# On mixed-$\omega$-sheaves

Osamu Fujino  Department of Mathematics, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan<EMAIL_ADDRESS>
(Date: 2019/7/29, version 0.40)

###### Abstract.

We introduce the notion of mixed-$\omega$-sheaves and use it for the study of
a relative version of Fujita’s freeness conjecture.

###### Key words and phrases:

weak positivity, big sheaves, Fujita’s freeness conjecture, mixed Hodge
structures, Nakayama’s numerical dimension

###### 2010 Mathematics Subject Classification:

Primary 14F17; Secondary 14E30

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
  3. 3 Preliminary lemmas
  4. 4 Weakly positive sheaves and big sheaves
  5. 5 Mixed-$\omega$-sheaves and mixed-$\omega$-big sheaves
  6. 6 Basic properties: Part 1
  7. 7 Basic properties: Part 2
  8. 8 A special case
  9. 9 Fundamental theorem
  10. 10 Proof of Theorems 1.5, 1.7, and 1.8
  11. 11 Some other applications

##  1\. Introduction

Let us recall Fujita’s famous freeness conjecture on adjoint bundles. Note
that everything is defined over $\mathbb{C}$, the complex number field, in
this paper.

######  Conjecture 1.1 (Takao Fujita).

Let $X$ be a smooth projective variety with $\dim X=n$ and let $\mathcal{L}$
be any ample invertible sheaf on $X$. Then
$\omega_{X}\otimes\mathcal{L}^{\otimes l}$ is generated by global sections for
every $l\geq n+1$.

Although there have already been many related results, Conjecture 1.1 is still
open. As a generalization of Conjecture 1.1, Popa and Schnell proposed the
following conjecture, which is a relative version of Fujita’s freeness
conjecture.

######  Conjecture 1.2 (Popa and Schnell, see [PS, Conjecture 1.3]).

Let $f:X\to Y$ be a surjective morphism between smooth projective varieties
with $\dim Y=n$. Let $\mathcal{L}$ be any ample invertible sheaf on $Y$. Then,
for every positive integer $k$, the sheaf

$f_{*}\omega^{\otimes k}_{X}\otimes\mathcal{L}^{\otimes l}$

is generated by global sections for $l\geq k(n+1)$.

We can find some interesting results on Conjecture 1.2 in [De], [Du], [DuM],
and [I]. In this paper, we do not directly treat Conjecture 1.2. We propose a
new conjecture similar to Conjecture 1.2. It is also a generalization of
Conjecture 1.1.

######  Conjecture 1.3.

Let $f:X\to Y$ be a surjective morphism between smooth projective varieties
with $\dim Y=n$. Let $\mathcal{L}$ be any ample invertible sheaf on $Y$. Then,
for every positive integer $k$, the sheaf

$f_{*}\omega^{\otimes k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{L}^{\otimes l}$

is generated by global sections for $l\geq n+1$.

If Conjecture 1.3 and Fujita’s original freeness conjecture (see Conjecture
1.1) hold true, then $f_{*}\omega^{\otimes k}_{X}\otimes\mathcal{L}^{\otimes
l}$ is generated by global sections for every $l\geq k(n+1)$ since

$f_{*}\omega^{\otimes k}_{X}\otimes\mathcal{L}^{\otimes l}\simeq
f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{L}^{\otimes\left(l-(k-1)(n+1)\right)}\otimes\left(\omega_{Y}\otimes\mathcal{L}^{\otimes
n+1}\right)^{\otimes k-1}.$

Therefore, Conjecture 1.3 is sharper than Conjecture 1.2.

It is well known that Conjecture 1.3 holds true when $Y$ is a curve. This
means that $f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{L}^{\otimes l}$ is generated by
global sections for every $l\geq 2$. More generally, we have:

######  Theorem 1.4.

Let $f:X\to C$ be a surjective morphism from a smooth projective variety $X$
onto a smooth projective curve $C$. Let $\mathcal{H}$ be an ample invertible
sheaf on $C$ with $\deg\mathcal{H}\geq 2$ and let $k$ be any positive integer.
Then the sheaf

$f_{*}\mathcal{\omega}^{\otimes k}_{X/C}\otimes\omega_{C}\otimes\mathcal{H}$

is generated by global sections.

Here, we give a detailed proof of Theorem 1.4 in order to explain our idea.

###### Proof of Theorem 1.4.

If $f_{*}\omega^{\otimes k}_{X/C}=0$, then there are nothing to prove. So we
assume that $f_{*}\omega^{\otimes k}_{X/C}\neq 0$. We take any closed point
$P$.

######  Claim.

$H^{1}(C,f_{*}\omega^{\otimes
k}_{X/C}\otimes\omega_{C}\otimes\mathcal{H}\otimes\mathcal{O}_{C}(-P))=0$.

###### Proof of Claim.

By Viehweg’s weak positivity theorem, $f_{*}\omega^{\otimes k}_{X/C}$ is a nef
locally free sheaf since $C$ is a smooth projective curve. Therefore,
$\mathcal{E}:=f_{*}\omega^{\otimes
k}_{X/C}\otimes\mathcal{H}\otimes\mathcal{O}_{C}(-P)$ is ample. If
$H^{1}(C,\mathcal{E}\otimes\omega_{C})\neq 0$, then we get
$H^{0}(C,\mathcal{E}^{*})\neq 0$ by Serre duality. This implies that there is
a nontrivial inclusion $0\to\mathcal{O}_{C}\to\mathcal{E}^{*}$. By taking the
dual of this inclusion, we have the following surjection
$\mathcal{E}\to\mathcal{O}_{C}\to 0$. This is a contradiction since
$\mathcal{E}$ is ample. Hence we have
$H^{1}(C,\mathcal{E}\otimes\omega_{C})=0$. ∎

By Claim, the natural restriction map

$H^{0}(C,f_{*}\omega^{\otimes k}_{X/C}\otimes\omega_{C}\otimes\mathcal{H})\to
f_{*}\omega^{\otimes
k}_{X/C}\otimes\omega_{C}\otimes\mathcal{H}\otimes\mathbb{C}(P)$

is surjective. This means that $f_{*}\omega^{\otimes
k}_{X/C}\otimes\omega_{C}\otimes\mathcal{H}$ is generated by global sections.
∎

The following theorem supports Conjecture 1.3.

######  Theorem 1.5.

Let $f:X\to Y$ be a surjective morphism between smooth projective varieties
and let $H$ be an ample divisor on $Y$ such that $|H|$ is free. We put $\dim
Y=n$. Then

$f_{*}\omega^{\otimes k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(lH)$

is generically generated by global sections for all integers $k\geq 1$ and
$l\geq n+1$. Moreover,

$\left(\bigotimes^{s}(f_{*}\omega^{\otimes
k}_{X/Y})\right)^{**}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(lH)$

is generically generated by global sections for all integers $k\geq 1$, $s\geq
1$, and $l\geq n+1$.

Let $H^{\dagger}$ be an ample divisor on $Y$ such that $|H^{\dagger}|$ is not
necessarily free. Then

$f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(lH^{\dagger})$

is generically generated by global sections for all integers $k\geq 1$ and
$l\geq n^{2}+\min\\{2,k\\}$. Moreover, the sheaf

$\left(\bigotimes^{s}(f_{*}\omega^{\otimes
k}_{X/Y})\right)^{**}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(lH^{\dagger})$

is generically generated by global sections for all integers $k\geq 1$, $s\geq
1$, and $l\geq n^{2}+\min\\{2,k\\}$.

The author learned the following remark from Masataka Iwai.

######  Remark 1.6.

Let $f:X\to Y$ and $H^{\dagger}$ be as in Theorem 1.5. Let $U$ be the largest
Zariski open set of $Y$ such that $f$ is smooth over $U$. Then, by the
argument in [I], we can check that the natural map

$H^{0}(Y,f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(lH^{\dagger}))\otimes\mathcal{O}_{Y}\to
f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(lH^{\dagger})$

is surjective on $U$ for all integers $k\geq 1$ and $l\geq\frac{n(n+1)}{2}+1$.

Theorem 1.5 is a special case of Theorems 1.7 and 1.8. In this paper, we will
establish Theorems 1.7 and 1.8 by our new theory of mixed-$\omega$-sheaves.

######  Theorem 1.7.

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$ with $\dim Y=n$. Let $\Delta$ be an
effective $\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier and that $(X,\Delta)$ is log canonical over a nonempty
Zariski open set of $Y$. Let $L$ be a Cartier divisor on $X$ with
$L\sim_{\mathbb{R}}k(K_{X/Y}+\Delta)$ for some positive integer $k$. Let $H$
be a big Cartier divisor on $Y$ such that $|H|$ is free. Then

$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y}+lH)$

is generically generated by global sections for every $l\geq n+1$. Moreover,

$\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}\otimes\mathcal{O}_{Y}(K_{Y}+lH)$

is generically generated by global sections for all integers $s\geq 1$ and
$l\geq n+1$.

######  Theorem 1.8.

In Theorem 1.7, we assume that $H^{\dagger}$ is a nef and big Cartier divisor
on $Y$ such that $|H^{\dagger}|$ is not necessarily free. Then we have that

$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})\quad\text{and}\quad\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}\otimes\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})$

are generically generated by global sections for all integers $s\geq 1$ and
$l\geq n^{2}+\min\\{2,k\\}$.

Let us quickly explain the idea of the proof of Theorem 1.5, which is mainly
due to Nakayama (see [N]). Let $f:X\to Y$ be a surjective morphism between
smooth projective varieties and let $H$ be an ample Cartier divisor on $Y$
such that $|H|$ is free. We fix a positive integer $k\geq 2$. Then we can
construct a surjective morphism $g:Z\to Y$ from a smooth projective variety
$Z$ and a direct summand $\mathcal{F}$ of $g_{*}\mathcal{O}_{Z}(K_{Z})$ such
that there exists a generically isomorphic injection

$\mathcal{F}\hookrightarrow f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(H).$

By Kollár’s vanishing theorem, we have

$H^{i}(Y,\mathcal{F}\otimes\mathcal{O}_{Y}((n+1-i)H))=0$

for every $i>0$, where $n=\dim Y$. Therefore, by Castelnuovo–Mumford
regularity, $\mathcal{F}\otimes\mathcal{O}_{Y}((n+1)H)$ is generated by global
sections. This implies that

$f_{*}\omega^{\otimes k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}((n+2)H)$

is generically generated by global sections. Note that we do not try to
establish any vanishing theorem for $f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(H)$ directly. Anyway, it is
natural to consider:

######  Definition 1.9 (Mixed-$\omega$-sheaf and pure-$\omega$-sheaf, see
Definition 5.1).

A torsion-free coherent sheaf $\mathcal{F}$ on a normal quasi-projective
variety $W$ is called a mixed-$\omega$-sheaf if there exist a projective
surjective morphism from a smooth quasi-projective variety $V$ and a simple
normal crossing divisor $D$ on $V$ such that $\mathcal{F}$ is a direct summand
of $f_{*}\mathcal{O}_{V}(K_{V}+D)$. When $D=0$, $\mathcal{F}$ is called a
pure-$\omega$-sheaf on $W$.

For the study of klt pairs, the notion of pure-$\omega$-sheaves is sufficient
and is essentially due to Nakayama (see [N]). In this paper, we study some
basic properties of mixed-$\omega$-sheaves. They are indispensable for the
study of log canonical pairs. Of course, the theory of mixed-$\omega$-sheaves
(resp. pure-$\omega$-sheaves) in this paper is based on that of mixed (resp.
pure) Hodge structures. Roughly speaking, Nakayama only treats
pure-$\omega$-sheaves in [N, Chapter V]. However, his theory of
$\omega$-sheaves is more sophisticated and some of his results are much
sharper than ours. We do not try to make the framework discussed in this paper
supersede Nakayama’s theory of $\omega$-sheaves in [N, Chapter V]. The main
purpose of this paper is to make Nakayama’s theory of $\omega$-sheaves more
accessible and make it applicable to the study of log canonical pairs. Theorem
9.3 (and Remark 9.4) is one of the main results of this paper, which we call a
fundamental theorem of the theory of mixed-$\omega$-sheaves.

######  Theorem 1.10 (see [N, Chapter V, 3.35. Theorem], Theorem 9.3, and
Remark 9.4).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and
let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $D$ be an $\mathbb{R}$-divisor on
$Y$. Let $k$ be a positive integer with $k\geq 2$. Assume the following
conditions:

  * (i)

$(X,\Delta)$ is log canonical (resp. klt) over a nonempty Zariski open set of
$Y$, and

  * (ii)

$L+f^{*}D-k(K_{X/Y}+\Delta)-f^{*}A$ is semi-ample for some big
$\mathbb{R}$-divisor $A$ on $Y$.

If $f_{*}\mathcal{O}_{Y}(L)\neq 0$, then there exist a mixed-$\omega$-big-
sheaf (resp. pure-$\omega$-big-sheaf) $\mathcal{F}$ on $Y$ and a generically
isomorphic injection

$\mathcal{F}\hookrightarrow\mathcal{O}_{Y}(K_{Y}+\lceil D\rceil)\otimes
f_{*}\mathcal{O}_{X}(L).$

For the precise definition of mixed-$\omega$-big-sheaves and
pure-$\omega$-big-sheaves, see Definition 5.3 below.

As an application of Theorem 1.10, we have:

######  Theorem 1.11 ([N, Chapter V, 4.1. Theorem (1)], [Fn10, Section 3], and
Theorem 11.3).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$ with connected fibers. Let $\Delta$ be an
effective $\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier and that $(X,\Delta)$ is log canonical over a nonempty
Zariski open set of $Y$. Let $D$ be an $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor on $X$ such that $D-(K_{X/Y}+\Delta)$ is nef. Then, for
any $\mathbb{R}$-divisor $Q$ on $Y$, we have

$\kappa_{\sigma}(X,D+f^{*}Q)\geq\kappa_{\sigma}(F,D|_{F})+\kappa(Y,Q)$

and

$\kappa_{\sigma}(X,D+f^{*}Q)\geq\kappa(F,D|_{F})+\kappa_{\sigma}(Y,Q)$

where $F$ is a sufficiently general fiber of $f:X\to Y$.

We note that $\kappa_{\sigma}(X,D)$ and $\kappa(X,D)$ denote Nakayama’s
numerical dimension and the Iitaka dimension of $D$, respectively. Theorem
1.11 already played a crucial role in the theory of minimal models.

We explain the organization of this paper. In Section 2, we collect some basic
definitions. In Section 3, we prepare some useful and important lemmas. They
will play a crucial role in this paper. In Section 4, we quickly explain some
basic properties of Viehweg’s weakly positive sheaves and big sheaves. In
Section 5, we introduce mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves.
In Sections 6 and 7, we prove some basic properties of mixed-$\omega$-sheaves
based on the theory of mixed Hodge structures. In Section 8, we treat a very
special but interesting case. Section 9 is the main part of this paper. We
establish a fundamental theorem of the theory of mixed-$\omega$-sheaves.
Section 10 is devoted to the proof of Theorems 1.5, 1.7, and 1.8. In Section
11, we treat Nakayama’s inequality on $\kappa_{\sigma}$, which has already
played a crucial role in the theory of minimal models, and a slight
generalization of the twisted weak positivity theorem.

######  Acknowledgments.

The author was partially supported by JSPS KAKENHI Grant Numbers JP16H03925,
JP16H06337. He thanks Masataka Iwai for useful comments.

We will work over $\mathbb{C}$, the complex number field, throughout this
paper. We note that a scheme is a separated scheme of finite type over
$\mathbb{C}$ and a variety is an integral scheme.

##  2\. Preliminary

In this section, we collect some basic definitions. For the details, see
[Fn2], [Fn6], and [Fn9].

Let us start with the definition of canonical sheaves and canonical divisors.

######  Definition 2.1 (Canonical sheaf and canonical divisor).

Let $X$ be an equidimensional scheme of dimension $n$ and let
$\omega^{\bullet}_{X}$ be the dualizing complex of $X$. Then we put

$\omega_{X}:=h^{-n}(\omega^{\bullet}_{X})$

and call it the canonical sheaf of $X$.

We further assume that $X$ is normal. Then a canonical divisor $K_{X}$ of $X$
is a Weil divisor on $X$ such that

$\mathcal{O}_{X_{\mathrm{sm}}}(K_{X})\simeq\Omega^{n}_{X_{\mathrm{sm}}}$

holds, where $X_{\mathrm{sm}}$ is the largest smooth Zariski open set of $X$.

It is well known that

$\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$

holds when $X$ is normal.

If $f:X\to Y$ is a morphism between Gorenstein schemes, then we put

$\omega_{X/Y}:=\omega_{X}\otimes f^{*}\omega^{\otimes-1}_{Y}.$

If $f:X\to Y$ is a morphism from a normal scheme $X$ to a normal Gorenstein
scheme $Y$, then we put

$K_{X/Y}:=K_{X}-f^{*}K_{Y}.$

Let us quickly see the definition of singularities of pairs.

######  Definition 2.2 (Singularities of pairs).

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $f:Y\to X$ be a resolution of singularities of $X$ such that
${\operatorname{Exc}}(f)\cup f^{-1}_{*}\Delta$ has a simple normal crossing
support, where ${\operatorname{Exc}}(f)$ is the exceptional locus of $f$ on
$Y$ and $f^{-1}_{*}\Delta$ is the strict transform of $\Delta$ on $Y$. Then we
can write

$K_{Y}=f^{*}(K_{X}+\Delta)+\sum_{i}a_{i}E_{i}$

with $f_{*}\left(\sum_{i}a_{i}E_{i}\right)=-\Delta$. We say that $(X,\Delta)$
is log canonical (resp. klt) if $a_{i}\geq-1$ (resp. $a_{i}>-1$) for every
$i$.

