Limits of Hodge structures in several variables, \RomII\footnotemark[0]
Taro Fujisawa
Tokyo Denki University
e-mail: fujisawa@mail.dendai.ac.jp
The aim of this article is to study degeneration of the variations of Hodge structure associated to a proper K\"ahler semistable morphism. We prove that the weight filtrations constructed in \cite{FujisawaLHSSV coincide with the monodromy weight filtrations on the relative log de Rham cohomology groups of a proper K\"ahler semistable morphism. Moreover, we show that the limiting mixed Hodge structures form admissible variations of mixed Hodge structure.
1506.02271.tex
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\begin{document}

\title{Limits of Hodge structures in several variables, \RomII\footnotemark[0]}
\author{Taro Fujisawa \\
 \\
Tokyo Denki University \\
e-mail: fujisawa@mail.dendai.ac.jp}
\footnotetext[0]
{\hspace{-18pt}2010 {\itshape Mathematics Subject Classification}.
Primary 14D07; Secondary 14C30, 32G20. \\
{\itshape Key words and phrases}.
variation of Hodge structures, monodromy weight filtrations.}

\maketitle

\begin{abstract}
The aim of this article is to study
degeneration of the variations of Hodge structure
associated to a proper K\"ahler semistable morphism.
We prove that the weight filtrations
constructed in \cite{FujisawaLHSSV}
coincide with the monodromy weight filtrations
on the relative log de Rham cohomology groups
of a proper K\"ahler semistable morphism.
Moreover, we show that
the limiting mixed Hodge structures
form admissible variations of mixed Hodge structure.
\end{abstract}

\section{Introduction}

\begin{para}
\label{question in the general form}
Let $Y$ be a complex manifold
and $E$ a \snc divisor on it.
A morphism $f: X \longrightarrow Y$ from a complex manifold $X$ to $Y$
is said to be semistable (along $E$)
if $D=f^{\ast}E$ is a {\itshape reduced} \snc divisor on $X$
and if $f$ is log smooth in the sense of K. Kato
\cite{KazuyaKato}
(for the precise definition,
see Definition \ref{definition for semistable morphism}).
A proper K\"ahler semistable morphism $f: X \longrightarrow Y$ along $E$
induces variations of Hodge structure
$R^qf_{\ast}\bQ_X|_{Y \setminus E}$
for all $q$,
because $f$ is smooth over $Y \setminus E$.
In this article,
we study the degeneration of these variations of Hodge structure
from the algebro-geometric viewpoint.
Because the question is of local nature,
we mainly treat the case of $Y=\Delta^k \times S$
and $E=\{t_1t_2 \cdots t_k=0\}$,
where $\Delta^k$ is the $k$-dimensional polydisc
with the coordinate functions $t_1, t_2, \dots, t_k$
and $S$ is a complex manifold.
\end{para}

\begin{para}
The degeneration of the variation of Hodge structure
$R^qf_{\ast}\bQ_X|_{Y \setminus E}$
is described by Deligne's canonical extension of
$\cO_{Y \setminus E} \otimes R^qf_{\ast}\bQ_X|_{Y \setminus E}$
and its decreasing filtration
which is the extension of the Hodge filtration of
$\cO_{Y \setminus E} \otimes R^qf_{\ast}\bQ_X|_{Y \setminus E}$.

For the case of $Y=\Delta$,
Steenbrink \cite{SteenbrinkLHS}
proved that
$R^qf_{\ast}\Omega_{X/\Delta}(\log D)$
is the canonical extension of
$\cO_{\pd} \otimes R^qf_{\ast}\bQ_X|_{\pd}$
and constructed a mixed Hodge structure on 
\begin{equation*}
\coh^q(D, \cO_D \otimes \Omega_{X/\Delta}(\log D))
\simeq
\bC(0) \otimes R^qf_{\ast}\Omega_{X/\Delta}(\log D)
\end{equation*}
whose Hodge filtration is induced
by the stupid filtration on $\Omega_{X/\Delta}(\log D)$.
He also proved
that the stupid filtration
induces the desired filtration on
$R^qf_{\ast}\Omega_{X/\Delta}(\log D)$.
Moreover, he claimed that its weight filtration
coincides with the monodromy weight filtration
(see also \cite[Remarque 3.18]{ElZeinCE},
\cite[4.2.5 Remarque]{SaitoMorihikoMHP},
\cite[(A.1)]{UsuiMTTS}).

For the case of $Y=\Delta^k$,
Usui stated in \cite{UsuiRVCSV} that
$R^qf_{\ast}\Omega_{X/\Delta^k}(\log D)$
is the canonical extension of
$\cO_{\Delta^k \setminus E} \otimes R^qf_{\ast}\bQ_X|_{\Delta^k \setminus E}$
(see also Theorem \ref{theorem on the canonical extension} below).
Then, in \cite{FujisawaLHSSV},
the author constructed a mixed Hodge structure on
\begin{equation}
\label{relative log de Rham cohomology over 0:eq}
\coh^q(X_0, \cO_{X_0} \otimes \Omega_{X/\Delta^k}(\log D))
\simeq
\bC(0) \otimes R^qf_{\ast}\Omega_{X/\Delta^k}(\log D)
\end{equation}
whose Hodge filtration is induced
by the stupid filtration on $\Omega_{X/\Delta^k}(\log D)$,
where $X_0=f^{-1}(0)$,
and proved that the stupid filtration on $\Omega_{X/\Delta^k}(\log D)$
induces the desired filtration on
$R^qf_{\ast}\Omega_{X/\Delta^k}(\log D)$.

An aim of this article
is to prove that the weight filtration
of the mixed Hodge structure
\eqref{relative log de Rham cohomology over 0:eq}
coincides with the monodromy weight filtration
as in the case of $k=1$.
\end{para}

\begin{para}
\label{problems for VMHS}
For $Y=\Delta^k \times S$, we set
\begin{equation*}
\begin{split}
&Y[I]=\bigcap_{i \in I}\{t_i=0\} \\
&Y[I]^{\ast}=Y[I] \setminus (Y[I] \cap \bigcup_{j \notin I}\{t_j=0\})
\end{split}
\end{equation*}
for a subset $I \subset \ski$
as in \ref{notation for Y and E} below.
We note that we have $Y=\Delta^I \times Y[I]$,
where $\Delta^I$ is the $|I|$-dimensional polydisc
with the coordinate functions $(t_i)_{i \in I}$.

Now we consider the cohomology group
\begin{equation}
\label{relative log de Rham cohomology over s:eq}
\coh^q(X_s, \cO_{X_s} \otimes \Omega_{X/Y}(\log D)) ,
\end{equation}
for any $s \in Y[I]^{\ast}$,
where $X_s=f^{-1}(s)$.
Because the restriction
of the morphism $f$ over the closed subspace
$\Delta^I \times \{s\} \subset Y$
is again semistable,
we have a mixed Hodge structure
on the relative log de Rham cohomology groups
\eqref{relative log de Rham cohomology over s:eq}
as in \cite{FujisawaLHSSV}.
Thus we obtain a family of the mixed Hodge structures
parametrized by the points $s \in Y[I]^{\ast}$.
Then two questions arise naturally.
The first is whether
this family forms a variation of mixed Hodge structure.
If this is the case,
then the second is whether it is admissible.
Another aim of this article
is to give the affirmative answer to these questions.
\end{para}

\begin{para}
For the case of $Y=\Delta^k$,
the degeneration of abstract polarized variations of Hodge structure
on $(\pd)^k=\Delta^k \setminus E$
is intensively studied
by Cattani, Kaplan, Kashiwara, Kawai, Schmid and others,
apart from the morphism $f$.
The existence of the limiting mixed Hodge structure
of an abstract polarized variation of Hodge structure
on $(\pd)^k$
is proved in \cite[(6.16) Theorem]{Schmid}.
Combining results in \cite{Schmid} and \cite{Cattani-Kaplan}
we can show that
the limiting mixed Hodge structures
form an admissible variation of mixed Hodge structure
on $Y[I]^{\ast}$
(see also \cite[Section 5]{Fujino-Fujisawa}).

Coincidence of our weight filtration
and the monodromy weight filtration
on \eqref{relative log de Rham cohomology over 0:eq}
implies that our mixed Hodge structure
on \eqref{relative log de Rham cohomology over 0:eq}
coincides with Schmid's limiting mixed Hodge structure.
Thus the results in this article
are the algebro-geometric counterpart of
\cite{Schmid} and \cite{Cattani-Kaplan}.
\end{para}

\begin{para}
It is expected that
the results in this article
give the alternative proof of the fact
that $R^qf_{\ast}\Omega_{X/Y}(\log D)$ underlies
a logarithmic Hodge structure of weight $q$
in the sense of Kato-Usui
(cf. \cite[Definition 2.6.5]{KatoUsuiAMS}).
Remaining task is to compare the integral or rational structures.
We will not discuss this question in this article.
\end{para}

\begin{para}
This article is organized as follows.
Section \ref{preliminaries}
presents some preliminaries.
In Section \ref{semistable morpshims},
we give the precise definition of the semistable morphism.
Moreover, we fix the notation
which is constantly used in this article.
In Section \ref{section for complex B},
we construct the complex $sB_I(f)$
which is the replacement for the relative log de Rham complex
on the closed subspace $Y[I]$ for $I \subset \ski$,
and study the properties of it in detail.
Section \ref{gauss-manin}
is devoted to the computation of the Gauss-Manin connection
in terms of the complex $sB_I(f)$.
In Section \ref{section for the rational structure},
we construct rational structures
for the complex $sB_I(f)$,
which play important roles
in the following sections.
Section \ref{section for the proper case}
deals with the case
where the semistable morphism $f$ is proper.
By using the rational structures
constructed in the last section,
we prove that
the higher direct image sheaves of $sB_I(f)$
are locally free of finite rank.
In Section \ref{section for the Kahler case},
we give the answer to the first question
in \ref{problems for VMHS}:
we show that the mixed Hodge structures
on \eqref{relative log de Rham cohomology over s:eq}
form a variation of mixed Hodge structure
on $Y[I]^{\ast}$.
In the final section, Section \ref{section for the main results},
the main results of this article are proved.
We prove that our weight filtrations coincide
with the monodromy weight filtrations
and that the variation of mixed Hodge structure on $Y[I]^{\ast}$ above
is admissible.
\end{para}

\subsection*{Notation}

\begin{para}
The cardinality of a finite set $A$
is denoted by $|A|$.
\end{para}

\begin{para}
Once we fix a set $A$, the complement of a subset $B \subset A$
is denoted by $\overline{B}$,
that is, $\overline{B}=A \setminus B$.
\end{para}

\begin{para}
\label{notation for ZI}
For a finite set $I$,
\begin{equation*}
\bZ^I=\bigoplus_{i \in I}\bZ e_i
\end{equation*}
is the free $\bZ$-module of rank $|I|$
generated by $\{e_i\}_{i \in I}$.
We set $e_I=\sum_{i \in I}e_i \in \bZ^I$.
For $q=\sum_{i \in I}q_ie_i \in \bZ^I$,
we set $|q|=\sum_{i \in I}q_i$.
We also use the convention
\begin{align*}
&\bnnZ^I
=\{q=\sum_{i \in I}q_ie_i \in \bZ^I; q_i \ge 0 \text{ for all $i$}\} \\
&\bpZ^I
=\{q=\sum_{i \in I}q_ie_i \in \bZ^I; q_i > 0 \text{ for all $i$}\} .
\end{align*}
For a subset $J$ of $I$,
we canonically have
\begin{equation*}
\bZ^I = \bZ^J \oplus \bZ^{I \setminus J}
\end{equation*}
which induces the canonical inclusion
$\bZ^J \longrightarrow \bZ^I$
and the canonical projection
$\bZ^I \longrightarrow \bZ^J$.
For the case of $I=\ski$
we use $\bZ^k$ instead of $\bZ^I$.
Similarly, we write $e$ instead of $e_I$.
As usual, we write $q=(q_1,q_2, \dots, q_k)$
for $q=\sum_{i=1}^{k}q_ie_i \in \bZ^k$.

We use the notation $\bQ^I$, $\bR^I$, $\bpR^I$, $\bC^I$ etc.
in the same way as above.
\end{para}

\begin{para}
Let $\Lambda$ be a set.
We denote the set of all the subsets of $\Lambda$
is denoted by $S(\Lambda)$.
Moreover, we set
\begin{equation*}
S_n(\Lambda)=\{\Gamma \in S(\Lambda); |\Gamma|=n\}
\end{equation*}
for $n \in \bZ$.
If a decomposition into disjoint union
\begin{equation*}
\Lambda=\coprod_{i=1}^k\Lambda_i
\end{equation*}
is given,
we set
\begin{align*}
&S^q(\Lambda)
=\{\Gamma \in S(\Lambda); |\Gamma \cap \Lambda_i|=q_i
\text{ for any $i \in J$}\} \\
&S^{\ge q}(\Lambda)
=\{\Gamma \in S(\Lambda); |\Gamma \cap \Lambda_i| \ge q_i
\text{ for any $i \in J$}\} \\
&S^{\le q}(\Lambda)
=\{\Gamma \in S(\Lambda); |\Gamma \cap \Lambda_i| \le q_i
\text{ for any $i \in J$}\}
\end{align*}
for $q \in \bZ^J$ and for $J \subset I$.
Moreover, we set
\begin{equation*}
S_n^q(\Lambda)=S_n(\Lambda) \cap S^q(\Lambda), \quad
S_n^{\ge q}(\Lambda)=S_n(\Lambda) \cap S^{\ge q}(\Lambda), \quad
S_n^{\le q}(\Lambda)=S_n(\Lambda) \cap S^{\le q}(\Lambda)
\end{equation*}
for $n \in \bZ$.
\end{para}

\section{Preliminaries}
\label{preliminaries}

\begin{para}
In this section $(X, \cO_X)$ denotes a commutative ringed space,
i.e. $\cO_X$ is a sheaf of commutative rings on $X$.
We usually use the symbol $X$ instead of $(X, \cO_X)$
if there is no danger of confusion.
\end{para}

\begin{defn}
Let $\cL$ and $\cM$ be $\cO_X$-modules on $X$
and $F$ a finite decreasing filtration on $\cM$.
Then we define a finite decreasing filtration $F$ on $\cL \otimes \cM$
by setting
\begin{equation*}
F^p(\cL \otimes \cM)=\image(\cL \otimes F^p\cM \longrightarrow \cL \otimes \cM)
\end{equation*} 
for all $p$.
\end{defn}

\begin{rmk}
We have the canonical surjective morphisms
\begin{align}
&\cL \otimes F^p\cM
\longrightarrow
F^p(\cL \otimes \cM)
\label{canonical surjection for tensor product and F:eq} \\
&\cL \otimes \gr_F^p\cM
\longrightarrow
\gr_F^p(\cL \otimes \cM)
\label{canonical surjection for tensor product and gr:eq}
\end{align}
by definition.
\end{rmk}

\begin{rmk}
If $\gr_F^p\cM$ is $\cO_X$-flat for all $p$,
then the canonical morphism
\eqref{canonical surjection for tensor product and F:eq}
is an isomorphism for all $p$.
Then the canonical morphism
\eqref{canonical surjection for tensor product and gr:eq}
is an isomorphism for all $p$.
\end{rmk}

\begin{lem}
\label{lemma on sub and tensor and gr}
Let $\cL, \cM$ and $F$ be as above
and $\cN$ an $\cO_X$-submodule of $\cM$.
We set
\begin{align*}
&\cN\gr_F^p\cM=\image(\cN \cap F^p\cM \longrightarrow \gr_F^p\cM) \\
&\cN(\cL \otimes \cM)
=\image(\cL \otimes \cN \longrightarrow \cL \otimes \cM) \\
&\cN(\cL \otimes \gr_F^p\cM)
=\image(\cL \otimes \cN\gr_F^p\cM
\longrightarrow \cL \otimes \gr_F^p\cM) \\
&\cN\gr_F^p(\cL \otimes \cM)
=\image(\cN(\cL \otimes \cM) \cap F^p(\cL \otimes \cM)
\longrightarrow \gr_F^p(\cL \otimes \cM))
\end{align*}
as before.
Then the image of $\cN(\cL \otimes \gr_F^p\cM)$
by the canonical surjection
\eqref{canonical surjection for tensor product and gr:eq}
is $\cN\gr_F^p(\cL \otimes \cM)$.
If $\gr_F^p\cM$ is $\cO_X$-flat for all $p$,
then $\cN(\cL \otimes \gr_F^p\cM)$
is identified with
$\cN\gr_F^p(\cL \otimes \cM)$
under the canonical isomorphism
\eqref{canonical surjection for tensor product and gr:eq}.
\end{lem}
\begin{proof}
Easy.
\end{proof}

\begin{defn}
Let $\cL, \cM$ be $\cO_X$-modules
and $F_1, F_2, \dots, F_l$ finite decreasing filtrations on $\cM$.
Then the canonical morphism
\begin{equation}
\label{the canonical morphism for tensor product and gr's:eq}
\cL \otimes \gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2} \gr_{F_1}^{p_1}\cM
\longrightarrow
\gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2} \gr_{F_1}^{p_1}(\cL \otimes \cM)
\end{equation}
is obtained by the composition
\begin{equation*}
\begin{split}
\cL \otimes \gr_{F_l}^{p_l} \gr_{F_{l-1}}^{p_{l-1}} \gr_{F_{l-2}}^{p_{l-2}}
\cdots \gr_{F_2}^{p_2}\gr_{F_1}^{p_1}\cM
&\longrightarrow
\gr_{F_l}^{p_l}(\cL \otimes \gr_{F_{l-1}}^{p_{l-1}} \gr_{F_{l-2}}^{p_{l-2}}
\cdots \gr_{F_2}^{p_2}\gr_{F_1}^{p_1}\cM) \\
&\longrightarrow
\gr_{F_l}^{p_l}\gr_{F_{l-1}}^{p_{l-1}} (\cL \otimes \gr_{F_{l-2}}^{p_{l-2}}
\cdots \gr_{F_2}^{p_2}\gr_{F_1}^{p_1}\cM) \\
&\cdots \\
&\longrightarrow
\gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2}
(\cL \otimes \gr_{F_1}^{p_1}\cM) \\
&\longrightarrow
\gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2}\gr_{F_1}^{p_1}(\cL \otimes \cM),
\end{split}
\end{equation*}
where each step is the morphism
induced by the canonical morphism
\eqref{canonical surjection for tensor product and gr:eq}.
The canonical morphism
\eqref{the canonical morphism for tensor product and gr's:eq}
is surjective by Lemma \ref{lemma on sub and tensor and gr}.
\end{defn}

\begin{lem}
\label{local freeness and the commutattivity of gr}
Let $\cL$ and $\cM$ be $\cO_X$-modules
and $F_1, F_2, \dots, F_l$ finite decreasing filtrations on $\cM$.
If $\gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2}\gr_{F_1}^{p_1}\cM$
is $\cO_X$-flat for all $p_1,p_2, \dots, p_l$,
then the canonical morphism
\eqref{the canonical morphism for tensor product and gr's:eq}
is an isomorphism
for all $p_1, p_2, \dots, p_l$.
For a submodule $\cN$ of $\cM$,
the canonical isomorphism
\eqref{the canonical morphism for tensor product and gr's:eq}
identifies
$\cN(\cL \otimes \gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2}\gr_{F_l}^{p_l}\cM)$
with
$\cN\gr_{F_l}^{p_l} \cdots \gr_{F_2}^{p_2}\gr_{F_l}^{p_l}(\cL \otimes \cM)$.
\end{lem}
\begin{proof}
Easy from Lemma \ref{lemma on sub and tensor and gr}.
\end{proof}

\begin{defn}
Let $K$ be a complex of $\cO_X$-modules on $X$
and $F$ a finite decreasing filtration on $K$.
We say that $(K,F)$ (or $F$ on $K$) is strict
if the differential
$d:K^p \longrightarrow K^{p+1}$
is strictly compatible with $F$ for all $p$.
\end{defn}

\begin{lem}
\label{lemma on successively strict filtrations}
Let $K$ be a complex of $\cO_X$-modules
and $F_1, F_2, \dots, F_l$ finite decreasing filtrations on $K$.
We assume that
$(\gr_{F_i}^{p_i}\gr_{F_{i-1}}^{p_{i-1}} \cdots \gr_{F_1}^{p_1}K, F_j)$
is strict for $0 \le i < j \le l$.
$($For $i=0$, we mean that $(K,F_j)$ is strict for all $j$.$)$
Then there exists an isomorphism
\begin{equation*}
\coh^a(\gr_{F_i}^{p_i}\gr_{F_{i-1}}^{p_{i-1}} \cdots \gr_{F_1}^{p_1}K)
\simeq
\gr_{F_i}^{p_i}\gr_{F_{i-1}}^{p_{i-1}} \cdots \gr_{F_1}^{p_1}\coh^a(K)
\end{equation*}
for any $i$ with $1 \le i \le l$
and for all $a,p_1,p_2, \dots, p_i$,
under which the filtrations $F_j$
on the both sides
are identified for $j$ with $i< j \le l$.
\end{lem}
\begin{proof}
From the assumption that $(K,F_1)$ is strict,
we have the isomorphism
\begin{equation*}
\coh^a(\gr_{F_1}^{p_1}K)
\simeq
\gr_{F_1}^{p_1}\coh^a(K)
\end{equation*}
for all $a,p_1$,
under which the filtrations $F_j$
on the both sides are identified
by the lemma on two filtrations
\cite[Proposition (7.2.5)]{DeligneIII}
and by the strictness of $F_j$ on $\gr_{F_1}^{p_1}K$
for $j=2,3, \dots, l$.
Thus we have an isomorphism
\begin{equation*}
\coh^a(\gr_{F_2}^{p_2}\gr_{F_1}^{p_1}K)
\simeq
\gr_{F_2}^{p_2}\coh^a(\gr_{F_1}^{p_1}K)
\simeq
\gr_{F_2}^{p_2}\gr_{F_1}^{p_1}\coh^a(K)
\end{equation*}
for all $p_2$
by the strictness of $F_2$ on $\gr_{F_1}^{p_1}K$.
Lemma on two filtrations implies that
the filtration $F_j$ on the first term
are identified with $F_j$ on the second term
under the first isomorphism
by the strictness of $F_j$ on $\gr_{F_2}^{p_2}\gr_{F_1}^{p_1}K$
for $j=3,4, \dots, l$.
Hence the filtrations $F_j$ on the first term
are identified with $F_j$ on the third
under these isomorphisms.
By applying this procedure repeatedly,
we obtain the conclusion.
\end{proof}

\begin{defn}
Let $K$ be a complex of $\cO_X$-modules on $X$
and $F$ a finite decreasing filtration on $K$.
We say that $(K,F)$ (or $F$ on $K$)
is strongly strict
if $(K,F)$ is strict
and if $\coh^a(\gr_F^pK)$ is a locally free $\cO_X$-module
of finite rank for all $a,p$.
\end{defn}

\begin{rmk}
If $(K,F)$ is strongly strict,
then we have the following:
\begin{mylist}
\itemno
The sequence
\begin{equation*}
\begin{CD}
0 @>>> \coh^a(F^{p-1}K)
  @>>> \coh^a(F^pK)
  @>>> \coh^a(\gr_F^pK) @>>> 0
\end{CD}
\end{equation*}
is exact for all $a,p$.
In particular,
the canonical morphism
\begin{equation*}
\coh^a(F^pK) \longrightarrow F^p\coh^a(K)
\end{equation*}
is an isomorphism for all $a,p$.
\itemno
The $\cO_X$-module $\coh^a(F^pK)$
is locally free of finite rank for all $a,p$.
\end{mylist}
\end{rmk}

\begin{rmk}
The notion of strong strictness is well-defined
in the filtered derived category
in the following sense.
If $(K_1,F)$ and $(K_2,F)$
are isomorphic in the filtered derived category,
then $(K_1,F)$ is strongly strict
if and only if $(K_2,F)$ is strongly strict.
\end{rmk}

\begin{lem}
\label{lemma on strong strictness and filtered tensor product}
Let $(K,F)$ be a filtered perfect complex of $\cO_X$-modules
and $\cF$ an $\cO_X$-module.
The filtered derived tensor product is denoted by $(\cF \otimes^L K,F)$.
If $(K,F)$ is strongly strict,
then $(\cF \otimes^L K,F)$ is strict
and the canonical morphisms
\begin{align}
&\cF \otimes \coh^a(F^pK)
\longrightarrow
\coh^a(\cF \otimes^L F^pK)
\label{canonical morphism for F:eq} \\
&\cF \otimes \coh^a(\gr_F^pK)
\longrightarrow
\coh^a(\cF \otimes^L \gr_F^pK)
\label{canonical morphism for grF:eq}
\end{align}
are isomorphisms for all $a,p$.
In particular,
$(\cA \otimes^L K, F)$ is strongly strict
as a filtered complex of $\cA$-modules
for a commutative $\cO_X$-algebra $\cA$.
\end{lem}
\begin{proof}
The canonical morphism
\eqref{canonical morphism for grF:eq} is an isomorphism
because $\coh^a(\gr_F^pK)$ is locally free for all $a,p$.
Then we have the commutative diagram
\begin{equation*}
\begin{CD}
0 @>>> \cF \otimes \coh^a(F^{p-1}K)
  @>>> \cF \otimes \coh^a(F^pK)
  @>>> \cF \otimes \coh^a(\gr_F^pK)
  @>>> 0 \\
@. @VVV @VVV @VV{\simeq}V @. \\
0 @>>> \coh^a(\cF \otimes^L F^{p-1}K)
  @>>> \coh^a(\cF \otimes^L F^pK)
  @>>> \coh^a(\cF \otimes^L \gr_F^pK)
  @>>> 0 ,
\end{CD}
\end{equation*}
in which the first row is exact
because $(K, F)$ is strongly strict.
Then we can see that the second row in the diagram above
is exact and the canonical morphism
\eqref{canonical morphism for F:eq} is an isomorphism for all $a,p$
by induction on $p$.
The injectivity of the morphism
\begin{equation*}
\coh^a(\cF \otimes^L F^{p-1}K)
\longrightarrow
\coh^a(\cF \otimes^L F^pK)
\end{equation*}
for all $p$
implies that $(\cF \otimes^L K, F)$ is strict.
\end{proof}

