# Variation of mixed Hodge structure and its applications

Osamu Fujino and Taro Fujisawa  Department of Mathematics, Graduate School of
Science, Kyoto University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>Department of Mathematics, School of Engineering, Tokyo Denki University,
Tokyo, Japan<EMAIL_ADDRESS>
(Date: 2024/3/5, version 0.48)

###### Abstract.

We treat generalizations of Kollár’s torsion-freeness, vanishing theorem, and
so on, for projective morphisms between complex analytic spaces as an
application of the theory of variations of mixed Hodge structure. The results
will play a crucial role in the theory of minimal models for projective
morphisms of complex analytic spaces. In this paper, we do not use Saito’s
theory of mixed Hodge modules.

###### Key words and phrases:

Hodge bundles, vanishing theorems, strict support condition, torsion-freeness,
injectivity theorem, variation of mixed Hodge structure, semipositivity
theorem, minimal model program

###### 2010 Mathematics Subject Classification:

Primary 32L20; Secondary 14D07, 14E30

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Lemmas on analytic simple normal crossing pairs
    2. 2.2 Complex analytic generalization of Kollár’s package
  3. 3 On variations of mixed Hodge structure
  4. 4 Proofs of Theorems 1.1, 1.3, and 1.4
  5. 5 Proof of Theorem 1.5
  6. 6 Proof of Theorem 1.7
  7. 7 Supplement to [St2]

##  1\. Introduction

We will establish the following theorem, which is an analytic generalization
of [FF1, Theorems 7.1 and 7.3]. Note that $f\colon(X,D)\to Y$ is assumed to be
algebraic in [FF1]. Our approach in this paper is slightly different from the
one in [FF1] (see Remark 1.6 below). We also note that we do not use Saito’s
theory of mixed Hodge modules (see [Sa1], [Sa2], [Sa3], [Sa4], [FFS], and
[Sa5]) in this paper.

######  Theorem 1.1 (Canonical extensions of Hodge bundles, see [FF1, Theorems
7.1 and 7.3]).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced and let $f\colon X\to Y$ be a proper surjective morphism onto a smooth
complex variety $Y$. Assume that every stratum of $(X,D)$ is dominant onto
$Y$. Let $\Sigma$ be a normal crossing divisor on $Y$ such that every stratum
of $(X,D)$ is smooth over $Y^{*}:=Y\setminus\Sigma$. We put
$X^{*}:=f^{-1}(Y^{*})$, $D^{*}:=D|_{X^{*}}$, and $d:=\dim X-\dim Y$. If we
assume that every stratum of $(X,D)$ is a Kähler manifold in addition, then we
have:

  * (i)

$R^{k}(f|_{X^{*}\setminus D^{*}})_{!}\mathbb{R}_{X^{*}\setminus D^{*}}$
underlies a graded polarizable variation of $\mathbb{R}$-mixed Hodge structure
on $Y^{*}$ for every $k$.

We put

$\mathcal{V}^{k}_{Y^{*}}:=R^{k}(f|_{X^{*}\setminus
D^{*}})_{!}\mathbb{R}_{X^{*}\setminus D^{*}}\otimes\mathcal{O}_{Y^{*}}$

for every $k$. The Hodge filtration and the weight filtration on
$\mathcal{V}^{k}_{Y^{*}}$ are denoted by $F$ and $L$ respectively. Moreover
the lower canonical extension of $\mathcal{V}^{k}_{Y^{*}}$ is denoted by
${}^{l}\mathcal{V}^{k}_{Y^{*}}$. The weight filtration $L$ on
$\mathcal{V}^{k}_{Y^{*}}$ is extended to ${}^{l}\mathcal{V}^{k}_{Y^{*}}$ by
$L_{m}({}^{l}\mathcal{V}^{k}_{Y^{*}})={}^{l}L_{m}(\mathcal{V}^{k}_{Y^{*}})$
for every $m$. Then we have the following:

  * (ii)

There exists a unique finite decreasing filtration $F$ on
${}^{l}\mathcal{V}^{k}_{Y^{*}}$ such that

    * $\bullet$

$F^{p}({}^{l}\mathcal{V}^{k}_{Y^{*}})|_{Y^{*}}\simeq
F^{p}(\mathcal{V}^{k}_{Y^{*}})$, and

    * $\bullet$

$\operatorname{Gr}_{F}^{p}\operatorname{Gr}_{m}^{L}({}^{l}\mathcal{V}^{k}_{Y^{*}})$
is a locally free $\mathcal{O}_{Y}$-module of finite rank

for every $k,m,p$.

  * (iii)

$R^{d-i}f_{*}\mathcal{O}_{X}(-D)$ is isomorphic to

$\operatorname{Gr}^{0}_{F}({}^{l}\mathcal{V}^{d-i}_{Y^{*}})=F^{0}({}^{l}\mathcal{V}^{d-i}_{Y^{*}})/F^{1}({}^{l}\mathcal{V}^{d-i}_{Y^{*}})$

for every $i$. In particular, $R^{d-i}f_{*}\mathcal{O}_{X}(-D)$ is locally
free for every $i$.

  * (iv)

$R^{i}f_{*}\omega_{X/Y}(D)$ is isomorphic to

$\left(\operatorname{Gr}^{0}_{F}({}^{l}\mathcal{V}^{d-i}_{Y^{*}})\right)^{*}=\mathcal{H}om_{\mathcal{O}_{Y}}(\operatorname{Gr}^{0}_{F}({}^{l}\mathcal{V}^{d-i}_{Y^{*}}),\mathcal{O}_{Y})$

for every $i$. In particular, $R^{i}f_{*}\omega_{X/Y}(D)$ is locally free for
every $i$.

For the precise definition of upper and lower canonical extensions in Theorem
1.1, see [FF1, Remark 7.4]. In Theorem 1.1, $X$ may be reducible, and we are
mainly interested in the case where $X$ is reducible.

######  Remark 1.2.

We do not need the relative monodromy weight filtration for applications in
the theory of minimal models (see, for example, [Fn10]). We are mainly
interested in Hodge bundles and their extensions. However, we can prove the
existence of the relative monodromy weight filtration in Theorem 1.1. In fact,
the variations of $\mathbb{R}$-mixed Hodge structure in (i) of the theorem
above are admissible. This can be checked by the same argument as in [FF1].
For the details, see Remark 4.8 below.

By Theorem 1.1, we can use the Fujita–Zucker–Kawamata semipositivity theorem
in the complex analytic setting.

######  Theorem 1.3 (Semipositivity).

In Theorem 1.1, we further assume that every local monodromy on the local
system $R^{d-i}(f|_{X^{*}\setminus D^{*}})_{!}\mathbb{R}_{X^{*}\setminus
D^{*}}$ around $\Sigma$ is unipotent. Let $\varphi\colon V\to X$ be any
morphism from a projective variety $V$. Then
$\varphi^{*}R^{i}f_{*}\omega_{X/Y}(D)$ is a nef locally free sheaf on $V$.

In order to prove Theorem 1.1, we will establish:

######  Theorem 1.4 (Weight spectral sequence).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced and let $f\colon X\to Y$ be a proper morphism between complex analytic
spaces. We assume that $Y$ is a smooth complex variety and that there exists a
normal crossing divisor $\Sigma$ on $Y$ such that every stratum of $(X,D)$ is
dominant onto $Y$, and smooth over $Y\setminus\Sigma$. If we assume that every
stratum of $(X,D)$ is a Kähler manifold in addition, then we have a spectral
sequence:

$E^{p,q}_{1}=\bigoplus_{S}R^{q}f_{*}\mathcal{O}_{S}\Rightarrow
R^{p+q}f_{*}\mathcal{O}_{X}(-D),$

where $S$ runs through all $(\dim X-p)$-dimensional strata of $(X,D)$, such
that it degenerates at $E_{2}$ and its $E_{1}$-differential $d_{1}$ splits.
Moreover, $R^{i}f_{*}\mathcal{O}_{X}(-D)$ is locally free of finite rank for
every $i$.

By combining Theorem 1.4 with Takegoshi’s results (see [T]), we can prove:

######  Theorem 1.5 (Torsion-freeness and vanishing theorem).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced and let $f\colon X\to Y$ be a projective morphism between complex
analytic spaces. We assume that $Y$ is a complex variety and that every
stratum of $(X,D)$ is dominant onto $Y$. Then we have the following
properties.

  * (i)

$($Torsion-freeness$)$. $R^{q}f_{*}\omega_{X}(D)$ is a torsion-free sheaf for
every $q$.

  * (ii)

$($Vanishing theorem$)$. Let $\pi\colon Y\to Z$ be a projective morphism
between complex analytic spaces and let $\mathcal{A}$ be a $\pi$-ample line
bundle on $Y$. Then

$R^{p}\pi_{*}\left(\mathcal{A}\otimes R^{q}f_{*}\omega_{X}(D)\right)=0$

holds for every $p>0$ and every $q$.

Of course, Theorem 1.5 is a generalization of Kollár’s torsion-freeness and
vanishing theorem (see [Ko1]) for reducible complex analytic spaces. We make a
remark on the relationship between [FF1] and this paper.

######  Remark 1.6.

In [FF1], we have already treated Theorems 1.1 and 1.5 when $X$ and $Y$ are
algebraic and $f\colon X\to Y$ is projective. Roughly speaking, in [FF1, §6],
we first establish Theorem 1.5 when $X$ is quasi-projective and $f\colon X\to
Y$ is algebraic. Then, by using it, we prove Theorem 1.1 under the assumption
that $X$ and $Y$ are algebraic and $f\colon X\to Y$ is projective in [FF1,
§7]. When $X$ is quasi-projective, we can use the theory of mixed Hodge
structures. Hence we can obtain desired vanishing theorems and torsion-
freeness without using the theory of variations of mixed Hodge structure (for
the details, see [Fn3, Chapter 5]). In this paper, we will directly prove
Theorems 1.1 and 1.4 with the aid of some results established for Kähler
manifolds (see [T]). Then, we will prove Theorem 1.5 as an application.
Theorem 1.4 is new even when $X$ and $Y$ are algebraic and $f\colon X\to Y$ is
projective.

By using Theorem 1.5, we have:

######  Theorem 1.7 (see [Fn9, Theorem 3.1]).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced and let $f\colon X\to Y$ be a projective morphism between complex
analytic spaces. Then we have the following properties.

  * (i)

$($Strict support condition$)$. Every associated subvariety of
$R^{q}f_{*}\omega_{X}(D)$ is the $f$-image of some stratum of $(X,D)$ for
every $q$.

  * (ii)

$($Vanishing theorem$)$. Let $\pi\colon Y\to Z$ be a projective morphism
between complex analytic spaces and let $\mathcal{A}$ be a $\pi$-ample line
bundle on $Y$. Then

$R^{p}\pi_{*}\left(\mathcal{A}\otimes R^{q}f_{*}\omega_{X}(D)\right)=0$

holds for every $p>0$ and every $q$.

  * (iii)

$($Injectivity theorem$)$. Let $\mathcal{L}$ be an $f$-semiample line bundle
on $X$. Let $s$ be a nonzero element of $H^{0}(X,\mathcal{L}^{\otimes k})$ for
some nonnegative integer $k$ such that the zero locus of $s$ does not contain
any strata of $(X,D)$. Then, for every $q$, the map

$\times s\colon R^{q}f_{*}\left(\omega_{X}(D)\otimes\mathcal{L}^{\otimes
l}\right)\to R^{q}f_{*}\left(\omega_{X}(D)\otimes\mathcal{L}^{\otimes
k+l}\right)$

induced by $\otimes s$ is injective for every positive integer $l$.

Note that Theorem 1.7 was first obtained in [Fn9, Theorem 3.1] under a weaker
assumption that $f\colon X\to Y$ is Kähler by using Saito’s theory of mixed
Hodge modules. Theorems 1.8 and 1.9 are the main results of [Fn9]. Although
they may look artificial and technical, they are very useful and indispensable
for the study of varieties and pairs whose singularities are worse than
kawamata log terminal (see [A], [Fn3, Chapter 6], [Fn6], [Fn7], [Fn10],
[Fn11], and so on). In [Fn9], we showed that Theorems 1.8 and 1.9 follow from
Theorem 1.7 (i) and (ii). Note that Theorem 1.7 (iii) is an easy consequence
of Theorem 1.7 (i) and (ii). Hence this paper gives an approach to Theorems
1.8 and 1.9 without using Saito’s theory of mixed Hodge modules.

######  Theorem 1.8 (see [Fn9, Theorem 1.1]).

Let $(X,\Delta)$ be an analytic simple normal crossing pair such that $\Delta$
is a boundary $\mathbb{R}$-divisor on $X$. Let $f\colon X\to Y$ be a
projective morphism to a complex analytic space $Y$ and let $\mathcal{L}$ be a
line bundle on $X$. Let $q$ be an arbitrary nonnegative integer. Then we have
the following properties.

  * (i)

$($Strict support condition$)$. If $\mathcal{L}-(\omega_{X}+\Delta)$ is
$f$-semiample, then every associated subvariety of $R^{q}f_{*}\mathcal{L}$ is
the $f$-image of some stratum of $(X,\Delta)$.

  * (ii)

$($Vanishing theorem$)$. If
$\mathcal{L}-(\omega_{X}+\Delta)\sim_{\mathbb{R}}f^{*}\mathcal{H}$ holds for
some $\pi$-ample $\mathbb{R}$-line bundle $\mathcal{H}$ on $Y$, where
$\pi\colon Y\to Z$ is a projective morphism to a complex analytic space $Z$,
then we have $R^{p}\pi_{*}R^{q}f_{*}\mathcal{L}=0$ for every $p>0$.

######  Theorem 1.9 (Vanishing theorem of Reid–Fukuda type, see [Fn9, Theorem
1.2]).

Let $(X,\Delta)$ be an analytic simple normal crossing pair such that $\Delta$
is a boundary $\mathbb{R}$-divisor on $X$. Let $f\colon X\to Y$ and $\pi\colon
Y\to Z$ be projective morphisms between complex analytic spaces and let
$\mathcal{L}$ be a line bundle on $X$. If
$\mathcal{L}-(\omega_{X}+\Delta)\sim_{\mathbb{R}}f^{*}\mathcal{H}$ holds such
that $\mathcal{H}$ is an $\mathbb{R}$-line bundle, which is nef and log big
over $Z$ with respect to $f\colon(X,\Delta)\to Y$, on $Y$, then
$R^{p}\pi_{*}R^{q}f_{*}\mathcal{L}=0$ holds for every $p>0$ and every $q$.

In this paper, we do not prove Theorems 1.8 and 1.9. For the details of
Theorems 1.8 and 1.9, see [Fn9]. Although the motivation of the first author
is obviously the minimal model theory for projective morphisms between complex
analytic spaces, we do not treat the minimal model program in this paper. We
recommend that the interested reader looks at [Fn8], [Fn10], [Fn11], and so
on. Theorems 1.8 and 1.9 have already played a crucial role in [Fn10] and
[Fn11], where we established the fundamental theorems of the theory of minimal
models for projective morphisms between complex analytic spaces. Anyway, by
this paper, [Fn10] and [Fn11] become free from Saito’s theory of mixed Hodge
modules. The relationship between [Fn9] and this paper is as follows.

######  Remark 1.10.

In [FFS, Corollary 1 and 4.7. Remark] (see [Fn9, Theorem 2.6]), we constructed
a weight spectral sequence of mixed Hodge modules. It is much more general
than Theorem 1.4 in some sense. By combining it with Takegoshi’s results (see
[T]), we proved Theorems 1.7, 1.8, 1.9, and so on, in [Fn9]. From the Hodge
theoretic viewpoint, one of the main ingredients of this paper is Steenbrink’s
result obtained in [St1] and [St2].

We look at the organization of this paper. In Section 2, we will briefly
explain basic definitions and results necessary for this paper. In Subsection
2.1, we will explain some useful lemmas on analytic simple normal crossing
pairs. In Subsection 2.2, we will briefly review Kollár’s package in the
complex analytic setting. In Section 3, we will present abstract arguments
which help us to obtain a variation of mixed Hodge structure whose Hodge
filtration can be extended to its canonical extension along a simple normal
crossing divisor. Section 4 is the main part of this paper, where we will
prove Theorems 1.1 and 1.4. We will also see that a generalization of the
Fujita–Zucker–Kawamata semipositivity theorem holds in the complex analytic
setting (see Theorem 1.3). In Section 5, we will prove Theorem 1.5. In Section
6, we will prove Theorem 1.7. Section 7 is a supplementary section, where we
will explain a new construction of the rational structure for the
cohomological $\mathbb{Q}$-mixed Hodge complex in [St2]. We hope that it will
help the reader to understand [St1] and [St2].

######  Acknowledgments.

The authors thank Yuta Kusakabe very much for answering their questions. The
first author was partially supported by JSPS KAKENHI Grant Numbers JP19H01787,
JP20H00111, JP21H00974, JP21H04994. The second author was partially supported
by JSPS KAKENHI Grant Number JP20K03542.

In this paper, every complex analytic space is assumed to be Hausdorff and
second-countable. Note that an irreducible and reduced complex analytic space
is called a complex variety. We will freely use the basic results on complex
analytic geometry in [BS] and [Fi].

##  2\. Preliminaries

In this section, we will collect some basic definitions. Let us start with the
definition of analytic simple normal crossing pairs.

######  Definition 2.1 (Analytic simple normal crossing pairs).

Let $X$ be a simple normal crossing divisor on a smooth complex analytic space
$M$ and let $B$ be an $\mathbb{R}$-divisor on $M$ such that the support of
$B+X$ is a simple normal crossing divisor on $M$ and that $B$ and $X$ have no
common irreducible components. Then we put $D:=B|_{X}$ and consider the pair
$(X,D)$. We call $(X,D)$ an analytic globally embedded simple normal crossing
pair and $M$ the ambient space of $(X,D)$. If the pair $(X,D)$ is locally
isomorphic to an analytic globally embedded simple normal crossing pair at any
point of $X$ and the irreducible components of $X$ and $D$ are all smooth,
then $(X,D)$ is called an analytic simple normal crossing pair.

When $(X,D)$ is an analytic simple normal crossing pair, $X$ has an invertible
dualizing sheaf $\omega_{X}$. We usually use the symbol $K_{X}$ as a formal
divisor class with an isomorphism $\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$ if
there is no danger of confusion. We note that we can not always define $K_{X}$
globally with $\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$. In general, it only
exists locally on $X$.

