# The homology growth for finite abelian covers of 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 epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow H$, where $H$ is a finitely generated
abelian group with $\mathrm{rank}H\geq 1$. In this paper, we study the
asymptotic behaviour of 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. When $\nu$ is orbifold effective, we
give explicit formulas of these invariants at degree 1. This generalizes the
authors' previous work for $H\cong\mathbb{Z}$.

###### Key words and phrases:

Mahler measure, jump loci, orbifold map, $L^{2}$-betti number

###### 2010 Mathematics Subject Classification:

14F45,32S20,14H30,57Q99

##  1\. Introduction

There is a general principle to consider a classical invariant of a finite CW
complex $X$ and to define its analogues 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 have been vast literatures for the
$L^{2}$-invariant theory, see [Luc02]. 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 CW complex can be found as limits of
normalized Betti numbers of finitely sheeted normal coverings. Then it becomes
a classical subject to study $L^{2}$-type invariants focusing on its
approximation by towers of finite coverings, see e.g. [Luc13, Luc16].

Let $X$ be a connected finite CW complex with a fixed epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow H$, where $H$ is a finitely generated
abelian group with $\mathrm{rank}H=n\geq 1$. We fix an isomorphism
$H\cong\mathbb{Z}^{n}\oplus T$, where $T$ is a finite abelian group. Consider
$\mathbb{Z}^{n}$ as a subgroup of $H$ under this isomorphism. For a subgroup
$\Gamma\subset\mathbb{Z}^{n}$ of finite index, we set

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

Let $X^{\nu,\Gamma}$ denote the covering space of $X$ associated to the
corresponding composition of $\nu$ and the quotient map $H\to H/\Gamma$.
Consider the following limits:

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

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

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

Here $|H_{i}(X^{\nu,\Gamma},\mathbb{Z})_{\mathrm{tor}}|$ denotes the order of
the torsion part of $H_{i}(X^{\nu,\Gamma},\mathbb{Z}).$ These two limits are
particular cases of $L^{2}$-type invariants. Such kind of $L^{2}$-type
invariants have been studied by many people (see [Luc13, Luc16] for more
results in this direction).

When $H$ is free abelian, Linnell, Lück, Sauer [LLS11, Theorem 0.2] (for
$\mathrm{char}(\mathbb{K})=0$) and Abért, Jaikin-Zapirain, Nikolov [AJN11,
Theorem 17] (for $\mathrm{char}(\mathbb{K})>0$) showed that the first limit
always exists. Meanwhile, the second limit also exists and can be computed by
the Mahler measure of the Alexander polynomial, see [Le14].

Moreover, if $H$ has non-trivial torsion part (i.e., $T\neq 0$), the
computation of these limits can be reduced to the free abelian case using a
finite cover trick, see section 3.

When $X$ is a complex smooth quasi-projective variety and $H\cong\mathbb{Z}$,
the authors [LL21] gave concrete formulas for these limits (at degree 1) in
terms of the geometric information of $X$. The main results in this paper are
to generalize these formulas to arbitrary finitely generated abelian group $H$
with $\mathrm{rank}H\geq 1$, when $\nu$ is an orbifold effective morphism.

We first give the definition of orbifold morphism.

######  Definition 1.1.

Let $X$ be a smooth complex quasi-projective variety. An algebraic map
$f\colon X\to\Sigma_{g,r}$ is called an orbifold map, if $f$ is surjective,
has connected generic fiber and $\Sigma_{g,r}$ is a smooth algebraic curve of
genus $g$ with $r$ points removed. We always assume that
$\Sigma_{g,r}\neq\mathbb{CP}^{1},\mathbb{C}^{1}$. 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 $\mu_{j}$ (the $\gcd$
of the coefficients of the divisor $f^{*}q_{j}$) of the fiber $f^{*}(q_{j})$
to the point $q_{j}$. Such orbifold map $f$ is called of type
$(g,r,{\mbox{\boldmath{$\mu$}}})$, where
${\mbox{\boldmath{$\mu$}}}=(\mu_{1},\ldots,\mu_{s})$. When $B=\emptyset$,
$\prod_{j=1}^{s}\mu_{j}=1$ by convention.

The orbiford group
$\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})$ 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 $q_{j}$. An orbifold map $f\colon
X\to\Sigma$ of type $(g,r,{\mbox{\boldmath{$\mu$}}})$ induces an surjective
map to the orbifold group (see [ACM10, Proposition 1.4])

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

When $\Sigma_{g,r}$ is clear in the context, we simply write $\Sigma$.

