#  $L^{2}$-type invariants and cohomology jump loci for complex smooth quasi-
projective varieties

Fenglin Li  Fenglin Li: School of Mathematical Sciences, University of Science
and Technology of China, 96 Jinzhai Road, Hefei Anhui 230026 China
<EMAIL_ADDRESS>and  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>
###### Abstract.

Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphsim
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. In this paper, we consider
the asymptotic behaviour of invariants such as Betti numbers with all possible
field coefficients and the order of the torsion subgroup of singular homology
associated to $\nu$, known as the $L^{2}$-type invariants. For degree 1, we
give concrete formulas to compute these limits by the geometric information of
$X$ when $\nu$ is orbifold effective. The proof relies on the theory of
cohomology jump loci. We give a detailed study about the degree 1 cohomology
jump loci of $X$ with arbitrary algebraically closed field coefficients. In
particular, we extend part of Arapura's result for degree 1 cohomology jump
loci of $X$ with complex field coefficients to the one with positive
characteristic field coefficients. As an application, when $X$ is a hyperplane
arrangement complement, an combinatoric upper bound is given for the number of
parallel positive dimension components in degree 1 cohomology jump loci with
complex coefficients. Another application is that we give a positive answer to
a question posed by Denham and Suciu: for any prime number $p\geq 2$, there
exists a central hyperplane arrangement such that its Milnor fiber has non-
trivial $p$-torsion in homology and $p$ does not divide the number of
hyperplanes in the arrangement.

###### Key words and phrases:

Alexander module, Mahler measure, jump loci, orbifold map, $L^{2}$-betti
number, hyperplane arrangement, Milnor fiber

###### 2010 Mathematics Subject Classification:

14F17, 14K12, 32S60

###### Contents

  1. 1 Introduction
    1. 1.1 $L^{2}$-type invariants
    2. 1.2 Main results
      1. 1.2.1 Cohomology jump loci and orbifold maps
      2. 1.2.2 Formulas for $L^{2}$-type invariants
    3. 1.3 Hyperplane arrangements
    4. 1.4 Structure of the paper
    5. 1.5 Notations
  2. 2 Alexander and $L^{2}$-type invariants
    1. 2.1 Algebraic Preliminaries
    2. 2.2 Alexander modules and Alexander polynomials
    3. 2.3 $L^{2}$-type invariants
      1. 2.3.1 $L^{2}$-betti number
      2. 2.3.2 Limits of torsion
      3. 2.3.3 Universal Coefficients Theorem
    4. 2.4 Aomoto complex
  3. 3 Cohomology jump loci and orbifold groups
    1. 3.1 Cohomology jump loci
    2. 3.2 Orbifold Groups
      1. 3.2.1 Non-compact case
      2. 3.2.2 Compact case
      3. 3.2.3 $L^{2}$-type invariants for orbifold group
  4. 4 Proof of the main results
    1. 4.1 Orbifold Maps
    2. 4.2 Proof of Theorem 1.6
    3. 4.3 Proof of Theorem 1.10
  5. 5 Hyperplane arrangement
    1. 5.1 Some properties and examples
    2. 5.2 Multinet
    3. 5.3 A question by Denham and Suciu

##  1\. Introduction

There is a general principal to consider a classical invariant of a finite CW
complex $X$ and to define its analog for some covering space of $X$. 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 literatures 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 finitely sheeted normal coverings.

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

Assume that $X$ is a connected finite CW-complex with a fixed group
epimorphism $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Let $X^{\nu}$
denote the corresponding infinite cyclic covering space of $X$. For any
positive integer $N$, let $X^{\nu,N}$ denote the covering space of $X$
associated to the following composition of maps

$\pi_{1}(X)\overset{\nu}{\twoheadrightarrow}\mathbb{Z}\twoheadrightarrow\mathbb{Z}/N\mathbb{Z}.$

One can study $X^{\nu}$ by its approximation $X^{\nu,N}$. Consider the
following two limits

(1) $\lim_{N\rightarrow\infty}\frac{\dim H_{i}(X^{\nu,N},\mathbb{K})}{N}$

for some field $\mathbb{K}$ and

(2)
$\lim_{N\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|}{N},$

where $|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|$ denote the cardinality of
the torsion part of $H_{i}(X^{\nu,N},\mathbb{Z})$. In prior we do not know if
these two limits exist.

When $\mathbb{K}=\mathbb{C}$, the well known Lück's approximation theorem
[Luc94] shows that the first limit exists and it coincides with the
$L^{2}$-betti number $b_{i}^{(2)}(X^{\nu},\mathbb{C})$ defined by the Von
Neumann algebra. (Lück's approximation theorem is proved with much more
generality.) Meanwhile, Abért, Jaikin-Zapirain and Nikolov [AJN11, Theorem 17]
showed that the first limit also exists when $\mathrm{char}(\mathbb{K})>0$. We
set

$\alpha_{i}(X^{\nu},\mathbb{K})\coloneqq\lim_{N\rightarrow\infty}\frac{\dim
H_{i}(X^{\nu,N},\mathbb{K})}{N}.$

Note that $b_{i}(X,\mathbb{K})$ only depend on $\mathrm{char}(\mathbb{K})$,
not on the specific choice of the field $\mathbb{K}$. So without loss of
generality, we always assume that $\mathbb{K}$ is algebraically closed in the
rest of this paper.

This limit is related to cohomology jump loci by the following claim, which is
folklore .

######  Proposition 1.1.

(see Proposition 2.7) With the assumptions and notations as above, for any
$i\geq 0$ we have

$\alpha_{i}(X^{\nu},\mathbb{K})=\dim H_{i}(X,L_{\rho})$ for
$\rho\in\mathbb{K}^{*}$ being general.

Here $L_{\rho}$ is the rank one $\mathbb{K}$-local system on $X$ corresponding
to $\rho$ pulling back by $\nu$. In particular,
$\alpha_{i}(X^{\nu},\mathbb{K})$ is always an integer.

When $X$ is a complex smooth projective variety and $\mathbb{K}=\mathbb{C}$,
similar formulas for betti numbers and Hodge numbers can be found in [CL19].
When $X$ is a complex smooth quasi-projective variety and
$\mathbb{K}=\mathbb{C}$, there are a lot of works to study the polynomial
periodicity for the betti numbers and Hodge numbers of congruence covers, see
[Suc01, Section 5] and [Bud09]. However we do not pursue this direction in
this paper.

On the other hand, the limit in (2) also exists, e.g. see [Le14, Theorem 5],
and it can be computed by the Mahler measure (see Definition 2.9) of the
$i$-th integral Alexander polynomial
$\Delta_{i}(X^{\nu})\in\mathbb{Z}[t,t^{-1}]$ (see Definition 2.3). We set

$\mathbf{M}_{i}(X^{\nu})\coloneqq\lim_{N\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|}{N}.$

When $X$ is a link complement, the multivariable version of this limit is
studied by Silver and Williams in [SW02, SW04].

The above discussion shows that these limits are determined by cohomology jump
loci and integral Alexander polynomials. When $X$ is a smooth complex quasi-
projective variety, under certain assumptions for $\nu$ we give concrete
formulas to compute these limits at degree 1 by geometric informations of $X$.
The formulas rely on a detailed study about the degree 1 cohomology jump loci
of $X$ that we discuss in next subsection.

###  1.2. Main results

####  1.2.1. Cohomology jump loci and orbifold maps

We first recall the definition of cohomology jump loci. Let $X$ be a smooth
complex quasi-projective variety with $\pi_{1}(X)=G$ and $\mathbb{K}$ be an
algebraically closed field. The group of $\mathbb{K}$-valued characters,
$\mathrm{Hom}(G,\mathbb{K}^{*})$, is a commutative affine algebraic group.
Each character $\rho\in\mathrm{Hom}(G,\mathbb{K}^{*})$ defines a rank one
local system on $X$, denoted by $L_{\rho}$.

######  Definition 1.2.

The cohomology jump loci of $X$ are defined as

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

When $k=1$, we simply write $\mathcal{V}^{i}(X,\mathbb{K})$.

When $\mathbb{K}=\mathbb{C}$, the cohomology jump loci of complex smooth
quasi-projective variety have been intensively studied. In particular, the
well-known structure theorem (see Theorem 3.5) for
$\mathcal{V}^{i}_{k}(X,\mathbb{C})$ put strong constraints for the homotopy
type of complex smooth quasi-projective variety. A celebrity result for degree
1 cohomology jump loci $\mathcal{V}^{1}_{k}(X,\mathbb{C})$ due to Arapura put
even stronger constraints for its fundamental group.

To explain Arapura's results, we recall the definition of orbifold maps. Let
$\Sigma_{g,r}$ be a smooth Riemann surface of genus $g\geq 0$ with $r\geq 0$
points removed. Assume $b_{1}(\Sigma_{g,r})=2g+r-1>0$, i.e.,
$\Sigma_{g,r}\neq\mathbb{CP}^{1},\mathbb{C}$.

######  Definition 1.3.

Let $X$ be a complex smooth quasi-projective variety. An algebraic map
$f:X\to\Sigma_{g,r}$ is called an orbifold map, if $f$ is surjective and has
connected generic fiber. There exists a maximal Zariski open subset
$U\subset\Sigma_{g,r}$ such that $f$ is a fibration over $U$. Say
$B=\Sigma_{g,r}-U$ (could be empty) has $s$ points, denoted by
$\\{q_{1},\ldots,q_{s}\\}$. We assign the multiplicity $\mu_{j}$ of the fiber
$f^{-1}(q_{j})$ to the point $q_{j}$. Such orbifold map $f$ is called of type
$(g,r,{\mbox{\boldmath{$\mu$}}})$, where
${\mbox{\boldmath{$\mu$}}}=(\mu_{1},\ldots,\mu_{s})$. Moreover, we say $f$ is

  * •

large if $2g+r-2>0$,

  * •

small if $2g+r-2=0$ and $\prod_{j=1}^{s}\mu_{j}>1$,

  * •

null if $2g+r-2=0$ and $\prod_{j=1}^{s}\mu_{j}=1$.

Here when $B=\emptyset$, $\prod_{j=1}^{s}\mu_{j}=1$ by convention. If
$\prod_{j=1}^{d}\mu_{j}=1$, we say $f$ has no multiple fiber. When
$\Sigma_{g,r}$ is clear in the context, we simply write $\Sigma$. The orbiford
group associated to these data is defined as

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

where $\gamma_{j}$ is a meridian of $q_{j}$.

Our definition of orbifold maps is a bit of different from the classical one,
e.g. see [DS14, Section 4.1], as we also include null orbifold maps. For
example, if $f\colon\mathbb{C}^{n}\to\mathbb{C}$ is a reduced weighted
homogeneous polynomial, then $f\colon\mathbb{C}^{n}\setminus
f^{-1}(0)\to\mathbb{C}^{*}$ gives the Milnor fibration, hence a null orbifold
map.

$f$ induces a surjective map on the fundamental group level (see Proposition
4.2):

$f_{*}\colon\pi_{1}(X)\twoheadrightarrow\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}}),$

hence it induces an embedding

$\mathcal{V}^{1}(\pi_{1}^{orb}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}}),\mathbb{K})\to\mathcal{V}^{1}(X,\mathbb{K})$

Arapura [Ara97] showed that every positive dimensional component of
$\mathcal{V}^{1}(X,\mathbb{C})$ can always be realized by the orbifold map in
this way. This result is further refined by Artal Bartolo, Cogolludo-Agustín
and Matei in [ACM13, Theorem 1]. We recall Arapura's theorem here and adjust
it for our needs.

######  Theorem 1.4.

[ACM13, Theorem 1], [Dim07A, Theorem 3.6] Let $X$ be a complex smooth quasi-
projective variety. Then the following two statements hold.

  * (a)

We have

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

where $Z$ is a finite set of torsion points and the union runs over all large
and small orbifold maps $f\colon X\rightarrow\Sigma_{g,r}$. Moreover,
$f^{*}\mathcal{V}^{1}(\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}}),\mathbb{C})$
has $\prod_{j=1}^{s}\mu_{j}$ many parallel components when $f$ is large and
has $(\prod_{j=1}^{s}\mu_{j}-1)$ many parallel components when $f$ is small.

  * (b)

Given an orbifold map $f\colon X\to\Sigma$ of type
$(g,r,{\mbox{\boldmath{$\mu$}}})$ and any
$\rho\in\mathrm{Hom}(\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}}),\mathbb{C}^{*})$

$\dim H^{1}(X,f^{*}L_{\rho})\geq 2g+r-2$

and equality holds with finitely many (torsion) exceptions.

In this paper, we consider the above theorem with $\mathbb{K}$-coefficient
case where $\mathrm{char}(\mathbb{K})>0$. For Theorem 1.4(a), Delzant [Delz08]
used Bieri-Neumann-Strebel invariants to generalize it to the
$\mathbb{K}$-coefficient case when $X$ is a smooth projective variety.

######  Theorem 1.5.

Let $X$ be a complex smooth projective variety with $b_{1}(X)>0$ and
$\mathbb{K}$ be an algebraically closed field. Then we have

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

where $Z^{\prime}$ is a finite set of points (depends on $\mathbb{K}$) and the
union runs over all large and small orbifold maps $f\colon
X\rightarrow\Sigma_{g,r}$.

We generalize Theorem 1.4(b) in Theorem 4.8. To avoid too many notations in
the introduction, we present here a simplified version of Theorem 4.8.

######  Theorem 1.6.

Let $X$ be a complex smooth quasi-projective variety with $b_{1}(X)>0$ and
$\mathbb{K}$ be an algebraically closed field with
$\mathrm{char}(\mathbb{K})=p\geq 0$. Consider an orbifold map $f\colon
X\to\Sigma$ of type $(g,r,{\mbox{\boldmath{$\mu$}}})$. Then for any
$\rho\in\mathrm{Hom}(\pi_{1}(\Sigma),\mathbb{K}^{*})$, we have

$\dim H^{1}(X,f^{*}L_{\rho})\geq 2g+r-2+\\#\\{j\mid p\text{ divides
}\mu_{j}\\}$

and equality holds with finitely many exceptions.

######  Remark 1.7.

Theorem 1.4 (see e.g. [Suc14A, Theorem 9.3]) and Theorem 1.5 ([Del74]) also
work for any compact Kähler manifold. Correspondingly, one can also prove
Theorem 1.6 for any compact Kähler manifold by the same proof presented for
smooth quasi-projective variety in this paper.

####  1.2.2. Formulas for $L^{2}$-type invariants

Next we connect the epimorphism $\nu$ to the orbifold maps.

######  Definition 1.8.

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

$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{f_{*}}$$\textstyle{\mathbb{Z},}$$\textstyle{\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

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

######  Remark 1.9.

This definition is well-defined, see Remark 4.4. Note that there are examples
where $\nu$ is not orbifold effective, see Example 4.6. On the other hand,
under certain Hodge structure assumptions for $\nu$ (considered as an element
in $H^{1}(X,\mathbb{Z})$), it is always orbifold effective. For example, if
$\nu$ is of type $(1,1)$, i.e., $\nu\in
F^{1}H^{1}(X,\mathbb{C})\cap\overline{F^{1}H^{1}(X,\mathbb{C})}$, then $\nu$
is orbifold effective. Here $F$ stands for the Hodge filtration. In
particular, if $H^{1}(X,\mathbb{Q})$ has pure Hodge structure of weight
$(1,1)$, then any epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ is orbifold effective of
type $(0,r,{\mbox{\boldmath{$\mu$}}})$. (This is why we include null orbifold
maps in our definition.) This assumption holds if $X$ has a smooth
compactification $\overline{X}$ such that $H^{1}(\overline{X},\mathbb{Q})=0$.
Typical examples are hyperplane arrangement complement, toric arrangement
complement and the complement of some algebraic curves in $\mathbb{CP}^{2}$.
For more details about these statements, see Remark 4.5.

######  Theorem 1.10.

Let $X$ be a complex smooth quasi-projective variety and $\mathbb{K}$ be an
field with $\mathrm{char}(\mathbb{K})=p\geq 0$. Consider an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Then we have the following.

  * (a)

If $\nu$ is orbifold effective of type $(g,r,{\mbox{\boldmath{$\mu$}}})$. Then
we have that

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

and

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

In particular, $\Delta_{1}(X^{\nu})$ is a product of $\prod_{j=1}^{s}\mu_{j}$
with some cyclotomic polynomials.

  * (b)

Assume that $X$ is either smooth projective or $H^{1}(X,\mathbb{Q})$ has pure
Hodge structure of weight $(1,1)$. If $\alpha_{1}(X^{\nu},\mathbb{K})>0$ for
some field $\mathbb{K}$ or $\mathbf{M}_{1}(X^{\nu})>0$, $\nu$ is orbifold
effective by a large or small orbifold map.

The above formula for $\alpha_{1}(X^{\nu},\mathbb{K})$ follows from Theorem
1.6 directly and the one for $\mathbf{M}_{1}(X^{\nu})$ follows by a similar
proof to Theorem 1.6. Theorem 1.10(b) follows from Theorem 1.5 and Remark 1.9.