The notion of strata plays a crucial role.

######  Definition 2.2 (Strata).

Let $(X,D)$ be an analytic simple normal crossing pair as in Definition 2.1.
Let $\nu\colon X^{\nu}\to X$ be the normalization. We put

$K_{X^{\nu}}+\Theta=\nu^{*}(K_{X}+D).$

This means that $\Theta$ is the union of $\nu^{-1}_{*}D$ and the inverse image
of the singular locus of $X$. We note that $X^{\nu}$ is smooth and the support
of $\Theta$ is a simple normal crossing divisor on $X^{\nu}$. If $W$ is an
irreducible component of $X$ or the $\nu$-image of some log canonical center
of $(X^{\nu},\Theta)$, then $W$ is called a stratum of $(X,D)$.

######  Remark 2.3.

In this paper, $D$ is always assumed to be reduced. Hence, $\Theta$ in
Definition 2.2 is a reduced simple normal crossing divisor on $X^{\nu}$. We do
not need $\mathbb{Q}$-divisors nor $\mathbb{R}$-divisors in this paper.

We recall Siu’s theorem on complex analytic sheaves, which is a special case
of [Si, Theorem 4]. We need it for Theorem 1.7 (i) and Theorem 1.8 (i).

######  Theorem 2.4.

Let $\mathcal{F}$ be a coherent sheaf on a complex analytic space $X$. Then
there exists a locally finite family $\\{Y_{i}\\}_{i\in I}$ of complex
analytic subvarieties of $X$ such that

$\operatorname{Ass}_{\mathcal{O}_{X,x}}(\mathcal{F}_{x})=\\{\mathfrak{p}_{x,1},\ldots,\mathfrak{p}_{x,r(x)}\\}$

holds for every point $x\in X$, where
$\mathfrak{p}_{x,1},\ldots,\mathfrak{p}_{x,r(x)}$ are the prime ideals of
$\mathcal{O}_{X,x}$ associated to the irreducible components of the germs
$Y_{i,x}$ of $Y_{i}$ at $x$ with $x\in Y_{i}$. We note that each $Y_{i}$ is
called an associated subvariety of $\mathcal{F}$.

######  Definition 2.5 (Relatively nef, ample, and big line bundles).

Let $f\colon X\to Y$ be a projective morphism of complex analytic spaces and
let $\mathcal{L}$ be a line bundle on $X$. Then we say that

  * •

$\mathcal{L}$ is $f$-nef if $\mathcal{L}\cdot C\geq 0$ holds for every curve
$C$ on $X$ such that $f(C)$ is a point, and

  * •

$\mathcal{L}$ is $f$-ample if $\mathcal{L}|_{f^{-1}(y)}$ is ample in the usual
sense for every $y\in Y$.

We further assume that $f\colon X\to Y$ is a projective surjective morphism of
complex varieties. Then we say that

  * •

$\mathcal{L}$ is $f$-big if there exists some positive real number $c$ such
that $\operatorname{rank}f_{*}\mathcal{L}^{\otimes m}>c\cdot m^{d}$ holds for
$m\gg 0$, where $d=\dim X-\dim Y$.

We need the notion of nef locally free sheaves in Theorem 1.3.

######  Definition 2.6 (Nef locally free sheaves).

Let $\mathcal{E}$ be a locally free sheaf of finite rank on a projective
variety $V$. If $\mathcal{O}_{\mathbb{P}_{V}(\mathcal{E})}(1)$ is nef, that
is, $\mathcal{O}_{\mathbb{P}_{V}(\mathcal{E})}(1)\cdot C\geq 0$ holds for
every curve $C$ on $\mathbb{P}_{V}(\mathcal{E})$, then $\mathcal{E}$ is called
a nef locally free sheaf on $V$.

A nef locally free sheaf is sometimes called a semipositive vector bundle or a
semipositive locally free sheaf in the literature.

###  2.1. Lemmas on analytic simple normal crossing pairs

In this subsection, we will collect some useful lemmas on analytic simple
normal crossing pairs. We will repeatedly use these lemmas in subsequent
sections.

######  Lemma 2.7 (see [Fn9, Lemmas 2.13 and 2.15]).

Let $(X,D)$ and $(X^{\prime},D^{\prime})$ be simple normal crossing pairs such
that $D$ and $D^{\prime}$ are reduced. Let $g\colon X^{\prime}\to X$ be a
projective bimeromorphic morphism. Assume that there exists a Zariski open
subset $U$ of $X$ such that $g\colon U^{\prime}:=g^{-1}(U)\to U$ is an
isomorphism and that $U$ (resp. $U^{\prime}$) intersects every stratum of
$(X,D)$ (resp. $(X^{\prime},D^{\prime})$). Then
$R^{i}g_{*}\mathcal{O}_{X^{\prime}}=0$ and
$R^{i}g_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D^{\prime})=0$ for every
$i>0$, and $g_{*}\mathcal{O}_{X^{\prime}}\simeq\mathcal{O}_{X}$ and
$g_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D^{\prime})\simeq\mathcal{O}_{X}(K_{X}+D)$
hold.

###### Proof.

By [Fn9, Lemma 2.15], we have $R^{i}g_{*}\mathcal{O}_{X^{\prime}}=0$ for every
$i>0$ and $g_{*}\mathcal{O}_{X^{\prime}}\simeq\mathcal{O}_{X}$. Since $D$ and
$D^{\prime}$ are reduced, we can easily check that

(2.1) $K_{X^{\prime}}+D^{\prime}=g^{*}(K_{X}+D)+E$

holds for some effective $g$-exceptional Cartier divisor $E$ on $X^{\prime}$
and that $D^{\prime}=g^{-1}_{*}D$ holds. By (2.1), we have
$g_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D^{\prime})\simeq\mathcal{O}_{X}(K_{X}+D)$.
By [Fn9, Lemma 2.13], we obtain
$R^{i}g_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D^{\prime})=0$ for every
$i>0$. We finish the proof. ∎

######  Lemma 2.8 (see [Fn9, Lemma 5.1]).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced and let $f\colon X\to Y$ be a projective morphism between complex
analytic spaces. Let $L$ be a Cartier divisor on $X$. We take an arbitrary
point $P\in Y$. Then, after shrinking $Y$ around $P$ suitably, we can
construct the following commutative diagram:

---
$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Y}}$$\textstyle{\Delta^{m}}$

such that

  * (i)

$\iota_{Y}\colon Y\hookrightarrow\Delta^{m}$ is a closed embedding into a
polydisc $\Delta^{m}$ with $\iota_{Y}(P)=0\in\Delta^{m}$,

  * (ii)

$(Z,D_{Z})$ is an analytic globally embedded simple normal crossing pair such
that $D_{Z}$ is reduced,

  * (iii)

$M$ is the ambient space of $(Z,D_{Z})$ and is projective over $\Delta^{m}$,

  * (iv)

there exists a Cartier divisor $L_{Z}$ on $Z$ satisfying

$L_{Z}-(K_{Z}+D_{Z})=p^{*}(L-(K_{X}+D)),$

$p_{*}\mathcal{O}_{Z}(L_{Z})\simeq\mathcal{O}_{X}(L)$, and
$R^{i}p_{*}\mathcal{O}_{Z}(L_{Z})=0$ for every $i>0$,

  * (v)

$p(W)$ is a stratum of $(X,D)$ for every stratum $W$ of $(Z,D_{Z})$,

  * (vi)

there exists a Zariski open subset $U$ of $X$, which intersects every stratum
of $X$, such that $p$ is an isomorphism over $U$,

  * (vii)

$p$ maps every stratum of $Z$ bimeromorphically onto some stratum of $X$, and

  * (viii)

for any stratum $S$ of $(X,D)$, there exists a stratum $W$ of $(Z,D_{Z})$ such
that $S=p(W)$.

In particular, we have:

  * (ix)

$p_{*}\mathcal{O}_{Z}(K_{Z}+D_{Z})\simeq\mathcal{O}_{X}(K_{X}+D)$ and
$R^{i}p_{*}\mathcal{O}_{Z}(K_{Z}+D_{Z})=0$ holds for every $i>0$.

###### Proof.

The proof of [Fn9, Lemma 5.1], where we allow $D$ to be a boundary
$\mathbb{R}$-divisor, works without any modifications. Thus we have the
desired commutative diagram satisfying (i)–(viii). When $L=K_{X}+D$, we have
$L_{Z}=K_{Z}+D_{Z}$ by (iv). Hence we obtain (ix). ∎

######  Lemma 2.9 (see [Fn9, Lemma 2.17]).

Let $(X,D)$ be an analytic globally embedded simple normal crossing pair such
that $D$ is reduced and let $M$ be the ambient space of $(X,D)$. Let $C$ be a
stratum of $(X,D)$, which is not an irreducible component of $X$. Let
$\sigma\colon M^{\prime}\to M$ be the blow-up along $C$ and let $X^{\prime}$
denote the reduced structure of the total transform of $X$ on $M^{\prime}$. We
put

$K_{X^{\prime}}+D^{\prime}:=g^{*}(K_{X}+D),$

where $g:=\sigma|_{X^{\prime}}$. Then we have the following properties:

  * (i)

$(X^{\prime},D^{\prime})$ is an analytic globally embedded simple normal
crossing pair such that $D^{\prime}$ is reduced,

  * (ii)

$M^{\prime}$ is the ambient space of $(X^{\prime},D^{\prime})$,

  * (iii)

$g_{*}\mathcal{O}_{X^{\prime}}\simeq\mathcal{O}_{X}$ holds and
$R^{i}g_{*}\mathcal{O}_{X^{\prime}}=0$ for every $i>0$,

  * (iv)

the strata of $(X,D)$ are exactly the images of the strata of
$(X^{\prime},D^{\prime})$, and

  * (v)

$\sigma^{-1}(C)$ is a maximal (with respect to the inclusion) stratum of
$(X^{\prime},D^{\prime})$, that is, $\sigma^{-1}(C)$ is an irreducible
component of $X^{\prime}$.

###### Proof.

The proof of [Fn9, Lemma 2.17], where we allow $D$ to be a boundary
$\mathbb{R}$-divisor, works without any modifications. ∎

###  2.2. Complex analytic generalization of Kollár’s package

Here, let us briefly review Kollár’s package (see [Ko1] and [Ko2]) in the
complex analytic setting. We recommend that the interested reader looks at
[Fn2], [N3, Chapter V. 3.7. Theorem], and [T].

Theorem 2.10 is a variant of Takegoshi’s vanishing theorem (see [T, Theorem IV
Relative vanishing Theorem] and [Fn2, Corollary 1.5]). We note that it is well
known when $f\colon X\to Y$ and $\pi\colon Y\to Z$ are projective morphisms of
algebraic varieties.

######  Theorem 2.10 (Vanishing theorem).

Let $f\colon X\to Y$ and $\pi\colon Y\to Z$ be projective surjective morphisms
between complex varieties such that $X$ is smooth. Let $\mathcal{M}$ be a line
bundle on $Y$. Assume that $\mathcal{M}$ is $\pi$-nef and $\pi$-big over $Z$.
Then

(2.2) $R^{p}\pi_{*}\left(\mathcal{M}\otimes R^{q}f_{*}\omega_{X}\right)=0$

holds for every $p>0$ and every $q$. In particular, if further $\pi$ is
bimeromorphic, then

(2.3) $R^{p}\pi_{*}R^{q}f_{*}\omega_{X}=0$

holds for every $p>0$ and every $q$.

###### Proof.

The vanishing theorem (2.2) is more or less well known to the experts. For the
details, see, for example, [Fn2, Corollary 1.5]. Note that (2.3) is a special
case of (2.2). This is because the trivial line bundle on $Y$ is $\pi$-nef and
$\pi$-big when $\pi$ is bimeromorphic. ∎

Lemma 2.11 is an easy consequence of Theorem 2.10.

######  Lemma 2.11.

Let $f_{i}\colon X_{i}\to Y$ be a projective surjective morphism of complex
varieties such that $X_{i}$ is smooth for every $1\leq i\leq k$. Let
$\pi\colon Y\to Z$ be a projective bimeromorphic morphism between complex
varieties. We put

$\mathcal{F}:=\bigoplus_{i=1}^{k}R^{q_{i}}f_{i*}\omega_{X_{i}},$

where $q_{i}$ is some nonnegative integer for every $i$. Let $\mathcal{G}$ be
a coherent sheaf on $Y$. Assume that $\mathcal{G}$ is a direct summand of
$\mathcal{F}$. Then we have $R^{p}\pi_{*}\mathcal{F}=0$ and
$R^{p}\pi_{*}\mathcal{G}=0$ for every $p>0$. In particular,
$\pi_{*}\mathcal{G}$ is a direct summand of

$\bigoplus_{i=1}^{k}R^{q_{i}}(\pi\circ f_{i})_{*}\omega_{X_{i}}.$

###### Proof.

By Lemma 2.10, $R^{p}\pi_{*}R^{q_{i}}f_{i*}\omega_{X_{i}}=0$ holds for every
$p>0$. Hence we have $R^{p}\pi_{*}\mathcal{F}=0$ for every $p>0$. Since
$\mathcal{G}$ is a direct summand of $\mathcal{F}$, we obtain
$R^{p}\pi_{*}\mathcal{G}=0$ for every $p>0$. It is obvious that
$\pi_{*}\mathcal{G}$ is a direct summand of $\pi_{*}\mathcal{F}$. Since

$\pi_{*}\mathcal{F}=\bigoplus_{i=1}^{k}\pi_{*}R^{q_{i}}f_{i*}\omega_{X_{i}}\simeq\bigoplus_{i=1}^{k}R^{q_{i}}(\pi\circ
f_{i})_{*}\omega_{X_{i}},$

$\pi_{*}\mathcal{G}$ is a direct summand of
$\bigoplus_{i=1}^{k}R^{q_{i}}(\pi\circ f_{i})_{*}\omega_{X_{i}}$. We finish
the proof. ∎

Theorem 2.12 below is a special case of Takegoshi’s torsion-freeness (see [T,
Theorem II Torsion freeness Theorem] and [Fn2, Corollary 1.2]). When $f\colon
X\to Y$ is a projective surjective morphism between projective varieties, it
is nothing but Kollár’s famous torsion-freeness (see [Ko1, Theorem 2.1 (i)]).

######  Theorem 2.12 (Torsion-freeness).

Let $f\colon X\to Y$ be a projective surjective morphism of complex varieties
such that $X$ is smooth. Then $R^{q}f_{*}\omega_{X}$ is torsion-free for every
$q$.

When $f\colon X\to Y$ is algebraic, Theorem 2.13 below was first obtained
independently by Kollár (see [Ko2, Theorem 2.6]) and Nakayama (see [N2,
Theorem 1]). When $f\colon X\to Y$ is a projective morphism of smooth complex
varieties, it was obtained by Moriwaki (see [Mo, Theorem (2.4)]).

######  Theorem 2.13 (Hodge filtration, see [T, Theorem V Local freeness
Theorem (ii)] and [N3, Chapter V, 3.7. Theorem (4)]).

Let $f\colon X\to Y$ be a proper surjective morphism between smooth complex
varieties and let $\Sigma$ be a normal crossing divisor on $Y$ such that $f$
is smooth over $Y^{*}:=Y\setminus\Sigma$. We assume that $X$ is a Kähler
manifold. Then $R^{q}f_{*}\omega_{X/Y}$ is locally free and is characterized
as the upper canonical extension of the corresponding bottom Hodge filtration
on $Y^{*}$ for every $q$.

We make a remark on the proof of Theorem 2.13.

######  Remark 2.14.

One of the main ingredients of [N2] is Steenbrink’s result established in
[St1] and [St2] (see [N2, Theorem 3]). Although it was explicitly stated only
for projective morphisms, it also holds for proper morphisms from Kähler
manifolds (see Remark 4.4 below). Hence the argument in [N2] works for Kähler
manifolds with the aid of [T]. We recommend that the interested reader looks
at [N1, Conjectures 7.2 and 7.3] and [N2].

##  3\. On variations of mixed Hodge structure

In this section, we present abstract arguments which help us to obtain a
variation of mixed Hodge structure whose Hodge filtration can be extended to
its canonical extension along a simple normal crossing divisor. Although the
arguments look rather technical, they give us an appropriate viewpoint for the
proofs of Theorems 1.1 and 1.4.

######  3.1.

Let $X$ be a complex manifold and

(3.1) $K=((K_{\mathbb{R}},W),(K_{\mathcal{O}},W,F),\alpha)$

be a triple consisting of

  * •

a bounded complex of $\mathbb{R}$-sheaves $K_{\mathbb{R}}$ on $X$ equipped
with a finite increasing filtration $W$,

  * •

a bounded complex of $\mathcal{O}_{Y}$-modules $K_{\mathcal{O}}$ on $X$
equipped with a finite increasing filtration $W$ and a finite decreasing
filtration $F$, and

  * •

a morphism of complexes of $\mathbb{R}$-sheaves $\alpha\colon
K_{\mathbb{R}}\to K_{\mathcal{O}}$ preserving the filtration $W$.

Such a triple $K$ yields a pair of spectral sequences

(3.2)
$E_{r}^{p,q}(K,W)=(E_{r}^{p,q}(K_{\mathbb{R}},W),(E_{r}^{p,q}(K_{\mathcal{O}},W),F),E_{r}^{p,q}(\alpha))$

where $F$ on $E_{r}^{p,q}(K_{\mathcal{O}},W)$ stands for the first direct
filtration (cf. [D2, (1.3.8)]), that is,

(3.3)
$F^{a}E_{r}^{p,q}(K_{\mathcal{O}},W)=\operatorname{Image}(E_{r}^{p,q}(F^{a}K_{\mathcal{O}},W)\to
E_{r}^{p,q}(K_{\mathcal{O}},W))$

for $r=0,1,2,\dots,\infty$ and for $a,p,q\in\mathbb{Z}$. Here we remark that
$F$ on $E_{r}^{p,q}(K_{\mathcal{O}},W)$ above is not necessarily a filtration
by subbundles. The morphism of $E_{r}$-terms is denoted by $d_{r}^{p,q}$ for
every $p,q,r$.