######  Definition 1.2.

Let $X$ be a smooth complex quasi-projective variety with an epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow H$. 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:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.67375pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.59573pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\nu}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 109.08684pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.99997pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
21.18637pt\raise-26.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 46.01506pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\hbox{\kern-1.68329pt\lower-1.07999pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
59.3803pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 109.08684pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H}$}}}}}}}{\hbox{\kern-3.0pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
39.67375pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}^{\mathrm{orb}}(\Sigma_{g,r},{\mbox{\boldmath{$\mu$}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
112.57999pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-1.60971pt\lower
1.18686pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
113.64932pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

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

Our main result is the following:

######  Theorem 1.3.

Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow H$, where $H$ is a finitely generated
abelian group with $\mathrm{rank}H\geq 1$. Suppose that $\nu$ is orbifold
effective of type $(g,r,{\mbox{\boldmath{$\mu$}}})$. Let $\mathbb{K}$ be a
field with $\mathrm{char}(\mathbb{K})=p\geq 0$.

  * (a)

If $H$ is a free abelian group, then

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

and

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

  * (b)

If $H$ has non-trivial torsion part, then we have

$\alpha_{1}(X^{\nu},\mathbb{K})=2g+r-2+\sum_{j=1}^{s}\left(1-\frac{1}{m_{j}}\right)+\sum_{1\leq
j\leq s,p\mid\frac{\mu_{j}}{m_{j}}}\frac{1}{m_{j}}.$

and

$\mathbf{M}_{1}(X^{\nu},\mathbb{K})=\sum_{j=1}^{s}\frac{1}{m_{j}}\log\frac{\mu_{j}}{m_{j}},$

where $m_{j}$ is a positive integer dividing $\mu_{j}$ and it only depends on
$X$ and $\nu$. For its definition, see Remark 3.2.

The proof of Theorem 1.3 is based on the theory of cohomogy jump loci. In
section 2, we recall some basic properties of cohomology jump loci and give
the proof of Theorem 1.3 for the free abelian case. Section 3 is devoted to
the proof of Theorem 1.3 for the non-free abelian case using a finite cover
trick.

Acknowledgments. The second named author is partially supported by National
Key Research and Development Project SQ2020YFA070080, NSFC-12001511, the
Project of Stable Support for Youth Team in Basic Research Field CAS
(YSBR-001), a research fund from University of Science and Technology of
China, the project “Analysis and Geometry on Bundles” of the Ministry of
Science and Technology of the People's Republic of China and Fundamental
Research Funds for the Central Universities.

##  2\. The Free Abelian Case

In this section we consider the case when $H$ is a free abelian group with
rank $n$, say $\mathbb{Z}^{n}$.

###  2.1. Limits of Betti numbers

We recall the definition of cohomology jump loci. Let $X$ be a connected
finite CW complex with $\pi_{1}(X)=G$. 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 $L_{\rho}$. Note that
$\mathrm{Hom}(G,\mathbb{K}^{*})$ only depends on $H_{1}(X,\mathbb{Z})$, the
abelianization of $G$.

######  Definition 2.1.

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 $\mathcal{V}^{i}(X,\mathbb{K})$.

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]).

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

######  Proposition 2.2.

Let $\mathbb{K}$ be an algebraically closed field. With the notations as
above, for any $i\geq 0$ and $\lambda\in(\mathbb{K}^{*})^{n}$ being general we
have

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

In particular, $\alpha_{i}(X^{\nu},\mathbb{K})$ is always an integer.

###### Proof.

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

$H^{i}(X^{\nu,\Gamma},\mathbb{K})=H^{i}(X,\pi_{*}\mathbb{K}),$

where $\pi_{*}\mathbb{K}$ is the push forward of the $\mathbb{K}$-constant
sheaf on $X^{\nu,\Gamma}$, hence a rank $|\mathbb{Z}^{n}/\Gamma|$ local
system.

