# Steenbrink isomorphism and crystals on tubular neighbourhoods

Yukiyoshi Nakkajima 2020 Mathematics subject classification number: 14F40.

Abstract.— For a locally nilpotent integrable connection on a proper (strict)
semistable family over a small polydisc with a relative horizontal simple
normal crossing divisor, we construct a canonical section in derived
categories inducing an isomorphism from the log de Rham cohomology of it on
the log special fiber of this family to the stalk of the higher direct image
of it at the origin modulo the maximal ideal of the localization of the
structure sheaf at the origin. As an application of the existence of this
section, we prove the following: (1): the locally freeness of the higher
direct image by a purely algebraic method; (2): the existence of a canonical
isomorphism between the base change of the log de Rham cohomology of the log
special fiber to the small polydisc and the higher direct image above. The
result (2) tells us that the higher direct image has the crystalline nature:
the invariance of the higher direct image.

##  1 Introduction

This article is a sequel of my article [N6]. The most important aim of this
article is to show that the higher direct image of the log de Rham complex of
a locally nilpotent integrable connection on a proper strict semistable
analytic family over a small unit polydisc is the resulting crystal of the log
de Rham cohomology of the connection on the log special fiber of this family.
To prove this, we construct an $\infty$-adic analogue of the Hyodo-Kato
section in the $p$-adic case ([HK], [EY]). And then we prove the locally
freeness of the higher direct image by a purely algebraic method. Lastly we
construct a canonical isomorphism between the base change of the log de Rham
cohomology to the small polydisc and the higher direct image, which implies
the existence of the crystal above. This isomorphism induces an $\infty$-adic
analogue of the Hyodo-Kato isomorphism in the $p$-adic case ([Bei], [N4],
[EY]). We call this $\infty$-adic analogue the Steenbrink isomorphism. The
existence of the crystal should have applications for certain classes of
concrete degenerate log analytic spaces and (local) moduli problems in
algebraic and analytic geometry.

First let us recall Steenbrink’s result.

Let $\overset{\circ}{\Delta}:=\\{z\in{\mathbb{C}}~{}|~{}|z|<1\\}$ be the unit
disc and let $f\colon\overset{\circ}{\cal
X}\longrightarrow\overset{\circ}{\Delta}$ be a proper (not necessarily strict)
semistable family of analytic spaces over ${\mathbb{C}}$. Endow
$\overset{\circ}{\cal X}$ and $\overset{\circ}{\Delta}$ with the canonical log
structures, respectively. Let ${\cal X}$ and $\Delta$ be the resulting log
analytic spaces, respectively. Let $s$ be the log point of ${\mathbb{C}}$: the
log analytic space whose underlying analytic space is the origin $\\{O\\}$ and
whose log structure is the inverse image of the log structure of $\Delta$ by
the inclusion
$\\{O\\}\overset{\subset}{\longrightarrow}\overset{\circ}{\Delta}$. Let $X/s$
be the log special fiber of ${\cal X}/\Delta$. Let $\Omega^{\bullet}_{{\cal
X}/\Delta}$ and $\Omega^{\bullet}_{X/s}$ be the relative log de Rham complexes
of ${\cal X}/\Delta$ and $X/s$, respectively. Set $H^{q}_{\rm
dR}(X/s)=H^{q}(X,\Omega^{\bullet}_{X/s})$ $(q\in{\mathbb{N}})$. Then
Steenbrink has proved the following theorem:

######  Theorem 1.1 ([S]).

The higher direct image $R^{q}f_{*}(\Omega^{\bullet}_{{\cal X}/\Delta})$
$(q\in{\mathbb{N}})$ is a coherent locally free ${\cal O}_{\Delta}$-module
$($and commutes with base change of a strict morphism of log analytic
spaces$)$.

Set ${\cal X}^{*}:={\cal X}\setminus X$ and
$\Delta^{*}:=\Delta\setminus\\{O\\}$. Let
${\mathfrak{H}}\longrightarrow\Delta^{*}$ be the universal cover of
$\Delta^{*}$. Set $\overline{{\cal X}^{*}}:={\cal
X}^{*}\times_{\Delta^{*}}{\mathfrak{H}}$ and ${\cal X}_{u}:={\cal
X}\times_{\Delta}u$ for a point $u\in\Delta^{*}$. Set $H^{q}_{\rm dR}({\cal
X}_{u}/{\mathbb{C}}):=H^{q}({\cal X}_{u},\Omega^{\bullet}_{{\cal
X}_{u}/{\mathbb{C}}})$ $(q\in{\mathbb{N}})$. The key point of the proof of
(1.1) is to prove that there exists the following isomorphism

(1.1.1) $\displaystyle H^{q}(\overline{{\cal
X}^{*}},{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{q}_{\rm dR}(X/s)$

depending on a parameter $z$ of $\Delta$. By using this isomorphism, we have
an isomorphism

(1.1.2) $\displaystyle\rho_{z}\colon H^{q}_{\rm
dR}(X/s)\overset{\sim}{\longrightarrow}H^{q}({\cal
X}_{u},{\mathbb{C}})=H^{q}_{\rm dR}({\cal X}_{u}/{\mathbb{C}})$

depending on a parameter $z$ of $\Delta$ for a point $u\in\Delta^{*}$ and we
obtain (1.1) by Grauert’s theorem ([Gra]) because $\dim_{\mathbb{C}}H^{q}_{\rm
dR}(X/s)$ does not jump. We call the isomorphism (1.1.2) Steenbrink’s
isomorphism. Let $\overline{j}\colon\overline{{\cal
X}^{*}}\longrightarrow{\cal X}$ and $i\colon
X\overset{\subset}{\longrightarrow}{\cal X}$ be the natural morphisms. Let
$R\Psi({\mathbb{C}})=i^{-1}R\overline{j}_{*}({\mathbb{C}})$ be the nearby
cycle sheaf on $X$. To prove (1.1.1), Steenbrink has proved that there exists
the following isomorphisms

(1.1.3) $\displaystyle
R\Psi({\mathbb{C}})\overset{\sim}{\longrightarrow}i^{-1}(\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}[\log
z])\overset{\sim}{\longrightarrow}\Omega^{\bullet}_{X/s}$

[S] (cf. [SGA 7-II]) depending on a parameter $z$ of $\Delta$. Here
$\Omega^{\bullet}_{{\cal X}/{\mathbb{C}}}[\log z]:={\mathbb{C}}[\log
z]\otimes_{\mathbb{C}}\Omega^{\bullet}_{{\cal X}/{\mathbb{C}}}$ is considered
as a subdga of $\overline{j}_{*}(\Omega^{\bullet}_{\overline{{\cal
X}^{*}}/{\mathbb{C}}})$. The complex $\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}[\log z]$ is the Hirsch extension of $\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}$ by a vector space ${\mathbb{C}}\log z$ over ${\mathbb{C}}$
of rank 1 with respect to a morphism $\varphi\colon{\mathbb{C}}\log z\owns\log
z\longmapsto d\log z=z^{-1}dz\in\Omega^{1}_{{\cal
X}/{\mathbb{C}}}=\Omega^{1}_{\overset{\circ}{\cal
X}/{\mathbb{C}}}(\log\overset{\circ}{X})$. The notion of the Hirsch extension
by a vector space appears in the definition of the minimal model of a dga in
Sullivan’s theory ([Su]). Conversely, if one knew (1.1) a priori, one could
construct a non-canonical isomorphism

(1.1.4) $\displaystyle H^{q}_{\rm
dR}(X/s)\overset{\sim}{\longrightarrow}H^{q}_{\rm dR}({\cal
X}_{u}/{\mathbb{C}})$

for a close point $u\in\Delta^{*}$ from $O$ and any point $u\in\Delta^{*}$.

Let $f\colon X\longrightarrow Y$ be a proper log smooth morphism of fs$($=fine
and saturated$)$ log analytic spaces over ${\mathbb{C}}$. Assume that $Y$ is
log smooth and that $f$ is exact. In [KF] F. Kato has proved that there exists
the following canonical isomorphism

(1.1.5) $\displaystyle{\cal
O}_{Y^{\log}}\otimes_{\mathbb{C}}Rf^{\log}_{*}({\mathbb{C}})\overset{\sim}{\longrightarrow}{\cal
O}_{Y^{\log}}\otimes_{\epsilon^{-1}_{Y}({\cal
O}_{Y})}\epsilon^{-1}_{Y}Rf_{*}(\Omega^{\bullet}_{X/Y}).$

Here $X^{\log}$ and $Y^{\log}$ are the Kato-Nakayama spaces of $X$ and $Y$,
respectively, defined in [KatN], ${\cal O}_{Y^{\log}}:={\cal O}_{Y}^{\log}$ in
[loc. cit.], $\epsilon_{Y}\colon Y^{\log}\longrightarrow Y$ is the natural
morphism of topological spaces defined in [KatN], which has been denoted by
$\tau$ in [loc. cit.] and $f^{\log}\colon X^{\log}\longrightarrow Y^{\log}$ is
the induced morphism by $f$. Using the isomorphism (1.1.5) in the case where
$Y$ is the log point of ${\mathbb{C}}$, F. Kato has reproved the existence of
the isomorphism (1.1.1) ([loc. cit.]).

In [IKN] L. Illusie, K. Kato and C. Nakayama have obtained a more general
theorem than the theorem (1.1) by using the Kato-Nakayama space:

######  Theorem 1.2 ([IKN, (6.4)]).

Let $f\colon X\longrightarrow Y$ be a proper log smooth morphism of fs$($=fine
and saturated$)$ log analytic spaces over ${\mathbb{C}}$. Assume that $Y$ is
log smooth or $f$ is exact. Assume also that ${\rm
Coker}(\overline{M}_{f(x)}^{\rm gp}\longrightarrow\overline{M}_{x}^{\rm gp})$
is torsion free for any exact point $x\in X$. Let $\nabla\colon{\cal
F}\longrightarrow{\cal F}\otimes_{{\cal
O}_{X}}\Omega^{\bullet}_{X/{\mathbb{C}}}$ be a locally nilpotent integrable
connection on $X$ in the sense of [KatN]. Let ${\cal F}\otimes_{{\cal
O}_{X}}\Omega^{\bullet}_{X/Y}$ be the log de Rham complex obtained by $({\cal
F},\nabla)$. Assume that any stalk of $\overline{M}_{Y}:=M_{Y}/{\cal
O}_{Y}^{*}$ is isomorphic to the finitely many direct sum of ${\mathbb{N}}$.
Then $R^{q}f_{*}({\cal F}\otimes_{{\cal O}_{X}}\Omega^{\bullet}_{X/Y})$
$(q\in{\mathbb{N}})$ is a locally free coherent ${\cal O}_{Y}$-module and
commutes with base change.

Let $V$ be a locally unipotent local system on $X^{\log}$ in the sense of
[KatN] corresponding to $({\cal F},\nabla)$ by the log Riemann-Hilbert
correspondence in [loc. cit.]: $V={\rm Ker}(\nabla\colon{\cal
F}^{\log}\longrightarrow{\cal F}^{\log}\otimes_{{\cal
O}_{X^{\log}}}\Omega^{1,\log}_{X/{\mathbb{C}}})$. In [loc. cit.] Illusie, K.
Kato and Nakayama have also proved that there exists a canonical isomorphism

(1.2.1) $\displaystyle{\cal
O}_{Y^{\log}}\otimes^{L}_{\mathbb{C}}Rf^{\log}_{*}(V)\overset{\sim}{\longrightarrow}{\cal
O}_{Y^{\log}}\otimes^{L}_{\epsilon^{-1}_{Y}({\cal
O}_{Y})}\epsilon^{-1}_{Y}Rf_{*}({\cal F}\otimes_{{\cal
O}_{X}}\Omega^{\bullet}_{X/Y}).$

By using (1.2), we see that, for each exact point $y\in Y$, we have a non-
canonical isomorphism

$H^{q}(X_{y},{\cal F}\otimes_{{\cal
O}_{X}}\Omega^{\bullet}_{X_{y}/y})\overset{\sim}{\longrightarrow}H^{q}(X_{u},{\cal
F}\otimes_{{\cal O}_{X}}\Omega^{\bullet}_{X_{u}/u})$

for a sufficiently close exact point $u\in Y$ from $y$.

On the other hand, the following $p$-adic case can be considered.

Let ${\cal V}$ be a complete discrete valuation ring of mixed characteristics
$(0,p)$ with perfect residue field $\kappa$. Let $K$ be the fraction field of
${\cal V}$. Endow ${\rm Spec}({\cal V})$ with the log structure ${\cal
V}\setminus\\{0\\}\overset{\subset}{\longrightarrow}{\cal V}$ and let $S$ be
the resulting log scheme. Let $s_{\kappa}$ be the exact closed point of $S$.
Let ${\cal X}$ be a proper log smooth scheme over $S$ whose log special fiber
$X/s_{\kappa}$ is of Cartier type. Let $\pi$ be a uniformizer of ${\cal V}$.
Let ${\cal W}$ be the Witt ring of $\kappa$ and $K_{0}$ the fraction field of
${\cal W}$. Let ${\cal W}(s_{\kappa})$ be the canonical lift of $s_{\kappa}$
over ${\rm Spf}({\cal W})$. Let ${\cal X}_{K}$ be the log generic fiber of
${\cal X}$ over $S$. In [HK] Hyodo and K. Kato have constructed the following
isomorphism

(1.2.2) $\displaystyle\rho_{\pi}\colon H^{q}_{\rm crys}(X/{\cal
W}(s_{\kappa}))\otimes_{\cal W}K\overset{\sim}{\longrightarrow}H^{q}_{\rm
dR}({\cal X}_{K}/K)\quad(q\in{\mathbb{N}})$

depending on the uniformizer $\pi$. (Because the proof of the existence of
this isomorphism in [loc. cit.] is not complete, see also [N2] and [N4].) The
key point of the proof of (1.2.2) is to prove that there exists a unique
section

(1.2.3) $\displaystyle K_{0}\otimes^{L}_{\cal W}R\Gamma_{\rm crys}(X/{\cal
W})\longrightarrow K_{0}\otimes^{L}_{\cal W}R\Gamma_{\rm
crys}(X/\widehat{{\cal W}\langle t\rangle})$

of the projection

$\displaystyle K_{0}\otimes^{L}_{\cal W}R\Gamma_{\rm crys}(X/\widehat{{\cal
W}\langle t\rangle})\longrightarrow K_{0}\otimes^{L}_{\cal W}R\Gamma_{\rm
crys}(X/{\cal W})$

which induces the following isomorphism:

(1.2.4) $\displaystyle K_{0}\otimes^{L}_{\cal W}\widehat{{\cal W}\langle
t\rangle}\otimes^{L}_{\cal W}R\Gamma_{\rm crys}(X/{\cal
W})\overset{\sim}{\longrightarrow}K_{0}\otimes_{\cal W}^{L}R\Gamma_{\rm
crys}(X/\widehat{{\cal W}\langle t\rangle}).$

Here ${\cal W}\langle t\rangle$ is the PD-polynomial algebra in one variable
over ${\cal W}$ and $\widehat{\quad}$ means the $p$-adic completion. The
$n$-th power of the Frobenius endomorphism on

${\rm``}{\rm Ker}(R\Gamma_{\rm crys}(X/\widehat{{\cal W}\langle
t\rangle})\longrightarrow R\Gamma_{\rm crys}(X/\widehat{\cal W})){\textrm{"}}$

tends to a zero morphism for $n\to\infty$, while the $n$-th power of the
Frobenius endomorphism on $K_{0}\otimes^{L}_{\cal W}R\Gamma_{\rm crys}(X/{\cal
W})$ is an isomorphism for all $n\in{\mathbb{N}}$. By this fact, we obtain the
canonical section (1.2.3) (as in the case of the Teichmüller lift) with
complicated calculations.

By modifying $\rho_{\pi}$, we have constructed a canonical contravariantly
functorial isomorphism

(1.2.5) $\displaystyle\rho_{p}\colon H^{q}_{\rm crys}(X/{\cal
W}(s_{\kappa}))\otimes_{\cal W}K\overset{\sim}{\longrightarrow}H^{q}_{\rm
dR}({\cal X}_{K}/K)\quad(q\in{\mathbb{N}})$

which is independent of the choice of $\pi$ ([N4]). We use the prime number
$p\in{\cal V}$ and the isomorphism “$\rho_{p}$” in spite that $p$ is not
necessarily a uniformizer of ${\cal V}$. The isomorphism $\rho_{p}$ is
compatible with any algebraic extension of $K$ ([loc. cit.]). We call this
isomorphism the Hyodo-Kato isomorphism. In [loc. cit.] we have conjectured
that there exists a canonical isomorphism

(1.2.6) $\displaystyle\rho_{p}\colon R\Gamma_{\rm crys}(X/{\cal
W}(s_{\kappa}))\otimes^{L}_{\cal
W}K\overset{\sim}{\longrightarrow}R\Gamma_{\rm dR}({\cal X}_{K}/K)$

in the derived category of bounded complexes of $K$-vector spaces, which we
call the derived Hyodo-Kato isomorphism (cf. [Bei]). Recently Ertl and Yamada
have proved this conjecture in an ingenious method when ${\cal X}$ is a proper
strictly semistable scheme over $S$ ([EY]). They have proved the existence of
the derived Hyodo-Kato isomorphism (1.2.6) in this case by using the Hirsch
extension.