######  3.2.

For a triple $K$ as in 3.1, we consider the following conditions:

  1. (3.2.1)

The triple

(3.4)
$\displaystyle(H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathbb{R}}),(H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}}),F),H^{n}(\operatorname{Gr}_{m}^{W}\alpha))$

is a variation of $\mathbb{R}$-Hodge structure of weight $n+m$ on $X$ for
every $m,n$.

  2. (3.2.2)

The spectral sequence associated to
$(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}},F)$ degenerates at $E_{1}$-terms
for every $m$.

The following lemma is a counterpart of [D3, Scholie (8.1.9)] for variations
of (mixed) Hodge structure.

######  Lemma 3.3.

If $K$ satisfies (3.2.1) and (3.2.2), then the following holds:

  1. (3.3.1)

$E_{r}^{p,q}(K,W)$ is a variation of $\mathbb{R}$-Hodge structure of weight
$q$ on $X$ for every $r=1,2,\dots,\infty$ and for every $p,q$.

  2. (3.3.2)

The triple

(3.5)
$(\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathbb{R}}),(\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}}),F),\operatorname{Gr}_{m}^{W}H^{n}(\alpha))$

is a variation of $\mathbb{R}$-Hodge structure of weight $n+m$ on $X$ for
every $m,n\in\mathbb{Z}$.

  3. (3.3.3)

The spectral sequence associated to $(K_{\mathcal{O}},W)$ degenerates at
$E_{2}$-terms.

  4. (3.3.4)

The spectral sequence associated to $(K_{\mathcal{O}},F)$ degenerates at
$E_{1}$-terms.

  5. (3.3.5)

There exists an isomorphism

(3.6)
$E_{r}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{F}^{a}E_{r}^{p,q}(K_{\mathcal{O}},W),$

under which the morphism of the $E_{r}$-terms of the left hand side coincides
with $\operatorname{Gr}_{F}^{a}d_{r}^{p,q}$ on the right hand side for every
$a,p,q,r$.

  6. (3.3.6)

The spectral sequence associated to
$(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ degenerates at $E_{2}$-terms
for every $a\in\mathbb{Z}$.

###### Proof.

From the assumption (3.2.2), the differential of
$\operatorname{Gr}_{m}^{W}K_{\mathcal{O}}$ is strictly compatible with $F$ for
every $m$. Then we obtain the conclusions from the assumption (3.2.1) by a
similar argument to [D3, Proposition (7.2.8) and Scholie (8.1.9)] and by Lemma
3.4 below. ∎

######  Lemma 3.4.

The category of the variations of $\mathbb{R}$-Hodge structure of a fixed
weight on a complex manifold is an abelian category.

###### Proof.

By Lemma 3.14 (i) of [FF1] (replaced $\mathbb{Q}$ by $\mathbb{R}$), it
suffices to prove that the kernel and the cokernel of a morphism of variations
of $\mathbb{R}$-Hodge structure satisfy the Griffiths transversality. Let
$\varphi\colon\mathbb{V}_{1}\to\mathbb{V}_{2}$ be a morphism of variations of
$\mathbb{R}$-Hodge structure of the same weight on a complex manifold $X$.
Then, from the commutative diagram

(3.7)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}\otimes\operatorname{Ker}(\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\otimes\operatorname{id}}$$\textstyle{\mathcal{O}_{X}\otimes\mathbb{V}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\otimes\operatorname{id}}$$\textstyle{\mathcal{O}_{X}\otimes\mathbb{V}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\otimes\operatorname{id}}$$\textstyle{\mathcal{O}_{X}\otimes\operatorname{Coker}(\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\otimes\operatorname{id}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X}\otimes\operatorname{Ker}(\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X}\otimes\mathbb{V}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X}\otimes\mathbb{V}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{X}\otimes\operatorname{Coker}(\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

we can easily check that the first and the last vertical arrows satisfy the
Griffiths transversality. Here we note that the Hodge filtrations on
$\mathcal{O}_{X}\otimes\operatorname{Ker}(\varphi)$ and
$\mathcal{O}_{X}\otimes\operatorname{Coker}(\varphi)$ are induced from the
Hodge filtrations on $\mathcal{O}_{X}\otimes\mathbb{V}_{1}$ and
$\mathcal{O}_{X}\otimes\mathbb{V}_{2}$ respectively. ∎

######  3.5.

Now we study the case of a pair $(X,\Sigma)$, where $X$ is a complex manifold
and $\Sigma$ a reduced simple normal crossing divisor on $X$ having finitely
many irreducible components. We set $X^{*}=X\setminus\Sigma$.

The following elementary lemma will be used several times in the present and
next sections.

######  Lemma 3.6.

Let $\mathcal{F}$ and $\mathcal{G}$ be locally free $\mathcal{O}_{X}$-modules
of finite rank on $X$ and $\varphi,\psi\colon\mathcal{F}\to\mathcal{G}$
morphisms of $\mathcal{O}_{X}$-modules. If $\varphi|_{X^{*}}=\psi|_{X^{*}}$,
then $\varphi=\psi$. In particular, if $\varphi|_{X^{*}}=0$ then $\varphi=0$.

###### Proof.

It is obvious. ∎

######  3.7.

For the case of $(X,\Sigma)$, we consider a triple

(3.8) $K=((K_{\mathbb{R}},W),(K_{\mathcal{O}},W,F),\alpha)$

consisting of

  * •

a bounded complex of $\mathbb{R}$-sheaves $K_{\mathbb{R}}$ on $X^{*}$ equipped
with a finite increasing filtration $W$,

  * •

a bounded complex of $\mathcal{O}_{X}$-modules $K_{\mathcal{O}}$ on $X$
equipped with a finite increasing filtration $W$ and a finite decreasing
filtration $F$, and

  * •

a morphism of complexes of $\mathbb{R}$-sheaves $\alpha\colon
K_{\mathbb{R}}\to K_{\mathcal{O}}|_{X^{*}}$ preserving the filtration $W$.

Note that only $(K_{\mathcal{O}},W,F)$ is defined over the whole $X$. We set

(3.9) $K|_{X^{*}}=((K_{\mathbb{R}},W),(K_{\mathcal{O}},W,F)|_{X^{*}},\alpha),$

which is a triple on $X^{*}$ considered in 3.1, and consider the following
conditions:

  1. (3.7.1)

$K|_{X^{*}}$ satisfies (3.2.1) and (3.2.2) on $X^{*}$.

  2. (3.7.2)

For every $m,n$, the $\mathbb{R}$-local system
$H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathbb{R}})$ is of quasi-unipotent local
monodromies around all the irreducible components of $\Sigma$.

  3. (3.7.3)

For every $m,n$, the variation of $\mathbb{R}$-Hodge structure on $X^{*}$ in
(3.2.1) for $K|_{X^{*}}$ is polarizable.

If $K$ satisfies these conditions, then the lower canonical extension of
$\mathcal{O}_{X^{*}}\otimes
H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathbb{R}})\simeq
H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}$ is denoted by
${}^{\ell}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}$ for every
$m,n$. By Schmid’s nilpotent orbit theorem [Sc, (4.12)], the Hodge filtration
$F$ on $H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}$ extends to a
filtration (by subbundles) on
${}^{\ell}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}$, which is
denoted by the same letter $F$. Moreover, the condition (3.2.2) for
$K|_{X^{*}}$ implies that there exists a natural isomorphism

(3.10)
$H^{n}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}\simeq\operatorname{Gr}_{F}^{a}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}$

for every $a,m,n$. Here we remark that this morphism induces the isomorphism
in (3.3.5) for $r=1$.

On the other hand, we conclude (3.3.1)–(3.3.6) for $K|_{X^{*}}$ by Lemma 3.3.
In addition, we can easily obtain the following:

  1. (3.7.4)

$H^{n}(K_{\mathbb{R}})$ is an $\mathbb{R}$-local system on $X^{*}$ of quasi-
unipotent local monodromy along $\Sigma$ for every $n$.

  2. (3.7.5)

$W_{m}H^{n}(K_{\mathcal{O}})|_{X^{*}}\simeq\mathcal{O}_{X^{*}}\otimes
W_{m}H^{n}(K_{\mathbb{R}})$ for every $m,n$. In particular, we have
$H^{n}(K_{\mathcal{O}})|_{X^{*}}\simeq\mathcal{O}_{X^{*}}\otimes
H^{n}(K_{\mathbb{R}})$ for every $n$.

  3. (3.7.6)

The variation of $\mathbb{R}$-Hodge structure $E_{r}^{p,q}(K|_{X^{*}},W)$ is
polarizable for every $p,q,r$. In particular, the variation of
$\mathbb{R}$-Hodge structure

(3.11)
$(\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathbb{R}}),(\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}}),F)|_{X^{*}},\operatorname{Gr}_{m}^{W}H^{n}(\alpha))$

is polarizable.

The following lemma and theorem play essential roles in the proofs of Theorems
1.1 and 1.4.

######  Lemma 3.8.

Let $(X,\Sigma)$ be as in 3.5 and
$K=((K_{\mathbb{R}},W),(K_{\mathcal{O}},W,F),\alpha)$ as in 3.7 satisfying the
conditions (3.7.1)–(3.7.3). For $a\in\mathbb{Z}$, if there exists an
isomorphism

(3.12)
$H^{n}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\overset{\simeq}{\longrightarrow}\operatorname{Gr}_{F}^{a}({}^{\ell}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}})$

whose restriction to $X^{*}$ coincides with the isomorphism (3.10) for every
$m,n$, then the spectral sequence associated to
$(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ degenerates at $E_{2}$-terms
and the morphism of $E_{1}$-terms splits. Moreover
$\operatorname{Gr}_{m}^{W}H^{n}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}})$ is
locally free of finite rank for every $m,n$. In particular, so is
$H^{n}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}})$ for every $n$.

###### Proof.

Because the variation of $\mathbb{R}$-Hodge structure
$E_{r}^{p,q}(K|_{X^{*}},W)$ is of quasi-unipotent local monodromy and
polarizable as mentioned in (3.7.5), we obtain a filtered
$\mathcal{O}_{X}$-module
$({}^{\ell}E_{r}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}},F)$ for every $p,q,r$ by
using Schmid’s nilpotent orbit theorem as in 3.7.

Because $d_{1}^{p,q}|_{X^{*}}$ is a morphism of variations of
$\mathbb{R}$-Hodge structure of weight $q$, we obtain a polarizable variation
of $\mathbb{R}$-Hodge structure $I^{p,q}$ of weight $q$ on $X^{*}$ by

(3.13)
$\begin{split}I^{p,q}&=(I_{\mathbb{R}}^{p,q},(I_{\mathcal{O}}^{p,q},F))\\\
&=\operatorname{Image}(d_{1}^{p-1,q}|_{X^{*}}\colon
E_{1}^{p-1,q}(K,W)|_{X^{*}}\to E_{1}^{p,q}(K,W)|_{X^{*}})\end{split}$

for every $p,q$. It is clear that $I_{\mathbb{R}}^{p,q}$ is of quasi-unipotent
local monodromy along $\Sigma$. Thus we obtain a filtered
$\mathcal{O}_{X}$-module $({}^{\ell}I_{\mathcal{O}}^{p,q},F)$ as in 3.7 again.

By the semisimplicity of the polarizable variations of $\mathbb{R}$-Hodge
structure, we have the direct sum decomposition

(3.14) $E_{1}^{p,q}(K|_{X^{*}},W)\simeq E_{2}^{p,q}(K|_{X^{*}},W)\oplus
I^{p,q}\oplus I^{p+1,q}$

as variations of $\mathbb{R}$-Hodge structure on $X^{*}$. Therefore we obtain
the direct sum decomposition

(3.15)
$({}^{\ell}E_{1}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}},F)\simeq({}^{\ell}E_{2}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}},F)\oplus({}^{\ell}I_{\mathcal{O}}^{p,q},F)\oplus({}^{\ell}I_{\mathcal{O}}^{p+1,q},F)$

as filtered $\mathcal{O}_{X}$-modules because the extension of the filtration
is unique by [FF1, Corollary 5.2]. It is clear that the morphism
${}^{\ell}d_{1}^{p,q}|_{X^{*}}$ is identified with the composite of the
projection

(3.16)
${}^{\ell}E_{2}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}}\oplus{}^{\ell}I_{\mathcal{O}}^{p,q}\oplus{}^{\ell}I_{\mathcal{O}}^{p+1,q}\to{}^{\ell}I_{\mathcal{O}}^{p+1,q}$

and the inclusion

(3.17)
${}^{\ell}I_{\mathcal{O}}^{p+1,q}\hookrightarrow{}^{\ell}E_{2}^{p+1,q}(K_{\mathcal{O}},W)|_{X^{*}}$

under the isomorphism (3.15).

We fix $a\in\mathbb{Z}$ satisfying the assumption, and consider the spectral
sequence

(3.18) $E_{r}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)\Rightarrow
H^{p+q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}})$

with the morphism of $E_{r}$-terms
$d_{r}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$. By the assumption,
there exists an isomorphism

(3.19)
$E_{1}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}E_{1}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}})$

for every $p,q$, because we have

(3.20)
$\begin{split}E_{1}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)&\simeq
H^{p+q}(\operatorname{Gr}_{-p}^{W}\operatorname{Gr}_{F}^{a}K_{\mathcal{O}})\\\
&\simeq
H^{p+q}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{-p}^{W}K_{\mathcal{O}})\\\
&\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}H^{p+q}(\operatorname{Gr}_{-p}^{W}K_{\mathcal{O}})|_{X^{*}})\\\
&\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}E_{1}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}}).\end{split}$

In particular, $E_{1}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ is
locally free of finite rank for every $p,q$. Moreover, the restriction of the
isomorphism (3.19) to $X^{*}$ identifies
$d_{1}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)|_{X^{*}}$ and
$\operatorname{Gr}_{F}^{a}d_{1}^{p,q}|_{X^{*}}$ by the assumption and by
(3.3.5) for $K|_{X^{*}}$. Then

(3.21)
$d_{1}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)=\operatorname{Gr}_{F}^{a}({}^{\ell}d_{1}^{p,q}|_{X^{*}})$

under the identification (3.19) by Lemma 3.6. Therefore there exists an
isomorphism

(3.22)
$E_{2}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}E_{2}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}})$

and $E_{2}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ is isomorphic to
a direct factor of $E_{1}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$
by the direct sum decomposition (3.15). In particular,
$E_{2}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ is locally free of
finite rank. Then $d_{2}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)=0$
because $d_{2}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)|_{X^{*}}=0$
by (3.3.6) of Lemma 3.3. Inductively,
$E_{r}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ is locally free of
finite rank, and $d_{r}^{p,q}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)=0$
for $r\geq 2$ by (3.3.6) of Lemma 3.3 again. Thus the spectral sequence (3.19)
degenerates at $E_{2}$-terms. Moreover,

(3.23)
$\operatorname{Gr}_{m}^{W}H^{n}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}})\simeq
E_{2}^{-m,n+m}(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}E_{2}^{-m,n+m}(K_{\mathcal{O}},W)|_{X^{*}})$

is locally free of finite rank for every $m,n$. ∎

######  Theorem 3.9.

Let $(X,\Sigma)$ be as in 3.5 and
$K=((K_{\mathbb{R}},W),(K_{\mathcal{O}},W,F),\alpha)$ as in 3.7 satisfying the
conditions (3.7.1)–(3.7.3). If there exists an isomorphism

(3.24)
$H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\simeq{}^{\ell}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}}$

whose restriction to $X^{*}$ is the identity for every $m,n$, then we have the
following:

  1. (3.9.1)

There exist isomorphisms

(3.25) $\displaystyle
H^{n}(K_{\mathcal{O}})\simeq{}^{\ell}H^{n}(K_{\mathcal{O}})|_{X^{*}},$ (3.26)
$\displaystyle
W_{m}H^{n}(K_{\mathcal{O}})\simeq{}^{\ell}W_{m}H^{n}(K_{\mathcal{O}})|_{X^{*}},$

whose restriction to $X^{*}$ coincide with the identities, for every
$m,n\in\mathbb{Z}$. In particular,
$\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}})$ is locally free of finite
rank on $X$ for every $m,n$.

  2. (3.9.2)

The spectral sequence associated to $(K_{\mathcal{O}},W)$ degenerates at
$E_{2}$-terms on $X$.

If we further assume that
$H^{n}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})$ is
locally free of finite rank for every $a,m,n$ and that $K_{\mathcal{O}}$
satisfies (3.2.2) on the whole $X$, then we have the following:

  1. (3.9.3)

The spectral sequence associated to $(K_{\mathcal{O}},F)$ degenerates at
$E_{1}$-terms.

  2. (3.9.4)

$\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}})$ is
a locally free $\mathcal{O}_{X}$-module of finite rank for every $a,m,n$.

  3. (3.9.5)

The spectral sequence associated to
$(\operatorname{Gr}_{F}^{a}K_{\mathcal{O}},W)$ degenerates at $E_{2}$-terms
for every $a$.

###### Proof.

We use the same notation as in the proof of Lemma 3.8.

From (3.24), we have

(3.27)
$E_{1}^{p,q}(K_{\mathcal{O}},W)\simeq{}^{\ell}E_{1}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}}$

for every $p,q$. In particular, $E_{1}^{p,q}(K_{\mathcal{O}},W)$ is locally
free of finite rank. Then, by Lemma 3.6, we have
$d_{1}^{p,q}={}^{\ell}d_{1}^{p,q}|_{X^{*}}$ under the isomorphism (3.27)
because $({}^{\ell}d_{1}^{p,q}|_{X^{*}})|_{X^{*}}=d_{1}^{p,q}|_{X^{*}}$.
Therefore

(3.28)
$E_{2}^{p,q}(K_{\mathcal{O}},W)\simeq{}^{\ell}E_{2}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}}$

by (3.15). In particular, $E_{2}^{p,q}(K_{\mathcal{O}},W)$ is locally free of
finite rank for every $p,q$. Lemma 3.6 implies $d_{2}^{p,q}=0$ because
$d_{2}^{p,q}|_{X^{*}}=0$ by (3.3.3) of Lemma 3.3. Inductively,
$E_{r}^{p,q}(K_{\mathcal{O}},W)$ is locally free of finite rank and
$d_{r}^{p,q}=0$ for $r\geq 2$. Thus we obtain (3.9.2). We have

(3.29) $\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}})\simeq
E_{2}^{-m,n+m}(K_{\mathcal{O}},W)\simeq{}^{\ell}E_{2}^{-m,n+m}(K_{\mathcal{O}},W)|_{X^{*}}\simeq{}^{\ell}(\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}})|_{X^{*}}),$

from which we obtain (3.9.1).