######  Remark 1.11.

Note that by Theorem 1.10, $\exp(\mathbf{M}_{1}(X^{\nu}))$ is an integer when
$\nu$ is orbifold effective. In fact, more is true. Let $X$ be a complex
smooth quasi-projective variety with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Then for any $i\geq 0$,
$\exp(\mathbf{M}_{i}(X^{\nu}))$ is a positive integer. This follows directly
from the structure theorem for $\mathcal{V}^{i}_{k}(X,\mathbb{C})$, see
Proposition 3.7.

On the other hand, $\exp(\mathbf{M}_{*}(X^{\nu}))$ is not always an integer
for general $X$. For example, the integral Alexander polynomial of the
complement $X$ of the knot $4_{1}$ in $S^{3}$ [SW02, p. 981] is given by

$\Delta_{1}(X^{\nu})=t^{2}-3t+1,$

hence $\exp(\mathbf{M}_{1}(X^{\nu}))=\frac{3+\sqrt{5}}{2}$.

###  1.3. Hyperplane arrangements

Let $X$ be the complement of some hyperplane arrangement in $\mathbb{P}^{n}$.
It is well known that the cohomology ring $H^{*}(X,\mathbb{Z})$ are determined
by the combinatoric data of the hyperplane arrangement.

Given an epimorphism $\nu\colon\pi_{1}(X)\rightarrow\mathbb{Z}$ and a 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 1.12.

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 $i$-th Aomoto torsion number 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.

Since the ring structure $H^{*}(X,\mathbb{Z})$ are combinatorially determined,
so are $\beta_{*}(X,\nu_{\mathbb{K}})$ and $\tau_{*}(X,\nu_{\mathbb{Z}})$ once
$\nu$, considered as an element in $H^{1}(X,\mathbb{Z})$, is fixed. Due to
Papadima and Suciu's work [PS10], for any field $\mathbb{K}$ we have

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

(Papadima and Suciu [PS10] indeed proved the above inequality for any finite
CW complex.) On the other hand, we give the following combinatorial upper
bound for $\mathbf{M}_{*}(X^{\nu})$.

######  Proposition 1.13.

Let $X$ be the complement of a hyperplane arrangement. For any epimorphsim
$\nu\colon\pi_{1}(X)\to\mathbb{Z}$ 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
$(g,r,{\mbox{\boldmath{$\mu$}}})$, then

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

The proof of this proposition is based on Papadima and Suciu's work [PS10,
Theorem 12.6].

######  Remark 1.14.

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 Example 5.3,
5.4.

Based on Theorem 1.4(1), we give a different interpretation of (3) as follows.

######  Corollary 1.15.

Let $X$ be a hyperplane arrangement complement. Let $V$ be a irreducible
positive dimensional component of $\mathcal{V}^{1}(X,\mathbb{C})$ and say it
has $m$ many parallel components (including itself) in
$\mathcal{V}^{1}(X,\mathbb{C})$. Any epimorphism
$\nu\colon\pi_{1}(X)\to\mathbb{Z}$ induces an embedding
$\nu^{*}\colon\mathbb{C}^{*}\to\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*}).$ If
there exists some $\rho\in\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})$ such that
$\mathrm{im}\nu^{*}\subset\rho\cdot V$, we have the following:

  * •

If $\dim V\geq 2$, then $m$ divides $\tau_{1}(X,\nu_{\mathbb{Z}})$,

  * •

If $\dim V=1$, then $m+1$ divides $\tau_{1}(X,\nu_{\mathbb{Z}})$.

######  Remark 1.16.

It is a long-standing question for hyperplane arrangement if
$\mathcal{V}^{1}(X,\mathbb{C})$ is combinatorially determined. By the tangent
cone equality it is known that the positive dimension components of
$\mathcal{V}^{1}(X,\mathbb{C})$ passing through the origin are combinatorially
determined (see e.g. [Dim17, Theorem 6.1]) and they can be described by the
multinet structure due to Falk and Yuzvinsky [FY07]. So one can get all
dimension $\geq 2$ components of $\mathcal{V}^{1}(X,\mathbb{C})$ passing
through the origin by combinatoric data. Then the above corollary gives an
combinatorial upper bound for the number of its parallel components, which is
new up to our knowledge. Under certain assumptions for the multinet (see
Assumption 5.7), we show that $\mathcal{V}^{1}(X,\mathbb{C})$ has no parallel
2-dimensional components associated to this multinet.

One approach to find the translated positive dimension component of
$\mathcal{V}^{1}(X,\mathbb{C})$ is by the pointed multinet structure,
introduced by Suciu, e.g., see [DS14, Definition].

Since $H_{*}(X,\mathbb{Z})$ is always torsion free for hyperplane arrangement
complement $X$, people was wondering if the Milnor fiber of a central
hyperplane arrangement also exhibits the same property. In [DS14], Denham and
Suciu provided a complete answer to this question by showing that for any
prime number $p\geq 2$ there is a central hyperplane arrangement whose Milnor
fiber has non-trivial $p$-torsion in homology. Their results hold under the
assumption that $p$ does not divide the number of hyperplanes and they asked
if this assumption can be dropped [DS14, Question 8.12]. As an application of
the $L^{2}$-type results, we follow Denham and Suciu's construction in [DS14]
and answer their question positively.

######  Theorem 1.17.

For every prime $p\geq 2$, there is a central hyperplane arrangement whose
Milnor fiber has non-trivial $p$-torsion in homology and $p$ divides the
number of hyperplanes in the arrangement.

###  1.4. Structure of the paper

In section 2, we introduce the Alexander modules and Alexander polynomials for
the pair $(X,\nu)$ and study their relations with the $L^{2}$-type invariants.
Section 3 is devoted to the cohomology jump loci and detailed calculations of
these invariants for the orbifold group. In section 4, we give the proof for
Theorem 1.6 and Theorem 1.10. Section 5 is devoted to hyperplane arrangements.
In section 5.1, we prove Proposition 1.13 and compute some examples; while in
section 5.2, we show that if the hyperplane arrangement admits certain kind of
combinatoric structure, then $\mathcal{V}^{1}(X,\mathbb{C})$ has no parallel
2-dimensional components. In section 5.3, we prove Theorem 1.17.

###  1.5. Notations

  1. (1)

$\mathbb{K}$ is assumed to be an algebraically closed filed in the rest of
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.

  3. (3)

Let $\mathbb{Z}_{>0}$ denote the collection of positive integers and
$\mathbb{Z}_{\geq 0}$ denote the collection of non-negative integers.

Acknowledgments. We thank Laurentiu Maxim, Botong Wang and Zhenjian Wang for
valuable discussions. The second named author is partially supported by
National Key Research and Development Project SQ2020YFA070080, the starting
grant KY2340000123 from University of Science and Technology of China,
National Natural Science Funds of China (Grant No. 12001511), the Project of
Stable Support for Youth Team in Basic Research Field, CAS (YSBR-001), 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.

##  2\. Alexander and $L^{2}$-type invariants

In this section, we always assume that $X$ is a connected finite CW-complex
with a fixed group epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Denote by $X^{\nu}$ and
$X^{\nu,N}$ the corresponding covering spaces of $X$ as introduced in Section
1.1. In this section, we study various topological properties of $X^{\nu}$ and
its approximation $X^{\nu,N}$.

###  2.1. Algebraic Preliminaries

Let $R$ be a Noetherian UFD and $M$ be a finitely generated $R$-module. The
rank of $M$, denoted by $\mathrm{rank}M$, is defined to be
$\dim_{Q}Q\otimes_{R}M$, where $Q$ is the fraction field of $R$. Consider a
finite presentation of $M$

$\textstyle{R^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{R^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $A$ is a $(p\times q)$ matrix in $R$. Assume that the matrix $A$ has
rank $r$. Let $I(M)$ denote the ideal generated by all possible $(r\times
r)$-minors of $A$. Two finite presentations of $M$ can be related by a
sequence of elementary operations, so this ideal does not depend on the choice
of presentation. Since $R$ is an UFD, we can define the greatest common
divisor of an ideal and set

$\Delta(M)\coloneqq\mathrm{gcd}(I(M)).$

Then $\Delta(M)$ is defined uniquely up to a multiplication with a unit of
$R$. When $A=0$, $\Delta(M)=1$ by convention.

######  Lemma 2.1.

[Tur01, Lemma 4.9] With the assumptions and notations as above, we have

$\Delta(M)=\Delta(M_{\mathrm{tor}}),$

where $M_{\mathrm{tor}}$ denote the torsion sub-module of $M$.

######  Lemma 2.2.

Let $0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$ be a short exact sequence
of finitely generated $R$-modules. Then we have

$\Delta(M)\mid\Delta(M^{\prime})\cdot\Delta(M^{\prime\prime}).$

Moreover, if $\mathrm{rank}M=\mathrm{rank}M^{\prime\prime}$ (e.g.
$\mathrm{rank}M=0$), we have

$\Delta(M)=\Delta(M^{\prime})\cdot\Delta(M^{\prime\prime}).$

###### Proof.

The first claim follows from Horseshoe Lemma [Wei94, Horseshoe Lemma 2.2.8].
In fact, for a finite presentation of $M^{\prime}$ with matrix $A^{\prime}$
and one for $M^{\prime\prime}$ with matrix $A^{\prime\prime}$, Horseshoe Lemma
gives a way to construct a finite presentation for $M$ with matrix

$\begin{pmatrix}A^{\prime}&*\\\ 0&A^{\prime\prime}\end{pmatrix}.$

Since localization is an exact functor, we get
$\mathrm{rank}M=\mathrm{rank}M^{\prime}+\mathrm{rank}M^{\prime\prime}$, which
implies

$\mathrm{rank}\begin{pmatrix}A^{\prime}&*\\\
0&A^{\prime\prime}\end{pmatrix}=\mathrm{rank}A^{\prime}+\mathrm{rank}A^{\prime\prime}.$

Then it is easy to see
$\Delta(M)\mid\Delta(M^{\prime})\cdot\Delta(M^{\prime\prime}).$

For the second claim, we have the following commutative diagram:

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\otimes_{R}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\prime\prime}\otimes_{R}Q}$

where $Q$ is the fraction field of $R$. The additional assumption
$\mathrm{rank}M=\mathrm{rank}M^{\prime\prime}$ implies that the bottom
horizontal map is an isomorphism. Then we claim the induced map
$M_{\mathrm{tor}}\to M^{\prime\prime}_{\mathrm{tor}}$ is surjective. For any
$x\in M^{\prime\prime}_{\mathrm{tor}}$, there exists $y\in M$ maps to $x$.
Note that $x=0$ in $M^{\prime\prime}\otimes_{R}Q$, hence so is $y$, which
implies $y\in M_{\mathrm{tor}}$. Taking torsion part is an left exact functor.
Putting all together we get a short exact sequence of finitely generated
$R$-modules:

$0\to M^{\prime}_{\mathrm{tor}}\to M_{\mathrm{tor}}\to
M^{\prime\prime}_{\mathrm{tor}}\to 0.$

It implies
$\Delta(M_{\mathrm{tor}})=\Delta(M^{\prime}_{\mathrm{tor}})\cdot\Delta(M^{\prime\prime}_{\mathrm{tor}})$,
e.g. see [SW77, Lemma 2.1 (i)]. Then the claim follows from Lemma 2.1
directly. ∎

###  2.2. Alexander modules and Alexander polynomials

Recall that $X$ is a connected finite CW-complex with a fixed group
epimorphism $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Then the group
of covering transformations of the covering space $X^{\nu}$ is isomorphic to
$\mathbb{Z}$ 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 $\Gamma_{R}$-modules, where $\Gamma_{R}=R[\mathbb{Z}]=R[t,t^{-1}]$. So the
cellular chain complex of $X^{\nu}$, $C_{*}(X^{\nu},R)$, is a bounded complex
of finitely generated free $\Gamma_{R}$-modules:

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

With the above free basis for $C_{*}(X^{\nu},R)$, $\partial^{R}_{i}$ can be
written down as a matrix with entries in $R[t,t^{-1}]$. Note that
$R[t,t^{-1}]$ is also a Notherian UFD.

######  Definition 2.3.

The $i$-th homology group $H_{i}(X^{\nu},R)$ of $C_{*}(X^{\nu},R)$, regarded
as a finitely generated $\Gamma_{R}$-module, is called the $i$-th homology
Alexander module of the pair $(X,\nu)$ with $R$-coefficients.
$\Delta(H_{i}(X^{\nu},R))$ is called the $i$-th Alexander polynomial of
$(X,\nu)$ with $R$-coefficients, denoted by $\Delta_{i}(X^{\nu},R)$.

Consider the case $R=\mathbb{Z}$. Note that $\Delta_{i}(X^{\nu},\mathbb{Z})$
is only defined up to multiplication by a unit in $\mathbb{Z}[t,t^{-1}]$. But
there is a unique representative of the associate class with no negative
powers of $t$, non-zero constant term and positive coefficient for the leading
term. We use the notation $\Delta_{i}(X^{\nu})$ to denote this representative.
In particular, $\Delta_{i}(X^{\nu})$ is an integer valued polynomial.

Note that the homology Alexander modules and Alexander polynomials are
homotopy invariants for the pair $(X,\nu)$. In degree 1,
$\Delta_{1}(X^{\nu},R)$ depends only on $\pi_{1}(X)$ and $\nu$.

For the Alexander polynomial, one can define $\Delta_{i}(X^{\nu},R)$ from the
map $\partial^{R}_{i}$ in (4) directly.

######  Proposition 2.4.

Suppose that the $R[t,t^{-1}]$-module map $\partial_{i}^{R}$ in (4) has rank
$r$. Then $\Delta_{i}(X^{\nu},R)$ equals to the greatest common divisor of all
possible $(r\times r)$-minors of $\partial_{i}^{R}$.

###### Proof.

Note that we have the following short exact sequence of $\Gamma_{R}$-modules

$0\rightarrow H_{i}(X^{\nu},R)\rightarrow
C_{i}(X^{\nu},R)/\mathrm{im}\partial_{i}^{R}\rightarrow\mathrm{im}\partial_{i-1}^{R}\rightarrow
0,$

where $\mathrm{im}$ denotes the image functor. Since
$\mathrm{im}\partial_{i-1}^{R}$ is torsion free, we get

$H_{i}(X^{\nu},R)_{\mathrm{tor}}\cong(C_{i}(X^{\nu},R)/\mathrm{im}\partial_{i}^{R})_{\mathrm{tor}}.$

Then Lemma 2.1 implies that

$\Delta_{i}(X^{\nu},R)=\Delta(H_{i}(X^{\nu},R)_{\mathrm{tor}})=\Delta((C_{i}(X^{\nu},R)/\mathrm{im}\partial_{i}^{R})_{\mathrm{tor}})=\Delta(C_{i}(X^{\nu},R)/\mathrm{im}\partial_{i}^{R}).$

Consider the finite presentation of
$C_{i}(X^{\nu},R)/\mathrm{im}\partial_{i}^{R}$

$C_{i+1}(X^{\nu},R)\overset{\partial_{i}^{R}}{\to}C_{i}(X^{\nu},R)\to
C_{i}(X^{\nu},R)/\mathrm{im}\partial_{i}^{R}\to 0$

Then the claim follows by definition. ∎

######  Definition 2.5.

A continuous map $f:Y\to X$ of connected finite CW complexes is said to be
$n$-connected ($n\geq 1$), if the induced homomorphisms

$\pi_{i}(Y)\to\pi_{i}(X)$

are isomorphisms for $i<n$ and epimorhism for $i=n$.

######  Proposition 2.6.

Let $f:Y\rightarrow X$ be a $n$-connected ($n\geq 1$) continuous map of
connected finite CW complexes. The composed map
$\pi_{1}(Y)\overset{f_{*}}{\twoheadrightarrow}\pi_{1}(X)\overset{\nu}{\twoheadrightarrow}\mathbb{Z}$
defines a $\mathbb{Z}$-cover of $Y$, denoted by $Y^{\nu}$. Then for $i<n$, we
have $\Delta_{i}(Y^{\nu},R)=\Delta_{i}(X^{\nu},R)$. Moreover, if
$\mathrm{rank}H_{n}(X^{\nu},R)=\mathrm{rank}H_{n}(Y^{\nu},R)$, then
$\Delta_{n}(X^{\nu},R)\mid\Delta_{n}(Y^{\nu},R).$

###### Proof.

It is clear that $f$ induces isomorphisms $H_{i}(Y^{\nu},R)\cong
H_{i}(X^{\nu},R)$ for $i<n$, hence
$\Delta_{i}(Y^{\nu},R)=\Delta_{i}(X^{\nu},R)$. On the other hand, the
assumption $f$ being $n$-connected implies $H_{n}(Y^{\nu},R)\twoheadrightarrow
H_{n}(X^{\nu},R)$. The claim follows from Lemma 2.2 by taking
$M=H_{n}(Y^{\nu},R)$ and $M^{\prime\prime}=H_{n}(X^{\nu},R)$. ∎

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

####  2.3.1. $L^{2}$-betti number

As mentioned in Section 1.1, for any field $\mathbb{K}$, the following $i$-th
$L^{2}$-type limit

$\alpha_{i}(X^{\nu},\mathbb{K})=\lim_{N\rightarrow\infty}\frac{\dim
H_{i}(X^{\nu,N},\mathbb{K})}{N}$

exists. In this subsection, we give a direct proof for the existence of the
limit and relate these limits with Alexander modules.

