# Remarks on the Kodaira dimension of base spaces of families of manifolds

Behrouz Taji  Behrouz Taji, University of Notre Dame, Department of
Mathematics, 278 Hurley, Notre Dame, IN 46556 USA<EMAIL_ADDRESS>http://sites.nd.edu/b-taji

###### Abstract.

We prove that the variation in a smooth projective family of varieties
admitting a good minimal model forms a lower bound for the Kodaira dimension
of the base, if the dimension of the base is at most five and its Kodaira
dimension is non-negative. This gives an affirmative answer to the conjecture
of Kebekus and Kovács for base spaces of dimension at most five.

###### Key words and phrases:

Families of manifolds, minimal models, Kodaira dimension, variation of Hodge
structures, moduli of polarized varieties.

###### 2010 Mathematics Subject Classification:

14D06, 14D23, 14E05, 14E30, 14D07.

##  1\. Introduction and main results

###  1.1. Introduction

A conjecture of Shafarevich [Shaf63] and Viehweg [Viehweg01] predicted that if
the geometric general fiber of a smooth projective family $f_{U}:U\to V$ is
canonically polarized, then $\kappa(V)=\dim(V)$, assuming that $f_{U}$ has
_maximal variation_. The problem was subsequently generalized to the case of
fibers with good minimal models, cf. [Vie-Zuo01] and [PS15]. This conjecture
is usually referred to as the _Viehweg Hyperbolicity Conjecture_ and has been
recently settled in full generality.

Generalizing Viehweg’s conjecture, in their groundbreaking series of papers,
Kebekus and Kovács predicted that variation in the smooth family $f_{U}$,
which we denote by $\operatorname{Var}(f_{U})$, should be closely connected to
$\kappa(V)$, even when it is _not_ maximal.

######  Conjecture 1.1 (Kebekus-Kovács Hyperbolicity Conjecture. I, cf. [KK08,
Conj. 1.6]).

Let $f_{U}:U\to V$ be a smooth projective family whose general fiber admits a
good minimal model 111The original conjecture of Kebekus and Kovács was
formulated for the case of canonically polarized fibers.. Then, either

  1. (1.1.1)

$\kappa(V)=-\infty$ and $\operatorname{Var}(f_{U})<\dim V$, or

  2. (1.1.2)

$\kappa(V)\geq 0$ and $\operatorname{Var}(f_{U})\leq\kappa(V)$.

Once the family $f_{U}$ arises from a moduli functor with an algebraic coarse
moduli scheme, then an even stronger version of Conjecture 1.1, that is due to
Campana, can be verified (see Section 5). However, the main focus of this
paper is to study Conjecture 1.1, when the family $f_{U}$ is not associated
with a well-behaved moduli functor, even after running a relative minimal
model program. Most families of higher dimensional projective manifolds that
are not of general type but have pseudo-effective canonical bundle belong to
this category.

The following theorem is the main result of this paper.

######  Theorem 1.2.

Conjecture 1.1 holds when $\dim(V)\leq 5$.

When dimension of the base and fibers are equal to one, Viehweg’s
hyperbolicity conjecture (or equivalently Conjecture 1.1) was proved by
Parshin [Parshin68], in the compact case, and in general by Arakelov
[Arakelov71]. For higher dimensional fibers and assuming that $\dim(V)=1$,
this conjecture was confirmed by Kovács [Kovacs00a], in the canonically
polarized case (see also [Kovacs02]), and by Viehweg and Zuo [Vie-Zuo01] in
general. Over Abelian varieties Viehweg’s conjecture was solved by Kovács
[Kovacs97c]. When $\dim(V)=2$ or $3$, it was resolved by Kebekus and Kovács,
cf. [KK08] and [KK10]. In the compact case it was settled by Patakfalvi
[MR2871152]. Viehweg’s conjecture was finally solved in complete generality by
the fundamental work of Campana and Păun [CP16] and more recently by Popa and
Schnell [PS15]. For the more analytic counterparts of these results please see
[Vie-Zuo03a], [Sch12], [TY15], [BPW], [TY16], [PTW] and [Deng].

By using the solution of Viehweg’s hyperbolicity conjecture, one can
reformulate Conjecture 1.1 as follows.

######  Conjecture 1.3 (Kebekus-Kovács Hyperbolicity Conjecture. II).

Let $f_{U}:U\to V$ be a smooth family of projective varieties and $(X,D)$ a
smooth compactification of $V$ with $V\cong X\smallsetminus D$. Assume that
the geometric general fiber of $f_{U}$ has a good minimal model. Then, the
inequality

$\operatorname{Var}(f_{U})\leq\kappa(X,D)$

holds, if $\kappa(X,D)\geq 0$.

When $f_{U}$ is canonically polarized, Conjecture 1.3 was confirmed by Kebekus
and Kovács in [KK08], assuming that $\dim(X)=2$, in [KK10], if $\dim(X)=3$,
and as a consequence of [taji16] in full generality. The latter result
establishes an independent, but closely related, conjecture of Campana; the
so-called _Isotriviality Conjecture_.

Campana’s conjecture (Conjecture 5.1) predicts that once the fibers of $f_{U}$
are canonically polarized (or more generally have semi-ample canonical bundle,
assuming that $f_{U}$ is polarized with a fixed Hilbert polynomial), then
$f_{U}$ is isotrivial, if $V$ is _special_. We refer to Section 5 for more
details on the notion of special varieties and various other particular cases
where the Isotriviality Conjecture and Conjecture 1.3 can be confirmed.

###  1.2. Brief review of the strategy of the proof

A key tool in proving Conjecture 1.3, in the canonically polarized case, is
the celebrated result of Viehweg and Zuo [Vie-Zuo01, Thm. 1.4.(i)] on the
existence of an invertible subsheaf $\scr{L}\subseteq\Omega^{\otimes
i}_{X}\log(D)$, for some $i\in\mathbb{N}$, whose Kodaira dimension verifies
the inequality:

(1.3.1) $\kappa(X,\scr{L})\geq\operatorname{Var}(f_{U}).$

The sheaf $\scr{L}$ is usually referred to as a _Viehweg-Zuo subsheaf_. In
general, once the canonically polarized condition is dropped, in the absence
of a well-behaved moduli functor associated to the family, the approach of
[Vie-Zuo01] cannot be directly applied. Nevertheless, we show that one can
still construct a subsheaf of $\Omega_{X}^{\otimes i}\log(D)$ arising from the
variation in $f_{U}$, as long as the geometric general fiber of $f_{U}$ has a
good minimal model. But in this more general context the sheaf $\scr{L}$
injects into $\Omega^{\otimes i}_{X}\log(D)$ only after it is twisted by some
pseudo-effective line bundle $\scr{B}$. As such we can no longer guarantee
that this (twisted) subsheaf verifies the inequality (1.3.1).

######  Theorem 1.4 (Existence of pseudo-effective Viehweg-Zuo subsheaves).

Let $f_{U}:U\to V$ be a smooth non-isotrivial family whose general fiber
admits a good minimal model. Let $(X,D)$ a smooth compactification of $V$.
There exist a positive integer $i$, a line bundle $\scr{L}$ on $X$, with
$\kappa(X,\scr{L})\geq\operatorname{Var}(f_{U})$, and a pseudo-effective line
bundle $\scr{B}$ with an inclusion

$\scr{L}\otimes\scr{B}\subseteq\Omega_{X}^{\otimes i}\log(D).$

To mark the difference between the two cases we refer to these newly
constructed sheaves as _pseudo-effective Viehewg-Zuo subsheaves_.

For the proof of Theorem 1.4 we follow the general strategy of [VZ02] and more
recently [PS15], where one combines the positivity properties of direct images
of relative dualizing sheaves with the negativity of curvature of Hodge
metrics along certain subsheaves of variation of Hodge structures to construct
subsheaves of $\Omega_{X}^{\otimes i}\log(D)$ that are sensitive to the
variation in the birational structure of the members of the family. We note
that, in our constructions, we do not make use of the theory of Hodge modules.
In particular Saito’s decomposition theorem will not be used.

It is worth pointing out that when $\dim(X)=\operatorname{Var}(f_{U})$,
$\scr{L}$ in Theorem 1.4 is big and we have
$\kappa(X,\scr{L}\otimes\scr{B})=\dim(X)$. Thus, in this case, Theorem 1.4
coincides with the result of [PS15, Thm. B] and [VZ02, Thm. 1.4.(i)] (the
latter in the canonically polarized case), while providing a simpler proof.

When variation is not maximal, Theorerm 1.4 is notably different from—and in
some sense weaker than—the theorem of Viehweg and Zuo [VZ02, Thm. 1.4.(i)] in
the canonically polarized case. The reason for this difference is due to the
fact that without a reasonable functor associated to the family, the existence
of Viehweg-Zuo subsheaves can no longer be extracted from their construction
at the level of moduli stacks, where the variation is maximal. We refer the
reader to Section 5 for more details and a brief review of the cases where
this difficulty can be overcome.

