi$ is a cyclotomic
polynomial in a single variable. The following theorem due to Schmidt
characterizes when could the Mahler measure be zero.

######  Theorem 2.5.

[Sch95, Theorem 19.5] For $h\in\mathbb{Z}[\mathbf{t}^{\pm}]$,
$\mathbf{M}(h)=0$ if and only if $h$ is a product of generalized cyclotomic
polynomials up to units.

Based on the work of Schmidt [Sch95, Theorem 21.1], Le showed that [Le14,
Theorem 5] $\mathbf{M}_{i}(X^{\nu})$ can be computed by the Mahler measure of
$\Delta_{i}(X^{\nu})$. When $n=1$, the following result is also proved by
Raimbault [Rai12, Theorem 0.2] independently.

######  Theorem 2.6.

[Le14, Theorem 5] Let $X$ be a connected finite CW complex with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Then for any $i$ we
have

$\mathbf{M}_{i}(X^{\nu})=\mathbf{M}(\Delta_{i}(X^{\nu})).$

If $n=1$ then $\limsup$ in the definition of $\mathbf{M}_{i}(X^{\nu})$ can be
replaced by the ordinary $\lim$.

Inspired by Theorem 2.5, we prove the following proposition.

######  Proposition 2.7.

Let $X$ be a connected finite CW complex with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. If
$\Delta_{i}(X^{\nu})$ is a product of some generalized cyclotomic polynomials
with a non-zero integer $c_{i}$, then $\limsup$ in the definition of
$\mathbf{M}_{i}(X^{\nu})$ can be replaced by the ordinary $\lim$ and
$\mathbf{M}_{i}(X^{\nu})=\log|c_{i}|$.

The proof of this proposition is given in section 5.

###  2.2. Cohomology jump loci

We first recall the definition of cohomology jump loci. For any algebraically
closed field $\mathbb{K}$, the group of $\mathbb{K}$-valued characters,
$\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})$, is a commutative affine algebraic
group. Each character $\rho$ in $\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})$
defines a rank one local system on $X$, denoted by $L_{\rho}$. Since
$\mathbb{K}^{*}$ is abelian, $\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})$ in fact
only depends on $H_{1}(X,\mathbb{Z})$, the abelianization of $\pi_{1}(X)$ and
$\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})=\mathrm{Hom}(H_{1}(X,\mathbb{Z}),\mathbb{K}^{*})$.

######  Definition 2.8.

The cohomology jump loci of $X$ are defined as

$\mathcal{V}^{i}_{k}(X,\mathbb{K})\coloneqq\\{\rho\in\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})\mid\dim_{\mathbb{K}}H^{i}(X,L_{\rho})\geq
k\\}.$

Cohomology jump loci are closed sub-varieties of
$\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})$ and homotopy invariants of $X$. In
cohomological degree one, $\mathcal{V}^{1}_{k}(X,\mathbb{K})$ depends only on
$\pi_{1}(X)$ (e.g. see [Suc11, Section 2.2]).

The map $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$ induces an
embedding
$\nu^{*}\colon(\mathbb{K}^{*})^{n}\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{K}^{*})$.
For a tuple $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in(\mathbb{K}^{*})^{n}$,
let $\nu^{-1}L_{\lambda}$ denote the corresponding rank one $\mathbb{K}$-local
system on $X$ whose monodromy representation factors through $\nu$.

######  Proposition 2.9.

With the notations and assumptions as above, for any $i\geq 0$ and
$\lambda\in(\mathbb{K}^{*})^{n}$ being general we have

$\alpha_{i}(X^{\nu},\mathbb{K})=\dim
H_{i}(X,\nu^{-1}L_{\lambda})=\mathrm{rank}H_{i}(X^{\nu},\mathbb{K}).$

Here $\mathrm{rank}H_{i}(X^{\nu},\mathbb{K})$ is the dimension of the vector
space $H_{i}(X^{\nu},\mathbb{K})\otimes\mathcal{F}$ over the fractional field
$\mathcal{F}$ of $\mathbb{K}[\mathbf{t}^{\pm}]$. In particular,
$\alpha_{i}(X^{\nu},\mathbb{K})$ is always an integer. Moreover, if
$\Delta_{i}(X^{\nu})$ is not a unit in $\mathbb{C}[\mathbf{t}^{\pm}]$, every
irreducible component of the zero locus of $\Delta_{i}(X^{\nu})$ in
$(\mathbb{C}^{*})^{n}$ is a irreducible component of
$\mathcal{V}^{i}_{k}(X,\mathbb{C})\cap(\mathbb{C}^{*})^{n}$ (via $\nu^{*}$)
with codimension one for $k=\mathrm{rank}H_{i}(X^{\nu},\mathbb{C})+1.$

This proposition is not new and the claims are either already proved in [LL23,
Propsition 2.2] or folklore. We give a brief proof in section 5 due to the
lack of precise references. For more relations between cohomology jump loci
and Alexander type invariants, we refer the readers to Suciu’s paper [Suc11].

We also list some folklore properties of $L^{2}$-type invariants, which would
be useful later.

######  Proposition 2.10.

Let $X$ be a connected finite CW complex with a fixed epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Then we have the
following properties.

  * (i)

$\alpha_{*}(X^{\nu},\mathbb{K})$ and $\mathbf{M}_{*}(X^{\nu})$ are homotopy
invariant of the pair $(X,\nu)$.

  * (ii)

Euler number: For any field $\mathbb{K}$,

$\sum_{i\geq 0}(-1)^{i}\alpha_{i}(X^{\nu},\mathbb{K})=\chi(X).$

  * (iii)

Poincare duality: If $X$ is an oriented compact manifold with
$\dim_{\mathbb{R}}X=N$, then

$\alpha_{i}(X^{\nu},\mathbb{K})=\alpha_{N-i}(X^{\nu},\mathbb{K})$ and
$\mathbf{M}_{i}(X^{\nu})=\mathbf{M}_{N-i-1}(X^{\nu})$.

  * (v)

Zero-th $L^{2}$-type invariants: $\alpha_{0}(X^{\nu},\mathbb{K})=0$ and
$\mathbf{M}_{0}(X^{\nu})=0$.

##  3\. Complex smooth quasi-projective variety

In this section, we always assume that $X$ is a complex smooth quasi-
projective variety.

###  3.1. Structure theorem

The following structure theorem for $\mathcal{V}^{i}_{k}(X,\mathbb{C})$ put
strong constraints for the homotopy type of complex smooth quasi-projective
variety. It is contributed by many people and we name a few here: Green and
Lazarsfeld [GL91], Simpson [Sim93A], Arapura [Ara97], Dimca, Papadima and
Suciu [DPS09], Artal Bartolo, Cogolludo-Agustín and Matei [ACM13], Dimca and
Papadima [DP14], Schnell [Sch15], etc. It is finalized by Budur and Wang in
[BW15, BW20].

######  Theorem 3.1.

[BW15, BW20] Let $X$ be a complex smooth quasi-projective variety. Then for
any $i$ and $k$, $\mathcal{V}^{i}_{k}(X,\mathbb{C})$ is a finite union of
torsion translated sub-tori of $\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})$.
Moreover, for every positive dimensional irreducible component $V$ of
$\mathcal{V}^{i}_{k}(X,\mathbb{C})$, there exists a torsion point
$\rho\in\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})$ and an algebraic map $f\colon
X\to G$ with $G$ being a complex semi-abelian variety such that

$V=\rho\cdot\mathrm{im}\\{f^{*}\colon\mathrm{Hom}(\pi_{1}(G),\mathbb{C}^{*})\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})\\}.$

If $X$ is a smooth projective variety, $G$ has to be an abelian variety, hence
every irreducible component of $\mathcal{V}^{i}_{k}(X,\mathbb{C})$ is even
dimensional.

