---
abstract: 'We introduce the notion of mixed-$\omega$-sheaves and use it for the study of a relative version of Fujita’s freeness conjecture.'
address: 'Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Osamu Fujino
date: '2019/7/29, version 0.40'
title: 'On mixed-$\omega$-sheaves'
---

Introduction {#z-sec1}
============

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.

\[z-conj1.1\] 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 \[z-conj1.1\] is still open. As a generalization of Conjecture \[z-conj1.1\], Popa and Schnell proposed the following conjecture, which is a relative version of Fujita’s freeness conjecture.

\[z-conj1.2\] 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 \[z-conj1.2\] in [@denf], [@dutta], [@dutta-murayama], and [@iwai]. In this paper, we do not directly treat Conjecture \[z-conj1.2\]. We propose a new conjecture similar to Conjecture \[z-conj1.2\]. It is also a generalization of Conjecture \[z-conj1.1\].

\[z-conj1.3\] 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 \[z-conj1.3\] and Fujita’s original freeness conjecture (see Conjecture \[z-conj1.1\]) 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 \[z-conj1.3\] is sharper than Conjecture \[z-conj1.2\].

It is well known that Conjecture \[z-conj1.3\] 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:

\[z-thm1.4\] 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 \[z-thm1.4\] in order to explain our idea.

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

\[z-claim\] $H^1(C, f_*\omega^{\otimes k}_{X/C}\otimes \omega_C
\otimes \mathcal H\otimes \mathcal O_C(-P))=0$.

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 $H^1(C, \mathcal E\otimes \omega_C)\ne 0$, then we get $H^0(C, \mathcal E^*)\ne 0$ 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 \[z-conj1.3\].

\[z-thm1.5\] 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 $H^\dag$ be an ample divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is generically generated by global sections for all integers $k\geq 1$ and $l\geq n^2+\min\{2, k\}$. Moreover, the sheaf $$\left(\bigotimes ^s (f_*\omega^{\otimes k}_{X/Y})\right)^{**} 
\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ 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.

\[y-rem1.6\] Let $f:X\to Y$ and $H^\dag$ be as in Theorem \[z-thm1.5\]. Let $U$ be the largest Zariski open set of $Y$ such that $f$ is smooth over $U$. Then, by the argument in [@iwai], we can check that the natural map $$H^0(Y, f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal O_Y(lH^\dag))\otimes 
\mathcal O_Y\to 
f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is surjective on $U$ for all integers $k\geq 1$ and $l\geq \frac{n(n+1)}{2}+1$.

Theorem \[z-thm1.5\] is a special case of Theorems \[y-thm1.7\] and \[y-thm1.8\]. In this paper, we will establish Theorems \[y-thm1.7\] and \[y-thm1.8\] by our new theory of mixed-$\omega$-sheaves.

\[y-thm1.7\] 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 $l\geq n+1$.

\[y-thm1.8\] In Theorem \[y-thm1.7\], we assume that $H^\dag$ is a nef and big Cartier divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then we have that $$f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH^\dag)
\quad \text{and}\quad 
\left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**} 
\otimes \mathcal O_Y(K_Y+lH^\dag)$$ 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 \[z-thm1.5\], which is mainly due to Nakayama (see [@nakayama]). 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:

\[y-def1.9\] 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 [@nakayama]). 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 [@nakayama 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 [@nakayama 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 \[x-thm9.3\] (and Remark \[x-rem9.4\]) is one of the main results of this paper, which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves.

\[y-thm1.10\] 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:

-   $(X, \Delta)$ is log canonical [*[(]{}*]{}resp. klt[*[)]{}*]{} over a nonempty Zariski open set of $Y$, and

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

If $f_*\mathcal O_Y(L)\ne 0$, 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 \[x-def5.3\] below.

As an application of Theorem \[y-thm1.10\], we have:

\[y-thm1.11\] 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 \[y-thm1.11\] already played a crucial role in the theory of minimal models.

We explain the organization of this paper. In Section \[x-sec2\], we collect some basic definitions. In Section \[x-sec3\], we prepare some useful and important lemmas. They will play a crucial role in this paper. In Section \[x-sec4\], we quickly explain some basic properties of Viehweg’s weakly positive sheaves and big sheaves. In Section \[x-sec5\], we introduce mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves. In Sections \[x-sec6\] and \[x-sec7\], we prove some basic properties of mixed-$\omega$-sheaves based on the theory of mixed Hodge structures. In Section \[x-sec8\], we treat a very special but interesting case. Section \[x-sec9\] is the main part of this paper. We establish a fundamental theorem of the theory of mixed-$\omega$-sheaves. Section \[x-sec10\] is devoted to the proof of Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\]. In Section \[x-sec11\], 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.

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.

Preliminary {#x-sec2}
===========

In this section, we collect some basic definitions. For the details, see [@fujino-fundamental], [@fujino-foundations], and [@fujino-iitaka].

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

\[x-def2.1\] 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.

\[x-def2.2\] 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 $(X, \Delta)$.

\[x-def2.3\] 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.

\[x-def2.4\] 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]{}*]{}).

\[x-def2.5\] 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 $i$, $D_i\ne D_j$ for $i\ne j$, 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.

\[x-def2.6\] 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 $\mathrm{codim}_Yf({{\operatorname{Supp}}}E)\geq 2$. We note that $f$ is not always assumed to be birational.

Preliminary lemmas {#x-sec3}
==================

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.

\[x-lem3.1\] 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 $B=\sum _{i\in I} b_iB_i$ where $b_i\in \mathbb R$ and $B_i$ is a prime divisor on $V$ for every $i$ with $B_i\ne B_j$ for $i\ne j$. 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

-   $A\sim _{\mathbb Q} C$,

-   $c_i=b_i$ holds if $b_i\in \mathbb Q$, and

-   $|c_i-b_i|\ll 1$ 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 ${{\operatorname{Supp}}}\{C\}={{\operatorname{Supp}}}\{B\}$.

It is an easy exercise. For the details, see, for example, the proof of [@fujino-fundamental Lemma 4.15].

\[x-lem3.2\] 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 $D_i\ne D_j$ for $i\ne j$. Assume that $D$ is semi-ample. Then we can construct a $\mathbb Q$-divisor $D^\dag=\sum _{i\in I} b_iD_i$ such that

-   $D^\dag$ is semi-ample,

-   $b_i=a_i$ holds if $a_i \in \mathbb Q$, and

-   $|b_i-a_i|\ll1$ holds for $a_i\in \mathbb R\setminus \mathbb Q$.

Since $D$ is semi-ample, we can write $D=\sum _{j\in J} m_jM_j$ 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 $D^\dag$ on $V$. For the details, see, for example, the proof of [@fujino-fundamental 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 \[x-thm9.3\].

\[x-lem3.3\] 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 $\tau:W'\to W$ from a smooth quasi-projective variety $W'$ and a Cartier divisor $H'$ on $W'$ such that $\tau^*H\sim dH'$ and that $V'=V\times _W W'$ is a smooth quasi-projective variety with $\omega_{V'/W'}=\rho^*\omega_{V/W}$, where $\rho:V'\to V$. By construction, we may assume that $\tau:W'\to W$ is Galois.