Now let us come back to the case where the base field is ${\mathbb{C}}$. It is
a natural problem to ask whether there exists a canonical isomorphism between
$R\Gamma_{\rm dR}(X/s)$ and $R\Gamma({\cal X}_{u},{\mathbb{C}})=R\Gamma_{\rm
dR}({\cal X}_{u}/{\mathbb{C}})$ for a sufficiently close point
$u\in\Delta^{*}$ from $O$. Let
$\iota_{O}\colon\\{O\\}\overset{\subset}{\longrightarrow}\Delta$ and
$\iota\colon X\overset{\subset}{\longrightarrow}{\cal X}$ be the closed
immersions. Though we do not know whether there exists a canonical
contravariantly functorial section

(1.2.7) $\displaystyle R\Gamma_{\rm dR}(X/s)\longrightarrow R\Gamma({\cal
X},\Omega^{\bullet}_{{\cal X}/\Delta})$

of the natural morphism $R\Gamma({\cal X},\Omega^{\bullet}_{{\cal
X}/\Delta})\longrightarrow R\Gamma_{\rm dR}(X/s)$ (this seems us impossible),
we construct a canonical contravariantly functorial section

(1.2.8) $\displaystyle\iota_{O*}R\Gamma_{\rm dR}(X/s)\longrightarrow
Rf_{*}\iota_{*}\iota^{-1}(\Omega^{\bullet}_{{\cal X}/\Delta})$

of the natural morphism $Rf_{*}\iota_{*}\iota^{-1}(\Omega^{\bullet}_{{\cal
X}/\Delta})\longrightarrow\iota_{O*}R\Gamma_{\rm dR}(X/s)$ induced by the
natural morphism

(1.2.9) $\displaystyle\iota^{-1}(\Omega^{\bullet}_{{\cal
X}/\Delta})\longrightarrow\iota^{!}(\Omega^{\bullet}_{{\cal
X}/\Delta})=\Omega^{\bullet}_{X/s}$

as an example of a more general section without using the topological spaces
$\overline{{\cal X}^{*}}$ nor $X^{\log}$. (In fact, the morphism (1.2.9) is a
quasi-isomorphism by [S].) Moreover, if $u\in\Delta^{*}$ is sufficiently close
from $O$, then we prove that there exists a canonical isomorphism

(1.2.10) $\displaystyle\rho\colon H^{q}_{\rm
dR}(X/s)\overset{\sim}{\longrightarrow}H^{q}_{\rm dR}({\cal
X}_{u}/{\mathbb{C}})\quad(q\in{\mathbb{N}})$

without using $\overline{{\cal X}^{\log}}$ nor ${\cal X}^{\log}$. Let $M_{s}$
be the log structure of $s$. The section (1.2.8) is independent of the choice
of a parameter $z$ of $\Delta$; the isomorphism (1.2.10) is also independent
of the choice of a parameter $z$ of $\Delta$ in our sense: we prepare a
parameter $z$ of $\Delta$ according to a global section of the log structure
of $s$ whose image in $\overline{M}_{s}:=M_{s}/{\cal O}_{s}^{*}={\mathbb{N}}$
is the generator. The choice of $z$ is determined up to ${\mathbb{C}}^{*}$ not
${\cal O}_{\Delta}^{*}$. If we replace $z$ by $cz$ for $c\in{\mathbb{C}}^{*}$,
then the isomorphism (1.2.10) does not change. One can consider a similar
isomorphism to (1.2.10) by using a convergent power series $\alpha(z)$ around
$\\{O\\}$ instead of $z$, then the similar isomorphism changes by ${\rm
exp}(-\alpha(0)N)$, where $N\colon H^{q}_{\rm dR}(X/s)\longrightarrow
H^{q}_{\rm dR}(X/s)(-1)$ is the monodromy operator. The existence of the
canonical section (1.2.8) and the canonical isomorphism (1.2.10) have not been
conjectured. The hope in [KU, 0.2.23] that the theory of log geometry in the
sense of Fontaine-Illusie-Kato is useful for the case where the base field is
of characteristic $0$ as in the case of characteristic $p>0$ or mixed
characteristics comes true in our setting. To construct the section (1.2.8),
we do not calculate the nearby cycle sheaf $R\Psi({\mathbb{C}})$ (especially
we do not use the space $\overline{{\cal X}^{*}}$). We use only the Hirsch
extension of a dga as in [EY] (and [N5]) by believing in a philosophy that
dga’s have informations of geometry (modulo torsion). Though
$\iota_{O*}R\Gamma_{\rm dR}(X/s)$ has only the information of the log special
fiber, $Rf_{*}\iota_{*}\iota^{-1}(\Omega^{\bullet}_{{\cal X}/\Delta})$ has an
information of the general fiber of ${\cal X}/\Delta$ near $X/s$. We have
constructed a canonical lift to
$Rf_{*}\iota_{*}\iota^{-1}(\Omega^{\bullet}_{{\cal X}/\Delta})$ from
$\iota_{O*}R_{\rm dR}(X/s)$. Since the topology of a (log) analytic space is
finer than the Zariski topology of an algebraic scheme and since the different
points in the analytic setting are more far than the different points in the
algebraic case, it remains a little surprising for us.

Our method is nothing but a log analytic version of Ertl-Yamada’s one in the
log rigid analytic case ([EY]). Though ${\cal O}_{\Delta}$ is not complete
unlike in the $p$-adic case, we heavily depend on their idea in this article.
We show that their method is also applicable for the log analytic case over
${\mathbb{C}}$ by paying careful attentions to the proof of the existence of
the canonical section (1.2.8) and the canonical isomorphism (1.2.10). (These
attentions are not necessary in the $p$-adic case because the underlying
scheme ${\rm Spec}({\cal V})$ of the base log scheme is complete and we have
only to consider (log) formal schemes by Grothendieck’s formal function
theorem.) Though we do not use any result in [EY] and [N4] in this article, we
are psychologically supported by the existence of the Hyodo-Kato section
(1.2.3). However, even if $u\in\Delta^{*}$ is sufficiently close from $O$, we
do not know whether there exists a canonical isomorphism

(1.2.11) $\displaystyle R\Gamma_{\rm
dR}(X/s)\overset{\sim}{\longrightarrow}R\Gamma_{\rm dR}({\cal
X}_{u}/\\{u\\}).$

We prove the following as a special case of our result:

######  Theorem 1.3.

Let $r_{1}$ be a positive integer and let $r_{2}$ be a nonnegative integer.
Set $B:=\Delta^{r_{1}}\times\overset{\circ}{\Delta}{}^{r_{2}}$. Let
$f\colon{\cal X}\longrightarrow B$ be a $($not necessarily proper$)$ $($not
necessarily strictly$)$-semistable analytic space with horizontal
NCD$($=normal crossing divisor$)$ ${\cal D}$. $($See the text for the notion
of the semistable analytic space. Let $({\cal E},\nabla)$ be a locally
nilpotent integrable connection on $({\cal X},{\cal D})$ in the sense of
[KatN]: $\nabla\colon{\cal E}\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{1}_{{\cal X}/{\mathbb{C}}}(\log{\cal D})$. Let ${\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D})$ be
the log de Rham complex obtained by $({\cal E},\nabla)$. Let $\\{O\\}$ be the
origin of $B$. Endow $\\{O\\}$ with the pull-back of the log structure of $B$.
Let $s$ be the resulting log analytic space. Set
$S:=s\times\overset{\circ}{\Delta}{}^{r_{2}}$, $X:={\cal X}\times_{B}S$ and
$D:={\cal D}\times_{B}S$. Let $\iota\colon
X\overset{\subset}{\longrightarrow}{\cal X}$ be the natural exact closed
immersion. Set $f_{S}:=f\circ\iota\colon X\longrightarrow B$. Then there
exists a canonical contravariantly functorial section

(1.3.1) $\displaystyle{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)\longrightarrow\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D}))$

of the natural morphism

(1.3.2) $\displaystyle\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D}))\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)$

in the derived category $D^{+}(\iota^{-1}f^{-1}({\cal O}_{B}))$ of bounded
below complexes of $\iota^{-1}f^{-1}({\cal O}_{B})$-modules. Consequently
there exists a canonical contravariantly functorial section

(1.3.3) $\displaystyle Rf_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow Rf_{S*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D}))$

of the natural morphism

(1.3.4) $\displaystyle Rf_{S*}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D}))\longrightarrow Rf_{S*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D))$

in the derived category $D^{+}({\cal O}_{B})$ of the bounded below complexes
of ${\cal O}_{B}$-modules.

To prove the theorem (1.3), we use the theory of log geometry essentially and
the Hirsch extension of a dga, more generally the Hirsch extension of a log
integrable connection. As a corollary of (1.3), we prove the following
corollary:

######  Corollary 1.4 (cf. [U2]).

Let $\overset{\circ}{\cal Y}$ be a smooth analytic space with an NCD $Y$. Let
${\cal Y}$ be the log analytic space obtained by a pair $(\overset{\circ}{\cal
Y},Y)$. Let $\overset{\circ}{f}\colon\overset{\circ}{\cal
X}\longrightarrow\overset{\circ}{\cal Y}$ be a proper morphism of smooth
schemes over $K$. Assume that $X:=f^{*}(Y)$ is a normal crossing divisor on
$\overset{\circ}{\cal X}$. Let ${\cal X}$ be the log analytic space obtained
by a pair $(\overset{\circ}{\cal X},X)$. Let $f\colon{\cal
X}\longrightarrow{\cal Y}$ be a proper strictly semistable morphism. Let
${\cal D}$ be a horizontal NCD on ${\cal X}/{\cal Y}$. Let $({\cal E},\nabla)$
be a locally nilpotent integrable connection on $({\cal X},{\cal D})$ in the
sense of [KatN]: $\nabla\colon{\cal E}\longrightarrow{\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{1}_{{\cal X}/{\mathbb{C}}}(\log{\cal D})$. Let ${\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D})$ be
the log de Rham complex obtained by $({\cal E},\nabla)$. Then
$R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/B}(\log{\cal D}))$ $(q\in{\mathbb{N}})$ is a coherent locally free ${\cal
O}_{\cal Y}$-module and commutes with base change. Here, for a morphism
$\overset{\circ}{\cal Z}\longrightarrow\overset{\circ}{\cal Y}$, we have
endowed $\overset{\circ}{\cal Z}$ with the inverse image of the log structure
of ${\cal Y}$.

Consider the case ${\cal Y}=B$. Let $B^{*}$ be the maximal open log analytic
space of $B$ whose log structure is trivial. By (1.4) we see that
$R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/B}(\log{\cal D}))$ is the canonical extension of $R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal
D}))|_{B^{*}}$ in the sense of [D]. In the case where ${\cal E}$ is trivial,
this has been already stated in [U2] without proof. In order to prove (1.4),
we use only log de Rham complexes; we do not use the Kummer étale topos nor
${\cal X}^{\log}$ unlike [KF] and [IKN]. Hence we can give a purely algebraic
proof of (1.4).

Using a variant of the section (1.2.8) (we call this variant the $\infty$-adic
Yamada section) and (1.3), we obtain the following most important result in
this article (we call this theorem the invariance theorem):

######  Theorem 1.5 (Invariance theorem).

Let the notations be as in (1.3). Assume that $f\colon\overset{\circ}{\cal
X}\longrightarrow\overset{\circ}{B}$ is proper. Let $p\colon
B\longrightarrow\overset{\circ}{\Delta}{}^{r_{2}}(\simeq\overset{\circ}{S})$
be the projection. For a positive integer $\epsilon$ less than $1$, let
$B(\epsilon)$ be an open log analytic space whose underlying space is
$\\{z\in\overset{\circ}{\Delta}~{}|~{}|z|<\epsilon\\}^{r_{1}}\times\overset{\circ}{\Delta}{}^{r_{2}}$
and whose log structure is the inverse image of $B$ $($a tubular neighborhood
of $S)$ in $B$ . Let $p(\epsilon)\colon
B(\epsilon)\longrightarrow\overset{\circ}{\Delta}{}^{r_{2}}$ be the
projection. Then there exists a canonical contravariantly functorial
isomorphism

(1.5.1) $\displaystyle\sigma\colon{\cal
O}_{B(\epsilon)}\otimes_{p(\epsilon)^{-1}({\cal
O}_{S})}p^{-1}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log
D))\overset{\sim}{\longrightarrow}R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon))).$

Here ${\cal X}(\epsilon):={\cal X}\times_{B}B(\epsilon)$, ${\cal
D}(\epsilon):={\cal D}\times_{B}B(\epsilon)$ and $f(\epsilon)\colon{\cal
X}(\epsilon)\longrightarrow B(\epsilon)$ is the structural morphism.

The isomorphism (1.5.1) tells us that $R^{q}f_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon)))$ depends only on $R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))$ and $\overset{\circ}{B}(\epsilon)$. Set
$X_{\overset{\circ}{B}(\epsilon)}:=X\times_{\overset{\circ}{S},\overset{\circ}{p}(\epsilon)}\overset{\circ}{B}$,
$D_{\overset{\circ}{B}(\epsilon)}:=D\times_{\overset{\circ}{S},\overset{\circ}{p}(\epsilon)}\overset{\circ}{B}$
and let $f_{\overset{\circ}{B}(\epsilon)}\colon
X_{\overset{\circ}{B}(\epsilon)}\longrightarrow\overset{\circ}{B}(\epsilon)$
be the structural morphism (“the constant family of $X/S$ over
$\overset{\circ}{B}(\epsilon)$”). In other words, the isomorphism (1.5.1) is
the following isomorphism

(1.5.2) $\displaystyle R^{q}f_{\overset{\circ}{B}(\epsilon)*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)\otimes_{{\cal
O}_{S}}{\cal O}_{B(\epsilon)})\overset{\sim}{\longrightarrow}R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}(\epsilon)/B(\epsilon)}(\log{\cal D}(\epsilon))).$

However the isomorphism (1.5.2) does not induce a filtered isomorphism with
respect to log Hodge filtrations in general. As usual, the variation of the
filtrations depends on the family $({\cal X}(\epsilon),{\cal D}(\epsilon))$
over $B(\epsilon)$. Note also that the log structures of the base log analytic
spaces of both hand sides in (1.5.2) are different. Though the following is
included in the theorem above, we would like to state it because of the
importance:

######  Corollary 1.6 (Invariance theorem of the pull-backs of morphisms).

Let $B^{\prime}$ and $S^{\prime}$ be analogous log analytic spaces to $B$ and
$S$, respectively. Let $({\cal X}^{\prime},{\cal D}^{\prime})/B^{\prime}$ be
an analogous log analytic space to $({\cal X},{\cal D})/B$ and let
$(X^{\prime},D^{\prime})$ be an analogous log analytic space to $(X,D)$ for
$({\cal X}^{\prime},{\cal D}^{\prime})/B^{\prime}$. Assume that we are given
the following commutative diagram

$\begin{CD}(X,D)@>{g}>{}>(X^{\prime},D^{\prime})\\\ @V{}V{}V@V{}V{}V\\\
S@>{u|_{S}}>{}>S^{\prime}\\\ @V{\bigcap}V{}V@V{}V{\bigcap}V\\\
<EMAIL_ADDRESS>
Let $({\cal E}^{\prime},\nabla^{\prime})$ be an analogous integrable
connection to $({\cal E},\nabla)$ for ${\cal
X}^{\prime}/\overset{\circ}{S}{}^{\prime}$. Set $E^{\square}:={\cal
E}^{\square}$, where $\square=$nothing or ′. Let $\Phi\colon
g^{*}(E^{\prime})\longrightarrow E$ be a morphism of ${\cal O}_{X}$ modules
fitting into the following commutative diagram$:$

$\begin{CD}g^{*}(E^{\prime})@>{\Phi}>{}>E\\\
@V{g^{*}(\nabla^{\prime})}V{}V@V{}V{\nabla}V\\\ g^{*}(E^{\prime}\otimes_{{\cal
O}_{X^{\prime}}}\Omega^{1}_{X^{\prime}/\overset{\circ}{S}{}^{\prime}}(\log
D^{\prime}))@>{\Phi\otimes g^{*}}>{}>E\otimes_{{\cal
O}_{X}}\Omega^{1}_{X/\overset{\circ}{S}{}}(\log D).\end{CD}$

Then there exist $\epsilon>0$ and $\epsilon^{\prime}>0$ such that
$u(B(\epsilon))\subset B^{\prime}(\epsilon^{\prime})$ such that there exists
the following canonical morphism

$\displaystyle g(\epsilon,\epsilon^{\prime})^{*}\colon$ $\displaystyle
u(\epsilon,\epsilon^{\prime})^{*}(R^{q}f^{\prime}(\epsilon^{\prime})_{*}({\cal
E}^{\prime}\otimes_{{\cal O}_{{\cal X}^{\prime}}}\Omega^{\bullet}_{{\cal
X}^{\prime}(\epsilon^{\prime})/B^{\prime}(\epsilon^{\prime})}(\log{\cal
D}^{\prime}(\epsilon^{\prime}))))$ $\displaystyle\longrightarrow
R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon))).$

Here $u(\epsilon,\epsilon^{\prime})\colon B(\epsilon)\longrightarrow
B^{\prime}(\epsilon^{\prime})$ is the induced morphism by $u$. The morphism
$g(\epsilon,\epsilon^{\prime})$ satisfies the usual transitive relation. If
the morphism $g$ is lifted to a morphism $\widetilde{g}\colon{\cal
X}(\epsilon)\longrightarrow{\cal X}^{\prime}(\epsilon^{\prime})$ and if the
morphism $\Phi$ is lifted to a morphism $\widetilde{\Phi}\colon g^{*}({\cal
E}^{\prime})\longrightarrow{\cal E}$ be a morphism of ${\cal O}_{\cal X}$
modules fitting into the following commutative diagram$:$

$\begin{CD}\widetilde{g}^{*}({\cal
E}^{\prime}(\epsilon^{\prime}))@>{\widetilde{\Phi}}>{}>{\cal E}(\epsilon)\\\
@V{\widetilde{g}{}^{*}(\nabla^{\prime})}V{}V@V{}V{\nabla}V\\\
\widetilde{g}^{*}({\cal E}^{\prime}\otimes_{{\cal O}_{{\cal
X}^{\prime}}}\Omega^{1}_{{\cal
X}^{\prime}(\epsilon^{\prime})/\overset{\circ}{S}{}^{\prime}}(\log{\cal
D}^{\prime}(\epsilon^{\prime})))@>{\widetilde{\Phi}\otimes\widetilde{g}{}^{*}}>{}>{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{1}_{{\cal
X}(\epsilon)/\overset{\circ}{S}{}}(\log{\cal D}(\epsilon)),\end{CD}$

where ${\cal E}^{\square}(\epsilon^{\square})={\cal E}^{\square}|_{{\cal
X}^{\square}(\epsilon^{\square})}$, then

$\displaystyle g(\epsilon,\epsilon^{\prime})^{*}=\widetilde{g}^{*}\colon$
$\displaystyle
u(\epsilon,\epsilon^{\prime})^{*}(R^{q}f^{\prime}(\epsilon^{\prime})_{*}({\cal
E}^{\prime}\otimes_{{\cal O}_{{\cal X}^{\prime}}}\Omega^{\bullet}_{{\cal
X}^{\prime}(\epsilon^{\prime})/B^{\prime}(\epsilon^{\prime})}(\log{\cal
D}^{\prime}(\epsilon^{\prime}))))$ $\displaystyle\longrightarrow
R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon))).$

The results (1.5) and (1.6) suggest that there may exist a theory of crystals
for an integrable connection on a proper log smooth morphism $Y\longrightarrow
S$ of fine or fs(=fine and saturated) log analytic spaces over $B(\epsilon)$
as in the algebraic case ([Gro], [KK]). The main theorem of [NO] (cf. [KajN])
which tells us that $f^{\log}\colon Y^{\log}\longrightarrow S^{\log}$ is a
fiber bundle (cf. [U1], [U2]) also suggests this. However the main theorem
does not imply (1.5): it only implies that there exists a canonical
isomorphism

(1.6.1) $\displaystyle{\cal
O}_{B(\epsilon)^{\log}}\otimes_{p(\epsilon)^{-1}({\cal
O}_{S})}p^{-1}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\overset{\sim}{\longrightarrow}$
$\displaystyle{\cal O}_{B(\epsilon)^{\log}}\otimes_{{\cal
O}_{B(\epsilon)}}R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon))).$

Moreover our theorem (1.5) has an advantage in the concrete calculation of the
isomorphism (1.5.1) by using Čech coverings of ${\cal X}(\epsilon)$ as in the
log rigid analytic case in [EY].