As a direct application of the above structure theorem, Theorem 2.6,
Proposition 2.7 and Proposition 2.9, we have the following results.

######  Corollary 3.2.

Let $X$ be a complex smooth quasi-projective variety with an epimorphism
$\nu\colon\pi_{1}(X)\to\mathbb{Z}^{n}$. Then for any $i$,
$\Delta_{i}(X^{\nu})$ is product of some generalized cyclotomic polynomials
and a non-zero integer $c_{i}$. Then $\limsup$ in the definition of
$\mathbf{M}_{i}(X^{\nu})$ can be replaced by the ordinary $\lim$ and we have

$\mathbf{M}_{i}(X^{\nu})=\log|c_{i}|$.

Let $X$ be a complex smooth projective variety. For any subgroup
$\Gamma\subset\mathbb{Z}^{n}$ with finite index, $X^{\nu,\Gamma}$ is also a
complex smooth projective variety. In particular,
$H^{i}(X^{\nu,\Gamma},\mathbb{Q})$ has pure Hodge structures. Instead of
studying the asymptotic behaviour of Betti number of
$H^{i}(X^{\nu,\Gamma},\mathbb{Q})$, one can also study the asymptotic
behaviour of the Hodge numbers of $H^{i}(X^{\nu,\Gamma},\mathbb{Q})$. Let
$h^{p,q}(X^{\nu,\Gamma})$ denotes the dimension of
$H^{q}(X^{\nu,\Gamma},\Omega^{p}_{X^{\nu,\Gamma}})$.

######  Proposition 3.3.

With the above notations and assumptions, for any $(p,q)$ we have

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\frac{h^{p,q}(X^{\nu,\Gamma})}{|\mathbb{Z}^{n}/\Gamma|}=\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}^{\prime}),$

where $\mathcal{L}^{\prime}=L^{\prime}\otimes\mathcal{O}_{X}$ and $L^{\prime}$
is a generic unitary rank one local system in $\mathrm{im}\nu^{*}$. Here
$\nu^{*}$ is the embedding
$(\mathbb{C}^{*})^{n}\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})$
induced by $\nu$.

The proof of this proposition is given in section 5. When $\nu$ is the natural
map to $H_{1}(X,\mathbb{Z})$ modulo torsions and the Albanese map of $X$ is
semi-small, Di Cerbo and Lombardi [DL19] give an effective estimate for the
asymptotic behaviour of the Hodge numbers and showed that

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\frac{h^{p,q}(X^{\nu,\Gamma})}{|\mathbb{Z}^{n}/\Gamma|}=\begin{cases}(-1)^{q}\chi(X,\Omega^{p}_{X}),&\mathrm{if}\
p+q=\dim_{\mathbb{C}}X,\\\ 0,&\mathrm{else}.\end{cases}$

Using the generic vanishing results proved by Popa and Schnell [PS13], Di
Cerbo and Lombardi’s results are compatible with Proposition 3.3.

One may ask how about Proposition 3.3 for smooth quasi-projective varieties.
In this case, $H^{i}(X^{\nu,\Gamma},\mathbb{Q})$ has mixed Hodge structures.

######  Question 3.4.

Doe the limit for the Hodge numbers of $H^{i}(X^{\nu,\Gamma},\mathbb{Q})$
exist?

Budur proved [Bud09, Theorem 1.8, Theorem 1.9] the polynomial periodicity for
certain Hodge numbers of congruence covers of complex smooth
(quasi-)projective variety based on a structure theorem for line bundles
[Bud09, Theorem 1.3, Theorem 1.4]. His work shed light on the above question
at least for the congruence covers. For more references about polynomial
periodicity of congruence covers, we refer the readers to Suciu’s survey paper
[Suc01, Section 5].

###  3.2. Degree one case

####  3.2.1. Cohomology jump loci

If we focus on cohomological degree one jump loci, a celebrity result due to
Beauville [Bea92], Arapura [Ara97] and Artal Bartolo, Cogolludo-Agustín and
Matei [ACM13] puts even stronger constraints for its fundamental group. To
explain their results, we introduce the notion of orbifold maps.

######  Definition 3.5.

Let $X$ be a smooth complex quasi-projective variety. An algebraic map
$f\colon X\to C_{g,r}$ is called an orbifold map, if $f$ is surjective, has
connected generic fiber and $C_{g,r}$ is a smooth algebraic curve of genus $g$
with $r$ points removed. We always assume that
$C_{g,r}\neq\mathbb{CP}^{1},\mathbb{C}$. There exists a maximal Zariski open
subset $U\subset C_{g,r}$ such that $f$ is a fibration over $U$. Say
$B=C_{g,r}-U$ (could be empty) has $s$ points, denoted by
$\\{q_{1},\ldots,q_{s}\\}$. We assign the multiplicity $\mu_{j}$ of the fiber
$f^{*}(q_{j})$ (the $\gcd$ of the coefficients of the divisor $f^{*}q_{j}$) to
the point $q_{j}$. Such orbifold map $f$ is called of type $(g,r,{\bm{\mu}})$,
where ${\bm{\mu}}=(\mu_{1},\ldots,\mu_{s})$. When $B=\emptyset$,
$\prod_{j=1}^{s}\mu_{j}=1$ by convention.

When $C_{g,r}$ is clear in the context, we simply write $C$. Our definition of
orbifold maps is a bit of different from the classical one, e.g. see [DS14,
Section 4.1]. The classical one will be called the hyperbolic orbifold map,
where we adopt the name from [Delz08].

######  Definition 3.6.

An orbifold map $f\colon X\to C$ of type $(g,r,{\bm{\mu}})$ is called
hyperbolic, if

$2g+r-2+\sum_{j=1}^{s}(1-\dfrac{1}{\mu_{j}})>0.$

This definition is used to rule out the case where $C$ is either
$\mathbb{C}^{*}$ or Elliptic curve and $f$ has no multiple fibers.

The orbifold group $\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}})$ associated to
these data is defined as

$\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}})\coloneqq\pi_{1}(C_{g,r}\backslash\\{q_{1},\ldots,q_{s}\\})/\langle\gamma_{j}^{\mu_{j}}=1\text{
for all }1\leq j\leq s\rangle,$

where $\gamma_{j}$ is a meridian of $q_{j}$. An orbifold map $f\colon X\to C$
of type $(g,r,{\bm{\mu}})$ induces an surjective map to the orbifold group
(see [ACM10, Proposition 1.4])

$f_{*}\colon\pi_{1}(X)\twoheadrightarrow\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}}).$

Hence it induces an embedding for any $k\geq 1$

$\mathcal{V}^{1}_{k}(\pi_{1}^{orb}(C_{g,r},{\bm{\mu}}),\mathbb{K})\to\mathcal{V}^{1}_{k}(X,\mathbb{K}).$

The following theorem shows that every positive-dimensional component of
$\mathcal{V}^{1}_{1}(X,\mathbb{C})$ is characterized by the orbifold map in
this way. The idea first appeared in Beauville’s work [Bea92] for the
projective case and was extended to the quasi-projective case by Arapura
[Ara97] and Artal Bartolo, Cogolludo-Agustín and Matei [ACM13]. Further
properties were found by Dimca [Dim07, Dim09], Dimca-Papadima-Suciu [DPS09],
etc.