If $(X,\Delta)$ is log canonical and there exist a resolution of singularities
$f:Y\to X$ as above and a prime divisor $E_{i}$ on $Y$ with $a_{i}=-1$, then
$f(E_{i})$ is called a log canonical center of $(X,\Delta)$.

######  Definition 2.3 (Dlt pairs).

Let $(X,\Delta)$ be a log canonical pair. If there exists a resolution of
singularities $f:Y\to X$ such that the exceptional locus
${\operatorname{Exc}}(f)$ of $f$ is a divisor on $Y$,
${\operatorname{Exc}}(f)\cup f^{-1}_{*}\Delta$ has a simple normal crossing
support, and

$K_{Y}=f^{*}(K_{X}+\Delta)+\sum_{i}a_{i}E_{i}$

with $a_{i}>-1$ for every $f$-exceptional divisor $E_{i}$, then the pair
$(X,\Delta)$ is called a dlt pair.

The following definitions are very useful in this paper.

######  Definition 2.4 (Horizontal and vertical divisors).

Let $f:X\to Y$ be a dominant morphism between normal varieties and let $D$ be
an $\mathbb{R}$-divisor on $X$. We can write

$D=D_{\mathrm{hor}}+D_{\mathrm{ver}}$

such that every irreducible component of $D_{\mathrm{hor}}$ (resp.
$D_{\mathrm{ver}}$) is mapped (resp. not mapped) onto $Y$. If
$D=D_{\mathrm{hor}}$ (resp. $D=D_{\mathrm{ver}}$), $D$ is said to be
horizontal (resp. vertical).

######  Definition 2.5 (Operations for $\mathbb{R}$-divisors).

Let $D=\sum_{i}d_{i}D_{i}$ be an $\mathbb{R}$-divisor on a normal variety $X$,
where $D_{i}$ is a prime divisor on $X$ for every $i$, $D_{i}\neq D_{j}$ for
$i\neq j$, and $d_{i}\in\mathbb{R}$ for every $i$. Then we put

$\lfloor D\rfloor=\sum_{i}\lfloor d_{i}\rfloor D_{i},\quad\\{D\\}=D-\lfloor
D\rfloor,\quad\text{and}\quad\lceil D\rceil=-\lfloor-D\rfloor.$

Note that $\lfloor d_{i}\rfloor$ is the integer which satisfies
$d_{i}-1<\lfloor d_{i}\rfloor\leq d_{i}$. We also note that $\lfloor
D\rfloor$, $\lceil D\rceil$, and $\\{D\\}$ are called the round-down, round-
up, and fractional part of $D$ respectively.

If $0\leq d_{i}\leq 1$ for every $i$, then we say that $D$ is a boundary
$\mathbb{R}$-divisor on $X$. We note that $\sim_{\mathbb{Q}}$ (resp.
$\sim_{\mathbb{R}}$) denotes the $\mathbb{Q}$-linear (resp.
$\mathbb{R}$-linear) equivalence of $\mathbb{Q}$-divisors (resp.
$\mathbb{R}$-divisors).

In this paper, we will repeatedly use the following notation:

$D^{=1}=\sum_{d_{i}=1}D_{i},\quad
D^{>1}=\sum_{d_{i}>1}d_{i}D_{i},\quad\text{and}\quad
D^{<0}=\sum_{d_{i}<0}d_{i}D_{i}.$

We close this section with the definition of exceptional divisors for proper
surjective morphisms between normal varieties.

######  Definition 2.6 (Exceptional divisors).

Let $f:X\to Y$ be a proper surjective morphism between normal varieties. Let
$E$ be a Weil divisor on $X$. We say that $E$ is $f$-exceptional if
$\mathrm{codim}_{Y}f({\operatorname{Supp}}E)\geq 2$. We note that $f$ is not
always assumed to be birational.

##  3\. Preliminary lemmas

In this section, we collect some useful and important lemmas for the reader’s
convenience. They are more or less well known to the experts.

Let us start with the following easy lemmas on $\mathbb{R}$-divisors. We will
use them repeatedly in this paper.

######  Lemma 3.1.

Let $A$ be a Cartier divisor on a normal variety $V$. Let $B$ be an
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor on $V$ such that $B=\sum_{i\in
I}b_{i}B_{i}$ where $b_{i}\in\mathbb{R}$ and $B_{i}$ is a prime divisor on $V$
for every $i$ with $B_{i}\neq B_{j}$ for $i\neq j$. Assume that
$A\sim_{\mathbb{R}}B$. Then we can take a $\mathbb{Q}$-Cartier
$\mathbb{Q}$-divisor $C=\sum_{i\in I}c_{i}B_{i}$ on $V$ such that

  * (i)

$A\sim_{\mathbb{Q}}C$,

  * (ii)

$c_{i}=b_{i}$ holds if $b_{i}\in\mathbb{Q}$, and

  * (iii)

$|c_{i}-b_{i}|\ll 1$ holds for $b_{i}\in\mathbb{R}\setminus\mathbb{Q}$.

In particular, ${\operatorname{Supp}}C={\operatorname{Supp}}B$, $\lfloor
C\rfloor=\lfloor B\rfloor$, $\lceil C\rceil=\lceil B\rceil$, and
${\operatorname{Supp}}\\{C\\}={\operatorname{Supp}}\\{B\\}$.

###### Proof.

It is an easy exercise. For the details, see, for example, the proof of [Fn2,
Lemma 4.15]. ∎

######  Lemma 3.2.

Let $D=\sum_{i\in I}a_{i}D_{i}$ be an $\mathbb{R}$-divisor on a smooth
projective variety $V$, where $a_{i}\in\mathbb{R}$ and $D_{i}$ is a prime
divisor on $V$ for every $i$ with $D_{i}\neq D_{j}$ for $i\neq j$. Assume that
$D$ is semi-ample. Then we can construct a $\mathbb{Q}$-divisor
$D^{\dagger}=\sum_{i\in I}b_{i}D_{i}$ such that

  * (i)

$D^{\dagger}$ is semi-ample,

  * (ii)

$b_{i}=a_{i}$ holds if $a_{i}\in\mathbb{Q}$, and

  * (iii)

$|b_{i}-a_{i}|\ll 1$ holds for $a_{i}\in\mathbb{R}\setminus\mathbb{Q}$.

###### Proof.

Since $D$ is semi-ample, we can write $D=\sum_{j\in J}m_{j}M_{j}$ where
$m_{j}\in\mathbb{R}$ and $M_{j}$ is a semi-ample Cartier divisor on $V$ for
every $j$. As usual, by perturbing $m_{j}$s suitably, we get a desired semi-
ample $\mathbb{Q}$-divisor $D^{\dagger}$ on $V$. For the details, see, for
example, the proof of [Fn2, Lemma 4.15]. ∎

Next, we treat a very useful covering trick, which is essentially due to
Yujiro Kawamata. We will use it in the proof of Theorem 9.3.

######  Lemma 3.3.

Let $f:V\to W$ be a projective surjective morphism between smooth quasi-
projective varieties and let $H$ be a Cartier divisor on $W$. Let $d$ be an
arbitrary positive integer. Then we can take a finite flat morphism
$\tau:W^{\prime}\to W$ from a smooth quasi-projective variety $W^{\prime}$ and
a Cartier divisor $H^{\prime}$ on $W^{\prime}$ such that $\tau^{*}H\sim
dH^{\prime}$ and that $V^{\prime}=V\times_{W}W^{\prime}$ is a smooth quasi-
projective variety with $\omega_{V^{\prime}/W^{\prime}}=\rho^{*}\omega_{V/W}$,
where $\rho:V^{\prime}\to V$. By construction, we may assume that
$\tau:W^{\prime}\to W$ is Galois.

###### Proof.

We take general very ample Cartier divisors $D_{1}$ and $D_{2}$ with the
following properties.

  * (i)

$H\sim D_{1}-D_{2}$,

  * (ii)

$D_{1}$, $D_{2}$, $f^{*}D_{1}$, and $f^{*}D_{2}$ are smooth,

  * (iii)

$D_{1}$ and $D_{2}$ have no common components, and

  * (iv)

${\operatorname{Supp}}(D_{1}+D_{2})$ and
${\operatorname{Supp}}(f^{*}D_{1}+f^{*}D_{2})$ are simple normal crossing
divisors.

We take a finite flat cover due to Kawamata with respect to $W$ and
$D_{1}+D_{2}$. Then we obtain $\tau:W^{\prime}\to W$ and $H^{\prime}$ such
that $\tau^{*}H\sim dH^{\prime}$. By the construction of the above Kawamata
cover $\tau:W^{\prime}\to W$, we may assume that the ramification locus
$\Sigma$ of $\tau$ in $W$ is a general simple normal crossing divisor. This
means that $f^{*}P$ is a smooth divisor for any irreducible component $P$ of
$\Sigma$ and that $f^{*}\Sigma$ is a simple normal crossing divisor on $V$. In
this situation, we can easily check that $V^{\prime}=V\times_{W}W^{\prime}$ is
a smooth quasi-projective variety.

$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{f^{\prime}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{W}$

By construction, we can also easily check that
$\omega_{V^{\prime}/W^{\prime}}=\rho^{*}\omega_{V/W}$ by the Hurwitz formula.

Let us see the construction of $f^{\prime}:V^{\prime}\to W^{\prime}$ more
precisely for the reader’s convenience. Let $\mathcal{A}$ be an ample
invertible sheaf on $W$ such that $\mathcal{A}^{\otimes
d}\otimes\mathcal{O}_{W}(-D_{i})$ is generated by global sections for $i=1,2$.
We put $n=\dim W$. We take smooth divisors

$H^{(1)}_{1},\ldots,H^{(1)}_{n},H^{(2)}_{1},\ldots,H^{(2)}_{n}$

on $W$ in general position such that $\mathcal{A}^{\otimes
d}=\mathcal{O}_{W}(D_{i}+H^{(i)}_{j})$ for $1\leq j\leq n$ and $i=1,2$. Let
$Z^{(i)}_{j}$ be the cyclic cover associated to $\mathcal{A}^{\otimes
d}=\mathcal{O}_{W}(D_{i}+H^{(i)}_{j})$ for $1\leq j\leq n$ and $i=1,2$. Then
$W^{\prime}$ is the normalization of

$\left(Z^{(1)}_{1}\times_{W}\cdots\times_{W}Z^{(1)}_{n}\right)\times_{W}\left(Z^{(2)}_{1}\times_{W}\cdots\times_{W}Z^{(2)}_{n}\right).$

For the details, see, for example, [EV, 3.19. Lemma] and [V3, Lemma 2.5]. Let
$S^{(i)}_{j}$ be the cyclic cover of $V$ associated to
$\left(f^{*}\mathcal{A}\right)^{\otimes
d}=\mathcal{O}_{V}(f^{*}D_{i}+f^{*}H^{(i)}_{j})$. Then we define $V^{\prime}$
as the normalization of

$\left(S^{(1)}_{1}\times_{V}\cdots\times_{V}S^{(1)}_{n}\right)\times_{V}\left(S^{(2)}_{1}\times_{V}\cdots\times_{V}S^{(2)}_{n}\right).$

Note that $\rho:V^{\prime}\to V$ is a finite flat morphism between smooth
quasi-projective varieties. Since $V\times_{W}W^{\prime}\to V$ is finite and
flat and $V$ is smooth, $V\times_{W}W^{\prime}$ is Cohen–Macaulay (see, for
example, [KM, Corollary 5.5]). By construction, we can easily see that
$V\times_{W}W^{\prime}$ is smooth in codimension one. Therefore,
$V\times_{W}W^{\prime}$ is normal. Since $\rho$ factors through
$V\times_{W}W^{\prime}$ by construction, we see that
$V^{\prime}=V\times_{W}W^{\prime}$ by Zariski’s main theorem. By the above
construction of $\tau:W^{\prime}\to W$, we see that $\tau:W^{\prime}\to W$ is
Galois. ∎

We give a very important remark on Lemma 3.3.

######  Remark 3.4.

In the proof of Lemma 3.3, let $S$ be any simple normal crossing divisor on
$V$. Then we can choose the ramification locus $\Sigma$ of $\tau$ such that
$f^{*}P\not\subset S$ for any irreducible component $P$ of $\Sigma$ and that
$f^{*}\Sigma\cup S$ is a simple normal crossing divisor on $V$. If we choose
$\Sigma$ as above, then we obtain that $\rho^{*}S$ is a simple normal crossing
divisor on $V^{\prime}$.

Finally, let us explain Viehweg’s fiber product trick. We include the proof
for the benefit of the reader. We will use it in the proof of Theorems 1.7 and
1.8 in Section 10.

######  Lemma 3.5.

Let $V$ be a reduced Gorenstein scheme. Note that $V$ may be reducible. We
consider

$V^{\prime}\overset{\delta}{\longrightarrow}V^{\nu}\overset{\nu}{\longrightarrow}V$

where $\nu:V^{\nu}\to V$ is the normalization and $\delta:V^{\prime}\to
V^{\nu}$ is a resolution of singularities. Then, for every positive integer
$n$, we have

(3.1) $\nu_{*}\mathcal{O}_{V^{\nu}}(nK_{V^{\nu}})\subset\omega^{\otimes
n}_{V}$

and

(3.2)
$\delta_{*}\mathcal{O}_{V^{\prime}}(nK_{V^{\prime}}+E)\subset\mathcal{O}_{V^{\nu}}(nK_{V^{\nu}})$

where $E$ is any $\delta$-exceptional divisor on $V^{\prime}$. In particular,
we have

(3.3)
$(\nu\circ\delta)_{*}\mathcal{O}_{V^{\prime}}(nK_{V^{\prime}}+E)\subset\omega^{\otimes
n}_{V}$

for every positive integer $n$. If $U$ is a Zariski open set of $V$ such that
$\nu\circ\delta$ is an isomorphism over $U$, then the inclusion (3.3) is an
isomorphism over $U$.

###### Proof.

In Steps 1 and 2, we will prove (3.2) and (3.1), respectively.

######  Step 1.

By taking the double dual of
$\delta_{*}\mathcal{O}_{V^{\prime}}(nK_{V^{\prime}}+E)$, we obtain
$\mathcal{O}_{V^{\nu}}(nK_{V^{\nu}})$. Therefore, we have

$\delta_{*}\mathcal{O}_{V^{\prime}}(nK_{V^{\prime}}+E)\subset\mathcal{O}_{V^{\nu}}(nK_{V^{\nu}})$

for every integer $n$.

######  Step 2.

Since $\nu$ is birational, the trace map
$\nu_{*}\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})\to\omega_{V}$ is a generically
isomorphic injection

(3.4) $\nu_{*}\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})\hookrightarrow\omega_{V}.$

Since $\nu$ is finite,

(3.5)
$\nu^{*}\nu_{*}\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})\to\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})$

is surjective and the kernel of (3.5) is the torsion part of
$\nu^{*}\nu_{*}\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})$. Therefore, by (3.4), we
get an inclusion

(3.6) $\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})\hookrightarrow\nu^{*}\omega_{V}.$

Let $n$ be a positive integer with $n\geq 2$. Then we have

$\mathcal{O}_{V^{\nu}}(nK_{V^{\nu}})=\mathcal{O}_{V^{\nu}}(K_{V^{\nu}}+(n-1)K_{V^{\nu}})\hookrightarrow\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})\otimes\nu^{*}\omega^{\otimes
n-1}_{V}$

by (3.6). Therefore, by taking $\nu_{*}$, we get

$\nu_{*}\mathcal{O}_{V^{\nu}}(nK_{V^{\nu}})\hookrightarrow\nu_{*}\mathcal{O}_{V^{\nu}}(K_{V^{\nu}})\otimes\omega^{\otimes
n-1}_{V}\hookrightarrow\omega^{\otimes n}_{V}$

by (3.4). This is what we wanted.