For the case where $X$ is a complex manifold,
we have the following results.

\begin{lem}
\label{strong strictness criteria by pointwise strictness}
Let $X$ be a complex manifold
and $(K,F)$ a filtered perfect complex
such that $\coh^a(K)$ is locally free of finite rank for all $a$.
Then $(K,F)$ is strongly strict
if and only if $(\bC(x) \otimes^L K,F)$ is strict for every point $x \in X$.
\end{lem}

This lemma is a direct consequence
of Lemma 3.4 (iv) in \cite{Fujino-Fujisawa}.
However, the proof of Lemma 3.4 (iv) in \cite{Fujino-Fujisawa}
is somewhat misleading.
So, we restate the result and complete the proof.

\begin{lem}[Lemma 3.4 (iv) of \cite{Fujino-Fujisawa}]
Let $(K, F)$ be a filtered perfect complex on a complex manifold $X$.
Assume that the function
$X \ni x \mapsto \dim \coh^q(\bC(x) \otimes^L K)$
is locally constant.
If the morphisms
\begin{equation}
\label{differentials on C(x) otimes K:eq}
\begin{split}
&d(x):
(\bC(x) \otimes^L K)^{q-1}
\longrightarrow
(\bC(x) \otimes^L K)^q, \\
&d(x):
(\bC(x) \otimes^L K)^q
\longrightarrow
(\bC(x) \otimes^L K)^{q+1}
\end{split}
\end{equation}
are strictly compatible with the filtration
$F(\bC(x) \otimes^L K)$ for all $x \in X$,
then $\coh^q(\gr_F^pK)$ and $\coh^q(F^pK)$ are locally free of finite rank,
the canonical morphisms
\begin{equation}
\label{canonical morphisms for grF and F:eq}
\begin{split}
&\bC(x) \otimes \coh^q(\gr_F^pK)
\longrightarrow
\coh^q(\bC(x) \otimes^L \gr_F^pK) \\
&\bC(x) \otimes \coh^q(F^pK)
\longrightarrow
\coh^q(\bC(x) \otimes^L F^pK)
\end{split}
\end{equation}
are isomorphisms for all $p$ and for all $x \in X$,
and the differentials
\begin{equation}
\label{differentials on K:eq}
\begin{split}
&d: K^{q-1} \longrightarrow K^q \\
&d: K^q \longrightarrow K^{q+1}
\end{split}
\end{equation}
are strictly compatible with the filtration $F$.
\end{lem}
\begin{proof}
Note that we have the canonical isomorphisms
\begin{equation*}
\begin{split}
&\bC(x) \otimes^L F^pK
\simeq
F^p(\bC(x) \otimes^L K) \\
&\bC(x) \otimes^L \gr_F^pK
\simeq
\gr_F^p(\bC(x) \otimes^L K)
\end{split}
\end{equation*}
by the definition of filtered derived tensor product.
We obtain the exact sequences
\begin{equation}
\label{exact sequence for K otimes C(x):eq}
\begin{CD}
0 @>>> \coh^q(\bC(x) \otimes^L F^{p+1}K)
  @>>> \coh^q(\bC(x) \otimes^L F^pK) \\
@. @>>> \coh^q(\bC(x) \otimes^L \gr_F^pK)
   @>>> 0
\end{CD}
\end{equation}
from the assumption that
the differentials \eqref{differentials on C(x) otimes K:eq}
are strictly compatible with the filtration $F$.
Thus we have the equality
\begin{equation}
\label{equality for the dimension of the cohomology:eq}
\dim \coh^q(\bC(x) \otimes^L F^pK)
=
\sum_{p' \ge p}\dim \coh^q(\bC(x) \otimes^L \gr_F^{p'}K)
\end{equation}
for all $p$ and for all $x \in X$.
Taking $p$ sufficiently small,
we obtain
\begin{equation*}
\dim \coh^q(\bC(x) \otimes^L K)
=
\sum_p\dim \coh^q(\bC(x) \otimes^L \gr_F^pK) ,
\end{equation*}
which implies that the function
$X \ni x \mapsto \dim \coh^q(\bC(x) \otimes^L \gr_F^pK)$
is locally constant
as in the proof of Lemma 3.4 (iv) in \cite{Fujino-Fujisawa}.
Then the function
$X \ni x \mapsto \dim \coh^q(\bC(x) \otimes^L F^pK)$
is locally constant from the equality
\eqref{equality for the dimension of the cohomology:eq}.
Therefore $\coh^q(F^pK)$ and $\coh^q(\gr_F^pK)$ are locally free
and the canonical morphisms
\eqref{canonical morphisms for grF and F:eq}
are isomorphisms for all $p$
by Lemma 3.4 (iii) in \cite{Fujino-Fujisawa}.
From the exact sequence
\eqref{exact sequence for K otimes C(x):eq},
we have the exact sequence
\begin{equation*}
\begin{CD}
0 @>>> \bC(x) \otimes \coh^q(F^{p+1}K)
  @>>> \bC(x) \otimes \coh^q(F^pK) \\
@. @>>> \bC(x) \otimes \coh^q(\gr_F^pK)
   @>>> 0 ,
\end{CD}
\end{equation*}
from which we obtain the exact sequence
\begin{equation*}
\begin{CD}
0 @>>> \coh^q(F^{p+1}K)
  @>>> \coh^q(F^pK)
  @>>> \coh^q(\gr_F^pK)
  @>>> 0
\end{CD}
\end{equation*}
by using the local freeness of $\coh^q(F^{p+1}K)$ for all $p$.
Therefore the differentials
\eqref{differentials on K:eq}
are strictly compatible with $F$ as desired.
\end{proof}

\section{Semistable morphism}
\label{semistable morpshims}

\begin{defn}
\label{definition for semistable morphism}
Let $Y$ be a complex manifold and $E$ a \snc divisor on $Y$.
A morphism of complex manifolds
$f: X \longrightarrow Y$
is said to be semistable along $E$
if the following two conditions are satisfied:
\begin{mylist}
\itemno
\label{condition for D being reduced}
The divisor $D=f^{\ast}E$
is a {\slshape reduced} \snc divisor on $X$.
\itemno
\label{second condition for a semistable morphism}
The morphism
\begin{equation*}
f^{\ast}\Omega_Y^1(\log E) \longrightarrow \Omega_X^1(\log D)
\end{equation*}
is injective
and the cokernel
\begin{equation*}
\Omega_{X/Y}^1(\log D)
=\Omega_X^1(\log D)/f^{\ast}\Omega_Y^1(\log E)
\end{equation*}
is locally free.
\end{mylist}
Once we fix the divisor $E$,
the words ``along $E$'' is omitted.
Moreover, we use the notation
\begin{equation*}
\omega^p_Y=\Omega^p_Y(\log E), \quad
\omega^p_X=\Omega^p_X(\log D), \quad
\omega^p_{X/Y}=\Omega^p_{X/Y}(\log D)
\end{equation*}
if there is no danger of confusion.

We set
$Y^{\ast}=Y \setminus E$
and $X^{\ast}=X \setminus D=f^{-1}(Y^{\ast})$.
\end{defn}

\begin{exa}
Let $\Delta$ be the unit disc in $\bC$
with the coordinate function $t$,
and $\Delta^n$ the polydisc in $\bC^n$ with the coordinate functions
$x_1, x_2, \dots, x_n$.
A morphism $f: \Delta^n \longrightarrow \Delta$ given by
\begin{equation}
\label{semistable over Delta}
f^{\ast}t=x_1x_2 \cdots x_n
\end{equation}
is semistable along $\{0\}$.

Let $f_i:\Delta^{n_i} \longrightarrow \Delta$ ($i=1,2,\dots, k$)
be morphisms given by the form \eqref{semistable over Delta}
and $g: T \longrightarrow S$ a smooth morphism of complex manifolds.
Set $Y=\Delta^k \times S$ and $E=\{t_1t_2 \cdots t_k=0\}$
where $t_1, t_2, \dots, t_k$ is the coordinate functions of $\Delta^k$.
Then the morphism
\begin{equation}
\label{local model of semistable morphism:eq}
f_1 \times f_2 \times \dots \times f_k \times g:
\Delta^{n_1} \times \Delta^{n_2} \times \dots \times \Delta^{n_k} \times T
\longrightarrow
\Delta^k \times S
\end{equation}
is also semistable along $E$.
\end{exa}

\begin{lem}
\label{local description of semistable morphism}
Locally on $X$ and $Y$,
any semistable morphism $f: X \longrightarrow Y$
is isomorphic to the morphism of the form
\eqref{local model of semistable morphism:eq}.
In particular, any semistable morphism is equi-dimensional and flat.
\end{lem}
\begin{proof}
Similar to the proof of Lemma (6.5) in \cite{FujisawaLHSSV}.
\end{proof}

\begin{defn}
\label{definition of the filtration G}
Let $f:X \longrightarrow Y$ be a semistable morphism along $E$.
A finite decreasing filtration $G$ on $\omega_X$
is defined by
\begin{equation*}
G^p\omega_X^n
=\image(f^{-1}\omega^p_Y \otimes_{f^{-1}\cO_Y} \omega^{n-p}_X
\overset{\wedge}{\longrightarrow} \omega^n_X)
\end{equation*}
for all $n,p$ as in \cite{Katz-Oda}.
Then we have the canonical isomorphism
\begin{equation*}
f^{-1}\omega^p_Y \otimes_{f^{-1}\cO_Y} \omega_{X/Y}[-p]
\overset{\simeq}{\longrightarrow}
\gr_G^p\omega_X
\end{equation*}
for every $p$.
Therefore the spectral sequence
$E_r^{a,b}(Rf_{\ast}\omega_X,G)$
satisfies the property
\begin{equation*}
\begin{split}
E_1^{a,b}(Rf_{\ast}\omega_X,G)
&\simeq
R^bf_{\ast}(f^{-1}\omega^a_Y \otimes_{f^{-1}\cO_Y}\omega_{X/Y}) \\
&\simeq
\omega^a_Y \otimes_{\cO_Y} R^bf_{\ast}\omega_{X/Y}
\end{split}
\end{equation*}
for all $a,b$
because $\omega^a_Y$ is a locally free $\cO_Y$-module of finite rank.
Then the morphism of $E_1$-terms
induces the morphism
\begin{equation}
\label{the Gauss-Manin connection:eq}
R^nf_{\ast}\omega_{X/Y}
\longrightarrow
\omega^1_Y \otimes_{\cO_Y} R^nf_{\ast}\omega_{X/Y}
\end{equation}
which is called the Gauss-Manin connection
and denoted by $\nabla$.
The residue of
the Gauss-Manin connection \eqref{the Gauss-Manin connection:eq}
along the divisor $E_i$
is denoted by
\begin{equation}
\label{the residue of nabla:eq}
\res_{E_i}(\nabla):
\cO_{E_i} \otimes_{\cO_Y} R^nf_{\ast}\omega_{X/Y}
\longrightarrow
\cO_{E_i} \otimes_{\cO_Y} R^nf_{\ast}\omega_{X/Y}
\end{equation}
for every $i=1,2,\dots,k$.
\end{defn}

\begin{rmk}
As in \cite{Katz-Oda},
$\nabla$ is an integrable logarithmic connection,
that is,
$\nabla$ satisfies the Leibniz rule
and $\nabla^2=0$.
Moreover, it satisfies the Griffiths transversality
\begin{equation*}
\nabla(F^pR^nf_{\ast}\omega_{X/Y})
\subset
\omega^1_Y \otimes F^{p-1}R^nf_{\ast}\omega_{X/Y}
\end{equation*}
where $F$ denotes the filtration
induced by the stupid filtration $F$ on $\omega_{X/Y}$.

On the other hand,
the restriction of $\nabla$ on $Y^{\ast}=Y \setminus E$
is identified with
\begin{equation*}
d \otimes \id:
\cO_{Y^{\ast}} \otimes_{\bC} R^nf_{\ast}\bC_{X^{\ast}}
\longrightarrow
\Omega^1_{Y^{\ast}} \otimes_{\bC} R^nf_{\ast}\bC_{X^{\ast}}
\end{equation*}
via the isomorphisms
\begin{equation*}
R^nf_{\ast}\omega_{X/Y}|_{Y^{\ast}}
=
R^nf_{\ast}\Omega_{X^{\ast}/Y^{\ast}}
\simeq
R^nf_{\ast}f^{-1}\cO_{Y^{\ast}}
\simeq
\cO_{Y^{\ast}} \otimes_{\bC} R^bf_{\ast}\bC_{X^{\ast}}
\end{equation*}
as in \cite{Katz-Oda}, \cite[4.3]{Fujino-Fujisawa}.
\end{rmk}

\begin{para}
\label{notation for Y and E}
Let $Y$ be a complex manifold,
$E$ a \snc divisor on $Y$
and
\begin{equation*}
E=\sum_{i=1}^{k}E_i
\end{equation*}
the irreducible decomposition of $E$.
We set
\begin{equation*}
E_I=\sum_{i \in I}E_i, \quad
Y[I]=\bigcap_{i \in I}E_i
\end{equation*}
for $I \subset \ski$.
(We set $E_{\emptyset}=0, Y[\emptyset]=Y$.)
We write $Y_0=Y[\ski]$ for short.
Since $E$ is a \snc divisor,
$Y[I]$ is a closed submanifold of $Y$ for any $I$.
We denote the closed immersion
$Y[I] \hookrightarrow Y$ by $\iota_I$
($Y_0=Y[\ski] \hookrightarrow Y$ by $\iota$).
By setting $\overline{I}=\ski \setminus I$,
we have the \snc divisor $E_{\overline{I}}$ on $Y$,
whose restriction to $Y[I]$
is denoted by $E_{\overline{I}} \cap Y[I]$.
We use the notation
\begin{equation*}
\omega_{Y[I]}^p=\Omega_{Y[I]}^p(\log E_{\overline{I}} \cap Y[I])
\end{equation*}
for all $p$.
We set
$Y[I]^{\ast}=Y[I] \setminus (E_{\overline{I}} \cap Y[I])$
for $I \subset \ski$.

We have a sequence of closed submanifolds
\begin{equation*}
Y_0=Y[\ski] \subset Y[I] \subset Y[J] \subset Y[\emptyset]=Y
\end{equation*}
for $J \subset I \subset \ski$.
\end{para}

\begin{para}
\label{setting for a semistable morphism}
Let $f:X \longrightarrow Y$ be a semistable morphism along $E$.
We set $D=f^{\ast}E$ and $D_i=f^{\ast}E_i$ for $i=\ki$.
Then $D$ and $D_i$'s are {\itshape reduced} \snc divisors on $X$
with $D=\sum_{i=1}^{k}D_i$.
We set
\begin{equation*}
D_I=\sum_{i \in I}D_i, \quad
X[I]=\bigcap_{i \in I}D_i
\end{equation*}
for $I \subset \ski$
($D_{\emptyset}=0, X[\emptyset]=X$ as above).
We write $X_0=X[\ski]$ as before.
The closed immersion
$X[I] \hookrightarrow X$
(resp. $X_0 \hookrightarrow X$) is also denoted by $\iota_I$
(resp. $\iota$)
if there is no danger of confusion.
We have the cartesian square
\begin{equation*}
\begin{CD}
X[I] @>{\iota_I}>> X \\
@V{f_I}VV @VV{f}V \\
Y[I] @>>{\iota_I}> Y
\end{CD}
\end{equation*}
in which the left vertical arrow is denoted by $f_I$.
We use the notation $f_{\ski}=f_0$,
which fits in the cartesian square
\begin{equation*}
\begin{CD}
X_0 @>{\iota}>> X \\
@V{f_0}VV @VV{f}V \\
Y_0 @>>{\iota}> Y
\end{CD}
\end{equation*}
by definition.
We set
\begin{equation*}
X[I]^{\ast}
=X[I] \setminus (D_{\overline{I}} \cap X[I])
=f_I^{-1}(Y[I]^{\ast})
\end{equation*}
for $I \subset \ski$.

Let
$D=\sum_{\lambda \in \Lambda}D_{\lambda}$
be the irreducible decomposition of $D$.
We set
\begin{equation*}
X[\Gamma]=\bigcap_{\lambda \in \Gamma}D_{\lambda}
\end{equation*}
for $\Gamma \subset \Lambda$
as before.
The closed immersion
$X[\Gamma] \hookrightarrow X$ is denoted by $\iota_{\Gamma}$.

From the equality $D=\sum_{i=1}^{k}D_i$ and the fact that $D$ is reduced,
the index set $\Lambda$ decomposes into a disjoint union
\begin{equation*}
\Lambda = \coprod_{i=1}^k\Lambda_i
\end{equation*}
such that
\begin{equation*}
D_i=\sum_{\lambda \in \Lambda_i}D_{\lambda}
\end{equation*}
for $i=\ki$.
We set
\begin{equation*}
\Lambda_I=\coprod_{i \in I}\Lambda_i \subset \Lambda
\end{equation*}
for $I \subset \ski$.
If $\Gamma \cap \Lambda_i \not= \emptyset$ for all $i \in I$,
then $X[\Gamma] \subset X[I]$.
Thus we have a commutative diagram
\begin{equation}
\label{square for Gamma:eq}
\begin{CD}
X[\Gamma] @>{\iota_{\Gamma}}>> X \\
@V{f_{\Gamma}}VV @VV{f}V \\
Y[I] @>>{\iota_I}> Y
\end{CD}
\end{equation}
in which the left vertical arrow is denoted by $f_{\Gamma}$,
once we fix $I \subset \ski$.
For $\Gamma \subset \Lambda_I$,
the restriction $D_{\overline{I}} \cap X[\Gamma]$
of the divisor $D_{\overline{I}}$ to $X[\Gamma]$
is a \snc divisor on $X[\Gamma]$.
The morphism $f_{\Gamma}:X[\Gamma] \longrightarrow Y[I]$ above
is semistable along $E_{\overline{I}} \cap Y[I]$
by Lemma \ref{local description of semistable morphism}.
\end{para}

\section{Construction of $sB_I(f)$}
\label{section for complex B}

\begin{para}
\label{local situation}
From now on,
we treat the case of $Y=\Delta^k \times S$
equipped with a \snc divisor $E=\{t_1t_2 \cdots t_k=0\}$,
where $t_1, t_2, \dots, t_k$ are the coordinate functions of $\Delta^k$.
By setting $E_i=\{t_i=0\}$ for $i=\ki$,
we have
$E=\sum_{i=1}^kE_i$.
We use the notation in \ref{notation for Y and E}.
We have $S=\{0\} \times S=Y[\ski]=Y_0$ by definition.

There exists the canonical projection
\begin{equation*}
\pi_I: Y \longrightarrow Y[I]
\end{equation*}
for every $I$.
We trivially have $\pi_I\iota_I=\id$.
When we consider $Y[I]$ as a complex manifold {\itshape below} $Y$
by the projection $\pi_I$ above,
we use the notation $Y_I$ instead of $Y[I]$.
The \snc divisor $E_{\overline{I}} \cap Y[I]$ on $Y[I]$
defines a \snc divisor on $Y_I$,
which we denote by $E_{\overline{I}} \cap Y_I$.
We always consider
that $Y_I$ is equipped with the divisor $E_{\overline{I}} \cap Y_I$,
unless the otherwise is mentioned.
Thus we use the notation
\begin{equation*}
\omega^p_{Y_I}=\Omega^p_{Y_I}(\log E_{\overline{I}} \cap Y_I)
\end{equation*}
as in Definition \ref{definition for semistable morphism}
and \ref{notation for Y and E}.
We set
$Y_I^{\ast}=Y_I \setminus (E_{\overline{I}} \cap Y_I)$
for $I \subset \ski$.
We trivially have $Y[I]^{\ast}=Y_I^{\ast}$ as complex manifolds.

For $J \subset I \subset \ski$,
we have a sequence of the projections
\begin{equation*}
Y=Y_{\emptyset}
\longrightarrow Y_J
\longrightarrow Y_I
\longrightarrow Y_{\ski}=S
\end{equation*}
by definition.
We use $\pi:Y \longrightarrow S$ instead of $\pi_{\ski}$.

