# An Upper Bound for the Depth of Some Classes of Polyhedra

Mojtaba Mohareri<EMAIL_ADDRESS>Department of Pure Mathematics,
Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University
of Mashhad, P.O.Box 1159-91775, Mashhad, Iran.  Behrooz Mashayekhy
<EMAIL_ADDRESS>Department of Pure Mathematics, Center of Excellence in
Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box
1159-91775, Mashhad, Iran.  Hanieh Mirebrahimi∗00footnotetext: ∗ Corresponding
author<EMAIL_ADDRESS>Department of Pure Mathematics, Center of
Excellence in Analysis on Algebraic Structures, Ferdowsi University of
Mashhad, P.O.Box 1159-91775, Mashhad, Iran.

(Month Day, Year; Month Day, Year)

###### Abstract

K. Borsuk in the seventies introduced the notions of capacity and depth of
compacta together with some relevant problems. In this paper, first, we
introduce the concepts of the (strong) capacity and the (strong) depth of an
object in an arbitrary category. Then in the category of groups, we compute
the (strong) capacity and the (strong) depth of some well-known groups.
Finally, we find an upper bound for the depth of some classes of finite
polyhedra which generalizes a result of D. Kolodziejczyk in this subject.

###### :

55P15, 55P55, 55P20,54E30, 55Q20.

###### keywords:

Homotopy domination, Homotopy type, Polyhedron, CW-complex, Compactum..

Month Day, Year 1 1

##  1 Introduction and Motivation

Throughout this paper, each polyhedron and CW-complex is assumed to be finite
and connected. Also, by a map between two CW-complexes we mean here a cellular
one.

K. Borsuk in [3] introduced concepts of the capacity and the depth of a
compactum together with some relevant problems. Indeed, the capacity $C(A)$ of
a compactum $A$ is the cardinality of the class of the shapes of all compacta
$X$ such that $Sh(X)\leqslant Sh(A)$, and the depth $D(A)$ of a compactum $A$
is the least upper bound of the lengths of all chains
$\mathcal{S}h(X_{k})<\cdots<\mathcal{S}h(X_{2})<\mathcal{S}h(X_{1})\leqslant\mathcal{S}h(A)$,
where $\mathcal{S}h(X)<\mathcal{S}h(Y)$ denotes that
$\mathcal{S}h(X)\leqslant\mathcal{S}h(Y)$ holds but
$\mathcal{S}h(Y)\leqslant\mathcal{S}h(X)$ fails. If this upper bound is
infinite, we write $D(A)=\mathcal{N}_{0}$ (for more details, see [21]). It is
clear that $D(A)\leq C(A)$ for each compactum $A$.

For polyhedra, the notions shape and shape domination in the above definition
can be replaced by the notions homotopy type and homotopy domination,
respectively. Consequently, the capacity and depth in shape category of
compacta are the same as in the homotopy category of CW-complexes and homotopy
classes of cellular maps between them (see [15]). S. Mather in [23] proved
that every polyhedron dominates only countably many different homotopy types
(see also [8]).

Note that there exist polyhedra with the same capacities but different depths.
For instance, one can consider $P_{1}=\mathbb{T}\\#\mathbb{T}$ and
$P_{2}=\mathbb{S}^{1}\times\mathbb{S}^{2}$, where $\\#$ denotes the connected
sum operation. One can observe that $C(P_{1})=C(P_{2})=4$ but $4=D(P_{1})\neq
D(P_{2})=3$ (see [1]).

The answers to most of problems of [3] are given by D. Kolodziejczyk in [12].
In [18], answering the question (4) of [3]: “Is it true that the capacity of
every finite polyhedron is finite? ”, Kolodziejczyk showed that there is a
polyhedron dominating infinitely many different homotopy types of polyhedra.
Kolodziejczyk in [14], gave an affirmative answer to question (8) of [3]: “Is
there a compactum with infinite capacity and finite depth?”. Indeed, by a
result of Kolodziejczyk, for every non-abelian poly-$\mathbb{Z}$-group $G$ and
an integer $n\geq 3$, there exists a polyhedron $P$ such that $\pi_{1}(P)\cong
G$, $\dim P=n$, $C(P)$ is infinite, but $D(P)$ is finite. Accordingly, the
examples of polyhedra with infinite capacity are even frequent. In [17],
Kolodziejczyk also proved that every polyhedron with virtually polycyclic
fundamental group has finite depth.

Moreover, it is investigated the finiteness conditions for capacities of some
polyhedra by Kolodziejczyk (see [16, 15]). For instance, a polyhedron $Q$ with
finite fundamental group $\pi_{1}(Q)$, and a polyhedron $P$ with abelian
fundamental group $\pi_{1}(P)$ and finitely generated homology groups
$H_{i}(\tilde{P})$, for $i\geq 2$, have finite capacities (hence finite
depths), where $\tilde{P}$ denotes the universal covering space of $P$.

Note that the capacity of a polyhedron is a homotopy invariant, i.e., for
polyhedra $X$ and $Y$ with the same homotopy type, $C(X)=C(Y)$. Hence it seems
interesting to find polyhedra with finite capacities and compute the
capacities of some of their well-known spaces.

Note that every compactum shape dominated by some polyhedron $P$ has the shape
of some $P$-like compactum (for the definition of $P$-like, see [20]). This is
a corollary of some more general results of Holsztynski, Mardesic and Segal
(see, for example, [21, Ch.II.9, Lemma 4 on p. 222] or [11, Lemma 7]).
Therefore, the capacities of some polyhedra $P$ may be easily obtained from
the classifications of P-like compacta. Such classifications are known for the
$n$-dimensional torus, $\mathbb{S}^{n}$, $\mathbb{CP}^{n}$, $\mathbb{RP}^{n}$
([4], [10], [22], [29], [6]) and for a few other spaces. In the paper [29] one
can find shape classifications of $P$-like compacta for finite wedges of
spheres $\mathbb{S}^{n}$ ($n\geq 2$). Two compacta of this kind,
$X=\bigvee_{i}\mathbb{S}^{n}$ and $Y=\bigvee_{j}\mathbb{S}^{n}$, have the same
shape if and only if $\check{H}^{n}(X;\mathbb{Z})=\check{H}^{n}(Y;\mathbb{Z}$)
(Cech cohomology groups) [29, Theorem 2]. As a consequence, the capacities of
finite wedges of spheres $\mathbb{S}^{n}$ ($n\geq 2$) can be easily computed.
It is also easily seen that the capacity of the wedge of $k$ copies of
$\mathbb{S}^{1}$ is $k+1$.

M. Abbasi and B. Mashayekhy in [1] computed the capacities of compact
2-dimensional manifolds. They showed that the capacities (and also the depths)
of a compact orientable surface of genus $g\geq 0$ and a compact non-
orientable surface of genus $g>0$ are equal to $g+2$ and $[\frac{g}{2}]+2$,
respectively. The authors, in [25] computed the capacities of finite wedges of
spheres of various dimensions. Indeed, we showed that the capacity (and also
the depth) of $\bigvee_{n\in I}(\vee_{i_{n}}\mathbb{S}^{n})$ is equal to
$\prod_{n\in I}(i_{n}+1)$, where $\vee_{i_{n}}\mathbb{S}^{n}$ denotes the
wedge of $i_{n}$ copies of $\mathbb{S}^{n}$, $I$ is a finite subset of
$\mathbb{N}$ and $i_{n}\in\mathbb{N}$. Also, in [24], we computed the capacity
of the product of two spheres of the same or different dimensions and the
capacities of lens spaces which are a class of closed orientable 3-manifolds.

In Section 2, we define concepts of the (strong) capacity and the (strong)
depth for an $A\in Obj\;(\mathcal{C})$, where $\mathcal{C}$ denotes an
arbitrary category. Also we prove some facts concerning relationships between
concepts of the capacity and the depth in both ordinary and strong cases. In
Section 3, in the category of groups, we compute the (strong) capacities and
(strong) depths of two well-known groups: the free groups of finite rank and
the finitely generated abelian groups. Also, we show that every virtually
polycyclic group has finite depth. Finally, in Section 4, we generalize a
theorem of Kolodziejczyk (see [17, Theorem 2]) which states that every
polyhedron with virtually polycyclic fundamental group has finite depth. Also,
we present an upper bound for the depth of some classes of polyhedra.

##  2 Strong capacity and strong depth in a category

In what follows, $\mathcal{C}$ denotes an arbitrary category.

Recall that a domination in a given category $\mathcal{C}$ is a morphism
$g:Y\longrightarrow X$, $X,Y\in Obj\mathcal{C}$, for which there exists a
morphism $f:X\longrightarrow Y$ of $\mathcal{C}$ such that $g\circ f=id_{X}$.
In this case, we say that $X$ is dominated by $Y$ and we write
$X\leqslant^{d}Y$ (rather than the notation “$\leqslant$”, see for example
[17]). Also, $X$ is called strongly dominated by $Y$ and is denoted by
$X<^{s}Y$ if $X\leqslant^{d}Y$ holds but $Y\leqslant^{d}X$ fails. Moreover, we
say that $X$ is properly dominated by $Y$ and we denote it by $X<^{p}Y$ if
$X\leqslant^{d}Y$ but $X\cong Y$ fails.

Clearly, if $X,A\in Obj\;\mathcal{C}$ and $X<^{s}A$, then $X<^{p}A$ but the
converse does not hold in general. For example, in the category of groups,
there exists a countable torsion-free abelian group $A$ such that $A\cong
A\oplus A\oplus A$ but $A\not\cong A\oplus A$ (see [5]). It follows that
$A\leqslant^{d}A\oplus A$ and $A\oplus A\leqslant^{d}A$ but $A\not\cong
A\oplus A$. Therefore, $A<^{p}A\oplus A$ but $A\not<^{s}A\oplus A$. Also, in
the homotopy category of CW-complexes and homotopy classes of cellular maps
between them, there exist two CW-complexes $X$ and $Y$ for which
$X\leqslant^{d}Y$ and $Y\leqslant^{d}X$ but $X\not\simeq Y$ (see [26]).
Accordingly, $X<^{p}Y$ but $X\not<^{s}Y$.

######  Definition 2.1.

The capacity $C(A)$ of an $A\in Obj\;\mathcal{C}$ is the cardinality of the
class of isomorphism classes of all the $X\in Obj\;\mathcal{C}$ such that
$X\leqslant^{d}A$. Also, the strong capacity $SC(A)$ of an $A\in
Obj\;\mathcal{C}$ is the cardinality of the class of isomorphism classes of
all $X\in Obj\;\mathcal{C}$ such that $X<^{s}A$. If there is no such $X$, we
write $SC(A)=0$.

The previous definition of the capacity of a compatum in the shape category of
compacta coincides with Borsuk’s definiton of the capacity (see [3]).

######  Definition 2.2.

  1. 1.

A chain $X_{k}<^{p}\cdots<^{p}X_{1}\leqslant^{d}A$, where $X_{i}\in
Obj\;\mathcal{C}$ for $i=1,\cdots,k$, is called a chain of length $k$ for
$A\in Obj\;\mathcal{C}$.

  2. 2.

The depth $D(A)$ of an $A\in Obj\;\mathcal{C}$ is the least upper bound of the
lengths of all chains for $A$. If this upper bound is infinite, we write
$D(A)=\mathcal{N}_{0}$.

  3. 3.

A chain $X_{k}<^{s}\cdots<^{s}X_{1}\leqslant^{d}A$, where $X_{i}\in
Obj\;\mathcal{C}$ for $i=1,\cdots,k$, is called an $s$-chain of length $k$ for
$A\in Obj\;\mathcal{C}$.

  4. 4.

The strong depth $SD(A)$ of $A$ is the least upper bound of the lengths of all
$s$-chains for $A$. If this upper bound is infinite, we write
$SD(A)=\mathcal{N}_{0}$.

Note that our definition of strong depth of a compactum in the shape category
of compacta coincides with Borsuk’s definition of the depth of a compactum
(for more details, see [3]).

It is easy to see that $A\in Obj\;\mathcal{C}$ has finite (strong) depth $k$
if and only if there exists (an) a $(s-)$chain of length $k$ for $A$ but there
is no $(s-)$chain of length $k+1$, equivalently, there exists (an) a
$(s-)$chain of length $k$ for $A$ and for every $(s-)$chain

$X_{k}<^{p}\cdots<^{p}X_{1}\leqslant^{d}X_{0}=A\quad(X_{k}<^{s}\cdots<^{s}X_{1}\leqslant^{d}X_{0}=A)$

of length of $k$ and for every $X_{k+1}\leqslant^{d}X_{k}$, we have
$X_{k+1}\cong X_{k}$ ($X_{k}\leqslant^{d}X_{k+1}$).