We take general very ample Cartier divisors $D_1$ and $D_2$ with the following properties.

-   $H\sim D_1-D_2$,

-   $D_1$, $D_2$, $f^*D_1$, and $f^*D_2$ are smooth,

-   $D_1$ and $D_2$ have no common components, and

-   ${{\operatorname{Supp}}}(D_1+D_2)$ 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 $\tau:W'\to W$ and $H'$ such that $\tau^*H\sim dH'$. By the construction of the above Kawamata cover $\tau:W'\to W$, 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 $V'=V\times _WW'$ is a smooth quasi-projective variety. $$\xymatrix{
 V' \ar[r]^{\rho} \ar[d]_{f'} & V\ar[d]^{f} \\
  W' \ar[r]_{\tau} & W
}$$ By construction, we can also easily check that $\omega_{V'/W'}=\rho^*\omega_{V/W}$ by the Hurwitz formula.

Let us see the construction of $f':V'\to W'$ 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 $W'$ is the normalization of $$\left(Z^{(1)}_1\times _W \cdots \times _WZ^{(1)}_n\right)
\times _W \left(Z^{(2)}_1\times _W \cdots \times _WZ^{(2)}_n\right).$$ For the details, see, for example, [@esnault-viehweg 3.19. Lemma] and [@viehweg3 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 $V'$ as the normalization of $$\left(S^{(1)}_1\times _V \cdots \times _VS^{(1)}_n\right)
\times _V \left(S^{(2)}_1\times _V \cdots \times _VS^{(2)}_n\right).$$ Note that $\rho:V'\to V$ is a finite flat morphism between smooth quasi-projective varieties. Since $V\times _WW'\to V$ is finite and flat and $V$ is smooth, $V\times _W W'$ is Cohen–Macaulay (see, for example, [@kollar-mori Corollary 5.5]). By construction, we can easily see that $V\times _W W'$ is smooth in codimension one. Therefore, $V\times _W W'$ is normal. Since $\rho$ factors through $V\times _W W'$ by construction, we see that $V'=V\times _W W'$ by Zariski’s main theorem. By the above construction of $\tau:W'\to W$, we see that $\tau:W'\to W$ is Galois.

We give a very important remark on Lemma \[x-lem3.3\].

\[x-rem3.4\] In the proof of Lemma \[x-lem3.3\], 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 $V'$.

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 \[y-thm1.7\] and \[y-thm1.8\] in Section \[x-sec10\].

\[x-rem3.5\] Let $V$ be a reduced Gorenstein scheme. Note that $V$ may be reducible. We consider $$V'\overset{\delta}\longrightarrow V^\nu\overset{\nu}\longrightarrow V$$ where $\nu:V^\nu\to V$ is the normalization and $\delta:V'\to V^\nu$ is a resolution of singularities. Then, for every positive integer $n$, we have $$\label{x-3.1} 
\nu_*\mathcal O_{V^\nu}(nK_{V^\nu})\subset \omega^{\otimes n}_V$$ and $$\label{x-3.2}
\delta_*\mathcal O_{V'}(nK_{V'}+E)\subset \mathcal O_{V^\nu}(nK_{V^\nu})$$ where $E$ is any $\delta$-exceptional divisor on $V'$. In particular, we have $$\label{x-3.3}
(\nu\circ \delta)_*\mathcal O_{V'}(nK_{V'}+E)\subset 
\omega^{\otimes n}_V$$ 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 is an isomorphism over $U$.

In Steps \[x-step3.5.1\] and \[x-step3.5.2\], we will prove and , respectively.

\[x-step3.5.1\] By taking the double dual of $\delta_*\mathcal O_{V'}(nK_{V'}+E)$, we obtain $\mathcal O_{V^\nu}(nK_{V^\nu})$. Therefore, we have $$\delta_*\mathcal O_{V'}(nK_{V'}+E)\subset \mathcal O_{V^\nu}(nK_{V^\nu})$$ for every integer $n$.

\[x-step3.5.2\] Since $\nu$ is birational, the trace map $\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\to \omega_V$ is a generically isomorphic injection $$\label{x-3.4}
\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\hookrightarrow \omega_V.$$ Since $\nu$ is finite, $$\label{x-3.5}
\nu^*\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\to \mathcal O_{V^\nu}(K_{V^\nu})$$ is surjective and the kernel of is the torsion part of $\nu^*\nu_*\mathcal O_{V^\nu}(K_{V^\nu})$. Therefore, by , we get an inclusion $$\label{x-3.6}
\mathcal O_{V^\nu}(K_{V^\nu})\hookrightarrow \nu^*\omega_V.$$ 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 . 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 . This is what we wanted.

By the above construction of and , it is obvious that the inclusion $$(\nu\circ \delta)_*\mathcal O_{V'}(nK_{V'}+E)\subset 
\omega^{\otimes n}_V$$ is an isomorphism over $U$.

\[x-lem3.6\] 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 $$f^{(s)}_*\mathcal O_{X^{(s)}_0}(L^{(s)}_0)\hookrightarrow 
\bigotimes _{i=1}^sf_*\mathcal O_{X_0}(L_0)$$ 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.

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 $$\label{x-3.7}
\begin{split}
L^{(s)}_0&=\sum _{i=1}^s \rho^*p^*_i (kK_{X_0/Y_0}+(L_0-kK_{X_0/Y_0}))+kR
\\& \sim kK_{X^{(s)}_0/Y_0}+\sum _{i=1}^s \pi^*_i (L_0-kK_{X_0/Y_0}). 
\end{split}$$

We have the following isomorphism of locally free sheaves: $$f^s_*\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_iL_0\right)\simeq 
\bigotimes ^s f_*\mathcal O_{X_0}(L_0).$$