By the above construction of (3.1) and (3.2), it is obvious that the inclusion

$(\nu\circ\delta)_{*}\mathcal{O}_{V^{\prime}}(nK_{V^{\prime}}+E)\subset\omega^{\otimes
n}_{V}$

is an isomorphism over $U$. ∎

######  Lemma 3.6.

Let $f:X_{0}\to Y_{0}$ be a projective surjective morphism between smooth
quasi-projective varieties and let $\Delta_{0}$ be an effective
$\mathbb{R}$-divisor on $X_{0}$ such that ${\operatorname{Supp}}\Delta_{0}$ is
a simple normal crossing divisor on $X_{0}$ and $(X_{0},\Delta_{0})$ is log
canonical over a nonempty Zariski open set of $Y_{0}$. Let $L_{0}$ be a
Cartier divisor on $X_{0}$ such that
$L_{0}\sim_{\mathbb{R}}k(K_{X_{0}/Y_{0}}+\Delta_{0})$ for some positive
integer $k$. Assume that $f$ is flat. We consider the $s$-fold fiber product

$X^{s}_{0}:=\underbrace{X_{0}\times_{Y_{0}}X_{0}\times_{Y_{0}}\cdots\times_{Y_{0}}X_{0}}_{s}$

of $X_{0}$ over $Y_{0}$ and let $f^{s}:X^{s}_{0}\to Y_{0}$ be the induced
morphism. We take a resolution of singularities $\rho:X^{(s)}_{0}\to
X^{s}_{0}$ which is an isomorphism over a nonempty Zariski open set of
$Y_{0}$. Then we can write

$\mathcal{O}_{X^{(s)}_{0}}(K_{X^{(s)}_{0}})=\rho^{*}\omega_{X^{s}_{0}}\otimes\mathcal{O}_{X^{(s)}_{0}}(R)$

where $R$ is an $(f^{s}\circ\rho)$-vertical Cartier divisor by construction.
Let $p_{i}:X^{s}_{0}\to X_{0}$ be the $i$-th projection. We put
$\pi_{i}=p_{i}\circ\rho:X^{(s)}_{0}\to X_{0}$. We consider

$L^{(s)}_{0}:=\sum_{i=1}^{s}\pi_{i}^{*}L_{0}+kR.$

We further assume that $f_{*}\mathcal{O}_{X_{0}}(L_{0})$ is locally free. Then
there exists a generically isomorphic injection

$f^{(s)}_{*}\mathcal{O}_{X^{(s)}_{0}}(L^{(s)}_{0})\hookrightarrow\bigotimes_{i=1}^{s}f_{*}\mathcal{O}_{X_{0}}(L_{0})$

with $f^{(s)}=f^{s}\circ\rho$. By construction, we have

$L^{(s)}_{0}\sim_{\mathbb{R}}k\left(K_{X^{(s)}_{0}/Y_{0}}+\sum_{i=1}^{s}\pi_{i}^{*}\Delta_{0}\right).$

Note that

$\left(X^{(s)}_{0},\sum_{i=1}^{s}\pi_{i}^{*}\Delta_{0}\right)$

is log canonical over a nonempty Zariski open set of $Y_{0}$. We also note
that $X^{(s)}_{0}$ may be reducible, that is, $X^{(s)}_{0}$ may be a disjoint
union of smooth varieties.

###### Proof.

By the flat base change theorem, we have

$\omega_{X^{s}_{0}/Y_{0}}=\bigotimes_{i=1}^{s}p^{*}_{i}\omega_{X_{0}/Y_{0}}.$

In particular, $X^{s}_{0}$ is Gorenstein. We note that

(3.7)
$\begin{split}L^{(s)}_{0}&=\sum_{i=1}^{s}\rho^{*}p^{*}_{i}(kK_{X_{0}/Y_{0}}+(L_{0}-kK_{X_{0}/Y_{0}}))+kR\\\
&\sim
kK_{X^{(s)}_{0}/Y_{0}}+\sum_{i=1}^{s}\pi^{*}_{i}(L_{0}-kK_{X_{0}/Y_{0}}).\end{split}$

######  Claim.

We have the following isomorphism of locally free sheaves:

$f^{s}_{*}\mathcal{O}_{X^{s}_{0}}\left(\sum_{i=1}^{s}p^{*}_{i}L_{0}\right)\simeq\bigotimes^{s}f_{*}\mathcal{O}_{X_{0}}(L_{0}).$

###### Proof of Claim.

We use induction on $s$. If $s=1$, then the statement is obvious. So we assume
that $s\geq 2$. We consider the following commutative diagram

$\textstyle{X^{s}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{s}}$$\scriptstyle{p_{s}}$$\scriptstyle{q}$$\textstyle{X^{s-1}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{s-1}}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y_{0}}$

where $q=(p_{1},\cdots,p_{s-1})$. Then we have

(3.8)
$\mathcal{O}_{X^{s}_{0}}\left(\sum_{i=1}^{s}p^{*}_{i}L_{0}\right)\simeq\mathcal{O}_{X^{s}_{0}}(p^{*}_{s}L_{0})\otimes
q^{*}\mathcal{O}_{X^{s-1}_{0}}\left(\sum_{i=1}^{s-1}p^{*}_{i}L_{0}\right).$

Therefore, we obtain

$\begin{split}f^{s}_{*}\mathcal{O}_{X^{s}_{0}}\left(\sum_{i=1}^{s}p^{*}_{i}L_{0}\right)&\simeq
f_{*}{p_{s}}_{*}\left(\mathcal{O}_{X^{s}_{0}}(p^{*}_{s}L_{0})\otimes
q^{*}\mathcal{O}_{X^{s-1}_{0}}\left(\sum_{i=1}^{s-1}p^{*}_{i}L_{0}\right)\right)\\\
&\simeq
f_{*}\left(\mathcal{O}_{X_{0}}(L_{0})\otimes{p_{s}}_{*}q^{*}\mathcal{O}_{X^{s-1}_{0}}\left(\sum_{i=1}^{s-1}p^{*}_{i}L_{0}\right)\right)\\\
&\simeq f_{*}\left(\mathcal{O}_{X_{0}}(L_{0})\otimes
f^{*}f^{s-1}_{*}\mathcal{O}_{X^{s-1}_{0}}\left(\sum_{i=1}^{s-1}p^{*}_{i}L_{0}\right)\right)\\\
&\simeq f_{*}\left(\mathcal{O}_{X_{0}}(L_{0})\otimes
f^{*}\left(\bigotimes^{s-1}f_{*}\mathcal{O}_{X_{0}}(L_{0})\right)\right)\\\
&\simeq
f_{*}\mathcal{O}_{X_{0}}(L_{0})\otimes\bigotimes^{s-1}f_{*}\mathcal{O}_{X_{0}}(L_{0})\\\
&\simeq\bigotimes^{s}f_{*}\mathcal{O}_{X_{0}}(L_{0}).\end{split}$

Note that the first isomorphism follows from (3.8), the second one is due to
the projection formula, the third one is obtained by the flat base change
theorem, the fourth one is due to induction on $s$, and the fifth one follows
from the projection formula. Anyway, we obtain the desired isomorphism. ∎

Let us go back to the proof of Lemma 3.6. We have an inclusion

$\begin{split}\rho_{*}\mathcal{O}_{X^{(s)}_{0}}(L^{(s)}_{0})&\subset\omega^{\otimes
k}_{X^{s}_{0}/Y_{0}}\otimes\mathcal{O}_{X^{s}_{0}}\left(\sum_{i=1}^{s}p^{*}_{i}(L_{0}-kK_{X_{0}/Y_{0}})\right)\\\
&\simeq\mathcal{O}_{X^{s}_{0}}\left(\sum_{i=1}^{s}p^{*}_{i}L_{0}\right)\end{split}$

by (3.7) and Lemma 3.5, which is an isomorphism over a nonempty Zariski open
set of $Y_{0}$. By taking $f^{s}_{*}$, we obtain a generically isomorphic
injection

$\begin{split}f^{(s)}_{*}\mathcal{O}_{X^{(s)}_{0}}(L^{(s)}_{0})&\subset
f^{s}_{*}\mathcal{O}_{X^{s}_{0}}\left(\sum_{i=1}^{s}p^{*}_{i}L_{0}\right)\\\
&\simeq\bigotimes_{i=1}^{s}f_{*}\mathcal{O}_{X_{0}}(L_{0}),\end{split}$

where $f^{(s)}=f^{s}\circ\rho:X^{(s)}_{0}\to Y_{0}$. By assumption,
$L_{0}-kK_{X_{0}/Y_{0}}\sim_{\mathbb{R}}k\Delta_{0}$. Therefore,

$L^{(s)}_{0}\sim_{\mathbb{R}}k\left(K_{X^{(s)}_{0}/Y_{0}}+\sum_{i=1}^{s}\pi^{*}_{i}\Delta_{0}\right)$

by (3.7). We can take a nonempty Zariski open set $U$ of $Y_{0}$ such that $f$
is smooth over $U$, ${\operatorname{Supp}}\Delta$ is relatively simple normal
crossing over $U$, and $\rho$ is an isomorphism over $U$. Then we see that
$\left(X^{(s)}_{0},\sum_{i=1}^{s}\pi^{*}_{i}\Delta_{0}\right)$ is log
canonical over $U$. ∎

##  4\. Weakly positive sheaves and big sheaves

We quickly see some basic properties of Viehweg’s weakly positive sheaves and
big sheaves. For the details, see [Fn9, Chapter 3], [V1], [V2], and [V3].

######  Definition 4.1 (Weak positivity and bigness).

Let $\mathcal{F}$ be a torsion-free coherent sheaf on a smooth quasi-
projective variety $W$. We say that $\mathcal{F}$ is weakly positive if, for
every positive integer $\alpha$ and every ample invertible sheaf
$\mathcal{H}$, there exists a positive integer $\beta$ such that
$\widehat{S}^{\alpha\beta}(\mathcal{F})\otimes\mathcal{H}^{\otimes\beta}$ is
generically generated by global sections. We say that a nonzero torsion-free
coherent sheaf $\mathcal{F}$ is big (in the sense of Viehweg) if, for every
ample invertible sheaf $\mathcal{H}$, there exists a positive integer $a$ such
that $\widehat{S}^{a}(\mathcal{F})\otimes\mathcal{H}^{\otimes-1}$ is weakly
positive.

For the reader’s convenience, let us recall the following basic properties of
big sheaves without proof.

######  Lemma 4.2 ([V2, Lemma 3.6] and [Fn9, Lemma 3.1.15]).

Let $\mathcal{F}$ be a nonzero torsion-free coherent sheaf on a smooth quasi-
projective variety $W$. Then the following three conditions are equivalent.

  * (i)

There exist an ample invertible sheaf $\mathcal{H}$ on $W$, some positive
integer $\nu$, and an inclusion
$\bigoplus\mathcal{H}\hookrightarrow\widehat{S}^{\nu}(\mathcal{F})$, which is
an isomorphism over a nonempty Zariski open set of $W$.

  * (ii)

For every invertible sheaf $\mathcal{M}$ on $W$, there exists some positive
integer $\gamma$ such that
$\widehat{S}^{\gamma}(\mathcal{F})\otimes\mathcal{M}^{\otimes-1}$ is weakly
positive. In particular, $\mathcal{F}$ is a big sheaf.

  * (iii)

There exist some positive integer $\gamma$ and an ample invertible sheaf
$\mathcal{M}$ on $W$ such that
$\widehat{S}^{\gamma}(\mathcal{F})\otimes\mathcal{M}^{\otimes-1}$ is weakly
positive.

We will use the following two easy lemmas on big sheaves in this paper. So we
explicitly state them here for the reader’s convenience.

######  Lemma 4.3.

Let $\mathcal{F}$ be a weakly positive sheaf and let $\mathcal{H}$ be an ample
invertible sheaf on a smooth quasi-projective variety $W$. Then
$\mathcal{F}\otimes\mathcal{H}$ is big.

We give a proof for the sake of completeness.

###### Proof of Lemma 4.3.

Since $\mathcal{F}$ is weakly positive,
$\widehat{S}^{2b}(\mathcal{F})\otimes\mathcal{H}^{\otimes b}$ is generically
generated by global sections for some positive integer $b$. By replacing $b$
with a multiple, we may assume that $\mathcal{H}^{\otimes b-1}$ is generated
by global sections. Then
$\widehat{S}^{2b}(\mathcal{F}\otimes\mathcal{H})\otimes\mathcal{H}^{\otimes-1}=\widehat{S}^{2b}(\mathcal{F})\otimes\mathcal{H}^{\otimes
2b-1}$ is generically generated by global sections. In particular,
$\widehat{S}^{2b}(\mathcal{F}\otimes\mathcal{H})\otimes\mathcal{H}^{\otimes-1}$
is weakly positive. This implies that $\mathcal{F}\otimes\mathcal{H}$ is big
by Lemma 4.2. ∎

######  Lemma 4.4.

Let $\mathcal{F}$ be a torsion-free coherent sheaf on a smooth quasi-
projective variety $W$ and let $\tau:W^{\prime}\to W$ be a finite surjective
morphism from a smooth quasi-projective variety $W^{\prime}$. Assume that
$\tau^{*}\mathcal{F}$ is big. Then $\mathcal{F}$ is a big sheaf on $W$.

We include the proof for the benefit of the reader.

###### Proof of Lemma 4.4.

We take an ample invertible sheaf $\mathcal{H}$ on $W$. By replacing $W$ with
$W\setminus\Sigma$ for some suitable closed subset $\Sigma$ of codimension
$\geq 2$ (see, for example, [Fn9, Lemma 3.1.12 (i)]), we may assume that
$\mathcal{F}$ is locally free. Since $\tau^{*}\mathcal{F}$ is big by
assumption, there exists a positive integer $a$ such that
$S^{a}(\tau^{*}\mathcal{F})\otimes\tau^{*}\mathcal{H}^{\otimes-1}=\tau^{*}\left(S^{a}(\mathcal{F})\otimes\mathcal{H}^{\otimes-1}\right)$
is weakly positive (see Lemma 4.2). Therefore,
$S^{a}(\mathcal{F})\otimes\mathcal{H}^{\otimes-1}$ is weakly positive since
$\tau$ is finite (see, for example, [Fn9, Lemma 3.1.12 (v)]). This means that
$\mathcal{F}$ is big by Lemma 4.2. ∎

##  5\. Mixed-$\omega$-sheaves and mixed-$\omega$-big sheaves

In this section, we introduce mixed-$\omega$-sheaves, mixed-$\omega$-big-
sheaves, mixed-$\widehat{\omega}$-sheaves, and mixed-$\widehat{\omega}$-big-
sheaves. We also treat some important examples in Lemmas 5.5, 5.8, and 5.9.

Let us start with the definition of mixed-$\omega$-sheaves and
pure-$\omega$-sheaves.

######  Definition 5.1 (Mixed-$\omega$-sheaf and pure-$\omega$-sheaf).

A torsion-free coherent sheaf $\mathcal{F}$ on a normal quasi-projective
variety $W$ is called a mixed-$\omega$-sheaf if there exist a projective
surjective morphism from a smooth quasi-projective variety $V$ and a simple
normal crossing divisor $D$ on $V$ such that $\mathcal{F}$ is a direct summand
of $f_{*}\mathcal{O}_{V}(K_{V}+D)$. When $D=0$, $\mathcal{F}$ is called a
pure-$\omega$-sheaf on $W$.

We give a very important remark on Definition 5.1.

######  Remark 5.2 (Pure-$\omega$-sheaves versus Nakayama’s $\omega$-sheaves).

The notion of pure-$\omega$-sheaves is essentially the same as that of
Nakayama’s $\omega$-sheaves in [N] when we treat torsion-free coherent sheaves
on normal projective varieties. However, the definition of
pure-$\omega$-sheaves in Definition 5.1 does not coincide with [N, Chapter V,
3.8. Definition]. Our definition seems to be more reasonable than Nakayama’s
from the mixed Hodge theoretic viewpoint.

For some geometric applications, the notion of mixed-$\omega$-big-sheaves and
pure-$\omega$-big-sheaves is very useful.

######  Definition 5.3 (Mixed-$\omega$-big-sheaf and pure-$\omega$-big-sheaf).

Let $\mathcal{F}$ be a torsion-free coherent sheaf on a normal quasi-
projective variety $W$. If there exist projective surjective morphisms $f:V\to
W$, $p:V\to Z$, and an ample divisor $A$ on $Z$ satisfying the following
conditions:

  * (i)

$V$ is a smooth quasi-projective variety,

  * (ii)

$Z$ is a normal quasi-projective variety,

  * (iii)

$D$ is a simple normal crossing divisor on $V$,

  * (iv)

there exists a projective surjective morphism $q:Z\to W$ such that $f=q\circ
p$, and

  * (v)

$\mathcal{F}$ is a direct summand of $f_{*}\mathcal{O}_{V}(K_{V}+D+P)$, where
$P$ is a Cartier divisor on $V$ such that $P\sim_{\mathbb{Q}}p^{*}A$,

---
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{p}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{W}$

then $\mathcal{F}$ is called a mixed-$\omega$-big-sheaf on $W$. As in
Definition 5.1, $\mathcal{F}$ is called a pure-$\omega$-big-sheaf on $W$ when
$D=0$.

######  Remark 5.4.

Of course, we defined mixed-$\omega$-big-sheaves and pure-$\omega$-big-sheaves
referring to [N, Chapter V, 3.16. Definition (1)]. However, Nakayama’s
definition of $\omega$-bigness is different from ours.

Lemma 5.5 gives a very basic example of mixed-$\omega$-sheaves.

######  Lemma 5.5.

Let $V$ be a smooth quasi-projective variety and let $D$ be a simple normal
crossing divisor on $V$. Let $L$ be a semi-ample Cartier divisor on $V$. Then
$\mathcal{O}_{V}(K_{V}+D+L)$ is a mixed-$\omega$-sheaf on $V$ and
$\mathcal{O}_{V}(K_{V}+L)$ is a pure-$\omega$-sheaf on $V$.

Although this lemma is well known, we give a proof for the sake of
completeness.

###### Proof of Lemma 5.5.

Let $m$ be a positive integer such that $|mL|$ is free. We take a general
section $s\in H^{0}(V,\mathcal{O}_{V}(mL))$, whose zero divisor is $B$. We may
assume that $B$ is a smooth divisor, $B$ and $D$ have no common irreducible
components, and ${\operatorname{Supp}}(B+D)$ is a simple normal crossing
divisor on $V$. The dual of

$s:\mathcal{O}_{V}\to\mathcal{O}_{V}(mL)$

defines an $\mathcal{O}_{V}$-algebra structure on

$\bigoplus_{i=0}^{m-1}\mathcal{O}_{V}(-iL).$

We put

$\pi:Z:={\operatorname{Spec}}_{V}\bigoplus_{i=0}^{m-1}\mathcal{O}_{V}(-iL)\to
V.$

Then $Z$ is a smooth quasi-projective variety and $\pi^{*}D$ is a simple
normal crossing divisor on $Z$ by construction. We can check that

$\pi_{*}\mathcal{O}_{Z}(K_{Z}+\pi^{*}D)\simeq\bigoplus_{i=0}^{m-1}\mathcal{O}_{V}(K_{V}+D+iL)$

since $\pi_{*}\mathcal{O}_{V}=\bigoplus_{i=0}^{m-1}\mathcal{O}_{V}(-iL)$. This
means that $\mathcal{O}_{V}(K_{V}+D+L)$ is a mixed-$\omega$-sheaf on $V$. We
put $D=0$ in the above argument. Then we see that $\mathcal{O}_{V}(K_{V}+L)$
is a pure-$\omega$-sheaf on $V$. ∎

We treat two elementary lemmas.