[LLS11, Theorem 0.2] and [AJN11, Theorem 17] show that the limit
$\alpha_{i}(X^{\nu},\mathbb{K})$ always exists. By choosing a sub-sequence, we
may assume that $\mathrm{char}(\mathbb{K})\nmid|\mathbb{Z}^{n}/\Gamma|$. Since
$\Gamma$ is a sub-group of $\mathbb{Z}^{n}$ with finite index, $\Gamma$ is
also a free abelian group with rank $n$. Using Smith normal form, without loss
of generality we say $\Gamma=N_{1}\mathbb{Z}\oplus\cdots\oplus
N_{n}\mathbb{Z}$ for a tuple of positive integers where $N_{n}\mid
N_{n-1}\mid\cdots\mid N_{1}$. In particular, $\mathrm{char}(\mathbb{K})\nmid
N_{1}$. Then the local system $\pi_{*}\mathbb{K}$ decomposes as the direct sum
of $|\mathbb{Z}^{n}/\Gamma|$-many rank one local systems. Say $L_{\lambda}$
with $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in(\mathbb{K}^{*})^{n}$ is one
of the direct sum factors. Then $\lambda_{j}^{N_{j}}=1$ for any $1\leq j\leq
n$. Suppose that $a$ is the number such that

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

Note that $(\mathbb{K}^{*})^{n}\cap\mathcal{V}^{i}_{a+1}(X,\mathbb{K})$ is a
subvariety of $(\mathbb{K}^{*})^{n}$ with less dimension. There exists a
hypersurface $V$ such that

$(\mathbb{K}^{*})^{n}\cap\mathcal{V}^{i}_{a+1}(X,\mathbb{K})\subset
V=\\{(t_{1},\ldots,t_{n})\in(\mathbb{K}^{*})^{n}|u(t_{1},\ldots,t_{n})=0\\},$

where $u$ is a polynomial. Without loss of generality, suppose the degree of
$t_{1}$ for $u$ is $d\geq 1$. Then for a fixed
$(\lambda_{2},\ldots,\lambda_{n})$, $u(t_{1},\lambda_{2},\ldots,\lambda_{n})$
has at most $d$-many solutions. Hence we have

$a\cdot\prod_{j=1}^{n}N_{j}\leq\dim H^{i}(X^{\nu,\Gamma},\mathbb{K})\leq
a\cdot\prod_{j=1}^{n}N_{j}+c\cdot d\cdot\prod_{j=2}^{n}N_{j},$

where $c$ is some constant number which only depends on $X$ (e.g. $c$ can be
taken as the number of $i$-cells in $X$). It implies

$a\leq\frac{\dim H^{i}(X^{\nu,\Gamma},\mathbb{K})}{\prod_{j=1}^{n}N_{j}}\leq
a+c\cdot\frac{d}{N_{1}},$

For field coefficients, $\dim H^{i}(X^{\nu,\Gamma},\mathbb{K})=\dim
H_{i}(X^{\nu,\Gamma},\mathbb{K})$. Taking
$\langle\Gamma\rangle\rightarrow\infty$, we are done since $N_{1}$ goes to
infinity. ∎

When $\mathbb{K}=\mathbb{C}$, the cohomology jump loci of complex smooth
quasi-projective variety have been intensively studied. In particular, the
following structure theorem for $\mathcal{V}^{i}_{k}(X,\mathbb{C})$ put strong
constraints for the homotopy type of complex smooth quasi-projective variety.
It is contributed by many people and we name a few here: Green-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 2.3.

[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 $\mathrm{Hom}(G,\mathbb{C}^{*})$.

###  2.2. Limits of torsion

The second type limit also exists, see [Le14, Theorem 5]. It can be computed
by the Mahler measure of the $i$-th integral Alexander polynomial
$\Delta_{i}(X^{\nu})\in\mathbb{Z}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]$. Let us
recall the definitions of multivariable Alexander polynomials and Mahler
measures.

Recall that $X$ is a connected finite CW-complex with a group epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Then the group of
covering transformations of the covering space $X^{\nu}$ is isomorphic to
$\mathbb{Z}^{n}$ and acts on it. By choosing fixed lifts of the cells of $X$
to $X^{\nu}$, we obtain a free basis for the cellular chain complex of
$X^{\nu}$ as $R_{n}$-modules, where
$R_{n}=\mathbb{Z}[\mathbb{Z}^{n}]=\mathbb{Z}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]$.
So the cellular chain complex of $X^{\nu}$, $C_{*}(X^{\nu},\mathbb{Z})$, is a
bounded complex of finitely generated free $R_{n}$-modules:

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

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

######  Definition 2.4.

$\Delta_{i}(X^{\nu})$ is called the $i$-th $n$-variable Alexander polynomial
of $(X,\nu)$. Then $\Delta_{i}(X^{\nu})$ is defined uniquely up to a
multiplication with a unit of $R_{n}$.

Let $h\in R_{n}$ be a nonzero polynomial. The Mahler measure of $h$ is defined
by

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

where $\mathrm{d}s$ indicates integration with respect to normalized Haar
measure, and $(S^{1})^{n}$ is the multiplicative subgroup of $n$-dimensional
complex space $\mathbb{C}^{n}$ consisting of all vectors
$(s_{1},\ldots,s_{n})$ with $|s_{1}|=\cdots=|s_{n}|=1$. Here $h$ is regarded
as a function on $\mathbb{C}^{n}$.