Once Theorem 1.4 is established, it remains to trace a connection between
$\kappa(\scr{L})$ and $\kappa(X,D)$. In the maximal variation case, Viehweg-
Zuo subsheaves are guaranteed to be big (as soon as the geometric general
fiber has a good minimal model). In this case a key result of Campana and Păun
then implies that $\kappa(X,D)=\dim(X)$, cf. [CP16, Thm 7.11]. But when
$\operatorname{Var}(f_{U})<\dim(X)$, as $\scr{L}$ is not big, this strategy
can no longer be applied. Instead, our proof of Theorem 1.2 relies on the
following vanishing result.

######  Theorem 1.5 (Vanishing for twisted logarithmic pluri-differential
forms).

Let $(X,D)$ be a pair consisting of a smooth projective variety $X$ of
dimension at most equal to $5$ and a reduced effective divisor $D$ with simple
normal crossing support. If $\kappa(X,D)\geq 0$, then the equality

$H^{0}\big{(}X,\Omega^{\otimes
i}_{X}\log(D)\otimes(\scr{L}\otimes\scr{B})^{-1}\big{)}=0$

holds, assuming that $\scr{B}$ is pseudo-effective and
$\kappa(\scr{L})>\kappa(X,D)$.

Theorem 1.2 is now an immediate consequence of Theorems 1.5 and 1.4.

In the presence of a flat Kähler-Einstein metric, or assuming that the main
conjectures of the minimal model program hold, various analogues of Theorem
1.5 can be verified. When $D=0$ and $c_{1}(X)=0$, and
$\scr{B}\cong\scr{O}_{X}$, Theorem 1.5 follows, in all dimensions, directly
from Yau’s solution to Calabi’s conjecture [MR0451180] and the Bochner
formula. When $\scr{B}\cong\scr{O}_{X}$ and $D=0$, the vanishing in Theorem
1.5 was conjectured by Campana in [Ca95] where he proved that it holds for an
$n$-dimensional variety $X$, if the Abundance Conjecture holds in dimension
$n$. As was shown by Campana, such vanishing results are closely related to
compactness properties of the universal cover of algebraic varieties. We refer
to [Ca95] for details of this very interesting subject (see also the book of
Kollár [Kollar95s] and [Kollar93]). We also invite the reader to consult
[CP16, Thm. 7.3] where the authors successfully deal with a similar problem
with $\kappa$ replaced by another invariant $\nu$; the latter being the
numerical Kodaira dimension.

###  1.3. Structure of the paper

In Section 2 we provide the preliminary constructions needed for the proof of
Theorem 1.4. The proof of Theorem 1.4 appears in Section 3. In Section 4 we
prove the vanishing result; Theorem 1.5. Section 5 is devoted to further
results and related problems, including a discussion on the connection between
Conjecture 1.3 and a conjecture of Campana.

###  1.4. Acknowledgements

I would like to thank Sándor Kovács, Frédéric Campana and Mihnea Popa for
their interest and suggestions. I am also grateful to Christian Schnell for
pointing out a mistake in an earlier draft of this paper.

##  2\. Preliminary results and constructions

Our aim in this section is to establish two ket background results that we
will need in order to construct the pseudo-effective Viehweg-Zuo subsheaves in
the proceeding section.

###  2.1. Positivity of direct images of relative dualizing sheaves

In [Kawamata85, Thm 1.1] Kawamata shows that, assuming that the general fiber
admits a good minimal model, for any algebraic fiber space $f:Y\to X$ of
smooth projective varieties $Y$ and $X$, the inequality

(2.0.1)
$\kappa\Big{(}X,\big{(}\det(f_{*}\omega^{m}_{Y/X})\big{)}^{**}\Big{)}\geq\operatorname{Var}(f),$

holds, for all sufficiently large integers $m\geq 1$.

One can use (2.0.1) to extract positivity results for $f_{*}\omega^{m}_{Y/X}$.
We refer the reader to [Vie-Zuo03a], [VZ02] and [PS15] for the case where
$\operatorname{Var}(f)=\dim(X)$ (see also [Kawamata85] and [Viehweg83]). The
key ingredient is the fiber product trick of Viehweg, where, given $f:Y\to X$
as above, one considers the $r$-fold fiber product

$Y^{r}:=\underbrace{Y\times_{X}Y\times_{X}\ldots\times_{X}Y}_{\text{$r$
times}}.$

We denote by $Y^{(r)}$ a desingularization of $Y^{r}$ and the resulting
morphism by $f^{(r)}:Y^{(r)}\to X$. The next proposition is the extension of
the arguments of [PS15, pp. 708–709] to the case where variation is not
maximal.

######  Proposition 2.1.

Let $f:Y\to X$ be a fiber space of smooth projective varieties $Y$ and $X$. If
the general geometric fiber admits a good minimal model, then, for every
sufficiently large $m\geq 1$, there exists $r:=r(m)\in\mathbb{N}$ and a line
bundle $\scr{L}$ on $X$, with $\kappa(X,\scr{L})\geq\operatorname{Var}(f)$,
and an inclusion

(2.1.1) $\scr{L}^{m}\subseteq f_{*}^{(r)}\big{(}\omega^{m}_{Y^{(r)}/X}\big{)}$

which holds in codimension one.

###### Proof.

According to [Viehweg83, Prop. 6.1] there is a be a finite, flat and Galois
morphism $\gamma:X_{1}\to X$, with $G:=\operatorname{Gal}(X_{1}/X)$, such that
the induced morphism $f_{1}:Y_{1}\to X_{1}$ from the $G$-equivariant
resolution $Y_{1}$ of the fiber product $Y\times_{X_{1}}X$ is semistable in
codimesnion one:

$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{Y\times_{X_{1}}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$flat
and Galois$\textstyle{X.}$

By [Viehweg83, Sect. 3] (see also [Mori87, Thm 4.10]), for any
$m\in\mathbb{N}$, there is an inclusion

$(f_{1})_{*}\omega^{m}_{Y_{1}/X_{1}}\subseteq\gamma^{*}(f_{*}\omega^{m}_{Y/X})$

and thus
$\det((f_{1})_{*}\omega^{m}_{Y_{1}/X_{1}})\subseteq\det(\gamma^{*}(f_{*}\omega^{m}_{Y/X}))$.
Let us define

$\scr{B}_{1}:=\det((f_{1})_{*}\omega^{m}_{Y_{1}/X_{1}})\;\;\;\text{and}\;\;\;\scr{B}:=\det(\gamma^{*}(f_{*}\omega^{m}_{Y/X})).$

We can allow ourselves to remove codimension two subsets from $X$. In
particular we may assume that $\scr{B}_{1}$ and $\scr{B}$ are locally free and
that

(2.1.2) $\scr{B}_{1}\subseteq\gamma^{*}(\scr{B}).$

Next, we observe that, as $\Omega^{1}_{X_{1}}$ and $\Omega^{1}_{Y_{1}}$ are
naturally equipped with the structure of $G$-sheaves (or linearized sheaves),
$\omega^{m}_{Y_{1}/X_{1}}$ is also a $G$-sheaf. It follows that
$(f_{1})_{*}\omega^{m}_{Y_{1}/X_{1}}$ is a $G$-sheaf. The inclusion (2.1.2)
then implies that $\scr{B}_{1}$ descends, that is there is a line bundle
$\scr{L}$ on $X$ such that $\scr{B}_{1}\cong\gamma^{*}(\scr{L})$, cf.
[MR2665168, Thm. 4.2.15]. Moreover, as $\gamma$ is Galois, it follows that
$\kappa(X_{1},\scr{B}_{1})=\kappa(X,\scr{L})$. Note that we also have
$\kappa(\scr{B}_{1})\geq\operatorname{Var}(f)$, again by using [Kawamata85,
Thm 1.1].