######  Proposition 2.7.

With the assumptions and notations as above, for any $i\geq 0$ the limit
$\lim_{N\rightarrow\infty}\frac{\dim H_{i}(X^{\nu,N},\mathbb{K})}{N}$ exists.
Moreover, the following three numbers coincide

  * (1)

$\alpha_{i}(X^{\nu},\mathbb{K})=\lim_{N\rightarrow\infty}\frac{\dim
H_{i}(X^{\nu,N},\mathbb{K})}{N}$,

  * (2)

$\mathrm{rank}H_{i}(X^{\nu},\mathbb{K})$,

  * (3)

$\dim H_{i}(X,L_{\rho})$ for $\rho\in\mathbb{K}^{*}$ being general. Here for
any $\rho\in\mathbb{K}^{*}$, $L_{\rho}$ is the corresponding rank one
$\mathbb{K}$-local system on $X$ pulling back by $\nu$.

For the proof of this proposition, we need the following lemma.

######  Lemma 2.8.

Given a nonzero polynomial $h(t)=t^{d}+\cdots+a_{1}t+a_{0}\in\mathbb{K}[t]$,
we denote by $J_{N}$ the following $N\times N$ ($N\geq d$) matrix

$J_{N}\coloneqq\begin{pmatrix}0&1&0&\cdots&0\\\ 0&0&1&\cdots&0\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\cdots&1\\\ 1&0&0&\cdots&0\\\
\end{pmatrix}.$

Then there exists a positive integer $c$ such that

$N-c\leq\mathrm{rank}h(J_{N})\leq N,$

where $c$ does not depend on $N$.

###### Proof.

Set $g(t)=\gcd(t^{N}-1,h(t))$. Then there exist
$u_{1}(t),u_{2}(t)\in\mathbb{K}[t]$ such that

$g(t)=u_{1}(t)(t^{N}-1)+u_{2}(t)h(t).$

Note that $J_{N}^{N}=\mathrm{Id}$, where $\mathrm{Id}$ is the $(N\times N)$
identity matrix. It gives that

$g(J_{N})=u_{1}(J_{N})(J_{N}^{N}-\mathrm{Id})+u_{2}(J_{N})h(J_{N})=u_{2}(J_{N})h(J_{N}),$

hence $\mathrm{rank}g(J_{N})\leq\mathrm{rank}h(J_{N})$. On the other hand,
since $g(t)\mid h(t)$, we have
$\mathrm{rank}g(J_{N})\geq\mathrm{rank}h(J_{N})$. Hence

$\mathrm{rank}g(J_{N})=\mathrm{rank}h(J_{N}).$

Since $\mathbb{K}$ is algebraically closed, we can write down
$h(t)=\prod_{j=1}^{d}(t-\alpha_{j}).$ Collect all $\alpha_{j}$ such that there
exists some positive integer $n_{j}$ such that $\alpha_{j}^{n_{j}}=1$. Then
there exists an positive integer $c$, which does not depend on $N$, such that

$g(t)\mid t^{c}-1.$

Here when $\mathrm{char}(\mathbb{K})=0$, $c$ can be taken as the product of
$n_{j}$; while when $\mathrm{char}(\mathbb{K})=p>0$, $c$ can be taken as
product of $n_{j}$ with some power of $p$. Note that the dimension of
$\mathrm{ker}(J^{c}_{N}-\mathrm{Id})$ is $\gcd(c,N)$. Hence for $N$ large
enough we have

$\mathrm{rank}h(J_{N})=\mathrm{rank}g(J_{N})\geq\mathrm{rank}(J^{c}_{N}-\mathrm{Id})\geq
N-c$

The other part of the inequality is obvious. ∎

###### Proof of Proposition 2.7.

Recall that $C_{*}(X^{\nu},\mathbb{K})$ is the cellular chain complex of
$X^{\nu}$ with the field coefficient $\mathbb{K}$. Each
$C_{i}(X^{\nu},\mathbb{K})$ is a free $\mathbb{K}[t^{\pm 1}]$-module with
finite rank. Since $\mathbb{K}[t^{\pm 1}]$ is a PID, by choosing suitable
basis we can assume that $\partial_{i}^{\mathbb{K}}$ has the following form

$\begin{pmatrix}h_{1}&&&&\\\ &\ddots&&&\\\ &&h_{m_{i}}&&\\\ &&&&\end{pmatrix}$

with $m_{i}=\mathrm{rank}\partial_{i}^{\mathbb{K}}$. In particular,

$\mathrm{rank}H_{i}(X^{\nu},\mathbb{K})=\mathrm{rank}C_{i}(X^{\nu},\mathbb{K})-m_{i}-m_{i-1}.$

By tensoring with $\mathbb{K}[t^{\pm 1}]/(t^{N}-1)$, we get the cellular chain
complex for $X^{\nu,N}$ with $\mathbb{K}$-coefficients. We use
$\partial_{i,N}^{\mathbb{K}}$ to denote the corresponding differential map.
Then

$\dim H_{i}(X^{\nu,N},\mathbb{K})=\mathrm{rank}C_{i}(X^{\nu},\mathbb{K})\cdot
N-\mathrm{rank}\partial_{i,N}^{\mathbb{K}}-\mathrm{rank}\partial_{i-1,N}^{\mathbb{K}}.$

Note that each polynomial $h_{j}(t)$ ($1\leq j\leq m_{i})$) becomes an
$N\times N$ matrix $h_{j}(J_{N})$ in $\partial_{i,N}^{\mathbb{K}}$. By Lemma
2.8, there exist constant numbers $c_{i}$ and $c_{i-1}$ such that

$\mathrm{rank}H_{i}(X^{\nu},\mathbb{K})\leq\frac{\dim
H_{i}(X^{\nu,N},\mathbb{K})}{N}\leq\mathrm{rank}H_{i}(X^{\nu},\mathbb{K})+\frac{c_{i}+c_{i-1}}{N}.$

Taking $N\rightarrow\infty$, we get

$\alpha_{i}(X^{\nu},\mathbb{K})=\lim_{N\to\infty}\frac{\dim
H_{i}(X^{\nu},\mathbb{K})}{N}=\mathrm{rank}H_{i}(X^{\nu},\mathbb{K}).$

Let $\pi_{N}\colon X^{\nu,N}\rightarrow X$ denote the covering map. Then

$H_{i}(X^{\nu,N},\mathbb{K})=H_{i}(X,(\pi_{N})_{*}\mathbb{K}),$

where $(\pi_{N})_{*}\mathbb{K}$ is the push forward of the
$\mathbb{K}$-constant sheaf on $X^{\nu,N}$, hence a rank $N$
$\mathbb{K}$-local system. Since we have already known that
$\lim_{N\rightarrow\infty}\frac{\dim H_{i}(X^{\nu},\mathbb{K})}{N}$ exists,
one can compute this limit with the subsequence
$\\{N\in\mathbb{Z}_{>0}\mid\mathrm{char}(\mathbb{K})\nmid N\\}$. In this case
$(\pi_{N})_{*}\mathbb{K}$ is a direct sum of $N$ many rank one local systems.
Since the function $\mathbb{K}^{*}\to\mathbb{Z}$ by sending
$\rho\in\mathbb{K}^{*}$ to $\dim H_{i}(X,L_{\rho})$ is semi-continuous, there
are at most finitely many rank one $\mathbb{K}$-local systems with larger
homology dimension than the general one. In particular, these finitely many
local systems only depends on the pair $(X,\nu)$, not on $N$. Hence we have

$\dim H_{i}(X,L_{\rho})\leq\frac{b_{i}(X^{\nu,N},\mathbb{K})}{N}\leq\dim
H_{i}(X,L_{\rho})+\frac{c^{\prime}}{N},$

with $\rho\in\mathbb{K}^{*}$ being general and $c^{\prime}$ being some
constant number. Taking $N\rightarrow\infty$, we are done. ∎

####  2.3.2. Limits of torsion

One can also study the following limit associated to the torsion part of
$H_{i}(X^{\nu,N},\mathbb{Z})$

$\lim_{N\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|}{N},$

where $|H_{i}(X^{N},\mathbb{Z})_{\mathrm{tor}}|$ denote the cardinality of the
torsion part of $H_{i}(X^{\nu,N},\mathbb{Z})$. This limit can be computed by
the Mahler measure of the Alexander polynomial $\Delta_{i}(X^{\nu})$. Let us
recall the definition of Mahler measure based on Jensen's formula [Ahl66, p.
208].

######  Definition 2.9.

[Mah60, Mah62] For a non-zero polynomial $h(t)\in\mathbb{Z}[t]$, 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 $a_{d}\neq 0$ and $\alpha_{j}\in\mathbb{C}$.

We define the Mahler measure of $h$ to be

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

We say that $h$ is of cyclotomic type, if all its roots $\alpha_{j}$ are roots
of unity. In particular, in this case $\mathbf{M}(h)=\log|a_{d}|$.

Note that the Mahler measure of the Alexander polynomial $\Delta_{i}(X^{\nu})$
is well defined since the units of $\mathbb{Z}[t,t^{-1}]$ are the monomials
$\pm t^{j}$ for $j\in\mathbb{Z}$. The following theorem is well known, e.g.
see [Le14, Theorem 5]

######  Theorem 2.10.

With the above assumptions and notations, for any $i\geq 0$ the limit
$\lim_{N\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|}{N}$
exists and we have that

$\lim_{N\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|}{N}=\mathbf{M}(\Delta_{i}(X^{\nu})).$

Based on the above theorem, we set

$\mathbf{M}_{i}(X^{\nu})\coloneqq\lim_{N\rightarrow\infty}\frac{\log|H_{i}(X^{\nu,N},\mathbb{Z})_{\mathrm{tor}}|}{N}=\mathbf{M}(\Delta_{i}(X^{\nu})).$

######  Proposition 2.11.

With the same assumptions as in Proposition 2.6, for any $i<n$ we have
$\mathbf{M}_{i}(Y^{\nu})=\mathbf{M}_{i}(X^{\nu})$. Moreover, if
$\alpha_{n}(X^{\nu},\mathbb{C})=\alpha_{n}(Y^{\nu},\mathbb{C})$, then
$\mathbf{M}_{n}(Y^{\nu})\geq\mathbf{M}_{n}(X^{\nu}).$

###### Proof.

Note that $\alpha_{n}(X^{\nu},\mathbb{C})=\alpha_{n}(Y^{\nu},\mathbb{C})$
implies
$\mathrm{rank}H_{n}(X^{\nu},\mathbb{Z})=\mathrm{rank}H_{n}(Y^{\nu},\mathbb{Z})$.
Then the claims follow from Proposition 2.6 and the definition of Mahler
measure directly. ∎

####  2.3.3. Universal Coefficients Theorem

By the Universal Coefficients Theorem,

$\dim H_{i}(X,\mathbb{C})\leq\dim H_{i}(X,\mathbb{K})$

for any filed $\mathbb{K}$. Moreover, if $\mathrm{char}(\mathbb{K})=p>0$ and
the inequality is strict, then either $H_{i}(X,\mathbb{Z})$ or
$H_{i-1}(X,\mathbb{Z})$ has non-trivial $p$-torsion. On the other hand, if
$H_{i}(X,\mathbb{Z})$ has non-trivial $p$-torsion, then

$\dim H_{i}(X,\mathbb{C})<\dim H_{i}(X,\overline{\mathbb{F}}_{p})$ and $\dim
H_{i+1}(X,\mathbb{C})<\dim H_{i+1}(X,\overline{\mathbb{F}}_{p})$

######  Proposition 2.12.

Let $X$ be a connected finite CW-complex with a fixed group epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Then we have
$\alpha_{i}(X^{\nu},\mathbb{C})\leq\alpha_{i}(X^{\nu},\bar{\mathbb{F}}_{p})$.
Moreover, if
$\alpha_{i}(X^{\nu},\mathbb{C})<\alpha_{i}(X^{\nu},\bar{\mathbb{F}}_{p})$,
then we have the following

  * (1)

For $N$ large enough either $H_{i}(X^{\nu,N},\mathbb{Z})$ or
$H_{i-1}(X^{\nu,N},\mathbb{Z})$ has non-trivial $p$-torsion.

  * (2)

$\Delta_{i}(X^{\nu})=0$ or $\Delta_{i-1}(X^{\nu})=0$ in
$\overline{\mathbb{F}}_{p}[t,t^{-1}]$, hence $\mathbf{M}_{i}(X^{\nu})>0$ or
$\mathbf{M}_{i-1}(X^{\nu})>0$, respectively.

###### Proof.

(1) Set
$\epsilon=|\alpha_{i}(X,\mathbb{C})-\alpha_{i}(X,\bar{\mathbb{F}}_{p})|/2$.
For $N$ large enough, we have

$\frac{\dim_{\mathbb{C}}H_{i}(X^{\nu,N},\mathbb{C})}{N}<\alpha_{i}(X^{\nu},\mathbb{C})+\epsilon=\alpha_{i}(X^{\nu},\bar{\mathbb{F}}_{p})-\epsilon<\frac{\dim_{\bar{\mathbb{F}}_{p}}H_{i}(X^{\nu,N},\bar{\mathbb{F}}_{p})}{N}.$

Hence
$\dim_{\mathbb{C}}H_{i}(X^{\nu,N},\mathbb{C})<\dim_{\bar{\mathbb{F}}_{p}}H_{i}(X^{\nu,N},\bar{\mathbb{F}}_{p})$
and the conclusion follows.

(2) $\alpha_{i}(X^{\nu},\mathbb{C})<\alpha_{i}(X^{\nu},\bar{\mathbb{F}}_{p})$
is equivalent to

$\mathrm{rank}C_{i}(X^{\nu},\mathbb{C})-\mathrm{rank}\partial_{i}^{\mathbb{C}}-\mathrm{rank}\partial_{i-1}^{\mathbb{C}}<\mathrm{rank}C_{i}(X^{\nu},\bar{\mathbb{F}}_{p})-\mathrm{rank}\partial_{i}^{\bar{\mathbb{F}}_{p}}-\mathrm{rank}\partial_{i-1}^{\bar{\mathbb{F}}_{p}},$

which implies
$\mathrm{rank}\partial_{i}^{\mathbb{Z}}=\mathrm{rank}\partial_{i}^{\mathbb{C}}>\mathrm{rank}\partial_{i}^{\bar{\mathbb{F}}_{p}}$
or the same inequality for degree $i-1$. Note that
$\mathrm{rank}\partial_{i}^{\mathbb{Z}}>\mathrm{rank}\partial_{i}^{\bar{\mathbb{F}}_{p}}$
if and only if $p\mid\Delta_{i}(X^{\nu})$ (i.e., $\Delta_{i}(X^{\nu})=0$ in
$\overline{\mathbb{F}}_{p}[t,t^{-1}]$). In particular,
$\mathbf{M}_{i}(X^{\nu})\geq\log p$. Then the claim follows. ∎

######  Remark 2.13.

If $\Delta_{i}(X^{\nu})$ is of cyclotomic type and
$\mathbf{M}_{i}(X^{\nu})\neq 0$, then there exists a prime number $p$ such
that $\Delta_{i}(X^{\nu})=0$ in $\overline{\mathbb{F}}_{p}[t,t^{-1}]$, hence

$\alpha_{i}(X^{\nu},\mathbb{C})<\alpha_{i}(X^{\nu},\overline{\mathbb{F}}_{p})$
and
$\alpha_{i+1}(X^{\nu},\mathbb{C})<\alpha_{i+1}(X^{\nu},\overline{\mathbb{F}}_{p})$.

###  2.4. Aomoto complex

Given an epimorphism $\nu\colon\pi_{1}(X)\rightarrow\mathbb{Z}$, we denote
$\nu_{\mathbb{Z}}$ the corresponding element in $H^{1}(X,\mathbb{Z})$. By
obstruction theory, there is a map $f\colon X\rightarrow S^{1}$ and a class
$\omega\in H^{1}(S^{1},\mathbb{Z})$ such that
$\nu_{\mathbb{Z}}=f^{\ast}(\omega)$. Hence we have

$\nu_{\mathbb{Z}}\cup v_{\mathbb{Z}}=f^{\ast}(\omega\cup\omega)=0.$

For any field $\mathbb{K}$, there is a coresponding class $\nu_{\mathbb{K}}\in
H^{1}(X,\mathbb{K})$ and we have $\nu_{\mathbb{K}}\cup\nu_{\mathbb{K}}=0$.
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}$

and

$\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}$

We call them the Aomoto complex of $H^{\ast}(X,\mathbb{Z})$ and
$H^{\ast}(X,\mathbb{K})$ with respect to $\nu_{\mathbb{Z}}$ and
$\nu_{\mathbb{K}}$, respectively.

######  Definition 2.14.

With the above assumptions and 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 $i$-th Aomoto torsion number 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.

The following theorem follows directly from Papadima-Suciu's work [PS10]. In
fact, they constructed a spectral sequence converges to
$H_{*}(X^{\nu},\mathbb{K})$ ([PS10, Proposition 8.1]) with the first page
being the Aomoto complex [PS10, Corollary 8.3].

######  Theorem 2.15.

Given the pair $(X,\nu)$ and any field $\mathbb{K}$, we have that

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

We recall the following theorem due to Papadima-Suciu [PS10, Theorem 12.6]
specialised to our case.