It is clear that $SC(A)\leq C(A)-1$, for every $A\in Obj\;\mathcal{C}$.

######  Definition 2.3.

Two objects $X$ and $Y$ in $\mathcal{C}$ are called $d$-equal and denoted by
$X=_{d}Y$ if $X\leqslant^{d}Y$ and $Y\leqslant^{d}X$.

######  Definition 2.4.

We say that $A\in Obj\;\mathcal{C}$ has domination equality ($d$-equlaity for
short) property if for every $X\in Obj\;\mathcal{C}$ with $X=_{d}A$, we have
$X\cong A$.

By [13, Theorem 3], if $X\in ANR$ and $\pi_{1}(X)$ is virtually polycyclic
fundamental group, then $X$ has $d$-equality property (also see [19]). Also,
every Hopfian group has $d$-equality property (see Prposition 3.3).

######  Lemma 2.5.

Suppose that $A\in Obj\;\mathcal{C}$ has $d$-equality property. If $X\in
Obj\;\mathcal{C}$ is properly dominated by $A$, then $X$ is strongly dominated
by $A$.

######  Proof 2.6.

By contrary, assume that $X$ is not strongly dominated by $A$, i.e.,
$X\leqslant^{d}A$ and $A\leqslant^{d}X$. Since $A$ has $d$-equality property,
we have $A\cong X$. But this is a contradiction to $X<^{p}A$.

######  Proposition 2.7.

If $A\in Obj\;\mathcal{C}$ has $d$-equality property, then $SC(A)=C(A)-1$.

######  Proof 2.8.

It follows from Lemma 2.5.

######  Lemma 2.9.

If $Y\leqslant^{d}X$ and $X<^{s}A$ for $A,X,Y\in Obj\;\mathcal{C}$, then
$Y<^{s}A$.

######  Proof 2.10.

Since $Y\leqslant^{d}X$ and $X\leqslant^{d}A$, then $Y\leqslant^{d}A$. By
contrary, assume that $A\leqslant^{d}Y$. Then $A\leqslant^{d}X$ which is a
contradiction to $X<^{s}A$. Thus we have $Y<^{s}A$.

######  Remark 2.11.

From the previous lemma, one can conclude that the relation “$<^{s}$” is a
transitive relation. Clearly, the relation “$\leqslant^{d}$” is also
transitive. But the relation “$<^{p}$” is not necessarily transitive. Indeed,
as mentioned before, there exist two CW-complexes $X$ and $Y$ for which
$X\leqslant^{d}Y$ and $Y\leqslant^{d}X$ but $X\not\simeq Y$. It follows that
$X<^{p}Y$ and $Y<^{p}X$ but $X\simeq X$.

######  Lemma 2.12.

Suppose that $A\in Obj\;\mathcal{C}$. If
$X_{k}<^{s}\cdots<^{s}X_{1}\leqslant^{d}X_{0}=A$ is an $s$-chain of length $k$
for $A$, then $X_{i}\not\cong X_{j}$ for all $1\leq i\neq j\leq k$.

######  Proof 2.13.

By Lemma 2.9, $X_{k}<^{s}X_{i}$ for all $1\leq i\leq k-1$. Then by definition
we have $X_{k}\not\cong X_{i}$ for all $1\leq i\leq k-1$. Similarly, one can
show that for each $k-1\leq i\leq 2$, we have $X_{i}\not\cong X_{j}$ for all
$i+1\leq j\leq 1$. Thus the proof is finished.

######  Proposition 2.14.

For each $A\in Obj\;\mathcal{C}$, $SD(A)\leq\min\\{D(A),C(A)\\}$.

######  Proof 2.15.

By definition, it is clear that $SD(A)\leq D(A)$. Suppose that
$C(A)=k<\infty$. By contrary, assume that there is the following $s$-chain for
$A$

$X_{k+1}<^{s}X_{k}<^{s}\cdots<^{s}X_{1}\leqslant^{d}X_{0}=A.$

Since the relation “$\leqslant^{d}$” is transitive, so $X_{i}\leqslant^{d}A$
for all $i=1,\cdots,k+1$. By Lemma 2.12, $X_{i}$’s are mutually non-isomorphic
for all $i=1,\cdots,k+1$. It follows that $C(A)\geq k+1$ which is a
contradiction to the hypothesis. Then there is no $s$-chain of length $k+1$
for $A$. This proves that $SD(A)\leq k$.

######  Proposition 2.16.

Suppose that $A\in Obj\;\mathcal{C}$. If $D(A)=k$, then each sequence
$\cdots\leqslant^{d}X_{i}\leqslant^{d}\cdots\leqslant^{d}X_{1}\leqslant^{d}A$
contains at most $k$ different objects (up to isomorphism).

######  Proof 2.17.

By contrary, assume that there exists a sequence
$\cdots\leqslant^{d}X_{i}\leqslant^{d}\cdots\leqslant^{d}X_{1}\leqslant^{d}A$
contains at least $k+1$ different objects (up to isomorphism), called
$X_{i_{k+1}},X_{i_{k}},\cdots,X_{i_{1}}$ with $i_{k+1}<i_{k}<\cdots<i_{1}$. It
follows that the sequence
$X_{i_{k+1}}<^{p}X_{i_{k}}<^{p}\cdots<^{p}X_{i_{1}}\leqslant^{d}A$ is a chain
of length $k+1$ for $A$ which is a contradiction to $D(A)=k$.

######  Proposition 2.18.

Suppose that $A\in Obj\;\mathcal{C}$. If there exists an integer $k$ such that
each sequence
$\cdots\leqslant^{d}X_{i}\leqslant^{d}\cdots\leqslant^{d}X_{1}\leqslant^{d}A$
contains at most $k$ different objects (up to isomorphism), then $SD(A)\leq
k$.

######  Proof 2.19.

By contrary, assume that there exists the following $s$-chain for $A$

$X_{k+1}<^{s}\cdots<^{s}X_{1}\leqslant^{d}A.$

By Lemma 2.12, we have $X_{i}\not\cong X_{j}$ for all $1\leq i\neq j\leq k+1$.
Then the sequence

$\cdots\leqslant^{d}X_{i}\leqslant^{d}\cdots\leqslant^{d}X_{1}\leqslant^{d}A$

with $X_{i}=X_{k+1}$ for all $i\geq k+1$, contains at least $k+1$ different
objects (up to isomorphism) which is a contradiction to the hypothesis.
Therefore, $SD(A)\leq k$.

######  Lemma 2.20.

Suppose that $A\in Obj\;\mathcal{C}$. If every $X\leqslant^{d}A$ has
$d$-equality property, then every chain for $A$ is an $s$-chain.

######  Proof 2.21.

Asume that $X_{k}<^{p}\cdots<^{p}X_{1}\leqslant^{d}A$ is a chain of length $k$
for $A$. Assume that $X_{i-1}\leq X_{i}$ for some $2\leq i\leq k$. Since
$X_{i}$ has $d$-equality property by the hypothesis, then $X_{i}\cong X_{i-1}$
which is a contradiction to $X_{i}<^{p}X_{i-1}$. So $X_{i}<^{s}X_{i-1}$ for
every $2\leq i\leq k+1$. Hence

$X_{k+1}<^{s}X_{k}<^{s}\cdots<^{s}X_{1}\leqslant^{d}A$

is an $s$-chain of length $k$ for $A$.

Note that by [13, Theorem 3], if $A\in ANR$ and $\pi_{1}(A)$ is virtually
polycyclic, then every $X\leqslant^{d}A$ has $d$-equality property.

######  Proposition 2.22.

Suppose that $A\in Obj\;\mathcal{C}$. If every $X\leqslant^{d}A$ has
$d$-equality property, then $D(A)=SD(A)$.

######  Proof 2.23.

It follows from Lemma 2.20.

######  Lemma 2.24.

Suppose that $A\in Obj\;\mathcal{C}$. If $D(A)<\infty$, then every
$X\leqslant^{d}A$ has $d$-equality property.

######  Proof 2.25.

Suppose that $D(A)=k$. First, we show that $A$ itself has $d$-equality
property. By contrary, assume that there exists $X\in Ob\;\mathcal{C}$ for
which $X\leqslant^{d}A$ and $A\leqslant^{d}X$ but $A\not\cong X$. Then the
sequence

$\underbrace{A<^{p}X<^{p}A<^{p}\cdots<^{p}X<^{p}A}_{\text{of length}\;k+1},$

is a chain of length $k+1$ for $A$ which is a contradiction to $D(A)=k$.

Now, we know that

$``Y\leqslant^{d}X\quad\text{implies}\quad D(Y)\leq D(X)\text{''}$

and so the result holds.

######  Corollary 2.26.

Suppose that $A\in Obj\;\mathcal{C}$. If $D(A)<\infty$, then $D(A)=SD(A)$.

######  Proof 2.27.

It is concluded from Lemma 2.24 and Proposition 2.22.

##  3 The (strong) capacities and (strong) depths of some classes of groups

Recall that a homomorphism $g:G\longrightarrow H$ of groups is an
$r$-homomorphism if there exists a (converse) homomorphism $f:H\longrightarrow
G$ such that $g\circ f=id_{H}$. Then $H$ is called an $r$-image of $G$. Note
that in the categroy of groups, concepts of the domination and the
$r$-homomorphism are the same. So the group $H$ is an $r$-image of the group
$G$ if and only if $H\leqslant^{d}G$. Also, if $H$ is (strognly) properly
dominated by $G$, then we say that $H$ is an ($s$-image) proper $r$-image of
$G$.

Recall that an endomorphism $h:G\longrightarrow G$ of a group $G$ is called an
idempotent if $h\circ h=h$. It is clear that if $H$ is an $r$-image of $G$
with $r$-homomorphism $g:G\longrightarrow H$ and converse homomorphism
$f:H\longrightarrow G$, then $f\circ g:G\longrightarrow G$ is an idempotent.
By [8, Corollary], $C(G)\leq e(G)$, where $e(G)$ denotes the number of
idempotent endomorphisms on $G$.

######  Proposition 3.1.

Let $G$ be a finite group of order $n$. Then $C(G)\leq
n^{(\frac{1}{2}+o(1))\log_{2}(n)}$.

######  Proof 3.2.

Clearly, $e(G)$ is equal to the number of pairs of subgroups $H,K$ such that
$G$ is the semidirect product of $H$ and $K$. By [2, Corollary 1.6], the
number of subgroups of $G$ is bounded by $n^{(\frac{1}{4}+o(1))\log_{2}(n)}$.
Squaring gives an upper bound on the total number of pairs $(H,K)$ and hence
on $e(G)$ as follows:

$e(G)\leq n^{(\frac{1}{2}+o(1))\log_{2}(n)}.$

By the fact that $C(G)\leq e(G)$, the proof is finished.

Recall that a group $G$ is Hopfian if every epimorphism $f:G\longrightarrow G$
is an automorphism (equivalently, $N=1$ is the only normal subgroup for which
$G/N\cong G$).

######  Proposition 3.3.

Every Hopfian group has $d$-equality property.

######  Proof 3.4.

Assume that $G$ is a Hopfian group. Also, assume that for a group $H$, we have
$G\leqslant^{r}H$ and $H\leqslant^{r}G$. Then there are homomorphisms
$f_{1}:H\longrightarrow G$ and $g_{1}:G\longrightarrow H$ with $g_{1}\circ
f_{1}=id_{H}$ and homomorphisms $f_{2}:G\longrightarrow H$ and
$g_{2}:H\longrightarrow G$ with $g_{2}\circ f_{2}=id_{G}$. It follows that
$g_{2}\circ g_{1}:G\longrightarrow G$ is an epimorphism and so an isomorphism
($G$ is Hopfian). Now one can conclude that $g_{1}$ is a monomorphism and so
an isomorphism, hence $G\cong H$.

######  Corollary 3.5.

For a Hopfian group $G$, $SC(G)=C(G)-1$.

######  Proof 3.6.

It follows from Propositions 2.7 and 3.3.

Now, we compute the (strong) capacities and (strong) depths of free groups of
finite ranks and finitely generated abelian groups.

######  Lemma 3.7.

Let $F$ be a free group of finite rank. If $H$ is an $r$-image (a proper
$r$-image) of $F$, then $H$ is a free group with $rank(H)\leq rank(F)$
$(rank(H)<rank(F))$.