Recall that an element $h\in R_{n}$ is called a generalized cyclotomic
polynomial [Sch95, page 47] if it is of the form
$h(t_{1},\ldots,t_{n})=t^{\mathbf{m}}\Phi(t^{\mathbf{n}})$, where
$\mathbf{m},\mathbf{n}\in\mathbb{Z}^{n},\mathbf{n}\neq 0$, and $\Phi$ is a
cyclotomic polynomial in a single variable.

######  Theorem 2.5.

[Sch95, Theorem 19.5] Let $h\in R_{n}$. Then $\mathbf{M}(h)=0$ if and only if
$\pm h$ is a product of generalized cyclotomic polynomials.

######  Theorem 2.6.

[Le14, Theorem 5] With the notations above, we have

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

When $n=1$, then $\limsup$ can be replaced by the ordinary $\lim$.

Theorem 2.3 implies the following property for the Alexander polynomial
associated to the pair $(X,\nu)$. For the one-variable version of the
following result, see [BLW18, Proposition 1.4] and [LL21, Proposition 3.7].

######  Proposition 2.7.

Let $X$ be a complex smooth variety with a fixed epimorphism
$\nu\colon\pi_{1}(X)\twoheadrightarrow\mathbb{Z}^{n}$. Then
$\Delta_{i}(X^{\nu})$ is a product of an positive integer $c_{i}$ with some
generalized cyclotomic polynomials, where $c_{i}$ is the leading coefficient
of $\Delta_{i}(X^{\nu})$. In particular, $\mathbf{M}(\Delta_{i}(X^{\nu}))=\log
c_{i}$.

###### Proof.

The structure theorem for cohomology jump loci
$\mathcal{V}^{i}_{k}(X,\mathbb{C})$ implies that
$(\mathbb{C}^{*})^{n}\cap\mathcal{V}^{i}_{k}(X,\mathbb{C})$ is a finite union
of torsion translated sub-tori. From now on, we consider $\Delta_{i}(X^{\nu})$
as an element in $\mathbb{C}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]$. We claim that
the irreducible factors of $\Delta_{i}(X^{\nu})$ can be observed by the
irreducible hyersurfaces in
$(\mathbb{C}^{*})^{n}\cap\mathcal{V}^{i}_{k}(X,\mathbb{C})$. Say $h$ is a
irreducible factor of $\Delta_{i}(X^{\nu})$. Then $h$ generates a prime ideal
with height 1. Let $\mathbb{C}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]_{(h)}$ be its
localization at the prime ideal $(h)$, which is a PID. Consider
$H_{i}(X^{\nu},\mathbb{C})$ as a finitely generated
$\mathbb{C}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]$-module and its localization
$H_{i}(X^{\nu},\mathbb{C})_{(h)}$. Then by a similar proof to [DN04, Theorem
4.2], we have

$\dim
H_{i}(X,\nu^{-1}L_{\lambda})=\mathrm{rank}H_{i}(X^{\nu},\mathbb{C})_{(h)}+J_{i}+J_{i-1},$

where $\lambda$ is a general point in the zero locus of $h$,
$\mathrm{rank}H_{i}(X^{\nu},\mathbb{C})_{(h)}$ is its rank as a finitely
generated module over the PID
$\mathbb{C}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]_{(h)}$ and $J_{i}$ is the number
of direct summands of the torsion part of $H_{i}(X^{\nu},\mathbb{C})_{(h)}$.
By Universal Coefficients Theorem over the PID
$\mathbb{C}[t^{\pm}_{1},\ldots,t^{\pm}_{n}]_{(h)}$, one can translate from
homology to cohomology. Then by structure theorem, we get that $h$ has the
form $\prod_{j=1}^{n}t_{j}^{d_{j}}=\epsilon$, where $(d_{1},\ldots,d_{n})$ is
a non-zero tuple of integers and $\epsilon$ is a roots of unity. Since
$\Delta_{i}(X^{\nu})$ has integer coefficients, $\Delta_{i}(X^{\nu})$ is a
product of generalized cyclotomic polynomial with some integer $c_{i}$ by
Galois theory. Then the claim follows. In particular, it follows from Theorem
2.5 that $\mathbf{M}(\Delta_{i}(X^{\nu}))=\log c_{i}$. ∎

###### Proof of Theorem 1.3(a).

Note that since $b_{i}(X,\mathbb{K})$ only depend on
$\mathrm{char}(\mathbb{K})$, not on the specific choice of the field
$\mathbb{K}$. So without loss of generality, we can assume that $\mathbb{K}$
is algebraically closed. The first equation follows from Proposition 2.2 and
[LL21, Theorem 1.7].