######  Theorem 2.16.

Let $X$ be a connected finite minimal CW-complex with a fixed epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$. Then the linearization of
the equivariant cochain complex of the cover $X^{\nu}$, with coefficients in
$\mathbb{Z}$ or a field $\mathbb{K}$, coincides with the Aomoto complex
$(H^{\ast}(X,\mathbb{Z}),\cdot\nu_{\mathbb{Z}})$ or
$(H^{\ast}(X,\mathbb{K}),\cdot\nu_{\mathbb{K}})$, respectively.

As an application, we prove the following result.

######  Proposition 2.17.

If $X$ is a connected finite minimal CW complex and the pair $(X,\nu)$
satisfies $\alpha_{i}(X^{\nu},\mathbb{C})=\beta_{i}(X,\nu_{\mathbb{C}})$, then
we have

$\Delta^{\prime}_{i}(X^{\nu})(1)\mid\tau_{i}(X,\nu_{\mathbb{Z}}),$

where $\Delta^{\prime}_{i}(X^{\nu})$ is obtained from $\Delta_{i}(X^{\nu})$ by
taking out $t-1$ factors. The same conclusion also holds for degree $i-1$.

###### Proof.

When $X$ is a minimal CW complex, up to homotopy and under suitable basis we
can assume $\partial_{i}^{\mathbb{Z}}(1)=0$ and write
$\partial_{i}^{\mathbb{Z}}$ in matrix form $(t-1)A_{i}$. Under the same basis,
the above theorem shows that the map
$H^{i}(X,\mathbb{Z})\overset{\nu_{\mathbb{Z}}}{\to}H^{i+1}(X,\mathbb{Z})$ can
be identified with $A_{i}(1)$ (up to dual operation).

Note that $\mathrm{rank}(t-1)A_{i}=\mathrm{rank}A_{i}$,

$\alpha_{i}(X^{\nu},\mathbb{C})=b_{i}(X)-\mathrm{rank}\partial_{i}^{\mathbb{Z}}-\mathrm{rank}\partial_{i-1}^{\mathbb{Z}}$

and

$\beta_{i}(X,\nu_{\mathbb{C}})=b_{i}(X)-\mathrm{rank}A_{i}(1)-\mathrm{rank}A_{i-1}(1).$

Then $\alpha_{i}(X^{\nu},\mathbb{C})=\beta_{i}(X,\nu_{\mathbb{C}})$ implies
$\mathrm{rank}A_{i}=\mathrm{rank}A_{i}(1)$, denoted by $r_{i}$. For a
$(r_{i}\times r_{i})$-minor of $A_{i}$, say $g$, it may happen $g(1)=0$. Hence
$\Delta(A_{i})(1)$ divides the greatest common divisor of all possible
$(r_{i}\times r_{i})$-minors of $A_{i}(1)$, which coincides with
$\tau_{i}(X,\nu_{\mathbb{Z}}).$ The conclusion follows. One can also get
$\mathrm{rank}A_{i-1}=\mathrm{rank}A_{i-1}(1)$, hence the claim for degree
$i-1$ follows by the same argument. ∎

Hyperplane arrangement complement satisfies the assumptions in the above
proposition. For more details, see Section 5.

##  3\. Cohomology jump loci and orbifold groups

###  3.1. Cohomology jump loci

Throughout this subsection, we always assume that $X$ is a connected finite
CW-complex with $\pi_{1}(X)=G$ and $\mathbb{K}$ is an algebraically closed
field, e.g. $\mathbb{C}$, $\overline{\mathbb{F}}_{p}$.

Let us recall the definition of cohomology jump loci. Each character
$\rho\in\mathrm{Hom}(G,\mathbb{K}^{*})$ defines a rank one local system on
$X$, denoted by $L_{\rho}$. Note that $\mathrm{Hom}(G,\mathbb{K}^{*})$ only
depends on $H_{1}(X,\mathbb{Z})$, the abelianization of $G$. The cohomology
jump loci of $X$ are defined as

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

######  Remark 3.1.

One can also define the homology jump loci of $X$ as follows

$\mathcal{W}_{i}^{k}(X,\mathbb{K})\coloneqq\\{\rho\in\mathrm{Hom}(G,\mathbb{K}^{*})\mid\dim_{\mathbb{K}}H_{i}(X,L_{\rho})\geq
k\\}.$

Let $\rho^{-1}$ denote the inverse of $\rho$ in
$\mathrm{Hom}(G,\mathbb{K}^{*})$. Then we have the following isomorphism
between $\mathbb{K}$-vector spaces [Dim04, p. 50]

$H^{i}(X,L_{\rho^{-1}})\cong\mathrm{Hom}_{\mathbb{K}}(H_{i}(X,L_{\rho}),\mathbb{K}),$

which gives

$\mathcal{V}^{i}_{k}(X,\mathbb{K})=\\{\rho\in\mathrm{Hom}(G,\mathbb{K}^{*})\mid\rho^{-1}\in\mathcal{W}_{i}^{k}(X,\mathbb{K})\\}.$

So $\mathcal{V}^{i}_{k}(X,\mathbb{K})$ and $\mathcal{W}^{k}_{i}(X,\mathbb{K})$
share the same information. Since we use sheaf theory to study jump loci, we
will focus on the cohomology jump loci in the rest of this paper.

######  Remark 3.2.

If $X$ is a finite connected CW complex, then
$\mathcal{V}_{1}^{0}(X,\mathbb{K})=\\{\mathbb{K}_{X}\\}$ consists of just one
point, the trivial rank-one local system, and
$V^{0}_{k}(X,\mathbb{K})=\emptyset$ for $k>1$.

We include here the following elementary examples.

######  Example 3.3.

Let $X=\Sigma_{g,r}$ be a Riemann surface of genues $g\geq 0$ and with $r\geq
0$ points removed. Then $b_{1}(X)=\begin{cases}2g,&\mathrm{if}\ r=0,\\\
2g+r-1,&\mathrm{if}\ r>0\end{cases}$ and $\chi(X)=2-2g-r$. To have
$b_{1}(X)>0$, we assume either $g>1$ or $g=0,r>1$, i.e.,
$X\neq\mathbb{CP}^{1},\mathbb{C}$. If $r=0$, Poincaré duality yields

$\mathcal{V}^{i}_{k}(X,\mathbb{K})=\begin{cases}\\{\mathbb{K}_{X}\\},&\mathrm{if}\
i=0,2,k=1\text{ or }i=1,k=2g-1,2g,\\\
\mathrm{Hom}(G,\mathbb{K}^{*})\cong(\mathbb{K}^{*})^{2g},&\mathrm{if}\
i=1,1\leq k\leq 2g-2,\\\ \emptyset,&\text{ otherwise}.\end{cases}$

On the other hand, if $r>0$, we get

$\mathcal{V}^{i}_{k}(X,\mathbb{K})=\begin{cases}\\{\mathbb{K}_{X}\\},&\mathrm{if}\
i=0,k=1\text{ or }i=1,k=2g+r-1,\\\
\mathrm{Hom}(G,\mathbb{K}^{*})\cong(\mathbb{K}^{*})^{2g+r-1},&\mathrm{if}\
i=1,1\leq k\leq 2g+r-2,\\\ \emptyset,&\text{ otherwise}.\end{cases}$

Cohomology jump loci are closed sub-varieties of
$\mathrm{Hom}(G,\mathbb{K}^{*})$ and homotopy invariants of $X$. In degree 1,
$\mathcal{V}^{1}_{k}(X,\mathbb{K})$ depends only on $\pi_{1}(X)$ (e.g. see
[Suc11, Section 2.2]). So for any finitely presented group $G$,
$\mathcal{V}^{1}_{k}(G,\mathbb{K})$ is well defined. The next proposition
describes a functoriality property enjoyed by the cohomology jump loci of
groups, e.g. see [Suc14B, Proposition A.1].

######  Proposition 3.4.

Let $\varphi\colon G\twoheadrightarrow Q$ be an epimorphism between finitely
presented groups $G$ and $Q$. Then the induced monomorphism
$\varphi^{*}\colon\mathrm{Hom}(Q,\mathbb{K}^{*})\hookrightarrow\mathrm{Hom}(G,\mathbb{K}^{*})$
restricts to an embedding
$\mathcal{V}^{1}_{k}(Q,\mathbb{K})\hookrightarrow\mathcal{V}^{1}_{k}(G,\mathbb{K})$.

The following structure theorem are contributed by many people and we name a
few here: Green-Lazarsfeld [GL91], Simpson [Sim93], Arapura [Ara97], Dimca-
Papadima [DP14], Dimca-Papadima-Suciu [DPS09], etc. It is finalized by Budur
and Wang in [BW15, BW20].

######  Theorem 3.5.

[BW15, BW20] If $X$ is a complex smooth variety, then
$\mathcal{V}^{i}_{k}(X,\mathbb{C})$ is a finite union of torsion translated
sub-tori of $\mathrm{Hom}(G,\mathbb{C}^{*})$.

When $\mathrm{char}(\mathbb{K})>0$, the following question abut the cohomology
jump loci is still open [Suc01, Section 3.5]:

######  Question 3.6.

Does the structure theorem hold for cohomology jump loci of a complex smooth
quasi-projective variety with $\mathrm{char}(\mathbb{K})>0$?

Theorem 3.5 implies the following property for the Alexander polynomial
associated to the pair $(X,\nu)$.

######  Proposition 3.7.

Let $X$ be a complex smooth variety. Then for any epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$, the $i$-th Alexander
polynomial $\Delta_{i}(X,\nu)$ is of cyclotomic type for any $i$. Hence
$\exp(\mathbf{M}_{i}(X^{\nu}))$ is a positive integer.

###### Proof.

The structure theorem for cohomology jump loci implies that [BLW18,
Proposition 1.4] all the roots of $\Delta_{i}(X^{\nu},\mathbb{C})$ are roots
of unity. Since $\Delta_{i}(X^{\nu},\mathbb{C})$ and $\Delta_{i}(X^{\nu})$
only differ by multiplication with a non-zero constant integer, the claim
follows. ∎

Chomology jump loci can be viewed as generalizations of
$\alpha_{*}(X^{\nu},\mathbb{K})$. In fact, consider an epimorphism
$\nu\colon\pi_{1}(X)\to\mathbb{Z}$. It induces an injective map

$\nu^{*}\colon\mathbb{K}^{*}\to\mathrm{Hom}(G,\mathbb{K}^{*})$

Then by Remark 3.1 it is easy to see that $\alpha_{i}(X^{\nu},\mathbb{K})=k$
if and only if

$\mathrm{im}\nu^{*}\subseteq\mathcal{V}^{i}_{k}(X,\mathbb{K})$ and
$\mathrm{im}\nu^{*}\nsubseteq\mathcal{V}^{i}_{k+1}(X,\mathbb{K})$.

The corresponding generalizations of $\beta_{*}(X,\nu_{\mathbb{K}})$ are the
resonance varieties. For its definition and properties, see e.g. [Suc11].

###  3.2. Orbifold Groups

In this subsection, we compute the degree 1 cohomology jump loci and
$L^{2}$-type invariants for the orbifold group.

Let $\Sigma_{g,r}$ be a Riemann surface of genues $g\geq 0$ and with $r\geq 0$
points removed. Then $\Sigma_{g,r}$ is a connected smooth quasi-projective
variety. To have $b_{1}(X)>0$, we always assume either $g>1$ or $g=0,r>1$,
i.e., $X\neq\mathbb{CP}^{1},\mathbb{C}$. The cohomology jump loci of
$\Sigma_{g,r}$ are already computed in Section 3.1. More generally, let us
consider as in [Dim07A, ACM13, Suc14A] an orbifold group as follows.

######  Definition 3.8.

Consider a Riemann Surface $\Sigma_{g,r}$ with $s$ marked points
$q_{1},\ldots,q_{s}$ and weight vector
${\mbox{\boldmath{$\mu$}}}=(\mu_{1},\cdots,\mu_{s})$, where
$\mu_{i}\in\mathbb{Z}_{>0}$. The orbiford group
$\Gamma=\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})$
associated to these data is defined as

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

where $\gamma_{j}$ is a meridian of $q_{j}$.

If $\mu_{j}=1$ for some $j$, we get the same orbifold group by omitting
$q_{j}$ and $\mu_{j}$. So without loss of generality, we assume $\mu_{j}>1$
for all $1\leq j\leq s$.

####  3.2.1. Non-compact case

If $\Sigma_{g,r}$ is not compact (i.e., $r>0$), then $\pi_{1}(\Sigma_{g,r})$
is a free group with rank $n=2g+r-1$, hence $\Gamma\cong
F_{n}\ast\mathbb{Z}_{\mu_{1}}\cdots\ast\mathbb{Z}_{\mu_{s}}.$ Then we have

$\mathrm{Hom}(\Gamma,\mathbb{K}^{*})\coloneqq\\{(t_{1},\ldots,t_{n},\lambda_{1},\ldots,\lambda_{s})\in(\mathbb{K}^{*})^{n+s}|\lambda_{j}^{\mu_{j}}=1\
\text{ for all }\ 1\leq j\leq s\\}.$

When $\mathrm{char}(\mathbb{K})=0$, we have the following short exact sequence
as in [ACM13, Proposition 2.7]:

$1\to(\mathbb{K}^{*})^{n}\to\mathrm{Hom}(\Gamma,\mathbb{K}^{*})\to\oplus_{j=1}^{s}C_{\mu_{j}}\to
1,$

where $C_{\mu_{j}}$ is the cyclic multiplicative subgroup of $\mathbb{K}^{*}$
with order $\mu_{j}$. Hence $\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$ has
$\prod_{j=1}^{s}\mu_{j}$ connected components, with every connected component
isomorphic to $(\mathbb{K}^{*})^{n}$.

On the other hand, when $\mathrm{char}(\mathbb{K})=p>0$, let
$\mu^{\prime}_{j}$ be the largest positive integer such that
$\gcd(p,\mu^{\prime}_{j})=1$ and $\mu^{\prime}_{j}\mid\mu_{j}$
($\mu^{\prime}_{j}$ could be 1). Then we have the following short exact
sequence:

$1\to(\mathbb{K}^{*})^{n}\to\mathrm{Hom}(\Gamma,\mathbb{K}^{*})\to\oplus_{j=1}^{s}C_{\mu^{\prime}_{j}}\to
1,$

where $C_{\mu^{\prime}_{j}}$ is the cyclic multiplicative subgroup of
$\mathbb{K}^{*}$ with order $\mu^{\prime}_{j}$. Hence
$\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$ has $\prod_{j=1}^{s}\mu^{\prime}_{j}$
connected components, with every connected component isomorphic to
$(\mathbb{K}^{*})^{n}$. In particular, if $\mu_{j}$ is a power of $p$ for all
$1\leq j\leq s$ (i.e., $\prod_{j=1}^{s}\mu^{\prime}_{j}=1$),
$\mathrm{Hom}(\Gamma,\mathbb{K}^{*})\cong(\mathbb{K}^{*})^{n}.$

In both cases, for any
$\rho=(t_{1},\ldots,t_{n},\lambda_{1},\ldots,\lambda_{s})\in\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$,
we set $\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})$ as the number of
trivial coordinates $\lambda_{j}=1$ such that
$\mathrm{char}(\mathbb{K})=p\nmid\mu_{j}$. When $\rho$ is the trivial
character, we simply write $\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}}).$

######  Proposition 3.9.

With the above notations, we have

$\dim
H^{1}(\Gamma,L_{\rho})=\begin{cases}n+s-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})-1,&\mathrm{if}\
L_{\rho}\neq\mathbb{K}_{\Gamma},\\\
n+s-\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}}),&\mathrm{if}\
L_{\rho}=\mathbb{K}_{\Gamma}.\end{cases}$

Here $\mathbb{K}_{\Gamma}$ denotes the trivial character on $\Gamma$.

###### Proof.

Consider the Eilenberg–MacLane space $K(\pi_{1},1)$ of $\Gamma$, say $Y$. The
homology group $H_{1}(\Gamma,L_{\rho})$ can be computed by the chain complex
$C_{*}(Y,L_{\rho})$, see [Dim04, p. 50]. By Fox calculus $\partial_{2}^{\rho}$
has the form of an $(n+s)\times s$ matrix as follows

(5)
$\begin{pmatrix}\frac{\lambda_{1}^{\mu_{1}}-1}{\lambda_{1}-1}&0&\cdots&0\\\
0&\frac{\lambda_{2}^{\mu_{2}}-1}{\lambda_{2}-1}&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&\frac{\lambda_{s}^{\mu_{s}}-1}{\lambda_{s}-1}\\\ 0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&0\\\ \end{pmatrix}.$

Note that for any $1\leq j\leq s$,
$\frac{\lambda_{j}^{\mu_{j}}-1}{\lambda_{j}-1}\neq 0$ in $\mathbb{K}$ if and
only if $\lambda_{j}=1$ and $p\nmid\mu_{j}$. Then we see that the above matrix
has rank $\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})$. If $L_{\rho}$ is
not the constant sheaf, then the rank of $\partial_{1}^{\rho}$ is 1, hence

$\dim
H_{1}(Y,L_{\rho})=n+s-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})-1.$

On the other hand, if $L_{\rho}$ is the constant sheaf, then the rank of
$\partial_{1}^{\rho}$ is 0, hence

$\dim H_{1}(Y,\mathbb{K})=n+s-\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}}).$

Then the claim follows from Remark 3.1. ∎

####  3.2.2. Compact case

If $\Sigma_{g,r}$ is compact (i.e., $r=0$), then

$\Gamma\cong\langle
x_{1},\ldots,x_{g},y_{1},\ldots,y_{g},\gamma_{1},\ldots,\gamma_{s}|\prod_{i=1}^{g}[x_{i},y_{i}]\prod_{j=1}^{s}\gamma_{j}=1,\gamma_{j}^{\mu_{j}}=1\text{
for all }1\leq j\leq s\rangle,$

hence

$\mathrm{Hom}(\Gamma,\mathbb{K}^{*})=\\{(t_{1},\ldots,t_{2g},\lambda_{1},\ldots,\lambda_{s})\in(\mathbb{K}^{*})^{2g+s}|\lambda_{1}\cdots\lambda_{s}=1,\lambda_{j}^{\mu_{j}}=1\
\mathrm{for}\ 1\leq j\leq s\\}.$

Since we assumed $\mu_{j}>1$ for all $1\leq j\leq s$, we get $s\geq 2$.