######  Proof 3.8.

Put $rank(F)=k$. Assume that $H$ is an $r$-image of $F$ with $r$-homomorphism
$g:F\longrightarrow H$ and converse homomorphism $f:H\longrightarrow F$.
Define homomorphisms
$\bar{g}:\frac{F}{F^{\prime}}\longrightarrow\frac{H}{H^{\prime}}$ and
$\bar{f}:\frac{H}{H^{\prime}}\longrightarrow\frac{F}{F^{\prime}}$ by
$\bar{g}(xF^{\prime})=g(x)H^{\prime}$ and
$\bar{f}(xH^{\prime})=f(x)F^{\prime}$, respectively. Since $g\circ f=id_{H}$,
then $\bar{g}\circ\bar{f}=id_{\frac{H}{H^{\prime}}}$. It follows that
$\frac{H}{H^{\prime}}$ is isomorphic to a subgroup of the free abelian group
$\frac{F}{F^{\prime}}$. But we know that the ranks of free groups $F$ and $H$
are equal to the ranks of free abelian groups $\frac{F}{F^{\prime}}$ and
$\frac{H}{H^{\prime}}$, respectively. On the other hand,
$rank(\frac{H}{H^{\prime}})$ is less than or equal to
$rank(\frac{F}{F^{\prime}})=k$. Therefore, $rank\;(H)\leq k$.

Now if $H$ is a proper $r$-image of $F$, then $H\not\cong F$ by definition. So
it follows that $rank(H)<k$.

######  Proposition 3.9.

Let $F$ be a free group of the rank $k$. Then the following statements
satisfy:

  1. 1.

The strong capacity and the capacity of $F$ are equal to $k$ and $k+1$,
respectively;

  2. 2.

The strong depth and the depth of $F$ are equal to $k+1$.

######  Proof 3.10.

(1) By Lemma 3.7, each $r$-image of $F$ is a free group of rank less than or
equal to $k$. Also, it is easy to show that each free group of rank $i$ is an
$r$-image of $F$ for all $0\leq i\leq k$. Therefore, we conclude that
$C(F)=k+1$. But $F$ is a Hopfian group which implies that $SC(F)=k$, by
Corollary 3.5.

(2) First, we show that $SD(F)=k+1$. By part $(i)$ and Proposition 2.14, we
have $SD(F)\leq C(F)=k+1$. Assume that $\\{x_{1},\cdots,x_{k}\\}$ is a
generator set of $F=F_{0}$. Then it is not hard to show that the free group
$F_{i}$ generated by $\\{x_{1},\cdots,x_{k-i+1}\\}$ is an $r$-image of the
free group $F_{i-1}$ generated by $\\{x_{1},\cdots,x_{k-i+2}\\}$, for all
$2\leq i\leq k+1$. Note that if $F_{i_{0}}\leqslant^{d}F_{i_{0}+1}$ for some
$i_{0}\in\\{1,\cdots,k-1\\}$, then by Proposition 3.3, we have $F_{i_{0}}\cong
F_{i_{0}+1}$ which is a contradiction. Hence, one can obtain the following
$s$-chain of length $k+1$ for $F$

$1=F_{k+1}<^{s}F_{k}<^{s}\cdots<^{s}F_{2}<^{s}F_{1}\leqslant^{d}F_{0}=F.$

Thus one can conclude that $SD(F)=k+1$.

Now we show that $D(F)=k+1$. Assume that there exists the following chain of
length $k+2$ for $F$

$H_{k+2}<^{p}H_{k+1}<^{p}\cdots<^{p}H_{2}<^{p}H_{1}\leqslant^{d}H_{0}=F.$

By Proposition 3.7, $H_{1}$ is a free group with $rank(H_{1})\leq rank(F)=k$.
Again by Proposition 3.7, $H_{2}$ is a free group with
$rank(H_{2})<rank(H_{1})$. By a similar argument, $H_{i}$ is a free group with
$rank(H_{i})<rank(H_{i-1})$ for all $2\leq i\leq k+2$. Then we have

$r_{k+2}<\cdots<r_{2}<r_{1}\leq k,$

where $r_{i}=rank(H_{i})$ for all $1\leq i\leq k+2$.

It is clear that $rank(H_{k+1})=0$. Thus we have $H_{k+2}=H_{k+1}=1$ which is
a contradiction to $H_{k+2}<^{p}H_{k+1}$. It follows that there is no chain of
length $k+2$, and so $D(F)\leq k+1$. On the other hand, $k+1=SD(F)\leq D(F)$
which implies that $D(F)=k+1$. Thus the proof is complete.

######  Lemma 3.11.

Let $G$ be a finitely generated abelian group of the form

$\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(k_{1})}\oplus\mathbb{Z}_{p_{2}^{\alpha_{2}}}^{(k_{2})}\oplus\cdots\oplus\mathbb{Z}_{p_{n}^{\alpha_{n}}}^{(k_{n})},$

where for $i\neq j$, $p_{i}^{\alpha_{i}}\neq p_{j}^{\alpha_{j}}$, $p_{i}$’s
are prime numbers, $\alpha_{i}$’s are non-negative integers,
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$ is the direct sum of $k_{i}$
copies of $\mathbb{Z}_{p_{i}^{\alpha_{i}}}$, and $\mathbb{Z}_{1}=\mathbb{Z}$.
Then the number of direct summands of $G$, up to isomorphism, is equal to

$(k_{1}+1)\times\cdots\times(k_{n}+1).$

######  Proof 3.12.

The proof is in three steps.

Step One. For each $1\leq i\leq n$, the number of direct summands of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$, up to isomorphism, is equal to
$k_{i}+1$.

For this, it is obviuos that for every $0\leq t\leq k_{i}$,
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(t)}$ is a direct summand of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$ and for each $0\leq t\neq
t^{\prime}\leq k_{i}$, we have
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(t)}\not\cong\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(t^{\prime})}$.
Now, suppose that $C$ is a direct summand of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$. There exists a subgroup $D$ of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$ such that
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\cong C\oplus D$. By [9, Corollary
2.1.7], $C$ is a finitely generated abelian group. Suppose that
$C\cong\mathbb{Z}_{q_{1}^{\beta_{1}}}^{(l_{1})}\oplus\cdots\oplus\mathbb{Z}_{q_{s}^{\beta_{s}}}^{(l_{s})}$.
Since $C$ is a direct summand of $\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$
and for every $1\leq j\leq s$, $\mathbb{Z}_{q_{j}^{\beta_{j}}}^{(l_{j})}$ is a
direct summand of $C$, so for every $1\leq j\leq s$,
$\mathbb{Z}_{q_{j}^{\beta_{j}}}^{(l_{j})}$ is a direct summand of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$. Now, by uniqueness of
decomposition of finitely generated abelian groups [9, Theorem 2.2.6, (iii)],
for any $j=1,\cdots,s$, there exists an $i=1,\cdots,n$ such that $q_{j}=p_{i}$
and $\beta_{j}=\alpha_{i}$. Hence,
$C\cong\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(t)}$ for some $0\leq t\leq k_{i}$.

Step Two. The number of direct summands of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(k_{j})}$
for $i\neq j$, up to isomorphism, is equal to $(k_{i}+1)(k_{j}+1)$.

It is easy to see that for every $0\leq t\leq k_{i}$ and $0\leq s\leq k_{j}$,
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(t)}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(s)}$
is a direct summand of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(k_{j})}$.
Now similar to Step One, suppose that $C$ is a direct summand of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(k_{j})}$
and $D$ is a subgroup of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(k_{j})}$
such that
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(k_{j})}\cong
C\oplus D$. Suppose
$C\cong\mathbb{Z}_{q_{1}^{\beta_{1}}}^{(l_{1})}\oplus\cdots\oplus\mathbb{Z}_{q_{s}^{\beta_{s}}}^{(l_{s})}$.
Since for every $1\leq m\leq s$, $\mathbb{Z}_{q_{m}^{\beta_{m}}}^{(l_{m})}$ is
a direct summand of
$\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(k_{j})}$,
so similar to the above argument,
$C\cong\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(t)}\oplus\mathbb{Z}_{p_{j}^{\alpha_{j}}}^{(s)}$
for some $0\leq t\leq k_{i}$ and $0\leq s\leq k_{j}$.

Step Three: the number of direct summands of
$\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(k_{1})}\oplus\mathbb{Z}_{p_{2}^{\alpha_{2}}}^{(k_{2})}\oplus\cdots\oplus\mathbb{Z}_{p_{n}^{\alpha_{n}}}^{(k_{n})}$,
up to isomorphism, is equal to $(k_{1}+1)(k_{2}+1)\cdots(k_{n}+1)$ which is
obtained by induction from Step Two.

######  Proposition 3.13.

Let $G$ be a finitely generated abelian group of the form

$\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(k_{1})}\oplus\mathbb{Z}_{p_{2}^{\alpha_{2}}}^{(k_{2})}\oplus\cdots\oplus\mathbb{Z}_{p_{n}^{\alpha_{n}}}^{(k_{n})},$

up to isomorphism, where for $i\neq j$, $p_{i}^{\alpha_{i}}\neq
p_{j}^{\alpha_{j}}$, $p_{i}$’s are prime numbers, $\alpha_{i}$’s are non-
negative integers, $\mathbb{Z}_{p_{i}^{\alpha_{i}}}^{(k_{i})}$ is the direct
sum of $k_{i}$ copies of $\mathbb{Z}_{p_{i}^{\alpha_{i}}}$, and
$\mathbb{Z}_{1}=\mathbb{Z}$. Then the following statements satisfy:

  1. 1.

The strong capacity and the capacity of $G$ are equal to
$\big{(}(k_{1}+1)\times\cdots\times(k_{n}+1)\big{)}-1$ and
$(k_{1}+1)\times\cdots\times(k_{n}+1)$, respectively.

  2. 2.

The strong depth and the depth of $G$ are equal to
$\big{(}\sum_{i=1}^{m}k_{i}\big{)}+1$.

######  Proof 3.14.

(1) It is easy to show that for every arbitrary abelian group $A$, there
exists a one-to-one corresponding between the class of isomorphism classes of
all $r$-images of $A$ and the class of isomorphism classes of all direct
summands of $A$. Accordingly, the capacity of $G$ is equal to
$(k_{1}+1)\times\cdots\times(k_{n}+1)$ by Lemma 3.11. Now, by the fact that
every finitely generated abelian group is Hopfian and by Corollary 3.5, we
have $SC(G)=\big{(}(k_{1}+1)\times\cdots\times(k_{n}+1)\big{)}-1$.

(2) First, we prove that $D(G)=\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$. Suppose
that $H_{1}$ is an arbitrary $r$-image of $H_{0}=G$ with $r$-homomorphism
$r_{0}:H_{0}\longrightarrow H_{1}$. Since $H_{1}$ is an $r$-image of abelian
group $G$, then it is isomorphic to a direct summand of $G$. One can easily
see that
$H_{1}\cong\mathbb{Z}_{p_{i_{1}}^{\alpha_{i_{1}}}}^{(l_{i_{1}})}\oplus\cdots\mathbb{Z}_{p_{i_{m}}^{\alpha_{i_{m}}}}^{(l_{i_{m}})}$
for $\\{i_{1},\cdots,i_{m}\\}\subseteq\\{1,\cdots,n\\}$, where $l_{i_{j}}\leq
k_{i_{j}}$ for all $j$. Suppose that $H_{2}$ is a proper $r$-image of $H_{1}$.
Since $H_{2}\not\cong H_{1}$, one can conclude that
$H_{2}\cong\mathbb{Z}_{p_{j_{1}}^{\alpha_{j_{1}}}}^{(s_{j_{1}})}\oplus\cdots\mathbb{Z}_{p_{j_{t}}^{\alpha_{j_{t}}}}^{(s_{j_{t}})}$
for $\\{j_{1},\cdots,j_{t}\\}\subsetneq\\{i_{1},\cdots,i_{m}\\}$, where
$s_{j_{r}}\leq l_{j_{r}}$ for all $r$. Without loss of generality, we can
suppose that

$H_{2}\cong\mathbb{Z}_{p_{i_{1}}^{\alpha_{i_{1}}}}^{(l_{i_{1}})}\oplus\cdots\mathbb{Z}_{p_{i_{m}}^{\alpha_{i_{m}}}}^{(l_{i_{m}}-1)}.$

Now if

$H_{l}<^{p}\cdots<^{p}H_{2}<^{p}H_{1}\leqslant^{d}H_{0}=G$

is an arbitrary $r$-chain for $G$ with $l=\sum_{j=1}^{m}l_{i_{j}}$, then with
a similar argument and without loss of generality, we can assume that
$H_{l}\cong\mathbb{Z}_{p_{1}^{\alpha_{1}}}$. But it is clear that every proper
$r$-image of $\mathbb{Z}_{p_{1}^{\alpha_{1}}}$ is trivial. This shows that
there is no chain of length $l+2$. Since $l\leq\sum_{i=1}^{n}k_{i}$, one can
conclude that there is no chain of length
$\big{(}\sum_{i=1}^{n}k_{i}\big{)}+2$ and so
$D(G)\leq\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$.