For $J \subset I \subset \ski$,
the coordinate functions $t_1, t_2, \dots, t_k$
induce the direct sum decomposition
\begin{equation*}
\omega^1_{Y[J]}
\simeq
\omega^1_{Y[J]/Y_I} \oplus \omega^1_{Y[I]}
\end{equation*}
which induces the projection
\begin{equation}
\pr_{J|I}:
\omega^p_{Y[J]} \longrightarrow \omega^p_{Y[I]}
\label{the projection between Omega from Y[J] to Y[I]:eq}
\end{equation}
for every $p$.
\end{para}

\begin{rmk}
For a semistable morphism $f:X \longrightarrow Y$ along $E$,
$\pi_If:X \longrightarrow Y_I$
and $\pi_If\iota_{\Gamma}:X[\Gamma] \longrightarrow Y_I$
are semistable along $E_{\overline{I}} \cap Y_I$
for $I \subset \ski$ and for $\Gamma \subset \Lambda_I$.
We can see these facts
by Lemma \ref{local description of semistable morphism}.
Moreover, we have
$\pi_If\iota_{\Gamma}=f_{\Gamma}$ via the identification
$Y_I=Y[I]$.
\end{rmk}

Now, we construct a complex $sB_I(f)$
which is a replacement for
$f_I^{-1}\cO_{Y[I]} \otimes^L_{f^{-1}\cO_Y} \omega_{X/Y}$
for $I \subset \ski$.

\begin{defn}
\label{definition of B_I(f)}
Let $Y$ and $E$ be as above
and $f:X \longrightarrow Y$ a semistable morphism along $E$.
For a subset $I \subset \ski$,
a complex $sB_I(f)$ is defined as follows.
\begin{mylist}
\itemno
For $p \in \bnnZ, q \in \bnnZ^I$, we set
\begin{equation*}
B_I(f)^{p,q}
=\omega_{X/Y_I}^{p+|q|+|I|}\left/
\sum_{i \in I}W(D_i)_{q_i} \right.
\end{equation*}
where $W(D_i)$ denotes the filtration
induced by $W(D_i)$ on $\omega_X$.
\itemno
We set
\begin{equation*}
d_0=(-1)^{|I|}d:B_I(f)^{p,q} \longrightarrow B_I(f)^{p+1,q}
\end{equation*}
where $d$ on the right hand side of the equality
is the morphism induced from $d$
on $\omega_{X/Y_I}$.
\itemno
We set
\begin{equation*}
d_i=(-1)^{|I|}\dlog t_i \wedge:B_I(f)^{p,q} \longrightarrow B_I(f)^{p,q+e_i}
\end{equation*}
where $\dlog t_i \wedge$ denotes the morphism defined by
\begin{equation*}
\omega
\mapsto
\dlog t_i \wedge \omega=\frac{dt_i}{t_i} \wedge \omega
\end{equation*}
as in \cite{FujisawaPLMHS}.
\end{mylist}
By setting
\begin{equation*}
sB_I(f)^n
=\bigoplus_{p+|q|=n}B_I(f)^{p,q}
=\bigoplus_{q \in \bnnZ^I}
\left(\omega_{X/Y_I}^{n+|I|}\left/
\sum_{i \in I}W(D_i)_{q_i} \right.\right)
\end{equation*}
with the differential
$d=d_0+\sum_{i \in I}d_i$,
we obtain the complex $sB_I(f)$.
We have
\begin{equation}
\label{support of B:eq}(
\supp(B_I(f)^{p,q}) \subset X[I]
\end{equation}
for any $p,q$.
Thus we identify
$B_I(f)^{p,q}$ and $\iota_I^{-1}B_I(f)^{p,q}$
for all $p,q$.
Moreover $B_I(f)^{p,q}(=\iota_I^{-1}B_I(f)^{p,q})$
is an $\cO_{X[I]}$-module for any $p,q$.
However $sB_I(f)$ is considered
as a complex of $f_I^{-1}\cO_{Y[I]}$-modules
because $d_0$ is not a morphism of $\cO_{X[I]}$-modules.
We use the symbol $sB(f)$ for the case of $I=\ski$.

We define a finite decreasing filtration $F$ on $sB_I(f)$ by
\begin{equation}
\label{definition of F on sBI(f):eq}
F^psB_I(f)^n
=\bigoplus_{\substack{p'+|q|=n \\
                      p' \ge p}}
B_I(f)^{p',q}
=\bigoplus_{\substack{q \in \bnnZ^I \\
                      |q| \le n-p}}
\left(\omega_{X/Y_I}^{n+|I|}\left/
\sum_{i \in I}W(D_i)_{q_i} \right.\right)
\end{equation}
for all $n,p$.

By definition,
$sB_{\emptyset}(f)$ equipped with $F$
is nothing but $\omega_{X/Y}$
equipped with the stupid filtration $F$.
\end{defn}

\begin{rmk}
In the case where $S$ is a point
and $I=\ski$,
the complex $sB(f)$ is same as
$sB_X(D_1, \dots, D_k)$ in \cite[(3.12)]{FujisawaLHSSV}
except the sign of the differentials.
\end{rmk}

\begin{rmk}
\label{remark for grF}
Because we have $d_0(F^psB_I(f)^n) \subset F^{p+1}sB_I(f)^{n+1}$
for all $n,p$,
the complex
$\gr_F^psB_I(f)$ turns out to be a complex of $\cO_{X[I]}$-modules
for all $p$.
We trivially have
\begin{equation}
\label{description of grF:eq}
\gr_F^psB_I(f)^n
=\bigoplus_{\substack{q \in \bnnZ^I \\
                      |q|=n-p}}
B_I(f)^{p,q}
=\bigoplus_{\substack{q \in \bnnZ^I \\
                      |q|=n-p}}
\left(\omega^{n+|I|}_{X/Y_I}\left/
\sum_{i \in I}W(D_i)_{q_i}\right.\right)
\end{equation}
for all $n, p$.
\end{rmk}

\begin{defn}
\label{definition of L(K) on BI}
In the same situation as in Definition \ref{definition of B_I(f)},
a finite increasing filtration $L(K)$ on $B_I(f)^{p,q}$ for $K \subset I$
is defined by
\begin{equation}
L(K)_mB_I(f)^{p,q}
=W(D_K)_{m+2|q_K|+|K|}\omega_{X/Y_I}^{p+|q|+|I|}\left/
\sum_{i \in I}W(D_i)_{q_i} \right.
\label{definition of L(K) on BI:eq}
\end{equation}
where $q_K$ is the image of $q$
by the projection $\bZ^I \longrightarrow \bZ^K$
as in \ref{notation for ZI}.
We can easily see that $L(K)$ is preserved by every $d_i$.
Thus we obtain a finite increasing filtration $L(K)$ on $sB_I(f)$.
We writhe $L=L(\ski)$ on $sB(f)$ for short.
By definition, $L(\emptyset)$ is the trivial filtration,
that is,
$L(\emptyset)_{-1}sB_I(f)=0$
and $L(\emptyset)_0sB_I(f)=sB_I(f)$.
\end{defn}

\begin{lem}
\label{lemma on grL(K)sB}
For any $K \subset I$,
an isomorphism of complexes
\begin{equation}
\label{grL(K)sB:eq}
\begin{split}
\gr_m^{L(K)}sB_I(f)
&\simeq
\bigoplus_{q \in \bnnZ^K}
\bigoplus_{\Gamma \in S_{m+2|q|+|K|}^{\ge q+e_K}(\Lambda_K)}
sB_{I \setminus K}(\pi_Kf\iota_{\Gamma})[-m-2|q|] \\
&=
\bigoplus_{q \in \bnnZ^K}
\bigoplus_{\Gamma \in S_{m+2|q|+|K|}^{\ge q+e_K}(\Lambda_K)}
sB_{I \setminus K}(f_{\Gamma})[-m-2|q|]
\end{split}
\end{equation}
is induced from the residue morphism for $W(D_K)$,
where $f_{\Gamma}:X[\Gamma] \longrightarrow Y[K]$
is the morphism
in the commutative diagram \eqref{square for Gamma:eq}.
Under the isomorphism \eqref{grL(K)sB:eq} above,
we have
\begin{equation*}
\begin{split}
&L(J)\gr_m^{L(K)}sB_I(f) \\
&\qquad
\simeq
\bigoplus
L(J \setminus K \cap J)
[\,|\Gamma \cap \Lambda_{K \cap J}|-2|q_{K \cap J}|-|K \cap J|\,]\,
sB_{I \setminus K}(f_{\Gamma})[-m-2|q|]
\end{split}
\end{equation*}
for any $J \subset I$ and
\begin{equation*}
F\gr_m^{L(K)}sB_I(f)
\simeq
\bigoplus F[-m-|q|]\,sB_{I \setminus K}(f_{\Gamma})[-m-2|q|] ,
\end{equation*}
where the index sets of the direct sums above
are the same as in \eqref{grL(K)sB:eq}.
In particular, we have
\begin{equation*}
L(J)\gr_m^{L(K)}sB_I(f)
\simeq
\bigoplus
L(J \setminus K)[m]
sB_{I \setminus K}(f_{\Gamma})[-m-2|q|]
\end{equation*}
for $J \supset K$
and
\begin{equation*}
L(J)\gr_m^{L(K)}sB_I(f)
\simeq
\bigoplus
L(J)
sB_{I \setminus K}(f_{\Gamma})[-m-2|q|]
\end{equation*}
for $J$ with $J \cap K=\emptyset$.
\end{lem}
\begin{proof}
We obtain the conclusion
by Lemma \ref{local description of semistable morphism}
and by the local computation
as in \cite[Proposition 3.6]{DeligneED}.
\end{proof}

\begin{rmk}
In the lemma above,
we use the residue morphism
defined as in \cite[\RomII.3.7]{DeligneED}.
It is different by the sign
from the residue morphism used in \cite{FujisawaLHSSV}.
\end{rmk}

\begin{lem}
\label{flatness lemma}
For $K \subset I$,
the $\cO_{X[I]}$-sheaf
$\gr_F^p\gr_m^{L(K)}sB_I(f)^n$ is $f_I^{-1}\cO_{Y[I]}$-flat
for all $m,n,p$.
\end{lem}
\begin{proof}
First, we treat the case of $K=I$.
We have an isomorphism
\begin{equation*}
\gr_F^p\gr_m^{L(I)}sB_I(f)^n
\simeq
\bigoplus
\omega^{n-m-2|q|}_{X[\Gamma]/Y[I]}
\end{equation*}
from Lemma \ref{lemma on grL(K)sB},
where the direct sum on the right hand side
is taken over the index set
\begin{equation*}
\{(q,\Gamma)| q \in \bnnZ^I \text{ with } |q|=n-p,
\Gamma \in S_{m+2|q|+|I|}^{\ge q+e_I}(\Lambda_I)\} .
\end{equation*}
Therefore $\gr_F^p\gr_m^{L(I)}sB_I(f)^n$ are $f_I^{-1}\cO_{Y[I]}$-flat
for all $m,n,p$
because the semistable morphism $f_{\Gamma}:X[\Gamma] \longrightarrow Y[I]$
is flat by Lemma \ref{local description of semistable morphism}.
In particular,
\begin{equation*}
\gr_F^psB_I(f)^n,\quad
\gr_m^{L(I)}sB_I(f)^n, \quad
sB_I(f)^n
\end{equation*}
are $f_I^{-1}\cO_{Y[I]}$-flat for all $m,n,p$.
Hence we obtain the conclusion for general $K \subset I$
from Lemma \ref{lemma on grL(K)sB} again.
\end{proof}

\begin{defn}
\label{definition of theta}
For $J \subset I \subset \ski$,
a morphism of $\cO_X$-modules
\begin{equation*}
\omega_X^{n+|J|}
\longrightarrow
\omega_X^{n+|I|}
\end{equation*}
is defined by sending $\omega \in \omega_X^{n+|J|}$ to
\begin{equation}
\dlog t_{i_1} \wedge \dlog t_{i_2}
\wedge \dots \wedge \dlog t_{i_l} \wedge \omega
\in \omega_X^{n+|I|}
\label{morphism on Omega for J|I:eq}
\end{equation}
where $I \setminus J=\{i_1,i_2,\dots, i_l\}$
with $1 \le i_1 < i_2 < \dots < i_l \le k$.
We can easily check that this morphism
induces a morphism of $\cO_{X[J]}$-modules
\begin{equation*}
\theta_{J|I}(f):
B_J(f)^{p,q}
\longrightarrow
B_I(f)^{p,q}
\end{equation*}
for $p \in \bnnZ$ and for $q \in \bnnZ^J \subset \bnnZ^I$.
Moreover, the equalities
\begin{align*}
&d_0\theta_{J|I}(f)=\theta_{J|I}(f)d_0 \\
&d_i\theta_{J|I}(f)=\theta_{J|I}(f)d_i \quad \text{for $i \in J$} \\
&d_i\theta_{J|I}(f)=0 \quad \text{for $i \in I \setminus J$}
\end{align*}
are easily seen.
Thus a morphism of complexes
\begin{equation}
\label{the morphism thetaJ|I:eq}
\theta_{J|I}(f): sB_J(f) \longrightarrow sB_I(f)
\end{equation}
is obtained.
It is easy to see that
$\theta_{J|I}(f)$ preserves the filtrations $F$ and $L(K)$ for $K \subset J$.
We use the notation
$\theta_{J|}(f)=\theta_{J|\ski}(f)$,
$\theta_{|I}(f)=\theta_{\emptyset|I}(f)$
and $\theta(f)=\theta_{\emptyset|\ski}(f)$ for short.
\end{defn}

\begin{lem}
\label{transitivity of theta}
For $K \subset J \subset I$,
the diagram
\begin{equation*}
\begin{CD}
sB_K(f) @>{\theta_{K|J}(f)}>> sB_J(f) \\
@| @VV{\theta_{J|I}(f)}V \\
sB_K(f) @>>{\theta_{K|I}(f)}> sB_I(f)
\end{CD}
\end{equation*}
commutes up to sign.
\end{lem}
\begin{proof}
Easy.
\end{proof}

\begin{lem}
\label{lemma on grLsBJ to grLsBI}
For $K \subset J \subset I$,
the morphism $\theta_{J|I}(f)$ induces a quasi-isomorphism
\begin{equation*}
f_I^{-1}\cO_{Y[I]}
\otimes_{f_J^{-1}\cO_{Y[J]}}
\gr_F^p\gr_m^{L(K)}sB_J(f)
\overset{\simeq}{\longrightarrow}
\gr_F^p\gr_m^{L(K)}sB_I(f)
\end{equation*}
for all $m, p$.
\end{lem}
\begin{proof}
Once we obtain the conclusion for $K=\emptyset$,
we obtain the conclusion for general $K$
by using Lemma \ref{lemma on grL(K)sB}.
So, we assume that $K=\emptyset$.
Namely,
we will prove that the morphism
\begin{equation}
\label{qis theta:eq}
f_I^{-1}\cO_{Y[I]} \otimes_{f_J^{-1}\cO_{Y[J]}} \gr_F^psB_J(f)
\longrightarrow
\gr_F^psB_I(f)
\end{equation}
induced by $\theta_{J|I}(f)$
is a quasi-isomorphism.

First, we treat the case
where $I=\ski$ and $S$ is a point.
If $J=\emptyset$,
the conclusion is already proved in \cite[Corollary (5.14)]{FujisawaLHSSV}.
Then the conclusion for $J=\emptyset$ above
and Lemma \ref{lemma on grL(K)sB}
imply that the morphism
$\theta_{J|}(f)$ induces a filtered quasi-isomorphism
\begin{equation*}
f_0^{-1}\cO_{Y_0}
\otimes_{f_J^{-1}\cO_{Y[J]}}
\gr_F^p\gr_m^{L(J)}sB_J(f)
\longrightarrow
\gr_F^p\gr_m^{L(J)}sB(f)
\end{equation*}
for all $m,p$ and for any $J$.
Thus we conclude that
the morphism \eqref{qis theta:eq}
is a quasi-isomorphism for $I=\ski$ and for $Y=\Delta^k$
by using the $f_J^{-1}\cO_{Y[J]}$-flatness
of $\gr_m^{L(J)}sB_J(f)^n$
in Lemma \ref{flatness lemma}.

Next, we treat the general case.
By definition,
we have $Y=\Delta^I \times Y_I$,
where $\Delta^I$ denotes the polydisc of dimension $|I|$
with the coordinate functions $(t_i)_{i \in I}$.
Since the problem is of local nature,
we may assume that
\begin{equation*}
X=\Delta^n \times Z, \qquad
f=g \times h,
\end{equation*}
where
$g: \Delta^n \longrightarrow \Delta^I$
is a product of the morphisms
of the form \eqref{semistable over Delta}
and where $h: Z \longrightarrow Y_I$ is a semistable morphism
along $E_{\overline{I}} \cap Y_I$.
We denote the projections
$X \longrightarrow \Delta^n$ and $X \longrightarrow Z$
by $\pr_1$ and $\pr_2$
and set $D_Z=h^{\ast}(E_{\overline{I}} \cap Y_I)$
for a while.
From the equality \eqref{description of grF:eq},
we have the identifications
\begin{equation*}
\begin{split}
&\gr_F^psB_I(f)
\simeq
\bigoplus_{q \ge 0}\pr_1^{\ast}\gr_F^{p-q}sB(g)[-q]
\otimes_{\cO_X} \pr_2^{\ast}\omega_{Z/Y_I}^q \\
&\gr_F^psB_J(f)
\simeq
\bigoplus_{q \ge 0}
\pr_1^{\ast}\gr_F^{p-q}sB_J(g)[-q]
\otimes_{\cO_X} \pr_2^{\ast}\omega_{Z/Y_I}^q ,
\end{split}
\end{equation*}
under which the morphism $\gr_F^p\theta_{J|I}(f)$
is identified with the direct sum
of the morphisms $\pr_1^{\ast}\gr_F^{p-q}\theta_{J|}(g) \otimes \id$
over all $q \ge 0$.
From what we have proved above,
$\pr_1^{\ast}\gr_F^{p-q}\theta_{J|}(g) \otimes \id$
induces a quasi-isomorphism
because $\pr_1:X \longrightarrow \Delta^n$ is flat
and $\pr_2^{\ast}\omega_{Z/Y_I}^q$ is a locally free $\cO_X$-module.
Thus we conclude that
$\theta_{J|I}(f)$ induces a quasi-isomorphism \eqref{qis theta:eq}.
\end{proof}

\begin{para}
\label{notation for a base change for S}
Let $S'$ be a complex manifold
and $g:S' \longrightarrow S$ a morphism of complex manifolds.
We set
$Y'=\Delta^k \times S'$,
$g=\id \times g: Y'=\Delta^k \times S' \longrightarrow Y=\Delta^k \times S$,
$E'_i=(\id \times g)^{\ast}E_i$
and $E'=(\id \times g)^{\ast}E$.
By the cartesian square
\begin{equation*}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
Y' @>>{\id \times g}> Y
\end{CD}
\end{equation*}
we define $f': X' \longrightarrow Y'$,
which is semistable along $E'$
by Lemma \ref{local description of semistable morphism}.
We set $D'=f'^{\ast}E'=g'^{\ast}D$
and $D'_i=f'^{\ast}E'_i=g'^{\ast}D_i$ for $i=1,2, \dots, k$.
We use the notation such as $Y'[I], Y'_I, X'[I], X'[\Gamma]$
as in the case of $f: X \longrightarrow Y$.
From the cartesian squares
\begin{equation*}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
Y' @>>{\id \times g}> Y \\
@V{\pi'_I}VV @VV{\pi_I}V \\
Y'_I @>>{\id \times g}> Y_I
\end{CD}
\end{equation*}
we have the canonical morphisms
\begin{equation*}
g'^{-1}\omega^p_{X/Y_I}
\longrightarrow
\omega^p_{X'/Y'_I}
\end{equation*}
for every $I$ and for all $p$,
which preserve the filtrations $W(D_i)$ and $W(D'_i)$
on the both sides for $i \in I$.
Thus we obtain the morphism of complexes
\begin{equation}
\label{base change morphism for sB:eq}
f'^{-1}\cO_{Y'[I]} \otimes_{g'^{-1}f^{-1}\cO_{Y[I]}} g'^{-1}sB_I(f)
\longrightarrow
sB_I(f')
\end{equation}
for all $I \subset \ski$,
which preserves the filtrations $F$ and $L(J)$ for all $J \subset I$.
\end{para}

\section{The Gauss-Manin connection}
\label{gauss-manin}

In this section,
we construct the Gauss-Manin connection
on $R^n(f_I)sB_I(f)$
as in the case on $R^nf_{\ast}\omega_{X/Y}$
in \eqref{the Gauss-Manin connection:eq}.
For this purpose,
we construct a complex $s\tB_I(f)$
which is a replacement of $f_I^{-1}\cO_{Y[I]} \otimes \omega_X$
for $I \subset \ski$
and follow the arguments
in Definition \ref{definition of the filtration G}.
Then we describe its residues
by a similar way to \cite{SteenbrinkLHS}.