As in the non-compact case, when $\mathrm{char}(\mathbb{K})=p>0$, we set
$\mu^{\prime}_{j}$ to be the largest positive integer such that
$\gcd(p,\mu^{\prime}_{j})=1$ and $\mu^{\prime}_{j}\mid\mu_{j}$. To unify the
notations, we set $\mu^{\prime}_{j}=\mu_{j}$ when
$\mathrm{char}(\mathbb{K})=0$. Then in both cases, we have the following short
exact sequence:

$1\to(\mathbb{K}^{*})^{2g}\to\mathrm{Hom}(\Gamma,\mathbb{K}^{*})\to\big{(}\oplus_{j=1}^{s}C_{\mu^{\prime}_{j}}\big{)}/C_{\mu^{\prime}}\to
1,$

where
$\mu^{\prime}\coloneqq\mathrm{lcm}(\mu^{\prime}_{1},\ldots,\mu^{\prime}_{s})$
and the last term is the cokernel of the natural mapping
$C_{\mu^{\prime}}\to\oplus_{j=1}^{s}C_{\mu^{\prime}_{j}}$. Hence
$\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$ has
$\frac{\prod_{j=1}^{s}\mu^{\prime}_{j}}{\mu^{\prime}}$ connected components,
with every connected component isomorphic to $(\mathbb{K}^{*})^{2g}$.

For any
$\rho=(t_{1},\ldots,t_{2g},\lambda_{1},\ldots,\lambda_{s})\in\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$,
we set $\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})$ as the number of
trivial coordinates $\lambda_{j}=1$ such that
$\mathrm{char}(\mathbb{K})=p\nmid\mu_{j}$. When $\rho$ is the trivial
character, we simply write $\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}}).$

######  Proposition 3.10.

With the notations above, we have

$\dim
H^{1}(\Gamma,L_{\rho})=\begin{cases}2g+s-2-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}}),&\mathrm{if}\
L_{\rho}\neq\mathbb{K}_{\Gamma},\\\
2g+s-1-\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}}),&\mathrm{if}\
L_{\rho}=\mathbb{K}_{\Gamma},\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})<s\\\
2g,&\mathrm{if}\
L_{\rho}=\mathbb{K}_{\Gamma},\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})=s.\end{cases}$

###### Proof.

Consider the Eilenberg–MacLane space $K(\pi_{1},1)$ of $\Gamma$, say $Y$. By
Fox calculus $\partial_{2}^{\rho}$ has the form of an $(2g+s)\times(s+1)$
matrix as follows

(6)
$\begin{pmatrix}\frac{\lambda_{1}^{\mu_{1}}-1}{\lambda_{1}-1}&0&\cdots&0&1\\\
0&\frac{\lambda_{2}^{\mu_{2}}-1}{\lambda_{2}-1}&\cdots&0&\lambda_{1}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&\frac{\lambda_{s}^{\mu_{s}}-1}{\lambda_{s}-1}&\prod_{j=1}^{s-1}\lambda_{j}\\\
0&0&\cdots&0&y_{1}-1\\\ 0&0&\cdots&0&1-x_{1}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&0&y_{g}-1\\\
0&0&\cdots&0&1-x_{g}\\\ \end{pmatrix}.$

If $L_{\rho}$ is not the constant sheaf, this matrix has rank
$\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})+1$ and
$\partial_{1}^{\rho}$ has rank 1, hence

$\dim
H_{1}(Y,L_{\rho})=2g+s-2-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}}).$

While for the constant sheaf case, if
$\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})=s$, the above matrix has rank
$s$; and if $\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})<s$ (i.e., there
exists some $\mu_{j}$ such that $\mathrm{char}(\mathbb{K})=p\mid\mu_{j}$), the
above matrix has rank $\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})+1$. Then
we have

$\dim
H_{1}(Y,\mathbb{K})=\begin{cases}2g+s-\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})-1,&\mathrm{if}\
\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})<s\\\ 2g,&\mathrm{if}\
\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})=s.\end{cases}$

Then the claim follows from Remark 3.1. ∎

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

Let $\Gamma=\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})$ be
the orbifold group associated to the data $(g,r,{\mbox{\boldmath{$\mu$}}}).$
As mentioned before, any epimorphism
$\nu\colon\Gamma\twoheadrightarrow\mathbb{Z}$ gives a $\mathbb{Z}$-cover of
the space $K(\Gamma,1)$. We set

$\Delta_{1}(\Gamma^{\nu})\coloneqq\Delta_{1}(K(\Gamma,1)^{\nu})$ and
$\alpha_{1}(\Gamma^{\nu},\mathbb{K})\coloneqq\alpha_{1}(K(G,1)^{\nu},\mathbb{K})$.

######  Proposition 3.11.

With the above notations, for any epimorphism
$\nu\colon\Gamma\twoheadrightarrow\mathbb{Z}$, we have

  * (1)

$\alpha_{1}(\Gamma^{\nu},\mathbb{K})=-\chi(\Sigma_{g,r})+\\#\\{j\mid\mathrm{char}(\mathbb{K})=p\text{
divides }\mu_{j}\\}.$

  * (2)

$\Delta_{1}(\Gamma^{\nu})=\begin{cases}\prod_{j=1}^{s}\mu_{j},&\mathrm{if}\
r>0\\\ (\prod_{j=1}^{s}\mu_{j})\cdot(t-1),&\mathrm{if}\ r=0.\end{cases}$

###### Proof.

(1) Note that for any $\rho\in\mathbb{K}^{*}$, the pulled character
$\nu^{*}\rho$ always have
$\ell_{\mathbb{K}}(\nu^{*}\rho,{\mbox{\boldmath{$\mu$}}})=\ell_{\mathbb{K}}({\mbox{\boldmath{$\mu$}}})$,
since the torsion part of the abelianization of $\Gamma$ has to map to zero by
$\nu$. Then the claim follows by computation.

(2) If $r>0$, $\lambda_{j}=1$ implies
$\dfrac{\lambda_{j}^{\mu_{j}}-1}{\lambda_{j}-1}=\mu_{j}$. Then the claim
follows from (5). If $r=0$, $x_{i}$ and $y_{i}$ in (6) will be replaced by
some power of $t$. $\nu$ being epimorphism implies that the greatest common
divisor of the non-zero powers is $\pm 1$. Then the claim follows. ∎

##  4\. Proof of the main results

In this section, we focus on the fundamental group of complex smooth quasi-
projective varieties.

###  4.1. Orbifold Maps

We already defined orbifold maps in 1.2.1. Note that if an orbifold map $f$ is
either small or null, then $\Sigma$ is either an Elliptic curve or
$\Sigma=\mathbb{C}^{*}$.

######  Definition 4.1.

Given two orbifold maps $f_{i}\colon X\rightarrow\Sigma_{i}$ for $i=1,2$. We
say that they are equivalent if there is an isomorphism
$g\colon\Sigma_{1}\rightarrow\Sigma_{2}$ such that
${\mbox{\boldmath{$\mu$}}}_{2}(g(B_{1}))={\mbox{\boldmath{$\mu$}}}_{1}(B_{1})$
and the following diagram is commutative

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$$\textstyle{\Sigma_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\Sigma_{2}.}$

The following result is well known.

######  Proposition 4.2.

Let $f\colon X\rightarrow\Sigma_{g,r}$ be an orbifold map. Set
$\Gamma=\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})$. Then
$f$ induces an epimorphism

(7) $f_{*}\colon\pi_{1}(X)\twoheadrightarrow\Gamma.$

In particular, for any $\rho\in\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$, $f$
induces an embedding
$\mathcal{V}^{1}_{k}(\Gamma,\mathbb{K})\hookrightarrow\mathcal{V}^{1}_{k}(X,\mathbb{K})$,

###### Proof.

The proof for $f_{*}$ being surjective can in found in [ACM10, Proposition
1.4]. As part of the proof is needed later, we repeat it here. Consider the
following commutative diagram

$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(f^{-1}(U))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(U)}$

All the vertical maps are surjective since they are induced by inclusions. All
the horizontal maps are surjective since the generic fiber is connected. For
any point $q\in B$, say $f^{-1}(q)\coloneqq\mu^{1}C_{1}+\cdots+\mu^{m}C_{m}$,
where $\\{C_{1},\cdots,C_{m}\\}$ are reduced irreducible components of
$f^{-1}(q)$. Then the point $q$ has the signed multiplicity
$\mu\coloneqq\gcd(\mu^{1},\cdots,\mu^{m})$. Then $f$ maps $\gamma_{C_{i}}$ to
$(\gamma_{q})^{\mu^{i}}$, where $\gamma_{C^{i}}$ and $\gamma_{q}$ are the
meridians for $C_{i}$ and $q$, respectively. Then it is easy to see that the
first horizontal map in the above diagram factors through
$\pi_{1}(X)\twoheadrightarrow\Gamma$.

The latter claim follows directly from Proposition 3.4. ∎

######  Remark 4.3.

Theorem 1.4 indeed implies that there are at most finitely many equivalent
large or small orbifold maps for $X$ [Ara97, Theorem 1.6], since
$\mathcal{V}^{1}(X,\mathbb{C})$ is algebraic. For any orbifold map $f\colon
X\to\Sigma$, the image of the induced embedding
$\mathrm{Hom}(\pi_{1}(\Sigma),\mathbb{C}^{*})\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})$
is called the shadow of $f$ as in [ACM13]. Then for any two either large or
small orbifold maps $f_{i}\colon X\to\Sigma_{i}$ with $i=1,2$, their shadows
are either equal or intersects at the trivial character [ACM13, Lemma 6.4].

The definition of $\nu$ being orbifold effective is given in Definition 1.8.
Note that for the last claim in Definition 1.8 it is possible that $\nu$
factors through two different orbifold maps.

######  Remark 4.4.

Assume that $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ is orbifold
effective. If the associated orbifold map $f$ in Definition 1.8 is either
large or small, $f$ is unique under the equivalence relationship given in
Definition 4.1. In fact, $\nu$ and $f$ induces two injective maps

$\nu^{*}_{\mathbb{C}}\colon\mathbb{C}^{*}\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*})$
and
$f^{*}_{\mathbb{C}}\colon\mathrm{Hom}(\pi_{1}(\Sigma),\mathbb{C}^{*})\hookrightarrow\mathrm{Hom}(\pi_{1}(X),\mathbb{C}^{*}).$

In particular,
$\mathrm{im}\nu_{\mathbb{C}}^{*}\subseteq\mathrm{im}f^{*}_{\mathbb{C}}$. Then
Remark 4.3 shows that $f$ is unique, hence it makes sense to call $\nu$ of
type $(g,r,{\mbox{\boldmath{$\mu$}}})$.

On the other hand, when $f$ is null, $\mu$ is trivial and we simply write 1.
Consider $\nu$ as an element in $H^{1}(X,\mathbb{Z})$. If
$\Sigma=\mathbb{C}^{*}$, $\nu$ has to be of type $(1,1)$; while if $\Sigma$ is
an Elliptic curve, $\nu$ is contained in a dimension 2 weight 1 pure sub-Hodge
structure of $H^{1}(X,\mathbb{Q})$. In these two cases, $\nu$ is of type
$(0,2,1)$ or $(1,0,1)$, respectively.

######  Remark 4.5.

Consider $\nu$ as an element in $H^{1}(X,\mathbb{Z})$. Under certain Hodge
structure assumptions for $\nu$, it is always orbifold effective.

  * (1)

$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ is induced by an algebraic
map $h\colon X\to\mathbb{C}^{*}$ if and only if, when considered as an element
in $H^{1}(X,\mathbb{Z})$, $\nu$ is of type $(1,1)$, i.e., $\nu\in
F^{1}H^{1}(X,\mathbb{C})\cap\overline{F^{1}H^{1}(X,\mathbb{C})}$. Here $F$
stands for the Hodge filtration. This follows from Deligne’s theory of
1-motives (cf. [Del74, (10.I.3)]). Then $\nu$ is orbifold effective in this
case. In fact, projectivising and resolving indeterminacy, we get a map
$\overline{h}\colon\overline{X}\to\mathbb{CP}^{1}$. Using Stein factorization,
we get the following commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\scriptstyle{\overline{h}}$$\textstyle{\mathrm{im}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime\prime}}$$\textstyle{\mathbb{C}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{CP}^{1}}$

where $h^{\prime\prime}$ is a finite map, $h^{\prime}$ has connected fiber and
$f\coloneqq h^{\prime}|_{X}\colon X\to\mathrm{im}(f)$. [Dim07A, Lemma 2.2]
shows that $f$ has connected generic fiber. Hence $f$ is an orbifold map and
it makes $\nu$ orbifold effective.

In particular, if $H^{1}(X,\mathbb{Q})$ has pure Hodge structure of weight
$(1,1)$, then any epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ is orbifold effective. This
assumption holds if $X$ has a smooth compactification $\overline{X}$ such that
$H^{1}(\overline{X},\mathbb{Q})=0$. Typical examples are hyperplane
arrangement complement, toric arrangement complement and the complement of
some algebraic curves in $\mathbb{CP}^{2}$. See [Dim07A, Example 2.3] for more
examples.

  * (2)

$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ factors through the induced
map on the fundamental group level by an algebraic map $h\colon X\to E$ with
$E$ being an Elliptic curve, if and only if, $\nu$, considered as an element
in $H^{1}(X,\mathbb{Z})$, is contained in a dimension 2 weight 1 pure sub-
Hodge structure of $H^{1}(X,\mathbb{Q})$. This also follows from Deligne’s
theory of 1-motives (cf. [Del74, (10.I.3)]). Using a similar proof as above,
$\nu$ is also orbifold effective in this case.

  * (3)

Let $a\colon X\to\mathrm{Alb}_{X}$ denote the Albanese map of $X$, where
$\mathrm{Alb}_{X}$ is a semi-abelian variety. Since $X$ has complex dimension
2, then $\mathrm{im}a$ has dimension either 1 or 2. If $\dim\mathrm{im}a=1$,
then any $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ is orbifold
effective by a similar proof as above.

######  Example 4.6.

In general $\nu$ is not necessarily orbifold effective. Consider
$X=E\times\mathbb{C}^{*}$. Here $E$ is an Elliptic curve. Then we have
$\pi_{1}(X,\mathbb{Z})\cong\pi_{1}(E,\mathbb{Z})\oplus\pi_{1}(\mathbb{C}^{*},\mathbb{Z})\cong\mathbb{Z}^{3}$.
Take a corresponding basis for this direct sum. Say
$\nu\colon\mathbb{Z}^{3}\twoheadrightarrow\mathbb{Z}$ is given by three
integers $(a,b,c)$. Then $\nu$ is orbifold effective, if and only if, either
$c=0$ or $a=0=b$.

###  4.2. Proof of Theorem 1.6

Following the notations in section 3.2, for any

$\rho=(t,\lambda_{1},\ldots,\lambda_{s})\in\mathrm{Hom}(\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}}),\mathbb{K}^{*})$

with $t\in\mathrm{Hom}(\pi_{1}(\Sigma_{g,r}),\mathbb{K}^{*})$, we have
$L_{\rho}=L_{t}\otimes_{\mathbb{K}}L_{\lambda}$ on
$\Gamma=\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})$, where
$t$ and $\lambda$ represent the corresponding coordinates. Note that $L_{t}$
is indeed a rank one local system on $\Sigma_{g,r}$.

To prove Theorem 1.6, we need the following lemma proved by Dimca in [Dim07A,
Proposition 4.3]. Dimca only stated the results when $\mathbb{K}=\mathbb{C}$,
but the proof works for general field $\mathbb{K}$.

######  Proposition 4.7.

Let $f\colon X\to\Sigma$ be an orbifold map. Then for any $\mathbb{K}$-local
system $L_{1}$ on $X$ and any rank one $\mathbb{K}$-local system $L_{t}$ on
$\Sigma$, one has the following exact sequence

$0\rightarrow H^{1}(\Sigma,R^{0}f_{\ast}L_{1}\otimes L_{t})\rightarrow
H^{1}(X,L_{1}\otimes f^{-1}L_{t})\rightarrow
H^{0}(\Sigma,(R^{1}f_{\ast}L_{1})\otimes L_{t}).$

######  Theorem 4.8.

Let $X$ be a complex smooth quasi-projective variety with $b_{1}(X)>0$ and
$\mathbb{K}$ be an algebraically closed field with
$\mathrm{char}(\mathbb{K})=p\geq 0$. Consider an orbifold map $f\colon
X\to\Sigma$ of type $(g,r,{\mbox{\boldmath{$\mu$}}})$. Then with the same
notations as in section 3.2, for any
$\rho\in\mathrm{Hom}(\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}}),\mathbb{K}^{*})$
we have

$\dim H^{1}(X,f^{*}L_{\rho})\geq
2g+r+s-2-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}})$

and equality holds with finitely many exceptions.