For the second equation, consider the following commutative diagram

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

where $\kappa$ is a surjective morphism to $\mathbb{Z}$. Say $\kappa$ is
represented by a tuple $(a_{1},\ldots,a_{n})$ with
$\gcd(a_{1},\ldots,a_{n})=1$. Then it is easy to see if we substitute
$(t_{1},\ldots,t_{n})$ by $(t^{a_{1}},\ldots,t^{a_{n}})$ in the matrix
$\partial_{1}$ of the chain complex (1), we get the corresponding matrix for
the chain complex of $X^{\kappa\circ\nu}$. By choosing a general tuple
$(a_{1},\ldots,a_{n})$, the rank of $\partial_{1}$ does not change after
substituting. Hence
$\Delta_{1}(X^{\kappa\circ\nu})=\Delta_{1}(X^{\nu})(t^{a_{1}},\ldots,t^{a_{n}})$
in this case. Note that $\kappa\circ\nu$ is an orbifold effective morphism of
the type $(g,r,{\mbox{\boldmath{$\mu$}}})$. By [LL21, Theorem 1.11] we know
that $\Delta_{1}(X^{\kappa\circ\nu})$ has leading coefficient
$\prod_{j=1}^{s}\mu_{j}$. It coincides with the leading coefficient of
$\Delta_{1}(X^{\nu})$. Then the claim follows from the above proposition. ∎

##  3\. The General Case

In this section, we study the case where $H$ has non-trivial torsion part. Say
$H\cong\mathbb{Z}^{n}\oplus T$, where $T$ is a finitely generated torsion
abelian group. Let $q$ be the quotient map $H\rightarrow T$ and $X^{T}$ denote
the corresponding finite cover of $X$ associated to $q\circ\nu$. Then it is
easy to see that there exists an epimorphism
$\nu^{T}\colon\pi_{1}(X^{T})\to\mathbb{Z}^{n}$ such that the following diagram
commutes

$\textstyle{\pi_{1}(X^{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{T}}$$\textstyle{\mathbb{Z}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}^{n}/\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\mathbb{Z}^{n}\oplus
T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}^{n}/\Gamma\oplus
T.}$

Due to the choice of $\Gamma$, we can identify the finite index covering
spaces associated to the two composed horizontal maps. Then we have the
following result, which reduces the computations to the free abelian case.

######  Lemma 3.1.

With the notations above, we have the following equations

(2)
$\alpha_{1}(X^{\nu},\mathbb{K})=\frac{1}{|T|}\alpha_{1}((X^{T})^{\nu^{T}},\mathbb{K})$

and

(3) $\mathbf{M}_{1}(X^{\nu})=\frac{1}{|T|}\mathbf{M}_{1}((X^{T})^{\nu^{T}}).$

We can choose $X^{T}$ to be a smooth quasi-projective variety and the
corresponding cover map $X^{T}\to X$ to be algebraic. Assume that the
epimorphism $\nu\colon\pi_{1}(X)\twoheadrightarrow H$ is orbifold effective by
some orbifold map $f\colon X\rightarrow C$ for some smooth curve $C$. By
projectivising, we get a map $h\colon\overline{X^{T}}\to\overline{C}$ for the
map $X^{T}\overset{f\circ\pi}{\longrightarrow}C$. Using Stein factorization,
we get the following commutative diagram

$\textstyle{X^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\overline{X^{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\scriptstyle{h}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime\prime}}$$\textstyle{\overline{C},}$

where $h^{\prime\prime}$ is a finite map, $h^{\prime}$ has connected fiber and
$g\coloneqq h^{\prime}|_{X^{T}}\colon X^{T}\to S\coloneqq\mathrm{im}(g)$. Then
[Dim07, Lemma 2.2] shows that $g$ has connected generic fiber. Hence $g$ is an
orbifold map and we have the following commutative diagram

$\textstyle{X^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\pi}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C,}$

where $\pi$ is the covering map and $\pi^{\prime}$ is obtained from
$h^{\prime\prime}$ by taking restrictions over $S$.