By (1.4) and (1.5) we obtain the following result:

######  Corollary 1.7.

For a sufficiently close point $u\in B$ from a point of $S$, there exists a
canonical contravariantly functorial isomorphism

(1.7.1) $\displaystyle{\cal O}_{\overline{S},u}\otimes_{{\cal
O}_{S,p(u)}}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal
D})))_{p(u)}\overset{\sim}{\longrightarrow}R^{q}f_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D})))_{u}.$

Consequently there exists a canonical contravariantly functorial isomorphism

(1.7.2) $\displaystyle H^{q}(X_{s},{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X_{s}/s}(\log
D_{s})))\overset{\sim}{\longrightarrow}H^{q}({\cal X}_{u},{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}_{u}/u}(\log{\cal
D}_{u}))).$

If $({\cal E}|_{{\cal U}_{u}},\nabla)=({\cal O}_{{\cal
U}_{u}}\otimes_{\mathbb{C}}V,d\otimes{\rm id}_{V})$ for a local system $V$ on
${\cal U}_{u}$ of finite dimensional vector spaces over ${\mathbb{C}}$, then
we obtain a canonical contravariantly functorial isomorphism

(1.7.3) $\displaystyle H^{q}(X_{s},{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X_{s}/s}(\log
D_{s}))\overset{\sim}{\longrightarrow}H^{q}({\cal U}_{u},V).$

This is a generalization of the isomorphism (1.2.10). We call this
generalization the Steenbrink isomorphism. This also tells us that
$H^{q}({\cal U}_{u},V)$ is determined by $(X_{s},D_{s})$ and ${\cal
E}|_{X_{s}}$.

We can also obtain the algebraic analogues of (1.3) and (1.4) by algebraic
proofs (without using the Lefschetz principle, GAGA nor (1.3)). This is also
one of advantages of our algebraic method. In a future article we would like
to discuss this.

As to the locally freeness in (1.4), our method is also applicable for a more
general family with worse reduction by virtue of the contravariant
functoriality of the section (1.3.1). This general case is not contained in
the case (1.2). We would also like to discuss this.

Our results (1.5) and (1.6) should be applied for (the automorphism groups of)
concrete degeneratation of analytic spaces, e. g., (log) $K3$ surfaces, more
generally (log) Calabi-Yau analytic spaces, degenerations of complex tori,
degenerations of hyperkähler manifolds and so on (cf. [Fr], [KawN], [Bea], …).

The contents of this article are as follows.

In §2 we recall results in [N6] which are necessary in this article.

In §3 we give additional results to those in [N6].

In §4 we give the definition of SNCL analytic spaces.

In §5 we consider the completed Hirsch extension of an integrable connection
on a semistable family over a polydisc.

In §6 we give the invariance theorems (1.5) and (1.6).

Acknowledgment. I would like to express my sincere gratitude to T. Fujisawa
for pointing out a non-minor mistake in a previous version of this article.

##  2 Recall of results in [N6]

In this section we recall results in [N6] which are necessary in this article.

Let $r$ be a positive integer. Let $S$ be an analytic family over
${\mathbb{C}}$ of log points of virtual dimension $r$, that is, locally on
$S$, the log structure $M_{S}$ of $S$ is isomorphic to
${\mathbb{N}}^{r}\oplus{\cal
O}_{S}^{*}\owns((n_{1},\ldots,n_{r}),u)\longmapsto
0^{\sum_{i=1}^{r}n_{i}}u\in{\cal O}_{S}$ (cf. [N4, §2]), where $0^{n}=0$ when
$n\not=0$ and $0^{0}:=1$. Let $g\colon Y\longrightarrow S$ be a log smooth
morphism of log analytic spaces over ${\mathbb{C}}$.

Locally on $S$, there exists a family $\\{t_{i}\\}_{i=1}^{r}$ of local
sections of $M_{S}$ giving a local basis of $\overline{M}_{S}:=M_{S}/{\cal
O}_{S}^{*}$. Let $M_{i}$ be the submonoid sheaf of $M_{S}$ generated by
$t_{i}$. For all $1\leq\forall i\leq r$, the submonoid sheaf
$\bigoplus_{j=1}^{i}M_{j}$ of $M$ and $\overset{\circ}{S}$ defines a family of
log points of virtual dimension $i$. Let
$S_{i}:=S(M_{1},\ldots,M_{i})=(\overset{\circ}{S},(\bigoplus_{j=1}^{i}M_{j}\longrightarrow{\cal
O}_{S})^{a})$ be the resulting local log analytic space. Set
$S_{0}:=\overset{\circ}{S}$. Then we have the following sequence of families
of log points of virtual dimensions:

(2.0.1) $\displaystyle S=S_{r}\longrightarrow S_{r-1}\longrightarrow
S_{r-2}\longrightarrow\cdots\longrightarrow S_{1}\longrightarrow
S_{0}=\overset{\circ}{S}.$

The one-form $d\log t_{i}\in\Omega^{1}_{S/\overset{\circ}{S}}$ is independent
of the choice of the generator $t_{i}$ of $M_{i}$. Denote also by $d\log
t_{i}\in\Omega^{1}_{Y/\overset{\circ}{S}}$ the image of $d\log
t_{i}\in\Omega^{1}_{S/\overset{\circ}{S}}$ in
$\Omega^{1}_{Y/\overset{\circ}{S}}$. Let ${\cal F}$ be a (not necessarily
coherent) locally free ${\cal O}_{Y}$-module and let

(2.0.2) $\displaystyle\nabla\colon{\cal F}\longrightarrow{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{1}_{Y/\overset{\circ}{S}}$

be an integrable connection. (In later sections we assume that ${\cal F}$ is
coherent.) Then we have the complex ${\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}$ of $g^{-1}({\cal
O}_{S})$-modules.

######  Lemma 2.1 ([N6, (3.1)]).

$(1)$ The sheaf $\Omega^{i}_{Y/\overset{\circ}{S}}$ $(i\in{\mathbb{N}})$ is a
locally free ${\cal O}_{Y}$-module.

$(2)$ Locally on $Y$, the following sequence

(2.1.1) $\displaystyle 0\longrightarrow{\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}[-1]\overset{{\rm id}_{\cal F}\otimes(d\log
t_{i}\wedge)}{\longrightarrow}{\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i-1}}\longrightarrow{\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}\longrightarrow 0\quad(1\leq i\leq r)$

is exact. $($Note that the morphism

$\displaystyle{\rm id}_{\cal F}\otimes(d\log t_{i}\wedge)\colon{\cal
F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}[-1]\longrightarrow{\cal
F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i-1}}$

is indeed a morphism of complexes of $g^{-1}({\cal O}_{S})$-modules.$)$

Let $\\{t_{1},\ldots,t_{r}\\}$ be a set of local sections of $M_{S}$ whose
images in $\overline{M}_{S}$ is a unique system of local generators of
$\overline{M}_{S}$. Set $L_{S}(t_{1},\ldots,t_{r}):=\bigoplus_{i=1}^{r}{\cal
O}_{S}t_{i}$ (a free ${\cal O}_{S}$-module with basis $t_{1},\ldots,t_{r}$).
This ${\cal O}_{S}$-module patches together to a locally free ${\cal
O}_{S}$-module $L_{S}$ by [N6, (3.3)]. In the following we denote $t_{i}$ in
$L_{S}$ by $u_{i}$. Because we prefer not to use non-intrinsic variables
$u_{1},\ldots,u_{r}$, we consider the following sheaf $U_{S}$ of sets of
$r$-pieces on $\overset{\circ}{S}$ defined by the following presheaf:

(2.1.2) $\displaystyle U_{S}\colon\\{{\rm open}~{}{\rm log}~{}{\rm
analytic}~{}{\rm spaces}~{}{\rm of}~{}S\\}\owns V\longmapsto\\{{\rm
generators}~{}{\rm of}~{}\Gamma(V,\overline{M}_{S})\\}\in({\rm Sets}).$

Let ${\rm Sym}_{{\cal O}_{S}}(L_{S})$ be the symmetric algebra of $L_{S}$ over
${\cal O}_{S}$ by $\Gamma_{{\cal O}_{S}}(L_{S})$. We consider the following
Hirsch extension

(2.1.3) ${\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[U_{S}]:=\Gamma_{{\cal
O}_{S}}(L_{S})\otimes_{{\cal O}_{S}}{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}$

of ${\cal F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}$ by
the morphism

(2.1.4) $\displaystyle d\log\colon g^{-1}(L_{S})\owns u_{i}\longmapsto d\log
t_{i}\in{\rm
Ker}(\Omega^{1}_{Y/\overset{\circ}{S}}\longrightarrow\Omega^{2}_{Y/\overset{\circ}{S}})$

([N5, §3]). (Here we have omitted to write $g^{-1}$ for $g^{-1}(\Gamma_{{\cal
O}_{S}}(L_{S}))\otimes_{g^{-1}({\cal O}_{S})}$ in (2.1.3). It may be better to
denote ${\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[U_{S}]$ by ${\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[L_{S}]$.) It
is easy to check that the morphism $g^{-1}(L_{S})\longrightarrow{\rm
Ker}(\Omega^{1}_{Y/\overset{\circ}{S}}\longrightarrow\Omega^{2}_{Y/\overset{\circ}{S}})$
is well-defined (cf. [N4, §2]). The boundary morphism

$\nabla\colon{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{i}_{Y/\overset{\circ}{S}}[U_{S}]\longrightarrow{\cal
F}\otimes_{{\cal
O}_{Y}}\Omega^{i+1}_{Y/\overset{\circ}{S}}[U_{S}]\quad(i\in{\mathbb{Z}}_{\geq
0})$

is, by definition, the following ${\cal O}_{S}$-linear morphism:

(2.1.5) $\displaystyle\nabla(m_{1}^{[e_{1}]}\cdots
m_{r}^{[e_{r}]}\otimes\omega)=\sum_{j=1}^{r}m_{1}^{[e_{1}]}\cdots
m_{j}^{[e_{j}-1]}\cdots m_{r}^{[e_{r}]}d\log
m_{j}\wedge\omega+m_{1}^{[e_{1}]}\cdots m_{r}^{[e_{r}]}\otimes\nabla(\omega)$
$\displaystyle\quad(~{}\omega\in{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{j}_{Y/\overset{\circ}{S}},m_{1},\ldots m_{r}\in
L_{S},e_{1},\ldots,e_{r}\in{\mathbb{Z}}_{\geq
1},m_{i}^{[e_{i}]}=((e_{i})!)^{-1}m^{e_{i}}_{i}~{}).$

It is easy to check an equality $\nabla^{2}=0$. We have the following equality

(2.1.6) $\displaystyle\Gamma_{{\cal O}_{S}}(L_{S})={\cal
O}_{S}[\overline{M}_{S}].$

Let $\epsilon\colon S^{\log}\longrightarrow S$ be the real blow up defined in
[KatN]. Then $\Gamma_{{\cal O}_{S}}(L_{S})$ is nothing but $\epsilon_{*}({\cal
O}_{S^{\log}})$. The sheaf ${\cal O}_{S}[\overline{M}_{S}]$ is locally
isomorphic to a sheaf ${\cal O}_{S}[\underline{u}]:={\cal
O}_{S}[u_{1},\ldots,u_{r}]$ of polynomial algebras over ${\cal O}_{S}$ of
independent $r$-variables. The projection $\Gamma_{{\cal O}_{S}}(L_{S})={\rm
Sym}_{{\cal O}_{S}}(L_{S})\longrightarrow{\rm Sym}^{0}_{{\cal
O}_{S}}(L_{S})={\cal O}_{S}$ induces the following natural morphism of
complexes:

(2.1.7) ${\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[U_{S}]\longrightarrow{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/S}.$

([N5, §3]). Let $\Gamma^{\geq n}_{{\cal O}_{S}}(L_{S})$ $(n\in{\mathbb{N}})$
be the degree $\geq n$-part of $\Gamma_{{\cal O}_{S}}(L_{S})$. We set

(2.1.8) $\displaystyle\Gamma_{{\cal
O}_{S}}(L_{S})^{\wedge}:=\varprojlim_{n}(\Gamma_{{\cal
O}_{S}}(L_{S})/\Gamma^{\geq n}_{{\cal O}_{S}}(L_{S})).$

Then we have the following “inclusion morphism”

(2.1.9) $\displaystyle\Gamma_{{\cal
O}_{S}}(L_{S})\overset{\subset}{\longrightarrow}\Gamma_{{\cal
O}_{S}}(L_{S})^{\wedge}.$

We also consider the following PD-Hirsch extension by completed powers:

${\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[[U_{S}]]:=\Gamma_{{\cal
O}_{S}}(L_{S})^{\wedge}\otimes_{{\cal O}_{S}}{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}$

of ${\cal F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}$;
the boundary morphism

(2.1.10) $\nabla\colon{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{j}_{Y/\overset{\circ}{S}}[[U_{S}]]\longrightarrow{\cal
F}\otimes_{{\cal
O}_{Y}}\Omega^{j+1}_{Y/\overset{\circ}{S}}[[U_{S}]]\quad(j\in{\mathbb{Z}}_{\geq
0})$

is an ${\cal O}_{S}$-linear morphism defined by the following

(2.1.11) $\displaystyle\nabla(m_{1}^{[e_{1}]}\cdots
m_{r}^{[e_{r}]}\otimes\omega)=$
$\displaystyle\sum_{j=1}^{r}m_{1}^{[e_{1}]}\cdots m_{j}^{[e_{j}-1]}\cdots
m_{r}^{[e_{r}]}d\log m_{j}\wedge\omega$ $\displaystyle+m_{1}^{[e_{1}]}\cdots
m_{r}^{[e_{r}]}\otimes\nabla(\omega)$

as in (2.1.5). It is easy to check an equality $\nabla^{2}=0$ for $\nabla$ in
(2.1.11). The projection

$\Gamma_{{\cal O}_{S}}(L_{S})/\Gamma^{\geq n}_{{\cal
O}_{S}}(L_{S})(L_{S})\longrightarrow\Gamma_{{\cal O}_{S}}(L_{S})/\Gamma^{\geq
1}_{{\cal O}_{S}}(L_{S})=\Gamma^{0}_{{\cal O}_{S}}(L_{S})={\cal
O}_{S}\quad(n\geq 1)$

induces the following natural morphism of complexes:

(2.1.12) ${\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[[U_{S}]]\longrightarrow{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/S}.$

The morphism (2.1.9) induces the following inclusion morphism

(2.1.13) $\displaystyle{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[U_{S}]\overset{\subset}{\longrightarrow}{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[[U_{S}]].$

We have the following commutative diagram

(2.1.14) $\begin{CD}{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[U_{S}]@>{\subset}>{}>{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/\overset{\circ}{S}}[[U_{S}]]\\\
@V{}V{}V@V{}V{}V\\\ {\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/S}@={\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/S}.\end{CD}$

The following is a (simpler) log analytic version of [N5, (4.3)]:

######  Definition 2.2 ([N6, (3.4)]).

(1) Take a local basis of $\Omega^{1}_{Y/\overset{\circ}{S}}$ containing
$\\{d\log t_{1},\ldots,d\log t_{r}\\}$. Let $(d\log
t_{i})^{*}\colon\Omega^{1}_{Y/\overset{\circ}{S}}\longrightarrow{\cal O}_{Y}$
be the local morphism defined by the local dual basis of $d\log t_{i}$. We say
that the connection $({\cal F},\nabla)$ has no poles along $S$ if the
composite morphism ${\cal F}\overset{\nabla}{\longrightarrow}{\cal
F}\otimes_{Y}\Omega^{1}_{Y/\overset{\circ}{S}}\overset{{\rm id}_{\cal
F}\otimes(d\log t_{i})^{*}}{\longrightarrow}{\cal F}$ vanishes for
$1\leq\forall i\leq r$.