###### Proof.

By Proposition 3.9, 3.10 and 4.2, we have

$\dim H^{1}(X,f^{*}L_{\rho})\geq
2g+r+s-2-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}}).$

Next we show that the reverse inequality holds with finitely many exceptions.
We first compute $H^{1}(\Sigma,R^{0}f_{*}f^{*}L_{\lambda}\otimes L_{t})$. For
any point $q\in B$, let $D_{q}$ be a small enough disk in $\Sigma$ centered at
$q$. Set $T_{q}=f^{-1}(D_{q}).$ If the corresponding monodromy
$\lambda_{q}\neq 1$, then $(f^{*}L_{\lambda})|T_{q}$ is not a constant sheaf
according to the proof of Proposition 4.2, hence
$(R^{0}f_{*}f^{*}L_{\lambda})_{q}=0$. On the other hand, if $\lambda_{q}=1$,
then $(R^{0}f_{*}f^{*}L_{\lambda})_{q}\neq 0$. Set $B^{\prime}=\\{q\in
B\mid\lambda_{q}\neq 1\\}$ and $U^{\prime}=\Sigma-B^{\prime}$. Let
$j^{\prime}$ denote the inclusion from $U^{\prime}$ to $\Sigma$. Then Dimca's
work [Dim07B, Proposition 1.8] showed that

$R^{0}f_{*}f^{*}L_{\lambda}=Rj^{\prime}_{*}j^{\prime-1}R^{0}f_{*}f^{*}L_{\lambda}$

and $j^{\prime-1}R^{0}f_{*}f^{*}L_{\lambda}$ is a local system on
$U^{\prime}$. Hence for all but finitely many
$t\in\mathrm{Hom}(\pi_{1}(\Sigma),\mathbb{K}^{*})$,

$\dim H^{1}(\Sigma,R^{0}f_{*}f^{*}L_{\lambda}\otimes L_{t})=\dim
H^{1}(U^{\prime},j^{\prime-1}(R^{0}f_{*}f^{*}L_{\lambda})\otimes
j^{\prime-1}L_{t})=2g+r-2+\\#(B^{\prime}).$

Meanwhile, for all but finitely many
$t\in\mathrm{Hom}(\pi_{1}(\Sigma),\mathbb{K}^{*})$, the above proposition
gives the following short exact sequence

$0\rightarrow H^{1}(\Sigma,(R^{0}f_{*}f^{*}L_{\lambda})\otimes
L_{t})\rightarrow H^{1}(X,f^{*}L_{\rho})\rightarrow
H^{0}(\Sigma,(R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t})$

Let $i$ and $j$ denote the inclusions from $B$ and $U$ to $\Sigma$. To compute
$H^{0}(\Sigma,(R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t})$, we have the
following short exact sequence

(8) $0\to H^{0}(B,i^{!}\big{(}(R^{1}f_{\ast}f^{*}L_{\lambda})\otimes
L_{t}\big{)})\to H^{0}(\Sigma,(R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t})\to
H^{0}(U,((R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t})|_{U})$

$(R^{1}f_{\ast}f^{*}L_{\lambda})|_{U}$ is a local system since $f|_{U}$ is a
fibration, hence so is $((R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t})|_{U}$.
Then for all but finitely many
$t\in\mathrm{Hom}(\pi_{1}(\Sigma),\mathbb{K}^{*}),$
$H^{0}(U,((R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t})|_{U})=0$. If
$B=\emptyset$, the proof is done. So we only need to compute
$H^{0}(B,i^{!}\big{(}(R^{1}f_{\ast}f^{*}L_{\lambda})\otimes L_{t}\big{)})$
when $B$ is non-empty. We mimic the proof given in [Ara97, Proposition 1.7]
and [Dim07A, Lemma 4.4].

For a point $q_{j}\in B$, let $D_{j}$ be a small enough disk in $\Sigma$
centered at $q_{j}$. Set $D_{j}^{*}=D_{j}-\\{q_{j}\\}.$ Note that
$L_{t}|_{D_{j}}$ is the constant sheaf. Then by excision we have

(9) $H^{0}(B,i^{!}\big{(}(R^{1}f_{\ast}f^{*}L_{\lambda})\otimes
L_{t}\big{)})\cong\oplus_{j=1}^{s}\ker\big{(}H^{0}(D_{j},(R^{1}f_{\ast}f^{*}L_{\lambda})|_{D_{j}})\to
H^{0}(D_{j}^{*},(R^{1}f_{\ast}f^{*}L_{\lambda})|_{D_{j}^{*}})\big{)}.$

So the computations reduce to a local computation for each point in $B$.

To simplify the notations, we do the computations for a point $q\in B$ with
small disk $D$ and punched disk $D^{*}$. Set $T\coloneqq f^{-1}(D)$,
$T^{*}\coloneqq f^{-1}(D^{*})$, $F$ being the generic fiber and
$F_{q}\coloneqq f^{-1}(q)$. Say
$f^{-1}(q)\coloneqq\mu^{1}C_{1}+\cdots+\mu^{m}C_{m}$, where
$\\{C_{1},\cdots,C_{m}\\}$ are reduced irreducible components of $f^{-1}(q)$.
Then the point $q$ has the signed multiplicity
$\mu\coloneqq\gcd(\mu^{1},\cdots,\mu^{m})$.

Let $\lambda_{q}$ denote the monodromy around the point $q$. We split the
proof to two cases:

  * •

Case I: $\lambda_{q}=1$.

Then $(R^{1}f_{*}f^{*}L_{\lambda})|_{D}=(R^{1}f_{*}\mathbb{K}_{X})|_{D}$. Note
that $H^{0}(D,(R^{1}f_{*}\mathbb{K}_{X})|_{D})\cong H^{1}(T,\mathbb{K})$ and
$H^{0}(D^{*},(R^{1}f_{*}\mathbb{K}_{X})|_{D^{*}})$ is isomorphic to the
invariant part of $H^{1}(F,\mathbb{K})$ with respect to the monodromy
associated to $D^{*}$, denoted by $H^{1}_{\mathrm{inv}}(F,\mathbb{K})$. Let
$i_{q}$ and $j_{q}$ be the inclusions from $F$ and $T^{*}$ to $T$,
respectively. Then we have the following commutative digram for the induced
maps:

(10)
$\textstyle{H^{1}(T,\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{q}^{*}}$$\scriptstyle{i_{q}^{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(D^{*},\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{H^{1}(T^{*},\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{inv}}(F,\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The horizontal row is exact (see [Mil68, p. 118]), since the restriction of
$f$ on $D^{*}$ is a locally trivial fibration with connected generic fiber
$F$. The vertical map is injective since it is induced by an inclusion. Note
the term on the right hand of (9) is exactly $\ker i_{q}^{*}$, hence we get

$\mathrm{ker}i_{q}^{*}=\mathrm{im}j_{q}^{*}\cap\mathrm{im}f^{*}.$

In particular, $\dim\ker i_{q}^{*}\leq 1$.

Consider the following commutative diagram

$\textstyle{\bigoplus_{i=1}^{m}\mathbb{Z}[\gamma_{C_{i}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{(\mu^{1},\ldots,\mu^{m})}$$\textstyle{H_{1}(T^{*},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\scriptstyle{j_{q,*}}$$\textstyle{H_{1}(D^{*},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H_{1}(T,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(D,\mathbb{Z}).}$

Here $\gamma_{C_{i}}$ denote the meridian along irreducible divisor $C_{i}$.
Note that $\mu=\gcd(\mu^{1},\ldots,\mu^{m})$. If $p\nmid\mu$, the dual map
$H^{1}(D^{*},\mathbb{K})\to\mathbb{K}^{m}$ for $(\mu^{1},\cdots,\mu^{m})$ is
injective, but the other dual map $H^{1}(T,\mathbb{K})\to\mathbb{K}^{m}$ is
zero, hence $\dim\mathrm{im}j_{q}^{*}\cap\mathrm{im}f^{*}=0.$ Then we get

$\dim\mathrm{ker}i_{q}^{*}=\begin{cases}0,&\mathrm{if}\ p\nmid\mu\\\ \leq
1,&\mathrm{if}\ p\mid\mu.\end{cases}$

  * •

Case II: $\lambda_{q}\neq 1$.

Then we have a similar diagram as above:

(11)
$\textstyle{H^{1}(T,f^{*}L_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{q}^{*}}$$\scriptstyle{i_{q}^{*}}$$\textstyle{0=H^{1}(D^{*},L_{\lambda_{q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{H^{1}(T^{*},f^{*}L_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\lambda_{q}^{-1}}(F,\mathbb{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The horizontal map is an isomorphism due to the Milnor long exact sequence,
where $H^{1}_{\lambda_{q}^{-1}}(F,\mathbb{K})$ denotes the $\lambda_{q}^{-1}$
sub-eigenspace of $H^{1}(F,\mathbb{K})$. On the other hand, $j_{q}^{*}$ is
injective since it is induced by an inclusion. Then it is easy to see that
$\ker i_{q}^{*}=0$ in this case.

Putting these two cases together, for all but finitely many
$\rho\in\mathrm{Hom}(\Gamma,\mathbb{K}^{*})$, we have

$\dim H^{1}(X,f^{*}L_{\rho})\leq
2g+r+s-2-\ell_{\mathbb{K}}(\rho,{\mbox{\boldmath{$\mu$}}}).$

Then the claim follows. ∎

###  4.3. Proof of Theorem 1.10

In Definition 1.8, one indeed gets the following sequence of epimorhisms:

$\pi_{1}(X)\twoheadrightarrow\Gamma\twoheadrightarrow\pi_{1}(\Sigma)\twoheadrightarrow\mathbb{Z}$

To simplify the notations, we use $\Gamma^{\nu}$ and $\Sigma^{\nu}$ to denote
the corresponding $\mathbb{Z}$-covers.

By Theorem 1.6
$\alpha_{1}(X^{\nu},\mathbb{C})=\alpha_{1}(\Gamma^{\nu},\mathbb{C})$, hence
$\mathrm{rank}H_{1}(X^{\nu},\mathbb{Z})=\mathrm{rank}H_{1}(\Gamma^{\nu},\mathbb{Z}).$
By Proposition 4.2, 2.6 and 3.11, we get

$\prod_{j=1}^{s}\mu_{j}\mid\Delta_{1}(X^{\nu})$ and
$\sum_{j=1}^{a}\log\mu_{j}\leq\mathbf{M}_{1}(X^{\nu})$.

Next we prove the converse inequality. Consider the following sequence of
epimorphisms:

$\pi_{1}(f^{-1}(U))\twoheadrightarrow\pi_{1}(U)\twoheadrightarrow\pi_{1}(\Sigma)\twoheadrightarrow\mathbb{Z}.$

We denote by $f^{-1}(U)^{\nu}$ and $U^{\nu}$ the corresponding
$\mathbb{Z}$-covers. Then we have the following commutative diagram of exact
sequences:

(12)
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(U^{\nu},f^{-1}(U)^{\nu},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(f^{-1}(U)^{\nu},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(U^{\nu},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2}(\Sigma^{\nu},X^{\nu},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(X^{\nu},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\Sigma^{\nu},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H_{1}\big{(}(\Sigma^{\nu},X^{\nu}),(U^{\nu},f^{-1}(U)^{\nu}),\mathbb{Z}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

The two horizontal rows are exact since $H_{2}(U^{\nu},\mathbb{Z})=0$ and
$H_{2}(\Sigma^{\nu},\mathbb{Z})=0$. The right two vertical maps are surjective
due to Proposition 4.2, since $f$ has connected generic fiber. Note that
$f|_{f^{-1}(U)}$ is a fibration, in particular an orbifold map with no
multiple fiber. By Theorem 1.10, we have

$\alpha_{1}(f^{-1}(U)^{\nu},\mathbb{K})=\alpha_{1}(U^{\nu},\mathbb{K})\text{
for any field }\mathbb{K}\text{ and }\mathbf{M}_{1}(f^{-1}(U)^{\nu})=0.$

By proposition 2.2, $\mathbf{M}_{2}(U^{\nu},f^{-1}(U)^{\nu})=0$.

Note that the first two terms on the first column of (12) have rank 0. By
Proposition 2.2 we have

$\mathbf{M}_{1}(X^{\nu})=\mathbf{M}_{2}(\Sigma^{\nu},X^{\nu})=\mathbf{M}_{1}\big{(}(\Sigma^{\nu},X^{\nu}),(U^{\nu},f^{-1}(U)^{\nu})\big{)}.$

So we only need to show
$\mathbf{M}_{1}\big{(}(\Sigma^{\nu},X^{\nu}),(U^{\nu},f^{-1}(U)^{\nu})\big{)}\leq\sum_{j=1}^{s}\log\mu_{j}$.

By excision, we have

$H_{1}\big{(}(\Sigma^{\nu},X^{\nu}),(U^{\nu},f^{-1}(U)^{\nu}),\mathbb{Z}\big{)}\cong\oplus_{j=1}^{s}H_{1}\big{(}(D_{j},f^{-1}(D_{j})),(D_{j}^{*},f^{-1}(D_{j}^{*})),\mathbb{Z}\big{)}\otimes_{\mathbb{Z}}\mathbb{Z}[t,t^{-1}].$

Then we do the computations for one point $q\in B$ and we use the same
notations as in the proof of Theorem 1.6. Set $H\coloneqq
H_{1}\big{(}(D,T),(D^{*},T^{*}),\mathbb{Z}\big{)}$. Consider the following
commutative diagrams:

$\textstyle{\bigoplus_{i=1}^{m}\mathbb{Z}[\gamma_{C_{i}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{(\mu^{1},\ldots,\mu^{m})}$$\textstyle{H_{2}(D^{*},T^{*},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{H_{1}(T^{*},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\scriptstyle{g_{2}}$$\textstyle{H_{1}(D^{*},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H_{2}(D,T,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H_{1}(T,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

Here $\gamma_{C_{i}}$ denote the meridian along irreducible divisor
$C_{i}\subseteq f^{-1}(q)$ and we identify $H_{2}(D,T,\mathbb{Z})$ with
$H_{1}(T,\mathbb{Z})$ since $D$ is contractible. Hence
$H\cong\mathrm{coker}(g_{2}\cdot g_{1}).$ Since $g_{2}$ is surjective and the
second horizontal row is exact, we get an epimorphism

$\mathbb{Z}\cong H_{1}(D^{*},\mathbb{Z})\cong
H_{1}(T^{*},\mathbb{Z})/\mathrm{im}(g_{1})\twoheadrightarrow\mathrm{coker}(g_{2}\cdot
g_{1}).$

On the other hand, $g_{2}\cdot h_{1}=0$ and $\mathrm{im}(h_{2}\cdot
h_{1})=\mu\mathbb{Z}$. Therefore

$|\mathrm{coker}g_{2}\circ g_{1}|\leq|\mathbb{Z}/\mu\mathbb{Z}|=\mu.$

This shows that
$\mathbf{M}(H\otimes_{\mathbb{Z}}\mathbb{Z}[t,t^{-1}])\leq\log\mu$ and the
claims in (a) follows.

For (b), if $X$ is projective and $\nu$ is not orbifold effective,
$\alpha_{1}(X^{\nu},\mathbb{K})=0$ for any field $\mathbb{K}$ by Theorem 1.5.
Then $\mathbf{M}_{1}(X^{\nu})=0$ follows by Remark 2.13 and Proposition 3.7.

##  5\. Hyperplane arrangement

In this section, we specialize to the case of complex hyperplane arrangements.
Let $\mathcal{A}$ be a hyperplane arrangement in $\mathbb{P}^{n}$, defined by
a product of degree 1 homogeneous polynomials

$Q(\mathcal{A})\coloneqq\prod_{H\in\mathcal{A}}^{d}f_{H}.$

This also gives a central hyperplane arrangement in $\mathbb{C}^{n+1}$,
denoted by $\overline{\mathcal{A}}$. Set

$X(\mathcal{A})\coloneqq\mathbb{P}^{n}-\mathcal{A}$ and
$M(\mathcal{A})\coloneqq\mathbb{C}^{n+1}\setminus\overline{\mathcal{A}}$.

Note that $M(\mathcal{A})=X(\mathcal{A})\times\mathbb{C}^{*}$, hence
$\mathcal{V}^{1}(X(\mathcal{A}),\mathbb{K})=\mathcal{V}^{1}(M(\mathcal{A}),\mathbb{K})$.
When the context is clear, we simply write $X(\mathcal{A})$ as $X$.

###  5.1. Some properties and examples

We list some facts for hyperplane arrangement complement $X$.

  * (1)

$X$ is homotopy equivalent to a minimal CW complex [DP03, Ran02], hence
$H^{i}(X,\mathbb{Z})$ is a free abelian group for any $0\leq i\leq n$.

  * (2)

The cohomology ring $H^{*}(X,\mathbb{Z})$ is determined by the combinatorial
data [Dim17, Chapter 3].