Our aim is now to show that $\scr{L}$ is the line bundle admitting the
inclusion (2.1.1). To this end, let
$t:=\mathrm{rank}\bigl{(}(f_{1})_{*}\omega^{m}_{Y_{1}/X_{1}}\bigr{)}$ and
consider the injection

$\scr{B}_{1}^{m}\lhook\joinrel\longrightarrow\bigotimes^{(t+m)}(f_{1})_{*}\omega^{m}_{Y_{1}/X_{1}}.$

Set $r:=(t+m)$. Let $Y_{1}^{(r)}$ be a desingularization of the $r$-fold fiber
product $Y_{1}\times_{Y_{1}}\ldots\times_{X_{1}}Y_{1}$ such that the resulting
morphism from $Y_{1}^{(r)}$ to $Y^{r}$ factors through the desingulariation
map $Y^{(r)}\to Y^{r}$. Let $f_{1}^{(r)}:Y_{1}^{(r)}\to X_{1}$ be the induced
map. Using the fact that $f_{1}$ is semistable in codimension one (and
remembering that we are arguing in codimension one), thanks to [Viehweg83,
Lem. 3.5], we have

$(f_{1}^{(r)})_{*}\big{(}\omega^{m}_{Y_{1}^{(r)}/X_{1}}\big{)}=\bigotimes^{r}(f_{1})_{*}\big{(}\omega^{m}_{Y_{1}/X_{1}}\big{)}.$

On the other hand, since $\gamma$ is flat, according to [Viehweg83, Lem. 3.2],
we have

$(f_{1}^{(r)})_{*}\big{(}\omega^{m}_{Y_{1}^{(r)}/X_{1}}\big{)}\subseteq\gamma^{*}\big{(}f^{(r)}_{*}\omega^{m}_{Y^{(r)}/X}\big{)},$

where $Y^{(r)}$ is a desingularization of $Y^{r}$ with the induced map
$f^{(r)}:Y^{(r)}\to X$. Therefore, there is an injection

$\scr{B}_{1}^{m}=\gamma^{*}(\scr{L}^{m})\lhook\joinrel\longrightarrow\gamma^{*}\big{(}f_{*}^{(r)}\omega^{m}_{Y^{(r)}/X}\big{)}.$

By applying the $G$-invariant section functor $\gamma_{*}(\cdot)^{G}$ to both
sides we find the required injection

$\scr{L}^{m}\lhook\joinrel\longrightarrow f_{*}^{(r)}\omega^{m}_{Y^{(r)}/X}.$

∎

###  2.2. Hodge theoretic constructions

Let $f:Y\to X$ be a surjective, projective morphism of smooth quasi-projective
varieties $Y$ and $X$ of relative dimension $n$ and let
$D=\operatorname{disc}(f)$ be the discriminant locus of $f$. Set
$\Delta:=f^{*}(D)$ and assume that the support of $D$ and $\Delta$ is simple
normal crossing.

######  Definition 2.2 (Systems with $\scr{W}$-valued operators).

Let $\scr{W}$ be a coherent sheaf on $X$. We call the graded torsion free
sheaf $\scr{F}=\bigoplus\scr{F}_{\bullet}$ a system with $\scr{W}$-valued
operator, if it can be equipped with a sheaf morphism
$\tau:\scr{F}\to\scr{F}\otimes\scr{W}$ satisfying the Griffiths transversality
condition
$\tau|_{\scr{F}_{\bullet}}:\scr{F}_{\bullet}\to\scr{F}_{\bullet+1}\otimes\scr{W}$.

Throughout the rest of this section we will allow ourselves to discard closed
subsets of $X$ of $\operatorname{codim}_{X}\geq 2$, whenever necessary.

Our goal is to construct a system $(\scr{F},\tau)$ with
$\Omega^{1}_{X}\log(D)$-operator $\tau$ which can be equipped with compatible
maps to a system of Hodge bundles underlying the variation of Hodge structures
of a second family that arises from $f$ via certain covering constructions.

To this end, let $\scr{M}$ be a line bundle on $Y$ with
$H^{0}(Y,\scr{M}^{m})\neq 0$. Let $\psi:Z\to Y$ be a desingularization of the
finite cyclic covering associated to taking roots of a non-zero section $s\in
H^{0}(Y,\scr{M}^{m})$ so that

(2.2.1) $H^{0}(Z,\psi^{*}\scr{M})\neq 0,$

cf. [Laz04-I, Prop. 4.1.6].

Now, consider the exact sequence of relative differential forms

(2.2.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{*}(\Omega^{1}_{X}\log(D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{Y}\log(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{Y/X}\log(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which is locally free over $X$ (after removing a codimension two subset of
$X$). Define $h:=f\circ\psi$. Using the pullback of (2.2.2) the bundle
$\psi^{*}(\Omega^{\bullet}_{Y}\log(\Delta))$ can be filtered by a decreasing
filtration $F_{i}^{\bullet}$, $i\geq 0$, with
$F^{\bullet}_{i}/F^{\bullet}_{i+1}\cong\psi^{*}(\Omega_{Y/X}^{\bullet-i}\log(\Delta))\otimes
h^{*}(\Omega^{i}_{X}\log(D))$. After taking the quotient by $F^{\bullet}_{2}$,
the exact sequence corresponding to
$F^{\bullet}_{0}/F^{\bullet}_{1}\cong\psi^{*}(\Omega^{\bullet}_{Y/X}\log(\Delta))$
reads as

(2.2.3)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\psi^{*}(\Omega_{Y/X}^{\bullet-1}\log(\Delta))}\otimes
h^{*}(\Omega^{1}_{X}\log(D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\bullet}_{0}/F^{\bullet}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\psi^{*}(\Omega^{\bullet}_{Y/X}\log(\Delta))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where ${\psi^{*}(\Omega_{Y/X}^{\bullet-1}\log(\Delta))}\otimes
h^{*}(\Omega^{1}_{X}\log(D))\cong F^{\bullet}_{1}/F^{\bullet}_{2}$ and

(2.2.4)
$F^{\bullet}_{0}/F^{\bullet}_{2}\cong\psi^{*}(\Omega^{\bullet}_{Y}\log(\Delta))/\big{(}h^{*}\Omega^{2}_{X}\log(D)\otimes\psi^{*}\Omega^{\bullet-2}_{Y/X}\log(\Delta)\big{)}.$

We tensor the sequence (2.2.3) by $\psi^{*}(\scr{M}^{-1})$ to get the exact
sequence

$\begin{multlined}A_{\bullet}:\;\;0\to\psi^{*}(\Omega_{Y/X}^{\bullet-1}\otimes\scr{M}^{-1})\otimes
h^{*}(\Omega^{1}_{X}\log(D))\to(F^{\bullet}_{0}/F^{\bullet}_{2})\otimes\psi^{*}(\scr{M}^{-1})\to\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\longrightarrow\psi^{*}(\Omega^{\bullet}_{Y/X}\log(\Delta)\otimes\scr{M}^{-1})\to
0.\end{multlined}A_{\bullet}:\;\;0\to\psi^{*}(\Omega_{Y/X}^{\bullet-1}\otimes\scr{M}^{-1})\otimes
h^{*}(\Omega^{1}_{X}\log(D))\to(F^{\bullet}_{0}/F^{\bullet}_{2})\otimes\psi^{*}(\scr{M}^{-1})\to\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\longrightarrow\psi^{*}(\Omega^{\bullet}_{Y/X}\log(\Delta)\otimes\scr{M}^{-1})\to
0.$

Now, define a $\Omega^{1}_{X}\log(D)$-valued system
$(\scr{F}=\bigoplus\scr{F}_{\bullet},\tau)$ of weight $n$ on $X$ by

(2.2.5)
$\scr{F}_{\bullet}:=\operatorname{R}^{\bullet}h_{*}\big{(}\psi^{*}(\Omega^{n-\bullet}_{Y/X}\log(\Delta)\otimes\scr{M}^{-1})\big{)}/{\text{torsion}},$

whose map $\tau$ is determined by
$\tau|_{\scr{F}_{\bullet}}:\scr{F}_{\bullet}\to\scr{F}_{\bullet+1}\otimes\Omega^{1}_{X}\log(D)$
being equal to the corresponding morphism in the relative log complex
$\operatorname{\mathbf{R}}h_{*}(A_{n-\bullet})$.

Next, let $\operatorname{disc}(h)=(D+S)$. By removing a subset of $X$ of
$\operatorname{codim}_{X}\geq 2$ we may assume that $(D+S)$ has simple normal
crossing support. We further assume that, after replacing $Z$ by a higher
birational model, if necessary, the divisor $h^{*}(D+S)$ has simple normal
crossing support, which we denote by $\Delta^{\prime}$.