######  Lemma 5.6.

Let $\mathcal{F}$ be a mixed-$\omega$-big-sheaf (resp. pure-$\omega$-big-
sheaf) on a normal quasi-projective variety $W$. Then $\mathcal{F}$ is a
mixed-$\omega$-sheaf (resp. pure-$\omega$-sheaf) on $W$.

###### Proof.

We may assume that $\mathcal{F}$ is a direct summand of
$f_{*}\mathcal{O}_{V}(K_{V}+D+P)$ as in Definition 5.3. By Lemma 5.5,
$\mathcal{O}_{V}(K_{V}+D+P)$ is a mixed-$\omega$-sheaf on $V$. Therefore, we
see that $\mathcal{F}$ is a mixed-$\omega$-sheaf on $W$. If we put $D=0$, then
we see that $\mathcal{F}$ is a pure-$\omega$-sheaf on $W$. ∎

######  Lemma 5.7.

Let $\mathcal{F}$ be a mixed-$\omega$-sheaf (resp. pure-$\omega$-sheaf) on a
normal quasi-projective variety $W$ and let $\mathcal{A}$ be an ample
invertible sheaf on $W$. Then $\mathcal{F}\otimes\mathcal{A}$ is a
mixed-$\omega$-big-sheaf (resp. pure-$\omega$-big-sheaf) on $W$.

###### Proof.

We may assume that $\mathcal{F}$ is a direct summand of
$f_{*}\mathcal{O}_{V}(K_{V}+D)$ as in Definition 5.1. We put $Z=W$. Let $A$ be
an ample divisor on $W$ such that $\mathcal{O}_{W}(A)=\mathcal{A}$. Then
$\mathcal{F}\otimes\mathcal{A}$ is a direct summand of
$f_{*}\mathcal{O}_{V}(K_{V}+D+f^{*}A)$. Therefore,
$\mathcal{F}\otimes\mathcal{A}$ is a mixed-$\omega$-big-sheaf on $W$. When
$D=0$, we see that $\mathcal{F}\otimes\mathcal{A}$ is a pure-$\omega$-big-
sheaf on $W$. ∎

Lemmas 5.8 and 5.9 give many nontrivial important examples of
mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves in the study of higher-
dimensional algebraic varieties.

######  Lemma 5.8.

Let $f:V\to W$ be a projective surjective morphism from a smooth projective
variety $V$ onto a normal projective variety $W$. Let $D$ be a simple normal
crossing divisor on $V$ and let $M$ be an $\mathbb{R}$-divisor on $V$ such
that $M-f^{*}H$ is semi-ample for some ample $\mathbb{Q}$-divisor $H$ on $W$.
We assume that $D$ and ${\operatorname{Supp}}\\{M\\}$ have no common
irreducible components and ${\operatorname{Supp}}(D+\\{M\\})$ is a simple
normal crossing divisor on $V$. Then $f_{*}\mathcal{O}_{V}(K_{V}+D+\lceil
M\rceil)$ is a mixed-$\omega$-big-sheaf on $W$ and
$f_{*}\mathcal{O}_{V}(K_{V}+\lceil M\rceil)$ is a pure-$\omega$-big-sheaf on
$W$.

###### Proof.

By Lemma 3.2, we can construct a $\mathbb{Q}$-divisor $M^{\dagger}$ on $V$
such that $M^{\dagger}-f^{*}H$ is semi-ample,
${\operatorname{Supp}}\\{M^{\dagger}\\}={\operatorname{Supp}}\\{M\\}$, and
$\lceil M^{\dagger}\rceil=\lceil M\rceil$. Therefore, we may assume that $M$
is a $\mathbb{Q}$-divisor by replacing $M$ with $M^{\dagger}$. By Kawamata’s
covering construction, we can construct a finite Galois cover
$\pi:V^{\prime}\to V$ from a smooth projective variety $V^{\prime}$ with the
following properties:

  * (i)

$\pi^{*}D$ is a simple normal crossing divisor on $V^{\prime}$,

  * (ii)

$\pi^{*}\\{M\\}$ is a $\mathbb{Z}$-divisor on $V^{\prime}$,

  * (iii)

${\operatorname{Supp}}(\pi^{*}D+\pi^{*}\\{M\\})$ is a simple normal crossing
divisor on $V^{\prime}$, and

  * (iv)

$\left(\pi_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+\pi^{*}D+\pi^{*}M)\right)^{G}\simeq\mathcal{O}_{V}(K_{V}+D+\lceil
M\rceil)$, where $G$ is the Galois group of $\pi:V^{\prime}\to V$.

By assumption, $\pi^{*}M$ is semi-ample. Since $\pi^{*}M-(f\circ\pi)^{*}H$ is
semi-ample, we have the following commutative diagram:

---
$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\circ\pi}$$\scriptstyle{p}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{W}$

such that

  * (a)

$Z$ is a normal projective variety, and

  * (b)

there is an ample $\mathbb{Q}$-divisor $A$ on $Z$ with
$\pi^{*}M\sim_{\mathbb{Q}}p^{*}A$.

Therefore, $f_{*}\mathcal{O}_{V}(K_{V}+D+\lceil M\rceil)$ is a
mixed-$\omega$-big-sheaf on $W$ since it is a direct summand of
$(f\circ\pi)_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+\pi^{*}D+\pi^{*}M)$.
We put $D=0$ in the above argument. Then $f_{*}\mathcal{O}_{V}(K_{V}+\lceil
M\rceil)$ is a pure-$\omega$-big-sheaf on $W$. ∎

######  Lemma 5.9.

Let $V$ be a smooth quasi-projective variety and let $D$ be a simple normal
crossing divisor on $V$. Let $B$ be a $\mathbb{Q}$-divisor on $V$ such that
$rB\sim 0$ for some positive integer $r$, ${\operatorname{Supp}}\\{B\\}$ and
$D$ have no common irreducible components, and
${\operatorname{Supp}}(\\{B\\}+D)$ is a simple normal crossing divisor on $V$.
Then there exist a generically finite morphism $\pi:V^{\prime}\to V$ from a
smooth quasi-projective variety $V^{\prime}$ and a simple normal crossing
divisor $D^{\prime}$ on $V^{\prime}$ such that $\mathcal{O}_{V}(K_{V}+D+\lceil
B\rceil)$ is a direct summand of
$\pi_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D^{\prime})$. In particular,
$\mathcal{O}_{V}(K_{V}+D+\lceil B\rceil)$ is a mixed-$\omega$-sheaf on $V$.
When $D=0$, $\mathcal{O}_{V}(K_{V}+\lceil B\rceil)$ is obviously a
pure-$\omega$-sheaf on $V$.

###### Proof.

If $B\sim 0$, then there are nothing to prove. By replacing $r$ suitably, we
may assume that $iB\not\sim 0$ for $1\leq i\leq r-1$ and that $r\geq 2$. We
consider the following $\mathcal{O}_{V}$-algebra

$\mathcal{A}=\bigoplus_{i=0}^{r-1}\mathcal{O}_{V}(\lfloor-iB\rfloor)$

defined by an isomorphism $\mathcal{O}_{V}(-rB)\simeq\mathcal{O}_{V}$. Let $Z$
be the normalization of ${\operatorname{Spec}}_{V}\mathcal{A}$. Then we have

$\tau_{*}\mathcal{O}_{Z}(K_{Z}+\tau^{*}D)\simeq\bigoplus_{i=0}^{r-1}(K_{V}+D+\lceil
iB\rceil)$

where $\tau:Z\to V$. By construction, we see that $(Z,\tau^{*}D)$ is dlt. We
take a suitable resolution of singularities $\rho:V^{\prime}\to Z$ and write

$K_{V^{\prime}}+D^{\prime}=\rho^{*}(K_{Z}+\tau^{*}D)+E$

where $D^{\prime}$ is a reduced simple normal crossing divisor on $V^{\prime}$
and $E$ is an effective $\rho$-exceptional $\mathbb{Q}$-divisor on
$V^{\prime}$. We put $\pi:=\tau\circ\rho:V^{\prime}\to V$. Then

$\begin{split}\pi_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D^{\prime})&\simeq\tau_{*}\mathcal{O}_{Z}(K_{Z}+\tau^{*}D)\\\
&\simeq\bigoplus_{i=0}^{r-1}\mathcal{O}_{V}(K_{V}+D+\lceil
iB\rceil).\end{split}$

Therefore, we have the desired statement. ∎

We close this section with the definition of mixed-$\widehat{\omega}$-sheaves,
mixed-$\widehat{\omega}$-big-sheaves, pure-$\widehat{\omega}$-sheaves, and
pure-$\widehat{\omega}$-big-sheaves.

######  Definition 5.10 (Mixed-$\widehat{\omega}$-sheaf,
mixed-$\widehat{\omega}$-big-sheaf, pure-$\widehat{\omega}$-sheaf, and
pure-$\widehat{\omega}$-big-sheaf).

A torsion-free coherent sheaf $\mathcal{G}$ on a normal quasi-projective
variety $W$ is called a mixed-$\widehat{\omega}$-sheaf (resp.
mixed-$\widehat{\omega}$-big-sheaf) if there exist a mixed-$\omega$-sheaf
(resp. mixed-$\omega$-big-sheaf) $\mathcal{F}$ on $W$ and a generically
isomorphic injection $\mathcal{F}\hookrightarrow\mathcal{G}^{**}$ into the
double dual $\mathcal{G}^{**}$ of $\mathcal{G}$. If $\mathcal{F}$ is a
pure-$\omega$-sheaf (resp. pure-$\omega$-big-sheaf) in the above inclusion
$\mathcal{F}\hookrightarrow\mathcal{G}^{**}$, then $\mathcal{G}$ is called a
pure-$\widehat{\omega}$-sheaf (resp. pure-$\widehat{\omega}$-big-sheaf).

Let $f:X\to Y$ be a surjective morphism between smooth projective varieties
and let $\Delta$ be a simple normal crossing divisor on $X$. Let $k$ be a
positive integer with $k\geq 2$ and let $H$ be an ample Cartier divisor on
$Y$. Then we will show that

$\mathcal{O}_{Y}(K_{Y}+H)\otimes f_{*}\mathcal{O}_{X}(k(K_{X/Y}+\Delta))$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$. This is a special case of
Theorem 9.3, which we call a fundamental theorem of the theory of
mixed-$\omega$-sheaves.

##  6\. Basic properties: Part 1

In this section, we treat the weak positivity and the bigness of
mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves, respectively.

Let us start with the following weak positivity theorem, which follows from
the theory of mixed Hodge structures.

######  Theorem 6.1.

Let $f:V\to W$ be a projective surjective morphism between smooth quasi-
projective varieties. Let $D$ be a simple normal crossing divisor on $V$. Then
$f_{*}\mathcal{O}_{V}(K_{V/W}+D)$ is weakly positive.

###### Proof.

We may assume that $V$ and $W$ are smooth projective varieties by
compactifying $f:V\to W$ suitably. Then this result is more or less well
known. For the proof based on the theory of variations of mixed Hodge
structure (see [Fn1], [FFS], [FF], [Fs], and so on), see [Fn8, Theorem 7.8 and
Corollary 7.11]. For the proof based on the vanishing theorem, see [Fn3,
Theorem 8.4]. ∎

As an easy consequence of Theorem 6.1, we have:

######  Theorem 6.2 (Weak positivity).

Let $\mathcal{F}$ be a mixed-$\omega$-sheaf on a smooth quasi-projective
variety $W$. Then $\mathcal{F}\otimes\omega^{\otimes-1}_{W}$ is weakly
positive.

###### Proof.

We may assume that $\mathcal{F}$ is a direct summand of
$f_{*}\mathcal{O}_{V}(K_{V}+D)$ as in Definition 5.1. By Theorem 6.1,
$f_{*}\mathcal{O}_{V}(K_{V/W}+D)$ is weakly positive. Then
$\mathcal{F}\otimes\omega^{\otimes-1}_{W}$ is weakly positive since it is a
direct summand of $f_{*}\mathcal{O}_{V}(K_{V/W}+D)$. ∎

When $\mathcal{F}$ is a mixed-$\omega$-big-sheaf on $W$ in Theorem 6.2,
$\mathcal{F}\otimes\omega^{\otimes-1}_{W}$ is not only weakly positive but
also big.

######  Theorem 6.3 (Bigness).

Let $\mathcal{F}$ be a mixed-$\omega$-big-sheaf on a smooth quasi-projective
variety $W$. Then $\mathcal{F}\otimes\omega^{\otimes-1}_{W}$ is big.

###### Proof.

Without loss of generality, we may assume that $\mathcal{F}$ is a direct
summand of $f_{*}\mathcal{O}_{V}(K_{V}+D+P)$ as in Definition 5.3. It is
sufficient to prove that $f_{*}\mathcal{O}_{V}(K_{V/W}+D+P)$ is big. Let

---
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{p}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{W}$

and $A$ be as in Definition 5.3. Let $H$ be an ample Cartier divisor on $W$.
We take a positive integer $m$ such that $mA-q^{*}H$ is ample. We can take a
finite surjective morphism $\tau:W^{\prime}\to W$ from a smooth quasi-
projective variety $W^{\prime}$ and get the following commutative diagram

$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\rho}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{W}$

such that $\tau^{*}H\sim mH^{\prime}$ for some Cartier divisor $H^{\prime}$,
$V^{\prime}=V\times_{W}W^{\prime}$ is a smooth quasi-projective variety,
$\rho^{*}D$ is a simple normal crossing divisor, and $\rho^{*}\omega^{\otimes
n}_{V/W}=\omega^{\otimes n}_{V/W}$ holds for every integer $n$ (see Lemma 3.3
and Remark 3.4). By Lemma 4.4, It is sufficient to prove that

$\tau^{*}f_{*}\mathcal{O}_{V}(K_{V/W}+D+P)\simeq
f^{\prime}_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}/W^{\prime}}+\rho^{*}D+\rho^{*}P)$

is a big sheaf on $W^{\prime}$. By construction, we see that
$\rho^{*}P-f^{\prime*}H^{\prime}$ is a semi-ample Cartier divisor on
$V^{\prime}$ since it is $\mathbb{Q}$-linearly equivalent to
$\rho^{*}p^{*}(A-(1/m)q^{*}H)$. Therefore, by Lemma 5.5,
$\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+\rho^{*}D+\rho^{*}P-f^{\prime*}H^{\prime})$
is a mixed-$\omega$-sheaf on $V^{\prime}$. Therefore,
$\mathcal{E}:=f^{\prime}_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+\rho^{*}D+\rho^{*}P-f^{\prime*}H^{\prime})$
is a mixed-$\omega$-sheaf on $W^{\prime}$. We note that

$f^{\prime}_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}/W^{\prime}}+\rho^{*}D+\rho^{*}P)\simeq\mathcal{E}\otimes\omega^{\otimes-1}_{W^{\prime}}\otimes\mathcal{O}_{W^{\prime}}(H^{\prime}).$

By Theorem 6.2, $\mathcal{E}\otimes\omega^{\otimes-1}_{W^{\prime}}$ is weakly
positive. By Lemma 4.3,
$\mathcal{E}\otimes\omega^{\otimes-1}_{W^{\prime}}\otimes\mathcal{O}_{W^{\prime}}(H^{\prime})$
is big since $H^{\prime}$ is ample. This is what we wanted. ∎

We close this section with an obvious corollary.

######  Corollary 6.4.

Let $\mathcal{F}$ be a mixed-$\widehat{\omega}$-sheaf (resp.
mixed-$\widehat{\omega}$-big-sheaf) on a smooth quasi-projective variety $W$.
Then $\mathcal{F}\otimes\omega^{\otimes-1}_{W}$ is weakly positive (resp.
big).

###### Proof.

We note that $\mathcal{F}\otimes\omega^{\otimes-1}_{W}$ is weakly positive
(resp. big) if and only if so is
$\mathcal{F}^{**}\otimes\omega^{\otimes-1}_{W}$. Therefore, the desired
statement follows from Theorems 6.2 and 6.3. ∎

##  7\. Basic properties: Part 2

In this section, we discuss some vanishing theorems for mixed-$\omega$-sheaves
and several related topics.

######  Lemma 7.1 (Vanishing theorem for mixed-$\omega$-big-sheaf).

Let $\mathcal{F}$ be a mixed-$\omega$-big-sheaf on a normal projective variety
$W$. Then $H^{i}(W,\mathcal{F}\otimes\mathcal{N})=0$ for every $i>0$ and every
nef invertible sheaf $\mathcal{N}$ on $W$.