######  Theorem 3.7.

[Ara97, ACM13, Dim07] Let $X$ be a complex smooth quasi-projective variety.

  * (a)

We have

$\mathcal{V}^{1}_{1}(X,\mathbb{C})=\bigcup_{f}{f^{*}}\mathcal{V}^{1}_{1}(\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}})),\mathbb{C})\cup
Z,$

where $Z$ is a finite set of torsion points and the union runs over all
hyperbolic orbifold maps $f$.

  * (b)

Fix an orbifold map $f\colon X\to C$ of type $(g,r,{\bm{\mu}})$. Then for any
$\rho\in\mathrm{Hom}(\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}}),\mathbb{C}^{*})$
we have

$\dim H^{1}(X,f^{*}L_{\rho})\geq\dim
H^{1}(\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}}),L_{\rho})$

and equality holds with finitely many (torsion) exceptions.

######  Remark 3.8.

This theorem indeed gives a complete description of all positive-dimensional
components contained in $\mathcal{V}^{1}_{k}(X,\mathbb{C})$ for all $k\geq 1$.

When $\mathrm{char}(\mathbb{K})=p>0$, not much is known for
$\mathcal{V}^{i}_{k}(X,\mathbb{K})$. So far all the known results concerns
about cohomological degree one jump loci. Delzant [Delz08] enabled to
generalize Theorem 3.7(a) to any algebraically closed $\mathbb{K}$-coefficient
case when $X$ is a complex smooth projective variety.

######  Theorem 3.9.

[Delz08] Let $X$ be a complex smooth projective variety and $\mathbb{K}$ be an
algebraically closed field. Then we have

$\mathcal{V}^{1}_{1}(X,\mathbb{K})=\bigcup_{f}{f^{*}}\mathcal{V}^{1}_{1}(\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\bm{\mu}})),\mathbb{K})\cup
Z^{\prime},$

where $Z^{\prime}$ is a finite set of torsion points (depends on $\mathbb{K}$)
and the union runs over all hyperbolic orbifold maps $f$. In particular, the
structure theorem holds for $\mathcal{V}^{1}_{1}(X,\mathbb{K})$.

The following theorem due to Li and the author generalizes Theorem 3.7(b) to
any algebraically closed field coefficients.

######  Theorem 3.10.

[LL21, Theorem 1.6] Let $X$ be a complex smooth quasi-projective variety and
$\mathbb{K}$ be an algebraically closed field. Given an orbifold map $f\colon
X\to C$ of type $(g,r,{\bm{\mu}})$ and any
$\rho\in\mathrm{Hom}(\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}}),\mathbb{K}^{*})$,
we have

$\dim H^{1}(X,f^{*}L_{\rho})\geq\dim
H^{1}(\pi_{1}^{\mathrm{orb}}(C_{g,r},{\bm{\mu}}),L_{\rho})$

and equality holds with finitely many (torsion) exceptions.

######  Remark 3.11.

As like in Remark 3.8, Theorem 3.9 and Theorem 3.10 together give a complete
description for all positive-dimensional components contained in
$\mathcal{V}^{1}_{k}(X,\mathbb{K})$ for all $k$ when $X$ is smooth projective.
Hence the structure theorem holds for $\mathcal{V}^{1}_{k}(X,\mathbb{K})$ for
all $k\geq 1$.

####  3.2.2. BNS-invariants

One may wonder how about Theorem 3.9 for smooth quasi-projective varieties.
Following Delzant’s approach in [Delz08, Delz10], one needs to study the BNS-
invariants of $X$. Let us recall its definition.

Pick a finite generating set of $\pi_{1}(X)$ and let $\Omega(X)$ be the
corresponding Cayley graph of $\pi_{1}(X)$. Given an additive real character
$\chi\colon\pi_{1}(X)\to\mathbb{R}$, let $\Omega_{\chi}(X)$ be the full
subgraph on vertex set $\\{g\in\pi_{1}(X)\mid\chi(g)\geq 0\\}$. Bieri,
Neumann, and Strebel defined [BNS87] a subset $\Sigma^{1}(X)$ of
$H^{1}(X,\mathbb{R})\setminus\\{0\\}$, nowadays called the BNS-invariant of
$X$ as follows, see e.g. [PS10, Definition 7.1].

######  Definition 3.12.

The set $\Sigma^{1}(X)$ consists of those non-zero homomorphisms
$\chi\colon\pi_{1}(X)\to\mathbb{R}$ for which the graph $\Omega_{\chi}(X)$ is
connected.

Papadima and Suciu studied the relations between $\Sigma^{1}(X)$ and the
orbifold maps in [PS10, Section 16], and the results are refined by Suciu
recently via tropical method [Suc21].

######  Theorem 3.13.

Let $X$ be a complex smooth quasi-projective variety. Then

$\Sigma^{1}(X)\subset\big{(}\bigcup_{f}\mathrm{im}(f^{*}\colon
H^{1}(C,\mathbb{R})\to H^{1}(X,\mathbb{R}))\big{)}^{c},$

where the finite union runs over all hyperbolic orbifold maps $f$.

In fact, a result of Delzant [Delz10] shows that the above inclusion holds as
an equality when $X$ is a smooth projective variety (or even a compact Kähler
manifold). Based on a deep theorem due to Simpson [Sim93B], Delzant gave a
complete description of BNS-invariants of complex compact Kähler manifold
[Delz10], which are characterized by the hyperbolic orbifold maps. Using this
description of BNS-invariants by orbifold maps, Delzant managed to prove
Theorem 3.9 in [Delz08].

######  Question 3.14.

Inspired by these work, one may ask the following questions:

  * (i)

Does [Sim93B, Theorem 1] hold for complex smooth quasi-projective varieties?

  * (ii)

Does the inclusion in Theorem 3.13 hold as an equality?

  * (iii)

Does Theorem 3.9 hold for complex smooth quasi-projective varieties?

Ideally a positive answer to question (i) (resp. (ii)) would imply a positive
answer to question (ii) (resp. (iii)). The question (ii) was first raised by
Papadima and Suciu in [PS10] and refined by Suciu in [Suc21, Question 13.7]
for hyperplane arrangement complements.

####  3.2.3. $L^{2}$-type invariants

To compute the $L^{2}$-type invariants of the pair $(X,\nu)$ at homological
degree one, we need to connect $\nu$ with the orbifold maps.

######  Definition 3.15.

Let $X$ be a smooth complex quasi-projective variety with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. We say that $\nu$ is
orbifold effective if there is an orbifold map $f\colon X\rightarrow C$ of
type $(g,r,{\bm{\mu}})$ such that $\nu$ factors through $f_{*}$ as follows:

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We say that $\nu$ is orbifold effective by $f$ and call $\nu$ being of type
$(g,r,{\bm{\mu}})$.

######  Remark 3.16.

As shown in [LL21, Remark 4.3], if $\nu$ is orbifold effective by two
different orbifold maps $f$ and $g$, then $f$ and $g$ must have the same type
$(g,r,{\bm{\mu}})$. So it is well defined to call $\nu$ being of type
$(g,r,{\bm{\mu}})$.

The following theorem by Li and the author gave concrete formulas for the
$L^{2}$-type invariants of $X$ at homological degree one when $\nu$ is
orbifold effective.