Similar to the above construction, we can consider the exact sequence

$B_{\bullet}:\;\;\Omega^{\bullet-1}_{Z/X}\log(\Delta^{\prime})\otimes
h^{*}(\Omega_{X}^{1}\log(D+S))\to
G^{\bullet}_{0}/{G^{\bullet}_{2}}\to\Omega^{\bullet}_{Z/X}\log(\Delta^{\prime})\to
0,$

with $G^{\bullet}_{i}$ being the decreasing filtration associated to the
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{h^{*}\Omega^{1}_{X}\log(D+S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{Z}\log(\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{Z/X}\log(\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

In this case we have

(2.2.6)
$G^{\bullet}_{0}/G^{\bullet}_{2}\cong\Omega_{Z}^{\bullet}\log(\Delta^{\prime})/\big{(}h^{*}\Omega^{2}_{X}\log(D+S)\otimes\Omega_{Z/X}^{\bullet-2}\log(\Delta^{\prime})\big{)}.$

Now, let $(\scr{E},\theta)$ with
$\theta:\scr{E}\to\scr{E}\otimes\Omega^{1}_{X}\log(D+S)$ be the logarithmic
Higgs bundles underlying the canonical extension of the local system
$\operatorname{R}^{n}h_{*}(\mathbb{C}_{Z\smallsetminus\Delta^{\prime}})$,
whose residues have eigenvalues contained in $[0,1)$, cf. [Deligne70, Prop.
I.5.4]. According to [Ste76, Thm. 2.18] the associated graded module of the
Hodge filtration induces a structure of a Hodge bundle on $(\scr{E},\theta)$
whose gradings $\scr{E}_{\bullet}$ are given by

(2.2.7)
$\scr{E}_{\bullet}=\operatorname{R}^{\bullet}h_{*}\big{(}\Omega^{n-\bullet}_{Z/X}\log(\Delta^{\prime})\big{)},$

with morphisms
$\theta:\scr{E}_{\bullet}\to\scr{E}_{\bullet+1}\otimes\Omega^{1}_{X}\log(D+S)$
determined by those in the complex
$\operatorname{\mathbf{R}}h_{*}(B_{n-\bullet})$.

Our aim is now to construct a morphism from $(\scr{F},\tau)$ to
$(\scr{E},\theta)$ that is compatible with $\tau$ and $\theta$. To this end,
we observe that the non-vanishing (2.2.1) implies that there is a natural
injection

$\psi^{*}\big{(}\Omega^{n-\bullet}_{Y/X}\log(\Delta)\otimes\scr{M}^{-1}\big{)}\lhook\joinrel\longrightarrow\Omega^{n-\bullet}_{Z/X}\log(\Delta^{\prime}).$

Together with the two isomorphisms (2.2.4) and (2.2.6) it then follows that
the induced map

$\frac{F^{n-\bullet}_{0}}{F^{n-\bullet}_{2}}\otimes\psi^{*}(\scr{M}^{-1})\longrightarrow\frac{G^{n-\bullet}_{0}}{G^{n-\bullet}_{2}}$

is an injection. Therefore, the sequence defined by $A_{n-\bullet}$ is a
subsequence of $B_{n-\bullet}$, inducing a morphism of complexes between
$\operatorname{\mathbf{R}}h_{*}(A_{n-\bullet})$ and
$\operatorname{\mathbf{R}}h_{*}(B_{n-\bullet})$. In particular there is a
sheaf morphism

$\Phi_{\bullet}:\underbrace{\operatorname{R}^{\bullet}h_{*}\big{(}\psi^{*}(\Omega^{n-\bullet}_{Y/X}\log(\Delta))\otimes\scr{M}^{-1}\big{)}/{\text{torsion}}}_{\scr{F}_{\bullet}}\longrightarrow\underbrace{\operatorname{R}^{\bullet}h_{*}\big{(}\Omega^{n-\bullet}_{Z/X}\log(\Delta^{\prime})\big{)}}_{\scr{E}_{\bullet}}.$

The compatibility of $\Phi_{\bullet}$, with respect to $\tau$ and $\theta$,
follows from the fact that, by construction, each $\tau|_{\scr{F}_{\bullet}}$
is defined by the corresponding morphism in the complex
$\operatorname{\mathbf{R}}h_{*}(A_{n-\bullet})$;

$\textstyle{\operatorname{\mathbf{R}}h_{*}(A_{n-\bullet})/\text{torsion}:}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scr{F}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\Phi_{\bullet}}$$\textstyle{\scr{F}_{\bullet+1}\otimes\Omega^{1}_{X}\log(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{\bullet+1}}$$\textstyle{...}$$\textstyle{\operatorname{\mathbf{R}}h_{*}(B_{n-\bullet}):}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scr{E}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\scr{E}_{\bullet+1}\otimes\Omega^{1}_{X}\log(D+S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...\;.}$

##  3\. Constructing pseudo-effective Viehweg-Zuo subsheaves

In the current section we will prove Theorem 1.4. We will be working in the
context of the following set-up.

######  Set-up 3.1.

Let $f:Y\to X$ be a smooth compactification of the smooth projective family
$f_{U}:U\to V$ whose general fiber has a good minimal model and
$\operatorname{Var}(f)\neq 0$. Set $D$ to be the divisor defined by
$X\smallsetminus D\cong V$ and $\Delta=\operatorname{Supp}(f^{*}D)$ (both $D$
and $\Delta$ are assumed to have simple normal crossing support).

######  Proposition 3.2.

In the situation of Set-up 3.1, after removing a subset of $X$ of
$\operatorname{codim}_{X}\geq 2$, the following constructions and properties
can be verified.

  1. (3.2.1)

There exists a system of logarithmic Hodge bundles
$(\scr{E}=\bigoplus\scr{E}_{\bullet},\theta)$ on $X$ with
$\theta:\scr{E}_{\bullet}\to\scr{E}_{\bullet+1}\otimes\Omega^{1}_{X}\log(D+S)$.

  2. (3.2.2)

The torsion free sheaf $\ker(\theta|_{\scr{E}_{\bullet}})$ is seminegatively
curved.

  3. (3.2.3)

There exists a subsystem $(\scr{G}=\bigoplus\scr{G}_{\bullet},\theta)$ of
$(\scr{E},\theta)$ such that
$\theta(\scr{G}_{\bullet})\subseteq\scr{G}_{\bullet+1}\otimes\Omega^{1}_{X}\log(D)$.

  4. (3.2.4)

There is a line bundle $\scr{L}$ on $X$, with
$\kappa(X,\scr{L})\geq\operatorname{Var}(f)$, equipped with an injection
$\scr{L}\lhook\joinrel\longrightarrow\scr{G}_{0}$.

Here, and following the terminology introduced in [MR2449950, p. 357] by
Berndtsson and Păun, we say that a torsion free sheaf $\scr{N}$ is
seminegatively curved if it carries a seminegatively curved (possibly
singular) metric $h$ over its smooth locus $X_{\scr{N}}$ (where $\scr{N}$ is
locally free). We refer to [MR2449950] for more details (see also the survey
paper [Pau16]). An immediate consequence of this property is the fact that,
once it is satisfied, then $\det(\scr{N})$ extends to a anti-pseudo-effective
line bundle on the projective variety $X$ and this is all that is needed for
our purposes in the current paper.

Granting Proposition 3.2 for the moment let us proceed with the proof of
Theorem 1.4.

_Proof of Theorem 1.4._ Let $X^{\circ}\subseteq X$ be the open subset over
which Items (3.2.1)–(3.2.4) in Proposition 3.2 are valid. By iterating the
morphism

$\theta\otimes\operatorname{id}:\scr{G}_{j}\otimes\Omega^{\otimes
j}_{X^{\circ}}\log(D)\longrightarrow\scr{G}_{j+1}\otimes\Omega^{\otimes(j+1)}_{X^{\circ}}\log(D),$

for every $j\in\mathbb{N}$, we can construct a map

$\theta^{j}:\scr{G}_{0}\longrightarrow\scr{G}_{j}\otimes\Omega^{\otimes
j}_{X^{\circ}}\log(D).$

Note that $\theta(\scr{G}_{0})\neq 0$. Otherwise, there is an injection

$\scr{L}|_{X^{\circ}}\lhook\joinrel\longrightarrow\ker(\theta|_{\scr{G}_{0}}).$

On the other hand, by construction we have
$\ker(\theta|_{\scr{G}_{0}})\subseteq\ker(\theta|_{\scr{E}_{0}})$ and
according to Item (3.2.2) $\ker(\theta|_{\scr{E}_{0}})$ is seminegatively
curved. This implies that $\scr{L}$ is anti-pseudo-effective on $X$ and
therefore $\kappa(X,\scr{L})\leq 0$, contradicting our assumption on
$\operatorname{Var}(f)$ not being equal to zero.