###### Proof.

We may assume that $\mathcal{F}$ is a direct summand of
$f_{*}\mathcal{O}_{V}(K_{V}+D+P)$ as in Definition 5.3. Let $N$ be a Cartier
divisor on $W$ such that $\mathcal{N}\simeq\mathcal{O}_{W}(N)$. It is
sufficient to prove that $H^{i}(W,f_{*}\mathcal{O}_{V}(K_{V}+D+P+f^{*}N))=0$
for every $i>0$. We take an ample $\mathbb{Q}$-divisor $H$ on $W$ such that
$A-q^{*}H$ is an ample $\mathbb{Q}$-divisor on $Z$, where $A$ and $q:Z\to W$
are as in Definition 5.3. Then we can take a boundary $\mathbb{Q}$-divisor
$\Delta$ on $V$ such that $\Delta\sim_{\mathbb{Q}}D+P-f^{*}H$ and that
${\operatorname{Supp}}\Delta$ is a simple normal crossing divisor on $V$. Then
we have

$K_{V}+D+P+f^{*}N-(K_{V}+\Delta)\sim_{\mathbb{Q}}f^{*}(H+N).$

We note that $H+N$ is ample. Therefore, by [Fn2, Theorem 6.3 (ii)] (see also
[Fn6, Theorem 3.16.3 (ii) and Theorem 5.6.2 (ii)], and so on), we obtain that
$H^{i}(W,f_{*}\mathcal{O}_{V}(K_{V}+D+P+f^{*}N))=0$ for every $i>0$. ∎

As an easy consequence of Lemma 7.1, we have:

######  Lemma 7.2.

Let $\mathcal{F}$ be a mixed-$\omega$-sheaf (resp. mixed-$\omega$-big-sheaf)
on a normal projective variety $W$ with $\dim W=n$. Let $\mathcal{A}$ be an
ample invertible sheaf on $W$ such that $|\mathcal{A}|$ is free. Then
$\mathcal{F}\otimes\mathcal{A}^{\otimes n+1}$ (resp.
$\mathcal{F}\otimes\mathcal{A}^{\otimes n}$) is generated by global sections.

###### Proof.

This is a direct consequence of Lemmas 5.7, 7.1, and Castelnuovo–Mumford
regularity. ∎

Let us recall a vanishing theorem for dlt pairs.

######  Lemma 7.3.

Let $f:V\to W$ be a surjective morphism from a smooth projective variety $V$
onto a normal projective variety $W$. Let $\Delta$ be an effective
$\mathbb{R}$-divisor on $V$ such that $(V,\Delta)$ is dlt and that every log
canonical center of $(V,\Delta)$ is dominant onto $W$. Let $L$ be a Cartier
divisor on $V$ such that $L-(K_{V}+\Delta)\sim_{\mathbb{R}}f^{*}H$ for some
nef and big $\mathbb{R}$-divisor $H$ on $W$. Then
$H^{i}(W,R^{j}f_{*}\mathcal{O}_{V}(L)\otimes\mathcal{N})=0$ for $i>0$, $j\geq
0$, and every nef invertible sheaf $\mathcal{N}$ on $W$.

###### Sketch of Proof.

By Kodaira’s lemma, we can write $H\sim_{\mathbb{R}}A+E$ such that $A$ is an
ample $\mathbb{R}$-divisor on $W$ and $E$ is an effective $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor on $W$. Since every log canonical center of $(V,\Delta)$
is dominant onto $W$, $(V,\Delta+\varepsilon f^{*}E)$ is dlt for
$0<\varepsilon\ll 1$. Let $N$ be a Cartier divisor on $W$ such that
$\mathcal{N}\simeq\mathcal{O}_{W}(N)$. We note that

$L+f^{*}N-(K_{V}+\Delta+\varepsilon
f^{*}E)\sim_{\mathbb{R}}f^{*}(N+(1-\varepsilon)H+\varepsilon A)$

and that $N+(1-\varepsilon)H+\varepsilon A$ is ample for $0<\varepsilon\ll 1$.
By [Fn8, Lemma 7.14],

$H^{i}(W,R^{j}f_{*}\mathcal{O}_{V}(L)\otimes\mathcal{N})=0$

for $i>0$ and $j\geq 0$. ∎

In Lemmas 7.4 and 7.5, we treat mixed-$\widehat{\omega}$-big-sheaves on smooth
projective curves.

######  Lemma 7.4.

Let $\mathcal{G}$ be a mixed-$\widehat{\omega}$-big-sheaf on a smooth
projective curve $C$. Then $H^{1}(C,\mathcal{G}\otimes\mathcal{N})=0$ holds
for every nef invertible sheaf $\mathcal{N}$ on $C$.

###### Proof.

We note that $\mathcal{G}$ is locally free since $C$ is a smooth curve. By
definition, we have a mixed-$\omega$-big-sheaf $\mathcal{F}$ on $C$ and a
generically isomorphic injection
$\iota:\mathcal{F}\hookrightarrow\mathcal{G}$. Note that the cokernel of
$\iota$ is a skyscraper sheaf on $C$. By Lemma 7.1,
$H^{1}(C,\mathcal{F}\otimes\mathcal{N})=0$ holds. Therefore, we have
$H^{1}(C,\mathcal{G}\otimes\mathcal{N})=0$ by the surjection
$H^{1}(C,\mathcal{F}\otimes\mathcal{N})\to
H^{1}(C,\mathcal{G}\otimes\mathcal{N})$. ∎

######  Lemma 7.5.

Let $\mathcal{E}$ be a locally free sheaf on a smooth projective curve $C$ and
let $P$ be a closed point of $C$. If
$\mathcal{E}\otimes\mathcal{O}_{C}(-P)\otimes\mathcal{N}^{\otimes-1}$ is a
mixed-$\widehat{\omega}$-big-sheaf on $C$ for some nef invertible sheaf
$\mathcal{N}$ on $C$, then $\mathcal{E}$ is generated by global sections at
$P$.

###### Proof.

By Lemma 7.4, $H^{1}(C,\mathcal{E}\otimes\mathcal{O}_{C}(-P))=0$. This means
that the natural restriction map

$H^{0}(C,\mathcal{E})\to\mathcal{E}\otimes\mathbb{C}(P)$

is surjective. Therefore, $\mathcal{E}$ is generated by global sections at
$P$. ∎

Let us discuss generically global generations of mixed-$\omega$-big-sheaves.

######  Lemma 7.6.

Let $\mathcal{F}$ be a mixed-$\omega$-sheaf on a normal projective variety $W$
with $\dim W=n$. Let $H$ be a big Cartier divisor on $W$ such that $|H|$ is
free. Then $\mathcal{F}\otimes\mathcal{O}_{W}((n+1)H)$ is generically
generated by global sections.

###### Proof.

We may assume that $\mathcal{F}=f_{*}\mathcal{O}_{V}(K_{V}+D)$ as in
Definition 5.1.

######  Step 1.

Let $\mu:\widetilde{V}\to V$ be a projective birational morphism from a smooth
projective variety $\widetilde{V}$ such that

$K_{\widetilde{V}}+\widetilde{D}=\mu^{*}(K_{V}+D)+E$

where $\widetilde{D}$ and $E$ are effective divisors and have no common
irreducible components. Since
$\mu_{*}\mathcal{O}_{\widetilde{V}}(K_{\widetilde{V}}+\widetilde{D})\simeq\mathcal{O}_{V}(K_{V}+D)$,
we may replace $(V,D)$ and $f:V\to W$ with $(\widetilde{V},\widetilde{D})$ and
$f\circ\mu:\widetilde{V}\to W$, respectively. By taking $\mu:\widetilde{V}\to
V$ suitably, we may assume that all the log canonical centers of
$(V,D_{\mathrm{hor}})$ are dominant onto $V$, where $D_{\mathrm{hor}}$ is the
horizontal part of $D$. Since
$f_{*}\mathcal{O}_{V}(K_{V}+D_{\mathrm{hor}})\hookrightarrow
f_{*}\mathcal{O}_{V}(K_{V}+D)$ is a generically isomorphic injection, we may
replace $D$ with $D_{\mathrm{hor}}$.

######  Step 2.

We will prove that
$f_{*}\mathcal{O}_{V}(K_{V}+D)\otimes\mathcal{O}_{V}((n+1)H)$ is generically
generated by global sections by induction on $n=\dim W$. We take a general
member $W^{\prime}$ of $|H|$. We put $f^{-1}(W^{\prime})=V^{\prime}$. Then we
have a short exact sequence

$0\to\mathcal{O}_{V}(K_{V}+D)\to\mathcal{O}_{V}(K_{V}+V^{\prime}+D)\to\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D|_{V^{\prime}})\to
0$

by adjunction. Since $W^{\prime}$ is a general member of $|H|$, we get a short
exact sequence

(7.1) $\begin{split}0&\to
f_{*}\mathcal{O}_{V}(K_{V}+D)\otimes\mathcal{O}_{W}(nH)\to
f_{*}\mathcal{O}_{V}(K_{V}+V^{\prime}+D)\otimes\mathcal{O}_{W}(nH)\\\ &\to
f_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D|_{V^{\prime}})\otimes\mathcal{O}_{W^{\prime}}(nH|_{W^{\prime}})\to
0.\end{split}$

By the vanishing theorem (see Lemma 7.3), we have

(7.2) $H^{1}(W,f_{*}\mathcal{O}_{V}(K_{V}+D)\otimes\mathcal{O}_{W}(nH))=0.$

Therefore, the restriction map

$H^{0}(W,f_{*}\mathcal{O}_{V}(K_{V}+D)\otimes\mathcal{O}_{W}((n+1)H))\to
H^{0}(W^{\prime},f_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D|_{V^{\prime}})\otimes\mathcal{O}_{W^{\prime}}(nH|_{W^{\prime}}))$

is surjective by (7.1) and (7.2). By induction on $n$,
$f_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D|_{V^{\prime}})\otimes\mathcal{O}_{W^{\prime}}(nH|_{W^{\prime}})$
is generically generated by global sections. This implies that so is
$f_{*}\mathcal{O}_{V}(K_{V}+D)\otimes\mathcal{O}_{W}((n+1)H)$.

We obtain the desired statement. ∎

Lemma 7.7 is similar to Lemma 7.6.

######  Lemma 7.7.

Let $\mathcal{F}$ be a mixed-$\omega$-big-sheaf on a normal projective variety
$W$ with $\dim W=n$. Let $H$ be a big Cartier divisor on $W$ such that $|H|$
is free. Then $\mathcal{F}\otimes\mathcal{O}_{W}(nH)$ is generically generated
by global sections.

The proof of Lemma 7.7 is essentially the same as that of Lemma 7.6.

###### Sketch of Proof of Lemma 7.7.

If $n=0$, then the statement is trivial. If $n=1$, then it follows from Lemma
7.5. Therefore, we assume that $n\geq 2$. As in the proof of Lemma 7.6, we may
assume that $\mathcal{F}=f_{*}\mathcal{O}_{V}(K_{V}+D+P)$ as in Definition
5.3. Moreover, we may further assume that $D=D_{\mathrm{hor}}$ and that every
log canonical center of $(V,D)$ is dominant onto $W$ (see Step 1 in the proof
of Lemma 7.6). We take a general member $W^{\prime}$ of $|H|$ and put
$V^{\prime}=f^{-1}(W^{\prime})$. Then the natural restriction map

$\begin{split}&H^{0}(W,f_{*}\mathcal{O}_{V}(K_{V}+D+P)\otimes\mathcal{O}_{W}(nH))\\\
&\to
H^{0}(W^{\prime},f_{*}\mathcal{O}_{V^{\prime}}(K_{V^{\prime}}+D|_{V^{\prime}}+P|_{V^{\prime}})\otimes\mathcal{O}_{W^{\prime}}((n-1)H|_{W^{\prime}}))\end{split}$

is surjective as in Step 2 in the proof of Lemma 7.6. By induction on
dimension, we see that

$f_{*}\mathcal{O}_{V}(K_{V}+D+P)\otimes\mathcal{O}_{W}(nH)$

is generically generated by global sections. ∎

We close this section with the following result, which is due to [DuM]. We
will use it in the proof of Theorem 1.8.

######  Lemma 7.8.

Let $\mathcal{F}$ be a mixed-$\omega$-sheaf on a normal projective variety $W$
and let $H$ be a nef and big Cartier divisor on $W$. We put $\dim W=n$. Then
$\mathcal{F}\otimes\mathcal{O}_{W}(lH)$ is generically generated by global
sections for $l\geq n^{2}+1$.

###### Proof.

We may assume that $\mathcal{F}=f_{*}\mathcal{O}_{V}(K_{V}+D)$ as in
Definition 5.1. Then, by [DuM, Theorems C and 2.20],
$\mathcal{F}\otimes\mathcal{O}_{W}(lH)$ is generically generated by global
sections for $l\geq n^{2}+1$. ∎

For the details of Lemma 7.8, we recommend the reader to see [DuM].

##  8\. A special case

In this section, we will freely use the standard notation and some basic
results in the theory of minimal models (see, for example, [Fn2], [Fn6], and
[Fn9]). The reader can skip this section.

######  Theorem 8.1.

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$ with connected fibers. Assume that $f$ is
weakly semistable in the sense of Abramovich–Karu and that the geometric
generic fiber $X_{\overline{\eta}}$ of $f:X\to Y$ has a good minimal model.
Let $H$ be an ample Cartier divisor on $Y$. Let $k$ be a positive integer such
that $k\geq 2$ and $f_{*}\omega^{\otimes k}_{X/Y}\neq 0$. Then

$f_{*}\omega^{\otimes k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(H)$

is locally free and is a pure-$\omega$-big-sheaf on $Y$. In particular,

$H^{i}(Y,f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(H))=0$

for every $i>0$.

###### Proof.

As mentioned above, we will freely use some basic results in the theory of
minimal models.

######  Step 1.

By the proof of [Fn4, Theorem 1.6] (see also [Fn5]), we have already known
that $f_{*}\omega^{\otimes m}_{X/Y}$ is a nef locally free sheaf on $Y$ for
every $m\geq 1$.

######  Step 2.

We consider a relative good minimal model $f^{\prime}:X^{\prime}\to Y$ of
$f:X\to Y$.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Y}$

Since

$f_{*}\omega^{\otimes m}_{X/Y}\simeq
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(mK_{X^{\prime}/Y})$

holds for every $m\geq 1$, it is sufficient to prove that

$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+(k-1)K_{X^{\prime}/Y}+f^{\prime*}H)$

is a pure-$\omega$-big-sheaf on $Y$.

######  Step 3.

In this step, we will prove:

######  Claim.

$K_{X^{\prime}/Y}$ is nef and $f^{\prime}$-semi-ample.

###### Proof of Claim.

Since $f^{\prime}:X^{\prime}\to Y$ is a relative good minimal model of $f:X\to
Y$, $K_{X^{\prime}/Y}$ is $f^{\prime}$-semi-ample. Therefore,

$f^{\prime*}f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(lK_{X^{\prime}/Y})\to\mathcal{O}_{X^{\prime}}(lK_{X^{\prime}/Y})$

is surjective for a sufficiently large and divisible positive integer $l$.
Since $f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(lK_{X^{\prime}/Y})\simeq
f_{*}\omega^{\otimes l}_{X/Y}$ is a nef locally free sheaf, $K_{X^{\prime}/Y}$
is nef by the above surjection. ∎

######  Step 4.

Since $K_{X^{\prime}/Y}$ is nef and $f^{\prime}$-semi-ample,
$(k-1)K_{X^{\prime}/Y}+af^{\prime*}H$ is semi-ample for every positive
rational number $a$. We take a birational morphism $\rho:\widetilde{X}\to
X^{\prime}$ from a smooth projective variety $\widetilde{X}$ such that the
exceptional locus ${\operatorname{Exc}}(\rho)$ of $\rho$ is a simple normal
crossing divisor on $\widetilde{X}$. Since $X^{\prime}$ has only canonical
singularities, we see that

$\rho_{*}\mathcal{O}_{\widetilde{X}}(K_{\widetilde{X}}+\lceil(k-1)\rho^{*}K_{X^{\prime}/Y}+\rho^{*}f^{\prime*}H\rceil)\simeq\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+(k-1)K_{X^{\prime}/Y}+f^{\prime*}H)$

holds. By Lemma 5.8,

$(f^{\prime}\circ\rho)_{*}\mathcal{O}_{\widetilde{X}}(K_{\widetilde{X}}+\lceil(k-1)\rho^{*}K_{X^{\prime}/Y}+\rho^{*}f^{\prime*}H\rceil)\simeq
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+(k-1)K_{X^{\prime}/Y}+f^{\prime*}H)$

is a pure-$\omega$-big-sheaf on $Y$ since

$(k-1)\rho^{*}K_{X^{\prime}/Y}+\rho^{*}f^{\prime*}H-\frac{1}{2}\rho^{*}f^{\prime*}H$

is semi-ample.

This means that

$f_{*}\omega^{\otimes k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(H)$

is locally free and a pure-$\omega$-big-sheaf on $Y$. The vanishing theorem
follows from Lemma 7.1. ∎

Theorem 8.1 predicts that $f_{*}\omega^{\otimes
k}_{X/Y}\otimes\omega_{Y}\otimes\mathcal{O}_{Y}(H)$ has good properties.

##  9\. Fundamental theorem

This section is the main part of this paper. The main result of this section
is Theorem 9.3, which we call a fundamental theorem of the theory of
mixed-$\omega$-sheaves.

Let us start with the following lemma.

######  Lemma 9.1 ([N, Chapter V, 3.34. Lemma]).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and
let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $k$ be a positive integer with
$k\geq 2$. We assume the following conditions:

  * (i)

$(X,\Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and

  * (ii)

$L-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample.