(2) If there exists a finite increasing filtration $\\{{\cal
F}_{i}\\}_{i\in{\mathbb{Z}}}$ on ${\cal F}$ locally on $Y$ such that ${\rm
gr}_{i}{\cal F}:={\cal F}_{i}/{\cal F}_{i-1}$ is a locally free ${\cal
O}_{Y}$-module and $\nabla$ induces a connection ${\cal
F}_{i}\longrightarrow{\cal F}_{i}\otimes_{{\cal
O}_{Y}}\Omega^{1}_{Y/\overset{\circ}{S}}$ whose induced connection ${\rm
gr}_{i}{\cal F}\longrightarrow({\rm gr}_{i}{\cal F})\otimes_{{\cal
O}_{Y}}\Omega^{1}_{Y/\overset{\circ}{S}}$ has no poles along $M_{S}$, then we
say that $({\cal F},\nabla)$ is a locally nilpotent integrable connection on
$Y$ with respect to $S$.

Let us recall the following result:

######  Theorem 2.3 ([N6, (3.5)]).

Assume that $(F,\nabla)$ is a locally nilpotent integrable connection on $Y$
with respect to $S$. Then the morphism (2.1.7) is a quasi-isomorphism.

In the rest of this section we consider the functoriality of the quasi-
isomorphism (2.1.7). Let us recall a formalism of the functoriality in [N6].

Let $S^{\prime}$ be an analytic family over ${\mathbb{C}}$ of log points of
virtual dimension $r^{\prime}$ and let $v\colon S\longrightarrow S^{\prime}$
be a morphism of log analytic spaces. This morphism induces a morphism
$v^{*}\colon\overline{M}_{S^{\prime}}\longrightarrow v_{*}(\overline{M}_{S})$.
Locally on $S$, this morphism is equal to a morphism
$v^{*}\colon{\mathbb{N}}^{r^{\prime}}\longrightarrow{\mathbb{N}}^{r}$. Set
$e_{i}=(0,\ldots,0,\overset{i}{1},0,\ldots,0)\in{\mathbb{N}}^{s}$ for $s=r$ or
$r^{\prime}$ $(1\leq i\leq r)$. Let $A=(a_{ij})_{1\leq i\leq r,1\leq j\leq
r^{\prime}}\in M_{rr^{\prime}}({\mathbb{N}})$ be the representing matrix of
$v^{*}$:

$v^{*}(e_{1},\ldots,e_{r^{\prime}})=(e_{1},\ldots,e_{r})A.$

Let $t_{1},\ldots,t_{r}$ and $t^{\prime}_{1},\ldots,t^{\prime}_{r^{\prime}}$
be local sections of $M_{S}$ and $M_{S^{\prime}}$ whose images in
$\overline{M}_{S}\overset{\sim}{\longrightarrow}{\mathbb{N}}^{r}$ and
$\overline{M}{}^{\prime}_{S^{\prime}}\overset{\sim}{\longrightarrow}{\mathbb{N}}^{r^{\prime}}$
are $e_{1},\ldots,e_{r}$ and $e_{1},\ldots,e_{r^{\prime}}$, respectively. Then
there exists a local section $b_{j}\in{\cal O}_{S}^{*}$ $(1\leq j\leq
r^{\prime})$ such that

(2.3.1) $\displaystyle v^{*}(t^{\prime}_{j})=b_{j}t^{a_{1j}}_{1}\cdots
t^{a_{rj}}_{r}.$

Let $u_{1},\ldots,u_{r}$ and $u^{\prime}_{1},\ldots,u^{\prime}_{r^{\prime}}$
be the corresponding local sections to $t_{1},\ldots,t_{r}$ and
$t^{\prime}_{1},\ldots,t^{\prime}_{r^{\prime}}$ of $L_{S}$ and
$L_{S^{\prime}}$, respectively. Then we define an ${\cal
O}_{S^{\prime}}$-linear morphism $v^{*}\colon\Gamma_{{\cal
O}_{S^{\prime}}}(L_{S^{\prime}})\longrightarrow v_{*}(\Gamma_{{\cal
O}_{S}}(L_{S}))$ by the following formula:

(2.3.2) $\displaystyle v^{*}(u_{j})=a_{1j}u_{1}+\cdots+a_{rj}u_{r}$

and by ${\rm Sym}(v^{*})$ for the $v^{*}$ in (2.3.2). Since ${\rm
Aut}({\mathbb{N}}^{s})={\mathfrak{S}}_{s}$ ([N3, p. 47]), the morphism
$v^{*}\colon\Gamma_{{\cal O}_{S^{\prime}}}(L_{S^{\prime}})\longrightarrow
v_{*}(\Gamma_{{\cal O}_{S}}(L_{S}))$ is independent of the choice of the local
isomorphisms $\overline{M}_{S^{\prime}}\simeq{\mathbb{N}}^{r^{\prime}}$ and
$\overline{M}_{S}\simeq{\mathbb{N}}^{r}$. In conclusion, we obtain the
following well-defined morphism

(2.3.3) $\displaystyle v^{*}\colon\Gamma_{{\cal
O}_{S^{\prime}}}(L_{S^{\prime}})\longrightarrow v_{*}(\Gamma_{{\cal
O}_{S}}(L_{S}))$

of sheaves of commutative rings of unit elements on $S^{\prime}$. This
morphism satisfies the usual transitive relation

(2.3.4) $\displaystyle{\textrm{`}`}(v\circ
v^{\prime})^{*}=v^{\prime}{}^{*}v^{*}{\textrm{'}}{\textrm{'}}.$

Let $g^{\prime}\colon Y^{\prime}\longrightarrow S^{\prime}$ be an analogous
morphism of log analytic spaces to $g\colon Y\longrightarrow S$. Assume that
we are given a commutative diagram

(2.3.5) $\begin{CD}Y@>{h}>{}>Y^{\prime}\\\ @V{g}V{}V@V{}V{g^{\prime}}V\\\
<EMAIL_ADDRESS>
######  Proposition 2.4 (Functoriality).

Assume that we are given a commutative diagram

(2.4.1) $\begin{CD}{\cal F}^{\prime}@>{\nabla^{\prime}}>{}>{\cal
F}^{\prime}\otimes_{{\cal
O}_{Y^{\prime}}}\Omega^{1}_{Y^{\prime}/\overset{\circ}{S}{}^{\prime}}\\\
@V{}V{}V@V{}V{}V\\\ h_{*}({\cal F})@>{h_{*}(\nabla)}>{}>h_{*}({\cal
F}\otimes_{{\cal O}_{X}}\Omega^{1}_{Y/\overset{\circ}{S}}),\end{CD}$

where $({\cal F}^{\prime},{\nabla^{\prime}})$ is an analogous connection to
$({\cal F},{\nabla})$ on $Y^{\prime}/\overset{\circ}{S}{}^{\prime}$. Then the
morphism (2.1.7) is contravariantly functorial for the commutative diagrams
(2.3.5) and (2.4.1). This contravariance satisfies the usual transitive
relation “$(i\circ h)^{*}=h^{*}\circ i^{*}$”.

###### Proof.

Obvious. ∎

##  3 Results on completed Hirsch extensions

In this section we give an easy result (3.2) on completed Hirsch extensions.
Using this result and the result (2.3) in the previous section, we see that
the morphism (2.1.13) is a quasi-isomorphism. Though we do not use this result
to obtain the main result in this article, it is of independent interest. If
one wishes to know the proof of the main result in this article, one can skip
this section and (4.3) and (4.4) in the next section.

######  Lemma 3.1.

Let the notations be as in (2.1). Let $\\{t_{1},\ldots,t_{r}\\}$ be a set of
local sections of $M_{S}$ whose images in $\overline{M}_{S}$ is a unique
system of local generators of $\overline{M}_{S}$. Denote $t_{i}$ in $L_{S}$ by
$u_{i}$. For $1\leq i\leq r$, let us consider the completed Hirsch extensions
${\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i-1}}[[u_{i+1},\ldots,u_{r}]]$ and ${\cal
F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}[[u_{i+1},\ldots,u_{r}]]$
of ${\cal F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i-1}}$ and ${\cal
F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}$ by the morphisms
$\bigoplus_{j=i+1}^{r}{\cal O}_{S}u_{j}\owns u_{j}\longmapsto d\log
t_{j}\in{\Omega}^{1}_{Y/S_{i-1}}$ and $\bigoplus_{j=i+1}^{r}{\cal
O}_{S}u_{j}\owns u_{j}\longmapsto d\log t_{j}\in{\Omega}^{1}_{Y/S_{i}}$,
respectively. Then the following sequence

(3.1.1) $\displaystyle 0$ $\displaystyle\longrightarrow{\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}[[u_{i+1},\ldots,u_{r}]][-1]\overset{{\rm
id}_{\cal F}\otimes(d\log t_{i}\wedge)}{\longrightarrow}{\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i-1}}[[u_{i+1},\ldots,u_{r}]]$
$\displaystyle\longrightarrow{\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{i}}[[u_{i+1},\ldots,u_{r}]]\longrightarrow
0\quad(1\leq i\leq r)$

is exact.

###### Proof.

This immediately follows from (2.1) by considering “coefficient forms” of each
$u_{i+1}^{e_{i+1}}\cdots u_{r}^{e_{r}}$ $(e_{i+1},\ldots
e_{r}\in{\mathbb{N}})$. ∎

######  Proposition 3.2.

The morphism (2.1.12) is a quasi-isomorphism.

###### Proof.

The problem is local. We may assume that there exists a sequence (2.0.1) and
that $L_{S}=\oplus_{i=1}^{r}{\cal O}_{S}u_{i}$. The complex ${\cal
F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{r-1}}[[u_{r}]]$ is, by
definition, the following complex:

(3.2.1) $\begin{CD}\cdots @>{}>{}>\cdots @>{}>{}>\cdots\\\ @A{{\rm
id}\otimes\nabla}A{}A@A{{\rm id}\otimes\nabla}A{}A@A{{\rm
id}\otimes\nabla}A{}A\\\ {\cal O}_{S}u^{2}_{r}\otimes_{{\cal O}_{S}}{\cal
F}@>{(?)^{\prime}d\log t_{r}\wedge}>{}>{\cal O}_{S}u_{r}\otimes_{{\cal
O}_{S}}{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{1}_{Y/S_{r-1}}@>{(?)^{\prime}d\log t_{r}\wedge}>{}>{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{2}_{Y/S_{r-1}}\\\ @A{{\rm
id}\otimes\nabla}A{}A@A{{\rm id}\otimes\nabla}A{}A\\\ {\cal
O}_{S}u_{r}\otimes_{{\cal O}_{S}}{\cal F}@>{(?)^{\prime}d\log
t_{r}\wedge}>{}>{\cal F}\otimes_{{\cal O}_{Y}}\Omega^{1}_{Y/S_{r-1}}\\\
@A{\nabla}A{}A\\\ {\cal F},\end{CD}$

where the horizontal arrow $(?)^{\prime}d\log t_{r}\wedge$ is defined by
$u^{[i]}_{r}\otimes f\otimes\omega\longmapsto u^{[i-1]}_{r}\otimes
f\otimes(d\log t_{r}\wedge\omega)$ $(f\in{\cal
F},\omega\in\Omega^{\bullet}_{Y/S_{r-1}})$. This is augmented to the complex
${\cal F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/S}$. By (2.1.1) we see that
this augmentation

(3.2.2) $\displaystyle{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{\bullet}_{Y/S_{r-1}}[[u_{r}]]\longrightarrow{\cal
F}\otimes_{{\cal O}_{Y}}\Omega^{\bullet}_{Y/S}$

is an isomorphism. The complex ${\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{r-2}}[[u_{r},u_{r-1}]]=({\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{r-2}}[[u_{r}]])[[u_{r-1}]]$ is, by definition,
the following complex:

(3.2.3) ${\footnotesize{\begin{CD}\cdots @>{}>{}>\cdots @>{}>{}>\cdots\\\
@A{{\rm id}\otimes\nabla}A{}A@A{{\rm id}\otimes\nabla}A{}A@A{{\rm
id}\otimes\nabla}A{}A\\\ {\cal O}_{S}u^{2}_{r-1}\otimes_{{\cal O}_{S}}{\cal
F}[[u_{r}]]@>{(?)^{\prime}d\log t_{r-1}\wedge}>{}>{\cal
O}_{S}u_{r-1}\otimes_{{\cal O}_{S}}{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{1}_{Y/S_{r-2}}[[u_{r}]]@>{(?)^{\prime}d\log
t_{r-1}\wedge}>{}>{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{2}_{Y/S_{r-2}}[[u_{r}]]\\\ @A{{\rm id}\otimes\nabla}A{}A@A{{\rm
id}\otimes\nabla}A{}A\\\ {\cal O}_{S}u_{r-1}\otimes_{{\cal O}_{S}}{\cal
F}[[u_{r}]]@>{(?)^{\prime}d\log t_{r-1}\wedge}>{}>{\cal F}\otimes_{{\cal
O}_{Y}}\Omega^{1}_{Y/S_{r-2}}[[u_{r}]]\\\ @A{\nabla}A{}A\\\ {\cal
F}.\end{CD}}}$

The augmentation

(3.2.4) $\displaystyle({\cal F}\otimes_{{\cal
O}_{Y}}{\Omega}^{\bullet}_{Y/S_{r-2}}[[u_{r}]])[[u_{r-1}]]\longrightarrow{\cal
F}\otimes_{{\cal O}_{Y}}{\Omega}^{\bullet}_{Y/S_{r-1}}[[u_{r}]]$

is an isomorphism by (3.1). Continuing this argument repeatedly, we see that
the morphism (2.1.12) is a quasi-isomorphism. ∎

Though we do not use the following in this article, this is of independent
interest as in [EY, (3.17) (4)]:

######  Corollary 3.3.

The morphism (2.1.13) is a quasi-isomorphism.

###### Proof.

This immediately follows from (2.1.14), (2.3) and (3.2). ∎

######  Proposition 3.4 (Functoriality).

Let the notations be as in (2.4). Then the morphism (2.1.13) is
contravariantly functorial for the commutative diagrams (2.3.5) and (2.4.1).
This contravariance satisfies the usual transitive relation “$(i\circ
h)^{*}=h^{*}\circ i^{*}$”.

###### Proof.

Obvious. ∎

##  4 Log analytic families

In this section we consider a “thicker” family than the family in the previous
two sections.

Let the notations be as in the previous section. Let $r$ be a positive
integer. Set
$\overset{\circ}{\Delta}{}^{r}_{\overset{\circ}{S}}:=\overset{\circ}{\Delta}{}^{r}\times\overset{\circ}{S}$.
The zero section
$\overset{\circ}{S}\overset{\subset}{\longrightarrow}\overset{\circ}{\Delta}_{\overset{\circ}{S}}$
defines an fs log structure on $\overset{\circ}{\Delta}_{\overset{\circ}{S}}$.
Let $\Delta_{\overset{\circ}{S}}$ be the resulting log analytic space and let
$\Delta^{r}_{\overset{\circ}{S}}$ be the $r$-times product of
$\Delta_{\overset{\circ}{S}}$ over $\overset{\circ}{S}$. Locally on $S$, we
have the following exact closed immersion

$\displaystyle
S\overset{\subset}{\longrightarrow}\Delta^{r}_{\overset{\circ}{S}}$

of log analytic spaces. This immersion fits into the following commutative
diagram

$\begin{CD}S@>{\subset}>{}>\Delta^{r}_{\overset{\circ}{S}}\\\
@V{}V{}V@V{}V{}V\\\ \overset{\circ}{S}@=\overset{\circ}{S},\end{CD}$

where the vertical morphisms are natural morphisms.

By replacing ${\mathbb{A}}^{r}_{\overset{\circ}{S}}$ in [N4, (1.1)] by
$\Delta^{r}_{\overset{\circ}{S}}$, we have a well-defined log analytic space
$\overline{S}$ over $\overset{\circ}{S}$ with an exact closed immersion
$S\overset{\subset}{\longrightarrow}\overline{S}$ fitting into the following
commutative diagram over $\overset{\circ}{S}$:

(4.0.1) $\begin{CD}S@>{\subset}>{}>\overline{S}\\\ @V{}V{}V@V{}V{p}V\\\
<EMAIL_ADDRESS>
Here $p\colon\overline{S}\longrightarrow\overset{\circ}{S}$ is the structural
morphism. Locally on $S$, $\overline{S}$ is isomorphic to a unit polydisc
$\Delta^{r}_{\overset{\circ}{S}}=\\{(t_{1},\ldots,t_{r})\in{\mathbb{C}}^{r}~{}|~{}|t_{1}|<1,\ldots,|t_{r}|<1\\}$
with log structures ${\mathbb{N}}^{r}\owns e_{i}\longmapsto t_{i}\in{\cal
O}_{\overline{S}}$ over $\overset{\circ}{S}$; $\overline{S}$ is obtained by
gluing
$\\{(t_{1},\ldots,t_{r})\in{\mathbb{C}}^{r}~{}|~{}|t_{1}|<1,\ldots,|t_{r}|<1\\}$
by the log structure $M_{S}$. It is important to note that, by the
construction of $\overline{S}$, we obtain well-defined closed 1-forms $d\log
t_{1},\ldots,d\log t_{r}\in\Omega^{1}_{\overline{S}/\overset{\circ}{S}}$ for
local sections $t_{1},\ldots,t_{r}$ of $M_{S}$ in §2; we do not permit to take
other differential forms $d\log(v_{1}t_{1}),\ldots,d\log(v_{r}t_{r})$ for
$v_{1},\ldots v_{r}\in{\cal O}_{\overline{S}}^{*}$; we permit to take only
$d\log(v_{i}t_{i})=d\log t_{i}\in\Omega^{1}_{\overline{S}/\overset{\circ}{S}}$
for $v_{i}\in{\cal O}_{S}^{*}$. This convention is similar to the convention
in [N4] in which to construct the isomorphism (1.2.5), we use the prime number
$p\in{\cal V}$ (note that $p$ is not necessarily a uniformizer of ${\cal V}$!)
and the isomorphism “$\rho_{p}$” in [N4, (6.3.14)].