Now, let $k$ be the positive integer defined by
$k=\max\\{j\in\mathbb{N}\;|\;\theta^{j}(\scr{G}_{0})\neq 0\\}$, so that

$\theta^{k}(\scr{G}_{0})\subset\underbrace{\ker(\theta|_{\scr{G}_{k}})}_{\scr{N}_{k}}\otimes\Omega^{\otimes
k}_{X^{\circ}}\log(D).$

From Item (3.2.4) it follows that there is a non-trivial morphism

$\scr{L}\longrightarrow\scr{N}_{k}\otimes\Omega^{\otimes
k}_{X^{\circ}}\log(D),$

which implies the existence of a non-zero map

(3.2.1)
$\scr{L}\otimes\big{(}\det(\scr{N}_{k})\big{)}^{-1}\longrightarrow\Omega^{\otimes
i}_{X^{\circ}}\log(D),$

for some $i\in\mathbb{N}$. Now, let $\scr{B}$ be the line bundle on $X$
defined by the extension of $\det(\scr{N}_{k})^{-1}$ so that (3.2.1) extends
to the injection

$\scr{L}\otimes\scr{B}\lhook\joinrel\longrightarrow\Omega^{\otimes
i}_{X}\log(D).$

It remains to verify that $\scr{B}$ is pseudo-effective. But again, according
to Item (3.2.2), the torsion free sheaf
$\scr{N}_{k}\subseteq\ker(\theta|_{\scr{E}_{k}})$ is seminegatively curved and
therefore so is $\det(\scr{N}_{k})$ and thus extends to a anti-pseudo-
effective line bundle $\scr{B}^{-1}$ on $X$. ∎

_Proof of Proposition 3.2._ The proof is a consequence of Proposition 2.1
combined with the Hodge theoretic constructions in Subsection 2.2. To lighten
the notation we will replace the initial family $f:Y\to X$ in Set-up 3.1 by
$f^{(r)}:Y^{(r)}\to X$, which was constructed in Proposition 2.1.

After removing a codimension two subset from $X$ over which the inclusion
(2.1.1) holds, define the line bundle

$\scr{M}=\omega_{Y/X}(\Delta)\otimes f^{*}(\scr{L}^{-1})$

so that $H^{0}(Y,\scr{M}^{m})\neq 0$. The arguments in Subsection 2.2 can now
be used to construct the two systems $(\scr{E},\theta)$ and $(\scr{F},\tau)$
defined in (2.2.5) and (2.2.7), with logarithmic poles along $(D+S)$ and $D$,
respectively. Here after deleting a codimension two subset we have assumed
that $\operatorname{Supp}(D+S)$ is simple normal crossing.

Item (3.2.2) follows from Zuo’s result [Zuo00]—based on Cattani, Kaplan and
Schmid’s work on asymptotic behaviour of Hodge metrics, cf. [CKS] and [PTW,
Lem 3.2]—and Brunbarbe [Bru15], and more generally from Fujino and Fujisawa
[FF17]. The reader may wish to consult Simpson [MR1040197] and [Bru17] where
these problems are dealt with in the more general setting of tame harmonic
metrics.

For (3.2.3), define
$(\scr{G}=\bigoplus\scr{G}_{\bullet},\theta)\subset(\scr{E},\theta)$ to be the
image of the system $(\scr{F}=\bigoplus\scr{F}_{\bullet},\tau)$ under the
morphism $\Phi_{\bullet}$, constructed in Subsection 2.2.

It remains to verify (3.2.4). According to the definition of $\Phi_{\bullet}$
we have

$\Phi_{0}:\operatorname{R}^{0}h_{*}\big{(}\psi^{*}(\omega_{Y/X}(\Delta)\otimes\scr{M}^{-1})\big{)}\longrightarrow\operatorname{R}^{0}h_{*}(\omega_{Z/X}(\Delta^{\prime})),$

which is an injection; for the map

$\psi^{*}\big{(}\omega_{Y/X}(\Delta)\otimes\scr{M}^{-1}\big{)}\longrightarrow\omega_{Z/X}(\Delta^{\prime})$

is injective. Item (3.2.4) now follows from the isomorphism

$h_{*}\big{(}\psi^{*}(\omega_{Y/X}(\Delta)\otimes\underbrace{\omega^{-1}_{Y/X}(\Delta)\otimes
f^{*}\scr{L}}_{\scr{M}^{-1}})\big{)}\cong\scr{L}.$

##  4\. Vanishing results

In this section we will prove Theorem 1.5. The methods will heavily rely on
birational techniques and results in the minimal model program. For an in-
depth discussions of preliminary notions and background we refer the reader to
the book of Kollár and Mori [KM98] and the references therein.

Not surprisingly an important construction that we will repeatedly make use of
is the Iitaka fibration. Please see [Mori87, Sect. 1] and [Laz04-I, Sect.
2.1.C] for the definition and a review of the basic properties.

######  Notation 4.1.

Let $\scr{L}$ be a line bundle on a normal projective variety $X$ with
$\kappa(X,\scr{L})>0$. By $\phi^{(I)}:X^{(I)}\to Y^{(I)}$ we denote the Iitaka
fibration of $\scr{L}$ with an induced birational morphism
$\pi^{(I)}:X^{(I)}\to X$.

###  4.1. Proof of Theorem 1.5

We begin by stating the following lemma concerning the behaviour of the
Kodaira dimension on fibers of the Iitaka fibration. The proof follows from
standard arguments; see for example [Laz04-I, pp. 136–137].

######  Lemma 4.2.

Let $X$ be a smooth projective variety and
${\pi^{(\operatorname{I})}}:X^{(\operatorname{I})}\to X$ a birational morphism
such that $\phi^{(\operatorname{I})}:X^{(\operatorname{I})}\to
Z^{(\operatorname{I})}$ is the Iitaka fibration of the line bundle $\scr{L}$
on $X$. Then, for any ${\pi^{(\operatorname{I})}}$-exceptional and effective
divisor $E$, we have

$\kappa\big{(}F^{(\operatorname{I})},({\pi^{(\operatorname{I})}}^{*}(\scr{L})\otimes\scr{O}_{X^{(\operatorname{I})}}(E))\big{|}_{F^{(\operatorname{I})}}\big{)}=0,$

where $F^{(\operatorname{I})}$ is a very general fibre of
$\phi^{(\operatorname{I})}$.

The next lemma is the final technical background that we need before we can
proceed to the proof of Theorem 1.5. It relies on the so-called flattening
lemma, due to Gruson and Raynaud, which we recall below.

Let $f:X\to Z$ be an algebraic fiber space of normal (quasi-)projective
varieties. There exists an equidimensional fiber space
$f^{\prime}:X^{\prime}\to Z^{\prime}$ of normal varieties that is birationally
equivalent to $f$ through birational morpshisms $\sigma:X^{\prime}\to X$ and
$\tau:Z^{\prime}\to Z$. We call $f^{\prime}$ the _flattening_ of $f$.

######  Lemma 4.3.

Let $f:X\to Z$ be a fiber space of normal projective varieties with a
flattening $f^{\prime}:X^{\prime}\to Z^{\prime}$ as above. Let $A$ be a
$f$-nef and effective $\mathbb{Q}$-divisor and assume that $A|_{F}\equiv 0$,
where $F$ is the general fiber of $f$. There exists
$A_{Z^{\prime}}\in\operatorname{Div}(Z^{\prime})_{\mathbb{Q}}$ such that
$\sigma^{*}(A)\sim_{\mathbb{Q}}(f^{\prime})^{*}(A_{Z^{\prime}})$.

###### Proof.

It suffices to show that $\sigma^{*}(A)$ is $f^{\prime}$-numerically trivial.
Aiming for a contradiction, assume that there exists a fiber $F_{z^{\prime}}$
of $f^{\prime}$ containing an irreducible contractible curve $C$ such that
$\sigma^{*}(A)\cdot C\neq 0$. Let $A_{1},\ldots,A_{d-1}$ and $A_{d}$ be a
collection of ample divisors in $X^{\prime}$ such that $(H_{1}\cdot\ldots\cdot
H_{d})\cap F_{z^{\prime}}$ defines the numerical cycle of $(C+\widetilde{C})$,
where $\widetilde{C}$ is an effective $1$-cycle in $F_{z^{\prime}}$. Since
$\sigma^{*}(A)|_{F^{\prime}}\equiv 0$, where $F^{\prime}$ is a general fiber
of $f^{\prime}$, we have

(4.3.1) $\sigma^{*}(A)\cdot(C+C^{\prime})=0.$

On the other hand, $\sigma^{*}(A)$ is $f^{\prime}$-nef. Combined with (4.3.1),
this implies that $\sigma^{*}(A)\cdot C=0$, which contradicts our initial
assumption.

∎

_Proof of Theorem 1.5. (Preparation)._ Let $\scr{L}$ and $\scr{B}$ be two line
bundles on $X$, with $\scr{B}$ being pseudo-effective, such that, for some
$i\in\mathbb{N}$, there is a non-trivial morphism

$\scr{L}\otimes\scr{B}\longrightarrow\Omega_{X}^{\otimes i}\log(D).$

######  Claim 4.4.

The line bundle $w_{X}^{i}(D)\otimes\scr{L}^{-1}$ is pseudo-effective.