  * (3)

$\mathcal{V}^{i}_{k}(X,\mathbb{C})$ is a finite union of torsion translated
sub-tori due to the structure theorem.

  * (4)

For any epimorphism $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$,
$\alpha_{i}(X^{\nu},\mathbb{C})=\beta_{i}(X,\nu_{\mathbb{C}})$ for any $i$.
This follows from the tangent cone equality between
$\mathcal{V}^{i}_{k}(X,\mathbb{C})$ and the corresponding resonance varieties
of $X$, see e.g. [CO00, Theorem 3.7]

Next we give an application of Proposition 2.17. For each positive integer
$k>1$, let $\Phi_{k}(t)$ be the irreducible cyclotomic polynomial of primitive
$k$-th root of unity. It is known that

$\Phi_{k}(1)=\begin{cases}1&\text{if }k\text{ is not a power of any prime
number},\\\ p&\text{if }k=p^{r}\text{ for some prime number }p.\end{cases}$

For any epimorphism $\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$, by
Proposition 3.7 $\Delta_{i}(X^{\nu})$ is of cyclotomic type. Let
$\Delta^{\prime}_{i}(X^{\nu})$ denote the polynomial obtained from
$\Delta_{i}(X^{\nu})$ by taking out $(t-1)$ factors and we write it down as

$\Delta^{\prime}_{i}(X^{\nu})=c_{i}\prod_{k>1}\Phi_{k}(t)^{e_{i}^{k}}.$

######  Proposition 5.1.

With the above assumptions and notations for the hyperplane arrangement
complement $X$, for any degree $i>0$, we have

(13)
$c_{i}(\prod_{p}p^{e_{i}^{p}+e_{i}^{p^{2}}+\cdots})\mid\tau_{i}(X,\nu_{\mathbb{Z}}),$

where the product runs over all prime number $p>0$. Moreover, we have

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

In particular, say $\nu$ is orbifold effective of type
$(0,r,{\mbox{\boldmath{$\mu$}}})$. Then we have

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

###### Proof.

Since $\Delta_{i}(X^{\nu})$ is of cyclotomic type, we get
$\mathbf{M}_{i}(X^{\nu})=\log c_{i}$, where $c_{i}$ is the leading coefficient
of $\Delta_{i}(X^{\nu})$. By the properties listed above, hyperplane
arrangement complement satisfies the assumptions in Proposition 2.17, hence
the first claim follows. It implies that
$\exp(c_{i})\mid\tau_{i}(X,\nu_{\mathbb{Z}})$. Combined with Theorem 1.10, the
third claim follows. ∎

One may compare (13) in the above proposition with the following result due to
Cohen and Orlik [CO00, Theorem 1.3] (also see [PS10] for its generalization).

######  Theorem 5.2.

Let $X$ be a hyperplane arrangement complement. Then for any epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}$ and any
$\lambda\in\mathbb{C}^{*}$ with order being some power of a prime number $p$,
we have

$\dim H^{i}(X,L_{\lambda})\leq\beta_{i}(X,\nu_{\overline{\mathbb{F}}_{p}}).$

Next two examples show that both inequalities could happen in Theorem 2.15 and
Proposition 5.1.

######  Example 5.3.

Let $\mathcal{A}\coloneqq\\{[x,y]\in\mathbb{P}^{1}|x^{d}+y^{d}=0\\}$. Take
$\nu\colon\pi_{1}(M(\mathcal{A}))\to\mathbb{Z}$ as the epimorphism that sends
the meridian for $H_{\ell}$ to an integer $n_{\ell}$, where
$\gcd\\{n_{\ell}\colon 1\leq\ell\leq d\\}=1$. It is easy to see that for any
field $\mathbb{K}$

$\alpha_{1}(M(\mathcal{A})^{\nu},\mathbb{K})=\begin{cases}0,&\mathrm{if}\
\sum_{\ell=1}^{d}n_{\ell}\neq 0,\\\ d-2,&\mathrm{if}\
\sum_{\ell=1}^{d}n_{\ell}=0,\end{cases}$

hence $\mathbf{M}_{1}(M(\mathcal{A})^{\nu})=0$.

We have a basis $\\{a_{1},\ldots,a_{d}\\}$ for
$H^{1}(M(\mathcal{A}),\mathbb{Z})$, where $a_{\ell}$ corresponds to the
$\ell$-th hyperplane, hence
$\nu_{\mathbb{Z}}=\sum_{\ell=1}^{d}n_{\ell}a_{\ell}$.
$H^{2}(M(\mathcal{A}),\mathbb{Z})$ is generated by $\\{a_{ij}\coloneqq
a_{i}\cup a_{j}\\}_{1\leq i<j\leq d}$ with the relation
$a_{ij}=a_{1j}-a_{1i}$. Hence $H^{2}(M(\mathcal{A}),\mathbb{Z})$ has a basis
$\\{a_{1\ell}\\}_{2\leq\ell\leq d}$. Let us write down the matrix of the map
$H^{1}(M(\mathcal{A}),Z)\overset{\nu_{\mathbb{Z}}}{\to}H^{2}(M(\mathcal{A}),\mathbb{Z})$
under these basis:

$\begin{pmatrix}n_{2}&n_{3}&n_{4}&\cdots&n_{d}\\\ -\sum_{i\neq
2}n_{i}&n_{3}&n_{4}&\cdots&n_{d}\\\ n_{2}&-\sum_{i\neq
3}&n_{4}&\cdots&n_{d}\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{2}&n_{3}&n_{4}&\cdots&-\sum_{i\neq d}\end{pmatrix}$

When $\sum_{\ell=1}^{d}n_{\ell}\neq 0$, this matrix has rank $d-1$ and
$\tau_{1}(M(\mathcal{A}),\nu_{\mathbb{Z}})=(\sum_{\ell=1}^{d}n_{\ell})^{d-2}$
(since $\gcd\\{n_{\ell}\colon 1\leq\ell\leq d\\}=1$). On the other hand, when
$\sum_{\ell=1}^{d}n_{\ell}=0$, it has rank $1$ and the matrix can be changed
to $\begin{pmatrix}1&0&\ldots&0\\\ \vdots&\vdots&\vdots&0\\\
1&0&\ldots&0\end{pmatrix}$ by elementary column operation since
$\gcd\\{n_{\ell}\colon 1\leq\ell\leq d\\}=1$, hence
$\tau_{1}(M(\mathcal{A}),\nu_{\mathbb{Z}})=1$. Then we have

$\beta_{1}(M(\mathcal{A}),\nu_{\mathbb{K}})=\begin{cases}d-2,&\mathrm{if}\
\sum_{\ell=1}^{d}n_{\ell}=0,\\\ d-2,&\mathrm{if}\
p\mid\sum_{\ell=1}^{d}n_{\ell}\neq 0,\\\ 0,&\mathrm{if}\
p\nmid\sum_{\ell=1}^{d}n_{\ell}\neq 0\end{cases}$ where
$p=\mathrm{char}(\mathbb{K}).$

Hence if $p\mid\sum_{\ell=1}^{d}n_{\ell}\neq 0$,
$\alpha_{1}(M(\mathcal{A})^{\nu},\mathbb{C})<\beta_{1}(M(\mathcal{A}),\nu_{\overline{\mathbb{F}}_{p}})$
and
$\exp(\mathbf{M}_{1}(M(\mathcal{A})^{\nu}))=1<(\sum_{\ell=1}^{d}n_{\ell})^{d-2}=\tau_{1}(M(\mathcal{A}),\nu_{\mathbb{Z}}).$

Under suitable basis $\partial_{2}^{\mathbb{Z}}$ can be written as form of
matrix of size $d\times(d-1)$

$\begin{pmatrix}1-t^{n_{1}}&1-t^{n_{2}}&\ldots&1-t^{n_{d-1}}\\\
t^{\sum_{\ell=1}^{d}n_{\ell}}-1&0&\ldots&0\\\
0&t^{\sum_{\ell=1}^{d}n_{\ell}}-1&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&t^{\sum_{\ell=1}^{d}n_{\ell}}-1\end{pmatrix}$

One can get

$\Delta_{1}(M(\mathcal{A})^{\nu})=\begin{cases}(t-1)(t^{\sum_{\ell=1}^{d}n_{\ell}}-1)^{d-2},&\mathrm{if}\
\sum_{\ell=1}^{d}n_{\ell}\neq 0,\\\ t-1,&\mathrm{if}\
\sum_{\ell=1}^{d}n_{\ell}=0.\end{cases}$

In particular, if $\sum_{\ell=1}^{d}n_{\ell}\neq 0$,
$(1+t+\ldots+t^{(\sum_{\ell=1}^{d}n_{\ell})-1})^{d-2}|_{t=1}=(\sum_{\ell=1}^{d}n_{\ell})^{d-2}$,
which coincides with $\tau_{1}(M(\mathcal{A}),\nu_{\mathbb{Z}})$.

######  Example 5.4.

Let $\mathcal{B}$ be the deleted $B_{3}$-arrangement in $\mathbb{P}^{2}$ 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
$X(\mathcal{A})$ be the complement of the arrangement. It was first discovered
by Suciu [Suc02] that $\mathcal{V}^{1}(X(\mathcal{A}),\mathbb{C})$ has a
transalated 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 $(0,2,2)$. This suggests us to 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 $\tau_{1}(X,\nu_{\mathbb{Z}})=4$,
$H^{2}(H^{*}(X,\mathbb{Z}),\cdot\nu_{\mathbb{Z}})_{\mathrm{tor}}\cong\mathbb{Z}/4\mathbb{Z}$
and $\beta_{1}(X,\nu_{\mathbb{K}})=\begin{cases}1,&\mathrm{if}\
\mathrm{char}(\mathbb{K})=2,\\\ 0,&\mathrm{if}\ \mathrm{char}(\mathbb{K})\neq
2.\end{cases}$

Using the formula given in [Suc02, Example 4.1] (be careful that in [Suc02,
Example 4.1] hyperplanes are ordered different from ours), we get
$\Delta^{\prime}_{1}(X^{\nu})=2(t+1)$, hence
$\Delta^{\prime}_{1}(X^{\nu})(1)=4$, which coincides with
$\tau_{1}(X,\nu_{Z})$.

It would be interesting to have an example where
$\Delta^{\prime}_{1}(X^{\nu})(1)\neq\tau_{1}(X,\nu_{\mathbb{Z}}).$

###  5.2. Multinet

We first recall the definition of multinet.

######  Definition 5.5.

Let $\mathcal{A}$ be an arrangement in $\mathbb{P}^{2}$. 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, $n\colon\mathcal{A}\to\mathbb{Z}_{>0}$, and a subset
$\mathfrak{X}$ of intersection points in $\mathcal{A}$ satisfying the
following conditions:

  * (a)

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

  * (d)

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

  * (e)

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

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

By results of Pereira-Yuzvinsky [PY08] and Yuzvinsky [Yuz09], $k=3$ or $4$ if
$|\mathfrak{X}|>1$. It was conjectured by Yuzvinksy that $k=4$ only happens
for Hessian arrangement and now confirmed in [BK20]. As shown by Falk, Pereira
and Yuzvinsky [FY07, PY08], every $(k,\kappa)$-multinet determines a large
orbifold map $f\colon M(\mathcal{A})\to\Sigma_{0,k}\subset\mathbb{P}^{1}$ by

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

######  Theorem 5.6.

Let $\mathcal{A}$ be a hyperplane arrangement in $\mathbb{P}^{2}$. Then every
positive-dimensional, non-local component of
$\mathcal{V}^{1}(X(\mathcal{A}),\mathbb{C})=\mathcal{V}^{1}(M(\mathcal{A}),\mathbb{C})$
is obtained from some orbifold map $f\colon M(\mathcal{A})\to\Sigma_{0,r}$
with either

  * •

$r=2$ and $f$ has at least one multiple fiber, or

  * •

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

As $k=4$ only happens for Hessian arrangement, we consider a multinet on
$\mathcal{A}$ with $k=3$. Under the following assumptions, we will show that
the corresponding orbifold map has no multiple fiber.

######  Assumption 5.7.

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

  * (a)

$n_{H}=1$ 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 in $\mathcal{A}_{3}$ left.

######  Proposition 5.8.

With the above assumptions,
$\mathcal{V}^{1}(X(\mathcal{A}),\mathbb{C})=\mathcal{V}^{1}(M(\mathcal{A}),\mathbb{C})$
does not have translated two dimensional component corresponding to this
multinet, i.e., the corresponding orbifold map has no multiple fiber.

###### Proof.

We take $\nu$ which maps the meridian of $H$ to $1$ for $H\in\mathcal{A}_{1}$
or $\mathcal{A}_{2}$ and $-2n_{H}$ for $H\in\mathcal{A}_{3}$. Then the claim
follows if we can show that $\tau_{1}(M(\mathcal{A})^{\nu})=1$.

Consider the map
$H^{1}(M(\mathcal{A}),\mathbb{Z})\overset{\nu_{\mathbb{Z}}}{\to}H^{2}(M(\mathcal{A}),\mathbb{Z})$.
Let $\mathfrak{Y}_{i}$ denote the intersection points in the sub-arrangement
$\mathcal{A}_{i}$. Due to Brieskorn Decomposition [Dim17, Theorem 3.2],

$H^{2}(M(\mathcal{A}),\mathbb{Z})\cong\oplus_{x\in\mathfrak{X}\cup\mathfrak{Y}_{1}\cup\mathfrak{Y}_{2}\cup\mathfrak{Y}_{3}}H^{2}(M(\mathcal{A}_{x}),\mathbb{Z}),$

where $\mathcal{A}_{x}$ is the central line arrangement for the point $x$. For
any $x\in\mathfrak{X}$, Definition 5.5(c) and the choice of $\nu$ give

$\sum_{H\in\mathcal{A}_{1},x\in H}n_{H}+\sum_{H\in\mathcal{A}_{2},x\in
H}n_{H}-\sum_{H\in\mathcal{A}_{3},x\in H}2n_{H}=0,$

and Definition 5.5(c) together with Assumption 5.7(a) implies that one can
find $H\in\mathcal{A}_{1}$ with $n_{H}=1$ and $x\in H$. Then the computations
in Example 5.3 show that one can replace the sub-matrix corresponding to
$\mathcal{A}_{i}\times\mathfrak{X}$ by the column obtained from the incidence
relation between the hyperplanes and $\mathfrak{X}$. We have the following
matrix

(14) $\vbox{\kern 12.4pt\hbox{$\kern 0.0pt\kern-6.66669pt\left(\kern
0.0pt\vbox{\kern-12.4pt\vbox{\halign{ \hfil$#$\hfil\kern 2\p@\kern\@leftbwd&
\thinspace\hfil$#$\hfil&& \quad\hfil$#$\hfil\cr$\hfil\kern 2.0pt\kern
6.66669pt&\mathfrak{X}&\mathfrak{Y}_{1}&\mathfrak{Y}_{2}&\mathfrak{Y}_{3}\crcr\kern
2.0pt\cr\mathcal{A}_{1}$\hfil\kern 2.0pt\kern
6.66669pt&C_{1}&B_{1}&0&0\cr\mathcal{A}_{2}$\hfil\kern 2.0pt\kern
6.66669pt&C_{2}&0&B_{2}&0\cr\mathcal{A}_{3}$\hfil\kern 2.0pt\kern
6.66669pt&C_{3}&0&0&B_{3}\crcr} }}\right)$}},$

Here $C_{i}$ is the $|\mathcal{A}_{i}|\times|\mathfrak{X}|$ incidence matrix,
where the entry $(H,x)$ is 1 precisely when $x\in H$. The incidence matrix
$\begin{pmatrix}C_{1}\\\ C_{2}\\\ C_{3}\end{pmatrix}$ has rank $d-2$ (see
[LY00, Theorem 3.2]), where $d=|\mathcal{A}|$. Together with Definition
5.5(a), one gets that $C_{3}$ is full row rank. Assumption 5.7(c) implies that
$C_{3}$ can be changed to $\begin{pmatrix}\mathrm{Id}&0\end{pmatrix}$ by
elementary column operations. Here $\mathrm{Id}$ is the identity matrix.

Next we compute the matrix $B_{1}$. Consider a graph, whose vertices
correspond to the hyperplanes in $\mathcal{A}_{1}$ and two vertices has an
unordered edge if the corresponding two hyperplanes intersect at a point in
$\mathfrak{Y}_{1}$. Definition 5.5(d) is equivalent to say that this graph is
connected. Assumption 5.7(b) implies that there is a one to one correspondence
between the edges and $\mathfrak{Y}_{1}$. Choose a spanning tree in this graph
and consider the submatirx corresponding to the vertices and the edges in it.
By Assumption 5.7(a),(b), we get a $\kappa\times(\kappa-1)$ matrix, where
$\kappa=|\mathcal{A}_{1}|$ and every column in this matrix has $1$ and $-1$
both for precisely one entry and $0$ for the rest. By easy computations, we
get that $B_{1}$ has rank $\kappa-1$ and has a $(\kappa-1)\times(\kappa-1)$
minor being $\pm 1$. The same computation works for $B_{2}$.