The local morphism

(4.0.2) $\displaystyle
L_{S}\owns\sum_{i=1}^{r}c_{i}u_{i}\longmapsto\sum_{i=1}^{r}c_{i}t_{i}\in{\cal
O}_{\overline{S}}$

induces the following well-defined global morphism

(4.0.3) $\displaystyle\Gamma_{{\cal O}_{S}}(L_{S})\longrightarrow{\cal
O}_{\overline{S}}.$

######  Remark 4.1.

Let $\epsilon>0$ be a real number. Set
$\overset{\circ}{\Delta}(\epsilon):=\\{z\in{\mathbb{C}}~{}|~{}|z|<\epsilon\\}$.
Replacing $\overset{\circ}{\Delta}$ by $\overset{\circ}{\Delta}(\epsilon)$, we
obtain a log anaytic space $\overline{S}(\epsilon)$ (a tubular neighborhood of
$S$ in $\overline{S}$). If $\overset{\circ}{S}$ is quasi-compact and if
$\epsilon$ is small enough, then we have a natural immersion

(4.1.1)
$\displaystyle\overline{S}(\epsilon)\overset{\subset}{\longrightarrow}\overline{S}.$

In this case, for a morphism ${\cal Z}\longrightarrow\overline{S}$, we denote
$f\times_{\overline{S}}\overline{S}(\epsilon)\colon{\cal
Z}\times_{\overline{S}}\overline{S}(\epsilon)\longrightarrow S(\epsilon)$ by

(4.1.2) $\displaystyle f(\epsilon)\colon{\cal Z}(\epsilon)\longrightarrow
S(\epsilon).$

In this section we assume that the morphism $g\colon Y\longrightarrow S$ in
the previous section fits into the following cartesian diagram

(4.1.2) $\begin{CD}Y@>{\subset}>{}>{\cal Y}\\\ @V{g}V{}V@V{}V{h}V\\\
S@>{\subset}>{}>{\overline{S}},\end{CD}$

where the morphism $h\colon{\cal Y}\longrightarrow\overline{S}$ is log smooth.
Let $U_{S}$ and $L_{S}$ be as in §2. By abuse of notation, denote by $d\log
t_{i}\in\Omega^{1}_{{\cal Y}/\overset{\circ}{S}}$ the image of $d\log t_{i}$
by the morphism
$h^{*}(\Omega^{1}_{\overline{S}/\overset{\circ}{S}})\longrightarrow\Omega^{1}_{{\cal
Y}/\overset{\circ}{S}}$. Let $\overline{\cal F}$ be a locally free ${\cal
O}_{\cal Y}$-module. Let

(4.1.3) $\displaystyle\overline{\nabla}\colon\overline{\cal
F}\longrightarrow\overline{\cal F}\otimes_{{\cal O}_{\cal Y}}\Omega^{1}_{{\cal
Y}/\overset{\circ}{S}}$

be an integrable connection. As in the previous section, we can define the
Hirsch extension $\overline{\cal F}\otimes_{{\cal O}_{\cal
Y}}\Omega^{\bullet}_{{\cal Y}/\overset{\circ}{S}}[[U_{S}]]$ of the log de Rham
complex $\overline{\cal F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal
Y}/\overset{\circ}{S}}$ by using the well-defined morphism $L_{S}\owns
u_{j}:=t_{j}\longmapsto d\log t_{j}\in\Omega^{1}_{{\cal
Y}/\overset{\circ}{S}}$:

(4.1.4) $\displaystyle\overline{\cal F}\otimes_{{\cal O}_{\cal
Y}}\Omega^{\bullet}_{{\cal Y}/\overset{\circ}{S}}[[U_{S}]]$
$\displaystyle:=\Gamma_{{\cal O}_{S}}(L_{S})\otimes_{{\cal
O}_{S}}\overline{\cal F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal
Y}/\overset{\circ}{S}}$ $\displaystyle:=(p\circ h)^{-1}(\Gamma_{{\cal
O}_{S}}(L_{S})\otimes_{(p\circ h)^{-1}({\cal O}_{S})}\overline{\cal
F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal Y}/\overset{\circ}{S}}.$

We also have the log de Rham complex $\overline{\cal F}\otimes_{{\cal O}_{\cal
Y}}\Omega^{\bullet}_{{\cal Y}/S}$.

The following remark is important:

######  Remark 4.2.

In the definition (4.1.4), we consider $L_{S}$ as a sheaf on
$\overset{\circ}{S}$ not on $S$. The reader should also note that we do not
consider $\overline{M}_{\overline{S}}:=M_{\overline{S}}/{\cal
O}_{\overline{S}}^{*}$, $U_{\overline{S}}$ nor $L_{\overline{S}}$ because the
sheaf $\overline{M}_{\overline{S}}$ is not locally constant.

######  Lemma 4.3.

Let the notations be as in (2.1). Let $\\{t_{1},\ldots,t_{r}\\}$ be a set of
local sections of $M_{S}$ whose images in $\overline{M}_{S}$ is a unique
system of local generators of $\overline{M}_{S}$. Denote $t_{i}$ in $L_{S}$ by
$u_{i}$. Then the following hold$:$

$(1)$ The sheaf $\Omega^{i}_{\overline{\cal Y}/\overset{\circ}{S}}$
$(i\in{\mathbb{N}})$ is a locally free ${\cal O}_{\overline{\cal Y}}$-module.

$(2)$ Locally on $\overline{S}$, the following sequence

(4.3.1) $\displaystyle 0\longrightarrow\overline{\cal F}\otimes_{{\cal
O}_{\cal Y}}{\Omega}^{\bullet}_{{\cal Y}/\overline{S}_{i}}[-1]\overset{{\rm
id}_{\cal F}\otimes(d\log t_{i}\wedge)}{\longrightarrow}\overline{\cal
F}\otimes_{{\cal O}_{\cal Y}}{\Omega}^{\bullet}_{{\cal
Y}/\overline{S}_{i-1}}\longrightarrow\overline{\cal F}\otimes_{{\cal O}_{\cal
Y}}{\Omega}^{\bullet}_{{\cal Y}/\overline{S}_{i}}\longrightarrow 0\quad(1\leq
i\leq r).$

is exact.

$(3)$ For $1\leq i\leq r$, let us consider the completed Hirsch extensions
$\overline{\cal F}\otimes_{{\cal O}_{\cal Y}}{\Omega}^{\bullet}_{{\cal
Y}/\overline{S}_{i-1}}[[u_{i+1},\ldots,u_{r}]]$ and $\overline{\cal
F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal
Y}/\overline{S}_{i}}[[u_{i+1},\ldots,u_{r}]]$ of $\overline{\cal
F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal Y}/\overline{S}_{i-1}}$
and $\overline{\cal F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal
Y}/\overline{S}_{i}}$ by the morphisms $\bigoplus_{j=i+1}^{r}{\cal
O}_{\overline{S}}u_{j}\owns u_{j}\longmapsto d\log t_{j}\in\Omega^{1}_{{\cal
Y}/\overline{S}_{i-1}}$ and $\bigoplus_{j=i+1}^{r}{\cal
O}_{\overline{S}}u_{j}\owns u_{j}\longmapsto d\log t_{j}\in\Omega^{1}_{{\cal
Y}/\overline{S}_{i}}$, respectively. Then the following sequence

(4.3.2) $\displaystyle 0$ $\displaystyle\longrightarrow\overline{\cal
F}\otimes_{{\cal O}_{\cal Y}}\Omega^{\bullet}_{{\cal
Y}/\overline{S}_{i}}[[u_{i+1},\ldots,u_{r}]][-1]\overset{{\rm id}_{\cal
F}\otimes(d\log t_{i}\wedge)}{\longrightarrow}\overline{\cal F}\otimes_{{\cal
O}_{\cal Y}}{\Omega}^{\bullet}_{{\cal
Y}/\overline{S}_{i-1}}[[u_{i+1},\ldots,u_{r}]]$
$\displaystyle\longrightarrow\overline{\cal F}\otimes_{{\cal O}_{\cal
Y}}\Omega^{\bullet}_{{\cal
Y}/\overline{S}_{i}}[[u_{i+1},\ldots,u_{r}]]\longrightarrow 0\quad(1\leq i\leq
r)$

is exact.

###### Proof.

The proof is the same as that of [N6, (3.1)] and (3.1); the proof is easy. ∎

######  Proposition 4.4.

The natural morphism

(4.4.1) $\displaystyle\overline{\cal F}\otimes_{{\cal O}_{\cal
Y}}\Omega^{\bullet}_{{\cal
Y}/\overset{\circ}{S}}[[U_{S}]]\longrightarrow\overline{\cal F}\otimes_{{\cal
O}_{\cal Y}}\Omega^{\bullet}_{{\cal Y}/\overline{S}}$

is a quasi-isomorphism. This quasi-isomorphism is contravariantly functorial
for a morphism from the commutative diagram (4.1.2) to a similar commutative
diagram and a morphism to the integrable connection (4.1.3) from a similar
integrable connection on the similar diagram.

###### Proof.

By using (4.3), the proof is the same as that of (3.2). ∎

Set ${\cal F}:=\overline{\cal F}\otimes_{{\cal O}_{\cal Y}}{\cal O}_{Y}$. Then
the connection (4.1.3) induces the following connection:

(4.4.2) $\displaystyle\nabla\colon{\cal F}\longrightarrow{\cal
F}\otimes_{{\cal O}_{\cal Y}}\Omega^{1}_{Y/\overset{\circ}{S}}.$

Indeed, this is a local question. We may assume that
$\overline{S}=\Delta^{r}_{\overset{\circ}{S}}$. Then
$\overline{\nabla}(t_{i}f)=dt_{i}f+t_{i}\overline{\nabla}(f)=t_{i}(fd\log
t_{i}+\overline{\nabla}(f))$ $(f\in\overline{\cal F})$. Hence
$\overline{\nabla}$ induces the connection $\nabla$ in (4.4.2).

##  5 SNCL analytic spaces

Let $r$ be a positive integer. Let $S$ be a log analytic family of log points
of virtual dimension $r$. In this section we give a definition of an
(S)NCL(=(simple) normal crossing log) analytic space over $S$. To define it,
we have only to mimic the definition of an (S)NCL(=(simple) normal crossing
log) analytic scheme defined in [N4] suitably. An (S)NCL analytic space over
$S$ is locally the finitely many product of (S)NCL analytic spaces in the
usual sense. The definition is an (obvious) generalization of the definition
of a (generalized) semistable analytic space over a log point of
${\mathbb{C}}$ virtual dimension $r$ in [Fu1] and [Fu4].

Let $B$ be an analytic space over ${\mathbb{C}}$. Let $a_{1}<\ldots<a_{r}$ be
nonnegative integers. Set ${\bf a}:=(a_{1},\ldots a_{r})$ and let $d$ be a
nonnegative integer such that $a_{r}\leq d+r$. Let $z_{1},\ldots,z_{d+r}$ be a
system of standard coordinates of $\overset{\circ}{\Delta}{}^{d+r}$. Let
$\overset{\circ}{\Delta}_{B}({\bf a},d+r)$ be a closed analytic space of
$\overset{\circ}{\Delta}{}^{d+r}\times B$ defined by an ideal sheaf

$(z_{1}\cdots z_{a_{1}},z_{a_{1}+1}\cdots z_{a_{2}},\ldots,z_{a_{r-1}+1}\cdots
z_{a_{r}}).$

######  Definition 5.1.

Let $Z$ be an analytic space over $B$ with structural morphism $g\colon
Z\longrightarrow B$. We call $Z$ an $($S$)$NC$($=$($simple$)$ normal
crossing$)$ analytic space over $B$ if $Z$ is a union of (smooth) analytic
space $\\{Z_{\lambda}\\}_{\lambda\in\Lambda}$ over $B$ ($\Lambda$ is a set)
and if, for any point of $x\in Z$, there exist an open neighborhood $V$ of $x$
and an open neighborhood $W$ of $g(x)$ such that there exists an étale
morphism $\pi\colon V\longrightarrow\overset{\circ}{\Delta}_{B}({\bf a},d+r)$
such that

(5.1.1)
$\\{Z_{\lambda}|_{V}\\}_{\lambda\in\Lambda}=\\{\pi^{*}(z_{b_{1}})=\cdots=\pi^{*}(z_{b_{r}})=0\\}_{1\leq
b_{1}\leq a_{1},\ldots,a_{r-1}+1\leq b_{r}\leq a_{r}}$

where $a_{1},\ldots,a_{r}$ and $d$ are nonnegative integers such that
$\sum_{i=1}^{r}a_{i}\leq d+r$, which depend on a local neighborhood of $z$ in
$Z$ and $Z_{\lambda}|_{V}:=Z_{\lambda}\cap V$. The strict meaning of the
equality (5.1.1) is as follows. There exists a subset $\Lambda(V)$ of
$\Lambda$ such that there exists a bijection
$i\colon\Lambda(V)\longrightarrow\\{1,\ldots,a_{1}\\}\times\cdots\times\\{a_{r-1}+1,\ldots,a_{r}\\}$
such that, if $\lambda\not\in\Lambda(V)$, then $Z_{\lambda}|_{V}=\emptyset$
and if $\lambda\in\Lambda(V)$, then $Z_{\lambda}$ is a closed analytic space
defined by the ideal sheaf $(\pi^{*}(z_{b_{1}}),\ldots,\pi^{*}(z_{b_{r}}))$.
We call the set $\\{Z_{\lambda}\\}_{\lambda\in\Lambda}$ a decomposition of $Z$
by $($smooth$)$ components of $Z$ over $B$. In this case, we call
$Z_{\lambda}$ a $($smooth$)$ component of $Z$ over $B$.

Let $Z$ be an SNC analytic space over $B$. For a nonnegative integer $k$, set

(5.1.2)
$Z_{\\{\lambda_{0},\lambda_{1},\ldots\lambda_{k}\\}}:=Z_{\lambda_{0}}\cap\cdots\cap
Z_{\lambda_{k}}\quad(\lambda_{i}\not=\lambda_{j}~{}{\rm if}~{}i\not=j)$

and set

(5.1.3)
$Z^{(k)}=\underset{\\{\lambda_{0},\ldots,\lambda_{k}~{}|~{}\lambda_{i}\not=\lambda_{j}~{}(i\not=j)\\}}{\coprod}Z_{\\{\lambda_{0},\lambda_{1},\ldots,\lambda_{k}\\}}.$

For a negative integer $k$, set $Z^{(k)}=\emptyset$. Set
$\Gamma:=\\{Z_{\lambda}\\}_{\lambda\in\Lambda}$ and
$\Gamma_{V}:=\\{Z_{\lambda}|_{V}\\}_{\lambda\in\Lambda}$ for an open analytic
space $V$ of $Z$.

First assume that $M_{S}$ is the free hollow log structure ([N4]). We fix an
isomorphism

(5.1.4) $\displaystyle(M_{S},\alpha_{S})\simeq({\mathbb{N}}^{r}\oplus{\cal
O}_{S}^{*}\longrightarrow{\cal O}_{S})$

globally on $S$. Let $M_{S}({\bf a},d+r)$ be the log structure on
$\overset{\circ}{\Delta}_{\overset{\circ}{S}}({\bf a},d+r)$ associated to the
following morphism

(5.1.5)
${\mathbb{N}}^{{\oplus}a_{r}}\owns(0,\ldots,0,\overset{i}{1},0,\ldots,0)\longmapsto
z_{i}\in{\cal O}_{\overset{\circ}{\Delta}_{\overset{\circ}{S}}({\bf a},d+r)}.$

Let ${\Delta}_{B}({\bf a},d+r)$ be the resulting log analytic space over $S$.
We have the “multi-diagonal” morphism
${\mathbb{N}}^{r}\longrightarrow{\mathbb{N}}^{a_{1}}\oplus{\mathbb{N}}^{a_{2}-a_{1}}\oplus\cdots\oplus{\mathbb{N}}^{a_{r}-a_{r-1}}={\mathbb{N}}^{a_{r}}$
induces a morphism ${\Delta}_{S}({\bf a},d+r)\longrightarrow S$ of log
analytic spaces. We call ${\Delta}_{S}({\bf a},d+r)$ the standard NCL analytic
space.

Let $S$ be a family of log points of virtual dimension $r$ ($M_{S}$ is not
necessarily free).