\begin{defn}
For a subset $I \subset \ski$,
we set
\begin{equation*}
\tB_I(f)^{p,q}=
\omega^{p+|q|+|I|}_X\left/\sum_{i \in I}W(D_i)_{q_i} \right.
\end{equation*}
for $p \in \bnnZ$ and $q \in \bnnZ^I$,
with the morphisms
\begin{equation*}
\begin{split}
&d_0=(-1)^{|I|}d:
\tB_I(f)^{p,q} \longrightarrow \tB_I(f)^{p+1,q} \\
&d_i=(-1)^{|I|}\dlog t_i \wedge:
\tB_I(f)^{p,q} \longrightarrow \tB_I(f)^{p,q+e_i}
\end{split}
\end{equation*}
as in Definition \ref{definition of B_I(f)}.
Then we obtain the complex $s\tB_I(f)$
by setting
\begin{equation*}
s\tB_I(f)^n
=\bigoplus_{p+|q|=n}\tB_I(f)^{p,q}
=\bigoplus_{q \in \bnnZ^I}
\left(\omega^{n+|I|}_X\left/\sum_{i \in I}W(D_i)_{q_i} \right.\right)
\end{equation*}
with the differential
$d=d_0+\sum_{i \in I}d_i$
as in Definition \ref{definition of B_I(f)}.
As in \eqref{support of B:eq},
we have
$\supp(\tB_I(f)^{p,q}) \subset X[I]$ for any $p,q$.
Thus we identify
$\tB_I(f)^{p,q}$ and $\iota_I^{-1}\tB_I(f)^{p,q}$ for all $p,q$
as before.
Although $\tB_I(f)^{p,q}$ is an $\cO_{X[I]}$-module for all $p,q$,
we consider $s\tB_I(f)$ as a complex of $\bC$-sheaves
because the morphism $d_0$ is not a morphism of $\cO_{X[I]}$-modules.
The filtration $F$ on $s\tB_I(f)$ is defined by
\begin{equation*}
F^ps\tB_I(f)^n
=\bigoplus_{\substack{p'+|q|=n \\
                      p' \ge p}}
\tB_I(f)^{p',q}
=\bigoplus_{\substack{q \in \bnnZ^I\\
                      |q| \le n-p}}
\left(\omega^{n+|I|}_X\left/\sum_{i \in I}W(D_i)_{q_i} \right.\right)
\end{equation*}
as in \eqref{definition of F on sBI(f):eq}.
For $K \subset I$,
the filtration $L(K)$ on $\tB_I(f)$ is defined by
\begin{equation*}
L(K)_m\tB_I(f)^{p,q}
=W(D_K)_{m+2|q_K|+|K|}\omega_X^{p+|q|+|I|}\left/
\sum_{i \in I}W(D_i)_{q_i} \right.
\end{equation*}
as in \eqref{definition of L(K) on BI:eq}.
The canonical projection
$\omega^n_X \longrightarrow \omega^n_{X/Y_I}$ induces
a morphism of complexes
\begin{equation*}
s\tB_I(f) \longrightarrow sB_I(f) ,
\end{equation*}
which is denoted by $\pr_I$ if there is no danger of confusion.
It is easy to see that $\pr_I$ preserves the filtrations $F$ and $L(K)$.
\end{defn}

\begin{defn}
For a subset $I \subset \ski$,
a finite decreasing filtration $G(I)$ on $\omega_X$ is defined by
\begin{equation*}
G(I)^p\omega^n_X
=\image((\pi_I f)^{-1}\omega^p_{Y_I}
\otimes_{(\pi_I f)^{-1}\cO_{Y_I}} \omega^{n-p}_X
\overset{\wedge}{\longrightarrow} \omega^n_X)
\end{equation*}
for all $n,p$.
By definition
$G(\emptyset)$ is nothing but the filtration $G$
defined in Definition \ref{definition of the filtration G}.
Because we have
\begin{equation*}
\dlog t_i \wedge G(I)^p\omega_X \subset G(I)^p\omega_X
\end{equation*}
for every $i \in I$ and for every $p$,
the filtration $G$ on $\tB_I(f)$ is defined by
\begin{equation*}
G^r\tB_I(f)^{p,q}
=G(I)^r\omega^{p+|q|+|I|}_X \left/
\sum_{i \in I}W(D_i)_{q_i} \right.
\end{equation*}
for all $p,q,r$.
Thus we obtain a finite decreasing filtration $G$ on $s\tB_I(f)$.
\end{defn}

\begin{lem}
\label{lemma on grGstBI}
We have the isomorphism
\begin{equation}
f_I^{-1}\omega^p_{Y[I]} \otimes_{f_I^{-1}\cO_{Y[I]}} sB_I(f)[-p]
\overset{\simeq}{\longrightarrow}
\gr_G^ps\tB_I(f)
\label{identification for grG:eq}
\end{equation}
for every $p$,
under which
\begin{align*}
&f_I^{-1}\omega^p_{Y[I]} \otimes_{f_I^{-1}\cO_{Y[I]}}
F^{r-p}sB_I(f)[-p]
\simeq
F^r\gr_G^ps\tB_I(f) \\
&f_I^{-1}\omega^p_{Y[I]} \otimes_{f_I^{-1}\cO_{Y[I]}}
L(K)_m\gr_G^ps\tB_I(f)
\simeq
L(K)_m\gr_G^ps\tB_I(f)
\end{align*}
for $K \subset I$
and for all $m,r$.
\end{lem}
\begin{proof}
Easy from $\iota_I^{-1}(\pi_If)^{-1}=(\pi_If\iota_I)^{-1}=f_I^{-1}$.
\end{proof}

\begin{defn}
The morphism of $E_1$-terms
\begin{equation*}
E_1^{p,n}(R(f_I)_{\ast}s\tB_I(f),G)
\longrightarrow
E_1^{p+1,n}(R(f_I)_{\ast}s\tB_I(f),G)
\end{equation*}
induces a morphism of $\bC$-sheaves
\begin{equation*}
\nabla(I):
\omega^p_{Y[I]} \otimes_{\cO_{Y[I]}} R^n(f_I)_{\ast}sB_I(f)
\longrightarrow
\omega^{p+1}_{Y[I]} \otimes_{\cO_{Y[I]}} R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
via the identification
\begin{equation*}
E_1^{p,n}(R(f_I)_{\ast}s\tB_I(f),G)
\simeq
\omega^p_{Y[I]} \otimes_{\cO_{Y[I]}} R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
which is given by Lemma \ref{lemma on grGstBI}
together with the local freeness of $\omega_{Y[I]}^p$.
We have
\begin{equation*}
\begin{split}
&\nabla(I)(\omega^p_{Y[I]}
\otimes_{\cO_{Y[I]}}
L(K)_mR^n(f_I)_{\ast}sB_I(f))
\subset
\omega^{p+1}_{Y[I]}
\otimes_{\cO_{Y[I]}}
L(K)_mR^n(f_I)_{\ast}sB_I(f) \\
&\nabla(I)(\omega^p_{Y[I]}
\otimes_{\cO_{Y[I]}}
F^rR^n(f_I)_{\ast}sB_I(f))
\subset
\omega^{p+1}_{Y[I]}
\otimes_{\cO_{Y[I]}}
F^{r-1}R^n(f_I)_{\ast}sB_I(f)
\end{split}
\end{equation*}
for all $m,r$ by Lemma \ref{lemma on grGstBI} again.
\end{defn}

\begin{rmk}
For the case of $I=\emptyset$,
the filtered complex $(s\tB_{\emptyset}(f),G)$
coincides with $(\omega_X,G)$
in Definition \ref{definition of the filtration G}.
Therefore we have $\nabla(\emptyset)=\nabla$.
\end{rmk}

\begin{rmk}
\label{wedge product for omegaY and B}
A morphism
\begin{equation*}
(\pi_If)^{-1}\omega_{Y_I}^p
\otimes_{(\pi_I f)^{-1}\cO_{Y_I}}
\omega_X^n
\longrightarrow
\omega_X^{n+p}
\end{equation*}
is defined by sending $\omega \otimes \eta$
to $(-1)^{p|I|}\omega \wedge \eta$.
Direct computation shows
that this morphism gives a morphism of complexes
\begin{equation*}
\iota_I^{-1}(\pi_If)^{-1}\omega_{Y_I}
\otimes_{\iota_I^{-1}(\pi_I f)^{-1}\cO_{Y_I}} s\tB_I(f)
=
f_I^{-1}\omega_{Y[I]}
\otimes_{f_I^{-1}\cO_{Y[I]}} s\tB_I(f)
\longrightarrow
s\tB_I(f) .
\end{equation*}
Then we can prove the equality
\begin{equation*}
\nabla(I)(\omega \otimes v)
=(-1)^{|I|}d\omega \otimes v+(-1)^p\omega \wedge \nabla(I)(v)
\end{equation*}
for $\omega \in \omega^p_{Y[I]}$ and $v \in R^n(f_I)_{\ast}sB_I(f)$
as in \cite{Katz-Oda}.
Therefore $(-1)^{|I|}\nabla(I)$ is
an integrable logarithmic connection
on $R^n(f_I)_{\ast}sB_I(f)$ for any $n$.
\end{rmk}

\begin{defn}
The filtration $G$ on $s\tB_I(f)$
induces a morphism
\begin{equation*}
\begin{split}
\gamma_I(f):
f_I^{-1}\omega^p_{Y[I]}
\otimes_{f_I^{-1}\cO_{Y[I]}} sB_I(f)[-p]
\longrightarrow
f_I^{-1}\omega^{p+1}_{Y[I]}
\otimes_{f_I^{-1}\cO_{Y[I]}} sB_I(f)[-p]
\end{split}
\end{equation*}
in the derived category
for every $p$.
Then we have
\begin{equation*}
\nabla(I)=R^{p+n}(f_I)_{\ast}(\gamma_I(f))
\end{equation*}
by definition.
\end{defn}

\begin{defn}
For $J \subset I \subset \ski$,
the morphism
defined by \eqref{morphism on Omega for J|I:eq}
induces a morphism of complexes
\begin{equation*}
\widetilde{\theta}_{J|I}(f):
s\tB_J(f)
\longrightarrow
s\tB_I(f)
\end{equation*}
as in Definition \ref{definition of theta}.
\end{defn}

\begin{lem}
The morphism $\widetilde{\theta}_{J|I}(f)$
preserves the filtration $G$ on the both sides.
The morphism
\begin{equation*}
\gr_G^p\widetilde{\theta}_{J|I}(f):
\gr_G^p\tB_J(f)
\longrightarrow
\gr_G^p\tB_I(f)
\end{equation*}
is identified with the morphism
\begin{equation*}
\begin{split}
(-1)^{p(|I|+|J|)}\pr_{J|I} \otimes \theta_{J|I}(f)[-p]:
f_J^{-1}\omega&^p_{Y[J]}
\otimes_{f_J^{-1}\cO_{Y[J]}} sB_J(f)[-p] \\
&\longrightarrow
f_I^{-1}\omega^p_{Y[I]}
\otimes_{f_I^{-1}\cO_{Y[I]}} sB_I(f)[-p]
\end{split}
\end{equation*}
for every $p$,
where $\pr_{J|I}$ is the morphism
\eqref{the projection between Omega from Y[J] to Y[I]:eq}.
\end{lem}
\begin{proof}
Easy by definition.
\end{proof}

\begin{cor}
\label{cor on compatibility of nabla(J) and nabla(I)}
The diagram
\begin{equation*}
\begin{CD}
\omega^p_{Y[J]}
\otimes_{\cO_{Y[J]}} R^n(f_J)_{\ast}sB_J(f)
@>{(-1)^{|J|}\nabla(J)}>>
\omega^{p+1}_{Y[J]}
\otimes_{\cO_{Y[J]}} R^n(f_J)_{\ast}sB_J(f) \\
@V{\pr_{J|I} \otimes \theta_{J|I}(f)}VV
@VV{\pr_{J|I} \otimes \theta_{J|I}(f)}V \\
\omega^p_{Y[I]}
\otimes_{\cO_{Y[I]}} R^n(f_I)_{\ast}sB_I(f)
@>>{(-1)^{|I|}\nabla(I)}>
\omega^{p+1}_{Y[I]}
\otimes_{\cO_{Y[I]}} R^n(f_I)_{\ast}sB_I(f)
\end{CD}
\end{equation*}
is commutative.
\end{cor}

\begin{defn}
\label{definition of nu}
For $j \in I$,
the canonical projection
\begin{equation*}
\begin{split}
B_I(f)^{p,q}
=\omega^{p+|q|+|I|}_{X/Y_I}&\left/\sum_{i \in I}W(D_i)_{q_i} \right. \\
&\longrightarrow
B_I(f)^{p-1,q+e_j}
=\omega^{p+|q|+|I|}_{X/Y_I} \left/
\sum_{i \in I, i \not= j}W(D_i)_{q_i}+W(D_j)_{q_j+1} \right.
\end{split}
\end{equation*}
induces a morphism of complexes
\begin{equation*}
\nu_{j|I}(f): sB_I(f) \longrightarrow sB_I(f)
\end{equation*}
for $j \in I$.
We set
\begin{equation*}
\nu_{J|I}(f)=\sum_{j \in J}\nu_{j|I}(f):
sB_I(f) \longrightarrow sB_I(f)
\end{equation*}
for $J \subset I$.
We use the convention
$\nu_I(f)=\nu_{I|I}(f)$
and $\nu(f)=\nu_{\ski}(f)$
as before.

On the other hand,
we define a morphism of complexes
$\mu_{J|I}(f)$ by
\begin{equation*}
\mu_{J|I}(f)=\sum_{j \in I \setminus J}\dlog t_j \otimes \nu_{j|I}(f):
sB_I(f)
\longrightarrow
(f_J)^{-1}\omega_{Y[J]}^1
\otimes_{(f_J)^{-1}\cO_{Y[J]}} sB_I(f)
\end{equation*}
for $J \subsetneq I$.

Similarly,
the projection
\begin{align*}
\tB_I(f)^{p,q}
=\omega^{p+|q|+|I|}_X&\left/\sum_{i \in I}W(D_i)_{q_i} \right. \\
&\longrightarrow
\tB_I(f)^{p-1,q+e_j}
=\omega^{p+|q|+|I|}_X\left/
\sum_{i \in I, i \not= j}W(D_i)_{q_i}+W(D_j)_{q_j+1} \right.
\end{align*}
induces a morphism of complexes
\begin{equation*}
\widetilde{\nu}_{j|I}(f): s\tB_I(f) \longrightarrow s\tB_I(f)
\end{equation*}
for $j \in I$.
\end{defn}

\begin{lem}
\label{lemma for nu and L}
For $K \subset I \subset \ski$,
we have
\begin{equation*}
\begin{split}
&\nu_{j|I}(f)(L(K)_msB_I(f)) \subset L(K)_msB_I(f), \\
&\nu_{j|I}(f)(F^psB_I(f)) \subset F^{p-1}sB_I(f)
\end{split}
\end{equation*}
for all $m,p$ and for every $j \in I$.
If $j \in K$ in addition,
we have
\begin{equation*}
\nu_{j|I}(f)(L(K)_msB_I(f)) \subset L(K)_{m-2}sB_I(f)
\end{equation*}
for every $m$.
Therefore we have
\begin{equation*}
\mu_{J|I}(f)(L(K)_msB_I(f))
\subset
(f_J)^{-1}\omega_{Y[J]}^1
\otimes_{(f_J)^{-1}\cO_{Y[J]}} L(K)_{m-2}sB_I(f)
\end{equation*}
for every $m$ if $K \cup J=I$.
\end{lem}
\begin{proof}
Easy.
\end{proof}

\begin{lem}
The morphism $\widetilde{\nu}_{j|I}(f)$ preserves the filtration $G$
on $s\tB_I(f)$ for every $j \in I$.
Moreover $\gr_G^p\widetilde{\nu}_{j|I}(f)$ coincides with
$\id \otimes \nu_{j|I}(f)[-p]$ for every $j \in I$
under the identification
\eqref{identification for grG:eq}.
\end{lem}
\begin{proof}
Easy by definition.
\end{proof}

\begin{cor}
\label{corollary for commutativity of nu and gamma}
The diagram
\begin{equation*}
\begin{CD}
sB_I(f)
@>{\gamma_I(f)}>>
f_I^{-1}\omega^1_{Y[I]}
\otimes_{f_I^{-1}\cO_{Y[I]}} sB_I(f) \\
@V{\nu_{j|I}(f)}VV @VV{\id \otimes \nu_{j|I}(f)}V \\
sB_I(f)
@>>{\gamma_I(f)}>
f_I^{-1}\omega^1_{Y[I]}
\otimes_{f_I^{-1}\cO_{Y[I]}} sB_I(f)
\end{CD}
\end{equation*}
is commutative in the derived category.
\end{cor}

\begin{lem}
\label{lemma for nu and residue of nabla}
For $J \subsetneq I$,
the diagram
\begin{equation*}
\begin{CD}
sB_J(f)
@>{\gamma_J(f)}>>
f_J^{-1}\omega_{Y[J]}^1
\otimes_{f_J^{-1}\cO_{Y[J]}} sB_J(f) \\
@V{\theta_{J|I}(f)}VV @VV{\id \otimes \theta_{J|I}(f)}V \\
sB_I(f)
@>>{(-1)^{|J|+1}\mu_{J|I}(f)}>
f_J^{-1}\omega_{Y[J]}^1
\otimes_{f_J^{-1}\cO_{Y[J]}} sB_I(f)
\end{CD}
\end{equation*}
is commutative.
\end{lem}
\begin{proof}
We define a complex of
$\bC$-sheaves
$C_{J|I}(f)$ by
\begin{equation*}
C_{J|I}(f)^p=
(f_J^{-1}\omega^1_{Y[J]}
\otimes_{f_J^{-1}\cO_{Y[J]}} sB_I(f)^{p-1})
\oplus sB_I(f)^p
\end{equation*}
with the differentials
\begin{equation*}
d(\omega \otimes x, y)
=(-\omega \otimes dx+(-1)^{|J|+1}\mu_{J|I}(f)(y), dy)
\end{equation*}
for $\omega \in f_J^{-1}\omega^1_{Y[J]}$,
$x \in sB_I(f)^{p-1}$
and $y \in sB_I(f)^p$.
We can easily check the equality $d^2=0$
from the fact that $\mu_{J|I}(f)$ is a morphism of complexes.
Then a morphism of complexes
\begin{equation*}
C_{J|I}(f) \longrightarrow sB_I(f)
\end{equation*}
is defined by the projection
$C_{J|I}(f)^p \longrightarrow sB_I(f)^p$.
Moreover, the injection
\begin{equation*}
f_J^{-1}\omega_{Y[J]}^1
\otimes_{f_J^{-1}\cO_{Y[J]}} sB_I(f)^{p-1}
\longrightarrow
C_{J|I}(f)^p
\end{equation*}
induces a morphism of complexes
\begin{equation*}
f_J^{-1}\omega_{Y[J]}^1
\otimes_{f_J^{-1}\cO_{Y[J]}} sB_I(f)[-1]
\longrightarrow
C_{J|I}(f)
\end{equation*}
which fit in the exact sequence
\begin{equation*}
\begin{CD}
0 @>>> f_J^{-1}\omega_{Y[J]}^1 \otimes_{f_J^{-1}\cO_{Y[J]}} sB_I(f)[-1]
  @>>> C_I(f)
  @>>> sB_I(f) @>>> 0
\end{CD}
\end{equation*}
by definition.

We denote by $I \setminus J=\{i_1,i_2,\dots,i_l\}$
with $1 \le i_1 < i_2 < \dots < i_l \le k$.
The morphism
\begin{equation*}
\omega_X^p \longrightarrow \omega_X^{p-1+l}
\end{equation*}
sending $\omega \in \omega_X^p$ to
\begin{equation*}
(-1)^{l-j}\dlog t_{i_1} \wedge \dots \wedge \dlog t_{i_{j-1}}
\wedge \dlog t_{i_{j+1}} \wedge \dots \wedge \dlog t_{i_l} \wedge \omega
\end{equation*}
induces a morphism
\begin{equation*}
\eta_j:
\tB_J(f)^{p,q}/G^2\tB_J(f)^{p,q}
\longrightarrow
B_I(f)^{p-1,q}
\end{equation*}
for $p \in \bnnZ$, $q \in \bnnZ^J \subset \bnnZ^I$.
Thus we obtain a morphism
\begin{equation*}
\eta_j:
s\tB_J(f)^p/G^2s\tB_J(f)^p
\longrightarrow
sB_I(f)^{p-1}
\end{equation*}
for $p \in \bZ$ for every $j=\li$.

Now a morphism
\begin{equation*}
\eta:
s\tB_J(f)^p/G^2s\tB_J(f)^p
\longrightarrow
C_{J|I}(f)^p
\end{equation*}
is defined by
\begin{equation*}
\eta(\omega)
=(\sum_{j=1}^{l}\dlog t_{i_j} \otimes \eta_j(\omega),
\pr_I \widetilde{\theta}_{J|I}(f)(\omega))
\end{equation*}
for $\omega \in s\tB_J(f)^p$.
It is easy to check that $\eta$ defines a morphism of complexes.
Then the diagram
\begin{equation*}
\begin{CD}
0 @. 0 \\
@VVV @VVV \\
f_J^{-1}\omega_{Y[J]}^1 \otimes_{f_J^{-1}\cO_{Y[J]}} sB_J(f)[-1]
@>{\id \otimes \theta_{J|I}(f)[-1]}>>
f_J^{-1}\omega_{Y[J]}^1 \otimes_{f_J^{-1}\cO_{Y[J]}} sB_I(f)[-1] \\
@VVV @VVV \\
s\tB_J(f)/G^2s\tB_J(f) @>{\eta}>> C_{J|I}(f) \\
@VVV @VVV \\
sB_J(f) @>{\theta_{J|I}(f)}>> sB_I(f) \\
@VVV @VVV \\
0 @. 0
\end{CD}
\end{equation*}
is commutative with exact columns.
Thus we obtain the conclusion.
\end{proof}

\begin{defn}
\label{definition of N}
The morphism
\begin{equation*}
R^n(f_I)_{\ast}(\nu_{j|I}(f)):
R^n(f_I)_{\ast}sB_I(f)
\longrightarrow
R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
induced by $\nu_{j|I}(f)$ for every $n$
is denoted by $N_{j|I}(f)$ for short.
We set
\begin{equation*}
N_{J|I}(f;c)=\sum_{j \in J}c_jN_{j|I}(f):
R^n(f_I)_{\ast}sB_I(f)
\longrightarrow
R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for $J \subset I$
and for $c=(c_j)_{j \in J} \in \bC^J$.
We simply denote $N_{J|I}(f)=N_{J|I}(f;e_J)$.
We use the convention
$N_I(f;c)=N_{I|I}(f;c)$,
$N_I(f)=N_I(f;e_I)$,
$N(f;c)=N_{\ski}(f;c)$
and $N(f)=N(f;e)$.
\end{defn}

\begin{lem}
For $K \subset I$, we have
\begin{align*}
&N_{j|I}(f)(F^pR^n(f_I)_{\ast}sB_I(f))
\subset F^{p-1}R^n(f_I)_{\ast}sB_I(f) \\
&N_{j|I}(f)(L(K)_mR^n(f_I)_{\ast}sB_I(f))
\subset L(K)_mR^n(f_I)_{\ast}sB_I(f)
\end{align*}
for all $m,n,p$.
Moreover,
we have
\begin{equation*}
N_{j|I}(f)(L(K)_mR^n(f_I)_{\ast}sB_I(f))
\subset L(K)_{m-2}R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for all $m, n$
if $j \in K$.
Therefore
\begin{equation*}
N_{J|I}(f;c)(L(K)_mR^n(f_I)_{\ast}sB_I(f))
\subset L(K)_{m-2}R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for $J \subset K$, for $c \in \bC^J$ and for all $m,n,p$.
\end{lem}
\begin{proof}
Easy by Lemma \ref{lemma for nu and L}.
\end{proof}

\begin{lem}
The diagram
\begin{equation*}
\begin{CD}
R^n(f_I)sB_I(f)
@>{\nabla(I)}>>
\omega_{Y[I]}^1 \otimes_{\cO_{Y[I]}} R^n(f_I)sB_I(f) \\
@V{N_{j|I}(f)}VV @VV{\id \otimes N_{j|I}(f)}V \\
R^n(f_I)sB_I(f)
@>{\nabla(I)}>>
\omega_{Y[I]}^1 \otimes_{\cO_{Y[I]}} R^n(f_I)sB_I(f)
\end{CD}
\end{equation*}
is commutative for every $j \in I$.
In particular,
the diagram
\begin{equation*}
\begin{CD}
R^n(f_I)sB_I(f)
@>{\nabla(I)}>>
\omega_{Y[I]}^1 \otimes_{\cO_{Y[I]}} R^n(f_I)sB_I(f) \\
@V{N_{J|I}(f)}VV @VV{\id \otimes N_{J|I}(f)}V \\
R^n(f_I)sB_I(f)
@>{\nabla(I)}>>
\omega_{Y[I]}^1 \otimes_{\cO_{Y[I]}} R^n(f_I)sB_I(f)
\end{CD}
\end{equation*}
is commutative for all $J \subset I$.
\end{lem}
\begin{proof}
Easy by Corollary \ref{corollary for commutativity of nu and gamma}.
\end{proof}

\begin{thm}
\label{thm on the compatibility of N and nu}
For the residue of $\nabla$
in \eqref{the residue of nabla:eq},
the diagram
\begin{equation*}
\begin{CD}
\cO_{E_i} \otimes_{\cO_Y} R^nf_{\ast}\omega_{X/Y}
@>{\res_j(\nabla)}>>
\cO_{E_i} \otimes_{\cO_Y} R^nf_{\ast}\omega_{X/Y} \\
@V{\theta_{|I}(f)}VV @VV{\theta_{|I}(f)}V \\
R^n(f_I)_{\ast}sB_I(f)
@>>{-N_{j|I}(f)}>
R^n(f_I)_{\ast}sB_I(f)
\end{CD}
\end{equation*}
is commutative for $j \in I$.
\end{thm}
\begin{proof}
By Lemma \ref{lemma for nu and residue of nabla}
for $J=\emptyset$,
we can easily obtain the conclusion.
\end{proof}

\begin{rmk}
\label{nu and base change for S}
Let $g: S' \longrightarrow S$ be a morphism of complex manifolds.
Then the construction of $\nu_{j|I}(f)$
is compatible with the base change by $g$.
More precisely, we have the commutative diagram
\begin{equation*}
\begin{CD}
f'^{-1}\cO_{Y'[I]} \otimes_{g'^{-1}f^{-1}\cO_{Y[I]}} g'^{-1}sB_I(f)
@>>>
sB_I(f') \\
@V{\id \otimes \nu_{j|I}(f)}VV @VV{\nu_{j|I}(f')}V \\
f'^{-1}\cO_{Y'[I]} \otimes_{g'^{-1}f^{-1}\cO_{Y[I]}} g'^{-1}sB_I(f)
@>>>
sB_I(f')
\end{CD}
\end{equation*}
where the horizontal arrows are the morphism
\eqref{base change morphism for sB:eq}.
\end{rmk}