Let $H$ be an ample divisor on $Y$. We assume that
$f_{*}\mathcal{O}_{Y}(L)\neq 0$. We take a positive integer $l$ such that

$\mathcal{O}_{Y}(lH)\otimes f_{*}\mathcal{O}_{X}(L)$

is big. Then

$\mathcal{O}_{Y}(K_{Y}+(l-\lfloor l/k\rfloor)H)\otimes
f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$. Hence we obtain that

$\mathcal{O}_{Y}(K_{Y}+(k-1)H)\otimes f_{*}\mathcal{O}_{X}(L)$

is always a mixed-$\widehat{\omega}$-big-sheaf on $Y$.

We include all the details although Lemma 9.1 is essentially the same as [N,
Chapter V, 3.34. Lemma].

###### Proof of Lemma 9.1.

We divide the proof into several small steps.

######  Step 1 (Resolution of singularities).

Let $\mu:\widetilde{X}\to X$ be a projective birational morphism from a smooth
projective variety $\widetilde{X}$ such that
$K_{\widetilde{X}}+\widetilde{\Delta}=\mu^{*}(K_{X}+\Delta)$ and that
${\operatorname{Supp}}\widetilde{\Delta}$ is a simple normal crossing divisor
on $\widetilde{X}$. We put $E=\lceil-(\widetilde{\Delta}^{<0})\rceil$. Then
$E$ is an effective $\mu$-exceptional divisor on $\widetilde{X}$,
$\widetilde{\Delta}+E$ is effective, and
$(\widetilde{X},\widetilde{\Delta}+E)$ is log canonical over a nonempty
Zariski open set of $Y$ by construction. We note that

$\mu^{*}L+kE-k(K_{\widetilde{X}/Y}+\widetilde{\Delta}+E)=\mu^{*}(L-k(K_{X/Y}+\Delta))$

and that
$\mu_{*}\mathcal{O}_{\widetilde{X}}(\mu^{*}L+kE)\simeq\mathcal{O}_{X}(L)$.
Therefore, by replacing $f:X\to Y$, $L$, and $\Delta$ with
$f\circ\mu:\widetilde{X}\to Y$, $\mu^{*}L+kE$, and $\widetilde{\Delta}+E$
respectively, we may assume that $X$ is smooth and
${\operatorname{Supp}}\Delta$ is a simple normal crossing divisor on $X$.

######  Step 2.

Since $\mathcal{O}_{Y}(lH)\otimes f_{*}\mathcal{O}_{X}(L)$ is big, we can take
a positive integer $a$ such that

$\widehat{S}^{a}(\mathcal{O}_{Y}(lH)\otimes
f_{*}\mathcal{O}_{X}(L))\otimes\mathcal{O}_{Y}(-H)=\mathcal{O}_{Y}((al-1)H)\otimes\widehat{S}^{a}(f_{*}\mathcal{O}_{X}(L))$

is generically generated by global sections by Lemma 4.2.

######  Step 3.

We take an effective $f$-exceptional divisor $E$ on $X$ such that

$\left(f_{*}\mathcal{O}_{X}(bL)\right)^{**}\simeq
f_{*}\mathcal{O}_{X}(b(L+E))$

holds for every $1\leq b\leq a$. By taking a resolution of singularities as in
Step 1, we may assume that ${\operatorname{Supp}}(\Delta+E)$ is a simple
normal crossing divisor on $X$. Since

$(L+E)-k(K_{X/Y}+\Delta+(1/k)E)=L-k(K_{X/Y}+\Delta),$

we may replace $L$ and $\Delta$ with $L+E$ and $\Delta+(1/k)E$, respectively.
This is because

$\mathcal{O}_{Y}(K_{Y}+(l-\lfloor l/k\rfloor)H)\otimes
f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$ if and only if so is

$\mathcal{O}_{Y}(K_{Y}+(l-\lfloor
l/k\rfloor)H)\otimes\left(f_{*}\mathcal{O}_{X}(L)\right)^{**}.$

Anyway, we may assume that $f_{*}\mathcal{O}_{X}(bL)$ is reflexive for every
$1\leq b\leq a$.

######  Step 4.

By taking a suitable birational modification of $X$ again (see Step 1), we may
further assume that the image of the natural map

$f^{*}f_{*}\mathcal{O}_{X}(L)\to\mathcal{O}_{X}(L)$

is invertible and can be written as $\mathcal{O}_{X}(L-B)$ such that
${\operatorname{Supp}}(\Delta+B)$ is a simple normal crossing divisor on $X$.
By the definition of $B$, we have
$f_{*}\mathcal{O}_{X}(L-B)=f_{*}\mathcal{O}_{X}(L)$.

######  Step 5.

We note that we can take an effective $f$-exceptional divisor $E$ on $X$ such
that the map $f^{*}f_{*}\mathcal{O}_{X}(L)\to\mathcal{O}_{X}(L-B)$ induces

$f^{*}\widehat{S}^{a}(f_{*}\mathcal{O}_{X}(L))\to\mathcal{O}_{X}(a(L-B)+E).$

Then we have the following map

(9.1)
$H^{0}(Y,\mathcal{O}_{Y}((al-1)H)\otimes\widehat{S}^{a}(f_{*}\mathcal{O}_{X}(L)))\otimes\mathcal{O}_{X}\to\mathcal{O}_{X}(a(L-B)+E+(al-1)f^{*}H).$

By taking a suitable birational modification of $X$ again (see Step 1), we may
assume that the image of (9.1) is

$\mathcal{O}_{X}(a(L-B)+E-F+(al-1)f^{*}H)$

for some effective $f$-vertical divisor $F$ on $X$. We may further assume that
${\operatorname{Supp}}(\Delta+B+E+F)$ is a simple normal crossing divisor on
$X$. We put

$N:=a(L-B)+E-F+(al-1)f^{*}H.$

Then $|N|$ is free by construction.

######  Step 6.

We take a positive number $\varepsilon$. Then $L-k(K_{X/Y}+\Delta)+\varepsilon
f^{*}H$ is semi-ample because $L-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample
by assumption. We put

$M:=L-(K_{X/Y}+\Delta)-\frac{k-1}{k}B+\frac{k-1}{ak}(E-F)+\left(l-\left\lfloor\frac{l}{k}\right\rfloor\right)f^{*}H.$

Then

$M-\frac{k-1}{ak}N-\frac{1}{k}(L-k(K_{X/Y}+\Delta)+\varepsilon f^{*}H)=\alpha
f^{*}H$

for some $\alpha>0$ if $\varepsilon$ is sufficiently small. We note that

$\frac{(al-1)(k-1)}{ak}<\frac{\lceil
l(k-1)\rceil}{k}=l-\left\lfloor\frac{l}{k}\right\rfloor.$

Thus $M$ and $M-\alpha f^{*}H$ are semi-ample. Without loss of generality, we
may assume that $\varepsilon$ and $\alpha$ are rational numbers since

$\left(l-\left\lfloor\frac{l}{k}\right\rfloor\right)-\frac{(al-1)(k-1)}{k}-\frac{\varepsilon}{k}=\alpha.$

######  Step 7.

We consider

$\left\lfloor\frac{k-1}{k}B+\Delta\right\rfloor.$

We put

$B_{0}=\max\left\\{T\left|\text{$T$ is a Weil divisor with $0\leq T\leq B$ and
$T\leq\left\lfloor\frac{k-1}{k}B+\Delta\right\rfloor$}\right.\right\\}.$

We write

$\left\lfloor\frac{k-1}{k}B+\Delta\right\rfloor-B_{0}=\Delta_{1}+\Delta_{2}$

where $\Delta_{1}$ is the horizontal part and $\Delta_{2}$ is the vertical
part. By assumption (i), $\Delta_{1}=\Delta^{=1}_{1}$. By construction, we see
that $\Delta_{1}\subset{\operatorname{Supp}}\Delta^{=1}$ and that $\Delta_{1}$
and ${\operatorname{Supp}}\\{M\\}$ have no common irreducible components.

######  Step 8.

We have the following generically isomorphic injections:

$\begin{split}f_{*}\mathcal{O}_{X}(K_{X}+\Delta_{1}+\lceil
M\rceil)&\hookrightarrow\omega_{Y}((l-\lfloor
l/k\rfloor)H)\otimes\left(f_{*}\mathcal{O}_{X}\left(L-\left\lfloor\frac{k-1}{k}B+\Delta\right\rfloor+\Delta_{1}\right)\right)^{**}\\\
&=\omega_{Y}((l-\lfloor
l/k\rfloor)H)\otimes\left(f_{*}\mathcal{O}_{X}\left(L-B_{0}-\Delta_{2}\right)\right)^{**}\\\
&\hookrightarrow\omega_{Y}((l-\lfloor l/k\rfloor)H)\otimes
f_{*}\mathcal{O}_{X}(L).\end{split}$

We note that

$f_{*}\mathcal{O}_{X}(L)=f_{*}\mathcal{O}_{X}(L-B)\subset
f_{*}\mathcal{O}_{X}(L-B_{0})\subset f_{*}\mathcal{O}_{X}(L).$

This implies that

$\mathcal{O}_{Y}(K_{Y}+(l-\lfloor l/k\rfloor)H)\otimes
f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$ because
$f_{*}\mathcal{O}_{X}(K_{X}+\Delta_{1}+\lceil M\rceil)$ is a
mixed-$\omega$-big-sheaf by Lemma 5.8.

######  Step 9.

Let $l_{0}$ be the minimum positive integer such that

$\mathcal{O}_{Y}(K_{Y}+l_{0}H)\otimes f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$. By Theorem 6.3,

$\mathcal{O}_{Y}(l_{0}H)\otimes f_{*}\mathcal{O}_{X}(L)$

is big. By the result obtained above,

$\mathcal{O}_{Y}(K_{Y}+(l_{0}-\lfloor l_{0}/k\rfloor)H)\otimes
f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$. This implies that
$l_{0}-\lfloor l_{0}/k\rfloor\geq l_{0}$. Thus we get $l_{0}\leq k-1$. Hence
we have

$\mathcal{O}_{Y}(K_{Y}+(k-1)H)\otimes f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$.

Thus we get the desired statements. ∎

######  Remark 9.2.

In Lemma 9.1, we further assume that $(X,\Delta)$ is klt over a nonempty
Zariski open set of $Y$. Then we can easily see that $\Delta_{1}=0$ in Step 7
in the proof of Lemma 9.1. Therefore, we obtain that

$\mathcal{O}_{Y}(K_{Y}+(l-\lfloor l/k\rfloor)H)\otimes
f_{*}\mathcal{O}_{X}(L)$

and

$\mathcal{O}_{Y}(K_{Y}+(k-1)H)\otimes f_{*}\mathcal{O}_{X}(L)$

are pure-$\widehat{\omega}$-big-sheaves on $Y$.

Theorem 9.3 is the most important result in the theory of
mixed-$\omega$-sheaves. So we call it a fundamental theorem of the theory of
mixed-$\omega$-sheaves.

######  Theorem 9.3 ([N, Chapter V, 3.35. Theorem]).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and
let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $D$ be an $\mathbb{R}$-divisor on
$Y$. Let $k$ be a positive integer with $k\geq 2$. Assume the following
conditions:

  * (i)

$(X,\Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and

  * (ii)

$L+f^{*}D-k(K_{X/Y}+\Delta)-f^{*}A$ is semi-ample for some big
$\mathbb{R}$-divisor $A$ on $Y$.

If $f_{*}\mathcal{O}_{Y}(L)\neq 0$, then

$\mathcal{O}_{Y}(K_{Y}+\lceil D\rceil)\otimes f_{*}\mathcal{O}_{X}(L)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y$.

###### Proof.

We divide the proof into several small steps.

######  Step 1 (Reductions).

By taking a resolution as in Step 1 in the proof of Lemma 9.1, we may assume
that $X$ is a smooth projective variety and that ${\operatorname{Supp}}\Delta$
is a simple normal crossing divisor on $X$. We note that

$L+f^{*}\lceil
D\rceil-k\left(K_{X/Y}+\Delta+\frac{1}{k}f^{*}\\{-D\\}\right)-f^{*}A=L+f^{*}D-k(K_{X/Y}+\Delta)-f^{*}A.$

Therefore, by replacing $L$ and $\Delta$ with $L+f^{*}\lceil D\rceil$ and
$\Delta+\frac{1}{k}f^{*}\\{-D\\}$, respectively, we may assume that $D=0$. By
Kodaira’s lemma, we have $A\sim_{\mathbb{R}}A_{1}+A_{2}$ such that $A_{1}$ is
an ample $\mathbb{R}$-divisor and $A_{2}$ is an effective
$\mathbb{R}$-divisor. By replacing $A$ and $\Delta$ with $A_{1}$ and
$\Delta+\frac{1}{k}f^{*}A_{2}$ respectively, we may further assume that $A$ is
an ample $\mathbb{R}$-divisor on $Y$. We take an ample Cartier divisor $H$ on
$Y$ and a positive integer $m$ such that $A-\frac{k-1}{m}H$ is ample. Then

$L-k(K_{X/Y}+\Delta)-\frac{k-1}{m}f^{*}H$

is semi-ample. Therefore, we may replace $A$ with $\frac{k-1}{m}H$. By taking
a resolution as in Step 1 in the proof of Lemma 9.1 again, we may assume that
$X$ is a smooth projective variety and that ${\operatorname{Supp}}\Delta$ is a
simple normal crossing divisor on $X$. By Lemma 3.2, we may further assume
that $\Delta$ is a $\mathbb{Q}$-divisor. We take an effective $f$-exceptional
divisor $E$ and replace $L$ and $\Delta$ with $L+E$ and $\Delta+(1/k)E$
respectively. Then we may assume that $f_{*}\mathcal{O}_{X}(L)$ is reflexive.
By taking a birational modification of $X$, we may assume that the image of

$f^{*}f_{*}\mathcal{O}_{X}(L)\to\mathcal{O}_{X}(L)$

is $\mathcal{O}_{X}(L-B)$ for some effective divisor $B$ such that
${\operatorname{Supp}}(\Delta+B)$ is a simple normal crossing divisor on $X$.
Let $S$ denote the union of all $f$-exceptional divisors on $X$. We may assume
that ${\operatorname{Supp}}(\Delta+B+S)$ is a simple normal crossing divisor
on $X$ by taking a suitable birational modification of $X$ again (see Step 1
in the proof of Lemma 9.1).

######  Step 2.

By Lemma 3.3 and Remark 3.4, we take a finite flat Galois cover
$\tau:Y^{\prime}\to Y$ from a smooth projective variety $Y^{\prime}$ and get
the following commutative diagram

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\rho}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{Y}$

such that $X^{\prime}=X\times_{Y}Y^{\prime}$ is a smooth projective variety,
$\tau^{*}H\sim mH^{\prime}$ for some ample Cartier divisor $H^{\prime}$ on
$Y^{\prime}$, and $\rho^{*}\omega^{\otimes n}_{X/Y}=\omega^{\otimes
n}_{X^{\prime}/Y^{\prime}}$ for every integer $n$. Let $G$ denote the Galois
group of $\tau:Y^{\prime}\to Y$. By construction (see the proof of Lemma 3.3),
we may assume that $H^{\prime}$ is $G$-invariant. We put
$L^{\prime}=\rho^{*}L$, $B^{\prime}=\rho^{*}B$,
$\Delta^{\prime}=\rho^{*}\Delta$, and $S^{\prime}=\rho^{*}S$. Without loss of
generality, we may assume that
${\operatorname{Supp}}(\Delta^{\prime}+B^{\prime}+S^{\prime})$ is a simple
normal crossing divisor on $X^{\prime}$ and that
$\rho^{*}(K_{X/Y}+\Delta)=K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime}$ holds
(see Remark 3.4). We note that

$L^{\prime}-(k-1)f^{\prime*}H^{\prime}-k(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})\sim_{\mathbb{Q}}\rho^{*}\left(L-k(K_{X/Y}+\Delta)-\frac{k-1}{m}f^{*}H\right)$

by construction. This implies that
$\left(L^{\prime}-(k-1)f^{\prime*}H^{\prime}\right)-k(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})$
is semi-ample.

######  Step 3.

We apply Lemma 9.1 to

$\left(L^{\prime}-(k-1)f^{\prime*}H^{\prime}\right)-k(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime}).$

Then we obtain that $\mathcal{O}_{Y^{\prime}}(K_{Y^{\prime}})\otimes
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})$ is a
mixed-$\widehat{\omega}$-big-sheaf on $Y^{\prime}$. Therefore,
$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})$ is a big sheaf on
$Y^{\prime}$ by Theorem 6.3. Thus we can take a positive integer $a$ such that
$\widehat{S}^{a}(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}))$ is
generically generated by global sections (see Lemma 4.2). Then we take an
effective $G$-invariant $f^{\prime}$-exceptional divisor $E^{\prime}$ on
$X^{\prime}$ such that

$\left(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(bL^{\prime})\right)^{**}\simeq
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(b(L^{\prime}+E^{\prime}))$

holds for every $1\leq b\leq a$. By replacing $L^{\prime}$, $\Delta^{\prime}$,
and $B^{\prime}$ with $L^{\prime}+E^{\prime}$,
$\Delta^{\prime}+(1/k)E^{\prime}$, and $B^{\prime}+E^{\prime}$ respectively,
we may assume that $f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(bL^{\prime})$ is
reflexive for every $1\leq b\leq a$.