######  Theorem 3.17.

[LL21, Theorem 1.10], [LL23, Theorem 1.3] Let $X$ be a complex smooth quasi-
projective variety with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Suppose that $\nu$ is
orbifold effective of type $(g,r,{\bm{\mu}})$. Let $\mathbb{K}$ be a field
with $\mathrm{char}(\mathbb{K})=p\geq 0$. Then

$\alpha_{1}(X^{\nu},\mathbb{K})=2g+r-2+\\#\\{j\mid p\ \mathrm{divides}\
\mu_{j}\\}$

and

$\mathbf{M}_{1}(X^{\nu})=\sum_{j=1}^{s}\log\mu_{j}.$

On the other hand, assume that either $X$ is projective or
$H^{1}(X,\mathbb{Q})$ has pure Hodge structure of type $(1,1)$. Then
$\alpha_{1}(X^{\nu},\mathbb{K})>0$ or $\mathbf{M}_{1}(X^{\nu})>0$ happens only
if $\nu$ is orbifold effective.

######  Remark 3.18.

A positive answer to Question 3.14(iii) would enable us to drop the assumption
that either $X$ is projective or $H^{1}(X,\mathbb{Q})$ has pure Hodge
structure of type $(1,1)$ in the last claim of the above theorem.

######  Remark 3.19.

Due to Proposition 2.9 and Corollary 3.2, the above theorem also implies the
corresponding formulas for the rank of the Alexander homology and the leading
coefficient of the integral Alexander polynomial at homological degree one.

The above theorem is just about $L^{2}$-type invariants at homological degree
one. How about higher homological degrees? Following the structure theorem, we
propose the following higher homological degree analogues. Let $V$ be a
positive-dimensional irreducible component in $\mathcal{V}^{i}(X,\mathbb{C})$.
According to the structure theorem there exists an algebraic map $f\colon X\to
G$ with $G$ being a complex semi-abelian variety such that $V$ is contained in
a translate of

$\mathrm{im}\\{f^{*}\colon\mathrm{Hom}(\pi_{1}(G),\mathbb{C}^{*})\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})\\}.$

######  Question 3.20.

Note that $\pi_{1}(G)$ is a finitely generated free abelian group. Consider
the epimorphism $\nu=f_{*}\colon\pi_{1}(X)\to\pi_{1}(G)$ induced by $f$. Can
one compute $\alpha_{i}(X^{\nu},\mathbb{K})$ and $\mathbf{M}_{i}(X^{\nu})$ by
the geometric information of $f$? One may need to take the Stein factorization
of $f$.

Using Theorem 3.17 and Proposition 2.10, we give a complete description of the
$L^{2}$-type invariants when $X$ is a complex smooth projective surface.

######  Proposition 3.21.

Let $X$ be a complex smooth projective surface with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Let $\mathbb{K}$ be a
field with $\mathrm{char}(\mathbb{K})=p\geq 0$. If $\nu$ is not orbifold
effective, then

$\alpha_{i}(X^{\nu},\mathbb{K})=\begin{cases}\chi(X),&\mathrm{if}\ i=2,\\\
0,&\mathrm{else}\end{cases}$

and

$\mathbf{M}_{i}(X^{\nu})=0$ for all $i$.

On other hand, if $\nu$ is orbifold effective, say of type $(g,r,{\bm{\mu}})$,
then

$\alpha_{i}(X^{\nu},\mathbb{K})=\begin{cases}2g+r-2+\\#\\{j\mid p\
\mathrm{divides}\ \mu_{j}\\},&\mathrm{if}\ i=1,3,\\\
\chi(X)+4g+2r-4+2\\#\\{j\mid p\ \mathrm{divides}\ \mu_{j}\\},&\mathrm{if}\
i=2,\\\ 0&\mathrm{else}\end{cases}$

and

$\mathbf{M}_{i}(X^{\nu})=\begin{cases}\sum_{j=1}^{s}\log\mu_{j},&\mathrm{if}\
i=1,2,\\\ 0,&\mathrm{else}.\end{cases}$

##  4\. Hyperplane arrangement

In this section, we always assume that $X$ is the complement of a hyperplane
arrangement $\mathcal{A}$ in $\mathbb{CP}^{N}$ for some $N$ and we focus on a
fixed epimorphism $\nu\colon\pi_{1}(X)\rightarrow\mathbb{Z}$ for $n=1$.

For any field $\mathbb{K}$, we denote $\nu_{\mathbb{Z}}$ and
$\nu_{\mathbb{K}}$ the corresponding element in $H^{1}(X,\mathbb{Z})$ and
$H^{1}(X,\mathbb{K})$, respectively. Note that $\nu_{\mathbb{Z}}\cup
v_{\mathbb{Z}}=0,$ hence $\nu_{\mathbb{K}}\cup\nu_{\mathbb{K}}=0$. Then we get
two chain complexes by cup product

$\textstyle{(H^{\ast}(X,\mathbb{Z}),\cdot\nu_{\mathbb{Z}})\colon}$$\textstyle{H^{0}(X,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\mathbb{Z}}}$$\textstyle{H^{1}(X,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\mathbb{Z}}}$$\textstyle{H^{2}(X,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{(H^{\ast}(X,\mathbb{K}),\cdot\nu_{\mathbb{K}})\colon}$$\textstyle{H^{0}(X,\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\mathbb{K}}}$$\textstyle{H^{1}(X,\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\mathbb{K}}}$$\textstyle{H^{2}(X,\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

which are called the Aomoto complex of $H^{\ast}(X,\mathbb{Z})$ and
$H^{\ast}(X,\mathbb{K})$ associated to $\nu_{\mathbb{Z}}$ and
$\nu_{\mathbb{K}}$, respectively.

######  Definition 4.1.

With the above notations, we define the $i$-th Aomoto Betti number with
$\mathbb{K}$-coefficients as

$\beta_{i}(X,\nu_{\mathbb{K}})\coloneqq\dim_{\mathbb{K}}H^{i}(H^{\ast}(X,\mathbb{K}),\cdot\nu_{\mathbb{K}}).$

and the order of the $i$-th Aomoto torsion as

$\tau_{i}(X,\nu_{\mathbb{Z}})\coloneqq|H^{i+1}(H^{*}(X,\mathbb{Z}),\cdot\nu_{\mathbb{Z}})_{\mathrm{tor}}|.$

Here the shift by 1 is due to the Universal Coefficient Theorem.

As shown by Orlik and Solomon [OS80] the cohomology ring $H^{*}(X,\mathbb{Z})$
are determined by the combinatorics data of the hyperplane arrangement. Then
$\beta_{*}(X,\nu_{\mathbb{K}})$ and $\tau_{*}(X,\nu_{\mathbb{Z}})$ are
combinatorially determined, once $\nu$, considered as an element in
$H^{1}(X,\mathbb{Z})$, is fixed. Papadima and Suciu discovered the following
results in [PS10].

######  Theorem 4.2.

With the above notations and assumptions, for any field $\mathbb{K}$ we have

$\alpha_{i}(X^{\nu},\mathbb{K})\leq\beta_{i}(X,\nu_{\mathbb{K}}).$

Moreover, when $\mathbb{K}=\mathbb{C}$, the above inequality holds as a
equality.

Papadima and Suciu [PS10] indeed proved the above inequality for any finite CW
complex. When $\mathbb{K}=\mathbb{C}$, the equality was first proved (by
different methods) by Cohen and Suciu [CS99], Libgober [Lib02], and Libgober
and Yuzvinsky [LY00] for $i=1$ and was generalized to the higher-degrees by
Cohen and Orlik [CO00, Theorem 3.7].