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 $$\xymatrix{
X^s_0 \ar[dr]^-{f^s}\ar[d]_-{p_s}\ar[r]^-q& X^{s-1}_0 \ar[d]^-{f^{s-1}}\\
X_0 \ar[r]_-f& Y_0
}$$ where $q=(p_1, \cdots, p_{s-1})$. Then we have $$\label{x-3.8}
\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_i L_0\right)\simeq 
\mathcal O_{X^s_0}(p^*_sL_0)\otimes 
q^*\mathcal O_{X^{s-1}_0}\left(\sum _{i=1}^{s-1}p^*_iL_0\right).$$ Therefore, we obtain $$\begin{split}
f^s_*\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_iL_0\right)
&\simeq  f_*{p_s}_*\left(\mathcal O_{X^s_0}(p^*_sL_0)\otimes 
q^*\mathcal O_{X^{s-1}_0}\left(\sum _{i=1}^{s-1}p^*_iL_0\right)\right)
\\
&\simeq 
f_*\left(\mathcal O_{X_0}(L_0)\otimes {p_s}_*q^*\mathcal O_{X^{s-1}_0}\left(
\sum _{i=1}^{s-1}p^*_iL_0\right)\right)
\\ 
&\simeq 
f_*\left(\mathcal O_{X_0}(L_0)\otimes f^*f^{s-1}_*\mathcal O_{X^{s-1}_0}\left(
\sum _{i=1}^{s-1}p^*_iL_0\right)\right)
\\ 
&\simeq 
f_*\left(\mathcal O_{X_0}(L_0)\otimes f^*\left(\bigotimes^{s-1}f_*\mathcal O_{X_0}(L_0)\right)\right) 
\\ 
&\simeq f_*\mathcal O_{X_0}(L_0)\otimes \bigotimes ^{s-1}f_*\mathcal O_{X_0}(L_0)
\\ &\simeq 
\bigotimes ^s f_*\mathcal O_{X_0}(L_0). 
\end{split}$$ Note that the first isomorphism follows from , 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 \[x-lem3.6\]. 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 and Lemma \[x-rem3.5\], 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 . 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 $U$.

Weakly positive sheaves and big sheaves {#x-sec4}
=======================================

We quickly see some basic properties of Viehweg’s weakly positive sheaves and big sheaves. For the details, see [@fujino-iitaka Chapter 3], [@viehweg1], [@viehweg2], and [@viehweg3].

\[x-def4.1\] 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.

\[x-lem4.2\] Let $\mathcal F$ be a nonzero torsion-free coherent sheaf on a smooth quasi-projective variety $W$. Then the following three conditions are equivalent.

-   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 $W$.

-   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.

-   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.

\[x-lem4.3\] 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.

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 \[x-lem4.2\].

\[x-lem4.4\] Let $\mathcal F$ be a torsion-free coherent sheaf on a smooth quasi-projective variety $W$ and let $\tau:W'\to W$ be a finite surjective morphism from a smooth quasi-projective variety $W'$. 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.

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, [@fujino-iitaka 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 \[x-lem4.2\]). Therefore, $S^a(\mathcal F)\otimes 
\mathcal H^{\otimes -1}$ is weakly positive since $\tau$ is finite (see, for example, [@fujino-iitaka Lemma 3.1.12 (v)]). This means that $\mathcal F$ is big by Lemma \[x-lem4.2\].

Mixed-$\omega$-sheaves and mixed-$\omega$-big sheaves {#x-sec5}
=====================================================

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 \[x-lem5.5\], \[x-lem5.8\], and \[x-lem5.9\].

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

\[x-def5.1\] 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 $W$.

We give a very important remark on Definition \[x-def5.1\].

\[x-rem5.2\] The notion of pure-$\omega$-sheaves is essentially the same as that of Nakayama’s $\omega$-sheaves in [@nakayama] when we treat torsion-free coherent sheaves on normal projective varieties. However, the definition of pure-$\omega$-sheaves in Definition \[x-def5.1\] does not coincide with [@nakayama 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.

\[x-def5.3\] 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:

-   $V$ is a smooth quasi-projective variety,

-   $Z$ is a normal quasi-projective variety,

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

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

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

$$\xymatrix{
V \ar[dd]_-f\ar[dr]^-p& \\ 
& Z \ar[dl]^-q\\
W
}$$ then $\mathcal F$ is called a [*[mixed-$\omega$-big-sheaf]{}*]{} on $W$. As in Definition \[x-def5.1\], $\mathcal F$ is called a [*[pure-$\omega$-big-sheaf]{}*]{} on $W$ when $D=0$.

\[x-rem5.4\] Of course, we defined mixed-$\omega$-big-sheaves and pure-$\omega$-big-sheaves referring to [@nakayama Chapter V, 3.16. Definition (1)]. However, Nakayama’s definition of $\omega$-bigness is different from ours.

Lemma \[x-lem5.5\] gives a very basic example of mixed-$\omega$-sheaves.

\[x-lem5.5\] 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.

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.

\[x-lem5.6\] 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 $W$.

We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. By Lemma \[x-lem5.5\], $\mathcal O_V(K_V+D+P)$ 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 $W$.

\[x-lem5.7\] 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 $W$.

We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. 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 \[x-lem5.8\] and \[x-lem5.9\] give many nontrivial important examples of mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves in the study of higher-dimensional algebraic varieties.

\[x-lem5.8\] 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 $W$.

By Lemma \[x-lem3.2\], we can construct a $\mathbb Q$-divisor $M^\dag$ on $V$ such that $M^\dag-f^*H$ is semi-ample, ${{\operatorname{Supp}}}\{M^\dag\}={{\operatorname{Supp}}}\{M\}$, and $\lceil M^\dag\rceil =\lceil M\rceil$. Therefore, we may assume that $M$ is a $\mathbb Q$-divisor by replacing $M$ with $M^\dag$. By Kawamata’s covering construction, we can construct a finite Galois cover $\pi:V'\to V$ from a smooth projective variety $V'$ with the following properties:

-   $\pi^*D$ is a simple normal crossing divisor on $V'$,

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

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

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

By assumption, $\pi^*M$ is semi-ample. Since $\pi^*M-(f\circ \pi)^*H$ is semi-ample, we have the following commutative diagram: $$\xymatrix{
V'\ar[dd]_-{f\circ\pi}\ar[dr]^-p & \\
& Z\ar[dl]^-q\\ 
W
}$$such that

-   $Z$ is a normal projective variety, and

-   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 $(f\circ \pi)_*\mathcal O_{V'}(K_{V'}+
\pi^*D+\pi^*M)$. 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 $W$.

\[x-lem5.9\] 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 $\pi:V'\to V$ from a smooth quasi-projective variety $V'$ and a simple normal crossing divisor $D'$ on $V'$ such that $\mathcal O_V(K_V+D+\lceil B\rceil)$ is a direct summand of $\pi_*\mathcal O_{V'}(K_{V'}+D')$. 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 $V$.

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 $\rho:V'\to Z$ and write $$K_{V'}+D'=\rho^*(K_Z+\tau^*D)+E$$ where $D'$ is a reduced simple normal crossing divisor on $V'$ and $E$ is an effective $\rho$-exceptional $\mathbb Q$-divisor on $V'$. We put $\pi:=\tau\circ \rho: V'\to V$. Then $$\begin{split}
\pi_*\mathcal O_{V'}(K_{V'}+D')&\simeq 
\tau_*\mathcal O_Z(K_Z+\tau^*D)\\ 
&\simeq \bigoplus_{i=0}^{r-1}\mathcal O_V(K_V+D+\lceil iB\rceil).  
\end{split}$$ 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.

\[x-def5.10\] 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 \[x-thm9.3\], which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves.

Basic properties: Part 1 {#x-sec6}
========================

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.

\[x-thm6.1\] 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.