_Proof of Claim 4.4._ Let $\scr{F}$ be the saturation of the image of the non-
trivial morphism

$\scr{L}\otimes\scr{B}\longrightarrow\Omega^{\otimes i}_{X}\log(D)$

and $\scr{Q}$ the torsion-free quotient resulting in the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scr{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\otimes
i}_{X}\log(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scr{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

After taking determinants we find that

(4.4.1) $\omega_{X}^{i}(D)\otimes\scr{F}^{-1}\cong\det(\scr{Q}).$

Thanks to [CP16, Thm. 1.3] the right-hand side of (4.4.1) is pseudo-effective
and thus so is the left-hand side. This implies that
$\omega^{i}_{X}(D)\otimes(\scr{L}\otimes\scr{B})^{-1}\in\overline{\operatorname{NE}}^{1}(X).$
But $\scr{B}$ is also assumed to be pseudo-effective and therefore
$\omega^{i}_{X}(D)\otimes\scr{L}^{-1}\in\overline{\operatorname{NE}}^{1}(X)$.
This finishes the proof of the claim.

Before we proceed further, notice that we may assume that $(i(K_{X}+D)+L)$ is
not big, where $\scr{L}=\scr{O}_{X}(L)$. Otherwise the divisor $K_{X}+D$ is
big, as it can be written as the sum of a pseudo-effective and big divisors:

$K_{X}+D=\frac{1}{2i}\big{(}(i\cdot(K_{X}+D)+L)+(i\cdot(K_{X}+D)-L)\big{)}.$

We can also assume that $\kappa(\scr{L})\geq 1$.

The first step in proving the theorem consists of replacing $X$ by a
birational model $Y$ where establishing the vanishing in Theorem 1.5 proves to
be easier.

######  Claim 4.5.

There exists a birational morphism $\pi:Y\to X$ from a smooth projective
variety $Y$ that can be equipped with a fiber space $f:Y\to Z$ over a smooth
projective variety $Z$. Furthermore, $Y$ contains a reduced divisor $D_{Y}$
such that $(Y,D_{Y})$ is log-smooth and a divisor $L_{Y}$ which satisfy the
following properties.

  1. (4.5.1)

$\dim(Z)=\kappa(i(K_{X}+D)+L)$,

  2. (4.5.2)

$\scr{O}_{Y}(L_{Y})\otimes\pi^{*}(\scr{B})\subseteq\Omega^{\otimes
i}_{Y}\log(D_{Y})$,

  3. (4.5.3)

$\kappa(K_{Y}+D_{Y})=\kappa(K_{X}+D)$,

  4. (4.5.4)

$\kappa(L_{Y})=\kappa(L)$,

  5. (4.5.5)

$\kappa(F,(i(K_{Y}+D_{Y})+L_{Y})|_{F})=0$ and
$\kappa(F,(K_{Y}+D_{Y})|_{F})=0$, where $F$ is a very general fiber of $f$,

  6. (4.5.6)

$\big{(}i(K_{Y}+D_{Y})-L_{Y}\big{)}\in\overline{\operatorname{NE}}^{1}(Y)$,
and

  7. (4.5.7)

$L_{Y}\sim_{\mathbb{Q}}f^{*}(L_{Z})$, for some
$L_{Z}\in\operatorname{Div}_{\mathbb{Q}}(Z)$.

_Proof of Claim 4.5._ Let $\psi:X\dashrightarrow Z$ be the rational mapping
associated to the linear system $|m\cdot(i(K_{X}+D)+L)|$, with $m$ being
sufficiently large so that $\dim(Z)=\kappa(i(K_{X}+D)+L)$. Note that as
$\kappa(L)\geq 1$, we have $\dim(Z)\geq 1$. Let $\psi_{1}:X_{1}\to Z_{1}$ be
the Iitaka fibration of $(i(K_{X}+D)+L)$ resulting in the commutative diagram

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\psi_{1}}$$\textstyle{Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{Z,}$

where $\pi_{1}:X_{1}\to X$ is a birational morphism. Define
$L_{1}:=\pi_{1}^{*}(L)$. Let $E_{1}$ and $E_{2}$ be two effective and
exceptional divisors such that

$K_{X_{1}}+\underbrace{\widetilde{D}+E_{2}}_{:=D_{1}}\sim\pi_{1}^{*}(K_{X}+D)+E_{1},$

where $\widetilde{D}$ is the birational transform of $D$. Note that Claim 4.4
implies that
$\big{(}i\cdot(K_{X_{1}}+D_{1})-L_{1}\big{)}\in\overline{\operatorname{NE}}^{1}(X_{1})$.
Furthermore, by Lemma 4.2 we have
$\kappa(F_{1},(i(K_{X_{1}}+D_{1})+L_{1})|_{F_{1}})=0$, where $F_{1}$ is a very
general fibre of $\psi_{1}$. On the other hand, we have

$\kappa(F_{1},L_{1}|_{F_{1}})\leq\kappa(F_{1},(i(K_{X_{1}}+D_{1})+L_{1})|_{F_{1}})=0.$

Therefore, we find that $\kappa(F_{1},L_{1}|_{F_{1}})=0$. As a result, and
thanks to [Mori87, Def–Thm. 1.11], the Iitaka fibration $\psi_{2}:X_{2}\to Y$
of $L_{1}$ factors through the fiber space $\psi_{1}:X_{1}\to Z_{1}$ via a
birational morphism $\pi_{2}:X_{2}\to X_{1}$ and a rational map
$\nu:Z_{1}\dashrightarrow Y$ (see Diagram 4.5.1 below). Let
$\widetilde{\nu}:Z_{2}\to Y$ be a desingularization of $\nu$ through the
birational morphism $\mu:Z_{2}\to Z_{1}$. Finally, let $\psi_{3}:X_{3}\to
Z_{2}$ be a desingularization of the rational map $X_{2}\dashrightarrow
Z_{2}$, defined by the composition of $\pi_{2}$, $\psi_{1}$ and $\mu^{-1}$
(where it is defined), via the biraitonal morphism $\pi_{3}:X_{3}\to X_{2}$.

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0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
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By construction, there is an effective $\mathbb{Q}$-divisor $E\subset X_{2}$
and a very ample $\mathbb{Q}$-divisor $L_{Y}$ in $Y$ such that

$\pi_{2}^{*}(L_{1})-E\sim_{\mathbb{Q}}\psi_{2}^{*}(L_{Y}).$

Define $L_{3}:=\pi_{3}^{*}\big{(}\pi_{2}^{*}(L_{1})-E\big{)}$. Let $E_{3}$ and
$E_{4}$ be two effective exceptional divisors for which we have

$K_{X_{3}}+\underbrace{\widetilde{D}_{3}+E_{3}}_{:=D_{3}}\sim\pi_{3}^{*}\big{(}\pi_{2}^{*}(K_{X_{1}}+D_{1})\big{)}+E_{4},$

where $\widetilde{D}_{3}$ is the birational transform of $D_{1}$.

We now claim that the two divisors $\big{(}i\cdot(K_{X_{3}}+D_{3})\big{)}$ and
$L_{3}$ together with the fiber space $\psi_{3}:X_{3}\to Z_{2}$ satisfy the
properties listed in Claim 4.5 for $Y$, $D$, $L_{Y}$, $Z$ and $f$.

To see this, first note that $\kappa(L_{3})=\kappa(L)$ and that

$L_{3}\sim\psi_{3}^{*}\big{(}\underbrace{\widetilde{\nu}^{*}(L_{Y})}_{:=L_{Z_{3}}}\big{)},$

thanks to the commutativity of Diagram 4.5.1.

Next, to verify Property (4.5.5), let $F_{3}$ to be a very general fiber of
$\psi_{3}$ and note that

$0\leq\kappa(F_{3},(K_{X_{3}}+D_{3})\big{|}_{F_{3}})=\kappa(F_{3},(i(K_{X_{3}}+D_{3})+L_{3})\big{|}_{F_{3}}),$

On the other hand we have

(4.5.2)
$\kappa\big{(}F_{3},(i(K_{X_{3}}+D_{3})+L_{3})|_{F_{3}}\big{)}\leq\kappa\big{(}F_{3},(i(K_{X_{3}}+D_{3})+\pi_{3}^{*}(\pi_{2}^{*}L_{1}))\big{|}_{F_{3}}\big{)}.$

The two equalities in (4.5.5) now follow form the fact that the right-hand
side of (4.5.2) is less than or equal to zero, cf. Lemma 4.2.

The pseudo-effectivity of $i\cdot(K_{X_{3}}+D_{3})-L_{3}$ (Property (4.5.6))
follows from the relation

$i\cdot(K_{X_{3}}+D_{3})-L_{3}\sim\pi_{3}^{*}\big{(}\pi_{2}^{*}(i\cdot(K_{X_{1}}+D_{1})-L_{1})\big{)}+\big{(}i\cdot
E_{4}+\pi_{3}^{*}(E)\big{)}$

and the fact that
$\big{(}i\cdot(K_{X_{1}}+D_{1})-L_{1}\big{)}\in\overline{\operatorname{NE}}^{1}(X_{1})$.
The remaining properties hold by construction. This concludes the proof of
Claim 4.5.