Next, we will prove the latter half of the theorem. By the assumption (3.2.2)
on the whole $X$, the sequence of the canonical morphisms

(3.30) $0\to H^{n}(F^{a+1}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\to
H^{n}(F^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\to
H^{n}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\to 0$

is exact for every $a,m,n$. Then we have

(3.31)
$\operatorname{Gr}_{F}^{a}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\simeq
H^{n}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})$

for every $a,m,n$, from which
$\operatorname{Gr}_{F}^{a}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})$ is
locally free of finite rank by the assumption. Thus
$F^{a}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})$ turns out to be a
subbundle for every $a$. Since the extension of the filtration $F$ on
$E_{1}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}}$ is unique by [FF1, Corollary 5.2],
the isomorphism (3.24) induces an isomorphism

(3.32)
$(H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}}),F)\simeq({}^{\ell}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}},F)$

of filtered $\mathcal{O}_{X}$-modules. In particular, we have isomorphisms

(3.33)
$H^{n}(\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\simeq\operatorname{Gr}_{F}^{a}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}H^{n}(\operatorname{Gr}_{m}^{W}K_{\mathcal{O}})|_{X^{*}})$

whose restriction to $X^{*}$ coincides with the natural isomorphism (3.10).
Thus we obtain (3.9.5) by Lemma 3.8. From (3.32), it turns out that (3.27) is,
in fact, an isomorphism of filtered $\mathcal{O}_{X}$-modules

(3.34)
$(E_{1}^{p,q}(K_{\mathcal{O}},W),F)\simeq({}^{\ell}E_{1}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}},F),$

under which $d_{1}^{p,q}={}^{\ell}d_{1}^{p,q}|_{X^{*}}$. Therefore
$d_{1}^{p,q}$ is strictly compatible with $F$ by the direct sum decomposition
(3.15). The assumption (3.2.2) implies that

(3.35) $d_{0}^{p,q}\colon E_{0}^{p,q}(K_{\mathcal{O}},W)\to
E_{0}^{p,q+1}(K_{\mathcal{O}},W)$

is strictly compatible with the filtration $F$. By (3.9.2), we already have
$d_{r}^{p,q}=0$ for all $r\geq 2$. Thus the morphism $d_{r}^{p,q}$ is strictly
compatible with the filtration $F$ for every $p,q,r$. Hence we conclude
(3.9.3) by the lemma on two filtrations (see e.g. [D3, Proposition (7.2.8)],
[PS, Theorem 3.12, 3)]).

From (3.34) and the direct sum decomposition (3.15), the isomorphism (3.28)
induces an isomorphism of filtered $\mathcal{O}_{X}$-modules

(3.36)
$(E_{2}^{p,q}(K_{\mathcal{O}},W),F_{\operatorname{rec}})\simeq({}^{\ell}E_{2}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}},F)$

where $F_{\operatorname{rec}}$ denotes the inductive filtration (la filtration
récurrente in [D2, (1.3.11)]). On the other hand, $F=F_{\operatorname{rec}}$
on $E_{2}^{p,q}(K_{\mathcal{O}},W)$ by the lemma on two filtrations again (see
e.g. [D3, Proposition (7.2.5)], [PS, Theorem 3.12, 1)]. Thus we have

(3.37)
$\operatorname{Gr}_{F}^{a}E_{2}^{p,q}(K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{F_{\operatorname{rec}}}^{a}E_{2}^{p,q}(K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}E_{2}^{p,q}(K_{\mathcal{O}},W)|_{X^{*}})$

for every $a,p,q$. In particular

(3.38) $\operatorname{Gr}_{F}^{a}E_{2}^{p,q}(K_{\mathcal{O}},W)$

is locally free of finite rank for every $a,p,q$. Moreover, the filtration $F$
on $E_{2}^{p,q}(K_{\mathcal{O}},W)$ coincides with the filtration $F$ on
$\operatorname{Gr}_{-p}^{W}H^{p+q}(K_{\mathcal{O}})$ under the isomorphism

(3.39)
$E_{2}^{p,q}(K_{\mathcal{O}},W)\simeq\operatorname{Gr}_{-p}^{W}H^{p+q}(K_{\mathcal{O}})$

by the lemma on two filtrations (see e.g. [PS, Theorem 3.12, 2)]). Thus we
obtain (3.9.4) from

(3.40)
$\operatorname{Gr}_{F}^{a}E_{2}^{-m,n+m}(K_{\mathcal{O}})\simeq\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{W}H^{n}(K_{\mathcal{O}})$

for every $a,m,n$. ∎

##  4\. Proofs of Theorems 1.1, 1.3, and 1.4

In this section, we will prove Theorems 1.1, 1.3, and 1.4. Our approach to
Theorem 1.1 (ii)–(iv) here is different from [FF1] (see also [Fn5, Section
13]) because we do not assume that $(X,D)$ is projective over $Y$ in this
section. We use the terminologies in [FF1, Section 4].

######  4.1.

First, we briefly recall several constructions and results in [FF1, Section
4], which are necessary for the proofs of Theorems 1.1 and 1.4.

Let $f\colon(X,D)\to Y$ be as in Theorems 1.1 and 1.4. Let

$X=\bigcup_{i\in I}X_{i}\quad{\text{and}}\quad
D=\bigcup_{\lambda\in\Lambda}D_{\lambda}$

be the irreducible decompositions of $X$ and $D$, respectively. Fixing orders
$<$ on $\Lambda$ and $I$, we put

$D_{k}\cap
X_{l}=\coprod_{\begin{subarray}{c}\lambda_{0}<\lambda_{1}<\cdots<\lambda_{k}\\\
i_{0}<i_{1}<\cdots<i_{l}\end{subarray}}D_{\lambda_{0}}\cap
D_{\lambda_{1}}\cap\cdots\cap D_{\lambda_{k}}\cap X_{i_{0}}\cap
X_{i_{1}}\cap\cdots\cap X_{i_{l}}$

for $k,l\geq 0$ (see [FF1, 4.14]). Here we use the convention

$\displaystyle D_{k}=D_{k}\cap
X_{-1}=\coprod_{\lambda_{0}<\lambda_{1}<\cdots<\lambda_{k}}D_{\lambda_{0}}\cap
D_{\lambda_{1}}\cap\cdots\cap D_{\lambda_{k}}$ $\displaystyle X_{l}=D_{-1}\cap
X_{l}=\coprod_{i_{0}<i_{1}<\cdots<i_{l}}X_{i_{0}}\cap X_{i_{1}}\cap\cdots\cap
X_{i_{l}}$

for $k,l\geq 0$. By setting

(4.1) $(X,D)_{n}:=(D\cap X)_{n}\setminus
D_{n}=\coprod_{\begin{subarray}{c}k+l+1=n\\\ l\geq 0\end{subarray}}D_{k}\cap
X_{l},$

we obtain an augmented semisimplicial variety
$\varepsilon\colon(X,D)_{\bullet}\to X$. Note that $(X,D)_{n}$ is the disjoint
union of all the strata of $(X,D)$ of dimension $\dim X-n$ for all
$n\in\mathbb{Z}_{\geq 0}$. We set $f_{n}:=f\varepsilon_{n}\colon(X,D)_{n}\to
Y$ for every $n$. Then $f_{n}$ is smooth over $Y^{*}=Y\setminus\Sigma$. Then
the complex $\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}}$ is given by

$(\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}})^{n}=(\varepsilon_{n})_{*}\mathbb{R}_{(X,D)_{n}}=\bigoplus_{l\geq
0}\mathbb{R}_{D_{n-l-1}\cap X_{l}}$

with the Čech type morphism $\delta$ as the differential. Note that this
complex is the single complex associated to the double complex obtained by
deleting the first vertical column of the double complex in [FF1, p.626,
4.14], and by replacing $\mathbb{Q}$ with $\mathbb{R}$. Then we have quasi-
isomorphisms

$i_{!}\mathbb{R}_{X\setminus
D}\xrightarrow{\simeq}(\mathbb{R}_{X}\to\mathbb{R}_{D_{0}}\xrightarrow{\delta}\mathbb{R}_{D_{1}}\xrightarrow{\delta}\cdots)\\\
\xrightarrow{\simeq}\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}}$

from the double complex in [FF1] mentioned above, where $i$ denotes the open
immersion $X\setminus D\hookrightarrow X$. By setting

$L_{m}(\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}})^{n}=\begin{cases}0\quad&n<-m\\\
(\varepsilon_{n})_{*}\mathbb{R}_{(X,D)_{n}}\quad&n\geq-m\end{cases}$

a finite increasing filtration $L$ is defined on
$\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}}$. We have the relative de Rham
complex $\Omega_{(X,D)_{\bullet}/Y}$ for the morphism
$f\varepsilon\colon(X,D)_{\bullet}\to Y$. Then the complex
$\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y}$ is given by

(4.2) $(\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y})^{n}=\bigoplus_{k\geq
0}(\varepsilon_{k})_{*}\Omega^{n-k}_{(X,D)_{k}/Y}$

with the differential $\delta+(-1)^{k}d$ on
$(\varepsilon_{k})_{*}\Omega^{n-k}_{(X,D)_{k}/Y}$, where $\delta$ denotes the
Čech type morphism for $(X,D)_{\bullet}$ and $d$ denotes the differential of
the relative de Rham complex $\Omega_{(X,D)_{n}/Y}$. By setting

(4.3) $\displaystyle
L_{m}(\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y})^{n}=\bigoplus_{k\geq-m}(\varepsilon_{k})_{*}\Omega^{n-k}_{(X,D)_{k}/Y}$
(4.4) $\displaystyle
F^{p}(\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y})^{n}=\bigoplus_{0\leq k\leq
n-p}(\varepsilon_{k})_{*}\Omega^{n-k}_{(X,D)_{k}/Y},$

a finite increasing filtration $L$ and a finite decreasing filtration $F$ on
$\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y}$ are defined. The canonical
morphism $\mathbb{R}_{(X,D)_{n}}\to\mathcal{O}_{(X,D)_{n}}$ induces a morphism
of complexes
$\iota\colon\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}}\to\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y}$.

By setting

(4.5) $\begin{split}K&=((K_{\mathbb{R}},L),(K_{\mathcal{O}},L,F),\alpha)\\\
&=((Rf_{*}\varepsilon_{*}\mathbb{R}_{(X,D)_{\bullet}},L)|_{Y^{*}},(Rf_{*}\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y},L,F),Rf_{*}\iota|_{Y^{*}})\end{split}$

(see [FF1, 4.1]), we obtain a triple $K$ as in 3.7 satisfying the following:

  1. (4.1.1)

There exists a quasi-isomorphism $R(f|_{X^{*}\setminus
D^{*}})_{!}\mathbb{R}_{X^{*}\setminus D^{*}}\simeq K_{\mathbb{R}}$.

  2. (4.1.2)

There exists a quasi-isomorphism
$\operatorname{Gr}_{F}^{p}K_{\mathcal{O}}\simeq
Rf_{*}\varepsilon_{*}\Omega^{p}_{(X,D)_{\bullet}/Y}[-p]$. for every $p$. In
particular,
$Rf_{*}\mathcal{O}_{X}(-D)\simeq\operatorname{Gr}_{F}^{0}K_{\mathcal{O}}$.

  3. (4.1.3)

For every $m\in\mathbb{Z}$,

$\begin{split}\operatorname{Gr}_{m}^{L}K&=(\operatorname{Gr}_{m}^{L}K_{\mathbb{R}},(\operatorname{Gr}_{m}^{L}K_{\mathcal{O}},F),\operatorname{Gr}_{m}^{L}\alpha)\\\
&\simeq\bigoplus_{S}(R(f_{S^{*}})_{*}\mathbb{R}_{S^{*}}[m],(R(f_{S})_{*}\Omega_{S/Y}[m],F),R(f_{S^{*}})_{*}\iota_{S^{*}}[m]),\end{split}$

where $S$ runs through all $(\dim X+m)$-dimensional strata of $(X,D)$,
$f_{S}=f|_{S}$, $S^{*}=f_{S}^{-1}(Y^{*})$, $f_{S^{*}}=(f_{S})|_{S^{*}}$, and
$\iota_{S^{*}}$ is the composite
$\mathbb{R}_{S^{*}}\hookrightarrow\mathbb{C}_{S^{*}}\to\Omega_{S^{*}/Y^{*}}$.

First, we will prove Theorem 1.4.

###### Proof of Theorem 1.4.

We use the notations and terminologies in 4.1. We will prove that the spectral
sequence

(4.6) $E_{r}^{p,q}(\operatorname{Gr}_{F}^{0}K_{\mathcal{O}},L)\Rightarrow
H^{p+q}(\operatorname{Gr}_{F}^{0}K_{\mathcal{O}})$

associated to the filtered complex
$(\operatorname{Gr}_{F}^{0}K_{\mathcal{O}},L)$ satisfies the desired
properties.

By (4.1.2), we have an isomorphism

(4.7) $H^{p+q}(\operatorname{Gr}_{F}^{0}K_{\mathcal{O}})\simeq
R^{p+q}f_{*}\mathcal{O}_{X}(-D)$

as desired.

By (4.1.3), $K$ satisfies (3.7.1)–(3.7.3) because $f_{S}\colon S\to Y$ is
smooth over $Y^{*}$ for every stratum $S$ of $(X,D)$. Moreover, we have

(4.8)
$H^{n}(\operatorname{Gr}_{F}^{0}\operatorname{Gr}_{m}^{L}K_{\mathcal{O}})\simeq\bigoplus_{S}R^{n+m}(f_{S})_{*}\mathcal{O}_{S}$

where $S$ runs through all $(\dim X+m)$-dimensional strata of $(X,D)$.
Therefore $K$ satisfies the assumption of Lemma 3.8 by the dual of Theorem
2.13. Thus we obtain the conclusion by Lemma 3.8. ∎

Next, we will prove Theorem 1.1 (i).

###### Proof of Theorem 1.1 (i).

As already mentioned above, $K$ satisfies the conditions (3.7.1)–(3.7.3). By
applying Lemma 3.3 together with (3.7.4)–(3.7.6), the triple

(4.9)
$(\operatorname{Gr}_{m}^{L}H^{k}(K_{\mathbb{R}}),(\operatorname{Gr}_{m}^{L}H^{k}(K_{\mathcal{O}}),F)|_{Y^{*}},\operatorname{Gr}_{m}^{L}H^{k}(\alpha))$

is a polarizable variation of $\mathbb{R}$-Hodge structure of weight $k+m$ for
every $k,m$. By (4.1.1), we have $R^{k}(f|_{X^{*}\setminus
D^{*}})_{!}\mathbb{R}_{X\setminus D}\simeq H^{k}(K_{\mathbb{R}})$, which
implies $\mathcal{V}^{k}_{Y^{*}}\simeq H^{k}(K_{\mathcal{O}})|_{Y^{*}}$ as in
(3.7.5) for all $k$. By using these isomorphism, we introduce filtrations $L$
on $R^{k}(f|_{X^{*}\setminus D^{*}})_{!}\mathbb{R}_{X\setminus D}$ and
$\mathcal{V}^{k}_{Y^{*}}$, $F$ on $\mathcal{V}^{k}_{Y^{*}}$, and then obtain a
polarizable variation of $\mathbb{R}$-Hodge structure

(4.10) $(\operatorname{Gr}_{m}^{L[k]}R^{k}(f|_{X^{*}\setminus
D^{*}})_{!}\mathbb{R}_{X\setminus
D},(\operatorname{Gr}_{m}^{L[k]}\mathcal{V}^{k}_{Y^{*}},F),\operatorname{Gr}_{m}^{L[k]}\beta)$

of weight $m$ on $Y^{*}$ for every $k,m$, where the natural morphism
$R^{k}(f|_{X^{*}\setminus D^{*}})_{!}\mathbb{R}_{X\setminus
D}\to\mathcal{V}^{k}_{Y^{*}}=R^{k}(f|_{X^{*}\setminus
D^{*}})_{!}\mathbb{R}_{X\setminus D}\otimes\mathcal{O}_{Y^{*}}$ is denoted by
$\beta$.

On the other hand, the Griffiths transversality for

(4.11)
$(\mathcal{V}^{k}_{Y^{*}},F)\simeq(H^{k}(K_{\mathcal{O}}),F)|_{Y^{*}}\simeq(R^{k}f_{*}\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y},F)|_{Y^{*}}$

can be easily seen by the same way as in the proof of Lemma 4.5 of [FF1].
Therefore

(4.12) $((R^{k}(f|_{X^{*}\setminus D^{*}})_{!}\mathbb{R}_{X\setminus
D},L[k]),(\mathcal{V}^{k}_{Y^{*}},L[k],F),\beta)$

is a graded polarizable variation of $\mathbb{R}$-mixed Hodge structure on
$Y^{*}$ as desired. ∎

In order to prove Theorem 1.1 (ii)–(iv), we recall results in [St1] and [St2]
in a slightly generalized form.

######  Definition 4.2.

Let $f\colon X\to Y$ be a surjective morphism of smooth complex varieties and
$\Sigma$ a simple normal crossing divisor on $Y$. We assume that
$E=(f^{*}\Sigma)_{\operatorname{red}}$ is a simple normal crossing divisor on
$X$. For such $f$, we set

$\Omega^{1}_{X/Y}(\log
E)=\operatorname{Coker}(f^{*}\Omega^{1}_{Y}(\log\Sigma)\to\Omega^{1}_{X}(\log
E))$

and

$\Omega^{p}_{X/Y}(\log E)=\bigwedge^{p}\Omega^{1}_{X/Y}(\log E)$

for every $p$. An $f^{-1}\mathcal{O}_{Y}$-differential
$d\colon\Omega^{p}_{X/Y}(\log E)\to\Omega^{p+1}_{X/Y}(\log E)$ can be uniquely
defined by the commutative diagram

$\textstyle{\Omega^{p}_{X}(\log
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega^{p}_{X/Y}(\log
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega^{p+1}_{X}(\log
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{p+1}_{X/Y}(\log
E),}$

where the horizontal arrows are the canonical surjections induced from the
surjection $\Omega^{1}_{X}(\log E)\to\Omega^{1}_{X/Y}(\log E)$. Thus we obtain
a complex of $f^{-1}\mathcal{O}_{Y}$-modules $\Omega_{X/Y}(\log E)$, which is
called the relative log de Rham complex of $f$.