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 [@fujino-higher], [@ffs], [@fujino-fujisawa], [@fujisawa], and so on), see [@fujino-weak Theorem 7.8 and Corollary 7.11]. For the proof based on the vanishing theorem, see [@fujino-quasi-alb Theorem 8.4].

As an easy consequence of Theorem \[x-thm6.1\], we have:

\[x-thm6.2\] 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.

We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. By Theorem \[x-thm6.1\], $f_*\mathcal O_V(K_{V/W}+D)$ 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 \[x-thm6.2\], $\mathcal F\otimes \omega^{\otimes -1}_W$ is not only weakly positive but also big.

\[x-thm6.3\] 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.

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 \[x-def5.3\]. It is sufficient to prove that $f_*\mathcal O_V(K_{V/W}+D+P)$ is big. Let $$\xymatrix{
V \ar[dd]_-f\ar[dr]^-p& \\ 
& Z \ar[dl]^-q\\
W
}$$ and $A$ be as in Definition \[x-def5.3\]. 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 $\tau:W'\to W$ from a smooth quasi-projective variety $W'$ and get the following commutative diagram $$\xymatrix{
V' \ar[d]_-{f'}\ar[r]^-\rho& V \ar[d]^-f\\
W' \ar[r]_-\tau&W
}$$ such that $\tau^*H\sim mH'$ for some Cartier divisor $H'$, $V'=V\times _W W'$ 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 \[x-lem3.3\] and Remark \[x-rem3.4\]). By Lemma \[x-lem4.4\], It is sufficient to prove that $$\tau^*f_*\mathcal O_V(K_{V/W}+D+P)\simeq 
f'_*\mathcal O_{V'}(K_{V'/W'}+
\rho^*D+\rho^*P)$$ is a big sheaf on $W'$. By construction, we see that $\rho^*P-f'^*H'$ is a semi-ample Cartier divisor on $V'$ since it is $\mathbb Q$-linearly equivalent to $\rho^*p^*(A-(1/m)q^*H)$. Therefore, by Lemma \[x-lem5.5\], $\mathcal O_{V'}(K_{V'}+\rho^*D+\rho^*P-f'^*H')$ is a mixed-$\omega$-sheaf on $V'$. Therefore, $\mathcal E:=f'_*\mathcal O_{V'}(K_{V'}+\rho^*D
+\rho^*P-f'^*H')$ is a mixed-$\omega$-sheaf on $W'$. We note that $$f'_*\mathcal O_{V'}(K_{V'/W'}+\rho^*D
+\rho^*P)\simeq \mathcal E \otimes \omega^{\otimes -1}_{W'}\otimes 
\mathcal O_{W'}(H').$$ By Theorem \[x-thm6.2\], $\mathcal E\otimes \omega^{\otimes -1}_{W'}$ is weakly positive. By Lemma \[x-lem4.3\], $\mathcal E\otimes 
\omega^{\otimes -1}_{W'}\otimes 
\mathcal O_{W'}(H')$ is big since $H'$ is ample. This is what we wanted.

We close this section with an obvious corollary.

\[x-cor6.4\] 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[*[)]{}*]{}.

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 \[x-thm6.2\] and \[x-thm6.3\].

Basic properties: Part 2 {#x-sec7}
========================

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

\[x-lem7.1\] 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 $W$.

We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. 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 \[x-def5.3\]. 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 [@fujino-fundamental Theorem 6.3 (ii)] (see also [@fujino-foundations 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 \[x-lem7.1\], we have:

\[x-lem7.2\] 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.

This is a direct consequence of Lemmas \[x-lem5.7\], \[x-lem7.1\], and Castelnuovo–Mumford regularity.

Let us recall a vanishing theorem for dlt pairs.

\[x-lem7.3\] 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 $H^i(W, R^jf_*\mathcal O_V(L)\otimes \mathcal N)=0$ for $i>0$, $j\geq 0$, and every nef invertible sheaf $\mathcal N$ on $W$.

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 [@fujino-weak Lemma 7.14], $$H^i(W, R^jf_*\mathcal O_V(L)\otimes \mathcal N)=0$$ for $i>0$ and $j\geq 0$.

In Lemmas \[x-lem7.4\] and \[x-lem7.5\], we treat mixed-$\widehat\omega$-big-sheaves on smooth projective curves.

\[x-lem7.4\] 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 $C$.

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 \[x-lem7.1\], $H^1(C, \mathcal F\otimes \mathcal N)=0$ holds. Therefore, we have $H^1(C, \mathcal G\otimes \mathcal N)=0$ by the surjection $H^1(C, \mathcal F\otimes \mathcal N)\to H^1(C, \mathcal G
\otimes \mathcal N)$.

\[x-lem7.5\] 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 $P$.

By Lemma \[x-lem7.4\], $H^1(C, \mathcal E\otimes \mathcal O_C(-P))=0$. 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.

\[x-lem7.6\] 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.

We may assume that $\mathcal F=f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\].

\[x-step7.6.1\] 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 $D_{\mathrm{hor}}$.

\[x-step7.6.2\] 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 $W'$ of $|H|$. We put $f^{-1}(W')=V'$. Then we have a short exact sequence $$0\to \mathcal O_V(K_V+D)\to \mathcal O_V(K_V+V'+D)\to 
\mathcal O_{V'}(K_{V'}+D|_{V'})\to 0$$ by adjunction. Since $W'$ is a general member of $|H|$, we get a short exact sequence $$\label{x-7.1}
\begin{split}
0&\to f_*\mathcal O_V(K_V+D)\otimes \mathcal O_W(nH)\to f_*\mathcal O_V(K_V+V'+D)\otimes 
\mathcal O_W(nH)\\&\to 
f_*\mathcal O_{V'}(K_{V'}+D|_{V'})\otimes 
\mathcal O_{W'}(nH|_{W'})\to 0. 
\end{split}$$ By the vanishing theorem (see Lemma \[x-lem7.3\]), we have $$\label{x-7.2}
H^1(W, f_*\mathcal O_V(K_V+D)\otimes 
\mathcal O_W(nH))=0.$$ Therefore, the restriction map $$H^0(W, f_*\mathcal O_V(K_V+D)\otimes 
\mathcal O_W((n+1)H))\to 
H^0(W', f_*\mathcal O_{V'}(K_{V'}+D|_{V'})\otimes 
\mathcal O_{W'}(nH|_{W'}))$$ is surjective by and . By induction on $n$, $f_*\mathcal O_{V'}(K_{V'}+D|_{V'})
\otimes \mathcal O_{W'}(nH|_{W'})$ 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 \[x-lem7.7\] is similar to Lemma \[x-lem7.6\].

\[x-lem7.7\] 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 \[x-lem7.7\] is essentially the same as that of Lemma \[x-lem7.6\].