\section{Rational structures}
\label{section for the rational structure}

In this section,
we construct a rational structure on $sB_I(f)$
as in \cite{SteenbrinkLE},
which is inspired by the log geometry of Fontaine-Illusie \cite{KazuyaKato}.
By using these rational structures,
we prove the local freeness of
$R^n(f_I)_{\ast}sB_I(f)$, $\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)$ etc.
for a proper semistable morphism $f$
in the next section.

\begin{para}
\label{situation for hodge theoretic properties}
Let $Y=\Delta^k \times S$ and $E$ be as in \ref{local situation}
and $f:X \longrightarrow Y$ a semistable morphism along $E$
as before.
We use the notation in \ref{notation for Y and E},
\ref{setting for a semistable morphism}
and \ref{local situation}.
\end{para}

\begin{para}
For $I \subset \ski$,
the open immersion
$X \setminus D_I \hookrightarrow X$
is denoted by $j_I$.
Then the sheaf of monoids
\begin{equation*}
M_X(D_I)=(j_I)_{\ast}\cO_{X \setminus D_I}^{\ast} \cap \cO_X
\end{equation*}
defines a log structure on $X$ (see \cite{KazuyaKato}).
We use the notation $M_X(D)$
for $I=\ski$ as before.
We have
\begin{equation*}
\cO_X^{\ast}=M_X(D_{\emptyset})
\subset M_X(D_J)
\subset M_X(D_I)
\subset M_X(D)
\end{equation*}
for $J \subset I \subset \ski$.
A morphism of abelian sheaves
\begin{equation*}
\mathbf{e}(f):\cO_X \longrightarrow M_X(D)\gp
\end{equation*}
is the composite of the exponential map
\begin{equation*}
\cO_X \ni f \mapsto \exp(2\pi\sqrt{-1}f) \in \cO_X^{\ast}
\end{equation*}
and the inclusion $\cO_X^{\ast} \hookrightarrow M_X(D)$.
We denote by
\begin{equation*}
\mathbf{e}(f)_{\bQ}:
\cO_X \longrightarrow M_X(D)\gp_{\bQ}=M_X(D)\gp \otimes_{\bZ} \bQ
\end{equation*}
the base extension of the morphism $\mathbf{e}(f)$.
For $I \subset \ski$,
a morphism of abelian sheaves
\begin{equation*}
\bZ^I \longrightarrow M_X(D)\gp
\end{equation*}
is defined by sending $e_i \in \bZ^I$
to the global section $t_i$ of $M_X(D)$ for $i \in I$.
Then a morphism of $\bQ$-sheaves
\begin{equation*}
\bQ^I \longrightarrow M_X(D)\gp_{\bQ}
\end{equation*}
is obtained by the base extension to $\bQ$.
The direct sum of the morphism above
and $\mathbf{e}(f)_{\bQ}$
gives us the morphism
\begin{equation*}
\bQ^I \oplus \cO_X \longrightarrow M_X(D)_{\bQ}\gp
\end{equation*}
denoted by $\varphi^I(f)$.
We write $\varphi(f)$ for $\varphi^{\ski}(f)$.
On the other hand,
$\varphi^{\emptyset}(f)$
is nothing but the morphism $\mathbf{e}(f)_{\bQ}$

A global section $1$ of $\cO_X$ can be considered
as a global section of $\bQ^I \oplus \cO_X$
by the canonical inclusion $\cO_X \subset \bQ^I \oplus \cO_X$.
Then the global section $1$
is contained in $\kernel(\varphi^I)$ for every $I \subset \ski$.
A complex of $\bQ$-sheaves
\begin{equation*}
\kos(\varphi^I(f);\infty;1)
\end{equation*}
is defined in \cite[Definition 1.8]{FujisawaMHSLSD}.
For $J \subset \overline{I}=\ski \setminus I$,
we have
\begin{equation*}
\image(\varphi^I(f))
\subset
M_X(D_{\overline{J}})_{\bQ}\gp
\end{equation*}
because $I \subset \overline{J}=\ski \setminus J$.
Then an increasing filtration
$W(J)$ on $\kos(\varphi^I(f);\infty;1)$
is defined as $W(M_X(D_{\overline{J}})\gp_{\bQ})$
in \cite[Definition 1.8]{FujisawaMHSLSD}.
For the case of $J=\{i\}$ with $i \notin I$,
we write $W(i)$ instead of $W(\{i\})$.
\end{para}

\begin{defn}
For $I \subset I' \subset \ski$,
we set
\begin{equation*}
A_{I|I'}(f)^{p,q}
=\kos(\varphi^{I' \setminus I}(f);\infty;1)^{p+|q|+|I|}
\left/\sum_{i \in I}W(i)_{q_i} \right.
\end{equation*}
for $p \in \bnnZ, q \in \bnnZ^I$.
We set
\begin{align*}
&d_0=(-1)^{|I|}d:
A_{I|I'}(f)^{p,q} \longrightarrow A_{I|I'}(f)^{p+1,q} \\
&d_i=(-1)^{|I|}t_i\wedge:
A_{I|I'}(f)^{p,q} \longrightarrow A_{I|I'}(f)^{p,q+e_i}
\qquad \text{$i \in I$,}
\end{align*}
where $d$ on the right hand side of the first equality
denotes the differential of the complex
$\kos(\varphi^{I' \setminus I}(f);\infty;1)$
and where $t_i \wedge$
denotes the morphism of complexes
\begin{equation*}
\kos(\varphi^{I' \setminus I}(f);\infty;1)
\longrightarrow
\kos(\varphi^{I' \setminus I}(f);\infty;1)[1]
\end{equation*}
induced by the wedge product $t_i \wedge$
as in \cite[(1.11)]{FujisawaMHSLSD}.
By setting
\begin{equation*}
sA_{I|I'}(f)^n=
\bigoplus_{p+|q|=n}A_{I|I'}(f)^{p,q}
\end{equation*}
with the differential
$d=d_0+\sum_{i \in I}d_i$,
we obtain a complex $sA_{I|I'}(f)$.
By definition,
$sA_{\emptyset|I'}(f)$ is nothing but $\kos(\varphi^{I'}(f);\infty;1)$.
We write $sA_I(f)$
for $sA_{I|I}(f)$.
Moreover, $sA(f)$ denotes $sA_{\ski}(f)$.
We use $sA_{I|}(f)$ and $sA_{|I'}(f)$
instead of $sA_{I|\ski}(f)$ and $sA_{\emptyset|I'}(f)$ respectively.

An increasing filtration
$L(K)$ on $A_{I|I'}(f)^{p,q}$ for $K \subset I$
is defined by
\begin{equation*}
L(K)_mA_{I|I'}(f)^{p,q}
=W(K)_{m+2|q_K|+|K|}\kos(\varphi^{I' \setminus I};\infty;1)^{p+|q|+|I|}
\left/\sum_{i \in I}W(i)_{q_i} \right.
\end{equation*}
for every $m$.
It is easy to check that this defines an increasing filtration
$L(K)$ on $sA_{I|I'}(f)$.
We use the notation $L=L(\ski)$ on $sA(f)$.
\end{defn}

\begin{rmk}
We can easily see the inclusion
\begin{equation*}
\supp(A_{I|I'}(f)^{p,q}) \subset X[I]
\end{equation*}
for all $p,q$.
Thus we identify
$A_{I|I'}(f)^{p,q}$ and $\iota_I^{-1}A_{I|I'}(f)^{p,q}$
for all $p,q$ as before.
\end{rmk}

\begin{lem}
\label{Lemma of grLsA}
We have an isomorphism of complexes
\begin{equation*}
\label{equality for grLsA:eq}
\gr_m^{L(K)}sA_{I|I'}(f)
\simeq
\bigoplus_{q \in \bnnZ^K}
\bigoplus_{\Gamma \in S_{m+2|q|+|K|}^{\ge q+e_K}(\Lambda_K)}
\iota_{\Gamma}^{-1}sA_{(I \setminus K)|(I' \setminus K)}(\pi_Kf)[-m-2|q|]
\end{equation*}
for every integer $m$,
under which
the filtration
$L(J)\gr_m^{L(K)}sA_{I|I'}(f)$
coincides with the direct sum of
\begin{equation*}
L(J \setminus K \cap J)
[\,|\Gamma \cap \Lambda_{K \cap J}|-2|q_{K \cap J}|-|K \cap J|\,]\,
\iota_{\Gamma}^{-1}sA_{(I \setminus K)|(I' \setminus K)}(\pi_Kf)[-m-2|q|]
\end{equation*}
over the same index set
as in \eqref{equality for grLsA:eq}.
\end{lem}
\begin{proof}
Easy from Proposition 1.10 in \cite{FujisawaMHSLSD}.
\end{proof}

\begin{para}
Under the situation
in Lemma \ref{Lemma of grLsA},
we have the commutative diagram of the canonical morphisms
\begin{equation*}
\begin{CD}
\iota_{\Gamma}^{-1}(\bQ^{I' \setminus I} \oplus \cO_X)
@>{\iota_{\Gamma}^{-1}\varphi^{I' \setminus I}(\pi_K f)}>>
\iota_{\Gamma}^{-1}M_X(D_{\overline{K}})\gp_{\bQ} \\
@VVV @VVV \\
\bQ^{I' \setminus I} \oplus \cO_{X[\Gamma]}
@>>{\varphi^{I' \setminus I}(f_{\Gamma})}>
M_{X[\Gamma]}(D_{\overline{K}} \cap X[\Gamma])\gp_{\bQ}
\end{CD}
\end{equation*}
which induces morphisms of complexes
\begin{equation*}
\label{the canonical morphism of kos for closed immersion:eq}
\iota_{\Gamma}^{-1}\kos(\varphi^{I' \setminus I}(\pi_K f);\infty;1)
\longrightarrow
\kos(\varphi^{I' \setminus I}(f_{\Gamma});\infty;1)
\end{equation*}
and
\begin{equation*}
\label{the canonical morphism of A for closed immersion:eq}
\iota_{\Gamma}^{-1}sA_{(I \setminus K)|(I' \setminus K)}(\pi_K f)
\longrightarrow
sA_{(I \setminus K)|(I' \setminus K)}(f_{\Gamma})
\end{equation*}
for $K \subset I \subset I' \subset \ski$.
\end{para}

\begin{lem}
\label{lemma for closed immersion}
The morphism
\eqref{the canonical morphism of A for closed immersion:eq}
is a filtered quasi-isomorphism
with respect to $L(J)$
for $J \subset I \setminus K$.
\end{lem}
\begin{proof}
Because the problem is of local nature,
Lemma \ref{local description of semistable morphism}
enables us to assume the following:
\begin{itemize}
\item
$X=X[\Gamma] \times \Delta^{\Gamma}$
where $\Delta^{\Gamma}$ denotes the polydisc
with the coordinate functions $(x_{\lambda})_{\lambda \in \Gamma}$.
\item
$\pi_K f$ coincides with the composite of the projection
$X=X[\Gamma] \times \Delta^{\Gamma} \longrightarrow X[\Gamma]$
and $f_{\Gamma}:X[\Gamma] \longrightarrow Y[K]$
via the identification $Y[K]=Y_K$.
\end{itemize}
The morphism \eqref{the canonical morphism of kos for closed immersion:eq}
is a quasi-isomorphism
by Lemma (1.4) in \cite{SteenbrinkLE}
because
\begin{equation*}
\kernel(\iota_{\Gamma}^{-1}\varphi^{I' \setminus I}(\pi_K f))
=\kernel(\varphi^{I' \setminus I}(f_{\Gamma}))
\end{equation*}
and
\begin{equation*}
\cok(\iota_{\Gamma}^{-1}\varphi^{I' \setminus I}(\pi_K f))
=\cok(\varphi^{I' \setminus I}(f_{\Gamma})) .
\end{equation*}
Therefore the morphism
\eqref{the canonical morphism of A for closed immersion:eq}
is a quasi-isomorphism for the case of $K=I$.
Then the morphism
\eqref{the canonical morphism of A for closed immersion:eq}
is a filtered quasi-isomorphism with respect to $L(I \setminus K)$
by Lemma \ref{Lemma of grLsA}.
In particular, the morphism
\eqref{the canonical morphism of A for closed immersion:eq}
is a quasi-isomorphism.
Then we obtain the conclusion by Lemma \ref{Lemma of grLsA} again.
\end{proof}

\begin{para}
Let $g: S' \longrightarrow S$ be a morphism of complex manifolds.
We set $f': X' \longrightarrow Y'=\Delta^k \times S'$ etc.
as in \ref{notation for a base change for S}.
Then we have the commutative diagram
\begin{equation*}
\begin{CD}
g'^{-1}(\bQ^{I' \setminus I} \oplus \cO_X)
@>{g'^{-1}\varphi^{I' \setminus I}(f)}>>
g'^{-1}M_X(D)\gp_{\bQ} \\
@VVV @VVV \\
\bQ^{I' \setminus I} \oplus \cO_{X'}
@>>{\varphi^{I' \setminus I}(f')}>
M_{X'}(D)\gp_{\bQ}
\end{CD}
\end{equation*}
which induces a morphism of complexes
\begin{equation*}
g'^{-1}\kos(\varphi^{I' \setminus I}(f);\infty;1)
\longrightarrow
\kos(\varphi^{I' \setminus I}(f');\infty;1)
\end{equation*}
for $I \subset I' \subset \ski$.
Then we obtain the morphism of complexes
\begin{equation}
\label{base change morphism for sA:eq}
g'^{-1}sA_{I|I'}(f) \longrightarrow sA_{I|I'}(f')
\end{equation}
which preserves the filtration $L(K)$ for $K \subset I$.
\end{para}

\begin{lem}
\label{lemma for base change isomorphism for sA}
The morphism \eqref{base change morphism for sA:eq}
is a filtered quasi-isomorphism
with respect to the filtration $L(K)$ for $K \subset I$.
\end{lem}
\begin{proof}
Because we have
\begin{equation*}
\kernel(g'^{-1}\varphi^{I' \setminus I}(f))
=\kernel(\varphi^{I' \setminus I}(f'))
\end{equation*}
and
\begin{equation*}
\cok(g'^{-1}\varphi^{I' \setminus I}(f))
=\cok(\varphi^{I' \setminus I}(f')) ,
\end{equation*}
we can proceed the same way as in Lemma \ref{lemma for closed immersion}.
\end{proof}

\begin{defn}
For $I \subset \ski$,
the projection
\begin{equation*}
\bQ^I \oplus \cO_X \longrightarrow \cO_X
\end{equation*}
is denoted by $\tau^I(f)$ for a while.
Then a morphism
\begin{equation*}
\psi^I(f): \kos(\varphi^I(f);n)^p \longrightarrow \omega^p_X
\end{equation*}
is defined by sending
\begin{equation*}
\psi^I(f)(x_1^{[n_1]}x_2^{[n_2]} \cdots x_k^{[n_k]} \otimes y)
=\frac{(2\pi\sqrt{-1})^{-p}}{n_1!n_2! \cdots n_k!}
\tau^I(x_1)^{n_1}\tau^I(x_2)^{n_2} \cdots \tau^I(x_k)^{n_k}
(\bigwedge^p\dlog)(y)
\end{equation*}
for $x_1, x_2, \dots, x_k \in \bQ^I \oplus \cO_X$, $y \in \bigwedge^pM_X(D)\gp$
and $n_1, n_2, \dots, n_k \in \bnnZ$ with $n_1+n_2+\dots+n_k=n-p$
as in \cite[(2.4)]{FujisawaMHSLSD}.
Note that this does not define a morphism of complexes.
\end{defn}

\begin{lem}
\label{lemma for t^ and dlog t ^}
The diagram
\begin{equation*}
\begin{CD}
\kos(\varphi^I(f);\infty;1)^p @>{\psi^I(f)}>> \omega^p_X \\
@V{t \wedge}VV @VV{\dlog t \wedge}V \\
\kos(\varphi^I(f);\infty;1)^{p+1}
@>>{(2\pi\sqrt{-1})\psi^I(f)}>
\omega^{p+1}_X
\end{CD}
\end{equation*}
is commutative
for any global section $t$ of $M_X(D)$.
\end{lem}
\begin{proof}
Easy by definition.
\end{proof}

\begin{defn}
\label{definition of alpha}
For $I \subset I' \subset \ski$
and for $p \in \bnnZ, q \in \bnnZ^I$,
the composite of the morphism
\begin{equation*}
(2\pi\sqrt{-1})^{|q|+|I|}\psi^{I' \setminus I}(f):
\kos(\varphi^{I' \setminus I}(f);\infty;1)^{p+|q|+|I|}
\longrightarrow
\omega_X^{p+|q|+|I|}
\end{equation*}
and the projection
\begin{equation*}
\omega_X^{p+|q|+|I|}
\longrightarrow
\omega_{X/Y_I}^{p+|q|+|I|}
\end{equation*}
induces a morphism
\begin{equation}
\label{morphism from AII' to BI:eq}
A_{I|I'}(f)^{p,q} \longrightarrow B_I(f)^{p,q}
\end{equation}
because we have
$\psi^{I' \setminus I}(f)(W(i)_m) \subset W(D_i)_m$ for all $m$.
It is easy to check that
the morphism \eqref{morphism from AII' to BI:eq}
is compatible with
$d_0$.
Moreover Lemma \ref{lemma for t^ and dlog t ^}
implies that the morphism \eqref{morphism from AII' to BI:eq}
is compatible with $d_i$ for any $i \in I$.
Thus a morphism of complexes
\begin{equation}
\label{definition of alpha:eq}
\alpha_{I|I'}(f): sA_{I|I'}(f) \longrightarrow sB_I(f)
\end{equation}
is obtained.
Clearly, $\alpha_{I|I'}(f)$
preserves the filtration $L(K)$ for $K \subset I$.
We use $\alpha_I(f)=\alpha_{I|I}(f)$,
$\alpha_{I|}(f)=\alpha_{I|\ski}(f)$,
$\alpha_{|I'}(f)=\alpha_{\emptyset|I'}(f)$,
$\alpha_0(f)=\alpha_{\emptyset|\ski}(f)$
and $\alpha(f)=\alpha_{\ski}(f)=\alpha_{\ski|\ski}(f)$ for short.
\end{defn}

\begin{lem}
\label{lemma for alpha0}
The morphism $\alpha_0(f)$ induces a quasi-isomorphism
\begin{equation*}
\cO_{Y,f(x)} \otimes_{\bQ} \kos(\varphi(f);\infty;1)_x
\longrightarrow
\omega_{X/Y,x}
\end{equation*}
for any $x \in X_0$.
\end{lem}
\begin{proof}
Since the question is of local nature,
we may assume that $f$ is a morphism of the form
\eqref{local model of semistable morphism:eq}.
By using Lemma (1.4) in \cite{SteenbrinkLE},
we can easily reduce the problem
to the case of $k=1$.
Thus we obtain the conclusion
by Lemma (1.4) in \cite{SteenbrinkLE}
and by Proposition (1.13) in \cite{SteenbrinkLHS}.
\end{proof}

\begin{prop}
\label{proposition for alpha being fqis}
The morphism $\alpha_{I|I'}(f)$ induces a filtered quasi-isomorphism
\begin{equation*}
(f_I^{-1}\cO_{Y[I]} \otimes_{\bQ} sA_{I|I'}(f))\left|_{X[I']^{\ast}}\right.
\longrightarrow
sB_I(f)\left|_{X[I']^{\ast}}\right.
\end{equation*}
with respect to $L(K)$ for $K \subset I$.
\end{prop}
\begin{proof}
We may assume $I'=\ski$,
because we consider the situation over $X[I']^{\ast}$.
Therefore it is sufficient to prove
that the morphism $\alpha_{I|}(f)$
induces a filtered quasi-isomorphism
\begin{equation*}
(f_I^{-1}\cO_{Y[I]} \otimes_{\bQ} sA_{I|}(f))\left|_{X_0}\right.
\longrightarrow
sB_I(f)\left|_{X_0}\right.
\end{equation*}
with respect to $L(K)$ for $K \subset I$.