On the other hand, it is easy to see that the chain

$1<^{p}\mathbb{Z}_{p_{1}^{\alpha_{1}}}<^{p}\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(2)}<^{p}\cdots<^{p}\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(k_{1})}<^{p}\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(k_{1})}\oplus\mathbb{Z}_{p_{2}^{\alpha_{2}}}<^{p}\cdots<^{p}\mathbb{Z}_{p_{1}^{\alpha_{1}}}^{(k_{1})}\oplus\cdots\oplus\mathbb{Z}_{p_{n}^{\alpha_{n}}}^{(k_{n}-1)}<^{p}G,\;(*)$

is an $r$-chain of length $\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$. Therefore,
$D(G)=\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$.

Now, we show that $SD(G)=\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$. Since
$SD(G)\leq D(G)$, we have $SD(G)\leq\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$. On
the other hand, since every finitely generated abelian group is Hopfian, by
Proposition 3.3, the chain $(*)$ is an $s$-chain of length
$\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$ for $G$. Accordingly,
$SD(G)=\big{(}\sum_{i=1}^{n}k_{i}\big{)}+1$.

Note that one can easily see that the equality of $SD(G)=C(G)$ does not hold,
in general (see, Proposition 2.14). It is enough to consider
$G=\mathbb{Z}_{2}\oplus\mathbb{Z}\oplus\mathbb{Z}$ with $C(G)=6$, $C(G)=5$,
$D(G)=SD(G)=4$.

######  Theorem 3.15.

The depth of every virtually polycyclic group is finite.

######  Proof 3.16.

By [17, Lemma 1], there exists an integer $k$ such that any sequence
$H_{l}\subseteq\cdots\subseteq H_{1}\subseteq H_{0}=G$ of subgroups of $G$
with $r$-homomorphisms $r_{i}:H_{i-1}\longrightarrow H_{i}$ for $i=1,\cdots,l$
contains at most $k$ distinct subgroups.

We claim that $D(G)\leq k$. By contrary, assume that there exists the
following chain of length $k+1$ for $G$

$H_{k+1}<^{p}H_{k}<^{p}\cdots<^{p}H_{1}\leqslant^{d}H_{0}=G,\quad(*)$

with $r$-homomorphisms $g_{i}:H_{i-1}\longrightarrow H_{i}$ and converse
homomorphisms $f_{i}:H_{i}\longrightarrow H_{i-1}$ for all $1\leq i\leq k+1$.
Then we have the following sequence of subgroups for $G$

$f_{1}f_{2}\cdots f_{k+1}(H_{k+1})\leqslant^{d}f_{1}f_{2}\cdots
f_{k}(H_{k})\leqslant^{d}\cdots\leqslant^{d}f_{2}f_{1}(H_{2})\leqslant^{d}f_{1}(H_{1})\leqslant^{d}H_{0}=G,\quad(**)$

with retractions $r_{i}:f_{1}f_{2}\cdots f_{i-1}(H_{i-1})\longrightarrow
f_{1}f_{2}\cdots f_{i}(H_{i})$ which are defined by

$r_{i}(f_{1}(f_{2}(\cdots(f_{i-1}(x)))))=f_{1}(f_{2}(\cdots(f_{i-1}(f_{i}(g_{i}(x)))))),$

for all $1\leq i\leq k+1$. Clearly, the retraction $r_{i}$ is well-defined for
all $1\leq i\leq k+1$.

Now we show that $f_{1}f_{2}\cdots f_{i}(H_{i})$’s are mutually distinct for
all $1\leq i\leq k+1$. Suppose that $f_{1}f_{2}\cdots
f_{i}(H_{i})=f_{1}f_{2}\cdots f_{j}(H_{j})$ for $1\leq j<i\leq k+1$. Then
since $f_{k}$’s are monomorphisms, we have $f_{j+1}\cdots f_{i}(H_{i})=H_{j}$.
It follows that the monomorphism $f_{j+1}\cdots f_{i}$ is onto and so is an
isomorphism. Now since $H_{j}=Im(f_{j+1}\cdots f_{i})\subseteq Im(f_{j+1})$,
$f_{j+1}$ is also an isomorphism, i.e. $H_{j+1}\cong H_{j}$ which is a
contradiction to the chain $(*)$. Consequently, we have the sequence $(**)$ of
subgroups of $G$ containing at least $k+1$ distinct subgroups which is a
contradiction. Thus $D(G)\leq k$.

######  Remark 3.17.

Note that the converse of Theorem 3.15 does not hold, in general. As an
example, a free group of finite rank $k$ has the strong depth $k+1$ by part
$(ii)$ of Proposition 3.9. But we know that free groups of rank $\geq 2$ are
not virtually polycyclic.

######  Proposition 3.18.

Let $G$ be a virtually polycyclic group.The depth of a finitely generated
module $M$ over the integral group ring $\mathbb{Z}G$ is finite.

######  Proof 3.19.

It is concluded from [17, Lemma 2] and a similar argument used in the proof of
Theorem 3.15.

##  4 A theorem concerning the depth of polyhedra

In this section, we generalize a theorem of Kolodziejczyk (see [17, Theorem
2]) which states that every polyhedron with virtually polycyclic fundamental
group has finite depth. Note that for polyhedra, the notions shape and shape
domination can be replaced by the notions homotopy type and homotopy
domination, respectively (see [15]).

######  Lemma 4.1.

Let $H_{k}\leqslant^{d}\cdots\leqslant^{d}H_{1}\leqslant^{d}H_{0}=G$ be a
chain of length $k$ for a group $G$ with $r$-homomorphisms
$g_{i}:H_{i-1}\longrightarrow H_{i}$, for all $1\leq i\leq k$. Also, suppose
that $H_{i}$ is a Hopfian group $(0\leqslant i\leqslant k)$. If for some
$1\leq j_{0}\leq k$, $H_{j_{0}}\not\cong H_{j_{0}-1}$, then
$H_{j_{0}}\not\cong H_{i}$ for all $0\leq i<j_{0}$.

######  Proof 4.2.

By contrary, assume for some $j_{0}\geq 2$, $H_{j_{0}}\not\cong H_{j_{0}-1}$,
but $H_{j_{0}}\cong H_{i}$ for some $i<j_{0}-1$. Then since $g_{j_{0}}\cdots
g_{i+1}:H_{i}\longrightarrow H_{j_{0}}$ is an epimorhism between two
isomorphic Hopfian groups, so one can conclude that $g_{j_{0}}\cdots g_{i+1}$
is an isomorphism. Since $Ker(g_{i+1})\subseteq Ker(g_{j_{0}}\cdots g_{i+1})$,
$g_{i+1}$ is also isomorphism. It follows that $g_{j_{0}}\cdots g_{i+2}$ is an
isomorphism. With a similar argument, we can conclude that
$g_{j_{0}}:H_{j_{0}-1}\longrightarrow H_{j_{0}}$ is an isomorphism which is a
contradiction to $H_{j_{0}}\not\cong H_{j_{0}-1}$. Thus the proof is complete.

Now we present the main result of the paper.

######  Theorem 4.3.

Let $P$ be a finite $n$-dimensional polyhedron. Suppose that $P$ satisfies the
following conditions:

  1. 1.

$D(\pi_{1}(P))=k_{1}<\infty$ and every retract of $\pi_{1}(P)$ is Hopfian;

  2. 2.

$D(H_{i}(\tilde{P}))=k_{i}<\infty$ and every retract of $H_{i}(\tilde{P})$ is
Hopfian, for all $2\leq i\leq n$.

Then $D(P)\leq\big{(}\sum_{i=1}^{n}k_{i}\big{)}-n+1$.

Here, $H_{i}(\tilde{P})$ may be considered as either $\mathbb{Z}$-module or
$\mathbb{Z}\pi_{1}(P)$-module. If it is considered as $\mathbb{Z}$-module,
then the condition “every retract of $H_{i}(\tilde{P})$ is Hopfian, for all
$2\leq i\leq n$” holds automatically.

######  Proof 4.4.

Put $k=\sum_{i=1}^{n}k_{i}$. We are going to show that $D(P)\leq k-n+1$. By
contrary, assume that there exists the following chain of length $k$ for $P$

$\displaystyle
X_{k-n+2}<^{p}X_{k-n+1}<^{p}\cdots<^{p}X_{2}<^{p}X_{1}\leqslant^{d}X_{0}=P,$
(1)

with domination maps $d_{j}:X_{j}\longrightarrow X_{j+1}$ and converse maps
$u_{j}:X_{j+1}\longrightarrow X_{j}$, for all $0\leq j\leq k-1$.

Without loss of generality, each $X_{i}$ may be assumed to be a CW-complex,
not necessarily finite (by the known results of J.H.C. Whitehead, each space
homotopy dominated by a polyhedron has the homotopy type of some CW-complex,
not necessarily finite (see also [28])).

From the chain (1), we have the following sequence of $r$-images of
$\pi_{1}(P)$

$\displaystyle\pi_{1}(X_{k-n+2})\leqslant^{d}\pi_{1}(X_{k-n+1})\leqslant^{d}\cdots\leqslant^{d}\pi_{1}(X_{2})\leqslant^{d}\pi_{1}(X_{1})\leqslant^{d}\pi_{1}(P),$
(2)

and the following sequence of $r$-images of $H_{i}(\tilde{P})$

$\displaystyle
H_{i}(\tilde{X}_{k-n+2})\leqslant^{d}H_{i}(\tilde{X}_{k-n+1})\leqslant^{d}\cdots\leqslant^{d}H_{i}(\tilde{X}_{2})\leqslant^{d}H_{i}(\tilde{X}_{1})\leqslant^{d}H_{i}(\tilde{P}),$
(3)

for all $2\leq i\leq n$.

Suppose that for some $1\leq j_{0}\leq k-n+1$, we have

$\pi_{1}(X_{j_{0}})\cong\pi_{1}(X_{j_{0}+1})\quad\text{and}\quad
H_{i}(\tilde{X}_{j_{0}})\cong H_{i}(\tilde{X}_{j_{0}+1})$

for all $2\leq i\leq n$. Since an epimorphism between two isomorphic Hopfian
groups is an isomorphism, one can easily see that the epimorphisms
$\pi_{1}(d_{j_{0}}):\pi_{1}(X_{j_{0}})\longrightarrow\pi_{1}(X_{j_{0}+1})$ and
$H_{i}(d_{j_{0}}):H_{i}(\tilde{X}_{j_{0}})\longrightarrow
H_{i}(\tilde{X}_{j_{0}+1})$ are isomorphisms. Now by the Whitehead Theorem,
$d_{j_{0}}:X_{j_{0}}\longrightarrow X_{j_{0}+1}$ is a homotopy equivalence
which implies that $X_{j_{0}}$ and $X_{j_{0}+1}$ have the same homotopy type
which is a contradiction to $X_{j_{0}+1}<^{p}X_{j_{0}}$. Accordingly, for each
$1\leq j\leq k-n+1$, one can conclude that either
$\pi_{1}(X_{j})\not\cong\pi_{1}(X_{j+1})$ or $H_{i}(\tilde{X}_{j})\not\cong
H_{i}(\tilde{X}_{j+1})$ at least for some $2\leq i\leq n$. Since
$D(\pi_{1}(P))=k_{1}$ and $D(H_{i}(\tilde{P}))=k_{i}$ $(2\leq i\leq n)$, by
Proposition 2.16 the chains (2) and (3) contain at most $k_{1}$ and $k_{i}$
non-isomorphic groups, respectively. This shows that there are at most
$k_{1}-1$ and $k_{i}-1$ copies of the sign “$\not\cong$” $(2\leq i\leq n)$ in
the chains (2) and (3), respectively. Consequently, there is at most
$\sum_{i=1}^{n}(k_{i}-1)=k-n$ copies of the sign “$\not\cong$” in all chains
(2) and (3) $(2\leq i\leq n)$. Accordingly, by Lemma 4.1, it is easy to see
that $\pi_{1}(X_{k-n+2})\cong\pi_{1}(X_{k-n+1})$ and
$H_{i}(\tilde{X}_{k-n+2})\cong H_{i}(\tilde{X}_{k-n+1})$ for all $2\leq i\leq
n$. By the Whitehead Theorem, one can conclude that $X_{k-n+2}$ and
$X_{k-n+1}$ have the same homotopy type which is a contradiction to the chain
(1). Consequently, we have $D(P)\leq k-n+1$.