######  Lemma 4.3.

Let $f\colon X\to Y$ be a proper surjective morphism from a Kähler manifold
$X$ to a smooth complex variety $Y$. Assume that there exists a smooth divisor
$\Sigma$ on $Y$ such that

  1. (4.3.1)

$f$ is smooth over $Y^{*}=Y\setminus\Sigma$,

  2. (4.3.2)

$E=(f^{*}\Sigma)_{\operatorname{red}}$ is a simple normal crossing divisor on
$X$ having finitely many irreducible components, and

  3. (4.3.3)

$\Omega_{X/Y}^{1}(\log E)$ is a locally free $\mathcal{O}_{X}$-module of
finite rank.

Then we have

$R^{i}f_{*}\Omega_{X/Y}(\log E)\simeq{}^{l}(R^{i}f_{*}\Omega_{X/Y}(\log
E)|_{Y^{*}})\simeq{}^{l}(\mathcal{O}_{Y^{*}}\otimes(R^{i}f_{*}\mathbb{C}_{X})|_{Y^{*}})$

for all $i$, where ${}^{l}(\cdot)$ stands for the lower canonical extension as
before. In particular, $R^{i}f_{*}\Omega_{X/Y}(\log E)$ is a locally free
$\mathcal{O}_{Y}$-module of finite rank for all $i$. Moreover,
$R^{i}f_{*}\Omega_{X/Y}^{p}(\log E)$ is also a locally free
$\mathcal{O}_{Y}$-module of finite rank, and the stupid filtration
$($filtration bête in [D2, (1.4.7)]$)$ $F$ on $\Omega_{X/Y}(\log E)$ induces
the natural exact sequence

(4.13) $0\to R^{i}f_{*}F^{p+1}\Omega_{X/Y}(\log E)\to
R^{i}f_{*}F^{p}\Omega_{X/Y}(\log E)\to R^{i}f_{*}\Omega^{p}_{X/Y}(\log E)\to
0$

for all $i,p$.

###### Proof.

We may assume $Y=\Delta^{k}$ with the coordinates $t_{1},\dots,t_{k}$ and
$\Sigma=\\{t_{1}=0\\}$. For any $x\in E$, we can take local coordinates
$x_{1},\dots,x_{n}$ centered at $x$ on $X$ with

(4.14) $f^{*}t_{1}=x_{1}^{a_{1}}\cdots x_{l}^{a_{l}}$

for some $a_{1},\dots,a_{l}\in\mathbb{Z}_{>0}$ by (4.3.2). We set
$f_{i}=f^{*}t_{i}$ for $i=2,\dots,k$. On the other hand, we have the canonical
exact sequence

(4.15) $0\to
f^{*}\Omega_{Y}^{1}(\log\Sigma)_{x}\otimes\mathbb{C}(x)\to\Omega_{X}^{1}(\log
E)_{x}\otimes\mathbb{C}(x)\to\Omega_{X/Y}^{1}(\log
E)_{x}\otimes\mathbb{C}(x)\to 0,$

where $\mathbb{C}(x)$ denotes the residue field at $x$, because
$\Omega_{X/Y}^{1}(\log E)$ is a locally free $\mathcal{O}_{X}$-module of rank
$\dim X-\dim Y$ by (4.3.3). Under the isomorphisms

(4.16)
$\displaystyle\Omega^{1}_{Y}(\log\Sigma)\simeq\mathcal{O}_{Y}\frac{dt_{1}}{t_{1}}\oplus(\bigoplus_{i=2}^{k}\mathcal{O}_{Y}dt_{i}),$
(4.17) $\displaystyle\Omega^{1}_{X}(\log
E)\simeq(\bigoplus_{i=1}^{l}\mathcal{O}_{X}\frac{dx_{i}}{x_{i}})\oplus(\bigoplus_{i=l+1}^{n}\mathcal{O}_{X}dx_{i})$

the morphism
$f^{*}\Omega_{Y}^{1}(\log\Sigma)_{x}\otimes\mathbb{C}(x)\to\Omega_{X}^{1}(\log
E)_{x}\otimes\mathbb{C}(x)$ is represented by the matrix

(4.18) $\left(\begin{array}[]{ccc|ccc}a_{1}&\dots&a_{l}&0&\dots&0\\\ \hline\cr
0&\dots&0&&&\\\
\vdots&\ddots&\vdots&\lx@intercol\hfil{\displaystyle\frac{\partial
f_{i}}{\partial x_{j}}(0)}\hfil\lx@intercol\\\ 0&\dots&0&&&\end{array}\right)$

where $i$ and $j$ run through $2,\dots,k$ and $l+1,\dots,n$ respectively. The
exactness of (4.15) implies that the matrix (4.18) is of rank $k$, and then we
may assume

(4.19) $\operatorname{rank}\left(\frac{\partial f_{i}}{\partial
x_{j}}(0)\right)_{2\leq i\leq k,l+1\leq j\leq l+k-1}=k-1$

by changing the order of $x_{l+1},\dots,x_{n}$. Replacing
$x_{l+1},\dots,x_{l+k-1}$ by $f_{2},\dots,f_{k}$, we obtain a new local
coordinates $(x_{1},\dots,x_{n})$ at $x$, under which the morphism $f$ is
given in the form

(4.20) $t_{1}=x_{1}^{a_{1}}\cdots
x_{l}^{a_{l}},t_{2}=x_{l+1},\dots,t_{k}=x_{l+k-1}$

around $x$. We set $f_{s}\colon X_{s}\to\Delta=\Delta\times\\{s\\}$ by the
Cartesian square

$\textstyle{X_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{s}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

for any $s=(t_{2},\dots,t_{k})\in\Delta^{k-1}$. Then $X_{s}$ is smooth,
$f_{s}$ is smooth over $\Delta^{*}=\Delta\setminus\\{0\\}$ and
$\operatorname{Supp}f_{s}^{-1}(0)$ is a simple normal crossing divisor on
$X_{s}$ by the local description (4.20). Hence
$R^{i}(f_{s})_{*}\Omega_{X_{s}/\Delta}(\log(E\cap X_{s}))$ and
$R^{i}(f_{s})_{*}\Omega^{p}_{X_{s}/\Delta}(\log(E\cap X_{s}))$ are locally
free of finite rank for every $i,p$ by [St1, (2.18) Theorem] and by [St2,
(2.11) Theorem]. Therefore $R^{i}f_{*}\Omega_{X/Y}(\log E)$ and
$R^{i}f_{*}\Omega_{X/Y}^{p}(\log E)$ are locally free
$\mathcal{O}_{Y}$-modules of finite rank for all $i,p$ by the base change
theorem. Once we know that $R^{i}f_{*}\Omega_{X/Y}(\log E)$ is locally free,
it is the lower canonical extension of its restriction to
$Y^{*}=Y\setminus\Sigma$ by [St1, (2.20) Proposition].

Next, we consider the spectral sequence

(4.21) $E_{r}^{p,q}(Rf_{*}\Omega_{X/Y}(\log E),F)\Rightarrow
E^{p+q}(Rf_{*}\Omega_{X/Y}(\log E))=R^{p+q}f_{*}\Omega_{X/Y}(\log E)$

and denote the morphism of $E_{r}$-terms by

$d_{r}^{p,q}\colon E_{r}^{p,q}(Rf_{*}\Omega_{X/Y}(\log E),F)\to
E_{r}^{p+r,q-r+1}(Rf_{*}\Omega_{X/Y}(\log E),F)$

for a while. Then $d_{r}^{p,q}|_{Y^{*}}=0$ for all $p,q$ and $r\geq 1$ because
the restriction of this spectral sequence to $Y^{*}$ degenerates at
$E_{1}$-terms. Since

$E_{1}^{p,q}(Rf_{*}\Omega_{X/Y}(\log E),F)\simeq
R^{q}f_{*}\Omega^{p}_{X/Y}(\log E)$

is a locally free $\mathcal{O}_{Y}$-module of finite rank for all $p,q$, we
have $d_{1}^{p,q}=0$ for all $p,q$ by Lemma 3.6. This implies that

$E_{2}^{p,q}(Rf_{*}\Omega_{X/Y}(\log E),F)\simeq
E_{1}^{p,q}(Rf_{*}\Omega_{X/Y}(\log E),F)$

is locally free for all $p,q$ and that $d_{2}^{p,q}=0$ for all $p,q$ by Lemma
3.6 again. Inductively, we obtain $d_{r}^{p,q}=0$ for all $p,q$ and $r\geq 1$.
Thus the spectral sequence (4.21) degenerates at $E_{1}$-terms, or
equivalently, (4.13) is exact. ∎

######  Remark 4.4.

In [St2], $f_{s}$ is assumed to be a projective morphism. However, we can
check that the proof of (2.11) Theorem in [St2] is also valid to a proper
morphism from a Kähler manifold by using results in [PS, I.2.5 Almost Kähler
$V$-manifolds]. See also Theorem 7.9 below.

######  Corollary 4.5.

In the situation of Lemma 4.3, we have the canonical isomorphisms

$\displaystyle R^{i}f_{*}F^{p}\Omega_{X/Y}(\log E)\simeq
F^{p}R^{i}f_{*}\Omega_{X/Y}(\log E),$ $\displaystyle
R^{i}f_{*}\Omega^{p}_{X/Y}(\log
E)\simeq\operatorname{Gr}_{F}^{p}R^{i}f_{*}\Omega_{X/Y}(\log E)$

for all $i,p$. In particular, $F^{p}R^{i}f_{*}\Omega_{X/Y}(\log E)$ is a
subbundle of $R^{i}f_{*}\Omega_{X/Y}(\log E)$.

######  Lemma 4.6.

Let $f\colon X\to Y$ be a proper surjective morphism between smooth complex
varieties. Assume that there exists a smooth divisor $\Sigma$ such that

  * •

$f$ is smooth over $Y^{*}=Y\setminus\Sigma$, and

  * •

$E=(f^{*}\Sigma)_{\operatorname{red}}$ is a simple normal crossing divisor on
$X$ having finitely many irreducible components.

Then there exists a closed analytic subset $\Sigma_{0}\subset\Sigma$ with
$\dim\Sigma_{0}\leq\dim Y-2$, such that $\Omega^{1}_{X/Y}(\log E)$ is locally
free on $f^{-1}(Y\setminus\Sigma_{0})$.

###### Proof.

We may assume that $\Sigma$ is irreducible. Let $E=\sum_{i=1}^{N}E_{i}$ be the
irreducible decomposition of $E$. For a nonempty subset
$I\subset\\{1,\dots,N\\}$, we set $E_{I}=\bigcap_{i\in I}E_{i}$, which is a
smooth closed subvariety of $X$. If $f(E_{I})\not=\Sigma$, we set
$\Sigma_{I}=f(E_{I})$, which is a closed analytic subset of $\Sigma$. If
$f(E_{I})=\Sigma$, then there exists a closed analytic subset
$\Sigma_{I}\subsetneq\Sigma$ such that $f|_{E_{I}}\colon E_{I}\to\Sigma$ is
smooth over $\Sigma\setminus\Sigma_{I}$. We are going to check that the closed
analytic subset

$\Sigma_{0}:=\bigcup_{\emptyset\not=I\subset\\{1,\dots,N\\}}\Sigma_{I}$

satisfies the desired property. We have $\Sigma_{0}\not=\Sigma$, by
definition. Therefore $\dim\Sigma_{0}\leq\dim Y-2$ because $\Sigma$ is
irreducible. Then, it suffices to prove that $\Omega^{1}_{X/Y}(\log E)$ is
locally free on $f^{-1}(Y\setminus\Sigma_{0})$. A point $x\in E\cap
f^{-1}(Y\setminus\Sigma_{0})$ defines a nonempty subset
$I\subset\\{1,\dots,N\\}$ by $I=\\{i\mid x\in E_{i}\\}$. Then $x\in E_{I}$ and
$f(E_{I})=\Sigma$. Take local coordinates $x_{1},\dots,x_{n}$ and
$t_{1},\dots,t_{k}$ centered at $x$ and $f(x)$ on $X$ and $Y$ respectively,
satisfying the following conditions:

  * •

$\Sigma=\\{t_{1}=0\\}$ on $Y$, and

  * •

$f^{*}t_{1}=x_{1}^{a_{1}}\cdots x_{l}^{a_{l}}$ for some
$a_{1},\dots,a_{l}\in\mathbb{Z}_{>0}$.

We set $f_{i}=f^{*}t_{i}$ for $i=2,\dots,k$. Then
$E_{I}=\\{x_{1}=\dots=x_{l}=0\\}$ and the morphism
$(f|_{E_{I}})^{*}\Omega^{1}_{\Sigma}\to\Omega^{1}_{E_{I}}$ is represented by
the matrix

$\left(\frac{\partial f_{i}}{\partial
x_{j}}(0,\dots,0,x_{l+1},\dots,x_{n})\right)_{2\leq i\leq k,l+1\leq j\leq n}$

via the isomorphisms
$(f|_{E_{I}})^{*}\Omega^{1}_{\Sigma}\simeq\bigoplus_{j=2}^{k}\mathcal{O}_{E_{I}}f^{*}dt_{j}$
and $\Omega^{1}_{E_{I}}\simeq\bigoplus_{i=l+1}^{n}\mathcal{O}_{E_{I}}dx_{i}$.
Since $x\in f^{-1}(\Sigma\setminus\Sigma_{I})$, the morphism $f|_{E_{I}}$ is
smooth at $x$. Then

$\operatorname{rank}\left(\frac{\partial f_{i}}{\partial
x_{j}}(0)\right)_{2\leq i\leq k,l+1\leq j\leq n}=k-1,$

which implies that the matrix (4.18) in the proof of Lemma 4.3 is of rank $k$.
Therefore the canonical morphism
$f^{*}\Omega^{1}_{Y}(\log\Sigma)_{x}\otimes\mathbb{C}(x)\to\Omega^{1}_{X}(\log
E)_{x}\otimes\mathbb{C}(x)$ is injective, by which we conclude that
$\Omega^{1}_{X/Y}(\log E)$ is locally free around $x$. ∎

######  4.7.

We return to the situation in 4.1. For the moment, we assume that there exist
another semisimplicial variety $Z_{\bullet}$ and a morphism of semisimplicial
varieties $\sigma\colon Z_{\bullet}\to(X,D)_{\bullet}$ satisfying the
conditions

  * •

$Z_{n}$ is smooth and Kähler,

  * •

$\sigma_{n}\colon Z_{n}\to(X,D)_{n}$ is a projective surjective morphism,

  * •

for $g_{n}:=f_{n}\sigma_{n}=f\varepsilon_{n}\sigma_{n}\colon Z_{n}\to Y$, the
divisor $E_{n}:=(g_{n}^{*}\Sigma)_{\operatorname{red}}$ is a simple normal
crossing divisor on $Z_{n}$ having finitely many irreducible components, and

  * •

$\sigma_{n}\colon Z_{n}\to(X,D)_{n}$ is isomorphic over $Y^{*}$

for every $n\in\mathbb{Z}_{\geq 0}$. We obtain an augmentation $\eta\colon
Z_{\bullet}\to X$ by setting $\eta=\varepsilon\sigma$. The relative log de
Rham complex of $Z_{n}$ over $Y$ is denoted by $\Omega_{Z_{n}/Y}(\log E_{n})$.
Then $\\{\Omega_{Z_{n}/Y}(\log E_{n})\\}_{n\in\mathbb{Z}_{\geq 0}}$ forms a
complex on the semisimplicial variety $Z_{\bullet}$.

For an augmentation of a semisimplicial variety, we can define the direct
image functor as in [FF1, 4.1, 4.2] (for the detail, see e.g. [D3, 5.1, 5.2],
[PS, 5.1.2]). The complex $R\varepsilon_{*}\Omega_{(X,D)_{\bullet}}$ is
isomorphic to $\varepsilon_{*}\Omega_{(X,D)_{\bullet}}$ defined in the proof
of Theorem 1.1 (i) in the derived category because
$\varepsilon_{n}\colon(X,D)_{n}\to X$ is a finite morphism for all $n$. On the
other hand, we obtain a complex $R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet})$ on $X$. Here, we briefly recall the definitions of this complex,
of the finite increasing filtration $L$, and of the finite decreasing
filtration $F$ on it. First, the complex $R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet})$ is given as the total single complex associated to the double
complex

---
$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(R(\eta_{p})_{*}\Omega_{Z_{p}/Y}(\log
E_{p}))^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{(-1)^{p}d}$$\textstyle{(R(\eta_{p+1})_{*}\Omega_{Z_{p+1}/Y}(\log
E_{p+1}))^{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1)^{p+1}d}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(R(\eta_{p})_{*}\Omega_{Z_{p}/Y}(\log
E_{p}))^{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{(R(\eta_{p+1})_{*}\Omega_{Z_{p+1}/Y}(\log
E_{p+1}))^{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$

that is,

$(R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet}))^{n}=\bigoplus_{p}(R(\eta_{p})_{*}\Omega_{Z_{p}/Y}(\log
E_{p}))^{n-p},$

where $R(\eta_{p})_{*}\Omega_{Z_{p}/Y}(\log E_{p})$ is regarded as a genuine
complex on $X$ by using the Godement resolutions (cf. [FF1, 4.1]). The
filtrations $L$ and $F$ are defined by

$\displaystyle L_{m}(R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet}))^{n}=\bigoplus_{p\geq-m}(R(\eta_{p})_{*}\Omega_{Z_{p}/Y}(\log
E_{p}))^{n-p},$ $\displaystyle F^{r}(R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet}))^{n}=\bigoplus_{p}F^{r}(R(\eta_{p})_{*}\Omega_{Z_{p}/Y}(\log
E_{p}))^{n-p}$

for all $m,n,r$. Therefore we have

(4.22) $(\operatorname{Gr}_{m}^{L}R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet}),F)\simeq(R(\eta_{-m})_{*}\Omega_{Z_{-m}/Y}(\log E_{-m})[m],F)$

in the derived category. From the morphism $\sigma\colon
Z_{\bullet}\to(X,D)_{\bullet}$, we obtain a morphism of bifiltered complexes

(4.23)
$(\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y},L,F)\to(R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet}),L,F),$

which induces a morphism

(4.24)
$\begin{split}\operatorname{Gr}_{m}^{L}\operatorname{Gr}_{F}^{0}\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y}&\simeq(\varepsilon_{-m})_{*}\mathcal{O}_{(X,D)_{-m}}\\\
&\to
R(\eta_{-m})_{*}\mathcal{O}_{Z_{-m}}\simeq\operatorname{Gr}_{m}^{L}\operatorname{Gr}_{F}^{0}R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet})\end{split}$

for all $m$. Because $\sigma_{n}$ induces the isomorphism
$\mathcal{O}_{(X,D)_{n}}\xrightarrow{\simeq}R(\sigma_{n})_{*}\mathcal{O}_{Z_{n}}$
for all $n$, we have the isomorphisms

$(\varepsilon_{-m})_{*}\mathcal{O}_{(X,D)_{-m}}\simeq
R(\varepsilon_{-m})_{*}\mathcal{O}_{(X,D)_{-m}}\simeq
R(\varepsilon_{-m})_{*}R(\sigma_{-m})_{*}\mathcal{O}_{Z_{-m}}\simeq
R(\eta_{-m})_{*}\mathcal{O}_{Z_{-m}}$

for all $m$. Therefore the morphism (4.24) is an isomorphism for all $m$ in
the derived category, which implies

(4.25)
$(\operatorname{Gr}_{F}^{0}\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y},L)\simeq(\operatorname{Gr}_{F}^{0}R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet}),L)$

in the filtered derived category.