######  Definition 5.2.

Let $f\colon X(=(\overset{\circ}{X},M_{X}))\longrightarrow S$ be a morphism of
log analytic spaces such that $\overset{\circ}{X}$ is an (S)NC analytic space
over $\overset{\circ}{S}$ with a decomposition
$\Delta:=\\{\overset{\circ}{X}_{\lambda}\\}_{\lambda\in\Lambda}$ of
$\overset{\circ}{X}/\overset{\circ}{S}$ by its smooth components. We call $f$
(or $X/S$) an NCL$($=normal crossing log$)$ analytic space if, for any point
of $x\in\overset{\circ}{X}$, there exist an open neighborhood
$\overset{\circ}{V}$ of $x$ and an open neighborhood $\overset{\circ}{W}$ of
$\overset{\circ}{f}(x)$ such that $M_{W}$ is the free hollow log structure of
rank $r$ and such that $f|_{V}$ factors through a strict étale morphism
$\pi\colon V{\longrightarrow}{\Delta}_{S}({\bf a},d+r)$ over $W$ for some
${\bf a},d$ depending on $\overset{\circ}{V}$ such that
$\Delta_{\overset{\circ}{V}}=\\{\pi^{*}(z_{b_{1}})=\cdots=\pi^{*}(z_{b_{r}})=0\\}_{1\leq
b_{1}\leq a_{1},\ldots,a_{r-1}+1\leq b_{r}\leq a_{r}}$ in
$\overset{\circ}{V}$. If $\overset{\circ}{X}/\overset{\circ}{S}$ is an SNC
analytic space, then we call $f$ or $X/S$ an SNCL analytic space.

In the case $r=1$, an SNCL analytic space is an SNCL analytic space in the
usual sense ([Fu2], [N5]).

######  Corollary 5.3.

Let $f\colon X\longrightarrow S$ be an SNCL analytic space with a
decomposition $\Delta:=\\{\overset{\circ}{X}_{\lambda}\\}_{\lambda\in\Lambda}$
of $\overset{\circ}{X}/\overset{\circ}{S}$ by its smooth components. Then,
locally on $X$, $X$ is a finitely many product of SNCL analytic spaces in the
usual sence with some decomposition of their smooth components. Consequently
$f$ is log smooth.

###### Proof.

Obvious. ∎

The following is an obvious imitation of [NS, (2.1.7)].

######  Definition 5.4.

Let $f\colon X\longrightarrow S$ be an SNCL analytic space with a
decomposition $\Delta:=\\{\overset{\circ}{X}_{\lambda}\\}_{\lambda\in\Lambda}$
of $\overset{\circ}{X}/\overset{\circ}{S}$ by its smooth components. We call
an effective Cartier divisor $D$ on $X/S$ is a relative simple normal crossing
divisor $(=:$relative SNCD$)$ on $X/S$ if there exists a set
$\Gamma:=\\{D_{\mu}\\}_{\mu}$ of non-zero effective Cartier divisors on $X/S$
of locally finite intersection such that

(5.4.1) $D=\sum_{\mu}D_{\mu}\quad\text{in}\quad{\rm Div}(X/S)_{\geq 0},$

such that $D_{\mu}|_{Z_{\lambda}}$ is a smooth divisor on $Z_{\lambda}$ for
any $\lambda$ and, for any point $x$ of $D$, there exist an open neighborhood
$V\simeq\Delta_{S}({\bf a},d+r)$ of $x$ in $X$ and
$D|_{V}=\bigcup_{c=a_{r}+1}^{b}\\{z_{c}=0\\}$ for some $a_{r}+1\leq b\leq
d+r$.

We call $\Gamma=\\{D_{\mu}\\}$ a decomposition of $D$ by smooth components of
$D$ over $S$. Note that $D_{\mu}$ itself is not necessarily smooth over
$\overset{\circ}{S}$; $D_{\mu}$ is a union of smooth analytic space over
$\overset{\circ}{S}$.

Let ${\rm Div}_{D}(X/S)_{\geq 0}$ be a submonoid of ${\rm Div}(X/S)_{\geq 0}$
consisting of effective Cartier divisors $E$’s on $X/S$ such that there exists
an open covering $X=\bigcup_{i\in I}V_{i}$ (depending on $E$) of $X$ such that
$E|_{V_{i}}$ is contained in the submonoid of ${\rm Div}(V_{i}/S)_{\geq 0}$
generated by $D_{\lambda}|_{V_{i}}$ $(\lambda\in\Lambda)$. As in [NS,
(A.0.1)], we see that the definition of ${\rm Div}_{D}(X/S)_{\geq 0}$ is
independent of the choice of $\Delta$.

Let $M(D)^{\prime}$ be a presheaf of monoids on $X$ defined as follows: for an
open sub analytic space $V$ of $X$,

(5.4.2) $\displaystyle\Gamma(V,M(D)^{\prime}):=\\{(E,a)\in$ $\displaystyle{\rm
Div}_{D|_{V}}(V/S)_{\geq 0}\times\Gamma(V,{\cal O}_{X})|$ $\displaystyle
a\text{ is a generator of }\Gamma(V,{\cal O}_{X}(-E))\\}$

with a monoid structure defined by
$(E,a)\cdot(E^{\prime},a^{\prime}):=(E+E^{\prime},aa^{\prime})$. The natural
morphism $M(D)^{\prime}\longrightarrow{\cal O}_{X}$ defined by the second
projection $(E,a)\mapsto a$ induces a morphism
$M(D)^{\prime}\longrightarrow({\cal O}_{X},*)$ of presheaves of monoids on
$X$. The log structure $M(D)$ is, by definition, the associated log structure
to the sheafification of $M(D)^{\prime}$. Because ${\rm
Div}_{D|_{V}}(V/S)_{\geq 0}$ is independent of the choice of the decomposition
of $D|_{V}$ by smooth components, $M(D)$ is independent of the choice of the
decomposition of $D$ by smooth components of $D$.

##  6 Main results

In [Fu1] Fujisawa has introduced the notion of a generalized semistable family
over a unit polydisc. In this article we give a generalization of this notion
and we call this a semistable family. In this section we give a main result in
this article.

Let the notations be as in the previous section. Let $f\colon{\cal
X}\longrightarrow\overline{S}$ be a log smooth morphism. We assume that
$\overset{\circ}{f}\colon\overset{\circ}{\cal
X}\longrightarrow\overset{\circ}{S}$ is smooth and that, locally on
$\overset{\circ}{\cal X}$, this is a product of semistable families of
analytic spaces in the usual sense. That is, locally on $\overset{\circ}{\cal
X}$, there exist a system of local coordinates $t_{1},\ldots,t_{r}$ of
$\overline{S}$ over $\overset{\circ}{S}$ and a system of local coordinates
$x_{1},\ldots,x_{d+r}$ such that $x_{1}\cdots
x_{a_{1}}=t_{1},x_{a_{1}+1}\cdots x_{a_{2}}=t_{2},\ldots,x_{a_{r-1}+1}\cdots
x_{a_{r}}=t_{r}$ $(0\leq i_{1}<i_{2}<\cdots<i_{r}\leq d+r)$ over
$\overset{\circ}{S}$ ([Fu1, (6.2), (6.5)]; see also [AK, (0.3)]). We also
assume that the morphism $f^{-1}(M_{\overline{S}})\longrightarrow M_{\cal X}$
of log structures of $f$ is locally isomorphic to the associated log structure
to the multi-diagonal morphism
${\mathbb{N}}^{r}\overset{\subset}{\longrightarrow}{\mathbb{N}}^{a_{1}}\oplus{\mathbb{N}}^{a_{2}-a_{1}}\oplus\cdots\oplus{\mathbb{N}}^{a_{r}-a_{r-1}}={\mathbb{N}}^{a_{r}}$,
where $M_{S}$ and $M_{\cal X}$ are the associated log structures to the
morphism ${\mathbb{N}}^{r}\owns e_{i}\longmapsto t_{i}\in{\cal
O}_{\overline{S}}$ $(1\leq i\leq r)$ and ${\mathbb{N}}^{a_{i+1}-a_{i}}\owns
e_{j}\longmapsto x_{j+a_{i}+1}\in{\cal O}_{\cal X}$ $(1\leq i\leq r)$ in the
local situation above. We call ${\cal X}/\overline{S}$ a semistable family
over $\overline{S}$. Let ${\cal D}$ be a horizontal NCD(=normal crossing
divisor) on ${\cal X}/\overline{S}$. That is, it is a relative NCD on
$\overset{\circ}{\cal X}/\overset{\circ}{\overline{S}}$ which is locally
defined by an equation $x_{a_{r}+1}\cdots x_{b}=0$ for some $a_{r}+1\leq b\leq
d+r$ in the local situation above. Set $X:={\cal X}\times_{\overline{S}}S$ and
$D:={\cal D}\times_{\overline{S}}S$. Let $f_{S}\colon X\longrightarrow S$ be
the structural morphism.

######  Lemma 6.1.

Let ${\cal E}$ be a coherent locally free ${\cal O}_{\cal X}$-module and let
$\nabla\colon{\cal E}\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{1}_{{\cal X}/{\mathbb{C}}}(\log D)$ be a locally nilpotent
integrable connection on ${\cal X}$ and let ${\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/{\mathbb{C}}}(\log D)$ be the associated log de
Rham complex to $\nabla$. Let $\iota\colon
X\overset{\subset}{\longrightarrow}{\cal X}$ be the natural exact closed
immersion. Then the following natural morphism

(6.1.1) $\displaystyle\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal
D}))\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)$

is a quasi-isomorphism of complexes of $f^{-1}({\cal O}_{S})$-modules. This
quasi-isomorphism is contravariantly functorial.

###### Proof.

(cf. [S, (2.10)], [EY, (3.17) (2)]) Since the problem is local, we may assume
that $\nabla$ is nilpotent by the five lemma. In this case, by [KatN, (4.6)
(2)], there exists a local system $V$ of finite dimensional
${\mathbb{C}}$-vector spaces on $\overset{\circ}{\cal X}$ such that ${\cal
E}:={\cal O}_{\cal X}\otimes_{\mathbb{C}}V$ and $\nabla:=d\otimes{\rm
id}\colon{\cal E}\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{1}_{{\cal X}/{\mathbb{C}}}(\log D)$. Indeed, we have only to set
$V:={\rm Ker}(\nabla)$. Let ${\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/{\mathbb{C}}}(\log D)$ be the associated log de
Rham complex to $\nabla$. We may assume that $V={\mathbb{C}}$. Moreover we may
assume that $\overset{\circ}{S}=\\{O\\}$ since the family ${\cal
X}/\overline{S}$ is locally obtained by the base change of a semistable family
in the case $\overset{\circ}{S}=\\{O\\}$. Since
$\Omega^{\bullet}_{X/{\mathbb{C}}}(\log D)=\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}(\log{\cal D})\otimes_{{\cal O}_{\cal X}}{\cal O}_{X}$, it
suffices to prove that the natural morphism

(6.1.2) $\displaystyle\iota^{-1}(\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}(\log{\cal D}))\longrightarrow\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}(\log{\cal D})\otimes_{{\cal O}_{\cal X}}{\cal O}_{X}$

is a quasi-isomorphism.

Let $(z_{1},\ldots,z_{m+1})$ $(|z_{i}|<1)$ be the local coordinate of a point
of ${\cal X}$. We may assume that ${\cal X}=\Delta^{m+1}$ and that the
morphism $f\colon{\cal X}\longrightarrow\Delta$ is defined by the following
equation

$f(z_{1},\ldots,z_{m})=(z_{1}\cdots z_{a_{1}},z_{a_{1}+1}\cdots
z_{a_{2}},\ldots,z_{a_{r-1}+1}\cdots
z_{a_{r}})=(t_{1},\ldots,t_{r})\in\Delta^{r}.$

We may also assume that ${\cal D}$ is a closed analytic space of
$\overset{\circ}{\cal X}$ defined by an ideal sheaf $(z_{a_{r}+1}\cdots
z_{b})$ $(a_{r}+1\leq b\leq m)$.

To prove that the morphism (6.1.2) is a quasi-isomorphism, it suffices to
prove that the complex $\iota^{-1}(t_{1}\cdots t_{r}\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}(\log{\cal D}))$ is acyclic. (This is indeed a complex.)
Indeed, the stalk of $\iota^{-1}(t_{1}\cdots t_{r}\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}(\log{\cal D}))$ at $S$ is the Koszul complex on
${\mathbb{C}}\\{z_{0},\ldots,z_{m}\\}$ with respect to the operators
$z_{i}\dfrac{d}{dz_{i}}+1$ $(1\leq i\leq a_{r})$, $z_{i}\dfrac{d}{dz_{i}}$
$(a_{r}+1\leq i\leq b)$ and $\dfrac{d}{dz_{i}}$ $(b+1\leq i\leq m)$. Since the
operator $z_{i}\dfrac{d}{dz_{i}}+1$ $(1\leq i\leq a_{r})$ is bijective, we see
that $\iota^{-1}(t_{1}\cdots t_{r}\Omega^{\bullet}_{{\cal
X}/{\mathbb{C}}}(\log{\cal D}))$ is acyclic by [S, (1.12)].

We can complete the proof of (6.1). ∎

######  Corollary 6.2.

The natural morphism

(6.2.1) $\displaystyle\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal
D}))[U_{S}]\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log{\cal D})[U_{S}]$

is a quasi-isomorphism of complexes of $f^{-1}({\cal O}_{S})$-modules. This
quasi-isomorphism is contravariantly functorial.

###### Proof.

In [N5, (3.14)] we have proved that the (PD-)Hirsch extension preserves the
quasi-isomorphism. Hence (6.2) follows from (6.1). ∎

######  Theorem 6.3.

Let the notations be as in (6.1). Then there exists a canonical section

(6.3.1) $\displaystyle\rho\colon{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)\longrightarrow\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))$

of complexes of $f^{-1}({\cal O}_{S})$-modules of the natural morphism

(6.3.2) $\displaystyle\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)$

in the derived category $D^{+}(\iota^{-1}f^{-1}({\cal O}_{\overline{S}}))$ of
bounded below complexes of $\iota^{-1}f^{-1}({\cal
O}_{\overline{S}})$-modules. The section (6.3.1) is contravariantly functorial
with respect to a morphism $({\cal X},{\cal D})\longrightarrow({\cal
X}^{\prime},{\cal D}^{\prime})$ over
$\overline{S}\longrightarrow\overline{S}{}^{\prime}$, where $({\cal
X}^{\prime},{\cal D}^{\prime})/\overline{S}{}^{\prime}$ is an analogous log
analytic space over ${\mathbb{C}}$.

###### Proof.

By (2.3), (4.4) and (6.2) we obtain the following commutative diagram (cf.
[EY, (3.17)]):

(6.3.3) $\begin{CD}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal
D}))[[U_{S}]]@>{}>{}>\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))\\\
@A{\bigcup}A{}A\Big{\|}\\\ \iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal
D}))[U_{S}]\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))\\\
@V{\simeq}V{}V\Big{\|}\\\ {\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)[U_{S}]\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))\\\ @V{\simeq}V{}V@V{}V{}V\\\ {\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)@={\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D).\end{CD}$

(By (4.4) the horizontal morphism in (6.3.3) is indeed an isomorphism.)

The contravariantly functoriality follows from (2.4), (4.4) and (6.2). ∎

######  Remark 6.4.

(1) In [N6, (2.9)] we have proved that the morphism

$\displaystyle{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)[U_{S}]\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)$

is an underlying isomorphism of filtered isomorphism.

(2) Because the following natural morphism

$\displaystyle\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet\geq i}_{{\cal X}/\overline{S}}(\log{\cal
D}))\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet\geq
i}_{X/S}(\log D)\quad(i\in{\mathbb{N}})$

is not a quasi-isomorphism in general, the morphism (6.3.1) cannot be an
underlying morphism of a filtered morphism with respect to the Hodge
filtration in general.

######  Corollary 6.5.

Let $\iota_{S}\colon S\overset{\subset}{\longrightarrow}\overline{S}$ be the
natural inclusion. Set $f_{S}:=f\times_{\overline{S}}S$. Then there exists a
canonical section

(6.5.1) $\displaystyle\rho\colon\iota_{S*}Rf_{S*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow
Rf_{*}\iota_{*}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))$

of complexes of $f^{-1}({\cal O}_{S})$-modules of the natural morphism

(6.5.2) $\displaystyle Rf_{*}\iota_{*}\iota^{-1}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal
D}))\longrightarrow\iota_{S*}Rf_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)).$

The section (6.5.1) is contravariantly functorial with respect to a morphism
$({\cal X},{\cal D})\longrightarrow({\cal X}^{\prime},{\cal D}^{\prime})$ over
$\overline{S}\longrightarrow\overline{S}{}^{\prime}$, where $({\cal
X}^{\prime},{\cal D}^{\prime})/\overline{S}{}^{\prime}$ is an analogous log
analytic space over ${\mathbb{C}}$.

######  Theorem 6.6.

Assume that $\overset{\circ}{S}$ is smooth and that
$\overset{\circ}{f}\colon\overset{\circ}{\cal
X}\longrightarrow\overset{\circ}{\overline{S}}$ is proper. Then the higher
direct image $R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))$ is a locally free
${\cal O}_{\overline{S}}$-module and commutes with base change.