In the next corollary, we present an upper bound for the depths of polyhedra
with finite fundamental groups.

######  Corollary 4.5.

Let $P$ be a finite $n$-dimensional polyhedron with finite fundamental group.
Then

$D(P)\leq D(\pi_{1}(P))+\sum_{i=2}^{n}t_{i},$

where $t_{i}$’s are introduced as follows

$H_{i}(\tilde{P},\mathbb{Z})\cong\mathbb{Z}_{p_{i_{1}}^{\alpha_{i_{1}}}}\oplus\cdots\oplus\mathbb{Z}_{p_{i_{t_{i}}}^{\alpha_{i_{t_{i}}}}}\;\text{for}\;2\leq
i\leq n,$

where $p_{i_{j}}^{\alpha_{i_{j}}}\neq
p_{i_{j^{\prime}}}^{\alpha_{i_{j^{\prime}}}}$ for $j\neq j^{\prime}$,
$p_{i_{j}}$’s are prime numbers, $\alpha_{i_{j}}$’s are non-negative integers,
and $\mathbb{Z}_{1}=\mathbb{Z}$.

######  Proof 4.6.

Since $\pi_{1}(P)<\infty$, the universal covering space $\tilde{P}$ of a
finite polyhedron $P$ is a finite polyhedron. Hence the homology groups
$H_{i}(\tilde{P},\mathbb{Z})$ are finitely generated abelian groups for all
$i$. Now by part $(ii)$ of Proposition 3.13, we have
$D(H_{i}(\tilde{P},\mathbb{Z}))=t_{i}+1$ for all $2\leq i\leq n$. Since all
finite groups and finitely generated abelian groups are Hopfian, we have
$D(P)\leq D(\pi_{1}(P))+\sum_{i=2}^{n}t_{i}$ by Theorem 4.3.

######  Corollary 4.7.

Le $P$ be an $n$-dimensional polyhedron with abelian fundamental group
$\pi_{1}(P)\cong\mathbb{Z}_{p_{1}^{\alpha_{1}}}\oplus\cdots\mathbb{Z}_{p_{t}^{\alpha_{t}}}$,
where for $i\neq j$, $p_{i}^{\alpha_{i}}\neq p_{j}^{\alpha_{j}}$, $p_{i}$’s
are prime numbers, $\alpha_{i}$’s are non-negative integers, and
$\mathbb{Z}_{1}=\mathbb{Z}$. If $D(H_{i}(\tilde{P}))=k_{i}<\infty$ (as
$\mathbb{Z}\pi_{1}(P)$-module) for all $2\leq i\leq n$, then
$D(P)\leq\big{(}\sum_{i=2}^{n}k_{i}\big{)}+t-n+1$.

######  Proof 4.8.

Note that $\pi_{1}(P)$ is a finitely generated abelian, so $\pi_{1}(P)$ is of
the form
$\mathbb{Z}_{p_{1}^{\alpha_{1}}}\oplus\cdots\mathbb{Z}_{p_{t_{1}}^{\alpha_{t_{1}}}}$,
up to isomophism, where for $i\neq j$, $p_{i}^{\alpha_{i}}\neq
p_{j}^{\alpha_{j}}$, $p_{i}$’s are prime numbers, $\alpha_{i}$’s are non-
negative integers, and $\mathbb{Z}_{1}=\mathbb{Z}$. We know that
$H_{i}(\tilde{P})$ is a finitely generated $\mathbb{Z}\pi_{1}(P)$-module for
all $2\leq i\leq n$. But since $\pi_{1}(P)$ is abelian, the group ring
$\mathbb{Z}\pi_{1}(P)$ is a commutative ring. It follows that
$H_{i}(\tilde{P})$ $(2\leq i\leq n)$ is a Hopfian module by [27, Proposition
1.2]. Now by part $(ii)$ of Proposition 3.13, Theorem 4.3 and the hypothesis,
the proof is complete.

######  Corollary 4.9.

Let $P$ be an $n$-dimensional polyhedron with the abelian fundamental group
$\pi_{1}(P)\cong\mathbb{Z}_{p_{1}^{\alpha_{1}}}\oplus\cdots\mathbb{Z}_{p_{t_{1}}^{\alpha_{t_{1}}}}$,
where for $i\neq j$, $p_{i}^{\alpha_{i}}\neq p_{j}^{\alpha_{j}}$, $p_{i}$’s
are prime numbers, $\alpha_{i}$’s are non-negative integers, and
$\mathbb{Z}_{1}=\mathbb{Z}$. Also,suppose that the homology groups
$H_{i}(\tilde{P};\mathbb{Z})$ are finitely generated of the form

$H_{i}(\tilde{P};\mathbb{Z})\cong\mathbb{Z}_{p_{i_{1}}^{\alpha_{i_{1}}}}\oplus\cdots\oplus\mathbb{Z}_{p_{i_{t_{i}}}^{\alpha_{i_{t_{i}}}}},$

where for $j\neq j^{\prime}$, $p_{i_{j}}^{\alpha_{i_{j}}}\neq
p_{i_{j^{\prime}}}^{\alpha_{i_{j^{\prime}}}}$, $p_{i_{j}}$’s are prime
numbers, $\alpha_{i_{j}}$’s are non-negative integers, and
$\mathbb{Z}_{1}=\mathbb{Z}$. Then the depth of $P$ is at most
$\big{(}\sum_{i=1}^{n}t_{i}\big{)}+1$.

######  Proof 4.10.

Since $\pi_{1}(P)$ and $H_{i}(\tilde{P},\mathbb{Z})$’s $(2\leq i\leq n)$ are
finitely generated abelian groups, we have $D(\pi_{1}(P))=t_{1}+1$ and
$D(H_{i}(\tilde{P},\mathbb{Z}))=t_{i}+1$ by part $(ii)$ of Proposition 3.13.
Now by Theorem 4.3, the proof is completed.

######  Remark 4.11.

Theorem 4.3 in comparison with recent works concerning the depth of polyhedra
(see, for example [17]) has two advantages: one that it presents an upper
bound for the depth of some polyhedra in terms of the depths of their
fundamental groups and the homology groups of their universal covering spaces,
and other one is that it contains wider class of polyhedra. For example, by
[17, Theorem 2], every polyhedron with virtually polycyclic fundamental group
has finite depth. But the polyhedron $\mathbb{S}^{1}\vee\mathbb{S}^{1}$ is an
example of a polyhedron with non-virtually polycyclic group but finite depth.
Note that free groups with finite rank $\geq 2$ are not virtually polycyclic.

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# Kodaira dimension of fibrations over abelian varieties

Fanjun Meng  Department of Mathematics, Northwestern University, 2033 Sheridan
Road, Evanston, IL 60208, USA<EMAIL_ADDRESS>and  Mihnea
Popa  Department of Mathematics, Harvard University, 1 Oxford Street,
Cambridge, MA 02138, USA<EMAIL_ADDRESS>
###### Abstract.

We prove a superadditivity result for the Kodaira dimension of algebraic fiber
spaces over abelian varieties, an additivity result for (almost) smooth
morphisms, as well as a subadditivity result for log pairs.

###### Key words and phrases:

Kodaira dimension; abelian varieties; positivity

MP was partially supported by the NSF grant DMS-2040378.

###  1\. Introduction

In this paper we show the validity, in the case of algebraic fiber spaces over
abelian varieties (and more generally over varieties of maximal Albanese
dimension), of some conjectures on the behavior of Kodaira dimension proposed
by the second author; see [Popa, Conjectures 3.1 and 3.4].

######  Theorem A.

Let $f\colon X\to A$ be an algebraic fiber space, with $X$ a smooth projective
variety and $A$ an abelian variety. Assume that $f$ is smooth over an open set
$V\subseteq Y$, and denote its general fiber by $F$. Then

  1. (1)

$\kappa(V)+\kappa(F)\geq\kappa(X)$.

  2. (2)

If $A\smallsetminus V$ has codimension at least $2$ in $A$ (e.g. if $f$ is
smooth), then $\kappa(X)=\kappa(F)$. More precisely, there exists an étale
cover $X^{\prime}$ of $X$ such that

$P_{m}(X^{\prime})=P_{m}(F)\geq P_{m}(X)\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,m\geq 0.$

Here, for a quasi-projective variety $V$, $\kappa(V)$ denotes the log Kodaira
dimension, defined as follows: if we take any smooth projective
compactification $Y$ of $V$ such that $D=Y\smallsetminus V$ is a divisor with
simple normal crossings, then $\kappa(V)=\kappa(Y,K_{Y}+D)$.

Item (2) recovers (and strengthens) in particular [PS2, Corollary 3.1],
stating that if $f$ is smooth, then $\dim X-\kappa(X)\geq\dim A$. As for the
proof, thanks to the structure of effective divisors on abelian varieties,
item (1) is in fact a consequence of (2). On the other hand, Iitaka’s
$C_{n,m}$ conjecture on the subadditivity of the Kodaira dimension is known
for algebraic fiber spaces over abelian varieties by [CP] (see also [HPS]),
giving $\kappa(X)\geq\kappa(F)$ for any such. Thus the key point is the last
statement in (2), for which we employ techniques from [LPS] regarding the
Chen-Jiang decomposition, as well as a hyperbolicity-type result from [PS3].
We prove in fact a stronger statement:

######  Theorem B.

Let $f\colon X\to A$ be a surjective morphism from a smooth projective variety
to an abelian variety. Assume that $f$ is smooth away from a closed set of
codimension at least $2$ in $A$, and denote its general fiber by $F$. Then for
every $m\geq 1$ we have

$f_{*}\omega_{X}^{\otimes m}\simeq\bigoplus_{i=1}^{P_{m}(F)}\alpha_{i},$

where $\alpha_{i}\in{\rm Pic}^{0}(A)$ are (possibly repeated) torsion line
bundles. In particular, if $f_{*}\omega_{X}^{\otimes m}$ is globally generated
for some $m$, then

$f_{*}\omega_{X}^{\otimes m}\simeq\mathscr{O}_{A}^{\oplus P_{m}(F)}.$

Note that assuming Viehweg’s $C_{n,m}^{+}$ conjecture, the result in Theorem
A(2) implies that a morphism $f\colon X\to A$ which is smooth in codimension
$1$ satisfies ${\rm Var}(f)=0$, i.e. $f$ is birationally isotrivial (cf. also
[PS2, Corollary 3.2]). If $f$ is smooth and one assumes that $F$ has a good
minimal model, this is also implied by the main result in [Taji], namely the
proof of the Kebekus-Kovács conjecture bounding ${\rm Var}(f)$ from above by
the Kodaira dimension of the base. By analogy with Ueno’s Conjecture $K$, we
propose the following strengthening:

######  Conjecture C.

Let $f\colon X\to A$ be an algebraic fiber space, with $X$ a smooth projective
variety and $A$ an abelian variety, and with general fiber $F$. If $f$ is
smooth away from a closed set of codimension at least $2$ in $A$ , then there
exists an isogeny $A^{\prime}\to A$ such that

$X\times_{A}A^{\prime}\sim A^{\prime}\times F,$

i.e. $X$ becomes birational to a product after an étale base change.

When $f$ is smooth with canonically polarized fibers, i.e. $K_{F}$ is ample,
this is proved (with isomorphism) in [Kovacs, §2].

It turns out that the result in Theorem A holds more generally over any
variety of maximal Albanese dimension, i.e. endowed with a (not necessarily
surjective) generically finite morphism to an abelian variety. We state this
separately, since the proof is a rather involved reduction to the case of
abelian varieties, combined with Theorem A.

######  Theorem D.

Let $f\colon X\to Y$ be an algebraic fiber space, with $X$ and $Y$ smooth and
projective, and $Y$ of maximal Albanese dimension. Assume that $f$ is smooth
over an open set $V\subseteq Y$, and denote its general fiber by $F$. Then

  1. (1)

$\kappa(V)+\kappa(F)\geq\kappa(X)$.

  2. (2)

If $Y\smallsetminus V$ has codimension at least $2$ in $Y$, then
$\kappa(X)=\kappa(F)+\kappa(Y)$.