On the other hand, the author jointly with Li gave the following combinatorial
upper bound for $\mathbf{M}_{*}(X^{\nu})$.

######  Proposition 4.3.

[LL21, Proposition 1.12] Let $X$ be the complement of a hyperplane
arrangement. For any epimorphism $\nu\colon\pi_{1}(X)\to\mathbb{Z}$ and any
$i\geq 0$, we have

$\exp(\mathbf{M}_{i}(X^{\nu}))\mid\tau_{i}(X,\nu_{\mathbb{Z}}).$

In particular, if $\nu$ is orbifold effective of type $(g,r,{\bm{\mu}})$, then

(4.1) $\prod_{j=1}^{s}\mu_{j}\mid\tau_{1}(X,\nu_{\mathbb{Z}}).$

The proof of this proposition is based on Papadima and Suciu’s work [PS10,
Theorem 12.6] and two facts about hyperplane arrangement complement:

  * •

the hyperplane arrangement complement $X$ is homotopy equivalent to a minimal
CW complex, see [DP03, Ran02].

  * •

$\alpha_{i}(X^{\nu},\mathbb{C})=\beta_{i}(X,\nu_{\mathbb{C}})$ for any $i$ as
in Theorem 4.2.

There are examples of hyperplane arrangements where
$\alpha_{i}(X,\mathbb{K})\neq\beta_{i}(X,\mathbb{K})$ and
$\prod_{j=1}^{s}\mu_{j}\neq\tau_{1}(X,\nu_{\mathbb{Z}})$, see [LL21, Example
5.3] and the next example.

######  Example 4.4.

Let $\mathcal{B}$ be the deleted $B_{3}$-arrangement in $\mathbb{CP}^{2}$ with
defining equations

$zxy(x-y)(x-z)(y-z)(x-y-z)(x-y+z).$

Order the hyperplanes as the factors of the defining polynomial. Let $X$ be
the complement of the arrangement. It was first discovered by Suciu [Suc02]
that $\mathcal{V}^{1}_{1}(X,\mathbb{C})$ has a translated component

$V\coloneqq\rho\otimes\\{t,t^{-1},1,t^{-1},t,t^{2},t^{-2}\mid
t\in\mathbb{C}^{*}\\}$

with $\rho=(1,-1,-1,-1,1,1,1)\in(\mathbb{C}^{*})^{7}$, e.g. see [Dim17,
Example 6.16]. Here we take $\\{z=0\\}$ as the hyperplane at infinity. This
component is induced by the orbifold map $f\colon X\rightarrow\mathbb{C}^{*}$
as following

$f([x,y,z])=\frac{x(y-z)(x-y-z)^{2}}{y(x-z)(x-y+z)^{2}}$

and $f$ is of type $(0,2,2)$. Consider $\nu=(1,-1,0,-1,1,2,-2)\in
H^{1}(X,\mathbb{Z})$ induced by $f$. Then

$\alpha_{1}(X^{\nu},\mathbb{K})=\begin{cases}1,&\mathrm{if}\
\mathrm{char}(\mathbb{K})=2,\\\ 0,&\mathrm{if}\ \mathrm{char}(\mathbb{K})\neq
2,\end{cases}$ and $\mathbf{M}_{1}(X^{\nu})=\log 2$.

On the other hand, the Aomoto complex for $\nu_{\mathbb{Z}}$ and
$\nu_{\mathbb{K}}$ can be computed by the formula given in [Dim17, p.119]. A
direct computation shows that

$\tau_{1}(X,\nu_{\mathbb{Z}})=4$ and
$\beta_{1}(X,\nu_{\mathbb{K}})=\begin{cases}1,&\mathrm{if}\
\mathrm{char}(\mathbb{K})=2,\\\ 0,&\mathrm{if}\ \mathrm{char}(\mathbb{K})\neq
2.\end{cases}$

It is a long standing open question for hyperplane arrangement if
$\mathcal{V}^{1}_{1}(X,\mathbb{C})$ is combinatorially determined, see e.g.
[Suc14, Problem 3.15]. It is known that the positive-dimensional components of
$\mathcal{V}^{1}_{1}(X,\mathbb{C})$ passing through the origin are
combinatorially determined (see e.g. [Dim17, Theorem 6.1]) and they can be
described by the multinet structure, for which we refer in particular to the
work of Falk and Yuzvinsky [FY07]. Since $\mathcal{V}^{1}_{1}(X,\mathbb{C})$
only depends on the the fundamental group of $X$, using Lefschetz hyperplane
section theorem, one can assume that $X$ is the complement of a line
arrangement in $\mathbb{CP}^{2}$. Let us now recall the definition of
multinet.

######  Definition 4.5.

Let $\mathcal{A}$ be an arrangement in $\mathbb{CP}^{2}$. A multinet on
$\mathcal{A}$ is a partition of $\mathcal{A}$ into $k\geq 3$ subsets
$\mathcal{A}_{1},\cdots,\mathcal{A}_{k}$, together with an assignment of
multiplicities, $m\colon\mathcal{A}\to\mathbb{Z}_{>0}$, and a subset
$\mathfrak{X}$ of intersection points in $\mathcal{A}$ satisfying the
following conditions:

  * (a)

$\sum_{H\in\mathcal{A}_{i}}m_{H}=\kappa$, independent of $i$;

  * (b)

for each $H\in\mathcal{A}_{i}$ and $H^{\prime}\in\mathcal{A}_{j}$ with $i\neq
j$, $H\cap H^{\prime}\in\mathfrak{X}$;

  * (c)

for each $x\in\mathfrak{X}$, the sum $\sum_{H\in\mathcal{A}_{i},x\in H}m_{H}$
is independent of $i$;

  * (d)

for every $1\leq i\leq k$ and $H,H^{\prime}\in\mathcal{A}_{i}$, there is a
sequence $\\{H_{0},H_{1},\cdots,H_{s}\\}$ with $H_{0}=H$ and
$H_{s}=H^{\prime}$ such that $H_{j-1}\cap H_{j}\notin\mathfrak{X}$ for all
$1\leq j\leq s$.

  * (e)

$\gcd(m_{H}\colon H\in\mathcal{A})$=1.

Such multinet is called a $(k,\kappa)$-multinet.

Note that the multinet only depends on the intersection lattice of
$\mathcal{A}$.

By results of Pereira-Yuzvinsky [PY08] and Yuzvinsky [Yuz09], $k=3$ or $4$ if
$|\mathfrak{X}|>1$. Moreover, Yuzvinsky conjectured that $k=4$ only happens
for Hessian arrangement of 12 planes in $\mathbb{CP}^{2}$, defined by the
polynomial

$xyz\prod_{j=0}^{2}\prod_{k=0}^{2}(x+\omega^{j}y+\omega^{k}z)=0$ with
$\omega=\exp{2\pi i/3}$.