To lighten the notation, from now on we will assume that $i=1$.

_Proof of Theorem 1.5_. After fixing the dimension of $X$, our proof will be
based on induction on $d:=\kappa(K_{X}+D+L)$, assuming that $\kappa(L)>0$. The
next claim provides the base case.

######  Claim 4.6.

If $d=1$, then $\kappa(\scr{L})=\kappa(\omega_{X}(D))$.

_Proof of Claim 4.6._ Using (4.5.5) we can see that the Iitaka fibration
$\phi^{(I)}:Y^{(I)}\to Z^{(I)}$ of $(K_{Y}+D_{Y}+L_{Y})$ factors through
$f:Y\to Z$ via a birational morphism $\pi^{(I)}:Y^{(I)}\to Y$ and a finite
morphism $\nu:Z\to Z^{(I)}$. As both $(\pi^{(I)}\circ f)$ and $\phi^{(I)}$ are
fiber spaces, the finite map $\nu$ must be trivial, that is the two maps
$\phi^{(I)}$ and $f$ coincide. In particular we have
$(K_{Y}+D_{Y}+L_{Y})\sim_{\mathbb{Q}}f^{*}(B_{Z})$, for some very ample
divisor $B_{Z}$ in $Z$. By using (4.5.7) it then follows that

(4.6.1) $(K_{Y}+D_{Y})\sim_{\mathbb{Q}}\frac{1}{2}\big{(}f^{*}(B_{Z}-2\cdot
L_{Z})+f^{*}(B_{Z})\big{)},$

Now, as $(B_{Z}-2\cdot L_{Z})\in\overline{\operatorname{NE}}^{1}(Z)$ by
(4.5.6), we conclude that the right-hand side of (4.6.1) is ample in $Z$.
Therefore $\kappa(K_{Y}+D_{Y})=\kappa(L_{Y})=1$, which establishes the claim.

_Inductive step._ We assume that Theorem 1.5 holds for any line bundle
$\scr{A}$ on a smooth projective variety $W$ (having the same dimension as
$X$) that satisfies the following two properties.

  1. (4.6.1)

There is a reduced divisor $D_{W}$ such that $(W,D_{W})$ is log-smooth and
$(\scr{A}\otimes\scr{M})\subseteq\Omega^{\otimes i}_{W}\log(D_{W})$, for some
pseudo-effective line bundle $\scr{M}$.

  2. (4.6.2)

$\kappa(W,\omega_{W}(D_{W})\otimes\scr{A})<d$.

Let $(Y,D_{Y})$, $L_{Y}$ and $f:Y\to Z$ be as in the setting of Claim 4.5. By
the inductive step we may assume that $\dim(Z)=\kappa(K_{Y}+D_{Y}+L_{Y})$.
Furthermore, we can use Claim 4.6 to exclude the possibility that
$\kappa(K_{Y}+D_{Y}+L_{Y})=1$.

We treat the case where the dimension of the fibers of $f$ is equal to $3$
(i.e. $\dim(Y)=5$ and $\kappa(K_{Y}+D_{Y}+L_{Y})=2$). The case of lower
dimensional fibers can be dealt with similarly.

Let $g:(Y,D_{Y})\dashrightarrow(Y_{n},D_{Y_{n}})$ be the birational map
associated to a relative minimal model program for $(Y,D_{Y})$ over $Z$,
consisting of $n$ number of divisorial and flipping contractions

$g_{j}:(Y_{j},D_{Y_{j}})\dashrightarrow(Y_{D_{j+1}},D_{Y_{j+1}}),$

cf. [SecondAsterisque, Chapt. 4]. Here we have set
$(Y_{0},D_{Y_{0}}):=(Y,D_{Y})$. For each $1\leq j\leq n$, let
$f_{j}:Y_{D_{j}}\to Z$ be the induced morphism resulting in the following
diagram.

|  |
---|---|---
$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{g_{1}}$$\scriptstyle{f}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\scriptstyle{f_{1}}$$\textstyle{....\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{n-1}}$$\textstyle{Y_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{n}}$$\scriptstyle{f_{n-1}}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}}$$\textstyle{Z.}$

######  Claim 4.7.

$\kappa(Y_{n},K_{Y_{n}}+D_{Y_{n}}+f_{n}^{*}L_{Z})=\kappa(Y,K_{Y}+D_{Y}+L_{Y})$.

######  Claim 4.8.

$\big{(}(K_{Y_{n}}+D_{Y_{n}})-(f_{n}^{*}L_{Z})\big{)}\in\overline{\operatorname{NE}}^{1}(Y_{n}).$

Let us for the moment assume that Claims 4.7 and 4.8 hold and proceed with the
proof of Theorem 1.5. Let $f_{n}^{\prime}:Y_{n}^{\prime}\to Z^{\prime}$ be a
flattening of $f_{n}$ (see the discussion preceding Lemma 4.3), with induced
biratinal maps $\tau:Z^{\prime}\to Z$ and $\sigma:Y_{n}^{\prime}\to Y_{n}$
leading to the following commutative diagram.

$\textstyle{Y_{n}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{f^{\prime}_{n}}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}}$$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{Z.}$

Thanks to the solution of the relative log-abundance problem in $\dim(X)=3$ by
Keel, Matsuki and McKernan [MR2057020], using (4.5.5), we have
$(K_{Y_{n}}+D_{Y_{n}})|_{F_{z}}\equiv 0$, where $F_{z}$ is the general fiber
of $f_{n}$. Therefore, by Lemma 4.3, there exists a $\mathbb{Q}$-divisor
$A_{Z^{\prime}}$ in $Z^{\prime}$ such that
$\sigma^{*}(K_{Y_{n}}+D_{Y_{n}})\sim_{\mathbb{Q}}(f_{n}^{\prime})^{*}(A_{Z^{\prime}})$
so that

$\sigma^{*}\big{(}(K_{Y_{n}}+D_{Y_{n}})\pm
f_{n}^{*}L_{Z}\big{)}\sim_{\mathbb{Q}}(f_{n}^{\prime})^{*}(A_{Z^{\prime}}\pm\tau^{*}L_{Z}).$

By Claim 4.7 it thus follows that
$\kappa(Z^{\prime},A_{Z^{\prime}}+\tau^{*}(L_{Z}))=\kappa(Y_{n},K_{Y_{n}}+D_{Y_{n}}+f_{n}^{*}L_{Z})=\dim(Z)$.
Moreover, by using Claim 4.8, we find
$(A_{Z^{\prime}}-\tau^{*}(L_{Z}))\in\overline{\operatorname{NE}}^{1}(Z^{\prime})$.
Therefore $A_{Z^{\prime}}$ is big in $Z^{\prime}$ and we have

(4.8.1)
$\kappa(Z^{\prime},A_{Z^{\prime}})\geq\kappa(Z^{\prime},\tau^{*}(L_{Z}))=\kappa(Z,L_{Z}).$

On the other hand, by using the negativity lemma, we have

(4.8.2)
$\kappa(Z^{\prime},A_{Z^{\prime}})=\kappa(Y_{n},K_{Y_{n}}+D_{Y_{n}})=\kappa(Y,K_{Y}+D_{Y}).$

By combining (4.8.1) and (4.8.2) we reach the inequality

$\kappa(Y,L_{Y})\leq\kappa(Y,K_{Y}+D_{Y}).$

We now turn to proving Claims 4.7 and 4.8.

_Proof of Claim 4.7._ Let $(\widetilde{Y},\widetilde{D}_{Y})$ be a common log-
smooth higher birational model for $(Y,D_{Y})$ and $(Y_{n},D_{Y_{n}})$, with
birational morphisms $\mu:\widetilde{Y}\to Y$ and $\mu_{n}:\widetilde{Y}\to
Y_{n}$. According to [KM98, Lem. 3.38] there is an effective $\mu$-exceptional
divisor $E_{\mu}$ and an effective $\mu_{n}$-exceptional divisor $E_{\mu_{n}}$
such that

$\mu^{*}(K_{Y}+D_{Y})+E_{\mu}\sim_{\mathbb{Q}}(\mu_{n})^{*}(K_{Y_{n}}+D_{Y_{n}})+E_{\mu_{n}}.$

Therefore, we have

$\mu^{*}\big{(}(K_{Y}+D_{Y})+f^{*}L_{Z}\big{)}+E_{\mu}\sim_{\mathbb{Q}}(\mu_{n})^{*}(K_{Y_{n}}+D_{Y_{n}}+f_{n}^{*}L_{Z})+E_{\mu_{n}},$

which establishes the claim.