The image of the following composed map

$\pi_{1}(X^{T})\to\pi_{1}(X)\to\pi_{1}^{\mathrm{orb}}(C)\to\mathbb{Z}^{n}\oplus
T$

is $\mathbb{Z}^{n}$ (the last map exists since $f$ is orbifold effective),
hence one has the following commutative diagram

$\textstyle{\pi_{1}(X^{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}^{n}.}$

In particular, the orbifold map $g$ makes
$\pi_{1}(X^{T})\overset{\nu^{T}}{\twoheadrightarrow}\mathbb{Z}^{n}$ orbifold
effective.

Using Lemma 3.1, one can reduce the computation to the free abelian case. All
we need to know is the orbifold information of $g$. For any point $b\in C$,
let $D_{b}$ be a small enough open disc of $b$. Set $T_{b}=f^{-1}(D_{b})$.
Next we study the point $b$ in two cases.

  1. Case 1:

$b\notin B$. Then $F=f^{-1}(b)$ is the generic fiber of $f$ and $T_{b}$ is
smooth over $D_{b}$, hence admits a trivial fibaration over $D_{b}$. Dimca
proved the following short exact sequence (see [Dim07, Section 5] and [Dim17,
Section 6.3])

$H_{1}(F,\mathbb{Z})\to H_{1}(X,\mathbb{Z})\to
H_{1}(\pi^{\mathrm{orb}}_{1}(C)),$

where $H_{1}(\pi_{1}(C)^{\mathrm{orb}})$ is the abelization of
$\pi^{\mathrm{orb}}_{1}(C)$. Then the following composed group homomorphism is
trivial

$\pi_{1}(F)\to\pi_{1}(X)\twoheadrightarrow\pi^{\mathrm{orb}}_{1}(C)\twoheadrightarrow
H\twoheadrightarrow T,$

since $T$ is abelian. Hence $\pi^{-1}(F)$ has $|T|$-many disjoint copies of
$F$. Note that $g$ is an orbifold map, which has connected generic fiber.
Therefore $(\pi^{\prime})^{-1}(b)=g(\pi^{-1}(F))$ consists of $|T|$-many
points. In particular, $F$ is also a generic fiber for $g$.

  2. Case 2:

$b\in B$. Consider the following composed group homomorphism

$\pi_{1}(T_{b})\to\pi_{1}(X)\twoheadrightarrow\pi_{1}^{\mathrm{orb}}(C)\twoheadrightarrow
H\twoheadrightarrow T.$

Let $m_{b}$ denote the order of the image group of the above composed group
homomorphism. Then $\pi^{-1}(T_{b})$ has $\frac{|T|}{m_{b}}$-many connected
components. Note that $g$ is an orbifold morphism, which has connected generic
fiber. Hence $g(\pi^{-1}(T_{b}))=(\pi^{\prime})^{-1}(D_{b})$ also has
$\frac{|T|}{m_{b}}$-many connected components, i.e., ${\pi^{\prime}}^{-1}(b)$
has $\frac{|T|}{m_{b}}$ many points. Therefore we have the following
commutative diagram

$\textstyle{\pi^{-1}(T_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{b}/m_{b}}$$\scriptstyle{g}$$\textstyle{(\pi^{\prime})^{-1}(D_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{m_{b}}$$\textstyle{T_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{b}}$$\scriptstyle{f}$$\textstyle{D_{b}.}$

Let us explain the integers appearing in the diagram. $\mu_{b}$ for the bottom
horizontal map means that $f$ has multiplicity $\mu_{b}$ over $b$. Note that
for any point $a\in D_{b}\setminus\\{b\\}$, $(\pi^{\prime})^{-1}(a)$ has
$|T|$-many points and $(\pi^{\prime})^{-1}(b)$ has $\frac{|T|}{m_{b}}$-many
points. Due to the construction of $g$, $(\pi^{\prime})^{*}b$ as a divisor has
the same coefficients for every point in $(\pi^{\prime})^{-1}(b)$ (since
composing with the deck transformation over $X^{T}$ does not change $g$). So
$\pi^{\prime}$ has multiplicity $m_{b}$ over $b$. On the other hand, since
$\pi$ is a finite cover map, the left vertical map has multiplicity 1. Putting
these together, we get that $g$ has multiplicity $\frac{\mu_{b}}{m_{b}}$ over
every point in $(\pi^{\prime})^{-1}(b)$.