As shown by Falk and Yuzvinsky [FY07], and Pereira and Yuzvinsky [PY08], every
$(k,\kappa)$-multinet gives an orbifold map $f\colon X\to
C_{0,k}\subset\mathbb{CP}^{1}$ with $X=\mathbb{CP}^{2}-\mathcal{A}$ by

$f(x)\coloneqq[\prod_{H\in\mathcal{A}_{1}}f_{H}^{m_{H}}(x),\prod_{H\in\mathcal{A}_{2}}f_{H}^{m_{H}}(x)].$

######  Definition 4.6.

Let $X=\mathbb{CP}^{2}-\mathcal{A}$ with $\mathcal{A}$ being a essential line
arrangement in $\mathbb{CP}^{2}$. We call a hyperbolic orbifold map $f\colon
X\to C_{0,k}$ non-local, if it is not the restriction of another hyperbolic
orbifold map over the complement of a sub-arrangement of $\mathcal{A}$.

######  Theorem 4.7.

[FY07] Let $\mathcal{A}$ be a essential hyperplane arrangement in
$\mathbb{CP}^{2}$ with $X=\mathbb{CP}^{2}-\mathcal{A}$. Then every non-local
hyperbolic orbifold map $f\colon X\to C_{0,k}$ satisfies

  * •

either $k=2$ and $f$ has at least one multiple fiber,

  * •

or $k=3,4$, and $f$ is obtained from a $(k,\kappa)$-multinet for some
$\kappa$.

The above theorem shows that the existence of the non-local hyperbolic
orbifold map $f\colon X\to C_{0,k}$ with $k=3$ or $4$ are combinatorially
determined (by the multinet structure).

######  Question 4.8.

We have the following questions for $X=\mathbb{CP}^{2}\setminus\mathcal{A}$
with $\mathcal{A}$ being a essential line arrangement in $\mathbb{CP}^{2}$.

  * (i)

Is the existence of non-local hyperbolic orbifold map $f\colon X\to
C_{0,2}=\mathbb{C}^{*}$ combinatorially determined?

  * (2)

Is the existence of non-local hyperbolic orbifold map $f\colon X\to C_{0,k}$
having multiple fibers for $k=3$ or $4$ combinatorially determined?

Assume that $\mathcal{A}$ has $(d+1)$-many lines and let $f_{i}$ be the degree
one defining equation for the $i$-th line $H_{i}\in\mathcal{A}$ with $0\leq
i\leq d$.

######  Assumption 4.9.

For $a=(a_{0},\cdots,a_{d})\in\mathbb{Z}^{d+1}$, we assume that
$\sum_{i=0}^{d}a_{i}=0$, $a_{i}\neq 0$ for all $i$ (this is for the non-local
requirement) and $\gcd(a_{0},\cdots,a_{d})=1$.

Then each $a\in\mathbb{Z}^{d+1}$ satisfying this assumption gives a well-
defined algebraic map

$f^{a}\coloneqq\prod_{i=0}^{d}f_{i}^{a_{i}}\colon X\to\mathbb{C}^{*}.$

Note that $f^{a}$ has connected generic fiber, since
$\gcd(a_{0},\cdots,a_{d})=1$. On the other hand, clearly $f^{a}$ is
surjective. So $f^{a}$ is an orbifold map. Moreover, any orbifold map from $X$
to $\mathbb{C}^{*}$ has to be realized in this way. So Question 4.8(i) is
equivalent to ask for which $a\in\mathbb{Z}^{d+1}$ satisfying Assumption 4.9
does $f^{a}$ have multiple fibers.

Note that $f^{a}$ induces a surjective map

$f^{a}_{*}\colon\pi_{1}(X)\to\pi_{1}(\mathbb{C}^{*})\cong\mathbb{Z}.$

This map only depends on $a$, which sends the meridian of $H_{i}$ to $a_{i}$.
It gives a cohomology class in $H^{1}(X,\mathbb{Z})$, say denoted by
$\nu^{a}_{\mathbb{Z}}$. Proposition 4.3 shows that $f^{a}$ has multiple fibers
only if $\tau_{1}(X,\nu^{a}_{\mathbb{Z}})>1$, which leads to the following
combinatorics question.

######  Question 4.10.

For which $a\in\mathbb{Z}^{d+1}$ satisfying Assumption 4.9, we have
$\tau_{1}(X,\nu^{a}_{\mathbb{Z}})>1$ ?

On the other hand, we can also use Proposition 4.3 to study Question 4.8(ii).
Let $f\colon X\to C_{0,k}$ be a non-local orbifold map with $k=3$ or $4$. Then
$f$ induces a surjective map

$f_{*}\colon H_{1}(X,\mathbb{Z})\to
H_{1}(C_{0,k},\mathbb{Z})\cong\mathbb{Z}^{k-1}.$

Choosing any epimorphism $\nu\colon\mathbb{Z}^{k-1}\to\mathbb{Z},$ we obtain a
composed epimorphism, still denoted by $\nu$. Denote corresponding cohomology
class in $H^{1}(X,\mathbb{Z})$ by $\nu_{\mathbb{Z}}$. Proposition 4.3 shows
that $f$ has multiple fiber only if $\tau_{1}(X,\nu_{\mathbb{Z}})>1$ for any
such $\nu$. Since $k=3$ or $4$, there are infinitely many choices for $\nu$,
which gives a great possibility to show that $f$ has no multi fiber in this
case. We confirm this guess under some assumptions for multinet when $k=3$.

######  Assumption 4.11.

Given a $(3,\kappa)$-multinet on $\mathcal{A}$ in $\mathbb{CP}^{2}$, we assume
that

  * (a)

$m_{H}=1$ for any $H\in\mathcal{A}_{1}$ or $\mathcal{A}_{2}$.

  * (b)

If $H,H^{\prime}\in\mathcal{A}_{1}$ and $H\cap H^{\prime}\notin\mathfrak{X}$,
then there is no third hyperplane in $\mathcal{A}$ passing through $H\cap
H^{\prime}$. So is $\mathcal{A}_{2}$.

  * (c)

Consider $\mathcal{A}_{3}$ and $\mathfrak{X}$. If there exists a point
$x\in\mathfrak{X}$ such that $x$ is contained in precisely one hyperplane
$H\in\mathcal{A}_{3}$, one replaces $\mathcal{A}_{3}$ and $\mathfrak{X}$ by
deleting the hyperplane $H$ and the point $x$, respectively. We assume that
one can run this procedure for $\mathcal{A}_{3}$ and $\mathfrak{X}$ until
there is no hyperplane left in $\mathcal{A}_{3}$.

Under this assumption, the author jointly with Li showed that the
corresponding orbifold map $f$ has no multiple fibers. The technical
conditions in Assumption 4.11 are used to prove
$\tau_{1}(X,\nu_{\mathbb{Z}})=1$ for some $\nu$.

######  Proposition 4.12.

[LL21, Proposition 5.9] Let $f\colon X\to C_{0,3}$ be an orbifold map
corresponding to a $(3,\kappa)$-multinet structure for $\mathcal{A}$ in
$\mathbb{CP}^{2}$ with $X=\mathbb{CP}^{2}\setminus\mathcal{A}$. If this
multinet satisfies Assumption 4.11, $f$ has no multiple fibers.

######  Example 4.13.

We give a list of examples which satisfies Assumption 4.11: Figure 1(a),
Figure 2, Figure 3(a),(b) in [FY07] with $A,B,C$ corresponding to
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$ and Figure 1 in [Suc14]. It
is also easy to find examples which do not satisfy Assumption 4.11, see Figure
1(b) in [FY07] or the monomial (alias Ceva) arrangement (with $m\geq 3$) in
[Dim17, Example 6.11].