###### Proof.

By Grauert’s theorem ([Gra, p. 62 Satz 3]), we have only to prove that, for
any point $y\in\overline{S}$, the natural morphism

(6.6.1) $\displaystyle R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))\otimes_{{\cal
O}_{\overline{S}}}\kappa(y)\longrightarrow H^{q}(X_{y},{\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}_{y}/y}(\log{\cal D}_{y}))$

is surjective. We may assume that $\overset{\circ}{y}$ is the origin
$\overset{\circ}{S}$. Indeed, if $\overset{\circ}{y}$ is not the origin, we
have only to consider the semistable family around $y$. By (6.3.1) we see that
the following natural morphism

(6.6.2) $\displaystyle R^{q}f_{*}(\iota_{*}\iota^{-1}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D})))$
$\displaystyle\longrightarrow R^{q}f_{*}(\iota_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)))$
$\displaystyle=\iota_{S*}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))$

is surjective. Set

${\cal K}^{\bullet}:={\rm Ker}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal
D})\longrightarrow\iota_{*}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D})))$

(${\cal K}^{\bullet}$ is equal to $j_{!}j^{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))$, where
$j\colon{\cal X}\setminus X\overset{\subset}{\longrightarrow}{\cal X}$ is the
natural open immersion.) Then we have the following exact sequence

$\displaystyle 0\longrightarrow{\cal K}^{\bullet}\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D})\longrightarrow\iota_{*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D})))\longrightarrow 0$

of $f^{-1}({\cal O}_{\overline{S}})$-modules. Hence we have the following
exact sequence

$\displaystyle\cdots$ $\displaystyle\longrightarrow R^{q}f_{*}({\cal
K}^{\bullet})\longrightarrow R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))$
$\displaystyle\longrightarrow R^{q}f_{*}\iota_{*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))\longrightarrow R^{q+1}f_{*}({\cal
K}^{\bullet})\longrightarrow\cdots$

of $f^{-1}({\cal O}_{\overline{S}})$-modules. By the proper base change
theorem and by the following cartesian diagrams:

$\begin{CD}X_{s}@=X@>{}>{}>{\cal X}\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\
s@=S@>{\subset}>{}>\overline{S},\end{CD}$

$R^{q}f_{*}({\cal K}^{\bullet})_{y}=R^{q}f_{S*}(\iota^{-1}({\cal
K}^{\bullet}))_{y}$. Since $\iota^{-1}({\cal K}^{\bullet})={\cal
K}^{\bullet}|_{X}=0$, $R^{q}f_{*}({\cal K}^{\bullet})_{y}=0$. Hence

(6.6.3) $\displaystyle R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal
D}))_{y}=R^{q}f_{*}\iota_{*}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))_{y}.$

Consequently the morphism

$\displaystyle R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal
D}))_{y}\longrightarrow\iota_{S*}Rf_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))_{y}=Rf_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))_{y}$

is surjective.

We complete the proof of this theorem. ∎

######  Remark 6.7.

(1) Though (6.6) is a special case of (1.2), our proof of (6.6) is purely
algebraic.

(2) Let $\pi\colon Y\longrightarrow Z$ be a proper morphism of complex
analytic spaces. Assume that $Z$ is reduced. Usually Grauert’s theorem has
been stated for a $\pi$-flat coherent sheaf. However the proof for this
theorem is valid for a bounded complex whose any component is a coherent
${\cal O}_{Y}$-module with $\pi^{-1}({\cal O}_{Y})$-linear boundary morphism.

We also consider the case where $\overset{\circ}{S}$ is not necessarily a
point.

######  Corollary 6.8.

Assume that $\overset{\circ}{S}$ is smooth. Then the higher direct image
$R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))$ is a locally free ${\cal
O}_{\overline{S}}$-module and commutes with base change.

###### Proof.

Take any point $y\in\overline{S}$. Then we have only to consider the
semistable family around $y$ and we may assume that $y$ is the origin of this
neighborhood of $y$. ∎

######  Corollary 6.9.

Assume that $\overset{\circ}{S}$ is smooth. Then $R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log{\cal D}))$ is a
locally free ${\cal O}_{\overline{S}}$-module and commutes with base change.

######  Remark 6.10.

In (6.8) and (6.9) it is not necessary to assume that $\overset{\circ}{S}$ is
smooth by [IKN, (6.4)].

In the following we do not assume that $\overset{\circ}{S}$ is smooth.

Let

(6.10.1) $\displaystyle\sigma_{({\cal X},{\cal D})/\overline{S}}\colon{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overset{\circ}{S}}(\log{\cal D})[U_{S}]\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D})$

be the induced morphism by the morphism

$\displaystyle{\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overset{\circ}{S}}(\log{\cal D})\overset{{\rm
proj}.}{\longrightarrow}{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D})$

and the morphism (4.0.3). The morphism $\sigma_{({\cal X},{\cal
D})/\overline{S}}$ is a morphism of complexes since “$d\log t_{i}=0=dt_{i}$”
in $\Omega^{1}_{{\cal X}/\overline{S}}(\log{\cal D})$.

######  Definition 6.11.

We call the following morphism

(6.11.1) $\displaystyle{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)$
$\displaystyle\overset{\sim}{\longleftarrow}{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log
D)[U_{S}]\overset{\sim}{\longleftarrow}\iota^{-1}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal
D}))[U_{S}]$ $\displaystyle\overset{\iota^{-1}(\sigma_{({\cal X},{\cal
D})/\overline{S}})}{\longrightarrow}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))$

the $\infty$-adic Yamada section. We denote this section by
$\iota^{-1}(\sigma_{({\cal X},{\cal D})/\overline{S}})$ by abuse of notation.

The $\infty$-adic Yamada section is indeed a section of the following natural
morphism

$\displaystyle\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)$

because the following diagram is commutative:

(6.11.2) $\begin{CD}\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal
D}))[U_{S}]@>{\iota^{-1}(\sigma_{({\cal X},{\cal
D})/\overline{S}})}>{}>\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))\\\
@V{\simeq}V{}V@V{}V{}V\\\ {\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)[U_{S}]{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D).\\\
@V{\simeq}V{}V\Big{\|}\\\ {\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)@={\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D).\end{CD}$

This commutative diagram is the crux in this article. The $\infty$-adic Yamada
section induces the following morphism

(6.11.3) $\displaystyle\iota_{S*}Rf_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log
D))\overset{\sim}{\longleftarrow}\iota_{S*}Rf_{S*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)[U_{S}])$
$\displaystyle\overset{\sim}{\longleftarrow}Rf_{*}(\iota_{*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overset{\circ}{S}}(\log{\cal
D}))[U_{S}])\overset{\iota^{-1}(\sigma_{({\cal X},{\cal
D})/\overline{S}})}{\longrightarrow}Rf_{*}(\iota_{*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))).$

Let $q$ be a nonnegative integer. Obviously the morphism (6.11.3) induces the
following isomorphism of ${\cal O}_{S}$-modues:

(6.11.4) $\displaystyle\iota_{S*}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log
D))\overset{\sim}{\longleftarrow}R^{q}f_{*}(\iota_{*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))).$

Let $y$ be any exact point of $S$. Because $R^{q}f_{S*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D))$ is a coherent ${\cal
O}_{S}$-module, there exists an open neighbourhood $V(y)$ of $y$ in
$\overline{S}$ such that the restriction of the isomorphism (6.11.4) to $V(y)$
factors through the natural morphism

(6.11.5) $\displaystyle R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal
D}))|_{V(y)}\longrightarrow R^{q}f_{*}\iota_{*}\iota^{-1}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))|_{V(y)}$

of ${\cal O}_{S}$-modules by (6.6.3). Consequently we have the following
injective morphism

(6.11.6) $\displaystyle\iota_{S*}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log
D))|_{V(y)}\overset{\subset}{\longrightarrow}R^{q}f_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))|_{V(y)}.$

Assume that $\overset{\circ}{S}$ is quasi-compact. If $\epsilon>0$ is small
enough, then we have a natural immersion
$\overline{S}(\epsilon)\overset{\subset}{\longrightarrow}\overline{S}$
((4.1.1)). Because $\overset{\circ}{S}$ is quasi-compact, there exists an open
neighbourhood $V$ of $S$ in $\overline{S}$ such that the morphism restricted
to $V\cap V(y)$ extends to the following morphism

(6.11.7) $\displaystyle\iota_{V*}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))|_{V}$

of ${\cal O}_{S}$-modules. Here $\iota_{V}\colon
S\overset{\subset}{\longrightarrow}V$ is the inclusion morphism. Let
$p_{V}\colon V\longrightarrow\overset{\circ}{S}$ be the projection. Applying
$p_{V*}$ to (6.11.7), we have the following morphism

(6.11.8) $\displaystyle R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow p_{V*}(R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))|_{V})$

(Because $\overline{S}(\epsilon)\longrightarrow S$ for
$\epsilon\longrightarrow 0$, we may assume that $\overline{S}(\epsilon)\subset
V$.) Hence we have the following morphism

$\displaystyle p^{-1}_{V}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))_{V}.$

of $p_{V}^{-1}({\cal O}_{S})$-modules. Consequently we have the following
morphism

(6.11.9) $\displaystyle\sigma_{V}:=\sigma(({\cal X},{\cal
D})/\overline{S})_{V}\colon p^{*}_{V}R^{q}f_{S*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal
X}/\overline{S}}(\log{\cal D}))_{V}$

of ${\cal O}_{V}$-modules.

The following is the main result in this article:

######  Theorem 6.12 (Invariance theorem).

If $V$ is small enough $($containing $S)$, then the morphism $\sigma_{V}$ is
an isomorphism.

###### Proof.

By (6.10) and (6.6) the source and the target of the morphism (6.11.9) are
locally free ${\cal O}_{\overline{S}}$-modules and commutes with base change.
Hence the morphism (6.11.9) induces the following isomorphism

(6.12.1) $\displaystyle{\mathbb{C}}\otimes_{{\mathbb{C}}}R^{q}f_{s*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}_{s}/s}(\log{\cal
D}_{s}))=R^{q}f_{s*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}_{s}/s}(\log{\cal D}_{s}))$

at any point $s\in S$, where $f_{s}\colon{\cal X}_{s}\longrightarrow s$ is the
structural morphism. Take an open neighborhood $W$ of $s$ in $\overline{S}$
such that $R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))|_{W}$ and
$R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log
D))|_{W\cap S}$ are free. Let $\\{v_{i}\\}_{i=1}^{d}$ and
$\\{w_{i}\\}_{i=1}^{d}$ be bases of $R^{q}f_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/\overline{S}}(\log{\cal D}))|_{W}$ and
$R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log
D))|_{W\cap S}$, respectively. Let $\overline{\mathfrak{m}}_{s}$ be the
maximal ideal of ${\cal O}_{\overline{S}}$ at $s$. Then

$\displaystyle v_{j}~{}{\rm
mod}~{}\overline{\mathfrak{m}}_{s}=\sum_{i=1}^{d}(f_{ij}~{}{\rm
mod}~{}\overline{\mathfrak{m}}_{s})(\sigma|_{V\cap\overline{S}(\epsilon)}(p^{-1}(w_{i}))~{}{\rm
mod}~{}\overline{\mathfrak{m}}_{s})$

for some local sections $f_{ij}$’s of ${\cal O}_{\overline{S}}$. By Nakayama’s
lemma, $v_{j}=\sum_{i=1}^{d}f_{ij}\sigma(\epsilon)(p^{-1}(w_{i}))$ for some
local sections $f_{ij}$’s of ${\cal O}_{\overline{S}}$. Hence there exist an
open log analytic space $V_{1}$ and a set $\\{f_{ij}\\}_{i,j=1}^{r}$ of local
sections of $\Gamma(V_{1},{\cal O}_{\overline{S}})$ such that
$v_{j}=\sum_{i=1}^{r}f_{ij}\sigma(p^{-1}(w_{i}))$ on $V_{1}$. This means that
the morphism

(6.12.2) $\displaystyle\Gamma(V_{1},{\cal
O}_{\overline{S}}\otimes_{p^{-1}({\cal O}_{S})}p^{-1}R^{q}f_{*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log
D)))\longrightarrow\Gamma(V_{1},R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/\overline{S}(\epsilon)}(\log{\cal
D}(\epsilon))))$

is surjective.

Because the source and the target of the morphism (6.12.2) are free
$\Gamma(V_{1},{\cal O}_{\overline{S}})$-modules of the same rank, the morphism
(6.12.2) is injective by [Bo, II §3 Corollaire to Théoreme 1] and [loc. cit.,
II §3 Corollaire to Proposition 6]. Because $\overset{\circ}{S}$ is quasi-
compact, we have only to consider a finitely many open covering $W$’s. ∎

######  Example 6.13.

Let $f\colon{\cal E}\longrightarrow\Delta$ be a log elliptic curve (Tate
curve) over $\Delta$ described in [KU, 0.2.10] defined as follows. (Strictly
speaking, the definition in [loc. cit.] is not perfect.)

Set $X=\\{(t_{1},t_{2})\in{\mathbb{C}}^{2}~{}|~{}|t_{1}t_{2}|<1\\}$ and let
the equivalence relation $\sim$ be as follows:
$(t_{1},t_{2})\sim(t^{\prime}_{1},t^{\prime}_{2})\underset{\rm
def}{\Longleftrightarrow}t_{1}t_{2}=t^{\prime}_{1}t^{\prime}_{2}$. Let
$\pi\colon X\longrightarrow X/\sim$ be the projection and let $g\colon
X/\sim\owns[(t_{1},t_{2})]\longmapsto t_{1}t_{2}\in\Delta$. We define ${\cal
E}$ as the quotient of $X/\sim$ by the following equivalence relation
$\approx$: for $q=t_{1}t_{2}=t_{1}^{\prime}t_{2}^{\prime}\in\Delta^{*}$,
$[(t_{1},t_{2})]\approx[(t^{\prime}_{1},t^{\prime}_{2})]\underset{\rm
def}{\Longleftrightarrow}t_{1}^{\prime}=q^{n}t_{1}$ and
$t_{2}^{\prime}=q^{-n}t_{2}$ for some $n\in{\mathbb{Z}}$; for $q=0$,
$[(t_{1},t_{2})]\approx[(t^{\prime}_{1},t^{\prime}_{2})]\underset{\rm
def}{\Longleftrightarrow}$ if $t_{1}=0$, take $t_{2}\not=0$. Then
$(t^{\prime}_{1},t^{\prime}_{2})=(0,t_{2})$ or $(t_{2}^{-1},0)$. If $t_{2}=0$,
take $t_{1}\not=0$. Then $(t^{\prime}_{1},t^{\prime}_{2})=(t_{1},0)$ or
$(0,t_{1}^{-1})$. Let $f\colon{\cal E}\longrightarrow\Delta$ be the induced
morphism by $g$. Let $O$ be the origin of $\Delta$. Then
$f^{-1}(q)={\mathbb{C}}^{*}/q^{\mathbb{Z}}$ and
$f^{-1}(O)={\mathbb{P}}^{1}_{\mathbb{C}}({\mathbb{C}})/(0\sim{\infty})$. Let
$E$ be the log special fiber of ${\cal E}$ at $O$.

Consider the map $h\colon
X\owns(t_{1},t_{2})\longmapsto|t_{1}t_{2}|\in{\mathbb{R}}$. Set
$W_{1}(q):=(|q|^{\frac{3}{2}}-\epsilon,|q|+\epsilon)$ and
$W_{2}(q):=(|q^{2}|-\epsilon,|q|^{\frac{3}{2}}+\epsilon)$ be intervals in
${\mathbb{R}}$ for a small $\epsilon>0$. Set
$W_{1}:=\cup_{|q|<1}h^{-1}(W_{1}(q))$ and
$W_{2}:=\cup_{|q|<1}h^{-1}(W_{2}(q))$. Let $U_{i}$ $(i=1,2)$ be the image of
$W_{i}$ in $E$. Then we obtain an open covering $E=U_{1}\cup U_{2}$. If
$\epsilon>0$ is small, then $U_{1}\cap U_{2}=V_{1}\coprod V_{2}$ for two
nonempty open sets $V_{1}$ and $V_{2}$. The notation $(0)$ stands for the
specialization $q=0$.

The cohomology $H^{1}_{\rm dR}(E/s)$ is calculated by the $H^{1}$ of the
following double complex:

$\begin{CD}\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(U_{j}(0),\Omega^{2}_{U_{j}(0)/{\mathbb{C}}}u^{[i]})@>{\partial}>{}>\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(V_{j}(0),\Omega^{2}_{V_{j}(0)/{\mathbb{C}}}u^{[i]})\\\
@A{d}A{}A@A{}A{-d}A\\\
\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(U_{j}(0),\Omega^{1}_{U_{j}(0)/{\mathbb{C}}}u^{[i]})@>{\partial}>{}>\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(V_{j}(0),\Omega^{1}_{V_{j}(0)/{\mathbb{C}}}u^{[i]})\\\
@A{d}A{}A@A{}A{-d}A\\\
\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(U_{j}(0),{\cal
O}_{U_{j}(0)}u^{[i]})@>{\partial}>{}>\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(V_{j}(0),{\cal
O}_{V_{j}(0)}u^{[i]}).\end{CD}$

Here $\partial$’s are the usual boundary morphsims in Čech cohomologies. Let
$\overline{\omega}(0)$ be the cohomology class of
$(0,1)\in\Gamma(V_{1}(0),{\cal O}_{V_{1}(0)})\oplus\Gamma(V_{2}(0),{\cal
O}_{V_{2}(0)})$. Let $\omega(0)$ be the cohomology class of $(d\log
t_{1},d\log
t_{2})\oplus(-u,u)\in\bigoplus_{j=1}^{2}\Gamma(U_{j}(0),\Omega^{1}_{U_{j}(0)/s})\oplus\bigoplus_{j=1}^{2}\Gamma(V_{j}(0)u,{\cal
O}_{V_{j}(0)}u)$.