######  Remark 3.2.

The above proof shows that $m_{b}$ divides $\mu_{b}$. To explain this fact, we
fist recall the proof of [ACM10, Proposition 1.4]. Consider the following
commutative diagram

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

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 $b\in B$, say
$f^{*}(b)\coloneqq\mu^{1}_{b}E_{1}+\cdots+\mu^{k}_{b}E_{k}$, where
$\\{E_{1},\cdots,E_{k}\\}$ are reduced irreducible components of $f^{-1}(b)$.
Then the point $b$ has the signed multiplicity
$\mu_{b}\coloneqq\gcd(\mu^{1}_{b},\cdots,\mu_{b}^{k})$. Since $f$ maps
$\gamma_{E_{i}}$ to $(\gamma_{b})^{\mu^{i}_{b}}$, where $\gamma_{E^{i}}$ and
$\gamma_{b}$ are the meridians for $E_{i}$ and $b$, respectively, it is easy
to see that the first horizontal map in the above diagram factors through
$\pi_{1}(X)\twoheadrightarrow\pi_{1}^{\mathrm{orb}}(C)$.

By the same proof, we get a surjective map
$\pi_{1}(T_{b})\twoheadrightarrow\mathbb{Z}/\mu_{b}\mathbb{Z}$. In particular,
we get the following commutative diagram

$\textstyle{\pi_{1}(T_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/\mu_{b}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\mathrm{orb}}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H,}$

where $\mathbb{Z}/\mu_{b}\mathbb{Z}\to\pi_{1}^{\mathrm{orb}}(C)$ is given by
the natural injective map. Hence $m_{b}$ can be defined as the order of the
image group for the composition of the bottom horizontal maps. In particular,
$m_{b}$ divides $\mu_{b}$.

###### Proof of Theorem 1.3(b).

To finish the proof, we need to compute $\chi(S)$. Note that

(4) $\chi(S)-\sum_{b\in B}\frac{|T|}{m_{b}}=|T|(\chi(C)-|B|),$

hence

(5) $\chi(S)=|T|(\chi(C)+\sum_{b\in B}(\frac{1}{m_{b}}-1)).$

Using the results in Section 2 and Lemma 3.1, we get

(6) $\alpha_{1}(X^{\nu},\mathbb{K})=-\chi(C)+\sum_{b\in
B}(1-\frac{1}{m_{b}})+\sum_{b\in
B,p\mid\frac{\mu_{b}}{m_{b}}}\frac{1}{m_{b}}.$

and

(7) $\mathbf{M}_{1}(X^{\nu},\mathbb{K})=\sum_{b\in
B}\frac{1}{m_{b}}\log\frac{\mu_{b}}{m_{b}}.$

∎

######  Remark 3.3.

Let $\mathbb{K}$ be an algebraically closed filed with characteristic $p\geq
0$. If $p$ does not divide $|T|$, then $R\pi_{*}\mathbb{K}_{X^{T}}$ is a
direct sum of $|T|$-many rank one local systems and
$\mathrm{Hom}(H,\mathbb{K}^{*})$ has exactly $|T|$ many connected components,
where each of them is a copy of $\mathrm{Hom}(\mathbb{Z}^{n},\mathbb{K}^{*})$.
By [LL21, Theorem 4.8], the formula (6) should be understood as the average of
$\dim H^{1}(X,\nu^{-1}L_{\lambda})$ for $\lambda$ being general in every
connected components.

######  Example 3.4.

[DS14, Example 6.12] Fix an integer $\mu\geq 2$. Let $\mathcal{A}_{\mu}$ be
the deleted monomial arrangement, where its defining equations in
$\mathbb{CP}^{2}$ 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 $X$ admits an orbifold map $X\to\mathbb{C}^{*}$ given by

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

which is of type $(0,2,\mu)$.

Let $m$ be a positive integer, which divides $\mu$. Consider $\nu$ as the
composition of the following maps

$\pi_{1}(X)\twoheadrightarrow\mathbb{Z}*\mathbb{Z}/\mu\mathbb{Z}\twoheadrightarrow\mathbb{Z}\oplus\mathbb{Z}/m\mathbb{Z},$

where the first map is induced by the orbifold map and the second map is the
quotient map. Let $\mathbb{K}$ be an algebraically closed field with
characteristic $p\geq 0$. Then we have

$\alpha_{1}(X^{\nu},\mathbb{K})=\begin{cases}1,&\mathrm{if}\
p\mid\frac{\mu}{m},\\\ 1-\frac{1}{m},&\mathrm{otherwise}\end{cases}$

and $\mathbf{M}_{1}(X^{\nu})=\frac{1}{m}\log\frac{\mu}{m}$.