If $n=0$, then the statement is trivial. If $n=1$, then it follows from Lemma \[x-lem7.5\]. Therefore, we assume that $n\geq 2$. As in the proof of Lemma \[x-lem7.6\], we may assume that $\mathcal F=f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. 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 \[x-step7.6.1\] in the proof of Lemma \[x-lem7.6\]). We take a general member $W'$ of $|H|$ and put $V'=f^{-1}(W')$. Then the natural restriction map $$\begin{split}
&H^0(W, f_*\mathcal O_V(K_V+D+P)\otimes 
\mathcal O_W(nH))\\ &\to 
H^0(W', f_*\mathcal O_{V'}(K_{V'}+D|_{V'}+P|_{V'})\otimes 
\mathcal O_{W'}((n-1)H|_{W'}))
\end{split}$$ is surjective as in Step \[x-step7.6.2\] in the proof of Lemma \[x-lem7.6\]. 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 [@dutta-murayama]. We will use it in the proof of Theorem \[y-thm1.8\].

\[x-lem7.8\] 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 $l\geq n^2+1$.

We may assume that $\mathcal F=f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. Then, by [@dutta-murayama 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 \[x-lem7.8\], we recommend the reader to see [@dutta-murayama].

A special case {#x-sec8}
==============

In this section, we will freely use the standard notation and some basic results in the theory of minimal models (see, for example, [@fujino-fundamental], [@fujino-foundations], and [@fujino-iitaka]). The reader can skip this section.

\[x-thm8.1\] 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 $f_*\omega^{\otimes k}_{X/Y}\ne 0$. 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 $i>0$.

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

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

We consider a relative good minimal model $f':X'\to Y$ of $f:X\to Y$. $$\xymatrix{
X \ar[dr]_-f\ar@{-->}^-\phi[rr]&& X'\ar[dl]^-{f'} \\ 
& Y& 
}$$ Since $$f_*\omega^{\otimes m}_{X/Y}\simeq f'_*\mathcal O_{X'}(mK_{X'/Y})$$ holds for every $m\geq 1$, it is sufficient to prove that $$f'_*\mathcal O_{X'}(K_{X'}+(k-1)K_{X'/Y}+f'^*H)$$ is a pure-$\omega$-big-sheaf on $Y$.

In this step, we will prove:

$K_{X'/Y}$ is nef and $f'$-semi-ample.

Since $f': X'\to Y$ is a relative good minimal model of $f:X\to Y$, $K_{X'/Y}$ is $f'$-semi-ample. Therefore, $$f'^*f'_*\mathcal O_{X'}(lK_{X'/Y})\to \mathcal O_{X'}(lK_{X'/Y})$$ is surjective for a sufficiently large and divisible positive integer $l$. Since $f'_*\mathcal O_{X'}(lK_{X'/Y})\simeq 
f_*\omega^{\otimes l}_{X/Y}$ is a nef locally free sheaf, $K_{X'/Y}$ is nef by the above surjection.

Since $K_{X'/Y}$ is nef and $f'$-semi-ample, $(k-1)K_{X'/Y}+af'^*H$ is semi-ample for every positive rational number $a$. We take a birational morphism $\rho:\widetilde X\to X'$ 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 $X'$ has only canonical singularities, we see that $$\rho_*\mathcal O_{\widetilde X}(K_{\widetilde X}+\lceil 
(k-1)\rho^*K_{X'/Y}+\rho^*f'^*H\rceil)
\simeq \mathcal O_{X'}(K_{X'}+(k-1)K_{X'/Y}+f'^*H)$$ holds. By Lemma \[x-lem5.8\], $$(f'\circ\rho)_*\mathcal O_{\widetilde X}(K_{\widetilde X}+\lceil 
(k-1)\rho^*K_{X'/Y}+\rho^*f'^*H\rceil)
\simeq f'_*\mathcal O_{X'}(K_{X'}+(k-1)K_{X'/Y}+f'^*H)$$ is a pure-$\omega$-big-sheaf on $Y$ since $$(k-1)\rho^*K_{X'/Y}+\rho^*f'^*H-\frac{1}{2}\rho^*f'^*H$$ 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 \[x-lem7.1\].

Theorem \[x-thm8.1\] predicts that $f_*\omega^{\otimes k}
_{X/Y}\otimes \omega_Y 
\otimes \mathcal O_Y(H)$ has good properties.

Fundamental theorem {#x-sec9}
===================

This section is the main part of this paper. The main result of this section is Theorem \[x-thm9.3\], which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves.

Let us start with the following lemma.

\[x-lem9.1\] 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:

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

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

Let $H$ be an ample divisor on $Y$. We assume that $f_*\mathcal O_Y(L)\ne 0$. 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 \[x-lem9.1\] is essentially the same as [@nakayama Chapter V, 3.34. Lemma].

We divide the proof into several small steps.

\[x-step9.1\] 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 $X$.

\[x-step9.2\] 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 \[x-lem4.2\].

\[x-step9.3\] 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 \[x-step9.1\], 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 $1\leq b\leq a$.

\[x-step9.4\] By taking a suitable birational modification of $X$ again (see Step \[x-step9.1\]), 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 $f_*\mathcal O_X(L-B)=f_*\mathcal O_X(L)$.

\[x-step9.5\] 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 $$\label{x-9.1}
H^0(Y, \mathcal O_Y((al-1)H)\otimes 
\widehat S^a(f_*\mathcal O_X(L)))\otimes \mathcal O_X\to 
\mathcal O_X(a(L-B)+E+(al-1)f^*H).$$ By taking a suitable birational modification of $X$ again (see Step \[x-step9.1\]), we may assume that the image of 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.

\[x-step9.6\] 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 $$\left(l-\left\lfloor \frac{l}{k}\right\rfloor\right)-\frac{(al-1)(k-1)}{k} 
-\frac{\varepsilon}{k}=\alpha.$$

\[x-step9.7\] 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.

\[x-step9.8\] 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 \[x-lem5.8\].

\[x-step9.9\] Let $l_0$ be the minimum positive integer such that $$\mathcal O_Y(K_Y+l_0H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. By Theorem \[x-thm6.3\], $$\mathcal O_Y(l_0H)\otimes f_*\mathcal O_X(L)$$ 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.

\[x-rem9.2\] In Lemma \[x-lem9.1\], 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 \[x-step9.7\] in the proof of Lemma \[x-lem9.1\]. 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 \[x-thm9.3\] 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.

\[x-thm9.3\] 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:

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

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

If $f_*\mathcal O_Y(L)\ne 0$, then $$\mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat \omega$-big-sheaf on $Y$.