Note that the subadditivity $\kappa(X)\geq\kappa(F)+\kappa(Y)$ is known to
hold for any algebraic fiber space over $Y$ of maximal Albanese dimension, by
[HPS, Theorem 1.1].

In a different but related direction, we remark that using the klt version
provided in [Meng] (see also [Jiang] for related results) of the global
generation result for direct images in [LPS] used for the results above, one
can go beyond the results of [CP] and [HPS]. Concretely, we show the validity
of the _log_ Iitaka conjecture over an abelian variety $A$, in the case of
divisors that do not dominate $A$ (though the statement can be phrased
slightly more generally). For a divisor $D$ on a smooth projective variety
$X$, we use the notation $P_{m}(X,D):=h^{0}(X,m(K_{X}+D))$.

######  Theorem E.

Let $f\colon X\to A$ be an algebraic fiber space, with $X$ a smooth projective
variety, and $A$ an abelian variety. Denote by $F$ the general fiber of $f$.
Let $D$ be a reduced effective divisor on $A$ and $E$ a reduced effective
divisor on $X$ such that ${\rm Supp}(f^{*}D)\subseteq E$. Then

$\kappa(X,K_{X}+E)\geq\kappa(F)+\kappa(A,D).$

More precisely, there exists an étale cover $\psi\colon X^{\prime}\to X$ and a
fixed integer $k\geq 1$ such that

$P_{mk}(X^{\prime},\psi^{*}E)\geq P_{mk}(F)\cdot P_{m}(A,D)\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,m\geq 0.$

The most general case of the log Iitaka conjecture allows for divisors $E$
that dominate $A$, in other words replacing $\kappa(F)$ by
$\kappa(F,K_{F}+E|_{F})$. It would be solved with similar methods if
extensions of the results in [LPS] were available in the setting of log
canonical pairs; this is however beyond our reach at the moment. Note also
that if $E=(f^{*}D)_{\rm red}$, it is predicted in [Popa, Conjecture 4.1] that
the inequality in Theorem E becomes an equality if $f$ is smooth away from
$D$, which would extend Theorem A(2).

One final word: in the results of this paper, morphisms that are smooth away
from a closed subset of codimension at least $2$ behave just like morphisms
that are smooth everywhere. However, working under this weaker hypothesis
leads to additional technical arguments. In order to isolate the main ideas,
it may be helpful to assume global smoothness at a first reading.

Acknowledgements. The first author would like to thank the Department of
Mathematics at Harvard University for its hospitality during the preparation
of this paper.

###  2\. Background

We will make use of the following two results from [LPS], which are shown in
_loc. cit._ to be equivalent.

######  Theorem 2.1 ([LPS, Theorem B]).

Let $f\colon X\to A$ be a morphism from a smooth projective variety to an
abelian variety. Then there exists an isogeny $\varphi\colon A^{\prime}\to A$
such that if

${X^{\prime}}$${X}$${A^{\prime}}$${A}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\varphi}$

is the fiber product, then $\varphi^{*}f_{*}\omega_{X}^{\otimes m}\simeq
f^{\prime}_{*}\omega_{X^{\prime}}^{\otimes m}$ is globally generated for every
$m\geq 1$.

Recall that an (algebraic) fiber space is a surjective projective morphism
with connected fibers. We remark, as could have been done already in [LPS],
that Theorem 2.1 implies a strengthening of the subadditivity of the Kodaira
dimension for algebraic fiber spaces over abelian varieties proved in [CP] and
later also in [HPS].

######  Corollary 2.2.

Let $f\colon X\to A$ be an algebraic fiber space over an abelian variety, with
$X$ smooth and projective, and with general fiber $F$. Then there exists an
étale cover $X^{\prime}\to X$ such that $P_{m}(X^{\prime})\geq P_{m}(F)$ for
all $m\geq 1$.

###### Proof.

We consider the construction in Theorem 2.1, so that
$f^{\prime}_{*}\omega_{X^{\prime}}^{\otimes m}$ is globally generated for
every $m\geq 1$. The number of independent global sections of this sheaf is
therefore at least equal to its rank, which is equivalent to the inequality

$P_{m}(X^{\prime})\geq P_{m}(F).$

∎

For the next statement, and for later use in the paper, recall that for a
coherent sheaf $\mathscr{F}$ on an abelian variety $A$, and for $i\geq 0$, we
consider the $i$-th cohomological support locus of $\mathscr{F}$ given by

$V^{i}(A,\mathscr{F})=\\{\alpha\in{\rm
Pic}^{0}(A)~{}|~{}H^{i}(A,\mathscr{F}\otimes\alpha)\neq 0\\}.$

We will use the following standard terminology:

  * •

$\mathscr{F}$ is a _GV-sheaf_ if $\operatorname{codim}_{{\rm
Pic}^{0}(A)}V^{i}(A,\mathscr{F})\geq i$ for all $i>0$.

  * •

$\mathscr{F}$ is an _M-regular sheaf_ if $\operatorname{codim}_{{\rm
Pic}^{0}(A)}V^{i}(A,\mathscr{F})>i$ for all $i>0$.

######  Theorem 2.3 ([LPS, Theorem C]).

In the setting of Theorem 2.1, there exists a finite direct sum decomposition

$f_{*}\omega_{X}^{\otimes m}\simeq\bigoplus_{i\in I}\bigl{(}\alpha_{i}\otimes
p^{\ast}_{i}\mathscr{F}_{i}\bigr{)},$

where each $p_{i}\colon A\to A_{i}$ is a quotient morphism with connected
fibers to an abelian variety, each $\mathscr{F}_{i}$ is an M-regular coherent
sheaf on $A_{i}$, and each $\alpha_{i}\in{\rm Pic}^{0}(A)$ is a line bundle
that becomes trivial when pulled back by the isogeny in Theorem 2.1.

The direct sum decomposition in this last theorem goes under the name of a
_Chen-Jiang decomposition_. The only special fact about $M$-regular sheaves
that we will use here is the fact that they are ample, as shown in [Debarre];
similarly, $GV$-sheaves are nef, as shown in [PP2].

Finally, we will need a hyperbolicity-type result, proved in this generality
in [PS3]; it relies on important ideas and results of Viehweg-Zuo and Campana-
Păun described in _loc. cit._ , as well as on the theory of Hodge modules. For
simplicity, in the following statement we combine two results, and only state
the special consequence needed in this paper.111The statement works in a more
general setting; it is comparatively much simpler however, avoiding semistable
reduction tricks, when the base is a priori known not to be uniruled.

######  Theorem 2.4 ([PS3, Theorem 4.1 and Theorem 3.5]).

Let $f\colon X\to Y$ be an algebraic fiber space between smooth projective
varieties, such that $Y$ is not uniruled. Assume that $f$ is smooth over the
complement of a closet subset $Z\subset Y$, and that there exists $m\geq 1$
such that ${\rm det}f_{*}\omega_{X/Y}^{\otimes m}$ is big. Denote by $D$ the
union of the divisorial components of $Z$. Then the pair $(Y,D)$ is of log
general type, i.e. the line bundle $\omega_{Y}(D)$ is big.

######  Remark 2.5.

The theorem above is stated in _loc. cit._ only when $Z=D$, but the proof
shows more generally the statement above, as all the objects it involves can
be constructed from $Y$ with any codimension $2$ subset removed.

###  3\. Proof of Theorem B

Let $f\colon X\to A$ be a surjective morphisms of smooth projective varieties,
with $A$ an abelian variety, and let $F$ be the general fiber of $f$. Assume
that $f$ is smooth over an open subset $V\subseteq A$ whose complement has
codimension at least $2$. We divide the proof of Theorem B into two steps:

_Step 1._ We first prove the last assertion; namely if
$f_{*}\omega_{X}^{\otimes m}$ is globally generated for some $m\geq 1$, then

(3.1) $f_{*}\omega_{X}^{\otimes m}\simeq\mathscr{O}_{A}^{\oplus P_{m}(F)}.$

We first prove this result when $f$ is a fiber space. We use the statement and
notation of Theorem 2.3, and we claim in fact that $\dim A_{i}=0$ for all
$i\in I$ appearing in the decomposition in that theorem. Since
$f_{*}\omega_{X}^{\otimes m}$ is globally generated, this immediately implies
(3.1).

Assume on the contrary that we have $\dim A_{k}>0$ for one of the quotients
$p_{k}\colon A\to A_{k}$. (This includes the case when $A_{k}=A$.) Denote the
kernel of $p_{k}$ by $C$; this is an abelian subvariety of $A$. By Poincaré’s
complete reducibility theorem, there exists an abelian variety $B\subseteq A$
such that $B+C=A$ and $B\cap C$ is finite, so that the natural morphism
$\varphi\colon B\times C\to A$ is an isogeny. We consider the following
commutative diagram, $q$ is the projection onto $C$, $c\in C$ is a general
point, and $f^{\prime}$ and $f^{\prime}_{c}$ are obtained by base change from
$f$ via $\varphi$ and the inclusion $i_{c}$ of the fiber $B_{c}$ of $q$ over
$c$ respectively:

${X^{\prime}_{c}}$${X^{\prime}}$${X}$${B_{c}}$${B\times
C}$${A}$${\\{c\\}}$${C}$${A_{k}}$$\scriptstyle{f^{\prime}_{c}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{i_{c}}$$\scriptstyle{q}$$\scriptstyle{\varphi}$$\scriptstyle{p_{k}}$

Note that by construction the composition

$\psi_{c}=p_{k}\circ\varphi\circ i_{c}\colon B_{c}\to A_{k}$

is an isogeny. Furthermore, $X^{\prime}$ is smooth, since $\varphi$ is étale.
We denote by $Z\subset A$ the closed subset of codimension $\geq 2$ such that
$f$ is smooth over $A\smallsetminus Z$. If $Z^{\prime}:=\varphi^{-1}(Z)$, then
$Z^{\prime}$ has codimension $\geq 2$ in $B\times C$ as well, and $f^{\prime}$
is smooth over its complement. Moreover, $c$ can be chosen sufficiently
general so that $X^{\prime}_{c}$ is smooth (by generic smoothness applied to
$q\circ f^{\prime}$) and
$\operatorname{codim}_{B_{c}}i_{c}^{-1}(Z^{\prime})\geq 2$ as well, hence
$f^{\prime}_{c}$ inherits the same property as $f$.

Claim. The line bundle
$\det\big{(}(f^{\prime}_{c})_{*}\omega_{X^{\prime}_{c}}^{\otimes m}\big{)}$ is
big (hence ample, as $B_{c}$ is an abelian variety).

Assuming the Claim, we can immediately conclude the proof. Indeed, by Theorem
2.4 this would imply that $\omega_{B_{c}}$ is big, which is a contradiction
and shows that we cannot have $\dim A_{k}>0$.

We are left with proving the Claim. To this end, applying the base change
theorem as in [LPS, Proposition 4.1], since $c$ is general we have

(3.2) $(f^{\prime}_{c})_{*}\omega_{X^{\prime}_{c}}^{\otimes
m}\simeq(f^{\prime}_{c})_{*}(\omega_{X^{\prime}}^{\otimes
m}|_{X^{\prime}_{c}})\simeq
i_{c}^{*}(f^{\prime}_{*}\omega_{X^{\prime}}^{\otimes m})\simeq
i_{c}^{*}\varphi^{*}(f_{*}\omega_{X}^{\otimes m}).$

To analyze this, we need to look more carefully at the decomposition of
$f_{*}\omega_{X}^{\otimes m}$ from Theorem 2.3. Since
$f_{*}\omega_{X}^{\otimes m}$ is globally generated, we deduce that
$\alpha_{k}\otimes p_{k}^{*}\mathscr{F}_{k}$ is globally generated, and in
particular $h^{0}(A,\alpha_{k}\otimes p_{k}^{*}\mathscr{F}_{k})>0$. It follows
that $\alpha_{k}$ is trivial on the general, hence every, fiber of $p_{k}$,
and so there exists a torsion line bundle $\beta_{k}\in{\rm Pic}^{0}(A_{k})$
such that $\alpha_{k}\simeq p_{k}^{*}\beta_{k}$. Moreover

${p_{k}}_{*}(\alpha_{k}\otimes
p_{k}^{*}\mathscr{F}_{k})\simeq\beta_{k}\otimes\mathscr{F}_{k}$

by the projection formula (applied to the $0$-th cohomology of
$\mathbf{R}{p_{k}}_{*}(\alpha_{k}\otimes p_{k}^{*}\mathscr{F}_{k})$). If we
denote $\mathscr{G}:=\bigoplus_{i\neq k}(\alpha_{i}\otimes
p_{i}^{*}\mathscr{F}_{i})$, we then have

(3.3) $f_{*}\omega_{X}^{\otimes m}\simeq
p_{k}^{*}(\beta_{k}\otimes\mathscr{F}_{k})\oplus\mathscr{G}.$

These summands have various positivity properties. Since
$\beta_{k}\otimes\mathscr{F}_{k}$ is an $M$-regular sheaf on $A_{k}$, it is
ample. On the other hand, since since $f_{*}\omega_{X}^{\otimes m}$ is a
$GV$-sheaf by [PS1, Theorem 1.10], it is nef, hence so is $\mathscr{G}$. (For
all this, see the comments after Theorem 2.3.) Since a priori they might not
be locally free, it is also useful to record that $\mathscr{G}$ is a weakly
positive sheaf, since $f_{*}\omega_{X}^{\otimes m}$ is so by [Viehweg, Theorem
III].