_Proof of Claim 4.8._ The proof will be based on induction on $n$. Assume that

$\big{(}(K_{Y_{n-1}}+D_{Y_{n-1}})-f_{n-1}^{*}(L_{Z})\big{)}\in\overline{\operatorname{NE}}^{1}(Y_{n-1}).$

Now, the map $g_{n}:Y_{n-1}\dashrightarrow Y_{n}$ is either a divisorial
contraction or a flip over $Z$. In the case of the latter the claim is easy to
check, so let us assume that $g_{n}$ is a divisorial contraction and let
$E_{1}$ and $E_{2}$ be two effective exceptional divisors such that the
equivalence

(4.8.3)
$K_{Y_{n-1}}+D_{Y_{n-1}}+E_{1}\sim_{\mathbb{Q}}g_{n}^{*}(K_{Y_{n}}+D_{Y_{n}})+E_{2}$

holds. After subtracting $f_{n-1}^{*}(L_{Z})=g_{n}^{*}(f_{n}^{*}L_{Z})$ from
both sides of (4.8.3) and by using the inductive hypothesis we find that

$\big{(}g_{n}^{*}(K_{Y_{n}}+D_{Y_{n}}-f_{n}^{*}L_{Z})+E_{2}\big{)}\in\overline{\operatorname{NE}}^{1}(Y_{n-1}),$

which implies
$\big{(}K_{Y_{n}}+D_{Y_{n}}-f_{n}^{*}(L_{Z})\big{)}\in\overline{\operatorname{NE}}^{1}(Y_{n})$.
This finishes the proof of Claim 4.8. ∎

######  Remark 4.9.

Theorem 1.5 is naturally related to a conjecture of Campana and Peternell,
where the authors predict that over a smooth projective variety $X$, if
$K_{X}\sim_{\mathbb{Q}}L+B$, where $L$ is effective and $B$ is pseudo-
effective, then $\kappa(X)\geq\kappa(L)$, cf. [CP11]. They also provide a
proof to this conjecture when $B$ is numerically trivial [CP11, Thm. 0.3].
(See also [CKP12] for the generalization to the logarithmic setting.)

##  5\. Concluding remarks and further questions

We recall that a quasi-projective variety $V$ of dimension $n$ is said to be
_special_ if, for every invertible subsheaf
$\scr{L}\subseteq\Omega^{p}_{X}\log(D)$, the inequality $\kappa(X,\scr{L})<p$
holds, for all $1\leq p\leq n$. Here, by $(X,D)$ we denote a log-smooth
compactification of $V$. We refer to [Cam04] for an in-depth discussion of
this notion. We note that while varieties of Kodaira dimension zero form an
important class of special varieties [Cam04, Thm 5.1], there are special
varieties of every possible (but not maximal) Kodaira dimension.

A conjecture of Campana predicts that a smooth projective family of
canonically polarized manifolds $f_{U}:U\to V$ parametrized by a special
quasi-projective variety $V$ is isotrivial. One can naturally extend this
conjecture to the following setting.

######  Conjecture 5.1.

Let $f_{U}:U\to V$ be a smooth projective family, where $V$ is equipped with a
morphism $\mu:V\to P_{h}$ to the coarse moduli scheme $P_{h}$ associated with
the moduli functor $\scr{P}_{h}(\cdot)$ of polarized projective manifolds with
semi-ample canonical bundle and fixed Hilbert polynomial $h$. If $V$ is
special, then $f_{V}$ is isotrivial.

By using the refinement of [VZ02], due to Jabbusch and Kebekus [MR2976311],
and the main result of [CP13], in the canonically polarized case, Conjecture
5.1 was established in [taji16]. We invite the reader to also consult [CKT16],
Claudon [Claudon15], [CP16] and Schnell [Sch17].

After a close inspection one can observe that the the strategy of [taji16] can
be extended to establish 5.1 in its full generality. More precisely, existence
of the functor $\scr{P}_{h}$ with is associated algebraic coarse moduli scheme
gives us a twofold advantage. It ensures the existence of “effective” Viehweg-
Zuo subsheaves $\scr{L}\subseteq\Omega^{\otimes i}_{X}\log(D)$ whose Kodaira
dimension verifies $\kappa(\scr{L})\geq\operatorname{Var}(f_{U})$, cf. [VZ02]
and is constructed at the level of moduli stacks, cf. [MR2976311]. After
imposing some additional orbifold structures naturally arising from the
induced moduli map $\mu:V\to P_{h}$, these two properties allow us to
essentially reduce the problem to the case where $\scr{L}$ is big, cf.
[taji16, Thm. 4.3].

######  Theorem 5.2.

Conjecture 5.1 holds.

Campana has kindly informed the author that, extending their current result
[AC17], in a joint work with Amerik, they have established Theorem 5.2 in a
more general context of projective families with orbifold base.

Using the results of [Cam04], it is not difficult to trace a connection
between the two Conjectures 5.1 and 1.3; a solution to Conjecture 5.1 leads to
a solution for Conjecture 1.3. We record this observation in the following
theorem.

######  Theorem 5.3.

For any quasi-projective variety $V$ equipped with $\mu_{V}:V\to P_{h}$,
induced by a family $f_{U}:U\to V$ of polarized manifolds, we have

(5.3.1) $\operatorname{Var}(f_{U})\leq\kappa(X,D),$

$(X,D)$ being a log-smooth compactification of $V$.

Before proceeding to the proof of Theorem 5.3, let us recall the notion of the
_core map_ defined by Campana. Given a smooth quasi-projective variety $V$
that is not of log-general type, the core map is a rational map
$c_{X}:X\dashrightarrow Z$ satisfying the following two key properties.

  1. (5.3.1)

$c_{X}$ is almost holomorphic with special geometric fibers.

  2. (5.3.2)

$c_{X}$ is birationally equivalent to a fiber space
$c_{\widetilde{X}}:(\widetilde{X},\widetilde{D})\to(\widetilde{Z},\Delta_{\widetilde{Z}})$,
where
$\Delta_{\widetilde{Z}}\in\operatorname{Div}_{\mathbb{Q}}(\widetilde{Z})$ and
the pair $(\widetilde{Z},\Delta_{\widetilde{Z}})$ is a log-smooth orbifold
base for $C_{\widetilde{X}}$ and is of log-general type.

###### Proof.

Assume that $f_{U}$ is not isotrivial and $V$ is not of log-general type.
Then, by Item (5.3.1) and Theorem 5.2, the compactification
$\overline{\mu}_{V}:X\to\overline{P_{h}}$ of $\mu_{V}$ factors through the
core map $c_{X}:X\dashrightarrow Z$ with positive dimensional fibers. In
particular we have

(5.3.2) $\operatorname{Var}(f_{U})\leq\dim(Z).$

The theorem then follows from Campana’s orbifold $C_{n,m}^{\rm{orb}}$ theorem
for any orbifold, log-general type fibration; an example of which is
$c_{\widetilde{X}}$. More precisely, by using (5.3.2) and [Cam04, Thm. 4.2] we
can conclude that the inequality

$\kappa(\widetilde{X},\widetilde{D})\geq\kappa(\widetilde{Z},\Delta_{\widetilde{Z}})$

holds, which, together with (5.3.2) and the inequality
$\kappa(X,D)\geq\kappa(\widetilde{X},\widetilde{D})$, establishes the theorem.
∎

When the fibers are of general type, it is conceivable that one may be able to
use the result of Birkar, Cascini, Hacon and McKernan [BCHM10] on the
existence of good minimal models for varieties of general type (and relative
base point freeness theorem) to reduce to the case of maximal variation. More
precisely, the arguments of [VZ02, Lem. 2.8] combined with Hodge theoretic
construction of [PS15], or the ones in Section 3 of the current paper, may
allow for the construction of a Viehweg-Zuo subsheaf $\scr{L}$ over a new base
where variation of the pulled back family is maximal. If so, in this case the
discussion prior to Theorem 5.2 again applies and Theorem 5.2 and consequently
Theorem 5.3 hold. But as pointed out from the outset, the main remaining
difficulty is solving the isotriviality problem in the absence of a well-
behaved functor, such as $\scr{P}_{h}$, detecting the variation in the family
(after running the minimal model program, if necessary). At the moment, it is
not clear to the author that one can expect Conjecture 5.1 to hold for all
smooth projective families whose geometric fiber has semi-ample canonical
bundle or—even more generally—admits a good minimal model.

######  Question 5.4.

Let $f_{U}:U\to V$ be a smooth projective family of manifolds admitting a good
minimal model. If $V$ is special, (apart from the cases discussed above) is it
true that $f_{U}$ is isotrivial?

## References