We divide the proof into several small steps.

\[x-step9.3.1\] By taking a resolution as in Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\], 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 \[x-step9.1\] in the proof of Lemma \[x-lem9.1\] 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 \[x-lem3.2\], 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 \[x-step9.1\] in the proof of Lemma \[x-lem9.1\]).

\[x-step9.3.2\] By Lemma \[x-lem3.3\] and Remark \[x-rem3.4\], we take a finite flat Galois cover $\tau:Y'\to Y$ from a smooth projective variety $Y'$ and get the following commutative diagram $$\xymatrix{
X' \ar[d]_-{f'}\ar[r]^-\rho& X\ar[d]^-f \\
Y' \ar[r]_-\tau&Y
}$$ such that $X'=X\times _YY'$ is a smooth projective variety, $\tau^*H\sim mH'$ for some ample Cartier divisor $H'$ on $Y'$, and $\rho^*\omega^{\otimes n}_{X/Y}=\omega^{\otimes n}_{X'/Y'}$ for every integer $n$. Let $G$ denote the Galois group of $\tau:Y'\to Y$. By construction (see the proof of Lemma \[x-lem3.3\]), we may assume that $H'$ is $G$-invariant. We put $L'=\rho^*L$, $B'=\rho^*B$, $\Delta'=\rho^*\Delta$, and $S'=\rho^*S$. Without loss of generality, we may assume that ${{\operatorname{Supp}}}(\Delta'+B'+S')$ is a simple normal crossing divisor on $X'$ and that $\rho^*(K_{X/Y}+\Delta)=K_{X'/Y'}+\Delta'$ holds (see Remark \[x-rem3.4\]). We note that $$L'-(k-1)f'^*H'-k(K_{X'/Y'}+\Delta')\sim 
_{\mathbb Q} \rho^*\left(L-k(K_{X/Y}+\Delta)-\frac{k-1}{m}
f^*H\right)$$ by construction. This implies that $\left(L'-(k-1)f'^*H'\right)-k(K_{X'/Y'}+\Delta')$ is semi-ample.

\[x-step9.3.3\] We apply Lemma \[x-lem9.1\] to $$\left(L'-(k-1)f'^*H'\right)-k(K_{X'/Y'}+\Delta').$$ Then we obtain that $\mathcal O_{Y'}(K_{Y'})\otimes 
f'_*\mathcal O_{X'}(L')$ is a mixed-$\widehat\omega$-big-sheaf on $Y'$. Therefore, $f'_*\mathcal O_{X'}(L')$ is a big sheaf on $Y'$ by Theorem \[x-thm6.3\]. Thus we can take a positive integer $a$ such that $\widehat S^a(f'_*\mathcal O_{X'}(L'))$ is generically generated by global sections (see Lemma \[x-lem4.2\]). Then we take an effective $G$-invariant $f'$-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 replacing $L'$, $\Delta'$, and $B'$ with $L'+E'$, $\Delta'+(1/k)E'$, and $B'+E'$ respectively, we may assume that $f'_*\mathcal O_{X'}(bL')$ is reflexive for every $1\leq b\leq a$.

\[x-step9.3.4\] We can take an effective $G$-invariant $f'$-exceptional divisor $E'$ on $X'$ such that the surjective 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 $$\label{x-9.2} 
H^0(Y', \widehat S^a(f'_*\mathcal O_{X'}(L')))
\otimes \mathcal O_{X'}\to \mathcal O_{X'}
(a(L'-B')+E').$$ By taking an equivariant resolution of singularities of $X'$, we may assume that the image of is $$\mathcal O_{X'}(a(L'-B')+E'-F')$$ for some effective $G$-invariant $f'$-vertical divisor $F'$ on $X'$. Of course, we may assume that ${{\operatorname{Supp}}}(\Delta'+B'+E'+F')$ is a simple normal crossing divisor on $X'$. We put $$N':=a(L'-B')+E'-F'.$$ Then $|N'|$ is free by construction. We put $$M':=L'-(K_{X'/Y'}+\Delta')-\frac{k-1}{k}B'+\frac{k-1}{ak}
(E'-F').$$ Then $$M'-\frac{k-1}{ak}N'-\frac{1}{k}(L'-k(K_{X'/Y'}+\Delta')-
(k-1)f'^*H')=\frac{k-1}{k}f'^*H'.$$ In particular, $M'$ and $M'-\frac{k-1}{k}f'^*H'$ are semi-ample.

\[x-step9.3.5\] We put $$\left\lfloor \frac{k-1}{k}B'+\Delta'\right\rfloor=B'_0+\Delta'_1+\Delta'_2$$ as in Step \[x-step9.7\] in the proof of Lemma \[x-lem9.1\]. Then $\Delta'_1$ is a $G$-invariant $f'$-horizontal simple normal crossing divisor on $X'$. As before, ${{\operatorname{Supp}}}\{M'\}$ and $\Delta'_1$ have no common irreducible components. Thus, by Lemma \[x-lem5.8\], $f'_*\mathcal O_{X'}(K_{X'}+\Delta'_1+\lceil M'\rceil)$ is a mixed-$\omega$-big-sheaf on $Y'$. Note that the Galois group $G$ acts on $f'_*\mathcal O_{X'}(K_{X'}+\Delta'_1+\lceil M'\rceil)$.

\[x-step9.3.6\] Therefore, we get the following generically isomorphic $G$-equivariant embedding: $$\label{x-9.3}
f'_*\mathcal O_{X'}(K_{X'}+\Delta'_1+\lceil M'\rceil)
\hookrightarrow \mathcal O_{Y'}(K_{Y'})
\otimes f'_*\mathcal O_{X'}(L')$$ as in Step \[x-step9.8\] in the proof of Lemma \[x-lem9.1\]. We note that $f'_*\mathcal O_{X'}(L')\simeq 
\tau^*f_*\mathcal O_X(L)$ by the flat base change theorem. We take $\tau_*$ of and then take the $G$-invariant parts. Thus, we get a mixed-$\omega$-big-sheaf $$\mathcal F:=\left(\tau_*f'_*\mathcal O_{X'}(K_{X'}
+\Delta'_1+\lceil M'\rceil)\right)^G$$ 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 $Y$.

\[x-rem9.4\] As in Remark \[x-rem9.2\], we further assume that $(X, \Delta)$ is klt over a nonempty Zariski open set of $Y$ in Theorem \[x-thm9.3\]. Then we see that $\Delta'_1=0$ in Step \[x-step9.3.5\] in the proof of Theorem \[x-thm9.3\]. 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 \[x-thm9.3\], we have:

\[x-cor9.5\] 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:

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

-   $L+f^*D-k(K_{X/Y}+\Delta)$ 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 $H^\dag$ be a nef and big Cartier divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then the sheaf $$\mathcal O_Y(K_Y+\lceil D\rceil+lH^\dag)\otimes 
f_*\mathcal O_X(L)$$ is generically generated by global sections for $l\geq n^2+2$.