By Lemma \ref{lemma on grL(K)sB},
Lemma \ref{Lemma of grLsA} and
Lemma \ref{lemma for closed immersion}
we obtain a commutative diagram
\begin{equation}
\begin{CD}
\gr_m^{L(K)}sA_{I|}(f)
@>{\gr_m^{L(K)}\alpha_{I|}(f)}>>
\gr_m^{L(K)}sB_I(f) \\
@A{\simeq}AA @| \\
\bigoplus
\iota_{\Gamma}^{-1}sA_{(I \setminus K)|}(\pi_Kf)[-m-2|q|]
@.
\gr_m^{L(K)}sB_I(f) \\
@VVV @AA{\simeq}A \\
\bigoplus
sA_{(I \setminus K)|}(f_{\Gamma})[-m-2|q|]
@>>>
\bigoplus
sB_{I \setminus K}(f_{\Gamma})[-m-2|q|],
\end{CD}\label{commutative diagram for grA and grB:eq}
\end{equation}
where the direct sums are taken over the index set
\begin{equation}
\label{index set for grL(K):eq}
\{(q, \Gamma); q \in \bnnZ^K, \Gamma \in S_{m+2|q|+|K|}^{\ge q+e_K}(\Lambda_K)\}
\end{equation}
as in \eqref{grL(K)sB:eq},
and where the bottom horizontal arrow is the direct sum
of the morphisms
\begin{equation*}
(2\pi\sqrt{-1})^{-m-|q|}
\alpha_{(I \setminus K)|}(f_{\Gamma})[-m-2|q|]
\end{equation*}
over the index set \eqref{index set for grL(K):eq}.
Therefore Lemma \ref{lemma for alpha0} implies that
$\alpha_{I|}(f)$ induces a filtered quasi-isomorphism
with respect to $L(I)$.
In particular, $\alpha_{I|}(f)$ induces a quasi-isomorphism.
Then we can conclude that
$\alpha_{I|}(f)$ induces a filtered quasi-isomorphism
with respect to $L(K)$ for $K \subset I$
by the commutative diagram
\eqref{commutative diagram for grA and grB:eq} again.
\end{proof}

\begin{cor}
\label{corollary on grL's for sA and sB}
The morphism $\alpha_{I|I'}(f)$ induces
a bifiltered quasi-isomorphism
\begin{equation*}
(f_I^{-1}\cO_{Y[I]} \otimes_{\bQ}
sA_{I|I'}(f), L(J), L(K))\left|_{X[I']^{\ast}}\right. \\
\overset{\simeq}{\longrightarrow}
(sB_I(f), L(J), L(K))\left|_{X[I']^{\ast}}\right.
\end{equation*}
for $J, K \subset I$.
\end{cor}
\begin{proof}
By the commutative diagram \eqref{commutative diagram for grA and grB:eq}
we can easily obtain the conclusion.
\end{proof}

\begin{defn}
From the morphism
$\alpha_{I|I'}(f): sA_{I|I'}(f) \longrightarrow sB_I(f)$,
we obtain morphisms
\begin{equation*}
R(f_I)_{\ast}sA_{I|I'}(f)
\longrightarrow
R(f_I)_{\ast}sB_I(f)
\end{equation*}
and
\begin{equation*}
\cO_{Y[I]} \otimes_{\bQ} R(f_I)_{\ast}sA_{I|I'}(f)
\longrightarrow
R(f_I)_{\ast}sB_I(f)
\end{equation*}
which we denote by the same letter $\alpha_{I|I'}(f)$
if there is no danger of confusion.
These morphisms preserves the filtration $L(K)$
for $K \subset I$.
\end{defn}

\begin{lem}
\label{lemma on base change for sB}
Under the situation in $\ref{notation for a base change for S}$,
the morphism \eqref{base change morphism for sB:eq}
is a filtered quasi-isomorphism with respect to $L(K)$
for all $K \subset I$.
\end{lem}
\begin{proof}
For any $I' \supset I$,
we have the commutative diagram
\begin{equation*}
\begin{CD}
(f_I'^{-1}\cO_{Y'[I]}
\otimes_{\bQ}
g'^{-1}sA_{I|I'}(f))|_{X'[I']^{\ast}}
@>>>
(f_I'^{-1}\cO_{Y'[I]}
\otimes_{g'^{-1}f_I^{-1}\cO_{Y[I]}}
g'^{-1}sB_I(f))|_{X'[I']^{\ast}} \\
@VVV @VVV \\
(f_I'^{-1}\cO_{Y'[I]}
\otimes_{\bQ}
sA_{I|I'}(f'))|_{X'[I']^{\ast}}
@>>>
sB_I(f')|_{X'[I']^{\ast}}
\end{CD}
\end{equation*}
where the horizontal arrows
and the left vertical arrow are filtered quasi-isomorphisms
with respect to $L(K)$
by Proposition \ref{proposition for alpha being fqis}
and Lemma \ref{lemma for base change isomorphism for sA}.
Thus we obtain the conclusion.
\end{proof}

Now we compare the canonical connection $d \otimes \id$
on $\cO_{Y[I]} \otimes R^n(f_I)_{\ast}sA_I(f)$
and the connection $(-1)^{|I|}\nabla(I)$ on $R^n(f_I)_{\ast}sB_I(f)$
via the morphism $\alpha_I(f)$.

\begin{defn}
For $I \subset \ski$,
the morphism
\begin{equation*}
(2\pi\sqrt{-1})^{|q|+|I|}\psi^{\emptyset}(f):
\kos(\e(f)_{\bQ},\infty,1)^{p+|q|+|I|}
\longrightarrow
\omega_X^{p+|q|+|I|}
\end{equation*}
induces a morphism
$A_I(f)^{p,q} \longrightarrow \tB_I(f)^{p,q}$
for $p \in \bnnZ, q \in \bnnZ^I$.
We define a morphism
\begin{equation*}
f_I^{-1}\omega^r_{Y[I]} \otimes_{\bQ} A_I(f)^{p,q}
\longrightarrow
\tB_I(f)^{p+r,q}
\end{equation*}
by sending $\omega \otimes \eta$
to $(-1)^{r|I|}\omega \wedge \alpha_I(f)(\eta)$
for $p \in \bnnZ, q \in \bnnZ^I, r \in \bnnZ$
as in Remark \ref{wedge product for omegaY and B}.
It is easy to check that these morphisms
induce a morphism of complexes
\begin{equation*}
f_I^{-1}\omega_{Y[I]} \otimes_{\bQ} sA_I(f)
\longrightarrow
s\tB_I(f)
\end{equation*}
which is denoted by $\widetilde{\alpha}_I(f)$.
A finite decreasing filtration $G$
on $f_I^{-1}\omega_{Y[I]} \otimes_{\bQ} sA_I(f)$
is induced from the stupid filtration on $\omega_{Y[I]}$.
Then $\widetilde{\alpha}_I(f)$ preserves the filtrations $G$
on the both sides.
\end{defn}

\begin{rmk}
\label{remark on the canonical and Gauss-Manin}
From the definition above,
the diagram
\begin{equation*}
\begin{CD}
\omega^p_{Y[I]} \otimes R^n(f_I)_{\ast}sA_I(f)
@>{\id \otimes \alpha_I(f)}>>
\omega^p_{Y[I]} \otimes R^n(f_I)_{\ast}sB_I(f) \\
@V{d \otimes \id}VV @VV{(-1)^{|I|}\nabla(I)}V \\
\omega^{p+1}_{Y[I]} \otimes R^n(f_I)_{\ast}sA_I(f)
@>>{\id \otimes \alpha_I(f)}>
\omega^{p+1}_{Y[I]} \otimes R^n(f_I)_{\ast}sB_I(f)
\end{CD}
\end{equation*}
is commutative for all $p$.
\end{rmk}

\begin{lem}
\label{lemma for alphaI to BI}
The restriction
\begin{equation*}
\widetilde{\alpha}_I(f)\left|_{X[I]^{\ast}}\right.:
(f_I^{-1}\omega_{Y[I]} \otimes_{\bQ} sA_I(f))\left|_{X[I]^{\ast}}\right.
\longrightarrow
s\tB_I(f)\left|_{X[I]^{\ast}}\right.
\end{equation*}
of the morphism $\widetilde{\alpha}_I(f)$ on $X[I]^{\ast}$
is a filtered quasi-isomorphism
with respect to the filtrations $G$ on the both sides.
\end{lem}
\begin{proof}
We have isomorphisms of complexes
\begin{equation*}
\begin{split}
&\gr_G^p(f_I^{-1}\omega_{Y[I]} \otimes sA_I(f))
\simeq
f_I^{-1}\omega^p_{Y[I]} \otimes sA_I(f)[-p] \\
&\gr_G^ps\tB_I(f)
\simeq
f_I^{-1}\omega^p_{Y[I]} \otimes sB_I(f)[-p]
\end{split}
\end{equation*}
for every $p$.
Under these identification,
$\gr_G^p\widetilde{\alpha}_I(f)$
is identified with $(-1)^{p|I|}\id \otimes \alpha_I(f)[-p]$
by definition.
Thus we obtain the conclusion
by Proposition \ref{proposition for alpha being fqis}.
\end{proof}

\section{Proper case}
\label{section for the proper case}

From now on,
we assume that the semistable morphism
$f:X \longrightarrow Y=\Delta^k \times S$
is proper.

\begin{lem}
\label{lemma for sA and sB}
For $K, J \subset I \subset I'$,
the morphism $\alpha_{I|I'}(f)$ in \eqref{definition of alpha:eq}
induces an isomorphism
\begin{equation*}
(\cO_{Y[I]} \otimes_{\bQ}
R(f_I)_{\ast}sA_{I|I'}(f), L(J), L(K))
\left|_{Y[I']^{\ast}}\right.
\overset{\simeq}{\longrightarrow}
(R(f_I)_{\ast}sB_I(f), L(J), L(K))\left|_{Y[I']^{\ast}}\right.
\end{equation*}
in the bifiltered derived category.
\end{lem}
\begin{proof}
It suffices to prove
that the morphism $\alpha_{I|I'}(f)$ induces
a quasi-isomorphism
\begin{equation*}
\cO_{Y[I]} \otimes_{\bQ}
\gr_l^{L(J)}\gr_m^{L(K)}
R(f_I)_{\ast}sA_{I|I'}(f)\left|_{Y[I']^{\ast}}\right.
\overset{\simeq}{\longrightarrow}
\gr_l^{L(J)}\gr_m^{L(K)}
R(f_I)_{\ast}sB_I(f)\left|_{Y[I']^{\ast}}\right.
\end{equation*}
for all $l,m$.
Since $f$ is proper,
we obtain the conclusion
from Corollary \ref{corollary on grL's for sA and sB},
by applying the projection formula
(cf. \cite[Proposition 2.6.6]{Kashiwara-SchapiraSM}).
\end{proof}

\begin{lem}
\label{lemma on theta for direct image complexes}
For $K, J \subset I$,
the morphism $\theta_{J|I}(f)$ in \eqref{the morphism thetaJ|I:eq}
induces an isomorphism
\begin{equation*}
(\cO_{Y[I]} \otimes^L_{\cO_{Y[J]}} R(f_J)_{\ast}sB_J(f), F, L(K)) \\
\overset{\simeq}{\longrightarrow}
(R(f_I)_{\ast}sB_I(f), F, L(K))
\end{equation*}
in the bifiltered derived category,
where $\otimes^L_{\cO_{Y[J]}}$ stands for
the bifiltered derived tensor product over $\cO_{Y[J]}$.
\end{lem}
\begin{proof}
It suffices to prove that
the morphism $\theta_{J|I}(f)$
induces an isomorphism
\begin{equation*}
\cO_{Y[I]}
\otimes^L_{\cO_{Y[J]}}
\gr_F^p\gr_m^{L(K)}
R(f_J)_{\ast}sB_J(f)
\overset{\simeq}{\longrightarrow}
\gr_F^p\gr_m^{L(K)}
R(f_I)_{\ast}sB_I(f)
\end{equation*}
in the derived category,
where $\otimes^L_{\cO_{Y[J]}}$
stands for the derived tensor product
over $\cO_{Y[J]}$.
By Lemma \ref{flatness lemma}
and Lemma \ref{lemma on grLsBJ to grLsBI},
we have the isomorphisms
\begin{equation*}
\begin{split}
f_I^{-1}\cO_{Y[I]}
\otimes^L_{(f_J)^{-1}\cO_{Y[J]}}
\gr_F^p&\gr_m^{L(K)}sB_J(f) \\
&\overset{\simeq}{\longrightarrow}
f_I^{-1}\cO_{Y[I]}
\otimes_{(f_J)^{-1}\cO_{Y[J]}}
\gr_F^p\gr_m^{L(K)}sB_J(f) \\
&\overset{\simeq}{\longrightarrow}
\gr_F^p\gr_m^{L(K)}sB_I(f)
\end{split}
\end{equation*}
for all $m, p$
in the derived category.
Then we obtain the conclusion by
the projection formula and the proper base change theorem
(cf. \cite[Proposition 2.6.7]{Kashiwara-SchapiraSM}).
\end{proof}

\begin{lem}
\label{filtered identification between ss of A and B}
For $K, J \subset I \subset I'$,
the morphism $\alpha_{I|I'}(f)$ induces an isomorphism
\begin{equation}
\label{rational structure of ss:eq}
\begin{split}
(\cO_{Y[I]}
\otimes_{\bQ}
E_r^{a,b}(R(f_I)_{\ast}sA_{I|I'}&(f),L(K))\left|_{Y[I']^{\ast}}\right. \\
&\overset{\simeq}{\longrightarrow}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))\left|_{Y[I']^{\ast}}\right.
\end{split}
\end{equation}
for all $a,b,r$,
under which we have
\begin{equation*}
\begin{split}
(\cO_{Y[I]}
\otimes_{\bQ}
(L(J)_{\rec})_mE_r^{a,b}(R(f_I)_{\ast}&sA_{I|I'}(f)),L(K))
\left|_{Y[I']^{\ast}}\right. \\
&\simeq
(L(J)_{\rec})_mE_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\left|_{Y[I']^{\ast}}\right.
\end{split}
\end{equation*}
for every $m$.
In particular,
we have
\begin{equation*}
\begin{split}
(\cO_{Y[I]}
\otimes_{\bQ}
\gr_{m_l}^{L(J_l)_{\rec}}
&\cdots
\gr_{m_2}^{L(J_2)_{\rec}}
\gr_{m_1}^{L(J_1)_{\rec}}
E_r^{a,b}(R(f_I)_{\ast}sA_{I|I'}(f)),L(K))\left|_{Y[I']^{\ast}}\right. \\
&\simeq
\gr_{m_l}^{L(J_l)_{\rec}}
\cdots
\gr_{m_2}^{L(J_2)_{\rec}}
\gr_{m_1}^{L(J_1)_{\rec}}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))\left|_{Y[I']^{\ast}}\right.
\end{split}
\end{equation*}
for $K, J_1,J_2, \dots, J_l \subset I$
and for all $a,b,m_1,m_2, \dots, m_l,r$.
\end{lem}
\begin{proof}
Easy from Lemma \ref{lemma for sA and sB}
and Lemma \ref{local freeness and the commutattivity of gr}.
\end{proof}

\begin{prop}
For $J \subset I \subset I' \subset \ski$,
the morphism $\alpha_{I|I'}(f)$ induces an isomorphism
\begin{equation}
\label{isom form sAI|I' to sBI:eq}
(\cO_{Y[I]}
\otimes_{\bQ}
R^n(f_I)_{\ast}sA_{I|I'}(f))\left|_{Y[I']^{\ast}}\right.
\simeq
R^n(f_I)_{\ast}sB_I(f)\left|_{Y[I']^{\ast}}\right.
\end{equation}
for all $n$,
under which we have
\begin{equation*}
(\cO_{Y[I]}
\otimes_{\bQ}
L(J)_mR^n(f_I)_{\ast}sA_{I|I'}(f))\left|_{Y[I']^{\ast}}\right.
\simeq
L(J)_m
R^n(f_I)_{\ast}sB_I(f)\left|_{Y[I']^{\ast}}\right.
\end{equation*}
for all $m$.
In particular,
the morphism $\alpha_{I|I'}(f)$ induces an isomorphism
\begin{equation*}
\begin{split}
(\cO_{Y[I]}
\otimes_{\bQ}
\gr_{m_l}^{L(J_l)}
&\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sA_{I|I'}(f))\left|_{Y[I']^{\ast}}\right. \\
&\simeq
\gr_{m_l}^{L(J_l)}
\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sB_I(f)\left|_{Y[I']^{\ast}}\right.
\end{split}
\end{equation*}
for $J_1, J_2, \dots, J_l \subset I$
and for all $m_1,m_2, \dots, m_l, n$.
\end{prop}
\begin{proof}
Take $K=\emptyset$ and $r=1$
in Lemma \ref{filtered identification between ss of A and B}.
\end{proof}

\begin{prop}
\label{local freeness of the spectral sequence}
For $K, J_1, J_2, \dots, J_l \subset I$,
the coherent $\cO_{Y[I]}$-module
\begin{equation*}
\gr_{m_l}^{L(J_l)_{\rec}}
\cdots
\gr_{m_2}^{L(J_2)_{\rec}}
\gr_{m_1}^{L(J_1)_{\rec}}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is locally free of finite rank
for all $a,b,m_1,m_2, \dots, m_l,r$.
In particular,
\begin{equation*}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is locally free $\cO_{Y[I]}$-module of finite rank
for $K \subset I$ and for all $a,b,r$.
\end{prop}
\begin{proof}
For any $y \in Y[I]$,
there exists $I' \supset I$
such that $y \in Y[I']^{\ast}$.
Then the stalk
\begin{equation*}
\gr_{m_l}^{L(J_l)_{\rec}}
\cdots
\gr_{m_2}^{L(J_2)_{\rec}}
\gr_{m_1}^{L(J_1)_{\rec}}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))_y
\end{equation*}
is a free $\cO_{Y[I],y}$-module
from Lemma \ref{filtered identification between ss of A and B}.
\end{proof}

\begin{prop}
\label{proposition on the local freeness of grLgrLsBI}
For $J_1, J_2, \dots, J_l \subset I$,
the coherent $\cO_{Y[I]}$-module
\begin{equation*}
\gr_{m_l}^{L(J_l)}
\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
is locally free of finite rank
for all $m_1, m_2, \dots, m_l,n$.
In particular,
\begin{equation*}
R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
is locally free of finite rank for all $n$.
\end{prop}
\begin{proof}
Take $K=\emptyset$ and $r=1$
in Proposition \ref{local freeness of the spectral sequence}.
\end{proof}

\begin{thm}
\label{theorem on the canonical extension}
If the semistable morphism $f:X \longrightarrow Y=\Delta^k \times S$
is proper, then
$R^n(f_I)_{\ast}sB_I(f)$
is the canonical extension of
$(\cO_{Y[I]} \otimes R^n(f_I)_{\ast}sA_I(f))|_{Y[I]^{\ast}}$
via the isomorphisms
\eqref{isom form sAI|I' to sBI:eq}
for $I=I'$.
\end{thm}
\begin{proof}
Considering Proposition \ref{proposition on the local freeness of grLgrLsBI}
and Remark \ref{remark on the canonical and Gauss-Manin},
it suffices to check
that the residues of the connection $(-1)^{|I|}\nabla(I)$
are nilpotent.
By Corollary \ref{cor on compatibility of nabla(J) and nabla(I)},
these residues
coincide with the residues of $\nabla$
on $R^nf_{\ast}\omega_{X/Y}$.
Therefore
Theorem \ref{thm on the compatibility of N and nu}
shows that
the residues of $(-1)^{|I|}\nabla(I)$
along $E_j \cap Y[I]$ for $j \notin I$
coincide with $-N_{j|I}(f)$,
which is nilpotent by definition.
Thus $R^n(f_I)_{\ast}sB_I(f)$ equipped with $(-1)^{|I|}\nabla(I)$
is the canonical extension as desired.
\end{proof}

\begin{rmk}
For $I=\emptyset$,
the theorem above tells us that
$R^nf_{\ast}\omega_{X/Y}$ is
the canonical extension of its restriction on $Y^{\ast}$.
This has been already claimed
by Usui \cite[p.138]{UsuiRVCSV}.
\end{rmk}

\begin{lem}
\label{lemma for ss being the canonical extension}
For $K \subset I$,
the locally free $\cO_{Y[I]}$-module
\begin{equation*}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is the canonical extension of
\begin{equation*}
(\cO_{Y[I]}
\otimes_{\bQ}
E_r^{a,b}(R(f_I)_{\ast}sA_I(f),L(K)))\left|_{Y[I]^{\ast}}\right.
\end{equation*}
via the isomorphism \eqref{rational structure of ss:eq} for $I'=I$.
\end{lem}
\begin{proof}
From the commutative diagram
\eqref{commutative diagram for grA and grB:eq},
Theorem \ref{theorem on the canonical extension}
implies the conclusion for $r=1$.
Then we obtain the conclusion for general $r$
because taking the canonical extension is the exact functor.
\end{proof}

\begin{rmk}
\label{remark on the canonical extension of ss}
For $K, J \subset I$,
the locally free $\cO_{Y[I]}$-module
\begin{equation*}
(L(J)_{\rec})_mE_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is the canonical extension of its restriction on $Y[I]^{\ast}$.
\end{rmk}

\begin{lem}
\label{lemma for theta and the spectral sequences}
For $K \subset J \subset I$,
the morphism $\theta_{J|I}(f)$ in \eqref{the morphism thetaJ|I:eq}
induces an isomorphism
\begin{equation}
\label{theta and ss:eq}
\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
E_r^{a,b}(R(f_J)_{\ast}sB_J(f),L(K))
\overset{\simeq}{\longrightarrow}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation}
for all $a,b,r$.
\end{lem}
\begin{proof}
Lemma \ref{lemma on theta for direct image complexes}
and the local freeness of $E_r^{a,b}(R(f_J)_{\ast}sB_J(f),L(K))$
implies the conclusion
from the base change theorem.
\end{proof}

\begin{prop}
For $K \subset J \subset I$,
the morphism $\theta_{J|I}(f)$
induces isomorphisms
\begin{align}
&\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
L(K)_mR^n(f_J)_{\ast}sB_J(f)
\overset{\simeq}{\longrightarrow}
L(K)_mR^n(f_I)_{\ast}sB_I(f)
\label{isom for L(K) induced by theta:eq} \\
&\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
\gr_m^{L(K)}R^n(f_J)_{\ast}sB_J(f)
\overset{\simeq}{\longrightarrow}
\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)
\label{isom for grL(K) induced by theta:eq}
\end{align}
for all $m,n$.
\end{prop}
\begin{proof}
Taking $r$ sufficiently large
in \eqref{theta and ss:eq},
we obtain the isomorphism
\eqref{isom for grL(K) induced by theta:eq}
for all $m,n$.
The local freeness of $\gr_m^{L(K)}R^n(f_J)_{\ast}sB_J(f)$
implies \eqref{isom for L(K) induced by theta:eq}.
\end{proof}

\begin{cor}
\label{corollary on grL's and theta}
For $J_1,J_2, \dots, J_l \subset J \subset I$,
the morphism $\theta_{J|I}(f)$
induces an isomorphism
\begin{equation}
\label{the morphism induced by theta on grL's:eq}
\begin{split}
\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
\gr_{m_l}^{L(J_l)} \cdots
&\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
R^n(f_J)_{\ast}sB_J(f) \\
&\overset{\simeq}{\longrightarrow}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sB_I(f)
\end{split}
\end{equation}
for all $m_1,m_2, \dots, m_l,n$.
\end{cor}
\begin{proof}
From the local freeness of
$\gr_{m_l}^{L(J_l)} \cdots \gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
R^n(f_J)_{\ast}sB_J(f)$
in Proposition \ref{proposition on the local freeness of grLgrLsBI},
we obtain the conclusion
by Lemma \ref{local freeness and the commutattivity of gr}.
\end{proof}


\section{Degeneration of Hodge structures}
\label{section for the Kahler case}

In this section,
we assume
that the semistable morphism
$f:X \longrightarrow Y=\Delta^k \times S$
is proper and K\"ahler.