Using (3.2) and (3.3), we deduce that

$(f^{\prime}_{c})_{*}\omega_{X^{\prime}_{c}}^{\otimes m}\simeq
i_{c}^{*}\varphi^{*}(f_{*}\omega_{X}^{\otimes
m})\simeq\psi_{c}^{*}(\beta_{k}\otimes\mathcal{F}_{k})\oplus
i_{c}^{*}\varphi^{*}\mathscr{G}.$

Note in passing that if $f$ is smooth, and so also $f^{\prime}$ and
$f^{\prime}_{c}$ are smooth to begin with, then all of the sheaves above are
locally free (by the deformation invariance of plurigenera), so by the
previous paragraph this a sum of an ample and a nef vector bundle, hence its
determinant is clearly ample. In general we have to be slightly more careful
in order to draw the same conclusion. Since $\psi_{c}$ is an isogeny, we
deduce that $\psi_{c}^{*}(\beta_{k}\otimes\mathcal{F}_{k})$ is an ample sheaf
as well, while $i_{c}^{*}\varphi^{*}\mathscr{G}$ is weakly positive just as
$\mathscr{G}$ (being a summand of
$(f^{\prime}_{c})_{*}\omega_{X^{\prime}_{c}}^{\otimes m}$). In other words, we
have that

$(f^{\prime}_{c})_{*}\omega_{X^{\prime}_{c}}^{\otimes
m}\simeq\mathcal{H}_{1}\oplus\mathcal{H}_{2},$

with $\mathcal{H}_{1}$ ample and $\mathcal{H}_{2}$ weakly positive. But an
ample sheaf is big in the sense of Viehweg (see e.g. [Debarre, §2 a)] and
[Mori, §5 p.293]), and so its determinant222Recall that the determinant of a
torsion-free sheaf $\mathscr{F}$ of generic rank $r$ is the line bundle
$(\wedge^{r}\mathscr{F})^{\vee\vee}$, i.e. the unique extension of the
determinant line bundle from the big open set on which $\mathscr{F}$ is
locally free. $\det\mathcal{H}_{1}$ is a big line bundle by [Viehweg2, Lemma
3.2(iii)] (see also [Mori, 5.1.1]). On the other hand, $\det\mathcal{H}_{2}$
is weakly positive, e.g. also by [Viehweg2, Lemma 3.2(iii)]; for a line bundle
this is the same as being pseudoeffective. Their tensor product is therefore
also big, hence finally the line bundle
$\det\big{(}(f^{\prime}_{c})_{*}\omega_{X^{\prime}_{c}}^{\otimes m}\big{)}$ is
big. This concludes the proof in the case of fiber spaces.

If $f$ is not a fiber space, we consider its Stein factorization $f=g\circ h$.
Here $B$ is a normal projective variety, $h\colon X\to B$ is a fiber space,
and $g\colon B\to A$ is a finite surjective morphism. Note that $h$ smooth
over $B\smallsetminus g^{-1}(Z)$ and $g$ is étale over $A\smallsetminus Z$;
see e.g. [FG, Lemma 2.4]. By the purity of branch locus, it follows that $g$
is actually étale over $A$, and thus $B$ is also an abelian variety. Moreover
the canonical morphism

$g^{*}g_{*}h_{*}\omega_{X}^{\otimes m}\to h_{*}\omega_{X}^{\otimes m}$

is surjective, which implies that $h_{*}\omega_{X}^{\otimes m}$ is globally
generated as well. Using what we showed above for fiber spaces that are smooth
in codimension $1$, we deduce that

$h_{*}\omega_{X}^{\otimes m}\simeq\mathscr{O}_{B}^{\oplus P_{m}(H)},$

where $H$ denotes the general fiber of $h$. Furthermore, we have

$g_{*}\mathscr{O}_{B}\simeq\bigoplus_{\alpha\in{\rm Ker}(\hat{g})}\alpha,$

hence $f_{*}\omega_{X}^{\otimes m}$ is a direct sum of torsion line bundles on
$A$. Since $f_{*}\omega_{X}^{\otimes m}$ is globally generated, we obtain the
same conclusion (3.1).

_Step 2._ We deduce next the general case from the statement for globally
generated sheaves proved in Step 1. By the same Stein factorization argument
as above, we may assume that $f$ is a fiber space.

We now use the statement and notation of Theorem 2.1. By Step 1, we have that

$\varphi^{*}f_{*}\omega_{X}^{\otimes m}\simeq
f^{\prime}_{*}\omega_{X^{\prime}}^{\otimes
m}\simeq\mathscr{O}_{A^{\prime}}^{\oplus P_{m}(F)},$

which implies that $f_{*}\omega_{X}^{\otimes m}$ is a direct summand of
$(\varphi_{*}\mathscr{O}_{A^{\prime}})^{\oplus P_{m}(F)}$. Since

$\varphi_{*}\mathscr{O}_{A^{\prime}}\simeq\bigoplus_{\beta\in{\rm
Ker}(\hat{\varphi})}\beta,$

we deduce that

$V^{0}(A,f_{*}\omega_{X}^{\otimes m})\subseteq{\rm Ker}(\hat{\varphi})$

and in particular $\dim V^{0}(A,f_{*}\omega_{X}^{\otimes m})=0$.

We consider again the Chen-Jiang decomposition of $f_{*}\omega_{X}^{\otimes
m}$ provided by Theorem 2.3. By [LPS, Lemma 3.3], we have

$V^{0}(A,f_{*}\omega_{X}^{\otimes m})=\bigcup_{i\in I}\alpha_{i}^{-1}\otimes
p_{i}^{*}{\rm Pic}^{0}(A_{i}).$

Since $\dim V^{0}(A,f_{*}\omega_{X}^{\otimes m})=0$, we deduce that $\dim
A_{i}=0$ for all $i\in I$, which in turn leads to a decomposition

$f_{*}\omega_{X}^{\otimes m}\simeq\bigoplus_{i=1}^{P_{m}(F)}\alpha_{i}$

with $\alpha_{i}$ torsion line bundles on $A$, as desired.

###  4\. Proof of Theorem A

We start with part (2), which follows immediately from Theorem B. Indeed, we
consider the étale base change $X^{\prime}\to X$ as in Theorem 2.1, so that
$f^{\prime}_{*}\omega_{X^{\prime}}^{\otimes m}$ is globally generated for
$m\geq 1$. According to Theorem B, it follows that

$f^{\prime}_{*}\omega_{X^{\prime}}^{\otimes
m}\simeq\mathscr{O}_{A^{\prime}}^{\oplus P_{m}(F)},$

which gives $P_{m}(X^{\prime})=P_{m}(F)$. We obviously have
$P_{m}(X^{\prime})\geq P_{m}(X)$, and in particular we deduce that
$\kappa(F)\geq\kappa(X)$. The opposite inequality is the subadditivity of the
Kodaira dimension over abelian varieties, proved in [CP] (see also [HPS]).

The rest of the section is devoted to proving (1), essentially by reducing it
to (2). We consider the closed subset $Z:=A\smallsetminus V$, and decompose it
as $Z=D\cup W$, where each irreducible component of $D$ has codimension $1$,
while $W$ has codimension at least $2$. We first show that

$\kappa(V)\geq\kappa(A,D).$

To this end, we take a log resolution $\mu\colon Y\to A$ of the pair $(X,Z)$;
thus $\mu^{-1}(Z)\cup{\rm Exc}(\mu)$ is a divisor with simple normal crossing
support. Denote $C=(\mu^{-1}(Z))_{{\rm red}}$. Since $K_{A}\sim 0$, we have
$K_{Y}\sim E$, where $E$ is an effective and $\mu$-exceptional divisor which
has the same support as ${\rm Exc}(\mu)$. Then, by definition,

$\kappa(V)=\kappa(Y,K_{Y}+C)=\kappa(Y,E+C).$

On the other hand $\operatorname{Supp}(\mu^{*}D)\subseteq C$, hence we deduce
that

$\kappa(V)=\kappa(Y,E+C)\geq\kappa(Y,\mu^{*}D)=\kappa(A,D).$

We conclude that in order to finish the proof, we only need to show that

$\kappa(A,D)+\kappa(F)\geq\kappa(X).$

If $D=\emptyset$, then we are done by (2), so we assume that $D$ is non-
trivial. There are two cases, according to whether $D$ is an ample divisor or
not.

If $D$ is ample, then $\kappa(A,D)=\dim A$, and the inequality

$\dim A+\kappa(F)\geq\kappa(X)$

is simply the Easy Addition formula (see e.g. [Mori, Corollary 2.3]).

If $D$ is not ample, a well-known structural fact says that there exist a
fibration $q\colon A\to B$ of abelian varieties and an ample effective divisor
$H$ on $B$ such that $\dim A>\dim B$ and $D=q^{*}H$.

${X}$${A}$${B}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{q}$

Denote the general fiber of $g$ by $G$, and the general fiber of $q$ by $Q$,
which is again an abelian variety. Since $\kappa(B,H)=\dim B$, again by Easy
Addition we have

$\kappa(B,H)+\kappa(G)\geq\kappa(X).$

Since $q$ is a fiber space, we also have

$\kappa(A,D)=\kappa(B,H).$

Let’s now assume more precisely that $G$ and $Q$ are fibers over a point in
$b\in B\smallsetminus H$. We can choose $b$ sufficiently general such that
$\operatorname{codim}_{Q}(Q\cap W)\geq 2$. Since $Q\cap D=\emptyset$, we then
also have

$\operatorname{codim}_{Q}(Q\cap Z)\geq 2.$

We obtain a fiber space $f_{b}\colon G\to Q$ induced by $f$ by restriction,
which is smooth over the complement of a closed subset of $Q$ whose
codimension is at least $2$. By part (2) we deduce that $\kappa(G)=\kappa(F)$,
and thus finally

$\kappa(A,D)+\kappa(F)\geq\kappa(X).$

###  5\. Proof of Theorem D

We start with a general set-up. Since $Y$ is of maximal Albanese dimension, we
can take a Stein factorization of the Albanese morphism $a_{Y}\colon
Y\to\operatorname{Alb}(Y)$ such that $a_{Y}=g\circ h$, where $h\colon Y\to Z$
is birational, $g\colon Z\to\operatorname{Alb}(Y)$ is finite onto its image,
and $Z$ is a normal projective variety.

By [Kawamata, Theorem 13], there exists an étale cover $\varphi\colon
Z^{\prime}\to Z$, an abelian variety $A$, and a normal projective variety $W$
such that

$Z^{\prime}\simeq W\times A\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\kappa(W)=\dim
W=\kappa(Z)=\kappa(Y).$

Denote the projection of $Z^{\prime}$ onto $W$ by $p$. We consider the
following commutative diagram, where the morphisms in the middle column are
obtained by base change from $h$ and $f$ via $\varphi$ and then
$\varphi^{\prime}$, while on the left we have the respective fibers over a
general point $w\in W$. (The horizontal maps between the two left columns are
all inclusions.)

(5.1)
${X^{\prime}_{w}}$${X^{\prime}}$${X}$${Y^{\prime}_{w}}$${Y^{\prime}}$${Y}$${A=A_{w}}$${Z^{\prime}}$${Z}$${\\{w\\}}$${W}$$\scriptstyle{f^{\prime}_{w}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{i_{w}}$$\scriptstyle{h^{\prime}_{w}}$$\scriptstyle{h^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{h}$$\scriptstyle{p}$$\scriptstyle{\varphi}$

The projective varieties $X^{\prime}$ and $Y^{\prime}$ are both smooth, since
$\varphi$ is étale. We can choose $w$ sufficiently general so that
$X^{\prime}_{w}$ and $Y^{\prime}_{w}$ are smooth, and $h^{\prime}_{w}$ is
birational. Moreover $f^{\prime}_{w}$ is a fiber space.