By taking a resolution of singularities as in Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\], 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 $\tau:Y'\to Y$ from a smooth projective variety $Y'$ such that the main component of $X\times _Y Y'$ is flat over $Y'$. Let $X'$ be a resolution of the main component of $X\times _Y Y'$. Then we get the following commutative diagram. $$\xymatrix{
X' \ar[d]_-{f'}\ar[r]^-\rho& X \ar[d]^-f\\
Y' \ar[r]_-\tau&Y
}$$ By construction, any $f'$-exceptional divisor is $\rho$-exceptional. We put $K_{X'}+B=\rho^*(K_X+\Delta)$. We may assume that ${{\operatorname{Supp}}}B$ is a simple normal crossing divisor on $X'$. We write $$K_{Y'}=\tau^*K_Y+R$$ where $R$ is an effective $\tau$-exceptional divisor on $Y'$. We put $$L':=\rho^*L+k\lceil -(B^{<0})\rceil -kf'^*R$$ and $\Delta'=B+\lceil -(B^{<0})\rceil$. Note that $\lceil -(B^{<0})\rceil$ is effective and $\rho$-exceptional. Then we have $$L'-k(K_{X'/Y'}+\Delta')=\rho^*(L-k(K_{X/Y}+\Delta)).$$ We take an effective $f'$-exceptional divisor $E$ on $X'$ such that $$\left(f'_*\mathcal O_{X'}(L')\right)^{**} 
\simeq f'_*\mathcal O_{X'}(L'+E).$$ Note that $E$ is $\rho$-exceptional and that there is a generically isomorphic injection $$\tau_*f'_*\mathcal O_{X'}(L'+E)
=f_*\rho_*\mathcal O_{X'}(L'+E)\subset 
f_*\mathcal O_X(L).$$ Therefore, we have a generically isomorphic injection $$\tau_*\left((f'_*\mathcal O_{X'}(L'))^{**}\right)
\subset f_*\mathcal O_X(L).$$ 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 $$\begin{split}
&L'+E+f'^*\tau^*H-k\left(K_{X'/Y'}+\Delta'+\frac{1}{k}E+
\frac{1}{k}f'^*B\right)-\frac{1}{2}f'^*A\\
&=L'-k(K_{X'/Y'}+\Delta')+\frac{1}{2}f'^*A
\end{split}$$ is semi-ample. Therefore, by Theorem \[x-thm9.3\], $$\mathcal O_{Y'} (K_{Y'}+\tau^*H)\otimes 
f'_*\mathcal O_{X'}(L'+E)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y'$. Thus, by Lemma \[x-lem7.7\], $$\mathcal O_{Y'}(K_{Y'}+(n+1)\tau^*H)
\otimes (f'_*\mathcal O_{X'}(L'))^{**}$$ 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 $$\mathcal O_{Y'}(K_{Y'}+\tau^*H^\dag)\otimes 
f'_*\mathcal O_{X'}(L'+E)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y'$. Hence the sheaf $$\mathcal O_{Y'}(K_{Y'}+l\tau^*H^\dag)
\otimes (f'_*\mathcal O_{X'}(L'))^{**}$$ is generically generated by global sections for $l\geq n^2+2$ by Lemma \[x-lem7.8\]. Therefore, so is $\mathcal O_Y(K_Y+lH^\dag)\otimes 
f_*\mathcal O_X(L)$. Anyway, we get the desired statements.

We note that [@nakayama 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 \[x-cor9.5\] can be applied to log canonical pairs. This is the main difference between [@nakayama Chapter V, 3.37. Corollary] and Corollary \[x-cor9.5\].

Proof of Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\] {#x-sec10}
==============================================================

In this section, we prove Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\] in Section \[z-sec1\].

Let us first prove Theorem \[y-thm1.7\].

We divide the proof into small steps.

\[x-step10-1\] 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 \[x-step9.1\] in the proof of Lemma \[x-lem9.1\]). We may further assume that every log canonical center of $(X, \Delta_{\mathrm{hor}})$ is dominant onto $Y$.

\[x-step10-2\] 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 \[x-step7.6.2\] in the Proof of Lemma \[x-lem7.6\], we see that $f_*\mathcal O_X(L)\otimes 
\mathcal O_Y(K_Y+lH)$ is generically generated by global sections.

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 \[x-cor9.5\]. More precisely, we put $D=0$ and apply Corollary \[x-cor9.5\].

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 \[x-lem3.6\] 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 \[x-thm9.3\], $$\mathcal O_Y(K_Y+H)\otimes f^{(s)}_*\mathcal O_{X^{(s)}}(L^{(s)})$$ 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 \[x-lem7.7\], $$\mathcal O_Y(K_Y+lH)\otimes \left(\bigotimes ^s 
f_*\mathcal O_X(L)\right)^{**}$$ 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 \[x-step10-1\] and \[x-step10-2\]. 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 \[y-thm1.8\].

It is not difficult to modify the proof of Theorem \[y-thm1.7\].

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 \[x-lem5.9\], $$\mathcal O_X(K_X+\lfloor \Delta\rfloor 
+\lceil L-K_{X/Y}-\Delta\rceil)\simeq 
\mathcal O_X(L)\otimes f^*\mathcal O_Y(K_Y)$$ 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 \[x-lem7.8\], $f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH^\dag)$ 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 \[y-thm1.7\] is a finite direct sum of mixed-$\omega$-sheaves on $Y$ when $k=1$. Therefore, $$\left(\bigotimes ^sf_*\mathcal O_X(L)\right)^{**}
\otimes \mathcal O_Y(K_Y+lH^\dag)$$ is generically generated by global sections for $l\geq n^2+1$.

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

If we use Lemma \[x-lem7.8\] instead of Lemma \[x-lem7.7\], then the proof of Theorem \[y-thm1.7\] implies that $$\mathcal O_Y(K_Y+lH^\dag)\otimes 
\left(\bigotimes ^sf_*\mathcal O_X(L)\right)^{**}$$ is generically generated by global sections for $l\geq n^2+2$. By Corollary \[x-cor9.5\], $$\mathcal O_Y(K_Y+lH^\dag)\otimes f_*\mathcal O_X(L)$$ is generically generated by global sections for $l\geq n^2+2$.

Thus we get the desired statements.

Finally, we prove Theorem \[z-thm1.5\].

We put $L=kK_{X/Y}$. Then this theorem directly follows from Theorems \[y-thm1.7\] and \[y-thm1.8\].

We close this section with an easy remark.

\[x-rem10.1\] 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 \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\].

Some other applications {#x-sec11}
=======================

In this final section, we treat Nakayama’s inequality on $\kappa _\sigma$ and a slight generalization of the twisted weak positivity theorem. Theorem \[x-thm11.3\] and a special case of Theorem \[x-thm11.7\] 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.

\[x-def11.1\] 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 $H^0(X, \mathcal O_X(\lfloor mD\rfloor +A))\ne 0$ 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 $H^0(X, \mathcal O_X(\lfloor mD\rfloor +A))\ne 0$ 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, [@nakayama 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 [@nakayama]. The following remark is easy but very useful.

\[x-rem11.2\] Let $X$ be a smooth projective variety and let $D$ be an $\mathbb 
R$-divisor on $X$. We put $$\sigma (D; A)'=\max \left\{ k\in \mathbb Z_{\geq 0}\cup 
\{-\infty\}\, \left|\, {\underset{m\to \infty}{\limsup}}
\frac{\dim H^0(X, \mathcal O_X(\lceil mD\rceil +A))}{m^k}>0 \right.\right\},$$ where $A$ is a divisor on $X$. Then we have the following equality $$\kappa _\sigma(X, D)=\max\{ \sigma(D; A)' \, |\, 
\text{$A$ is a divisor}\}.$$ We will use this characterization of $\kappa_\sigma$ in the proof of Theorem \[x-thm11.3\] 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 [@fujino-corri Remark 2.2]), which will be useful in the proof of Theorem \[x-thm11.3\] below.