The cohomology sheaf $H^{1}_{\rm dR}({\cal E}/\Delta)$ is calculated by the
$H^{1}$ of the following double complex:

$\begin{CD}\bigoplus_{j=1}^{2}\Gamma(U_{j},\Omega^{1}_{U_{j}/\Delta})@>{\partial}>{}>\bigoplus_{j=1}^{2}\Gamma(V_{j},\Omega^{1}_{V_{j}/\Delta})\\\
@A{d}A{}A@A{}A{-d}A\\\
\bigoplus_{j=1}^{2}\bigoplus_{i=0}^{\infty}\Gamma(U_{j},{\cal
O}_{U_{j}})@>{\partial}>{}>\bigoplus_{j=1}^{2}\Gamma(V_{j},{\cal
O}_{V_{j}}).\end{CD}$

Let $\overline{\omega}$ be the cohomology class of $(0,1)\in\Gamma(V_{1},{\cal
O}_{V_{1}})\oplus\Gamma(V_{2},{\cal O}_{V_{2}})$ and let $\omega$ be the
cohomology class of $(d\log t_{1},d\log
t_{2})\in\bigoplus_{j=1}^{2}\Gamma(V_{j},\Omega^{1}_{V_{j}/\Delta})$. Then the
morphism

$\displaystyle\sigma_{\Delta(\epsilon)}\colon H^{1}_{\rm
dR}(E/s)\longrightarrow H^{1}_{\rm dR}({\cal E}/\Delta)$

is given by the following:

$\displaystyle\sigma_{\Delta(\epsilon)}(\overline{\omega}(0))=\overline{\omega},\quad\sigma_{\Delta(\epsilon)}(\omega(0))=\omega+q\overline{\omega}.$

In particular, $\sigma_{\Delta(\epsilon)}$ is not a filtered morphism on
$\Delta^{*}=\\{q\in{\mathbb{C}}~{}|~{}0<|q|<1\\}$. However the following
inclusion holds:

$\displaystyle\sigma_{\Delta(\epsilon)}(F^{i}_{H}H^{1}_{\rm
dR}(E/s))\subset\sum_{j=0}^{i}q^{i-j}F^{i}_{H}H^{1}_{\rm dR}({\cal
E}/\Delta)\quad(i,j\in{\mathbb{N}}),$

where $F_{H}$ means the Hodge filtration.

######  Remark 6.14.

See [EY, §5] for the Hyodo-Kato section between the rigid cohomology of a Tate
curve over a complete discrete valuation ring of mixed characteristics.

As in [N5, (10.2.4)] consider the morphism

(6.14.1) $d/du_{k}:{\cal O}_{S}[\underline{u}]\owns u^{[i]}_{k}\longmapsto
u^{[i-1]}_{k}\in{\cal O}_{S}[\underline{u}]\quad(1\leq k\leq r).$

Because this morphism is independent of the choice of $\underline{u}$, we have
the following morphism

(6.14.2) $D_{k}\colon\Gamma_{{\cal O}_{T}}(L_{S})\longrightarrow\Gamma_{{\cal
O}_{T}}(L_{S}).$

This morphism induces the following morphism

(6.14.3) $D_{k}:{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)[U_{S}]\longrightarrow{\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log
D)[U_{S}].$

Let the notations be as in the beginning of §2. Set
$S^{k}:=(\overset{\circ}{S},(\oplus_{j\not=k}M_{j}\longrightarrow{\cal
O}_{S})^{a})$. Then we have the following exact sequence

(6.14.4) $\displaystyle 0\longrightarrow{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)[-1]\overset{d\log
t_{k}\wedge}{\longrightarrow}{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S^{k}}(\log D)\longrightarrow{\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D)\longrightarrow 0.$

Hence we have the following boundary morphism

(6.14.5) $\displaystyle N_{k}\colon{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)\longrightarrow{\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D).$

######  Proposition 6.15.

The following diagram is commutative:

(6.15.1) $\begin{CD}{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)@>{N_{k}}>{}>{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)\\\ @A{\simeq}A{}A@A{}A{\simeq}A\\\ {\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log
D)[U_{S}]@>{-D_{k}}>{}>{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)[U_{S}].\end{CD}$

###### Proof.

Because the proof is standard, we omit the proof (cf. [N5, (10.3)]). ∎

Consider the local situation in the beginning of §2. and the following
morphism

(6.15.2) $\displaystyle
L_{S}\owns\sum_{i=1}^{r}c_{i}u_{i}\longmapsto\sum_{i=1}^{r}c_{i}(t_{i}+\alpha_{i}(\underline{t}))\in{\cal
O}_{\overline{S}}$

instead of the morphism (4.0.2), where
$\alpha(\underline{t})\in\Gamma(\overline{S},{\cal O}_{\overline{S}})$. Here
$\underline{t}=(t_{1},\ldots,t_{r})$. Let

(6.15.3)
$\displaystyle\sigma_{V,\\{\alpha_{i}(\underline{t})\\}_{i=1}^{r}}\colon
p^{*}_{V}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log
D))\overset{\sim}{\longrightarrow}R^{q}f_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/S}(\log{\cal D}))_{V}$

be the isomorphism by the use of the morphism (6.15.2) instead of the morphism
(4.0.2).

######  Proposition 6.16.

The isomorphism

(6.16.1)
$\displaystyle\sigma_{V,\\{\alpha_{i}(\underline{t})\\}_{i=1}^{r}}\circ\sigma_{V}^{-1}\colon
p^{*}_{V}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\longrightarrow p^{*}_{V}R^{q}f_{S*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{X/S}(\log D))$

is equal to ${\rm id}_{V}\otimes{\rm
exp}(-\sum_{l=1}^{r}\alpha_{l}(\underline{0})N_{l})$.

###### Proof.

We have only to give a generalization of the argument in [S, p. 248].

We use the multi-index notation
$\underline{u}^{\underline{k}}:=u_{1}^{k_{1}}\cdots u_{r}^{k_{r}}$
$(k_{1},\ldots,k_{r}\in{\mathbb{N}})$. Let ${\mathfrak{U}}$ be a Stein
covering of $X$. Set $|\underline{k}|=\sum_{j=1}^{r}k_{j}$. Let
$C^{\bullet}({\mathfrak{U}},\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/\overset{\circ}{S}}(\log D)))$ be the Čech complex. Let
$\sum_{|\underline{k}|=0}^{|\underline{s}|}\underline{u}^{\underline{k}}\omega_{\underline{k}}\in
C^{q}({\mathfrak{U}},\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/\overset{\circ}{S}}(\log{\cal D})))$
$(q\in{\mathbb{N}})$ be a cocycle. Set
$e_{l}:=(0,\ldots,0,\underset{l}{1},0,\ldots,0)$. Then
$d\omega_{\underline{s}}=0$ and
$d\omega_{\underline{k}}+\sum_{l=1}^{r}(k_{l}+1)d\log
t_{l}\wedge\omega_{\underline{k}+e_{l}}=0$ $(\underline{k}<\underline{s})$.
The last equation implies that $d\omega_{\underline{k}}+(k_{l}+1)d\log
t_{l}\wedge\omega_{\underline{k}+e_{l}}=0$ in
$C^{q}({\mathfrak{U}},\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}/S^{l}}(\log{\cal D})))$. Hence the image
$\overline{\omega}_{\underline{k}}$ of $\omega_{\underline{k}}$ in
$C^{q}({\mathfrak{U}},\iota^{-1}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D)))$ satisfies the following relations:
$N_{l}^{s_{l}+1}(\overline{\omega}_{\underline{0}})=0$ and
$N_{l}(\overline{\omega}_{\underline{k}})=-(k_{l}+1)\overline{\omega}_{\underline{k}+e_{l}}$
$(\underline{k}<\underline{s})$. Hence
$\overline{\omega}_{\underline{k}}=-\sum_{l=1}^{r}\dfrac{N^{k_{1}}_{1}\cdots
N^{k_{r}}_{r}(\overline{\omega}_{\underline{0}})}{k_{1}!\cdots k_{r}!}$. Then

$\displaystyle\sigma_{V,\\{\alpha_{i}(\underline{t})\\}_{i=1}^{r}}\circ\sigma_{V}^{-1}(\overline{\omega}_{\underline{0}})=\sigma_{V,\\{\alpha_{i}(\underline{t})\\}_{i=1}^{r}}(\sum_{|\underline{k}|=0}^{|\underline{s}|}\underline{u}^{\underline{k}}\overline{\omega}_{\underline{k}})=\sum_{|\underline{k}|=0}^{|\underline{s}|}(\underline{t}+\underline{\alpha}(\underline{t}))^{\underline{k}}\overline{\omega}_{\underline{k}}|_{\underline{t}=0}$
$\displaystyle=-\sum_{l=1}^{r}\sum_{k_{1},\ldots,k_{r}\geq
0}\alpha_{1}(\underline{0})^{k_{1}}\cdots\alpha_{r}(\underline{0})^{k_{r}}\dfrac{N^{k_{1}}_{1}\cdots
N^{k_{r}}_{r}(\overline{\omega}_{\underline{0}})}{k_{1}!\cdots k_{l}!}$
$\displaystyle={\rm
exp}(-\sum_{l=1}^{r}\alpha_{l}(\underline{0})N_{l})(\overline{\omega}_{\underline{0}}).$

∎

######  Corollary 6.17 (Invariance theorem of the pull-backs of morphisms).

Let $B^{\prime}$ and $S^{\prime}$ be an analogous log analytic spaces to $B$
and $S$, respectively. Let $({\cal X}^{\prime},{\cal D}^{\prime})/B^{\prime}$
be an analogous log analytic space to $({\cal X},{\cal D})/B$ and let
$(X^{\prime},D^{\prime})$ be an analogous log analytic space to $(X,D)$ for
$({\cal X}^{\prime},{\cal D}^{\prime})/B^{\prime}$. Assume that we are given
the following diagram

$\begin{CD}(X,D)@>{g}>{}>(X^{\prime},D^{\prime})\\\ @V{}V{}V@V{}V{}V\\\
S@>{u|_{S}}>{}>S^{\prime}\\\ @V{\bigcap}V{}V@V{}V{\bigcap}V\\\
<EMAIL_ADDRESS>
Let $({\cal E}^{\prime},\nabla^{\prime})$ be an analogous integrable
connection to $({\cal E},\nabla)$ for ${\cal
X}^{\prime}/\overset{\circ}{S}{}^{\prime}$. Set $E^{\square}:={\cal
E}^{\square}$, where $\square=$nothing or ′. Let $\Phi\colon
g^{*}(E^{\prime})\longrightarrow E$ be a morphism of ${\cal O}_{X}$ modules
fitting into the following commutative diagram$:$

$\begin{CD}g^{*}(E^{\prime})@>{}>{}>E\\\
@V{g^{*}(\nabla^{\prime})}V{}V@V{}V{\nabla}V\\\ g^{*}(E^{\prime}\otimes_{{\cal
O}_{X^{\prime}}}\Omega^{1}_{X^{\prime}/\overset{\circ}{S}{}^{\prime}}(\log
D^{\prime}))@>{}>{}>E\otimes_{{\cal
O}_{X}}\Omega^{1}_{X/\overset{\circ}{S}{}}(\log D).\end{CD}$

Then there exist $\epsilon>0$ and $\epsilon^{\prime}>0$ such that
$u(B(\epsilon))\subset B^{\prime}(\epsilon^{\prime})$ such that there exists
the following canonical morphism

$\displaystyle g(\epsilon,\epsilon^{\prime})^{*}\colon$ $\displaystyle
u(\epsilon,\epsilon^{\prime})^{*}(R^{q}f^{\prime}(\epsilon^{\prime})_{*}({\cal
E}^{\prime}\otimes_{{\cal O}_{{\cal X}^{\prime}}}\Omega^{\bullet}_{{\cal
X}^{\prime}(\epsilon^{\prime})/B^{\prime}(\epsilon^{\prime})}(\log{\cal
D}^{\prime}(\epsilon^{\prime}))))$ $\displaystyle\longrightarrow
R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon))).$

Here $u(\epsilon,\epsilon^{\prime})\colon B(\epsilon)\longrightarrow
B^{\prime}(\epsilon^{\prime})$ is the induced morphism by $u$. The morphism
$g(\epsilon,\epsilon^{\prime})$ satisfies the usual transitive relation. If
the morphism $g$ is lifted to a morphism $\widetilde{g}\colon{\cal
X}(\epsilon)\longrightarrow{\cal X}^{\prime}(\epsilon^{\prime})$ and if the
morphism $\Phi$ is lifted to a morphism $\widetilde{\Phi}\colon g^{*}({\cal
E}^{\prime})\longrightarrow{\cal E}$ be a morphism of ${\cal O}_{\cal X}$
modules fitting into the following commutative diagram$:$

$\begin{CD}\widetilde{g}^{*}({\cal
E}^{\prime}(\epsilon^{\prime}))@>{}>{}>{\cal E}(\epsilon)\\\
@V{g^{*}(\nabla^{\prime})}V{}V@V{}V{\nabla}V\\\ \widetilde{g}^{*}({\cal
E}^{\prime}\otimes_{{\cal O}_{{\cal X}^{\prime}}}\Omega^{1}_{{\cal
X}^{\prime}(\epsilon^{\prime})/\overset{\circ}{S}{}^{\prime}}(\log{\cal
D}^{\prime}(\epsilon^{\prime})))@>{}>{}>{\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{1}_{{\cal X}(\epsilon)/\overset{\circ}{S}{}}(\log{\cal
D}(\epsilon)),\end{CD}$

where ${\cal E}^{\square}(\epsilon^{\square})={\cal E}^{\square}|_{{\cal
X}^{\square}(\epsilon^{\square})}$, then

$\displaystyle g(\epsilon,\epsilon^{\prime})^{*}=\widetilde{g}^{*}\colon$
$\displaystyle
u(\epsilon,\epsilon^{\prime})^{*}(R^{q}f^{\prime}(\epsilon^{\prime})_{*}({\cal
E}^{\prime}\otimes_{{\cal O}_{{\cal X}^{\prime}}}\Omega^{\bullet}_{{\cal
X}^{\prime}(\epsilon^{\prime})/B^{\prime}(\epsilon^{\prime})}(\log{\cal
D}^{\prime}(\epsilon^{\prime}))))$ $\displaystyle\longrightarrow
R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/B(\epsilon)}(\log{\cal
D}(\epsilon))).$

######  Corollary 6.18 (Invariance of relative log de Rham complexes).

Let the notations be as in (6.17). Assume that $g_{1}=g_{2}$ and
$\Phi_{1}=\Phi_{2}$. Assume also that
$(X^{\prime},D^{\prime})/S^{\prime}=(X,D)/S$, that $u_{S}={\rm id}_{S}$,
$g_{1}|_{(X,D)}={\rm id}_{(X,D)}$, ${\cal E}|_{X_{S}}={\cal
E}^{\prime}_{X_{S}}$ and $\Phi_{1}|_{{\cal E}|_{X_{S}}}={\rm id}|_{{\cal
E}|_{X_{S}}}$. Then there exists a positive integer $\epsilon$ such that

(6.18.1) $\displaystyle R^{q}f^{\prime}(\epsilon)_{*}({\cal
E}^{\prime}\otimes_{{\cal O}_{{\cal X}^{\prime}}}\Omega^{\bullet}_{{\cal
X}^{\prime}(\epsilon)/\overline{S}^{\prime}(\epsilon)}(\log{\cal
D}^{\prime}(\epsilon))))=R^{q}f(\epsilon)_{*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{{\cal X}(\epsilon)/\overline{S}(\epsilon)}(\log{\cal
D}(\epsilon))).$

By the proof of (6.12) we obtain the following:

######  Corollary 6.19.

Let $s$ be an exact point of $S$. For a sufficiently close point
$u\in\overline{S}=\Delta^{r}$ from $s$ such that $p(u)=\overset{\circ}{s}$,
there exists a canonical contravariantly functorial isomorphism

(6.19.1) $\displaystyle{\cal O}_{\overline{S},u}\otimes_{{\cal O}_{{\cal
O}_{S,s}}}R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log
D))_{s}\overset{\sim}{\longrightarrow}R^{q}f_{*}({\cal E}\otimes_{{\cal
O}_{\cal X}}\Omega^{\bullet}_{{\cal X}/B}(\log{\cal D})))_{u}.$

Consequently there exists a canonical contravariantly functorial isomorphism

(6.19.2) $\displaystyle R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\otimes_{{\cal
O}_{S}}\kappa(s)\overset{\sim}{\longrightarrow}R^{q}f_{u*}({\cal
E}\otimes_{{\cal O}_{\cal X}}\Omega^{\bullet}_{{\cal X}_{u}/u}(\log{\cal
D}_{u}))).$

Here $f_{u}\colon({\cal X}_{u},{\cal D}_{u})\longrightarrow\\{u\\}$ is the
structural morphism.

######  Corollary 6.20.

Let the notations be as in (6.19). Assume that $u\in\overline{S}\setminus S$.
Set ${\cal U}:={\cal X}\setminus{\cal D}$ and $V:={\rm Ker}(\nabla\colon{\cal
E}|_{{\cal U}_{u}}\longrightarrow{\cal E}|_{{\cal U}_{u}}\otimes_{{\cal
O}_{{\cal U}_{u}}}\Omega^{1}_{{{\cal U}_{u}}/{\mathbb{C}}})$. If $u$ is
sufficiently close from $s$, then there exists the following canonical
isomorphisms

(6.20.1) $\displaystyle R^{q}f_{S*}({\cal E}\otimes_{{\cal O}_{\cal
X}}\Omega^{\bullet}_{X/S}(\log D))\otimes_{{\cal
O}_{S}}\kappa(s)\overset{\sim}{\longrightarrow}R^{q}f_{u*}(\Omega^{\bullet}_{{\cal
U}_{u}/u})\overset{\sim}{\longleftarrow}R^{q}f_{u*}(V).$

###### Proof.

We obtain the first isomorphism in (6.20.1) by [D, II (3.13) (ii)] and the
second isomorphism in (6.20.1) by [loc. cit., I (2.27.2)]. ∎

######  Corollary 6.21.

(1.4) holds.

###### Proof.

The problem is local. Hence (1.4) follows from (6.6). ∎

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Yukiyoshi Nakkajima

Department of Mathematics, Tokyo Denki University, 5 Asahi-cho Senju Adachi-
ku, Tokyo 120–8551, Japan.

E-mail address<EMAIL_ADDRESS>