Now, we complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1 (ii)–(iv).

First, we prove (ii). The uniqueness of the filtration $F$ on
${}^{l}\mathcal{V}^{k}_{Y^{*}}$ follows from [FF1, Corollary 5.2]. Therefore
we may work locally on $Y$. Then after shrinking $Y$ to a relatively compact
open subset, we can take $Z_{\bullet}$ and $\sigma_{\bullet}\colon
Z_{\bullet}\to(X,D)_{\bullet}$ in 4.7 by the theorem of resolution of
singularities (see [BM, Section 13]). By Lemma 4.6, there exists a closed
analytic subset $\Sigma_{0}\subset\Sigma$ with $\dim\Sigma_{0}\leq\dim Y-2$
such that $\Sigma\setminus\Sigma_{0}$ is a smooth divisor in
$Y\setminus\Sigma_{0}$, and that $\Omega^{1}_{Z_{n}/Y}(\log E_{n})$ is locally
free over $g_{n}^{-1}(Y\setminus\Sigma_{0})$ for all $n\in\mathbb{Z}_{\geq
0}$. By setting $Y_{0}:=Y\setminus\Sigma_{0}$, we trivially have $Y^{*}\subset
Y_{0}\subset Y$.

Now we set

$K(\log)_{\mathcal{O}}=Rf_{*}R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet})$

equipped with the induced filtrations $L$ and $F$. Because $\sigma$ is
isomorphic over $Y^{*}$, the morphism (4.23) induces an isomorphism

(4.26)
$(K_{\mathcal{O}},L,F)|_{Y^{*}}\xrightarrow{\simeq}(K(\log)_{\mathcal{O}},L,F)|_{Y^{*}},$

and then a morphism of complexes $\beta\colon K_{\mathbb{R}}\to
K(\log)_{\mathcal{O}}|_{Y^{*}}$ is defined by

(4.27) $\beta\colon
K_{\mathbb{R}}\xrightarrow{\alpha}K_{\mathcal{O}}|_{Y^{*}}\xrightarrow{\simeq}K(\log)_{\mathcal{O}}|_{Y^{*}},$

where $K_{\mathbb{R}}$ and $\alpha$ are given in (4.5). A triple

(4.28) $K(\log)=((K_{\mathbb{R}},L),(K(\log)_{\mathcal{O}},L,F),\beta)$

satisfies the conditions (3.7.1)–(3.7.3) because $K|_{Y^{*}}\simeq
K(\log)|_{Y^{*}}$ as above. Then the triple on $Y_{0}$

(4.29)
$K(\log)|_{Y_{0}}=((K_{\mathbb{R}},L),(K(\log)_{\mathcal{O}},L,F)|_{Y_{0}},\beta)$

satisfies all the assumptions in Theorem 3.9 by (4.22) and Lemma 4.3. Applying
Theorem 3.9 to $K(\log)|_{Y_{0}}$, we conclude

(4.30) $\displaystyle
H^{n}(K(\log)_{\mathcal{O}})|_{Y_{0}}\simeq({}^{\ell}H^{n}(K(\log)_{\mathcal{O}})|_{Y^{*}})|_{Y_{0}}\simeq({}^{\ell}H^{n}(K_{\mathcal{O}})|_{Y^{*}})|_{Y_{0}}$
(4.31) $\displaystyle
L_{m}H^{n}(K(\log)_{\mathcal{O}})|_{Y_{0}}\simeq({}^{\ell}L_{m}H^{n}(K(\log)_{\mathcal{O}})|_{Y^{*}})|_{Y_{0}}\simeq({}^{\ell}L_{m}H^{n}(K_{\mathcal{O}})|_{Y^{*}})|_{Y_{0}}$

from (4.26), and that
$\operatorname{Gr}_{F}^{a}\operatorname{Gr}_{m}^{L}H^{n}(K(\log)_{\mathcal{O}})|_{Y_{0}}$
is locally free of finite rank for every $a,m,n$. By the isomorphism
$\mathcal{V}^{k}_{Y^{*}}\simeq H^{k}(K_{\mathcal{O}})|_{Y^{*}}$ as in the
proof of Theorem 1.1 (i) above, we obtain a filtration $F$ on
$({}^{\ell}\mathcal{V}^{k}_{Y^{*}})|_{Y_{0}}$ satisfying the two conditions in
Theorem 1.1 (ii) on $Y_{0}$. Then Lemma 1.11.2 in [Ka] together with Schmid’s
nilpotent orbit theorem (see [Sc, (4.12)]) for each
$\operatorname{Gr}_{m}^{L}\mathcal{V}^{k}_{Y^{*}}$ implies the conclusion of
Theorem 1.1 (ii) on the whole $Y$.

Next, we will prove (iii). By (3.9.3) of Theorem 3.9 for $K(\log)|_{Y_{0}}$,
we have

(4.32)
$H^{k}(\operatorname{Gr}_{F}^{a}K(\log)_{\mathcal{O}})|_{Y_{0}}\simeq\operatorname{Gr}_{F}^{a}H^{k}(K(\log)_{\mathcal{O}})|_{Y_{0}}\simeq\operatorname{Gr}_{F}^{a}({}^{\ell}\mathcal{V}^{k}_{Y^{*}})|_{Y_{0}}$

for every $a,k$. On the other hand,

(4.33)
$\begin{split}\operatorname{Gr}_{F}^{0}K(\log)_{\mathcal{O}}&=\operatorname{Gr}_{F}^{0}Rf_{*}R\eta_{*}\Omega_{Z_{\bullet}/Y}(\log
E_{\bullet})\\\
&\simeq\operatorname{Gr}_{F}^{0}Rf_{*}\varepsilon_{*}\Omega_{(X,D)_{\bullet}/Y}=\operatorname{Gr}_{F}^{0}K_{\mathcal{O}}\simeq
Rf_{*}\mathcal{O}_{X}(-D)\end{split}$

by (4.25) and (4.1.2). Thus we obtain an isomorphism

(4.34) $R^{d-i}f_{*}\mathcal{O}_{X}(-D)|_{Y_{0}}\simeq
H^{d-i}(\operatorname{Gr}_{F}^{0}K(\log)_{\mathcal{O}})|_{Y_{0}}\simeq\operatorname{Gr}_{F}^{0}({}^{\ell}\mathcal{V}^{d-i}_{Y^{*}})|_{Y_{0}}$

for every $i$. Because $R^{d-i}f_{*}\mathcal{O}_{X}(-D)$ is locally free of
finite rank by Theorem 1.4, and because $\Sigma_{0}=Y\setminus Y_{0}$ has the
codimension at least two, the isomorphism above extends uniquely to the whole
$Y$.

By Grothendieck duality (see [RRV]), we obtain (iv) from (iii). ∎

######  Remark 4.8.

As already mentioned in Remark 1.2, the local system $R^{k}(f|_{X^{*}\setminus
D^{*}})_{!}\mathbb{R}_{X^{*}\setminus D^{*}}$ underlies an admissible graded
polarizable variation of $\mathbb{R}$-mixed Hodge structure on $Y^{*}$. In
order to check the admissibility, we may assume that
$(Y,\Sigma)=(\Delta,\\{0\\})$ from the beginning. We may further assume that
all the monodromy automorphisms of
$R^{i}(f_{S})_{\ast}\mathbb{R}_{S}|_{\Delta^{*}}$ around the origin are
unipotent for all strata $S$ and for all $i\in\mathbb{Z}$ by [Ka, Lemma
1.9.1]. Then the extension of the Hodge filtration has been already given by
Theorem 1.1 (ii). On the other hand, the existence of the relative monodromy
weight filtration can be proved by the same way as the proof of Lemma 4.10 of
[FF1]. Here we remark that the coincidence of the monodromy weight filtration
and the weight filtration on the limit mixed Hodge structure in [St1, (5.9)
Theorem] holds true for a proper surjective morphism $f\colon X\to\Delta$ from
a Kähler manifold $X$ to the unit disc $\Delta$ such that $f$ is smooth over
$\Delta^{*}$, that the local system $R^{i}f_{*}\mathbb{R}|_{\Delta^{*}}$ is of
unipotent monodromy for every $i$, and that $f^{-1}(0)_{\operatorname{red}}$
is a simple normal crossing divisor (cf. [Sa1, 4.2.5 Remarque], [GN, (5.2)
Théorème]).

The following theorem is an easy consequence of the proof of Theorem 1.4. We
will use it in the proof of Theorem 1.5.

######  Theorem 4.9.

In Theorem 1.1, for every $i$, there exists a finite filtration of locally
free sheaves

$0=\mathcal{E}^{i}_{0}\subset\mathcal{E}^{i}_{1}\subset\cdots\subset\mathcal{E}^{i}_{l_{i}}=R^{i}f_{*}\omega_{X/Y}(D)$

such that

$\mathcal{E}^{i}_{j+1}/\mathcal{E}^{i}_{j}$

is isomorphic to a direct summand of

$\bigoplus_{\mathrm{finite}}R^{\alpha}f_{*}\omega_{S_{\beta}/Y},$

where $\alpha$ is a nonnegative integer and $S_{\beta}$ is a stratum of
$(X,D)$, for every $j$.

Moreover, if $\pi\colon Y\to Z$ is a projective bimeromorphic morphism of
complex varieties, then

$R^{p}\pi_{*}R^{i}f_{*}\omega_{X}(D)=0$

holds for every $p>0$. In particular, we have

$\pi_{*}R^{i}f_{*}\omega_{X}(D)\simeq R^{i}(\pi\circ f)_{*}\omega_{X}(D).$

###### Proof.

By Theorem 1.4, there exists a finite filtration of locally free sheaves

$0=\mathcal{F}^{d-i}_{0}\subset\mathcal{F}^{d-i}_{1}\subset\cdots\subset\mathcal{F}^{d-i}_{l_{i}}=R^{d-i}f_{*}\mathcal{O}_{X}(-D)$

such that

$\mathcal{F}^{d-i}_{j+1}/\mathcal{F}^{d-i}_{j}$

is isomorphic to a direct summand of

$\bigoplus_{\mathrm{finite}}R^{r}f_{*}\mathcal{O}_{S_{\beta}},$

where $S_{\beta}$ is a stratum of $(X,D)$ and $r$ is a nonnegative integer,
for every $j$. We put

$\mathcal{E}^{i}_{j}:=\mathcal{H}om_{\mathcal{O}_{Y}}(\mathcal{F}^{d-i}_{l_{i}}/\mathcal{F}^{d-i}_{l_{i}-j},\mathcal{O}_{Y})$

for every $j$. Then, by Grothendieck duality (see [RRV]), we obtain a desired
filtration of $R^{i}f_{*}\omega_{X/Y}(D)$. By Theorem 2.10, we have
$R^{p}\pi_{*}R^{\alpha}f_{*}\omega_{S_{\beta}}=0$ for every $p>0$. This
implies that $R^{p}\pi_{*}\left(\mathcal{E}^{i}_{j}\otimes\omega_{Y}\right)=0$
holds for every $p>0$ and every $j$. Thus we obtain
$R^{p}\pi_{*}R^{i}f_{*}\omega_{X}(D)=0$ for every $p>0$. Hence we have
$\pi_{*}R^{i}f_{*}\omega_{X}(D)\simeq R^{i}(\pi\circ f)_{*}\omega_{X}(D)$. We
finish the proof. ∎

We close this section with the proof of Theorem 1.3.

###### Proof of Theorem 1.3.

This theorem is obvious by Theorem 1.1 (iv) and the Fujita–Zucker–Kawamata
semipositivity theorem. For the details of the Fujita–Zucker–Kawamata
semipositivity theorem, see, for example, [FF1, Section 5], [FFS, Corollary
2], [FF2], and so on. ∎

We note that Theorems 1.1 and 1.3 have already played a crucial role when
$f\colon(X,D)\to Y$ is algebraic. We recommend that the interested reader
looks at [Fn4], [Fn5], [Fn6], [Fn7], [FFL], [FH], and so on.

##  5\. Proof of Theorem 1.5

In this section, we will prove Theorem 1.5 by using Theorem 4.9. In Section 6,
we will see that Theorem 1.7 follows from Theorem 1.5.

###### Proof of Theorem 1.5.

In Step 1 and Step 2, we will prove (i) and (ii), respectively.

######  Step 1.

In this step, we will prove (i).

We take an arbitrary point $P\in Y$. It is sufficient to prove (i) around $P$.
By Lemma 2.8, we may assume that $(X,D)$ is an analytic globally embedded
simple normal crossing pair and that there exists the following commutative
diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{M}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Y}}$$\textstyle{\Delta^{m},}$

where $M$ is the ambient space of $(X,D)$, such that $q_{M}$ is projective and
$\iota_{Y}(P)=0\in\Delta^{m}$. By taking a suitable resolution of
singularities of $Y$ (see [BM, Sections 12 and 13]), there exist a projective
bimeromorphic morphism $\psi\colon Y^{\prime}\to Y$ from a smooth complex
variety $Y^{\prime}$ and a simple normal crossing divisor $\Sigma^{\prime}$ on
$Y^{\prime}$ such that every stratum of $(X,D)$ is smooth over
$Y\setminus\psi(\Sigma^{\prime})$. Then, by taking a suitable resolution of
singularities of $M$ (see [BM, Sections 12 and 13]) and applying Lemma 2.7, we
may assume that

$f^{\prime}\colon
X\overset{f}{\longrightarrow}Y\overset{\psi^{-1}}{\dashrightarrow}Y^{\prime}$

is a projective morphism. Hence we have the following commutative diagram:

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

such that every stratum of $(X,D)$ is smooth over
$Y^{\prime}\setminus\Sigma^{\prime}$. By Theorem 4.9,
$R^{q}f^{\prime}_{*}\omega_{X/Y^{\prime}}(D)$ is locally free and has a finite
filtration as in Theorem 4.9. Since $\psi$ is projective bimeromorphic and
$R^{q}f_{*}\omega_{X}(D)\simeq\psi_{*}R^{q}f^{\prime}_{*}\omega_{X}(D)$ by
Theorem 4.9, $R^{q}f_{*}\omega_{X}(D)$ is torsion-free. This is what we
wanted.

######  Step 2.

In this step, we will prove (ii).

We take an arbitrary point $P\in Z$. It is sufficient to prove (ii) around
$P$. As in Step 1, after shrinking $Z$ suitably, by Lemma 2.8, a suitable
resolution of singularities (see [BM, Sections 12 and 13]), and Lemma 2.7, we
may assume that there exists the following commutative diagram:

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{M}}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Z}}$$\textstyle{\Delta^{m}}$

such that $\iota_{Z}(P)=0\in\Delta^{m}$. By Theorem 4.9, we have

$R^{p}\pi_{*}\left(\mathcal{A}\otimes R^{q}f_{*}\omega_{X}(D)\right)\simeq
R^{p}\pi_{*}\left(\mathcal{A}\otimes\psi_{*}R^{q}f^{\prime}_{*}\omega_{X}(D)\right)\simeq
R^{p}(\pi\circ\psi)_{*}\left(\psi^{*}\mathcal{A}\otimes
R^{q}f^{\prime}_{*}\omega_{X}(D)\right).$

By Theorem 4.9 again, $R^{q}f^{\prime}_{*}\omega_{X/{Y^{\prime}}}(D)$ has a
finite filtration as in Theorem 4.9. Thus we can reduce the problem to the
case where $X$ is smooth and $D=0$. Since $\psi^{*}\mathcal{A}$ is
$(\pi\circ\psi)$-nef and $(\pi\circ\psi)$-big over $Z$, we get the desired
vanishing theorem by Theorem 2.10. We finish the proof of (ii).

We finish the proof of Theorem 1.5. ∎

######  Remark 5.1.

By the above proof, we see that Theorem 1.5 (ii) holds under a weaker
assumption that $\mathcal{A}$ is $\pi$-nef and $\pi$-big over $Z$ (see Theorem
2.10).

##  6\. Proof of Theorem 1.7

In this section, we will prove Theorem 1.7 by using Theorem 1.5. As we
mentioned before, Theorem 1.7 (iii) is an easy consequence of Theorem 1.7 (i)
and (ii).

###### Proof of Theorem 1.7.

In Step 1, we will prove Theorem 1.7 (i). Then, in Steps 2 and 3, we will
prove Theorem 1.7 (ii) and (iii), respectively.

######  Step 1.

In this step, we will prove Theorem 1.7 (i).