Consider a non-local orbifold map $f\colon X\to C_{0,k}$ with $k=2,3,4$. When
$k=2$, there are plenty of examples where $f$ has multiple fibers, e.g. see
Example 4.4. In fact, Suciu discovered a combinatorial construction to produce
such orbifold maps having multiple fibers from a given multinet, called the
pointed multinet structure, see e.g. [Suc14, Section 3.6]. He asked if all
such orbifold maps appear in this way, see [Suc14, Problem 3.12].

On the other hand, to the author’s knowledge, no examples with $f$ having
multiple fibers are known when $k=3,4$.

######  Question 4.14.

Is it possible to show that $f$ always has no multiple fibers when $k=3$ or
$k=4$?

All the questions in this section should be considered as sub-questions of the
aforementioned question:

######  Question 4.15.

Is $\mathcal{V}^{1}_{1}(X,\mathbb{C})$ combinatorially determined for
hyperplane arrangement complement $X$?

##  5\. Proof of results

###  5.1. Proof of Proposition 2.7

Set $\mathbb{Z}[\mathbf{t}^{\pm}]=\mathbb{Z}[\mathbb{Z}^{n}].$ Let $M$ be a
finitely generated $\mathbb{Z}[\mathbf{t}^{\pm}]$-module. Consider a
presentation of $M$ by an exact sequence

$\mathbb{Z}[\mathbf{t}^{\pm}]^{b^{\prime}}\overset{\partial}{\to}\mathbb{Z}[\mathbf{t}^{\pm}]^{b}\to
M\to 0$

where $\partial$ is given by a matrix of size $b^{\prime}\times b$ with
entries in $\mathbb{Z}[\mathbf{t}^{\pm}]$. For any $j\geq 0$, let
$\Delta^{j}(M)$ denote the the greatest common divisor of all the
$(b-j)\times(b-j)$ minors of $\partial$ with the convention that
$\Delta^{j}(M)=1$ if $j\geq b$. Clearly $\Delta^{j}(M)$ divides
$\Delta^{j-1}(M)$ for every $j\geq 1$. It is known that $\Delta^{j}(M)$ only
depends on $M$, not on any particular presentation. The rank of the module $M$
over $\mathbb{Z}[\mathbf{t}^{\pm}]$ is the dimension of the vector space
$M\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathcal{F}^{\prime}$ over the
fractional field $\mathcal{F}^{\prime}$ of $\mathbb{Z}[\mathbf{t}^{\pm}]$. If
M has rank $r$, then $\Delta^{j}(M)=0$ if $j<r$, and
$\Delta^{j}(M)=\Delta^{j-r}(\mathrm{Tor}(M))$, where

$\mathrm{Tor}(M)=\\{x\in M\mid fx=0\text{ for some non-zero
}f\in\mathbb{Z}[\mathbf{t}^{\pm}]\\}.$

As in [Le14, section 1.3], a $\mathbb{Z}[\mathbf{t}^{\pm}]$-module $M$ is
called pseudo-zero if for every prime ideal $P$ of height 1, the localization
$M_{P}$ is 0. A $\mathbb{Z}[\mathbf{t}^{\pm}]$-module morphism $M_{1}\to
M_{2}$ is a pseudo-isomorphism if the kernel and co-kernel are pseudo-zero.
Moreover, two finitely generated torsion
$\mathbb{Z}[\mathbf{t}^{\pm}]$-modules $M_{1}$ and $M_{2}$ are pseudo-
isomorphic if and only if $\Delta^{j}(M_{1})=\Delta^{j}(M_{2})$ for any $j\geq
0$.

To prove Proposition 2.7, let us first recall a theorem proved by Le.

######  Theorem 5.1.

[Le14, Theorem 6] Assume that $M_{1}$ and $M_{2}$ are two pseudo-isomorphic
finitely generated $\mathbb{Z}[\mathbf{t}^{\pm}]$ modules. Then
$|\mathrm{tor}_{\mathbb{Z}}\big{(}M_{1}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|$
and
$|\mathrm{tor}_{\mathbb{Z}}\big{(}M_{2}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|$
have the same growth rate in the sense that

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\big{(}\frac{\log|\mathrm{tor}_{\mathbb{Z}}\big{(}M_{1}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|}{|\mathbb{Z}^{n}/\Gamma|}-\frac{\log|\mathrm{tor}_{\mathbb{Z}}\big{(}M_{2}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|}{|\mathbb{Z}^{n}/\Gamma|}\big{)}=0.$

Here
$|\mathrm{tor}_{\mathbb{Z}}\big{(}M_{i}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|$
denotes the order of the torsion part of
$M_{i}\otimes_{\mathbb{Z}[t^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]$ as an
abelian group.

Note that the limit in this theorem is the ordinary limit, not the upper
limit. The following result is a direct application of this theorem and
Theorem 2.5.

######  Proposition 5.2.

Let $M$ be a finitely generated $\mathbb{Z}[\mathbf{t}^{\pm}]$-modules with
rank $r$. If $\Delta^{r}(M)$ is a product of some generalized cyclotomic
polynomials with a non-zero integer $c$, then

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\frac{\log|\mathrm{tor}_{\mathbb{Z}}\big{(}M\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|}{|\mathbb{Z}^{n}/\Gamma|}=\log|c|.$

###### Proof.

Consider the sub-module $\mathrm{Tor}(M)$ of $M$. We have that

$\Delta^{j}(M)=\Delta^{j-r}(\mathrm{Tor}(M)).$

Moreover, it follows from Theorem 2.5 and [Le14, Theorem 1.2, Proposirion 3.4]
that

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\big{(}\frac{\log|\mathrm{tor}_{\mathbb{Z}}\big{(}M\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|}{|\mathbb{Z}^{n}/\Gamma|}-\frac{\log|\mathrm{tor}_{\mathbb{Z}}\big{(}\mathrm{Tor}(M)\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|}{|\mathbb{Z}^{n}/\Gamma|}\big{)}=0.$

So without loss of generality, we assume that $M$ is a finitely generated
torsion $\mathbb{Z}[\mathbf{t}^{\pm}]$-module. Assume that $\Delta^{j}(M)=1$
for $j>k$. Consider a finitely generated torsion
$\mathbb{Z}[\mathbf{t}^{\pm}]$-module

$M^{\prime}\coloneqq\bigoplus_{j=0}^{k}\frac{\mathbb{Z}[\mathbf{t}^{\pm}]}{\big{(}\Delta^{j}(M)/\Delta^{j+1}(M)\big{)}}.$

It is easy to see that $\Delta^{j}(M)=\Delta^{j}(M^{\prime})$ for any $j\geq
0$. Then $M$ is pseudo-isomorphic to $M^{\prime}$ by [Hil02, Theorem 3.5].
Using Theorem 5.1, the proof is reduced to $M^{\prime}$. Note that
$\Delta^{j+1}(M)$ divides $\Delta^{j}(M)$ for all $j\geq 0$, hence
$\Delta^{j}(M)/\Delta^{j+1}(M)$ is a also product of some generalized
cyclotomic polynomials and some non-zero integer. Since $M^{\prime}$ is a
direct sum, the proof is further reduced to the module with the type
$N=\mathbb{Z}[\mathbf{t}^{\pm}]/(h)$, where $h$ a product of some generalized
cyclotomic polynomials and some non-zero integer $d$.