\begin{lem}
\label{lemma on strong strictness for F}
If the semistable morphism $f:X \longrightarrow Y=\Delta^k \times S$
is proper and K\"ahler,
$(R(f_I)_{\ast}sB_I(f),F)$ is strongly strict
for all $I \subset \ski$.
In particular,
the coherent $\cO_{Y[I]}$-module
$\gr_F^pR^n(f_I)_{\ast}sB_I(f)$
is locally free of finite rank
for all $n,p$.
\end{lem}
\begin{proof}
For the case of $I=\emptyset$,
the strong strictness of $(Rf_{\ast}\omega_{X/Y},F)$ is proved
in \cite[Theorem (6.10)]{FujisawaLHSSV}.
The morphism $\theta_{|I}(f)$ induces an isomorphism
\begin{equation*}
(\cO_{Y[I]} \otimes_{\cO_Y}^L Rf_{\ast}\omega_{X/Y},F)
\simeq
(R(f_I)_{\ast}sB_I(f),F)
\end{equation*}
in the filtered derived category
by Lemma \ref{lemma on theta for direct image complexes}.
Therefore we obtain the conclusion
by Lemma \ref{lemma on strong strictness and filtered tensor product}.
\end{proof}

\begin{prop}
The morphism $\theta_{J|I}(f)$ induces an isomorphism
\begin{equation*}
\cO_{Y[I]} \otimes_{\cO_{Y[J]}} R^n(f_J)_{\ast}sB_J(f)
\simeq
R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for every $n$,
under which we have
\begin{equation*}
\cO_{Y[I]} \otimes_{\cO_{Y[J]}} F^pR^n(f_J)_{\ast}sB_J(f)
\simeq
F^pR^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for all $p$.
\end{prop}
\begin{proof}
Lemma \ref{lemma on theta for direct image complexes}
and the strong strictness of $(R(f_J)_{\ast}sB_J(f),F)$
implies the conclusion
as in Lemma \ref{lemma on strong strictness and filtered tensor product}.
\end{proof}

\begin{prop}
\label{isom induced by theta on FgrL's}
The isomorphism \eqref{the morphism induced by theta on grL's:eq}
induces an isomorphism
\begin{equation*}
\begin{split}
F^p(\cO_{Y[I]} \otimes
\gr_{m_l}^{L(J_l)}
\cdots
&\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_J)_{\ast}sB_J(f)) \\
&\overset{\simeq}{\longrightarrow}
F^p\gr_{m_l}^{L(J_l)}
\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sB_I(f)
\end{split}
\end{equation*}
for all $p$.
\end{prop}
\begin{proof}
Easy by Lemma \ref{local freeness and the commutattivity of gr}.
\end{proof}

\begin{lem}
\label{lemma on VMHS for I}
If the semistable morphism $f:X \longrightarrow Y=\Delta^k \times S$
is proper and K\"ahler,
then the data
\begin{equation*}
((R^n(f_I)_{\ast}L(J)_msA_I(f),L(I)[n]),
(R^n(f_I)_{\ast}L(J)_msB_I(f),L(I)[n],F), \alpha_I(f))|_{Y[I]^{\ast}}
\end{equation*}
is a graded polarizable variation of $\bQ$-mixed Hodge structure
on $Y[I]^{\ast}$
for $J \subset I$ and for all $m,n$.
\end{lem}
\begin{proof}
We may assume that $I=\ski$.
Then
\begin{equation*}
Y[\ski]^{\ast}=Y[\ski]=\{0\} \times S=S
\end{equation*}
by definition.
As in the proof of Proposition \ref{proposition for alpha being fqis},
we have the isomorphism
\begin{equation*}
(\gr_{-a}^LL(J)_msB(f),F)
\simeq
\bigoplus
(\Omega_{X[\Gamma]/S}[a-2|q|],F[a-|q|])
\end{equation*}
and the quasi-isomorphism
\begin{equation*}
\gr_{-a}^LL(J)_msA(f)
\simeq
\bigoplus
\kos(\mathbf{e}(f_{\Gamma});\infty;1)[a-2|q|]
\end{equation*}
by Lemma \ref{lemma on grL(K)sB},
Lemma \ref{Lemma of grLsA} and
Lemma \ref{lemma for closed immersion},
where the direct sums are taken over the same index set
\begin{equation*}
\{(q, \Gamma) ; {q \in \bnnZ^k}, 
\Gamma \in S_{-a+2|q|+k}^{\ge q+e}(\Lambda)
\text{ with }
|\Gamma \cap \Lambda_J| \le m+2|q_J|+|J|\}.
\end{equation*}
Under these identifications
the morphism $\gr_{-a}^L\alpha(f)$
is identified with the direct sum of the morphisms
$(2\pi\sqrt{-1})^{a-|q|}\psi(f_{\Gamma})[a-2|q|]$.
Because
\begin{equation*}
f_{\Gamma}: X[\Gamma] \longrightarrow S
\end{equation*}
is a proper K\"ahler smooth morphism,
we have the following:
\begin{mylist}
\itemno
\label{VHS condition:eq}
The data
\begin{equation*}
(R^{a+b}f_{0\ast}\gr_{-a}^LL(J)_msA(f),
(R^{a+b}f_{0\ast}\gr_{-a}^LL(J)_msB(f),F),
R^{a+b}f_{0\ast}\gr_{-a}^L\alpha(f))
\end{equation*}
is a polarizable variation of $\bQ$-Hodge structure of weight $b$
on $S$.
\itemno
\label{strictness condition for E0:eq}
The filtered complex
$(Rf_{0\ast}\gr_{-a}^LL(J)_msB(f),F)$
is strongly strict for all $a$.
\end{mylist}
The morphism
\begin{equation*}
d_0:
E_0^{a,b}(Rf_{0\ast}L(J)_msB(f),L)
\longrightarrow
E_0^{a,b+1}(Rf_{0\ast}L(J)_msB(f),L)
\end{equation*}
is strictly compatible with respect to the filtration
$F=F_{\rec}=F_{d}=F_{d^{\ast}}$
by \eqref{strictness condition for E0:eq}.
Then we have $F_{\rec}=F_{d}=F_{d^{\ast}}$
on $E_1^{a,b}(Rf_{0\ast}L(J)_msB(f),L)$
and on $E_2^{a,b}(Rf_{0\ast}L(J)_msB(f),L)$
by the lemma on two filtration
\cite[Proposition (7.2.5)]{DeligneIII}.
Moreover
\begin{equation*}
(E_1^{a,b}(Rf_{0\ast}L(J)_msA(f),L),
(E_1^{a,b}(Rf_{0\ast}L(J)_msB(f),L),F_{\rec}),
E_1^{a,b}(\alpha(f)))
\end{equation*}
is a polarizable variation of $\bQ$-Hodge structure of weight $b$ on $S$
by \eqref{VHS condition:eq}.
Because the morphism
\begin{equation*}
d_1:
E_1^{a,b}(Rf_{0\ast}L(J)_msB(f),L)
\longrightarrow
E_1^{a+1,b}(Rf_{0\ast}L(J)_msB(f),L)
\end{equation*}
underlies a morphism of variations of $\bQ$-Hodge structure,
the morphism $d_1$ is strictly compatible with the filtration $F_{\rec}$ and
\begin{equation*}
(E_2^{a,b}(Rf_{0\ast}L(J)_msA(f),L),
(E_2^{a,b}(Rf_{0\ast}L(J)_msB(f),L),F_{\rec}),
E_2^{a,b}(\alpha(f)))
\end{equation*}
is a polarizable variation
of $\bQ$-Hodge structures of weight $b$ on $S$.
Because the morphism
\begin{equation*}
d_2:
E_2^{a,b}(Rf_{0\ast}L(J)_msB(f),L)
\longrightarrow
E_2^{a+2,b-1}(Rf_{0\ast}L(J)_msB(f),L)
\end{equation*}
underlies a morphism of variations of $\bQ$-Hodge structure again,
we obtain $d_2=0$.
Inductively, we obtain $d_r=0$ for $r \ge 2$.
Then we have
\begin{equation*}
\begin{split}
&E_2^{a,b}(Rf_{0\ast}L(J)_msA(f),L)
\simeq
E_{\infty}^{a,b}(Rf_{0\ast}L(J)_msA(f),L) \\
&E_2^{a,b}(Rf_{0\ast}L(J)_msB(f),L)
\simeq
E_{\infty}^{a,b}(Rf_{0\ast}L(J)_msB(f),L)
\end{split}
\end{equation*}
and $F=F_{\rec}=F_{d}=F_{d^{\ast}}$
on $E_{\infty}^{a,b}(Rf_{0\ast}L(J)_msB(f),L)$
for all $a,b$.
Thus the data
\begin{equation*}
(\gr_{-a}^LR^{a+b}f_{0\ast}L(J)_msA(f),
(\gr_{-a}^LR^{a+b}f_{0\ast}L(J)_msB(f),F), \gr_{-a}^L\alpha_0(f))
\end{equation*}
is a polarizable variation of Hodge structures of weight $b$ on $S$.
By Remark \ref{remark on the canonical and Gauss-Manin}
the canonical connection $d \otimes \id$
on $\cO_{S} \otimes R^qf_{0\ast}sA(f)$ is identified
with the Gauss-Manin connection $(-1)^k\nabla(\ski)$,
which satisfies the Griffiths transversality.
Thus we obtain the conclusion.
\end{proof}

The lemma above implies the following theorem.

\begin{thm}
\label{theorem on VMHS for I}
In the situation above,
the data
\begin{equation}
\label{VMHS for I:eq}
((R^n(f_I)_{\ast}sA_I(f),L(I)[n]),
(R^n(f_I)_{\ast}sB_I(f),L(I)[n],F), \alpha_I(f))|_{Y[I]^{\ast}}
\end{equation}
is a graded polarizable variation
of $\bQ$-mixed Hodge structure on $Y[I]^{\ast}$.
Moreover, $L(J)_m$
gives a subvariation of $\bQ$-mixed Hodge structure
for every $J \subset I$ and for every $m$.
\end{thm}

\begin{thm}
\label{theorem for E2-degeneracy}
If the semistable morphism $f:X \longrightarrow Y=\Delta^k \times S$
is proper and K\"ahler,
the spectral sequence
$E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))$
degenerates at $E_2$-terms
for $K \subset I$.
In other words,
we have the canonical isomorphism
\begin{equation}
\label{E2-degeneration for L(K):eq}
\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)
\overset{\simeq}{\longrightarrow}
E_2^{-m,n+m}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation}
for all $m,n$.
\end{thm}
\begin{proof}
We first treat the case of $I=K$.
As in the proof of Theorem \ref{theorem on VMHS for I},
the morphisms of $E_r$-terms
\begin{equation*}
E_r^{a,b}(R(f_K)_{\ast}sB_K(f),L(K))
\longrightarrow
E_r^{a+r,b-r+1}(R(f_K)_{\ast}sB_K(f),L(K))
\end{equation*}
are the zero morphisms over $Y[K]^{\ast}$ for all $a,b$
and for all $r \ge 2$.
Therefore the morphisms of $E_r$-terms are
the zero morphisms over the whole $Y[K]$
for $r \ge 2$
because $E_r^{a,b}(R(f_K)_{\ast}sB_K(f),L(K))$
are locally free over $Y[K]$
by Proposition \ref{local freeness of the spectral sequence}.
Thus we obtain the desired $E_2$-degeneracy.

For the general case,
we have the isomorphism
\eqref{theta and ss:eq} for $J=K$
\begin{equation*}
\cO_{Y[I]} \otimes_{\cO_{Y[K]}} E_r^{a,b}(R(f_K)_{\ast}sB_K(f),L(K))
\overset{\simeq}{\longrightarrow}
E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
for all $a,b,r$,
which is compatible with the morphisms $d_r$.
Then the $E_2$-degeneracy
of $E_r^{a,b}(R(f_K)_{\ast}sB_K(f),L(K))$
implies that of $E_r^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))$.
\end{proof}

\begin{rmk}
By taking $I=\ski$ in Theorem \ref{theorem for E2-degeneracy},
we obtain Theorem (4.1) in \cite{FujisawaDWSS}.
\end{rmk}

\begin{lem}
\label{coincidence for F via E2-degeneracy}
Under the identification
\eqref{E2-degeneration for L(K):eq},
the filtration $F$ on the left hand side
coincides with the filtration $F_{\rec}$ on the right.
\end{lem}
\begin{proof}
From Lemma \ref{lemma on strong strictness for F},
the filtration $F$ on the complex
$R(f_I)_{\ast}\gr_m^{L(K)}sB_I(f)$
is strongly strict for all $m$.
Therefore the lemma on two filtrations
\cite[Proposition (7.2.5)]{DeligneIII}
implies that $F_{\rec}=F_d=F_{d^\ast}$
on $E^{a,b}_2(R(f_I)_{\ast}sB_I(f),L(K))$ for all $a,b$
as in the proof of Lemma \ref{lemma on VMHS for I}.
Then the $E_2$-degeneracy above implies the conclusion.
\end{proof}

\begin{lem}
\label{MHS on E1-terms}
For $K \subset I$,
the data
\begin{equation*}
\begin{split}
((E_1^{a,b}(R(f_I)_{\ast}sA_I(f),&L(K)), L(I \setminus K)[b]), \\
&(E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K)),L(I \setminus K)[b],F),
E_1^{a,b}(\alpha_I(f)))
\left|_{Y[I]^{\ast}}\right.
\end{split}
\end{equation*}
is a variation of $\bQ$-mixed Hodge structure on $Y[I]^{\ast}$.
For $J \subset I$,
$L(J)_m$ gives us a subvariation of $\bQ$-mixed Hodge structure
for all $m$.
\end{lem}
\begin{proof}
By the identifications
\begin{align}
&E_1^{a,b}(R(f_I)_{\ast}sA_I(f),L(K))
\simeq
R^{a+b}(f_I)_{\ast}\gr_{-a}^{L(K)}sA_I(f) \\
&E_1^{a,b}(R(f_I)_{\ast}sB(f),L(K))
\simeq
R^{a+b}(f_I)_{\ast}\gr_{-a}^{L(K)}sB_I(f)
\label{identification of E_1 for sBI:eq}
\end{align}
we obtain the conclusion by Lemma \ref{lemma on grL(K)sB},
Lemma \ref{Lemma of grLsA}, Lemma \ref{lemma for closed immersion}
and Theorem \ref{theorem on VMHS for I}.
\end{proof}

\begin{para}
Now we fix an integer $b$ and consider the complex
\begin{equation*}
(E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)), d_1) ,
\end{equation*}
where
\begin{equation*}
d_1:
E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\longrightarrow
E_1^{a+1,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is the morphism of $E_1$-terms for all $a$.
For any $J \subset I$,
the filtration $L(J)_{\rec}=L(J)_d=L(J)_{d^\ast}$
on $E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))$
is simply denoted by $L(J)$ for a while,
which forms an increasing filtration of the complex
$E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))$.
\end{para}

\begin{lem}
\label{lemma on the isom induced by theta on E1}
For $K, J_1, J_2, \dots, J_l \subset J \subset I$,
the morphism $\theta_{J|I}(f)$ induces an isomorphism of complexes
\begin{equation*}
\begin{split}
\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
\gr_{m_l}^{L(J_l)}
\cdots
&\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_J)_{\ast}sB_J(f),L(K)) \\
&\overset{\simeq}{\longrightarrow}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{split}
\end{equation*}
for all $m_1, m_2, \dots, m_l$,
under which we have the identification
\begin{equation*}
\begin{split}
F^p(\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
\gr_{m_l}^{L(J_l)}&\cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_J)_{\ast}sB_J(f),L(K))) \\
&\overset{\simeq}{\longrightarrow}
F^p\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{split}
\end{equation*}
for all $p$.
\end{lem}
\begin{proof}
From the identification \eqref{identification of E_1 for sBI:eq},
we easily obtain the conclusion
by Lemma \ref{lemma on grL(K)sB},
Corollary \ref{corollary on grL's and theta}
and Proposition \ref{isom induced by theta on FgrL's}.
\end{proof}

\begin{lem}
\label{strictness for L(J setminus K) on ss}
For $K, J_1, J_2, \dots, J_l \subset J \subset I$,
the filtration $L(J \setminus K)$ on the complex
\begin{equation*}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is strongly strict
for all $m_1, m_2, \dots, m_l$.
\end{lem}
\begin{proof}
By Lemma \ref{filtered identification between ss of A and B},
we have the isomorphism
\begin{equation*}
\begin{split}
\cO_{Y[I]}
\otimes_{\bQ}
\coh^a&(\gr_m^{L(J \setminus K)}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sA_{I|I'}(f),L(K)))
\left|_{Y[I']^{\ast}}\right. \\
&\simeq
\coh^a(\gr_m^{L(J \setminus K)}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)))
\left|_{Y[I']^{\ast}}\right.
\end{split}
\end{equation*}
for all $a, m, m_1, m_2, \dots, m_l$
and for any $I' \supset I$.
Therefore the coherent $\cO_{Y[I]}$-module
\begin{equation*}
\coh^a(\gr_m^{L(J \setminus K)}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)))
\end{equation*}
is locally free for all $a, m, m_1, m_2, \dots, m_l$
as Proposition \ref{proposition on the local freeness of grLgrLsBI}.
Hence it suffices to prove the strictness
of $L(J \setminus K)$.

First we treat the case of $J=I$.
On the open subset $Y[I]^{\ast}$,
\begin{equation*}
(\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K)), L(I \setminus K)[b], F)
\left|_{Y[I]^{\ast}}\right.
\end{equation*}
underlies a variation of $\bQ$-mixed Hodge structures for all $a$
and the morphism $d_1$ underlies a morphism of variations
from Lemma \ref{MHS on E1-terms}.
Therefore the filtration $L(I \setminus K)$
on the complex
\begin{equation*}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\left|_{Y[I]^{\ast}}\right.
\end{equation*}
is strict,
that is,
the canonical morphism
\begin{equation}
\label{the canonical morphism for strictness of L:eq}
\begin{split}
\coh^a(L(I \setminus K)_m
&\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))) \\
&\longrightarrow
\coh^a(
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)))
\end{split}
\end{equation}
is injective over $Y[I]^{\ast}$.
Since
\begin{equation*}
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K)))
\end{equation*}
and
\begin{equation*}
L(I \setminus K)_m
\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K)))
\end{equation*}
are the canonical extensions of their restrictions on $Y[I]^{\ast}$
by Lemma \ref{lemma for ss being the canonical extension}
(see also Remark \ref{remark on the canonical extension of ss})
for all $a$,
the morphism \eqref{the canonical morphism for strictness of L:eq}
is injective over the whole $Y[I]$.
Thus we obtain the strictness of $L(I \setminus K)$ over $Y[I]$.

From the case of $J=I$ above,
the filtration $L(J \setminus K)$ on the complex
\begin{equation*}
\gr_{m_l}^{L(J_l)}
\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_J)_{\ast}sB_J(f),L(K)))
\end{equation*}
is strongly strict.
On the other hand,
the morphism $\theta_{J|I}(f)$ induces the isomorphism
\begin{equation*}
\begin{split}
(\cO_{Y[I]} \otimes_{\cO_{Y[J]}}^L
&\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_J)_{\ast}sB_J(f),L(K)), L(J \setminus K)) \\
&\simeq
(\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)), L(J \setminus K))
\end{split}
\end{equation*}
in the filtered derived category
from Lemma \ref{lemma on the isom induced by theta on E1}.
Therefore we obtain the conclusion
from Lemma \ref{lemma on strong strictness and filtered tensor product}.
\end{proof}

\begin{lem}
\label{lemma on theta and E2}
For $K \subset J \subset I$,
the morphism $\theta_{J|I}(f)$ induces an isomorphism
\begin{equation*}
\begin{split}
\cO_{Y[I]} \otimes_{\cO_{Y[J]}}
\gr_m^{L(J \setminus K)_{\rec}}
&E_2^{a,b}(R(f_J)_sB_J(f),L(K)) \\
&\overset{\simeq}{\longrightarrow}
\gr_m^{L(J \setminus K)_{\rec}}
E_2^{a,b}(R(f_I)_sB_I(f),L(K))
\end{split}
\end{equation*}
for all $a, b, m$.
\end{lem}
\begin{proof}
From Lemma \ref{lemma on the isom induced by theta on E1},
we have the isomorphism
\begin{equation*}
\begin{split}
(\cO_{Y[I]} \otimes_{\cO_{Y[J]}}^L
E_1^{a,b}(R(f_J)_sB_J(f),&L(K)), L(J \setminus K)) \\
&\simeq
E_1^{a,b}(R(f_I)_sB_I(f),L(K)), L(J \setminus K))
\end{split}
\end{equation*}
in the filtered derived category.
Because $L(J \setminus K)$
is strongly strict
on the complex
$E_1^{\bullet,b}(R(f_J)_sB_J(f),L(K))$,
we obtain the conclusion
by Lemma \ref{lemma on strong strictness and filtered tensor product}.
\end{proof}

\begin{prop}
\label{compatiblity of L(J) and L(J setminus K) on grLK}
We assume that
the semistable morphism $f:X \longrightarrow Y=\Delta^k \times S$
is proper and K\"ahler.
Under the identification
\begin{equation*}
\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)
\overset{\simeq}{\longrightarrow}
E_2^{-m,n+m}(R(f_I)_{\ast}sB_I(f),L(K)) ,
\end{equation*}
in \eqref{E2-degeneration for L(K):eq},
the filtration $L(J)[-m]$ on the left hand side
coincides with the filtration $L(J \setminus K)_{\rec}$ on the right
for all $K \subset J \subset I$.
\end{prop}
\begin{proof}
First we treat the case $J=I$.
It suffices to prove the desired coincidence over $Y[I]^{\ast}$
because 
$\gr_l^{L(J)}\gr_m^{L(K)}R^n(f_I)sB_I(f)$ and 
$\gr_l^{L(J \setminus K)_{\rec}}E_2^{-m,n+m}(R(f_I)_{\ast}sB_I(f),L(K))$
are local free for all $l$.
Therefore we may assume that $J=I=\ski$.
Thus what we have to prove is
the coincidence of $L[-m]$ and $L(\overline{K})_{\rec}$
under the identification
\begin{equation}
\label{identification for f0:eq}
\gr_m^{L(K)}R^nf_{0\ast}sB(f)
\overset{\simeq}{\longrightarrow}
E_2^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\end{equation}
in Theorem \ref{theorem for E2-degeneracy},
where $\overline{K}=\ski \setminus K$
as in the proof of Lemma \ref{MHS on E1-terms}.
The following argument is similar to that of El Zein
\cite[6.1.12 Th\'eor\`em]{ElZeinbook}.
So we only give a sketch of the proof here.

From the $E_2$-degeneracy in Theorem \ref{theorem for E2-degeneracy},
the identification \eqref{identification for f0:eq}
fits in the commutative diagram
\begin{equation}
\label{commutative diagram from E2-degeneracy:eq}
\begin{CD}
R^nf_{0\ast}L(K)_msB(f)
@>>>
\kernel(d_1) \\
@VVV @VVV \\
\gr_m^{L(K)}R^nf_{0\ast}sB(f)
@>{\simeq}>>
E_2^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\end{CD}
\end{equation}
where $d_1$ in the right hand side of the top horizontal arrow
is the morphism of $E_1$-terms
\begin{equation*}
d_1:
E_1^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\longrightarrow
E_1^{-m+1,n+m}(Rf_{0\ast}sB(f),L(K))
\end{equation*}
and the top horizontal arrow
is induced from the projection
\begin{equation}
\label{projection from LKsB to grLKsB:eq}
L(K)_msB(f) \longrightarrow \gr_m^{L(K)}sB(f)
\end{equation}
by the identification
$R^nf_{0\ast}\gr_m^{L(K)}sB(f) \simeq E_1^{-m,n+m}(Rf_{0\ast}sB(f),L(K))$.
It is easy to see that
the projection \eqref{projection from LKsB to grLKsB:eq}
sends $L_l$ to $L(\overline{K})_{l-m}=L(\overline{K})[m]_l$ for all $l$.
Lemma \ref{lemma on VMHS for I},
Theorem \ref{theorem on VMHS for I}
and Lemma \ref{MHS on E1-terms} imply that
\begin{equation*}
\begin{split}
&(R^nf_{0\ast}L(K)_msB(f),L[n],F), \\
&(\gr_m^{L(K)}R^nf_{0\ast}sB(f),L[n],F), \\
&(\kernel(d_1),L(\overline{K})[n+m],F), \\
&(E_2^{-m,n+m}(Rf_{0\ast}sB(f),L(K)),L(\overline{K})_{\rec}[n+m],F_{\rec})
\end{split}
\end{equation*}
underlie variations of $\bQ$-mixed Hodge structure
on $Y[\ski]=S$.
Moreover the top horizontal arrow and two vertical arrows
in the commutative diagram
\eqref{commutative diagram from E2-degeneracy:eq}
underlie morphisms of variations of $\bQ$-mixed Hodge structure
by definition.
Thus we can easily see that
the identification \eqref{identification for f0:eq},
the bottom horizontal arrow
in the commutative diagram \eqref{commutative diagram from E2-degeneracy:eq},
underlies a morphism of variations of $\bQ$-mixed Hodge structure
by using the fact that
the left vertical arrow is surjective and strictly compatible
with the filtrations $L[n]$ and $F$.
Therefore we obtain the desired coincidence.