By replacing $Y$ with $f(X)$, we may assume that $f(X)=Y$. Let $P\in Y$ be an
arbitrary point. It is sufficient to prove the statement after shrinking $Y$
around $P$ suitably. By Lemma 2.8, we may assume that $(X,D)$ is an analytic
globally embedded simple normal crossing pair and that there exists the
following commutative diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{M}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Y}}$$\textstyle{\Delta^{m},}$

where $M$ is the ambient space of $(X,D)$, such that $q_{M}$ is projective and
$\iota_{Y}(P)=0\in\Delta^{m}$. By using Lemma 2.9 finitely many times, we can
decompose $X=X^{\prime}+X^{\prime\prime}$ as follows: $X^{\prime}$ is the
union of all strata of $(X,D)$ that are not mapped onto irreducible components
of $Y=f(X)$ and $X^{\prime\prime}=X-X^{\prime}$. We put

$K_{X^{\prime}}+D_{X^{\prime}}:=(K_{X}+D)|_{X^{\prime}}$

and

$K_{X^{\prime\prime}}+D_{X^{\prime\prime}}:=(K_{X}+D)|_{X^{\prime\prime}}-X^{\prime}|_{X^{\prime\prime}}.$

We note that $(X^{\prime\prime},D_{X^{\prime\prime}})$ is an analytic globally
embedded simple normal crossing pair such that $D_{X^{\prime\prime}}$ is
reduced and that every stratum of $(X^{\prime\prime},D_{X^{\prime\prime}})$ is
mapped onto some irreducible component of $Y$. We consider the following short
exact sequence:

$0\to\mathcal{O}_{X^{\prime\prime}}(K_{X^{\prime\prime}}+D_{X^{\prime\prime}})\to\mathcal{O}_{X}(K_{X}+D)\to\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})\to
0.$

By Theorem 1.5 (i), every associated subvariety of
$R^{q}f_{*}\mathcal{O}_{X^{\prime\prime}}(K_{X^{\prime\prime}}+D_{X^{\prime\prime}})$
is an irreducible component of $Y$ for every $q$. Note that every associated
subvariety of
$R^{q}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})$ is
contained in $f(X^{\prime})$ for every $q$. Thus, the connecting homomorphisms

$\delta\colon
R^{q}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})\to
R^{q+1}f_{*}\mathcal{O}_{X^{\prime\prime}}(K_{X^{\prime\prime}}+D_{X^{\prime\prime}})$

are zero for all $q$. Hence we obtain the following short exact sequence

(6.1) $0\to
R^{q}f_{*}\mathcal{O}_{X^{\prime\prime}}(K_{X^{\prime\prime}}+D_{X^{\prime\prime}})\to
R^{q}f_{*}\mathcal{O}_{X}(K_{X}+D)\to
R^{q}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})\to 0$

for every $q$. By induction on $\dim f(X)$, every associated subvariety of
$R^{q}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})$ is the
$f$-image of some stratum of $(X^{\prime},D_{X^{\prime}})$ for every $q$.
Therefore, every associated subvariety of $R^{q}f_{*}\mathcal{O}_{X}(K_{X}+D)$
is the $f$-image of some stratum of $(X,D)$ for every $q$ by (6.1).

######  Step 2.

In this step, we will prove Theorem 1.7 (ii).

We may assume that $f(X)=Y$ and $\pi\circ f(X)=Z$. Let $P\in Z$ be an
arbitrary point. It is sufficient to prove the desired vanishing theorem after
shrinking $Z$ around $P$ suitably. As in Step 1, by Lemma 2.8, we have the
following commutative diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\circ
f}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{M}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Z}}$$\textstyle{\Delta^{m},}$

where $M$ is the ambient space of $(X,D)$, such that $q_{M}$ is projective and
$\iota_{Z}(P)=0\in\Delta^{m}$. By the same argument as in Step 1, we obtain

(6.2) $0\to
R^{q}f_{*}\mathcal{O}_{X^{\prime\prime}}(K_{X^{\prime\prime}}+D_{X^{\prime\prime}})\to
R^{q}f_{*}\mathcal{O}_{X}(K_{X}+D)\to
R^{q}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})\to 0$

for every $q$. By applying Theorem 1.5 (ii) to every connected component of
$X^{\prime\prime}$, we see that

$R^{p}\pi_{*}\left(\mathcal{A}\otimes
R^{q}f_{*}\mathcal{O}_{X^{\prime\prime}}(K_{X^{\prime\prime}}+D_{X^{\prime\prime}})\right)=0$

holds for every $p>0$. By induction on $\dim f(X)$, we obtain

$R^{p}\pi_{*}\left(\mathcal{A}\otimes
R^{q}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D_{X^{\prime}})\right)=0$

for every $p>0$. This implies

$R^{p}\pi_{*}\left(\mathcal{A}\otimes
R^{q}f_{*}\mathcal{O}_{X}(K_{X}+D)\right)=0$

for every $p>0$. This is what we wanted.

######  Step 3.

In this step, we will prove Theorem 1.7 (iii).

Since we have already proved the strict support condition (see (i)) and the
vanishing theorem (see (ii)) in Steps 1 and 2, respectively, the proof of
[Fn9, Theorem 3.1 (iii)] works. Hence we obtain the desired injectivity in
(iii).

We finish the proof of Theorem 1.7. ∎

######  Remark 6.1.

Theorem 1.7 (ii) holds under a weaker assumption that $\mathcal{A}$ is nef and
log big over $Z$ with respect to $f\colon(X,D)\to Y$. We can easily check it
by the above proof of Theorem 1.7 (ii) and Remark 5.1. We do not discuss the
details here because we have already known a more general statement, that is,
the vanishing theorem of Reid–Fukuda type (see Theorem 1.9).

##  7\. Supplement to [St2]

In this section, we give a remark on the construction of the cohomological
$\mathbb{Q}$-mixed Hodge complex $((A_{\mathbb{Q}},W),(A_{\mathbb{C}},W,F))$
in [St2, p.536]. More precisely, we will present a new construction of
$(A_{\mathbb{Q}},W)$ here. In the context of log geometry, such a construction
is originated in [St3] and used in other articles (e.g. [FN], [Fs2] and so
on). For the case of a semistable reduction, a new construction of
$(A_{\mathbb{Q}},W)$, which is similar to [St3], is given in [PS, 11.2.6 The
Rational Structure]. (For the case of a semistable morphism over the polydisc,
see e.g. [Fs1].) Here we will see that the construction in [Fs2] works in the
situation of [St2].

######  7.1.

Let $f\colon X\to\Delta$ be a proper surjective morphism from a smooth complex
variety $X$ to the unit disc $\Delta$ satisfying the conditions

  * •

$f$ is smooth over $\Delta^{*}=\Delta\setminus\\{0\\}$, and

  * •

$\operatorname{Supp}f^{-1}(0)$ is a simple normal crossing divisor on $X$

as in [St2, (2.1) Notations]. Note that $f^{-1}(0)$ is not assumed to be
reduced. We fix $N\in\mathbb{Z}_{>0}$, which is a multiple of all the
multiplicities of the irreducible components of
$\operatorname{Supp}f^{-1}(0)$, and consider the morphism
$\sigma\colon\Delta\to\Delta$ given by $\sigma(t)=t^{N}$. We define
$\widetilde{X},\pi$ and $\widetilde{f}$ by the commutative diagram

(7.1)
$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{\pi}$$\scriptstyle{\widetilde{f}}$$\textstyle{X\times_{\Delta}\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\Delta}$

where $\nu$ is the normalization. We set
$E=\operatorname{Supp}\widetilde{f}^{-1}(0)$, which is an effective Cartier
divisor on $\widetilde{X}$. The irreducible decomposition of $E$ is written in
$E=\bigcup_{i=1}^{l}E_{i}$. The closed immersion
$E_{i}\hookrightarrow\widetilde{X}$ is denoted by $a_{i}$.

######  7.2.

We recall the local description of $\widetilde{X}$ and $\widetilde{f}$ given
in the proof of [St2, (2.2) Lemma]. For any point of $\widetilde{X}$, there
exist an open neighborhood $\widetilde{U}$ in $\widetilde{X}$,
$d_{1},\dots,d_{k}\in\mathbb{Z}_{>0}$ with $\gcd(d_{1},\dots,d_{k})=1$, and
$e\in\mathbb{Z}_{>0}\cap(\bigcap_{i=1}^{k}d_{i}\mathbb{Z})$ with $N\in
e\mathbb{Z}$ such that $\widetilde{U}$ and $\widetilde{f}|_{\widetilde{U}}$
are described by using $d_{1},\dots,d_{k},e$ as follows. By setting
$c_{i}:=e/d_{i}\in\mathbb{Z}_{>0}$ and
$G:=\bigoplus_{i=1}^{k}\mathbb{Z}/c_{i}\mathbb{Z}$, the kernel of the morphism

(7.2)
$G=\bigoplus_{i=1}^{k}\mathbb{Z}/c_{i}\mathbb{Z}\ni(b_{1},\dots,b_{k})\mapsto\sum_{i=1}^{k}d_{i}b_{i}\in\mathbb{Z}/e\mathbb{Z}$

is denoted by $H$. The finite abelian group $G$ acts on the polydisc
$\Delta^{n}$ by

(7.3) $(b_{1},\dots,b_{k})\cdot
y_{i}=\begin{cases}\exp(2\pi\sqrt{-1}b_{i}/c_{i})y_{i}&\quad\text{for $1\leq
i\leq k$}\\\ y_{i}&\quad\text{for $k+1\leq i\leq n$},\end{cases}$

where $(y_{1},\dots,y_{n})$ is the coordinate of $\Delta^{n}$. Then
$\widetilde{U}\simeq\Delta^{n}/H$ and $\widetilde{f}^{*}t=y_{1}\cdots y_{k}$,
where $t$ is the coordinate of $\Delta$. Note that $y_{1}\cdots y_{k}$ is
$H$-invariant. Moreover, $U=\pi(\widetilde{U})$ is an open subset of $X$, and
we also have $U\simeq\Delta^{n}/G$ and $f^{*}t=(y_{1}\cdots y_{k})^{N}$. Here
we note that $(y_{1}\cdots y_{k})^{N}$ is $G$-invariant because $N\in
e\mathbb{Z}$. The $G$-invariant functions
$y_{1}^{c_{1}},\dots,y_{k}^{c_{k}},y_{k+1},\dots,y_{n}$ give us a coordinate
on $U$.

From the local description above, $\widetilde{X}$ is trivially a $V$-manifold.
We can easily see that $E_{i}$ is a reduced Cartier divisor on
$X\setminus\bigcup_{j\not=i}E_{j}$. Moreover, $E_{i}$ is locally irreducible
at any point because $\pi(E_{i})$ is an irreducible component of
$\operatorname{Supp}f^{-1}(0)$ and because $\operatorname{Supp}f^{-1}(0)$ is a
simple normal crossing divisor on $X$.

######  7.3.

In the situation 7.1, the log structure on $\widetilde{X}$ associated to the
effective divisor $E$ is denoted by $\mathcal{M}$, that is,

(7.4) $\mathcal{M}:=\mathcal{O}_{\widetilde{X}}\cap
j_{*}\mathcal{O}^{*}_{\widetilde{X}\setminus E}$

in $j_{*}\mathcal{O}_{\widetilde{X}\setminus E}$, where $j$ denotes the open
immersion $\widetilde{X}\setminus E\hookrightarrow\widetilde{X}$. The abelian
sheaf associated to the monoid sheaf $\mathcal{M}$ is denoted by
$\mathcal{M}^{\rm gp}$. By using the fact that $E_{i}$ is locally irreducible,
a morphism of monoid sheaves $\mathcal{M}\to(a_{i})_{*}\mathbb{N}_{E_{i}}$ can
be defined by

(7.5) $\mathcal{M}=\mathcal{O}_{\widetilde{X}}\cap
j_{*}\mathcal{O}^{*}_{\widetilde{X}\setminus E}\ni
a\mapsto\operatorname{ord}_{E_{i}}(a)\in(a_{i})_{*}\mathbb{N}_{E_{i}}$

for any $i$, where $\operatorname{ord}_{E_{i}}$ denotes the vanishing order of
a holomorphic function on $\widetilde{X}$ along the divisor $E_{i}$. The
direct sum of the morphisms (7.5) for all $i$ induces a morphism

(7.6) $\mathcal{M}^{\rm
gp}\to\bigoplus_{i=1}^{l}(a_{i})_{*}\mathbb{Z}_{E_{i}},$

which fits in an exact sequence

(7.7) $0\to\mathcal{O}^{*}_{\widetilde{X}}\to\mathcal{M}^{\rm
gp}\to\bigoplus_{i=1}^{l}(a_{i})_{*}\mathbb{Z}_{E_{i}}$

by definition.

The following is a key lemma for the construction of $(A_{\mathbb{Q}},W)$.

######  Lemma 7.4.

We obtain the exact sequence

(7.8)
$0\to\mathcal{O}^{*}_{\widetilde{X}}\otimes_{\mathbb{Z}}\mathbb{Q}\to\mathcal{M}^{\rm
gp}\otimes_{\mathbb{Z}}\mathbb{Q}\to\bigoplus_{i=1}^{l}(a_{i})_{*}\mathbb{Q}_{E_{i}}\to
0$

by tensoring $\mathbb{Q}$ to (7.7).

###### Proof.

We may work in the local situation described in 7.2. Since $y_{i}^{c_{i}}$ is
$H$-invariant, it gives us a holomorphic function on $\widetilde{U}$ for
$i=1,\dots,k$. We may assume that $E_{i}=\operatorname{Supp}\\{y^{c_{i}}=0\\}$
for $1\leq i\leq k$ and $E_{i}\cap\widetilde{U}=\emptyset$ for $k+1\leq i\leq
l$ by changing the indices. Because $E_{i}$ is the zero set of
$\widetilde{f}^{*}t=y_{1}\cdots y_{k}$ on
$\widetilde{U}\setminus\bigcup_{j\not=i}(E_{j}\cap\widetilde{U})$, the image
of $y_{i}^{c_{i}}\in\mathcal{M}\subset\mathcal{M}^{\rm gp}$ by the morphism
(7.6) is
$(0,\dots,0,c_{i},0\dots,0)\in\bigoplus_{j=1}^{l}(a_{j})_{*}\mathbb{Z}_{E_{j}}$,
where $c_{i}$ is on the $i$-th entry. Thus we obtain the conclusion. ∎

######  7.5.

We briefly recall the constructions of the Koszul complexes and related
objects. For the detail, see [Fs2, Sections 1 and 2] and [PS, §4.4 and
§11.2.6] (cf. [I], [St3] and so on).

A morphism of abelian sheaves
$\mathbf{e}\colon\mathcal{O}_{\widetilde{X}}\to\mathcal{M}^{\rm gp}$ is
defined as the composite of the exponential map

(7.9) $\mathcal{O}_{\widetilde{X}}\ni a\mapsto
e^{2\pi\sqrt{-1}a}\in\mathcal{O}^{*}_{\widetilde{X}}$

and the inclusion
$\mathcal{O}^{*}_{\widetilde{X}}\hookrightarrow\mathcal{M}^{\rm gp}$. Then the
morphism
$\mathbf{e}\otimes\operatorname{id}\colon\mathcal{O}_{\widetilde{X}}\simeq\mathcal{O}_{\widetilde{X}}\otimes\mathbb{Q}\to\mathcal{M}^{\rm
gp}\otimes\mathbb{Q}$ is obtained. Note that
$1\in\Gamma(X,\mathcal{O}_{\widetilde{X}})$ is contained in the kernel of
$\mathbf{e}\otimes\operatorname{id}$. We set $\mathcal{M}^{\rm
gp}_{\mathbb{Q}}=\mathcal{M}^{\rm gp}\otimes\mathbb{Q}$ for short. For
$p\in\mathbb{Z}$, a $\mathbb{Q}$-sheaf $\operatorname{Kos}(\mathcal{M})^{p}$
on $\widetilde{X}$ is defined by

(7.10)
$\operatorname{Kos}(\mathcal{M})^{p}:=\varinjlim_{n}\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}},$

where $\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})$
denotes the symmetric tensor product of degree $n-p$ of
$\mathcal{O}_{\widetilde{X}}$ over $\mathbb{Q}$, and where the inductive limit
is taken over the inductive system defined by the morphisms

(7.11)
$\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}\ni a\otimes b\mapsto(1\cdot a)\otimes
b\in\operatorname{Sym}_{\mathbb{Q}}^{n+1-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}$

for all $n\geq p$. A morphism of $\mathbb{Q}$-sheaves

(7.12)
$\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}\to\operatorname{Sym}_{\mathbb{Q}}^{n-p-1}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p+1}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}$

is defined by

(7.13) $a_{1}^{n_{1}}\cdots a_{k}^{n_{k}}\otimes
b\mapsto\sum_{j=1}^{k}n_{j}a_{1}^{n_{1}}\cdots a_{j}^{n_{j}-1}\cdots
a_{k}^{n_{k}}\otimes(\mathbf{e}\otimes\operatorname{id})(a_{j})\wedge b$

where $n_{1},\dots,n_{k}$ are positive integers with $n_{1}+\cdots+n_{k}=n-p$.
These morphisms form a morphism of inductive systems defined by (7.11) for $p$
and $p+1$ because $1\in\Gamma(X,\mathcal{O}_{\widetilde{X}})$ is contained in
the kernel of $\mathbf{e}\otimes\operatorname{id}$. Then a morphism of
$\mathbb{Q}$-sheaves

(7.14)
$d\colon\operatorname{Kos}(\mathcal{M})^{p}\to\operatorname{Kos}(\mathcal{M})^{p+1}$

is induced. We can easily see the equality $d^{2}=0$. Thus we obtain a complex
of $\mathbb{Q}$-sheaves $\operatorname{Kos}(\mathcal{M})$ on $\widetilde{X}$.
Replacing $\mathcal{M}^{\rm gp}$ by $\mathcal{O}^{*}_{\widetilde{X}}$, we
obtain a complex of $\mathbb{Q}$-sheaves
$\operatorname{Kos}(\mathcal{O}^{*}_{\widetilde{X}})$.

We set

(7.15)
$W_{m}(\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}})=\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p-m}(\mathcal{O}^{*}_{\widetilde{X}}\otimes\mathbb{Q})\otimes_{\mathbb{Q}}\bigwedge^{m}\mathcal{M}^{\rm
gp}_{\mathbb{Q}},$

which is a $\mathbb{Q}$-subsheaf of
$\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}$ for every $m\in\mathbb{Z}$. Since the morphism (7.11)
trivially preserves $W_{m}$ on the both sides, a subsheaf
$W_{m}\operatorname{Kos}(\mathcal{M})^{p}$ of
$\operatorname{Kos}(\mathcal{M})^{p}$ is obtained by

(7.16)
$W_{m}\operatorname{Kos}(\mathcal{M})^{p}=\varinjlim_{n}W_{m}(\operatorname{Sym}_{\mathbb{Q}}^{n-p}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}})$

for every $m$. It can be easily checked that they define a finite increasing
filtration $W$ on the complex $\operatorname{Kos}(\mathcal{M})$.