Set $h^{\prime}=\frac{h}{d}$ and
$N^{\prime}=\mathbb{Z}[\mathbf{t}^{\pm}]/(h^{\prime})$. Clearly we have

$|\mathrm{tor}_{\mathbb{Z}}\big{(}N\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|=|d|^{|\mathbb{Z}^{n}/\Gamma|}\cdot|\mathrm{tor}_{\mathbb{Z}}\big{(}N^{\prime}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|.$

Since $h^{\prime}$ is a product of some generalized cyclotomic polynomials,
using Theorem 2.5 and Theorem [Le14, Theorem 3] we have

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\frac{\log|\mathrm{tor}_{\mathbb{Z}}\big{(}N^{\prime}\otimes_{\mathbb{Z}[\mathbf{t}^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|}{|\mathbb{Z}^{n}/\Gamma|}=0.$

Then the claim follows. ∎

###### Proof of Proposition 2.7.

As shown in the proof of [Le14, Theorem 4], we have

$|H_{i}(X^{\nu,\Gamma},\mathbb{Z})_{\mathrm{tor}}|=|\mathrm{tor}_{\mathbb{Z}}\big{(}M\otimes_{\mathbb{Z}[t^{\pm}]}\mathbb{Z}[\mathbb{Z}^{n}/\Gamma]\big{)}|$

with $M=\mathrm{coker}\partial_{i}$, see [Le14, (54) on page 745]. Clearly $M$
has a presentation by the exact sequence

$C_{i+1}(X^{\nu},\mathbb{Z})\overset{\partial_{i}}{\to}C_{i}(X^{\nu},\mathbb{Z})\to
M\to 0$

and
$\mathrm{rank}M=\mathrm{rank}C_{i}(X^{\nu},\mathbb{Z})-\mathrm{rank}\partial_{i}$.
Then we have $\Delta^{\mathrm{rank}M}(M)=\Delta_{i}(X^{\nu})$ by definition.
Therefore the claim follows from Proposition 5.2. ∎

###  5.2. Proof of Proposition 2.9

The first equality follows from the proof of [LL23, Propsition 2.2]. The
second equality follows directly from the fact that computing
$\mathrm{rank}H_{i}(X^{\nu},\mathbb{K})$ over the fraction field is same as
computing the corresponding homology over a general point.

The second claim follows from the fact that every irreducible component of the
zero locus of $\Delta_{i}(X^{\nu})$ in $(\mathbb{C}^{*})^{n}$ gives a prime
ideal with height one in $\mathbb{C}[\mathbf{t}^{\pm}]$. Taking localization
of $\mathbb{C}[\mathbf{t}^{\pm}]$ over this primed ideal, one gets a PID, say
denoted by $R$. Consider the localized complex
$C_{*}(X^{\nu},\mathbb{C})\otimes_{\mathbb{C}[\mathbf{t}^{\pm}]}R$. Then the
second claim follows easily by analysing this localized complex of modules
over PID, see e.g. [BLW18, Section 2.2].

###  5.3. Proof of Proposition 3.3

Let $\mathrm{Pic}^{0}(X)$ denote the identity component of the Picard variety
of $X$, which parameterise topologically trivial line bundles on $X$. Set

$S^{p,q}_{k}(X)\coloneqq\\{\mathcal{L}\in\mathrm{Pic}^{0}(X)\mid\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L})\geq k\\}.$

Green and Lazarsfeld showed [GL91] that $S^{p,q}_{k}(X)$ is a finite union of
torsion translated sub-abelian variety of $\mathrm{Pic}^{0}(X)$. In
particular, these are sub-varieties of $\mathrm{Pic}^{0}(X)$.

Consider the moduli space of unitary rank one $\mathbb{C}$-local system on $X$
defined as

$\mathrm{Uchar}^{0}(X)\coloneqq\mathrm{Hom}(H_{1}(X,\mathbb{Z})/\\{\mathrm{torsion}\\},U(1)).$

Any unitary local system $L\in\mathrm{Uchar}^{0}(X)$ corresponds to a
topologically trivial line bundle
$\mathcal{L}=L\otimes_{\mathbb{C}}\mathcal{O}_{X}$. This gives a one-to-one
correspondence between $\mathrm{Uchar}^{0}(X)$ and $\mathrm{Pic}^{0}(X)$ as
real torus. By slight modification of the usual Hodge theoretic arguments, one
gets the following Hodge decomposition

(5.1)
$H^{i}(X,L)\cong\bigoplus_{p+q=i}H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}).$

The epimorphism $\nu$ induces an embedding
$\nu^{\\#}\colon(U(1))^{n}\hookrightarrow\mathrm{Uchar}^{0}(X).$ Let
$\mathrm{im}\nu^{\\#}$ denote the corresponding image in $\mathrm{Pic}^{0}(X)$
by the above identification. Since $S^{p,q}_{k}(X)$ are sub-varieties of
$\mathrm{Pic}^{0}(X)$, we can pick a generic unitary local system
$L^{\prime}\in\mathrm{im}\nu^{\\#}$ with
$\mathcal{L}^{\prime}=L^{\prime}\otimes\mathcal{O}_{X}$ (to be precise,
$L^{\prime}$ is generic with respect to all degrees $(p,q)$ with $0\leq
p,q\leq\dim X$). By semi-continuity, for any unitary local system $L$ and its
corresponding line bundle $\mathcal{L}$ we have

$\dim H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}^{\prime})\leq\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}).$

Let $\pi\colon X^{\nu,\Gamma}\to X$ denote the covering map. Then

$\pi_{*}\mathcal{O}_{X}=\bigoplus\mathcal{L}_{\lambda}$

where $\mathcal{L}_{\lambda}=L_{\lambda}\otimes\mathcal{O}_{X}$ and the direct
sum runs over all unitary local system $L_{\lambda}$ with
$\lambda\in\mathrm{Hom}(\mathbb{Z}^{n}/\Gamma,\mathbb{C}^{*})$. We have

$\dim H^{q}(X^{\Gamma},\Omega^{p}_{X^{\Gamma}})=\sum_{\lambda}\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}_{\lambda})\geq|\mathbb{Z}^{n}/\Gamma|\cdot\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}^{\prime}).$

Meanwhile, the Hodge decomposition gives us

$\sum_{p+q=i}\dim H^{q}(X^{\Gamma},\Omega^{p}_{X^{\Gamma}})=\dim
H^{i}(X^{\Gamma},\mathbb{C}).$

Putting all together, we have that

$\sum_{p+q=i}\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}^{\prime})\leq\sum_{p+q=i}\dfrac{\dim
H^{q}(X^{\Gamma},\Omega^{p}_{X^{\Gamma}})}{|\mathbb{Z}^{n}/\Gamma|}=\dfrac{\dim
H^{i}(X^{\Gamma},\mathbb{C})}{|\mathbb{Z}^{n}/\Gamma|}.$

$\mathcal{V}^{i}_{k}(X)$ are also closed sub-varieties. By further requiring
the unitary local system $L^{\prime}$ generic with respect to the jump loci,
Proposition 2.9 and Universal Coefficients Theorem imply that the right hand
of the above inequality converges to $\dim H^{i}(X,L^{\prime})$ when
$\langle\Gamma\rangle\to\infty$. Note that the Hodge decomposition (5.1) gives
us

$\sum_{p+q=i}\dim H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}^{\prime})=\dim
H^{i}(X,L^{\prime}).$

Therefore we have

$\lim_{\langle\Gamma\rangle\rightarrow\infty}\frac{h^{p,q}(X^{\nu,\Gamma})}{|\mathbb{Z}^{n}/\Gamma|}=\dim
H^{q}(X,\Omega^{p}_{X}\otimes\mathcal{L}^{\prime}).$

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