For the case $J \subsetneq I$,
we have the commutative diagram
\begin{equation*}
\begin{CD}
\cO_{Y[I]} \otimes \gr_m^{L(K)}R^q(f_J)_{\ast}sB_J(f)
@>>>
\gr_m^{L(K)}R^q(f_I)_{\ast}sB_I(f) \\
@VVV @VVV \\
\cO_{Y[I]} \otimes E_2^{-m,q+m}(R(f_J)_{\ast}sB_J(f),L(K))
@>>>
E_2^{-m,q+m}(R(f_I)_{\ast}sB_I(f),L(K)) .
\end{CD}
\end{equation*}
where the vertical arrows are the identifications
\eqref{E2-degeneration for L(K):eq}
for $J$ and $I$,
and the horizontal arrows
are the morphisms induced by $\theta_{J|I}(f)$,
which are filtered isomorphisms
with respect to $L(J)$ and $L(J \setminus K)_{\rec}$
by Corollary \ref{corollary on grL's and theta}
and Lemma \ref{lemma on theta and E2}.
Then the coincidence of $L(J)[m]$ and $L(J \setminus K)_{\rec}$
under the left vertical arrow
implies the desired coincidence under the right vertical arrow.
\end{proof}

\begin{thm}
\label{strictness for F on ss and local freeness}
If the semistable morphism $f:X \longrightarrow Y=\Delta^k \times S$
is proper and K\"ahler,
we have the following$:$
\begin{itemize}
\item[$(A_l)$]
For $J_1 \subset J_2 \subset \dots \subset J_l \subset I \subset \ski$,
the coherent $\cO_{Y[I]}$-module
\begin{equation*}
\gr_F^p\gr_{m_l}^{L(J_l)} \cdots
\gr_{m_2}^{L(J_2)}\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
is locally free of finite rank
for all $m_1,m_2, \dots, m_l,n,p$.
\item[$(B_l)$]
For
$K \subset J_1 \subset J_2 \subset \dots \subset J_l \subset I \subset \ski$,
the filtration $F$ on the complex
\begin{equation*}
\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is strongly strict
for all $b, m_1, m_2, \dots, m_l$.
\end{itemize}
\end{thm}
\begin{proof}
We already proved $(A_0)$
in Lemma \ref{lemma on strong strictness for F}.
So it suffices to prove the implications
$(A_l) \Rightarrow (B_l)$ and $(B_0)+(B_l) \Rightarrow (A_{l+1})$.

First, we prove $(A_l) \Rightarrow (B_l)$.
From the identification
\eqref{identification of E_1 for sBI:eq},
the assumption $(A_l)$ implies that
\begin{equation*}
\gr_F^p\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
is locally free of finite rank.
Thus the filtered complex
\begin{equation*}
(\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)), F)
\end{equation*}
is filtered perfect.
Therefore it is sufficient to prove that
the filtered complex
\begin{equation}
\label{the complex in question:eq}
(\bC(x) \otimes_{\cO_{Y[I]}}^L
\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)),F)
\end{equation}
is strict for all $x \in Y[I]$
by Lemma \ref{strong strictness criteria by pointwise strictness}
because
\begin{equation*}
\coh^a(
\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)))
\end{equation*}
is locally free as in the proof
of Lemma \ref{strictness for L(J setminus K) on ss}.
Because the filtration $F$ on the complex
\begin{equation*}
\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\left|_{Y[I]^{\ast}}\right.
\end{equation*}
is strongly strict
as in the proof of Lemma \ref{MHS on E1-terms},
the filtered complex
\eqref{the complex in question:eq}
is strict for $x \in Y[I]^{\ast}$
by Lemma \ref{strong strictness criteria by pointwise strictness}.
For a point $x \in Y[I] \setminus Y[I]^{\ast}$,
we can find $I' \supset I$ such that $x \in Y[I']^{\ast}$.
By Lemma \ref{lemma on the isom induced by theta on E1},
we have the isomorphism in the filtered derived category
\begin{equation*}
\begin{split}
(\cO_{Y[I']} \otimes_{\cO_{Y[I]}}^L
&\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)),F) \\
&\simeq
(\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_{I'})_{\ast}sB_{I'}(f),L(K)),F)
\end{split}
\end{equation*}
induced by the morphism $\theta_{I|I'}(f)$.
Then we have an isomorphism
\begin{equation*}
\begin{split}
(\bC(x) &\otimes_{\cO_{Y[I]}}^L
\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K)),F) \\
&\simeq
(\bC(x) \otimes_{\cO_{Y[I']}}^L
\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
\gr_{m_1}^{L(J_1 \setminus K)}
E_1^{\bullet,b}(R(f_{I'})_{\ast}sB_{I'}(f),L(K)),F)
\end{split}
\end{equation*}
in the filtered derived category.
Therefore the complex \eqref{the complex in question:eq}
is strict as desired.

Next, we prove $(B_0)+(B_l) \Rightarrow (A_{l+1})$.
We set $K=J_1$.
Then the complex
\begin{equation*}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{equation*}
equipped with the filtrations
\begin{equation*}
F, L(J_2 \setminus K), \dots, L(J_l \setminus K), L(J_{l+1} \setminus K)
\end{equation*}
satisfies the assumption
in Lemma \ref{lemma on successively strict filtrations}
by the assumptions $(B_0)$, $(B_l)$
and Lemma \ref{strictness for L(J setminus K) on ss}.
Therefore we have the isomorphism
\begin{equation*}
\begin{split}
\coh^a(
\gr_F^p&\gr_{m_{l+1}}^{L(J_{l+1} \setminus K)}
\gr_F^p\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
E_1^{\bullet,b}(R(f_I)_{\ast}sB_I(f),L(K))) \\
&\simeq
\gr_{F_{\rec}}^p
\gr_{m_{l+1}}^{L(J_{l+1} \setminus K)}
\gr_F^p\gr_{m_l}^{L(J_l \setminus K)}
\cdots
\gr_{m_2}^{L(J_2 \setminus K)}
E_2^{a,b}(R(f_I)_{\ast}sB_I(f),L(K))
\end{split}
\end{equation*}
for all $a, b, m_2, \dots, m_l, m_{l+1}, p$.
Moreover, it is locally free of finite rank
by the strong strictness of $F$ in the assumption $(B_l)$.
Then the identification \eqref{E2-degeneration for L(K):eq},
Lemma \ref{coincidence for F via E2-degeneracy}
and Proposition \ref{compatiblity of L(J) and L(J setminus K) on grLK}
give us $(A_{l+1})$.
\end{proof}

\begin{cor}
For $J_1 \subset J_2 \subset \dots \subset J_l \subset J \subset I$,
the morphism $\theta_{J|I}(f)$ induces an isomorphism
\begin{equation*}
\begin{split}
\cO_{Y[I]} \otimes
\gr_F^p
\gr_{m_l}^{L(J_l)}
&\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_J)_{\ast}sB_J(f) \\
&\overset{\simeq}{\longrightarrow}
\gr_F^p
\gr_{m_l}^{L(J_l)}
\cdots
\gr_{m_2}^{L(J_2)}
\gr_{m_1}^{L(J_1)}
R^n(f_I)_{\ast}sB_I(f)
\end{split}
\end{equation*}
for all $m_1, m_2, \dots, m_l, n, p$.
\end{cor}
\begin{proof}
Easy from Lemma \ref{local freeness and the commutattivity of gr}.
\end{proof}

\section{Main result}
\label{section for the main results}

\begin{para}
Let $f:X \longrightarrow Y=\Delta^k \times S$ be
a proper K\"ahler semistable morphism as in the last section.
For $i=1,2, \dots, k$,
we write $Y[i]=Y[\{i\}](=E_i)$ for short.
We use the similar convention for other objects,
$X[i]=X[\{i\}]$, $sB_i(f)=sB_{\{i\}}(f)$, $L(i)=L(\{i\})$ and so on.
\end{para}

\begin{lem}
\label{lemma for the main result for the case of |J|=1}
The morphism $N_{j|I}(f)^l$ induces an isomorphism
\begin{equation}
\label{isomorphism for Nj:eq}
\gr_l^{L(j)}R^n(f_I)_{\ast}sB_I(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l}^{L(j)}R^n(f_I)_{\ast}sB_I(f)
\end{equation}
for all $l \ge 0$ and for all $j \in I$.
In other words,
the filtration $L(j)$ coincides with
the monodromy weight filtration $W(N_{j|I}(f))$
on $R^n(f_I)_{\ast}sB_I(f)$ for $j \in I$.
\end{lem}
\begin{proof}
The morphism $\theta_{j|I}(f)$ induces the isomorphism
\begin{equation*}
\cO_{Y[I]}
\otimes
\gr_l^{L(j)}R^n(f_j)_{\ast}sB_j(f)
\overset{\simeq}{\longrightarrow}
\gr_l^{L(j)}R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for all $l$.
The morphism $\theta_{j|I}(f)$ is compatible with
the morphism $N_{j|I}(f)$ and $N_j(f)$ for all $j \in J$.
Therefore we may assume $I=\{j\}$.
Moreover,
the morphism $N_j(f)$ is compatible with
the logarithmic connection $-\nabla(j)$
on $R^n(f_j)_{\ast}sB_j(f)$,
it is sufficient to prove
that the restriction $N_j(f)|_{Y[j]^{\ast}}$
induces the isomorphism \eqref{isomorphism for Nj:eq}
over $Y[j]^{\ast}$
because $(R^n(f_j)_{\ast}sB_j(f), -\nabla(j))$
is the canonical extension of its restriction over $Y[j]^{\ast}$.
Thus we may assume that $k=1$.
Because $N_j(f)$ is compatible with the base change
as in Remark \ref{nu and base change for S},
we can reduce the problem
to the case that $S$ is a point
by Lemma \ref{lemma on base change for sB}.
Hence we obtain the conclusion
by \cite{SaitoMorihikoMHP}, \cite{Guillen-NavarroAznarCI}
and \cite{UsuiMTTS}.
\end{proof}

The following theorem is the main result of this article.

\begin{thm}
\label{theorem on the monodromy weight filtrations}
Let $f:X \longrightarrow Y=\Delta^k \times S$ be
a proper K\"ahler semistable morphism.
For all $c \in \bpR^{J \setminus K}$
and for a non-negative integer $l$,
the morphism $N_{(J \setminus K)|I}(f;c)^l$
induces an isomorphism
\begin{equation*}
\gr_{l+m}^{L(J)}\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l+m}^{L(J)}\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for all $m,n \in \bZ$
and for all $K \subset J \subset I \subset \ski$.
For the case of $K=\emptyset$,
$N_{J|I}(f;c)^l$ induces the isomorphism
\begin{equation*}
\gr_l^{L(J)}R^n(f_I)_{\ast}sB_I(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l}^{L(J)}R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for all $J \subset I$.
\end{thm}
\begin{proof}
The morphism $\theta_{J|I}(f)$ induces the isomorphism
\begin{equation*}
\cO_{Y[I]}
\otimes
\gr_{l+m}^{L(J)}\gr_m^{L(K)}R^n(f_J)_{\ast}sB_J(f)
\overset{\simeq}{\longrightarrow}
\gr_{l+m}^{L(J)}\gr_m^{L(K)}R^n(f_I)_{\ast}sB_I(f)
\end{equation*}
for all $l,m,n$.
Moreover the morphism $\theta_{J|I}(f)$ is compatible with
the morphism $N_{j|I}(f)$ for all $j \in J$.
Therefore we may assume $J=I$.
Moreover, we may assume that $I=\ski$ as before
by using the fact that
$(R^n(f_I)_{\ast}sB_I(f),(-1)^{|I|}\nabla(I))$ is the canonical extension
of its restriction over $Y[I]^{\ast}$.
Hence it suffices to prove that the morphism
$N_{\overline{K}|}(f;c)^l$ induces an isomorphism
\begin{equation*}
\gr_{l+m}^L\gr_m^{L(K)}R^nf_{0\ast}sB(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l+m}^L\gr_m^{L(K)}R^nf_{0\ast}sB(f)
\end{equation*}
for all $l,m,n$ with $l \ge 0$
and for $c \in \bpR^{\overline{K}}$,
where $\overline{K}=\ski \setminus K$.

Now we proceed by induction on $k$.
The case of $k=1$ is already proved
in Lemma \ref{lemma for the main result for the case of |J|=1}.
So we assume $k \ge 2$.

For a non-empty subset $K \subset \ski$,
we have the identification
\begin{equation}
\label{(grLKsBI,LJ) for nu:eq}
(\gr_m^{L(K)}sB(f),L(\overline{K}))
\simeq
\bigoplus_{(q,\Gamma)}
(sB(f_{\Gamma})[-m-2|q|],L)
\end{equation}
by Lemma \ref{lemma on grL(K)sB},
where the index $(q,\Gamma)$ runs through the set
\begin{equation}
\label{index set for grLKsBI for nu:eq}
\{(q,\Gamma); q \in \bnnZ^K,
\Gamma \in S_{m+2|q|+|K|}^{\ge q+e_K}(\Lambda_K)\}
\end{equation}
as in Lemma \ref{lemma on grL(K)sB}.
Under the identification
\eqref{(grLKsBI,LJ) for nu:eq} above,
the morphism
\begin{equation*}
\gr_m^{L(K)}\nu_{j|}(f):
\gr_m^{L(K)}sB(f)
\longrightarrow
\gr_m^{L(K)}sB(f)
\end{equation*}
is identified with
\begin{equation*}
\bigoplus \nu_{j|}(f_{\Gamma})[-m-2|q|]
\end{equation*}
for any $j \in \overline{K}$,
where the direct sum is taken over the same index set
as \eqref{index set for grLKsBI for nu:eq}.
Thus the morphism $\gr_m^{L(K)}N_{\overline{K}|}(f;c)^l$
induces an isomorphism
\begin{equation*}
\gr_l^{L(\overline{K})}E_1^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\overset{\simeq}{\longrightarrow}
\gr_{-l}^{L(\overline{K})}E_1^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\end{equation*}
for all $l,m,n$ with $l \ge 0$
because of the identifications
\eqref{(grLKsBI,LJ) for nu:eq} and
\begin{equation*}
E_1^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\simeq
R^nf_{0\ast}\gr_m^{L(K)}sB(f),
\end{equation*}
together with the induction hypothesis.
Then $\gr_m^{L(K)}N_{\overline{K}|}(f;c)^l$ induces an isomorphism
\begin{equation*}
\gr_l^{L(\overline{K})_{\rec}}E_2^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\overset{\simeq}{\longrightarrow}
\gr_{-l}^{L(\overline{K})_{\rec}}E_2^{-m,n+m}(Rf_{0\ast}sB(f),L(K))
\end{equation*}
because $L(\overline{K})$ is strictly compatible
with the morphism $d_1$ of the $E_1$-terms
by Lemma \ref{strictness for L(J setminus K) on ss}.
Therefore we obtain the desired isomorphism
by Proposition \ref{compatiblity of L(J) and L(J setminus K) on grLK}.

Now the case $K=\emptyset$ is remained.
The morphism $N_{\overline{\{j\}}|}(f)^l$ induces an isomorphism
\begin{equation*}
\gr_{l+m}^L\gr_m^{L(j)}R^nf_{0\ast}sB(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l+m}^L\gr_m^{L(j)}R^nf_{0\ast}sB(f)
\end{equation*}
for all $l,m,n$ with $l \ge 0$ as proved above.
Since we know the coincidence $L(j)=W(N_{j|}(f))$
by Lemma \ref{lemma for the main result for the case of |J|=1},
the monodromy weight filtration $W(N(f;c))$
satisfies the same property, that is,
the morphism $N_{\{\overline{j}\}|}(f)^l$ induces an isomorphism
\begin{equation*}
\gr_{l+m}^{W(N(f;c))}\gr_m^{L(j)}R^nf_{0\ast}sB(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l+m}^{W(N(f;c))}\gr_m^{L(j)}R^nf_{0\ast}sB(f)
\end{equation*}
for all $l,m,n$ with $l \ge 0$
by \cite[(3.3) Theorem]{Cattani-Kaplan}
(see also \cite[(3.12) Theorem]{Steenbrink-Zucker}).
Then the uniqueness of the relative monodromy weight filtrations
implies the coincidence $L=W(N(f;c))$.
Then we conclude that the morphism $N(f;c)^l$ induces an isomorphism
\begin{equation*}
\gr_l^LR^nf_{0\ast}sB(f)
\overset{\simeq}{\longrightarrow}
\gr_{-l}^LR^nf_{0\ast}sB(f)
\end{equation*}
for all $l \ge 0$ by the definition of $W(N(f;c))$.
\end{proof}

\begin{thm}
The variation of $\bQ$-mixed Hodge structure
\eqref{VMHS for I:eq} on $Y[I]^{\ast}$
is admissible in $Y[I]$.
\end{thm}
\begin{proof}
Let $\varphi: \Delta \longrightarrow Y[I]$ be a holomorphic map
such that $\varphi^{-1}(Y[I]^{\ast})=\pd$.
Take $I' \supsetneq I$
such that $\varphi(0) \in Y[I']^{\ast}$.
For every $j \in I' \setminus I$,
$c_j$ denotes the order of zero of the holomorphic function
$\varphi^{\ast}t_j$
at the origin $0 \in \Delta$.
Then $c_j$ is a positive integer for all $j \in I' \setminus I$.

Now we consider
a locally free $\cO_{\Delta}$-module of finite rank
with an integrable logarithmic connection
$(\varphi^{\ast}R^n(f_I)_{\ast}sB_I(f), (-1)^{|I|}\varphi^{\ast}\nabla(I))$
on $\Delta$.
Since $\gr_F^p\gr_m^{L(I)}R^n(f_I)_{\ast}sB_I(f)$
is locally free of finite rank,
we have
\begin{equation*}
\gr_F^p\gr_m^{L(J)}\varphi^{\ast}R^n(f_I)_{\ast}sB_I(f)
\simeq
\varphi^{\ast}\gr_F^p\gr_m^{L(J)}R^n(f_I)_{\ast}sB_I(f) ,
\end{equation*}
which is locally free $\cO_{\Delta}$-module of finite rank.
By using the morphism $\theta_{I'|I}(f)$,
we have the identification
\begin{equation*}
\bC(0) \otimes \varphi^{\ast}R^n(f_I)_{\ast}sB_I(f)
\simeq
\bC(\varphi(0)) \otimes R^n(f_I)_{\ast}sB_I(f)
\simeq
\bC(\varphi(0)) \otimes R^n(f_{I'})_{\ast}sB_{I'}(f)
\end{equation*}
under which 
the equality
\begin{equation*}
\res_{0}((-1)^{|I|}\varphi^{\ast}\nabla(I))
=\sum_{j \in I' \setminus I}c_j\res_j((-1)^{|I|}\nabla(I))
=(-1)^{|I|}N_{(I' \setminus I)|I'}(f;c)
\end{equation*}
is easily seen.
Thus $\res_{0}((-1)^{|I|}\varphi^{\ast}\nabla(I))$ is nilpotent.
Therefore
\begin{equation*}
(\varphi^{\ast}R^n(f_I)_{\ast}sB_I(f),(-1)^{|I|}\varphi^{\ast}\nabla(I))
\end{equation*}
is of unipotent monodromy around $0 \in \Delta$
and the canonical extension of its restriction over $\pd$.
Moreover, $L(I')$ gives us
the relative monodromy weight filtration
of $\res_0((-1)^{|I|}\varphi^{\ast}\nabla(I))$
with respect to $L(I)$ on
\begin{equation*}
\bC(0) \otimes \varphi^{\ast}R^n(f_I)_{\ast}sB_I(f)
\simeq
\bC(\varphi(0)) \otimes R^n(f_{I'})_{\ast}sB_{I'}(f)
\end{equation*}
by Theorem \ref{theorem on the monodromy weight filtrations}.
\end{proof}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
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\end{document}

1506.02271.extracted_bib
{Cattani-Kaplan E.~Cattani and A.~Kaplan, {Polarized Mixed Hodge Structures and the Local Monodromy of a Variation of Hodge Structure, Invent. Math. 67 (1982), 101--115.
1506.02271.extracted_bib
{DeligneED P.~Deligne, {Equations Diff\'erentielles \`a Points Singuliers R\'eguliers, Lecture Notes in Math., vol. 163, Springer-Verlag, 1970.
1506.02271.extracted_bib
{DeligneIII \bysame, {Th\'eorie de Hodge {\rm \RomIII, Inst. Hautes {\'Etudes Sci. Publ. Math. 44 (1974), 5--77.
1506.02271.extracted_bib
{ElZeinCE F.~{El Zein, {Th\'eorie de Hodge des cycles \'evanscents, Ann. Sci. {\'Ecole Norm. Sup. (4) 19 (1986), 107--184.
1506.02271.extracted_bib
{ElZeinbook \bysame, {Introduction \`a la th\'eorie de Hodge mixte, Hermann, 1991.
1506.02271.extracted_bib
{Fujino-Fujisawa O.~Fujino and T.~Fujisawa, {Variations of Mixed Hodge Structure and Semipositivity Theorems, Publ. RIMS Kyoto Univ. 50 (2014), no.~4, 589--661.
1506.02271.extracted_bib
{FujisawaLHSSV T.~Fujisawa, {Limits of Hodge structures in several variables, Compositio Math. 115 (1999), 129--183.
1506.02271.extracted_bib
{FujisawaDWSS \bysame, {Degeneration of weight spectral sequences, Manuscripta Math. 108 (2002), 91--121.
1506.02271.extracted_bib
{FujisawaMHSLSD \bysame, Mixed hodge structures on log smooth degenerations, Tohoku Math. J. (2) 60 (2008), no.~1, 71--100.
1506.02271.extracted_bib
{FujisawaPLMHS \bysame, {Polarizations on limiting mixed Hodge structures, Journal of Singularities 8 (2014), 146--193.
1506.02271.extracted_bib
{Guillen-NavarroAznarCI F.~Guill\'en and V.~{Navarro Aznar, {Sur le th\'eor\`eme local des cycles invariants, Duke Math. J. 61 (1990), 133--155.
1506.02271.extracted_bib
{Kashiwara-SchapiraSM M.~Kashiwara and P.~Schapira, {Sheaves on manifolds, {Grundlehren der mathematischen Wissenschaften, vol. 292, Springer, 1990.
1506.02271.extracted_bib
{KazuyaKato K.~Kato, {Logarithmic structures of Fontaine-Illusie, {Algebraic Analysis, Geometry and Number Theory (J.-I. Igusa, ed.), Johns Hopkins Univ., 1988, pp.~191--224.
1506.02271.extracted_bib
{KatoUsuiAMS K.~Kato and S.~Usui, {Classifying Spaces of Degenerating Polarized Hodge Structures, Annales of Math. Stud., vol. 169, Princeton Univ. Press, 2009.
1506.02271.extracted_bib
{Katz-Oda N.~Katz and T.~Oda, {On the differentiation of De Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199--213.
1506.02271.extracted_bib
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