######  Step 4.

We can take an effective $G$-invariant $f^{\prime}$-exceptional divisor
$E^{\prime}$ on $X^{\prime}$ such that the surjective map

$f^{\prime*}f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})\to\mathcal{O}_{X^{\prime}}(L^{\prime}-B^{\prime})$

induces

$f^{\prime*}\widehat{S}^{a}(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}))\to\mathcal{O}_{X^{\prime}}(a(L^{\prime}-B^{\prime})+E^{\prime}).$

Then we have the following map

(9.2)
$H^{0}(Y^{\prime},\widehat{S}^{a}(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})))\otimes\mathcal{O}_{X^{\prime}}\to\mathcal{O}_{X^{\prime}}(a(L^{\prime}-B^{\prime})+E^{\prime}).$

By taking an equivariant resolution of singularities of $X^{\prime}$, we may
assume that the image of (9.2) is

$\mathcal{O}_{X^{\prime}}(a(L^{\prime}-B^{\prime})+E^{\prime}-F^{\prime})$

for some effective $G$-invariant $f^{\prime}$-vertical divisor $F^{\prime}$ on
$X^{\prime}$. Of course, we may assume that
${\operatorname{Supp}}(\Delta^{\prime}+B^{\prime}+E^{\prime}+F^{\prime})$ is a
simple normal crossing divisor on $X^{\prime}$. We put

$N^{\prime}:=a(L^{\prime}-B^{\prime})+E^{\prime}-F^{\prime}.$

Then $|N^{\prime}|$ is free by construction. We put

$M^{\prime}:=L^{\prime}-(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})-\frac{k-1}{k}B^{\prime}+\frac{k-1}{ak}(E^{\prime}-F^{\prime}).$

Then

$M^{\prime}-\frac{k-1}{ak}N^{\prime}-\frac{1}{k}(L^{\prime}-k(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})-(k-1)f^{\prime*}H^{\prime})=\frac{k-1}{k}f^{\prime*}H^{\prime}.$

In particular, $M^{\prime}$ and
$M^{\prime}-\frac{k-1}{k}f^{\prime*}H^{\prime}$ are semi-ample.

######  Step 5.

We put

$\left\lfloor\frac{k-1}{k}B^{\prime}+\Delta^{\prime}\right\rfloor=B^{\prime}_{0}+\Delta^{\prime}_{1}+\Delta^{\prime}_{2}$

as in Step 7 in the proof of Lemma 9.1. Then $\Delta^{\prime}_{1}$ is a
$G$-invariant $f^{\prime}$-horizontal simple normal crossing divisor on
$X^{\prime}$. As before, ${\operatorname{Supp}}\\{M^{\prime}\\}$ and
$\Delta^{\prime}_{1}$ have no common irreducible components. Thus, by Lemma
5.8,
$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+\Delta^{\prime}_{1}+\lceil
M^{\prime}\rceil)$ is a mixed-$\omega$-big-sheaf on $Y^{\prime}$. Note that
the Galois group $G$ acts on
$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+\Delta^{\prime}_{1}+\lceil
M^{\prime}\rceil)$.

######  Step 6.

Therefore, we get the following generically isomorphic $G$-equivariant
embedding:

(9.3)
$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+\Delta^{\prime}_{1}+\lceil
M^{\prime}\rceil)\hookrightarrow\mathcal{O}_{Y^{\prime}}(K_{Y^{\prime}})\otimes
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})$

as in Step 8 in the proof of Lemma 9.1. We note that
$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})\simeq\tau^{*}f_{*}\mathcal{O}_{X}(L)$
by the flat base change theorem. We take $\tau_{*}$ of (9.3) and then take the
$G$-invariant parts. Thus, we get a mixed-$\omega$-big-sheaf

$\mathcal{F}:=\left(\tau_{*}f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+\Delta^{\prime}_{1}+\lceil
M^{\prime}\rceil)\right)^{G}$

on $Y$ and a generically isomorphic injection

$\mathcal{F}\hookrightarrow\mathcal{O}_{Y}(K_{Y})\otimes
f_{*}\mathcal{O}_{X}(L).$

This means that $\mathcal{O}_{Y}(K_{Y})\otimes f_{*}\mathcal{O}_{X}(L)$ is a
mixed-$\widehat{\omega}$-big-sheaf on $Y$.

Anyway, we obtain that $\mathcal{O}_{Y}(K_{Y}+\lceil D\rceil)\otimes
f_{*}\mathcal{O}_{X}(L)$ is a mixed-$\widehat{\omega}$-big-sheaf on $Y$. ∎

######  Remark 9.4.

As in Remark 9.2, we further assume that $(X,\Delta)$ is klt over a nonempty
Zariski open set of $Y$ in Theorem 9.3. Then we see that
$\Delta^{\prime}_{1}=0$ in Step 5 in the proof of Theorem 9.3. Hence we obtain
that

$\mathcal{O}_{Y}(K_{Y}+\lceil D\rceil)\otimes f_{*}\mathcal{O}_{X}(L)$

is a pure-$\widehat{\omega}$-big-sheaf on $Y$.

As a corollary of Theorem 9.3, we have:

######  Corollary 9.5 ([N, Chapter V. 3.37. Corollary]).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$ with $\dim Y=n$. Let $L$ be a Cartier
divisor on $X$ and let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$
such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $D$ be an
$\mathbb{R}$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 2$.
Assume the following conditions:

  * (i)

$(X,\Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and

  * (ii)

$L+f^{*}D-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample.

Let $H$ be a big Cartier divisor on $Y$ such that $|H|$ is free. Then

$\mathcal{O}_{Y}(K_{Y}+\lceil D\rceil+(n+1)H)\otimes f_{*}\mathcal{O}_{X}(L)$

is generically generated by global sections.

Let $H^{\dagger}$ be a nef and big Cartier divisor on $Y$ such that
$|H^{\dagger}|$ is not necessarily free. Then the sheaf

$\mathcal{O}_{Y}(K_{Y}+\lceil D\rceil+lH^{\dagger})\otimes
f_{*}\mathcal{O}_{X}(L)$

is generically generated by global sections for $l\geq n^{2}+2$.

###### Proof.

By taking a resolution of singularities as in Step 1 in the proof of Lemma
9.1, we may assume that $X$ is smooth. By replacing $L$ and $\Delta$ with
$L+f^{*}\lceil D\rceil$ and $\Delta+\frac{1}{k}f^{*}\\{-D\\}$ respectively, we
may assume that $D=0$. By the flattening theorem, there is a birational
morphism $\tau:Y^{\prime}\to Y$ from a smooth projective variety $Y^{\prime}$
such that the main component of $X\times_{Y}Y^{\prime}$ is flat over
$Y^{\prime}$. Let $X^{\prime}$ be a resolution of the main component of
$X\times_{Y}Y^{\prime}$. Then we get the following commutative diagram.

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\rho}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{Y}$

By construction, any $f^{\prime}$-exceptional divisor is $\rho$-exceptional.
We put $K_{X^{\prime}}+B=\rho^{*}(K_{X}+\Delta)$. We may assume that
${\operatorname{Supp}}B$ is a simple normal crossing divisor on $X^{\prime}$.
We write

$K_{Y^{\prime}}=\tau^{*}K_{Y}+R$

where $R$ is an effective $\tau$-exceptional divisor on $Y^{\prime}$. We put

$L^{\prime}:=\rho^{*}L+k\lceil-(B^{<0})\rceil-kf^{\prime*}R$

and $\Delta^{\prime}=B+\lceil-(B^{<0})\rceil$. Note that
$\lceil-(B^{<0})\rceil$ is effective and $\rho$-exceptional. Then we have

$L^{\prime}-k(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})=\rho^{*}(L-k(K_{X/Y}+\Delta)).$

We take an effective $f^{\prime}$-exceptional divisor $E$ on $X^{\prime}$ such
that

$\left(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime})\right)^{**}\simeq
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}+E).$

Note that $E$ is $\rho$-exceptional and that there is a generically isomorphic
injection

$\tau_{*}f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}+E)=f_{*}\rho_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}+E)\subset
f_{*}\mathcal{O}_{X}(L).$

Therefore, we have a generically isomorphic injection

$\tau_{*}\left((f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}))^{**}\right)\subset
f_{*}\mathcal{O}_{X}(L).$

By Kodaira’s lemma, we have $\tau^{*}H\sim_{\mathbb{Q}}A+B$ such that $A$ is
an ample $\mathbb{Q}$-divisor and $B$ is an effective $\mathbb{Q}$-divisor.
Note that

$\begin{split}&L^{\prime}+E+f^{\prime*}\tau^{*}H-k\left(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime}+\frac{1}{k}E+\frac{1}{k}f^{\prime*}B\right)-\frac{1}{2}f^{\prime*}A\\\
&=L^{\prime}-k(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})+\frac{1}{2}f^{\prime*}A\end{split}$

is semi-ample. Therefore, by Theorem 9.3,

$\mathcal{O}_{Y^{\prime}}(K_{Y^{\prime}}+\tau^{*}H)\otimes
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}+E)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y^{\prime}$. Thus, by Lemma 7.7,

$\mathcal{O}_{Y^{\prime}}(K_{Y^{\prime}}+(n+1)\tau^{*}H)\otimes(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}))^{**}$

is generically generated by global sections. Therefore, so is
$\mathcal{O}_{Y}(K_{Y}+(n+1)H)\otimes f_{*}\mathcal{O}_{X}(L)$. By the same
argument, we see that

$\mathcal{O}_{Y^{\prime}}(K_{Y^{\prime}}+\tau^{*}H^{\dagger})\otimes
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}+E)$

is a mixed-$\widehat{\omega}$-big-sheaf on $Y^{\prime}$. Hence the sheaf

$\mathcal{O}_{Y^{\prime}}(K_{Y^{\prime}}+l\tau^{*}H^{\dagger})\otimes(f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(L^{\prime}))^{**}$

is generically generated by global sections for $l\geq n^{2}+2$ by Lemma 7.8.
Therefore, so is $\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})\otimes
f_{*}\mathcal{O}_{X}(L)$. Anyway, we get the desired statements. ∎

We note that [N, Chapter V, 3.37. Corollary] needs the assumption that
$(X,\Delta)$ is klt over a nonempty Zariski open set of $Y$. On the other
hand, Corollary 9.5 can be applied to log canonical pairs. This is the main
difference between [N, Chapter V, 3.37. Corollary] and Corollary 9.5.

##  10\. Proof of Theorems 1.5, 1.7, and 1.8

In this section, we prove Theorems 1.5, 1.7, and 1.8 in Section 1.

Let us first prove Theorem 1.7.

###### Proof of Theorem 1.7.

We divide the proof into small steps.

######  Step 1.

By taking a suitable resolution of singularities of $X$, we may assume that
$X$ is a smooth projective variety and ${\operatorname{Supp}}\Delta$ is a
simple normal crossing divisor on $X$ (see Step 1 in the proof of Lemma 9.1).
We may further assume that every log canonical center of
$(X,\Delta_{\mathrm{hor}})$ is dominant onto $Y$.

######  Step 2.

In this step, we will prove the generically generation of
$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y}+lH)$ when $k=1$.

By replacing $L$ and $\Delta$ with $L-\lfloor\Delta_{\mathrm{ver}}\rfloor$ and
$\Delta-\lfloor\Delta_{\mathrm{ver}}\rfloor$ respectively, we may further
assume that $(X,\Delta)$ is dlt and that every log canonical center of
$(X,\Delta)$ is dominant onto $Y$. By the arguments in Step 2 in the Proof of
Lemma 7.6, we see that
$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y}+lH)$ is generically
generated by global sections.

######  Step 3.

In this step, we will see that
$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y}+lH)$ is generically
generated by global sections when $k\geq 2$.

This follows directly from Corollary 9.5. More precisely, we put $D=0$ and
apply Corollary 9.5.

######  Step 4.

In this final step, we treat the case when $s\geq 2$. We take the $s$-fold
fiber product

$X^{s}:=\underbrace{X\times_{Y}X\times_{Y}\cdots\times_{Y}X}_{s}$

of $X$ over $Y$. Let $f^{s}:X^{s}\to Y$ be the induced morphism. Let
$\rho:X^{(s)}\to X^{s}$ be a resolution of singularities of the dominant
components of $X^{s}$ such that $\rho$ is an isomorphism over a nonempty
Zariski open set of $Y$. We put $f^{(s)}=f^{s}\circ\rho:X^{(s)}\to Y$. We note
that $X^{(s)}$ may be reducible, that is, a disjoint union of some smooth
projective varieties. We can take a Zariski open set $U$ of $Y$ such that
$\mathrm{codim}_{Y}(Y\setminus U)\geq 2$, $f_{*}\mathcal{O}_{X}(L)$ is locally
free on $U$, and $f$ is flat over $U$. By applying Lemma 3.6 to $f^{-1}(U)\to
U$, we can construct a Cartier divisor $L^{(s)}$ on $X^{(s)}$ and an effective
$\mathbb{R}$-divisor $\Delta^{(s)}$ on $X^{(s)}$ such that

$L^{(s)}\sim_{\mathbb{R}}k(K_{X^{(s)}/Y}+\Delta^{(s)}),$

$(X^{(s)},\Delta^{(s)})$ is log canonical over a nonempty Zariski open set of
$Y$, and there exists a generically isomorphic injection

$\left(f^{(s)}_{*}\mathcal{O}_{X^{(s)}}(L^{(s)})\right)^{**}\subset\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}.$

By Theorem 9.3,

$\mathcal{O}_{Y}(K_{Y}+H)\otimes f^{(s)}_{*}\mathcal{O}_{X^{(s)}}(L^{(s)})$

is a finite direct sum of mixed-$\widehat{\omega}$-big-sheaves when $k\geq 2$.
Note that $X^{(s)}$ may be reducible. Therefore,

$\mathcal{O}_{Y}(K_{Y}+H)\otimes\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}$

is also a finite direct sum of mixed-$\widehat{\omega}$-big-sheaves. Thus, by
Lemma 7.7,

$\mathcal{O}_{Y}(K_{Y}+lH)\otimes\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}$

is generically generated by global sections for $l\geq n+1$ when $k\geq 2$.

If $k=1$, then we can check that $\mathcal{O}_{Y}(K_{Y}+lH)\otimes
f^{(s)}_{*}\mathcal{O}_{X^{(s)}}(L^{(s)})$ is generically generated by global
sections for $l\geq n+1$ by the arguments in Steps 1 and 2. Therefore,

$\mathcal{O}_{Y}(K_{Y}+lH)\otimes\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}$

is generically generated by global sections for $l\geq n+1$ when $k=1$.

Anyway, we have obtained the desired statements. ∎

Next we prove Theorem 1.8.

###### Sketch of Proof of Theorem 1.8.

It is not difficult to modify the proof of Theorem 1.7.

######  Step 1.

In this step, we will treat the case when $k=1$.

As usual, by taking a suitable birational modification of $X$, we may assume
that $X$ is smooth and ${\operatorname{Supp}}\Delta$ is a simple normal
crossing divisor on $X$. By replacing $L$ and $\Delta$ with
$L-\lfloor\Delta^{>1}\rfloor$ and $\Delta-\lfloor\Delta^{>1}\rfloor$
respectively, we may assume that $\Delta$ is a boundary $\mathbb{R}$-divisor
on $X$. Note that $\Delta^{>1}$ is $f$-vertical. By perturbing the
coefficients of $\Delta$, we may further assume that $\Delta$ is a
$\mathbb{Q}$-divisor with $L\sim_{\mathbb{Q}}K_{X/Y}+\Delta$. By Lemma 5.9,

$\mathcal{O}_{X}(K_{X}+\lfloor\Delta\rfloor+\lceil
L-K_{X/Y}-\Delta\rceil)\simeq\mathcal{O}_{X}(L)\otimes
f^{*}\mathcal{O}_{Y}(K_{Y})$

is a mixed-$\omega$-sheaf on $X$. Therefore,
$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y})$ is a
mixed-$\omega$-sheaf on $Y$. Thus, by Lemma 7.8,
$f_{*}\mathcal{O}_{X}(L)\otimes\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})$ is
generically generated by global sections for $l\geq n^{2}+1$. Similarly, we
may assume that the sheaf
$f^{(s)}_{*}\mathcal{O}_{X^{(s)}}(L^{(s)})\otimes\mathcal{O}_{Y}(K_{Y})$ in
the proof of Theorem 1.7 is a finite direct sum of mixed-$\omega$-sheaves on
$Y$ when $k=1$. Therefore,

$\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}\otimes\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})$

is generically generated by global sections for $l\geq n^{2}+1$.

######  Step 2.

In this step, we will treat the case when $k\geq 2$.

If we use Lemma 7.8 instead of Lemma 7.7, then the proof of Theorem 1.7
implies that

$\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})\otimes\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}$

is generically generated by global sections for $l\geq n^{2}+2$. By Corollary
9.5,

$\mathcal{O}_{Y}(K_{Y}+lH^{\dagger})\otimes f_{*}\mathcal{O}_{X}(L)$

is generically generated by global sections for $l\geq n^{2}+2$.

Thus we get the desired statements. ∎

Finally, we prove Theorem 1.5.

###### Proof of Theorem 1.5.

We put $L=kK_{X/Y}$. Then this theorem directly follows from Theorems 1.7 and
1.8. ∎

We close this section with an easy remark.