On the other hand, if $p$ does not divide $m$, then [LL21, Theorem 4.8] shows
that for general
$\lambda\in\mathrm{Hom}(\mathbb{Z}\oplus\mathbb{Z}/m\mathbb{Z},\mathbb{K}^{*})$,

$\dim H_{1}(X,\nu^{-1}L_{\lambda})=\begin{cases}0,&\mathrm{if}\
p\nmid\mu\text{ and }\lambda\text{ is general in
}\mathrm{Hom}(\mathbb{Z},\mathbb{K}^{*}),\\\
1,&\mathrm{otherwise},\end{cases}$

where its average is same as the formula given above.

## References

  * [AJN11] M. Abért, A. Jaikin-Zapirain, N. Nikolov, The rank gradient from a combinatorial viewpoint. Groups Geom. Dyn. 5 (2011), no. 2, 213-230.
  * [Ara97] D. Arapura, Geometry of cohomology support loci for local systems. I. J. Algebraic Geom. 6 (1997), no. 3, 563-597.
  * [ACM10] E. Artal Bartolo, J. I. Cogolludo-Agustín, D. Matei, On the connection between fundamental groups and pencils with multiple fibers. J. Singul. 2 (2010), 1-18.
  * [Ati76] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras. Astérisque, 32-33:43-72, 1976.
  * [BLW18] N. Budur, Y. Liu, B. Wang, The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties. Math. Ann. 371 (2018), no. 3-4, 1069–1086.
  * [BW15] N. Budur, B. Wang, Cohomology jump loci of quasi-projective varieties. Ann. Sci. Éc Norm. Supér. (4) 48 (2015), no. 1, 227-236.
  * [BW20] N. Budur, B. Wang, Absolute sets and the decomposition theorem. Ann. Sci. Éc Norm. Supér. (4) 53 (2020), no. 2, 469-536.
  * [DS14] G. Denham, A. Suciu, Multinets, parallel connections, and Milnor fibrations of arrangements. Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1435-1470.
  * [Dim07] A. Dimca, Characteristic varieties and constructible sheaves. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 18 (2007), no. 4, 365-389.
  * [Dim17] A. Dimca, Hyperplane arrangements. An introduction. Universitext. Springer, Cham, 2017. xii+200 pp. ISBN: 978-3-319-56220-9; 978-3-319-56221-6.
  * [DN04] A. Dimca, A. Némethi, Hypersurface complements, Alexander modules and monodromy. Real and complex singularities, 19–43, Contemp. Math. 354, Amer. Math. Soc., Providence, RI, 2004.
  * [DP14] A. Dimca, S. Papadima, Non-abelian cohomology jump loci from an analytic viewpoint. Commun. Contemp. Math. 16 (2014), no. 4, 1350025, 47 pp.
  * [DPS09] A. Dimca, S. Papadima, A. I. Suciu, Topology and geometry of cohomology jump loci. Duke Math. J. 148 (2009) 405-457.
  * [GL91] M. Green, R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles. J. Amer. Math. Soc. 4 (1991), no. 1, 87-103.
  * [Le14] T. Le, Homology torsion growth and Mahler measure. Comment. Math. Helv. 89 (2014), no. 3, 719-757.
  * [LL21] F. Li, Y. Liu, $L^{2}$-type invariants and cohomology jump loci for smooth complex quasi-projective varieties. arXiv:2110.03356
  * [LLS11] P. Linnell, W. Lück, R. Sauer. The limit of $\mathbb{F}_{p}$-Betti numbers of a tower of finite covers with amenable fundamental groups. Proc. Amer. Math. Soc., 139(2):421-434, 2011
  * [Luc94] W. Lück, Approximating $L^{2}$-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4 (1994), no. 4, 455-481.
  * [Luc02] W. Lück, $L^{2}$-invariants: theory and applications to geometry and $K$-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 44. Springer-Verlag, Berlin, 2002. xvi+595 pp. ISBN: 3-540-43566-2.
  * [Luc13] W. Lück, Approximating $L^{2}$-invariants and homology growth.  Geom. Funct. Anal. 23 (2013), no. 2, 622-663.
  * [Luc16] W. Lück, Approximating $L^{2}$-invariants by their classical counterparts. EMS Surv. Math. Sci. 3 (2016), no. 2, 269-344.
  * [Sch95] K. Schmidt, Dynamical systems of algebraic origin. Progr. Math. 128, Birkhäuser Verlag, Basel 1995.
  * [Sim93] C. Simpson, Subspaces of moduli spaces of rank one local systems. Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 361-401.
  * [Suc11] A. I. Suciu, Fundamental groups, Alexander invariants, and cohomology jumping loci. (English summary) Topology of algebraic varieties and singularities. 179-223, Contemp. Math., 538, Amer. Math. Soc., Providence, RI, 2011.