After this preparation, we are ready to prove Theorem D. We start again with
part (2), which is somewhat less involved.

Recall that in (2) we are assuming that $f$ is smooth over an open set
$V\subseteq Y$, and denote $T=Y\smallsetminus V$. The morphism $f^{\prime}$ is
smooth over $Y^{\prime}\smallsetminus\varphi^{\prime-1}(T)$, and
$\operatorname{codim}\varphi^{\prime-1}(T)\geq 2$. If
$T^{\prime}_{w}:=(\varphi^{\prime}\circ i_{w})^{-1}(T)$, we also that
$f^{\prime}_{w}$ is smooth over $Y^{\prime}_{w}\smallsetminus T^{\prime}_{w}$
as well, while choosing $w$ carefully we can also ensure that
$\operatorname{codim}_{Y^{\prime}_{w}}T^{\prime}_{w}\geq 2$. Since
$h^{\prime}_{w}$ is a birational morphism, it is immediate to check that
$h^{\prime}_{w}\circ f^{\prime}_{w}$ is also smooth over the complement in $A$
of a closed subset of codimension at least $2$, and its general fiber is $F$.

At this stage, by Theorem A(2) we have $\kappa(X^{\prime}_{w})=\kappa(F)$. On
the other hand, since $W$ is of general type, the additivity of the Kodaira
dimension holds for the fiber space $X^{\prime}\to W$ with fiber
$X^{\prime}_{w}$ (see [Viehweg, Corollary IV]),333Note that $W$ being normal
suffices, since for this statement one can pass to a base change by a
resolution of $W$. to finally give

$\kappa(X)=\kappa(X^{\prime})=\kappa(X^{\prime}_{w})+\kappa(W)=\kappa(F)+\kappa(Y).$

We now prove (1). We note first that we can assume $Y\smallsetminus V$ to be a
simple normal crossing divisor. Let $\mu\colon\widetilde{Y}\to Y$ be a log
resolution of $Y\smallsetminus V$ which is an isomorphism over $V$. In
particular $\mu^{-1}(Y\smallsetminus V)$ is a divisor with simple normal
crossing support, and $\widetilde{Y}$ is still of maximal Albanese dimension.
We consider the commutative diagram

${\widetilde{X}}$${(X\times_{Y}\widetilde{Y})_{\rm{main}}}$${X}$${\widetilde{Y}}$${Y}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{m}$$\scriptstyle{f}$$\scriptstyle{\mu}$

where $(X\times_{Y}\widetilde{Y})_{\rm{main}}$ is the main component of
$X\times_{Y}\widetilde{Y}$, and $\widetilde{X}$ is a resolution which is an
isomorphism over $(\mu\circ m)^{-1}(V)$. The algebraic fiber space
$\widetilde{f}\colon\widetilde{X}\to\widetilde{Y}$ is smooth over
$\mu^{-1}(V)\simeq V$, with general fiber $F$. Thus we can assume from the
start that $D:=Y\smallsetminus V$ is a simple normal crossing divisor, and

$\kappa(V)=\kappa(Y,K_{Y}+D).$

We next show that we can also assume that there exists a birational morphism
$h\colon Y\to A$, where $A$ is an abelian variety. We consider again the
diagram (5.1) in our set-up, but we now keep track of the divisor $D$ and its
pullbacks as well. Since $\varphi^{\prime}$ is étale,
$D^{\prime}:=\varphi^{\prime*}D$ is also a simple normal crossing divisor. For
$w$ sufficiently general, in addition to all the properties described in the
set-up, we can also assume that $D^{\prime}_{w}:=D^{\prime}|_{Y^{\prime}_{w}}$
is a simple normal crossing divisor. Since $W$ is of general type, as above we
have

$\kappa(X)=\kappa(X^{\prime})=\kappa(X^{\prime}_{w})+\dim W,$

but we also have the additivity of log Kodaira dimension

$\kappa(Y,K_{Y}+D)=\kappa(Y^{\prime},K_{Y^{\prime}}+D^{\prime})=\kappa(Y^{\prime}_{w},K_{Y^{\prime}_{w}}+D^{\prime}_{w})+\dim
W.$

(See [Fujino, Theorem 1.7] and the references therein.) Now the general fiber
of $f^{\prime}_{w}$ is $F$. Thus to obtain the conclusion
$\kappa(V)+\kappa(F)\geq\kappa(X)$, we only need to prove the inequality

$\kappa(Y^{\prime}_{w},K_{Y^{\prime}_{w}}+D^{\prime}_{w})+\kappa(F)\geq\kappa(X^{\prime}_{w}).$

Noting that the morphism $f^{\prime}_{w}$ is smooth over
$V^{\prime}_{w}:=(\varphi^{\prime}\circ
i_{w})^{-1}(V)=Y^{\prime}_{w}\smallsetminus D^{\prime}_{w}$, and

$\kappa(V^{\prime}_{w})=\kappa(Y^{\prime}_{w},K_{Y^{\prime}_{w}}+D^{\prime}_{w}),$

this allows us indeed to assume from the start (replacing $f\colon X\to Y$ by
$f^{\prime}_{w}\colon X^{\prime}_{w}\to Y^{\prime}_{w}$) that there exists a
birational morphism $h\colon Y\to A$, with $A$ is an abelian variety.

The picture is now

${X}$${Y}$${A}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$

At this stage we can forget about the previous steps in the proof, and reuse
some of the notation symbols. The birational morphism $h$ is an isomorphism
away from $Z:=h\big{(}\operatorname{Supp}(D)\cup{\rm Exc}(h)\big{)}$ and thus
$g$ is smooth over $A\smallsetminus Z$. By Theorem A(1), we then have

$\kappa(A\smallsetminus Z)+\kappa(F)\geq\kappa(X).$

We can take a log resolution $\mu\colon Y^{\prime}\to Y$ which is an
isomorphism over $Y\smallsetminus\big{(}\operatorname{Supp}(D)\cup{\rm
Exc}(h)\big{)}$, such that the support of
$\mu^{-1}\big{(}\operatorname{Supp}(D)\cup{\rm Exc}(h)\big{)}\cup{\rm
Exc}(\mu)$ is a simple normal crossing divisor $D^{\prime}$. Since
$h^{-1}(Z)=\operatorname{Supp}(D)\cup{\rm Exc}(h)$, we deduce that

$\kappa(A\smallsetminus Z)=\kappa(Y^{\prime},K_{Y^{\prime}}+D^{\prime}).$

Now $K_{Y^{\prime}}\sim\mu^{*}K_{Y}+E^{\prime}$, where $E^{\prime}$ is an
effective and $\mu$-exceptional divisor. Since $A$ is an abelian variety, we
also have $K_{Y}\sim E$, where $E$ is an effective and $h$-exceptional divisor
supported on ${\rm Exc}(h)$. We deduce that

$K_{Y^{\prime}}+D^{\prime}\sim\mu^{*}E+E^{\prime}+D^{\prime}\geq 0,$

and since
$\operatorname{Supp}(\mu^{*}E+E^{\prime}+D^{\prime})=\operatorname{Supp}(\mu^{*}E+E^{\prime}+\mu^{*}D)$,
we finally have

$\kappa(A\smallsetminus
Z)=\kappa(Y^{\prime},K_{Y^{\prime}}+D^{\prime})=\kappa(Y^{\prime},\mu^{*}E+E^{\prime}+D^{\prime})$
$=\kappa(Y^{\prime},\mu^{*}E+E^{\prime}+\mu^{*}D)=\kappa(Y^{\prime},\mu^{*}(K_{Y}+D)+E^{\prime})=\kappa(Y,K_{Y}+D)=\kappa(V).$

Putting everything together, we obtain

$\kappa(V)+\kappa(F)\geq\kappa(X).$

###  6\. Proof of Theorem E

It is clear that

$\kappa(X,K_{X}+E)\geq\kappa(X,K_{X}+(f^{*}D)_{\rm red}),$

hence to prove the theorem it suffices to assume $E=(f^{*}D)_{\rm red}$.

We first show that we can reduce to the case when $E$ has simple normal
crossings. Take a log resolution $\mu\colon Y\to X$ of the pair $(X,E)$, so
that $\mu^{*}E$ has simple normal crossing support, and let
$K_{Y}=\mu^{*}K_{X}+G$ with $G$ an effective exceptional divisor on $Y$. We
may assume that $\mu$ is an isomorphism over $X\smallsetminus E$. Thus the
general fiber of $g=f\circ\mu$ is isomorphic to $F$.

${Y}$${X}$${A}$$\scriptstyle{\mu}$$\scriptstyle{g}$$\scriptstyle{f}$

We have that $\mu^{*}E\geq(g^{*}D)_{\rm red}$, and since $G$ is effective and
exceptional, we deduce that

$\kappa(X,K_{X}+E)=\kappa(Y,\mu^{*}(K_{X}+E)+G)$
$=\kappa(Y,K_{Y}+\mu^{*}E)\geq\kappa(Y,K_{Y}+(g^{*}D)_{\rm
red})\geq\kappa(F)+\kappa(A,D),$

where the last inequality holds if we assume the simple normal crossings case.

We proceed therefore assuming that $E$ has simple normal crossings. Let $k\geq
1$ be the maximal coefficient of the irreducible components of $f^{*}D$. Thus
the coefficients of the divisor $E-\frac{1}{k}f^{*}D$ are nonnegative and
strictly smaller than $1$. Since it has simple normal crossing support, we
deduce that the log pair $(X,E-\frac{1}{k}f^{*}D)$ is klt. We consider the
divisor class

$T:=kK_{X}+kE-f^{*}D\sim_{\mathbb{Q}}k(K_{X}+E-\frac{1}{k}f^{*}D).$

By [Meng, Theorem 1.1], generalizing Theorem 2.1, there exists an isogeny
$\varphi\colon A^{\prime}\to A$ and an induced fiber diagram

${X^{\prime}}$${X}$${A^{\prime}}$${A}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\varphi}$

such that
$f^{\prime}_{*}\mathscr{O}_{X^{\prime}}(\varphi^{\prime*}(mT))\simeq\varphi^{*}f_{*}\mathscr{O}_{X}(mT)$
is globally generated for all $m\geq 1$. Note that by the base change theorem

${\rm rk}\big{(}f_{*}\mathscr{O}_{X}(mT)\big{)}=P_{mk}(F).$

We conclude that for each $m\geq 1$ there exist injective coherent sheaf
homomorphisms

$\mathscr{O}_{A^{\prime}}^{\oplus P_{mk}(F)}\hookrightarrow
f^{\prime}_{*}\mathscr{O}_{X^{\prime}}(\varphi^{\prime*}(mT)).$

Tensoring with $\mathscr{O}_{A^{\prime}}(\varphi^{*}(mD))$, and noting that

$f^{\prime}_{*}\mathscr{O}_{X^{\prime}}(\varphi^{\prime*}(mT))\otimes\mathscr{O}_{A^{\prime}}(\varphi^{*}(mD))\simeq
f^{\prime}_{*}\mathscr{O}_{X^{\prime}}\big{(}\varphi^{\prime*}(mk(K_{X}+E))\big{)},$

this leads to injections

$\mathscr{O}_{A^{\prime}}(\varphi^{*}(mD))^{\oplus P_{mk}(F)}\hookrightarrow
f^{\prime}_{*}\mathscr{O}_{X^{\prime}}\big{(}mk(K_{X^{\prime}}+\varphi^{\prime*}E)\big{)}.$

Thus we deduce that

$P_{mk}(X^{\prime},\varphi^{\prime*}E)\geq P_{mk}(F)\cdot
P_{m}(A^{\prime},\varphi^{*}D)\geq P_{mk}(F)\cdot P_{m}(A,D)$

for all $m$. Since $k$ is fixed and
$\varphi^{\prime*}\omega_{X}\simeq\omega_{X^{\prime}}$, it is immediate to see
that this implies

$\kappa(X^{\prime},K_{X^{\prime}}+\varphi^{\prime*}E)\geq\kappa(F)+\kappa(A,D).$

Now $\varphi^{\prime}$ is surjective, hence
$\kappa\big{(}X^{\prime},\varphi^{\prime*}(K_{X}+E)\big{)}=\kappa(X,K_{X}+E)$,
and so finally

$\kappa(X,K_{X}+E)\geq\kappa(F)+\kappa(A,D).$

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