Since
$\beta_{i}(M(\mathcal{A}),\nu_{\mathbb{C}})=\alpha_{i}(M(\mathcal{A})^{\nu},\mathbb{C})=-\chi(\Sigma_{0,2})=1$,
the matrix (14) has rank $d-2$. Putting all the computations for $C_{3}$,
$B_{1}$ and $B_{2}$ together, we find a $(d-2)\times(d-2)$ minor being $\pm
1$. Then the proof is done. ∎

######  Example 5.9.

We give a list of examples which satisfies Assumption 5.7: 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 [Suc14B]. It
is also easy to find examples which do not satisfy Assumption 5.7, see Figure
1(b) in [FY07] or the monomial (alias Ceva) arrangement (with $m\geq 3$) in
[Dim17, Example 6.11].

###  5.3. A question by Denham and Suciu

Let $\mathcal{A}$ be a hyperplane arrangement in $\mathbb{P}^{n}$, defined by
a product of degree 1 homogeneous polynomials

$Q(\mathcal{A})\coloneqq\prod_{1\leq\ell\leq d}^{d}f_{\ell}.$

This also gives a central hyperplane arrangement in $\mathbb{C}^{n+1}$,
denoted by $\overline{\mathcal{A}}$. Let

$F(\mathcal{A})\coloneqq\\{x\in\mathbb{C}^{n+1}\mid Q(\mathcal{A})(x)=1\\}$

denote the Milnor fiber of the arrangement $\mathcal{A}$. Then
$F(\mathcal{A})$ admits a finite cover to $X(\mathcal{A})$.

For the proof of Theorem 1.17, we use the same construction introduced by
Denham and Suciu in [DS14]. Their construction focusses entirely on `non-
modular torsion`: that is, $p$-torsion in the homology of cyclic covers, where
$p$ does not divide the order of the cover, hence part of results in [DS14]
are proved under this assumption. We modify all the related results in [DS14]
without this assumption and list them as follows.

The next theorem, corresponding to [DS14, Theorem 4.3], follows from Theorem
1.10 and Proposition 2.12.

######  Theorem 5.10.

Let $X$ be a complex smooth quasi-projective variety. Suppose that there is a
orbifold fibration $f\colon X\to\Sigma$ of type
$(g,r,{\mbox{\boldmath{$\mu$}}})$ with $\prod_{j=1}^{s}\mu_{j}>1$. Take a
prime number $p$ dividing $\prod_{j=1}^{s}\mu_{j}$. Then, for all sufficiently
large integer $N$, there exists a regular, $N$-fold cyclic cover $X^{N}\to X$
such that $H_{1}(X^{N},\mathbb{Z})$ has $p$-torsion.

###### Proof.

Let $\Gamma$ be the corresponding orbifold group. Take any epimorphism
$\nu\colon\Gamma\twoheadrightarrow\mathbb{Z}$ and consider the covering space
$X^{\nu}$ for the composed map

$\pi_{1}(X)\overset{f_{*}}{\twoheadrightarrow}\Gamma\overset{\nu}{\twoheadrightarrow}\mathbb{Z}.$

Let $X^{\nu,N}$ denote the corresponding $N$-fold cover. Theorem 1.10 shows
$\alpha_{1}(X^{\nu},\mathbb{C})<\alpha_{1}(X^{\nu},\overline{\mathbb{F}}_{p})$.
Then the claim follows from Proposition 2.12 by taking $i=1$. ∎

For a technical reason, we need to allow multiplicities on the hyperplanes: if
$\mathbf{m}\coloneqq(m_{1},\ldots,m_{d})\in\mathbb{Z}^{d}_{>0}$ is a positive
lattice vector, then the pair $(\mathcal{A},\mathbf{m})$ is called a
multiarrangement. A defining polynomial for the multiarrangement is the
product

$Q(\mathcal{A},\mathbf{m})\coloneqq\prod_{\ell=1}^{d}f_{\ell}^{m_{\ell}}$

and

$F(\mathcal{A},\mathbf{m})\coloneqq\\{x\in\mathbb{C}^{n+1}\mid
Q(\mathcal{A},\mathbf{m})(x)=1\\}$

is called the Milnor fiber of the multiarrangement $(\mathcal{A},\mathbf{m})$.
Set $m\coloneqq\sum_{\ell=1}^{d}m_{\ell}$. Then $F(\mathcal{A},\mathbf{m})$
admits a regular $m$-fold cover of $X(\mathcal{A})$ given by the homomorphism

$\delta\colon\pi_{1}(X(\mathcal{A}))\to\mathbb{Z}/m\mathbb{Z}$,
$\gamma_{\ell}\mapsto m_{\ell}\mod m,$

where $\gamma_{\ell}$ is the meridian for $H_{\ell}$ in $X(\mathcal{A})$.

The following lemma corresponds to [DS14, Proposition 6.7].

######  Proposition 5.11.

Let $\mathcal{A}$ be an hyperplane arrangement and let $X^{N}\rightarrow X$ be
a connected regular $\mathbb{Z}_{N}$ cover of $X=X(\mathcal{A})$ given by an
epimorphism $\chi\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}/N\mathbb{Z}$.
Then there exists infinitely many multiplicity vectors
$\mathbf{m}\in\mathbb{Z}^{d}_{>0}$ for which the covering projection
$F(\mathcal{A},\mathbf{m})\to X$ factors through $X^{N}$

$\textstyle{F(\mathcal{A},\mathbf{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$

Moreover, for any prime $p>0$, we may choose $\mathbf{m}$ such that
$\frac{m}{N}$ is not divisible by $p$, where $m=\sum_{\ell=1}^{d}m_{\ell}$.

###### Proof.

For any $1\leq\ell\leq d$, say $\chi$ sends $\gamma_{\ell}$ to
$\chi_{\ell}\mod N$, where $\chi_{\ell}\in[0,N-1]$ is an integer. In
particular, $N$ divides $\sum_{\ell=1}^{d}\chi_{\ell}$. Choose any
multiplicity vector $(q_{1},\ldots,q_{d})\in\mathbb{Z}^{d}_{\geq 0}$ such that
$m_{\ell}=\chi_{\ell}+N\cdot q_{\ell}>0$ and $\frac{m}{N}$ is not divisible by
$p$, where $m=\sum_{\ell=1}^{d}m_{\ell}$. Then it is clear
$F(\mathcal{A},\mathbf{m})\to X$ factors through $X^{N}$. ∎

######  Example 5.12.

[DS14, Example 6.12] Fix an integer $\mu\geq 2$. Let $\mathcal{A}_{\mu}$ be
the deleted monomial arrangement, where its defining equations in
$\mathbb{P}^{2}$ are $yz(x^{\mu}-y^{\mu})(x^{\mu}-z^{\mu})(y^{\mu}-z^{\mu})$.
Ordering the hyperplanes as the factors of the defining polynomial. Its
projective complement $X(\mathcal{A}_{\mu})$ admits an orbifold map
$X(\mathcal{A}_{\mu})\to\mathbb{C}^{*}$ given by

$\dfrac{x^{\mu}(y^{\mu}-z^{\mu})}{y^{\mu}(x^{\mu}-z^{\mu})},$

which is of type $(0,2,\mu)$. Moreover,
$\mathcal{V}^{1}(M(\mathcal{A}),\mathbb{C})$ has a translated component

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

with $\rho=(1,\zeta^{-1},\ldots,\zeta^{-1},1,\ldots,1,\zeta,\ldots,\zeta)$.
Here $\zeta$ is a primitive $\mu$-th root of unity.

For any fixed prime number $p$ and large enough $N$, we can choose the
multiplicity vector as

$\mathbf{m}=(\mu,kN-\mu,\overbrace{1,\ldots,1}^{\mu},\overbrace{N-1,\ldots,N-1}^{\mu},\overbrace{N,\ldots,N}^{\mu})$

and $m=(2\mu+k)N$ such that $p$ is coprime with $2\mu+k$. Then this
multiplicity vector satisfies the requirement in Proposition 5.11.

Next we go to the construction considered in [DS14]. For $m_{1}>2$, by [DS14,
Definition 7.8] we consider the parallel connection of the multiarrangement
$(\mathcal{A},\mathbf{m})$ with a central line arrangement in
$\mathbb{C}^{2}$, denoted by $\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}$,
where the line arrangement has exactly $m_{1}$ hyperplanes. Up to change of
coordinates, we can assume that $H_{1}$ for $\mathcal{A}$ is defined by
$\\{x_{1}=0\\}$ and the defining equations for the line arrangement are
$\prod_{j=0}^{m_{1}-1}(y_{1}-j\cdot y_{2})$. Then
$\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}$ can be viewed as a central
multiarrangement in $\mathbb{C}^{n+2}$ with the following defining equations
by the coordinates $(x,y_{2})\in\mathbb{C}^{n+2}$

$x_{1}\cdot\big{(}\prod_{\ell=2}^{d}f_{\ell}^{m_{\ell}}(x)\big{)}\cdot\prod_{j=1}^{m_{1}-1}(x_{1}-j\cdot
y_{2}).$

In particular, $X(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})\cong
X(\mathcal{A})\times\Sigma_{0,m_{1}}$, where
$\Sigma_{0,m_{1}}\coloneqq\mathbb{CP}^{1}\setminus\\{m_{1}\text{ points}\\}$.
If $m_{1}=1$, then $\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}$ is simply the
original arrangement $\mathcal{A}$.

######  Proposition 5.13.

With the above notations, we have a pullback diagram between the respective
Milnor covers,

$\textstyle{F(\mathcal{A},\mathbf{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X(\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{X(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})}$

where $j$ is the map given by choosing a basepoint in $\mathcal{P}_{m_{1}}$.
Moreover, if $m_{1}\geq 3$ and $i\geq 1$, there exists a surjective group
homomorphism

$H_{i+1}\big{(}F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}),\mathbb{Z}\big{)}_{\mathrm{tor}}\twoheadrightarrow
H_{i}(F(\mathcal{A},\mathbf{m}),\mathbb{Z})_{\mathrm{tor}}.$

###### Proof.

The first claim is already proved in [DS14, Lemma 8.4] and we only need to
prove the second one.

$\Sigma_{0,m_{1}}$ has a simple cell structure with a single $0$-dimensional
cell, say $a$, and $(m_{1}-1)$ many $1$-dimensional cell, say
$\\{b_{1},\ldots,b_{m_{1}-1}\\}$. The product structure
$X(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})\cong
X(\mathcal{A})\times\Sigma_{0,m_{1}}$ gives a cell structure for
$X(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})$. Note that
$F(\mathcal{A},\mathbf{m})$ and
$F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})$ are
$\mathbb{Z}/m\mathbb{Z}$-fold covers of $X(\mathcal{A})$ and
$X(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})$, respectively. By choosing
fixed lifts of the cells of $F(\mathcal{A},\mathbf{m})$ and
$F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})$ respectively, we obtain free
basis for the chain complex
$C_{*}\big{(}F(\mathcal{A},\mathbf{m}),\mathbb{Z}\big{)}$ and
$C_{*}\big{(}F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}),\mathbb{Z}\big{)}$
as $R$-modules, where
$R=\mathbb{Z}[t]/(t^{m}-1)\cong\mathbb{Z}[\mathbb{Z}/m\mathbb{Z}]$. The
$R$-module map between finitely generated free $R$-modules

$C_{i}(F(\mathcal{A},\mathbf{m}),\mathbb{Z})\overset{\partial_{i}}{\to}C_{i-1}(F(\mathcal{A},\mathbf{m}),\mathbb{Z})$

can be written down as a matrix, say $A$, with entries in
$\mathbb{Z}[t]/(t^{m}-1)$. By the product structure
$X(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}})\cong
X(\mathcal{A})\times\Sigma_{0,m_{1}}$, we have the following matrix for the
map
$C_{i+1}\big{(}F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}),\mathbb{Z}\big{)}\to
C_{i}\big{(}F(\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}),\mathbb{Z}\big{)}$:

$\kern 0.0pt\kern-6.66669pt\left(\kern 0.0pt\vbox{\kern-12.4pt\vbox{\halign{
\hfil$#$\hfil\kern 2\p@\kern\@leftbwd& \thinspace\hfil$#$\hfil&&
\quad\hfil$#$\hfil\cr$\hfil\kern 2.0pt\kern
6.66669pt&(i-1,b_{1})&(i-1,b_{2})&\cdots&(i-1,b_{m_{1}-1})&(i,a)\crcr\kern
2.0pt\cr(i,b_{1})$\hfil\kern 2.0pt\kern
6.66669pt&A&0&\cdots&0&(t-1)\mathrm{id}\cr(i,b_{2})$\hfil\kern 2.0pt\kern
6.66669pt&0&A&\cdots&0&(t-1)\mathrm{id}\cr\vdots$\hfil\kern 2.0pt\kern
6.66669pt&\vdots&\vdots&\vdots&\vdots&(t-1)\mathrm{id}\cr(i,b_{m_{1}-1})$\hfil\kern
2.0pt\kern 6.66669pt&0&0&\cdots&A&(t-1)\mathrm{id}\cr(i+1,a)$\hfil\kern
2.0pt\kern 6.66669pt&0&0&\cdots&0&*\crcr} }}\right)$

Here for example $(i,b_{1})$ stands for the basis
$C_{i}(X(\mathcal{A}),R)\otimes_{R}\langle b_{1}\rangle$, where $\langle
b_{1}\rangle$ is the free $R$-module generated by $b_{1}$. When $m_{1}\geq 3$,
by elementary operations the above matrix becomes

$\begin{pmatrix}A&0&\cdots&0&(t-1)\mathrm{id}\\\ 0&A&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&A&0\\\
0&0&\cdots&0&*\end{pmatrix}$

Consider $\mathbb{Z}[t]/(t^{m}-1)\cong\mathbb{Z}^{m}$ as an abelian group.
Then $A$ can be also viewed as a matrix (with different size) with entries in
$\mathbb{Z}$ and the torsion part of
$H_{i}(F(\mathcal{A},\mathbf{m}),\mathbb{Z})$ is determined by $A$. Then the
above computation gives the claim. ∎

Denham and Suciu iterated the parallel connections of the multiarrangement
$(\mathcal{A},\mathbf{m})$ with a collection of line arrangements and obtained
the following definition.

######  Definition 5.14.

[DS14, Definition 8.1] The polarization of a multiarrangement
$(\mathcal{A},\mathbf{m})$, denoted by $\mathcal{A}||\mathbf{m}$, is the
arrangement of $m=\sum_{\ell=1}^{d}m_{\ell}$ hyperplanes given by

$\mathcal{A}||\mathbf{m}=\mathcal{A}\circ_{H_{1}}\mathcal{P}_{m_{1}}\circ_{H_{2}}\mathcal{P}_{m_{2}}\circ_{H_{3}}\cdots\circ_{H_{d}}\mathcal{P}_{m_{d}}.$

Note that the polarization of a multiarrangement is a reduced hyperplane
arrangement with $m=\sum_{\ell=1}^{d}m_{\ell}$ many hyperplanes. Let
$F(\mathcal{A}||\mathbf{m})$ denote the Minor fiber for the arrangement
$(\mathcal{A}||\mathbf{m})$. A similar diagram to the one presented in
Proposition 5.13 is obtained in [DS14, Lemma 8.4].

###### Proof of Theorem 1.17.

We take $\mathcal{A}_{\mu}$ to be the arrangement considered in Example 5.12
with $X$ being its complement. There exists an orbifold map $f$ of type
$(0,2,\mu)$ for $X$. By choosing $\mu$ such that $p$ divides $\mu$, for any
integer $N$ big enough Theorem 5.10 gives $X^{N}$, a
$\mathbb{Z}/N\mathbb{Z}$-cover of $X$, such that $H_{1}(X^{N},\mathbb{Z})$ has
non-trivial $p$-torsion. Take the multiplicity vector given in Example 5.12.
Since $F(\mathcal{A}_{\mu},\mathbf{m})$ is a $\frac{m}{N}$-fold cover of
$X^{N}$ and $p$ does not divides $\frac{m}{N}$, by [DS14, Lemma 2.4]
$H_{1}(F(\mathcal{A}_{\mu},\mathbf{m}),\mathbb{Z})$ also has non-trivial
$p$-torsion. Using Proposition 5.13 repetitively, we get that
$H_{2\mu+3}(F(\mathcal{A}_{\mu}||\mathbf{m}),\mathbb{Z})$ has non-trivial
$p$-torsion (sine there are exactly $\mu$ many $m_{\ell}=1$).

Then we take $\mathcal{B}=\mathcal{A}_{\mu}||\mathbf{m}$, which is a reduced
hyperplane arrangement with $|\mathcal{B}|=m$. For the additional requirement
$p$ dividing $|\mathcal{B}|$, we can choose $N$ such that $p$ divides $N$,
hence $p$ divides $m$. Then the claim follows. ∎

######  Remark 5.15.

As we used the same construction as in [DS14], for any prime number $p\geq 2$,
the Milnor fiber of the central arrangement we construed can have non-trivial
$p$-torsion in homology at degree $2p+3$, same as in [DS14, Corollary 8.8].
Denham and Suciu asked if there is a hyperplane arrangement $\mathcal{A}$
whose Milnor fiber $F(\mathcal{A})$ has torsion in
$H_{1}(F(\mathcal{A}),\mathbb{Z})$ [DS14, Question 8.10]. Yoshinaga gave the
first example where $H_{1}(F(\mathcal{A}),\mathbb{Z})$ has non-trivial
$2$-torsion, see [Yos20].

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