######  Remark 10.1.

Let $Y$ be a smooth projective variety and let $H$ be an ample Cartier divisor
on $Y$. Let $m$ be any positive integer. Then we can construct a finite cover
$f:X\to Y$ from a smooth projective variety $X$ such that
$\mathcal{O}_{Y}(-mH)$ is a direct summand of $f_{*}\mathcal{O}_{X}$.
Therefore, we need the condition $k\geq 1$ in Theorems 1.5, 1.7, and 1.8.

##  11\. Some other applications

In this final section, we treat Nakayama’s inequality on $\kappa_{\sigma}$ and
a slight generalization of the twisted weak positivity theorem. Theorem 11.3
and a special case of Theorem 11.7 have already played a crucial role in the
theory of minimal models.

Let us first recall the definition of $\kappa_{\sigma}$ for the reader’s
convenience.

######  Definition 11.1 (Nakayama’s numerical dimension, see [N, Chapter
V.2.5. Definition]).

Let $D$ be a pseudo-effective $\mathbb{R}$-Cartier divisor on a normal
projective variety $X$ and let $A$ be a Cartier divisor on $X$. If
$H^{0}(X,\mathcal{O}_{X}(\lfloor mD\rfloor+A))\neq 0$ for infinitely many
positive integers $m$, then we set

$\sigma(D;A)=\max\left\\{k\in\mathbb{Z}_{\geq
0}\,\left|\,{\underset{m\to\infty}{\limsup}}\frac{\dim
H^{0}(X,\mathcal{O}_{X}(\lfloor mD\rfloor+A))}{m^{k}}>0\right.\right\\}.$

If $H^{0}(X,\mathcal{O}_{X}(\lfloor mD\rfloor+A))\neq 0$ only for finitely
many $m\in\mathbb{Z}_{\geq 0}$, then we set $\sigma(D;A)=-\infty$. We define
Nakayama’s numerical dimension $\kappa_{\sigma}$ by

$\kappa_{\sigma}(X,D)=\max\\{\sigma(D;A)\,|\,A\ {\text{is a Cartier divisor
on}}\ X\\}.$

It is well known that $\kappa_{\sigma}(X,D)\geq 0$ (see, for example, [N,
Chapter V. 2.7. Proposition]). If $D$ is not pseudo-effective, then we put
$\kappa_{\sigma}(X,D)=-\infty$. By this convention, we can define
$\kappa_{\sigma}(X,D)$ for every $\mathbb{R}$-Cartier divisor $D$ on $X$. It
is obvious that

$\kappa_{\sigma}(X,D)\geq\kappa(X,D)$

always holds for every $\mathbb{R}$-Cartier divisor $D$ on $X$ by definition,
where $\kappa(X,D)$ denotes the Iitaka dimension of $D$.

For the details of $\kappa_{\sigma}(X,D)$ and $\kappa(X,D)$, we recommend the
reader to see [N]. The following remark is easy but very useful.

######  Remark 11.2 ([N, Chapter V, 2.6. Remark (6)]).

Let $X$ be a smooth projective variety and let $D$ be an $\mathbb{R}$-divisor
on $X$. We put

$\sigma(D;A)^{\prime}=\max\left\\{k\in\mathbb{Z}_{\geq
0}\cup\\{-\infty\\}\,\left|\,{\underset{m\to\infty}{\limsup}}\frac{\dim
H^{0}(X,\mathcal{O}_{X}(\lceil mD\rceil+A))}{m^{k}}>0\right.\right\\},$

where $A$ is a divisor on $X$. Then we have the following equality

$\kappa_{\sigma}(X,D)=\max\\{\sigma(D;A)^{\prime}\,|\,\text{$A$ is a
divisor}\\}.$

We will use this characterization of $\kappa_{\sigma}$ in the proof of Theorem
11.3 below.

We note the following easy but important fact that
$\kappa_{\sigma}(X,lD)=\kappa_{\sigma}(X,D)$ holds for every positive integer
$l$ (see [Fn10, Remark 2.2]), which will be useful in the proof of Theorem
11.3 below.

The inequalities in Theorem 11.3 are indispensable in the theory of minimal
models (see Remarks 11.4 and 11.5).

######  Theorem 11.3 ([N, Chapter V, 4.1. Theorem (1)] and [Fn10, Section 3]).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$ with connected fibers. Let $\Delta$ be an
effective $\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier and that $(X,\Delta)$ is log canonical over a nonempty
Zariski open set of $Y$. Let $D$ be an $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor on $X$ such that $D-(K_{X/Y}+\Delta)$ is nef. Then, for
any $\mathbb{R}$-divisor $Q$ on $Y$, we have

$\kappa_{\sigma}(X,D+f^{*}Q)\geq\kappa_{\sigma}(F,D|_{F})+\kappa(Y,Q)$

and

$\kappa_{\sigma}(X,D+f^{*}Q)\geq\kappa(F,D|_{F})+\kappa_{\sigma}(Y,Q)$

where $F$ is a sufficiently general fiber of $f:X\to Y$.

Before we prove Theorem 11.3, we give two important remarks.

######  Remark 11.4.

We think that one of the most important results of Nakayama’s theory of
$\omega$-sheaves is the inequality on $\kappa_{\sigma}$ in [N, Chapter V, 4.1.
Theorem (1)]. However, as we explained in [Fn7, Remark 3.8] and [Fn10, Section
3], the proof of [N, Chapter V, 4.1. Theorem (1)] is incomplete. For the
details, see, for example, [Fn10, Section 1]. So, in Theorem 11.3, we claim
two weaker inequalities than Nakayama’s original one (see [Fn10, (3.3) and
(3.4)]). Anyway, the first inequality in Theorem 11.3 is still sufficiently
powerful for some geometric applications (see [Fn10, Section 3]).

######  Remark 11.5 (see [Fn10, Section 3]).

The troubles in the proof of [DHP, Remark 2.6] and [GL, Theorem 4.3] caused by
the incompleteness of [N, Chapter V, 4.1. Theorem (1)] can be corrected by
using the first inequality in Theorem 11.3. For the details, we recommend the
reader to see [HH, Lemma 2.10].

Let us prove Theorem 11.3.

###### Proof of Theorem 11.3.

If $Q$ is not pseudo-effective, then the desired inequalities are obviously
true. So we may assume that $Q$ is pseudo-effective. Similarly, we may further
assume that $D|_{F}$ is pseudo-effective. As usual (see Step 1 in the proof of
Lemma 9.1), we may assume that $X$ is smooth and ${\operatorname{Supp}}\Delta$
is a simple normal crossing divisor on $X$ by the basic properties of
$\kappa_{\sigma}$ and $\kappa$. We take a sufficiently ample Cartier divisor
$A$ on $X$ such that $A+\\{-mD\\}$ is ample for every integer $m$. Then

$\lceil mD\rceil+A-m(K_{X/Y}+\Delta)=m(D-(K_{X/Y}+\Delta))+A+\\{-mD\\}$

is ample for every positive integer $m$. Then we can take an ample Cartier
divisor $H$ on $Y$ such that $\mathcal{O}_{Y}(H)\otimes
f_{*}\mathcal{O}_{X}(\lceil mD\rceil+A)$ is generically generated by global
sections for every positive integer $m$ by Corollary 9.5. Thus there exists a
generically isomorphic injection

$\mathcal{O}_{Y}^{\oplus r(mD;A)}\hookrightarrow\mathcal{O}_{Y}(H)\otimes
f_{*}\mathcal{O}_{X}(\lceil mD\rceil+A),$

where $r(mD;A):={\operatorname{rank}}f_{*}\mathcal{O}_{X}(\lceil mD\rceil+A)$.
This induces the following injection

$\mathcal{O}_{Y}(\lfloor mQ\rfloor+H)^{\oplus
r(mD;A)}\hookrightarrow\mathcal{O}_{Y}(\lfloor mQ\rfloor+2H)\otimes
f_{*}\mathcal{O}_{X}(\lceil mD\rceil+A).$

Therefore, we have

(11.1) $\begin{split}&\dim_{\mathbb{C}}H^{0}(X,\mathcal{O}_{X}(\lceil
m(D+f^{*}Q)\rceil+A+2f^{*}H)\\\
&\geq\dim_{\mathbb{C}}H^{0}(X,\mathcal{O}_{X}(\lceil mD\rceil+f^{*}(\lfloor
mQ\rfloor)+A+2f^{*}H))\\\ &\geq
r(mD;A)\cdot\dim_{\mathbb{C}}H^{0}(Y,\mathcal{O}_{Y}(\lfloor
mQ\rfloor+H))\end{split}$

for every positive integer $m$. We can take a positive integer $m_{0}$ and a
positive real number $C_{0}$ such that

(11.2) $C_{0}m^{\kappa(F,D|_{F})}\leq r(mm_{0}D;A)$

for every large positive integer $m$ (see, for example, [N, Chapter II, 3.7.
Theorem]). Thus we have

(11.3) $\begin{split}&\dim H^{0}(X,\mathcal{O}_{X}(\lceil
mm_{0}(D+f^{*}Q)\rceil+A+2f^{*}H))\\\ &\geq C_{0}m^{\kappa(F,D|_{F})}\cdot\dim
H^{0}(Y,\mathcal{O}_{Y}(\lfloor mm_{0}Q\rfloor+H))\end{split}$

for every large positive integer $m$ by (11.1) and (11.2). We may assume that
$H$ is sufficiently ample. Then we get

(11.4) $\limsup_{m\to\infty}\frac{\dim H^{0}(X,\mathcal{O}_{X}(\lceil
mm_{0}(D+f^{*}Q)\rceil+A+2f^{*}H))}{m^{\kappa(F,D|_{F})+\kappa_{\sigma}(Y,Q)}}>0$

by (11.3) and the definition of $\kappa_{\sigma}(Y,Q)$. This means that the
following inequality

(11.5) $\kappa_{\sigma}(X,D+f^{*}Q)\geq\kappa(F,D|_{F})+\kappa_{\sigma}(Y,Q)$

holds.

Similarly, we can take a positive integer $m_{1}$ and a positive real number
$C_{1}$ such that

(11.6) $\begin{split}C_{1}m^{\kappa(Y,Q)}&\leq\dim
H^{0}(Y,\mathcal{O}_{Y}(\lfloor mm_{1}Q\rfloor))\\\ &\leq\dim
H^{0}(Y,\mathcal{O}_{Y}(\lfloor mm_{1}Q\rfloor+H))\end{split}$

for every large positive integer $m$ (see, for example, [N, Chapter II, 3.7.
Theorem]) if $H$ is a sufficiently ample Cartier divisor. Then, by (11.1) and
(11.6), we have

(11.7) $\begin{split}&\dim H^{0}(X,\mathcal{O}_{X}(\lceil
mm_{1}(D+f^{*}Q)\rceil+A+2f^{*}H))\\\ &\geq C_{1}m^{\kappa(Y,Q)}\cdot
r(mm_{1}D;A)\end{split}$

for every large positive integer $m$. Therefore, we get

(11.8) $\limsup_{m\to\infty}\frac{\dim H^{0}(X,\mathcal{O}_{X}(\lceil
mm_{1}(D+f^{*}Q)\rceil+A+2f^{*}H))}{m^{\kappa_{\sigma}(F,D|_{F})+\kappa(Y,Q)}}>0$

when $A$ is sufficiently ample. Note that

(11.9) $\sigma(m_{1}D|_{F};A|_{F})^{\prime}=\max\left\\{k\in\mathbb{Z}_{\geq
0}\cup\\{-\infty\\}\,\left|\,\underset{m\to\infty}{\limsup}\frac{r(mm_{1}D;A)}{m^{k}}>0\right.\right\\}$

for a sufficiently general fiber $F$ of $f:X\to Y$ and that

(11.10)
$\begin{split}\kappa_{\sigma}(F,D|_{F})&=\kappa_{\sigma}(F,m_{1}D|_{F})\\\
&=\max\\{\sigma(m_{1}D|_{F};A|_{F})^{\prime}\,|\,{\text{$A$ is very
ample}}\\}.\end{split}$

Hence we have the inequality

(11.11) $\kappa_{\sigma}(X,D+f^{*}Q)\geq\kappa_{\sigma}(F,D|_{F})+\kappa(Y,Q)$

by (11.8). ∎

It is highly desirable to solve the following conjecture. As we explained in
[Fn10], Nakayama’s original inequality on $\kappa_{\sigma}$ (see [N, Chapter
V, 4.1. Theorem (1)]) follows from Conjecture 11.6 and the argument in the
proof of Theorem 11.3.

######  Conjecture 11.6 ([Fn10, Conjecture 1.3]).

Let $X$ be a smooth projective variety and let $D$ be a pseudo-effective
$\mathbb{R}$-divisor on $X$. Then there exist a positive integer $m_{0}$, a
positive rational number $C$, and an ample Cartier divisor $A$ on $X$ such
that

$Cm^{\kappa_{\sigma}(X,D)}\leq\dim H^{0}(X,\mathcal{O}_{X}(\lfloor
mm_{0}D\rfloor+A))$

holds for every large positive integer $m$.

Finally, we treat a slight generalization of the twisted weak positivity
theorem.

######  Theorem 11.7 (Twisted weak positivity theorem).

Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$
onto a smooth projective variety $Y$. Let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier
and that $(X,\Delta)$ is log canonical over a nonempty Zariski open set of
$Y$. Then the sheaf $f_{*}\mathcal{O}_{X}(L)$ is weakly positive.

###### Proof.

Let $\alpha$ be a positive integer and let $\mathcal{H}$ be an ample
invertible sheaf on $Y$. By Theorem 1.7 or Theorem 1.8, we can take a positive
integer $\beta$ which depends only on $Y$ such that

$\left(\bigotimes^{s}f_{*}\mathcal{O}_{X}(L)\right)^{**}\otimes\mathcal{H}^{\otimes\beta}$

is generically generated by global sections for every positive integer $s$.
This implies that

$\widehat{S}^{\alpha\beta}(f_{*}\mathcal{O}_{X}(L))\otimes\mathcal{H}^{\otimes\beta}$

is generically generated by global sections. This means that
$f_{*}\mathcal{O}_{X}(L)$ is weakly positive. ∎

## References

  * [DHP] J.-P. Demailly, C. D. Hacon, M. Păun, Extension theorems, non-vanishing and the existence of good minimal models, Acta Math. 210 (2013), no. 2, 203–259.
  * [De] Y. Deng, Applications of the Ohsawa–Takegoshi extension theorem to direct image problems, preprint (2017). arXiv:1703.07279 [math.AG]
  * [Du] Y. Dutta, On the effective freeness of the direct images of pluricanonical bundles, preprint (2017). arXiv:1701.08830 [math.AG]
  * [DuM] Y. Dutta, T. Murayama, Effective generation and twisted weak positivity of direct images, Algebra Number Theory 13 (2019), no. 2, 425–454.
  * [EV] H. Esnault, E. Viehweg, Lectures on vanishing theorems, DMV Seminar, 20. Birkhäuser Verlag, Basel, 1992.
  * [Fn1] O. Fujino, Higher direct images of log canonical divisors, J. Differential Geom. 66 (2004), no. 3, 453–479.
  * [Fn2] O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789.
  * [Fn3] O. Fujino, On quasi-Albanese maps, preprint (2014).
  * [Fn4] O. Fujino, Direct images of relative pluricanonical bundles, Algebr. Geom. 3 (2016), no. 1, 50–62.
  * [Fn5] O. Fujino, Corrigendum: Direct images of relative pluricanonical bundles (Algebraic Geometry 3, no. 1, (2016), 50–62), Algebr. Geom. 3 (2016), no. 2, 261–263.
  * [Fn6] O. Fujino, Foundations of the minimal model program, MSJ Memoirs, 35. Mathematical Society of Japan, Tokyo, 2017.
  * [Fn7] O. Fujino, On subadditivity of the logarithmic Kodaira dimension, J. Math. Soc. Japan 69 (2017), no. 4, 1565–1581.
  * [Fn8] O. Fujino, Notes on the weak positivity theorems, Algebraic varieties and automorphism groups, 73–118, Adv. Stud. Pure Math., 75, Math. Soc. Japan, Tokyo, 2017.
  * [Fn9] O. Fujino, Iitaka conjecture: An introduction, to appear.
  * [Fn10] O. Fujino, Corrigendum: On subadditivity of the logarithmic Kodaira dimension, preprint (2019). arXiv:1904.11639 [math.AG]
  * [FF] O. Fujino, T. Fujisawa, Variations of mixed Hodge structure and semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, 589–661.
  * [FFS] O. Fujino, T. Fujisawa, M. Saito, Some remarks on the semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 1, 85–112.
  * [Fs] T. Fujisawa, A remark on semipositivity theorems, preprint (2017). arXiv:1710.01008 [math.AG]
  * [GL] Y. Gongyo, B. Lehmann, Reduction maps and minimal model theory, Compos. Math. 149 (2013), no. 2, 295–308.
  * [HH] K. Hashizume, Z. Hu, On minimal model theory for log abundant lc pairs, preprint (2019). arXiv:1906.00769 [math.AG]
  * [I] M. Iwai, On the global generation of direct images of pluri-adjoint line bundles, to appear in Math. Z.
  * [KM] J. Kollár, S. Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.
  * [N] N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004.
  * [PS] M. Popa, C. Schnell, On direct images of pluricanonical bundles, Algebra Number Theory 8 (2014), no. 9, 2273–2295.
  * [V1] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981), 329–353, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983.
  * [V2] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension. II. The local Torelli map, Classification of algebraic and analytic manifolds (Katata, 1982), 567–589, Progr. Math., 39, Birkhäuser Boston, Boston, MA, 1983.
  * [V3] E. Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 30. Springer-Verlag, Berlin, 1995.