The inequalities in Theorem \[x-thm11.3\] are indispensable in the theory of minimal models (see Remarks \[x-rem11.4\] and \[x-rem11.5\]).

\[x-thm11.3\] 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 \[x-thm11.3\], we give two important remarks.

\[x-rem11.4\] We think that one of the most important results of Nakayama’s theory of $\omega$-sheaves is the inequality on $\kappa_\sigma$ in [@nakayama Chapter V, 4.1. Theorem (1)]. However, as we explained in [@fujino-subadd Remark 3.8] and [@fujino-corri Section 3], the proof of [@nakayama Chapter V, 4.1. Theorem (1)] is incomplete. For the details, see, for example, [@fujino-corri Section 1]. So, in Theorem \[x-thm11.3\], we claim two weaker inequalities than Nakayama’s original one (see [@fujino-corri (3.3) and (3.4)]). Anyway, the first inequality in Theorem \[x-thm11.3\] is still sufficiently powerful for some geometric applications (see [@fujino-corri Section 3]).

\[x-rem11.5\] The troubles in the proof of [@dhp Remark 2.6] and [@gongyo-lehmann Theorem 4.3] caused by the incompleteness of [@nakayama Chapter V, 4.1. Theorem (1)] can be corrected by using the first inequality in Theorem \[x-thm11.3\]. For the details, we recommend the reader to see [@hashizume-hu Lemma 2.10].

Let us prove Theorem \[x-thm11.3\].

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 \[x-step9.1\] in the proof of Lemma \[x-lem9.1\]), 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 \[x-cor9.5\]. 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 $$\label{eq1.1}
\begin{split}
&\dim _{\mathbb C} H^0(X, \mathcal O_X(\lceil m(D+f^*Q)\rceil 
+A+2f^*H)
\\&\geq \dim _{\mathbb C} H^0(X, \mathcal O_X(\lceil mD\rceil+f^*(\lfloor 
mQ\rfloor)+A+2f^*H)) 
\\ & \geq 
r(mD; A)\cdot \dim _{\mathbb C}
H^0(Y, \mathcal O_Y(\lfloor mQ\rfloor +H))
\end{split}$$ for every positive integer $m$. We can take a positive integer $m_0$ and a positive real number $C_0$ such that $$\label{eq1.2}
C_0m^{\kappa (F, D|_F)}\leq r(mm_0D; A)$$ for every large positive integer $m$ (see, for example, ). Thus we have $$\label{eq1.3}
\begin{split}
&\dim H^0(X, \mathcal O_X(\lceil mm_0(D+f^*Q)\rceil+A+2f^*H))\\
&\geq C_0m^{\kappa(F, D|_F)}
\cdot \dim H^0(Y, \mathcal O_Y(\lfloor mm_0Q\rfloor +H)) 
\end{split}$$ for every large positive integer $m$ by and . We may assume that $H$ is sufficiently ample. Then we get $$\label{eq1.4}
\limsup_{m\to \infty} \frac{\dim H^0(X, 
\mathcal O_X(\lceil mm_0(D+f^*Q)\rceil+A+2f^*H))}
{m^{\kappa (F, D|_F)+\kappa _{\sigma}(Y, Q)}}>0$$ by and the definition of $\kappa _\sigma(Y, Q)$. This means that the following inequality $$\label{eq1.5}
\kappa _\sigma(X, D+f^*Q)\geq \kappa (F, D|_F)+
\kappa _\sigma(Y, Q)$$ holds.

Similarly, we can take a positive integer $m_1$ and a positive real number $C_1$ such that $$\label{eq1.6}
\begin{split}
C_1m^{\kappa (Y, Q)}&\leq 
\dim H^0(Y, \mathcal O_Y(\lfloor mm_1Q\rfloor))\\ 
&\leq \dim H^0(Y, \mathcal O_Y(\lfloor mm_1Q\rfloor+H))
\end{split}$$ for every large positive integer $m$ (see, for example, ) if $H$ is a sufficiently ample Cartier divisor. Then, by and , we have $$\label{eq1.7}
\begin{split}
&\dim H^0(X, \mathcal O_X(\lceil mm_1 (D+f^*Q)\rceil 
+A+2f^*H))\\
&\geq C_1m^{\kappa(Y, Q)}
\cdot r(mm_1D; A)
\end{split}$$ for every large positive integer $m$. Therefore, we get $$\label{eq1.8}
\limsup_{m\to \infty} \frac{\dim H^0(X, 
\mathcal O_X(\lceil mm_1 (D+f^*Q)\rceil +A+2f^*H))}
{m^{\kappa _\sigma(F, D|_F)+\kappa(Y, Q)}}>0$$ when $A$ is sufficiently ample. Note that $$\label{eq1.9}
\sigma (m_1D|_F; A|_F)'=\max \left\{
k\in \mathbb Z_{\geq 0} \cup \{-\infty\}\, 
\left|\, \underset{m\to \infty}{\limsup}\frac{r(mm_1D; A)}{m^k}>0\right.\right\}$$ for a sufficiently general fiber $F$ of $f:X\to Y$ and that $$\label{eq1.10}
\begin{split}
\kappa_\sigma(F, D|_F)
&=\kappa_\sigma (F, m_1D|_F)
\\&=\max\{\sigma(m_1D|_F; A|_F)'\, |\, {\text{$A$ is very ample}}\}. 
\end{split}$$ Hence we have the inequality $$\label{eq1.11}
\kappa _\sigma(X, D+f^*Q)\geq \kappa_\sigma (F, D|_F)+
\kappa (Y, Q)$$ by .

It is highly desirable to solve the following conjecture. As we explained in [@fujino-corri], Nakayama’s original inequality on $\kappa_\sigma$ (see [@nakayama Chapter V, 4.1. Theorem (1)]) follows from Conjecture \[x-conj11.6\] and the argument in the proof of Theorem \[x-thm11.3\].

\[x-conj11.6\] 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 $$Cm^{\kappa_\sigma(X, D)}\leq \dim H^0(X, \mathcal O_X
(\lfloor mm_0D\rfloor +A))$$ holds for every large positive integer $m$.

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

\[x-thm11.7\] 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.

Let $\alpha$ be a positive integer and let $\mathcal H$ be an ample invertible sheaf on $Y$. By Theorem \[y-thm1.7\] or Theorem \[y-thm1.8\], 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.

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