The singular locus of $\widetilde{X}$ is denoted by
$\operatorname{Sing}(\widetilde{X})$ and the smooth locus $\widetilde{X}_{\rm
sm}$ is defined by $\widetilde{X}_{\rm
sm}=\widetilde{X}\setminus\operatorname{Sing}(\widetilde{X})$. Note that the
restriction $E\cap\widetilde{X}_{\rm sm}$ of $E$ to $\widetilde{X}_{\rm sm}$
is a simple normal crossing divisor on $\widetilde{X}_{\rm sm}$. Then we have
a morphism of monoid sheaves

(7.17) $\operatorname{dlog}\colon\mathcal{M}|_{\widetilde{X}_{\rm
sm}}\to\Omega^{1}_{\widetilde{X}_{\rm sm}}(\log E\cap\widetilde{X}_{\rm sm})$

defined by $\operatorname{dlog}(a):=a^{-1}da$ for
$a\in\mathcal{M}|_{\widetilde{X}_{\rm
sm}}\subset\mathcal{O}_{\widetilde{X}_{\rm sm}}$. Thus we obtain a morphism

(7.18) $\mathcal{M}\to\widetilde{\Omega}^{1}_{\widetilde{X}}(\log E)$

denoted by the same letter $\operatorname{dlog}$ as above. This morphism
induces a morphism of $\mathbb{Q}$-sheaves

(7.19) $\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}\to\widetilde{\Omega}^{p}_{\widetilde{X}}(\log E)$

denoted by $\bigwedge^{p}\operatorname{dlog}$ for every $p$. A morphism of
$\mathbb{Q}$-sheaves

(7.20)
$\operatorname{Sym}^{n-p}_{\mathbb{Q}}(\mathcal{O}_{\widetilde{X}})\otimes_{\mathbb{Q}}\bigwedge^{p}\mathcal{M}^{\rm
gp}_{\mathbb{Q}}\to\widetilde{\Omega}^{p}_{\widetilde{X}}(\log E),$

defined by

(7.21) $a_{1}^{n_{1}}\cdots a_{k}^{n_{k}}\otimes
b\mapsto(2\pi\sqrt{-1})^{-p}a_{1}^{n_{1}}\cdots
a_{k}^{n_{k}}\bigwedge^{p}\operatorname{dlog}(b)$

for positive integers $n_{1},\dots,n_{k}$ with $n_{1}+\dots+n_{k}=n-p$, is
compatible with the morphisms (7.11) and (7.13). Therefore we have a morphism
of complexes of $\mathbb{Q}$-sheaves

(7.22)
$\operatorname{Kos}(\mathcal{M})\to\widetilde{\Omega}_{\widetilde{X}}(\log
E),$

which is denoted by $\psi$ as in [Fs2, (2.4)]. It can be easily seen that the
morphism $\psi$ preserves the filtration $W$ on the both sides.

The global section $\widetilde{f}^{*}t\in\Gamma(\widetilde{X},\mathcal{M})$
defines a morphism of complexes

(7.23)
$(\widetilde{f}^{*}t)\wedge\colon\operatorname{Kos}(\mathcal{M})\to\operatorname{Kos}(\mathcal{M})[1],$

which sends $W_{m}\operatorname{Kos}(\mathcal{M})^{n}$ to
$W_{m+1}\operatorname{Kos}(\mathcal{M})^{n+1}$ as in [Fs2, (1.11) and (1.12)].
It can be easily checked that the diagram

(7.24)
$\textstyle{\operatorname{Kos}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\widetilde{f}^{*}t)\wedge}$$\scriptstyle{\psi}$$\textstyle{\widetilde{\Omega}_{\widetilde{X}}(\log
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta\wedge}$$\textstyle{\operatorname{Kos}(\mathcal{M})[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(2\pi\sqrt{-1})\psi}$$\textstyle{\widetilde{\Omega}_{\widetilde{X}}(\log
E)[1]}$

is commutative, where
$\theta=\widetilde{f}^{*}(dt/t)\in\widetilde{\Omega}_{\widetilde{X}}^{1}(\log
E)$.

For $\operatorname{Kos}(\mathcal{M})$ and $\psi$ above, we have the following
lemmas.

######  Lemma 7.6.

In the situation above, we set

(7.25) $E^{(m)}=\coprod_{1\leq i_{1}<\dots<i_{m}\leq l}E_{i_{1}}\cap\cdots\cap
E_{i_{m}}$

for $m\in\mathbb{Z}_{>0}$. Moreover, we set $E^{(0)}=\widetilde{X}$. The
natural morphism $E^{(m)}\to\widetilde{X}$ is denoted by $a_{m}$ for
$m\in\mathbb{Z}_{\geq 0}$. Then there exists a quasi-isomorphism

(7.26)
$(a_{m})_{*}\mathbb{Q}_{E^{(m)}}[-m]\to\operatorname{Gr}_{m}^{W}\operatorname{Kos}(\mathcal{M})$

for all $m\in\mathbb{Z}$.

###### Proof.

We have an isomorphism

(7.27) $\bigwedge^{m}(\mathcal{M}^{\rm
gp}\otimes\mathbb{Q}/\mathcal{O}_{\widetilde{X}}^{*}\otimes\mathbb{Q})\otimes\operatorname{Kos}(\mathcal{O}_{\widetilde{X}}^{*})[-m]\simeq\operatorname{Gr}_{m}^{W}\operatorname{Kos}(\mathcal{M})$

by [Fs2, Proposition 1.10], and a quasi-isomorphism
$\mathbb{Q}_{\widetilde{X}}\to\operatorname{Kos}(\mathcal{O}_{\widetilde{X}}^{*})$
by [Fs2, Corollary 1.15]. Therefore we obtain the conclusion by Lemma 7.4. ∎

######  Lemma 7.7.

In the situation above, we have the commutative diagram

(7.28)
$\textstyle{(a_{m})_{*}\mathbb{Q}_{E^{(m)}}[-m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(2\pi\sqrt{-1})^{-m}\iota[-m]}$$\textstyle{(a_{m})_{*}\widetilde{\Omega}_{E^{(m)}}[-m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Gr}_{m}^{W}\operatorname{Kos}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Gr}_{m}^{W}\psi}$$\textstyle{\operatorname{Gr}_{m}^{W}\widetilde{\Omega}_{\widetilde{X}}(\log
E)}$

where $\iota$ is the natural morphism induced from the inclusion
$\mathbb{Q}\to\mathcal{O}_{E^{(m)}}$, the left vertical arrow is the quasi-
isomorphism in Lemma 7.6, and the right vertical arrow is the inverse of the
residue isomorphism in [St2, (1.18) Definition and (1.19) Lemma] $($see also
[D1, 3.5]$)$. In particular, the morphism

(7.29)
$\operatorname{Kos}(\mathcal{M})\otimes\mathbb{C}\to\widetilde{\Omega}_{\widetilde{X}}(\log
E)$

induced by $\psi$ is a filtered quasi-isomorphism with respect to $W$ on the
both sides.

###### Proof.

The commutativity of the diagram (7.28) can be checked by the direct
computation from the definition of $\psi$ (cf. [Fs2, (2.4)]). Then the latter
conclusion follows from [St2, (1.9) Corollary]. ∎

Once we obtain these two lemmas, it is more or less clear that the
construction, parallel to $A_{\mathbb{C}}$ in [St1, (4.14) and (4.17)] and
[St2, (2.8)], works for $A_{\mathbb{Q}}$.

######  Definition 7.8.

In the situation 7.1, a bifiltered complex of $\mathbb{C}$-sheaves
$(A_{\mathbb{C}},W,F)$ on $\widetilde{X}$ is defined by

(7.30) $\displaystyle A^{n}_{\mathbb{C}}:=\bigoplus_{q\geq
0}\widetilde{\Omega}^{n+1}_{\widetilde{X}}(\log
E)/W_{q}\widetilde{\Omega}^{n+1}_{\widetilde{X}}(\log E),$ (7.31)
$\displaystyle W_{m}A^{n}_{\mathbb{C}}:=\bigoplus_{q\geq
0}W_{m+2q+1}\widetilde{\Omega}^{n+1}_{\widetilde{X}}(\log
E)/W_{q}\widetilde{\Omega}^{n+1}_{\widetilde{X}}(\log E),$ (7.32)
$\displaystyle F^{p}A^{n}_{\mathbb{C}}:=\bigoplus_{0\leq q\leq
n-p}\widetilde{\Omega}_{\widetilde{X}}^{n+1}(\log
E)/W_{q}\widetilde{\Omega}_{\widetilde{X}}^{n+1}(\log E)$

with the differential $-d-\theta\wedge$, where $d$ denotes the differential of
the complex $\widetilde{\Omega}_{\widetilde{X}}(\log E)$ as in [St1, (4.17)].
Similarly, a filtered complex of $\mathbb{Q}$-sheaves $(A_{\mathbb{Q}},W)$ on
$\widetilde{X}$ is defined by

(7.33) $\displaystyle A^{n}_{\mathbb{Q}}:=\bigoplus_{q\geq
0}\operatorname{Kos}(\mathcal{M})^{n+1}/W_{q}\operatorname{Kos}(\mathcal{M})^{n+1}$
(7.34) $\displaystyle W_{m}A^{n}_{\mathbb{Q}}:=\bigoplus_{q\geq
0}W_{m+2q+1}\operatorname{Kos}(\mathcal{M})^{n+1}/W_{q}\operatorname{Kos}(\mathcal{M})^{n+1}$

with the differential $-d-(\widetilde{f}^{*}t)\wedge$, where $d$ denotes the
differential of the complex $\operatorname{Kos}(\mathcal{M})$. The direct sum
of the morphisms of $\mathbb{Q}$-sheaves

(7.35)
$(2\pi\sqrt{-1})^{q+1}\psi\colon\operatorname{Kos}(\mathcal{M})^{n+1}/W_{q}\operatorname{Kos}(\mathcal{M})^{n+1}\to\widetilde{\Omega}_{\widetilde{X}}^{n+1}(\log
E)/W_{q}\widetilde{\Omega}_{\widetilde{X}}^{n+1}(\log E)$

gives us a morphism of $\mathbb{Q}$-sheaves

(7.36) $A^{n}_{\mathbb{Q}}=\bigoplus_{q\geq
0}\operatorname{Kos}(\mathcal{M})^{n+1}/W_{q}\operatorname{Kos}(\mathcal{M})^{n+1}\to\bigoplus_{q\geq
0}\widetilde{\Omega}_{\widetilde{X}}^{n+1}(\log
E)/W_{q}\widetilde{\Omega}_{\widetilde{X}}^{n+1}(\log E)=A^{n}_{\mathbb{C}}$

which is compatible with the differentials by the commutativity of the diagram
(7.24). Thus we obtain a morphism of filtered complexes of
$\mathbb{Q}$-sheaves $\alpha\colon(A_{\mathbb{Q}},W)\to(A_{\mathbb{C}},W)$.
Note that the supports of $A^{n}_{\mathbb{C}}$ and $A^{n}_{\mathbb{Q}}$ are
contained in $E$ for every $n$. Therefore they are the (bi)filtered complexes
on $E$.

######  Theorem 7.9 (cf. [St2, (2.8)]).

Let $f\colon X\to\Delta$ be as in 7.1. If we assume that $X$ is Kähler, then
$((A_{\mathbb{Q}},W),(A_{\mathbb{C}},W,F),\alpha)$ is a cohomological
$\mathbb{Q}$-mixed Hodge complex on $E$.

###### Proof.

By Lemmas 7.6 and 7.7,
$(\operatorname{Gr}_{m}^{W}A_{\mathbb{Q}},(\operatorname{Gr}_{m}^{W}A_{\mathbb{C}},F),\operatorname{Gr}_{m}^{W}\alpha)$
is identified with the direct sum of the direct images of

(7.37)
$(\mathbb{Q}(-m-q)[-m-2q],(\widetilde{\Omega}_{E^{(m+2q+1)}}[-m-2q],F[-m-q]))$

by the finite morphism $a_{m+2q+1}$ for all $q\geq\max(0,-m)$. Since
$\widetilde{X}$ is an almost Kähler $V$-manifold as in [PS, I.2.5] by the
assumption for $X$ being Kähler, we obtain the conclusion by Theorem 2.43 of
[PS]. ∎

## References

  * [A] F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 220–239; reprinted in Proc. Steklov Inst. Math. 2003, no. 1(240), 214–233.
  * [BS] C. Bănică, O. Stănăşilă, Algebraic methods in the global theory of complex spaces, Translated from the Romanian. Editura Academiei, Bucharest; John Wiley & Sons, London–New York–Sydney, 1976.
  * [BM] E. Bierstone, P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302.
  * [D1] P. Deligne, Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin–New York, 1970.
  * [D2] P. Deligne, Théorie de Hodge II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–58.
  * [D3] P. Deligne, Théorie de Hodge III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.
  * [Fi] G. Fischer, Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538. Springer-Verlag, Berlin–New York, 1976.
  * [Fn1] O. Fujino, Higher direct images of log canonical divisors, J. Differential Geom. 66 (2004), no. 3, 453–479.
  * [Fn2] O. Fujino, A transcendental approach to Kollár’s injectivity theorem II, J. Reine Angew. Math. 681 (2013), 149–174.
  * [Fn3] O. Fujino, Foundations of the minimal model program, MSJ Memoirs, 35. Mathematical Society of Japan, Tokyo, 2017.
  * [Fn4] O. Fujino, Semipositivity theorems for moduli problems, Ann. of Math. (2) 187 (2018), no. 3, 639–665.
  * [Fn5] O. Fujino, Fundamental properties of basic slc-trivial fibrations I, Publ. Res. Inst. Math. Sci. 58 (2022), no. 3, 473–526.
  * [Fn6] O. Fujino, Cone theorem and Mori hyperbolicity, preprint (2020). arXiv:2102.11986 [math.AG]
  * [Fn7] O. Fujino, On quasi-log schemes, J. Math. Soc. Japan 75 (2023), no. 3, 829–856.
  * [Fn8] O. Fujino, Minimal model program for projective morphisms between complex analytic spaces, preprint (2022). arXiv:2201.11315 [math.AG]
  * [Fn9] O. Fujino, Vanishing theorems for projective morphisms between complex analytic spaces, to appear in Math. Res. Lett.
  * [Fn10] O. Fujino, Cone and contraction theorem for projective morphisms between complex analytic spaces, preprint (2022). arXiv:2209.06382 [math.AG]
  * [Fn11] O. Fujino, On quasi-log structures for complex analytic spaces, preprint (2022). arXiv:2209.11401 [math.AG]
  * [FF1] O. Fujino, T. Fujisawa, Variations of mixed Hodge structure and semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, 589–661.
  * [FF2] O. Fujino, T. Fujisawa, On semipositivity theorems, Math. Res. Lett. 26 (2019), no. 5, 1359–1382.
  * [FFL] O. Fujino, T. Fujisawa, H. Liu, Fundamental properties of basic slc-trivial fibrations II, Publ. Res. Inst. Math. Sci. 58 (2022), no. 3, 527–549.
  * [FFS] O. Fujino, T. Fujisawa, M. Saito, Some remarks on the semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 1, 85–112.
  * [FH] O. Fujino, K. Hashizume, Adjunction and inversion of adjunction, Nagoya Math. J. 249 (2023), 119–147.
  * [Fs1] T. Fujisawa, Limits of Hodge structures in several variables, Compositio Math. 115 (1999), 129–183.
  * [Fs2] T. Fujisawa, Mixed Hodge structures on log smooth degenerations, Tohoku Math. J. (2) 60 (2008), no. 1, 71–100.
  * [FN] T. Fujisawa, C. Nakayama, Mixed Hodge structures on log deformations, Rend. Sem. Mat. Univ. Padova 110 (2003), 221–268.
  * [GN] F. Guillén and V. Navarro Aznar, _Sur le théorème local des cycles invariants_ , Duke Math. J. 61 (1990), 133–155.
  * [I] L. Illusie, Complexe cotangent et déformations. I, Lecture Notes in Math., vol. 239. Springer-Verlag, Berlin–New York, 1971.
  * [Ka] M. Kashiwara, A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci. 22 (1986), no. 5, 991–1024.
  * [Ko1] J. Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42.
  * [Ko2] J. Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171–202.
  * [Mo] A. Moriwaki, Torsion freeness of higher direct images of canonical bundles, Math. Ann. 276 (1987), no. 3, 385–398.
  * [N1] N. Nakayama, The lower semicontinuity of the plurigenera of complex varieties, Algebraic geometry, Sendai, 1985, 551–590, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.
  * [N2] N. Nakayama, Hodge filtrations and the higher direct images of canonical sheaves, Invent. Math. 85 (1986), no. 1, 217–221.
  * [N3] N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004.
  * [PS] C. A. M. Peters, J. H. M. Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52. Springer-Verlag, Berlin, 2008.
  * [RRV] J. P. Ramis, G. Ruget, J. L. Verdier, Dualité relative en géométrie analytique complexe, Invent. Math. 13 (1971), 261–283.
  * [Sa1] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995.
  * [Sa2] M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333.
  * [Sa3] M. Saito, Decomposition theorem for proper Kähler morphisms, Tohoku Math. J. (2) 42 (1990), no. 2, 127–147.
  * [Sa4] M. Saito, On Kollár’s conjecture, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 509–517, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.
  * [Sa5] M. Saito, Some remarks on decomposition theorem for proper Kähler morphisms, preprint (2022). arXiv:2204.09026 [math.AG]
  * [Sc] W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.
  * [Si] Y.-T. Siu, Noether–Lasker decomposition of coherent analytic subsheaves, Trans. Amer. Math. Soc. 135 (1969), 375–385.
  * [St1] J. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229–257.
  * [St2] J. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 525–563. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.
  * [St3] J. Steenbrink, Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures, Math. Ann. 301 (1995), no. 1, 105–118.
  * [T] K. Takegoshi, Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms, Math. Ann. 303 (1995), no. 3, 389–416.